LMFDB - Embedded newform 4012.2.a.j.1.18

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			1.18
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)

$f(q)$	$=$	$q+2.45518 q^{3} -1.72896 q^{5} -4.96392 q^{7} +3.02793 q^{9} +O(q^{10})$	$q+2.45518 q^{3} -1.72896 q^{5} -4.96392 q^{7} +3.02793 q^{9} +3.66145 q^{11} -4.13229 q^{13} -4.24491 q^{15} +1.00000 q^{17} +3.33290 q^{19} -12.1873 q^{21} +2.88363 q^{23} -2.01070 q^{25} +0.0685696 q^{27} +0.607824 q^{29} -1.49455 q^{31} +8.98953 q^{33} +8.58241 q^{35} +8.68630 q^{37} -10.1455 q^{39} +4.95804 q^{41} +6.72183 q^{43} -5.23516 q^{45} +0.874202 q^{47} +17.6405 q^{49} +2.45518 q^{51} +3.88436 q^{53} -6.33049 q^{55} +8.18288 q^{57} +1.00000 q^{59} +14.6865 q^{61} -15.0304 q^{63} +7.14456 q^{65} +15.7886 q^{67} +7.07985 q^{69} -4.83238 q^{71} +1.47694 q^{73} -4.93665 q^{75} -18.1751 q^{77} -7.91124 q^{79} -8.91543 q^{81} +2.71579 q^{83} -1.72896 q^{85} +1.49232 q^{87} -16.6246 q^{89} +20.5124 q^{91} -3.66940 q^{93} -5.76244 q^{95} +11.0269 q^{97} +11.0866 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.45518	1.41750	0.708751	−	0.705459i	$-0.249259\pi$
$3$	2.45518	1.41750	0.708751	+	0.705459i	$0.249259\pi$
$4$	0	0
$5$	−1.72896	−0.773213	−0.386607	−	0.922245i	$-0.626353\pi$
$5$	−1.72896	−0.773213	−0.386607	+	0.922245i	$0.626353\pi$
$6$	0	0
$7$	−4.96392	−1.87619	−0.938093	−	0.346383i	$-0.887410\pi$
$7$	−4.96392	−1.87619	−0.938093	+	0.346383i	$0.887410\pi$
$8$	0	0
$9$	3.02793	1.00931
$10$	0	0
$11$	3.66145	1.10397	0.551984	−	0.833855i	$-0.313871\pi$
$11$	3.66145	1.10397	0.551984	+	0.833855i	$0.313871\pi$
$12$	0	0
$13$	−4.13229	−1.14609	−0.573046	−	0.819523i	$-0.694238\pi$
$13$	−4.13229	−1.14609	−0.573046	+	0.819523i	$0.694238\pi$
$14$	0	0
$15$	−4.24491	−1.09603
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	3.33290	0.764619	0.382310	−	0.924034i	$-0.375129\pi$
$19$	3.33290	0.764619	0.382310	+	0.924034i	$0.375129\pi$
$20$	0	0
$21$	−12.1873	−2.65950
$22$	0	0
$23$	2.88363	0.601279	0.300639	−	0.953738i	$-0.402800\pi$
$23$	2.88363	0.601279	0.300639	+	0.953738i	$0.402800\pi$
$24$	0	0
$25$	−2.01070	−0.402141
$26$	0	0
$27$	0.0685696	0.0131962
$28$	0	0
$29$	0.607824	0.112870	0.0564351	−	0.998406i	$-0.482027\pi$
$29$	0.607824	0.112870	0.0564351	+	0.998406i	$0.482027\pi$
$30$	0	0
$31$	−1.49455	−0.268429	−0.134215	−	0.990952i	$-0.542851\pi$
$31$	−1.49455	−0.268429	−0.134215	+	0.990952i	$0.542851\pi$
$32$	0	0
$33$	8.98953	1.56488
$34$	0	0
$35$	8.58241	1.45069
$36$	0	0
$37$	8.68630	1.42802	0.714009	−	0.700136i	$-0.246877\pi$
$37$	8.68630	1.42802	0.714009	+	0.700136i	$0.246877\pi$
$38$	0	0
$39$	−10.1455	−1.62459
$40$	0	0
$41$	4.95804	0.774316	0.387158	−	0.922013i	$-0.373457\pi$
$41$	4.95804	0.774316	0.387158	+	0.922013i	$0.373457\pi$
$42$	0	0
$43$	6.72183	1.02507	0.512534	−	0.858667i	$-0.328707\pi$
$43$	6.72183	1.02507	0.512534	+	0.858667i	$0.328707\pi$
$44$	0	0
$45$	−5.23516	−0.780412
$46$	0	0
$47$	0.874202	0.127515	0.0637577	−	0.997965i	$-0.479692\pi$
$47$	0.874202	0.127515	0.0637577	+	0.997965i	$0.479692\pi$
$48$	0	0
$49$	17.6405	2.52008
$50$	0	0
$51$	2.45518	0.343795
$52$	0	0
$53$	3.88436	0.533558	0.266779	−	0.963758i	$-0.414041\pi$
$53$	3.88436	0.533558	0.266779	+	0.963758i	$0.414041\pi$
$54$	0	0
$55$	−6.33049	−0.853603
$56$	0	0
$57$	8.18288	1.08385
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	14.6865	1.88041	0.940204	−	0.340611i	$-0.110634\pi$
$61$	14.6865	1.88041	0.940204	+	0.340611i	$0.110634\pi$
$62$	0	0
$63$	−15.0304	−1.89365
$64$	0	0
$65$	7.14456	0.886173
$66$	0	0
$67$	15.7886	1.92888	0.964440	−	0.264301i	$-0.0851412\pi$
$67$	15.7886	1.92888	0.964440	+	0.264301i	$0.0851412\pi$
$68$	0	0
$69$	7.07985	0.852314
$70$	0	0
$71$	−4.83238	−0.573497	−0.286749	−	0.958006i	$-0.592574\pi$
$71$	−4.83238	−0.573497	−0.286749	+	0.958006i	$0.592574\pi$
$72$	0	0
$73$	1.47694	0.172863	0.0864313	−	0.996258i	$-0.472454\pi$
$73$	1.47694	0.172863	0.0864313	+	0.996258i	$0.472454\pi$
$74$	0	0
$75$	−4.93665	−0.570035
$76$	0	0
$77$	−18.1751	−2.07125
$78$	0	0
$79$	−7.91124	−0.890084	−0.445042	−	0.895510i	$-0.646811\pi$
$79$	−7.91124	−0.890084	−0.445042	+	0.895510i	$0.646811\pi$
$80$	0	0
$81$	−8.91543	−0.990604
$82$	0	0
$83$	2.71579	0.298097	0.149049	−	0.988830i	$-0.452379\pi$
$83$	2.71579	0.298097	0.149049	+	0.988830i	$0.452379\pi$
$84$	0	0
$85$	−1.72896	−0.187532
$86$	0	0
$87$	1.49232	0.159994
$88$	0	0
$89$	−16.6246	−1.76221	−0.881104	−	0.472922i	$-0.843199\pi$
$89$	−16.6246	−1.76221	−0.881104	+	0.472922i	$0.843199\pi$
$90$	0	0
$91$	20.5124	2.15028
$92$	0	0
$93$	−3.66940	−0.380499
$94$	0	0
$95$	−5.76244	−0.591214
$96$	0	0
$97$	11.0269	1.11961	0.559804	−	0.828625i	$-0.310876\pi$
$97$	11.0269	1.11961	0.559804	+	0.828625i	$0.310876\pi$
$98$	0	0
$99$	11.0866	1.11425
$100$	0	0
$101$	3.14697	0.313135	0.156568	−	0.987667i	$-0.449957\pi$
$101$	3.14697	0.313135	0.156568	+	0.987667i	$0.449957\pi$
$102$	0	0
$103$	9.02301	0.889063	0.444532	−	0.895763i	$-0.353370\pi$
$103$	9.02301	0.889063	0.444532	+	0.895763i	$0.353370\pi$
$104$	0	0
$105$	21.0714	2.05636
$106$	0	0
$107$	−6.66702	−0.644525	−0.322262	−	0.946650i	$-0.604443\pi$
$107$	−6.66702	−0.644525	−0.322262	+	0.946650i	$0.604443\pi$
$108$	0	0
$109$	4.96409	0.475474	0.237737	−	0.971330i	$-0.423594\pi$
$109$	4.96409	0.475474	0.237737	+	0.971330i	$0.423594\pi$
$110$	0	0
$111$	21.3265	2.02422
$112$	0	0
$113$	1.51366	0.142393	0.0711965	−	0.997462i	$-0.477318\pi$
$113$	1.51366	0.142393	0.0711965	+	0.997462i	$0.477318\pi$
$114$	0	0
$115$	−4.98568	−0.464917
$116$	0	0
$117$	−12.5123	−1.15676
$118$	0	0
$119$	−4.96392	−0.455042
$120$	0	0
$121$	2.40620	0.218745
$122$	0	0
$123$	12.1729	1.09759
$124$	0	0
$125$	12.1212	1.08415
$126$	0	0
$127$	−13.5730	−1.20441	−0.602206	−	0.798341i	$-0.705711\pi$
$127$	−13.5730	−1.20441	−0.602206	+	0.798341i	$0.705711\pi$
$128$	0	0
$129$	16.5033	1.45304
$130$	0	0
$131$	−12.9970	−1.13555	−0.567777	−	0.823183i	$-0.692196\pi$
$131$	−12.9970	−1.13555	−0.567777	+	0.823183i	$0.692196\pi$
$132$	0	0
$133$	−16.5443	−1.43457
$134$	0	0
$135$	−0.118554	−0.0102035
$136$	0	0
$137$	6.83663	0.584093	0.292046	−	0.956404i	$-0.405664\pi$
$137$	6.83663	0.584093	0.292046	+	0.956404i	$0.405664\pi$
$138$	0	0
$139$	−2.13802	−0.181344	−0.0906720	−	0.995881i	$-0.528902\pi$
$139$	−2.13802	−0.181344	−0.0906720	+	0.995881i	$0.528902\pi$
$140$	0	0
$141$	2.14633	0.180753
$142$	0	0
$143$	−15.1302	−1.26525
$144$	0	0
$145$	−1.05090	−0.0872727
$146$	0	0
$147$	43.3108	3.57221
$148$	0	0
$149$	−9.41517	−0.771321	−0.385660	−	0.922641i	$-0.626026\pi$
$149$	−9.41517	−0.771321	−0.385660	+	0.922641i	$0.626026\pi$
$150$	0	0
$151$	0.325079	0.0264546	0.0132273	−	0.999913i	$-0.495790\pi$
$151$	0.325079	0.0264546	0.0132273	+	0.999913i	$0.495790\pi$
$152$	0	0
$153$	3.02793	0.244794
$154$	0	0
$155$	2.58402	0.207553
$156$	0	0
$157$	14.8718	1.18690	0.593448	−	0.804872i	$-0.297766\pi$
$157$	14.8718	1.18690	0.593448	+	0.804872i	$0.297766\pi$
$158$	0	0
$159$	9.53682	0.756319
$160$	0	0
$161$	−14.3141	−1.12811
$162$	0	0
$163$	2.22024	0.173902	0.0869511	−	0.996213i	$-0.472288\pi$
$163$	2.22024	0.173902	0.0869511	+	0.996213i	$0.472288\pi$
$164$	0	0
$165$	−15.5425	−1.20998
$166$	0	0
$167$	12.5098	0.968040	0.484020	−	0.875057i	$-0.339176\pi$
$167$	12.5098	0.968040	0.484020	+	0.875057i	$0.339176\pi$
$168$	0	0
$169$	4.07584	0.313526
$170$	0	0
$171$	10.0918	0.771738
$172$	0	0
$173$	−19.0402	−1.44760	−0.723800	−	0.690010i	$-0.757606\pi$
$173$	−19.0402	−1.44760	−0.723800	+	0.690010i	$0.757606\pi$
$174$	0	0
$175$	9.98098	0.754492
$176$	0	0
$177$	2.45518	0.184543
$178$	0	0
$179$	15.5099	1.15927	0.579634	−	0.814877i	$-0.303196\pi$
$179$	15.5099	1.15927	0.579634	+	0.814877i	$0.303196\pi$
$180$	0	0
$181$	−22.8754	−1.70031	−0.850157	−	0.526529i	$-0.823493\pi$
$181$	−22.8754	−1.70031	−0.850157	+	0.526529i	$0.823493\pi$
$182$	0	0
$183$	36.0580	2.66548
$184$	0	0
$185$	−15.0182	−1.10416
$186$	0	0
$187$	3.66145	0.267752
$188$	0	0
$189$	−0.340374	−0.0247586
$190$	0	0
$191$	18.4318	1.33368	0.666838	−	0.745203i	$-0.267647\pi$
$191$	18.4318	1.33368	0.666838	+	0.745203i	$0.267647\pi$
$192$	0	0
$193$	6.22354	0.447980	0.223990	−	0.974591i	$-0.428092\pi$
$193$	6.22354	0.447980	0.223990	+	0.974591i	$0.428092\pi$
$194$	0	0
$195$	17.5412	1.25615
$196$	0	0
$197$	15.8851	1.13177	0.565884	−	0.824485i	$-0.308535\pi$
$197$	15.8851	1.13177	0.565884	+	0.824485i	$0.308535\pi$
$198$	0	0
$199$	20.5451	1.45641	0.728203	−	0.685362i	$-0.240356\pi$
$199$	20.5451	1.45641	0.728203	+	0.685362i	$0.240356\pi$
$200$	0	0
$201$	38.7638	2.73419
$202$	0	0
$203$	−3.01719	−0.211765
$204$	0	0
$205$	−8.57224	−0.598711
$206$	0	0
$207$	8.73143	0.606876
$208$	0	0
$209$	12.2032	0.844115
$210$	0	0
$211$	−18.7762	−1.29261	−0.646303	−	0.763081i	$-0.723686\pi$
$211$	−18.7762	−1.29261	−0.646303	+	0.763081i	$0.723686\pi$
$212$	0	0
$213$	−11.8644	−0.812933
$214$	0	0
$215$	−11.6218	−0.792597
$216$	0	0
$217$	7.41884	0.503623
$218$	0	0
$219$	3.62615	0.245033
$220$	0	0
$221$	−4.13229	−0.277968
$222$	0	0
$223$	−13.4871	−0.903166	−0.451583	−	0.892229i	$-0.649140\pi$
$223$	−13.4871	−0.903166	−0.451583	+	0.892229i	$0.649140\pi$
$224$	0	0
$225$	−6.08827	−0.405885
$226$	0	0
$227$	25.5224	1.69398	0.846992	−	0.531605i	$-0.178411\pi$
$227$	25.5224	1.69398	0.846992	+	0.531605i	$0.178411\pi$
$228$	0	0
$229$	17.9249	1.18451	0.592257	−	0.805749i	$-0.298237\pi$
$229$	17.9249	1.18451	0.592257	+	0.805749i	$0.298237\pi$
$230$	0	0
$231$	−44.6233	−2.93600
$232$	0	0
$233$	−25.5384	−1.67308	−0.836539	−	0.547908i	$-0.815425\pi$
$233$	−25.5384	−1.67308	−0.836539	+	0.547908i	$0.815425\pi$
$234$	0	0
$235$	−1.51146	−0.0985967
$236$	0	0
$237$	−19.4236	−1.26170
$238$	0	0
$239$	−3.56213	−0.230415	−0.115208	−	0.993341i	$-0.536753\pi$
$239$	−3.56213	−0.230415	−0.115208	+	0.993341i	$0.536753\pi$
$240$	0	0
$241$	−3.17085	−0.204252	−0.102126	−	0.994771i	$-0.532565\pi$
$241$	−3.17085	−0.204252	−0.102126	+	0.994771i	$0.532565\pi$
$242$	0	0
$243$	−22.0947	−1.41738
$244$	0	0
$245$	−30.4997	−1.94856
$246$	0	0
$247$	−13.7725	−0.876324
$248$	0	0
$249$	6.66777	0.422553
$250$	0	0
$251$	27.1578	1.71419	0.857094	−	0.515160i	$-0.172267\pi$
$251$	27.1578	1.71419	0.857094	+	0.515160i	$0.172267\pi$
$252$	0	0
$253$	10.5583	0.663793
$254$	0	0
$255$	−4.24491	−0.265827
$256$	0	0
$257$	19.2597	1.20139	0.600693	−	0.799480i	$-0.294892\pi$
$257$	19.2597	1.20139	0.600693	+	0.799480i	$0.294892\pi$
$258$	0	0
$259$	−43.1181	−2.67923
$260$	0	0
$261$	1.84045	0.113921
$262$	0	0
$263$	−7.21278	−0.444759	−0.222380	−	0.974960i	$-0.571382\pi$
$263$	−7.21278	−0.444759	−0.222380	+	0.974960i	$0.571382\pi$
$264$	0	0
$265$	−6.71589	−0.412554
$266$	0	0
$267$	−40.8166	−2.49793
$268$	0	0
$269$	18.7351	1.14230	0.571150	−	0.820846i	$-0.306497\pi$
$269$	18.7351	1.14230	0.571150	+	0.820846i	$0.306497\pi$
$270$	0	0
$271$	−14.0783	−0.855198	−0.427599	−	0.903968i	$-0.640641\pi$
$271$	−14.0783	−0.855198	−0.427599	+	0.903968i	$0.640641\pi$
$272$	0	0
$273$	50.3617	3.04803
$274$	0	0
$275$	−7.36209	−0.443951
$276$	0	0
$277$	27.1390	1.63063	0.815313	−	0.579021i	$-0.196565\pi$
$277$	27.1390	1.63063	0.815313	+	0.579021i	$0.196565\pi$
$278$	0	0
$279$	−4.52539	−0.270928
$280$	0	0
$281$	25.2806	1.50811	0.754057	−	0.656809i	$-0.228094\pi$
$281$	25.2806	1.50811	0.754057	+	0.656809i	$0.228094\pi$
$282$	0	0
$283$	0.0521765	0.00310157	0.00155079	−	0.999999i	$-0.499506\pi$
$283$	0.0521765	0.00310157	0.00155079	+	0.999999i	$0.499506\pi$
$284$	0	0
$285$	−14.1479	−0.838047
$286$	0	0
$287$	−24.6113	−1.45276
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	27.0730	1.58705
$292$	0	0
$293$	4.90891	0.286782	0.143391	−	0.989666i	$-0.454199\pi$
$293$	4.90891	0.286782	0.143391	+	0.989666i	$0.454199\pi$
$294$	0	0
$295$	−1.72896	−0.100664
$296$	0	0
$297$	0.251064	0.0145682
$298$	0	0
$299$	−11.9160	−0.689121
$300$	0	0
$301$	−33.3666	−1.92322
$302$	0	0
$303$	7.72639	0.443870
$304$	0	0
$305$	−25.3923	−1.45396
$306$	0	0
$307$	−7.23986	−0.413201	−0.206600	−	0.978425i	$-0.566240\pi$
$307$	−7.23986	−0.413201	−0.206600	+	0.978425i	$0.566240\pi$
$308$	0	0
$309$	22.1531	1.26025
$310$	0	0
$311$	−1.10884	−0.0628768	−0.0314384	−	0.999506i	$-0.510009\pi$
$311$	−1.10884	−0.0628768	−0.0314384	+	0.999506i	$0.510009\pi$
$312$	0	0
$313$	−33.5344	−1.89548	−0.947739	−	0.319046i	$-0.896638\pi$
$313$	−33.5344	−1.89548	−0.947739	+	0.319046i	$0.896638\pi$
$314$	0	0
$315$	25.9869	1.46420
$316$	0	0
$317$	−6.18007	−0.347107	−0.173554	−	0.984824i	$-0.555525\pi$
$317$	−6.18007	−0.347107	−0.173554	+	0.984824i	$0.555525\pi$
$318$	0	0
$319$	2.22552	0.124605
$320$	0	0
$321$	−16.3688	−0.913615
$322$	0	0
$323$	3.33290	0.185447
$324$	0	0
$325$	8.30882	0.460890
$326$	0	0
$327$	12.1878	0.673985
$328$	0	0
$329$	−4.33947	−0.239243
$330$	0	0
$331$	−23.9647	−1.31722	−0.658609	−	0.752485i	$-0.728855\pi$
$331$	−23.9647	−1.31722	−0.658609	+	0.752485i	$0.728855\pi$
$332$	0	0
$333$	26.3015	1.44131
$334$	0	0
$335$	−27.2978	−1.49144
$336$	0	0
$337$	−13.3587	−0.727696	−0.363848	−	0.931458i	$-0.618537\pi$
$337$	−13.3587	−0.727696	−0.363848	+	0.931458i	$0.618537\pi$
$338$	0	0
$339$	3.71631	0.201842
$340$	0	0
$341$	−5.47222	−0.296337
$342$	0	0
$343$	−52.8188	−2.85195
$344$	0	0
$345$	−12.2408	−0.659020
$346$	0	0
$347$	21.1365	1.13467	0.567334	−	0.823488i	$-0.307975\pi$
$347$	21.1365	1.13467	0.567334	+	0.823488i	$0.307975\pi$
$348$	0	0
$349$	11.8984	0.636904	0.318452	−	0.947939i	$-0.396837\pi$
$349$	11.8984	0.636904	0.318452	+	0.947939i	$0.396837\pi$
$350$	0	0
$351$	−0.283350	−0.0151241
$352$	0	0
$353$	18.5375	0.986653	0.493326	−	0.869844i	$-0.335781\pi$
$353$	18.5375	0.986653	0.493326	+	0.869844i	$0.335781\pi$
$354$	0	0
$355$	8.35497	0.443436
$356$	0	0
$357$	−12.1873	−0.645023
$358$	0	0
$359$	−24.9390	−1.31623	−0.658116	−	0.752917i	$-0.728646\pi$
$359$	−24.9390	−1.31623	−0.658116	+	0.752917i	$0.728646\pi$
$360$	0	0
$361$	−7.89179	−0.415357
$362$	0	0
$363$	5.90766	0.310072
$364$	0	0
$365$	−2.55356	−0.133660
$366$	0	0
$367$	13.1456	0.686194	0.343097	−	0.939300i	$-0.388524\pi$
$367$	13.1456	0.686194	0.343097	+	0.939300i	$0.388524\pi$
$368$	0	0
$369$	15.0126	0.781524
$370$	0	0
$371$	−19.2817	−1.00105
$372$	0	0
$373$	−14.0122	−0.725525	−0.362762	−	0.931882i	$-0.618166\pi$
$373$	−14.0122	−0.725525	−0.362762	+	0.931882i	$0.618166\pi$
$374$	0	0
$375$	29.7598	1.53679
$376$	0	0
$377$	−2.51171	−0.129360
$378$	0	0
$379$	−16.1110	−0.827569	−0.413784	−	0.910375i	$-0.635793\pi$
$379$	−16.1110	−0.827569	−0.413784	+	0.910375i	$0.635793\pi$
$380$	0	0
$381$	−33.3243	−1.70726
$382$	0	0
$383$	−21.2446	−1.08555	−0.542775	−	0.839878i	$-0.682627\pi$
$383$	−21.2446	−1.08555	−0.542775	+	0.839878i	$0.682627\pi$
$384$	0	0
$385$	31.4241	1.60152
$386$	0	0
$387$	20.3532	1.03461
$388$	0	0
$389$	−37.3114	−1.89176	−0.945882	−	0.324511i	$-0.894800\pi$
$389$	−37.3114	−1.89176	−0.945882	+	0.324511i	$0.894800\pi$
$390$	0	0
$391$	2.88363	0.145832
$392$	0	0
$393$	−31.9100	−1.60965
$394$	0	0
$395$	13.6782	0.688225
$396$	0	0
$397$	27.8644	1.39847	0.699237	−	0.714890i	$-0.253523\pi$
$397$	27.8644	1.39847	0.699237	+	0.714890i	$0.253523\pi$
$398$	0	0
$399$	−40.6192	−2.03350
$400$	0	0
$401$	−23.2771	−1.16240	−0.581202	−	0.813759i	$-0.697418\pi$
$401$	−23.2771	−1.16240	−0.581202	+	0.813759i	$0.697418\pi$
$402$	0	0
$403$	6.17592	0.307645
$404$	0	0
$405$	15.4144	0.765948
$406$	0	0
$407$	31.8044	1.57649
$408$	0	0
$409$	16.6789	0.824719	0.412359	−	0.911021i	$-0.364705\pi$
$409$	16.6789	0.824719	0.412359	+	0.911021i	$0.364705\pi$
$410$	0	0
$411$	16.7852	0.827952
$412$	0	0
$413$	−4.96392	−0.244259
$414$	0	0
$415$	−4.69549	−0.230493
$416$	0	0
$417$	−5.24922	−0.257055
$418$	0	0
$419$	17.2966	0.844995	0.422498	−	0.906364i	$-0.361153\pi$
$419$	17.2966	0.844995	0.422498	+	0.906364i	$0.361153\pi$
$420$	0	0
$421$	−12.6645	−0.617230	−0.308615	−	0.951187i	$-0.599866\pi$
$421$	−12.6645	−0.617230	−0.308615	+	0.951187i	$0.599866\pi$
$422$	0	0
$423$	2.64702	0.128703
$424$	0	0
$425$	−2.01070	−0.0975335
$426$	0	0
$427$	−72.9025	−3.52800
$428$	0	0
$429$	−37.1474	−1.79349
$430$	0	0
$431$	−21.0558	−1.01422	−0.507112	−	0.861880i	$-0.669287\pi$
$431$	−21.0558	−1.01422	−0.507112	+	0.861880i	$0.669287\pi$
$432$	0	0
$433$	−14.7090	−0.706869	−0.353435	−	0.935459i	$-0.614986\pi$
$433$	−14.7090	−0.706869	−0.353435	+	0.935459i	$0.614986\pi$
$434$	0	0
$435$	−2.58016	−0.123709
$436$	0	0
$437$	9.61085	0.459750
$438$	0	0
$439$	14.6690	0.700113	0.350056	−	0.936729i	$-0.386162\pi$
$439$	14.6690	0.700113	0.350056	+	0.936729i	$0.386162\pi$
$440$	0	0
$441$	53.4143	2.54354
$442$	0	0
$443$	12.0436	0.572211	0.286106	−	0.958198i	$-0.407639\pi$
$443$	12.0436	0.572211	0.286106	+	0.958198i	$0.407639\pi$
$444$	0	0
$445$	28.7433	1.36256
$446$	0	0
$447$	−23.1160	−1.09335
$448$	0	0
$449$	6.14716	0.290102	0.145051	−	0.989424i	$-0.453665\pi$
$449$	6.14716	0.290102	0.145051	+	0.989424i	$0.453665\pi$
$450$	0	0
$451$	18.1536	0.854820
$452$	0	0
$453$	0.798129	0.0374994
$454$	0	0
$455$	−35.4650	−1.66263
$456$	0	0
$457$	15.3876	0.719802	0.359901	−	0.932991i	$-0.382811\pi$
$457$	15.3876	0.719802	0.359901	+	0.932991i	$0.382811\pi$
$458$	0	0
$459$	0.0685696	0.00320056
$460$	0	0
$461$	21.8567	1.01797	0.508984	−	0.860776i	$-0.330021\pi$
$461$	21.8567	1.01797	0.508984	+	0.860776i	$0.330021\pi$
$462$	0	0
$463$	4.64249	0.215755	0.107877	−	0.994164i	$-0.465595\pi$
$463$	4.64249	0.215755	0.107877	+	0.994164i	$0.465595\pi$
$464$	0	0
$465$	6.34423	0.294207
$466$	0	0
$467$	24.6610	1.14117	0.570587	−	0.821237i	$-0.306716\pi$
$467$	24.6610	1.14117	0.570587	+	0.821237i	$0.306716\pi$
$468$	0	0
$469$	−78.3732	−3.61894
$470$	0	0
$471$	36.5129	1.68243
$472$	0	0
$473$	24.6116	1.13164
$474$	0	0
$475$	−6.70148	−0.307485
$476$	0	0
$477$	11.7616	0.538525
$478$	0	0
$479$	−27.8086	−1.27061	−0.635303	−	0.772263i	$-0.719125\pi$
$479$	−27.8086	−1.27061	−0.635303	+	0.772263i	$0.719125\pi$
$480$	0	0
$481$	−35.8943	−1.63664
$482$	0	0
$483$	−35.1438	−1.59910
$484$	0	0
$485$	−19.0650	−0.865697
$486$	0	0
$487$	16.7930	0.760965	0.380482	−	0.924788i	$-0.375758\pi$
$487$	16.7930	0.760965	0.380482	+	0.924788i	$0.375758\pi$
$488$	0	0
$489$	5.45109	0.246507
$490$	0	0
$491$	2.35483	0.106272	0.0531361	−	0.998587i	$-0.483078\pi$
$491$	2.35483	0.106272	0.0531361	+	0.998587i	$0.483078\pi$
$492$	0	0
$493$	0.607824	0.0273750
$494$	0	0
$495$	−19.1683	−0.861549
$496$	0	0
$497$	23.9875	1.07599
$498$	0	0
$499$	−9.24602	−0.413909	−0.206954	−	0.978351i	$-0.566355\pi$
$499$	−9.24602	−0.413909	−0.206954	+	0.978351i	$0.566355\pi$
$500$	0	0
$501$	30.7140	1.37220
$502$	0	0
$503$	8.12541	0.362294	0.181147	−	0.983456i	$-0.442019\pi$
$503$	8.12541	0.362294	0.181147	+	0.983456i	$0.442019\pi$
$504$	0	0
$505$	−5.44098	−0.242120
$506$	0	0
$507$	10.0069	0.444424
$508$	0	0
$509$	6.35347	0.281612	0.140806	−	0.990037i	$-0.455031\pi$
$509$	6.35347	0.281612	0.140806	+	0.990037i	$0.455031\pi$
$510$	0	0
$511$	−7.33141	−0.324322
$512$	0	0
$513$	0.228536	0.0100901
$514$	0	0
$515$	−15.6004	−0.687436
$516$	0	0
$517$	3.20085	0.140773
$518$	0	0
$519$	−46.7472	−2.05197
$520$	0	0
$521$	−20.8481	−0.913371	−0.456686	−	0.889628i	$-0.650964\pi$
$521$	−20.8481	−0.913371	−0.456686	+	0.889628i	$0.650964\pi$
$522$	0	0
$523$	−3.02813	−0.132411	−0.0662054	−	0.997806i	$-0.521089\pi$
$523$	−3.02813	−0.132411	−0.0662054	+	0.997806i	$0.521089\pi$
$524$	0	0
$525$	24.5052	1.06949
$526$	0	0
$527$	−1.49455	−0.0651037
$528$	0	0
$529$	−14.6847	−0.638464
$530$	0	0
$531$	3.02793	0.131401
$532$	0	0
$533$	−20.4881	−0.887437
$534$	0	0
$535$	11.5270	0.498355
$536$	0	0
$537$	38.0798	1.64326
$538$	0	0
$539$	64.5899	2.78208
$540$	0	0
$541$	−16.2351	−0.698002	−0.349001	−	0.937122i	$-0.613479\pi$
$541$	−16.2351	−0.698002	−0.349001	+	0.937122i	$0.613479\pi$
$542$	0	0
$543$	−56.1633	−2.41020
$544$	0	0
$545$	−8.58271	−0.367643
$546$	0	0
$547$	19.0461	0.814354	0.407177	−	0.913349i	$-0.366513\pi$
$547$	19.0461	0.814354	0.407177	+	0.913349i	$0.366513\pi$
$548$	0	0
$549$	44.4696	1.89791
$550$	0	0
$551$	2.02582	0.0863027
$552$	0	0
$553$	39.2708	1.66996
$554$	0	0
$555$	−36.8726	−1.56515
$556$	0	0
$557$	43.1265	1.82733	0.913664	−	0.406470i	$-0.133240\pi$
$557$	43.1265	1.82733	0.913664	+	0.406470i	$0.133240\pi$
$558$	0	0
$559$	−27.7765	−1.17482
$560$	0	0
$561$	8.98953	0.379538
$562$	0	0
$563$	36.5128	1.53883	0.769416	−	0.638748i	$-0.220547\pi$
$563$	36.5128	1.53883	0.769416	+	0.638748i	$0.220547\pi$
$564$	0	0
$565$	−2.61705	−0.110100
$566$	0	0
$567$	44.2555	1.85856
$568$	0	0
$569$	−33.1787	−1.39092	−0.695461	−	0.718564i	$-0.744800\pi$
$569$	−33.1787	−1.39092	−0.695461	+	0.718564i	$0.744800\pi$
$570$	0	0
$571$	43.2023	1.80796	0.903980	−	0.427575i	$-0.140632\pi$
$571$	43.2023	1.80796	0.903980	+	0.427575i	$0.140632\pi$
$572$	0	0
$573$	45.2534	1.89049
$574$	0	0
$575$	−5.79813	−0.241799
$576$	0	0
$577$	−33.5889	−1.39832	−0.699162	−	0.714963i	$-0.746444\pi$
$577$	−33.5889	−1.39832	−0.699162	+	0.714963i	$0.746444\pi$
$578$	0	0
$579$	15.2799	0.635013
$580$	0	0
$581$	−13.4810	−0.559286
$582$	0	0
$583$	14.2224	0.589031
$584$	0	0
$585$	21.6332	0.894423
$586$	0	0
$587$	11.1606	0.460646	0.230323	−	0.973114i	$-0.426022\pi$
$587$	11.1606	0.460646	0.230323	+	0.973114i	$0.426022\pi$
$588$	0	0
$589$	−4.98119	−0.205246
$590$	0	0
$591$	39.0009	1.60428
$592$	0	0
$593$	−37.0819	−1.52277	−0.761387	−	0.648298i	$-0.775481\pi$
$593$	−37.0819	−1.52277	−0.761387	+	0.648298i	$0.775481\pi$
$594$	0	0
$595$	8.58241	0.351845
$596$	0	0
$597$	50.4421	2.06446
$598$	0	0
$599$	−20.0196	−0.817978	−0.408989	−	0.912539i	$-0.634119\pi$
$599$	−20.0196	−0.817978	−0.408989	+	0.912539i	$0.634119\pi$
$600$	0	0
$601$	−15.3246	−0.625103	−0.312551	−	0.949901i	$-0.601184\pi$
$601$	−15.3246	−0.625103	−0.312551	+	0.949901i	$0.601184\pi$
$602$	0	0
$603$	47.8067	1.94684
$604$	0	0
$605$	−4.16022	−0.169137
$606$	0	0
$607$	27.5357	1.11764	0.558820	−	0.829289i	$-0.311254\pi$
$607$	27.5357	1.11764	0.558820	+	0.829289i	$0.311254\pi$
$608$	0	0
$609$	−7.40776	−0.300178
$610$	0	0
$611$	−3.61246	−0.146144
$612$	0	0
$613$	23.0044	0.929141	0.464570	−	0.885536i	$-0.346209\pi$
$613$	23.0044	0.929141	0.464570	+	0.885536i	$0.346209\pi$
$614$	0	0
$615$	−21.0464	−0.848674
$616$	0	0
$617$	42.0691	1.69364	0.846820	−	0.531880i	$-0.178514\pi$
$617$	42.0691	1.69364	0.846820	+	0.531880i	$0.178514\pi$
$618$	0	0
$619$	−4.41917	−0.177622	−0.0888108	−	0.996049i	$-0.528307\pi$
$619$	−4.41917	−0.177622	−0.0888108	+	0.996049i	$0.528307\pi$
$620$	0	0
$621$	0.197730	0.00793461
$622$	0	0
$623$	82.5234	3.30623
$624$	0	0
$625$	−10.9035	−0.436142
$626$	0	0
$627$	29.9612	1.19653
$628$	0	0
$629$	8.68630	0.346345
$630$	0	0
$631$	−45.4190	−1.80810	−0.904051	−	0.427424i	$-0.859421\pi$
$631$	−45.4190	−1.80810	−0.904051	+	0.427424i	$0.859421\pi$
$632$	0	0
$633$	−46.0990	−1.83227
$634$	0	0
$635$	23.4672	0.931267
$636$	0	0
$637$	−72.8958	−2.88824
$638$	0	0
$639$	−14.6321	−0.578836
$640$	0	0
$641$	23.6578	0.934425	0.467213	−	0.884145i	$-0.345258\pi$
$641$	23.6578	0.934425	0.467213	+	0.884145i	$0.345258\pi$
$642$	0	0
$643$	33.9085	1.33722	0.668611	−	0.743612i	$-0.266889\pi$
$643$	33.9085	1.33722	0.668611	+	0.743612i	$0.266889\pi$
$644$	0	0
$645$	−28.5335	−1.12351
$646$	0	0
$647$	−3.53562	−0.139000	−0.0694999	−	0.997582i	$-0.522140\pi$
$647$	−3.53562	−0.139000	−0.0694999	+	0.997582i	$0.522140\pi$
$648$	0	0
$649$	3.66145	0.143724
$650$	0	0
$651$	18.2146	0.713887
$652$	0	0
$653$	46.9284	1.83645	0.918225	−	0.396060i	$-0.129623\pi$
$653$	46.9284	1.83645	0.918225	+	0.396060i	$0.129623\pi$
$654$	0	0
$655$	22.4713	0.878025
$656$	0	0
$657$	4.47206	0.174472
$658$	0	0
$659$	−1.89367	−0.0737668	−0.0368834	−	0.999320i	$-0.511743\pi$
$659$	−1.89367	−0.0737668	−0.0368834	+	0.999320i	$0.511743\pi$
$660$	0	0
$661$	−1.65798	−0.0644880	−0.0322440	−	0.999480i	$-0.510265\pi$
$661$	−1.65798	−0.0644880	−0.0322440	+	0.999480i	$0.510265\pi$
$662$	0	0
$663$	−10.1455	−0.394020
$664$	0	0
$665$	28.6043	1.10923
$666$	0	0
$667$	1.75274	0.0678664
$668$	0	0
$669$	−33.1134	−1.28024
$670$	0	0
$671$	53.7737	2.07591
$672$	0	0
$673$	15.6581	0.603574	0.301787	−	0.953375i	$-0.402417\pi$
$673$	15.6581	0.603574	0.301787	+	0.953375i	$0.402417\pi$
$674$	0	0
$675$	−0.137873	−0.00530674
$676$	0	0
$677$	−4.52858	−0.174048	−0.0870238	−	0.996206i	$-0.527736\pi$
$677$	−4.52858	−0.174048	−0.0870238	+	0.996206i	$0.527736\pi$
$678$	0	0
$679$	−54.7365	−2.10060
$680$	0	0
$681$	62.6623	2.40122
$682$	0	0
$683$	13.2970	0.508797	0.254398	−	0.967100i	$-0.418123\pi$
$683$	13.2970	0.508797	0.254398	+	0.967100i	$0.418123\pi$
$684$	0	0
$685$	−11.8202	−0.451628
$686$	0	0
$687$	44.0090	1.67905
$688$	0	0
$689$	−16.0513	−0.611506
$690$	0	0
$691$	0.113928	0.00433404	0.00216702	−	0.999998i	$-0.499310\pi$
$691$	0.113928	0.00433404	0.00216702	+	0.999998i	$0.499310\pi$
$692$	0	0
$693$	−55.0330	−2.09053
$694$	0	0
$695$	3.69654	0.140218
$696$	0	0
$697$	4.95804	0.187799
$698$	0	0
$699$	−62.7015	−2.37159
$700$	0	0
$701$	−37.5651	−1.41881	−0.709407	−	0.704800i	$-0.751037\pi$
$701$	−37.5651	−1.41881	−0.709407	+	0.704800i	$0.751037\pi$
$702$	0	0
$703$	28.9506	1.09189
$704$	0	0
$705$	−3.71091	−0.139761
$706$	0	0
$707$	−15.6213	−0.587500
$708$	0	0
$709$	−5.07498	−0.190595	−0.0952974	−	0.995449i	$-0.530380\pi$
$709$	−5.07498	−0.190595	−0.0952974	+	0.995449i	$0.530380\pi$
$710$	0	0
$711$	−23.9547	−0.898370
$712$	0	0
$713$	−4.30974	−0.161401
$714$	0	0
$715$	26.1594	0.978307
$716$	0	0
$717$	−8.74569	−0.326614
$718$	0	0
$719$	14.1273	0.526861	0.263430	−	0.964678i	$-0.415146\pi$
$719$	14.1273	0.526861	0.263430	+	0.964678i	$0.415146\pi$
$720$	0	0
$721$	−44.7895	−1.66805
$722$	0	0
$723$	−7.78501	−0.289528
$724$	0	0
$725$	−1.22216	−0.0453897
$726$	0	0
$727$	−12.6092	−0.467650	−0.233825	−	0.972279i	$-0.575124\pi$
$727$	−12.6092	−0.467650	−0.233825	+	0.972279i	$0.575124\pi$
$728$	0	0
$729$	−27.5004	−1.01853
$730$	0	0
$731$	6.72183	0.248616
$732$	0	0
$733$	4.40552	0.162722	0.0813608	−	0.996685i	$-0.474073\pi$
$733$	4.40552	0.162722	0.0813608	+	0.996685i	$0.474073\pi$
$734$	0	0
$735$	−74.8825	−2.76208
$736$	0	0
$737$	57.8090	2.12942
$738$	0	0
$739$	11.5560	0.425093	0.212546	−	0.977151i	$-0.431824\pi$
$739$	11.5560	0.425093	0.212546	+	0.977151i	$0.431824\pi$
$740$	0	0
$741$	−33.8141	−1.24219
$742$	0	0
$743$	−13.0616	−0.479185	−0.239593	−	0.970874i	$-0.577014\pi$
$743$	−13.0616	−0.479185	−0.239593	+	0.970874i	$0.577014\pi$
$744$	0	0
$745$	16.2784	0.596395
$746$	0	0
$747$	8.22323	0.300872
$748$	0	0
$749$	33.0946	1.20925
$750$	0	0
$751$	27.9003	1.01810	0.509048	−	0.860738i	$-0.329998\pi$
$751$	27.9003	1.01810	0.509048	+	0.860738i	$0.329998\pi$
$752$	0	0
$753$	66.6775	2.42986
$754$	0	0
$755$	−0.562048	−0.0204550
$756$	0	0
$757$	−40.0406	−1.45530	−0.727651	−	0.685948i	$-0.759388\pi$
$757$	−40.0406	−1.45530	−0.727651	+	0.685948i	$0.759388\pi$
$758$	0	0
$759$	25.9225	0.940927
$760$	0	0
$761$	−28.4983	−1.03306	−0.516532	−	0.856268i	$-0.672777\pi$
$761$	−28.4983	−1.03306	−0.516532	+	0.856268i	$0.672777\pi$
$762$	0	0
$763$	−24.6414	−0.892078
$764$	0	0
$765$	−5.23516	−0.189278
$766$	0	0
$767$	−4.13229	−0.149208
$768$	0	0
$769$	21.8105	0.786506	0.393253	−	0.919430i	$-0.371350\pi$
$769$	21.8105	0.786506	0.393253	+	0.919430i	$0.371350\pi$
$770$	0	0
$771$	47.2860	1.70297
$772$	0	0
$773$	−19.9076	−0.716025	−0.358013	−	0.933717i	$-0.616546\pi$
$773$	−19.9076	−0.716025	−0.358013	+	0.933717i	$0.616546\pi$
$774$	0	0
$775$	3.00510	0.107946
$776$	0	0
$777$	−105.863	−3.79781
$778$	0	0
$779$	16.5246	0.592057
$780$	0	0
$781$	−17.6935	−0.633123
$782$	0	0
$783$	0.0416783	0.00148946
$784$	0	0
$785$	−25.7127	−0.917724
$786$	0	0
$787$	11.4842	0.409369	0.204684	−	0.978828i	$-0.434383\pi$
$787$	11.4842	0.409369	0.204684	+	0.978828i	$0.434383\pi$
$788$	0	0
$789$	−17.7087	−0.630446
$790$	0	0
$791$	−7.51369	−0.267156
$792$	0	0
$793$	−60.6887	−2.15512
$794$	0	0
$795$	−16.4888	−0.584796
$796$	0	0
$797$	−3.27174	−0.115891	−0.0579455	−	0.998320i	$-0.518455\pi$
$797$	−3.27174	−0.115891	−0.0579455	+	0.998320i	$0.518455\pi$
$798$	0	0
$799$	0.874202	0.0309270
$800$	0	0
$801$	−50.3382	−1.77861
$802$	0	0
$803$	5.40773	0.190835
$804$	0	0
$805$	24.7485	0.872271
$806$	0	0
$807$	45.9981	1.61921
$808$	0	0
$809$	−52.1988	−1.83521	−0.917606	−	0.397492i	$-0.869881\pi$
$809$	−52.1988	−1.83521	−0.917606	+	0.397492i	$0.869881\pi$
$810$	0	0
$811$	48.3536	1.69792	0.848962	−	0.528454i	$-0.177228\pi$
$811$	48.3536	1.69792	0.848962	+	0.528454i	$0.177228\pi$
$812$	0	0
$813$	−34.5649	−1.21224
$814$	0	0
$815$	−3.83869	−0.134464
$816$	0	0
$817$	22.4032	0.783788
$818$	0	0
$819$	62.1100	2.17030
$820$	0	0
$821$	−11.4031	−0.397970	−0.198985	−	0.980003i	$-0.563765\pi$
$821$	−11.4031	−0.397970	−0.198985	+	0.980003i	$0.563765\pi$
$822$	0	0
$823$	−18.0296	−0.628473	−0.314237	−	0.949345i	$-0.601749\pi$
$823$	−18.0296	−0.628473	−0.314237	+	0.949345i	$0.601749\pi$
$824$	0	0
$825$	−18.0753	−0.629301
$826$	0	0
$827$	−8.02861	−0.279182	−0.139591	−	0.990209i	$-0.544579\pi$
$827$	−8.02861	−0.279182	−0.139591	+	0.990209i	$0.544579\pi$
$828$	0	0
$829$	−11.2256	−0.389883	−0.194941	−	0.980815i	$-0.562452\pi$
$829$	−11.2256	−0.389883	−0.194941	+	0.980815i	$0.562452\pi$
$830$	0	0
$831$	66.6313	2.31141
$832$	0	0
$833$	17.6405	0.611208
$834$	0	0
$835$	−21.6290	−0.748502
$836$	0	0
$837$	−0.102481	−0.00354225
$838$	0	0
$839$	18.6560	0.644078	0.322039	−	0.946726i	$-0.395632\pi$
$839$	18.6560	0.644078	0.322039	+	0.946726i	$0.395632\pi$
$840$	0	0
$841$	−28.6305	−0.987260
$842$	0	0
$843$	62.0685	2.13775
$844$	0	0
$845$	−7.04695	−0.242423
$846$	0	0
$847$	−11.9442	−0.410407
$848$	0	0
$849$	0.128103	0.00439648
$850$	0	0
$851$	25.0481	0.858638
$852$	0	0
$853$	27.7189	0.949078	0.474539	−	0.880234i	$-0.342615\pi$
$853$	27.7189	0.949078	0.474539	+	0.880234i	$0.342615\pi$
$854$	0	0
$855$	−17.4483	−0.596718
$856$	0	0
$857$	−31.9770	−1.09231	−0.546156	−	0.837683i	$-0.683910\pi$
$857$	−31.9770	−1.09231	−0.546156	+	0.837683i	$0.683910\pi$
$858$	0	0
$859$	16.0513	0.547665	0.273832	−	0.961777i	$-0.411709\pi$
$859$	16.0513	0.547665	0.273832	+	0.961777i	$0.411709\pi$
$860$	0	0
$861$	−60.4253	−2.05929
$862$	0	0
$863$	43.3207	1.47465	0.737326	−	0.675537i	$-0.236088\pi$
$863$	43.3207	1.47465	0.737326	+	0.675537i	$0.236088\pi$
$864$	0	0
$865$	32.9197	1.11930
$866$	0	0
$867$	2.45518	0.0833824
$868$	0	0
$869$	−28.9666	−0.982624
$870$	0	0
$871$	−65.2430	−2.21067
$872$	0	0
$873$	33.3886	1.13003
$874$	0	0
$875$	−60.1688	−2.03408
$876$	0	0
$877$	−36.8532	−1.24444	−0.622222	−	0.782841i	$-0.713770\pi$
$877$	−36.8532	−1.24444	−0.622222	+	0.782841i	$0.713770\pi$
$878$	0	0
$879$	12.0523	0.406514
$880$	0	0
$881$	−29.7708	−1.00300	−0.501501	−	0.865157i	$-0.667219\pi$
$881$	−29.7708	−1.00300	−0.501501	+	0.865157i	$0.667219\pi$
$882$	0	0
$883$	−53.0833	−1.78640	−0.893198	−	0.449664i	$-0.851544\pi$
$883$	−53.0833	−1.78640	−0.893198	+	0.449664i	$0.851544\pi$
$884$	0	0
$885$	−4.24491	−0.142691
$886$	0	0
$887$	−0.101562	−0.00341011	−0.00170506	−	0.999999i	$-0.500543\pi$
$887$	−0.101562	−0.00341011	−0.00170506	+	0.999999i	$0.500543\pi$
$888$	0	0
$889$	67.3755	2.25970
$890$	0	0
$891$	−32.6434	−1.09359
$892$	0	0
$893$	2.91363	0.0975008
$894$	0	0
$895$	−26.8160	−0.896361
$896$	0	0
$897$	−29.2560	−0.976829
$898$	0	0
$899$	−0.908424	−0.0302976
$900$	0	0
$901$	3.88436	0.129407
$902$	0	0
$903$	−81.9212	−2.72617
$904$	0	0
$905$	39.5506	1.31471
$906$	0	0
$907$	−41.2978	−1.37127	−0.685635	−	0.727946i	$-0.740475\pi$
$907$	−41.2978	−1.37127	−0.685635	+	0.727946i	$0.740475\pi$
$908$	0	0
$909$	9.52880	0.316050
$910$	0	0
$911$	10.1921	0.337678	0.168839	−	0.985644i	$-0.445998\pi$
$911$	10.1921	0.337678	0.168839	+	0.985644i	$0.445998\pi$
$912$	0	0
$913$	9.94374	0.329090
$914$	0	0
$915$	−62.3427	−2.06099
$916$	0	0
$917$	64.5161	2.13051
$918$	0	0
$919$	−40.8729	−1.34827	−0.674136	−	0.738607i	$-0.735484\pi$
$919$	−40.8729	−1.34827	−0.674136	+	0.738607i	$0.735484\pi$
$920$	0	0
$921$	−17.7752	−0.585712
$922$	0	0
$923$	19.9688	0.657281
$924$	0	0
$925$	−17.4656	−0.574265
$926$	0	0
$927$	27.3210	0.897340
$928$	0	0
$929$	0.278040	0.00912221	0.00456111	−	0.999990i	$-0.498548\pi$
$929$	0.278040	0.00912221	0.00456111	+	0.999990i	$0.498548\pi$
$930$	0	0
$931$	58.7941	1.92690
$932$	0	0
$933$	−2.72242	−0.0891279
$934$	0	0
$935$	−6.33049	−0.207029
$936$	0	0
$937$	−17.2182	−0.562493	−0.281246	−	0.959636i	$-0.590748\pi$
$937$	−17.2182	−0.562493	−0.281246	+	0.959636i	$0.590748\pi$
$938$	0	0
$939$	−82.3332	−2.68684
$940$	0	0
$941$	−24.3054	−0.792333	−0.396167	−	0.918179i	$-0.629660\pi$
$941$	−24.3054	−0.792333	−0.396167	+	0.918179i	$0.629660\pi$
$942$	0	0
$943$	14.2972	0.465580
$944$	0	0
$945$	0.588493	0.0191437
$946$	0	0
$947$	38.1976	1.24126	0.620628	−	0.784105i	$-0.286878\pi$
$947$	38.1976	1.24126	0.620628	+	0.784105i	$0.286878\pi$
$948$	0	0
$949$	−6.10314	−0.198116
$950$	0	0
$951$	−15.1732	−0.492025
$952$	0	0
$953$	−53.7934	−1.74254	−0.871270	−	0.490804i	$-0.836703\pi$
$953$	−53.7934	−1.74254	−0.871270	+	0.490804i	$0.836703\pi$
$954$	0	0
$955$	−31.8677	−1.03122
$956$	0	0
$957$	5.46405	0.176628
$958$	0	0
$959$	−33.9365	−1.09587
$960$	0	0
$961$	−28.7663	−0.927946
$962$	0	0
$963$	−20.1873	−0.650525
$964$	0	0
$965$	−10.7602	−0.346384
$966$	0	0
$967$	−5.61884	−0.180690	−0.0903448	−	0.995911i	$-0.528797\pi$
$967$	−5.61884	−0.180690	−0.0903448	+	0.995911i	$0.528797\pi$
$968$	0	0
$969$	8.18288	0.262872
$970$	0	0
$971$	−42.0473	−1.34936	−0.674681	−	0.738110i	$-0.735719\pi$
$971$	−42.0473	−1.34936	−0.674681	+	0.738110i	$0.735719\pi$
$972$	0	0
$973$	10.6129	0.340235
$974$	0	0
$975$	20.3997	0.653313
$976$	0	0
$977$	−44.1238	−1.41165	−0.705823	−	0.708388i	$-0.749423\pi$
$977$	−44.1238	−1.41165	−0.705823	+	0.708388i	$0.749423\pi$
$978$	0	0
$979$	−60.8703	−1.94542
$980$	0	0
$981$	15.0309	0.479900
$982$	0	0
$983$	2.79265	0.0890716	0.0445358	−	0.999008i	$-0.485819\pi$
$983$	2.79265	0.0890716	0.0445358	+	0.999008i	$0.485819\pi$
$984$	0	0
$985$	−27.4647	−0.875099
$986$	0	0
$987$	−10.6542	−0.339127
$988$	0	0
$989$	19.3833	0.616352
$990$	0	0
$991$	9.38666	0.298177	0.149088	−	0.988824i	$-0.452366\pi$
$991$	9.38666	0.298177	0.149088	+	0.988824i	$0.452366\pi$
$992$	0	0
$993$	−58.8377	−1.86716
$994$	0	0
$995$	−35.5217	−1.12611
$996$	0	0
$997$	−29.0674	−0.920574	−0.460287	−	0.887770i	$-0.652254\pi$
$997$	−29.0674	−0.920574	−0.460287	+	0.887770i	$0.652254\pi$
$998$	0	0
$999$	0.595616	0.0188445

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.18	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.18	✓	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)

\(f(q)\)	\(=\)	\(q+2.45518 q^{3} -1.72896 q^{5} -4.96392 q^{7} +3.02793 q^{9} +O(q^{10})\)	\(q+2.45518 q^{3} -1.72896 q^{5} -4.96392 q^{7} +3.02793 q^{9} +3.66145 q^{11} -4.13229 q^{13} -4.24491 q^{15} +1.00000 q^{17} +3.33290 q^{19} -12.1873 q^{21} +2.88363 q^{23} -2.01070 q^{25} +0.0685696 q^{27} +0.607824 q^{29} -1.49455 q^{31} +8.98953 q^{33} +8.58241 q^{35} +8.68630 q^{37} -10.1455 q^{39} +4.95804 q^{41} +6.72183 q^{43} -5.23516 q^{45} +0.874202 q^{47} +17.6405 q^{49} +2.45518 q^{51} +3.88436 q^{53} -6.33049 q^{55} +8.18288 q^{57} +1.00000 q^{59} +14.6865 q^{61} -15.0304 q^{63} +7.14456 q^{65} +15.7886 q^{67} +7.07985 q^{69} -4.83238 q^{71} +1.47694 q^{73} -4.93665 q^{75} -18.1751 q^{77} -7.91124 q^{79} -8.91543 q^{81} +2.71579 q^{83} -1.72896 q^{85} +1.49232 q^{87} -16.6246 q^{89} +20.5124 q^{91} -3.66940 q^{93} -5.76244 q^{95} +11.0269 q^{97} +11.0866 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 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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