LMFDB - Embedded newform 448.3.c.e.321.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,3,Mod(321,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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("448.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.c (of order $2$, degree $1$, 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:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.2
Root			$-0.765367i$ of defining polynomial
Character	$\chi$	$=$	448.321
Dual form			448.3.c.e.321.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53073i q^{3} +8.92177i q^{5} +(-3.82843 - 5.86030i) q^{7} +6.65685 q^{9} +O(q^{10})$	$q-1.53073i q^{3} +8.92177i q^{5} +(-3.82843 - 5.86030i) q^{7} +6.65685 q^{9} -9.31371 q^{11} -0.262632i q^{13} +13.6569 q^{15} +14.7821i q^{17} +25.4972i q^{19} +(-8.97056 + 5.86030i) q^{21} -16.6274 q^{23} -54.5980 q^{25} -23.9665i q^{27} -14.6863 q^{29} +8.65914i q^{31} +14.2568i q^{33} +(52.2843 - 34.1563i) q^{35} -43.9411 q^{37} -0.402020 q^{39} +22.9159i q^{41} -46.0000 q^{43} +59.3909i q^{45} +86.1562i q^{47} +(-19.6863 + 44.8715i) q^{49} +22.6274 q^{51} +60.6274 q^{53} -83.0948i q^{55} +39.0294 q^{57} +58.1228i q^{59} -9.97230i q^{61} +(-25.4853 - 39.0112i) q^{63} +2.34315 q^{65} -20.6274 q^{67} +25.4521i q^{69} +78.9706 q^{71} +21.4303i q^{73} +83.5750i q^{75} +(35.6569 + 54.5811i) q^{77} +90.2843 q^{79} +23.2254 q^{81} -71.8544i q^{83} -131.882 q^{85} +22.4808i q^{87} -2.01094i q^{89} +(-1.53911 + 1.00547i) q^{91} +13.2548 q^{93} -227.480 q^{95} -158.581i q^{97} -62.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{7} + 4 q^{9} + 8 q^{11} + 32 q^{15} + 32 q^{21} + 24 q^{23} - 60 q^{25} - 104 q^{29} + 96 q^{35} - 40 q^{37} - 160 q^{39} - 184 q^{43} - 124 q^{49} + 152 q^{53} + 224 q^{57} - 68 q^{63} + 32 q^{65} + 8 q^{67} + 248 q^{71} + 120 q^{77} + 248 q^{79} - 156 q^{81} - 256 q^{85} + 288 q^{91} - 128 q^{93} - 480 q^{95} - 248 q^{99}+O(q^{100}) $ 4 * q - 4 * q^7 + 4 * q^9 + 8 * q^11 + 32 * q^15 + 32 * q^21 + 24 * q^23 - 60 * q^25 - 104 * q^29 + 96 * q^35 - 40 * q^37 - 160 * q^39 - 184 * q^43 - 124 * q^49 + 152 * q^53 + 224 * q^57 - 68 * q^63 + 32 * q^65 + 8 * q^67 + 248 * q^71 + 120 * q^77 + 248 * q^79 - 156 * q^81 - 256 * q^85 + 288 * q^91 - 128 * q^93 - 480 * q^95 - 248 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	1.53073i		−	0.510245i	−0.966909	−	0.255122i	$-0.917884\pi$
$3$		−	1.53073i		−	0.510245i	0.966909	−	0.255122i	$-0.0821157\pi$
$4$		0			0
$5$			8.92177i			1.78435i	0.451686	+	0.892177i	$0.350823\pi$
$5$			8.92177i			1.78435i	−0.451686	+	0.892177i	$0.649177\pi$
$6$		0			0
$7$	−3.82843	−	5.86030i	−0.546918	−	0.837186i
$7$	−3.82843	−	5.86030i	−0.546918	−	0.837186i
$8$		0			0
$9$	6.65685			0.739650
$10$		0			0
$11$	−9.31371			−0.846701			−0.423350	−	0.905966i	$-0.639146\pi$
$11$	−9.31371			−0.846701			−0.423350	+	0.905966i	$0.639146\pi$
$12$		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
$17$			14.7821i			0.869534i	0.900543	+	0.434767i	$0.143169\pi$
$17$			14.7821i			0.869534i	−0.900543	+	0.434767i	$0.856831\pi$
$18$		0			0
$19$			25.4972i			1.34196i	0.741476	+	0.670979i	$0.234126\pi$
$19$			25.4972i			1.34196i	−0.741476	+	0.670979i	$0.765874\pi$
$20$		0			0
$21$	−8.97056	+	5.86030i	−0.427170	+	0.279062i
$22$		0			0
$23$	−16.6274			−0.722931			−0.361466	−	0.932385i	$-0.617724\pi$
$23$	−16.6274			−0.722931			−0.361466	+	0.932385i	$0.617724\pi$
$24$		0			0
$25$	−54.5980			−2.18392
$26$		0			0
$27$		−	23.9665i		−	0.887647i
$28$		0			0
$29$	−14.6863			−0.506424			−0.253212	−	0.967411i	$-0.581487\pi$
$29$	−14.6863			−0.506424			−0.253212	+	0.967411i	$0.581487\pi$
$30$		0			0
$31$			8.65914i			0.279327i	0.990199	+	0.139664i	$0.0446021\pi$
$31$			8.65914i			0.279327i	−0.990199	+	0.139664i	$0.955398\pi$
$32$		0			0
$33$			14.2568i			0.432024i
$34$		0			0
$35$	52.2843	−	34.1563i	1.49384	−	0.975896i
$36$		0			0
$37$	−43.9411			−1.18760			−0.593799	−	0.804613i	$-0.702373\pi$
$37$	−43.9411			−1.18760			−0.593799	+	0.804613i	$0.702373\pi$
$38$		0			0
$39$	−0.402020			−0.0103082
$40$		0			0
$41$			22.9159i			0.558925i	0.960157	+	0.279463i	$0.0901563\pi$
$41$			22.9159i			0.558925i	−0.960157	+	0.279463i	$0.909844\pi$
$42$		0			0
$43$	−46.0000			−1.06977			−0.534884	−	0.844926i	$-0.679645\pi$
$43$	−46.0000			−1.06977			−0.534884	+	0.844926i	$0.679645\pi$
$44$		0			0
$45$			59.3909i			1.31980i
$46$		0			0
$47$			86.1562i			1.83311i	0.399907	+	0.916556i	$0.369042\pi$
$47$			86.1562i			1.83311i	−0.399907	+	0.916556i	$0.630958\pi$
$48$		0			0
$49$	−19.6863	+	44.8715i	−0.401761	+	0.915745i
$50$		0			0
$51$	22.6274			0.443675
$52$		0			0
$53$	60.6274			1.14391			0.571957	−	0.820284i	$-0.306185\pi$
$53$	60.6274			1.14391			0.571957	+	0.820284i	$0.306185\pi$
$54$		0			0
$55$		−	83.0948i		−	1.51081i
$56$		0			0
$57$	39.0294			0.684727
$58$		0			0
$59$			58.1228i			0.985133i	0.870275	+	0.492566i	$0.163941\pi$
$59$			58.1228i			0.985133i	−0.870275	+	0.492566i	$0.836059\pi$
$60$		0			0
$61$		−	9.97230i		−	0.163480i	−0.996654	−	0.0817402i	$-0.973952\pi$
$61$		−	9.97230i		−	0.163480i	0.996654	−	0.0817402i	$-0.0260478\pi$
$62$		0			0
$63$	−25.4853	−	39.0112i	−0.404528	−	0.619225i
$64$		0			0
$65$	2.34315			0.0360484
$66$		0			0
$67$	−20.6274			−0.307872			−0.153936	−	0.988081i	$-0.549195\pi$
$67$	−20.6274			−0.307872			−0.153936	+	0.988081i	$0.549195\pi$
$68$		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
$73$			21.4303i			0.293565i	0.989169	+	0.146783i	$0.0468918\pi$
$73$			21.4303i			0.293565i	−0.989169	+	0.146783i	$0.953108\pi$
$74$		0			0
$75$			83.5750i			1.11433i
$76$		0			0
$77$	35.6569	+	54.5811i	0.463076	+	0.708846i
$78$		0			0
$79$	90.2843			1.14284			0.571419	−	0.820658i	$-0.306393\pi$
$79$	90.2843			1.14284			0.571419	+	0.820658i	$0.306393\pi$
$80$		0			0
$81$	23.2254			0.286733
$82$		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
$87$			22.4808i			0.258400i
$88$		0			0
$89$		−	2.01094i		−	0.0225948i	−0.999936	−	0.0112974i	$-0.996404\pi$
$89$		−	2.01094i		−	0.0225948i	0.999936	−	0.0112974i	$-0.00359615\pi$
$90$		0			0
$91$	−1.53911	+	1.00547i	−0.0169132	+	0.0110491i
$92$		0			0
$93$	13.2548			0.142525
$94$		0			0
$95$	−227.480			−2.39453
$96$		0			0
$97$		−	158.581i		−	1.63485i	−0.576032	−	0.817427i	$-0.695399\pi$
$97$		−	158.581i		−	1.63485i	0.576032	−	0.817427i	$-0.304601\pi$
$98$		0			0
$99$	−62.0000			−0.626263
$100$		0			0
$101$		−	93.5022i		−	0.925764i	−0.886420	−	0.462882i	$-0.846815\pi$
$101$		−	93.5022i		−	0.925764i	0.886420	−	0.462882i	$-0.153185\pi$
$102$		0			0
$103$		−	20.3797i		−	0.197862i	−0.995094	−	0.0989308i	$-0.968458\pi$
$103$		−	20.3797i		−	0.197862i	0.995094	−	0.0989308i	$-0.0315422\pi$
$104$		0			0
$105$	−52.2843	−	80.0333i	−0.497945	−	0.762222i
$106$		0			0
$107$	14.1177			0.131942			0.0659708	−	0.997822i	$-0.478986\pi$
$107$	14.1177			0.131942			0.0659708	+	0.997822i	$0.478986\pi$
$108$		0			0
$109$	−23.2548			−0.213347			−0.106674	−	0.994294i	$-0.534020\pi$
$109$	−23.2548			−0.213347			−0.106674	+	0.994294i	$0.534020\pi$
$110$		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
$115$		−	148.346i		−	1.28997i
$116$		0			0
$117$		−	1.74831i		−	0.0149428i
$118$		0			0
$119$	86.6274	−	56.5921i	0.727961	−	0.475564i
$120$		0			0
$121$	−34.2548			−0.283098
$122$		0			0
$123$	35.0782			0.285189
$124$		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
$129$			70.4138i			0.545843i
$130$		0			0
$131$		−	112.614i		−	0.859648i	−0.902913	−	0.429824i	$-0.858576\pi$
$131$		−	112.614i		−	0.859648i	0.902913	−	0.429824i	$-0.141424\pi$
$132$		0			0
$133$	149.421	−	97.6142i	1.12347	−	0.733941i
$134$		0			0
$135$	213.823			1.58388
$136$		0			0
$137$	121.765			0.888792			0.444396	−	0.895830i	$-0.353418\pi$
$137$	121.765			0.888792			0.444396	+	0.895830i	$0.353418\pi$
$138$		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
$143$			2.44608i			0.0171055i
$144$		0			0
$145$		−	131.028i		−	0.903639i
$146$		0			0
$147$	68.6863	+	30.1345i	0.467254	+	0.204996i
$148$		0			0
$149$	237.765			1.59573			0.797867	−	0.602833i	$-0.205961\pi$
$149$	237.765			1.59573			0.797867	+	0.602833i	$0.205961\pi$
$150$		0			0
$151$	−99.3726			−0.658097			−0.329048	−	0.944313i	$-0.606728\pi$
$151$	−99.3726			−0.658097			−0.329048	+	0.944313i	$0.606728\pi$
$152$		0			0
$153$			98.4021i			0.643151i
$154$		0			0
$155$	−77.2548			−0.498418
$156$		0			0
$157$			96.6538i			0.615629i	0.951446	+	0.307815i	$0.0995977\pi$
$157$			96.6538i			0.615629i	−0.951446	+	0.307815i	$0.900402\pi$
$158$		0			0
$159$		−	92.8044i		−	0.583676i
$160$		0			0
$161$	63.6569	+	97.4417i	0.395384	+	0.605228i
$162$		0			0
$163$	170.569			1.04643			0.523216	−	0.852200i	$-0.324732\pi$
$163$	170.569			1.04643			0.523216	+	0.852200i	$0.324732\pi$
$164$		0			0
$165$	−127.196			−0.770885
$166$		0			0
$167$			94.9055i			0.568296i	0.958780	+	0.284148i	$0.0917108\pi$
$167$			94.9055i			0.568296i	−0.958780	+	0.284148i	$0.908289\pi$
$168$		0			0
$169$	168.931			0.999592
$170$		0			0
$171$			169.731i			0.992580i
$172$		0			0
$173$		−	182.720i		−	1.05618i	−0.849187	−	0.528092i	$-0.822908\pi$
$173$		−	182.720i		−	1.05618i	0.849187	−	0.528092i	$-0.177092\pi$
$174$		0			0
$175$	209.024	+	319.961i	1.19443	+	1.82835i
$176$		0			0
$177$	88.9706			0.502659
$178$		0			0
$179$	−204.059			−1.13999			−0.569997	−	0.821647i	$-0.693056\pi$
$179$	−204.059			−1.13999			−0.569997	+	0.821647i	$0.693056\pi$
$180$		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
$185$		−	392.033i		−	2.11910i
$186$		0			0
$187$		−	137.676i		−	0.736235i
$188$		0			0
$189$	−140.451	+	91.7539i	−0.743126	+	0.485470i
$190$		0			0
$191$	−9.59798			−0.0502512			−0.0251256	−	0.999684i	$-0.507999\pi$
$191$	−9.59798			−0.0502512			−0.0251256	+	0.999684i	$0.507999\pi$
$192$		0			0
$193$	−3.42136			−0.0177272			−0.00886362	−	0.999961i	$-0.502821\pi$
$193$	−3.42136			−0.0177272			−0.00886362	+	0.999961i	$0.502821\pi$
$194$		0			0
$195$		−	3.58673i		−	0.0183935i
$196$		0			0
$197$	146.451			0.743405			0.371703	−	0.928352i	$-0.378774\pi$
$197$	146.451			0.743405			0.371703	+	0.928352i	$0.378774\pi$
$198$		0			0
$199$		−	244.647i		−	1.22938i	−0.788768	−	0.614691i	$-0.789281\pi$
$199$		−	244.647i		−	1.22938i	0.788768	−	0.614691i	$-0.210719\pi$
$200$		0			0
$201$			31.5751i			0.157090i
$202$		0			0
$203$	56.2254	+	86.0661i	0.276972	+	0.423971i
$204$		0			0
$205$	−204.451			−0.997321
$206$		0			0
$207$	−110.686			−0.534716
$208$		0			0
$209$		−	237.474i		−	1.13624i
$210$		0			0
$211$	311.019			1.47403			0.737013	−	0.675879i	$-0.236236\pi$
$211$	311.019			1.47403			0.737013	+	0.675879i	$0.236236\pi$
$212$		0			0
$213$		−	120.883i		−	0.567525i
$214$		0			0
$215$		−	410.401i		−	1.90884i
$216$		0			0
$217$	50.7452	−	33.1509i	0.233849	−	0.152769i
$218$		0			0
$219$	32.8040			0.149790
$220$		0			0
$221$	3.88225			0.0175667
$222$		0			0
$223$			116.156i			0.520877i	0.965490	+	0.260438i	$0.0838671\pi$
$223$			116.156i			0.520877i	−0.965490	+	0.260438i	$0.916133\pi$
$224$		0			0
$225$	−363.451			−1.61534
$226$		0			0
$227$		−	59.1734i		−	0.260676i	−0.991470	−	0.130338i	$-0.958394\pi$
$227$		−	59.1734i		−	0.260676i	0.991470	−	0.130338i	$-0.0416062\pi$
$228$		0			0
$229$			128.664i			0.561852i	0.959729	+	0.280926i	$0.0906415\pi$
$229$			128.664i			0.561852i	−0.959729	+	0.280926i	$0.909359\pi$
$230$		0			0
$231$	83.5492	−	54.5811i	0.361685	−	0.236282i
$232$		0			0
$233$	−88.9807			−0.381891			−0.190946	−	0.981601i	$-0.561155\pi$
$233$	−88.9807			−0.381891			−0.190946	+	0.981601i	$0.561155\pi$
$234$		0			0
$235$	−768.666			−3.27092
$236$		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
$241$			367.451i			1.52469i	0.647170	+	0.762346i	$0.275953\pi$
$241$			367.451i			1.52469i	−0.647170	+	0.762346i	$0.724047\pi$
$242$		0			0
$243$		−	251.250i		−	1.03395i
$244$		0			0
$245$	−400.333	−	175.637i	−1.63401	−	0.716884i
$246$		0			0
$247$	6.69639			0.0271109
$248$		0			0
$249$	−109.990			−0.441727
$250$		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
$255$			201.877i			0.791673i
$256$		0			0
$257$		−	27.4631i		−	0.106860i	−0.998572	−	0.0534301i	$-0.982985\pi$
$257$		−	27.4631i		−	0.106860i	0.998572	−	0.0534301i	$-0.0170154\pi$
$258$		0			0
$259$	168.225	+	257.508i	0.649519	+	0.994240i
$260$		0			0
$261$	−97.7645			−0.374577
$262$		0			0
$263$	−77.1472			−0.293335			−0.146668	−	0.989186i	$-0.546855\pi$
$263$	−77.1472			−0.293335			−0.146668	+	0.989186i	$0.546855\pi$
$264$		0			0
$265$			540.904i			2.04115i
$266$		0			0
$267$	−3.07821			−0.0115289
$268$		0			0
$269$			34.3739i			0.127784i	0.997957	+	0.0638920i	$0.0203513\pi$
$269$			34.3739i			0.127784i	−0.997957	+	0.0638920i	$0.979649\pi$
$270$		0			0
$271$			202.927i			0.748809i	0.927266	+	0.374404i	$0.122153\pi$
$271$			202.927i			0.748809i	−0.927266	+	0.374404i	$0.877847\pi$
$272$		0			0
$273$	1.53911	+	2.35596i	0.00563775	+	0.00862989i
$274$		0			0
$275$	508.510			1.84913
$276$		0			0
$277$	254.098			0.917320			0.458660	−	0.888612i	$-0.348330\pi$
$277$	254.098			0.917320			0.458660	+	0.888612i	$0.348330\pi$
$278$		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
$283$		−	283.786i		−	1.00278i	−0.865223	−	0.501388i	$-0.832823\pi$
$283$		−	283.786i		−	1.00278i	0.865223	−	0.501388i	$-0.167177\pi$
$284$		0			0
$285$			348.212i			1.22180i
$286$		0			0
$287$	134.294	−	87.7320i	0.467925	−	0.305687i
$288$		0			0
$289$	70.4903			0.243911
$290$		0			0
$291$	−242.745			−0.834176
$292$		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
$297$			223.217i			0.751572i
$298$		0			0
$299$			4.36690i			0.0146050i
$300$		0			0
$301$	176.108	+	269.574i	0.585075	+	0.895594i
$302$		0			0
$303$	−143.127			−0.472366
$304$		0			0
$305$	88.9706			0.291707
$306$		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
$311$			284.101i			0.913508i	0.889593	+	0.456754i	$0.150988\pi$
$311$			284.101i			0.913508i	−0.889593	+	0.456754i	$0.849012\pi$
$312$		0			0
$313$			514.311i			1.64317i	0.570089	+	0.821583i	$0.306909\pi$
$313$			514.311i			1.64317i	−0.570089	+	0.821583i	$0.693091\pi$
$314$		0			0
$315$	348.049	−	227.374i	1.10492	−	0.721822i
$316$		0			0
$317$	−18.5685			−0.0585758			−0.0292879	−	0.999571i	$-0.509324\pi$
$317$	−18.5685			−0.0585758			−0.0292879	+	0.999571i	$0.509324\pi$
$318$		0			0
$319$	136.784			0.428789
$320$		0			0
$321$		−	21.6105i		−	0.0673225i
$322$		0			0
$323$	−376.902			−1.16688
$324$		0			0
$325$			14.3392i			0.0441206i
$326$		0			0
$327$			35.5970i			0.108859i
$328$		0			0
$329$	504.902	−	329.843i	1.53466	−	1.00256i
$330$		0			0
$331$	194.333			0.587109			0.293554	−	0.955942i	$-0.405162\pi$
$331$	194.333			0.587109			0.293554	+	0.955942i	$0.405162\pi$
$332$		0			0
$333$	−292.510			−0.878407
$334$		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
$339$		−	20.9951i		−	0.0619325i
$340$		0			0
$341$		−	80.6487i		−	0.236506i
$342$		0			0
$343$	338.328	−	56.4196i	0.986379	−	0.164489i
$344$		0			0
$345$	−227.078			−0.658198
$346$		0			0
$347$	−160.745			−0.463243			−0.231621	−	0.972806i	$-0.574403\pi$
$347$	−160.745			−0.463243			−0.231621	+	0.972806i	$0.574403\pi$
$348$		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
$353$		−	189.631i		−	0.537198i	−0.963252	−	0.268599i	$-0.913439\pi$
$353$		−	189.631i		−	0.537198i	0.963252	−	0.268599i	$-0.0865606\pi$
$354$		0			0
$355$			704.557i			1.98467i
$356$		0			0
$357$	−86.6274	−	132.604i	−0.242654	−	0.371438i
$358$		0			0
$359$	494.431			1.37724			0.688622	−	0.725121i	$-0.258216\pi$
$359$	494.431			1.37724			0.688622	+	0.725121i	$0.258216\pi$
$360$		0			0
$361$	−289.108			−0.800852
$362$		0			0
$363$			52.4350i			0.144449i
$364$		0			0
$365$	−191.196			−0.523825
$366$		0			0
$367$		−	718.994i		−	1.95911i	−0.201171	−	0.979556i	$-0.564475\pi$
$367$		−	718.994i		−	1.95911i	0.201171	−	0.979556i	$-0.435525\pi$
$368$		0			0
$369$			152.548i			0.413410i
$370$		0			0
$371$	−232.108	−	355.295i	−0.625627	−	0.957668i
$372$		0			0
$373$	−132.843			−0.356147			−0.178073	−	0.984017i	$-0.556986\pi$
$373$	−132.843			−0.356147			−0.178073	+	0.984017i	$0.556986\pi$
$374$		0			0
$375$	−404.215			−1.07791
$376$		0			0
$377$			3.85710i			0.0102310i
$378$		0			0
$379$	324.607			0.856483			0.428242	−	0.903664i	$-0.359133\pi$
$379$	324.607			0.856483			0.428242	+	0.903664i	$0.359133\pi$
$380$		0			0
$381$			272.110i			0.714200i
$382$		0			0
$383$			391.252i			1.02155i	0.859715	+	0.510773i	$0.170641\pi$
$383$			391.252i			1.02155i	−0.859715	+	0.510773i	$0.829359\pi$
$384$		0			0
$385$	−486.960	+	318.122i	−1.26483	+	0.826292i
$386$		0			0
$387$	−306.215			−0.791254
$388$		0			0
$389$	−223.352			−0.574171			−0.287085	−	0.957905i	$-0.592686\pi$
$389$	−223.352			−0.574171			−0.287085	+	0.957905i	$0.592686\pi$
$390$		0			0
$391$		−	245.788i		−	0.628613i
$392$		0			0
$393$	−172.382			−0.438631
$394$		0			0
$395$			805.496i			2.03923i
$396$		0			0
$397$		−	223.044i		−	0.561824i	−0.959733	−	0.280912i	$-0.909363\pi$
$397$		−	223.044i		−	0.561824i	0.959733	−	0.280912i	$-0.0906370\pi$
$398$		0			0
$399$	−149.421	−	228.724i	−0.374490	−	0.573244i
$400$		0			0
$401$	136.225			0.339714			0.169857	−	0.985469i	$-0.445669\pi$
$401$	136.225			0.339714			0.169857	+	0.985469i	$0.445669\pi$
$402$		0			0
$403$	2.27417			0.00564310
$404$		0			0
$405$			207.212i			0.511634i
$406$		0			0
$407$	409.255			1.00554
$408$		0			0
$409$		−	671.587i		−	1.64202i	−0.570913	−	0.821010i	$-0.693411\pi$
$409$		−	671.587i		−	1.64202i	0.570913	−	0.821010i	$-0.306589\pi$
$410$		0			0
$411$		−	186.389i		−	0.453501i
$412$		0			0
$413$	340.617	−	222.519i	0.824739	−	0.538787i
$414$		0			0
$415$	641.068			1.54474
$416$		0			0
$417$	268.382			0.643601
$418$		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	$-0.666949\pi$
$421$	−421.647			−1.00154			−0.500768	+	0.865581i	$0.666949\pi$
$422$		0			0
$423$			573.529i			1.35586i
$424$		0			0
$425$		−	807.071i		−	1.89899i
$426$		0			0
$427$	−58.4407	+	38.1782i	−0.136863	+	0.0894104i
$428$		0			0
$429$	3.74430			0.00872797
$430$		0			0
$431$	−793.529			−1.84113			−0.920567	−	0.390584i	$-0.872273\pi$
$431$	−793.529			−1.84113			−0.920567	+	0.390584i	$0.872273\pi$
$432$		0			0
$433$			254.627i			0.588053i	0.955797	+	0.294027i	$0.0949954\pi$
$433$			254.627i			0.588053i	−0.955797	+	0.294027i	$0.905005\pi$
$434$		0			0
$435$	−200.569			−0.461077
$436$		0			0
$437$		−	423.953i		−	0.970144i
$438$		0			0
$439$			839.517i			1.91234i	0.292814	+	0.956169i	$0.405408\pi$
$439$			839.517i			1.91234i	−0.292814	+	0.956169i	$0.594592\pi$
$440$		0			0
$441$	−131.049	+	298.703i	−0.297163	+	0.677331i
$442$		0			0
$443$	774.215			1.74766			0.873832	−	0.486228i	$-0.161627\pi$
$443$	774.215			1.74766			0.873832	+	0.486228i	$0.161627\pi$
$444$		0			0
$445$	17.9411			0.0403171
$446$		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
$451$		−	213.432i		−	0.473243i
$452$		0			0
$453$			152.113i			0.335790i
$454$		0			0
$455$	−8.97056	−	13.7315i	−0.0197155	−	0.0301792i
$456$		0			0
$457$	−251.990			−0.551400			−0.275700	−	0.961244i	$-0.588910\pi$
$457$	−251.990			−0.551400			−0.275700	+	0.961244i	$0.588910\pi$
$458$		0			0
$459$	354.274			0.771839
$460$		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
$465$			118.257i			0.254315i
$466$		0			0
$467$			364.179i			0.779828i	0.920851	+	0.389914i	$0.127495\pi$
$467$			364.179i			0.779828i	−0.920851	+	0.389914i	$0.872505\pi$
$468$		0			0
$469$	78.9706	+	120.883i	0.168381	+	0.257746i
$470$		0			0
$471$	147.951			0.314122
$472$		0			0
$473$	428.431			0.905773
$474$		0			0
$475$		−	1392.10i		−	2.93073i
$476$		0			0
$477$	403.588			0.846096
$478$		0			0
$479$			137.061i			0.286139i	0.989713	+	0.143069i	$0.0456972\pi$
$479$			137.061i			0.286139i	−0.989713	+	0.143069i	$0.954303\pi$
$480$		0			0
$481$			11.5404i			0.0239924i
$482$		0			0
$483$	149.157	−	97.4417i	0.308814	−	0.201743i
$484$		0			0
$485$	1414.82			2.91716
$486$		0			0
$487$	−557.411			−1.14458			−0.572291	−	0.820051i	$-0.693945\pi$
$487$	−557.411			−1.14458			−0.572291	+	0.820051i	$0.693945\pi$
$488$		0			0
$489$		−	261.095i		−	0.533937i
$490$		0			0
$491$	−537.647			−1.09500			−0.547502	−	0.836805i	$-0.684421\pi$
$491$	−537.647			−1.09500			−0.547502	+	0.836805i	$0.684421\pi$
$492$		0			0
$493$		−	217.094i		−	0.440353i
$494$		0			0
$495$		−	553.150i		−	1.11747i
$496$		0			0
$497$	−302.333	−	462.791i	−0.608316	−	0.931170i
$498$		0			0
$499$	−633.980			−1.27050			−0.635250	−	0.772306i	$-0.719103\pi$
$499$	−633.980			−1.27050			−0.635250	+	0.772306i	$0.719103\pi$
$500$		0			0
$501$	145.275			0.289970
$502$		0			0
$503$			91.1385i			0.181190i	0.995888	+	0.0905949i	$0.0288769\pi$
$503$			91.1385i			0.181190i	−0.995888	+	0.0905949i	$0.971123\pi$
$504$		0			0
$505$	834.205			1.65189
$506$		0			0
$507$		−	258.588i		−	0.510036i
$508$		0			0
$509$		−	353.366i		−	0.694237i	−0.937821	−	0.347118i	$-0.887160\pi$
$509$		−	353.366i		−	0.694237i	0.937821	−	0.347118i	$-0.112840\pi$
$510$		0			0
$511$	125.588	−	82.0442i	0.245769	−	0.160556i
$512$		0			0
$513$	611.078			1.19119
$514$		0			0
$515$	181.823			0.353055
$516$		0			0
$517$		−	802.434i		−	1.55210i
$518$		0			0
$519$	−279.696			−0.538912
$520$		0			0
$521$			949.039i			1.82157i	0.412878	+	0.910786i	$0.364524\pi$
$521$			949.039i			1.82157i	−0.412878	+	0.910786i	$0.635476\pi$
$522$		0			0
$523$			662.702i			1.26712i	0.773695	+	0.633558i	$0.218406\pi$
$523$			662.702i			1.26712i	−0.773695	+	0.633558i	$0.781594\pi$
$524$		0			0
$525$	489.775	−	319.961i	0.932904	−	0.609449i
$526$		0			0
$527$	−128.000			−0.242884
$528$		0			0
$529$	−252.529			−0.477371
$530$		0			0
$531$			386.915i			0.728654i
$532$		0			0
$533$	6.01847			0.0112917
$534$		0			0
$535$			125.955i			0.235430i
$536$		0			0
$537$			312.360i			0.581676i
$538$		0			0
$539$	183.352	−	417.920i	0.340171	−	0.775362i
$540$		0			0
$541$	−1026.90			−1.89815			−0.949077	−	0.315043i	$-0.897981\pi$
$541$	−1026.90			−1.89815			−0.949077	+	0.315043i	$0.897981\pi$
$542$		0			0
$543$	350.794			0.646029
$544$		0			0
$545$		−	207.474i		−	0.380687i
$546$		0			0
$547$	−258.686			−0.472918			−0.236459	−	0.971641i	$-0.575987\pi$
$547$	−258.686			−0.472918			−0.236459	+	0.971641i	$0.575987\pi$
$548$		0			0
$549$		−	66.3841i		−	0.120918i
$550$		0			0
$551$		−	374.459i		−	0.679600i
$552$		0			0
$553$	−345.647	−	529.093i	−0.625039	−	0.956769i
$554$		0			0
$555$	−600.098			−1.08126
$556$		0			0
$557$	741.529			1.33129			0.665645	−	0.746268i	$-0.268156\pi$
$557$	741.529			1.33129			0.665645	+	0.746268i	$0.268156\pi$
$558$		0			0
$559$			12.0811i			0.0216120i
$560$		0			0
$561$	−210.745			−0.375660
$562$		0			0
$563$		−	80.5135i		−	0.143008i	−0.997440	−	0.0715040i	$-0.977220\pi$
$563$		−	80.5135i		−	0.143008i	0.997440	−	0.0715040i	$-0.0227799\pi$
$564$		0			0
$565$			122.369i			0.216582i
$566$		0			0
$567$	−88.9167	−	136.108i	−0.156820	−	0.240049i
$568$		0			0
$569$	−742.971			−1.30575			−0.652874	−	0.757467i	$-0.726437\pi$
$569$	−742.971			−1.30575			−0.652874	+	0.757467i	$0.726437\pi$
$570$		0			0
$571$	597.882			1.04708			0.523540	−	0.852001i	$-0.324611\pi$
$571$	597.882			1.04708			0.523540	+	0.852001i	$0.324611\pi$
$572$		0			0
$573$			14.6920i			0.0256404i
$574$		0			0
$575$	907.823			1.57882
$576$		0			0
$577$		−	830.767i		−	1.43980i	−0.694075	−	0.719902i	$-0.744187\pi$
$577$		−	830.767i		−	1.43980i	0.694075	−	0.719902i	$-0.255813\pi$
$578$		0			0
$579$			5.23719i			0.00904523i
$580$		0			0
$581$	−421.088	+	275.089i	−0.724765	+	0.473475i
$582$		0			0
$583$	−564.666			−0.968552
$584$		0			0
$585$	15.5980			0.0266632
$586$		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
$591$		−	224.177i		−	0.379318i
$592$		0			0
$593$		−	417.050i		−	0.703288i	−0.936134	−	0.351644i	$-0.885623\pi$
$593$		−	417.050i		−	0.703288i	0.936134	−	0.351644i	$-0.114377\pi$
$594$		0			0
$595$	504.902	+	772.870i	0.848574	+	1.29894i
$596$		0			0
$597$	−374.489			−0.627286
$598$		0			0
$599$	−223.892			−0.373777			−0.186888	−	0.982381i	$-0.559840\pi$
$599$	−223.892			−0.373777			−0.186888	+	0.982381i	$0.559840\pi$
$600$		0			0
$601$		−	721.185i		−	1.19998i	−0.800009	−	0.599988i	$-0.795172\pi$
$601$		−	721.185i		−	1.19998i	0.800009	−	0.599988i	$-0.204828\pi$
$602$		0			0
$603$	−137.314			−0.227718
$604$		0			0
$605$		−	305.614i		−	0.505147i
$606$		0			0
$607$			320.494i			0.527996i	0.964523	+	0.263998i	$0.0850413\pi$
$607$			320.494i			0.527996i	−0.964523	+	0.263998i	$0.914959\pi$
$608$		0			0
$609$	131.744	−	86.0661i	0.216329	−	0.141324i
$610$		0			0
$611$	22.6274			0.0370334
$612$		0			0
$613$	−195.373			−0.318715			−0.159358	−	0.987221i	$-0.550942\pi$
$613$	−195.373			−0.318715			−0.159358	+	0.987221i	$0.550942\pi$
$614$		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
$619$		−	515.407i		−	0.832644i	−0.909217	−	0.416322i	$-0.863319\pi$
$619$		−	515.407i		−	0.832644i	0.909217	−	0.416322i	$-0.136681\pi$
$620$		0			0
$621$			398.501i			0.641708i
$622$		0			0
$623$	−11.7847	+	7.69873i	−0.0189161	+	0.0123575i
$624$		0			0
$625$	990.990			1.58558
$626$		0			0
$627$	−363.509			−0.579759
$628$		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
$633$		−	476.088i		−	0.752113i
$634$		0			0
$635$		−	1585.97i		−	2.49760i
$636$		0			0
$637$	11.7847	+	5.17026i	0.0185003	+	0.00811657i
$638$		0			0
$639$	525.696			0.822685
$640$		0			0
$641$	−813.460			−1.26905			−0.634524	−	0.772903i	$-0.718804\pi$
$641$	−813.460			−1.26905			−0.634524	+	0.772903i	$0.718804\pi$
$642$		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
$647$			114.325i			0.176700i	0.996089	+	0.0883499i	$0.0281594\pi$
$647$			114.325i			0.176700i	−0.996089	+	0.0883499i	$0.971841\pi$
$648$		0			0
$649$		−	541.339i		−	0.834113i
$650$		0			0
$651$	−50.7452	−	77.6773i	−0.0779496	−	0.119320i
$652$		0			0
$653$	−415.685			−0.636578			−0.318289	−	0.947994i	$-0.603108\pi$
$653$	−415.685			−0.636578			−0.318289	+	0.947994i	$0.603108\pi$
$654$		0			0
$655$	1004.71			1.53392
$656$		0			0
$657$			142.658i			0.217136i
$658$		0			0
$659$	650.431			0.986996			0.493498	−	0.869747i	$-0.335718\pi$
$659$	650.431			0.986996			0.493498	+	0.869747i	$0.335718\pi$
$660$		0			0
$661$			1023.81i			1.54888i	0.632645	+	0.774442i	$0.281969\pi$
$661$			1023.81i			1.54888i	−0.632645	+	0.774442i	$0.718031\pi$
$662$		0			0
$663$		−	5.94269i		−	0.00896334i
$664$		0			0
$665$	870.891	+	1333.10i	1.30961	+	2.00467i
$666$		0			0
$667$	244.195			0.366110
$668$		0			0
$669$	177.803			0.265775
$670$		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
$675$			1308.52i			1.93855i
$676$		0			0
$677$		−	905.826i		−	1.33800i	−0.743262	−	0.669000i	$-0.766723\pi$
$677$		−	905.826i		−	1.33800i	0.743262	−	0.669000i	$-0.233277\pi$
$678$		0			0
$679$	−929.332	+	607.116i	−1.36868	+	0.894132i
$680$		0			0
$681$	−90.5786			−0.133008
$682$		0			0
$683$	−315.255			−0.461574			−0.230787	−	0.973004i	$-0.574130\pi$
$683$	−315.255			−0.461574			−0.230787	+	0.973004i	$0.574130\pi$
$684$		0			0
$685$			1086.35i			1.58592i
$686$		0			0
$687$	196.950			0.286682
$688$		0			0
$689$		−	15.9227i		−	0.0231099i
$690$		0			0
$691$		−	334.960i		−	0.484747i	−0.970183	−	0.242374i	$-0.922074\pi$
$691$		−	334.960i		−	0.484747i	0.970183	−	0.242374i	$-0.0779260\pi$
$692$		0			0
$693$	237.362	+	363.339i	0.342514	+	0.524298i
$694$		0			0
$695$	−1564.24			−2.25071
$696$		0			0
$697$	−338.745			−0.486005
$698$		0			0
$699$			136.206i			0.194858i
$700$		0			0
$701$	362.353			0.516909			0.258455	−	0.966023i	$-0.416787\pi$
$701$	362.353			0.516909			0.258455	+	0.966023i	$0.416787\pi$
$702$		0			0
$703$		−	1120.38i		−	1.59371i
$704$		0			0
$705$			1176.62i			1.66897i
$706$		0			0
$707$	−547.951	+	357.966i	−0.775037	+	0.506317i
$708$		0			0
$709$	1117.96			1.57681			0.788406	−	0.615155i	$-0.210907\pi$
$709$	1117.96			1.57681			0.788406	+	0.615155i	$0.210907\pi$
$710$		0			0
$711$	601.009			0.845301
$712$		0			0
$713$		−	143.979i		−	0.201934i
$714$		0			0
$715$	−21.8234			−0.0305222
$716$		0			0
$717$		−	10.9250i		−	0.0152371i
$718$		0			0
$719$		−	972.571i		−	1.35267i	−0.736593	−	0.676336i	$-0.763567\pi$
$719$		−	972.571i		−	1.35267i	0.736593	−	0.676336i	$-0.236433\pi$
$720$		0			0
$721$	−119.431	+	78.0224i	−0.165647	+	0.108214i
$722$		0			0
$723$	562.469			0.777966
$724$		0			0
$725$	801.842			1.10599
$726$		0			0
$727$			660.646i			0.908729i	0.890816	+	0.454365i	$0.150134\pi$
$727$			660.646i			0.908729i	−0.890816	+	0.454365i	$0.849866\pi$
$728$		0			0
$729$	−175.569			−0.240835
$730$		0			0
$731$		−	679.975i		−	0.930199i
$732$		0			0
$733$		−	258.222i		−	0.352280i	−0.984365	−	0.176140i	$-0.943639\pi$
$733$		−	258.222i		−	0.352280i	0.984365	−	0.176140i	$-0.0563612\pi$
$734$		0			0
$735$	−268.853	+	612.803i	−0.365786	+	0.833746i
$736$		0			0
$737$	192.118			0.260675
$738$		0			0
$739$	−333.391			−0.451138			−0.225569	−	0.974227i	$-0.572424\pi$
$739$	−333.391			−0.451138			−0.225569	+	0.974227i	$0.572424\pi$
$740$		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
$745$			2121.28i			2.84736i
$746$		0			0
$747$		−	478.324i		−	0.640327i
$748$		0			0
$749$	−54.0488	−	82.7343i	−0.0721612	−	0.110460i
$750$		0			0
$751$	18.1177			0.0241248			0.0120624	−	0.999927i	$-0.496160\pi$
$751$	18.1177			0.0241248			0.0120624	+	0.999927i	$0.496160\pi$
$752$		0			0
$753$	270.932			0.359803
$754$		0			0
$755$		−	886.579i		−	1.17428i
$756$		0			0
$757$	−107.137			−0.141529			−0.0707643	−	0.997493i	$-0.522544\pi$
$757$	−107.137			−0.141529			−0.0707643	+	0.997493i	$0.522544\pi$
$758$		0			0
$759$		−	237.054i		−	0.312324i
$760$		0			0
$761$			898.284i			1.18040i	0.807257	+	0.590200i	$0.200951\pi$
$761$			898.284i			1.18040i	−0.807257	+	0.590200i	$0.799049\pi$
$762$		0			0
$763$	89.0294	+	136.280i	0.116683	+	0.178611i
$764$		0			0
$765$	−877.921			−1.14761
$766$		0			0
$767$	15.2649			0.0199021
$768$		0			0
$769$			929.350i			1.20852i	0.796788	+	0.604259i	$0.206531\pi$
$769$			929.350i			1.20852i	−0.796788	+	0.604259i	$0.793469\pi$
$770$		0			0
$771$	−42.0387			−0.0545249
$772$		0			0
$773$			914.395i			1.18292i	0.806335	+	0.591459i	$0.201448\pi$
$773$			914.395i			1.18292i	−0.806335	+	0.591459i	$0.798552\pi$
$774$		0			0
$775$		−	472.771i		−	0.610028i
$776$		0			0
$777$	394.177	−	257.508i	0.507306	−	0.331413i
$778$		0			0
$779$	−584.293			−0.750055
$780$		0			0
$781$	−735.509			−0.941753
$782$		0			0
$783$			351.979i			0.449526i
$784$		0			0
$785$	−862.323			−1.09850
$786$		0			0
$787$		−	761.720i		−	0.967878i	−0.875102	−	0.483939i	$-0.839206\pi$
$787$		−	761.720i		−	0.967878i	0.875102	−	0.483939i	$-0.160794\pi$
$788$		0			0
$789$			118.092i			0.149673i
$790$		0			0
$791$	−52.5097	−	80.3783i	−0.0663839	−	0.101616i
$792$		0			0
$793$	−2.61905			−0.00330271
$794$		0			0
$795$	827.980			1.04148
$796$		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
$801$		−	13.3865i		−	0.0167123i
$802$		0			0
$803$		−	199.595i		−	0.248562i
$804$		0			0
$805$	−869.352	+	567.932i	−1.07994	+	0.705505i
$806$		0			0
$807$	52.6173			0.0652011
$808$		0			0
$809$	−877.793			−1.08503			−0.542517	−	0.840045i	$-0.682529\pi$
$809$	−877.793			−1.08503			−0.542517	+	0.840045i	$0.682529\pi$
$810$		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
$815$			1521.77i			1.86721i
$816$		0			0
$817$		−	1172.87i		−	1.43558i
$818$		0			0
$819$	−10.2456	+	6.69326i	−0.0125099	+	0.00817248i
$820$		0			0
$821$	−437.038			−0.532324			−0.266162	−	0.963928i	$-0.585756\pi$
$821$	−437.038			−0.532324			−0.266162	+	0.963928i	$0.585756\pi$
$822$		0			0
$823$	1592.34			1.93480			0.967402	−	0.253247i	$-0.0814985\pi$
$823$	1592.34			1.93480			0.967402	+	0.253247i	$0.0814985\pi$
$824$		0			0
$825$		−	778.393i		−	0.943507i
$826$		0			0
$827$	1182.45			1.42981			0.714904	−	0.699223i	$-0.246470\pi$
$827$	1182.45			1.42981			0.714904	+	0.699223i	$0.246470\pi$
$828$		0			0
$829$		−	1048.05i		−	1.26423i	−0.774873	−	0.632117i	$-0.782186\pi$
$829$		−	1048.05i		−	1.26423i	0.774873	−	0.632117i	$-0.217814\pi$
$830$		0			0
$831$		−	388.956i		−	0.468057i
$832$		0			0
$833$	−663.294	−	291.004i	−0.796271	−	0.349345i
$834$		0			0
$835$	−846.725			−1.01404
$836$		0			0
$837$	207.529			0.247944
$838$		0			0
$839$			585.611i			0.697986i	0.937125	+	0.348993i	$0.113476\pi$
$839$			585.611i			0.697986i	−0.937125	+	0.348993i	$0.886524\pi$
$840$		0			0
$841$	−625.313			−0.743535
$842$		0			0
$843$			339.102i			0.402256i
$844$		0			0
$845$			1507.16i			1.78363i
$846$		0			0
$847$	131.142	+	200.744i	0.154831	+	0.237006i
$848$		0			0
$849$	−434.400			−0.511661
$850$		0			0
$851$	730.627			0.858552
$852$		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
$857$		−	965.997i		−	1.12718i	−0.826053	−	0.563592i	$-0.809419\pi$
$857$		−	965.997i		−	1.12718i	0.826053	−	0.563592i	$-0.190581\pi$
$858$		0			0
$859$			250.830i			0.292003i	0.989284	+	0.146001i	$0.0466404\pi$
$859$			250.830i			0.292003i	−0.989284	+	0.146001i	$0.953360\pi$
$860$		0			0
$861$	−134.294	−	205.569i	−0.155975	−	0.238756i
$862$		0			0
$863$	−1497.13			−1.73479			−0.867397	−	0.497617i	$-0.834209\pi$
$863$	−1497.13			−1.73479			−0.867397	+	0.497617i	$0.834209\pi$
$864$		0			0
$865$	1630.18			1.88461
$866$		0			0
$867$		−	107.902i		−	0.124454i
$868$		0			0
$869$	−840.881			−0.967643
$870$		0			0
$871$			5.41743i			0.00621978i
$872$		0			0
$873$		−	1055.65i		−	1.20922i
$874$		0			0
$875$	−1547.51	+	1010.96i	−1.76858	+	1.15538i
$876$		0			0
$877$	−141.354			−0.161179			−0.0805896	−	0.996747i	$-0.525680\pi$
$877$	−141.354			−0.161179			−0.0805896	+	0.996747i	$0.525680\pi$
$878$		0			0
$879$	609.344			0.693224
$880$		0			0
$881$			1413.83i			1.60480i	0.596788	+	0.802399i	$0.296443\pi$
$881$			1413.83i			1.60480i	−0.596788	+	0.802399i	$0.703557\pi$
$882$		0			0
$883$	−1386.04			−1.56969			−0.784845	−	0.619692i	$-0.787258\pi$
$883$	−1386.04			−1.56969			−0.784845	+	0.619692i	$0.787258\pi$
$884$		0			0
$885$			793.775i			0.896921i
$886$		0			0
$887$		−	303.595i		−	0.342272i	−0.985247	−	0.171136i	$-0.945256\pi$
$887$		−	303.595i		−	0.342272i	0.985247	−	0.171136i	$-0.0547437\pi$
$888$		0			0
$889$	680.558	+	1041.75i	0.765533	+	1.17183i
$890$		0			0
$891$	−216.315			−0.242777
$892$		0			0
$893$	−2196.74			−2.45996
$894$		0			0
$895$		−	1820.57i		−	2.03415i
$896$		0			0
$897$	6.68456			0.00745213
$898$		0			0
$899$		−	127.171i		−	0.141458i
$900$		0			0
$901$			896.199i			0.994671i
$902$		0			0
$903$	412.646	−	269.574i	0.456972	−	0.298531i
$904$		0			0
$905$	−2044.58			−2.25920
$906$		0			0
$907$	1304.02			1.43773			0.718864	−	0.695151i	$-0.244663\pi$
$907$	1304.02			1.43773			0.718864	+	0.695151i	$0.244663\pi$
$908$		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
$913$			669.231i			0.733002i
$914$		0			0
$915$		−	136.190i		−	0.148842i
$916$		0			0
$917$	−659.951	+	431.134i	−0.719685	+	0.470157i
$918$		0			0
$919$	1178.34			1.28220			0.641100	−	0.767458i	$-0.278478\pi$
$919$	1178.34			1.28220			0.641100	+	0.767458i	$0.278478\pi$
$920$		0			0
$921$	−256.735			−0.278757
$922$		0			0
$923$		−	20.7402i		−	0.0224705i
$924$		0			0
$925$	2399.10			2.59362
$926$		0			0
$927$		−	135.665i		−	0.146348i
$928$		0			0
$929$			911.161i			0.980798i	0.871498	+	0.490399i	$0.163149\pi$
$929$			911.161i			0.980798i	−0.871498	+	0.490399i	$0.836851\pi$
$930$		0			0
$931$	−1144.10	−	501.945i	−1.22889	−	0.539147i
$932$		0			0
$933$	434.883			0.466113
$934$		0			0
$935$	1228.31			1.31370
$936$		0			0
$937$		−	1035.35i		−	1.10496i	−0.833527	−	0.552479i	$-0.813682\pi$
$937$		−	1035.35i		−	1.10496i	0.833527	−	0.552479i	$-0.186318\pi$
$938$		0			0
$939$	787.273			0.838417
$940$		0			0
$941$			64.4479i			0.0684887i	0.999413	+	0.0342443i	$0.0109024\pi$
$941$			64.4479i			0.0684887i	−0.999413	+	0.0342443i	$0.989098\pi$
$942$		0			0
$943$		−	381.033i		−	0.404065i
$944$		0			0
$945$	−818.607	−	1253.07i	−0.866251	−	1.32600i
$946$		0			0
$947$	132.274			0.139677			0.0698385	−	0.997558i	$-0.477752\pi$
$947$	132.274			0.139677			0.0698385	+	0.997558i	$0.477752\pi$
$948$		0			0
$949$	5.62828			0.00593075
$950$		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
$955$		−	85.6310i		−	0.0896659i
$956$		0			0
$957$		−	209.380i		−	0.218787i
$958$		0			0
$959$	−466.167	−	713.577i	−0.486096	−	0.744084i
$960$		0			0
$961$	886.019			0.921976
$962$		0			0
$963$	93.9798			0.0975907
$964$		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
$969$			576.936i			0.595393i
$970$		0			0
$971$			1799.26i			1.85299i	0.376304	+	0.926496i	$0.377195\pi$
$971$			1799.26i			1.85299i	−0.376304	+	0.926496i	$0.622805\pi$
$972$		0			0
$973$	1027.48	−	671.234i	1.05599	−	0.689860i
$974$		0			0
$975$	21.9495			0.0225123
$976$		0			0
$977$	1381.61			1.41413			0.707066	−	0.707148i	$-0.250018\pi$
$977$	1381.61			1.41413			0.707066	+	0.707148i	$0.250018\pi$
$978$		0			0
$979$			18.7293i			0.0191310i
$980$		0			0
$981$	−154.804			−0.157802
$982$		0			0
$983$		−	1306.67i		−	1.32927i	−0.747167	−	0.664636i	$-0.768587\pi$
$983$		−	1306.67i		−	1.32927i	0.747167	−	0.664636i	$-0.231413\pi$
$984$		0			0
$985$			1306.60i			1.32650i
$986$		0			0
$987$	−504.902	−	772.870i	−0.511552	−	0.783050i
$988$		0			0
$989$	764.861			0.773368
$990$		0			0
$991$	−1033.99			−1.04338			−0.521689	−	0.853136i	$-0.674698\pi$
$991$	−1033.99			−1.04338			−0.521689	+	0.853136i	$0.674698\pi$
$992$		0			0
$993$		−	297.472i		−	0.299569i
$994$		0			0
$995$	2182.68			2.19365
$996$		0			0
$997$		−	391.500i		−	0.392678i	−0.980536	−	0.196339i	$-0.937095\pi$
$997$		−	391.500i		−	0.392678i	0.980536	−	0.196339i	$-0.0629052\pi$
$998$		0			0
$999$			1053.11i			1.05417i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.c.e.321.2		4
4.3	odd	2		448.3.c.f.321.3		4
7.6	odd	2	inner	448.3.c.e.321.3		4
8.3	odd	2		56.3.c.a.41.2	✓	4
8.5	even	2		112.3.c.c.97.3		4
24.5	odd	2		1008.3.f.h.433.4		4
24.11	even	2		504.3.f.a.433.4		4
28.27	even	2		448.3.c.f.321.2		4
40.3	even	4		1400.3.p.a.1049.5		8
40.19	odd	2		1400.3.f.a.601.3		4
40.27	even	4		1400.3.p.a.1049.4		8
56.3	even	6		392.3.o.b.313.3		8
56.5	odd	6		784.3.s.f.129.3		8
56.11	odd	6		392.3.o.b.313.2		8
56.13	odd	2		112.3.c.c.97.2		4
56.19	even	6		392.3.o.b.129.2		8
56.27	even	2		56.3.c.a.41.3	yes	4
56.37	even	6		784.3.s.f.129.2		8
56.45	odd	6		784.3.s.f.705.2		8
56.51	odd	6		392.3.o.b.129.3		8
56.53	even	6		784.3.s.f.705.3		8
168.83	odd	2		504.3.f.a.433.1		4
168.125	even	2		1008.3.f.h.433.1		4
280.27	odd	4		1400.3.p.a.1049.6		8
280.83	odd	4		1400.3.p.a.1049.3		8
280.139	even	2		1400.3.f.a.601.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.c.a.41.2	✓	4	8.3	odd	2
56.3.c.a.41.3	yes	4	56.27	even	2
112.3.c.c.97.2		4	56.13	odd	2
112.3.c.c.97.3		4	8.5	even	2
392.3.o.b.129.2		8	56.19	even	6
392.3.o.b.129.3		8	56.51	odd	6
392.3.o.b.313.2		8	56.11	odd	6
392.3.o.b.313.3		8	56.3	even	6
448.3.c.e.321.2		4	1.1	even	1	trivial
448.3.c.e.321.3		4	7.6	odd	2	inner
448.3.c.f.321.2		4	28.27	even	2
448.3.c.f.321.3		4	4.3	odd	2
504.3.f.a.433.1		4	168.83	odd	2
504.3.f.a.433.4		4	24.11	even	2
784.3.s.f.129.2		8	56.37	even	6
784.3.s.f.129.3		8	56.5	odd	6
784.3.s.f.705.2		8	56.45	odd	6
784.3.s.f.705.3		8	56.53	even	6
1008.3.f.h.433.1		4	168.125	even	2
1008.3.f.h.433.4		4	24.5	odd	2
1400.3.f.a.601.2		4	280.139	even	2
1400.3.f.a.601.3		4	40.19	odd	2
1400.3.p.a.1049.3		8	280.83	odd	4
1400.3.p.a.1049.4		8	40.27	even	4
1400.3.p.a.1049.5		8	40.3	even	4
1400.3.p.a.1049.6		8	280.27	odd	4

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.53073i q^{3} +8.92177i q^{5} +(-3.82843 - 5.86030i) q^{7} +6.65685 q^{9} +O(q^{10})\)	\(q-1.53073i q^{3} +8.92177i q^{5} +(-3.82843 - 5.86030i) q^{7} +6.65685 q^{9} -9.31371 q^{11} -0.262632i q^{13} +13.6569 q^{15} +14.7821i q^{17} +25.4972i q^{19} +(-8.97056 + 5.86030i) q^{21} -16.6274 q^{23} -54.5980 q^{25} -23.9665i q^{27} -14.6863 q^{29} +8.65914i q^{31} +14.2568i q^{33} +(52.2843 - 34.1563i) q^{35} -43.9411 q^{37} -0.402020 q^{39} +22.9159i q^{41} -46.0000 q^{43} +59.3909i q^{45} +86.1562i q^{47} +(-19.6863 + 44.8715i) q^{49} +22.6274 q^{51} +60.6274 q^{53} -83.0948i q^{55} +39.0294 q^{57} +58.1228i q^{59} -9.97230i q^{61} +(-25.4853 - 39.0112i) q^{63} +2.34315 q^{65} -20.6274 q^{67} +25.4521i q^{69} +78.9706 q^{71} +21.4303i q^{73} +83.5750i q^{75} +(35.6569 + 54.5811i) q^{77} +90.2843 q^{79} +23.2254 q^{81} -71.8544i q^{83} -131.882 q^{85} +22.4808i q^{87} -2.01094i q^{89} +(-1.53911 + 1.00547i) q^{91} +13.2548 q^{93} -227.480 q^{95} -158.581i q^{97} -62.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{7} + 4 q^{9} + 8 q^{11} + 32 q^{15} + 32 q^{21} + 24 q^{23} - 60 q^{25} - 104 q^{29} + 96 q^{35} - 40 q^{37} - 160 q^{39} - 184 q^{43} - 124 q^{49} + 152 q^{53} + 224 q^{57} - 68 q^{63} + 32 q^{65} + 8 q^{67} + 248 q^{71} + 120 q^{77} + 248 q^{79} - 156 q^{81} - 256 q^{85} + 288 q^{91} - 128 q^{93} - 480 q^{95} - 248 q^{99}+O(q^{100}) \) 4 * q - 4 * q^7 + 4 * q^9 + 8 * q^11 + 32 * q^15 + 32 * q^21 + 24 * q^23 - 60 * q^25 - 104 * q^29 + 96 * q^35 - 40 * q^37 - 160 * q^39 - 184 * q^43 - 124 * q^49 + 152 * q^53 + 224 * q^57 - 68 * q^63 + 32 * q^65 + 8 * q^67 + 248 * q^71 + 120 * q^77 + 248 * q^79 - 156 * q^81 - 256 * q^85 + 288 * q^91 - 128 * q^93 - 480 * q^95 - 248 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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