LMFDB - Embedded newform 1216.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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([0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.a (trivial)

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:	yes
Analytic conductor:	$71.7463225670$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 76)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.37228 q^{3} -11.8614 q^{5} +26.4891 q^{7} +1.86141 q^{9} +O(q^{10})$	$q-5.37228 q^{3} -11.8614 q^{5} +26.4891 q^{7} +1.86141 q^{9} -49.8614 q^{11} +49.0951 q^{13} +63.7228 q^{15} +17.2337 q^{17} -19.0000 q^{19} -142.307 q^{21} -166.965 q^{23} +15.6930 q^{25} +135.052 q^{27} +109.198 q^{29} +273.783 q^{31} +267.870 q^{33} -314.198 q^{35} -167.022 q^{37} -263.753 q^{39} +15.1684 q^{41} +413.557 q^{43} -22.0789 q^{45} +161.438 q^{47} +358.674 q^{49} -92.5842 q^{51} +490.791 q^{53} +591.426 q^{55} +102.073 q^{57} -335.970 q^{59} -725.149 q^{61} +49.3070 q^{63} -582.337 q^{65} +497.709 q^{67} +896.981 q^{69} +798.554 q^{71} -311.174 q^{73} -84.3070 q^{75} -1320.79 q^{77} +665.723 q^{79} -775.793 q^{81} -372.440 q^{83} -204.416 q^{85} -586.644 q^{87} -673.783 q^{89} +1300.49 q^{91} -1470.84 q^{93} +225.367 q^{95} -960.505 q^{97} -92.8124 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{3} + 5 q^{5} + 30 q^{7} - 25 q^{9}+O(q^{10}) $ 2 * q - 5 * q^3 + 5 * q^5 + 30 * q^7 - 25 * q^9	$ 2 q - 5 q^{3} + 5 q^{5} + 30 q^{7} - 25 q^{9} - 71 q^{11} + 35 q^{13} + 70 q^{15} - 38 q^{19} - 141 q^{21} + 5 q^{23} + 175 q^{25} + 115 q^{27} - 155 q^{29} + 88 q^{31} + 260 q^{33} - 255 q^{35} - 380 q^{37} - 269 q^{39} - 142 q^{41} + 155 q^{43} - 475 q^{45} + 455 q^{47} + 28 q^{49} - 99 q^{51} + 275 q^{53} + 235 q^{55} + 95 q^{57} - 873 q^{59} - 445 q^{61} - 45 q^{63} - 820 q^{65} + 645 q^{67} + 961 q^{69} + 1712 q^{71} - 990 q^{73} - 25 q^{75} - 1395 q^{77} + 1274 q^{79} - 58 q^{81} - 90 q^{83} - 495 q^{85} - 685 q^{87} - 888 q^{89} + 1251 q^{91} - 1540 q^{93} - 95 q^{95} + 710 q^{97} + 475 q^{99}+O(q^{100}) $ 2 * q - 5 * q^3 + 5 * q^5 + 30 * q^7 - 25 * q^9 - 71 * q^11 + 35 * q^13 + 70 * q^15 - 38 * q^19 - 141 * q^21 + 5 * q^23 + 175 * q^25 + 115 * q^27 - 155 * q^29 + 88 * q^31 + 260 * q^33 - 255 * q^35 - 380 * q^37 - 269 * q^39 - 142 * q^41 + 155 * q^43 - 475 * q^45 + 455 * q^47 + 28 * q^49 - 99 * q^51 + 275 * q^53 + 235 * q^55 + 95 * q^57 - 873 * q^59 - 445 * q^61 - 45 * q^63 - 820 * q^65 + 645 * q^67 + 961 * q^69 + 1712 * q^71 - 990 * q^73 - 25 * q^75 - 1395 * q^77 + 1274 * q^79 - 58 * q^81 - 90 * q^83 - 495 * q^85 - 685 * q^87 - 888 * q^89 + 1251 * q^91 - 1540 * q^93 - 95 * q^95 + 710 * q^97 + 475 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−5.37228	−1.03390	−0.516948	−	0.856017i	$-0.672932\pi$
$3$	−5.37228	−1.03390	−0.516948	+	0.856017i	$0.672932\pi$
$4$	0	0
$5$	−11.8614	−1.06092	−0.530458	−	0.847711i	$-0.677980\pi$
$5$	−11.8614	−1.06092	−0.530458	+	0.847711i	$0.677980\pi$
$6$	0	0
$7$	26.4891	1.43028	0.715139	−	0.698982i	$-0.246363\pi$
$7$	26.4891	1.43028	0.715139	+	0.698982i	$0.246363\pi$
$8$	0	0
$9$	1.86141	0.0689410
$10$	0	0
$11$	−49.8614	−1.36671	−0.683354	−	0.730088i	$-0.739479\pi$
$11$	−49.8614	−1.36671	−0.683354	+	0.730088i	$0.739479\pi$
$12$	0	0
$13$	49.0951	1.04743	0.523713	−	0.851895i	$-0.324547\pi$
$13$	49.0951	1.04743	0.523713	+	0.851895i	$0.324547\pi$
$14$	0	0
$15$	63.7228	1.09688
$16$	0	0
$17$	17.2337	0.245870	0.122935	−	0.992415i	$-0.460769\pi$
$17$	17.2337	0.245870	0.122935	+	0.992415i	$0.460769\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−142.307	−1.47876
$22$	0	0
$23$	−166.965	−1.51368	−0.756838	−	0.653603i	$-0.773257\pi$
$23$	−166.965	−1.51368	−0.756838	+	0.653603i	$0.773257\pi$
$24$	0	0
$25$	15.6930	0.125544
$26$	0	0
$27$	135.052	0.962618
$28$	0	0
$29$	109.198	0.699228	0.349614	−	0.936894i	$-0.386313\pi$
$29$	109.198	0.699228	0.349614	+	0.936894i	$0.386313\pi$
$30$	0	0
$31$	273.783	1.58622	0.793110	−	0.609079i	$-0.208461\pi$
$31$	273.783	1.58622	0.793110	+	0.609079i	$0.208461\pi$
$32$	0	0
$33$	267.870	1.41303
$34$	0	0
$35$	−314.198	−1.51741
$36$	0	0
$37$	−167.022	−0.742114	−0.371057	−	0.928610i	$-0.621005\pi$
$37$	−167.022	−0.742114	−0.371057	+	0.928610i	$0.621005\pi$
$38$	0	0
$39$	−263.753	−1.08293
$40$	0	0
$41$	15.1684	0.0577783	0.0288892	−	0.999583i	$-0.490803\pi$
$41$	15.1684	0.0577783	0.0288892	+	0.999583i	$0.490803\pi$
$42$	0	0
$43$	413.557	1.46667	0.733335	−	0.679867i	$-0.237963\pi$
$43$	413.557	1.46667	0.733335	+	0.679867i	$0.237963\pi$
$44$	0	0
$45$	−22.0789	−0.0731406
$46$	0	0
$47$	161.438	0.501023	0.250512	−	0.968114i	$-0.419401\pi$
$47$	161.438	0.501023	0.250512	+	0.968114i	$0.419401\pi$
$48$	0	0
$49$	358.674	1.04570
$50$	0	0
$51$	−92.5842	−0.254204
$52$	0	0
$53$	490.791	1.27199	0.635993	−	0.771695i	$-0.280591\pi$
$53$	490.791	1.27199	0.635993	+	0.771695i	$0.280591\pi$
$54$	0	0
$55$	591.426	1.44996
$56$	0	0
$57$	102.073	0.237192
$58$	0	0
$59$	−335.970	−0.741349	−0.370674	−	0.928763i	$-0.620873\pi$
$59$	−335.970	−0.741349	−0.370674	+	0.928763i	$0.620873\pi$
$60$	0	0
$61$	−725.149	−1.52206	−0.761032	−	0.648715i	$-0.775307\pi$
$61$	−725.149	−1.52206	−0.761032	+	0.648715i	$0.775307\pi$
$62$	0	0
$63$	49.3070	0.0986048
$64$	0	0
$65$	−582.337	−1.11123
$66$	0	0
$67$	497.709	0.907535	0.453768	−	0.891120i	$-0.350080\pi$
$67$	497.709	0.907535	0.453768	+	0.891120i	$0.350080\pi$
$68$	0	0
$69$	896.981	1.56498
$70$	0	0
$71$	798.554	1.33480	0.667401	−	0.744698i	$-0.267407\pi$
$71$	798.554	1.33480	0.667401	+	0.744698i	$0.267407\pi$
$72$	0	0
$73$	−311.174	−0.498906	−0.249453	−	0.968387i	$-0.580251\pi$
$73$	−311.174	−0.498906	−0.249453	+	0.968387i	$0.580251\pi$
$74$	0	0
$75$	−84.3070	−0.129799
$76$	0	0
$77$	−1320.79	−1.95477
$78$	0	0
$79$	665.723	0.948097	0.474049	−	0.880499i	$-0.342792\pi$
$79$	665.723	0.948097	0.474049	+	0.880499i	$0.342792\pi$
$80$	0	0
$81$	−775.793	−1.06419
$82$	0	0
$83$	−372.440	−0.492537	−0.246269	−	0.969202i	$-0.579205\pi$
$83$	−372.440	−0.492537	−0.246269	+	0.969202i	$0.579205\pi$
$84$	0	0
$85$	−204.416	−0.260847
$86$	0	0
$87$	−586.644	−0.722929
$88$	0	0
$89$	−673.783	−0.802481	−0.401240	−	0.915973i	$-0.631421\pi$
$89$	−673.783	−0.802481	−0.401240	+	0.915973i	$0.631421\pi$
$90$	0	0
$91$	1300.49	1.49811
$92$	0	0
$93$	−1470.84	−1.63999
$94$	0	0
$95$	225.367	0.243391
$96$	0	0
$97$	−960.505	−1.00541	−0.502704	−	0.864459i	$-0.667661\pi$
$97$	−960.505	−1.00541	−0.502704	+	0.864459i	$0.667661\pi$
$98$	0	0
$99$	−92.8124	−0.0942221
$100$	0	0
$101$	−1889.68	−1.86169	−0.930845	−	0.365415i	$-0.880927\pi$
$101$	−1889.68	−1.86169	−0.930845	+	0.365415i	$0.880927\pi$
$102$	0	0
$103$	−760.217	−0.727247	−0.363624	−	0.931546i	$-0.618461\pi$
$103$	−760.217	−0.727247	−0.363624	+	0.931546i	$0.618461\pi$
$104$	0	0
$105$	1687.96	1.56884
$106$	0	0
$107$	681.730	0.615938	0.307969	−	0.951396i	$-0.400351\pi$
$107$	681.730	0.615938	0.307969	+	0.951396i	$0.400351\pi$
$108$	0	0
$109$	1310.76	1.15182	0.575910	−	0.817513i	$-0.304648\pi$
$109$	1310.76	1.15182	0.575910	+	0.817513i	$0.304648\pi$
$110$	0	0
$111$	897.288	0.767268
$112$	0	0
$113$	−713.413	−0.593914	−0.296957	−	0.954891i	$-0.595972\pi$
$113$	−713.413	−0.593914	−0.296957	+	0.954891i	$0.595972\pi$
$114$	0	0
$115$	1980.43	1.60588
$116$	0	0
$117$	91.3859	0.0722105
$118$	0	0
$119$	456.505	0.351662
$120$	0	0
$121$	1155.16	0.867889
$122$	0	0
$123$	−81.4891	−0.0597368
$124$	0	0
$125$	1296.54	0.927725
$126$	0	0
$127$	107.847	0.0753535	0.0376768	−	0.999290i	$-0.488004\pi$
$127$	107.847	0.0753535	0.0376768	+	0.999290i	$0.488004\pi$
$128$	0	0
$129$	−2221.74	−1.51638
$130$	0	0
$131$	373.247	0.248937	0.124469	−	0.992224i	$-0.460277\pi$
$131$	373.247	0.248937	0.124469	+	0.992224i	$0.460277\pi$
$132$	0	0
$133$	−503.293	−0.328128
$134$	0	0
$135$	−1601.90	−1.02126
$136$	0	0
$137$	−3118.25	−1.94460	−0.972301	−	0.233732i	$-0.924906\pi$
$137$	−3118.25	−1.94460	−0.972301	+	0.233732i	$0.924906\pi$
$138$	0	0
$139$	−56.3774	−0.0344019	−0.0172010	−	0.999852i	$-0.505476\pi$
$139$	−56.3774	−0.0344019	−0.0172010	+	0.999852i	$0.505476\pi$
$140$	0	0
$141$	−867.288	−0.518006
$142$	0	0
$143$	−2447.95	−1.43152
$144$	0	0
$145$	−1295.25	−0.741823
$146$	0	0
$147$	−1926.90	−1.08114
$148$	0	0
$149$	1810.51	0.995457	0.497728	−	0.867333i	$-0.334168\pi$
$149$	1810.51	0.995457	0.497728	+	0.867333i	$0.334168\pi$
$150$	0	0
$151$	−2894.72	−1.56006	−0.780029	−	0.625744i	$-0.784796\pi$
$151$	−2894.72	−1.56006	−0.780029	+	0.625744i	$0.784796\pi$
$152$	0	0
$153$	32.0789	0.0169505
$154$	0	0
$155$	−3247.45	−1.68285
$156$	0	0
$157$	−1381.71	−0.702371	−0.351185	−	0.936306i	$-0.614221\pi$
$157$	−1381.71	−0.702371	−0.351185	+	0.936306i	$0.614221\pi$
$158$	0	0
$159$	−2636.67	−1.31510
$160$	0	0
$161$	−4422.75	−2.16498
$162$	0	0
$163$	−3740.83	−1.79757	−0.898785	−	0.438389i	$-0.855549\pi$
$163$	−3740.83	−1.79757	−0.898785	+	0.438389i	$0.855549\pi$
$164$	0	0
$165$	−3177.31	−1.49911
$166$	0	0
$167$	−3085.17	−1.42957	−0.714783	−	0.699347i	$-0.753474\pi$
$167$	−3085.17	−1.42957	−0.714783	+	0.699347i	$0.753474\pi$
$168$	0	0
$169$	213.328	0.0970998
$170$	0	0
$171$	−35.3667	−0.0158161
$172$	0	0
$173$	−293.000	−0.128765	−0.0643827	−	0.997925i	$-0.520508\pi$
$173$	−293.000	−0.128765	−0.0643827	+	0.997925i	$0.520508\pi$
$174$	0	0
$175$	415.693	0.179562
$176$	0	0
$177$	1804.93	0.766478
$178$	0	0
$179$	−2996.32	−1.25115	−0.625574	−	0.780165i	$-0.715135\pi$
$179$	−2996.32	−1.25115	−0.625574	+	0.780165i	$0.715135\pi$
$180$	0	0
$181$	3265.41	1.34097	0.670487	−	0.741922i	$-0.266085\pi$
$181$	3265.41	1.34097	0.670487	+	0.741922i	$0.266085\pi$
$182$	0	0
$183$	3895.71	1.57365
$184$	0	0
$185$	1981.11	0.787321
$186$	0	0
$187$	−859.296	−0.336032
$188$	0	0
$189$	3577.40	1.37681
$190$	0	0
$191$	1502.27	0.569113	0.284556	−	0.958659i	$-0.408154\pi$
$191$	1502.27	0.569113	0.284556	+	0.958659i	$0.408154\pi$
$192$	0	0
$193$	−4103.30	−1.53037	−0.765186	−	0.643809i	$-0.777353\pi$
$193$	−4103.30	−1.53037	−0.765186	+	0.643809i	$0.777353\pi$
$194$	0	0
$195$	3128.48	1.14890
$196$	0	0
$197$	−5420.95	−1.96054	−0.980271	−	0.197658i	$-0.936666\pi$
$197$	−5420.95	−1.96054	−0.980271	+	0.197658i	$0.936666\pi$
$198$	0	0
$199$	−1666.05	−0.593484	−0.296742	−	0.954958i	$-0.595900\pi$
$199$	−1666.05	−0.593484	−0.296742	+	0.954958i	$0.595900\pi$
$200$	0	0
$201$	−2673.83	−0.938297
$202$	0	0
$203$	2892.57	1.00009
$204$	0	0
$205$	−179.919	−0.0612980
$206$	0	0
$207$	−310.789	−0.104354
$208$	0	0
$209$	947.367	0.313544
$210$	0	0
$211$	5865.90	1.91386	0.956931	−	0.290314i	$-0.0937598\pi$
$211$	5865.90	1.91386	0.956931	+	0.290314i	$0.0937598\pi$
$212$	0	0
$213$	−4290.06	−1.38005
$214$	0	0
$215$	−4905.37	−1.55602
$216$	0	0
$217$	7252.26	2.26873
$218$	0	0
$219$	1671.71	0.515817
$220$	0	0
$221$	846.090	0.257530
$222$	0	0
$223$	5402.94	1.62246	0.811229	−	0.584729i	$-0.198799\pi$
$223$	5402.94	1.62246	0.811229	+	0.584729i	$0.198799\pi$
$224$	0	0
$225$	29.2110	0.00865511
$226$	0	0
$227$	−2571.34	−0.751833	−0.375916	−	0.926654i	$-0.622672\pi$
$227$	−2571.34	−0.751833	−0.375916	+	0.926654i	$0.622672\pi$
$228$	0	0
$229$	−3256.08	−0.939597	−0.469799	−	0.882774i	$-0.655673\pi$
$229$	−3256.08	−0.939597	−0.469799	+	0.882774i	$0.655673\pi$
$230$	0	0
$231$	7095.63	2.02103
$232$	0	0
$233$	1205.58	0.338972	0.169486	−	0.985533i	$-0.445789\pi$
$233$	1205.58	0.338972	0.169486	+	0.985533i	$0.445789\pi$
$234$	0	0
$235$	−1914.88	−0.531544
$236$	0	0
$237$	−3576.45	−0.980234
$238$	0	0
$239$	3404.52	0.921424	0.460712	−	0.887550i	$-0.347594\pi$
$239$	3404.52	0.921424	0.460712	+	0.887550i	$0.347594\pi$
$240$	0	0
$241$	3301.51	0.882443	0.441221	−	0.897398i	$-0.354545\pi$
$241$	3301.51	0.882443	0.441221	+	0.897398i	$0.354545\pi$
$242$	0	0
$243$	521.386	0.137642
$244$	0	0
$245$	−4254.38	−1.10940
$246$	0	0
$247$	−932.807	−0.240296
$248$	0	0
$249$	2000.85	0.509233
$250$	0	0
$251$	−5330.81	−1.34055	−0.670275	−	0.742113i	$-0.733824\pi$
$251$	−5330.81	−1.34055	−0.670275	+	0.742113i	$0.733824\pi$
$252$	0	0
$253$	8325.09	2.06875
$254$	0	0
$255$	1098.18	0.269689
$256$	0	0
$257$	3812.14	0.925272	0.462636	−	0.886548i	$-0.346904\pi$
$257$	3812.14	0.925272	0.462636	+	0.886548i	$0.346904\pi$
$258$	0	0
$259$	−4424.26	−1.06143
$260$	0	0
$261$	203.262	0.0482055
$262$	0	0
$263$	−899.159	−0.210816	−0.105408	−	0.994429i	$-0.533615\pi$
$263$	−899.159	−0.210816	−0.105408	+	0.994429i	$0.533615\pi$
$264$	0	0
$265$	−5821.47	−1.34947
$266$	0	0
$267$	3619.75	0.829682
$268$	0	0
$269$	−2645.77	−0.599685	−0.299842	−	0.953989i	$-0.596934\pi$
$269$	−2645.77	−0.599685	−0.299842	+	0.953989i	$0.596934\pi$
$270$	0	0
$271$	−516.529	−0.115782	−0.0578909	−	0.998323i	$-0.518438\pi$
$271$	−516.529	−0.115782	−0.0578909	+	0.998323i	$0.518438\pi$
$272$	0	0
$273$	−6986.58	−1.54889
$274$	0	0
$275$	−782.473	−0.171582
$276$	0	0
$277$	−2365.14	−0.513023	−0.256512	−	0.966541i	$-0.582573\pi$
$277$	−2365.14	−0.513023	−0.256512	+	0.966541i	$0.582573\pi$
$278$	0	0
$279$	509.621	0.109356
$280$	0	0
$281$	−1526.54	−0.324077	−0.162038	−	0.986784i	$-0.551807\pi$
$281$	−1526.54	−0.324077	−0.162038	+	0.986784i	$0.551807\pi$
$282$	0	0
$283$	4764.01	1.00067	0.500337	−	0.865831i	$-0.333209\pi$
$283$	4764.01	1.00067	0.500337	+	0.865831i	$0.333209\pi$
$284$	0	0
$285$	−1210.73	−0.251641
$286$	0	0
$287$	401.799	0.0826391
$288$	0	0
$289$	−4616.00	−0.939548
$290$	0	0
$291$	5160.10	1.03949
$292$	0	0
$293$	−3189.07	−0.635862	−0.317931	−	0.948114i	$-0.602988\pi$
$293$	−3189.07	−0.635862	−0.317931	+	0.948114i	$0.602988\pi$
$294$	0	0
$295$	3985.08	0.786509
$296$	0	0
$297$	−6733.86	−1.31562
$298$	0	0
$299$	−8197.14	−1.58546
$300$	0	0
$301$	10954.8	2.09775
$302$	0	0
$303$	10151.9	1.92479
$304$	0	0
$305$	8601.29	1.61478
$306$	0	0
$307$	5998.56	1.11517	0.557583	−	0.830121i	$-0.311729\pi$
$307$	5998.56	1.11517	0.557583	+	0.830121i	$0.311729\pi$
$308$	0	0
$309$	4084.10	0.751898
$310$	0	0
$311$	−5114.96	−0.932614	−0.466307	−	0.884623i	$-0.654416\pi$
$311$	−5114.96	−0.932614	−0.466307	+	0.884623i	$0.654416\pi$
$312$	0	0
$313$	2131.63	0.384942	0.192471	−	0.981303i	$-0.438350\pi$
$313$	2131.63	0.384942	0.192471	+	0.981303i	$0.438350\pi$
$314$	0	0
$315$	−584.851	−0.104611
$316$	0	0
$317$	−4457.03	−0.789691	−0.394845	−	0.918748i	$-0.629202\pi$
$317$	−4457.03	−0.789691	−0.394845	+	0.918748i	$0.629202\pi$
$318$	0	0
$319$	−5444.78	−0.955640
$320$	0	0
$321$	−3662.45	−0.636816
$322$	0	0
$323$	−327.440	−0.0564064
$324$	0	0
$325$	770.448	0.131498
$326$	0	0
$327$	−7041.79	−1.19086
$328$	0	0
$329$	4276.34	0.716602
$330$	0	0
$331$	−4143.68	−0.688088	−0.344044	−	0.938954i	$-0.611797\pi$
$331$	−4143.68	−0.688088	−0.344044	+	0.938954i	$0.611797\pi$
$332$	0	0
$333$	−310.895	−0.0511620
$334$	0	0
$335$	−5903.53	−0.962819
$336$	0	0
$337$	−5148.60	−0.832231	−0.416115	−	0.909312i	$-0.636609\pi$
$337$	−5148.60	−0.832231	−0.416115	+	0.909312i	$0.636609\pi$
$338$	0	0
$339$	3832.66	0.614045
$340$	0	0
$341$	−13651.2	−2.16790
$342$	0	0
$343$	415.184	0.0653581
$344$	0	0
$345$	−10639.5	−1.66032
$346$	0	0
$347$	−5873.56	−0.908672	−0.454336	−	0.890830i	$-0.650123\pi$
$347$	−5873.56	−0.908672	−0.454336	+	0.890830i	$0.650123\pi$
$348$	0	0
$349$	−5167.58	−0.792590	−0.396295	−	0.918123i	$-0.629704\pi$
$349$	−5167.58	−0.792590	−0.396295	+	0.918123i	$0.629704\pi$
$350$	0	0
$351$	6630.37	1.00827
$352$	0	0
$353$	7191.17	1.08427	0.542135	−	0.840291i	$-0.317616\pi$
$353$	7191.17	1.08427	0.542135	+	0.840291i	$0.317616\pi$
$354$	0	0
$355$	−9471.98	−1.41611
$356$	0	0
$357$	−2452.47	−0.363582
$358$	0	0
$359$	7049.62	1.03639	0.518196	−	0.855262i	$-0.326604\pi$
$359$	7049.62	1.03639	0.518196	+	0.855262i	$0.326604\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−6205.84	−0.897307
$364$	0	0
$365$	3690.96	0.529298
$366$	0	0
$367$	−6935.53	−0.986463	−0.493231	−	0.869898i	$-0.664184\pi$
$367$	−6935.53	−0.986463	−0.493231	+	0.869898i	$0.664184\pi$
$368$	0	0
$369$	28.2346	0.00398330
$370$	0	0
$371$	13000.6	1.81929
$372$	0	0
$373$	6346.92	0.881048	0.440524	−	0.897741i	$-0.354793\pi$
$373$	6346.92	0.881048	0.440524	+	0.897741i	$0.354793\pi$
$374$	0	0
$375$	−6965.35	−0.959171
$376$	0	0
$377$	5361.10	0.732389
$378$	0	0
$379$	11363.0	1.54004	0.770022	−	0.638017i	$-0.220245\pi$
$379$	11363.0	1.54004	0.770022	+	0.638017i	$0.220245\pi$
$380$	0	0
$381$	−579.386	−0.0779077
$382$	0	0
$383$	3848.09	0.513390	0.256695	−	0.966493i	$-0.417366\pi$
$383$	3848.09	0.513390	0.256695	+	0.966493i	$0.417366\pi$
$384$	0	0
$385$	15666.4	2.07385
$386$	0	0
$387$	769.798	0.101114
$388$	0	0
$389$	−1936.75	−0.252435	−0.126217	−	0.992003i	$-0.540284\pi$
$389$	−1936.75	−0.252435	−0.126217	+	0.992003i	$0.540284\pi$
$390$	0	0
$391$	−2877.42	−0.372167
$392$	0	0
$393$	−2005.19	−0.257375
$394$	0	0
$395$	−7896.41	−1.00585
$396$	0	0
$397$	−3851.40	−0.486892	−0.243446	−	0.969914i	$-0.578278\pi$
$397$	−3851.40	−0.486892	−0.243446	+	0.969914i	$0.578278\pi$
$398$	0	0
$399$	2703.83	0.339251
$400$	0	0
$401$	−1381.47	−0.172038	−0.0860189	−	0.996294i	$-0.527415\pi$
$401$	−1381.47	−0.172038	−0.0860189	+	0.996294i	$0.527415\pi$
$402$	0	0
$403$	13441.4	1.66145
$404$	0	0
$405$	9202.00	1.12901
$406$	0	0
$407$	8327.94	1.01425
$408$	0	0
$409$	368.878	0.0445962	0.0222981	−	0.999751i	$-0.492902\pi$
$409$	368.878	0.0445962	0.0222981	+	0.999751i	$0.492902\pi$
$410$	0	0
$411$	16752.1	2.01052
$412$	0	0
$413$	−8899.56	−1.06034
$414$	0	0
$415$	4417.66	0.522541
$416$	0	0
$417$	302.875	0.0355680
$418$	0	0
$419$	−1814.85	−0.211602	−0.105801	−	0.994387i	$-0.533741\pi$
$419$	−1814.85	−0.211602	−0.105801	+	0.994387i	$0.533741\pi$
$420$	0	0
$421$	−4614.01	−0.534141	−0.267070	−	0.963677i	$-0.586056\pi$
$421$	−4614.01	−0.534141	−0.267070	+	0.963677i	$0.586056\pi$
$422$	0	0
$423$	300.501	0.0345410
$424$	0	0
$425$	270.448	0.0308674
$426$	0	0
$427$	−19208.6	−2.17697
$428$	0	0
$429$	13151.1	1.48005
$430$	0	0
$431$	2739.14	0.306125	0.153062	−	0.988217i	$-0.451086\pi$
$431$	2739.14	0.306125	0.153062	+	0.988217i	$0.451086\pi$
$432$	0	0
$433$	−10827.9	−1.20174	−0.600872	−	0.799345i	$-0.705180\pi$
$433$	−10827.9	−1.20174	−0.600872	+	0.799345i	$0.705180\pi$
$434$	0	0
$435$	6958.42	0.766967
$436$	0	0
$437$	3172.33	0.347261
$438$	0	0
$439$	10359.5	1.12627	0.563137	−	0.826364i	$-0.309594\pi$
$439$	10359.5	1.12627	0.563137	+	0.826364i	$0.309594\pi$
$440$	0	0
$441$	667.638	0.0720913
$442$	0	0
$443$	8040.36	0.862322	0.431161	−	0.902275i	$-0.358104\pi$
$443$	8040.36	0.862322	0.431161	+	0.902275i	$0.358104\pi$
$444$	0	0
$445$	7992.01	0.851365
$446$	0	0
$447$	−9726.59	−1.02920
$448$	0	0
$449$	−14071.8	−1.47904	−0.739522	−	0.673132i	$-0.764948\pi$
$449$	−14071.8	−1.47904	−0.739522	+	0.673132i	$0.764948\pi$
$450$	0	0
$451$	−756.320	−0.0789661
$452$	0	0
$453$	15551.2	1.61294
$454$	0	0
$455$	−15425.6	−1.58937
$456$	0	0
$457$	−4372.74	−0.447589	−0.223795	−	0.974636i	$-0.571845\pi$
$457$	−4372.74	−0.447589	−0.223795	+	0.974636i	$0.571845\pi$
$458$	0	0
$459$	2327.44	0.236679
$460$	0	0
$461$	17819.4	1.80029	0.900146	−	0.435589i	$-0.143460\pi$
$461$	17819.4	1.80029	0.900146	+	0.435589i	$0.143460\pi$
$462$	0	0
$463$	−7114.51	−0.714124	−0.357062	−	0.934081i	$-0.616222\pi$
$463$	−7114.51	−0.714124	−0.357062	+	0.934081i	$0.616222\pi$
$464$	0	0
$465$	17446.2	1.73989
$466$	0	0
$467$	−8208.41	−0.813361	−0.406681	−	0.913570i	$-0.633314\pi$
$467$	−8208.41	−0.813361	−0.406681	+	0.913570i	$0.633314\pi$
$468$	0	0
$469$	13183.9	1.29803
$470$	0	0
$471$	7422.92	0.726178
$472$	0	0
$473$	−20620.5	−2.00451
$474$	0	0
$475$	−298.166	−0.0288017
$476$	0	0
$477$	913.561	0.0876920
$478$	0	0
$479$	11889.5	1.13413	0.567064	−	0.823674i	$-0.308079\pi$
$479$	11889.5	1.13413	0.567064	+	0.823674i	$0.308079\pi$
$480$	0	0
$481$	−8199.95	−0.777309
$482$	0	0
$483$	23760.2	2.23836
$484$	0	0
$485$	11392.9	1.06665
$486$	0	0
$487$	−13922.4	−1.29545	−0.647725	−	0.761875i	$-0.724279\pi$
$487$	−13922.4	−1.29545	−0.647725	+	0.761875i	$0.724279\pi$
$488$	0	0
$489$	20096.8	1.85850
$490$	0	0
$491$	2718.22	0.249840	0.124920	−	0.992167i	$-0.460133\pi$
$491$	2718.22	0.249840	0.124920	+	0.992167i	$0.460133\pi$
$492$	0	0
$493$	1881.89	0.171919
$494$	0	0
$495$	1100.89	0.0999618
$496$	0	0
$497$	21153.0	1.90914
$498$	0	0
$499$	−7830.51	−0.702489	−0.351244	−	0.936284i	$-0.614241\pi$
$499$	−7830.51	−0.702489	−0.351244	+	0.936284i	$0.614241\pi$
$500$	0	0
$501$	16574.4	1.47802
$502$	0	0
$503$	6554.35	0.581002	0.290501	−	0.956875i	$-0.406178\pi$
$503$	6554.35	0.581002	0.290501	+	0.956875i	$0.406178\pi$
$504$	0	0
$505$	22414.3	1.97510
$506$	0	0
$507$	−1146.06	−0.100391
$508$	0	0
$509$	−12611.4	−1.09821	−0.549105	−	0.835753i	$-0.685031\pi$
$509$	−12611.4	−1.09821	−0.549105	+	0.835753i	$0.685031\pi$
$510$	0	0
$511$	−8242.73	−0.713575
$512$	0	0
$513$	−2565.98	−0.220840
$514$	0	0
$515$	9017.25	0.771548
$516$	0	0
$517$	−8049.50	−0.684752
$518$	0	0
$519$	1574.08	0.133130
$520$	0	0
$521$	1695.01	0.142533	0.0712666	−	0.997457i	$-0.477296\pi$
$521$	1695.01	0.142533	0.0712666	+	0.997457i	$0.477296\pi$
$522$	0	0
$523$	−21001.4	−1.75589	−0.877944	−	0.478764i	$-0.841085\pi$
$523$	−21001.4	−1.75589	−0.877944	+	0.478764i	$0.841085\pi$
$524$	0	0
$525$	−2233.22	−0.185649
$526$	0	0
$527$	4718.28	0.390003
$528$	0	0
$529$	15710.2	1.29121
$530$	0	0
$531$	−625.377	−0.0511093
$532$	0	0
$533$	744.696	0.0605185
$534$	0	0
$535$	−8086.28	−0.653459
$536$	0	0
$537$	16097.1	1.29356
$538$	0	0
$539$	−17884.0	−1.42916
$540$	0	0
$541$	555.994	0.0441849	0.0220925	−	0.999756i	$-0.492967\pi$
$541$	555.994	0.0441849	0.0220925	+	0.999756i	$0.492967\pi$
$542$	0	0
$543$	−17542.7	−1.38643
$544$	0	0
$545$	−15547.5	−1.22198
$546$	0	0
$547$	−2987.93	−0.233555	−0.116778	−	0.993158i	$-0.537256\pi$
$547$	−2987.93	−0.233555	−0.116778	+	0.993158i	$0.537256\pi$
$548$	0	0
$549$	−1349.80	−0.104933
$550$	0	0
$551$	−2074.77	−0.160414
$552$	0	0
$553$	17634.4	1.35604
$554$	0	0
$555$	−10643.1	−0.814008
$556$	0	0
$557$	4357.38	0.331469	0.165735	−	0.986170i	$-0.447001\pi$
$557$	4357.38	0.331469	0.165735	+	0.986170i	$0.447001\pi$
$558$	0	0
$559$	20303.6	1.53623
$560$	0	0
$561$	4616.38	0.347422
$562$	0	0
$563$	1669.67	0.124988	0.0624941	−	0.998045i	$-0.480095\pi$
$563$	1669.67	0.124988	0.0624941	+	0.998045i	$0.480095\pi$
$564$	0	0
$565$	8462.08	0.630093
$566$	0	0
$567$	−20550.1	−1.52209
$568$	0	0
$569$	−20440.1	−1.50596	−0.752982	−	0.658041i	$-0.771385\pi$
$569$	−20440.1	−1.50596	−0.752982	+	0.658041i	$0.771385\pi$
$570$	0	0
$571$	−6920.98	−0.507240	−0.253620	−	0.967304i	$-0.581621\pi$
$571$	−6920.98	−0.507240	−0.253620	+	0.967304i	$0.581621\pi$
$572$	0	0
$573$	−8070.62	−0.588403
$574$	0	0
$575$	−2620.17	−0.190032
$576$	0	0
$577$	−5021.59	−0.362307	−0.181154	−	0.983455i	$-0.557983\pi$
$577$	−5021.59	−0.362307	−0.181154	+	0.983455i	$0.557983\pi$
$578$	0	0
$579$	22044.1	1.58225
$580$	0	0
$581$	−9865.61	−0.704466
$582$	0	0
$583$	−24471.5	−1.73843
$584$	0	0
$585$	−1083.97	−0.0766093
$586$	0	0
$587$	−6784.80	−0.477068	−0.238534	−	0.971134i	$-0.576667\pi$
$587$	−6784.80	−0.477068	−0.238534	+	0.971134i	$0.576667\pi$
$588$	0	0
$589$	−5201.87	−0.363904
$590$	0	0
$591$	29122.9	2.02700
$592$	0	0
$593$	22058.5	1.52754	0.763771	−	0.645487i	$-0.223346\pi$
$593$	22058.5	1.52754	0.763771	+	0.645487i	$0.223346\pi$
$594$	0	0
$595$	−5414.80	−0.373084
$596$	0	0
$597$	8950.51	0.613601
$598$	0	0
$599$	15616.7	1.06524	0.532620	−	0.846354i	$-0.321207\pi$
$599$	15616.7	1.06524	0.532620	+	0.846354i	$0.321207\pi$
$600$	0	0
$601$	26769.6	1.81690	0.908448	−	0.417998i	$-0.137268\pi$
$601$	26769.6	1.81690	0.908448	+	0.417998i	$0.137268\pi$
$602$	0	0
$603$	926.439	0.0625664
$604$	0	0
$605$	−13701.8	−0.920757
$606$	0	0
$607$	−5164.26	−0.345323	−0.172661	−	0.984981i	$-0.555237\pi$
$607$	−5164.26	−0.345323	−0.172661	+	0.984981i	$0.555237\pi$
$608$	0	0
$609$	−15539.7	−1.03399
$610$	0	0
$611$	7925.79	0.524784
$612$	0	0
$613$	−4226.03	−0.278447	−0.139223	−	0.990261i	$-0.544461\pi$
$613$	−4226.03	−0.278447	−0.139223	+	0.990261i	$0.544461\pi$
$614$	0	0
$615$	966.576	0.0633758
$616$	0	0
$617$	14233.5	0.928717	0.464358	−	0.885647i	$-0.346285\pi$
$617$	14233.5	0.928717	0.464358	+	0.885647i	$0.346285\pi$
$618$	0	0
$619$	−3737.82	−0.242707	−0.121353	−	0.992609i	$-0.538723\pi$
$619$	−3737.82	−0.242707	−0.121353	+	0.992609i	$0.538723\pi$
$620$	0	0
$621$	−22548.8	−1.45709
$622$	0	0
$623$	−17847.9	−1.14777
$624$	0	0
$625$	−17340.4	−1.10978
$626$	0	0
$627$	−5089.52	−0.324172
$628$	0	0
$629$	−2878.40	−0.182463
$630$	0	0
$631$	2893.31	0.182537	0.0912684	−	0.995826i	$-0.470908\pi$
$631$	2893.31	0.182537	0.0912684	+	0.995826i	$0.470908\pi$
$632$	0	0
$633$	−31513.3	−1.97874
$634$	0	0
$635$	−1279.22	−0.0799438
$636$	0	0
$637$	17609.1	1.09529
$638$	0	0
$639$	1486.43	0.0920226
$640$	0	0
$641$	6867.26	0.423152	0.211576	−	0.977362i	$-0.432140\pi$
$641$	6867.26	0.423152	0.211576	+	0.977362i	$0.432140\pi$
$642$	0	0
$643$	4975.32	0.305144	0.152572	−	0.988292i	$-0.451244\pi$
$643$	4975.32	0.305144	0.152572	+	0.988292i	$0.451244\pi$
$644$	0	0
$645$	26353.0	1.60876
$646$	0	0
$647$	24033.2	1.46035	0.730174	−	0.683262i	$-0.239439\pi$
$647$	24033.2	1.46035	0.730174	+	0.683262i	$0.239439\pi$
$648$	0	0
$649$	16751.9	1.01321
$650$	0	0
$651$	−38961.2	−2.34564
$652$	0	0
$653$	25718.4	1.54125	0.770626	−	0.637288i	$-0.219944\pi$
$653$	25718.4	1.54125	0.770626	+	0.637288i	$0.219944\pi$
$654$	0	0
$655$	−4427.24	−0.264102
$656$	0	0
$657$	−579.221	−0.0343951
$658$	0	0
$659$	−7970.84	−0.471168	−0.235584	−	0.971854i	$-0.575700\pi$
$659$	−7970.84	−0.471168	−0.235584	+	0.971854i	$0.575700\pi$
$660$	0	0
$661$	−14207.8	−0.836038	−0.418019	−	0.908438i	$-0.637275\pi$
$661$	−14207.8	−0.836038	−0.418019	+	0.908438i	$0.637275\pi$
$662$	0	0
$663$	−4545.43	−0.266259
$664$	0	0
$665$	5969.77	0.348117
$666$	0	0
$667$	−18232.2	−1.05840
$668$	0	0
$669$	−29026.1	−1.67745
$670$	0	0
$671$	36157.0	2.08021
$672$	0	0
$673$	17080.0	0.978287	0.489143	−	0.872203i	$-0.337309\pi$
$673$	17080.0	0.978287	0.489143	+	0.872203i	$0.337309\pi$
$674$	0	0
$675$	2119.36	0.120851
$676$	0	0
$677$	16410.9	0.931641	0.465821	−	0.884879i	$-0.345759\pi$
$677$	16410.9	0.931641	0.465821	+	0.884879i	$0.345759\pi$
$678$	0	0
$679$	−25442.9	−1.43801
$680$	0	0
$681$	13814.0	0.777317
$682$	0	0
$683$	−183.169	−0.0102617	−0.00513086	−	0.999987i	$-0.501633\pi$
$683$	−183.169	−0.0102617	−0.00513086	+	0.999987i	$0.501633\pi$
$684$	0	0
$685$	36986.9	2.06306
$686$	0	0
$687$	17492.6	0.971446
$688$	0	0
$689$	24095.4	1.33231
$690$	0	0
$691$	14974.0	0.824366	0.412183	−	0.911101i	$-0.364766\pi$
$691$	14974.0	0.824366	0.412183	+	0.911101i	$0.364766\pi$
$692$	0	0
$693$	−2458.52	−0.134764
$694$	0	0
$695$	668.715	0.0364976
$696$	0	0
$697$	261.408	0.0142059
$698$	0	0
$699$	−6476.74	−0.350462
$700$	0	0
$701$	−4341.08	−0.233895	−0.116947	−	0.993138i	$-0.537311\pi$
$701$	−4341.08	−0.233895	−0.116947	+	0.993138i	$0.537311\pi$
$702$	0	0
$703$	3173.41	0.170253
$704$	0	0
$705$	10287.3	0.549561
$706$	0	0
$707$	−50056.1	−2.66273
$708$	0	0
$709$	−23931.9	−1.26767	−0.633837	−	0.773467i	$-0.718521\pi$
$709$	−23931.9	−1.26767	−0.633837	+	0.773467i	$0.718521\pi$
$710$	0	0
$711$	1239.18	0.0653627
$712$	0	0
$713$	−45712.0	−2.40102
$714$	0	0
$715$	29036.1	1.51873
$716$	0	0
$717$	−18290.1	−0.952656
$718$	0	0
$719$	−16258.6	−0.843314	−0.421657	−	0.906756i	$-0.638551\pi$
$719$	−16258.6	−0.843314	−0.421657	+	0.906756i	$0.638551\pi$
$720$	0	0
$721$	−20137.5	−1.04017
$722$	0	0
$723$	−17736.6	−0.912354
$724$	0	0
$725$	1713.65	0.0877837
$726$	0	0
$727$	2608.95	0.133096	0.0665479	−	0.997783i	$-0.478801\pi$
$727$	2608.95	0.133096	0.0665479	+	0.997783i	$0.478801\pi$
$728$	0	0
$729$	18145.4	0.921881
$730$	0	0
$731$	7127.11	0.360610
$732$	0	0
$733$	−9843.63	−0.496020	−0.248010	−	0.968757i	$-0.579777\pi$
$733$	−9843.63	−0.496020	−0.248010	+	0.968757i	$0.579777\pi$
$734$	0	0
$735$	22855.7	1.14700
$736$	0	0
$737$	−24816.5	−1.24033
$738$	0	0
$739$	2419.46	0.120435	0.0602174	−	0.998185i	$-0.480821\pi$
$739$	2419.46	0.120435	0.0602174	+	0.998185i	$0.480821\pi$
$740$	0	0
$741$	5011.30	0.248441
$742$	0	0
$743$	−10442.9	−0.515631	−0.257816	−	0.966194i	$-0.583003\pi$
$743$	−10442.9	−0.515631	−0.257816	+	0.966194i	$0.583003\pi$
$744$	0	0
$745$	−21475.2	−1.05610
$746$	0	0
$747$	−693.262	−0.0339560
$748$	0	0
$749$	18058.4	0.880963
$750$	0	0
$751$	−29434.6	−1.43021	−0.715103	−	0.699019i	$-0.753620\pi$
$751$	−29434.6	−1.43021	−0.715103	+	0.699019i	$0.753620\pi$
$752$	0	0
$753$	28638.6	1.38599
$754$	0	0
$755$	34335.4	1.65509
$756$	0	0
$757$	20638.3	0.990902	0.495451	−	0.868636i	$-0.335003\pi$
$757$	20638.3	0.990902	0.495451	+	0.868636i	$0.335003\pi$
$758$	0	0
$759$	−44724.7	−2.13887
$760$	0	0
$761$	4103.09	0.195449	0.0977246	−	0.995213i	$-0.468844\pi$
$761$	4103.09	0.195449	0.0977246	+	0.995213i	$0.468844\pi$
$762$	0	0
$763$	34721.0	1.64742
$764$	0	0
$765$	−380.501	−0.0179831
$766$	0	0
$767$	−16494.5	−0.776508
$768$	0	0
$769$	−14020.0	−0.657444	−0.328722	−	0.944427i	$-0.606618\pi$
$769$	−14020.0	−0.657444	−0.328722	+	0.944427i	$0.606618\pi$
$770$	0	0
$771$	−20479.9	−0.956635
$772$	0	0
$773$	−32823.8	−1.52728	−0.763642	−	0.645640i	$-0.776591\pi$
$773$	−32823.8	−1.52728	−0.763642	+	0.645640i	$0.776591\pi$
$774$	0	0
$775$	4296.46	0.199140
$776$	0	0
$777$	23768.4	1.09741
$778$	0	0
$779$	−288.200	−0.0132553
$780$	0	0
$781$	−39817.0	−1.82428
$782$	0	0
$783$	14747.4	0.673090
$784$	0	0
$785$	16389.0	0.745157
$786$	0	0
$787$	33903.0	1.53559	0.767796	−	0.640694i	$-0.221353\pi$
$787$	33903.0	1.53559	0.767796	+	0.640694i	$0.221353\pi$
$788$	0	0
$789$	4830.54	0.217962
$790$	0	0
$791$	−18897.7	−0.849462
$792$	0	0
$793$	−35601.3	−1.59425
$794$	0	0
$795$	31274.6	1.39521
$796$	0	0
$797$	−20830.2	−0.925778	−0.462889	−	0.886416i	$-0.653187\pi$
$797$	−20830.2	−0.925778	−0.462889	+	0.886416i	$0.653187\pi$
$798$	0	0
$799$	2782.16	0.123186
$800$	0	0
$801$	−1254.18	−0.0553238
$802$	0	0
$803$	15515.6	0.681859
$804$	0	0
$805$	52460.0	2.29686
$806$	0	0
$807$	14213.8	0.620012
$808$	0	0
$809$	−26505.7	−1.15190	−0.575952	−	0.817484i	$-0.695368\pi$
$809$	−26505.7	−1.15190	−0.575952	+	0.817484i	$0.695368\pi$
$810$	0	0
$811$	−10829.0	−0.468876	−0.234438	−	0.972131i	$-0.575325\pi$
$811$	−10829.0	−0.468876	−0.234438	+	0.972131i	$0.575325\pi$
$812$	0	0
$813$	2774.94	0.119706
$814$	0	0
$815$	44371.5	1.90707
$816$	0	0
$817$	−7857.58	−0.336477
$818$	0	0
$819$	2420.73	0.103281
$820$	0	0
$821$	−42046.8	−1.78738	−0.893692	−	0.448681i	$-0.851894\pi$
$821$	−42046.8	−1.78738	−0.893692	+	0.448681i	$0.851894\pi$
$822$	0	0
$823$	12949.0	0.548448	0.274224	−	0.961666i	$-0.411579\pi$
$823$	12949.0	0.548448	0.274224	+	0.961666i	$0.411579\pi$
$824$	0	0
$825$	4203.67	0.177397
$826$	0	0
$827$	−13874.0	−0.583368	−0.291684	−	0.956515i	$-0.594216\pi$
$827$	−13874.0	−0.583368	−0.291684	+	0.956515i	$0.594216\pi$
$828$	0	0
$829$	2387.20	0.100013	0.0500065	−	0.998749i	$-0.484076\pi$
$829$	2387.20	0.100013	0.0500065	+	0.998749i	$0.484076\pi$
$830$	0	0
$831$	12706.2	0.530412
$832$	0	0
$833$	6181.27	0.257105
$834$	0	0
$835$	36594.4	1.51665
$836$	0	0
$837$	36974.8	1.52692
$838$	0	0
$839$	−47308.8	−1.94670	−0.973350	−	0.229324i	$-0.926349\pi$
$839$	−47308.8	−1.94670	−0.973350	+	0.229324i	$0.926349\pi$
$840$	0	0
$841$	−12464.7	−0.511080
$842$	0	0
$843$	8200.99	0.335062
$844$	0	0
$845$	−2530.37	−0.103015
$846$	0	0
$847$	30599.2	1.24132
$848$	0	0
$849$	−25593.6	−1.03459
$850$	0	0
$851$	27886.7	1.12332
$852$	0	0
$853$	18211.3	0.730998	0.365499	−	0.930812i	$-0.380898\pi$
$853$	18211.3	0.730998	0.365499	+	0.930812i	$0.380898\pi$
$854$	0	0
$855$	419.499	0.0167796
$856$	0	0
$857$	−14824.5	−0.590893	−0.295447	−	0.955359i	$-0.595468\pi$
$857$	−14824.5	−0.590893	−0.295447	+	0.955359i	$0.595468\pi$
$858$	0	0
$859$	−44191.1	−1.75527	−0.877637	−	0.479326i	$-0.840881\pi$
$859$	−44191.1	−1.75527	−0.877637	+	0.479326i	$0.840881\pi$
$860$	0	0
$861$	−2158.58	−0.0854403
$862$	0	0
$863$	34927.0	1.37767	0.688835	−	0.724918i	$-0.258122\pi$
$863$	34927.0	1.37767	0.688835	+	0.724918i	$0.258122\pi$
$864$	0	0
$865$	3475.40	0.136609
$866$	0	0
$867$	24798.5	0.971395
$868$	0	0
$869$	−33193.9	−1.29577
$870$	0	0
$871$	24435.1	0.950575
$872$	0	0
$873$	−1787.89	−0.0693138
$874$	0	0
$875$	34344.1	1.32691
$876$	0	0
$877$	16914.8	0.651278	0.325639	−	0.945494i	$-0.394421\pi$
$877$	16914.8	0.651278	0.325639	+	0.945494i	$0.394421\pi$
$878$	0	0
$879$	17132.6	0.657415
$880$	0	0
$881$	12634.8	0.483177	0.241588	−	0.970379i	$-0.422332\pi$
$881$	12634.8	0.483177	0.241588	+	0.970379i	$0.422332\pi$
$882$	0	0
$883$	−47260.3	−1.80117	−0.900586	−	0.434678i	$-0.856862\pi$
$883$	−47260.3	−1.80117	−0.900586	+	0.434678i	$0.856862\pi$
$884$	0	0
$885$	−21409.0	−0.813169
$886$	0	0
$887$	4758.84	0.180142	0.0900711	−	0.995935i	$-0.471291\pi$
$887$	4758.84	0.180142	0.0900711	+	0.995935i	$0.471291\pi$
$888$	0	0
$889$	2856.78	0.107777
$890$	0	0
$891$	38682.1	1.45443
$892$	0	0
$893$	−3067.31	−0.114943
$894$	0	0
$895$	35540.6	1.32736
$896$	0	0
$897$	44037.4	1.63920
$898$	0	0
$899$	29896.6	1.10913
$900$	0	0
$901$	8458.13	0.312743
$902$	0	0
$903$	−58852.1	−2.16885
$904$	0	0
$905$	−38732.4	−1.42266
$906$	0	0
$907$	−4111.61	−0.150522	−0.0752612	−	0.997164i	$-0.523979\pi$
$907$	−4111.61	−0.150522	−0.0752612	+	0.997164i	$0.523979\pi$
$908$	0	0
$909$	−3517.47	−0.128347
$910$	0	0
$911$	35113.1	1.27700	0.638501	−	0.769621i	$-0.279555\pi$
$911$	35113.1	1.27700	0.638501	+	0.769621i	$0.279555\pi$
$912$	0	0
$913$	18570.4	0.673155
$914$	0	0
$915$	−46208.5	−1.66952
$916$	0	0
$917$	9887.00	0.356049
$918$	0	0
$919$	−37509.1	−1.34637	−0.673184	−	0.739475i	$-0.735074\pi$
$919$	−37509.1	−1.34637	−0.673184	+	0.739475i	$0.735074\pi$
$920$	0	0
$921$	−32226.0	−1.15297
$922$	0	0
$923$	39205.1	1.39811
$924$	0	0
$925$	−2621.07	−0.0931677
$926$	0	0
$927$	−1415.07	−0.0501371
$928$	0	0
$929$	18393.8	0.649604	0.324802	−	0.945782i	$-0.394702\pi$
$929$	18393.8	0.649604	0.324802	+	0.945782i	$0.394702\pi$
$930$	0	0
$931$	−6814.80	−0.239899
$932$	0	0
$933$	27479.0	0.964226
$934$	0	0
$935$	10192.5	0.356502
$936$	0	0
$937$	−17106.7	−0.596427	−0.298214	−	0.954499i	$-0.596391\pi$
$937$	−17106.7	−0.596427	−0.298214	+	0.954499i	$0.596391\pi$
$938$	0	0
$939$	−11451.7	−0.397990
$940$	0	0
$941$	38100.7	1.31992	0.659961	−	0.751300i	$-0.270573\pi$
$941$	38100.7	1.31992	0.659961	+	0.751300i	$0.270573\pi$
$942$	0	0
$943$	−2532.59	−0.0874576
$944$	0	0
$945$	−42433.0	−1.46068
$946$	0	0
$947$	−52712.7	−1.80880	−0.904400	−	0.426686i	$-0.859681\pi$
$947$	−52712.7	−1.80880	−0.904400	+	0.426686i	$0.859681\pi$
$948$	0	0
$949$	−15277.1	−0.522567
$950$	0	0
$951$	23944.4	0.816458
$952$	0	0
$953$	−6744.38	−0.229246	−0.114623	−	0.993409i	$-0.536566\pi$
$953$	−6744.38	−0.229246	−0.114623	+	0.993409i	$0.536566\pi$
$954$	0	0
$955$	−17819.0	−0.603781
$956$	0	0
$957$	29250.9	0.988032
$958$	0	0
$959$	−82599.8	−2.78132
$960$	0	0
$961$	45165.9	1.51609
$962$	0	0
$963$	1268.98	0.0424634
$964$	0	0
$965$	48670.9	1.62360
$966$	0	0
$967$	28449.8	0.946104	0.473052	−	0.881034i	$-0.343152\pi$
$967$	28449.8	0.946104	0.473052	+	0.881034i	$0.343152\pi$
$968$	0	0
$969$	1759.10	0.0583183
$970$	0	0
$971$	−13018.7	−0.430266	−0.215133	−	0.976585i	$-0.569019\pi$
$971$	−13018.7	−0.430266	−0.215133	+	0.976585i	$0.569019\pi$
$972$	0	0
$973$	−1493.39	−0.0492043
$974$	0	0
$975$	−4139.06	−0.135955
$976$	0	0
$977$	−20043.8	−0.656355	−0.328178	−	0.944616i	$-0.606434\pi$
$977$	−20043.8	−0.656355	−0.328178	+	0.944616i	$0.606434\pi$
$978$	0	0
$979$	33595.7	1.09676
$980$	0	0
$981$	2439.86	0.0794076
$982$	0	0
$983$	1823.72	0.0591735	0.0295868	−	0.999562i	$-0.490581\pi$
$983$	1823.72	0.0591735	0.0295868	+	0.999562i	$0.490581\pi$
$984$	0	0
$985$	64300.1	2.07997
$986$	0	0
$987$	−22973.7	−0.740892
$988$	0	0
$989$	−69049.4	−2.22006
$990$	0	0
$991$	26635.4	0.853786	0.426893	−	0.904302i	$-0.359608\pi$
$991$	26635.4	0.853786	0.426893	+	0.904302i	$0.359608\pi$
$992$	0	0
$993$	22261.0	0.711411
$994$	0	0
$995$	19761.7	0.629637
$996$	0	0
$997$	−50874.6	−1.61606	−0.808031	−	0.589139i	$-0.799467\pi$
$997$	−50874.6	−1.61606	−0.808031	+	0.589139i	$0.799467\pi$
$998$	0	0
$999$	−22556.6	−0.714372

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.h.1.1		2
4.3	odd	2		1216.4.a.o.1.2		2
8.3	odd	2		76.4.a.a.1.1	✓	2
8.5	even	2		304.4.a.f.1.2		2
24.11	even	2		684.4.a.g.1.1		2
40.3	even	4		1900.4.c.b.1749.1		4
40.19	odd	2		1900.4.a.b.1.2		2
40.27	even	4		1900.4.c.b.1749.4		4
152.75	even	2		1444.4.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.a.a.1.1	✓	2	8.3	odd	2
304.4.a.f.1.2		2	8.5	even	2
684.4.a.g.1.1		2	24.11	even	2
1216.4.a.h.1.1		2	1.1	even	1	trivial
1216.4.a.o.1.2		2	4.3	odd	2
1444.4.a.d.1.2		2	152.75	even	2
1900.4.a.b.1.2		2	40.19	odd	2
1900.4.c.b.1749.1		4	40.3	even	4
1900.4.c.b.1749.4		4	40.27	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q-5.37228 q^{3} -11.8614 q^{5} +26.4891 q^{7} +1.86141 q^{9} +O(q^{10})\)	\(q-5.37228 q^{3} -11.8614 q^{5} +26.4891 q^{7} +1.86141 q^{9} -49.8614 q^{11} +49.0951 q^{13} +63.7228 q^{15} +17.2337 q^{17} -19.0000 q^{19} -142.307 q^{21} -166.965 q^{23} +15.6930 q^{25} +135.052 q^{27} +109.198 q^{29} +273.783 q^{31} +267.870 q^{33} -314.198 q^{35} -167.022 q^{37} -263.753 q^{39} +15.1684 q^{41} +413.557 q^{43} -22.0789 q^{45} +161.438 q^{47} +358.674 q^{49} -92.5842 q^{51} +490.791 q^{53} +591.426 q^{55} +102.073 q^{57} -335.970 q^{59} -725.149 q^{61} +49.3070 q^{63} -582.337 q^{65} +497.709 q^{67} +896.981 q^{69} +798.554 q^{71} -311.174 q^{73} -84.3070 q^{75} -1320.79 q^{77} +665.723 q^{79} -775.793 q^{81} -372.440 q^{83} -204.416 q^{85} -586.644 q^{87} -673.783 q^{89} +1300.49 q^{91} -1470.84 q^{93} +225.367 q^{95} -960.505 q^{97} -92.8124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{3} + 5 q^{5} + 30 q^{7} - 25 q^{9}+O(q^{10}) \) 2 * q - 5 * q^3 + 5 * q^5 + 30 * q^7 - 25 * q^9	\( 2 q - 5 q^{3} + 5 q^{5} + 30 q^{7} - 25 q^{9} - 71 q^{11} + 35 q^{13} + 70 q^{15} - 38 q^{19} - 141 q^{21} + 5 q^{23} + 175 q^{25} + 115 q^{27} - 155 q^{29} + 88 q^{31} + 260 q^{33} - 255 q^{35} - 380 q^{37} - 269 q^{39} - 142 q^{41} + 155 q^{43} - 475 q^{45} + 455 q^{47} + 28 q^{49} - 99 q^{51} + 275 q^{53} + 235 q^{55} + 95 q^{57} - 873 q^{59} - 445 q^{61} - 45 q^{63} - 820 q^{65} + 645 q^{67} + 961 q^{69} + 1712 q^{71} - 990 q^{73} - 25 q^{75} - 1395 q^{77} + 1274 q^{79} - 58 q^{81} - 90 q^{83} - 495 q^{85} - 685 q^{87} - 888 q^{89} + 1251 q^{91} - 1540 q^{93} - 95 q^{95} + 710 q^{97} + 475 q^{99}+O(q^{100}) \) 2 * q - 5 * q^3 + 5 * q^5 + 30 * q^7 - 25 * q^9 - 71 * q^11 + 35 * q^13 + 70 * q^15 - 38 * q^19 - 141 * q^21 + 5 * q^23 + 175 * q^25 + 115 * q^27 - 155 * q^29 + 88 * q^31 + 260 * q^33 - 255 * q^35 - 380 * q^37 - 269 * q^39 - 142 * q^41 + 155 * q^43 - 475 * q^45 + 455 * q^47 + 28 * q^49 - 99 * q^51 + 275 * q^53 + 235 * q^55 + 95 * q^57 - 873 * q^59 - 445 * q^61 - 45 * q^63 - 820 * q^65 + 645 * q^67 + 961 * q^69 + 1712 * q^71 - 990 * q^73 - 25 * q^75 - 1395 * q^77 + 1274 * q^79 - 58 * q^81 - 90 * q^83 - 495 * q^85 - 685 * q^87 - 888 * q^89 + 1251 * q^91 - 1540 * q^93 - 95 * q^95 + 710 * q^97 + 475 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more