LMFDB - Embedded newform 784.3.s.f.705.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,3,Mod(129,784)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("784.129");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	784.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$21.3624527258$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			705.2
Root			$-1.60021 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	784.705
Dual form			784.3.s.f.129.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.32565 + 0.765367i) q^{3} +(-7.72648 - 4.46088i) q^{5} +(-3.32843 + 5.76500i) q^{9} +O(q^{10})$	\(q+(-1.32565 + 0.765367i) q^{3} +(-7.72648 - 4.46088i) q^{5} +(-3.32843 + 5.76500i) q^{9} +(-4.65685 - 8.06591i) q^{11} -0.262632i q^{13} +13.6569 q^{15} +(-12.8017 + 7.39104i) q^{17} +(-22.0812 - 12.7486i) q^{19} +(8.31371 - 14.3998i) q^{23} +(27.2990 + 47.2832i) q^{25} -23.9665i q^{27} +14.6863 q^{29} +(-7.49903 + 4.32957i) q^{31} +(12.3468 + 7.12840i) q^{33} +(-21.9706 + 38.0541i) q^{37} +(0.201010 + 0.348160i) q^{39} -22.9159i q^{41} +46.0000 q^{43} +(51.4340 - 29.6955i) q^{45} +(74.6135 + 43.0781i) q^{47} +(11.3137 - 19.5959i) q^{51} +(30.3137 + 52.5049i) q^{53} +83.0948i q^{55} +39.0294 q^{57} +(50.3358 - 29.0614i) q^{59} +(8.63626 + 4.98615i) q^{61} +(-1.17157 + 2.02922i) q^{65} +(-10.3137 - 17.8639i) q^{67} +25.4521i q^{69} +78.9706 q^{71} +(-18.5592 + 10.7151i) q^{73} +(-72.3780 - 41.7875i) q^{75} +(-45.1421 + 78.1885i) q^{79} +(-11.6127 - 20.1138i) q^{81} -71.8544i q^{83} +131.882 q^{85} +(-19.4689 + 11.2404i) q^{87} +(-1.74152 - 1.00547i) q^{89} +(6.62742 - 11.4790i) q^{93} +(113.740 + 197.004i) q^{95} +158.581i q^{97} +62.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^9	$ 8 q - 4 q^{9} + 8 q^{11} + 64 q^{15} - 24 q^{23} + 60 q^{25} + 208 q^{29} - 40 q^{37} + 160 q^{39} + 368 q^{43} + 152 q^{53} + 448 q^{57} - 32 q^{65} + 8 q^{67} + 496 q^{71} - 248 q^{79} + 156 q^{81} + 512 q^{85} - 128 q^{93} + 480 q^{95} + 496 q^{99}+O(q^{100}) $ 8 * q - 4 * q^9 + 8 * q^11 + 64 * q^15 - 24 * q^23 + 60 * q^25 + 208 * q^29 - 40 * q^37 + 160 * q^39 + 368 * q^43 + 152 * q^53 + 448 * q^57 - 32 * q^65 + 8 * q^67 + 496 * q^71 - 248 * q^79 + 156 * q^81 + 512 * q^85 - 128 * q^93 + 480 * q^95 + 496 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.32565	+	0.765367i	−0.441885	+	0.255122i	−0.704397	−	0.709806i	$-0.748782\pi$
$3$	−1.32565	+	0.765367i	−0.441885	+	0.255122i	0.262512	+	0.964929i	$0.415449\pi$
$4$		0			0
$5$	−7.72648	−	4.46088i	−1.54530	−	0.892177i	−0.998491	−	0.0549170i	$-0.982511\pi$
$5$	−7.72648	−	4.46088i	−1.54530	−	0.892177i	−0.546805	−	0.837260i	$-0.684156\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−3.32843	+	5.76500i	−0.369825	+	0.640556i
$10$		0			0
$11$	−4.65685	−	8.06591i	−0.423350	−	0.733264i	0.572914	−	0.819615i	$-0.305813\pi$
$11$	−4.65685	−	8.06591i	−0.423350	−	0.733264i	−0.996265	+	0.0863508i	$0.972479\pi$
$12$		0			0
$13$		−	0.262632i		−	0.0202025i	−0.999949	−	0.0101012i	$-0.996785\pi$
$13$		−	0.262632i		−	0.0202025i	0.999949	−	0.0101012i	$-0.00321538\pi$
$14$		0			0
$15$	13.6569			0.910457
$16$		0			0
$17$	−12.8017	+	7.39104i	−0.753038	+	0.434767i	−0.826791	−	0.562510i	$-0.809836\pi$
$17$	−12.8017	+	7.39104i	−0.753038	+	0.434767i	0.0737524	+	0.997277i	$0.476503\pi$
$18$		0			0
$19$	−22.0812	−	12.7486i	−1.16217	−	0.670979i	−0.210347	−	0.977627i	$-0.567459\pi$
$19$	−22.0812	−	12.7486i	−1.16217	−	0.670979i	−0.951823	+	0.306648i	$0.900793\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	8.31371	−	14.3998i	0.361466	−	0.626077i	−0.626737	−	0.779231i	$-0.715610\pi$
$23$	8.31371	−	14.3998i	0.361466	−	0.626077i	0.988202	+	0.153154i	$0.0489432\pi$
$24$		0			0
$25$	27.2990	+	47.2832i	1.09196	+	1.89133i
$26$		0			0
$27$		−	23.9665i		−	0.887647i
$28$		0			0
$29$	14.6863			0.506424			0.253212	−	0.967411i	$-0.418513\pi$
$29$	14.6863			0.506424			0.253212	+	0.967411i	$0.418513\pi$
$30$		0			0
$31$	−7.49903	+	4.32957i	−0.241904	+	0.139664i	−0.616052	−	0.787706i	$-0.711269\pi$
$31$	−7.49903	+	4.32957i	−0.241904	+	0.139664i	0.374147	+	0.927369i	$0.377935\pi$
$32$		0			0
$33$	12.3468	+	7.12840i	0.374144	+	0.216012i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−21.9706	+	38.0541i	−0.593799	+	1.02849i	0.399916	+	0.916552i	$0.369039\pi$
$37$	−21.9706	+	38.0541i	−0.593799	+	1.02849i	−0.993715	+	0.111938i	$0.964294\pi$
$38$		0			0
$39$	0.201010	+	0.348160i	0.00515411	+	0.00892717i
$40$		0			0
$41$		−	22.9159i		−	0.558925i	−0.960157	−	0.279463i	$-0.909844\pi$
$41$		−	22.9159i		−	0.558925i	0.960157	−	0.279463i	$-0.0901563\pi$
$42$		0			0
$43$	46.0000			1.06977			0.534884	−	0.844926i	$-0.320355\pi$
$43$	46.0000			1.06977			0.534884	+	0.844926i	$0.320355\pi$
$44$		0			0
$45$	51.4340	−	29.6955i	1.14298	−	0.659899i
$46$		0			0
$47$	74.6135	+	43.0781i	1.58752	+	0.916556i	0.993714	+	0.111948i	$0.0357090\pi$
$47$	74.6135	+	43.0781i	1.58752	+	0.916556i	0.593807	+	0.804608i	$0.297624\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	11.3137	−	19.5959i	0.221837	−	0.384234i
$52$		0			0
$53$	30.3137	+	52.5049i	0.571957	+	0.990658i	0.996365	+	0.0851872i	$0.0271488\pi$
$53$	30.3137	+	52.5049i	0.571957	+	0.990658i	−0.424408	+	0.905471i	$0.639518\pi$
$54$		0			0
$55$			83.0948i			1.51081i
$56$		0			0
$57$	39.0294			0.684727
$58$		0			0
$59$	50.3358	−	29.0614i	0.853150	−	0.492566i	−0.00856255	−	0.999963i	$-0.502726\pi$
$59$	50.3358	−	29.0614i	0.853150	−	0.492566i	0.861712	+	0.507397i	$0.169392\pi$
$60$		0			0
$61$	8.63626	+	4.98615i	0.141578	+	0.0817402i	0.569116	−	0.822257i	$-0.307286\pi$
$61$	8.63626	+	4.98615i	0.141578	+	0.0817402i	−0.427538	+	0.903997i	$0.640619\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.17157	+	2.02922i	−0.0180242	+	0.0312188i
$66$		0			0
$67$	−10.3137	−	17.8639i	−0.153936	−	0.266625i	0.778735	−	0.627353i	$-0.215862\pi$
$67$	−10.3137	−	17.8639i	−0.153936	−	0.266625i	−0.932671	+	0.360728i	$0.882528\pi$
$68$		0			0
$69$			25.4521i			0.368872i
$70$		0			0
$71$	78.9706			1.11226			0.556131	−	0.831095i	$-0.312285\pi$
$71$	78.9706			1.11226			0.556131	+	0.831095i	$0.312285\pi$
$72$		0			0
$73$	−18.5592	+	10.7151i	−0.254235	+	0.146783i	−0.621702	−	0.783254i	$-0.713558\pi$
$73$	−18.5592	+	10.7151i	−0.254235	+	0.146783i	0.367467	+	0.930037i	$0.380225\pi$
$74$		0			0
$75$	−72.3780	−	41.7875i	−0.965041	−	0.557166i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−45.1421	+	78.1885i	−0.571419	+	0.989727i	0.425001	+	0.905193i	$0.360274\pi$
$79$	−45.1421	+	78.1885i	−0.571419	+	0.989727i	−0.996421	+	0.0845347i	$0.973060\pi$
$80$		0			0
$81$	−11.6127	−	20.1138i	−0.143367	−	0.248318i
$82$		0			0
$83$		−	71.8544i		−	0.865715i	−0.901462	−	0.432858i	$-0.857505\pi$
$83$		−	71.8544i		−	0.865715i	0.901462	−	0.432858i	$-0.142495\pi$
$84$		0			0
$85$	131.882			1.55156
$86$		0			0
$87$	−19.4689	+	11.2404i	−0.223781	+	0.129200i
$88$		0			0
$89$	−1.74152	−	1.00547i	−0.0195677	−	0.0112974i	0.490184	−	0.871619i	$-0.336929\pi$
$89$	−1.74152	−	1.00547i	−0.0195677	−	0.0112974i	−0.509752	+	0.860321i	$0.670263\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	6.62742	−	11.4790i	0.0712625	−	0.123430i
$94$		0			0
$95$	113.740	+	197.004i	1.19726	+	2.07372i
$96$		0			0
$97$			158.581i			1.63485i	0.576032	+	0.817427i	$0.304601\pi$
$97$			158.581i			1.63485i	−0.576032	+	0.817427i	$0.695399\pi$
$98$		0			0
$99$	62.0000			0.626263
$100$		0			0
$101$	−80.9753	+	46.7511i	−0.801736	+	0.462882i	−0.844078	−	0.536221i	$-0.819851\pi$
$101$	−80.9753	+	46.7511i	−0.801736	+	0.462882i	0.0423421	+	0.999103i	$0.486518\pi$
$102$		0			0
$103$	−17.6494	−	10.1899i	−0.171353	−	0.0989308i	0.411871	−	0.911242i	$-0.364876\pi$
$103$	−17.6494	−	10.1899i	−0.171353	−	0.0989308i	−0.583224	+	0.812312i	$0.698209\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.05887	−	12.2263i	0.0659708	−	0.114265i	−0.831153	−	0.556043i	$-0.812319\pi$
$107$	7.05887	−	12.2263i	0.0659708	−	0.114265i	0.897124	+	0.441778i	$0.145652\pi$
$108$		0			0
$109$	−11.6274	−	20.1393i	−0.106674	−	0.184764i	0.807747	−	0.589529i	$-0.200687\pi$
$109$	−11.6274	−	20.1393i	−0.106674	−	0.184764i	−0.914421	+	0.404765i	$0.867353\pi$
$110$		0			0
$111$		−	67.2622i		−	0.605965i
$112$		0			0
$113$	13.7157			0.121378			0.0606891	−	0.998157i	$-0.480670\pi$
$113$	13.7157			0.121378			0.0606891	+	0.998157i	$0.480670\pi$
$114$		0			0
$115$	−128.471	+	74.1730i	−1.11714	+	0.644983i
$116$		0			0
$117$	1.51408	+	0.874153i	0.0129408	+	0.00747139i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	17.1274	−	29.6656i	0.141549	−	0.245170i
$122$		0			0
$123$	17.5391	+	30.3786i	0.142594	+	0.246981i
$124$		0			0
$125$		−	264.066i		−	2.11253i
$126$		0			0
$127$	−177.765			−1.39972			−0.699860	−	0.714280i	$-0.746754\pi$
$127$	−177.765			−1.39972			−0.699860	+	0.714280i	$0.746754\pi$
$128$		0			0
$129$	−60.9801	+	35.2069i	−0.472714	+	0.272922i
$130$		0			0
$131$	97.5265	+	56.3069i	0.744477	+	0.429824i	0.823695	−	0.567033i	$-0.191909\pi$
$131$	97.5265	+	56.3069i	0.744477	+	0.429824i	−0.0792180	+	0.996857i	$0.525242\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−106.912	+	185.176i	−0.791938	+	1.37168i
$136$		0			0
$137$	−60.8823	−	105.451i	−0.444396	−	0.769716i	0.553614	−	0.832773i	$-0.313248\pi$
$137$	−60.8823	−	105.451i	−0.444396	−	0.769716i	−0.998010	+	0.0630570i	$0.979915\pi$
$138$		0			0
$139$			175.329i			1.26136i	0.776043	+	0.630679i	$0.217224\pi$
$139$			175.329i			1.26136i	−0.776043	+	0.630679i	$0.782776\pi$
$140$		0			0
$141$	−131.882			−0.935335
$142$		0			0
$143$	−2.11837	+	1.22304i	−0.0148138	+	0.00855273i
$144$		0			0
$145$	−113.473	−	65.5139i	−0.782575	−	0.451820i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	118.882	−	205.910i	0.797867	−	1.38195i	−0.123135	−	0.992390i	$-0.539295\pi$
$149$	118.882	−	205.910i	0.797867	−	1.38195i	0.921002	−	0.389557i	$-0.127372\pi$
$150$		0			0
$151$	49.6863	+	86.0592i	0.329048	+	0.569928i	0.982323	−	0.187192i	$-0.0599388\pi$
$151$	49.6863	+	86.0592i	0.329048	+	0.569928i	−0.653275	+	0.757121i	$0.726605\pi$
$152$		0			0
$153$		−	98.4021i		−	0.643151i
$154$		0			0
$155$	77.2548			0.498418
$156$		0			0
$157$	83.7046	−	48.3269i	0.533151	−	0.307815i	−0.209148	−	0.977884i	$-0.567069\pi$
$157$	83.7046	−	48.3269i	0.533151	−	0.307815i	0.742298	+	0.670069i	$0.233736\pi$
$158$		0			0
$159$	−80.3710	−	46.4022i	−0.505478	−	0.291838i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	85.2843	−	147.717i	0.523216	−	0.906237i	−0.476419	−	0.879219i	$-0.658065\pi$
$163$	85.2843	−	147.717i	0.523216	−	0.906237i	0.999635	−	0.0270188i	$-0.00860138\pi$
$164$		0			0
$165$	−63.5980	−	110.155i	−0.385442	−	0.667606i
$166$		0			0
$167$		−	94.9055i		−	0.568296i	−0.958780	−	0.284148i	$-0.908289\pi$
$167$		−	94.9055i		−	0.568296i	0.958780	−	0.284148i	$-0.0917108\pi$
$168$		0			0
$169$	168.931			0.999592
$170$		0			0
$171$	146.992	−	84.8656i	0.859600	−	0.496290i
$172$		0			0
$173$	158.240	+	91.3600i	0.914683	+	0.528092i	0.881935	−	0.471371i	$-0.156241\pi$
$173$	158.240	+	91.3600i	0.914683	+	0.528092i	0.0327478	+	0.999464i	$0.489574\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−44.4853	+	77.0508i	−0.251329	+	0.435315i
$178$		0			0
$179$	−102.029	−	176.720i	−0.569997	−	0.987264i	−0.996565	−	0.0828083i	$-0.973611\pi$
$179$	−102.029	−	176.720i	−0.569997	−	0.987264i	0.426569	−	0.904455i	$-0.359722\pi$
$180$		0			0
$181$			229.167i			1.26612i	0.774104	+	0.633059i	$0.218201\pi$
$181$			229.167i			1.26612i	−0.774104	+	0.633059i	$0.781799\pi$
$182$		0			0
$183$	−15.2649			−0.0834149
$184$		0			0
$185$	339.510	−	196.016i	1.83519	−	1.05955i
$186$		0			0
$187$	119.231	+	68.8380i	0.637598	+	0.368117i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.79899	−	8.31209i	0.0251256	−	0.0435188i	−0.853189	−	0.521602i	$-0.825335\pi$
$191$	4.79899	−	8.31209i	0.0251256	−	0.0435188i	0.878315	+	0.478083i	$0.158668\pi$
$192$		0			0
$193$	1.71068	+	2.96298i	0.00886362	+	0.0153522i	0.870423	−	0.492304i	$-0.163845\pi$
$193$	1.71068	+	2.96298i	0.00886362	+	0.0153522i	−0.861560	+	0.507656i	$0.830512\pi$
$194$		0			0
$195$		−	3.58673i		−	0.0183935i
$196$		0			0
$197$	−146.451			−0.743405			−0.371703	−	0.928352i	$-0.621226\pi$
$197$	−146.451			−0.743405			−0.371703	+	0.928352i	$0.621226\pi$
$198$		0			0
$199$	211.871	−	122.324i	1.06468	−	0.614691i	0.137954	−	0.990439i	$-0.455947\pi$
$199$	211.871	−	122.324i	1.06468	−	0.614691i	0.926722	+	0.375748i	$0.122614\pi$
$200$		0			0
$201$	27.3448	+	15.7875i	0.136044	+	0.0785450i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−102.225	+	177.060i	−0.498660	+	0.863705i
$206$		0			0
$207$	55.3431	+	95.8571i	0.267358	+	0.463078i
$208$		0			0
$209$			237.474i			1.13624i
$210$		0			0
$211$	−311.019			−1.47403			−0.737013	−	0.675879i	$-0.763764\pi$
$211$	−311.019			−1.47403			−0.737013	+	0.675879i	$0.763764\pi$
$212$		0			0
$213$	−104.688	+	60.4415i	−0.491491	+	0.283763i
$214$		0			0
$215$	−355.418	−	205.201i	−1.65311	−	0.954422i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	16.4020	−	28.4091i	0.0748951	−	0.129722i
$220$		0			0
$221$	1.94113	+	3.36213i	0.00878337	+	0.0152132i
$222$		0			0
$223$		−	116.156i		−	0.520877i	−0.965490	−	0.260438i	$-0.916133\pi$
$223$		−	116.156i		−	0.520877i	0.965490	−	0.260438i	$-0.0838671\pi$
$224$		0			0
$225$	−363.451			−1.61534
$226$		0			0
$227$	−51.2456	+	29.5867i	−0.225752	+	0.130338i	−0.608611	−	0.793469i	$-0.708273\pi$
$227$	−51.2456	+	29.5867i	−0.225752	+	0.130338i	0.382859	+	0.923807i	$0.374940\pi$
$228$		0			0
$229$	−111.426	−	64.3320i	−0.486578	−	0.280926i	0.236576	−	0.971613i	$-0.423975\pi$
$229$	−111.426	−	64.3320i	−0.486578	−	0.280926i	−0.723154	+	0.690687i	$0.757308\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	44.4903	−	77.0595i	0.190946	−	0.330728i	−0.754618	−	0.656164i	$-0.772178\pi$
$233$	44.4903	−	77.0595i	0.190946	−	0.330728i	0.945564	+	0.325437i	$0.105511\pi$
$234$		0			0
$235$	−384.333	−	665.684i	−1.63546	−	2.83270i
$236$		0			0
$237$		−	138.201i		−	0.583127i
$238$		0			0
$239$	7.13708			0.0298623			0.0149311	−	0.999889i	$-0.495247\pi$
$239$	7.13708			0.0298623			0.0149311	+	0.999889i	$0.495247\pi$
$240$		0			0
$241$	−318.222	+	183.725i	−1.32042	+	0.762346i	−0.983796	−	0.179292i	$-0.942619\pi$
$241$	−318.222	+	183.725i	−1.32042	+	0.762346i	−0.336626	+	0.941638i	$0.609286\pi$
$242$		0			0
$243$	217.589	+	125.625i	0.895428	+	0.516976i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3.34820	+	5.79925i	−0.0135555	+	0.0234787i
$248$		0			0
$249$	54.9949	+	95.2540i	0.220863	+	0.382546i
$250$		0			0
$251$			176.995i			0.705159i	0.935782	+	0.352579i	$0.114695\pi$
$251$			176.995i			0.705159i	−0.935782	+	0.352579i	$0.885305\pi$
$252$		0			0
$253$	−154.863			−0.612106
$254$		0			0
$255$	−174.830	+	100.938i	−0.685609	+	0.395836i
$256$		0			0
$257$	−23.7837	−	13.7315i	−0.0925437	−	0.0534301i	0.453014	−	0.891503i	$-0.350349\pi$
$257$	−23.7837	−	13.7315i	−0.0925437	−	0.0534301i	−0.545558	+	0.838073i	$0.683682\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−48.8823	+	84.6665i	−0.187288	+	0.324393i
$262$		0			0
$263$	38.5736	+	66.8114i	0.146668	+	0.254036i	0.929994	−	0.367575i	$-0.119812\pi$
$263$	38.5736	+	66.8114i	0.146668	+	0.254036i	−0.783326	+	0.621611i	$0.786479\pi$
$264$		0			0
$265$		−	540.904i		−	2.04115i
$266$		0			0
$267$	3.07821			0.0115289
$268$		0			0
$269$	29.7687	−	17.1870i	0.110664	−	0.0638920i	−0.443646	−	0.896202i	$-0.646315\pi$
$269$	29.7687	−	17.1870i	0.110664	−	0.0638920i	0.554311	+	0.832310i	$0.312982\pi$
$270$		0			0
$271$	175.740	+	101.464i	0.648487	+	0.374404i	0.787876	−	0.615833i	$-0.211181\pi$
$271$	175.740	+	101.464i	0.648487	+	0.374404i	−0.139389	+	0.990238i	$0.544514\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	254.255	−	440.382i	0.924563	−	1.60139i
$276$		0			0
$277$	127.049	+	220.055i	0.458660	+	0.794422i	0.998890	−	0.0470949i	$-0.0149963\pi$
$277$	127.049	+	220.055i	0.458660	+	0.794422i	−0.540231	+	0.841517i	$0.681663\pi$
$278$		0			0
$279$		−	57.6426i		−	0.206604i
$280$		0			0
$281$	−221.529			−0.788359			−0.394180	−	0.919033i	$-0.628971\pi$
$281$	−221.529			−0.788359			−0.394180	+	0.919033i	$0.628971\pi$
$282$		0			0
$283$	−245.766	+	141.893i	−0.868430	+	0.501388i	−0.866826	−	0.498611i	$-0.833844\pi$
$283$	−245.766	+	141.893i	−0.868430	+	0.501388i	−0.00160358	+	0.999999i	$0.500510\pi$
$284$		0			0
$285$	−301.560	−	174.106i	−1.05811	−	0.610898i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−35.2452	+	61.0464i	−0.121956	+	0.211233i
$290$		0			0
$291$	−121.373	−	210.223i	−0.417088	−	0.722417i
$292$		0			0
$293$			398.073i			1.35861i	0.733855	+	0.679306i	$0.237719\pi$
$293$			398.073i			1.35861i	−0.733855	+	0.679306i	$0.762281\pi$
$294$		0			0
$295$	−518.558			−1.75783
$296$		0			0
$297$	−193.311	+	111.608i	−0.650880	+	0.375786i
$298$		0			0
$299$	−3.78184	−	2.18345i	−0.0126483	−	0.00730251i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	71.5635	−	123.952i	0.236183	−	0.409081i
$304$		0			0
$305$	−44.4853	−	77.0508i	−0.145853	−	0.252625i
$306$		0			0
$307$		−	167.720i		−	0.546320i	−0.961969	−	0.273160i	$-0.911931\pi$
$307$		−	167.720i		−	0.546320i	0.961969	−	0.273160i	$-0.0880689\pi$
$308$		0			0
$309$	31.1960			0.100958
$310$		0			0
$311$	−246.039	+	142.051i	−0.791121	+	0.456754i	−0.840357	−	0.542033i	$-0.817655\pi$
$311$	−246.039	+	142.051i	−0.791121	+	0.456754i	0.0492358	+	0.998787i	$0.484321\pi$
$312$		0			0
$313$	445.406	+	257.156i	1.42302	+	0.821583i	0.996556	−	0.0829197i	$-0.0264245\pi$
$313$	445.406	+	257.156i	1.42302	+	0.821583i	0.426468	+	0.904503i	$0.359758\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.28427	+	16.0808i	−0.0292879	+	0.0507282i	−0.880298	−	0.474421i	$-0.842657\pi$
$317$	−9.28427	+	16.0808i	−0.0292879	+	0.0507282i	0.851010	+	0.525150i	$0.175991\pi$
$318$		0			0
$319$	−68.3919	−	118.458i	−0.214395	−	0.371343i
$320$		0			0
$321$			21.6105i			0.0673225i
$322$		0			0
$323$	376.902			1.16688
$324$		0			0
$325$	12.4181	−	7.16960i	0.0382096	−	0.0220603i
$326$		0			0
$327$	30.8279	+	17.7985i	0.0942748	+	0.0544296i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	97.1665	−	168.297i	0.293554	−	0.508451i	−0.681093	−	0.732197i	$-0.738495\pi$
$331$	97.1665	−	168.297i	0.293554	−	0.508451i	0.974648	+	0.223746i	$0.0718285\pi$
$332$		0			0
$333$	−146.255	−	253.321i	−0.439204	−	0.760723i
$334$		0			0
$335$			184.033i			0.549352i
$336$		0			0
$337$	−125.265			−0.371706			−0.185853	−	0.982578i	$-0.559505\pi$
$337$	−125.265			−0.371706			−0.185853	+	0.982578i	$0.559505\pi$
$338$		0			0
$339$	−18.1823	+	10.4976i	−0.0536351	+	0.0309663i
$340$		0			0
$341$	69.8438	+	40.3243i	0.204821	+	0.118253i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	113.539	−	196.655i	0.329099	−	0.570016i
$346$		0			0
$347$	−80.3726	−	139.209i	−0.231621	−	0.401180i	0.726664	−	0.686993i	$-0.241070\pi$
$347$	−80.3726	−	139.209i	−0.231621	−	0.401180i	−0.958285	+	0.285813i	$0.907736\pi$
$348$		0			0
$349$			93.0671i			0.266668i	0.991071	+	0.133334i	$0.0425683\pi$
$349$			93.0671i			0.266668i	−0.991071	+	0.133334i	$0.957432\pi$
$350$		0			0
$351$	−6.29437			−0.0179327
$352$		0			0
$353$	164.225	−	94.8154i	0.465227	−	0.268599i	−0.249013	−	0.968500i	$-0.580106\pi$
$353$	164.225	−	94.8154i	0.465227	−	0.268599i	0.714239	+	0.699901i	$0.246773\pi$
$354$		0			0
$355$	−610.164	−	352.279i	−1.71877	−	0.992334i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−247.215	+	428.189i	−0.688622	+	1.19273i	0.283662	+	0.958924i	$0.408451\pi$
$359$	−247.215	+	428.189i	−0.688622	+	1.19273i	−0.972284	+	0.233804i	$0.924883\pi$
$360$		0			0
$361$	144.554	+	250.375i	0.400426	+	0.693558i
$362$		0			0
$363$			52.4350i			0.144449i
$364$		0			0
$365$	191.196			0.523825
$366$		0			0
$367$	622.667	−	359.497i	1.69664	−	0.979556i	0.747735	−	0.663997i	$-0.231141\pi$
$367$	622.667	−	359.497i	1.69664	−	0.979556i	0.948906	−	0.315559i	$-0.102192\pi$
$368$		0			0
$369$	132.111	+	76.2741i	0.358023	+	0.206705i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−66.4214	+	115.045i	−0.178073	+	0.308432i	−0.941221	−	0.337793i	$-0.890320\pi$
$373$	−66.4214	+	115.045i	−0.178073	+	0.308432i	0.763147	+	0.646225i	$0.223653\pi$
$374$		0			0
$375$	202.108	+	350.061i	0.538954	+	0.933495i
$376$		0			0
$377$		−	3.85710i		−	0.0102310i
$378$		0			0
$379$	−324.607			−0.856483			−0.428242	−	0.903664i	$-0.640867\pi$
$379$	−324.607			−0.856483			−0.428242	+	0.903664i	$0.640867\pi$
$380$		0			0
$381$	235.654	−	136.055i	0.618515	−	0.357100i
$382$		0			0
$383$	338.835	+	195.626i	0.884686	+	0.510773i	0.872200	−	0.489149i	$-0.162692\pi$
$383$	338.835	+	195.626i	0.884686	+	0.510773i	0.0124851	+	0.999922i	$0.496026\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−153.108	+	265.190i	−0.395627	+	0.685246i
$388$		0			0
$389$	−111.676	−	193.429i	−0.287085	−	0.497246i	0.686027	−	0.727576i	$-0.259353\pi$
$389$	−111.676	−	193.429i	−0.287085	−	0.497246i	−0.973113	+	0.230329i	$0.926020\pi$
$390$		0			0
$391$			245.788i			0.628613i
$392$		0			0
$393$	−172.382			−0.438631
$394$		0			0
$395$	697.580	−	402.748i	1.76602	−	1.01961i
$396$		0			0
$397$	193.162	+	111.522i	0.486554	+	0.280912i	0.723144	−	0.690698i	$-0.242696\pi$
$397$	193.162	+	111.522i	0.486554	+	0.280912i	−0.236590	+	0.971610i	$0.576030\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−68.1127	+	117.975i	−0.169857	+	0.294201i	−0.938369	−	0.345634i	$-0.887664\pi$
$401$	−68.1127	+	117.975i	−0.169857	+	0.294201i	0.768512	+	0.639835i	$0.220997\pi$
$402$		0			0
$403$	1.13708	+	1.96949i	0.00282155	+	0.00488707i
$404$		0			0
$405$			207.212i			0.511634i
$406$		0			0
$407$	409.255			1.00554
$408$		0			0
$409$	581.611	−	335.793i	1.42203	−	0.821010i	0.425559	−	0.904931i	$-0.360077\pi$
$409$	581.611	−	335.793i	1.42203	−	0.821010i	0.996472	+	0.0839200i	$0.0267440\pi$
$410$		0			0
$411$	161.418	+	93.1945i	0.392744	+	0.226751i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−320.534	+	555.181i	−0.772371	+	1.33779i
$416$		0			0
$417$	−134.191	−	232.425i	−0.321801	−	0.557375i
$418$		0			0
$419$		−	10.8053i		−	0.0257882i	−0.999917	−	0.0128941i	$-0.995896\pi$
$419$		−	10.8053i		−	0.0257882i	0.999917	−	0.0128941i	$-0.00410443\pi$
$420$		0			0
$421$	421.647			1.00154			0.500768	−	0.865581i	$-0.333051\pi$
$421$	421.647			1.00154			0.500768	+	0.865581i	$0.333051\pi$
$422$		0			0
$423$	−496.691	+	286.765i	−1.17421	+	0.677931i
$424$		0			0
$425$	−698.944	−	403.536i	−1.64457	−	0.949496i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	1.87215	−	3.24266i	0.00436399	−	0.00755864i
$430$		0			0
$431$	396.765	+	687.216i	0.920567	+	1.59447i	0.798539	+	0.601943i	$0.205606\pi$
$431$	396.765	+	687.216i	0.920567	+	1.59447i	0.122028	+	0.992527i	$0.461060\pi$
$432$		0			0
$433$		−	254.627i		−	0.588053i	−0.955797	−	0.294027i	$-0.905005\pi$
$433$		−	254.627i		−	0.588053i	0.955797	−	0.294027i	$-0.0949954\pi$
$434$		0			0
$435$	200.569			0.461077
$436$		0			0
$437$	−367.154	+	211.976i	−0.840169	+	0.485072i
$438$		0			0
$439$	727.043	+	419.758i	1.65613	+	0.956169i	0.974474	+	0.224500i	$0.0720749\pi$
$439$	727.043	+	419.758i	1.65613	+	0.956169i	0.681660	+	0.731669i	$0.261258\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	387.108	−	670.490i	0.873832	−	1.51352i	0.0158306	−	0.999875i	$-0.494961\pi$
$443$	387.108	−	670.490i	0.873832	−	1.51352i	0.858002	−	0.513647i	$-0.171706\pi$
$444$		0			0
$445$	8.97056	+	15.5375i	0.0201586	+	0.0349157i
$446$		0			0
$447$			363.954i			0.814215i
$448$		0			0
$449$	751.921			1.67466			0.837328	−	0.546700i	$-0.184116\pi$
$449$	751.921			1.67466			0.837328	+	0.546700i	$0.184116\pi$
$450$		0			0
$451$	−184.838	+	106.716i	−0.409840	+	0.236621i
$452$		0			0
$453$	−131.734	−	76.0565i	−0.290803	−	0.167895i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	125.995	−	218.230i	0.275700	−	0.477527i	−0.694611	−	0.719385i	$-0.744424\pi$
$457$	125.995	−	218.230i	0.275700	−	0.477527i	0.970312	+	0.241859i	$0.0777570\pi$
$458$		0			0
$459$	177.137	+	306.810i	0.385920	+	0.668432i
$460$		0			0
$461$		−	93.4121i		−	0.202629i	−0.994854	−	0.101315i	$-0.967695\pi$
$461$		−	93.4121i		−	0.202629i	0.994854	−	0.101315i	$-0.0323049\pi$
$462$		0			0
$463$	−167.990			−0.362829			−0.181415	−	0.983407i	$-0.558068\pi$
$463$	−167.990			−0.362829			−0.181415	+	0.983407i	$0.558068\pi$
$464$		0			0
$465$	−102.413	+	59.1283i	−0.220243	+	0.127158i
$466$		0			0
$467$	−315.389	−	182.090i	−0.675350	−	0.389914i	0.122750	−	0.992438i	$-0.460829\pi$
$467$	−315.389	−	182.090i	−0.675350	−	0.389914i	−0.798101	+	0.602524i	$0.794162\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−73.9756	+	128.130i	−0.157061	+	0.272037i
$472$		0			0
$473$	−214.215	−	371.032i	−0.452886	−	0.784422i
$474$		0			0
$475$		−	1392.10i		−	2.93073i
$476$		0			0
$477$	−403.588			−0.846096
$478$		0			0
$479$	−118.698	+	68.5303i	−0.247804	+	0.143069i	−0.618758	−	0.785582i	$-0.712364\pi$
$479$	−118.698	+	68.5303i	−0.247804	+	0.143069i	0.370955	+	0.928651i	$0.379031\pi$
$480$		0			0
$481$	9.99425	+	5.77018i	0.0207781	+	0.0119962i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	707.411	−	1225.27i	1.45858	−	2.52633i
$486$		0			0
$487$	278.706	+	482.732i	0.572291	+	0.991237i	0.996330	+	0.0855930i	$0.0272785\pi$
$487$	278.706	+	482.732i	0.572291	+	0.991237i	−0.424039	+	0.905644i	$0.639388\pi$
$488$		0			0
$489$			261.095i			0.533937i
$490$		0			0
$491$	537.647			1.09500			0.547502	−	0.836805i	$-0.315579\pi$
$491$	537.647			1.09500			0.547502	+	0.836805i	$0.315579\pi$
$492$		0			0
$493$	−188.009	+	108.547i	−0.381357	+	0.220176i
$494$		0			0
$495$	−479.042	−	276.575i	−0.967761	−	0.558737i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−316.990	+	549.043i	−0.635250	+	1.10029i	0.351212	+	0.936296i	$0.385770\pi$
$499$	−316.990	+	549.043i	−0.635250	+	1.10029i	−0.986462	+	0.163990i	$0.947564\pi$
$500$		0			0
$501$	72.6375	+	125.812i	0.144985	+	0.251122i
$502$		0			0
$503$		−	91.1385i		−	0.181190i	−0.995888	−	0.0905949i	$-0.971123\pi$
$503$		−	91.1385i		−	0.181190i	0.995888	−	0.0905949i	$-0.0288769\pi$
$504$		0			0
$505$	834.205			1.65189
$506$		0			0
$507$	−223.944	+	129.294i	−0.441704	+	0.255018i
$508$		0			0
$509$	306.024	+	176.683i	0.601227	+	0.347118i	0.769524	−	0.638618i	$-0.220494\pi$
$509$	306.024	+	176.683i	0.601227	+	0.347118i	−0.168297	+	0.985736i	$0.553827\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−305.539	+	529.209i	−0.595593	+	1.03160i
$514$		0			0
$515$	90.9117	+	157.464i	0.176528	+	0.305755i
$516$		0			0
$517$		−	802.434i		−	1.55210i
$518$		0			0
$519$	−279.696			−0.538912
$520$		0			0
$521$	−821.892	+	474.520i	−1.57753	+	0.910786i	−0.582325	+	0.812956i	$0.697857\pi$
$521$	−821.892	+	474.520i	−1.57753	+	0.910786i	−0.995203	+	0.0978305i	$0.968810\pi$
$522$		0			0
$523$	−573.917	−	331.351i	−1.09736	−	0.633558i	−0.161830	−	0.986819i	$-0.551740\pi$
$523$	−573.917	−	331.351i	−1.09736	−	0.633558i	−0.935525	+	0.353260i	$0.885073\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	64.0000	−	110.851i	0.121442	−	0.210344i
$528$		0			0
$529$	126.265	+	218.697i	0.238685	+	0.413415i
$530$		0			0
$531$			386.915i			0.728654i
$532$		0			0
$533$	−6.01847			−0.0112917
$534$		0			0
$535$	−109.080	+	62.9777i	−0.203889	+	0.117715i
$536$		0			0
$537$	270.512	+	156.180i	0.503746	+	0.290838i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−513.451	+	889.323i	−0.949077	+	1.64385i	−0.201703	+	0.979447i	$0.564648\pi$
$541$	−513.451	+	889.323i	−0.949077	+	1.64385i	−0.747374	+	0.664403i	$0.768686\pi$
$542$		0			0
$543$	−175.397	−	303.796i	−0.323015	−	0.559478i
$544$		0			0
$545$			207.474i			0.380687i
$546$		0			0
$547$	258.686			0.472918			0.236459	−	0.971641i	$-0.424013\pi$
$547$	258.686			0.472918			0.236459	+	0.971641i	$0.424013\pi$
$548$		0			0
$549$	−57.4904	+	33.1921i	−0.104718	+	0.0604591i
$550$		0			0
$551$	−324.291	−	187.230i	−0.588551	−	0.339800i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−300.049	+	519.700i	−0.540628	+	0.936396i
$556$		0			0
$557$	370.765	+	642.183i	0.665645	+	1.15293i	0.979110	+	0.203332i	$0.0651770\pi$
$557$	370.765	+	642.183i	0.665645	+	1.15293i	−0.313464	+	0.949600i	$0.601490\pi$
$558$		0			0
$559$		−	12.0811i		−	0.0216120i
$560$		0			0
$561$	−210.745			−0.375660
$562$		0			0
$563$	−69.7267	+	40.2568i	−0.123849	+	0.0715040i	−0.560644	−	0.828057i	$-0.689447\pi$
$563$	−69.7267	+	40.2568i	−0.123849	+	0.0715040i	0.436796	+	0.899561i	$0.356113\pi$
$564$		0			0
$565$	−105.974	−	61.1843i	−0.187565	−	0.108291i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	371.485	−	643.431i	0.652874	−	1.13081i	−0.329548	−	0.944139i	$-0.606896\pi$
$569$	371.485	−	643.431i	0.652874	−	1.13081i	0.982422	−	0.186672i	$-0.0597702\pi$
$570$		0			0
$571$	298.941	+	517.781i	0.523540	+	0.906797i	0.999625	+	0.0273980i	$0.00872213\pi$
$571$	298.941	+	517.781i	0.523540	+	0.906797i	−0.476085	+	0.879399i	$0.657945\pi$
$572$		0			0
$573$			14.6920i			0.0256404i
$574$		0			0
$575$	907.823			1.57882
$576$		0			0
$577$	719.466	−	415.384i	1.24691	−	0.719902i	0.276416	−	0.961038i	$-0.410853\pi$
$577$	719.466	−	415.384i	1.24691	−	0.719902i	0.970491	+	0.241136i	$0.0775199\pi$
$578$		0			0
$579$	−4.53554	−	2.61859i	−0.00783339	−	0.00452261i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	282.333	−	489.015i	0.484276	−	0.838791i
$584$		0			0
$585$	−7.79899	−	13.5082i	−0.0133316	−	0.0230910i
$586$		0			0
$587$			700.310i			1.19303i	0.802601	+	0.596516i	$0.203449\pi$
$587$			700.310i			1.19303i	−0.802601	+	0.596516i	$0.796551\pi$
$588$		0			0
$589$	220.784			0.374845
$590$		0			0
$591$	194.143	−	112.089i	0.328499	−	0.189659i
$592$		0			0
$593$	−361.176	−	208.525i	−0.609065	−	0.351644i	0.163534	−	0.986538i	$-0.447711\pi$
$593$	−361.176	−	208.525i	−0.609065	−	0.351644i	−0.772599	+	0.634894i	$0.781044\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−187.245	+	324.317i	−0.313643	+	0.543245i
$598$		0			0
$599$	111.946	+	193.896i	0.186888	+	0.323700i	0.944211	−	0.329340i	$-0.106826\pi$
$599$	111.946	+	193.896i	0.186888	+	0.323700i	−0.757323	+	0.653041i	$0.773493\pi$
$600$		0			0
$601$			721.185i			1.19998i	0.800009	+	0.599988i	$0.204828\pi$
$601$			721.185i			1.19998i	−0.800009	+	0.599988i	$0.795172\pi$
$602$		0			0
$603$	137.314			0.227718
$604$		0			0
$605$	−264.669	+	152.807i	−0.437470	+	0.252573i
$606$		0			0
$607$	277.556	+	160.247i	0.457258	+	0.263998i	0.710891	−	0.703303i	$-0.248292\pi$
$607$	277.556	+	160.247i	0.457258	+	0.263998i	−0.253633	+	0.967301i	$0.581625\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11.3137	−	19.5959i	0.0185167	−	0.0320719i
$612$		0			0
$613$	−97.6863	−	169.198i	−0.159358	−	0.276016i	0.775280	−	0.631618i	$-0.217609\pi$
$613$	−97.6863	−	169.198i	−0.159358	−	0.276016i	−0.934637	+	0.355603i	$0.884276\pi$
$614$		0			0
$615$		−	312.960i		−	0.508878i
$616$		0			0
$617$	838.284			1.35865			0.679323	−	0.733840i	$-0.262274\pi$
$617$	838.284			1.35865			0.679323	+	0.733840i	$0.262274\pi$
$618$		0			0
$619$	−446.355	+	257.703i	−0.721091	+	0.416322i	−0.815154	−	0.579244i	$-0.803348\pi$
$619$	−446.355	+	257.703i	−0.721091	+	0.416322i	0.0940632	+	0.995566i	$0.470014\pi$
$620$		0			0
$621$	−345.112	−	199.250i	−0.555735	−	0.320854i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−495.495	+	858.222i	−0.792792	+	1.37316i
$626$		0			0
$627$	−181.754	−	314.808i	−0.289879	−	0.502086i
$628$		0			0
$629$		−	649.541i		−	1.03266i
$630$		0			0
$631$	392.538			0.622089			0.311045	−	0.950395i	$-0.399321\pi$
$631$	392.538			0.622089			0.311045	+	0.950395i	$0.399321\pi$
$632$		0			0
$633$	412.304	−	238.044i	0.651349	−	0.376057i
$634$		0			0
$635$	1373.49	+	792.987i	2.16298	+	1.24880i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−262.848	+	455.266i	−0.411342	+	0.712466i
$640$		0			0
$641$	406.730	+	704.477i	0.634524	+	1.09903i	0.986616	+	0.163063i	$0.0521372\pi$
$641$	406.730	+	704.477i	0.634524	+	1.09903i	−0.352092	+	0.935966i	$0.614529\pi$
$642$		0			0
$643$			906.299i			1.40948i	0.709463	+	0.704742i	$0.248937\pi$
$643$			906.299i			1.40948i	−0.709463	+	0.704742i	$0.751063\pi$
$644$		0			0
$645$	628.215			0.973977
$646$		0			0
$647$	−99.0082	+	57.1624i	−0.153027	+	0.0883499i	−0.574558	−	0.818464i	$-0.694826\pi$
$647$	−99.0082	+	57.1624i	−0.153027	+	0.0883499i	0.421531	+	0.906814i	$0.361493\pi$
$648$		0			0
$649$	−468.813	−	270.670i	−0.722363	−	0.417056i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−207.843	+	359.994i	−0.318289	+	0.551293i	−0.980131	−	0.198351i	$-0.936442\pi$
$653$	−207.843	+	359.994i	−0.318289	+	0.551293i	0.661842	+	0.749643i	$0.269775\pi$
$654$		0			0
$655$	−502.357	−	870.109i	−0.766958	−	1.32841i
$656$		0			0
$657$		−	142.658i		−	0.217136i
$658$		0			0
$659$	−650.431			−0.986996			−0.493498	−	0.869747i	$-0.664282\pi$
$659$	−650.431			−0.986996			−0.493498	+	0.869747i	$0.664282\pi$
$660$		0			0
$661$	886.648	−	511.906i	1.34137	−	0.774442i	0.354364	−	0.935108i	$-0.384697\pi$
$661$	886.648	−	511.906i	1.34137	−	0.774442i	0.987009	+	0.160666i	$0.0513640\pi$
$662$		0			0
$663$	−5.14652	−	2.97135i	−0.00776248	−	0.00448167i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	122.098	−	211.479i	0.183055	−	0.317060i
$668$		0			0
$669$	88.9016	+	153.982i	0.132887	+	0.230168i
$670$		0			0
$671$		−	92.8791i		−	0.138419i
$672$		0			0
$673$	−1187.21			−1.76406			−0.882032	−	0.471190i	$-0.843824\pi$
$673$	−1187.21			−1.76406			−0.882032	+	0.471190i	$0.843824\pi$
$674$		0			0
$675$	1133.21	−	654.261i	1.67883	−	0.969275i
$676$		0			0
$677$	784.468	+	452.913i	1.15874	+	0.669000i	0.951002	−	0.309184i	$-0.100056\pi$
$677$	784.468	+	452.913i	1.15874	+	0.669000i	0.207740	+	0.978184i	$0.433389\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	45.2893	−	78.4434i	0.0665041	−	0.115189i
$682$		0			0
$683$	−157.627	−	273.019i	−0.230787	−	0.399735i	0.727253	−	0.686369i	$-0.240797\pi$
$683$	−157.627	−	273.019i	−0.230787	−	0.399735i	−0.958040	+	0.286635i	$0.907463\pi$
$684$		0			0
$685$			1086.35i			1.58592i
$686$		0			0
$687$	196.950			0.286682
$688$		0			0
$689$	13.7895	−	7.96136i	0.0200138	−	0.0115550i
$690$		0			0
$691$	290.084	+	167.480i	0.419803	+	0.242374i	0.694993	−	0.719016i	$-0.255407\pi$
$691$	290.084	+	167.480i	0.419803	+	0.242374i	−0.275190	+	0.961390i	$0.588741\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	782.122	−	1354.67i	1.12536	−	1.94917i
$696$		0			0
$697$	169.373	+	293.362i	0.243002	+	0.420892i
$698$		0			0
$699$			136.206i			0.194858i
$700$		0			0
$701$	−362.353			−0.516909			−0.258455	−	0.966023i	$-0.583213\pi$
$701$	−362.353			−0.516909			−0.258455	+	0.966023i	$0.583213\pi$
$702$		0			0
$703$	970.274	−	560.188i	1.38019	−	0.796854i
$704$		0			0
$705$	1018.99	+	588.312i	1.44537	+	0.834484i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	558.980	−	968.181i	0.788406	−	1.36556i	−0.138537	−	0.990357i	$-0.544240\pi$
$709$	558.980	−	968.181i	0.788406	−	1.36556i	0.926943	−	0.375202i	$-0.122427\pi$
$710$		0			0
$711$	−300.505	−	520.489i	−0.422651	−	0.732052i
$712$		0			0
$713$			143.979i			0.201934i
$714$		0			0
$715$	21.8234			0.0305222
$716$		0			0
$717$	−9.46131	+	5.46249i	−0.0131957	+	0.00761853i
$718$		0			0
$719$	−842.271	−	486.285i	−1.17145	−	0.676336i	−0.217427	−	0.976077i	$-0.569766\pi$
$719$	−842.271	−	486.285i	−1.17145	−	0.676336i	−0.954021	+	0.299741i	$0.903100\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	281.235	−	487.113i	0.388983	−	0.673738i
$724$		0			0
$725$	400.921	+	694.415i	0.552994	+	0.957814i
$726$		0			0
$727$		−	660.646i		−	0.908729i	−0.890816	−	0.454365i	$-0.849866\pi$
$727$		−	660.646i		−	0.908729i	0.890816	−	0.454365i	$-0.150134\pi$
$728$		0			0
$729$	−175.569			−0.240835
$730$		0			0
$731$	−588.876	+	339.988i	−0.805576	+	0.465099i
$732$		0			0
$733$	223.626	+	129.111i	0.305084	+	0.176140i	0.644724	−	0.764415i	$-0.276972\pi$
$733$	223.626	+	129.111i	0.305084	+	0.176140i	−0.339641	+	0.940555i	$0.610305\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−96.0589	+	166.379i	−0.130338	+	0.225751i
$738$		0			0
$739$	−166.696	−	288.725i	−0.225569	−	0.390697i	0.730921	−	0.682462i	$-0.239091\pi$
$739$	−166.696	−	288.725i	−0.225569	−	0.390697i	−0.956490	+	0.291765i	$0.905757\pi$
$740$		0			0
$741$		−	10.2504i		−	0.0138332i
$742$		0			0
$743$	610.118			0.821154			0.410577	−	0.911826i	$-0.365327\pi$
$743$	610.118			0.821154			0.410577	+	0.911826i	$0.365327\pi$
$744$		0			0
$745$	−1837.08	+	1060.64i	−2.46588	+	1.42368i
$746$		0			0
$747$	414.241	+	239.162i	0.554539	+	0.320163i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−9.05887	+	15.6904i	−0.0120624	+	0.0208927i	−0.871994	−	0.489517i	$-0.837173\pi$
$751$	−9.05887	+	15.6904i	−0.0120624	+	0.0208927i	0.859931	+	0.510410i	$0.170506\pi$
$752$		0			0
$753$	−135.466	−	234.634i	−0.179902	−	0.311599i
$754$		0			0
$755$		−	886.579i		−	1.17428i
$756$		0			0
$757$	107.137			0.141529			0.0707643	−	0.997493i	$-0.477456\pi$
$757$	107.137			0.141529			0.0707643	+	0.997493i	$0.477456\pi$
$758$		0			0
$759$	205.295	−	118.527i	0.270480	−	0.156162i
$760$		0			0
$761$	777.937	+	449.142i	1.02226	+	0.590200i	0.914757	−	0.404005i	$-0.132382\pi$
$761$	777.937	+	449.142i	1.02226	+	0.590200i	0.107500	+	0.994205i	$0.465716\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−438.960	+	760.302i	−0.573805	+	0.993859i
$766$		0			0
$767$	−7.63247	−	13.2198i	−0.00995107	−	0.0172358i
$768$		0			0
$769$		−	929.350i		−	1.20852i	−0.796788	−	0.604259i	$-0.793469\pi$
$769$		−	929.350i		−	1.20852i	0.796788	−	0.604259i	$-0.206531\pi$
$770$		0			0
$771$	42.0387			0.0545249
$772$		0			0
$773$	791.889	−	457.198i	1.02444	−	0.591459i	0.109051	−	0.994036i	$-0.465219\pi$
$773$	791.889	−	457.198i	1.02444	−	0.591459i	0.915386	+	0.402577i	$0.131886\pi$
$774$		0			0
$775$	−409.432	−	236.386i	−0.528299	−	0.305014i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−292.146	+	506.012i	−0.375027	+	0.649566i
$780$		0			0
$781$	−367.754	−	636.969i	−0.470876	−	0.815582i
$782$		0			0
$783$		−	351.979i		−	0.449526i
$784$		0			0
$785$	−862.323			−1.09850
$786$		0			0
$787$	−659.669	+	380.860i	−0.838207	+	0.483939i	−0.856654	−	0.515891i	$-0.827461\pi$
$787$	−659.669	+	380.860i	−0.838207	+	0.483939i	0.0184477	+	0.999830i	$0.494128\pi$
$788$		0			0
$789$	−102.270	−	59.0459i	−0.129620	−	0.0748364i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.30952	−	2.26816i	0.00165135	−	0.00286023i
$794$		0			0
$795$	413.990	+	717.052i	0.520742	+	0.901952i
$796$		0			0
$797$			395.267i			0.495943i	0.968767	+	0.247972i	$0.0797639\pi$
$797$			395.267i			0.495943i	−0.968767	+	0.247972i	$0.920236\pi$
$798$		0			0
$799$	−1273.57			−1.59395
$800$		0			0
$801$	11.5931	−	6.69326i	0.0144732	−	0.00835613i
$802$		0			0
$803$	172.855	+	99.7977i	0.215261	+	0.124281i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−26.3087	+	45.5679i	−0.0326006	+	0.0564658i
$808$		0			0
$809$	438.897	+	760.191i	0.542517	+	0.939668i	0.998759	+	0.0498115i	$0.0158621\pi$
$809$	438.897	+	760.191i	0.542517	+	0.939668i	−0.456241	+	0.889856i	$0.650805\pi$
$810$		0			0
$811$			851.717i			1.05021i	0.851039	+	0.525103i	$0.175973\pi$
$811$			851.717i			1.05021i	−0.851039	+	0.525103i	$0.824027\pi$
$812$		0			0
$813$	−310.627			−0.382076
$814$		0			0
$815$	−1317.89	+	760.887i	−1.61705	+	0.933603i
$816$		0			0
$817$	−1015.74	−	586.436i	−1.24325	−	0.717792i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−218.519	+	378.486i	−0.266162	+	0.461006i	−0.967867	−	0.251461i	$-0.919089\pi$
$821$	−218.519	+	378.486i	−0.266162	+	0.461006i	0.701705	+	0.712467i	$0.252422\pi$
$822$		0			0
$823$	−796.172	−	1379.01i	−0.967402	−	1.67559i	−0.703019	−	0.711171i	$-0.748165\pi$
$823$	−796.172	−	1379.01i	−0.967402	−	1.67559i	−0.264383	−	0.964418i	$-0.585168\pi$
$824$		0			0
$825$			778.393i			0.943507i
$826$		0			0
$827$	−1182.45			−1.42981			−0.714904	−	0.699223i	$-0.753530\pi$
$827$	−1182.45			−1.42981			−0.714904	+	0.699223i	$0.753530\pi$
$828$		0			0
$829$	−907.637	+	524.025i	−1.09486	+	0.632117i	−0.934866	−	0.355002i	$-0.884480\pi$
$829$	−907.637	+	524.025i	−1.09486	+	0.632117i	−0.159992	+	0.987118i	$0.551147\pi$
$830$		0			0
$831$	−336.846	−	194.478i	−0.405350	−	0.234029i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−423.362	+	733.285i	−0.507021	+	0.878186i
$836$		0			0
$837$	103.765	+	179.725i	0.123972	+	0.214726i
$838$		0			0
$839$		−	585.611i		−	0.697986i	−0.937125	−	0.348993i	$-0.886524\pi$
$839$		−	585.611i		−	0.697986i	0.937125	−	0.348993i	$-0.113476\pi$
$840$		0			0
$841$	−625.313			−0.743535
$842$		0			0
$843$	293.671	−	169.551i	0.348364	−	0.201128i
$844$		0			0
$845$	−1305.24	−	753.582i	−1.54467	−	0.891813i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	217.200	−	376.202i	0.255831	−	0.443112i
$850$		0			0
$851$	365.314	+	632.742i	0.429276	+	0.743528i
$852$		0			0
$853$		−	901.189i		−	1.05649i	−0.849091	−	0.528247i	$-0.822850\pi$
$853$		−	901.189i		−	1.05649i	0.849091	−	0.528247i	$-0.177150\pi$
$854$		0			0
$855$	−1514.30			−1.77111
$856$		0			0
$857$	836.578	−	482.999i	0.976171	−	0.563592i	0.0750587	−	0.997179i	$-0.476086\pi$
$857$	836.578	−	482.999i	0.976171	−	0.563592i	0.901112	+	0.433587i	$0.142752\pi$
$858$		0			0
$859$	−217.226	−	125.415i	−0.252882	−	0.146001i	0.368201	−	0.929746i	$-0.379974\pi$
$859$	−217.226	−	125.415i	−0.252882	−	0.146001i	−0.621083	+	0.783745i	$0.713307\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	748.563	−	1296.55i	0.867397	−	1.50238i	0.00274946	−	0.999996i	$-0.499125\pi$
$863$	748.563	−	1296.55i	0.867397	−	1.50238i	0.864647	−	0.502379i	$-0.167542\pi$
$864$		0			0
$865$	−815.092	−	1411.78i	−0.942303	−	1.63212i
$866$		0			0
$867$		−	107.902i		−	0.124454i
$868$		0			0
$869$	840.881			0.967643
$870$		0			0
$871$	−4.69163	+	2.70871i	−0.00538649	+	0.00310989i
$872$		0			0
$873$	−914.220	−	527.825i	−1.04722	−	0.604611i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−70.6771	+	122.416i	−0.0805896	+	0.139585i	−0.903503	−	0.428581i	$-0.859014\pi$
$877$	−70.6771	+	122.416i	−0.0805896	+	0.139585i	0.822914	+	0.568166i	$0.192347\pi$
$878$		0			0
$879$	−304.672	−	527.707i	−0.346612	−	0.600350i
$880$		0			0
$881$		−	1413.83i		−	1.60480i	−0.596788	−	0.802399i	$-0.703557\pi$
$881$		−	1413.83i		−	1.60480i	0.596788	−	0.802399i	$-0.296443\pi$
$882$		0			0
$883$	1386.04			1.56969			0.784845	−	0.619692i	$-0.212742\pi$
$883$	1386.04			1.56969			0.784845	+	0.619692i	$0.212742\pi$
$884$		0			0
$885$	687.429	−	396.887i	0.776756	−	0.448460i
$886$		0			0
$887$	−262.921	−	151.798i	−0.296416	−	0.171136i	0.344416	−	0.938817i	$-0.388077\pi$
$887$	−262.921	−	151.798i	−0.296416	−	0.171136i	−0.640832	+	0.767681i	$0.721410\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−108.157	+	187.334i	−0.121389	+	0.210251i
$892$		0			0
$893$	−1098.37	−	1902.44i	−1.22998	−	2.13039i
$894$		0			0
$895$			1820.57i			2.03415i
$896$		0			0
$897$	6.68456			0.00745213
$898$		0			0
$899$	−110.133	+	63.5853i	−0.122506	+	0.0707289i
$900$		0			0
$901$	−776.131	−	448.099i	−0.861411	−	0.497336i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1022.29	−	1770.66i	1.12960	−	1.95653i
$906$		0			0
$907$	652.009	+	1129.31i	0.718864	+	1.24511i	0.961450	+	0.274978i	$0.0886707\pi$
$907$	652.009	+	1129.31i	0.718864	+	1.24511i	−0.242587	+	0.970130i	$0.577996\pi$
$908$		0			0
$909$		−	622.431i		−	0.684742i
$910$		0			0
$911$	466.118			0.511655			0.255828	−	0.966722i	$-0.417652\pi$
$911$	466.118			0.511655			0.255828	+	0.966722i	$0.417652\pi$
$912$		0			0
$913$	−579.571	+	334.615i	−0.634798	+	0.366501i
$914$		0			0
$915$	117.944	+	68.0951i	0.128901	+	0.0744209i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−589.171	+	1020.47i	−0.641100	+	1.11042i	0.344088	+	0.938937i	$0.388188\pi$
$919$	−589.171	+	1020.47i	−0.641100	+	1.11042i	−0.985188	+	0.171480i	$0.945145\pi$
$920$		0			0
$921$	128.368	+	222.339i	0.139378	+	0.241411i
$922$		0			0
$923$		−	20.7402i		−	0.0224705i
$924$		0			0
$925$	−2399.10			−2.59362
$926$		0			0
$927$	117.489	−	67.8325i	0.126741	−	0.0731742i
$928$		0			0
$929$	789.089	+	455.581i	0.849396	+	0.490399i	0.860447	−	0.509540i	$-0.170184\pi$
$929$	789.089	+	455.581i	0.849396	+	0.490399i	−0.0110511	+	0.999939i	$0.503518\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	217.442	−	376.620i	0.233056	−	0.403665i
$934$		0			0
$935$	−614.156	−	1063.75i	−0.656852	−	1.13770i
$936$		0			0
$937$			1035.35i			1.10496i	0.833527	+	0.552479i	$0.186318\pi$
$937$			1035.35i			1.10496i	−0.833527	+	0.552479i	$0.813682\pi$
$938$		0			0
$939$	−787.273			−0.838417
$940$		0			0
$941$	55.8135	−	32.2239i	0.0593130	−	0.0342443i	−0.470050	−	0.882640i	$-0.655764\pi$
$941$	55.8135	−	32.2239i	0.0593130	−	0.0342443i	0.529363	+	0.848395i	$0.322431\pi$
$942$		0			0
$943$	−329.984	−	190.516i	−0.349930	−	0.202032i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	66.1371	−	114.553i	0.0698385	−	0.120964i	−0.828992	−	0.559261i	$-0.811085\pi$
$947$	66.1371	−	114.553i	0.0698385	−	0.120964i	0.898830	+	0.438297i	$0.144418\pi$
$948$		0			0
$949$	2.81414	+	4.87424i	0.00296538	+	0.00513618i
$950$		0			0
$951$		−	28.4235i		−	0.0298880i
$952$		0			0
$953$	1346.86			1.41329			0.706643	−	0.707570i	$-0.250209\pi$
$953$	1346.86			1.41329			0.706643	+	0.707570i	$0.250209\pi$
$954$		0			0
$955$	−74.1586	+	42.8155i	−0.0776530	+	0.0448330i
$956$		0			0
$957$	181.328	+	104.690i	0.189476	+	0.109394i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−443.010	+	767.315i	−0.460988	+	0.798455i
$962$		0			0
$963$	46.9899	+	81.3889i	0.0487953	+	0.0845160i
$964$		0			0
$965$		−	30.5246i		−	0.0316317i
$966$		0			0
$967$	498.979			0.516007			0.258004	−	0.966144i	$-0.416935\pi$
$967$	498.979			0.516007			0.258004	+	0.966144i	$0.416935\pi$
$968$		0			0
$969$	−499.641	+	288.468i	−0.515626	+	0.297697i
$970$		0			0
$971$	−1558.20	−	899.628i	−1.60474	−	0.926496i	−0.990521	−	0.137359i	$-0.956138\pi$
$971$	−1558.20	−	899.628i	−1.60474	−	0.926496i	−0.614217	−	0.789137i	$-0.710528\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−10.9747	+	19.0088i	−0.0112562	+	0.0194962i
$976$		0			0
$977$	−690.803	−	1196.51i	−0.707066	−	1.22467i	−0.965941	−	0.258763i	$-0.916685\pi$
$977$	−690.803	−	1196.51i	−0.707066	−	1.22467i	0.258875	−	0.965911i	$-0.416648\pi$
$978$		0			0
$979$			18.7293i			0.0191310i
$980$		0			0
$981$	154.804			0.157802
$982$		0			0
$983$	1131.61	−	653.337i	1.15118	−	0.664636i	0.202008	−	0.979384i	$-0.435253\pi$
$983$	1131.61	−	653.337i	1.15118	−	0.664636i	0.949175	+	0.314747i	$0.101920\pi$
$984$		0			0
$985$	1131.55	+	653.300i	1.14878	+	0.663249i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	382.431	−	662.389i	0.386684	−	0.669757i
$990$		0			0
$991$	516.994	+	895.460i	0.521689	+	0.903592i	0.999682	+	0.0252284i	$0.00803130\pi$
$991$	516.994	+	895.460i	0.521689	+	0.903592i	−0.477992	+	0.878364i	$0.658635\pi$
$992$		0			0
$993$			297.472i			0.299569i
$994$		0			0
$995$	−2182.68			−2.19365
$996$		0			0
$997$	−339.049	+	195.750i	−0.340069	+	0.196339i	−0.660302	−	0.751000i	$-0.729572\pi$
$997$	−339.049	+	195.750i	−0.340069	+	0.196339i	0.320234	+	0.947339i	$0.396239\pi$
$998$		0			0
$999$	912.023	+	526.557i	0.912936	+	0.527084i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.3.s.f.705.2		8
4.3	odd	2		392.3.o.b.313.3		8
7.2	even	3		112.3.c.c.97.2		4
7.3	odd	6	inner	784.3.s.f.129.2		8
7.4	even	3	inner	784.3.s.f.129.3		8
7.5	odd	6		112.3.c.c.97.3		4
7.6	odd	2	inner	784.3.s.f.705.3		8
21.2	odd	6		1008.3.f.h.433.1		4
21.5	even	6		1008.3.f.h.433.4		4
28.3	even	6		392.3.o.b.129.3		8
28.11	odd	6		392.3.o.b.129.2		8
28.19	even	6		56.3.c.a.41.2	✓	4
28.23	odd	6		56.3.c.a.41.3	yes	4
28.27	even	2		392.3.o.b.313.2		8
56.5	odd	6		448.3.c.e.321.2		4
56.19	even	6		448.3.c.f.321.3		4
56.37	even	6		448.3.c.e.321.3		4
56.51	odd	6		448.3.c.f.321.2		4
84.23	even	6		504.3.f.a.433.1		4
84.47	odd	6		504.3.f.a.433.4		4
140.19	even	6		1400.3.f.a.601.3		4
140.23	even	12		1400.3.p.a.1049.3		8
140.47	odd	12		1400.3.p.a.1049.4		8
140.79	odd	6		1400.3.f.a.601.2		4
140.103	odd	12		1400.3.p.a.1049.5		8
140.107	even	12		1400.3.p.a.1049.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.c.a.41.2	✓	4	28.19	even	6
56.3.c.a.41.3	yes	4	28.23	odd	6
112.3.c.c.97.2		4	7.2	even	3
112.3.c.c.97.3		4	7.5	odd	6
392.3.o.b.129.2		8	28.11	odd	6
392.3.o.b.129.3		8	28.3	even	6
392.3.o.b.313.2		8	28.27	even	2
392.3.o.b.313.3		8	4.3	odd	2
448.3.c.e.321.2		4	56.5	odd	6
448.3.c.e.321.3		4	56.37	even	6
448.3.c.f.321.2		4	56.51	odd	6
448.3.c.f.321.3		4	56.19	even	6
504.3.f.a.433.1		4	84.23	even	6
504.3.f.a.433.4		4	84.47	odd	6
784.3.s.f.129.2		8	7.3	odd	6	inner
784.3.s.f.129.3		8	7.4	even	3	inner
784.3.s.f.705.2		8	1.1	even	1	trivial
784.3.s.f.705.3		8	7.6	odd	2	inner
1008.3.f.h.433.1		4	21.2	odd	6
1008.3.f.h.433.4		4	21.5	even	6
1400.3.f.a.601.2		4	140.79	odd	6
1400.3.f.a.601.3		4	140.19	even	6
1400.3.p.a.1049.3		8	140.23	even	12
1400.3.p.a.1049.4		8	140.47	odd	12
1400.3.p.a.1049.5		8	140.103	odd	12
1400.3.p.a.1049.6		8	140.107	even	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	784.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.32565 + 0.765367i) q^{3} +(-7.72648 - 4.46088i) q^{5} +(-3.32843 + 5.76500i) q^{9} +O(q^{10})\)	\(q+(-1.32565 + 0.765367i) q^{3} +(-7.72648 - 4.46088i) q^{5} +(-3.32843 + 5.76500i) q^{9} +(-4.65685 - 8.06591i) q^{11} -0.262632i q^{13} +13.6569 q^{15} +(-12.8017 + 7.39104i) q^{17} +(-22.0812 - 12.7486i) q^{19} +(8.31371 - 14.3998i) q^{23} +(27.2990 + 47.2832i) q^{25} -23.9665i q^{27} +14.6863 q^{29} +(-7.49903 + 4.32957i) q^{31} +(12.3468 + 7.12840i) q^{33} +(-21.9706 + 38.0541i) q^{37} +(0.201010 + 0.348160i) q^{39} -22.9159i q^{41} +46.0000 q^{43} +(51.4340 - 29.6955i) q^{45} +(74.6135 + 43.0781i) q^{47} +(11.3137 - 19.5959i) q^{51} +(30.3137 + 52.5049i) q^{53} +83.0948i q^{55} +39.0294 q^{57} +(50.3358 - 29.0614i) q^{59} +(8.63626 + 4.98615i) q^{61} +(-1.17157 + 2.02922i) q^{65} +(-10.3137 - 17.8639i) q^{67} +25.4521i q^{69} +78.9706 q^{71} +(-18.5592 + 10.7151i) q^{73} +(-72.3780 - 41.7875i) q^{75} +(-45.1421 + 78.1885i) q^{79} +(-11.6127 - 20.1138i) q^{81} -71.8544i q^{83} +131.882 q^{85} +(-19.4689 + 11.2404i) q^{87} +(-1.74152 - 1.00547i) q^{89} +(6.62742 - 11.4790i) q^{93} +(113.740 + 197.004i) q^{95} +158.581i q^{97} +62.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^9	\( 8 q - 4 q^{9} + 8 q^{11} + 64 q^{15} - 24 q^{23} + 60 q^{25} + 208 q^{29} - 40 q^{37} + 160 q^{39} + 368 q^{43} + 152 q^{53} + 448 q^{57} - 32 q^{65} + 8 q^{67} + 496 q^{71} - 248 q^{79} + 156 q^{81} + 512 q^{85} - 128 q^{93} + 480 q^{95} + 496 q^{99}+O(q^{100}) \) 8 * q - 4 * q^9 + 8 * q^11 + 64 * q^15 - 24 * q^23 + 60 * q^25 + 208 * q^29 - 40 * q^37 + 160 * q^39 + 368 * q^43 + 152 * q^53 + 448 * q^57 - 32 * q^65 + 8 * q^67 + 496 * q^71 - 248 * q^79 + 156 * q^81 + 512 * q^85 - 128 * q^93 + 480 * q^95 + 496 * q^99

Introduction

L-functions

Modular forms

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Database

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