LMFDB - Embedded newform 1863.2.a.f.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1863,2,Mod(1,1863)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1863, 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("1863.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1863 = 3^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1863.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:	$14.8761298966$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10x^{6} + 7x^{5} + 30x^{4} - 13x^{3} - 27x^{2} + 3x + 1 $ x^8 - x^7 - 10x^6 + 7x^5 + 30x^4 - 13x^3 - 27x^2 + 3x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 207)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.60803$ of defining polynomial
Character	$\chi$	$=$	1863.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.60803 q^{2} +4.80181 q^{4} -3.21048 q^{5} -2.97298 q^{7} +7.30721 q^{8} +O(q^{10})$	\(q+2.60803 q^{2} +4.80181 q^{4} -3.21048 q^{5} -2.97298 q^{7} +7.30721 q^{8} -8.37303 q^{10} -1.16055 q^{11} -6.43686 q^{13} -7.75362 q^{14} +9.45378 q^{16} -4.73790 q^{17} -1.71414 q^{19} -15.4161 q^{20} -3.02675 q^{22} -1.00000 q^{23} +5.30721 q^{25} -16.7875 q^{26} -14.2757 q^{28} +4.79082 q^{29} +4.73978 q^{31} +10.0413 q^{32} -12.3566 q^{34} +9.54471 q^{35} -2.59497 q^{37} -4.47052 q^{38} -23.4597 q^{40} +4.31183 q^{41} -4.45847 q^{43} -5.57274 q^{44} -2.60803 q^{46} +0.322437 q^{47} +1.83862 q^{49} +13.8414 q^{50} -30.9086 q^{52} +11.6411 q^{53} +3.72593 q^{55} -21.7242 q^{56} +12.4946 q^{58} +5.00187 q^{59} +0.863729 q^{61} +12.3615 q^{62} +7.28048 q^{64} +20.6654 q^{65} +1.19471 q^{67} -22.7505 q^{68} +24.8929 q^{70} -9.14295 q^{71} -8.15065 q^{73} -6.76776 q^{74} -8.23096 q^{76} +3.45029 q^{77} -3.80357 q^{79} -30.3512 q^{80} +11.2454 q^{82} +0.0913053 q^{83} +15.2110 q^{85} -11.6278 q^{86} -8.48038 q^{88} -9.39887 q^{89} +19.1367 q^{91} -4.80181 q^{92} +0.840925 q^{94} +5.50321 q^{95} -14.4386 q^{97} +4.79517 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 5 q^{4} - 7 q^{7} + 6 q^{8}+O(q^{10}) $ 8 * q + q^2 + 5 * q^4 - 7 * q^7 + 6 * q^8	$ 8 q + q^{2} + 5 q^{4} - 7 q^{7} + 6 q^{8} - 10 q^{10} - 3 q^{11} - 15 q^{13} - 5 q^{14} - q^{16} - 6 q^{17} - 12 q^{19} - 7 q^{20} - 17 q^{22} - 8 q^{23} - 10 q^{25} - 26 q^{26} - 20 q^{28} + 10 q^{29} - 8 q^{31} + 13 q^{32} - 15 q^{34} + 15 q^{35} - 19 q^{37} + 7 q^{38} - 30 q^{40} + q^{41} - 17 q^{43} - 6 q^{44} - q^{46} + 5 q^{47} - 15 q^{49} + 7 q^{50} - 39 q^{52} + 4 q^{53} - 10 q^{55} - 24 q^{56} - 4 q^{58} + 20 q^{59} - 16 q^{61} + 39 q^{62} - 4 q^{64} + 2 q^{65} - 7 q^{67} - 24 q^{68} + 16 q^{70} - 5 q^{71} - 48 q^{73} + 21 q^{74} - 6 q^{76} - 16 q^{77} - 12 q^{79} - 47 q^{80} - 8 q^{82} - 12 q^{83} - 6 q^{85} - 19 q^{86} - 7 q^{88} + 11 q^{89} - 2 q^{91} - 5 q^{92} - q^{94} + 4 q^{95} - 43 q^{97} + 28 q^{98}+O(q^{100}) $ 8 * q + q^2 + 5 * q^4 - 7 * q^7 + 6 * q^8 - 10 * q^10 - 3 * q^11 - 15 * q^13 - 5 * q^14 - q^16 - 6 * q^17 - 12 * q^19 - 7 * q^20 - 17 * q^22 - 8 * q^23 - 10 * q^25 - 26 * q^26 - 20 * q^28 + 10 * q^29 - 8 * q^31 + 13 * q^32 - 15 * q^34 + 15 * q^35 - 19 * q^37 + 7 * q^38 - 30 * q^40 + q^41 - 17 * q^43 - 6 * q^44 - q^46 + 5 * q^47 - 15 * q^49 + 7 * q^50 - 39 * q^52 + 4 * q^53 - 10 * q^55 - 24 * q^56 - 4 * q^58 + 20 * q^59 - 16 * q^61 + 39 * q^62 - 4 * q^64 + 2 * q^65 - 7 * q^67 - 24 * q^68 + 16 * q^70 - 5 * q^71 - 48 * q^73 + 21 * q^74 - 6 * q^76 - 16 * q^77 - 12 * q^79 - 47 * q^80 - 8 * q^82 - 12 * q^83 - 6 * q^85 - 19 * q^86 - 7 * q^88 + 11 * q^89 - 2 * q^91 - 5 * q^92 - q^94 + 4 * q^95 - 43 * q^97 + 28 * 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 $

$2$	2.60803	1.84415	0.922077	−	0.387006i	$-0.126491\pi$
$2$	2.60803	1.84415	0.922077	+	0.387006i	$0.126491\pi$
$3$	0	0
$3$	0	0
$4$	4.80181	2.40091
$5$	−3.21048	−1.43577	−0.717886	−	0.696161i	$-0.754890\pi$
$5$	−3.21048	−1.43577	−0.717886	+	0.696161i	$0.754890\pi$
$6$	0	0
$7$	−2.97298	−1.12368	−0.561841	−	0.827245i	$-0.689907\pi$
$7$	−2.97298	−1.12368	−0.561841	+	0.827245i	$0.689907\pi$
$8$	7.30721	2.58349
$9$	0	0
$10$	−8.37303	−2.64779
$11$	−1.16055	−0.349919	−0.174959	−	0.984576i	$-0.555979\pi$
$11$	−1.16055	−0.349919	−0.174959	+	0.984576i	$0.555979\pi$
$12$	0	0
$13$	−6.43686	−1.78526	−0.892632	−	0.450786i	$-0.851144\pi$
$13$	−6.43686	−1.78526	−0.892632	+	0.450786i	$0.851144\pi$
$14$	−7.75362	−2.07224
$15$	0	0
$16$	9.45378	2.36345
$17$	−4.73790	−1.14911	−0.574555	−	0.818466i	$-0.694825\pi$
$17$	−4.73790	−1.14911	−0.574555	+	0.818466i	$0.694825\pi$
$18$	0	0
$19$	−1.71414	−0.393250	−0.196625	−	0.980479i	$-0.562998\pi$
$19$	−1.71414	−0.393250	−0.196625	+	0.980479i	$0.562998\pi$
$20$	−15.4161	−3.44715
$21$	0	0
$22$	−3.02675	−0.645305
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	5.30721	1.06144
$26$	−16.7875	−3.29230
$27$	0	0
$28$	−14.2757	−2.69785
$29$	4.79082	0.889632	0.444816	−	0.895622i	$-0.353269\pi$
$29$	4.79082	0.889632	0.444816	+	0.895622i	$0.353269\pi$
$30$	0	0
$31$	4.73978	0.851290	0.425645	−	0.904890i	$-0.360047\pi$
$31$	4.73978	0.851290	0.425645	+	0.904890i	$0.360047\pi$
$32$	10.0413	1.77507
$33$	0	0
$34$	−12.3566	−2.11914
$35$	9.54471	1.61335
$36$	0	0
$37$	−2.59497	−0.426611	−0.213305	−	0.976986i	$-0.568423\pi$
$37$	−2.59497	−0.426611	−0.213305	+	0.976986i	$0.568423\pi$
$38$	−4.47052	−0.725214
$39$	0	0
$40$	−23.4597	−3.70930
$41$	4.31183	0.673395	0.336698	−	0.941613i	$-0.390690\pi$
$41$	4.31183	0.673395	0.336698	+	0.941613i	$0.390690\pi$
$42$	0	0
$43$	−4.45847	−0.679911	−0.339955	−	0.940442i	$-0.610412\pi$
$43$	−4.45847	−0.679911	−0.339955	+	0.940442i	$0.610412\pi$
$44$	−5.57274	−0.840122
$45$	0	0
$46$	−2.60803	−0.384533
$47$	0.322437	0.0470323	0.0235161	−	0.999723i	$-0.492514\pi$
$47$	0.322437	0.0470323	0.0235161	+	0.999723i	$0.492514\pi$
$48$	0	0
$49$	1.83862	0.262660
$50$	13.8414	1.95746
$51$	0	0
$52$	−30.9086	−4.28625
$53$	11.6411	1.59902	0.799512	−	0.600650i	$-0.205091\pi$
$53$	11.6411	1.59902	0.799512	+	0.600650i	$0.205091\pi$
$54$	0	0
$55$	3.72593	0.502404
$56$	−21.7242	−2.90302
$57$	0	0
$58$	12.4946	1.64062
$59$	5.00187	0.651187	0.325594	−	0.945510i	$-0.394436\pi$
$59$	5.00187	0.651187	0.325594	+	0.945510i	$0.394436\pi$
$60$	0	0
$61$	0.863729	0.110589	0.0552946	−	0.998470i	$-0.482390\pi$
$61$	0.863729	0.110589	0.0552946	+	0.998470i	$0.482390\pi$
$62$	12.3615	1.56991
$63$	0	0
$64$	7.28048	0.910059
$65$	20.6654	2.56323
$66$	0	0
$67$	1.19471	0.145957	0.0729787	−	0.997333i	$-0.476749\pi$
$67$	1.19471	0.145957	0.0729787	+	0.997333i	$0.476749\pi$
$68$	−22.7505	−2.75891
$69$	0	0
$70$	24.8929	2.97527
$71$	−9.14295	−1.08507	−0.542534	−	0.840034i	$-0.682535\pi$
$71$	−9.14295	−1.08507	−0.542534	+	0.840034i	$0.682535\pi$
$72$	0	0
$73$	−8.15065	−0.953962	−0.476981	−	0.878914i	$-0.658269\pi$
$73$	−8.15065	−0.953962	−0.476981	+	0.878914i	$0.658269\pi$
$74$	−6.76776	−0.786736
$75$	0	0
$76$	−8.23096	−0.944156
$77$	3.45029	0.393197
$78$	0	0
$79$	−3.80357	−0.427936	−0.213968	−	0.976841i	$-0.568639\pi$
$79$	−3.80357	−0.427936	−0.213968	+	0.976841i	$0.568639\pi$
$80$	−30.3512	−3.39337
$81$	0	0
$82$	11.2454	1.24184
$83$	0.0913053	0.0100221	0.00501103	−	0.999987i	$-0.498405\pi$
$83$	0.0913053	0.0100221	0.00501103	+	0.999987i	$0.498405\pi$
$84$	0	0
$85$	15.2110	1.64986
$86$	−11.6278	−1.25386
$87$	0	0
$88$	−8.48038	−0.904011
$89$	−9.39887	−0.996278	−0.498139	−	0.867097i	$-0.665983\pi$
$89$	−9.39887	−0.996278	−0.498139	+	0.867097i	$0.665983\pi$
$90$	0	0
$91$	19.1367	2.00607
$92$	−4.80181	−0.500624
$93$	0	0
$94$	0.840925	0.0867347
$95$	5.50321	0.564617
$96$	0	0
$97$	−14.4386	−1.46601	−0.733007	−	0.680221i	$-0.761884\pi$
$97$	−14.4386	−1.46601	−0.733007	+	0.680221i	$0.761884\pi$
$98$	4.79517	0.484386
$99$	0	0
$100$	25.4842	2.54842
$101$	5.62160	0.559370	0.279685	−	0.960092i	$-0.409770\pi$
$101$	5.62160	0.559370	0.279685	+	0.960092i	$0.409770\pi$
$102$	0	0
$103$	2.83887	0.279722	0.139861	−	0.990171i	$-0.455334\pi$
$103$	2.83887	0.279722	0.139861	+	0.990171i	$0.455334\pi$
$104$	−47.0355	−4.61221
$105$	0	0
$106$	30.3603	2.94885
$107$	−17.6357	−1.70491	−0.852456	−	0.522799i	$-0.824888\pi$
$107$	−17.6357	−1.70491	−0.852456	+	0.522799i	$0.824888\pi$
$108$	0	0
$109$	2.64839	0.253669	0.126835	−	0.991924i	$-0.459518\pi$
$109$	2.64839	0.253669	0.126835	+	0.991924i	$0.459518\pi$
$110$	9.71732	0.926510
$111$	0	0
$112$	−28.1059	−2.65576
$113$	15.5145	1.45948	0.729740	−	0.683725i	$-0.239641\pi$
$113$	15.5145	1.45948	0.729740	+	0.683725i	$0.239641\pi$
$114$	0	0
$115$	3.21048	0.299379
$116$	23.0046	2.13592
$117$	0	0
$118$	13.0450	1.20089
$119$	14.0857	1.29123
$120$	0	0
$121$	−9.65312	−0.877557
$122$	2.25263	0.203943
$123$	0	0
$124$	22.7595	2.04387
$125$	−0.986287	−0.0882162
$126$	0	0
$127$	−12.4652	−1.10611	−0.553054	−	0.833146i	$-0.686538\pi$
$127$	−12.4652	−1.10611	−0.553054	+	0.833146i	$0.686538\pi$
$128$	−1.09494	−0.0967803
$129$	0	0
$130$	53.8960	4.72700
$131$	−15.3849	−1.34418	−0.672092	−	0.740468i	$-0.734604\pi$
$131$	−15.3849	−1.34418	−0.672092	+	0.740468i	$0.734604\pi$
$132$	0	0
$133$	5.09610	0.441888
$134$	3.11585	0.269168
$135$	0	0
$136$	−34.6208	−2.96871
$137$	5.69225	0.486322	0.243161	−	0.969986i	$-0.421816\pi$
$137$	5.69225	0.486322	0.243161	+	0.969986i	$0.421816\pi$
$138$	0	0
$139$	−18.5839	−1.57626	−0.788132	−	0.615506i	$-0.788952\pi$
$139$	−18.5839	−1.57626	−0.788132	+	0.615506i	$0.788952\pi$
$140$	45.8319	3.87350
$141$	0	0
$142$	−23.8451	−2.00103
$143$	7.47029	0.624697
$144$	0	0
$145$	−15.3808	−1.27731
$146$	−21.2571	−1.75925
$147$	0	0
$148$	−12.4606	−1.02425
$149$	1.73071	0.141786	0.0708928	−	0.997484i	$-0.477415\pi$
$149$	1.73071	0.141786	0.0708928	+	0.997484i	$0.477415\pi$
$150$	0	0
$151$	8.05296	0.655341	0.327670	−	0.944792i	$-0.393736\pi$
$151$	8.05296	0.655341	0.327670	+	0.944792i	$0.393736\pi$
$152$	−12.5256	−1.01596
$153$	0	0
$154$	8.99846	0.725117
$155$	−15.2170	−1.22226
$156$	0	0
$157$	0.538658	0.0429896	0.0214948	−	0.999769i	$-0.493157\pi$
$157$	0.538658	0.0429896	0.0214948	+	0.999769i	$0.493157\pi$
$158$	−9.91983	−0.789179
$159$	0	0
$160$	−32.2375	−2.54860
$161$	2.97298	0.234304
$162$	0	0
$163$	3.01163	0.235889	0.117944	−	0.993020i	$-0.462370\pi$
$163$	3.01163	0.235889	0.117944	+	0.993020i	$0.462370\pi$
$164$	20.7046	1.61676
$165$	0	0
$166$	0.238127	0.0184822
$167$	9.81510	0.759515	0.379757	−	0.925086i	$-0.376007\pi$
$167$	9.81510	0.759515	0.379757	+	0.925086i	$0.376007\pi$
$168$	0	0
$169$	28.4332	2.18717
$170$	39.6706	3.04260
$171$	0	0
$172$	−21.4088	−1.63240
$173$	−15.1494	−1.15179	−0.575895	−	0.817524i	$-0.695346\pi$
$173$	−15.1494	−1.15179	−0.575895	+	0.817524i	$0.695346\pi$
$174$	0	0
$175$	−15.7782	−1.19272
$176$	−10.9716	−0.827014
$177$	0	0
$178$	−24.5125	−1.83729
$179$	7.72994	0.577763	0.288881	−	0.957365i	$-0.406717\pi$
$179$	7.72994	0.577763	0.288881	+	0.957365i	$0.406717\pi$
$180$	0	0
$181$	−3.73899	−0.277917	−0.138958	−	0.990298i	$-0.544375\pi$
$181$	−3.73899	−0.277917	−0.138958	+	0.990298i	$0.544375\pi$
$182$	49.9090	3.69950
$183$	0	0
$184$	−7.30721	−0.538695
$185$	8.33112	0.612516
$186$	0	0
$187$	5.49857	0.402095
$188$	1.54828	0.112920
$189$	0	0
$190$	14.3525	1.04124
$191$	21.0917	1.52614	0.763072	−	0.646313i	$-0.223690\pi$
$191$	21.0917	1.52614	0.763072	+	0.646313i	$0.223690\pi$
$192$	0	0
$193$	−23.0926	−1.66224	−0.831122	−	0.556090i	$-0.812301\pi$
$193$	−23.0926	−1.66224	−0.831122	+	0.556090i	$0.812301\pi$
$194$	−37.6562	−2.70356
$195$	0	0
$196$	8.82871	0.630622
$197$	12.7284	0.906862	0.453431	−	0.891291i	$-0.350200\pi$
$197$	12.7284	0.906862	0.453431	+	0.891291i	$0.350200\pi$
$198$	0	0
$199$	19.5735	1.38753	0.693763	−	0.720204i	$-0.255952\pi$
$199$	19.5735	1.38753	0.693763	+	0.720204i	$0.255952\pi$
$200$	38.7809	2.74222
$201$	0	0
$202$	14.6613	1.03156
$203$	−14.2430	−0.999664
$204$	0	0
$205$	−13.8431	−0.966842
$206$	7.40386	0.515851
$207$	0	0
$208$	−60.8527	−4.21937
$209$	1.98934	0.137606
$210$	0	0
$211$	−8.97995	−0.618205	−0.309102	−	0.951029i	$-0.600029\pi$
$211$	−8.97995	−0.618205	−0.309102	+	0.951029i	$0.600029\pi$
$212$	55.8983	3.83911
$213$	0	0
$214$	−45.9945	−3.14412
$215$	14.3139	0.976197
$216$	0	0
$217$	−14.0913	−0.956579
$218$	6.90707	0.467806
$219$	0	0
$220$	17.8912	1.20622
$221$	30.4972	2.05146
$222$	0	0
$223$	−16.6340	−1.11390	−0.556948	−	0.830547i	$-0.688028\pi$
$223$	−16.6340	−1.11390	−0.556948	+	0.830547i	$0.688028\pi$
$224$	−29.8526	−1.99461
$225$	0	0
$226$	40.4622	2.69151
$227$	7.02734	0.466421	0.233210	−	0.972426i	$-0.425077\pi$
$227$	7.02734	0.466421	0.233210	+	0.972426i	$0.425077\pi$
$228$	0	0
$229$	21.9524	1.45065	0.725326	−	0.688405i	$-0.241689\pi$
$229$	21.9524	1.45065	0.725326	+	0.688405i	$0.241689\pi$
$230$	8.37303	0.552102
$231$	0	0
$232$	35.0075	2.29835
$233$	13.7997	0.904046	0.452023	−	0.892006i	$-0.350703\pi$
$233$	13.7997	0.904046	0.452023	+	0.892006i	$0.350703\pi$
$234$	0	0
$235$	−1.03518	−0.0675276
$236$	24.0180	1.56344
$237$	0	0
$238$	36.7359	2.38123
$239$	−0.949071	−0.0613903	−0.0306952	−	0.999529i	$-0.509772\pi$
$239$	−0.949071	−0.0613903	−0.0306952	+	0.999529i	$0.509772\pi$
$240$	0	0
$241$	−12.9672	−0.835289	−0.417644	−	0.908611i	$-0.637144\pi$
$241$	−12.9672	−0.835289	−0.417644	+	0.908611i	$0.637144\pi$
$242$	−25.1756	−1.61835
$243$	0	0
$244$	4.14746	0.265514
$245$	−5.90286	−0.377120
$246$	0	0
$247$	11.0337	0.702055
$248$	34.6346	2.19930
$249$	0	0
$250$	−2.57226	−0.162684
$251$	21.4989	1.35700	0.678500	−	0.734600i	$-0.262630\pi$
$251$	21.4989	1.35700	0.678500	+	0.734600i	$0.262630\pi$
$252$	0	0
$253$	1.16055	0.0729631
$254$	−32.5096	−2.03983
$255$	0	0
$256$	−17.4166	−1.08854
$257$	2.60013	0.162192	0.0810960	−	0.996706i	$-0.474158\pi$
$257$	2.60013	0.162192	0.0810960	+	0.996706i	$0.474158\pi$
$258$	0	0
$259$	7.71481	0.479375
$260$	99.2316	6.15408
$261$	0	0
$262$	−40.1242	−2.47888
$263$	−13.0294	−0.803425	−0.401712	−	0.915766i	$-0.631585\pi$
$263$	−13.0294	−0.803425	−0.401712	+	0.915766i	$0.631585\pi$
$264$	0	0
$265$	−37.3735	−2.29584
$266$	13.2908	0.814909
$267$	0	0
$268$	5.73679	0.350430
$269$	−23.9547	−1.46054	−0.730271	−	0.683158i	$-0.760606\pi$
$269$	−23.9547	−1.46054	−0.730271	+	0.683158i	$0.760606\pi$
$270$	0	0
$271$	7.81616	0.474798	0.237399	−	0.971412i	$-0.423705\pi$
$271$	7.81616	0.474798	0.237399	+	0.971412i	$0.423705\pi$
$272$	−44.7911	−2.71586
$273$	0	0
$274$	14.8456	0.896853
$275$	−6.15928	−0.371418
$276$	0	0
$277$	23.5439	1.41462	0.707309	−	0.706904i	$-0.249909\pi$
$277$	23.5439	1.41462	0.707309	+	0.706904i	$0.249909\pi$
$278$	−48.4673	−2.90688
$279$	0	0
$280$	69.7452	4.16807
$281$	−9.88360	−0.589606	−0.294803	−	0.955558i	$-0.595254\pi$
$281$	−9.88360	−0.589606	−0.294803	+	0.955558i	$0.595254\pi$
$282$	0	0
$283$	0.400468	0.0238053	0.0119027	−	0.999929i	$-0.496211\pi$
$283$	0.400468	0.0238053	0.0119027	+	0.999929i	$0.496211\pi$
$284$	−43.9027	−2.60515
$285$	0	0
$286$	19.4827	1.15204
$287$	−12.8190	−0.756682
$288$	0	0
$289$	5.44772	0.320454
$290$	−40.1137	−2.35556
$291$	0	0
$292$	−39.1379	−2.29037
$293$	6.13408	0.358357	0.179178	−	0.983817i	$-0.442656\pi$
$293$	6.13408	0.358357	0.179178	+	0.983817i	$0.442656\pi$
$294$	0	0
$295$	−16.0584	−0.934957
$296$	−18.9620	−1.10214
$297$	0	0
$298$	4.51375	0.261475
$299$	6.43686	0.372253
$300$	0	0
$301$	13.2550	0.764003
$302$	21.0024	1.20855
$303$	0	0
$304$	−16.2051	−0.929425
$305$	−2.77299	−0.158781
$306$	0	0
$307$	14.3572	0.819408	0.409704	−	0.912218i	$-0.365632\pi$
$307$	14.3572	0.819408	0.409704	+	0.912218i	$0.365632\pi$
$308$	16.5677	0.944030
$309$	0	0
$310$	−39.6864	−2.25403
$311$	−14.8545	−0.842319	−0.421160	−	0.906987i	$-0.638377\pi$
$311$	−14.8545	−0.842319	−0.421160	+	0.906987i	$0.638377\pi$
$312$	0	0
$313$	−33.8614	−1.91396	−0.956979	−	0.290156i	$-0.906293\pi$
$313$	−33.8614	−1.91396	−0.956979	+	0.290156i	$0.906293\pi$
$314$	1.40483	0.0792794
$315$	0	0
$316$	−18.2641	−1.02743
$317$	−12.2867	−0.690092	−0.345046	−	0.938586i	$-0.612137\pi$
$317$	−12.2867	−0.690092	−0.345046	+	0.938586i	$0.612137\pi$
$318$	0	0
$319$	−5.55998	−0.311299
$320$	−23.3739	−1.30664
$321$	0	0
$322$	7.75362	0.432092
$323$	8.12141	0.451887
$324$	0	0
$325$	−34.1618	−1.89495
$326$	7.85441	0.435016
$327$	0	0
$328$	31.5075	1.73971
$329$	−0.958599	−0.0528493
$330$	0	0
$331$	15.5421	0.854271	0.427135	−	0.904188i	$-0.359523\pi$
$331$	15.5421	0.854271	0.427135	+	0.904188i	$0.359523\pi$
$332$	0.438431	0.0240620
$333$	0	0
$334$	25.5981	1.40066
$335$	−3.83561	−0.209562
$336$	0	0
$337$	−31.5900	−1.72082	−0.860409	−	0.509603i	$-0.829792\pi$
$337$	−31.5900	−1.72082	−0.860409	+	0.509603i	$0.829792\pi$
$338$	74.1545	4.03347
$339$	0	0
$340$	73.0402	3.96116
$341$	−5.50075	−0.297882
$342$	0	0
$343$	15.3447	0.828535
$344$	−32.5790	−1.75654
$345$	0	0
$346$	−39.5102	−2.12408
$347$	−22.5040	−1.20808	−0.604040	−	0.796954i	$-0.706443\pi$
$347$	−22.5040	−1.20808	−0.604040	+	0.796954i	$0.706443\pi$
$348$	0	0
$349$	23.9021	1.27945	0.639724	−	0.768604i	$-0.279049\pi$
$349$	23.9021	1.27945	0.639724	+	0.768604i	$0.279049\pi$
$350$	−41.1501	−2.19956
$351$	0	0
$352$	−11.6534	−0.621131
$353$	−2.38540	−0.126962	−0.0634809	−	0.997983i	$-0.520220\pi$
$353$	−2.38540	−0.126962	−0.0634809	+	0.997983i	$0.520220\pi$
$354$	0	0
$355$	29.3533	1.55791
$356$	−45.1316	−2.39197
$357$	0	0
$358$	20.1599	1.06548
$359$	−0.497590	−0.0262618	−0.0131309	−	0.999914i	$-0.504180\pi$
$359$	−0.497590	−0.0262618	−0.0131309	+	0.999914i	$0.504180\pi$
$360$	0	0
$361$	−16.0617	−0.845355
$362$	−9.75139	−0.512522
$363$	0	0
$364$	91.8907	4.81638
$365$	26.1675	1.36967
$366$	0	0
$367$	−20.3981	−1.06477	−0.532386	−	0.846502i	$-0.678704\pi$
$367$	−20.3981	−1.06477	−0.532386	+	0.846502i	$0.678704\pi$
$368$	−9.45378	−0.492812
$369$	0	0
$370$	21.7278	1.12957
$371$	−34.6087	−1.79679
$372$	0	0
$373$	−35.9243	−1.86009	−0.930045	−	0.367445i	$-0.880233\pi$
$373$	−35.9243	−1.86009	−0.930045	+	0.367445i	$0.880233\pi$
$374$	14.3404	0.741526
$375$	0	0
$376$	2.35611	0.121507
$377$	−30.8378	−1.58823
$378$	0	0
$379$	−37.2450	−1.91315	−0.956573	−	0.291492i	$-0.905848\pi$
$379$	−37.2450	−1.91315	−0.956573	+	0.291492i	$0.905848\pi$
$380$	26.4254	1.35559
$381$	0	0
$382$	55.0078	2.81445
$383$	−30.6688	−1.56710	−0.783552	−	0.621326i	$-0.786594\pi$
$383$	−30.6688	−1.56710	−0.783552	+	0.621326i	$0.786594\pi$
$384$	0	0
$385$	−11.0771	−0.564542
$386$	−60.2262	−3.06543
$387$	0	0
$388$	−69.3313	−3.51976
$389$	−7.15846	−0.362948	−0.181474	−	0.983396i	$-0.558087\pi$
$389$	−7.15846	−0.362948	−0.181474	+	0.983396i	$0.558087\pi$
$390$	0	0
$391$	4.73790	0.239606
$392$	13.4352	0.678579
$393$	0	0
$394$	33.1961	1.67239
$395$	12.2113	0.614418
$396$	0	0
$397$	32.3003	1.62111	0.810553	−	0.585665i	$-0.199166\pi$
$397$	32.3003	1.62111	0.810553	+	0.585665i	$0.199166\pi$
$398$	51.0481	2.55881
$399$	0	0
$400$	50.1732	2.50866
$401$	9.08784	0.453825	0.226912	−	0.973915i	$-0.427137\pi$
$401$	9.08784	0.453825	0.226912	+	0.973915i	$0.427137\pi$
$402$	0	0
$403$	−30.5093	−1.51978
$404$	26.9939	1.34299
$405$	0	0
$406$	−37.1462	−1.84353
$407$	3.01159	0.149279
$408$	0	0
$409$	−34.1156	−1.68691	−0.843453	−	0.537202i	$-0.819481\pi$
$409$	−34.1156	−1.68691	−0.843453	+	0.537202i	$0.819481\pi$
$410$	−36.1031	−1.78301
$411$	0	0
$412$	13.6317	0.671587
$413$	−14.8705	−0.731727
$414$	0	0
$415$	−0.293134	−0.0143894
$416$	−64.6345	−3.16897
$417$	0	0
$418$	5.18826	0.253766
$419$	28.9962	1.41656	0.708278	−	0.705933i	$-0.249472\pi$
$419$	28.9962	1.41656	0.708278	+	0.705933i	$0.249472\pi$
$420$	0	0
$421$	−4.54709	−0.221612	−0.110806	−	0.993842i	$-0.535343\pi$
$421$	−4.54709	−0.221612	−0.110806	+	0.993842i	$0.535343\pi$
$422$	−23.4200	−1.14007
$423$	0	0
$424$	85.0638	4.13106
$425$	−25.1450	−1.21971
$426$	0	0
$427$	−2.56785	−0.124267
$428$	−84.6835	−4.09333
$429$	0	0
$430$	37.3310	1.80026
$431$	5.64346	0.271836	0.135918	−	0.990720i	$-0.456602\pi$
$431$	5.64346	0.271836	0.135918	+	0.990720i	$0.456602\pi$
$432$	0	0
$433$	−8.32441	−0.400046	−0.200023	−	0.979791i	$-0.564102\pi$
$433$	−8.32441	−0.400046	−0.200023	+	0.979791i	$0.564102\pi$
$434$	−36.7505	−1.76408
$435$	0	0
$436$	12.7171	0.609037
$437$	1.71414	0.0819983
$438$	0	0
$439$	−11.0435	−0.527080	−0.263540	−	0.964648i	$-0.584890\pi$
$439$	−11.0435	−0.527080	−0.263540	+	0.964648i	$0.584890\pi$
$440$	27.2261	1.29795
$441$	0	0
$442$	79.5376	3.78322
$443$	17.7009	0.840997	0.420499	−	0.907293i	$-0.361855\pi$
$443$	17.7009	0.840997	0.420499	+	0.907293i	$0.361855\pi$
$444$	0	0
$445$	30.1749	1.43043
$446$	−43.3820	−2.05420
$447$	0	0
$448$	−21.6447	−1.02262
$449$	15.9726	0.753793	0.376896	−	0.926255i	$-0.376991\pi$
$449$	15.9726	0.753793	0.376896	+	0.926255i	$0.376991\pi$
$450$	0	0
$451$	−5.00410	−0.235634
$452$	74.4977	3.50408
$453$	0	0
$454$	18.3275	0.860152
$455$	−61.4380	−2.88026
$456$	0	0
$457$	12.1863	0.570052	0.285026	−	0.958520i	$-0.407998\pi$
$457$	12.1863	0.570052	0.285026	+	0.958520i	$0.407998\pi$
$458$	57.2524	2.67523
$459$	0	0
$460$	15.4161	0.718781
$461$	14.2441	0.663415	0.331708	−	0.943382i	$-0.392375\pi$
$461$	14.2441	0.663415	0.331708	+	0.943382i	$0.392375\pi$
$462$	0	0
$463$	34.8700	1.62055	0.810273	−	0.586053i	$-0.199319\pi$
$463$	34.8700	1.62055	0.810273	+	0.586053i	$0.199319\pi$
$464$	45.2913	2.10260
$465$	0	0
$466$	35.9899	1.66720
$467$	6.73985	0.311883	0.155942	−	0.987766i	$-0.450159\pi$
$467$	6.73985	0.311883	0.155942	+	0.987766i	$0.450159\pi$
$468$	0	0
$469$	−3.55186	−0.164010
$470$	−2.69978	−0.124531
$471$	0	0
$472$	36.5497	1.68233
$473$	5.17428	0.237914
$474$	0	0
$475$	−9.09728	−0.417412
$476$	67.6369	3.10013
$477$	0	0
$478$	−2.47520	−0.113213
$479$	3.92229	0.179214	0.0896070	−	0.995977i	$-0.471439\pi$
$479$	3.92229	0.179214	0.0896070	+	0.995977i	$0.471439\pi$
$480$	0	0
$481$	16.7035	0.761613
$482$	−33.8187	−1.54040
$483$	0	0
$484$	−46.3525	−2.10693
$485$	46.3548	2.10486
$486$	0	0
$487$	−5.94494	−0.269391	−0.134696	−	0.990887i	$-0.543006\pi$
$487$	−5.94494	−0.269391	−0.134696	+	0.990887i	$0.543006\pi$
$488$	6.31145	0.285706
$489$	0	0
$490$	−15.3948	−0.695468
$491$	13.6253	0.614900	0.307450	−	0.951564i	$-0.400524\pi$
$491$	13.6253	0.614900	0.307450	+	0.951564i	$0.400524\pi$
$492$	0	0
$493$	−22.6984	−1.02229
$494$	28.7761	1.29470
$495$	0	0
$496$	44.8089	2.01198
$497$	27.1818	1.21927
$498$	0	0
$499$	24.2653	1.08627	0.543133	−	0.839647i	$-0.317238\pi$
$499$	24.2653	1.08627	0.543133	+	0.839647i	$0.317238\pi$
$500$	−4.73596	−0.211799
$501$	0	0
$502$	56.0698	2.50252
$503$	−31.2131	−1.39172	−0.695861	−	0.718177i	$-0.744977\pi$
$503$	−31.2131	−1.39172	−0.695861	+	0.718177i	$0.744977\pi$
$504$	0	0
$505$	−18.0481	−0.803128
$506$	3.02675	0.134555
$507$	0	0
$508$	−59.8555	−2.65566
$509$	8.22331	0.364492	0.182246	−	0.983253i	$-0.441663\pi$
$509$	8.22331	0.364492	0.182246	+	0.983253i	$0.441663\pi$
$510$	0	0
$511$	24.2317	1.07195
$512$	−43.2331	−1.91065
$513$	0	0
$514$	6.78122	0.299107
$515$	−9.11416	−0.401618
$516$	0	0
$517$	−0.374204	−0.0164575
$518$	20.1204	0.884041
$519$	0	0
$520$	151.007	6.62208
$521$	8.60438	0.376965	0.188482	−	0.982077i	$-0.439643\pi$
$521$	8.60438	0.376965	0.188482	+	0.982077i	$0.439643\pi$
$522$	0	0
$523$	−1.51553	−0.0662696	−0.0331348	−	0.999451i	$-0.510549\pi$
$523$	−1.51553	−0.0662696	−0.0331348	+	0.999451i	$0.510549\pi$
$524$	−73.8754	−3.22726
$525$	0	0
$526$	−33.9809	−1.48164
$527$	−22.4566	−0.978226
$528$	0	0
$529$	1.00000	0.0434783
$530$	−97.4711	−4.23388
$531$	0	0
$532$	24.4705	1.06093
$533$	−27.7547	−1.20219
$534$	0	0
$535$	56.6193	2.44786
$536$	8.73002	0.377079
$537$	0	0
$538$	−62.4745	−2.69347
$539$	−2.13381	−0.0919097
$540$	0	0
$541$	−19.7783	−0.850336	−0.425168	−	0.905115i	$-0.639785\pi$
$541$	−19.7783	−0.850336	−0.425168	+	0.905115i	$0.639785\pi$
$542$	20.3848	0.875601
$543$	0	0
$544$	−47.5748	−2.03975
$545$	−8.50260	−0.364212
$546$	0	0
$547$	4.04097	0.172779	0.0863897	−	0.996261i	$-0.472467\pi$
$547$	4.04097	0.172779	0.0863897	+	0.996261i	$0.472467\pi$
$548$	27.3331	1.16761
$549$	0	0
$550$	−16.0636	−0.684953
$551$	−8.21212	−0.349848
$552$	0	0
$553$	11.3080	0.480863
$554$	61.4033	2.60878
$555$	0	0
$556$	−89.2363	−3.78446
$557$	−9.69006	−0.410581	−0.205290	−	0.978701i	$-0.565814\pi$
$557$	−9.69006	−0.410581	−0.205290	+	0.978701i	$0.565814\pi$
$558$	0	0
$559$	28.6986	1.21382
$560$	90.2336	3.81307
$561$	0	0
$562$	−25.7767	−1.08732
$563$	39.5348	1.66619	0.833096	−	0.553128i	$-0.186566\pi$
$563$	39.5348	1.66619	0.833096	+	0.553128i	$0.186566\pi$
$564$	0	0
$565$	−49.8090	−2.09548
$566$	1.04443	0.0439007
$567$	0	0
$568$	−66.8094	−2.80326
$569$	8.50299	0.356464	0.178232	−	0.983988i	$-0.442962\pi$
$569$	8.50299	0.356464	0.178232	+	0.983988i	$0.442962\pi$
$570$	0	0
$571$	−14.3216	−0.599342	−0.299671	−	0.954043i	$-0.596877\pi$
$571$	−14.3216	−0.599342	−0.299671	+	0.954043i	$0.596877\pi$
$572$	35.8710	1.49984
$573$	0	0
$574$	−33.4323	−1.39544
$575$	−5.30721	−0.221326
$576$	0	0
$577$	16.9073	0.703859	0.351929	−	0.936027i	$-0.385526\pi$
$577$	16.9073	0.703859	0.351929	+	0.936027i	$0.385526\pi$
$578$	14.2078	0.590967
$579$	0	0
$580$	−73.8559	−3.06670
$581$	−0.271449	−0.0112616
$582$	0	0
$583$	−13.5100	−0.559529
$584$	−59.5585	−2.46455
$585$	0	0
$586$	15.9979	0.660865
$587$	−10.4802	−0.432565	−0.216283	−	0.976331i	$-0.569393\pi$
$587$	−10.4802	−0.432565	−0.216283	+	0.976331i	$0.569393\pi$
$588$	0	0
$589$	−8.12463	−0.334770
$590$	−41.8808	−1.72420
$591$	0	0
$592$	−24.5323	−1.00827
$593$	−14.0368	−0.576423	−0.288212	−	0.957567i	$-0.593061\pi$
$593$	−14.0368	−0.576423	−0.288212	+	0.957567i	$0.593061\pi$
$594$	0	0
$595$	−45.2219	−1.85392
$596$	8.31056	0.340414
$597$	0	0
$598$	16.7875	0.686492
$599$	−23.5175	−0.960898	−0.480449	−	0.877023i	$-0.659526\pi$
$599$	−23.5175	−0.960898	−0.480449	+	0.877023i	$0.659526\pi$
$600$	0	0
$601$	−11.6719	−0.476109	−0.238054	−	0.971252i	$-0.576510\pi$
$601$	−11.6719	−0.476109	−0.238054	+	0.971252i	$0.576510\pi$
$602$	34.5693	1.40894
$603$	0	0
$604$	38.6688	1.57341
$605$	30.9912	1.25997
$606$	0	0
$607$	5.48939	0.222808	0.111404	−	0.993775i	$-0.464465\pi$
$607$	5.48939	0.222808	0.111404	+	0.993775i	$0.464465\pi$
$608$	−17.2122	−0.698046
$609$	0	0
$610$	−7.23203	−0.292816
$611$	−2.07548	−0.0839650
$612$	0	0
$613$	35.5993	1.43784	0.718920	−	0.695092i	$-0.244637\pi$
$613$	35.5993	1.43784	0.718920	+	0.695092i	$0.244637\pi$
$614$	37.4440	1.51112
$615$	0	0
$616$	25.2120	1.01582
$617$	−34.8741	−1.40398	−0.701989	−	0.712188i	$-0.747704\pi$
$617$	−34.8741	−1.40398	−0.701989	+	0.712188i	$0.747704\pi$
$618$	0	0
$619$	44.3472	1.78246	0.891232	−	0.453547i	$-0.149842\pi$
$619$	44.3472	1.78246	0.891232	+	0.453547i	$0.149842\pi$
$620$	−73.0692	−2.93453
$621$	0	0
$622$	−38.7409	−1.55337
$623$	27.9427	1.11950
$624$	0	0
$625$	−23.3696	−0.934783
$626$	−88.3115	−3.52964
$627$	0	0
$628$	2.58653	0.103214
$629$	12.2947	0.490223
$630$	0	0
$631$	−20.3008	−0.808164	−0.404082	−	0.914723i	$-0.632409\pi$
$631$	−20.3008	−0.808164	−0.404082	+	0.914723i	$0.632409\pi$
$632$	−27.7935	−1.10557
$633$	0	0
$634$	−32.0442	−1.27264
$635$	40.0193	1.58812
$636$	0	0
$637$	−11.8349	−0.468917
$638$	−14.5006	−0.574084
$639$	0	0
$640$	3.51530	0.138954
$641$	−4.96248	−0.196006	−0.0980031	−	0.995186i	$-0.531246\pi$
$641$	−4.96248	−0.196006	−0.0980031	+	0.995186i	$0.531246\pi$
$642$	0	0
$643$	−17.4631	−0.688679	−0.344340	−	0.938845i	$-0.611897\pi$
$643$	−17.4631	−0.688679	−0.344340	+	0.938845i	$0.611897\pi$
$644$	14.2757	0.562541
$645$	0	0
$646$	21.1809	0.833350
$647$	23.7292	0.932891	0.466446	−	0.884550i	$-0.345534\pi$
$647$	23.7292	0.932891	0.466446	+	0.884550i	$0.345534\pi$
$648$	0	0
$649$	−5.80491	−0.227863
$650$	−89.0948	−3.49459
$651$	0	0
$652$	14.4613	0.566347
$653$	−20.7573	−0.812295	−0.406147	−	0.913808i	$-0.633128\pi$
$653$	−20.7573	−0.812295	−0.406147	+	0.913808i	$0.633128\pi$
$654$	0	0
$655$	49.3930	1.92994
$656$	40.7631	1.59153
$657$	0	0
$658$	−2.50005	−0.0974622
$659$	−42.8447	−1.66899	−0.834497	−	0.551012i	$-0.814242\pi$
$659$	−42.8447	−1.66899	−0.834497	+	0.551012i	$0.814242\pi$
$660$	0	0
$661$	−28.1639	−1.09545	−0.547724	−	0.836659i	$-0.684505\pi$
$661$	−28.1639	−1.09545	−0.547724	+	0.836659i	$0.684505\pi$
$662$	40.5342	1.57541
$663$	0	0
$664$	0.667187	0.0258919
$665$	−16.3609	−0.634450
$666$	0	0
$667$	−4.79082	−0.185501
$668$	47.1303	1.82352
$669$	0	0
$670$	−10.0034	−0.386464
$671$	−1.00240	−0.0386972
$672$	0	0
$673$	39.2795	1.51412	0.757058	−	0.653348i	$-0.226636\pi$
$673$	39.2795	1.51412	0.757058	+	0.653348i	$0.226636\pi$
$674$	−82.3877	−3.17346
$675$	0	0
$676$	136.531	5.25118
$677$	−44.7748	−1.72084	−0.860418	−	0.509588i	$-0.829798\pi$
$677$	−44.7748	−1.72084	−0.860418	+	0.509588i	$0.829798\pi$
$678$	0	0
$679$	42.9256	1.64733
$680$	111.150	4.26239
$681$	0	0
$682$	−14.3461	−0.549341
$683$	−26.0050	−0.995054	−0.497527	−	0.867449i	$-0.665758\pi$
$683$	−26.0050	−0.995054	−0.497527	+	0.867449i	$0.665758\pi$
$684$	0	0
$685$	−18.2749	−0.698247
$686$	40.0194	1.52795
$687$	0	0
$688$	−42.1494	−1.60693
$689$	−74.9320	−2.85468
$690$	0	0
$691$	−19.8323	−0.754455	−0.377227	−	0.926121i	$-0.623122\pi$
$691$	−19.8323	−0.754455	−0.377227	+	0.926121i	$0.623122\pi$
$692$	−72.7448	−2.76534
$693$	0	0
$694$	−58.6912	−2.22789
$695$	59.6633	2.26316
$696$	0	0
$697$	−20.4290	−0.773805
$698$	62.3373	2.35950
$699$	0	0
$700$	−75.7641	−2.86361
$701$	13.9631	0.527381	0.263690	−	0.964607i	$-0.415060\pi$
$701$	13.9631	0.527381	0.263690	+	0.964607i	$0.415060\pi$
$702$	0	0
$703$	4.44814	0.167765
$704$	−8.44935	−0.318447
$705$	0	0
$706$	−6.22118	−0.234137
$707$	−16.7129	−0.628554
$708$	0	0
$709$	3.32048	0.124703	0.0623516	−	0.998054i	$-0.480140\pi$
$709$	3.32048	0.124703	0.0623516	+	0.998054i	$0.480140\pi$
$710$	76.5542	2.87303
$711$	0	0
$712$	−68.6795	−2.57387
$713$	−4.73978	−0.177506
$714$	0	0
$715$	−23.9833	−0.896923
$716$	37.1177	1.38715
$717$	0	0
$718$	−1.29773	−0.0484309
$719$	45.2541	1.68769	0.843847	−	0.536584i	$-0.180286\pi$
$719$	45.2541	1.68769	0.843847	+	0.536584i	$0.180286\pi$
$720$	0	0
$721$	−8.43992	−0.314319
$722$	−41.8895	−1.55896
$723$	0	0
$724$	−17.9539	−0.667253
$725$	25.4259	0.944293
$726$	0	0
$727$	−8.77227	−0.325345	−0.162673	−	0.986680i	$-0.552011\pi$
$727$	−8.77227	−0.325345	−0.162673	+	0.986680i	$0.552011\pi$
$728$	139.836	5.18265
$729$	0	0
$730$	68.2457	2.52589
$731$	21.1238	0.781293
$732$	0	0
$733$	23.8365	0.880421	0.440210	−	0.897895i	$-0.354904\pi$
$733$	23.8365	0.880421	0.440210	+	0.897895i	$0.354904\pi$
$734$	−53.1988	−1.96360
$735$	0	0
$736$	−10.0413	−0.370128
$737$	−1.38652	−0.0510733
$738$	0	0
$739$	15.3419	0.564362	0.282181	−	0.959361i	$-0.408942\pi$
$739$	15.3419	0.564362	0.282181	+	0.959361i	$0.408942\pi$
$740$	40.0045	1.47059
$741$	0	0
$742$	−90.2605	−3.31357
$743$	10.4473	0.383274	0.191637	−	0.981466i	$-0.438620\pi$
$743$	10.4473	0.383274	0.191637	+	0.981466i	$0.438620\pi$
$744$	0	0
$745$	−5.55643	−0.203572
$746$	−93.6916	−3.43029
$747$	0	0
$748$	26.4031	0.965393
$749$	52.4307	1.91578
$750$	0	0
$751$	27.8789	1.01732	0.508658	−	0.860969i	$-0.330142\pi$
$751$	27.8789	1.01732	0.508658	+	0.860969i	$0.330142\pi$
$752$	3.04825	0.111158
$753$	0	0
$754$	−80.4259	−2.92894
$755$	−25.8539	−0.940920
$756$	0	0
$757$	−9.30099	−0.338050	−0.169025	−	0.985612i	$-0.554062\pi$
$757$	−9.30099	−0.338050	−0.169025	+	0.985612i	$0.554062\pi$
$758$	−97.1360	−3.52814
$759$	0	0
$760$	40.2131	1.45868
$761$	5.94534	0.215519	0.107759	−	0.994177i	$-0.465632\pi$
$761$	5.94534	0.215519	0.107759	+	0.994177i	$0.465632\pi$
$762$	0	0
$763$	−7.87361	−0.285044
$764$	101.279	3.66413
$765$	0	0
$766$	−79.9852	−2.88998
$767$	−32.1963	−1.16254
$768$	0	0
$769$	28.5763	1.03049	0.515244	−	0.857043i	$-0.327701\pi$
$769$	28.5763	1.03049	0.515244	+	0.857043i	$0.327701\pi$
$770$	−28.8894	−1.04110
$771$	0	0
$772$	−110.886	−3.99089
$773$	28.3191	1.01857	0.509283	−	0.860599i	$-0.329911\pi$
$773$	28.3191	1.01857	0.509283	+	0.860599i	$0.329911\pi$
$774$	0	0
$775$	25.1550	0.903594
$776$	−105.506	−3.78743
$777$	0	0
$778$	−18.6695	−0.669333
$779$	−7.39107	−0.264813
$780$	0	0
$781$	10.6108	0.379686
$782$	12.3566	0.441871
$783$	0	0
$784$	17.3819	0.620783
$785$	−1.72935	−0.0617232
$786$	0	0
$787$	52.2102	1.86109	0.930547	−	0.366173i	$-0.119332\pi$
$787$	52.2102	1.86109	0.930547	+	0.366173i	$0.119332\pi$
$788$	61.1194	2.17729
$789$	0	0
$790$	31.8475	1.13308
$791$	−46.1243	−1.63999
$792$	0	0
$793$	−5.55970	−0.197431
$794$	84.2402	2.98957
$795$	0	0
$796$	93.9881	3.33132
$797$	−20.1132	−0.712446	−0.356223	−	0.934401i	$-0.615936\pi$
$797$	−20.1132	−0.712446	−0.356223	+	0.934401i	$0.615936\pi$
$798$	0	0
$799$	−1.52767	−0.0540452
$800$	53.2914	1.88413
$801$	0	0
$802$	23.7013	0.836923
$803$	9.45924	0.333809
$804$	0	0
$805$	−9.54471	−0.336407
$806$	−79.5691	−2.80270
$807$	0	0
$808$	41.0782	1.44513
$809$	−0.925066	−0.0325236	−0.0162618	−	0.999868i	$-0.505177\pi$
$809$	−0.925066	−0.0325236	−0.0162618	+	0.999868i	$0.505177\pi$
$810$	0	0
$811$	−37.0990	−1.30272	−0.651360	−	0.758768i	$-0.725801\pi$
$811$	−37.0990	−1.30272	−0.651360	+	0.758768i	$0.725801\pi$
$812$	−68.3923	−2.40010
$813$	0	0
$814$	7.85432	0.275294
$815$	−9.66878	−0.338683
$816$	0	0
$817$	7.64243	0.267375
$818$	−88.9744	−3.11092
$819$	0	0
$820$	−66.4718	−2.32130
$821$	−49.3751	−1.72320	−0.861601	−	0.507586i	$-0.830538\pi$
$821$	−49.3751	−1.72320	−0.861601	+	0.507586i	$0.830538\pi$
$822$	0	0
$823$	−20.7979	−0.724970	−0.362485	−	0.931990i	$-0.618072\pi$
$823$	−20.7979	−0.724970	−0.362485	+	0.931990i	$0.618072\pi$
$824$	20.7442	0.722660
$825$	0	0
$826$	−38.7826	−1.34942
$827$	1.88619	0.0655893	0.0327947	−	0.999462i	$-0.489559\pi$
$827$	1.88619	0.0655893	0.0327947	+	0.999462i	$0.489559\pi$
$828$	0	0
$829$	0.759943	0.0263939	0.0131970	−	0.999913i	$-0.495799\pi$
$829$	0.759943	0.0263939	0.0131970	+	0.999913i	$0.495799\pi$
$830$	−0.764502	−0.0265363
$831$	0	0
$832$	−46.8634	−1.62470
$833$	−8.71120	−0.301825
$834$	0	0
$835$	−31.5112	−1.09049
$836$	9.55244	0.330378
$837$	0	0
$838$	75.6229	2.61235
$839$	5.61780	0.193948	0.0969739	−	0.995287i	$-0.469084\pi$
$839$	5.61780	0.193948	0.0969739	+	0.995287i	$0.469084\pi$
$840$	0	0
$841$	−6.04807	−0.208554
$842$	−11.8589	−0.408686
$843$	0	0
$844$	−43.1200	−1.48425
$845$	−91.2842	−3.14027
$846$	0	0
$847$	28.6986	0.986094
$848$	110.052	3.77921
$849$	0	0
$850$	−65.5790	−2.24934
$851$	2.59497	0.0889545
$852$	0	0
$853$	4.86782	0.166671	0.0833356	−	0.996522i	$-0.473443\pi$
$853$	4.86782	0.166671	0.0833356	+	0.996522i	$0.473443\pi$
$854$	−6.69703	−0.229167
$855$	0	0
$856$	−128.868	−4.40462
$857$	−19.5274	−0.667045	−0.333522	−	0.942742i	$-0.608237\pi$
$857$	−19.5274	−0.667045	−0.333522	+	0.942742i	$0.608237\pi$
$858$	0	0
$859$	12.0950	0.412675	0.206338	−	0.978481i	$-0.433845\pi$
$859$	12.0950	0.412675	0.206338	+	0.978481i	$0.433845\pi$
$860$	68.7325	2.34376
$861$	0	0
$862$	14.7183	0.501307
$863$	−11.6847	−0.397752	−0.198876	−	0.980025i	$-0.563729\pi$
$863$	−11.6847	−0.397752	−0.198876	+	0.980025i	$0.563729\pi$
$864$	0	0
$865$	48.6370	1.65371
$866$	−21.7103	−0.737746
$867$	0	0
$868$	−67.6637	−2.29666
$869$	4.41424	0.149743
$870$	0	0
$871$	−7.69020	−0.260573
$872$	19.3523	0.655352
$873$	0	0
$874$	4.47052	0.151217
$875$	2.93221	0.0991269
$876$	0	0
$877$	−8.91577	−0.301064	−0.150532	−	0.988605i	$-0.548099\pi$
$877$	−8.91577	−0.301064	−0.150532	+	0.988605i	$0.548099\pi$
$878$	−28.8019	−0.972017
$879$	0	0
$880$	35.2241	1.18740
$881$	0.306690	0.0103327	0.00516633	−	0.999987i	$-0.498355\pi$
$881$	0.306690	0.0103327	0.00516633	+	0.999987i	$0.498355\pi$
$882$	0	0
$883$	51.2396	1.72435	0.862175	−	0.506610i	$-0.169102\pi$
$883$	51.2396	1.72435	0.862175	+	0.506610i	$0.169102\pi$
$884$	146.442	4.92537
$885$	0	0
$886$	46.1646	1.55093
$887$	1.91073	0.0641559	0.0320780	−	0.999485i	$-0.489788\pi$
$887$	1.91073	0.0641559	0.0320780	+	0.999485i	$0.489788\pi$
$888$	0	0
$889$	37.0588	1.24291
$890$	78.6970	2.63793
$891$	0	0
$892$	−79.8734	−2.67436
$893$	−0.552701	−0.0184954
$894$	0	0
$895$	−24.8168	−0.829536
$896$	3.25525	0.108750
$897$	0	0
$898$	41.6569	1.39011
$899$	22.7074	0.757335
$900$	0	0
$901$	−55.1543	−1.83746
$902$	−13.0508	−0.434545
$903$	0	0
$904$	113.368	3.77055
$905$	12.0040	0.399025
$906$	0	0
$907$	20.3285	0.674997	0.337499	−	0.941326i	$-0.390419\pi$
$907$	20.3285	0.674997	0.337499	+	0.941326i	$0.390419\pi$
$908$	33.7440	1.11983
$909$	0	0
$910$	−160.232	−5.31164
$911$	−26.2906	−0.871046	−0.435523	−	0.900178i	$-0.643437\pi$
$911$	−26.2906	−0.871046	−0.435523	+	0.900178i	$0.643437\pi$
$912$	0	0
$913$	−0.105964	−0.00350691
$914$	31.7823	1.05126
$915$	0	0
$916$	105.411	3.48288
$917$	45.7390	1.51043
$918$	0	0
$919$	−21.2514	−0.701019	−0.350509	−	0.936559i	$-0.613992\pi$
$919$	−21.2514	−0.701019	−0.350509	+	0.936559i	$0.613992\pi$
$920$	23.4597	0.773443
$921$	0	0
$922$	37.1491	1.22344
$923$	58.8519	1.93713
$924$	0	0
$925$	−13.7721	−0.452823
$926$	90.9419	2.98854
$927$	0	0
$928$	48.1061	1.57916
$929$	27.3715	0.898029	0.449015	−	0.893524i	$-0.351775\pi$
$929$	27.3715	0.898029	0.449015	+	0.893524i	$0.351775\pi$
$930$	0	0
$931$	−3.15165	−0.103291
$932$	66.2634	2.17053
$933$	0	0
$934$	17.5777	0.575161
$935$	−17.6531	−0.577317
$936$	0	0
$937$	−51.2264	−1.67349	−0.836746	−	0.547591i	$-0.815545\pi$
$937$	−51.2264	−1.67349	−0.836746	+	0.547591i	$0.815545\pi$
$938$	−9.26335	−0.302459
$939$	0	0
$940$	−4.97073	−0.162127
$941$	47.0238	1.53293	0.766466	−	0.642285i	$-0.222014\pi$
$941$	47.0238	1.53293	0.766466	+	0.642285i	$0.222014\pi$
$942$	0	0
$943$	−4.31183	−0.140413
$944$	47.2865	1.53905
$945$	0	0
$946$	13.4947	0.438750
$947$	−56.9190	−1.84962	−0.924810	−	0.380429i	$-0.875776\pi$
$947$	−56.9190	−1.84962	−0.924810	+	0.380429i	$0.875776\pi$
$948$	0	0
$949$	52.4646	1.70307
$950$	−23.7260	−0.769772
$951$	0	0
$952$	102.927	3.33589
$953$	0.150027	0.00485984	0.00242992	−	0.999997i	$-0.499227\pi$
$953$	0.150027	0.00485984	0.00242992	+	0.999997i	$0.499227\pi$
$954$	0	0
$955$	−67.7147	−2.19120
$956$	−4.55726	−0.147392
$957$	0	0
$958$	10.2294	0.330498
$959$	−16.9230	−0.546471
$960$	0	0
$961$	−8.53447	−0.275306
$962$	43.5631	1.40453
$963$	0	0
$964$	−62.2659	−2.00545
$965$	74.1385	2.38660
$966$	0	0
$967$	32.0847	1.03177	0.515886	−	0.856657i	$-0.327463\pi$
$967$	32.0847	1.03177	0.515886	+	0.856657i	$0.327463\pi$
$968$	−70.5374	−2.26716
$969$	0	0
$970$	120.895	3.88169
$971$	10.8639	0.348638	0.174319	−	0.984689i	$-0.444228\pi$
$971$	10.8639	0.348638	0.174319	+	0.984689i	$0.444228\pi$
$972$	0	0
$973$	55.2495	1.77122
$974$	−15.5046	−0.496799
$975$	0	0
$976$	8.16550	0.261371
$977$	−42.7187	−1.36669	−0.683346	−	0.730095i	$-0.739476\pi$
$977$	−42.7187	−1.36669	−0.683346	+	0.730095i	$0.739476\pi$
$978$	0	0
$979$	10.9079	0.348616
$980$	−28.3444	−0.905430
$981$	0	0
$982$	35.5351	1.13397
$983$	43.7234	1.39456	0.697280	−	0.716799i	$-0.254393\pi$
$983$	43.7234	1.39456	0.697280	+	0.716799i	$0.254393\pi$
$984$	0	0
$985$	−40.8644	−1.30205
$986$	−59.1981	−1.88525
$987$	0	0
$988$	52.9816	1.68557
$989$	4.45847	0.141771
$990$	0	0
$991$	48.2749	1.53350	0.766751	−	0.641944i	$-0.221872\pi$
$991$	48.2749	1.53350	0.766751	+	0.641944i	$0.221872\pi$
$992$	47.5936	1.51110
$993$	0	0
$994$	70.8910	2.24853
$995$	−62.8403	−1.99217
$996$	0	0
$997$	41.3100	1.30830	0.654150	−	0.756365i	$-0.273026\pi$
$997$	41.3100	1.30830	0.654150	+	0.756365i	$0.273026\pi$
$998$	63.2847	2.00324
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1863.2.a.f.1.8		8
3.2	odd	2		1863.2.a.e.1.1		8
9.2	odd	6		621.2.e.a.415.8		16
9.4	even	3		207.2.e.a.70.1	✓	16
9.5	odd	6		621.2.e.a.208.8		16
9.7	even	3		207.2.e.a.139.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.2.e.a.70.1	✓	16	9.4	even	3
207.2.e.a.139.1	yes	16	9.7	even	3
621.2.e.a.208.8		16	9.5	odd	6
621.2.e.a.415.8		16	9.2	odd	6
1863.2.a.e.1.1		8	3.2	odd	2
1863.2.a.f.1.8		8	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:	\( N \)	\(=\)	\( 1863 = 3^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1863.a (trivial)

\(f(q)\)	\(=\)	\(q+2.60803 q^{2} +4.80181 q^{4} -3.21048 q^{5} -2.97298 q^{7} +7.30721 q^{8} +O(q^{10})\)	\(q+2.60803 q^{2} +4.80181 q^{4} -3.21048 q^{5} -2.97298 q^{7} +7.30721 q^{8} -8.37303 q^{10} -1.16055 q^{11} -6.43686 q^{13} -7.75362 q^{14} +9.45378 q^{16} -4.73790 q^{17} -1.71414 q^{19} -15.4161 q^{20} -3.02675 q^{22} -1.00000 q^{23} +5.30721 q^{25} -16.7875 q^{26} -14.2757 q^{28} +4.79082 q^{29} +4.73978 q^{31} +10.0413 q^{32} -12.3566 q^{34} +9.54471 q^{35} -2.59497 q^{37} -4.47052 q^{38} -23.4597 q^{40} +4.31183 q^{41} -4.45847 q^{43} -5.57274 q^{44} -2.60803 q^{46} +0.322437 q^{47} +1.83862 q^{49} +13.8414 q^{50} -30.9086 q^{52} +11.6411 q^{53} +3.72593 q^{55} -21.7242 q^{56} +12.4946 q^{58} +5.00187 q^{59} +0.863729 q^{61} +12.3615 q^{62} +7.28048 q^{64} +20.6654 q^{65} +1.19471 q^{67} -22.7505 q^{68} +24.8929 q^{70} -9.14295 q^{71} -8.15065 q^{73} -6.76776 q^{74} -8.23096 q^{76} +3.45029 q^{77} -3.80357 q^{79} -30.3512 q^{80} +11.2454 q^{82} +0.0913053 q^{83} +15.2110 q^{85} -11.6278 q^{86} -8.48038 q^{88} -9.39887 q^{89} +19.1367 q^{91} -4.80181 q^{92} +0.840925 q^{94} +5.50321 q^{95} -14.4386 q^{97} +4.79517 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 5 q^{4} - 7 q^{7} + 6 q^{8}+O(q^{10}) \) 8 * q + q^2 + 5 * q^4 - 7 * q^7 + 6 * q^8	\( 8 q + q^{2} + 5 q^{4} - 7 q^{7} + 6 q^{8} - 10 q^{10} - 3 q^{11} - 15 q^{13} - 5 q^{14} - q^{16} - 6 q^{17} - 12 q^{19} - 7 q^{20} - 17 q^{22} - 8 q^{23} - 10 q^{25} - 26 q^{26} - 20 q^{28} + 10 q^{29} - 8 q^{31} + 13 q^{32} - 15 q^{34} + 15 q^{35} - 19 q^{37} + 7 q^{38} - 30 q^{40} + q^{41} - 17 q^{43} - 6 q^{44} - q^{46} + 5 q^{47} - 15 q^{49} + 7 q^{50} - 39 q^{52} + 4 q^{53} - 10 q^{55} - 24 q^{56} - 4 q^{58} + 20 q^{59} - 16 q^{61} + 39 q^{62} - 4 q^{64} + 2 q^{65} - 7 q^{67} - 24 q^{68} + 16 q^{70} - 5 q^{71} - 48 q^{73} + 21 q^{74} - 6 q^{76} - 16 q^{77} - 12 q^{79} - 47 q^{80} - 8 q^{82} - 12 q^{83} - 6 q^{85} - 19 q^{86} - 7 q^{88} + 11 q^{89} - 2 q^{91} - 5 q^{92} - q^{94} + 4 q^{95} - 43 q^{97} + 28 q^{98}+O(q^{100}) \) 8 * q + q^2 + 5 * q^4 - 7 * q^7 + 6 * q^8 - 10 * q^10 - 3 * q^11 - 15 * q^13 - 5 * q^14 - q^16 - 6 * q^17 - 12 * q^19 - 7 * q^20 - 17 * q^22 - 8 * q^23 - 10 * q^25 - 26 * q^26 - 20 * q^28 + 10 * q^29 - 8 * q^31 + 13 * q^32 - 15 * q^34 + 15 * q^35 - 19 * q^37 + 7 * q^38 - 30 * q^40 + q^41 - 17 * q^43 - 6 * q^44 - q^46 + 5 * q^47 - 15 * q^49 + 7 * q^50 - 39 * q^52 + 4 * q^53 - 10 * q^55 - 24 * q^56 - 4 * q^58 + 20 * q^59 - 16 * q^61 + 39 * q^62 - 4 * q^64 + 2 * q^65 - 7 * q^67 - 24 * q^68 + 16 * q^70 - 5 * q^71 - 48 * q^73 + 21 * q^74 - 6 * q^76 - 16 * q^77 - 12 * q^79 - 47 * q^80 - 8 * q^82 - 12 * q^83 - 6 * q^85 - 19 * q^86 - 7 * q^88 + 11 * q^89 - 2 * q^91 - 5 * q^92 - q^94 + 4 * q^95 - 43 * q^97 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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