LMFDB - Embedded newform 4004.2.e.b.3849.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(3849,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4004.3849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.e (of order $2$, degree $1$, 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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3849.14
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.b.3849.35

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.33626i q^{3} +2.57665i q^{5} +(-2.48110 - 0.918779i) q^{7} +1.21441 q^{9} +O(q^{10})$	$q-1.33626i q^{3} +2.57665i q^{5} +(-2.48110 - 0.918779i) q^{7} +1.21441 q^{9} +(3.23240 + 0.742713i) q^{11} -1.00000 q^{13} +3.44307 q^{15} -4.02710 q^{17} -2.61318 q^{19} +(-1.22773 + 3.31539i) q^{21} -3.29441 q^{23} -1.63912 q^{25} -5.63155i q^{27} +1.57419i q^{29} -6.91184i q^{31} +(0.992457 - 4.31932i) q^{33} +(2.36737 - 6.39292i) q^{35} +7.23876 q^{37} +1.33626i q^{39} +4.62693 q^{41} -3.28129i q^{43} +3.12911i q^{45} -11.0576i q^{47} +(5.31169 + 4.55916i) q^{49} +5.38125i q^{51} +10.3482 q^{53} +(-1.91371 + 8.32875i) q^{55} +3.49189i q^{57} +11.2281i q^{59} +11.5917 q^{61} +(-3.01307 - 1.11578i) q^{63} -2.57665i q^{65} -11.9180 q^{67} +4.40218i q^{69} +0.0198406 q^{71} +13.4667 q^{73} +2.19029i q^{75} +(-7.33750 - 4.81260i) q^{77} +8.42685i q^{79} -3.88197 q^{81} +5.30101 q^{83} -10.3764i q^{85} +2.10353 q^{87} +12.3588i q^{89} +(2.48110 + 0.918779i) q^{91} -9.23602 q^{93} -6.73326i q^{95} -10.1697i q^{97} +(3.92546 + 0.901958i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 4 q^{7} - 48 q^{9}+O(q^{10}) $ 48 * q + 4 * q^7 - 48 * q^9	$ 48 q + 4 q^{7} - 48 q^{9} + 2 q^{11} - 48 q^{13} + 8 q^{15} - 4 q^{17} - 10 q^{21} + 4 q^{23} - 44 q^{25} - 10 q^{33} + 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} + 12 q^{55} - 16 q^{61} - 16 q^{63} + 4 q^{67} + 16 q^{73} + 22 q^{77} + 64 q^{81} + 4 q^{83} + 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) $ 48 * q + 4 * q^7 - 48 * q^9 + 2 * q^11 - 48 * q^13 + 8 * q^15 - 4 * q^17 - 10 * q^21 + 4 * q^23 - 44 * q^25 - 10 * q^33 + 14 * q^35 - 16 * q^37 + 10 * q^49 - 8 * q^53 + 12 * q^55 - 16 * q^61 - 16 * q^63 + 4 * q^67 + 16 * q^73 + 22 * q^77 + 64 * q^81 + 4 * q^83 + 12 * q^87 - 4 * q^91 + 36 * q^93 - 40 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times$.

$n$	$365$	$925$	$2003$	$3433$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$		−	1.33626i		−	0.771490i	−0.922606	−	0.385745i	$-0.873945\pi$
$3$		−	1.33626i		−	0.771490i	0.922606	−	0.385745i	$-0.126055\pi$
$4$		0			0
$5$			2.57665i			1.15231i	0.817339	+	0.576156i	$0.195448\pi$
$5$			2.57665i			1.15231i	−0.817339	+	0.576156i	$0.804552\pi$
$6$		0			0
$7$	−2.48110	−	0.918779i	−0.937767	−	0.347266i
$7$	−2.48110	−	0.918779i	−0.937767	−	0.347266i
$8$		0			0
$9$	1.21441			0.404804
$10$		0			0
$11$	3.23240	+	0.742713i	0.974604	+	0.223936i
$11$	3.23240	+	0.742713i	0.974604	+	0.223936i
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	3.44307			0.888997
$16$		0			0
$17$	−4.02710			−0.976715			−0.488357	−	0.872644i	$-0.662404\pi$
$17$	−4.02710			−0.976715			−0.488357	+	0.872644i	$0.662404\pi$
$18$		0			0
$19$	−2.61318			−0.599505			−0.299753	−	0.954017i	$-0.596904\pi$
$19$	−2.61318			−0.599505			−0.299753	+	0.954017i	$0.596904\pi$
$20$		0			0
$21$	−1.22773	+	3.31539i	−0.267912	+	0.723477i
$22$		0			0
$23$	−3.29441			−0.686932			−0.343466	−	0.939165i	$-0.611601\pi$
$23$	−3.29441			−0.686932			−0.343466	+	0.939165i	$0.611601\pi$
$24$		0			0
$25$	−1.63912			−0.327825
$26$		0			0
$27$		−	5.63155i		−	1.08379i
$28$		0			0
$29$			1.57419i			0.292321i	0.989261	+	0.146160i	$0.0466915\pi$
$29$			1.57419i			0.292321i	−0.989261	+	0.146160i	$0.953308\pi$
$30$		0			0
$31$		−	6.91184i		−	1.24140i	−0.784047	−	0.620702i	$-0.786848\pi$
$31$		−	6.91184i		−	1.24140i	0.784047	−	0.620702i	$-0.213152\pi$
$32$		0			0
$33$	0.992457	−	4.31932i	0.172765	−	0.751897i
$34$		0			0
$35$	2.36737	−	6.39292i	0.400159	−	1.08060i
$36$		0			0
$37$	7.23876			1.19004			0.595022	−	0.803709i	$-0.297143\pi$
$37$	7.23876			1.19004			0.595022	+	0.803709i	$0.297143\pi$
$38$		0			0
$39$			1.33626i			0.213973i
$40$		0			0
$41$	4.62693			0.722606			0.361303	−	0.932449i	$-0.382332\pi$
$41$	4.62693			0.722606			0.361303	+	0.932449i	$0.382332\pi$
$42$		0			0
$43$		−	3.28129i		−	0.500391i	−0.968195	−	0.250196i	$-0.919505\pi$
$43$		−	3.28129i		−	0.500391i	0.968195	−	0.250196i	$-0.0804949\pi$
$44$		0			0
$45$			3.12911i			0.466460i
$46$		0			0
$47$		−	11.0576i		−	1.61292i	−0.591290	−	0.806459i	$-0.701381\pi$
$47$		−	11.0576i		−	1.61292i	0.591290	−	0.806459i	$-0.298619\pi$
$48$		0			0
$49$	5.31169	+	4.55916i	0.758813	+	0.651309i
$50$		0			0
$51$			5.38125i			0.753525i
$52$		0			0
$53$	10.3482			1.42143			0.710714	−	0.703481i	$-0.248372\pi$
$53$	10.3482			1.42143			0.710714	+	0.703481i	$0.248372\pi$
$54$		0			0
$55$	−1.91371	+	8.32875i	−0.258045	+	1.12305i
$56$		0			0
$57$			3.49189i			0.462512i
$58$		0			0
$59$			11.2281i			1.46178i	0.682496	+	0.730889i	$0.260894\pi$
$59$			11.2281i			1.46178i	−0.682496	+	0.730889i	$0.739106\pi$
$60$		0			0
$61$	11.5917			1.48417			0.742084	−	0.670307i	$-0.233838\pi$
$61$	11.5917			1.48417			0.742084	+	0.670307i	$0.233838\pi$
$62$		0			0
$63$	−3.01307	−	1.11578i	−0.379611	−	0.140574i
$64$		0			0
$65$		−	2.57665i		−	0.319594i
$66$		0			0
$67$	−11.9180			−1.45602			−0.728009	−	0.685567i	$-0.759554\pi$
$67$	−11.9180			−1.45602			−0.728009	+	0.685567i	$0.759554\pi$
$68$		0			0
$69$			4.40218i			0.529961i
$70$		0			0
$71$	0.0198406			0.00235465			0.00117732	−	0.999999i	$-0.499625\pi$
$71$	0.0198406			0.00235465			0.00117732	+	0.999999i	$0.499625\pi$
$72$		0			0
$73$	13.4667			1.57616			0.788079	−	0.615575i	$-0.211076\pi$
$73$	13.4667			1.57616			0.788079	+	0.615575i	$0.211076\pi$
$74$		0			0
$75$			2.19029i			0.252913i
$76$		0			0
$77$	−7.33750	−	4.81260i	−0.836186	−	0.548447i
$78$		0			0
$79$			8.42685i			0.948095i	0.880499	+	0.474047i	$0.157207\pi$
$79$			8.42685i			0.948095i	−0.880499	+	0.474047i	$0.842793\pi$
$80$		0			0
$81$	−3.88197			−0.431330
$82$		0			0
$83$	5.30101			0.581861			0.290931	−	0.956744i	$-0.406035\pi$
$83$	5.30101			0.581861			0.290931	+	0.956744i	$0.406035\pi$
$84$		0			0
$85$		−	10.3764i		−	1.12548i
$86$		0			0
$87$	2.10353			0.225522
$88$		0			0
$89$			12.3588i			1.31003i	0.755617	+	0.655014i	$0.227337\pi$
$89$			12.3588i			1.31003i	−0.755617	+	0.655014i	$0.772663\pi$
$90$		0			0
$91$	2.48110	+	0.918779i	0.260090	+	0.0963142i
$92$		0			0
$93$	−9.23602			−0.957730
$94$		0			0
$95$		−	6.73326i		−	0.690818i
$96$		0			0
$97$		−	10.1697i		−	1.03258i	−0.856414	−	0.516289i	$-0.827313\pi$
$97$		−	10.1697i		−	1.03258i	0.856414	−	0.516289i	$-0.172687\pi$
$98$		0			0
$99$	3.92546	+	0.901958i	0.394523	+	0.0906502i
$100$		0			0
$101$	−17.4634			−1.73767			−0.868835	−	0.495101i	$-0.835131\pi$
$101$	−17.4634			−1.73767			−0.868835	+	0.495101i	$0.835131\pi$
$102$		0			0
$103$		−	1.68390i		−	0.165920i	−0.996553	−	0.0829600i	$-0.973563\pi$
$103$		−	1.68390i		−	0.165920i	0.996553	−	0.0829600i	$-0.0264374\pi$
$104$		0			0
$105$	−8.54260	−	3.16342i	−0.833672	−	0.308718i
$106$		0			0
$107$			11.5677i			1.11829i	0.829070	+	0.559144i	$0.188870\pi$
$107$			11.5677i			1.11829i	−0.829070	+	0.559144i	$0.811130\pi$
$108$		0			0
$109$		−	11.9686i		−	1.14638i	−0.819422	−	0.573191i	$-0.805705\pi$
$109$		−	11.9686i		−	1.14638i	0.819422	−	0.573191i	$-0.194295\pi$
$110$		0			0
$111$		−	9.67286i		−	0.918107i
$112$		0			0
$113$	12.4765			1.17369			0.586844	−	0.809700i	$-0.300370\pi$
$113$	12.4765			1.17369			0.586844	+	0.809700i	$0.300370\pi$
$114$		0			0
$115$		−	8.48854i		−	0.791560i
$116$		0			0
$117$	−1.21441			−0.112272
$118$		0			0
$119$	9.99163	+	3.70001i	0.915931	+	0.339180i
$120$		0			0
$121$	9.89676	+	4.80148i	0.899705	+	0.436498i
$122$		0			0
$123$		−	6.18278i		−	0.557483i
$124$		0			0
$125$			8.65980i			0.774556i
$126$		0			0
$127$		−	10.1108i		−	0.897188i	−0.893736	−	0.448594i	$-0.851925\pi$
$127$		−	10.1108i		−	0.897188i	0.893736	−	0.448594i	$-0.148075\pi$
$128$		0			0
$129$	−4.38465			−0.386047
$130$		0			0
$131$	5.16524			0.451289			0.225644	−	0.974210i	$-0.427551\pi$
$131$	5.16524			0.451289			0.225644	+	0.974210i	$0.427551\pi$
$132$		0			0
$133$	6.48356	+	2.40094i	0.562196	+	0.208188i
$134$		0			0
$135$	14.5105			1.24887
$136$		0			0
$137$	2.31951			0.198169			0.0990845	−	0.995079i	$-0.468409\pi$
$137$	2.31951			0.198169			0.0990845	+	0.995079i	$0.468409\pi$
$138$		0			0
$139$	9.97595			0.846149			0.423075	−	0.906095i	$-0.360951\pi$
$139$	9.97595			0.846149			0.423075	+	0.906095i	$0.360951\pi$
$140$		0			0
$141$	−14.7758			−1.24435
$142$		0			0
$143$	−3.23240	−	0.742713i	−0.270306	−	0.0621087i
$144$		0			0
$145$	−4.05615			−0.336845
$146$		0			0
$147$	6.09222	−	7.09780i	0.502478	−	0.585416i
$148$		0			0
$149$			17.0323i			1.39534i	0.716417	+	0.697672i	$0.245781\pi$
$149$			17.0323i			1.39534i	−0.716417	+	0.697672i	$0.754219\pi$
$150$		0			0
$151$		−	17.9351i		−	1.45954i	−0.683694	−	0.729769i	$-0.739628\pi$
$151$		−	17.9351i		−	1.45954i	0.683694	−	0.729769i	$-0.260372\pi$
$152$		0			0
$153$	−4.89055			−0.395378
$154$		0			0
$155$	17.8094			1.43049
$156$		0			0
$157$		−	0.100223i		−	0.00799868i	−0.999992	−	0.00399934i	$-0.998727\pi$
$157$		−	0.100223i		−	0.00799868i	0.999992	−	0.00399934i	$-0.00127303\pi$
$158$		0			0
$159$		−	13.8278i		−	1.09662i
$160$		0			0
$161$	8.17375	+	3.02683i	0.644182	+	0.238548i
$162$		0			0
$163$	10.6079			0.830871			0.415436	−	0.909623i	$-0.363629\pi$
$163$	10.6079			0.830871			0.415436	+	0.909623i	$0.363629\pi$
$164$		0			0
$165$	11.1294	+	2.55721i	0.866420	+	0.199079i
$166$		0			0
$167$	18.1812			1.40690			0.703452	−	0.710743i	$-0.251641\pi$
$167$	18.1812			1.40690			0.703452	+	0.710743i	$0.251641\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−3.17348			−0.242682
$172$		0			0
$173$	25.0488			1.90443			0.952213	−	0.305436i	$-0.0988022\pi$
$173$	25.0488			1.90443			0.952213	+	0.305436i	$0.0988022\pi$
$174$		0			0
$175$	4.06683	+	1.50599i	0.307423	+	0.113842i
$176$		0			0
$177$	15.0037			1.12775
$178$		0			0
$179$	13.5158			1.01022			0.505110	−	0.863055i	$-0.331452\pi$
$179$	13.5158			1.01022			0.505110	+	0.863055i	$0.331452\pi$
$180$		0			0
$181$			9.66163i			0.718143i	0.933310	+	0.359071i	$0.116907\pi$
$181$			9.66163i			0.718143i	−0.933310	+	0.359071i	$0.883093\pi$
$182$		0			0
$183$		−	15.4895i		−	1.14502i
$184$		0			0
$185$			18.6517i			1.37130i
$186$		0			0
$187$	−13.0172	−	2.99098i	−0.951910	−	0.218722i
$188$		0			0
$189$	−5.17414	+	13.9724i	−0.376364	+	1.01634i
$190$		0			0
$191$	−1.10813			−0.0801818			−0.0400909	−	0.999196i	$-0.512765\pi$
$191$	−1.10813			−0.0801818			−0.0400909	+	0.999196i	$0.512765\pi$
$192$		0			0
$193$		−	25.7712i		−	1.85505i	−0.373757	−	0.927527i	$-0.621931\pi$
$193$		−	25.7712i		−	1.85505i	0.373757	−	0.927527i	$-0.378069\pi$
$194$		0			0
$195$	−3.44307			−0.246564
$196$		0			0
$197$		−	13.5745i		−	0.967142i	−0.875305	−	0.483571i	$-0.839340\pi$
$197$		−	13.5745i		−	0.967142i	0.875305	−	0.483571i	$-0.160660\pi$
$198$		0			0
$199$		−	9.84385i		−	0.697812i	−0.937158	−	0.348906i	$-0.886553\pi$
$199$		−	9.84385i		−	0.697812i	0.937158	−	0.348906i	$-0.113447\pi$
$200$		0			0
$201$			15.9256i			1.12330i
$202$		0			0
$203$	1.44634	−	3.90573i	0.101513	−	0.274129i
$204$		0			0
$205$			11.9220i			0.832668i
$206$		0			0
$207$	−4.00077			−0.278072
$208$		0			0
$209$	−8.44684	−	1.94084i	−0.584280	−	0.134251i
$210$		0			0
$211$			21.8995i			1.50763i	0.657088	+	0.753814i	$0.271788\pi$
$211$			21.8995i			1.50763i	−0.657088	+	0.753814i	$0.728212\pi$
$212$		0			0
$213$		−	0.0265122i		−	0.00181659i
$214$		0			0
$215$	8.45472			0.576607
$216$		0			0
$217$	−6.35046	+	17.1490i	−0.431097	+	1.16415i
$218$		0			0
$219$		−	17.9950i		−	1.21599i
$220$		0			0
$221$	4.02710			0.270892
$222$		0			0
$223$		−	23.0567i		−	1.54399i	−0.635629	−	0.771994i	$-0.719259\pi$
$223$		−	23.0567i		−	1.54399i	0.635629	−	0.771994i	$-0.280741\pi$
$224$		0			0
$225$	−1.99057			−0.132705
$226$		0			0
$227$	−8.49411			−0.563774			−0.281887	−	0.959448i	$-0.590960\pi$
$227$	−8.49411			−0.563774			−0.281887	+	0.959448i	$0.590960\pi$
$228$		0			0
$229$		−	17.7740i		−	1.17454i	−0.809392	−	0.587269i	$-0.800203\pi$
$229$		−	17.7740i		−	1.17454i	0.809392	−	0.587269i	$-0.199797\pi$
$230$		0			0
$231$	−6.43088	+	9.80480i	−0.423121	+	0.645109i
$232$		0			0
$233$		−	20.7234i		−	1.35763i	−0.734308	−	0.678816i	$-0.762493\pi$
$233$		−	20.7234i		−	1.35763i	0.734308	−	0.678816i	$-0.237507\pi$
$234$		0			0
$235$	28.4916			1.85859
$236$		0			0
$237$	11.2605			0.731445
$238$		0			0
$239$		−	18.3850i		−	1.18922i	−0.804013	−	0.594612i	$-0.797306\pi$
$239$		−	18.3850i		−	1.18922i	0.804013	−	0.594612i	$-0.202694\pi$
$240$		0			0
$241$	20.9388			1.34878			0.674392	−	0.738374i	$-0.264406\pi$
$241$	20.9388			1.34878			0.674392	+	0.738374i	$0.264406\pi$
$242$		0			0
$243$		−	11.7073i		−	0.751025i
$244$		0			0
$245$	−11.7474	+	13.6864i	−0.750511	+	0.874390i
$246$		0			0
$247$	2.61318			0.166273
$248$		0			0
$249$		−	7.08352i		−	0.448900i
$250$		0			0
$251$		−	3.52520i		−	0.222509i	−0.993792	−	0.111254i	$-0.964513\pi$
$251$		−	3.52520i		−	0.222509i	0.993792	−	0.111254i	$-0.0354868\pi$
$252$		0			0
$253$	−10.6488	−	2.44680i	−0.669486	−	0.153829i
$254$		0			0
$255$	−13.8656			−0.868297
$256$		0			0
$257$			20.3157i			1.26726i	0.773638	+	0.633628i	$0.218435\pi$
$257$			20.3157i			1.26726i	−0.773638	+	0.633628i	$0.781565\pi$
$258$		0			0
$259$	−17.9601	−	6.65082i	−1.11598	−	0.413262i
$260$		0			0
$261$			1.91172i			0.118332i
$262$		0			0
$263$			19.3129i			1.19088i	0.803399	+	0.595442i	$0.203023\pi$
$263$			19.3129i			1.19088i	−0.803399	+	0.595442i	$0.796977\pi$
$264$		0			0
$265$			26.6636i			1.63793i
$266$		0			0
$267$	16.5145			1.01067
$268$		0			0
$269$		−	19.4601i		−	1.18650i	−0.805017	−	0.593252i	$-0.797844\pi$
$269$		−	19.4601i		−	1.18650i	0.805017	−	0.593252i	$-0.202156\pi$
$270$		0			0
$271$	−17.9888			−1.09274			−0.546370	−	0.837544i	$-0.683991\pi$
$271$	−17.9888			−1.09274			−0.546370	+	0.837544i	$0.683991\pi$
$272$		0			0
$273$	1.22773	−	3.31539i	0.0743054	−	0.200657i
$274$		0			0
$275$	−5.29830	−	1.21740i	−0.319499	−	0.0734118i
$276$		0			0
$277$			16.8131i			1.01020i	0.863060	+	0.505101i	$0.168545\pi$
$277$			16.8131i			1.01020i	−0.863060	+	0.505101i	$0.831455\pi$
$278$		0			0
$279$		−	8.39382i		−	0.502525i
$280$		0			0
$281$		−	7.34222i		−	0.438000i	−0.975725	−	0.219000i	$-0.929721\pi$
$281$		−	7.34222i		−	0.438000i	0.975725	−	0.219000i	$-0.0702794\pi$
$282$		0			0
$283$	11.3934			0.677269			0.338634	−	0.940918i	$-0.390035\pi$
$283$	11.3934			0.677269			0.338634	+	0.940918i	$0.390035\pi$
$284$		0			0
$285$	−8.99738			−0.532959
$286$		0			0
$287$	−11.4799	−	4.25113i	−0.677636	−	0.250936i
$288$		0			0
$289$	−0.782476			−0.0460280
$290$		0			0
$291$	−13.5894			−0.796624
$292$		0			0
$293$	−32.0078			−1.86991			−0.934957	−	0.354762i	$-0.884562\pi$
$293$	−32.0078			−1.86991			−0.934957	+	0.354762i	$0.884562\pi$
$294$		0			0
$295$	−28.9310			−1.68443
$296$		0			0
$297$	4.18262	−	18.2034i	0.242700	−	1.05627i
$298$		0			0
$299$	3.29441			0.190521
$300$		0			0
$301$	−3.01478	+	8.14119i	−0.173769	+	0.469250i
$302$		0			0
$303$			23.3356i			1.34060i
$304$		0			0
$305$			29.8678i			1.71023i
$306$		0			0
$307$	11.3661			0.648700			0.324350	−	0.945937i	$-0.394854\pi$
$307$	11.3661			0.648700			0.324350	+	0.945937i	$0.394854\pi$
$308$		0			0
$309$	−2.25013			−0.128006
$310$		0			0
$311$		−	4.11245i		−	0.233196i	−0.993179	−	0.116598i	$-0.962801\pi$
$311$		−	4.11245i		−	0.233196i	0.993179	−	0.116598i	$-0.0371988\pi$
$312$		0			0
$313$		−	25.4969i		−	1.44117i	−0.693367	−	0.720585i	$-0.743873\pi$
$313$		−	25.4969i		−	1.44117i	0.693367	−	0.720585i	$-0.256127\pi$
$314$		0			0
$315$	2.87496	−	7.76363i	0.161986	−	0.437431i
$316$		0			0
$317$	−0.101242			−0.00568631			−0.00284315	−	0.999996i	$-0.500905\pi$
$317$	−0.101242			−0.00568631			−0.00284315	+	0.999996i	$0.500905\pi$
$318$		0			0
$319$	−1.16917	+	5.08842i	−0.0654612	+	0.284897i
$320$		0			0
$321$	15.4574			0.862748
$322$		0			0
$323$	10.5235			0.585546
$324$		0			0
$325$	1.63912			0.0909222
$326$		0			0
$327$	−15.9931			−0.884421
$328$		0			0
$329$	−10.1595	+	27.4350i	−0.560111	+	1.51254i
$330$		0			0
$331$	18.5214			1.01803			0.509013	−	0.860759i	$-0.330010\pi$
$331$	18.5214			1.01803			0.509013	+	0.860759i	$0.330010\pi$
$332$		0			0
$333$	8.79083			0.481734
$334$		0			0
$335$		−	30.7086i		−	1.67779i
$336$		0			0
$337$		−	30.7440i		−	1.67473i	−0.546642	−	0.837367i	$-0.684094\pi$
$337$		−	30.7440i		−	1.67473i	0.546642	−	0.837367i	$-0.315906\pi$
$338$		0			0
$339$		−	16.6718i		−	0.905488i
$340$		0			0
$341$	5.13351	−	22.3418i	0.277995	−	1.20988i
$342$		0			0
$343$	−8.98997	−	16.1920i	−0.485412	−	0.874285i
$344$		0			0
$345$	−11.3429			−0.610680
$346$		0			0
$347$		−	29.7844i		−	1.59891i	−0.600726	−	0.799455i	$-0.705122\pi$
$347$		−	29.7844i		−	1.59891i	0.600726	−	0.799455i	$-0.294878\pi$
$348$		0			0
$349$	−16.7055			−0.894227			−0.447113	−	0.894477i	$-0.647548\pi$
$349$	−16.7055			−0.894227			−0.447113	+	0.894477i	$0.647548\pi$
$350$		0			0
$351$			5.63155i			0.300590i
$352$		0			0
$353$			14.1599i			0.753658i	0.926283	+	0.376829i	$0.122985\pi$
$353$			14.1599i			0.753658i	−0.926283	+	0.376829i	$0.877015\pi$
$354$		0			0
$355$			0.0511223i			0.00271329i
$356$		0			0
$357$	4.94418	−	13.3514i	0.261674	−	0.706631i
$358$		0			0
$359$			27.4770i			1.45018i	0.688654	+	0.725090i	$0.258202\pi$
$359$			27.4770i			1.45018i	−0.688654	+	0.725090i	$0.741798\pi$
$360$		0			0
$361$	−12.1713			−0.640593
$362$		0			0
$363$	6.41602	−	13.2246i	0.336754	−	0.694113i
$364$		0			0
$365$			34.6989i			1.81623i
$366$		0			0
$367$			22.5441i			1.17679i	0.808573	+	0.588396i	$0.200240\pi$
$367$			22.5441i			1.17679i	−0.808573	+	0.588396i	$0.799760\pi$
$368$		0			0
$369$	5.61900			0.292513
$370$		0			0
$371$	−25.6748	−	9.50767i	−1.33297	−	0.493613i
$372$		0			0
$373$			15.9419i			0.825438i	0.910858	+	0.412719i	$0.135421\pi$
$373$			15.9419i			0.825438i	−0.910858	+	0.412719i	$0.864579\pi$
$374$		0			0
$375$	11.5717			0.597562
$376$		0			0
$377$		−	1.57419i		−	0.0810752i
$378$		0			0
$379$	32.4202			1.66531			0.832656	−	0.553790i	$-0.186819\pi$
$379$	32.4202			1.66531			0.832656	+	0.553790i	$0.186819\pi$
$380$		0			0
$381$	−13.5106			−0.692171
$382$		0			0
$383$			25.4553i			1.30071i	0.759632	+	0.650353i	$0.225379\pi$
$383$			25.4553i			1.30071i	−0.759632	+	0.650353i	$0.774621\pi$
$384$		0			0
$385$	12.4004	−	18.9062i	0.631982	−	0.963547i
$386$		0			0
$387$		−	3.98483i		−	0.202560i
$388$		0			0
$389$	−8.02472			−0.406869			−0.203435	−	0.979089i	$-0.565210\pi$
$389$	−8.02472			−0.406869			−0.203435	+	0.979089i	$0.565210\pi$
$390$		0			0
$391$	13.2669			0.670936
$392$		0			0
$393$		−	6.90209i		−	0.348165i
$394$		0			0
$395$	−21.7130			−1.09250
$396$		0			0
$397$			22.1685i			1.11260i	0.830980	+	0.556302i	$0.187780\pi$
$397$			22.1685i			1.11260i	−0.830980	+	0.556302i	$0.812220\pi$
$398$		0			0
$399$	3.20827	−	8.66372i	0.160615	−	0.433729i
$400$		0			0
$401$	−12.9439			−0.646385			−0.323193	−	0.946333i	$-0.604756\pi$
$401$	−12.9439			−0.646385			−0.323193	+	0.946333i	$0.604756\pi$
$402$		0			0
$403$			6.91184i			0.344303i
$404$		0			0
$405$		−	10.0025i		−	0.497027i
$406$		0			0
$407$	23.3985	+	5.37632i	1.15982	+	0.266494i
$408$		0			0
$409$	−7.27023			−0.359490			−0.179745	−	0.983713i	$-0.557527\pi$
$409$	−7.27023			−0.359490			−0.179745	+	0.983713i	$0.557527\pi$
$410$		0			0
$411$		−	3.09947i		−	0.152885i
$412$		0			0
$413$	10.3162	−	27.8581i	0.507626	−	1.37081i
$414$		0			0
$415$			13.6588i			0.670486i
$416$		0			0
$417$		−	13.3305i		−	0.652795i
$418$		0			0
$419$			13.4085i			0.655048i	0.944843	+	0.327524i	$0.106214\pi$
$419$			13.4085i			0.655048i	−0.944843	+	0.327524i	$0.893786\pi$
$420$		0			0
$421$	−31.7729			−1.54852			−0.774259	−	0.632869i	$-0.781877\pi$
$421$	−31.7729			−1.54852			−0.774259	+	0.632869i	$0.781877\pi$
$422$		0			0
$423$		−	13.4285i		−	0.652915i
$424$		0			0
$425$	6.60091			0.320191
$426$		0			0
$427$	−28.7602	−	10.6502i	−1.39180	−	0.515400i
$428$		0			0
$429$	−0.992457	+	4.31932i	−0.0479163	+	0.208539i
$430$		0			0
$431$		−	21.1211i		−	1.01737i	−0.860953	−	0.508684i	$-0.830132\pi$
$431$		−	21.1211i		−	1.01737i	0.860953	−	0.508684i	$-0.169868\pi$
$432$		0			0
$433$			12.9161i			0.620710i	0.950621	+	0.310355i	$0.100448\pi$
$433$			12.9161i			0.620710i	−0.950621	+	0.310355i	$0.899552\pi$
$434$		0			0
$435$			5.42007i			0.259872i
$436$		0			0
$437$	8.60889			0.411819
$438$		0			0
$439$	7.24208			0.345645			0.172823	−	0.984953i	$-0.444711\pi$
$439$	7.24208			0.345645			0.172823	+	0.984953i	$0.444711\pi$
$440$		0			0
$441$	6.45058	+	5.53669i	0.307170	+	0.263652i
$442$		0			0
$443$	37.1609			1.76557			0.882783	−	0.469781i	$-0.155667\pi$
$443$	37.1609			1.76557			0.882783	+	0.469781i	$0.155667\pi$
$444$		0			0
$445$	−31.8442			−1.50956
$446$		0			0
$447$	22.7596			1.07649
$448$		0			0
$449$	−34.3981			−1.62335			−0.811674	−	0.584110i	$-0.801444\pi$
$449$	−34.3981			−1.62335			−0.811674	+	0.584110i	$0.801444\pi$
$450$		0			0
$451$	14.9561	+	3.43648i	0.704254	+	0.161818i
$452$		0			0
$453$	−23.9659			−1.12602
$454$		0			0
$455$	−2.36737	+	6.39292i	−0.110984	+	0.299705i
$456$		0			0
$457$			36.8369i			1.72316i	0.507625	+	0.861578i	$0.330524\pi$
$457$			36.8369i			1.72316i	−0.507625	+	0.861578i	$0.669476\pi$
$458$		0			0
$459$			22.6788i			1.05856i
$460$		0			0
$461$	28.5506			1.32973			0.664867	−	0.746962i	$-0.268488\pi$
$461$	28.5506			1.32973			0.664867	+	0.746962i	$0.268488\pi$
$462$		0			0
$463$	−7.77127			−0.361161			−0.180581	−	0.983560i	$-0.557798\pi$
$463$	−7.77127			−0.361161			−0.180581	+	0.983560i	$0.557798\pi$
$464$		0			0
$465$		−	23.7980i		−	1.10360i
$466$		0			0
$467$		−	12.9533i		−	0.599409i	−0.954032	−	0.299704i	$-0.903112\pi$
$467$		−	12.9533i		−	0.599409i	0.954032	−	0.299704i	$-0.0968880\pi$
$468$		0			0
$469$	29.5698	+	10.9500i	1.36541	+	0.505625i
$470$		0			0
$471$	−0.133924			−0.00617090
$472$		0			0
$473$	2.43705	−	10.6064i	0.112056	−	0.487683i
$474$		0			0
$475$	4.28333			0.196533
$476$		0			0
$477$	12.5669			0.575400
$478$		0			0
$479$	−22.6696			−1.03580			−0.517901	−	0.855440i	$-0.673287\pi$
$479$	−22.6696			−1.03580			−0.517901	+	0.855440i	$0.673287\pi$
$480$		0			0
$481$	−7.23876			−0.330059
$482$		0			0
$483$	4.04463	−	10.9222i	0.184037	−	0.496979i
$484$		0			0
$485$	26.2038			1.18985
$486$		0			0
$487$	−20.1382			−0.912547			−0.456273	−	0.889840i	$-0.650816\pi$
$487$	−20.1382			−0.912547			−0.456273	+	0.889840i	$0.650816\pi$
$488$		0			0
$489$		−	14.1748i		−	0.641009i
$490$		0			0
$491$		−	7.62960i		−	0.344319i	−0.985069	−	0.172159i	$-0.944926\pi$
$491$		−	7.62960i		−	0.344319i	0.985069	−	0.172159i	$-0.0550744\pi$
$492$		0			0
$493$		−	6.33944i		−	0.285514i
$494$		0			0
$495$	−2.32403	+	10.1145i	−0.104457	+	0.454614i
$496$		0			0
$497$	−0.0492265	−	0.0182291i	−0.00220811	−	0.000817689i
$498$		0			0
$499$	−40.1012			−1.79518			−0.897588	−	0.440835i	$-0.854682\pi$
$499$	−40.1012			−1.79518			−0.897588	+	0.440835i	$0.854682\pi$
$500$		0			0
$501$		−	24.2948i		−	1.08541i
$502$		0			0
$503$	4.09763			0.182704			0.0913521	−	0.995819i	$-0.470881\pi$
$503$	4.09763			0.182704			0.0913521	+	0.995819i	$0.470881\pi$
$504$		0			0
$505$		−	44.9970i		−	2.00234i
$506$		0			0
$507$		−	1.33626i		−	0.0593454i
$508$		0			0
$509$		−	22.0166i		−	0.975869i	−0.872880	−	0.487935i	$-0.837750\pi$
$509$		−	22.0166i		−	0.975869i	0.872880	−	0.487935i	$-0.162250\pi$
$510$		0			0
$511$	−33.4122	−	12.3729i	−1.47807	−	0.547345i
$512$		0			0
$513$			14.7163i			0.649739i
$514$		0			0
$515$	4.33883			0.191192
$516$		0			0
$517$	8.21263	−	35.7426i	0.361191	−	1.57196i
$518$		0			0
$519$		−	33.4717i		−	1.46924i
$520$		0			0
$521$			37.7543i			1.65405i	0.562168	+	0.827023i	$0.309967\pi$
$521$			37.7543i			1.65405i	−0.562168	+	0.827023i	$0.690033\pi$
$522$		0			0
$523$	19.6864			0.860828			0.430414	−	0.902632i	$-0.358368\pi$
$523$	19.6864			0.860828			0.430414	+	0.902632i	$0.358368\pi$
$524$		0			0
$525$	2.01240	−	5.43433i	0.0878282	−	0.237174i
$526$		0			0
$527$			27.8347i			1.21250i
$528$		0			0
$529$	−12.1469			−0.528125
$530$		0			0
$531$			13.6356i			0.591733i
$532$		0			0
$533$	−4.62693			−0.200415
$534$		0			0
$535$	−29.8058			−1.28862
$536$		0			0
$537$		−	18.0606i		−	0.779374i
$538$		0			0
$539$	13.7833	+	18.6821i	0.593690	+	0.804694i
$540$		0			0
$541$		−	13.1582i		−	0.565716i	−0.959162	−	0.282858i	$-0.908718\pi$
$541$		−	13.1582i		−	0.565716i	0.959162	−	0.282858i	$-0.0912824\pi$
$542$		0			0
$543$	12.9104			0.554040
$544$		0			0
$545$	30.8388			1.32099
$546$		0			0
$547$			22.6649i			0.969081i	0.874769	+	0.484540i	$0.161013\pi$
$547$			22.6649i			0.969081i	−0.874769	+	0.484540i	$0.838987\pi$
$548$		0			0
$549$	14.0771			0.600796
$550$		0			0
$551$		−	4.11366i		−	0.175248i
$552$		0			0
$553$	7.74241	−	20.9078i	0.329241	−	0.889092i
$554$		0			0
$555$	24.9236			1.05795
$556$		0			0
$557$		−	4.52008i		−	0.191522i	−0.995404	−	0.0957610i	$-0.969472\pi$
$557$		−	4.52008i		−	0.191522i	0.995404	−	0.0957610i	$-0.0305285\pi$
$558$		0			0
$559$			3.28129i			0.138784i
$560$		0			0
$561$	−3.99672	+	17.3943i	−0.168742	+	0.734389i
$562$		0			0
$563$	−22.6601			−0.955008			−0.477504	−	0.878629i	$-0.658458\pi$
$563$	−22.6601			−0.955008			−0.477504	+	0.878629i	$0.658458\pi$
$564$		0			0
$565$			32.1475i			1.35246i
$566$		0			0
$567$	9.63155	+	3.56667i	0.404487	+	0.149786i
$568$		0			0
$569$		−	38.2735i		−	1.60451i	−0.596984	−	0.802253i	$-0.703634\pi$
$569$		−	38.2735i		−	1.60451i	0.596984	−	0.802253i	$-0.296366\pi$
$570$		0			0
$571$		−	26.1638i		−	1.09492i	−0.836832	−	0.547460i	$-0.815595\pi$
$571$		−	26.1638i		−	1.09492i	0.836832	−	0.547460i	$-0.184405\pi$
$572$		0			0
$573$			1.48076i			0.0618594i
$574$		0			0
$575$	5.39994			0.225193
$576$		0			0
$577$		−	4.08141i		−	0.169911i	−0.996385	−	0.0849557i	$-0.972925\pi$
$577$		−	4.08141i		−	0.169911i	0.996385	−	0.0849557i	$-0.0270749\pi$
$578$		0			0
$579$	−34.4370			−1.43115
$580$		0			0
$581$	−13.1523	−	4.87046i	−0.545650	−	0.202061i
$582$		0			0
$583$	33.4493	+	7.68571i	1.38533	+	0.318309i
$584$		0			0
$585$		−	3.12911i		−	0.129373i
$586$		0			0
$587$			24.3864i			1.00654i	0.864131	+	0.503268i	$0.167869\pi$
$587$			24.3864i			1.00654i	−0.864131	+	0.503268i	$0.832131\pi$
$588$		0			0
$589$			18.0619i			0.744228i
$590$		0			0
$591$	−18.1390			−0.746140
$592$		0			0
$593$	−5.83213			−0.239497			−0.119748	−	0.992804i	$-0.538209\pi$
$593$	−5.83213			−0.239497			−0.119748	+	0.992804i	$0.538209\pi$
$594$		0			0
$595$	−9.53364	+	25.7449i	−0.390841	+	1.05544i
$596$		0			0
$597$	−13.1539			−0.538355
$598$		0			0
$599$	−2.93724			−0.120012			−0.0600062	−	0.998198i	$-0.519112\pi$
$599$	−2.93724			−0.120012			−0.0600062	+	0.998198i	$0.519112\pi$
$600$		0			0
$601$	27.0411			1.10303			0.551515	−	0.834165i	$-0.314050\pi$
$601$	27.0411			1.10303			0.551515	+	0.834165i	$0.314050\pi$
$602$		0			0
$603$	−14.4734			−0.589402
$604$		0			0
$605$	−12.3717	+	25.5005i	−0.502983	+	1.03674i
$606$		0			0
$607$	15.9589			0.647751			0.323876	−	0.946100i	$-0.395014\pi$
$607$	15.9589			0.647751			0.323876	+	0.946100i	$0.395014\pi$
$608$		0			0
$609$	−5.21907	−	1.93268i	−0.211487	−	0.0783162i
$610$		0			0
$611$			11.0576i			0.447343i
$612$		0			0
$613$			20.4180i			0.824674i	0.911031	+	0.412337i	$0.135287\pi$
$613$			20.4180i			0.824674i	−0.911031	+	0.412337i	$0.864713\pi$
$614$		0			0
$615$	15.9309			0.642395
$616$		0			0
$617$	11.1925			0.450592			0.225296	−	0.974290i	$-0.427665\pi$
$617$	11.1925			0.450592			0.225296	+	0.974290i	$0.427665\pi$
$618$		0			0
$619$			18.5158i			0.744212i	0.928190	+	0.372106i	$0.121364\pi$
$619$			18.5158i			0.744212i	−0.928190	+	0.372106i	$0.878636\pi$
$620$		0			0
$621$			18.5526i			0.744491i
$622$		0			0
$623$	11.3550	−	30.6633i	0.454928	−	1.22850i
$624$		0			0
$625$	−30.5089			−1.22036
$626$		0			0
$627$	−2.59347	+	11.2872i	−0.103573	+	0.450766i
$628$		0			0
$629$	−29.1512			−1.16233
$630$		0			0
$631$	−17.2349			−0.686110			−0.343055	−	0.939315i	$-0.611462\pi$
$631$	−17.2349			−0.686110			−0.343055	+	0.939315i	$0.611462\pi$
$632$		0			0
$633$	29.2635			1.16312
$634$		0			0
$635$	26.0520			1.03384
$636$		0			0
$637$	−5.31169	−	4.55916i	−0.210457	−	0.180640i
$638$		0			0
$639$	0.0240947			0.000953170
$640$		0			0
$641$	34.0424			1.34460			0.672298	−	0.740281i	$-0.265308\pi$
$641$	34.0424			1.34460			0.672298	+	0.740281i	$0.265308\pi$
$642$		0			0
$643$		−	9.51717i		−	0.375321i	−0.982234	−	0.187660i	$-0.939910\pi$
$643$		−	9.51717i		−	0.375321i	0.982234	−	0.187660i	$-0.0600904\pi$
$644$		0			0
$645$		−	11.2977i		−	0.444847i
$646$		0			0
$647$			28.7905i			1.13187i	0.824449	+	0.565936i	$0.191485\pi$
$647$			28.7905i			1.13187i	−0.824449	+	0.565936i	$0.808515\pi$
$648$		0			0
$649$	−8.33928	+	36.2938i	−0.327345	+	1.42465i
$650$		0			0
$651$	22.9155	+	8.48586i	0.898128	+	0.332587i
$652$		0			0
$653$	11.8219			0.462628			0.231314	−	0.972879i	$-0.425698\pi$
$653$	11.8219			0.462628			0.231314	+	0.972879i	$0.425698\pi$
$654$		0			0
$655$			13.3090i			0.520026i
$656$		0			0
$657$	16.3541			0.638034
$658$		0			0
$659$		−	23.3630i		−	0.910094i	−0.890467	−	0.455047i	$-0.849622\pi$
$659$		−	23.3630i		−	0.910094i	0.890467	−	0.455047i	$-0.150378\pi$
$660$		0			0
$661$			32.4345i			1.26155i	0.775964	+	0.630777i	$0.217264\pi$
$661$			32.4345i			1.26155i	−0.775964	+	0.630777i	$0.782736\pi$
$662$		0			0
$663$		−	5.38125i		−	0.208990i
$664$		0			0
$665$	−6.18637	+	16.7059i	−0.239897	+	0.647826i
$666$		0			0
$667$		−	5.18604i		−	0.200804i
$668$		0			0
$669$	−30.8097			−1.19117
$670$		0			0
$671$	37.4690	+	8.60932i	1.44648	+	0.332359i
$672$		0			0
$673$		−	8.33000i		−	0.321098i	−0.987028	−	0.160549i	$-0.948674\pi$
$673$		−	8.33000i		−	0.321098i	0.987028	−	0.160549i	$-0.0513264\pi$
$674$		0			0
$675$			9.23080i			0.355294i
$676$		0			0
$677$	−32.7192			−1.25750			−0.628751	−	0.777607i	$-0.716433\pi$
$677$	−32.7192			−1.25750			−0.628751	+	0.777607i	$0.716433\pi$
$678$		0			0
$679$	−9.34372	+	25.2321i	−0.358579	+	0.968318i
$680$		0			0
$681$			11.3503i			0.434946i
$682$		0			0
$683$	28.1115			1.07566			0.537828	−	0.843055i	$-0.319245\pi$
$683$	28.1115			1.07566			0.537828	+	0.843055i	$0.319245\pi$
$684$		0			0
$685$			5.97656i			0.228353i
$686$		0			0
$687$	−23.7506			−0.906144
$688$		0			0
$689$	−10.3482			−0.394233
$690$		0			0
$691$			33.5232i			1.27528i	0.770333	+	0.637641i	$0.220090\pi$
$691$			33.5232i			1.27528i	−0.770333	+	0.637641i	$0.779910\pi$
$692$		0			0
$693$	−8.91074	−	5.84447i	−0.338491	−	0.222013i
$694$		0			0
$695$			25.7045i			0.975028i
$696$		0			0
$697$	−18.6331			−0.705780
$698$		0			0
$699$	−27.6918			−1.04740
$700$		0			0
$701$			46.3603i			1.75101i	0.483213	+	0.875503i	$0.339470\pi$
$701$			46.3603i			1.75101i	−0.483213	+	0.875503i	$0.660530\pi$
$702$		0			0
$703$	−18.9162			−0.713438
$704$		0			0
$705$		−	38.0722i		−	1.43388i
$706$		0			0
$707$	43.3283	+	16.0450i	1.62953	+	0.603434i
$708$		0			0
$709$	−19.9661			−0.749844			−0.374922	−	0.927056i	$-0.622330\pi$
$709$	−19.9661			−0.749844			−0.374922	+	0.927056i	$0.622330\pi$
$710$		0			0
$711$			10.2337i			0.383792i
$712$		0			0
$713$			22.7704i			0.852760i
$714$		0			0
$715$	1.91371	−	8.32875i	0.0715687	−	0.311478i
$716$		0			0
$717$	−24.5671			−0.917474
$718$		0			0
$719$			18.9717i			0.707525i	0.935335	+	0.353762i	$0.115098\pi$
$719$			18.9717i			0.707525i	−0.935335	+	0.353762i	$0.884902\pi$
$720$		0			0
$721$	−1.54714	+	4.17793i	−0.0576183	+	0.155594i
$722$		0			0
$723$		−	27.9796i		−	1.04057i
$724$		0			0
$725$		−	2.58030i		−	0.0958299i
$726$		0			0
$727$		−	22.9890i		−	0.852614i	−0.904579	−	0.426307i	$-0.859814\pi$
$727$		−	22.9890i		−	0.852614i	0.904579	−	0.426307i	$-0.140186\pi$
$728$		0			0
$729$	−27.2899			−1.01074
$730$		0			0
$731$			13.2141i			0.488740i
$732$		0			0
$733$	20.7925			0.767987			0.383994	−	0.923336i	$-0.374548\pi$
$733$	20.7925			0.767987			0.383994	+	0.923336i	$0.374548\pi$
$734$		0			0
$735$	18.2885	+	15.6975i	0.674583	+	0.579012i
$736$		0			0
$737$	−38.5238	−	8.85167i	−1.41904	−	0.326055i
$738$		0			0
$739$			13.2156i			0.486144i	0.970008	+	0.243072i	$0.0781551\pi$
$739$			13.2156i			0.486144i	−0.970008	+	0.243072i	$0.921845\pi$
$740$		0			0
$741$		−	3.49189i		−	0.128278i
$742$		0			0
$743$		−	30.4520i		−	1.11718i	−0.829445	−	0.558588i	$-0.811343\pi$
$743$		−	30.4520i		−	1.11718i	0.829445	−	0.558588i	$-0.188657\pi$
$744$		0			0
$745$	−43.8864			−1.60787
$746$		0			0
$747$	6.43760			0.235540
$748$		0			0
$749$	10.6281	−	28.7005i	0.388343	−	1.04869i
$750$		0			0
$751$	−17.5306			−0.639702			−0.319851	−	0.947468i	$-0.603633\pi$
$751$	−17.5306			−0.639702			−0.319851	+	0.947468i	$0.603633\pi$
$752$		0			0
$753$	−4.71058			−0.171663
$754$		0			0
$755$	46.2125			1.68184
$756$		0			0
$757$	12.5242			0.455200			0.227600	−	0.973755i	$-0.426912\pi$
$757$	12.5242			0.455200			0.227600	+	0.973755i	$0.426912\pi$
$758$		0			0
$759$	−3.26956	+	14.2296i	−0.118677	+	0.516502i
$760$		0			0
$761$	34.9298			1.26620			0.633102	−	0.774069i	$-0.281782\pi$
$761$	34.9298			1.26620			0.633102	+	0.774069i	$0.281782\pi$
$762$		0			0
$763$	−10.9965	+	29.6952i	−0.398099	+	1.07504i
$764$		0			0
$765$		−	12.6012i		−	0.455599i
$766$		0			0
$767$		−	11.2281i		−	0.405424i
$768$		0			0
$769$	39.2987			1.41715			0.708574	−	0.705637i	$-0.249339\pi$
$769$	39.2987			1.41715			0.708574	+	0.705637i	$0.249339\pi$
$770$		0			0
$771$	27.1470			0.977675
$772$		0			0
$773$		−	43.3196i		−	1.55810i	−0.626964	−	0.779048i	$-0.715703\pi$
$773$		−	43.3196i		−	1.55810i	0.626964	−	0.779048i	$-0.284297\pi$
$774$		0			0
$775$			11.3294i			0.406963i
$776$		0			0
$777$	−8.88721	+	23.9993i	−0.318827	+	0.860970i
$778$		0			0
$779$	−12.0910			−0.433206
$780$		0			0
$781$	0.0641327	+	0.0147359i	0.00229485	+	0.000527291i
$782$		0			0
$783$	8.86515			0.316815
$784$		0			0
$785$	0.258240			0.00921699
$786$		0			0
$787$	−43.2976			−1.54339			−0.771697	−	0.635990i	$-0.780592\pi$
$787$	−43.2976			−1.54339			−0.771697	+	0.635990i	$0.780592\pi$
$788$		0			0
$789$	25.8070			0.918754
$790$		0			0
$791$	−30.9553	−	11.4631i	−1.10065	−	0.407582i
$792$		0			0
$793$	−11.5917			−0.411634
$794$		0			0
$795$	35.6295			1.26365
$796$		0			0
$797$			7.31674i			0.259172i	0.991568	+	0.129586i	$0.0413649\pi$
$797$			7.31674i			0.259172i	−0.991568	+	0.129586i	$0.958635\pi$
$798$		0			0
$799$			44.5301i			1.57536i
$800$		0			0
$801$			15.0086i			0.530304i
$802$		0			0
$803$	43.5297	+	10.0019i	1.53613	+	0.352959i
$804$		0			0
$805$	−7.79909	+	21.0609i	−0.274882	+	0.742299i
$806$		0			0
$807$	−26.0037			−0.915375
$808$		0			0
$809$			26.7725i			0.941270i	0.882328	+	0.470635i	$0.155975\pi$
$809$			26.7725i			0.941270i	−0.882328	+	0.470635i	$0.844025\pi$
$810$		0			0
$811$	−15.7141			−0.551798			−0.275899	−	0.961187i	$-0.588975\pi$
$811$	−15.7141			−0.551798			−0.275899	+	0.961187i	$0.588975\pi$
$812$		0			0
$813$			24.0376i			0.843037i
$814$		0			0
$815$			27.3327i			0.957423i
$816$		0			0
$817$			8.57460i			0.299987i
$818$		0			0
$819$	3.01307	+	1.11578i	0.105285	+	0.0389883i
$820$		0			0
$821$		−	28.9455i		−	1.01021i	−0.863059	−	0.505103i	$-0.831455\pi$
$821$		−	28.9455i		−	1.01021i	0.863059	−	0.505103i	$-0.168545\pi$
$822$		0			0
$823$	−32.0243			−1.11630			−0.558149	−	0.829741i	$-0.688488\pi$
$823$	−32.0243			−1.11630			−0.558149	+	0.829741i	$0.688488\pi$
$824$		0			0
$825$	−1.62676	+	7.07990i	−0.0566365	+	0.246490i
$826$		0			0
$827$		−	6.95018i		−	0.241682i	−0.992672	−	0.120841i	$-0.961441\pi$
$827$		−	6.95018i		−	0.241682i	0.992672	−	0.120841i	$-0.0385590\pi$
$828$		0			0
$829$			3.48716i			0.121114i	0.998165	+	0.0605571i	$0.0192877\pi$
$829$			3.48716i			0.121114i	−0.998165	+	0.0605571i	$0.980712\pi$
$830$		0			0
$831$	22.4667			0.779361
$832$		0			0
$833$	−21.3907	−	18.3602i	−0.741144	−	0.636143i
$834$		0			0
$835$			46.8466i			1.62119i
$836$		0			0
$837$	−38.9244			−1.34542
$838$		0			0
$839$			0.268087i			0.00925541i	0.999989	+	0.00462770i	$0.00147305\pi$
$839$			0.268087i			0.00925541i	−0.999989	+	0.00462770i	$0.998527\pi$
$840$		0			0
$841$	26.5219			0.914549
$842$		0			0
$843$	−9.81110			−0.337912
$844$		0			0
$845$			2.57665i			0.0886394i
$846$		0			0
$847$	−20.1433	−	21.0059i	−0.692133	−	0.721770i
$848$		0			0
$849$		−	15.2246i		−	0.522506i
$850$		0			0
$851$	−23.8474			−0.817479
$852$		0			0
$853$	13.5894			0.465291			0.232646	−	0.972562i	$-0.425262\pi$
$853$	13.5894			0.465291			0.232646	+	0.972562i	$0.425262\pi$
$854$		0			0
$855$		−	8.17694i		−	0.279646i
$856$		0			0
$857$	−12.6572			−0.432361			−0.216180	−	0.976353i	$-0.569360\pi$
$857$	−12.6572			−0.432361			−0.216180	+	0.976353i	$0.569360\pi$
$858$		0			0
$859$			4.82686i			0.164690i	0.996604	+	0.0823451i	$0.0262410\pi$
$859$			4.82686i			0.164690i	−0.996604	+	0.0823451i	$0.973759\pi$
$860$		0			0
$861$	−5.68061	+	15.3401i	−0.193595	+	0.522789i
$862$		0			0
$863$	−7.33401			−0.249653			−0.124826	−	0.992179i	$-0.539837\pi$
$863$	−7.33401			−0.249653			−0.124826	+	0.992179i	$0.539837\pi$
$864$		0			0
$865$			64.5420i			2.19449i
$866$		0			0
$867$			1.04559i			0.0355101i
$868$		0			0
$869$	−6.25873	+	27.2389i	−0.212313	+	0.924017i
$870$		0			0
$871$	11.9180			0.403827
$872$		0			0
$873$		−	12.3502i		−	0.417992i
$874$		0			0
$875$	7.95644	−	21.4858i	0.268977	−	0.726353i
$876$		0			0
$877$		−	57.1623i		−	1.93023i	−0.261821	−	0.965117i	$-0.584323\pi$
$877$		−	57.1623i		−	1.93023i	0.261821	−	0.965117i	$-0.415677\pi$
$878$		0			0
$879$			42.7707i			1.44262i
$880$		0			0
$881$		−	8.01752i		−	0.270117i	−0.990838	−	0.135058i	$-0.956878\pi$
$881$		−	8.01752i		−	0.270117i	0.990838	−	0.135058i	$-0.0431222\pi$
$882$		0			0
$883$	19.8323			0.667410			0.333705	−	0.942678i	$-0.391701\pi$
$883$	19.8323			0.667410			0.333705	+	0.942678i	$0.391701\pi$
$884$		0			0
$885$			38.6593i			1.29952i
$886$		0			0
$887$	27.7517			0.931810			0.465905	−	0.884835i	$-0.345729\pi$
$887$	27.7517			0.931810			0.465905	+	0.884835i	$0.345729\pi$
$888$		0			0
$889$	−9.28959	+	25.0859i	−0.311563	+	0.841353i
$890$		0			0
$891$	−12.5481	−	2.88319i	−0.420376	−	0.0965905i
$892$		0			0
$893$			28.8956i			0.966953i
$894$		0			0
$895$			34.8255i			1.16409i
$896$		0			0
$897$		−	4.40218i		−	0.146985i
$898$		0			0
$899$	10.8806			0.362888
$900$		0			0
$901$	−41.6731			−1.38833
$902$		0			0
$903$	10.8787	+	4.02852i	0.362022	+	0.134061i
$904$		0			0
$905$	−24.8946			−0.827525
$906$		0			0
$907$	−18.7783			−0.623523			−0.311761	−	0.950160i	$-0.600919\pi$
$907$	−18.7783			−0.623523			−0.311761	+	0.950160i	$0.600919\pi$
$908$		0			0
$909$	−21.2077			−0.703416
$910$		0			0
$911$	−3.80583			−0.126093			−0.0630464	−	0.998011i	$-0.520082\pi$
$911$	−3.80583			−0.126093			−0.0630464	+	0.998011i	$0.520082\pi$
$912$		0			0
$913$	17.1350	+	3.93713i	0.567084	+	0.130300i
$914$		0			0
$915$	39.9111			1.31942
$916$		0			0
$917$	−12.8155	−	4.74571i	−0.423204	−	0.156717i
$918$		0			0
$919$			10.6298i			0.350646i	0.984511	+	0.175323i	$0.0560970\pi$
$919$			10.6298i			0.350646i	−0.984511	+	0.175323i	$0.943903\pi$
$920$		0			0
$921$		−	15.1881i		−	0.500465i
$922$		0			0
$923$	−0.0198406			−0.000653062
$924$		0			0
$925$	−11.8652			−0.390126
$926$		0			0
$927$		−	2.04495i		−	0.0671650i
$928$		0			0
$929$			37.8292i			1.24114i	0.784152	+	0.620568i	$0.213098\pi$
$929$			37.8292i			1.24114i	−0.784152	+	0.620568i	$0.786902\pi$
$930$		0			0
$931$	−13.8804	−	11.9139i	−0.454912	−	0.390463i
$932$		0			0
$933$	−5.49530			−0.179908
$934$		0			0
$935$	7.70670	−	33.5407i	0.252036	−	1.09690i
$936$		0			0
$937$	19.3778			0.633045			0.316522	−	0.948585i	$-0.397485\pi$
$937$	19.3778			0.633045			0.316522	+	0.948585i	$0.397485\pi$
$938$		0			0
$939$	−34.0705			−1.11185
$940$		0			0
$941$	−28.2704			−0.921588			−0.460794	−	0.887507i	$-0.652435\pi$
$941$	−28.2704			−0.921588			−0.460794	+	0.887507i	$0.652435\pi$
$942$		0			0
$943$	−15.2430			−0.496381
$944$		0			0
$945$	−36.0020	−	13.3320i	−1.17115	−	0.433689i
$946$		0			0
$947$	0.960657			0.0312171			0.0156086	−	0.999878i	$-0.495031\pi$
$947$	0.960657			0.0312171			0.0156086	+	0.999878i	$0.495031\pi$
$948$		0			0
$949$	−13.4667			−0.437147
$950$		0			0
$951$			0.135285i			0.00438693i
$952$		0			0
$953$			40.9702i			1.32715i	0.748108	+	0.663577i	$0.230963\pi$
$953$			40.9702i			1.32715i	−0.748108	+	0.663577i	$0.769037\pi$
$954$		0			0
$955$		−	2.85527i		−	0.0923945i
$956$		0			0
$957$	6.79945	+	1.56232i	0.219795	+	0.0505026i
$958$		0			0
$959$	−5.75493	−	2.13112i	−0.185836	−	0.0688173i
$960$		0			0
$961$	−16.7736			−0.541084
$962$		0			0
$963$			14.0479i			0.452687i
$964$		0			0
$965$	66.4034			2.13760
$966$		0			0
$967$		−	3.99863i		−	0.128587i	−0.997931	−	0.0642937i	$-0.979521\pi$
$967$		−	3.99863i		−	0.128587i	0.997931	−	0.0642937i	$-0.0204794\pi$
$968$		0			0
$969$		−	14.0622i		−	0.451743i
$970$		0			0
$971$		−	23.3189i		−	0.748338i	−0.927360	−	0.374169i	$-0.877928\pi$
$971$		−	23.3189i		−	0.748338i	0.927360	−	0.374169i	$-0.122072\pi$
$972$		0			0
$973$	−24.7513	−	9.16569i	−0.793491	−	0.293839i
$974$		0			0
$975$		−	2.19029i		−	0.0701456i
$976$		0			0
$977$	−24.5279			−0.784716			−0.392358	−	0.919813i	$-0.628340\pi$
$977$	−24.5279			−0.784716			−0.392358	+	0.919813i	$0.628340\pi$
$978$		0			0
$979$	−9.17902	+	39.9484i	−0.293363	+	1.27676i
$980$		0			0
$981$		−	14.5348i		−	0.464059i
$982$		0			0
$983$		−	43.2182i		−	1.37845i	−0.724549	−	0.689223i	$-0.757952\pi$
$983$		−	43.2182i		−	1.37845i	0.724549	−	0.689223i	$-0.242048\pi$
$984$		0			0
$985$	34.9767			1.11445
$986$		0			0
$987$	36.6603	+	13.5757i	1.16691	+	0.432120i
$988$		0			0
$989$			10.8099i			0.343735i
$990$		0			0
$991$	23.7374			0.754042			0.377021	−	0.926205i	$-0.376948\pi$
$991$	23.7374			0.754042			0.377021	+	0.926205i	$0.376948\pi$
$992$		0			0
$993$		−	24.7494i		−	0.785397i
$994$		0			0
$995$	25.3641			0.804097
$996$		0			0
$997$	−27.8976			−0.883526			−0.441763	−	0.897132i	$-0.645647\pi$
$997$	−27.8976			−0.883526			−0.441763	+	0.897132i	$0.645647\pi$
$998$		0			0
$999$		−	40.7654i		−	1.28976i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.b.3849.14	yes	48
7.6	odd	2		4004.2.e.a.3849.35	yes	48
11.10	odd	2		4004.2.e.a.3849.14	✓	48
77.76	even	2	inner	4004.2.e.b.3849.35	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.14	✓	48	11.10	odd	2
4004.2.e.a.3849.35	yes	48	7.6	odd	2
4004.2.e.b.3849.14	yes	48	1.1	even	1	trivial
4004.2.e.b.3849.35	yes	48	77.76	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.33626i q^{3} +2.57665i q^{5} +(-2.48110 - 0.918779i) q^{7} +1.21441 q^{9} +O(q^{10})\)	\(q-1.33626i q^{3} +2.57665i q^{5} +(-2.48110 - 0.918779i) q^{7} +1.21441 q^{9} +(3.23240 + 0.742713i) q^{11} -1.00000 q^{13} +3.44307 q^{15} -4.02710 q^{17} -2.61318 q^{19} +(-1.22773 + 3.31539i) q^{21} -3.29441 q^{23} -1.63912 q^{25} -5.63155i q^{27} +1.57419i q^{29} -6.91184i q^{31} +(0.992457 - 4.31932i) q^{33} +(2.36737 - 6.39292i) q^{35} +7.23876 q^{37} +1.33626i q^{39} +4.62693 q^{41} -3.28129i q^{43} +3.12911i q^{45} -11.0576i q^{47} +(5.31169 + 4.55916i) q^{49} +5.38125i q^{51} +10.3482 q^{53} +(-1.91371 + 8.32875i) q^{55} +3.49189i q^{57} +11.2281i q^{59} +11.5917 q^{61} +(-3.01307 - 1.11578i) q^{63} -2.57665i q^{65} -11.9180 q^{67} +4.40218i q^{69} +0.0198406 q^{71} +13.4667 q^{73} +2.19029i q^{75} +(-7.33750 - 4.81260i) q^{77} +8.42685i q^{79} -3.88197 q^{81} +5.30101 q^{83} -10.3764i q^{85} +2.10353 q^{87} +12.3588i q^{89} +(2.48110 + 0.918779i) q^{91} -9.23602 q^{93} -6.73326i q^{95} -10.1697i q^{97} +(3.92546 + 0.901958i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{7} - 48 q^{9}+O(q^{10}) \) 48 * q + 4 * q^7 - 48 * q^9	\( 48 q + 4 q^{7} - 48 q^{9} + 2 q^{11} - 48 q^{13} + 8 q^{15} - 4 q^{17} - 10 q^{21} + 4 q^{23} - 44 q^{25} - 10 q^{33} + 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} + 12 q^{55} - 16 q^{61} - 16 q^{63} + 4 q^{67} + 16 q^{73} + 22 q^{77} + 64 q^{81} + 4 q^{83} + 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) \) 48 * q + 4 * q^7 - 48 * q^9 + 2 * q^11 - 48 * q^13 + 8 * q^15 - 4 * q^17 - 10 * q^21 + 4 * q^23 - 44 * q^25 - 10 * q^33 + 14 * q^35 - 16 * q^37 + 10 * q^49 - 8 * q^53 + 12 * q^55 - 16 * q^61 - 16 * q^63 + 4 * q^67 + 16 * q^73 + 22 * q^77 + 64 * q^81 + 4 * q^83 + 12 * q^87 - 4 * q^91 + 36 * q^93 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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