LMFDB - Embedded newform 8880.2.a.bh.1.2

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			1.2
Root			$5.81507$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +6.81507 q^{13} -1.00000 q^{15} +1.00000 q^{17} +2.81507 q^{19} +3.00000 q^{21} +4.81507 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.81507 q^{29} +3.81507 q^{31} -1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} -6.81507 q^{39} +5.81507 q^{41} -5.81507 q^{43} +1.00000 q^{45} +8.00000 q^{47} +2.00000 q^{49} -1.00000 q^{51} -10.6301 q^{53} +1.00000 q^{55} -2.81507 q^{57} -2.00000 q^{59} -3.81507 q^{61} -3.00000 q^{63} +6.81507 q^{65} +13.6301 q^{67} -4.81507 q^{69} -9.63015 q^{71} +4.81507 q^{73} -1.00000 q^{75} -3.00000 q^{77} -12.0000 q^{79} +1.00000 q^{81} -6.81507 q^{83} +1.00000 q^{85} -3.81507 q^{87} +1.18493 q^{89} -20.4452 q^{91} -3.81507 q^{93} +2.81507 q^{95} +5.81507 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	6.81507	1.89016	0.945081	−	0.326837i	$-0.105983\pi$
$13$	6.81507	1.89016	0.945081	+	0.326837i	$0.105983\pi$
$14$	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
$19$	2.81507	0.645822	0.322911	−	0.946429i	$-0.395339\pi$
$19$	2.81507	0.645822	0.322911	+	0.946429i	$0.395339\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	4.81507	1.00401	0.502006	−	0.864864i	$-0.332596\pi$
$23$	4.81507	1.00401	0.502006	+	0.864864i	$0.332596\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.81507	0.708441	0.354221	−	0.935162i	$-0.384746\pi$
$29$	3.81507	0.708441	0.354221	+	0.935162i	$0.384746\pi$
$30$	0	0
$31$	3.81507	0.685207	0.342604	−	0.939480i	$-0.388691\pi$
$31$	3.81507	0.685207	0.342604	+	0.939480i	$0.388691\pi$
$32$	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
$39$	−6.81507	−1.09129
$40$	0	0
$41$	5.81507	0.908162	0.454081	−	0.890960i	$-0.349968\pi$
$41$	5.81507	0.908162	0.454081	+	0.890960i	$0.349968\pi$
$42$	0	0
$43$	−5.81507	−0.886790	−0.443395	−	0.896326i	$-0.646226\pi$
$43$	−5.81507	−0.886790	−0.443395	+	0.896326i	$0.646226\pi$
$44$	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	$-0.760518\pi$
$53$	−10.6301	−1.46016	−0.730081	+	0.683360i	$0.760518\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−2.81507	−0.372866
$58$	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
$61$	−3.81507	−0.488470	−0.244235	−	0.969716i	$-0.578537\pi$
$61$	−3.81507	−0.488470	−0.244235	+	0.969716i	$0.578537\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	6.81507	0.845306
$66$	0	0
$67$	13.6301	1.66519	0.832594	−	0.553884i	$-0.186855\pi$
$67$	13.6301	1.66519	0.832594	+	0.553884i	$0.186855\pi$
$68$	0	0
$69$	−4.81507	−0.579667
$70$	0	0
$71$	−9.63015	−1.14289	−0.571444	−	0.820641i	$-0.693617\pi$
$71$	−9.63015	−1.14289	−0.571444	+	0.820641i	$0.693617\pi$
$72$	0	0
$73$	4.81507	0.563562	0.281781	−	0.959479i	$-0.409075\pi$
$73$	4.81507	0.563562	0.281781	+	0.959479i	$0.409075\pi$
$74$	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
$83$	−6.81507	−0.748051	−0.374026	−	0.927418i	$-0.622023\pi$
$83$	−6.81507	−0.748051	−0.374026	+	0.927418i	$0.622023\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	−3.81507	−0.409019
$88$	0	0
$89$	1.18493	0.125602	0.0628010	−	0.998026i	$-0.479997\pi$
$89$	1.18493	0.125602	0.0628010	+	0.998026i	$0.479997\pi$
$90$	0	0
$91$	−20.4452	−2.14324
$92$	0	0
$93$	−3.81507	−0.395605
$94$	0	0
$95$	2.81507	0.288820
$96$	0	0
$97$	5.81507	0.590431	0.295216	−	0.955431i	$-0.404609\pi$
$97$	5.81507	0.590431	0.295216	+	0.955431i	$0.404609\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−9.63015	−0.958235	−0.479118	−	0.877751i	$-0.659043\pi$
$101$	−9.63015	−0.958235	−0.479118	+	0.877751i	$0.659043\pi$
$102$	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
$107$	18.8151	1.81892	0.909461	−	0.415790i	$-0.136495\pi$
$107$	18.8151	1.81892	0.909461	+	0.415790i	$0.136495\pi$
$108$	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
$113$	3.81507	0.358892	0.179446	−	0.983768i	$-0.442570\pi$
$113$	3.81507	0.358892	0.179446	+	0.983768i	$0.442570\pi$
$114$	0	0
$115$	4.81507	0.449008
$116$	0	0
$117$	6.81507	0.630054
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−5.81507	−0.524327
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.4452	−0.926863	−0.463432	−	0.886133i	$-0.653382\pi$
$127$	−10.4452	−0.926863	−0.463432	+	0.886133i	$0.653382\pi$
$128$	0	0
$129$	5.81507	0.511989
$130$	0	0
$131$	−11.6301	−1.01613	−0.508065	−	0.861319i	$-0.669639\pi$
$131$	−11.6301	−1.01613	−0.508065	+	0.861319i	$0.669639\pi$
$132$	0	0
$133$	−8.44522	−0.732293
$134$	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
$139$	−12.1849	−1.03351	−0.516756	−	0.856133i	$-0.672861\pi$
$139$	−12.1849	−1.03351	−0.516756	+	0.856133i	$0.672861\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	6.81507	0.569905
$144$	0	0
$145$	3.81507	0.316825
$146$	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
$151$	6.81507	0.554603	0.277301	−	0.960783i	$-0.410560\pi$
$151$	6.81507	0.554603	0.277301	+	0.960783i	$0.410560\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	3.81507	0.306434
$156$	0	0
$157$	−2.18493	−0.174376	−0.0871881	−	0.996192i	$-0.527788\pi$
$157$	−2.18493	−0.174376	−0.0871881	+	0.996192i	$0.527788\pi$
$158$	0	0
$159$	10.6301	0.843025
$160$	0	0
$161$	−14.4452	−1.13844
$162$	0	0
$163$	4.63015	0.362661	0.181331	−	0.983422i	$-0.441960\pi$
$163$	4.63015	0.362661	0.181331	+	0.983422i	$0.441960\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	11.1849	0.865516	0.432758	−	0.901510i	$-0.357541\pi$
$167$	11.1849	0.865516	0.432758	+	0.901510i	$0.357541\pi$
$168$	0	0
$169$	33.4452	2.57271
$170$	0	0
$171$	2.81507	0.215274
$172$	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
$183$	3.81507	0.282018
$184$	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
$191$	16.6301	1.20332	0.601658	−	0.798754i	$-0.294507\pi$
$191$	16.6301	1.20332	0.601658	+	0.798754i	$0.294507\pi$
$192$	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
$195$	−6.81507	−0.488038
$196$	0	0
$197$	−14.8151	−1.05553	−0.527765	−	0.849390i	$-0.676970\pi$
$197$	−14.8151	−1.05553	−0.527765	+	0.849390i	$0.676970\pi$
$198$	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
$201$	−13.6301	−0.961396
$202$	0	0
$203$	−11.4452	−0.803297
$204$	0	0
$205$	5.81507	0.406142
$206$	0	0
$207$	4.81507	0.334671
$208$	0	0
$209$	2.81507	0.194723
$210$	0	0
$211$	23.4452	1.61404	0.807018	−	0.590527i	$-0.201080\pi$
$211$	23.4452	1.61404	0.807018	+	0.590527i	$0.201080\pi$
$212$	0	0
$213$	9.63015	0.659847
$214$	0	0
$215$	−5.81507	−0.396585
$216$	0	0
$217$	−11.4452	−0.776952
$218$	0	0
$219$	−4.81507	−0.325372
$220$	0	0
$221$	6.81507	0.458431
$222$	0	0
$223$	6.18493	0.414173	0.207087	−	0.978323i	$-0.433602\pi$
$223$	6.18493	0.414173	0.207087	+	0.978323i	$0.433602\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	14.1849	0.941487	0.470743	−	0.882270i	$-0.343986\pi$
$227$	14.1849	0.941487	0.470743	+	0.882270i	$0.343986\pi$
$228$	0	0
$229$	21.6301	1.42936	0.714680	−	0.699451i	$-0.246572\pi$
$229$	21.6301	1.42936	0.714680	+	0.699451i	$0.246572\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	13.6301	0.892941	0.446470	−	0.894798i	$-0.352681\pi$
$233$	13.6301	0.892941	0.446470	+	0.894798i	$0.352681\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	12.0000	0.779484
$238$	0	0
$239$	25.4452	1.64591	0.822957	−	0.568103i	$-0.192323\pi$
$239$	25.4452	1.64591	0.822957	+	0.568103i	$0.192323\pi$
$240$	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
$247$	19.1849	1.22071
$248$	0	0
$249$	6.81507	0.431888
$250$	0	0
$251$	−9.63015	−0.607849	−0.303925	−	0.952696i	$-0.598297\pi$
$251$	−9.63015	−0.607849	−0.303925	+	0.952696i	$0.598297\pi$
$252$	0	0
$253$	4.81507	0.302721
$254$	0	0
$255$	−1.00000	−0.0626224
$256$	0	0
$257$	6.81507	0.425113	0.212556	−	0.977149i	$-0.431821\pi$
$257$	6.81507	0.425113	0.212556	+	0.977149i	$0.431821\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	3.81507	0.236147
$262$	0	0
$263$	−1.44522	−0.0891160	−0.0445580	−	0.999007i	$-0.514188\pi$
$263$	−1.44522	−0.0891160	−0.0445580	+	0.999007i	$0.514188\pi$
$264$	0	0
$265$	−10.6301	−0.653005
$266$	0	0
$267$	−1.18493	−0.0725164
$268$	0	0
$269$	−0.815073	−0.0496959	−0.0248479	−	0.999691i	$-0.507910\pi$
$269$	−0.815073	−0.0496959	−0.0248479	+	0.999691i	$0.507910\pi$
$270$	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
$273$	20.4452	1.23740
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−10.4452	−0.627592	−0.313796	−	0.949490i	$-0.601601\pi$
$277$	−10.4452	−0.627592	−0.313796	+	0.949490i	$0.601601\pi$
$278$	0	0
$279$	3.81507	0.228402
$280$	0	0
$281$	12.4452	0.742420	0.371210	−	0.928549i	$-0.378943\pi$
$281$	12.4452	0.742420	0.371210	+	0.928549i	$0.378943\pi$
$282$	0	0
$283$	11.1849	0.664875	0.332437	−	0.943125i	$-0.392129\pi$
$283$	11.1849	0.664875	0.332437	+	0.943125i	$0.392129\pi$
$284$	0	0
$285$	−2.81507	−0.166751
$286$	0	0
$287$	−17.4452	−1.02976
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−5.81507	−0.340886
$292$	0	0
$293$	−14.2603	−0.833095	−0.416548	−	0.909114i	$-0.636760\pi$
$293$	−14.2603	−0.833095	−0.416548	+	0.909114i	$0.636760\pi$
$294$	0	0
$295$	−2.00000	−0.116445
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	32.8151	1.89774
$300$	0	0
$301$	17.4452	1.00553
$302$	0	0
$303$	9.63015	0.553237
$304$	0	0
$305$	−3.81507	−0.218450
$306$	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
$311$	21.4452	1.21605	0.608023	−	0.793919i	$-0.291963\pi$
$311$	21.4452	1.21605	0.608023	+	0.793919i	$0.291963\pi$
$312$	0	0
$313$	−30.8904	−1.74603	−0.873015	−	0.487693i	$-0.837839\pi$
$313$	−30.8904	−1.74603	−0.873015	+	0.487693i	$0.837839\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	7.81507	0.438938	0.219469	−	0.975619i	$-0.429567\pi$
$317$	7.81507	0.438938	0.219469	+	0.975619i	$0.429567\pi$
$318$	0	0
$319$	3.81507	0.213603
$320$	0	0
$321$	−18.8151	−1.05015
$322$	0	0
$323$	2.81507	0.156635
$324$	0	0
$325$	6.81507	0.378032
$326$	0	0
$327$	−13.0000	−0.718902
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−19.2603	−1.05864	−0.529321	−	0.848422i	$-0.677553\pi$
$331$	−19.2603	−1.05864	−0.529321	+	0.848422i	$0.677553\pi$
$332$	0	0
$333$	1.00000	0.0547997
$334$	0	0
$335$	13.6301	0.744694
$336$	0	0
$337$	−14.8151	−0.807028	−0.403514	−	0.914973i	$-0.632211\pi$
$337$	−14.8151	−0.807028	−0.403514	+	0.914973i	$0.632211\pi$
$338$	0	0
$339$	−3.81507	−0.207206
$340$	0	0
$341$	3.81507	0.206598
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	−4.81507	−0.259235
$346$	0	0
$347$	31.2603	1.67814	0.839070	−	0.544023i	$-0.183100\pi$
$347$	31.2603	1.67814	0.839070	+	0.544023i	$0.183100\pi$
$348$	0	0
$349$	−35.2603	−1.88744	−0.943720	−	0.330745i	$-0.892700\pi$
$349$	−35.2603	−1.88744	−0.943720	+	0.330745i	$0.892700\pi$
$350$	0	0
$351$	−6.81507	−0.363762
$352$	0	0
$353$	−23.8151	−1.26755	−0.633774	−	0.773518i	$-0.718495\pi$
$353$	−23.8151	−1.26755	−0.633774	+	0.773518i	$0.718495\pi$
$354$	0	0
$355$	−9.63015	−0.511115
$356$	0	0
$357$	3.00000	0.158777
$358$	0	0
$359$	−27.6301	−1.45826	−0.729132	−	0.684373i	$-0.760076\pi$
$359$	−27.6301	−1.45826	−0.729132	+	0.684373i	$0.760076\pi$
$360$	0	0
$361$	−11.0754	−0.582914
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	4.81507	0.252032
$366$	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
$369$	5.81507	0.302721
$370$	0	0
$371$	31.8904	1.65567
$372$	0	0
$373$	25.2603	1.30793	0.653964	−	0.756526i	$-0.273105\pi$
$373$	25.2603	1.30793	0.653964	+	0.756526i	$0.273105\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	26.0000	1.33907
$378$	0	0
$379$	−1.63015	−0.0837350	−0.0418675	−	0.999123i	$-0.513331\pi$
$379$	−1.63015	−0.0837350	−0.0418675	+	0.999123i	$0.513331\pi$
$380$	0	0
$381$	10.4452	0.535125
$382$	0	0
$383$	−1.55478	−0.0794456	−0.0397228	−	0.999211i	$-0.512647\pi$
$383$	−1.55478	−0.0794456	−0.0397228	+	0.999211i	$0.512647\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	−5.81507	−0.295597
$388$	0	0
$389$	38.7055	1.96245	0.981224	−	0.192873i	$-0.0617807\pi$
$389$	38.7055	1.96245	0.981224	+	0.192873i	$0.0617807\pi$
$390$	0	0
$391$	4.81507	0.243509
$392$	0	0
$393$	11.6301	0.586663
$394$	0	0
$395$	−12.0000	−0.603786
$396$	0	0
$397$	15.6301	0.784455	0.392227	−	0.919868i	$-0.371705\pi$
$397$	15.6301	0.784455	0.392227	+	0.919868i	$0.371705\pi$
$398$	0	0
$399$	8.44522	0.422790
$400$	0	0
$401$	32.4452	1.62024	0.810118	−	0.586266i	$-0.199403\pi$
$401$	32.4452	1.62024	0.810118	+	0.586266i	$0.199403\pi$
$402$	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
$409$	27.6301	1.36622	0.683111	−	0.730314i	$-0.260626\pi$
$409$	27.6301	1.36622	0.683111	+	0.730314i	$0.260626\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	6.00000	0.295241
$414$	0	0
$415$	−6.81507	−0.334539
$416$	0	0
$417$	12.1849	0.596698
$418$	0	0
$419$	−6.44522	−0.314870	−0.157435	−	0.987529i	$-0.550322\pi$
$419$	−6.44522	−0.314870	−0.157435	+	0.987529i	$0.550322\pi$
$420$	0	0
$421$	13.2603	0.646267	0.323134	−	0.946353i	$-0.395264\pi$
$421$	13.2603	0.646267	0.323134	+	0.946353i	$0.395264\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	11.4452	0.553873
$428$	0	0
$429$	−6.81507	−0.329035
$430$	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
$433$	−8.81507	−0.423625	−0.211813	−	0.977310i	$-0.567937\pi$
$433$	−8.81507	−0.423625	−0.211813	+	0.977310i	$0.567937\pi$
$434$	0	0
$435$	−3.81507	−0.182919
$436$	0	0
$437$	13.5548	0.648413
$438$	0	0
$439$	−13.4452	−0.641705	−0.320853	−	0.947129i	$-0.603969\pi$
$439$	−13.4452	−0.641705	−0.320853	+	0.947129i	$0.603969\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	7.63015	0.362519	0.181260	−	0.983435i	$-0.441983\pi$
$443$	7.63015	0.362519	0.181260	+	0.983435i	$0.441983\pi$
$444$	0	0
$445$	1.18493	0.0561709
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	−32.8904	−1.55220	−0.776098	−	0.630612i	$-0.782804\pi$
$449$	−32.8904	−1.55220	−0.776098	+	0.630612i	$0.782804\pi$
$450$	0	0
$451$	5.81507	0.273821
$452$	0	0
$453$	−6.81507	−0.320200
$454$	0	0
$455$	−20.4452	−0.958487
$456$	0	0
$457$	25.0754	1.17298	0.586488	−	0.809958i	$-0.300510\pi$
$457$	25.0754	1.17298	0.586488	+	0.809958i	$0.300510\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	24.1849	1.12640	0.563202	−	0.826319i	$-0.309569\pi$
$461$	24.1849	1.12640	0.563202	+	0.826319i	$0.309569\pi$
$462$	0	0
$463$	−3.63015	−0.168707	−0.0843536	−	0.996436i	$-0.526883\pi$
$463$	−3.63015	−0.168707	−0.0843536	+	0.996436i	$0.526883\pi$
$464$	0	0
$465$	−3.81507	−0.176920
$466$	0	0
$467$	−18.1849	−0.841498	−0.420749	−	0.907177i	$-0.638233\pi$
$467$	−18.1849	−0.841498	−0.420749	+	0.907177i	$0.638233\pi$
$468$	0	0
$469$	−40.8904	−1.88814
$470$	0	0
$471$	2.18493	0.100676
$472$	0	0
$473$	−5.81507	−0.267377
$474$	0	0
$475$	2.81507	0.129164
$476$	0	0
$477$	−10.6301	−0.486721
$478$	0	0
$479$	0.815073	0.0372416	0.0186208	−	0.999827i	$-0.494072\pi$
$479$	0.815073	0.0372416	0.0186208	+	0.999827i	$0.494072\pi$
$480$	0	0
$481$	6.81507	0.310741
$482$	0	0
$483$	14.4452	0.657280
$484$	0	0
$485$	5.81507	0.264049
$486$	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
$489$	−4.63015	−0.209382
$490$	0	0
$491$	−12.8151	−0.578336	−0.289168	−	0.957278i	$-0.593379\pi$
$491$	−12.8151	−0.578336	−0.289168	+	0.957278i	$0.593379\pi$
$492$	0	0
$493$	3.81507	0.171822
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	28.8904	1.29591
$498$	0	0
$499$	−14.4452	−0.646657	−0.323328	−	0.946287i	$-0.604802\pi$
$499$	−14.4452	−0.646657	−0.323328	+	0.946287i	$0.604802\pi$
$500$	0	0
$501$	−11.1849	−0.499706
$502$	0	0
$503$	−43.2603	−1.92888	−0.964441	−	0.264300i	$-0.914859\pi$
$503$	−43.2603	−1.92888	−0.964441	+	0.264300i	$0.914859\pi$
$504$	0	0
$505$	−9.63015	−0.428536
$506$	0	0
$507$	−33.4452	−1.48535
$508$	0	0
$509$	−14.4452	−0.640273	−0.320137	−	0.947371i	$-0.603729\pi$
$509$	−14.4452	−0.640273	−0.320137	+	0.947371i	$0.603729\pi$
$510$	0	0
$511$	−14.4452	−0.639019
$512$	0	0
$513$	−2.81507	−0.124289
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	1.00000	0.0438951
$520$	0	0
$521$	−9.81507	−0.430006	−0.215003	−	0.976613i	$-0.568976\pi$
$521$	−9.81507	−0.430006	−0.215003	+	0.976613i	$0.568976\pi$
$522$	0	0
$523$	43.2603	1.89164	0.945820	−	0.324691i	$-0.105260\pi$
$523$	43.2603	1.89164	0.945820	+	0.324691i	$0.105260\pi$
$524$	0	0
$525$	3.00000	0.130931
$526$	0	0
$527$	3.81507	0.166187
$528$	0	0
$529$	0.184927	0.00804031
$530$	0	0
$531$	−2.00000	−0.0867926
$532$	0	0
$533$	39.6301	1.71657
$534$	0	0
$535$	18.8151	0.813447
$536$	0	0
$537$	16.0000	0.690451
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	17.1849	0.738838	0.369419	−	0.929263i	$-0.379557\pi$
$541$	17.1849	0.738838	0.369419	+	0.929263i	$0.379557\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	13.0000	0.556859
$546$	0	0
$547$	39.8904	1.70559	0.852796	−	0.522244i	$-0.174905\pi$
$547$	39.8904	1.70559	0.852796	+	0.522244i	$0.174905\pi$
$548$	0	0
$549$	−3.81507	−0.162823
$550$	0	0
$551$	10.7397	0.457527
$552$	0	0
$553$	36.0000	1.53088
$554$	0	0
$555$	−1.00000	−0.0424476
$556$	0	0
$557$	−0.369854	−0.0156712	−0.00783561	−	0.999969i	$-0.502494\pi$
$557$	−0.369854	−0.0156712	−0.00783561	+	0.999969i	$0.502494\pi$
$558$	0	0
$559$	−39.6301	−1.67618
$560$	0	0
$561$	−1.00000	−0.0422200
$562$	0	0
$563$	−23.4452	−0.988098	−0.494049	−	0.869434i	$-0.664484\pi$
$563$	−23.4452	−0.988098	−0.494049	+	0.869434i	$0.664484\pi$
$564$	0	0
$565$	3.81507	0.160501
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	−25.1849	−1.05581	−0.527904	−	0.849304i	$-0.677022\pi$
$569$	−25.1849	−1.05581	−0.527904	+	0.849304i	$0.677022\pi$
$570$	0	0
$571$	4.18493	0.175134	0.0875669	−	0.996159i	$-0.472091\pi$
$571$	4.18493	0.175134	0.0875669	+	0.996159i	$0.472091\pi$
$572$	0	0
$573$	−16.6301	−0.694734
$574$	0	0
$575$	4.81507	0.200802
$576$	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
$581$	20.4452	0.848211
$582$	0	0
$583$	−10.6301	−0.440256
$584$	0	0
$585$	6.81507	0.281769
$586$	0	0
$587$	−10.1849	−0.420377	−0.210188	−	0.977661i	$-0.567408\pi$
$587$	−10.1849	−0.420377	−0.210188	+	0.977661i	$0.567408\pi$
$588$	0	0
$589$	10.7397	0.442522
$590$	0	0
$591$	14.8151	0.609411
$592$	0	0
$593$	7.63015	0.313333	0.156666	−	0.987652i	$-0.449925\pi$
$593$	7.63015	0.313333	0.156666	+	0.987652i	$0.449925\pi$
$594$	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
$601$	−40.6301	−1.65734	−0.828669	−	0.559739i	$-0.810901\pi$
$601$	−40.6301	−1.65734	−0.828669	+	0.559739i	$0.810901\pi$
$602$	0	0
$603$	13.6301	0.555062
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	−5.26029	−0.213509	−0.106754	−	0.994285i	$-0.534046\pi$
$607$	−5.26029	−0.213509	−0.106754	+	0.994285i	$0.534046\pi$
$608$	0	0
$609$	11.4452	0.463784
$610$	0	0
$611$	54.5206	2.20567
$612$	0	0
$613$	47.0754	1.90136	0.950678	−	0.310179i	$-0.100389\pi$
$613$	47.0754	1.90136	0.950678	+	0.310179i	$0.100389\pi$
$614$	0	0
$615$	−5.81507	−0.234486
$616$	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
$619$	45.4452	1.82660	0.913299	−	0.407290i	$-0.133526\pi$
$619$	45.4452	1.82660	0.913299	+	0.407290i	$0.133526\pi$
$620$	0	0
$621$	−4.81507	−0.193222
$622$	0	0
$623$	−3.55478	−0.142419
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.81507	−0.112423
$628$	0	0
$629$	1.00000	0.0398726
$630$	0	0
$631$	40.7055	1.62046	0.810230	−	0.586112i	$-0.199342\pi$
$631$	40.7055	1.62046	0.810230	+	0.586112i	$0.199342\pi$
$632$	0	0
$633$	−23.4452	−0.931864
$634$	0	0
$635$	−10.4452	−0.414506
$636$	0	0
$637$	13.6301	0.540046
$638$	0	0
$639$	−9.63015	−0.380963
$640$	0	0
$641$	−45.4452	−1.79498	−0.897489	−	0.441037i	$-0.854611\pi$
$641$	−45.4452	−1.79498	−0.897489	+	0.441037i	$0.854611\pi$
$642$	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
$645$	5.81507	0.228968
$646$	0	0
$647$	28.0754	1.10376	0.551878	−	0.833925i	$-0.313911\pi$
$647$	28.0754	1.10376	0.551878	+	0.833925i	$0.313911\pi$
$648$	0	0
$649$	−2.00000	−0.0785069
$650$	0	0
$651$	11.4452	0.448573
$652$	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
$655$	−11.6301	−0.454427
$656$	0	0
$657$	4.81507	0.187854
$658$	0	0
$659$	−11.2603	−0.438639	−0.219319	−	0.975653i	$-0.570384\pi$
$659$	−11.2603	−0.438639	−0.219319	+	0.975653i	$0.570384\pi$
$660$	0	0
$661$	46.2603	1.79932	0.899658	−	0.436595i	$-0.143816\pi$
$661$	46.2603	1.79932	0.899658	+	0.436595i	$0.143816\pi$
$662$	0	0
$663$	−6.81507	−0.264675
$664$	0	0
$665$	−8.44522	−0.327492
$666$	0	0
$667$	18.3699	0.711284
$668$	0	0
$669$	−6.18493	−0.239123
$670$	0	0
$671$	−3.81507	−0.147279
$672$	0	0
$673$	16.8151	0.648173	0.324087	−	0.946027i	$-0.394943\pi$
$673$	16.8151	0.648173	0.324087	+	0.946027i	$0.394943\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−6.07536	−0.233495	−0.116748	−	0.993162i	$-0.537247\pi$
$677$	−6.07536	−0.233495	−0.116748	+	0.993162i	$0.537247\pi$
$678$	0	0
$679$	−17.4452	−0.669486
$680$	0	0
$681$	−14.1849	−0.543568
$682$	0	0
$683$	−48.3357	−1.84951	−0.924756	−	0.380560i	$-0.875731\pi$
$683$	−48.3357	−1.84951	−0.924756	+	0.380560i	$0.875731\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	−21.6301	−0.825242
$688$	0	0
$689$	−72.4452	−2.75994
$690$	0	0
$691$	−19.0754	−0.725661	−0.362831	−	0.931855i	$-0.618190\pi$
$691$	−19.0754	−0.725661	−0.362831	+	0.931855i	$0.618190\pi$
$692$	0	0
$693$	−3.00000	−0.113961
$694$	0	0
$695$	−12.1849	−0.462201
$696$	0	0
$697$	5.81507	0.220262
$698$	0	0
$699$	−13.6301	−0.515539
$700$	0	0
$701$	12.3699	0.467203	0.233601	−	0.972332i	$-0.424949\pi$
$701$	12.3699	0.467203	0.233601	+	0.972332i	$0.424949\pi$
$702$	0	0
$703$	2.81507	0.106172
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	28.8904	1.08654
$708$	0	0
$709$	−0.630146	−0.0236656	−0.0118328	−	0.999930i	$-0.503767\pi$
$709$	−0.630146	−0.0236656	−0.0118328	+	0.999930i	$0.503767\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	0	0
$713$	18.3699	0.687956
$714$	0	0
$715$	6.81507	0.254869
$716$	0	0
$717$	−25.4452	−0.950269
$718$	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
$725$	3.81507	0.141688
$726$	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
$731$	−5.81507	−0.215078
$732$	0	0
$733$	8.55478	0.315978	0.157989	−	0.987441i	$-0.449499\pi$
$733$	8.55478	0.315978	0.157989	+	0.987441i	$0.449499\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	13.6301	0.502073
$738$	0	0
$739$	19.0754	0.701699	0.350849	−	0.936432i	$-0.385893\pi$
$739$	19.0754	0.701699	0.350849	+	0.936432i	$0.385893\pi$
$740$	0	0
$741$	−19.1849	−0.704776
$742$	0	0
$743$	27.4452	1.00687	0.503434	−	0.864034i	$-0.332070\pi$
$743$	27.4452	1.00687	0.503434	+	0.864034i	$0.332070\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	−6.81507	−0.249350
$748$	0	0
$749$	−56.4452	−2.06246
$750$	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
$753$	9.63015	0.350942
$754$	0	0
$755$	6.81507	0.248026
$756$	0	0
$757$	13.1849	0.479214	0.239607	−	0.970870i	$-0.422981\pi$
$757$	13.1849	0.479214	0.239607	+	0.970870i	$0.422981\pi$
$758$	0	0
$759$	−4.81507	−0.174776
$760$	0	0
$761$	−19.8151	−0.718296	−0.359148	−	0.933281i	$-0.616933\pi$
$761$	−19.8151	−0.718296	−0.359148	+	0.933281i	$0.616933\pi$
$762$	0	0
$763$	−39.0000	−1.41189
$764$	0	0
$765$	1.00000	0.0361551
$766$	0	0
$767$	−13.6301	−0.492156
$768$	0	0
$769$	6.73971	0.243040	0.121520	−	0.992589i	$-0.461223\pi$
$769$	6.73971	0.243040	0.121520	+	0.992589i	$0.461223\pi$
$770$	0	0
$771$	−6.81507	−0.245439
$772$	0	0
$773$	−28.6301	−1.02975	−0.514877	−	0.857264i	$-0.672163\pi$
$773$	−28.6301	−1.02975	−0.514877	+	0.857264i	$0.672163\pi$
$774$	0	0
$775$	3.81507	0.137041
$776$	0	0
$777$	3.00000	0.107624
$778$	0	0
$779$	16.3699	0.586511
$780$	0	0
$781$	−9.63015	−0.344594
$782$	0	0
$783$	−3.81507	−0.136340
$784$	0	0
$785$	−2.18493	−0.0779834
$786$	0	0
$787$	18.3699	0.654815	0.327407	−	0.944883i	$-0.393825\pi$
$787$	18.3699	0.654815	0.327407	+	0.944883i	$0.393825\pi$
$788$	0	0
$789$	1.44522	0.0514511
$790$	0	0
$791$	−11.4452	−0.406945
$792$	0	0
$793$	−26.0000	−0.923287
$794$	0	0
$795$	10.6301	0.377012
$796$	0	0
$797$	28.8904	1.02335	0.511676	−	0.859179i	$-0.329025\pi$
$797$	28.8904	1.02335	0.511676	+	0.859179i	$0.329025\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	1.18493	0.0418673
$802$	0	0
$803$	4.81507	0.169920
$804$	0	0
$805$	−14.4452	−0.509127
$806$	0	0
$807$	0.815073	0.0286919
$808$	0	0
$809$	40.0754	1.40897	0.704487	−	0.709717i	$-0.251177\pi$
$809$	40.0754	1.40897	0.704487	+	0.709717i	$0.251177\pi$
$810$	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
$815$	4.63015	0.162187
$816$	0	0
$817$	−16.3699	−0.572709
$818$	0	0
$819$	−20.4452	−0.714414
$820$	0	0
$821$	−42.0754	−1.46844	−0.734220	−	0.678911i	$-0.762452\pi$
$821$	−42.0754	−1.46844	−0.734220	+	0.678911i	$0.762452\pi$
$822$	0	0
$823$	12.0754	0.420921	0.210460	−	0.977602i	$-0.432504\pi$
$823$	12.0754	0.420921	0.210460	+	0.977602i	$0.432504\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	−25.0754	−0.871956	−0.435978	−	0.899957i	$-0.643597\pi$
$827$	−25.0754	−0.871956	−0.435978	+	0.899957i	$0.643597\pi$
$828$	0	0
$829$	31.5206	1.09476	0.547378	−	0.836886i	$-0.315626\pi$
$829$	31.5206	1.09476	0.547378	+	0.836886i	$0.315626\pi$
$830$	0	0
$831$	10.4452	0.362341
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	11.1849	0.387070
$836$	0	0
$837$	−3.81507	−0.131868
$838$	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
$841$	−14.4452	−0.498111
$842$	0	0
$843$	−12.4452	−0.428636
$844$	0	0
$845$	33.4452	1.15055
$846$	0	0
$847$	30.0000	1.03081
$848$	0	0
$849$	−11.1849	−0.383866
$850$	0	0
$851$	4.81507	0.165059
$852$	0	0
$853$	42.0754	1.44063	0.720317	−	0.693646i	$-0.243997\pi$
$853$	42.0754	1.44063	0.720317	+	0.693646i	$0.243997\pi$
$854$	0	0
$855$	2.81507	0.0962735
$856$	0	0
$857$	−12.2603	−0.418804	−0.209402	−	0.977830i	$-0.567152\pi$
$857$	−12.2603	−0.418804	−0.209402	+	0.977830i	$0.567152\pi$
$858$	0	0
$859$	0.445219	0.0151907	0.00759533	−	0.999971i	$-0.497582\pi$
$859$	0.445219	0.0151907	0.00759533	+	0.999971i	$0.497582\pi$
$860$	0	0
$861$	17.4452	0.594531
$862$	0	0
$863$	33.4452	1.13849	0.569244	−	0.822168i	$-0.307236\pi$
$863$	33.4452	1.13849	0.569244	+	0.822168i	$0.307236\pi$
$864$	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
$871$	92.8904	3.14747
$872$	0	0
$873$	5.81507	0.196810
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−24.7055	−0.834246	−0.417123	−	0.908850i	$-0.636962\pi$
$877$	−24.7055	−0.834246	−0.417123	+	0.908850i	$0.636962\pi$
$878$	0	0
$879$	14.2603	0.480988
$880$	0	0
$881$	53.8151	1.81308	0.906538	−	0.422124i	$-0.138715\pi$
$881$	53.8151	1.81308	0.906538	+	0.422124i	$0.138715\pi$
$882$	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
$887$	−39.0754	−1.31202	−0.656011	−	0.754751i	$-0.727758\pi$
$887$	−39.0754	−1.31202	−0.656011	+	0.754751i	$0.727758\pi$
$888$	0	0
$889$	31.3357	1.05096
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	22.5206	0.753623
$894$	0	0
$895$	−16.0000	−0.534821
$896$	0	0
$897$	−32.8151	−1.09566
$898$	0	0
$899$	14.5548	0.485429
$900$	0	0
$901$	−10.6301	−0.354142
$902$	0	0
$903$	−17.4452	−0.580541
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	35.1849	1.16830	0.584148	−	0.811647i	$-0.301429\pi$
$907$	35.1849	1.16830	0.584148	+	0.811647i	$0.301429\pi$
$908$	0	0
$909$	−9.63015	−0.319412
$910$	0	0
$911$	−30.5206	−1.01119	−0.505596	−	0.862770i	$-0.668727\pi$
$911$	−30.5206	−1.01119	−0.505596	+	0.862770i	$0.668727\pi$
$912$	0	0
$913$	−6.81507	−0.225546
$914$	0	0
$915$	3.81507	0.126122
$916$	0	0
$917$	34.8904	1.15218
$918$	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
$923$	−65.6301	−2.16024
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−7.44522	−0.244270	−0.122135	−	0.992514i	$-0.538974\pi$
$929$	−7.44522	−0.244270	−0.122135	+	0.992514i	$0.538974\pi$
$930$	0	0
$931$	5.63015	0.184521
$932$	0	0
$933$	−21.4452	−0.702085
$934$	0	0
$935$	1.00000	0.0327035
$936$	0	0
$937$	9.63015	0.314603	0.157302	−	0.987551i	$-0.449721\pi$
$937$	9.63015	0.314603	0.157302	+	0.987551i	$0.449721\pi$
$938$	0	0
$939$	30.8904	1.00807
$940$	0	0
$941$	41.2603	1.34505	0.672524	−	0.740076i	$-0.265210\pi$
$941$	41.2603	1.34505	0.672524	+	0.740076i	$0.265210\pi$
$942$	0	0
$943$	28.0000	0.911805
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	57.4452	1.86672	0.933359	−	0.358943i	$-0.116863\pi$
$947$	57.4452	1.86672	0.933359	+	0.358943i	$0.116863\pi$
$948$	0	0
$949$	32.8151	1.06522
$950$	0	0
$951$	−7.81507	−0.253421
$952$	0	0
$953$	−48.8904	−1.58372	−0.791858	−	0.610705i	$-0.790886\pi$
$953$	−48.8904	−1.58372	−0.791858	+	0.610705i	$0.790886\pi$
$954$	0	0
$955$	16.6301	0.538139
$956$	0	0
$957$	−3.81507	−0.123324
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−16.4452	−0.530491
$962$	0	0
$963$	18.8151	0.606307
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	13.6301	0.438316	0.219158	−	0.975689i	$-0.429669\pi$
$967$	13.6301	0.438316	0.219158	+	0.975689i	$0.429669\pi$
$968$	0	0
$969$	−2.81507	−0.0904332
$970$	0	0
$971$	−5.07536	−0.162876	−0.0814381	−	0.996678i	$-0.525951\pi$
$971$	−5.07536	−0.162876	−0.0814381	+	0.996678i	$0.525951\pi$
$972$	0	0
$973$	36.5548	1.17189
$974$	0	0
$975$	−6.81507	−0.218257
$976$	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
$979$	1.18493	0.0378704
$980$	0	0
$981$	13.0000	0.415058
$982$	0	0
$983$	38.3357	1.22272	0.611359	−	0.791354i	$-0.290623\pi$
$983$	38.3357	1.22272	0.611359	+	0.791354i	$0.290623\pi$
$984$	0	0
$985$	−14.8151	−0.472047
$986$	0	0
$987$	24.0000	0.763928
$988$	0	0
$989$	−28.0000	−0.890348
$990$	0	0
$991$	6.55478	0.208219	0.104110	−	0.994566i	$-0.466801\pi$
$991$	6.55478	0.208219	0.104110	+	0.994566i	$0.466801\pi$
$992$	0	0
$993$	19.2603	0.611207
$994$	0	0
$995$	4.00000	0.126809
$996$	0	0
$997$	1.92464	0.0609538	0.0304769	−	0.999535i	$-0.490297\pi$
$997$	1.92464	0.0609538	0.0304769	+	0.999535i	$0.490297\pi$
$998$	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.1.2		2
4.3	odd	2		1110.2.a.q.1.2	✓	2
12.11	even	2		3330.2.a.bf.1.2		2
20.19	odd	2		5550.2.a.bx.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.q.1.2	✓	2	4.3	odd	2
3330.2.a.bf.1.2		2	12.11	even	2
5550.2.a.bx.1.1		2	20.19	odd	2
8880.2.a.bh.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +6.81507 q^{13} -1.00000 q^{15} +1.00000 q^{17} +2.81507 q^{19} +3.00000 q^{21} +4.81507 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.81507 q^{29} +3.81507 q^{31} -1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} -6.81507 q^{39} +5.81507 q^{41} -5.81507 q^{43} +1.00000 q^{45} +8.00000 q^{47} +2.00000 q^{49} -1.00000 q^{51} -10.6301 q^{53} +1.00000 q^{55} -2.81507 q^{57} -2.00000 q^{59} -3.81507 q^{61} -3.00000 q^{63} +6.81507 q^{65} +13.6301 q^{67} -4.81507 q^{69} -9.63015 q^{71} +4.81507 q^{73} -1.00000 q^{75} -3.00000 q^{77} -12.0000 q^{79} +1.00000 q^{81} -6.81507 q^{83} +1.00000 q^{85} -3.81507 q^{87} +1.18493 q^{89} -20.4452 q^{91} -3.81507 q^{93} +2.81507 q^{95} +5.81507 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 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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