LMFDB - Embedded newform 624.4.c.e.337.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(337,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("624.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.5054412.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 29x^{2} + 48 $ x^4 + 29*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$5.21898i$ of defining polynomial
Character	$\chi$	$=$	624.337
Dual form			624.4.c.e.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +5.83936i q^{5} +31.3139i q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +5.83936i q^{5} +31.3139i q^{7} +9.00000 q^{9} +16.2773i q^{11} +(-43.4755 - 17.5181i) q^{13} +17.5181i q^{15} +54.0000 q^{17} +66.3500i q^{19} +93.9416i q^{21} -182.853 q^{23} +90.9019 q^{25} +27.0000 q^{27} -164.853 q^{29} -58.9055i q^{31} +48.8319i q^{33} -182.853 q^{35} -110.366i q^{37} +(-130.426 - 52.5542i) q^{39} -55.0357i q^{41} -113.147 q^{43} +52.5542i q^{45} -514.089i q^{47} -637.559 q^{49} +162.000 q^{51} +242.559 q^{53} -95.0490 q^{55} +199.050i q^{57} +265.036i q^{59} -468.098 q^{61} +281.825i q^{63} +(102.294 - 253.869i) q^{65} +852.919i q^{67} -548.559 q^{69} +165.619i q^{71} +315.325i q^{73} +272.706 q^{75} -509.706 q^{77} -479.608 q^{79} +81.0000 q^{81} +574.235i q^{83} +315.325i q^{85} -494.559 q^{87} +66.7144i q^{89} +(548.559 - 1361.39i) q^{91} -176.716i q^{93} -387.441 q^{95} +1438.25i q^{97} +146.496i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 36 * q^9	$ 4 q + 12 q^{3} + 36 q^{9} - 72 q^{13} + 216 q^{17} - 120 q^{23} - 44 q^{25} + 108 q^{27} - 48 q^{29} - 120 q^{35} - 216 q^{39} - 1064 q^{43} - 716 q^{49} + 648 q^{51} - 864 q^{53} - 584 q^{55} - 2280 q^{61} + 1632 q^{65} - 360 q^{69} - 132 q^{75} - 816 q^{77} - 288 q^{79} + 324 q^{81} - 144 q^{87} + 360 q^{91} - 3384 q^{95}+O(q^{100}) $ 4 * q + 12 * q^3 + 36 * q^9 - 72 * q^13 + 216 * q^17 - 120 * q^23 - 44 * q^25 + 108 * q^27 - 48 * q^29 - 120 * q^35 - 216 * q^39 - 1064 * q^43 - 716 * q^49 + 648 * q^51 - 864 * q^53 - 584 * q^55 - 2280 * q^61 + 1632 * q^65 - 360 * q^69 - 132 * q^75 - 816 * q^77 - 288 * q^79 + 324 * q^81 - 144 * q^87 + 360 * q^91 - 3384 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	3.00000			0.577350
$3$	3.00000			0.577350
$4$		0			0
$5$			5.83936i			0.522288i	0.965300	+	0.261144i	$0.0840997\pi$
$5$			5.83936i			0.522288i	−0.965300	+	0.261144i	$0.915900\pi$
$6$		0			0
$7$			31.3139i			1.69079i	0.534141	+	0.845395i	$0.320635\pi$
$7$			31.3139i			1.69079i	−0.534141	+	0.845395i	$0.679365\pi$
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			16.2773i			0.446163i	0.974800	+	0.223082i	$0.0716116\pi$
$11$			16.2773i			0.446163i	−0.974800	+	0.223082i	$0.928388\pi$
$12$		0			0
$13$	−43.4755	−	17.5181i	−0.927533	−	0.373741i
$13$	−43.4755	−	17.5181i	−0.927533	−	0.373741i
$14$		0			0
$15$			17.5181i			0.301543i
$16$		0			0
$17$	54.0000			0.770407			0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000			0.770407			0.385204	+	0.922832i	$0.374131\pi$
$18$		0			0
$19$			66.3500i			0.801144i	0.916265	+	0.400572i	$0.131189\pi$
$19$			66.3500i			0.801144i	−0.916265	+	0.400572i	$0.868811\pi$
$20$		0			0
$21$			93.9416i			0.976178i
$22$		0			0
$23$	−182.853			−1.65772			−0.828858	−	0.559459i	$-0.811009\pi$
$23$	−182.853			−1.65772			−0.828858	+	0.559459i	$0.811009\pi$
$24$		0			0
$25$	90.9019			0.727215
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	−164.853			−1.05560			−0.527800	−	0.849369i	$-0.676983\pi$
$29$	−164.853			−1.05560			−0.527800	+	0.849369i	$0.676983\pi$
$30$		0			0
$31$		−	58.9055i		−	0.341282i	−0.985333	−	0.170641i	$-0.945416\pi$
$31$		−	58.9055i		−	0.341282i	0.985333	−	0.170641i	$-0.0545838\pi$
$32$		0			0
$33$			48.8319i			0.257592i
$34$		0			0
$35$	−182.853			−0.883079
$36$		0			0
$37$		−	110.366i		−	0.490382i	−0.969475	−	0.245191i	$-0.921149\pi$
$37$		−	110.366i		−	0.490382i	0.969475	−	0.245191i	$-0.0788506\pi$
$38$		0			0
$39$	−130.426	−	52.5542i	−0.535511	−	0.215780i
$40$		0			0
$41$		−	55.0357i		−	0.209637i	−0.994491	−	0.104819i	$-0.966574\pi$
$41$		−	55.0357i		−	0.209637i	0.994491	−	0.104819i	$-0.0334262\pi$
$42$		0			0
$43$	−113.147			−0.401274			−0.200637	−	0.979666i	$-0.564301\pi$
$43$	−113.147			−0.401274			−0.200637	+	0.979666i	$0.564301\pi$
$44$		0			0
$45$			52.5542i			0.174096i
$46$		0			0
$47$		−	514.089i		−	1.59548i	−0.603001	−	0.797740i	$-0.706029\pi$
$47$		−	514.089i		−	1.59548i	0.603001	−	0.797740i	$-0.293971\pi$
$48$		0			0
$49$	−637.559			−1.85877
$50$		0			0
$51$	162.000			0.444795
$52$		0			0
$53$	242.559			0.628641			0.314321	−	0.949317i	$-0.398223\pi$
$53$	242.559			0.628641			0.314321	+	0.949317i	$0.398223\pi$
$54$		0			0
$55$	−95.0490			−0.233026
$56$		0			0
$57$			199.050i			0.462541i
$58$		0			0
$59$			265.036i			0.584825i	0.956292	+	0.292413i	$0.0944581\pi$
$59$			265.036i			0.584825i	−0.956292	+	0.292413i	$0.905542\pi$
$60$		0			0
$61$	−468.098			−0.982522			−0.491261	−	0.871013i	$-0.663464\pi$
$61$	−468.098			−0.982522			−0.491261	+	0.871013i	$0.663464\pi$
$62$		0			0
$63$			281.825i			0.563597i
$64$		0			0
$65$	102.294	−	253.869i	0.195201	−	0.484439i
$66$		0			0
$67$			852.919i			1.55523i	0.628739	+	0.777617i	$0.283572\pi$
$67$			852.919i			1.55523i	−0.628739	+	0.777617i	$0.716428\pi$
$68$		0			0
$69$	−548.559			−0.957083
$70$		0			0
$71$			165.619i			0.276836i	0.990374	+	0.138418i	$0.0442018\pi$
$71$			165.619i			0.276836i	−0.990374	+	0.138418i	$0.955798\pi$
$72$		0			0
$73$			315.325i			0.505562i	0.967524	+	0.252781i	$0.0813452\pi$
$73$			315.325i			0.505562i	−0.967524	+	0.252781i	$0.918655\pi$
$74$		0			0
$75$	272.706			0.419858
$76$		0			0
$77$	−509.706			−0.754368
$78$		0			0
$79$	−479.608			−0.683039			−0.341519	−	0.939875i	$-0.610942\pi$
$79$	−479.608			−0.683039			−0.341519	+	0.939875i	$0.610942\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			574.235i			0.759404i	0.925109	+	0.379702i	$0.123973\pi$
$83$			574.235i			0.759404i	−0.925109	+	0.379702i	$0.876027\pi$
$84$		0			0
$85$			315.325i			0.402374i
$86$		0			0
$87$	−494.559			−0.609451
$88$		0			0
$89$			66.7144i			0.0794575i	0.999211	+	0.0397287i	$0.0126494\pi$
$89$			66.7144i			0.0794575i	−0.999211	+	0.0397287i	$0.987351\pi$
$90$		0			0
$91$	548.559	−	1361.39i	0.631918	−	1.56826i
$92$		0			0
$93$		−	176.716i		−	0.197039i
$94$		0			0
$95$	−387.441			−0.418428
$96$		0			0
$97$			1438.25i			1.50549i	0.658313	+	0.752744i	$0.271270\pi$
$97$			1438.25i			1.50549i	−0.658313	+	0.752744i	$0.728730\pi$
$98$		0			0
$99$			146.496i			0.148721i
$100$		0			0
$101$	896.264			0.882986			0.441493	−	0.897265i	$-0.354449\pi$
$101$	896.264			0.882986			0.441493	+	0.897265i	$0.354449\pi$
$102$		0			0
$103$	−22.2644			−0.0212988			−0.0106494	−	0.999943i	$-0.503390\pi$
$103$	−22.2644			−0.0212988			−0.0106494	+	0.999943i	$0.503390\pi$
$104$		0			0
$105$	−548.559			−0.509846
$106$		0			0
$107$	351.441			0.317524			0.158762	−	0.987317i	$-0.449250\pi$
$107$	351.441			0.317524			0.158762	+	0.987317i	$0.449250\pi$
$108$		0			0
$109$		−	967.008i		−	0.849748i	−0.905252	−	0.424874i	$-0.860318\pi$
$109$		−	967.008i		−	0.849748i	0.905252	−	0.424874i	$-0.139682\pi$
$110$		0			0
$111$		−	331.099i		−	0.283122i
$112$		0			0
$113$	48.2943			0.0402048			0.0201024	−	0.999798i	$-0.493601\pi$
$113$	48.2943			0.0402048			0.0201024	+	0.999798i	$0.493601\pi$
$114$		0			0
$115$		−	1067.74i		−	0.865805i
$116$		0			0
$117$	−391.279	−	157.663i	−0.309178	−	0.124580i
$118$		0			0
$119$			1690.95i			1.30260i
$120$		0			0
$121$	1066.05			0.800938
$122$		0			0
$123$		−	165.107i		−	0.121034i
$124$		0			0
$125$			1260.73i			0.902104i
$126$		0			0
$127$	−1763.02			−1.23183			−0.615916	−	0.787812i	$-0.711214\pi$
$127$	−1763.02			−1.23183			−0.615916	+	0.787812i	$0.711214\pi$
$128$		0			0
$129$	−339.441			−0.231676
$130$		0			0
$131$	−955.970			−0.637584			−0.318792	−	0.947825i	$-0.603277\pi$
$131$	−955.970			−0.637584			−0.318792	+	0.947825i	$0.603277\pi$
$132$		0			0
$133$	−2077.68			−1.35457
$134$		0			0
$135$			157.663i			0.100514i
$136$		0			0
$137$			15.9215i			0.00992894i	0.999988	+	0.00496447i	$0.00158025\pi$
$137$			15.9215i			0.00992894i	−0.999988	+	0.00496447i	$0.998420\pi$
$138$		0			0
$139$	−2074.26			−1.26573			−0.632866	−	0.774261i	$-0.718122\pi$
$139$	−2074.26			−1.26573			−0.632866	+	0.774261i	$0.718122\pi$
$140$		0			0
$141$		−	1542.27i		−	0.921151i
$142$		0			0
$143$	285.147	−	707.664i	0.166750	−	0.413831i
$144$		0			0
$145$		−	962.635i		−	0.551327i
$146$		0			0
$147$	−1912.68			−1.07316
$148$		0			0
$149$			2764.08i			1.51975i	0.650071	+	0.759873i	$0.274739\pi$
$149$			2764.08i			1.51975i	−0.650071	+	0.759873i	$0.725261\pi$
$150$		0			0
$151$			1618.46i			0.872239i	0.899889	+	0.436120i	$0.143648\pi$
$151$			1618.46i			0.872239i	−0.899889	+	0.436120i	$0.856352\pi$
$152$		0			0
$153$	486.000			0.256802
$154$		0			0
$155$	343.970			0.178247
$156$		0			0
$157$	−1109.97			−0.564237			−0.282119	−	0.959380i	$-0.591037\pi$
$157$	−1109.97			−0.564237			−0.282119	+	0.959380i	$0.591037\pi$
$158$		0			0
$159$	727.676			0.362946
$160$		0			0
$161$		−	5725.83i		−	2.80285i
$162$		0			0
$163$		−	233.201i		−	0.112060i	−0.998429	−	0.0560299i	$-0.982156\pi$
$163$		−	233.201i		−	0.112060i	0.998429	−	0.0560299i	$-0.0178442\pi$
$164$		0			0
$165$	−285.147			−0.134537
$166$		0			0
$167$		−	215.405i		−	0.0998118i	−0.998754	−	0.0499059i	$-0.984108\pi$
$167$		−	215.405i		−	0.0998118i	0.998754	−	0.0499059i	$-0.0158921\pi$
$168$		0			0
$169$	1583.23	+	1523.21i	0.720635	+	0.693315i
$170$		0			0
$171$			597.150i			0.267048i
$172$		0			0
$173$	1383.15			0.607854			0.303927	−	0.952695i	$-0.401702\pi$
$173$	1383.15			0.607854			0.303927	+	0.952695i	$0.401702\pi$
$174$		0			0
$175$			2846.49i			1.22957i
$176$		0			0
$177$			795.107i			0.337649i
$178$		0			0
$179$	3642.79			1.52109			0.760545	−	0.649285i	$-0.224932\pi$
$179$	3642.79			1.52109			0.760545	+	0.649285i	$0.224932\pi$
$180$		0			0
$181$	−2621.97			−1.07674			−0.538369	−	0.842709i	$-0.680959\pi$
$181$	−2621.97			−1.07674			−0.538369	+	0.842709i	$0.680959\pi$
$182$		0			0
$183$	−1404.29			−0.567259
$184$		0			0
$185$	644.469			0.256121
$186$		0			0
$187$			878.975i			0.343727i
$188$		0			0
$189$			845.475i			0.325393i
$190$		0			0
$191$	3419.32			1.29536			0.647679	−	0.761913i	$-0.275740\pi$
$191$	3419.32			1.29536			0.647679	+	0.761913i	$0.275740\pi$
$192$		0			0
$193$		−	1698.39i		−	0.633435i	−0.948520	−	0.316718i	$-0.897419\pi$
$193$		−	1698.39i		−	0.633435i	0.948520	−	0.316718i	$-0.102581\pi$
$194$		0			0
$195$	306.883	−	761.606i	0.112699	−	0.279691i
$196$		0			0
$197$			2293.72i			0.829548i	0.909925	+	0.414774i	$0.136139\pi$
$197$			2293.72i			0.829548i	−0.909925	+	0.414774i	$0.863861\pi$
$198$		0			0
$199$	900.981			0.320949			0.160474	−	0.987040i	$-0.448698\pi$
$199$	900.981			0.320949			0.160474	+	0.987040i	$0.448698\pi$
$200$		0			0
$201$			2558.76i			0.897915i
$202$		0			0
$203$		−	5162.18i		−	1.78480i
$204$		0			0
$205$	321.373			0.109491
$206$		0			0
$207$	−1645.68			−0.552572
$208$		0			0
$209$	−1080.00			−0.357441
$210$		0			0
$211$	−431.019			−0.140628			−0.0703142	−	0.997525i	$-0.522400\pi$
$211$	−431.019			−0.140628			−0.0703142	+	0.997525i	$0.522400\pi$
$212$		0			0
$213$			496.857i			0.159831i
$214$		0			0
$215$		−	660.706i		−	0.209580i
$216$		0			0
$217$	1844.56			0.577036
$218$		0			0
$219$			945.976i			0.291886i
$220$		0			0
$221$	−2347.68	−	945.976i	−0.714578	−	0.287933i
$222$		0			0
$223$			4104.30i			1.23249i	0.787556	+	0.616243i	$0.211346\pi$
$223$			4104.30i			1.23249i	−0.787556	+	0.616243i	$0.788654\pi$
$224$		0			0
$225$	818.117			0.242405
$226$		0			0
$227$		−	1809.11i		−	0.528963i	−0.964391	−	0.264482i	$-0.914799\pi$
$227$		−	1809.11i		−	0.528963i	0.964391	−	0.264482i	$-0.0852009\pi$
$228$		0			0
$229$			5249.33i			1.51478i	0.652961	+	0.757392i	$0.273527\pi$
$229$			5249.33i			1.51478i	−0.652961	+	0.757392i	$0.726473\pi$
$230$		0			0
$231$	−1529.12			−0.435535
$232$		0			0
$233$	2808.88			0.789768			0.394884	−	0.918731i	$-0.370785\pi$
$233$	2808.88			0.789768			0.394884	+	0.918731i	$0.370785\pi$
$234$		0			0
$235$	3001.95			0.833300
$236$		0			0
$237$	−1438.82			−0.394353
$238$		0			0
$239$		−	6712.01i		−	1.81659i	−0.418334	−	0.908293i	$-0.637386\pi$
$239$		−	6712.01i		−	1.81659i	0.418334	−	0.908293i	$-0.362614\pi$
$240$		0			0
$241$		−	2519.11i		−	0.673321i	−0.941626	−	0.336661i	$-0.890703\pi$
$241$		−	2519.11i		−	0.673321i	0.941626	−	0.336661i	$-0.109297\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$		−	3722.93i		−	0.970814i
$246$		0			0
$247$	1162.32	−	2884.60i	0.299421	−	0.743087i
$248$		0			0
$249$			1722.71i			0.438442i
$250$		0			0
$251$	828.000			0.208219			0.104109	−	0.994566i	$-0.466801\pi$
$251$	828.000			0.208219			0.104109	+	0.994566i	$0.466801\pi$
$252$		0			0
$253$		−	2976.35i		−	0.739612i
$254$		0			0
$255$			945.976i			0.232311i
$256$		0			0
$257$	5840.76			1.41765			0.708826	−	0.705383i	$-0.249225\pi$
$257$	5840.76			1.41765			0.708826	+	0.705383i	$0.249225\pi$
$258$		0			0
$259$	3456.00			0.829133
$260$		0			0
$261$	−1483.68			−0.351867
$262$		0			0
$263$	4064.06			0.952854			0.476427	−	0.879214i	$-0.341932\pi$
$263$	4064.06			0.952854			0.476427	+	0.879214i	$0.341932\pi$
$264$		0			0
$265$			1416.39i			0.328332i
$266$		0			0
$267$			200.143i			0.0458748i
$268$		0			0
$269$	1845.44			0.418285			0.209142	−	0.977885i	$-0.432933\pi$
$269$	1845.44			0.418285			0.209142	+	0.977885i	$0.432933\pi$
$270$		0			0
$271$		−	2106.78i		−	0.472242i	−0.971724	−	0.236121i	$-0.924124\pi$
$271$		−	2106.78i		−	0.472242i	0.971724	−	0.236121i	$-0.0758761\pi$
$272$		0			0
$273$	1645.68	−	4084.16i	0.364838	−	0.905437i
$274$		0			0
$275$			1479.64i			0.324457i
$276$		0			0
$277$	4781.94			1.03725			0.518626	−	0.855001i	$-0.326444\pi$
$277$	4781.94			1.03725			0.518626	+	0.855001i	$0.326444\pi$
$278$		0			0
$279$		−	530.149i		−	0.113761i
$280$		0			0
$281$		−	5865.81i		−	1.24528i	−0.782507	−	0.622642i	$-0.786059\pi$
$281$		−	5865.81i		−	1.24528i	0.782507	−	0.622642i	$-0.213941\pi$
$282$		0			0
$283$	−6407.02			−1.34579			−0.672894	−	0.739739i	$-0.734949\pi$
$283$	−6407.02			−1.34579			−0.672894	+	0.739739i	$0.734949\pi$
$284$		0			0
$285$	−1162.32			−0.241579
$286$		0			0
$287$	1723.38			0.354453
$288$		0			0
$289$	−1997.00			−0.406473
$290$		0			0
$291$			4314.75i			0.869194i
$292$		0			0
$293$			3010.24i			0.600204i	0.953907	+	0.300102i	$0.0970208\pi$
$293$			3010.24i			0.600204i	−0.953907	+	0.300102i	$0.902979\pi$
$294$		0			0
$295$	−1547.64			−0.305447
$296$		0			0
$297$			439.487i			0.0858641i
$298$		0			0
$299$	7949.62	+	3203.23i	1.53759	+	0.619557i
$300$		0			0
$301$		−	3543.07i		−	0.678470i
$302$		0			0
$303$	2688.79			0.509792
$304$		0			0
$305$		−	2733.39i		−	0.513159i
$306$		0			0
$307$		−	3341.84i		−	0.621266i	−0.950530	−	0.310633i	$-0.899459\pi$
$307$		−	3341.84i		−	0.621266i	0.950530	−	0.310633i	$-0.100541\pi$
$308$		0			0
$309$	−66.7931			−0.0122968
$310$		0			0
$311$	8755.20			1.59634			0.798170	−	0.602432i	$-0.205801\pi$
$311$	8755.20			1.59634			0.798170	+	0.602432i	$0.205801\pi$
$312$		0			0
$313$	1948.93			0.351949			0.175974	−	0.984395i	$-0.443692\pi$
$313$	1948.93			0.351949			0.175974	+	0.984395i	$0.443692\pi$
$314$		0			0
$315$	−1645.68			−0.294360
$316$		0			0
$317$		−	1940.43i		−	0.343802i	−0.985114	−	0.171901i	$-0.945009\pi$
$317$		−	1940.43i		−	0.343802i	0.985114	−	0.171901i	$-0.0549909\pi$
$318$		0			0
$319$		−	2683.36i		−	0.470970i
$320$		0			0
$321$	1054.32			0.183323
$322$		0			0
$323$			3582.90i			0.617207i
$324$		0			0
$325$	−3952.00	−	1592.43i	−0.674516	−	0.271790i
$326$		0			0
$327$		−	2901.02i		−	0.490602i
$328$		0			0
$329$	16098.1			2.69762
$330$		0			0
$331$			7402.13i			1.22918i	0.788848	+	0.614589i	$0.210678\pi$
$331$			7402.13i			1.22918i	−0.788848	+	0.614589i	$0.789322\pi$
$332$		0			0
$333$		−	993.298i		−	0.163461i
$334$		0			0
$335$	−4980.50			−0.812280
$336$		0			0
$337$	5494.05			0.888071			0.444035	−	0.896009i	$-0.353546\pi$
$337$	5494.05			0.888071			0.444035	+	0.896009i	$0.353546\pi$
$338$		0			0
$339$	144.883			0.0232122
$340$		0			0
$341$	958.823			0.152267
$342$		0			0
$343$		−	9223.77i		−	1.45200i
$344$		0			0
$345$		−	3203.23i		−	0.499873i
$346$		0			0
$347$	−3410.76			−0.527664			−0.263832	−	0.964569i	$-0.584986\pi$
$347$	−3410.76			−0.527664			−0.263832	+	0.964569i	$0.584986\pi$
$348$		0			0
$349$			12629.8i			1.93712i	0.248773	+	0.968562i	$0.419973\pi$
$349$			12629.8i			1.93712i	−0.248773	+	0.968562i	$0.580027\pi$
$350$		0			0
$351$	−1173.84	−	472.988i	−0.178504	−	0.0719266i
$352$		0			0
$353$			2981.78i			0.449586i	0.974407	+	0.224793i	$0.0721706\pi$
$353$			2981.78i			0.449586i	−0.974407	+	0.224793i	$0.927829\pi$
$354$		0			0
$355$	−967.109			−0.144588
$356$		0			0
$357$			5072.85i			0.752055i
$358$		0			0
$359$			8942.30i			1.31464i	0.753611	+	0.657321i	$0.228310\pi$
$359$			8942.30i			1.31464i	−0.753611	+	0.657321i	$0.771690\pi$
$360$		0			0
$361$	2456.68			0.358168
$362$		0			0
$363$	3198.15			0.462422
$364$		0			0
$365$	−1841.30			−0.264049
$366$		0			0
$367$	4735.26			0.673511			0.336756	−	0.941592i	$-0.390670\pi$
$367$	4735.26			0.673511			0.336756	+	0.941592i	$0.390670\pi$
$368$		0			0
$369$		−	495.321i		−	0.0698792i
$370$		0			0
$371$			7595.45i			1.06290i
$372$		0			0
$373$	−8304.01			−1.15272			−0.576361	−	0.817195i	$-0.695528\pi$
$373$	−8304.01			−1.15272			−0.576361	+	0.817195i	$0.695528\pi$
$374$		0			0
$375$			3782.18i			0.520830i
$376$		0			0
$377$	7167.06	+	2887.90i	0.979104	+	0.394522i
$378$		0			0
$379$		−	4088.11i		−	0.554069i	−0.960860	−	0.277035i	$-0.910648\pi$
$379$		−	4088.11i		−	0.554069i	0.960860	−	0.277035i	$-0.0893517\pi$
$380$		0			0
$381$	−5289.06			−0.711198
$382$		0			0
$383$		−	13951.5i		−	1.86132i	−0.365879	−	0.930662i	$-0.619232\pi$
$383$		−	13951.5i		−	1.86132i	0.365879	−	0.930662i	$-0.380768\pi$
$384$		0			0
$385$		−	2976.35i		−	0.393997i
$386$		0			0
$387$	−1018.32			−0.133758
$388$		0			0
$389$	2804.26			0.365506			0.182753	−	0.983159i	$-0.441499\pi$
$389$	2804.26			0.365506			0.182753	+	0.983159i	$0.441499\pi$
$390$		0			0
$391$	−9874.06			−1.27712
$392$		0			0
$393$	−2867.91			−0.368109
$394$		0			0
$395$		−	2800.60i		−	0.356743i
$396$		0			0
$397$		−	6556.18i		−	0.828830i	−0.910088	−	0.414415i	$-0.863986\pi$
$397$		−	6556.18i		−	0.828830i	0.910088	−	0.414415i	$-0.136014\pi$
$398$		0			0
$399$	−6233.03			−0.782059
$400$		0			0
$401$			4730.95i			0.589157i	0.955627	+	0.294579i	$0.0951793\pi$
$401$			4730.95i			0.589157i	−0.955627	+	0.294579i	$0.904821\pi$
$402$		0			0
$403$	−1031.91	+	2560.94i	−0.127551	+	0.316550i
$404$		0			0
$405$			472.988i			0.0580320i
$406$		0			0
$407$	1796.47			0.218790
$408$		0			0
$409$		−	12314.4i		−	1.48878i	−0.667746	−	0.744389i	$-0.732741\pi$
$409$		−	12314.4i		−	1.48878i	0.667746	−	0.744389i	$-0.267259\pi$
$410$		0			0
$411$			47.7645i			0.00573247i
$412$		0			0
$413$	−8299.29			−0.988817
$414$		0			0
$415$	−3353.16			−0.396627
$416$		0			0
$417$	−6222.79			−0.730771
$418$		0			0
$419$	5499.85			0.641254			0.320627	−	0.947206i	$-0.396106\pi$
$419$	5499.85			0.641254			0.320627	+	0.947206i	$0.396106\pi$
$420$		0			0
$421$			12629.7i			1.46208i	0.682336	+	0.731039i	$0.260964\pi$
$421$			12629.7i			1.46208i	−0.682336	+	0.731039i	$0.739036\pi$
$422$		0			0
$423$		−	4626.80i		−	0.531827i
$424$		0			0
$425$	4908.70			0.560252
$426$		0			0
$427$		−	14658.0i		−	1.66124i
$428$		0			0
$429$	855.441	−	2122.99i	0.0962730	−	0.238925i
$430$		0			0
$431$		−	7191.15i		−	0.803679i	−0.915710	−	0.401840i	$-0.868371\pi$
$431$		−	7191.15i		−	0.803679i	0.915710	−	0.401840i	$-0.131629\pi$
$432$		0			0
$433$	−6062.68			−0.672873			−0.336436	−	0.941706i	$-0.609222\pi$
$433$	−6062.68			−0.672873			−0.336436	+	0.941706i	$0.609222\pi$
$434$		0			0
$435$		−	2887.90i		−	0.318309i
$436$		0			0
$437$		−	12132.3i		−	1.32807i
$438$		0			0
$439$	11864.3			1.28986			0.644932	−	0.764240i	$-0.276886\pi$
$439$	11864.3			1.28986			0.644932	+	0.764240i	$0.276886\pi$
$440$		0			0
$441$	−5738.03			−0.619590
$442$		0			0
$443$	−10560.5			−1.13261			−0.566303	−	0.824197i	$-0.691627\pi$
$443$	−10560.5			−1.13261			−0.566303	+	0.824197i	$0.691627\pi$
$444$		0			0
$445$	−389.569			−0.0414997
$446$		0			0
$447$			8292.24i			0.877426i
$448$		0			0
$449$			12659.7i			1.33062i	0.746569	+	0.665308i	$0.231700\pi$
$449$			12659.7i			1.33062i	−0.746569	+	0.665308i	$0.768300\pi$
$450$		0			0
$451$	895.834			0.0935325
$452$		0			0
$453$			4855.37i			0.503587i
$454$		0			0
$455$	7949.62	+	3203.23i	0.819085	+	0.330043i
$456$		0			0
$457$		−	1544.24i		−	0.158067i	−0.996872	−	0.0790336i	$-0.974817\pi$
$457$		−	1544.24i		−	0.158067i	0.996872	−	0.0790336i	$-0.0251834\pi$
$458$		0			0
$459$	1458.00			0.148265
$460$		0			0
$461$		−	13196.8i		−	1.33327i	−0.745384	−	0.666635i	$-0.767734\pi$
$461$		−	13196.8i		−	1.33327i	0.745384	−	0.666635i	$-0.232266\pi$
$462$		0			0
$463$			16309.2i			1.63705i	0.574472	+	0.818524i	$0.305208\pi$
$463$			16309.2i			1.63705i	−0.574472	+	0.818524i	$0.694792\pi$
$464$		0			0
$465$	1031.91			0.102911
$466$		0			0
$467$	−14260.8			−1.41308			−0.706541	−	0.707672i	$-0.749745\pi$
$467$	−14260.8			−1.41308			−0.706541	+	0.707672i	$0.749745\pi$
$468$		0			0
$469$	−26708.2			−2.62957
$470$		0			0
$471$	−3329.91			−0.325763
$472$		0			0
$473$		−	1841.73i		−	0.179034i
$474$		0			0
$475$			6031.34i			0.582604i
$476$		0			0
$477$	2183.03			0.209547
$478$		0			0
$479$			18011.5i			1.71809i	0.511899	+	0.859046i	$0.328942\pi$
$479$			18011.5i			1.71809i	−0.511899	+	0.859046i	$0.671058\pi$
$480$		0			0
$481$	−1933.41	+	4798.23i	−0.183276	+	0.454845i
$482$		0			0
$483$		−	17177.5i		−	1.61823i
$484$		0			0
$485$	−8398.46			−0.786298
$486$		0			0
$487$			14043.3i			1.30670i	0.757055	+	0.653351i	$0.226637\pi$
$487$			14043.3i			1.30670i	−0.757055	+	0.653351i	$0.773363\pi$
$488$		0			0
$489$		−	699.604i		−	0.0646977i
$490$		0			0
$491$	−12966.1			−1.19176			−0.595878	−	0.803075i	$-0.703196\pi$
$491$	−12966.1			−1.19176			−0.595878	+	0.803075i	$0.703196\pi$
$492$		0			0
$493$	−8902.06			−0.813242
$494$		0			0
$495$	−855.441			−0.0776752
$496$		0			0
$497$	−5186.17			−0.468072
$498$		0			0
$499$			10215.5i			0.916453i	0.888835	+	0.458227i	$0.151515\pi$
$499$			10215.5i			0.916453i	−0.888835	+	0.458227i	$0.848485\pi$
$500$		0			0
$501$		−	646.216i		−	0.0576264i
$502$		0			0
$503$	16632.0			1.47432			0.737161	−	0.675717i	$-0.236166\pi$
$503$	16632.0			1.47432			0.737161	+	0.675717i	$0.236166\pi$
$504$		0			0
$505$			5233.61i			0.461173i
$506$		0			0
$507$	4749.70	+	4569.64i	0.416059	+	0.400286i
$508$		0			0
$509$			15235.3i			1.32671i	0.748306	+	0.663354i	$0.230868\pi$
$509$			15235.3i			1.32671i	−0.748306	+	0.663354i	$0.769132\pi$
$510$		0			0
$511$	−9874.06			−0.854799
$512$		0			0
$513$			1791.45i			0.154180i
$514$		0			0
$515$		−	130.009i		−	0.0111241i
$516$		0			0
$517$	8367.99			0.711845
$518$		0			0
$519$	4149.44			0.350945
$520$		0			0
$521$	2680.23			0.225380			0.112690	−	0.993630i	$-0.464053\pi$
$521$	2680.23			0.225380			0.112690	+	0.993630i	$0.464053\pi$
$522$		0			0
$523$	2410.38			0.201527			0.100764	−	0.994910i	$-0.467871\pi$
$523$	2410.38			0.201527			0.100764	+	0.994910i	$0.467871\pi$
$524$		0			0
$525$			8539.47i			0.709892i
$526$		0			0
$527$		−	3180.90i		−	0.262926i
$528$		0			0
$529$	21268.2			1.74802
$530$		0			0
$531$			2385.32i			0.194942i
$532$		0			0
$533$	−964.120	+	2392.70i	−0.0783502	+	0.194446i
$534$		0			0
$535$			2052.19i			0.165839i
$536$		0			0
$537$	10928.4			0.878202
$538$		0			0
$539$		−	10377.7i		−	0.829315i
$540$		0			0
$541$			9969.58i			0.792284i	0.918189	+	0.396142i	$0.129651\pi$
$541$			9969.58i			0.792284i	−0.918189	+	0.396142i	$0.870349\pi$
$542$		0			0
$543$	−7865.91			−0.621655
$544$		0			0
$545$	5646.70			0.443813
$546$		0			0
$547$	−16848.8			−1.31701			−0.658505	−	0.752576i	$-0.728811\pi$
$547$	−16848.8			−1.31701			−0.658505	+	0.752576i	$0.728811\pi$
$548$		0			0
$549$	−4212.88			−0.327507
$550$		0			0
$551$		−	10938.0i		−	0.845688i
$552$		0			0
$553$		−	15018.4i		−	1.15488i
$554$		0			0
$555$	1933.41			0.147871
$556$		0			0
$557$		−	3800.83i		−	0.289132i	−0.989495	−	0.144566i	$-0.953821\pi$
$557$		−	3800.83i		−	0.289132i	0.989495	−	0.144566i	$-0.0461786\pi$
$558$		0			0
$559$	4919.13	+	1982.12i	0.372195	+	0.149973i
$560$		0			0
$561$			2636.92i			0.198451i
$562$		0			0
$563$	−15750.2			−1.17903			−0.589513	−	0.807759i	$-0.700680\pi$
$563$	−15750.2			−1.17903			−0.589513	+	0.807759i	$0.700680\pi$
$564$		0			0
$565$			282.007i			0.0209985i
$566$		0			0
$567$			2536.42i			0.187866i
$568$		0			0
$569$	−17753.2			−1.30800			−0.654002	−	0.756493i	$-0.726911\pi$
$569$	−17753.2			−1.30800			−0.654002	+	0.756493i	$0.726911\pi$
$570$		0			0
$571$	25293.1			1.85374			0.926868	−	0.375388i	$-0.122491\pi$
$571$	25293.1			1.85374			0.926868	+	0.375388i	$0.122491\pi$
$572$		0			0
$573$	10258.0			0.747875
$574$		0			0
$575$	−16621.7			−1.20552
$576$		0			0
$577$			18488.8i			1.33396i	0.745073	+	0.666982i	$0.232414\pi$
$577$			18488.8i			1.33396i	−0.745073	+	0.666982i	$0.767586\pi$
$578$		0			0
$579$		−	5095.18i		−	0.365714i
$580$		0			0
$581$	−17981.5			−1.28399
$582$		0			0
$583$			3948.20i			0.280477i
$584$		0			0
$585$	920.648	−	2284.82i	0.0650669	−	0.161480i
$586$		0			0
$587$		−	17376.7i		−	1.22183i	−0.791697	−	0.610914i	$-0.790802\pi$
$587$		−	17376.7i		−	1.22183i	0.791697	−	0.610914i	$-0.209198\pi$
$588$		0			0
$589$	3908.38			0.273416
$590$		0			0
$591$			6881.16i			0.478940i
$592$		0			0
$593$			7991.09i			0.553381i	0.960959	+	0.276690i	$0.0892377\pi$
$593$			7991.09i			0.553381i	−0.960959	+	0.276690i	$0.910762\pi$
$594$		0			0
$595$	−9874.06			−0.680331
$596$		0			0
$597$	2702.94			0.185300
$598$		0			0
$599$	−10386.5			−0.708480			−0.354240	−	0.935154i	$-0.615260\pi$
$599$	−10386.5			−0.708480			−0.354240	+	0.935154i	$0.615260\pi$
$600$		0			0
$601$	−9241.77			−0.627254			−0.313627	−	0.949546i	$-0.601544\pi$
$601$	−9241.77			−0.627254			−0.313627	+	0.949546i	$0.601544\pi$
$602$		0			0
$603$			7676.27i			0.518411i
$604$		0			0
$605$			6225.04i			0.418320i
$606$		0			0
$607$	−18921.5			−1.26524			−0.632619	−	0.774463i	$-0.718020\pi$
$607$	−18921.5			−1.26524			−0.632619	+	0.774463i	$0.718020\pi$
$608$		0			0
$609$		−	15486.5i		−	1.03045i
$610$		0			0
$611$	−9005.85	+	22350.3i	−0.596297	+	1.47986i
$612$		0			0
$613$		−	17138.1i		−	1.12921i	−0.825363	−	0.564603i	$-0.809029\pi$
$613$		−	17138.1i		−	1.12921i	0.825363	−	0.564603i	$-0.190971\pi$
$614$		0			0
$615$	964.120			0.0632147
$616$		0			0
$617$		−	18825.8i		−	1.22836i	−0.789166	−	0.614180i	$-0.789487\pi$
$617$		−	18825.8i		−	1.22836i	0.789166	−	0.614180i	$-0.210513\pi$
$618$		0			0
$619$		−	1392.83i		−	0.0904404i	−0.998977	−	0.0452202i	$-0.985601\pi$
$619$		−	1392.83i		−	0.0904404i	0.998977	−	0.0452202i	$-0.0143989\pi$
$620$		0			0
$621$	−4937.03			−0.319028
$622$		0			0
$623$	−2089.09			−0.134346
$624$		0			0
$625$	4000.90			0.256057
$626$		0			0
$627$	−3240.00			−0.206369
$628$		0			0
$629$		−	5959.79i		−	0.377794i
$630$		0			0
$631$		−	25488.4i		−	1.60804i	−0.594599	−	0.804022i	$-0.702689\pi$
$631$		−	25488.4i		−	1.60804i	0.594599	−	0.804022i	$-0.297311\pi$
$632$		0			0
$633$	−1293.06			−0.0811918
$634$		0			0
$635$		−	10294.9i		−	0.643371i
$636$		0			0
$637$	27718.2	+	11168.8i	1.72407	+	0.694700i
$638$		0			0
$639$			1490.57i			0.0922787i
$640$		0			0
$641$	6066.41			0.373805			0.186902	−	0.982379i	$-0.440155\pi$
$641$	6066.41			0.373805			0.186902	+	0.982379i	$0.440155\pi$
$642$		0			0
$643$		−	1598.78i		−	0.0980554i	−0.998797	−	0.0490277i	$-0.984388\pi$
$643$		−	1598.78i		−	0.0980554i	0.998797	−	0.0490277i	$-0.0156123\pi$
$644$		0			0
$645$		−	1982.12i		−	0.121001i
$646$		0			0
$647$	23067.2			1.40164			0.700822	−	0.713336i	$-0.252817\pi$
$647$	23067.2			1.40164			0.700822	+	0.713336i	$0.252817\pi$
$648$		0			0
$649$	−4314.07			−0.260928
$650$		0			0
$651$	5533.68			0.333152
$652$		0			0
$653$	−23743.9			−1.42293			−0.711463	−	0.702723i	$-0.751967\pi$
$653$	−23743.9			−1.42293			−0.711463	+	0.702723i	$0.751967\pi$
$654$		0			0
$655$		−	5582.25i		−	0.333002i
$656$		0			0
$657$			2837.93i			0.168521i
$658$		0			0
$659$	7497.65			0.443197			0.221598	−	0.975138i	$-0.428873\pi$
$659$	7497.65			0.443197			0.221598	+	0.975138i	$0.428873\pi$
$660$		0			0
$661$			1255.26i			0.0738638i	0.999318	+	0.0369319i	$0.0117585\pi$
$661$			1255.26i			0.0738638i	−0.999318	+	0.0369319i	$0.988242\pi$
$662$		0			0
$663$	−7043.03	−	2837.93i	−0.412562	−	0.166238i
$664$		0			0
$665$		−	12132.3i		−	0.707474i
$666$		0			0
$667$	30143.8			1.74989
$668$		0			0
$669$			12312.9i			0.711576i
$670$		0			0
$671$		−	7619.38i		−	0.438365i
$672$		0			0
$673$	−1505.97			−0.0862569			−0.0431284	−	0.999070i	$-0.513732\pi$
$673$	−1505.97			−0.0862569			−0.0431284	+	0.999070i	$0.513732\pi$
$674$		0			0
$675$	2454.35			0.139953
$676$		0			0
$677$	−16201.4			−0.919751			−0.459876	−	0.887983i	$-0.652106\pi$
$677$	−16201.4			−0.919751			−0.459876	+	0.887983i	$0.652106\pi$
$678$		0			0
$679$	−45037.2			−2.54546
$680$		0			0
$681$		−	5427.32i		−	0.305397i
$682$		0			0
$683$			29090.4i			1.62974i	0.579644	+	0.814870i	$0.303192\pi$
$683$			29090.4i			1.62974i	−0.579644	+	0.814870i	$0.696808\pi$
$684$		0			0
$685$	−92.9712			−0.00518576
$686$		0			0
$687$			15748.0i			0.874561i
$688$		0			0
$689$	−10545.4	−	4249.16i	−0.583085	−	0.234949i
$690$		0			0
$691$		−	940.952i		−	0.0518025i	−0.999665	−	0.0259012i	$-0.991754\pi$
$691$		−	940.952i		−	0.0518025i	0.999665	−	0.0259012i	$-0.00824554\pi$
$692$		0			0
$693$	−4587.35			−0.251456
$694$		0			0
$695$		−	12112.4i		−	0.661077i
$696$		0			0
$697$		−	2971.93i		−	0.161506i
$698$		0			0
$699$	8426.65			0.455973
$700$		0			0
$701$	30713.9			1.65485			0.827424	−	0.561578i	$-0.189805\pi$
$701$	30713.9			1.65485			0.827424	+	0.561578i	$0.189805\pi$
$702$		0			0
$703$	7322.81			0.392866
$704$		0			0
$705$	9005.85			0.481106
$706$		0			0
$707$			28065.5i			1.49294i
$708$		0			0
$709$		−	25640.5i		−	1.35818i	−0.734056	−	0.679089i	$-0.762375\pi$
$709$		−	25640.5i		−	1.35818i	0.734056	−	0.679089i	$-0.237625\pi$
$710$		0			0
$711$	−4316.47			−0.227680
$712$		0			0
$713$			10771.0i			0.565748i
$714$		0			0
$715$	4132.30	+	1665.08i	0.216139	+	0.0870913i
$716$		0			0
$717$		−	20136.0i		−	1.04881i
$718$		0			0
$719$	20842.5			1.08108			0.540538	−	0.841319i	$-0.318221\pi$
$719$	20842.5			1.08108			0.540538	+	0.841319i	$0.318221\pi$
$720$		0			0
$721$		−	697.183i		−	0.0360117i
$722$		0			0
$723$		−	7557.34i		−	0.388742i
$724$		0			0
$725$	−14985.4			−0.767649
$726$		0			0
$727$	−263.608			−0.0134480			−0.00672398	−	0.999977i	$-0.502140\pi$
$727$	−263.608			−0.0134480			−0.00672398	+	0.999977i	$0.502140\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−6109.94			−0.309144
$732$		0			0
$733$			21835.5i			1.10029i	0.835069	+	0.550146i	$0.185428\pi$
$733$			21835.5i			1.10029i	−0.835069	+	0.550146i	$0.814572\pi$
$734$		0			0
$735$		−	11168.8i		−	0.560500i
$736$		0			0
$737$	−13883.2			−0.693888
$738$		0			0
$739$		−	19536.4i		−	0.972476i	−0.873826	−	0.486238i	$-0.838369\pi$
$739$		−	19536.4i		−	0.972476i	0.873826	−	0.486238i	$-0.161631\pi$
$740$		0			0
$741$	3486.97	−	8653.80i	0.172871	−	0.429022i
$742$		0			0
$743$			30353.1i			1.49872i	0.662163	+	0.749360i	$0.269639\pi$
$743$			30353.1i			1.49872i	−0.662163	+	0.749360i	$0.730361\pi$
$744$		0			0
$745$	−16140.5			−0.793746
$746$		0			0
$747$			5168.12i			0.253135i
$748$		0			0
$749$			11005.0i			0.536867i
$750$		0			0
$751$	−3904.20			−0.189702			−0.0948510	−	0.995491i	$-0.530237\pi$
$751$	−3904.20			−0.189702			−0.0948510	+	0.995491i	$0.530237\pi$
$752$		0			0
$753$	2484.00			0.120215
$754$		0			0
$755$	−9450.74			−0.455560
$756$		0			0
$757$	−2900.52			−0.139262			−0.0696308	−	0.997573i	$-0.522182\pi$
$757$	−2900.52			−0.139262			−0.0696308	+	0.997573i	$0.522182\pi$
$758$		0			0
$759$		−	8929.06i		−	0.427015i
$760$		0			0
$761$			33518.7i			1.59665i	0.602227	+	0.798325i	$0.294280\pi$
$761$			33518.7i			1.59665i	−0.602227	+	0.798325i	$0.705720\pi$
$762$		0			0
$763$	30280.8			1.43675
$764$		0			0
$765$			2837.93i			0.134125i
$766$		0			0
$767$	4642.91	−	11522.6i	0.218573	−	0.542445i
$768$		0			0
$769$		−	552.769i		−	0.0259211i	−0.999916	−	0.0129606i	$-0.995874\pi$
$769$		−	552.769i		−	0.0259211i	0.999916	−	0.0129606i	$-0.00412559\pi$
$770$		0			0
$771$	17522.3			0.818482
$772$		0			0
$773$			36498.8i			1.69828i	0.528166	+	0.849141i	$0.322880\pi$
$773$			36498.8i			1.69828i	−0.528166	+	0.849141i	$0.677120\pi$
$774$		0			0
$775$		−	5354.62i		−	0.248185i
$776$		0			0
$777$	10368.0			0.478700
$778$		0			0
$779$	3651.62			0.167950
$780$		0			0
$781$	−2695.83			−0.123514
$782$		0			0
$783$	−4451.03			−0.203150
$784$		0			0
$785$		−	6481.51i		−	0.294694i
$786$		0			0
$787$			4284.79i			0.194074i	0.995281	+	0.0970371i	$0.0309366\pi$
$787$			4284.79i			0.194074i	−0.995281	+	0.0970371i	$0.969063\pi$
$788$		0			0
$789$	12192.2			0.550131
$790$		0			0
$791$			1512.28i			0.0679779i
$792$		0			0
$793$	20350.8	+	8200.18i	0.911321	+	0.367209i
$794$		0			0
$795$			4249.16i			0.189562i
$796$		0			0
$797$	29538.5			1.31281			0.656405	−	0.754409i	$-0.272076\pi$
$797$	29538.5			1.31281			0.656405	+	0.754409i	$0.272076\pi$
$798$		0			0
$799$		−	27760.8i		−	1.22917i
$800$		0			0
$801$			600.430i			0.0264858i
$802$		0			0
$803$	−5132.65			−0.225563
$804$		0			0
$805$	33435.2			1.46389
$806$		0			0
$807$	5536.32			0.241497
$808$		0			0
$809$	−895.586			−0.0389211			−0.0194605	−	0.999811i	$-0.506195\pi$
$809$	−895.586			−0.0389211			−0.0194605	+	0.999811i	$0.506195\pi$
$810$		0			0
$811$			20139.7i			0.872011i	0.899944	+	0.436006i	$0.143607\pi$
$811$			20139.7i			0.872011i	−0.899944	+	0.436006i	$0.856393\pi$
$812$		0			0
$813$		−	6320.33i		−	0.272649i
$814$		0			0
$815$	1361.75			0.0585274
$816$		0			0
$817$		−	7507.31i		−	0.321478i
$818$		0			0
$819$	4937.03	−	12252.5i	0.210639	−	0.522755i
$820$		0			0
$821$			17263.2i			0.733848i	0.930251	+	0.366924i	$0.119589\pi$
$821$			17263.2i			0.733848i	−0.930251	+	0.366924i	$0.880411\pi$
$822$		0			0
$823$	12114.5			0.513104			0.256552	−	0.966530i	$-0.417413\pi$
$823$	12114.5			0.513104			0.256552	+	0.966530i	$0.417413\pi$
$824$		0			0
$825$			4438.92i			0.187325i
$826$		0			0
$827$		−	31450.9i		−	1.32243i	−0.750194	−	0.661217i	$-0.770040\pi$
$827$		−	31450.9i		−	1.32243i	0.750194	−	0.661217i	$-0.229960\pi$
$828$		0			0
$829$	−13760.4			−0.576499			−0.288250	−	0.957555i	$-0.593073\pi$
$829$	−13760.4			−0.576499			−0.288250	+	0.957555i	$0.593073\pi$
$830$		0			0
$831$	14345.8			0.598858
$832$		0			0
$833$	−34428.2			−1.43201
$834$		0			0
$835$	1257.83			0.0521305
$836$		0			0
$837$		−	1590.45i		−	0.0656797i
$838$		0			0
$839$			9846.21i			0.405160i	0.979266	+	0.202580i	$0.0649326\pi$
$839$			9846.21i			0.405160i	−0.979266	+	0.202580i	$0.935067\pi$
$840$		0			0
$841$	2787.47			0.114292
$842$		0			0
$843$		−	17597.4i		−	0.718965i
$844$		0			0
$845$	−8894.58	+	9245.07i	−0.362110	+	0.376379i
$846$		0			0
$847$			33382.1i			1.35422i
$848$		0			0
$849$	−19221.1			−0.776991
$850$		0			0
$851$			20180.8i			0.812914i
$852$		0			0
$853$		−	27574.5i		−	1.10684i	−0.832903	−	0.553420i	$-0.813323\pi$
$853$		−	27574.5i		−	1.10684i	0.832903	−	0.553420i	$-0.186677\pi$
$854$		0			0
$855$	−3486.97			−0.139476
$856$		0			0
$857$	8046.95			0.320745			0.160373	−	0.987057i	$-0.448730\pi$
$857$	8046.95			0.320745			0.160373	+	0.987057i	$0.448730\pi$
$858$		0			0
$859$	−2898.13			−0.115114			−0.0575570	−	0.998342i	$-0.518331\pi$
$859$	−2898.13			−0.115114			−0.0575570	+	0.998342i	$0.518331\pi$
$860$		0			0
$861$	5170.14			0.204644
$862$		0			0
$863$		−	4961.16i		−	0.195689i	−0.995202	−	0.0978447i	$-0.968805\pi$
$863$		−	4961.16i		−	0.195689i	0.995202	−	0.0978447i	$-0.0311948\pi$
$864$		0			0
$865$			8076.69i			0.317475i
$866$		0			0
$867$	−5991.00			−0.234677
$868$		0			0
$869$		−	7806.72i		−	0.304747i
$870$		0			0
$871$	14941.5	−	37081.1i	0.581255	−	1.44253i
$872$		0			0
$873$			12944.3i			0.501829i
$874$		0			0
$875$	−39478.3			−1.52527
$876$		0			0
$877$			1386.66i			0.0533913i	0.999644	+	0.0266957i	$0.00849850\pi$
$877$			1386.66i			0.0533913i	−0.999644	+	0.0266957i	$0.991502\pi$
$878$		0			0
$879$			9030.71i			0.346528i
$880$		0			0
$881$	−9030.36			−0.345335			−0.172668	−	0.984980i	$-0.555239\pi$
$881$	−9030.36			−0.345335			−0.172668	+	0.984980i	$0.555239\pi$
$882$		0			0
$883$	15512.7			0.591216			0.295608	−	0.955309i	$-0.404478\pi$
$883$	15512.7			0.591216			0.295608	+	0.955309i	$0.404478\pi$
$884$		0			0
$885$	−4642.91			−0.176350
$886$		0			0
$887$	−7431.21			−0.281303			−0.140651	−	0.990059i	$-0.544920\pi$
$887$	−7431.21			−0.281303			−0.140651	+	0.990059i	$0.544920\pi$
$888$		0			0
$889$		−	55207.0i		−	2.08277i
$890$		0			0
$891$			1318.46i			0.0495737i
$892$		0			0
$893$	34109.8			1.27821
$894$		0			0
$895$			21271.6i			0.794447i
$896$		0			0
$897$	23848.8	+	9609.69i	0.887726	+	0.357701i
$898$		0			0
$899$			9710.74i			0.360257i
$900$		0			0
$901$	13098.2			0.484310
$902$		0			0
$903$		−	10629.2i		−	0.391715i
$904$		0			0
$905$		−	15310.6i		−	0.562367i
$906$		0			0
$907$	10550.2			0.386234			0.193117	−	0.981176i	$-0.438140\pi$
$907$	10550.2			0.386234			0.193117	+	0.981176i	$0.438140\pi$
$908$		0			0
$909$	8066.38			0.294329
$910$		0			0
$911$	−35703.3			−1.29847			−0.649234	−	0.760589i	$-0.724910\pi$
$911$	−35703.3			−1.29847			−0.649234	+	0.760589i	$0.724910\pi$
$912$		0			0
$913$	−9347.01			−0.338818
$914$		0			0
$915$		−	8200.18i		−	0.296273i
$916$		0			0
$917$		−	29935.1i		−	1.07802i
$918$		0			0
$919$	42896.6			1.53975			0.769873	−	0.638197i	$-0.220319\pi$
$919$	42896.6			1.53975			0.769873	+	0.638197i	$0.220319\pi$
$920$		0			0
$921$		−	10025.5i		−	0.358688i
$922$		0			0
$923$	2901.33	−	7200.37i	0.103465	−	0.256775i
$924$		0			0
$925$		−	10032.5i		−	0.356613i
$926$		0			0
$927$	−200.379			−0.00709959
$928$		0			0
$929$			10366.7i			0.366114i	0.983102	+	0.183057i	$0.0585993\pi$
$929$			10366.7i			0.366114i	−0.983102	+	0.183057i	$0.941401\pi$
$930$		0			0
$931$		−	42302.0i		−	1.48914i
$932$		0			0
$933$	26265.6			0.921648
$934$		0			0
$935$	−5132.65			−0.179525
$936$		0			0
$937$	20289.8			0.707405			0.353702	−	0.935358i	$-0.384923\pi$
$937$	20289.8			0.707405			0.353702	+	0.935358i	$0.384923\pi$
$938$		0			0
$939$	5846.79			0.203198
$940$		0			0
$941$		−	37089.1i		−	1.28488i	−0.766337	−	0.642438i	$-0.777923\pi$
$941$		−	37089.1i		−	1.28488i	0.766337	−	0.642438i	$-0.222077\pi$
$942$		0			0
$943$			10063.4i			0.347519i
$944$		0			0
$945$	−4937.03			−0.169949
$946$		0			0
$947$		−	23458.2i		−	0.804952i	−0.915430	−	0.402476i	$-0.868150\pi$
$947$		−	23458.2i		−	0.804952i	0.915430	−	0.402476i	$-0.131850\pi$
$948$		0			0
$949$	5523.89	−	13708.9i	0.188949	−	0.468925i
$950$		0			0
$951$		−	5821.28i		−	0.198494i
$952$		0			0
$953$	−34695.5			−1.17933			−0.589663	−	0.807649i	$-0.700740\pi$
$953$	−34695.5			−1.17933			−0.589663	+	0.807649i	$0.700740\pi$
$954$		0			0
$955$			19966.6i			0.676550i
$956$		0			0
$957$		−	8050.09i		−	0.271915i
$958$		0			0
$959$	−498.563			−0.0167878
$960$		0			0
$961$	26321.1			0.883527
$962$		0			0
$963$	3162.97			0.105841
$964$		0			0
$965$	9917.53			0.330836
$966$		0			0
$967$			6289.66i			0.209164i	0.994516	+	0.104582i	$0.0333505\pi$
$967$			6289.66i			0.209164i	−0.994516	+	0.104582i	$0.966649\pi$
$968$		0			0
$969$			10748.7i			0.356345i
$970$		0			0
$971$	20185.9			0.667145			0.333573	−	0.942724i	$-0.391746\pi$
$971$	20185.9			0.667145			0.333573	+	0.942724i	$0.391746\pi$
$972$		0			0
$973$		−	64953.2i		−	2.14009i
$974$		0			0
$975$	−11856.0	−	4777.28i	−0.389432	−	0.156918i
$976$		0			0
$977$		−	44244.0i		−	1.44881i	−0.689373	−	0.724406i	$-0.742114\pi$
$977$		−	44244.0i		−	1.44881i	0.689373	−	0.724406i	$-0.257886\pi$
$978$		0			0
$979$	−1085.93			−0.0354510
$980$		0			0
$981$		−	8703.07i		−	0.283249i
$982$		0			0
$983$		−	8835.11i		−	0.286670i	−0.989674	−	0.143335i	$-0.954217\pi$
$983$		−	8835.11i		−	0.286670i	0.989674	−	0.143335i	$-0.0457826\pi$
$984$		0			0
$985$	−13393.9			−0.433263
$986$		0			0
$987$	48294.3			1.55747
$988$		0			0
$989$	20689.3			0.665198
$990$		0			0
$991$	−34915.1			−1.11919			−0.559594	−	0.828767i	$-0.689043\pi$
$991$	−34915.1			−1.11919			−0.559594	+	0.828767i	$0.689043\pi$
$992$		0			0
$993$			22206.4i			0.709666i
$994$		0			0
$995$			5261.15i			0.167628i
$996$		0			0
$997$	37962.8			1.20591			0.602956	−	0.797774i	$-0.293989\pi$
$997$	37962.8			1.20591			0.602956	+	0.797774i	$0.293989\pi$
$998$		0			0
$999$		−	2979.89i		−	0.0943740i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.c.e.337.3		4
4.3	odd	2		39.4.b.a.25.1	✓	4
12.11	even	2		117.4.b.d.64.4		4
13.12	even	2	inner	624.4.c.e.337.2		4
52.31	even	4		507.4.a.j.1.1		4
52.47	even	4		507.4.a.j.1.4		4
52.51	odd	2		39.4.b.a.25.4	yes	4
156.47	odd	4		1521.4.a.x.1.1		4
156.83	odd	4		1521.4.a.x.1.4		4
156.155	even	2		117.4.b.d.64.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.a.25.1	✓	4	4.3	odd	2
39.4.b.a.25.4	yes	4	52.51	odd	2
117.4.b.d.64.1		4	156.155	even	2
117.4.b.d.64.4		4	12.11	even	2
507.4.a.j.1.1		4	52.31	even	4
507.4.a.j.1.4		4	52.47	even	4
624.4.c.e.337.2		4	13.12	even	2	inner
624.4.c.e.337.3		4	1.1	even	1	trivial
1521.4.a.x.1.1		4	156.47	odd	4
1521.4.a.x.1.4		4	156.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.83936i q^{5} +31.3139i q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +5.83936i q^{5} +31.3139i q^{7} +9.00000 q^{9} +16.2773i q^{11} +(-43.4755 - 17.5181i) q^{13} +17.5181i q^{15} +54.0000 q^{17} +66.3500i q^{19} +93.9416i q^{21} -182.853 q^{23} +90.9019 q^{25} +27.0000 q^{27} -164.853 q^{29} -58.9055i q^{31} +48.8319i q^{33} -182.853 q^{35} -110.366i q^{37} +(-130.426 - 52.5542i) q^{39} -55.0357i q^{41} -113.147 q^{43} +52.5542i q^{45} -514.089i q^{47} -637.559 q^{49} +162.000 q^{51} +242.559 q^{53} -95.0490 q^{55} +199.050i q^{57} +265.036i q^{59} -468.098 q^{61} +281.825i q^{63} +(102.294 - 253.869i) q^{65} +852.919i q^{67} -548.559 q^{69} +165.619i q^{71} +315.325i q^{73} +272.706 q^{75} -509.706 q^{77} -479.608 q^{79} +81.0000 q^{81} +574.235i q^{83} +315.325i q^{85} -494.559 q^{87} +66.7144i q^{89} +(548.559 - 1361.39i) q^{91} -176.716i q^{93} -387.441 q^{95} +1438.25i q^{97} +146.496i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 36 * q^9	\( 4 q + 12 q^{3} + 36 q^{9} - 72 q^{13} + 216 q^{17} - 120 q^{23} - 44 q^{25} + 108 q^{27} - 48 q^{29} - 120 q^{35} - 216 q^{39} - 1064 q^{43} - 716 q^{49} + 648 q^{51} - 864 q^{53} - 584 q^{55} - 2280 q^{61} + 1632 q^{65} - 360 q^{69} - 132 q^{75} - 816 q^{77} - 288 q^{79} + 324 q^{81} - 144 q^{87} + 360 q^{91} - 3384 q^{95}+O(q^{100}) \) 4 * q + 12 * q^3 + 36 * q^9 - 72 * q^13 + 216 * q^17 - 120 * q^23 - 44 * q^25 + 108 * q^27 - 48 * q^29 - 120 * q^35 - 216 * q^39 - 1064 * q^43 - 716 * q^49 + 648 * q^51 - 864 * q^53 - 584 * q^55 - 2280 * q^61 + 1632 * q^65 - 360 * q^69 - 132 * q^75 - 816 * q^77 - 288 * q^79 + 324 * q^81 - 144 * q^87 + 360 * q^91 - 3384 * q^95

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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