LMFDB - Embedded newform 8880.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8880,2,Mod(1,8880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8880, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8880.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8880.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:	$70.9071569949$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{113}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 28 $ x^2 - x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.81507$ of defining polynomial
Character	$\chi$	$=$	8880.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.81507 q^{13} -1.00000 q^{15} +1.00000 q^{17} -7.81507 q^{19} +3.00000 q^{21} -5.81507 q^{23} +1.00000 q^{25} -1.00000 q^{27} -6.81507 q^{29} -6.81507 q^{31} -1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} +3.81507 q^{39} -4.81507 q^{41} +4.81507 q^{43} +1.00000 q^{45} +8.00000 q^{47} +2.00000 q^{49} -1.00000 q^{51} +10.6301 q^{53} +1.00000 q^{55} +7.81507 q^{57} -2.00000 q^{59} +6.81507 q^{61} -3.00000 q^{63} -3.81507 q^{65} -7.63015 q^{67} +5.81507 q^{69} +11.6301 q^{71} -5.81507 q^{73} -1.00000 q^{75} -3.00000 q^{77} -12.0000 q^{79} +1.00000 q^{81} +3.81507 q^{83} +1.00000 q^{85} +6.81507 q^{87} +11.8151 q^{89} +11.4452 q^{91} +6.81507 q^{93} -7.81507 q^{95} -4.81507 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} + 2 q^{11} + 3 q^{13} - 2 q^{15} + 2 q^{17} - 5 q^{19} + 6 q^{21} - q^{23} + 2 q^{25} - 2 q^{27} - 3 q^{29} - 3 q^{31} - 2 q^{33} - 6 q^{35} + 2 q^{37} - 3 q^{39} + q^{41} - q^{43} + 2 q^{45} + 16 q^{47} + 4 q^{49} - 2 q^{51} + 2 q^{55} + 5 q^{57} - 4 q^{59} + 3 q^{61} - 6 q^{63} + 3 q^{65} + 6 q^{67} + q^{69} + 2 q^{71} - q^{73} - 2 q^{75} - 6 q^{77} - 24 q^{79} + 2 q^{81} - 3 q^{83} + 2 q^{85} + 3 q^{87} + 13 q^{89} - 9 q^{91} + 3 q^{93} - 5 q^{95} + q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 + 2 * q^11 + 3 * q^13 - 2 * q^15 + 2 * q^17 - 5 * q^19 + 6 * q^21 - q^23 + 2 * q^25 - 2 * q^27 - 3 * q^29 - 3 * q^31 - 2 * q^33 - 6 * q^35 + 2 * q^37 - 3 * q^39 + q^41 - q^43 + 2 * q^45 + 16 * q^47 + 4 * q^49 - 2 * q^51 + 2 * q^55 + 5 * q^57 - 4 * q^59 + 3 * q^61 - 6 * q^63 + 3 * q^65 + 6 * q^67 + q^69 + 2 * q^71 - q^73 - 2 * q^75 - 6 * q^77 - 24 * q^79 + 2 * q^81 - 3 * q^83 + 2 * q^85 + 3 * q^87 + 13 * q^89 - 9 * q^91 + 3 * q^93 - 5 * q^95 + q^97 + 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−3.81507	−1.05811	−0.529055	−	0.848587i	$-0.677454\pi$
$13$	−3.81507	−1.05811	−0.529055	+	0.848587i	$0.677454\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−7.81507	−1.79290	−0.896450	−	0.443144i	$-0.853863\pi$
$19$	−7.81507	−1.79290	−0.896450	+	0.443144i	$0.853863\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	−5.81507	−1.21253	−0.606263	−	0.795264i	$-0.707332\pi$
$23$	−5.81507	−1.21253	−0.606263	+	0.795264i	$0.707332\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.81507	−1.26553	−0.632764	−	0.774345i	$-0.718080\pi$
$29$	−6.81507	−1.26553	−0.632764	+	0.774345i	$0.718080\pi$
$30$	0	0
$31$	−6.81507	−1.22402	−0.612012	−	0.790849i	$-0.709639\pi$
$31$	−6.81507	−1.22402	−0.612012	+	0.790849i	$0.709639\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	3.81507	0.610901
$40$	0	0
$41$	−4.81507	−0.751988	−0.375994	−	0.926622i	$-0.622699\pi$
$41$	−4.81507	−0.751988	−0.375994	+	0.926622i	$0.622699\pi$
$42$	0	0
$43$	4.81507	0.734292	0.367146	−	0.930163i	$-0.380335\pi$
$43$	4.81507	0.734292	0.367146	+	0.930163i	$0.380335\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	10.6301	1.46016	0.730081	−	0.683360i	$-0.239482\pi$
$53$	10.6301	1.46016	0.730081	+	0.683360i	$0.239482\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	7.81507	1.03513
$58$	0	0
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	6.81507	0.872581	0.436290	−	0.899806i	$-0.356292\pi$
$61$	6.81507	0.872581	0.436290	+	0.899806i	$0.356292\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	−3.81507	−0.473202
$66$	0	0
$67$	−7.63015	−0.932171	−0.466085	−	0.884740i	$-0.654336\pi$
$67$	−7.63015	−0.932171	−0.466085	+	0.884740i	$0.654336\pi$
$68$	0	0
$69$	5.81507	0.700053
$70$	0	0
$71$	11.6301	1.38024	0.690122	−	0.723693i	$-0.257557\pi$
$71$	11.6301	1.38024	0.690122	+	0.723693i	$0.257557\pi$
$72$	0	0
$73$	−5.81507	−0.680603	−0.340301	−	0.940316i	$-0.610529\pi$
$73$	−5.81507	−0.680603	−0.340301	+	0.940316i	$0.610529\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.81507	0.418759	0.209379	−	0.977834i	$-0.432856\pi$
$83$	3.81507	0.418759	0.209379	+	0.977834i	$0.432856\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	6.81507	0.730653
$88$	0	0
$89$	11.8151	1.25240	0.626198	−	0.779664i	$-0.284610\pi$
$89$	11.8151	1.25240	0.626198	+	0.779664i	$0.284610\pi$
$90$	0	0
$91$	11.4452	1.19978
$92$	0	0
$93$	6.81507	0.706690
$94$	0	0
$95$	−7.81507	−0.801810
$96$	0	0
$97$	−4.81507	−0.488897	−0.244448	−	0.969662i	$-0.578607\pi$
$97$	−4.81507	−0.488897	−0.244448	+	0.969662i	$0.578607\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	11.6301	1.15724	0.578621	−	0.815596i	$-0.303591\pi$
$101$	11.6301	1.15724	0.578621	+	0.815596i	$0.303591\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	8.18493	0.791267	0.395633	−	0.918409i	$-0.370525\pi$
$107$	8.18493	0.791267	0.395633	+	0.918409i	$0.370525\pi$
$108$	0	0
$109$	13.0000	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	−6.81507	−0.641108	−0.320554	−	0.947230i	$-0.603869\pi$
$113$	−6.81507	−0.641108	−0.320554	+	0.947230i	$0.603869\pi$
$114$	0	0
$115$	−5.81507	−0.542258
$116$	0	0
$117$	−3.81507	−0.352704
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	4.81507	0.434161
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	21.4452	1.90296	0.951478	−	0.307718i	$-0.0995652\pi$
$127$	21.4452	1.90296	0.951478	+	0.307718i	$0.0995652\pi$
$128$	0	0
$129$	−4.81507	−0.423944
$130$	0	0
$131$	9.63015	0.841390	0.420695	−	0.907202i	$-0.361786\pi$
$131$	9.63015	0.841390	0.420695	+	0.907202i	$0.361786\pi$
$132$	0	0
$133$	23.4452	2.03296
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−22.8151	−1.93515	−0.967575	−	0.252585i	$-0.918719\pi$
$139$	−22.8151	−1.93515	−0.967575	+	0.252585i	$0.918719\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−3.81507	−0.319032
$144$	0	0
$145$	−6.81507	−0.565961
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−3.81507	−0.310466	−0.155233	−	0.987878i	$-0.549613\pi$
$151$	−3.81507	−0.310466	−0.155233	+	0.987878i	$0.549613\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−6.81507	−0.547400
$156$	0	0
$157$	−12.8151	−1.02275	−0.511377	−	0.859356i	$-0.670864\pi$
$157$	−12.8151	−1.02275	−0.511377	+	0.859356i	$0.670864\pi$
$158$	0	0
$159$	−10.6301	−0.843025
$160$	0	0
$161$	17.4452	1.37488
$162$	0	0
$163$	−16.6301	−1.30257	−0.651287	−	0.758832i	$-0.725770\pi$
$163$	−16.6301	−1.30257	−0.651287	+	0.758832i	$0.725770\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	21.8151	1.68810	0.844051	−	0.536264i	$-0.180165\pi$
$167$	21.8151	1.68810	0.844051	+	0.536264i	$0.180165\pi$
$168$	0	0
$169$	1.55478	0.119599
$170$	0	0
$171$	−7.81507	−0.597634
$172$	0	0
$173$	−1.00000	−0.0760286	−0.0380143	−	0.999277i	$-0.512103\pi$
$173$	−1.00000	−0.0760286	−0.0380143	+	0.999277i	$0.512103\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	2.00000	0.150329
$178$	0	0
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−6.81507	−0.503785
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	1.00000	0.0731272
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	−4.63015	−0.335026	−0.167513	−	0.985870i	$-0.553574\pi$
$191$	−4.63015	−0.335026	−0.167513	+	0.985870i	$0.553574\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	3.81507	0.273203
$196$	0	0
$197$	−4.18493	−0.298164	−0.149082	−	0.988825i	$-0.547632\pi$
$197$	−4.18493	−0.298164	−0.149082	+	0.988825i	$0.547632\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	7.63015	0.538189
$202$	0	0
$203$	20.4452	1.43497
$204$	0	0
$205$	−4.81507	−0.336299
$206$	0	0
$207$	−5.81507	−0.404176
$208$	0	0
$209$	−7.81507	−0.540580
$210$	0	0
$211$	−8.44522	−0.581393	−0.290696	−	0.956815i	$-0.593887\pi$
$211$	−8.44522	−0.581393	−0.290696	+	0.956815i	$0.593887\pi$
$212$	0	0
$213$	−11.6301	−0.796884
$214$	0	0
$215$	4.81507	0.328385
$216$	0	0
$217$	20.4452	1.38791
$218$	0	0
$219$	5.81507	0.392946
$220$	0	0
$221$	−3.81507	−0.256630
$222$	0	0
$223$	16.8151	1.12602	0.563010	−	0.826450i	$-0.309643\pi$
$223$	16.8151	1.12602	0.563010	+	0.826450i	$0.309643\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	24.8151	1.64703	0.823517	−	0.567291i	$-0.192009\pi$
$227$	24.8151	1.64703	0.823517	+	0.567291i	$0.192009\pi$
$228$	0	0
$229$	0.369854	0.0244407	0.0122203	−	0.999925i	$-0.496110\pi$
$229$	0.369854	0.0244407	0.0122203	+	0.999925i	$0.496110\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	−7.63015	−0.499867	−0.249934	−	0.968263i	$-0.580409\pi$
$233$	−7.63015	−0.499867	−0.249934	+	0.968263i	$0.580409\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	12.0000	0.779484
$238$	0	0
$239$	−6.44522	−0.416907	−0.208453	−	0.978032i	$-0.566843\pi$
$239$	−6.44522	−0.416907	−0.208453	+	0.978032i	$0.566843\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	29.8151	1.89709
$248$	0	0
$249$	−3.81507	−0.241770
$250$	0	0
$251$	11.6301	0.734088	0.367044	−	0.930204i	$-0.380370\pi$
$251$	11.6301	0.734088	0.367044	+	0.930204i	$0.380370\pi$
$252$	0	0
$253$	−5.81507	−0.365591
$254$	0	0
$255$	−1.00000	−0.0626224
$256$	0	0
$257$	−3.81507	−0.237978	−0.118989	−	0.992896i	$-0.537965\pi$
$257$	−3.81507	−0.237978	−0.118989	+	0.992896i	$0.537965\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	−6.81507	−0.421842
$262$	0	0
$263$	30.4452	1.87733	0.938666	−	0.344827i	$-0.112062\pi$
$263$	30.4452	1.87733	0.938666	+	0.344827i	$0.112062\pi$
$264$	0	0
$265$	10.6301	0.653005
$266$	0	0
$267$	−11.8151	−0.723071
$268$	0	0
$269$	9.81507	0.598436	0.299218	−	0.954185i	$-0.403274\pi$
$269$	9.81507	0.598436	0.299218	+	0.954185i	$0.403274\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	−11.4452	−0.692696
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	21.4452	1.28852	0.644259	−	0.764807i	$-0.277166\pi$
$277$	21.4452	1.28852	0.644259	+	0.764807i	$0.277166\pi$
$278$	0	0
$279$	−6.81507	−0.408008
$280$	0	0
$281$	−19.4452	−1.16000	−0.580002	−	0.814615i	$-0.696948\pi$
$281$	−19.4452	−1.16000	−0.580002	+	0.814615i	$0.696948\pi$
$282$	0	0
$283$	21.8151	1.29677	0.648386	−	0.761312i	$-0.275444\pi$
$283$	21.8151	1.29677	0.648386	+	0.761312i	$0.275444\pi$
$284$	0	0
$285$	7.81507	0.462925
$286$	0	0
$287$	14.4452	0.852674
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	4.81507	0.282265
$292$	0	0
$293$	28.2603	1.65098	0.825492	−	0.564414i	$-0.190898\pi$
$293$	28.2603	1.65098	0.825492	+	0.564414i	$0.190898\pi$
$294$	0	0
$295$	−2.00000	−0.116445
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	22.1849	1.28299
$300$	0	0
$301$	−14.4452	−0.832609
$302$	0	0
$303$	−11.6301	−0.668134
$304$	0	0
$305$	6.81507	0.390230
$306$	0	0
$307$	18.0000	1.02731	0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000	1.02731	0.513657	+	0.857996i	$0.328290\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.4452	−0.592294	−0.296147	−	0.955142i	$-0.595702\pi$
$311$	−10.4452	−0.592294	−0.296147	+	0.955142i	$0.595702\pi$
$312$	0	0
$313$	32.8904	1.85908	0.929539	−	0.368725i	$-0.120205\pi$
$313$	32.8904	1.85908	0.929539	+	0.368725i	$0.120205\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	−2.81507	−0.158110	−0.0790551	−	0.996870i	$-0.525190\pi$
$317$	−2.81507	−0.158110	−0.0790551	+	0.996870i	$0.525190\pi$
$318$	0	0
$319$	−6.81507	−0.381571
$320$	0	0
$321$	−8.18493	−0.456838
$322$	0	0
$323$	−7.81507	−0.434842
$324$	0	0
$325$	−3.81507	−0.211622
$326$	0	0
$327$	−13.0000	−0.718902
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	23.2603	1.27850	0.639251	−	0.768998i	$-0.279245\pi$
$331$	23.2603	1.27850	0.639251	+	0.768998i	$0.279245\pi$
$332$	0	0
$333$	1.00000	0.0547997
$334$	0	0
$335$	−7.63015	−0.416879
$336$	0	0
$337$	−4.18493	−0.227968	−0.113984	−	0.993483i	$-0.536361\pi$
$337$	−4.18493	−0.227968	−0.113984	+	0.993483i	$0.536361\pi$
$338$	0	0
$339$	6.81507	0.370144
$340$	0	0
$341$	−6.81507	−0.369057
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	5.81507	0.313073
$346$	0	0
$347$	−11.2603	−0.604484	−0.302242	−	0.953231i	$-0.597735\pi$
$347$	−11.2603	−0.604484	−0.302242	+	0.953231i	$0.597735\pi$
$348$	0	0
$349$	7.26029	0.388635	0.194317	−	0.980939i	$-0.437751\pi$
$349$	7.26029	0.388635	0.194317	+	0.980939i	$0.437751\pi$
$350$	0	0
$351$	3.81507	0.203634
$352$	0	0
$353$	−13.1849	−0.701763	−0.350881	−	0.936420i	$-0.614118\pi$
$353$	−13.1849	−0.701763	−0.350881	+	0.936420i	$0.614118\pi$
$354$	0	0
$355$	11.6301	0.617264
$356$	0	0
$357$	3.00000	0.158777
$358$	0	0
$359$	−6.36985	−0.336188	−0.168094	−	0.985771i	$-0.553761\pi$
$359$	−6.36985	−0.336188	−0.168094	+	0.985771i	$0.553761\pi$
$360$	0	0
$361$	42.0754	2.21449
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	−5.81507	−0.304375
$366$	0	0
$367$	21.0000	1.09619	0.548096	−	0.836416i	$-0.315353\pi$
$367$	21.0000	1.09619	0.548096	+	0.836416i	$0.315353\pi$
$368$	0	0
$369$	−4.81507	−0.250663
$370$	0	0
$371$	−31.8904	−1.65567
$372$	0	0
$373$	−17.2603	−0.893704	−0.446852	−	0.894608i	$-0.647455\pi$
$373$	−17.2603	−0.893704	−0.446852	+	0.894608i	$0.647455\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	26.0000	1.33907
$378$	0	0
$379$	19.6301	1.00833	0.504166	−	0.863607i	$-0.331800\pi$
$379$	19.6301	1.00833	0.504166	+	0.863607i	$0.331800\pi$
$380$	0	0
$381$	−21.4452	−1.09867
$382$	0	0
$383$	−33.4452	−1.70897	−0.854485	−	0.519475i	$-0.826127\pi$
$383$	−33.4452	−1.70897	−0.854485	+	0.519475i	$0.826127\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	4.81507	0.244764
$388$	0	0
$389$	−35.7055	−1.81034	−0.905171	−	0.425048i	$-0.860257\pi$
$389$	−35.7055	−1.81034	−0.905171	+	0.425048i	$0.860257\pi$
$390$	0	0
$391$	−5.81507	−0.294081
$392$	0	0
$393$	−9.63015	−0.485777
$394$	0	0
$395$	−12.0000	−0.603786
$396$	0	0
$397$	−5.63015	−0.282569	−0.141284	−	0.989969i	$-0.545123\pi$
$397$	−5.63015	−0.282569	−0.141284	+	0.989969i	$0.545123\pi$
$398$	0	0
$399$	−23.4452	−1.17373
$400$	0	0
$401$	0.554781	0.0277045	0.0138522	−	0.999904i	$-0.495591\pi$
$401$	0.554781	0.0277045	0.0138522	+	0.999904i	$0.495591\pi$
$402$	0	0
$403$	26.0000	1.29515
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	1.00000	0.0495682
$408$	0	0
$409$	6.36985	0.314969	0.157485	−	0.987521i	$-0.449662\pi$
$409$	6.36985	0.314969	0.157485	+	0.987521i	$0.449662\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	6.00000	0.295241
$414$	0	0
$415$	3.81507	0.187275
$416$	0	0
$417$	22.8151	1.11726
$418$	0	0
$419$	25.4452	1.24308	0.621540	−	0.783382i	$-0.286507\pi$
$419$	25.4452	1.24308	0.621540	+	0.783382i	$0.286507\pi$
$420$	0	0
$421$	−29.2603	−1.42606	−0.713030	−	0.701134i	$-0.752678\pi$
$421$	−29.2603	−1.42606	−0.713030	+	0.701134i	$0.752678\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	−20.4452	−0.989413
$428$	0	0
$429$	3.81507	0.184193
$430$	0	0
$431$	−21.0000	−1.01153	−0.505767	−	0.862670i	$-0.668791\pi$
$431$	−21.0000	−1.01153	−0.505767	+	0.862670i	$0.668791\pi$
$432$	0	0
$433$	1.81507	0.0872268	0.0436134	−	0.999048i	$-0.486113\pi$
$433$	1.81507	0.0872268	0.0436134	+	0.999048i	$0.486113\pi$
$434$	0	0
$435$	6.81507	0.326758
$436$	0	0
$437$	45.4452	2.17394
$438$	0	0
$439$	18.4452	0.880342	0.440171	−	0.897914i	$-0.354918\pi$
$439$	18.4452	0.880342	0.440171	+	0.897914i	$0.354918\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−13.6301	−0.647588	−0.323794	−	0.946128i	$-0.604958\pi$
$443$	−13.6301	−0.647588	−0.323794	+	0.946128i	$0.604958\pi$
$444$	0	0
$445$	11.8151	0.560088
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	30.8904	1.45781	0.728905	−	0.684615i	$-0.240030\pi$
$449$	30.8904	1.45781	0.728905	+	0.684615i	$0.240030\pi$
$450$	0	0
$451$	−4.81507	−0.226733
$452$	0	0
$453$	3.81507	0.179248
$454$	0	0
$455$	11.4452	0.536560
$456$	0	0
$457$	−28.0754	−1.31331	−0.656655	−	0.754191i	$-0.728029\pi$
$457$	−28.0754	−1.31331	−0.656655	+	0.754191i	$0.728029\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	34.8151	1.62150	0.810750	−	0.585393i	$-0.199060\pi$
$461$	34.8151	1.62150	0.810750	+	0.585393i	$0.199060\pi$
$462$	0	0
$463$	17.6301	0.819342	0.409671	−	0.912233i	$-0.365643\pi$
$463$	17.6301	0.819342	0.409671	+	0.912233i	$0.365643\pi$
$464$	0	0
$465$	6.81507	0.316041
$466$	0	0
$467$	−28.8151	−1.33340	−0.666701	−	0.745325i	$-0.732294\pi$
$467$	−28.8151	−1.33340	−0.666701	+	0.745325i	$0.732294\pi$
$468$	0	0
$469$	22.8904	1.05698
$470$	0	0
$471$	12.8151	0.590487
$472$	0	0
$473$	4.81507	0.221397
$474$	0	0
$475$	−7.81507	−0.358580
$476$	0	0
$477$	10.6301	0.486721
$478$	0	0
$479$	−9.81507	−0.448462	−0.224231	−	0.974536i	$-0.571987\pi$
$479$	−9.81507	−0.448462	−0.224231	+	0.974536i	$0.571987\pi$
$480$	0	0
$481$	−3.81507	−0.173952
$482$	0	0
$483$	−17.4452	−0.793785
$484$	0	0
$485$	−4.81507	−0.218641
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	16.6301	0.752041
$490$	0	0
$491$	−2.18493	−0.0986044	−0.0493022	−	0.998784i	$-0.515700\pi$
$491$	−2.18493	−0.0986044	−0.0493022	+	0.998784i	$0.515700\pi$
$492$	0	0
$493$	−6.81507	−0.306935
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	−34.8904	−1.56505
$498$	0	0
$499$	17.4452	0.780955	0.390478	−	0.920612i	$-0.372310\pi$
$499$	17.4452	0.780955	0.390478	+	0.920612i	$0.372310\pi$
$500$	0	0
$501$	−21.8151	−0.974626
$502$	0	0
$503$	−0.739708	−0.0329820	−0.0164910	−	0.999864i	$-0.505249\pi$
$503$	−0.739708	−0.0329820	−0.0164910	+	0.999864i	$0.505249\pi$
$504$	0	0
$505$	11.6301	0.517535
$506$	0	0
$507$	−1.55478	−0.0690503
$508$	0	0
$509$	17.4452	0.773246	0.386623	−	0.922238i	$-0.373642\pi$
$509$	17.4452	0.773246	0.386623	+	0.922238i	$0.373642\pi$
$510$	0	0
$511$	17.4452	0.771731
$512$	0	0
$513$	7.81507	0.345044
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	1.00000	0.0438951
$520$	0	0
$521$	0.815073	0.0357090	0.0178545	−	0.999841i	$-0.494316\pi$
$521$	0.815073	0.0357090	0.0178545	+	0.999841i	$0.494316\pi$
$522$	0	0
$523$	0.739708	0.0323452	0.0161726	−	0.999869i	$-0.494852\pi$
$523$	0.739708	0.0323452	0.0161726	+	0.999869i	$0.494852\pi$
$524$	0	0
$525$	3.00000	0.130931
$526$	0	0
$527$	−6.81507	−0.296869
$528$	0	0
$529$	10.8151	0.470221
$530$	0	0
$531$	−2.00000	−0.0867926
$532$	0	0
$533$	18.3699	0.795687
$534$	0	0
$535$	8.18493	0.353865
$536$	0	0
$537$	16.0000	0.690451
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	27.8151	1.19586	0.597932	−	0.801547i	$-0.295989\pi$
$541$	27.8151	1.19586	0.597932	+	0.801547i	$0.295989\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	13.0000	0.556859
$546$	0	0
$547$	−23.8904	−1.02148	−0.510741	−	0.859735i	$-0.670629\pi$
$547$	−23.8904	−1.02148	−0.510741	+	0.859735i	$0.670629\pi$
$548$	0	0
$549$	6.81507	0.290860
$550$	0	0
$551$	53.2603	2.26896
$552$	0	0
$553$	36.0000	1.53088
$554$	0	0
$555$	−1.00000	−0.0424476
$556$	0	0
$557$	−21.6301	−0.916499	−0.458249	−	0.888824i	$-0.651523\pi$
$557$	−21.6301	−0.916499	−0.458249	+	0.888824i	$0.651523\pi$
$558$	0	0
$559$	−18.3699	−0.776962
$560$	0	0
$561$	−1.00000	−0.0422200
$562$	0	0
$563$	8.44522	0.355924	0.177962	−	0.984037i	$-0.443050\pi$
$563$	8.44522	0.355924	0.177962	+	0.984037i	$0.443050\pi$
$564$	0	0
$565$	−6.81507	−0.286712
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	−35.8151	−1.50145	−0.750723	−	0.660617i	$-0.770295\pi$
$569$	−35.8151	−1.50145	−0.750723	+	0.660617i	$0.770295\pi$
$570$	0	0
$571$	14.8151	0.619992	0.309996	−	0.950738i	$-0.399672\pi$
$571$	14.8151	0.619992	0.309996	+	0.950738i	$0.399672\pi$
$572$	0	0
$573$	4.63015	0.193427
$574$	0	0
$575$	−5.81507	−0.242505
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	0	0
$581$	−11.4452	−0.474828
$582$	0	0
$583$	10.6301	0.440256
$584$	0	0
$585$	−3.81507	−0.157734
$586$	0	0
$587$	−20.8151	−0.859130	−0.429565	−	0.903036i	$-0.641333\pi$
$587$	−20.8151	−0.859130	−0.429565	+	0.903036i	$0.641333\pi$
$588$	0	0
$589$	53.2603	2.19455
$590$	0	0
$591$	4.18493	0.172145
$592$	0	0
$593$	−13.6301	−0.559723	−0.279862	−	0.960040i	$-0.590289\pi$
$593$	−13.6301	−0.559723	−0.279862	+	0.960040i	$0.590289\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−19.3699	−0.790113	−0.395056	−	0.918657i	$-0.629275\pi$
$601$	−19.3699	−0.790113	−0.395056	+	0.918657i	$0.629275\pi$
$602$	0	0
$603$	−7.63015	−0.310724
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	37.2603	1.51235	0.756174	−	0.654370i	$-0.227066\pi$
$607$	37.2603	1.51235	0.756174	+	0.654370i	$0.227066\pi$
$608$	0	0
$609$	−20.4452	−0.828482
$610$	0	0
$611$	−30.5206	−1.23473
$612$	0	0
$613$	−6.07536	−0.245382	−0.122691	−	0.992445i	$-0.539152\pi$
$613$	−6.07536	−0.245382	−0.122691	+	0.992445i	$0.539152\pi$
$614$	0	0
$615$	4.81507	0.194162
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	13.5548	0.544813	0.272406	−	0.962182i	$-0.412181\pi$
$619$	13.5548	0.544813	0.272406	+	0.962182i	$0.412181\pi$
$620$	0	0
$621$	5.81507	0.233351
$622$	0	0
$623$	−35.4452	−1.42008
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.81507	0.312104
$628$	0	0
$629$	1.00000	0.0398726
$630$	0	0
$631$	−33.7055	−1.34180	−0.670898	−	0.741550i	$-0.734091\pi$
$631$	−33.7055	−1.34180	−0.670898	+	0.741550i	$0.734091\pi$
$632$	0	0
$633$	8.44522	0.335667
$634$	0	0
$635$	21.4452	0.851028
$636$	0	0
$637$	−7.63015	−0.302317
$638$	0	0
$639$	11.6301	0.460081
$640$	0	0
$641$	−13.5548	−0.535382	−0.267691	−	0.963505i	$-0.586261\pi$
$641$	−13.5548	−0.535382	−0.267691	+	0.963505i	$0.586261\pi$
$642$	0	0
$643$	−17.0000	−0.670415	−0.335207	−	0.942144i	$-0.608806\pi$
$643$	−17.0000	−0.670415	−0.335207	+	0.942144i	$0.608806\pi$
$644$	0	0
$645$	−4.81507	−0.189593
$646$	0	0
$647$	−25.0754	−0.985814	−0.492907	−	0.870082i	$-0.664066\pi$
$647$	−25.0754	−0.985814	−0.492907	+	0.870082i	$0.664066\pi$
$648$	0	0
$649$	−2.00000	−0.0785069
$650$	0	0
$651$	−20.4452	−0.801311
$652$	0	0
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	0	0
$655$	9.63015	0.376281
$656$	0	0
$657$	−5.81507	−0.226868
$658$	0	0
$659$	31.2603	1.21773	0.608864	−	0.793275i	$-0.291625\pi$
$659$	31.2603	1.21773	0.608864	+	0.793275i	$0.291625\pi$
$660$	0	0
$661$	3.73971	0.145458	0.0727289	−	0.997352i	$-0.476829\pi$
$661$	3.73971	0.145458	0.0727289	+	0.997352i	$0.476829\pi$
$662$	0	0
$663$	3.81507	0.148165
$664$	0	0
$665$	23.4452	0.909167
$666$	0	0
$667$	39.6301	1.53449
$668$	0	0
$669$	−16.8151	−0.650108
$670$	0	0
$671$	6.81507	0.263093
$672$	0	0
$673$	6.18493	0.238411	0.119206	−	0.992870i	$-0.461965\pi$
$673$	6.18493	0.238411	0.119206	+	0.992870i	$0.461965\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	47.0754	1.80925	0.904627	−	0.426205i	$-0.140150\pi$
$677$	47.0754	1.80925	0.904627	+	0.426205i	$0.140150\pi$
$678$	0	0
$679$	14.4452	0.554357
$680$	0	0
$681$	−24.8151	−0.950916
$682$	0	0
$683$	47.3357	1.81125	0.905624	−	0.424081i	$-0.139403\pi$
$683$	47.3357	1.81125	0.905624	+	0.424081i	$0.139403\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	−0.369854	−0.0141108
$688$	0	0
$689$	−40.5548	−1.54501
$690$	0	0
$691$	34.0754	1.29629	0.648144	−	0.761518i	$-0.275545\pi$
$691$	34.0754	1.29629	0.648144	+	0.761518i	$0.275545\pi$
$692$	0	0
$693$	−3.00000	−0.113961
$694$	0	0
$695$	−22.8151	−0.865425
$696$	0	0
$697$	−4.81507	−0.182384
$698$	0	0
$699$	7.63015	0.288599
$700$	0	0
$701$	33.6301	1.27019	0.635097	−	0.772433i	$-0.280960\pi$
$701$	33.6301	1.27019	0.635097	+	0.772433i	$0.280960\pi$
$702$	0	0
$703$	−7.81507	−0.294751
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	−34.8904	−1.31219
$708$	0	0
$709$	20.6301	0.774781	0.387391	−	0.921916i	$-0.373376\pi$
$709$	20.6301	0.774781	0.387391	+	0.921916i	$0.373376\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	0	0
$713$	39.6301	1.48416
$714$	0	0
$715$	−3.81507	−0.142676
$716$	0	0
$717$	6.44522	0.240701
$718$	0	0
$719$	−34.0000	−1.26799	−0.633993	−	0.773339i	$-0.718585\pi$
$719$	−34.0000	−1.26799	−0.633993	+	0.773339i	$0.718585\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	26.0000	0.966950
$724$	0	0
$725$	−6.81507	−0.253105
$726$	0	0
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.81507	0.178092
$732$	0	0
$733$	40.4452	1.49388	0.746939	−	0.664892i	$-0.231523\pi$
$733$	40.4452	1.49388	0.746939	+	0.664892i	$0.231523\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	−7.63015	−0.281060
$738$	0	0
$739$	−34.0754	−1.25348	−0.626741	−	0.779227i	$-0.715612\pi$
$739$	−34.0754	−1.25348	−0.626741	+	0.779227i	$0.715612\pi$
$740$	0	0
$741$	−29.8151	−1.09528
$742$	0	0
$743$	−4.44522	−0.163079	−0.0815396	−	0.996670i	$-0.525984\pi$
$743$	−4.44522	−0.163079	−0.0815396	+	0.996670i	$0.525984\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	3.81507	0.139586
$748$	0	0
$749$	−24.5548	−0.897212
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	−11.6301	−0.423826
$754$	0	0
$755$	−3.81507	−0.138845
$756$	0	0
$757$	23.8151	0.865574	0.432787	−	0.901496i	$-0.357530\pi$
$757$	23.8151	0.865574	0.432787	+	0.901496i	$0.357530\pi$
$758$	0	0
$759$	5.81507	0.211074
$760$	0	0
$761$	−9.18493	−0.332953	−0.166477	−	0.986045i	$-0.553239\pi$
$761$	−9.18493	−0.332953	−0.166477	+	0.986045i	$0.553239\pi$
$762$	0	0
$763$	−39.0000	−1.41189
$764$	0	0
$765$	1.00000	0.0361551
$766$	0	0
$767$	7.63015	0.275509
$768$	0	0
$769$	49.2603	1.77637	0.888186	−	0.459485i	$-0.151966\pi$
$769$	49.2603	1.77637	0.888186	+	0.459485i	$0.151966\pi$
$770$	0	0
$771$	3.81507	0.137396
$772$	0	0
$773$	−7.36985	−0.265075	−0.132538	−	0.991178i	$-0.542313\pi$
$773$	−7.36985	−0.265075	−0.132538	+	0.991178i	$0.542313\pi$
$774$	0	0
$775$	−6.81507	−0.244805
$776$	0	0
$777$	3.00000	0.107624
$778$	0	0
$779$	37.6301	1.34824
$780$	0	0
$781$	11.6301	0.416159
$782$	0	0
$783$	6.81507	0.243551
$784$	0	0
$785$	−12.8151	−0.457390
$786$	0	0
$787$	39.6301	1.41266	0.706331	−	0.707882i	$-0.250349\pi$
$787$	39.6301	1.41266	0.706331	+	0.707882i	$0.250349\pi$
$788$	0	0
$789$	−30.4452	−1.08388
$790$	0	0
$791$	20.4452	0.726948
$792$	0	0
$793$	−26.0000	−0.923287
$794$	0	0
$795$	−10.6301	−0.377012
$796$	0	0
$797$	−34.8904	−1.23588	−0.617941	−	0.786224i	$-0.712033\pi$
$797$	−34.8904	−1.23588	−0.617941	+	0.786224i	$0.712033\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	11.8151	0.417465
$802$	0	0
$803$	−5.81507	−0.205209
$804$	0	0
$805$	17.4452	0.614863
$806$	0	0
$807$	−9.81507	−0.345507
$808$	0	0
$809$	−13.0754	−0.459705	−0.229853	−	0.973225i	$-0.573824\pi$
$809$	−13.0754	−0.459705	−0.229853	+	0.973225i	$0.573824\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	−20.0000	−0.701431
$814$	0	0
$815$	−16.6301	−0.582529
$816$	0	0
$817$	−37.6301	−1.31651
$818$	0	0
$819$	11.4452	0.399928
$820$	0	0
$821$	11.0754	0.386533	0.193266	−	0.981146i	$-0.438092\pi$
$821$	11.0754	0.386533	0.193266	+	0.981146i	$0.438092\pi$
$822$	0	0
$823$	−41.0754	−1.43180	−0.715899	−	0.698204i	$-0.753983\pi$
$823$	−41.0754	−1.43180	−0.715899	+	0.698204i	$0.753983\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	28.0754	0.976276	0.488138	−	0.872766i	$-0.337676\pi$
$827$	28.0754	0.976276	0.488138	+	0.872766i	$0.337676\pi$
$828$	0	0
$829$	−53.5206	−1.85885	−0.929423	−	0.369015i	$-0.879695\pi$
$829$	−53.5206	−1.85885	−0.929423	+	0.369015i	$0.879695\pi$
$830$	0	0
$831$	−21.4452	−0.743926
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	21.8151	0.754942
$836$	0	0
$837$	6.81507	0.235563
$838$	0	0
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	0	0
$841$	17.4452	0.601559
$842$	0	0
$843$	19.4452	0.669729
$844$	0	0
$845$	1.55478	0.0534861
$846$	0	0
$847$	30.0000	1.03081
$848$	0	0
$849$	−21.8151	−0.748691
$850$	0	0
$851$	−5.81507	−0.199338
$852$	0	0
$853$	−11.0754	−0.379213	−0.189607	−	0.981860i	$-0.560721\pi$
$853$	−11.0754	−0.379213	−0.189607	+	0.981860i	$0.560721\pi$
$854$	0	0
$855$	−7.81507	−0.267270
$856$	0	0
$857$	30.2603	1.03367	0.516836	−	0.856084i	$-0.327110\pi$
$857$	30.2603	1.03367	0.516836	+	0.856084i	$0.327110\pi$
$858$	0	0
$859$	−31.4452	−1.07290	−0.536449	−	0.843933i	$-0.680234\pi$
$859$	−31.4452	−1.07290	−0.536449	+	0.843933i	$0.680234\pi$
$860$	0	0
$861$	−14.4452	−0.492292
$862$	0	0
$863$	1.55478	0.0529254	0.0264627	−	0.999650i	$-0.491576\pi$
$863$	1.55478	0.0529254	0.0264627	+	0.999650i	$0.491576\pi$
$864$	0	0
$865$	−1.00000	−0.0340010
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	29.1096	0.986340
$872$	0	0
$873$	−4.81507	−0.162966
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	49.7055	1.67844	0.839218	−	0.543795i	$-0.183013\pi$
$877$	49.7055	1.67844	0.839218	+	0.543795i	$0.183013\pi$
$878$	0	0
$879$	−28.2603	−0.953196
$880$	0	0
$881$	43.1849	1.45494	0.727469	−	0.686141i	$-0.240697\pi$
$881$	43.1849	1.45494	0.727469	+	0.686141i	$0.240697\pi$
$882$	0	0
$883$	−27.0000	−0.908622	−0.454311	−	0.890843i	$-0.650115\pi$
$883$	−27.0000	−0.908622	−0.454311	+	0.890843i	$0.650115\pi$
$884$	0	0
$885$	2.00000	0.0672293
$886$	0	0
$887$	14.0754	0.472604	0.236302	−	0.971680i	$-0.424064\pi$
$887$	14.0754	0.472604	0.236302	+	0.971680i	$0.424064\pi$
$888$	0	0
$889$	−64.3357	−2.15775
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−62.5206	−2.09217
$894$	0	0
$895$	−16.0000	−0.534821
$896$	0	0
$897$	−22.1849	−0.740733
$898$	0	0
$899$	46.4452	1.54903
$900$	0	0
$901$	10.6301	0.354142
$902$	0	0
$903$	14.4452	0.480707
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	45.8151	1.52126	0.760632	−	0.649183i	$-0.224889\pi$
$907$	45.8151	1.52126	0.760632	+	0.649183i	$0.224889\pi$
$908$	0	0
$909$	11.6301	0.385748
$910$	0	0
$911$	54.5206	1.80635	0.903174	−	0.429275i	$-0.141231\pi$
$911$	54.5206	1.80635	0.903174	+	0.429275i	$0.141231\pi$
$912$	0	0
$913$	3.81507	0.126260
$914$	0	0
$915$	−6.81507	−0.225299
$916$	0	0
$917$	−28.8904	−0.954046
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	−18.0000	−0.593120
$922$	0	0
$923$	−44.3699	−1.46045
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24.4452	0.802022	0.401011	−	0.916073i	$-0.368659\pi$
$929$	24.4452	0.802022	0.401011	+	0.916073i	$0.368659\pi$
$930$	0	0
$931$	−15.6301	−0.512257
$932$	0	0
$933$	10.4452	0.341961
$934$	0	0
$935$	1.00000	0.0327035
$936$	0	0
$937$	−11.6301	−0.379940	−0.189970	−	0.981790i	$-0.560839\pi$
$937$	−11.6301	−0.379940	−0.189970	+	0.981790i	$0.560839\pi$
$938$	0	0
$939$	−32.8904	−1.07334
$940$	0	0
$941$	−1.26029	−0.0410843	−0.0205422	−	0.999789i	$-0.506539\pi$
$941$	−1.26029	−0.0410843	−0.0205422	+	0.999789i	$0.506539\pi$
$942$	0	0
$943$	28.0000	0.911805
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	25.5548	0.830419	0.415209	−	0.909726i	$-0.363708\pi$
$947$	25.5548	0.830419	0.415209	+	0.909726i	$0.363708\pi$
$948$	0	0
$949$	22.1849	0.720153
$950$	0	0
$951$	2.81507	0.0912850
$952$	0	0
$953$	14.8904	0.482349	0.241174	−	0.970482i	$-0.422467\pi$
$953$	14.8904	0.482349	0.241174	+	0.970482i	$0.422467\pi$
$954$	0	0
$955$	−4.63015	−0.149828
$956$	0	0
$957$	6.81507	0.220300
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	15.4452	0.498233
$962$	0	0
$963$	8.18493	0.263756
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−7.63015	−0.245369	−0.122684	−	0.992446i	$-0.539150\pi$
$967$	−7.63015	−0.245369	−0.122684	+	0.992446i	$0.539150\pi$
$968$	0	0
$969$	7.81507	0.251056
$970$	0	0
$971$	48.0754	1.54281	0.771406	−	0.636343i	$-0.219554\pi$
$971$	48.0754	1.54281	0.771406	+	0.636343i	$0.219554\pi$
$972$	0	0
$973$	68.4452	2.19425
$974$	0	0
$975$	3.81507	0.122180
$976$	0	0
$977$	21.0000	0.671850	0.335925	−	0.941889i	$-0.390951\pi$
$977$	21.0000	0.671850	0.335925	+	0.941889i	$0.390951\pi$
$978$	0	0
$979$	11.8151	0.377611
$980$	0	0
$981$	13.0000	0.415058
$982$	0	0
$983$	−57.3357	−1.82872	−0.914362	−	0.404898i	$-0.867307\pi$
$983$	−57.3357	−1.82872	−0.914362	+	0.404898i	$0.867307\pi$
$984$	0	0
$985$	−4.18493	−0.133343
$986$	0	0
$987$	24.0000	0.763928
$988$	0	0
$989$	−28.0000	−0.890348
$990$	0	0
$991$	38.4452	1.22125	0.610626	−	0.791919i	$-0.290918\pi$
$991$	38.4452	1.22125	0.610626	+	0.791919i	$0.290918\pi$
$992$	0	0
$993$	−23.2603	−0.738143
$994$	0	0
$995$	4.00000	0.126809
$996$	0	0
$997$	55.0754	1.74425	0.872127	−	0.489279i	$-0.162740\pi$
$997$	55.0754	1.74425	0.872127	+	0.489279i	$0.162740\pi$
$998$	0	0
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8880.2.a.bh.1.1		2
4.3	odd	2		1110.2.a.q.1.1	✓	2
12.11	even	2		3330.2.a.bf.1.1		2
20.19	odd	2		5550.2.a.bx.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.q.1.1	✓	2	4.3	odd	2
3330.2.a.bf.1.1		2	12.11	even	2
5550.2.a.bx.1.2		2	20.19	odd	2
8880.2.a.bh.1.1		2	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:	\( N \)	\(=\)	\( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8880.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.81507 q^{13} -1.00000 q^{15} +1.00000 q^{17} -7.81507 q^{19} +3.00000 q^{21} -5.81507 q^{23} +1.00000 q^{25} -1.00000 q^{27} -6.81507 q^{29} -6.81507 q^{31} -1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} +3.81507 q^{39} -4.81507 q^{41} +4.81507 q^{43} +1.00000 q^{45} +8.00000 q^{47} +2.00000 q^{49} -1.00000 q^{51} +10.6301 q^{53} +1.00000 q^{55} +7.81507 q^{57} -2.00000 q^{59} +6.81507 q^{61} -3.00000 q^{63} -3.81507 q^{65} -7.63015 q^{67} +5.81507 q^{69} +11.6301 q^{71} -5.81507 q^{73} -1.00000 q^{75} -3.00000 q^{77} -12.0000 q^{79} +1.00000 q^{81} +3.81507 q^{83} +1.00000 q^{85} +6.81507 q^{87} +11.8151 q^{89} +11.4452 q^{91} +6.81507 q^{93} -7.81507 q^{95} -4.81507 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} + 2 q^{11} + 3 q^{13} - 2 q^{15} + 2 q^{17} - 5 q^{19} + 6 q^{21} - q^{23} + 2 q^{25} - 2 q^{27} - 3 q^{29} - 3 q^{31} - 2 q^{33} - 6 q^{35} + 2 q^{37} - 3 q^{39} + q^{41} - q^{43} + 2 q^{45} + 16 q^{47} + 4 q^{49} - 2 q^{51} + 2 q^{55} + 5 q^{57} - 4 q^{59} + 3 q^{61} - 6 q^{63} + 3 q^{65} + 6 q^{67} + q^{69} + 2 q^{71} - q^{73} - 2 q^{75} - 6 q^{77} - 24 q^{79} + 2 q^{81} - 3 q^{83} + 2 q^{85} + 3 q^{87} + 13 q^{89} - 9 q^{91} + 3 q^{93} - 5 q^{95} + q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 + 2 * q^11 + 3 * q^13 - 2 * q^15 + 2 * q^17 - 5 * q^19 + 6 * q^21 - q^23 + 2 * q^25 - 2 * q^27 - 3 * q^29 - 3 * q^31 - 2 * q^33 - 6 * q^35 + 2 * q^37 - 3 * q^39 + q^41 - q^43 + 2 * q^45 + 16 * q^47 + 4 * q^49 - 2 * q^51 + 2 * q^55 + 5 * q^57 - 4 * q^59 + 3 * q^61 - 6 * q^63 + 3 * q^65 + 6 * q^67 + q^69 + 2 * q^71 - q^73 - 2 * q^75 - 6 * q^77 - 24 * q^79 + 2 * q^81 - 3 * q^83 + 2 * q^85 + 3 * q^87 + 13 * q^89 - 9 * q^91 + 3 * q^93 - 5 * q^95 + q^97 + 2 * q^99

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