LMFDB - Embedded newform 4100.2.d.g.1149.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4100,2,Mod(1149,4100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4100.1149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4100 = 2^{2} \cdot 5^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4100.d (of order $2$, degree $1$, not 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:	$32.7386648287$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 261x^{10} + 1141x^{8} + 2289x^{6} + 1896x^{4} + 549x^{2} + 9 $ x^14 + 27x^12 + 261x^10 + 1141x^8 + 2289x^6 + 1896x^4 + 549x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.8
Root			$-0.131983i$ of defining polynomial
Character	$\chi$	$=$	4100.1149
Dual form			4100.2.d.g.1149.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.131983i q^{3} -4.26732i q^{7} +2.98258 q^{9} +O(q^{10})$	$q+0.131983i q^{3} -4.26732i q^{7} +2.98258 q^{9} -3.36237 q^{11} +2.80259i q^{13} +3.75289i q^{17} +8.24990 q^{19} +0.563214 q^{21} +6.46174i q^{23} +0.789600i q^{27} -3.69855 q^{29} -2.96885 q^{31} -0.443776i q^{33} +10.9850i q^{37} -0.369894 q^{39} +1.00000 q^{41} +2.11438i q^{43} +2.84891i q^{47} -11.2100 q^{49} -0.495318 q^{51} -7.79547i q^{53} +1.08885i q^{57} +12.8873 q^{59} -11.8662 q^{61} -12.7276i q^{63} +8.49048i q^{67} -0.852841 q^{69} +16.0922 q^{71} -2.28061i q^{73} +14.3483i q^{77} +9.65859 q^{79} +8.84353 q^{81} -1.03693i q^{83} -0.488146i q^{87} +1.50455 q^{89} +11.9595 q^{91} -0.391838i q^{93} -15.5974i q^{97} -10.0285 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 12 q^{9}+O(q^{10}) $ 14 * q - 12 * q^9	$ 14 q - 12 q^{9} - 2 q^{11} + 2 q^{19} - 2 q^{21} + 10 q^{29} + 8 q^{31} + 22 q^{39} + 14 q^{41} + 6 q^{49} + 54 q^{51} + 32 q^{59} + 20 q^{61} + 38 q^{69} + 54 q^{71} + 2 q^{79} + 54 q^{81} + 32 q^{89} + 18 q^{91} + 80 q^{99}+O(q^{100}) $ 14 * q - 12 * q^9 - 2 * q^11 + 2 * q^19 - 2 * q^21 + 10 * q^29 + 8 * q^31 + 22 * q^39 + 14 * q^41 + 6 * q^49 + 54 * q^51 + 32 * q^59 + 20 * q^61 + 38 * q^69 + 54 * q^71 + 2 * q^79 + 54 * q^81 + 32 * q^89 + 18 * q^91 + 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times$.

$n$	$1477$	$2051$	$3901$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0.131983i	0.0762005i	0.999274	+	0.0381002i	$0.0121306\pi$
$3$	0.131983i	0.0762005i	−0.999274	+	0.0381002i	$0.987869\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.26732i	− 1.61289i	−0.591307	−	0.806447i	$-0.701388\pi$
$7$	− 4.26732i	− 1.61289i	0.591307	−	0.806447i	$-0.298612\pi$
$8$	0	0
$9$	2.98258	0.994193
$10$	0	0
$11$	−3.36237	−1.01379	−0.506897	−	0.862007i	$-0.669207\pi$
$11$	−3.36237	−1.01379	−0.506897	+	0.862007i	$0.669207\pi$
$12$	0	0
$13$	2.80259i	0.777298i	0.921386	+	0.388649i	$0.127058\pi$
$13$	2.80259i	0.777298i	−0.921386	+	0.388649i	$0.872942\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.75289i	0.910210i	0.890438	+	0.455105i	$0.150398\pi$
$17$	3.75289i	0.910210i	−0.890438	+	0.455105i	$0.849602\pi$
$18$	0	0
$19$	8.24990	1.89266	0.946328	−	0.323208i	$-0.104761\pi$
$19$	8.24990	1.89266	0.946328	+	0.323208i	$0.104761\pi$
$20$	0	0
$21$	0.563214	0.122903
$22$	0	0
$23$	6.46174i	1.34737i	0.739020	+	0.673683i	$0.235289\pi$
$23$	6.46174i	1.34737i	−0.739020	+	0.673683i	$0.764711\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0.789600i	0.151959i
$28$	0	0
$29$	−3.69855	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$29$	−3.69855	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$30$	0	0
$31$	−2.96885	−0.533221	−0.266610	−	0.963804i	$-0.585904\pi$
$31$	−2.96885	−0.533221	−0.266610	+	0.963804i	$0.585904\pi$
$32$	0	0
$33$	− 0.443776i	− 0.0772515i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.9850i	1.80592i	0.429727	+	0.902959i	$0.358610\pi$
$37$	10.9850i	1.80592i	−0.429727	+	0.902959i	$0.641390\pi$
$38$	0	0
$39$	−0.369894	−0.0592305
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	2.11438i	0.322440i	0.986918	+	0.161220i	$0.0515429\pi$
$43$	2.11438i	0.322440i	−0.986918	+	0.161220i	$0.948457\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.84891i	0.415557i	0.978176	+	0.207778i	$0.0666233\pi$
$47$	2.84891i	0.415557i	−0.978176	+	0.207778i	$0.933377\pi$
$48$	0	0
$49$	−11.2100	−1.60143
$50$	0	0
$51$	−0.495318	−0.0693585
$52$	0	0
$53$	− 7.79547i	− 1.07079i	−0.844602	−	0.535395i	$-0.820163\pi$
$53$	− 7.79547i	− 1.07079i	0.844602	−	0.535395i	$-0.179837\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.08885i	0.144221i
$58$	0	0
$59$	12.8873	1.67779	0.838893	−	0.544297i	$-0.183203\pi$
$59$	12.8873	1.67779	0.838893	+	0.544297i	$0.183203\pi$
$60$	0	0
$61$	−11.8662	−1.51931	−0.759653	−	0.650329i	$-0.774631\pi$
$61$	−11.8662	−1.51931	−0.759653	+	0.650329i	$0.774631\pi$
$62$	0	0
$63$	− 12.7276i	− 1.60353i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.49048i	1.03728i	0.854993	+	0.518639i	$0.173561\pi$
$67$	8.49048i	1.03728i	−0.854993	+	0.518639i	$0.826439\pi$
$68$	0	0
$69$	−0.852841	−0.102670
$70$	0	0
$71$	16.0922	1.90980	0.954898	−	0.296933i	$-0.0959640\pi$
$71$	16.0922	1.90980	0.954898	+	0.296933i	$0.0959640\pi$
$72$	0	0
$73$	− 2.28061i	− 0.266925i	−0.991054	−	0.133462i	$-0.957390\pi$
$73$	− 2.28061i	− 0.266925i	0.991054	−	0.133462i	$-0.0426095\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.3483i	1.63514i
$78$	0	0
$79$	9.65859	1.08668	0.543338	−	0.839514i	$-0.317160\pi$
$79$	9.65859	1.08668	0.543338	+	0.839514i	$0.317160\pi$
$80$	0	0
$81$	8.84353	0.982614
$82$	0	0
$83$	− 1.03693i	− 0.113818i	−0.998379	−	0.0569088i	$-0.981876\pi$
$83$	− 1.03693i	− 0.113818i	0.998379	−	0.0569088i	$-0.0181244\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 0.488146i	− 0.0523347i
$88$	0	0
$89$	1.50455	0.159482	0.0797412	−	0.996816i	$-0.474591\pi$
$89$	1.50455	0.159482	0.0797412	+	0.996816i	$0.474591\pi$
$90$	0	0
$91$	11.9595	1.25370
$92$	0	0
$93$	− 0.391838i	− 0.0406317i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 15.5974i	− 1.58368i	−0.610730	−	0.791839i	$-0.709124\pi$
$97$	− 15.5974i	− 1.58368i	0.610730	−	0.791839i	$-0.290876\pi$
$98$	0	0
$99$	−10.0285	−1.00791
$100$	0	0
$101$	5.79603	0.576726	0.288363	−	0.957521i	$-0.406889\pi$
$101$	5.79603	0.576726	0.288363	+	0.957521i	$0.406889\pi$
$102$	0	0
$103$	− 9.40964i	− 0.927159i	−0.886055	−	0.463580i	$-0.846565\pi$
$103$	− 9.40964i	− 0.927159i	0.886055	−	0.463580i	$-0.153435\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.93369i	0.573631i	0.957986	+	0.286816i	$0.0925967\pi$
$107$	5.93369i	0.573631i	−0.957986	+	0.286816i	$0.907403\pi$
$108$	0	0
$109$	6.29507	0.602959	0.301479	−	0.953473i	$-0.402520\pi$
$109$	6.29507	0.602959	0.301479	+	0.953473i	$0.402520\pi$
$110$	0	0
$111$	−1.44983	−0.137612
$112$	0	0
$113$	14.7388i	1.38651i	0.720691	+	0.693257i	$0.243825\pi$
$113$	14.7388i	1.38651i	−0.720691	+	0.693257i	$0.756175\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.35894i	0.772785i
$118$	0	0
$119$	16.0148	1.46807
$120$	0	0
$121$	0.305544	0.0277768
$122$	0	0
$123$	0.131983i	0.0119005i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 3.07424i	− 0.272795i	−0.990654	−	0.136398i	$-0.956448\pi$
$127$	− 3.07424i	− 0.272795i	0.990654	−	0.136398i	$-0.0435524\pi$
$128$	0	0
$129$	−0.279063	−0.0245701
$130$	0	0
$131$	18.2988	1.59878	0.799388	−	0.600815i	$-0.205157\pi$
$131$	18.2988	1.59878	0.799388	+	0.600815i	$0.205157\pi$
$132$	0	0
$133$	− 35.2049i	− 3.05265i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.9365i	1.10524i	0.833435	+	0.552618i	$0.186371\pi$
$137$	12.9365i	1.10524i	−0.833435	+	0.552618i	$0.813629\pi$
$138$	0	0
$139$	0.861606	0.0730805	0.0365402	−	0.999332i	$-0.488366\pi$
$139$	0.861606	0.0730805	0.0365402	+	0.999332i	$0.488366\pi$
$140$	0	0
$141$	−0.376008	−0.0316656
$142$	0	0
$143$	− 9.42334i	− 0.788019i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.47953i	− 0.122029i
$148$	0	0
$149$	−9.97407	−0.817107	−0.408554	−	0.912734i	$-0.633967\pi$
$149$	−9.97407	−0.817107	−0.408554	+	0.912734i	$0.633967\pi$
$150$	0	0
$151$	1.46014	0.118824	0.0594121	−	0.998234i	$-0.481077\pi$
$151$	1.46014	0.118824	0.0594121	+	0.998234i	$0.481077\pi$
$152$	0	0
$153$	11.1933i	0.904925i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 12.4706i	− 0.995259i	−0.867390	−	0.497629i	$-0.834204\pi$
$157$	− 12.4706i	− 0.995259i	0.867390	−	0.497629i	$-0.165796\pi$
$158$	0	0
$159$	1.02887	0.0815947
$160$	0	0
$161$	27.5743	2.17316
$162$	0	0
$163$	− 11.8887i	− 0.931197i	−0.884996	−	0.465598i	$-0.845839\pi$
$163$	− 11.8887i	− 0.931197i	0.884996	−	0.465598i	$-0.154161\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.25120i	0.406350i	0.979142	+	0.203175i	$0.0651260\pi$
$167$	5.25120i	0.406350i	−0.979142	+	0.203175i	$0.934874\pi$
$168$	0	0
$169$	5.14550	0.395808
$170$	0	0
$171$	24.6060	1.88167
$172$	0	0
$173$	12.0852i	0.918822i	0.888224	+	0.459411i	$0.151940\pi$
$173$	12.0852i	0.918822i	−0.888224	+	0.459411i	$0.848060\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.70091i	0.127848i
$178$	0	0
$179$	4.03830	0.301836	0.150918	−	0.988546i	$-0.451777\pi$
$179$	4.03830	0.301836	0.150918	+	0.988546i	$0.451777\pi$
$180$	0	0
$181$	−12.8601	−0.955884	−0.477942	−	0.878391i	$-0.658617\pi$
$181$	−12.8601	−0.955884	−0.477942	+	0.878391i	$0.658617\pi$
$182$	0	0
$183$	− 1.56613i	− 0.115772i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.6186i	− 0.922765i
$188$	0	0
$189$	3.36947	0.245093
$190$	0	0
$191$	−24.2184	−1.75238	−0.876190	−	0.481967i	$-0.839923\pi$
$191$	−24.2184	−1.75238	−0.876190	+	0.481967i	$0.839923\pi$
$192$	0	0
$193$	− 20.3095i	− 1.46191i	−0.682427	−	0.730953i	$-0.739076\pi$
$193$	− 20.3095i	− 1.46191i	0.682427	−	0.730953i	$-0.260924\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.00228i	0.213904i	0.994264	+	0.106952i	$0.0341091\pi$
$197$	3.00228i	0.213904i	−0.994264	+	0.106952i	$0.965891\pi$
$198$	0	0
$199$	−1.23588	−0.0876090	−0.0438045	−	0.999040i	$-0.513948\pi$
$199$	−1.23588	−0.0876090	−0.0438045	+	0.999040i	$0.513948\pi$
$200$	0	0
$201$	−1.12060	−0.0790411
$202$	0	0
$203$	15.7829i	1.10774i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	19.2727i	1.33954i
$208$	0	0
$209$	−27.7392	−1.91876
$210$	0	0
$211$	−16.5137	−1.13685	−0.568426	−	0.822734i	$-0.692448\pi$
$211$	−16.5137	−1.13685	−0.568426	+	0.822734i	$0.692448\pi$
$212$	0	0
$213$	2.12390i	0.145527i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.6690i	0.860029i
$218$	0	0
$219$	0.301001	0.0203398
$220$	0	0
$221$	−10.5178	−0.707504
$222$	0	0
$223$	29.1235i	1.95025i	0.221649	+	0.975127i	$0.428856\pi$
$223$	29.1235i	1.95025i	−0.221649	+	0.975127i	$0.571144\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.85406i	0.521292i	0.965434	+	0.260646i	$0.0839355\pi$
$227$	7.85406i	0.521292i	−0.965434	+	0.260646i	$0.916064\pi$
$228$	0	0
$229$	7.29751	0.482233	0.241117	−	0.970496i	$-0.422486\pi$
$229$	7.29751	0.482233	0.241117	+	0.970496i	$0.422486\pi$
$230$	0	0
$231$	−1.89373	−0.124599
$232$	0	0
$233$	3.74915i	0.245615i	0.992431	+	0.122807i	$0.0391897\pi$
$233$	3.74915i	0.245615i	−0.992431	+	0.122807i	$0.960810\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.27477i	0.0828053i
$238$	0	0
$239$	17.6427	1.14121	0.570605	−	0.821224i	$-0.306709\pi$
$239$	17.6427	1.14121	0.570605	+	0.821224i	$0.306709\pi$
$240$	0	0
$241$	−13.9801	−0.900538	−0.450269	−	0.892893i	$-0.648672\pi$
$241$	−13.9801	−0.900538	−0.450269	+	0.892893i	$0.648672\pi$
$242$	0	0
$243$	3.53600i	0.226834i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	23.1211i	1.47116i
$248$	0	0
$249$	0.136857	0.00867295
$250$	0	0
$251$	28.4994	1.79886	0.899432	−	0.437061i	$-0.143981\pi$
$251$	28.4994	1.79886	0.899432	+	0.437061i	$0.143981\pi$
$252$	0	0
$253$	− 21.7268i	− 1.36595i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.46306i	− 0.590289i	−0.955453	−	0.295145i	$-0.904632\pi$
$257$	− 9.46306i	− 0.590289i	0.955453	−	0.295145i	$-0.0953678\pi$
$258$	0	0
$259$	46.8763	2.91275
$260$	0	0
$261$	−11.0312	−0.682815
$262$	0	0
$263$	− 21.7474i	− 1.34100i	−0.741910	−	0.670500i	$-0.766080\pi$
$263$	− 21.7474i	− 1.34100i	0.741910	−	0.670500i	$-0.233920\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.198576i	0.0121526i
$268$	0	0
$269$	21.3654	1.30267	0.651336	−	0.758789i	$-0.274209\pi$
$269$	21.3654	1.30267	0.651336	+	0.758789i	$0.274209\pi$
$270$	0	0
$271$	−7.50280	−0.455762	−0.227881	−	0.973689i	$-0.573180\pi$
$271$	−7.50280	−0.455762	−0.227881	+	0.973689i	$0.573180\pi$
$272$	0	0
$273$	1.57846i	0.0955325i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.2823i	1.21865i	0.792922	+	0.609323i	$0.208559\pi$
$277$	20.2823i	1.21865i	−0.792922	+	0.609323i	$0.791441\pi$
$278$	0	0
$279$	−8.85483	−0.530125
$280$	0	0
$281$	13.1326	0.783423	0.391712	−	0.920088i	$-0.371883\pi$
$281$	13.1326	0.783423	0.391712	+	0.920088i	$0.371883\pi$
$282$	0	0
$283$	19.1643i	1.13920i	0.821922	+	0.569600i	$0.192902\pi$
$283$	19.1643i	1.13920i	−0.821922	+	0.569600i	$0.807098\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.26732i	− 0.251892i
$288$	0	0
$289$	2.91580	0.171518
$290$	0	0
$291$	2.05860	0.120677
$292$	0	0
$293$	27.1851i	1.58817i	0.607806	+	0.794086i	$0.292050\pi$
$293$	27.1851i	1.58817i	−0.607806	+	0.794086i	$0.707950\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.65493i	− 0.154055i
$298$	0	0
$299$	−18.1096	−1.04730
$300$	0	0
$301$	9.02273	0.520061
$302$	0	0
$303$	0.764978i	0.0439468i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.4664i	0.996862i	0.866929	+	0.498431i	$0.166090\pi$
$307$	17.4664i	0.996862i	−0.866929	+	0.498431i	$0.833910\pi$
$308$	0	0
$309$	1.24191	0.0706500
$310$	0	0
$311$	6.64928	0.377046	0.188523	−	0.982069i	$-0.439630\pi$
$311$	6.64928	0.377046	0.188523	+	0.982069i	$0.439630\pi$
$312$	0	0
$313$	− 27.7980i	− 1.57123i	−0.618714	−	0.785617i	$-0.712346\pi$
$313$	− 27.7980i	− 1.57123i	0.618714	−	0.785617i	$-0.287654\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.8017i	1.28067i	0.768096	+	0.640334i	$0.221204\pi$
$317$	22.8017i	1.28067i	−0.768096	+	0.640334i	$0.778796\pi$
$318$	0	0
$319$	12.4359	0.696276
$320$	0	0
$321$	−0.783146	−0.0437110
$322$	0	0
$323$	30.9610i	1.72271i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.830844i	0.0459458i
$328$	0	0
$329$	12.1572	0.670249
$330$	0	0
$331$	6.17416	0.339362	0.169681	−	0.985499i	$-0.445726\pi$
$331$	6.17416	0.339362	0.169681	+	0.985499i	$0.445726\pi$
$332$	0	0
$333$	32.7635i	1.79543i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.26838i	0.178040i	0.996030	+	0.0890199i	$0.0283735\pi$
$337$	3.26838i	0.178040i	−0.996030	+	0.0890199i	$0.971627\pi$
$338$	0	0
$339$	−1.94528	−0.105653
$340$	0	0
$341$	9.98237	0.540576
$342$	0	0
$343$	17.9653i	0.970036i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 21.7209i	− 1.16604i	−0.812458	−	0.583019i	$-0.801871\pi$
$347$	− 21.7209i	− 1.16604i	0.812458	−	0.583019i	$-0.198129\pi$
$348$	0	0
$349$	18.4343	0.986767	0.493383	−	0.869812i	$-0.335760\pi$
$349$	18.4343	0.986767	0.493383	+	0.869812i	$0.335760\pi$
$350$	0	0
$351$	−2.21292	−0.118117
$352$	0	0
$353$	− 21.9093i	− 1.16612i	−0.812430	−	0.583058i	$-0.801856\pi$
$353$	− 21.9093i	− 1.16612i	0.812430	−	0.583058i	$-0.198144\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.11368i	0.111868i
$358$	0	0
$359$	−28.2357	−1.49022	−0.745110	−	0.666941i	$-0.767603\pi$
$359$	−28.2357	−1.49022	−0.745110	+	0.666941i	$0.767603\pi$
$360$	0	0
$361$	49.0608	2.58215
$362$	0	0
$363$	0.0403267i	0.00211660i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	15.4061i	0.804193i	0.915597	+	0.402096i	$0.131718\pi$
$367$	15.4061i	0.804193i	−0.915597	+	0.402096i	$0.868282\pi$
$368$	0	0
$369$	2.98258	0.155267
$370$	0	0
$371$	−33.2657	−1.72707
$372$	0	0
$373$	− 1.93447i	− 0.100163i	−0.998745	−	0.0500816i	$-0.984052\pi$
$373$	− 1.93447i	− 0.100163i	0.998745	−	0.0500816i	$-0.0159481\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 10.3655i	− 0.533850i
$378$	0	0
$379$	−4.89384	−0.251380	−0.125690	−	0.992070i	$-0.540114\pi$
$379$	−4.89384	−0.251380	−0.125690	+	0.992070i	$0.540114\pi$
$380$	0	0
$381$	0.405748	0.0207871
$382$	0	0
$383$	− 8.24959i	− 0.421535i	−0.977536	−	0.210767i	$-0.932404\pi$
$383$	− 8.24959i	− 0.421535i	0.977536	−	0.210767i	$-0.0675962\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.30631i	0.320568i
$388$	0	0
$389$	−7.22750	−0.366449	−0.183224	−	0.983071i	$-0.558653\pi$
$389$	−7.22750	−0.366449	−0.183224	+	0.983071i	$0.558653\pi$
$390$	0	0
$391$	−24.2502	−1.22639
$392$	0	0
$393$	2.41514i	0.121827i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.8332i	0.744457i	0.928141	+	0.372229i	$0.121406\pi$
$397$	14.8332i	0.744457i	−0.928141	+	0.372229i	$0.878594\pi$
$398$	0	0
$399$	4.64645	0.232614
$400$	0	0
$401$	−13.8248	−0.690378	−0.345189	−	0.938533i	$-0.612185\pi$
$401$	−13.8248	−0.690378	−0.345189	+	0.938533i	$0.612185\pi$
$402$	0	0
$403$	− 8.32046i	− 0.414472i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 36.9355i	− 1.83083i
$408$	0	0
$409$	−17.3659	−0.858688	−0.429344	−	0.903141i	$-0.641255\pi$
$409$	−17.3659	−0.858688	−0.429344	+	0.903141i	$0.641255\pi$
$410$	0	0
$411$	−1.70739	−0.0842195
$412$	0	0
$413$	− 54.9942i	− 2.70609i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.113717i	0.00556877i
$418$	0	0
$419$	−23.7099	−1.15831	−0.579153	−	0.815219i	$-0.696616\pi$
$419$	−23.7099	−1.15831	−0.579153	+	0.815219i	$0.696616\pi$
$420$	0	0
$421$	15.6490	0.762685	0.381342	−	0.924434i	$-0.375462\pi$
$421$	15.6490	0.762685	0.381342	+	0.924434i	$0.375462\pi$
$422$	0	0
$423$	8.49711i	0.413144i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	50.6366i	2.45048i
$428$	0	0
$429$	1.24372	0.0600475
$430$	0	0
$431$	17.2019	0.828586	0.414293	−	0.910143i	$-0.364029\pi$
$431$	17.2019	0.828586	0.414293	+	0.910143i	$0.364029\pi$
$432$	0	0
$433$	− 40.2912i	− 1.93627i	−0.250430	−	0.968135i	$-0.580572\pi$
$433$	− 40.2912i	− 1.93627i	0.250430	−	0.968135i	$-0.419428\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	53.3087i	2.55010i
$438$	0	0
$439$	−35.6171	−1.69991	−0.849956	−	0.526854i	$-0.823371\pi$
$439$	−35.6171	−1.69991	−0.849956	+	0.526854i	$0.823371\pi$
$440$	0	0
$441$	−33.4347	−1.59213
$442$	0	0
$443$	− 5.48419i	− 0.260562i	−0.991477	−	0.130281i	$-0.958412\pi$
$443$	− 5.48419i	− 0.260562i	0.991477	−	0.130281i	$-0.0415879\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.31641i	− 0.0622640i
$448$	0	0
$449$	11.7168	0.552950	0.276475	−	0.961021i	$-0.410834\pi$
$449$	11.7168	0.552950	0.276475	+	0.961021i	$0.410834\pi$
$450$	0	0
$451$	−3.36237	−0.158328
$452$	0	0
$453$	0.192713i	0.00905446i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.02360i	0.234994i	0.993073	+	0.117497i	$0.0374871\pi$
$457$	5.02360i	0.234994i	−0.993073	+	0.117497i	$0.962513\pi$
$458$	0	0
$459$	−2.96328	−0.138314
$460$	0	0
$461$	−39.2893	−1.82988	−0.914942	−	0.403586i	$-0.867764\pi$
$461$	−39.2893	−1.82988	−0.914942	+	0.403586i	$0.867764\pi$
$462$	0	0
$463$	− 13.7045i	− 0.636902i	−0.947939	−	0.318451i	$-0.896837\pi$
$463$	− 13.7045i	− 0.636902i	0.947939	−	0.318451i	$-0.103163\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 35.3092i	− 1.63391i	−0.576700	−	0.816956i	$-0.695660\pi$
$467$	− 35.3092i	− 1.63391i	0.576700	−	0.816956i	$-0.304340\pi$
$468$	0	0
$469$	36.2316	1.67302
$470$	0	0
$471$	1.64590	0.0758392
$472$	0	0
$473$	− 7.10933i	− 0.326888i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 23.2506i	− 1.06457i
$478$	0	0
$479$	12.5899	0.575249	0.287624	−	0.957743i	$-0.407135\pi$
$479$	12.5899	0.575249	0.287624	+	0.957743i	$0.407135\pi$
$480$	0	0
$481$	−30.7863	−1.40374
$482$	0	0
$483$	3.63934i	0.165596i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 35.1185i	− 1.59137i	−0.605709	−	0.795686i	$-0.707110\pi$
$487$	− 35.1185i	− 1.59137i	0.605709	−	0.795686i	$-0.292890\pi$
$488$	0	0
$489$	1.56911	0.0709576
$490$	0	0
$491$	16.8238	0.759250	0.379625	−	0.925141i	$-0.376053\pi$
$491$	16.8238	0.759250	0.379625	+	0.925141i	$0.376053\pi$
$492$	0	0
$493$	− 13.8802i	− 0.625135i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 68.6706i	− 3.08030i
$498$	0	0
$499$	−33.1274	−1.48299	−0.741493	−	0.670961i	$-0.765882\pi$
$499$	−33.1274	−1.48299	−0.741493	+	0.670961i	$0.765882\pi$
$500$	0	0
$501$	−0.693069	−0.0309641
$502$	0	0
$503$	35.6997i	1.59177i	0.605446	+	0.795886i	$0.292995\pi$
$503$	35.6997i	1.59177i	−0.605446	+	0.795886i	$0.707005\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.679120i	0.0301608i
$508$	0	0
$509$	9.05319	0.401276	0.200638	−	0.979665i	$-0.435699\pi$
$509$	9.05319	0.401276	0.200638	+	0.979665i	$0.435699\pi$
$510$	0	0
$511$	−9.73206	−0.430521
$512$	0	0
$513$	6.51411i	0.287605i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 9.57910i	− 0.421288i
$518$	0	0
$519$	−1.59504	−0.0700147
$520$	0	0
$521$	35.3384	1.54820	0.774101	−	0.633062i	$-0.218202\pi$
$521$	35.3384	1.54820	0.774101	+	0.633062i	$0.218202\pi$
$522$	0	0
$523$	− 3.16997i	− 0.138613i	−0.997595	−	0.0693066i	$-0.977921\pi$
$523$	− 3.16997i	− 0.138613i	0.997595	−	0.0693066i	$-0.0220787\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 11.1418i	− 0.485343i
$528$	0	0
$529$	−18.7541	−0.815395
$530$	0	0
$531$	38.4375	1.66804
$532$	0	0
$533$	2.80259i	0.121394i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.532987i	0.0230001i
$538$	0	0
$539$	37.6921	1.62351
$540$	0	0
$541$	12.5302	0.538717	0.269359	−	0.963040i	$-0.413188\pi$
$541$	12.5302	0.538717	0.269359	+	0.963040i	$0.413188\pi$
$542$	0	0
$543$	− 1.69732i	− 0.0728388i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 18.8269i	− 0.804978i	−0.915425	−	0.402489i	$-0.868145\pi$
$547$	− 18.8269i	− 0.804978i	0.915425	−	0.402489i	$-0.131855\pi$
$548$	0	0
$549$	−35.3918	−1.51048
$550$	0	0
$551$	−30.5126	−1.29988
$552$	0	0
$553$	− 41.2163i	− 1.75269i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.2390i	0.815182i	0.913164	+	0.407591i	$0.133631\pi$
$557$	19.2390i	0.815182i	−0.913164	+	0.407591i	$0.866369\pi$
$558$	0	0
$559$	−5.92574	−0.250632
$560$	0	0
$561$	1.66544	0.0703151
$562$	0	0
$563$	25.5849i	1.07827i	0.842218	+	0.539137i	$0.181250\pi$
$563$	25.5849i	1.07827i	−0.842218	+	0.539137i	$0.818750\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 37.7381i	− 1.58485i
$568$	0	0
$569$	14.0613	0.589482	0.294741	−	0.955577i	$-0.404767\pi$
$569$	14.0613	0.589482	0.294741	+	0.955577i	$0.404767\pi$
$570$	0	0
$571$	46.5404	1.94765	0.973827	−	0.227292i	$-0.0729873\pi$
$571$	46.5404	1.94765	0.973827	+	0.227292i	$0.0729873\pi$
$572$	0	0
$573$	− 3.19641i	− 0.133532i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.43550i	0.143022i	0.997440	+	0.0715109i	$0.0227821\pi$
$577$	3.43550i	0.143022i	−0.997440	+	0.0715109i	$0.977218\pi$
$578$	0	0
$579$	2.68051	0.111398
$580$	0	0
$581$	−4.42489	−0.183576
$582$	0	0
$583$	26.2113i	1.08556i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.8697i	1.60433i	0.597105	+	0.802163i	$0.296317\pi$
$587$	38.8697i	1.60433i	−0.597105	+	0.802163i	$0.703683\pi$
$588$	0	0
$589$	−24.4927	−1.00920
$590$	0	0
$591$	−0.396250	−0.0162996
$592$	0	0
$593$	1.44010i	0.0591380i	0.999563	+	0.0295690i	$0.00941347\pi$
$593$	1.44010i	0.0591380i	−0.999563	+	0.0295690i	$0.990587\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 0.163115i	− 0.00667584i
$598$	0	0
$599$	−1.25613	−0.0513239	−0.0256620	−	0.999671i	$-0.508169\pi$
$599$	−1.25613	−0.0513239	−0.0256620	+	0.999671i	$0.508169\pi$
$600$	0	0
$601$	−35.4144	−1.44458	−0.722291	−	0.691589i	$-0.756911\pi$
$601$	−35.4144	−1.44458	−0.722291	+	0.691589i	$0.756911\pi$
$602$	0	0
$603$	25.3236i	1.03125i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 42.6632i	− 1.73165i	−0.500350	−	0.865823i	$-0.666795\pi$
$607$	− 42.6632i	− 1.73165i	0.500350	−	0.865823i	$-0.333205\pi$
$608$	0	0
$609$	−2.08307	−0.0844103
$610$	0	0
$611$	−7.98433	−0.323011
$612$	0	0
$613$	2.12954i	0.0860114i	0.999075	+	0.0430057i	$0.0136934\pi$
$613$	2.12954i	0.0860114i	−0.999075	+	0.0430057i	$0.986307\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 16.2343i	− 0.653567i	−0.945099	−	0.326784i	$-0.894035\pi$
$617$	− 16.2343i	− 0.653567i	0.945099	−	0.326784i	$-0.105965\pi$
$618$	0	0
$619$	19.3177	0.776446	0.388223	−	0.921565i	$-0.373089\pi$
$619$	19.3177	0.776446	0.388223	+	0.921565i	$0.373089\pi$
$620$	0	0
$621$	−5.10219	−0.204744
$622$	0	0
$623$	− 6.42041i	− 0.257228i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.66111i	− 0.146211i
$628$	0	0
$629$	−41.2254	−1.64376
$630$	0	0
$631$	−8.88822	−0.353834	−0.176917	−	0.984226i	$-0.556612\pi$
$631$	−8.88822	−0.353834	−0.176917	+	0.984226i	$0.556612\pi$
$632$	0	0
$633$	− 2.17953i	− 0.0866287i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 31.4170i	− 1.24479i
$638$	0	0
$639$	47.9964	1.89871
$640$	0	0
$641$	23.4019	0.924321	0.462161	−	0.886796i	$-0.347074\pi$
$641$	23.4019	0.924321	0.462161	+	0.886796i	$0.347074\pi$
$642$	0	0
$643$	− 39.4807i	− 1.55697i	−0.627666	−	0.778483i	$-0.715990\pi$
$643$	− 39.4807i	− 1.55697i	0.627666	−	0.778483i	$-0.284010\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 7.38265i	− 0.290242i	−0.989414	−	0.145121i	$-0.953643\pi$
$647$	− 7.38265i	− 0.290242i	0.989414	−	0.145121i	$-0.0463571\pi$
$648$	0	0
$649$	−43.3320	−1.70093
$650$	0	0
$651$	−1.67210	−0.0655346
$652$	0	0
$653$	33.3826i	1.30636i	0.757202	+	0.653181i	$0.226566\pi$
$653$	33.3826i	1.30636i	−0.757202	+	0.653181i	$0.773434\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.80209i	− 0.265375i
$658$	0	0
$659$	8.75682	0.341117	0.170559	−	0.985348i	$-0.445443\pi$
$659$	8.75682	0.341117	0.170559	+	0.985348i	$0.445443\pi$
$660$	0	0
$661$	13.6695	0.531684	0.265842	−	0.964017i	$-0.414350\pi$
$661$	13.6695	0.531684	0.265842	+	0.964017i	$0.414350\pi$
$662$	0	0
$663$	− 1.38817i	− 0.0539122i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 23.8990i	− 0.925375i
$668$	0	0
$669$	−3.84381	−0.148610
$670$	0	0
$671$	39.8984	1.54026
$672$	0	0
$673$	44.5615i	1.71772i	0.512210	+	0.858860i	$0.328827\pi$
$673$	44.5615i	1.71772i	−0.512210	+	0.858860i	$0.671173\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26.5067i	1.01873i	0.860550	+	0.509367i	$0.170120\pi$
$677$	26.5067i	1.01873i	−0.860550	+	0.509367i	$0.829880\pi$
$678$	0	0
$679$	−66.5591	−2.55430
$680$	0	0
$681$	−1.03660	−0.0397227
$682$	0	0
$683$	5.63446i	0.215596i	0.994173	+	0.107798i	$0.0343800\pi$
$683$	5.63446i	0.215596i	−0.994173	+	0.107798i	$0.965620\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.963148i	0.0367464i
$688$	0	0
$689$	21.8475	0.832323
$690$	0	0
$691$	12.6396	0.480832	0.240416	−	0.970670i	$-0.422716\pi$
$691$	12.6396	0.480832	0.240416	+	0.970670i	$0.422716\pi$
$692$	0	0
$693$	42.7950i	1.62565i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.75289i	0.142151i
$698$	0	0
$699$	−0.494824	−0.0187160
$700$	0	0
$701$	−48.3130	−1.82476	−0.912379	−	0.409346i	$-0.865757\pi$
$701$	−48.3130	−1.82476	−0.912379	+	0.409346i	$0.865757\pi$
$702$	0	0
$703$	90.6248i	3.41798i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 24.7335i	− 0.930198i
$708$	0	0
$709$	−17.8824	−0.671588	−0.335794	−	0.941935i	$-0.609005\pi$
$709$	−17.8824	−0.671588	−0.335794	+	0.941935i	$0.609005\pi$
$710$	0	0
$711$	28.8075	1.08037
$712$	0	0
$713$	− 19.1839i	− 0.718444i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.32854i	0.0869608i
$718$	0	0
$719$	−39.5310	−1.47426	−0.737129	−	0.675752i	$-0.763819\pi$
$719$	−39.5310	−1.47426	−0.737129	+	0.675752i	$0.763819\pi$
$720$	0	0
$721$	−40.1539	−1.49541
$722$	0	0
$723$	− 1.84514i	− 0.0686214i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14.3442i	0.531996i	0.963974	+	0.265998i	$0.0857015\pi$
$727$	14.3442i	0.531996i	−0.963974	+	0.265998i	$0.914298\pi$
$728$	0	0
$729$	26.0639	0.965329
$730$	0	0
$731$	−7.93504	−0.293488
$732$	0	0
$733$	15.3226i	0.565951i	0.959127	+	0.282976i	$0.0913216\pi$
$733$	15.3226i	0.565951i	−0.959127	+	0.282976i	$0.908678\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 28.5482i	− 1.05159i
$738$	0	0
$739$	41.4581	1.52506	0.762530	−	0.646953i	$-0.223957\pi$
$739$	41.4581	1.52506	0.762530	+	0.646953i	$0.223957\pi$
$740$	0	0
$741$	−3.05159	−0.112103
$742$	0	0
$743$	− 20.2194i	− 0.741778i	−0.928677	−	0.370889i	$-0.879053\pi$
$743$	− 20.2194i	− 0.741778i	0.928677	−	0.370889i	$-0.120947\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3.09272i	− 0.113157i
$748$	0	0
$749$	25.3209	0.925206
$750$	0	0
$751$	31.0944	1.13465	0.567325	−	0.823494i	$-0.307978\pi$
$751$	31.0944	1.13465	0.567325	+	0.823494i	$0.307978\pi$
$752$	0	0
$753$	3.76143i	0.137074i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 25.6322i	− 0.931618i	−0.884885	−	0.465809i	$-0.845763\pi$
$757$	− 25.6322i	− 0.931618i	0.884885	−	0.465809i	$-0.154237\pi$
$758$	0	0
$759$	2.86757	0.104086
$760$	0	0
$761$	−42.1170	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$761$	−42.1170	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$762$	0	0
$763$	− 26.8631i	− 0.972508i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.1178i	1.30414i
$768$	0	0
$769$	−33.0827	−1.19299	−0.596497	−	0.802615i	$-0.703441\pi$
$769$	−33.0827	−1.19299	−0.596497	+	0.802615i	$0.703441\pi$
$770$	0	0
$771$	1.24896	0.0449803
$772$	0	0
$773$	− 10.3664i	− 0.372852i	−0.982469	−	0.186426i	$-0.940309\pi$
$773$	− 10.3664i	− 0.372852i	0.982469	−	0.186426i	$-0.0596905\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.18688i	0.221953i
$778$	0	0
$779$	8.24990	0.295583
$780$	0	0
$781$	−54.1081	−1.93614
$782$	0	0
$783$	− 2.92037i	− 0.104366i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 28.5572i	− 1.01795i	−0.860780	−	0.508977i	$-0.830024\pi$
$787$	− 28.5572i	− 1.01795i	0.860780	−	0.508977i	$-0.169976\pi$
$788$	0	0
$789$	2.87028	0.102185
$790$	0	0
$791$	62.8953	2.23630
$792$	0	0
$793$	− 33.2559i	− 1.18095i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.24548i	0.292070i	0.989279	+	0.146035i	$0.0466512\pi$
$797$	8.24548i	0.292070i	−0.989279	+	0.146035i	$0.953349\pi$
$798$	0	0
$799$	−10.6917	−0.378244
$800$	0	0
$801$	4.48745	0.158556
$802$	0	0
$803$	7.66824i	0.270606i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.81987i	0.0992643i
$808$	0	0
$809$	−1.40366	−0.0493502	−0.0246751	−	0.999696i	$-0.507855\pi$
$809$	−1.40366	−0.0493502	−0.0246751	+	0.999696i	$0.507855\pi$
$810$	0	0
$811$	29.9928	1.05319	0.526595	−	0.850116i	$-0.323468\pi$
$811$	29.9928	1.05319	0.526595	+	0.850116i	$0.323468\pi$
$812$	0	0
$813$	− 0.990243i	− 0.0347293i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	17.4434i	0.610268i
$818$	0	0
$819$	35.6702	1.24642
$820$	0	0
$821$	−44.0881	−1.53868	−0.769342	−	0.638837i	$-0.779416\pi$
$821$	−44.0881	−1.53868	−0.769342	+	0.638837i	$0.779416\pi$
$822$	0	0
$823$	24.5554i	0.855949i	0.903791	+	0.427974i	$0.140773\pi$
$823$	24.5554i	0.855949i	−0.903791	+	0.427974i	$0.859227\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44.3387i	1.54181i	0.636952	+	0.770903i	$0.280195\pi$
$827$	44.3387i	1.54181i	−0.636952	+	0.770903i	$0.719805\pi$
$828$	0	0
$829$	−12.2850	−0.426677	−0.213338	−	0.976978i	$-0.568434\pi$
$829$	−12.2850	−0.426677	−0.213338	+	0.976978i	$0.568434\pi$
$830$	0	0
$831$	−2.67692	−0.0928613
$832$	0	0
$833$	− 42.0699i	− 1.45763i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.34420i	− 0.0810275i
$838$	0	0
$839$	−4.53956	−0.156723	−0.0783615	−	0.996925i	$-0.524969\pi$
$839$	−4.53956	−0.156723	−0.0783615	+	0.996925i	$0.524969\pi$
$840$	0	0
$841$	−15.3208	−0.528302
$842$	0	0
$843$	1.73328i	0.0596972i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.30385i	− 0.0448010i
$848$	0	0
$849$	−2.52937	−0.0868076
$850$	0	0
$851$	−70.9820	−2.43323
$852$	0	0
$853$	− 23.0671i	− 0.789802i	−0.918724	−	0.394901i	$-0.870779\pi$
$853$	− 23.0671i	− 0.789802i	0.918724	−	0.394901i	$-0.129221\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.7724i	0.572935i	0.958090	+	0.286467i	$0.0924810\pi$
$857$	16.7724i	0.572935i	−0.958090	+	0.286467i	$0.907519\pi$
$858$	0	0
$859$	−25.7862	−0.879814	−0.439907	−	0.898043i	$-0.644989\pi$
$859$	−25.7862	−0.879814	−0.439907	+	0.898043i	$0.644989\pi$
$860$	0	0
$861$	0.563214	0.0191943
$862$	0	0
$863$	19.5823i	0.666591i	0.942822	+	0.333295i	$0.108161\pi$
$863$	19.5823i	0.666591i	−0.942822	+	0.333295i	$0.891839\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0.384836i	0.0130697i
$868$	0	0
$869$	−32.4758	−1.10167
$870$	0	0
$871$	−23.7953	−0.806274
$872$	0	0
$873$	− 46.5206i	− 1.57448i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 37.5098i	− 1.26662i	−0.773900	−	0.633308i	$-0.781697\pi$
$877$	− 37.5098i	− 1.26662i	0.773900	−	0.633308i	$-0.218303\pi$
$878$	0	0
$879$	−3.58798	−0.121019
$880$	0	0
$881$	−25.7777	−0.868471	−0.434236	−	0.900799i	$-0.642981\pi$
$881$	−25.7777	−0.868471	−0.434236	+	0.900799i	$0.642981\pi$
$882$	0	0
$883$	− 22.9829i	− 0.773435i	−0.922198	−	0.386717i	$-0.873609\pi$
$883$	− 22.9829i	− 0.773435i	0.922198	−	0.386717i	$-0.126391\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 0.922602i	− 0.0309780i	−0.999880	−	0.0154890i	$-0.995070\pi$
$887$	− 0.922602i	− 0.0309780i	0.999880	−	0.0154890i	$-0.00493049\pi$
$888$	0	0
$889$	−13.1188	−0.439989
$890$	0	0
$891$	−29.7352	−0.996168
$892$	0	0
$893$	23.5032i	0.786506i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.39016i	− 0.0798051i
$898$	0	0
$899$	10.9804	0.366218
$900$	0	0
$901$	29.2556	0.974644
$902$	0	0
$903$	1.19085i	0.0396289i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	33.6087i	1.11596i	0.829855	+	0.557980i	$0.188423\pi$
$907$	33.6087i	1.11596i	−0.829855	+	0.557980i	$0.811577\pi$
$908$	0	0
$909$	17.2871	0.573378
$910$	0	0
$911$	−52.5327	−1.74049	−0.870243	−	0.492623i	$-0.836038\pi$
$911$	−52.5327	−1.74049	−0.870243	+	0.492623i	$0.836038\pi$
$912$	0	0
$913$	3.48653i	0.115387i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 78.0869i	− 2.57866i
$918$	0	0
$919$	8.36159	0.275823	0.137912	−	0.990445i	$-0.455961\pi$
$919$	8.36159	0.275823	0.137912	+	0.990445i	$0.455961\pi$
$920$	0	0
$921$	−2.30528	−0.0759614
$922$	0	0
$923$	45.0999i	1.48448i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 28.0650i	− 0.921776i
$928$	0	0
$929$	3.34005	0.109583	0.0547917	−	0.998498i	$-0.482551\pi$
$929$	3.34005	0.109583	0.0547917	+	0.998498i	$0.482551\pi$
$930$	0	0
$931$	−92.4812	−3.03095
$932$	0	0
$933$	0.877593i	0.0287311i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.92101i	0.128094i	0.997947	+	0.0640469i	$0.0204007\pi$
$937$	3.92101i	0.128094i	−0.997947	+	0.0640469i	$0.979599\pi$
$938$	0	0
$939$	3.66886	0.119729
$940$	0	0
$941$	−11.8957	−0.387790	−0.193895	−	0.981022i	$-0.562112\pi$
$941$	−11.8957	−0.387790	−0.193895	+	0.981022i	$0.562112\pi$
$942$	0	0
$943$	6.46174i	0.210423i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 13.6402i	− 0.443245i	−0.975132	−	0.221623i	$-0.928865\pi$
$947$	− 13.6402i	− 0.443245i	0.975132	−	0.221623i	$-0.0711353\pi$
$948$	0	0
$949$	6.39160	0.207480
$950$	0	0
$951$	−3.00943	−0.0975876
$952$	0	0
$953$	44.5674i	1.44368i	0.692061	+	0.721839i	$0.256703\pi$
$953$	44.5674i	1.44368i	−0.692061	+	0.721839i	$0.743297\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.64133i	0.0530566i
$958$	0	0
$959$	55.2039	1.78263
$960$	0	0
$961$	−22.1859	−0.715675
$962$	0	0
$963$	17.6977i	0.570300i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 9.63452i	− 0.309825i	−0.987928	−	0.154913i	$-0.950490\pi$
$967$	− 9.63452i	− 0.309825i	0.987928	−	0.154913i	$-0.0495096\pi$
$968$	0	0
$969$	−4.08633	−0.131272
$970$	0	0
$971$	25.1511	0.807136	0.403568	−	0.914950i	$-0.367770\pi$
$971$	25.1511	0.807136	0.403568	+	0.914950i	$0.367770\pi$
$972$	0	0
$973$	− 3.67675i	− 0.117871i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.6505i	0.852626i	0.904576	+	0.426313i	$0.140188\pi$
$977$	26.6505i	0.852626i	−0.904576	+	0.426313i	$0.859812\pi$
$978$	0	0
$979$	−5.05887	−0.161682
$980$	0	0
$981$	18.7756	0.599458
$982$	0	0
$983$	− 3.12245i	− 0.0995908i	−0.998759	−	0.0497954i	$-0.984143\pi$
$983$	− 3.12245i	− 0.0995908i	0.998759	−	0.0497954i	$-0.0158569\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.60455i	0.0510733i
$988$	0	0
$989$	−13.6626	−0.434445
$990$	0	0
$991$	24.5447	0.779687	0.389844	−	0.920881i	$-0.372529\pi$
$991$	24.5447	0.779687	0.389844	+	0.920881i	$0.372529\pi$
$992$	0	0
$993$	0.814884i	0.0258596i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 9.71132i	− 0.307561i	−0.988105	−	0.153780i	$-0.950855\pi$
$997$	− 9.71132i	− 0.307561i	0.988105	−	0.153780i	$-0.0491448\pi$
$998$	0	0
$999$	−8.67372	−0.274425

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4100.2.d.g.1149.8		14
5.2	odd	4		4100.2.a.g.1.4	✓	7
5.3	odd	4		4100.2.a.j.1.4	yes	7
5.4	even	2	inner	4100.2.d.g.1149.7		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4100.2.a.g.1.4	✓	7	5.2	odd	4
4100.2.a.j.1.4	yes	7	5.3	odd	4
4100.2.d.g.1149.7		14	5.4	even	2	inner
4100.2.d.g.1149.8		14	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.131983i q^{3} -4.26732i q^{7} +2.98258 q^{9} +O(q^{10})\)	\(q+0.131983i q^{3} -4.26732i q^{7} +2.98258 q^{9} -3.36237 q^{11} +2.80259i q^{13} +3.75289i q^{17} +8.24990 q^{19} +0.563214 q^{21} +6.46174i q^{23} +0.789600i q^{27} -3.69855 q^{29} -2.96885 q^{31} -0.443776i q^{33} +10.9850i q^{37} -0.369894 q^{39} +1.00000 q^{41} +2.11438i q^{43} +2.84891i q^{47} -11.2100 q^{49} -0.495318 q^{51} -7.79547i q^{53} +1.08885i q^{57} +12.8873 q^{59} -11.8662 q^{61} -12.7276i q^{63} +8.49048i q^{67} -0.852841 q^{69} +16.0922 q^{71} -2.28061i q^{73} +14.3483i q^{77} +9.65859 q^{79} +8.84353 q^{81} -1.03693i q^{83} -0.488146i q^{87} +1.50455 q^{89} +11.9595 q^{91} -0.391838i q^{93} -15.5974i q^{97} -10.0285 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 12 q^{9}+O(q^{10}) \) 14 * q - 12 * q^9	\( 14 q - 12 q^{9} - 2 q^{11} + 2 q^{19} - 2 q^{21} + 10 q^{29} + 8 q^{31} + 22 q^{39} + 14 q^{41} + 6 q^{49} + 54 q^{51} + 32 q^{59} + 20 q^{61} + 38 q^{69} + 54 q^{71} + 2 q^{79} + 54 q^{81} + 32 q^{89} + 18 q^{91} + 80 q^{99}+O(q^{100}) \) 14 * q - 12 * q^9 - 2 * q^11 + 2 * q^19 - 2 * q^21 + 10 * q^29 + 8 * q^31 + 22 * q^39 + 14 * q^41 + 6 * q^49 + 54 * q^51 + 32 * q^59 + 20 * q^61 + 38 * q^69 + 54 * q^71 + 2 * q^79 + 54 * q^81 + 32 * q^89 + 18 * q^91 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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