LMFDB - Embedded newform 7623.2.a.ct.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
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} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 $ x^8 - x^7 - 11x^6 + 10x^5 + 35x^4 - 30x^3 - 30x^2 + 30x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.43045$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.43045 q^{2} +3.90710 q^{4} -1.25863 q^{5} +1.00000 q^{7} +4.63512 q^{8} +O(q^{10})$	$q+2.43045 q^{2} +3.90710 q^{4} -1.25863 q^{5} +1.00000 q^{7} +4.63512 q^{8} -3.05904 q^{10} -3.17831 q^{13} +2.43045 q^{14} +3.45123 q^{16} -5.92418 q^{17} +2.86222 q^{19} -4.91760 q^{20} -6.76343 q^{23} -3.41585 q^{25} -7.72473 q^{26} +3.90710 q^{28} -4.49549 q^{29} +9.72751 q^{31} -0.882184 q^{32} -14.3984 q^{34} -1.25863 q^{35} -5.45495 q^{37} +6.95648 q^{38} -5.83390 q^{40} +0.314484 q^{41} -0.132562 q^{43} -16.4382 q^{46} -9.37505 q^{47} +1.00000 q^{49} -8.30206 q^{50} -12.4180 q^{52} -4.35192 q^{53} +4.63512 q^{56} -10.9261 q^{58} +6.94685 q^{59} -2.45215 q^{61} +23.6423 q^{62} -9.04656 q^{64} +4.00032 q^{65} -9.41987 q^{67} -23.1464 q^{68} -3.05904 q^{70} +0.116610 q^{71} +0.615032 q^{73} -13.2580 q^{74} +11.1830 q^{76} +8.52928 q^{79} -4.34382 q^{80} +0.764339 q^{82} -0.950532 q^{83} +7.45636 q^{85} -0.322186 q^{86} -10.0552 q^{89} -3.17831 q^{91} -26.4254 q^{92} -22.7856 q^{94} -3.60248 q^{95} +17.5770 q^{97} +2.43045 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7}+O(q^{10}) $ 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7	$ 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7} + 6 q^{10} - 6 q^{13} - q^{14} + q^{16} + 5 q^{17} - 13 q^{19} - 23 q^{20} - 16 q^{23} + 16 q^{25} + 6 q^{26} + 7 q^{28} - 9 q^{29} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} + 7 q^{37} + 10 q^{38} + 5 q^{40} + 10 q^{41} - 4 q^{43} + 4 q^{46} - 16 q^{47} + 8 q^{49} - 6 q^{50} - 41 q^{52} - 37 q^{53} - 15 q^{58} - q^{59} + 19 q^{61} + 18 q^{62} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 6 q^{70} - 13 q^{71} - 25 q^{73} - 33 q^{74} + 26 q^{76} - 4 q^{80} - 13 q^{82} + 25 q^{83} + 3 q^{85} - 4 q^{86} - 37 q^{89} - 6 q^{91} - 35 q^{92} - 42 q^{94} - 21 q^{95} + 15 q^{97} - q^{98}+O(q^{100}) $ 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7 + 6 * q^10 - 6 * q^13 - q^14 + q^16 + 5 * q^17 - 13 * q^19 - 23 * q^20 - 16 * q^23 + 16 * q^25 + 6 * q^26 + 7 * q^28 - 9 * q^29 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 + 7 * q^37 + 10 * q^38 + 5 * q^40 + 10 * q^41 - 4 * q^43 + 4 * q^46 - 16 * q^47 + 8 * q^49 - 6 * q^50 - 41 * q^52 - 37 * q^53 - 15 * q^58 - q^59 + 19 * q^61 + 18 * q^62 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 6 * q^70 - 13 * q^71 - 25 * q^73 - 33 * q^74 + 26 * q^76 - 4 * q^80 - 13 * q^82 + 25 * q^83 + 3 * q^85 - 4 * q^86 - 37 * q^89 - 6 * q^91 - 35 * q^92 - 42 * q^94 - 21 * q^95 + 15 * q^97 - 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 $

$2$	2.43045	1.71859	0.859295	−	0.511481i	$-0.170903\pi$
$2$	2.43045	1.71859	0.859295	+	0.511481i	$0.170903\pi$
$3$	0	0
$3$	0	0
$4$	3.90710	1.95355
$5$	−1.25863	−0.562877	−0.281438	−	0.959579i	$-0.590812\pi$
$5$	−1.25863	−0.562877	−0.281438	+	0.959579i	$0.590812\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	4.63512	1.63876
$9$	0	0
$10$	−3.05904	−0.967354
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.17831	−0.881504	−0.440752	−	0.897629i	$-0.645288\pi$
$13$	−3.17831	−0.881504	−0.440752	+	0.897629i	$0.645288\pi$
$14$	2.43045	0.649566
$15$	0	0
$16$	3.45123	0.862807
$17$	−5.92418	−1.43682	−0.718412	−	0.695617i	$-0.755131\pi$
$17$	−5.92418	−1.43682	−0.718412	+	0.695617i	$0.755131\pi$
$18$	0	0
$19$	2.86222	0.656638	0.328319	−	0.944567i	$-0.393518\pi$
$19$	2.86222	0.656638	0.328319	+	0.944567i	$0.393518\pi$
$20$	−4.91760	−1.09961
$21$	0	0
$22$	0	0
$23$	−6.76343	−1.41027	−0.705136	−	0.709072i	$-0.749114\pi$
$23$	−6.76343	−1.41027	−0.705136	+	0.709072i	$0.749114\pi$
$24$	0	0
$25$	−3.41585	−0.683170
$26$	−7.72473	−1.51494
$27$	0	0
$28$	3.90710	0.738372
$29$	−4.49549	−0.834792	−0.417396	−	0.908725i	$-0.637057\pi$
$29$	−4.49549	−0.834792	−0.417396	+	0.908725i	$0.637057\pi$
$30$	0	0
$31$	9.72751	1.74711	0.873556	−	0.486723i	$-0.161808\pi$
$31$	9.72751	1.74711	0.873556	+	0.486723i	$0.161808\pi$
$32$	−0.882184	−0.155950
$33$	0	0
$34$	−14.3984	−2.46931
$35$	−1.25863	−0.212747
$36$	0	0
$37$	−5.45495	−0.896789	−0.448394	−	0.893836i	$-0.648004\pi$
$37$	−5.45495	−0.896789	−0.448394	+	0.893836i	$0.648004\pi$
$38$	6.95648	1.12849
$39$	0	0
$40$	−5.83390	−0.922421
$41$	0.314484	0.0491142	0.0245571	−	0.999698i	$-0.492182\pi$
$41$	0.314484	0.0491142	0.0245571	+	0.999698i	$0.492182\pi$
$42$	0	0
$43$	−0.132562	−0.0202155	−0.0101078	−	0.999949i	$-0.503217\pi$
$43$	−0.132562	−0.0202155	−0.0101078	+	0.999949i	$0.503217\pi$
$44$	0	0
$45$	0	0
$46$	−16.4382	−2.42368
$47$	−9.37505	−1.36749	−0.683746	−	0.729720i	$-0.739650\pi$
$47$	−9.37505	−1.36749	−0.683746	+	0.729720i	$0.739650\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−8.30206	−1.17409
$51$	0	0
$52$	−12.4180	−1.72206
$53$	−4.35192	−0.597783	−0.298891	−	0.954287i	$-0.596617\pi$
$53$	−4.35192	−0.597783	−0.298891	+	0.954287i	$0.596617\pi$
$54$	0	0
$55$	0	0
$56$	4.63512	0.619393
$57$	0	0
$58$	−10.9261	−1.43466
$59$	6.94685	0.904403	0.452202	−	0.891916i	$-0.350639\pi$
$59$	6.94685	0.904403	0.452202	+	0.891916i	$0.350639\pi$
$60$	0	0
$61$	−2.45215	−0.313966	−0.156983	−	0.987601i	$-0.550177\pi$
$61$	−2.45215	−0.313966	−0.156983	+	0.987601i	$0.550177\pi$
$62$	23.6423	3.00257
$63$	0	0
$64$	−9.04656	−1.13082
$65$	4.00032	0.496179
$66$	0	0
$67$	−9.41987	−1.15082	−0.575410	−	0.817865i	$-0.695158\pi$
$67$	−9.41987	−1.15082	−0.575410	+	0.817865i	$0.695158\pi$
$68$	−23.1464	−2.80691
$69$	0	0
$70$	−3.05904	−0.365626
$71$	0.116610	0.0138391	0.00691954	−	0.999976i	$-0.497797\pi$
$71$	0.116610	0.0138391	0.00691954	+	0.999976i	$0.497797\pi$
$72$	0	0
$73$	0.615032	0.0719840	0.0359920	−	0.999352i	$-0.488541\pi$
$73$	0.615032	0.0719840	0.0359920	+	0.999352i	$0.488541\pi$
$74$	−13.2580	−1.54121
$75$	0	0
$76$	11.1830	1.28277
$77$	0	0
$78$	0	0
$79$	8.52928	0.959619	0.479810	−	0.877373i	$-0.340706\pi$
$79$	8.52928	0.959619	0.479810	+	0.877373i	$0.340706\pi$
$80$	−4.34382	−0.485654
$81$	0	0
$82$	0.764339	0.0844072
$83$	−0.950532	−0.104334	−0.0521672	−	0.998638i	$-0.516613\pi$
$83$	−0.950532	−0.104334	−0.0521672	+	0.998638i	$0.516613\pi$
$84$	0	0
$85$	7.45636	0.808756
$86$	−0.322186	−0.0347422
$87$	0	0
$88$	0	0
$89$	−10.0552	−1.06585	−0.532923	−	0.846164i	$-0.678906\pi$
$89$	−10.0552	−1.06585	−0.532923	+	0.846164i	$0.678906\pi$
$90$	0	0
$91$	−3.17831	−0.333177
$92$	−26.4254	−2.75504
$93$	0	0
$94$	−22.7856	−2.35016
$95$	−3.60248	−0.369606
$96$	0	0
$97$	17.5770	1.78467	0.892337	−	0.451369i	$-0.149064\pi$
$97$	17.5770	1.78467	0.892337	+	0.451369i	$0.149064\pi$
$98$	2.43045	0.245513
$99$	0	0
$100$	−13.3461	−1.33461
$101$	12.4761	1.24142	0.620711	−	0.784039i	$-0.286844\pi$
$101$	12.4761	1.24142	0.620711	+	0.784039i	$0.286844\pi$
$102$	0	0
$103$	8.21802	0.809746	0.404873	−	0.914373i	$-0.367316\pi$
$103$	8.21802	0.809746	0.404873	+	0.914373i	$0.367316\pi$
$104$	−14.7318	−1.44457
$105$	0	0
$106$	−10.5771	−1.02734
$107$	−12.1885	−1.17831	−0.589154	−	0.808020i	$-0.700539\pi$
$107$	−12.1885	−1.17831	−0.589154	+	0.808020i	$0.700539\pi$
$108$	0	0
$109$	0.886088	0.0848718	0.0424359	−	0.999099i	$-0.486488\pi$
$109$	0.886088	0.0848718	0.0424359	+	0.999099i	$0.486488\pi$
$110$	0	0
$111$	0	0
$112$	3.45123	0.326110
$113$	−4.54502	−0.427559	−0.213780	−	0.976882i	$-0.568577\pi$
$113$	−4.54502	−0.427559	−0.213780	+	0.976882i	$0.568577\pi$
$114$	0	0
$115$	8.51266	0.793810
$116$	−17.5643	−1.63081
$117$	0	0
$118$	16.8840	1.55430
$119$	−5.92418	−0.543069
$120$	0	0
$121$	0	0
$122$	−5.95984	−0.539579
$123$	0	0
$124$	38.0064	3.41307
$125$	10.5924	0.947417
$126$	0	0
$127$	−8.02779	−0.712351	−0.356175	−	0.934419i	$-0.615919\pi$
$127$	−8.02779	−0.712351	−0.356175	+	0.934419i	$0.615919\pi$
$128$	−20.2229	−1.78747
$129$	0	0
$130$	9.72259	0.852727
$131$	−0.101461	−0.00886466	−0.00443233	−	0.999990i	$-0.501411\pi$
$131$	−0.101461	−0.00886466	−0.00443233	+	0.999990i	$0.501411\pi$
$132$	0	0
$133$	2.86222	0.248186
$134$	−22.8945	−1.97779
$135$	0	0
$136$	−27.4593	−2.35461
$137$	4.56409	0.389937	0.194968	−	0.980810i	$-0.437540\pi$
$137$	4.56409	0.389937	0.194968	+	0.980810i	$0.437540\pi$
$138$	0	0
$139$	−3.82533	−0.324460	−0.162230	−	0.986753i	$-0.551869\pi$
$139$	−3.82533	−0.324460	−0.162230	+	0.986753i	$0.551869\pi$
$140$	−4.91760	−0.415613
$141$	0	0
$142$	0.283415	0.0237837
$143$	0	0
$144$	0	0
$145$	5.65817	0.469885
$146$	1.49480	0.123711
$147$	0	0
$148$	−21.3130	−1.75192
$149$	4.87223	0.399149	0.199574	−	0.979883i	$-0.436044\pi$
$149$	4.87223	0.399149	0.199574	+	0.979883i	$0.436044\pi$
$150$	0	0
$151$	−16.9739	−1.38131	−0.690657	−	0.723183i	$-0.742678\pi$
$151$	−16.9739	−1.38131	−0.690657	+	0.723183i	$0.742678\pi$
$152$	13.2667	1.07607
$153$	0	0
$154$	0	0
$155$	−12.2434	−0.983410
$156$	0	0
$157$	2.24483	0.179157	0.0895785	−	0.995980i	$-0.471448\pi$
$157$	2.24483	0.179157	0.0895785	+	0.995980i	$0.471448\pi$
$158$	20.7300	1.64919
$159$	0	0
$160$	1.11034	0.0877804
$161$	−6.76343	−0.533033
$162$	0	0
$163$	−15.8372	−1.24047	−0.620235	−	0.784416i	$-0.712963\pi$
$163$	−15.8372	−1.24047	−0.620235	+	0.784416i	$0.712963\pi$
$164$	1.22872	0.0959470
$165$	0	0
$166$	−2.31022	−0.179308
$167$	−20.7416	−1.60503	−0.802515	−	0.596632i	$-0.796505\pi$
$167$	−20.7416	−1.60503	−0.802515	+	0.596632i	$0.796505\pi$
$168$	0	0
$169$	−2.89835	−0.222950
$170$	18.1223	1.38992
$171$	0	0
$172$	−0.517933	−0.0394920
$173$	21.5554	1.63883	0.819414	−	0.573202i	$-0.194299\pi$
$173$	21.5554	1.63883	0.819414	+	0.573202i	$0.194299\pi$
$174$	0	0
$175$	−3.41585	−0.258214
$176$	0	0
$177$	0	0
$178$	−24.4386	−1.83175
$179$	−4.78161	−0.357394	−0.178697	−	0.983904i	$-0.557188\pi$
$179$	−4.78161	−0.357394	−0.178697	+	0.983904i	$0.557188\pi$
$180$	0	0
$181$	−7.85284	−0.583697	−0.291849	−	0.956465i	$-0.594270\pi$
$181$	−7.85284	−0.583697	−0.291849	+	0.956465i	$0.594270\pi$
$182$	−7.72473	−0.572595
$183$	0	0
$184$	−31.3493	−2.31110
$185$	6.86578	0.504782
$186$	0	0
$187$	0	0
$188$	−36.6293	−2.67146
$189$	0	0
$190$	−8.75565	−0.635201
$191$	8.83029	0.638937	0.319469	−	0.947597i	$-0.396496\pi$
$191$	8.83029	0.638937	0.319469	+	0.947597i	$0.396496\pi$
$192$	0	0
$193$	25.6023	1.84289	0.921446	−	0.388507i	$-0.127009\pi$
$193$	25.6023	1.84289	0.921446	+	0.388507i	$0.127009\pi$
$194$	42.7201	3.06712
$195$	0	0
$196$	3.90710	0.279079
$197$	11.1977	0.797802	0.398901	−	0.916994i	$-0.369392\pi$
$197$	11.1977	0.797802	0.398901	+	0.916994i	$0.369392\pi$
$198$	0	0
$199$	−12.2503	−0.868400	−0.434200	−	0.900817i	$-0.642969\pi$
$199$	−12.2503	−0.868400	−0.434200	+	0.900817i	$0.642969\pi$
$200$	−15.8328	−1.11955
$201$	0	0
$202$	30.3227	2.13350
$203$	−4.49549	−0.315522
$204$	0	0
$205$	−0.395820	−0.0276453
$206$	19.9735	1.39162
$207$	0	0
$208$	−10.9691	−0.760568
$209$	0	0
$210$	0	0
$211$	−14.2636	−0.981946	−0.490973	−	0.871175i	$-0.663359\pi$
$211$	−14.2636	−0.981946	−0.490973	+	0.871175i	$0.663359\pi$
$212$	−17.0034	−1.16780
$213$	0	0
$214$	−29.6236	−2.02503
$215$	0.166847	0.0113788
$216$	0	0
$217$	9.72751	0.660347
$218$	2.15359	0.145860
$219$	0	0
$220$	0	0
$221$	18.8289	1.26657
$222$	0	0
$223$	−2.96400	−0.198484	−0.0992421	−	0.995063i	$-0.531642\pi$
$223$	−2.96400	−0.198484	−0.0992421	+	0.995063i	$0.531642\pi$
$224$	−0.882184	−0.0589434
$225$	0	0
$226$	−11.0465	−0.734799
$227$	−8.25605	−0.547974	−0.273987	−	0.961733i	$-0.588342\pi$
$227$	−8.25605	−0.547974	−0.273987	+	0.961733i	$0.588342\pi$
$228$	0	0
$229$	13.1081	0.866208	0.433104	−	0.901344i	$-0.357418\pi$
$229$	13.1081	0.866208	0.433104	+	0.901344i	$0.357418\pi$
$230$	20.6896	1.36423
$231$	0	0
$232$	−20.8371	−1.36802
$233$	12.8277	0.840369	0.420185	−	0.907439i	$-0.361965\pi$
$233$	12.8277	0.840369	0.420185	+	0.907439i	$0.361965\pi$
$234$	0	0
$235$	11.7997	0.769730
$236$	27.1420	1.76680
$237$	0	0
$238$	−14.3984	−0.933312
$239$	4.98109	0.322200	0.161100	−	0.986938i	$-0.448496\pi$
$239$	4.98109	0.322200	0.161100	+	0.986938i	$0.448496\pi$
$240$	0	0
$241$	−2.62686	−0.169211	−0.0846053	−	0.996415i	$-0.526963\pi$
$241$	−2.62686	−0.169211	−0.0846053	+	0.996415i	$0.526963\pi$
$242$	0	0
$243$	0	0
$244$	−9.58081	−0.613349
$245$	−1.25863	−0.0804110
$246$	0	0
$247$	−9.09701	−0.578829
$248$	45.0881	2.86310
$249$	0	0
$250$	25.7444	1.62822
$251$	26.2229	1.65518	0.827588	−	0.561337i	$-0.189713\pi$
$251$	26.2229	1.65518	0.827588	+	0.561337i	$0.189713\pi$
$252$	0	0
$253$	0	0
$254$	−19.5112	−1.22424
$255$	0	0
$256$	−31.0576	−1.94110
$257$	−29.9518	−1.86834	−0.934170	−	0.356827i	$-0.883859\pi$
$257$	−29.9518	−1.86834	−0.934170	+	0.356827i	$0.883859\pi$
$258$	0	0
$259$	−5.45495	−0.338954
$260$	15.6296	0.969309
$261$	0	0
$262$	−0.246595	−0.0152347
$263$	−3.33709	−0.205774	−0.102887	−	0.994693i	$-0.532808\pi$
$263$	−3.33709	−0.205774	−0.102887	+	0.994693i	$0.532808\pi$
$264$	0	0
$265$	5.47747	0.336478
$266$	6.95648	0.426529
$267$	0	0
$268$	−36.8044	−2.24818
$269$	1.73581	0.105834	0.0529172	−	0.998599i	$-0.483148\pi$
$269$	1.73581	0.105834	0.0529172	+	0.998599i	$0.483148\pi$
$270$	0	0
$271$	−2.47596	−0.150404	−0.0752019	−	0.997168i	$-0.523960\pi$
$271$	−2.47596	−0.150404	−0.0752019	+	0.997168i	$0.523960\pi$
$272$	−20.4457	−1.23970
$273$	0	0
$274$	11.0928	0.670141
$275$	0	0
$276$	0	0
$277$	5.15571	0.309776	0.154888	−	0.987932i	$-0.450498\pi$
$277$	5.15571	0.309776	0.154888	+	0.987932i	$0.450498\pi$
$278$	−9.29728	−0.557614
$279$	0	0
$280$	−5.83390	−0.348642
$281$	22.5705	1.34644	0.673222	−	0.739441i	$-0.264910\pi$
$281$	22.5705	1.34644	0.673222	+	0.739441i	$0.264910\pi$
$282$	0	0
$283$	−8.48757	−0.504534	−0.252267	−	0.967658i	$-0.581176\pi$
$283$	−8.48757	−0.504534	−0.252267	+	0.967658i	$0.581176\pi$
$284$	0.455608	0.0270353
$285$	0	0
$286$	0	0
$287$	0.314484	0.0185634
$288$	0	0
$289$	18.0959	1.06447
$290$	13.7519	0.807540
$291$	0	0
$292$	2.40299	0.140624
$293$	−24.6261	−1.43867	−0.719336	−	0.694662i	$-0.755554\pi$
$293$	−24.6261	−1.43867	−0.719336	+	0.694662i	$0.755554\pi$
$294$	0	0
$295$	−8.74353	−0.509068
$296$	−25.2843	−1.46962
$297$	0	0
$298$	11.8417	0.685973
$299$	21.4963	1.24316
$300$	0	0
$301$	−0.132562	−0.00764075
$302$	−41.2542	−2.37391
$303$	0	0
$304$	9.87817	0.566552
$305$	3.08636	0.176724
$306$	0	0
$307$	4.59391	0.262188	0.131094	−	0.991370i	$-0.458151\pi$
$307$	4.59391	0.262188	0.131094	+	0.991370i	$0.458151\pi$
$308$	0	0
$309$	0	0
$310$	−29.7569	−1.69008
$311$	2.21073	0.125359	0.0626796	−	0.998034i	$-0.480035\pi$
$311$	2.21073	0.125359	0.0626796	+	0.998034i	$0.480035\pi$
$312$	0	0
$313$	−9.69928	−0.548236	−0.274118	−	0.961696i	$-0.588386\pi$
$313$	−9.69928	−0.548236	−0.274118	+	0.961696i	$0.588386\pi$
$314$	5.45595	0.307897
$315$	0	0
$316$	33.3248	1.87466
$317$	−6.45539	−0.362571	−0.181286	−	0.983431i	$-0.558026\pi$
$317$	−6.45539	−0.362571	−0.181286	+	0.983431i	$0.558026\pi$
$318$	0	0
$319$	0	0
$320$	11.3863	0.636513
$321$	0	0
$322$	−16.4382	−0.916065
$323$	−16.9563	−0.943474
$324$	0	0
$325$	10.8566	0.602217
$326$	−38.4917	−2.13186
$327$	0	0
$328$	1.45767	0.0804864
$329$	−9.37505	−0.516863
$330$	0	0
$331$	3.62076	0.199015	0.0995075	−	0.995037i	$-0.468273\pi$
$331$	3.62076	0.199015	0.0995075	+	0.995037i	$0.468273\pi$
$332$	−3.71382	−0.203823
$333$	0	0
$334$	−50.4114	−2.75839
$335$	11.8561	0.647770
$336$	0	0
$337$	−6.40510	−0.348908	−0.174454	−	0.984665i	$-0.555816\pi$
$337$	−6.40510	−0.348908	−0.174454	+	0.984665i	$0.555816\pi$
$338$	−7.04430	−0.383159
$339$	0	0
$340$	29.1327	1.57994
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−0.614440	−0.0331284
$345$	0	0
$346$	52.3894	2.81647
$347$	17.4600	0.937302	0.468651	−	0.883383i	$-0.344740\pi$
$347$	17.4600	0.937302	0.468651	+	0.883383i	$0.344740\pi$
$348$	0	0
$349$	−21.6249	−1.15755	−0.578777	−	0.815486i	$-0.696470\pi$
$349$	−21.6249	−1.15755	−0.578777	+	0.815486i	$0.696470\pi$
$350$	−8.30206	−0.443764
$351$	0	0
$352$	0	0
$353$	−4.40150	−0.234268	−0.117134	−	0.993116i	$-0.537371\pi$
$353$	−4.40150	−0.234268	−0.117134	+	0.993116i	$0.537371\pi$
$354$	0	0
$355$	−0.146769	−0.00778970
$356$	−39.2865	−2.08218
$357$	0	0
$358$	−11.6215	−0.614214
$359$	10.6817	0.563760	0.281880	−	0.959450i	$-0.409042\pi$
$359$	10.6817	0.563760	0.281880	+	0.959450i	$0.409042\pi$
$360$	0	0
$361$	−10.8077	−0.568827
$362$	−19.0860	−1.00314
$363$	0	0
$364$	−12.4180	−0.650879
$365$	−0.774098	−0.0405181
$366$	0	0
$367$	10.2167	0.533309	0.266655	−	0.963792i	$-0.414082\pi$
$367$	10.2167	0.533309	0.266655	+	0.963792i	$0.414082\pi$
$368$	−23.3421	−1.21679
$369$	0	0
$370$	16.6869	0.867513
$371$	−4.35192	−0.225941
$372$	0	0
$373$	−27.8851	−1.44383	−0.721917	−	0.691980i	$-0.756739\pi$
$373$	−27.8851	−1.44383	−0.721917	+	0.691980i	$0.756739\pi$
$374$	0	0
$375$	0	0
$376$	−43.4544	−2.24099
$377$	14.2881	0.735873
$378$	0	0
$379$	1.14803	0.0589703	0.0294852	−	0.999565i	$-0.490613\pi$
$379$	1.14803	0.0589703	0.0294852	+	0.999565i	$0.490613\pi$
$380$	−14.0752	−0.722044
$381$	0	0
$382$	21.4616	1.09807
$383$	19.9516	1.01948	0.509740	−	0.860328i	$-0.329742\pi$
$383$	19.9516	1.01948	0.509740	+	0.860328i	$0.329742\pi$
$384$	0	0
$385$	0	0
$386$	62.2251	3.16717
$387$	0	0
$388$	68.6751	3.48645
$389$	−2.60000	−0.131825	−0.0659127	−	0.997825i	$-0.520996\pi$
$389$	−2.60000	−0.131825	−0.0659127	+	0.997825i	$0.520996\pi$
$390$	0	0
$391$	40.0678	2.02631
$392$	4.63512	0.234109
$393$	0	0
$394$	27.2154	1.37109
$395$	−10.7352	−0.540148
$396$	0	0
$397$	8.77237	0.440272	0.220136	−	0.975469i	$-0.429350\pi$
$397$	8.77237	0.440272	0.220136	+	0.975469i	$0.429350\pi$
$398$	−29.7737	−1.49242
$399$	0	0
$400$	−11.7889	−0.589444
$401$	−28.3535	−1.41591	−0.707953	−	0.706260i	$-0.750381\pi$
$401$	−28.3535	−1.41591	−0.707953	+	0.706260i	$0.750381\pi$
$402$	0	0
$403$	−30.9170	−1.54009
$404$	48.7455	2.42518
$405$	0	0
$406$	−10.9261	−0.542252
$407$	0	0
$408$	0	0
$409$	19.9055	0.984264	0.492132	−	0.870521i	$-0.336218\pi$
$409$	19.9055	0.984264	0.492132	+	0.870521i	$0.336218\pi$
$410$	−0.962021	−0.0475108
$411$	0	0
$412$	32.1086	1.58188
$413$	6.94685	0.341832
$414$	0	0
$415$	1.19637	0.0587275
$416$	2.80385	0.137470
$417$	0	0
$418$	0	0
$419$	30.8957	1.50935	0.754676	−	0.656097i	$-0.227794\pi$
$419$	30.8957	1.50935	0.754676	+	0.656097i	$0.227794\pi$
$420$	0	0
$421$	−24.1931	−1.17910	−0.589551	−	0.807732i	$-0.700695\pi$
$421$	−24.1931	−1.17910	−0.589551	+	0.807732i	$0.700695\pi$
$422$	−34.6670	−1.68756
$423$	0	0
$424$	−20.1717	−0.979623
$425$	20.2361	0.981595
$426$	0	0
$427$	−2.45215	−0.118668
$428$	−47.6218	−2.30188
$429$	0	0
$430$	0.405513	0.0195556
$431$	−19.4001	−0.934468	−0.467234	−	0.884134i	$-0.654749\pi$
$431$	−19.4001	−0.934468	−0.467234	+	0.884134i	$0.654749\pi$
$432$	0	0
$433$	−21.3877	−1.02783	−0.513914	−	0.857842i	$-0.671805\pi$
$433$	−21.3877	−1.02783	−0.513914	+	0.857842i	$0.671805\pi$
$434$	23.6423	1.13486
$435$	0	0
$436$	3.46203	0.165801
$437$	−19.3584	−0.926038
$438$	0	0
$439$	−31.7315	−1.51446	−0.757232	−	0.653146i	$-0.773449\pi$
$439$	−31.7315	−1.51446	−0.757232	+	0.653146i	$0.773449\pi$
$440$	0	0
$441$	0	0
$442$	45.7627	2.17671
$443$	1.66610	0.0791590	0.0395795	−	0.999216i	$-0.487398\pi$
$443$	1.66610	0.0791590	0.0395795	+	0.999216i	$0.487398\pi$
$444$	0	0
$445$	12.6557	0.599940
$446$	−7.20386	−0.341113
$447$	0	0
$448$	−9.04656	−0.427410
$449$	17.1858	0.811047	0.405524	−	0.914085i	$-0.367089\pi$
$449$	17.1858	0.811047	0.405524	+	0.914085i	$0.367089\pi$
$450$	0	0
$451$	0	0
$452$	−17.7578	−0.835258
$453$	0	0
$454$	−20.0659	−0.941742
$455$	4.00032	0.187538
$456$	0	0
$457$	10.2206	0.478101	0.239051	−	0.971007i	$-0.423164\pi$
$457$	10.2206	0.478101	0.239051	+	0.971007i	$0.423164\pi$
$458$	31.8586	1.48866
$459$	0	0
$460$	33.2598	1.55075
$461$	22.1160	1.03004	0.515022	−	0.857177i	$-0.327784\pi$
$461$	22.1160	1.03004	0.515022	+	0.857177i	$0.327784\pi$
$462$	0	0
$463$	−30.3717	−1.41149	−0.705747	−	0.708464i	$-0.749389\pi$
$463$	−30.3717	−1.41149	−0.705747	+	0.708464i	$0.749389\pi$
$464$	−15.5150	−0.720265
$465$	0	0
$466$	31.1771	1.44425
$467$	−25.1909	−1.16570	−0.582848	−	0.812581i	$-0.698062\pi$
$467$	−25.1909	−1.16570	−0.582848	+	0.812581i	$0.698062\pi$
$468$	0	0
$469$	−9.41987	−0.434969
$470$	28.6787	1.32285
$471$	0	0
$472$	32.1995	1.48210
$473$	0	0
$474$	0	0
$475$	−9.77690	−0.448595
$476$	−23.1464	−1.06091
$477$	0	0
$478$	12.1063	0.553729
$479$	21.2040	0.968835	0.484417	−	0.874837i	$-0.339032\pi$
$479$	21.2040	0.968835	0.484417	+	0.874837i	$0.339032\pi$
$480$	0	0
$481$	17.3375	0.790523
$482$	−6.38445	−0.290804
$483$	0	0
$484$	0	0
$485$	−22.1230	−1.00455
$486$	0	0
$487$	−16.8198	−0.762176	−0.381088	−	0.924539i	$-0.624451\pi$
$487$	−16.8198	−0.762176	−0.381088	+	0.924539i	$0.624451\pi$
$488$	−11.3660	−0.514515
$489$	0	0
$490$	−3.05904	−0.138193
$491$	−4.83804	−0.218338	−0.109169	−	0.994023i	$-0.534819\pi$
$491$	−4.83804	−0.218338	−0.109169	+	0.994023i	$0.534819\pi$
$492$	0	0
$493$	26.6321	1.19945
$494$	−22.1099	−0.994770
$495$	0	0
$496$	33.5719	1.50742
$497$	0.116610	0.00523068
$498$	0	0
$499$	30.5836	1.36911	0.684554	−	0.728962i	$-0.259997\pi$
$499$	30.5836	1.36911	0.684554	+	0.728962i	$0.259997\pi$
$500$	41.3858	1.85083
$501$	0	0
$502$	63.7335	2.84457
$503$	28.2407	1.25919	0.629594	−	0.776924i	$-0.283221\pi$
$503$	28.2407	1.25919	0.629594	+	0.776924i	$0.283221\pi$
$504$	0	0
$505$	−15.7029	−0.698768
$506$	0	0
$507$	0	0
$508$	−31.3654	−1.39161
$509$	4.20065	0.186190	0.0930952	−	0.995657i	$-0.470324\pi$
$509$	4.20065	0.186190	0.0930952	+	0.995657i	$0.470324\pi$
$510$	0	0
$511$	0.615032	0.0272074
$512$	−35.0383	−1.54849
$513$	0	0
$514$	−72.7964	−3.21091
$515$	−10.3435	−0.455787
$516$	0	0
$517$	0	0
$518$	−13.2580	−0.582523
$519$	0	0
$520$	18.5419	0.813118
$521$	20.6470	0.904560	0.452280	−	0.891876i	$-0.350611\pi$
$521$	20.6470	0.904560	0.452280	+	0.891876i	$0.350611\pi$
$522$	0	0
$523$	−22.6341	−0.989719	−0.494859	−	0.868973i	$-0.664780\pi$
$523$	−22.6341	−0.989719	−0.494859	+	0.868973i	$0.664780\pi$
$524$	−0.396417	−0.0173176
$525$	0	0
$526$	−8.11063	−0.353640
$527$	−57.6275	−2.51030
$528$	0	0
$529$	22.7440	0.988869
$530$	13.3127	0.578268
$531$	0	0
$532$	11.1830	0.484843
$533$	−0.999529	−0.0432944
$534$	0	0
$535$	15.3409	0.663243
$536$	−43.6622	−1.88592
$537$	0	0
$538$	4.21881	0.181886
$539$	0	0
$540$	0	0
$541$	42.7163	1.83652	0.918258	−	0.395982i	$-0.129596\pi$
$541$	42.7163	1.83652	0.918258	+	0.395982i	$0.129596\pi$
$542$	−6.01770	−0.258482
$543$	0	0
$544$	5.22622	0.224072
$545$	−1.11526	−0.0477724
$546$	0	0
$547$	44.4827	1.90194	0.950972	−	0.309278i	$-0.100087\pi$
$547$	44.4827	1.90194	0.950972	+	0.309278i	$0.100087\pi$
$548$	17.8324	0.761761
$549$	0	0
$550$	0	0
$551$	−12.8671	−0.548156
$552$	0	0
$553$	8.52928	0.362702
$554$	12.5307	0.532379
$555$	0	0
$556$	−14.9459	−0.633849
$557$	24.9911	1.05891	0.529454	−	0.848339i	$-0.322397\pi$
$557$	24.9911	1.05891	0.529454	+	0.848339i	$0.322397\pi$
$558$	0	0
$559$	0.421323	0.0178201
$560$	−4.34382	−0.183560
$561$	0	0
$562$	54.8565	2.31398
$563$	−9.01005	−0.379729	−0.189864	−	0.981810i	$-0.560805\pi$
$563$	−9.01005	−0.379729	−0.189864	+	0.981810i	$0.560805\pi$
$564$	0	0
$565$	5.72050	0.240663
$566$	−20.6286	−0.867086
$567$	0	0
$568$	0.540502	0.0226789
$569$	−29.1738	−1.22303	−0.611516	−	0.791232i	$-0.709440\pi$
$569$	−29.1738	−1.22303	−0.611516	+	0.791232i	$0.709440\pi$
$570$	0	0
$571$	1.78994	0.0749067	0.0374533	−	0.999298i	$-0.488075\pi$
$571$	1.78994	0.0749067	0.0374533	+	0.999298i	$0.488075\pi$
$572$	0	0
$573$	0	0
$574$	0.764339	0.0319029
$575$	23.1028	0.963455
$576$	0	0
$577$	−21.5850	−0.898594	−0.449297	−	0.893382i	$-0.648325\pi$
$577$	−21.5850	−0.898594	−0.449297	+	0.893382i	$0.648325\pi$
$578$	43.9813	1.82938
$579$	0	0
$580$	22.1070	0.917944
$581$	−0.950532	−0.0394347
$582$	0	0
$583$	0	0
$584$	2.85074	0.117965
$585$	0	0
$586$	−59.8526	−2.47249
$587$	−5.52355	−0.227981	−0.113991	−	0.993482i	$-0.536363\pi$
$587$	−5.52355	−0.227981	−0.113991	+	0.993482i	$0.536363\pi$
$588$	0	0
$589$	27.8423	1.14722
$590$	−21.2507	−0.874879
$591$	0	0
$592$	−18.8263	−0.773756
$593$	40.7867	1.67491	0.837454	−	0.546508i	$-0.184043\pi$
$593$	40.7867	1.67491	0.837454	+	0.546508i	$0.184043\pi$
$594$	0	0
$595$	7.45636	0.305681
$596$	19.0363	0.779757
$597$	0	0
$598$	52.2457	2.13648
$599$	26.5149	1.08337	0.541685	−	0.840582i	$-0.317787\pi$
$599$	26.5149	1.08337	0.541685	+	0.840582i	$0.317787\pi$
$600$	0	0
$601$	−37.7699	−1.54067	−0.770333	−	0.637641i	$-0.779910\pi$
$601$	−37.7699	−1.54067	−0.770333	+	0.637641i	$0.779910\pi$
$602$	−0.322186	−0.0131313
$603$	0	0
$604$	−66.3186	−2.69847
$605$	0	0
$606$	0	0
$607$	−27.7244	−1.12530	−0.562649	−	0.826696i	$-0.690218\pi$
$607$	−27.7244	−1.12530	−0.562649	+	0.826696i	$0.690218\pi$
$608$	−2.52500	−0.102402
$609$	0	0
$610$	7.50125	0.303717
$611$	29.7968	1.20545
$612$	0	0
$613$	36.2264	1.46317	0.731586	−	0.681749i	$-0.238780\pi$
$613$	36.2264	1.46317	0.731586	+	0.681749i	$0.238780\pi$
$614$	11.1653	0.450594
$615$	0	0
$616$	0	0
$617$	−41.1920	−1.65833	−0.829163	−	0.559007i	$-0.811183\pi$
$617$	−41.1920	−1.65833	−0.829163	+	0.559007i	$0.811183\pi$
$618$	0	0
$619$	−35.7395	−1.43649	−0.718247	−	0.695789i	$-0.755055\pi$
$619$	−35.7395	−1.43649	−0.718247	+	0.695789i	$0.755055\pi$
$620$	−47.8360	−1.92114
$621$	0	0
$622$	5.37308	0.215441
$623$	−10.0552	−0.402852
$624$	0	0
$625$	3.74725	0.149890
$626$	−23.5736	−0.942192
$627$	0	0
$628$	8.77077	0.349992
$629$	32.3161	1.28853
$630$	0	0
$631$	2.76508	0.110076	0.0550380	−	0.998484i	$-0.482472\pi$
$631$	2.76508	0.110076	0.0550380	+	0.998484i	$0.482472\pi$
$632$	39.5342	1.57259
$633$	0	0
$634$	−15.6895	−0.623111
$635$	10.1040	0.400966
$636$	0	0
$637$	−3.17831	−0.125929
$638$	0	0
$639$	0	0
$640$	25.4531	1.00612
$641$	−38.4864	−1.52012	−0.760061	−	0.649851i	$-0.774831\pi$
$641$	−38.4864	−1.52012	−0.760061	+	0.649851i	$0.774831\pi$
$642$	0	0
$643$	−26.3781	−1.04025	−0.520126	−	0.854089i	$-0.674115\pi$
$643$	−26.3781	−1.04025	−0.520126	+	0.854089i	$0.674115\pi$
$644$	−26.4254	−1.04131
$645$	0	0
$646$	−41.2115	−1.62144
$647$	27.3211	1.07410	0.537051	−	0.843549i	$-0.319538\pi$
$647$	27.3211	1.07410	0.537051	+	0.843549i	$0.319538\pi$
$648$	0	0
$649$	0	0
$650$	26.3865	1.03496
$651$	0	0
$652$	−61.8777	−2.42332
$653$	8.46165	0.331130	0.165565	−	0.986199i	$-0.447055\pi$
$653$	8.46165	0.331130	0.165565	+	0.986199i	$0.447055\pi$
$654$	0	0
$655$	0.127702	0.00498971
$656$	1.08536	0.0423761
$657$	0	0
$658$	−22.7856	−0.888276
$659$	5.29247	0.206165	0.103083	−	0.994673i	$-0.467129\pi$
$659$	5.29247	0.206165	0.103083	+	0.994673i	$0.467129\pi$
$660$	0	0
$661$	−19.2700	−0.749517	−0.374759	−	0.927122i	$-0.622274\pi$
$661$	−19.2700	−0.749517	−0.374759	+	0.927122i	$0.622274\pi$
$662$	8.80008	0.342025
$663$	0	0
$664$	−4.40583	−0.170979
$665$	−3.60248	−0.139698
$666$	0	0
$667$	30.4050	1.17728
$668$	−81.0393	−3.13551
$669$	0	0
$670$	28.8158	1.11325
$671$	0	0
$672$	0	0
$673$	−18.9922	−0.732096	−0.366048	−	0.930596i	$-0.619289\pi$
$673$	−18.9922	−0.732096	−0.366048	+	0.930596i	$0.619289\pi$
$674$	−15.5673	−0.599629
$675$	0	0
$676$	−11.3241	−0.435544
$677$	−32.4219	−1.24607	−0.623037	−	0.782192i	$-0.714101\pi$
$677$	−32.4219	−1.24607	−0.623037	+	0.782192i	$0.714101\pi$
$678$	0	0
$679$	17.5770	0.674544
$680$	34.5611	1.32536
$681$	0	0
$682$	0	0
$683$	−15.1260	−0.578779	−0.289389	−	0.957211i	$-0.593452\pi$
$683$	−15.1260	−0.578779	−0.289389	+	0.957211i	$0.593452\pi$
$684$	0	0
$685$	−5.74451	−0.219486
$686$	2.43045	0.0927951
$687$	0	0
$688$	−0.457502	−0.0174421
$689$	13.8318	0.526948
$690$	0	0
$691$	−31.4108	−1.19492	−0.597461	−	0.801898i	$-0.703824\pi$
$691$	−31.4108	−1.19492	−0.597461	+	0.801898i	$0.703824\pi$
$692$	84.2192	3.20153
$693$	0	0
$694$	42.4357	1.61084
$695$	4.81468	0.182631
$696$	0	0
$697$	−1.86306	−0.0705685
$698$	−52.5583	−1.98936
$699$	0	0
$700$	−13.3461	−0.504434
$701$	16.0647	0.606755	0.303377	−	0.952870i	$-0.401886\pi$
$701$	16.0647	0.606755	0.303377	+	0.952870i	$0.401886\pi$
$702$	0	0
$703$	−15.6133	−0.588865
$704$	0	0
$705$	0	0
$706$	−10.6976	−0.402611
$707$	12.4761	0.469214
$708$	0	0
$709$	39.6978	1.49088	0.745441	−	0.666572i	$-0.232239\pi$
$709$	39.6978	1.49088	0.745441	+	0.666572i	$0.232239\pi$
$710$	−0.356716	−0.0133873
$711$	0	0
$712$	−46.6069	−1.74667
$713$	−65.7913	−2.46391
$714$	0	0
$715$	0	0
$716$	−18.6822	−0.698187
$717$	0	0
$718$	25.9614	0.968873
$719$	9.86241	0.367806	0.183903	−	0.982944i	$-0.441127\pi$
$719$	9.86241	0.367806	0.183903	+	0.982944i	$0.441127\pi$
$720$	0	0
$721$	8.21802	0.306055
$722$	−26.2676	−0.977580
$723$	0	0
$724$	−30.6818	−1.14028
$725$	15.3559	0.570305
$726$	0	0
$727$	−31.5764	−1.17111	−0.585553	−	0.810634i	$-0.699122\pi$
$727$	−31.5764	−1.17111	−0.585553	+	0.810634i	$0.699122\pi$
$728$	−14.7318	−0.545998
$729$	0	0
$730$	−1.88141	−0.0696340
$731$	0.785321	0.0290462
$732$	0	0
$733$	41.2926	1.52518	0.762588	−	0.646885i	$-0.223928\pi$
$733$	41.2926	1.52518	0.762588	+	0.646885i	$0.223928\pi$
$734$	24.8313	0.916540
$735$	0	0
$736$	5.96659	0.219931
$737$	0	0
$738$	0	0
$739$	35.4095	1.30256	0.651281	−	0.758837i	$-0.274232\pi$
$739$	35.4095	1.30256	0.651281	+	0.758837i	$0.274232\pi$
$740$	26.8253	0.986116
$741$	0	0
$742$	−10.5771	−0.388299
$743$	4.02408	0.147629	0.0738146	−	0.997272i	$-0.476483\pi$
$743$	4.02408	0.147629	0.0738146	+	0.997272i	$0.476483\pi$
$744$	0	0
$745$	−6.13235	−0.224672
$746$	−67.7733	−2.48136
$747$	0	0
$748$	0	0
$749$	−12.1885	−0.445359
$750$	0	0
$751$	24.4190	0.891062	0.445531	−	0.895267i	$-0.353015\pi$
$751$	24.4190	0.891062	0.445531	+	0.895267i	$0.353015\pi$
$752$	−32.3554	−1.17988
$753$	0	0
$754$	34.7265	1.26466
$755$	21.3638	0.777510
$756$	0	0
$757$	−1.31723	−0.0478755	−0.0239377	−	0.999713i	$-0.507620\pi$
$757$	−1.31723	−0.0478755	−0.0239377	+	0.999713i	$0.507620\pi$
$758$	2.79023	0.101346
$759$	0	0
$760$	−16.6979	−0.605696
$761$	7.24404	0.262596	0.131298	−	0.991343i	$-0.458085\pi$
$761$	7.24404	0.262596	0.131298	+	0.991343i	$0.458085\pi$
$762$	0	0
$763$	0.886088	0.0320785
$764$	34.5008	1.24820
$765$	0	0
$766$	48.4915	1.75207
$767$	−22.0792	−0.797235
$768$	0	0
$769$	44.3139	1.59800	0.798999	−	0.601332i	$-0.205363\pi$
$769$	44.3139	1.59800	0.798999	+	0.601332i	$0.205363\pi$
$770$	0	0
$771$	0	0
$772$	100.031	3.60018
$773$	−19.0559	−0.685395	−0.342697	−	0.939446i	$-0.611341\pi$
$773$	−19.0559	−0.685395	−0.342697	+	0.939446i	$0.611341\pi$
$774$	0	0
$775$	−33.2277	−1.19357
$776$	81.4714	2.92465
$777$	0	0
$778$	−6.31918	−0.226554
$779$	0.900123	0.0322502
$780$	0	0
$781$	0	0
$782$	97.3828	3.48240
$783$	0	0
$784$	3.45123	0.123258
$785$	−2.82541	−0.100843
$786$	0	0
$787$	31.4600	1.12143	0.560714	−	0.828009i	$-0.310527\pi$
$787$	31.4600	1.12143	0.560714	+	0.828009i	$0.310527\pi$
$788$	43.7505	1.55855
$789$	0	0
$790$	−26.0914	−0.928292
$791$	−4.54502	−0.161602
$792$	0	0
$793$	7.79370	0.276763
$794$	21.3208	0.756648
$795$	0	0
$796$	−47.8631	−1.69646
$797$	12.5453	0.444376	0.222188	−	0.975004i	$-0.428680\pi$
$797$	12.5453	0.444376	0.222188	+	0.975004i	$0.428680\pi$
$798$	0	0
$799$	55.5395	1.96485
$800$	3.01340	0.106540
$801$	0	0
$802$	−68.9118	−2.43336
$803$	0	0
$804$	0	0
$805$	8.51266	0.300032
$806$	−75.1424	−2.64678
$807$	0	0
$808$	57.8284	2.03440
$809$	5.38024	0.189159	0.0945796	−	0.995517i	$-0.469849\pi$
$809$	5.38024	0.189159	0.0945796	+	0.995517i	$0.469849\pi$
$810$	0	0
$811$	13.6999	0.481070	0.240535	−	0.970641i	$-0.422677\pi$
$811$	13.6999	0.481070	0.240535	+	0.970641i	$0.422677\pi$
$812$	−17.5643	−0.616387
$813$	0	0
$814$	0	0
$815$	19.9333	0.698231
$816$	0	0
$817$	−0.379421	−0.0132743
$818$	48.3794	1.69155
$819$	0	0
$820$	−1.54651	−0.0540064
$821$	9.23116	0.322170	0.161085	−	0.986941i	$-0.448501\pi$
$821$	9.23116	0.322170	0.161085	+	0.986941i	$0.448501\pi$
$822$	0	0
$823$	−12.4427	−0.433724	−0.216862	−	0.976202i	$-0.569582\pi$
$823$	−12.4427	−0.433724	−0.216862	+	0.976202i	$0.569582\pi$
$824$	38.0915	1.32698
$825$	0	0
$826$	16.8840	0.587469
$827$	4.48387	0.155919	0.0779597	−	0.996957i	$-0.475159\pi$
$827$	4.48387	0.155919	0.0779597	+	0.996957i	$0.475159\pi$
$828$	0	0
$829$	−36.8776	−1.28081	−0.640405	−	0.768037i	$-0.721234\pi$
$829$	−36.8776	−1.28081	−0.640405	+	0.768037i	$0.721234\pi$
$830$	2.90772	0.100928
$831$	0	0
$832$	28.7528	0.996823
$833$	−5.92418	−0.205261
$834$	0	0
$835$	26.1060	0.903435
$836$	0	0
$837$	0	0
$838$	75.0905	2.59396
$839$	−9.16407	−0.316379	−0.158189	−	0.987409i	$-0.550566\pi$
$839$	−9.16407	−0.316379	−0.158189	+	0.987409i	$0.550566\pi$
$840$	0	0
$841$	−8.79054	−0.303122
$842$	−58.8003	−2.02639
$843$	0	0
$844$	−55.7293	−1.91828
$845$	3.64795	0.125493
$846$	0	0
$847$	0	0
$848$	−15.0195	−0.515771
$849$	0	0
$850$	49.1829	1.68696
$851$	36.8942	1.26472
$852$	0	0
$853$	6.52049	0.223257	0.111629	−	0.993750i	$-0.464393\pi$
$853$	6.52049	0.223257	0.111629	+	0.993750i	$0.464393\pi$
$854$	−5.95984	−0.203942
$855$	0	0
$856$	−56.4952	−1.93097
$857$	−48.0736	−1.64216	−0.821082	−	0.570810i	$-0.806629\pi$
$857$	−48.0736	−1.64216	−0.821082	+	0.570810i	$0.806629\pi$
$858$	0	0
$859$	−0.316298	−0.0107920	−0.00539598	−	0.999985i	$-0.501718\pi$
$859$	−0.316298	−0.0107920	−0.00539598	+	0.999985i	$0.501718\pi$
$860$	0.651887	0.0222291
$861$	0	0
$862$	−47.1509	−1.60597
$863$	−3.62693	−0.123462	−0.0617311	−	0.998093i	$-0.519662\pi$
$863$	−3.62693	−0.123462	−0.0617311	+	0.998093i	$0.519662\pi$
$864$	0	0
$865$	−27.1303	−0.922458
$866$	−51.9818	−1.76641
$867$	0	0
$868$	38.0064	1.29002
$869$	0	0
$870$	0	0
$871$	29.9393	1.01445
$872$	4.10712	0.139085
$873$	0	0
$874$	−47.0497	−1.59148
$875$	10.5924	0.358090
$876$	0	0
$877$	26.7076	0.901852	0.450926	−	0.892561i	$-0.351094\pi$
$877$	26.7076	0.901852	0.450926	+	0.892561i	$0.351094\pi$
$878$	−77.1220	−2.60274
$879$	0	0
$880$	0	0
$881$	−2.91937	−0.0983560	−0.0491780	−	0.998790i	$-0.515660\pi$
$881$	−2.91937	−0.0983560	−0.0491780	+	0.998790i	$0.515660\pi$
$882$	0	0
$883$	−45.1574	−1.51967	−0.759834	−	0.650117i	$-0.774720\pi$
$883$	−45.1574	−1.51967	−0.759834	+	0.650117i	$0.774720\pi$
$884$	73.5663	2.47430
$885$	0	0
$886$	4.04939	0.136042
$887$	24.0249	0.806676	0.403338	−	0.915051i	$-0.367850\pi$
$887$	24.0249	0.806676	0.403338	+	0.915051i	$0.367850\pi$
$888$	0	0
$889$	−8.02779	−0.269243
$890$	30.7592	1.03105
$891$	0	0
$892$	−11.5806	−0.387749
$893$	−26.8334	−0.897947
$894$	0	0
$895$	6.01828	0.201169
$896$	−20.2229	−0.675599
$897$	0	0
$898$	41.7692	1.39386
$899$	−43.7300	−1.45848
$900$	0	0
$901$	25.7816	0.858909
$902$	0	0
$903$	0	0
$904$	−21.0667	−0.700667
$905$	9.88384	0.328550
$906$	0	0
$907$	−16.0270	−0.532166	−0.266083	−	0.963950i	$-0.585730\pi$
$907$	−16.0270	−0.532166	−0.266083	+	0.963950i	$0.585730\pi$
$908$	−32.2572	−1.07049
$909$	0	0
$910$	9.72259	0.322301
$911$	16.4043	0.543500	0.271750	−	0.962368i	$-0.412398\pi$
$911$	16.4043	0.543500	0.271750	+	0.962368i	$0.412398\pi$
$912$	0	0
$913$	0	0
$914$	24.8408	0.821660
$915$	0	0
$916$	51.2147	1.69218
$917$	−0.101461	−0.00335053
$918$	0	0
$919$	32.3201	1.06614	0.533071	−	0.846070i	$-0.321038\pi$
$919$	32.3201	1.06614	0.533071	+	0.846070i	$0.321038\pi$
$920$	39.4572	1.30086
$921$	0	0
$922$	53.7518	1.77022
$923$	−0.370623	−0.0121992
$924$	0	0
$925$	18.6333	0.612659
$926$	−73.8171	−2.42578
$927$	0	0
$928$	3.96585	0.130185
$929$	8.19531	0.268879	0.134440	−	0.990922i	$-0.457077\pi$
$929$	8.19531	0.268879	0.134440	+	0.990922i	$0.457077\pi$
$930$	0	0
$931$	2.86222	0.0938054
$932$	50.1190	1.64170
$933$	0	0
$934$	−61.2253	−2.00335
$935$	0	0
$936$	0	0
$937$	34.1374	1.11522	0.557610	−	0.830103i	$-0.311718\pi$
$937$	34.1374	1.11522	0.557610	+	0.830103i	$0.311718\pi$
$938$	−22.8945	−0.747533
$939$	0	0
$940$	46.1027	1.50371
$941$	−0.979371	−0.0319266	−0.0159633	−	0.999873i	$-0.505081\pi$
$941$	−0.979371	−0.0319266	−0.0159633	+	0.999873i	$0.505081\pi$
$942$	0	0
$943$	−2.12699	−0.0692644
$944$	23.9752	0.780326
$945$	0	0
$946$	0	0
$947$	−0.935599	−0.0304029	−0.0152014	−	0.999884i	$-0.504839\pi$
$947$	−0.935599	−0.0304029	−0.0152014	+	0.999884i	$0.504839\pi$
$948$	0	0
$949$	−1.95476	−0.0634542
$950$	−23.7623	−0.770950
$951$	0	0
$952$	−27.4593	−0.889960
$953$	−16.7321	−0.542004	−0.271002	−	0.962579i	$-0.587355\pi$
$953$	−16.7321	−0.542004	−0.271002	+	0.962579i	$0.587355\pi$
$954$	0	0
$955$	−11.1141	−0.359643
$956$	19.4616	0.629434
$957$	0	0
$958$	51.5353	1.66503
$959$	4.56409	0.147382
$960$	0	0
$961$	63.6245	2.05240
$962$	42.1380	1.35859
$963$	0	0
$964$	−10.2634	−0.330561
$965$	−32.2238	−1.03732
$966$	0	0
$967$	36.4439	1.17196	0.585978	−	0.810327i	$-0.300710\pi$
$967$	36.4439	1.17196	0.585978	+	0.810327i	$0.300710\pi$
$968$	0	0
$969$	0	0
$970$	−53.7688	−1.72641
$971$	33.2417	1.06678	0.533389	−	0.845870i	$-0.320918\pi$
$971$	33.2417	1.06678	0.533389	+	0.845870i	$0.320918\pi$
$972$	0	0
$973$	−3.82533	−0.122634
$974$	−40.8796	−1.30987
$975$	0	0
$976$	−8.46294	−0.270892
$977$	−22.5666	−0.721970	−0.360985	−	0.932572i	$-0.617559\pi$
$977$	−22.5666	−0.721970	−0.360985	+	0.932572i	$0.617559\pi$
$978$	0	0
$979$	0	0
$980$	−4.91760	−0.157087
$981$	0	0
$982$	−11.7586	−0.375233
$983$	15.1048	0.481767	0.240884	−	0.970554i	$-0.422563\pi$
$983$	15.1048	0.481767	0.240884	+	0.970554i	$0.422563\pi$
$984$	0	0
$985$	−14.0938	−0.449064
$986$	64.7281	2.06136
$987$	0	0
$988$	−35.5429	−1.13077
$989$	0.896574	0.0285094
$990$	0	0
$991$	−55.1534	−1.75201	−0.876003	−	0.482305i	$-0.839800\pi$
$991$	−55.1534	−1.75201	−0.876003	+	0.482305i	$0.839800\pi$
$992$	−8.58145	−0.272461
$993$	0	0
$994$	0.283415	0.00898939
$995$	15.4186	0.488802
$996$	0	0
$997$	49.7048	1.57417	0.787083	−	0.616847i	$-0.211590\pi$
$997$	49.7048	1.57417	0.787083	+	0.616847i	$0.211590\pi$
$998$	74.3319	2.35293
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ct.1.8		8
3.2	odd	2		847.2.a.p.1.1		8
11.3	even	5		693.2.m.i.64.4		16
11.4	even	5		693.2.m.i.379.4		16
11.10	odd	2		7623.2.a.cw.1.1		8
21.20	even	2		5929.2.a.bt.1.1		8
33.2	even	10		847.2.f.v.323.1		16
33.5	odd	10		847.2.f.w.729.4		16
33.8	even	10		847.2.f.x.372.4		16
33.14	odd	10		77.2.f.b.64.1	✓	16
33.17	even	10		847.2.f.v.729.1		16
33.20	odd	10		847.2.f.w.323.4		16
33.26	odd	10		77.2.f.b.71.1	yes	16
33.29	even	10		847.2.f.x.148.4		16
33.32	even	2		847.2.a.o.1.8		8
231.26	even	30		539.2.q.f.214.1		32
231.47	even	30		539.2.q.f.361.4		32
231.59	even	30		539.2.q.f.324.4		32
231.80	even	30		539.2.q.f.471.1		32
231.125	even	10		539.2.f.e.148.1		16
231.146	even	10		539.2.f.e.295.1		16
231.158	odd	30		539.2.q.g.324.4		32
231.179	odd	30		539.2.q.g.471.1		32
231.191	odd	30		539.2.q.g.214.1		32
231.212	odd	30		539.2.q.g.361.4		32
231.230	odd	2		5929.2.a.bs.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.64.1	✓	16	33.14	odd	10
77.2.f.b.71.1	yes	16	33.26	odd	10
539.2.f.e.148.1		16	231.125	even	10
539.2.f.e.295.1		16	231.146	even	10
539.2.q.f.214.1		32	231.26	even	30
539.2.q.f.324.4		32	231.59	even	30
539.2.q.f.361.4		32	231.47	even	30
539.2.q.f.471.1		32	231.80	even	30
539.2.q.g.214.1		32	231.191	odd	30
539.2.q.g.324.4		32	231.158	odd	30
539.2.q.g.361.4		32	231.212	odd	30
539.2.q.g.471.1		32	231.179	odd	30
693.2.m.i.64.4		16	11.3	even	5
693.2.m.i.379.4		16	11.4	even	5
847.2.a.o.1.8		8	33.32	even	2
847.2.a.p.1.1		8	3.2	odd	2
847.2.f.v.323.1		16	33.2	even	10
847.2.f.v.729.1		16	33.17	even	10
847.2.f.w.323.4		16	33.20	odd	10
847.2.f.w.729.4		16	33.5	odd	10
847.2.f.x.148.4		16	33.29	even	10
847.2.f.x.372.4		16	33.8	even	10
5929.2.a.bs.1.8		8	231.230	odd	2
5929.2.a.bt.1.1		8	21.20	even	2
7623.2.a.ct.1.8		8	1.1	even	1	trivial
7623.2.a.cw.1.1		8	11.10	odd	2

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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+2.43045 q^{2} +3.90710 q^{4} -1.25863 q^{5} +1.00000 q^{7} +4.63512 q^{8} +O(q^{10})\)	\(q+2.43045 q^{2} +3.90710 q^{4} -1.25863 q^{5} +1.00000 q^{7} +4.63512 q^{8} -3.05904 q^{10} -3.17831 q^{13} +2.43045 q^{14} +3.45123 q^{16} -5.92418 q^{17} +2.86222 q^{19} -4.91760 q^{20} -6.76343 q^{23} -3.41585 q^{25} -7.72473 q^{26} +3.90710 q^{28} -4.49549 q^{29} +9.72751 q^{31} -0.882184 q^{32} -14.3984 q^{34} -1.25863 q^{35} -5.45495 q^{37} +6.95648 q^{38} -5.83390 q^{40} +0.314484 q^{41} -0.132562 q^{43} -16.4382 q^{46} -9.37505 q^{47} +1.00000 q^{49} -8.30206 q^{50} -12.4180 q^{52} -4.35192 q^{53} +4.63512 q^{56} -10.9261 q^{58} +6.94685 q^{59} -2.45215 q^{61} +23.6423 q^{62} -9.04656 q^{64} +4.00032 q^{65} -9.41987 q^{67} -23.1464 q^{68} -3.05904 q^{70} +0.116610 q^{71} +0.615032 q^{73} -13.2580 q^{74} +11.1830 q^{76} +8.52928 q^{79} -4.34382 q^{80} +0.764339 q^{82} -0.950532 q^{83} +7.45636 q^{85} -0.322186 q^{86} -10.0552 q^{89} -3.17831 q^{91} -26.4254 q^{92} -22.7856 q^{94} -3.60248 q^{95} +17.5770 q^{97} +2.43045 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7}+O(q^{10}) \) 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7	\( 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7} + 6 q^{10} - 6 q^{13} - q^{14} + q^{16} + 5 q^{17} - 13 q^{19} - 23 q^{20} - 16 q^{23} + 16 q^{25} + 6 q^{26} + 7 q^{28} - 9 q^{29} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} + 7 q^{37} + 10 q^{38} + 5 q^{40} + 10 q^{41} - 4 q^{43} + 4 q^{46} - 16 q^{47} + 8 q^{49} - 6 q^{50} - 41 q^{52} - 37 q^{53} - 15 q^{58} - q^{59} + 19 q^{61} + 18 q^{62} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 6 q^{70} - 13 q^{71} - 25 q^{73} - 33 q^{74} + 26 q^{76} - 4 q^{80} - 13 q^{82} + 25 q^{83} + 3 q^{85} - 4 q^{86} - 37 q^{89} - 6 q^{91} - 35 q^{92} - 42 q^{94} - 21 q^{95} + 15 q^{97} - q^{98}+O(q^{100}) \) 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7 + 6 * q^10 - 6 * q^13 - q^14 + q^16 + 5 * q^17 - 13 * q^19 - 23 * q^20 - 16 * q^23 + 16 * q^25 + 6 * q^26 + 7 * q^28 - 9 * q^29 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 + 7 * q^37 + 10 * q^38 + 5 * q^40 + 10 * q^41 - 4 * q^43 + 4 * q^46 - 16 * q^47 + 8 * q^49 - 6 * q^50 - 41 * q^52 - 37 * q^53 - 15 * q^58 - q^59 + 19 * q^61 + 18 * q^62 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 6 * q^70 - 13 * q^71 - 25 * q^73 - 33 * q^74 + 26 * q^76 - 4 * q^80 - 13 * q^82 + 25 * q^83 + 3 * q^85 - 4 * q^86 - 37 * q^89 - 6 * q^91 - 35 * q^92 - 42 * q^94 - 21 * q^95 + 15 * q^97 - q^98

Introduction

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