LMFDB - Embedded newform 1040.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(1,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, 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("1040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.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:	$8.30444181021$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} -1.00000 q^{5} -2.00000 q^{7} +4.46410 q^{9} +4.73205 q^{11} +1.00000 q^{13} +2.73205 q^{15} -3.46410 q^{17} +6.19615 q^{19} +5.46410 q^{21} -1.26795 q^{23} +1.00000 q^{25} -4.00000 q^{27} -2.53590 q^{29} -10.1962 q^{31} -12.9282 q^{33} +2.00000 q^{35} -4.00000 q^{37} -2.73205 q^{39} +3.46410 q^{41} +0.196152 q^{43} -4.46410 q^{45} -6.00000 q^{47} -3.00000 q^{49} +9.46410 q^{51} +10.3923 q^{53} -4.73205 q^{55} -16.9282 q^{57} -9.12436 q^{59} -8.39230 q^{61} -8.92820 q^{63} -1.00000 q^{65} -6.39230 q^{67} +3.46410 q^{69} -4.73205 q^{71} -4.00000 q^{73} -2.73205 q^{75} -9.46410 q^{77} +8.39230 q^{79} -2.46410 q^{81} +6.00000 q^{83} +3.46410 q^{85} +6.92820 q^{87} -12.9282 q^{89} -2.00000 q^{91} +27.8564 q^{93} -6.19615 q^{95} +2.00000 q^{97} +21.1244 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{13} + 2 q^{15} + 2 q^{19} + 4 q^{21} - 6 q^{23} + 2 q^{25} - 8 q^{27} - 12 q^{29} - 10 q^{31} - 12 q^{33} + 4 q^{35} - 8 q^{37} - 2 q^{39} - 10 q^{43}+ \cdots + 18 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 4 * q^7 + 2 * q^9 + 6 * q^11 + 2 * q^13 + 2 * q^15 + 2 * q^19 + 4 * q^21 - 6 * q^23 + 2 * q^25 - 8 * q^27 - 12 * q^29 - 10 * q^31 - 12 * q^33 + 4 * q^35 - 8 * q^37 - 2 * q^39 - 10 * q^43 - 2 * q^45 - 12 * q^47 - 6 * q^49 + 12 * q^51 - 6 * q^55 - 20 * q^57 + 6 * q^59 + 4 * q^61 - 4 * q^63 - 2 * q^65 + 8 * q^67 - 6 * q^71 - 8 * q^73 - 2 * q^75 - 12 * q^77 - 4 * q^79 + 2 * q^81 + 12 * q^83 - 12 * q^89 - 4 * q^91 + 28 * q^93 - 2 * q^95 + 4 * q^97 + 18 * q^99

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
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.73205	0.705412
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	6.19615	1.42149	0.710747	−	0.703447i	$-0.248357\pi$
$19$	6.19615	1.42149	0.710747	+	0.703447i	$0.248357\pi$
$20$	0	0
$21$	5.46410	1.19236
$22$	0	0
$23$	−1.26795	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$23$	−1.26795	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−2.53590	−0.470905	−0.235452	−	0.971886i	$-0.575657\pi$
$29$	−2.53590	−0.470905	−0.235452	+	0.971886i	$0.575657\pi$
$30$	0	0
$31$	−10.1962	−1.83128	−0.915642	−	0.401996i	$-0.868317\pi$
$31$	−10.1962	−1.83128	−0.915642	+	0.401996i	$0.868317\pi$
$32$	0	0
$33$	−12.9282	−2.25051
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	−2.73205	−0.437478
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	0.196152	0.0299130	0.0149565	−	0.999888i	$-0.495239\pi$
$43$	0.196152	0.0299130	0.0149565	+	0.999888i	$0.495239\pi$
$44$	0	0
$45$	−4.46410	−0.665469
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	9.46410	1.32524
$52$	0	0
$53$	10.3923	1.42749	0.713746	−	0.700404i	$-0.246997\pi$
$53$	10.3923	1.42749	0.713746	+	0.700404i	$0.246997\pi$
$54$	0	0
$55$	−4.73205	−0.638070
$56$	0	0
$57$	−16.9282	−2.24220
$58$	0	0
$59$	−9.12436	−1.18789	−0.593945	−	0.804506i	$-0.702430\pi$
$59$	−9.12436	−1.18789	−0.593945	+	0.804506i	$0.702430\pi$
$60$	0	0
$61$	−8.39230	−1.07452	−0.537262	−	0.843415i	$-0.680541\pi$
$61$	−8.39230	−1.07452	−0.537262	+	0.843415i	$0.680541\pi$
$62$	0	0
$63$	−8.92820	−1.12485
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−6.39230	−0.780944	−0.390472	−	0.920615i	$-0.627688\pi$
$67$	−6.39230	−0.780944	−0.390472	+	0.920615i	$0.627688\pi$
$68$	0	0
$69$	3.46410	0.417029
$70$	0	0
$71$	−4.73205	−0.561591	−0.280796	−	0.959768i	$-0.590598\pi$
$71$	−4.73205	−0.561591	−0.280796	+	0.959768i	$0.590598\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	−2.73205	−0.315470
$76$	0	0
$77$	−9.46410	−1.07853
$78$	0	0
$79$	8.39230	0.944208	0.472104	−	0.881543i	$-0.343495\pi$
$79$	8.39230	0.944208	0.472104	+	0.881543i	$0.343495\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	0	0
$87$	6.92820	0.742781
$88$	0	0
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	27.8564	2.88857
$94$	0	0
$95$	−6.19615	−0.635712
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	21.1244	2.12308
$100$	0	0
$101$	−0.928203	−0.0923597	−0.0461798	−	0.998933i	$-0.514705\pi$
$101$	−0.928203	−0.0923597	−0.0461798	+	0.998933i	$0.514705\pi$
$102$	0	0
$103$	0.196152	0.0193275	0.00966374	−	0.999953i	$-0.496924\pi$
$103$	0.196152	0.0193275	0.00966374	+	0.999953i	$0.496924\pi$
$104$	0	0
$105$	−5.46410	−0.533242
$106$	0	0
$107$	−17.6603	−1.70728	−0.853641	−	0.520862i	$-0.825610\pi$
$107$	−17.6603	−1.70728	−0.853641	+	0.520862i	$0.825610\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	10.9282	1.03726
$112$	0	0
$113$	8.53590	0.802990	0.401495	−	0.915861i	$-0.368491\pi$
$113$	8.53590	0.802990	0.401495	+	0.915861i	$0.368491\pi$
$114$	0	0
$115$	1.26795	0.118237
$116$	0	0
$117$	4.46410	0.412706
$118$	0	0
$119$	6.92820	0.635107
$120$	0	0
$121$	11.3923	1.03566
$122$	0	0
$123$	−9.46410	−0.853349
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.1962	−1.43718	−0.718588	−	0.695436i	$-0.755211\pi$
$127$	−16.1962	−1.43718	−0.718588	+	0.695436i	$0.755211\pi$
$128$	0	0
$129$	−0.535898	−0.0471832
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−12.3923	−1.07455
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	0.928203	0.0793018	0.0396509	−	0.999214i	$-0.487375\pi$
$137$	0.928203	0.0793018	0.0396509	+	0.999214i	$0.487375\pi$
$138$	0	0
$139$	−12.3923	−1.05110	−0.525551	−	0.850762i	$-0.676141\pi$
$139$	−12.3923	−1.05110	−0.525551	+	0.850762i	$0.676141\pi$
$140$	0	0
$141$	16.3923	1.38048
$142$	0	0
$143$	4.73205	0.395714
$144$	0	0
$145$	2.53590	0.210595
$146$	0	0
$147$	8.19615	0.676007
$148$	0	0
$149$	−7.85641	−0.643622	−0.321811	−	0.946804i	$-0.604292\pi$
$149$	−7.85641	−0.643622	−0.321811	+	0.946804i	$0.604292\pi$
$150$	0	0
$151$	1.80385	0.146795	0.0733975	−	0.997303i	$-0.476616\pi$
$151$	1.80385	0.146795	0.0733975	+	0.997303i	$0.476616\pi$
$152$	0	0
$153$	−15.4641	−1.25020
$154$	0	0
$155$	10.1962	0.818975
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	−28.3923	−2.25166
$160$	0	0
$161$	2.53590	0.199857
$162$	0	0
$163$	14.3923	1.12729	0.563646	−	0.826016i	$-0.309398\pi$
$163$	14.3923	1.12729	0.563646	+	0.826016i	$0.309398\pi$
$164$	0	0
$165$	12.9282	1.00646
$166$	0	0
$167$	0.928203	0.0718265	0.0359133	−	0.999355i	$-0.488566\pi$
$167$	0.928203	0.0718265	0.0359133	+	0.999355i	$0.488566\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	27.6603	2.11523
$172$	0	0
$173$	8.53590	0.648972	0.324486	−	0.945890i	$-0.394809\pi$
$173$	8.53590	0.648972	0.324486	+	0.945890i	$0.394809\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	24.9282	1.87372
$178$	0	0
$179$	18.9282	1.41476	0.707380	−	0.706833i	$-0.249877\pi$
$179$	18.9282	1.41476	0.707380	+	0.706833i	$0.249877\pi$
$180$	0	0
$181$	0.392305	0.0291598	0.0145799	−	0.999894i	$-0.495359\pi$
$181$	0.392305	0.0291598	0.0145799	+	0.999894i	$0.495359\pi$
$182$	0	0
$183$	22.9282	1.69490
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	−16.3923	−1.19872
$188$	0	0
$189$	8.00000	0.581914
$190$	0	0
$191$	5.07180	0.366982	0.183491	−	0.983021i	$-0.441260\pi$
$191$	5.07180	0.366982	0.183491	+	0.983021i	$0.441260\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	2.73205	0.195646
$196$	0	0
$197$	−12.9282	−0.921096	−0.460548	−	0.887635i	$-0.652347\pi$
$197$	−12.9282	−0.921096	−0.460548	+	0.887635i	$0.652347\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	17.4641	1.23182
$202$	0	0
$203$	5.07180	0.355970
$204$	0	0
$205$	−3.46410	−0.241943
$206$	0	0
$207$	−5.66025	−0.393415
$208$	0	0
$209$	29.3205	2.02814
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	12.9282	0.885826
$214$	0	0
$215$	−0.196152	−0.0133775
$216$	0	0
$217$	20.3923	1.38432
$218$	0	0
$219$	10.9282	0.738460
$220$	0	0
$221$	−3.46410	−0.233021
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	4.46410	0.297607
$226$	0	0
$227$	3.46410	0.229920	0.114960	−	0.993370i	$-0.463326\pi$
$227$	3.46410	0.229920	0.114960	+	0.993370i	$0.463326\pi$
$228$	0	0
$229$	6.39230	0.422415	0.211208	−	0.977441i	$-0.432260\pi$
$229$	6.39230	0.422415	0.211208	+	0.977441i	$0.432260\pi$
$230$	0	0
$231$	25.8564	1.70123
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	0	0
$237$	−22.9282	−1.48935
$238$	0	0
$239$	14.1962	0.918273	0.459136	−	0.888366i	$-0.348159\pi$
$239$	14.1962	0.918273	0.459136	+	0.888366i	$0.348159\pi$
$240$	0	0
$241$	−2.39230	−0.154102	−0.0770510	−	0.997027i	$-0.524550\pi$
$241$	−2.39230	−0.154102	−0.0770510	+	0.997027i	$0.524550\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	6.19615	0.394252
$248$	0	0
$249$	−16.3923	−1.03882
$250$	0	0
$251$	21.4641	1.35480	0.677401	−	0.735614i	$-0.263106\pi$
$251$	21.4641	1.35480	0.677401	+	0.735614i	$0.263106\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	−9.46410	−0.592665
$256$	0	0
$257$	19.8564	1.23861	0.619304	−	0.785151i	$-0.287415\pi$
$257$	19.8564	1.23861	0.619304	+	0.785151i	$0.287415\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	−11.3205	−0.700722
$262$	0	0
$263$	−1.26795	−0.0781851	−0.0390925	−	0.999236i	$-0.512447\pi$
$263$	−1.26795	−0.0781851	−0.0390925	+	0.999236i	$0.512447\pi$
$264$	0	0
$265$	−10.3923	−0.638394
$266$	0	0
$267$	35.3205	2.16158
$268$	0	0
$269$	−19.8564	−1.21067	−0.605333	−	0.795972i	$-0.706960\pi$
$269$	−19.8564	−1.21067	−0.605333	+	0.795972i	$0.706960\pi$
$270$	0	0
$271$	−30.9808	−1.88195	−0.940974	−	0.338480i	$-0.890087\pi$
$271$	−30.9808	−1.88195	−0.940974	+	0.338480i	$0.890087\pi$
$272$	0	0
$273$	5.46410	0.330702
$274$	0	0
$275$	4.73205	0.285353
$276$	0	0
$277$	−26.3923	−1.58576	−0.792880	−	0.609378i	$-0.791419\pi$
$277$	−26.3923	−1.58576	−0.792880	+	0.609378i	$0.791419\pi$
$278$	0	0
$279$	−45.5167	−2.72501
$280$	0	0
$281$	22.3923	1.33581	0.667906	−	0.744245i	$-0.267191\pi$
$281$	22.3923	1.33581	0.667906	+	0.744245i	$0.267191\pi$
$282$	0	0
$283$	−32.5885	−1.93718	−0.968591	−	0.248658i	$-0.920010\pi$
$283$	−32.5885	−1.93718	−0.968591	+	0.248658i	$0.920010\pi$
$284$	0	0
$285$	16.9282	1.00274
$286$	0	0
$287$	−6.92820	−0.408959
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	−5.46410	−0.320311
$292$	0	0
$293$	−5.07180	−0.296298	−0.148149	−	0.988965i	$-0.547331\pi$
$293$	−5.07180	−0.296298	−0.148149	+	0.988965i	$0.547331\pi$
$294$	0	0
$295$	9.12436	0.531241
$296$	0	0
$297$	−18.9282	−1.09833
$298$	0	0
$299$	−1.26795	−0.0733274
$300$	0	0
$301$	−0.392305	−0.0226121
$302$	0	0
$303$	2.53590	0.145684
$304$	0	0
$305$	8.39230	0.480542
$306$	0	0
$307$	18.7846	1.07209	0.536047	−	0.844188i	$-0.319917\pi$
$307$	18.7846	1.07209	0.536047	+	0.844188i	$0.319917\pi$
$308$	0	0
$309$	−0.535898	−0.0304862
$310$	0	0
$311$	16.3923	0.929522	0.464761	−	0.885436i	$-0.346140\pi$
$311$	16.3923	0.929522	0.464761	+	0.885436i	$0.346140\pi$
$312$	0	0
$313$	−14.3923	−0.813501	−0.406751	−	0.913539i	$-0.633338\pi$
$313$	−14.3923	−0.813501	−0.406751	+	0.913539i	$0.633338\pi$
$314$	0	0
$315$	8.92820	0.503047
$316$	0	0
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	48.2487	2.69298
$322$	0	0
$323$	−21.4641	−1.19429
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−5.46410	−0.302166
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	−2.58846	−0.142274	−0.0711372	−	0.997467i	$-0.522663\pi$
$331$	−2.58846	−0.142274	−0.0711372	+	0.997467i	$0.522663\pi$
$332$	0	0
$333$	−17.8564	−0.978525
$334$	0	0
$335$	6.39230	0.349249
$336$	0	0
$337$	−26.3923	−1.43768	−0.718840	−	0.695175i	$-0.755327\pi$
$337$	−26.3923	−1.43768	−0.718840	+	0.695175i	$0.755327\pi$
$338$	0	0
$339$	−23.3205	−1.26660
$340$	0	0
$341$	−48.2487	−2.61281
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	−3.46410	−0.186501
$346$	0	0
$347$	5.66025	0.303858	0.151929	−	0.988391i	$-0.451451\pi$
$347$	5.66025	0.303858	0.151929	+	0.988391i	$0.451451\pi$
$348$	0	0
$349$	−14.3923	−0.770402	−0.385201	−	0.922833i	$-0.625868\pi$
$349$	−14.3923	−0.770402	−0.385201	+	0.922833i	$0.625868\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−27.7128	−1.47500	−0.737502	−	0.675345i	$-0.763995\pi$
$353$	−27.7128	−1.47500	−0.737502	+	0.675345i	$0.763995\pi$
$354$	0	0
$355$	4.73205	0.251151
$356$	0	0
$357$	−18.9282	−1.00179
$358$	0	0
$359$	2.19615	0.115908	0.0579542	−	0.998319i	$-0.481542\pi$
$359$	2.19615	0.115908	0.0579542	+	0.998319i	$0.481542\pi$
$360$	0	0
$361$	19.3923	1.02065
$362$	0	0
$363$	−31.1244	−1.63361
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	−11.8038	−0.616156	−0.308078	−	0.951361i	$-0.599686\pi$
$367$	−11.8038	−0.616156	−0.308078	+	0.951361i	$0.599686\pi$
$368$	0	0
$369$	15.4641	0.805029
$370$	0	0
$371$	−20.7846	−1.07908
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	2.73205	0.141082
$376$	0	0
$377$	−2.53590	−0.130605
$378$	0	0
$379$	−18.9808	−0.974976	−0.487488	−	0.873130i	$-0.662087\pi$
$379$	−18.9808	−0.974976	−0.487488	+	0.873130i	$0.662087\pi$
$380$	0	0
$381$	44.2487	2.26693
$382$	0	0
$383$	−12.9282	−0.660600	−0.330300	−	0.943876i	$-0.607150\pi$
$383$	−12.9282	−0.660600	−0.330300	+	0.943876i	$0.607150\pi$
$384$	0	0
$385$	9.46410	0.482335
$386$	0	0
$387$	0.875644	0.0445115
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	4.39230	0.222128
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.39230	−0.422263
$396$	0	0
$397$	28.7846	1.44466	0.722329	−	0.691549i	$-0.243072\pi$
$397$	28.7846	1.44466	0.722329	+	0.691549i	$0.243072\pi$
$398$	0	0
$399$	33.8564	1.69494
$400$	0	0
$401$	−36.9282	−1.84411	−0.922053	−	0.387063i	$-0.873490\pi$
$401$	−36.9282	−1.84411	−0.922053	+	0.387063i	$0.873490\pi$
$402$	0	0
$403$	−10.1962	−0.507907
$404$	0	0
$405$	2.46410	0.122442
$406$	0	0
$407$	−18.9282	−0.938236
$408$	0	0
$409$	−17.6077	−0.870644	−0.435322	−	0.900275i	$-0.643366\pi$
$409$	−17.6077	−0.870644	−0.435322	+	0.900275i	$0.643366\pi$
$410$	0	0
$411$	−2.53590	−0.125087
$412$	0	0
$413$	18.2487	0.897960
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	33.8564	1.65796
$418$	0	0
$419$	2.53590	0.123887	0.0619434	−	0.998080i	$-0.480270\pi$
$419$	2.53590	0.123887	0.0619434	+	0.998080i	$0.480270\pi$
$420$	0	0
$421$	−30.7846	−1.50035	−0.750175	−	0.661239i	$-0.770031\pi$
$421$	−30.7846	−1.50035	−0.750175	+	0.661239i	$0.770031\pi$
$422$	0	0
$423$	−26.7846	−1.30231
$424$	0	0
$425$	−3.46410	−0.168034
$426$	0	0
$427$	16.7846	0.812264
$428$	0	0
$429$	−12.9282	−0.624180
$430$	0	0
$431$	25.5167	1.22909	0.614547	−	0.788880i	$-0.289339\pi$
$431$	25.5167	1.22909	0.614547	+	0.788880i	$0.289339\pi$
$432$	0	0
$433$	34.7846	1.67164	0.835821	−	0.549002i	$-0.184992\pi$
$433$	34.7846	1.67164	0.835821	+	0.549002i	$0.184992\pi$
$434$	0	0
$435$	−6.92820	−0.332182
$436$	0	0
$437$	−7.85641	−0.375823
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	−13.3923	−0.637729
$442$	0	0
$443$	16.9808	0.806780	0.403390	−	0.915028i	$-0.367832\pi$
$443$	16.9808	0.806780	0.403390	+	0.915028i	$0.367832\pi$
$444$	0	0
$445$	12.9282	0.612856
$446$	0	0
$447$	21.4641	1.01522
$448$	0	0
$449$	20.5359	0.969149	0.484574	−	0.874750i	$-0.338974\pi$
$449$	20.5359	0.969149	0.484574	+	0.874750i	$0.338974\pi$
$450$	0	0
$451$	16.3923	0.771883
$452$	0	0
$453$	−4.92820	−0.231547
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	10.7846	0.504483	0.252241	−	0.967664i	$-0.418832\pi$
$457$	10.7846	0.504483	0.252241	+	0.967664i	$0.418832\pi$
$458$	0	0
$459$	13.8564	0.646762
$460$	0	0
$461$	−3.46410	−0.161339	−0.0806696	−	0.996741i	$-0.525706\pi$
$461$	−3.46410	−0.161339	−0.0806696	+	0.996741i	$0.525706\pi$
$462$	0	0
$463$	2.39230	0.111180	0.0555899	−	0.998454i	$-0.482296\pi$
$463$	2.39230	0.111180	0.0555899	+	0.998454i	$0.482296\pi$
$464$	0	0
$465$	−27.8564	−1.29181
$466$	0	0
$467$	−27.8038	−1.28661	−0.643304	−	0.765611i	$-0.722437\pi$
$467$	−27.8038	−1.28661	−0.643304	+	0.765611i	$0.722437\pi$
$468$	0	0
$469$	12.7846	0.590338
$470$	0	0
$471$	27.3205	1.25886
$472$	0	0
$473$	0.928203	0.0426788
$474$	0	0
$475$	6.19615	0.284299
$476$	0	0
$477$	46.3923	2.12416
$478$	0	0
$479$	−35.6603	−1.62936	−0.814679	−	0.579912i	$-0.803087\pi$
$479$	−35.6603	−1.62936	−0.814679	+	0.579912i	$0.803087\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	−6.92820	−0.315244
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	26.3923	1.19595	0.597975	−	0.801515i	$-0.295972\pi$
$487$	26.3923	1.19595	0.597975	+	0.801515i	$0.295972\pi$
$488$	0	0
$489$	−39.3205	−1.77813
$490$	0	0
$491$	2.53590	0.114443	0.0572217	−	0.998361i	$-0.481776\pi$
$491$	2.53590	0.114443	0.0572217	+	0.998361i	$0.481776\pi$
$492$	0	0
$493$	8.78461	0.395639
$494$	0	0
$495$	−21.1244	−0.949469
$496$	0	0
$497$	9.46410	0.424523
$498$	0	0
$499$	38.9808	1.74502	0.872509	−	0.488598i	$-0.162491\pi$
$499$	38.9808	1.74502	0.872509	+	0.488598i	$0.162491\pi$
$500$	0	0
$501$	−2.53590	−0.113296
$502$	0	0
$503$	−19.5167	−0.870205	−0.435102	−	0.900381i	$-0.643288\pi$
$503$	−19.5167	−0.870205	−0.435102	+	0.900381i	$0.643288\pi$
$504$	0	0
$505$	0.928203	0.0413045
$506$	0	0
$507$	−2.73205	−0.121335
$508$	0	0
$509$	39.4641	1.74922	0.874608	−	0.484831i	$-0.161119\pi$
$509$	39.4641	1.74922	0.874608	+	0.484831i	$0.161119\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	0	0
$513$	−24.7846	−1.09427
$514$	0	0
$515$	−0.196152	−0.00864351
$516$	0	0
$517$	−28.3923	−1.24869
$518$	0	0
$519$	−23.3205	−1.02366
$520$	0	0
$521$	−28.3923	−1.24389	−0.621945	−	0.783061i	$-0.713657\pi$
$521$	−28.3923	−1.24389	−0.621945	+	0.783061i	$0.713657\pi$
$522$	0	0
$523$	24.1962	1.05802	0.529012	−	0.848614i	$-0.322563\pi$
$523$	24.1962	1.05802	0.529012	+	0.848614i	$0.322563\pi$
$524$	0	0
$525$	5.46410	0.238473
$526$	0	0
$527$	35.3205	1.53859
$528$	0	0
$529$	−21.3923	−0.930100
$530$	0	0
$531$	−40.7321	−1.76762
$532$	0	0
$533$	3.46410	0.150047
$534$	0	0
$535$	17.6603	0.763519
$536$	0	0
$537$	−51.7128	−2.23157
$538$	0	0
$539$	−14.1962	−0.611472
$540$	0	0
$541$	−26.3923	−1.13469	−0.567347	−	0.823479i	$-0.692030\pi$
$541$	−26.3923	−1.13469	−0.567347	+	0.823479i	$0.692030\pi$
$542$	0	0
$543$	−1.07180	−0.0459952
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	12.1962	0.521470	0.260735	−	0.965410i	$-0.416035\pi$
$547$	12.1962	0.521470	0.260735	+	0.965410i	$0.416035\pi$
$548$	0	0
$549$	−37.4641	−1.59893
$550$	0	0
$551$	−15.7128	−0.669388
$552$	0	0
$553$	−16.7846	−0.713754
$554$	0	0
$555$	−10.9282	−0.463876
$556$	0	0
$557$	1.85641	0.0786585	0.0393292	−	0.999226i	$-0.487478\pi$
$557$	1.85641	0.0786585	0.0393292	+	0.999226i	$0.487478\pi$
$558$	0	0
$559$	0.196152	0.00829636
$560$	0	0
$561$	44.7846	1.89081
$562$	0	0
$563$	22.0526	0.929405	0.464702	−	0.885467i	$-0.346161\pi$
$563$	22.0526	0.929405	0.464702	+	0.885467i	$0.346161\pi$
$564$	0	0
$565$	−8.53590	−0.359108
$566$	0	0
$567$	4.92820	0.206965
$568$	0	0
$569$	−2.53590	−0.106310	−0.0531552	−	0.998586i	$-0.516928\pi$
$569$	−2.53590	−0.106310	−0.0531552	+	0.998586i	$0.516928\pi$
$570$	0	0
$571$	−36.3923	−1.52297	−0.761485	−	0.648182i	$-0.775530\pi$
$571$	−36.3923	−1.52297	−0.761485	+	0.648182i	$0.775530\pi$
$572$	0	0
$573$	−13.8564	−0.578860
$574$	0	0
$575$	−1.26795	−0.0528771
$576$	0	0
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	0	0
$579$	27.3205	1.13540
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	49.1769	2.03670
$584$	0	0
$585$	−4.46410	−0.184568
$586$	0	0
$587$	8.53590	0.352314	0.176157	−	0.984362i	$-0.443633\pi$
$587$	8.53590	0.352314	0.176157	+	0.984362i	$0.443633\pi$
$588$	0	0
$589$	−63.1769	−2.60316
$590$	0	0
$591$	35.3205	1.45289
$592$	0	0
$593$	26.7846	1.09991	0.549956	−	0.835194i	$-0.314644\pi$
$593$	26.7846	1.09991	0.549956	+	0.835194i	$0.314644\pi$
$594$	0	0
$595$	−6.92820	−0.284029
$596$	0	0
$597$	54.6410	2.23631
$598$	0	0
$599$	7.60770	0.310842	0.155421	−	0.987848i	$-0.450327\pi$
$599$	7.60770	0.310842	0.155421	+	0.987848i	$0.450327\pi$
$600$	0	0
$601$	43.5692	1.77723	0.888613	−	0.458658i	$-0.151670\pi$
$601$	43.5692	1.77723	0.888613	+	0.458658i	$0.151670\pi$
$602$	0	0
$603$	−28.5359	−1.16207
$604$	0	0
$605$	−11.3923	−0.463163
$606$	0	0
$607$	−24.9808	−1.01394	−0.506969	−	0.861964i	$-0.669234\pi$
$607$	−24.9808	−1.01394	−0.506969	+	0.861964i	$0.669234\pi$
$608$	0	0
$609$	−13.8564	−0.561490
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	9.46410	0.381629
$616$	0	0
$617$	33.7128	1.35723	0.678613	−	0.734496i	$-0.262581\pi$
$617$	33.7128	1.35723	0.678613	+	0.734496i	$0.262581\pi$
$618$	0	0
$619$	−6.98076	−0.280581	−0.140290	−	0.990110i	$-0.544804\pi$
$619$	−6.98076	−0.280581	−0.140290	+	0.990110i	$0.544804\pi$
$620$	0	0
$621$	5.07180	0.203524
$622$	0	0
$623$	25.8564	1.03592
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−80.1051	−3.19909
$628$	0	0
$629$	13.8564	0.552491
$630$	0	0
$631$	−5.80385	−0.231048	−0.115524	−	0.993305i	$-0.536855\pi$
$631$	−5.80385	−0.231048	−0.115524	+	0.993305i	$0.536855\pi$
$632$	0	0
$633$	21.8564	0.868714
$634$	0	0
$635$	16.1962	0.642725
$636$	0	0
$637$	−3.00000	−0.118864
$638$	0	0
$639$	−21.1244	−0.835667
$640$	0	0
$641$	12.9282	0.510633	0.255317	−	0.966857i	$-0.417820\pi$
$641$	12.9282	0.510633	0.255317	+	0.966857i	$0.417820\pi$
$642$	0	0
$643$	6.78461	0.267559	0.133779	−	0.991011i	$-0.457289\pi$
$643$	6.78461	0.267559	0.133779	+	0.991011i	$0.457289\pi$
$644$	0	0
$645$	0.535898	0.0211010
$646$	0	0
$647$	−22.0526	−0.866976	−0.433488	−	0.901159i	$-0.642717\pi$
$647$	−22.0526	−0.866976	−0.433488	+	0.901159i	$0.642717\pi$
$648$	0	0
$649$	−43.1769	−1.69484
$650$	0	0
$651$	−55.7128	−2.18356
$652$	0	0
$653$	7.85641	0.307445	0.153722	−	0.988114i	$-0.450874\pi$
$653$	7.85641	0.307445	0.153722	+	0.988114i	$0.450874\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−17.8564	−0.696645
$658$	0	0
$659$	21.4641	0.836123	0.418061	−	0.908419i	$-0.362710\pi$
$659$	21.4641	0.836123	0.418061	+	0.908419i	$0.362710\pi$
$660$	0	0
$661$	10.7846	0.419473	0.209736	−	0.977758i	$-0.432739\pi$
$661$	10.7846	0.419473	0.209736	+	0.977758i	$0.432739\pi$
$662$	0	0
$663$	9.46410	0.367555
$664$	0	0
$665$	12.3923	0.480553
$666$	0	0
$667$	3.21539	0.124500
$668$	0	0
$669$	5.46410	0.211254
$670$	0	0
$671$	−39.7128	−1.53310
$672$	0	0
$673$	−14.3923	−0.554783	−0.277391	−	0.960757i	$-0.589470\pi$
$673$	−14.3923	−0.554783	−0.277391	+	0.960757i	$0.589470\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	10.3923	0.399409	0.199704	−	0.979856i	$-0.436002\pi$
$677$	10.3923	0.399409	0.199704	+	0.979856i	$0.436002\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	−9.46410	−0.362665
$682$	0	0
$683$	−32.5359	−1.24495	−0.622476	−	0.782639i	$-0.713873\pi$
$683$	−32.5359	−1.24495	−0.622476	+	0.782639i	$0.713873\pi$
$684$	0	0
$685$	−0.928203	−0.0354648
$686$	0	0
$687$	−17.4641	−0.666297
$688$	0	0
$689$	10.3923	0.395915
$690$	0	0
$691$	47.7654	1.81708	0.908540	−	0.417797i	$-0.137198\pi$
$691$	47.7654	1.81708	0.908540	+	0.417797i	$0.137198\pi$
$692$	0	0
$693$	−42.2487	−1.60490
$694$	0	0
$695$	12.3923	0.470067
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	16.3923	0.620014
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	−24.7846	−0.934769
$704$	0	0
$705$	−16.3923	−0.617370
$706$	0	0
$707$	1.85641	0.0698174
$708$	0	0
$709$	30.3923	1.14141	0.570703	−	0.821156i	$-0.306671\pi$
$709$	30.3923	1.14141	0.570703	+	0.821156i	$0.306671\pi$
$710$	0	0
$711$	37.4641	1.40501
$712$	0	0
$713$	12.9282	0.484165
$714$	0	0
$715$	−4.73205	−0.176969
$716$	0	0
$717$	−38.7846	−1.44844
$718$	0	0
$719$	25.8564	0.964281	0.482141	−	0.876094i	$-0.339859\pi$
$719$	25.8564	0.964281	0.482141	+	0.876094i	$0.339859\pi$
$720$	0	0
$721$	−0.392305	−0.0146102
$722$	0	0
$723$	6.53590	0.243073
$724$	0	0
$725$	−2.53590	−0.0941809
$726$	0	0
$727$	−44.5885	−1.65369	−0.826847	−	0.562427i	$-0.809868\pi$
$727$	−44.5885	−1.65369	−0.826847	+	0.562427i	$0.809868\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	−0.679492	−0.0251319
$732$	0	0
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	0	0
$735$	−8.19615	−0.302320
$736$	0	0
$737$	−30.2487	−1.11423
$738$	0	0
$739$	18.1962	0.669356	0.334678	−	0.942332i	$-0.391372\pi$
$739$	18.1962	0.669356	0.334678	+	0.942332i	$0.391372\pi$
$740$	0	0
$741$	−16.9282	−0.621873
$742$	0	0
$743$	16.1436	0.592251	0.296126	−	0.955149i	$-0.404305\pi$
$743$	16.1436	0.592251	0.296126	+	0.955149i	$0.404305\pi$
$744$	0	0
$745$	7.85641	0.287836
$746$	0	0
$747$	26.7846	0.979998
$748$	0	0
$749$	35.3205	1.29058
$750$	0	0
$751$	−36.3923	−1.32797	−0.663987	−	0.747744i	$-0.731137\pi$
$751$	−36.3923	−1.32797	−0.663987	+	0.747744i	$0.731137\pi$
$752$	0	0
$753$	−58.6410	−2.13700
$754$	0	0
$755$	−1.80385	−0.0656487
$756$	0	0
$757$	−2.39230	−0.0869498	−0.0434749	−	0.999055i	$-0.513843\pi$
$757$	−2.39230	−0.0869498	−0.0434749	+	0.999055i	$0.513843\pi$
$758$	0	0
$759$	16.3923	0.595003
$760$	0	0
$761$	−19.8564	−0.719794	−0.359897	−	0.932992i	$-0.617188\pi$
$761$	−19.8564	−0.719794	−0.359897	+	0.932992i	$0.617188\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	0	0
$765$	15.4641	0.559106
$766$	0	0
$767$	−9.12436	−0.329461
$768$	0	0
$769$	34.7846	1.25437	0.627183	−	0.778872i	$-0.284208\pi$
$769$	34.7846	1.25437	0.627183	+	0.778872i	$0.284208\pi$
$770$	0	0
$771$	−54.2487	−1.95372
$772$	0	0
$773$	6.92820	0.249190	0.124595	−	0.992208i	$-0.460237\pi$
$773$	6.92820	0.249190	0.124595	+	0.992208i	$0.460237\pi$
$774$	0	0
$775$	−10.1962	−0.366257
$776$	0	0
$777$	−21.8564	−0.784094
$778$	0	0
$779$	21.4641	0.769031
$780$	0	0
$781$	−22.3923	−0.801260
$782$	0	0
$783$	10.1436	0.362502
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	−31.5692	−1.12532	−0.562661	−	0.826688i	$-0.690222\pi$
$787$	−31.5692	−1.12532	−0.562661	+	0.826688i	$0.690222\pi$
$788$	0	0
$789$	3.46410	0.123325
$790$	0	0
$791$	−17.0718	−0.607003
$792$	0	0
$793$	−8.39230	−0.298019
$794$	0	0
$795$	28.3923	1.00697
$796$	0	0
$797$	−40.6410	−1.43958	−0.719789	−	0.694193i	$-0.755762\pi$
$797$	−40.6410	−1.43958	−0.719789	+	0.694193i	$0.755762\pi$
$798$	0	0
$799$	20.7846	0.735307
$800$	0	0
$801$	−57.7128	−2.03918
$802$	0	0
$803$	−18.9282	−0.667962
$804$	0	0
$805$	−2.53590	−0.0893787
$806$	0	0
$807$	54.2487	1.90965
$808$	0	0
$809$	−2.53590	−0.0891574	−0.0445787	−	0.999006i	$-0.514195\pi$
$809$	−2.53590	−0.0891574	−0.0445787	+	0.999006i	$0.514195\pi$
$810$	0	0
$811$	−17.8038	−0.625178	−0.312589	−	0.949889i	$-0.601196\pi$
$811$	−17.8038	−0.625178	−0.312589	+	0.949889i	$0.601196\pi$
$812$	0	0
$813$	84.6410	2.96849
$814$	0	0
$815$	−14.3923	−0.504140
$816$	0	0
$817$	1.21539	0.0425211
$818$	0	0
$819$	−8.92820	−0.311977
$820$	0	0
$821$	28.6410	0.999578	0.499789	−	0.866147i	$-0.333411\pi$
$821$	28.6410	0.999578	0.499789	+	0.866147i	$0.333411\pi$
$822$	0	0
$823$	15.4115	0.537213	0.268606	−	0.963250i	$-0.413437\pi$
$823$	15.4115	0.537213	0.268606	+	0.963250i	$0.413437\pi$
$824$	0	0
$825$	−12.9282	−0.450102
$826$	0	0
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	0	0
$829$	0.392305	0.0136253	0.00681266	−	0.999977i	$-0.497831\pi$
$829$	0.392305	0.0136253	0.00681266	+	0.999977i	$0.497831\pi$
$830$	0	0
$831$	72.1051	2.50130
$832$	0	0
$833$	10.3923	0.360072
$834$	0	0
$835$	−0.928203	−0.0321218
$836$	0	0
$837$	40.7846	1.40972
$838$	0	0
$839$	−0.339746	−0.0117293	−0.00586467	−	0.999983i	$-0.501867\pi$
$839$	−0.339746	−0.0117293	−0.00586467	+	0.999983i	$0.501867\pi$
$840$	0	0
$841$	−22.5692	−0.778249
$842$	0	0
$843$	−61.1769	−2.10704
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−22.7846	−0.782888
$848$	0	0
$849$	89.0333	3.05562
$850$	0	0
$851$	5.07180	0.173859
$852$	0	0
$853$	8.00000	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$853$	8.00000	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$854$	0	0
$855$	−27.6603	−0.945961
$856$	0	0
$857$	−35.5692	−1.21502	−0.607511	−	0.794311i	$-0.707832\pi$
$857$	−35.5692	−1.21502	−0.607511	+	0.794311i	$0.707832\pi$
$858$	0	0
$859$	17.1769	0.586069	0.293034	−	0.956102i	$-0.405335\pi$
$859$	17.1769	0.586069	0.293034	+	0.956102i	$0.405335\pi$
$860$	0	0
$861$	18.9282	0.645071
$862$	0	0
$863$	38.7846	1.32024	0.660122	−	0.751159i	$-0.270505\pi$
$863$	38.7846	1.32024	0.660122	+	0.751159i	$0.270505\pi$
$864$	0	0
$865$	−8.53590	−0.290229
$866$	0	0
$867$	13.6603	0.463927
$868$	0	0
$869$	39.7128	1.34716
$870$	0	0
$871$	−6.39230	−0.216595
$872$	0	0
$873$	8.92820	0.302174
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	13.8564	0.467365
$880$	0	0
$881$	−47.3205	−1.59427	−0.797134	−	0.603802i	$-0.793652\pi$
$881$	−47.3205	−1.59427	−0.797134	+	0.603802i	$0.793652\pi$
$882$	0	0
$883$	−23.8038	−0.801063	−0.400532	−	0.916283i	$-0.631175\pi$
$883$	−23.8038	−0.801063	−0.400532	+	0.916283i	$0.631175\pi$
$884$	0	0
$885$	−24.9282	−0.837952
$886$	0	0
$887$	47.9090	1.60863	0.804313	−	0.594206i	$-0.202534\pi$
$887$	47.9090	1.60863	0.804313	+	0.594206i	$0.202534\pi$
$888$	0	0
$889$	32.3923	1.08640
$890$	0	0
$891$	−11.6603	−0.390633
$892$	0	0
$893$	−37.1769	−1.24408
$894$	0	0
$895$	−18.9282	−0.632700
$896$	0	0
$897$	3.46410	0.115663
$898$	0	0
$899$	25.8564	0.862359
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	1.07180	0.0356672
$904$	0	0
$905$	−0.392305	−0.0130407
$906$	0	0
$907$	53.7654	1.78525	0.892625	−	0.450800i	$-0.148861\pi$
$907$	53.7654	1.78525	0.892625	+	0.450800i	$0.148861\pi$
$908$	0	0
$909$	−4.14359	−0.137434
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	28.3923	0.939648
$914$	0	0
$915$	−22.9282	−0.757983
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−9.17691	−0.302718	−0.151359	−	0.988479i	$-0.548365\pi$
$919$	−9.17691	−0.302718	−0.151359	+	0.988479i	$0.548365\pi$
$920$	0	0
$921$	−51.3205	−1.69107
$922$	0	0
$923$	−4.73205	−0.155757
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	0.875644	0.0287599
$928$	0	0
$929$	44.5359	1.46118	0.730588	−	0.682819i	$-0.239246\pi$
$929$	44.5359	1.46118	0.730588	+	0.682819i	$0.239246\pi$
$930$	0	0
$931$	−18.5885	−0.609212
$932$	0	0
$933$	−44.7846	−1.46618
$934$	0	0
$935$	16.3923	0.536086
$936$	0	0
$937$	34.7846	1.13636	0.568182	−	0.822903i	$-0.307647\pi$
$937$	34.7846	1.13636	0.568182	+	0.822903i	$0.307647\pi$
$938$	0	0
$939$	39.3205	1.28318
$940$	0	0
$941$	31.1769	1.01634	0.508169	−	0.861257i	$-0.330322\pi$
$941$	31.1769	1.01634	0.508169	+	0.861257i	$0.330322\pi$
$942$	0	0
$943$	−4.39230	−0.143033
$944$	0	0
$945$	−8.00000	−0.260240
$946$	0	0
$947$	−40.6410	−1.32066	−0.660328	−	0.750978i	$-0.729583\pi$
$947$	−40.6410	−1.32066	−0.660328	+	0.750978i	$0.729583\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−65.5692	−2.12623
$952$	0	0
$953$	0.928203	0.0300675	0.0150337	−	0.999887i	$-0.495214\pi$
$953$	0.928203	0.0300675	0.0150337	+	0.999887i	$0.495214\pi$
$954$	0	0
$955$	−5.07180	−0.164119
$956$	0	0
$957$	32.7846	1.05978
$958$	0	0
$959$	−1.85641	−0.0599465
$960$	0	0
$961$	72.9615	2.35360
$962$	0	0
$963$	−78.8372	−2.54049
$964$	0	0
$965$	10.0000	0.321911
$966$	0	0
$967$	50.3923	1.62051	0.810254	−	0.586079i	$-0.199329\pi$
$967$	50.3923	1.62051	0.810254	+	0.586079i	$0.199329\pi$
$968$	0	0
$969$	58.6410	1.88382
$970$	0	0
$971$	−18.9282	−0.607435	−0.303717	−	0.952762i	$-0.598228\pi$
$971$	−18.9282	−0.607435	−0.303717	+	0.952762i	$0.598228\pi$
$972$	0	0
$973$	24.7846	0.794558
$974$	0	0
$975$	−2.73205	−0.0874957
$976$	0	0
$977$	−15.7128	−0.502697	−0.251349	−	0.967897i	$-0.580874\pi$
$977$	−15.7128	−0.502697	−0.251349	+	0.967897i	$0.580874\pi$
$978$	0	0
$979$	−61.1769	−1.95522
$980$	0	0
$981$	8.92820	0.285056
$982$	0	0
$983$	34.3923	1.09694	0.548472	−	0.836169i	$-0.315210\pi$
$983$	34.3923	1.09694	0.548472	+	0.836169i	$0.315210\pi$
$984$	0	0
$985$	12.9282	0.411927
$986$	0	0
$987$	−32.7846	−1.04355
$988$	0	0
$989$	−0.248711	−0.00790856
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	7.07180	0.224417
$994$	0	0
$995$	20.0000	0.634043
$996$	0	0
$997$	33.6077	1.06437	0.532183	−	0.846629i	$-0.321372\pi$
$997$	33.6077	1.06437	0.532183	+	0.846629i	$0.321372\pi$
$998$	0	0
$999$	16.0000	0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.a.h.1.1		2
3.2	odd	2		9360.2.a.cm.1.1		2
4.3	odd	2		65.2.a.c.1.1	✓	2
5.4	even	2		5200.2.a.ca.1.2		2
8.3	odd	2		4160.2.a.y.1.1		2
8.5	even	2		4160.2.a.bj.1.2		2
12.11	even	2		585.2.a.k.1.2		2
20.3	even	4		325.2.b.e.274.4		4
20.7	even	4		325.2.b.e.274.1		4
20.19	odd	2		325.2.a.g.1.2		2
28.27	even	2		3185.2.a.k.1.1		2
44.43	even	2		7865.2.a.h.1.2		2
52.3	odd	6		845.2.e.e.191.2		4
52.7	even	12		845.2.m.a.361.1		4
52.11	even	12		845.2.m.c.316.1		4
52.15	even	12		845.2.m.a.316.1		4
52.19	even	12		845.2.m.c.361.1		4
52.23	odd	6		845.2.e.f.191.1		4
52.31	even	4		845.2.c.e.506.4		4
52.35	odd	6		845.2.e.e.146.2		4
52.43	odd	6		845.2.e.f.146.1		4
52.47	even	4		845.2.c.e.506.2		4
52.51	odd	2		845.2.a.d.1.2		2
60.23	odd	4		2925.2.c.v.2224.1		4
60.47	odd	4		2925.2.c.v.2224.4		4
60.59	even	2		2925.2.a.z.1.1		2
156.155	even	2		7605.2.a.be.1.1		2
260.259	odd	2		4225.2.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.c.1.1	✓	2	4.3	odd	2
325.2.a.g.1.2		2	20.19	odd	2
325.2.b.e.274.1		4	20.7	even	4
325.2.b.e.274.4		4	20.3	even	4
585.2.a.k.1.2		2	12.11	even	2
845.2.a.d.1.2		2	52.51	odd	2
845.2.c.e.506.2		4	52.47	even	4
845.2.c.e.506.4		4	52.31	even	4
845.2.e.e.146.2		4	52.35	odd	6
845.2.e.e.191.2		4	52.3	odd	6
845.2.e.f.146.1		4	52.43	odd	6
845.2.e.f.191.1		4	52.23	odd	6
845.2.m.a.316.1		4	52.15	even	12
845.2.m.a.361.1		4	52.7	even	12
845.2.m.c.316.1		4	52.11	even	12
845.2.m.c.361.1		4	52.19	even	12
1040.2.a.h.1.1		2	1.1	even	1	trivial
2925.2.a.z.1.1		2	60.59	even	2
2925.2.c.v.2224.1		4	60.23	odd	4
2925.2.c.v.2224.4		4	60.47	odd	4
3185.2.a.k.1.1		2	28.27	even	2
4160.2.a.y.1.1		2	8.3	odd	2
4160.2.a.bj.1.2		2	8.5	even	2
4225.2.a.w.1.1		2	260.259	odd	2
5200.2.a.ca.1.2		2	5.4	even	2
7605.2.a.be.1.1		2	156.155	even	2
7865.2.a.h.1.2		2	44.43	even	2
9360.2.a.cm.1.1		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} -1.00000 q^{5} -2.00000 q^{7} +4.46410 q^{9} +4.73205 q^{11} +1.00000 q^{13} +2.73205 q^{15} -3.46410 q^{17} +6.19615 q^{19} +5.46410 q^{21} -1.26795 q^{23} +1.00000 q^{25} -4.00000 q^{27} -2.53590 q^{29} -10.1962 q^{31} -12.9282 q^{33} +2.00000 q^{35} -4.00000 q^{37} -2.73205 q^{39} +3.46410 q^{41} +0.196152 q^{43} -4.46410 q^{45} -6.00000 q^{47} -3.00000 q^{49} +9.46410 q^{51} +10.3923 q^{53} -4.73205 q^{55} -16.9282 q^{57} -9.12436 q^{59} -8.39230 q^{61} -8.92820 q^{63} -1.00000 q^{65} -6.39230 q^{67} +3.46410 q^{69} -4.73205 q^{71} -4.00000 q^{73} -2.73205 q^{75} -9.46410 q^{77} +8.39230 q^{79} -2.46410 q^{81} +6.00000 q^{83} +3.46410 q^{85} +6.92820 q^{87} -12.9282 q^{89} -2.00000 q^{91} +27.8564 q^{93} -6.19615 q^{95} +2.00000 q^{97} +21.1244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{13} + 2 q^{15} + 2 q^{19} + 4 q^{21} - 6 q^{23} + 2 q^{25} - 8 q^{27} - 12 q^{29} - 10 q^{31} - 12 q^{33} + 4 q^{35} - 8 q^{37} - 2 q^{39} - 10 q^{43}+ \cdots + 18 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 4 * q^7 + 2 * q^9 + 6 * q^11 + 2 * q^13 + 2 * q^15 + 2 * q^19 + 4 * q^21 - 6 * q^23 + 2 * q^25 - 8 * q^27 - 12 * q^29 - 10 * q^31 - 12 * q^33 + 4 * q^35 - 8 * q^37 - 2 * q^39 - 10 * q^43 - 2 * q^45 - 12 * q^47 - 6 * q^49 + 12 * q^51 - 6 * q^55 - 20 * q^57 + 6 * q^59 + 4 * q^61 - 4 * q^63 - 2 * q^65 + 8 * q^67 - 6 * q^71 - 8 * q^73 - 2 * q^75 - 12 * q^77 - 4 * q^79 + 2 * q^81 + 12 * q^83 - 12 * q^89 - 4 * q^91 + 28 * q^93 - 2 * q^95 + 4 * q^97 + 18 * q^99

Introduction

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