LMFDB - Embedded newform 324.4.e.c.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(109,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("324.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.e (of order $3$, 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:	$19.1166188419$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	324.109
Dual form			324.4.e.c.217.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 + 2.59808i) q^{5} +(2.00000 + 3.46410i) q^{7} +(12.0000 + 20.7846i) q^{11} +(12.5000 - 21.6506i) q^{13} -21.0000 q^{17} -52.0000 q^{19} +(-84.0000 + 145.492i) q^{23} +(58.0000 + 100.459i) q^{25} +(88.5000 + 153.286i) q^{29} +(62.0000 - 107.387i) q^{31} -12.0000 q^{35} -265.000 q^{37} +(-213.000 + 368.927i) q^{41} +(80.0000 + 138.564i) q^{43} +(270.000 + 467.654i) q^{47} +(163.500 - 283.190i) q^{49} -258.000 q^{53} -72.0000 q^{55} +(-264.000 + 457.261i) q^{59} +(252.500 + 437.343i) q^{61} +(37.5000 + 64.9519i) q^{65} +(122.000 - 211.310i) q^{67} +204.000 q^{71} -397.000 q^{73} +(-48.0000 + 83.1384i) q^{77} +(-100.000 - 173.205i) q^{79} +(270.000 + 467.654i) q^{83} +(31.5000 - 54.5596i) q^{85} -453.000 q^{89} +100.000 q^{91} +(78.0000 - 135.100i) q^{95} +(-145.000 - 251.147i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} + 4 q^{7} + 24 q^{11} + 25 q^{13} - 42 q^{17} - 104 q^{19} - 168 q^{23} + 116 q^{25} + 177 q^{29} + 124 q^{31} - 24 q^{35} - 530 q^{37} - 426 q^{41} + 160 q^{43} + 540 q^{47} + 327 q^{49}+ \cdots - 290 q^{97}+O(q^{100}) $ 2 * q - 3 * q^5 + 4 * q^7 + 24 * q^11 + 25 * q^13 - 42 * q^17 - 104 * q^19 - 168 * q^23 + 116 * q^25 + 177 * q^29 + 124 * q^31 - 24 * q^35 - 530 * q^37 - 426 * q^41 + 160 * q^43 + 540 * q^47 + 327 * q^49 - 516 * q^53 - 144 * q^55 - 528 * q^59 + 505 * q^61 + 75 * q^65 + 244 * q^67 + 408 * q^71 - 794 * q^73 - 96 * q^77 - 200 * q^79 + 540 * q^83 + 63 * q^85 - 906 * q^89 + 200 * q^91 + 156 * q^95 - 290 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.50000	+	2.59808i	−0.134164	+	0.232379i	−0.925278	−	0.379290i	$-0.876168\pi$
$5$	−1.50000	+	2.59808i	−0.134164	+	0.232379i	0.791114	+	0.611669i	$0.209502\pi$
$6$		0			0
$7$	2.00000	+	3.46410i	0.107990	+	0.187044i	0.914956	−	0.403554i	$-0.132225\pi$
$7$	2.00000	+	3.46410i	0.107990	+	0.187044i	−0.806966	+	0.590598i	$0.798892\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	12.0000	+	20.7846i	0.328921	+	0.569709i	0.982298	−	0.187324i	$-0.0599815\pi$
$11$	12.0000	+	20.7846i	0.328921	+	0.569709i	−0.653377	+	0.757033i	$0.726648\pi$
$12$		0			0
$13$	12.5000	−	21.6506i	0.266683	−	0.461908i	−0.701320	−	0.712846i	$-0.747406\pi$
$13$	12.5000	−	21.6506i	0.266683	−	0.461908i	0.968003	+	0.250938i	$0.0807391\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−21.0000			−0.299603			−0.149801	−	0.988716i	$-0.547863\pi$
$17$	−21.0000			−0.299603			−0.149801	+	0.988716i	$0.547863\pi$
$18$		0			0
$19$	−52.0000			−0.627875			−0.313937	−	0.949444i	$-0.601648\pi$
$19$	−52.0000			−0.627875			−0.313937	+	0.949444i	$0.601648\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−84.0000	+	145.492i	−0.761531	+	1.31901i	0.180530	+	0.983569i	$0.442219\pi$
$23$	−84.0000	+	145.492i	−0.761531	+	1.31901i	−0.942061	+	0.335441i	$0.891115\pi$
$24$		0			0
$25$	58.0000	+	100.459i	0.464000	+	0.803672i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	88.5000	+	153.286i	0.566691	+	0.981538i	0.996890	+	0.0788035i	$0.0251100\pi$
$29$	88.5000	+	153.286i	0.566691	+	0.981538i	−0.430199	+	0.902734i	$0.641557\pi$
$30$		0			0
$31$	62.0000	−	107.387i	0.359211	−	0.622171i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	62.0000	−	107.387i	0.359211	−	0.622171i	0.987829	+	0.155543i	$0.0497126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−12.0000			−0.0579534
$36$		0			0
$37$	−265.000			−1.17745			−0.588726	−	0.808333i	$-0.700370\pi$
$37$	−265.000			−1.17745			−0.588726	+	0.808333i	$0.700370\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−213.000	+	368.927i	−0.811342	+	1.40529i	0.100583	+	0.994929i	$0.467929\pi$
$41$	−213.000	+	368.927i	−0.811342	+	1.40529i	−0.911925	+	0.410356i	$0.865404\pi$
$42$		0			0
$43$	80.0000	+	138.564i	0.283718	+	0.491414i	0.972298	−	0.233747i	$-0.0750986\pi$
$43$	80.0000	+	138.564i	0.283718	+	0.491414i	−0.688579	+	0.725161i	$0.741765\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	270.000	+	467.654i	0.837948	+	1.45137i	0.891608	+	0.452809i	$0.149578\pi$
$47$	270.000	+	467.654i	0.837948	+	1.45137i	−0.0536598	+	0.998559i	$0.517089\pi$
$48$		0			0
$49$	163.500	−	283.190i	0.476676	−	0.825628i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−258.000			−0.668661			−0.334330	−	0.942456i	$-0.608510\pi$
$53$	−258.000			−0.668661			−0.334330	+	0.942456i	$0.608510\pi$
$54$		0			0
$55$	−72.0000			−0.176518
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−264.000	+	457.261i	−0.582540	+	1.00899i	0.412637	+	0.910896i	$0.364608\pi$
$59$	−264.000	+	457.261i	−0.582540	+	1.00899i	−0.995177	+	0.0980937i	$0.968726\pi$
$60$		0			0
$61$	252.500	+	437.343i	0.529989	+	0.917967i	0.999388	+	0.0349814i	$0.0111372\pi$
$61$	252.500	+	437.343i	0.529989	+	0.917967i	−0.469399	+	0.882986i	$0.655529\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	37.5000	+	64.9519i	0.0715585	+	0.123943i
$66$		0			0
$67$	122.000	−	211.310i	0.222458	−	0.385308i	−0.733096	−	0.680125i	$-0.761925\pi$
$67$	122.000	−	211.310i	0.222458	−	0.385308i	0.955554	+	0.294817i	$0.0952587\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	204.000			0.340991			0.170495	−	0.985358i	$-0.445463\pi$
$71$	204.000			0.340991			0.170495	+	0.985358i	$0.445463\pi$
$72$		0			0
$73$	−397.000			−0.636511			−0.318256	−	0.948005i	$-0.603097\pi$
$73$	−397.000			−0.636511			−0.318256	+	0.948005i	$0.603097\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−48.0000	+	83.1384i	−0.0710404	+	0.123046i
$78$		0			0
$79$	−100.000	−	173.205i	−0.142416	−	0.246672i	0.785990	−	0.618239i	$-0.212154\pi$
$79$	−100.000	−	173.205i	−0.142416	−	0.246672i	−0.928406	+	0.371567i	$0.878820\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	270.000	+	467.654i	0.357064	+	0.618454i	0.987469	−	0.157813i	$-0.0504443\pi$
$83$	270.000	+	467.654i	0.357064	+	0.618454i	−0.630405	+	0.776267i	$0.717111\pi$
$84$		0			0
$85$	31.5000	−	54.5596i	0.0401959	−	0.0696214i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−453.000			−0.539527			−0.269764	−	0.962927i	$-0.586946\pi$
$89$	−453.000			−0.539527			−0.269764	+	0.962927i	$0.586946\pi$
$90$		0			0
$91$	100.000			0.115196
$92$		0			0
$93$		0			0
$94$		0			0
$95$	78.0000	−	135.100i	0.0842382	−	0.145905i
$96$		0			0
$97$	−145.000	−	251.147i	−0.151779	−	0.262888i	0.780103	−	0.625651i	$-0.215167\pi$
$97$	−145.000	−	251.147i	−0.151779	−	0.262888i	−0.931881	+	0.362763i	$0.881833\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−687.000	−	1189.92i	−0.676822	−	1.17229i	−0.975933	−	0.218072i	$-0.930023\pi$
$101$	−687.000	−	1189.92i	−0.676822	−	1.17229i	0.299110	−	0.954219i	$-0.403310\pi$
$102$		0			0
$103$	−634.000	+	1098.12i	−0.606504	+	1.05050i	0.385308	+	0.922788i	$0.374095\pi$
$103$	−634.000	+	1098.12i	−0.606504	+	1.05050i	−0.991812	+	0.127707i	$0.959238\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1164.00			1.05166			0.525832	−	0.850588i	$-0.323754\pi$
$107$	1164.00			1.05166			0.525832	+	0.850588i	$0.323754\pi$
$108$		0			0
$109$	587.000			0.515820			0.257910	−	0.966169i	$-0.416966\pi$
$109$	587.000			0.515820			0.257910	+	0.966169i	$0.416966\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	616.500	−	1067.81i	0.513234	−	0.888947i	−0.486648	−	0.873598i	$-0.661781\pi$
$113$	616.500	−	1067.81i	0.513234	−	0.888947i	0.999882	−	0.0153493i	$-0.00488603\pi$
$114$		0			0
$115$	−252.000	−	436.477i	−0.204340	−	0.353928i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−42.0000	−	72.7461i	−0.0323541	−	0.0560389i
$120$		0			0
$121$	377.500	−	653.849i	0.283621	−	0.491247i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−723.000			−0.517337
$126$		0			0
$127$	1496.00			1.04526			0.522632	−	0.852558i	$-0.324950\pi$
$127$	1496.00			1.04526			0.522632	+	0.852558i	$0.324950\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1002.00	−	1735.51i	0.668284	−	1.15750i	−0.310100	−	0.950704i	$-0.600363\pi$
$131$	1002.00	−	1735.51i	0.668284	−	1.15750i	0.978384	−	0.206797i	$-0.0663040\pi$
$132$		0			0
$133$	−104.000	−	180.133i	−0.0678041	−	0.117440i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1387.50	−	2403.22i	−0.865271	−	1.49869i	−0.866778	−	0.498694i	$-0.833813\pi$
$137$	−1387.50	−	2403.22i	−0.865271	−	1.49869i	0.00150687	−	0.999999i	$-0.499520\pi$
$138$		0			0
$139$	−1006.00	+	1742.44i	−0.613869	+	1.06325i	0.376713	+	0.926330i	$0.377054\pi$
$139$	−1006.00	+	1742.44i	−0.613869	+	1.06325i	−0.990582	+	0.136922i	$0.956279\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	600.000			0.350871
$144$		0			0
$145$	−531.000			−0.304118
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1072.50	−	1857.62i	0.589682	−	1.02136i	−0.404592	−	0.914497i	$-0.632586\pi$
$149$	1072.50	−	1857.62i	0.589682	−	1.02136i	0.994274	−	0.106862i	$-0.0340803\pi$
$150$		0			0
$151$	−1588.00	−	2750.50i	−0.855825	−	1.48233i	−0.875877	−	0.482534i	$-0.839717\pi$
$151$	−1588.00	−	2750.50i	−0.855825	−	1.48233i	0.0200521	−	0.999799i	$-0.493617\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	186.000	+	322.161i	0.0963863	+	0.166946i
$156$		0			0
$157$	−257.500	+	446.003i	−0.130896	+	0.226719i	−0.924022	−	0.382338i	$-0.875119\pi$
$157$	−257.500	+	446.003i	−0.130896	+	0.226719i	0.793126	+	0.609058i	$0.208452\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−672.000			−0.328950
$162$		0			0
$163$	3272.00			1.57229			0.786144	−	0.618044i	$-0.212075\pi$
$163$	3272.00			1.57229			0.786144	+	0.618044i	$0.212075\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	342.000	−	592.361i	0.158472	−	0.274481i	−0.775846	−	0.630922i	$-0.782677\pi$
$167$	342.000	−	592.361i	0.158472	−	0.274481i	0.934318	+	0.356441i	$0.116010\pi$
$168$		0			0
$169$	786.000	+	1361.39i	0.357761	+	0.619660i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1561.50	−	2704.60i	−0.686235	−	1.18859i	−0.973047	−	0.230607i	$-0.925929\pi$
$173$	−1561.50	−	2704.60i	−0.686235	−	1.18859i	0.286812	−	0.957987i	$-0.407404\pi$
$174$		0			0
$175$	−232.000	+	401.836i	−0.100215	+	0.173577i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2892.00			1.20759			0.603794	−	0.797140i	$-0.293655\pi$
$179$	2892.00			1.20759			0.603794	+	0.797140i	$0.293655\pi$
$180$		0			0
$181$	−1426.00			−0.585601			−0.292800	−	0.956174i	$-0.594587\pi$
$181$	−1426.00			−0.585601			−0.292800	+	0.956174i	$0.594587\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	397.500	−	688.490i	0.157972	−	0.273615i
$186$		0			0
$187$	−252.000	−	436.477i	−0.0985458	−	0.170686i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	798.000	+	1382.18i	0.302310	+	0.523617i	0.976659	−	0.214796i	$-0.0689088\pi$
$191$	798.000	+	1382.18i	0.302310	+	0.523617i	−0.674349	+	0.738413i	$0.735575\pi$
$192$		0			0
$193$	−557.500	+	965.618i	−0.207926	+	0.360138i	−0.951061	−	0.309003i	$-0.900005\pi$
$193$	−557.500	+	965.618i	−0.207926	+	0.360138i	0.743135	+	0.669142i	$0.233338\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1245.00			−0.450267			−0.225133	−	0.974328i	$-0.572282\pi$
$197$	−1245.00			−0.450267			−0.225133	+	0.974328i	$0.572282\pi$
$198$		0			0
$199$	−700.000			−0.249355			−0.124678	−	0.992197i	$-0.539790\pi$
$199$	−700.000			−0.249355			−0.124678	+	0.992197i	$0.539790\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−354.000	+	613.146i	−0.122394	+	0.211992i
$204$		0			0
$205$	−639.000	−	1106.78i	−0.217706	−	0.377078i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−624.000	−	1080.80i	−0.206521	−	0.357706i
$210$		0			0
$211$	−352.000	+	609.682i	−0.114847	+	0.198921i	−0.917719	−	0.397231i	$-0.869971\pi$
$211$	−352.000	+	609.682i	−0.114847	+	0.198921i	0.802872	+	0.596152i	$0.203304\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−480.000			−0.152259
$216$		0			0
$217$	496.000			0.155164
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−262.500	+	454.663i	−0.0798989	+	0.138389i
$222$		0			0
$223$	440.000	+	762.102i	0.132128	+	0.228853i	0.924497	−	0.381190i	$-0.124486\pi$
$223$	440.000	+	762.102i	0.132128	+	0.228853i	−0.792369	+	0.610043i	$0.791152\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1962.00	+	3398.28i	0.573667	+	0.993621i	0.996185	+	0.0872664i	$0.0278132\pi$
$227$	1962.00	+	3398.28i	0.573667	+	0.993621i	−0.422518	+	0.906355i	$0.638854\pi$
$228$		0			0
$229$	3072.50	−	5321.73i	0.886622	−	1.53567i	0.0427795	−	0.999085i	$-0.486379\pi$
$229$	3072.50	−	5321.73i	0.886622	−	1.53567i	0.843843	−	0.536590i	$-0.180288\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3885.00			−1.09234			−0.546169	−	0.837675i	$-0.683914\pi$
$233$	−3885.00			−1.09234			−0.546169	+	0.837675i	$0.683914\pi$
$234$		0			0
$235$	−1620.00			−0.449690
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2430.00	+	4208.88i	−0.657672	+	1.13912i	0.323545	+	0.946213i	$0.395125\pi$
$239$	−2430.00	+	4208.88i	−0.657672	+	1.13912i	−0.981217	+	0.192909i	$0.938208\pi$
$240$		0			0
$241$	2274.50	+	3939.55i	0.607940	+	1.05298i	0.991580	+	0.129499i	$0.0413370\pi$
$241$	2274.50	+	3939.55i	0.607940	+	1.05298i	−0.383640	+	0.923483i	$0.625330\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	490.500	+	849.571i	0.127906	+	0.221539i
$246$		0			0
$247$	−650.000	+	1125.83i	−0.167443	+	0.290020i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4800.00			1.20706			0.603532	−	0.797338i	$-0.293759\pi$
$251$	4800.00			1.20706			0.603532	+	0.797338i	$0.293759\pi$
$252$		0			0
$253$	−4032.00			−1.00194
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1255.50	+	2174.59i	−0.304731	+	0.527810i	−0.977201	−	0.212314i	$-0.931900\pi$
$257$	−1255.50	+	2174.59i	−0.304731	+	0.527810i	0.672470	+	0.740124i	$0.265233\pi$
$258$		0			0
$259$	−530.000	−	917.987i	−0.127153	−	0.220235i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2274.00	+	3938.68i	0.533159	+	0.923459i	0.999250	+	0.0387218i	$0.0123286\pi$
$263$	2274.00	+	3938.68i	0.533159	+	0.923459i	−0.466091	+	0.884737i	$0.654338\pi$
$264$		0			0
$265$	387.000	−	670.304i	0.0897103	−	0.155383i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2517.00			−0.570499			−0.285249	−	0.958453i	$-0.592076\pi$
$269$	−2517.00			−0.570499			−0.285249	+	0.958453i	$0.592076\pi$
$270$		0			0
$271$	4040.00			0.905581			0.452791	−	0.891617i	$-0.350429\pi$
$271$	4040.00			0.905581			0.452791	+	0.891617i	$0.350429\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1392.00	+	2411.01i	−0.305239	+	0.528690i
$276$		0			0
$277$	3545.00	+	6140.12i	0.768947	+	1.33186i	0.938134	+	0.346272i	$0.112553\pi$
$277$	3545.00	+	6140.12i	0.768947	+	1.33186i	−0.169187	+	0.985584i	$0.554114\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1582.50	+	2740.97i	0.335957	+	0.581895i	0.983668	−	0.179991i	$-0.0576068\pi$
$281$	1582.50	+	2740.97i	0.335957	+	0.581895i	−0.647711	+	0.761886i	$0.724274\pi$
$282$		0			0
$283$	3374.00	−	5843.94i	0.708705	−	1.22751i	−0.256633	−	0.966509i	$-0.582613\pi$
$283$	3374.00	−	5843.94i	0.708705	−	1.22751i	0.965338	−	0.261004i	$-0.0840535\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1704.00			−0.350467
$288$		0			0
$289$	−4472.00			−0.910238
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3247.50	+	5624.83i	−0.647512	+	1.12152i	0.336203	+	0.941789i	$0.390857\pi$
$293$	−3247.50	+	5624.83i	−0.647512	+	1.12152i	−0.983715	+	0.179734i	$0.942476\pi$
$294$		0			0
$295$	−792.000	−	1371.78i	−0.156312	−	0.270740i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2100.00	+	3637.31i	0.406174	+	0.703515i
$300$		0			0
$301$	−320.000	+	554.256i	−0.0612774	+	0.106136i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1515.00			−0.284422
$306$		0			0
$307$	−8980.00			−1.66943			−0.834716	−	0.550681i	$-0.814368\pi$
$307$	−8980.00			−1.66943			−0.834716	+	0.550681i	$0.814368\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1932.00	+	3346.32i	−0.352263	+	0.610137i	−0.986646	−	0.162882i	$-0.947921\pi$
$311$	−1932.00	+	3346.32i	−0.352263	+	0.610137i	0.634383	+	0.773019i	$0.281254\pi$
$312$		0			0
$313$	−2747.50	−	4758.81i	−0.496159	−	0.859373i	0.503831	−	0.863802i	$-0.331923\pi$
$313$	−2747.50	−	4758.81i	−0.496159	−	0.859373i	−0.999990	+	0.00442912i	$0.998590\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	442.500	+	766.432i	0.0784015	+	0.135795i	0.902560	−	0.430563i	$-0.141685\pi$
$317$	442.500	+	766.432i	0.0784015	+	0.135795i	−0.824159	+	0.566359i	$0.808352\pi$
$318$		0			0
$319$	−2124.00	+	3678.88i	−0.372794	+	0.645698i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1092.00			0.188113
$324$		0			0
$325$	2900.00			0.494963
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1080.00	+	1870.61i	−0.180980	+	0.313466i
$330$		0			0
$331$	5462.00	+	9460.46i	0.907005	+	1.57098i	0.818203	+	0.574930i	$0.194971\pi$
$331$	5462.00	+	9460.46i	0.907005	+	1.57098i	0.0888021	+	0.996049i	$0.471696\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	366.000	+	633.931i	0.0596917	+	0.103389i
$336$		0			0
$337$	3935.00	−	6815.62i	0.636063	−	1.10169i	−0.350226	−	0.936665i	$-0.613895\pi$
$337$	3935.00	−	6815.62i	0.636063	−	1.10169i	0.986289	−	0.165028i	$-0.0527713\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2976.00			0.472608
$342$		0			0
$343$	2680.00			0.421885
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2358.00	−	4084.18i	0.364796	−	0.631845i	−0.623948	−	0.781466i	$-0.714472\pi$
$347$	2358.00	−	4084.18i	0.364796	−	0.631845i	0.988743	+	0.149622i	$0.0478055\pi$
$348$		0			0
$349$	821.000	+	1422.01i	0.125923	+	0.218105i	0.922093	−	0.386968i	$-0.126477\pi$
$349$	821.000	+	1422.01i	0.125923	+	0.218105i	−0.796170	+	0.605073i	$0.793144\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1545.00	−	2676.02i	−0.232952	−	0.403485i	0.725724	−	0.687986i	$-0.241505\pi$
$353$	−1545.00	−	2676.02i	−0.232952	−	0.403485i	−0.958676	+	0.284502i	$0.908172\pi$
$354$		0			0
$355$	−306.000	+	530.008i	−0.0457487	+	0.0792391i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5688.00			−0.836215			−0.418107	−	0.908398i	$-0.637306\pi$
$359$	−5688.00			−0.836215			−0.418107	+	0.908398i	$0.637306\pi$
$360$		0			0
$361$	−4155.00			−0.605773
$362$		0			0
$363$		0			0
$364$		0			0
$365$	595.500	−	1031.44i	0.0853970	−	0.147912i
$366$		0			0
$367$	−3388.00	−	5868.19i	−0.481886	−	0.834651i	0.517898	−	0.855442i	$-0.326715\pi$
$367$	−3388.00	−	5868.19i	−0.481886	−	0.834651i	−0.999784	+	0.0207914i	$0.993381\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−516.000	−	893.738i	−0.0722086	−	0.125069i
$372$		0			0
$373$	−3079.00	+	5332.98i	−0.427412	+	0.740299i	−0.996642	−	0.0818791i	$-0.973908\pi$
$373$	−3079.00	+	5332.98i	−0.427412	+	0.740299i	0.569231	+	0.822178i	$0.307241\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4425.00			0.604507
$378$		0			0
$379$	7484.00			1.01432			0.507160	−	0.861852i	$-0.330695\pi$
$379$	7484.00			1.01432			0.507160	+	0.861852i	$0.330695\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2310.00	+	4001.04i	−0.308187	+	0.533795i	−0.977966	−	0.208765i	$-0.933055\pi$
$383$	−2310.00	+	4001.04i	−0.308187	+	0.533795i	0.669779	+	0.742560i	$0.266389\pi$
$384$		0			0
$385$	−144.000	−	249.415i	−0.0190621	−	0.0330166i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	465.000	+	805.404i	0.0606078	+	0.104976i	0.894737	−	0.446593i	$-0.147363\pi$
$389$	465.000	+	805.404i	0.0606078	+	0.104976i	−0.834129	+	0.551569i	$0.814029\pi$
$390$		0			0
$391$	1764.00	−	3055.34i	0.228157	−	0.395179i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	600.000			0.0764285
$396$		0			0
$397$	1919.00			0.242599			0.121300	−	0.992616i	$-0.461294\pi$
$397$	1919.00			0.242599			0.121300	+	0.992616i	$0.461294\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5002.50	−	8664.58i	0.622975	−	1.07902i	−0.365954	−	0.930633i	$-0.619257\pi$
$401$	5002.50	−	8664.58i	0.622975	−	1.07902i	0.988929	−	0.148391i	$-0.0474095\pi$
$402$		0			0
$403$	−1550.00	−	2684.68i	−0.191591	−	0.331845i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3180.00	−	5507.92i	−0.387289	−	0.670805i
$408$		0			0
$409$	−3827.50	+	6629.42i	−0.462733	+	0.801477i	−0.999096	−	0.0425106i	$-0.986464\pi$
$409$	−3827.50	+	6629.42i	−0.462733	+	0.801477i	0.536363	+	0.843987i	$0.319798\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2112.00			−0.251634
$414$		0			0
$415$	−1620.00			−0.191621
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7020.00	−	12159.0i	0.818495	−	1.41768i	−0.0882958	−	0.996094i	$-0.528142\pi$
$419$	7020.00	−	12159.0i	0.818495	−	1.41768i	0.906791	−	0.421581i	$-0.138525\pi$
$420$		0			0
$421$	−5609.50	−	9715.94i	−0.649383	−	1.12476i	−0.983270	−	0.182152i	$-0.941694\pi$
$421$	−5609.50	−	9715.94i	−0.649383	−	1.12476i	0.333887	−	0.942613i	$-0.391639\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1218.00	−	2109.64i	−0.139016	−	0.240782i
$426$		0			0
$427$	−1010.00	+	1749.37i	−0.114467	+	0.198262i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1884.00			−0.210555			−0.105277	−	0.994443i	$-0.533573\pi$
$431$	−1884.00			−0.210555			−0.105277	+	0.994443i	$0.533573\pi$
$432$		0			0
$433$	17507.0			1.94303			0.971516	−	0.236975i	$-0.0761558\pi$
$433$	17507.0			1.94303			0.971516	+	0.236975i	$0.0761558\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4368.00	−	7565.60i	0.478146	−	0.828173i
$438$		0			0
$439$	1754.00	+	3038.02i	0.190692	+	0.330288i	0.945480	−	0.325681i	$-0.105593\pi$
$439$	1754.00	+	3038.02i	0.190692	+	0.330288i	−0.754788	+	0.655969i	$0.772260\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6210.00	−	10756.0i	−0.666018	−	1.15358i	−0.979008	−	0.203821i	$-0.934664\pi$
$443$	−6210.00	−	10756.0i	−0.666018	−	1.15358i	0.312990	−	0.949756i	$-0.398669\pi$
$444$		0			0
$445$	679.500	−	1176.93i	0.0723851	−	0.125375i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8658.00			0.910014			0.455007	−	0.890488i	$-0.349637\pi$
$449$	8658.00			0.910014			0.455007	+	0.890488i	$0.349637\pi$
$450$		0			0
$451$	−10224.0			−1.06747
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−150.000	+	259.808i	−0.0154552	+	0.0267692i
$456$		0			0
$457$	−167.500	−	290.119i	−0.0171451	−	0.0296962i	0.857326	−	0.514775i	$-0.172124\pi$
$457$	−167.500	−	290.119i	−0.0171451	−	0.0296962i	−0.874471	+	0.485078i	$0.838791\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−123.000	−	213.042i	−0.0124266	−	0.0215236i	0.859745	−	0.510723i	$-0.170622\pi$
$461$	−123.000	−	213.042i	−0.0124266	−	0.0215236i	−0.872172	+	0.489200i	$0.837289\pi$
$462$		0			0
$463$	−7156.00	+	12394.6i	−0.718288	+	1.24411i	0.243389	+	0.969929i	$0.421741\pi$
$463$	−7156.00	+	12394.6i	−0.718288	+	1.24411i	−0.961677	+	0.274183i	$0.911593\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8184.00			0.810943			0.405471	−	0.914108i	$-0.367107\pi$
$467$	8184.00			0.810943			0.405471	+	0.914108i	$0.367107\pi$
$468$		0			0
$469$	976.000			0.0960927
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1920.00	+	3325.54i	−0.186642	+	0.323274i
$474$		0			0
$475$	−3016.00	−	5223.87i	−0.291334	−	0.504605i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9054.00	−	15682.0i	−0.863649	−	1.49588i	−0.868383	−	0.495895i	$-0.834840\pi$
$479$	−9054.00	−	15682.0i	−0.863649	−	1.49588i	0.00473362	−	0.999989i	$-0.498493\pi$
$480$		0			0
$481$	−3312.50	+	5737.42i	−0.314006	+	0.543875i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	870.000			0.0814529
$486$		0			0
$487$	13700.0			1.27476			0.637378	−	0.770551i	$-0.280019\pi$
$487$	13700.0			1.27476			0.637378	+	0.770551i	$0.280019\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10068.0	−	17438.3i	0.925382	−	1.60281i	0.134436	−	0.990922i	$-0.457078\pi$
$491$	10068.0	−	17438.3i	0.925382	−	1.60281i	0.790946	−	0.611886i	$-0.209589\pi$
$492$		0			0
$493$	−1858.50	−	3219.02i	−0.169782	−	0.294071i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	408.000	+	706.677i	0.0368235	+	0.0637802i
$498$		0			0
$499$	6704.00	−	11611.7i	0.601427	−	1.04170i	−0.391178	−	0.920315i	$-0.627932\pi$
$499$	6704.00	−	11611.7i	0.601427	−	1.04170i	0.992605	−	0.121388i	$-0.0387344\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−21600.0			−1.91470			−0.957352	−	0.288923i	$-0.906703\pi$
$503$	−21600.0			−1.91470			−0.957352	+	0.288923i	$0.906703\pi$
$504$		0			0
$505$	4122.00			0.363221
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2973.00	−	5149.39i	0.258892	−	0.448414i	−0.707054	−	0.707160i	$-0.749976\pi$
$509$	2973.00	−	5149.39i	0.258892	−	0.448414i	0.965945	+	0.258746i	$0.0833094\pi$
$510$		0			0
$511$	−794.000	−	1375.25i	−0.0687368	−	0.119056i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1902.00	−	3294.36i	−0.162742	−	0.281877i
$516$		0			0
$517$	−6480.00	+	11223.7i	−0.551238	+	0.954772i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8970.00			0.754286			0.377143	−	0.926155i	$-0.376907\pi$
$521$	8970.00			0.754286			0.377143	+	0.926155i	$0.376907\pi$
$522$		0			0
$523$	−12520.0			−1.04677			−0.523386	−	0.852096i	$-0.675331\pi$
$523$	−12520.0			−1.04677			−0.523386	+	0.852096i	$0.675331\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1302.00	+	2255.13i	−0.107621	+	0.186404i
$528$		0			0
$529$	−8028.50	−	13905.8i	−0.659859	−	1.14291i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	5325.00	+	9223.17i	0.432742	+	0.749531i
$534$		0			0
$535$	−1746.00	+	3024.16i	−0.141096	+	0.244385i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7848.00			0.627156
$540$		0			0
$541$	3911.00			0.310808			0.155404	−	0.987851i	$-0.450332\pi$
$541$	3911.00			0.310808			0.155404	+	0.987851i	$0.450332\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−880.500	+	1525.07i	−0.0692045	+	0.119866i
$546$		0			0
$547$	8510.00	+	14739.8i	0.665194	+	1.15215i	0.979233	+	0.202740i	$0.0649847\pi$
$547$	8510.00	+	14739.8i	0.665194	+	1.15215i	−0.314038	+	0.949410i	$0.601682\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4602.00	−	7970.90i	−0.355811	−	0.616283i
$552$		0			0
$553$	400.000	−	692.820i	0.0307590	−	0.0532762i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	13875.0			1.05548			0.527740	−	0.849406i	$-0.323039\pi$
$557$	13875.0			1.05548			0.527740	+	0.849406i	$0.323039\pi$
$558$		0			0
$559$	4000.00			0.302651
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−270.000	+	467.654i	−0.0202116	+	0.0350076i	−0.875954	−	0.482394i	$-0.839767\pi$
$563$	−270.000	+	467.654i	−0.0202116	+	0.0350076i	0.855743	+	0.517402i	$0.173101\pi$
$564$		0			0
$565$	1849.50	+	3203.43i	0.137715	+	0.238530i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6256.50	+	10836.6i	0.460960	+	0.798406i	0.999009	−	0.0445076i	$-0.0141719\pi$
$569$	6256.50	+	10836.6i	0.460960	+	0.798406i	−0.538049	+	0.842913i	$0.680839\pi$
$570$		0			0
$571$	10208.0	−	17680.8i	0.748146	−	1.29583i	−0.200564	−	0.979681i	$-0.564277\pi$
$571$	10208.0	−	17680.8i	0.748146	−	1.29583i	0.948710	−	0.316147i	$-0.102389\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−19488.0			−1.41340
$576$		0			0
$577$	−1801.00			−0.129942			−0.0649711	−	0.997887i	$-0.520696\pi$
$577$	−1801.00			−0.129942			−0.0649711	+	0.997887i	$0.520696\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1080.00	+	1870.61i	−0.0771187	+	0.133573i
$582$		0			0
$583$	−3096.00	−	5362.43i	−0.219937	−	0.380942i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2760.00	+	4780.46i	0.194067	+	0.336134i	0.946594	−	0.322427i	$-0.104499\pi$
$587$	2760.00	+	4780.46i	0.194067	+	0.336134i	−0.752527	+	0.658561i	$0.771165\pi$
$588$		0			0
$589$	−3224.00	+	5584.13i	−0.225539	+	0.390645i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11937.0			−0.826634			−0.413317	−	0.910587i	$-0.635630\pi$
$593$	−11937.0			−0.826634			−0.413317	+	0.910587i	$0.635630\pi$
$594$		0			0
$595$	252.000			0.0173630
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8094.00	+	14019.2i	−0.552107	+	0.956277i	0.446016	+	0.895025i	$0.352843\pi$
$599$	−8094.00	+	14019.2i	−0.552107	+	0.956277i	−0.998122	+	0.0612516i	$0.980491\pi$
$600$		0			0
$601$	5970.50	+	10341.2i	0.405228	+	0.701875i	0.994348	−	0.106171i	$-0.0338589\pi$
$601$	5970.50	+	10341.2i	0.405228	+	0.701875i	−0.589120	+	0.808045i	$0.700526\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1132.50	+	1961.55i	0.0761036	+	0.131815i
$606$		0			0
$607$	6230.00	−	10790.7i	0.416586	−	0.721549i	−0.579007	−	0.815322i	$-0.696560\pi$
$607$	6230.00	−	10790.7i	0.416586	−	0.721549i	0.995594	+	0.0937737i	$0.0298930\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13500.0			0.893865
$612$		0			0
$613$	4670.00			0.307699			0.153850	−	0.988094i	$-0.450833\pi$
$613$	4670.00			0.307699			0.153850	+	0.988094i	$0.450833\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4717.50	+	8170.95i	−0.307811	+	0.533144i	−0.977883	−	0.209152i	$-0.932930\pi$
$617$	−4717.50	+	8170.95i	−0.307811	+	0.533144i	0.670072	+	0.742296i	$0.266263\pi$
$618$		0			0
$619$	6980.00	+	12089.7i	0.453231	+	0.785019i	0.998585	−	0.0531872i	$-0.0169380\pi$
$619$	6980.00	+	12089.7i	0.453231	+	0.785019i	−0.545354	+	0.838206i	$0.683605\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−906.000	−	1569.24i	−0.0582634	−	0.100915i
$624$		0			0
$625$	−6165.50	+	10679.0i	−0.394592	+	0.683453i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5565.00			0.352768
$630$		0			0
$631$	−18904.0			−1.19264			−0.596320	−	0.802747i	$-0.703371\pi$
$631$	−18904.0			−1.19264			−0.596320	+	0.802747i	$0.703371\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2244.00	+	3886.72i	−0.140237	+	0.242897i
$636$		0			0
$637$	−4087.50	−	7079.76i	−0.254243	−	0.440361i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10690.5	+	18516.5i	0.658735	+	1.14096i	0.980943	+	0.194294i	$0.0622416\pi$
$641$	10690.5	+	18516.5i	0.658735	+	1.14096i	−0.322208	+	0.946669i	$0.604425\pi$
$642$		0			0
$643$	710.000	−	1229.76i	0.0435454	−	0.0754228i	−0.843431	−	0.537237i	$-0.819468\pi$
$643$	710.000	−	1229.76i	0.0435454	−	0.0754228i	0.886977	+	0.461814i	$0.152801\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21204.0			−1.28843			−0.644216	−	0.764844i	$-0.722816\pi$
$647$	−21204.0			−1.28843			−0.644216	+	0.764844i	$0.722816\pi$
$648$		0			0
$649$	−12672.0			−0.766440
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11595.0	+	20083.1i	−0.694866	+	1.20354i	0.275360	+	0.961341i	$0.411203\pi$
$653$	−11595.0	+	20083.1i	−0.694866	+	1.20354i	−0.970226	+	0.242202i	$0.922130\pi$
$654$		0			0
$655$	3006.00	+	5206.54i	0.179319	+	0.310590i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4902.00	+	8490.51i	0.289765	+	0.501887i	0.973753	−	0.227606i	$-0.0730897\pi$
$659$	4902.00	+	8490.51i	0.289765	+	0.501887i	−0.683989	+	0.729492i	$0.739756\pi$
$660$		0			0
$661$	15120.5	−	26189.5i	0.889742	−	1.54108i	0.0495612	−	0.998771i	$-0.484218\pi$
$661$	15120.5	−	26189.5i	0.889742	−	1.54108i	0.840181	−	0.542307i	$-0.182449\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	624.000			0.0363875
$666$		0			0
$667$	−29736.0			−1.72621
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6060.00	+	10496.2i	−0.348649	+	0.603878i
$672$		0			0
$673$	−401.500	−	695.418i	−0.0229966	−	0.0398312i	0.854298	−	0.519783i	$-0.173987\pi$
$673$	−401.500	−	695.418i	−0.0229966	−	0.0398312i	−0.877295	+	0.479952i	$0.840654\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7233.00	+	12527.9i	0.410616	+	0.711207i	0.994957	−	0.100301i	$-0.0319805\pi$
$677$	7233.00	+	12527.9i	0.410616	+	0.711207i	−0.584342	+	0.811508i	$0.698647\pi$
$678$		0			0
$679$	580.000	−	1004.59i	0.0327811	−	0.0567785i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7020.00			−0.393284			−0.196642	−	0.980475i	$-0.563004\pi$
$683$	−7020.00			−0.393284			−0.196642	+	0.980475i	$0.563004\pi$
$684$		0			0
$685$	8325.00			0.464353
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3225.00	+	5585.86i	−0.178320	+	0.308860i
$690$		0			0
$691$	12230.0	+	21183.0i	0.673301	+	1.16619i	0.976962	+	0.213412i	$0.0684576\pi$
$691$	12230.0	+	21183.0i	0.673301	+	1.16619i	−0.303661	+	0.952780i	$0.598209\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3018.00	−	5227.33i	−0.164718	−	0.285301i
$696$		0			0
$697$	4473.00	−	7747.46i	0.243080	−	0.421027i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27651.0			1.48982			0.744910	−	0.667165i	$-0.232492\pi$
$701$	27651.0			1.48982			0.744910	+	0.667165i	$0.232492\pi$
$702$		0			0
$703$	13780.0			0.739292
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2748.00	−	4759.68i	0.146180	−	0.253191i
$708$		0			0
$709$	2460.50	+	4261.71i	0.130333	+	0.225743i	0.923805	−	0.382864i	$-0.125062\pi$
$709$	2460.50	+	4261.71i	0.130333	+	0.225743i	−0.793472	+	0.608607i	$0.791729\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10416.0	+	18041.0i	0.547100	+	0.947605i
$714$		0			0
$715$	−900.000	+	1558.85i	−0.0470743	+	0.0815350i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20232.0			−1.04941			−0.524705	−	0.851284i	$-0.675824\pi$
$719$	−20232.0			−1.04941			−0.524705	+	0.851284i	$0.675824\pi$
$720$		0			0
$721$	−5072.00			−0.261985
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−10266.0	+	17781.2i	−0.525889	+	0.910867i
$726$		0			0
$727$	−3250.00	−	5629.17i	−0.165799	−	0.287172i	0.771140	−	0.636666i	$-0.219687\pi$
$727$	−3250.00	−	5629.17i	−0.165799	−	0.287172i	−0.936939	+	0.349494i	$0.886354\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1680.00	−	2909.85i	−0.0850028	−	0.147229i
$732$		0			0
$733$	−6859.00	+	11880.1i	−0.345625	+	0.598640i	−0.985467	−	0.169867i	$-0.945666\pi$
$733$	−6859.00	+	11880.1i	−0.345625	+	0.598640i	0.639842	+	0.768506i	$0.279000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5856.00			0.292685
$738$		0			0
$739$	−13588.0			−0.676377			−0.338189	−	0.941078i	$-0.609814\pi$
$739$	−13588.0			−0.676377			−0.338189	+	0.941078i	$0.609814\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−17046.0	+	29524.5i	−0.841665	+	1.45781i	0.0468213	+	0.998903i	$0.485091\pi$
$743$	−17046.0	+	29524.5i	−0.841665	+	1.45781i	−0.888486	+	0.458903i	$0.848242\pi$
$744$		0			0
$745$	3217.50	+	5572.87i	0.158228	+	0.274059i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2328.00	+	4032.21i	0.113569	+	0.196707i
$750$		0			0
$751$	470.000	−	814.064i	0.0228369	−	0.0395547i	−0.854381	−	0.519647i	$-0.826063\pi$
$751$	470.000	−	814.064i	0.0228369	−	0.0395547i	0.877218	+	0.480092i	$0.159397\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9528.00			0.459284
$756$		0			0
$757$	−5410.00			−0.259749			−0.129874	−	0.991530i	$-0.541457\pi$
$757$	−5410.00			−0.259749			−0.129874	+	0.991530i	$0.541457\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3424.50	−	5931.41i	0.163125	−	0.282541i	−0.772863	−	0.634573i	$-0.781176\pi$
$761$	3424.50	−	5931.41i	0.163125	−	0.282541i	0.935988	+	0.352032i	$0.114509\pi$
$762$		0			0
$763$	1174.00	+	2033.43i	0.0557033	+	0.0964810i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6600.00	+	11431.5i	0.310707	+	0.538160i
$768$		0			0
$769$	4566.50	−	7909.41i	0.214138	−	0.370898i	−0.738868	−	0.673851i	$-0.764639\pi$
$769$	4566.50	−	7909.41i	0.214138	−	0.370898i	0.953006	+	0.302953i	$0.0979724\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−37845.0			−1.76092			−0.880459	−	0.474122i	$-0.842766\pi$
$773$	−37845.0			−1.76092			−0.880459	+	0.474122i	$0.842766\pi$
$774$		0			0
$775$	14384.0			0.666695
$776$		0			0
$777$		0			0
$778$		0			0
$779$	11076.0	−	19184.2i	0.509421	−	0.882343i
$780$		0			0
$781$	2448.00	+	4240.06i	0.112159	+	0.194265i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−772.500	−	1338.01i	−0.0351232	−	0.0608352i
$786$		0			0
$787$	16988.0	−	29424.1i	0.769450	−	1.33273i	−0.168412	−	0.985717i	$-0.553864\pi$
$787$	16988.0	−	29424.1i	0.769450	−	1.33273i	0.937862	−	0.347009i	$-0.112803\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4932.00			0.221696
$792$		0			0
$793$	12625.0			0.565355
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11902.5	−	20615.7i	0.528994	−	0.916244i	−0.470434	−	0.882435i	$-0.655903\pi$
$797$	11902.5	−	20615.7i	0.528994	−	0.916244i	0.999428	−	0.0338094i	$-0.0107639\pi$
$798$		0			0
$799$	−5670.00	−	9820.73i	−0.251052	−	0.434834i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4764.00	−	8251.49i	−0.209362	−	0.362626i
$804$		0			0
$805$	1008.00	−	1745.91i	0.0441333	−	0.0764412i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6567.00			0.285394			0.142697	−	0.989766i	$-0.454423\pi$
$809$	6567.00			0.285394			0.142697	+	0.989766i	$0.454423\pi$
$810$		0			0
$811$	−8656.00			−0.374788			−0.187394	−	0.982285i	$-0.560004\pi$
$811$	−8656.00			−0.374788			−0.187394	+	0.982285i	$0.560004\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4908.00	+	8500.91i	−0.210944	+	0.365367i
$816$		0			0
$817$	−4160.00	−	7205.33i	−0.178140	−	0.308547i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2995.50	−	5188.36i	−0.127337	−	0.220554i	0.795307	−	0.606207i	$-0.207310\pi$
$821$	−2995.50	−	5188.36i	−0.127337	−	0.220554i	−0.922644	+	0.385653i	$0.873976\pi$
$822$		0			0
$823$	−22906.0	+	39674.4i	−0.970174	+	1.68039i	−0.275153	+	0.961400i	$0.588729\pi$
$823$	−22906.0	+	39674.4i	−0.970174	+	1.68039i	−0.695021	+	0.718990i	$0.744605\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	35160.0			1.47840			0.739198	−	0.673489i	$-0.235205\pi$
$827$	35160.0			1.47840			0.739198	+	0.673489i	$0.235205\pi$
$828$		0			0
$829$	−9898.00			−0.414682			−0.207341	−	0.978269i	$-0.566481\pi$
$829$	−9898.00			−0.414682			−0.207341	+	0.978269i	$0.566481\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3433.50	+	5947.00i	−0.142814	+	0.247360i
$834$		0			0
$835$	1026.00	+	1777.08i	0.0425224	+	0.0736509i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2016.00	−	3491.81i	−0.0829560	−	0.143684i	0.821562	−	0.570119i	$-0.193103\pi$
$839$	−2016.00	−	3491.81i	−0.0829560	−	0.143684i	−0.904518	+	0.426435i	$0.859769\pi$
$840$		0			0
$841$	−3470.00	+	6010.22i	−0.142277	+	0.246431i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4716.00			−0.191994
$846$		0			0
$847$	3020.00			0.122513
$848$		0			0
$849$		0			0
$850$		0			0
$851$	22260.0	−	38555.5i	0.896666	−	1.55307i
$852$		0			0
$853$	−9319.00	−	16141.0i	−0.374064	−	0.647898i	0.616123	−	0.787650i	$-0.288703\pi$
$853$	−9319.00	−	16141.0i	−0.374064	−	0.647898i	−0.990186	+	0.139753i	$0.955369\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−18787.5	−	32540.9i	−0.748855	−	1.29705i	−0.948372	−	0.317161i	$-0.897271\pi$
$857$	−18787.5	−	32540.9i	−0.748855	−	1.29705i	0.199517	−	0.979894i	$-0.436063\pi$
$858$		0			0
$859$	3134.00	−	5428.25i	0.124483	−	0.215610i	−0.797048	−	0.603916i	$-0.793606\pi$
$859$	3134.00	−	5428.25i	0.124483	−	0.215610i	0.921531	+	0.388306i	$0.126940\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−44832.0			−1.76837			−0.884183	−	0.467142i	$-0.845284\pi$
$863$	−44832.0			−1.76837			−0.884183	+	0.467142i	$0.845284\pi$
$864$		0			0
$865$	9369.00			0.368272
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2400.00	−	4156.92i	0.0936875	−	0.162271i
$870$		0			0
$871$	−3050.00	−	5282.75i	−0.118651	−	0.205510i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1446.00	−	2504.55i	−0.0558671	−	0.0967647i
$876$		0			0
$877$	−7247.50	+	12553.0i	−0.279054	+	0.483336i	−0.971150	−	0.238469i	$-0.923354\pi$
$877$	−7247.50	+	12553.0i	−0.279054	+	0.483336i	0.692096	+	0.721806i	$0.256688\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	25986.0			0.993746			0.496873	−	0.867823i	$-0.334481\pi$
$881$	25986.0			0.993746			0.496873	+	0.867823i	$0.334481\pi$
$882$		0			0
$883$	34868.0			1.32888			0.664440	−	0.747341i	$-0.268670\pi$
$883$	34868.0			1.32888			0.664440	+	0.747341i	$0.268670\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3360.00	−	5819.69i	0.127190	−	0.220300i	−0.795397	−	0.606089i	$-0.792737\pi$
$887$	3360.00	−	5819.69i	0.127190	−	0.220300i	0.922587	+	0.385789i	$0.126071\pi$
$888$		0			0
$889$	2992.00	+	5182.30i	0.112878	+	0.195510i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−14040.0	−	24318.0i	−0.526126	−	0.911277i
$894$		0			0
$895$	−4338.00	+	7513.64i	−0.162015	+	0.280618i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	21948.0			0.814246
$900$		0			0
$901$	5418.00			0.200333
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2139.00	−	3704.86i	0.0785666	−	0.136081i
$906$		0			0
$907$	8648.00	+	14978.8i	0.316596	+	0.548360i	0.979775	−	0.200101i	$-0.0641269\pi$
$907$	8648.00	+	14978.8i	0.316596	+	0.548360i	−0.663180	+	0.748460i	$0.730794\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	7128.00	+	12346.1i	0.259233	+	0.449005i	0.966037	−	0.258405i	$-0.0831969\pi$
$911$	7128.00	+	12346.1i	0.259233	+	0.449005i	−0.706804	+	0.707410i	$0.749864\pi$
$912$		0			0
$913$	−6480.00	+	11223.7i	−0.234892	+	0.406845i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8016.00			0.288671
$918$		0			0
$919$	15752.0			0.565409			0.282704	−	0.959207i	$-0.408768\pi$
$919$	15752.0			0.565409			0.282704	+	0.959207i	$0.408768\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2550.00	−	4416.73i	0.0909364	−	0.157506i
$924$		0			0
$925$	−15370.0	−	26621.6i	−0.546338	−	0.946285i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15781.5	−	27334.4i	−0.557346	−	0.965351i	−0.997717	−	0.0675353i	$-0.978486\pi$
$929$	−15781.5	−	27334.4i	−0.557346	−	0.965351i	0.440371	−	0.897816i	$-0.354847\pi$
$930$		0			0
$931$	−8502.00	+	14725.9i	−0.299293	+	0.518391i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1512.00			0.0528852
$936$		0			0
$937$	−3901.00			−0.136009			−0.0680043	−	0.997685i	$-0.521663\pi$
$937$	−3901.00			−0.136009			−0.0680043	+	0.997685i	$0.521663\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	26722.5	−	46284.7i	0.925748	−	1.60344i	0.135394	−	0.990792i	$-0.456770\pi$
$941$	26722.5	−	46284.7i	0.925748	−	1.60344i	0.790354	−	0.612650i	$-0.209897\pi$
$942$		0			0
$943$	−35784.0	−	61979.7i	−1.23572	−	2.14034i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4518.00	−	7825.41i	−0.155032	−	0.268523i	0.778039	−	0.628216i	$-0.216215\pi$
$947$	−4518.00	−	7825.41i	−0.155032	−	0.268523i	−0.933071	+	0.359693i	$0.882881\pi$
$948$		0			0
$949$	−4962.50	+	8595.30i	−0.169747	+	0.294010i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−25065.0			−0.851978			−0.425989	−	0.904728i	$-0.640074\pi$
$953$	−25065.0			−0.851978			−0.425989	+	0.904728i	$0.640074\pi$
$954$		0			0
$955$	−4788.00			−0.162237
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5550.00	−	9612.88i	0.186881	−	0.323687i
$960$		0			0
$961$	7207.50	+	12483.8i	0.241935	+	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1672.50	−	2896.85i	−0.0557924	−	0.0966353i
$966$		0			0
$967$	−7222.00	+	12508.9i	−0.240169	+	0.415986i	−0.960762	−	0.277373i	$-0.910536\pi$
$967$	−7222.00	+	12508.9i	−0.240169	+	0.415986i	0.720593	+	0.693358i	$0.243870\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	50004.0			1.65263			0.826316	−	0.563207i	$-0.190433\pi$
$971$	50004.0			1.65263			0.826316	+	0.563207i	$0.190433\pi$
$972$		0			0
$973$	−8048.00			−0.265167
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22695.0	−	39308.9i	0.743170	−	1.28721i	−0.207874	−	0.978156i	$-0.566655\pi$
$977$	22695.0	−	39308.9i	0.743170	−	1.28721i	0.951045	−	0.309053i	$-0.100012\pi$
$978$		0			0
$979$	−5436.00	−	9415.43i	−0.177462	−	0.307373i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17976.0	−	31135.3i	−0.583261	−	1.01024i	−0.995090	−	0.0989762i	$-0.968443\pi$
$983$	−17976.0	−	31135.3i	−0.583261	−	1.01024i	0.411829	−	0.911261i	$-0.364890\pi$
$984$		0			0
$985$	1867.50	−	3234.60i	0.0604096	−	0.104633i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−26880.0			−0.864241
$990$		0			0
$991$	14156.0			0.453764			0.226882	−	0.973922i	$-0.427147\pi$
$991$	14156.0			0.453764			0.226882	+	0.973922i	$0.427147\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1050.00	−	1818.65i	0.0334545	−	0.0579449i
$996$		0			0
$997$	25770.5	+	44635.8i	0.818616	+	1.41788i	0.906702	+	0.421771i	$0.138591\pi$
$997$	25770.5	+	44635.8i	0.818616	+	1.41788i	−0.0880865	+	0.996113i	$0.528075\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.e.c.109.1		2
3.2	odd	2		324.4.e.f.109.1		2
9.2	odd	6		324.4.e.f.217.1		2
9.4	even	3		324.4.a.b.1.1	yes	1
9.5	odd	6		324.4.a.a.1.1	✓	1
9.7	even	3	inner	324.4.e.c.217.1		2
36.23	even	6		1296.4.a.d.1.1		1
36.31	odd	6		1296.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.4.a.a.1.1	✓	1	9.5	odd	6
324.4.a.b.1.1	yes	1	9.4	even	3
324.4.e.c.109.1		2	1.1	even	1	trivial
324.4.e.c.217.1		2	9.7	even	3	inner
324.4.e.f.109.1		2	3.2	odd	2
324.4.e.f.217.1		2	9.2	odd	6
1296.4.a.d.1.1		1	36.23	even	6
1296.4.a.e.1.1		1	36.31	odd	6

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 + 2.59808i) q^{5} +(2.00000 + 3.46410i) q^{7} +(12.0000 + 20.7846i) q^{11} +(12.5000 - 21.6506i) q^{13} -21.0000 q^{17} -52.0000 q^{19} +(-84.0000 + 145.492i) q^{23} +(58.0000 + 100.459i) q^{25} +(88.5000 + 153.286i) q^{29} +(62.0000 - 107.387i) q^{31} -12.0000 q^{35} -265.000 q^{37} +(-213.000 + 368.927i) q^{41} +(80.0000 + 138.564i) q^{43} +(270.000 + 467.654i) q^{47} +(163.500 - 283.190i) q^{49} -258.000 q^{53} -72.0000 q^{55} +(-264.000 + 457.261i) q^{59} +(252.500 + 437.343i) q^{61} +(37.5000 + 64.9519i) q^{65} +(122.000 - 211.310i) q^{67} +204.000 q^{71} -397.000 q^{73} +(-48.0000 + 83.1384i) q^{77} +(-100.000 - 173.205i) q^{79} +(270.000 + 467.654i) q^{83} +(31.5000 - 54.5596i) q^{85} -453.000 q^{89} +100.000 q^{91} +(78.0000 - 135.100i) q^{95} +(-145.000 - 251.147i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} + 4 q^{7} + 24 q^{11} + 25 q^{13} - 42 q^{17} - 104 q^{19} - 168 q^{23} + 116 q^{25} + 177 q^{29} + 124 q^{31} - 24 q^{35} - 530 q^{37} - 426 q^{41} + 160 q^{43} + 540 q^{47} + 327 q^{49}+ \cdots - 290 q^{97}+O(q^{100}) \) 2 * q - 3 * q^5 + 4 * q^7 + 24 * q^11 + 25 * q^13 - 42 * q^17 - 104 * q^19 - 168 * q^23 + 116 * q^25 + 177 * q^29 + 124 * q^31 - 24 * q^35 - 530 * q^37 - 426 * q^41 + 160 * q^43 + 540 * q^47 + 327 * q^49 - 516 * q^53 - 144 * q^55 - 528 * q^59 + 505 * q^61 + 75 * q^65 + 244 * q^67 + 408 * q^71 - 794 * q^73 - 96 * q^77 - 200 * q^79 + 540 * q^83 + 63 * q^85 - 906 * q^89 + 200 * q^91 + 156 * q^95 - 290 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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