LMFDB - Embedded newform 756.4.x.a.125.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,4,Mod(125,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 3]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("756.125");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.x (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			125.1
Character	$\chi$	$=$	756.125
Dual form			756.4.x.a.629.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-10.5571 + 18.2854i) q^{5} +(17.6959 + 5.46406i) q^{7} +O(q^{10})$	$q+(-10.5571 + 18.2854i) q^{5} +(17.6959 + 5.46406i) q^{7} +(21.5235 - 12.4266i) q^{11} +(52.5651 + 30.3485i) q^{13} +117.397 q^{17} -104.946i q^{19} +(17.4885 + 10.0970i) q^{23} +(-160.403 - 277.826i) q^{25} +(24.2671 - 14.0106i) q^{29} +(216.679 + 125.100i) q^{31} +(-286.729 + 265.891i) q^{35} +18.2279 q^{37} +(153.058 - 265.105i) q^{41} +(74.5402 + 129.107i) q^{43} +(108.676 + 188.233i) q^{47} +(283.288 + 193.383i) q^{49} -116.324i q^{53} +524.754i q^{55} +(-38.3919 + 66.4967i) q^{59} +(-493.712 + 285.045i) q^{61} +(-1109.87 + 640.781i) q^{65} +(-33.8430 + 58.6179i) q^{67} +796.967i q^{71} -710.481i q^{73} +(448.778 - 102.294i) q^{77} +(-40.0640 - 69.3928i) q^{79} +(-57.6203 - 99.8013i) q^{83} +(-1239.37 + 2146.65i) q^{85} +1055.26 q^{89} +(764.359 + 824.261i) q^{91} +(1918.98 + 1107.92i) q^{95} +(-444.295 + 256.514i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 6 q^{7}+O(q^{10}) $ 48 * q + 6 * q^7	$ 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) $ 48 * q + 6 * q^7 + 12 * q^11 + 408 * q^23 - 600 * q^25 + 84 * q^29 + 336 * q^37 + 84 * q^43 + 318 * q^49 - 2964 * q^65 - 588 * q^67 - 2400 * q^77 + 204 * q^79 - 360 * q^85 - 1080 * q^91 - 300 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$

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
$5$	−10.5571	+	18.2854i	−0.944252	+	1.63549i	−0.187010	+	0.982358i	$0.559880\pi$
$5$	−10.5571	+	18.2854i	−0.944252	+	1.63549i	−0.757242	+	0.653135i	$0.773454\pi$
$6$		0			0
$7$	17.6959	+	5.46406i	0.955487	+	0.295032i
$7$	17.6959	+	5.46406i	0.955487	+	0.295032i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	21.5235	−	12.4266i	0.589963	−	0.340615i	−0.175120	−	0.984547i	$-0.556031\pi$
$11$	21.5235	−	12.4266i	0.589963	−	0.340615i	0.765083	+	0.643932i	$0.222698\pi$
$12$		0			0
$13$	52.5651	+	30.3485i	1.12146	+	0.647473i	0.941772	−	0.336251i	$-0.109159\pi$
$13$	52.5651	+	30.3485i	1.12146	+	0.647473i	0.179684	+	0.983724i	$0.442493\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	117.397			1.67489			0.837443	−	0.546525i	$-0.184050\pi$
$17$	117.397			1.67489			0.837443	+	0.546525i	$0.184050\pi$
$18$		0			0
$19$		−	104.946i		−	1.26717i	−0.773672	−	0.633586i	$-0.781582\pi$
$19$		−	104.946i		−	1.26717i	0.773672	−	0.633586i	$-0.218418\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	17.4885	+	10.0970i	0.158548	+	0.0915379i	0.577175	−	0.816621i	$-0.304155\pi$
$23$	17.4885	+	10.0970i	0.158548	+	0.0915379i	−0.418627	+	0.908158i	$0.637488\pi$
$24$		0			0
$25$	−160.403	−	277.826i	−1.28322	−	2.22261i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	24.2671	−	14.0106i	0.155390	−	0.0897142i	−0.420289	−	0.907390i	$-0.638071\pi$
$29$	24.2671	−	14.0106i	0.155390	−	0.0897142i	0.575679	+	0.817676i	$0.304738\pi$
$30$		0			0
$31$	216.679	+	125.100i	1.25538	+	0.724794i	0.972173	−	0.234265i	$-0.0752683\pi$
$31$	216.679	+	125.100i	1.25538	+	0.724794i	0.283207	+	0.959059i	$0.408602\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−286.729	+	265.891i	−1.38474	+	1.28411i
$36$		0			0
$37$	18.2279			0.0809904			0.0404952	−	0.999180i	$-0.487106\pi$
$37$	18.2279			0.0809904			0.0404952	+	0.999180i	$0.487106\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	153.058	−	265.105i	0.583016	−	1.00981i	−0.412103	−	0.911137i	$-0.635206\pi$
$41$	153.058	−	265.105i	0.583016	−	1.00981i	0.995120	−	0.0986769i	$-0.0314610\pi$
$42$		0			0
$43$	74.5402	+	129.107i	0.264355	+	0.457877i	0.967395	−	0.253274i	$-0.0815075\pi$
$43$	74.5402	+	129.107i	0.264355	+	0.457877i	−0.703039	+	0.711151i	$0.748174\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	108.676	+	188.233i	0.337278	+	0.584182i	0.983920	−	0.178611i	$-0.0571605\pi$
$47$	108.676	+	188.233i	0.337278	+	0.584182i	−0.646642	+	0.762794i	$0.723827\pi$
$48$		0			0
$49$	283.288	+	193.383i	0.825913	+	0.563798i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	116.324i		−	0.301479i	−0.988574	−	0.150739i	$-0.951835\pi$
$53$		−	116.324i		−	0.301479i	0.988574	−	0.150739i	$-0.0481655\pi$
$54$		0			0
$55$			524.754i			1.28651i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−38.3919	+	66.4967i	−0.0847153	+	0.146731i	−0.905270	−	0.424837i	$-0.860331\pi$
$59$	−38.3919	+	66.4967i	−0.0847153	+	0.146731i	0.820555	+	0.571568i	$0.193665\pi$
$60$		0			0
$61$	−493.712	+	285.045i	−1.03629	+	0.598300i	−0.918779	−	0.394773i	$-0.870823\pi$
$61$	−493.712	+	285.045i	−1.03629	+	0.598300i	−0.117506	+	0.993072i	$0.537490\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1109.87	+	640.781i	−2.11787	+	1.22276i
$66$		0			0
$67$	−33.8430	+	58.6179i	−0.0617102	+	0.106885i	−0.895230	−	0.445604i	$-0.852989\pi$
$67$	−33.8430	+	58.6179i	−0.0617102	+	0.106885i	0.833520	+	0.552490i	$0.186322\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			796.967i			1.33215i	0.745886	+	0.666074i	$0.232027\pi$
$71$			796.967i			1.33215i	−0.745886	+	0.666074i	$0.767973\pi$
$72$		0			0
$73$		−	710.481i		−	1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$73$		−	710.481i		−	1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	448.778	−	102.294i	0.664194	−	0.151396i
$78$		0			0
$79$	−40.0640	−	69.3928i	−0.0570576	−	0.0988266i	0.836086	−	0.548599i	$-0.184839\pi$
$79$	−40.0640	−	69.3928i	−0.0570576	−	0.0988266i	−0.893143	+	0.449772i	$0.851505\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−57.6203	−	99.8013i	−0.0762006	−	0.131983i	0.825407	−	0.564538i	$-0.190946\pi$
$83$	−57.6203	−	99.8013i	−0.0762006	−	0.131983i	−0.901608	+	0.432555i	$0.857612\pi$
$84$		0			0
$85$	−1239.37	+	2146.65i	−1.58151	+	2.73926i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1055.26			1.25682			0.628409	−	0.777883i	$-0.283706\pi$
$89$	1055.26			1.25682			0.628409	+	0.777883i	$0.283706\pi$
$90$		0			0
$91$	764.359	+	824.261i	0.880512	+	0.949517i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1918.98	+	1107.92i	2.07245	+	1.19653i
$96$		0			0
$97$	−444.295	+	256.514i	−0.465065	+	0.268505i	−0.714172	−	0.699971i	$-0.753196\pi$
$97$	−444.295	+	256.514i	−0.465065	+	0.268505i	0.249106	+	0.968476i	$0.419863\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−83.3542	−	144.374i	−0.0821193	−	0.142235i	0.822041	−	0.569429i	$-0.192836\pi$
$101$	−83.3542	−	144.374i	−0.0821193	−	0.142235i	−0.904160	+	0.427194i	$0.859502\pi$
$102$		0			0
$103$	1304.16	+	752.955i	1.24760	+	0.720300i	0.970629	−	0.240579i	$-0.0773375\pi$
$103$	1304.16	+	752.955i	1.24760	+	0.720300i	0.276967	+	0.960879i	$0.410671\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	776.966i		−	0.701983i	−0.936379	−	0.350991i	$-0.885845\pi$
$107$		−	776.966i		−	0.701983i	0.936379	−	0.350991i	$-0.114155\pi$
$108$		0			0
$109$	−562.383			−0.494189			−0.247094	−	0.968991i	$-0.579476\pi$
$109$	−562.383			−0.494189			−0.247094	+	0.968991i	$0.579476\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1648.89	−	951.987i	−1.37270	−	0.792526i	−0.381429	−	0.924398i	$-0.624568\pi$
$113$	−1648.89	−	951.987i	−1.37270	−	0.792526i	−0.991267	+	0.131872i	$0.957901\pi$
$114$		0			0
$115$	−369.255	+	213.189i	−0.299419	+	0.172870i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2077.45	+	641.467i	1.60033	+	0.494144i
$120$		0			0
$121$	−356.658	+	617.750i	−0.267963	+	0.464125i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4134.27			2.95824
$126$		0			0
$127$	−2470.46			−1.72612			−0.863062	−	0.505098i	$-0.831456\pi$
$127$	−2470.46			−1.72612			−0.863062	+	0.505098i	$0.831456\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	199.978	−	346.373i	0.133375	−	0.231013i	−0.791600	−	0.611039i	$-0.790752\pi$
$131$	199.978	−	346.373i	0.133375	−	0.231013i	0.924976	+	0.380026i	$0.124085\pi$
$132$		0			0
$133$	573.431	−	1857.11i	0.373856	−	1.21077i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2065.15	−	1192.31i	1.28787	−	0.743550i	0.309592	−	0.950869i	$-0.399807\pi$
$137$	2065.15	−	1192.31i	1.28787	−	0.743550i	0.978273	+	0.207320i	$0.0664741\pi$
$138$		0			0
$139$	−570.066	−	329.128i	−0.347859	−	0.200836i	0.315883	−	0.948798i	$-0.397699\pi$
$139$	−570.066	−	329.128i	−0.347859	−	0.200836i	−0.663742	+	0.747962i	$0.731033\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1508.52			0.882156
$144$		0			0
$145$			591.645i			0.338851i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1116.26	+	644.476i	0.613745	+	0.354346i	0.774430	−	0.632660i	$-0.218037\pi$
$149$	1116.26	+	644.476i	0.613745	+	0.354346i	−0.160685	+	0.987006i	$0.551370\pi$
$150$		0			0
$151$	−98.0311	−	169.795i	−0.0528322	−	0.0915080i	0.838400	−	0.545056i	$-0.183492\pi$
$151$	−98.0311	−	169.795i	−0.0528322	−	0.0915080i	−0.891232	+	0.453548i	$0.850158\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4575.00	+	2641.38i	−2.37079	+	1.36878i
$156$		0			0
$157$	2865.72	+	1654.52i	1.45675	+	0.841053i	0.998850	−	0.0479519i	$-0.0152694\pi$
$157$	2865.72	+	1654.52i	1.45675	+	0.841053i	0.457897	+	0.889005i	$0.348603\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	254.304	+	274.234i	0.124484	+	0.134240i
$162$		0			0
$163$	−1864.65			−0.896018			−0.448009	−	0.894029i	$-0.647867\pi$
$163$	−1864.65			−0.896018			−0.448009	+	0.894029i	$0.647867\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1863.24	+	3227.22i	−0.863363	+	1.49539i	0.00530083	+	0.999986i	$0.498313\pi$
$167$	−1863.24	+	3227.22i	−0.863363	+	1.49539i	−0.868664	+	0.495402i	$0.835021\pi$
$168$		0			0
$169$	743.558	+	1287.88i	0.338443	+	0.586200i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1798.60	−	3115.26i	−0.790432	−	1.36907i	−0.925700	−	0.378259i	$-0.876523\pi$
$173$	−1798.60	−	3115.26i	−0.790432	−	1.36907i	0.135268	−	0.990809i	$-0.456810\pi$
$174$		0			0
$175$	−1320.41	−	5792.83i	−0.570365	−	2.50227i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			598.233i			0.249799i	0.992169	+	0.124899i	$0.0398608\pi$
$179$			598.233i			0.249799i	−0.992169	+	0.124899i	$0.960139\pi$
$180$		0			0
$181$			1669.69i			0.685673i	0.939395	+	0.342837i	$0.111388\pi$
$181$			1669.69i			0.685673i	−0.939395	+	0.342837i	$0.888612\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−192.433	+	333.304i	−0.0764754	+	0.132459i
$186$		0			0
$187$	2526.81	−	1458.85i	0.988120	−	0.570491i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2927.86	+	1690.40i	−1.10918	+	0.640383i	−0.938616	−	0.344964i	$-0.887891\pi$
$191$	−2927.86	+	1690.40i	−1.10918	+	0.640383i	−0.170560	+	0.985347i	$0.554558\pi$
$192$		0			0
$193$	476.298	−	824.972i	0.177641	−	0.307683i	−0.763431	−	0.645889i	$-0.776487\pi$
$193$	476.298	−	824.972i	0.177641	−	0.307683i	0.941072	+	0.338206i	$0.109820\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2192.96i		−	0.793106i	−0.918012	−	0.396553i	$-0.870206\pi$
$197$		−	2192.96i		−	0.793106i	0.918012	−	0.396553i	$-0.129794\pi$
$198$		0			0
$199$			1581.79i			0.563469i	0.959492	+	0.281735i	$0.0909098\pi$
$199$			1581.79i			0.563469i	−0.959492	+	0.281735i	$0.909090\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	505.983	−	115.333i	0.174941	−	0.0398760i
$204$		0			0
$205$	3231.69	+	5597.45i	1.10103	+	1.90704i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1304.12	−	2258.81i	−0.431618	−	0.747584i
$210$		0			0
$211$	−1547.26	+	2679.93i	−0.504823	+	0.874379i	0.495161	+	0.868801i	$0.335109\pi$
$211$	−1547.26	+	2679.93i	−0.504823	+	0.874379i	−0.999984	+	0.00557832i	$0.998224\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3147.70			−0.998472
$216$		0			0
$217$	3150.78	+	3397.70i	0.985663	+	1.06291i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6171.00	+	3562.83i	1.87831	+	1.08444i
$222$		0			0
$223$	1320.52	−	762.403i	0.396541	−	0.228943i	−0.288450	−	0.957495i	$-0.593140\pi$
$223$	1320.52	−	762.403i	0.396541	−	0.228943i	0.684990	+	0.728552i	$0.259806\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−836.909	−	1449.57i	−0.244703	−	0.423838i	0.717345	−	0.696718i	$-0.245357\pi$
$227$	−836.909	−	1449.57i	−0.244703	−	0.423838i	−0.962048	+	0.272880i	$0.912024\pi$
$228$		0			0
$229$	−2751.35	−	1588.49i	−0.793949	−	0.458387i	0.0474019	−	0.998876i	$-0.484906\pi$
$229$	−2751.35	−	1588.49i	−0.793949	−	0.458387i	−0.841351	+	0.540489i	$0.818239\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3225.60i			0.906936i	0.891272	+	0.453468i	$0.149813\pi$
$233$			3225.60i			0.906936i	−0.891272	+	0.453468i	$0.850187\pi$
$234$		0			0
$235$	−4589.21			−1.27390
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1535.18	+	886.337i	0.415492	+	0.239884i	0.693147	−	0.720797i	$-0.256224\pi$
$239$	1535.18	+	886.337i	0.415492	+	0.239884i	−0.277655	+	0.960681i	$0.589557\pi$
$240$		0			0
$241$	−3202.47	+	1848.95i	−0.855972	+	0.494196i	−0.862661	−	0.505782i	$-0.831204\pi$
$241$	−3202.47	+	1848.95i	−0.855972	+	0.494196i	0.00668945	+	0.999978i	$0.497871\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6526.76	+	3138.47i	−1.70196	+	0.818407i
$246$		0			0
$247$	3184.95	−	5516.49i	0.820460	−	1.42108i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2501.61			−0.629084			−0.314542	−	0.949244i	$-0.601851\pi$
$251$	−2501.61			−0.629084			−0.314542	+	0.949244i	$0.601851\pi$
$252$		0			0
$253$	501.887			0.124717
$254$		0			0
$255$		0			0
$256$		0			0
$257$	472.215	−	817.900i	0.114615	−	0.198518i	−0.803011	−	0.595964i	$-0.796770\pi$
$257$	472.215	−	817.900i	0.114615	−	0.198518i	0.917626	+	0.397446i	$0.130103\pi$
$258$		0			0
$259$	322.558	+	99.5983i	0.0773854	+	0.0238947i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4896.74	−	2827.13i	1.14808	−	0.662846i	0.199664	−	0.979864i	$-0.436015\pi$
$263$	4896.74	−	2827.13i	1.14808	−	0.662846i	0.948420	+	0.317018i	$0.102682\pi$
$264$		0			0
$265$	2127.03	+	1228.04i	0.493067	+	0.284672i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4010.37			−0.908983			−0.454491	−	0.890751i	$-0.650179\pi$
$269$	−4010.37			−0.908983			−0.454491	+	0.890751i	$0.650179\pi$
$270$		0			0
$271$		−	3596.81i		−	0.806239i	−0.915147	−	0.403120i	$-0.867926\pi$
$271$		−	3596.81i		−	0.806239i	0.915147	−	0.403120i	$-0.132074\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6904.88	−	3986.53i	−1.51411	−	0.874171i
$276$		0			0
$277$	−3175.77	−	5500.59i	−0.688856	−	1.19313i	−0.972208	−	0.234118i	$-0.924780\pi$
$277$	−3175.77	−	5500.59i	−0.688856	−	1.19313i	0.283352	−	0.959016i	$-0.408554\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1591.00	−	918.563i	0.337761	−	0.195007i	−0.321520	−	0.946903i	$-0.604194\pi$
$281$	1591.00	−	918.563i	0.337761	−	0.195007i	0.659282	+	0.751896i	$0.270860\pi$
$282$		0			0
$283$	−2653.87	−	1532.21i	−0.557442	−	0.321839i	0.194676	−	0.980868i	$-0.437634\pi$
$283$	−2653.87	−	1532.21i	−0.557442	−	0.321839i	−0.752118	+	0.659028i	$0.770968\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4157.05	−	3854.94i	0.854992	−	0.792856i
$288$		0			0
$289$	8869.15			1.80524
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3473.03	+	6015.47i	−0.692481	+	1.19941i	0.278542	+	0.960424i	$0.410149\pi$
$293$	−3473.03	+	6015.47i	−0.692481	+	1.19941i	−0.971023	+	0.238988i	$0.923184\pi$
$294$		0			0
$295$	−810.611	−	1404.02i	−0.159985	−	0.277102i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	612.857	+	1061.50i	0.118537	+	0.205311i
$300$		0			0
$301$	613.604	+	2691.96i	0.117500	+	0.515489i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	12036.9i		−	2.25978i
$306$		0			0
$307$		−	2168.45i		−	0.403128i	−0.979475	−	0.201564i	$-0.935398\pi$
$307$		−	2168.45i		−	0.403128i	0.979475	−	0.201564i	$-0.0646023\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	142.507	−	246.830i	0.0259834	−	0.0450046i	−0.852741	−	0.522333i	$-0.825062\pi$
$311$	142.507	−	246.830i	0.0259834	−	0.0450046i	0.878725	+	0.477329i	$0.158395\pi$
$312$		0			0
$313$	9429.66	−	5444.22i	1.70286	−	0.983148i	0.760025	−	0.649894i	$-0.225187\pi$
$313$	9429.66	−	5444.22i	1.70286	−	0.983148i	0.942837	−	0.333253i	$-0.108146\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−598.240	+	345.394i	−0.105995	+	0.0611964i	−0.552060	−	0.833804i	$-0.686158\pi$
$317$	−598.240	+	345.394i	−0.105995	+	0.0611964i	0.446065	+	0.895000i	$0.352825\pi$
$318$		0			0
$319$	348.210	−	603.117i	0.0611160	−	0.105856i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	12320.4i		−	2.12237i
$324$		0			0
$325$		−	19471.9i		−	3.32341i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	894.606	+	3924.76i	0.149912	+	0.657686i
$330$		0			0
$331$	−4246.17	−	7354.58i	−0.705108	−	1.22128i	−0.966653	−	0.256091i	$-0.917565\pi$
$331$	−4246.17	−	7354.58i	−0.705108	−	1.22128i	0.261545	−	0.965191i	$-0.415768\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−714.566	−	1237.66i	−0.116540	−	0.201853i
$336$		0			0
$337$	1799.53	−	3116.88i	0.290881	−	0.503820i	−0.683138	−	0.730290i	$-0.739385\pi$
$337$	1799.53	−	3116.88i	0.290881	−	0.503820i	0.974018	+	0.226470i	$0.0727184\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6218.28			0.987503
$342$		0			0
$343$	3956.38	+	4969.98i	0.622811	+	0.782372i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5923.82	−	3420.12i	−0.916448	−	0.529112i	−0.0339480	−	0.999424i	$-0.510808\pi$
$347$	−5923.82	−	3420.12i	−0.916448	−	0.529112i	−0.882500	+	0.470312i	$0.844141\pi$
$348$		0			0
$349$	−5589.41	+	3227.04i	−0.857289	+	0.494956i	−0.863104	−	0.505027i	$-0.831483\pi$
$349$	−5589.41	+	3227.04i	−0.857289	+	0.494956i	0.00581419	+	0.999983i	$0.498149\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	294.916	+	510.809i	0.0444668	+	0.0770188i	0.887402	−	0.460996i	$-0.152508\pi$
$353$	294.916	+	510.809i	0.0444668	+	0.0770188i	−0.842935	+	0.538015i	$0.819174\pi$
$354$		0			0
$355$	−14572.8	−	8413.62i	−2.17872	−	1.25788i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	8812.02i		−	1.29549i	−0.761857	−	0.647745i	$-0.775712\pi$
$359$		−	8812.02i		−	1.29549i	0.761857	−	0.647745i	$-0.224288\pi$
$360$		0			0
$361$	−4154.66			−0.605724
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12991.4	+	7500.59i	1.86302	+	1.07561i
$366$		0			0
$367$	−9716.89	+	5610.05i	−1.38206	+	0.797935i	−0.992404	−	0.123023i	$-0.960741\pi$
$367$	−9716.89	+	5610.05i	−1.38206	+	0.797935i	−0.389661	+	0.920958i	$0.627408\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	635.604	−	2058.46i	0.0889458	−	0.288059i
$372$		0			0
$373$	−403.305	+	698.546i	−0.0559849	+	0.0969687i	−0.892660	−	0.450731i	$-0.851163\pi$
$373$	−403.305	+	698.546i	−0.0559849	+	0.0969687i	0.836675	+	0.547700i	$0.184497\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1700.81			0.232350
$378$		0			0
$379$	7667.10			1.03914			0.519568	−	0.854429i	$-0.326093\pi$
$379$	7667.10			1.03914			0.519568	+	0.854429i	$0.326093\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5599.27	+	9698.23i	−0.747022	+	1.29388i	0.202222	+	0.979340i	$0.435184\pi$
$383$	−5599.27	+	9698.23i	−0.747022	+	1.29388i	−0.949244	+	0.314541i	$0.898149\pi$
$384$		0			0
$385$	−2867.29	+	9285.98i	−0.379560	+	1.22924i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1300.72	+	750.970i	−0.169535	+	0.0978810i	−0.582366	−	0.812926i	$-0.697873\pi$
$389$	−1300.72	+	750.970i	−0.169535	+	0.0978810i	0.412832	+	0.910807i	$0.364540\pi$
$390$		0			0
$391$	2053.11	+	1185.36i	0.265550	+	0.153315i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1691.83			0.215507
$396$		0			0
$397$			1509.70i			0.190856i	0.995436	+	0.0954280i	$0.0304220\pi$
$397$			1509.70i			0.190856i	−0.995436	+	0.0954280i	$0.969578\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	780.767	+	450.776i	0.0972311	+	0.0561364i	0.547827	−	0.836592i	$-0.315455\pi$
$401$	780.767	+	450.776i	0.0972311	+	0.0561364i	−0.450596	+	0.892728i	$0.648788\pi$
$402$		0			0
$403$	7593.18	+	13151.8i	0.938569	+	1.62565i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	392.329	−	226.511i	0.0477813	−	0.0275866i
$408$		0			0
$409$	13891.4	+	8020.20i	1.67943	+	0.969617i	0.962027	+	0.272955i	$0.0880009\pi$
$409$	13891.4	+	8020.20i	1.67943	+	0.969617i	0.717399	+	0.696662i	$0.245332\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1042.72	+	966.942i	−0.124235	+	0.115206i
$414$		0			0
$415$	2433.20			0.287810
$416$		0			0
$417$		0			0
$418$		0			0
$419$	424.053	−	734.482i	0.0494424	−	0.0856367i	−0.840245	−	0.542207i	$-0.817589\pi$
$419$	424.053	−	734.482i	0.0494424	−	0.0856367i	0.889687	+	0.456570i	$0.150922\pi$
$420$		0			0
$421$	−3933.58	−	6813.16i	−0.455370	−	0.788725i	0.543339	−	0.839513i	$-0.317160\pi$
$421$	−3933.58	−	6813.16i	−0.455370	−	0.788725i	−0.998709	+	0.0507886i	$0.983827\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−18830.9	−	32616.1i	−2.14925	−	3.72262i
$426$		0			0
$427$	−10294.2	+	2346.45i	−1.16667	+	0.265931i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			9164.19i			1.02418i	0.858930	+	0.512092i	$0.171129\pi$
$431$			9164.19i			1.02418i	−0.858930	+	0.512092i	$0.828871\pi$
$432$		0			0
$433$		−	5634.15i		−	0.625312i	−0.949866	−	0.312656i	$-0.898781\pi$
$433$		−	5634.15i		−	0.625312i	0.949866	−	0.312656i	$-0.101219\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1059.64	−	1835.35i	0.115994	−	0.200908i
$438$		0			0
$439$	11030.6	−	6368.50i	1.19923	−	0.692373i	0.238843	−	0.971058i	$-0.423232\pi$
$439$	11030.6	−	6368.50i	1.19923	−	0.692373i	0.960383	+	0.278685i	$0.0898985\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1685.37	+	973.047i	−0.180754	+	0.104359i	−0.587647	−	0.809117i	$-0.699946\pi$
$443$	−1685.37	+	973.047i	−0.180754	+	0.104359i	0.406893	+	0.913476i	$0.366612\pi$
$444$		0			0
$445$	−11140.4	+	19295.7i	−1.18675	+	2.05552i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			7602.87i			0.799113i	0.916709	+	0.399556i	$0.130836\pi$
$449$			7602.87i			0.799113i	−0.916709	+	0.399556i	$0.869164\pi$
$450$		0			0
$451$		−	7607.98i		−	0.794337i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−23141.3	+	5274.81i	−2.38435	+	0.543488i
$456$		0			0
$457$	−102.789	−	178.036i	−0.0105214	−	0.0182236i	0.860717	−	0.509084i	$-0.170016\pi$
$457$	−102.789	−	178.036i	−0.0105214	−	0.0182236i	−0.871238	+	0.490861i	$0.836682\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9165.82	+	15875.7i	0.926019	+	1.60391i	0.789914	+	0.613218i	$0.210125\pi$
$461$	9165.82	+	15875.7i	0.926019	+	1.60391i	0.136105	+	0.990694i	$0.456542\pi$
$462$		0			0
$463$	6817.76	−	11808.7i	0.684338	−	1.18531i	−0.289307	−	0.957236i	$-0.593425\pi$
$463$	6817.76	−	11808.7i	0.684338	−	1.18531i	0.973644	−	0.228071i	$-0.0732419\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−15609.8			−1.54676			−0.773378	−	0.633946i	$-0.781434\pi$
$467$	−15609.8			−1.54676			−0.773378	+	0.633946i	$0.781434\pi$
$468$		0			0
$469$	−919.174	+	852.374i	−0.0904979	+	0.0839211i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3208.74	+	1852.57i	0.311920	+	0.180087i
$474$		0			0
$475$	−29156.7	+	16833.7i	−2.81643	+	1.62607i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7412.67	−	12839.1i	−0.707084	−	1.22471i	−0.965934	−	0.258788i	$-0.916677\pi$
$479$	−7412.67	−	12839.1i	−0.707084	−	1.22471i	0.258850	−	0.965918i	$-0.416657\pi$
$480$		0			0
$481$	958.150	+	553.188i	0.0908272	+	0.0524391i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	10832.1i		−	1.01415i
$486$		0			0
$487$	12830.1			1.19381			0.596906	−	0.802311i	$-0.296396\pi$
$487$	12830.1			1.19381			0.596906	+	0.802311i	$0.296396\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7054.53	+	4072.93i	0.648404	+	0.374356i	0.787844	−	0.615874i	$-0.211197\pi$
$491$	7054.53	+	4072.93i	0.648404	+	0.374356i	−0.139441	+	0.990230i	$0.544530\pi$
$492$		0			0
$493$	2848.90	−	1644.81i	0.260260	−	0.150261i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4354.67	+	14103.0i	−0.393026	+	1.27285i
$498$		0			0
$499$	8727.94	−	15117.2i	0.782999	−	1.35619i	−0.147189	−	0.989108i	$-0.547022\pi$
$499$	8727.94	−	15117.2i	0.782999	−	1.35619i	0.930187	−	0.367085i	$-0.119644\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	13372.9			1.18543			0.592714	−	0.805413i	$-0.298057\pi$
$503$	13372.9			1.18543			0.592714	+	0.805413i	$0.298057\pi$
$504$		0			0
$505$	3519.90			0.310165
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2910.35	−	5040.87i	0.253436	−	0.438964i	−0.711033	−	0.703158i	$-0.751773\pi$
$509$	2910.35	−	5040.87i	0.253436	−	0.438964i	0.964470	+	0.264194i	$0.0851059\pi$
$510$		0			0
$511$	3882.11	−	12572.6i	0.336075	−	1.08841i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−27536.1	+	15898.0i	−2.35609	+	1.36029i
$516$		0			0
$517$	4678.19	+	2700.96i	0.397963	+	0.229764i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6546.17			0.550466			0.275233	−	0.961378i	$-0.411245\pi$
$521$	6546.17			0.550466			0.275233	+	0.961378i	$0.411245\pi$
$522$		0			0
$523$			7475.29i			0.624994i	0.949919	+	0.312497i	$0.101165\pi$
$523$			7475.29i			0.624994i	−0.949919	+	0.312497i	$0.898835\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	25437.6	+	14686.4i	2.10262	+	1.21395i
$528$		0			0
$529$	−5879.60	−	10183.8i	−0.483242	−	0.836999i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	16091.0	−	9290.16i	1.30765	−	0.754975i
$534$		0			0
$535$	14207.1	+	8202.48i	1.14809	+	0.662849i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	8500.45	+	641.967i	0.679296	+	0.0513014i
$540$		0			0
$541$	−6578.64			−0.522806			−0.261403	−	0.965230i	$-0.584185\pi$
$541$	−6578.64			−0.522806			−0.261403	+	0.965230i	$0.584185\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5937.11	−	10283.4i	0.466639	−	0.808242i
$546$		0			0
$547$	−2357.36	−	4083.06i	−0.184266	−	0.319158i	0.759063	−	0.651017i	$-0.225657\pi$
$547$	−2357.36	−	4083.06i	−0.184266	−	0.319158i	−0.943329	+	0.331859i	$0.892324\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1470.36	−	2546.74i	−0.113683	−	0.196905i
$552$		0			0
$553$	−329.800	−	1446.88i	−0.0253608	−	0.111261i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	5985.03i		−	0.455286i	−0.973745	−	0.227643i	$-0.926898\pi$
$557$		−	5985.03i		−	0.455286i	0.973745	−	0.227643i	$-0.0731018\pi$
$558$		0			0
$559$			9048.73i			0.684652i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10574.8	+	18316.0i	−0.791605	+	1.37110i	0.133368	+	0.991067i	$0.457421\pi$
$563$	−10574.8	+	18316.0i	−0.791605	+	1.37110i	−0.924973	+	0.380033i	$0.875913\pi$
$564$		0			0
$565$	34814.9	−	20100.4i	2.59234	−	1.49669i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9600.92	+	5543.09i	−0.707366	+	0.408398i	−0.810085	−	0.586312i	$-0.800579\pi$
$569$	−9600.92	+	5543.09i	−0.707366	+	0.408398i	0.102719	+	0.994710i	$0.467246\pi$
$570$		0			0
$571$	−889.103	+	1539.97i	−0.0651626	+	0.112865i	−0.896766	−	0.442505i	$-0.854090\pi$
$571$	−889.103	+	1539.97i	−0.0651626	+	0.112865i	0.831604	+	0.555370i	$0.187423\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	6478.36i		−	0.469854i
$576$		0			0
$577$		−	9282.23i		−	0.669713i	−0.942269	−	0.334856i	$-0.891312\pi$
$577$		−	9282.23i		−	0.669713i	0.942269	−	0.334856i	$-0.108688\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−474.321	−	2080.91i	−0.0338695	−	0.148590i
$582$		0			0
$583$	−1445.52	−	2503.71i	−0.102688	−	0.177861i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12177.5	+	21092.1i	0.856253	+	1.48307i	0.875477	+	0.483259i	$0.160547\pi$
$587$	12177.5	+	21092.1i	0.856253	+	1.48307i	−0.0192240	+	0.999815i	$0.506120\pi$
$588$		0			0
$589$	13128.7	−	22739.6i	0.918439	−	1.59078i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3634.37			0.251679			0.125840	−	0.992051i	$-0.459838\pi$
$593$	3634.37			0.251679			0.125840	+	0.992051i	$0.459838\pi$
$594$		0			0
$595$	−33661.2	+	31214.9i	−2.31929	+	2.15073i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−16689.7	−	9635.82i	−1.13844	−	0.657277i	−0.192394	−	0.981318i	$-0.561625\pi$
$599$	−16689.7	−	9635.82i	−1.13844	−	0.657277i	−0.946043	+	0.324041i	$0.894959\pi$
$600$		0			0
$601$	9993.33	−	5769.65i	0.678264	−	0.391596i	−0.120937	−	0.992660i	$-0.538590\pi$
$601$	9993.33	−	5769.65i	0.678264	−	0.391596i	0.799201	+	0.601064i	$0.205256\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7530.53	−	13043.3i	−0.506049	−	0.876502i
$606$		0			0
$607$	3327.05	+	1920.88i	0.222473	+	0.128445i	0.607095	−	0.794629i	$-0.292335\pi$
$607$	3327.05	+	1920.88i	0.222473	+	0.128445i	−0.384622	+	0.923074i	$0.625668\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			13192.6i			0.873513i
$612$		0			0
$613$	−26792.1			−1.76529			−0.882644	−	0.470043i	$-0.844238\pi$
$613$	−26792.1			−1.76529			−0.882644	+	0.470043i	$0.844238\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2434.46	+	1405.54i	0.158846	+	0.0917095i	0.577316	−	0.816521i	$-0.304100\pi$
$617$	2434.46	+	1405.54i	0.158846	+	0.0917095i	−0.418470	+	0.908231i	$0.637434\pi$
$618$		0			0
$619$	2964.61	−	1711.62i	0.192500	−	0.111140i	−0.400652	−	0.916230i	$-0.631216\pi$
$619$	2964.61	−	1711.62i	0.192500	−	0.111140i	0.593152	+	0.805090i	$0.297883\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18673.7	+	5765.98i	1.20087	+	0.370801i
$624$		0			0
$625$	−23595.4	+	40868.4i	−1.51010	+	2.61558i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2139.91			0.135650
$630$		0			0
$631$	−4476.28			−0.282406			−0.141203	−	0.989981i	$-0.545097\pi$
$631$	−4476.28			−0.282406			−0.141203	+	0.989981i	$0.545097\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	26080.8	−	45173.2i	1.62990	−	2.82306i
$636$		0			0
$637$	9022.19	+	18762.5i	0.561181	+	1.16703i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15150.6	−	8747.20i	0.933561	−	0.538992i	0.0456248	−	0.998959i	$-0.485472\pi$
$641$	15150.6	−	8747.20i	0.933561	−	0.538992i	0.887936	+	0.459967i	$0.152139\pi$
$642$		0			0
$643$	−4602.79	−	2657.42i	−0.282296	−	0.162984i	0.352166	−	0.935937i	$-0.385445\pi$
$643$	−4602.79	−	2657.42i	−0.282296	−	0.162984i	−0.634462	+	0.772954i	$0.718778\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4660.14			−0.283167			−0.141584	−	0.989926i	$-0.545219\pi$
$647$	−4660.14			−0.283167			−0.141584	+	0.989926i	$0.545219\pi$
$648$		0			0
$649$			1908.33i			0.115421i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	16836.5	+	9720.58i	1.00898	+	0.582536i	0.910893	−	0.412642i	$-0.135394\pi$
$653$	16836.5	+	9720.58i	1.00898	+	0.582536i	0.0980880	+	0.995178i	$0.468727\pi$
$654$		0			0
$655$	4222.36	+	7313.35i	0.251880	+	0.436269i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4244.21	+	2450.39i	−0.250881	+	0.144846i	−0.620168	−	0.784469i	$-0.712935\pi$
$659$	−4244.21	+	2450.39i	−0.250881	+	0.144846i	0.369286	+	0.929316i	$0.379602\pi$
$660$		0			0
$661$	−6520.45	−	3764.58i	−0.383685	−	0.221521i	0.295735	−	0.955270i	$-0.404435\pi$
$661$	−6520.45	−	3764.58i	−0.383685	−	0.221521i	−0.679420	+	0.733749i	$0.737769\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	27904.2	+	30091.0i	1.62719	+	1.75471i
$666$		0			0
$667$	565.862			0.0328490
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−7084.29	+	12270.4i	−0.407580	+	0.705949i
$672$		0			0
$673$	5380.01	+	9318.45i	0.308149	+	0.533729i	0.977957	−	0.208804i	$-0.0669572\pi$
$673$	5380.01	+	9318.45i	0.308149	+	0.533729i	−0.669809	+	0.742534i	$0.733624\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7477.31	+	12951.1i	0.424485	+	0.735230i	0.996372	−	0.0851025i	$-0.0271218\pi$
$677$	7477.31	+	12951.1i	0.424485	+	0.735230i	−0.571887	+	0.820332i	$0.693788\pi$
$678$		0			0
$679$	−9263.80	+	2111.58i	−0.523582	+	0.119345i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			28835.1i			1.61544i	0.589566	+	0.807720i	$0.299299\pi$
$683$			28835.1i			1.61544i	−0.589566	+	0.807720i	$0.700701\pi$
$684$		0			0
$685$			50349.3i			2.80839i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3530.27	−	6114.60i	0.195199	−	0.338095i
$690$		0			0
$691$	845.405	−	488.095i	0.0465423	−	0.0268712i	−0.476548	−	0.879148i	$-0.658112\pi$
$691$	845.405	−	488.095i	0.0465423	−	0.0268712i	0.523091	+	0.852277i	$0.324779\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	12036.4	−	6949.24i	0.656933	−	0.379280i
$696$		0			0
$697$	17968.6	−	31122.6i	0.976486	−	1.69132i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			6963.25i			0.375176i	0.982248	+	0.187588i	$0.0600669\pi$
$701$			6963.25i			0.375176i	−0.982248	+	0.187588i	$0.939933\pi$
$702$		0			0
$703$		−	1912.94i		−	0.102629i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−686.159	−	3010.27i	−0.0365002	−	0.160131i
$708$		0			0
$709$	14914.8	+	25833.2i	0.790039	+	1.36839i	0.925942	+	0.377665i	$0.123273\pi$
$709$	14914.8	+	25833.2i	0.790039	+	1.36839i	−0.135903	+	0.990722i	$0.543394\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2526.27	+	4375.63i	0.132692	+	0.229830i
$714$		0			0
$715$	−15925.5	+	27583.7i	−0.832978	+	1.44276i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−14089.4			−0.730803			−0.365402	−	0.930850i	$-0.619068\pi$
$719$	−14089.4			−0.730803			−0.365402	+	0.930850i	$0.619068\pi$
$720$		0			0
$721$	18964.0	+	20450.2i	0.979551	+	1.05632i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7785.05	−	4494.70i	−0.398799	−	0.230247i
$726$		0			0
$727$	−16592.3	+	9579.56i	−0.846457	+	0.488702i	−0.859454	−	0.511214i	$-0.829196\pi$
$727$	−16592.3	+	9579.56i	−0.846457	+	0.488702i	0.0129971	+	0.999916i	$0.495863\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8750.83	+	15156.9i	0.442765	+	0.766891i
$732$		0			0
$733$	−31099.2	−	17955.1i	−1.56709	−	0.904757i	−0.996507	−	0.0835145i	$-0.973386\pi$
$733$	−31099.2	−	17955.1i	−1.56709	−	0.904757i	−0.570579	−	0.821243i	$-0.693281\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1682.22i			0.0840778i
$738$		0			0
$739$	15603.3			0.776694			0.388347	−	0.921513i	$-0.373046\pi$
$739$	15603.3			0.776694			0.388347	+	0.921513i	$0.373046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9187.85	−	5304.61i	−0.453660	−	0.261921i	0.255715	−	0.966752i	$-0.417689\pi$
$743$	−9187.85	−	5304.61i	−0.453660	−	0.261921i	−0.709375	+	0.704831i	$0.751023\pi$
$744$		0			0
$745$	−23569.0	+	13607.5i	−1.15906	+	0.669183i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4245.39	−	13749.1i	0.207107	−	0.670736i
$750$		0			0
$751$	−3593.39	+	6223.93i	−0.174600	+	0.302416i	−0.940023	−	0.341112i	$-0.889197\pi$
$751$	−3593.39	+	6223.93i	−0.174600	+	0.302416i	0.765423	+	0.643528i	$0.222530\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4139.68			0.199548
$756$		0			0
$757$	−30230.3			−1.45144			−0.725720	−	0.687990i	$-0.758493\pi$
$757$	−30230.3			−1.45144			−0.725720	+	0.687990i	$0.758493\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4067.61	+	7045.32i	−0.193759	+	0.335601i	−0.946493	−	0.322724i	$-0.895401\pi$
$761$	−4067.61	+	7045.32i	−0.193759	+	0.335601i	0.752734	+	0.658325i	$0.228735\pi$
$762$		0			0
$763$	−9951.87	−	3072.90i	−0.472191	−	0.145801i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4036.15	+	2330.27i	−0.190009	+	0.109702i
$768$		0			0
$769$	19458.9	+	11234.6i	0.912490	+	0.526826i	0.881231	−	0.472685i	$-0.156715\pi$
$769$	19458.9	+	11234.6i	0.912490	+	0.526826i	0.0312583	+	0.999511i	$0.490049\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18079.1			0.841217			0.420609	−	0.907242i	$-0.361817\pi$
$773$	18079.1			0.841217			0.420609	+	0.907242i	$0.361817\pi$
$774$		0			0
$775$		−	80265.6i		−	3.72029i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27821.7	−	16062.8i	−1.27961	−	0.738782i
$780$		0			0
$781$	9903.60	+	17153.5i	0.453750	+	0.785918i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−60507.1	+	34933.8i	−2.75107	+	1.58833i
$786$		0			0
$787$	24672.8	+	14244.8i	1.11752	+	0.645202i	0.940767	−	0.339053i	$-0.110107\pi$
$787$	24672.8	+	14244.8i	1.11752	+	0.645202i	0.176755	+	0.984255i	$0.443440\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−23976.8	−	25855.9i	−1.07777	−	1.16224i
$792$		0			0
$793$	−34602.7			−1.54953
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1556.36	−	2695.69i	0.0691708	−	0.119807i	−0.829366	−	0.558706i	$-0.811298\pi$
$797$	1556.36	−	2695.69i	0.0691708	−	0.119807i	0.898537	+	0.438899i	$0.144631\pi$
$798$		0			0
$799$	12758.3	+	22098.0i	0.564902	+	0.978438i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−8828.88	−	15292.1i	−0.388000	−	0.672036i
$804$		0			0
$805$	−7699.17	+	1754.94i	−0.337093	+	0.0768367i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	8901.87i		−	0.386864i	−0.981114	−	0.193432i	$-0.938038\pi$
$809$		−	8901.87i		−	0.386864i	0.981114	−	0.193432i	$-0.0619619\pi$
$810$		0			0
$811$			6818.31i			0.295220i	0.989046	+	0.147610i	$0.0471580\pi$
$811$			6818.31i			0.295220i	−0.989046	+	0.147610i	$0.952842\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	19685.3	−	34095.9i	0.846067	−	1.46543i
$816$		0			0
$817$	13549.3	−	7822.70i	0.580209	−	0.334984i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	14035.7	−	8103.52i	0.596650	−	0.344476i	−0.171073	−	0.985258i	$-0.554723\pi$
$821$	14035.7	−	8103.52i	0.596650	−	0.344476i	0.767722	+	0.640783i	$0.221390\pi$
$822$		0			0
$823$	−3635.33	+	6296.57i	−0.153973	+	0.266689i	−0.932684	−	0.360693i	$-0.882540\pi$
$823$	−3635.33	+	6296.57i	−0.153973	+	0.266689i	0.778712	+	0.627382i	$0.215873\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			7040.86i			0.296052i	0.988983	+	0.148026i	$0.0472919\pi$
$827$			7040.86i			0.296052i	−0.988983	+	0.148026i	$0.952708\pi$
$828$		0			0
$829$			17047.4i			0.714212i	0.934064	+	0.357106i	$0.116236\pi$
$829$			17047.4i			0.714212i	−0.934064	+	0.357106i	$0.883764\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	33257.3	+	22702.6i	1.38331	+	0.944297i
$834$		0			0
$835$	−39340.6	−	68139.9i	−1.63046	−	2.82405i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6588.67	+	11411.9i	0.271116	+	0.469587i	0.969148	−	0.246480i	$-0.0792740\pi$
$839$	6588.67	+	11411.9i	0.271116	+	0.469587i	−0.698032	+	0.716067i	$0.745941\pi$
$840$		0			0
$841$	−11801.9	+	20441.5i	−0.483903	+	0.838144i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−31399.2			−1.27830
$846$		0			0
$847$	−9686.81	+	8982.83i	−0.392967	+	0.364408i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	318.779	+	184.047i	0.0128409	+	0.00741369i
$852$		0			0
$853$	11459.1	−	6615.91i	0.459967	−	0.265562i	−0.252063	−	0.967711i	$-0.581109\pi$
$853$	11459.1	−	6615.91i	0.459967	−	0.265562i	0.712030	+	0.702149i	$0.247776\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4162.07	+	7208.91i	0.165897	+	0.287342i	0.936973	−	0.349401i	$-0.113615\pi$
$857$	4162.07	+	7208.91i	0.165897	+	0.287342i	−0.771077	+	0.636742i	$0.780282\pi$
$858$		0			0
$859$	36607.1	+	21135.1i	1.45404	+	0.839488i	0.998707	−	0.0508353i	$-0.0161884\pi$
$859$	36607.1	+	21135.1i	1.45404	+	0.839488i	0.455329	+	0.890323i	$0.349522\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			4503.10i			0.177621i	0.996049	+	0.0888107i	$0.0283066\pi$
$863$			4503.10i			0.177621i	−0.996049	+	0.0888107i	$0.971693\pi$
$864$		0			0
$865$	75951.5			2.98547
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1724.64	−	995.719i	−0.0673237	−	0.0388693i
$870$		0			0
$871$	−3557.93	+	2054.17i	−0.138411	+	0.0799114i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	73159.6	+	22589.9i	2.82656	+	0.872775i
$876$		0			0
$877$	12266.9	−	21246.9i	0.472320	−	0.818082i	−0.527178	−	0.849755i	$-0.676750\pi$
$877$	12266.9	−	21246.9i	0.472320	−	0.818082i	0.999498	+	0.0316724i	$0.0100833\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	6446.14			0.246511			0.123255	−	0.992375i	$-0.460667\pi$
$881$	6446.14			0.246511			0.123255	+	0.992375i	$0.460667\pi$
$882$		0			0
$883$	13490.2			0.514136			0.257068	−	0.966393i	$-0.417244\pi$
$883$	13490.2			0.514136			0.257068	+	0.966393i	$0.417244\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10349.3	−	17925.5i	0.391765	−	0.678556i	−0.600918	−	0.799311i	$-0.705198\pi$
$887$	10349.3	−	17925.5i	0.391765	−	0.678556i	0.992682	+	0.120755i	$0.0385314\pi$
$888$		0			0
$889$	−43716.9	−	13498.7i	−1.64929	−	0.509261i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	19754.3	−	11405.1i	0.740259	−	0.427389i
$894$		0			0
$895$	−10938.9	−	6315.58i	−0.408544	−	0.235873i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	7010.92			0.260097
$900$		0			0
$901$		−	13656.2i		−	0.504943i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−30530.8	−	17627.0i	−1.12141	−	0.647449i
$906$		0			0
$907$	−14361.2	−	24874.3i	−0.525750	−	0.910626i	−0.999550	−	0.0299935i	$-0.990451\pi$
$907$	−14361.2	−	24874.3i	−0.525750	−	0.910626i	0.473800	−	0.880633i	$-0.342882\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6255.03	−	3611.35i	0.227485	−	0.131338i	−0.381926	−	0.924193i	$-0.624739\pi$
$911$	6255.03	−	3611.35i	0.227485	−	0.131338i	0.609411	+	0.792854i	$0.291406\pi$
$912$		0			0
$913$	−2480.38	−	1432.05i	−0.0899110	−	0.0519102i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5431.39	−	5036.67i	0.195595	−	0.181380i
$918$		0			0
$919$	−9875.41			−0.354472			−0.177236	−	0.984168i	$-0.556716\pi$
$919$	−9875.41			−0.354472			−0.177236	+	0.984168i	$0.556716\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−24186.7	+	41892.6i	−0.862530	+	1.49395i
$924$		0			0
$925$	−2923.81	−	5064.18i	−0.103929	−	0.180010i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−22240.7	−	38522.0i	−0.785461	−	1.36046i	−0.928723	−	0.370774i	$-0.879092\pi$
$929$	−22240.7	−	38522.0i	−0.785461	−	1.36046i	0.143262	−	0.989685i	$-0.454241\pi$
$930$		0			0
$931$	20294.7	−	29729.9i	0.714429	−	1.04657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			61604.8i			2.15475i
$936$		0			0
$937$			1613.46i			0.0562532i	0.999604	+	0.0281266i	$0.00895416\pi$
$937$			1613.46i			0.0562532i	−0.999604	+	0.0281266i	$0.991046\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−835.528	+	1447.18i	−0.0289452	+	0.0501345i	−0.880135	−	0.474723i	$-0.842548\pi$
$941$	−835.528	+	1447.18i	−0.0289452	+	0.0501345i	0.851190	+	0.524858i	$0.175881\pi$
$942$		0			0
$943$	5353.53	−	3090.86i	0.184872	−	0.106736i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	31755.5	−	18334.1i	1.08967	−	0.629121i	0.156181	−	0.987729i	$-0.450082\pi$
$947$	31755.5	−	18334.1i	1.08967	−	0.629121i	0.933488	+	0.358608i	$0.116748\pi$
$948$		0			0
$949$	21562.0	−	37346.5i	0.737547	−	1.27747i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	38341.6i		−	1.30326i	−0.758537	−	0.651629i	$-0.774086\pi$
$953$		−	38341.6i		−	1.30326i	0.758537	−	0.651629i	$-0.225914\pi$
$954$		0			0
$955$		−	71382.7i		−	2.41873i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	43059.5	−	9814.94i	1.44991	−	0.330491i
$960$		0			0
$961$	16404.5	+	28413.4i	0.550653	+	0.953758i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	10056.6	+	17418.5i	0.335475	+	0.581060i
$966$		0			0
$967$	−24342.3	+	42162.0i	−0.809508	+	1.40211i	0.103697	+	0.994609i	$0.466933\pi$
$967$	−24342.3	+	42162.0i	−0.809508	+	1.40211i	−0.913205	+	0.407501i	$0.866400\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−41514.9			−1.37207			−0.686033	−	0.727570i	$-0.740650\pi$
$971$	−41514.9			−1.37207			−0.686033	+	0.727570i	$0.740650\pi$
$972$		0			0
$973$	−8289.44	−	8939.08i	−0.273122	−	0.294526i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−26952.7	−	15561.1i	−0.882592	−	0.509565i	−0.0110801	−	0.999939i	$-0.503527\pi$
$977$	−26952.7	−	15561.1i	−0.882592	−	0.509565i	−0.871512	+	0.490374i	$0.836860\pi$
$978$		0			0
$979$	22712.8	−	13113.3i	0.741476	−	0.428091i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13264.6	−	22974.9i	−0.430391	−	0.745460i	0.566516	−	0.824051i	$-0.308291\pi$
$983$	−13264.6	−	22974.9i	−0.430391	−	0.745460i	−0.996907	+	0.0785914i	$0.974958\pi$
$984$		0			0
$985$	40099.1	+	23151.2i	1.29712	+	0.748892i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			3010.53i			0.0967941i
$990$		0			0
$991$	21125.0			0.677153			0.338577	−	0.940939i	$-0.390055\pi$
$991$	21125.0			0.677153			0.338577	+	0.940939i	$0.390055\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−28923.7	−	16699.1i	−0.921550	−	0.532057i
$996$		0			0
$997$	34141.1	−	19711.4i	1.08451	−	0.626144i	0.152403	−	0.988318i	$-0.451299\pi$
$997$	34141.1	−	19711.4i	1.08451	−	0.626144i	0.932110	+	0.362175i	$0.117966\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.x.a.125.1		48
3.2	odd	2		252.4.x.a.41.23	yes	48
7.6	odd	2	inner	756.4.x.a.125.24		48
9.2	odd	6	inner	756.4.x.a.629.24		48
9.4	even	3		2268.4.f.a.1133.48		48
9.5	odd	6		2268.4.f.a.1133.1		48
9.7	even	3		252.4.x.a.209.2	yes	48
21.20	even	2		252.4.x.a.41.2	✓	48
63.13	odd	6		2268.4.f.a.1133.2		48
63.20	even	6	inner	756.4.x.a.629.1		48
63.34	odd	6		252.4.x.a.209.23	yes	48
63.41	even	6		2268.4.f.a.1133.47		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.2	✓	48	21.20	even	2
252.4.x.a.41.23	yes	48	3.2	odd	2
252.4.x.a.209.2	yes	48	9.7	even	3
252.4.x.a.209.23	yes	48	63.34	odd	6
756.4.x.a.125.1		48	1.1	even	1	trivial
756.4.x.a.125.24		48	7.6	odd	2	inner
756.4.x.a.629.1		48	63.20	even	6	inner
756.4.x.a.629.24		48	9.2	odd	6	inner
2268.4.f.a.1133.1		48	9.5	odd	6
2268.4.f.a.1133.2		48	63.13	odd	6
2268.4.f.a.1133.47		48	63.41	even	6
2268.4.f.a.1133.48		48	9.4	even	3

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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.x (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-10.5571 + 18.2854i) q^{5} +(17.6959 + 5.46406i) q^{7} +O(q^{10})\)	\(q+(-10.5571 + 18.2854i) q^{5} +(17.6959 + 5.46406i) q^{7} +(21.5235 - 12.4266i) q^{11} +(52.5651 + 30.3485i) q^{13} +117.397 q^{17} -104.946i q^{19} +(17.4885 + 10.0970i) q^{23} +(-160.403 - 277.826i) q^{25} +(24.2671 - 14.0106i) q^{29} +(216.679 + 125.100i) q^{31} +(-286.729 + 265.891i) q^{35} +18.2279 q^{37} +(153.058 - 265.105i) q^{41} +(74.5402 + 129.107i) q^{43} +(108.676 + 188.233i) q^{47} +(283.288 + 193.383i) q^{49} -116.324i q^{53} +524.754i q^{55} +(-38.3919 + 66.4967i) q^{59} +(-493.712 + 285.045i) q^{61} +(-1109.87 + 640.781i) q^{65} +(-33.8430 + 58.6179i) q^{67} +796.967i q^{71} -710.481i q^{73} +(448.778 - 102.294i) q^{77} +(-40.0640 - 69.3928i) q^{79} +(-57.6203 - 99.8013i) q^{83} +(-1239.37 + 2146.65i) q^{85} +1055.26 q^{89} +(764.359 + 824.261i) q^{91} +(1918.98 + 1107.92i) q^{95} +(-444.295 + 256.514i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 6 q^{7}+O(q^{10}) \) 48 * q + 6 * q^7	\( 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) \) 48 * q + 6 * q^7 + 12 * q^11 + 408 * q^23 - 600 * q^25 + 84 * q^29 + 336 * q^37 + 84 * q^43 + 318 * q^49 - 2964 * q^65 - 588 * q^67 - 2400 * q^77 + 204 * q^79 - 360 * q^85 - 1080 * q^91 - 300 * q^95

Introduction

L-functions

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