LMFDB - Embedded newform 8024.2.a.y.1.12

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			1.12
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)

$f(q)$	$=$	$q-0.725294 q^{3} -2.56147 q^{5} -0.934698 q^{7} -2.47395 q^{9} +O(q^{10})$	$q-0.725294 q^{3} -2.56147 q^{5} -0.934698 q^{7} -2.47395 q^{9} +0.446734 q^{11} +5.91850 q^{13} +1.85782 q^{15} -1.00000 q^{17} -3.85324 q^{19} +0.677931 q^{21} -7.09996 q^{23} +1.56115 q^{25} +3.97022 q^{27} +9.38960 q^{29} -8.41148 q^{31} -0.324014 q^{33} +2.39421 q^{35} +5.65988 q^{37} -4.29266 q^{39} +8.92656 q^{41} +9.40875 q^{43} +6.33696 q^{45} +9.71641 q^{47} -6.12634 q^{49} +0.725294 q^{51} -4.68648 q^{53} -1.14430 q^{55} +2.79473 q^{57} -1.00000 q^{59} +3.41446 q^{61} +2.31240 q^{63} -15.1601 q^{65} +1.82375 q^{67} +5.14956 q^{69} +0.929049 q^{71} -12.9737 q^{73} -1.13230 q^{75} -0.417561 q^{77} +10.8484 q^{79} +4.54226 q^{81} -13.5254 q^{83} +2.56147 q^{85} -6.81023 q^{87} +8.98484 q^{89} -5.53202 q^{91} +6.10080 q^{93} +9.86997 q^{95} -0.541858 q^{97} -1.10520 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.725294	−0.418749	−0.209374	−	0.977836i	$-0.567143\pi$
$3$	−0.725294	−0.418749	−0.209374	+	0.977836i	$0.567143\pi$
$4$	0	0
$5$	−2.56147	−1.14553	−0.572763	−	0.819721i	$-0.694129\pi$
$5$	−2.56147	−1.14553	−0.572763	+	0.819721i	$0.694129\pi$
$6$	0	0
$7$	−0.934698	−0.353283	−0.176641	−	0.984275i	$-0.556523\pi$
$7$	−0.934698	−0.353283	−0.176641	+	0.984275i	$0.556523\pi$
$8$	0	0
$9$	−2.47395	−0.824649
$10$	0	0
$11$	0.446734	0.134695	0.0673477	−	0.997730i	$-0.478546\pi$
$11$	0.446734	0.134695	0.0673477	+	0.997730i	$0.478546\pi$
$12$	0	0
$13$	5.91850	1.64150	0.820749	−	0.571289i	$-0.193556\pi$
$13$	5.91850	1.64150	0.820749	+	0.571289i	$0.193556\pi$
$14$	0	0
$15$	1.85782	0.479688
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−3.85324	−0.883993	−0.441997	−	0.897017i	$-0.645730\pi$
$19$	−3.85324	−0.883993	−0.441997	+	0.897017i	$0.645730\pi$
$20$	0	0
$21$	0.677931	0.147937
$22$	0	0
$23$	−7.09996	−1.48044	−0.740222	−	0.672362i	$-0.765280\pi$
$23$	−7.09996	−1.48044	−0.740222	+	0.672362i	$0.765280\pi$
$24$	0	0
$25$	1.56115	0.312231
$26$	0	0
$27$	3.97022	0.764070
$28$	0	0
$29$	9.38960	1.74361	0.871803	−	0.489857i	$-0.162951\pi$
$29$	9.38960	1.74361	0.871803	+	0.489857i	$0.162951\pi$
$30$	0	0
$31$	−8.41148	−1.51075	−0.755373	−	0.655295i	$-0.772544\pi$
$31$	−8.41148	−1.51075	−0.755373	+	0.655295i	$0.772544\pi$
$32$	0	0
$33$	−0.324014	−0.0564035
$34$	0	0
$35$	2.39421	0.404695
$36$	0	0
$37$	5.65988	0.930478	0.465239	−	0.885185i	$-0.345968\pi$
$37$	5.65988	0.930478	0.465239	+	0.885185i	$0.345968\pi$
$38$	0	0
$39$	−4.29266	−0.687376
$40$	0	0
$41$	8.92656	1.39409	0.697047	−	0.717025i	$-0.254497\pi$
$41$	8.92656	1.39409	0.697047	+	0.717025i	$0.254497\pi$
$42$	0	0
$43$	9.40875	1.43482	0.717411	−	0.696651i	$-0.245327\pi$
$43$	9.40875	1.43482	0.717411	+	0.696651i	$0.245327\pi$
$44$	0	0
$45$	6.33696	0.944658
$46$	0	0
$47$	9.71641	1.41728	0.708642	−	0.705569i	$-0.249308\pi$
$47$	9.71641	1.41728	0.708642	+	0.705569i	$0.249308\pi$
$48$	0	0
$49$	−6.12634	−0.875191
$50$	0	0
$51$	0.725294	0.101562
$52$	0	0
$53$	−4.68648	−0.643738	−0.321869	−	0.946784i	$-0.604311\pi$
$53$	−4.68648	−0.643738	−0.321869	+	0.946784i	$0.604311\pi$
$54$	0	0
$55$	−1.14430	−0.154297
$56$	0	0
$57$	2.79473	0.370171
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	3.41446	0.437176	0.218588	−	0.975817i	$-0.429855\pi$
$61$	3.41446	0.437176	0.218588	+	0.975817i	$0.429855\pi$
$62$	0	0
$63$	2.31240	0.291334
$64$	0	0
$65$	−15.1601	−1.88038
$66$	0	0
$67$	1.82375	0.222807	0.111403	−	0.993775i	$-0.464465\pi$
$67$	1.82375	0.222807	0.111403	+	0.993775i	$0.464465\pi$
$68$	0	0
$69$	5.14956	0.619935
$70$	0	0
$71$	0.929049	0.110258	0.0551289	−	0.998479i	$-0.482443\pi$
$71$	0.929049	0.110258	0.0551289	+	0.998479i	$0.482443\pi$
$72$	0	0
$73$	−12.9737	−1.51846	−0.759230	−	0.650822i	$-0.774424\pi$
$73$	−12.9737	−1.51846	−0.759230	+	0.650822i	$0.774424\pi$
$74$	0	0
$75$	−1.13230	−0.130746
$76$	0	0
$77$	−0.417561	−0.0475855
$78$	0	0
$79$	10.8484	1.22054	0.610268	−	0.792195i	$-0.291062\pi$
$79$	10.8484	1.22054	0.610268	+	0.792195i	$0.291062\pi$
$80$	0	0
$81$	4.54226	0.504696
$82$	0	0
$83$	−13.5254	−1.48461	−0.742305	−	0.670062i	$-0.766267\pi$
$83$	−13.5254	−1.48461	−0.742305	+	0.670062i	$0.766267\pi$
$84$	0	0
$85$	2.56147	0.277831
$86$	0	0
$87$	−6.81023	−0.730133
$88$	0	0
$89$	8.98484	0.952392	0.476196	−	0.879339i	$-0.342015\pi$
$89$	8.98484	0.952392	0.476196	+	0.879339i	$0.342015\pi$
$90$	0	0
$91$	−5.53202	−0.579913
$92$	0	0
$93$	6.10080	0.632623
$94$	0	0
$95$	9.86997	1.01264
$96$	0	0
$97$	−0.541858	−0.0550174	−0.0275087	−	0.999622i	$-0.508757\pi$
$97$	−0.541858	−0.0550174	−0.0275087	+	0.999622i	$0.508757\pi$
$98$	0	0
$99$	−1.10520	−0.111076
$100$	0	0
$101$	−0.720239	−0.0716664	−0.0358332	−	0.999358i	$-0.511409\pi$
$101$	−0.720239	−0.0716664	−0.0358332	+	0.999358i	$0.511409\pi$
$102$	0	0
$103$	−0.458495	−0.0451769	−0.0225884	−	0.999745i	$-0.507191\pi$
$103$	−0.458495	−0.0451769	−0.0225884	+	0.999745i	$0.507191\pi$
$104$	0	0
$105$	−1.73650	−0.169465
$106$	0	0
$107$	−8.74047	−0.844973	−0.422487	−	0.906369i	$-0.638843\pi$
$107$	−8.74047	−0.844973	−0.422487	+	0.906369i	$0.638843\pi$
$108$	0	0
$109$	−5.58873	−0.535304	−0.267652	−	0.963516i	$-0.586248\pi$
$109$	−5.58873	−0.535304	−0.267652	+	0.963516i	$0.586248\pi$
$110$	0	0
$111$	−4.10508	−0.389637
$112$	0	0
$113$	4.04937	0.380933	0.190466	−	0.981694i	$-0.439000\pi$
$113$	4.04937	0.380933	0.190466	+	0.981694i	$0.439000\pi$
$114$	0	0
$115$	18.1864	1.69589
$116$	0	0
$117$	−14.6421	−1.35366
$118$	0	0
$119$	0.934698	0.0856837
$120$	0	0
$121$	−10.8004	−0.981857
$122$	0	0
$123$	−6.47439	−0.583776
$124$	0	0
$125$	8.80852	0.787858
$126$	0	0
$127$	−4.01400	−0.356185	−0.178093	−	0.984014i	$-0.556993\pi$
$127$	−4.01400	−0.356185	−0.178093	+	0.984014i	$0.556993\pi$
$128$	0	0
$129$	−6.82412	−0.600830
$130$	0	0
$131$	21.4968	1.87818	0.939092	−	0.343665i	$-0.111669\pi$
$131$	21.4968	1.87818	0.939092	+	0.343665i	$0.111669\pi$
$132$	0	0
$133$	3.60161	0.312300
$134$	0	0
$135$	−10.1696	−0.875262
$136$	0	0
$137$	−7.82761	−0.668758	−0.334379	−	0.942439i	$-0.608526\pi$
$137$	−7.82761	−0.668758	−0.334379	+	0.942439i	$0.608526\pi$
$138$	0	0
$139$	−2.27916	−0.193316	−0.0966578	−	0.995318i	$-0.530815\pi$
$139$	−2.27916	−0.193316	−0.0966578	+	0.995318i	$0.530815\pi$
$140$	0	0
$141$	−7.04725	−0.593486
$142$	0	0
$143$	2.64400	0.221102
$144$	0	0
$145$	−24.0512	−1.99735
$146$	0	0
$147$	4.44340	0.366485
$148$	0	0
$149$	−14.8308	−1.21499	−0.607493	−	0.794325i	$-0.707825\pi$
$149$	−14.8308	−1.21499	−0.607493	+	0.794325i	$0.707825\pi$
$150$	0	0
$151$	−5.33764	−0.434371	−0.217186	−	0.976130i	$-0.569688\pi$
$151$	−5.33764	−0.434371	−0.217186	+	0.976130i	$0.569688\pi$
$152$	0	0
$153$	2.47395	0.200007
$154$	0	0
$155$	21.5458	1.73060
$156$	0	0
$157$	1.53179	0.122250	0.0611250	−	0.998130i	$-0.480531\pi$
$157$	1.53179	0.122250	0.0611250	+	0.998130i	$0.480531\pi$
$158$	0	0
$159$	3.39908	0.269565
$160$	0	0
$161$	6.63632	0.523016
$162$	0	0
$163$	20.2166	1.58348	0.791742	−	0.610855i	$-0.209174\pi$
$163$	20.2166	1.58348	0.791742	+	0.610855i	$0.209174\pi$
$164$	0	0
$165$	0.829953	0.0646117
$166$	0	0
$167$	0.942783	0.0729547	0.0364774	−	0.999334i	$-0.488386\pi$
$167$	0.942783	0.0729547	0.0364774	+	0.999334i	$0.488386\pi$
$168$	0	0
$169$	22.0287	1.69452
$170$	0	0
$171$	9.53271	0.728985
$172$	0	0
$173$	7.22155	0.549045	0.274522	−	0.961581i	$-0.411480\pi$
$173$	7.22155	0.549045	0.274522	+	0.961581i	$0.411480\pi$
$174$	0	0
$175$	−1.45921	−0.110306
$176$	0	0
$177$	0.725294	0.0545165
$178$	0	0
$179$	5.76976	0.431252	0.215626	−	0.976476i	$-0.430821\pi$
$179$	5.76976	0.431252	0.215626	+	0.976476i	$0.430821\pi$
$180$	0	0
$181$	−12.5556	−0.933253	−0.466626	−	0.884454i	$-0.654531\pi$
$181$	−12.5556	−0.933253	−0.466626	+	0.884454i	$0.654531\pi$
$182$	0	0
$183$	−2.47649	−0.183067
$184$	0	0
$185$	−14.4976	−1.06589
$186$	0	0
$187$	−0.446734	−0.0326684
$188$	0	0
$189$	−3.71096	−0.269933
$190$	0	0
$191$	−11.3447	−0.820876	−0.410438	−	0.911888i	$-0.634624\pi$
$191$	−11.3447	−0.820876	−0.410438	+	0.911888i	$0.634624\pi$
$192$	0	0
$193$	19.2719	1.38722	0.693612	−	0.720349i	$-0.256018\pi$
$193$	19.2719	1.38722	0.693612	+	0.720349i	$0.256018\pi$
$194$	0	0
$195$	10.9955	0.787407
$196$	0	0
$197$	23.7447	1.69174	0.845868	−	0.533392i	$-0.179083\pi$
$197$	23.7447	1.69174	0.845868	+	0.533392i	$0.179083\pi$
$198$	0	0
$199$	4.02230	0.285133	0.142567	−	0.989785i	$-0.454464\pi$
$199$	4.02230	0.285133	0.142567	+	0.989785i	$0.454464\pi$
$200$	0	0
$201$	−1.32276	−0.0933001
$202$	0	0
$203$	−8.77645	−0.615986
$204$	0	0
$205$	−22.8652	−1.59697
$206$	0	0
$207$	17.5649	1.22085
$208$	0	0
$209$	−1.72137	−0.119070
$210$	0	0
$211$	−22.5620	−1.55323	−0.776614	−	0.629976i	$-0.783065\pi$
$211$	−22.5620	−1.55323	−0.776614	+	0.629976i	$0.783065\pi$
$212$	0	0
$213$	−0.673834	−0.0461703
$214$	0	0
$215$	−24.1003	−1.64363
$216$	0	0
$217$	7.86219	0.533720
$218$	0	0
$219$	9.40977	0.635854
$220$	0	0
$221$	−5.91850	−0.398122
$222$	0	0
$223$	−10.1726	−0.681207	−0.340603	−	0.940207i	$-0.610631\pi$
$223$	−10.1726	−0.681207	−0.340603	+	0.940207i	$0.610631\pi$
$224$	0	0
$225$	−3.86221	−0.257481
$226$	0	0
$227$	−15.1402	−1.00489	−0.502444	−	0.864610i	$-0.667566\pi$
$227$	−15.1402	−1.00489	−0.502444	+	0.864610i	$0.667566\pi$
$228$	0	0
$229$	17.4377	1.15231	0.576157	−	0.817339i	$-0.304552\pi$
$229$	17.4377	1.15231	0.576157	+	0.817339i	$0.304552\pi$
$230$	0	0
$231$	0.302855	0.0199264
$232$	0	0
$233$	19.8800	1.30238	0.651192	−	0.758913i	$-0.274269\pi$
$233$	19.8800	1.30238	0.651192	+	0.758913i	$0.274269\pi$
$234$	0	0
$235$	−24.8883	−1.62354
$236$	0	0
$237$	−7.86826	−0.511098
$238$	0	0
$239$	−23.4315	−1.51566	−0.757830	−	0.652452i	$-0.773740\pi$
$239$	−23.4315	−1.51566	−0.757830	+	0.652452i	$0.773740\pi$
$240$	0	0
$241$	−29.5422	−1.90298	−0.951491	−	0.307677i	$-0.900448\pi$
$241$	−29.5422	−1.90298	−0.951491	+	0.307677i	$0.900448\pi$
$242$	0	0
$243$	−15.2051	−0.975411
$244$	0	0
$245$	15.6925	1.00255
$246$	0	0
$247$	−22.8054	−1.45107
$248$	0	0
$249$	9.80993	0.621679
$250$	0	0
$251$	5.79689	0.365896	0.182948	−	0.983123i	$-0.441436\pi$
$251$	5.79689	0.365896	0.182948	+	0.983123i	$0.441436\pi$
$252$	0	0
$253$	−3.17179	−0.199409
$254$	0	0
$255$	−1.85782	−0.116341
$256$	0	0
$257$	−18.0796	−1.12778	−0.563888	−	0.825851i	$-0.690695\pi$
$257$	−18.0796	−1.12778	−0.563888	+	0.825851i	$0.690695\pi$
$258$	0	0
$259$	−5.29028	−0.328722
$260$	0	0
$261$	−23.2294	−1.43786
$262$	0	0
$263$	−26.5121	−1.63481	−0.817404	−	0.576065i	$-0.804588\pi$
$263$	−26.5121	−1.63481	−0.817404	+	0.576065i	$0.804588\pi$
$264$	0	0
$265$	12.0043	0.737419
$266$	0	0
$267$	−6.51666	−0.398813
$268$	0	0
$269$	30.1028	1.83540	0.917699	−	0.397277i	$-0.130045\pi$
$269$	30.1028	1.83540	0.917699	+	0.397277i	$0.130045\pi$
$270$	0	0
$271$	10.0017	0.607560	0.303780	−	0.952742i	$-0.401751\pi$
$271$	10.0017	0.607560	0.303780	+	0.952742i	$0.401751\pi$
$272$	0	0
$273$	4.01234	0.242838
$274$	0	0
$275$	0.697420	0.0420560
$276$	0	0
$277$	−14.0171	−0.842209	−0.421104	−	0.907012i	$-0.638357\pi$
$277$	−14.0171	−0.842209	−0.421104	+	0.907012i	$0.638357\pi$
$278$	0	0
$279$	20.8096	1.24584
$280$	0	0
$281$	−1.81149	−0.108064	−0.0540322	−	0.998539i	$-0.517207\pi$
$281$	−1.81149	−0.108064	−0.0540322	+	0.998539i	$0.517207\pi$
$282$	0	0
$283$	−18.5608	−1.10333	−0.551664	−	0.834067i	$-0.686007\pi$
$283$	−18.5608	−1.10333	−0.551664	+	0.834067i	$0.686007\pi$
$284$	0	0
$285$	−7.15864	−0.424041
$286$	0	0
$287$	−8.34364	−0.492510
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0.393007	0.0230385
$292$	0	0
$293$	−19.6608	−1.14859	−0.574297	−	0.818647i	$-0.694724\pi$
$293$	−19.6608	−1.14859	−0.574297	+	0.818647i	$0.694724\pi$
$294$	0	0
$295$	2.56147	0.149135
$296$	0	0
$297$	1.77363	0.102917
$298$	0	0
$299$	−42.0212	−2.43015
$300$	0	0
$301$	−8.79435	−0.506898
$302$	0	0
$303$	0.522385	0.0300102
$304$	0	0
$305$	−8.74605	−0.500797
$306$	0	0
$307$	−16.5537	−0.944770	−0.472385	−	0.881392i	$-0.656607\pi$
$307$	−16.5537	−0.944770	−0.472385	+	0.881392i	$0.656607\pi$
$308$	0	0
$309$	0.332544	0.0189178
$310$	0	0
$311$	−35.0470	−1.98733	−0.993667	−	0.112362i	$-0.964158\pi$
$311$	−35.0470	−1.98733	−0.993667	+	0.112362i	$0.964158\pi$
$312$	0	0
$313$	15.3844	0.869579	0.434790	−	0.900532i	$-0.356823\pi$
$313$	15.3844	0.869579	0.434790	+	0.900532i	$0.356823\pi$
$314$	0	0
$315$	−5.92314	−0.333731
$316$	0	0
$317$	8.24780	0.463243	0.231621	−	0.972806i	$-0.425597\pi$
$317$	8.24780	0.463243	0.231621	+	0.972806i	$0.425597\pi$
$318$	0	0
$319$	4.19465	0.234856
$320$	0	0
$321$	6.33941	0.353832
$322$	0	0
$323$	3.85324	0.214400
$324$	0	0
$325$	9.23970	0.512526
$326$	0	0
$327$	4.05348	0.224158
$328$	0	0
$329$	−9.08191	−0.500702
$330$	0	0
$331$	−36.3521	−1.99809	−0.999046	−	0.0436765i	$-0.986093\pi$
$331$	−36.3521	−1.99809	−0.999046	+	0.0436765i	$0.986093\pi$
$332$	0	0
$333$	−14.0022	−0.767318
$334$	0	0
$335$	−4.67150	−0.255231
$336$	0	0
$337$	−12.2859	−0.669255	−0.334627	−	0.942351i	$-0.608610\pi$
$337$	−12.2859	−0.669255	−0.334627	+	0.942351i	$0.608610\pi$
$338$	0	0
$339$	−2.93698	−0.159515
$340$	0	0
$341$	−3.75769	−0.203490
$342$	0	0
$343$	12.2692	0.662473
$344$	0	0
$345$	−13.1905	−0.710152
$346$	0	0
$347$	−8.40762	−0.451345	−0.225672	−	0.974203i	$-0.572458\pi$
$347$	−8.40762	−0.451345	−0.225672	+	0.974203i	$0.572458\pi$
$348$	0	0
$349$	−17.6090	−0.942586	−0.471293	−	0.881977i	$-0.656213\pi$
$349$	−17.6090	−0.942586	−0.471293	+	0.881977i	$0.656213\pi$
$350$	0	0
$351$	23.4978	1.25422
$352$	0	0
$353$	−2.29316	−0.122052	−0.0610262	−	0.998136i	$-0.519437\pi$
$353$	−2.29316	−0.122052	−0.0610262	+	0.998136i	$0.519437\pi$
$354$	0	0
$355$	−2.37974	−0.126303
$356$	0	0
$357$	−0.677931	−0.0358799
$358$	0	0
$359$	6.10348	0.322129	0.161065	−	0.986944i	$-0.448507\pi$
$359$	6.10348	0.322129	0.161065	+	0.986944i	$0.448507\pi$
$360$	0	0
$361$	−4.15256	−0.218556
$362$	0	0
$363$	7.83349	0.411152
$364$	0	0
$365$	33.2319	1.73944
$366$	0	0
$367$	−28.4841	−1.48686	−0.743428	−	0.668816i	$-0.766802\pi$
$367$	−28.4841	−1.48686	−0.743428	+	0.668816i	$0.766802\pi$
$368$	0	0
$369$	−22.0839	−1.14964
$370$	0	0
$371$	4.38045	0.227421
$372$	0	0
$373$	−32.6127	−1.68862	−0.844311	−	0.535854i	$-0.819990\pi$
$373$	−32.6127	−1.68862	−0.844311	+	0.535854i	$0.819990\pi$
$374$	0	0
$375$	−6.38877	−0.329915
$376$	0	0
$377$	55.5724	2.86212
$378$	0	0
$379$	7.35687	0.377897	0.188948	−	0.981987i	$-0.439492\pi$
$379$	7.35687	0.377897	0.188948	+	0.981987i	$0.439492\pi$
$380$	0	0
$381$	2.91134	0.149152
$382$	0	0
$383$	24.4030	1.24693	0.623467	−	0.781850i	$-0.285724\pi$
$383$	24.4030	1.24693	0.623467	+	0.781850i	$0.285724\pi$
$384$	0	0
$385$	1.06957	0.0545105
$386$	0	0
$387$	−23.2768	−1.18322
$388$	0	0
$389$	20.4230	1.03549	0.517744	−	0.855536i	$-0.326772\pi$
$389$	20.4230	1.03549	0.517744	+	0.855536i	$0.326772\pi$
$390$	0	0
$391$	7.09996	0.359061
$392$	0	0
$393$	−15.5915	−0.786488
$394$	0	0
$395$	−27.7878	−1.39816
$396$	0	0
$397$	−38.1133	−1.91285	−0.956426	−	0.291975i	$-0.905688\pi$
$397$	−38.1133	−1.91285	−0.956426	+	0.291975i	$0.905688\pi$
$398$	0	0
$399$	−2.61223	−0.130775
$400$	0	0
$401$	−9.01541	−0.450208	−0.225104	−	0.974335i	$-0.572272\pi$
$401$	−9.01541	−0.450208	−0.225104	+	0.974335i	$0.572272\pi$
$402$	0	0
$403$	−49.7834	−2.47989
$404$	0	0
$405$	−11.6349	−0.578142
$406$	0	0
$407$	2.52846	0.125331
$408$	0	0
$409$	12.9479	0.640233	0.320116	−	0.947378i	$-0.396278\pi$
$409$	12.9479	0.640233	0.320116	+	0.947378i	$0.396278\pi$
$410$	0	0
$411$	5.67732	0.280042
$412$	0	0
$413$	0.934698	0.0459935
$414$	0	0
$415$	34.6451	1.70066
$416$	0	0
$417$	1.65306	0.0809507
$418$	0	0
$419$	−17.1407	−0.837378	−0.418689	−	0.908130i	$-0.637510\pi$
$419$	−17.1407	−0.837378	−0.418689	+	0.908130i	$0.637510\pi$
$420$	0	0
$421$	11.0403	0.538073	0.269036	−	0.963130i	$-0.413295\pi$
$421$	11.0403	0.538073	0.269036	+	0.963130i	$0.413295\pi$
$422$	0	0
$423$	−24.0379	−1.16876
$424$	0	0
$425$	−1.56115	−0.0757271
$426$	0	0
$427$	−3.19149	−0.154447
$428$	0	0
$429$	−1.91768	−0.0925863
$430$	0	0
$431$	40.0431	1.92881	0.964404	−	0.264431i	$-0.0851843\pi$
$431$	40.0431	1.92881	0.964404	+	0.264431i	$0.0851843\pi$
$432$	0	0
$433$	−13.6185	−0.654464	−0.327232	−	0.944944i	$-0.606116\pi$
$433$	−13.6185	−0.654464	−0.327232	+	0.944944i	$0.606116\pi$
$434$	0	0
$435$	17.4442	0.836387
$436$	0	0
$437$	27.3579	1.30870
$438$	0	0
$439$	3.86864	0.184640	0.0923200	−	0.995729i	$-0.470572\pi$
$439$	3.86864	0.184640	0.0923200	+	0.995729i	$0.470572\pi$
$440$	0	0
$441$	15.1562	0.721726
$442$	0	0
$443$	−11.4390	−0.543486	−0.271743	−	0.962370i	$-0.587600\pi$
$443$	−11.4390	−0.543486	−0.271743	+	0.962370i	$0.587600\pi$
$444$	0	0
$445$	−23.0145	−1.09099
$446$	0	0
$447$	10.7567	0.508774
$448$	0	0
$449$	8.08590	0.381597	0.190799	−	0.981629i	$-0.438892\pi$
$449$	8.08590	0.381597	0.190799	+	0.981629i	$0.438892\pi$
$450$	0	0
$451$	3.98780	0.187778
$452$	0	0
$453$	3.87136	0.181893
$454$	0	0
$455$	14.1701	0.664306
$456$	0	0
$457$	−2.33261	−0.109115	−0.0545574	−	0.998511i	$-0.517375\pi$
$457$	−2.33261	−0.109115	−0.0545574	+	0.998511i	$0.517375\pi$
$458$	0	0
$459$	−3.97022	−0.185314
$460$	0	0
$461$	0.470093	0.0218944	0.0109472	−	0.999940i	$-0.496515\pi$
$461$	0.470093	0.0218944	0.0109472	+	0.999940i	$0.496515\pi$
$462$	0	0
$463$	−13.4557	−0.625341	−0.312671	−	0.949862i	$-0.601224\pi$
$463$	−13.4557	−0.625341	−0.312671	+	0.949862i	$0.601224\pi$
$464$	0	0
$465$	−15.6270	−0.724687
$466$	0	0
$467$	15.5956	0.721679	0.360839	−	0.932628i	$-0.382490\pi$
$467$	15.5956	0.721679	0.360839	+	0.932628i	$0.382490\pi$
$468$	0	0
$469$	−1.70466	−0.0787138
$470$	0	0
$471$	−1.11100	−0.0511921
$472$	0	0
$473$	4.20321	0.193264
$474$	0	0
$475$	−6.01550	−0.276010
$476$	0	0
$477$	11.5941	0.530858
$478$	0	0
$479$	−11.1953	−0.511528	−0.255764	−	0.966739i	$-0.582327\pi$
$479$	−11.1953	−0.511528	−0.255764	+	0.966739i	$0.582327\pi$
$480$	0	0
$481$	33.4980	1.52738
$482$	0	0
$483$	−4.81329	−0.219012
$484$	0	0
$485$	1.38796	0.0630239
$486$	0	0
$487$	−30.9430	−1.40216	−0.701080	−	0.713082i	$-0.747299\pi$
$487$	−30.9430	−1.40216	−0.701080	+	0.713082i	$0.747299\pi$
$488$	0	0
$489$	−14.6630	−0.663083
$490$	0	0
$491$	−15.1379	−0.683165	−0.341582	−	0.939852i	$-0.610963\pi$
$491$	−15.1379	−0.683165	−0.341582	+	0.939852i	$0.610963\pi$
$492$	0	0
$493$	−9.38960	−0.422886
$494$	0	0
$495$	2.83093	0.127241
$496$	0	0
$497$	−0.868380	−0.0389522
$498$	0	0
$499$	−39.0950	−1.75013	−0.875065	−	0.484005i	$-0.839182\pi$
$499$	−39.0950	−1.75013	−0.875065	+	0.484005i	$0.839182\pi$
$500$	0	0
$501$	−0.683795	−0.0305497
$502$	0	0
$503$	−22.6860	−1.01152	−0.505759	−	0.862675i	$-0.668787\pi$
$503$	−22.6860	−1.01152	−0.505759	+	0.862675i	$0.668787\pi$
$504$	0	0
$505$	1.84487	0.0820958
$506$	0	0
$507$	−15.9773	−0.709577
$508$	0	0
$509$	−31.0842	−1.37778	−0.688891	−	0.724865i	$-0.741902\pi$
$509$	−31.0842	−1.37778	−0.688891	+	0.724865i	$0.741902\pi$
$510$	0	0
$511$	12.1265	0.536446
$512$	0	0
$513$	−15.2982	−0.675433
$514$	0	0
$515$	1.17442	0.0517513
$516$	0	0
$517$	4.34065	0.190901
$518$	0	0
$519$	−5.23775	−0.229912
$520$	0	0
$521$	41.9487	1.83780	0.918902	−	0.394486i	$-0.129077\pi$
$521$	41.9487	1.83780	0.918902	+	0.394486i	$0.129077\pi$
$522$	0	0
$523$	2.75418	0.120432	0.0602160	−	0.998185i	$-0.480821\pi$
$523$	2.75418	0.120432	0.0602160	+	0.998185i	$0.480821\pi$
$524$	0	0
$525$	1.05836	0.0461904
$526$	0	0
$527$	8.41148	0.366410
$528$	0	0
$529$	27.4095	1.19172
$530$	0	0
$531$	2.47395	0.107360
$532$	0	0
$533$	52.8319	2.28840
$534$	0	0
$535$	22.3885	0.967939
$536$	0	0
$537$	−4.18477	−0.180586
$538$	0	0
$539$	−2.73684	−0.117884
$540$	0	0
$541$	−19.8206	−0.852153	−0.426076	−	0.904687i	$-0.640104\pi$
$541$	−19.8206	−0.852153	−0.426076	+	0.904687i	$0.640104\pi$
$542$	0	0
$543$	9.10653	0.390799
$544$	0	0
$545$	14.3154	0.613204
$546$	0	0
$547$	−11.1238	−0.475619	−0.237809	−	0.971312i	$-0.576429\pi$
$547$	−11.1238	−0.475619	−0.237809	+	0.971312i	$0.576429\pi$
$548$	0	0
$549$	−8.44719	−0.360517
$550$	0	0
$551$	−36.1804	−1.54134
$552$	0	0
$553$	−10.1399	−0.431194
$554$	0	0
$555$	10.5151	0.446339
$556$	0	0
$557$	−15.3609	−0.650861	−0.325430	−	0.945566i	$-0.605509\pi$
$557$	−15.3609	−0.650861	−0.325430	+	0.945566i	$0.605509\pi$
$558$	0	0
$559$	55.6857	2.35526
$560$	0	0
$561$	0.324014	0.0136799
$562$	0	0
$563$	1.02740	0.0432996	0.0216498	−	0.999766i	$-0.493108\pi$
$563$	1.02740	0.0432996	0.0216498	+	0.999766i	$0.493108\pi$
$564$	0	0
$565$	−10.3724	−0.436368
$566$	0	0
$567$	−4.24565	−0.178300
$568$	0	0
$569$	−2.07739	−0.0870888	−0.0435444	−	0.999051i	$-0.513865\pi$
$569$	−2.07739	−0.0870888	−0.0435444	+	0.999051i	$0.513865\pi$
$570$	0	0
$571$	23.5776	0.986690	0.493345	−	0.869834i	$-0.335774\pi$
$571$	23.5776	0.986690	0.493345	+	0.869834i	$0.335774\pi$
$572$	0	0
$573$	8.22828	0.343741
$574$	0	0
$575$	−11.0841	−0.462240
$576$	0	0
$577$	10.1090	0.420843	0.210421	−	0.977611i	$-0.432516\pi$
$577$	10.1090	0.420843	0.210421	+	0.977611i	$0.432516\pi$
$578$	0	0
$579$	−13.9778	−0.580898
$580$	0	0
$581$	12.6422	0.524487
$582$	0	0
$583$	−2.09361	−0.0867085
$584$	0	0
$585$	37.5053	1.55065
$586$	0	0
$587$	−31.6779	−1.30749	−0.653743	−	0.756717i	$-0.726802\pi$
$587$	−31.6779	−1.30749	−0.653743	+	0.756717i	$0.726802\pi$
$588$	0	0
$589$	32.4114	1.33549
$590$	0	0
$591$	−17.2219	−0.708413
$592$	0	0
$593$	42.3183	1.73780	0.868902	−	0.494983i	$-0.164826\pi$
$593$	42.3183	1.73780	0.868902	+	0.494983i	$0.164826\pi$
$594$	0	0
$595$	−2.39421	−0.0981529
$596$	0	0
$597$	−2.91735	−0.119399
$598$	0	0
$599$	−17.0153	−0.695225	−0.347613	−	0.937638i	$-0.613008\pi$
$599$	−17.0153	−0.695225	−0.347613	+	0.937638i	$0.613008\pi$
$600$	0	0
$601$	−13.5937	−0.554499	−0.277250	−	0.960798i	$-0.589423\pi$
$601$	−13.5937	−0.554499	−0.277250	+	0.960798i	$0.589423\pi$
$602$	0	0
$603$	−4.51187	−0.183738
$604$	0	0
$605$	27.6650	1.12474
$606$	0	0
$607$	−33.1495	−1.34550	−0.672748	−	0.739872i	$-0.734886\pi$
$607$	−33.1495	−1.34550	−0.672748	+	0.739872i	$0.734886\pi$
$608$	0	0
$609$	6.36551	0.257943
$610$	0	0
$611$	57.5066	2.32647
$612$	0	0
$613$	0.654037	0.0264163	0.0132081	−	0.999913i	$-0.495796\pi$
$613$	0.654037	0.0264163	0.0132081	+	0.999913i	$0.495796\pi$
$614$	0	0
$615$	16.5840	0.668731
$616$	0	0
$617$	−3.54471	−0.142705	−0.0713524	−	0.997451i	$-0.522731\pi$
$617$	−3.54471	−0.142705	−0.0713524	+	0.997451i	$0.522731\pi$
$618$	0	0
$619$	37.2912	1.49886	0.749430	−	0.662084i	$-0.230328\pi$
$619$	37.2912	1.49886	0.749430	+	0.662084i	$0.230328\pi$
$620$	0	0
$621$	−28.1884	−1.13116
$622$	0	0
$623$	−8.39812	−0.336464
$624$	0	0
$625$	−30.3686	−1.21474
$626$	0	0
$627$	1.24850	0.0498603
$628$	0	0
$629$	−5.65988	−0.225674
$630$	0	0
$631$	32.2740	1.28481	0.642404	−	0.766366i	$-0.277937\pi$
$631$	32.2740	1.28481	0.642404	+	0.766366i	$0.277937\pi$
$632$	0	0
$633$	16.3641	0.650413
$634$	0	0
$635$	10.2818	0.408020
$636$	0	0
$637$	−36.2588	−1.43662
$638$	0	0
$639$	−2.29842	−0.0909240
$640$	0	0
$641$	−4.17268	−0.164811	−0.0824055	−	0.996599i	$-0.526260\pi$
$641$	−4.17268	−0.164811	−0.0824055	+	0.996599i	$0.526260\pi$
$642$	0	0
$643$	6.73630	0.265654	0.132827	−	0.991139i	$-0.457595\pi$
$643$	6.73630	0.265654	0.132827	+	0.991139i	$0.457595\pi$
$644$	0	0
$645$	17.4798	0.688267
$646$	0	0
$647$	16.9090	0.664762	0.332381	−	0.943145i	$-0.392148\pi$
$647$	16.9090	0.664762	0.332381	+	0.943145i	$0.392148\pi$
$648$	0	0
$649$	−0.446734	−0.0175358
$650$	0	0
$651$	−5.70240	−0.223495
$652$	0	0
$653$	32.2794	1.26319	0.631595	−	0.775298i	$-0.282401\pi$
$653$	32.2794	1.26319	0.631595	+	0.775298i	$0.282401\pi$
$654$	0	0
$655$	−55.0635	−2.15151
$656$	0	0
$657$	32.0963	1.25220
$658$	0	0
$659$	47.1252	1.83574	0.917869	−	0.396883i	$-0.129908\pi$
$659$	47.1252	1.83574	0.917869	+	0.396883i	$0.129908\pi$
$660$	0	0
$661$	−47.4873	−1.84704	−0.923521	−	0.383549i	$-0.874702\pi$
$661$	−47.4873	−1.84704	−0.923521	+	0.383549i	$0.874702\pi$
$662$	0	0
$663$	4.29266	0.166713
$664$	0	0
$665$	−9.22545	−0.357747
$666$	0	0
$667$	−66.6658	−2.58131
$668$	0	0
$669$	7.37812	0.285255
$670$	0	0
$671$	1.52535	0.0588856
$672$	0	0
$673$	13.2602	0.511144	0.255572	−	0.966790i	$-0.417736\pi$
$673$	13.2602	0.511144	0.255572	+	0.966790i	$0.417736\pi$
$674$	0	0
$675$	6.19813	0.238566
$676$	0	0
$677$	−26.9261	−1.03485	−0.517426	−	0.855728i	$-0.673110\pi$
$677$	−26.9261	−1.03485	−0.517426	+	0.855728i	$0.673110\pi$
$678$	0	0
$679$	0.506474	0.0194367
$680$	0	0
$681$	10.9811	0.420796
$682$	0	0
$683$	0.674417	0.0258059	0.0129029	−	0.999917i	$-0.495893\pi$
$683$	0.674417	0.0258059	0.0129029	+	0.999917i	$0.495893\pi$
$684$	0	0
$685$	20.0502	0.766080
$686$	0	0
$687$	−12.6474	−0.482530
$688$	0	0
$689$	−27.7370	−1.05669
$690$	0	0
$691$	−12.2692	−0.466744	−0.233372	−	0.972388i	$-0.574976\pi$
$691$	−12.2692	−0.466744	−0.233372	+	0.972388i	$0.574976\pi$
$692$	0	0
$693$	1.03303	0.0392414
$694$	0	0
$695$	5.83800	0.221448
$696$	0	0
$697$	−8.92656	−0.338118
$698$	0	0
$699$	−14.4189	−0.545372
$700$	0	0
$701$	10.7165	0.404758	0.202379	−	0.979307i	$-0.435133\pi$
$701$	10.7165	0.404758	0.202379	+	0.979307i	$0.435133\pi$
$702$	0	0
$703$	−21.8089	−0.822537
$704$	0	0
$705$	18.0514	0.679854
$706$	0	0
$707$	0.673206	0.0253185
$708$	0	0
$709$	5.26472	0.197721	0.0988604	−	0.995101i	$-0.468480\pi$
$709$	5.26472	0.197721	0.0988604	+	0.995101i	$0.468480\pi$
$710$	0	0
$711$	−26.8383	−1.00651
$712$	0	0
$713$	59.7212	2.23658
$714$	0	0
$715$	−6.77253	−0.253278
$716$	0	0
$717$	16.9948	0.634681
$718$	0	0
$719$	7.34566	0.273947	0.136973	−	0.990575i	$-0.456262\pi$
$719$	7.34566	0.273947	0.136973	+	0.990575i	$0.456262\pi$
$720$	0	0
$721$	0.428555	0.0159602
$722$	0	0
$723$	21.4268	0.796872
$724$	0	0
$725$	14.6586	0.544407
$726$	0	0
$727$	5.05292	0.187403	0.0937013	−	0.995600i	$-0.470130\pi$
$727$	5.05292	0.187403	0.0937013	+	0.995600i	$0.470130\pi$
$728$	0	0
$729$	−2.59858	−0.0962436
$730$	0	0
$731$	−9.40875	−0.347995
$732$	0	0
$733$	−26.3381	−0.972821	−0.486410	−	0.873730i	$-0.661694\pi$
$733$	−26.3381	−0.972821	−0.486410	+	0.873730i	$0.661694\pi$
$734$	0	0
$735$	−11.3817	−0.419819
$736$	0	0
$737$	0.814732	0.0300110
$738$	0	0
$739$	12.0392	0.442870	0.221435	−	0.975175i	$-0.428926\pi$
$739$	12.0392	0.442870	0.221435	+	0.975175i	$0.428926\pi$
$740$	0	0
$741$	16.5406	0.607635
$742$	0	0
$743$	−10.0430	−0.368442	−0.184221	−	0.982885i	$-0.558976\pi$
$743$	−10.0430	−0.368442	−0.184221	+	0.982885i	$0.558976\pi$
$744$	0	0
$745$	37.9887	1.39180
$746$	0	0
$747$	33.4612	1.22428
$748$	0	0
$749$	8.16970	0.298514
$750$	0	0
$751$	−1.21544	−0.0443522	−0.0221761	−	0.999754i	$-0.507059\pi$
$751$	−1.21544	−0.0443522	−0.0221761	+	0.999754i	$0.507059\pi$
$752$	0	0
$753$	−4.20445	−0.153219
$754$	0	0
$755$	13.6722	0.497584
$756$	0	0
$757$	28.2954	1.02842	0.514208	−	0.857666i	$-0.328086\pi$
$757$	28.2954	1.02842	0.514208	+	0.857666i	$0.328086\pi$
$758$	0	0
$759$	2.30048	0.0835023
$760$	0	0
$761$	−4.19969	−0.152239	−0.0761193	−	0.997099i	$-0.524253\pi$
$761$	−4.19969	−0.152239	−0.0761193	+	0.997099i	$0.524253\pi$
$762$	0	0
$763$	5.22378	0.189114
$764$	0	0
$765$	−6.33696	−0.229113
$766$	0	0
$767$	−5.91850	−0.213705
$768$	0	0
$769$	36.8364	1.32836	0.664178	−	0.747574i	$-0.268782\pi$
$769$	36.8364	1.32836	0.664178	+	0.747574i	$0.268782\pi$
$770$	0	0
$771$	13.1131	0.472255
$772$	0	0
$773$	−14.6491	−0.526891	−0.263446	−	0.964674i	$-0.584859\pi$
$773$	−14.6491	−0.526891	−0.263446	+	0.964674i	$0.584859\pi$
$774$	0	0
$775$	−13.1316	−0.471701
$776$	0	0
$777$	3.83701	0.137652
$778$	0	0
$779$	−34.3962	−1.23237
$780$	0	0
$781$	0.415038	0.0148512
$782$	0	0
$783$	37.2788	1.33224
$784$	0	0
$785$	−3.92364	−0.140041
$786$	0	0
$787$	31.7905	1.13321	0.566604	−	0.823990i	$-0.308257\pi$
$787$	31.7905	1.13321	0.566604	+	0.823990i	$0.308257\pi$
$788$	0	0
$789$	19.2291	0.684574
$790$	0	0
$791$	−3.78494	−0.134577
$792$	0	0
$793$	20.2085	0.717624
$794$	0	0
$795$	−8.70666	−0.308793
$796$	0	0
$797$	−37.4627	−1.32700	−0.663498	−	0.748178i	$-0.730929\pi$
$797$	−37.4627	−1.32700	−0.663498	+	0.748178i	$0.730929\pi$
$798$	0	0
$799$	−9.71641	−0.343742
$800$	0	0
$801$	−22.2280	−0.785389
$802$	0	0
$803$	−5.79580	−0.204529
$804$	0	0
$805$	−16.9988	−0.599128
$806$	0	0
$807$	−21.8334	−0.768571
$808$	0	0
$809$	−40.8574	−1.43647	−0.718235	−	0.695801i	$-0.755050\pi$
$809$	−40.8574	−1.43647	−0.718235	+	0.695801i	$0.755050\pi$
$810$	0	0
$811$	15.1495	0.531970	0.265985	−	0.963977i	$-0.414303\pi$
$811$	15.1495	0.531970	0.265985	+	0.963977i	$0.414303\pi$
$812$	0	0
$813$	−7.25418	−0.254415
$814$	0	0
$815$	−51.7843	−1.81392
$816$	0	0
$817$	−36.2542	−1.26837
$818$	0	0
$819$	13.6859	0.478225
$820$	0	0
$821$	15.7527	0.549772	0.274886	−	0.961477i	$-0.411360\pi$
$821$	15.7527	0.549772	0.274886	+	0.961477i	$0.411360\pi$
$822$	0	0
$823$	20.6716	0.720567	0.360284	−	0.932843i	$-0.382680\pi$
$823$	20.6716	0.720567	0.360284	+	0.932843i	$0.382680\pi$
$824$	0	0
$825$	−0.505835	−0.0176109
$826$	0	0
$827$	4.44020	0.154401	0.0772004	−	0.997016i	$-0.475402\pi$
$827$	4.44020	0.154401	0.0772004	+	0.997016i	$0.475402\pi$
$828$	0	0
$829$	−43.7937	−1.52102	−0.760510	−	0.649326i	$-0.775051\pi$
$829$	−43.7937	−1.52102	−0.760510	+	0.649326i	$0.775051\pi$
$830$	0	0
$831$	10.1666	0.352674
$832$	0	0
$833$	6.12634	0.212265
$834$	0	0
$835$	−2.41491	−0.0835715
$836$	0	0
$837$	−33.3954	−1.15432
$838$	0	0
$839$	−4.32417	−0.149287	−0.0746435	−	0.997210i	$-0.523782\pi$
$839$	−4.32417	−0.149287	−0.0746435	+	0.997210i	$0.523782\pi$
$840$	0	0
$841$	59.1646	2.04016
$842$	0	0
$843$	1.31386	0.0452519
$844$	0	0
$845$	−56.4260	−1.94111
$846$	0	0
$847$	10.0951	0.346873
$848$	0	0
$849$	13.4621	0.462017
$850$	0	0
$851$	−40.1849	−1.37752
$852$	0	0
$853$	5.33669	0.182725	0.0913623	−	0.995818i	$-0.470878\pi$
$853$	5.33669	0.182725	0.0913623	+	0.995818i	$0.470878\pi$
$854$	0	0
$855$	−24.4178	−0.835071
$856$	0	0
$857$	16.7413	0.571872	0.285936	−	0.958249i	$-0.407695\pi$
$857$	16.7413	0.571872	0.285936	+	0.958249i	$0.407695\pi$
$858$	0	0
$859$	−44.1369	−1.50593	−0.752965	−	0.658060i	$-0.771377\pi$
$859$	−44.1369	−1.50593	−0.752965	+	0.658060i	$0.771377\pi$
$860$	0	0
$861$	6.05160	0.206238
$862$	0	0
$863$	−6.59314	−0.224433	−0.112217	−	0.993684i	$-0.535795\pi$
$863$	−6.59314	−0.224433	−0.112217	+	0.993684i	$0.535795\pi$
$864$	0	0
$865$	−18.4978	−0.628945
$866$	0	0
$867$	−0.725294	−0.0246323
$868$	0	0
$869$	4.84633	0.164401
$870$	0	0
$871$	10.7939	0.365737
$872$	0	0
$873$	1.34053	0.0453700
$874$	0	0
$875$	−8.23331	−0.278337
$876$	0	0
$877$	17.7407	0.599061	0.299530	−	0.954087i	$-0.403170\pi$
$877$	17.7407	0.599061	0.299530	+	0.954087i	$0.403170\pi$
$878$	0	0
$879$	14.2598	0.480972
$880$	0	0
$881$	−28.6508	−0.965270	−0.482635	−	0.875822i	$-0.660320\pi$
$881$	−28.6508	−0.965270	−0.482635	+	0.875822i	$0.660320\pi$
$882$	0	0
$883$	−15.7856	−0.531229	−0.265615	−	0.964079i	$-0.585575\pi$
$883$	−15.7856	−0.531229	−0.265615	+	0.964079i	$0.585575\pi$
$884$	0	0
$885$	−1.85782	−0.0624501
$886$	0	0
$887$	−19.4723	−0.653817	−0.326909	−	0.945056i	$-0.606007\pi$
$887$	−19.4723	−0.653817	−0.326909	+	0.945056i	$0.606007\pi$
$888$	0	0
$889$	3.75188	0.125834
$890$	0	0
$891$	2.02918	0.0679802
$892$	0	0
$893$	−37.4396	−1.25287
$894$	0	0
$895$	−14.7791	−0.494011
$896$	0	0
$897$	30.4777	1.01762
$898$	0	0
$899$	−78.9804	−2.63414
$900$	0	0
$901$	4.68648	0.156129
$902$	0	0
$903$	6.37849	0.212263
$904$	0	0
$905$	32.1609	1.06907
$906$	0	0
$907$	−11.4770	−0.381089	−0.190544	−	0.981679i	$-0.561025\pi$
$907$	−11.4770	−0.381089	−0.190544	+	0.981679i	$0.561025\pi$
$908$	0	0
$909$	1.78183	0.0590997
$910$	0	0
$911$	−55.2982	−1.83211	−0.916056	−	0.401050i	$-0.868645\pi$
$911$	−55.2982	−1.83211	−0.916056	+	0.401050i	$0.868645\pi$
$912$	0	0
$913$	−6.04227	−0.199970
$914$	0	0
$915$	6.34346	0.209708
$916$	0	0
$917$	−20.0930	−0.663530
$918$	0	0
$919$	−51.3671	−1.69444	−0.847222	−	0.531239i	$-0.821727\pi$
$919$	−51.3671	−1.69444	−0.847222	+	0.531239i	$0.821727\pi$
$920$	0	0
$921$	12.0063	0.395621
$922$	0	0
$923$	5.49858	0.180988
$924$	0	0
$925$	8.83594	0.290524
$926$	0	0
$927$	1.13429	0.0372551
$928$	0	0
$929$	24.9838	0.819693	0.409847	−	0.912154i	$-0.365582\pi$
$929$	24.9838	0.819693	0.409847	+	0.912154i	$0.365582\pi$
$930$	0	0
$931$	23.6062	0.773663
$932$	0	0
$933$	25.4194	0.832194
$934$	0	0
$935$	1.14430	0.0374225
$936$	0	0
$937$	−33.0250	−1.07888	−0.539440	−	0.842024i	$-0.681364\pi$
$937$	−33.0250	−1.07888	−0.539440	+	0.842024i	$0.681364\pi$
$938$	0	0
$939$	−11.1582	−0.364135
$940$	0	0
$941$	56.3908	1.83829	0.919144	−	0.393921i	$-0.128882\pi$
$941$	56.3908	1.83829	0.919144	+	0.393921i	$0.128882\pi$
$942$	0	0
$943$	−63.3783	−2.06388
$944$	0	0
$945$	9.50554	0.309215
$946$	0	0
$947$	8.07846	0.262515	0.131257	−	0.991348i	$-0.458099\pi$
$947$	8.07846	0.262515	0.131257	+	0.991348i	$0.458099\pi$
$948$	0	0
$949$	−76.7851	−2.49255
$950$	0	0
$951$	−5.98209	−0.193982
$952$	0	0
$953$	43.6028	1.41243	0.706216	−	0.707997i	$-0.250401\pi$
$953$	43.6028	1.41243	0.706216	+	0.707997i	$0.250401\pi$
$954$	0	0
$955$	29.0593	0.940336
$956$	0	0
$957$	−3.04236	−0.0983455
$958$	0	0
$959$	7.31645	0.236261
$960$	0	0
$961$	39.7529	1.28235
$962$	0	0
$963$	21.6235	0.696807
$964$	0	0
$965$	−49.3646	−1.58910
$966$	0	0
$967$	29.9869	0.964314	0.482157	−	0.876085i	$-0.339853\pi$
$967$	29.9869	0.964314	0.482157	+	0.876085i	$0.339853\pi$
$968$	0	0
$969$	−2.79473	−0.0897797
$970$	0	0
$971$	−18.1930	−0.583841	−0.291920	−	0.956443i	$-0.594294\pi$
$971$	−18.1930	−0.583841	−0.291920	+	0.956443i	$0.594294\pi$
$972$	0	0
$973$	2.13032	0.0682950
$974$	0	0
$975$	−6.70150	−0.214620
$976$	0	0
$977$	−37.6344	−1.20403	−0.602015	−	0.798485i	$-0.705635\pi$
$977$	−37.6344	−1.20403	−0.602015	+	0.798485i	$0.705635\pi$
$978$	0	0
$979$	4.01383	0.128283
$980$	0	0
$981$	13.8262	0.441438
$982$	0	0
$983$	30.4375	0.970804	0.485402	−	0.874291i	$-0.338673\pi$
$983$	30.4375	0.970804	0.485402	+	0.874291i	$0.338673\pi$
$984$	0	0
$985$	−60.8213	−1.93793
$986$	0	0
$987$	6.58706	0.209668
$988$	0	0
$989$	−66.8018	−2.12417
$990$	0	0
$991$	35.7667	1.13617	0.568083	−	0.822971i	$-0.307685\pi$
$991$	35.7667	1.13617	0.568083	+	0.822971i	$0.307685\pi$
$992$	0	0
$993$	26.3660	0.836699
$994$	0	0
$995$	−10.3030	−0.326628
$996$	0	0
$997$	−50.7437	−1.60707	−0.803534	−	0.595259i	$-0.797050\pi$
$997$	−50.7437	−1.60707	−0.803534	+	0.595259i	$0.797050\pi$
$998$	0	0
$999$	22.4710	0.710951

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.12	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.12	✓	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)

\(f(q)\)	\(=\)	\(q-0.725294 q^{3} -2.56147 q^{5} -0.934698 q^{7} -2.47395 q^{9} +O(q^{10})\)	\(q-0.725294 q^{3} -2.56147 q^{5} -0.934698 q^{7} -2.47395 q^{9} +0.446734 q^{11} +5.91850 q^{13} +1.85782 q^{15} -1.00000 q^{17} -3.85324 q^{19} +0.677931 q^{21} -7.09996 q^{23} +1.56115 q^{25} +3.97022 q^{27} +9.38960 q^{29} -8.41148 q^{31} -0.324014 q^{33} +2.39421 q^{35} +5.65988 q^{37} -4.29266 q^{39} +8.92656 q^{41} +9.40875 q^{43} +6.33696 q^{45} +9.71641 q^{47} -6.12634 q^{49} +0.725294 q^{51} -4.68648 q^{53} -1.14430 q^{55} +2.79473 q^{57} -1.00000 q^{59} +3.41446 q^{61} +2.31240 q^{63} -15.1601 q^{65} +1.82375 q^{67} +5.14956 q^{69} +0.929049 q^{71} -12.9737 q^{73} -1.13230 q^{75} -0.417561 q^{77} +10.8484 q^{79} +4.54226 q^{81} -13.5254 q^{83} +2.56147 q^{85} -6.81023 q^{87} +8.98484 q^{89} -5.53202 q^{91} +6.10080 q^{93} +9.86997 q^{95} -0.541858 q^{97} -1.10520 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 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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