LMFDB - Embedded newform 1216.3.d.d.191.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(191,1216)]

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

lf = mfeigenbasis(mf)

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")

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

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

Self dual:	no
Analytic conductor:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + \cdots + 16384 $ x^14 - 2x^13 + x^12 + 14x^11 - 42x^10 + 28x^9 + 132x^8 - 440x^7 + 528x^6 + 448x^5 - 2688x^4 + 3584x^3 + 1024x^2 - 8192x + 16384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{25} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.6
Root			$0.645572 - 1.89294i$ of defining polynomial
Character	$\chi$	$=$	1216.191
Dual form			1216.3.d.d.191.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.820457i q^{3} +2.38184 q^{5} +12.3764i q^{7} +8.32685 q^{9} +O(q^{10})$	$q-0.820457i q^{3} +2.38184 q^{5} +12.3764i q^{7} +8.32685 q^{9} +9.15034i q^{11} -0.940214 q^{13} -1.95419i q^{15} +27.1792 q^{17} +4.35890i q^{19} +10.1543 q^{21} +13.8004i q^{23} -19.3269 q^{25} -14.2159i q^{27} -49.8488 q^{29} -24.6752i q^{31} +7.50746 q^{33} +29.4787i q^{35} +27.3757 q^{37} +0.771405i q^{39} -38.4024 q^{41} +41.2108i q^{43} +19.8332 q^{45} +45.1842i q^{47} -104.176 q^{49} -22.2994i q^{51} +19.9964 q^{53} +21.7946i q^{55} +3.57629 q^{57} +34.7054i q^{59} -33.2008 q^{61} +103.057i q^{63} -2.23944 q^{65} -3.48192i q^{67} +11.3226 q^{69} -88.8887i q^{71} -19.8573 q^{73} +15.8568i q^{75} -113.249 q^{77} -51.7431i q^{79} +63.2781 q^{81} +6.62943i q^{83} +64.7365 q^{85} +40.8988i q^{87} +31.5105 q^{89} -11.6365i q^{91} -20.2449 q^{93} +10.3822i q^{95} +159.282 q^{97} +76.1935i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 68 q^{9}+O(q^{10}) $ 14 * q - 68 * q^9	$ 14 q - 68 q^{9} - 54 q^{13} + 34 q^{17} + 38 q^{21} - 86 q^{25} - 54 q^{29} + 20 q^{33} - 100 q^{37} + 224 q^{41} + 168 q^{45} - 220 q^{49} - 14 q^{53} + 38 q^{57} - 28 q^{61} - 472 q^{65} - 122 q^{69} + 70 q^{73} - 228 q^{77} + 334 q^{81} - 48 q^{85} + 176 q^{93} + 308 q^{97}+O(q^{100}) $ 14 * q - 68 * q^9 - 54 * q^13 + 34 * q^17 + 38 * q^21 - 86 * q^25 - 54 * q^29 + 20 * q^33 - 100 * q^37 + 224 * q^41 + 168 * q^45 - 220 * q^49 - 14 * q^53 + 38 * q^57 - 28 * q^61 - 472 * q^65 - 122 * q^69 + 70 * q^73 - 228 * q^77 + 334 * q^81 - 48 * q^85 + 176 * q^93 + 308 * q^97

Display 100 coefficients 10 coefficients

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
$3$	− 0.820457i	− 0.273486i	−0.990607	−	0.136743i	$-0.956337\pi$
$3$	− 0.820457i	− 0.273486i	0.990607	−	0.136743i	$-0.0436634\pi$
$4$	0	0
$5$	2.38184	0.476367	0.238184	−	0.971220i	$-0.423448\pi$
$5$	2.38184	0.476367	0.238184	+	0.971220i	$0.423448\pi$
$6$	0	0
$7$	12.3764i	1.76806i	0.467427	+	0.884032i	$0.345181\pi$
$7$	12.3764i	1.76806i	−0.467427	+	0.884032i	$0.654819\pi$
$8$	0	0
$9$	8.32685	0.925206
$10$	0	0
$11$	9.15034i	0.831849i	0.909399	+	0.415925i	$0.136542\pi$
$11$	9.15034i	0.831849i	−0.909399	+	0.415925i	$0.863458\pi$
$12$	0	0
$13$	−0.940214	−0.0723242	−0.0361621	−	0.999346i	$-0.511513\pi$
$13$	−0.940214	−0.0723242	−0.0361621	+	0.999346i	$0.511513\pi$
$14$	0	0
$15$	− 1.95419i	− 0.130280i
$16$	0	0
$17$	27.1792	1.59878	0.799388	−	0.600814i	$-0.205157\pi$
$17$	27.1792	1.59878	0.799388	+	0.600814i	$0.205157\pi$
$18$	0	0
$19$	4.35890i	0.229416i
$19$	4.35890i	0.229416i
$20$	0	0
$21$	10.1543	0.483540
$22$	0	0
$23$	13.8004i	0.600018i	0.953936	+	0.300009i	$0.0969898\pi$
$23$	13.8004i	0.600018i	−0.953936	+	0.300009i	$0.903010\pi$
$24$	0	0
$25$	−19.3269	−0.773074
$26$	0	0
$27$	− 14.2159i	− 0.526516i
$28$	0	0
$29$	−49.8488	−1.71892	−0.859462	−	0.511200i	$-0.829201\pi$
$29$	−49.8488	−1.71892	−0.859462	+	0.511200i	$0.829201\pi$
$30$	0	0
$31$	− 24.6752i	− 0.795974i	−0.917391	−	0.397987i	$-0.869709\pi$
$31$	− 24.6752i	− 0.795974i	0.917391	−	0.397987i	$-0.130291\pi$
$32$	0	0
$33$	7.50746	0.227499
$34$	0	0
$35$	29.4787i	0.842248i
$36$	0	0
$37$	27.3757	0.739883	0.369942	−	0.929055i	$-0.379378\pi$
$37$	27.3757	0.739883	0.369942	+	0.929055i	$0.379378\pi$
$38$	0	0
$39$	0.771405i	0.0197796i
$40$	0	0
$41$	−38.4024	−0.936644	−0.468322	−	0.883558i	$-0.655141\pi$
$41$	−38.4024	−0.936644	−0.468322	+	0.883558i	$0.655141\pi$
$42$	0	0
$43$	41.2108i	0.958391i	0.877708	+	0.479195i	$0.159071\pi$
$43$	41.2108i	0.958391i	−0.877708	+	0.479195i	$0.840929\pi$
$44$	0	0
$45$	19.8332	0.440738
$46$	0	0
$47$	45.1842i	0.961367i	0.876894	+	0.480683i	$0.159611\pi$
$47$	45.1842i	0.961367i	−0.876894	+	0.480683i	$0.840389\pi$
$48$	0	0
$49$	−104.176	−2.12605
$50$	0	0
$51$	− 22.2994i	− 0.437242i
$52$	0	0
$53$	19.9964	0.377291	0.188646	−	0.982045i	$-0.439590\pi$
$53$	19.9964	0.377291	0.188646	+	0.982045i	$0.439590\pi$
$54$	0	0
$55$	21.7946i	0.396266i
$56$	0	0
$57$	3.57629	0.0627419
$58$	0	0
$59$	34.7054i	0.588228i	0.955770	+	0.294114i	$0.0950245\pi$
$59$	34.7054i	0.588228i	−0.955770	+	0.294114i	$0.904976\pi$
$60$	0	0
$61$	−33.2008	−0.544276	−0.272138	−	0.962258i	$-0.587731\pi$
$61$	−33.2008	−0.544276	−0.272138	+	0.962258i	$0.587731\pi$
$62$	0	0
$63$	103.057i	1.63582i
$64$	0	0
$65$	−2.23944	−0.0344529
$66$	0	0
$67$	− 3.48192i	− 0.0519690i	−0.999662	−	0.0259845i	$-0.991728\pi$
$67$	− 3.48192i	− 0.0519690i	0.999662	−	0.0259845i	$-0.00827205\pi$
$68$	0	0
$69$	11.3226	0.164096
$70$	0	0
$71$	− 88.8887i	− 1.25195i	−0.779842	−	0.625977i	$-0.784700\pi$
$71$	− 88.8887i	− 1.25195i	0.779842	−	0.625977i	$-0.215300\pi$
$72$	0	0
$73$	−19.8573	−0.272018	−0.136009	−	0.990708i	$-0.543428\pi$
$73$	−19.8573	−0.272018	−0.136009	+	0.990708i	$0.543428\pi$
$74$	0	0
$75$	15.8568i	0.211425i
$76$	0	0
$77$	−113.249	−1.47076
$78$	0	0
$79$	− 51.7431i	− 0.654976i	−0.944855	−	0.327488i	$-0.893798\pi$
$79$	− 51.7431i	− 0.654976i	0.944855	−	0.327488i	$-0.106202\pi$
$80$	0	0
$81$	63.2781	0.781211
$82$	0	0
$83$	6.62943i	0.0798727i	0.999202	+	0.0399363i	$0.0127155\pi$
$83$	6.62943i	0.0798727i	−0.999202	+	0.0399363i	$0.987284\pi$
$84$	0	0
$85$	64.7365	0.761605
$86$	0	0
$87$	40.8988i	0.470101i
$88$	0	0
$89$	31.5105	0.354051	0.177025	−	0.984206i	$-0.443353\pi$
$89$	31.5105	0.354051	0.177025	+	0.984206i	$0.443353\pi$
$90$	0	0
$91$	− 11.6365i	− 0.127874i
$92$	0	0
$93$	−20.2449	−0.217687
$94$	0	0
$95$	10.3822i	0.109286i
$96$	0	0
$97$	159.282	1.64208	0.821039	−	0.570873i	$-0.193395\pi$
$97$	159.282	1.64208	0.821039	+	0.570873i	$0.193395\pi$
$98$	0	0
$99$	76.1935i	0.769632i
$100$	0	0
$101$	−35.1573	−0.348092	−0.174046	−	0.984738i	$-0.555684\pi$
$101$	−35.1573	−0.348092	−0.174046	+	0.984738i	$0.555684\pi$
$102$	0	0
$103$	117.214i	1.13800i	0.822339	+	0.568998i	$0.192669\pi$
$103$	117.214i	1.13800i	−0.822339	+	0.568998i	$0.807331\pi$
$104$	0	0
$105$	24.1860	0.230343
$106$	0	0
$107$	191.832i	1.79282i	0.443228	+	0.896409i	$0.353833\pi$
$107$	191.832i	1.79282i	−0.443228	+	0.896409i	$0.646167\pi$
$108$	0	0
$109$	183.807	1.68630	0.843152	−	0.537675i	$-0.180697\pi$
$109$	183.807	1.68630	0.843152	+	0.537675i	$0.180697\pi$
$110$	0	0
$111$	− 22.4606i	− 0.202347i
$112$	0	0
$113$	72.9718	0.645768	0.322884	−	0.946439i	$-0.395348\pi$
$113$	72.9718	0.645768	0.322884	+	0.946439i	$0.395348\pi$
$114$	0	0
$115$	32.8704i	0.285829i
$116$	0	0
$117$	−7.82902	−0.0669147
$118$	0	0
$119$	336.382i	2.82674i
$120$	0	0
$121$	37.2713	0.308027
$122$	0	0
$123$	31.5075i	0.256159i
$124$	0	0
$125$	−105.579	−0.844635
$126$	0	0
$127$	11.4215i	0.0899331i	0.998988	+	0.0449665i	$0.0143181\pi$
$127$	11.4215i	0.0899331i	−0.998988	+	0.0449665i	$0.985682\pi$
$128$	0	0
$129$	33.8117	0.262106
$130$	0	0
$131$	− 36.9512i	− 0.282070i	−0.990005	−	0.141035i	$-0.954957\pi$
$131$	− 36.9512i	− 0.282070i	0.990005	−	0.141035i	$-0.0450430\pi$
$132$	0	0
$133$	−53.9477	−0.405622
$134$	0	0
$135$	− 33.8600i	− 0.250815i
$136$	0	0
$137$	−162.786	−1.18822	−0.594109	−	0.804385i	$-0.702495\pi$
$137$	−162.786	−1.18822	−0.594109	+	0.804385i	$0.702495\pi$
$138$	0	0
$139$	− 229.233i	− 1.64916i	−0.565744	−	0.824581i	$-0.691411\pi$
$139$	− 229.233i	− 1.64916i	0.565744	−	0.824581i	$-0.308589\pi$
$140$	0	0
$141$	37.0717	0.262920
$142$	0	0
$143$	− 8.60328i	− 0.0601628i
$144$	0	0
$145$	−118.732	−0.818839
$146$	0	0
$147$	85.4722i	0.581444i
$148$	0	0
$149$	110.328	0.740459	0.370229	−	0.928940i	$-0.379279\pi$
$149$	110.328	0.740459	0.370229	+	0.928940i	$0.379279\pi$
$150$	0	0
$151$	104.737i	0.693621i	0.937935	+	0.346810i	$0.112735\pi$
$151$	104.737i	0.693621i	−0.937935	+	0.346810i	$0.887265\pi$
$152$	0	0
$153$	226.317	1.47920
$154$	0	0
$155$	− 58.7723i	− 0.379176i
$156$	0	0
$157$	147.695	0.940732	0.470366	−	0.882471i	$-0.344122\pi$
$157$	147.695	0.940732	0.470366	+	0.882471i	$0.344122\pi$
$158$	0	0
$159$	− 16.4062i	− 0.103184i
$160$	0	0
$161$	−170.800	−1.06087
$162$	0	0
$163$	183.104i	1.12334i	0.827363	+	0.561668i	$0.189840\pi$
$163$	183.104i	1.12334i	−0.827363	+	0.561668i	$0.810160\pi$
$164$	0	0
$165$	17.8815	0.108373
$166$	0	0
$167$	− 214.501i	− 1.28444i	−0.766522	−	0.642218i	$-0.778014\pi$
$167$	− 214.501i	− 1.28444i	0.766522	−	0.642218i	$-0.221986\pi$
$168$	0	0
$169$	−168.116	−0.994769
$170$	0	0
$171$	36.2959i	0.212257i
$172$	0	0
$173$	−27.1849	−0.157138	−0.0785692	−	0.996909i	$-0.525035\pi$
$173$	−27.1849	−0.157138	−0.0785692	+	0.996909i	$0.525035\pi$
$174$	0	0
$175$	− 239.198i	− 1.36684i
$176$	0	0
$177$	28.4743	0.160872
$178$	0	0
$179$	185.284i	1.03511i	0.855651	+	0.517553i	$0.173157\pi$
$179$	185.284i	1.03511i	−0.855651	+	0.517553i	$0.826843\pi$
$180$	0	0
$181$	−17.0150	−0.0940053	−0.0470026	−	0.998895i	$-0.514967\pi$
$181$	−17.0150	−0.0940053	−0.0470026	+	0.998895i	$0.514967\pi$
$182$	0	0
$183$	27.2398i	0.148852i
$184$	0	0
$185$	65.2044	0.352456
$186$	0	0
$187$	248.699i	1.32994i
$188$	0	0
$189$	175.943	0.930914
$190$	0	0
$191$	2.41797i	0.0126595i	0.999980	+	0.00632976i	$0.00201484\pi$
$191$	2.41797i	0.0126595i	−0.999980	+	0.00632976i	$0.997985\pi$
$192$	0	0
$193$	−150.712	−0.780892	−0.390446	−	0.920626i	$-0.627679\pi$
$193$	−150.712	−0.780892	−0.390446	+	0.920626i	$0.627679\pi$
$194$	0	0
$195$	1.83736i	0.00942236i
$196$	0	0
$197$	−268.203	−1.36144	−0.680718	−	0.732545i	$-0.738332\pi$
$197$	−268.203	−1.36144	−0.680718	+	0.732545i	$0.738332\pi$
$198$	0	0
$199$	308.615i	1.55083i	0.631452	+	0.775415i	$0.282459\pi$
$199$	308.615i	1.55083i	−0.631452	+	0.775415i	$0.717541\pi$
$200$	0	0
$201$	−2.85676	−0.0142128
$202$	0	0
$203$	− 616.951i	− 3.03917i
$204$	0	0
$205$	−91.4683	−0.446187
$206$	0	0
$207$	114.914i	0.555140i
$208$	0	0
$209$	−39.8854	−0.190839
$210$	0	0
$211$	334.011i	1.58299i	0.611176	+	0.791495i	$0.290697\pi$
$211$	334.011i	1.58299i	−0.611176	+	0.791495i	$0.709303\pi$
$212$	0	0
$213$	−72.9293	−0.342391
$214$	0	0
$215$	98.1574i	0.456546i
$216$	0	0
$217$	305.391	1.40733
$218$	0	0
$219$	16.2920i	0.0743929i
$220$	0	0
$221$	−25.5543	−0.115630
$222$	0	0
$223$	− 290.142i	− 1.30108i	−0.759470	−	0.650542i	$-0.774542\pi$
$223$	− 290.142i	− 1.30108i	0.759470	−	0.650542i	$-0.225458\pi$
$224$	0	0
$225$	−160.932	−0.715252
$226$	0	0
$227$	76.2561i	0.335930i	0.985793	+	0.167965i	$0.0537195\pi$
$227$	76.2561i	0.335930i	−0.985793	+	0.167965i	$0.946280\pi$
$228$	0	0
$229$	212.780	0.929169	0.464585	−	0.885529i	$-0.346204\pi$
$229$	212.780	0.929169	0.464585	+	0.885529i	$0.346204\pi$
$230$	0	0
$231$	92.9156i	0.402232i
$232$	0	0
$233$	−108.504	−0.465680	−0.232840	−	0.972515i	$-0.574802\pi$
$233$	−108.504	−0.465680	−0.232840	+	0.972515i	$0.574802\pi$
$234$	0	0
$235$	107.622i	0.457964i
$236$	0	0
$237$	−42.4530	−0.179127
$238$	0	0
$239$	− 14.5126i	− 0.0607222i	−0.999539	−	0.0303611i	$-0.990334\pi$
$239$	− 14.5126i	− 0.0607222i	0.999539	−	0.0303611i	$-0.00966573\pi$
$240$	0	0
$241$	−278.238	−1.15451	−0.577257	−	0.816563i	$-0.695877\pi$
$241$	−278.238	−1.15451	−0.577257	+	0.816563i	$0.695877\pi$
$242$	0	0
$243$	− 179.860i	− 0.740166i
$244$	0	0
$245$	−248.131	−1.01278
$246$	0	0
$247$	− 4.09830i	− 0.0165923i
$248$	0	0
$249$	5.43916	0.0218440
$250$	0	0
$251$	− 344.252i	− 1.37152i	−0.727828	−	0.685760i	$-0.759470\pi$
$251$	− 344.252i	− 1.37152i	0.727828	−	0.685760i	$-0.240530\pi$
$252$	0	0
$253$	−126.279	−0.499125
$254$	0	0
$255$	− 53.1134i	− 0.208288i
$256$	0	0
$257$	50.0048	0.194571	0.0972856	−	0.995257i	$-0.468984\pi$
$257$	50.0048	0.194571	0.0972856	+	0.995257i	$0.468984\pi$
$258$	0	0
$259$	338.814i	1.30816i
$260$	0	0
$261$	−415.084	−1.59036
$262$	0	0
$263$	155.273i	0.590391i	0.955437	+	0.295195i	$0.0953848\pi$
$263$	155.273i	0.590391i	−0.955437	+	0.295195i	$0.904615\pi$
$264$	0	0
$265$	47.6283	0.179729
$266$	0	0
$267$	− 25.8530i	− 0.0968277i
$268$	0	0
$269$	457.281	1.69993	0.849964	−	0.526840i	$-0.176623\pi$
$269$	457.281	1.69993	0.849964	+	0.526840i	$0.176623\pi$
$270$	0	0
$271$	349.279i	1.28885i	0.764666	+	0.644427i	$0.222904\pi$
$271$	349.279i	1.28885i	−0.764666	+	0.644427i	$0.777096\pi$
$272$	0	0
$273$	−9.54725	−0.0349716
$274$	0	0
$275$	− 176.847i	− 0.643081i
$276$	0	0
$277$	108.548	0.391870	0.195935	−	0.980617i	$-0.437226\pi$
$277$	108.548	0.391870	0.195935	+	0.980617i	$0.437226\pi$
$278$	0	0
$279$	− 205.467i	− 0.736440i
$280$	0	0
$281$	190.452	0.677767	0.338883	−	0.940828i	$-0.389951\pi$
$281$	190.452	0.677767	0.338883	+	0.940828i	$0.389951\pi$
$282$	0	0
$283$	− 289.950i	− 1.02456i	−0.858819	−	0.512279i	$-0.828802\pi$
$283$	− 289.950i	− 1.02456i	0.858819	−	0.512279i	$-0.171198\pi$
$284$	0	0
$285$	8.51813	0.0298882
$286$	0	0
$287$	− 475.285i	− 1.65605i
$288$	0	0
$289$	449.709	1.55609
$290$	0	0
$291$	− 130.684i	− 0.449084i
$292$	0	0
$293$	177.881	0.607102	0.303551	−	0.952815i	$-0.401828\pi$
$293$	177.881	0.607102	0.303551	+	0.952815i	$0.401828\pi$
$294$	0	0
$295$	82.6627i	0.280212i
$296$	0	0
$297$	130.081	0.437982
$298$	0	0
$299$	− 12.9754i	− 0.0433958i
$300$	0	0
$301$	−510.043	−1.69450
$302$	0	0
$303$	28.8450i	0.0951981i
$304$	0	0
$305$	−79.0790	−0.259275
$306$	0	0
$307$	− 554.295i	− 1.80552i	−0.430143	−	0.902761i	$-0.641537\pi$
$307$	− 554.295i	− 1.80552i	0.430143	−	0.902761i	$-0.358463\pi$
$308$	0	0
$309$	96.1686	0.311225
$310$	0	0
$311$	− 67.7563i	− 0.217866i	−0.994049	−	0.108933i	$-0.965257\pi$
$311$	− 67.7563i	− 0.217866i	0.994049	−	0.108933i	$-0.0347434\pi$
$312$	0	0
$313$	447.248	1.42891	0.714454	−	0.699683i	$-0.246675\pi$
$313$	447.248	1.42891	0.714454	+	0.699683i	$0.246675\pi$
$314$	0	0
$315$	245.465i	0.779253i
$316$	0	0
$317$	−67.2154	−0.212036	−0.106018	−	0.994364i	$-0.533810\pi$
$317$	−67.2154	−0.212036	−0.106018	+	0.994364i	$0.533810\pi$
$318$	0	0
$319$	− 456.133i	− 1.42989i
$320$	0	0
$321$	157.389	0.490310
$322$	0	0
$323$	118.471i	0.366785i
$324$	0	0
$325$	18.1714	0.0559119
$326$	0	0
$327$	− 150.806i	− 0.461180i
$328$	0	0
$329$	−559.220	−1.69976
$330$	0	0
$331$	− 178.449i	− 0.539122i	−0.962983	−	0.269561i	$-0.913121\pi$
$331$	− 178.449i	− 0.539122i	0.962983	−	0.269561i	$-0.0868785\pi$
$332$	0	0
$333$	227.953	0.684544
$334$	0	0
$335$	− 8.29337i	− 0.0247563i
$336$	0	0
$337$	−280.146	−0.831292	−0.415646	−	0.909526i	$-0.636445\pi$
$337$	−280.146	−0.831292	−0.415646	+	0.909526i	$0.636445\pi$
$338$	0	0
$339$	− 59.8702i	− 0.176608i
$340$	0	0
$341$	225.787	0.662131
$342$	0	0
$343$	− 682.888i	− 1.99093i
$344$	0	0
$345$	26.9687	0.0781702
$346$	0	0
$347$	− 89.6003i	− 0.258214i	−0.991631	−	0.129107i	$-0.958789\pi$
$347$	− 89.6003i	− 0.258214i	0.991631	−	0.129107i	$-0.0412111\pi$
$348$	0	0
$349$	−5.38420	−0.0154275	−0.00771376	−	0.999970i	$-0.502455\pi$
$349$	−5.38420	−0.0154275	−0.00771376	+	0.999970i	$0.502455\pi$
$350$	0	0
$351$	13.3660i	0.0380798i
$352$	0	0
$353$	357.844	1.01372	0.506861	−	0.862028i	$-0.330806\pi$
$353$	357.844	1.01372	0.506861	+	0.862028i	$0.330806\pi$
$354$	0	0
$355$	− 211.718i	− 0.596390i
$356$	0	0
$357$	275.987	0.773072
$358$	0	0
$359$	− 415.792i	− 1.15820i	−0.815258	−	0.579098i	$-0.803405\pi$
$359$	− 415.792i	− 1.15820i	0.815258	−	0.579098i	$-0.196595\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	− 30.5795i	− 0.0842409i
$364$	0	0
$365$	−47.2968	−0.129580
$366$	0	0
$367$	262.695i	0.715789i	0.933762	+	0.357894i	$0.116505\pi$
$367$	262.695i	0.715789i	−0.933762	+	0.357894i	$0.883495\pi$
$368$	0	0
$369$	−319.771	−0.866588
$370$	0	0
$371$	247.485i	0.667075i
$372$	0	0
$373$	−158.344	−0.424516	−0.212258	−	0.977214i	$-0.568082\pi$
$373$	−158.344	−0.424516	−0.212258	+	0.977214i	$0.568082\pi$
$374$	0	0
$375$	86.6233i	0.230995i
$376$	0	0
$377$	46.8685	0.124320
$378$	0	0
$379$	− 539.136i	− 1.42252i	−0.702928	−	0.711261i	$-0.748124\pi$
$379$	− 539.136i	− 1.42252i	0.702928	−	0.711261i	$-0.251876\pi$
$380$	0	0
$381$	9.37084	0.0245954
$382$	0	0
$383$	− 422.810i	− 1.10394i	−0.833863	−	0.551971i	$-0.813876\pi$
$383$	− 422.810i	− 1.10394i	0.833863	−	0.551971i	$-0.186124\pi$
$384$	0	0
$385$	−269.740	−0.700623
$386$	0	0
$387$	343.156i	0.886709i
$388$	0	0
$389$	487.091	1.25216	0.626081	−	0.779758i	$-0.284658\pi$
$389$	487.091	1.25216	0.626081	+	0.779758i	$0.284658\pi$
$390$	0	0
$391$	375.085i	0.959296i
$392$	0	0
$393$	−30.3168	−0.0771420
$394$	0	0
$395$	− 123.244i	− 0.312009i
$396$	0	0
$397$	646.684	1.62893	0.814464	−	0.580215i	$-0.197031\pi$
$397$	646.684	1.62893	0.814464	+	0.580215i	$0.197031\pi$
$398$	0	0
$399$	44.2617i	0.110932i
$400$	0	0
$401$	−14.0555	−0.0350512	−0.0175256	−	0.999846i	$-0.505579\pi$
$401$	−14.0555	−0.0350512	−0.0175256	+	0.999846i	$0.505579\pi$
$402$	0	0
$403$	23.2000i	0.0575682i
$404$	0	0
$405$	150.718	0.372144
$406$	0	0
$407$	250.497i	0.615471i
$408$	0	0
$409$	143.668	0.351266	0.175633	−	0.984456i	$-0.443803\pi$
$409$	143.668	0.351266	0.175633	+	0.984456i	$0.443803\pi$
$410$	0	0
$411$	133.559i	0.324960i
$412$	0	0
$413$	−429.530	−1.04002
$414$	0	0
$415$	15.7902i	0.0380488i
$416$	0	0
$417$	−188.076	−0.451022
$418$	0	0
$419$	572.449i	1.36623i	0.730312	+	0.683114i	$0.239375\pi$
$419$	572.449i	1.36623i	−0.730312	+	0.683114i	$0.760625\pi$
$420$	0	0
$421$	−718.562	−1.70680	−0.853399	−	0.521258i	$-0.825463\pi$
$421$	−718.562	−1.70680	−0.853399	+	0.521258i	$0.825463\pi$
$422$	0	0
$423$	376.242i	0.889462i
$424$	0	0
$425$	−525.289	−1.23597
$426$	0	0
$427$	− 410.908i	− 0.962314i
$428$	0	0
$429$	−7.05862	−0.0164537
$430$	0	0
$431$	− 252.198i	− 0.585147i	−0.956243	−	0.292573i	$-0.905488\pi$
$431$	− 252.198i	− 0.585147i	0.956243	−	0.292573i	$-0.0945116\pi$
$432$	0	0
$433$	239.008	0.551982	0.275991	−	0.961160i	$-0.410994\pi$
$433$	239.008	0.551982	0.275991	+	0.961160i	$0.410994\pi$
$434$	0	0
$435$	97.4142i	0.223941i
$436$	0	0
$437$	−60.1547	−0.137654
$438$	0	0
$439$	17.2750i	0.0393508i	0.999806	+	0.0196754i	$0.00626327\pi$
$439$	17.2750i	0.0393508i	−0.999806	+	0.0196754i	$0.993737\pi$
$440$	0	0
$441$	−867.461	−1.96703
$442$	0	0
$443$	− 161.249i	− 0.363993i	−0.983299	−	0.181997i	$-0.941744\pi$
$443$	− 161.249i	− 0.363993i	0.983299	−	0.181997i	$-0.0582560\pi$
$444$	0	0
$445$	75.0529	0.168658
$446$	0	0
$447$	− 90.5196i	− 0.202505i
$448$	0	0
$449$	561.952	1.25156	0.625782	−	0.779998i	$-0.284780\pi$
$449$	561.952	1.25156	0.625782	+	0.779998i	$0.284780\pi$
$450$	0	0
$451$	− 351.395i	− 0.779147i
$452$	0	0
$453$	85.9319	0.189695
$454$	0	0
$455$	− 27.7163i	− 0.0609149i
$456$	0	0
$457$	−107.375	−0.234957	−0.117479	−	0.993075i	$-0.537481\pi$
$457$	−107.375	−0.234957	−0.117479	+	0.993075i	$0.537481\pi$
$458$	0	0
$459$	− 386.378i	− 0.841781i
$460$	0	0
$461$	−666.303	−1.44534	−0.722672	−	0.691191i	$-0.757086\pi$
$461$	−666.303	−1.44534	−0.722672	+	0.691191i	$0.757086\pi$
$462$	0	0
$463$	− 272.358i	− 0.588246i	−0.955768	−	0.294123i	$-0.904972\pi$
$463$	− 272.358i	− 0.588246i	0.955768	−	0.294123i	$-0.0950276\pi$
$464$	0	0
$465$	−48.2201	−0.103699
$466$	0	0
$467$	− 703.386i	− 1.50618i	−0.657917	−	0.753090i	$-0.728562\pi$
$467$	− 703.386i	− 1.50618i	0.657917	−	0.753090i	$-0.271438\pi$
$468$	0	0
$469$	43.0938	0.0918844
$470$	0	0
$471$	− 121.177i	− 0.257277i
$472$	0	0
$473$	−377.093	−0.797237
$474$	0	0
$475$	− 84.2438i	− 0.177355i
$476$	0	0
$477$	166.507	0.349072
$478$	0	0
$479$	− 451.971i	− 0.943572i	−0.881713	−	0.471786i	$-0.843609\pi$
$479$	− 451.971i	− 0.943572i	0.881713	−	0.471786i	$-0.156391\pi$
$480$	0	0
$481$	−25.7390	−0.0535114
$482$	0	0
$483$	140.134i	0.290133i
$484$	0	0
$485$	379.383	0.782232
$486$	0	0
$487$	293.894i	0.603478i	0.953391	+	0.301739i	$0.0975671\pi$
$487$	293.894i	0.603478i	−0.953391	+	0.301739i	$0.902433\pi$
$488$	0	0
$489$	150.229	0.307216
$490$	0	0
$491$	91.3499i	0.186049i	0.995664	+	0.0930243i	$0.0296534\pi$
$491$	91.3499i	0.186049i	−0.995664	+	0.0930243i	$0.970347\pi$
$492$	0	0
$493$	−1354.85	−2.74818
$494$	0	0
$495$	181.481i	0.366627i
$496$	0	0
$497$	1100.13	2.21353
$498$	0	0
$499$	758.796i	1.52063i	0.649553	+	0.760316i	$0.274956\pi$
$499$	758.796i	1.52063i	−0.649553	+	0.760316i	$0.725044\pi$
$500$	0	0
$501$	−175.989	−0.351275
$502$	0	0
$503$	− 55.4556i	− 0.110250i	−0.998479	−	0.0551248i	$-0.982444\pi$
$503$	− 55.4556i	− 0.110250i	0.998479	−	0.0551248i	$-0.0175557\pi$
$504$	0	0
$505$	−83.7389	−0.165820
$506$	0	0
$507$	137.932i	0.272055i
$508$	0	0
$509$	479.828	0.942688	0.471344	−	0.881949i	$-0.343769\pi$
$509$	479.828	0.942688	0.471344	+	0.881949i	$0.343769\pi$
$510$	0	0
$511$	− 245.763i	− 0.480945i
$512$	0	0
$513$	61.9658	0.120791
$514$	0	0
$515$	279.184i	0.542104i
$516$	0	0
$517$	−413.451	−0.799712
$518$	0	0
$519$	22.3041i	0.0429751i
$520$	0	0
$521$	29.8875	0.0573656	0.0286828	−	0.999589i	$-0.490869\pi$
$521$	29.8875	0.0573656	0.0286828	+	0.999589i	$0.490869\pi$
$522$	0	0
$523$	324.051i	0.619600i	0.950802	+	0.309800i	$0.100262\pi$
$523$	324.051i	0.619600i	−0.950802	+	0.309800i	$0.899738\pi$
$524$	0	0
$525$	−196.251	−0.373812
$526$	0	0
$527$	− 670.652i	− 1.27259i
$528$	0	0
$529$	338.548	0.639978
$530$	0	0
$531$	288.987i	0.544232i
$532$	0	0
$533$	36.1065	0.0677420
$534$	0	0
$535$	456.912i	0.854040i
$536$	0	0
$537$	152.018	0.283087
$538$	0	0
$539$	− 953.250i	− 1.76855i
$540$	0	0
$541$	584.274	1.07999	0.539994	−	0.841669i	$-0.318426\pi$
$541$	584.274	1.07999	0.539994	+	0.841669i	$0.318426\pi$
$542$	0	0
$543$	13.9600i	0.0257091i
$544$	0	0
$545$	437.799	0.803301
$546$	0	0
$547$	− 173.706i	− 0.317562i	−0.987314	−	0.158781i	$-0.949244\pi$
$547$	− 173.706i	− 0.317562i	0.987314	−	0.158781i	$-0.0507564\pi$
$548$	0	0
$549$	−276.458	−0.503567
$550$	0	0
$551$	− 217.286i	− 0.394348i
$552$	0	0
$553$	640.396	1.15804
$554$	0	0
$555$	− 53.4974i	− 0.0963917i
$556$	0	0
$557$	590.355	1.05988	0.529942	−	0.848034i	$-0.322214\pi$
$557$	590.355	1.05988	0.529942	+	0.848034i	$0.322214\pi$
$558$	0	0
$559$	− 38.7470i	− 0.0693148i
$560$	0	0
$561$	204.047	0.363720
$562$	0	0
$563$	− 1015.03i	− 1.80290i	−0.432882	−	0.901450i	$-0.642503\pi$
$563$	− 1015.03i	− 1.80290i	0.432882	−	0.901450i	$-0.357497\pi$
$564$	0	0
$565$	173.807	0.307623
$566$	0	0
$567$	783.158i	1.38123i
$568$	0	0
$569$	−98.2869	−0.172736	−0.0863681	−	0.996263i	$-0.527526\pi$
$569$	−98.2869	−0.172736	−0.0863681	+	0.996263i	$0.527526\pi$
$570$	0	0
$571$	− 366.213i	− 0.641354i	−0.947189	−	0.320677i	$-0.896090\pi$
$571$	− 366.213i	− 0.641354i	0.947189	−	0.320677i	$-0.103910\pi$
$572$	0	0
$573$	1.98384	0.00346219
$574$	0	0
$575$	− 266.719i	− 0.463859i
$576$	0	0
$577$	−267.138	−0.462977	−0.231489	−	0.972838i	$-0.574360\pi$
$577$	−267.138	−0.462977	−0.231489	+	0.972838i	$0.574360\pi$
$578$	0	0
$579$	123.653i	0.213563i
$580$	0	0
$581$	−82.0488	−0.141220
$582$	0	0
$583$	182.974i	0.313850i
$584$	0	0
$585$	−18.6475	−0.0318760
$586$	0	0
$587$	− 245.452i	− 0.418147i	−0.977900	−	0.209074i	$-0.932955\pi$
$587$	− 245.452i	− 0.418147i	0.977900	−	0.209074i	$-0.0670448\pi$
$588$	0	0
$589$	107.557	0.182609
$590$	0	0
$591$	220.049i	0.372333i
$592$	0	0
$593$	33.3697	0.0562726	0.0281363	−	0.999604i	$-0.491043\pi$
$593$	33.3697	0.0562726	0.0281363	+	0.999604i	$0.491043\pi$
$594$	0	0
$595$	801.207i	1.34657i
$596$	0	0
$597$	253.205	0.424129
$598$	0	0
$599$	− 427.260i	− 0.713288i	−0.934240	−	0.356644i	$-0.883921\pi$
$599$	− 427.260i	− 0.713288i	0.934240	−	0.356644i	$-0.116079\pi$
$600$	0	0
$601$	−316.625	−0.526831	−0.263415	−	0.964683i	$-0.584849\pi$
$601$	−316.625	−0.526831	−0.263415	+	0.964683i	$0.584849\pi$
$602$	0	0
$603$	− 28.9934i	− 0.0480820i
$604$	0	0
$605$	88.7741	0.146734
$606$	0	0
$607$	− 467.840i	− 0.770741i	−0.922762	−	0.385370i	$-0.874074\pi$
$607$	− 467.840i	− 0.770741i	0.922762	−	0.385370i	$-0.125926\pi$
$608$	0	0
$609$	−506.181	−0.831168
$610$	0	0
$611$	− 42.4829i	− 0.0695300i
$612$	0	0
$613$	636.118	1.03771	0.518856	−	0.854861i	$-0.326358\pi$
$613$	636.118	1.03771	0.518856	+	0.854861i	$0.326358\pi$
$614$	0	0
$615$	75.0458i	0.122026i
$616$	0	0
$617$	331.487	0.537256	0.268628	−	0.963244i	$-0.413430\pi$
$617$	331.487	0.537256	0.268628	+	0.963244i	$0.413430\pi$
$618$	0	0
$619$	479.724i	0.774999i	0.921870	+	0.387500i	$0.126661\pi$
$619$	479.724i	0.774999i	−0.921870	+	0.387500i	$0.873339\pi$
$620$	0	0
$621$	196.186	0.315919
$622$	0	0
$623$	389.988i	0.625984i
$624$	0	0
$625$	231.698	0.370718
$626$	0	0
$627$	32.7242i	0.0521918i
$628$	0	0
$629$	744.049	1.18291
$630$	0	0
$631$	389.892i	0.617896i	0.951079	+	0.308948i	$0.0999768\pi$
$631$	389.892i	0.617896i	−0.951079	+	0.308948i	$0.900023\pi$
$632$	0	0
$633$	274.041	0.432925
$634$	0	0
$635$	27.2042i	0.0428412i
$636$	0	0
$637$	97.9481	0.153765
$638$	0	0
$639$	− 740.163i	− 1.15831i
$640$	0	0
$641$	−170.264	−0.265623	−0.132811	−	0.991141i	$-0.542400\pi$
$641$	−170.264	−0.265623	−0.132811	+	0.991141i	$0.542400\pi$
$642$	0	0
$643$	420.651i	0.654200i	0.944990	+	0.327100i	$0.106071\pi$
$643$	420.651i	0.654200i	−0.944990	+	0.327100i	$0.893929\pi$
$644$	0	0
$645$	80.5339	0.124859
$646$	0	0
$647$	178.658i	0.276133i	0.990423	+	0.138067i	$0.0440888\pi$
$647$	178.658i	0.276133i	−0.990423	+	0.138067i	$0.955911\pi$
$648$	0	0
$649$	−317.566	−0.489317
$650$	0	0
$651$	− 250.560i	− 0.384885i
$652$	0	0
$653$	950.370	1.45539	0.727696	−	0.685900i	$-0.240591\pi$
$653$	950.370	1.45539	0.727696	+	0.685900i	$0.240591\pi$
$654$	0	0
$655$	− 88.0117i	− 0.134369i
$656$	0	0
$657$	−165.349	−0.251672
$658$	0	0
$659$	327.948i	0.497645i	0.968549	+	0.248823i	$0.0800436\pi$
$659$	327.948i	0.497645i	−0.968549	+	0.248823i	$0.919956\pi$
$660$	0	0
$661$	−256.403	−0.387902	−0.193951	−	0.981011i	$-0.562130\pi$
$661$	−256.403	−0.387902	−0.193951	+	0.981011i	$0.562130\pi$
$662$	0	0
$663$	20.9662i	0.0316232i
$664$	0	0
$665$	−128.495	−0.193225
$666$	0	0
$667$	− 687.935i	− 1.03139i
$668$	0	0
$669$	−238.049	−0.355828
$670$	0	0
$671$	− 303.799i	− 0.452755i
$672$	0	0
$673$	−624.347	−0.927707	−0.463853	−	0.885912i	$-0.653534\pi$
$673$	−624.347	−0.927707	−0.463853	+	0.885912i	$0.653534\pi$
$674$	0	0
$675$	274.749i	0.407036i
$676$	0	0
$677$	135.259	0.199791	0.0998955	−	0.994998i	$-0.468149\pi$
$677$	135.259	0.199791	0.0998955	+	0.994998i	$0.468149\pi$
$678$	0	0
$679$	1971.34i	2.90330i
$680$	0	0
$681$	62.5648	0.0918720
$682$	0	0
$683$	− 84.9635i	− 0.124398i	−0.998064	−	0.0621988i	$-0.980189\pi$
$683$	− 84.9635i	− 0.124398i	0.998064	−	0.0621988i	$-0.0198113\pi$
$684$	0	0
$685$	−387.729	−0.566028
$686$	0	0
$687$	− 174.577i	− 0.254114i
$688$	0	0
$689$	−18.8009	−0.0272873
$690$	0	0
$691$	673.647i	0.974887i	0.873155	+	0.487443i	$0.162070\pi$
$691$	673.647i	0.974887i	−0.873155	+	0.487443i	$0.837930\pi$
$692$	0	0
$693$	−943.005	−1.36076
$694$	0	0
$695$	− 545.997i	− 0.785607i
$696$	0	0
$697$	−1043.75	−1.49749
$698$	0	0
$699$	89.0225i	0.127357i
$700$	0	0
$701$	−406.227	−0.579496	−0.289748	−	0.957103i	$-0.593571\pi$
$701$	−406.227	−0.579496	−0.289748	+	0.957103i	$0.593571\pi$
$702$	0	0
$703$	119.328i	0.169741i
$704$	0	0
$705$	88.2988	0.125246
$706$	0	0
$707$	− 435.122i	− 0.615448i
$708$	0	0
$709$	−668.985	−0.943561	−0.471780	−	0.881716i	$-0.656389\pi$
$709$	−668.985	−0.943561	−0.471780	+	0.881716i	$0.656389\pi$
$710$	0	0
$711$	− 430.857i	− 0.605988i
$712$	0	0
$713$	340.528	0.477599
$714$	0	0
$715$	− 20.4916i	− 0.0286596i
$716$	0	0
$717$	−11.9070	−0.0166066
$718$	0	0
$719$	− 357.444i	− 0.497140i	−0.968614	−	0.248570i	$-0.920039\pi$
$719$	− 357.444i	− 0.497140i	0.968614	−	0.248570i	$-0.0799606\pi$
$720$	0	0
$721$	−1450.69	−2.01205
$722$	0	0
$723$	228.282i	0.315743i
$724$	0	0
$725$	963.420	1.32886
$726$	0	0
$727$	− 931.519i	− 1.28132i	−0.767825	−	0.640659i	$-0.778661\pi$
$727$	− 931.519i	− 1.28132i	0.767825	−	0.640659i	$-0.221339\pi$
$728$	0	0
$729$	421.935	0.578787
$730$	0	0
$731$	1120.08i	1.53225i
$732$	0	0
$733$	342.315	0.467006	0.233503	−	0.972356i	$-0.424981\pi$
$733$	342.315	0.467006	0.233503	+	0.972356i	$0.424981\pi$
$734$	0	0
$735$	203.581i	0.276981i
$736$	0	0
$737$	31.8608	0.0432303
$738$	0	0
$739$	− 233.486i	− 0.315949i	−0.987443	−	0.157975i	$-0.949504\pi$
$739$	− 233.486i	− 0.315949i	0.987443	−	0.157975i	$-0.0504964\pi$
$740$	0	0
$741$	−3.36248	−0.00453775
$742$	0	0
$743$	684.601i	0.921401i	0.887556	+	0.460701i	$0.152402\pi$
$743$	684.601i	0.921401i	−0.887556	+	0.460701i	$0.847598\pi$
$744$	0	0
$745$	262.784	0.352730
$746$	0	0
$747$	55.2023i	0.0738987i
$748$	0	0
$749$	−2374.19	−3.16982
$750$	0	0
$751$	985.017i	1.31161i	0.754931	+	0.655804i	$0.227670\pi$
$751$	985.017i	1.31161i	−0.754931	+	0.655804i	$0.772330\pi$
$752$	0	0
$753$	−282.444	−0.375091
$754$	0	0
$755$	249.466i	0.330418i
$756$	0	0
$757$	−1162.89	−1.53619	−0.768094	−	0.640338i	$-0.778794\pi$
$757$	−1162.89	−1.53619	−0.768094	+	0.640338i	$0.778794\pi$
$758$	0	0
$759$	103.606i	0.136503i
$760$	0	0
$761$	−460.254	−0.604802	−0.302401	−	0.953181i	$-0.597788\pi$
$761$	−460.254	−0.604802	−0.302401	+	0.953181i	$0.597788\pi$
$762$	0	0
$763$	2274.88i	2.98149i
$764$	0	0
$765$	539.051	0.704642
$766$	0	0
$767$	− 32.6305i	− 0.0425431i
$768$	0	0
$769$	757.145	0.984584	0.492292	−	0.870430i	$-0.336159\pi$
$769$	757.145	0.984584	0.492292	+	0.870430i	$0.336159\pi$
$770$	0	0
$771$	− 41.0268i	− 0.0532124i
$772$	0	0
$773$	−1071.48	−1.38614	−0.693068	−	0.720872i	$-0.743742\pi$
$773$	−1071.48	−1.38614	−0.693068	+	0.720872i	$0.743742\pi$
$774$	0	0
$775$	476.894i	0.615347i
$776$	0	0
$777$	277.982	0.357763
$778$	0	0
$779$	− 167.392i	− 0.214881i
$780$	0	0
$781$	813.362	1.04144
$782$	0	0
$783$	708.647i	0.905041i
$784$	0	0
$785$	351.785	0.448134
$786$	0	0
$787$	625.691i	0.795033i	0.917595	+	0.397517i	$0.130128\pi$
$787$	625.691i	0.795033i	−0.917595	+	0.397517i	$0.869872\pi$
$788$	0	0
$789$	127.395	0.161463
$790$	0	0
$791$	903.132i	1.14176i
$792$	0	0
$793$	31.2159	0.0393643
$794$	0	0
$795$	− 39.0769i	− 0.0491534i
$796$	0	0
$797$	549.357	0.689281	0.344641	−	0.938735i	$-0.388001\pi$
$797$	549.357	0.689281	0.344641	+	0.938735i	$0.388001\pi$
$798$	0	0
$799$	1228.07i	1.53701i
$800$	0	0
$801$	262.383	0.327570
$802$	0	0
$803$	− 181.701i	− 0.226278i
$804$	0	0
$805$	−406.818	−0.505364
$806$	0	0
$807$	− 375.179i	− 0.464906i
$808$	0	0
$809$	−7.38519	−0.00912879	−0.00456440	−	0.999990i	$-0.501453\pi$
$809$	−7.38519	−0.00912879	−0.00456440	+	0.999990i	$0.501453\pi$
$810$	0	0
$811$	1100.76i	1.35729i	0.734466	+	0.678645i	$0.237433\pi$
$811$	1100.76i	1.35729i	−0.734466	+	0.678645i	$0.762567\pi$
$812$	0	0
$813$	286.569	0.352483
$814$	0	0
$815$	436.123i	0.535121i
$816$	0	0
$817$	−179.634	−0.219870
$818$	0	0
$819$	− 96.8955i	− 0.118309i
$820$	0	0
$821$	−80.5795	−0.0981480	−0.0490740	−	0.998795i	$-0.515627\pi$
$821$	−80.5795	−0.0981480	−0.0490740	+	0.998795i	$0.515627\pi$
$822$	0	0
$823$	− 980.615i	− 1.19151i	−0.803165	−	0.595756i	$-0.796852\pi$
$823$	− 980.615i	− 1.19151i	0.803165	−	0.595756i	$-0.203148\pi$
$824$	0	0
$825$	−145.096	−0.175873
$826$	0	0
$827$	− 1517.68i	− 1.83516i	−0.397553	−	0.917579i	$-0.630141\pi$
$827$	− 1517.68i	− 1.83516i	0.397553	−	0.917579i	$-0.369859\pi$
$828$	0	0
$829$	229.487	0.276823	0.138412	−	0.990375i	$-0.455800\pi$
$829$	229.487	0.276823	0.138412	+	0.990375i	$0.455800\pi$
$830$	0	0
$831$	− 89.0589i	− 0.107171i
$832$	0	0
$833$	−2831.43	−3.39908
$834$	0	0
$835$	− 510.906i	− 0.611863i
$836$	0	0
$837$	−350.781	−0.419093
$838$	0	0
$839$	949.272i	1.13143i	0.824600	+	0.565716i	$0.191400\pi$
$839$	949.272i	1.13143i	−0.824600	+	0.565716i	$0.808600\pi$
$840$	0	0
$841$	1643.90	1.95470
$842$	0	0
$843$	− 156.258i	− 0.185359i
$844$	0	0
$845$	−400.425	−0.473876
$846$	0	0
$847$	461.286i	0.544611i
$848$	0	0
$849$	−237.891	−0.280202
$850$	0	0
$851$	377.796i	0.443944i
$852$	0	0
$853$	−245.110	−0.287350	−0.143675	−	0.989625i	$-0.545892\pi$
$853$	−245.110	−0.287350	−0.143675	+	0.989625i	$0.545892\pi$
$854$	0	0
$855$	86.4509i	0.101112i
$856$	0	0
$857$	659.731	0.769814	0.384907	−	0.922955i	$-0.374234\pi$
$857$	659.731	0.769814	0.384907	+	0.922955i	$0.374234\pi$
$858$	0	0
$859$	1240.81i	1.44449i	0.691639	+	0.722244i	$0.256889\pi$
$859$	1240.81i	1.44449i	−0.691639	+	0.722244i	$0.743111\pi$
$860$	0	0
$861$	−389.951	−0.452905
$862$	0	0
$863$	1063.44i	1.23226i	0.787646	+	0.616129i	$0.211300\pi$
$863$	1063.44i	1.23226i	−0.787646	+	0.616129i	$0.788700\pi$
$864$	0	0
$865$	−64.7501	−0.0748556
$866$	0	0
$867$	− 368.967i	− 0.425568i
$868$	0	0
$869$	473.467	0.544841
$870$	0	0
$871$	3.27375i	0.00375861i
$872$	0	0
$873$	1326.31	1.51926
$874$	0	0
$875$	− 1306.70i	− 1.49337i
$876$	0	0
$877$	867.685	0.989379	0.494689	−	0.869070i	$-0.335282\pi$
$877$	867.685	0.989379	0.494689	+	0.869070i	$0.335282\pi$
$878$	0	0
$879$	− 145.944i	− 0.166034i
$880$	0	0
$881$	1540.22	1.74826	0.874132	−	0.485688i	$-0.161431\pi$
$881$	1540.22	1.74826	0.874132	+	0.485688i	$0.161431\pi$
$882$	0	0
$883$	− 590.651i	− 0.668914i	−0.942411	−	0.334457i	$-0.891447\pi$
$883$	− 590.651i	− 0.668914i	0.942411	−	0.334457i	$-0.108553\pi$
$884$	0	0
$885$	67.8211	0.0766341
$886$	0	0
$887$	− 336.467i	− 0.379332i	−0.981849	−	0.189666i	$-0.939259\pi$
$887$	− 336.467i	− 0.379332i	0.981849	−	0.189666i	$-0.0607405\pi$
$888$	0	0
$889$	−141.358	−0.159007
$890$	0	0
$891$	579.016i	0.649850i
$892$	0	0
$893$	−196.954	−0.220553
$894$	0	0
$895$	441.316i	0.493091i
$896$	0	0
$897$	−10.6457	−0.0118681
$898$	0	0
$899$	1230.03i	1.36822i
$900$	0	0
$901$	543.488	0.603205
$902$	0	0
$903$	418.468i	0.463420i
$904$	0	0
$905$	−40.5269	−0.0447811
$906$	0	0
$907$	850.338i	0.937528i	0.883323	+	0.468764i	$0.155301\pi$
$907$	850.338i	0.937528i	−0.883323	+	0.468764i	$0.844699\pi$
$908$	0	0
$909$	−292.749	−0.322056
$910$	0	0
$911$	− 1658.66i	− 1.82070i	−0.413837	−	0.910351i	$-0.635812\pi$
$911$	− 1658.66i	− 1.82070i	0.413837	−	0.910351i	$-0.364188\pi$
$912$	0	0
$913$	−60.6616	−0.0664420
$914$	0	0
$915$	64.8809i	0.0709080i
$916$	0	0
$917$	457.324	0.498718
$918$	0	0
$919$	684.608i	0.744948i	0.928043	+	0.372474i	$0.121490\pi$
$919$	684.608i	0.744948i	−0.928043	+	0.372474i	$0.878510\pi$
$920$	0	0
$921$	−454.775	−0.493784
$922$	0	0
$923$	83.5744i	0.0905465i
$924$	0	0
$925$	−529.086	−0.571985
$926$	0	0
$927$	976.020i	1.05288i
$928$	0	0
$929$	−191.705	−0.206356	−0.103178	−	0.994663i	$-0.532901\pi$
$929$	−191.705	−0.206356	−0.103178	+	0.994663i	$0.532901\pi$
$930$	0	0
$931$	− 454.094i	− 0.487749i
$932$	0	0
$933$	−55.5911	−0.0595832
$934$	0	0
$935$	592.361i	0.633541i
$936$	0	0
$937$	1030.61	1.09990	0.549951	−	0.835197i	$-0.314646\pi$
$937$	1030.61	1.09990	0.549951	+	0.835197i	$0.314646\pi$
$938$	0	0
$939$	− 366.948i	− 0.390785i
$940$	0	0
$941$	551.115	0.585669	0.292835	−	0.956163i	$-0.405401\pi$
$941$	551.115	0.585669	0.292835	+	0.956163i	$0.405401\pi$
$942$	0	0
$943$	− 529.970i	− 0.562004i
$944$	0	0
$945$	419.067	0.443457
$946$	0	0
$947$	− 1465.05i	− 1.54704i	−0.633773	−	0.773519i	$-0.718494\pi$
$947$	− 1465.05i	− 1.54704i	0.633773	−	0.773519i	$-0.281506\pi$
$948$	0	0
$949$	18.6701	0.0196734
$950$	0	0
$951$	55.1473i	0.0579888i
$952$	0	0
$953$	−572.440	−0.600671	−0.300336	−	0.953834i	$-0.597099\pi$
$953$	−572.440	−0.600671	−0.300336	+	0.953834i	$0.597099\pi$
$954$	0	0
$955$	5.75920i	0.00603058i
$956$	0	0
$957$	−374.238	−0.391053
$958$	0	0
$959$	− 2014.71i	− 2.10084i
$960$	0	0
$961$	352.134	0.366425
$962$	0	0
$963$	1597.35i	1.65873i
$964$	0	0
$965$	−358.972	−0.371992
$966$	0	0
$967$	− 228.290i	− 0.236080i	−0.993009	−	0.118040i	$-0.962339\pi$
$967$	− 228.290i	− 0.236080i	0.993009	−	0.118040i	$-0.0376612\pi$
$968$	0	0
$969$	97.2007	0.100310
$970$	0	0
$971$	− 1338.46i	− 1.37843i	−0.724555	−	0.689217i	$-0.757955\pi$
$971$	− 1338.46i	− 1.37843i	0.724555	−	0.689217i	$-0.242045\pi$
$972$	0	0
$973$	2837.10	2.91582
$974$	0	0
$975$	− 14.9088i	− 0.0152911i
$976$	0	0
$977$	396.018	0.405341	0.202671	−	0.979247i	$-0.435038\pi$
$977$	396.018	0.405341	0.202671	+	0.979247i	$0.435038\pi$
$978$	0	0
$979$	288.332i	0.294517i
$980$	0	0
$981$	1530.54	1.56018
$982$	0	0
$983$	− 169.385i	− 0.172315i	−0.996282	−	0.0861573i	$-0.972541\pi$
$983$	− 169.385i	− 0.172315i	0.996282	−	0.0861573i	$-0.0274588\pi$
$984$	0	0
$985$	−638.816	−0.648544
$986$	0	0
$987$	458.816i	0.464859i
$988$	0	0
$989$	−568.727	−0.575052
$990$	0	0
$991$	− 899.988i	− 0.908161i	−0.890961	−	0.454080i	$-0.849968\pi$
$991$	− 899.988i	− 0.908161i	0.890961	−	0.454080i	$-0.150032\pi$
$992$	0	0
$993$	−146.410	−0.147442
$994$	0	0
$995$	735.071i	0.738765i
$996$	0	0
$997$	1265.57	1.26938	0.634689	−	0.772768i	$-0.281128\pi$
$997$	1265.57	1.26938	0.634689	+	0.772768i	$0.281128\pi$
$998$	0	0
$999$	− 389.171i	− 0.389560i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.d.d.191.6		14
4.3	odd	2	inner	1216.3.d.d.191.9		14
8.3	odd	2		76.3.b.b.39.7	✓	14
8.5	even	2		76.3.b.b.39.8	yes	14
24.5	odd	2		684.3.g.b.343.7		14
24.11	even	2		684.3.g.b.343.8		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.b.b.39.7	✓	14	8.3	odd	2
76.3.b.b.39.8	yes	14	8.5	even	2
684.3.g.b.343.7		14	24.5	odd	2
684.3.g.b.343.8		14	24.11	even	2
1216.3.d.d.191.6		14	1.1	even	1	trivial
1216.3.d.d.191.9		14	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.820457i q^{3} +2.38184 q^{5} +12.3764i q^{7} +8.32685 q^{9} +O(q^{10})\)	\(q-0.820457i q^{3} +2.38184 q^{5} +12.3764i q^{7} +8.32685 q^{9} +9.15034i q^{11} -0.940214 q^{13} -1.95419i q^{15} +27.1792 q^{17} +4.35890i q^{19} +10.1543 q^{21} +13.8004i q^{23} -19.3269 q^{25} -14.2159i q^{27} -49.8488 q^{29} -24.6752i q^{31} +7.50746 q^{33} +29.4787i q^{35} +27.3757 q^{37} +0.771405i q^{39} -38.4024 q^{41} +41.2108i q^{43} +19.8332 q^{45} +45.1842i q^{47} -104.176 q^{49} -22.2994i q^{51} +19.9964 q^{53} +21.7946i q^{55} +3.57629 q^{57} +34.7054i q^{59} -33.2008 q^{61} +103.057i q^{63} -2.23944 q^{65} -3.48192i q^{67} +11.3226 q^{69} -88.8887i q^{71} -19.8573 q^{73} +15.8568i q^{75} -113.249 q^{77} -51.7431i q^{79} +63.2781 q^{81} +6.62943i q^{83} +64.7365 q^{85} +40.8988i q^{87} +31.5105 q^{89} -11.6365i q^{91} -20.2449 q^{93} +10.3822i q^{95} +159.282 q^{97} +76.1935i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 68 q^{9}+O(q^{10}) \) 14 * q - 68 * q^9	\( 14 q - 68 q^{9} - 54 q^{13} + 34 q^{17} + 38 q^{21} - 86 q^{25} - 54 q^{29} + 20 q^{33} - 100 q^{37} + 224 q^{41} + 168 q^{45} - 220 q^{49} - 14 q^{53} + 38 q^{57} - 28 q^{61} - 472 q^{65} - 122 q^{69} + 70 q^{73} - 228 q^{77} + 334 q^{81} - 48 q^{85} + 176 q^{93} + 308 q^{97}+O(q^{100}) \) 14 * q - 68 * q^9 - 54 * q^13 + 34 * q^17 + 38 * q^21 - 86 * q^25 - 54 * q^29 + 20 * q^33 - 100 * q^37 + 224 * q^41 + 168 * q^45 - 220 * q^49 - 14 * q^53 + 38 * q^57 - 28 * q^61 - 472 * q^65 - 122 * q^69 + 70 * q^73 - 228 * q^77 + 334 * q^81 - 48 * q^85 + 176 * q^93 + 308 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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