LMFDB - Embedded newform 4012.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52360 q^{3} -3.46979 q^{5} +1.96229 q^{7} +3.36854 q^{9} +O(q^{10})$	$q-2.52360 q^{3} -3.46979 q^{5} +1.96229 q^{7} +3.36854 q^{9} +1.48545 q^{11} +5.95205 q^{13} +8.75635 q^{15} +1.00000 q^{17} -3.32813 q^{19} -4.95203 q^{21} +6.67517 q^{23} +7.03944 q^{25} -0.930045 q^{27} +4.76054 q^{29} -0.104974 q^{31} -3.74869 q^{33} -6.80873 q^{35} -10.5436 q^{37} -15.0206 q^{39} -5.65382 q^{41} -10.0816 q^{43} -11.6881 q^{45} +6.74139 q^{47} -3.14942 q^{49} -2.52360 q^{51} -3.21194 q^{53} -5.15421 q^{55} +8.39885 q^{57} +1.00000 q^{59} +12.2417 q^{61} +6.61005 q^{63} -20.6524 q^{65} -11.8520 q^{67} -16.8454 q^{69} +12.9536 q^{71} +2.55556 q^{73} -17.7647 q^{75} +2.91489 q^{77} +7.41722 q^{79} -7.75856 q^{81} -4.50032 q^{83} -3.46979 q^{85} -12.0137 q^{87} +11.7305 q^{89} +11.6797 q^{91} +0.264912 q^{93} +11.5479 q^{95} +4.10295 q^{97} +5.00381 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * 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
$3$	−2.52360	−1.45700	−0.728500	−	0.685046i	$-0.759782\pi$
$3$	−2.52360	−1.45700	−0.728500	+	0.685046i	$0.759782\pi$
$4$	0	0
$5$	−3.46979	−1.55174	−0.775869	−	0.630895i	$-0.782688\pi$
$5$	−3.46979	−1.55174	−0.775869	+	0.630895i	$0.782688\pi$
$6$	0	0
$7$	1.96229	0.741676	0.370838	−	0.928698i	$-0.379070\pi$
$7$	1.96229	0.741676	0.370838	+	0.928698i	$0.379070\pi$
$8$	0	0
$9$	3.36854	1.12285
$10$	0	0
$11$	1.48545	0.447881	0.223941	−	0.974603i	$-0.428108\pi$
$11$	1.48545	0.447881	0.223941	+	0.974603i	$0.428108\pi$
$12$	0	0
$13$	5.95205	1.65080	0.825401	−	0.564547i	$-0.190949\pi$
$13$	5.95205	1.65080	0.825401	+	0.564547i	$0.190949\pi$
$14$	0	0
$15$	8.75635	2.26088
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−3.32813	−0.763524	−0.381762	−	0.924261i	$-0.624683\pi$
$19$	−3.32813	−0.763524	−0.381762	+	0.924261i	$0.624683\pi$
$20$	0	0
$21$	−4.95203	−1.08062
$22$	0	0
$23$	6.67517	1.39187	0.695934	−	0.718105i	$-0.254990\pi$
$23$	6.67517	1.39187	0.695934	+	0.718105i	$0.254990\pi$
$24$	0	0
$25$	7.03944	1.40789
$26$	0	0
$27$	−0.930045	−0.178987
$28$	0	0
$29$	4.76054	0.884009	0.442005	−	0.897013i	$-0.354267\pi$
$29$	4.76054	0.884009	0.442005	+	0.897013i	$0.354267\pi$
$30$	0	0
$31$	−0.104974	−0.0188539	−0.00942694	−	0.999956i	$-0.503001\pi$
$31$	−0.104974	−0.0188539	−0.00942694	+	0.999956i	$0.503001\pi$
$32$	0	0
$33$	−3.74869	−0.652563
$34$	0	0
$35$	−6.80873	−1.15089
$36$	0	0
$37$	−10.5436	−1.73335	−0.866676	−	0.498871i	$-0.833748\pi$
$37$	−10.5436	−1.73335	−0.866676	+	0.498871i	$0.833748\pi$
$38$	0	0
$39$	−15.0206	−2.40522
$40$	0	0
$41$	−5.65382	−0.882978	−0.441489	−	0.897267i	$-0.645550\pi$
$41$	−5.65382	−0.882978	−0.441489	+	0.897267i	$0.645550\pi$
$42$	0	0
$43$	−10.0816	−1.53743	−0.768713	−	0.639594i	$-0.779102\pi$
$43$	−10.0816	−1.53743	−0.768713	+	0.639594i	$0.779102\pi$
$44$	0	0
$45$	−11.6881	−1.74236
$46$	0	0
$47$	6.74139	0.983333	0.491666	−	0.870784i	$-0.336388\pi$
$47$	6.74139	0.983333	0.491666	+	0.870784i	$0.336388\pi$
$48$	0	0
$49$	−3.14942	−0.449917
$50$	0	0
$51$	−2.52360	−0.353374
$52$	0	0
$53$	−3.21194	−0.441194	−0.220597	−	0.975365i	$-0.570800\pi$
$53$	−3.21194	−0.441194	−0.220597	+	0.975365i	$0.570800\pi$
$54$	0	0
$55$	−5.15421	−0.694994
$56$	0	0
$57$	8.39885	1.11245
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	12.2417	1.56739	0.783694	−	0.621147i	$-0.213333\pi$
$61$	12.2417	1.56739	0.783694	+	0.621147i	$0.213333\pi$
$62$	0	0
$63$	6.61005	0.832788
$64$	0	0
$65$	−20.6524	−2.56161
$66$	0	0
$67$	−11.8520	−1.44796	−0.723978	−	0.689823i	$-0.757688\pi$
$67$	−11.8520	−1.44796	−0.723978	+	0.689823i	$0.757688\pi$
$68$	0	0
$69$	−16.8454	−2.02795
$70$	0	0
$71$	12.9536	1.53731	0.768654	−	0.639664i	$-0.220927\pi$
$71$	12.9536	1.53731	0.768654	+	0.639664i	$0.220927\pi$
$72$	0	0
$73$	2.55556	0.299106	0.149553	−	0.988754i	$-0.452217\pi$
$73$	2.55556	0.299106	0.149553	+	0.988754i	$0.452217\pi$
$74$	0	0
$75$	−17.7647	−2.05129
$76$	0	0
$77$	2.91489	0.332183
$78$	0	0
$79$	7.41722	0.834502	0.417251	−	0.908791i	$-0.362994\pi$
$79$	7.41722	0.834502	0.417251	+	0.908791i	$0.362994\pi$
$80$	0	0
$81$	−7.75856	−0.862062
$82$	0	0
$83$	−4.50032	−0.493974	−0.246987	−	0.969019i	$-0.579440\pi$
$83$	−4.50032	−0.493974	−0.246987	+	0.969019i	$0.579440\pi$
$84$	0	0
$85$	−3.46979	−0.376352
$86$	0	0
$87$	−12.0137	−1.28800
$88$	0	0
$89$	11.7305	1.24343	0.621713	−	0.783245i	$-0.286437\pi$
$89$	11.7305	1.24343	0.621713	+	0.783245i	$0.286437\pi$
$90$	0	0
$91$	11.6797	1.22436
$92$	0	0
$93$	0.264912	0.0274701
$94$	0	0
$95$	11.5479	1.18479
$96$	0	0
$97$	4.10295	0.416591	0.208296	−	0.978066i	$-0.433208\pi$
$97$	4.10295	0.416591	0.208296	+	0.978066i	$0.433208\pi$
$98$	0	0
$99$	5.00381	0.502902
$100$	0	0
$101$	5.97892	0.594924	0.297462	−	0.954734i	$-0.403860\pi$
$101$	5.97892	0.594924	0.297462	+	0.954734i	$0.403860\pi$
$102$	0	0
$103$	−4.99769	−0.492437	−0.246219	−	0.969214i	$-0.579188\pi$
$103$	−4.99769	−0.492437	−0.246219	+	0.969214i	$0.579188\pi$
$104$	0	0
$105$	17.1825	1.67684
$106$	0	0
$107$	3.40456	0.329131	0.164566	−	0.986366i	$-0.447378\pi$
$107$	3.40456	0.329131	0.164566	+	0.986366i	$0.447378\pi$
$108$	0	0
$109$	3.29708	0.315803	0.157902	−	0.987455i	$-0.449527\pi$
$109$	3.29708	0.315803	0.157902	+	0.987455i	$0.449527\pi$
$110$	0	0
$111$	26.6077	2.52549
$112$	0	0
$113$	5.52954	0.520176	0.260088	−	0.965585i	$-0.416248\pi$
$113$	5.52954	0.520176	0.260088	+	0.965585i	$0.416248\pi$
$114$	0	0
$115$	−23.1614	−2.15981
$116$	0	0
$117$	20.0497	1.85360
$118$	0	0
$119$	1.96229	0.179883
$120$	0	0
$121$	−8.79342	−0.799402
$122$	0	0
$123$	14.2680	1.28650
$124$	0	0
$125$	−7.07642	−0.632935
$126$	0	0
$127$	−0.230808	−0.0204809	−0.0102405	−	0.999948i	$-0.503260\pi$
$127$	−0.230808	−0.0204809	−0.0102405	+	0.999948i	$0.503260\pi$
$128$	0	0
$129$	25.4418	2.24003
$130$	0	0
$131$	6.58469	0.575307	0.287654	−	0.957735i	$-0.407125\pi$
$131$	6.58469	0.575307	0.287654	+	0.957735i	$0.407125\pi$
$132$	0	0
$133$	−6.53075	−0.566288
$134$	0	0
$135$	3.22706	0.277741
$136$	0	0
$137$	−22.5209	−1.92409	−0.962047	−	0.272884i	$-0.912023\pi$
$137$	−22.5209	−1.92409	−0.962047	+	0.272884i	$0.912023\pi$
$138$	0	0
$139$	−10.6871	−0.906465	−0.453232	−	0.891392i	$-0.649729\pi$
$139$	−10.6871	−0.906465	−0.453232	+	0.891392i	$0.649729\pi$
$140$	0	0
$141$	−17.0125	−1.43271
$142$	0	0
$143$	8.84150	0.739363
$144$	0	0
$145$	−16.5181	−1.37175
$146$	0	0
$147$	7.94786	0.655528
$148$	0	0
$149$	−17.9204	−1.46810	−0.734049	−	0.679097i	$-0.762372\pi$
$149$	−17.9204	−1.46810	−0.734049	+	0.679097i	$0.762372\pi$
$150$	0	0
$151$	−14.0540	−1.14370	−0.571851	−	0.820358i	$-0.693774\pi$
$151$	−14.0540	−1.14370	−0.571851	+	0.820358i	$0.693774\pi$
$152$	0	0
$153$	3.36854	0.272330
$154$	0	0
$155$	0.364238	0.0292563
$156$	0	0
$157$	4.44083	0.354417	0.177208	−	0.984173i	$-0.443293\pi$
$157$	4.44083	0.354417	0.177208	+	0.984173i	$0.443293\pi$
$158$	0	0
$159$	8.10563	0.642819
$160$	0	0
$161$	13.0986	1.03232
$162$	0	0
$163$	11.3356	0.887876	0.443938	−	0.896058i	$-0.353581\pi$
$163$	11.3356	0.887876	0.443938	+	0.896058i	$0.353581\pi$
$164$	0	0
$165$	13.0072	1.01261
$166$	0	0
$167$	14.9584	1.15752	0.578760	−	0.815498i	$-0.303537\pi$
$167$	14.9584	1.15752	0.578760	+	0.815498i	$0.303537\pi$
$168$	0	0
$169$	22.4269	1.72515
$170$	0	0
$171$	−11.2109	−0.857321
$172$	0	0
$173$	14.4121	1.09573	0.547865	−	0.836567i	$-0.315441\pi$
$173$	14.4121	1.09573	0.547865	+	0.836567i	$0.315441\pi$
$174$	0	0
$175$	13.8134	1.04420
$176$	0	0
$177$	−2.52360	−0.189685
$178$	0	0
$179$	−3.95418	−0.295550	−0.147775	−	0.989021i	$-0.547211\pi$
$179$	−3.95418	−0.295550	−0.147775	+	0.989021i	$0.547211\pi$
$180$	0	0
$181$	−3.56181	−0.264747	−0.132374	−	0.991200i	$-0.542260\pi$
$181$	−3.56181	−0.264747	−0.132374	+	0.991200i	$0.542260\pi$
$182$	0	0
$183$	−30.8931	−2.28368
$184$	0	0
$185$	36.5840	2.68971
$186$	0	0
$187$	1.48545	0.108627
$188$	0	0
$189$	−1.82502	−0.132751
$190$	0	0
$191$	−7.47956	−0.541202	−0.270601	−	0.962692i	$-0.587222\pi$
$191$	−7.47956	−0.541202	−0.270601	+	0.962692i	$0.587222\pi$
$192$	0	0
$193$	−0.867219	−0.0624238	−0.0312119	−	0.999513i	$-0.509937\pi$
$193$	−0.867219	−0.0624238	−0.0312119	+	0.999513i	$0.509937\pi$
$194$	0	0
$195$	52.1182	3.73226
$196$	0	0
$197$	5.48973	0.391127	0.195564	−	0.980691i	$-0.437346\pi$
$197$	5.48973	0.391127	0.195564	+	0.980691i	$0.437346\pi$
$198$	0	0
$199$	15.1889	1.07671	0.538357	−	0.842717i	$-0.319045\pi$
$199$	15.1889	1.07671	0.538357	+	0.842717i	$0.319045\pi$
$200$	0	0
$201$	29.9097	2.10967
$202$	0	0
$203$	9.34155	0.655649
$204$	0	0
$205$	19.6176	1.37015
$206$	0	0
$207$	22.4856	1.56285
$208$	0	0
$209$	−4.94378	−0.341968
$210$	0	0
$211$	19.8970	1.36977	0.684884	−	0.728652i	$-0.259853\pi$
$211$	19.8970	1.36977	0.684884	+	0.728652i	$0.259853\pi$
$212$	0	0
$213$	−32.6896	−2.23986
$214$	0	0
$215$	34.9809	2.38568
$216$	0	0
$217$	−0.205989	−0.0139835
$218$	0	0
$219$	−6.44920	−0.435797
$220$	0	0
$221$	5.95205	0.400378
$222$	0	0
$223$	19.8285	1.32782	0.663908	−	0.747815i	$-0.268897\pi$
$223$	19.8285	1.32782	0.663908	+	0.747815i	$0.268897\pi$
$224$	0	0
$225$	23.7126	1.58084
$226$	0	0
$227$	21.7530	1.44380	0.721899	−	0.691999i	$-0.243270\pi$
$227$	21.7530	1.44380	0.721899	+	0.691999i	$0.243270\pi$
$228$	0	0
$229$	−10.3402	−0.683298	−0.341649	−	0.939828i	$-0.610985\pi$
$229$	−10.3402	−0.683298	−0.341649	+	0.939828i	$0.610985\pi$
$230$	0	0
$231$	−7.35601	−0.483990
$232$	0	0
$233$	−2.20268	−0.144302	−0.0721511	−	0.997394i	$-0.522986\pi$
$233$	−2.20268	−0.144302	−0.0721511	+	0.997394i	$0.522986\pi$
$234$	0	0
$235$	−23.3912	−1.52587
$236$	0	0
$237$	−18.7181	−1.21587
$238$	0	0
$239$	26.4403	1.71028	0.855142	−	0.518394i	$-0.173470\pi$
$239$	26.4403	1.71028	0.855142	+	0.518394i	$0.173470\pi$
$240$	0	0
$241$	−19.6772	−1.26752	−0.633760	−	0.773529i	$-0.718489\pi$
$241$	−19.6772	−1.26752	−0.633760	+	0.773529i	$0.718489\pi$
$242$	0	0
$243$	22.3696	1.43501
$244$	0	0
$245$	10.9278	0.698152
$246$	0	0
$247$	−19.8092	−1.26043
$248$	0	0
$249$	11.3570	0.719719
$250$	0	0
$251$	13.0037	0.820785	0.410393	−	0.911909i	$-0.365392\pi$
$251$	13.0037	0.820785	0.410393	+	0.911909i	$0.365392\pi$
$252$	0	0
$253$	9.91566	0.623392
$254$	0	0
$255$	8.75635	0.548344
$256$	0	0
$257$	15.2426	0.950804	0.475402	−	0.879769i	$-0.342303\pi$
$257$	15.2426	0.950804	0.475402	+	0.879769i	$0.342303\pi$
$258$	0	0
$259$	−20.6895	−1.28559
$260$	0	0
$261$	16.0361	0.992607
$262$	0	0
$263$	22.2739	1.37347	0.686735	−	0.726908i	$-0.259043\pi$
$263$	22.2739	1.37347	0.686735	+	0.726908i	$0.259043\pi$
$264$	0	0
$265$	11.1447	0.684616
$266$	0	0
$267$	−29.6030	−1.81167
$268$	0	0
$269$	−15.3076	−0.933319	−0.466660	−	0.884437i	$-0.654543\pi$
$269$	−15.3076	−0.933319	−0.466660	+	0.884437i	$0.654543\pi$
$270$	0	0
$271$	28.0529	1.70410	0.852048	−	0.523464i	$-0.175361\pi$
$271$	28.0529	1.70410	0.852048	+	0.523464i	$0.175361\pi$
$272$	0	0
$273$	−29.4747	−1.78389
$274$	0	0
$275$	10.4568	0.630567
$276$	0	0
$277$	−10.6554	−0.640223	−0.320111	−	0.947380i	$-0.603720\pi$
$277$	−10.6554	−0.640223	−0.320111	+	0.947380i	$0.603720\pi$
$278$	0	0
$279$	−0.353609	−0.0211700
$280$	0	0
$281$	−22.0032	−1.31260	−0.656301	−	0.754499i	$-0.727880\pi$
$281$	−22.0032	−1.31260	−0.656301	+	0.754499i	$0.727880\pi$
$282$	0	0
$283$	−23.7525	−1.41194	−0.705970	−	0.708241i	$-0.749489\pi$
$283$	−23.7525	−1.41194	−0.705970	+	0.708241i	$0.749489\pi$
$284$	0	0
$285$	−29.1422	−1.72624
$286$	0	0
$287$	−11.0944	−0.654883
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−10.3542	−0.606973
$292$	0	0
$293$	4.27196	0.249571	0.124785	−	0.992184i	$-0.460176\pi$
$293$	4.27196	0.249571	0.124785	+	0.992184i	$0.460176\pi$
$294$	0	0
$295$	−3.46979	−0.202019
$296$	0	0
$297$	−1.38154	−0.0801651
$298$	0	0
$299$	39.7309	2.29770
$300$	0	0
$301$	−19.7830	−1.14027
$302$	0	0
$303$	−15.0884	−0.866804
$304$	0	0
$305$	−42.4761	−2.43218
$306$	0	0
$307$	−4.07404	−0.232518	−0.116259	−	0.993219i	$-0.537090\pi$
$307$	−4.07404	−0.232518	−0.116259	+	0.993219i	$0.537090\pi$
$308$	0	0
$309$	12.6122	0.717481
$310$	0	0
$311$	−14.2127	−0.805927	−0.402964	−	0.915216i	$-0.632020\pi$
$311$	−14.2127	−0.805927	−0.402964	+	0.915216i	$0.632020\pi$
$312$	0	0
$313$	−28.4155	−1.60614	−0.803069	−	0.595887i	$-0.796801\pi$
$313$	−28.4155	−1.60614	−0.803069	+	0.595887i	$0.796801\pi$
$314$	0	0
$315$	−22.9355	−1.29227
$316$	0	0
$317$	10.5466	0.592354	0.296177	−	0.955133i	$-0.404288\pi$
$317$	10.5466	0.592354	0.296177	+	0.955133i	$0.404288\pi$
$318$	0	0
$319$	7.07156	0.395931
$320$	0	0
$321$	−8.59173	−0.479544
$322$	0	0
$323$	−3.32813	−0.185182
$324$	0	0
$325$	41.8991	2.32414
$326$	0	0
$327$	−8.32050	−0.460125
$328$	0	0
$329$	13.2286	0.729314
$330$	0	0
$331$	12.6291	0.694156	0.347078	−	0.937836i	$-0.387174\pi$
$331$	12.6291	0.694156	0.347078	+	0.937836i	$0.387174\pi$
$332$	0	0
$333$	−35.5164	−1.94629
$334$	0	0
$335$	41.1240	2.24685
$336$	0	0
$337$	−3.00542	−0.163716	−0.0818578	−	0.996644i	$-0.526085\pi$
$337$	−3.00542	−0.163716	−0.0818578	+	0.996644i	$0.526085\pi$
$338$	0	0
$339$	−13.9543	−0.757896
$340$	0	0
$341$	−0.155934	−0.00844430
$342$	0	0
$343$	−19.9161	−1.07537
$344$	0	0
$345$	58.4501	3.14685
$346$	0	0
$347$	5.45247	0.292704	0.146352	−	0.989233i	$-0.453247\pi$
$347$	5.45247	0.292704	0.146352	+	0.989233i	$0.453247\pi$
$348$	0	0
$349$	28.5383	1.52762	0.763810	−	0.645441i	$-0.223327\pi$
$349$	28.5383	1.52762	0.763810	+	0.645441i	$0.223327\pi$
$350$	0	0
$351$	−5.53568	−0.295473
$352$	0	0
$353$	35.1955	1.87326	0.936632	−	0.350314i	$-0.113925\pi$
$353$	35.1955	1.87326	0.936632	+	0.350314i	$0.113925\pi$
$354$	0	0
$355$	−44.9462	−2.38550
$356$	0	0
$357$	−4.95203	−0.262089
$358$	0	0
$359$	16.1939	0.854682	0.427341	−	0.904091i	$-0.359450\pi$
$359$	16.1939	0.854682	0.427341	+	0.904091i	$0.359450\pi$
$360$	0	0
$361$	−7.92358	−0.417030
$362$	0	0
$363$	22.1911	1.16473
$364$	0	0
$365$	−8.86725	−0.464133
$366$	0	0
$367$	7.68569	0.401190	0.200595	−	0.979674i	$-0.435712\pi$
$367$	7.68569	0.401190	0.200595	+	0.979674i	$0.435712\pi$
$368$	0	0
$369$	−19.0451	−0.991448
$370$	0	0
$371$	−6.30275	−0.327223
$372$	0	0
$373$	−29.5352	−1.52927	−0.764637	−	0.644461i	$-0.777082\pi$
$373$	−29.5352	−1.52927	−0.764637	+	0.644461i	$0.777082\pi$
$374$	0	0
$375$	17.8580	0.922185
$376$	0	0
$377$	28.3350	1.45932
$378$	0	0
$379$	−6.54478	−0.336183	−0.168091	−	0.985771i	$-0.553760\pi$
$379$	−6.54478	−0.336183	−0.168091	+	0.985771i	$0.553760\pi$
$380$	0	0
$381$	0.582467	0.0298407
$382$	0	0
$383$	0.382545	0.0195472	0.00977358	−	0.999952i	$-0.496889\pi$
$383$	0.382545	0.0195472	0.00977358	+	0.999952i	$0.496889\pi$
$384$	0	0
$385$	−10.1141	−0.515461
$386$	0	0
$387$	−33.9602	−1.72629
$388$	0	0
$389$	14.2478	0.722391	0.361195	−	0.932490i	$-0.382369\pi$
$389$	14.2478	0.722391	0.361195	+	0.932490i	$0.382369\pi$
$390$	0	0
$391$	6.67517	0.337578
$392$	0	0
$393$	−16.6171	−0.838222
$394$	0	0
$395$	−25.7362	−1.29493
$396$	0	0
$397$	−9.13240	−0.458342	−0.229171	−	0.973386i	$-0.573601\pi$
$397$	−9.13240	−0.458342	−0.229171	+	0.973386i	$0.573601\pi$
$398$	0	0
$399$	16.4810	0.825081
$400$	0	0
$401$	−31.6130	−1.57868	−0.789338	−	0.613959i	$-0.789576\pi$
$401$	−31.6130	−1.57868	−0.789338	+	0.613959i	$0.789576\pi$
$402$	0	0
$403$	−0.624810	−0.0311240
$404$	0	0
$405$	26.9206	1.33769
$406$	0	0
$407$	−15.6620	−0.776336
$408$	0	0
$409$	4.33278	0.214242	0.107121	−	0.994246i	$-0.465837\pi$
$409$	4.33278	0.214242	0.107121	+	0.994246i	$0.465837\pi$
$410$	0	0
$411$	56.8338	2.80340
$412$	0	0
$413$	1.96229	0.0965580
$414$	0	0
$415$	15.6151	0.766518
$416$	0	0
$417$	26.9698	1.32072
$418$	0	0
$419$	−23.3872	−1.14254	−0.571270	−	0.820762i	$-0.693549\pi$
$419$	−23.3872	−1.14254	−0.571270	+	0.820762i	$0.693549\pi$
$420$	0	0
$421$	−3.25973	−0.158869	−0.0794347	−	0.996840i	$-0.525312\pi$
$421$	−3.25973	−0.158869	−0.0794347	+	0.996840i	$0.525312\pi$
$422$	0	0
$423$	22.7086	1.10413
$424$	0	0
$425$	7.03944	0.341463
$426$	0	0
$427$	24.0218	1.16249
$428$	0	0
$429$	−22.3124	−1.07725
$430$	0	0
$431$	−5.87411	−0.282946	−0.141473	−	0.989942i	$-0.545184\pi$
$431$	−5.87411	−0.282946	−0.141473	+	0.989942i	$0.545184\pi$
$432$	0	0
$433$	17.1656	0.824928	0.412464	−	0.910974i	$-0.364668\pi$
$433$	17.1656	0.824928	0.412464	+	0.910974i	$0.364668\pi$
$434$	0	0
$435$	41.6849	1.99864
$436$	0	0
$437$	−22.2158	−1.06273
$438$	0	0
$439$	9.45779	0.451396	0.225698	−	0.974197i	$-0.427534\pi$
$439$	9.45779	0.451396	0.225698	+	0.974197i	$0.427534\pi$
$440$	0	0
$441$	−10.6089	−0.505187
$442$	0	0
$443$	10.5884	0.503069	0.251534	−	0.967848i	$-0.419065\pi$
$443$	10.5884	0.503069	0.251534	+	0.967848i	$0.419065\pi$
$444$	0	0
$445$	−40.7022	−1.92947
$446$	0	0
$447$	45.2239	2.13902
$448$	0	0
$449$	7.25177	0.342232	0.171116	−	0.985251i	$-0.445263\pi$
$449$	7.25177	0.342232	0.171116	+	0.985251i	$0.445263\pi$
$450$	0	0
$451$	−8.39849	−0.395469
$452$	0	0
$453$	35.4667	1.66637
$454$	0	0
$455$	−40.5259	−1.89988
$456$	0	0
$457$	31.9153	1.49294	0.746468	−	0.665421i	$-0.231748\pi$
$457$	31.9153	1.49294	0.746468	+	0.665421i	$0.231748\pi$
$458$	0	0
$459$	−0.930045	−0.0434108
$460$	0	0
$461$	−22.8763	−1.06546	−0.532728	−	0.846287i	$-0.678833\pi$
$461$	−22.8763	−1.06546	−0.532728	+	0.846287i	$0.678833\pi$
$462$	0	0
$463$	24.5713	1.14193	0.570963	−	0.820976i	$-0.306570\pi$
$463$	24.5713	1.14193	0.570963	+	0.820976i	$0.306570\pi$
$464$	0	0
$465$	−0.919189	−0.0426264
$466$	0	0
$467$	18.3884	0.850913	0.425456	−	0.904979i	$-0.360114\pi$
$467$	18.3884	0.850913	0.425456	+	0.904979i	$0.360114\pi$
$468$	0	0
$469$	−23.2571	−1.07391
$470$	0	0
$471$	−11.2069	−0.516385
$472$	0	0
$473$	−14.9757	−0.688584
$474$	0	0
$475$	−23.4281	−1.07496
$476$	0	0
$477$	−10.8195	−0.495393
$478$	0	0
$479$	−5.42577	−0.247910	−0.123955	−	0.992288i	$-0.539558\pi$
$479$	−5.42577	−0.247910	−0.123955	+	0.992288i	$0.539558\pi$
$480$	0	0
$481$	−62.7558	−2.86142
$482$	0	0
$483$	−33.0556	−1.50408
$484$	0	0
$485$	−14.2364	−0.646440
$486$	0	0
$487$	19.1195	0.866389	0.433195	−	0.901300i	$-0.357386\pi$
$487$	19.1195	0.866389	0.433195	+	0.901300i	$0.357386\pi$
$488$	0	0
$489$	−28.6066	−1.29363
$490$	0	0
$491$	1.45665	0.0657375	0.0328688	−	0.999460i	$-0.489536\pi$
$491$	1.45665	0.0657375	0.0328688	+	0.999460i	$0.489536\pi$
$492$	0	0
$493$	4.76054	0.214404
$494$	0	0
$495$	−17.3622	−0.780372
$496$	0	0
$497$	25.4187	1.14019
$498$	0	0
$499$	29.3225	1.31265	0.656327	−	0.754476i	$-0.272109\pi$
$499$	29.3225	1.31265	0.656327	+	0.754476i	$0.272109\pi$
$500$	0	0
$501$	−37.7491	−1.68650
$502$	0	0
$503$	23.1485	1.03214	0.516070	−	0.856546i	$-0.327394\pi$
$503$	23.1485	1.03214	0.516070	+	0.856546i	$0.327394\pi$
$504$	0	0
$505$	−20.7456	−0.923166
$506$	0	0
$507$	−56.5965	−2.51354
$508$	0	0
$509$	−0.0354775	−0.00157251	−0.000786255	−	1.00000i	$-0.500250\pi$
$509$	−0.0354775	−0.00157251	−0.000786255		1.00000i	$0.500250\pi$
$510$	0	0
$511$	5.01475	0.221839
$512$	0	0
$513$	3.09531	0.136661
$514$	0	0
$515$	17.3409	0.764133
$516$	0	0
$517$	10.0140	0.440416
$518$	0	0
$519$	−36.3703	−1.59648
$520$	0	0
$521$	24.4904	1.07295	0.536473	−	0.843918i	$-0.319756\pi$
$521$	24.4904	1.07295	0.536473	+	0.843918i	$0.319756\pi$
$522$	0	0
$523$	35.9674	1.57275	0.786373	−	0.617752i	$-0.211956\pi$
$523$	35.9674	1.57275	0.786373	+	0.617752i	$0.211956\pi$
$524$	0	0
$525$	−34.8595	−1.52139
$526$	0	0
$527$	−0.104974	−0.00457274
$528$	0	0
$529$	21.5579	0.937298
$530$	0	0
$531$	3.36854	0.146182
$532$	0	0
$533$	−33.6518	−1.45762
$534$	0	0
$535$	−11.8131	−0.510725
$536$	0	0
$537$	9.97876	0.430615
$538$	0	0
$539$	−4.67831	−0.201509
$540$	0	0
$541$	39.5147	1.69887	0.849436	−	0.527692i	$-0.176943\pi$
$541$	39.5147	1.69887	0.849436	+	0.527692i	$0.176943\pi$
$542$	0	0
$543$	8.98857	0.385736
$544$	0	0
$545$	−11.4402	−0.490043
$546$	0	0
$547$	19.0955	0.816466	0.408233	−	0.912878i	$-0.366145\pi$
$547$	19.0955	0.816466	0.408233	+	0.912878i	$0.366145\pi$
$548$	0	0
$549$	41.2366	1.75994
$550$	0	0
$551$	−15.8437	−0.674963
$552$	0	0
$553$	14.5547	0.618930
$554$	0	0
$555$	−92.3232	−3.91890
$556$	0	0
$557$	−3.30079	−0.139859	−0.0699295	−	0.997552i	$-0.522277\pi$
$557$	−3.30079	−0.139859	−0.0699295	+	0.997552i	$0.522277\pi$
$558$	0	0
$559$	−60.0060	−2.53798
$560$	0	0
$561$	−3.74869	−0.158270
$562$	0	0
$563$	−14.2726	−0.601517	−0.300758	−	0.953700i	$-0.597240\pi$
$563$	−14.2726	−0.601517	−0.300758	+	0.953700i	$0.597240\pi$
$564$	0	0
$565$	−19.1864	−0.807176
$566$	0	0
$567$	−15.2245	−0.639371
$568$	0	0
$569$	−10.9037	−0.457105	−0.228552	−	0.973532i	$-0.573399\pi$
$569$	−10.9037	−0.457105	−0.228552	+	0.973532i	$0.573399\pi$
$570$	0	0
$571$	−24.4198	−1.02194	−0.510969	−	0.859599i	$-0.670713\pi$
$571$	−24.4198	−1.02194	−0.510969	+	0.859599i	$0.670713\pi$
$572$	0	0
$573$	18.8754	0.788531
$574$	0	0
$575$	46.9894	1.95959
$576$	0	0
$577$	20.0634	0.835249	0.417625	−	0.908620i	$-0.362863\pi$
$577$	20.0634	0.835249	0.417625	+	0.908620i	$0.362863\pi$
$578$	0	0
$579$	2.18851	0.0909514
$580$	0	0
$581$	−8.83093	−0.366369
$582$	0	0
$583$	−4.77119	−0.197602
$584$	0	0
$585$	−69.5683	−2.87630
$586$	0	0
$587$	20.9288	0.863824	0.431912	−	0.901916i	$-0.357839\pi$
$587$	20.9288	0.863824	0.431912	+	0.901916i	$0.357839\pi$
$588$	0	0
$589$	0.349367	0.0143954
$590$	0	0
$591$	−13.8539	−0.569872
$592$	0	0
$593$	43.1442	1.77172	0.885860	−	0.463953i	$-0.153569\pi$
$593$	43.1442	1.77172	0.885860	+	0.463953i	$0.153569\pi$
$594$	0	0
$595$	−6.80873	−0.279131
$596$	0	0
$597$	−38.3307	−1.56877
$598$	0	0
$599$	−25.3845	−1.03718	−0.518591	−	0.855022i	$-0.673543\pi$
$599$	−25.3845	−1.03718	−0.518591	+	0.855022i	$0.673543\pi$
$600$	0	0
$601$	−23.2137	−0.946907	−0.473453	−	0.880819i	$-0.656993\pi$
$601$	−23.2137	−0.946907	−0.473453	+	0.880819i	$0.656993\pi$
$602$	0	0
$603$	−39.9240	−1.62583
$604$	0	0
$605$	30.5113	1.24046
$606$	0	0
$607$	−12.8130	−0.520064	−0.260032	−	0.965600i	$-0.583733\pi$
$607$	−12.8130	−0.520064	−0.260032	+	0.965600i	$0.583733\pi$
$608$	0	0
$609$	−23.5743	−0.955279
$610$	0	0
$611$	40.1251	1.62329
$612$	0	0
$613$	−47.2926	−1.91013	−0.955066	−	0.296393i	$-0.904216\pi$
$613$	−47.2926	−1.91013	−0.955066	+	0.296393i	$0.904216\pi$
$614$	0	0
$615$	−49.5068	−1.99631
$616$	0	0
$617$	−37.6095	−1.51410	−0.757051	−	0.653356i	$-0.773361\pi$
$617$	−37.6095	−1.51410	−0.757051	+	0.653356i	$0.773361\pi$
$618$	0	0
$619$	−36.7120	−1.47558	−0.737790	−	0.675031i	$-0.764130\pi$
$619$	−36.7120	−1.47558	−0.737790	+	0.675031i	$0.764130\pi$
$620$	0	0
$621$	−6.20821	−0.249127
$622$	0	0
$623$	23.0186	0.922220
$624$	0	0
$625$	−10.6435	−0.425740
$626$	0	0
$627$	12.4761	0.498248
$628$	0	0
$629$	−10.5436	−0.420400
$630$	0	0
$631$	41.8636	1.66657	0.833283	−	0.552847i	$-0.186459\pi$
$631$	41.8636	1.66657	0.833283	+	0.552847i	$0.186459\pi$
$632$	0	0
$633$	−50.2121	−1.99575
$634$	0	0
$635$	0.800857	0.0317810
$636$	0	0
$637$	−18.7455	−0.742723
$638$	0	0
$639$	43.6347	1.72616
$640$	0	0
$641$	28.0350	1.10732	0.553658	−	0.832744i	$-0.313231\pi$
$641$	28.0350	1.10732	0.553658	+	0.832744i	$0.313231\pi$
$642$	0	0
$643$	47.4187	1.87001	0.935005	−	0.354635i	$-0.115395\pi$
$643$	47.4187	1.87001	0.935005	+	0.354635i	$0.115395\pi$
$644$	0	0
$645$	−88.2777	−3.47593
$646$	0	0
$647$	−21.9018	−0.861049	−0.430524	−	0.902579i	$-0.641671\pi$
$647$	−21.9018	−0.861049	−0.430524	+	0.902579i	$0.641671\pi$
$648$	0	0
$649$	1.48545	0.0583092
$650$	0	0
$651$	0.519834	0.0203739
$652$	0	0
$653$	24.9440	0.976134	0.488067	−	0.872806i	$-0.337702\pi$
$653$	24.9440	0.976134	0.488067	+	0.872806i	$0.337702\pi$
$654$	0	0
$655$	−22.8475	−0.892725
$656$	0	0
$657$	8.60850	0.335850
$658$	0	0
$659$	20.5506	0.800537	0.400268	−	0.916398i	$-0.368917\pi$
$659$	20.5506	0.800537	0.400268	+	0.916398i	$0.368917\pi$
$660$	0	0
$661$	22.2379	0.864955	0.432478	−	0.901645i	$-0.357639\pi$
$661$	22.2379	0.864955	0.432478	+	0.901645i	$0.357639\pi$
$662$	0	0
$663$	−15.0206	−0.583351
$664$	0	0
$665$	22.6603	0.878730
$666$	0	0
$667$	31.7774	1.23042
$668$	0	0
$669$	−50.0392	−1.93463
$670$	0	0
$671$	18.1845	0.702004
$672$	0	0
$673$	6.48388	0.249935	0.124968	−	0.992161i	$-0.460117\pi$
$673$	6.48388	0.249935	0.124968	+	0.992161i	$0.460117\pi$
$674$	0	0
$675$	−6.54700	−0.251994
$676$	0	0
$677$	−38.5733	−1.48249	−0.741247	−	0.671232i	$-0.765765\pi$
$677$	−38.5733	−1.48249	−0.741247	+	0.671232i	$0.765765\pi$
$678$	0	0
$679$	8.05118	0.308976
$680$	0	0
$681$	−54.8958	−2.10361
$682$	0	0
$683$	15.5328	0.594347	0.297174	−	0.954823i	$-0.403956\pi$
$683$	15.5328	0.594347	0.297174	+	0.954823i	$0.403956\pi$
$684$	0	0
$685$	78.1429	2.98569
$686$	0	0
$687$	26.0944	0.995565
$688$	0	0
$689$	−19.1176	−0.728323
$690$	0	0
$691$	−18.4544	−0.702038	−0.351019	−	0.936368i	$-0.614165\pi$
$691$	−18.4544	−0.702038	−0.351019	+	0.936368i	$0.614165\pi$
$692$	0	0
$693$	9.81893	0.372990
$694$	0	0
$695$	37.0819	1.40659
$696$	0	0
$697$	−5.65382	−0.214154
$698$	0	0
$699$	5.55867	0.210248
$700$	0	0
$701$	−47.3306	−1.78765	−0.893825	−	0.448416i	$-0.851988\pi$
$701$	−47.3306	−1.78765	−0.893825	+	0.448416i	$0.851988\pi$
$702$	0	0
$703$	35.0903	1.32346
$704$	0	0
$705$	59.0299	2.22320
$706$	0	0
$707$	11.7324	0.441241
$708$	0	0
$709$	1.40345	0.0527075	0.0263537	−	0.999653i	$-0.491610\pi$
$709$	1.40345	0.0527075	0.0263537	+	0.999653i	$0.491610\pi$
$710$	0	0
$711$	24.9852	0.937018
$712$	0	0
$713$	−0.700719	−0.0262421
$714$	0	0
$715$	−30.6781	−1.14730
$716$	0	0
$717$	−66.7248	−2.49188
$718$	0	0
$719$	3.94257	0.147033	0.0735165	−	0.997294i	$-0.476578\pi$
$719$	3.94257	0.147033	0.0735165	+	0.997294i	$0.476578\pi$
$720$	0	0
$721$	−9.80692	−0.365229
$722$	0	0
$723$	49.6574	1.84678
$724$	0	0
$725$	33.5115	1.24459
$726$	0	0
$727$	−45.0084	−1.66927	−0.834634	−	0.550805i	$-0.814321\pi$
$727$	−45.0084	−1.66927	−0.834634	+	0.550805i	$0.814321\pi$
$728$	0	0
$729$	−33.1762	−1.22875
$730$	0	0
$731$	−10.0816	−0.372880
$732$	0	0
$733$	30.1382	1.11318	0.556590	−	0.830787i	$-0.312109\pi$
$733$	30.1382	1.11318	0.556590	+	0.830787i	$0.312109\pi$
$734$	0	0
$735$	−27.5774	−1.01721
$736$	0	0
$737$	−17.6056	−0.648512
$738$	0	0
$739$	−0.322636	−0.0118684	−0.00593418	−	0.999982i	$-0.501889\pi$
$739$	−0.322636	−0.0118684	−0.00593418	+	0.999982i	$0.501889\pi$
$740$	0	0
$741$	49.9904	1.83644
$742$	0	0
$743$	11.9826	0.439598	0.219799	−	0.975545i	$-0.429460\pi$
$743$	11.9826	0.439598	0.219799	+	0.975545i	$0.429460\pi$
$744$	0	0
$745$	62.1801	2.27810
$746$	0	0
$747$	−15.1595	−0.554657
$748$	0	0
$749$	6.68073	0.244109
$750$	0	0
$751$	−15.5453	−0.567257	−0.283629	−	0.958934i	$-0.591538\pi$
$751$	−15.5453	−0.567257	−0.283629	+	0.958934i	$0.591538\pi$
$752$	0	0
$753$	−32.8160	−1.19588
$754$	0	0
$755$	48.7646	1.77472
$756$	0	0
$757$	−31.1153	−1.13091	−0.565453	−	0.824780i	$-0.691299\pi$
$757$	−31.1153	−1.13091	−0.565453	+	0.824780i	$0.691299\pi$
$758$	0	0
$759$	−25.0231	−0.908282
$760$	0	0
$761$	−31.9584	−1.15849	−0.579246	−	0.815153i	$-0.696653\pi$
$761$	−31.9584	−1.15849	−0.579246	+	0.815153i	$0.696653\pi$
$762$	0	0
$763$	6.46983	0.234224
$764$	0	0
$765$	−11.6881	−0.422585
$766$	0	0
$767$	5.95205	0.214916
$768$	0	0
$769$	−31.3766	−1.13147	−0.565735	−	0.824587i	$-0.691408\pi$
$769$	−31.3766	−1.13147	−0.565735	+	0.824587i	$0.691408\pi$
$770$	0	0
$771$	−38.4661	−1.38532
$772$	0	0
$773$	−2.39920	−0.0862933	−0.0431466	−	0.999069i	$-0.513738\pi$
$773$	−2.39920	−0.0862933	−0.0431466	+	0.999069i	$0.513738\pi$
$774$	0	0
$775$	−0.738958	−0.0265441
$776$	0	0
$777$	52.2121	1.87310
$778$	0	0
$779$	18.8166	0.674175
$780$	0	0
$781$	19.2420	0.688532
$782$	0	0
$783$	−4.42752	−0.158226
$784$	0	0
$785$	−15.4088	−0.549962
$786$	0	0
$787$	24.1768	0.861810	0.430905	−	0.902397i	$-0.358194\pi$
$787$	24.1768	0.861810	0.430905	+	0.902397i	$0.358194\pi$
$788$	0	0
$789$	−56.2105	−2.00115
$790$	0	0
$791$	10.8506	0.385802
$792$	0	0
$793$	72.8632	2.58745
$794$	0	0
$795$	−28.1248	−0.997485
$796$	0	0
$797$	54.9190	1.94533	0.972665	−	0.232212i	$-0.0745962\pi$
$797$	54.9190	1.94533	0.972665	+	0.232212i	$0.0745962\pi$
$798$	0	0
$799$	6.74139	0.238493
$800$	0	0
$801$	39.5145	1.39618
$802$	0	0
$803$	3.79617	0.133964
$804$	0	0
$805$	−45.4494	−1.60188
$806$	0	0
$807$	38.6301	1.35985
$808$	0	0
$809$	35.2767	1.24026	0.620131	−	0.784499i	$-0.287080\pi$
$809$	35.2767	1.24026	0.620131	+	0.784499i	$0.287080\pi$
$810$	0	0
$811$	34.4373	1.20926	0.604629	−	0.796507i	$-0.293321\pi$
$811$	34.4373	1.20926	0.604629	+	0.796507i	$0.293321\pi$
$812$	0	0
$813$	−70.7943	−2.48287
$814$	0	0
$815$	−39.3323	−1.37775
$816$	0	0
$817$	33.5527	1.17386
$818$	0	0
$819$	39.3434	1.37477
$820$	0	0
$821$	9.84802	0.343698	0.171849	−	0.985123i	$-0.445026\pi$
$821$	9.84802	0.343698	0.171849	+	0.985123i	$0.445026\pi$
$822$	0	0
$823$	34.4204	1.19982	0.599909	−	0.800068i	$-0.295203\pi$
$823$	34.4204	1.19982	0.599909	+	0.800068i	$0.295203\pi$
$824$	0	0
$825$	−26.3887	−0.918735
$826$	0	0
$827$	−34.5644	−1.20192	−0.600961	−	0.799279i	$-0.705215\pi$
$827$	−34.5644	−1.20192	−0.600961	+	0.799279i	$0.705215\pi$
$828$	0	0
$829$	−26.3454	−0.915014	−0.457507	−	0.889206i	$-0.651258\pi$
$829$	−26.3454	−0.915014	−0.457507	+	0.889206i	$0.651258\pi$
$830$	0	0
$831$	26.8900	0.932804
$832$	0	0
$833$	−3.14942	−0.109121
$834$	0	0
$835$	−51.9027	−1.79617
$836$	0	0
$837$	0.0976306	0.00337461
$838$	0	0
$839$	4.53419	0.156538	0.0782689	−	0.996932i	$-0.475061\pi$
$839$	4.53419	0.156538	0.0782689	+	0.996932i	$0.475061\pi$
$840$	0	0
$841$	−6.33729	−0.218527
$842$	0	0
$843$	55.5273	1.91246
$844$	0	0
$845$	−77.8166	−2.67697
$846$	0	0
$847$	−17.2553	−0.592898
$848$	0	0
$849$	59.9418	2.05720
$850$	0	0
$851$	−70.3801	−2.41260
$852$	0	0
$853$	31.1473	1.06646	0.533232	−	0.845969i	$-0.320977\pi$
$853$	31.1473	1.06646	0.533232	+	0.845969i	$0.320977\pi$
$854$	0	0
$855$	38.8995	1.33034
$856$	0	0
$857$	−30.9551	−1.05741	−0.528704	−	0.848806i	$-0.677322\pi$
$857$	−30.9551	−1.05741	−0.528704	+	0.848806i	$0.677322\pi$
$858$	0	0
$859$	−38.2768	−1.30599	−0.652994	−	0.757363i	$-0.726487\pi$
$859$	−38.2768	−1.30599	−0.652994	+	0.757363i	$0.726487\pi$
$860$	0	0
$861$	27.9979	0.954165
$862$	0	0
$863$	58.1505	1.97946	0.989732	−	0.142932i	$-0.0456532\pi$
$863$	58.1505	1.97946	0.989732	+	0.142932i	$0.0456532\pi$
$864$	0	0
$865$	−50.0069	−1.70028
$866$	0	0
$867$	−2.52360	−0.0857058
$868$	0	0
$869$	11.0179	0.373758
$870$	0	0
$871$	−70.5439	−2.39029
$872$	0	0
$873$	13.8210	0.467768
$874$	0	0
$875$	−13.8860	−0.469432
$876$	0	0
$877$	−11.5772	−0.390936	−0.195468	−	0.980710i	$-0.562623\pi$
$877$	−11.5772	−0.390936	−0.195468	+	0.980710i	$0.562623\pi$
$878$	0	0
$879$	−10.7807	−0.363624
$880$	0	0
$881$	25.5585	0.861089	0.430545	−	0.902569i	$-0.358321\pi$
$881$	25.5585	0.861089	0.430545	+	0.902569i	$0.358321\pi$
$882$	0	0
$883$	2.44482	0.0822748	0.0411374	−	0.999153i	$-0.486902\pi$
$883$	2.44482	0.0822748	0.0411374	+	0.999153i	$0.486902\pi$
$884$	0	0
$885$	8.75635	0.294341
$886$	0	0
$887$	27.7155	0.930595	0.465298	−	0.885154i	$-0.345947\pi$
$887$	27.7155	0.930595	0.465298	+	0.885154i	$0.345947\pi$
$888$	0	0
$889$	−0.452913	−0.0151902
$890$	0	0
$891$	−11.5250	−0.386102
$892$	0	0
$893$	−22.4362	−0.750798
$894$	0	0
$895$	13.7202	0.458615
$896$	0	0
$897$	−100.265	−3.34775
$898$	0	0
$899$	−0.499732	−0.0166670
$900$	0	0
$901$	−3.21194	−0.107005
$902$	0	0
$903$	49.9242	1.66137
$904$	0	0
$905$	12.3587	0.410818
$906$	0	0
$907$	−3.04174	−0.100999	−0.0504997	−	0.998724i	$-0.516081\pi$
$907$	−3.04174	−0.100999	−0.0504997	+	0.998724i	$0.516081\pi$
$908$	0	0
$909$	20.1402	0.668009
$910$	0	0
$911$	0.477022	0.0158044	0.00790222	−	0.999969i	$-0.497485\pi$
$911$	0.477022	0.0158044	0.00790222	+	0.999969i	$0.497485\pi$
$912$	0	0
$913$	−6.68502	−0.221242
$914$	0	0
$915$	107.193	3.54368
$916$	0	0
$917$	12.9211	0.426692
$918$	0	0
$919$	−32.0199	−1.05624	−0.528120	−	0.849170i	$-0.677103\pi$
$919$	−32.0199	−1.05624	−0.528120	+	0.849170i	$0.677103\pi$
$920$	0	0
$921$	10.2812	0.338778
$922$	0	0
$923$	77.1004	2.53779
$924$	0	0
$925$	−74.2208	−2.44037
$926$	0	0
$927$	−16.8349	−0.552931
$928$	0	0
$929$	55.9424	1.83541	0.917705	−	0.397263i	$-0.130040\pi$
$929$	55.9424	1.83541	0.917705	+	0.397263i	$0.130040\pi$
$930$	0	0
$931$	10.4817	0.343522
$932$	0	0
$933$	35.8671	1.17424
$934$	0	0
$935$	−5.15421	−0.168561
$936$	0	0
$937$	34.0741	1.11315	0.556575	−	0.830797i	$-0.312115\pi$
$937$	34.0741	1.11315	0.556575	+	0.830797i	$0.312115\pi$
$938$	0	0
$939$	71.7092	2.34014
$940$	0	0
$941$	−54.0945	−1.76343	−0.881716	−	0.471781i	$-0.843611\pi$
$941$	−54.0945	−1.76343	−0.881716	+	0.471781i	$0.843611\pi$
$942$	0	0
$943$	−37.7402	−1.22899
$944$	0	0
$945$	6.33243	0.205994
$946$	0	0
$947$	−7.60901	−0.247259	−0.123630	−	0.992328i	$-0.539454\pi$
$947$	−7.60901	−0.247259	−0.123630	+	0.992328i	$0.539454\pi$
$948$	0	0
$949$	15.2108	0.493764
$950$	0	0
$951$	−26.6153	−0.863059
$952$	0	0
$953$	23.9562	0.776017	0.388008	−	0.921656i	$-0.373163\pi$
$953$	23.9562	0.776017	0.388008	+	0.921656i	$0.373163\pi$
$954$	0	0
$955$	25.9525	0.839803
$956$	0	0
$957$	−17.8458	−0.576872
$958$	0	0
$959$	−44.1926	−1.42705
$960$	0	0
$961$	−30.9890	−0.999645
$962$	0	0
$963$	11.4684	0.369564
$964$	0	0
$965$	3.00907	0.0968653
$966$	0	0
$967$	−7.96313	−0.256077	−0.128039	−	0.991769i	$-0.540868\pi$
$967$	−7.96313	−0.256077	−0.128039	+	0.991769i	$0.540868\pi$
$968$	0	0
$969$	8.39885	0.269810
$970$	0	0
$971$	−27.8309	−0.893135	−0.446567	−	0.894750i	$-0.647354\pi$
$971$	−27.8309	−0.893135	−0.446567	+	0.894750i	$0.647354\pi$
$972$	0	0
$973$	−20.9711	−0.672303
$974$	0	0
$975$	−105.736	−3.38628
$976$	0	0
$977$	23.9490	0.766196	0.383098	−	0.923708i	$-0.374857\pi$
$977$	23.9490	0.766196	0.383098	+	0.923708i	$0.374857\pi$
$978$	0	0
$979$	17.4251	0.556908
$980$	0	0
$981$	11.1063	0.354598
$982$	0	0
$983$	8.47362	0.270266	0.135133	−	0.990827i	$-0.456854\pi$
$983$	8.47362	0.270266	0.135133	+	0.990827i	$0.456854\pi$
$984$	0	0
$985$	−19.0482	−0.606927
$986$	0	0
$987$	−33.3836	−1.06261
$988$	0	0
$989$	−67.2962	−2.13989
$990$	0	0
$991$	46.9457	1.49128	0.745639	−	0.666350i	$-0.232144\pi$
$991$	46.9457	1.49128	0.745639	+	0.666350i	$0.232144\pi$
$992$	0	0
$993$	−31.8706	−1.01138
$994$	0	0
$995$	−52.7023	−1.67078
$996$	0	0
$997$	−5.88004	−0.186223	−0.0931113	−	0.995656i	$-0.529681\pi$
$997$	−5.88004	−0.186223	−0.0931113	+	0.995656i	$0.529681\pi$
$998$	0	0
$999$	9.80600	0.310248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.2	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.2	✓	21	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.52360 q^{3} -3.46979 q^{5} +1.96229 q^{7} +3.36854 q^{9} +O(q^{10})\)	\(q-2.52360 q^{3} -3.46979 q^{5} +1.96229 q^{7} +3.36854 q^{9} +1.48545 q^{11} +5.95205 q^{13} +8.75635 q^{15} +1.00000 q^{17} -3.32813 q^{19} -4.95203 q^{21} +6.67517 q^{23} +7.03944 q^{25} -0.930045 q^{27} +4.76054 q^{29} -0.104974 q^{31} -3.74869 q^{33} -6.80873 q^{35} -10.5436 q^{37} -15.0206 q^{39} -5.65382 q^{41} -10.0816 q^{43} -11.6881 q^{45} +6.74139 q^{47} -3.14942 q^{49} -2.52360 q^{51} -3.21194 q^{53} -5.15421 q^{55} +8.39885 q^{57} +1.00000 q^{59} +12.2417 q^{61} +6.61005 q^{63} -20.6524 q^{65} -11.8520 q^{67} -16.8454 q^{69} +12.9536 q^{71} +2.55556 q^{73} -17.7647 q^{75} +2.91489 q^{77} +7.41722 q^{79} -7.75856 q^{81} -4.50032 q^{83} -3.46979 q^{85} -12.0137 q^{87} +11.7305 q^{89} +11.6797 q^{91} +0.264912 q^{93} +11.5479 q^{95} +4.10295 q^{97} +5.00381 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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