LMFDB - Embedded newform 272.3.d.b.239.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [272,3,Mod(239,272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(272, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("272.239");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	272.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$7.41146319060$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} + 25 x^{10} - 16 x^{9} + 294 x^{8} - 156 x^{7} + 1960 x^{6} + 2136 x^{5} + 2980 x^{4} + \cdots + 64 $ x^12 - 3x^11 + 25x^10 - 16x^9 + 294x^8 - 156x^7 + 1960x^6 + 2136x^5 + 2980x^4 + 976x^3 + 528x^2 - 64*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.9
Root			$0.146455 - 0.253667i$ of defining polynomial
Character	$\chi$	$=$	272.239
Dual form			272.3.d.b.239.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.14999i q^{3} -1.41418 q^{5} -7.84997i q^{7} +4.37755 q^{9} +O(q^{10})$	$q+2.14999i q^{3} -1.41418 q^{5} -7.84997i q^{7} +4.37755 q^{9} +20.2823i q^{11} +21.9347 q^{13} -3.04047i q^{15} +4.12311 q^{17} +8.96036i q^{19} +16.8773 q^{21} +33.7008i q^{23} -23.0001 q^{25} +28.7616i q^{27} -36.1283 q^{29} -0.236222i q^{31} -43.6067 q^{33} +11.1013i q^{35} +71.2055 q^{37} +47.1593i q^{39} +67.1309 q^{41} -62.6485i q^{43} -6.19064 q^{45} -33.9679i q^{47} -12.6220 q^{49} +8.86463i q^{51} +20.7435 q^{53} -28.6828i q^{55} -19.2647 q^{57} +6.21939i q^{59} -8.71117 q^{61} -34.3636i q^{63} -31.0196 q^{65} +75.5392i q^{67} -72.4564 q^{69} -30.8397i q^{71} -51.4240 q^{73} -49.4500i q^{75} +159.215 q^{77} -10.3304i q^{79} -22.4392 q^{81} -105.018i q^{83} -5.83082 q^{85} -77.6755i q^{87} -132.156 q^{89} -172.186i q^{91} +0.507875 q^{93} -12.6716i q^{95} -53.4567 q^{97} +88.7866i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{5} - 28 q^{9}+O(q^{10}) $ 12 * q - 12 * q^5 - 28 * q^9	$ 12 q - 12 q^{5} - 28 q^{9} + 32 q^{13} - 16 q^{21} + 28 q^{25} - 92 q^{29} + 168 q^{33} - 116 q^{37} + 32 q^{41} + 212 q^{45} - 260 q^{49} - 80 q^{57} + 76 q^{61} - 40 q^{65} - 80 q^{69} + 184 q^{73} + 160 q^{77} - 252 q^{81} + 68 q^{85} + 72 q^{89} - 160 q^{93} + 144 q^{97}+O(q^{100}) $ 12 * q - 12 * q^5 - 28 * q^9 + 32 * q^13 - 16 * q^21 + 28 * q^25 - 92 * q^29 + 168 * q^33 - 116 * q^37 + 32 * q^41 + 212 * q^45 - 260 * q^49 - 80 * q^57 + 76 * q^61 - 40 * q^65 - 80 * q^69 + 184 * q^73 + 160 * q^77 - 252 * q^81 + 68 * q^85 + 72 * q^89 - 160 * q^93 + 144 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/272\mathbb{Z}\right)^\times$.

$n$	$69$	$239$	$241$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	2.14999i	0.716663i	0.933594	+	0.358332i	$0.116654\pi$
$3$	2.14999i	0.716663i	−0.933594	+	0.358332i	$0.883346\pi$
$4$	0	0
$5$	−1.41418	−0.282836	−0.141418	−	0.989950i	$-0.545166\pi$
$5$	−1.41418	−0.282836	−0.141418	+	0.989950i	$0.545166\pi$
$6$	0	0
$7$	− 7.84997i	− 1.12142i	−0.828011	−	0.560712i	$-0.810528\pi$
$7$	− 7.84997i	− 1.12142i	0.828011	−	0.560712i	$-0.189472\pi$
$8$	0	0
$9$	4.37755	0.486394
$10$	0	0
$11$	20.2823i	1.84384i	0.387376	+	0.921922i	$0.373381\pi$
$11$	20.2823i	1.84384i	−0.387376	+	0.921922i	$0.626619\pi$
$12$	0	0
$13$	21.9347	1.68728	0.843641	−	0.536908i	$-0.180408\pi$
$13$	21.9347	1.68728	0.843641	+	0.536908i	$0.180408\pi$
$14$	0	0
$15$	− 3.04047i	− 0.202698i
$16$	0	0
$17$	4.12311	0.242536
$17$	4.12311	0.242536
$18$	0	0
$19$	8.96036i	0.471598i	0.971802	+	0.235799i	$0.0757707\pi$
$19$	8.96036i	0.471598i	−0.971802	+	0.235799i	$0.924229\pi$
$20$	0	0
$21$	16.8773	0.803683
$22$	0	0
$23$	33.7008i	1.46525i	0.680631	+	0.732626i	$0.261706\pi$
$23$	33.7008i	1.46525i	−0.680631	+	0.732626i	$0.738294\pi$
$24$	0	0
$25$	−23.0001	−0.920004
$26$	0	0
$27$	28.7616i	1.06524i
$28$	0	0
$29$	−36.1283	−1.24580	−0.622902	−	0.782300i	$-0.714046\pi$
$29$	−36.1283	−1.24580	−0.622902	+	0.782300i	$0.714046\pi$
$30$	0	0
$31$	− 0.236222i	− 0.00762007i	−0.999993	−	0.00381004i	$-0.998787\pi$
$31$	− 0.236222i	− 0.00762007i	0.999993	−	0.00381004i	$-0.00121278\pi$
$32$	0	0
$33$	−43.6067	−1.32141
$34$	0	0
$35$	11.1013i	0.317179i
$36$	0	0
$37$	71.2055	1.92447	0.962236	−	0.272215i	$-0.0877563\pi$
$37$	71.2055	1.92447	0.962236	+	0.272215i	$0.0877563\pi$
$38$	0	0
$39$	47.1593i	1.20921i
$40$	0	0
$41$	67.1309	1.63734	0.818669	−	0.574266i	$-0.194712\pi$
$41$	67.1309	1.63734	0.818669	+	0.574266i	$0.194712\pi$
$42$	0	0
$43$	− 62.6485i	− 1.45694i	−0.685077	−	0.728471i	$-0.740232\pi$
$43$	− 62.6485i	− 1.45694i	0.685077	−	0.728471i	$-0.259768\pi$
$44$	0	0
$45$	−6.19064	−0.137570
$46$	0	0
$47$	− 33.9679i	− 0.722721i	−0.932426	−	0.361361i	$-0.882312\pi$
$47$	− 33.9679i	− 0.722721i	0.932426	−	0.361361i	$-0.117688\pi$
$48$	0	0
$49$	−12.6220	−0.257591
$50$	0	0
$51$	8.86463i	0.173816i
$52$	0	0
$53$	20.7435	0.391386	0.195693	−	0.980665i	$-0.437304\pi$
$53$	20.7435	0.391386	0.195693	+	0.980665i	$0.437304\pi$
$54$	0	0
$55$	− 28.6828i	− 0.521506i
$56$	0	0
$57$	−19.2647	−0.337977
$58$	0	0
$59$	6.21939i	0.105413i	0.998610	+	0.0527067i	$0.0167848\pi$
$59$	6.21939i	0.105413i	−0.998610	+	0.0527067i	$0.983215\pi$
$60$	0	0
$61$	−8.71117	−0.142806	−0.0714031	−	0.997448i	$-0.522748\pi$
$61$	−8.71117	−0.142806	−0.0714031	+	0.997448i	$0.522748\pi$
$62$	0	0
$63$	− 34.3636i	− 0.545454i
$64$	0	0
$65$	−31.0196	−0.477224
$66$	0	0
$67$	75.5392i	1.12745i	0.825962	+	0.563726i	$0.190632\pi$
$67$	75.5392i	1.12745i	−0.825962	+	0.563726i	$0.809368\pi$
$68$	0	0
$69$	−72.4564	−1.05009
$70$	0	0
$71$	− 30.8397i	− 0.434362i	−0.976131	−	0.217181i	$-0.930314\pi$
$71$	− 30.8397i	− 0.434362i	0.976131	−	0.217181i	$-0.0696861\pi$
$72$	0	0
$73$	−51.4240	−0.704438	−0.352219	−	0.935918i	$-0.614573\pi$
$73$	−51.4240	−0.704438	−0.352219	+	0.935918i	$0.614573\pi$
$74$	0	0
$75$	− 49.4500i	− 0.659333i
$76$	0	0
$77$	159.215	2.06773
$78$	0	0
$79$	− 10.3304i	− 0.130765i	−0.997860	−	0.0653824i	$-0.979173\pi$
$79$	− 10.3304i	− 0.130765i	0.997860	−	0.0653824i	$-0.0208267\pi$
$80$	0	0
$81$	−22.4392	−0.277027
$82$	0	0
$83$	− 105.018i	− 1.26528i	−0.774447	−	0.632639i	$-0.781972\pi$
$83$	− 105.018i	− 1.26528i	0.774447	−	0.632639i	$-0.218028\pi$
$84$	0	0
$85$	−5.83082	−0.0685979
$86$	0	0
$87$	− 77.6755i	− 0.892822i
$88$	0	0
$89$	−132.156	−1.48490	−0.742451	−	0.669901i	$-0.766337\pi$
$89$	−132.156	−1.48490	−0.742451	+	0.669901i	$0.766337\pi$
$90$	0	0
$91$	− 172.186i	− 1.89216i
$92$	0	0
$93$	0.507875	0.00546103
$94$	0	0
$95$	− 12.6716i	− 0.133385i
$96$	0	0
$97$	−53.4567	−0.551100	−0.275550	−	0.961287i	$-0.588860\pi$
$97$	−53.4567	−0.551100	−0.275550	+	0.961287i	$0.588860\pi$
$98$	0	0
$99$	88.7866i	0.896834i
$100$	0	0
$101$	53.0685	0.525430	0.262715	−	0.964873i	$-0.415382\pi$
$101$	53.0685	0.525430	0.262715	+	0.964873i	$0.415382\pi$
$102$	0	0
$103$	15.0783i	0.146391i	0.997318	+	0.0731956i	$0.0233197\pi$
$103$	15.0783i	0.146391i	−0.997318	+	0.0731956i	$0.976680\pi$
$104$	0	0
$105$	−23.8676	−0.227311
$106$	0	0
$107$	− 9.92258i	− 0.0927344i	−0.998924	−	0.0463672i	$-0.985236\pi$
$107$	− 9.92258i	− 0.0927344i	0.998924	−	0.0463672i	$-0.0147644\pi$
$108$	0	0
$109$	−140.923	−1.29287	−0.646435	−	0.762969i	$-0.723741\pi$
$109$	−140.923	−1.29287	−0.646435	+	0.762969i	$0.723741\pi$
$110$	0	0
$111$	153.091i	1.37920i
$112$	0	0
$113$	2.44314	0.0216207	0.0108103	−	0.999942i	$-0.496559\pi$
$113$	2.44314	0.0216207	0.0108103	+	0.999942i	$0.496559\pi$
$114$	0	0
$115$	− 47.6590i	− 0.414426i
$116$	0	0
$117$	96.0200	0.820683
$118$	0	0
$119$	− 32.3662i	− 0.271985i
$120$	0	0
$121$	−290.371	−2.39976
$122$	0	0
$123$	144.331i	1.17342i
$124$	0	0
$125$	67.8808	0.543047
$126$	0	0
$127$	− 174.319i	− 1.37259i	−0.727322	−	0.686296i	$-0.759235\pi$
$127$	− 174.319i	− 1.37259i	0.727322	−	0.686296i	$-0.240765\pi$
$128$	0	0
$129$	134.694	1.04414
$130$	0	0
$131$	23.0907i	0.176265i	0.996109	+	0.0881324i	$0.0280899\pi$
$131$	23.0907i	0.176265i	−0.996109	+	0.0881324i	$0.971910\pi$
$132$	0	0
$133$	70.3385	0.528861
$134$	0	0
$135$	− 40.6741i	− 0.301290i
$136$	0	0
$137$	−71.2727	−0.520238	−0.260119	−	0.965577i	$-0.583762\pi$
$137$	−71.2727	−0.520238	−0.260119	+	0.965577i	$0.583762\pi$
$138$	0	0
$139$	− 237.315i	− 1.70730i	−0.520845	−	0.853651i	$-0.674383\pi$
$139$	− 237.315i	− 1.70730i	0.520845	−	0.853651i	$-0.325617\pi$
$140$	0	0
$141$	73.0306	0.517948
$142$	0	0
$143$	444.885i	3.11108i
$144$	0	0
$145$	51.0920	0.352358
$146$	0	0
$147$	− 27.1371i	− 0.184606i
$148$	0	0
$149$	18.9223	0.126995	0.0634975	−	0.997982i	$-0.479775\pi$
$149$	18.9223	0.126995	0.0634975	+	0.997982i	$0.479775\pi$
$150$	0	0
$151$	− 120.416i	− 0.797460i	−0.917068	−	0.398730i	$-0.869451\pi$
$151$	− 120.416i	− 0.797460i	0.917068	−	0.398730i	$-0.130549\pi$
$152$	0	0
$153$	18.0491	0.117968
$154$	0	0
$155$	0.334061i	0.00215523i
$156$	0	0
$157$	−117.644	−0.749324	−0.374662	−	0.927161i	$-0.622241\pi$
$157$	−117.644	−0.749324	−0.374662	+	0.927161i	$0.622241\pi$
$158$	0	0
$159$	44.5982i	0.280492i
$160$	0	0
$161$	264.550	1.64317
$162$	0	0
$163$	213.980i	1.31276i	0.754431	+	0.656379i	$0.227913\pi$
$163$	213.980i	1.31276i	−0.754431	+	0.656379i	$0.772087\pi$
$164$	0	0
$165$	61.6677	0.373744
$166$	0	0
$167$	10.8114i	0.0647387i	0.999476	+	0.0323694i	$0.0103053\pi$
$167$	10.8114i	0.0647387i	−0.999476	+	0.0323694i	$0.989695\pi$
$168$	0	0
$169$	312.129	1.84692
$170$	0	0
$171$	39.2244i	0.229382i
$172$	0	0
$173$	−213.101	−1.23180	−0.615898	−	0.787826i	$-0.711207\pi$
$173$	−213.101	−1.23180	−0.615898	+	0.787826i	$0.711207\pi$
$174$	0	0
$175$	180.550i	1.03171i
$176$	0	0
$177$	−13.3716	−0.0755459
$178$	0	0
$179$	51.3170i	0.286687i	0.989673	+	0.143344i	$0.0457854\pi$
$179$	51.3170i	0.286687i	−0.989673	+	0.143344i	$0.954215\pi$
$180$	0	0
$181$	322.672	1.78272	0.891360	−	0.453297i	$-0.149752\pi$
$181$	322.672	1.78272	0.891360	+	0.453297i	$0.149752\pi$
$182$	0	0
$183$	− 18.7289i	− 0.102344i
$184$	0	0
$185$	−100.697	−0.544311
$186$	0	0
$187$	83.6260i	0.447198i
$188$	0	0
$189$	225.777	1.19459
$190$	0	0
$191$	− 76.7564i	− 0.401866i	−0.979605	−	0.200933i	$-0.935603\pi$
$191$	− 76.7564i	− 0.401866i	0.979605	−	0.200933i	$-0.0643974\pi$
$192$	0	0
$193$	−128.759	−0.667146	−0.333573	−	0.942724i	$-0.608254\pi$
$193$	−128.759	−0.667146	−0.333573	+	0.942724i	$0.608254\pi$
$194$	0	0
$195$	− 66.6918i	− 0.342009i
$196$	0	0
$197$	87.2916	0.443104	0.221552	−	0.975149i	$-0.428888\pi$
$197$	87.2916	0.443104	0.221552	+	0.975149i	$0.428888\pi$
$198$	0	0
$199$	− 259.225i	− 1.30264i	−0.758804	−	0.651319i	$-0.774216\pi$
$199$	− 259.225i	− 1.30264i	0.758804	−	0.651319i	$-0.225784\pi$
$200$	0	0
$201$	−162.409	−0.808003
$202$	0	0
$203$	283.606i	1.39707i
$204$	0	0
$205$	−94.9352	−0.463099
$206$	0	0
$207$	147.527i	0.712690i
$208$	0	0
$209$	−181.737	−0.869553
$210$	0	0
$211$	− 58.5888i	− 0.277672i	−0.990315	−	0.138836i	$-0.955664\pi$
$211$	− 58.5888i	− 0.277672i	0.990315	−	0.138836i	$-0.0443361\pi$
$212$	0	0
$213$	66.3050	0.311291
$214$	0	0
$215$	88.5963i	0.412076i
$216$	0	0
$217$	−1.85434	−0.00854533
$218$	0	0
$219$	− 110.561i	− 0.504845i
$220$	0	0
$221$	90.4389	0.409226
$222$	0	0
$223$	171.076i	0.767158i	0.923508	+	0.383579i	$0.125309\pi$
$223$	171.076i	0.767158i	−0.923508	+	0.383579i	$0.874691\pi$
$224$	0	0
$225$	−100.684	−0.447484
$226$	0	0
$227$	265.907i	1.17140i	0.810529	+	0.585698i	$0.199180\pi$
$227$	265.907i	1.17140i	−0.810529	+	0.585698i	$0.800820\pi$
$228$	0	0
$229$	−182.247	−0.795840	−0.397920	−	0.917420i	$-0.630268\pi$
$229$	−182.247	−0.795840	−0.397920	+	0.917420i	$0.630268\pi$
$230$	0	0
$231$	342.311i	1.48187i
$232$	0	0
$233$	−53.6367	−0.230200	−0.115100	−	0.993354i	$-0.536719\pi$
$233$	−53.6367	−0.230200	−0.115100	+	0.993354i	$0.536719\pi$
$234$	0	0
$235$	48.0368i	0.204412i
$236$	0	0
$237$	22.2103	0.0937142
$238$	0	0
$239$	− 346.889i	− 1.45142i	−0.688001	−	0.725710i	$-0.741511\pi$
$239$	− 346.889i	− 1.45142i	0.688001	−	0.725710i	$-0.258489\pi$
$240$	0	0
$241$	196.292	0.814489	0.407244	−	0.913319i	$-0.366490\pi$
$241$	196.292	0.814489	0.407244	+	0.913319i	$0.366490\pi$
$242$	0	0
$243$	210.610i	0.866709i
$244$	0	0
$245$	17.8497	0.0728560
$246$	0	0
$247$	196.542i	0.795718i
$248$	0	0
$249$	225.788	0.906778
$250$	0	0
$251$	370.244i	1.47507i	0.675306	+	0.737537i	$0.264011\pi$
$251$	370.244i	1.47507i	−0.675306	+	0.737537i	$0.735989\pi$
$252$	0	0
$253$	−683.529	−2.70170
$254$	0	0
$255$	− 12.5362i	− 0.0491616i
$256$	0	0
$257$	352.927	1.37326	0.686628	−	0.727009i	$-0.259090\pi$
$257$	352.927	1.37326	0.686628	+	0.727009i	$0.259090\pi$
$258$	0	0
$259$	− 558.961i	− 2.15815i
$260$	0	0
$261$	−158.153	−0.605951
$262$	0	0
$263$	− 206.194i	− 0.784006i	−0.919964	−	0.392003i	$-0.871782\pi$
$263$	− 206.194i	− 0.784006i	0.919964	−	0.392003i	$-0.128218\pi$
$264$	0	0
$265$	−29.3350	−0.110698
$266$	0	0
$267$	− 284.134i	− 1.06417i
$268$	0	0
$269$	−103.707	−0.385527	−0.192764	−	0.981245i	$-0.561745\pi$
$269$	−103.707	−0.385527	−0.192764	+	0.981245i	$0.561745\pi$
$270$	0	0
$271$	− 326.358i	− 1.20427i	−0.798393	−	0.602136i	$-0.794316\pi$
$271$	− 326.358i	− 1.20427i	0.798393	−	0.602136i	$-0.205684\pi$
$272$	0	0
$273$	370.199	1.35604
$274$	0	0
$275$	− 466.494i	− 1.69634i
$276$	0	0
$277$	101.124	0.365070	0.182535	−	0.983199i	$-0.441570\pi$
$277$	101.124	0.365070	0.182535	+	0.983199i	$0.441570\pi$
$278$	0	0
$279$	− 1.03407i	− 0.00370636i
$280$	0	0
$281$	220.052	0.783104	0.391552	−	0.920156i	$-0.371938\pi$
$281$	220.052	0.783104	0.391552	+	0.920156i	$0.371938\pi$
$282$	0	0
$283$	− 134.095i	− 0.473833i	−0.971530	−	0.236916i	$-0.923863\pi$
$283$	− 134.095i	− 0.473833i	0.971530	−	0.236916i	$-0.0761368\pi$
$284$	0	0
$285$	27.2438	0.0955921
$286$	0	0
$287$	− 526.975i	− 1.83615i
$288$	0	0
$289$	17.0000	0.0588235
$290$	0	0
$291$	− 114.931i	− 0.394953i
$292$	0	0
$293$	291.198	0.993850	0.496925	−	0.867793i	$-0.334462\pi$
$293$	291.198	0.993850	0.496925	+	0.867793i	$0.334462\pi$
$294$	0	0
$295$	− 8.79535i	− 0.0298147i
$296$	0	0
$297$	−583.350	−1.96414
$298$	0	0
$299$	739.215i	2.47229i
$300$	0	0
$301$	−491.788	−1.63385
$302$	0	0
$303$	114.097i	0.376557i
$304$	0	0
$305$	12.3192	0.0403907
$306$	0	0
$307$	143.879i	0.468661i	0.972157	+	0.234331i	$0.0752899\pi$
$307$	143.879i	0.468661i	−0.972157	+	0.234331i	$0.924710\pi$
$308$	0	0
$309$	−32.4182	−0.104913
$310$	0	0
$311$	194.149i	0.624272i	0.950037	+	0.312136i	$0.101045\pi$
$311$	194.149i	0.624272i	−0.950037	+	0.312136i	$0.898955\pi$
$312$	0	0
$313$	276.673	0.883938	0.441969	−	0.897030i	$-0.354280\pi$
$313$	276.673	0.883938	0.441969	+	0.897030i	$0.354280\pi$
$314$	0	0
$315$	48.5963i	0.154274i
$316$	0	0
$317$	−452.929	−1.42880	−0.714400	−	0.699738i	$-0.753300\pi$
$317$	−452.929	−1.42880	−0.714400	+	0.699738i	$0.753300\pi$
$318$	0	0
$319$	− 732.764i	− 2.29707i
$320$	0	0
$321$	21.3334	0.0664593
$322$	0	0
$323$	36.9445i	0.114379i
$324$	0	0
$325$	−504.499	−1.55230
$326$	0	0
$327$	− 302.982i	− 0.926552i
$328$	0	0
$329$	−266.647	−0.810476
$330$	0	0
$331$	− 285.346i	− 0.862072i	−0.902335	−	0.431036i	$-0.858148\pi$
$331$	− 285.346i	− 0.862072i	0.902335	−	0.431036i	$-0.141852\pi$
$332$	0	0
$333$	311.705	0.936052
$334$	0	0
$335$	− 106.826i	− 0.318884i
$336$	0	0
$337$	276.490	0.820446	0.410223	−	0.911985i	$-0.365451\pi$
$337$	276.490	0.820446	0.410223	+	0.911985i	$0.365451\pi$
$338$	0	0
$339$	5.25272i	0.0154947i
$340$	0	0
$341$	4.79113	0.0140502
$342$	0	0
$343$	− 285.566i	− 0.832555i
$344$	0	0
$345$	102.466	0.297004
$346$	0	0
$347$	122.985i	0.354423i	0.984173	+	0.177212i	$0.0567077\pi$
$347$	122.985i	0.354423i	−0.984173	+	0.177212i	$0.943292\pi$
$348$	0	0
$349$	−186.113	−0.533274	−0.266637	−	0.963797i	$-0.585912\pi$
$349$	−186.113	−0.533274	−0.266637	+	0.963797i	$0.585912\pi$
$350$	0	0
$351$	630.875i	1.79737i
$352$	0	0
$353$	88.5985	0.250987	0.125494	−	0.992094i	$-0.459949\pi$
$353$	88.5985	0.250987	0.125494	+	0.992094i	$0.459949\pi$
$354$	0	0
$355$	43.6129i	0.122853i
$356$	0	0
$357$	69.5871	0.194922
$358$	0	0
$359$	− 470.130i	− 1.30955i	−0.755822	−	0.654777i	$-0.772762\pi$
$359$	− 470.130i	− 1.30955i	0.755822	−	0.654777i	$-0.227238\pi$
$360$	0	0
$361$	280.712	0.777595
$362$	0	0
$363$	− 624.294i	− 1.71982i
$364$	0	0
$365$	72.7228	0.199241
$366$	0	0
$367$	66.7793i	0.181960i	0.995853	+	0.0909799i	$0.0289999\pi$
$367$	66.7793i	0.181960i	−0.995853	+	0.0909799i	$0.971000\pi$
$368$	0	0
$369$	293.868	0.796391
$370$	0	0
$371$	− 162.835i	− 0.438909i
$372$	0	0
$373$	−106.764	−0.286231	−0.143116	−	0.989706i	$-0.545712\pi$
$373$	−106.764	−0.286231	−0.143116	+	0.989706i	$0.545712\pi$
$374$	0	0
$375$	145.943i	0.389181i
$376$	0	0
$377$	−792.462	−2.10202
$378$	0	0
$379$	− 233.855i	− 0.617032i	−0.951219	−	0.308516i	$-0.900168\pi$
$379$	− 233.855i	− 0.617032i	0.951219	−	0.308516i	$-0.0998323\pi$
$380$	0	0
$381$	374.785	0.983687
$382$	0	0
$383$	570.095i	1.48850i	0.667902	+	0.744249i	$0.267193\pi$
$383$	570.095i	1.48850i	−0.667902	+	0.744249i	$0.732807\pi$
$384$	0	0
$385$	−225.159	−0.584829
$386$	0	0
$387$	− 274.247i	− 0.708647i
$388$	0	0
$389$	−11.9068	−0.0306087	−0.0153043	−	0.999883i	$-0.504872\pi$
$389$	−11.9068	−0.0306087	−0.0153043	+	0.999883i	$0.504872\pi$
$390$	0	0
$391$	138.952i	0.355376i
$392$	0	0
$393$	−49.6447	−0.126322
$394$	0	0
$395$	14.6091i	0.0369850i
$396$	0	0
$397$	68.4901	0.172519	0.0862596	−	0.996273i	$-0.472509\pi$
$397$	68.4901	0.172519	0.0862596	+	0.996273i	$0.472509\pi$
$398$	0	0
$399$	151.227i	0.379015i
$400$	0	0
$401$	512.212	1.27734	0.638668	−	0.769482i	$-0.279486\pi$
$401$	512.212	1.27734	0.638668	+	0.769482i	$0.279486\pi$
$402$	0	0
$403$	− 5.18145i	− 0.0128572i
$404$	0	0
$405$	31.7331	0.0783532
$406$	0	0
$407$	1444.21i	3.54843i
$408$	0	0
$409$	−524.532	−1.28247	−0.641237	−	0.767343i	$-0.721578\pi$
$409$	−524.532	−1.28247	−0.641237	+	0.767343i	$0.721578\pi$
$410$	0	0
$411$	− 153.235i	− 0.372836i
$412$	0	0
$413$	48.8220	0.118213
$414$	0	0
$415$	148.515i	0.357867i
$416$	0	0
$417$	510.225	1.22356
$418$	0	0
$419$	− 6.81698i	− 0.0162696i	−0.999967	−	0.00813482i	$-0.997411\pi$
$419$	− 6.81698i	− 0.0162696i	0.999967	−	0.00813482i	$-0.00258942\pi$
$420$	0	0
$421$	−432.282	−1.02680	−0.513399	−	0.858150i	$-0.671614\pi$
$421$	−432.282	−1.02680	−0.513399	+	0.858150i	$0.671614\pi$
$422$	0	0
$423$	− 148.696i	− 0.351527i
$424$	0	0
$425$	−94.8318	−0.223134
$426$	0	0
$427$	68.3824i	0.160146i
$428$	0	0
$429$	−956.497	−2.22960
$430$	0	0
$431$	− 615.188i	− 1.42735i	−0.700477	−	0.713675i	$-0.747029\pi$
$431$	− 615.188i	− 1.42735i	0.700477	−	0.713675i	$-0.252971\pi$
$432$	0	0
$433$	−277.588	−0.641081	−0.320540	−	0.947235i	$-0.603864\pi$
$433$	−277.588	−0.641081	−0.320540	+	0.947235i	$0.603864\pi$
$434$	0	0
$435$	109.847i	0.252522i
$436$	0	0
$437$	−301.971	−0.691010
$438$	0	0
$439$	317.273i	0.722717i	0.932427	+	0.361359i	$0.117687\pi$
$439$	317.273i	0.722717i	−0.932427	+	0.361359i	$0.882313\pi$
$440$	0	0
$441$	−55.2532	−0.125291
$442$	0	0
$443$	− 446.833i	− 1.00865i	−0.863513	−	0.504327i	$-0.831741\pi$
$443$	− 446.833i	− 1.00865i	0.863513	−	0.504327i	$-0.168259\pi$
$444$	0	0
$445$	186.893	0.419984
$446$	0	0
$447$	40.6826i	0.0910126i
$448$	0	0
$449$	107.074	0.238472	0.119236	−	0.992866i	$-0.461955\pi$
$449$	107.074	0.238472	0.119236	+	0.992866i	$0.461955\pi$
$450$	0	0
$451$	1361.57i	3.01899i
$452$	0	0
$453$	258.894	0.571510
$454$	0	0
$455$	243.503i	0.535171i
$456$	0	0
$457$	484.298	1.05973	0.529867	−	0.848081i	$-0.322242\pi$
$457$	484.298	1.05973	0.529867	+	0.848081i	$0.322242\pi$
$458$	0	0
$459$	118.587i	0.258360i
$460$	0	0
$461$	133.372	0.289311	0.144655	−	0.989482i	$-0.453793\pi$
$461$	133.372	0.289311	0.144655	+	0.989482i	$0.453793\pi$
$462$	0	0
$463$	55.7240i	0.120354i	0.998188	+	0.0601771i	$0.0191665\pi$
$463$	55.7240i	0.120354i	−0.998188	+	0.0601771i	$0.980833\pi$
$464$	0	0
$465$	−0.718228	−0.00154458
$466$	0	0
$467$	− 521.290i	− 1.11625i	−0.829756	−	0.558127i	$-0.811520\pi$
$467$	− 521.290i	− 1.11625i	0.829756	−	0.558127i	$-0.188480\pi$
$468$	0	0
$469$	592.980	1.26435
$470$	0	0
$471$	− 252.933i	− 0.537013i
$472$	0	0
$473$	1270.65	2.68637
$474$	0	0
$475$	− 206.089i	− 0.433872i
$476$	0	0
$477$	90.8054	0.190368
$478$	0	0
$479$	26.0734i	0.0544330i	0.999630	+	0.0272165i	$0.00866435\pi$
$479$	26.0734i	0.0544330i	−0.999630	+	0.0272165i	$0.991336\pi$
$480$	0	0
$481$	1561.87	3.24713
$482$	0	0
$483$	568.780i	1.17760i
$484$	0	0
$485$	75.5974	0.155871
$486$	0	0
$487$	152.308i	0.312747i	0.987698	+	0.156374i	$0.0499804\pi$
$487$	152.308i	0.312747i	−0.987698	+	0.156374i	$0.950020\pi$
$488$	0	0
$489$	−460.054	−0.940806
$490$	0	0
$491$	− 406.124i	− 0.827137i	−0.910473	−	0.413569i	$-0.864282\pi$
$491$	− 406.124i	− 0.827137i	0.910473	−	0.413569i	$-0.135718\pi$
$492$	0	0
$493$	−148.961	−0.302152
$494$	0	0
$495$	− 125.560i	− 0.253657i
$496$	0	0
$497$	−242.090	−0.487103
$498$	0	0
$499$	− 35.6035i	− 0.0713497i	−0.999363	−	0.0356749i	$-0.988642\pi$
$499$	− 35.6035i	− 0.0713497i	0.999363	−	0.0356749i	$-0.0113581\pi$
$500$	0	0
$501$	−23.2443	−0.0463959
$502$	0	0
$503$	271.350i	0.539463i	0.962936	+	0.269731i	$0.0869349\pi$
$503$	271.350i	0.539463i	−0.962936	+	0.269731i	$0.913065\pi$
$504$	0	0
$505$	−75.0484	−0.148611
$506$	0	0
$507$	671.074i	1.32362i
$508$	0	0
$509$	−265.151	−0.520926	−0.260463	−	0.965484i	$-0.583875\pi$
$509$	−265.151	−0.520926	−0.260463	+	0.965484i	$0.583875\pi$
$510$	0	0
$511$	403.676i	0.789974i
$512$	0	0
$513$	−257.714	−0.502367
$514$	0	0
$515$	− 21.3234i	− 0.0414047i
$516$	0	0
$517$	688.946	1.33258
$518$	0	0
$519$	− 458.164i	− 0.882782i
$520$	0	0
$521$	−186.168	−0.357328	−0.178664	−	0.983910i	$-0.557178\pi$
$521$	−186.168	−0.357328	−0.178664	+	0.983910i	$0.557178\pi$
$522$	0	0
$523$	676.053i	1.29264i	0.763065	+	0.646322i	$0.223694\pi$
$523$	676.053i	1.29264i	−0.763065	+	0.646322i	$0.776306\pi$
$524$	0	0
$525$	−388.180	−0.739391
$526$	0	0
$527$	− 0.973970i	− 0.00184814i
$528$	0	0
$529$	−606.744	−1.14696
$530$	0	0
$531$	27.2257i	0.0512725i
$532$	0	0
$533$	1472.49	2.76265
$534$	0	0
$535$	14.0323i	0.0262286i
$536$	0	0
$537$	−110.331	−0.205458
$538$	0	0
$539$	− 256.002i	− 0.474957i
$540$	0	0
$541$	−467.214	−0.863611	−0.431806	−	0.901967i	$-0.642123\pi$
$541$	−467.214	−0.863611	−0.431806	+	0.901967i	$0.642123\pi$
$542$	0	0
$543$	693.742i	1.27761i
$544$	0	0
$545$	199.290	0.365670
$546$	0	0
$547$	1.04686i	0.00191382i	1.00000		0.000956910i	$0.000304594\pi$
$547$	1.04686i	0.00191382i	−1.00000		0.000956910i	$0.999695\pi$
$548$	0	0
$549$	−38.1336	−0.0694600
$550$	0	0
$551$	− 323.723i	− 0.587519i
$552$	0	0
$553$	−81.0934	−0.146643
$554$	0	0
$555$	− 216.498i	− 0.390087i
$556$	0	0
$557$	−118.899	−0.213463	−0.106731	−	0.994288i	$-0.534038\pi$
$557$	−118.899	−0.213463	−0.106731	+	0.994288i	$0.534038\pi$
$558$	0	0
$559$	− 1374.17i	− 2.45827i
$560$	0	0
$561$	−179.795	−0.320490
$562$	0	0
$563$	614.354i	1.09122i	0.838041	+	0.545608i	$0.183701\pi$
$563$	614.354i	1.09122i	−0.838041	+	0.545608i	$0.816299\pi$
$564$	0	0
$565$	−3.45504	−0.00611511
$566$	0	0
$567$	176.147i	0.310665i
$568$	0	0
$569$	−141.788	−0.249189	−0.124594	−	0.992208i	$-0.539763\pi$
$569$	−141.788	−0.249189	−0.124594	+	0.992208i	$0.539763\pi$
$570$	0	0
$571$	1121.61i	1.96430i	0.188103	+	0.982149i	$0.439766\pi$
$571$	1121.61i	1.96430i	−0.188103	+	0.982149i	$0.560234\pi$
$572$	0	0
$573$	165.025	0.288003
$574$	0	0
$575$	− 775.122i	− 1.34804i
$576$	0	0
$577$	803.100	1.39185	0.695927	−	0.718112i	$-0.254994\pi$
$577$	803.100	1.39185	0.695927	+	0.718112i	$0.254994\pi$
$578$	0	0
$579$	− 276.831i	− 0.478119i
$580$	0	0
$581$	−824.388	−1.41891
$582$	0	0
$583$	420.724i	0.721654i
$584$	0	0
$585$	−135.790	−0.232119
$586$	0	0
$587$	80.5627i	0.137245i	0.997643	+	0.0686224i	$0.0218604\pi$
$587$	80.5627i	0.137245i	−0.997643	+	0.0686224i	$0.978140\pi$
$588$	0	0
$589$	2.11664	0.00359361
$590$	0	0
$591$	187.676i	0.317557i
$592$	0	0
$593$	−955.930	−1.61202	−0.806012	−	0.591899i	$-0.798378\pi$
$593$	−955.930	−1.61202	−0.806012	+	0.591899i	$0.798378\pi$
$594$	0	0
$595$	45.7717i	0.0769273i
$596$	0	0
$597$	557.331	0.933553
$598$	0	0
$599$	49.2252i	0.0821790i	0.999155	+	0.0410895i	$0.0130829\pi$
$599$	49.2252i	0.0821790i	−0.999155	+	0.0410895i	$0.986917\pi$
$600$	0	0
$601$	−372.047	−0.619046	−0.309523	−	0.950892i	$-0.600169\pi$
$601$	−372.047	−0.619046	−0.309523	+	0.950892i	$0.600169\pi$
$602$	0	0
$603$	330.676i	0.548385i
$604$	0	0
$605$	410.637	0.678739
$606$	0	0
$607$	− 409.110i	− 0.673987i	−0.941507	−	0.336993i	$-0.890590\pi$
$607$	− 409.110i	− 0.673987i	0.941507	−	0.336993i	$-0.109410\pi$
$608$	0	0
$609$	−609.750	−1.00123
$610$	0	0
$611$	− 745.074i	− 1.21943i
$612$	0	0
$613$	450.831	0.735450	0.367725	−	0.929934i	$-0.380137\pi$
$613$	450.831	0.735450	0.367725	+	0.929934i	$0.380137\pi$
$614$	0	0
$615$	− 204.110i	− 0.331886i
$616$	0	0
$617$	−373.012	−0.604557	−0.302279	−	0.953220i	$-0.597747\pi$
$617$	−373.012	−0.604557	−0.302279	+	0.953220i	$0.597747\pi$
$618$	0	0
$619$	562.981i	0.909501i	0.890619	+	0.454751i	$0.150272\pi$
$619$	562.981i	0.909501i	−0.890619	+	0.454751i	$0.849728\pi$
$620$	0	0
$621$	−969.288	−1.56085
$622$	0	0
$623$	1037.42i	1.66520i
$624$	0	0
$625$	479.006	0.766410
$626$	0	0
$627$	− 390.732i	− 0.623176i
$628$	0	0
$629$	293.588	0.466753
$630$	0	0
$631$	− 851.669i	− 1.34971i	−0.737949	−	0.674857i	$-0.764205\pi$
$631$	− 851.669i	− 1.34971i	0.737949	−	0.674857i	$-0.235795\pi$
$632$	0	0
$633$	125.965	0.198997
$634$	0	0
$635$	246.519i	0.388219i
$636$	0	0
$637$	−276.858	−0.434628
$638$	0	0
$639$	− 135.002i	− 0.211271i
$640$	0	0
$641$	−836.982	−1.30574	−0.652872	−	0.757468i	$-0.726436\pi$
$641$	−836.982	−1.30574	−0.652872	+	0.757468i	$0.726436\pi$
$642$	0	0
$643$	− 925.891i	− 1.43995i	−0.693998	−	0.719977i	$-0.744152\pi$
$643$	− 925.891i	− 1.43995i	0.693998	−	0.719977i	$-0.255848\pi$
$644$	0	0
$645$	−190.481	−0.295319
$646$	0	0
$647$	− 612.829i	− 0.947186i	−0.880744	−	0.473593i	$-0.842957\pi$
$647$	− 612.829i	− 0.947186i	0.880744	−	0.473593i	$-0.157043\pi$
$648$	0	0
$649$	−126.143	−0.194366
$650$	0	0
$651$	− 3.98680i	− 0.00612412i
$652$	0	0
$653$	−439.454	−0.672977	−0.336489	−	0.941688i	$-0.609239\pi$
$653$	−439.454	−0.672977	−0.336489	+	0.941688i	$0.609239\pi$
$654$	0	0
$655$	− 32.6544i	− 0.0498541i
$656$	0	0
$657$	−225.111	−0.342634
$658$	0	0
$659$	− 1148.12i	− 1.74222i	−0.491087	−	0.871110i	$-0.663400\pi$
$659$	− 1148.12i	− 1.74222i	0.491087	−	0.871110i	$-0.336600\pi$
$660$	0	0
$661$	73.2077	0.110753	0.0553765	−	0.998466i	$-0.482364\pi$
$661$	73.2077	0.110753	0.0553765	+	0.998466i	$0.482364\pi$
$662$	0	0
$663$	194.443i	0.293277i
$664$	0	0
$665$	−99.4714	−0.149581
$666$	0	0
$667$	− 1217.55i	− 1.82542i
$668$	0	0
$669$	−367.812	−0.549794
$670$	0	0
$671$	− 176.682i	− 0.263312i
$672$	0	0
$673$	289.275	0.429829	0.214915	−	0.976633i	$-0.431053\pi$
$673$	289.275	0.429829	0.214915	+	0.976633i	$0.431053\pi$
$674$	0	0
$675$	− 661.519i	− 0.980028i
$676$	0	0
$677$	34.0355	0.0502741	0.0251370	−	0.999684i	$-0.491998\pi$
$677$	34.0355	0.0502741	0.0251370	+	0.999684i	$0.491998\pi$
$678$	0	0
$679$	419.633i	0.618016i
$680$	0	0
$681$	−571.697	−0.839496
$682$	0	0
$683$	208.723i	0.305598i	0.988257	+	0.152799i	$0.0488287\pi$
$683$	208.723i	0.305598i	−0.988257	+	0.152799i	$0.951171\pi$
$684$	0	0
$685$	100.792	0.147142
$686$	0	0
$687$	− 391.830i	− 0.570349i
$688$	0	0
$689$	455.001	0.660378
$690$	0	0
$691$	717.508i	1.03836i	0.854665	+	0.519181i	$0.173763\pi$
$691$	717.508i	1.03836i	−0.854665	+	0.519181i	$0.826237\pi$
$692$	0	0
$693$	696.972	1.00573
$694$	0	0
$695$	335.606i	0.482887i
$696$	0	0
$697$	276.788	0.397113
$698$	0	0
$699$	− 115.318i	− 0.164976i
$700$	0	0
$701$	1226.55	1.74972	0.874860	−	0.484376i	$-0.160953\pi$
$701$	1226.55	1.74972	0.874860	+	0.484376i	$0.160953\pi$
$702$	0	0
$703$	638.027i	0.907578i
$704$	0	0
$705$	−103.279	−0.146494
$706$	0	0
$707$	− 416.586i	− 0.589230i
$708$	0	0
$709$	−456.115	−0.643322	−0.321661	−	0.946855i	$-0.604241\pi$
$709$	−456.115	−0.643322	−0.321661	+	0.946855i	$0.604241\pi$
$710$	0	0
$711$	− 45.2219i	− 0.0636032i
$712$	0	0
$713$	7.96088	0.0111653
$714$	0	0
$715$	− 629.148i	− 0.879927i
$716$	0	0
$717$	745.809	1.04018
$718$	0	0
$719$	− 37.0100i	− 0.0514742i	−0.999669	−	0.0257371i	$-0.991807\pi$
$719$	− 37.0100i	− 0.0514742i	0.999669	−	0.0257371i	$-0.00819328\pi$
$720$	0	0
$721$	118.364	0.164166
$722$	0	0
$723$	422.025i	0.583714i
$724$	0	0
$725$	830.954	1.14614
$726$	0	0
$727$	799.664i	1.09995i	0.835181	+	0.549975i	$0.185363\pi$
$727$	799.664i	1.09995i	−0.835181	+	0.549975i	$0.814637\pi$
$728$	0	0
$729$	−654.762	−0.898165
$730$	0	0
$731$	− 258.306i	− 0.353360i
$732$	0	0
$733$	694.060	0.946876	0.473438	−	0.880827i	$-0.343013\pi$
$733$	694.060	0.946876	0.473438	+	0.880827i	$0.343013\pi$
$734$	0	0
$735$	38.3767i	0.0522132i
$736$	0	0
$737$	−1532.11	−2.07884
$738$	0	0
$739$	− 585.040i	− 0.791665i	−0.918323	−	0.395832i	$-0.870456\pi$
$739$	− 585.040i	− 0.791665i	0.918323	−	0.395832i	$-0.129544\pi$
$740$	0	0
$741$	−422.564	−0.570262
$742$	0	0
$743$	− 842.319i	− 1.13367i	−0.823830	−	0.566836i	$-0.808167\pi$
$743$	− 842.319i	− 1.13367i	0.823830	−	0.566836i	$-0.191833\pi$
$744$	0	0
$745$	−26.7595	−0.0359188
$746$	0	0
$747$	− 459.721i	− 0.615424i
$748$	0	0
$749$	−77.8919	−0.103995
$750$	0	0
$751$	− 469.037i	− 0.624550i	−0.949992	−	0.312275i	$-0.898909\pi$
$751$	− 469.037i	− 0.624550i	0.949992	−	0.312275i	$-0.101091\pi$
$752$	0	0
$753$	−796.020	−1.05713
$754$	0	0
$755$	170.291i	0.225551i
$756$	0	0
$757$	−761.467	−1.00590	−0.502951	−	0.864315i	$-0.667752\pi$
$757$	−761.467	−1.00590	−0.502951	+	0.864315i	$0.667752\pi$
$758$	0	0
$759$	− 1469.58i	− 1.93621i
$760$	0	0
$761$	−1211.94	−1.59257	−0.796283	−	0.604925i	$-0.793203\pi$
$761$	−1211.94	−1.59257	−0.796283	+	0.604925i	$0.793203\pi$
$762$	0	0
$763$	1106.24i	1.44985i
$764$	0	0
$765$	−25.5247	−0.0333656
$766$	0	0
$767$	136.420i	0.177862i
$768$	0	0
$769$	505.719	0.657632	0.328816	−	0.944394i	$-0.393350\pi$
$769$	505.719	0.657632	0.328816	+	0.944394i	$0.393350\pi$
$770$	0	0
$771$	758.789i	0.984162i
$772$	0	0
$773$	−375.312	−0.485526	−0.242763	−	0.970086i	$-0.578054\pi$
$773$	−375.312	−0.485526	−0.242763	+	0.970086i	$0.578054\pi$
$774$	0	0
$775$	5.43313i	0.00701050i
$776$	0	0
$777$	1201.76	1.54667
$778$	0	0
$779$	601.517i	0.772165i
$780$	0	0
$781$	625.499	0.800895
$782$	0	0
$783$	− 1039.11i	− 1.32708i
$784$	0	0
$785$	166.370	0.211936
$786$	0	0
$787$	757.545i	0.962574i	0.876563	+	0.481287i	$0.159830\pi$
$787$	757.545i	0.962574i	−0.876563	+	0.481287i	$0.840170\pi$
$788$	0	0
$789$	443.314	0.561868
$790$	0	0
$791$	− 19.1785i	− 0.0242459i
$792$	0	0
$793$	−191.077	−0.240954
$794$	0	0
$795$	− 63.0699i	− 0.0793333i
$796$	0	0
$797$	−644.741	−0.808959	−0.404480	−	0.914547i	$-0.632547\pi$
$797$	−644.741	−0.808959	−0.404480	+	0.914547i	$0.632547\pi$
$798$	0	0
$799$	− 140.053i	− 0.175286i
$800$	0	0
$801$	−578.520	−0.722247
$802$	0	0
$803$	− 1043.00i	− 1.29887i
$804$	0	0
$805$	−374.122	−0.464748
$806$	0	0
$807$	− 222.969i	− 0.276293i
$808$	0	0
$809$	−976.519	−1.20707	−0.603535	−	0.797337i	$-0.706241\pi$
$809$	−976.519	−1.20707	−0.603535	+	0.797337i	$0.706241\pi$
$810$	0	0
$811$	1081.56i	1.33361i	0.745232	+	0.666805i	$0.232339\pi$
$811$	1081.56i	1.33361i	−0.745232	+	0.666805i	$0.767661\pi$
$812$	0	0
$813$	701.666	0.863058
$814$	0	0
$815$	− 302.606i	− 0.371296i
$816$	0	0
$817$	561.353	0.687091
$818$	0	0
$819$	− 753.753i	− 0.920334i
$820$	0	0
$821$	414.904	0.505364	0.252682	−	0.967549i	$-0.418687\pi$
$821$	414.904	0.505364	0.252682	+	0.967549i	$0.418687\pi$
$822$	0	0
$823$	406.864i	0.494368i	0.968969	+	0.247184i	$0.0795051\pi$
$823$	406.864i	0.494368i	−0.968969	+	0.247184i	$0.920495\pi$
$824$	0	0
$825$	1002.96	1.21571
$826$	0	0
$827$	− 438.426i	− 0.530140i	−0.964229	−	0.265070i	$-0.914605\pi$
$827$	− 438.426i	− 0.530140i	0.964229	−	0.265070i	$-0.0853951\pi$
$828$	0	0
$829$	1387.83	1.67410	0.837049	−	0.547128i	$-0.184279\pi$
$829$	1387.83	1.67410	0.837049	+	0.547128i	$0.184279\pi$
$830$	0	0
$831$	217.416i	0.261632i
$832$	0	0
$833$	−52.0416	−0.0624750
$834$	0	0
$835$	− 15.2892i	− 0.0183105i
$836$	0	0
$837$	6.79413	0.00811724
$838$	0	0
$839$	− 744.454i	− 0.887312i	−0.896197	−	0.443656i	$-0.853681\pi$
$839$	− 744.454i	− 0.887312i	0.896197	−	0.443656i	$-0.146319\pi$
$840$	0	0
$841$	464.255	0.552027
$842$	0	0
$843$	473.110i	0.561222i
$844$	0	0
$845$	−441.407	−0.522375
$846$	0	0
$847$	2279.40i	2.69115i
$848$	0	0
$849$	288.302	0.339579
$850$	0	0
$851$	2399.68i	2.81984i
$852$	0	0
$853$	−252.952	−0.296544	−0.148272	−	0.988947i	$-0.547371\pi$
$853$	−252.952	−0.296544	−0.148272	+	0.988947i	$0.547371\pi$
$854$	0	0
$855$	− 55.4704i	− 0.0648777i
$856$	0	0
$857$	50.0796	0.0584360	0.0292180	−	0.999573i	$-0.490698\pi$
$857$	50.0796	0.0584360	0.0292180	+	0.999573i	$0.490698\pi$
$858$	0	0
$859$	− 322.852i	− 0.375846i	−0.982184	−	0.187923i	$-0.939824\pi$
$859$	− 322.852i	− 0.375846i	0.982184	−	0.187923i	$-0.0601755\pi$
$860$	0	0
$861$	1132.99	1.31590
$862$	0	0
$863$	− 1364.23i	− 1.58080i	−0.612588	−	0.790402i	$-0.709872\pi$
$863$	− 1364.23i	− 1.58080i	0.612588	−	0.790402i	$-0.290128\pi$
$864$	0	0
$865$	301.363	0.348396
$866$	0	0
$867$	36.5498i	0.0421567i
$868$	0	0
$869$	209.524	0.241110
$870$	0	0
$871$	1656.93i	1.90233i
$872$	0	0
$873$	−234.009	−0.268052
$874$	0	0
$875$	− 532.862i	− 0.608985i
$876$	0	0
$877$	−509.762	−0.581257	−0.290628	−	0.956836i	$-0.593864\pi$
$877$	−509.762	−0.581257	−0.290628	+	0.956836i	$0.593864\pi$
$878$	0	0
$879$	626.073i	0.712256i
$880$	0	0
$881$	−283.329	−0.321599	−0.160799	−	0.986987i	$-0.551407\pi$
$881$	−283.329	−0.321599	−0.160799	+	0.986987i	$0.551407\pi$
$882$	0	0
$883$	1194.43i	1.35270i	0.736582	+	0.676349i	$0.236439\pi$
$883$	1194.43i	1.35270i	−0.736582	+	0.676349i	$0.763561\pi$
$884$	0	0
$885$	18.9099	0.0213671
$886$	0	0
$887$	1379.33i	1.55505i	0.628849	+	0.777527i	$0.283526\pi$
$887$	1379.33i	1.55505i	−0.628849	+	0.777527i	$0.716474\pi$
$888$	0	0
$889$	−1368.40	−1.53926
$890$	0	0
$891$	− 455.118i	− 0.510794i
$892$	0	0
$893$	304.365	0.340834
$894$	0	0
$895$	− 72.5716i	− 0.0810856i
$896$	0	0
$897$	−1589.31	−1.77180
$898$	0	0
$899$	8.53431i	0.00949312i
$900$	0	0
$901$	85.5275	0.0949250
$902$	0	0
$903$	− 1057.34i	− 1.17092i
$904$	0	0
$905$	−456.317	−0.504218
$906$	0	0
$907$	− 645.154i	− 0.711305i	−0.934618	−	0.355653i	$-0.884259\pi$
$907$	− 645.154i	− 0.711305i	0.934618	−	0.355653i	$-0.115741\pi$
$908$	0	0
$909$	232.310	0.255566
$910$	0	0
$911$	1572.67i	1.72631i	0.504941	+	0.863154i	$0.331514\pi$
$911$	1572.67i	1.72631i	−0.504941	+	0.863154i	$0.668486\pi$
$912$	0	0
$913$	2130.01	2.33297
$914$	0	0
$915$	26.4861i	0.0289466i
$916$	0	0
$917$	181.261	0.197667
$918$	0	0
$919$	886.584i	0.964727i	0.875971	+	0.482364i	$0.160222\pi$
$919$	886.584i	0.964727i	−0.875971	+	0.482364i	$0.839778\pi$
$920$	0	0
$921$	−309.338	−0.335872
$922$	0	0
$923$	− 676.458i	− 0.732890i
$924$	0	0
$925$	−1637.73	−1.77052
$926$	0	0
$927$	66.0059i	0.0712038i
$928$	0	0
$929$	−571.048	−0.614691	−0.307345	−	0.951598i	$-0.599441\pi$
$929$	−571.048	−0.614691	−0.307345	+	0.951598i	$0.599441\pi$
$930$	0	0
$931$	− 113.097i	− 0.121479i
$932$	0	0
$933$	−417.418	−0.447393
$934$	0	0
$935$	− 118.262i	− 0.126484i
$936$	0	0
$937$	−102.132	−0.108999	−0.0544997	−	0.998514i	$-0.517356\pi$
$937$	−102.132	−0.108999	−0.0544997	+	0.998514i	$0.517356\pi$
$938$	0	0
$939$	594.843i	0.633486i
$940$	0	0
$941$	745.924	0.792693	0.396346	−	0.918101i	$-0.370278\pi$
$941$	745.924	0.792693	0.396346	+	0.918101i	$0.370278\pi$
$942$	0	0
$943$	2262.36i	2.39911i
$944$	0	0
$945$	−319.290	−0.337873
$946$	0	0
$947$	1070.99i	1.13093i	0.824772	+	0.565466i	$0.191304\pi$
$947$	1070.99i	1.13093i	−0.824772	+	0.565466i	$0.808696\pi$
$948$	0	0
$949$	−1127.97	−1.18859
$950$	0	0
$951$	− 973.793i	− 1.02397i
$952$	0	0
$953$	1287.70	1.35121	0.675605	−	0.737264i	$-0.263883\pi$
$953$	1287.70	1.35121	0.675605	+	0.737264i	$0.263883\pi$
$954$	0	0
$955$	108.547i	0.113662i
$956$	0	0
$957$	1575.44	1.64622
$958$	0	0
$959$	559.488i	0.583408i
$960$	0	0
$961$	960.944	0.999942
$962$	0	0
$963$	− 43.4365i	− 0.0451054i
$964$	0	0
$965$	182.089	0.188693
$966$	0	0
$967$	479.166i	0.495518i	0.968822	+	0.247759i	$0.0796941\pi$
$967$	479.166i	0.495518i	−0.968822	+	0.247759i	$0.920306\pi$
$968$	0	0
$969$	−79.4303	−0.0819714
$970$	0	0
$971$	345.017i	0.355322i	0.984092	+	0.177661i	$0.0568530\pi$
$971$	345.017i	0.355322i	−0.984092	+	0.177661i	$0.943147\pi$
$972$	0	0
$973$	−1862.91	−1.91461
$974$	0	0
$975$	− 1084.67i	− 1.11248i
$976$	0	0
$977$	1603.19	1.64093	0.820466	−	0.571695i	$-0.193714\pi$
$977$	1603.19	1.64093	0.820466	+	0.571695i	$0.193714\pi$
$978$	0	0
$979$	− 2680.43i	− 2.73793i
$980$	0	0
$981$	−616.896	−0.628844
$982$	0	0
$983$	− 707.538i	− 0.719774i	−0.932996	−	0.359887i	$-0.882815\pi$
$983$	− 707.538i	− 0.719774i	0.932996	−	0.359887i	$-0.117185\pi$
$984$	0	0
$985$	−123.446	−0.125326
$986$	0	0
$987$	− 573.288i	− 0.580839i
$988$	0	0
$989$	2111.30	2.13479
$990$	0	0
$991$	54.3052i	0.0547984i	0.999625	+	0.0273992i	$0.00872253\pi$
$991$	54.3052i	0.0547984i	−0.999625	+	0.0273992i	$0.991277\pi$
$992$	0	0
$993$	613.490	0.617815
$994$	0	0
$995$	366.591i	0.368433i
$996$	0	0
$997$	−17.9257	−0.0179797	−0.00898983	−	0.999960i	$-0.502862\pi$
$997$	−17.9257	−0.0179797	−0.00898983	+	0.999960i	$0.502862\pi$
$998$	0	0
$999$	2047.98i	2.05003i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.3.d.b.239.9	yes	12
3.2	odd	2		2448.3.k.f.2143.5		12
4.3	odd	2	inner	272.3.d.b.239.4	✓	12
8.3	odd	2		1088.3.d.b.511.9		12
8.5	even	2		1088.3.d.b.511.4		12
12.11	even	2		2448.3.k.f.2143.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
272.3.d.b.239.4	✓	12	4.3	odd	2	inner
272.3.d.b.239.9	yes	12	1.1	even	1	trivial
1088.3.d.b.511.4		12	8.5	even	2
1088.3.d.b.511.9		12	8.3	odd	2
2448.3.k.f.2143.5		12	3.2	odd	2
2448.3.k.f.2143.6		12	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	272.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.14999i q^{3} -1.41418 q^{5} -7.84997i q^{7} +4.37755 q^{9} +O(q^{10})\)	\(q+2.14999i q^{3} -1.41418 q^{5} -7.84997i q^{7} +4.37755 q^{9} +20.2823i q^{11} +21.9347 q^{13} -3.04047i q^{15} +4.12311 q^{17} +8.96036i q^{19} +16.8773 q^{21} +33.7008i q^{23} -23.0001 q^{25} +28.7616i q^{27} -36.1283 q^{29} -0.236222i q^{31} -43.6067 q^{33} +11.1013i q^{35} +71.2055 q^{37} +47.1593i q^{39} +67.1309 q^{41} -62.6485i q^{43} -6.19064 q^{45} -33.9679i q^{47} -12.6220 q^{49} +8.86463i q^{51} +20.7435 q^{53} -28.6828i q^{55} -19.2647 q^{57} +6.21939i q^{59} -8.71117 q^{61} -34.3636i q^{63} -31.0196 q^{65} +75.5392i q^{67} -72.4564 q^{69} -30.8397i q^{71} -51.4240 q^{73} -49.4500i q^{75} +159.215 q^{77} -10.3304i q^{79} -22.4392 q^{81} -105.018i q^{83} -5.83082 q^{85} -77.6755i q^{87} -132.156 q^{89} -172.186i q^{91} +0.507875 q^{93} -12.6716i q^{95} -53.4567 q^{97} +88.7866i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{5} - 28 q^{9}+O(q^{10}) \) 12 * q - 12 * q^5 - 28 * q^9	\( 12 q - 12 q^{5} - 28 q^{9} + 32 q^{13} - 16 q^{21} + 28 q^{25} - 92 q^{29} + 168 q^{33} - 116 q^{37} + 32 q^{41} + 212 q^{45} - 260 q^{49} - 80 q^{57} + 76 q^{61} - 40 q^{65} - 80 q^{69} + 184 q^{73} + 160 q^{77} - 252 q^{81} + 68 q^{85} + 72 q^{89} - 160 q^{93} + 144 q^{97}+O(q^{100}) \) 12 * q - 12 * q^5 - 28 * q^9 + 32 * q^13 - 16 * q^21 + 28 * q^25 - 92 * q^29 + 168 * q^33 - 116 * q^37 + 32 * q^41 + 212 * q^45 - 260 * q^49 - 80 * q^57 + 76 * q^61 - 40 * q^65 - 80 * q^69 + 184 * q^73 + 160 * q^77 - 252 * q^81 + 68 * q^85 + 72 * q^89 - 160 * q^93 + 144 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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