LMFDB - Embedded newform 8024.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.24951 q^{3} -3.27317 q^{5} +3.90567 q^{7} +7.55931 q^{9} +O(q^{10})$	$q-3.24951 q^{3} -3.27317 q^{5} +3.90567 q^{7} +7.55931 q^{9} +3.39948 q^{11} +0.485234 q^{13} +10.6362 q^{15} -1.00000 q^{17} -3.70343 q^{19} -12.6915 q^{21} +6.21767 q^{23} +5.71363 q^{25} -14.8155 q^{27} +3.88250 q^{29} -4.41681 q^{31} -11.0466 q^{33} -12.7839 q^{35} -10.0631 q^{37} -1.57677 q^{39} -1.01838 q^{41} +10.8066 q^{43} -24.7429 q^{45} -11.2963 q^{47} +8.25428 q^{49} +3.24951 q^{51} -9.21459 q^{53} -11.1271 q^{55} +12.0343 q^{57} -1.00000 q^{59} -10.4523 q^{61} +29.5242 q^{63} -1.58825 q^{65} +0.0650174 q^{67} -20.2044 q^{69} +6.83545 q^{71} +12.0474 q^{73} -18.5665 q^{75} +13.2772 q^{77} -12.5012 q^{79} +25.4652 q^{81} -0.493623 q^{83} +3.27317 q^{85} -12.6162 q^{87} +4.81090 q^{89} +1.89516 q^{91} +14.3525 q^{93} +12.1219 q^{95} +4.67707 q^{97} +25.6977 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.24951	−1.87610	−0.938052	−	0.346494i	$-0.887372\pi$
$3$	−3.24951	−1.87610	−0.938052	+	0.346494i	$0.887372\pi$
$4$	0	0
$5$	−3.27317	−1.46381	−0.731903	−	0.681409i	$-0.761367\pi$
$5$	−3.27317	−1.46381	−0.731903	+	0.681409i	$0.761367\pi$
$6$	0	0
$7$	3.90567	1.47621	0.738103	−	0.674688i	$-0.235722\pi$
$7$	3.90567	1.47621	0.738103	+	0.674688i	$0.235722\pi$
$8$	0	0
$9$	7.55931	2.51977
$10$	0	0
$11$	3.39948	1.02498	0.512491	−	0.858693i	$-0.328723\pi$
$11$	3.39948	1.02498	0.512491	+	0.858693i	$0.328723\pi$
$12$	0	0
$13$	0.485234	0.134580	0.0672898	−	0.997733i	$-0.478565\pi$
$13$	0.485234	0.134580	0.0672898	+	0.997733i	$0.478565\pi$
$14$	0	0
$15$	10.6362	2.74625
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−3.70343	−0.849625	−0.424813	−	0.905281i	$-0.639660\pi$
$19$	−3.70343	−0.849625	−0.424813	+	0.905281i	$0.639660\pi$
$20$	0	0
$21$	−12.6915	−2.76952
$22$	0	0
$23$	6.21767	1.29647	0.648237	−	0.761439i	$-0.275507\pi$
$23$	6.21767	1.29647	0.648237	+	0.761439i	$0.275507\pi$
$24$	0	0
$25$	5.71363	1.14273
$26$	0	0
$27$	−14.8155	−2.85125
$28$	0	0
$29$	3.88250	0.720963	0.360482	−	0.932766i	$-0.382612\pi$
$29$	3.88250	0.720963	0.360482	+	0.932766i	$0.382612\pi$
$30$	0	0
$31$	−4.41681	−0.793283	−0.396642	−	0.917974i	$-0.629824\pi$
$31$	−4.41681	−0.793283	−0.396642	+	0.917974i	$0.629824\pi$
$32$	0	0
$33$	−11.0466	−1.92297
$34$	0	0
$35$	−12.7839	−2.16088
$36$	0	0
$37$	−10.0631	−1.65437	−0.827186	−	0.561929i	$-0.810059\pi$
$37$	−10.0631	−1.65437	−0.827186	+	0.561929i	$0.810059\pi$
$38$	0	0
$39$	−1.57677	−0.252486
$40$	0	0
$41$	−1.01838	−0.159044	−0.0795220	−	0.996833i	$-0.525339\pi$
$41$	−1.01838	−0.159044	−0.0795220	+	0.996833i	$0.525339\pi$
$42$	0	0
$43$	10.8066	1.64799	0.823996	−	0.566595i	$-0.191740\pi$
$43$	10.8066	1.64799	0.823996	+	0.566595i	$0.191740\pi$
$44$	0	0
$45$	−24.7429	−3.68845
$46$	0	0
$47$	−11.2963	−1.64774	−0.823869	−	0.566781i	$-0.808189\pi$
$47$	−11.2963	−1.64774	−0.823869	+	0.566781i	$0.808189\pi$
$48$	0	0
$49$	8.25428	1.17918
$50$	0	0
$51$	3.24951	0.455022
$52$	0	0
$53$	−9.21459	−1.26572	−0.632861	−	0.774266i	$-0.718119\pi$
$53$	−9.21459	−1.26572	−0.632861	+	0.774266i	$0.718119\pi$
$54$	0	0
$55$	−11.1271	−1.50037
$56$	0	0
$57$	12.0343	1.59399
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−10.4523	−1.33828	−0.669139	−	0.743138i	$-0.733337\pi$
$61$	−10.4523	−1.33828	−0.669139	+	0.743138i	$0.733337\pi$
$62$	0	0
$63$	29.5242	3.71970
$64$	0	0
$65$	−1.58825	−0.196998
$66$	0	0
$67$	0.0650174	0.00794314	0.00397157	−	0.999992i	$-0.498736\pi$
$67$	0.0650174	0.00794314	0.00397157	+	0.999992i	$0.498736\pi$
$68$	0	0
$69$	−20.2044	−2.43232
$70$	0	0
$71$	6.83545	0.811219	0.405610	−	0.914046i	$-0.367059\pi$
$71$	6.83545	0.811219	0.405610	+	0.914046i	$0.367059\pi$
$72$	0	0
$73$	12.0474	1.41004	0.705022	−	0.709185i	$-0.250937\pi$
$73$	12.0474	1.41004	0.705022	+	0.709185i	$0.250937\pi$
$74$	0	0
$75$	−18.5665	−2.14387
$76$	0	0
$77$	13.2772	1.51308
$78$	0	0
$79$	−12.5012	−1.40649	−0.703245	−	0.710947i	$-0.748266\pi$
$79$	−12.5012	−1.40649	−0.703245	+	0.710947i	$0.748266\pi$
$80$	0	0
$81$	25.4652	2.82947
$82$	0	0
$83$	−0.493623	−0.0541822	−0.0270911	−	0.999633i	$-0.508624\pi$
$83$	−0.493623	−0.0541822	−0.0270911	+	0.999633i	$0.508624\pi$
$84$	0	0
$85$	3.27317	0.355025
$86$	0	0
$87$	−12.6162	−1.35260
$88$	0	0
$89$	4.81090	0.509954	0.254977	−	0.966947i	$-0.417932\pi$
$89$	4.81090	0.509954	0.254977	+	0.966947i	$0.417932\pi$
$90$	0	0
$91$	1.89516	0.198667
$92$	0	0
$93$	14.3525	1.48828
$94$	0	0
$95$	12.1219	1.24369
$96$	0	0
$97$	4.67707	0.474884	0.237442	−	0.971402i	$-0.423691\pi$
$97$	4.67707	0.474884	0.237442	+	0.971402i	$0.423691\pi$
$98$	0	0
$99$	25.6977	2.58272
$100$	0	0
$101$	−0.466630	−0.0464314	−0.0232157	−	0.999730i	$-0.507390\pi$
$101$	−0.466630	−0.0464314	−0.0232157	+	0.999730i	$0.507390\pi$
$102$	0	0
$103$	−11.0563	−1.08941	−0.544703	−	0.838629i	$-0.683358\pi$
$103$	−11.0563	−1.08941	−0.544703	+	0.838629i	$0.683358\pi$
$104$	0	0
$105$	41.5415	4.05403
$106$	0	0
$107$	−9.42189	−0.910848	−0.455424	−	0.890275i	$-0.650512\pi$
$107$	−9.42189	−0.910848	−0.455424	+	0.890275i	$0.650512\pi$
$108$	0	0
$109$	−2.12000	−0.203059	−0.101529	−	0.994833i	$-0.532374\pi$
$109$	−2.12000	−0.203059	−0.101529	+	0.994833i	$0.532374\pi$
$110$	0	0
$111$	32.7003	3.10377
$112$	0	0
$113$	6.33823	0.596250	0.298125	−	0.954527i	$-0.403639\pi$
$113$	6.33823	0.596250	0.298125	+	0.954527i	$0.403639\pi$
$114$	0	0
$115$	−20.3515	−1.89778
$116$	0	0
$117$	3.66803	0.339110
$118$	0	0
$119$	−3.90567	−0.358032
$120$	0	0
$121$	0.556448	0.0505862
$122$	0	0
$123$	3.30923	0.298383
$124$	0	0
$125$	−2.33582	−0.208922
$126$	0	0
$127$	9.95463	0.883330	0.441665	−	0.897180i	$-0.354388\pi$
$127$	9.95463	0.883330	0.441665	+	0.897180i	$0.354388\pi$
$128$	0	0
$129$	−35.1162	−3.09181
$130$	0	0
$131$	12.8434	1.12214	0.561068	−	0.827769i	$-0.310390\pi$
$131$	12.8434	1.12214	0.561068	+	0.827769i	$0.310390\pi$
$132$	0	0
$133$	−14.4644	−1.25422
$134$	0	0
$135$	48.4936	4.17367
$136$	0	0
$137$	16.5641	1.41517	0.707585	−	0.706628i	$-0.249785\pi$
$137$	16.5641	1.41517	0.707585	+	0.706628i	$0.249785\pi$
$138$	0	0
$139$	11.6558	0.988632	0.494316	−	0.869282i	$-0.335419\pi$
$139$	11.6558	0.988632	0.494316	+	0.869282i	$0.335419\pi$
$140$	0	0
$141$	36.7075	3.09133
$142$	0	0
$143$	1.64954	0.137942
$144$	0	0
$145$	−12.7081	−1.05535
$146$	0	0
$147$	−26.8224	−2.21227
$148$	0	0
$149$	11.6689	0.955956	0.477978	−	0.878372i	$-0.341370\pi$
$149$	11.6689	0.955956	0.477978	+	0.878372i	$0.341370\pi$
$150$	0	0
$151$	−19.5354	−1.58977	−0.794886	−	0.606759i	$-0.792469\pi$
$151$	−19.5354	−1.58977	−0.794886	+	0.606759i	$0.792469\pi$
$152$	0	0
$153$	−7.55931	−0.611134
$154$	0	0
$155$	14.4570	1.16121
$156$	0	0
$157$	−6.19012	−0.494025	−0.247013	−	0.969012i	$-0.579449\pi$
$157$	−6.19012	−0.494025	−0.247013	+	0.969012i	$0.579449\pi$
$158$	0	0
$159$	29.9429	2.37463
$160$	0	0
$161$	24.2842	1.91386
$162$	0	0
$163$	−2.24487	−0.175832	−0.0879159	−	0.996128i	$-0.528021\pi$
$163$	−2.24487	−0.175832	−0.0879159	+	0.996128i	$0.528021\pi$
$164$	0	0
$165$	36.1575	2.81486
$166$	0	0
$167$	−12.0477	−0.932282	−0.466141	−	0.884710i	$-0.654356\pi$
$167$	−12.0477	−0.932282	−0.466141	+	0.884710i	$0.654356\pi$
$168$	0	0
$169$	−12.7645	−0.981888
$170$	0	0
$171$	−27.9954	−2.14086
$172$	0	0
$173$	−2.66403	−0.202543	−0.101271	−	0.994859i	$-0.532291\pi$
$173$	−2.66403	−0.202543	−0.101271	+	0.994859i	$0.532291\pi$
$174$	0	0
$175$	22.3156	1.68690
$176$	0	0
$177$	3.24951	0.244248
$178$	0	0
$179$	−19.9484	−1.49101	−0.745507	−	0.666497i	$-0.767793\pi$
$179$	−19.9484	−1.49101	−0.745507	+	0.666497i	$0.767793\pi$
$180$	0	0
$181$	−0.601740	−0.0447270	−0.0223635	−	0.999750i	$-0.507119\pi$
$181$	−0.601740	−0.0447270	−0.0223635	+	0.999750i	$0.507119\pi$
$182$	0	0
$183$	33.9648	2.51075
$184$	0	0
$185$	32.9384	2.42168
$186$	0	0
$187$	−3.39948	−0.248594
$188$	0	0
$189$	−57.8645	−4.20902
$190$	0	0
$191$	−7.21241	−0.521872	−0.260936	−	0.965356i	$-0.584031\pi$
$191$	−7.21241	−0.521872	−0.260936	+	0.965356i	$0.584031\pi$
$192$	0	0
$193$	9.44119	0.679592	0.339796	−	0.940499i	$-0.389642\pi$
$193$	9.44119	0.679592	0.339796	+	0.940499i	$0.389642\pi$
$194$	0	0
$195$	5.16104	0.369590
$196$	0	0
$197$	20.7921	1.48138	0.740689	−	0.671848i	$-0.234499\pi$
$197$	20.7921	1.48138	0.740689	+	0.671848i	$0.234499\pi$
$198$	0	0
$199$	−3.31070	−0.234690	−0.117345	−	0.993091i	$-0.537438\pi$
$199$	−3.31070	−0.234690	−0.117345	+	0.993091i	$0.537438\pi$
$200$	0	0
$201$	−0.211275	−0.0149022
$202$	0	0
$203$	15.1638	1.06429
$204$	0	0
$205$	3.33332	0.232809
$206$	0	0
$207$	47.0013	3.26681
$208$	0	0
$209$	−12.5897	−0.870850
$210$	0	0
$211$	−6.41954	−0.441939	−0.220970	−	0.975281i	$-0.570922\pi$
$211$	−6.41954	−0.441939	−0.220970	+	0.975281i	$0.570922\pi$
$212$	0	0
$213$	−22.2119	−1.52193
$214$	0	0
$215$	−35.3719	−2.41234
$216$	0	0
$217$	−17.2506	−1.17105
$218$	0	0
$219$	−39.1482	−2.64539
$220$	0	0
$221$	−0.485234	−0.0326404
$222$	0	0
$223$	16.8176	1.12619	0.563093	−	0.826393i	$-0.309611\pi$
$223$	16.8176	1.12619	0.563093	+	0.826393i	$0.309611\pi$
$224$	0	0
$225$	43.1911	2.87940
$226$	0	0
$227$	−11.9033	−0.790047	−0.395023	−	0.918671i	$-0.629263\pi$
$227$	−11.9033	−0.790047	−0.395023	+	0.918671i	$0.629263\pi$
$228$	0	0
$229$	14.4389	0.954149	0.477074	−	0.878863i	$-0.341697\pi$
$229$	14.4389	0.954149	0.477074	+	0.878863i	$0.341697\pi$
$230$	0	0
$231$	−43.1445	−2.83870
$232$	0	0
$233$	−27.4209	−1.79640	−0.898202	−	0.439584i	$-0.855126\pi$
$233$	−27.4209	−1.79640	−0.898202	+	0.439584i	$0.855126\pi$
$234$	0	0
$235$	36.9747	2.41197
$236$	0	0
$237$	40.6226	2.63872
$238$	0	0
$239$	−12.1198	−0.783965	−0.391982	−	0.919973i	$-0.628211\pi$
$239$	−12.1198	−0.783965	−0.391982	+	0.919973i	$0.628211\pi$
$240$	0	0
$241$	−23.4014	−1.50741	−0.753707	−	0.657211i	$-0.771736\pi$
$241$	−23.4014	−1.50741	−0.753707	+	0.657211i	$0.771736\pi$
$242$	0	0
$243$	−38.3029	−2.45713
$244$	0	0
$245$	−27.0177	−1.72609
$246$	0	0
$247$	−1.79703	−0.114342
$248$	0	0
$249$	1.60403	0.101651
$250$	0	0
$251$	−21.3389	−1.34690	−0.673449	−	0.739234i	$-0.735188\pi$
$251$	−21.3389	−1.34690	−0.673449	+	0.739234i	$0.735188\pi$
$252$	0	0
$253$	21.1368	1.32886
$254$	0	0
$255$	−10.6362	−0.666064
$256$	0	0
$257$	22.6204	1.41102	0.705510	−	0.708700i	$-0.250718\pi$
$257$	22.6204	1.41102	0.705510	+	0.708700i	$0.250718\pi$
$258$	0	0
$259$	−39.3034	−2.44219
$260$	0	0
$261$	29.3490	1.81666
$262$	0	0
$263$	−8.05937	−0.496962	−0.248481	−	0.968637i	$-0.579931\pi$
$263$	−8.05937	−0.496962	−0.248481	+	0.968637i	$0.579931\pi$
$264$	0	0
$265$	30.1609	1.85277
$266$	0	0
$267$	−15.6331	−0.956728
$268$	0	0
$269$	25.9244	1.58064	0.790318	−	0.612697i	$-0.209915\pi$
$269$	25.9244	1.58064	0.790318	+	0.612697i	$0.209915\pi$
$270$	0	0
$271$	6.69681	0.406802	0.203401	−	0.979095i	$-0.434800\pi$
$271$	6.69681	0.406802	0.203401	+	0.979095i	$0.434800\pi$
$272$	0	0
$273$	−6.15835	−0.372721
$274$	0	0
$275$	19.4233	1.17127
$276$	0	0
$277$	4.39800	0.264250	0.132125	−	0.991233i	$-0.457820\pi$
$277$	4.39800	0.264250	0.132125	+	0.991233i	$0.457820\pi$
$278$	0	0
$279$	−33.3881	−1.99889
$280$	0	0
$281$	12.1510	0.724869	0.362435	−	0.932009i	$-0.381946\pi$
$281$	12.1510	0.724869	0.362435	+	0.932009i	$0.381946\pi$
$282$	0	0
$283$	−5.19026	−0.308529	−0.154264	−	0.988030i	$-0.549301\pi$
$283$	−5.19026	−0.308529	−0.154264	+	0.988030i	$0.549301\pi$
$284$	0	0
$285$	−39.3904	−2.33328
$286$	0	0
$287$	−3.97745	−0.234782
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−15.1982	−0.890932
$292$	0	0
$293$	−20.9446	−1.22360	−0.611799	−	0.791013i	$-0.709554\pi$
$293$	−20.9446	−1.22360	−0.611799	+	0.791013i	$0.709554\pi$
$294$	0	0
$295$	3.27317	0.190571
$296$	0	0
$297$	−50.3650	−2.92247
$298$	0	0
$299$	3.01702	0.174479
$300$	0	0
$301$	42.2071	2.43278
$302$	0	0
$303$	1.51632	0.0871103
$304$	0	0
$305$	34.2121	1.95898
$306$	0	0
$307$	−11.9091	−0.679689	−0.339844	−	0.940482i	$-0.610374\pi$
$307$	−11.9091	−0.679689	−0.339844	+	0.940482i	$0.610374\pi$
$308$	0	0
$309$	35.9274	2.04384
$310$	0	0
$311$	20.5688	1.16635	0.583174	−	0.812348i	$-0.301811\pi$
$311$	20.5688	1.16635	0.583174	+	0.812348i	$0.301811\pi$
$312$	0	0
$313$	−12.0134	−0.679039	−0.339519	−	0.940599i	$-0.610264\pi$
$313$	−12.0134	−0.679039	−0.339519	+	0.940599i	$0.610264\pi$
$314$	0	0
$315$	−96.6376	−5.44491
$316$	0	0
$317$	3.01385	0.169274	0.0846372	−	0.996412i	$-0.473027\pi$
$317$	3.01385	0.169274	0.0846372	+	0.996412i	$0.473027\pi$
$318$	0	0
$319$	13.1985	0.738973
$320$	0	0
$321$	30.6165	1.70885
$322$	0	0
$323$	3.70343	0.206064
$324$	0	0
$325$	2.77244	0.153788
$326$	0	0
$327$	6.88894	0.380959
$328$	0	0
$329$	−44.1197	−2.43240
$330$	0	0
$331$	−25.0259	−1.37555	−0.687774	−	0.725925i	$-0.741412\pi$
$331$	−25.0259	−1.37555	−0.687774	+	0.725925i	$0.741412\pi$
$332$	0	0
$333$	−76.0704	−4.16863
$334$	0	0
$335$	−0.212813	−0.0116272
$336$	0	0
$337$	22.9880	1.25224	0.626118	−	0.779728i	$-0.284643\pi$
$337$	22.9880	1.25224	0.626118	+	0.779728i	$0.284643\pi$
$338$	0	0
$339$	−20.5961	−1.11863
$340$	0	0
$341$	−15.0149	−0.813100
$342$	0	0
$343$	4.89882	0.264512
$344$	0	0
$345$	66.1323	3.56044
$346$	0	0
$347$	30.6131	1.64340	0.821699	−	0.569922i	$-0.193026\pi$
$347$	30.6131	1.64340	0.821699	+	0.569922i	$0.193026\pi$
$348$	0	0
$349$	−10.4253	−0.558052	−0.279026	−	0.960284i	$-0.590012\pi$
$349$	−10.4253	−0.558052	−0.279026	+	0.960284i	$0.590012\pi$
$350$	0	0
$351$	−7.18898	−0.383720
$352$	0	0
$353$	34.3238	1.82687	0.913435	−	0.406984i	$-0.133420\pi$
$353$	34.3238	1.82687	0.913435	+	0.406984i	$0.133420\pi$
$354$	0	0
$355$	−22.3736	−1.18747
$356$	0	0
$357$	12.6915	0.671706
$358$	0	0
$359$	−22.7419	−1.20027	−0.600135	−	0.799899i	$-0.704886\pi$
$359$	−22.7419	−1.20027	−0.600135	+	0.799899i	$0.704886\pi$
$360$	0	0
$361$	−5.28461	−0.278137
$362$	0	0
$363$	−1.80818	−0.0949050
$364$	0	0
$365$	−39.4332	−2.06403
$366$	0	0
$367$	−9.85462	−0.514407	−0.257203	−	0.966357i	$-0.582801\pi$
$367$	−9.85462	−0.514407	−0.257203	+	0.966357i	$0.582801\pi$
$368$	0	0
$369$	−7.69823	−0.400754
$370$	0	0
$371$	−35.9892	−1.86847
$372$	0	0
$373$	−9.13888	−0.473193	−0.236597	−	0.971608i	$-0.576032\pi$
$373$	−9.13888	−0.473193	−0.236597	+	0.971608i	$0.576032\pi$
$374$	0	0
$375$	7.59027	0.391960
$376$	0	0
$377$	1.88392	0.0970270
$378$	0	0
$379$	−2.44145	−0.125409	−0.0627044	−	0.998032i	$-0.519973\pi$
$379$	−2.44145	−0.125409	−0.0627044	+	0.998032i	$0.519973\pi$
$380$	0	0
$381$	−32.3476	−1.65722
$382$	0	0
$383$	−34.9624	−1.78650	−0.893248	−	0.449563i	$-0.851580\pi$
$383$	−34.9624	−1.78650	−0.893248	+	0.449563i	$0.851580\pi$
$384$	0	0
$385$	−43.4587	−2.21486
$386$	0	0
$387$	81.6905	4.15256
$388$	0	0
$389$	−14.4956	−0.734957	−0.367479	−	0.930032i	$-0.619779\pi$
$389$	−14.4956	−0.734957	−0.367479	+	0.930032i	$0.619779\pi$
$390$	0	0
$391$	−6.21767	−0.314441
$392$	0	0
$393$	−41.7349	−2.10525
$394$	0	0
$395$	40.9184	2.05883
$396$	0	0
$397$	5.90766	0.296497	0.148249	−	0.988950i	$-0.452636\pi$
$397$	5.90766	0.296497	0.148249	+	0.988950i	$0.452636\pi$
$398$	0	0
$399$	47.0022	2.35305
$400$	0	0
$401$	−35.3278	−1.76419	−0.882094	−	0.471074i	$-0.843866\pi$
$401$	−35.3278	−1.76419	−0.882094	+	0.471074i	$0.843866\pi$
$402$	0	0
$403$	−2.14319	−0.106760
$404$	0	0
$405$	−83.3519	−4.14179
$406$	0	0
$407$	−34.2094	−1.69570
$408$	0	0
$409$	−23.1393	−1.14416	−0.572082	−	0.820197i	$-0.693864\pi$
$409$	−23.1393	−1.14416	−0.572082	+	0.820197i	$0.693864\pi$
$410$	0	0
$411$	−53.8253	−2.65501
$412$	0	0
$413$	−3.90567	−0.192186
$414$	0	0
$415$	1.61571	0.0793121
$416$	0	0
$417$	−37.8756	−1.85478
$418$	0	0
$419$	7.38925	0.360988	0.180494	−	0.983576i	$-0.442230\pi$
$419$	7.38925	0.360988	0.180494	+	0.983576i	$0.442230\pi$
$420$	0	0
$421$	−14.8612	−0.724289	−0.362145	−	0.932122i	$-0.617955\pi$
$421$	−14.8612	−0.724289	−0.362145	+	0.932122i	$0.617955\pi$
$422$	0	0
$423$	−85.3923	−4.15192
$424$	0	0
$425$	−5.71363	−0.277152
$426$	0	0
$427$	−40.8232	−1.97557
$428$	0	0
$429$	−5.36020	−0.258793
$430$	0	0
$431$	32.6450	1.57245	0.786227	−	0.617938i	$-0.212032\pi$
$431$	32.6450	1.57245	0.786227	+	0.617938i	$0.212032\pi$
$432$	0	0
$433$	−28.4878	−1.36903	−0.684517	−	0.728997i	$-0.739987\pi$
$433$	−28.4878	−1.36903	−0.684517	+	0.728997i	$0.739987\pi$
$434$	0	0
$435$	41.2950	1.97995
$436$	0	0
$437$	−23.0267	−1.10152
$438$	0	0
$439$	8.24123	0.393332	0.196666	−	0.980471i	$-0.436988\pi$
$439$	8.24123	0.393332	0.196666	+	0.980471i	$0.436988\pi$
$440$	0	0
$441$	62.3967	2.97127
$442$	0	0
$443$	7.15433	0.339912	0.169956	−	0.985452i	$-0.445637\pi$
$443$	7.15433	0.339912	0.169956	+	0.985452i	$0.445637\pi$
$444$	0	0
$445$	−15.7469	−0.746474
$446$	0	0
$447$	−37.9183	−1.79347
$448$	0	0
$449$	17.6625	0.833546	0.416773	−	0.909011i	$-0.363161\pi$
$449$	17.6625	0.833546	0.416773	+	0.909011i	$0.363161\pi$
$450$	0	0
$451$	−3.46195	−0.163017
$452$	0	0
$453$	63.4806	2.98258
$454$	0	0
$455$	−6.20319	−0.290810
$456$	0	0
$457$	33.3504	1.56006	0.780032	−	0.625740i	$-0.215203\pi$
$457$	33.3504	1.56006	0.780032	+	0.625740i	$0.215203\pi$
$458$	0	0
$459$	14.8155	0.691529
$460$	0	0
$461$	29.7900	1.38746	0.693730	−	0.720236i	$-0.255966\pi$
$461$	29.7900	1.38746	0.693730	+	0.720236i	$0.255966\pi$
$462$	0	0
$463$	−34.1663	−1.58784	−0.793922	−	0.608020i	$-0.791964\pi$
$463$	−34.1663	−1.58784	−0.793922	+	0.608020i	$0.791964\pi$
$464$	0	0
$465$	−46.9781	−2.17856
$466$	0	0
$467$	34.1411	1.57986	0.789931	−	0.613196i	$-0.210116\pi$
$467$	34.1411	1.57986	0.789931	+	0.613196i	$0.210116\pi$
$468$	0	0
$469$	0.253937	0.0117257
$470$	0	0
$471$	20.1148	0.926843
$472$	0	0
$473$	36.7368	1.68916
$474$	0	0
$475$	−21.1600	−0.970888
$476$	0	0
$477$	−69.6559	−3.18933
$478$	0	0
$479$	−6.59355	−0.301267	−0.150633	−	0.988590i	$-0.548131\pi$
$479$	−6.59355	−0.301267	−0.150633	+	0.988590i	$0.548131\pi$
$480$	0	0
$481$	−4.88298	−0.222645
$482$	0	0
$483$	−78.9116	−3.59060
$484$	0	0
$485$	−15.3088	−0.695138
$486$	0	0
$487$	−28.4263	−1.28812	−0.644059	−	0.764976i	$-0.722751\pi$
$487$	−28.4263	−1.28812	−0.644059	+	0.764976i	$0.722751\pi$
$488$	0	0
$489$	7.29473	0.329879
$490$	0	0
$491$	−38.6046	−1.74220	−0.871100	−	0.491106i	$-0.836593\pi$
$491$	−38.6046	−1.74220	−0.871100	+	0.491106i	$0.836593\pi$
$492$	0	0
$493$	−3.88250	−0.174859
$494$	0	0
$495$	−84.1129	−3.78059
$496$	0	0
$497$	26.6971	1.19753
$498$	0	0
$499$	11.5655	0.517741	0.258870	−	0.965912i	$-0.416650\pi$
$499$	11.5655	0.517741	0.258870	+	0.965912i	$0.416650\pi$
$500$	0	0
$501$	39.1492	1.74906
$502$	0	0
$503$	−5.10352	−0.227555	−0.113777	−	0.993506i	$-0.536295\pi$
$503$	−5.10352	−0.227555	−0.113777	+	0.993506i	$0.536295\pi$
$504$	0	0
$505$	1.52736	0.0679666
$506$	0	0
$507$	41.4785	1.84213
$508$	0	0
$509$	−15.0021	−0.664957	−0.332478	−	0.943111i	$-0.607885\pi$
$509$	−15.0021	−0.664957	−0.332478	+	0.943111i	$0.607885\pi$
$510$	0	0
$511$	47.0533	2.08152
$512$	0	0
$513$	54.8682	2.42249
$514$	0	0
$515$	36.1890	1.59468
$516$	0	0
$517$	−38.4016	−1.68890
$518$	0	0
$519$	8.65679	0.379991
$520$	0	0
$521$	27.3367	1.19764	0.598820	−	0.800883i	$-0.295636\pi$
$521$	27.3367	1.19764	0.598820	+	0.800883i	$0.295636\pi$
$522$	0	0
$523$	−11.1570	−0.487863	−0.243932	−	0.969792i	$-0.578437\pi$
$523$	−11.1570	−0.487863	−0.243932	+	0.969792i	$0.578437\pi$
$524$	0	0
$525$	−72.5146	−3.16480
$526$	0	0
$527$	4.41681	0.192399
$528$	0	0
$529$	15.6594	0.680843
$530$	0	0
$531$	−7.55931	−0.328046
$532$	0	0
$533$	−0.494152	−0.0214041
$534$	0	0
$535$	30.8394	1.33330
$536$	0	0
$537$	64.8226	2.79730
$538$	0	0
$539$	28.0603	1.20864
$540$	0	0
$541$	17.1623	0.737863	0.368932	−	0.929457i	$-0.379724\pi$
$541$	17.1623	0.737863	0.368932	+	0.929457i	$0.379724\pi$
$542$	0	0
$543$	1.95536	0.0839125
$544$	0	0
$545$	6.93910	0.297238
$546$	0	0
$547$	−41.1503	−1.75946	−0.879730	−	0.475474i	$-0.842277\pi$
$547$	−41.1503	−1.75946	−0.879730	+	0.475474i	$0.842277\pi$
$548$	0	0
$549$	−79.0120	−3.37215
$550$	0	0
$551$	−14.3786	−0.612548
$552$	0	0
$553$	−48.8254	−2.07627
$554$	0	0
$555$	−107.034	−4.54332
$556$	0	0
$557$	−15.4273	−0.653674	−0.326837	−	0.945081i	$-0.605983\pi$
$557$	−15.4273	−0.653674	−0.326837	+	0.945081i	$0.605983\pi$
$558$	0	0
$559$	5.24373	0.221786
$560$	0	0
$561$	11.0466	0.466389
$562$	0	0
$563$	−0.443135	−0.0186759	−0.00933795	−	0.999956i	$-0.502972\pi$
$563$	−0.443135	−0.0186759	−0.00933795	+	0.999956i	$0.502972\pi$
$564$	0	0
$565$	−20.7461	−0.872794
$566$	0	0
$567$	99.4587	4.17687
$568$	0	0
$569$	−42.7667	−1.79287	−0.896436	−	0.443173i	$-0.853853\pi$
$569$	−42.7667	−1.79287	−0.896436	+	0.443173i	$0.853853\pi$
$570$	0	0
$571$	18.3309	0.767124	0.383562	−	0.923515i	$-0.374697\pi$
$571$	18.3309	0.767124	0.383562	+	0.923515i	$0.374697\pi$
$572$	0	0
$573$	23.4368	0.979086
$574$	0	0
$575$	35.5254	1.48151
$576$	0	0
$577$	−24.7504	−1.03037	−0.515186	−	0.857078i	$-0.672277\pi$
$577$	−24.7504	−1.03037	−0.515186	+	0.857078i	$0.672277\pi$
$578$	0	0
$579$	−30.6792	−1.27499
$580$	0	0
$581$	−1.92793	−0.0799840
$582$	0	0
$583$	−31.3248	−1.29734
$584$	0	0
$585$	−12.0061	−0.496390
$586$	0	0
$587$	−25.2350	−1.04156	−0.520781	−	0.853691i	$-0.674359\pi$
$587$	−25.2350	−1.04156	−0.520781	+	0.853691i	$0.674359\pi$
$588$	0	0
$589$	16.3574	0.673993
$590$	0	0
$591$	−67.5642	−2.77922
$592$	0	0
$593$	19.9986	0.821245	0.410622	−	0.911806i	$-0.365311\pi$
$593$	19.9986	0.821245	0.410622	+	0.911806i	$0.365311\pi$
$594$	0	0
$595$	12.7839	0.524090
$596$	0	0
$597$	10.7582	0.440302
$598$	0	0
$599$	34.3922	1.40523	0.702615	−	0.711571i	$-0.252016\pi$
$599$	34.3922	1.40523	0.702615	+	0.711571i	$0.252016\pi$
$600$	0	0
$601$	−9.24784	−0.377227	−0.188614	−	0.982051i	$-0.560399\pi$
$601$	−9.24784	−0.377227	−0.188614	+	0.982051i	$0.560399\pi$
$602$	0	0
$603$	0.491486	0.0200149
$604$	0	0
$605$	−1.82135	−0.0740483
$606$	0	0
$607$	17.4645	0.708861	0.354430	−	0.935082i	$-0.384675\pi$
$607$	17.4645	0.708861	0.354430	+	0.935082i	$0.384675\pi$
$608$	0	0
$609$	−49.2749	−1.99672
$610$	0	0
$611$	−5.48136	−0.221752
$612$	0	0
$613$	−14.0627	−0.567986	−0.283993	−	0.958826i	$-0.591659\pi$
$613$	−14.0627	−0.567986	−0.283993	+	0.958826i	$0.591659\pi$
$614$	0	0
$615$	−10.8317	−0.436775
$616$	0	0
$617$	13.9872	0.563103	0.281551	−	0.959546i	$-0.409151\pi$
$617$	13.9872	0.563103	0.281551	+	0.959546i	$0.409151\pi$
$618$	0	0
$619$	31.2197	1.25483	0.627413	−	0.778686i	$-0.284114\pi$
$619$	31.2197	1.25483	0.627413	+	0.778686i	$0.284114\pi$
$620$	0	0
$621$	−92.1179	−3.69656
$622$	0	0
$623$	18.7898	0.752798
$624$	0	0
$625$	−20.9226	−0.836904
$626$	0	0
$627$	40.9104	1.63381
$628$	0	0
$629$	10.0631	0.401244
$630$	0	0
$631$	−21.7763	−0.866899	−0.433450	−	0.901178i	$-0.642704\pi$
$631$	−21.7763	−0.866899	−0.433450	+	0.901178i	$0.642704\pi$
$632$	0	0
$633$	20.8603	0.829124
$634$	0	0
$635$	−32.5832	−1.29302
$636$	0	0
$637$	4.00526	0.158694
$638$	0	0
$639$	51.6713	2.04408
$640$	0	0
$641$	27.9602	1.10436	0.552181	−	0.833725i	$-0.313796\pi$
$641$	27.9602	1.10436	0.552181	+	0.833725i	$0.313796\pi$
$642$	0	0
$643$	−24.2376	−0.955838	−0.477919	−	0.878404i	$-0.658609\pi$
$643$	−24.2376	−0.955838	−0.477919	+	0.878404i	$0.658609\pi$
$644$	0	0
$645$	114.941	4.52580
$646$	0	0
$647$	−20.8486	−0.819643	−0.409822	−	0.912166i	$-0.634409\pi$
$647$	−20.8486	−0.819643	−0.409822	+	0.912166i	$0.634409\pi$
$648$	0	0
$649$	−3.39948	−0.133441
$650$	0	0
$651$	56.0561	2.19701
$652$	0	0
$653$	10.0708	0.394101	0.197050	−	0.980393i	$-0.436864\pi$
$653$	10.0708	0.394101	0.197050	+	0.980393i	$0.436864\pi$
$654$	0	0
$655$	−42.0387	−1.64259
$656$	0	0
$657$	91.0702	3.55299
$658$	0	0
$659$	−40.3746	−1.57277	−0.786386	−	0.617735i	$-0.788050\pi$
$659$	−40.3746	−1.57277	−0.786386	+	0.617735i	$0.788050\pi$
$660$	0	0
$661$	−16.2897	−0.633594	−0.316797	−	0.948493i	$-0.602607\pi$
$661$	−16.2897	−0.633594	−0.316797	+	0.948493i	$0.602607\pi$
$662$	0	0
$663$	1.57677	0.0612367
$664$	0	0
$665$	47.3444	1.83594
$666$	0	0
$667$	24.1401	0.934709
$668$	0	0
$669$	−54.6488	−2.11284
$670$	0	0
$671$	−35.5323	−1.37171
$672$	0	0
$673$	−16.4101	−0.632564	−0.316282	−	0.948665i	$-0.602435\pi$
$673$	−16.4101	−0.632564	−0.316282	+	0.948665i	$0.602435\pi$
$674$	0	0
$675$	−84.6503	−3.25819
$676$	0	0
$677$	34.1140	1.31111	0.655554	−	0.755148i	$-0.272435\pi$
$677$	34.1140	1.31111	0.655554	+	0.755148i	$0.272435\pi$
$678$	0	0
$679$	18.2671	0.701027
$680$	0	0
$681$	38.6797	1.48221
$682$	0	0
$683$	20.3349	0.778095	0.389048	−	0.921218i	$-0.372804\pi$
$683$	20.3349	0.778095	0.389048	+	0.921218i	$0.372804\pi$
$684$	0	0
$685$	−54.2172	−2.07153
$686$	0	0
$687$	−46.9193	−1.79008
$688$	0	0
$689$	−4.47123	−0.170340
$690$	0	0
$691$	−6.31432	−0.240208	−0.120104	−	0.992761i	$-0.538323\pi$
$691$	−6.31432	−0.240208	−0.120104	+	0.992761i	$0.538323\pi$
$692$	0	0
$693$	100.367	3.81262
$694$	0	0
$695$	−38.1514	−1.44716
$696$	0	0
$697$	1.01838	0.0385738
$698$	0	0
$699$	89.1045	3.37024
$700$	0	0
$701$	−5.90253	−0.222936	−0.111468	−	0.993768i	$-0.535555\pi$
$701$	−5.90253	−0.222936	−0.111468	+	0.993768i	$0.535555\pi$
$702$	0	0
$703$	37.2682	1.40560
$704$	0	0
$705$	−120.150	−4.52510
$706$	0	0
$707$	−1.82251	−0.0685424
$708$	0	0
$709$	7.40979	0.278281	0.139140	−	0.990273i	$-0.455566\pi$
$709$	7.40979	0.278281	0.139140	+	0.990273i	$0.455566\pi$
$710$	0	0
$711$	−94.5001	−3.54403
$712$	0	0
$713$	−27.4623	−1.02847
$714$	0	0
$715$	−5.39923	−0.201920
$716$	0	0
$717$	39.3834	1.47080
$718$	0	0
$719$	−6.39091	−0.238341	−0.119170	−	0.992874i	$-0.538023\pi$
$719$	−6.39091	−0.238341	−0.119170	+	0.992874i	$0.538023\pi$
$720$	0	0
$721$	−43.1821	−1.60819
$722$	0	0
$723$	76.0429	2.82807
$724$	0	0
$725$	22.1832	0.823863
$726$	0	0
$727$	51.9050	1.92505	0.962524	−	0.271196i	$-0.0874190\pi$
$727$	51.9050	1.92505	0.962524	+	0.271196i	$0.0874190\pi$
$728$	0	0
$729$	48.0699	1.78037
$730$	0	0
$731$	−10.8066	−0.399697
$732$	0	0
$733$	42.8155	1.58143	0.790713	−	0.612187i	$-0.209710\pi$
$733$	42.8155	1.58143	0.790713	+	0.612187i	$0.209710\pi$
$734$	0	0
$735$	87.7941	3.23833
$736$	0	0
$737$	0.221025	0.00814156
$738$	0	0
$739$	30.8483	1.13477	0.567387	−	0.823451i	$-0.307954\pi$
$739$	30.8483	1.13477	0.567387	+	0.823451i	$0.307954\pi$
$740$	0	0
$741$	5.83946	0.214518
$742$	0	0
$743$	−12.9551	−0.475277	−0.237638	−	0.971354i	$-0.576373\pi$
$743$	−12.9551	−0.475277	−0.237638	+	0.971354i	$0.576373\pi$
$744$	0	0
$745$	−38.1944	−1.39933
$746$	0	0
$747$	−3.73145	−0.136527
$748$	0	0
$749$	−36.7988	−1.34460
$750$	0	0
$751$	−0.866086	−0.0316039	−0.0158020	−	0.999875i	$-0.505030\pi$
$751$	−0.866086	−0.0316039	−0.0158020	+	0.999875i	$0.505030\pi$
$752$	0	0
$753$	69.3408	2.52692
$754$	0	0
$755$	63.9428	2.32712
$756$	0	0
$757$	27.0762	0.984103	0.492051	−	0.870566i	$-0.336247\pi$
$757$	27.0762	0.984103	0.492051	+	0.870566i	$0.336247\pi$
$758$	0	0
$759$	−68.6843	−2.49308
$760$	0	0
$761$	−9.88322	−0.358266	−0.179133	−	0.983825i	$-0.557329\pi$
$761$	−9.88322	−0.358266	−0.179133	+	0.983825i	$0.557329\pi$
$762$	0	0
$763$	−8.28001	−0.299756
$764$	0	0
$765$	24.7429	0.894581
$766$	0	0
$767$	−0.485234	−0.0175208
$768$	0	0
$769$	−40.2120	−1.45008	−0.725040	−	0.688706i	$-0.758179\pi$
$769$	−40.2120	−1.45008	−0.725040	+	0.688706i	$0.758179\pi$
$770$	0	0
$771$	−73.5051	−2.64722
$772$	0	0
$773$	20.2285	0.727568	0.363784	−	0.931483i	$-0.381485\pi$
$773$	20.2285	0.727568	0.363784	+	0.931483i	$0.381485\pi$
$774$	0	0
$775$	−25.2360	−0.906505
$776$	0	0
$777$	127.717	4.58181
$778$	0	0
$779$	3.77149	0.135128
$780$	0	0
$781$	23.2370	0.831484
$782$	0	0
$783$	−57.5213	−2.05564
$784$	0	0
$785$	20.2613	0.723156
$786$	0	0
$787$	10.1726	0.362615	0.181307	−	0.983426i	$-0.441967\pi$
$787$	10.1726	0.362615	0.181307	+	0.983426i	$0.441967\pi$
$788$	0	0
$789$	26.1890	0.932353
$790$	0	0
$791$	24.7550	0.880188
$792$	0	0
$793$	−5.07180	−0.180105
$794$	0	0
$795$	−98.0081	−3.47599
$796$	0	0
$797$	36.5236	1.29373	0.646866	−	0.762603i	$-0.276079\pi$
$797$	36.5236	1.29373	0.646866	+	0.762603i	$0.276079\pi$
$798$	0	0
$799$	11.2963	0.399635
$800$	0	0
$801$	36.3671	1.28497
$802$	0	0
$803$	40.9549	1.44527
$804$	0	0
$805$	−79.4862	−2.80152
$806$	0	0
$807$	−84.2414	−2.96544
$808$	0	0
$809$	−20.0498	−0.704912	−0.352456	−	0.935828i	$-0.614653\pi$
$809$	−20.0498	−0.704912	−0.352456	+	0.935828i	$0.614653\pi$
$810$	0	0
$811$	−8.67887	−0.304756	−0.152378	−	0.988322i	$-0.548693\pi$
$811$	−8.67887	−0.304756	−0.152378	+	0.988322i	$0.548693\pi$
$812$	0	0
$813$	−21.7613	−0.763204
$814$	0	0
$815$	7.34784	0.257384
$816$	0	0
$817$	−40.0215	−1.40018
$818$	0	0
$819$	14.3261	0.500596
$820$	0	0
$821$	11.5975	0.404755	0.202378	−	0.979308i	$-0.435133\pi$
$821$	11.5975	0.404755	0.202378	+	0.979308i	$0.435133\pi$
$822$	0	0
$823$	−41.3897	−1.44275	−0.721377	−	0.692543i	$-0.756490\pi$
$823$	−41.3897	−1.44275	−0.721377	+	0.692543i	$0.756490\pi$
$824$	0	0
$825$	−63.1163	−2.19743
$826$	0	0
$827$	19.1665	0.666484	0.333242	−	0.942841i	$-0.391857\pi$
$827$	19.1665	0.666484	0.333242	+	0.942841i	$0.391857\pi$
$828$	0	0
$829$	−14.0287	−0.487238	−0.243619	−	0.969871i	$-0.578335\pi$
$829$	−14.0287	−0.487238	−0.243619	+	0.969871i	$0.578335\pi$
$830$	0	0
$831$	−14.2913	−0.495761
$832$	0	0
$833$	−8.25428	−0.285994
$834$	0	0
$835$	39.4343	1.36468
$836$	0	0
$837$	65.4373	2.26185
$838$	0	0
$839$	12.1966	0.421072	0.210536	−	0.977586i	$-0.432479\pi$
$839$	12.1966	0.421072	0.210536	+	0.977586i	$0.432479\pi$
$840$	0	0
$841$	−13.9262	−0.480212
$842$	0	0
$843$	−39.4849	−1.35993
$844$	0	0
$845$	41.7805	1.43729
$846$	0	0
$847$	2.17330	0.0746756
$848$	0	0
$849$	16.8658	0.578833
$850$	0	0
$851$	−62.5693	−2.14485
$852$	0	0
$853$	−34.2167	−1.17156	−0.585779	−	0.810471i	$-0.699211\pi$
$853$	−34.2167	−1.17156	−0.585779	+	0.810471i	$0.699211\pi$
$854$	0	0
$855$	91.6335	3.13380
$856$	0	0
$857$	32.4959	1.11004	0.555019	−	0.831838i	$-0.312711\pi$
$857$	32.4959	1.11004	0.555019	+	0.831838i	$0.312711\pi$
$858$	0	0
$859$	−12.1451	−0.414387	−0.207194	−	0.978300i	$-0.566433\pi$
$859$	−12.1451	−0.414387	−0.207194	+	0.978300i	$0.566433\pi$
$860$	0	0
$861$	12.9248	0.440475
$862$	0	0
$863$	−0.775147	−0.0263863	−0.0131931	−	0.999913i	$-0.504200\pi$
$863$	−0.775147	−0.0263863	−0.0131931	+	0.999913i	$0.504200\pi$
$864$	0	0
$865$	8.71982	0.296483
$866$	0	0
$867$	−3.24951	−0.110359
$868$	0	0
$869$	−42.4974	−1.44163
$870$	0	0
$871$	0.0315486	0.00106898
$872$	0	0
$873$	35.3554	1.19660
$874$	0	0
$875$	−9.12295	−0.308412
$876$	0	0
$877$	−32.2510	−1.08904	−0.544520	−	0.838748i	$-0.683288\pi$
$877$	−32.2510	−1.08904	−0.544520	+	0.838748i	$0.683288\pi$
$878$	0	0
$879$	68.0598	2.29560
$880$	0	0
$881$	13.4840	0.454287	0.227143	−	0.973861i	$-0.427061\pi$
$881$	13.4840	0.454287	0.227143	+	0.973861i	$0.427061\pi$
$882$	0	0
$883$	31.3597	1.05534	0.527669	−	0.849450i	$-0.323066\pi$
$883$	31.3597	1.05534	0.527669	+	0.849450i	$0.323066\pi$
$884$	0	0
$885$	−10.6362	−0.357532
$886$	0	0
$887$	−19.4434	−0.652846	−0.326423	−	0.945224i	$-0.605843\pi$
$887$	−19.4434	−0.652846	−0.326423	+	0.945224i	$0.605843\pi$
$888$	0	0
$889$	38.8795	1.30398
$890$	0	0
$891$	86.5684	2.90015
$892$	0	0
$893$	41.8351	1.39996
$894$	0	0
$895$	65.2945	2.18256
$896$	0	0
$897$	−9.80384	−0.327341
$898$	0	0
$899$	−17.1483	−0.571928
$900$	0	0
$901$	9.21459	0.306983
$902$	0	0
$903$	−137.152	−4.56414
$904$	0	0
$905$	1.96960	0.0654716
$906$	0	0
$907$	−32.8247	−1.08993	−0.544963	−	0.838460i	$-0.683456\pi$
$907$	−32.8247	−1.08993	−0.544963	+	0.838460i	$0.683456\pi$
$908$	0	0
$909$	−3.52740	−0.116997
$910$	0	0
$911$	50.5267	1.67402	0.837012	−	0.547184i	$-0.184300\pi$
$911$	50.5267	1.67402	0.837012	+	0.547184i	$0.184300\pi$
$912$	0	0
$913$	−1.67806	−0.0555357
$914$	0	0
$915$	−111.172	−3.67525
$916$	0	0
$917$	50.1623	1.65651
$918$	0	0
$919$	−25.2507	−0.832944	−0.416472	−	0.909148i	$-0.636734\pi$
$919$	−25.2507	−0.832944	−0.416472	+	0.909148i	$0.636734\pi$
$920$	0	0
$921$	38.6987	1.27517
$922$	0	0
$923$	3.31679	0.109174
$924$	0	0
$925$	−57.4971	−1.89049
$926$	0	0
$927$	−83.5777	−2.74505
$928$	0	0
$929$	17.0317	0.558793	0.279396	−	0.960176i	$-0.409866\pi$
$929$	17.0317	0.558793	0.279396	+	0.960176i	$0.409866\pi$
$930$	0	0
$931$	−30.5692	−1.00186
$932$	0	0
$933$	−66.8384	−2.18819
$934$	0	0
$935$	11.1271	0.363894
$936$	0	0
$937$	−22.4932	−0.734822	−0.367411	−	0.930059i	$-0.619756\pi$
$937$	−22.4932	−0.734822	−0.367411	+	0.930059i	$0.619756\pi$
$938$	0	0
$939$	39.0377	1.27395
$940$	0	0
$941$	2.17649	0.0709516	0.0354758	−	0.999371i	$-0.488705\pi$
$941$	2.17649	0.0709516	0.0354758	+	0.999371i	$0.488705\pi$
$942$	0	0
$943$	−6.33194	−0.206196
$944$	0	0
$945$	189.400	6.16119
$946$	0	0
$947$	8.36870	0.271946	0.135973	−	0.990713i	$-0.456584\pi$
$947$	8.36870	0.271946	0.135973	+	0.990713i	$0.456584\pi$
$948$	0	0
$949$	5.84582	0.189763
$950$	0	0
$951$	−9.79352	−0.317577
$952$	0	0
$953$	33.0999	1.07221	0.536105	−	0.844151i	$-0.319895\pi$
$953$	33.0999	1.07221	0.536105	+	0.844151i	$0.319895\pi$
$954$	0	0
$955$	23.6074	0.763918
$956$	0	0
$957$	−42.8886	−1.38639
$958$	0	0
$959$	64.6941	2.08908
$960$	0	0
$961$	−11.4917	−0.370702
$962$	0	0
$963$	−71.2229	−2.29513
$964$	0	0
$965$	−30.9026	−0.994790
$966$	0	0
$967$	−60.9480	−1.95996	−0.979979	−	0.199103i	$-0.936197\pi$
$967$	−60.9480	−1.95996	−0.979979	+	0.199103i	$0.936197\pi$
$968$	0	0
$969$	−12.0343	−0.386598
$970$	0	0
$971$	18.2388	0.585311	0.292656	−	0.956218i	$-0.405461\pi$
$971$	18.2388	0.585311	0.292656	+	0.956218i	$0.405461\pi$
$972$	0	0
$973$	45.5238	1.45942
$974$	0	0
$975$	−9.00908	−0.288522
$976$	0	0
$977$	19.5927	0.626828	0.313414	−	0.949617i	$-0.398527\pi$
$977$	19.5927	0.626828	0.313414	+	0.949617i	$0.398527\pi$
$978$	0	0
$979$	16.3545	0.522694
$980$	0	0
$981$	−16.0257	−0.511661
$982$	0	0
$983$	−44.1384	−1.40780	−0.703898	−	0.710301i	$-0.748559\pi$
$983$	−44.1384	−1.40780	−0.703898	+	0.710301i	$0.748559\pi$
$984$	0	0
$985$	−68.0561	−2.16845
$986$	0	0
$987$	143.367	4.56344
$988$	0	0
$989$	67.1919	2.13658
$990$	0	0
$991$	−34.0662	−1.08215	−0.541074	−	0.840975i	$-0.681982\pi$
$991$	−34.0662	−1.08215	−0.541074	+	0.840975i	$0.681982\pi$
$992$	0	0
$993$	81.3219	2.58067
$994$	0	0
$995$	10.8365	0.343540
$996$	0	0
$997$	28.3098	0.896579	0.448289	−	0.893888i	$-0.352033\pi$
$997$	28.3098	0.896579	0.448289	+	0.893888i	$0.352033\pi$
$998$	0	0
$999$	149.091	4.71702

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.1	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.1	✓	23	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-3.24951 q^{3} -3.27317 q^{5} +3.90567 q^{7} +7.55931 q^{9} +O(q^{10})\)	\(q-3.24951 q^{3} -3.27317 q^{5} +3.90567 q^{7} +7.55931 q^{9} +3.39948 q^{11} +0.485234 q^{13} +10.6362 q^{15} -1.00000 q^{17} -3.70343 q^{19} -12.6915 q^{21} +6.21767 q^{23} +5.71363 q^{25} -14.8155 q^{27} +3.88250 q^{29} -4.41681 q^{31} -11.0466 q^{33} -12.7839 q^{35} -10.0631 q^{37} -1.57677 q^{39} -1.01838 q^{41} +10.8066 q^{43} -24.7429 q^{45} -11.2963 q^{47} +8.25428 q^{49} +3.24951 q^{51} -9.21459 q^{53} -11.1271 q^{55} +12.0343 q^{57} -1.00000 q^{59} -10.4523 q^{61} +29.5242 q^{63} -1.58825 q^{65} +0.0650174 q^{67} -20.2044 q^{69} +6.83545 q^{71} +12.0474 q^{73} -18.5665 q^{75} +13.2772 q^{77} -12.5012 q^{79} +25.4652 q^{81} -0.493623 q^{83} +3.27317 q^{85} -12.6162 q^{87} +4.81090 q^{89} +1.89516 q^{91} +14.3525 q^{93} +12.1219 q^{95} +4.67707 q^{97} +25.6977 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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