LMFDB - Embedded newform 2900.2.c.h.349.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2900,2,Mod(349,2900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2900.349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0,0,0,0,-14,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$23.1566165862$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.9689973693776896.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 22x^{8} + 179x^{6} + 639x^{4} + 847x^{2} + 9 $ x^10 + 22x^8 + 179x^6 + 639x^4 + 847x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.9
Root			$2.40772i$ of defining polynomial
Character	$\chi$	$=$	2900.349
Dual form			2900.2.c.h.349.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.40772i q^{3} +4.71632i q^{7} -2.79711 q^{9} +4.05164 q^{11} -4.25453i q^{13} +5.20483i q^{17} -5.68524 q^{19} -11.3556 q^{21} -4.74304i q^{23} +0.488503i q^{27} -1.00000 q^{29} -10.0159 q^{31} +9.75522i q^{33} +7.01833i q^{37} +10.2437 q^{39} -2.17618 q^{41} -1.68768i q^{43} +6.89379i q^{47} -15.2437 q^{49} -12.5318 q^{51} +12.6514i q^{53} -13.6885i q^{57} +12.8379 q^{59} -0.542577 q^{61} -13.1921i q^{63} -13.2068i q^{67} +11.4199 q^{69} +2.84875 q^{71} +4.66419i q^{73} +19.1089i q^{77} +6.55847 q^{79} -9.56751 q^{81} -7.14267i q^{83} -2.40772i q^{87} -4.04941 q^{89} +20.0658 q^{91} -24.1155i q^{93} +9.36173i q^{97} -11.3329 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 14 q^{9} - 4 q^{19} - 26 q^{21} - 10 q^{29} - 14 q^{31} - 24 q^{39} - 22 q^{41} - 26 q^{49} - 38 q^{51} - 26 q^{59} - 18 q^{61} - 12 q^{69} - 26 q^{71} - 8 q^{79} - 54 q^{81} - 20 q^{89} + 6 q^{91}+ \cdots - 60 q^{99}+O(q^{100}) $ 10 * q - 14 * q^9 - 4 * q^19 - 26 * q^21 - 10 * q^29 - 14 * q^31 - 24 * q^39 - 22 * q^41 - 26 * q^49 - 38 * q^51 - 26 * q^59 - 18 * q^61 - 12 * q^69 - 26 * q^71 - 8 * q^79 - 54 * q^81 - 20 * q^89 + 6 * q^91 - 60 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	2.40772i	1.39010i	0.718963	+	0.695049i	$0.244617\pi$
$3$	2.40772i	1.39010i	−0.718963	+	0.695049i	$0.755383\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.71632i	1.78260i	0.453411	+	0.891302i	$0.350207\pi$
$7$	4.71632i	1.78260i	−0.453411	+	0.891302i	$0.649793\pi$
$8$	0	0
$9$	−2.79711	−0.932370
$10$	0	0
$11$	4.05164	1.22162	0.610808	−	0.791779i	$-0.290845\pi$
$11$	4.05164	1.22162	0.610808	+	0.791779i	$0.290845\pi$
$12$	0	0
$13$	− 4.25453i	− 1.18000i	−0.807405	−	0.589998i	$-0.799129\pi$
$13$	− 4.25453i	− 1.18000i	0.807405	−	0.589998i	$-0.200871\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.20483i	1.26236i	0.775638	+	0.631178i	$0.217428\pi$
$17$	5.20483i	1.26236i	−0.775638	+	0.631178i	$0.782572\pi$
$18$	0	0
$19$	−5.68524	−1.30428	−0.652142	−	0.758097i	$-0.726130\pi$
$19$	−5.68524	−1.30428	−0.652142	+	0.758097i	$0.726130\pi$
$20$	0	0
$21$	−11.3556	−2.47799
$22$	0	0
$23$	− 4.74304i	− 0.988991i	−0.869180	−	0.494496i	$-0.835353\pi$
$23$	− 4.74304i	− 0.988991i	0.869180	−	0.494496i	$-0.164647\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0.488503i	0.0940125i
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−10.0159	−1.79891	−0.899454	−	0.437015i	$-0.856036\pi$
$31$	−10.0159	−1.79891	−0.899454	+	0.437015i	$0.856036\pi$
$32$	0	0
$33$	9.75522i	1.69816i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.01833i	1.15381i	0.816813	+	0.576903i	$0.195739\pi$
$37$	7.01833i	1.15381i	−0.816813	+	0.576903i	$0.804261\pi$
$38$	0	0
$39$	10.2437	1.64031
$40$	0	0
$41$	−2.17618	−0.339862	−0.169931	−	0.985456i	$-0.554354\pi$
$41$	−2.17618	−0.339862	−0.169931	+	0.985456i	$0.554354\pi$
$42$	0	0
$43$	− 1.68768i	− 0.257368i	−0.991686	−	0.128684i	$-0.958925\pi$
$43$	− 1.68768i	− 0.257368i	0.991686	−	0.128684i	$-0.0410753\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.89379i	1.00556i	0.864414	+	0.502781i	$0.167690\pi$
$47$	6.89379i	1.00556i	−0.864414	+	0.502781i	$0.832310\pi$
$48$	0	0
$49$	−15.2437	−2.17767
$50$	0	0
$51$	−12.5318	−1.75480
$52$	0	0
$53$	12.6514i	1.73781i	0.494980	+	0.868904i	$0.335175\pi$
$53$	12.6514i	1.73781i	−0.494980	+	0.868904i	$0.664825\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 13.6885i	− 1.81308i
$58$	0	0
$59$	12.8379	1.67136	0.835679	−	0.549219i	$-0.185075\pi$
$59$	12.8379	1.67136	0.835679	+	0.549219i	$0.185075\pi$
$60$	0	0
$61$	−0.542577	−0.0694698	−0.0347349	−	0.999397i	$-0.511059\pi$
$61$	−0.542577	−0.0694698	−0.0347349	+	0.999397i	$0.511059\pi$
$62$	0	0
$63$	− 13.1921i	− 1.66205i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.2068i	− 1.61346i	−0.590918	−	0.806732i	$-0.701234\pi$
$67$	− 13.2068i	− 1.61346i	0.590918	−	0.806732i	$-0.298766\pi$
$68$	0	0
$69$	11.4199	1.37479
$70$	0	0
$71$	2.84875	0.338085	0.169042	−	0.985609i	$-0.445933\pi$
$71$	2.84875	0.338085	0.169042	+	0.985609i	$0.445933\pi$
$72$	0	0
$73$	4.66419i	0.545902i	0.962028	+	0.272951i	$0.0879997\pi$
$73$	4.66419i	0.545902i	−0.962028	+	0.272951i	$0.912000\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	19.1089i	2.17766i
$78$	0	0
$79$	6.55847	0.737886	0.368943	−	0.929452i	$-0.379720\pi$
$79$	6.55847	0.737886	0.368943	+	0.929452i	$0.379720\pi$
$80$	0	0
$81$	−9.56751	−1.06306
$82$	0	0
$83$	− 7.14267i	− 0.784010i	−0.919963	−	0.392005i	$-0.871782\pi$
$83$	− 7.14267i	− 0.784010i	0.919963	−	0.392005i	$-0.128218\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.40772i	− 0.258135i
$88$	0	0
$89$	−4.04941	−0.429236	−0.214618	−	0.976698i	$-0.568851\pi$
$89$	−4.04941	−0.429236	−0.214618	+	0.976698i	$0.568851\pi$
$90$	0	0
$91$	20.0658	2.10346
$92$	0	0
$93$	− 24.1155i	− 2.50066i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.36173i	0.950540i	0.879840	+	0.475270i	$0.157650\pi$
$97$	9.36173i	0.950540i	−0.879840	+	0.475270i	$0.842350\pi$
$98$	0	0
$99$	−11.3329	−1.13900
$100$	0	0
$101$	−13.4339	−1.33673	−0.668363	−	0.743835i	$-0.733005\pi$
$101$	−13.4339	−1.33673	−0.668363	+	0.743835i	$0.733005\pi$
$102$	0	0
$103$	− 0.165938i	− 0.0163503i	−0.999967	−	0.00817517i	$-0.997398\pi$
$103$	− 0.165938i	− 0.0163503i	0.999967	−	0.00817517i	$-0.00260227\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.1216i	− 1.55854i	−0.626691	−	0.779268i	$-0.715591\pi$
$107$	− 16.1216i	− 1.55854i	0.626691	−	0.779268i	$-0.284409\pi$
$108$	0	0
$109$	−14.0943	−1.34998	−0.674992	−	0.737825i	$-0.735853\pi$
$109$	−14.0943	−1.34998	−0.674992	+	0.737825i	$0.735853\pi$
$110$	0	0
$111$	−16.8982	−1.60390
$112$	0	0
$113$	− 5.48850i	− 0.516315i	−0.966103	−	0.258157i	$-0.916885\pi$
$113$	− 5.48850i	− 0.516315i	0.966103	−	0.258157i	$-0.0831153\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	11.9004i	1.10019i
$118$	0	0
$119$	−24.5477	−2.25028
$120$	0	0
$121$	5.41581	0.492346
$122$	0	0
$123$	− 5.23963i	− 0.472441i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.85870i	− 0.786082i	−0.919521	−	0.393041i	$-0.871423\pi$
$127$	− 8.85870i	− 0.786082i	0.919521	−	0.393041i	$-0.128577\pi$
$128$	0	0
$129$	4.06345	0.357767
$130$	0	0
$131$	−15.6708	−1.36916	−0.684582	−	0.728936i	$-0.740015\pi$
$131$	−15.6708	−1.36916	−0.684582	+	0.728936i	$0.740015\pi$
$132$	0	0
$133$	− 26.8135i	− 2.32502i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.5962i	1.16160i	0.814047	+	0.580799i	$0.197260\pi$
$137$	13.5962i	1.16160i	−0.814047	+	0.580799i	$0.802740\pi$
$138$	0	0
$139$	19.7170	1.67238	0.836189	−	0.548442i	$-0.184779\pi$
$139$	19.7170	1.67238	0.836189	+	0.548442i	$0.184779\pi$
$140$	0	0
$141$	−16.5983	−1.39783
$142$	0	0
$143$	− 17.2378i	− 1.44150i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 36.7026i	− 3.02718i
$148$	0	0
$149$	6.59645	0.540403	0.270201	−	0.962804i	$-0.412910\pi$
$149$	6.59645	0.540403	0.270201	+	0.962804i	$0.412910\pi$
$150$	0	0
$151$	8.66856	0.705437	0.352719	−	0.935729i	$-0.385257\pi$
$151$	8.66856	0.705437	0.352719	+	0.935729i	$0.385257\pi$
$152$	0	0
$153$	− 14.5585i	− 1.17698i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.7117i	1.41354i	0.707441	+	0.706772i	$0.249849\pi$
$157$	17.7117i	1.41354i	−0.707441	+	0.706772i	$0.750151\pi$
$158$	0	0
$159$	−30.4611	−2.41572
$160$	0	0
$161$	22.3697	1.76298
$162$	0	0
$163$	0.778781i	0.0609989i	0.999535	+	0.0304994i	$0.00970978\pi$
$163$	0.778781i	0.0609989i	−0.999535	+	0.0304994i	$0.990290\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 0.581014i	− 0.0449602i	−0.999747	−	0.0224801i	$-0.992844\pi$
$167$	− 0.581014i	− 0.0449602i	0.999747	−	0.0224801i	$-0.00715624\pi$
$168$	0	0
$169$	−5.10105	−0.392388
$170$	0	0
$171$	15.9023	1.21608
$172$	0	0
$173$	9.15706i	0.696198i	0.937458	+	0.348099i	$0.113173\pi$
$173$	9.15706i	0.696198i	−0.937458	+	0.348099i	$0.886827\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	30.9101i	2.32335i
$178$	0	0
$179$	15.7447	1.17681	0.588406	−	0.808565i	$-0.299756\pi$
$179$	15.7447	1.17681	0.588406	+	0.808565i	$0.299756\pi$
$180$	0	0
$181$	4.42168	0.328661	0.164330	−	0.986405i	$-0.447454\pi$
$181$	4.42168	0.328661	0.164330	+	0.986405i	$0.447454\pi$
$182$	0	0
$183$	− 1.30637i	− 0.0965698i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	21.0881i	1.54211i
$188$	0	0
$189$	−2.30394	−0.167587
$190$	0	0
$191$	−6.24451	−0.451837	−0.225918	−	0.974146i	$-0.572538\pi$
$191$	−6.24451	−0.451837	−0.225918	+	0.974146i	$0.572538\pi$
$192$	0	0
$193$	− 4.70972i	− 0.339013i	−0.985529	−	0.169507i	$-0.945783\pi$
$193$	− 4.70972i	− 0.339013i	0.985529	−	0.169507i	$-0.0542174\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 19.1089i	− 1.36145i	−0.732539	−	0.680725i	$-0.761665\pi$
$197$	− 19.1089i	− 1.36145i	0.732539	−	0.680725i	$-0.238335\pi$
$198$	0	0
$199$	−11.2415	−0.796888	−0.398444	−	0.917193i	$-0.630450\pi$
$199$	−11.2415	−0.796888	−0.398444	+	0.917193i	$0.630450\pi$
$200$	0	0
$201$	31.7982	2.24287
$202$	0	0
$203$	− 4.71632i	− 0.331021i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	13.2668i	0.922106i
$208$	0	0
$209$	−23.0346	−1.59334
$210$	0	0
$211$	12.9625	0.892374	0.446187	−	0.894940i	$-0.352782\pi$
$211$	12.9625	0.892374	0.446187	+	0.894940i	$0.352782\pi$
$212$	0	0
$213$	6.85899i	0.469970i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 47.2382i	− 3.20674i
$218$	0	0
$219$	−11.2301	−0.758857
$220$	0	0
$221$	22.1441	1.48957
$222$	0	0
$223$	15.0282i	1.00636i	0.864182	+	0.503180i	$0.167837\pi$
$223$	15.0282i	1.00636i	−0.864182	+	0.503180i	$0.832163\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.9699i	− 0.860843i	−0.902628	−	0.430422i	$-0.858365\pi$
$227$	− 12.9699i	− 0.860843i	0.902628	−	0.430422i	$-0.141635\pi$
$228$	0	0
$229$	−9.21477	−0.608929	−0.304465	−	0.952524i	$-0.598478\pi$
$229$	−9.21477	−0.608929	−0.304465	+	0.952524i	$0.598478\pi$
$230$	0	0
$231$	−46.0088	−3.02715
$232$	0	0
$233$	13.3793i	0.876507i	0.898851	+	0.438254i	$0.144403\pi$
$233$	13.3793i	0.876507i	−0.898851	+	0.438254i	$0.855597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	15.7910i	1.02573i
$238$	0	0
$239$	−10.5693	−0.683670	−0.341835	−	0.939760i	$-0.611048\pi$
$239$	−10.5693	−0.683670	−0.341835	+	0.939760i	$0.611048\pi$
$240$	0	0
$241$	18.5816	1.19695	0.598474	−	0.801142i	$-0.295774\pi$
$241$	18.5816	1.19695	0.598474	+	0.801142i	$0.295774\pi$
$242$	0	0
$243$	− 21.5704i	− 1.38374i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	24.1881i	1.53905i
$248$	0	0
$249$	17.1975	1.08985
$250$	0	0
$251$	−2.32788	−0.146935	−0.0734673	−	0.997298i	$-0.523406\pi$
$251$	−2.32788	−0.146935	−0.0734673	+	0.997298i	$0.523406\pi$
$252$	0	0
$253$	− 19.2171i	− 1.20817i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.0038i	1.06067i	0.847789	+	0.530334i	$0.177933\pi$
$257$	17.0038i	1.06067i	−0.847789	+	0.530334i	$0.822067\pi$
$258$	0	0
$259$	−33.1007	−2.05678
$260$	0	0
$261$	2.79711	0.173137
$262$	0	0
$263$	24.9399i	1.53786i	0.639332	+	0.768931i	$0.279211\pi$
$263$	24.9399i	1.53786i	−0.639332	+	0.768931i	$0.720789\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 9.74984i	− 0.596680i
$268$	0	0
$269$	31.0961	1.89596	0.947982	−	0.318325i	$-0.103120\pi$
$269$	31.0961	1.89596	0.947982	+	0.318325i	$0.103120\pi$
$270$	0	0
$271$	17.4109	1.05763	0.528817	−	0.848736i	$-0.322636\pi$
$271$	17.4109	1.05763	0.528817	+	0.848736i	$0.322636\pi$
$272$	0	0
$273$	48.3127i	2.92402i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.81107i	0.349153i	0.984644	+	0.174577i	$0.0558557\pi$
$277$	5.81107i	0.349153i	−0.984644	+	0.174577i	$0.944144\pi$
$278$	0	0
$279$	28.0156	1.67725
$280$	0	0
$281$	−11.4112	−0.680738	−0.340369	−	0.940292i	$-0.610552\pi$
$281$	−11.4112	−0.680738	−0.340369	+	0.940292i	$0.610552\pi$
$282$	0	0
$283$	28.0863i	1.66956i	0.550585	+	0.834779i	$0.314405\pi$
$283$	28.0863i	1.66956i	−0.550585	+	0.834779i	$0.685595\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 10.2636i	− 0.605839i
$288$	0	0
$289$	−10.0902	−0.593543
$290$	0	0
$291$	−22.5404	−1.32134
$292$	0	0
$293$	− 7.21610i	− 0.421569i	−0.977533	−	0.210784i	$-0.932398\pi$
$293$	− 7.21610i	− 0.421569i	0.977533	−	0.210784i	$-0.0676018\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.97924i	0.114847i
$298$	0	0
$299$	−20.1794	−1.16701
$300$	0	0
$301$	7.95963	0.458785
$302$	0	0
$303$	− 32.3451i	− 1.85818i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.0531i	0.687906i	0.938987	+	0.343953i	$0.111766\pi$
$307$	12.0531i	0.687906i	−0.938987	+	0.343953i	$0.888234\pi$
$308$	0	0
$309$	0.399531	0.0227285
$310$	0	0
$311$	−0.0425440	−0.00241245	−0.00120623	−	0.999999i	$-0.500384\pi$
$311$	−0.0425440	−0.00241245	−0.00120623	+	0.999999i	$0.500384\pi$
$312$	0	0
$313$	− 31.1500i	− 1.76070i	−0.474325	−	0.880350i	$-0.657308\pi$
$313$	− 31.1500i	− 1.76070i	0.474325	−	0.880350i	$-0.342692\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 7.45981i	− 0.418985i	−0.977810	−	0.209492i	$-0.932819\pi$
$317$	− 7.45981i	− 0.418985i	0.977810	−	0.209492i	$-0.0671812\pi$
$318$	0	0
$319$	−4.05164	−0.226848
$320$	0	0
$321$	38.8163	2.16652
$322$	0	0
$323$	− 29.5907i	− 1.64647i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 33.9350i	− 1.87661i
$328$	0	0
$329$	−32.5134	−1.79252
$330$	0	0
$331$	−4.81160	−0.264470	−0.132235	−	0.991218i	$-0.542215\pi$
$331$	−4.81160	−0.264470	−0.132235	+	0.991218i	$0.542215\pi$
$332$	0	0
$333$	− 19.6310i	− 1.07577i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 13.8049i	− 0.752002i	−0.926619	−	0.376001i	$-0.877299\pi$
$337$	− 13.8049i	− 0.752002i	0.926619	−	0.376001i	$-0.122701\pi$
$338$	0	0
$339$	13.2148	0.717728
$340$	0	0
$341$	−40.5808	−2.19758
$342$	0	0
$343$	− 38.8801i	− 2.09933i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.8783i	− 1.01344i	−0.862110	−	0.506721i	$-0.830857\pi$
$347$	− 18.8783i	− 1.01344i	0.862110	−	0.506721i	$-0.169143\pi$
$348$	0	0
$349$	12.4371	0.665743	0.332871	−	0.942972i	$-0.391983\pi$
$349$	12.4371	0.665743	0.332871	+	0.942972i	$0.391983\pi$
$350$	0	0
$351$	2.07835	0.110934
$352$	0	0
$353$	3.34435i	0.178002i	0.996032	+	0.0890009i	$0.0283674\pi$
$353$	3.34435i	0.178002i	−0.996032	+	0.0890009i	$0.971633\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 59.1039i	− 3.12811i
$358$	0	0
$359$	3.19801	0.168784	0.0843922	−	0.996433i	$-0.473105\pi$
$359$	3.19801	0.168784	0.0843922	+	0.996433i	$0.473105\pi$
$360$	0	0
$361$	13.3220	0.701158
$362$	0	0
$363$	13.0397i	0.684409i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 22.1845i	− 1.15802i	−0.815320	−	0.579010i	$-0.803439\pi$
$367$	− 22.1845i	− 1.15802i	0.815320	−	0.579010i	$-0.196561\pi$
$368$	0	0
$369$	6.08701	0.316877
$370$	0	0
$371$	−59.6683	−3.09782
$372$	0	0
$373$	21.0632i	1.09061i	0.838237	+	0.545306i	$0.183587\pi$
$373$	21.0632i	1.09061i	−0.838237	+	0.545306i	$0.816413\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.25453i	0.219120i
$378$	0	0
$379$	20.2033	1.03778	0.518888	−	0.854842i	$-0.326346\pi$
$379$	20.2033	1.03778	0.518888	+	0.854842i	$0.326346\pi$
$380$	0	0
$381$	21.3292	1.09273
$382$	0	0
$383$	31.2400i	1.59629i	0.602467	+	0.798144i	$0.294185\pi$
$383$	31.2400i	1.59629i	−0.602467	+	0.798144i	$0.705815\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.72061i	0.239962i
$388$	0	0
$389$	−25.3239	−1.28397	−0.641987	−	0.766715i	$-0.721890\pi$
$389$	−25.3239	−1.28397	−0.641987	+	0.766715i	$0.721890\pi$
$390$	0	0
$391$	24.6867	1.24846
$392$	0	0
$393$	− 37.7309i	− 1.90327i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.46691i	0.374754i	0.982288	+	0.187377i	$0.0599986\pi$
$397$	7.46691i	0.374754i	−0.982288	+	0.187377i	$0.940001\pi$
$398$	0	0
$399$	64.5593	3.23201
$400$	0	0
$401$	−17.2669	−0.862266	−0.431133	−	0.902288i	$-0.641886\pi$
$401$	−17.2669	−0.862266	−0.431133	+	0.902288i	$0.641886\pi$
$402$	0	0
$403$	42.6130i	2.12270i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	28.4358i	1.40951i
$408$	0	0
$409$	9.63807	0.476572	0.238286	−	0.971195i	$-0.423414\pi$
$409$	9.63807	0.476572	0.238286	+	0.971195i	$0.423414\pi$
$410$	0	0
$411$	−32.7357	−1.61473
$412$	0	0
$413$	60.5479i	2.97937i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	47.4731i	2.32477i
$418$	0	0
$419$	21.1636	1.03391	0.516954	−	0.856013i	$-0.327066\pi$
$419$	21.1636	1.03391	0.516954	+	0.856013i	$0.327066\pi$
$420$	0	0
$421$	−21.4947	−1.04759	−0.523794	−	0.851845i	$-0.675484\pi$
$421$	−21.4947	−1.04759	−0.523794	+	0.851845i	$0.675484\pi$
$422$	0	0
$423$	− 19.2827i	− 0.937556i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2.55897i	− 0.123837i
$428$	0	0
$429$	41.5039	2.00383
$430$	0	0
$431$	13.0802	0.630052	0.315026	−	0.949083i	$-0.397987\pi$
$431$	13.0802	0.630052	0.315026	+	0.949083i	$0.397987\pi$
$432$	0	0
$433$	− 28.7151i	− 1.37996i	−0.723829	−	0.689979i	$-0.757620\pi$
$433$	− 28.7151i	− 1.37996i	0.723829	−	0.689979i	$-0.242380\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.9653i	1.28993i
$438$	0	0
$439$	24.9420	1.19042	0.595209	−	0.803571i	$-0.297069\pi$
$439$	24.9420	1.19042	0.595209	+	0.803571i	$0.297069\pi$
$440$	0	0
$441$	42.6383	2.03040
$442$	0	0
$443$	7.68460i	0.365106i	0.983196	+	0.182553i	$0.0584361\pi$
$443$	7.68460i	0.365106i	−0.983196	+	0.182553i	$0.941564\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	15.8824i	0.751212i
$448$	0	0
$449$	−10.7998	−0.509674	−0.254837	−	0.966984i	$-0.582022\pi$
$449$	−10.7998	−0.509674	−0.254837	+	0.966984i	$0.582022\pi$
$450$	0	0
$451$	−8.81710	−0.415181
$452$	0	0
$453$	20.8714i	0.980626i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.0620i	0.611013i	0.952190	+	0.305506i	$0.0988258\pi$
$457$	13.0620i	0.611013i	−0.952190	+	0.305506i	$0.901174\pi$
$458$	0	0
$459$	−2.54258	−0.118677
$460$	0	0
$461$	10.1708	0.473699	0.236850	−	0.971546i	$-0.423885\pi$
$461$	10.1708	0.473699	0.236850	+	0.971546i	$0.423885\pi$
$462$	0	0
$463$	16.7839i	0.780016i	0.920811	+	0.390008i	$0.127528\pi$
$463$	16.7839i	0.780016i	−0.920811	+	0.390008i	$0.872472\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.2638i	0.752597i	0.926498	+	0.376299i	$0.122803\pi$
$467$	16.2638i	0.752597i	−0.926498	+	0.376299i	$0.877197\pi$
$468$	0	0
$469$	62.2874	2.87616
$470$	0	0
$471$	−42.6447	−1.96496
$472$	0	0
$473$	− 6.83786i	− 0.314405i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 35.3875i	− 1.62028i
$478$	0	0
$479$	−1.00508	−0.0459234	−0.0229617	−	0.999736i	$-0.507310\pi$
$479$	−1.00508	−0.0459234	−0.0229617	+	0.999736i	$0.507310\pi$
$480$	0	0
$481$	29.8597	1.36149
$482$	0	0
$483$	53.8599i	2.45071i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.3505i	1.19406i	0.802220	+	0.597028i	$0.203652\pi$
$487$	26.3505i	1.19406i	−0.802220	+	0.597028i	$0.796348\pi$
$488$	0	0
$489$	−1.87509	−0.0847943
$490$	0	0
$491$	8.41713	0.379860	0.189930	−	0.981798i	$-0.439174\pi$
$491$	8.41713	0.379860	0.189930	+	0.981798i	$0.439174\pi$
$492$	0	0
$493$	− 5.20483i	− 0.234414i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.4356i	0.602671i
$498$	0	0
$499$	−2.61832	−0.117212	−0.0586060	−	0.998281i	$-0.518666\pi$
$499$	−2.61832	−0.117212	−0.0586060	+	0.998281i	$0.518666\pi$
$500$	0	0
$501$	1.39892	0.0624990
$502$	0	0
$503$	18.5498i	0.827094i	0.910483	+	0.413547i	$0.135710\pi$
$503$	18.5498i	0.827094i	−0.910483	+	0.413547i	$0.864290\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.2819i	− 0.545458i
$508$	0	0
$509$	0.167599	0.00742869	0.00371434	−	0.999993i	$-0.498818\pi$
$509$	0.167599	0.00742869	0.00371434	+	0.999993i	$0.498818\pi$
$510$	0	0
$511$	−21.9978	−0.973127
$512$	0	0
$513$	− 2.77726i	− 0.122619i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	27.9312i	1.22841i
$518$	0	0
$519$	−22.0476	−0.967783
$520$	0	0
$521$	28.6635	1.25577	0.627886	−	0.778305i	$-0.283920\pi$
$521$	28.6635	1.25577	0.627886	+	0.778305i	$0.283920\pi$
$522$	0	0
$523$	29.0019i	1.26816i	0.773266	+	0.634081i	$0.218622\pi$
$523$	29.0019i	1.26816i	−0.773266	+	0.634081i	$0.781378\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 52.1310i	− 2.27086i
$528$	0	0
$529$	0.503606	0.0218959
$530$	0	0
$531$	−35.9091	−1.55832
$532$	0	0
$533$	9.25862i	0.401036i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	37.9088i	1.63588i
$538$	0	0
$539$	−61.7621	−2.66028
$540$	0	0
$541$	−35.5967	−1.53042	−0.765211	−	0.643779i	$-0.777366\pi$
$541$	−35.5967	−1.53042	−0.765211	+	0.643779i	$0.777366\pi$
$542$	0	0
$543$	10.6462i	0.456870i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.4296i	0.959022i	0.877536	+	0.479511i	$0.159186\pi$
$547$	22.4296i	0.959022i	−0.877536	+	0.479511i	$0.840814\pi$
$548$	0	0
$549$	1.51765	0.0647715
$550$	0	0
$551$	5.68524	0.242200
$552$	0	0
$553$	30.9319i	1.31536i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.61419i	0.322623i	0.986904	+	0.161312i	$0.0515725\pi$
$557$	7.61419i	0.322623i	−0.986904	+	0.161312i	$0.948428\pi$
$558$	0	0
$559$	−7.18027	−0.303693
$560$	0	0
$561$	−50.7742	−2.14369
$562$	0	0
$563$	− 22.9232i	− 0.966096i	−0.875594	−	0.483048i	$-0.839530\pi$
$563$	− 22.9232i	− 0.966096i	0.875594	−	0.483048i	$-0.160470\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 45.1235i	− 1.89501i
$568$	0	0
$569$	−34.3702	−1.44087	−0.720436	−	0.693522i	$-0.756058\pi$
$569$	−34.3702	−1.44087	−0.720436	+	0.693522i	$0.756058\pi$
$570$	0	0
$571$	22.8542	0.956419	0.478209	−	0.878246i	$-0.341286\pi$
$571$	22.8542	0.956419	0.478209	+	0.878246i	$0.341286\pi$
$572$	0	0
$573$	− 15.0350i	− 0.628097i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.19265i	0.216173i	0.994142	+	0.108086i	$0.0344723\pi$
$577$	5.19265i	0.216173i	−0.994142	+	0.108086i	$0.965528\pi$
$578$	0	0
$579$	11.3397	0.471261
$580$	0	0
$581$	33.6871	1.39758
$582$	0	0
$583$	51.2591i	2.12293i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.9091i	1.31703i	0.752569	+	0.658514i	$0.228815\pi$
$587$	31.9091i	1.31703i	−0.752569	+	0.658514i	$0.771185\pi$
$588$	0	0
$589$	56.9428	2.34629
$590$	0	0
$591$	46.0088	1.89255
$592$	0	0
$593$	5.76390i	0.236695i	0.992972	+	0.118347i	$0.0377596\pi$
$593$	5.76390i	0.236695i	−0.992972	+	0.118347i	$0.962240\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 27.0663i	− 1.10775i
$598$	0	0
$599$	10.6572	0.435442	0.217721	−	0.976011i	$-0.430138\pi$
$599$	10.6572	0.435442	0.217721	+	0.976011i	$0.430138\pi$
$600$	0	0
$601$	−28.2411	−1.15198	−0.575990	−	0.817457i	$-0.695383\pi$
$601$	−28.2411	−1.15198	−0.575990	+	0.817457i	$0.695383\pi$
$602$	0	0
$603$	36.9408i	1.50434i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	19.1612i	0.777729i	0.921295	+	0.388864i	$0.127133\pi$
$607$	19.1612i	0.777729i	−0.921295	+	0.388864i	$0.872867\pi$
$608$	0	0
$609$	11.3556	0.460151
$610$	0	0
$611$	29.3299	1.18656
$612$	0	0
$613$	3.76209i	0.151950i	0.997110	+	0.0759748i	$0.0242068\pi$
$613$	3.76209i	0.151950i	−0.997110	+	0.0759748i	$0.975793\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 20.9158i	− 0.842038i	−0.907052	−	0.421019i	$-0.861673\pi$
$617$	− 20.9158i	− 0.842038i	0.907052	−	0.421019i	$-0.138327\pi$
$618$	0	0
$619$	−23.7289	−0.953746	−0.476873	−	0.878972i	$-0.658230\pi$
$619$	−23.7289	−0.953746	−0.476873	+	0.878972i	$0.658230\pi$
$620$	0	0
$621$	2.31699	0.0929776
$622$	0	0
$623$	− 19.0983i	− 0.765158i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 55.4608i	− 2.21489i
$628$	0	0
$629$	−36.5292	−1.45651
$630$	0	0
$631$	8.17439	0.325417	0.162709	−	0.986674i	$-0.447977\pi$
$631$	8.17439	0.325417	0.162709	+	0.986674i	$0.447977\pi$
$632$	0	0
$633$	31.2100i	1.24049i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	64.8549i	2.56964i
$638$	0	0
$639$	−7.96827	−0.315220
$640$	0	0
$641$	2.95121	0.116566	0.0582828	−	0.998300i	$-0.481437\pi$
$641$	2.95121	0.116566	0.0582828	+	0.998300i	$0.481437\pi$
$642$	0	0
$643$	5.59525i	0.220655i	0.993895	+	0.110327i	$0.0351900\pi$
$643$	5.59525i	0.220655i	−0.993895	+	0.110327i	$0.964810\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 48.9228i	− 1.92335i	−0.274184	−	0.961677i	$-0.588408\pi$
$647$	− 48.9228i	− 1.92335i	0.274184	−	0.961677i	$-0.411592\pi$
$648$	0	0
$649$	52.0147	2.04176
$650$	0	0
$651$	113.736	4.45768
$652$	0	0
$653$	44.2109i	1.73010i	0.501682	+	0.865052i	$0.332715\pi$
$653$	44.2109i	1.73010i	−0.501682	+	0.865052i	$0.667285\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 13.0462i	− 0.508983i
$658$	0	0
$659$	46.9029	1.82708	0.913539	−	0.406750i	$-0.133338\pi$
$659$	46.9029	1.82708	0.913539	+	0.406750i	$0.133338\pi$
$660$	0	0
$661$	−36.0310	−1.40144	−0.700722	−	0.713434i	$-0.747139\pi$
$661$	−36.0310	−1.40144	−0.700722	+	0.713434i	$0.747139\pi$
$662$	0	0
$663$	53.3168i	2.07065i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.74304i	0.183651i
$668$	0	0
$669$	−36.1836	−1.39894
$670$	0	0
$671$	−2.19833	−0.0848654
$672$	0	0
$673$	− 19.5142i	− 0.752218i	−0.926575	−	0.376109i	$-0.877262\pi$
$673$	− 19.5142i	− 0.752218i	0.926575	−	0.376109i	$-0.122738\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25.8569i	0.993762i	0.867819	+	0.496881i	$0.165521\pi$
$677$	25.8569i	0.993762i	−0.867819	+	0.496881i	$0.834479\pi$
$678$	0	0
$679$	−44.1530	−1.69444
$680$	0	0
$681$	31.2279	1.19666
$682$	0	0
$683$	27.9887i	1.07096i	0.844548	+	0.535480i	$0.179869\pi$
$683$	27.9887i	1.07096i	−0.844548	+	0.535480i	$0.820131\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 22.1866i	− 0.846471i
$688$	0	0
$689$	53.8260	2.05061
$690$	0	0
$691$	12.9747	0.493581	0.246790	−	0.969069i	$-0.420624\pi$
$691$	12.9747	0.493581	0.246790	+	0.969069i	$0.420624\pi$
$692$	0	0
$693$	− 53.4496i	− 2.03038i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 11.3266i	− 0.429027i
$698$	0	0
$699$	−32.2136	−1.21843
$700$	0	0
$701$	2.54337	0.0960616	0.0480308	−	0.998846i	$-0.484705\pi$
$701$	2.54337	0.0960616	0.0480308	+	0.998846i	$0.484705\pi$
$702$	0	0
$703$	− 39.9009i	− 1.50489i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 63.3588i	− 2.38285i
$708$	0	0
$709$	0.951996	0.0357530	0.0178765	−	0.999840i	$-0.494309\pi$
$709$	0.951996	0.0357530	0.0178765	+	0.999840i	$0.494309\pi$
$710$	0	0
$711$	−18.3448	−0.687983
$712$	0	0
$713$	47.5058i	1.77910i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 25.4479i	− 0.950368i
$718$	0	0
$719$	−46.6803	−1.74088	−0.870442	−	0.492272i	$-0.836167\pi$
$719$	−46.6803	−1.74088	−0.870442	+	0.492272i	$0.836167\pi$
$720$	0	0
$721$	0.782616	0.0291462
$722$	0	0
$723$	44.7393i	1.66387i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	48.4373i	1.79644i	0.439546	+	0.898220i	$0.355139\pi$
$727$	48.4373i	1.79644i	−0.439546	+	0.898220i	$0.644861\pi$
$728$	0	0
$729$	23.2328	0.860475
$730$	0	0
$731$	8.78406	0.324890
$732$	0	0
$733$	− 16.3891i	− 0.605346i	−0.953095	−	0.302673i	$-0.902121\pi$
$733$	− 16.3891i	− 0.605346i	0.953095	−	0.302673i	$-0.0978790\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 53.5091i	− 1.97103i
$738$	0	0
$739$	27.5595	1.01379	0.506896	−	0.862007i	$-0.330793\pi$
$739$	27.5595	1.01379	0.506896	+	0.862007i	$0.330793\pi$
$740$	0	0
$741$	−58.2380	−2.13943
$742$	0	0
$743$	− 1.14739i	− 0.0420936i	−0.999778	−	0.0210468i	$-0.993300\pi$
$743$	− 1.14739i	− 0.0420936i	0.999778	−	0.0210468i	$-0.00669990\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	19.9788i	0.730987i
$748$	0	0
$749$	76.0348	2.77825
$750$	0	0
$751$	28.7547	1.04927	0.524637	−	0.851326i	$-0.324201\pi$
$751$	28.7547	1.04927	0.524637	+	0.851326i	$0.324201\pi$
$752$	0	0
$753$	− 5.60488i	− 0.204253i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33.1589i	1.20518i	0.798050	+	0.602591i	$0.205865\pi$
$757$	33.1589i	1.20518i	−0.798050	+	0.602591i	$0.794135\pi$
$758$	0	0
$759$	46.2693	1.67947
$760$	0	0
$761$	−12.2164	−0.442844	−0.221422	−	0.975178i	$-0.571070\pi$
$761$	−12.2164	−0.442844	−0.221422	+	0.975178i	$0.571070\pi$
$762$	0	0
$763$	− 66.4731i	− 2.40649i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 54.6194i	− 1.97219i
$768$	0	0
$769$	3.31823	0.119659	0.0598293	−	0.998209i	$-0.480944\pi$
$769$	3.31823	0.119659	0.0598293	+	0.998209i	$0.480944\pi$
$770$	0	0
$771$	−40.9404	−1.47443
$772$	0	0
$773$	− 13.9200i	− 0.500668i	−0.968160	−	0.250334i	$-0.919460\pi$
$773$	− 13.9200i	− 0.500668i	0.968160	−	0.250334i	$-0.0805404\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 79.6972i	− 2.85912i
$778$	0	0
$779$	12.3721	0.443277
$780$	0	0
$781$	11.5421	0.413010
$782$	0	0
$783$	− 0.488503i	− 0.0174577i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	42.2043i	1.50442i	0.658922	+	0.752211i	$0.271013\pi$
$787$	42.2043i	1.50442i	−0.658922	+	0.752211i	$0.728987\pi$
$788$	0	0
$789$	−60.0483	−2.13778
$790$	0	0
$791$	25.8856	0.920385
$792$	0	0
$793$	2.30841i	0.0819740i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.0302i	1.34710i	0.739143	+	0.673549i	$0.235231\pi$
$797$	38.0302i	1.34710i	−0.739143	+	0.673549i	$0.764769\pi$
$798$	0	0
$799$	−35.8810	−1.26938
$800$	0	0
$801$	11.3266	0.400207
$802$	0	0
$803$	18.8976i	0.666883i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	74.8707i	2.63557i
$808$	0	0
$809$	−5.88187	−0.206796	−0.103398	−	0.994640i	$-0.532971\pi$
$809$	−5.88187	−0.206796	−0.103398	+	0.994640i	$0.532971\pi$
$810$	0	0
$811$	55.2223	1.93912	0.969558	−	0.244860i	$-0.0787420\pi$
$811$	55.2223	1.93912	0.969558	+	0.244860i	$0.0787420\pi$
$812$	0	0
$813$	41.9205i	1.47021i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.59485i	0.335681i
$818$	0	0
$819$	−56.1261	−1.96121
$820$	0	0
$821$	42.9309	1.49830	0.749149	−	0.662402i	$-0.230463\pi$
$821$	42.9309	1.49830	0.749149	+	0.662402i	$0.230463\pi$
$822$	0	0
$823$	− 4.24562i	− 0.147993i	−0.997259	−	0.0739964i	$-0.976425\pi$
$823$	− 4.24562i	− 0.147993i	0.997259	−	0.0739964i	$-0.0235753\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.4998i	0.643302i	0.946858	+	0.321651i	$0.104238\pi$
$827$	18.4998i	0.643302i	−0.946858	+	0.321651i	$0.895762\pi$
$828$	0	0
$829$	10.6785	0.370878	0.185439	−	0.982656i	$-0.440629\pi$
$829$	10.6785	0.370878	0.185439	+	0.982656i	$0.440629\pi$
$830$	0	0
$831$	−13.9914	−0.485357
$832$	0	0
$833$	− 79.3409i	− 2.74900i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4.89280i	− 0.169120i
$838$	0	0
$839$	−51.6172	−1.78202	−0.891012	−	0.453980i	$-0.850004\pi$
$839$	−51.6172	−1.78202	−0.891012	+	0.453980i	$0.850004\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	− 27.4751i	− 0.946291i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	25.5427i	0.877657i
$848$	0	0
$849$	−67.6240	−2.32085
$850$	0	0
$851$	33.2882	1.14110
$852$	0	0
$853$	12.2449i	0.419258i	0.977781	+	0.209629i	$0.0672256\pi$
$853$	12.2449i	0.419258i	−0.977781	+	0.209629i	$0.932774\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 8.28316i	− 0.282947i	−0.989942	−	0.141474i	$-0.954816\pi$
$857$	− 8.28316i	− 0.282947i	0.989942	−	0.141474i	$-0.0451841\pi$
$858$	0	0
$859$	10.2156	0.348552	0.174276	−	0.984697i	$-0.444242\pi$
$859$	10.2156	0.348552	0.174276	+	0.984697i	$0.444242\pi$
$860$	0	0
$861$	24.7118	0.842175
$862$	0	0
$863$	− 47.5651i	− 1.61914i	−0.587027	−	0.809568i	$-0.699702\pi$
$863$	− 47.5651i	− 1.61914i	0.587027	−	0.809568i	$-0.300298\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 24.2944i	− 0.825083i
$868$	0	0
$869$	26.5726	0.901413
$870$	0	0
$871$	−56.1886	−1.90388
$872$	0	0
$873$	− 26.1858i	− 0.886255i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.33812i	0.0789528i	0.999221	+	0.0394764i	$0.0125690\pi$
$877$	2.33812i	0.0789528i	−0.999221	+	0.0394764i	$0.987431\pi$
$878$	0	0
$879$	17.3743	0.586022
$880$	0	0
$881$	−26.1420	−0.880745	−0.440372	−	0.897815i	$-0.645154\pi$
$881$	−26.1420	−0.880745	−0.440372	+	0.897815i	$0.645154\pi$
$882$	0	0
$883$	− 22.3132i	− 0.750898i	−0.926843	−	0.375449i	$-0.877488\pi$
$883$	− 22.3132i	− 0.750898i	0.926843	−	0.375449i	$-0.122512\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 28.3164i	− 0.950772i	−0.879777	−	0.475386i	$-0.842308\pi$
$887$	− 28.3164i	− 0.950772i	0.879777	−	0.475386i	$-0.157692\pi$
$888$	0	0
$889$	41.7805	1.40127
$890$	0	0
$891$	−38.7641	−1.29865
$892$	0	0
$893$	− 39.1929i	− 1.31154i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 48.5863i	− 1.62225i
$898$	0	0
$899$	10.0159	0.334049
$900$	0	0
$901$	−65.8486	−2.19373
$902$	0	0
$903$	19.1645i	0.637756i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5.19781i	0.172590i	0.996270	+	0.0862952i	$0.0275028\pi$
$907$	5.19781i	0.172590i	−0.996270	+	0.0862952i	$0.972497\pi$
$908$	0	0
$909$	37.5762	1.24632
$910$	0	0
$911$	−12.2911	−0.407221	−0.203611	−	0.979052i	$-0.565268\pi$
$911$	−12.2911	−0.407221	−0.203611	+	0.979052i	$0.565268\pi$
$912$	0	0
$913$	− 28.9395i	− 0.957759i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 73.9085i	− 2.44068i
$918$	0	0
$919$	40.2170	1.32664	0.663319	−	0.748337i	$-0.269147\pi$
$919$	40.2170	1.32664	0.663319	+	0.748337i	$0.269147\pi$
$920$	0	0
$921$	−29.0204	−0.956256
$922$	0	0
$923$	− 12.1201i	− 0.398938i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.464146i	0.0152446i
$928$	0	0
$929$	−13.3053	−0.436533	−0.218267	−	0.975889i	$-0.570040\pi$
$929$	−13.3053	−0.436533	−0.218267	+	0.975889i	$0.570040\pi$
$930$	0	0
$931$	86.6643	2.84031
$932$	0	0
$933$	− 0.102434i	− 0.00335354i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	32.6270i	1.06588i	0.846154	+	0.532939i	$0.178912\pi$
$937$	32.6270i	1.06588i	−0.846154	+	0.532939i	$0.821088\pi$
$938$	0	0
$939$	75.0004	2.44754
$940$	0	0
$941$	−34.7167	−1.13173	−0.565865	−	0.824498i	$-0.691458\pi$
$941$	−34.7167	−1.13173	−0.565865	+	0.824498i	$0.691458\pi$
$942$	0	0
$943$	10.3217i	0.336121i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.0192i	1.30045i	0.759742	+	0.650225i	$0.225325\pi$
$947$	40.0192i	1.30045i	−0.759742	+	0.650225i	$0.774675\pi$
$948$	0	0
$949$	19.8439	0.644162
$950$	0	0
$951$	17.9611	0.582430
$952$	0	0
$953$	50.2871i	1.62896i	0.580193	+	0.814479i	$0.302977\pi$
$953$	50.2871i	1.62896i	−0.580193	+	0.814479i	$0.697023\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 9.75522i	− 0.315341i
$958$	0	0
$959$	−64.1239	−2.07067
$960$	0	0
$961$	69.3182	2.23607
$962$	0	0
$963$	45.0939i	1.45313i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 30.7539i	− 0.988980i	−0.869183	−	0.494490i	$-0.835355\pi$
$967$	− 30.7539i	− 0.988980i	0.869183	−	0.494490i	$-0.164645\pi$
$968$	0	0
$969$	71.2461	2.28876
$970$	0	0
$971$	32.9631	1.05784	0.528918	−	0.848673i	$-0.322598\pi$
$971$	32.9631	1.05784	0.528918	+	0.848673i	$0.322598\pi$
$972$	0	0
$973$	92.9920i	2.98119i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 35.1814i	− 1.12555i	−0.826609	−	0.562776i	$-0.809733\pi$
$977$	− 35.1814i	− 1.12555i	0.826609	−	0.562776i	$-0.190267\pi$
$978$	0	0
$979$	−16.4068	−0.524362
$980$	0	0
$981$	39.4232	1.25868
$982$	0	0
$983$	− 39.4219i	− 1.25736i	−0.777663	−	0.628681i	$-0.783595\pi$
$983$	− 39.4219i	− 1.25736i	0.777663	−	0.628681i	$-0.216405\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 78.2830i	− 2.49178i
$988$	0	0
$989$	−8.00471	−0.254535
$990$	0	0
$991$	−50.0470	−1.58980	−0.794898	−	0.606743i	$-0.792476\pi$
$991$	−50.0470	−1.58980	−0.794898	+	0.606743i	$0.792476\pi$
$992$	0	0
$993$	− 11.5850i	− 0.367638i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	19.0094i	0.602033i	0.953619	+	0.301016i	$0.0973259\pi$
$997$	19.0094i	0.602033i	−0.953619	+	0.301016i	$0.902674\pi$
$998$	0	0
$999$	−3.42848	−0.108472

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.c.h.349.9		10
5.2	odd	4		2900.2.a.j.1.5	✓	5
5.3	odd	4		2900.2.a.l.1.1	yes	5
5.4	even	2	inner	2900.2.c.h.349.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2900.2.a.j.1.5	✓	5	5.2	odd	4
2900.2.a.l.1.1	yes	5	5.3	odd	4
2900.2.c.h.349.2		10	5.4	even	2	inner
2900.2.c.h.349.9		10	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.40772i q^{3} +4.71632i q^{7} -2.79711 q^{9} +4.05164 q^{11} -4.25453i q^{13} +5.20483i q^{17} -5.68524 q^{19} -11.3556 q^{21} -4.74304i q^{23} +0.488503i q^{27} -1.00000 q^{29} -10.0159 q^{31} +9.75522i q^{33} +7.01833i q^{37} +10.2437 q^{39} -2.17618 q^{41} -1.68768i q^{43} +6.89379i q^{47} -15.2437 q^{49} -12.5318 q^{51} +12.6514i q^{53} -13.6885i q^{57} +12.8379 q^{59} -0.542577 q^{61} -13.1921i q^{63} -13.2068i q^{67} +11.4199 q^{69} +2.84875 q^{71} +4.66419i q^{73} +19.1089i q^{77} +6.55847 q^{79} -9.56751 q^{81} -7.14267i q^{83} -2.40772i q^{87} -4.04941 q^{89} +20.0658 q^{91} -24.1155i q^{93} +9.36173i q^{97} -11.3329 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 14 q^{9} - 4 q^{19} - 26 q^{21} - 10 q^{29} - 14 q^{31} - 24 q^{39} - 22 q^{41} - 26 q^{49} - 38 q^{51} - 26 q^{59} - 18 q^{61} - 12 q^{69} - 26 q^{71} - 8 q^{79} - 54 q^{81} - 20 q^{89} + 6 q^{91}+ \cdots - 60 q^{99}+O(q^{100}) \) 10 * q - 14 * q^9 - 4 * q^19 - 26 * q^21 - 10 * q^29 - 14 * q^31 - 24 * q^39 - 22 * q^41 - 26 * q^49 - 38 * q^51 - 26 * q^59 - 18 * q^61 - 12 * q^69 - 26 * q^71 - 8 * q^79 - 54 * q^81 - 20 * q^89 + 6 * q^91 - 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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