LMFDB - Embedded newform 1216.4.a.bc.1.5

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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 20x^{3} + 24x^{2} + 2x - 4 $ x^5 - x^4 - 20x^3 + 24x^2 + 2*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4.32701$ 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)

$f(q)$	$=$	$q+10.0529 q^{3} -16.1852 q^{5} +6.15678 q^{7} +74.0603 q^{9} +O(q^{10})$	$q+10.0529 q^{3} -16.1852 q^{5} +6.15678 q^{7} +74.0603 q^{9} -56.4194 q^{11} -42.6330 q^{13} -162.708 q^{15} +21.7751 q^{17} +19.0000 q^{19} +61.8934 q^{21} +155.562 q^{23} +136.961 q^{25} +473.092 q^{27} -119.791 q^{29} -292.567 q^{31} -567.177 q^{33} -99.6489 q^{35} -130.110 q^{37} -428.585 q^{39} -326.866 q^{41} +220.998 q^{43} -1198.68 q^{45} +146.383 q^{47} -305.094 q^{49} +218.902 q^{51} -272.878 q^{53} +913.160 q^{55} +191.005 q^{57} -752.429 q^{59} -510.875 q^{61} +455.973 q^{63} +690.025 q^{65} -412.776 q^{67} +1563.85 q^{69} +218.017 q^{71} -451.880 q^{73} +1376.86 q^{75} -347.362 q^{77} -151.669 q^{79} +2756.30 q^{81} -58.3732 q^{83} -352.435 q^{85} -1204.24 q^{87} -1061.58 q^{89} -262.482 q^{91} -2941.14 q^{93} -307.519 q^{95} -1265.17 q^{97} -4178.44 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 27 q^{5} + 20 q^{7} + 154 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 - 27 * q^5 + 20 * q^7 + 154 * q^9	$ 5 q + 3 q^{3} - 27 q^{5} + 20 q^{7} + 154 q^{9} - 67 q^{11} - 191 q^{13} - 36 q^{15} - 52 q^{17} + 95 q^{19} + 29 q^{21} + 35 q^{23} - 2 q^{25} - 27 q^{27} + 33 q^{29} - 694 q^{31} + 298 q^{33} - 387 q^{35} - 108 q^{37} - 31 q^{39} + 54 q^{41} + 427 q^{43} - 1699 q^{45} + 79 q^{47} + 1033 q^{49} - 363 q^{51} - 1025 q^{53} + 1431 q^{55} + 57 q^{57} + 649 q^{59} - 1413 q^{61} - 1449 q^{63} + 548 q^{65} - 915 q^{67} - 407 q^{69} + 1444 q^{71} + 2682 q^{73} + 465 q^{75} - 169 q^{77} - 836 q^{79} + 2925 q^{81} + 502 q^{83} - 373 q^{85} - 3491 q^{87} + 996 q^{89} - 2919 q^{91} - 2492 q^{93} - 513 q^{95} + 424 q^{97} - 7249 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 27 * q^5 + 20 * q^7 + 154 * q^9 - 67 * q^11 - 191 * q^13 - 36 * q^15 - 52 * q^17 + 95 * q^19 + 29 * q^21 + 35 * q^23 - 2 * q^25 - 27 * q^27 + 33 * q^29 - 694 * q^31 + 298 * q^33 - 387 * q^35 - 108 * q^37 - 31 * q^39 + 54 * q^41 + 427 * q^43 - 1699 * q^45 + 79 * q^47 + 1033 * q^49 - 363 * q^51 - 1025 * q^53 + 1431 * q^55 + 57 * q^57 + 649 * q^59 - 1413 * q^61 - 1449 * q^63 + 548 * q^65 - 915 * q^67 - 407 * q^69 + 1444 * q^71 + 2682 * q^73 + 465 * q^75 - 169 * q^77 - 836 * q^79 + 2925 * q^81 + 502 * q^83 - 373 * q^85 - 3491 * q^87 + 996 * q^89 - 2919 * q^91 - 2492 * q^93 - 513 * q^95 + 424 * q^97 - 7249 * 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
$3$	10.0529	1.93468	0.967339	−	0.253488i	$-0.0815778\pi$
$3$	10.0529	1.93468	0.967339	+	0.253488i	$0.0815778\pi$
$4$	0	0
$5$	−16.1852	−1.44765	−0.723825	−	0.689984i	$-0.757618\pi$
$5$	−16.1852	−1.44765	−0.723825	+	0.689984i	$0.757618\pi$
$6$	0	0
$7$	6.15678	0.332435	0.166217	−	0.986089i	$-0.446845\pi$
$7$	6.15678	0.332435	0.166217	+	0.986089i	$0.446845\pi$
$8$	0	0
$9$	74.0603	2.74298
$10$	0	0
$11$	−56.4194	−1.54646	−0.773231	−	0.634124i	$-0.781361\pi$
$11$	−56.4194	−1.54646	−0.773231	+	0.634124i	$0.781361\pi$
$12$	0	0
$13$	−42.6330	−0.909560	−0.454780	−	0.890604i	$-0.650282\pi$
$13$	−42.6330	−0.909560	−0.454780	+	0.890604i	$0.650282\pi$
$14$	0	0
$15$	−162.708	−2.80074
$16$	0	0
$17$	21.7751	0.310661	0.155331	−	0.987863i	$-0.450356\pi$
$17$	21.7751	0.310661	0.155331	+	0.987863i	$0.450356\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	61.8934	0.643154
$22$	0	0
$23$	155.562	1.41030	0.705150	−	0.709058i	$-0.250880\pi$
$23$	155.562	1.41030	0.705150	+	0.709058i	$0.250880\pi$
$24$	0	0
$25$	136.961	1.09569
$26$	0	0
$27$	473.092	3.37209
$28$	0	0
$29$	−119.791	−0.767056	−0.383528	−	0.923529i	$-0.625291\pi$
$29$	−119.791	−0.767056	−0.383528	+	0.923529i	$0.625291\pi$
$30$	0	0
$31$	−292.567	−1.69505	−0.847527	−	0.530753i	$-0.821909\pi$
$31$	−292.567	−1.69505	−0.847527	+	0.530753i	$0.821909\pi$
$32$	0	0
$33$	−567.177	−2.99191
$34$	0	0
$35$	−99.6489	−0.481249
$36$	0	0
$37$	−130.110	−0.578106	−0.289053	−	0.957313i	$-0.593340\pi$
$37$	−130.110	−0.578106	−0.289053	+	0.957313i	$0.593340\pi$
$38$	0	0
$39$	−428.585	−1.75970
$40$	0	0
$41$	−326.866	−1.24507	−0.622536	−	0.782591i	$-0.713898\pi$
$41$	−326.866	−1.24507	−0.622536	+	0.782591i	$0.713898\pi$
$42$	0	0
$43$	220.998	0.783766	0.391883	−	0.920015i	$-0.371824\pi$
$43$	220.998	0.783766	0.391883	+	0.920015i	$0.371824\pi$
$44$	0	0
$45$	−1198.68	−3.97087
$46$	0	0
$47$	146.383	0.454300	0.227150	−	0.973860i	$-0.427059\pi$
$47$	146.383	0.454300	0.227150	+	0.973860i	$0.427059\pi$
$48$	0	0
$49$	−305.094	−0.889487
$50$	0	0
$51$	218.902	0.601029
$52$	0	0
$53$	−272.878	−0.707220	−0.353610	−	0.935393i	$-0.615046\pi$
$53$	−272.878	−0.707220	−0.353610	+	0.935393i	$0.615046\pi$
$54$	0	0
$55$	913.160	2.23874
$56$	0	0
$57$	191.005	0.443845
$58$	0	0
$59$	−752.429	−1.66030	−0.830152	−	0.557537i	$-0.811746\pi$
$59$	−752.429	−1.66030	−0.830152	+	0.557537i	$0.811746\pi$
$60$	0	0
$61$	−510.875	−1.07231	−0.536155	−	0.844120i	$-0.680124\pi$
$61$	−510.875	−1.07231	−0.536155	+	0.844120i	$0.680124\pi$
$62$	0	0
$63$	455.973	0.911861
$64$	0	0
$65$	690.025	1.31672
$66$	0	0
$67$	−412.776	−0.752665	−0.376333	−	0.926485i	$-0.622815\pi$
$67$	−412.776	−0.752665	−0.376333	+	0.926485i	$0.622815\pi$
$68$	0	0
$69$	1563.85	2.72848
$70$	0	0
$71$	218.017	0.364421	0.182211	−	0.983260i	$-0.441675\pi$
$71$	218.017	0.364421	0.182211	+	0.983260i	$0.441675\pi$
$72$	0	0
$73$	−451.880	−0.724501	−0.362251	−	0.932081i	$-0.617992\pi$
$73$	−451.880	−0.724501	−0.362251	+	0.932081i	$0.617992\pi$
$74$	0	0
$75$	1376.86	2.11981
$76$	0	0
$77$	−347.362	−0.514098
$78$	0	0
$79$	−151.669	−0.216002	−0.108001	−	0.994151i	$-0.534445\pi$
$79$	−151.669	−0.216002	−0.108001	+	0.994151i	$0.534445\pi$
$80$	0	0
$81$	2756.30	3.78094
$82$	0	0
$83$	−58.3732	−0.0771962	−0.0385981	−	0.999255i	$-0.512289\pi$
$83$	−58.3732	−0.0771962	−0.0385981	+	0.999255i	$0.512289\pi$
$84$	0	0
$85$	−352.435	−0.449729
$86$	0	0
$87$	−1204.24	−1.48401
$88$	0	0
$89$	−1061.58	−1.26435	−0.632177	−	0.774824i	$-0.717838\pi$
$89$	−1061.58	−1.26435	−0.632177	+	0.774824i	$0.717838\pi$
$90$	0	0
$91$	−262.482	−0.302369
$92$	0	0
$93$	−2941.14	−3.27938
$94$	0	0
$95$	−307.519	−0.332114
$96$	0	0
$97$	−1265.17	−1.32432	−0.662158	−	0.749364i	$-0.730359\pi$
$97$	−1265.17	−1.32432	−0.662158	+	0.749364i	$0.730359\pi$
$98$	0	0
$99$	−4178.44	−4.24191
$100$	0	0
$101$	−93.1524	−0.0917723	−0.0458862	−	0.998947i	$-0.514611\pi$
$101$	−93.1524	−0.0917723	−0.0458862	+	0.998947i	$0.514611\pi$
$102$	0	0
$103$	1300.37	1.24397	0.621987	−	0.783027i	$-0.286325\pi$
$103$	1300.37	1.24397	0.621987	+	0.783027i	$0.286325\pi$
$104$	0	0
$105$	−1001.76	−0.931062
$106$	0	0
$107$	−1378.54	−1.24550	−0.622751	−	0.782420i	$-0.713985\pi$
$107$	−1378.54	−1.24550	−0.622751	+	0.782420i	$0.713985\pi$
$108$	0	0
$109$	1835.37	1.61281	0.806407	−	0.591361i	$-0.201409\pi$
$109$	1835.37	1.61281	0.806407	+	0.591361i	$0.201409\pi$
$110$	0	0
$111$	−1307.98	−1.11845
$112$	0	0
$113$	1149.47	0.956928	0.478464	−	0.878107i	$-0.341194\pi$
$113$	1149.47	0.956928	0.478464	+	0.878107i	$0.341194\pi$
$114$	0	0
$115$	−2517.80	−2.04162
$116$	0	0
$117$	−3157.42	−2.49490
$118$	0	0
$119$	134.065	0.103275
$120$	0	0
$121$	1852.15	1.39155
$122$	0	0
$123$	−3285.95	−2.40881
$124$	0	0
$125$	−193.596	−0.138526
$126$	0	0
$127$	−1675.24	−1.17050	−0.585249	−	0.810854i	$-0.699003\pi$
$127$	−1675.24	−1.17050	−0.585249	+	0.810854i	$0.699003\pi$
$128$	0	0
$129$	2221.67	1.51633
$130$	0	0
$131$	819.566	0.546609	0.273305	−	0.961928i	$-0.411883\pi$
$131$	819.566	0.546609	0.273305	+	0.961928i	$0.411883\pi$
$132$	0	0
$133$	116.979	0.0762658
$134$	0	0
$135$	−7657.09	−4.88161
$136$	0	0
$137$	−703.345	−0.438619	−0.219310	−	0.975655i	$-0.570381\pi$
$137$	−703.345	−0.438619	−0.219310	+	0.975655i	$0.570381\pi$
$138$	0	0
$139$	3148.52	1.92125	0.960627	−	0.277841i	$-0.0896190\pi$
$139$	3148.52	1.92125	0.960627	+	0.277841i	$0.0896190\pi$
$140$	0	0
$141$	1471.57	0.878923
$142$	0	0
$143$	2405.33	1.40660
$144$	0	0
$145$	1938.84	1.11043
$146$	0	0
$147$	−3067.07	−1.72087
$148$	0	0
$149$	−3178.18	−1.74743	−0.873714	−	0.486440i	$-0.838295\pi$
$149$	−3178.18	−1.74743	−0.873714	+	0.486440i	$0.838295\pi$
$150$	0	0
$151$	723.050	0.389676	0.194838	−	0.980835i	$-0.437582\pi$
$151$	723.050	0.389676	0.194838	+	0.980835i	$0.437582\pi$
$152$	0	0
$153$	1612.67	0.852136
$154$	0	0
$155$	4735.27	2.45384
$156$	0	0
$157$	−273.470	−0.139014	−0.0695072	−	0.997581i	$-0.522143\pi$
$157$	−273.470	−0.139014	−0.0695072	+	0.997581i	$0.522143\pi$
$158$	0	0
$159$	−2743.21	−1.36824
$160$	0	0
$161$	957.761	0.468833
$162$	0	0
$163$	1872.36	0.899722	0.449861	−	0.893099i	$-0.351474\pi$
$163$	1872.36	0.899722	0.449861	+	0.893099i	$0.351474\pi$
$164$	0	0
$165$	9179.89	4.33123
$166$	0	0
$167$	1295.11	0.600112	0.300056	−	0.953922i	$-0.402995\pi$
$167$	1295.11	0.600112	0.300056	+	0.953922i	$0.402995\pi$
$168$	0	0
$169$	−379.425	−0.172701
$170$	0	0
$171$	1407.15	0.629282
$172$	0	0
$173$	−2022.23	−0.888711	−0.444356	−	0.895850i	$-0.646567\pi$
$173$	−2022.23	−0.888711	−0.444356	+	0.895850i	$0.646567\pi$
$174$	0	0
$175$	843.241	0.364246
$176$	0	0
$177$	−7564.08	−3.21215
$178$	0	0
$179$	−1230.63	−0.513864	−0.256932	−	0.966429i	$-0.582712\pi$
$179$	−1230.63	−0.513864	−0.256932	+	0.966429i	$0.582712\pi$
$180$	0	0
$181$	3404.40	1.39805	0.699026	−	0.715096i	$-0.253617\pi$
$181$	3404.40	1.39805	0.699026	+	0.715096i	$0.253617\pi$
$182$	0	0
$183$	−5135.77	−2.07457
$184$	0	0
$185$	2105.86	0.836895
$186$	0	0
$187$	−1228.54	−0.480426
$188$	0	0
$189$	2912.72	1.12100
$190$	0	0
$191$	−2198.25	−0.832773	−0.416386	−	0.909188i	$-0.636704\pi$
$191$	−2198.25	−0.832773	−0.416386	+	0.909188i	$0.636704\pi$
$192$	0	0
$193$	4588.16	1.71121	0.855604	−	0.517631i	$-0.173186\pi$
$193$	4588.16	1.71121	0.855604	+	0.517631i	$0.173186\pi$
$194$	0	0
$195$	6936.74	2.54744
$196$	0	0
$197$	−1595.22	−0.576927	−0.288463	−	0.957491i	$-0.593144\pi$
$197$	−1595.22	−0.576927	−0.288463	+	0.957491i	$0.593144\pi$
$198$	0	0
$199$	2159.96	0.769423	0.384712	−	0.923037i	$-0.374301\pi$
$199$	2159.96	0.769423	0.384712	+	0.923037i	$0.374301\pi$
$200$	0	0
$201$	−4149.58	−1.45616
$202$	0	0
$203$	−737.527	−0.254996
$204$	0	0
$205$	5290.40	1.80243
$206$	0	0
$207$	11521.0	3.86842
$208$	0	0
$209$	−1071.97	−0.354783
$210$	0	0
$211$	−187.626	−0.0612165	−0.0306083	−	0.999531i	$-0.509744\pi$
$211$	−187.626	−0.0612165	−0.0306083	+	0.999531i	$0.509744\pi$
$212$	0	0
$213$	2191.70	0.705038
$214$	0	0
$215$	−3576.91	−1.13462
$216$	0	0
$217$	−1801.27	−0.563495
$218$	0	0
$219$	−4542.70	−1.40168
$220$	0	0
$221$	−928.339	−0.282565
$222$	0	0
$223$	3103.40	0.931925	0.465962	−	0.884804i	$-0.345708\pi$
$223$	3103.40	0.931925	0.465962	+	0.884804i	$0.345708\pi$
$224$	0	0
$225$	10143.4	3.00545
$226$	0	0
$227$	−629.900	−0.184176	−0.0920880	−	0.995751i	$-0.529354\pi$
$227$	−629.900	−0.184176	−0.0920880	+	0.995751i	$0.529354\pi$
$228$	0	0
$229$	6366.11	1.83705	0.918525	−	0.395363i	$-0.129381\pi$
$229$	6366.11	1.83705	0.918525	+	0.395363i	$0.129381\pi$
$230$	0	0
$231$	−3491.99	−0.994614
$232$	0	0
$233$	−4721.12	−1.32743	−0.663714	−	0.747987i	$-0.731021\pi$
$233$	−4721.12	−1.32743	−0.663714	+	0.747987i	$0.731021\pi$
$234$	0	0
$235$	−2369.23	−0.657667
$236$	0	0
$237$	−1524.71	−0.417894
$238$	0	0
$239$	1528.90	0.413793	0.206896	−	0.978363i	$-0.433664\pi$
$239$	1528.90	0.413793	0.206896	+	0.978363i	$0.433664\pi$
$240$	0	0
$241$	453.241	0.121145	0.0605723	−	0.998164i	$-0.480707\pi$
$241$	453.241	0.121145	0.0605723	+	0.998164i	$0.480707\pi$
$242$	0	0
$243$	14935.3	3.94280
$244$	0	0
$245$	4938.01	1.28767
$246$	0	0
$247$	−810.028	−0.208667
$248$	0	0
$249$	−586.818	−0.149350
$250$	0	0
$251$	−4603.75	−1.15771	−0.578857	−	0.815429i	$-0.696501\pi$
$251$	−4603.75	−1.15771	−0.578857	+	0.815429i	$0.696501\pi$
$252$	0	0
$253$	−8776.71	−2.18098
$254$	0	0
$255$	−3542.98	−0.870079
$256$	0	0
$257$	1408.31	0.341821	0.170910	−	0.985287i	$-0.445329\pi$
$257$	1408.31	0.341821	0.170910	+	0.985287i	$0.445329\pi$
$258$	0	0
$259$	−801.058	−0.192183
$260$	0	0
$261$	−8871.77	−2.10402
$262$	0	0
$263$	−1567.72	−0.367567	−0.183783	−	0.982967i	$-0.558834\pi$
$263$	−1567.72	−0.367567	−0.183783	+	0.982967i	$0.558834\pi$
$264$	0	0
$265$	4416.59	1.02381
$266$	0	0
$267$	−10672.0	−2.44612
$268$	0	0
$269$	−782.679	−0.177401	−0.0887004	−	0.996058i	$-0.528271\pi$
$269$	−782.679	−0.177401	−0.0887004	+	0.996058i	$0.528271\pi$
$270$	0	0
$271$	3020.33	0.677019	0.338510	−	0.940963i	$-0.390077\pi$
$271$	3020.33	0.677019	0.338510	+	0.940963i	$0.390077\pi$
$272$	0	0
$273$	−2638.70	−0.584987
$274$	0	0
$275$	−7727.27	−1.69444
$276$	0	0
$277$	−791.413	−0.171666	−0.0858328	−	0.996310i	$-0.527355\pi$
$277$	−791.413	−0.171666	−0.0858328	+	0.996310i	$0.527355\pi$
$278$	0	0
$279$	−21667.6	−4.64949
$280$	0	0
$281$	6344.21	1.34685	0.673423	−	0.739258i	$-0.264823\pi$
$281$	6344.21	1.34685	0.673423	+	0.739258i	$0.264823\pi$
$282$	0	0
$283$	−2171.54	−0.456129	−0.228065	−	0.973646i	$-0.573240\pi$
$283$	−2171.54	−0.456129	−0.228065	+	0.973646i	$0.573240\pi$
$284$	0	0
$285$	−3091.45	−0.642533
$286$	0	0
$287$	−2012.45	−0.413906
$288$	0	0
$289$	−4438.84	−0.903490
$290$	0	0
$291$	−12718.6	−2.56213
$292$	0	0
$293$	6608.17	1.31759	0.658794	−	0.752323i	$-0.271067\pi$
$293$	6608.17	1.31759	0.658794	+	0.752323i	$0.271067\pi$
$294$	0	0
$295$	12178.2	2.40354
$296$	0	0
$297$	−26691.6	−5.21482
$298$	0	0
$299$	−6632.08	−1.28275
$300$	0	0
$301$	1360.64	0.260551
$302$	0	0
$303$	−936.449	−0.177550
$304$	0	0
$305$	8268.63	1.55233
$306$	0	0
$307$	−2935.95	−0.545810	−0.272905	−	0.962041i	$-0.587984\pi$
$307$	−2935.95	−0.545810	−0.272905	+	0.962041i	$0.587984\pi$
$308$	0	0
$309$	13072.5	2.40669
$310$	0	0
$311$	1750.18	0.319112	0.159556	−	0.987189i	$-0.448994\pi$
$311$	1750.18	0.319112	0.159556	+	0.987189i	$0.448994\pi$
$312$	0	0
$313$	4978.48	0.899043	0.449521	−	0.893270i	$-0.351595\pi$
$313$	4978.48	0.899043	0.449521	+	0.893270i	$0.351595\pi$
$314$	0	0
$315$	−7380.03	−1.32006
$316$	0	0
$317$	−885.389	−0.156872	−0.0784360	−	0.996919i	$-0.524993\pi$
$317$	−885.389	−0.156872	−0.0784360	+	0.996919i	$0.524993\pi$
$318$	0	0
$319$	6758.54	1.18622
$320$	0	0
$321$	−13858.3	−2.40964
$322$	0	0
$323$	413.727	0.0712705
$324$	0	0
$325$	−5839.07	−0.996596
$326$	0	0
$327$	18450.8	3.12027
$328$	0	0
$329$	901.245	0.151025
$330$	0	0
$331$	8615.98	1.43075	0.715374	−	0.698742i	$-0.246257\pi$
$331$	8615.98	1.43075	0.715374	+	0.698742i	$0.246257\pi$
$332$	0	0
$333$	−9635.98	−1.58573
$334$	0	0
$335$	6680.86	1.08960
$336$	0	0
$337$	5605.43	0.906075	0.453038	−	0.891491i	$-0.350340\pi$
$337$	5605.43	0.906075	0.453038	+	0.891491i	$0.350340\pi$
$338$	0	0
$339$	11555.5	1.85135
$340$	0	0
$341$	16506.5	2.62134
$342$	0	0
$343$	−3990.17	−0.628132
$344$	0	0
$345$	−25311.2	−3.94988
$346$	0	0
$347$	−11019.8	−1.70482	−0.852411	−	0.522873i	$-0.824860\pi$
$347$	−11019.8	−1.70482	−0.852411	+	0.522873i	$0.824860\pi$
$348$	0	0
$349$	538.748	0.0826318	0.0413159	−	0.999146i	$-0.486845\pi$
$349$	538.748	0.0826318	0.0413159	+	0.999146i	$0.486845\pi$
$350$	0	0
$351$	−20169.3	−3.06712
$352$	0	0
$353$	−4571.72	−0.689315	−0.344658	−	0.938728i	$-0.612005\pi$
$353$	−4571.72	−0.689315	−0.344658	+	0.938728i	$0.612005\pi$
$354$	0	0
$355$	−3528.66	−0.527555
$356$	0	0
$357$	1347.73	0.199803
$358$	0	0
$359$	−11925.1	−1.75316	−0.876580	−	0.481257i	$-0.840181\pi$
$359$	−11925.1	−1.75316	−0.876580	+	0.481257i	$0.840181\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	18619.4	2.69219
$364$	0	0
$365$	7313.78	1.04882
$366$	0	0
$367$	−257.392	−0.0366097	−0.0183049	−	0.999832i	$-0.505827\pi$
$367$	−257.392	−0.0366097	−0.0183049	+	0.999832i	$0.505827\pi$
$368$	0	0
$369$	−24207.8	−3.41520
$370$	0	0
$371$	−1680.05	−0.235105
$372$	0	0
$373$	4540.80	0.630332	0.315166	−	0.949037i	$-0.397940\pi$
$373$	4540.80	0.630332	0.315166	+	0.949037i	$0.397940\pi$
$374$	0	0
$375$	−1946.20	−0.268003
$376$	0	0
$377$	5107.05	0.697684
$378$	0	0
$379$	1744.78	0.236473	0.118236	−	0.992985i	$-0.462276\pi$
$379$	1744.78	0.236473	0.118236	+	0.992985i	$0.462276\pi$
$380$	0	0
$381$	−16840.9	−2.26453
$382$	0	0
$383$	4642.52	0.619377	0.309689	−	0.950838i	$-0.399775\pi$
$383$	4642.52	0.619377	0.309689	+	0.950838i	$0.399775\pi$
$384$	0	0
$385$	5622.13	0.744234
$386$	0	0
$387$	16367.2	2.14985
$388$	0	0
$389$	−1793.93	−0.233819	−0.116910	−	0.993143i	$-0.537299\pi$
$389$	−1793.93	−0.233819	−0.116910	+	0.993143i	$0.537299\pi$
$390$	0	0
$391$	3387.38	0.438126
$392$	0	0
$393$	8239.00	1.05751
$394$	0	0
$395$	2454.80	0.312695
$396$	0	0
$397$	−12440.0	−1.57266	−0.786332	−	0.617805i	$-0.788022\pi$
$397$	−12440.0	−1.57266	−0.786332	+	0.617805i	$0.788022\pi$
$398$	0	0
$399$	1175.97	0.147550
$400$	0	0
$401$	−8214.16	−1.02293	−0.511466	−	0.859304i	$-0.670897\pi$
$401$	−8214.16	−1.02293	−0.511466	+	0.859304i	$0.670897\pi$
$402$	0	0
$403$	12473.0	1.54175
$404$	0	0
$405$	−44611.4	−5.47348
$406$	0	0
$407$	7340.72	0.894019
$408$	0	0
$409$	5335.24	0.645014	0.322507	−	0.946567i	$-0.395474\pi$
$409$	5335.24	0.645014	0.322507	+	0.946567i	$0.395474\pi$
$410$	0	0
$411$	−7070.65	−0.848587
$412$	0	0
$413$	−4632.54	−0.551943
$414$	0	0
$415$	944.782	0.111753
$416$	0	0
$417$	31651.7	3.71701
$418$	0	0
$419$	3040.69	0.354529	0.177264	−	0.984163i	$-0.443275\pi$
$419$	3040.69	0.354529	0.177264	+	0.984163i	$0.443275\pi$
$420$	0	0
$421$	−16481.8	−1.90802	−0.954008	−	0.299782i	$-0.903086\pi$
$421$	−16481.8	−1.90802	−0.954008	+	0.299782i	$0.903086\pi$
$422$	0	0
$423$	10841.1	1.24613
$424$	0	0
$425$	2982.35	0.340388
$426$	0	0
$427$	−3145.35	−0.356473
$428$	0	0
$429$	24180.5	2.72132
$430$	0	0
$431$	−6865.68	−0.767304	−0.383652	−	0.923478i	$-0.625334\pi$
$431$	−6865.68	−0.767304	−0.383652	+	0.923478i	$0.625334\pi$
$432$	0	0
$433$	−3537.16	−0.392575	−0.196288	−	0.980546i	$-0.562889\pi$
$433$	−3537.16	−0.392575	−0.196288	+	0.980546i	$0.562889\pi$
$434$	0	0
$435$	19491.0	2.14832
$436$	0	0
$437$	2955.68	0.323545
$438$	0	0
$439$	15342.3	1.66799	0.833997	−	0.551769i	$-0.186047\pi$
$439$	15342.3	1.66799	0.833997	+	0.551769i	$0.186047\pi$
$440$	0	0
$441$	−22595.4	−2.43984
$442$	0	0
$443$	−7592.16	−0.814254	−0.407127	−	0.913372i	$-0.633469\pi$
$443$	−7592.16	−0.814254	−0.407127	+	0.913372i	$0.633469\pi$
$444$	0	0
$445$	17181.9	1.83034
$446$	0	0
$447$	−31949.9	−3.38071
$448$	0	0
$449$	−3840.44	−0.403657	−0.201828	−	0.979421i	$-0.564688\pi$
$449$	−3840.44	−0.403657	−0.201828	+	0.979421i	$0.564688\pi$
$450$	0	0
$451$	18441.6	1.92546
$452$	0	0
$453$	7268.74	0.753896
$454$	0	0
$455$	4248.33	0.437725
$456$	0	0
$457$	9924.31	1.01584	0.507921	−	0.861404i	$-0.330414\pi$
$457$	9924.31	1.01584	0.507921	+	0.861404i	$0.330414\pi$
$458$	0	0
$459$	10301.6	1.04758
$460$	0	0
$461$	1832.00	0.185086	0.0925429	−	0.995709i	$-0.470500\pi$
$461$	1832.00	0.185086	0.0925429	+	0.995709i	$0.470500\pi$
$462$	0	0
$463$	−3477.49	−0.349055	−0.174528	−	0.984652i	$-0.555840\pi$
$463$	−3477.49	−0.349055	−0.174528	+	0.984652i	$0.555840\pi$
$464$	0	0
$465$	47603.1	4.74740
$466$	0	0
$467$	−14109.1	−1.39805	−0.699027	−	0.715095i	$-0.746383\pi$
$467$	−14109.1	−1.39805	−0.699027	+	0.715095i	$0.746383\pi$
$468$	0	0
$469$	−2541.37	−0.250212
$470$	0	0
$471$	−2749.16	−0.268948
$472$	0	0
$473$	−12468.6	−1.21206
$474$	0	0
$475$	2602.26	0.251369
$476$	0	0
$477$	−20209.4	−1.93989
$478$	0	0
$479$	−12145.8	−1.15857	−0.579285	−	0.815125i	$-0.696668\pi$
$479$	−12145.8	−1.15857	−0.579285	+	0.815125i	$0.696668\pi$
$480$	0	0
$481$	5546.98	0.525822
$482$	0	0
$483$	9628.26	0.907041
$484$	0	0
$485$	20477.1	1.91715
$486$	0	0
$487$	−19965.9	−1.85779	−0.928893	−	0.370349i	$-0.879238\pi$
$487$	−19965.9	−1.85779	−0.928893	+	0.370349i	$0.879238\pi$
$488$	0	0
$489$	18822.6	1.74067
$490$	0	0
$491$	8244.96	0.757820	0.378910	−	0.925434i	$-0.376299\pi$
$491$	8244.96	0.757820	0.378910	+	0.925434i	$0.376299\pi$
$492$	0	0
$493$	−2608.46	−0.238295
$494$	0	0
$495$	67629.0	6.14080
$496$	0	0
$497$	1342.29	0.121146
$498$	0	0
$499$	12060.0	1.08192	0.540962	−	0.841047i	$-0.318060\pi$
$499$	12060.0	1.08192	0.540962	+	0.841047i	$0.318060\pi$
$500$	0	0
$501$	13019.6	1.16102
$502$	0	0
$503$	1699.89	0.150685	0.0753423	−	0.997158i	$-0.475995\pi$
$503$	1699.89	0.150685	0.0753423	+	0.997158i	$0.475995\pi$
$504$	0	0
$505$	1507.69	0.132854
$506$	0	0
$507$	−3814.31	−0.334121
$508$	0	0
$509$	15289.1	1.33139	0.665695	−	0.746224i	$-0.268135\pi$
$509$	15289.1	1.33139	0.665695	+	0.746224i	$0.268135\pi$
$510$	0	0
$511$	−2782.13	−0.240850
$512$	0	0
$513$	8988.74	0.773612
$514$	0	0
$515$	−21046.8	−1.80084
$516$	0	0
$517$	−8258.81	−0.702557
$518$	0	0
$519$	−20329.2	−1.71937
$520$	0	0
$521$	4927.63	0.414363	0.207182	−	0.978302i	$-0.433571\pi$
$521$	4927.63	0.414363	0.207182	+	0.978302i	$0.433571\pi$
$522$	0	0
$523$	−3932.07	−0.328752	−0.164376	−	0.986398i	$-0.552561\pi$
$523$	−3932.07	−0.328752	−0.164376	+	0.986398i	$0.552561\pi$
$524$	0	0
$525$	8477.00	0.704698
$526$	0	0
$527$	−6370.69	−0.526587
$528$	0	0
$529$	12032.5	0.988948
$530$	0	0
$531$	−55725.2	−4.55417
$532$	0	0
$533$	13935.3	1.13247
$534$	0	0
$535$	22312.0	1.80305
$536$	0	0
$537$	−12371.4	−0.994162
$538$	0	0
$539$	17213.2	1.37556
$540$	0	0
$541$	−21797.0	−1.73221	−0.866105	−	0.499862i	$-0.833384\pi$
$541$	−21797.0	−1.73221	−0.866105	+	0.499862i	$0.833384\pi$
$542$	0	0
$543$	34224.1	2.70478
$544$	0	0
$545$	−29705.9	−2.33479
$546$	0	0
$547$	−2353.06	−0.183930	−0.0919649	−	0.995762i	$-0.529315\pi$
$547$	−2353.06	−0.183930	−0.0919649	+	0.995762i	$0.529315\pi$
$548$	0	0
$549$	−37835.6	−2.94132
$550$	0	0
$551$	−2276.03	−0.175975
$552$	0	0
$553$	−933.795	−0.0718065
$554$	0	0
$555$	21169.9	1.61912
$556$	0	0
$557$	19378.0	1.47410	0.737050	−	0.675839i	$-0.236218\pi$
$557$	19378.0	1.47410	0.737050	+	0.675839i	$0.236218\pi$
$558$	0	0
$559$	−9421.83	−0.712882
$560$	0	0
$561$	−12350.3	−0.929469
$562$	0	0
$563$	18661.9	1.39699	0.698495	−	0.715615i	$-0.253853\pi$
$563$	18661.9	1.39699	0.698495	+	0.715615i	$0.253853\pi$
$564$	0	0
$565$	−18604.4	−1.38530
$566$	0	0
$567$	16970.0	1.25692
$568$	0	0
$569$	11784.8	0.868267	0.434133	−	0.900849i	$-0.357055\pi$
$569$	11784.8	0.868267	0.434133	+	0.900849i	$0.357055\pi$
$570$	0	0
$571$	−461.266	−0.0338063	−0.0169031	−	0.999857i	$-0.505381\pi$
$571$	−461.266	−0.0338063	−0.0169031	+	0.999857i	$0.505381\pi$
$572$	0	0
$573$	−22098.7	−1.61115
$574$	0	0
$575$	21306.0	1.54525
$576$	0	0
$577$	−7046.52	−0.508406	−0.254203	−	0.967151i	$-0.581813\pi$
$577$	−7046.52	−0.508406	−0.254203	+	0.967151i	$0.581813\pi$
$578$	0	0
$579$	46124.2	3.31063
$580$	0	0
$581$	−359.391	−0.0256627
$582$	0	0
$583$	15395.6	1.09369
$584$	0	0
$585$	51103.5	3.61174
$586$	0	0
$587$	−8869.36	−0.623642	−0.311821	−	0.950141i	$-0.600939\pi$
$587$	−8869.36	−0.623642	−0.311821	+	0.950141i	$0.600939\pi$
$588$	0	0
$589$	−5558.78	−0.388872
$590$	0	0
$591$	−16036.5	−1.11617
$592$	0	0
$593$	8672.37	0.600559	0.300280	−	0.953851i	$-0.402920\pi$
$593$	8672.37	0.600559	0.300280	+	0.953851i	$0.402920\pi$
$594$	0	0
$595$	−2169.86	−0.149505
$596$	0	0
$597$	21713.8	1.48859
$598$	0	0
$599$	−2789.39	−0.190269	−0.0951347	−	0.995464i	$-0.530328\pi$
$599$	−2789.39	−0.190269	−0.0951347	+	0.995464i	$0.530328\pi$
$600$	0	0
$601$	24897.8	1.68986	0.844928	−	0.534880i	$-0.179643\pi$
$601$	24897.8	1.68986	0.844928	+	0.534880i	$0.179643\pi$
$602$	0	0
$603$	−30570.3	−2.06454
$604$	0	0
$605$	−29977.4	−2.01447
$606$	0	0
$607$	−1136.81	−0.0760158	−0.0380079	−	0.999277i	$-0.512101\pi$
$607$	−1136.81	−0.0760158	−0.0380079	+	0.999277i	$0.512101\pi$
$608$	0	0
$609$	−7414.27	−0.493336
$610$	0	0
$611$	−6240.73	−0.413213
$612$	0	0
$613$	19706.5	1.29843	0.649216	−	0.760604i	$-0.275097\pi$
$613$	19706.5	1.29843	0.649216	+	0.760604i	$0.275097\pi$
$614$	0	0
$615$	53183.8	3.48712
$616$	0	0
$617$	16107.6	1.05100	0.525502	−	0.850793i	$-0.323878\pi$
$617$	16107.6	1.05100	0.525502	+	0.850793i	$0.323878\pi$
$618$	0	0
$619$	−20620.9	−1.33898	−0.669488	−	0.742823i	$-0.733486\pi$
$619$	−20620.9	−1.33898	−0.669488	+	0.742823i	$0.733486\pi$
$620$	0	0
$621$	73595.1	4.75567
$622$	0	0
$623$	−6535.93	−0.420315
$624$	0	0
$625$	−13986.8	−0.895153
$626$	0	0
$627$	−10776.4	−0.686390
$628$	0	0
$629$	−2833.15	−0.179595
$630$	0	0
$631$	−12428.3	−0.784096	−0.392048	−	0.919945i	$-0.628233\pi$
$631$	−12428.3	−0.784096	−0.392048	+	0.919945i	$0.628233\pi$
$632$	0	0
$633$	−1886.18	−0.118434
$634$	0	0
$635$	27114.1	1.69447
$636$	0	0
$637$	13007.1	0.809041
$638$	0	0
$639$	16146.4	0.999599
$640$	0	0
$641$	2330.10	0.143578	0.0717890	−	0.997420i	$-0.477129\pi$
$641$	2330.10	0.143578	0.0717890	+	0.997420i	$0.477129\pi$
$642$	0	0
$643$	21706.8	1.33131	0.665654	−	0.746260i	$-0.268153\pi$
$643$	21706.8	1.33131	0.665654	+	0.746260i	$0.268153\pi$
$644$	0	0
$645$	−35958.2	−2.19512
$646$	0	0
$647$	−17992.7	−1.09330	−0.546650	−	0.837361i	$-0.684097\pi$
$647$	−17992.7	−1.09330	−0.546650	+	0.837361i	$0.684097\pi$
$648$	0	0
$649$	42451.6	2.56760
$650$	0	0
$651$	−18108.0	−1.09018
$652$	0	0
$653$	−3007.79	−0.180251	−0.0901257	−	0.995930i	$-0.528727\pi$
$653$	−3007.79	−0.180251	−0.0901257	+	0.995930i	$0.528727\pi$
$654$	0	0
$655$	−13264.9	−0.791299
$656$	0	0
$657$	−33466.4	−1.98729
$658$	0	0
$659$	374.268	0.0221235	0.0110618	−	0.999939i	$-0.496479\pi$
$659$	374.268	0.0221235	0.0110618	+	0.999939i	$0.496479\pi$
$660$	0	0
$661$	−931.802	−0.0548304	−0.0274152	−	0.999624i	$-0.508728\pi$
$661$	−931.802	−0.0548304	−0.0274152	+	0.999624i	$0.508728\pi$
$662$	0	0
$663$	−9332.47	−0.546672
$664$	0	0
$665$	−1893.33	−0.110406
$666$	0	0
$667$	−18634.9	−1.08178
$668$	0	0
$669$	31198.1	1.80297
$670$	0	0
$671$	28823.3	1.65829
$672$	0	0
$673$	−23276.5	−1.33320	−0.666599	−	0.745416i	$-0.732251\pi$
$673$	−23276.5	−1.33320	−0.666599	+	0.745416i	$0.732251\pi$
$674$	0	0
$675$	64795.3	3.69477
$676$	0	0
$677$	7070.67	0.401400	0.200700	−	0.979653i	$-0.435678\pi$
$677$	7070.67	0.401400	0.200700	+	0.979653i	$0.435678\pi$
$678$	0	0
$679$	−7789.39	−0.440249
$680$	0	0
$681$	−6332.31	−0.356321
$682$	0	0
$683$	24737.3	1.38586	0.692932	−	0.721003i	$-0.256319\pi$
$683$	24737.3	1.38586	0.692932	+	0.721003i	$0.256319\pi$
$684$	0	0
$685$	11383.8	0.634967
$686$	0	0
$687$	63997.7	3.55410
$688$	0	0
$689$	11633.6	0.643258
$690$	0	0
$691$	14654.9	0.806799	0.403399	−	0.915024i	$-0.367829\pi$
$691$	14654.9	0.806799	0.403399	+	0.915024i	$0.367829\pi$
$692$	0	0
$693$	−25725.7	−1.41016
$694$	0	0
$695$	−50959.5	−2.78130
$696$	0	0
$697$	−7117.55	−0.386795
$698$	0	0
$699$	−47460.8	−2.56814
$700$	0	0
$701$	4849.19	0.261272	0.130636	−	0.991430i	$-0.458298\pi$
$701$	4849.19	0.261272	0.130636	+	0.991430i	$0.458298\pi$
$702$	0	0
$703$	−2472.09	−0.132627
$704$	0	0
$705$	−23817.6	−1.27237
$706$	0	0
$707$	−573.519	−0.0305083
$708$	0	0
$709$	8974.31	0.475370	0.237685	−	0.971342i	$-0.423611\pi$
$709$	8974.31	0.475370	0.237685	+	0.971342i	$0.423611\pi$
$710$	0	0
$711$	−11232.7	−0.592488
$712$	0	0
$713$	−45512.4	−2.39053
$714$	0	0
$715$	−38930.8	−2.03626
$716$	0	0
$717$	15369.9	0.800556
$718$	0	0
$719$	−18554.1	−0.962379	−0.481190	−	0.876617i	$-0.659795\pi$
$719$	−18554.1	−0.962379	−0.481190	+	0.876617i	$0.659795\pi$
$720$	0	0
$721$	8006.10	0.413541
$722$	0	0
$723$	4556.38	0.234376
$724$	0	0
$725$	−16406.7	−0.840456
$726$	0	0
$727$	1358.92	0.0693256	0.0346628	−	0.999399i	$-0.488964\pi$
$727$	1358.92	0.0693256	0.0346628	+	0.999399i	$0.488964\pi$
$728$	0	0
$729$	75722.6	3.84711
$730$	0	0
$731$	4812.26	0.243485
$732$	0	0
$733$	3728.20	0.187864	0.0939320	−	0.995579i	$-0.470056\pi$
$733$	3728.20	0.187864	0.0939320	+	0.995579i	$0.470056\pi$
$734$	0	0
$735$	49641.2	2.49122
$736$	0	0
$737$	23288.5	1.16397
$738$	0	0
$739$	297.299	0.0147988	0.00739941	−	0.999973i	$-0.497645\pi$
$739$	297.299	0.0147988	0.00739941	+	0.999973i	$0.497645\pi$
$740$	0	0
$741$	−8143.11	−0.403704
$742$	0	0
$743$	−28401.0	−1.40233	−0.701166	−	0.712998i	$-0.747337\pi$
$743$	−28401.0	−1.40233	−0.701166	+	0.712998i	$0.747337\pi$
$744$	0	0
$745$	51439.6	2.52966
$746$	0	0
$747$	−4323.14	−0.211747
$748$	0	0
$749$	−8487.38	−0.414048
$750$	0	0
$751$	−2901.64	−0.140989	−0.0704943	−	0.997512i	$-0.522458\pi$
$751$	−2901.64	−0.140989	−0.0704943	+	0.997512i	$0.522458\pi$
$752$	0	0
$753$	−46281.0	−2.23980
$754$	0	0
$755$	−11702.7	−0.564114
$756$	0	0
$757$	−7247.35	−0.347965	−0.173983	−	0.984749i	$-0.555664\pi$
$757$	−7247.35	−0.347965	−0.173983	+	0.984749i	$0.555664\pi$
$758$	0	0
$759$	−88231.2	−4.21949
$760$	0	0
$761$	−20012.4	−0.953285	−0.476642	−	0.879097i	$-0.658146\pi$
$761$	−20012.4	−0.953285	−0.476642	+	0.879097i	$0.658146\pi$
$762$	0	0
$763$	11300.0	0.536156
$764$	0	0
$765$	−26101.4	−1.23359
$766$	0	0
$767$	32078.3	1.51015
$768$	0	0
$769$	−12606.3	−0.591151	−0.295575	−	0.955319i	$-0.595511\pi$
$769$	−12606.3	−0.591151	−0.295575	+	0.955319i	$0.595511\pi$
$770$	0	0
$771$	14157.6	0.661313
$772$	0	0
$773$	41454.4	1.92886	0.964431	−	0.264334i	$-0.0851522\pi$
$773$	41454.4	1.92886	0.964431	+	0.264334i	$0.0851522\pi$
$774$	0	0
$775$	−40070.4	−1.85725
$776$	0	0
$777$	−8052.93	−0.371811
$778$	0	0
$779$	−6210.46	−0.285639
$780$	0	0
$781$	−12300.4	−0.563564
$782$	0	0
$783$	−56672.2	−2.58659
$784$	0	0
$785$	4426.17	0.201244
$786$	0	0
$787$	−1091.42	−0.0494345	−0.0247173	−	0.999694i	$-0.507869\pi$
$787$	−1091.42	−0.0494345	−0.0247173	+	0.999694i	$0.507869\pi$
$788$	0	0
$789$	−15760.1	−0.711123
$790$	0	0
$791$	7077.03	0.318116
$792$	0	0
$793$	21780.2	0.975329
$794$	0	0
$795$	44399.4	1.98073
$796$	0	0
$797$	−6451.15	−0.286714	−0.143357	−	0.989671i	$-0.545790\pi$
$797$	−6451.15	−0.286714	−0.143357	+	0.989671i	$0.545790\pi$
$798$	0	0
$799$	3187.49	0.141133
$800$	0	0
$801$	−78621.1	−3.46809
$802$	0	0
$803$	25494.8	1.12041
$804$	0	0
$805$	−15501.6	−0.678706
$806$	0	0
$807$	−7868.18	−0.343213
$808$	0	0
$809$	10349.6	0.449782	0.224891	−	0.974384i	$-0.427797\pi$
$809$	10349.6	0.449782	0.224891	+	0.974384i	$0.427797\pi$
$810$	0	0
$811$	30515.8	1.32127	0.660637	−	0.750706i	$-0.270286\pi$
$811$	30515.8	1.32127	0.660637	+	0.750706i	$0.270286\pi$
$812$	0	0
$813$	30363.1	1.30981
$814$	0	0
$815$	−30304.6	−1.30248
$816$	0	0
$817$	4198.97	0.179808
$818$	0	0
$819$	−19439.5	−0.829392
$820$	0	0
$821$	22704.9	0.965174	0.482587	−	0.875848i	$-0.339697\pi$
$821$	22704.9	0.965174	0.482587	+	0.875848i	$0.339697\pi$
$822$	0	0
$823$	16385.1	0.693983	0.346992	−	0.937868i	$-0.387203\pi$
$823$	16385.1	0.693983	0.346992	+	0.937868i	$0.387203\pi$
$824$	0	0
$825$	−77681.3	−3.27820
$826$	0	0
$827$	−33152.0	−1.39396	−0.696981	−	0.717089i	$-0.745474\pi$
$827$	−33152.0	−1.39396	−0.696981	+	0.717089i	$0.745474\pi$
$828$	0	0
$829$	−35124.7	−1.47157	−0.735785	−	0.677216i	$-0.763186\pi$
$829$	−35124.7	−1.47157	−0.735785	+	0.677216i	$0.763186\pi$
$830$	0	0
$831$	−7955.97	−0.332118
$832$	0	0
$833$	−6643.45	−0.276329
$834$	0	0
$835$	−20961.7	−0.868752
$836$	0	0
$837$	−138411.	−5.71588
$838$	0	0
$839$	−17722.3	−0.729251	−0.364626	−	0.931154i	$-0.618803\pi$
$839$	−17722.3	−0.729251	−0.364626	+	0.931154i	$0.618803\pi$
$840$	0	0
$841$	−10039.1	−0.411624
$842$	0	0
$843$	63777.5	2.60571
$844$	0	0
$845$	6141.08	0.250011
$846$	0	0
$847$	11403.3	0.462599
$848$	0	0
$849$	−21830.2	−0.882463
$850$	0	0
$851$	−20240.1	−0.815303
$852$	0	0
$853$	1538.42	0.0617519	0.0308760	−	0.999523i	$-0.490170\pi$
$853$	1538.42	0.0617519	0.0308760	+	0.999523i	$0.490170\pi$
$854$	0	0
$855$	−22775.0	−0.910980
$856$	0	0
$857$	−29482.0	−1.17513	−0.587565	−	0.809177i	$-0.699913\pi$
$857$	−29482.0	−1.17513	−0.587565	+	0.809177i	$0.699913\pi$
$858$	0	0
$859$	−18917.8	−0.751418	−0.375709	−	0.926738i	$-0.622601\pi$
$859$	−18917.8	−0.751418	−0.375709	+	0.926738i	$0.622601\pi$
$860$	0	0
$861$	−20230.9	−0.800774
$862$	0	0
$863$	−10119.6	−0.399161	−0.199580	−	0.979881i	$-0.563958\pi$
$863$	−10119.6	−0.399161	−0.199580	+	0.979881i	$0.563958\pi$
$864$	0	0
$865$	32730.2	1.28654
$866$	0	0
$867$	−44623.2	−1.74796
$868$	0	0
$869$	8557.10	0.334039
$870$	0	0
$871$	17597.9	0.684594
$872$	0	0
$873$	−93699.0	−3.63257
$874$	0	0
$875$	−1191.93	−0.0460509
$876$	0	0
$877$	24288.4	0.935189	0.467595	−	0.883943i	$-0.345121\pi$
$877$	24288.4	0.935189	0.467595	+	0.883943i	$0.345121\pi$
$878$	0	0
$879$	66431.1	2.54911
$880$	0	0
$881$	48672.6	1.86132	0.930659	−	0.365888i	$-0.119235\pi$
$881$	48672.6	1.86132	0.930659	+	0.365888i	$0.119235\pi$
$882$	0	0
$883$	15128.2	0.576562	0.288281	−	0.957546i	$-0.406916\pi$
$883$	15128.2	0.576562	0.288281	+	0.957546i	$0.406916\pi$
$884$	0	0
$885$	122426.	4.65007
$886$	0	0
$887$	−2509.30	−0.0949876	−0.0474938	−	0.998872i	$-0.515123\pi$
$887$	−2509.30	−0.0949876	−0.0474938	+	0.998872i	$0.515123\pi$
$888$	0	0
$889$	−10314.1	−0.389114
$890$	0	0
$891$	−155509.	−5.84708
$892$	0	0
$893$	2781.27	0.104223
$894$	0	0
$895$	19918.0	0.743896
$896$	0	0
$897$	−66671.5	−2.48171
$898$	0	0
$899$	35047.0	1.30020
$900$	0	0
$901$	−5941.94	−0.219706
$902$	0	0
$903$	13678.3	0.504082
$904$	0	0
$905$	−55101.0	−2.02389
$906$	0	0
$907$	−12268.2	−0.449129	−0.224564	−	0.974459i	$-0.572096\pi$
$907$	−12268.2	−0.449129	−0.224564	+	0.974459i	$0.572096\pi$
$908$	0	0
$909$	−6898.89	−0.251729
$910$	0	0
$911$	−7546.52	−0.274454	−0.137227	−	0.990540i	$-0.543819\pi$
$911$	−7546.52	−0.274454	−0.137227	+	0.990540i	$0.543819\pi$
$912$	0	0
$913$	3293.38	0.119381
$914$	0	0
$915$	83123.5	3.00325
$916$	0	0
$917$	5045.89	0.181712
$918$	0	0
$919$	−3497.39	−0.125537	−0.0627684	−	0.998028i	$-0.519993\pi$
$919$	−3497.39	−0.125537	−0.0627684	+	0.998028i	$0.519993\pi$
$920$	0	0
$921$	−29514.8	−1.05597
$922$	0	0
$923$	−9294.75	−0.331463
$924$	0	0
$925$	−17820.0	−0.633425
$926$	0	0
$927$	96306.0	3.41219
$928$	0	0
$929$	11612.4	0.410108	0.205054	−	0.978751i	$-0.434263\pi$
$929$	11612.4	0.410108	0.205054	+	0.978751i	$0.434263\pi$
$930$	0	0
$931$	−5796.79	−0.204062
$932$	0	0
$933$	17594.4	0.617378
$934$	0	0
$935$	19884.2	0.695488
$936$	0	0
$937$	−17890.6	−0.623756	−0.311878	−	0.950122i	$-0.600958\pi$
$937$	−17890.6	−0.623756	−0.311878	+	0.950122i	$0.600958\pi$
$938$	0	0
$939$	50048.1	1.73936
$940$	0	0
$941$	25999.4	0.900696	0.450348	−	0.892853i	$-0.351300\pi$
$941$	25999.4	0.900696	0.450348	+	0.892853i	$0.351300\pi$
$942$	0	0
$943$	−50848.0	−1.75593
$944$	0	0
$945$	−47143.1	−1.62282
$946$	0	0
$947$	−22773.4	−0.781453	−0.390726	−	0.920507i	$-0.627776\pi$
$947$	−22773.4	−0.781453	−0.390726	+	0.920507i	$0.627776\pi$
$948$	0	0
$949$	19265.0	0.658977
$950$	0	0
$951$	−8900.71	−0.303497
$952$	0	0
$953$	50535.8	1.71775	0.858875	−	0.512186i	$-0.171164\pi$
$953$	50535.8	1.71775	0.858875	+	0.512186i	$0.171164\pi$
$954$	0	0
$955$	35579.1	1.20556
$956$	0	0
$957$	67942.8	2.29496
$958$	0	0
$959$	−4330.34	−0.145812
$960$	0	0
$961$	55804.7	1.87321
$962$	0	0
$963$	−102095.	−3.41638
$964$	0	0
$965$	−74260.4	−2.47723
$966$	0	0
$967$	32916.9	1.09466	0.547330	−	0.836917i	$-0.315644\pi$
$967$	32916.9	1.09466	0.547330	+	0.836917i	$0.315644\pi$
$968$	0	0
$969$	4159.15	0.137885
$970$	0	0
$971$	17791.6	0.588010	0.294005	−	0.955804i	$-0.405012\pi$
$971$	17791.6	0.588010	0.294005	+	0.955804i	$0.405012\pi$
$972$	0	0
$973$	19384.8	0.638692
$974$	0	0
$975$	−58699.5	−1.92809
$976$	0	0
$977$	30781.8	1.00798	0.503991	−	0.863709i	$-0.331865\pi$
$977$	30781.8	1.00798	0.503991	+	0.863709i	$0.331865\pi$
$978$	0	0
$979$	59893.8	1.95528
$980$	0	0
$981$	135928.	4.42391
$982$	0	0
$983$	42533.9	1.38008	0.690041	−	0.723770i	$-0.257592\pi$
$983$	42533.9	1.38008	0.690041	+	0.723770i	$0.257592\pi$
$984$	0	0
$985$	25819.0	0.835188
$986$	0	0
$987$	9060.11	0.292185
$988$	0	0
$989$	34378.9	1.10535
$990$	0	0
$991$	−37748.5	−1.21001	−0.605006	−	0.796221i	$-0.706829\pi$
$991$	−37748.5	−1.21001	−0.605006	+	0.796221i	$0.706829\pi$
$992$	0	0
$993$	86615.4	2.76803
$994$	0	0
$995$	−34959.4	−1.11386
$996$	0	0
$997$	−21873.7	−0.694833	−0.347416	−	0.937711i	$-0.612941\pi$
$997$	−21873.7	−0.694833	−0.347416	+	0.937711i	$0.612941\pi$
$998$	0	0
$999$	−61553.9	−1.94943

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.bc.1.5		5
4.3	odd	2		1216.4.a.z.1.1		5
8.3	odd	2		608.4.a.g.1.5	yes	5
8.5	even	2		608.4.a.f.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.f.1.1	✓	5	8.5	even	2
608.4.a.g.1.5	yes	5	8.3	odd	2
1216.4.a.z.1.1		5	4.3	odd	2
1216.4.a.bc.1.5		5	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+10.0529 q^{3} -16.1852 q^{5} +6.15678 q^{7} +74.0603 q^{9} +O(q^{10})\)	\(q+10.0529 q^{3} -16.1852 q^{5} +6.15678 q^{7} +74.0603 q^{9} -56.4194 q^{11} -42.6330 q^{13} -162.708 q^{15} +21.7751 q^{17} +19.0000 q^{19} +61.8934 q^{21} +155.562 q^{23} +136.961 q^{25} +473.092 q^{27} -119.791 q^{29} -292.567 q^{31} -567.177 q^{33} -99.6489 q^{35} -130.110 q^{37} -428.585 q^{39} -326.866 q^{41} +220.998 q^{43} -1198.68 q^{45} +146.383 q^{47} -305.094 q^{49} +218.902 q^{51} -272.878 q^{53} +913.160 q^{55} +191.005 q^{57} -752.429 q^{59} -510.875 q^{61} +455.973 q^{63} +690.025 q^{65} -412.776 q^{67} +1563.85 q^{69} +218.017 q^{71} -451.880 q^{73} +1376.86 q^{75} -347.362 q^{77} -151.669 q^{79} +2756.30 q^{81} -58.3732 q^{83} -352.435 q^{85} -1204.24 q^{87} -1061.58 q^{89} -262.482 q^{91} -2941.14 q^{93} -307.519 q^{95} -1265.17 q^{97} -4178.44 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 27 q^{5} + 20 q^{7} + 154 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 - 27 * q^5 + 20 * q^7 + 154 * q^9	\( 5 q + 3 q^{3} - 27 q^{5} + 20 q^{7} + 154 q^{9} - 67 q^{11} - 191 q^{13} - 36 q^{15} - 52 q^{17} + 95 q^{19} + 29 q^{21} + 35 q^{23} - 2 q^{25} - 27 q^{27} + 33 q^{29} - 694 q^{31} + 298 q^{33} - 387 q^{35} - 108 q^{37} - 31 q^{39} + 54 q^{41} + 427 q^{43} - 1699 q^{45} + 79 q^{47} + 1033 q^{49} - 363 q^{51} - 1025 q^{53} + 1431 q^{55} + 57 q^{57} + 649 q^{59} - 1413 q^{61} - 1449 q^{63} + 548 q^{65} - 915 q^{67} - 407 q^{69} + 1444 q^{71} + 2682 q^{73} + 465 q^{75} - 169 q^{77} - 836 q^{79} + 2925 q^{81} + 502 q^{83} - 373 q^{85} - 3491 q^{87} + 996 q^{89} - 2919 q^{91} - 2492 q^{93} - 513 q^{95} + 424 q^{97} - 7249 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 27 * q^5 + 20 * q^7 + 154 * q^9 - 67 * q^11 - 191 * q^13 - 36 * q^15 - 52 * q^17 + 95 * q^19 + 29 * q^21 + 35 * q^23 - 2 * q^25 - 27 * q^27 + 33 * q^29 - 694 * q^31 + 298 * q^33 - 387 * q^35 - 108 * q^37 - 31 * q^39 + 54 * q^41 + 427 * q^43 - 1699 * q^45 + 79 * q^47 + 1033 * q^49 - 363 * q^51 - 1025 * q^53 + 1431 * q^55 + 57 * q^57 + 649 * q^59 - 1413 * q^61 - 1449 * q^63 + 548 * q^65 - 915 * q^67 - 407 * q^69 + 1444 * q^71 + 2682 * q^73 + 465 * q^75 - 169 * q^77 - 836 * q^79 + 2925 * q^81 + 502 * q^83 - 373 * q^85 - 3491 * q^87 + 996 * q^89 - 2919 * q^91 - 2492 * q^93 - 513 * q^95 + 424 * q^97 - 7249 * q^99

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