LMFDB - Embedded newform 684.5.y.f.145.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,5,Mod(145,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("684.145");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	684.y (of order $6$, degree $2$, 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:	$70.7050547493$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.2
Character	$\chi$	$=$	684.145
Dual form			684.5.y.f.217.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-21.2055 - 36.7290i) q^{5} -85.9865 q^{7} +O(q^{10})$	$q+(-21.2055 - 36.7290i) q^{5} -85.9865 q^{7} +212.838 q^{11} +(112.873 + 65.1673i) q^{13} +(17.9046 + 31.0117i) q^{17} +(239.434 + 270.171i) q^{19} +(78.2072 - 135.459i) q^{23} +(-586.846 + 1016.45i) q^{25} +(1136.73 + 656.293i) q^{29} +722.519i q^{31} +(1823.39 + 3158.20i) q^{35} -2089.04i q^{37} +(-1386.42 + 800.449i) q^{41} +(-493.646 - 855.020i) q^{43} +(817.463 - 1415.89i) q^{47} +4992.67 q^{49} +(-2586.77 - 1493.47i) q^{53} +(-4513.33 - 7817.32i) q^{55} +(486.331 - 280.784i) q^{59} +(1181.56 - 2046.52i) q^{61} -5527.62i q^{65} +(-5309.40 - 3065.38i) q^{67} +(7179.99 - 4145.37i) q^{71} +(2255.23 + 3906.17i) q^{73} -18301.2 q^{77} +(10595.8 - 6117.47i) q^{79} -3734.68 q^{83} +(759.352 - 1315.24i) q^{85} +(-1171.84 - 676.565i) q^{89} +(-9705.56 - 5603.51i) q^{91} +(4845.82 - 14523.3i) q^{95} +(1840.23 - 1062.46i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 304 q^{7}+O(q^{10}) $ 24 * q + 304 * q^7	$ 24 q + 304 q^{7} + 516 q^{13} + 416 q^{19} - 5108 q^{25} - 720 q^{43} + 25440 q^{49} - 12536 q^{55} + 5020 q^{61} - 13080 q^{67} + 22572 q^{73} + 63096 q^{79} - 32624 q^{85} - 89568 q^{91} - 3888 q^{97}+O(q^{100}) $ 24 * q + 304 * q^7 + 516 * q^13 + 416 * q^19 - 5108 * q^25 - 720 * q^43 + 25440 * q^49 - 12536 * q^55 + 5020 * q^61 - 13080 * q^67 + 22572 * q^73 + 63096 * q^79 - 32624 * q^85 - 89568 * q^91 - 3888 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−21.2055	−	36.7290i	−0.848220	−	1.46916i	−0.882796	−	0.469757i	$-0.844341\pi$
$5$	−21.2055	−	36.7290i	−0.848220	−	1.46916i	0.0345759	−	0.999402i	$-0.488992\pi$
$6$		0			0
$7$	−85.9865			−1.75483			−0.877413	−	0.479736i	$-0.840732\pi$
$7$	−85.9865			−1.75483			−0.877413	+	0.479736i	$0.840732\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	212.838			1.75899			0.879496	−	0.475906i	$-0.157880\pi$
$11$	212.838			1.75899			0.879496	+	0.475906i	$0.157880\pi$
$12$		0			0
$13$	112.873	+	65.1673i	0.667888	+	0.385605i	0.795276	−	0.606248i	$-0.207326\pi$
$13$	112.873	+	65.1673i	0.667888	+	0.385605i	−0.127388	+	0.991853i	$0.540659\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	17.9046	+	31.0117i	0.0619537	+	0.107307i	0.895339	−	0.445386i	$-0.146934\pi$
$17$	17.9046	+	31.0117i	0.0619537	+	0.107307i	−0.833385	+	0.552693i	$0.813600\pi$
$18$		0			0
$19$	239.434	+	270.171i	0.663251	+	0.748397i
$19$	239.434	+	270.171i	0.663251	+	0.748397i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	78.2072	−	135.459i	0.147840	−	0.256066i	−0.782589	−	0.622539i	$-0.786101\pi$
$23$	78.2072	−	135.459i	0.147840	−	0.256066i	0.930429	+	0.366473i	$0.119435\pi$
$24$		0			0
$25$	−586.846	+	1016.45i	−0.938953	+	1.62631i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1136.73	+	656.293i	1.35164	+	0.780372i	0.988480	−	0.151352i	$-0.0483628\pi$
$29$	1136.73	+	656.293i	1.35164	+	0.780372i	0.363165	+	0.931725i	$0.381696\pi$
$30$		0			0
$31$			722.519i			0.751841i	0.926652	+	0.375920i	$0.122673\pi$
$31$			722.519i			0.751841i	−0.926652	+	0.375920i	$0.877327\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1823.39	+	3158.20i	1.48848	+	2.57812i
$36$		0			0
$37$		−	2089.04i		−	1.52596i	−0.646420	−	0.762982i	$-0.723735\pi$
$37$		−	2089.04i		−	1.52596i	0.646420	−	0.762982i	$-0.276265\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1386.42	+	800.449i	−0.824759	+	0.476175i	−0.852055	−	0.523453i	$-0.824644\pi$
$41$	−1386.42	+	800.449i	−0.824759	+	0.476175i	0.0272961	+	0.999627i	$0.491310\pi$
$42$		0			0
$43$	−493.646	−	855.020i	−0.266980	−	0.462423i	0.701101	−	0.713062i	$-0.252692\pi$
$43$	−493.646	−	855.020i	−0.266980	−	0.462423i	−0.968080	+	0.250640i	$0.919359\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	817.463	−	1415.89i	0.370060	−	0.640963i	−0.619514	−	0.784985i	$-0.712670\pi$
$47$	817.463	−	1415.89i	0.370060	−	0.640963i	0.989574	+	0.144022i	$0.0460037\pi$
$48$		0			0
$49$	4992.67			2.07941
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2586.77	−	1493.47i	−0.920886	−	0.531674i	−0.0369684	−	0.999316i	$-0.511770\pi$
$53$	−2586.77	−	1493.47i	−0.920886	−	0.531674i	−0.883918	+	0.467643i	$0.845103\pi$
$54$		0			0
$55$	−4513.33	−	7817.32i	−1.49201	−	2.58424i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	486.331	−	280.784i	0.139710	−	0.0806617i	−0.428516	−	0.903534i	$-0.640963\pi$
$59$	486.331	−	280.784i	0.139710	−	0.0806617i	0.568226	+	0.822873i	$0.307630\pi$
$60$		0			0
$61$	1181.56	−	2046.52i	0.317539	−	0.549993i	−0.662435	−	0.749119i	$-0.730477\pi$
$61$	1181.56	−	2046.52i	0.317539	−	0.549993i	0.979974	+	0.199126i	$0.0638104\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	5527.62i		−	1.30831i
$66$		0			0
$67$	−5309.40	−	3065.38i	−1.18276	−	0.682866i	−0.226107	−	0.974102i	$-0.572600\pi$
$67$	−5309.40	−	3065.38i	−1.18276	−	0.682866i	−0.956651	+	0.291237i	$0.905933\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7179.99	−	4145.37i	1.42432	−	0.822331i	0.427655	−	0.903942i	$-0.359340\pi$
$71$	7179.99	−	4145.37i	1.42432	−	0.822331i	0.996664	+	0.0816113i	$0.0260066\pi$
$72$		0			0
$73$	2255.23	+	3906.17i	0.423199	+	0.733002i	0.996250	−	0.0865171i	$-0.0275737\pi$
$73$	2255.23	+	3906.17i	0.423199	+	0.733002i	−0.573051	+	0.819520i	$0.694240\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−18301.2			−3.08672
$78$		0			0
$79$	10595.8	−	6117.47i	1.69777	−	0.980207i	0.749896	−	0.661556i	$-0.230104\pi$
$79$	10595.8	−	6117.47i	1.69777	−	0.980207i	0.947872	−	0.318651i	$-0.103230\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3734.68			−0.542123			−0.271061	−	0.962562i	$-0.587375\pi$
$83$	−3734.68			−0.542123			−0.271061	+	0.962562i	$0.587375\pi$
$84$		0			0
$85$	759.352	−	1315.24i	0.105101	−	0.182040i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1171.84	−	676.565i	−0.147941	−	0.0854141i	0.424202	−	0.905568i	$-0.360555\pi$
$89$	−1171.84	−	676.565i	−0.147941	−	0.0854141i	−0.572144	+	0.820154i	$0.693888\pi$
$90$		0			0
$91$	−9705.56	−	5603.51i	−1.17203	−	0.676670i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4845.82	−	14523.3i	0.536932	−	1.60923i
$96$		0			0
$97$	1840.23	−	1062.46i	0.195582	−	0.112919i	−0.399011	−	0.916946i	$-0.630647\pi$
$97$	1840.23	−	1062.46i	0.195582	−	0.112919i	0.594593	+	0.804027i	$0.297313\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6052.19	+	10482.7i	−0.593294	+	1.02762i	0.400491	+	0.916301i	$0.368840\pi$
$101$	−6052.19	+	10482.7i	−0.593294	+	1.02762i	−0.993785	+	0.111315i	$0.964494\pi$
$102$		0			0
$103$			6446.83i			0.607676i	0.952724	+	0.303838i	$0.0982681\pi$
$103$			6446.83i			0.607676i	−0.952724	+	0.303838i	$0.901732\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			12983.4i			1.13402i	0.823711	+	0.567010i	$0.191900\pi$
$107$			12983.4i			1.13402i	−0.823711	+	0.567010i	$0.808100\pi$
$108$		0			0
$109$	−2629.04	+	1517.87i	−0.221281	+	0.127756i	−0.606543	−	0.795051i	$-0.707444\pi$
$109$	−2629.04	+	1517.87i	−0.221281	+	0.127756i	0.385262	+	0.922807i	$0.374111\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			17730.0i			1.38852i	0.719723	+	0.694261i	$0.244269\pi$
$113$			17730.0i			1.38852i	−0.719723	+	0.694261i	$0.755731\pi$
$114$		0			0
$115$	−6633.68			−0.501602
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1539.55	−	2666.59i	−0.108718	−	0.188305i
$120$		0			0
$121$	30659.0			2.09405
$122$		0			0
$123$		0			0
$124$		0			0
$125$	23270.5			1.48931
$126$		0			0
$127$	−11755.7	−	6787.18i	−0.728858	−	0.420806i	0.0891465	−	0.996019i	$-0.471586\pi$
$127$	−11755.7	−	6787.18i	−0.728858	−	0.420806i	−0.818004	+	0.575212i	$0.804919\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6358.78	−	11013.7i	−0.370537	−	0.641788i	0.619111	−	0.785303i	$-0.287493\pi$
$131$	−6358.78	−	11013.7i	−0.370537	−	0.641788i	−0.989648	+	0.143515i	$0.954160\pi$
$132$		0			0
$133$	−20588.0	−	23231.1i	−1.16389	−	1.31331i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3871.84	−	6706.22i	0.206289	−	0.357303i	−0.744254	−	0.667897i	$-0.767195\pi$
$137$	3871.84	−	6706.22i	0.206289	−	0.357303i	0.950543	+	0.310594i	$0.100528\pi$
$138$		0			0
$139$	14255.9	−	24691.9i	0.737845	−	1.27798i	−0.215619	−	0.976478i	$-0.569177\pi$
$139$	14255.9	−	24691.9i	0.737845	−	1.27798i	0.953464	−	0.301507i	$-0.0974897\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	24023.7	+	13870.1i	1.17481	+	0.678277i
$144$		0			0
$145$		−	55668.1i		−	2.64771i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1307.72	+	2265.04i	0.0589037	+	0.102024i	0.893974	−	0.448120i	$-0.147906\pi$
$149$	1307.72	+	2265.04i	0.0589037	+	0.102024i	−0.835070	+	0.550144i	$0.814573\pi$
$150$		0			0
$151$		−	23463.3i		−	1.02905i	−0.857476	−	0.514524i	$-0.827969\pi$
$151$		−	23463.3i		−	1.02905i	0.857476	−	0.514524i	$-0.172031\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	26537.4	−	15321.4i	1.10457	−	0.637726i
$156$		0			0
$157$	10082.2	+	17463.0i	0.409033	+	0.708465i	0.994782	−	0.102027i	$-0.0325329\pi$
$157$	10082.2	+	17463.0i	0.409033	+	0.708465i	−0.585749	+	0.810493i	$0.699200\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6724.76	+	11647.6i	−0.259433	+	0.449351i
$162$		0			0
$163$	1556.44			0.0585811			0.0292905	−	0.999571i	$-0.490675\pi$
$163$	1556.44			0.0585811			0.0292905	+	0.999571i	$0.490675\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−645.702	−	372.796i	−0.0231526	−	0.0133672i	0.488379	−	0.872632i	$-0.337588\pi$
$167$	−645.702	−	372.796i	−0.0231526	−	0.0133672i	−0.511532	+	0.859264i	$0.670922\pi$
$168$		0			0
$169$	−5786.94	−	10023.3i	−0.202617	−	0.350943i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	22155.3	−	12791.4i	0.740262	−	0.427390i	−0.0819027	−	0.996640i	$-0.526100\pi$
$173$	22155.3	−	12791.4i	0.740262	−	0.427390i	0.822165	+	0.569250i	$0.192766\pi$
$174$		0			0
$175$	50460.8	−	87400.6i	1.64770	−	2.85390i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	931.256i		−	0.0290645i	−0.999894	−	0.0145323i	$-0.995374\pi$
$179$		−	931.256i		−	0.0290645i	0.999894	−	0.0145323i	$-0.00462592\pi$
$180$		0			0
$181$	−40671.9	−	23481.9i	−1.24147	−	0.716764i	−0.272078	−	0.962275i	$-0.587711\pi$
$181$	−40671.9	−	23481.9i	−1.24147	−	0.716764i	−0.969394	+	0.245511i	$0.921044\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−76728.5	+	44299.2i	−2.24188	+	1.29435i
$186$		0			0
$187$	3810.78	+	6600.47i	0.108976	+	0.188752i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	68782.0			1.88542			0.942710	−	0.333614i	$-0.108268\pi$
$191$	68782.0			1.88542			0.942710	+	0.333614i	$0.108268\pi$
$192$		0			0
$193$	−19313.3	+	11150.5i	−0.518492	+	0.299352i	−0.736317	−	0.676636i	$-0.763437\pi$
$193$	−19313.3	+	11150.5i	−0.518492	+	0.299352i	0.217825	+	0.975988i	$0.430104\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2106.62			−0.0542818			−0.0271409	−	0.999632i	$-0.508640\pi$
$197$	−2106.62			−0.0542818			−0.0271409	+	0.999632i	$0.508640\pi$
$198$		0			0
$199$	3553.15	−	6154.24i	0.0897239	−	0.155406i	−0.817671	−	0.575686i	$-0.804735\pi$
$199$	3553.15	−	6154.24i	0.0897239	−	0.155406i	0.907394	+	0.420280i	$0.138068\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−97743.7	−	56432.3i	−2.37190	−	1.36942i
$204$		0			0
$205$	58799.4	+	33947.8i	1.39915	+	0.807801i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	50960.6	+	57502.7i	1.16665	+	1.31642i
$210$		0			0
$211$	−58118.6	+	33554.8i	−1.30542	+	0.753685i	−0.981328	−	0.192340i	$-0.938392\pi$
$211$	−58118.6	+	33554.8i	−1.30542	+	0.753685i	−0.324093	+	0.946025i	$0.605059\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−20936.0	+	36262.2i	−0.452915	+	0.784472i
$216$		0			0
$217$		−	62126.9i		−	1.31935i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			4667.18i			0.0955587i
$222$		0			0
$223$	44903.7	−	25925.2i	0.902969	−	0.521329i	0.0248067	−	0.999692i	$-0.492103\pi$
$223$	44903.7	−	25925.2i	0.902969	−	0.521329i	0.878162	+	0.478363i	$0.158770\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			18295.0i			0.355043i	0.984117	+	0.177522i	$0.0568079\pi$
$227$			18295.0i			0.355043i	−0.984117	+	0.177522i	$0.943192\pi$
$228$		0			0
$229$	52130.2			0.994074			0.497037	−	0.867729i	$-0.334421\pi$
$229$	52130.2			0.994074			0.497037	+	0.867729i	$0.334421\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−48911.6	−	84717.4i	−0.900949	−	1.56049i	−0.826264	−	0.563283i	$-0.809538\pi$
$233$	−48911.6	−	84717.4i	−0.900949	−	1.56049i	−0.0746852	−	0.997207i	$-0.523795\pi$
$234$		0			0
$235$	−69338.8			−1.25557
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−24074.5			−0.421464			−0.210732	−	0.977544i	$-0.567585\pi$
$239$	−24074.5			−0.421464			−0.210732	+	0.977544i	$0.567585\pi$
$240$		0			0
$241$	−67696.5	−	39084.6i	−1.16555	−	0.672933i	−0.212925	−	0.977069i	$-0.568299\pi$
$241$	−67696.5	−	39084.6i	−1.16555	−	0.672933i	−0.952629	+	0.304136i	$0.901632\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−105872.	−	183376.i	−1.76380	−	3.05499i
$246$		0			0
$247$	9419.26	+	46098.3i	0.154391	+	0.755599i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	62.5067	−	108.265i	0.000992154	−	0.00171846i	−0.865529	−	0.500859i	$-0.833018\pi$
$251$	62.5067	−	108.265i	0.000992154	−	0.00171846i	0.866521	+	0.499141i	$0.166351\pi$
$252$		0			0
$253$	16645.5	−	28830.8i	0.260049	−	0.450417i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	65981.9	+	38094.6i	0.998983	+	0.576763i	0.907947	−	0.419084i	$-0.137649\pi$
$257$	65981.9	+	38094.6i	0.998983	+	0.576763i	0.0910360	+	0.995848i	$0.470982\pi$
$258$		0			0
$259$			179629.i			2.67780i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−49122.8	−	85083.2i	−0.710186	−	1.23008i	−0.964787	−	0.263031i	$-0.915278\pi$
$263$	−49122.8	−	85083.2i	−0.710186	−	1.23008i	0.254602	−	0.967046i	$-0.418056\pi$
$264$		0			0
$265$			126679.i			1.80390i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	92239.9	−	53254.7i	1.27472	−	0.735959i	0.298846	−	0.954301i	$-0.403398\pi$
$269$	92239.9	−	53254.7i	1.27472	−	0.735959i	0.975872	+	0.218342i	$0.0700649\pi$
$270$		0			0
$271$	−5022.37	−	8699.01i	−0.0683865	−	0.118449i	0.829805	−	0.558054i	$-0.188452\pi$
$271$	−5022.37	−	8699.01i	−0.0683865	−	0.118449i	−0.898191	+	0.439605i	$0.855118\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−124903.	+	216338.i	−1.65161	+	2.86067i
$276$		0			0
$277$	10119.5			0.131887			0.0659434	−	0.997823i	$-0.478994\pi$
$277$	10119.5			0.131887			0.0659434	+	0.997823i	$0.478994\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	86418.9	+	49894.0i	1.09445	+	0.631881i	0.934758	−	0.355285i	$-0.115616\pi$
$281$	86418.9	+	49894.0i	1.09445	+	0.631881i	0.159693	+	0.987167i	$0.448950\pi$
$282$		0			0
$283$	−63814.3	−	110530.i	−0.796792	−	1.38008i	−0.921695	−	0.387916i	$-0.873195\pi$
$283$	−63814.3	−	110530.i	−0.796792	−	1.38008i	0.124903	−	0.992169i	$-0.460138\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	119213.	−	68827.8i	1.44731	−	0.835603i
$288$		0			0
$289$	41119.3	−	71220.8i	0.492323	−	0.852729i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	71974.2i		−	0.838381i	−0.907898	−	0.419191i	$-0.862314\pi$
$293$		−	71974.2i		−	0.838381i	0.907898	−	0.419191i	$-0.137686\pi$
$294$		0			0
$295$	−20625.8	−	11908.3i	−0.237010	−	0.136838i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	17655.0	−	10193.1i	0.197481	−	0.114015i
$300$		0			0
$301$	42446.9	+	73520.1i	0.468503	+	0.811471i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−100222.			−1.07737
$306$		0			0
$307$	59584.2	−	34401.0i	0.632200	−	0.365001i	−0.149403	−	0.988776i	$-0.547735\pi$
$307$	59584.2	−	34401.0i	0.632200	−	0.365001i	0.781604	+	0.623775i	$0.214402\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	129104.			1.33481			0.667406	−	0.744694i	$-0.267404\pi$
$311$	129104.			1.33481			0.667406	+	0.744694i	$0.267404\pi$
$312$		0			0
$313$	58809.0	−	101860.i	0.600282	−	1.03972i	−0.392496	−	0.919754i	$-0.628388\pi$
$313$	58809.0	−	101860.i	0.600282	−	1.03972i	0.992778	−	0.119965i	$-0.0382783\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	69133.5	+	39914.3i	0.687971	+	0.397200i	0.802852	−	0.596179i	$-0.203315\pi$
$317$	69133.5	+	39914.3i	0.687971	+	0.397200i	−0.114880	+	0.993379i	$0.536648\pi$
$318$		0			0
$319$	241940.	+	139684.i	2.37753	+	1.37267i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4091.51	+	12262.6i	−0.0392174	+	0.117537i
$324$		0			0
$325$	−132478.	+	76486.3i	−1.25423	+	0.724131i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−70290.8	+	121747.i	−0.649391	+	1.12478i
$330$		0			0
$331$		−	140871.i		−	1.28578i	−0.765959	−	0.642890i	$-0.777735\pi$
$331$		−	140871.i		−	1.28578i	0.765959	−	0.642890i	$-0.222265\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			260012.i			2.31688i
$336$		0			0
$337$	−129026.	+	74493.0i	−1.13610	+	0.655927i	−0.945462	−	0.325734i	$-0.894389\pi$
$337$	−129026.	+	74493.0i	−1.13610	+	0.655927i	−0.190637	+	0.981661i	$0.561055\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			153780.i			1.32248i
$342$		0			0
$343$	−222849.			−1.89418
$344$		0			0
$345$		0			0
$346$		0			0
$347$	51640.3	+	89443.6i	0.428874	+	0.742831i	0.996773	−	0.0802665i	$-0.0255771\pi$
$347$	51640.3	+	89443.6i	0.428874	+	0.742831i	−0.567900	+	0.823098i	$0.692244\pi$
$348$		0			0
$349$	93959.5			0.771418			0.385709	−	0.922620i	$-0.373957\pi$
$349$	93959.5			0.771418			0.385709	+	0.922620i	$0.373957\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	166106.			1.33301			0.666507	−	0.745499i	$-0.267789\pi$
$353$	166106.			1.33301			0.666507	+	0.745499i	$0.267789\pi$
$354$		0			0
$355$	−304510.	−	175809.i	−2.41627	−	1.39503i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−97076.6	−	168142.i	−0.753227	−	1.30463i	−0.946251	−	0.323433i	$-0.895163\pi$
$359$	−97076.6	−	168142.i	−0.753227	−	1.30463i	0.193024	−	0.981194i	$-0.438170\pi$
$360$		0			0
$361$	−15664.1	+	129376.i	−0.120197	+	0.992750i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	95646.4	−	165664.i	0.717932	−	1.24349i
$366$		0			0
$367$	50098.6	−	86773.3i	0.371958	−	0.644250i	−0.617909	−	0.786250i	$-0.712020\pi$
$367$	50098.6	−	86773.3i	0.371958	−	0.644250i	0.989867	+	0.142000i	$0.0453533\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	222427.	+	128418.i	1.61599	+	0.932995i
$372$		0			0
$373$		−	9316.12i		−	0.0669603i	−0.999439	−	0.0334801i	$-0.989341\pi$
$373$		−	9316.12i		−	0.0669603i	0.999439	−	0.0334801i	$-0.0106591\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	85537.7	+	148156.i	0.601832	+	1.04240i
$378$		0			0
$379$			166594.i			1.15979i	0.814691	+	0.579895i	$0.196907\pi$
$379$			166594.i			1.15979i	−0.814691	+	0.579895i	$0.803093\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−209543.	+	120980.i	−1.42848	+	0.824735i	−0.997001	−	0.0773862i	$-0.975343\pi$
$383$	−209543.	+	120980.i	−1.42848	+	0.824735i	−0.431482	+	0.902121i	$0.642009\pi$
$384$		0			0
$385$	388086.	+	672184.i	2.61822	+	4.53489i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	84413.1	−	146208.i	0.557841	−	0.966210i	−0.439835	−	0.898079i	$-0.644963\pi$
$389$	84413.1	−	146208.i	0.557841	−	0.966210i	0.997676	−	0.0681311i	$-0.0217036\pi$
$390$		0			0
$391$	5601.08			0.0366368
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−449377.	−	259448.i	−2.88016	−	1.66286i
$396$		0			0
$397$	107021.	+	185365.i	0.679026	+	1.17611i	0.975274	+	0.220997i	$0.0709311\pi$
$397$	107021.	+	185365.i	0.679026	+	1.17611i	−0.296248	+	0.955111i	$0.595736\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	206344.	−	119133.i	1.28323	−	0.740871i	0.305790	−	0.952099i	$-0.401080\pi$
$401$	206344.	−	119133.i	1.28323	−	0.740871i	0.977437	+	0.211228i	$0.0677463\pi$
$402$		0			0
$403$	−47084.6	+	81553.0i	−0.289914	+	0.502146i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	444628.i		−	2.68416i
$408$		0			0
$409$	−32444.6	−	18731.9i	−0.193953	−	0.111979i	0.399879	−	0.916568i	$-0.369052\pi$
$409$	−32444.6	−	18731.9i	−0.193953	−	0.111979i	−0.593832	+	0.804589i	$0.702386\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−41817.9	+	24143.6i	−0.245167	+	0.141547i
$414$		0			0
$415$	79195.8	+	137171.i	0.459839	+	0.796465i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−245774.			−1.39994			−0.699968	−	0.714174i	$-0.746802\pi$
$419$	−245774.			−1.39994			−0.699968	+	0.714174i	$0.746802\pi$
$420$		0			0
$421$	−135629.	+	78305.6i	−0.765225	+	0.441803i	−0.831169	−	0.556021i	$-0.812328\pi$
$421$	−135629.	+	78305.6i	−0.765225	+	0.441803i	0.0659437	+	0.997823i	$0.478994\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−42029.0			−0.232686
$426$		0			0
$427$	−101598.	+	175973.i	−0.557225	+	0.965142i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−190477.	−	109972.i	−1.02539	−	0.592008i	−0.109727	−	0.993962i	$-0.534998\pi$
$431$	−190477.	−	109972.i	−1.02539	−	0.592008i	−0.915660	+	0.401954i	$0.868331\pi$
$432$		0			0
$433$	143336.	+	82755.1i	0.764504	+	0.441386i	0.830910	−	0.556406i	$-0.187820\pi$
$433$	143336.	+	82755.1i	0.764504	+	0.441386i	−0.0664068	+	0.997793i	$0.521153\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	55322.5	−	11304.0i	0.289694	−	0.0591931i
$438$		0			0
$439$	−150964.	+	87159.3i	−0.783331	+	0.452256i	−0.837609	−	0.546270i	$-0.816047\pi$
$439$	−150964.	+	87159.3i	−0.783331	+	0.452256i	0.0542787	+	0.998526i	$0.482714\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6887.80	+	11930.0i	−0.0350972	+	0.0607902i	−0.883040	−	0.469297i	$-0.844507\pi$
$443$	−6887.80	+	11930.0i	−0.0350972	+	0.0607902i	0.847943	+	0.530087i	$0.177841\pi$
$444$		0			0
$445$			57387.6i			0.289800i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			82485.2i			0.409151i	0.978851	+	0.204575i	$0.0655813\pi$
$449$			82485.2i			0.409151i	−0.978851	+	0.204575i	$0.934419\pi$
$450$		0			0
$451$	−295083.	+	170366.i	−1.45074	+	0.837587i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			475300.i			2.29586i
$456$		0			0
$457$	−115009.			−0.550682			−0.275341	−	0.961347i	$-0.588791\pi$
$457$	−115009.			−0.550682			−0.275341	+	0.961347i	$0.588791\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−84197.0	−	145834.i	−0.396182	−	0.686208i	0.597069	−	0.802190i	$-0.296332\pi$
$461$	−84197.0	−	145834.i	−0.396182	−	0.686208i	−0.993251	+	0.115982i	$0.962998\pi$
$462$		0			0
$463$	−88084.1			−0.410899			−0.205450	−	0.978668i	$-0.565866\pi$
$463$	−88084.1			−0.410899			−0.205450	+	0.978668i	$0.565866\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	375649.			1.72246			0.861229	−	0.508217i	$-0.169695\pi$
$467$	375649.			1.72246			0.861229	+	0.508217i	$0.169695\pi$
$468$		0			0
$469$	456537.	+	263582.i	2.07553	+	1.19831i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−105067.	−	181981.i	−0.469616	−	0.813398i
$474$		0			0
$475$	−415125.	+	84822.5i	−1.83989	+	0.375945i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	53241.1	−	92216.4i	0.232047	−	0.401918i	−0.726363	−	0.687311i	$-0.758791\pi$
$479$	53241.1	−	92216.4i	0.232047	−	0.401918i	0.958410	+	0.285394i	$0.0921243\pi$
$480$		0			0
$481$	136137.	−	235797.i	0.588420	−	1.01917i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−78045.9	−	45059.8i	−0.331792	−	0.191560i
$486$		0			0
$487$		−	290053.i		−	1.22298i	−0.791253	−	0.611489i	$-0.790571\pi$
$487$		−	290053.i		−	1.22298i	0.791253	−	0.611489i	$-0.209429\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−173009.	−	299660.i	−0.717638	−	1.24299i	−0.961933	−	0.273285i	$-0.911890\pi$
$491$	−173009.	−	299660.i	−0.717638	−	1.24299i	0.244295	−	0.969701i	$-0.421443\pi$
$492$		0			0
$493$			47002.7i			0.193388i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−617382.	+	356446.i	−2.49943	+	1.44305i
$498$		0			0
$499$	−36221.7	−	62737.8i	−0.145468	−	0.251958i	0.784079	−	0.620661i	$-0.213135\pi$
$499$	−36221.7	−	62737.8i	−0.145468	−	0.251958i	−0.929547	+	0.368702i	$0.879802\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−64792.1	+	112223.i	−0.256086	+	0.443555i	−0.965190	−	0.261550i	$-0.915766\pi$
$503$	−64792.1	+	112223.i	−0.256086	+	0.443555i	0.709104	+	0.705104i	$0.249100\pi$
$504$		0			0
$505$	513359.			2.01298
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−102521.	−	59190.8i	−0.395712	−	0.228465i	0.288920	−	0.957353i	$-0.406704\pi$
$509$	−102521.	−	59190.8i	−0.395712	−	0.228465i	−0.684632	+	0.728889i	$0.740037\pi$
$510$		0			0
$511$	−193919.	−	335878.i	−0.742641	−	1.28629i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	236786.	−	136708.i	0.892773	−	0.515443i
$516$		0			0
$517$	173987.	−	301355.i	0.650933	−	1.12745i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			474111.i			1.74665i	0.487142	+	0.873323i	$0.338039\pi$
$521$			474111.i			1.74665i	−0.487142	+	0.873323i	$0.661961\pi$
$522$		0			0
$523$	−285050.	−	164574.i	−1.04212	−	0.601669i	−0.121688	−	0.992568i	$-0.538831\pi$
$523$	−285050.	−	164574.i	−1.04212	−	0.601669i	−0.920433	+	0.390899i	$0.872164\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−22406.6	+	12936.4i	−0.0806778	+	0.0465793i
$528$		0			0
$529$	127688.	+	221162.i	0.456287	+	0.790312i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−208653.			−0.734462
$534$		0			0
$535$	476867.	−	275319.i	1.66606	−	0.961897i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.06263e6			3.65767
$540$		0			0
$541$	−73319.7	+	126993.i	−0.250510	+	0.433897i	−0.963666	−	0.267108i	$-0.913932\pi$
$541$	−73319.7	+	126993.i	−0.250510	+	0.433897i	0.713156	+	0.701005i	$0.247265\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	111500.	+	64374.6i	0.375389	+	0.216731i
$546$		0			0
$547$	62940.1	+	36338.5i	0.210355	+	0.121448i	0.601476	−	0.798891i	$-0.294579\pi$
$547$	62940.1	+	36338.5i	0.210355	+	0.121448i	−0.391121	+	0.920339i	$0.627913\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	94860.4	+	464251.i	0.312451	+	1.52915i
$552$		0			0
$553$	−911093.	+	526020.i	−2.97929	+	1.72009i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	120662.	−	208992.i	0.388919	−	0.673627i	−0.603386	−	0.797449i	$-0.706182\pi$
$557$	120662.	−	208992.i	0.388919	−	0.673627i	0.992304	+	0.123823i	$0.0395154\pi$
$558$		0			0
$559$		−	128678.i		−	0.411796i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	586835.i		−	1.85139i	−0.378265	−	0.925697i	$-0.623479\pi$
$563$		−	586835.i		−	1.85139i	0.378265	−	0.925697i	$-0.376521\pi$
$564$		0			0
$565$	651207.	−	375974.i	2.03996	−	1.17777i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			600667.i			1.85528i	0.373477	+	0.927639i	$0.378165\pi$
$569$			600667.i			1.85528i	−0.373477	+	0.927639i	$0.621835\pi$
$570$		0			0
$571$	199533.			0.611987			0.305993	−	0.952034i	$-0.401011\pi$
$571$	199533.			0.611987			0.305993	+	0.952034i	$0.401011\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	91791.1	+	158987.i	0.277629	+	0.480867i
$576$		0			0
$577$	−8748.67			−0.0262779			−0.0131389	−	0.999914i	$-0.504182\pi$
$577$	−8748.67			−0.0262779			−0.0131389	+	0.999914i	$0.504182\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	321132.			0.951331
$582$		0			0
$583$	−550563.	−	317868.i	−1.61983	−	0.935210i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	107952.	+	186979.i	0.313297	+	0.542646i	0.979074	−	0.203505i	$-0.0652332\pi$
$587$	107952.	+	186979.i	0.313297	+	0.542646i	−0.665777	+	0.746151i	$0.731900\pi$
$588$		0			0
$589$	−195204.	+	172995.i	−0.562676	+	0.498659i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	76256.0	−	132079.i	0.216852	−	0.375599i	−0.736992	−	0.675902i	$-0.763754\pi$
$593$	76256.0	−	132079.i	0.216852	−	0.375599i	0.953844	+	0.300302i	$0.0970876\pi$
$594$		0			0
$595$	−65294.0	+	113093.i	−0.184433	+	0.319448i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2045.51	+	1180.98i	0.00570097	+	0.00329145i	0.502848	−	0.864375i	$-0.332286\pi$
$599$	2045.51	+	1180.98i	0.00570097	+	0.00329145i	−0.497147	+	0.867666i	$0.665619\pi$
$600$		0			0
$601$			162048.i			0.448636i	0.974516	+	0.224318i	$0.0720154\pi$
$601$			162048.i			0.448636i	−0.974516	+	0.224318i	$0.927985\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−650139.	−	1.12607e6i	−1.77622	−	3.07650i
$606$		0			0
$607$			231367.i			0.627948i	0.949432	+	0.313974i	$0.101661\pi$
$607$			231367.i			0.627948i	−0.949432	+	0.313974i	$0.898339\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	184539.	−	106544.i	0.494318	−	0.285394i
$612$		0			0
$613$	−251898.	−	436300.i	−0.670353	−	1.16109i	−0.977804	−	0.209521i	$-0.932809\pi$
$613$	−251898.	−	436300.i	−0.670353	−	1.16109i	0.307451	−	0.951564i	$-0.400524\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	108343.	−	187655.i	0.284596	−	0.492935i	−0.687915	−	0.725791i	$-0.741474\pi$
$617$	108343.	−	187655.i	0.284596	−	0.492935i	0.972511	+	0.232856i	$0.0748072\pi$
$618$		0			0
$619$	733476.			1.91428			0.957138	−	0.289632i	$-0.0935328\pi$
$619$	733476.			1.91428			0.957138	+	0.289632i	$0.0935328\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	100763.	+	58175.4i	0.259612	+	0.149887i
$624$		0			0
$625$	−126685.	−	219424.i	−0.324313	−	0.561726i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	64784.8	−	37403.5i	0.163746	−	0.0945391i
$630$		0			0
$631$	36952.4	−	64003.4i	0.0928076	−	0.160747i	−0.815884	−	0.578216i	$-0.803749\pi$
$631$	36952.4	−	64003.4i	0.0928076	−	0.160747i	0.908691	+	0.417468i	$0.137083\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			575702.i			1.42774i
$636$		0			0
$637$	563538.	+	325359.i	1.38882	+	0.801833i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−303233.	+	175072.i	−0.738007	+	0.426088i	−0.821344	−	0.570433i	$-0.806775\pi$
$641$	−303233.	+	175072.i	−0.738007	+	0.426088i	0.0833373	+	0.996521i	$0.473442\pi$
$642$		0			0
$643$	−190422.	−	329821.i	−0.460569	−	0.797730i	0.538420	−	0.842677i	$-0.319021\pi$
$643$	−190422.	−	329821.i	−0.460569	−	0.797730i	−0.998989	+	0.0449471i	$0.985688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−56799.4			−0.135686			−0.0678430	−	0.997696i	$-0.521612\pi$
$647$	−56799.4			−0.135686			−0.0678430	+	0.997696i	$0.521612\pi$
$648$		0			0
$649$	103510.	−	59761.4i	0.245749	−	0.141883i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	830516.			1.94770			0.973850	−	0.227194i	$-0.0729551\pi$
$653$	830516.			1.94770			0.973850	+	0.227194i	$0.0729551\pi$
$654$		0			0
$655$	−269682.	+	467103.i	−0.628593	+	1.08875i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	264748.	+	152853.i	0.609624	+	0.351967i	0.772818	−	0.634627i	$-0.218846\pi$
$659$	264748.	+	152853.i	0.609624	+	0.351967i	−0.163194	+	0.986594i	$0.552180\pi$
$660$		0			0
$661$	14657.8	+	8462.68i	0.0335479	+	0.0193689i	0.516680	−	0.856178i	$-0.327168\pi$
$661$	14657.8	+	8462.68i	0.0335479	+	0.0193689i	−0.483132	+	0.875547i	$0.660501\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−416674.	+	1.24880e6i	−0.942223	+	2.82391i
$666$		0			0
$667$	177801.	−	102654.i	0.399653	−	0.230740i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	251481.	−	435578.i	0.558548	−	0.967433i
$672$		0			0
$673$		−	354532.i		−	0.782754i	−0.920230	−	0.391377i	$-0.871999\pi$
$673$		−	354532.i		−	0.782754i	0.920230	−	0.391377i	$-0.128001\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			615414.i			1.34273i	0.741125	+	0.671367i	$0.234292\pi$
$677$			615414.i			1.34273i	−0.741125	+	0.671367i	$0.765708\pi$
$678$		0			0
$679$	−158235.	+	91356.8i	−0.343212	+	0.198153i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			209603.i			0.449320i	0.974437	+	0.224660i	$0.0721271\pi$
$683$			209603.i			0.449320i	−0.974437	+	0.224660i	$0.927873\pi$
$684$		0			0
$685$	−328417.			−0.699914
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−194651.	−	337146.i	−0.410033	−	0.710197i
$690$		0			0
$691$	−238346.			−0.499174			−0.249587	−	0.968352i	$-0.580295\pi$
$691$	−238346.			−0.499174			−0.249587	+	0.968352i	$0.580295\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.20921e6			−2.50342
$696$		0			0
$697$	−49646.6	−	28663.5i	−0.102194	−	0.0590016i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−312166.	−	540687.i	−0.635256	−	1.10030i	−0.986461	−	0.163997i	$-0.947561\pi$
$701$	−312166.	−	540687.i	−0.635256	−	1.10030i	0.351204	−	0.936299i	$-0.385772\pi$
$702$		0			0
$703$	564400.	−	500187.i	1.14203	−	1.01210i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	520407.	−	901371.i	1.04113	−	1.80329i
$708$		0			0
$709$	−27489.4	+	47613.0i	−0.0546855	+	0.0947180i	−0.892072	−	0.451893i	$-0.850749\pi$
$709$	−27489.4	+	47613.0i	−0.0546855	+	0.0947180i	0.837387	+	0.546611i	$0.184082\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	97871.5	+	56506.2i	0.192521	+	0.111152i
$714$		0			0
$715$		−	1.17649e6i		−	2.30131i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8474.63	+	14678.5i	0.0163932	+	0.0283938i	0.874106	−	0.485736i	$-0.161448\pi$
$719$	8474.63	+	14678.5i	0.0163932	+	0.0283938i	−0.857712	+	0.514130i	$0.828115\pi$
$720$		0			0
$721$		−	554340.i		−	1.06637i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.33417e6	+	770286.i	−2.53826	+	1.46547i
$726$		0			0
$727$	146369.	+	253518.i	0.276936	+	0.479668i	0.970622	−	0.240610i	$-0.0773475\pi$
$727$	146369.	+	253518.i	0.276936	+	0.479668i	−0.693685	+	0.720278i	$0.744014\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	17677.1	−	30617.6i	0.0330808	−	0.0572976i
$732$		0			0
$733$	−246112.			−0.458062			−0.229031	−	0.973419i	$-0.573556\pi$
$733$	−246112.			−0.458062			−0.229031	+	0.973419i	$0.573556\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.13004e6	−	652430.i	−2.08046	−	1.20116i
$738$		0			0
$739$	18872.5	+	32688.2i	0.0345574	+	0.0598552i	0.882787	−	0.469774i	$-0.155665\pi$
$739$	18872.5	+	32688.2i	0.0345574	+	0.0598552i	−0.848229	+	0.529629i	$0.822331\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	512608.	−	295954.i	0.928555	−	0.536101i	0.0422007	−	0.999109i	$-0.486563\pi$
$743$	512608.	−	295954.i	0.928555	−	0.536101i	0.886354	+	0.463008i	$0.153230\pi$
$744$		0			0
$745$	55461.7	−	96062.5i	0.0999265	−	0.173078i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	1.11640e6i		−	1.99001i
$750$		0			0
$751$	−248355.	−	143388.i	−0.440346	−	0.254234i	0.263399	−	0.964687i	$-0.415157\pi$
$751$	−248355.	−	143388.i	−0.440346	−	0.254234i	−0.703744	+	0.710453i	$0.748490\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−861785.	+	497552.i	−1.51184	+	0.872859i
$756$		0			0
$757$	−474233.	−	821396.i	−0.827562	−	1.43338i	−0.899946	−	0.436002i	$-0.856394\pi$
$757$	−474233.	−	821396.i	−0.827562	−	1.43338i	0.0723842	−	0.997377i	$-0.476939\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	931294.			1.60812			0.804058	−	0.594551i	$-0.202670\pi$
$761$	931294.			1.60812			0.804058	+	0.594551i	$0.202670\pi$
$762$		0			0
$763$	226062.	−	130517.i	0.388309	−	0.224190i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	73191.6			0.124414
$768$		0			0
$769$	296275.	−	513164.i	0.501006	−	0.867768i	−0.498993	−	0.866606i	$-0.666297\pi$
$769$	296275.	−	513164.i	0.501006	−	0.867768i	0.999999	−	0.00116213i	$-0.000369919\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	689452.	+	398055.i	1.15384	+	0.666169i	0.949820	−	0.312798i	$-0.101266\pi$
$773$	689452.	+	398055.i	1.15384	+	0.666169i	0.204019	+	0.978967i	$0.434600\pi$
$774$		0			0
$775$	−734402.	−	424007.i	−1.22273	−	0.705943i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−548214.	−	182916.i	−0.903390	−	0.301424i
$780$		0			0
$781$	1.52818e6	−	882292.i	2.50537	−	1.44647i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	427598.	−	740621.i	0.693899	−	1.20187i
$786$		0			0
$787$		−	819529.i		−	1.32317i	−0.749872	−	0.661583i	$-0.769885\pi$
$787$		−	819529.i		−	1.32317i	0.749872	−	0.661583i	$-0.230115\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1.52454e6i		−	2.43662i
$792$		0			0
$793$	266733.	−	153998.i	0.424161	−	0.244889i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			270222.i			0.425407i	0.977117	+	0.212703i	$0.0682268\pi$
$797$			270222.i			0.425407i	−0.977117	+	0.212703i	$0.931773\pi$
$798$		0			0
$799$	58545.5			0.0917064
$800$		0			0
$801$		0			0
$802$		0			0
$803$	479998.	+	831381.i	0.744404	+	1.28935i
$804$		0			0
$805$	570407.			0.880224
$806$		0			0
$807$		0			0
$808$		0			0
$809$	619454.			0.946482			0.473241	−	0.880933i	$-0.343084\pi$
$809$	619454.			0.946482			0.473241	+	0.880933i	$0.343084\pi$
$810$		0			0
$811$	−74022.6	−	42737.0i	−0.112544	−	0.0649774i	0.442671	−	0.896684i	$-0.354031\pi$
$811$	−74022.6	−	42737.0i	−0.112544	−	0.0649774i	−0.555215	+	0.831707i	$0.687364\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−33005.1	−	57166.5i	−0.0496896	−	0.0860649i
$816$		0			0
$817$	112806.	−	338089.i	0.169001	−	0.506509i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−281272.	+	487177.i	−0.417292	+	0.722770i	−0.995666	−	0.0930013i	$-0.970354\pi$
$821$	−281272.	+	487177.i	−0.417292	+	0.722770i	0.578374	+	0.815771i	$0.303687\pi$
$822$		0			0
$823$	579245.	−	1.00328e6i	0.855190	−	1.48123i	−0.0212783	−	0.999774i	$-0.506774\pi$
$823$	579245.	−	1.00328e6i	0.855190	−	1.48123i	0.876468	−	0.481459i	$-0.159893\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−609102.	−	351665.i	−0.890593	−	0.514184i	−0.0164563	−	0.999865i	$-0.505238\pi$
$827$	−609102.	−	351665.i	−0.890593	−	0.514184i	−0.874136	+	0.485681i	$0.838572\pi$
$828$		0			0
$829$			515714.i			0.750413i	0.926941	+	0.375206i	$0.122428\pi$
$829$			515714.i			0.750413i	−0.926941	+	0.375206i	$0.877572\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	89391.9	+	154831.i	0.128827	+	0.223136i
$834$		0			0
$835$			31621.3i			0.0453531i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−920199.	+	531277.i	−1.30725	+	0.754739i	−0.981636	−	0.190764i	$-0.938903\pi$
$839$	−920199.	+	531277.i	−1.30725	+	0.754739i	−0.325611	+	0.945504i	$0.605570\pi$
$840$		0			0
$841$	507801.	+	879537.i	0.717962	+	1.24355i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−245430.	+	425097.i	−0.343727	+	0.595353i
$846$		0			0
$847$	−2.63626e6			−3.67470
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−282979.	−	163378.i	−0.390747	−	0.225598i
$852$		0			0
$853$	344968.	+	597503.i	0.474112	+	0.821186i	0.999561	−	0.0296390i	$-0.00943577\pi$
$853$	344968.	+	597503.i	0.474112	+	0.821186i	−0.525448	+	0.850825i	$0.676102\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	645961.	−	372946.i	0.879518	−	0.507790i	0.00901841	−	0.999959i	$-0.497129\pi$
$857$	645961.	−	372946.i	0.879518	−	0.507790i	0.870499	+	0.492169i	$0.163796\pi$
$858$		0			0
$859$	−421515.	+	730085.i	−0.571250	+	0.989435i	0.425187	+	0.905105i	$0.360208\pi$
$859$	−421515.	+	730085.i	−0.571250	+	0.989435i	−0.996438	+	0.0843295i	$0.973125\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			164924.i			0.221444i	0.993851	+	0.110722i	$0.0353162\pi$
$863$			164924.i			0.221444i	−0.993851	+	0.110722i	$0.964684\pi$
$864$		0			0
$865$	−939628.	−	542494.i	−1.25581	−	0.725042i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.25518e6	−	1.30203e6i	2.98636	−	1.72418i
$870$		0			0
$871$	−399526.	−	691999.i	−0.526633	−	0.912156i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.00095e6			−2.61349
$876$		0			0
$877$	−285312.	+	164725.i	−0.370954	+	0.214170i	−0.673875	−	0.738845i	$-0.735371\pi$
$877$	−285312.	+	164725.i	−0.370954	+	0.214170i	0.302921	+	0.953016i	$0.402038\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	535550.			0.689999			0.344999	−	0.938603i	$-0.387879\pi$
$881$	535550.			0.689999			0.344999	+	0.938603i	$0.387879\pi$
$882$		0			0
$883$	507210.	−	878513.i	0.650528	−	1.12675i	−0.332466	−	0.943115i	$-0.607881\pi$
$883$	507210.	−	878513.i	0.650528	−	1.12675i	0.982995	−	0.183633i	$-0.0587859\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.34482e6	+	776433.i	1.70930	+	0.986863i	0.935429	+	0.353515i	$0.115014\pi$
$887$	1.34482e6	+	776433.i	1.70930	+	0.986863i	0.773867	+	0.633348i	$0.218320\pi$
$888$		0			0
$889$	1.01083e6	+	583606.i	1.27902	+	0.738441i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	578260.	−	118156.i	0.725138	−	0.148167i
$894$		0			0
$895$	−34204.1	+	19747.7i	−0.0427004	+	0.0246531i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−474184.	+	821311.i	−0.586716	+	1.01622i
$900$		0			0
$901$		−	106960.i		−	0.131757i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			1.99178e6i			2.43189i
$906$		0			0
$907$	487748.	−	281601.i	0.592899	−	0.342310i	−0.173344	−	0.984861i	$-0.555457\pi$
$907$	487748.	−	281601.i	0.592899	−	0.342310i	0.766243	+	0.642551i	$0.222124\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1.03690e6i			1.24939i	0.780868	+	0.624696i	$0.214777\pi$
$911$			1.03690e6i			1.24939i	−0.780868	+	0.624696i	$0.785223\pi$
$912$		0			0
$913$	−794883.			−0.953589
$914$		0			0
$915$		0			0
$916$		0			0
$917$	546769.	+	947032.i	0.650227	+	1.12623i
$918$		0			0
$919$	109237.			0.129342			0.0646708	−	0.997907i	$-0.479400\pi$
$919$	109237.			0.129342			0.0646708	+	0.997907i	$0.479400\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.08057e6			1.26838
$924$		0			0
$925$	2.12340e6	+	1.22595e6i	2.48170	+	1.43281i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	389990.	+	675482.i	0.451879	+	0.782677i	0.998503	−	0.0547012i	$-0.0174206\pi$
$929$	389990.	+	675482.i	0.451879	+	0.782677i	−0.546624	+	0.837378i	$0.684087\pi$
$930$		0			0
$931$	1.19541e6	+	1.34888e6i	1.37917	+	1.55623i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	161619.	−	279932.i	0.184871	−	0.320206i
$936$		0			0
$937$	−781290.	+	1.35323e6i	−0.889883	+	1.54132i	−0.0498708	+	0.998756i	$0.515881\pi$
$937$	−781290.	+	1.35323e6i	−0.889883	+	1.54132i	−0.840012	+	0.542567i	$0.817452\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−588604.	−	339831.i	−0.664728	−	0.383781i	0.129348	−	0.991599i	$-0.458711\pi$
$941$	−588604.	−	339831.i	−0.664728	−	0.383781i	−0.794076	+	0.607819i	$0.792045\pi$
$942$		0			0
$943$			250403.i			0.281590i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	271328.	+	469955.i	0.302549	+	0.524030i	0.976713	−	0.214552i	$-0.0688292\pi$
$947$	271328.	+	469955.i	0.302549	+	0.524030i	−0.674164	+	0.738582i	$0.735496\pi$
$948$		0			0
$949$			587869.i			0.652751i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	251919.	−	145445.i	0.277379	−	0.160145i	−0.354857	−	0.934921i	$-0.615470\pi$
$953$	251919.	−	145445.i	0.277379	−	0.160145i	0.632236	+	0.774775i	$0.282137\pi$
$954$		0			0
$955$	−1.45856e6	−	2.52629e6i	−1.59925	−	2.76998i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−332926.	+	576644.i	−0.362001	+	0.627005i
$960$		0			0
$961$	401487.			0.434735
$962$		0			0
$963$		0			0
$964$		0			0
$965$	819097.	+	472906.i	0.879590	+	0.507832i
$966$		0			0
$967$	−255285.	−	442167.i	−0.273006	−	0.472861i	0.696624	−	0.717437i	$-0.254685\pi$
$967$	−255285.	−	442167.i	−0.273006	−	0.472861i	−0.969630	+	0.244576i	$0.921351\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	519420.	−	299887.i	0.550909	−	0.318067i	−0.198580	−	0.980085i	$-0.563633\pi$
$971$	519420.	−	299887.i	0.550909	−	0.318067i	0.749489	+	0.662017i	$0.230299\pi$
$972$		0			0
$973$	−1.22581e6	+	2.12317e6i	−1.29479	+	2.24264i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.64637e6i		−	1.72480i	−0.506228	−	0.862400i	$-0.668960\pi$
$977$		−	1.64637e6i		−	1.72480i	0.506228	−	0.862400i	$-0.331040\pi$
$978$		0			0
$979$	−249413.	−	143999.i	−0.260228	−	0.150243i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−659632.	+	380839.i	−0.682645	+	0.394125i	−0.800851	−	0.598864i	$-0.795619\pi$
$983$	−659632.	+	380839.i	−0.682645	+	0.394125i	0.118206	+	0.992989i	$0.462286\pi$
$984$		0			0
$985$	44672.0	+	77374.1i	0.0460429	+	0.0797486i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−154427.			−0.157881
$990$		0			0
$991$	−1.63008e6	+	941127.i	−1.65982	+	0.958299i	−0.687027	+	0.726632i	$0.741085\pi$
$991$	−1.63008e6	+	941127.i	−1.65982	+	0.958299i	−0.972795	+	0.231667i	$0.925582\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−301386.			−0.304422
$996$		0			0
$997$	116896.	−	202470.i	0.117601	−	0.203690i	−0.801216	−	0.598376i	$-0.795813\pi$
$997$	116896.	−	202470.i	0.117601	−	0.203690i	0.918816	+	0.394685i	$0.129146\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.5.y.f.145.2	✓	24
3.2	odd	2	inner	684.5.y.f.145.11	yes	24
19.8	odd	6	inner	684.5.y.f.217.2	yes	24
57.8	even	6	inner	684.5.y.f.217.11	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.5.y.f.145.2	✓	24	1.1	even	1	trivial
684.5.y.f.145.11	yes	24	3.2	odd	2	inner
684.5.y.f.217.2	yes	24	19.8	odd	6	inner
684.5.y.f.217.11	yes	24	57.8	even	6	inner

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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-21.2055 - 36.7290i) q^{5} -85.9865 q^{7} +O(q^{10})\)	\(q+(-21.2055 - 36.7290i) q^{5} -85.9865 q^{7} +212.838 q^{11} +(112.873 + 65.1673i) q^{13} +(17.9046 + 31.0117i) q^{17} +(239.434 + 270.171i) q^{19} +(78.2072 - 135.459i) q^{23} +(-586.846 + 1016.45i) q^{25} +(1136.73 + 656.293i) q^{29} +722.519i q^{31} +(1823.39 + 3158.20i) q^{35} -2089.04i q^{37} +(-1386.42 + 800.449i) q^{41} +(-493.646 - 855.020i) q^{43} +(817.463 - 1415.89i) q^{47} +4992.67 q^{49} +(-2586.77 - 1493.47i) q^{53} +(-4513.33 - 7817.32i) q^{55} +(486.331 - 280.784i) q^{59} +(1181.56 - 2046.52i) q^{61} -5527.62i q^{65} +(-5309.40 - 3065.38i) q^{67} +(7179.99 - 4145.37i) q^{71} +(2255.23 + 3906.17i) q^{73} -18301.2 q^{77} +(10595.8 - 6117.47i) q^{79} -3734.68 q^{83} +(759.352 - 1315.24i) q^{85} +(-1171.84 - 676.565i) q^{89} +(-9705.56 - 5603.51i) q^{91} +(4845.82 - 14523.3i) q^{95} +(1840.23 - 1062.46i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 304 q^{7}+O(q^{10}) \) 24 * q + 304 * q^7	\( 24 q + 304 q^{7} + 516 q^{13} + 416 q^{19} - 5108 q^{25} - 720 q^{43} + 25440 q^{49} - 12536 q^{55} + 5020 q^{61} - 13080 q^{67} + 22572 q^{73} + 63096 q^{79} - 32624 q^{85} - 89568 q^{91} - 3888 q^{97}+O(q^{100}) \) 24 * q + 304 * q^7 + 516 * q^13 + 416 * q^19 - 5108 * q^25 - 720 * q^43 + 25440 * q^49 - 12536 * q^55 + 5020 * q^61 - 13080 * q^67 + 22572 * q^73 + 63096 * q^79 - 32624 * q^85 - 89568 * q^91 - 3888 * q^97

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