LMFDB - Embedded newform 1216.3.d.e.191.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,3,Mod(191,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1216.191"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$33.1336001462$
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} - 30 x^{14} + 116 x^{13} + 707 x^{12} - 2372 x^{11} - 7342 x^{10} + 12048 x^{9} + \cdots + 19859428 $ x^16 - 4x^15 - 30x^14 + 116x^13 + 707x^12 - 2372x^11 - 7342x^10 + 12048x^9 + 66793x^8 + 65416x^7 - 419708x^6 - 1201304x^5 + 619800x^4 + 4221760x^3 + 6110408x^2 + 10041872*x + 19859428
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{26} $
Twist minimal:	no (minimal twist has level 608)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.10
Root			$-2.45546 - 0.645214i$ of defining polynomial
Character	$\chi$	$=$	1216.191
Dual form			1216.3.d.e.191.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.29043i q^{3} +5.91093 q^{5} +0.267056i q^{7} +7.33480 q^{9} -7.46730i q^{11} +6.08281 q^{13} +7.62763i q^{15} +19.9294 q^{17} -4.35890i q^{19} -0.344616 q^{21} -10.7394i q^{23} +9.93909 q^{25} +21.0789i q^{27} -3.72524 q^{29} -43.7175i q^{31} +9.63601 q^{33} +1.57855i q^{35} -39.3891 q^{37} +7.84943i q^{39} +56.2327 q^{41} +17.5075i q^{43} +43.3555 q^{45} -70.1950i q^{47} +48.9287 q^{49} +25.7175i q^{51} -34.4075 q^{53} -44.1387i q^{55} +5.62484 q^{57} +53.2026i q^{59} -20.1622 q^{61} +1.95880i q^{63} +35.9551 q^{65} -41.5979i q^{67} +13.8584 q^{69} +20.3326i q^{71} -76.1222 q^{73} +12.8257i q^{75} +1.99419 q^{77} +60.8144i q^{79} +38.8124 q^{81} +27.1475i q^{83} +117.801 q^{85} -4.80715i q^{87} +17.0268 q^{89} +1.62445i q^{91} +56.4143 q^{93} -25.7651i q^{95} -13.3618 q^{97} -54.7711i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{5} - 40 q^{9} - 40 q^{13} - 32 q^{17} + 96 q^{21} + 88 q^{25} - 144 q^{29} + 88 q^{33} + 56 q^{37} - 104 q^{41} - 40 q^{45} - 144 q^{49} + 320 q^{53} - 8 q^{61} + 336 q^{65} - 392 q^{69} + 72 q^{77}+ \cdots - 240 q^{97}+O(q^{100}) $ 16 * q + 8 * q^5 - 40 * q^9 - 40 * q^13 - 32 * q^17 + 96 * q^21 + 88 * q^25 - 144 * q^29 + 88 * q^33 + 56 * q^37 - 104 * q^41 - 40 * q^45 - 144 * q^49 + 320 * q^53 - 8 * q^61 + 336 * q^65 - 392 * q^69 + 72 * q^77 - 272 * q^81 - 552 * q^85 - 136 * q^89 + 752 * q^93 - 240 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.29043i	0.430143i	0.976598	+	0.215071i	$0.0689984\pi$
$3$	1.29043i	0.430143i	−0.976598	+	0.215071i	$0.931002\pi$
$4$	0	0
$5$	5.91093	1.18219	0.591093	−	0.806603i	$-0.298697\pi$
$5$	5.91093	1.18219	0.591093	+	0.806603i	$0.298697\pi$
$6$	0	0
$7$	0.267056i	0.0381509i	0.999818	+	0.0190754i	$0.00607227\pi$
$7$	0.267056i	0.0381509i	−0.999818	+	0.0190754i	$0.993928\pi$
$8$	0	0
$9$	7.33480	0.814977
$10$	0	0
$11$	− 7.46730i	− 0.678846i	−0.940634	−	0.339423i	$-0.889768\pi$
$11$	− 7.46730i	− 0.678846i	0.940634	−	0.339423i	$-0.110232\pi$
$12$	0	0
$13$	6.08281	0.467909	0.233954	−	0.972248i	$-0.424833\pi$
$13$	6.08281	0.467909	0.233954	+	0.972248i	$0.424833\pi$
$14$	0	0
$15$	7.62763i	0.508508i
$16$	0	0
$17$	19.9294	1.17232	0.586159	−	0.810196i	$-0.300639\pi$
$17$	19.9294	1.17232	0.586159	+	0.810196i	$0.300639\pi$
$18$	0	0
$19$	− 4.35890i	− 0.229416i
$19$	− 4.35890i	− 0.229416i
$20$	0	0
$21$	−0.344616	−0.0164103
$22$	0	0
$23$	− 10.7394i	− 0.466931i	−0.972365	−	0.233465i	$-0.924993\pi$
$23$	− 10.7394i	− 0.466931i	0.972365	−	0.233465i	$-0.0750065\pi$
$24$	0	0
$25$	9.93909	0.397564
$26$	0	0
$27$	21.0789i	0.780699i
$28$	0	0
$29$	−3.72524	−0.128457	−0.0642283	−	0.997935i	$-0.520459\pi$
$29$	−3.72524	−0.128457	−0.0642283	+	0.997935i	$0.520459\pi$
$30$	0	0
$31$	− 43.7175i	− 1.41024i	−0.709087	−	0.705121i	$-0.750893\pi$
$31$	− 43.7175i	− 1.41024i	0.709087	−	0.705121i	$-0.249107\pi$
$32$	0	0
$33$	9.63601	0.292000
$34$	0	0
$35$	1.57855i	0.0451014i
$36$	0	0
$37$	−39.3891	−1.06457	−0.532286	−	0.846565i	$-0.678667\pi$
$37$	−39.3891	−1.06457	−0.532286	+	0.846565i	$0.678667\pi$
$38$	0	0
$39$	7.84943i	0.201267i
$40$	0	0
$41$	56.2327	1.37153	0.685764	−	0.727824i	$-0.259468\pi$
$41$	56.2327	1.37153	0.685764	+	0.727824i	$0.259468\pi$
$42$	0	0
$43$	17.5075i	0.407152i	0.979059	+	0.203576i	$0.0652564\pi$
$43$	17.5075i	0.407152i	−0.979059	+	0.203576i	$0.934744\pi$
$44$	0	0
$45$	43.3555	0.963455
$46$	0	0
$47$	− 70.1950i	− 1.49351i	−0.665099	−	0.746755i	$-0.731611\pi$
$47$	− 70.1950i	− 1.49351i	0.665099	−	0.746755i	$-0.268389\pi$
$48$	0	0
$49$	48.9287	0.998545
$50$	0	0
$51$	25.7175i	0.504264i
$52$	0	0
$53$	−34.4075	−0.649198	−0.324599	−	0.945852i	$-0.605229\pi$
$53$	−34.4075	−0.649198	−0.324599	+	0.945852i	$0.605229\pi$
$54$	0	0
$55$	− 44.1387i	− 0.802522i
$56$	0	0
$57$	5.62484	0.0986815
$58$	0	0
$59$	53.2026i	0.901739i	0.892590	+	0.450870i	$0.148886\pi$
$59$	53.2026i	0.901739i	−0.892590	+	0.450870i	$0.851114\pi$
$60$	0	0
$61$	−20.1622	−0.330527	−0.165264	−	0.986249i	$-0.552848\pi$
$61$	−20.1622	−0.330527	−0.165264	+	0.986249i	$0.552848\pi$
$62$	0	0
$63$	1.95880i	0.0310921i
$64$	0	0
$65$	35.9551	0.553155
$66$	0	0
$67$	− 41.5979i	− 0.620865i	−0.950595	−	0.310432i	$-0.899526\pi$
$67$	− 41.5979i	− 0.620865i	0.950595	−	0.310432i	$-0.100474\pi$
$68$	0	0
$69$	13.8584	0.200847
$70$	0	0
$71$	20.3326i	0.286375i	0.989696	+	0.143187i	$0.0457352\pi$
$71$	20.3326i	0.286375i	−0.989696	+	0.143187i	$0.954265\pi$
$72$	0	0
$73$	−76.1222	−1.04277	−0.521385	−	0.853322i	$-0.674584\pi$
$73$	−76.1222	−1.04277	−0.521385	+	0.853322i	$0.674584\pi$
$74$	0	0
$75$	12.8257i	0.171009i
$76$	0	0
$77$	1.99419	0.0258985
$78$	0	0
$79$	60.8144i	0.769803i	0.922958	+	0.384901i	$0.125765\pi$
$79$	60.8144i	0.769803i	−0.922958	+	0.384901i	$0.874235\pi$
$80$	0	0
$81$	38.8124	0.479166
$82$	0	0
$83$	27.1475i	0.327078i	0.986537	+	0.163539i	$0.0522910\pi$
$83$	27.1475i	0.327078i	−0.986537	+	0.163539i	$0.947709\pi$
$84$	0	0
$85$	117.801	1.38590
$86$	0	0
$87$	− 4.80715i	− 0.0552546i
$88$	0	0
$89$	17.0268	0.191312	0.0956560	−	0.995414i	$-0.469505\pi$
$89$	17.0268	0.191312	0.0956560	+	0.995414i	$0.469505\pi$
$90$	0	0
$91$	1.62445i	0.0178511i
$92$	0	0
$93$	56.4143	0.606605
$94$	0	0
$95$	− 25.7651i	− 0.271212i
$96$	0	0
$97$	−13.3618	−0.137751	−0.0688753	−	0.997625i	$-0.521941\pi$
$97$	−13.3618	−0.137751	−0.0688753	+	0.997625i	$0.521941\pi$
$98$	0	0
$99$	− 54.7711i	− 0.553244i
$100$	0	0
$101$	−38.2500	−0.378712	−0.189356	−	0.981908i	$-0.560640\pi$
$101$	−38.2500	−0.378712	−0.189356	+	0.981908i	$0.560640\pi$
$102$	0	0
$103$	96.1718i	0.933707i	0.884335	+	0.466854i	$0.154613\pi$
$103$	96.1718i	0.933707i	−0.884335	+	0.466854i	$0.845387\pi$
$104$	0	0
$105$	−2.03700	−0.0194000
$106$	0	0
$107$	87.7396i	0.819996i	0.912086	+	0.409998i	$0.134471\pi$
$107$	87.7396i	0.819996i	−0.912086	+	0.409998i	$0.865529\pi$
$108$	0	0
$109$	21.3065	0.195472	0.0977361	−	0.995212i	$-0.468840\pi$
$109$	21.3065	0.195472	0.0977361	+	0.995212i	$0.468840\pi$
$110$	0	0
$111$	− 50.8288i	− 0.457917i
$112$	0	0
$113$	173.285	1.53349	0.766746	−	0.641951i	$-0.221875\pi$
$113$	173.285	1.53349	0.766746	+	0.641951i	$0.221875\pi$
$114$	0	0
$115$	− 63.4799i	− 0.551999i
$116$	0	0
$117$	44.6162	0.381335
$118$	0	0
$119$	5.32227i	0.0447249i
$120$	0	0
$121$	65.2394	0.539169
$122$	0	0
$123$	72.5642i	0.589953i
$124$	0	0
$125$	−89.0240	−0.712192
$126$	0	0
$127$	− 152.379i	− 1.19984i	−0.800061	−	0.599919i	$-0.795199\pi$
$127$	− 152.379i	− 1.19984i	0.800061	−	0.599919i	$-0.204801\pi$
$128$	0	0
$129$	−22.5922	−0.175133
$130$	0	0
$131$	88.0861i	0.672413i	0.941788	+	0.336207i	$0.109144\pi$
$131$	88.0861i	0.672413i	−0.941788	+	0.336207i	$0.890856\pi$
$132$	0	0
$133$	1.16407	0.00875241
$134$	0	0
$135$	124.596i	0.922931i
$136$	0	0
$137$	182.128	1.32940	0.664701	−	0.747109i	$-0.268559\pi$
$137$	182.128	1.32940	0.664701	+	0.747109i	$0.268559\pi$
$138$	0	0
$139$	− 89.8380i	− 0.646316i	−0.946345	−	0.323158i	$-0.895255\pi$
$139$	− 89.8380i	− 0.646316i	0.946345	−	0.323158i	$-0.104745\pi$
$140$	0	0
$141$	90.5816	0.642422
$142$	0	0
$143$	− 45.4222i	− 0.317638i
$144$	0	0
$145$	−22.0196	−0.151860
$146$	0	0
$147$	63.1389i	0.429516i
$148$	0	0
$149$	163.173	1.09512	0.547559	−	0.836767i	$-0.315557\pi$
$149$	163.173	1.09512	0.547559	+	0.836767i	$0.315557\pi$
$150$	0	0
$151$	276.245i	1.82944i	0.404092	+	0.914718i	$0.367588\pi$
$151$	276.245i	1.82944i	−0.404092	+	0.914718i	$0.632412\pi$
$152$	0	0
$153$	146.178	0.955412
$154$	0	0
$155$	− 258.411i	− 1.66717i
$156$	0	0
$157$	192.553	1.22645	0.613227	−	0.789907i	$-0.289871\pi$
$157$	192.553	1.22645	0.613227	+	0.789907i	$0.289871\pi$
$158$	0	0
$159$	− 44.4003i	− 0.279247i
$160$	0	0
$161$	2.86802	0.0178138
$162$	0	0
$163$	− 194.586i	− 1.19378i	−0.802323	−	0.596890i	$-0.796403\pi$
$163$	− 194.586i	− 1.19378i	0.802323	−	0.596890i	$-0.203597\pi$
$164$	0	0
$165$	56.9578	0.345199
$166$	0	0
$167$	− 61.3776i	− 0.367530i	−0.982970	−	0.183765i	$-0.941171\pi$
$167$	− 61.3776i	− 0.367530i	0.982970	−	0.183765i	$-0.0588286\pi$
$168$	0	0
$169$	−131.999	−0.781062
$170$	0	0
$171$	− 31.9716i	− 0.186969i
$172$	0	0
$173$	50.3767	0.291195	0.145597	−	0.989344i	$-0.453490\pi$
$173$	50.3767	0.291195	0.145597	+	0.989344i	$0.453490\pi$
$174$	0	0
$175$	2.65429i	0.0151674i
$176$	0	0
$177$	−68.6541	−0.387876
$178$	0	0
$179$	267.545i	1.49467i	0.664450	+	0.747333i	$0.268666\pi$
$179$	267.545i	1.49467i	−0.664450	+	0.747333i	$0.731334\pi$
$180$	0	0
$181$	−125.695	−0.694449	−0.347224	−	0.937782i	$-0.612876\pi$
$181$	−125.695	−0.694449	−0.347224	+	0.937782i	$0.612876\pi$
$182$	0	0
$183$	− 26.0178i	− 0.142174i
$184$	0	0
$185$	−232.826	−1.25852
$186$	0	0
$187$	− 148.819i	− 0.795823i
$188$	0	0
$189$	−5.62924	−0.0297843
$190$	0	0
$191$	226.780i	1.18733i	0.804713	+	0.593665i	$0.202319\pi$
$191$	226.780i	1.18733i	−0.804713	+	0.593665i	$0.797681\pi$
$192$	0	0
$193$	22.0673	0.114339	0.0571693	−	0.998365i	$-0.481793\pi$
$193$	22.0673	0.114339	0.0571693	+	0.998365i	$0.481793\pi$
$194$	0	0
$195$	46.3974i	0.237935i
$196$	0	0
$197$	161.881	0.821733	0.410866	−	0.911696i	$-0.365226\pi$
$197$	161.881	0.821733	0.410866	+	0.911696i	$0.365226\pi$
$198$	0	0
$199$	339.840i	1.70774i	0.520488	+	0.853869i	$0.325750\pi$
$199$	339.840i	1.70774i	−0.520488	+	0.853869i	$0.674250\pi$
$200$	0	0
$201$	53.6791	0.267060
$202$	0	0
$203$	− 0.994848i	− 0.00490073i
$204$	0	0
$205$	332.387	1.62140
$206$	0	0
$207$	− 78.7713i	− 0.380538i
$208$	0	0
$209$	−32.5492	−0.155738
$210$	0	0
$211$	228.731i	1.08403i	0.840367	+	0.542017i	$0.182339\pi$
$211$	228.731i	1.08403i	−0.840367	+	0.542017i	$0.817661\pi$
$212$	0	0
$213$	−26.2378	−0.123182
$214$	0	0
$215$	103.486i	0.481329i
$216$	0	0
$217$	11.6750	0.0538020
$218$	0	0
$219$	− 98.2302i	− 0.448540i
$220$	0	0
$221$	121.227	0.548538
$222$	0	0
$223$	− 87.6629i	− 0.393107i	−0.980493	−	0.196554i	$-0.937025\pi$
$223$	− 87.6629i	− 0.393107i	0.980493	−	0.196554i	$-0.0629749\pi$
$224$	0	0
$225$	72.9012	0.324005
$226$	0	0
$227$	− 120.628i	− 0.531401i	−0.964056	−	0.265701i	$-0.914397\pi$
$227$	− 120.628i	− 0.531401i	0.964056	−	0.265701i	$-0.0856033\pi$
$228$	0	0
$229$	−66.0305	−0.288343	−0.144171	−	0.989553i	$-0.546052\pi$
$229$	−66.0305	−0.288343	−0.144171	+	0.989553i	$0.546052\pi$
$230$	0	0
$231$	2.57335i	0.0111401i
$232$	0	0
$233$	−132.586	−0.569039	−0.284520	−	0.958670i	$-0.591834\pi$
$233$	−132.586	−0.569039	−0.284520	+	0.958670i	$0.591834\pi$
$234$	0	0
$235$	− 414.918i	− 1.76561i
$236$	0	0
$237$	−78.4766	−0.331125
$238$	0	0
$239$	− 55.7666i	− 0.233333i	−0.993171	−	0.116666i	$-0.962779\pi$
$239$	− 55.7666i	− 0.233333i	0.993171	−	0.116666i	$-0.0372209\pi$
$240$	0	0
$241$	−318.825	−1.32292	−0.661462	−	0.749979i	$-0.730064\pi$
$241$	−318.825	−1.32292	−0.661462	+	0.749979i	$0.730064\pi$
$242$	0	0
$243$	239.794i	0.986808i
$244$	0	0
$245$	289.214	1.18047
$246$	0	0
$247$	− 26.5144i	− 0.107346i
$248$	0	0
$249$	−35.0318	−0.140690
$250$	0	0
$251$	− 426.271i	− 1.69829i	−0.528160	−	0.849145i	$-0.677118\pi$
$251$	− 426.271i	− 1.69829i	0.528160	−	0.849145i	$-0.322882\pi$
$252$	0	0
$253$	−80.1944	−0.316974
$254$	0	0
$255$	152.014i	0.596133i
$256$	0	0
$257$	41.3425	0.160866	0.0804330	−	0.996760i	$-0.474370\pi$
$257$	41.3425	0.160866	0.0804330	+	0.996760i	$0.474370\pi$
$258$	0	0
$259$	− 10.5191i	− 0.0406143i
$260$	0	0
$261$	−27.3239	−0.104689
$262$	0	0
$263$	− 160.784i	− 0.611345i	−0.952137	−	0.305672i	$-0.901119\pi$
$263$	− 160.784i	− 0.611345i	0.952137	−	0.305672i	$-0.0988812\pi$
$264$	0	0
$265$	−203.380	−0.767472
$266$	0	0
$267$	21.9718i	0.0822914i
$268$	0	0
$269$	−242.404	−0.901131	−0.450565	−	0.892743i	$-0.648778\pi$
$269$	−242.404	−0.901131	−0.450565	+	0.892743i	$0.648778\pi$
$270$	0	0
$271$	− 241.924i	− 0.892708i	−0.894856	−	0.446354i	$-0.852722\pi$
$271$	− 241.924i	− 0.892708i	0.894856	−	0.446354i	$-0.147278\pi$
$272$	0	0
$273$	−2.09624	−0.00767852
$274$	0	0
$275$	− 74.2182i	− 0.269884i
$276$	0	0
$277$	−153.558	−0.554361	−0.277180	−	0.960818i	$-0.589400\pi$
$277$	−153.558	−0.554361	−0.277180	+	0.960818i	$0.589400\pi$
$278$	0	0
$279$	− 320.659i	− 1.14932i
$280$	0	0
$281$	−204.158	−0.726540	−0.363270	−	0.931684i	$-0.618340\pi$
$281$	−204.158	−0.726540	−0.363270	+	0.931684i	$0.618340\pi$
$282$	0	0
$283$	170.889i	0.603846i	0.953332	+	0.301923i	$0.0976286\pi$
$283$	170.889i	0.603846i	−0.953332	+	0.301923i	$0.902371\pi$
$284$	0	0
$285$	33.2481	0.116660
$286$	0	0
$287$	15.0173i	0.0523250i
$288$	0	0
$289$	108.181	0.374329
$290$	0	0
$291$	− 17.2425i	− 0.0592524i
$292$	0	0
$293$	−573.341	−1.95680	−0.978398	−	0.206732i	$-0.933717\pi$
$293$	−573.341	−1.95680	−0.978398	+	0.206732i	$0.933717\pi$
$294$	0	0
$295$	314.477i	1.06602i
$296$	0	0
$297$	157.402	0.529974
$298$	0	0
$299$	− 65.3258i	− 0.218481i
$300$	0	0
$301$	−4.67549	−0.0155332
$302$	0	0
$303$	− 49.3588i	− 0.162900i
$304$	0	0
$305$	−119.177	−0.390745
$306$	0	0
$307$	500.022i	1.62874i	0.580348	+	0.814369i	$0.302917\pi$
$307$	500.022i	1.62874i	−0.580348	+	0.814369i	$0.697083\pi$
$308$	0	0
$309$	−124.103	−0.401627
$310$	0	0
$311$	− 260.716i	− 0.838316i	−0.907913	−	0.419158i	$-0.862325\pi$
$311$	− 260.716i	− 0.838316i	0.907913	−	0.419158i	$-0.137675\pi$
$312$	0	0
$313$	−356.299	−1.13833	−0.569167	−	0.822222i	$-0.692734\pi$
$313$	−356.299	−1.13833	−0.569167	+	0.822222i	$0.692734\pi$
$314$	0	0
$315$	11.5783i	0.0367566i
$316$	0	0
$317$	−253.417	−0.799422	−0.399711	−	0.916641i	$-0.630890\pi$
$317$	−253.417	−0.799422	−0.399711	+	0.916641i	$0.630890\pi$
$318$	0	0
$319$	27.8175i	0.0872022i
$320$	0	0
$321$	−113.222	−0.352715
$322$	0	0
$323$	− 86.8702i	− 0.268948i
$324$	0	0
$325$	60.4576	0.186023
$326$	0	0
$327$	27.4945i	0.0840809i
$328$	0	0
$329$	18.7460	0.0569787
$330$	0	0
$331$	− 97.6458i	− 0.295003i	−0.989062	−	0.147501i	$-0.952877\pi$
$331$	− 97.6458i	− 0.295003i	0.989062	−	0.147501i	$-0.0471230\pi$
$332$	0	0
$333$	−288.911	−0.867601
$334$	0	0
$335$	− 245.883i	− 0.733978i
$336$	0	0
$337$	−177.673	−0.527218	−0.263609	−	0.964630i	$-0.584913\pi$
$337$	−177.673	−0.527218	−0.263609	+	0.964630i	$0.584913\pi$
$338$	0	0
$339$	223.611i	0.659620i
$340$	0	0
$341$	−326.452	−0.957337
$342$	0	0
$343$	26.1524i	0.0762462i
$344$	0	0
$345$	81.9162	0.237438
$346$	0	0
$347$	45.2992i	0.130545i	0.997867	+	0.0652726i	$0.0207917\pi$
$347$	45.2992i	0.130545i	−0.997867	+	0.0652726i	$0.979208\pi$
$348$	0	0
$349$	−165.193	−0.473332	−0.236666	−	0.971591i	$-0.576055\pi$
$349$	−165.193	−0.473332	−0.236666	+	0.971591i	$0.576055\pi$
$350$	0	0
$351$	128.219i	0.365296i
$352$	0	0
$353$	−4.32169	−0.0122428	−0.00612138	−	0.999981i	$-0.501949\pi$
$353$	−4.32169	−0.0122428	−0.00612138	+	0.999981i	$0.501949\pi$
$354$	0	0
$355$	120.185i	0.338548i
$356$	0	0
$357$	−6.86800	−0.0192381
$358$	0	0
$359$	− 11.7180i	− 0.0326407i	−0.999867	−	0.0163204i	$-0.994805\pi$
$359$	− 11.7180i	− 0.0326407i	0.999867	−	0.0163204i	$-0.00519516\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	84.1867i	0.231919i
$364$	0	0
$365$	−449.953	−1.23275
$366$	0	0
$367$	− 169.090i	− 0.460736i	−0.973104	−	0.230368i	$-0.926007\pi$
$367$	− 169.090i	− 0.460736i	0.973104	−	0.230368i	$-0.0739931\pi$
$368$	0	0
$369$	412.455	1.11776
$370$	0	0
$371$	− 9.18872i	− 0.0247674i
$372$	0	0
$373$	−127.448	−0.341683	−0.170841	−	0.985299i	$-0.554649\pi$
$373$	−127.448	−0.341683	−0.170841	+	0.985299i	$0.554649\pi$
$374$	0	0
$375$	− 114.879i	− 0.306344i
$376$	0	0
$377$	−22.6599	−0.0601059
$378$	0	0
$379$	− 553.575i	− 1.46062i	−0.683116	−	0.730310i	$-0.739376\pi$
$379$	− 553.575i	− 1.46062i	0.683116	−	0.730310i	$-0.260624\pi$
$380$	0	0
$381$	196.635	0.516101
$382$	0	0
$383$	437.773i	1.14301i	0.820598	+	0.571506i	$0.193641\pi$
$383$	437.773i	1.14301i	−0.820598	+	0.571506i	$0.806359\pi$
$384$	0	0
$385$	11.7875	0.0306169
$386$	0	0
$387$	128.414i	0.331820i
$388$	0	0
$389$	−331.056	−0.851043	−0.425522	−	0.904948i	$-0.639909\pi$
$389$	−331.056	−0.851043	−0.425522	+	0.904948i	$0.639909\pi$
$390$	0	0
$391$	− 214.030i	− 0.547391i
$392$	0	0
$393$	−113.669	−0.289234
$394$	0	0
$395$	359.470i	0.910050i
$396$	0	0
$397$	−242.339	−0.610426	−0.305213	−	0.952284i	$-0.598728\pi$
$397$	−242.339	−0.610426	−0.305213	+	0.952284i	$0.598728\pi$
$398$	0	0
$399$	1.50215i	0.00376478i
$400$	0	0
$401$	−249.083	−0.621155	−0.310578	−	0.950548i	$-0.600522\pi$
$401$	−249.083	−0.621155	−0.310578	+	0.950548i	$0.600522\pi$
$402$	0	0
$403$	− 265.925i	− 0.659865i
$404$	0	0
$405$	229.417	0.566463
$406$	0	0
$407$	294.131i	0.722679i
$408$	0	0
$409$	601.314	1.47021	0.735103	−	0.677955i	$-0.237134\pi$
$409$	601.314	1.47021	0.735103	+	0.677955i	$0.237134\pi$
$410$	0	0
$411$	235.023i	0.571833i
$412$	0	0
$413$	−14.2081	−0.0344021
$414$	0	0
$415$	160.467i	0.386667i
$416$	0	0
$417$	115.929	0.278008
$418$	0	0
$419$	788.737i	1.88243i	0.337811	+	0.941214i	$0.390314\pi$
$419$	788.737i	1.88243i	−0.337811	+	0.941214i	$0.609686\pi$
$420$	0	0
$421$	487.237	1.15733	0.578666	−	0.815564i	$-0.303573\pi$
$421$	487.237	1.15733	0.578666	+	0.815564i	$0.303573\pi$
$422$	0	0
$423$	− 514.866i	− 1.21718i
$424$	0	0
$425$	198.080	0.466071
$426$	0	0
$427$	− 5.38443i	− 0.0126099i
$428$	0	0
$429$	58.6140	0.136629
$430$	0	0
$431$	− 389.242i	− 0.903113i	−0.892242	−	0.451557i	$-0.850869\pi$
$431$	− 389.242i	− 0.903113i	0.892242	−	0.451557i	$-0.149131\pi$
$432$	0	0
$433$	49.0481	0.113275	0.0566375	−	0.998395i	$-0.481962\pi$
$433$	49.0481	0.113275	0.0566375	+	0.998395i	$0.481962\pi$
$434$	0	0
$435$	− 28.4147i	− 0.0653212i
$436$	0	0
$437$	−46.8120	−0.107121
$438$	0	0
$439$	629.258i	1.43339i	0.697387	+	0.716694i	$0.254346\pi$
$439$	629.258i	1.43339i	−0.697387	+	0.716694i	$0.745654\pi$
$440$	0	0
$441$	358.882	0.813791
$442$	0	0
$443$	− 261.774i	− 0.590912i	−0.955356	−	0.295456i	$-0.904528\pi$
$443$	− 261.774i	− 0.590912i	0.955356	−	0.295456i	$-0.0954716\pi$
$444$	0	0
$445$	100.644	0.226166
$446$	0	0
$447$	210.562i	0.471057i
$448$	0	0
$449$	−768.178	−1.71086	−0.855432	−	0.517915i	$-0.826708\pi$
$449$	−768.178	−1.71086	−0.855432	+	0.517915i	$0.826708\pi$
$450$	0	0
$451$	− 419.906i	− 0.931056i
$452$	0	0
$453$	−356.474	−0.786919
$454$	0	0
$455$	9.60202i	0.0211033i
$456$	0	0
$457$	−753.743	−1.64933	−0.824665	−	0.565622i	$-0.808636\pi$
$457$	−753.743	−1.64933	−0.824665	+	0.565622i	$0.808636\pi$
$458$	0	0
$459$	420.089i	0.915227i
$460$	0	0
$461$	204.768	0.444183	0.222091	−	0.975026i	$-0.428712\pi$
$461$	204.768	0.444183	0.222091	+	0.975026i	$0.428712\pi$
$462$	0	0
$463$	− 617.825i	− 1.33439i	−0.744881	−	0.667197i	$-0.767494\pi$
$463$	− 617.825i	− 1.33439i	0.744881	−	0.667197i	$-0.232506\pi$
$464$	0	0
$465$	333.461	0.717120
$466$	0	0
$467$	425.268i	0.910638i	0.890328	+	0.455319i	$0.150475\pi$
$467$	425.268i	0.910638i	−0.890328	+	0.455319i	$0.849525\pi$
$468$	0	0
$469$	11.1090	0.0236865
$470$	0	0
$471$	248.476i	0.527550i
$472$	0	0
$473$	130.734	0.276393
$474$	0	0
$475$	− 43.3235i	− 0.0912073i
$476$	0	0
$477$	−252.372	−0.529081
$478$	0	0
$479$	− 740.431i	− 1.54578i	−0.634537	−	0.772892i	$-0.718809\pi$
$479$	− 740.431i	− 1.54578i	0.634537	−	0.772892i	$-0.281191\pi$
$480$	0	0
$481$	−239.597	−0.498122
$482$	0	0
$483$	3.70097i	0.00766247i
$484$	0	0
$485$	−78.9807	−0.162847
$486$	0	0
$487$	− 51.8620i	− 0.106493i	−0.998581	−	0.0532465i	$-0.983043\pi$
$487$	− 51.8620i	− 0.106493i	0.998581	−	0.0532465i	$-0.0169569\pi$
$488$	0	0
$489$	251.099	0.513495
$490$	0	0
$491$	− 668.327i	− 1.36115i	−0.732676	−	0.680577i	$-0.761729\pi$
$491$	− 668.327i	− 1.36115i	0.732676	−	0.680577i	$-0.238271\pi$
$492$	0	0
$493$	−74.2418	−0.150592
$494$	0	0
$495$	− 323.748i	− 0.654037i
$496$	0	0
$497$	−5.42995	−0.0109254
$498$	0	0
$499$	695.701i	1.39419i	0.716979	+	0.697095i	$0.245524\pi$
$499$	695.701i	1.39419i	−0.716979	+	0.697095i	$0.754476\pi$
$500$	0	0
$501$	79.2033	0.158090
$502$	0	0
$503$	480.062i	0.954397i	0.878795	+	0.477199i	$0.158348\pi$
$503$	480.062i	0.954397i	−0.878795	+	0.477199i	$0.841652\pi$
$504$	0	0
$505$	−226.093	−0.447709
$506$	0	0
$507$	− 170.336i	− 0.335968i
$508$	0	0
$509$	−535.267	−1.05161	−0.525803	−	0.850607i	$-0.676235\pi$
$509$	−535.267	−1.05161	−0.525803	+	0.850607i	$0.676235\pi$
$510$	0	0
$511$	− 20.3289i	− 0.0397826i
$512$	0	0
$513$	91.8807	0.179105
$514$	0	0
$515$	568.465i	1.10382i
$516$	0	0
$517$	−524.167	−1.01386
$518$	0	0
$519$	65.0074i	0.125255i
$520$	0	0
$521$	636.008	1.22074	0.610372	−	0.792115i	$-0.291020\pi$
$521$	636.008	1.22074	0.610372	+	0.792115i	$0.291020\pi$
$522$	0	0
$523$	106.122i	0.202910i	0.994840	+	0.101455i	$0.0323498\pi$
$523$	106.122i	0.202910i	−0.994840	+	0.101455i	$0.967650\pi$
$524$	0	0
$525$	−3.42517	−0.00652414
$526$	0	0
$527$	− 871.264i	− 1.65325i
$528$	0	0
$529$	413.665	0.781976
$530$	0	0
$531$	390.230i	0.734897i
$532$	0	0
$533$	342.053	0.641750
$534$	0	0
$535$	518.622i	0.969388i
$536$	0	0
$537$	−345.248	−0.642919
$538$	0	0
$539$	− 365.365i	− 0.677858i
$540$	0	0
$541$	−36.7012	−0.0678396	−0.0339198	−	0.999425i	$-0.510799\pi$
$541$	−36.7012	−0.0678396	−0.0339198	+	0.999425i	$0.510799\pi$
$542$	0	0
$543$	− 162.201i	− 0.298712i
$544$	0	0
$545$	125.941	0.231085
$546$	0	0
$547$	− 501.855i	− 0.917468i	−0.888574	−	0.458734i	$-0.848303\pi$
$547$	− 501.855i	− 0.917468i	0.888574	−	0.458734i	$-0.151697\pi$
$548$	0	0
$549$	−147.885	−0.269372
$550$	0	0
$551$	16.2379i	0.0294700i
$552$	0	0
$553$	−16.2409	−0.0293686
$554$	0	0
$555$	− 300.446i	− 0.541343i
$556$	0	0
$557$	−763.382	−1.37052	−0.685262	−	0.728297i	$-0.740312\pi$
$557$	−763.382	−1.37052	−0.685262	+	0.728297i	$0.740312\pi$
$558$	0	0
$559$	106.495i	0.190510i
$560$	0	0
$561$	192.040	0.342317
$562$	0	0
$563$	734.286i	1.30424i	0.758117	+	0.652119i	$0.226120\pi$
$563$	734.286i	1.30424i	−0.758117	+	0.652119i	$0.773880\pi$
$564$	0	0
$565$	1024.27	1.81287
$566$	0	0
$567$	10.3651i	0.0182806i
$568$	0	0
$569$	−257.303	−0.452201	−0.226101	−	0.974104i	$-0.572598\pi$
$569$	−257.303	−0.452201	−0.226101	+	0.974104i	$0.572598\pi$
$570$	0	0
$571$	315.894i	0.553230i	0.960981	+	0.276615i	$0.0892126\pi$
$571$	315.894i	0.553230i	−0.960981	+	0.276615i	$0.910787\pi$
$572$	0	0
$573$	−292.643	−0.510721
$574$	0	0
$575$	− 106.740i	− 0.185635i
$576$	0	0
$577$	−110.448	−0.191419	−0.0957093	−	0.995409i	$-0.530512\pi$
$577$	−110.448	−0.191419	−0.0957093	+	0.995409i	$0.530512\pi$
$578$	0	0
$579$	28.4763i	0.0491819i
$580$	0	0
$581$	−7.24990	−0.0124783
$582$	0	0
$583$	256.931i	0.440705i
$584$	0	0
$585$	263.723	0.450809
$586$	0	0
$587$	756.449i	1.28867i	0.764744	+	0.644335i	$0.222866\pi$
$587$	756.449i	1.28867i	−0.764744	+	0.644335i	$0.777134\pi$
$588$	0	0
$589$	−190.560	−0.323532
$590$	0	0
$591$	208.896i	0.353462i
$592$	0	0
$593$	−878.684	−1.48176	−0.740880	−	0.671637i	$-0.765591\pi$
$593$	−878.684	−1.48176	−0.740880	+	0.671637i	$0.765591\pi$
$594$	0	0
$595$	31.4595i	0.0528732i
$596$	0	0
$597$	−438.539	−0.734571
$598$	0	0
$599$	764.979i	1.27709i	0.769583	+	0.638547i	$0.220464\pi$
$599$	764.979i	1.27709i	−0.769583	+	0.638547i	$0.779536\pi$
$600$	0	0
$601$	−260.131	−0.432830	−0.216415	−	0.976301i	$-0.569436\pi$
$601$	−260.131	−0.432830	−0.216415	+	0.976301i	$0.569436\pi$
$602$	0	0
$603$	− 305.112i	− 0.505991i
$604$	0	0
$605$	385.626	0.637398
$606$	0	0
$607$	− 556.060i	− 0.916078i	−0.888932	−	0.458039i	$-0.848552\pi$
$607$	− 556.060i	− 0.916078i	0.888932	−	0.458039i	$-0.151448\pi$
$608$	0	0
$609$	1.28378	0.00210801
$610$	0	0
$611$	− 426.983i	− 0.698826i
$612$	0	0
$613$	1030.55	1.68117	0.840583	−	0.541682i	$-0.182212\pi$
$613$	1030.55	1.68117	0.840583	+	0.541682i	$0.182212\pi$
$614$	0	0
$615$	428.922i	0.697434i
$616$	0	0
$617$	−568.547	−0.921470	−0.460735	−	0.887538i	$-0.652414\pi$
$617$	−568.547	−0.921470	−0.460735	+	0.887538i	$0.652414\pi$
$618$	0	0
$619$	173.986i	0.281075i	0.990075	+	0.140538i	$0.0448831\pi$
$619$	173.986i	0.281075i	−0.990075	+	0.140538i	$0.955117\pi$
$620$	0	0
$621$	226.375	0.364532
$622$	0	0
$623$	4.54710i	0.00729871i
$624$	0	0
$625$	−774.692	−1.23951
$626$	0	0
$627$	− 42.0024i	− 0.0669895i
$628$	0	0
$629$	−785.002	−1.24802
$630$	0	0
$631$	969.221i	1.53601i	0.640446	+	0.768004i	$0.278750\pi$
$631$	969.221i	1.53601i	−0.640446	+	0.768004i	$0.721250\pi$
$632$	0	0
$633$	−295.161	−0.466289
$634$	0	0
$635$	− 900.704i	− 1.41843i
$636$	0	0
$637$	297.624	0.467228
$638$	0	0
$639$	149.136i	0.233389i
$640$	0	0
$641$	−718.485	−1.12088	−0.560441	−	0.828195i	$-0.689368\pi$
$641$	−718.485	−1.12088	−0.560441	+	0.828195i	$0.689368\pi$
$642$	0	0
$643$	− 1117.36i	− 1.73773i	−0.495051	−	0.868864i	$-0.664851\pi$
$643$	− 1117.36i	− 1.73773i	0.495051	−	0.868864i	$-0.335149\pi$
$644$	0	0
$645$	−133.541	−0.207040
$646$	0	0
$647$	− 979.031i	− 1.51319i	−0.653887	−	0.756593i	$-0.726863\pi$
$647$	− 979.031i	− 1.51319i	0.653887	−	0.756593i	$-0.273137\pi$
$648$	0	0
$649$	397.280	0.612142
$650$	0	0
$651$	15.0658i	0.0231425i
$652$	0	0
$653$	−258.364	−0.395657	−0.197829	−	0.980237i	$-0.563389\pi$
$653$	−258.364	−0.395657	−0.197829	+	0.980237i	$0.563389\pi$
$654$	0	0
$655$	520.671i	0.794917i
$656$	0	0
$657$	−558.341	−0.849834
$658$	0	0
$659$	− 722.618i	− 1.09654i	−0.836302	−	0.548269i	$-0.815287\pi$
$659$	− 722.618i	− 1.09654i	0.836302	−	0.548269i	$-0.184713\pi$
$660$	0	0
$661$	552.320	0.835583	0.417792	−	0.908543i	$-0.362804\pi$
$661$	552.320	0.835583	0.417792	+	0.908543i	$0.362804\pi$
$662$	0	0
$663$	156.434i	0.235949i
$664$	0	0
$665$	6.88074	0.0103470
$666$	0	0
$667$	40.0069i	0.0599803i
$668$	0	0
$669$	113.123	0.169092
$670$	0	0
$671$	150.557i	0.224377i
$672$	0	0
$673$	500.006	0.742952	0.371476	−	0.928443i	$-0.378852\pi$
$673$	500.006	0.742952	0.371476	+	0.928443i	$0.378852\pi$
$674$	0	0
$675$	209.505i	0.310377i
$676$	0	0
$677$	−889.777	−1.31429	−0.657147	−	0.753762i	$-0.728237\pi$
$677$	−889.777	−1.31429	−0.657147	+	0.753762i	$0.728237\pi$
$678$	0	0
$679$	− 3.56835i	− 0.00525531i
$680$	0	0
$681$	155.662	0.228578
$682$	0	0
$683$	− 774.698i	− 1.13426i	−0.823629	−	0.567129i	$-0.808054\pi$
$683$	− 774.698i	− 1.13426i	0.823629	−	0.567129i	$-0.191946\pi$
$684$	0	0
$685$	1076.55	1.57160
$686$	0	0
$687$	− 85.2075i	− 0.124028i
$688$	0	0
$689$	−209.294	−0.303765
$690$	0	0
$691$	− 1138.21i	− 1.64719i	−0.567181	−	0.823593i	$-0.691966\pi$
$691$	− 1138.21i	− 1.64719i	0.567181	−	0.823593i	$-0.308034\pi$
$692$	0	0
$693$	14.6270	0.0211067
$694$	0	0
$695$	− 531.026i	− 0.764066i
$696$	0	0
$697$	1120.68	1.60787
$698$	0	0
$699$	− 171.093i	− 0.244768i
$700$	0	0
$701$	390.090	0.556476	0.278238	−	0.960512i	$-0.410250\pi$
$701$	390.090	0.556476	0.278238	+	0.960512i	$0.410250\pi$
$702$	0	0
$703$	171.693i	0.244229i
$704$	0	0
$705$	535.421	0.759463
$706$	0	0
$707$	− 10.2149i	− 0.0144482i
$708$	0	0
$709$	1367.27	1.92844	0.964222	−	0.265096i	$-0.0854036\pi$
$709$	1367.27	1.92844	0.964222	+	0.265096i	$0.0854036\pi$
$710$	0	0
$711$	446.061i	0.627372i
$712$	0	0
$713$	−469.500	−0.658485
$714$	0	0
$715$	− 268.487i	− 0.375507i
$716$	0	0
$717$	71.9627	0.100366
$718$	0	0
$719$	− 361.674i	− 0.503023i	−0.967854	−	0.251512i	$-0.919072\pi$
$719$	− 361.674i	− 0.503023i	0.967854	−	0.251512i	$-0.0809276\pi$
$720$	0	0
$721$	−25.6833	−0.0356217
$722$	0	0
$723$	− 411.420i	− 0.569046i
$724$	0	0
$725$	−37.0255	−0.0510696
$726$	0	0
$727$	217.765i	0.299539i	0.988721	+	0.149769i	$0.0478531\pi$
$727$	217.765i	0.299539i	−0.988721	+	0.149769i	$0.952147\pi$
$728$	0	0
$729$	39.8743	0.0546973
$730$	0	0
$731$	348.915i	0.477311i
$732$	0	0
$733$	1400.53	1.91068	0.955338	−	0.295514i	$-0.0954909\pi$
$733$	1400.53	1.91068	0.955338	+	0.295514i	$0.0954909\pi$
$734$	0	0
$735$	373.210i	0.507768i
$736$	0	0
$737$	−310.624	−0.421471
$738$	0	0
$739$	296.229i	0.400851i	0.979709	+	0.200426i	$0.0642325\pi$
$739$	296.229i	0.400851i	−0.979709	+	0.200426i	$0.935767\pi$
$740$	0	0
$741$	34.2149	0.0461739
$742$	0	0
$743$	1104.49i	1.48653i	0.668998	+	0.743265i	$0.266724\pi$
$743$	1104.49i	1.48653i	−0.668998	+	0.743265i	$0.733276\pi$
$744$	0	0
$745$	964.502	1.29463
$746$	0	0
$747$	199.121i	0.266561i
$748$	0	0
$749$	−23.4314	−0.0312836
$750$	0	0
$751$	526.925i	0.701631i	0.936445	+	0.350815i	$0.114096\pi$
$751$	526.925i	0.701631i	−0.936445	+	0.350815i	$0.885904\pi$
$752$	0	0
$753$	550.072	0.730507
$754$	0	0
$755$	1632.86i	2.16273i
$756$	0	0
$757$	437.862	0.578418	0.289209	−	0.957266i	$-0.406608\pi$
$757$	437.862	0.578418	0.289209	+	0.957266i	$0.406608\pi$
$758$	0	0
$759$	− 103.485i	− 0.136344i
$760$	0	0
$761$	−1020.46	−1.34095	−0.670474	−	0.741933i	$-0.733909\pi$
$761$	−1020.46	−1.34095	−0.670474	+	0.741933i	$0.733909\pi$
$762$	0	0
$763$	5.69002i	0.00745743i
$764$	0	0
$765$	864.049	1.12948
$766$	0	0
$767$	323.622i	0.421932i
$768$	0	0
$769$	1317.04	1.71266	0.856331	−	0.516428i	$-0.172738\pi$
$769$	1317.04	1.71266	0.856331	+	0.516428i	$0.172738\pi$
$770$	0	0
$771$	53.3496i	0.0691953i
$772$	0	0
$773$	−691.054	−0.893989	−0.446995	−	0.894537i	$-0.647506\pi$
$773$	−691.054	−0.893989	−0.446995	+	0.894537i	$0.647506\pi$
$774$	0	0
$775$	− 434.512i	− 0.560661i
$776$	0	0
$777$	13.5741	0.0174699
$778$	0	0
$779$	− 245.113i	− 0.314650i
$780$	0	0
$781$	151.830	0.194404
$782$	0	0
$783$	− 78.5239i	− 0.100286i
$784$	0	0
$785$	1138.17	1.44990
$786$	0	0
$787$	− 1253.37i	− 1.59259i	−0.604910	−	0.796294i	$-0.706791\pi$
$787$	− 1253.37i	− 1.59259i	0.604910	−	0.796294i	$-0.293209\pi$
$788$	0	0
$789$	207.480	0.262965
$790$	0	0
$791$	46.2767i	0.0585040i
$792$	0	0
$793$	−122.643	−0.154657
$794$	0	0
$795$	− 262.447i	− 0.330122i
$796$	0	0
$797$	−849.270	−1.06558	−0.532792	−	0.846246i	$-0.678857\pi$
$797$	−849.270	−1.06558	−0.532792	+	0.846246i	$0.678857\pi$
$798$	0	0
$799$	− 1398.94i	− 1.75087i
$800$	0	0
$801$	124.888	0.155915
$802$	0	0
$803$	568.427i	0.707880i
$804$	0	0
$805$	16.9527	0.0210592
$806$	0	0
$807$	− 312.805i	− 0.387615i
$808$	0	0
$809$	579.347	0.716127	0.358063	−	0.933697i	$-0.383437\pi$
$809$	579.347	0.716127	0.358063	+	0.933697i	$0.383437\pi$
$810$	0	0
$811$	− 737.929i	− 0.909901i	−0.890517	−	0.454950i	$-0.849657\pi$
$811$	− 737.929i	− 0.909901i	0.890517	−	0.454950i	$-0.150343\pi$
$812$	0	0
$813$	312.185	0.383992
$814$	0	0
$815$	− 1150.18i	− 1.41127i
$816$	0	0
$817$	76.3136	0.0934071
$818$	0	0
$819$	11.9150i	0.0145483i
$820$	0	0
$821$	1070.41	1.30379	0.651893	−	0.758311i	$-0.273975\pi$
$821$	1070.41	1.30379	0.651893	+	0.758311i	$0.273975\pi$
$822$	0	0
$823$	− 242.250i	− 0.294350i	−0.989110	−	0.147175i	$-0.952982\pi$
$823$	− 242.250i	− 0.294350i	0.989110	−	0.147175i	$-0.0470181\pi$
$824$	0	0
$825$	95.7732	0.116089
$826$	0	0
$827$	1036.50i	1.25333i	0.779289	+	0.626665i	$0.215580\pi$
$827$	1036.50i	1.25333i	−0.779289	+	0.626665i	$0.784420\pi$
$828$	0	0
$829$	1444.93	1.74298	0.871492	−	0.490409i	$-0.163153\pi$
$829$	1444.93	1.74298	0.871492	+	0.490409i	$0.163153\pi$
$830$	0	0
$831$	− 198.155i	− 0.238454i
$832$	0	0
$833$	975.119	1.17061
$834$	0	0
$835$	− 362.799i	− 0.434489i
$836$	0	0
$837$	921.516	1.10098
$838$	0	0
$839$	969.665i	1.15574i	0.816129	+	0.577870i	$0.196116\pi$
$839$	969.665i	1.15574i	−0.816129	+	0.577870i	$0.803884\pi$
$840$	0	0
$841$	−827.123	−0.983499
$842$	0	0
$843$	− 263.451i	− 0.312516i
$844$	0	0
$845$	−780.239	−0.923360
$846$	0	0
$847$	17.4226i	0.0205697i
$848$	0	0
$849$	−220.519	−0.259740
$850$	0	0
$851$	423.016i	0.497081i
$852$	0	0
$853$	866.287	1.01558	0.507788	−	0.861482i	$-0.330463\pi$
$853$	866.287	1.01558	0.507788	+	0.861482i	$0.330463\pi$
$854$	0	0
$855$	− 188.982i	− 0.221032i
$856$	0	0
$857$	986.495	1.15110	0.575551	−	0.817766i	$-0.304788\pi$
$857$	986.495	1.15110	0.575551	+	0.817766i	$0.304788\pi$
$858$	0	0
$859$	673.745i	0.784336i	0.919894	+	0.392168i	$0.128275\pi$
$859$	673.745i	0.784336i	−0.919894	+	0.392168i	$0.871725\pi$
$860$	0	0
$861$	−19.3787	−0.0225072
$862$	0	0
$863$	135.833i	0.157396i	0.996898	+	0.0786982i	$0.0250764\pi$
$863$	135.833i	0.157396i	−0.996898	+	0.0786982i	$0.974924\pi$
$864$	0	0
$865$	297.773	0.344246
$866$	0	0
$867$	139.600i	0.161015i
$868$	0	0
$869$	454.120	0.522577
$870$	0	0
$871$	− 253.032i	− 0.290508i
$872$	0	0
$873$	−98.0062	−0.112264
$874$	0	0
$875$	− 23.7744i	− 0.0271707i
$876$	0	0
$877$	144.984	0.165318	0.0826592	−	0.996578i	$-0.473659\pi$
$877$	144.984	0.165318	0.0826592	+	0.996578i	$0.473659\pi$
$878$	0	0
$879$	− 739.855i	− 0.841701i
$880$	0	0
$881$	769.504	0.873444	0.436722	−	0.899597i	$-0.356139\pi$
$881$	769.504	0.873444	0.436722	+	0.899597i	$0.356139\pi$
$882$	0	0
$883$	− 422.222i	− 0.478168i	−0.970999	−	0.239084i	$-0.923153\pi$
$883$	− 422.222i	− 0.478168i	0.970999	−	0.239084i	$-0.0768471\pi$
$884$	0	0
$885$	−405.810	−0.458542
$886$	0	0
$887$	− 74.4386i	− 0.0839218i	−0.999119	−	0.0419609i	$-0.986640\pi$
$887$	− 74.4386i	− 0.0839218i	0.999119	−	0.0419609i	$-0.0133605\pi$
$888$	0	0
$889$	40.6938	0.0457748
$890$	0	0
$891$	− 289.824i	− 0.325279i
$892$	0	0
$893$	−305.973	−0.342635
$894$	0	0
$895$	1581.44i	1.76697i
$896$	0	0
$897$	84.2982	0.0939779
$898$	0	0
$899$	162.858i	0.181155i
$900$	0	0
$901$	−685.720	−0.761066
$902$	0	0
$903$	− 6.03338i	− 0.00668149i
$904$	0	0
$905$	−742.976	−0.820968
$906$	0	0
$907$	1027.96i	1.13336i	0.823936	+	0.566682i	$0.191773\pi$
$907$	1027.96i	1.13336i	−0.823936	+	0.566682i	$0.808227\pi$
$908$	0	0
$909$	−280.556	−0.308642
$910$	0	0
$911$	758.359i	0.832446i	0.909262	+	0.416223i	$0.136647\pi$
$911$	758.359i	0.832446i	−0.909262	+	0.416223i	$0.863353\pi$
$912$	0	0
$913$	202.718	0.222035
$914$	0	0
$915$	− 153.790i	− 0.168076i
$916$	0	0
$917$	−23.5239	−0.0256531
$918$	0	0
$919$	1498.27i	1.63033i	0.579230	+	0.815164i	$0.303353\pi$
$919$	1498.27i	1.63033i	−0.579230	+	0.815164i	$0.696647\pi$
$920$	0	0
$921$	−645.243	−0.700589
$922$	0	0
$923$	123.679i	0.133997i
$924$	0	0
$925$	−391.492	−0.423235
$926$	0	0
$927$	705.401i	0.760950i
$928$	0	0
$929$	−1481.21	−1.59441	−0.797206	−	0.603707i	$-0.793690\pi$
$929$	−1481.21	−1.59441	−0.797206	+	0.603707i	$0.793690\pi$
$930$	0	0
$931$	− 213.275i	− 0.229082i
$932$	0	0
$933$	336.436	0.360595
$934$	0	0
$935$	− 879.658i	− 0.940810i
$936$	0	0
$937$	−111.015	−0.118479	−0.0592394	−	0.998244i	$-0.518868\pi$
$937$	−111.015	−0.118479	−0.0592394	+	0.998244i	$0.518868\pi$
$938$	0	0
$939$	− 459.778i	− 0.489646i
$940$	0	0
$941$	208.566	0.221643	0.110821	−	0.993840i	$-0.464652\pi$
$941$	208.566	0.221643	0.110821	+	0.993840i	$0.464652\pi$
$942$	0	0
$943$	− 603.905i	− 0.640409i
$944$	0	0
$945$	−33.2740	−0.0352106
$946$	0	0
$947$	− 1239.20i	− 1.30855i	−0.756255	−	0.654277i	$-0.772973\pi$
$947$	− 1239.20i	− 1.30855i	0.756255	−	0.654277i	$-0.227027\pi$
$948$	0	0
$949$	−463.037	−0.487921
$950$	0	0
$951$	− 327.016i	− 0.343865i
$952$	0	0
$953$	662.495	0.695168	0.347584	−	0.937649i	$-0.387002\pi$
$953$	662.495	0.695168	0.347584	+	0.937649i	$0.387002\pi$
$954$	0	0
$955$	1340.48i	1.40364i
$956$	0	0
$957$	−35.8965	−0.0375094
$958$	0	0
$959$	48.6384i	0.0507179i
$960$	0	0
$961$	−950.222	−0.988784
$962$	0	0
$963$	643.552i	0.668278i
$964$	0	0
$965$	130.438	0.135169
$966$	0	0
$967$	− 1213.54i	− 1.25495i	−0.778636	−	0.627476i	$-0.784088\pi$
$967$	− 1213.54i	− 1.25495i	0.778636	−	0.627476i	$-0.215912\pi$
$968$	0	0
$969$	112.100	0.115686
$970$	0	0
$971$	− 416.270i	− 0.428702i	−0.976757	−	0.214351i	$-0.931236\pi$
$971$	− 416.270i	− 0.428702i	0.976757	−	0.214351i	$-0.0687637\pi$
$972$	0	0
$973$	23.9918	0.0246575
$974$	0	0
$975$	78.0162i	0.0800166i
$976$	0	0
$977$	44.4465	0.0454929	0.0227464	−	0.999741i	$-0.492759\pi$
$977$	44.4465	0.0454929	0.0227464	+	0.999741i	$0.492759\pi$
$978$	0	0
$979$	− 127.144i	− 0.129871i
$980$	0	0
$981$	156.279	0.159305
$982$	0	0
$983$	− 691.898i	− 0.703863i	−0.936026	−	0.351932i	$-0.885525\pi$
$983$	− 691.898i	− 0.703863i	0.936026	−	0.351932i	$-0.114475\pi$
$984$	0	0
$985$	956.869	0.971441
$986$	0	0
$987$	24.1904i	0.0245090i
$988$	0	0
$989$	188.020	0.190112
$990$	0	0
$991$	655.977i	0.661934i	0.943642	+	0.330967i	$0.107375\pi$
$991$	655.977i	0.661934i	−0.943642	+	0.330967i	$0.892625\pi$
$992$	0	0
$993$	126.005	0.126893
$994$	0	0
$995$	2008.77i	2.01886i
$996$	0	0
$997$	−274.866	−0.275693	−0.137846	−	0.990454i	$-0.544018\pi$
$997$	−274.866	−0.275693	−0.137846	+	0.990454i	$0.544018\pi$
$998$	0	0
$999$	− 830.279i	− 0.831110i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.d.e.191.10		16
4.3	odd	2	inner	1216.3.d.e.191.7		16
8.3	odd	2		608.3.d.a.191.10	yes	16
8.5	even	2		608.3.d.a.191.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.d.a.191.7	✓	16	8.5	even	2
608.3.d.a.191.10	yes	16	8.3	odd	2
1216.3.d.e.191.7		16	4.3	odd	2	inner
1216.3.d.e.191.10		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.29043i q^{3} +5.91093 q^{5} +0.267056i q^{7} +7.33480 q^{9} -7.46730i q^{11} +6.08281 q^{13} +7.62763i q^{15} +19.9294 q^{17} -4.35890i q^{19} -0.344616 q^{21} -10.7394i q^{23} +9.93909 q^{25} +21.0789i q^{27} -3.72524 q^{29} -43.7175i q^{31} +9.63601 q^{33} +1.57855i q^{35} -39.3891 q^{37} +7.84943i q^{39} +56.2327 q^{41} +17.5075i q^{43} +43.3555 q^{45} -70.1950i q^{47} +48.9287 q^{49} +25.7175i q^{51} -34.4075 q^{53} -44.1387i q^{55} +5.62484 q^{57} +53.2026i q^{59} -20.1622 q^{61} +1.95880i q^{63} +35.9551 q^{65} -41.5979i q^{67} +13.8584 q^{69} +20.3326i q^{71} -76.1222 q^{73} +12.8257i q^{75} +1.99419 q^{77} +60.8144i q^{79} +38.8124 q^{81} +27.1475i q^{83} +117.801 q^{85} -4.80715i q^{87} +17.0268 q^{89} +1.62445i q^{91} +56.4143 q^{93} -25.7651i q^{95} -13.3618 q^{97} -54.7711i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{5} - 40 q^{9} - 40 q^{13} - 32 q^{17} + 96 q^{21} + 88 q^{25} - 144 q^{29} + 88 q^{33} + 56 q^{37} - 104 q^{41} - 40 q^{45} - 144 q^{49} + 320 q^{53} - 8 q^{61} + 336 q^{65} - 392 q^{69} + 72 q^{77}+ \cdots - 240 q^{97}+O(q^{100}) \) 16 * q + 8 * q^5 - 40 * q^9 - 40 * q^13 - 32 * q^17 + 96 * q^21 + 88 * q^25 - 144 * q^29 + 88 * q^33 + 56 * q^37 - 104 * q^41 - 40 * q^45 - 144 * q^49 + 320 * q^53 - 8 * q^61 + 336 * q^65 - 392 * q^69 + 72 * q^77 - 272 * q^81 - 552 * q^85 - 136 * q^89 + 752 * q^93 - 240 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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