LMFDB - Embedded newform 4140.2.f.c.829.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.f (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4 $ x^14 + 27x^12 + 283x^10 + 1441x^8 + 3596x^6 + 3740x^4 + 772x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.13
Root			$-2.39625i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.c.829.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.22987 - 0.166381i) q^{5} -1.74202i q^{7} +O(q^{10})$	$q+(2.22987 - 0.166381i) q^{5} -1.74202i q^{7} -2.39958 q^{11} +3.50266i q^{13} +5.88762i q^{17} -1.93352 q^{19} +1.00000i q^{23} +(4.94463 - 0.742018i) q^{25} -4.22038 q^{29} +1.61463 q^{31} +(-0.289839 - 3.88447i) q^{35} +4.06846i q^{37} -8.46771 q^{41} +2.70706i q^{43} +8.18788i q^{47} +3.96537 q^{49} +2.78453i q^{53} +(-5.35074 + 0.399245i) q^{55} -12.5884 q^{59} +3.35310 q^{61} +(0.582778 + 7.81048i) q^{65} +2.16022i q^{67} -13.5711 q^{71} +4.98670i q^{73} +4.18010i q^{77} +6.25469 q^{79} +1.55486i q^{83} +(0.979591 + 13.1286i) q^{85} +7.14462 q^{89} +6.10170 q^{91} +(-4.31149 + 0.321701i) q^{95} +5.93042i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q+O(q^{10}) $ 14 * q	$ 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) $ 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.22987	−	0.166381i	0.997228	−	0.0744080i
$5$	2.22987	−	0.166381i	0.997228	−	0.0744080i
$6$		0			0
$7$		−	1.74202i		−	0.658421i	−0.944257	−	0.329210i	$-0.893217\pi$
$7$		−	1.74202i		−	0.658421i	0.944257	−	0.329210i	$-0.106783\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.39958			−0.723499			−0.361750	−	0.932275i	$-0.617820\pi$
$11$	−2.39958			−0.723499			−0.361750	+	0.932275i	$0.617820\pi$
$12$		0			0
$13$			3.50266i			0.971464i	0.874108	+	0.485732i	$0.161447\pi$
$13$			3.50266i			0.971464i	−0.874108	+	0.485732i	$0.838553\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			5.88762i			1.42796i	0.700167	+	0.713979i	$0.253109\pi$
$17$			5.88762i			1.42796i	−0.700167	+	0.713979i	$0.746891\pi$
$18$		0			0
$19$	−1.93352			−0.443579			−0.221790	−	0.975095i	$-0.571190\pi$
$19$	−1.93352			−0.443579			−0.221790	+	0.975095i	$0.571190\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	4.94463	−	0.742018i	0.988927	−	0.148404i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.22038			−0.783705			−0.391853	−	0.920028i	$-0.628166\pi$
$29$	−4.22038			−0.783705			−0.391853	+	0.920028i	$0.628166\pi$
$30$		0			0
$31$	1.61463			0.289997			0.144998	−	0.989432i	$-0.453682\pi$
$31$	1.61463			0.289997			0.144998	+	0.989432i	$0.453682\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.289839	−	3.88447i	−0.0489918	−	0.656596i
$36$		0			0
$37$			4.06846i			0.668851i	0.942422	+	0.334425i	$0.108542\pi$
$37$			4.06846i			0.668851i	−0.942422	+	0.334425i	$0.891458\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.46771			−1.32243			−0.661217	−	0.750195i	$-0.729960\pi$
$41$	−8.46771			−1.32243			−0.661217	+	0.750195i	$0.729960\pi$
$42$		0			0
$43$			2.70706i			0.412823i	0.978465	+	0.206412i	$0.0661786\pi$
$43$			2.70706i			0.412823i	−0.978465	+	0.206412i	$0.933821\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.18788i			1.19432i	0.802120	+	0.597162i	$0.203705\pi$
$47$			8.18788i			1.19432i	−0.802120	+	0.597162i	$0.796295\pi$
$48$		0			0
$49$	3.96537			0.566482
$50$		0			0
$51$		0			0
$52$		0			0
$53$			2.78453i			0.382485i	0.981543	+	0.191243i	$0.0612517\pi$
$53$			2.78453i			0.382485i	−0.981543	+	0.191243i	$0.938748\pi$
$54$		0			0
$55$	−5.35074	+	0.399245i	−0.721494	+	0.0538342i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.5884			−1.63887			−0.819433	−	0.573176i	$-0.805711\pi$
$59$	−12.5884			−1.63887			−0.819433	+	0.573176i	$0.805711\pi$
$60$		0			0
$61$	3.35310			0.429321			0.214660	−	0.976689i	$-0.431136\pi$
$61$	3.35310			0.429321			0.214660	+	0.976689i	$0.431136\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.582778	+	7.81048i	0.0722847	+	0.968771i
$66$		0			0
$67$			2.16022i			0.263913i	0.991256	+	0.131956i	$0.0421259\pi$
$67$			2.16022i			0.263913i	−0.991256	+	0.131956i	$0.957874\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−13.5711			−1.61060			−0.805298	−	0.592870i	$-0.797995\pi$
$71$	−13.5711			−1.61060			−0.805298	+	0.592870i	$0.797995\pi$
$72$		0			0
$73$			4.98670i			0.583649i	0.956472	+	0.291824i	$0.0942623\pi$
$73$			4.98670i			0.583649i	−0.956472	+	0.291824i	$0.905738\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			4.18010i			0.476367i
$78$		0			0
$79$	6.25469			0.703708			0.351854	−	0.936055i	$-0.385551\pi$
$79$	6.25469			0.703708			0.351854	+	0.936055i	$0.385551\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1.55486i			0.170668i	0.996352	+	0.0853339i	$0.0271957\pi$
$83$			1.55486i			0.170668i	−0.996352	+	0.0853339i	$0.972804\pi$
$84$		0			0
$85$	0.979591	+	13.1286i	0.106252	+	1.42400i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.14462			0.757328			0.378664	−	0.925534i	$-0.376384\pi$
$89$	7.14462			0.757328			0.378664	+	0.925534i	$0.376384\pi$
$90$		0			0
$91$	6.10170			0.639632
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.31149	+	0.321701i	−0.442350	+	0.0330059i
$96$		0			0
$97$			5.93042i			0.602143i	0.953602	+	0.301071i	$0.0973443\pi$
$97$			5.93042i			0.602143i	−0.953602	+	0.301071i	$0.902656\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.66922			0.365101			0.182550	−	0.983197i	$-0.441565\pi$
$101$	3.66922			0.365101			0.182550	+	0.983197i	$0.441565\pi$
$102$		0			0
$103$		−	4.24771i		−	0.418539i	−0.977858	−	0.209270i	$-0.932891\pi$
$103$		−	4.24771i		−	0.418539i	0.977858	−	0.209270i	$-0.0671086\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	17.4604i		−	1.68796i	−0.536375	−	0.843980i	$-0.680207\pi$
$107$		−	17.4604i		−	1.68796i	0.536375	−	0.843980i	$-0.319793\pi$
$108$		0			0
$109$	−1.16286			−0.111382			−0.0556911	−	0.998448i	$-0.517736\pi$
$109$	−1.16286			−0.111382			−0.0556911	+	0.998448i	$0.517736\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			12.3695i			1.16363i	0.813322	+	0.581814i	$0.197657\pi$
$113$			12.3695i			1.16363i	−0.813322	+	0.581814i	$0.802343\pi$
$114$		0			0
$115$	0.166381	+	2.22987i	0.0155151	+	0.207936i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	10.2563			0.940197
$120$		0			0
$121$	−5.24204			−0.476549
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.9024	−	2.47730i	0.975143	−	0.221576i
$126$		0			0
$127$			7.70884i			0.684048i	0.939691	+	0.342024i	$0.111113\pi$
$127$			7.70884i			0.684048i	−0.939691	+	0.342024i	$0.888887\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.0980931			−0.00857043			−0.00428522	−	0.999991i	$-0.501364\pi$
$131$	−0.0980931			−0.00857043			−0.00428522	+	0.999991i	$0.501364\pi$
$132$		0			0
$133$			3.36822i			0.292062i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			17.4869i			1.49401i	0.664819	+	0.747004i	$0.268509\pi$
$137$			17.4869i			1.49401i	−0.664819	+	0.747004i	$0.731491\pi$
$138$		0			0
$139$	18.6911			1.58536			0.792680	−	0.609638i	$-0.208685\pi$
$139$	18.6911			1.58536			0.792680	+	0.609638i	$0.208685\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	8.40490i		−	0.702853i
$144$		0			0
$145$	−9.41090	+	0.702193i	−0.781533	+	0.0583140i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	16.2670			1.33264			0.666322	−	0.745664i	$-0.267868\pi$
$149$	16.2670			1.33264			0.666322	+	0.745664i	$0.267868\pi$
$150$		0			0
$151$	11.2367			0.914431			0.457215	−	0.889356i	$-0.348847\pi$
$151$	11.2367			0.914431			0.457215	+	0.889356i	$0.348847\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.60042	−	0.268645i	0.289193	−	0.0215781i
$156$		0			0
$157$		−	7.83707i		−	0.625467i	−0.949841	−	0.312733i	$-0.898755\pi$
$157$		−	7.83707i		−	0.625467i	0.949841	−	0.312733i	$-0.101245\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.74202			0.137290
$162$		0			0
$163$		−	15.1257i		−	1.18473i	−0.805669	−	0.592366i	$-0.798194\pi$
$163$		−	15.1257i		−	1.18473i	0.805669	−	0.592366i	$-0.201806\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			10.6535i			0.824395i	0.911095	+	0.412197i	$0.135239\pi$
$167$			10.6535i			0.824395i	−0.911095	+	0.412197i	$0.864761\pi$
$168$		0			0
$169$	0.731358			0.0562583
$170$		0			0
$171$		0			0
$172$		0			0
$173$			9.96562i			0.757672i	0.925464	+	0.378836i	$0.123676\pi$
$173$			9.96562i			0.757672i	−0.925464	+	0.378836i	$0.876324\pi$
$174$		0			0
$175$	−1.29261	−	8.61364i	−0.0977119	−	0.651130i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.1579			−0.759238			−0.379619	−	0.925143i	$-0.623945\pi$
$179$	−10.1579			−0.759238			−0.379619	+	0.925143i	$0.623945\pi$
$180$		0			0
$181$	−11.1894			−0.831699			−0.415849	−	0.909433i	$-0.636516\pi$
$181$	−11.1894			−0.831699			−0.415849	+	0.909433i	$0.636516\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.676916	+	9.07214i	0.0497679	+	0.666997i
$186$		0			0
$187$		−	14.1278i		−	1.03313i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.52056			0.471811			0.235906	−	0.971776i	$-0.424194\pi$
$191$	6.52056			0.471811			0.235906	+	0.971776i	$0.424194\pi$
$192$		0			0
$193$		−	18.0166i		−	1.29686i	−0.761273	−	0.648432i	$-0.775425\pi$
$193$		−	18.0166i		−	1.29686i	0.761273	−	0.648432i	$-0.224575\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3.73169i		−	0.265872i	−0.991125	−	0.132936i	$-0.957560\pi$
$197$		−	3.73169i		−	0.265872i	0.991125	−	0.132936i	$-0.0424404\pi$
$198$		0			0
$199$	−14.4415			−1.02373			−0.511866	−	0.859066i	$-0.671045\pi$
$199$	−14.4415			−1.02373			−0.511866	+	0.859066i	$0.671045\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			7.35198i			0.516008i
$204$		0			0
$205$	−18.8819	+	1.40887i	−1.31877	+	0.0983997i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.63962			0.320929
$210$		0			0
$211$	14.5575			1.00218			0.501089	−	0.865395i	$-0.332933\pi$
$211$	14.5575			1.00218			0.501089	+	0.865395i	$0.332933\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.450405	+	6.03639i	0.0307173	+	0.411679i
$216$		0			0
$217$		−	2.81272i		−	0.190940i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−20.6223			−1.38721
$222$		0			0
$223$			24.0458i			1.61023i	0.593122	+	0.805113i	$0.297895\pi$
$223$			24.0458i			1.61023i	−0.593122	+	0.805113i	$0.702105\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			11.1383i			0.739274i	0.929176	+	0.369637i	$0.120518\pi$
$227$			11.1383i			0.739274i	−0.929176	+	0.369637i	$0.879482\pi$
$228$		0			0
$229$	−26.4281			−1.74642			−0.873210	−	0.487344i	$-0.837966\pi$
$229$	−26.4281			−1.74642			−0.873210	+	0.487344i	$0.837966\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2.30427i		−	0.150957i	−0.997147	−	0.0754787i	$-0.975951\pi$
$233$		−	2.30427i		−	0.150957i	0.997147	−	0.0754787i	$-0.0240485\pi$
$234$		0			0
$235$	1.36231	+	18.2579i	0.0888673	+	1.19101i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.30509			0.537211			0.268606	−	0.963250i	$-0.413437\pi$
$239$	8.30509			0.537211			0.268606	+	0.963250i	$0.413437\pi$
$240$		0			0
$241$	9.70884			0.625401			0.312701	−	0.949852i	$-0.398766\pi$
$241$	9.70884			0.625401			0.312701	+	0.949852i	$0.398766\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.84227	−	0.659765i	0.564912	−	0.0421508i
$246$		0			0
$247$		−	6.77246i		−	0.430921i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7.24520			−0.457313			−0.228657	−	0.973507i	$-0.573433\pi$
$251$	−7.24520			−0.457313			−0.228657	+	0.973507i	$0.573433\pi$
$252$		0			0
$253$		−	2.39958i		−	0.150860i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			7.88649i			0.491945i	0.969277	+	0.245973i	$0.0791074\pi$
$257$			7.88649i			0.491945i	−0.969277	+	0.245973i	$0.920893\pi$
$258$		0			0
$259$	7.08733			0.440385
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	11.5834i		−	0.714261i	−0.934054	−	0.357131i	$-0.883755\pi$
$263$		−	11.5834i		−	0.714261i	0.934054	−	0.357131i	$-0.116245\pi$
$264$		0			0
$265$	0.463295	+	6.20915i	0.0284600	+	0.381425i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	20.3478			1.24063			0.620314	−	0.784354i	$-0.287005\pi$
$269$	20.3478			1.24063			0.620314	+	0.784354i	$0.287005\pi$
$270$		0			0
$271$	−10.4230			−0.633155			−0.316578	−	0.948567i	$-0.602534\pi$
$271$	−10.4230			−0.633155			−0.316578	+	0.948567i	$0.602534\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11.8650	+	1.78053i	−0.715488	+	0.107370i
$276$		0			0
$277$			24.0451i			1.44473i	0.691511	+	0.722366i	$0.256945\pi$
$277$			24.0451i			1.44473i	−0.691511	+	0.722366i	$0.743055\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−19.9078			−1.18760			−0.593801	−	0.804612i	$-0.702373\pi$
$281$	−19.9078			−1.18760			−0.593801	+	0.804612i	$0.702373\pi$
$282$		0			0
$283$		−	15.0350i		−	0.893738i	−0.894599	−	0.446869i	$-0.852539\pi$
$283$		−	15.0350i		−	0.893738i	0.894599	−	0.446869i	$-0.147461\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			14.7509i			0.870718i
$288$		0			0
$289$	−17.6641			−1.03906
$290$		0			0
$291$		0			0
$292$		0			0
$293$			19.5439i			1.14177i	0.821032	+	0.570883i	$0.193399\pi$
$293$			19.5439i			1.14177i	−0.821032	+	0.570883i	$0.806601\pi$
$294$		0			0
$295$	−28.0704	+	2.09447i	−1.63432	+	0.121945i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.50266			−0.202564
$300$		0			0
$301$	4.71575			0.271811
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.47698	−	0.557894i	0.428131	−	0.0319449i
$306$		0			0
$307$		−	7.97609i		−	0.455219i	−0.973752	−	0.227610i	$-0.926909\pi$
$307$		−	7.97609i		−	0.455219i	0.973752	−	0.227610i	$-0.0730910\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.416573			0.0236217			0.0118108	−	0.999930i	$-0.496240\pi$
$311$	0.416573			0.0236217			0.0118108	+	0.999930i	$0.496240\pi$
$312$		0			0
$313$		−	19.5898i		−	1.10728i	−0.832757	−	0.553639i	$-0.813239\pi$
$313$		−	19.5898i		−	1.10728i	0.832757	−	0.553639i	$-0.186761\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	15.8200i		−	0.888540i	−0.895893	−	0.444270i	$-0.853463\pi$
$317$		−	15.8200i		−	0.888540i	0.895893	−	0.444270i	$-0.146537\pi$
$318$		0			0
$319$	10.1271			0.567010
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	11.3838i		−	0.633413i
$324$		0			0
$325$	2.59904	+	17.3194i	0.144169	+	0.960707i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	14.2634			0.786368
$330$		0			0
$331$	33.6604			1.85014			0.925072	−	0.379791i	$-0.124004\pi$
$331$	33.6604			1.85014			0.925072	+	0.379791i	$0.124004\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.359420	+	4.81701i	0.0196372	+	0.263181i
$336$		0			0
$337$			12.6725i			0.690314i	0.938545	+	0.345157i	$0.112174\pi$
$337$			12.6725i			0.690314i	−0.938545	+	0.345157i	$0.887826\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.87444			−0.209813
$342$		0			0
$343$		−	19.1019i		−	1.03140i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			24.8467i			1.33384i	0.745130	+	0.666919i	$0.232387\pi$
$347$			24.8467i			1.33384i	−0.745130	+	0.666919i	$0.767613\pi$
$348$		0			0
$349$	−25.0038			−1.33842			−0.669211	−	0.743073i	$-0.733368\pi$
$349$	−25.0038			−1.33842			−0.669211	+	0.743073i	$0.733368\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			14.0262i			0.746538i	0.927723	+	0.373269i	$0.121763\pi$
$353$			14.0262i			0.746538i	−0.927723	+	0.373269i	$0.878237\pi$
$354$		0			0
$355$	−30.2618	+	2.25798i	−1.60613	+	0.119841i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.75873			0.356712			0.178356	−	0.983966i	$-0.442922\pi$
$359$	6.75873			0.356712			0.178356	+	0.983966i	$0.442922\pi$
$360$		0			0
$361$	−15.2615			−0.803237
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.829694	+	11.1197i	0.0434282	+	0.582031i
$366$		0			0
$367$			33.2652i			1.73643i	0.496188	+	0.868215i	$0.334733\pi$
$367$			33.2652i			1.73643i	−0.496188	+	0.868215i	$0.665267\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.85071			0.251836
$372$		0			0
$373$			15.1293i			0.783364i	0.920101	+	0.391682i	$0.128107\pi$
$373$			15.1293i			0.783364i	−0.920101	+	0.391682i	$0.871893\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	14.7826i		−	0.761341i
$378$		0			0
$379$	24.2386			1.24505			0.622527	−	0.782598i	$-0.286106\pi$
$379$	24.2386			1.24505			0.622527	+	0.782598i	$0.286106\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	36.0221i		−	1.84064i	−0.391163	−	0.920321i	$-0.627927\pi$
$383$		−	36.0221i		−	1.84064i	0.391163	−	0.920321i	$-0.372073\pi$
$384$		0			0
$385$	0.695491	+	9.32108i	0.0354455	+	0.475046i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.5588			−0.890266			−0.445133	−	0.895465i	$-0.646844\pi$
$389$	−17.5588			−0.890266			−0.445133	+	0.895465i	$0.646844\pi$
$390$		0			0
$391$	−5.88762			−0.297750
$392$		0			0
$393$		0			0
$394$		0			0
$395$	13.9471	−	1.04066i	0.701757	−	0.0523615i
$396$		0			0
$397$			6.31056i			0.316718i	0.987382	+	0.158359i	$0.0506204\pi$
$397$			6.31056i			0.316718i	−0.987382	+	0.158359i	$0.949380\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	22.3779			1.11750			0.558748	−	0.829337i	$-0.311282\pi$
$401$	22.3779			1.11750			0.558748	+	0.829337i	$0.311282\pi$
$402$		0			0
$403$			5.65552i			0.281722i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	9.76258i		−	0.483913i
$408$		0			0
$409$	−21.4334			−1.05981			−0.529907	−	0.848056i	$-0.677773\pi$
$409$	−21.4334			−1.05981			−0.529907	+	0.848056i	$0.677773\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			21.9291i			1.07906i
$414$		0			0
$415$	0.258699	+	3.46713i	0.0126991	+	0.170195i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.376413			0.0183890			0.00919449	−	0.999958i	$-0.497073\pi$
$419$	0.376413			0.0183890			0.00919449	+	0.999958i	$0.497073\pi$
$420$		0			0
$421$	20.8051			1.01398			0.506990	−	0.861952i	$-0.330758\pi$
$421$	20.8051			1.01398			0.506990	+	0.861952i	$0.330758\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.36872	+	29.1121i	0.211914	+	1.41215i
$426$		0			0
$427$		−	5.84116i		−	0.282674i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.47802			0.215699			0.107849	−	0.994167i	$-0.465604\pi$
$431$	4.47802			0.215699			0.107849	+	0.994167i	$0.465604\pi$
$432$		0			0
$433$			3.25198i			0.156280i	0.996942	+	0.0781402i	$0.0248982\pi$
$433$			3.25198i			0.156280i	−0.996942	+	0.0781402i	$0.975102\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1.93352i		−	0.0924927i
$438$		0			0
$439$	−32.4223			−1.54743			−0.773715	−	0.633533i	$-0.781604\pi$
$439$	−32.4223			−1.54743			−0.773715	+	0.633533i	$0.781604\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	21.2879i		−	1.01142i	−0.862704	−	0.505709i	$-0.831231\pi$
$443$		−	21.2879i		−	1.01142i	0.862704	−	0.505709i	$-0.168769\pi$
$444$		0			0
$445$	15.9316	−	1.18873i	0.755229	−	0.0563513i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	26.6732			1.25878			0.629392	−	0.777088i	$-0.283304\pi$
$449$	26.6732			1.25878			0.629392	+	0.777088i	$0.283304\pi$
$450$		0			0
$451$	20.3189			0.956780
$452$		0			0
$453$		0			0
$454$		0			0
$455$	13.6060	−	1.01521i	0.637859	−	0.0475937i
$456$		0			0
$457$			40.8011i			1.90859i	0.298859	+	0.954297i	$0.403394\pi$
$457$			40.8011i			1.90859i	−0.298859	+	0.954297i	$0.596606\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.3218			−0.806758			−0.403379	−	0.915033i	$-0.632164\pi$
$461$	−17.3218			−0.806758			−0.403379	+	0.915033i	$0.632164\pi$
$462$		0			0
$463$		−	25.8238i		−	1.20013i	−0.799950	−	0.600066i	$-0.795141\pi$
$463$		−	25.8238i		−	1.20013i	0.799950	−	0.600066i	$-0.204859\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			2.32525i			0.107600i	0.998552	+	0.0537998i	$0.0171333\pi$
$467$			2.32525i			0.107600i	−0.998552	+	0.0537998i	$0.982867\pi$
$468$		0			0
$469$	3.76314			0.173766
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	6.49580i		−	0.298677i
$474$		0			0
$475$	−9.56054	+	1.43470i	−0.438668	+	0.0658287i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.92961			−0.316622			−0.158311	−	0.987389i	$-0.550605\pi$
$479$	−6.92961			−0.316622			−0.158311	+	0.987389i	$0.550605\pi$
$480$		0			0
$481$	−14.2504			−0.649764
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.986711	+	13.2241i	0.0448043	+	0.600474i
$486$		0			0
$487$			21.3848i			0.969039i	0.874781	+	0.484519i	$0.161005\pi$
$487$			21.3848i			0.969039i	−0.874781	+	0.484519i	$0.838995\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	12.0791			0.545121			0.272560	−	0.962139i	$-0.412130\pi$
$491$	12.0791			0.545121			0.272560	+	0.962139i	$0.412130\pi$
$492$		0			0
$493$		−	24.8480i		−	1.11910i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			23.6411i			1.06045i
$498$		0			0
$499$	1.61463			0.0722810			0.0361405	−	0.999347i	$-0.488494\pi$
$499$	1.61463			0.0722810			0.0361405	+	0.999347i	$0.488494\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			0.525453i			0.0234288i	0.999931	+	0.0117144i	$0.00372889\pi$
$503$			0.525453i			0.0234288i	−0.999931	+	0.0117144i	$0.996271\pi$
$504$		0			0
$505$	8.18187	−	0.610489i	0.364089	−	0.0271664i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	32.9180			1.45906			0.729531	−	0.683947i	$-0.239738\pi$
$509$	32.9180			1.45906			0.729531	+	0.683947i	$0.239738\pi$
$510$		0			0
$511$	8.68691			0.384286
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.706740	−	9.47184i	−0.0311427	−	0.417379i
$516$		0			0
$517$		−	19.6474i		−	0.864093i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−43.1838			−1.89192			−0.945959	−	0.324287i	$-0.894876\pi$
$521$	−43.1838			−1.89192			−0.945959	+	0.324287i	$0.894876\pi$
$522$		0			0
$523$		−	8.09218i		−	0.353846i	−0.984225	−	0.176923i	$-0.943386\pi$
$523$		−	8.09218i		−	0.353846i	0.984225	−	0.176923i	$-0.0566144\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			9.50636i			0.414103i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	29.6595i		−	1.28470i
$534$		0			0
$535$	−2.90508	−	38.9344i	−0.125598	−	1.68328i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9.51522			−0.409849
$540$		0			0
$541$	13.5834			0.583995			0.291997	−	0.956419i	$-0.405680\pi$
$541$	13.5834			0.583995			0.291997	+	0.956419i	$0.405680\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.59303	+	0.193479i	−0.111073	+	0.00828772i
$546$		0			0
$547$			26.1855i			1.11961i	0.828624	+	0.559806i	$0.189124\pi$
$547$			26.1855i			1.11961i	−0.828624	+	0.559806i	$0.810876\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.16019			0.347636
$552$		0			0
$553$		−	10.8958i		−	0.463336i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	13.2053i		−	0.559526i	−0.960069	−	0.279763i	$-0.909744\pi$
$557$		−	13.2053i		−	0.559526i	0.960069	−	0.279763i	$-0.0902558\pi$
$558$		0			0
$559$	−9.48192			−0.401043
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	7.08877i		−	0.298756i	−0.988780	−	0.149378i	$-0.952273\pi$
$563$		−	7.08877i		−	0.298756i	0.988780	−	0.149378i	$-0.0477271\pi$
$564$		0			0
$565$	2.05806	+	27.5825i	0.0865833	+	1.16040i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−30.8739			−1.29430			−0.647151	−	0.762362i	$-0.724040\pi$
$569$	−30.8739			−1.29430			−0.647151	+	0.762362i	$0.724040\pi$
$570$		0			0
$571$	−4.15299			−0.173797			−0.0868986	−	0.996217i	$-0.527696\pi$
$571$	−4.15299			−0.173797			−0.0868986	+	0.996217i	$0.527696\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.742018	+	4.94463i	0.0309443	+	0.206206i
$576$		0			0
$577$		−	29.2537i		−	1.21785i	−0.793228	−	0.608925i	$-0.791601\pi$
$577$		−	29.2537i		−	1.21785i	0.793228	−	0.608925i	$-0.208399\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.70859			0.112371
$582$		0			0
$583$		−	6.68170i		−	0.276728i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	6.83774i		−	0.282224i	−0.989994	−	0.141112i	$-0.954932\pi$
$587$		−	6.83774i		−	0.282224i	0.989994	−	0.141112i	$-0.0450677\pi$
$588$		0			0
$589$	−3.12193			−0.128637
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	44.0408i		−	1.80854i	−0.426963	−	0.904269i	$-0.640417\pi$
$593$		−	44.0408i		−	1.80854i	0.426963	−	0.904269i	$-0.359583\pi$
$594$		0			0
$595$	22.8703	−	1.70646i	0.937591	−	0.0699582i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−22.5794			−0.922569			−0.461284	−	0.887252i	$-0.652611\pi$
$599$	−22.5794			−0.922569			−0.461284	+	0.887252i	$0.652611\pi$
$600$		0			0
$601$	22.3516			0.911741			0.455871	−	0.890046i	$-0.349328\pi$
$601$	22.3516			0.911741			0.455871	+	0.890046i	$0.349328\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.6891	+	0.872177i	−0.475228	+	0.0354591i
$606$		0			0
$607$			25.0246i			1.01572i	0.861440	+	0.507859i	$0.169563\pi$
$607$			25.0246i			1.01572i	−0.861440	+	0.507859i	$0.830437\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−28.6794			−1.16024
$612$		0			0
$613$			12.5826i			0.508205i	0.967177	+	0.254102i	$0.0817800\pi$
$613$			12.5826i			0.508205i	−0.967177	+	0.254102i	$0.918220\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	33.9407i		−	1.36640i	−0.730231	−	0.683200i	$-0.760588\pi$
$617$		−	33.9407i		−	1.36640i	0.730231	−	0.683200i	$-0.239412\pi$
$618$		0			0
$619$	29.5425			1.18741			0.593707	−	0.804681i	$-0.297664\pi$
$619$	29.5425			1.18741			0.593707	+	0.804681i	$0.297664\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	12.4461i		−	0.498641i
$624$		0			0
$625$	23.8988	−	7.33801i	0.955953	−	0.293520i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−23.9536			−0.955091
$630$		0			0
$631$	1.95895			0.0779845			0.0389922	−	0.999240i	$-0.487585\pi$
$631$	1.95895			0.0779845			0.0389922	+	0.999240i	$0.487585\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.28261	+	17.1897i	0.0508987	+	0.682152i
$636$		0			0
$637$			13.8894i			0.550317i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	31.0768			1.22746			0.613730	−	0.789516i	$-0.289668\pi$
$641$	31.0768			1.22746			0.613730	+	0.789516i	$0.289668\pi$
$642$		0			0
$643$		−	16.4373i		−	0.648225i	−0.946019	−	0.324113i	$-0.894934\pi$
$643$		−	16.4373i		−	0.648225i	0.946019	−	0.324113i	$-0.105066\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	21.2561i		−	0.835664i	−0.908524	−	0.417832i	$-0.862790\pi$
$647$		−	21.2561i		−	0.835664i	0.908524	−	0.417832i	$-0.137210\pi$
$648$		0			0
$649$	30.2067			1.18572
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	2.62063i		−	0.102553i	−0.998684	−	0.0512766i	$-0.983671\pi$
$653$		−	2.62063i		−	0.102553i	0.998684	−	0.0512766i	$-0.0163290\pi$
$654$		0			0
$655$	−0.218735	+	0.0163209i	−0.00854668	+	0.000637709i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−32.1165			−1.25108			−0.625541	−	0.780191i	$-0.715122\pi$
$659$	−32.1165			−1.25108			−0.625541	+	0.780191i	$0.715122\pi$
$660$		0			0
$661$	31.5098			1.22559			0.612795	−	0.790242i	$-0.290045\pi$
$661$	31.5098			1.22559			0.612795	+	0.790242i	$0.290045\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.560410	+	7.51070i	0.0217317	+	0.291252i
$666$		0			0
$667$		−	4.22038i		−	0.163414i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−8.04602			−0.310613
$672$		0			0
$673$		−	37.5158i		−	1.44613i	−0.690781	−	0.723064i	$-0.742733\pi$
$673$		−	37.5158i		−	1.44613i	0.690781	−	0.723064i	$-0.257267\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			12.6869i			0.487596i	0.969826	+	0.243798i	$0.0783934\pi$
$677$			12.6869i			0.487596i	−0.969826	+	0.243798i	$0.921607\pi$
$678$		0			0
$679$	10.3309			0.396463
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	45.0528i		−	1.72390i	−0.506995	−	0.861949i	$-0.669244\pi$
$683$		−	45.0528i		−	1.72390i	0.506995	−	0.861949i	$-0.330756\pi$
$684$		0			0
$685$	2.90950	+	38.9936i	0.111166	+	1.48987i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.75328			−0.371570
$690$		0			0
$691$	−35.9799			−1.36874			−0.684370	−	0.729135i	$-0.739923\pi$
$691$	−35.9799			−1.36874			−0.684370	+	0.729135i	$0.739923\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	41.6787	−	3.10985i	1.58096	−	0.117963i
$696$		0			0
$697$		−	49.8546i		−	1.88838i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	19.5731			0.739264			0.369632	−	0.929178i	$-0.379484\pi$
$701$	19.5731			0.739264			0.369632	+	0.929178i	$0.379484\pi$
$702$		0			0
$703$		−	7.86644i		−	0.296689i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	6.39184i		−	0.240390i
$708$		0			0
$709$	−2.22794			−0.0836721			−0.0418361	−	0.999124i	$-0.513321\pi$
$709$	−2.22794			−0.0836721			−0.0418361	+	0.999124i	$0.513321\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			1.61463i			0.0604685i
$714$		0			0
$715$	−1.39842	−	18.7418i	−0.0522979	−	0.700905i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−31.6309			−1.17964			−0.589818	−	0.807537i	$-0.700800\pi$
$719$	−31.6309			−1.17964			−0.589818	+	0.807537i	$0.700800\pi$
$720$		0			0
$721$	−7.39958			−0.275575
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−20.8683	+	3.13160i	−0.775027	+	0.116305i
$726$		0			0
$727$		−	50.2224i		−	1.86264i	−0.364196	−	0.931322i	$-0.618656\pi$
$727$		−	50.2224i		−	1.86264i	0.364196	−	0.931322i	$-0.381344\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−15.9382			−0.589494
$732$		0			0
$733$		−	32.1706i		−	1.18825i	−0.804374	−	0.594124i	$-0.797499\pi$
$733$		−	32.1706i		−	1.18825i	0.804374	−	0.594124i	$-0.202501\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	5.18361i		−	0.190941i
$738$		0			0
$739$	10.7707			0.396206			0.198103	−	0.980181i	$-0.436522\pi$
$739$	10.7707			0.396206			0.198103	+	0.980181i	$0.436522\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			45.4354i			1.66686i	0.552622	+	0.833432i	$0.313627\pi$
$743$			45.4354i			1.66686i	−0.552622	+	0.833432i	$0.686373\pi$
$744$		0			0
$745$	36.2733	−	2.70653i	1.32895	−	0.0991594i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−30.4163			−1.11139
$750$		0			0
$751$	2.04253			0.0745329			0.0372664	−	0.999305i	$-0.488135\pi$
$751$	2.04253			0.0745329			0.0372664	+	0.999305i	$0.488135\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	25.0564	−	1.86958i	0.911896	−	0.0680410i
$756$		0			0
$757$		−	28.7771i		−	1.04592i	−0.852357	−	0.522960i	$-0.824828\pi$
$757$		−	28.7771i		−	1.04592i	0.852357	−	0.522960i	$-0.175172\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13.2842			−0.481551			−0.240776	−	0.970581i	$-0.577402\pi$
$761$	−13.2842			−0.481551			−0.240776	+	0.970581i	$0.577402\pi$
$762$		0			0
$763$			2.02573i			0.0733363i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	44.0928i		−	1.59210i
$768$		0			0
$769$	−22.7986			−0.822137			−0.411069	−	0.911604i	$-0.634844\pi$
$769$	−22.7986			−0.822137			−0.411069	+	0.911604i	$0.634844\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			2.40585i			0.0865323i	0.999064	+	0.0432662i	$0.0137763\pi$
$773$			2.40585i			0.0865323i	−0.999064	+	0.0432662i	$0.986224\pi$
$774$		0			0
$775$	7.98378	−	1.19809i	0.286786	−	0.0430366i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	16.3725			0.586604
$780$		0			0
$781$	32.5649			1.16527
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.30394	−	17.4756i	−0.0465397	−	0.623733i
$786$		0			0
$787$		−	5.69270i		−	0.202923i	−0.994839	−	0.101461i	$-0.967648\pi$
$787$		−	5.69270i		−	0.202923i	0.994839	−	0.101461i	$-0.0323519\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	21.5480			0.766157
$792$		0			0
$793$			11.7448i			0.417069i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			8.19237i			0.290189i	0.989418	+	0.145094i	$0.0463486\pi$
$797$			8.19237i			0.290189i	−0.989418	+	0.145094i	$0.953651\pi$
$798$		0			0
$799$	−48.2071			−1.70544
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	11.9660i		−	0.422269i
$804$		0			0
$805$	3.88447	−	0.289839i	0.136910	−	0.0102155i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.42195			0.190626			0.0953129	−	0.995447i	$-0.469615\pi$
$809$	5.42195			0.190626			0.0953129	+	0.995447i	$0.469615\pi$
$810$		0			0
$811$	−26.8462			−0.942697			−0.471348	−	0.881947i	$-0.656233\pi$
$811$	−26.8462			−0.942697			−0.471348	+	0.881947i	$0.656233\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.51663	−	33.7282i	−0.0881536	−	1.18145i
$816$		0			0
$817$		−	5.23415i		−	0.183120i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	17.4428			0.608756			0.304378	−	0.952551i	$-0.401551\pi$
$821$	17.4428			0.608756			0.304378	+	0.952551i	$0.401551\pi$
$822$		0			0
$823$		−	22.8108i		−	0.795136i	−0.917573	−	0.397568i	$-0.869854\pi$
$823$		−	22.8108i		−	0.795136i	0.917573	−	0.397568i	$-0.130146\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	19.4023i		−	0.674684i	−0.941382	−	0.337342i	$-0.890472\pi$
$827$		−	19.4023i		−	0.674684i	0.941382	−	0.337342i	$-0.109528\pi$
$828$		0			0
$829$	1.12685			0.0391371			0.0195686	−	0.999809i	$-0.493771\pi$
$829$	1.12685			0.0391371			0.0195686	+	0.999809i	$0.493771\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			23.3466i			0.808913i
$834$		0			0
$835$	1.77255	+	23.7560i	0.0613416	+	0.822109i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	11.7398			0.405303			0.202652	−	0.979251i	$-0.435044\pi$
$839$	11.7398			0.405303			0.202652	+	0.979251i	$0.435044\pi$
$840$		0			0
$841$	−11.1884			−0.385806
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.63083	−	0.121684i	0.0561023	−	0.00418607i
$846$		0			0
$847$			9.13172i			0.313770i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.06846			−0.139465
$852$		0			0
$853$		−	16.1382i		−	0.552562i	−0.961077	−	0.276281i	$-0.910898\pi$
$853$		−	16.1382i		−	0.552562i	0.961077	−	0.276281i	$-0.0891020\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			36.7856i			1.25657i	0.777983	+	0.628285i	$0.216243\pi$
$857$			36.7856i			1.25657i	−0.777983	+	0.628285i	$0.783757\pi$
$858$		0			0
$859$	32.5839			1.11175			0.555875	−	0.831266i	$-0.312383\pi$
$859$	32.5839			1.11175			0.555875	+	0.831266i	$0.312383\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			38.1553i			1.29882i	0.760437	+	0.649412i	$0.224985\pi$
$863$			38.1553i			1.29882i	−0.760437	+	0.649412i	$0.775015\pi$
$864$		0			0
$865$	1.65809	+	22.2220i	0.0563769	+	0.755572i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−15.0086			−0.509132
$870$		0			0
$871$	−7.56652			−0.256382
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.31550	−	18.9922i	−0.145890	−	0.642054i
$876$		0			0
$877$			31.3496i			1.05860i	0.848435	+	0.529300i	$0.177545\pi$
$877$			31.3496i			1.05860i	−0.848435	+	0.529300i	$0.822455\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−37.7277			−1.27108			−0.635539	−	0.772069i	$-0.719222\pi$
$881$	−37.7277			−1.27108			−0.635539	+	0.772069i	$0.719222\pi$
$882$		0			0
$883$		−	11.5400i		−	0.388353i	−0.980967	−	0.194177i	$-0.937796\pi$
$883$		−	11.5400i		−	0.388353i	0.980967	−	0.194177i	$-0.0622035\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	36.5554i		−	1.22741i	−0.789535	−	0.613705i	$-0.789678\pi$
$887$		−	36.5554i		−	1.22741i	0.789535	−	0.613705i	$-0.210322\pi$
$888$		0			0
$889$	13.4289			0.450392
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	15.8314i		−	0.529778i
$894$		0			0
$895$	−22.6508	+	1.69009i	−0.757133	+	0.0564934i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−6.81438			−0.227272
$900$		0			0
$901$	−16.3943			−0.546173
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−24.9508	+	1.86170i	−0.829393	+	0.0618851i
$906$		0			0
$907$			57.6553i			1.91441i	0.289408	+	0.957206i	$0.406542\pi$
$907$			57.6553i			1.91441i	−0.289408	+	0.957206i	$0.593458\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−17.7441			−0.587887			−0.293943	−	0.955823i	$-0.594968\pi$
$911$	−17.7441			−0.587887			−0.293943	+	0.955823i	$0.594968\pi$
$912$		0			0
$913$		−	3.73100i		−	0.123478i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			0.170880i			0.00564295i
$918$		0			0
$919$	51.0199			1.68299			0.841495	−	0.540265i	$-0.181676\pi$
$919$	51.0199			1.68299			0.841495	+	0.540265i	$0.181676\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	47.5351i		−	1.56464i
$924$		0			0
$925$	3.01887	+	20.1171i	0.0992598	+	0.661445i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	46.8225			1.53620			0.768099	−	0.640331i	$-0.221203\pi$
$929$	46.8225			1.53620			0.768099	+	0.640331i	$0.221203\pi$
$930$		0			0
$931$	−7.66712			−0.251280
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.35060	−	31.5031i	−0.0768729	−	1.03026i
$936$		0			0
$937$		−	2.43665i		−	0.0796018i	−0.999208	−	0.0398009i	$-0.987328\pi$
$937$		−	2.43665i		−	0.0796018i	0.999208	−	0.0398009i	$-0.0126724\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.1291			−0.460597			−0.230298	−	0.973120i	$-0.573970\pi$
$941$	−14.1291			−0.460597			−0.230298	+	0.973120i	$0.573970\pi$
$942$		0			0
$943$		−	8.46771i		−	0.275746i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	7.60118i		−	0.247005i	−0.992344	−	0.123503i	$-0.960587\pi$
$947$		−	7.60118i		−	0.247005i	0.992344	−	0.123503i	$-0.0394127\pi$
$948$		0			0
$949$	−17.4667			−0.566994
$950$		0			0
$951$		0			0
$952$		0			0
$953$			37.3732i			1.21064i	0.795983	+	0.605319i	$0.206954\pi$
$953$			37.3732i			1.21064i	−0.795983	+	0.605319i	$0.793046\pi$
$954$		0			0
$955$	14.5400	−	1.08490i	0.470503	−	0.0351066i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	30.4625			0.983686
$960$		0			0
$961$	−28.3930			−0.915902
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.99763	−	40.1747i	−0.0964970	−	1.29327i
$966$		0			0
$967$			31.9761i			1.02828i	0.857706	+	0.514141i	$0.171889\pi$
$967$			31.9761i			1.02828i	−0.857706	+	0.514141i	$0.828111\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	30.1480			0.967494			0.483747	−	0.875208i	$-0.339275\pi$
$971$	30.1480			0.967494			0.483747	+	0.875208i	$0.339275\pi$
$972$		0			0
$973$		−	32.5603i		−	1.04383i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	2.90839i		−	0.0930476i	−0.998917	−	0.0465238i	$-0.985186\pi$
$977$		−	2.90839i		−	0.0930476i	0.998917	−	0.0465238i	$-0.0148143\pi$
$978$		0			0
$979$	−17.1441			−0.547927
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	50.3154i		−	1.60481i	−0.596778	−	0.802407i	$-0.703553\pi$
$983$		−	50.3154i		−	1.60481i	0.596778	−	0.802407i	$-0.296447\pi$
$984$		0			0
$985$	−0.620884	−	8.32118i	−0.0197830	−	0.265135i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.70706			−0.0860795
$990$		0			0
$991$	34.1637			1.08524			0.542622	−	0.839977i	$-0.317431\pi$
$991$	34.1637			1.08524			0.542622	+	0.839977i	$0.317431\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−32.2027	+	2.40280i	−1.02089	+	0.0761738i
$996$		0			0
$997$		−	60.0051i		−	1.90038i	−0.311672	−	0.950190i	$-0.600889\pi$
$997$		−	60.0051i		−	1.90038i	0.311672	−	0.950190i	$-0.399111\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.c.829.13		14
3.2	odd	2		1380.2.f.b.829.8	yes	14
5.4	even	2	inner	4140.2.f.c.829.14		14
15.2	even	4		6900.2.a.bd.1.5		7
15.8	even	4		6900.2.a.bc.1.3		7
15.14	odd	2		1380.2.f.b.829.1	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.b.829.1	✓	14	15.14	odd	2
1380.2.f.b.829.8	yes	14	3.2	odd	2
4140.2.f.c.829.13		14	1.1	even	1	trivial
4140.2.f.c.829.14		14	5.4	even	2	inner
6900.2.a.bc.1.3		7	15.8	even	4
6900.2.a.bd.1.5		7	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.22987 - 0.166381i) q^{5} -1.74202i q^{7} +O(q^{10})\)	\(q+(2.22987 - 0.166381i) q^{5} -1.74202i q^{7} -2.39958 q^{11} +3.50266i q^{13} +5.88762i q^{17} -1.93352 q^{19} +1.00000i q^{23} +(4.94463 - 0.742018i) q^{25} -4.22038 q^{29} +1.61463 q^{31} +(-0.289839 - 3.88447i) q^{35} +4.06846i q^{37} -8.46771 q^{41} +2.70706i q^{43} +8.18788i q^{47} +3.96537 q^{49} +2.78453i q^{53} +(-5.35074 + 0.399245i) q^{55} -12.5884 q^{59} +3.35310 q^{61} +(0.582778 + 7.81048i) q^{65} +2.16022i q^{67} -13.5711 q^{71} +4.98670i q^{73} +4.18010i q^{77} +6.25469 q^{79} +1.55486i q^{83} +(0.979591 + 13.1286i) q^{85} +7.14462 q^{89} +6.10170 q^{91} +(-4.31149 + 0.321701i) q^{95} +5.93042i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q+O(q^{10}) \) 14 * q	\( 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) \) 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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