LMFDB - Embedded newform 2432.2.c.i.1217.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2432,2,Mod(1217,2432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2432, 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("2432.1217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2432 = 2^{7} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2432.c (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:	$19.4196177716$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 8 x^{14} - 8 x^{13} - 3 x^{12} + 20 x^{11} - 24 x^{10} + 28 x^{8} - 96 x^{6} + \cdots + 256 $ x^16 - 4x^15 + 8x^14 - 8x^13 - 3x^12 + 20x^11 - 24x^10 + 28x^8 - 96x^6 + 160x^5 - 48x^4 - 256x^3 + 512x^2 - 512*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.3
Root			$1.41341 - 0.0477852i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.i.1217.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.06644i q^{3} -1.45635i q^{5} +0.196723 q^{7} -1.27016 q^{9} +O(q^{10})$	$q-2.06644i q^{3} -1.45635i q^{5} +0.196723 q^{7} -1.27016 q^{9} -3.27016i q^{11} -6.72211i q^{13} -3.00945 q^{15} -3.82843 q^{17} -1.00000i q^{19} -0.406516i q^{21} +6.72211 q^{23} +2.87905 q^{25} -3.57461i q^{27} +4.26907i q^{29} +5.92215 q^{31} -6.75758 q^{33} -0.286497i q^{35} +10.1782i q^{37} -13.8908 q^{39} -7.16502 q^{41} -10.4194i q^{43} +1.84979i q^{45} -3.61269 q^{47} -6.96130 q^{49} +7.91120i q^{51} -8.52508i q^{53} -4.76249 q^{55} -2.06644 q^{57} -0.137287i q^{59} +10.3879i q^{61} -0.249870 q^{63} -9.78973 q^{65} +2.22167i q^{67} -13.8908i q^{69} +8.11856 q^{71} +2.01634 q^{73} -5.94938i q^{75} -0.643316i q^{77} +13.6473 q^{79} -11.1972 q^{81} -1.14921i q^{83} +5.57552i q^{85} +8.82177 q^{87} +6.11706 q^{89} -1.32239i q^{91} -12.2377i q^{93} -1.45635 q^{95} -0.132873 q^{97} +4.15362i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^9	$ 16 q - 8 q^{9} - 16 q^{17} - 40 q^{25} + 48 q^{33} - 16 q^{41} + 16 q^{49} + 8 q^{57} + 16 q^{65} + 16 q^{73} - 64 q^{81} + 16 q^{89} + 80 q^{97}+O(q^{100}) $ 16 * q - 8 * q^9 - 16 * q^17 - 40 * q^25 + 48 * q^33 - 16 * q^41 + 16 * q^49 + 8 * q^57 + 16 * q^65 + 16 * q^73 - 64 * q^81 + 16 * q^89 + 80 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times$.

$n$	$1407$	$1921$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 2.06644i	− 1.19306i	−0.802592	−	0.596529i	$-0.796546\pi$
$3$	− 2.06644i	− 1.19306i	0.802592	−	0.596529i	$-0.203454\pi$
$4$	0	0
$5$	− 1.45635i	− 0.651299i	−0.945491	−	0.325649i	$-0.894417\pi$
$5$	− 1.45635i	− 0.651299i	0.945491	−	0.325649i	$-0.105583\pi$
$6$	0	0
$7$	0.196723	0.0743544	0.0371772	−	0.999309i	$-0.488163\pi$
$7$	0.196723	0.0743544	0.0371772	+	0.999309i	$0.488163\pi$
$8$	0	0
$9$	−1.27016	−0.423387
$10$	0	0
$11$	− 3.27016i	− 0.985990i	−0.870032	−	0.492995i	$-0.835902\pi$
$11$	− 3.27016i	− 0.985990i	0.870032	−	0.492995i	$-0.164098\pi$
$12$	0	0
$13$	− 6.72211i	− 1.86438i	−0.361973	−	0.932189i	$-0.617897\pi$
$13$	− 6.72211i	− 1.86438i	0.361973	−	0.932189i	$-0.382103\pi$
$14$	0	0
$15$	−3.00945	−0.777037
$16$	0	0
$17$	−3.82843	−0.928530	−0.464265	−	0.885696i	$-0.653681\pi$
$17$	−3.82843	−0.928530	−0.464265	+	0.885696i	$0.653681\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	− 0.406516i	− 0.0887090i
$22$	0	0
$23$	6.72211	1.40166	0.700828	−	0.713330i	$-0.252814\pi$
$23$	6.72211	1.40166	0.700828	+	0.713330i	$0.252814\pi$
$24$	0	0
$25$	2.87905	0.575810
$26$	0	0
$27$	− 3.57461i	− 0.687933i
$28$	0	0
$29$	4.26907i	0.792747i	0.918089	+	0.396374i	$0.129732\pi$
$29$	4.26907i	0.792747i	−0.918089	+	0.396374i	$0.870268\pi$
$30$	0	0
$31$	5.92215	1.06365	0.531824	−	0.846855i	$-0.321507\pi$
$31$	5.92215	1.06365	0.531824	+	0.846855i	$0.321507\pi$
$32$	0	0
$33$	−6.75758	−1.17634
$34$	0	0
$35$	− 0.286497i	− 0.0484269i
$36$	0	0
$37$	10.1782i	1.67328i	0.547755	+	0.836639i	$0.315483\pi$
$37$	10.1782i	1.67328i	−0.547755	+	0.836639i	$0.684517\pi$
$38$	0	0
$39$	−13.8908	−2.22431
$40$	0	0
$41$	−7.16502	−1.11899	−0.559494	−	0.828834i	$-0.689005\pi$
$41$	−7.16502	−1.11899	−0.559494	+	0.828834i	$0.689005\pi$
$42$	0	0
$43$	− 10.4194i	− 1.58894i	−0.607304	−	0.794470i	$-0.707749\pi$
$43$	− 10.4194i	− 1.58894i	0.607304	−	0.794470i	$-0.292251\pi$
$44$	0	0
$45$	1.84979i	0.275751i
$46$	0	0
$47$	−3.61269	−0.526965	−0.263482	−	0.964664i	$-0.584871\pi$
$47$	−3.61269	−0.526965	−0.263482	+	0.964664i	$0.584871\pi$
$48$	0	0
$49$	−6.96130	−0.994471
$50$	0	0
$51$	7.91120i	1.10779i
$52$	0	0
$53$	− 8.52508i	− 1.17101i	−0.810669	−	0.585505i	$-0.800896\pi$
$53$	− 8.52508i	− 1.17101i	0.810669	−	0.585505i	$-0.199104\pi$
$54$	0	0
$55$	−4.76249	−0.642174
$56$	0	0
$57$	−2.06644	−0.273706
$58$	0	0
$59$	− 0.137287i	− 0.0178732i	−0.999960	−	0.00893660i	$-0.997155\pi$
$59$	− 0.137287i	− 0.0178732i	0.999960	−	0.00893660i	$-0.00284465\pi$
$60$	0	0
$61$	10.3879i	1.33004i	0.746826	+	0.665020i	$0.231577\pi$
$61$	10.3879i	1.33004i	−0.746826	+	0.665020i	$0.768423\pi$
$62$	0	0
$63$	−0.249870	−0.0314806
$64$	0	0
$65$	−9.78973	−1.21427
$66$	0	0
$67$	2.22167i	0.271420i	0.990749	+	0.135710i	$0.0433316\pi$
$67$	2.22167i	0.271420i	−0.990749	+	0.135710i	$0.956668\pi$
$68$	0	0
$69$	− 13.8908i	− 1.67226i
$70$	0	0
$71$	8.11856	0.963496	0.481748	−	0.876310i	$-0.340002\pi$
$71$	8.11856	0.963496	0.481748	+	0.876310i	$0.340002\pi$
$72$	0	0
$73$	2.01634	0.235994	0.117997	−	0.993014i	$-0.462353\pi$
$73$	2.01634	0.235994	0.117997	+	0.993014i	$0.462353\pi$
$74$	0	0
$75$	− 5.94938i	− 0.686975i
$76$	0	0
$77$	− 0.643316i	− 0.0733127i
$78$	0	0
$79$	13.6473	1.53544	0.767719	−	0.640787i	$-0.221392\pi$
$79$	13.6473	1.53544	0.767719	+	0.640787i	$0.221392\pi$
$80$	0	0
$81$	−11.1972	−1.24413
$82$	0	0
$83$	− 1.14921i	− 0.126142i	−0.998009	−	0.0630711i	$-0.979911\pi$
$83$	− 1.14921i	− 0.126142i	0.998009	−	0.0630711i	$-0.0200895\pi$
$84$	0	0
$85$	5.57552i	0.604750i
$86$	0	0
$87$	8.82177	0.945793
$88$	0	0
$89$	6.11706	0.648407	0.324204	−	0.945987i	$-0.394904\pi$
$89$	6.11706	0.648407	0.324204	+	0.945987i	$0.394904\pi$
$90$	0	0
$91$	− 1.32239i	− 0.138625i
$92$	0	0
$93$	− 12.2377i	− 1.26899i
$94$	0	0
$95$	−1.45635	−0.149418
$96$	0	0
$97$	−0.132873	−0.0134912	−0.00674560	−	0.999977i	$-0.502147\pi$
$97$	−0.132873	−0.0134912	−0.00674560	+	0.999977i	$0.502147\pi$
$98$	0	0
$99$	4.15362i	0.417455i
$100$	0	0
$101$	7.81876i	0.777996i	0.921239	+	0.388998i	$0.127179\pi$
$101$	7.81876i	0.777996i	−0.921239	+	0.388998i	$0.872821\pi$
$102$	0	0
$103$	−19.6631	−1.93746	−0.968729	−	0.248121i	$-0.920187\pi$
$103$	−19.6631	−1.93746	−0.968729	+	0.248121i	$0.920187\pi$
$104$	0	0
$105$	−0.592028	−0.0577761
$106$	0	0
$107$	− 3.18952i	− 0.308343i	−0.988044	−	0.154171i	$-0.950729\pi$
$107$	− 3.18952i	− 0.308343i	0.988044	−	0.154171i	$-0.0492708\pi$
$108$	0	0
$109$	5.44941i	0.521959i	0.965344	+	0.260980i	$0.0840455\pi$
$109$	5.44941i	0.521959i	−0.965344	+	0.260980i	$0.915954\pi$
$110$	0	0
$111$	21.0325	1.99632
$112$	0	0
$113$	−12.6890	−1.19368	−0.596841	−	0.802360i	$-0.703578\pi$
$113$	−12.6890	−1.19368	−0.596841	+	0.802360i	$0.703578\pi$
$114$	0	0
$115$	− 9.78973i	− 0.912897i
$116$	0	0
$117$	8.53815i	0.789352i
$118$	0	0
$119$	−0.753140	−0.0690403
$120$	0	0
$121$	0.306056	0.0278233
$122$	0	0
$123$	14.8061i	1.33502i
$124$	0	0
$125$	− 11.4746i	− 1.02632i
$126$	0	0
$127$	−8.83484	−0.783965	−0.391983	−	0.919973i	$-0.628211\pi$
$127$	−8.83484	−0.783965	−0.391983	+	0.919973i	$0.628211\pi$
$128$	0	0
$129$	−21.5310	−1.89570
$130$	0	0
$131$	21.1285i	1.84600i	0.384798	+	0.923001i	$0.374271\pi$
$131$	21.1285i	1.84600i	−0.384798	+	0.923001i	$0.625729\pi$
$132$	0	0
$133$	− 0.196723i	− 0.0170581i
$134$	0	0
$135$	−5.20587	−0.448050
$136$	0	0
$137$	−12.3464	−1.05482	−0.527411	−	0.849610i	$-0.676837\pi$
$137$	−12.3464	−1.05482	−0.527411	+	0.849610i	$0.676837\pi$
$138$	0	0
$139$	− 9.84638i	− 0.835159i	−0.908640	−	0.417579i	$-0.862879\pi$
$139$	− 9.84638i	− 0.835159i	0.908640	−	0.417579i	$-0.137121\pi$
$140$	0	0
$141$	7.46539i	0.628699i
$142$	0	0
$143$	−21.9824	−1.83826
$144$	0	0
$145$	6.21726	0.516315
$146$	0	0
$147$	14.3851i	1.18646i
$148$	0	0
$149$	1.06290i	0.0870763i	0.999052	+	0.0435381i	$0.0138630\pi$
$149$	1.06290i	0.0870763i	−0.999052	+	0.0435381i	$0.986137\pi$
$150$	0	0
$151$	2.91270	0.237032	0.118516	−	0.992952i	$-0.462186\pi$
$151$	2.91270	0.237032	0.118516	+	0.992952i	$0.462186\pi$
$152$	0	0
$153$	4.86271	0.393127
$154$	0	0
$155$	− 8.62470i	− 0.692753i
$156$	0	0
$157$	− 8.63179i	− 0.688892i	−0.938806	−	0.344446i	$-0.888067\pi$
$157$	− 8.63179i	− 0.688892i	0.938806	−	0.344446i	$-0.111933\pi$
$158$	0	0
$159$	−17.6165	−1.39708
$160$	0	0
$161$	1.32239	0.104219
$162$	0	0
$163$	− 1.45541i	− 0.113996i	−0.998374	−	0.0569982i	$-0.981847\pi$
$163$	− 1.45541i	− 0.113996i	0.998374	−	0.0569982i	$-0.0181529\pi$
$164$	0	0
$165$	9.84138i	0.766151i
$166$	0	0
$167$	14.5539	1.12622	0.563109	−	0.826383i	$-0.309605\pi$
$167$	14.5539	1.12622	0.563109	+	0.826383i	$0.309605\pi$
$168$	0	0
$169$	−32.1867	−2.47590
$170$	0	0
$171$	1.27016i	0.0971315i
$172$	0	0
$173$	17.7002i	1.34572i	0.739769	+	0.672861i	$0.234935\pi$
$173$	17.7002i	1.34572i	−0.739769	+	0.672861i	$0.765065\pi$
$174$	0	0
$175$	0.566376	0.0428140
$176$	0	0
$177$	−0.283694	−0.0213238
$178$	0	0
$179$	11.5725i	0.864967i	0.901642	+	0.432483i	$0.142363\pi$
$179$	11.5725i	0.864967i	−0.901642	+	0.432483i	$0.857637\pi$
$180$	0	0
$181$	− 3.95620i	− 0.294062i	−0.989132	−	0.147031i	$-0.953028\pi$
$181$	− 3.95620i	− 0.294062i	0.989132	−	0.147031i	$-0.0469717\pi$
$182$	0	0
$183$	21.4660	1.58681
$184$	0	0
$185$	14.8229	1.08980
$186$	0	0
$187$	12.5196i	0.915521i
$188$	0	0
$189$	− 0.703208i	− 0.0511508i
$190$	0	0
$191$	−4.02874	−0.291510	−0.145755	−	0.989321i	$-0.546561\pi$
$191$	−4.02874	−0.291510	−0.145755	+	0.989321i	$0.546561\pi$
$192$	0	0
$193$	13.2979	0.957203	0.478602	−	0.878032i	$-0.341144\pi$
$193$	13.2979	0.957203	0.478602	+	0.878032i	$0.341144\pi$
$194$	0	0
$195$	20.2298i	1.44869i
$196$	0	0
$197$	6.61228i	0.471106i	0.971862	+	0.235553i	$0.0756901\pi$
$197$	6.61228i	0.471106i	−0.971862	+	0.235553i	$0.924310\pi$
$198$	0	0
$199$	22.3159	1.58193	0.790967	−	0.611859i	$-0.209578\pi$
$199$	22.3159	1.58193	0.790967	+	0.611859i	$0.209578\pi$
$200$	0	0
$201$	4.59094	0.323820
$202$	0	0
$203$	0.839826i	0.0589442i
$204$	0	0
$205$	10.4348i	0.728796i
$206$	0	0
$207$	−8.53815	−0.593442
$208$	0	0
$209$	−3.27016	−0.226202
$210$	0	0
$211$	15.2315i	1.04858i	0.851541	+	0.524288i	$0.175669\pi$
$211$	15.2315i	1.04858i	−0.851541	+	0.524288i	$0.824331\pi$
$212$	0	0
$213$	− 16.7765i	− 1.14951i
$214$	0	0
$215$	−15.1742	−1.03487
$216$	0	0
$217$	1.16502	0.0790869
$218$	0	0
$219$	− 4.16663i	− 0.281555i
$220$	0	0
$221$	25.7351i	1.73113i
$222$	0	0
$223$	−1.00343	−0.0671947	−0.0335973	−	0.999435i	$-0.510696\pi$
$223$	−1.00343	−0.0671947	−0.0335973	+	0.999435i	$0.510696\pi$
$224$	0	0
$225$	−3.65685	−0.243790
$226$	0	0
$227$	8.20586i	0.544642i	0.962206	+	0.272321i	$0.0877913\pi$
$227$	8.20586i	0.544642i	−0.962206	+	0.272321i	$0.912209\pi$
$228$	0	0
$229$	− 25.4321i	− 1.68060i	−0.542122	−	0.840300i	$-0.682379\pi$
$229$	− 25.4321i	− 1.68060i	0.542122	−	0.840300i	$-0.317621\pi$
$230$	0	0
$231$	−1.32937	−0.0874662
$232$	0	0
$233$	−17.8702	−1.17072	−0.585359	−	0.810774i	$-0.699046\pi$
$233$	−17.8702	−1.17072	−0.585359	+	0.810774i	$0.699046\pi$
$234$	0	0
$235$	5.26133i	0.343211i
$236$	0	0
$237$	− 28.2012i	− 1.83186i
$238$	0	0
$239$	5.17212	0.334556	0.167278	−	0.985910i	$-0.446502\pi$
$239$	5.17212	0.334556	0.167278	+	0.985910i	$0.446502\pi$
$240$	0	0
$241$	24.9189	1.60516	0.802582	−	0.596542i	$-0.203459\pi$
$241$	24.9189	1.60516	0.802582	+	0.596542i	$0.203459\pi$
$242$	0	0
$243$	12.4144i	0.796386i
$244$	0	0
$245$	10.1381i	0.647698i
$246$	0	0
$247$	−6.72211	−0.427717
$248$	0	0
$249$	−2.37477	−0.150495
$250$	0	0
$251$	− 13.4673i	− 0.850051i	−0.905181	−	0.425025i	$-0.860265\pi$
$251$	− 13.4673i	− 0.850051i	0.905181	−	0.425025i	$-0.139735\pi$
$252$	0	0
$253$	− 21.9824i	− 1.38202i
$254$	0	0
$255$	11.5215	0.721502
$256$	0	0
$257$	7.55666	0.471371	0.235686	−	0.971829i	$-0.424266\pi$
$257$	7.55666	0.471371	0.235686	+	0.971829i	$0.424266\pi$
$258$	0	0
$259$	2.00228i	0.124415i
$260$	0	0
$261$	− 5.42241i	− 0.335639i
$262$	0	0
$263$	14.3441	0.884498	0.442249	−	0.896892i	$-0.354181\pi$
$263$	14.3441	0.884498	0.442249	+	0.896892i	$0.354181\pi$
$264$	0	0
$265$	−12.4155	−0.762677
$266$	0	0
$267$	− 12.6405i	− 0.773587i
$268$	0	0
$269$	− 22.1161i	− 1.34844i	−0.738530	−	0.674221i	$-0.764480\pi$
$269$	− 22.1161i	− 1.34844i	0.738530	−	0.674221i	$-0.235520\pi$
$270$	0	0
$271$	−5.51563	−0.335051	−0.167525	−	0.985868i	$-0.553578\pi$
$271$	−5.51563	−0.335051	−0.167525	+	0.985868i	$0.553578\pi$
$272$	0	0
$273$	−2.73264	−0.165387
$274$	0	0
$275$	− 9.41496i	− 0.567743i
$276$	0	0
$277$	− 14.4135i	− 0.866022i	−0.901389	−	0.433011i	$-0.857451\pi$
$277$	− 14.4135i	− 0.866022i	0.901389	−	0.433011i	$-0.142549\pi$
$278$	0	0
$279$	−7.52207	−0.450335
$280$	0	0
$281$	−18.4956	−1.10335	−0.551677	−	0.834058i	$-0.686012\pi$
$281$	−18.4956	−1.10335	−0.551677	+	0.834058i	$0.686012\pi$
$282$	0	0
$283$	4.83004i	0.287116i	0.989642	+	0.143558i	$0.0458544\pi$
$283$	4.83004i	0.287116i	−0.989642	+	0.143558i	$0.954146\pi$
$284$	0	0
$285$	3.00945i	0.178264i
$286$	0	0
$287$	−1.40953	−0.0832017
$288$	0	0
$289$	−2.34315	−0.137832
$290$	0	0
$291$	0.274573i	0.0160958i
$292$	0	0
$293$	1.94022i	0.113349i	0.998393	+	0.0566745i	$0.0180497\pi$
$293$	1.94022i	0.113349i	−0.998393	+	0.0566745i	$0.981950\pi$
$294$	0	0
$295$	−0.199937	−0.0116408
$296$	0	0
$297$	−11.6895	−0.678295
$298$	0	0
$299$	− 45.1867i	− 2.61322i
$300$	0	0
$301$	− 2.04973i	− 0.118145i
$302$	0	0
$303$	16.1570	0.928194
$304$	0	0
$305$	15.1285	0.866253
$306$	0	0
$307$	− 13.7053i	− 0.782205i	−0.920347	−	0.391103i	$-0.872094\pi$
$307$	− 13.7053i	− 0.782205i	0.920347	−	0.391103i	$-0.127906\pi$
$308$	0	0
$309$	40.6325i	2.31150i
$310$	0	0
$311$	−29.8349	−1.69178	−0.845890	−	0.533357i	$-0.820930\pi$
$311$	−29.8349	−1.69178	−0.845890	+	0.533357i	$0.820930\pi$
$312$	0	0
$313$	20.2244	1.14315	0.571575	−	0.820550i	$-0.306333\pi$
$313$	20.2244	1.14315	0.571575	+	0.820550i	$0.306333\pi$
$314$	0	0
$315$	0.363897i	0.0205033i
$316$	0	0
$317$	13.6037i	0.764057i	0.924150	+	0.382029i	$0.124774\pi$
$317$	13.6037i	0.764057i	−0.924150	+	0.382029i	$0.875226\pi$
$318$	0	0
$319$	13.9606	0.781641
$320$	0	0
$321$	−6.59094	−0.367871
$322$	0	0
$323$	3.82843i	0.213019i
$324$	0	0
$325$	− 19.3533i	− 1.07353i
$326$	0	0
$327$	11.2609	0.622727
$328$	0	0
$329$	−0.710700	−0.0391821
$330$	0	0
$331$	− 6.66501i	− 0.366342i	−0.983081	−	0.183171i	$-0.941364\pi$
$331$	− 6.66501i	− 0.366342i	0.983081	−	0.183171i	$-0.0586363\pi$
$332$	0	0
$333$	− 12.9279i	− 0.708443i
$334$	0	0
$335$	3.23553	0.176776
$336$	0	0
$337$	33.8420	1.84349	0.921745	−	0.387796i	$-0.126764\pi$
$337$	33.8420	1.84349	0.921745	+	0.387796i	$0.126764\pi$
$338$	0	0
$339$	26.2210i	1.42413i
$340$	0	0
$341$	− 19.3664i	− 1.04875i
$342$	0	0
$343$	−2.74651	−0.148298
$344$	0	0
$345$	−20.2298	−1.08914
$346$	0	0
$347$	− 4.69716i	− 0.252157i	−0.992020	−	0.126079i	$-0.959761\pi$
$347$	− 4.69716i	− 0.252157i	0.992020	−	0.126079i	$-0.0402391\pi$
$348$	0	0
$349$	− 4.27540i	− 0.228857i	−0.993431	−	0.114428i	$-0.963496\pi$
$349$	− 4.27540i	− 0.228857i	0.993431	−	0.114428i	$-0.0365037\pi$
$350$	0	0
$351$	−24.0289	−1.28257
$352$	0	0
$353$	4.82401	0.256756	0.128378	−	0.991725i	$-0.459023\pi$
$353$	4.82401	0.256756	0.128378	+	0.991725i	$0.459023\pi$
$354$	0	0
$355$	− 11.8235i	− 0.627524i
$356$	0	0
$357$	1.55632i	0.0823690i
$358$	0	0
$359$	−5.52237	−0.291460	−0.145730	−	0.989324i	$-0.546553\pi$
$359$	−5.52237	−0.291460	−0.145730	+	0.989324i	$0.546553\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 0.632446i	− 0.0331948i
$364$	0	0
$365$	− 2.93649i	− 0.153703i
$366$	0	0
$367$	10.1750	0.531133	0.265566	−	0.964093i	$-0.414441\pi$
$367$	10.1750	0.531133	0.265566	+	0.964093i	$0.414441\pi$
$368$	0	0
$369$	9.10072	0.473765
$370$	0	0
$371$	− 1.67708i	− 0.0870697i
$372$	0	0
$373$	0.640102i	0.0331432i	0.999863	+	0.0165716i	$0.00527515\pi$
$373$	0.640102i	0.0331432i	−0.999863	+	0.0165716i	$0.994725\pi$
$374$	0	0
$375$	−23.7116	−1.22446
$376$	0	0
$377$	28.6972	1.47798
$378$	0	0
$379$	− 32.9406i	− 1.69204i	−0.533149	−	0.846022i	$-0.678991\pi$
$379$	− 32.9406i	− 1.69204i	0.533149	−	0.846022i	$-0.321009\pi$
$380$	0	0
$381$	18.2566i	0.935316i
$382$	0	0
$383$	−8.11545	−0.414680	−0.207340	−	0.978269i	$-0.566481\pi$
$383$	−8.11545	−0.414680	−0.207340	+	0.978269i	$0.566481\pi$
$384$	0	0
$385$	−0.936892	−0.0477484
$386$	0	0
$387$	13.2343i	0.672735i
$388$	0	0
$389$	32.5702i	1.65138i	0.564126	+	0.825689i	$0.309213\pi$
$389$	32.5702i	1.65138i	−0.564126	+	0.825689i	$0.690787\pi$
$390$	0	0
$391$	−25.7351	−1.30148
$392$	0	0
$393$	43.6606	2.20239
$394$	0	0
$395$	− 19.8752i	− 1.00003i
$396$	0	0
$397$	27.6380i	1.38711i	0.720402	+	0.693557i	$0.243957\pi$
$397$	27.6380i	1.38711i	−0.720402	+	0.693557i	$0.756043\pi$
$398$	0	0
$399$	−0.406516	−0.0203512
$400$	0	0
$401$	37.4219	1.86876	0.934381	−	0.356275i	$-0.115953\pi$
$401$	37.4219	1.86876	0.934381	+	0.356275i	$0.115953\pi$
$402$	0	0
$403$	− 39.8093i	− 1.98304i
$404$	0	0
$405$	16.3070i	0.810300i
$406$	0	0
$407$	33.2842	1.64984
$408$	0	0
$409$	26.5114	1.31090	0.655452	−	0.755237i	$-0.272478\pi$
$409$	26.5114	1.31090	0.655452	+	0.755237i	$0.272478\pi$
$410$	0	0
$411$	25.5130i	1.25846i
$412$	0	0
$413$	− 0.0270075i	− 0.00132895i
$414$	0	0
$415$	−1.67365	−0.0821563
$416$	0	0
$417$	−20.3469	−0.996392
$418$	0	0
$419$	− 29.3006i	− 1.43143i	−0.698393	−	0.715714i	$-0.746101\pi$
$419$	− 29.3006i	− 1.43143i	0.698393	−	0.715714i	$-0.253899\pi$
$420$	0	0
$421$	4.10610i	0.200119i	0.994981	+	0.100060i	$0.0319034\pi$
$421$	4.10610i	0.200119i	−0.994981	+	0.100060i	$0.968097\pi$
$422$	0	0
$423$	4.58869	0.223110
$424$	0	0
$425$	−11.0222	−0.534657
$426$	0	0
$427$	2.04355i	0.0988943i
$428$	0	0
$429$	45.4252i	2.19315i
$430$	0	0
$431$	17.0502	0.821277	0.410639	−	0.911798i	$-0.365306\pi$
$431$	17.0502	0.821277	0.410639	+	0.911798i	$0.365306\pi$
$432$	0	0
$433$	16.1356	0.775427	0.387713	−	0.921780i	$-0.373265\pi$
$433$	16.1356	0.775427	0.387713	+	0.921780i	$0.373265\pi$
$434$	0	0
$435$	− 12.8476i	− 0.615994i
$436$	0	0
$437$	− 6.72211i	− 0.321562i
$438$	0	0
$439$	37.3232	1.78134	0.890670	−	0.454651i	$-0.150236\pi$
$439$	37.3232	1.78134	0.890670	+	0.454651i	$0.150236\pi$
$440$	0	0
$441$	8.84196	0.421046
$442$	0	0
$443$	− 11.2254i	− 0.533337i	−0.963788	−	0.266668i	$-0.914077\pi$
$443$	− 11.2254i	− 0.533337i	0.963788	−	0.266668i	$-0.0859228\pi$
$444$	0	0
$445$	− 8.90857i	− 0.422307i
$446$	0	0
$447$	2.19642	0.103887
$448$	0	0
$449$	−15.3791	−0.725783	−0.362891	−	0.931831i	$-0.618210\pi$
$449$	−15.3791	−0.725783	−0.362891	+	0.931831i	$0.618210\pi$
$450$	0	0
$451$	23.4308i	1.10331i
$452$	0	0
$453$	− 6.01890i	− 0.282793i
$454$	0	0
$455$	−1.92587	−0.0902860
$456$	0	0
$457$	24.8093	1.16053	0.580265	−	0.814428i	$-0.302949\pi$
$457$	24.8093	1.16053	0.580265	+	0.814428i	$0.302949\pi$
$458$	0	0
$459$	13.6851i	0.638767i
$460$	0	0
$461$	− 5.74289i	− 0.267473i	−0.991017	−	0.133737i	$-0.957302\pi$
$461$	− 5.74289i	− 0.267473i	0.991017	−	0.133737i	$-0.0426976\pi$
$462$	0	0
$463$	23.5494	1.09443	0.547217	−	0.836991i	$-0.315687\pi$
$463$	23.5494	1.09443	0.547217	+	0.836991i	$0.315687\pi$
$464$	0	0
$465$	−17.8224	−0.826494
$466$	0	0
$467$	− 26.2375i	− 1.21413i	−0.794653	−	0.607063i	$-0.792347\pi$
$467$	− 26.2375i	− 1.21413i	0.794653	−	0.607063i	$-0.207653\pi$
$468$	0	0
$469$	0.437054i	0.0201813i
$470$	0	0
$471$	−17.8371	−0.821888
$472$	0	0
$473$	−34.0730	−1.56668
$474$	0	0
$475$	− 2.87905i	− 0.132100i
$476$	0	0
$477$	10.8282i	0.495790i
$478$	0	0
$479$	19.5262	0.892176	0.446088	−	0.894989i	$-0.352817\pi$
$479$	19.5262	0.892176	0.446088	+	0.894989i	$0.352817\pi$
$480$	0	0
$481$	68.4186	3.11962
$482$	0	0
$483$	− 2.73264i	− 0.124340i
$484$	0	0
$485$	0.193509i	0.00878680i
$486$	0	0
$487$	−27.5206	−1.24708	−0.623539	−	0.781792i	$-0.714306\pi$
$487$	−27.5206	−1.24708	−0.623539	+	0.781792i	$0.714306\pi$
$488$	0	0
$489$	−3.00751	−0.136004
$490$	0	0
$491$	0.870365i	0.0392790i	0.999807	+	0.0196395i	$0.00625186\pi$
$491$	0.870365i	0.0392790i	−0.999807	+	0.0196395i	$0.993748\pi$
$492$	0	0
$493$	− 16.3438i	− 0.736090i
$494$	0	0
$495$	6.04912	0.271888
$496$	0	0
$497$	1.59711	0.0716402
$498$	0	0
$499$	1.37036i	0.0613456i	0.999529	+	0.0306728i	$0.00976499\pi$
$499$	1.37036i	0.0613456i	−0.999529	+	0.0306728i	$0.990235\pi$
$500$	0	0
$501$	− 30.0748i	− 1.34364i
$502$	0	0
$503$	−22.2354	−0.991430	−0.495715	−	0.868485i	$-0.665094\pi$
$503$	−22.2354	−0.991430	−0.495715	+	0.868485i	$0.665094\pi$
$504$	0	0
$505$	11.3868	0.506708
$506$	0	0
$507$	66.5118i	2.95389i
$508$	0	0
$509$	− 19.0391i	− 0.843895i	−0.906620	−	0.421947i	$-0.861347\pi$
$509$	− 19.0391i	− 0.843895i	0.906620	−	0.421947i	$-0.138653\pi$
$510$	0	0
$511$	0.396660	0.0175472
$512$	0	0
$513$	−3.57461	−0.157823
$514$	0	0
$515$	28.6362i	1.26186i
$516$	0	0
$517$	11.8141i	0.519582i
$518$	0	0
$519$	36.5764	1.60552
$520$	0	0
$521$	−17.8425	−0.781694	−0.390847	−	0.920456i	$-0.627818\pi$
$521$	−17.8425	−0.781694	−0.390847	+	0.920456i	$0.627818\pi$
$522$	0	0
$523$	− 39.9928i	− 1.74876i	−0.485239	−	0.874382i	$-0.661267\pi$
$523$	− 39.9928i	− 1.74876i	0.485239	−	0.874382i	$-0.338733\pi$
$524$	0	0
$525$	− 1.17038i	− 0.0510796i
$526$	0	0
$527$	−22.6725	−0.987630
$528$	0	0
$529$	22.1867	0.964641
$530$	0	0
$531$	0.174376i	0.00756727i
$532$	0	0
$533$	48.1641i	2.08622i
$534$	0	0
$535$	−4.64505	−0.200823
$536$	0	0
$537$	23.9138	1.03196
$538$	0	0
$539$	22.7646i	0.980539i
$540$	0	0
$541$	− 25.3449i	− 1.08966i	−0.838546	−	0.544830i	$-0.816594\pi$
$541$	− 25.3449i	− 1.08966i	0.838546	−	0.544830i	$-0.183406\pi$
$542$	0	0
$543$	−8.17524	−0.350833
$544$	0	0
$545$	7.93624	0.339951
$546$	0	0
$547$	22.4541i	0.960068i	0.877250	+	0.480034i	$0.159376\pi$
$547$	22.4541i	0.960068i	−0.877250	+	0.480034i	$0.840624\pi$
$548$	0	0
$549$	− 13.1943i	− 0.563121i
$550$	0	0
$551$	4.26907	0.181869
$552$	0	0
$553$	2.68473	0.114166
$554$	0	0
$555$	− 30.6306i	− 1.30020i
$556$	0	0
$557$	− 26.8511i	− 1.13772i	−0.822434	−	0.568860i	$-0.807385\pi$
$557$	− 26.8511i	− 1.13772i	0.822434	−	0.568860i	$-0.192615\pi$
$558$	0	0
$559$	−70.0401	−2.96238
$560$	0	0
$561$	25.8709	1.09227
$562$	0	0
$563$	− 37.0560i	− 1.56172i	−0.624703	−	0.780862i	$-0.714780\pi$
$563$	− 37.0560i	− 1.56172i	0.624703	−	0.780862i	$-0.285220\pi$
$564$	0	0
$565$	18.4796i	0.777443i
$566$	0	0
$567$	−2.20274	−0.0925065
$568$	0	0
$569$	−2.20600	−0.0924804	−0.0462402	−	0.998930i	$-0.514724\pi$
$569$	−2.20600	−0.0924804	−0.0462402	+	0.998930i	$0.514724\pi$
$570$	0	0
$571$	20.4444i	0.855571i	0.903880	+	0.427786i	$0.140706\pi$
$571$	20.4444i	0.855571i	−0.903880	+	0.427786i	$0.859294\pi$
$572$	0	0
$573$	8.32514i	0.347788i
$574$	0	0
$575$	19.3533	0.807088
$576$	0	0
$577$	−41.8850	−1.74369	−0.871847	−	0.489779i	$-0.837077\pi$
$577$	−41.8850	−1.74369	−0.871847	+	0.489779i	$0.837077\pi$
$578$	0	0
$579$	− 27.4793i	− 1.14200i
$580$	0	0
$581$	− 0.226076i	− 0.00937923i
$582$	0	0
$583$	−27.8784	−1.15460
$584$	0	0
$585$	12.4345	0.514104
$586$	0	0
$587$	− 33.4301i	− 1.37981i	−0.723901	−	0.689904i	$-0.757653\pi$
$587$	− 33.4301i	− 1.37981i	0.723901	−	0.689904i	$-0.242347\pi$
$588$	0	0
$589$	− 5.92215i	− 0.244018i
$590$	0	0
$591$	13.6639	0.562056
$592$	0	0
$593$	4.83769	0.198660	0.0993301	−	0.995055i	$-0.468330\pi$
$593$	4.83769	0.198660	0.0993301	+	0.995055i	$0.468330\pi$
$594$	0	0
$595$	1.09683i	0.0449658i
$596$	0	0
$597$	− 46.1144i	− 1.88734i
$598$	0	0
$599$	4.64816	0.189919	0.0949594	−	0.995481i	$-0.469728\pi$
$599$	4.64816	0.189919	0.0949594	+	0.995481i	$0.469728\pi$
$600$	0	0
$601$	12.8661	0.524819	0.262409	−	0.964957i	$-0.415483\pi$
$601$	12.8661	0.524819	0.262409	+	0.964957i	$0.415483\pi$
$602$	0	0
$603$	− 2.82188i	− 0.114916i
$604$	0	0
$605$	− 0.445725i	− 0.0181213i
$606$	0	0
$607$	−14.4246	−0.585477	−0.292739	−	0.956193i	$-0.594567\pi$
$607$	−14.4246	−0.585477	−0.292739	+	0.956193i	$0.594567\pi$
$608$	0	0
$609$	1.73545	0.0703238
$610$	0	0
$611$	24.2849i	0.982461i
$612$	0	0
$613$	− 37.2502i	− 1.50452i	−0.658865	−	0.752261i	$-0.728963\pi$
$613$	− 37.2502i	− 1.50452i	0.658865	−	0.752261i	$-0.271037\pi$
$614$	0	0
$615$	21.5628	0.869495
$616$	0	0
$617$	−28.6841	−1.15478	−0.577388	−	0.816470i	$-0.695928\pi$
$617$	−28.6841	−1.15478	−0.577388	+	0.816470i	$0.695928\pi$
$618$	0	0
$619$	− 4.26470i	− 0.171413i	−0.996320	−	0.0857063i	$-0.972685\pi$
$619$	− 4.26470i	− 0.171413i	0.996320	−	0.0857063i	$-0.0273147\pi$
$620$	0	0
$621$	− 24.0289i	− 0.964246i
$622$	0	0
$623$	1.20337	0.0482119
$624$	0	0
$625$	−2.31581	−0.0926324
$626$	0	0
$627$	6.75758i	0.269872i
$628$	0	0
$629$	− 38.9663i	− 1.55369i
$630$	0	0
$631$	−9.34444	−0.371996	−0.185998	−	0.982550i	$-0.559552\pi$
$631$	−9.34444	−0.371996	−0.185998	+	0.982550i	$0.559552\pi$
$632$	0	0
$633$	31.4748	1.25101
$634$	0	0
$635$	12.8666i	0.510596i
$636$	0	0
$637$	46.7946i	1.85407i
$638$	0	0
$639$	−10.3119	−0.407931
$640$	0	0
$641$	33.0359	1.30484	0.652420	−	0.757858i	$-0.273754\pi$
$641$	33.0359	1.30484	0.652420	+	0.757858i	$0.273754\pi$
$642$	0	0
$643$	3.53485i	0.139401i	0.997568	+	0.0697005i	$0.0222044\pi$
$643$	3.53485i	0.139401i	−0.997568	+	0.0697005i	$0.977796\pi$
$644$	0	0
$645$	31.3566i	1.23466i
$646$	0	0
$647$	36.0925	1.41894	0.709471	−	0.704734i	$-0.248934\pi$
$647$	36.0925	1.41894	0.709471	+	0.704734i	$0.248934\pi$
$648$	0	0
$649$	−0.448949	−0.0176228
$650$	0	0
$651$	− 2.40745i	− 0.0943553i
$652$	0	0
$653$	19.8193i	0.775588i	0.921746	+	0.387794i	$0.126763\pi$
$653$	19.8193i	0.775588i	−0.921746	+	0.387794i	$0.873237\pi$
$654$	0	0
$655$	30.7704	1.20230
$656$	0	0
$657$	−2.56107	−0.0999169
$658$	0	0
$659$	40.8393i	1.59087i	0.606036	+	0.795437i	$0.292759\pi$
$659$	40.8393i	1.59087i	−0.606036	+	0.795437i	$0.707241\pi$
$660$	0	0
$661$	26.8218i	1.04325i	0.853176	+	0.521623i	$0.174673\pi$
$661$	26.8218i	1.04325i	−0.853176	+	0.521623i	$0.825327\pi$
$662$	0	0
$663$	53.1799	2.06534
$664$	0	0
$665$	−0.286497	−0.0111099
$666$	0	0
$667$	28.6972i	1.11116i
$668$	0	0
$669$	2.07352i	0.0801671i
$670$	0	0
$671$	33.9702	1.31141
$672$	0	0
$673$	28.6633	1.10489	0.552445	−	0.833549i	$-0.313695\pi$
$673$	28.6633	1.10489	0.552445	+	0.833549i	$0.313695\pi$
$674$	0	0
$675$	− 10.2915i	− 0.396119i
$676$	0	0
$677$	− 7.30155i	− 0.280621i	−0.990108	−	0.140311i	$-0.955190\pi$
$677$	− 7.30155i	− 0.280621i	0.990108	−	0.140311i	$-0.0448102\pi$
$678$	0	0
$679$	−0.0261392	−0.00100313
$680$	0	0
$681$	16.9569	0.649789
$682$	0	0
$683$	− 25.3780i	− 0.971063i	−0.874219	−	0.485531i	$-0.838626\pi$
$683$	− 25.3780i	− 0.971063i	0.874219	−	0.485531i	$-0.161374\pi$
$684$	0	0
$685$	17.9806i	0.687005i
$686$	0	0
$687$	−52.5538	−2.00505
$688$	0	0
$689$	−57.3065	−2.18320
$690$	0	0
$691$	− 32.3136i	− 1.22927i	−0.788813	−	0.614633i	$-0.789304\pi$
$691$	− 32.3136i	− 1.22927i	0.788813	−	0.614633i	$-0.210696\pi$
$692$	0	0
$693$	0.817114i	0.0310396i
$694$	0	0
$695$	−14.3397	−0.543938
$696$	0	0
$697$	27.4308	1.03901
$698$	0	0
$699$	36.9277i	1.39673i
$700$	0	0
$701$	− 12.8573i	− 0.485612i	−0.970075	−	0.242806i	$-0.921932\pi$
$701$	− 12.8573i	− 0.485612i	0.970075	−	0.242806i	$-0.0780678\pi$
$702$	0	0
$703$	10.1782	0.383876
$704$	0	0
$705$	10.8722	0.409471
$706$	0	0
$707$	1.53813i	0.0578474i
$708$	0	0
$709$	41.1583i	1.54573i	0.634568	+	0.772867i	$0.281178\pi$
$709$	41.1583i	1.54573i	−0.634568	+	0.772867i	$0.718822\pi$
$710$	0	0
$711$	−17.3342	−0.650083
$712$	0	0
$713$	39.8093	1.49087
$714$	0	0
$715$	32.0140i	1.19725i
$716$	0	0
$717$	− 10.6878i	− 0.399145i
$718$	0	0
$719$	−36.3230	−1.35462	−0.677309	−	0.735698i	$-0.736854\pi$
$719$	−36.3230	−1.35462	−0.677309	+	0.735698i	$0.736854\pi$
$720$	0	0
$721$	−3.86818	−0.144058
$722$	0	0
$723$	− 51.4932i	− 1.91505i
$724$	0	0
$725$	12.2909i	0.456472i
$726$	0	0
$727$	1.34652	0.0499398	0.0249699	−	0.999688i	$-0.492051\pi$
$727$	1.34652	0.0499398	0.0249699	+	0.999688i	$0.492051\pi$
$728$	0	0
$729$	−7.93789	−0.293996
$730$	0	0
$731$	39.8898i	1.47538i
$732$	0	0
$733$	9.03789i	0.333822i	0.985972	+	0.166911i	$0.0533793\pi$
$733$	9.03789i	0.333822i	−0.985972	+	0.166911i	$0.946621\pi$
$734$	0	0
$735$	20.9497	0.772741
$736$	0	0
$737$	7.26522	0.267618
$738$	0	0
$739$	36.7004i	1.35005i	0.737796	+	0.675023i	$0.235866\pi$
$739$	36.7004i	1.35005i	−0.737796	+	0.675023i	$0.764134\pi$
$740$	0	0
$741$	13.8908i	0.510292i
$742$	0	0
$743$	−36.7973	−1.34996	−0.674981	−	0.737835i	$-0.735848\pi$
$743$	−36.7973	−1.34996	−0.674981	+	0.737835i	$0.735848\pi$
$744$	0	0
$745$	1.54795	0.0567127
$746$	0	0
$747$	1.45968i	0.0534069i
$748$	0	0
$749$	− 0.627453i	− 0.0229266i
$750$	0	0
$751$	42.2062	1.54013	0.770064	−	0.637967i	$-0.220224\pi$
$751$	42.2062	1.54013	0.770064	+	0.637967i	$0.220224\pi$
$752$	0	0
$753$	−27.8294	−1.01416
$754$	0	0
$755$	− 4.24190i	− 0.154378i
$756$	0	0
$757$	8.89431i	0.323269i	0.986851	+	0.161635i	$0.0516766\pi$
$757$	8.89431i	0.323269i	−0.986851	+	0.161635i	$0.948323\pi$
$758$	0	0
$759$	−45.4252	−1.64883
$760$	0	0
$761$	−36.3702	−1.31842	−0.659210	−	0.751959i	$-0.729109\pi$
$761$	−36.3702	−1.31842	−0.659210	+	0.751959i	$0.729109\pi$
$762$	0	0
$763$	1.07203i	0.0388099i
$764$	0	0
$765$	− 7.08180i	− 0.256043i
$766$	0	0
$767$	−0.922856	−0.0333224
$768$	0	0
$769$	8.11654	0.292690	0.146345	−	0.989234i	$-0.453249\pi$
$769$	8.11654	0.292690	0.146345	+	0.989234i	$0.453249\pi$
$770$	0	0
$771$	− 15.6154i	− 0.562373i
$772$	0	0
$773$	− 41.0053i	− 1.47486i	−0.675424	−	0.737429i	$-0.736039\pi$
$773$	− 41.0053i	− 1.47486i	0.675424	−	0.737429i	$-0.263961\pi$
$774$	0	0
$775$	17.0502	0.612460
$776$	0	0
$777$	4.13758	0.148435
$778$	0	0
$779$	7.16502i	0.256714i
$780$	0	0
$781$	− 26.5490i	− 0.949998i
$782$	0	0
$783$	15.2603	0.545357
$784$	0	0
$785$	−12.5709	−0.448674
$786$	0	0
$787$	27.1644i	0.968305i	0.874984	+	0.484153i	$0.160872\pi$
$787$	27.1644i	0.968305i	−0.874984	+	0.484153i	$0.839128\pi$
$788$	0	0
$789$	− 29.6413i	− 1.05526i
$790$	0	0
$791$	−2.49622	−0.0887554
$792$	0	0
$793$	69.8289	2.47970
$794$	0	0
$795$	25.6558i	0.909918i
$796$	0	0
$797$	4.55623i	0.161390i	0.996739	+	0.0806949i	$0.0257140\pi$
$797$	4.55623i	0.161390i	−0.996739	+	0.0806949i	$0.974286\pi$
$798$	0	0
$799$	13.8309	0.489303
$800$	0	0
$801$	−7.76964	−0.274527
$802$	0	0
$803$	− 6.59375i	− 0.232688i
$804$	0	0
$805$	− 1.92587i	− 0.0678779i
$806$	0	0
$807$	−45.7015	−1.60877
$808$	0	0
$809$	22.5386	0.792414	0.396207	−	0.918161i	$-0.370326\pi$
$809$	22.5386	0.792414	0.396207	+	0.918161i	$0.370326\pi$
$810$	0	0
$811$	36.0887i	1.26725i	0.773642	+	0.633623i	$0.218433\pi$
$811$	36.0887i	1.26725i	−0.773642	+	0.633623i	$0.781567\pi$
$812$	0	0
$813$	11.3977i	0.399735i
$814$	0	0
$815$	−2.11958	−0.0742457
$816$	0	0
$817$	−10.4194	−0.364528
$818$	0	0
$819$	1.67965i	0.0586918i
$820$	0	0
$821$	− 14.0875i	− 0.491658i	−0.969313	−	0.245829i	$-0.920940\pi$
$821$	− 14.0875i	− 0.491658i	0.969313	−	0.245829i	$-0.0790602\pi$
$822$	0	0
$823$	37.0424	1.29122	0.645608	−	0.763669i	$-0.276604\pi$
$823$	37.0424	1.29122	0.645608	+	0.763669i	$0.276604\pi$
$824$	0	0
$825$	−19.4554	−0.677350
$826$	0	0
$827$	− 5.03700i	− 0.175154i	−0.996158	−	0.0875768i	$-0.972088\pi$
$827$	− 5.03700i	− 0.175154i	0.996158	−	0.0875768i	$-0.0279123\pi$
$828$	0	0
$829$	− 28.9611i	− 1.00586i	−0.864327	−	0.502930i	$-0.832255\pi$
$829$	− 28.9611i	− 1.00586i	0.864327	−	0.502930i	$-0.167745\pi$
$830$	0	0
$831$	−29.7845	−1.03321
$832$	0	0
$833$	26.6508	0.923397
$834$	0	0
$835$	− 21.1956i	− 0.733504i
$836$	0	0
$837$	− 21.1693i	− 0.731719i
$838$	0	0
$839$	−36.5197	−1.26080	−0.630400	−	0.776270i	$-0.717109\pi$
$839$	−36.5197	−1.26080	−0.630400	+	0.776270i	$0.717109\pi$
$840$	0	0
$841$	10.7750	0.371552
$842$	0	0
$843$	38.2200i	1.31637i
$844$	0	0
$845$	46.8751i	1.61255i
$846$	0	0
$847$	0.0602084	0.00206878
$848$	0	0
$849$	9.98097	0.342546
$850$	0	0
$851$	68.4186i	2.34536i
$852$	0	0
$853$	5.19322i	0.177812i	0.996040	+	0.0889062i	$0.0283371\pi$
$853$	5.19322i	0.177812i	−0.996040	+	0.0889062i	$0.971663\pi$
$854$	0	0
$855$	1.84979	0.0632616
$856$	0	0
$857$	−21.1808	−0.723524	−0.361762	−	0.932271i	$-0.617825\pi$
$857$	−21.1808	−0.723524	−0.361762	+	0.932271i	$0.617825\pi$
$858$	0	0
$859$	13.3430i	0.455257i	0.973748	+	0.227629i	$0.0730972\pi$
$859$	13.3430i	0.455257i	−0.973748	+	0.227629i	$0.926903\pi$
$860$	0	0
$861$	2.91270i	0.0992644i
$862$	0	0
$863$	36.8361	1.25392	0.626958	−	0.779053i	$-0.284300\pi$
$863$	36.8361	1.25392	0.626958	+	0.779053i	$0.284300\pi$
$864$	0	0
$865$	25.7777	0.876467
$866$	0	0
$867$	4.84196i	0.164442i
$868$	0	0
$869$	− 44.6287i	− 1.51393i
$870$	0	0
$871$	14.9343	0.506030
$872$	0	0
$873$	0.168770	0.00571199
$874$	0	0
$875$	− 2.25733i	− 0.0763116i
$876$	0	0
$877$	15.1343i	0.511047i	0.966803	+	0.255524i	$0.0822479\pi$
$877$	15.1343i	0.511047i	−0.966803	+	0.255524i	$0.917752\pi$
$878$	0	0
$879$	4.00935	0.135232
$880$	0	0
$881$	−51.9345	−1.74972	−0.874859	−	0.484377i	$-0.839046\pi$
$881$	−51.9345	−1.74972	−0.874859	+	0.484377i	$0.839046\pi$
$882$	0	0
$883$	− 24.5238i	− 0.825293i	−0.910891	−	0.412646i	$-0.864605\pi$
$883$	− 24.5238i	− 0.825293i	0.910891	−	0.412646i	$-0.135395\pi$
$884$	0	0
$885$	0.413157i	0.0138881i
$886$	0	0
$887$	45.2252	1.51851	0.759257	−	0.650791i	$-0.225562\pi$
$887$	45.2252	1.51851	0.759257	+	0.650791i	$0.225562\pi$
$888$	0	0
$889$	−1.73802	−0.0582912
$890$	0	0
$891$	36.6165i	1.22670i
$892$	0	0
$893$	3.61269i	0.120894i
$894$	0	0
$895$	16.8535	0.563352
$896$	0	0
$897$	−93.3755	−3.11772
$898$	0	0
$899$	25.2821i	0.843205i
$900$	0	0
$901$	32.6376i	1.08732i
$902$	0	0
$903$	−4.23564	−0.140953
$904$	0	0
$905$	−5.76161	−0.191522
$906$	0	0
$907$	5.25715i	0.174561i	0.996184	+	0.0872804i	$0.0278176\pi$
$907$	5.25715i	0.174561i	−0.996184	+	0.0872804i	$0.972182\pi$
$908$	0	0
$909$	− 9.93108i	− 0.329393i
$910$	0	0
$911$	−8.95774	−0.296783	−0.148392	−	0.988929i	$-0.547410\pi$
$911$	−8.95774	−0.296783	−0.148392	+	0.988929i	$0.547410\pi$
$912$	0	0
$913$	−3.75810	−0.124375
$914$	0	0
$915$	− 31.2620i	− 1.03349i
$916$	0	0
$917$	4.15646i	0.137258i
$918$	0	0
$919$	29.1737	0.962351	0.481175	−	0.876624i	$-0.340210\pi$
$919$	29.1737	0.962351	0.481175	+	0.876624i	$0.340210\pi$
$920$	0	0
$921$	−28.3212	−0.933216
$922$	0	0
$923$	− 54.5739i	− 1.79632i
$924$	0	0
$925$	29.3034i	0.963490i
$926$	0	0
$927$	24.9752	0.820294
$928$	0	0
$929$	−39.2930	−1.28916	−0.644580	−	0.764537i	$-0.722968\pi$
$929$	−39.2930	−1.28916	−0.644580	+	0.764537i	$0.722968\pi$
$930$	0	0
$931$	6.96130i	0.228147i
$932$	0	0
$933$	61.6519i	2.01839i
$934$	0	0
$935$	18.2328	0.596278
$936$	0	0
$937$	12.4973	0.408271	0.204135	−	0.978943i	$-0.434562\pi$
$937$	12.4973	0.408271	0.204135	+	0.978943i	$0.434562\pi$
$938$	0	0
$939$	− 41.7924i	− 1.36384i
$940$	0	0
$941$	− 23.1052i	− 0.753207i	−0.926375	−	0.376603i	$-0.877092\pi$
$941$	− 23.1052i	− 0.753207i	0.926375	−	0.376603i	$-0.122908\pi$
$942$	0	0
$943$	−48.1641	−1.56844
$944$	0	0
$945$	−1.02411	−0.0333145
$946$	0	0
$947$	− 3.97265i	− 0.129094i	−0.997915	−	0.0645468i	$-0.979440\pi$
$947$	− 3.97265i	− 0.129094i	0.997915	−	0.0645468i	$-0.0205602\pi$
$948$	0	0
$949$	− 13.5540i	− 0.439983i
$950$	0	0
$951$	28.1111	0.911565
$952$	0	0
$953$	−10.4951	−0.339968	−0.169984	−	0.985447i	$-0.554372\pi$
$953$	−10.4951	−0.339968	−0.169984	+	0.985447i	$0.554372\pi$
$954$	0	0
$955$	5.86725i	0.189860i
$956$	0	0
$957$	− 28.8486i	− 0.932543i
$958$	0	0
$959$	−2.42882	−0.0784307
$960$	0	0
$961$	4.07181	0.131349
$962$	0	0
$963$	4.05120i	0.130548i
$964$	0	0
$965$	− 19.3664i	− 0.623425i
$966$	0	0
$967$	−36.6501	−1.17859	−0.589294	−	0.807919i	$-0.700594\pi$
$967$	−36.6501	−1.17859	−0.589294	+	0.807919i	$0.700594\pi$
$968$	0	0
$969$	7.91120	0.254144
$970$	0	0
$971$	45.3202i	1.45439i	0.686429	+	0.727197i	$0.259177\pi$
$971$	45.3202i	1.45439i	−0.686429	+	0.727197i	$0.740823\pi$
$972$	0	0
$973$	− 1.93701i	− 0.0620977i
$974$	0	0
$975$	−39.9923	−1.28078
$976$	0	0
$977$	22.1618	0.709018	0.354509	−	0.935053i	$-0.384648\pi$
$977$	22.1618	0.709018	0.354509	+	0.935053i	$0.384648\pi$
$978$	0	0
$979$	− 20.0038i	− 0.639323i
$980$	0	0
$981$	− 6.92162i	− 0.220990i
$982$	0	0
$983$	−42.6385	−1.35996	−0.679978	−	0.733233i	$-0.738011\pi$
$983$	−42.6385	−1.35996	−0.679978	+	0.733233i	$0.738011\pi$
$984$	0	0
$985$	9.62978	0.306830
$986$	0	0
$987$	1.46862i	0.0467465i
$988$	0	0
$989$	− 70.0401i	− 2.22715i
$990$	0	0
$991$	−3.20296	−0.101745	−0.0508727	−	0.998705i	$-0.516200\pi$
$991$	−3.20296	−0.101745	−0.0508727	+	0.998705i	$0.516200\pi$
$992$	0	0
$993$	−13.7728	−0.437068
$994$	0	0
$995$	− 32.4997i	− 1.03031i
$996$	0	0
$997$	13.5267i	0.428395i	0.976790	+	0.214198i	$0.0687137\pi$
$997$	13.5267i	0.428395i	−0.976790	+	0.214198i	$0.931286\pi$
$998$	0	0
$999$	36.3829	1.15110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.i.1217.3	✓	16
4.3	odd	2	inner	2432.2.c.i.1217.13	yes	16
8.3	odd	2	inner	2432.2.c.i.1217.4	yes	16
8.5	even	2	inner	2432.2.c.i.1217.14	yes	16
16.3	odd	4		4864.2.a.br.1.1		8
16.5	even	4		4864.2.a.br.1.2		8
16.11	odd	4		4864.2.a.bm.1.8		8
16.13	even	4		4864.2.a.bm.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.i.1217.3	✓	16	1.1	even	1	trivial
2432.2.c.i.1217.4	yes	16	8.3	odd	2	inner
2432.2.c.i.1217.13	yes	16	4.3	odd	2	inner
2432.2.c.i.1217.14	yes	16	8.5	even	2	inner
4864.2.a.bm.1.7		8	16.13	even	4
4864.2.a.bm.1.8		8	16.11	odd	4
4864.2.a.br.1.1		8	16.3	odd	4
4864.2.a.br.1.2		8	16.5	even	4

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:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.06644i q^{3} -1.45635i q^{5} +0.196723 q^{7} -1.27016 q^{9} +O(q^{10})\)	\(q-2.06644i q^{3} -1.45635i q^{5} +0.196723 q^{7} -1.27016 q^{9} -3.27016i q^{11} -6.72211i q^{13} -3.00945 q^{15} -3.82843 q^{17} -1.00000i q^{19} -0.406516i q^{21} +6.72211 q^{23} +2.87905 q^{25} -3.57461i q^{27} +4.26907i q^{29} +5.92215 q^{31} -6.75758 q^{33} -0.286497i q^{35} +10.1782i q^{37} -13.8908 q^{39} -7.16502 q^{41} -10.4194i q^{43} +1.84979i q^{45} -3.61269 q^{47} -6.96130 q^{49} +7.91120i q^{51} -8.52508i q^{53} -4.76249 q^{55} -2.06644 q^{57} -0.137287i q^{59} +10.3879i q^{61} -0.249870 q^{63} -9.78973 q^{65} +2.22167i q^{67} -13.8908i q^{69} +8.11856 q^{71} +2.01634 q^{73} -5.94938i q^{75} -0.643316i q^{77} +13.6473 q^{79} -11.1972 q^{81} -1.14921i q^{83} +5.57552i q^{85} +8.82177 q^{87} +6.11706 q^{89} -1.32239i q^{91} -12.2377i q^{93} -1.45635 q^{95} -0.132873 q^{97} +4.15362i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^9	\( 16 q - 8 q^{9} - 16 q^{17} - 40 q^{25} + 48 q^{33} - 16 q^{41} + 16 q^{49} + 8 q^{57} + 16 q^{65} + 16 q^{73} - 64 q^{81} + 16 q^{89} + 80 q^{97}+O(q^{100}) \) 16 * q - 8 * q^9 - 16 * q^17 - 40 * q^25 + 48 * q^33 - 16 * q^41 + 16 * q^49 + 8 * q^57 + 16 * q^65 + 16 * q^73 - 64 * q^81 + 16 * q^89 + 80 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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