LMFDB - Embedded newform 2900.2.c.h.349.1

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			349.1
Root			$-2.67880i$ of defining polynomial
Character	$\chi$	$=$	2900.349
Dual form			2900.2.c.h.349.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67880i q^{3} -1.65303i q^{7} -4.17598 q^{9} +4.63554 q^{11} -3.45956i q^{13} +1.49718i q^{17} +5.51965 q^{19} -4.42815 q^{21} -6.60977i q^{23} +3.15021i q^{27} -1.00000 q^{29} +0.464244 q^{31} -12.4177i q^{33} -4.53358i q^{37} -9.26748 q^{39} +7.43878 q^{41} +10.5890i q^{43} +5.54074i q^{47} +4.26748 q^{49} +4.01064 q^{51} -11.9463i q^{53} -14.7861i q^{57} -3.91553 q^{59} -2.71641 q^{61} +6.90302i q^{63} -7.17033i q^{67} -17.7063 q^{69} +4.81152 q^{71} -3.54609i q^{73} -7.66270i q^{77} -1.74783 q^{79} -4.08915 q^{81} +6.23607i q^{83} +2.67880i q^{87} +2.66696 q^{89} -5.71877 q^{91} -1.24362i q^{93} +14.9220i q^{97} -19.3579 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 2.67880i	− 1.54661i	−0.634036	−	0.773303i	$-0.718603\pi$
$3$	− 2.67880i	− 1.54661i	0.634036	−	0.773303i	$-0.281397\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.65303i	− 0.624788i	−0.949953	−	0.312394i	$-0.898869\pi$
$7$	− 1.65303i	− 0.624788i	0.949953	−	0.312394i	$-0.101131\pi$
$8$	0	0
$9$	−4.17598	−1.39199
$10$	0	0
$11$	4.63554	1.39767	0.698834	−	0.715284i	$-0.253703\pi$
$11$	4.63554	1.39767	0.698834	+	0.715284i	$0.253703\pi$
$12$	0	0
$13$	− 3.45956i	− 0.959510i	−0.877402	−	0.479755i	$-0.840725\pi$
$13$	− 3.45956i	− 0.959510i	0.877402	−	0.479755i	$-0.159275\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.49718i	0.363118i	0.983380	+	0.181559i	$0.0581144\pi$
$17$	1.49718i	0.363118i	−0.983380	+	0.181559i	$0.941886\pi$
$18$	0	0
$19$	5.51965	1.26630	0.633148	−	0.774031i	$-0.281763\pi$
$19$	5.51965	1.26630	0.633148	+	0.774031i	$0.281763\pi$
$20$	0	0
$21$	−4.42815	−0.966301
$22$	0	0
$23$	− 6.60977i	− 1.37823i	−0.724651	−	0.689116i	$-0.757999\pi$
$23$	− 6.60977i	− 1.37823i	0.724651	−	0.689116i	$-0.242001\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.15021i	0.606258i
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0.464244	0.0833807	0.0416904	−	0.999131i	$-0.486726\pi$
$31$	0.464244	0.0833807	0.0416904	+	0.999131i	$0.486726\pi$
$32$	0	0
$33$	− 12.4177i	− 2.16164i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 4.53358i	− 0.745316i	−0.927969	−	0.372658i	$-0.878446\pi$
$37$	− 4.53358i	− 0.745316i	0.927969	−	0.372658i	$-0.121554\pi$
$38$	0	0
$39$	−9.26748	−1.48399
$40$	0	0
$41$	7.43878	1.16174	0.580871	−	0.813996i	$-0.302712\pi$
$41$	7.43878	1.16174	0.580871	+	0.813996i	$0.302712\pi$
$42$	0	0
$43$	10.5890i	1.61481i	0.590001	+	0.807403i	$0.299127\pi$
$43$	10.5890i	1.61481i	−0.590001	+	0.807403i	$0.700873\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.54074i	0.808200i	0.914715	+	0.404100i	$0.132415\pi$
$47$	5.54074i	0.808200i	−0.914715	+	0.404100i	$0.867585\pi$
$48$	0	0
$49$	4.26748	0.609641
$50$	0	0
$51$	4.01064	0.561601
$52$	0	0
$53$	− 11.9463i	− 1.64095i	−0.571683	−	0.820474i	$-0.693709\pi$
$53$	− 11.9463i	− 1.64095i	0.571683	−	0.820474i	$-0.306291\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 14.7861i	− 1.95846i
$58$	0	0
$59$	−3.91553	−0.509759	−0.254879	−	0.966973i	$-0.582036\pi$
$59$	−3.91553	−0.509759	−0.254879	+	0.966973i	$0.582036\pi$
$60$	0	0
$61$	−2.71641	−0.347801	−0.173901	−	0.984763i	$-0.555637\pi$
$61$	−2.71641	−0.347801	−0.173901	+	0.984763i	$0.555637\pi$
$62$	0	0
$63$	6.90302i	0.869699i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.17033i	− 0.875995i	−0.898976	−	0.437997i	$-0.855688\pi$
$67$	− 7.17033i	− 0.875995i	0.898976	−	0.437997i	$-0.144312\pi$
$68$	0	0
$69$	−17.7063	−2.13158
$70$	0	0
$71$	4.81152	0.571022	0.285511	−	0.958375i	$-0.407837\pi$
$71$	4.81152	0.571022	0.285511	+	0.958375i	$0.407837\pi$
$72$	0	0
$73$	− 3.54609i	− 0.415038i	−0.978231	−	0.207519i	$-0.933461\pi$
$73$	− 3.54609i	− 0.415038i	0.978231	−	0.207519i	$-0.0665389\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 7.66270i	− 0.873245i
$78$	0	0
$79$	−1.74783	−0.196646	−0.0983232	−	0.995155i	$-0.531348\pi$
$79$	−1.74783	−0.196646	−0.0983232	+	0.995155i	$0.531348\pi$
$80$	0	0
$81$	−4.08915	−0.454350
$82$	0	0
$83$	6.23607i	0.684497i	0.939609	+	0.342249i	$0.111188\pi$
$83$	6.23607i	0.684497i	−0.939609	+	0.342249i	$0.888812\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.67880i	0.287198i
$88$	0	0
$89$	2.66696	0.282697	0.141349	−	0.989960i	$-0.454856\pi$
$89$	2.66696	0.282697	0.141349	+	0.989960i	$0.454856\pi$
$90$	0	0
$91$	−5.71877	−0.599490
$92$	0	0
$93$	− 1.24362i	− 0.128957i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.9220i	1.51510i	0.652776	+	0.757551i	$0.273604\pi$
$97$	14.9220i	1.51510i	−0.652776	+	0.757551i	$0.726396\pi$
$98$	0	0
$99$	−19.3579	−1.94554
$100$	0	0
$101$	−15.3265	−1.52504	−0.762521	−	0.646963i	$-0.776039\pi$
$101$	−15.3265	−1.52504	−0.762521	+	0.646963i	$0.776039\pi$
$102$	0	0
$103$	− 12.2618i	− 1.20819i	−0.796911	−	0.604097i	$-0.793534\pi$
$103$	− 12.2618i	− 1.20819i	0.796911	−	0.604097i	$-0.206466\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.73750i	− 0.554665i	−0.960774	−	0.277333i	$-0.910550\pi$
$107$	− 5.73750i	− 0.554665i	0.960774	−	0.277333i	$-0.0894504\pi$
$108$	0	0
$109$	−12.4341	−1.19097	−0.595485	−	0.803366i	$-0.703040\pi$
$109$	−12.4341	−1.19097	−0.595485	+	0.803366i	$0.703040\pi$
$110$	0	0
$111$	−12.1446	−1.15271
$112$	0	0
$113$	− 8.15021i	− 0.766707i	−0.923602	−	0.383354i	$-0.874769\pi$
$113$	− 8.15021i	− 0.766707i	0.923602	−	0.383354i	$-0.125231\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	14.4471i	1.33563i
$118$	0	0
$119$	2.47488	0.226872
$120$	0	0
$121$	10.4882	0.953476
$122$	0	0
$123$	− 19.9270i	− 1.79676i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.5184i	1.82072i	0.413820	+	0.910359i	$0.364194\pi$
$127$	20.5184i	1.82072i	−0.413820	+	0.910359i	$0.635806\pi$
$128$	0	0
$129$	28.3658	2.49747
$130$	0	0
$131$	−11.3602	−0.992548	−0.496274	−	0.868166i	$-0.665299\pi$
$131$	−11.3602	−0.992548	−0.496274	+	0.868166i	$0.665299\pi$
$132$	0	0
$133$	− 9.12416i	− 0.791165i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.0251i	1.19825i	0.800657	+	0.599123i	$0.204484\pi$
$137$	14.0251i	1.19825i	−0.800657	+	0.599123i	$0.795516\pi$
$138$	0	0
$139$	−12.4481	−1.05584	−0.527919	−	0.849295i	$-0.677027\pi$
$139$	−12.4481	−1.05584	−0.527919	+	0.849295i	$0.677027\pi$
$140$	0	0
$141$	14.8425	1.24997
$142$	0	0
$143$	− 16.0369i	− 1.34108i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 11.4317i	− 0.942874i
$148$	0	0
$149$	16.6545	1.36439	0.682193	−	0.731172i	$-0.261026\pi$
$149$	16.6545	1.36439	0.682193	+	0.731172i	$0.261026\pi$
$150$	0	0
$151$	−2.94227	−0.239438	−0.119719	−	0.992808i	$-0.538199\pi$
$151$	−2.94227	−0.239438	−0.119719	+	0.992808i	$0.538199\pi$
$152$	0	0
$153$	− 6.25217i	− 0.505458i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.11380i	0.408126i	0.978958	+	0.204063i	$0.0654147\pi$
$157$	5.11380i	0.408126i	−0.978958	+	0.204063i	$0.934585\pi$
$158$	0	0
$159$	−32.0017	−2.53790
$160$	0	0
$161$	−10.9262	−0.861103
$162$	0	0
$163$	13.7096i	1.07382i	0.843641	+	0.536908i	$0.180408\pi$
$163$	13.7096i	1.07382i	−0.843641	+	0.536908i	$0.819592\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.46068i	0.345178i	0.984994	+	0.172589i	$0.0552132\pi$
$167$	4.46068i	0.345178i	−0.984994	+	0.172589i	$0.944787\pi$
$168$	0	0
$169$	1.03142	0.0793399
$170$	0	0
$171$	−23.0499	−1.76267
$172$	0	0
$173$	0.207939i	0.0158093i	0.999969	+	0.00790464i	$0.00251615\pi$
$173$	0.207939i	0.0158093i	−0.999969	+	0.00790464i	$0.997484\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.4889i	0.788396i
$178$	0	0
$179$	5.65358	0.422568	0.211284	−	0.977425i	$-0.432235\pi$
$179$	5.65358	0.422568	0.211284	+	0.977425i	$0.432235\pi$
$180$	0	0
$181$	−8.81620	−0.655303	−0.327651	−	0.944799i	$-0.606257\pi$
$181$	−8.81620	−0.655303	−0.327651	+	0.944799i	$0.606257\pi$
$182$	0	0
$183$	7.27673i	0.537911i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.94022i	0.507519i
$188$	0	0
$189$	5.20740	0.378782
$190$	0	0
$191$	2.38062	0.172256	0.0861279	−	0.996284i	$-0.472551\pi$
$191$	2.38062	0.172256	0.0861279	+	0.996284i	$0.472551\pi$
$192$	0	0
$193$	5.55935i	0.400171i	0.979778	+	0.200085i	$0.0641219\pi$
$193$	5.55935i	0.400171i	−0.979778	+	0.200085i	$0.935878\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.66270i	0.545945i	0.962022	+	0.272972i	$0.0880067\pi$
$197$	7.66270i	0.545945i	−0.962022	+	0.272972i	$0.911993\pi$
$198$	0	0
$199$	15.5700	1.10373	0.551863	−	0.833935i	$-0.313917\pi$
$199$	15.5700	1.10373	0.551863	+	0.833935i	$0.313917\pi$
$200$	0	0
$201$	−19.2079	−1.35482
$202$	0	0
$203$	1.65303i	0.116020i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	27.6023i	1.91849i
$208$	0	0
$209$	25.5866	1.76986
$210$	0	0
$211$	−13.9899	−0.963101	−0.481551	−	0.876418i	$-0.659926\pi$
$211$	−13.9899	−0.963101	−0.481551	+	0.876418i	$0.659926\pi$
$212$	0	0
$213$	− 12.8891i	− 0.883146i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.767411i	− 0.0520952i
$218$	0	0
$219$	−9.49926	−0.641900
$220$	0	0
$221$	5.17957	0.348416
$222$	0	0
$223$	19.6784i	1.31777i	0.752245	+	0.658883i	$0.228971\pi$
$223$	19.6784i	1.31777i	−0.752245	+	0.658883i	$0.771029\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 13.1366i	− 0.871906i	−0.899969	−	0.435953i	$-0.856411\pi$
$227$	− 13.1366i	− 0.871906i	0.899969	−	0.435953i	$-0.143589\pi$
$228$	0	0
$229$	25.8328	1.70708	0.853539	−	0.521028i	$-0.174451\pi$
$229$	25.8328	1.70708	0.853539	+	0.521028i	$0.174451\pi$
$230$	0	0
$231$	−20.5268	−1.35057
$232$	0	0
$233$	23.3385i	1.52895i	0.644651	+	0.764477i	$0.277003\pi$
$233$	23.3385i	1.52895i	−0.644651	+	0.764477i	$0.722997\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.68209i	0.304135i
$238$	0	0
$239$	−20.9792	−1.35703	−0.678516	−	0.734585i	$-0.737377\pi$
$239$	−20.9792	−1.35703	−0.678516	+	0.734585i	$0.737377\pi$
$240$	0	0
$241$	−13.2652	−0.854484	−0.427242	−	0.904137i	$-0.640515\pi$
$241$	−13.2652	−0.854484	−0.427242	+	0.904137i	$0.640515\pi$
$242$	0	0
$243$	20.4046i	1.30896i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 19.0956i	− 1.21502i
$248$	0	0
$249$	16.7052	1.05865
$250$	0	0
$251$	17.8379	1.12592	0.562958	−	0.826485i	$-0.309663\pi$
$251$	17.8379	1.12592	0.562958	+	0.826485i	$0.309663\pi$
$252$	0	0
$253$	− 30.6399i	− 1.92631i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 26.8238i	− 1.67323i	−0.547795	−	0.836613i	$-0.684533\pi$
$257$	− 26.8238i	− 1.67323i	0.547795	−	0.836613i	$-0.315467\pi$
$258$	0	0
$259$	−7.49415	−0.465664
$260$	0	0
$261$	4.17598	0.258486
$262$	0	0
$263$	− 5.27700i	− 0.325394i	−0.986676	−	0.162697i	$-0.947981\pi$
$263$	− 5.27700i	− 0.325394i	0.986676	−	0.162697i	$-0.0520193\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 7.14425i	− 0.437221i
$268$	0	0
$269$	−12.0629	−0.735488	−0.367744	−	0.929927i	$-0.619870\pi$
$269$	−12.0629	−0.735488	−0.367744	+	0.929927i	$0.619870\pi$
$270$	0	0
$271$	−14.5432	−0.883439	−0.441720	−	0.897153i	$-0.645631\pi$
$271$	−14.5432	−0.883439	−0.441720	+	0.897153i	$0.645631\pi$
$272$	0	0
$273$	15.3195i	0.927175i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 0.961419i	− 0.0577661i	−0.999583	−	0.0288830i	$-0.990805\pi$
$277$	− 0.961419i	− 0.0577661i	0.999583	−	0.0288830i	$-0.00919504\pi$
$278$	0	0
$279$	−1.93867	−0.116065
$280$	0	0
$281$	−28.2563	−1.68563	−0.842813	−	0.538206i	$-0.819102\pi$
$281$	−28.2563	−1.68563	−0.842813	+	0.538206i	$0.819102\pi$
$282$	0	0
$283$	− 1.94985i	− 0.115907i	−0.998319	−	0.0579533i	$-0.981543\pi$
$283$	− 1.94985i	− 0.115907i	0.998319	−	0.0579533i	$-0.0184575\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 12.2965i	− 0.725842i
$288$	0	0
$289$	14.7585	0.868145
$290$	0	0
$291$	39.9732	2.34327
$292$	0	0
$293$	− 13.6367i	− 0.796662i	−0.917242	−	0.398331i	$-0.869590\pi$
$293$	− 13.6367i	− 0.796662i	0.917242	−	0.398331i	$-0.130410\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	14.6029i	0.847347i
$298$	0	0
$299$	−22.8669	−1.32243
$300$	0	0
$301$	17.5039	1.00891
$302$	0	0
$303$	41.0566i	2.35864i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.05118i	0.516578i	0.966068	+	0.258289i	$0.0831587\pi$
$307$	9.05118i	0.516578i	−0.966068	+	0.258289i	$0.916841\pi$
$308$	0	0
$309$	−32.8470	−1.86860
$310$	0	0
$311$	−6.63427	−0.376195	−0.188097	−	0.982150i	$-0.560232\pi$
$311$	−6.63427	−0.376195	−0.188097	+	0.982150i	$0.560232\pi$
$312$	0	0
$313$	− 17.0441i	− 0.963390i	−0.876339	−	0.481695i	$-0.840021\pi$
$313$	− 17.0441i	− 0.963390i	0.876339	−	0.481695i	$-0.159979\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.63083i	0.316259i	0.987418	+	0.158129i	$0.0505463\pi$
$317$	5.63083i	0.316259i	−0.987418	+	0.158129i	$0.949454\pi$
$318$	0	0
$319$	−4.63554	−0.259540
$320$	0	0
$321$	−15.3696	−0.857849
$322$	0	0
$323$	8.26389i	0.459815i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	33.3085i	1.84196i
$328$	0	0
$329$	9.15902	0.504953
$330$	0	0
$331$	−28.6212	−1.57316	−0.786581	−	0.617488i	$-0.788151\pi$
$331$	−28.6212	−1.57316	−0.786581	+	0.617488i	$0.788151\pi$
$332$	0	0
$333$	18.9321i	1.03747i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5.45530i	0.297169i	0.988900	+	0.148585i	$0.0474717\pi$
$337$	5.45530i	0.297169i	−0.988900	+	0.148585i	$0.952528\pi$
$338$	0	0
$339$	−21.8328	−1.18579
$340$	0	0
$341$	2.15202	0.116539
$342$	0	0
$343$	− 18.6255i	− 1.00568i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 19.4174i	− 1.04238i	−0.853440	−	0.521191i	$-0.825488\pi$
$347$	− 19.4174i	− 1.04238i	0.853440	−	0.521191i	$-0.174512\pi$
$348$	0	0
$349$	35.4679	1.89855	0.949277	−	0.314442i	$-0.101817\pi$
$349$	35.4679	1.89855	0.949277	+	0.314442i	$0.101817\pi$
$350$	0	0
$351$	10.8983	0.581711
$352$	0	0
$353$	13.1256i	0.698603i	0.937010	+	0.349301i	$0.113581\pi$
$353$	13.1256i	0.698603i	−0.937010	+	0.349301i	$0.886419\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6.62971i	− 0.350881i
$358$	0	0
$359$	25.1317	1.32640	0.663199	−	0.748443i	$-0.269198\pi$
$359$	25.1317	1.32640	0.663199	+	0.748443i	$0.269198\pi$
$360$	0	0
$361$	11.4666	0.603503
$362$	0	0
$363$	− 28.0959i	− 1.47465i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.32437i	0.225730i	0.993610	+	0.112865i	$0.0360028\pi$
$367$	4.32437i	0.225730i	−0.993610	+	0.112865i	$0.963997\pi$
$368$	0	0
$369$	−31.0642	−1.61714
$370$	0	0
$371$	−19.7476	−1.02524
$372$	0	0
$373$	− 21.6506i	− 1.12103i	−0.828145	−	0.560513i	$-0.810604\pi$
$373$	− 21.6506i	− 1.12103i	0.828145	−	0.560513i	$-0.189396\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.45956i	0.178177i
$378$	0	0
$379$	10.2365	0.525811	0.262906	−	0.964822i	$-0.415319\pi$
$379$	10.2365	0.525811	0.262906	+	0.964822i	$0.415319\pi$
$380$	0	0
$381$	54.9648	2.81593
$382$	0	0
$383$	− 28.6676i	− 1.46485i	−0.680850	−	0.732423i	$-0.738390\pi$
$383$	− 28.6676i	− 1.46485i	0.680850	−	0.732423i	$-0.261610\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 44.2194i	− 2.24780i
$388$	0	0
$389$	26.8661	1.36217	0.681084	−	0.732206i	$-0.261509\pi$
$389$	26.8661	1.36217	0.681084	+	0.732206i	$0.261509\pi$
$390$	0	0
$391$	9.89599	0.500462
$392$	0	0
$393$	30.4318i	1.53508i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 17.4588i	− 0.876232i	−0.898918	−	0.438116i	$-0.855646\pi$
$397$	− 17.4588i	− 0.876232i	0.898918	−	0.438116i	$-0.144354\pi$
$398$	0	0
$399$	−24.4418	−1.22362
$400$	0	0
$401$	25.7848	1.28763	0.643816	−	0.765180i	$-0.277350\pi$
$401$	25.7848	1.28763	0.643816	+	0.765180i	$0.277350\pi$
$402$	0	0
$403$	− 1.60608i	− 0.0800046i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 21.0156i	− 1.04170i
$408$	0	0
$409$	12.4498	0.615603	0.307802	−	0.951451i	$-0.400407\pi$
$409$	12.4498	0.615603	0.307802	+	0.951451i	$0.400407\pi$
$410$	0	0
$411$	37.5705	1.85321
$412$	0	0
$413$	6.47250i	0.318491i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	33.3461i	1.63297i
$418$	0	0
$419$	−23.0559	−1.12635	−0.563177	−	0.826336i	$-0.690421\pi$
$419$	−23.0559	−1.12635	−0.563177	+	0.826336i	$0.690421\pi$
$420$	0	0
$421$	30.5881	1.49077	0.745385	−	0.666634i	$-0.232266\pi$
$421$	30.5881	1.49077	0.745385	+	0.666634i	$0.232266\pi$
$422$	0	0
$423$	− 23.1380i	− 1.12501i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.49032i	0.217302i
$428$	0	0
$429$	−42.9598	−2.07412
$430$	0	0
$431$	−19.5987	−0.944034	−0.472017	−	0.881589i	$-0.656474\pi$
$431$	−19.5987	−0.944034	−0.472017	+	0.881589i	$0.656474\pi$
$432$	0	0
$433$	− 37.0395i	− 1.78000i	−0.455957	−	0.890002i	$-0.650703\pi$
$433$	− 37.0395i	− 1.78000i	0.455957	−	0.890002i	$-0.349297\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 36.4836i	− 1.74525i
$438$	0	0
$439$	20.2424	0.966118	0.483059	−	0.875588i	$-0.339526\pi$
$439$	20.2424	0.966118	0.483059	+	0.875588i	$0.339526\pi$
$440$	0	0
$441$	−17.8209	−0.848615
$442$	0	0
$443$	13.4397i	0.638542i	0.947663	+	0.319271i	$0.103438\pi$
$443$	13.4397i	0.638542i	−0.947663	+	0.319271i	$0.896562\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 44.6140i	− 2.11017i
$448$	0	0
$449$	−10.8909	−0.513974	−0.256987	−	0.966415i	$-0.582730\pi$
$449$	−10.8909	−0.513974	−0.256987	+	0.966415i	$0.582730\pi$
$450$	0	0
$451$	34.4828	1.62373
$452$	0	0
$453$	7.88175i	0.370317i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.1051i	1.45503i	0.686090	+	0.727517i	$0.259326\pi$
$457$	31.1051i	1.45503i	−0.686090	+	0.727517i	$0.740674\pi$
$458$	0	0
$459$	−4.71641	−0.220143
$460$	0	0
$461$	10.2781	0.478698	0.239349	−	0.970934i	$-0.423066\pi$
$461$	10.2781	0.478698	0.239349	+	0.970934i	$0.423066\pi$
$462$	0	0
$463$	39.6884i	1.84448i	0.386621	+	0.922238i	$0.373642\pi$
$463$	39.6884i	1.84448i	−0.386621	+	0.922238i	$0.626358\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.4129i	0.574400i	0.957871	+	0.287200i	$0.0927244\pi$
$467$	12.4129i	0.574400i	−0.957871	+	0.287200i	$0.907276\pi$
$468$	0	0
$469$	−11.8528	−0.547311
$470$	0	0
$471$	13.6989	0.631210
$472$	0	0
$473$	49.0857i	2.25696i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	49.8874i	2.28419i
$478$	0	0
$479$	28.1913	1.28809	0.644047	−	0.764986i	$-0.277254\pi$
$479$	28.1913	1.28809	0.644047	+	0.764986i	$0.277254\pi$
$480$	0	0
$481$	−15.6842	−0.715138
$482$	0	0
$483$	29.2690i	1.33179i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 18.6127i	− 0.843422i	−0.906730	−	0.421711i	$-0.861430\pi$
$487$	− 18.6127i	− 0.843422i	0.906730	−	0.421711i	$-0.138570\pi$
$488$	0	0
$489$	36.7252	1.66077
$490$	0	0
$491$	−41.5995	−1.87736	−0.938680	−	0.344788i	$-0.887951\pi$
$491$	−41.5995	−1.87736	−0.938680	+	0.344788i	$0.887951\pi$
$492$	0	0
$493$	− 1.49718i	− 0.0674294i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.95359i	− 0.356767i
$498$	0	0
$499$	42.4872	1.90199	0.950995	−	0.309206i	$-0.100063\pi$
$499$	42.4872	1.90199	0.950995	+	0.309206i	$0.100063\pi$
$500$	0	0
$501$	11.9493	0.533854
$502$	0	0
$503$	− 19.1736i	− 0.854907i	−0.904037	−	0.427453i	$-0.859411\pi$
$503$	− 19.1736i	− 0.854907i	0.904037	−	0.427453i	$-0.140589\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.76297i	− 0.122708i
$508$	0	0
$509$	−20.8633	−0.924751	−0.462375	−	0.886684i	$-0.653003\pi$
$509$	−20.8633	−0.924751	−0.462375	+	0.886684i	$0.653003\pi$
$510$	0	0
$511$	−5.86179	−0.259311
$512$	0	0
$513$	17.3881i	0.767701i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	25.6843i	1.12960i
$518$	0	0
$519$	0.557026	0.0244507
$520$	0	0
$521$	37.4133	1.63911	0.819554	−	0.573003i	$-0.194221\pi$
$521$	37.4133	1.63911	0.819554	+	0.573003i	$0.194221\pi$
$522$	0	0
$523$	− 12.4970i	− 0.546456i	−0.961949	−	0.273228i	$-0.911909\pi$
$523$	− 12.4970i	− 0.546456i	0.961949	−	0.273228i	$-0.0880913\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.695055i	0.0302771i
$528$	0	0
$529$	−20.6891	−0.899525
$530$	0	0
$531$	16.3512	0.709580
$532$	0	0
$533$	− 25.7349i	− 1.11470i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 15.1448i	− 0.653547i
$538$	0	0
$539$	19.7821	0.852075
$540$	0	0
$541$	17.6163	0.757382	0.378691	−	0.925523i	$-0.376374\pi$
$541$	17.6163	0.757382	0.378691	+	0.925523i	$0.376374\pi$
$542$	0	0
$543$	23.6168i	1.01350i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 8.49542i	− 0.363238i	−0.983369	−	0.181619i	$-0.941866\pi$
$547$	− 8.49542i	− 0.363238i	0.983369	−	0.181619i	$-0.0581338\pi$
$548$	0	0
$549$	11.3437	0.484136
$550$	0	0
$551$	−5.51965	−0.235145
$552$	0	0
$553$	2.88922i	0.122862i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 13.1378i	− 0.556667i	−0.960484	−	0.278334i	$-0.910218\pi$
$557$	− 13.1378i	− 0.556667i	0.960484	−	0.278334i	$-0.0897821\pi$
$558$	0	0
$559$	36.6333	1.54942
$560$	0	0
$561$	18.5915	0.784932
$562$	0	0
$563$	− 1.68159i	− 0.0708705i	−0.999372	−	0.0354352i	$-0.988718\pi$
$563$	− 1.68159i	− 0.0708705i	0.999372	−	0.0354352i	$-0.0112818\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.75950i	0.283872i
$568$	0	0
$569$	7.51373	0.314992	0.157496	−	0.987520i	$-0.449658\pi$
$569$	7.51373	0.314992	0.157496	+	0.987520i	$0.449658\pi$
$570$	0	0
$571$	44.4197	1.85891	0.929454	−	0.368939i	$-0.120279\pi$
$571$	44.4197	1.85891	0.929454	+	0.368939i	$0.120279\pi$
$572$	0	0
$573$	− 6.37722i	− 0.266412i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25.5246i	1.06260i	0.847182	+	0.531302i	$0.178297\pi$
$577$	25.5246i	1.06260i	−0.847182	+	0.531302i	$0.821703\pi$
$578$	0	0
$579$	14.8924	0.618907
$580$	0	0
$581$	10.3084	0.427665
$582$	0	0
$583$	− 55.3775i	− 2.29350i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.02728i	− 0.372596i	−0.982493	−	0.186298i	$-0.940351\pi$
$587$	− 9.02728i	− 0.372596i	0.982493	−	0.186298i	$-0.0596489\pi$
$588$	0	0
$589$	2.56247	0.105585
$590$	0	0
$591$	20.5268	0.844362
$592$	0	0
$593$	13.0080i	0.534176i	0.963672	+	0.267088i	$0.0860615\pi$
$593$	13.0080i	0.534176i	−0.963672	+	0.267088i	$0.913939\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 41.7089i	− 1.70703i
$598$	0	0
$599$	−33.2601	−1.35897	−0.679485	−	0.733689i	$-0.737797\pi$
$599$	−33.2601	−1.35897	−0.679485	+	0.733689i	$0.737797\pi$
$600$	0	0
$601$	18.0444	0.736047	0.368023	−	0.929817i	$-0.380035\pi$
$601$	18.0444	0.736047	0.368023	+	0.929817i	$0.380035\pi$
$602$	0	0
$603$	29.9431i	1.21938i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 14.1415i	− 0.573985i	−0.957933	−	0.286992i	$-0.907345\pi$
$607$	− 14.1415i	− 0.573985i	0.957933	−	0.286992i	$-0.0926555\pi$
$608$	0	0
$609$	4.42815	0.179438
$610$	0	0
$611$	19.1685	0.775476
$612$	0	0
$613$	16.2869i	0.657823i	0.944361	+	0.328912i	$0.106682\pi$
$613$	16.2869i	0.657823i	−0.944361	+	0.328912i	$0.893318\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 32.8943i	− 1.32427i	−0.749383	−	0.662137i	$-0.769650\pi$
$617$	− 32.8943i	− 1.32427i	0.749383	−	0.662137i	$-0.230350\pi$
$618$	0	0
$619$	42.1050	1.69234	0.846170	−	0.532912i	$-0.178902\pi$
$619$	42.1050	1.69234	0.846170	+	0.532912i	$0.178902\pi$
$620$	0	0
$621$	20.8222	0.835564
$622$	0	0
$623$	− 4.40857i	− 0.176626i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 68.5413i	− 2.73728i
$628$	0	0
$629$	6.78756	0.270638
$630$	0	0
$631$	31.2225	1.24295	0.621474	−	0.783435i	$-0.286534\pi$
$631$	31.2225	1.24295	0.621474	+	0.783435i	$0.286534\pi$
$632$	0	0
$633$	37.4760i	1.48954i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 14.7636i	− 0.584956i
$638$	0	0
$639$	−20.0928	−0.794858
$640$	0	0
$641$	−34.1293	−1.34803	−0.674014	−	0.738719i	$-0.735431\pi$
$641$	−34.1293	−1.34803	−0.674014	+	0.738719i	$0.735431\pi$
$642$	0	0
$643$	40.1920i	1.58502i	0.609859	+	0.792510i	$0.291226\pi$
$643$	40.1920i	1.58502i	−0.609859	+	0.792510i	$0.708774\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 5.71898i	− 0.224836i	−0.993661	−	0.112418i	$-0.964140\pi$
$647$	− 5.71898i	− 0.224836i	0.993661	−	0.112418i	$-0.0358596\pi$
$648$	0	0
$649$	−18.1506	−0.712473
$650$	0	0
$651$	−2.05574	−0.0805708
$652$	0	0
$653$	6.46401i	0.252956i	0.991969	+	0.126478i	$0.0403673\pi$
$653$	6.46401i	0.252956i	−0.991969	+	0.126478i	$0.959633\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.8084i	0.577730i
$658$	0	0
$659$	−44.6819	−1.74056	−0.870281	−	0.492556i	$-0.836063\pi$
$659$	−44.6819	−1.74056	−0.870281	+	0.492556i	$0.836063\pi$
$660$	0	0
$661$	44.3667	1.72566	0.862832	−	0.505490i	$-0.168688\pi$
$661$	44.3667	1.72566	0.862832	+	0.505490i	$0.168688\pi$
$662$	0	0
$663$	− 13.8750i	− 0.538862i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.60977i	0.255931i
$668$	0	0
$669$	52.7146	2.03807
$670$	0	0
$671$	−12.5920	−0.486110
$672$	0	0
$673$	− 27.8980i	− 1.07539i	−0.843140	−	0.537694i	$-0.819296\pi$
$673$	− 27.8980i	− 1.07539i	0.843140	−	0.537694i	$-0.180704\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 19.4085i	− 0.745930i	−0.927845	−	0.372965i	$-0.878341\pi$
$677$	− 19.4085i	− 0.745930i	0.927845	−	0.372965i	$-0.121659\pi$
$678$	0	0
$679$	24.6666	0.946617
$680$	0	0
$681$	−35.1903	−1.34850
$682$	0	0
$683$	17.8605i	0.683414i	0.939807	+	0.341707i	$0.111005\pi$
$683$	17.8605i	0.683414i	−0.939807	+	0.341707i	$0.888995\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 69.2009i	− 2.64018i
$688$	0	0
$689$	−41.3289	−1.57451
$690$	0	0
$691$	−28.1722	−1.07172	−0.535861	−	0.844306i	$-0.680013\pi$
$691$	−28.1722	−1.07172	−0.535861	+	0.844306i	$0.680013\pi$
$692$	0	0
$693$	31.9992i	1.21555i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.1372i	0.421850i
$698$	0	0
$699$	62.5191	2.36469
$700$	0	0
$701$	15.6033	0.589327	0.294664	−	0.955601i	$-0.404792\pi$
$701$	15.6033	0.589327	0.294664	+	0.955601i	$0.404792\pi$
$702$	0	0
$703$	− 25.0238i	− 0.943790i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25.3352i	0.952828i
$708$	0	0
$709$	−25.2425	−0.948001	−0.474000	−	0.880525i	$-0.657190\pi$
$709$	−25.2425	−0.948001	−0.474000	+	0.880525i	$0.657190\pi$
$710$	0	0
$711$	7.29890	0.273730
$712$	0	0
$713$	− 3.06855i	− 0.114918i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	56.1992i	2.09880i
$718$	0	0
$719$	−27.7739	−1.03579	−0.517897	−	0.855443i	$-0.673285\pi$
$719$	−27.7739	−1.03579	−0.517897	+	0.855443i	$0.673285\pi$
$720$	0	0
$721$	−20.2692	−0.754865
$722$	0	0
$723$	35.5347i	1.32155i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 3.59058i	− 0.133167i	−0.997781	−	0.0665835i	$-0.978790\pi$
$727$	− 3.59058i	− 0.133167i	0.997781	−	0.0665835i	$-0.0212099\pi$
$728$	0	0
$729$	42.3925	1.57009
$730$	0	0
$731$	−15.8536	−0.586365
$732$	0	0
$733$	7.94354i	0.293401i	0.989181	+	0.146701i	$0.0468654\pi$
$733$	7.94354i	0.293401i	−0.989181	+	0.146701i	$0.953135\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 33.2383i	− 1.22435i
$738$	0	0
$739$	24.2554	0.892248	0.446124	−	0.894971i	$-0.352804\pi$
$739$	24.2554	0.892248	0.446124	+	0.894971i	$0.352804\pi$
$740$	0	0
$741$	−51.1533	−1.87916
$742$	0	0
$743$	42.2210i	1.54894i	0.632612	+	0.774469i	$0.281983\pi$
$743$	42.2210i	1.54894i	−0.632612	+	0.774469i	$0.718017\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 26.0417i	− 0.952815i
$748$	0	0
$749$	−9.48428	−0.346548
$750$	0	0
$751$	26.4938	0.966771	0.483386	−	0.875408i	$-0.339407\pi$
$751$	26.4938	0.966771	0.483386	+	0.875408i	$0.339407\pi$
$752$	0	0
$753$	− 47.7841i	− 1.74135i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 17.2891i	− 0.628382i	−0.949360	−	0.314191i	$-0.898267\pi$
$757$	− 17.2891i	− 0.628382i	0.949360	−	0.314191i	$-0.101733\pi$
$758$	0	0
$759$	−82.0781	−2.97925
$760$	0	0
$761$	−28.2660	−1.02464	−0.512321	−	0.858794i	$-0.671214\pi$
$761$	−28.2660	−1.02464	−0.512321	+	0.858794i	$0.671214\pi$
$762$	0	0
$763$	20.5540i	0.744104i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.5460i	0.489119i
$768$	0	0
$769$	24.1214	0.869842	0.434921	−	0.900469i	$-0.356776\pi$
$769$	24.1214	0.869842	0.434921	+	0.900469i	$0.356776\pi$
$770$	0	0
$771$	−71.8558	−2.58782
$772$	0	0
$773$	− 19.5460i	− 0.703022i	−0.936184	−	0.351511i	$-0.885668\pi$
$773$	− 19.5460i	− 0.703022i	0.936184	−	0.351511i	$-0.114332\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20.0753i	0.720199i
$778$	0	0
$779$	41.0595	1.47111
$780$	0	0
$781$	22.3040	0.798099
$782$	0	0
$783$	− 3.15021i	− 0.112579i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	37.2397i	1.32745i	0.747976	+	0.663725i	$0.231026\pi$
$787$	37.2397i	1.32745i	−0.747976	+	0.663725i	$0.768974\pi$
$788$	0	0
$789$	−14.1360	−0.503257
$790$	0	0
$791$	−13.4726	−0.479029
$792$	0	0
$793$	9.39760i	0.333719i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.5159i	1.11635i	0.829723	+	0.558175i	$0.188498\pi$
$797$	31.5159i	1.11635i	−0.829723	+	0.558175i	$0.811502\pi$
$798$	0	0
$799$	−8.29546	−0.293472
$800$	0	0
$801$	−11.1372	−0.393512
$802$	0	0
$803$	− 16.4380i	− 0.580085i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	32.3141i	1.13751i
$808$	0	0
$809$	−11.3607	−0.399420	−0.199710	−	0.979855i	$-0.564000\pi$
$809$	−11.3607	−0.399420	−0.199710	+	0.979855i	$0.564000\pi$
$810$	0	0
$811$	−7.60523	−0.267056	−0.133528	−	0.991045i	$-0.542631\pi$
$811$	−7.60523	−0.267056	−0.133528	+	0.991045i	$0.542631\pi$
$812$	0	0
$813$	38.9585i	1.36633i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	58.4475i	2.04482i
$818$	0	0
$819$	23.8815	0.834486
$820$	0	0
$821$	0.0409341	0.00142861	0.000714304	−	1.00000i	$-0.499773\pi$
$821$	0.0409341	0.00142861	0.000714304		1.00000i	$0.499773\pi$
$822$	0	0
$823$	46.9194i	1.63551i	0.575568	+	0.817754i	$0.304781\pi$
$823$	46.9194i	1.63551i	−0.575568	+	0.817754i	$0.695219\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.6229i	1.51692i	0.651720	+	0.758459i	$0.274048\pi$
$827$	43.6229i	1.51692i	−0.651720	+	0.758459i	$0.725952\pi$
$828$	0	0
$829$	−44.6055	−1.54921	−0.774606	−	0.632444i	$-0.782052\pi$
$829$	−44.6055	−1.54921	−0.774606	+	0.632444i	$0.782052\pi$
$830$	0	0
$831$	−2.57545	−0.0893414
$832$	0	0
$833$	6.38917i	0.221372i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.46247i	0.0505502i
$838$	0	0
$839$	23.8485	0.823343	0.411671	−	0.911332i	$-0.364945\pi$
$839$	23.8485	0.823343	0.411671	+	0.911332i	$0.364945\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	75.6929i	2.60700i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 17.3374i	− 0.595720i
$848$	0	0
$849$	−5.22326	−0.179262
$850$	0	0
$851$	−29.9659	−1.02722
$852$	0	0
$853$	− 10.6361i	− 0.364174i	−0.983282	−	0.182087i	$-0.941715\pi$
$853$	− 10.6361i	− 0.364174i	0.983282	−	0.182087i	$-0.0582852\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 34.5645i	− 1.18070i	−0.807147	−	0.590350i	$-0.798990\pi$
$857$	− 34.5645i	− 1.18070i	0.807147	−	0.590350i	$-0.201010\pi$
$858$	0	0
$859$	15.3791	0.524730	0.262365	−	0.964969i	$-0.415498\pi$
$859$	15.3791	0.524730	0.262365	+	0.964969i	$0.415498\pi$
$860$	0	0
$861$	−32.9400	−1.12259
$862$	0	0
$863$	41.0655i	1.39788i	0.715178	+	0.698942i	$0.246346\pi$
$863$	41.0655i	1.39788i	−0.715178	+	0.698942i	$0.753654\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 39.5350i	− 1.34268i
$868$	0	0
$869$	−8.10214	−0.274846
$870$	0	0
$871$	−24.8062	−0.840526
$872$	0	0
$873$	− 62.3140i	− 2.10901i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 39.5385i	− 1.33512i	−0.744556	−	0.667560i	$-0.767339\pi$
$877$	− 39.5385i	− 1.33512i	0.744556	−	0.667560i	$-0.232661\pi$
$878$	0	0
$879$	−36.5299	−1.23212
$880$	0	0
$881$	55.5100	1.87018	0.935090	−	0.354412i	$-0.115319\pi$
$881$	55.5100	1.87018	0.935090	+	0.354412i	$0.115319\pi$
$882$	0	0
$883$	− 11.7459i	− 0.395282i	−0.980274	−	0.197641i	$-0.936672\pi$
$883$	− 11.7459i	− 0.395282i	0.980274	−	0.197641i	$-0.0633280\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 6.39449i	− 0.214706i	−0.994221	−	0.107353i	$-0.965763\pi$
$887$	− 6.39449i	− 0.214706i	0.994221	−	0.107353i	$-0.0342375\pi$
$888$	0	0
$889$	33.9177	1.13756
$890$	0	0
$891$	−18.9554	−0.635030
$892$	0	0
$893$	30.5830i	1.02342i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	61.2560i	2.04528i
$898$	0	0
$899$	−0.464244	−0.0154834
$900$	0	0
$901$	17.8857	0.595858
$902$	0	0
$903$	− 46.8896i	− 1.56039i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	33.0153i	1.09626i	0.836395	+	0.548128i	$0.184659\pi$
$907$	33.0153i	1.09626i	−0.836395	+	0.548128i	$0.815341\pi$
$908$	0	0
$909$	64.0031	2.12285
$910$	0	0
$911$	−33.4462	−1.10812	−0.554061	−	0.832476i	$-0.686923\pi$
$911$	−33.4462	−1.10812	−0.554061	+	0.832476i	$0.686923\pi$
$912$	0	0
$913$	28.9075i	0.956700i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.7788i	0.620132i
$918$	0	0
$919$	41.7948	1.37868	0.689341	−	0.724437i	$-0.257900\pi$
$919$	41.7948	1.37868	0.689341	+	0.724437i	$0.257900\pi$
$920$	0	0
$921$	24.2463	0.798943
$922$	0	0
$923$	− 16.6457i	− 0.547902i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	51.2051i	1.68180i
$928$	0	0
$929$	−11.0440	−0.362340	−0.181170	−	0.983452i	$-0.557988\pi$
$929$	−11.0440	−0.362340	−0.181170	+	0.983452i	$0.557988\pi$
$930$	0	0
$931$	23.5550	0.771985
$932$	0	0
$933$	17.7719i	0.581825i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	36.4185i	1.18974i	0.803822	+	0.594870i	$0.202796\pi$
$937$	36.4185i	1.18974i	−0.803822	+	0.594870i	$0.797204\pi$
$938$	0	0
$939$	−45.6578	−1.48999
$940$	0	0
$941$	46.2476	1.50763	0.753815	−	0.657087i	$-0.228211\pi$
$941$	46.2476	1.50763	0.753815	+	0.657087i	$0.228211\pi$
$942$	0	0
$943$	− 49.1686i	− 1.60115i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 15.5456i	− 0.505164i	−0.967575	−	0.252582i	$-0.918720\pi$
$947$	− 15.5456i	− 0.505164i	0.967575	−	0.252582i	$-0.0812798\pi$
$948$	0	0
$949$	−12.2679	−0.398233
$950$	0	0
$951$	15.0839	0.489128
$952$	0	0
$953$	− 45.8154i	− 1.48411i	−0.670341	−	0.742053i	$-0.733852\pi$
$953$	− 45.8154i	− 1.48411i	0.670341	−	0.742053i	$-0.266148\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	12.4177i	0.401407i
$958$	0	0
$959$	23.1840	0.748649
$960$	0	0
$961$	−30.7845	−0.993048
$962$	0	0
$963$	23.9597i	0.772090i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.56207i	0.114548i	0.998358	+	0.0572741i	$0.0182409\pi$
$967$	3.56207i	0.114548i	−0.998358	+	0.0572741i	$0.981759\pi$
$968$	0	0
$969$	22.1373	0.711153
$970$	0	0
$971$	0.374633	0.0120225	0.00601127	−	0.999982i	$-0.498087\pi$
$971$	0.374633	0.0120225	0.00601127	+	0.999982i	$0.498087\pi$
$972$	0	0
$973$	20.5772i	0.659674i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	51.1721i	1.63714i	0.574407	+	0.818570i	$0.305233\pi$
$977$	51.1721i	1.63714i	−0.574407	+	0.818570i	$0.694767\pi$
$978$	0	0
$979$	12.3628	0.395117
$980$	0	0
$981$	51.9245	1.65782
$982$	0	0
$983$	13.2021i	0.421081i	0.977585	+	0.210540i	$0.0675224\pi$
$983$	13.2021i	0.421081i	−0.977585	+	0.210540i	$0.932478\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 24.5352i	− 0.780964i
$988$	0	0
$989$	69.9908	2.22558
$990$	0	0
$991$	−10.6725	−0.339022	−0.169511	−	0.985528i	$-0.554219\pi$
$991$	−10.6725	−0.339022	−0.169511	+	0.985528i	$0.554219\pi$
$992$	0	0
$993$	76.6704i	2.43306i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	41.3114i	1.30835i	0.756345	+	0.654173i	$0.226983\pi$
$997$	41.3114i	1.30835i	−0.756345	+	0.654173i	$0.773017\pi$
$998$	0	0
$999$	14.2817	0.451853

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.c.h.349.1		10
5.2	odd	4		2900.2.a.j.1.1	✓	5
5.3	odd	4		2900.2.a.l.1.5	yes	5
5.4	even	2	inner	2900.2.c.h.349.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2900.2.a.j.1.1	✓	5	5.2	odd	4
2900.2.a.l.1.5	yes	5	5.3	odd	4
2900.2.c.h.349.1		10	1.1	even	1	trivial
2900.2.c.h.349.10		10	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

\(f(q)\)	\(=\)	\(q-2.67880i q^{3} -1.65303i q^{7} -4.17598 q^{9} +4.63554 q^{11} -3.45956i q^{13} +1.49718i q^{17} +5.51965 q^{19} -4.42815 q^{21} -6.60977i q^{23} +3.15021i q^{27} -1.00000 q^{29} +0.464244 q^{31} -12.4177i q^{33} -4.53358i q^{37} -9.26748 q^{39} +7.43878 q^{41} +10.5890i q^{43} +5.54074i q^{47} +4.26748 q^{49} +4.01064 q^{51} -11.9463i q^{53} -14.7861i q^{57} -3.91553 q^{59} -2.71641 q^{61} +6.90302i q^{63} -7.17033i q^{67} -17.7063 q^{69} +4.81152 q^{71} -3.54609i q^{73} -7.66270i q^{77} -1.74783 q^{79} -4.08915 q^{81} +6.23607i q^{83} +2.67880i q^{87} +2.66696 q^{89} -5.71877 q^{91} -1.24362i q^{93} +14.9220i q^{97} -19.3579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 14 q^{9} - 4 q^{19} - 26 q^{21} - 10 q^{29} - 14 q^{31} - 24 q^{39} - 22 q^{41} - 26 q^{49} - 38 q^{51} - 26 q^{59} - 18 q^{61} - 12 q^{69} - 26 q^{71} - 8 q^{79} - 54 q^{81} - 20 q^{89} + 6 q^{91}+ \cdots - 60 q^{99}+O(q^{100}) \) 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

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Database

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