LMFDB - Embedded newform 1024.4.b.j.513.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,4,Mod(513,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1024.513");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1024.b (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:	$60.4179558459$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 36x^{8} + 405x^{6} + 1380x^{4} + 420x^{2} + 32 $ x^10 + 36x^8 + 405x^6 + 1380x^4 + 420x^2 + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			513.2
Root			$0.357936i$ of defining polynomial
Character	$\chi$	$=$	1024.513
Dual form			1024.4.b.j.513.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.77277i q^{3} +6.59550i q^{5} -24.8965 q^{7} -33.4160 q^{9} +O(q^{10})$	$q-7.77277i q^{3} +6.59550i q^{5} -24.8965 q^{7} -33.4160 q^{9} +31.5979i q^{11} -15.9402i q^{13} +51.2653 q^{15} +88.4846 q^{17} +53.4838i q^{19} +193.515i q^{21} -48.1224 q^{23} +81.4994 q^{25} +49.8702i q^{27} -14.7689i q^{29} -96.9578 q^{31} +245.604 q^{33} -164.205i q^{35} +230.911i q^{37} -123.899 q^{39} +360.519 q^{41} -141.774i q^{43} -220.395i q^{45} -220.669 q^{47} +276.837 q^{49} -687.771i q^{51} -248.551i q^{53} -208.404 q^{55} +415.717 q^{57} -572.767i q^{59} -939.852i q^{61} +831.942 q^{63} +105.133 q^{65} +151.854i q^{67} +374.045i q^{69} +215.050 q^{71} +668.587 q^{73} -633.476i q^{75} -786.679i q^{77} +822.956 q^{79} -514.603 q^{81} +462.269i q^{83} +583.600i q^{85} -114.795 q^{87} -262.733 q^{89} +396.855i q^{91} +753.631i q^{93} -352.752 q^{95} -150.801 q^{97} -1055.88i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 28 q^{7} - 54 q^{9}+O(q^{10}) $ 10 * q - 28 * q^7 - 54 * q^9	$ 10 q - 28 q^{7} - 54 q^{9} + 124 q^{15} + 4 q^{17} - 276 q^{23} - 50 q^{25} + 368 q^{31} - 4 q^{33} - 732 q^{39} + 944 q^{47} - 94 q^{49} - 1380 q^{55} - 108 q^{57} + 2628 q^{63} - 492 q^{65} - 3468 q^{71} + 296 q^{73} + 4416 q^{79} - 482 q^{81} - 6036 q^{87} - 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) $ 10 * q - 28 * q^7 - 54 * q^9 + 124 * q^15 + 4 * q^17 - 276 * q^23 - 50 * q^25 + 368 * q^31 - 4 * q^33 - 732 * q^39 + 944 * q^47 - 94 * q^49 - 1380 * q^55 - 108 * q^57 + 2628 * q^63 - 492 * q^65 - 3468 * q^71 + 296 * q^73 + 4416 * q^79 - 482 * q^81 - 6036 * q^87 - 88 * q^89 + 6900 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$-1$	$1$

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
$3$	− 7.77277i	− 1.49587i	−0.663771	−	0.747935i	$-0.731045\pi$
$3$	− 7.77277i	− 1.49587i	0.663771	−	0.747935i	$-0.268955\pi$
$4$	0	0
$5$	6.59550i	0.589919i	0.955510	+	0.294960i	$0.0953062\pi$
$5$	6.59550i	0.589919i	−0.955510	+	0.294960i	$0.904694\pi$
$6$	0	0
$7$	−24.8965	−1.34429	−0.672143	−	0.740422i	$-0.734626\pi$
$7$	−24.8965	−1.34429	−0.672143	+	0.740422i	$0.734626\pi$
$8$	0	0
$9$	−33.4160	−1.23763
$10$	0	0
$11$	31.5979i	0.866103i	0.901369	+	0.433052i	$0.142563\pi$
$11$	31.5979i	0.866103i	−0.901369	+	0.433052i	$0.857437\pi$
$12$	0	0
$13$	− 15.9402i	− 0.340078i	−0.985437	−	0.170039i	$-0.945611\pi$
$13$	− 15.9402i	− 0.340078i	0.985437	−	0.170039i	$-0.0543893\pi$
$14$	0	0
$15$	51.2653	0.882443
$16$	0	0
$17$	88.4846	1.26239	0.631196	−	0.775623i	$-0.282564\pi$
$17$	88.4846	1.26239	0.631196	+	0.775623i	$0.282564\pi$
$18$	0	0
$19$	53.4838i	0.645790i	0.946435	+	0.322895i	$0.104656\pi$
$19$	53.4838i	0.645790i	−0.946435	+	0.322895i	$0.895344\pi$
$20$	0	0
$21$	193.515i	2.01088i
$22$	0	0
$23$	−48.1224	−0.436270	−0.218135	−	0.975919i	$-0.569997\pi$
$23$	−48.1224	−0.436270	−0.218135	+	0.975919i	$0.569997\pi$
$24$	0	0
$25$	81.4994	0.651995
$26$	0	0
$27$	49.8702i	0.355464i
$28$	0	0
$29$	− 14.7689i	− 0.0945692i	−0.998881	−	0.0472846i	$-0.984943\pi$
$29$	− 14.7689i	− 0.0945692i	0.998881	−	0.0472846i	$-0.0150568\pi$
$30$	0	0
$31$	−96.9578	−0.561746	−0.280873	−	0.959745i	$-0.590624\pi$
$31$	−96.9578	−0.561746	−0.280873	+	0.959745i	$0.590624\pi$
$32$	0	0
$33$	245.604	1.29558
$34$	0	0
$35$	− 164.205i	− 0.793020i
$36$	0	0
$37$	230.911i	1.02599i	0.858392	+	0.512994i	$0.171464\pi$
$37$	230.911i	1.02599i	−0.858392	+	0.512994i	$0.828536\pi$
$38$	0	0
$39$	−123.899	−0.508713
$40$	0	0
$41$	360.519	1.37326	0.686629	−	0.727008i	$-0.259090\pi$
$41$	360.519	1.37326	0.686629	+	0.727008i	$0.259090\pi$
$42$	0	0
$43$	− 141.774i	− 0.502797i	−0.967884	−	0.251398i	$-0.919110\pi$
$43$	− 141.774i	− 0.502797i	0.967884	−	0.251398i	$-0.0808905\pi$
$44$	0	0
$45$	− 220.395i	− 0.730102i
$46$	0	0
$47$	−220.669	−0.684849	−0.342425	−	0.939545i	$-0.611248\pi$
$47$	−220.669	−0.684849	−0.342425	+	0.939545i	$0.611248\pi$
$48$	0	0
$49$	276.837	0.807104
$50$	0	0
$51$	− 687.771i	− 1.88838i
$52$	0	0
$53$	− 248.551i	− 0.644172i	−0.946710	−	0.322086i	$-0.895616\pi$
$53$	− 248.551i	− 0.644172i	0.946710	−	0.322086i	$-0.104384\pi$
$54$	0	0
$55$	−208.404	−0.510931
$56$	0	0
$57$	415.717	0.966019
$58$	0	0
$59$	− 572.767i	− 1.26386i	−0.775024	−	0.631932i	$-0.782262\pi$
$59$	− 572.767i	− 1.26386i	0.775024	−	0.631932i	$-0.217738\pi$
$60$	0	0
$61$	− 939.852i	− 1.97272i	−0.164614	−	0.986358i	$-0.552638\pi$
$61$	− 939.852i	− 1.97272i	0.164614	−	0.986358i	$-0.447362\pi$
$62$	0	0
$63$	831.942	1.66373
$64$	0	0
$65$	105.133	0.200619
$66$	0	0
$67$	151.854i	0.276894i	0.990370	+	0.138447i	$0.0442111\pi$
$67$	151.854i	0.276894i	−0.990370	+	0.138447i	$0.955789\pi$
$68$	0	0
$69$	374.045i	0.652604i
$70$	0	0
$71$	215.050	0.359461	0.179731	−	0.983716i	$-0.442477\pi$
$71$	215.050	0.359461	0.179731	+	0.983716i	$0.442477\pi$
$72$	0	0
$73$	668.587	1.07195	0.535974	−	0.844235i	$-0.319945\pi$
$73$	668.587	1.07195	0.535974	+	0.844235i	$0.319945\pi$
$74$	0	0
$75$	− 633.476i	− 0.975300i
$76$	0	0
$77$	− 786.679i	− 1.16429i
$78$	0	0
$79$	822.956	1.17202	0.586012	−	0.810303i	$-0.300697\pi$
$79$	822.956	1.17202	0.586012	+	0.810303i	$0.300697\pi$
$80$	0	0
$81$	−514.603	−0.705902
$82$	0	0
$83$	462.269i	0.611332i	0.952139	+	0.305666i	$0.0988791\pi$
$83$	462.269i	0.611332i	−0.952139	+	0.305666i	$0.901121\pi$
$84$	0	0
$85$	583.600i	0.744710i
$86$	0	0
$87$	−114.795	−0.141463
$88$	0	0
$89$	−262.733	−0.312918	−0.156459	−	0.987684i	$-0.550008\pi$
$89$	−262.733	−0.312918	−0.156459	+	0.987684i	$0.550008\pi$
$90$	0	0
$91$	396.855i	0.457162i
$92$	0	0
$93$	753.631i	0.840300i
$94$	0	0
$95$	−352.752	−0.380964
$96$	0	0
$97$	−150.801	−0.157850	−0.0789251	−	0.996881i	$-0.525149\pi$
$97$	−150.801	−0.157850	−0.0789251	+	0.996881i	$0.525149\pi$
$98$	0	0
$99$	− 1055.88i	− 1.07192i
$100$	0	0
$101$	− 690.115i	− 0.679891i	−0.940445	−	0.339945i	$-0.889591\pi$
$101$	− 690.115i	− 0.679891i	0.940445	−	0.339945i	$-0.110409\pi$
$102$	0	0
$103$	−1840.58	−1.76075	−0.880377	−	0.474275i	$-0.842710\pi$
$103$	−1840.58	−1.76075	−0.880377	+	0.474275i	$0.842710\pi$
$104$	0	0
$105$	−1276.33	−1.18626
$106$	0	0
$107$	− 112.302i	− 0.101464i	−0.998712	−	0.0507322i	$-0.983845\pi$
$107$	− 112.302i	− 0.101464i	0.998712	−	0.0507322i	$-0.0161555\pi$
$108$	0	0
$109$	1347.72i	1.18429i	0.805831	+	0.592146i	$0.201719\pi$
$109$	1347.72i	1.18429i	−0.805831	+	0.592146i	$0.798281\pi$
$110$	0	0
$111$	1794.82	1.53475
$112$	0	0
$113$	720.469	0.599788	0.299894	−	0.953973i	$-0.403049\pi$
$113$	720.469	0.599788	0.299894	+	0.953973i	$0.403049\pi$
$114$	0	0
$115$	− 317.391i	− 0.257364i
$116$	0	0
$117$	532.657i	0.420890i
$118$	0	0
$119$	−2202.96	−1.69702
$120$	0	0
$121$	332.571	0.249865
$122$	0	0
$123$	− 2802.23i	− 2.05422i
$124$	0	0
$125$	1361.97i	0.974544i
$126$	0	0
$127$	2622.35	1.83225	0.916124	−	0.400895i	$-0.131301\pi$
$127$	2622.35	1.83225	0.916124	+	0.400895i	$0.131301\pi$
$128$	0	0
$129$	−1101.97	−0.752119
$130$	0	0
$131$	− 929.950i	− 0.620230i	−0.950699	−	0.310115i	$-0.899632\pi$
$131$	− 929.950i	− 0.620230i	0.950699	−	0.310115i	$-0.100368\pi$
$132$	0	0
$133$	− 1331.56i	− 0.868127i
$134$	0	0
$135$	−328.919	−0.209695
$136$	0	0
$137$	−2511.52	−1.56623	−0.783117	−	0.621874i	$-0.786371\pi$
$137$	−2511.52	−1.56623	−0.783117	+	0.621874i	$0.786371\pi$
$138$	0	0
$139$	− 1535.86i	− 0.937195i	−0.883412	−	0.468598i	$-0.844759\pi$
$139$	− 1535.86i	− 0.937195i	0.883412	−	0.468598i	$-0.155241\pi$
$140$	0	0
$141$	1715.21i	1.02445i
$142$	0	0
$143$	503.677	0.294543
$144$	0	0
$145$	97.4080	0.0557882
$146$	0	0
$147$	− 2151.79i	− 1.20732i
$148$	0	0
$149$	− 3230.96i	− 1.77645i	−0.459413	−	0.888223i	$-0.651940\pi$
$149$	− 3230.96i	− 1.77645i	0.459413	−	0.888223i	$-0.348060\pi$
$150$	0	0
$151$	2814.39	1.51677	0.758383	−	0.651809i	$-0.225990\pi$
$151$	2814.39	1.51677	0.758383	+	0.651809i	$0.225990\pi$
$152$	0	0
$153$	−2956.80	−1.56237
$154$	0	0
$155$	− 639.485i	− 0.331385i
$156$	0	0
$157$	1281.71i	0.651541i	0.945449	+	0.325770i	$0.105624\pi$
$157$	1281.71i	0.651541i	−0.945449	+	0.325770i	$0.894376\pi$
$158$	0	0
$159$	−1931.93	−0.963598
$160$	0	0
$161$	1198.08	0.586472
$162$	0	0
$163$	− 1969.09i	− 0.946204i	−0.881008	−	0.473102i	$-0.843134\pi$
$163$	− 1969.09i	− 0.946204i	0.881008	−	0.473102i	$-0.156866\pi$
$164$	0	0
$165$	1619.88i	0.764287i
$166$	0	0
$167$	−1221.66	−0.566075	−0.283038	−	0.959109i	$-0.591342\pi$
$167$	−1221.66	−0.566075	−0.283038	+	0.959109i	$0.591342\pi$
$168$	0	0
$169$	1942.91	0.884347
$170$	0	0
$171$	− 1787.21i	− 0.799249i
$172$	0	0
$173$	− 796.794i	− 0.350168i	−0.984553	−	0.175084i	$-0.943980\pi$
$173$	− 796.794i	− 0.350168i	0.984553	−	0.175084i	$-0.0560198\pi$
$174$	0	0
$175$	−2029.05	−0.876468
$176$	0	0
$177$	−4451.99	−1.89058
$178$	0	0
$179$	− 3114.42i	− 1.30046i	−0.759737	−	0.650230i	$-0.774672\pi$
$179$	− 3114.42i	− 1.30046i	0.759737	−	0.650230i	$-0.225328\pi$
$180$	0	0
$181$	171.535i	0.0704425i	0.999380	+	0.0352213i	$0.0112136\pi$
$181$	171.535i	0.0704425i	−0.999380	+	0.0352213i	$0.988786\pi$
$182$	0	0
$183$	−7305.26	−2.95093
$184$	0	0
$185$	−1522.98	−0.605251
$186$	0	0
$187$	2795.93i	1.09336i
$188$	0	0
$189$	− 1241.59i	− 0.477845i
$190$	0	0
$191$	3927.65	1.48793	0.743966	−	0.668218i	$-0.232942\pi$
$191$	3927.65	1.48793	0.743966	+	0.668218i	$0.232942\pi$
$192$	0	0
$193$	−3249.02	−1.21176	−0.605880	−	0.795556i	$-0.707179\pi$
$193$	−3249.02	−1.21176	−0.605880	+	0.795556i	$0.707179\pi$
$194$	0	0
$195$	− 817.179i	− 0.300099i
$196$	0	0
$197$	− 3423.68i	− 1.23821i	−0.785309	−	0.619103i	$-0.787496\pi$
$197$	− 3423.68i	− 1.23821i	0.785309	−	0.619103i	$-0.212504\pi$
$198$	0	0
$199$	1371.30	0.488488	0.244244	−	0.969714i	$-0.421460\pi$
$199$	1371.30	0.488488	0.244244	+	0.969714i	$0.421460\pi$
$200$	0	0
$201$	1180.33	0.414198
$202$	0	0
$203$	367.693i	0.127128i
$204$	0	0
$205$	2377.80i	0.810112i
$206$	0	0
$207$	1608.06	0.539941
$208$	0	0
$209$	−1689.98	−0.559321
$210$	0	0
$211$	− 2291.73i	− 0.747721i	−0.927485	−	0.373861i	$-0.878034\pi$
$211$	− 2291.73i	− 0.747721i	0.927485	−	0.373861i	$-0.121966\pi$
$212$	0	0
$213$	− 1671.54i	− 0.537708i
$214$	0	0
$215$	935.068	0.296610
$216$	0	0
$217$	2413.91	0.755148
$218$	0	0
$219$	− 5196.77i	− 1.60349i
$220$	0	0
$221$	− 1410.46i	− 0.429312i
$222$	0	0
$223$	−419.617	−0.126007	−0.0630036	−	0.998013i	$-0.520068\pi$
$223$	−419.617	−0.126007	−0.0630036	+	0.998013i	$0.520068\pi$
$224$	0	0
$225$	−2723.38	−0.806929
$226$	0	0
$227$	3017.42i	0.882262i	0.897443	+	0.441131i	$0.145423\pi$
$227$	3017.42i	0.882262i	−0.897443	+	0.441131i	$0.854577\pi$
$228$	0	0
$229$	2226.56i	0.642513i	0.946992	+	0.321256i	$0.104105\pi$
$229$	2226.56i	0.642513i	−0.946992	+	0.321256i	$0.895895\pi$
$230$	0	0
$231$	−6114.67	−1.74163
$232$	0	0
$233$	−1194.86	−0.335957	−0.167978	−	0.985791i	$-0.553724\pi$
$233$	−1194.86	−0.335957	−0.167978	+	0.985791i	$0.553724\pi$
$234$	0	0
$235$	− 1455.42i	− 0.404006i
$236$	0	0
$237$	− 6396.65i	− 1.75320i
$238$	0	0
$239$	4241.03	1.14782	0.573911	−	0.818917i	$-0.305425\pi$
$239$	4241.03	1.14782	0.573911	+	0.818917i	$0.305425\pi$
$240$	0	0
$241$	5571.19	1.48910	0.744548	−	0.667569i	$-0.232665\pi$
$241$	5571.19	1.48910	0.744548	+	0.667569i	$0.232665\pi$
$242$	0	0
$243$	5346.38i	1.41140i
$244$	0	0
$245$	1825.88i	0.476126i
$246$	0	0
$247$	852.541	0.219619
$248$	0	0
$249$	3593.11	0.914474
$250$	0	0
$251$	682.681i	0.171675i	0.996309	+	0.0858375i	$0.0273566\pi$
$251$	682.681i	0.171675i	−0.996309	+	0.0858375i	$0.972643\pi$
$252$	0	0
$253$	− 1520.57i	− 0.377855i
$254$	0	0
$255$	4536.19	1.11399
$256$	0	0
$257$	8093.12	1.96434	0.982169	−	0.188002i	$-0.0602013\pi$
$257$	8093.12	1.96434	0.982169	+	0.188002i	$0.0602013\pi$
$258$	0	0
$259$	− 5748.89i	− 1.37922i
$260$	0	0
$261$	493.516i	0.117042i
$262$	0	0
$263$	−410.300	−0.0961984	−0.0480992	−	0.998843i	$-0.515316\pi$
$263$	−410.300	−0.0961984	−0.0480992	+	0.998843i	$0.515316\pi$
$264$	0	0
$265$	1639.32	0.380009
$266$	0	0
$267$	2042.17i	0.468085i
$268$	0	0
$269$	− 6.75940i	− 0.00153207i	−1.00000		0.000766037i	$-0.999756\pi$
$269$	− 6.75940i	− 0.00153207i	1.00000		0.000766037i	$-0.000243837\pi$
$270$	0	0
$271$	2833.98	0.635247	0.317623	−	0.948217i	$-0.397115\pi$
$271$	2833.98	0.635247	0.317623	+	0.948217i	$0.397115\pi$
$272$	0	0
$273$	3084.67	0.683855
$274$	0	0
$275$	2575.21i	0.564695i
$276$	0	0
$277$	2157.81i	0.468052i	0.972230	+	0.234026i	$0.0751900\pi$
$277$	2157.81i	0.468052i	−0.972230	+	0.234026i	$0.924810\pi$
$278$	0	0
$279$	3239.94	0.695234
$280$	0	0
$281$	4750.23	1.00845	0.504226	−	0.863572i	$-0.331778\pi$
$281$	4750.23	1.00845	0.504226	+	0.863572i	$0.331778\pi$
$282$	0	0
$283$	− 910.900i	− 0.191334i	−0.995413	−	0.0956668i	$-0.969502\pi$
$283$	− 910.900i	− 0.191334i	0.995413	−	0.0956668i	$-0.0304983\pi$
$284$	0	0
$285$	2741.86i	0.569873i
$286$	0	0
$287$	−8975.67	−1.84605
$288$	0	0
$289$	2916.53	0.593635
$290$	0	0
$291$	1172.14i	0.236124i
$292$	0	0
$293$	2026.80i	0.404119i	0.979373	+	0.202060i	$0.0647635\pi$
$293$	2026.80i	0.404119i	−0.979373	+	0.202060i	$0.935237\pi$
$294$	0	0
$295$	3777.69	0.745578
$296$	0	0
$297$	−1575.79	−0.307868
$298$	0	0
$299$	767.080i	0.148366i
$300$	0	0
$301$	3529.67i	0.675903i
$302$	0	0
$303$	−5364.10	−1.01703
$304$	0	0
$305$	6198.79	1.16374
$306$	0	0
$307$	− 326.981i	− 0.0607877i	−0.999538	−	0.0303938i	$-0.990324\pi$
$307$	− 326.981i	− 0.0607877i	0.999538	−	0.0303938i	$-0.00967615\pi$
$308$	0	0
$309$	14306.4i	2.63386i
$310$	0	0
$311$	−871.410	−0.158885	−0.0794423	−	0.996839i	$-0.525314\pi$
$311$	−871.410	−0.158885	−0.0794423	+	0.996839i	$0.525314\pi$
$312$	0	0
$313$	−3515.02	−0.634762	−0.317381	−	0.948298i	$-0.602803\pi$
$313$	−3515.02	−0.634762	−0.317381	+	0.948298i	$0.602803\pi$
$314$	0	0
$315$	5487.07i	0.981465i
$316$	0	0
$317$	− 6680.42i	− 1.18363i	−0.806075	−	0.591814i	$-0.798412\pi$
$317$	− 6680.42i	− 1.18363i	0.806075	−	0.591814i	$-0.201588\pi$
$318$	0	0
$319$	466.665	0.0819067
$320$	0	0
$321$	−872.901	−0.151778
$322$	0	0
$323$	4732.49i	0.815241i
$324$	0	0
$325$	− 1299.12i	− 0.221729i
$326$	0	0
$327$	10475.5	1.77155
$328$	0	0
$329$	5493.90	0.920633
$330$	0	0
$331$	− 2574.24i	− 0.427471i	−0.976892	−	0.213736i	$-0.931437\pi$
$331$	− 2574.24i	− 0.427471i	0.976892	−	0.213736i	$-0.0685631\pi$
$332$	0	0
$333$	− 7716.13i	− 1.26979i
$334$	0	0
$335$	−1001.55	−0.163345
$336$	0	0
$337$	−74.0970	−0.0119772	−0.00598861	−	0.999982i	$-0.501906\pi$
$337$	−74.0970	−0.0119772	−0.00598861	+	0.999982i	$0.501906\pi$
$338$	0	0
$339$	− 5600.04i	− 0.897205i
$340$	0	0
$341$	− 3063.67i	− 0.486530i
$342$	0	0
$343$	1647.24	0.259307
$344$	0	0
$345$	−2467.01	−0.384984
$346$	0	0
$347$	− 2973.72i	− 0.460050i	−0.973185	−	0.230025i	$-0.926119\pi$
$347$	− 2973.72i	− 0.460050i	0.973185	−	0.230025i	$-0.0738808\pi$
$348$	0	0
$349$	9351.98i	1.43438i	0.696875	+	0.717192i	$0.254573\pi$
$349$	9351.98i	1.43438i	−0.696875	+	0.717192i	$0.745427\pi$
$350$	0	0
$351$	794.940	0.120885
$352$	0	0
$353$	2216.90	0.334259	0.167130	−	0.985935i	$-0.446550\pi$
$353$	2216.90	0.334259	0.167130	+	0.985935i	$0.446550\pi$
$354$	0	0
$355$	1418.36i	0.212053i
$356$	0	0
$357$	17123.1i	2.53852i
$358$	0	0
$359$	−2082.23	−0.306117	−0.153059	−	0.988217i	$-0.548912\pi$
$359$	−2082.23	−0.306117	−0.153059	+	0.988217i	$0.548912\pi$
$360$	0	0
$361$	3998.49	0.582955
$362$	0	0
$363$	− 2585.00i	− 0.373766i
$364$	0	0
$365$	4409.66i	0.632362i
$366$	0	0
$367$	−4509.22	−0.641360	−0.320680	−	0.947188i	$-0.603911\pi$
$367$	−4509.22	−0.641360	−0.320680	+	0.947188i	$0.603911\pi$
$368$	0	0
$369$	−12047.1	−1.69959
$370$	0	0
$371$	6188.05i	0.865951i
$372$	0	0
$373$	− 12249.3i	− 1.70039i	−0.526470	−	0.850194i	$-0.676485\pi$
$373$	− 12249.3i	− 1.70039i	0.526470	−	0.850194i	$-0.323515\pi$
$374$	0	0
$375$	10586.3	1.45779
$376$	0	0
$377$	−235.418	−0.0321609
$378$	0	0
$379$	− 4981.51i	− 0.675153i	−0.941298	−	0.337576i	$-0.890393\pi$
$379$	− 4981.51i	− 0.675153i	0.941298	−	0.337576i	$-0.109607\pi$
$380$	0	0
$381$	− 20382.9i	− 2.74081i
$382$	0	0
$383$	3044.88	0.406229	0.203115	−	0.979155i	$-0.434894\pi$
$383$	3044.88	0.406229	0.203115	+	0.979155i	$0.434894\pi$
$384$	0	0
$385$	5188.54	0.686837
$386$	0	0
$387$	4737.51i	0.622277i
$388$	0	0
$389$	− 2733.30i	− 0.356256i	−0.984007	−	0.178128i	$-0.942996\pi$
$389$	− 2733.30i	− 0.356256i	0.984007	−	0.178128i	$-0.0570042\pi$
$390$	0	0
$391$	−4258.09	−0.550744
$392$	0	0
$393$	−7228.29	−0.927784
$394$	0	0
$395$	5427.81i	0.691399i
$396$	0	0
$397$	− 950.997i	− 0.120225i	−0.998192	−	0.0601123i	$-0.980854\pi$
$397$	− 950.997i	− 0.120225i	0.998192	−	0.0601123i	$-0.0191459\pi$
$398$	0	0
$399$	−10349.9	−1.29861
$400$	0	0
$401$	7606.74	0.947288	0.473644	−	0.880716i	$-0.342938\pi$
$401$	7606.74	0.947288	0.473644	+	0.880716i	$0.342938\pi$
$402$	0	0
$403$	1545.53i	0.191037i
$404$	0	0
$405$	− 3394.06i	− 0.416425i
$406$	0	0
$407$	−7296.32	−0.888612
$408$	0	0
$409$	4981.58	0.602257	0.301129	−	0.953584i	$-0.402637\pi$
$409$	4981.58	0.602257	0.301129	+	0.953584i	$0.402637\pi$
$410$	0	0
$411$	19521.5i	2.34288i
$412$	0	0
$413$	14259.9i	1.69899i
$414$	0	0
$415$	−3048.89	−0.360637
$416$	0	0
$417$	−11937.9	−1.40192
$418$	0	0
$419$	4855.53i	0.566129i	0.959101	+	0.283065i	$0.0913511\pi$
$419$	4855.53i	0.566129i	−0.959101	+	0.283065i	$0.908649\pi$
$420$	0	0
$421$	− 11276.5i	− 1.30543i	−0.757604	−	0.652714i	$-0.773630\pi$
$421$	− 11276.5i	− 1.30543i	0.757604	−	0.652714i	$-0.226370\pi$
$422$	0	0
$423$	7373.88	0.847590
$424$	0	0
$425$	7211.44	0.823074
$426$	0	0
$427$	23399.0i	2.65189i
$428$	0	0
$429$	− 3914.97i	− 0.440598i
$430$	0	0
$431$	−4800.16	−0.536463	−0.268232	−	0.963354i	$-0.586439\pi$
$431$	−4800.16	−0.536463	−0.268232	+	0.963354i	$0.586439\pi$
$432$	0	0
$433$	−6242.32	−0.692810	−0.346405	−	0.938085i	$-0.612598\pi$
$433$	−6242.32	−0.692810	−0.346405	+	0.938085i	$0.612598\pi$
$434$	0	0
$435$	− 757.130i	− 0.0834520i
$436$	0	0
$437$	− 2573.77i	− 0.281739i
$438$	0	0
$439$	−4929.27	−0.535903	−0.267951	−	0.963432i	$-0.586347\pi$
$439$	−4929.27	−0.535903	−0.267951	+	0.963432i	$0.586347\pi$
$440$	0	0
$441$	−9250.77	−0.998896
$442$	0	0
$443$	10848.0i	1.16344i	0.813390	+	0.581718i	$0.197619\pi$
$443$	10848.0i	1.16344i	−0.813390	+	0.581718i	$0.802381\pi$
$444$	0	0
$445$	− 1732.86i	− 0.184596i
$446$	0	0
$447$	−25113.5	−2.65733
$448$	0	0
$449$	−11515.2	−1.21032	−0.605162	−	0.796102i	$-0.706892\pi$
$449$	−11515.2	−1.21032	−0.605162	+	0.796102i	$0.706892\pi$
$450$	0	0
$451$	11391.7i	1.18938i
$452$	0	0
$453$	− 21875.6i	− 2.26889i
$454$	0	0
$455$	−2617.46	−0.269689
$456$	0	0
$457$	−4829.89	−0.494383	−0.247191	−	0.968967i	$-0.579508\pi$
$457$	−4829.89	−0.494383	−0.247191	+	0.968967i	$0.579508\pi$
$458$	0	0
$459$	4412.74i	0.448735i
$460$	0	0
$461$	11689.6i	1.18100i	0.807039	+	0.590498i	$0.201069\pi$
$461$	11689.6i	1.18100i	−0.807039	+	0.590498i	$0.798931\pi$
$462$	0	0
$463$	5043.86	0.506281	0.253141	−	0.967430i	$-0.418536\pi$
$463$	5043.86	0.506281	0.253141	+	0.967430i	$0.418536\pi$
$464$	0	0
$465$	−4970.57	−0.495709
$466$	0	0
$467$	17591.1i	1.74308i	0.490321	+	0.871542i	$0.336880\pi$
$467$	17591.1i	1.74308i	−0.490321	+	0.871542i	$0.663120\pi$
$468$	0	0
$469$	− 3780.63i	− 0.372225i
$470$	0	0
$471$	9962.47	0.974621
$472$	0	0
$473$	4479.75	0.435474
$474$	0	0
$475$	4358.89i	0.421052i
$476$	0	0
$477$	8305.58i	0.797246i
$478$	0	0
$479$	13059.7	1.24575	0.622875	−	0.782321i	$-0.285964\pi$
$479$	13059.7	1.24575	0.622875	+	0.782321i	$0.285964\pi$
$480$	0	0
$481$	3680.77	0.348916
$482$	0	0
$483$	− 9312.41i	− 0.877286i
$484$	0	0
$485$	− 994.605i	− 0.0931189i
$486$	0	0
$487$	15549.3	1.44683	0.723414	−	0.690414i	$-0.242572\pi$
$487$	15549.3	1.44683	0.723414	+	0.690414i	$0.242572\pi$
$488$	0	0
$489$	−15305.3	−1.41540
$490$	0	0
$491$	− 12202.3i	− 1.12155i	−0.827967	−	0.560777i	$-0.810503\pi$
$491$	− 12202.3i	− 1.12155i	0.827967	−	0.560777i	$-0.189497\pi$
$492$	0	0
$493$	− 1306.82i	− 0.119383i
$494$	0	0
$495$	6964.03	0.632344
$496$	0	0
$497$	−5354.00	−0.483219
$498$	0	0
$499$	2450.01i	0.219794i	0.993943	+	0.109897i	$0.0350521\pi$
$499$	2450.01i	0.219794i	−0.993943	+	0.109897i	$0.964948\pi$
$500$	0	0
$501$	9495.65i	0.846775i
$502$	0	0
$503$	16579.6	1.46968	0.734839	−	0.678241i	$-0.237258\pi$
$503$	16579.6	1.46968	0.734839	+	0.678241i	$0.237258\pi$
$504$	0	0
$505$	4551.65	0.401081
$506$	0	0
$507$	− 15101.8i	− 1.32287i
$508$	0	0
$509$	11074.6i	0.964387i	0.876065	+	0.482193i	$0.160160\pi$
$509$	11074.6i	0.964387i	−0.876065	+	0.482193i	$0.839840\pi$
$510$	0	0
$511$	−16645.5	−1.44100
$512$	0	0
$513$	−2667.24	−0.229555
$514$	0	0
$515$	− 12139.5i	− 1.03870i
$516$	0	0
$517$	− 6972.69i	− 0.593150i
$518$	0	0
$519$	−6193.30	−0.523807
$520$	0	0
$521$	3400.02	0.285907	0.142953	−	0.989729i	$-0.454340\pi$
$521$	3400.02	0.285907	0.142953	+	0.989729i	$0.454340\pi$
$522$	0	0
$523$	− 2856.01i	− 0.238785i	−0.992847	−	0.119393i	$-0.961905\pi$
$523$	− 2856.01i	− 0.238785i	0.992847	−	0.119393i	$-0.0380947\pi$
$524$	0	0
$525$	15771.4i	1.31108i
$526$	0	0
$527$	−8579.28	−0.709144
$528$	0	0
$529$	−9851.23	−0.809668
$530$	0	0
$531$	19139.6i	1.56420i
$532$	0	0
$533$	− 5746.74i	− 0.467015i
$534$	0	0
$535$	740.691	0.0598558
$536$	0	0
$537$	−24207.7	−1.94532
$538$	0	0
$539$	8747.47i	0.699035i
$540$	0	0
$541$	18996.6i	1.50966i	0.655921	+	0.754829i	$0.272280\pi$
$541$	18996.6i	1.50966i	−0.655921	+	0.754829i	$0.727720\pi$
$542$	0	0
$543$	1333.30	0.105373
$544$	0	0
$545$	−8888.86	−0.698637
$546$	0	0
$547$	7603.46i	0.594333i	0.954826	+	0.297167i	$0.0960417\pi$
$547$	7603.46i	0.594333i	−0.954826	+	0.297167i	$0.903958\pi$
$548$	0	0
$549$	31406.1i	2.44149i
$550$	0	0
$551$	789.894	0.0610719
$552$	0	0
$553$	−20488.7	−1.57553
$554$	0	0
$555$	11837.7i	0.905377i
$556$	0	0
$557$	− 2936.30i	− 0.223367i	−0.993744	−	0.111683i	$-0.964376\pi$
$557$	− 2936.30i	− 0.223367i	0.993744	−	0.111683i	$-0.0356242\pi$
$558$	0	0
$559$	−2259.90	−0.170990
$560$	0	0
$561$	21732.1	1.63553
$562$	0	0
$563$	− 23536.9i	− 1.76192i	−0.473188	−	0.880961i	$-0.656897\pi$
$563$	− 23536.9i	− 1.76192i	0.473188	−	0.880961i	$-0.343103\pi$
$564$	0	0
$565$	4751.85i	0.353826i
$566$	0	0
$567$	12811.8	0.948934
$568$	0	0
$569$	5659.60	0.416982	0.208491	−	0.978024i	$-0.433145\pi$
$569$	5659.60	0.416982	0.208491	+	0.978024i	$0.433145\pi$
$570$	0	0
$571$	− 6891.00i	− 0.505043i	−0.967591	−	0.252521i	$-0.918740\pi$
$571$	− 6891.00i	− 0.505043i	0.967591	−	0.252521i	$-0.0812598\pi$
$572$	0	0
$573$	− 30528.8i	− 2.22575i
$574$	0	0
$575$	−3921.95	−0.284446
$576$	0	0
$577$	10652.2	0.768556	0.384278	−	0.923217i	$-0.374450\pi$
$577$	10652.2	0.768556	0.384278	+	0.923217i	$0.374450\pi$
$578$	0	0
$579$	25253.9i	1.81264i
$580$	0	0
$581$	− 11508.9i	− 0.821805i
$582$	0	0
$583$	7853.69	0.557919
$584$	0	0
$585$	−3513.14	−0.248291
$586$	0	0
$587$	14006.3i	0.984841i	0.870357	+	0.492421i	$0.163888\pi$
$587$	14006.3i	0.984841i	−0.870357	+	0.492421i	$0.836112\pi$
$588$	0	0
$589$	− 5185.67i	− 0.362770i
$590$	0	0
$591$	−26611.5	−1.85220
$592$	0	0
$593$	3528.04	0.244316	0.122158	−	0.992511i	$-0.461019\pi$
$593$	3528.04	0.244316	0.122158	+	0.992511i	$0.461019\pi$
$594$	0	0
$595$	− 14529.6i	− 1.00110i
$596$	0	0
$597$	− 10658.8i	− 0.730715i
$598$	0	0
$599$	−19024.9	−1.29772	−0.648861	−	0.760907i	$-0.724754\pi$
$599$	−19024.9	−1.29772	−0.648861	+	0.760907i	$0.724754\pi$
$600$	0	0
$601$	1065.92	0.0723460	0.0361730	−	0.999346i	$-0.488483\pi$
$601$	1065.92	0.0723460	0.0361730	+	0.999346i	$0.488483\pi$
$602$	0	0
$603$	− 5074.35i	− 0.342692i
$604$	0	0
$605$	2193.47i	0.147400i
$606$	0	0
$607$	12909.7	0.863242	0.431621	−	0.902055i	$-0.357942\pi$
$607$	12909.7	0.863242	0.431621	+	0.902055i	$0.357942\pi$
$608$	0	0
$609$	2858.00	0.190167
$610$	0	0
$611$	3517.51i	0.232902i
$612$	0	0
$613$	12393.3i	0.816578i	0.912853	+	0.408289i	$0.133874\pi$
$613$	12393.3i	0.816578i	−0.912853	+	0.408289i	$0.866126\pi$
$614$	0	0
$615$	18482.1	1.21182
$616$	0	0
$617$	18921.2	1.23459	0.617293	−	0.786733i	$-0.288229\pi$
$617$	18921.2	1.23459	0.617293	+	0.786733i	$0.288229\pi$
$618$	0	0
$619$	− 21378.2i	− 1.38814i	−0.719906	−	0.694072i	$-0.755815\pi$
$619$	− 21378.2i	− 1.38814i	0.719906	−	0.694072i	$-0.244185\pi$
$620$	0	0
$621$	− 2399.87i	− 0.155078i
$622$	0	0
$623$	6541.15	0.420651
$624$	0	0
$625$	1204.57	0.0770926
$626$	0	0
$627$	13135.8i	0.836672i
$628$	0	0
$629$	20432.1i	1.29520i
$630$	0	0
$631$	9602.80	0.605834	0.302917	−	0.953017i	$-0.402039\pi$
$631$	9602.80	0.605834	0.302917	+	0.953017i	$0.402039\pi$
$632$	0	0
$633$	−17813.1	−1.11849
$634$	0	0
$635$	17295.7i	1.08088i
$636$	0	0
$637$	− 4412.83i	− 0.274478i
$638$	0	0
$639$	−7186.12	−0.444880
$640$	0	0
$641$	4450.84	0.274256	0.137128	−	0.990553i	$-0.456213\pi$
$641$	4450.84	0.274256	0.137128	+	0.990553i	$0.456213\pi$
$642$	0	0
$643$	− 9180.04i	− 0.563026i	−0.959557	−	0.281513i	$-0.909164\pi$
$643$	− 9180.04i	− 0.563026i	0.959557	−	0.281513i	$-0.0908362\pi$
$644$	0	0
$645$	− 7268.07i	− 0.443690i
$646$	0	0
$647$	−5546.17	−0.337005	−0.168503	−	0.985701i	$-0.553893\pi$
$647$	−5546.17	−0.337005	−0.168503	+	0.985701i	$0.553893\pi$
$648$	0	0
$649$	18098.3	1.09464
$650$	0	0
$651$	− 18762.8i	− 1.12960i
$652$	0	0
$653$	8948.56i	0.536270i	0.963381	+	0.268135i	$0.0864073\pi$
$653$	8948.56i	0.536270i	−0.963381	+	0.268135i	$0.913593\pi$
$654$	0	0
$655$	6133.49	0.365886
$656$	0	0
$657$	−22341.5	−1.32667
$658$	0	0
$659$	21404.4i	1.26525i	0.774460	+	0.632623i	$0.218022\pi$
$659$	21404.4i	1.26525i	−0.774460	+	0.632623i	$0.781978\pi$
$660$	0	0
$661$	− 33177.6i	− 1.95228i	−0.217140	−	0.976140i	$-0.569673\pi$
$661$	− 33177.6i	− 1.95228i	0.217140	−	0.976140i	$-0.430327\pi$
$662$	0	0
$663$	−10963.2	−0.642195
$664$	0	0
$665$	8782.30	0.512125
$666$	0	0
$667$	710.713i	0.0412577i
$668$	0	0
$669$	3261.59i	0.188491i
$670$	0	0
$671$	29697.4	1.70858
$672$	0	0
$673$	30638.5	1.75487	0.877436	−	0.479694i	$-0.159252\pi$
$673$	30638.5	1.75487	0.877436	+	0.479694i	$0.159252\pi$
$674$	0	0
$675$	4064.39i	0.231760i
$676$	0	0
$677$	17633.7i	1.00106i	0.865720	+	0.500529i	$0.166861\pi$
$677$	17633.7i	1.00106i	−0.865720	+	0.500529i	$0.833139\pi$
$678$	0	0
$679$	3754.41	0.212196
$680$	0	0
$681$	23453.8	1.31975
$682$	0	0
$683$	19570.6i	1.09641i	0.836344	+	0.548206i	$0.184689\pi$
$683$	19570.6i	1.09641i	−0.836344	+	0.548206i	$0.815311\pi$
$684$	0	0
$685$	− 16564.8i	− 0.923952i
$686$	0	0
$687$	17306.6	0.961116
$688$	0	0
$689$	−3961.95	−0.219068
$690$	0	0
$691$	− 149.923i	− 0.00825375i	−0.999991	−	0.00412688i	$-0.998686\pi$
$691$	− 149.923i	− 0.00825375i	0.999991	−	0.00412688i	$-0.00131363\pi$
$692$	0	0
$693$	26287.7i	1.44096i
$694$	0	0
$695$	10129.8	0.552870
$696$	0	0
$697$	31900.4	1.73359
$698$	0	0
$699$	9287.38i	0.502548i
$700$	0	0
$701$	− 11086.5i	− 0.597335i	−0.954357	−	0.298667i	$-0.903458\pi$
$701$	− 11086.5i	− 0.597335i	0.954357	−	0.298667i	$-0.0965421\pi$
$702$	0	0
$703$	−12350.0	−0.662574
$704$	0	0
$705$	−11312.7	−0.604341
$706$	0	0
$707$	17181.5i	0.913968i
$708$	0	0
$709$	− 3858.40i	− 0.204380i	−0.994765	−	0.102190i	$-0.967415\pi$
$709$	− 3858.40i	− 0.204380i	0.994765	−	0.102190i	$-0.0325850\pi$
$710$	0	0
$711$	−27499.9	−1.45053
$712$	0	0
$713$	4665.84	0.245073
$714$	0	0
$715$	3322.00i	0.173756i
$716$	0	0
$717$	− 32964.6i	− 1.71699i
$718$	0	0
$719$	32717.8	1.69704	0.848519	−	0.529166i	$-0.177495\pi$
$719$	32717.8	1.69704	0.848519	+	0.529166i	$0.177495\pi$
$720$	0	0
$721$	45824.0	2.36696
$722$	0	0
$723$	− 43303.6i	− 2.22750i
$724$	0	0
$725$	− 1203.65i	− 0.0616587i
$726$	0	0
$727$	−25847.2	−1.31859	−0.659297	−	0.751883i	$-0.729146\pi$
$727$	−25847.2	−1.31859	−0.659297	+	0.751883i	$0.729146\pi$
$728$	0	0
$729$	27662.0	1.40537
$730$	0	0
$731$	− 12544.8i	− 0.634727i
$732$	0	0
$733$	− 14328.7i	− 0.722023i	−0.932561	−	0.361011i	$-0.882432\pi$
$733$	− 14328.7i	− 0.722023i	0.932561	−	0.361011i	$-0.117568\pi$
$734$	0	0
$735$	14192.1	0.712223
$736$	0	0
$737$	−4798.27	−0.239819
$738$	0	0
$739$	− 27101.5i	− 1.34905i	−0.738253	−	0.674524i	$-0.764349\pi$
$739$	− 27101.5i	− 1.34905i	0.738253	−	0.674524i	$-0.235651\pi$
$740$	0	0
$741$	− 6626.61i	− 0.328522i
$742$	0	0
$743$	−23322.1	−1.15155	−0.575777	−	0.817607i	$-0.695301\pi$
$743$	−23322.1	−1.15155	−0.575777	+	0.817607i	$0.695301\pi$
$744$	0	0
$745$	21309.8	1.04796
$746$	0	0
$747$	− 15447.2i	− 0.756603i
$748$	0	0
$749$	2795.94i	0.136397i
$750$	0	0
$751$	−25994.0	−1.26303	−0.631515	−	0.775364i	$-0.717566\pi$
$751$	−25994.0	−1.26303	−0.631515	+	0.775364i	$0.717566\pi$
$752$	0	0
$753$	5306.32	0.256804
$754$	0	0
$755$	18562.3i	0.894770i
$756$	0	0
$757$	31317.8i	1.50365i	0.659363	+	0.751825i	$0.270826\pi$
$757$	31317.8i	1.50365i	−0.659363	+	0.751825i	$0.729174\pi$
$758$	0	0
$759$	−11819.0	−0.565222
$760$	0	0
$761$	16497.5	0.785853	0.392926	−	0.919570i	$-0.371463\pi$
$761$	16497.5	0.785853	0.392926	+	0.919570i	$0.371463\pi$
$762$	0	0
$763$	− 33553.4i	− 1.59203i
$764$	0	0
$765$	− 19501.6i	− 0.921675i
$766$	0	0
$767$	−9130.02	−0.429812
$768$	0	0
$769$	−24867.3	−1.16611	−0.583055	−	0.812433i	$-0.698143\pi$
$769$	−24867.3	−1.16611	−0.583055	+	0.812433i	$0.698143\pi$
$770$	0	0
$771$	− 62905.9i	− 2.93839i
$772$	0	0
$773$	− 2661.15i	− 0.123823i	−0.998082	−	0.0619114i	$-0.980280\pi$
$773$	− 2661.15i	− 0.123823i	0.998082	−	0.0619114i	$-0.0197196\pi$
$774$	0	0
$775$	−7902.00	−0.366256
$776$	0	0
$777$	−44684.8	−2.06314
$778$	0	0
$779$	19281.9i	0.886837i
$780$	0	0
$781$	6795.14i	0.311331i
$782$	0	0
$783$	736.525	0.0336159
$784$	0	0
$785$	−8453.54	−0.384356
$786$	0	0
$787$	− 11018.1i	− 0.499049i	−0.968369	−	0.249524i	$-0.919726\pi$
$787$	− 11018.1i	− 0.499049i	0.968369	−	0.249524i	$-0.0802742\pi$
$788$	0	0
$789$	3189.17i	0.143900i
$790$	0	0
$791$	−17937.2	−0.806286
$792$	0	0
$793$	−14981.4	−0.670877
$794$	0	0
$795$	− 12742.0i	− 0.568445i
$796$	0	0
$797$	5953.61i	0.264602i	0.991210	+	0.132301i	$0.0422365\pi$
$797$	5953.61i	0.264602i	−0.991210	+	0.132301i	$0.957763\pi$
$798$	0	0
$799$	−19525.8	−0.864549
$800$	0	0
$801$	8779.50	0.387276
$802$	0	0
$803$	21126.0i	0.928417i
$804$	0	0
$805$	7901.94i	0.345971i
$806$	0	0
$807$	−52.5393	−0.00229178
$808$	0	0
$809$	−27554.3	−1.19747	−0.598737	−	0.800946i	$-0.704330\pi$
$809$	−27554.3	−1.19747	−0.598737	+	0.800946i	$0.704330\pi$
$810$	0	0
$811$	4817.57i	0.208592i	0.994546	+	0.104296i	$0.0332589\pi$
$811$	4817.57i	0.208592i	−0.994546	+	0.104296i	$0.966741\pi$
$812$	0	0
$813$	− 22027.9i	− 0.950247i
$814$	0	0
$815$	12987.1	0.558184
$816$	0	0
$817$	7582.59	0.324701
$818$	0	0
$819$	− 13261.3i	− 0.565797i
$820$	0	0
$821$	− 13914.6i	− 0.591504i	−0.955265	−	0.295752i	$-0.904430\pi$
$821$	− 13914.6i	− 0.591504i	0.955265	−	0.295752i	$-0.0955701\pi$
$822$	0	0
$823$	36653.5	1.55244	0.776221	−	0.630461i	$-0.217134\pi$
$823$	36653.5	1.55244	0.776221	+	0.630461i	$0.217134\pi$
$824$	0	0
$825$	20016.5	0.844711
$826$	0	0
$827$	31428.3i	1.32149i	0.750612	+	0.660743i	$0.229759\pi$
$827$	31428.3i	1.32149i	−0.750612	+	0.660743i	$0.770241\pi$
$828$	0	0
$829$	− 20810.9i	− 0.871885i	−0.899975	−	0.435942i	$-0.856415\pi$
$829$	− 20810.9i	− 0.871885i	0.899975	−	0.435942i	$-0.143585\pi$
$830$	0	0
$831$	16772.2	0.700145
$832$	0	0
$833$	24495.8	1.01888
$834$	0	0
$835$	− 8057.43i	− 0.333939i
$836$	0	0
$837$	− 4835.30i	− 0.199680i
$838$	0	0
$839$	−11010.0	−0.453050	−0.226525	−	0.974005i	$-0.572737\pi$
$839$	−11010.0	−0.453050	−0.226525	+	0.974005i	$0.572737\pi$
$840$	0	0
$841$	24170.9	0.991057
$842$	0	0
$843$	− 36922.5i	− 1.50852i
$844$	0	0
$845$	12814.5i	0.521694i
$846$	0	0
$847$	−8279.85	−0.335890
$848$	0	0
$849$	−7080.22	−0.286210
$850$	0	0
$851$	− 11112.0i	− 0.447608i
$852$	0	0
$853$	18116.0i	0.727175i	0.931560	+	0.363587i	$0.118448\pi$
$853$	18116.0i	0.727175i	−0.931560	+	0.363587i	$0.881552\pi$
$854$	0	0
$855$	11787.6	0.471493
$856$	0	0
$857$	−38510.8	−1.53501	−0.767505	−	0.641043i	$-0.778502\pi$
$857$	−38510.8	−1.53501	−0.767505	+	0.641043i	$0.778502\pi$
$858$	0	0
$859$	− 32858.7i	− 1.30515i	−0.757723	−	0.652576i	$-0.773688\pi$
$859$	− 32858.7i	− 1.30515i	0.757723	−	0.652576i	$-0.226312\pi$
$860$	0	0
$861$	69765.8i	2.76146i
$862$	0	0
$863$	−22079.5	−0.870911	−0.435456	−	0.900210i	$-0.643413\pi$
$863$	−22079.5	−0.870911	−0.435456	+	0.900210i	$0.643413\pi$
$864$	0	0
$865$	5255.26	0.206571
$866$	0	0
$867$	− 22669.5i	− 0.888001i
$868$	0	0
$869$	26003.7i	1.01509i
$870$	0	0
$871$	2420.58	0.0941656
$872$	0	0
$873$	5039.15	0.195360
$874$	0	0
$875$	− 33908.2i	− 1.31007i
$876$	0	0
$877$	5773.00i	0.222281i	0.993805	+	0.111140i	$0.0354504\pi$
$877$	5773.00i	0.222281i	−0.993805	+	0.111140i	$0.964550\pi$
$878$	0	0
$879$	15753.9	0.604510
$880$	0	0
$881$	−7132.59	−0.272762	−0.136381	−	0.990656i	$-0.543547\pi$
$881$	−7132.59	−0.272762	−0.136381	+	0.990656i	$0.543547\pi$
$882$	0	0
$883$	27110.4i	1.03323i	0.856219	+	0.516613i	$0.172807\pi$
$883$	27110.4i	1.03323i	−0.856219	+	0.516613i	$0.827193\pi$
$884$	0	0
$885$	− 29363.1i	− 1.11529i
$886$	0	0
$887$	−45045.7	−1.70517	−0.852585	−	0.522589i	$-0.824966\pi$
$887$	−45045.7	−1.70517	−0.852585	+	0.522589i	$0.824966\pi$
$888$	0	0
$889$	−65287.3	−2.46306
$890$	0	0
$891$	− 16260.4i	− 0.611384i
$892$	0	0
$893$	− 11802.2i	− 0.442269i
$894$	0	0
$895$	20541.1	0.767167
$896$	0	0
$897$	5962.34	0.221936
$898$	0	0
$899$	1431.96i	0.0531239i
$900$	0	0
$901$	− 21992.9i	− 0.813197i
$902$	0	0
$903$	27435.3	1.01106
$904$	0	0
$905$	−1131.36	−0.0415554
$906$	0	0
$907$	− 47599.9i	− 1.74259i	−0.490760	−	0.871295i	$-0.663281\pi$
$907$	− 47599.9i	− 1.74259i	0.490760	−	0.871295i	$-0.336719\pi$
$908$	0	0
$909$	23060.9i	0.841453i
$910$	0	0
$911$	42503.8	1.54579	0.772895	−	0.634534i	$-0.218808\pi$
$911$	42503.8	1.54579	0.772895	+	0.634534i	$0.218808\pi$
$912$	0	0
$913$	−14606.7	−0.529477
$914$	0	0
$915$	− 48181.8i	− 1.74081i
$916$	0	0
$917$	23152.5i	0.833766i
$918$	0	0
$919$	−8819.41	−0.316568	−0.158284	−	0.987394i	$-0.550596\pi$
$919$	−8819.41	−0.316568	−0.158284	+	0.987394i	$0.550596\pi$
$920$	0	0
$921$	−2541.55	−0.0909305
$922$	0	0
$923$	− 3427.94i	− 0.122245i
$924$	0	0
$925$	18819.1i	0.668940i
$926$	0	0
$927$	61504.8	2.17916
$928$	0	0
$929$	−14155.6	−0.499925	−0.249963	−	0.968256i	$-0.580418\pi$
$929$	−14155.6	−0.499925	−0.249963	+	0.968256i	$0.580418\pi$
$930$	0	0
$931$	14806.3i	0.521220i
$932$	0	0
$933$	6773.27i	0.237671i
$934$	0	0
$935$	−18440.6	−0.644996
$936$	0	0
$937$	−38518.7	−1.34296	−0.671479	−	0.741023i	$-0.734341\pi$
$937$	−38518.7	−1.34296	−0.671479	+	0.741023i	$0.734341\pi$
$938$	0	0
$939$	27321.4i	0.949523i
$940$	0	0
$941$	39595.4i	1.37170i	0.727741	+	0.685852i	$0.240570\pi$
$941$	39595.4i	1.37170i	−0.727741	+	0.685852i	$0.759430\pi$
$942$	0	0
$943$	−17349.0	−0.599112
$944$	0	0
$945$	8188.93	0.281890
$946$	0	0
$947$	− 46442.1i	− 1.59363i	−0.604225	−	0.796814i	$-0.706517\pi$
$947$	− 46442.1i	− 1.59363i	0.604225	−	0.796814i	$-0.293483\pi$
$948$	0	0
$949$	− 10657.4i	− 0.364545i
$950$	0	0
$951$	−51925.4	−1.77055
$952$	0	0
$953$	20600.1	0.700211	0.350106	−	0.936710i	$-0.386146\pi$
$953$	20600.1	0.700211	0.350106	+	0.936710i	$0.386146\pi$
$954$	0	0
$955$	25904.8i	0.877760i
$956$	0	0
$957$	− 3627.28i	− 0.122522i
$958$	0	0
$959$	62528.2	2.10547
$960$	0	0
$961$	−20390.2	−0.684441
$962$	0	0
$963$	3752.70i	0.125575i
$964$	0	0
$965$	− 21428.9i	− 0.714841i
$966$	0	0
$967$	−38210.9	−1.27071	−0.635356	−	0.772219i	$-0.719147\pi$
$967$	−38210.9	−1.27071	−0.635356	+	0.772219i	$0.719147\pi$
$968$	0	0
$969$	36784.6	1.21950
$970$	0	0
$971$	52643.9i	1.73988i	0.493157	+	0.869941i	$0.335843\pi$
$971$	52643.9i	1.73988i	−0.493157	+	0.869941i	$0.664157\pi$
$972$	0	0
$973$	38237.6i	1.25986i
$974$	0	0
$975$	−10097.7	−0.331678
$976$	0	0
$977$	−7985.95	−0.261508	−0.130754	−	0.991415i	$-0.541740\pi$
$977$	−7985.95	−0.261508	−0.130754	+	0.991415i	$0.541740\pi$
$978$	0	0
$979$	− 8301.83i	− 0.271019i
$980$	0	0
$981$	− 45035.3i	− 1.46571i
$982$	0	0
$983$	−10703.1	−0.347279	−0.173639	−	0.984809i	$-0.555553\pi$
$983$	−10703.1	−0.347279	−0.173639	+	0.984809i	$0.555553\pi$
$984$	0	0
$985$	22580.9	0.730442
$986$	0	0
$987$	− 42702.8i	− 1.37715i
$988$	0	0
$989$	6822.49i	0.219355i
$990$	0	0
$991$	−23945.4	−0.767558	−0.383779	−	0.923425i	$-0.625377\pi$
$991$	−23945.4	−0.767558	−0.383779	+	0.923425i	$0.625377\pi$
$992$	0	0
$993$	−20009.0	−0.639442
$994$	0	0
$995$	9044.42i	0.288168i
$996$	0	0
$997$	20212.7i	0.642068i	0.947068	+	0.321034i	$0.104030\pi$
$997$	20212.7i	0.642068i	−0.947068	+	0.321034i	$0.895970\pi$
$998$	0	0
$999$	−11515.6	−0.364702

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.4.b.j.513.2		10
4.3	odd	2		1024.4.b.k.513.9		10
8.3	odd	2		1024.4.b.k.513.2		10
8.5	even	2	inner	1024.4.b.j.513.9		10
16.3	odd	4		1024.4.a.m.1.2		10
16.5	even	4		1024.4.a.n.1.2		10
16.11	odd	4		1024.4.a.m.1.9		10
16.13	even	4		1024.4.a.n.1.9		10
32.3	odd	8		64.4.e.a.49.1		10
32.5	even	8		16.4.e.a.13.1	yes	10
32.11	odd	8		128.4.e.a.33.5		10
32.13	even	8		128.4.e.b.97.1		10
32.19	odd	8		128.4.e.a.97.5		10
32.21	even	8		128.4.e.b.33.1		10
32.27	odd	8		64.4.e.a.17.1		10
32.29	even	8		16.4.e.a.5.1	✓	10
96.5	odd	8		144.4.k.a.109.5		10
96.29	odd	8		144.4.k.a.37.5		10
96.35	even	8		576.4.k.a.433.4		10
96.59	even	8		576.4.k.a.145.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.1	✓	10	32.29	even	8
16.4.e.a.13.1	yes	10	32.5	even	8
64.4.e.a.17.1		10	32.27	odd	8
64.4.e.a.49.1		10	32.3	odd	8
128.4.e.a.33.5		10	32.11	odd	8
128.4.e.a.97.5		10	32.19	odd	8
128.4.e.b.33.1		10	32.21	even	8
128.4.e.b.97.1		10	32.13	even	8
144.4.k.a.37.5		10	96.29	odd	8
144.4.k.a.109.5		10	96.5	odd	8
576.4.k.a.145.4		10	96.59	even	8
576.4.k.a.433.4		10	96.35	even	8
1024.4.a.m.1.2		10	16.3	odd	4
1024.4.a.m.1.9		10	16.11	odd	4
1024.4.a.n.1.2		10	16.5	even	4
1024.4.a.n.1.9		10	16.13	even	4
1024.4.b.j.513.2		10	1.1	even	1	trivial
1024.4.b.j.513.9		10	8.5	even	2	inner
1024.4.b.k.513.2		10	8.3	odd	2
1024.4.b.k.513.9		10	4.3	odd	2

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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1024.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.77277i q^{3} +6.59550i q^{5} -24.8965 q^{7} -33.4160 q^{9} +O(q^{10})\)	\(q-7.77277i q^{3} +6.59550i q^{5} -24.8965 q^{7} -33.4160 q^{9} +31.5979i q^{11} -15.9402i q^{13} +51.2653 q^{15} +88.4846 q^{17} +53.4838i q^{19} +193.515i q^{21} -48.1224 q^{23} +81.4994 q^{25} +49.8702i q^{27} -14.7689i q^{29} -96.9578 q^{31} +245.604 q^{33} -164.205i q^{35} +230.911i q^{37} -123.899 q^{39} +360.519 q^{41} -141.774i q^{43} -220.395i q^{45} -220.669 q^{47} +276.837 q^{49} -687.771i q^{51} -248.551i q^{53} -208.404 q^{55} +415.717 q^{57} -572.767i q^{59} -939.852i q^{61} +831.942 q^{63} +105.133 q^{65} +151.854i q^{67} +374.045i q^{69} +215.050 q^{71} +668.587 q^{73} -633.476i q^{75} -786.679i q^{77} +822.956 q^{79} -514.603 q^{81} +462.269i q^{83} +583.600i q^{85} -114.795 q^{87} -262.733 q^{89} +396.855i q^{91} +753.631i q^{93} -352.752 q^{95} -150.801 q^{97} -1055.88i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 28 q^{7} - 54 q^{9}+O(q^{10}) \) 10 * q - 28 * q^7 - 54 * q^9	\( 10 q - 28 q^{7} - 54 q^{9} + 124 q^{15} + 4 q^{17} - 276 q^{23} - 50 q^{25} + 368 q^{31} - 4 q^{33} - 732 q^{39} + 944 q^{47} - 94 q^{49} - 1380 q^{55} - 108 q^{57} + 2628 q^{63} - 492 q^{65} - 3468 q^{71} + 296 q^{73} + 4416 q^{79} - 482 q^{81} - 6036 q^{87} - 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) \) 10 * q - 28 * q^7 - 54 * q^9 + 124 * q^15 + 4 * q^17 - 276 * q^23 - 50 * q^25 + 368 * q^31 - 4 * q^33 - 732 * q^39 + 944 * q^47 - 94 * q^49 - 1380 * q^55 - 108 * q^57 + 2628 * q^63 - 492 * q^65 - 3468 * q^71 + 296 * q^73 + 4416 * q^79 - 482 * q^81 - 6036 * q^87 - 88 * q^89 + 6900 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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