LMFDB - Embedded newform 7800.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.30056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 26 $ x^4 - 11*x^2 + 26
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.85431$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.09417 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.09417 q^{7} +1.00000 q^{9} +5.51003 q^{11} +1.00000 q^{13} -6.21728 q^{17} +3.09417 q^{21} +4.21728 q^{23} -1.00000 q^{27} -1.70861 q^{29} -5.51003 q^{33} -4.02893 q^{37} -1.00000 q^{39} +0.198578 q^{41} -5.70861 q^{43} -6.41586 q^{47} +2.57391 q^{49} +6.21728 q^{51} +4.02893 q^{53} -7.53896 q^{59} +11.9259 q^{61} -3.09417 q^{63} +4.83172 q^{67} -4.21728 q^{69} +7.98977 q^{71} +9.31145 q^{73} -17.0490 q^{77} +8.51139 q^{79} +1.00000 q^{81} -14.7926 q^{83} +1.70861 q^{87} -9.40699 q^{89} -3.09417 q^{91} +14.3026 q^{97} +5.51003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 7 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 7 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 7 q^{7} + 4 q^{9} + q^{11} + 4 q^{13} - 3 q^{17} + 7 q^{21} - 5 q^{23} - 4 q^{27} + 8 q^{29} - q^{33} - 5 q^{37} - 4 q^{39} + 7 q^{41} - 8 q^{43} - 10 q^{47} + 9 q^{49} + 3 q^{51} + 5 q^{53} + 2 q^{59} + 11 q^{61} - 7 q^{63} - 12 q^{67} + 5 q^{69} + 15 q^{71} + 10 q^{73} - 15 q^{77} - q^{79} + 4 q^{81} - 22 q^{83} - 8 q^{87} + 9 q^{89} - 7 q^{91} - q^{97} + q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 7 * q^7 + 4 * q^9 + q^11 + 4 * q^13 - 3 * q^17 + 7 * q^21 - 5 * q^23 - 4 * q^27 + 8 * q^29 - q^33 - 5 * q^37 - 4 * q^39 + 7 * q^41 - 8 * q^43 - 10 * q^47 + 9 * q^49 + 3 * q^51 + 5 * q^53 + 2 * q^59 + 11 * q^61 - 7 * q^63 - 12 * q^67 + 5 * q^69 + 15 * q^71 + 10 * q^73 - 15 * q^77 - q^79 + 4 * q^81 - 22 * q^83 - 8 * q^87 + 9 * q^89 - 7 * q^91 - q^97 + 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.09417	−1.16949	−0.584744	−	0.811218i	$-0.698805\pi$
$7$	−3.09417	−1.16949	−0.584744	+	0.811218i	$0.698805\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.51003	1.66134	0.830669	−	0.556767i	$-0.187958\pi$
$11$	5.51003	1.66134	0.830669	+	0.556767i	$0.187958\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.21728	−1.50791	−0.753956	−	0.656925i	$-0.771857\pi$
$17$	−6.21728	−1.50791	−0.753956	+	0.656925i	$0.771857\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	3.09417	0.675204
$22$	0	0
$23$	4.21728	0.879364	0.439682	−	0.898154i	$-0.355091\pi$
$23$	4.21728	0.879364	0.439682	+	0.898154i	$0.355091\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.70861	−0.317281	−0.158640	−	0.987336i	$-0.550711\pi$
$29$	−1.70861	−0.317281	−0.158640	+	0.987336i	$0.550711\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−5.51003	−0.959173
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.02893	−0.662352	−0.331176	−	0.943569i	$-0.607445\pi$
$37$	−4.02893	−0.662352	−0.331176	+	0.943569i	$0.607445\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	0.198578	0.0310127	0.0155064	−	0.999880i	$-0.495064\pi$
$41$	0.198578	0.0310127	0.0155064	+	0.999880i	$0.495064\pi$
$42$	0	0
$43$	−5.70861	−0.870555	−0.435277	−	0.900296i	$-0.643350\pi$
$43$	−5.70861	−0.870555	−0.435277	+	0.900296i	$0.643350\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.41586	−0.935849	−0.467925	−	0.883768i	$-0.654998\pi$
$47$	−6.41586	−0.935849	−0.467925	+	0.883768i	$0.654998\pi$
$48$	0	0
$49$	2.57391	0.367702
$50$	0	0
$51$	6.21728	0.870593
$52$	0	0
$53$	4.02893	0.553416	0.276708	−	0.960954i	$-0.410756\pi$
$53$	4.02893	0.553416	0.276708	+	0.960954i	$0.410756\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.53896	−0.981489	−0.490745	−	0.871303i	$-0.663275\pi$
$59$	−7.53896	−0.981489	−0.490745	+	0.871303i	$0.663275\pi$
$60$	0	0
$61$	11.9259	1.52695	0.763477	−	0.645835i	$-0.223491\pi$
$61$	11.9259	1.52695	0.763477	+	0.645835i	$0.223491\pi$
$62$	0	0
$63$	−3.09417	−0.389829
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.83172	0.590288	0.295144	−	0.955453i	$-0.404632\pi$
$67$	4.83172	0.590288	0.295144	+	0.955453i	$0.404632\pi$
$68$	0	0
$69$	−4.21728	−0.507701
$70$	0	0
$71$	7.98977	0.948211	0.474106	−	0.880468i	$-0.342772\pi$
$71$	7.98977	0.948211	0.474106	+	0.880468i	$0.342772\pi$
$72$	0	0
$73$	9.31145	1.08982	0.544912	−	0.838494i	$-0.316563\pi$
$73$	9.31145	1.08982	0.544912	+	0.838494i	$0.316563\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−17.0490	−1.94291
$78$	0	0
$79$	8.51139	0.957607	0.478803	−	0.877922i	$-0.341071\pi$
$79$	8.51139	0.957607	0.478803	+	0.877922i	$0.341071\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.7926	−1.62369	−0.811847	−	0.583871i	$-0.801538\pi$
$83$	−14.7926	−1.62369	−0.811847	+	0.583871i	$0.801538\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.70861	0.183182
$88$	0	0
$89$	−9.40699	−0.997139	−0.498569	−	0.866850i	$-0.666141\pi$
$89$	−9.40699	−0.997139	−0.498569	+	0.866850i	$0.666141\pi$
$90$	0	0
$91$	−3.09417	−0.324358
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.3026	1.45221	0.726104	−	0.687585i	$-0.241329\pi$
$97$	14.3026	1.45221	0.726104	+	0.687585i	$0.241329\pi$
$98$	0	0
$99$	5.51003	0.553779
$100$	0	0
$101$	−6.72595	−0.669257	−0.334628	−	0.942350i	$-0.608611\pi$
$101$	−6.72595	−0.669257	−0.334628	+	0.942350i	$0.608611\pi$
$102$	0	0
$103$	−17.0779	−1.68274	−0.841369	−	0.540461i	$-0.818250\pi$
$103$	−17.0779	−1.68274	−0.841369	+	0.540461i	$0.818250\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.63450	0.738055	0.369027	−	0.929418i	$-0.379691\pi$
$107$	7.63450	0.738055	0.369027	+	0.929418i	$0.379691\pi$
$108$	0	0
$109$	−5.02006	−0.480835	−0.240417	−	0.970670i	$-0.577284\pi$
$109$	−5.02006	−0.480835	−0.240417	+	0.970670i	$0.577284\pi$
$110$	0	0
$111$	4.02893	0.382409
$112$	0	0
$113$	16.6229	1.56375	0.781876	−	0.623434i	$-0.214263\pi$
$113$	16.6229	1.56375	0.781876	+	0.623434i	$0.214263\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	19.2373	1.76348
$120$	0	0
$121$	19.3604	1.76004
$122$	0	0
$123$	−0.198578	−0.0179052
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.87689	−0.255283	−0.127642	−	0.991820i	$-0.540741\pi$
$127$	−2.87689	−0.255283	−0.127642	+	0.991820i	$0.540741\pi$
$128$	0	0
$129$	5.70861	0.502615
$130$	0	0
$131$	−16.5199	−1.44335	−0.721674	−	0.692233i	$-0.756627\pi$
$131$	−16.5199	−1.44335	−0.721674	+	0.692233i	$0.756627\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.8504	1.09789	0.548943	−	0.835860i	$-0.315031\pi$
$137$	12.8504	1.09789	0.548943	+	0.835860i	$0.315031\pi$
$138$	0	0
$139$	−17.0490	−1.44608	−0.723038	−	0.690808i	$-0.757255\pi$
$139$	−17.0490	−1.44608	−0.723038	+	0.690808i	$0.757255\pi$
$140$	0	0
$141$	6.41586	0.540313
$142$	0	0
$143$	5.51003	0.460772
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2.57391	−0.212293
$148$	0	0
$149$	19.5679	1.60306	0.801532	−	0.597952i	$-0.204019\pi$
$149$	19.5679	1.60306	0.801532	+	0.597952i	$0.204019\pi$
$150$	0	0
$151$	−9.22887	−0.751035	−0.375518	−	0.926815i	$-0.622535\pi$
$151$	−9.22887	−0.751035	−0.375518	+	0.926815i	$0.622535\pi$
$152$	0	0
$153$	−6.21728	−0.502637
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.89423	−0.310794	−0.155397	−	0.987852i	$-0.549666\pi$
$157$	−3.89423	−0.310794	−0.155397	+	0.987852i	$0.549666\pi$
$158$	0	0
$159$	−4.02893	−0.319515
$160$	0	0
$161$	−13.0490	−1.02840
$162$	0	0
$163$	−0.237741	−0.0186213	−0.00931067	−	0.999957i	$-0.502964\pi$
$163$	−0.237741	−0.0186213	−0.00931067	+	0.999957i	$0.502964\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.8304	−0.915460	−0.457730	−	0.889091i	$-0.651337\pi$
$167$	−11.8304	−0.915460	−0.457730	+	0.889091i	$0.651337\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.6634	0.886754	0.443377	−	0.896335i	$-0.353780\pi$
$173$	11.6634	0.886754	0.443377	+	0.896335i	$0.353780\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.53896	0.566663
$178$	0	0
$179$	−12.5403	−0.937308	−0.468654	−	0.883382i	$-0.655261\pi$
$179$	−12.5403	−0.937308	−0.468654	+	0.883382i	$0.655261\pi$
$180$	0	0
$181$	−5.44343	−0.404607	−0.202303	−	0.979323i	$-0.564843\pi$
$181$	−5.44343	−0.404607	−0.202303	+	0.979323i	$0.564843\pi$
$182$	0	0
$183$	−11.9259	−0.881587
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−34.2574	−2.50515
$188$	0	0
$189$	3.09417	0.225068
$190$	0	0
$191$	5.22887	0.378348	0.189174	−	0.981944i	$-0.439419\pi$
$191$	5.22887	0.378348	0.189174	+	0.981944i	$0.439419\pi$
$192$	0	0
$193$	17.5287	1.26175	0.630873	−	0.775886i	$-0.282697\pi$
$193$	17.5287	1.26175	0.630873	+	0.775886i	$0.282697\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−22.4764	−1.60138	−0.800690	−	0.599079i	$-0.795534\pi$
$197$	−22.4764	−1.60138	−0.800690	+	0.599079i	$0.795534\pi$
$198$	0	0
$199$	−16.9143	−1.19902	−0.599511	−	0.800366i	$-0.704638\pi$
$199$	−16.9143	−1.19902	−0.599511	+	0.800366i	$0.704638\pi$
$200$	0	0
$201$	−4.83172	−0.340803
$202$	0	0
$203$	5.28674	0.371056
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.21728	0.293121
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−11.8391	−0.815037	−0.407518	−	0.913197i	$-0.633606\pi$
$211$	−11.8391	−0.815037	−0.407518	+	0.913197i	$0.633606\pi$
$212$	0	0
$213$	−7.98977	−0.547450
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−9.31145	−0.629210
$220$	0	0
$221$	−6.21728	−0.418219
$222$	0	0
$223$	4.48246	0.300168	0.150084	−	0.988673i	$-0.452046\pi$
$223$	4.48246	0.300168	0.150084	+	0.988673i	$0.452046\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.35799	−0.421995	−0.210997	−	0.977487i	$-0.567671\pi$
$227$	−6.35799	−0.421995	−0.210997	+	0.977487i	$0.567671\pi$
$228$	0	0
$229$	6.45502	0.426560	0.213280	−	0.976991i	$-0.431585\pi$
$229$	6.45502	0.426560	0.213280	+	0.976991i	$0.431585\pi$
$230$	0	0
$231$	17.0490	1.12174
$232$	0	0
$233$	−6.40290	−0.419468	−0.209734	−	0.977758i	$-0.567260\pi$
$233$	−6.40290	−0.419468	−0.209734	+	0.977758i	$0.567260\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.51139	−0.552874
$238$	0	0
$239$	−1.55793	−0.100774	−0.0503872	−	0.998730i	$-0.516046\pi$
$239$	−1.55793	−0.100774	−0.0503872	+	0.998730i	$0.516046\pi$
$240$	0	0
$241$	−14.1911	−0.914127	−0.457064	−	0.889434i	$-0.651099\pi$
$241$	−14.1911	−0.914127	−0.457064	+	0.889434i	$0.651099\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	14.7926	0.937440
$250$	0	0
$251$	22.7082	1.43333	0.716665	−	0.697418i	$-0.245668\pi$
$251$	22.7082	1.43333	0.716665	+	0.697418i	$0.245668\pi$
$252$	0	0
$253$	23.2373	1.46092
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.4172	−1.33597	−0.667985	−	0.744175i	$-0.732843\pi$
$257$	−21.4172	−1.33597	−0.667985	+	0.744175i	$0.732843\pi$
$258$	0	0
$259$	12.4662	0.774613
$260$	0	0
$261$	−1.70861	−0.105760
$262$	0	0
$263$	−26.4951	−1.63376	−0.816880	−	0.576807i	$-0.804298\pi$
$263$	−26.4951	−1.63376	−0.816880	+	0.576807i	$0.804298\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.40699	0.575698
$268$	0	0
$269$	14.7259	0.897857	0.448928	−	0.893568i	$-0.351806\pi$
$269$	14.7259	0.897857	0.448928	+	0.893568i	$0.351806\pi$
$270$	0	0
$271$	−25.8491	−1.57022	−0.785109	−	0.619357i	$-0.787393\pi$
$271$	−25.8491	−1.57022	−0.785109	+	0.619357i	$0.787393\pi$
$272$	0	0
$273$	3.09417	0.187268
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.1227	1.20906	0.604528	−	0.796584i	$-0.293362\pi$
$277$	20.1227	1.20906	0.604528	+	0.796584i	$0.293362\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.3954	−0.978067	−0.489034	−	0.872265i	$-0.662650\pi$
$281$	−16.3954	−0.978067	−0.489034	+	0.872265i	$0.662650\pi$
$282$	0	0
$283$	−14.5650	−0.865802	−0.432901	−	0.901441i	$-0.642510\pi$
$283$	−14.5650	−0.865802	−0.432901	+	0.901441i	$0.642510\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.614436	−0.0362690
$288$	0	0
$289$	21.6546	1.27380
$290$	0	0
$291$	−14.3026	−0.838432
$292$	0	0
$293$	−26.6995	−1.55980	−0.779900	−	0.625904i	$-0.784730\pi$
$293$	−26.6995	−1.55980	−0.779900	+	0.625904i	$0.784730\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.51003	−0.319724
$298$	0	0
$299$	4.21728	0.243892
$300$	0	0
$301$	17.6634	1.01810
$302$	0	0
$303$	6.72595	0.386396
$304$	0	0
$305$	0	0
$306$	0	0
$307$	11.4230	0.651943	0.325972	−	0.945380i	$-0.394309\pi$
$307$	11.4230	0.651943	0.325972	+	0.945380i	$0.394309\pi$
$308$	0	0
$309$	17.0779	0.971529
$310$	0	0
$311$	19.4546	1.10317	0.551585	−	0.834119i	$-0.314023\pi$
$311$	19.4546	1.10317	0.551585	+	0.834119i	$0.314023\pi$
$312$	0	0
$313$	−32.5805	−1.84156	−0.920778	−	0.390087i	$-0.872445\pi$
$313$	−32.5805	−1.84156	−0.920778	+	0.390087i	$0.872445\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.3781	−1.76237	−0.881184	−	0.472774i	$-0.843253\pi$
$317$	−31.3781	−1.76237	−0.881184	+	0.472774i	$0.843253\pi$
$318$	0	0
$319$	−9.41450	−0.527111
$320$	0	0
$321$	−7.63450	−0.426116
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.02006	0.277610
$328$	0	0
$329$	19.8518	1.09446
$330$	0	0
$331$	−30.4951	−1.67616	−0.838082	−	0.545544i	$-0.816323\pi$
$331$	−30.4951	−1.67616	−0.838082	+	0.545544i	$0.816323\pi$
$332$	0	0
$333$	−4.02893	−0.220784
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−15.8765	−0.864848	−0.432424	−	0.901670i	$-0.642342\pi$
$337$	−15.8765	−0.864848	−0.432424	+	0.901670i	$0.642342\pi$
$338$	0	0
$339$	−16.6229	−0.902832
$340$	0	0
$341$	0	0
$342$	0	0
$343$	13.6951	0.739465
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.1069	−1.13308	−0.566538	−	0.824036i	$-0.691717\pi$
$347$	−21.1069	−1.13308	−0.566538	+	0.824036i	$0.691717\pi$
$348$	0	0
$349$	4.82899	0.258490	0.129245	−	0.991613i	$-0.458745\pi$
$349$	4.82899	0.258490	0.129245	+	0.991613i	$0.458745\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−19.4359	−1.03447	−0.517235	−	0.855844i	$-0.673039\pi$
$353$	−19.4359	−1.03447	−0.517235	+	0.855844i	$0.673039\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−19.2373	−1.01815
$358$	0	0
$359$	−27.2271	−1.43699	−0.718496	−	0.695531i	$-0.755169\pi$
$359$	−27.2271	−1.43699	−0.718496	+	0.695531i	$0.755169\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−19.3604	−1.01616
$364$	0	0
$365$	0	0
$366$	0	0
$367$	15.7008	0.819577	0.409788	−	0.912181i	$-0.365603\pi$
$367$	15.7008	0.819577	0.409788	+	0.912181i	$0.365603\pi$
$368$	0	0
$369$	0.198578	0.0103376
$370$	0	0
$371$	−12.4662	−0.647214
$372$	0	0
$373$	−3.86952	−0.200356	−0.100178	−	0.994970i	$-0.531941\pi$
$373$	−3.86952	−0.200356	−0.100178	+	0.994970i	$0.531941\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.70861	−0.0879979
$378$	0	0
$379$	−7.30149	−0.375052	−0.187526	−	0.982260i	$-0.560047\pi$
$379$	−7.30149	−0.375052	−0.187526	+	0.982260i	$0.560047\pi$
$380$	0	0
$381$	2.87689	0.147388
$382$	0	0
$383$	37.2094	1.90131	0.950655	−	0.310250i	$-0.100413\pi$
$383$	37.2094	1.90131	0.950655	+	0.310250i	$0.100413\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.70861	−0.290185
$388$	0	0
$389$	30.1227	1.52728	0.763641	−	0.645641i	$-0.223410\pi$
$389$	30.1227	1.52728	0.763641	+	0.645641i	$0.223410\pi$
$390$	0	0
$391$	−26.2200	−1.32600
$392$	0	0
$393$	16.5199	0.833317
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.25780	0.0631274	0.0315637	−	0.999502i	$-0.489951\pi$
$397$	1.25780	0.0631274	0.0315637	+	0.999502i	$0.489951\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.6443	1.03093	0.515464	−	0.856911i	$-0.327619\pi$
$401$	20.6443	1.03093	0.515464	+	0.856911i	$0.327619\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−22.1995	−1.10039
$408$	0	0
$409$	−31.3820	−1.55174	−0.775870	−	0.630893i	$-0.782689\pi$
$409$	−31.3820	−1.55174	−0.775870	+	0.630893i	$0.782689\pi$
$410$	0	0
$411$	−12.8504	−0.633864
$412$	0	0
$413$	23.3269	1.14784
$414$	0	0
$415$	0	0
$416$	0	0
$417$	17.0490	0.834893
$418$	0	0
$419$	−19.5549	−0.955321	−0.477661	−	0.878544i	$-0.658515\pi$
$419$	−19.5549	−0.955321	−0.477661	+	0.878544i	$0.658515\pi$
$420$	0	0
$421$	−5.07793	−0.247483	−0.123742	−	0.992314i	$-0.539489\pi$
$421$	−5.07793	−0.247483	−0.123742	+	0.992314i	$0.539489\pi$
$422$	0	0
$423$	−6.41586	−0.311950
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−36.9008	−1.78575
$428$	0	0
$429$	−5.51003	−0.266027
$430$	0	0
$431$	7.52900	0.362659	0.181330	−	0.983422i	$-0.441960\pi$
$431$	7.52900	0.362659	0.181330	+	0.983422i	$0.441960\pi$
$432$	0	0
$433$	−37.4821	−1.80127	−0.900637	−	0.434573i	$-0.856899\pi$
$433$	−37.4821	−1.80127	−0.900637	+	0.434573i	$0.856899\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	13.4836	0.643535	0.321767	−	0.946819i	$-0.395723\pi$
$439$	13.4836	0.643535	0.321767	+	0.946819i	$0.395723\pi$
$440$	0	0
$441$	2.57391	0.122567
$442$	0	0
$443$	−6.61171	−0.314132	−0.157066	−	0.987588i	$-0.550204\pi$
$443$	−6.61171	−0.314132	−0.157066	+	0.987588i	$0.550204\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−19.5679	−0.925530
$448$	0	0
$449$	5.24183	0.247377	0.123689	−	0.992321i	$-0.460528\pi$
$449$	5.24183	0.247377	0.123689	+	0.992321i	$0.460528\pi$
$450$	0	0
$451$	1.09417	0.0515226
$452$	0	0
$453$	9.22887	0.433610
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.69702	−0.406829	−0.203415	−	0.979093i	$-0.565204\pi$
$457$	−8.69702	−0.406829	−0.203415	+	0.979093i	$0.565204\pi$
$458$	0	0
$459$	6.21728	0.290198
$460$	0	0
$461$	−3.11287	−0.144981	−0.0724905	−	0.997369i	$-0.523095\pi$
$461$	−3.11287	−0.144981	−0.0724905	+	0.997369i	$0.523095\pi$
$462$	0	0
$463$	−33.8328	−1.57234	−0.786172	−	0.618008i	$-0.787940\pi$
$463$	−33.8328	−1.57234	−0.786172	+	0.618008i	$0.787940\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.0112	1.29620	0.648102	−	0.761554i	$-0.275563\pi$
$467$	28.0112	1.29620	0.648102	+	0.761554i	$0.275563\pi$
$468$	0	0
$469$	−14.9502	−0.690335
$470$	0	0
$471$	3.89423	0.179437
$472$	0	0
$473$	−31.4546	−1.44629
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.02893	0.184472
$478$	0	0
$479$	0.424328	0.0193880	0.00969401	−	0.999953i	$-0.496914\pi$
$479$	0.424328	0.0193880	0.00969401	+	0.999953i	$0.496914\pi$
$480$	0	0
$481$	−4.02893	−0.183703
$482$	0	0
$483$	13.0490	0.593750
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−32.5720	−1.47598	−0.737989	−	0.674813i	$-0.764224\pi$
$487$	−32.5720	−1.47598	−0.737989	+	0.674813i	$0.764224\pi$
$488$	0	0
$489$	0.237741	0.0107510
$490$	0	0
$491$	−1.36659	−0.0616734	−0.0308367	−	0.999524i	$-0.509817\pi$
$491$	−1.36659	−0.0616734	−0.0308367	+	0.999524i	$0.509817\pi$
$492$	0	0
$493$	10.6229	0.478432
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.7217	−1.10892
$498$	0	0
$499$	17.6430	0.789808	0.394904	−	0.918722i	$-0.370778\pi$
$499$	17.6430	0.789808	0.394904	+	0.918722i	$0.370778\pi$
$500$	0	0
$501$	11.8304	0.528541
$502$	0	0
$503$	24.6202	1.09776	0.548880	−	0.835901i	$-0.315054\pi$
$503$	24.6202	1.09776	0.548880	+	0.835901i	$0.315054\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	1.41204	0.0625876	0.0312938	−	0.999510i	$-0.490037\pi$
$509$	1.41204	0.0625876	0.0312938	+	0.999510i	$0.490037\pi$
$510$	0	0
$511$	−28.8113	−1.27453
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−35.3516	−1.55476
$518$	0	0
$519$	−11.6634	−0.511968
$520$	0	0
$521$	38.5298	1.68802	0.844011	−	0.536326i	$-0.180188\pi$
$521$	38.5298	1.68802	0.844011	+	0.536326i	$0.180188\pi$
$522$	0	0
$523$	−2.23353	−0.0976653	−0.0488326	−	0.998807i	$-0.515550\pi$
$523$	−2.23353	−0.0976653	−0.0488326	+	0.998807i	$0.515550\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−5.21455	−0.226720
$530$	0	0
$531$	−7.53896	−0.327163
$532$	0	0
$533$	0.198578	0.00860139
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.5403	0.541155
$538$	0	0
$539$	14.1823	0.610876
$540$	0	0
$541$	−37.8892	−1.62898	−0.814492	−	0.580175i	$-0.802984\pi$
$541$	−37.8892	−1.62898	−0.814492	+	0.580175i	$0.802984\pi$
$542$	0	0
$543$	5.44343	0.233600
$544$	0	0
$545$	0	0
$546$	0	0
$547$	43.8645	1.87551	0.937755	−	0.347299i	$-0.112901\pi$
$547$	43.8645	1.87551	0.937755	+	0.347299i	$0.112901\pi$
$548$	0	0
$549$	11.9259	0.508985
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−26.3357	−1.11991
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13.4537	−0.570050	−0.285025	−	0.958520i	$-0.592002\pi$
$557$	−13.4537	−0.570050	−0.285025	+	0.958520i	$0.592002\pi$
$558$	0	0
$559$	−5.70861	−0.241448
$560$	0	0
$561$	34.2574	1.44635
$562$	0	0
$563$	−31.6140	−1.33237	−0.666186	−	0.745785i	$-0.732074\pi$
$563$	−31.6140	−1.33237	−0.666186	+	0.745785i	$0.732074\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.09417	−0.129943
$568$	0	0
$569$	−22.3562	−0.937222	−0.468611	−	0.883405i	$-0.655245\pi$
$569$	−22.3562	−0.937222	−0.468611	+	0.883405i	$0.655245\pi$
$570$	0	0
$571$	−12.6596	−0.529788	−0.264894	−	0.964277i	$-0.585337\pi$
$571$	−12.6596	−0.529788	−0.264894	+	0.964277i	$0.585337\pi$
$572$	0	0
$573$	−5.22887	−0.218439
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.48861	−0.311755	−0.155877	−	0.987776i	$-0.549820\pi$
$577$	−7.48861	−0.311755	−0.155877	+	0.987776i	$0.549820\pi$
$578$	0	0
$579$	−17.5287	−0.728469
$580$	0	0
$581$	45.7707	1.89889
$582$	0	0
$583$	22.1995	0.919411
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.41858	−0.182374	−0.0911872	−	0.995834i	$-0.529066\pi$
$587$	−4.41858	−0.182374	−0.0911872	+	0.995834i	$0.529066\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	22.4764	0.924557
$592$	0	0
$593$	12.5810	0.516641	0.258320	−	0.966059i	$-0.416831\pi$
$593$	12.5810	0.516641	0.258320	+	0.966059i	$0.416831\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.9143	0.692256
$598$	0	0
$599$	−2.64337	−0.108005	−0.0540025	−	0.998541i	$-0.517198\pi$
$599$	−2.64337	−0.108005	−0.0540025	+	0.998541i	$0.517198\pi$
$600$	0	0
$601$	−10.4740	−0.427243	−0.213622	−	0.976916i	$-0.568526\pi$
$601$	−10.4740	−0.427243	−0.213622	+	0.976916i	$0.568526\pi$
$602$	0	0
$603$	4.83172	0.196763
$604$	0	0
$605$	0	0
$606$	0	0
$607$	12.0197	0.487863	0.243932	−	0.969792i	$-0.421563\pi$
$607$	12.0197	0.487863	0.243932	+	0.969792i	$0.421563\pi$
$608$	0	0
$609$	−5.28674	−0.214229
$610$	0	0
$611$	−6.41586	−0.259558
$612$	0	0
$613$	34.7676	1.40425	0.702124	−	0.712054i	$-0.252235\pi$
$613$	34.7676	1.40425	0.702124	+	0.712054i	$0.252235\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.6644	1.27476	0.637380	−	0.770549i	$-0.280018\pi$
$617$	31.6644	1.27476	0.637380	+	0.770549i	$0.280018\pi$
$618$	0	0
$619$	22.6024	0.908469	0.454234	−	0.890882i	$-0.349913\pi$
$619$	22.6024	0.908469	0.454234	+	0.890882i	$0.349913\pi$
$620$	0	0
$621$	−4.21728	−0.169234
$622$	0	0
$623$	29.1069	1.16614
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	25.0490	0.998769
$630$	0	0
$631$	18.6229	0.741366	0.370683	−	0.928759i	$-0.379124\pi$
$631$	18.6229	0.741366	0.370683	+	0.928759i	$0.379124\pi$
$632$	0	0
$633$	11.8391	0.470562
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.57391	0.101982
$638$	0	0
$639$	7.98977	0.316070
$640$	0	0
$641$	−8.77658	−0.346654	−0.173327	−	0.984864i	$-0.555452\pi$
$641$	−8.77658	−0.346654	−0.173327	+	0.984864i	$0.555452\pi$
$642$	0	0
$643$	−30.6920	−1.21037	−0.605186	−	0.796084i	$-0.706901\pi$
$643$	−30.6920	−1.21037	−0.605186	+	0.796084i	$0.706901\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.2023	1.42326	0.711629	−	0.702555i	$-0.247958\pi$
$647$	36.2023	1.42326	0.711629	+	0.702555i	$0.247958\pi$
$648$	0	0
$649$	−41.5399	−1.63058
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−42.8746	−1.67781	−0.838906	−	0.544277i	$-0.816804\pi$
$653$	−42.8746	−1.67781	−0.838906	+	0.544277i	$0.816804\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.31145	0.363274
$658$	0	0
$659$	−3.72907	−0.145264	−0.0726320	−	0.997359i	$-0.523140\pi$
$659$	−3.72907	−0.145264	−0.0726320	+	0.997359i	$0.523140\pi$
$660$	0	0
$661$	−13.2057	−0.513642	−0.256821	−	0.966459i	$-0.582675\pi$
$661$	−13.2057	−0.513642	−0.256821	+	0.966459i	$0.582675\pi$
$662$	0	0
$663$	6.21728	0.241459
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.20569	−0.279005
$668$	0	0
$669$	−4.48246	−0.173302
$670$	0	0
$671$	65.7120	2.53678
$672$	0	0
$673$	−0.901611	−0.0347546	−0.0173773	−	0.999849i	$-0.505532\pi$
$673$	−0.901611	−0.0347546	−0.0173773	+	0.999849i	$0.505532\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.0057	−0.384552	−0.192276	−	0.981341i	$-0.561587\pi$
$677$	−10.0057	−0.384552	−0.192276	+	0.981341i	$0.561587\pi$
$678$	0	0
$679$	−44.2547	−1.69834
$680$	0	0
$681$	6.35799	0.243639
$682$	0	0
$683$	−16.9110	−0.647082	−0.323541	−	0.946214i	$-0.604873\pi$
$683$	−16.9110	−0.647082	−0.323541	+	0.946214i	$0.604873\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6.45502	−0.246274
$688$	0	0
$689$	4.02893	0.153490
$690$	0	0
$691$	43.1003	1.63961	0.819807	−	0.572640i	$-0.194081\pi$
$691$	43.1003	1.63961	0.819807	+	0.572640i	$0.194081\pi$
$692$	0	0
$693$	−17.0490	−0.647638
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.23462	−0.0467645
$698$	0	0
$699$	6.40290	0.242180
$700$	0	0
$701$	34.9170	1.31880	0.659399	−	0.751793i	$-0.270811\pi$
$701$	34.9170	1.31880	0.659399	+	0.751793i	$0.270811\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.8113	0.782688
$708$	0	0
$709$	20.8140	0.781685	0.390843	−	0.920457i	$-0.372184\pi$
$709$	20.8140	0.781685	0.390843	+	0.920457i	$0.372184\pi$
$710$	0	0
$711$	8.51139	0.319202
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.55793	0.0581821
$718$	0	0
$719$	10.6024	0.395404	0.197702	−	0.980262i	$-0.436652\pi$
$719$	10.6024	0.395404	0.197702	+	0.980262i	$0.436652\pi$
$720$	0	0
$721$	52.8421	1.96794
$722$	0	0
$723$	14.1911	0.527772
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.06524	−0.0395076	−0.0197538	−	0.999805i	$-0.506288\pi$
$727$	−1.06524	−0.0395076	−0.0197538	+	0.999805i	$0.506288\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	35.4920	1.31272
$732$	0	0
$733$	−29.1216	−1.07563	−0.537816	−	0.843062i	$-0.680750\pi$
$733$	−29.1216	−1.07563	−0.537816	+	0.843062i	$0.680750\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.6229	0.980667
$738$	0	0
$739$	−41.7966	−1.53751	−0.768757	−	0.639541i	$-0.779125\pi$
$739$	−41.7966	−1.53751	−0.768757	+	0.639541i	$0.779125\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.8531	1.20526	0.602632	−	0.798019i	$-0.294119\pi$
$743$	32.8531	1.20526	0.602632	+	0.798019i	$0.294119\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−14.7926	−0.541231
$748$	0	0
$749$	−23.6225	−0.863146
$750$	0	0
$751$	−20.2956	−0.740597	−0.370299	−	0.928913i	$-0.620745\pi$
$751$	−20.2956	−0.740597	−0.370299	+	0.928913i	$0.620745\pi$
$752$	0	0
$753$	−22.7082	−0.827533
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.62331	−0.0590000	−0.0295000	−	0.999565i	$-0.509392\pi$
$757$	−1.62331	−0.0590000	−0.0295000	+	0.999565i	$0.509392\pi$
$758$	0	0
$759$	−23.2373	−0.843462
$760$	0	0
$761$	−50.4410	−1.82848	−0.914242	−	0.405169i	$-0.867213\pi$
$761$	−50.4410	−1.82848	−0.914242	+	0.405169i	$0.867213\pi$
$762$	0	0
$763$	15.5329	0.562330
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.53896	−0.272216
$768$	0	0
$769$	−6.75339	−0.243533	−0.121767	−	0.992559i	$-0.538856\pi$
$769$	−6.75339	−0.243533	−0.121767	+	0.992559i	$0.538856\pi$
$770$	0	0
$771$	21.4172	0.771322
$772$	0	0
$773$	36.5660	1.31519	0.657594	−	0.753373i	$-0.271574\pi$
$773$	36.5660	1.31519	0.657594	+	0.753373i	$0.271574\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−12.4662	−0.447223
$778$	0	0
$779$	0	0
$780$	0	0
$781$	44.0239	1.57530
$782$	0	0
$783$	1.70861	0.0610607
$784$	0	0
$785$	0	0
$786$	0	0
$787$	6.21426	0.221514	0.110757	−	0.993847i	$-0.464672\pi$
$787$	6.21426	0.221514	0.110757	+	0.993847i	$0.464672\pi$
$788$	0	0
$789$	26.4951	0.943252
$790$	0	0
$791$	−51.4342	−1.82879
$792$	0	0
$793$	11.9259	0.423501
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.5788	1.40195	0.700977	−	0.713184i	$-0.252748\pi$
$797$	39.5788	1.40195	0.700977	+	0.713184i	$0.252748\pi$
$798$	0	0
$799$	39.8892	1.41118
$800$	0	0
$801$	−9.40699	−0.332380
$802$	0	0
$803$	51.3064	1.81056
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.7259	−0.518378
$808$	0	0
$809$	−3.47236	−0.122082	−0.0610408	−	0.998135i	$-0.519442\pi$
$809$	−3.47236	−0.122082	−0.0610408	+	0.998135i	$0.519442\pi$
$810$	0	0
$811$	−29.2404	−1.02677	−0.513384	−	0.858159i	$-0.671608\pi$
$811$	−29.2404	−1.02677	−0.513384	+	0.858159i	$0.671608\pi$
$812$	0	0
$813$	25.8491	0.906566
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−3.09417	−0.108119
$820$	0	0
$821$	48.3417	1.68714	0.843569	−	0.537020i	$-0.180450\pi$
$821$	48.3417	1.68714	0.843569	+	0.537020i	$0.180450\pi$
$822$	0	0
$823$	−9.33889	−0.325533	−0.162767	−	0.986665i	$-0.552042\pi$
$823$	−9.33889	−0.325533	−0.162767	+	0.986665i	$0.552042\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−45.8175	−1.59323	−0.796616	−	0.604486i	$-0.793379\pi$
$827$	−45.8175	−1.59323	−0.796616	+	0.604486i	$0.793379\pi$
$828$	0	0
$829$	−37.8819	−1.31569	−0.657847	−	0.753151i	$-0.728533\pi$
$829$	−37.8819	−1.31569	−0.657847	+	0.753151i	$0.728533\pi$
$830$	0	0
$831$	−20.1227	−0.698049
$832$	0	0
$833$	−16.0027	−0.554462
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32.6678	−1.12782	−0.563909	−	0.825837i	$-0.690703\pi$
$839$	−32.6678	−1.12782	−0.563909	+	0.825837i	$0.690703\pi$
$840$	0	0
$841$	−26.0807	−0.899333
$842$	0	0
$843$	16.3954	0.564687
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−59.9046	−2.05835
$848$	0	0
$849$	14.5650	0.499871
$850$	0	0
$851$	−16.9911	−0.582448
$852$	0	0
$853$	26.4867	0.906887	0.453443	−	0.891285i	$-0.350195\pi$
$853$	26.4867	0.906887	0.453443	+	0.891285i	$0.350195\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.1292	1.47327	0.736634	−	0.676292i	$-0.236414\pi$
$857$	43.1292	1.47327	0.736634	+	0.676292i	$0.236414\pi$
$858$	0	0
$859$	−41.1690	−1.40467	−0.702334	−	0.711848i	$-0.747859\pi$
$859$	−41.1690	−1.40467	−0.702334	+	0.711848i	$0.747859\pi$
$860$	0	0
$861$	0.614436	0.0209399
$862$	0	0
$863$	7.88822	0.268518	0.134259	−	0.990946i	$-0.457135\pi$
$863$	7.88822	0.268518	0.134259	+	0.990946i	$0.457135\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−21.6546	−0.735428
$868$	0	0
$869$	46.8980	1.59091
$870$	0	0
$871$	4.83172	0.163716
$872$	0	0
$873$	14.3026	0.484069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.03740	−0.203869	−0.101934	−	0.994791i	$-0.532503\pi$
$877$	−6.03740	−0.203869	−0.101934	+	0.994791i	$0.532503\pi$
$878$	0	0
$879$	26.6995	0.900551
$880$	0	0
$881$	−41.4546	−1.39664	−0.698321	−	0.715785i	$-0.746069\pi$
$881$	−41.4546	−1.39664	−0.698321	+	0.715785i	$0.746069\pi$
$882$	0	0
$883$	35.0575	1.17978	0.589889	−	0.807484i	$-0.299172\pi$
$883$	35.0575	1.17978	0.589889	+	0.807484i	$0.299172\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.8661	0.902075	0.451038	−	0.892505i	$-0.351054\pi$
$887$	26.8661	0.902075	0.451038	+	0.892505i	$0.351054\pi$
$888$	0	0
$889$	8.90161	0.298550
$890$	0	0
$891$	5.51003	0.184593
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.21728	−0.140811
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−25.0490	−0.834503
$902$	0	0
$903$	−17.6634	−0.587802
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−38.2137	−1.26887	−0.634433	−	0.772978i	$-0.718766\pi$
$907$	−38.2137	−1.26887	−0.634433	+	0.772978i	$0.718766\pi$
$908$	0	0
$909$	−6.72595	−0.223086
$910$	0	0
$911$	−49.8919	−1.65299	−0.826496	−	0.562942i	$-0.809669\pi$
$911$	−49.8919	−1.65299	−0.826496	+	0.562942i	$0.809669\pi$
$912$	0	0
$913$	−81.5074	−2.69750
$914$	0	0
$915$	0	0
$916$	0	0
$917$	51.1153	1.68798
$918$	0	0
$919$	5.41599	0.178657	0.0893284	−	0.996002i	$-0.471528\pi$
$919$	5.41599	0.178657	0.0893284	+	0.996002i	$0.471528\pi$
$920$	0	0
$921$	−11.4230	−0.376400
$922$	0	0
$923$	7.98977	0.262986
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−17.0779	−0.560913
$928$	0	0
$929$	−58.7311	−1.92691	−0.963453	−	0.267878i	$-0.913678\pi$
$929$	−58.7311	−1.92691	−0.963453	+	0.267878i	$0.913678\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−19.4546	−0.636916
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−6.01541	−0.196515	−0.0982574	−	0.995161i	$-0.531327\pi$
$937$	−6.01541	−0.196515	−0.0982574	+	0.995161i	$0.531327\pi$
$938$	0	0
$939$	32.5805	1.06322
$940$	0	0
$941$	40.7389	1.32805	0.664025	−	0.747710i	$-0.268847\pi$
$941$	40.7389	1.32805	0.664025	+	0.747710i	$0.268847\pi$
$942$	0	0
$943$	0.837461	0.0272715
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.69715	0.120141	0.0600705	−	0.998194i	$-0.480867\pi$
$947$	3.69715	0.120141	0.0600705	+	0.998194i	$0.480867\pi$
$948$	0	0
$949$	9.31145	0.302263
$950$	0	0
$951$	31.3781	1.01750
$952$	0	0
$953$	−1.12460	−0.0364292	−0.0182146	−	0.999834i	$-0.505798\pi$
$953$	−1.12460	−0.0364292	−0.0182146	+	0.999834i	$0.505798\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.41450	0.304327
$958$	0	0
$959$	−39.7614	−1.28396
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	7.63450	0.246018
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5.85956	−0.188431	−0.0942153	−	0.995552i	$-0.530034\pi$
$967$	−5.85956	−0.188431	−0.0942153	+	0.995552i	$0.530034\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.0737	1.06138	0.530692	−	0.847565i	$-0.321932\pi$
$971$	33.0737	1.06138	0.530692	+	0.847565i	$0.321932\pi$
$972$	0	0
$973$	52.7526	1.69117
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21.1488	−0.676610	−0.338305	−	0.941037i	$-0.609853\pi$
$977$	−21.1488	−0.676610	−0.338305	+	0.941037i	$0.609853\pi$
$978$	0	0
$979$	−51.8328	−1.65658
$980$	0	0
$981$	−5.02006	−0.160278
$982$	0	0
$983$	−32.8960	−1.04922	−0.524610	−	0.851343i	$-0.675789\pi$
$983$	−32.8960	−1.04922	−0.524610	+	0.851343i	$0.675789\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−19.8518	−0.631889
$988$	0	0
$989$	−24.0748	−0.765534
$990$	0	0
$991$	−18.6322	−0.591871	−0.295935	−	0.955208i	$-0.595631\pi$
$991$	−18.6322	−0.591871	−0.295935	+	0.955208i	$0.595631\pi$
$992$	0	0
$993$	30.4951	0.967734
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.08298	0.0976389	0.0488195	−	0.998808i	$-0.484454\pi$
$997$	3.08298	0.0976389	0.0488195	+	0.998808i	$0.484454\pi$
$998$	0	0
$999$	4.02893	0.127470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bt.1.2		4
5.2	odd	4		1560.2.l.d.1249.7	yes	8
5.3	odd	4		1560.2.l.d.1249.3	✓	8
5.4	even	2		7800.2.a.by.1.3		4
15.2	even	4		4680.2.l.g.2809.3		8
15.8	even	4		4680.2.l.g.2809.4		8
20.3	even	4		3120.2.l.n.1249.7		8
20.7	even	4		3120.2.l.n.1249.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.d.1249.3	✓	8	5.3	odd	4
1560.2.l.d.1249.7	yes	8	5.2	odd	4
3120.2.l.n.1249.3		8	20.7	even	4
3120.2.l.n.1249.7		8	20.3	even	4
4680.2.l.g.2809.3		8	15.2	even	4
4680.2.l.g.2809.4		8	15.8	even	4
7800.2.a.bt.1.2		4	1.1	even	1	trivial
7800.2.a.by.1.3		4	5.4	even	2

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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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.09417 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.09417 q^{7} +1.00000 q^{9} +5.51003 q^{11} +1.00000 q^{13} -6.21728 q^{17} +3.09417 q^{21} +4.21728 q^{23} -1.00000 q^{27} -1.70861 q^{29} -5.51003 q^{33} -4.02893 q^{37} -1.00000 q^{39} +0.198578 q^{41} -5.70861 q^{43} -6.41586 q^{47} +2.57391 q^{49} +6.21728 q^{51} +4.02893 q^{53} -7.53896 q^{59} +11.9259 q^{61} -3.09417 q^{63} +4.83172 q^{67} -4.21728 q^{69} +7.98977 q^{71} +9.31145 q^{73} -17.0490 q^{77} +8.51139 q^{79} +1.00000 q^{81} -14.7926 q^{83} +1.70861 q^{87} -9.40699 q^{89} -3.09417 q^{91} +14.3026 q^{97} +5.51003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 7 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 7 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 7 q^{7} + 4 q^{9} + q^{11} + 4 q^{13} - 3 q^{17} + 7 q^{21} - 5 q^{23} - 4 q^{27} + 8 q^{29} - q^{33} - 5 q^{37} - 4 q^{39} + 7 q^{41} - 8 q^{43} - 10 q^{47} + 9 q^{49} + 3 q^{51} + 5 q^{53} + 2 q^{59} + 11 q^{61} - 7 q^{63} - 12 q^{67} + 5 q^{69} + 15 q^{71} + 10 q^{73} - 15 q^{77} - q^{79} + 4 q^{81} - 22 q^{83} - 8 q^{87} + 9 q^{89} - 7 q^{91} - q^{97} + q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 7 * q^7 + 4 * q^9 + q^11 + 4 * q^13 - 3 * q^17 + 7 * q^21 - 5 * q^23 - 4 * q^27 + 8 * q^29 - q^33 - 5 * q^37 - 4 * q^39 + 7 * q^41 - 8 * q^43 - 10 * q^47 + 9 * q^49 + 3 * q^51 + 5 * q^53 + 2 * q^59 + 11 * q^61 - 7 * q^63 - 12 * q^67 + 5 * q^69 + 15 * q^71 + 10 * q^73 - 15 * q^77 - q^79 + 4 * q^81 - 22 * q^83 - 8 * q^87 + 9 * q^89 - 7 * q^91 - q^97 + q^99

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