LMFDB - Embedded newform 8649.2.a.bv.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 2883)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.294138 q^{2} -1.91348 q^{4} +1.97505 q^{5} -3.94391 q^{7} +1.15111 q^{8} +O(q^{10})$	$q-0.294138 q^{2} -1.91348 q^{4} +1.97505 q^{5} -3.94391 q^{7} +1.15111 q^{8} -0.580938 q^{10} -3.45259 q^{11} -4.05458 q^{13} +1.16005 q^{14} +3.48838 q^{16} -5.61029 q^{17} +2.86283 q^{19} -3.77922 q^{20} +1.01554 q^{22} -9.04908 q^{23} -1.09918 q^{25} +1.19261 q^{26} +7.54659 q^{28} -3.02709 q^{29} -3.32828 q^{32} +1.65020 q^{34} -7.78941 q^{35} -6.66407 q^{37} -0.842069 q^{38} +2.27349 q^{40} -4.11108 q^{41} -6.48949 q^{43} +6.60646 q^{44} +2.66168 q^{46} +7.29819 q^{47} +8.55439 q^{49} +0.323310 q^{50} +7.75837 q^{52} +5.96000 q^{53} -6.81903 q^{55} -4.53985 q^{56} +0.890383 q^{58} +1.56201 q^{59} +1.20786 q^{61} -5.99779 q^{64} -8.00800 q^{65} -9.99250 q^{67} +10.7352 q^{68} +2.29117 q^{70} -13.9367 q^{71} +1.87929 q^{73} +1.96016 q^{74} -5.47798 q^{76} +13.6167 q^{77} -11.2376 q^{79} +6.88973 q^{80} +1.20923 q^{82} +11.7045 q^{83} -11.0806 q^{85} +1.90881 q^{86} -3.97429 q^{88} -4.43701 q^{89} +15.9909 q^{91} +17.3153 q^{92} -2.14668 q^{94} +5.65424 q^{95} +9.53985 q^{97} -2.51618 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8}+O(q^{10}) $ 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8	$ 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} + 16 q^{11} - 32 q^{13} + 24 q^{14} + 48 q^{16} + 32 q^{17} + 32 q^{19} + 24 q^{20} - 32 q^{22} + 32 q^{23} + 40 q^{25} + 16 q^{26} + 8 q^{28} + 48 q^{29} + 48 q^{32} + 48 q^{35} - 64 q^{37} + 24 q^{38} - 32 q^{43} + 48 q^{44} - 32 q^{46} + 48 q^{47} + 56 q^{49} + 24 q^{50} - 64 q^{52} + 80 q^{53} + 48 q^{56} - 32 q^{58} - 32 q^{61} + 56 q^{64} + 16 q^{65} - 16 q^{67} + 80 q^{68} + 8 q^{70} - 32 q^{73} + 56 q^{76} + 96 q^{77} - 32 q^{79} + 72 q^{80} + 8 q^{82} + 48 q^{83} - 96 q^{85} - 32 q^{86} - 96 q^{88} - 16 q^{89} + 32 q^{92} + 48 q^{94} + 48 q^{95} + 16 q^{97} + 24 q^{98}+O(q^{100}) $ 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 + 16 * q^11 - 32 * q^13 + 24 * q^14 + 48 * q^16 + 32 * q^17 + 32 * q^19 + 24 * q^20 - 32 * q^22 + 32 * q^23 + 40 * q^25 + 16 * q^26 + 8 * q^28 + 48 * q^29 + 48 * q^32 + 48 * q^35 - 64 * q^37 + 24 * q^38 - 32 * q^43 + 48 * q^44 - 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 - 64 * q^52 + 80 * q^53 + 48 * q^56 - 32 * q^58 - 32 * q^61 + 56 * q^64 + 16 * q^65 - 16 * q^67 + 80 * q^68 + 8 * q^70 - 32 * q^73 + 56 * q^76 + 96 * q^77 - 32 * q^79 + 72 * q^80 + 8 * q^82 + 48 * q^83 - 96 * q^85 - 32 * q^86 - 96 * q^88 - 16 * q^89 + 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * q^98

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 $

$2$	−0.294138	−0.207987	−0.103994	−	0.994578i	$-0.533162\pi$
$2$	−0.294138	−0.207987	−0.103994	+	0.994578i	$0.533162\pi$
$3$	0	0
$3$	0	0
$4$	−1.91348	−0.956741
$5$	1.97505	0.883269	0.441635	−	0.897195i	$-0.354399\pi$
$5$	1.97505	0.883269	0.441635	+	0.897195i	$0.354399\pi$
$6$	0	0
$7$	−3.94391	−1.49066	−0.745328	−	0.666698i	$-0.767707\pi$
$7$	−3.94391	−1.49066	−0.745328	+	0.666698i	$0.767707\pi$
$8$	1.15111	0.406977
$9$	0	0
$10$	−0.580938	−0.183709
$11$	−3.45259	−1.04099	−0.520497	−	0.853864i	$-0.674253\pi$
$11$	−3.45259	−1.04099	−0.520497	+	0.853864i	$0.674253\pi$
$12$	0	0
$13$	−4.05458	−1.12454	−0.562269	−	0.826954i	$-0.690071\pi$
$13$	−4.05458	−1.12454	−0.562269	+	0.826954i	$0.690071\pi$
$14$	1.16005	0.310038
$15$	0	0
$16$	3.48838	0.872095
$17$	−5.61029	−1.36070	−0.680348	−	0.732889i	$-0.738171\pi$
$17$	−5.61029	−1.36070	−0.680348	+	0.732889i	$0.738171\pi$
$18$	0	0
$19$	2.86283	0.656779	0.328389	−	0.944542i	$-0.393494\pi$
$19$	2.86283	0.656779	0.328389	+	0.944542i	$0.393494\pi$
$20$	−3.77922	−0.845060
$21$	0	0
$22$	1.01554	0.216513
$23$	−9.04908	−1.88686	−0.943432	−	0.331566i	$-0.892423\pi$
$23$	−9.04908	−1.88686	−0.943432	+	0.331566i	$0.892423\pi$
$24$	0	0
$25$	−1.09918	−0.219835
$26$	1.19261	0.233890
$27$	0	0
$28$	7.54659	1.42617
$29$	−3.02709	−0.562116	−0.281058	−	0.959691i	$-0.590685\pi$
$29$	−3.02709	−0.562116	−0.281058	+	0.959691i	$0.590685\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−3.32828	−0.588362
$33$	0	0
$34$	1.65020	0.283007
$35$	−7.78941	−1.31665
$36$	0	0
$37$	−6.66407	−1.09557	−0.547783	−	0.836620i	$-0.684528\pi$
$37$	−6.66407	−1.09557	−0.547783	+	0.836620i	$0.684528\pi$
$38$	−0.842069	−0.136602
$39$	0	0
$40$	2.27349	0.359471
$41$	−4.11108	−0.642042	−0.321021	−	0.947072i	$-0.604026\pi$
$41$	−4.11108	−0.642042	−0.321021	+	0.947072i	$0.604026\pi$
$42$	0	0
$43$	−6.48949	−0.989637	−0.494819	−	0.868996i	$-0.664766\pi$
$43$	−6.48949	−0.989637	−0.494819	+	0.868996i	$0.664766\pi$
$44$	6.60646	0.995962
$45$	0	0
$46$	2.66168	0.392444
$47$	7.29819	1.06455	0.532276	−	0.846571i	$-0.321337\pi$
$47$	7.29819	1.06455	0.532276	+	0.846571i	$0.321337\pi$
$48$	0	0
$49$	8.55439	1.22206
$50$	0.323310	0.0457229
$51$	0	0
$52$	7.75837	1.07589
$53$	5.96000	0.818669	0.409335	−	0.912384i	$-0.365761\pi$
$53$	5.96000	0.818669	0.409335	+	0.912384i	$0.365761\pi$
$54$	0	0
$55$	−6.81903	−0.919478
$56$	−4.53985	−0.606663
$57$	0	0
$58$	0.890383	0.116913
$59$	1.56201	0.203356	0.101678	−	0.994817i	$-0.467579\pi$
$59$	1.56201	0.203356	0.101678	+	0.994817i	$0.467579\pi$
$60$	0	0
$61$	1.20786	0.154651	0.0773256	−	0.997006i	$-0.475362\pi$
$61$	1.20786	0.154651	0.0773256	+	0.997006i	$0.475362\pi$
$62$	0	0
$63$	0	0
$64$	−5.99779	−0.749723
$65$	−8.00800	−0.993270
$66$	0	0
$67$	−9.99250	−1.22078	−0.610389	−	0.792102i	$-0.708987\pi$
$67$	−9.99250	−1.22078	−0.610389	+	0.792102i	$0.708987\pi$
$68$	10.7352	1.30183
$69$	0	0
$70$	2.29117	0.273847
$71$	−13.9367	−1.65398	−0.826992	−	0.562214i	$-0.809950\pi$
$71$	−13.9367	−1.65398	−0.826992	+	0.562214i	$0.809950\pi$
$72$	0	0
$73$	1.87929	0.219954	0.109977	−	0.993934i	$-0.464922\pi$
$73$	1.87929	0.219954	0.109977	+	0.993934i	$0.464922\pi$
$74$	1.96016	0.227864
$75$	0	0
$76$	−5.47798	−0.628368
$77$	13.6167	1.55176
$78$	0	0
$79$	−11.2376	−1.26433	−0.632166	−	0.774833i	$-0.717834\pi$
$79$	−11.2376	−1.26433	−0.632166	+	0.774833i	$0.717834\pi$
$80$	6.88973	0.770295
$81$	0	0
$82$	1.20923	0.133537
$83$	11.7045	1.28473	0.642365	−	0.766398i	$-0.277953\pi$
$83$	11.7045	1.28473	0.642365	+	0.766398i	$0.277953\pi$
$84$	0	0
$85$	−11.0806	−1.20186
$86$	1.90881	0.205832
$87$	0	0
$88$	−3.97429	−0.423661
$89$	−4.43701	−0.470322	−0.235161	−	0.971956i	$-0.575562\pi$
$89$	−4.43701	−0.470322	−0.235161	+	0.971956i	$0.575562\pi$
$90$	0	0
$91$	15.9909	1.67630
$92$	17.3153	1.80524
$93$	0	0
$94$	−2.14668	−0.221413
$95$	5.65424	0.580113
$96$	0	0
$97$	9.53985	0.968625	0.484312	−	0.874895i	$-0.339070\pi$
$97$	9.53985	0.968625	0.484312	+	0.874895i	$0.339070\pi$
$98$	−2.51618	−0.254172
$99$	0	0
$100$	2.10325	0.210325
$101$	−0.934185	−0.0929548	−0.0464774	−	0.998919i	$-0.514800\pi$
$101$	−0.934185	−0.0929548	−0.0464774	+	0.998919i	$0.514800\pi$
$102$	0	0
$103$	12.7352	1.25484	0.627420	−	0.778681i	$-0.284111\pi$
$103$	12.7352	1.25484	0.627420	+	0.778681i	$0.284111\pi$
$104$	−4.66725	−0.457662
$105$	0	0
$106$	−1.75307	−0.170273
$107$	8.70168	0.841223	0.420612	−	0.907241i	$-0.361815\pi$
$107$	8.70168	0.841223	0.420612	+	0.907241i	$0.361815\pi$
$108$	0	0
$109$	4.87197	0.466650	0.233325	−	0.972399i	$-0.425039\pi$
$109$	4.87197	0.466650	0.233325	+	0.972399i	$0.425039\pi$
$110$	2.00574	0.191240
$111$	0	0
$112$	−13.7578	−1.29999
$113$	−19.3083	−1.81637	−0.908186	−	0.418567i	$-0.862533\pi$
$113$	−19.3083	−1.81637	−0.908186	+	0.418567i	$0.862533\pi$
$114$	0	0
$115$	−17.8724	−1.66661
$116$	5.79228	0.537800
$117$	0	0
$118$	−0.459447	−0.0422955
$119$	22.1265	2.02833
$120$	0	0
$121$	0.920345	0.0836677
$122$	−0.355279	−0.0321655
$123$	0	0
$124$	0	0
$125$	−12.0462	−1.07744
$126$	0	0
$127$	−7.51074	−0.666471	−0.333235	−	0.942844i	$-0.608140\pi$
$127$	−7.51074	−0.666471	−0.333235	+	0.942844i	$0.608140\pi$
$128$	8.42074	0.744295
$129$	0	0
$130$	2.35546	0.206588
$131$	−0.418866	−0.0365965	−0.0182983	−	0.999833i	$-0.505825\pi$
$131$	−0.418866	−0.0365965	−0.0182983	+	0.999833i	$0.505825\pi$
$132$	0	0
$133$	−11.2907	−0.979032
$134$	2.93918	0.253906
$135$	0	0
$136$	−6.45804	−0.553772
$137$	−6.96623	−0.595166	−0.297583	−	0.954696i	$-0.596180\pi$
$137$	−6.96623	−0.595166	−0.297583	+	0.954696i	$0.596180\pi$
$138$	0	0
$139$	10.7352	0.910547	0.455273	−	0.890352i	$-0.349542\pi$
$139$	10.7352	0.910547	0.455273	+	0.890352i	$0.349542\pi$
$140$	14.9049	1.25969
$141$	0	0
$142$	4.09932	0.344008
$143$	13.9988	1.17064
$144$	0	0
$145$	−5.97865	−0.496500
$146$	−0.552771	−0.0457476
$147$	0	0
$148$	12.7516	1.04817
$149$	0.261979	0.0214622	0.0107311	−	0.999942i	$-0.496584\pi$
$149$	0.261979	0.0214622	0.0107311	+	0.999942i	$0.496584\pi$
$150$	0	0
$151$	2.39989	0.195300	0.0976502	−	0.995221i	$-0.468867\pi$
$151$	2.39989	0.195300	0.0976502	+	0.995221i	$0.468867\pi$
$152$	3.29542	0.267294
$153$	0	0
$154$	−4.00519	−0.322747
$155$	0	0
$156$	0	0
$157$	−21.3951	−1.70751	−0.853756	−	0.520674i	$-0.825681\pi$
$157$	−21.3951	−1.70751	−0.853756	+	0.520674i	$0.825681\pi$
$158$	3.30542	0.262965
$159$	0	0
$160$	−6.57352	−0.519682
$161$	35.6887	2.81267
$162$	0	0
$163$	14.6877	1.15043	0.575215	−	0.818002i	$-0.304918\pi$
$163$	14.6877	1.15043	0.575215	+	0.818002i	$0.304918\pi$
$164$	7.86647	0.614268
$165$	0	0
$166$	−3.44273	−0.267208
$167$	−4.86141	−0.376187	−0.188093	−	0.982151i	$-0.560231\pi$
$167$	−4.86141	−0.376187	−0.188093	+	0.982151i	$0.560231\pi$
$168$	0	0
$169$	3.43962	0.264586
$170$	3.25923	0.249972
$171$	0	0
$172$	12.4175	0.946827
$173$	−8.74476	−0.664852	−0.332426	−	0.943129i	$-0.607867\pi$
$173$	−8.74476	−0.664852	−0.332426	+	0.943129i	$0.607867\pi$
$174$	0	0
$175$	4.33505	0.327699
$176$	−12.0439	−0.907845
$177$	0	0
$178$	1.30510	0.0978211
$179$	−8.68384	−0.649061	−0.324530	−	0.945875i	$-0.605206\pi$
$179$	−8.68384	−0.649061	−0.324530	+	0.945875i	$0.605206\pi$
$180$	0	0
$181$	−4.97464	−0.369762	−0.184881	−	0.982761i	$-0.559190\pi$
$181$	−4.97464	−0.369762	−0.184881	+	0.982761i	$0.559190\pi$
$182$	−4.70353	−0.348649
$183$	0	0
$184$	−10.4164	−0.767911
$185$	−13.1619	−0.967680
$186$	0	0
$187$	19.3700	1.41648
$188$	−13.9650	−1.01850
$189$	0	0
$190$	−1.66313	−0.120656
$191$	22.4386	1.62360	0.811801	−	0.583935i	$-0.198488\pi$
$191$	22.4386	1.62360	0.811801	+	0.583935i	$0.198488\pi$
$192$	0	0
$193$	−5.47817	−0.394328	−0.197164	−	0.980371i	$-0.563173\pi$
$193$	−5.47817	−0.394328	−0.197164	+	0.980371i	$0.563173\pi$
$194$	−2.80604	−0.201462
$195$	0	0
$196$	−16.3687	−1.16919
$197$	26.7745	1.90761	0.953804	−	0.300431i	$-0.0971305\pi$
$197$	26.7745	1.90761	0.953804	+	0.300431i	$0.0971305\pi$
$198$	0	0
$199$	−9.16973	−0.650025	−0.325013	−	0.945710i	$-0.605369\pi$
$199$	−9.16973	−0.650025	−0.325013	+	0.945710i	$0.605369\pi$
$200$	−1.26527	−0.0894679
$201$	0	0
$202$	0.274780	0.0193334
$203$	11.9386	0.837922
$204$	0	0
$205$	−8.11958	−0.567096
$206$	−3.74593	−0.260991
$207$	0	0
$208$	−14.1439	−0.980704
$209$	−9.88418	−0.683703
$210$	0	0
$211$	−22.8396	−1.57235	−0.786173	−	0.618007i	$-0.787940\pi$
$211$	−22.8396	−1.57235	−0.786173	+	0.618007i	$0.787940\pi$
$212$	−11.4044	−0.783255
$213$	0	0
$214$	−2.55950	−0.174964
$215$	−12.8171	−0.874117
$216$	0	0
$217$	0	0
$218$	−1.43303	−0.0970572
$219$	0	0
$220$	13.0481	0.879702
$221$	22.7474	1.53015
$222$	0	0
$223$	9.73668	0.652017	0.326008	−	0.945367i	$-0.394296\pi$
$223$	9.73668	0.652017	0.326008	+	0.945367i	$0.394296\pi$
$224$	13.1264	0.877046
$225$	0	0
$226$	5.67931	0.377782
$227$	20.0195	1.32874	0.664370	−	0.747404i	$-0.268700\pi$
$227$	20.0195	1.32874	0.664370	+	0.747404i	$0.268700\pi$
$228$	0	0
$229$	5.63145	0.372137	0.186068	−	0.982537i	$-0.440425\pi$
$229$	5.63145	0.372137	0.186068	+	0.982537i	$0.440425\pi$
$230$	5.25696	0.346634
$231$	0	0
$232$	−3.48450	−0.228769
$233$	16.8968	1.10694	0.553472	−	0.832868i	$-0.313302\pi$
$233$	16.8968	1.10694	0.553472	+	0.832868i	$0.313302\pi$
$234$	0	0
$235$	14.4143	0.940285
$236$	−2.98888	−0.194559
$237$	0	0
$238$	−6.50824	−0.421867
$239$	1.97666	0.127860	0.0639298	−	0.997954i	$-0.479637\pi$
$239$	1.97666	0.127860	0.0639298	+	0.997954i	$0.479637\pi$
$240$	0	0
$241$	−24.7460	−1.59403	−0.797017	−	0.603957i	$-0.793590\pi$
$241$	−24.7460	−1.59403	−0.797017	+	0.603957i	$0.793590\pi$
$242$	−0.270709	−0.0174018
$243$	0	0
$244$	−2.31123	−0.147961
$245$	16.8954	1.07940
$246$	0	0
$247$	−11.6076	−0.738573
$248$	0	0
$249$	0	0
$250$	3.54325	0.224094
$251$	−12.7560	−0.805150	−0.402575	−	0.915387i	$-0.631885\pi$
$251$	−12.7560	−0.805150	−0.402575	+	0.915387i	$0.631885\pi$
$252$	0	0
$253$	31.2427	1.96421
$254$	2.20920	0.138617
$255$	0	0
$256$	9.51871	0.594919
$257$	−17.6767	−1.10264	−0.551322	−	0.834292i	$-0.685876\pi$
$257$	−17.6767	−1.10264	−0.551322	+	0.834292i	$0.685876\pi$
$258$	0	0
$259$	26.2825	1.63311
$260$	15.3232	0.950303
$261$	0	0
$262$	0.123205	0.00761161
$263$	1.61901	0.0998324	0.0499162	−	0.998753i	$-0.484105\pi$
$263$	1.61901	0.0998324	0.0499162	+	0.998753i	$0.484105\pi$
$264$	0	0
$265$	11.7713	0.723105
$266$	3.32104	0.203626
$267$	0	0
$268$	19.1205	1.16797
$269$	2.88150	0.175688	0.0878441	−	0.996134i	$-0.472002\pi$
$269$	2.88150	0.175688	0.0878441	+	0.996134i	$0.472002\pi$
$270$	0	0
$271$	17.4612	1.06069	0.530346	−	0.847781i	$-0.322062\pi$
$271$	17.4612	1.06069	0.530346	+	0.847781i	$0.322062\pi$
$272$	−19.5708	−1.18666
$273$	0	0
$274$	2.04904	0.123787
$275$	3.79500	0.228847
$276$	0	0
$277$	−18.5518	−1.11467	−0.557334	−	0.830288i	$-0.688176\pi$
$277$	−18.5518	−1.11467	−0.557334	+	0.830288i	$0.688176\pi$
$278$	−3.15763	−0.189382
$279$	0	0
$280$	−8.96644	−0.535847
$281$	11.6011	0.692066	0.346033	−	0.938222i	$-0.387529\pi$
$281$	11.6011	0.692066	0.346033	+	0.938222i	$0.387529\pi$
$282$	0	0
$283$	−2.26878	−0.134865	−0.0674324	−	0.997724i	$-0.521481\pi$
$283$	−2.26878	−0.134865	−0.0674324	+	0.997724i	$0.521481\pi$
$284$	26.6677	1.58243
$285$	0	0
$286$	−4.11758	−0.243478
$287$	16.2137	0.957064
$288$	0	0
$289$	14.4754	0.851493
$290$	1.75855	0.103266
$291$	0	0
$292$	−3.59598	−0.210439
$293$	5.99637	0.350312	0.175156	−	0.984541i	$-0.443957\pi$
$293$	5.99637	0.350312	0.175156	+	0.984541i	$0.443957\pi$
$294$	0	0
$295$	3.08505	0.179618
$296$	−7.67105	−0.445871
$297$	0	0
$298$	−0.0770581	−0.00446386
$299$	36.6902	2.12185
$300$	0	0
$301$	25.5939	1.47521
$302$	−0.705900	−0.0406200
$303$	0	0
$304$	9.98665	0.572774
$305$	2.38559	0.136599
$306$	0	0
$307$	7.49278	0.427636	0.213818	−	0.976874i	$-0.431410\pi$
$307$	7.49278	0.427636	0.213818	+	0.976874i	$0.431410\pi$
$308$	−26.0553	−1.48464
$309$	0	0
$310$	0	0
$311$	−15.3565	−0.870784	−0.435392	−	0.900241i	$-0.643390\pi$
$311$	−15.3565	−0.870784	−0.435392	+	0.900241i	$0.643390\pi$
$312$	0	0
$313$	−19.4123	−1.09725	−0.548624	−	0.836069i	$-0.684848\pi$
$313$	−19.4123	−1.09725	−0.548624	+	0.836069i	$0.684848\pi$
$314$	6.29311	0.355141
$315$	0	0
$316$	21.5030	1.20964
$317$	8.81093	0.494871	0.247436	−	0.968904i	$-0.420412\pi$
$317$	8.81093	0.494871	0.247436	+	0.968904i	$0.420412\pi$
$318$	0	0
$319$	10.4513	0.585159
$320$	−11.8459	−0.662208
$321$	0	0
$322$	−10.4974	−0.584999
$323$	−16.0613	−0.893676
$324$	0	0
$325$	4.45670	0.247213
$326$	−4.32022	−0.239275
$327$	0	0
$328$	−4.73228	−0.261297
$329$	−28.7834	−1.58688
$330$	0	0
$331$	28.5653	1.57009	0.785046	−	0.619437i	$-0.212639\pi$
$331$	28.5653	1.57009	0.785046	+	0.619437i	$0.212639\pi$
$332$	−22.3963	−1.22916
$333$	0	0
$334$	1.42993	0.0782421
$335$	−19.7357	−1.07828
$336$	0	0
$337$	−21.2178	−1.15581	−0.577904	−	0.816105i	$-0.696129\pi$
$337$	−21.2178	−1.15581	−0.577904	+	0.816105i	$0.696129\pi$
$338$	−1.01172	−0.0550306
$339$	0	0
$340$	21.2026	1.14987
$341$	0	0
$342$	0	0
$343$	−6.13038	−0.331009
$344$	−7.47009	−0.402760
$345$	0	0
$346$	2.57217	0.138281
$347$	−10.1625	−0.545552	−0.272776	−	0.962078i	$-0.587942\pi$
$347$	−10.1625	−0.545552	−0.272776	+	0.962078i	$0.587942\pi$
$348$	0	0
$349$	−31.6483	−1.69409	−0.847047	−	0.531518i	$-0.821622\pi$
$349$	−31.6483	−1.69409	−0.847047	+	0.531518i	$0.821622\pi$
$350$	−1.27510	−0.0681572
$351$	0	0
$352$	11.4912	0.612481
$353$	−7.54591	−0.401628	−0.200814	−	0.979629i	$-0.564359\pi$
$353$	−7.54591	−0.401628	−0.200814	+	0.979629i	$0.564359\pi$
$354$	0	0
$355$	−27.5257	−1.46091
$356$	8.49014	0.449977
$357$	0	0
$358$	2.55425	0.134996
$359$	25.0521	1.32220	0.661099	−	0.750298i	$-0.270090\pi$
$359$	25.0521	1.32220	0.661099	+	0.750298i	$0.270090\pi$
$360$	0	0
$361$	−10.8042	−0.568641
$362$	1.46323	0.0769058
$363$	0	0
$364$	−30.5983	−1.60379
$365$	3.71169	0.194279
$366$	0	0
$367$	−27.9753	−1.46030	−0.730148	−	0.683289i	$-0.760549\pi$
$367$	−27.9753	−1.46030	−0.730148	+	0.683289i	$0.760549\pi$
$368$	−31.5666	−1.64552
$369$	0	0
$370$	3.87141	0.201265
$371$	−23.5057	−1.22035
$372$	0	0
$373$	−28.2318	−1.46179	−0.730893	−	0.682492i	$-0.760896\pi$
$373$	−28.2318	−1.46179	−0.730893	+	0.682492i	$0.760896\pi$
$374$	−5.69747	−0.294609
$375$	0	0
$376$	8.40099	0.433248
$377$	12.2736	0.632121
$378$	0	0
$379$	20.2787	1.04165	0.520824	−	0.853664i	$-0.325625\pi$
$379$	20.2787	1.04165	0.520824	+	0.853664i	$0.325625\pi$
$380$	−10.8193	−0.555018
$381$	0	0
$382$	−6.60006	−0.337688
$383$	29.0305	1.48339	0.741696	−	0.670736i	$-0.234022\pi$
$383$	29.0305	1.48339	0.741696	+	0.670736i	$0.234022\pi$
$384$	0	0
$385$	26.8936	1.37063
$386$	1.61134	0.0820151
$387$	0	0
$388$	−18.2543	−0.926723
$389$	33.0706	1.67675	0.838373	−	0.545097i	$-0.183507\pi$
$389$	33.0706	1.67675	0.838373	+	0.545097i	$0.183507\pi$
$390$	0	0
$391$	50.7680	2.56745
$392$	9.84701	0.497349
$393$	0	0
$394$	−7.87542	−0.396758
$395$	−22.1949	−1.11675
$396$	0	0
$397$	22.9993	1.15430	0.577152	−	0.816637i	$-0.304164\pi$
$397$	22.9993	1.15430	0.577152	+	0.816637i	$0.304164\pi$
$398$	2.69717	0.135197
$399$	0	0
$400$	−3.83434	−0.191717
$401$	0.0298135	0.00148881	0.000744407	−	1.00000i	$-0.499763\pi$
$401$	0.0298135	0.00148881	0.000744407		1.00000i	$0.499763\pi$
$402$	0	0
$403$	0	0
$404$	1.78755	0.0889337
$405$	0	0
$406$	−3.51159	−0.174277
$407$	23.0083	1.14048
$408$	0	0
$409$	−16.7574	−0.828601	−0.414300	−	0.910140i	$-0.635974\pi$
$409$	−16.7574	−0.828601	−0.414300	+	0.910140i	$0.635974\pi$
$410$	2.38828	0.117949
$411$	0	0
$412$	−24.3687	−1.20056
$413$	−6.16042	−0.303134
$414$	0	0
$415$	23.1169	1.13476
$416$	13.4948	0.661636
$417$	0	0
$418$	2.90732	0.142201
$419$	5.60603	0.273873	0.136936	−	0.990580i	$-0.456274\pi$
$419$	5.60603	0.273873	0.136936	+	0.990580i	$0.456274\pi$
$420$	0	0
$421$	−19.5366	−0.952155	−0.476077	−	0.879403i	$-0.657942\pi$
$421$	−19.5366	−0.952155	−0.476077	+	0.879403i	$0.657942\pi$
$422$	6.71802	0.327028
$423$	0	0
$424$	6.86059	0.333180
$425$	6.16670	0.299129
$426$	0	0
$427$	−4.76370	−0.230532
$428$	−16.6505	−0.804833
$429$	0	0
$430$	3.76999	0.181805
$431$	0.547349	0.0263649	0.0131824	−	0.999913i	$-0.495804\pi$
$431$	0.547349	0.0263649	0.0131824	+	0.999913i	$0.495804\pi$
$432$	0	0
$433$	−29.8625	−1.43510	−0.717551	−	0.696506i	$-0.754737\pi$
$433$	−29.8625	−1.43510	−0.717551	+	0.696506i	$0.754737\pi$
$434$	0	0
$435$	0	0
$436$	−9.32242	−0.446463
$437$	−25.9060	−1.23925
$438$	0	0
$439$	4.10095	0.195728	0.0978639	−	0.995200i	$-0.468799\pi$
$439$	4.10095	0.195728	0.0978639	+	0.995200i	$0.468799\pi$
$440$	−7.84943	−0.374207
$441$	0	0
$442$	−6.69088	−0.318253
$443$	13.6085	0.646558	0.323279	−	0.946304i	$-0.395215\pi$
$443$	13.6085	0.646558	0.323279	+	0.946304i	$0.395215\pi$
$444$	0	0
$445$	−8.76332	−0.415421
$446$	−2.86393	−0.135611
$447$	0	0
$448$	23.6547	1.11758
$449$	10.6669	0.503401	0.251700	−	0.967805i	$-0.419010\pi$
$449$	10.6669	0.503401	0.251700	+	0.967805i	$0.419010\pi$
$450$	0	0
$451$	14.1938	0.668362
$452$	36.9461	1.73780
$453$	0	0
$454$	−5.88850	−0.276361
$455$	31.5828	1.48062
$456$	0	0
$457$	1.05264	0.0492406	0.0246203	−	0.999697i	$-0.492162\pi$
$457$	1.05264	0.0492406	0.0246203	+	0.999697i	$0.492162\pi$
$458$	−1.65643	−0.0773997
$459$	0	0
$460$	34.1985	1.59451
$461$	17.0113	0.792293	0.396147	−	0.918187i	$-0.370347\pi$
$461$	17.0113	0.792293	0.396147	+	0.918187i	$0.370347\pi$
$462$	0	0
$463$	−18.0693	−0.839751	−0.419876	−	0.907582i	$-0.637926\pi$
$463$	−18.0693	−0.839751	−0.419876	+	0.907582i	$0.637926\pi$
$464$	−10.5596	−0.490219
$465$	0	0
$466$	−4.96999	−0.230230
$467$	−17.2750	−0.799389	−0.399695	−	0.916648i	$-0.630884\pi$
$467$	−17.2750	−0.799389	−0.399695	+	0.916648i	$0.630884\pi$
$468$	0	0
$469$	39.4095	1.81976
$470$	−4.23980	−0.195567
$471$	0	0
$472$	1.79804	0.0827614
$473$	22.4055	1.03021
$474$	0	0
$475$	−3.14676	−0.144383
$476$	−42.3386	−1.94059
$477$	0	0
$478$	−0.581412	−0.0265932
$479$	−31.3490	−1.43237	−0.716186	−	0.697910i	$-0.754114\pi$
$479$	−31.3490	−1.43237	−0.716186	+	0.697910i	$0.754114\pi$
$480$	0	0
$481$	27.0200	1.23201
$482$	7.27877	0.331539
$483$	0	0
$484$	−1.76106	−0.0800484
$485$	18.8417	0.855557
$486$	0	0
$487$	15.9682	0.723587	0.361794	−	0.932258i	$-0.382164\pi$
$487$	15.9682	0.723587	0.361794	+	0.932258i	$0.382164\pi$
$488$	1.39038	0.0629395
$489$	0	0
$490$	−4.96958	−0.224503
$491$	7.86684	0.355025	0.177513	−	0.984119i	$-0.443195\pi$
$491$	7.86684	0.355025	0.177513	+	0.984119i	$0.443195\pi$
$492$	0	0
$493$	16.9828	0.764869
$494$	3.41424	0.153614
$495$	0	0
$496$	0	0
$497$	54.9651	2.46552
$498$	0	0
$499$	−6.56356	−0.293825	−0.146913	−	0.989149i	$-0.546934\pi$
$499$	−6.56356	−0.293825	−0.146913	+	0.989149i	$0.546934\pi$
$500$	23.0502	1.03083
$501$	0	0
$502$	3.75202	0.167461
$503$	−40.6886	−1.81422	−0.907108	−	0.420898i	$-0.861715\pi$
$503$	−40.6886	−1.81422	−0.907108	+	0.420898i	$0.861715\pi$
$504$	0	0
$505$	−1.84506	−0.0821042
$506$	−9.18969	−0.408531
$507$	0	0
$508$	14.3717	0.637640
$509$	27.4219	1.21545	0.607726	−	0.794146i	$-0.292082\pi$
$509$	27.4219	1.21545	0.607726	+	0.794146i	$0.292082\pi$
$510$	0	0
$511$	−7.41173	−0.327876
$512$	−19.6413	−0.868031
$513$	0	0
$514$	5.19941	0.229336
$515$	25.1527	1.10836
$516$	0	0
$517$	−25.1976	−1.10819
$518$	−7.73069	−0.339667
$519$	0	0
$520$	−9.21806	−0.404238
$521$	−1.20179	−0.0526513	−0.0263257	−	0.999653i	$-0.508381\pi$
$521$	−1.20179	−0.0526513	−0.0263257	+	0.999653i	$0.508381\pi$
$522$	0	0
$523$	−1.67858	−0.0733993	−0.0366996	−	0.999326i	$-0.511684\pi$
$523$	−1.67858	−0.0733993	−0.0366996	+	0.999326i	$0.511684\pi$
$524$	0.801494	0.0350134
$525$	0	0
$526$	−0.476213	−0.0207639
$527$	0	0
$528$	0	0
$529$	58.8859	2.56025
$530$	−3.46239	−0.150397
$531$	0	0
$532$	21.6046	0.936680
$533$	16.6687	0.722001
$534$	0	0
$535$	17.1863	0.743027
$536$	−11.5024	−0.496829
$537$	0	0
$538$	−0.847560	−0.0365409
$539$	−29.5348	−1.27215
$540$	0	0
$541$	4.07216	0.175076	0.0875379	−	0.996161i	$-0.472100\pi$
$541$	4.07216	0.175076	0.0875379	+	0.996161i	$0.472100\pi$
$542$	−5.13601	−0.220610
$543$	0	0
$544$	18.6726	0.800582
$545$	9.62238	0.412177
$546$	0	0
$547$	−27.6261	−1.18121	−0.590603	−	0.806963i	$-0.701110\pi$
$547$	−27.6261	−1.18121	−0.590603	+	0.806963i	$0.701110\pi$
$548$	13.3298	0.569419
$549$	0	0
$550$	−1.11625	−0.0475973
$551$	−8.66605	−0.369186
$552$	0	0
$553$	44.3201	1.88468
$554$	5.45679	0.231837
$555$	0	0
$556$	−20.5416	−0.871157
$557$	11.6673	0.494361	0.247181	−	0.968969i	$-0.420496\pi$
$557$	11.6673	0.494361	0.247181	+	0.968969i	$0.420496\pi$
$558$	0	0
$559$	26.3121	1.11289
$560$	−27.1724	−1.14825
$561$	0	0
$562$	−3.41234	−0.143941
$563$	−39.8082	−1.67772	−0.838858	−	0.544350i	$-0.816776\pi$
$563$	−39.8082	−1.67772	−0.838858	+	0.544350i	$0.816776\pi$
$564$	0	0
$565$	−38.1349	−1.60435
$566$	0.667335	0.0280502
$567$	0	0
$568$	−16.0426	−0.673134
$569$	43.3349	1.81670	0.908348	−	0.418216i	$-0.137344\pi$
$569$	43.3349	1.81670	0.908348	+	0.418216i	$0.137344\pi$
$570$	0	0
$571$	32.8400	1.37431	0.687155	−	0.726511i	$-0.258859\pi$
$571$	32.8400	1.37431	0.687155	+	0.726511i	$0.258859\pi$
$572$	−26.7864	−1.12000
$573$	0	0
$574$	−4.76907	−0.199057
$575$	9.94653	0.414799
$576$	0	0
$577$	−23.2906	−0.969599	−0.484799	−	0.874625i	$-0.661107\pi$
$577$	−23.2906	−0.969599	−0.484799	+	0.874625i	$0.661107\pi$
$578$	−4.25777	−0.177100
$579$	0	0
$580$	11.4400	0.475022
$581$	−46.1613	−1.91509
$582$	0	0
$583$	−20.5774	−0.852229
$584$	2.16326	0.0895163
$585$	0	0
$586$	−1.76376	−0.0728604
$587$	−21.1258	−0.871955	−0.435977	−	0.899958i	$-0.643597\pi$
$587$	−21.1258	−0.871955	−0.435977	+	0.899958i	$0.643597\pi$
$588$	0	0
$589$	0	0
$590$	−0.907431	−0.0373584
$591$	0	0
$592$	−23.2468	−0.955438
$593$	1.65769	0.0680733	0.0340367	−	0.999421i	$-0.489164\pi$
$593$	1.65769	0.0680733	0.0340367	+	0.999421i	$0.489164\pi$
$594$	0	0
$595$	43.7009	1.79156
$596$	−0.501292	−0.0205337
$597$	0	0
$598$	−10.7920	−0.441318
$599$	26.1988	1.07045	0.535227	−	0.844708i	$-0.320226\pi$
$599$	26.1988	1.07045	0.535227	+	0.844708i	$0.320226\pi$
$600$	0	0
$601$	7.40626	0.302108	0.151054	−	0.988526i	$-0.451733\pi$
$601$	7.40626	0.302108	0.151054	+	0.988526i	$0.451733\pi$
$602$	−7.52816	−0.306825
$603$	0	0
$604$	−4.59215	−0.186852
$605$	1.81773	0.0739011
$606$	0	0
$607$	19.1718	0.778158	0.389079	−	0.921204i	$-0.372793\pi$
$607$	19.1718	0.778158	0.389079	+	0.921204i	$0.372793\pi$
$608$	−9.52831	−0.386424
$609$	0	0
$610$	−0.701695	−0.0284108
$611$	−29.5911	−1.19713
$612$	0	0
$613$	33.5998	1.35708	0.678542	−	0.734562i	$-0.262612\pi$
$613$	33.5998	1.35708	0.678542	+	0.734562i	$0.262612\pi$
$614$	−2.20392	−0.0889428
$615$	0	0
$616$	15.6742	0.631533
$617$	19.5305	0.786268	0.393134	−	0.919481i	$-0.371391\pi$
$617$	19.5305	0.786268	0.393134	+	0.919481i	$0.371391\pi$
$618$	0	0
$619$	15.9023	0.639169	0.319584	−	0.947558i	$-0.396457\pi$
$619$	15.9023	0.639169	0.319584	+	0.947558i	$0.396457\pi$
$620$	0	0
$621$	0	0
$622$	4.51692	0.181112
$623$	17.4992	0.701089
$624$	0	0
$625$	−18.2959	−0.731837
$626$	5.70990	0.228214
$627$	0	0
$628$	40.9391	1.63365
$629$	37.3874	1.49073
$630$	0	0
$631$	−24.0646	−0.957996	−0.478998	−	0.877816i	$-0.659000\pi$
$631$	−24.0646	−0.957996	−0.478998	+	0.877816i	$0.659000\pi$
$632$	−12.9357	−0.514555
$633$	0	0
$634$	−2.59163	−0.102927
$635$	−14.8341	−0.588673
$636$	0	0
$637$	−34.6845	−1.37425
$638$	−3.07412	−0.121706
$639$	0	0
$640$	16.6314	0.657413
$641$	5.80809	0.229406	0.114703	−	0.993400i	$-0.463408\pi$
$641$	5.80809	0.229406	0.114703	+	0.993400i	$0.463408\pi$
$642$	0	0
$643$	−29.4661	−1.16203	−0.581015	−	0.813893i	$-0.697344\pi$
$643$	−29.4661	−1.16203	−0.581015	+	0.813893i	$0.697344\pi$
$644$	−68.2897	−2.69099
$645$	0	0
$646$	4.72426	0.185873
$647$	15.2651	0.600135	0.300067	−	0.953918i	$-0.402991\pi$
$647$	15.2651	0.600135	0.300067	+	0.953918i	$0.402991\pi$
$648$	0	0
$649$	−5.39297	−0.211693
$650$	−1.31089	−0.0514172
$651$	0	0
$652$	−28.1047	−1.10066
$653$	−22.0501	−0.862887	−0.431443	−	0.902140i	$-0.641995\pi$
$653$	−22.0501	−0.862887	−0.431443	+	0.902140i	$0.641995\pi$
$654$	0	0
$655$	−0.827282	−0.0323246
$656$	−14.3410	−0.559922
$657$	0	0
$658$	8.46630	0.330051
$659$	33.7220	1.31362	0.656812	−	0.754054i	$-0.271904\pi$
$659$	33.7220	1.31362	0.656812	+	0.754054i	$0.271904\pi$
$660$	0	0
$661$	26.9435	1.04798	0.523991	−	0.851724i	$-0.324443\pi$
$661$	26.9435	1.04798	0.523991	+	0.851724i	$0.324443\pi$
$662$	−8.40216	−0.326559
$663$	0	0
$664$	13.4731	0.522856
$665$	−22.2998	−0.864749
$666$	0	0
$667$	27.3924	1.06064
$668$	9.30222	0.359914
$669$	0	0
$670$	5.80503	0.224268
$671$	−4.17025	−0.160991
$672$	0	0
$673$	12.5546	0.483944	0.241972	−	0.970283i	$-0.422206\pi$
$673$	12.5546	0.483944	0.241972	+	0.970283i	$0.422206\pi$
$674$	6.24097	0.240393
$675$	0	0
$676$	−6.58165	−0.253140
$677$	0.317270	0.0121937	0.00609684	−	0.999981i	$-0.498059\pi$
$677$	0.317270	0.0121937	0.00609684	+	0.999981i	$0.498059\pi$
$678$	0	0
$679$	−37.6243	−1.44389
$680$	−12.7550	−0.489130
$681$	0	0
$682$	0	0
$683$	−33.9305	−1.29831	−0.649157	−	0.760654i	$-0.724878\pi$
$683$	−33.9305	−1.29831	−0.649157	+	0.760654i	$0.724878\pi$
$684$	0	0
$685$	−13.7587	−0.525692
$686$	1.80318	0.0688458
$687$	0	0
$688$	−22.6378	−0.863058
$689$	−24.1653	−0.920625
$690$	0	0
$691$	1.85903	0.0707209	0.0353604	−	0.999375i	$-0.488742\pi$
$691$	1.85903	0.0707209	0.0353604	+	0.999375i	$0.488742\pi$
$692$	16.7329	0.636091
$693$	0	0
$694$	2.98919	0.113468
$695$	21.2025	0.804258
$696$	0	0
$697$	23.0643	0.873624
$698$	9.30898	0.352350
$699$	0	0
$700$	−8.29503	−0.313523
$701$	−23.3456	−0.881751	−0.440875	−	0.897568i	$-0.645332\pi$
$701$	−23.3456	−0.881751	−0.440875	+	0.897568i	$0.645332\pi$
$702$	0	0
$703$	−19.0781	−0.719545
$704$	20.7079	0.780457
$705$	0	0
$706$	2.21954	0.0835336
$707$	3.68434	0.138564
$708$	0	0
$709$	15.4550	0.580425	0.290212	−	0.956962i	$-0.406274\pi$
$709$	15.4550	0.580425	0.290212	+	0.956962i	$0.406274\pi$
$710$	8.09637	0.303851
$711$	0	0
$712$	−5.10747	−0.191411
$713$	0	0
$714$	0	0
$715$	27.6483	1.03399
$716$	16.6164	0.620983
$717$	0	0
$718$	−7.36878	−0.275001
$719$	−18.8424	−0.702703	−0.351351	−	0.936244i	$-0.614278\pi$
$719$	−18.8424	−0.702703	−0.351351	+	0.936244i	$0.614278\pi$
$720$	0	0
$721$	−50.2266	−1.87054
$722$	3.17793	0.118270
$723$	0	0
$724$	9.51888	0.353766
$725$	3.32730	0.123573
$726$	0	0
$727$	17.3820	0.644663	0.322331	−	0.946627i	$-0.395533\pi$
$727$	17.3820	0.644663	0.322331	+	0.946627i	$0.395533\pi$
$728$	18.4072	0.682216
$729$	0	0
$730$	−1.09175	−0.0404075
$731$	36.4079	1.34660
$732$	0	0
$733$	−28.1539	−1.03989	−0.519944	−	0.854200i	$-0.674047\pi$
$733$	−28.1539	−1.03989	−0.519944	+	0.854200i	$0.674047\pi$
$734$	8.22860	0.303723
$735$	0	0
$736$	30.1179	1.11016
$737$	34.5000	1.27082
$738$	0	0
$739$	−3.19924	−0.117686	−0.0588429	−	0.998267i	$-0.518741\pi$
$739$	−3.19924	−0.117686	−0.0588429	+	0.998267i	$0.518741\pi$
$740$	25.1850	0.925820
$741$	0	0
$742$	6.91393	0.253818
$743$	−1.52247	−0.0558541	−0.0279270	−	0.999610i	$-0.508891\pi$
$743$	−1.52247	−0.0558541	−0.0279270	+	0.999610i	$0.508891\pi$
$744$	0	0
$745$	0.517422	0.0189569
$746$	8.30406	0.304033
$747$	0	0
$748$	−37.0642	−1.35520
$749$	−34.3186	−1.25397
$750$	0	0
$751$	18.7964	0.685892	0.342946	−	0.939355i	$-0.388575\pi$
$751$	18.7964	0.685892	0.342946	+	0.939355i	$0.388575\pi$
$752$	25.4589	0.928390
$753$	0	0
$754$	−3.61013	−0.131473
$755$	4.73991	0.172503
$756$	0	0
$757$	−41.1966	−1.49731	−0.748657	−	0.662957i	$-0.769301\pi$
$757$	−41.1966	−1.49731	−0.748657	+	0.662957i	$0.769301\pi$
$758$	−5.96475	−0.216649
$759$	0	0
$760$	6.50863	0.236093
$761$	−41.8230	−1.51608	−0.758042	−	0.652206i	$-0.773844\pi$
$761$	−41.8230	−1.51608	−0.758042	+	0.652206i	$0.773844\pi$
$762$	0	0
$763$	−19.2146	−0.695614
$764$	−42.9359	−1.55337
$765$	0	0
$766$	−8.53900	−0.308527
$767$	−6.33329	−0.228682
$768$	0	0
$769$	−14.4609	−0.521475	−0.260738	−	0.965410i	$-0.583966\pi$
$769$	−14.4609	−0.521475	−0.260738	+	0.965410i	$0.583966\pi$
$770$	−7.91045	−0.285073
$771$	0	0
$772$	10.4824	0.377269
$773$	29.2571	1.05231	0.526153	−	0.850390i	$-0.323634\pi$
$773$	29.2571	1.05231	0.526153	+	0.850390i	$0.323634\pi$
$774$	0	0
$775$	0	0
$776$	10.9814	0.394208
$777$	0	0
$778$	−9.72733	−0.348742
$779$	−11.7693	−0.421680
$780$	0	0
$781$	48.1177	1.72179
$782$	−14.9328	−0.533997
$783$	0	0
$784$	29.8410	1.06575
$785$	−42.2563	−1.50819
$786$	0	0
$787$	−2.26533	−0.0807503	−0.0403752	−	0.999185i	$-0.512855\pi$
$787$	−2.26533	−0.0807503	−0.0403752	+	0.999185i	$0.512855\pi$
$788$	−51.2326	−1.82509
$789$	0	0
$790$	6.52837	0.232269
$791$	76.1501	2.70759
$792$	0	0
$793$	−4.89738	−0.173911
$794$	−6.76499	−0.240080
$795$	0	0
$796$	17.5461	0.621906
$797$	10.3106	0.365220	0.182610	−	0.983185i	$-0.441545\pi$
$797$	10.3106	0.365220	0.182610	+	0.983185i	$0.441545\pi$
$798$	0	0
$799$	−40.9450	−1.44853
$800$	3.65836	0.129343
$801$	0	0
$802$	−0.00876929	−0.000309655 0
$803$	−6.48840	−0.228971
$804$	0	0
$805$	70.4870	2.48434
$806$	0	0
$807$	0	0
$808$	−1.07535	−0.0378305
$809$	−1.91813	−0.0674379	−0.0337189	−	0.999431i	$-0.510735\pi$
$809$	−1.91813	−0.0674379	−0.0337189	+	0.999431i	$0.510735\pi$
$810$	0	0
$811$	53.9722	1.89522	0.947610	−	0.319429i	$-0.103491\pi$
$811$	53.9722	1.89522	0.947610	+	0.319429i	$0.103491\pi$
$812$	−22.8442	−0.801675
$813$	0	0
$814$	−6.76762	−0.237205
$815$	29.0090	1.01614
$816$	0	0
$817$	−18.5783	−0.649973
$818$	4.92900	0.172338
$819$	0	0
$820$	15.5367	0.542564
$821$	18.8341	0.657313	0.328656	−	0.944450i	$-0.393404\pi$
$821$	18.8341	0.657313	0.328656	+	0.944450i	$0.393404\pi$
$822$	0	0
$823$	−29.0127	−1.01132	−0.505659	−	0.862733i	$-0.668751\pi$
$823$	−29.0127	−1.01132	−0.505659	+	0.862733i	$0.668751\pi$
$824$	14.6596	0.510692
$825$	0	0
$826$	1.81202	0.0630481
$827$	23.6817	0.823494	0.411747	−	0.911298i	$-0.364919\pi$
$827$	23.6817	0.823494	0.411747	+	0.911298i	$0.364919\pi$
$828$	0	0
$829$	−21.2581	−0.738325	−0.369163	−	0.929365i	$-0.620355\pi$
$829$	−21.2581	−0.738325	−0.369163	+	0.929365i	$0.620355\pi$
$830$	−6.79957	−0.236016
$831$	0	0
$832$	24.3185	0.843092
$833$	−47.9926	−1.66285
$834$	0	0
$835$	−9.60152	−0.332274
$836$	18.9132	0.654127
$837$	0	0
$838$	−1.64895	−0.0569620
$839$	36.4951	1.25995	0.629975	−	0.776615i	$-0.283065\pi$
$839$	36.4951	1.25995	0.629975	+	0.776615i	$0.283065\pi$
$840$	0	0
$841$	−19.8367	−0.684025
$842$	5.74646	0.198036
$843$	0	0
$844$	43.7033	1.50433
$845$	6.79342	0.233701
$846$	0	0
$847$	−3.62975	−0.124720
$848$	20.7908	0.713957
$849$	0	0
$850$	−1.81386	−0.0622150
$851$	60.3037	2.06718
$852$	0	0
$853$	6.15365	0.210697	0.105349	−	0.994435i	$-0.466404\pi$
$853$	6.15365	0.210697	0.105349	+	0.994435i	$0.466404\pi$
$854$	1.40119	0.0479477
$855$	0	0
$856$	10.0166	0.342359
$857$	28.1278	0.960829	0.480414	−	0.877042i	$-0.340486\pi$
$857$	28.1278	0.960829	0.480414	+	0.877042i	$0.340486\pi$
$858$	0	0
$859$	−4.49335	−0.153311	−0.0766555	−	0.997058i	$-0.524424\pi$
$859$	−4.49335	−0.153311	−0.0766555	+	0.997058i	$0.524424\pi$
$860$	24.5252	0.836303
$861$	0	0
$862$	−0.160996	−0.00548356
$863$	−30.4584	−1.03681	−0.518407	−	0.855134i	$-0.673475\pi$
$863$	−30.4584	−1.03681	−0.518407	+	0.855134i	$0.673475\pi$
$864$	0	0
$865$	−17.2713	−0.587243
$866$	8.78372	0.298483
$867$	0	0
$868$	0	0
$869$	38.7989	1.31616
$870$	0	0
$871$	40.5154	1.37281
$872$	5.60815	0.189916
$873$	0	0
$874$	7.61995	0.257749
$875$	47.5090	1.60610
$876$	0	0
$877$	28.2703	0.954622	0.477311	−	0.878735i	$-0.341612\pi$
$877$	28.2703	0.954622	0.477311	+	0.878735i	$0.341612\pi$
$878$	−1.20625	−0.0407089
$879$	0	0
$880$	−23.7874	−0.801872
$881$	2.63632	0.0888200	0.0444100	−	0.999013i	$-0.485859\pi$
$881$	2.63632	0.0888200	0.0444100	+	0.999013i	$0.485859\pi$
$882$	0	0
$883$	−7.09201	−0.238665	−0.119333	−	0.992854i	$-0.538075\pi$
$883$	−7.09201	−0.238665	−0.119333	+	0.992854i	$0.538075\pi$
$884$	−43.5267	−1.46396
$885$	0	0
$886$	−4.00278	−0.134476
$887$	28.2230	0.947637	0.473819	−	0.880622i	$-0.342875\pi$
$887$	28.2230	0.947637	0.473819	+	0.880622i	$0.342875\pi$
$888$	0	0
$889$	29.6217	0.993479
$890$	2.57763	0.0864024
$891$	0	0
$892$	−18.6310	−0.623811
$893$	20.8935	0.699175
$894$	0	0
$895$	−17.1510	−0.573296
$896$	−33.2106	−1.10949
$897$	0	0
$898$	−3.13754	−0.104701
$899$	0	0
$900$	0	0
$901$	−33.4373	−1.11396
$902$	−4.17496	−0.139011
$903$	0	0
$904$	−22.2259	−0.739222
$905$	−9.82516	−0.326599
$906$	0	0
$907$	21.1065	0.700828	0.350414	−	0.936595i	$-0.386041\pi$
$907$	21.1065	0.700828	0.350414	+	0.936595i	$0.386041\pi$
$908$	−38.3069	−1.27126
$909$	0	0
$910$	−9.28972	−0.307951
$911$	17.9317	0.594104	0.297052	−	0.954861i	$-0.403997\pi$
$911$	17.9317	0.594104	0.297052	+	0.954861i	$0.403997\pi$
$912$	0	0
$913$	−40.4106	−1.33740
$914$	−0.309623	−0.0102414
$915$	0	0
$916$	−10.7757	−0.356038
$917$	1.65197	0.0545529
$918$	0	0
$919$	30.0004	0.989623	0.494811	−	0.869000i	$-0.335237\pi$
$919$	30.0004	0.989623	0.494811	+	0.869000i	$0.335237\pi$
$920$	−20.5730	−0.678272
$921$	0	0
$922$	−5.00366	−0.164787
$923$	56.5075	1.85997
$924$	0	0
$925$	7.32499	0.240844
$926$	5.31487	0.174658
$927$	0	0
$928$	10.0750	0.330728
$929$	−45.7734	−1.50178	−0.750888	−	0.660430i	$-0.770374\pi$
$929$	−45.7734	−1.50178	−0.750888	+	0.660430i	$0.770374\pi$
$930$	0	0
$931$	24.4898	0.802621
$932$	−32.3317	−1.05906
$933$	0	0
$934$	5.08123	0.166263
$935$	38.2568	1.25113
$936$	0	0
$937$	34.3560	1.12236	0.561181	−	0.827693i	$-0.310347\pi$
$937$	34.3560	1.12236	0.561181	+	0.827693i	$0.310347\pi$
$938$	−11.5918	−0.378487
$939$	0	0
$940$	−27.5815	−0.899610
$941$	1.41320	0.0460689	0.0230345	−	0.999735i	$-0.492667\pi$
$941$	1.41320	0.0460689	0.0230345	+	0.999735i	$0.492667\pi$
$942$	0	0
$943$	37.2015	1.21145
$944$	5.44889	0.177346
$945$	0	0
$946$	−6.59032	−0.214270
$947$	−5.03874	−0.163737	−0.0818685	−	0.996643i	$-0.526089\pi$
$947$	−5.03874	−0.163737	−0.0818685	+	0.996643i	$0.526089\pi$
$948$	0	0
$949$	−7.61972	−0.247347
$950$	0.925582	0.0300299
$951$	0	0
$952$	25.4699	0.825484
$953$	1.65808	0.0537106	0.0268553	−	0.999639i	$-0.491451\pi$
$953$	1.65808	0.0537106	0.0268553	+	0.999639i	$0.491451\pi$
$954$	0	0
$955$	44.3174	1.43408
$956$	−3.78231	−0.122328
$957$	0	0
$958$	9.22094	0.297915
$959$	27.4742	0.887187
$960$	0	0
$961$	0	0
$962$	−7.94762	−0.256242
$963$	0	0
$964$	47.3511	1.52508
$965$	−10.8197	−0.348297
$966$	0	0
$967$	−31.6889	−1.01905	−0.509524	−	0.860457i	$-0.670178\pi$
$967$	−31.6889	−1.01905	−0.509524	+	0.860457i	$0.670178\pi$
$968$	1.05941	0.0340509
$969$	0	0
$970$	−5.54206	−0.177945
$971$	23.3669	0.749879	0.374939	−	0.927049i	$-0.377663\pi$
$971$	23.3669	0.749879	0.374939	+	0.927049i	$0.377663\pi$
$972$	0	0
$973$	−42.3386	−1.35731
$974$	−4.69686	−0.150497
$975$	0	0
$976$	4.21349	0.134871
$977$	−11.0820	−0.354545	−0.177272	−	0.984162i	$-0.556727\pi$
$977$	−11.0820	−0.354545	−0.177272	+	0.984162i	$0.556727\pi$
$978$	0	0
$979$	15.3192	0.489602
$980$	−32.3290	−1.03271
$981$	0	0
$982$	−2.31394	−0.0738408
$983$	−53.6801	−1.71213	−0.856065	−	0.516868i	$-0.827098\pi$
$983$	−53.6801	−1.71213	−0.856065	+	0.516868i	$0.827098\pi$
$984$	0	0
$985$	52.8811	1.68493
$986$	−4.99531	−0.159083
$987$	0	0
$988$	22.2109	0.706623
$989$	58.7239	1.86731
$990$	0	0
$991$	−19.6272	−0.623479	−0.311739	−	0.950168i	$-0.600912\pi$
$991$	−19.6272	−0.623479	−0.311739	+	0.950168i	$0.600912\pi$
$992$	0	0
$993$	0	0
$994$	−16.1674	−0.512797
$995$	−18.1107	−0.574147
$996$	0	0
$997$	−20.3322	−0.643928	−0.321964	−	0.946752i	$-0.604343\pi$
$997$	−20.3322	−0.643928	−0.321964	+	0.946752i	$0.604343\pi$
$998$	1.93059	0.0611119
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bv.1.12		24
3.2	odd	2		2883.2.a.u.1.13	✓	24
31.30	odd	2		8649.2.a.bu.1.12		24
93.92	even	2		2883.2.a.v.1.13	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2883.2.a.u.1.13	✓	24	3.2	odd	2
2883.2.a.v.1.13	yes	24	93.92	even	2
8649.2.a.bu.1.12		24	31.30	odd	2
8649.2.a.bv.1.12		24	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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-0.294138 q^{2} -1.91348 q^{4} +1.97505 q^{5} -3.94391 q^{7} +1.15111 q^{8} +O(q^{10})\)	\(q-0.294138 q^{2} -1.91348 q^{4} +1.97505 q^{5} -3.94391 q^{7} +1.15111 q^{8} -0.580938 q^{10} -3.45259 q^{11} -4.05458 q^{13} +1.16005 q^{14} +3.48838 q^{16} -5.61029 q^{17} +2.86283 q^{19} -3.77922 q^{20} +1.01554 q^{22} -9.04908 q^{23} -1.09918 q^{25} +1.19261 q^{26} +7.54659 q^{28} -3.02709 q^{29} -3.32828 q^{32} +1.65020 q^{34} -7.78941 q^{35} -6.66407 q^{37} -0.842069 q^{38} +2.27349 q^{40} -4.11108 q^{41} -6.48949 q^{43} +6.60646 q^{44} +2.66168 q^{46} +7.29819 q^{47} +8.55439 q^{49} +0.323310 q^{50} +7.75837 q^{52} +5.96000 q^{53} -6.81903 q^{55} -4.53985 q^{56} +0.890383 q^{58} +1.56201 q^{59} +1.20786 q^{61} -5.99779 q^{64} -8.00800 q^{65} -9.99250 q^{67} +10.7352 q^{68} +2.29117 q^{70} -13.9367 q^{71} +1.87929 q^{73} +1.96016 q^{74} -5.47798 q^{76} +13.6167 q^{77} -11.2376 q^{79} +6.88973 q^{80} +1.20923 q^{82} +11.7045 q^{83} -11.0806 q^{85} +1.90881 q^{86} -3.97429 q^{88} -4.43701 q^{89} +15.9909 q^{91} +17.3153 q^{92} -2.14668 q^{94} +5.65424 q^{95} +9.53985 q^{97} -2.51618 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8}+O(q^{10}) \) 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} + 16 q^{11} - 32 q^{13} + 24 q^{14} + 48 q^{16} + 32 q^{17} + 32 q^{19} + 24 q^{20} - 32 q^{22} + 32 q^{23} + 40 q^{25} + 16 q^{26} + 8 q^{28} + 48 q^{29} + 48 q^{32} + 48 q^{35} - 64 q^{37} + 24 q^{38} - 32 q^{43} + 48 q^{44} - 32 q^{46} + 48 q^{47} + 56 q^{49} + 24 q^{50} - 64 q^{52} + 80 q^{53} + 48 q^{56} - 32 q^{58} - 32 q^{61} + 56 q^{64} + 16 q^{65} - 16 q^{67} + 80 q^{68} + 8 q^{70} - 32 q^{73} + 56 q^{76} + 96 q^{77} - 32 q^{79} + 72 q^{80} + 8 q^{82} + 48 q^{83} - 96 q^{85} - 32 q^{86} - 96 q^{88} - 16 q^{89} + 32 q^{92} + 48 q^{94} + 48 q^{95} + 16 q^{97} + 24 q^{98}+O(q^{100}) \) 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 + 16 * q^11 - 32 * q^13 + 24 * q^14 + 48 * q^16 + 32 * q^17 + 32 * q^19 + 24 * q^20 - 32 * q^22 + 32 * q^23 + 40 * q^25 + 16 * q^26 + 8 * q^28 + 48 * q^29 + 48 * q^32 + 48 * q^35 - 64 * q^37 + 24 * q^38 - 32 * q^43 + 48 * q^44 - 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 - 64 * q^52 + 80 * q^53 + 48 * q^56 - 32 * q^58 - 32 * q^61 + 56 * q^64 + 16 * q^65 - 16 * q^67 + 80 * q^68 + 8 * q^70 - 32 * q^73 + 56 * q^76 + 96 * q^77 - 32 * q^79 + 72 * q^80 + 8 * q^82 + 48 * q^83 - 96 * q^85 - 32 * q^86 - 96 * q^88 - 16 * q^89 + 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * q^98

Introduction

L-functions

Modular forms

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Database

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