LMFDB - Embedded newform 72.7.e.b.17.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,7,Mod(17,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("72.17");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	72.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:	$16.5638940206$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{145})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 67x^{2} + 68x + 1446 $ x^4 - 2x^3 - 67x^2 + 68*x + 1446
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.3
Root			$-5.52080 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	72.17
Dual form			72.7.e.b.17.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+168.997i q^{5} +180.998 q^{7} +O(q^{10})$	$q+168.997i q^{5} +180.998 q^{7} -485.068i q^{11} -3110.99 q^{13} +2718.12i q^{17} -4424.01 q^{19} +23211.4i q^{23} -12935.1 q^{25} -2715.92i q^{29} -51902.8 q^{31} +30588.2i q^{35} -81213.9 q^{37} +1725.03i q^{41} +149257. q^{43} +132021. i q^{47} -84888.6 q^{49} -216746. i q^{53} +81975.2 q^{55} -98714.1i q^{59} +197389. q^{61} -525749. i q^{65} +299275. q^{67} +388685. i q^{71} +233475. q^{73} -87796.5i q^{77} +622981. q^{79} -38949.7i q^{83} -459354. q^{85} +38826.5i q^{89} -563083. q^{91} -747646. i q^{95} -1.00974e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 432 q^{7}+O(q^{10}) $ 4 * q - 432 * q^7	$ 4 q - 432 q^{7} - 4352 q^{13} - 26944 q^{19} - 109540 q^{25} - 56176 q^{31} - 237000 q^{37} + 2848 q^{43} - 89860 q^{49} + 1107040 q^{55} + 146824 q^{61} + 1546208 q^{67} + 281920 q^{73} + 1258480 q^{79} - 7480 q^{85} - 1868544 q^{91} - 1322368 q^{97}+O(q^{100}) $ 4 * q - 432 * q^7 - 4352 * q^13 - 26944 * q^19 - 109540 * q^25 - 56176 * q^31 - 237000 * q^37 + 2848 * q^43 - 89860 * q^49 + 1107040 * q^55 + 146824 * q^61 + 1546208 * q^67 + 281920 * q^73 + 1258480 * q^79 - 7480 * q^85 - 1868544 * q^91 - 1322368 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	168.997i	1.35198i	0.736912	+	0.675989i	$0.236283\pi$
$5$	168.997i	1.35198i	−0.736912	+	0.675989i	$0.763717\pi$
$6$	0	0
$7$	180.998	0.527692	0.263846	−	0.964565i	$-0.415009\pi$
$7$	180.998	0.527692	0.263846	+	0.964565i	$0.415009\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 485.068i	− 0.364439i	−0.983258	−	0.182219i	$-0.941672\pi$
$11$	− 485.068i	− 0.364439i	0.983258	−	0.182219i	$-0.0583281\pi$
$12$	0	0
$13$	−3110.99	−1.41602	−0.708008	−	0.706204i	$-0.750406\pi$
$13$	−3110.99	−1.41602	−0.708008	+	0.706204i	$0.750406\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2718.12i	0.553250i	0.960978	+	0.276625i	$0.0892159\pi$
$17$	2718.12i	0.553250i	−0.960978	+	0.276625i	$0.910784\pi$
$18$	0	0
$19$	−4424.01	−0.644994	−0.322497	−	0.946570i	$-0.604522\pi$
$19$	−4424.01	−0.644994	−0.322497	+	0.946570i	$0.604522\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	23211.4i	1.90774i	0.300226	+	0.953868i	$0.402938\pi$
$23$	23211.4i	1.90774i	−0.300226	+	0.953868i	$0.597062\pi$
$24$	0	0
$25$	−12935.1	−0.827846
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2715.92i	− 0.111358i	−0.998449	−	0.0556791i	$-0.982268\pi$
$29$	− 2715.92i	− 0.111358i	0.998449	−	0.0556791i	$-0.0177324\pi$
$30$	0	0
$31$	−51902.8	−1.74223	−0.871115	−	0.491079i	$-0.836603\pi$
$31$	−51902.8	−1.74223	−0.871115	+	0.491079i	$0.836603\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	30588.2i	0.713428i
$36$	0	0
$37$	−81213.9	−1.60334	−0.801669	−	0.597768i	$-0.796054\pi$
$37$	−81213.9	−1.60334	−0.801669	+	0.597768i	$0.796054\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1725.03i	0.0250292i	0.999922	+	0.0125146i	$0.00398362\pi$
$41$	1725.03i	0.0250292i	−0.999922	+	0.0125146i	$0.996016\pi$
$42$	0	0
$43$	149257.	1.87728	0.938641	−	0.344895i	$-0.112085\pi$
$43$	149257.	1.87728	0.938641	+	0.344895i	$0.112085\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	132021.i	1.27159i	0.771856	+	0.635797i	$0.219328\pi$
$47$	132021.i	1.27159i	−0.771856	+	0.635797i	$0.780672\pi$
$48$	0	0
$49$	−84888.6	−0.721541
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 216746.i	− 1.45587i	−0.685646	−	0.727935i	$-0.740480\pi$
$53$	− 216746.i	− 1.45587i	0.685646	−	0.727935i	$-0.259520\pi$
$54$	0	0
$55$	81975.2	0.492713
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 98714.1i	− 0.480643i	−0.970693	−	0.240322i	$-0.922747\pi$
$59$	− 98714.1i	− 0.480643i	0.970693	−	0.240322i	$-0.0772529\pi$
$60$	0	0
$61$	197389.	0.869628	0.434814	−	0.900520i	$-0.356814\pi$
$61$	197389.	0.869628	0.434814	+	0.900520i	$0.356814\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 525749.i	− 1.91442i
$66$	0	0
$67$	299275.	0.995051	0.497525	−	0.867449i	$-0.334242\pi$
$67$	299275.	0.995051	0.497525	+	0.867449i	$0.334242\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	388685.i	1.08598i	0.839738	+	0.542991i	$0.182708\pi$
$71$	388685.i	1.08598i	−0.839738	+	0.542991i	$0.817292\pi$
$72$	0	0
$73$	233475.	0.600167	0.300083	−	0.953913i	$-0.402986\pi$
$73$	233475.	0.600167	0.300083	+	0.953913i	$0.402986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 87796.5i	− 0.192311i
$78$	0	0
$79$	622981.	1.26355	0.631777	−	0.775150i	$-0.282326\pi$
$79$	622981.	1.26355	0.631777	+	0.775150i	$0.282326\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 38949.7i	− 0.0681192i	−0.999420	−	0.0340596i	$-0.989156\pi$
$83$	− 38949.7i	− 0.0681192i	0.999420	−	0.0340596i	$-0.0108436\pi$
$84$	0	0
$85$	−459354.	−0.747982
$86$	0	0
$87$	0	0
$88$	0	0
$89$	38826.5i	0.0550755i	0.999621	+	0.0275377i	$0.00876664\pi$
$89$	38826.5i	0.0550755i	−0.999621	+	0.0275377i	$0.991233\pi$
$90$	0	0
$91$	−563083.	−0.747220
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 747646.i	− 0.872018i
$96$	0	0
$97$	−1.00974e6	−1.10635	−0.553176	−	0.833064i	$-0.686584\pi$
$97$	−1.00974e6	−1.10635	−0.553176	+	0.833064i	$0.686584\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	682986.i	0.662899i	0.943473	+	0.331450i	$0.107538\pi$
$101$	682986.i	0.662899i	−0.943473	+	0.331450i	$0.892462\pi$
$102$	0	0
$103$	−51914.6	−0.0475092	−0.0237546	−	0.999718i	$-0.507562\pi$
$103$	−51914.6	−0.0475092	−0.0237546	+	0.999718i	$0.507562\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.29346e6i	1.05585i	0.849290	+	0.527926i	$0.177030\pi$
$107$	1.29346e6i	1.05585i	−0.849290	+	0.527926i	$0.822970\pi$
$108$	0	0
$109$	422140.	0.325969	0.162985	−	0.986629i	$-0.447888\pi$
$109$	422140.	0.325969	0.162985	+	0.986629i	$0.447888\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.37979e6i	1.64932i	0.565632	+	0.824658i	$0.308632\pi$
$113$	2.37979e6i	1.64932i	−0.565632	+	0.824658i	$0.691368\pi$
$114$	0	0
$115$	−3.92267e6	−2.57922
$116$	0	0
$117$	0	0
$118$	0	0
$119$	491974.i	0.291945i
$120$	0	0
$121$	1.53627e6	0.867184
$122$	0	0
$123$	0	0
$124$	0	0
$125$	454588.i	0.232749i
$126$	0	0
$127$	−306053.	−0.149412	−0.0747059	−	0.997206i	$-0.523802\pi$
$127$	−306053.	−0.149412	−0.0747059	+	0.997206i	$0.523802\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 4.06776e6i	− 1.80943i	−0.426018	−	0.904715i	$-0.640084\pi$
$131$	− 4.06776e6i	− 1.80943i	0.426018	−	0.904715i	$-0.359916\pi$
$132$	0	0
$133$	−800739.	−0.340358
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 729562.i	− 0.283727i	−0.989886	−	0.141864i	$-0.954691\pi$
$137$	− 729562.i	− 0.283727i	0.989886	−	0.141864i	$-0.0453094\pi$
$138$	0	0
$139$	−758683.	−0.282498	−0.141249	−	0.989974i	$-0.545112\pi$
$139$	−758683.	−0.282498	−0.141249	+	0.989974i	$0.545112\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.50904e6i	0.516051i
$144$	0	0
$145$	458982.	0.150554
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.73416e6i	0.826543i	0.910608	+	0.413271i	$0.135614\pi$
$149$	2.73416e6i	0.826543i	−0.910608	+	0.413271i	$0.864386\pi$
$150$	0	0
$151$	4.25336e6	1.23538	0.617691	−	0.786421i	$-0.288068\pi$
$151$	4.25336e6	1.23538	0.617691	+	0.786421i	$0.288068\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 8.77143e6i	− 2.35546i
$156$	0	0
$157$	4.95708e6	1.28093	0.640467	−	0.767985i	$-0.278741\pi$
$157$	4.95708e6	1.28093	0.640467	+	0.767985i	$0.278741\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.20123e6i	1.00670i
$162$	0	0
$163$	3.16283e6	0.730321	0.365160	−	0.930945i	$-0.381014\pi$
$163$	3.16283e6	0.730321	0.365160	+	0.930945i	$0.381014\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.99796e6i	0.428980i	0.976726	+	0.214490i	$0.0688089\pi$
$167$	1.99796e6i	0.428980i	−0.976726	+	0.214490i	$0.931191\pi$
$168$	0	0
$169$	4.85144e6	1.00510
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 7.63829e6i	− 1.47522i	−0.675225	−	0.737612i	$-0.735954\pi$
$173$	− 7.63829e6i	− 1.47522i	0.675225	−	0.737612i	$-0.264046\pi$
$174$	0	0
$175$	−2.34123e6	−0.436847
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1.95941e6i	− 0.341639i	−0.985302	−	0.170819i	$-0.945359\pi$
$179$	− 1.95941e6i	− 0.341639i	0.985302	−	0.170819i	$-0.0546414\pi$
$180$	0	0
$181$	−1.04105e6	−0.175564	−0.0877821	−	0.996140i	$-0.527978\pi$
$181$	−1.04105e6	−0.175564	−0.0877821	+	0.996140i	$0.527978\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1.37249e7i	− 2.16768i
$186$	0	0
$187$	1.31847e6	0.201626
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.12572e6i	1.30969i	0.755765	+	0.654843i	$0.227265\pi$
$191$	9.12572e6i	1.30969i	−0.755765	+	0.654843i	$0.772735\pi$
$192$	0	0
$193$	−5.19028e6	−0.721970	−0.360985	−	0.932572i	$-0.617559\pi$
$193$	−5.19028e6	−0.721970	−0.360985	+	0.932572i	$0.617559\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.38622e6i	1.22770i	0.789423	+	0.613850i	$0.210380\pi$
$197$	9.38622e6i	1.22770i	−0.789423	+	0.613850i	$0.789620\pi$
$198$	0	0
$199$	1.53784e6	0.195143	0.0975715	−	0.995229i	$-0.468893\pi$
$199$	1.53784e6	0.195143	0.0975715	+	0.995229i	$0.468893\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 491576.i	− 0.0587628i
$204$	0	0
$205$	−291526.	−0.0338389
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.14595e6i	0.235061i
$210$	0	0
$211$	−5.42827e6	−0.577849	−0.288924	−	0.957352i	$-0.593298\pi$
$211$	−5.42827e6	−0.577849	−0.288924	+	0.957352i	$0.593298\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.52240e7i	2.53805i
$216$	0	0
$217$	−9.39431e6	−0.919360
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 8.45603e6i	− 0.783411i
$222$	0	0
$223$	1.69263e7	1.52632	0.763162	−	0.646208i	$-0.223646\pi$
$223$	1.69263e7	1.52632	0.763162	+	0.646208i	$0.223646\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.55170e7i	− 1.32657i	−0.748366	−	0.663286i	$-0.769161\pi$
$227$	− 1.55170e7i	− 1.32657i	0.748366	−	0.663286i	$-0.230839\pi$
$228$	0	0
$229$	−2.30906e7	−1.92277	−0.961387	−	0.275200i	$-0.911256\pi$
$229$	−2.30906e7	−1.92277	−0.961387	+	0.275200i	$0.911256\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.34236e7i	1.06121i	0.847620	+	0.530603i	$0.178035\pi$
$233$	1.34236e7i	1.06121i	−0.847620	+	0.530603i	$0.821965\pi$
$234$	0	0
$235$	−2.23111e7	−1.71917
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.40267e7i	1.02745i	0.857955	+	0.513725i	$0.171735\pi$
$239$	1.40267e7i	1.02745i	−0.857955	+	0.513725i	$0.828265\pi$
$240$	0	0
$241$	−2.04836e7	−1.46337	−0.731686	−	0.681642i	$-0.761266\pi$
$241$	−2.04836e7	−1.46337	−0.731686	+	0.681642i	$0.761266\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.43459e7i	− 0.975508i
$246$	0	0
$247$	1.37631e7	0.913322
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.10216e7i	0.696986i	0.937311	+	0.348493i	$0.113307\pi$
$251$	1.10216e7i	0.696986i	−0.937311	+	0.348493i	$0.886693\pi$
$252$	0	0
$253$	1.12591e7	0.695253
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.25146e7i	− 0.737257i	−0.929577	−	0.368628i	$-0.879828\pi$
$257$	− 1.25146e7i	− 0.737257i	0.929577	−	0.368628i	$-0.120172\pi$
$258$	0	0
$259$	−1.46996e7	−0.846068
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.73610e6i	− 0.150406i	−0.997168	−	0.0752029i	$-0.976040\pi$
$263$	− 2.73610e6i	− 0.150406i	0.997168	−	0.0752029i	$-0.0239605\pi$
$264$	0	0
$265$	3.66294e7	1.96831
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.85252e7i	0.951712i	0.879523	+	0.475856i	$0.157862\pi$
$269$	1.85252e7i	0.951712i	−0.879523	+	0.475856i	$0.842138\pi$
$270$	0	0
$271$	2.25588e7	1.13346	0.566732	−	0.823902i	$-0.308207\pi$
$271$	2.25588e7	1.13346	0.566732	+	0.823902i	$0.308207\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.27440e6i	0.301699i
$276$	0	0
$277$	−5.15195e6	−0.242400	−0.121200	−	0.992628i	$-0.538674\pi$
$277$	−5.15195e6	−0.242400	−0.121200	+	0.992628i	$0.538674\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 2.00914e7i	− 0.905505i	−0.891636	−	0.452753i	$-0.850442\pi$
$281$	− 2.00914e7i	− 0.905505i	0.891636	−	0.452753i	$-0.149558\pi$
$282$	0	0
$283$	−1.19830e7	−0.528696	−0.264348	−	0.964427i	$-0.585157\pi$
$283$	−1.19830e7	−0.528696	−0.264348	+	0.964427i	$0.585157\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	312228.i	0.0132077i
$288$	0	0
$289$	1.67494e7	0.693915
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.83911e7i	− 0.731147i	−0.930782	−	0.365573i	$-0.880873\pi$
$293$	− 1.83911e7i	− 0.731147i	0.930782	−	0.365573i	$-0.119127\pi$
$294$	0	0
$295$	1.66824e7	0.649820
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 7.22105e7i	− 2.70139i
$300$	0	0
$301$	2.70153e7	0.990627
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.33582e7i	1.17572i
$306$	0	0
$307$	1.58218e6	0.0546816	0.0273408	−	0.999626i	$-0.491296\pi$
$307$	1.58218e6	0.0546816	0.0273408	+	0.999626i	$0.491296\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 4.21576e7i	− 1.40150i	−0.713405	−	0.700752i	$-0.752848\pi$
$311$	− 4.21576e7i	− 1.40150i	0.713405	−	0.700752i	$-0.247152\pi$
$312$	0	0
$313$	−2.33332e7	−0.760926	−0.380463	−	0.924796i	$-0.624235\pi$
$313$	−2.33332e7	−0.760926	−0.380463	+	0.924796i	$0.624235\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.84387e7i	0.578831i	0.957204	+	0.289416i	$0.0934610\pi$
$317$	1.84387e7i	0.578831i	−0.957204	+	0.289416i	$0.906539\pi$
$318$	0	0
$319$	−1.31740e6	−0.0405832
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1.20250e7i	− 0.356843i
$324$	0	0
$325$	4.02409e7	1.17224
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.38955e7i	0.671010i
$330$	0	0
$331$	3.18480e7	0.878209	0.439104	−	0.898436i	$-0.355296\pi$
$331$	3.18480e7	0.878209	0.439104	+	0.898436i	$0.355296\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.05766e7i	1.34529i
$336$	0	0
$337$	3.09382e7	0.808361	0.404181	−	0.914679i	$-0.367557\pi$
$337$	3.09382e7	0.808361	0.404181	+	0.914679i	$0.367557\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.51764e7i	0.634936i
$342$	0	0
$343$	−3.66590e7	−0.908443
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.63459e7i	− 0.869896i	−0.900455	−	0.434948i	$-0.856767\pi$
$347$	− 3.63459e7i	− 0.869896i	0.900455	−	0.434948i	$-0.143233\pi$
$348$	0	0
$349$	−4.10721e7	−0.966208	−0.483104	−	0.875563i	$-0.660491\pi$
$349$	−4.10721e7	−0.966208	−0.483104	+	0.875563i	$0.660491\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2.53325e7i	− 0.575910i	−0.957644	−	0.287955i	$-0.907025\pi$
$353$	− 2.53325e7i	− 0.575910i	0.957644	−	0.287955i	$-0.0929753\pi$
$354$	0	0
$355$	−6.56867e7	−1.46822
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.26577e7i	1.35423i	0.735879	+	0.677113i	$0.236769\pi$
$359$	6.26577e7i	1.35423i	−0.735879	+	0.677113i	$0.763231\pi$
$360$	0	0
$361$	−2.74740e7	−0.583983
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.94566e7i	0.811412i
$366$	0	0
$367$	−4.87723e7	−0.986677	−0.493338	−	0.869837i	$-0.664224\pi$
$367$	−4.87723e7	−0.986677	−0.493338	+	0.869837i	$0.664224\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 3.92306e7i	− 0.768251i
$372$	0	0
$373$	2.14563e7	0.413454	0.206727	−	0.978399i	$-0.433719\pi$
$373$	2.14563e7	0.413454	0.206727	+	0.978399i	$0.433719\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.44918e6i	0.157685i
$378$	0	0
$379$	−4.78716e7	−0.879347	−0.439674	−	0.898158i	$-0.644906\pi$
$379$	−4.78716e7	−0.879347	−0.439674	+	0.898158i	$0.644906\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 4.60632e7i	− 0.819894i	−0.912109	−	0.409947i	$-0.865547\pi$
$383$	− 4.60632e7i	− 0.819894i	0.912109	−	0.409947i	$-0.134453\pi$
$384$	0	0
$385$	1.48374e7	0.260001
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 7.96521e6i	− 0.135316i	−0.997709	−	0.0676579i	$-0.978447\pi$
$389$	− 7.96521e6i	− 0.135316i	0.997709	−	0.0676579i	$-0.0215526\pi$
$390$	0	0
$391$	−6.30913e7	−1.05545
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.05282e8i	1.70830i
$396$	0	0
$397$	−5.77919e7	−0.923625	−0.461813	−	0.886977i	$-0.652801\pi$
$397$	−5.77919e7	−0.923625	−0.461813	+	0.886977i	$0.652801\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.07572e7i	0.942247i	0.882067	+	0.471124i	$0.156151\pi$
$401$	6.07572e7i	0.942247i	−0.882067	+	0.471124i	$0.843849\pi$
$402$	0	0
$403$	1.61469e8	2.46703
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.93942e7i	0.584318i
$408$	0	0
$409$	−9.98427e7	−1.45931	−0.729653	−	0.683818i	$-0.760318\pi$
$409$	−9.98427e7	−1.45931	−0.729653	+	0.683818i	$0.760318\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.78671e7i	− 0.253632i
$414$	0	0
$415$	6.58239e6	0.0920957
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.78526e7i	1.05835i	0.848511	+	0.529177i	$0.177499\pi$
$419$	7.78526e7i	1.05835i	−0.848511	+	0.529177i	$0.822501\pi$
$420$	0	0
$421$	−3.16370e7	−0.423984	−0.211992	−	0.977271i	$-0.567995\pi$
$421$	−3.16370e7	−0.423984	−0.211992	+	0.977271i	$0.567995\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 3.51591e7i	− 0.458005i
$426$	0	0
$427$	3.57271e7	0.458896
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 1.08160e8i	− 1.35094i	−0.737387	−	0.675471i	$-0.763940\pi$
$431$	− 1.08160e8i	− 1.35094i	0.737387	−	0.675471i	$-0.236060\pi$
$432$	0	0
$433$	8.52784e7	1.05045	0.525225	−	0.850963i	$-0.323981\pi$
$433$	8.52784e7	1.05045	0.525225	+	0.850963i	$0.323981\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.02688e8i	− 1.23048i
$438$	0	0
$439$	9.54232e7	1.12787	0.563937	−	0.825818i	$-0.309286\pi$
$439$	9.54232e7	1.12787	0.563937	+	0.825818i	$0.309286\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.74043e7i	0.315216i	0.987502	+	0.157608i	$0.0503782\pi$
$443$	2.74043e7i	0.315216i	−0.987502	+	0.157608i	$0.949622\pi$
$444$	0	0
$445$	−6.56157e6	−0.0744609
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 1.13018e8i	− 1.24856i	−0.781201	−	0.624280i	$-0.785392\pi$
$449$	− 1.13018e8i	− 1.24856i	0.781201	−	0.624280i	$-0.214608\pi$
$450$	0	0
$451$	836759.	0.00912159
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 9.51596e7i	− 1.01023i
$456$	0	0
$457$	−7.30658e7	−0.765536	−0.382768	−	0.923844i	$-0.625029\pi$
$457$	−7.30658e7	−0.765536	−0.382768	+	0.923844i	$0.625029\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 2.79729e7i	− 0.285519i	−0.989757	−	0.142759i	$-0.954402\pi$
$461$	− 2.79729e7i	− 0.285519i	0.989757	−	0.142759i	$-0.0455975\pi$
$462$	0	0
$463$	4.47324e7	0.450692	0.225346	−	0.974279i	$-0.427649\pi$
$463$	4.47324e7	0.450692	0.225346	+	0.974279i	$0.427649\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.37576e7i	0.822382i	0.911549	+	0.411191i	$0.134887\pi$
$467$	8.37576e7i	0.822382i	−0.911549	+	0.411191i	$0.865113\pi$
$468$	0	0
$469$	5.41682e7	0.525080
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 7.23998e7i	− 0.684154i
$474$	0	0
$475$	5.72250e7	0.533955
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.04717e8i	0.952820i	0.879223	+	0.476410i	$0.158062\pi$
$479$	1.04717e8i	0.952820i	−0.879223	+	0.476410i	$0.841938\pi$
$480$	0	0
$481$	2.52655e8	2.27035
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.70643e8i	− 1.49576i
$486$	0	0
$487$	8.20131e7	0.710062	0.355031	−	0.934855i	$-0.384470\pi$
$487$	8.20131e7	0.710062	0.355031	+	0.934855i	$0.384470\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.91913e8i	1.62129i	0.585538	+	0.810645i	$0.300883\pi$
$491$	1.91913e8i	1.62129i	−0.585538	+	0.810645i	$0.699117\pi$
$492$	0	0
$493$	7.38217e6	0.0616089
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.03513e7i	0.573064i
$498$	0	0
$499$	−1.00704e8	−0.810488	−0.405244	−	0.914209i	$-0.632813\pi$
$499$	−1.00704e8	−0.810488	−0.405244	+	0.914209i	$0.632813\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 2.28028e8i	− 1.79178i	−0.444276	−	0.895890i	$-0.646539\pi$
$503$	− 2.28028e8i	− 1.79178i	0.444276	−	0.895890i	$-0.353461\pi$
$504$	0	0
$505$	−1.15423e8	−0.896226
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.96573e8i	1.49063i	0.666712	+	0.745315i	$0.267701\pi$
$509$	1.96573e8i	1.49063i	−0.666712	+	0.745315i	$0.732299\pi$
$510$	0	0
$511$	4.22586e7	0.316703
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 8.77342e6i	− 0.0642314i
$516$	0	0
$517$	6.40390e7	0.463418
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.00487e8i	− 0.710550i	−0.934762	−	0.355275i	$-0.884387\pi$
$521$	− 1.00487e8i	− 0.710550i	0.934762	−	0.355275i	$-0.115613\pi$
$522$	0	0
$523$	1.23329e8	0.862107	0.431054	−	0.902326i	$-0.358142\pi$
$523$	1.23329e8	0.862107	0.431054	+	0.902326i	$0.358142\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.41078e8i	− 0.963888i
$528$	0	0
$529$	−3.90734e8	−2.63946
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 5.36656e6i	− 0.0354417i
$534$	0	0
$535$	−2.18592e8	−1.42749
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.11767e7i	0.262958i
$540$	0	0
$541$	5.00184e7	0.315891	0.157946	−	0.987448i	$-0.449513\pi$
$541$	5.00184e7	0.315891	0.157946	+	0.987448i	$0.449513\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.13405e7i	0.440704i
$546$	0	0
$547$	1.13629e8	0.694271	0.347135	−	0.937815i	$-0.387154\pi$
$547$	1.13629e8	0.694271	0.347135	+	0.937815i	$0.387154\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.20152e7i	0.0718254i
$552$	0	0
$553$	1.12759e8	0.666767
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.28112e8i	1.32003i	0.751254	+	0.660013i	$0.229449\pi$
$557$	2.28112e8i	1.32003i	−0.751254	+	0.660013i	$0.770551\pi$
$558$	0	0
$559$	−4.64337e8	−2.65826
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.69680e8i	1.51121i	0.655030	+	0.755603i	$0.272656\pi$
$563$	2.69680e8i	1.51121i	−0.655030	+	0.755603i	$0.727344\pi$
$564$	0	0
$565$	−4.02178e8	−2.22984
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.14748e8i	1.70854i	0.519828	+	0.854271i	$0.325996\pi$
$569$	3.14748e8i	1.70854i	−0.519828	+	0.854271i	$0.674004\pi$
$570$	0	0
$571$	8.56187e7	0.459897	0.229948	−	0.973203i	$-0.426144\pi$
$571$	8.56187e7	0.459897	0.229948	+	0.973203i	$0.426144\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 3.00242e8i	− 1.57931i
$576$	0	0
$577$	3.29347e8	1.71446	0.857228	−	0.514936i	$-0.172184\pi$
$577$	3.29347e8	1.71446	0.857228	+	0.514936i	$0.172184\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 7.04983e6i	− 0.0359460i
$582$	0	0
$583$	−1.05136e8	−0.530576
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.64541e7i	0.130791i	0.997859	+	0.0653956i	$0.0208309\pi$
$587$	2.64541e7i	0.130791i	−0.997859	+	0.0653956i	$0.979169\pi$
$588$	0	0
$589$	2.29619e8	1.12373
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.58567e7i	0.363773i	0.983320	+	0.181886i	$0.0582203\pi$
$593$	7.58567e7i	0.363773i	−0.983320	+	0.181886i	$0.941780\pi$
$594$	0	0
$595$	−8.31423e7	−0.394704
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.71776e8i	0.799249i	0.916679	+	0.399625i	$0.130860\pi$
$599$	1.71776e8i	0.799249i	−0.916679	+	0.399625i	$0.869140\pi$
$600$	0	0
$601$	1.22182e8	0.562838	0.281419	−	0.959585i	$-0.409195\pi$
$601$	1.22182e8	0.562838	0.281419	+	0.959585i	$0.409195\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.59625e8i	1.17241i
$606$	0	0
$607$	−3.71122e8	−1.65940	−0.829699	−	0.558212i	$-0.811488\pi$
$607$	−3.71122e8	−1.65940	−0.829699	+	0.558212i	$0.811488\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 4.10715e8i	− 1.80060i
$612$	0	0
$613$	5.59703e7	0.242983	0.121492	−	0.992592i	$-0.461232\pi$
$613$	5.59703e7	0.242983	0.121492	+	0.992592i	$0.461232\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3.23673e8i	− 1.37800i	−0.724760	−	0.689002i	$-0.758049\pi$
$617$	− 3.23673e8i	− 1.37800i	0.724760	−	0.689002i	$-0.241951\pi$
$618$	0	0
$619$	−4.63762e7	−0.195534	−0.0977672	−	0.995209i	$-0.531170\pi$
$619$	−4.63762e7	−0.195534	−0.0977672	+	0.995209i	$0.531170\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.02753e6i	0.0290629i
$624$	0	0
$625$	−2.78935e8	−1.14252
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 2.20749e8i	− 0.887046i
$630$	0	0
$631$	−8.84920e7	−0.352222	−0.176111	−	0.984370i	$-0.556352\pi$
$631$	−8.84920e7	−0.352222	−0.176111	+	0.984370i	$0.556352\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 5.17221e7i	− 0.202002i
$636$	0	0
$637$	2.64087e8	1.02171
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.95879e8i	1.12342i	0.827335	+	0.561708i	$0.189856\pi$
$641$	2.95879e8i	1.12342i	−0.827335	+	0.561708i	$0.810144\pi$
$642$	0	0
$643$	3.33798e8	1.25560	0.627799	−	0.778375i	$-0.283956\pi$
$643$	3.33798e8	1.25560	0.627799	+	0.778375i	$0.283956\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.23169e8i	− 0.823988i	−0.911187	−	0.411994i	$-0.864832\pi$
$647$	− 2.23169e8i	− 0.823988i	0.911187	−	0.411994i	$-0.135168\pi$
$648$	0	0
$649$	−4.78830e7	−0.175165
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.07175e8i	− 0.744044i	−0.928224	−	0.372022i	$-0.878665\pi$
$653$	− 2.07175e8i	− 0.744044i	0.928224	−	0.372022i	$-0.121335\pi$
$654$	0	0
$655$	6.87441e8	2.44631
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.54536e8i	0.539976i	0.962864	+	0.269988i	$0.0870198\pi$
$659$	1.54536e8i	0.539976i	−0.962864	+	0.269988i	$0.912980\pi$
$660$	0	0
$661$	1.96283e7	0.0679640	0.0339820	−	0.999422i	$-0.489181\pi$
$661$	1.96283e7	0.0679640	0.0339820	+	0.999422i	$0.489181\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.35323e8i	− 0.460157i
$666$	0	0
$667$	6.30403e7	0.212442
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 9.57471e7i	− 0.316926i
$672$	0	0
$673$	4.71561e8	1.54701	0.773504	−	0.633791i	$-0.218502\pi$
$673$	4.71561e8	1.54701	0.773504	+	0.633791i	$0.218502\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.07589e8i	0.346738i	0.984857	+	0.173369i	$0.0554654\pi$
$677$	1.07589e8i	0.346738i	−0.984857	+	0.173369i	$0.944535\pi$
$678$	0	0
$679$	−1.82761e8	−0.583813
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 2.94096e8i	− 0.923052i	−0.887127	−	0.461526i	$-0.847302\pi$
$683$	− 2.94096e8i	− 0.923052i	0.887127	−	0.461526i	$-0.152698\pi$
$684$	0	0
$685$	1.23294e8	0.383593
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.74293e8i	2.06154i
$690$	0	0
$691$	3.21718e8	0.975082	0.487541	−	0.873100i	$-0.337894\pi$
$691$	3.21718e8	0.975082	0.487541	+	0.873100i	$0.337894\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.28215e8i	− 0.381931i
$696$	0	0
$697$	−4.68884e6	−0.0138474
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 3.97297e8i	− 1.15335i	−0.816974	−	0.576675i	$-0.804350\pi$
$701$	− 3.97297e8i	− 1.15335i	0.816974	−	0.576675i	$-0.195650\pi$
$702$	0	0
$703$	3.59291e8	1.03414
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.23619e8i	0.349807i
$708$	0	0
$709$	−7.01450e8	−1.96815	−0.984075	−	0.177754i	$-0.943117\pi$
$709$	−7.01450e8	−1.96815	−0.984075	+	0.177754i	$0.943117\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1.20474e9i	− 3.32372i
$714$	0	0
$715$	−2.55024e8	−0.697690
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.75290e8i	0.740633i	0.928906	+	0.370317i	$0.120751\pi$
$719$	2.75290e8i	0.740633i	−0.928906	+	0.370317i	$0.879249\pi$
$720$	0	0
$721$	−9.39645e6	−0.0250702
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.51306e7i	0.0921874i
$726$	0	0
$727$	8.39495e7	0.218482	0.109241	−	0.994015i	$-0.465158\pi$
$727$	8.39495e7	0.218482	0.109241	+	0.994015i	$0.465158\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.05698e8i	1.03861i
$732$	0	0
$733$	4.73091e8	1.20125	0.600624	−	0.799532i	$-0.294919\pi$
$733$	4.73091e8	1.20125	0.600624	+	0.799532i	$0.294919\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.45168e8i	− 0.362635i
$738$	0	0
$739$	−6.15675e8	−1.52552	−0.762761	−	0.646680i	$-0.776157\pi$
$739$	−6.15675e8	−1.52552	−0.762761	+	0.646680i	$0.776157\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.17331e7i	0.223645i	0.993728	+	0.111823i	$0.0356688\pi$
$743$	9.17331e7i	0.223645i	−0.993728	+	0.111823i	$0.964331\pi$
$744$	0	0
$745$	−4.62066e8	−1.11747
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.34115e8i	0.557165i
$750$	0	0
$751$	1.28332e7	0.0302981	0.0151490	−	0.999885i	$-0.495178\pi$
$751$	1.28332e7	0.0302981	0.0151490	+	0.999885i	$0.495178\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.18806e8i	1.67021i
$756$	0	0
$757$	−2.94852e8	−0.679698	−0.339849	−	0.940480i	$-0.610376\pi$
$757$	−2.94852e8	−0.679698	−0.339849	+	0.940480i	$0.610376\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1.97094e8i	− 0.447219i	−0.974679	−	0.223609i	$-0.928216\pi$
$761$	− 1.97094e8i	− 0.447219i	0.974679	−	0.223609i	$-0.0717840\pi$
$762$	0	0
$763$	7.64066e7	0.172011
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.07098e8i	0.680599i
$768$	0	0
$769$	−7.92473e7	−0.174263	−0.0871316	−	0.996197i	$-0.527770\pi$
$769$	−7.92473e7	−0.174263	−0.0871316	+	0.996197i	$0.527770\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 3.46436e8i	− 0.750039i	−0.927017	−	0.375020i	$-0.877636\pi$
$773$	− 3.46436e8i	− 0.750039i	0.927017	−	0.375020i	$-0.122364\pi$
$774$	0	0
$775$	6.71367e8	1.44230
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 7.63158e6i	− 0.0161437i
$780$	0	0
$781$	1.88539e8	0.395774
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.37733e8i	1.73180i
$786$	0	0
$787$	−4.96459e8	−1.01850	−0.509248	−	0.860620i	$-0.670076\pi$
$787$	−4.96459e8	−1.01850	−0.509248	+	0.860620i	$0.670076\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.30738e8i	0.870330i
$792$	0	0
$793$	−6.14075e8	−1.23141
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.83204e8i	0.954454i	0.878780	+	0.477227i	$0.158358\pi$
$797$	4.83204e8i	0.954454i	−0.878780	+	0.477227i	$0.841642\pi$
$798$	0	0
$799$	−3.58848e8	−0.703509
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.13251e8i	− 0.218724i
$804$	0	0
$805$	−7.09996e8	−1.36103
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 2.45581e8i	− 0.463819i	−0.972737	−	0.231909i	$-0.925503\pi$
$809$	− 2.45581e8i	− 0.463819i	0.972737	−	0.231909i	$-0.0744973\pi$
$810$	0	0
$811$	−2.62696e8	−0.492483	−0.246241	−	0.969209i	$-0.579196\pi$
$811$	−2.62696e8	−0.492483	−0.246241	+	0.969209i	$0.579196\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.34511e8i	0.987378i
$816$	0	0
$817$	−6.60316e8	−1.21084
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 4.45061e8i	− 0.804248i	−0.915585	−	0.402124i	$-0.868272\pi$
$821$	− 4.45061e8i	− 0.804248i	0.915585	−	0.402124i	$-0.131728\pi$
$822$	0	0
$823$	8.74881e8	1.56946	0.784728	−	0.619840i	$-0.212802\pi$
$823$	8.74881e8	1.56946	0.784728	+	0.619840i	$0.212802\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 7.62945e8i	− 1.34889i	−0.738325	−	0.674445i	$-0.764383\pi$
$827$	− 7.62945e8i	− 1.34889i	0.738325	−	0.674445i	$-0.235617\pi$
$828$	0	0
$829$	−4.16920e8	−0.731794	−0.365897	−	0.930655i	$-0.619238\pi$
$829$	−4.16920e8	−0.731794	−0.365897	+	0.930655i	$0.619238\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.30737e8i	− 0.399193i
$834$	0	0
$835$	−3.37649e8	−0.579971
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7.44317e7i	0.126029i	0.998013	+	0.0630147i	$0.0200715\pi$
$839$	7.44317e7i	0.126029i	−0.998013	+	0.0630147i	$0.979928\pi$
$840$	0	0
$841$	5.87447e8	0.987599
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.19880e8i	1.35888i
$846$	0	0
$847$	2.78062e8	0.457606
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.88509e9i	− 3.05875i
$852$	0	0
$853$	1.38357e8	0.222923	0.111461	−	0.993769i	$-0.464447\pi$
$853$	1.38357e8	0.222923	0.111461	+	0.993769i	$0.464447\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.05160e8i	− 0.325950i	−0.986630	−	0.162975i	$-0.947891\pi$
$857$	− 2.05160e8i	− 0.325950i	0.986630	−	0.162975i	$-0.0521090\pi$
$858$	0	0
$859$	−3.45411e8	−0.544950	−0.272475	−	0.962163i	$-0.587842\pi$
$859$	−3.45411e8	−0.544950	−0.272475	+	0.962163i	$0.587842\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.22088e9i	1.89950i	0.313007	+	0.949751i	$0.398664\pi$
$863$	1.22088e9i	1.89950i	−0.313007	+	0.949751i	$0.601336\pi$
$864$	0	0
$865$	1.29085e9	1.99447
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 3.02188e8i	− 0.460488i
$870$	0	0
$871$	−9.31039e8	−1.40901
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.22797e7i	0.122820i
$876$	0	0
$877$	8.65697e8	1.28341	0.641707	−	0.766950i	$-0.278226\pi$
$877$	8.65697e8	1.28341	0.641707	+	0.766950i	$0.278226\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.00489e9i	1.46957i	0.678301	+	0.734784i	$0.262717\pi$
$881$	1.00489e9i	1.46957i	−0.678301	+	0.734784i	$0.737283\pi$
$882$	0	0
$883$	7.84891e8	1.14006	0.570029	−	0.821624i	$-0.306932\pi$
$883$	7.84891e8	1.14006	0.570029	+	0.821624i	$0.306932\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.95521e8i	0.566759i	0.959008	+	0.283379i	$0.0914556\pi$
$887$	3.95521e8i	0.566759i	−0.959008	+	0.283379i	$0.908544\pi$
$888$	0	0
$889$	−5.53950e7	−0.0788434
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 5.84061e8i	− 0.820171i
$894$	0	0
$895$	3.31136e8	0.461888
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.40964e8i	0.194012i
$900$	0	0
$901$	5.89140e8	0.805460
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.75935e8i	− 0.237359i
$906$	0	0
$907$	4.68516e8	0.627917	0.313959	−	0.949437i	$-0.398345\pi$
$907$	4.68516e8	0.627917	0.313959	+	0.949437i	$0.398345\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1.36250e9i	− 1.80211i	−0.433708	−	0.901053i	$-0.642795\pi$
$911$	− 1.36250e9i	− 1.80211i	0.433708	−	0.901053i	$-0.357205\pi$
$912$	0	0
$913$	−1.88932e7	−0.0248253
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7.36258e8i	− 0.954821i
$918$	0	0
$919$	−5.86555e8	−0.755722	−0.377861	−	0.925862i	$-0.623340\pi$
$919$	−5.86555e8	−0.755722	−0.377861	+	0.925862i	$0.623340\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.20919e9i	− 1.53777i
$924$	0	0
$925$	1.05051e9	1.32732
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 5.94468e7i	− 0.0741449i	−0.999313	−	0.0370725i	$-0.988197\pi$
$929$	− 5.94468e7i	− 0.0741449i	0.999313	−	0.0370725i	$-0.0118032\pi$
$930$	0	0
$931$	3.75548e8	0.465390
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.22818e8i	0.272593i
$936$	0	0
$937$	1.30966e8	0.159199	0.0795996	−	0.996827i	$-0.474636\pi$
$937$	1.30966e8	0.159199	0.0795996	+	0.996827i	$0.474636\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.21488e9i	1.45803i	0.684498	+	0.729014i	$0.260021\pi$
$941$	1.21488e9i	1.45803i	−0.684498	+	0.729014i	$0.739979\pi$
$942$	0	0
$943$	−4.00405e7	−0.0477490
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.28318e9i	− 1.51091i	−0.655202	−	0.755454i	$-0.727416\pi$
$947$	− 1.28318e9i	− 1.51091i	0.655202	−	0.755454i	$-0.272584\pi$
$948$	0	0
$949$	−7.26338e8	−0.849846
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.79458e8i	0.207340i	0.994612	+	0.103670i	$0.0330586\pi$
$953$	1.79458e8i	0.207340i	−0.994612	+	0.103670i	$0.966941\pi$
$954$	0	0
$955$	−1.54222e9	−1.77067
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.32050e8i	− 0.149720i
$960$	0	0
$961$	1.80639e9	2.03537
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 8.77144e8i	− 0.976088i
$966$	0	0
$967$	6.66401e8	0.736981	0.368491	−	0.929631i	$-0.379875\pi$
$967$	6.66401e8	0.736981	0.368491	+	0.929631i	$0.379875\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 9.05457e8i	− 0.989032i	−0.869169	−	0.494516i	$-0.835345\pi$
$971$	− 9.05457e8i	− 0.989032i	0.869169	−	0.494516i	$-0.164655\pi$
$972$	0	0
$973$	−1.37320e8	−0.149072
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.14731e8i	0.444716i	0.974965	+	0.222358i	$0.0713754\pi$
$977$	4.14731e8i	0.444716i	−0.974965	+	0.222358i	$0.928625\pi$
$978$	0	0
$979$	1.88335e7	0.0200716
$980$	0	0
$981$	0	0
$982$	0	0
$983$	9.81870e7i	0.103370i	0.998663	+	0.0516849i	$0.0164591\pi$
$983$	9.81870e7i	0.103370i	−0.998663	+	0.0516849i	$0.983541\pi$
$984$	0	0
$985$	−1.58625e9	−1.65982
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.46447e9i	3.58136i
$990$	0	0
$991$	8.91123e8	0.915623	0.457812	−	0.889049i	$-0.348633\pi$
$991$	8.91123e8	0.915623	0.457812	+	0.889049i	$0.348633\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.59891e8i	0.263829i
$996$	0	0
$997$	−1.48348e8	−0.149691	−0.0748454	−	0.997195i	$-0.523846\pi$
$997$	−1.48348e8	−0.149691	−0.0748454	+	0.997195i	$0.523846\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.7.e.b.17.3	yes	4
3.2	odd	2	inner	72.7.e.b.17.2	✓	4
4.3	odd	2		144.7.e.e.17.3		4
8.3	odd	2		576.7.e.r.449.2		4
8.5	even	2		576.7.e.m.449.2		4
12.11	even	2		144.7.e.e.17.2		4
24.5	odd	2		576.7.e.m.449.3		4
24.11	even	2		576.7.e.r.449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.e.b.17.2	✓	4	3.2	odd	2	inner
72.7.e.b.17.3	yes	4	1.1	even	1	trivial
144.7.e.e.17.2		4	12.11	even	2
144.7.e.e.17.3		4	4.3	odd	2
576.7.e.m.449.2		4	8.5	even	2
576.7.e.m.449.3		4	24.5	odd	2
576.7.e.r.449.2		4	8.3	odd	2
576.7.e.r.449.3		4	24.11	even	2

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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	72.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+168.997i q^{5} +180.998 q^{7} +O(q^{10})\)	\(q+168.997i q^{5} +180.998 q^{7} -485.068i q^{11} -3110.99 q^{13} +2718.12i q^{17} -4424.01 q^{19} +23211.4i q^{23} -12935.1 q^{25} -2715.92i q^{29} -51902.8 q^{31} +30588.2i q^{35} -81213.9 q^{37} +1725.03i q^{41} +149257. q^{43} +132021. i q^{47} -84888.6 q^{49} -216746. i q^{53} +81975.2 q^{55} -98714.1i q^{59} +197389. q^{61} -525749. i q^{65} +299275. q^{67} +388685. i q^{71} +233475. q^{73} -87796.5i q^{77} +622981. q^{79} -38949.7i q^{83} -459354. q^{85} +38826.5i q^{89} -563083. q^{91} -747646. i q^{95} -1.00974e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 432 q^{7}+O(q^{10}) \) 4 * q - 432 * q^7	\( 4 q - 432 q^{7} - 4352 q^{13} - 26944 q^{19} - 109540 q^{25} - 56176 q^{31} - 237000 q^{37} + 2848 q^{43} - 89860 q^{49} + 1107040 q^{55} + 146824 q^{61} + 1546208 q^{67} + 281920 q^{73} + 1258480 q^{79} - 7480 q^{85} - 1868544 q^{91} - 1322368 q^{97}+O(q^{100}) \) 4 * q - 432 * q^7 - 4352 * q^13 - 26944 * q^19 - 109540 * q^25 - 56176 * q^31 - 237000 * q^37 + 2848 * q^43 - 89860 * q^49 + 1107040 * q^55 + 146824 * q^61 + 1546208 * q^67 + 281920 * q^73 + 1258480 * q^79 - 7480 * q^85 - 1868544 * q^91 - 1322368 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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