LMFDB - Embedded newform 96.4.d.a.49.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,4,Mod(49,96)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("96.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$96 = 2^{5} \cdot 3$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	96.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:	$5.66418336055$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.8248384.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64$ x^6 + x^4 - 12x^3 + 4x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{12}\cdot 3^{2}$
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.5
Root			$1.88322 - 0.673417i$ of defining polynomial
Character	$\chi$	$=$	96.49
Dual form			96.4.d.a.49.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -0.612661i q^{5} +22.7441 q^{7} -9.00000 q^{9} +60.2630i q^{11} +52.9062i q^{13} +1.83798 q^{15} +47.1643 q^{17} -29.1643i q^{19} +68.2324i q^{21} -109.488 q^{23} +124.625 q^{25} -27.0000i q^{27} -10.4250i q^{29} +220.881 q^{31} -180.789 q^{33} -13.9345i q^{35} -408.348i q^{37} -158.718 q^{39} -360.742 q^{41} -236.414i q^{43} +5.51395i q^{45} -129.113 q^{47} +174.296 q^{49} +141.493i q^{51} -117.819i q^{53} +36.9208 q^{55} +87.4928 q^{57} -262.854i q^{59} +273.465i q^{61} -204.697 q^{63} +32.4135 q^{65} -89.4077i q^{67} -328.465i q^{69} +350.521 q^{71} +532.610 q^{73} +373.874i q^{75} +1370.63i q^{77} +166.561 q^{79} +81.0000 q^{81} +361.934i q^{83} -28.8957i q^{85} +31.2750 q^{87} +40.3285 q^{89} +1203.31i q^{91} +662.642i q^{93} -17.8678 q^{95} -614.921 q^{97} -542.367i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 28 q^{7} - 54 q^{9} + 60 q^{15} + 52 q^{17} - 328 q^{23} - 106 q^{25} + 636 q^{31} - 312 q^{39} + 236 q^{41} + 408 q^{47} + 654 q^{49} - 1024 q^{55} - 168 q^{57} + 252 q^{63} - 1744 q^{65} + 1704 q^{71}+ \cdots - 2444 q^{97}+O(q^{100})$ 6 * q - 28 * q^7 - 54 * q^9 + 60 * q^15 + 52 * q^17 - 328 * q^23 - 106 * q^25 + 636 * q^31 - 312 * q^39 + 236 * q^41 + 408 * q^47 + 654 * q^49 - 1024 * q^55 - 168 * q^57 + 252 * q^63 - 1744 * q^65 + 1704 * q^71 + 956 * q^73 + 44 * q^79 + 486 * q^81 - 1044 * q^87 - 220 * q^89 - 5104 * q^95 - 2444 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$ .

$n$	$31$	$37$	$65$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	− 0.612661i	− 0.0547981i	−0.999625	−	0.0273990i	$-0.991278\pi$
$5$	− 0.612661i	− 0.0547981i	0.999625	−	0.0273990i	$-0.00872248\pi$
$6$	0	0
$7$	22.7441	1.22807	0.614034	−	0.789279i	$-0.289546\pi$
$7$	22.7441	1.22807	0.614034	+	0.789279i	$0.289546\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	60.2630i	1.65182i	0.563805	+	0.825908i	$0.309337\pi$
$11$	60.2630i	1.65182i	−0.563805	+	0.825908i	$0.690663\pi$
$12$	0	0
$13$	52.9062i	1.12873i	0.825524	+	0.564367i	$0.190880\pi$
$13$	52.9062i	1.12873i	−0.825524	+	0.564367i	$0.809120\pi$
$14$	0	0
$15$	1.83798	0.0316377
$16$	0	0
$17$	47.1643	0.672883	0.336442	−	0.941704i	$-0.390777\pi$
$17$	47.1643	0.672883	0.336442	+	0.941704i	$0.390777\pi$
$18$	0	0
$19$	− 29.1643i	− 0.352144i	−0.984377	−	0.176072i	$-0.943661\pi$
$19$	− 29.1643i	− 0.352144i	0.984377	−	0.176072i	$-0.0563392\pi$
$20$	0	0
$21$	68.2324i	0.709026i
$22$	0	0
$23$	−109.488	−0.992604	−0.496302	−	0.868150i	$-0.665309\pi$
$23$	−109.488	−0.992604	−0.496302	+	0.868150i	$0.665309\pi$
$24$	0	0
$25$	124.625	0.996997
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	− 10.4250i	− 0.0667542i	−0.999443	−	0.0333771i	$-0.989374\pi$
$29$	− 10.4250i	− 0.0667542i	0.999443	−	0.0333771i	$-0.0106262\pi$
$30$	0	0
$31$	220.881	1.27972	0.639860	−	0.768492i	$-0.278992\pi$
$31$	220.881	1.27972	0.639860	+	0.768492i	$0.278992\pi$
$32$	0	0
$33$	−180.789	−0.953676
$34$	0	0
$35$	− 13.9345i	− 0.0672958i
$36$	0	0
$37$	− 408.348i	− 1.81438i	−0.420725	−	0.907188i	$-0.638224\pi$
$37$	− 408.348i	− 1.81438i	0.420725	−	0.907188i	$-0.361776\pi$
$38$	0	0
$39$	−158.718	−0.651674
$40$	0	0
$41$	−360.742	−1.37411	−0.687054	−	0.726606i	$-0.741097\pi$
$41$	−360.742	−1.37411	−0.687054	+	0.726606i	$0.741097\pi$
$42$	0	0
$43$	− 236.414i	− 0.838436i	−0.907886	−	0.419218i	$-0.862304\pi$
$43$	− 236.414i	− 0.838436i	0.907886	−	0.419218i	$-0.137696\pi$
$44$	0	0
$45$	5.51395i	0.0182660i
$46$	0	0
$47$	−129.113	−0.400703	−0.200352	−	0.979724i	$-0.564208\pi$
$47$	−129.113	−0.400703	−0.200352	+	0.979724i	$0.564208\pi$
$48$	0	0
$49$	174.296	0.508152
$50$	0	0
$51$	141.493i	0.388489i
$52$	0	0
$53$	− 117.819i	− 0.305353i	−0.988276	−	0.152677i	$-0.951211\pi$
$53$	− 117.819i	− 0.305353i	0.988276	−	0.152677i	$-0.0487893\pi$
$54$	0	0
$55$	36.9208	0.0905163
$56$	0	0
$57$	87.4928	0.203311
$58$	0	0
$59$	− 262.854i	− 0.580012i	−0.957025	−	0.290006i	$-0.906343\pi$
$59$	− 262.854i	− 0.580012i	0.957025	−	0.290006i	$-0.0936574\pi$
$60$	0	0
$61$	273.465i	0.573993i	0.957932	+	0.286996i	$0.0926568\pi$
$61$	273.465i	0.573993i	−0.957932	+	0.286996i	$0.907343\pi$
$62$	0	0
$63$	−204.697	−0.409356
$64$	0	0
$65$	32.4135	0.0618524
$66$	0	0
$67$	− 89.4077i	− 0.163028i	−0.996672	−	0.0815141i	$-0.974024\pi$
$67$	− 89.4077i	− 0.163028i	0.996672	−	0.0815141i	$-0.0259756\pi$
$68$	0	0
$69$	− 328.465i	− 0.573080i
$70$	0	0
$71$	350.521	0.585904	0.292952	−	0.956127i	$-0.405362\pi$
$71$	350.521	0.585904	0.292952	+	0.956127i	$0.405362\pi$
$72$	0	0
$73$	532.610	0.853936	0.426968	−	0.904267i	$-0.359582\pi$
$73$	532.610	0.853936	0.426968	+	0.904267i	$0.359582\pi$
$74$	0	0
$75$	373.874i	0.575617i
$76$	0	0
$77$	1370.63i	2.02854i
$78$	0	0
$79$	166.561	0.237210	0.118605	−	0.992942i	$-0.462158\pi$
$79$	166.561	0.237210	0.118605	+	0.992942i	$0.462158\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	361.934i	0.478644i	0.970940	+	0.239322i	$0.0769252\pi$
$83$	361.934i	0.478644i	−0.970940	+	0.239322i	$0.923075\pi$
$84$	0	0
$85$	− 28.8957i	− 0.0368727i
$86$	0	0
$87$	31.2750	0.0385405
$88$	0	0
$89$	40.3285	0.0480316	0.0240158	−	0.999712i	$-0.492355\pi$
$89$	40.3285	0.0480316	0.0240158	+	0.999712i	$0.492355\pi$
$90$	0	0
$91$	1203.31i	1.38616i
$92$	0	0
$93$	662.642i	0.738846i
$94$	0	0
$95$	−17.8678	−0.0192968
$96$	0	0
$97$	−614.921	−0.643667	−0.321834	−	0.946796i	$-0.604299\pi$
$97$	−614.921	−0.643667	−0.321834	+	0.946796i	$0.604299\pi$
$98$	0	0
$99$	− 542.367i	− 0.550605i
$100$	0	0
$101$	− 1664.99i	− 1.64033i	−0.572130	−	0.820163i	$-0.693883\pi$
$101$	− 1664.99i	− 1.64033i	0.572130	−	0.820163i	$-0.306117\pi$
$102$	0	0
$103$	−396.858	−0.379647	−0.189823	−	0.981818i	$-0.560792\pi$
$103$	−396.858	−0.379647	−0.189823	+	0.981818i	$0.560792\pi$
$104$	0	0
$105$	41.8034	0.0388532
$106$	0	0
$107$	350.630i	0.316791i	0.987376	+	0.158396i	$0.0506321\pi$
$107$	350.630i	0.316791i	−0.987376	+	0.158396i	$0.949368\pi$
$108$	0	0
$109$	597.009i	0.524615i	0.964984	+	0.262308i	$0.0844835\pi$
$109$	597.009i	0.524615i	−0.964984	+	0.262308i	$0.915516\pi$
$110$	0	0
$111$	1225.04	1.04753
$112$	0	0
$113$	496.422	0.413270	0.206635	−	0.978418i	$-0.433749\pi$
$113$	496.422	0.413270	0.206635	+	0.978418i	$0.433749\pi$
$114$	0	0
$115$	67.0792i	0.0543928i
$116$	0	0
$117$	− 476.155i	− 0.376244i
$118$	0	0
$119$	1072.71	0.826346
$120$	0	0
$121$	−2300.63	−1.72849
$122$	0	0
$123$	− 1082.23i	− 0.793342i
$124$	0	0
$125$	− 152.935i	− 0.109432i
$126$	0	0
$127$	−1799.85	−1.25756	−0.628782	−	0.777581i	$-0.716446\pi$
$127$	−1799.85	−1.25756	−0.628782	+	0.777581i	$0.716446\pi$
$128$	0	0
$129$	709.241	0.484071
$130$	0	0
$131$	− 1121.45i	− 0.747949i	−0.927439	−	0.373974i	$-0.877995\pi$
$131$	− 1121.45i	− 0.747949i	0.927439	−	0.373974i	$-0.122005\pi$
$132$	0	0
$133$	− 663.316i	− 0.432457i
$134$	0	0
$135$	−16.5419	−0.0105459
$136$	0	0
$137$	2449.55	1.52759	0.763793	−	0.645461i	$-0.223335\pi$
$137$	2449.55	1.52759	0.763793	+	0.645461i	$0.223335\pi$
$138$	0	0
$139$	− 2457.56i	− 1.49962i	−0.661652	−	0.749811i	$-0.730144\pi$
$139$	− 2457.56i	− 1.49962i	0.661652	−	0.749811i	$-0.269856\pi$
$140$	0	0
$141$	− 387.339i	− 0.231346i
$142$	0	0
$143$	−3188.28	−1.86446
$144$	0	0
$145$	−6.38698	−0.00365800
$146$	0	0
$147$	522.888i	0.293382i
$148$	0	0
$149$	− 2084.96i	− 1.14635i	−0.819432	−	0.573177i	$-0.805711\pi$
$149$	− 2084.96i	− 1.14635i	0.819432	−	0.573177i	$-0.194289\pi$
$150$	0	0
$151$	−1057.80	−0.570084	−0.285042	−	0.958515i	$-0.592008\pi$
$151$	−1057.80	−0.570084	−0.285042	+	0.958515i	$0.592008\pi$
$152$	0	0
$153$	−424.478	−0.224294
$154$	0	0
$155$	− 135.325i	− 0.0701262i
$156$	0	0
$157$	3193.01i	1.62312i	0.584270	+	0.811559i	$0.301381\pi$
$157$	3193.01i	1.62312i	−0.584270	+	0.811559i	$0.698619\pi$
$158$	0	0
$159$	353.458	0.176296
$160$	0	0
$161$	−2490.22	−1.21899
$162$	0	0
$163$	846.854i	0.406937i	0.979081	+	0.203469i	$0.0652215\pi$
$163$	846.854i	0.406937i	−0.979081	+	0.203469i	$0.934779\pi$
$164$	0	0
$165$	110.762i	0.0522596i
$166$	0	0
$167$	2630.15	1.21873	0.609363	−	0.792892i	$-0.291425\pi$
$167$	2630.15	1.21873	0.609363	+	0.792892i	$0.291425\pi$
$168$	0	0
$169$	−602.062	−0.274038
$170$	0	0
$171$	262.478i	0.117381i
$172$	0	0
$173$	429.843i	0.188904i	0.995529	+	0.0944519i	$0.0301099\pi$
$173$	429.843i	0.188904i	−0.995529	+	0.0944519i	$0.969890\pi$
$174$	0	0
$175$	2834.48	1.22438
$176$	0	0
$177$	788.563	0.334870
$178$	0	0
$179$	1516.30i	0.633149i	0.948568	+	0.316574i	$0.102533\pi$
$179$	1516.30i	0.633149i	−0.948568	+	0.316574i	$0.897467\pi$
$180$	0	0
$181$	− 3380.20i	− 1.38811i	−0.719921	−	0.694056i	$-0.755822\pi$
$181$	− 3380.20i	− 1.38811i	0.719921	−	0.694056i	$-0.244178\pi$
$182$	0	0
$183$	−820.394	−0.331395
$184$	0	0
$185$	−250.179	−0.0994244
$186$	0	0
$187$	2842.26i	1.11148i
$188$	0	0
$189$	− 614.092i	− 0.236342i
$190$	0	0
$191$	2799.71	1.06063	0.530314	−	0.847801i	$-0.322074\pi$
$191$	2799.71	1.06063	0.530314	+	0.847801i	$0.322074\pi$
$192$	0	0
$193$	624.106	0.232768	0.116384	−	0.993204i	$-0.462870\pi$
$193$	624.106	0.232768	0.116384	+	0.993204i	$0.462870\pi$
$194$	0	0
$195$	97.2406i	0.0357105i
$196$	0	0
$197$	4779.25i	1.72846i	0.503094	+	0.864232i	$0.332195\pi$
$197$	4779.25i	1.72846i	−0.503094	+	0.864232i	$0.667805\pi$
$198$	0	0
$199$	−2615.92	−0.931846	−0.465923	−	0.884825i	$-0.654278\pi$
$199$	−2615.92	−0.931846	−0.465923	+	0.884825i	$0.654278\pi$
$200$	0	0
$201$	268.223	0.0941244
$202$	0	0
$203$	− 237.107i	− 0.0819787i
$204$	0	0
$205$	221.013i	0.0752985i
$206$	0	0
$207$	985.395	0.330868
$208$	0	0
$209$	1757.52	0.581677
$210$	0	0
$211$	1745.78i	0.569595i	0.958588	+	0.284798i	$0.0919264\pi$
$211$	1745.78i	0.569595i	−0.958588	+	0.284798i	$0.908074\pi$
$212$	0	0
$213$	1051.56i	0.338272i
$214$	0	0
$215$	−144.841	−0.0459447
$216$	0	0
$217$	5023.74	1.57158
$218$	0	0
$219$	1597.83i	0.493020i
$220$	0	0
$221$	2495.28i	0.759505i
$222$	0	0
$223$	3385.60	1.01667	0.508333	−	0.861161i	$-0.330262\pi$
$223$	3385.60	1.01667	0.508333	+	0.861161i	$0.330262\pi$
$224$	0	0
$225$	−1121.62	−0.332332
$226$	0	0
$227$	− 3847.72i	− 1.12503i	−0.826787	−	0.562515i	$-0.809834\pi$
$227$	− 3847.72i	− 1.12503i	0.826787	−	0.562515i	$-0.190166\pi$
$228$	0	0
$229$	1335.15i	0.385279i	0.981270	+	0.192640i	$0.0617049\pi$
$229$	1335.15i	0.385279i	−0.981270	+	0.192640i	$0.938295\pi$
$230$	0	0
$231$	−4111.89	−1.17118
$232$	0	0
$233$	−5146.38	−1.44700	−0.723499	−	0.690325i	$-0.757468\pi$
$233$	−5146.38	−1.44700	−0.723499	+	0.690325i	$0.757468\pi$
$234$	0	0
$235$	79.1025i	0.0219578i
$236$	0	0
$237$	499.683i	0.136953i
$238$	0	0
$239$	−7085.07	−1.91755	−0.958777	−	0.284160i	$-0.908285\pi$
$239$	−7085.07	−1.91755	−0.958777	+	0.284160i	$0.908285\pi$
$240$	0	0
$241$	2538.40	0.678476	0.339238	−	0.940701i	$-0.389831\pi$
$241$	2538.40	0.678476	0.339238	+	0.940701i	$0.389831\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	− 106.784i	− 0.0278458i
$246$	0	0
$247$	1542.97	0.397477
$248$	0	0
$249$	−1085.80	−0.276345
$250$	0	0
$251$	1696.99i	0.426746i	0.976971	+	0.213373i	$0.0684449\pi$
$251$	1696.99i	0.426746i	−0.976971	+	0.213373i	$0.931555\pi$
$252$	0	0
$253$	− 6598.09i	− 1.63960i
$254$	0	0
$255$	86.6871	0.0212885
$256$	0	0
$257$	−382.902	−0.0929369	−0.0464685	−	0.998920i	$-0.514797\pi$
$257$	−382.902	−0.0929369	−0.0464685	+	0.998920i	$0.514797\pi$
$258$	0	0
$259$	− 9287.52i	− 2.22818i
$260$	0	0
$261$	93.8249i	0.0222514i
$262$	0	0
$263$	5002.02	1.17277	0.586383	−	0.810034i	$-0.300551\pi$
$263$	5002.02	1.17277	0.586383	+	0.810034i	$0.300551\pi$
$264$	0	0
$265$	−72.1833	−0.0167328
$266$	0	0
$267$	120.986i	0.0277311i
$268$	0	0
$269$	− 6117.47i	− 1.38658i	−0.720661	−	0.693288i	$-0.756161\pi$
$269$	− 6117.47i	− 1.38658i	0.720661	−	0.693288i	$-0.243839\pi$
$270$	0	0
$271$	3956.12	0.886780	0.443390	−	0.896329i	$-0.353776\pi$
$271$	3956.12	0.886780	0.443390	+	0.896329i	$0.353776\pi$
$272$	0	0
$273$	−3609.92	−0.800301
$274$	0	0
$275$	7510.25i	1.64686i
$276$	0	0
$277$	4842.17i	1.05032i	0.851004	+	0.525158i	$0.175994\pi$
$277$	4842.17i	1.05032i	−0.851004	+	0.525158i	$0.824006\pi$
$278$	0	0
$279$	−1987.92	−0.426573
$280$	0	0
$281$	1878.68	0.398835	0.199417	−	0.979915i	$-0.436095\pi$
$281$	1878.68	0.398835	0.199417	+	0.979915i	$0.436095\pi$
$282$	0	0
$283$	− 5724.87i	− 1.20250i	−0.799060	−	0.601251i	$-0.794669\pi$
$283$	− 5724.87i	− 1.20250i	0.799060	−	0.601251i	$-0.205331\pi$
$284$	0	0
$285$	− 53.6034i	− 0.0111410i
$286$	0	0
$287$	−8204.77	−1.68750
$288$	0	0
$289$	−2688.53	−0.547228
$290$	0	0
$291$	− 1844.76i	− 0.371622i
$292$	0	0
$293$	− 5088.75i	− 1.01464i	−0.861759	−	0.507318i	$-0.830637\pi$
$293$	− 5088.75i	− 1.01464i	0.861759	−	0.507318i	$-0.169363\pi$
$294$	0	0
$295$	−161.041	−0.0317836
$296$	0	0
$297$	1627.10	0.317892
$298$	0	0
$299$	− 5792.61i	− 1.12038i
$300$	0	0
$301$	− 5377.02i	− 1.02966i
$302$	0	0
$303$	4994.98	0.947043
$304$	0	0
$305$	167.541	0.0314537
$306$	0	0
$307$	7219.21i	1.34209i	0.741416	+	0.671046i	$0.234155\pi$
$307$	7219.21i	1.34209i	−0.741416	+	0.671046i	$0.765845\pi$
$308$	0	0
$309$	− 1190.58i	− 0.219189i
$310$	0	0
$311$	1537.06	0.280252	0.140126	−	0.990134i	$-0.455249\pi$
$311$	1537.06	0.280252	0.140126	+	0.990134i	$0.455249\pi$
$312$	0	0
$313$	−2200.93	−0.397456	−0.198728	−	0.980055i	$-0.563681\pi$
$313$	−2200.93	−0.397456	−0.198728	+	0.980055i	$0.563681\pi$
$314$	0	0
$315$	125.410i	0.0224319i
$316$	0	0
$317$	− 2840.41i	− 0.503260i	−0.967824	−	0.251630i	$-0.919033\pi$
$317$	− 2840.41i	− 0.503260i	0.967824	−	0.251630i	$-0.0809665\pi$
$318$	0	0
$319$	628.240	0.110266
$320$	0	0
$321$	−1051.89	−0.182899
$322$	0	0
$323$	− 1375.51i	− 0.236952i
$324$	0	0
$325$	6593.41i	1.12534i
$326$	0	0
$327$	−1791.03	−0.302887
$328$	0	0
$329$	−2936.56	−0.492091
$330$	0	0
$331$	2118.52i	0.351795i	0.984408	+	0.175898i	$0.0562828\pi$
$331$	2118.52i	0.351795i	−0.984408	+	0.175898i	$0.943717\pi$
$332$	0	0
$333$	3675.13i	0.604792i
$334$	0	0
$335$	−54.7766	−0.00893364
$336$	0	0
$337$	−659.599	−0.106619	−0.0533096	−	0.998578i	$-0.516977\pi$
$337$	−659.599	−0.106619	−0.0533096	+	0.998578i	$0.516977\pi$
$338$	0	0
$339$	1489.27i	0.238601i
$340$	0	0
$341$	13310.9i	2.11386i
$342$	0	0
$343$	−3837.03	−0.604023
$344$	0	0
$345$	−201.238	−0.0314037
$346$	0	0
$347$	8377.42i	1.29603i	0.761626	+	0.648017i	$0.224401\pi$
$347$	8377.42i	1.29603i	−0.761626	+	0.648017i	$0.775599\pi$
$348$	0	0
$349$	− 3254.18i	− 0.499119i	−0.968360	−	0.249559i	$-0.919714\pi$
$349$	− 3254.18i	− 0.499119i	0.968360	−	0.249559i	$-0.0802857\pi$
$350$	0	0
$351$	1428.47	0.217225
$352$	0	0
$353$	11117.5	1.67627	0.838137	−	0.545459i	$-0.183645\pi$
$353$	11117.5	1.67627	0.838137	+	0.545459i	$0.183645\pi$
$354$	0	0
$355$	− 214.750i	− 0.0321064i
$356$	0	0
$357$	3218.13i	0.477091i
$358$	0	0
$359$	4756.56	0.699281	0.349640	−	0.936884i	$-0.386304\pi$
$359$	4756.56	0.699281	0.349640	+	0.936884i	$0.386304\pi$
$360$	0	0
$361$	6008.45	0.875994
$362$	0	0
$363$	− 6901.88i	− 0.997946i
$364$	0	0
$365$	− 326.310i	− 0.0467940i
$366$	0	0
$367$	−1837.40	−0.261339	−0.130670	−	0.991426i	$-0.541713\pi$
$367$	−1837.40	−0.261339	−0.130670	+	0.991426i	$0.541713\pi$
$368$	0	0
$369$	3246.68	0.458036
$370$	0	0
$371$	− 2679.70i	− 0.374995i
$372$	0	0
$373$	5598.07i	0.777097i	0.921428	+	0.388549i	$0.127023\pi$
$373$	5598.07i	0.777097i	−0.921428	+	0.388549i	$0.872977\pi$
$374$	0	0
$375$	458.806	0.0631804
$376$	0	0
$377$	551.546	0.0753476
$378$	0	0
$379$	3460.18i	0.468965i	0.972120	+	0.234482i	$0.0753395\pi$
$379$	3460.18i	0.468965i	−0.972120	+	0.234482i	$0.924661\pi$
$380$	0	0
$381$	− 5399.55i	− 0.726055i
$382$	0	0
$383$	−5059.63	−0.675027	−0.337513	−	0.941321i	$-0.609586\pi$
$383$	−5059.63	−0.675027	−0.337513	+	0.941321i	$0.609586\pi$
$384$	0	0
$385$	839.732	0.111160
$386$	0	0
$387$	2127.72i	0.279479i
$388$	0	0
$389$	2192.22i	0.285732i	0.989742	+	0.142866i	$0.0456318\pi$
$389$	2192.22i	0.285732i	−0.989742	+	0.142866i	$0.954368\pi$
$390$	0	0
$391$	−5163.93	−0.667906
$392$	0	0
$393$	3364.34	0.431828
$394$	0	0
$395$	− 102.045i	− 0.0129986i
$396$	0	0
$397$	5519.94i	0.697828i	0.937155	+	0.348914i	$0.113449\pi$
$397$	5519.94i	0.697828i	−0.937155	+	0.348914i	$0.886551\pi$
$398$	0	0
$399$	1989.95	0.249679
$400$	0	0
$401$	−7352.64	−0.915645	−0.457822	−	0.889044i	$-0.651370\pi$
$401$	−7352.64	−0.915645	−0.457822	+	0.889044i	$0.651370\pi$
$402$	0	0
$403$	11685.9i	1.44446i
$404$	0	0
$405$	− 49.6256i	− 0.00608868i
$406$	0	0
$407$	24608.2	2.99702
$408$	0	0
$409$	−11311.2	−1.36749	−0.683745	−	0.729721i	$-0.739650\pi$
$409$	−11311.2	−1.36749	−0.683745	+	0.729721i	$0.739650\pi$
$410$	0	0
$411$	7348.66i	0.881952i
$412$	0	0
$413$	− 5978.40i	− 0.712295i
$414$	0	0
$415$	221.743	0.0262288
$416$	0	0
$417$	7372.68	0.865808
$418$	0	0
$419$	− 13042.2i	− 1.52065i	−0.649543	−	0.760325i	$-0.725040\pi$
$419$	− 13042.2i	− 1.52065i	0.649543	−	0.760325i	$-0.274960\pi$
$420$	0	0
$421$	− 4544.38i	− 0.526080i	−0.964785	−	0.263040i	$-0.915275\pi$
$421$	− 4544.38i	− 0.526080i	0.964785	−	0.263040i	$-0.0847251\pi$
$422$	0	0
$423$	1162.02	0.133568
$424$	0	0
$425$	5877.83	0.670863
$426$	0	0
$427$	6219.72i	0.704903i
$428$	0	0
$429$	− 9564.85i	− 1.07645i
$430$	0	0
$431$	−7713.83	−0.862093	−0.431047	−	0.902330i	$-0.641856\pi$
$431$	−7713.83	−0.862093	−0.431047	+	0.902330i	$0.641856\pi$
$432$	0	0
$433$	−15068.3	−1.67237	−0.836183	−	0.548451i	$-0.815218\pi$
$433$	−15068.3	−1.67237	−0.836183	+	0.548451i	$0.815218\pi$
$434$	0	0
$435$	− 19.1609i	− 0.00211195i
$436$	0	0
$437$	3193.14i	0.349540i
$438$	0	0
$439$	−11004.7	−1.19642	−0.598208	−	0.801341i	$-0.704120\pi$
$439$	−11004.7	−1.19642	−0.598208	+	0.801341i	$0.704120\pi$
$440$	0	0
$441$	−1568.67	−0.169384
$442$	0	0
$443$	− 2513.04i	− 0.269522i	−0.990878	−	0.134761i	$-0.956973\pi$
$443$	− 2513.04i	− 0.269522i	0.990878	−	0.134761i	$-0.0430266\pi$
$444$	0	0
$445$	− 24.7077i	− 0.00263204i
$446$	0	0
$447$	6254.89	0.661848
$448$	0	0
$449$	−15752.7	−1.65571	−0.827855	−	0.560942i	$-0.810439\pi$
$449$	−15752.7	−1.65571	−0.827855	+	0.560942i	$0.810439\pi$
$450$	0	0
$451$	− 21739.4i	− 2.26977i
$452$	0	0
$453$	− 3173.40i	− 0.329138i
$454$	0	0
$455$	737.218	0.0759590
$456$	0	0
$457$	−5257.06	−0.538107	−0.269053	−	0.963125i	$-0.586711\pi$
$457$	−5257.06	−0.538107	−0.269053	+	0.963125i	$0.586711\pi$
$458$	0	0
$459$	− 1273.43i	− 0.129496i
$460$	0	0
$461$	− 8066.31i	− 0.814936i	−0.913220	−	0.407468i	$-0.866412\pi$
$461$	− 8066.31i	− 0.814936i	0.913220	−	0.407468i	$-0.133588\pi$
$462$	0	0
$463$	−5683.43	−0.570478	−0.285239	−	0.958456i	$-0.592073\pi$
$463$	−5683.43	−0.570478	−0.285239	+	0.958456i	$0.592073\pi$
$464$	0	0
$465$	405.975	0.0404874
$466$	0	0
$467$	11139.3i	1.10378i	0.833916	+	0.551891i	$0.186094\pi$
$467$	11139.3i	1.10378i	−0.833916	+	0.551891i	$0.813906\pi$
$468$	0	0
$469$	− 2033.50i	− 0.200210i
$470$	0	0
$471$	−9579.02	−0.937108
$472$	0	0
$473$	14247.0	1.38494
$474$	0	0
$475$	− 3634.59i	− 0.351087i
$476$	0	0
$477$	1060.37i	0.101784i
$478$	0	0
$479$	−3477.35	−0.331699	−0.165850	−	0.986151i	$-0.553037\pi$
$479$	−3477.35	−0.331699	−0.165850	+	0.986151i	$0.553037\pi$
$480$	0	0
$481$	21604.1	2.04795
$482$	0	0
$483$	− 7470.65i	− 0.703782i
$484$	0	0
$485$	376.738i	0.0352717i
$486$	0	0
$487$	478.797	0.0445510	0.0222755	−	0.999752i	$-0.492909\pi$
$487$	478.797	0.0445510	0.0222755	+	0.999752i	$0.492909\pi$
$488$	0	0
$489$	−2540.56	−0.234945
$490$	0	0
$491$	− 16601.8i	− 1.52592i	−0.646444	−	0.762961i	$-0.723745\pi$
$491$	− 16601.8i	− 1.52592i	0.646444	−	0.762961i	$-0.276255\pi$
$492$	0	0
$493$	− 491.687i	− 0.0449178i
$494$	0	0
$495$	−332.287	−0.0301721
$496$	0	0
$497$	7972.29	0.719530
$498$	0	0
$499$	9482.20i	0.850664i	0.905037	+	0.425332i	$0.139843\pi$
$499$	9482.20i	0.850664i	−0.905037	+	0.425332i	$0.860157\pi$
$500$	0	0
$501$	7890.45i	0.703631i
$502$	0	0
$503$	−16561.2	−1.46805	−0.734023	−	0.679124i	$-0.762360\pi$
$503$	−16561.2	−1.46805	−0.734023	+	0.679124i	$0.762360\pi$
$504$	0	0
$505$	−1020.08	−0.0898867
$506$	0	0
$507$	− 1806.19i	− 0.158216i
$508$	0	0
$509$	4197.35i	0.365509i	0.983159	+	0.182755i	$0.0585014\pi$
$509$	4197.35i	0.365509i	−0.983159	+	0.182755i	$0.941499\pi$
$510$	0	0
$511$	12113.8	1.04869
$512$	0	0
$513$	−787.435	−0.0677702
$514$	0	0
$515$	243.140i	0.0208039i
$516$	0	0
$517$	− 7780.73i	− 0.661888i
$518$	0	0
$519$	−1289.53	−0.109064
$520$	0	0
$521$	−15755.5	−1.32488	−0.662440	−	0.749115i	$-0.730479\pi$
$521$	−15755.5	−1.32488	−0.662440	+	0.749115i	$0.730479\pi$
$522$	0	0
$523$	− 11555.1i	− 0.966098i	−0.875593	−	0.483049i	$-0.839529\pi$
$523$	− 11555.1i	− 0.966098i	0.875593	−	0.483049i	$-0.160471\pi$
$524$	0	0
$525$	8503.44i	0.706897i
$526$	0	0
$527$	10417.7	0.861102
$528$	0	0
$529$	−179.314	−0.0147378
$530$	0	0
$531$	2365.69i	0.193337i
$532$	0	0
$533$	− 19085.5i	− 1.55100i
$534$	0	0
$535$	214.817	0.0173595
$536$	0	0
$537$	−4548.90	−0.365549
$538$	0	0
$539$	10503.6i	0.839373i
$540$	0	0
$541$	7475.65i	0.594091i	0.954863	+	0.297045i	$0.0960013\pi$
$541$	7475.65i	0.594091i	−0.954863	+	0.297045i	$0.903999\pi$
$542$	0	0
$543$	10140.6	0.801427
$544$	0	0
$545$	365.764	0.0287479
$546$	0	0
$547$	6028.08i	0.471192i	0.971851	+	0.235596i	$0.0757043\pi$
$547$	6028.08i	0.471192i	−0.971851	+	0.235596i	$0.924296\pi$
$548$	0	0
$549$	− 2461.18i	− 0.191331i
$550$	0	0
$551$	−304.037	−0.0235071
$552$	0	0
$553$	3788.29	0.291310
$554$	0	0
$555$	− 750.536i	− 0.0574027i
$556$	0	0
$557$	19381.5i	1.47436i	0.675696	+	0.737181i	$0.263843\pi$
$557$	19381.5i	1.47436i	−0.675696	+	0.737181i	$0.736157\pi$
$558$	0	0
$559$	12507.7	0.946370
$560$	0	0
$561$	−8526.77	−0.641712
$562$	0	0
$563$	20565.0i	1.53946i	0.638372	+	0.769728i	$0.279608\pi$
$563$	20565.0i	1.53946i	−0.638372	+	0.769728i	$0.720392\pi$
$564$	0	0
$565$	− 304.139i	− 0.0226464i
$566$	0	0
$567$	1842.28	0.136452
$568$	0	0
$569$	15252.9	1.12379	0.561895	−	0.827209i	$-0.310073\pi$
$569$	15252.9	1.12379	0.561895	+	0.827209i	$0.310073\pi$
$570$	0	0
$571$	16492.8i	1.20876i	0.796697	+	0.604379i	$0.206579\pi$
$571$	16492.8i	1.20876i	−0.796697	+	0.604379i	$0.793421\pi$
$572$	0	0
$573$	8399.13i	0.612354i
$574$	0	0
$575$	−13644.9	−0.989623
$576$	0	0
$577$	−10298.2	−0.743016	−0.371508	−	0.928430i	$-0.621159\pi$
$577$	−10298.2	−0.743016	−0.371508	+	0.928430i	$0.621159\pi$
$578$	0	0
$579$	1872.32i	0.134388i
$580$	0	0
$581$	8231.89i	0.587808i
$582$	0	0
$583$	7100.14	0.504387
$584$	0	0
$585$	−291.722	−0.0206175
$586$	0	0
$587$	13104.8i	0.921453i	0.887542	+	0.460727i	$0.152411\pi$
$587$	13104.8i	0.921453i	−0.887542	+	0.460727i	$0.847589\pi$
$588$	0	0
$589$	− 6441.82i	− 0.450646i
$590$	0	0
$591$	−14337.7	−0.997929
$592$	0	0
$593$	4163.34	0.288310	0.144155	−	0.989555i	$-0.453954\pi$
$593$	4163.34	0.288310	0.144155	+	0.989555i	$0.453954\pi$
$594$	0	0
$595$	− 657.208i	− 0.0452822i
$596$	0	0
$597$	− 7847.75i	− 0.538001i
$598$	0	0
$599$	5718.60	0.390076	0.195038	−	0.980796i	$-0.437517\pi$
$599$	5718.60	0.390076	0.195038	+	0.980796i	$0.437517\pi$
$600$	0	0
$601$	17473.0	1.18592	0.592959	−	0.805233i	$-0.297960\pi$
$601$	17473.0	1.18592	0.592959	+	0.805233i	$0.297960\pi$
$602$	0	0
$603$	804.670i	0.0543428i
$604$	0	0
$605$	1409.50i	0.0947181i
$606$	0	0
$607$	5647.60	0.377643	0.188821	−	0.982011i	$-0.439533\pi$
$607$	5647.60	0.377643	0.188821	+	0.982011i	$0.439533\pi$
$608$	0	0
$609$	711.322	0.0473304
$610$	0	0
$611$	− 6830.87i	− 0.452287i
$612$	0	0
$613$	16023.6i	1.05577i	0.849317	+	0.527884i	$0.177014\pi$
$613$	16023.6i	1.05577i	−0.849317	+	0.527884i	$0.822986\pi$
$614$	0	0
$615$	−663.038	−0.0434736
$616$	0	0
$617$	−21022.1	−1.37167	−0.685834	−	0.727758i	$-0.740562\pi$
$617$	−21022.1	−1.37167	−0.685834	+	0.727758i	$0.740562\pi$
$618$	0	0
$619$	− 17824.2i	− 1.15737i	−0.815550	−	0.578686i	$-0.803566\pi$
$619$	− 17824.2i	− 1.15737i	0.815550	−	0.578686i	$-0.196434\pi$
$620$	0	0
$621$	2956.18i	0.191027i
$622$	0	0
$623$	917.238	0.0589861
$624$	0	0
$625$	15484.4	0.991001
$626$	0	0
$627$	5272.57i	0.335831i
$628$	0	0
$629$	− 19259.4i	− 1.22086i
$630$	0	0
$631$	22339.0	1.40935	0.704677	−	0.709528i	$-0.251092\pi$
$631$	22339.0	1.40935	0.704677	+	0.709528i	$0.251092\pi$
$632$	0	0
$633$	−5237.35	−0.328856
$634$	0	0
$635$	1102.70i	0.0689121i
$636$	0	0
$637$	9221.34i	0.573568i
$638$	0	0
$639$	−3154.69	−0.195301
$640$	0	0
$641$	−5268.43	−0.324634	−0.162317	−	0.986739i	$-0.551897\pi$
$641$	−5268.43	−0.324634	−0.162317	+	0.986739i	$0.551897\pi$
$642$	0	0
$643$	− 21965.8i	− 1.34719i	−0.739099	−	0.673597i	$-0.764748\pi$
$643$	− 21965.8i	− 1.34719i	0.739099	−	0.673597i	$-0.235252\pi$
$644$	0	0
$645$	− 434.524i	− 0.0265262i
$646$	0	0
$647$	−3165.40	−0.192341	−0.0961706	−	0.995365i	$-0.530659\pi$
$647$	−3165.40	−0.192341	−0.0961706	+	0.995365i	$0.530659\pi$
$648$	0	0
$649$	15840.4	0.958073
$650$	0	0
$651$	15071.2i	0.907354i
$652$	0	0
$653$	12094.7i	0.724814i	0.932020	+	0.362407i	$0.118045\pi$
$653$	12094.7i	0.724814i	−0.932020	+	0.362407i	$0.881955\pi$
$654$	0	0
$655$	−687.067	−0.0409861
$656$	0	0
$657$	−4793.49	−0.284645
$658$	0	0
$659$	− 7523.17i	− 0.444706i	−0.974966	−	0.222353i	$-0.928626\pi$
$659$	− 7523.17i	− 0.444706i	0.974966	−	0.222353i	$-0.0713737\pi$
$660$	0	0
$661$	− 24141.1i	− 1.42054i	−0.703928	−	0.710271i	$-0.748572\pi$
$661$	− 24141.1i	− 1.42054i	0.703928	−	0.710271i	$-0.251428\pi$
$662$	0	0
$663$	−7485.84	−0.438501
$664$	0	0
$665$	−406.388	−0.0236978
$666$	0	0
$667$	1141.41i	0.0662604i
$668$	0	0
$669$	10156.8i	0.586972i
$670$	0	0
$671$	−16479.8	−0.948130
$672$	0	0
$673$	944.143	0.0540773	0.0270387	−	0.999634i	$-0.491392\pi$
$673$	944.143	0.0540773	0.0270387	+	0.999634i	$0.491392\pi$
$674$	0	0
$675$	− 3364.87i	− 0.191872i
$676$	0	0
$677$	− 4450.51i	− 0.252654i	−0.991989	−	0.126327i	$-0.959681\pi$
$677$	− 4450.51i	− 0.252654i	0.991989	−	0.126327i	$-0.0403189\pi$
$678$	0	0
$679$	−13985.8	−0.790468
$680$	0	0
$681$	11543.1	0.649536
$682$	0	0
$683$	− 1726.40i	− 0.0967189i	−0.998830	−	0.0483594i	$-0.984601\pi$
$683$	− 1726.40i	− 0.0967189i	0.998830	−	0.0483594i	$-0.0153993\pi$
$684$	0	0
$685$	− 1500.75i	− 0.0837088i
$686$	0	0
$687$	−4005.44	−0.222441
$688$	0	0
$689$	6233.37	0.344662
$690$	0	0
$691$	− 683.143i	− 0.0376092i	−0.999823	−	0.0188046i	$-0.994014\pi$
$691$	− 683.143i	− 0.0376092i	0.999823	−	0.0188046i	$-0.00598605\pi$
$692$	0	0
$693$	− 12335.7i	− 0.676181i
$694$	0	0
$695$	−1505.65	−0.0821764
$696$	0	0
$697$	−17014.1	−0.924614
$698$	0	0
$699$	− 15439.1i	− 0.835425i
$700$	0	0
$701$	19533.2i	1.05244i	0.850349	+	0.526220i	$0.176391\pi$
$701$	19533.2i	1.05244i	−0.850349	+	0.526220i	$0.823609\pi$
$702$	0	0
$703$	−11909.2	−0.638922
$704$	0	0
$705$	−237.307	−0.0126773
$706$	0	0
$707$	− 37868.8i	− 2.01443i
$708$	0	0
$709$	− 26081.9i	− 1.38156i	−0.723066	−	0.690779i	$-0.757268\pi$
$709$	− 26081.9i	− 1.38156i	0.723066	−	0.690779i	$-0.242732\pi$
$710$	0	0
$711$	−1499.05	−0.0790699
$712$	0	0
$713$	−24183.8	−1.27025
$714$	0	0
$715$	1953.34i	0.102169i
$716$	0	0
$717$	− 21255.2i	− 1.10710i
$718$	0	0
$719$	30077.3	1.56007	0.780036	−	0.625734i	$-0.215200\pi$
$719$	30077.3	1.56007	0.780036	+	0.625734i	$0.215200\pi$
$720$	0	0
$721$	−9026.21	−0.466232
$722$	0	0
$723$	7615.20i	0.391718i
$724$	0	0
$725$	− 1299.21i	− 0.0665537i
$726$	0	0
$727$	−23049.9	−1.17589	−0.587946	−	0.808900i	$-0.700063\pi$
$727$	−23049.9	−1.17589	−0.587946	+	0.808900i	$0.700063\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 11150.3i	− 0.564169i
$732$	0	0
$733$	4444.57i	0.223962i	0.993710	+	0.111981i	$0.0357195\pi$
$733$	4444.57i	0.223962i	−0.993710	+	0.111981i	$0.964280\pi$
$734$	0	0
$735$	320.353	0.0160768
$736$	0	0
$737$	5387.98	0.269293
$738$	0	0
$739$	28465.1i	1.41692i	0.705749	+	0.708462i	$0.250610\pi$
$739$	28465.1i	1.41692i	−0.705749	+	0.708462i	$0.749390\pi$
$740$	0	0
$741$	4628.91i	0.229483i
$742$	0	0
$743$	−4389.76	−0.216749	−0.108374	−	0.994110i	$-0.534565\pi$
$743$	−4389.76	−0.216749	−0.108374	+	0.994110i	$0.534565\pi$
$744$	0	0
$745$	−1277.38	−0.0628180
$746$	0	0
$747$	− 3257.41i	− 0.159548i
$748$	0	0
$749$	7974.77i	0.389041i
$750$	0	0
$751$	−14121.9	−0.686171	−0.343085	−	0.939304i	$-0.611472\pi$
$751$	−14121.9	−0.686171	−0.343085	+	0.939304i	$0.611472\pi$
$752$	0	0
$753$	−5090.98	−0.246382
$754$	0	0
$755$	648.074i	0.0312395i
$756$	0	0
$757$	− 17006.3i	− 0.816516i	−0.912866	−	0.408258i	$-0.866136\pi$
$757$	− 17006.3i	− 0.816516i	0.912866	−	0.408258i	$-0.133864\pi$
$758$	0	0
$759$	19794.3	0.946622
$760$	0	0
$761$	−4603.38	−0.219280	−0.109640	−	0.993971i	$-0.534970\pi$
$761$	−4603.38	−0.219280	−0.109640	+	0.993971i	$0.534970\pi$
$762$	0	0
$763$	13578.5i	0.644264i
$764$	0	0
$765$	260.061i	0.0122909i
$766$	0	0
$767$	13906.6	0.654679
$768$	0	0
$769$	−12459.1	−0.584248	−0.292124	−	0.956380i	$-0.594362\pi$
$769$	−12459.1	−0.584248	−0.292124	+	0.956380i	$0.594362\pi$
$770$	0	0
$771$	− 1148.71i	− 0.0536571i
$772$	0	0
$773$	− 27068.1i	− 1.25947i	−0.776808	−	0.629737i	$-0.783163\pi$
$773$	− 27068.1i	− 1.25947i	0.776808	−	0.629737i	$-0.216837\pi$
$774$	0	0
$775$	27527.2	1.27588
$776$	0	0
$777$	27862.6	1.28644
$778$	0	0
$779$	10520.8i	0.483884i
$780$	0	0
$781$	21123.4i	0.967804i
$782$	0	0
$783$	−281.475	−0.0128468
$784$	0	0
$785$	1956.23	0.0889438
$786$	0	0
$787$	6986.86i	0.316461i	0.987402	+	0.158230i	$0.0505789\pi$
$787$	6986.86i	0.316461i	−0.987402	+	0.158230i	$0.949421\pi$
$788$	0	0
$789$	15006.1i	0.677097i
$790$	0	0
$791$	11290.7	0.507523
$792$	0	0
$793$	−14468.0	−0.647885
$794$	0	0
$795$	− 216.550i	− 0.00966067i
$796$	0	0
$797$	− 27271.5i	− 1.21205i	−0.795444	−	0.606027i	$-0.792762\pi$
$797$	− 27271.5i	− 1.21205i	0.795444	−	0.606027i	$-0.207238\pi$
$798$	0	0
$799$	−6089.52	−0.269627
$800$	0	0
$801$	−362.957	−0.0160105
$802$	0	0
$803$	32096.7i	1.41054i
$804$	0	0
$805$	1525.66i	0.0667981i
$806$	0	0
$807$	18352.4	0.800540
$808$	0	0
$809$	23785.9	1.03371	0.516853	−	0.856074i	$-0.327103\pi$
$809$	23785.9	1.03371	0.516853	+	0.856074i	$0.327103\pi$
$810$	0	0
$811$	− 21703.5i	− 0.939718i	−0.882741	−	0.469859i	$-0.844305\pi$
$811$	− 21703.5i	− 0.939718i	0.882741	−	0.469859i	$-0.155695\pi$
$812$	0	0
$813$	11868.4i	0.511982i
$814$	0	0
$815$	518.835	0.0222994
$816$	0	0
$817$	−6894.83	−0.295250
$818$	0	0
$819$	− 10829.7i	− 0.462054i
$820$	0	0
$821$	33240.4i	1.41303i	0.707698	+	0.706515i	$0.249734\pi$
$821$	33240.4i	1.41303i	−0.707698	+	0.706515i	$0.750266\pi$
$822$	0	0
$823$	17227.5	0.729665	0.364832	−	0.931073i	$-0.381126\pi$
$823$	17227.5	0.729665	0.364832	+	0.931073i	$0.381126\pi$
$824$	0	0
$825$	−22530.8	−0.950812
$826$	0	0
$827$	− 21678.4i	− 0.911528i	−0.890101	−	0.455764i	$-0.849366\pi$
$827$	− 21678.4i	− 0.911528i	0.890101	−	0.455764i	$-0.150634\pi$
$828$	0	0
$829$	34269.8i	1.43575i	0.696170	+	0.717877i	$0.254886\pi$
$829$	34269.8i	1.43575i	−0.696170	+	0.717877i	$0.745114\pi$
$830$	0	0
$831$	−14526.5	−0.606401
$832$	0	0
$833$	8220.55	0.341927
$834$	0	0
$835$	− 1611.39i	− 0.0667838i
$836$	0	0
$837$	− 5963.77i	− 0.246282i
$838$	0	0
$839$	−33444.1	−1.37618	−0.688092	−	0.725623i	$-0.741552\pi$
$839$	−33444.1	−1.37618	−0.688092	+	0.725623i	$0.741552\pi$
$840$	0	0
$841$	24280.3	0.995544
$842$	0	0
$843$	5636.04i	0.230267i
$844$	0	0
$845$	368.860i	0.0150168i
$846$	0	0
$847$	−52325.8	−2.12271
$848$	0	0
$849$	17174.6	0.694265
$850$	0	0
$851$	44709.3i	1.80096i
$852$	0	0
$853$	37037.3i	1.48667i	0.668917	+	0.743337i	$0.266758\pi$
$853$	37037.3i	1.48667i	−0.668917	+	0.743337i	$0.733242\pi$
$854$	0	0
$855$	160.810	0.00643227
$856$	0	0
$857$	35794.2	1.42673	0.713365	−	0.700793i	$-0.247170\pi$
$857$	35794.2	1.42673	0.713365	+	0.700793i	$0.247170\pi$
$858$	0	0
$859$	20582.1i	0.817522i	0.912641	+	0.408761i	$0.134039\pi$
$859$	20582.1i	0.817522i	−0.912641	+	0.408761i	$0.865961\pi$
$860$	0	0
$861$	− 24614.3i	− 0.974278i
$862$	0	0
$863$	25677.7	1.01284	0.506420	−	0.862287i	$-0.330969\pi$
$863$	25677.7	1.01284	0.506420	+	0.862287i	$0.330969\pi$
$864$	0	0
$865$	263.348	0.0103516
$866$	0	0
$867$	− 8065.60i	− 0.315942i
$868$	0	0
$869$	10037.5i	0.391827i
$870$	0	0
$871$	4730.22	0.184015
$872$	0	0
$873$	5534.29	0.214556
$874$	0	0
$875$	− 3478.38i	− 0.134389i
$876$	0	0
$877$	32783.0i	1.26226i	0.775676	+	0.631131i	$0.217409\pi$
$877$	32783.0i	1.26226i	−0.775676	+	0.631131i	$0.782591\pi$
$878$	0	0
$879$	15266.3	0.585800
$880$	0	0
$881$	26615.7	1.01783	0.508914	−	0.860818i	$-0.330047\pi$
$881$	26615.7	1.01783	0.508914	+	0.860818i	$0.330047\pi$
$882$	0	0
$883$	− 19203.4i	− 0.731875i	−0.930639	−	0.365937i	$-0.880748\pi$
$883$	− 19203.4i	− 0.731875i	0.930639	−	0.365937i	$-0.119252\pi$
$884$	0	0
$885$	− 483.122i	− 0.0183503i
$886$	0	0
$887$	−23198.0	−0.878144	−0.439072	−	0.898452i	$-0.644693\pi$
$887$	−23198.0	−0.878144	−0.439072	+	0.898452i	$0.644693\pi$
$888$	0	0
$889$	−40936.0	−1.54438
$890$	0	0
$891$	4881.30i	0.183535i
$892$	0	0
$893$	3765.48i	0.141105i
$894$	0	0
$895$	928.979	0.0346953
$896$	0	0
$897$	17377.8	0.646854
$898$	0	0
$899$	− 2302.68i	− 0.0854266i
$900$	0	0
$901$	− 5556.86i	− 0.205467i
$902$	0	0
$903$	16131.1	0.594472
$904$	0	0
$905$	−2070.92	−0.0760659
$906$	0	0
$907$	16151.3i	0.591285i	0.955299	+	0.295643i	$0.0955337\pi$
$907$	16151.3i	0.591285i	−0.955299	+	0.295643i	$0.904466\pi$
$908$	0	0
$909$	14984.9i	0.546776i
$910$	0	0
$911$	3487.34	0.126829	0.0634143	−	0.997987i	$-0.479801\pi$
$911$	3487.34	0.126829	0.0634143	+	0.997987i	$0.479801\pi$
$912$	0	0
$913$	−21811.2	−0.790632
$914$	0	0
$915$	502.624i	0.0181598i
$916$	0	0
$917$	− 25506.3i	− 0.918532i
$918$	0	0
$919$	−17055.5	−0.612198	−0.306099	−	0.952000i	$-0.599024\pi$
$919$	−17055.5	−0.612198	−0.306099	+	0.952000i	$0.599024\pi$
$920$	0	0
$921$	−21657.6	−0.774857
$922$	0	0
$923$	18544.7i	0.661329i
$924$	0	0
$925$	− 50890.2i	− 1.80893i
$926$	0	0
$927$	3571.73	0.126549
$928$	0	0
$929$	−55339.3	−1.95438	−0.977192	−	0.212359i	$-0.931885\pi$
$929$	−55339.3	−1.95438	−0.977192	+	0.212359i	$0.931885\pi$
$930$	0	0
$931$	− 5083.22i	− 0.178943i
$932$	0	0
$933$	4611.17i	0.161804i
$934$	0	0
$935$	1741.34	0.0609069
$936$	0	0
$937$	−20457.6	−0.713255	−0.356627	−	0.934247i	$-0.616073\pi$
$937$	−20457.6	−0.713255	−0.356627	+	0.934247i	$0.616073\pi$
$938$	0	0
$939$	− 6602.78i	− 0.229471i
$940$	0	0
$941$	− 55891.0i	− 1.93623i	−0.250502	−	0.968116i	$-0.580596\pi$
$941$	− 55891.0i	− 1.93623i	0.250502	−	0.968116i	$-0.419404\pi$
$942$	0	0
$943$	39497.0	1.36395
$944$	0	0
$945$	−376.230	−0.0129511
$946$	0	0
$947$	− 54727.0i	− 1.87792i	−0.344027	−	0.938960i	$-0.611791\pi$
$947$	− 54727.0i	− 1.87792i	0.344027	−	0.938960i	$-0.388209\pi$
$948$	0	0
$949$	28178.4i	0.963865i
$950$	0	0
$951$	8521.23	0.290557
$952$	0	0
$953$	29958.8	1.01832	0.509160	−	0.860672i	$-0.329956\pi$
$953$	29958.8	1.01832	0.509160	+	0.860672i	$0.329956\pi$
$954$	0	0
$955$	− 1715.27i	− 0.0581204i
$956$	0	0
$957$	1884.72i	0.0636619i
$958$	0	0
$959$	55713.0	1.87598
$960$	0	0
$961$	18997.2	0.637682
$962$	0	0
$963$	− 3155.67i	− 0.105597i
$964$	0	0
$965$	− 382.365i	− 0.0127552i
$966$	0	0
$967$	−13498.5	−0.448896	−0.224448	−	0.974486i	$-0.572058\pi$
$967$	−13498.5	−0.448896	−0.224448	+	0.974486i	$0.572058\pi$
$968$	0	0
$969$	4126.53	0.136804
$970$	0	0
$971$	9652.36i	0.319010i	0.987197	+	0.159505i	$0.0509899\pi$
$971$	9652.36i	0.319010i	−0.987197	+	0.159505i	$0.949010\pi$
$972$	0	0
$973$	− 55895.1i	− 1.84164i
$974$	0	0
$975$	−19780.2	−0.649717
$976$	0	0
$977$	12169.8	0.398511	0.199256	−	0.979948i	$-0.436148\pi$
$977$	12169.8	0.398511	0.199256	+	0.979948i	$0.436148\pi$
$978$	0	0
$979$	2430.32i	0.0793394i
$980$	0	0
$981$	− 5373.08i	− 0.174872i
$982$	0	0
$983$	−47545.2	−1.54268	−0.771341	−	0.636423i	$-0.780413\pi$
$983$	−47545.2	−1.54268	−0.771341	+	0.636423i	$0.780413\pi$
$984$	0	0
$985$	2928.06	0.0947165
$986$	0	0
$987$	− 8809.69i	− 0.284109i
$988$	0	0
$989$	25884.5i	0.832234i
$990$	0	0
$991$	892.350	0.0286039	0.0143019	−	0.999898i	$-0.495447\pi$
$991$	892.350	0.0286039	0.0143019	+	0.999898i	$0.495447\pi$
$992$	0	0
$993$	−6355.55	−0.203109
$994$	0	0
$995$	1602.67i	0.0510634i
$996$	0	0
$997$	− 4458.57i	− 0.141629i	−0.997489	−	0.0708146i	$-0.977440\pi$
$997$	− 4458.57i	− 0.141629i	0.997489	−	0.0708146i	$-0.0225599\pi$
$998$	0	0
$999$	−11025.4	−0.349177

Display

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p

up to: 50 250 1000 (See

a_n

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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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.4.d.a.49.5		6
3.2	odd	2		288.4.d.d.145.4		6
4.3	odd	2		24.4.d.a.13.6	yes	6
8.3	odd	2		24.4.d.a.13.5	✓	6
8.5	even	2	inner	96.4.d.a.49.2		6
12.11	even	2		72.4.d.d.37.1		6
16.3	odd	4		768.4.a.s.1.2		3
16.5	even	4		768.4.a.t.1.2		3
16.11	odd	4		768.4.a.r.1.2		3
16.13	even	4		768.4.a.q.1.2		3
24.5	odd	2		288.4.d.d.145.3		6
24.11	even	2		72.4.d.d.37.2		6
48.5	odd	4		2304.4.a.bu.1.2		3
48.11	even	4		2304.4.a.bt.1.2		3
48.29	odd	4		2304.4.a.bw.1.2		3
48.35	even	4		2304.4.a.bv.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.4.d.a.13.5	✓	6	8.3	odd	2
24.4.d.a.13.6	yes	6	4.3	odd	2
72.4.d.d.37.1		6	12.11	even	2
72.4.d.d.37.2		6	24.11	even	2
96.4.d.a.49.2		6	8.5	even	2	inner
96.4.d.a.49.5		6	1.1	even	1	trivial
288.4.d.d.145.3		6	24.5	odd	2
288.4.d.d.145.4		6	3.2	odd	2
768.4.a.q.1.2		3	16.13	even	4
768.4.a.r.1.2		3	16.11	odd	4
768.4.a.s.1.2		3	16.3	odd	4
768.4.a.t.1.2		3	16.5	even	4
2304.4.a.bt.1.2		3	48.11	even	4
2304.4.a.bu.1.2		3	48.5	odd	4
2304.4.a.bv.1.2		3	48.35	even	4
2304.4.a.bw.1.2		3	48.29	odd	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	96.4.d.a.49.5

Level	$96$
Weight	$4$
Character	96.49
Analytic conductor	$5.664$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$