LMFDB - Embedded newform 7623.2.a.cl.1.4

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.477260$ 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)

$f(q)$	$=$	$q+1.77222 q^{2} +1.14077 q^{4} -0.589926 q^{5} +1.00000 q^{7} -1.52274 q^{8} +O(q^{10})$	$q+1.77222 q^{2} +1.14077 q^{4} -0.589926 q^{5} +1.00000 q^{7} -1.52274 q^{8} -1.04548 q^{10} +1.54445 q^{13} +1.77222 q^{14} -4.98018 q^{16} +7.14511 q^{17} +6.19059 q^{19} -0.672972 q^{20} -3.76314 q^{23} -4.65199 q^{25} +2.73710 q^{26} +1.14077 q^{28} +0.607298 q^{29} -6.87147 q^{31} -5.78051 q^{32} +12.6627 q^{34} -0.589926 q^{35} -7.70693 q^{37} +10.9711 q^{38} +0.898304 q^{40} +7.48287 q^{41} +10.9537 q^{43} -6.66913 q^{46} +10.2072 q^{47} +1.00000 q^{49} -8.24436 q^{50} +1.76186 q^{52} -7.04469 q^{53} -1.52274 q^{56} +1.07627 q^{58} +3.35386 q^{59} +3.37044 q^{61} -12.1778 q^{62} -0.283993 q^{64} -0.911108 q^{65} -2.19138 q^{67} +8.15095 q^{68} -1.04548 q^{70} +13.9430 q^{71} +2.58993 q^{73} -13.6584 q^{74} +7.06206 q^{76} +4.66351 q^{79} +2.93794 q^{80} +13.2613 q^{82} -3.50970 q^{83} -4.21508 q^{85} +19.4125 q^{86} -8.15095 q^{89} +1.54445 q^{91} -4.29289 q^{92} +18.0894 q^{94} -3.65199 q^{95} -4.79710 q^{97} +1.77222 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 3 q^{4} + 4 q^{5} + 4 q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q + q^2 + 3 * q^4 + 4 * q^5 + 4 * q^7 - 9 * q^8	$ 4 q + q^{2} + 3 q^{4} + 4 q^{5} + 4 q^{7} - 9 q^{8} - 10 q^{10} - 6 q^{13} + q^{14} - 3 q^{16} + 8 q^{17} + 10 q^{19} + 10 q^{23} + 12 q^{25} + 20 q^{26} + 3 q^{28} - 18 q^{31} - 2 q^{32} + 18 q^{34} + 4 q^{35} - 2 q^{37} + 8 q^{38} - 6 q^{40} + 10 q^{41} + 4 q^{43} - 11 q^{46} - 4 q^{47} + 4 q^{49} - 9 q^{50} - 20 q^{52} - 9 q^{56} + 14 q^{58} + 16 q^{59} - 14 q^{61} - 11 q^{64} - 28 q^{65} - 28 q^{67} - 16 q^{68} - 10 q^{70} + 18 q^{71} + 4 q^{73} - 41 q^{74} + 4 q^{76} + 20 q^{79} + 36 q^{80} - 24 q^{82} + 6 q^{83} + 20 q^{85} + 20 q^{86} + 16 q^{89} - 6 q^{91} + 22 q^{92} + 16 q^{94} + 16 q^{95} + 32 q^{97} + q^{98}+O(q^{100}) $ 4 * q + q^2 + 3 * q^4 + 4 * q^5 + 4 * q^7 - 9 * q^8 - 10 * q^10 - 6 * q^13 + q^14 - 3 * q^16 + 8 * q^17 + 10 * q^19 + 10 * q^23 + 12 * q^25 + 20 * q^26 + 3 * q^28 - 18 * q^31 - 2 * q^32 + 18 * q^34 + 4 * q^35 - 2 * q^37 + 8 * q^38 - 6 * q^40 + 10 * q^41 + 4 * q^43 - 11 * q^46 - 4 * q^47 + 4 * q^49 - 9 * q^50 - 20 * q^52 - 9 * q^56 + 14 * q^58 + 16 * q^59 - 14 * q^61 - 11 * q^64 - 28 * q^65 - 28 * q^67 - 16 * q^68 - 10 * q^70 + 18 * q^71 + 4 * q^73 - 41 * q^74 + 4 * q^76 + 20 * q^79 + 36 * q^80 - 24 * q^82 + 6 * q^83 + 20 * q^85 + 20 * q^86 + 16 * q^89 - 6 * q^91 + 22 * q^92 + 16 * q^94 + 16 * q^95 + 32 * 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 $

$2$	1.77222	1.25315	0.626575	−	0.779361i	$-0.284456\pi$
$2$	1.77222	1.25315	0.626575	+	0.779361i	$0.284456\pi$
$3$	0	0
$3$	0	0
$4$	1.14077	0.570387
$5$	−0.589926	−0.263823	−0.131911	−	0.991262i	$-0.542111\pi$
$5$	−0.589926	−0.263823	−0.131911	+	0.991262i	$0.542111\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.52274	−0.538370
$9$	0	0
$10$	−1.04548	−0.330610
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.54445	0.428352	0.214176	−	0.976795i	$-0.431293\pi$
$13$	1.54445	0.428352	0.214176	+	0.976795i	$0.431293\pi$
$14$	1.77222	0.473646
$15$	0	0
$16$	−4.98018	−1.24505
$17$	7.14511	1.73294	0.866472	−	0.499226i	$-0.166382\pi$
$17$	7.14511	1.73294	0.866472	+	0.499226i	$0.166382\pi$
$18$	0	0
$19$	6.19059	1.42022	0.710109	−	0.704092i	$-0.248646\pi$
$19$	6.19059	1.42022	0.710109	+	0.704092i	$0.248646\pi$
$20$	−0.672972	−0.150481
$21$	0	0
$22$	0	0
$23$	−3.76314	−0.784669	−0.392335	−	0.919823i	$-0.628332\pi$
$23$	−3.76314	−0.784669	−0.392335	+	0.919823i	$0.628332\pi$
$24$	0	0
$25$	−4.65199	−0.930398
$26$	2.73710	0.536790
$27$	0	0
$28$	1.14077	0.215586
$29$	0.607298	0.112772	0.0563862	−	0.998409i	$-0.482042\pi$
$29$	0.607298	0.112772	0.0563862	+	0.998409i	$0.482042\pi$
$30$	0	0
$31$	−6.87147	−1.23415	−0.617077	−	0.786903i	$-0.711683\pi$
$31$	−6.87147	−1.23415	−0.617077	+	0.786903i	$0.711683\pi$
$32$	−5.78051	−1.02186
$33$	0	0
$34$	12.6627	2.17164
$35$	−0.589926	−0.0997157
$36$	0	0
$37$	−7.70693	−1.26701	−0.633505	−	0.773738i	$-0.718385\pi$
$37$	−7.70693	−1.26701	−0.633505	+	0.773738i	$0.718385\pi$
$38$	10.9711	1.77975
$39$	0	0
$40$	0.898304	0.142034
$41$	7.48287	1.16863	0.584314	−	0.811528i	$-0.301364\pi$
$41$	7.48287	1.16863	0.584314	+	0.811528i	$0.301364\pi$
$42$	0	0
$43$	10.9537	1.67043	0.835214	−	0.549925i	$-0.185344\pi$
$43$	10.9537	1.67043	0.835214	+	0.549925i	$0.185344\pi$
$44$	0	0
$45$	0	0
$46$	−6.66913	−0.983309
$47$	10.2072	1.48887	0.744434	−	0.667696i	$-0.232719\pi$
$47$	10.2072	1.48887	0.744434	+	0.667696i	$0.232719\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−8.24436	−1.16593
$51$	0	0
$52$	1.76186	0.244327
$53$	−7.04469	−0.967663	−0.483831	−	0.875161i	$-0.660755\pi$
$53$	−7.04469	−0.967663	−0.483831	+	0.875161i	$0.660755\pi$
$54$	0	0
$55$	0	0
$56$	−1.52274	−0.203485
$57$	0	0
$58$	1.07627	0.141321
$59$	3.35386	0.436635	0.218318	−	0.975878i	$-0.429943\pi$
$59$	3.35386	0.436635	0.218318	+	0.975878i	$0.429943\pi$
$60$	0	0
$61$	3.37044	0.431541	0.215770	−	0.976444i	$-0.430774\pi$
$61$	3.37044	0.431541	0.215770	+	0.976444i	$0.430774\pi$
$62$	−12.1778	−1.54658
$63$	0	0
$64$	−0.283993	−0.0354992
$65$	−0.911108	−0.113009
$66$	0	0
$67$	−2.19138	−0.267719	−0.133860	−	0.991000i	$-0.542737\pi$
$67$	−2.19138	−0.267719	−0.133860	+	0.991000i	$0.542737\pi$
$68$	8.15095	0.988448
$69$	0	0
$70$	−1.04548	−0.124959
$71$	13.9430	1.65473	0.827364	−	0.561665i	$-0.189839\pi$
$71$	13.9430	1.65473	0.827364	+	0.561665i	$0.189839\pi$
$72$	0	0
$73$	2.58993	0.303128	0.151564	−	0.988447i	$-0.451569\pi$
$73$	2.58993	0.303128	0.151564	+	0.988447i	$0.451569\pi$
$74$	−13.6584	−1.58776
$75$	0	0
$76$	7.06206	0.810074
$77$	0	0
$78$	0	0
$79$	4.66351	0.524686	0.262343	−	0.964975i	$-0.415505\pi$
$79$	4.66351	0.524686	0.262343	+	0.964975i	$0.415505\pi$
$80$	2.93794	0.328471
$81$	0	0
$82$	13.2613	1.46447
$83$	−3.50970	−0.385240	−0.192620	−	0.981273i	$-0.561698\pi$
$83$	−3.50970	−0.385240	−0.192620	+	0.981273i	$0.561698\pi$
$84$	0	0
$85$	−4.21508	−0.457190
$86$	19.4125	2.09330
$87$	0	0
$88$	0	0
$89$	−8.15095	−0.863999	−0.432000	−	0.901874i	$-0.642192\pi$
$89$	−8.15095	−0.863999	−0.432000	+	0.901874i	$0.642192\pi$
$90$	0	0
$91$	1.54445	0.161902
$92$	−4.29289	−0.447565
$93$	0	0
$94$	18.0894	1.86578
$95$	−3.65199	−0.374686
$96$	0	0
$97$	−4.79710	−0.487071	−0.243536	−	0.969892i	$-0.578307\pi$
$97$	−4.79710	−0.487071	−0.243536	+	0.969892i	$0.578307\pi$
$98$	1.77222	0.179022
$99$	0	0
$100$	−5.30687	−0.530687
$101$	−1.42666	−0.141958	−0.0709788	−	0.997478i	$-0.522612\pi$
$101$	−1.42666	−0.141958	−0.0709788	+	0.997478i	$0.522612\pi$
$102$	0	0
$103$	17.6172	1.73588	0.867939	−	0.496670i	$-0.165444\pi$
$103$	17.6172	1.73588	0.867939	+	0.496670i	$0.165444\pi$
$104$	−2.35179	−0.230612
$105$	0	0
$106$	−12.4848	−1.21263
$107$	−7.20005	−0.696055	−0.348027	−	0.937484i	$-0.613148\pi$
$107$	−7.20005	−0.696055	−0.348027	+	0.937484i	$0.613148\pi$
$108$	0	0
$109$	8.88678	0.851199	0.425599	−	0.904912i	$-0.360063\pi$
$109$	8.88678	0.851199	0.425599	+	0.904912i	$0.360063\pi$
$110$	0	0
$111$	0	0
$112$	−4.98018	−0.470583
$113$	7.80862	0.734573	0.367287	−	0.930108i	$-0.380287\pi$
$113$	7.80862	0.734573	0.367287	+	0.930108i	$0.380287\pi$
$114$	0	0
$115$	2.21997	0.207014
$116$	0.692789	0.0643239
$117$	0	0
$118$	5.94378	0.547170
$119$	7.14511	0.654991
$120$	0	0
$121$	0	0
$122$	5.97317	0.540785
$123$	0	0
$124$	−7.83880	−0.703945
$125$	5.69396	0.509283
$126$	0	0
$127$	−8.25186	−0.732234	−0.366117	−	0.930569i	$-0.619313\pi$
$127$	−8.25186	−0.732234	−0.366117	+	0.930569i	$0.619313\pi$
$128$	11.0577	0.977374
$129$	0	0
$130$	−1.61469	−0.141617
$131$	12.1162	1.05860	0.529299	−	0.848435i	$-0.322455\pi$
$131$	12.1162	1.05860	0.529299	+	0.848435i	$0.322455\pi$
$132$	0	0
$133$	6.19059	0.536792
$134$	−3.88361	−0.335493
$135$	0	0
$136$	−10.8801	−0.932964
$137$	17.4548	1.49126	0.745631	−	0.666360i	$-0.232148\pi$
$137$	17.4548	1.49126	0.745631	+	0.666360i	$0.232148\pi$
$138$	0	0
$139$	−14.7624	−1.25213	−0.626063	−	0.779772i	$-0.715335\pi$
$139$	−14.7624	−1.25213	−0.626063	+	0.779772i	$0.715335\pi$
$140$	−0.672972	−0.0568765
$141$	0	0
$142$	24.7101	2.07362
$143$	0	0
$144$	0	0
$145$	−0.358261	−0.0297519
$146$	4.58993	0.379865
$147$	0	0
$148$	−8.79186	−0.722686
$149$	12.5258	1.02615	0.513077	−	0.858343i	$-0.328506\pi$
$149$	12.5258	1.02615	0.513077	+	0.858343i	$0.328506\pi$
$150$	0	0
$151$	−0.328572	−0.0267388	−0.0133694	−	0.999911i	$-0.504256\pi$
$151$	−0.328572	−0.0267388	−0.0133694	+	0.999911i	$0.504256\pi$
$152$	−9.42666	−0.764603
$153$	0	0
$154$	0	0
$155$	4.05366	0.325598
$156$	0	0
$157$	−2.68089	−0.213958	−0.106979	−	0.994261i	$-0.534118\pi$
$157$	−2.68089	−0.213958	−0.106979	+	0.994261i	$0.534118\pi$
$158$	8.26479	0.657511
$159$	0	0
$160$	3.41007	0.269590
$161$	−3.76314	−0.296577
$162$	0	0
$163$	−0.891668	−0.0698408	−0.0349204	−	0.999390i	$-0.511118\pi$
$163$	−0.891668	−0.0698408	−0.0349204	+	0.999390i	$0.511118\pi$
$164$	8.53627	0.666570
$165$	0	0
$166$	−6.21997	−0.482764
$167$	11.5610	0.894619	0.447310	−	0.894379i	$-0.352382\pi$
$167$	11.5610	0.894619	0.447310	+	0.894379i	$0.352382\pi$
$168$	0	0
$169$	−10.6147	−0.816514
$170$	−7.47007	−0.572928
$171$	0	0
$172$	12.4957	0.952790
$173$	8.98926	0.683441	0.341721	−	0.939802i	$-0.388990\pi$
$173$	8.98926	0.683441	0.341721	+	0.939802i	$0.388990\pi$
$174$	0	0
$175$	−4.65199	−0.351657
$176$	0	0
$177$	0	0
$178$	−14.4453	−1.08272
$179$	6.14590	0.459366	0.229683	−	0.973265i	$-0.426231\pi$
$179$	6.14590	0.459366	0.229683	+	0.973265i	$0.426231\pi$
$180$	0	0
$181$	14.9662	1.11243	0.556215	−	0.831039i	$-0.312253\pi$
$181$	14.9662	1.11243	0.556215	+	0.831039i	$0.312253\pi$
$182$	2.73710	0.202888
$183$	0	0
$184$	5.73029	0.422442
$185$	4.54651	0.334266
$186$	0	0
$187$	0	0
$188$	11.6441	0.849231
$189$	0	0
$190$	−6.47214	−0.469538
$191$	−9.87351	−0.714422	−0.357211	−	0.934024i	$-0.616272\pi$
$191$	−9.87351	−0.714422	−0.357211	+	0.934024i	$0.616272\pi$
$192$	0	0
$193$	6.18113	0.444927	0.222464	−	0.974941i	$-0.428590\pi$
$193$	6.18113	0.444927	0.222464	+	0.974941i	$0.428590\pi$
$194$	−8.50152	−0.610374
$195$	0	0
$196$	1.14077	0.0814839
$197$	−19.0720	−1.35882	−0.679412	−	0.733757i	$-0.737765\pi$
$197$	−19.0720	−1.35882	−0.679412	+	0.733757i	$0.737765\pi$
$198$	0	0
$199$	−11.8479	−0.839878	−0.419939	−	0.907552i	$-0.637948\pi$
$199$	−11.8479	−0.839878	−0.419939	+	0.907552i	$0.637948\pi$
$200$	7.08377	0.500898
$201$	0	0
$202$	−2.52835	−0.177894
$203$	0.607298	0.0426239
$204$	0	0
$205$	−4.41434	−0.308311
$206$	31.2217	2.17532
$207$	0	0
$208$	−7.69162	−0.533318
$209$	0	0
$210$	0	0
$211$	20.9810	1.44439	0.722197	−	0.691687i	$-0.243132\pi$
$211$	20.9810	1.44439	0.722197	+	0.691687i	$0.243132\pi$
$212$	−8.03640	−0.551942
$213$	0	0
$214$	−12.7601	−0.872262
$215$	−6.46189	−0.440697
$216$	0	0
$217$	−6.87147	−0.466466
$218$	15.7493	1.06668
$219$	0	0
$220$	0	0
$221$	11.0352	0.742310
$222$	0	0
$223$	3.38860	0.226918	0.113459	−	0.993543i	$-0.463807\pi$
$223$	3.38860	0.226918	0.113459	+	0.993543i	$0.463807\pi$
$224$	−5.78051	−0.386227
$225$	0	0
$226$	13.8386	0.920531
$227$	19.3973	1.28744	0.643721	−	0.765260i	$-0.277390\pi$
$227$	19.3973	1.28744	0.643721	+	0.765260i	$0.277390\pi$
$228$	0	0
$229$	−9.86405	−0.651835	−0.325917	−	0.945398i	$-0.605673\pi$
$229$	−9.86405	−0.651835	−0.325917	+	0.945398i	$0.605673\pi$
$230$	3.93429	0.259419
$231$	0	0
$232$	−0.924756	−0.0607132
$233$	7.12619	0.466852	0.233426	−	0.972375i	$-0.425006\pi$
$233$	7.12619	0.466852	0.233426	+	0.972375i	$0.425006\pi$
$234$	0	0
$235$	−6.02147	−0.392798
$236$	3.82599	0.249051
$237$	0	0
$238$	12.6627	0.820802
$239$	22.2894	1.44178	0.720892	−	0.693048i	$-0.243733\pi$
$239$	22.2894	1.44178	0.720892	+	0.693048i	$0.243733\pi$
$240$	0	0
$241$	−14.8753	−0.958199	−0.479100	−	0.877761i	$-0.659037\pi$
$241$	−14.8753	−0.958199	−0.479100	+	0.877761i	$0.659037\pi$
$242$	0	0
$243$	0	0
$244$	3.84491	0.246145
$245$	−0.589926	−0.0376890
$246$	0	0
$247$	9.56103	0.608354
$248$	10.4635	0.664431
$249$	0	0
$250$	10.0910	0.638208
$251$	25.9233	1.63626	0.818132	−	0.575031i	$-0.195010\pi$
$251$	25.9233	1.63626	0.818132	+	0.575031i	$0.195010\pi$
$252$	0	0
$253$	0	0
$254$	−14.6241	−0.917600
$255$	0	0
$256$	20.1647	1.26030
$257$	−0.871474	−0.0543610	−0.0271805	−	0.999631i	$-0.508653\pi$
$257$	−0.871474	−0.0543610	−0.0271805	+	0.999631i	$0.508653\pi$
$258$	0	0
$259$	−7.70693	−0.478885
$260$	−1.03937	−0.0644589
$261$	0	0
$262$	21.4726	1.32658
$263$	26.8873	1.65794	0.828970	−	0.559293i	$-0.188927\pi$
$263$	26.8873	1.65794	0.828970	+	0.559293i	$0.188927\pi$
$264$	0	0
$265$	4.15584	0.255291
$266$	10.9711	0.672681
$267$	0	0
$268$	−2.49987	−0.152704
$269$	0.992574	0.0605183	0.0302592	−	0.999542i	$-0.490367\pi$
$269$	0.992574	0.0605183	0.0302592	+	0.999542i	$0.490367\pi$
$270$	0	0
$271$	−26.8650	−1.63193	−0.815966	−	0.578099i	$-0.803795\pi$
$271$	−26.8650	−1.63193	−0.815966	+	0.578099i	$0.803795\pi$
$272$	−35.5839	−2.15759
$273$	0	0
$274$	30.9337	1.86878
$275$	0	0
$276$	0	0
$277$	27.6302	1.66014	0.830069	−	0.557660i	$-0.188301\pi$
$277$	27.6302	1.66014	0.830069	+	0.557660i	$0.188301\pi$
$278$	−26.1622	−1.56910
$279$	0	0
$280$	0.898304	0.0536839
$281$	23.8418	1.42228	0.711141	−	0.703050i	$-0.248179\pi$
$281$	23.8418	1.42228	0.711141	+	0.703050i	$0.248179\pi$
$282$	0	0
$283$	−8.74735	−0.519976	−0.259988	−	0.965612i	$-0.583719\pi$
$283$	−8.74735	−0.519976	−0.259988	+	0.965612i	$0.583719\pi$
$284$	15.9058	0.943836
$285$	0	0
$286$	0	0
$287$	7.48287	0.441700
$288$	0	0
$289$	34.0526	2.00309
$290$	−0.634918	−0.0372836
$291$	0	0
$292$	2.95452	0.172900
$293$	−19.6252	−1.14651	−0.573257	−	0.819376i	$-0.694320\pi$
$293$	−19.6252	−1.14651	−0.573257	+	0.819376i	$0.694320\pi$
$294$	0	0
$295$	−1.97853	−0.115194
$296$	11.7356	0.682120
$297$	0	0
$298$	22.1985	1.28592
$299$	−5.81197	−0.336115
$300$	0	0
$301$	10.9537	0.631362
$302$	−0.582303	−0.0335078
$303$	0	0
$304$	−30.8303	−1.76824
$305$	−1.98831	−0.113850
$306$	0	0
$307$	17.0835	0.975009	0.487504	−	0.873121i	$-0.337907\pi$
$307$	17.0835	0.975009	0.487504	+	0.873121i	$0.337907\pi$
$308$	0	0
$309$	0	0
$310$	7.18399	0.408023
$311$	7.23351	0.410175	0.205087	−	0.978744i	$-0.434252\pi$
$311$	7.23351	0.410175	0.205087	+	0.978744i	$0.434252\pi$
$312$	0	0
$313$	13.7784	0.778804	0.389402	−	0.921068i	$-0.372682\pi$
$313$	13.7784	0.778804	0.389402	+	0.921068i	$0.372682\pi$
$314$	−4.75113	−0.268122
$315$	0	0
$316$	5.32002	0.299274
$317$	6.95629	0.390704	0.195352	−	0.980733i	$-0.437415\pi$
$317$	6.95629	0.390704	0.195352	+	0.980733i	$0.437415\pi$
$318$	0	0
$319$	0	0
$320$	0.167535	0.00936550
$321$	0	0
$322$	−6.66913	−0.371656
$323$	44.2324	2.46116
$324$	0	0
$325$	−7.18474	−0.398538
$326$	−1.58023	−0.0875211
$327$	0	0
$328$	−11.3945	−0.629154
$329$	10.2072	0.562739
$330$	0	0
$331$	−29.6698	−1.63080	−0.815401	−	0.578896i	$-0.803484\pi$
$331$	−29.6698	−1.63080	−0.815401	+	0.578896i	$0.803484\pi$
$332$	−4.00378	−0.219736
$333$	0	0
$334$	20.4887	1.12109
$335$	1.29275	0.0706305
$336$	0	0
$337$	−19.5714	−1.06612	−0.533062	−	0.846076i	$-0.678959\pi$
$337$	−19.5714	−1.06612	−0.533062	+	0.846076i	$0.678959\pi$
$338$	−18.8116	−1.02322
$339$	0	0
$340$	−4.80846	−0.260775
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−16.6797	−0.899308
$345$	0	0
$346$	15.9310	0.856455
$347$	−6.49184	−0.348500	−0.174250	−	0.984701i	$-0.555750\pi$
$347$	−6.49184	−0.348500	−0.174250	+	0.984701i	$0.555750\pi$
$348$	0	0
$349$	−7.87965	−0.421788	−0.210894	−	0.977509i	$-0.567638\pi$
$349$	−7.87965	−0.421788	−0.210894	+	0.977509i	$0.567638\pi$
$350$	−8.24436	−0.440680
$351$	0	0
$352$	0	0
$353$	5.34594	0.284536	0.142268	−	0.989828i	$-0.454561\pi$
$353$	5.34594	0.284536	0.142268	+	0.989828i	$0.454561\pi$
$354$	0	0
$355$	−8.22533	−0.436555
$356$	−9.29840	−0.492814
$357$	0	0
$358$	10.8919	0.575655
$359$	−17.8515	−0.942169	−0.471084	−	0.882088i	$-0.656137\pi$
$359$	−17.8515	−0.942169	−0.471084	+	0.882088i	$0.656137\pi$
$360$	0	0
$361$	19.3234	1.01702
$362$	26.5235	1.39404
$363$	0	0
$364$	1.76186	0.0923467
$365$	−1.52786	−0.0799721
$366$	0	0
$367$	−29.9357	−1.56263	−0.781316	−	0.624135i	$-0.785451\pi$
$367$	−29.9357	−1.56263	−0.781316	+	0.624135i	$0.785451\pi$
$368$	18.7411	0.976949
$369$	0	0
$370$	8.05744	0.418886
$371$	−7.04469	−0.365742
$372$	0	0
$373$	−23.4915	−1.21635	−0.608173	−	0.793805i	$-0.708097\pi$
$373$	−23.4915	−1.21635	−0.608173	+	0.793805i	$0.708097\pi$
$374$	0	0
$375$	0	0
$376$	−15.5429	−0.801562
$377$	0.937938	0.0483063
$378$	0	0
$379$	−31.4532	−1.61564	−0.807821	−	0.589428i	$-0.799353\pi$
$379$	−31.4532	−1.61564	−0.807821	+	0.589428i	$0.799353\pi$
$380$	−4.16609	−0.213716
$381$	0	0
$382$	−17.4981	−0.895278
$383$	5.62749	0.287551	0.143776	−	0.989610i	$-0.454076\pi$
$383$	5.62749	0.287551	0.143776	+	0.989610i	$0.454076\pi$
$384$	0	0
$385$	0	0
$386$	10.9543	0.557561
$387$	0	0
$388$	−5.47240	−0.277819
$389$	−26.1072	−1.32369	−0.661845	−	0.749641i	$-0.730226\pi$
$389$	−26.1072	−1.32369	−0.661845	+	0.749641i	$0.730226\pi$
$390$	0	0
$391$	−26.8881	−1.35979
$392$	−1.52274	−0.0769100
$393$	0	0
$394$	−33.7998	−1.70281
$395$	−2.75113	−0.138424
$396$	0	0
$397$	10.3828	0.521096	0.260548	−	0.965461i	$-0.416097\pi$
$397$	10.3828	0.521096	0.260548	+	0.965461i	$0.416097\pi$
$398$	−20.9972	−1.05249
$399$	0	0
$400$	23.1677	1.15839
$401$	−3.68876	−0.184208	−0.0921040	−	0.995749i	$-0.529359\pi$
$401$	−3.68876	−0.184208	−0.0921040	+	0.995749i	$0.529359\pi$
$402$	0	0
$403$	−10.6126	−0.528652
$404$	−1.62749	−0.0809708
$405$	0	0
$406$	1.07627	0.0534142
$407$	0	0
$408$	0	0
$409$	−1.08022	−0.0534136	−0.0267068	−	0.999643i	$-0.508502\pi$
$409$	−1.08022	−0.0534136	−0.0267068	+	0.999643i	$0.508502\pi$
$410$	−7.82319	−0.386360
$411$	0	0
$412$	20.0973	0.990123
$413$	3.35386	0.165033
$414$	0	0
$415$	2.07046	0.101635
$416$	−8.92769	−0.437716
$417$	0	0
$418$	0	0
$419$	−10.0766	−0.492273	−0.246136	−	0.969235i	$-0.579161\pi$
$419$	−10.0766	−0.492273	−0.246136	+	0.969235i	$0.579161\pi$
$420$	0	0
$421$	11.9757	0.583658	0.291829	−	0.956470i	$-0.405736\pi$
$421$	11.9757	0.583658	0.291829	+	0.956470i	$0.405736\pi$
$422$	37.1831	1.81004
$423$	0	0
$424$	10.7272	0.520960
$425$	−33.2390	−1.61233
$426$	0	0
$427$	3.37044	0.163107
$428$	−8.21363	−0.397021
$429$	0	0
$430$	−11.4519	−0.552260
$431$	−0.548547	−0.0264226	−0.0132113	−	0.999913i	$-0.504205\pi$
$431$	−0.548547	−0.0264226	−0.0132113	+	0.999913i	$0.504205\pi$
$432$	0	0
$433$	19.5879	0.941332	0.470666	−	0.882311i	$-0.344014\pi$
$433$	19.5879	0.941332	0.470666	+	0.882311i	$0.344014\pi$
$434$	−12.1778	−0.584552
$435$	0	0
$436$	10.1378	0.485513
$437$	−23.2961	−1.11440
$438$	0	0
$439$	−37.4069	−1.78533	−0.892667	−	0.450716i	$-0.851169\pi$
$439$	−37.4069	−1.78533	−0.892667	+	0.450716i	$0.851169\pi$
$440$	0	0
$441$	0	0
$442$	19.5569	0.930226
$443$	−21.4478	−1.01902	−0.509508	−	0.860466i	$-0.670173\pi$
$443$	−21.4478	−1.01902	−0.509508	+	0.860466i	$0.670173\pi$
$444$	0	0
$445$	4.80846	0.227943
$446$	6.00536	0.284362
$447$	0	0
$448$	−0.283993	−0.0134174
$449$	−6.39904	−0.301989	−0.150995	−	0.988535i	$-0.548248\pi$
$449$	−6.39904	−0.301989	−0.150995	+	0.988535i	$0.548248\pi$
$450$	0	0
$451$	0	0
$452$	8.90787	0.418991
$453$	0	0
$454$	34.3763	1.61336
$455$	−0.911108	−0.0427134
$456$	0	0
$457$	36.9430	1.72812	0.864060	−	0.503389i	$-0.167914\pi$
$457$	36.9430	1.72812	0.864060	+	0.503389i	$0.167914\pi$
$458$	−17.4813	−0.816847
$459$	0	0
$460$	2.53249	0.118078
$461$	7.39825	0.344571	0.172285	−	0.985047i	$-0.444885\pi$
$461$	7.39825	0.344571	0.172285	+	0.985047i	$0.444885\pi$
$462$	0	0
$463$	−14.3717	−0.667910	−0.333955	−	0.942589i	$-0.608383\pi$
$463$	−14.3717	−0.667910	−0.333955	+	0.942589i	$0.608383\pi$
$464$	−3.02445	−0.140407
$465$	0	0
$466$	12.6292	0.585036
$467$	−36.1626	−1.67341	−0.836704	−	0.547656i	$-0.815520\pi$
$467$	−36.1626	−1.67341	−0.836704	+	0.547656i	$0.815520\pi$
$468$	0	0
$469$	−2.19138	−0.101188
$470$	−10.6714	−0.492235
$471$	0	0
$472$	−5.10705	−0.235071
$473$	0	0
$474$	0	0
$475$	−28.7985	−1.32137
$476$	8.15095	0.373598
$477$	0	0
$478$	39.5018	1.80677
$479$	16.0252	0.732212	0.366106	−	0.930573i	$-0.380691\pi$
$479$	16.0252	0.732212	0.366106	+	0.930573i	$0.380691\pi$
$480$	0	0
$481$	−11.9029	−0.542727
$482$	−26.3623	−1.20077
$483$	0	0
$484$	0	0
$485$	2.82993	0.128501
$486$	0	0
$487$	1.55059	0.0702641	0.0351320	−	0.999383i	$-0.488815\pi$
$487$	1.55059	0.0702641	0.0351320	+	0.999383i	$0.488815\pi$
$488$	−5.13230	−0.232328
$489$	0	0
$490$	−1.04548	−0.0472300
$491$	−0.0331635	−0.00149665	−0.000748324	−	1.00000i	$-0.500238\pi$
$491$	−0.0331635	−0.00149665	−0.000748324		1.00000i	$0.500238\pi$
$492$	0	0
$493$	4.33921	0.195428
$494$	16.9443	0.762359
$495$	0	0
$496$	34.2212	1.53658
$497$	13.9430	0.625429
$498$	0	0
$499$	−27.0314	−1.21009	−0.605046	−	0.796190i	$-0.706845\pi$
$499$	−27.0314	−1.21009	−0.605046	+	0.796190i	$0.706845\pi$
$500$	6.49552	0.290488
$501$	0	0
$502$	45.9418	2.05048
$503$	−29.3341	−1.30794	−0.653972	−	0.756519i	$-0.726899\pi$
$503$	−29.3341	−1.30794	−0.653972	+	0.756519i	$0.726899\pi$
$504$	0	0
$505$	0.841621	0.0374516
$506$	0	0
$507$	0	0
$508$	−9.41351	−0.417657
$509$	16.6329	0.737239	0.368620	−	0.929580i	$-0.379831\pi$
$509$	16.6329	0.737239	0.368620	+	0.929580i	$0.379831\pi$
$510$	0	0
$511$	2.58993	0.114572
$512$	13.6210	0.601967
$513$	0	0
$514$	−1.54445	−0.0681226
$515$	−10.3929	−0.457964
$516$	0	0
$517$	0	0
$518$	−13.6584	−0.600115
$519$	0	0
$520$	1.38738	0.0608407
$521$	−35.8923	−1.57247	−0.786236	−	0.617927i	$-0.787973\pi$
$521$	−35.8923	−1.57247	−0.786236	+	0.617927i	$0.787973\pi$
$522$	0	0
$523$	−7.27873	−0.318276	−0.159138	−	0.987256i	$-0.550872\pi$
$523$	−7.27873	−0.318276	−0.159138	+	0.987256i	$0.550872\pi$
$524$	13.8219	0.603811
$525$	0	0
$526$	47.6502	2.07765
$527$	−49.0974	−2.13872
$528$	0	0
$529$	−8.83876	−0.384294
$530$	7.36508	0.319919
$531$	0	0
$532$	7.06206	0.306179
$533$	11.5569	0.500585
$534$	0	0
$535$	4.24749	0.183635
$536$	3.33690	0.144132
$537$	0	0
$538$	1.75906	0.0758386
$539$	0	0
$540$	0	0
$541$	10.4709	0.450177	0.225089	−	0.974338i	$-0.427733\pi$
$541$	10.4709	0.450177	0.225089	+	0.974338i	$0.427733\pi$
$542$	−47.6108	−2.04506
$543$	0	0
$544$	−41.3024	−1.77083
$545$	−5.24254	−0.224566
$546$	0	0
$547$	−8.16609	−0.349157	−0.174578	−	0.984643i	$-0.555856\pi$
$547$	−8.16609	−0.349157	−0.174578	+	0.984643i	$0.555856\pi$
$548$	19.9119	0.850596
$549$	0	0
$550$	0	0
$551$	3.75953	0.160161
$552$	0	0
$553$	4.66351	0.198313
$554$	48.9669	2.08040
$555$	0	0
$556$	−16.8405	−0.714197
$557$	11.8765	0.503225	0.251612	−	0.967828i	$-0.419039\pi$
$557$	11.8765	0.503225	0.251612	+	0.967828i	$0.419039\pi$
$558$	0	0
$559$	16.9174	0.715532
$560$	2.93794	0.124151
$561$	0	0
$562$	42.2530	1.78233
$563$	4.83929	0.203952	0.101976	−	0.994787i	$-0.467484\pi$
$563$	4.83929	0.203952	0.101976	+	0.994787i	$0.467484\pi$
$564$	0	0
$565$	−4.60651	−0.193797
$566$	−15.5023	−0.651608
$567$	0	0
$568$	−21.2316	−0.890856
$569$	6.52993	0.273749	0.136875	−	0.990588i	$-0.456294\pi$
$569$	6.52993	0.273749	0.136875	+	0.990588i	$0.456294\pi$
$570$	0	0
$571$	33.8691	1.41738	0.708689	−	0.705521i	$-0.249287\pi$
$571$	33.8691	1.41738	0.708689	+	0.705521i	$0.249287\pi$
$572$	0	0
$573$	0	0
$574$	13.2613	0.553517
$575$	17.5061	0.730054
$576$	0	0
$577$	−37.3609	−1.55536	−0.777678	−	0.628662i	$-0.783603\pi$
$577$	−37.3609	−1.55536	−0.777678	+	0.628662i	$0.783603\pi$
$578$	60.3487	2.51018
$579$	0	0
$580$	−0.408694	−0.0169701
$581$	−3.50970	−0.145607
$582$	0	0
$583$	0	0
$584$	−3.94378	−0.163195
$585$	0	0
$586$	−34.7802	−1.43675
$587$	2.07974	0.0858399	0.0429199	−	0.999079i	$-0.486334\pi$
$587$	2.07974	0.0858399	0.0429199	+	0.999079i	$0.486334\pi$
$588$	0	0
$589$	−42.5385	−1.75277
$590$	−3.50639	−0.144356
$591$	0	0
$592$	38.3819	1.57749
$593$	6.54983	0.268969	0.134485	−	0.990916i	$-0.457062\pi$
$593$	6.54983	0.268969	0.134485	+	0.990916i	$0.457062\pi$
$594$	0	0
$595$	−4.21508	−0.172802
$596$	14.2891	0.585304
$597$	0	0
$598$	−10.3001	−0.421203
$599$	4.98187	0.203554	0.101777	−	0.994807i	$-0.467547\pi$
$599$	4.98187	0.203554	0.101777	+	0.994807i	$0.467547\pi$
$600$	0	0
$601$	32.4713	1.32453	0.662266	−	0.749269i	$-0.269595\pi$
$601$	32.4713	1.32453	0.662266	+	0.749269i	$0.269595\pi$
$602$	19.4125	0.791192
$603$	0	0
$604$	−0.374827	−0.0152515
$605$	0	0
$606$	0	0
$607$	−3.13388	−0.127200	−0.0636002	−	0.997975i	$-0.520258\pi$
$607$	−3.13388	−0.127200	−0.0636002	+	0.997975i	$0.520258\pi$
$608$	−35.7848	−1.45126
$609$	0	0
$610$	−3.52373	−0.142672
$611$	15.7644	0.637760
$612$	0	0
$613$	−46.8484	−1.89219	−0.946094	−	0.323892i	$-0.895009\pi$
$613$	−46.8484	−1.89219	−0.946094	+	0.323892i	$0.895009\pi$
$614$	30.2758	1.22183
$615$	0	0
$616$	0	0
$617$	−26.5924	−1.07057	−0.535286	−	0.844671i	$-0.679796\pi$
$617$	−26.5924	−1.07057	−0.535286	+	0.844671i	$0.679796\pi$
$618$	0	0
$619$	−32.9765	−1.32544	−0.662718	−	0.748869i	$-0.730597\pi$
$619$	−32.9765	−1.32544	−0.662718	+	0.748869i	$0.730597\pi$
$620$	4.62431	0.185717
$621$	0	0
$622$	12.8194	0.514011
$623$	−8.15095	−0.326561
$624$	0	0
$625$	19.9009	0.796037
$626$	24.4185	0.975959
$627$	0	0
$628$	−3.05828	−0.122039
$629$	−55.0668	−2.19566
$630$	0	0
$631$	11.1721	0.444754	0.222377	−	0.974961i	$-0.428618\pi$
$631$	11.1721	0.444754	0.222377	+	0.974961i	$0.428618\pi$
$632$	−7.10132	−0.282475
$633$	0	0
$634$	12.3281	0.489611
$635$	4.86798	0.193180
$636$	0	0
$637$	1.54445	0.0611932
$638$	0	0
$639$	0	0
$640$	−6.52324	−0.257854
$641$	41.9153	1.65556	0.827778	−	0.561056i	$-0.189605\pi$
$641$	41.9153	1.65556	0.827778	+	0.561056i	$0.189605\pi$
$642$	0	0
$643$	29.0873	1.14709	0.573546	−	0.819173i	$-0.305567\pi$
$643$	29.0873	1.14709	0.573546	+	0.819173i	$0.305567\pi$
$644$	−4.29289	−0.169164
$645$	0	0
$646$	78.3897	3.08420
$647$	−1.89804	−0.0746196	−0.0373098	−	0.999304i	$-0.511879\pi$
$647$	−1.89804	−0.0746196	−0.0373098	+	0.999304i	$0.511879\pi$
$648$	0	0
$649$	0	0
$650$	−12.7330	−0.499428
$651$	0	0
$652$	−1.01719	−0.0398363
$653$	−20.2519	−0.792517	−0.396258	−	0.918139i	$-0.629692\pi$
$653$	−20.2519	−0.792517	−0.396258	+	0.918139i	$0.629692\pi$
$654$	0	0
$655$	−7.14766	−0.279282
$656$	−37.2661	−1.45500
$657$	0	0
$658$	18.0894	0.705197
$659$	−11.5620	−0.450392	−0.225196	−	0.974314i	$-0.572302\pi$
$659$	−11.5620	−0.450392	−0.225196	+	0.974314i	$0.572302\pi$
$660$	0	0
$661$	−32.6894	−1.27147	−0.635735	−	0.771907i	$-0.719303\pi$
$661$	−32.6894	−1.27147	−0.635735	+	0.771907i	$0.719303\pi$
$662$	−52.5816	−2.04364
$663$	0	0
$664$	5.34436	0.207402
$665$	−3.65199	−0.141618
$666$	0	0
$667$	−2.28535	−0.0884890
$668$	13.1885	0.510279
$669$	0	0
$670$	2.29104	0.0885107
$671$	0	0
$672$	0	0
$673$	20.1249	0.775758	0.387879	−	0.921710i	$-0.373208\pi$
$673$	20.1249	0.775758	0.387879	+	0.921710i	$0.373208\pi$
$674$	−34.6850	−1.33601
$675$	0	0
$676$	−12.1090	−0.465729
$677$	23.5746	0.906045	0.453022	−	0.891499i	$-0.350346\pi$
$677$	23.5746	0.906045	0.453022	+	0.891499i	$0.350346\pi$
$678$	0	0
$679$	−4.79710	−0.184096
$680$	6.41848	0.246137
$681$	0	0
$682$	0	0
$683$	−11.5812	−0.443143	−0.221572	−	0.975144i	$-0.571119\pi$
$683$	−11.5812	−0.443143	−0.221572	+	0.975144i	$0.571119\pi$
$684$	0	0
$685$	−10.2970	−0.393429
$686$	1.77222	0.0676638
$687$	0	0
$688$	−54.5516	−2.07976
$689$	−10.8801	−0.414500
$690$	0	0
$691$	43.8164	1.66685	0.833426	−	0.552630i	$-0.186376\pi$
$691$	43.8164	1.66685	0.833426	+	0.552630i	$0.186376\pi$
$692$	10.2547	0.389826
$693$	0	0
$694$	−11.5050	−0.436723
$695$	8.70869	0.330340
$696$	0	0
$697$	53.4659	2.02517
$698$	−13.9645	−0.528564
$699$	0	0
$700$	−5.30687	−0.200581
$701$	−22.8165	−0.861769	−0.430884	−	0.902407i	$-0.641798\pi$
$701$	−22.8165	−0.861769	−0.430884	+	0.902407i	$0.641798\pi$
$702$	0	0
$703$	−47.7104	−1.79943
$704$	0	0
$705$	0	0
$706$	9.47420	0.356566
$707$	−1.42666	−0.0536549
$708$	0	0
$709$	39.9746	1.50128	0.750638	−	0.660713i	$-0.229746\pi$
$709$	39.9746	1.50128	0.750638	+	0.660713i	$0.229746\pi$
$710$	−14.5771	−0.547070
$711$	0	0
$712$	12.4118	0.465151
$713$	25.8583	0.968402
$714$	0	0
$715$	0	0
$716$	7.01108	0.262016
$717$	0	0
$718$	−31.6369	−1.18068
$719$	26.2200	0.977840	0.488920	−	0.872329i	$-0.337391\pi$
$719$	26.2200	0.977840	0.488920	+	0.872329i	$0.337391\pi$
$720$	0	0
$721$	17.6172	0.656100
$722$	34.2453	1.27448
$723$	0	0
$724$	17.0731	0.634515
$725$	−2.82514	−0.104923
$726$	0	0
$727$	−22.2366	−0.824708	−0.412354	−	0.911024i	$-0.635293\pi$
$727$	−22.2366	−0.824708	−0.412354	+	0.911024i	$0.635293\pi$
$728$	−2.35179	−0.0871631
$729$	0	0
$730$	−2.70772	−0.100217
$731$	78.2656	2.89476
$732$	0	0
$733$	21.1284	0.780395	0.390197	−	0.920731i	$-0.372407\pi$
$733$	21.1284	0.780395	0.390197	+	0.920731i	$0.372407\pi$
$734$	−53.0528	−1.95821
$735$	0	0
$736$	21.7529	0.801822
$737$	0	0
$738$	0	0
$739$	−37.5036	−1.37959	−0.689795	−	0.724004i	$-0.742299\pi$
$739$	−37.5036	−1.37959	−0.689795	+	0.724004i	$0.742299\pi$
$740$	5.18655	0.190661
$741$	0	0
$742$	−12.4848	−0.458330
$743$	−36.1464	−1.32608	−0.663041	−	0.748583i	$-0.730735\pi$
$743$	−36.1464	−1.32608	−0.663041	+	0.748583i	$0.730735\pi$
$744$	0	0
$745$	−7.38929	−0.270723
$746$	−41.6322	−1.52426
$747$	0	0
$748$	0	0
$749$	−7.20005	−0.263084
$750$	0	0
$751$	−43.5533	−1.58928	−0.794641	−	0.607080i	$-0.792341\pi$
$751$	−43.5533	−1.58928	−0.794641	+	0.607080i	$0.792341\pi$
$752$	−50.8336	−1.85371
$753$	0	0
$754$	1.66224	0.0605351
$755$	0.193833	0.00705431
$756$	0	0
$757$	−44.4442	−1.61535	−0.807675	−	0.589628i	$-0.799274\pi$
$757$	−44.4442	−1.61535	−0.807675	+	0.589628i	$0.799274\pi$
$758$	−55.7421	−2.02464
$759$	0	0
$760$	5.56103	0.201720
$761$	−15.9613	−0.578598	−0.289299	−	0.957239i	$-0.593422\pi$
$761$	−15.9613	−0.578598	−0.289299	+	0.957239i	$0.593422\pi$
$762$	0	0
$763$	8.88678	0.321723
$764$	−11.2634	−0.407497
$765$	0	0
$766$	9.97317	0.360345
$767$	5.17985	0.187034
$768$	0	0
$769$	−5.19899	−0.187480	−0.0937402	−	0.995597i	$-0.529882\pi$
$769$	−5.19899	−0.187480	−0.0937402	+	0.995597i	$0.529882\pi$
$770$	0	0
$771$	0	0
$772$	7.05127	0.253781
$773$	12.1614	0.437416	0.218708	−	0.975790i	$-0.429816\pi$
$773$	12.1614	0.437416	0.218708	+	0.975790i	$0.429816\pi$
$774$	0	0
$775$	31.9660	1.14825
$776$	7.30473	0.262224
$777$	0	0
$778$	−46.2678	−1.65878
$779$	46.3234	1.65971
$780$	0	0
$781$	0	0
$782$	−47.6516	−1.70402
$783$	0	0
$784$	−4.98018	−0.177864
$785$	1.58152	0.0564470
$786$	0	0
$787$	39.6430	1.41312	0.706559	−	0.707654i	$-0.250246\pi$
$787$	39.6430	1.41312	0.706559	+	0.707654i	$0.250246\pi$
$788$	−21.7569	−0.775056
$789$	0	0
$790$	−4.87561	−0.173466
$791$	7.80862	0.277643
$792$	0	0
$793$	5.20546	0.184851
$794$	18.4006	0.653011
$795$	0	0
$796$	−13.5158	−0.479055
$797$	35.1118	1.24372	0.621862	−	0.783127i	$-0.286376\pi$
$797$	35.1118	1.24372	0.621862	+	0.783127i	$0.286376\pi$
$798$	0	0
$799$	72.9313	2.58012
$800$	26.8909	0.950736
$801$	0	0
$802$	−6.53731	−0.230840
$803$	0	0
$804$	0	0
$805$	2.21997	0.0782438
$806$	−18.8079	−0.662481
$807$	0	0
$808$	2.17243	0.0764257
$809$	−41.7873	−1.46916	−0.734581	−	0.678521i	$-0.762621\pi$
$809$	−41.7873	−1.46916	−0.734581	+	0.678521i	$0.762621\pi$
$810$	0	0
$811$	9.89879	0.347594	0.173797	−	0.984782i	$-0.444396\pi$
$811$	9.89879	0.347594	0.173797	+	0.984782i	$0.444396\pi$
$812$	0.692789	0.0243121
$813$	0	0
$814$	0	0
$815$	0.526018	0.0184256
$816$	0	0
$817$	67.8100	2.37237
$818$	−1.91440	−0.0669353
$819$	0	0
$820$	−5.03576	−0.175856
$821$	16.1936	0.565161	0.282580	−	0.959244i	$-0.408810\pi$
$821$	16.1936	0.565161	0.282580	+	0.959244i	$0.408810\pi$
$822$	0	0
$823$	−3.86642	−0.134775	−0.0673874	−	0.997727i	$-0.521466\pi$
$823$	−3.86642	−0.134775	−0.0673874	+	0.997727i	$0.521466\pi$
$824$	−26.8265	−0.934545
$825$	0	0
$826$	5.94378	0.206811
$827$	−46.8179	−1.62802	−0.814009	−	0.580852i	$-0.802720\pi$
$827$	−46.8179	−1.62802	−0.814009	+	0.580852i	$0.802720\pi$
$828$	0	0
$829$	−41.5599	−1.44344	−0.721718	−	0.692187i	$-0.756647\pi$
$829$	−41.5599	−1.44344	−0.721718	+	0.692187i	$0.756647\pi$
$830$	3.66932	0.127364
$831$	0	0
$832$	−0.438613	−0.0152062
$833$	7.14511	0.247563
$834$	0	0
$835$	−6.82015	−0.236021
$836$	0	0
$837$	0	0
$838$	−17.8579	−0.616892
$839$	−5.14119	−0.177494	−0.0887469	−	0.996054i	$-0.528286\pi$
$839$	−5.14119	−0.177494	−0.0887469	+	0.996054i	$0.528286\pi$
$840$	0	0
$841$	−28.6312	−0.987282
$842$	21.2236	0.731412
$843$	0	0
$844$	23.9346	0.823864
$845$	6.26188	0.215415
$846$	0	0
$847$	0	0
$848$	35.0838	1.20478
$849$	0	0
$850$	−58.9068	−2.02049
$851$	29.0023	0.994184
$852$	0	0
$853$	−6.69649	−0.229283	−0.114642	−	0.993407i	$-0.536572\pi$
$853$	−6.69649	−0.229283	−0.114642	+	0.993407i	$0.536572\pi$
$854$	5.97317	0.204398
$855$	0	0
$856$	10.9638	0.374735
$857$	−36.1064	−1.23337	−0.616686	−	0.787209i	$-0.711525\pi$
$857$	−36.1064	−1.23337	−0.616686	+	0.787209i	$0.711525\pi$
$858$	0	0
$859$	−37.1034	−1.26595	−0.632976	−	0.774171i	$-0.718167\pi$
$859$	−37.1034	−1.26595	−0.632976	+	0.774171i	$0.718167\pi$
$860$	−7.37155	−0.251368
$861$	0	0
$862$	−0.972147	−0.0331115
$863$	−38.3724	−1.30621	−0.653105	−	0.757267i	$-0.726534\pi$
$863$	−38.3724	−1.30621	−0.653105	+	0.757267i	$0.726534\pi$
$864$	0	0
$865$	−5.30300	−0.180307
$866$	34.7140	1.17963
$867$	0	0
$868$	−7.83880	−0.266066
$869$	0	0
$870$	0	0
$871$	−3.38446	−0.114678
$872$	−13.5323	−0.458260
$873$	0	0
$874$	−41.2858	−1.39651
$875$	5.69396	0.192491
$876$	0	0
$877$	0.811156	0.0273908	0.0136954	−	0.999906i	$-0.495640\pi$
$877$	0.811156	0.0273908	0.0136954	+	0.999906i	$0.495640\pi$
$878$	−66.2934	−2.23729
$879$	0	0
$880$	0	0
$881$	32.5076	1.09521	0.547605	−	0.836737i	$-0.315540\pi$
$881$	32.5076	1.09521	0.547605	+	0.836737i	$0.315540\pi$
$882$	0	0
$883$	48.0995	1.61868	0.809338	−	0.587343i	$-0.199826\pi$
$883$	48.0995	1.61868	0.809338	+	0.587343i	$0.199826\pi$
$884$	12.5887	0.423404
$885$	0	0
$886$	−38.0103	−1.27698
$887$	11.3847	0.382260	0.191130	−	0.981565i	$-0.438785\pi$
$887$	11.3847	0.382260	0.191130	+	0.981565i	$0.438785\pi$
$888$	0	0
$889$	−8.25186	−0.276758
$890$	8.52166	0.285647
$891$	0	0
$892$	3.86563	0.129431
$893$	63.1884	2.11452
$894$	0	0
$895$	−3.62562	−0.121191
$896$	11.0577	0.369413
$897$	0	0
$898$	−11.3405	−0.378438
$899$	−4.17303	−0.139178
$900$	0	0
$901$	−50.3351	−1.67690
$902$	0	0
$903$	0	0
$904$	−11.8905	−0.395472
$905$	−8.82895	−0.293484
$906$	0	0
$907$	13.4259	0.445799	0.222899	−	0.974841i	$-0.428448\pi$
$907$	13.4259	0.445799	0.222899	+	0.974841i	$0.428448\pi$
$908$	22.1279	0.734340
$909$	0	0
$910$	−1.61469	−0.0535264
$911$	−3.39303	−0.112416	−0.0562080	−	0.998419i	$-0.517901\pi$
$911$	−3.39303	−0.112416	−0.0562080	+	0.998419i	$0.517901\pi$
$912$	0	0
$913$	0	0
$914$	65.4712	2.16560
$915$	0	0
$916$	−11.2526	−0.371798
$917$	12.1162	0.400112
$918$	0	0
$919$	44.5668	1.47012	0.735062	−	0.678000i	$-0.237153\pi$
$919$	44.5668	1.47012	0.735062	+	0.678000i	$0.237153\pi$
$920$	−3.38044	−0.111450
$921$	0	0
$922$	13.1113	0.431799
$923$	21.5342	0.708807
$924$	0	0
$925$	35.8525	1.17882
$926$	−25.4699	−0.836992
$927$	0	0
$928$	−3.51049	−0.115238
$929$	7.15205	0.234651	0.117325	−	0.993094i	$-0.462568\pi$
$929$	7.15205	0.234651	0.117325	+	0.993094i	$0.462568\pi$
$930$	0	0
$931$	6.19059	0.202888
$932$	8.12937	0.266287
$933$	0	0
$934$	−64.0883	−2.09703
$935$	0	0
$936$	0	0
$937$	1.30811	0.0427342	0.0213671	−	0.999772i	$-0.493198\pi$
$937$	1.30811	0.0427342	0.0213671	+	0.999772i	$0.493198\pi$
$938$	−3.88361	−0.126804
$939$	0	0
$940$	−6.86914	−0.224047
$941$	−22.1225	−0.721174	−0.360587	−	0.932726i	$-0.617424\pi$
$941$	−22.1225	−0.721174	−0.360587	+	0.932726i	$0.617424\pi$
$942$	0	0
$943$	−28.1591	−0.916987
$944$	−16.7028	−0.543631
$945$	0	0
$946$	0	0
$947$	17.6842	0.574658	0.287329	−	0.957832i	$-0.407233\pi$
$947$	17.6842	0.574658	0.287329	+	0.957832i	$0.407233\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	−51.0374	−1.65587
$951$	0	0
$952$	−10.8801	−0.352627
$953$	−23.7741	−0.770118	−0.385059	−	0.922892i	$-0.625819\pi$
$953$	−23.7741	−0.770118	−0.385059	+	0.922892i	$0.625819\pi$
$954$	0	0
$955$	5.82464	0.188481
$956$	25.4272	0.822374
$957$	0	0
$958$	28.4003	0.917573
$959$	17.4548	0.563644
$960$	0	0
$961$	16.2172	0.523134
$962$	−21.0946	−0.680119
$963$	0	0
$964$	−16.9693	−0.546544
$965$	−3.64641	−0.117382
$966$	0	0
$967$	−21.1829	−0.681195	−0.340597	−	0.940209i	$-0.610629\pi$
$967$	−21.1829	−0.681195	−0.340597	+	0.940209i	$0.610629\pi$
$968$	0	0
$969$	0	0
$970$	5.01527	0.161031
$971$	47.4360	1.52229	0.761147	−	0.648579i	$-0.224637\pi$
$971$	47.4360	1.52229	0.761147	+	0.648579i	$0.224637\pi$
$972$	0	0
$973$	−14.7624	−0.473259
$974$	2.74800	0.0880515
$975$	0	0
$976$	−16.7854	−0.537288
$977$	13.1860	0.421858	0.210929	−	0.977501i	$-0.432351\pi$
$977$	13.1860	0.421858	0.210929	+	0.977501i	$0.432351\pi$
$978$	0	0
$979$	0	0
$980$	−0.672972	−0.0214973
$981$	0	0
$982$	−0.0587731	−0.00187553
$983$	4.07740	0.130049	0.0650244	−	0.997884i	$-0.479287\pi$
$983$	4.07740	0.130049	0.0650244	+	0.997884i	$0.479287\pi$
$984$	0	0
$985$	11.2511	0.358489
$986$	7.69004	0.244901
$987$	0	0
$988$	10.9070	0.346997
$989$	−41.2204	−1.31073
$990$	0	0
$991$	−5.49154	−0.174445	−0.0872223	−	0.996189i	$-0.527799\pi$
$991$	−5.49154	−0.174445	−0.0872223	+	0.996189i	$0.527799\pi$
$992$	39.7206	1.26113
$993$	0	0
$994$	24.7101	0.783757
$995$	6.98940	0.221579
$996$	0	0
$997$	−22.9725	−0.727548	−0.363774	−	0.931487i	$-0.618512\pi$
$997$	−22.9725	−0.727548	−0.363774	+	0.931487i	$0.618512\pi$
$998$	−47.9057	−1.51643
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cl.1.4		4
3.2	odd	2		2541.2.a.bm.1.1		4
11.2	odd	10		693.2.m.f.631.2		8
11.6	odd	10		693.2.m.f.190.2		8
11.10	odd	2		7623.2.a.ci.1.1		4
33.2	even	10		231.2.j.f.169.1	✓	8
33.17	even	10		231.2.j.f.190.1	yes	8
33.32	even	2		2541.2.a.bn.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.f.169.1	✓	8	33.2	even	10
231.2.j.f.190.1	yes	8	33.17	even	10
693.2.m.f.190.2		8	11.6	odd	10
693.2.m.f.631.2		8	11.2	odd	10
2541.2.a.bm.1.1		4	3.2	odd	2
2541.2.a.bn.1.4		4	33.32	even	2
7623.2.a.ci.1.1		4	11.10	odd	2
7623.2.a.cl.1.4		4	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+1.77222 q^{2} +1.14077 q^{4} -0.589926 q^{5} +1.00000 q^{7} -1.52274 q^{8} +O(q^{10})\)	\(q+1.77222 q^{2} +1.14077 q^{4} -0.589926 q^{5} +1.00000 q^{7} -1.52274 q^{8} -1.04548 q^{10} +1.54445 q^{13} +1.77222 q^{14} -4.98018 q^{16} +7.14511 q^{17} +6.19059 q^{19} -0.672972 q^{20} -3.76314 q^{23} -4.65199 q^{25} +2.73710 q^{26} +1.14077 q^{28} +0.607298 q^{29} -6.87147 q^{31} -5.78051 q^{32} +12.6627 q^{34} -0.589926 q^{35} -7.70693 q^{37} +10.9711 q^{38} +0.898304 q^{40} +7.48287 q^{41} +10.9537 q^{43} -6.66913 q^{46} +10.2072 q^{47} +1.00000 q^{49} -8.24436 q^{50} +1.76186 q^{52} -7.04469 q^{53} -1.52274 q^{56} +1.07627 q^{58} +3.35386 q^{59} +3.37044 q^{61} -12.1778 q^{62} -0.283993 q^{64} -0.911108 q^{65} -2.19138 q^{67} +8.15095 q^{68} -1.04548 q^{70} +13.9430 q^{71} +2.58993 q^{73} -13.6584 q^{74} +7.06206 q^{76} +4.66351 q^{79} +2.93794 q^{80} +13.2613 q^{82} -3.50970 q^{83} -4.21508 q^{85} +19.4125 q^{86} -8.15095 q^{89} +1.54445 q^{91} -4.29289 q^{92} +18.0894 q^{94} -3.65199 q^{95} -4.79710 q^{97} +1.77222 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 3 q^{4} + 4 q^{5} + 4 q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q + q^2 + 3 * q^4 + 4 * q^5 + 4 * q^7 - 9 * q^8	\( 4 q + q^{2} + 3 q^{4} + 4 q^{5} + 4 q^{7} - 9 q^{8} - 10 q^{10} - 6 q^{13} + q^{14} - 3 q^{16} + 8 q^{17} + 10 q^{19} + 10 q^{23} + 12 q^{25} + 20 q^{26} + 3 q^{28} - 18 q^{31} - 2 q^{32} + 18 q^{34} + 4 q^{35} - 2 q^{37} + 8 q^{38} - 6 q^{40} + 10 q^{41} + 4 q^{43} - 11 q^{46} - 4 q^{47} + 4 q^{49} - 9 q^{50} - 20 q^{52} - 9 q^{56} + 14 q^{58} + 16 q^{59} - 14 q^{61} - 11 q^{64} - 28 q^{65} - 28 q^{67} - 16 q^{68} - 10 q^{70} + 18 q^{71} + 4 q^{73} - 41 q^{74} + 4 q^{76} + 20 q^{79} + 36 q^{80} - 24 q^{82} + 6 q^{83} + 20 q^{85} + 20 q^{86} + 16 q^{89} - 6 q^{91} + 22 q^{92} + 16 q^{94} + 16 q^{95} + 32 q^{97} + q^{98}+O(q^{100}) \) 4 * q + q^2 + 3 * q^4 + 4 * q^5 + 4 * q^7 - 9 * q^8 - 10 * q^10 - 6 * q^13 + q^14 - 3 * q^16 + 8 * q^17 + 10 * q^19 + 10 * q^23 + 12 * q^25 + 20 * q^26 + 3 * q^28 - 18 * q^31 - 2 * q^32 + 18 * q^34 + 4 * q^35 - 2 * q^37 + 8 * q^38 - 6 * q^40 + 10 * q^41 + 4 * q^43 - 11 * q^46 - 4 * q^47 + 4 * q^49 - 9 * q^50 - 20 * q^52 - 9 * q^56 + 14 * q^58 + 16 * q^59 - 14 * q^61 - 11 * q^64 - 28 * q^65 - 28 * q^67 - 16 * q^68 - 10 * q^70 + 18 * q^71 + 4 * q^73 - 41 * q^74 + 4 * q^76 + 20 * q^79 + 36 * q^80 - 24 * q^82 + 6 * q^83 + 20 * q^85 + 20 * q^86 + 16 * q^89 - 6 * q^91 + 22 * q^92 + 16 * q^94 + 16 * q^95 + 32 * q^97 + q^98

Introduction

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