LMFDB - Embedded newform 1024.4.b.k.513.7

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			513.7
Root			$3.82089i$ of defining polynomial
Character	$\chi$	$=$	1024.513
Dual form			1024.4.b.k.513.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.80518i q^{3} +0.844070i q^{5} -29.0828 q^{7} +19.1310 q^{9} +O(q^{10})$	$q+2.80518i q^{3} +0.844070i q^{5} -29.0828 q^{7} +19.1310 q^{9} -17.1532i q^{11} -68.6824i q^{13} -2.36777 q^{15} -86.7193 q^{17} +77.5614i q^{19} -81.5826i q^{21} +70.2145 q^{23} +124.288 q^{25} +129.406i q^{27} -89.6641i q^{29} +8.86868 q^{31} +48.1178 q^{33} -24.5480i q^{35} +30.7089i q^{37} +192.667 q^{39} +153.274 q^{41} +171.050i q^{43} +16.1479i q^{45} -99.9792 q^{47} +502.812 q^{49} -243.263i q^{51} +550.315i q^{53} +14.4785 q^{55} -217.574 q^{57} -459.364i q^{59} +0.479693i q^{61} -556.383 q^{63} +57.9728 q^{65} +799.438i q^{67} +196.964i q^{69} +419.500 q^{71} -374.833 q^{73} +348.649i q^{75} +498.864i q^{77} -705.750 q^{79} +153.530 q^{81} +1339.39i q^{83} -73.1972i q^{85} +251.524 q^{87} -4.72918 q^{89} +1997.48i q^{91} +24.8782i q^{93} -65.4673 q^{95} +379.542 q^{97} -328.157i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.80518i	0.539857i	0.962880	+	0.269929i	$0.0870001\pi$
$3$	2.80518i	0.539857i	−0.962880	+	0.269929i	$0.913000\pi$
$4$	0	0
$5$	0.844070i	0.0754959i	0.999287	+	0.0377480i	$0.0120184\pi$
$5$	0.844070i	0.0754959i	−0.999287	+	0.0377480i	$0.987982\pi$
$6$	0	0
$7$	−29.0828	−1.57033	−0.785163	−	0.619289i	$-0.787421\pi$
$7$	−29.0828	−1.57033	−0.785163	+	0.619289i	$0.787421\pi$
$8$	0	0
$9$	19.1310	0.708554
$10$	0	0
$11$	− 17.1532i	− 0.470171i	−0.971975	−	0.235086i	$-0.924463\pi$
$11$	− 17.1532i	− 0.470171i	0.971975	−	0.235086i	$-0.0755370\pi$
$12$	0	0
$13$	− 68.6824i	− 1.46531i	−0.680598	−	0.732657i	$-0.738280\pi$
$13$	− 68.6824i	− 1.46531i	0.680598	−	0.732657i	$-0.261720\pi$
$14$	0	0
$15$	−2.36777	−0.0407570
$16$	0	0
$17$	−86.7193	−1.23721	−0.618604	−	0.785703i	$-0.712301\pi$
$17$	−86.7193	−1.23721	−0.618604	+	0.785703i	$0.712301\pi$
$18$	0	0
$19$	77.5614i	0.936517i	0.883592	+	0.468258i	$0.155118\pi$
$19$	77.5614i	0.936517i	−0.883592	+	0.468258i	$0.844882\pi$
$20$	0	0
$21$	− 81.5826i	− 0.847752i
$22$	0	0
$23$	70.2145	0.636554	0.318277	−	0.947998i	$-0.396896\pi$
$23$	70.2145	0.636554	0.318277	+	0.947998i	$0.396896\pi$
$24$	0	0
$25$	124.288	0.994300
$26$	0	0
$27$	129.406i	0.922375i
$28$	0	0
$29$	− 89.6641i	− 0.574145i	−0.957909	−	0.287072i	$-0.907318\pi$
$29$	− 89.6641i	− 0.574145i	0.957909	−	0.287072i	$-0.0926820\pi$
$30$	0	0
$31$	8.86868	0.0513826	0.0256913	−	0.999670i	$-0.491821\pi$
$31$	8.86868	0.0513826	0.0256913	+	0.999670i	$0.491821\pi$
$32$	0	0
$33$	48.1178	0.253825
$34$	0	0
$35$	− 24.5480i	− 0.118553i
$36$	0	0
$37$	30.7089i	0.136446i	0.997670	+	0.0682231i	$0.0217330\pi$
$37$	30.7089i	0.136446i	−0.997670	+	0.0682231i	$0.978267\pi$
$38$	0	0
$39$	192.667	0.791060
$40$	0	0
$41$	153.274	0.583840	0.291920	−	0.956443i	$-0.405706\pi$
$41$	153.274	0.583840	0.291920	+	0.956443i	$0.405706\pi$
$42$	0	0
$43$	171.050i	0.606625i	0.952891	+	0.303312i	$0.0980926\pi$
$43$	171.050i	0.606625i	−0.952891	+	0.303312i	$0.901907\pi$
$44$	0	0
$45$	16.1479i	0.0534930i
$46$	0	0
$47$	−99.9792	−0.310286	−0.155143	−	0.987892i	$-0.549584\pi$
$47$	−99.9792	−0.310286	−0.155143	+	0.987892i	$0.549584\pi$
$48$	0	0
$49$	502.812	1.46592
$50$	0	0
$51$	− 243.263i	− 0.667915i
$52$	0	0
$53$	550.315i	1.42626i	0.701034	+	0.713128i	$0.252722\pi$
$53$	550.315i	1.42626i	−0.701034	+	0.713128i	$0.747278\pi$
$54$	0	0
$55$	14.4785	0.0354960
$56$	0	0
$57$	−217.574	−0.505585
$58$	0	0
$59$	− 459.364i	− 1.01363i	−0.862055	−	0.506814i	$-0.830823\pi$
$59$	− 459.364i	− 1.01363i	0.862055	−	0.506814i	$-0.169177\pi$
$60$	0	0
$61$	0.479693i	0.00100686i	1.00000		0.000503430i	$0.000160247\pi$
$61$	0.479693i	0.00100686i	−1.00000		0.000503430i	$0.999840\pi$
$62$	0	0
$63$	−556.383	−1.11266
$64$	0	0
$65$	57.9728	0.110625
$66$	0	0
$67$	799.438i	1.45771i	0.684666	+	0.728857i	$0.259948\pi$
$67$	799.438i	1.45771i	−0.684666	+	0.728857i	$0.740052\pi$
$68$	0	0
$69$	196.964i	0.343648i
$70$	0	0
$71$	419.500	0.701205	0.350602	−	0.936524i	$-0.385977\pi$
$71$	419.500	0.701205	0.350602	+	0.936524i	$0.385977\pi$
$72$	0	0
$73$	−374.833	−0.600971	−0.300485	−	0.953786i	$-0.597149\pi$
$73$	−374.833	−0.600971	−0.300485	+	0.953786i	$0.597149\pi$
$74$	0	0
$75$	348.649i	0.536780i
$76$	0	0
$77$	498.864i	0.738322i
$78$	0	0
$79$	−705.750	−1.00510	−0.502551	−	0.864547i	$-0.667605\pi$
$79$	−705.750	−1.00510	−0.502551	+	0.864547i	$0.667605\pi$
$80$	0	0
$81$	153.530	0.210604
$82$	0	0
$83$	1339.39i	1.77129i	0.464361	+	0.885646i	$0.346284\pi$
$83$	1339.39i	1.77129i	−0.464361	+	0.885646i	$0.653716\pi$
$84$	0	0
$85$	− 73.1972i	− 0.0934041i
$86$	0	0
$87$	251.524	0.309956
$88$	0	0
$89$	−4.72918	−0.00563249	−0.00281625	−	0.999996i	$-0.500896\pi$
$89$	−4.72918	−0.00563249	−0.00281625	+	0.999996i	$0.500896\pi$
$90$	0	0
$91$	1997.48i	2.30102i
$92$	0	0
$93$	24.8782i	0.0277393i
$94$	0	0
$95$	−65.4673	−0.0707032
$96$	0	0
$97$	379.542	0.397285	0.198643	−	0.980072i	$-0.436347\pi$
$97$	379.542	0.397285	0.198643	+	0.980072i	$0.436347\pi$
$98$	0	0
$99$	− 328.157i	− 0.333142i
$100$	0	0
$101$	552.964i	0.544772i	0.962188	+	0.272386i	$0.0878127\pi$
$101$	552.964i	0.544772i	−0.962188	+	0.272386i	$0.912187\pi$
$102$	0	0
$103$	307.935	0.294580	0.147290	−	0.989093i	$-0.452945\pi$
$103$	307.935	0.294580	0.147290	+	0.989093i	$0.452945\pi$
$104$	0	0
$105$	68.8615	0.0640018
$106$	0	0
$107$	850.801i	0.768692i	0.923189	+	0.384346i	$0.125573\pi$
$107$	850.801i	0.768692i	−0.923189	+	0.384346i	$0.874427\pi$
$108$	0	0
$109$	− 1341.93i	− 1.17921i	−0.807692	−	0.589605i	$-0.799283\pi$
$109$	− 1341.93i	− 1.17921i	0.807692	−	0.589605i	$-0.200717\pi$
$110$	0	0
$111$	−86.1439	−0.0736614
$112$	0	0
$113$	−1824.02	−1.51849	−0.759244	−	0.650807i	$-0.774431\pi$
$113$	−1824.02	−1.51849	−0.759244	+	0.650807i	$0.774431\pi$
$114$	0	0
$115$	59.2660i	0.0480573i
$116$	0	0
$117$	− 1313.96i	− 1.03825i
$118$	0	0
$119$	2522.04	1.94282
$120$	0	0
$121$	1036.77	0.778939
$122$	0	0
$123$	429.962i	0.315190i
$124$	0	0
$125$	210.416i	0.150562i
$126$	0	0
$127$	−988.748	−0.690844	−0.345422	−	0.938447i	$-0.612264\pi$
$127$	−988.748	−0.690844	−0.345422	+	0.938447i	$0.612264\pi$
$128$	0	0
$129$	−479.826	−0.327491
$130$	0	0
$131$	1122.28i	0.748505i	0.927327	+	0.374252i	$0.122101\pi$
$131$	1122.28i	0.748505i	−0.927327	+	0.374252i	$0.877899\pi$
$132$	0	0
$133$	− 2255.71i	− 1.47064i
$134$	0	0
$135$	−109.227	−0.0696356
$136$	0	0
$137$	1595.30	0.994856	0.497428	−	0.867505i	$-0.334278\pi$
$137$	1595.30	0.994856	0.497428	+	0.867505i	$0.334278\pi$
$138$	0	0
$139$	392.721i	0.239641i	0.992796	+	0.119821i	$0.0382320\pi$
$139$	392.721i	0.239641i	−0.992796	+	0.119821i	$0.961768\pi$
$140$	0	0
$141$	− 280.460i	− 0.167510i
$142$	0	0
$143$	−1178.12	−0.688948
$144$	0	0
$145$	75.6828	0.0433456
$146$	0	0
$147$	1410.48i	0.791390i
$148$	0	0
$149$	− 839.014i	− 0.461307i	−0.973036	−	0.230653i	$-0.925914\pi$
$149$	− 839.014i	− 0.461307i	0.973036	−	0.230653i	$-0.0740863\pi$
$150$	0	0
$151$	−160.655	−0.0865821	−0.0432911	−	0.999063i	$-0.513784\pi$
$151$	−160.655	−0.0865821	−0.0432911	+	0.999063i	$0.513784\pi$
$152$	0	0
$153$	−1659.02	−0.876629
$154$	0	0
$155$	7.48579i	0.00387918i
$156$	0	0
$157$	998.098i	0.507369i	0.967287	+	0.253684i	$0.0816425\pi$
$157$	998.098i	0.507369i	−0.967287	+	0.253684i	$0.918358\pi$
$158$	0	0
$159$	−1543.73	−0.769975
$160$	0	0
$161$	−2042.04	−0.999598
$162$	0	0
$163$	2648.31i	1.27259i	0.771447	+	0.636294i	$0.219533\pi$
$163$	2648.31i	1.27259i	−0.771447	+	0.636294i	$0.780467\pi$
$164$	0	0
$165$	40.6148i	0.0191628i
$166$	0	0
$167$	3852.19	1.78498	0.892490	−	0.451066i	$-0.148956\pi$
$167$	3852.19	1.78498	0.892490	+	0.451066i	$0.148956\pi$
$168$	0	0
$169$	−2520.28	−1.14715
$170$	0	0
$171$	1483.83i	0.663573i
$172$	0	0
$173$	3713.17i	1.63183i	0.578170	+	0.815917i	$0.303767\pi$
$173$	3713.17i	1.63183i	−0.578170	+	0.815917i	$0.696233\pi$
$174$	0	0
$175$	−3614.64	−1.56138
$176$	0	0
$177$	1288.60	0.547215
$178$	0	0
$179$	1749.01i	0.730318i	0.930945	+	0.365159i	$0.118985\pi$
$179$	1749.01i	0.730318i	−0.930945	+	0.365159i	$0.881015\pi$
$180$	0	0
$181$	2227.25i	0.914640i	0.889302	+	0.457320i	$0.151191\pi$
$181$	2227.25i	0.914640i	−0.889302	+	0.457320i	$0.848809\pi$
$182$	0	0
$183$	−1.34563	−0.000543560 0
$184$	0	0
$185$	−25.9204	−0.0103011
$186$	0	0
$187$	1487.51i	0.581699i
$188$	0	0
$189$	− 3763.48i	− 1.44843i
$190$	0	0
$191$	−3585.92	−1.35847	−0.679236	−	0.733920i	$-0.737689\pi$
$191$	−3585.92	−1.35847	−0.679236	+	0.733920i	$0.737689\pi$
$192$	0	0
$193$	523.601	0.195283	0.0976415	−	0.995222i	$-0.468870\pi$
$193$	523.601	0.195283	0.0976415	+	0.995222i	$0.468870\pi$
$194$	0	0
$195$	162.624i	0.0597218i
$196$	0	0
$197$	1591.89i	0.575724i	0.957672	+	0.287862i	$0.0929444\pi$
$197$	1591.89i	0.575724i	−0.957672	+	0.287862i	$0.907056\pi$
$198$	0	0
$199$	2312.48	0.823757	0.411878	−	0.911239i	$-0.364873\pi$
$199$	2312.48	0.823757	0.411878	+	0.911239i	$0.364873\pi$
$200$	0	0
$201$	−2242.57	−0.786957
$202$	0	0
$203$	2607.69i	0.901595i
$204$	0	0
$205$	129.374i	0.0440775i
$206$	0	0
$207$	1343.27	0.451033
$208$	0	0
$209$	1330.43	0.440323
$210$	0	0
$211$	− 2006.19i	− 0.654558i	−0.944928	−	0.327279i	$-0.893868\pi$
$211$	− 2006.19i	− 0.654558i	0.944928	−	0.327279i	$-0.106132\pi$
$212$	0	0
$213$	1176.77i	0.378550i
$214$	0	0
$215$	−144.378	−0.0457977
$216$	0	0
$217$	−257.926	−0.0806875
$218$	0	0
$219$	− 1051.47i	− 0.324438i
$220$	0	0
$221$	5956.10i	1.81290i
$222$	0	0
$223$	4315.08	1.29578	0.647890	−	0.761734i	$-0.275651\pi$
$223$	4315.08	1.29578	0.647890	+	0.761734i	$0.275651\pi$
$224$	0	0
$225$	2377.74	0.704516
$226$	0	0
$227$	991.651i	0.289948i	0.989435	+	0.144974i	$0.0463099\pi$
$227$	991.651i	0.289948i	−0.989435	+	0.144974i	$0.953690\pi$
$228$	0	0
$229$	938.121i	0.270711i	0.990797	+	0.135355i	$0.0432176\pi$
$229$	938.121i	0.270711i	−0.990797	+	0.135355i	$0.956782\pi$
$230$	0	0
$231$	−1399.40	−0.398588
$232$	0	0
$233$	3490.15	0.981318	0.490659	−	0.871352i	$-0.336756\pi$
$233$	3490.15	0.981318	0.490659	+	0.871352i	$0.336756\pi$
$234$	0	0
$235$	− 84.3895i	− 0.0234254i
$236$	0	0
$237$	− 1979.76i	− 0.542612i
$238$	0	0
$239$	2950.43	0.798525	0.399263	−	0.916837i	$-0.369266\pi$
$239$	2950.43	0.798525	0.399263	+	0.916837i	$0.369266\pi$
$240$	0	0
$241$	1128.96	0.301755	0.150877	−	0.988552i	$-0.451790\pi$
$241$	1128.96	0.301755	0.150877	+	0.988552i	$0.451790\pi$
$242$	0	0
$243$	3924.63i	1.03607i
$244$	0	0
$245$	424.409i	0.110671i
$246$	0	0
$247$	5327.11	1.37229
$248$	0	0
$249$	−3757.23	−0.956244
$250$	0	0
$251$	6536.30i	1.64370i	0.569706	+	0.821848i	$0.307057\pi$
$251$	6536.30i	1.64370i	−0.569706	+	0.821848i	$0.692943\pi$
$252$	0	0
$253$	− 1204.40i	− 0.299289i
$254$	0	0
$255$	205.331	0.0504249
$256$	0	0
$257$	610.977	0.148295	0.0741473	−	0.997247i	$-0.476377\pi$
$257$	610.977	0.148295	0.0741473	+	0.997247i	$0.476377\pi$
$258$	0	0
$259$	− 893.101i	− 0.214265i
$260$	0	0
$261$	− 1715.36i	− 0.406813i
$262$	0	0
$263$	−4973.57	−1.16610	−0.583048	−	0.812438i	$-0.698140\pi$
$263$	−4973.57	−1.16610	−0.583048	+	0.812438i	$0.698140\pi$
$264$	0	0
$265$	−464.505	−0.107677
$266$	0	0
$267$	− 13.2662i	− 0.00304074i
$268$	0	0
$269$	− 1327.12i	− 0.300803i	−0.988625	−	0.150401i	$-0.951943\pi$
$269$	− 1327.12i	− 0.300803i	0.988625	−	0.150401i	$-0.0480566\pi$
$270$	0	0
$271$	−4010.64	−0.898999	−0.449500	−	0.893280i	$-0.648398\pi$
$271$	−4010.64	−0.898999	−0.449500	+	0.893280i	$0.648398\pi$
$272$	0	0
$273$	−5603.29	−1.24222
$274$	0	0
$275$	− 2131.93i	− 0.467491i
$276$	0	0
$277$	− 4999.23i	− 1.08439i	−0.840254	−	0.542193i	$-0.817594\pi$
$277$	− 4999.23i	− 1.08439i	0.840254	−	0.542193i	$-0.182406\pi$
$278$	0	0
$279$	169.666	0.0364074
$280$	0	0
$281$	7468.35	1.58550	0.792748	−	0.609550i	$-0.208650\pi$
$281$	7468.35	1.58550	0.792748	+	0.609550i	$0.208650\pi$
$282$	0	0
$283$	− 3180.88i	− 0.668140i	−0.942548	−	0.334070i	$-0.891578\pi$
$283$	− 3180.88i	− 0.668140i	0.942548	−	0.334070i	$-0.108422\pi$
$284$	0	0
$285$	− 183.648i	− 0.0381696i
$286$	0	0
$287$	−4457.66	−0.916819
$288$	0	0
$289$	2607.24	0.530682
$290$	0	0
$291$	1064.68i	0.214477i
$292$	0	0
$293$	5590.09i	1.11460i	0.830312	+	0.557298i	$0.188162\pi$
$293$	5590.09i	1.11460i	−0.830312	+	0.557298i	$0.811838\pi$
$294$	0	0
$295$	387.736	0.0765249
$296$	0	0
$297$	2219.72	0.433674
$298$	0	0
$299$	− 4822.51i	− 0.932752i
$300$	0	0
$301$	− 4974.62i	− 0.952599i
$302$	0	0
$303$	−1551.16	−0.294099
$304$	0	0
$305$	−0.404895	−7.60138e−5 0
$306$	0	0
$307$	5452.13i	1.01358i	0.862069	+	0.506791i	$0.169168\pi$
$307$	5452.13i	1.01358i	−0.862069	+	0.506791i	$0.830832\pi$
$308$	0	0
$309$	863.814i	0.159031i
$310$	0	0
$311$	−5194.39	−0.947096	−0.473548	−	0.880768i	$-0.657027\pi$
$311$	−5194.39	−0.947096	−0.473548	+	0.880768i	$0.657027\pi$
$312$	0	0
$313$	4710.01	0.850561	0.425281	−	0.905062i	$-0.360175\pi$
$313$	4710.01	0.850561	0.425281	+	0.905062i	$0.360175\pi$
$314$	0	0
$315$	− 469.626i	− 0.0840014i
$316$	0	0
$317$	− 8033.04i	− 1.42328i	−0.702544	−	0.711641i	$-0.747952\pi$
$317$	− 8033.04i	− 1.42328i	0.702544	−	0.711641i	$-0.252048\pi$
$318$	0	0
$319$	−1538.03	−0.269946
$320$	0	0
$321$	−2386.65	−0.414984
$322$	0	0
$323$	− 6726.08i	− 1.15867i
$324$	0	0
$325$	− 8536.37i	− 1.45696i
$326$	0	0
$327$	3764.36	0.636605
$328$	0	0
$329$	2907.68	0.487251
$330$	0	0
$331$	− 2567.92i	− 0.426423i	−0.977006	−	0.213211i	$-0.931608\pi$
$331$	− 2567.92i	− 0.426423i	0.977006	−	0.213211i	$-0.0683923\pi$
$332$	0	0
$333$	587.490i	0.0966795i
$334$	0	0
$335$	−674.782	−0.110052
$336$	0	0
$337$	2683.29	0.433733	0.216867	−	0.976201i	$-0.430416\pi$
$337$	2683.29	0.433733	0.216867	+	0.976201i	$0.430416\pi$
$338$	0	0
$339$	− 5116.69i	− 0.819766i
$340$	0	0
$341$	− 152.126i	− 0.0241586i
$342$	0	0
$343$	−4647.79	−0.731653
$344$	0	0
$345$	−166.252	−0.0259440
$346$	0	0
$347$	− 7483.74i	− 1.15778i	−0.815407	−	0.578888i	$-0.803487\pi$
$347$	− 7483.74i	− 1.15778i	0.815407	−	0.578888i	$-0.196513\pi$
$348$	0	0
$349$	− 104.239i	− 0.0159880i	−0.999968	−	0.00799400i	$-0.997455\pi$
$349$	− 104.239i	− 0.0159880i	0.999968	−	0.00799400i	$-0.00254460\pi$
$350$	0	0
$351$	8887.90	1.35157
$352$	0	0
$353$	−5067.25	−0.764030	−0.382015	−	0.924156i	$-0.624770\pi$
$353$	−5067.25	−0.764030	−0.382015	+	0.924156i	$0.624770\pi$
$354$	0	0
$355$	354.088i	0.0529381i
$356$	0	0
$357$	7074.79i	1.04884i
$358$	0	0
$359$	−970.230	−0.142637	−0.0713186	−	0.997454i	$-0.522721\pi$
$359$	−970.230	−0.142637	−0.0713186	+	0.997454i	$0.522721\pi$
$360$	0	0
$361$	843.224	0.122937
$362$	0	0
$363$	2908.32i	0.420516i
$364$	0	0
$365$	− 316.385i	− 0.0453709i
$366$	0	0
$367$	13451.4	1.91323	0.956617	−	0.291347i	$-0.0941035\pi$
$367$	13451.4	1.91323	0.956617	+	0.291347i	$0.0941035\pi$
$368$	0	0
$369$	2932.29	0.413682
$370$	0	0
$371$	− 16004.7i	− 2.23969i
$372$	0	0
$373$	8341.34i	1.15790i	0.815362	+	0.578952i	$0.196538\pi$
$373$	8341.34i	1.15790i	−0.815362	+	0.578952i	$0.803462\pi$
$374$	0	0
$375$	−590.255	−0.0812817
$376$	0	0
$377$	−6158.35	−0.841302
$378$	0	0
$379$	− 6288.62i	− 0.852308i	−0.904651	−	0.426154i	$-0.859868\pi$
$379$	− 6288.62i	− 0.852308i	0.904651	−	0.426154i	$-0.140132\pi$
$380$	0	0
$381$	− 2773.62i	− 0.372957i
$382$	0	0
$383$	6417.68	0.856209	0.428105	−	0.903729i	$-0.359181\pi$
$383$	6417.68	0.856209	0.428105	+	0.903729i	$0.359181\pi$
$384$	0	0
$385$	−421.076	−0.0557403
$386$	0	0
$387$	3272.35i	0.429827i
$388$	0	0
$389$	− 9271.04i	− 1.20838i	−0.796840	−	0.604191i	$-0.793497\pi$
$389$	− 9271.04i	− 1.20838i	0.796840	−	0.604191i	$-0.206503\pi$
$390$	0	0
$391$	−6088.96	−0.787549
$392$	0	0
$393$	−3148.20	−0.404086
$394$	0	0
$395$	− 595.703i	− 0.0758812i
$396$	0	0
$397$	12590.0i	1.59163i	0.605541	+	0.795814i	$0.292957\pi$
$397$	12590.0i	1.59163i	−0.605541	+	0.795814i	$0.707043\pi$
$398$	0	0
$399$	6327.66	0.793933
$400$	0	0
$401$	6425.77	0.800218	0.400109	−	0.916468i	$-0.368972\pi$
$401$	6425.77	0.800218	0.400109	+	0.916468i	$0.368972\pi$
$402$	0	0
$403$	− 609.122i	− 0.0752917i
$404$	0	0
$405$	129.590i	0.0158997i
$406$	0	0
$407$	526.755	0.0641530
$408$	0	0
$409$	−12796.0	−1.54699	−0.773496	−	0.633801i	$-0.781494\pi$
$409$	−12796.0	−1.54699	−0.773496	+	0.633801i	$0.781494\pi$
$410$	0	0
$411$	4475.09i	0.537080i
$412$	0	0
$413$	13359.6i	1.59173i
$414$	0	0
$415$	−1130.54	−0.133725
$416$	0	0
$417$	−1101.65	−0.129372
$418$	0	0
$419$	− 9256.33i	− 1.07924i	−0.841909	−	0.539620i	$-0.818568\pi$
$419$	− 9256.33i	− 1.07924i	0.841909	−	0.539620i	$-0.181432\pi$
$420$	0	0
$421$	9036.83i	1.04615i	0.852287	+	0.523074i	$0.175215\pi$
$421$	9036.83i	1.04615i	−0.852287	+	0.523074i	$0.824785\pi$
$422$	0	0
$423$	−1912.70	−0.219855
$424$	0	0
$425$	−10778.1	−1.23016
$426$	0	0
$427$	− 13.9508i	− 0.00158110i
$428$	0	0
$429$	− 3304.85i	− 0.371934i
$430$	0	0
$431$	−10639.3	−1.18904	−0.594519	−	0.804081i	$-0.702658\pi$
$431$	−10639.3	−1.18904	−0.594519	+	0.804081i	$0.702658\pi$
$432$	0	0
$433$	−3806.14	−0.422428	−0.211214	−	0.977440i	$-0.567742\pi$
$433$	−3806.14	−0.422428	−0.211214	+	0.977440i	$0.567742\pi$
$434$	0	0
$435$	212.304i	0.0234004i
$436$	0	0
$437$	5445.94i	0.596143i
$438$	0	0
$439$	−14102.8	−1.53323	−0.766616	−	0.642106i	$-0.778061\pi$
$439$	−14102.8	−1.53323	−0.766616	+	0.642106i	$0.778061\pi$
$440$	0	0
$441$	9619.28	1.03869
$442$	0	0
$443$	− 10836.3i	− 1.16219i	−0.813836	−	0.581095i	$-0.802625\pi$
$443$	− 10836.3i	− 1.16219i	0.813836	−	0.581095i	$-0.197375\pi$
$444$	0	0
$445$	− 3.99176i	0 0.000425230i
$446$	0	0
$447$	2353.58	0.249040
$448$	0	0
$449$	13679.4	1.43779	0.718897	−	0.695117i	$-0.244647\pi$
$449$	13679.4	1.43779	0.718897	+	0.695117i	$0.244647\pi$
$450$	0	0
$451$	− 2629.14i	− 0.274505i
$452$	0	0
$453$	− 450.666i	− 0.0467420i
$454$	0	0
$455$	−1686.01	−0.173718
$456$	0	0
$457$	7913.48	0.810016	0.405008	−	0.914313i	$-0.367269\pi$
$457$	7913.48	0.810016	0.405008	+	0.914313i	$0.367269\pi$
$458$	0	0
$459$	− 11222.0i	− 1.14117i
$460$	0	0
$461$	− 820.548i	− 0.0828996i	−0.999141	−	0.0414498i	$-0.986802\pi$
$461$	− 820.548i	− 0.0828996i	0.999141	−	0.0414498i	$-0.0131977\pi$
$462$	0	0
$463$	14236.5	1.42899	0.714497	−	0.699638i	$-0.246656\pi$
$463$	14236.5	1.42899	0.714497	+	0.699638i	$0.246656\pi$
$464$	0	0
$465$	−20.9990	−0.00209420
$466$	0	0
$467$	11801.0i	1.16935i	0.811269	+	0.584674i	$0.198777\pi$
$467$	11801.0i	1.16935i	−0.811269	+	0.584674i	$0.801223\pi$
$468$	0	0
$469$	− 23249.9i	− 2.28909i
$470$	0	0
$471$	−2799.85	−0.273907
$472$	0	0
$473$	2934.05	0.285217
$474$	0	0
$475$	9639.92i	0.931179i
$476$	0	0
$477$	10528.1i	1.01058i
$478$	0	0
$479$	5563.77	0.530720	0.265360	−	0.964149i	$-0.414509\pi$
$479$	5563.77	0.530720	0.265360	+	0.964149i	$0.414509\pi$
$480$	0	0
$481$	2109.16	0.199936
$482$	0	0
$483$	− 5728.29i	− 0.539640i
$484$	0	0
$485$	320.360i	0.0299934i
$486$	0	0
$487$	−18150.5	−1.68886	−0.844432	−	0.535662i	$-0.820062\pi$
$487$	−18150.5	−1.68886	−0.844432	+	0.535662i	$0.820062\pi$
$488$	0	0
$489$	−7428.99	−0.687015
$490$	0	0
$491$	− 16395.0i	− 1.50692i	−0.657495	−	0.753459i	$-0.728384\pi$
$491$	− 16395.0i	− 1.50692i	0.657495	−	0.753459i	$-0.271616\pi$
$492$	0	0
$493$	7775.61i	0.710336i
$494$	0	0
$495$	276.988	0.0251509
$496$	0	0
$497$	−12200.3	−1.10112
$498$	0	0
$499$	− 4397.61i	− 0.394517i	−0.980351	−	0.197259i	$-0.936796\pi$
$499$	− 4397.61i	− 0.394517i	0.980351	−	0.197259i	$-0.0632039\pi$
$500$	0	0
$501$	10806.1i	0.963635i
$502$	0	0
$503$	6221.21	0.551471	0.275736	−	0.961233i	$-0.411079\pi$
$503$	6221.21	0.551471	0.275736	+	0.961233i	$0.411079\pi$
$504$	0	0
$505$	−466.740	−0.0411281
$506$	0	0
$507$	− 7069.83i	− 0.619295i
$508$	0	0
$509$	− 19516.8i	− 1.69954i	−0.527154	−	0.849770i	$-0.676741\pi$
$509$	− 19516.8i	− 1.69954i	0.527154	−	0.849770i	$-0.323259\pi$
$510$	0	0
$511$	10901.2	0.943720
$512$	0	0
$513$	−10036.9	−0.863820
$514$	0	0
$515$	259.919i	0.0222396i
$516$	0	0
$517$	1714.96i	0.145888i
$518$	0	0
$519$	−10416.1	−0.880957
$520$	0	0
$521$	−6874.63	−0.578086	−0.289043	−	0.957316i	$-0.593337\pi$
$521$	−6874.63	−0.578086	−0.289043	+	0.957316i	$0.593337\pi$
$522$	0	0
$523$	− 3261.91i	− 0.272722i	−0.990659	−	0.136361i	$-0.956459\pi$
$523$	− 3261.91i	− 0.272722i	0.990659	−	0.136361i	$-0.0435407\pi$
$524$	0	0
$525$	− 10139.7i	− 0.842920i
$526$	0	0
$527$	−769.086	−0.0635710
$528$	0	0
$529$	−7236.92	−0.594799
$530$	0	0
$531$	− 8788.08i	− 0.718211i
$532$	0	0
$533$	− 10527.3i	− 0.855509i
$534$	0	0
$535$	−718.136	−0.0580332
$536$	0	0
$537$	−4906.28	−0.394267
$538$	0	0
$539$	− 8624.83i	− 0.689235i
$540$	0	0
$541$	18724.1i	1.48801i	0.668174	+	0.744005i	$0.267076\pi$
$541$	18724.1i	1.48801i	−0.668174	+	0.744005i	$0.732924\pi$
$542$	0	0
$543$	−6247.82	−0.493775
$544$	0	0
$545$	1132.69	0.0890256
$546$	0	0
$547$	− 18768.5i	− 1.46706i	−0.679657	−	0.733530i	$-0.737871\pi$
$547$	− 18768.5i	− 1.46706i	0.679657	−	0.733530i	$-0.262129\pi$
$548$	0	0
$549$	9.17700i	0 0.000713415i
$550$	0	0
$551$	6954.47	0.537696
$552$	0	0
$553$	20525.2	1.57834
$554$	0	0
$555$	− 72.7115i	− 0.00556114i
$556$	0	0
$557$	− 12021.7i	− 0.914497i	−0.889339	−	0.457249i	$-0.848835\pi$
$557$	− 12021.7i	− 0.914497i	0.889339	−	0.457249i	$-0.151165\pi$
$558$	0	0
$559$	11748.1	0.888896
$560$	0	0
$561$	−4172.74	−0.314034
$562$	0	0
$563$	− 24504.4i	− 1.83434i	−0.398491	−	0.917172i	$-0.630466\pi$
$563$	− 24504.4i	− 1.83434i	0.398491	−	0.917172i	$-0.369534\pi$
$564$	0	0
$565$	− 1539.60i	− 0.114640i
$566$	0	0
$567$	−4465.09	−0.330716
$568$	0	0
$569$	−8998.54	−0.662985	−0.331492	−	0.943458i	$-0.607552\pi$
$569$	−8998.54	−0.662985	−0.331492	+	0.943458i	$0.607552\pi$
$570$	0	0
$571$	− 13928.9i	− 1.02085i	−0.859921	−	0.510427i	$-0.829487\pi$
$571$	− 13928.9i	− 1.02085i	0.859921	−	0.510427i	$-0.170513\pi$
$572$	0	0
$573$	− 10059.1i	− 0.733380i
$574$	0	0
$575$	8726.79	0.632926
$576$	0	0
$577$	−20584.4	−1.48516	−0.742580	−	0.669757i	$-0.766398\pi$
$577$	−20584.4	−1.48516	−0.742580	+	0.669757i	$0.766398\pi$
$578$	0	0
$579$	1468.79i	0.105425i
$580$	0	0
$581$	− 38953.3i	− 2.78151i
$582$	0	0
$583$	9439.66	0.670585
$584$	0	0
$585$	1109.08	0.0783840
$586$	0	0
$587$	3658.25i	0.257227i	0.991695	+	0.128613i	$0.0410526\pi$
$587$	3658.25i	0.257227i	−0.991695	+	0.128613i	$0.958947\pi$
$588$	0	0
$589$	687.867i	0.0481207i
$590$	0	0
$591$	−4465.54	−0.310809
$592$	0	0
$593$	6035.89	0.417984	0.208992	−	0.977917i	$-0.432982\pi$
$593$	6035.89	0.417984	0.208992	+	0.977917i	$0.432982\pi$
$594$	0	0
$595$	2128.78i	0.146675i
$596$	0	0
$597$	6486.93i	0.444711i
$598$	0	0
$599$	5427.20	0.370199	0.185100	−	0.982720i	$-0.440739\pi$
$599$	5427.20	0.370199	0.185100	+	0.982720i	$0.440739\pi$
$600$	0	0
$601$	17725.7	1.20307	0.601535	−	0.798847i	$-0.294556\pi$
$601$	17725.7	1.20307	0.601535	+	0.798847i	$0.294556\pi$
$602$	0	0
$603$	15294.0i	1.03287i
$604$	0	0
$605$	875.105i	0.0588067i
$606$	0	0
$607$	−13487.6	−0.901884	−0.450942	−	0.892553i	$-0.648912\pi$
$607$	−13487.6	−0.901884	−0.450942	+	0.892553i	$0.648912\pi$
$608$	0	0
$609$	−7315.03	−0.486732
$610$	0	0
$611$	6866.81i	0.454667i
$612$	0	0
$613$	23830.0i	1.57012i	0.619417	+	0.785062i	$0.287369\pi$
$613$	23830.0i	1.57012i	−0.619417	+	0.785062i	$0.712631\pi$
$614$	0	0
$615$	−362.918	−0.0237956
$616$	0	0
$617$	535.243	0.0349239	0.0174620	−	0.999848i	$-0.494441\pi$
$617$	535.243	0.0349239	0.0174620	+	0.999848i	$0.494441\pi$
$618$	0	0
$619$	27847.2i	1.80820i	0.427324	+	0.904098i	$0.359456\pi$
$619$	27847.2i	1.80820i	−0.427324	+	0.904098i	$0.640544\pi$
$620$	0	0
$621$	9086.16i	0.587142i
$622$	0	0
$623$	137.538	0.00884485
$624$	0	0
$625$	15358.3	0.982934
$626$	0	0
$627$	3732.08i	0.237711i
$628$	0	0
$629$	− 2663.05i	− 0.168812i
$630$	0	0
$631$	−11880.2	−0.749511	−0.374755	−	0.927124i	$-0.622273\pi$
$631$	−11880.2	−0.749511	−0.374755	+	0.927124i	$0.622273\pi$
$632$	0	0
$633$	5627.72	0.353368
$634$	0	0
$635$	− 834.573i	− 0.0521560i
$636$	0	0
$637$	− 34534.4i	− 2.14804i
$638$	0	0
$639$	8025.45	0.496842
$640$	0	0
$641$	18341.0	1.13015	0.565074	−	0.825040i	$-0.308848\pi$
$641$	18341.0	1.13015	0.565074	+	0.825040i	$0.308848\pi$
$642$	0	0
$643$	9936.56i	0.609424i	0.952445	+	0.304712i	$0.0985602\pi$
$643$	9936.56i	0.609424i	−0.952445	+	0.304712i	$0.901440\pi$
$644$	0	0
$645$	− 405.007i	− 0.0247242i
$646$	0	0
$647$	21429.7	1.30215	0.651073	−	0.759015i	$-0.274319\pi$
$647$	21429.7	1.30215	0.651073	+	0.759015i	$0.274319\pi$
$648$	0	0
$649$	−7879.56	−0.476579
$650$	0	0
$651$	− 723.530i	− 0.0435597i
$652$	0	0
$653$	12277.1i	0.735742i	0.929877	+	0.367871i	$0.119913\pi$
$653$	12277.1i	0.735742i	−0.929877	+	0.367871i	$0.880087\pi$
$654$	0	0
$655$	−947.284	−0.0565091
$656$	0	0
$657$	−7170.92	−0.425821
$658$	0	0
$659$	5871.25i	0.347058i	0.984829	+	0.173529i	$0.0555171\pi$
$659$	5871.25i	0.347058i	−0.984829	+	0.173529i	$0.944483\pi$
$660$	0	0
$661$	17308.5i	1.01849i	0.860621	+	0.509246i	$0.170076\pi$
$661$	17308.5i	1.01849i	−0.860621	+	0.509246i	$0.829924\pi$
$662$	0	0
$663$	−16707.9	−0.978706
$664$	0	0
$665$	1903.98	0.111027
$666$	0	0
$667$	− 6295.72i	− 0.365474i
$668$	0	0
$669$	12104.6i	0.699536i
$670$	0	0
$671$	8.22827	0.000473396 0
$672$	0	0
$673$	6528.62	0.373937	0.186969	−	0.982366i	$-0.440134\pi$
$673$	6528.62	0.373937	0.186969	+	0.982366i	$0.440134\pi$
$674$	0	0
$675$	16083.5i	0.917118i
$676$	0	0
$677$	20110.9i	1.14169i	0.821057	+	0.570847i	$0.193385\pi$
$677$	20110.9i	1.14169i	−0.821057	+	0.570847i	$0.806615\pi$
$678$	0	0
$679$	−11038.2	−0.623867
$680$	0	0
$681$	−2781.76	−0.156530
$682$	0	0
$683$	− 30291.8i	− 1.69705i	−0.529157	−	0.848524i	$-0.677492\pi$
$683$	− 30291.8i	− 1.69705i	0.529157	−	0.848524i	$-0.322508\pi$
$684$	0	0
$685$	1346.54i	0.0751076i
$686$	0	0
$687$	−2631.60	−0.146145
$688$	0	0
$689$	37797.0	2.08991
$690$	0	0
$691$	23387.1i	1.28753i	0.765222	+	0.643767i	$0.222629\pi$
$691$	23387.1i	1.28753i	−0.765222	+	0.643767i	$0.777371\pi$
$692$	0	0
$693$	9543.74i	0.523141i
$694$	0	0
$695$	−331.484	−0.0180920
$696$	0	0
$697$	−13291.8	−0.722331
$698$	0	0
$699$	9790.49i	0.529772i
$700$	0	0
$701$	4280.14i	0.230612i	0.993330	+	0.115306i	$0.0367848\pi$
$701$	4280.14i	0.230612i	−0.993330	+	0.115306i	$0.963215\pi$
$702$	0	0
$703$	−2381.82	−0.127784
$704$	0	0
$705$	236.728	0.0126464
$706$	0	0
$707$	− 16081.8i	− 0.855470i
$708$	0	0
$709$	− 5630.18i	− 0.298231i	−0.988820	−	0.149115i	$-0.952357\pi$
$709$	− 5630.18i	− 0.298231i	0.988820	−	0.149115i	$-0.0476426\pi$
$710$	0	0
$711$	−13501.7	−0.712170
$712$	0	0
$713$	622.710	0.0327078
$714$	0	0
$715$	− 994.419i	− 0.0520128i
$716$	0	0
$717$	8276.49i	0.431090i
$718$	0	0
$719$	−5682.25	−0.294732	−0.147366	−	0.989082i	$-0.547079\pi$
$719$	−5682.25	−0.294732	−0.147366	+	0.989082i	$0.547079\pi$
$720$	0	0
$721$	−8955.64	−0.462587
$722$	0	0
$723$	3166.94i	0.162904i
$724$	0	0
$725$	− 11144.1i	− 0.570872i
$726$	0	0
$727$	18883.0	0.963317	0.481658	−	0.876359i	$-0.340035\pi$
$727$	18883.0	0.963317	0.481658	+	0.876359i	$0.340035\pi$
$728$	0	0
$729$	−6863.99	−0.348727
$730$	0	0
$731$	− 14833.3i	− 0.750521i
$732$	0	0
$733$	35302.3i	1.77888i	0.457049	+	0.889441i	$0.348906\pi$
$733$	35302.3i	1.77888i	−0.457049	+	0.889441i	$0.651094\pi$
$734$	0	0
$735$	−1190.54	−0.0597467
$736$	0	0
$737$	13712.9	0.685375
$738$	0	0
$739$	8771.48i	0.436623i	0.975879	+	0.218311i	$0.0700548\pi$
$739$	8771.48i	0.436623i	−0.975879	+	0.218311i	$0.929945\pi$
$740$	0	0
$741$	14943.5i	0.740841i
$742$	0	0
$743$	30.9140	0.00152641	0.000763205	−	1.00000i	$-0.499757\pi$
$743$	30.9140	0.00152641	0.000763205		1.00000i	$0.499757\pi$
$744$	0	0
$745$	708.187	0.0348268
$746$	0	0
$747$	25623.8i	1.25506i
$748$	0	0
$749$	− 24743.7i	− 1.20710i
$750$	0	0
$751$	−16318.5	−0.792905	−0.396453	−	0.918055i	$-0.629759\pi$
$751$	−16318.5	−0.792905	−0.396453	+	0.918055i	$0.629759\pi$
$752$	0	0
$753$	−18335.5	−0.887361
$754$	0	0
$755$	− 135.604i	− 0.00653660i
$756$	0	0
$757$	13936.5i	0.669129i	0.942373	+	0.334564i	$0.108589\pi$
$757$	13936.5i	0.669129i	−0.942373	+	0.334564i	$0.891411\pi$
$758$	0	0
$759$	3378.57	0.161573
$760$	0	0
$761$	3823.42	0.182127	0.0910637	−	0.995845i	$-0.470973\pi$
$761$	3823.42	0.182127	0.0910637	+	0.995845i	$0.470973\pi$
$762$	0	0
$763$	39027.2i	1.85174i
$764$	0	0
$765$	− 1400.33i	− 0.0661819i
$766$	0	0
$767$	−31550.2	−1.48528
$768$	0	0
$769$	31689.1	1.48601	0.743003	−	0.669288i	$-0.233401\pi$
$769$	31689.1	1.48601	0.743003	+	0.669288i	$0.233401\pi$
$770$	0	0
$771$	1713.90i	0.0800578i
$772$	0	0
$773$	1846.74i	0.0859282i	0.999077	+	0.0429641i	$0.0136801\pi$
$773$	1846.74i	0.0859282i	−0.999077	+	0.0429641i	$0.986320\pi$
$774$	0	0
$775$	1102.27	0.0510898
$776$	0	0
$777$	2505.31	0.115672
$778$	0	0
$779$	11888.2i	0.546776i
$780$	0	0
$781$	− 7195.77i	− 0.329686i
$782$	0	0
$783$	11603.0	0.529577
$784$	0	0
$785$	−842.465	−0.0383043
$786$	0	0
$787$	− 20364.0i	− 0.922363i	−0.887306	−	0.461181i	$-0.847426\pi$
$787$	− 20364.0i	− 0.922363i	0.887306	−	0.461181i	$-0.152574\pi$
$788$	0	0
$789$	− 13951.7i	− 0.629525i
$790$	0	0
$791$	53047.6	2.38452
$792$	0	0
$793$	32.9465	0.00147537
$794$	0	0
$795$	− 1303.02i	− 0.0581300i
$796$	0	0
$797$	7880.27i	0.350230i	0.984548	+	0.175115i	$0.0560297\pi$
$797$	7880.27i	0.350230i	−0.984548	+	0.175115i	$0.943970\pi$
$798$	0	0
$799$	8670.13	0.383889
$800$	0	0
$801$	−90.4737	−0.00399093
$802$	0	0
$803$	6429.58i	0.282559i
$804$	0	0
$805$	− 1723.62i	− 0.0754656i
$806$	0	0
$807$	3722.81	0.162390
$808$	0	0
$809$	−5081.28	−0.220826	−0.110413	−	0.993886i	$-0.535217\pi$
$809$	−5081.28	−0.220826	−0.110413	+	0.993886i	$0.535217\pi$
$810$	0	0
$811$	− 6354.33i	− 0.275130i	−0.990493	−	0.137565i	$-0.956072\pi$
$811$	− 6354.33i	− 0.275130i	0.990493	−	0.137565i	$-0.0439276\pi$
$812$	0	0
$813$	− 11250.6i	− 0.485331i
$814$	0	0
$815$	−2235.36	−0.0960752
$816$	0	0
$817$	−13266.9	−0.568114
$818$	0	0
$819$	38213.7i	1.63040i
$820$	0	0
$821$	2991.83i	0.127181i	0.997976	+	0.0635904i	$0.0202551\pi$
$821$	2991.83i	0.127181i	−0.997976	+	0.0635904i	$0.979745\pi$
$822$	0	0
$823$	24432.6	1.03483	0.517416	−	0.855734i	$-0.326894\pi$
$823$	24432.6	1.03483	0.517416	+	0.855734i	$0.326894\pi$
$824$	0	0
$825$	5980.44	0.252378
$826$	0	0
$827$	29771.2i	1.25181i	0.779900	+	0.625904i	$0.215270\pi$
$827$	29771.2i	1.25181i	−0.779900	+	0.625904i	$0.784730\pi$
$828$	0	0
$829$	− 35980.5i	− 1.50742i	−0.657206	−	0.753711i	$-0.728262\pi$
$829$	− 35980.5i	− 1.50742i	0.657206	−	0.753711i	$-0.271738\pi$
$830$	0	0
$831$	14023.7	0.585413
$832$	0	0
$833$	−43603.5	−1.81365
$834$	0	0
$835$	3251.52i	0.134759i
$836$	0	0
$837$	1147.66i	0.0473941i
$838$	0	0
$839$	18757.2	0.771837	0.385919	−	0.922533i	$-0.373885\pi$
$839$	18757.2	0.771837	0.385919	+	0.922533i	$0.373885\pi$
$840$	0	0
$841$	16349.4	0.670358
$842$	0	0
$843$	20950.1i	0.855941i
$844$	0	0
$845$	− 2127.29i	− 0.0866048i
$846$	0	0
$847$	−30152.2	−1.22319
$848$	0	0
$849$	8922.94	0.360700
$850$	0	0
$851$	2156.21i	0.0868553i
$852$	0	0
$853$	− 8626.94i	− 0.346285i	−0.984897	−	0.173142i	$-0.944608\pi$
$853$	− 8626.94i	− 0.346285i	0.984897	−	0.173142i	$-0.0553921\pi$
$854$	0	0
$855$	−1252.45	−0.0500971
$856$	0	0
$857$	2079.04	0.0828687	0.0414344	−	0.999141i	$-0.486807\pi$
$857$	2079.04	0.0828687	0.0414344	+	0.999141i	$0.486807\pi$
$858$	0	0
$859$	11740.7i	0.466343i	0.972436	+	0.233171i	$0.0749103\pi$
$859$	11740.7i	0.466343i	−0.972436	+	0.233171i	$0.925090\pi$
$860$	0	0
$861$	− 12504.5i	− 0.494951i
$862$	0	0
$863$	−43830.8	−1.72887	−0.864436	−	0.502743i	$-0.832324\pi$
$863$	−43830.8	−1.72887	−0.864436	+	0.502743i	$0.832324\pi$
$864$	0	0
$865$	−3134.18	−0.123197
$866$	0	0
$867$	7313.78i	0.286493i
$868$	0	0
$869$	12105.9i	0.472570i
$870$	0	0
$871$	54907.3	2.13601
$872$	0	0
$873$	7261.01	0.281498
$874$	0	0
$875$	− 6119.50i	− 0.236431i
$876$	0	0
$877$	− 3628.71i	− 0.139718i	−0.997557	−	0.0698591i	$-0.977745\pi$
$877$	− 3628.71i	− 0.139718i	0.997557	−	0.0698591i	$-0.0222550\pi$
$878$	0	0
$879$	−15681.2	−0.601723
$880$	0	0
$881$	−26429.8	−1.01072	−0.505360	−	0.862909i	$-0.668640\pi$
$881$	−26429.8	−1.01072	−0.505360	+	0.862909i	$0.668640\pi$
$882$	0	0
$883$	29286.1i	1.11614i	0.829793	+	0.558071i	$0.188458\pi$
$883$	29286.1i	1.11614i	−0.829793	+	0.558071i	$0.811542\pi$
$884$	0	0
$885$	1087.67i	0.0413125i
$886$	0	0
$887$	−22350.0	−0.846042	−0.423021	−	0.906120i	$-0.639030\pi$
$887$	−22350.0	−0.846042	−0.423021	+	0.906120i	$0.639030\pi$
$888$	0	0
$889$	28755.6	1.08485
$890$	0	0
$891$	− 2633.53i	− 0.0990197i
$892$	0	0
$893$	− 7754.53i	− 0.290588i
$894$	0	0
$895$	−1476.28	−0.0551360
$896$	0	0
$897$	13528.0	0.503553
$898$	0	0
$899$	− 795.202i	− 0.0295011i
$900$	0	0
$901$	− 47723.0i	− 1.76457i
$902$	0	0
$903$	13954.7	0.514267
$904$	0	0
$905$	−1879.95	−0.0690516
$906$	0	0
$907$	5779.79i	0.211593i	0.994388	+	0.105796i	$0.0337392\pi$
$907$	5779.79i	0.211593i	−0.994388	+	0.105796i	$0.966261\pi$
$908$	0	0
$909$	10578.7i	0.386000i
$910$	0	0
$911$	−20024.6	−0.728259	−0.364130	−	0.931348i	$-0.618633\pi$
$911$	−20024.6	−0.728259	−0.364130	+	0.931348i	$0.618633\pi$
$912$	0	0
$913$	22974.8	0.832810
$914$	0	0
$915$	− 1.13580i	0 4.10366e-5i
$916$	0	0
$917$	− 32639.1i	− 1.17540i
$918$	0	0
$919$	−26977.7	−0.968349	−0.484175	−	0.874971i	$-0.660880\pi$
$919$	−26977.7	−0.968349	−0.484175	+	0.874971i	$0.660880\pi$
$920$	0	0
$921$	−15294.2	−0.547189
$922$	0	0
$923$	− 28812.3i	− 1.02748i
$924$	0	0
$925$	3816.73i	0.135668i
$926$	0	0
$927$	5891.10	0.208726
$928$	0	0
$929$	−34371.4	−1.21387	−0.606937	−	0.794750i	$-0.707602\pi$
$929$	−34371.4	−1.21387	−0.606937	+	0.794750i	$0.707602\pi$
$930$	0	0
$931$	38998.8i	1.37286i
$932$	0	0
$933$	− 14571.2i	− 0.511297i
$934$	0	0
$935$	−1255.57	−0.0439159
$936$	0	0
$937$	−32901.8	−1.14712	−0.573561	−	0.819163i	$-0.694439\pi$
$937$	−32901.8	−1.14712	−0.573561	+	0.819163i	$0.694439\pi$
$938$	0	0
$939$	13212.4i	0.459181i
$940$	0	0
$941$	47208.7i	1.63545i	0.575607	+	0.817726i	$0.304766\pi$
$941$	47208.7i	1.63545i	−0.575607	+	0.817726i	$0.695234\pi$
$942$	0	0
$943$	10762.1	0.371646
$944$	0	0
$945$	3176.65	0.109351
$946$	0	0
$947$	− 19250.2i	− 0.660557i	−0.943884	−	0.330279i	$-0.892857\pi$
$947$	− 19250.2i	− 0.660557i	0.943884	−	0.330279i	$-0.107143\pi$
$948$	0	0
$949$	25744.4i	0.880611i
$950$	0	0
$951$	22534.1	0.768369
$952$	0	0
$953$	4572.49	0.155422	0.0777112	−	0.996976i	$-0.475239\pi$
$953$	4572.49	0.155422	0.0777112	+	0.996976i	$0.475239\pi$
$954$	0	0
$955$	− 3026.77i	− 0.102559i
$956$	0	0
$957$	− 4314.44i	− 0.145732i
$958$	0	0
$959$	−46395.7	−1.56225
$960$	0	0
$961$	−29712.3	−0.997360
$962$	0	0
$963$	16276.7i	0.544660i
$964$	0	0
$965$	441.956i	0.0147431i
$966$	0	0
$967$	−33060.9	−1.09945	−0.549725	−	0.835346i	$-0.685268\pi$
$967$	−33060.9	−1.09945	−0.549725	+	0.835346i	$0.685268\pi$
$968$	0	0
$969$	18867.9	0.625514
$970$	0	0
$971$	21995.9i	0.726964i	0.931601	+	0.363482i	$0.118412\pi$
$971$	21995.9i	0.726964i	−0.931601	+	0.363482i	$0.881588\pi$
$972$	0	0
$973$	− 11421.4i	− 0.376315i
$974$	0	0
$975$	23946.1	0.786551
$976$	0	0
$977$	−1241.54	−0.0406555	−0.0203277	−	0.999793i	$-0.506471\pi$
$977$	−1241.54	−0.0406555	−0.0203277	+	0.999793i	$0.506471\pi$
$978$	0	0
$979$	81.1205i	0.00264823i
$980$	0	0
$981$	− 25672.5i	− 0.835534i
$982$	0	0
$983$	−10797.7	−0.350349	−0.175175	−	0.984537i	$-0.556049\pi$
$983$	−10797.7	−0.350349	−0.175175	+	0.984537i	$0.556049\pi$
$984$	0	0
$985$	−1343.67	−0.0434648
$986$	0	0
$987$	8156.56i	0.263046i
$988$	0	0
$989$	12010.2i	0.386149i
$990$	0	0
$991$	10204.8	0.327112	0.163556	−	0.986534i	$-0.447704\pi$
$991$	10204.8	0.327112	0.163556	+	0.986534i	$0.447704\pi$
$992$	0	0
$993$	7203.49	0.230207
$994$	0	0
$995$	1951.90i	0.0621903i
$996$	0	0
$997$	− 21279.2i	− 0.675947i	−0.941156	−	0.337973i	$-0.890259\pi$
$997$	− 21279.2i	− 0.675947i	0.941156	−	0.337973i	$-0.109741\pi$
$998$	0	0
$999$	−3973.90	−0.125855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.4.b.k.513.7		10
4.3	odd	2		1024.4.b.j.513.4		10
8.3	odd	2		1024.4.b.j.513.7		10
8.5	even	2	inner	1024.4.b.k.513.4		10
16.3	odd	4		1024.4.a.n.1.7		10
16.5	even	4		1024.4.a.m.1.7		10
16.11	odd	4		1024.4.a.n.1.4		10
16.13	even	4		1024.4.a.m.1.4		10
32.3	odd	8		16.4.e.a.5.4	✓	10
32.5	even	8		64.4.e.a.17.2		10
32.11	odd	8		128.4.e.b.33.2		10
32.13	even	8		128.4.e.a.97.4		10
32.19	odd	8		128.4.e.b.97.2		10
32.21	even	8		128.4.e.a.33.4		10
32.27	odd	8		16.4.e.a.13.4	yes	10
32.29	even	8		64.4.e.a.49.2		10
96.5	odd	8		576.4.k.a.145.3		10
96.29	odd	8		576.4.k.a.433.3		10
96.35	even	8		144.4.k.a.37.2		10
96.59	even	8		144.4.k.a.109.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.4	✓	10	32.3	odd	8
16.4.e.a.13.4	yes	10	32.27	odd	8
64.4.e.a.17.2		10	32.5	even	8
64.4.e.a.49.2		10	32.29	even	8
128.4.e.a.33.4		10	32.21	even	8
128.4.e.a.97.4		10	32.13	even	8
128.4.e.b.33.2		10	32.11	odd	8
128.4.e.b.97.2		10	32.19	odd	8
144.4.k.a.37.2		10	96.35	even	8
144.4.k.a.109.2		10	96.59	even	8
576.4.k.a.145.3		10	96.5	odd	8
576.4.k.a.433.3		10	96.29	odd	8
1024.4.a.m.1.4		10	16.13	even	4
1024.4.a.m.1.7		10	16.5	even	4
1024.4.a.n.1.4		10	16.11	odd	4
1024.4.a.n.1.7		10	16.3	odd	4
1024.4.b.j.513.4		10	4.3	odd	2
1024.4.b.j.513.7		10	8.3	odd	2
1024.4.b.k.513.4		10	8.5	even	2	inner
1024.4.b.k.513.7		10	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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1024.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.80518i q^{3} +0.844070i q^{5} -29.0828 q^{7} +19.1310 q^{9} +O(q^{10})\)	\(q+2.80518i q^{3} +0.844070i q^{5} -29.0828 q^{7} +19.1310 q^{9} -17.1532i q^{11} -68.6824i q^{13} -2.36777 q^{15} -86.7193 q^{17} +77.5614i q^{19} -81.5826i q^{21} +70.2145 q^{23} +124.288 q^{25} +129.406i q^{27} -89.6641i q^{29} +8.86868 q^{31} +48.1178 q^{33} -24.5480i q^{35} +30.7089i q^{37} +192.667 q^{39} +153.274 q^{41} +171.050i q^{43} +16.1479i q^{45} -99.9792 q^{47} +502.812 q^{49} -243.263i q^{51} +550.315i q^{53} +14.4785 q^{55} -217.574 q^{57} -459.364i q^{59} +0.479693i q^{61} -556.383 q^{63} +57.9728 q^{65} +799.438i q^{67} +196.964i q^{69} +419.500 q^{71} -374.833 q^{73} +348.649i q^{75} +498.864i q^{77} -705.750 q^{79} +153.530 q^{81} +1339.39i q^{83} -73.1972i q^{85} +251.524 q^{87} -4.72918 q^{89} +1997.48i q^{91} +24.8782i q^{93} -65.4673 q^{95} +379.542 q^{97} -328.157i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 28 q^{7} - 54 q^{9}+O(q^{10}) \) 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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