LMFDB - Embedded newform 7623.2.a.cj.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:	$\Q(\sqrt{2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 9 $ x^4 - 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.16372$ 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)

$f(q)$	$=$	$q+1.41421 q^{2} +2.64575 q^{5} -1.00000 q^{7} -2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +2.64575 q^{5} -1.00000 q^{7} -2.82843 q^{8} +3.74166 q^{10} +5.74166 q^{13} -1.41421 q^{14} -4.00000 q^{16} +5.47418 q^{17} -5.74166 q^{19} -3.87729 q^{23} +2.00000 q^{25} +8.11993 q^{26} +3.87729 q^{29} +5.48331 q^{31} +7.74166 q^{34} -2.64575 q^{35} +3.48331 q^{37} -8.11993 q^{38} -7.48331 q^{40} -5.65685 q^{41} +1.00000 q^{43} -5.48331 q^{46} -5.47418 q^{47} +1.00000 q^{49} +2.82843 q^{50} +13.4114 q^{53} +2.82843 q^{56} +5.48331 q^{58} +8.30261 q^{59} -1.74166 q^{61} +7.75458 q^{62} +8.00000 q^{64} +15.1910 q^{65} -6.48331 q^{67} -3.74166 q^{70} -1.41421 q^{71} +15.7417 q^{73} +4.92615 q^{74} +13.4833 q^{79} -10.5830 q^{80} -8.00000 q^{82} +11.1310 q^{83} +14.4833 q^{85} +1.41421 q^{86} +5.83953 q^{89} -5.74166 q^{91} -7.74166 q^{94} -15.1910 q^{95} -7.22497 q^{97} +1.41421 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^7	$ 4 q - 4 q^{7} + 8 q^{13} - 16 q^{16} - 8 q^{19} + 8 q^{25} - 8 q^{31} + 16 q^{34} - 16 q^{37} + 4 q^{43} + 8 q^{46} + 4 q^{49} - 8 q^{58} + 8 q^{61} + 32 q^{64} + 4 q^{67} + 48 q^{73} + 24 q^{79} - 32 q^{82} + 28 q^{85} - 8 q^{91} - 16 q^{94} + 16 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 + 8 * q^13 - 16 * q^16 - 8 * q^19 + 8 * q^25 - 8 * q^31 + 16 * q^34 - 16 * q^37 + 4 * q^43 + 8 * q^46 + 4 * q^49 - 8 * q^58 + 8 * q^61 + 32 * q^64 + 4 * q^67 + 48 * q^73 + 24 * q^79 - 32 * q^82 + 28 * q^85 - 8 * q^91 - 16 * q^94 + 16 * q^97

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 $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.64575	1.18322	0.591608	−	0.806226i	$-0.298493\pi$
$5$	2.64575	1.18322	0.591608	+	0.806226i	$0.298493\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.82843	−1.00000
$9$	0	0
$10$	3.74166	1.18322
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.74166	1.59245	0.796225	−	0.605001i	$-0.206827\pi$
$13$	5.74166	1.59245	0.796225	+	0.605001i	$0.206827\pi$
$14$	−1.41421	−0.377964
$15$	0	0
$16$	−4.00000	−1.00000
$17$	5.47418	1.32768	0.663842	−	0.747873i	$-0.268925\pi$
$17$	5.47418	1.32768	0.663842	+	0.747873i	$0.268925\pi$
$18$	0	0
$19$	−5.74166	−1.31723	−0.658613	−	0.752482i	$-0.728857\pi$
$19$	−5.74166	−1.31723	−0.658613	+	0.752482i	$0.728857\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.87729	−0.808471	−0.404235	−	0.914655i	$-0.632462\pi$
$23$	−3.87729	−0.808471	−0.404235	+	0.914655i	$0.632462\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	8.11993	1.59245
$27$	0	0
$28$	0	0
$29$	3.87729	0.719995	0.359997	−	0.932953i	$-0.382778\pi$
$29$	3.87729	0.719995	0.359997	+	0.932953i	$0.382778\pi$
$30$	0	0
$31$	5.48331	0.984832	0.492416	−	0.870360i	$-0.336114\pi$
$31$	5.48331	0.984832	0.492416	+	0.870360i	$0.336114\pi$
$32$	0	0
$33$	0	0
$34$	7.74166	1.32768
$35$	−2.64575	−0.447214
$36$	0	0
$37$	3.48331	0.572653	0.286327	−	0.958132i	$-0.407566\pi$
$37$	3.48331	0.572653	0.286327	+	0.958132i	$0.407566\pi$
$38$	−8.11993	−1.31723
$39$	0	0
$40$	−7.48331	−1.18322
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	0	0
$46$	−5.48331	−0.808471
$47$	−5.47418	−0.798491	−0.399245	−	0.916844i	$-0.630728\pi$
$47$	−5.47418	−0.798491	−0.399245	+	0.916844i	$0.630728\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.82843	0.400000
$51$	0	0
$52$	0	0
$53$	13.4114	1.84220	0.921101	−	0.389324i	$-0.127291\pi$
$53$	13.4114	1.84220	0.921101	+	0.389324i	$0.127291\pi$
$54$	0	0
$55$	0	0
$56$	2.82843	0.377964
$57$	0	0
$58$	5.48331	0.719995
$59$	8.30261	1.08091	0.540454	−	0.841374i	$-0.318253\pi$
$59$	8.30261	1.08091	0.540454	+	0.841374i	$0.318253\pi$
$60$	0	0
$61$	−1.74166	−0.222996	−0.111498	−	0.993765i	$-0.535565\pi$
$61$	−1.74166	−0.222996	−0.111498	+	0.993765i	$0.535565\pi$
$62$	7.75458	0.984832
$63$	0	0
$64$	8.00000	1.00000
$65$	15.1910	1.88421
$66$	0	0
$67$	−6.48331	−0.792063	−0.396031	−	0.918237i	$-0.629613\pi$
$67$	−6.48331	−0.792063	−0.396031	+	0.918237i	$0.629613\pi$
$68$	0	0
$69$	0	0
$70$	−3.74166	−0.447214
$71$	−1.41421	−0.167836	−0.0839181	−	0.996473i	$-0.526743\pi$
$71$	−1.41421	−0.167836	−0.0839181	+	0.996473i	$0.526743\pi$
$72$	0	0
$73$	15.7417	1.84242	0.921211	−	0.389064i	$-0.127201\pi$
$73$	15.7417	1.84242	0.921211	+	0.389064i	$0.127201\pi$
$74$	4.92615	0.572653
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.4833	1.51699	0.758496	−	0.651678i	$-0.225935\pi$
$79$	13.4833	1.51699	0.758496	+	0.651678i	$0.225935\pi$
$80$	−10.5830	−1.18322
$81$	0	0
$82$	−8.00000	−0.883452
$83$	11.1310	1.22179	0.610895	−	0.791712i	$-0.290810\pi$
$83$	11.1310	1.22179	0.610895	+	0.791712i	$0.290810\pi$
$84$	0	0
$85$	14.4833	1.57094
$86$	1.41421	0.152499
$87$	0	0
$88$	0	0
$89$	5.83953	0.618989	0.309494	−	0.950901i	$-0.399840\pi$
$89$	5.83953	0.618989	0.309494	+	0.950901i	$0.399840\pi$
$90$	0	0
$91$	−5.74166	−0.601889
$92$	0	0
$93$	0	0
$94$	−7.74166	−0.798491
$95$	−15.1910	−1.55856
$96$	0	0
$97$	−7.22497	−0.733585	−0.366792	−	0.930303i	$-0.619544\pi$
$97$	−7.22497	−0.733585	−0.366792	+	0.930303i	$0.619544\pi$
$98$	1.41421	0.142857
$99$	0	0
$100$	0	0
$101$	−8.66796	−0.862494	−0.431247	−	0.902234i	$-0.641926\pi$
$101$	−8.66796	−0.862494	−0.431247	+	0.902234i	$0.641926\pi$
$102$	0	0
$103$	−13.2250	−1.30310	−0.651548	−	0.758608i	$-0.725880\pi$
$103$	−13.2250	−1.30310	−0.651548	+	0.758608i	$0.725880\pi$
$104$	−16.2399	−1.59245
$105$	0	0
$106$	18.9666	1.84220
$107$	−9.53414	−0.921700	−0.460850	−	0.887478i	$-0.652455\pi$
$107$	−9.53414	−0.921700	−0.460850	+	0.887478i	$0.652455\pi$
$108$	0	0
$109$	−16.4833	−1.57882	−0.789408	−	0.613869i	$-0.789612\pi$
$109$	−16.4833	−1.57882	−0.789408	+	0.613869i	$0.789612\pi$
$110$	0	0
$111$	0	0
$112$	4.00000	0.377964
$113$	5.29150	0.497783	0.248891	−	0.968531i	$-0.419934\pi$
$113$	5.29150	0.497783	0.248891	+	0.968531i	$0.419934\pi$
$114$	0	0
$115$	−10.2583	−0.956595
$116$	0	0
$117$	0	0
$118$	11.7417	1.08091
$119$	−5.47418	−0.501817
$120$	0	0
$121$	0	0
$122$	−2.46308	−0.222996
$123$	0	0
$124$	0	0
$125$	−7.93725	−0.709930
$126$	0	0
$127$	16.4833	1.46266	0.731329	−	0.682025i	$-0.238900\pi$
$127$	16.4833	1.46266	0.731329	+	0.682025i	$0.238900\pi$
$128$	11.3137	1.00000
$129$	0	0
$130$	21.4833	1.88421
$131$	8.66796	0.757323	0.378661	−	0.925535i	$-0.376384\pi$
$131$	8.66796	0.757323	0.378661	+	0.925535i	$0.376384\pi$
$132$	0	0
$133$	5.74166	0.497865
$134$	−9.16879	−0.792063
$135$	0	0
$136$	−15.4833	−1.32768
$137$	8.80344	0.752129	0.376064	−	0.926594i	$-0.377277\pi$
$137$	8.80344	0.752129	0.376064	+	0.926594i	$0.377277\pi$
$138$	0	0
$139$	−3.74166	−0.317363	−0.158682	−	0.987330i	$-0.550724\pi$
$139$	−3.74166	−0.317363	−0.158682	+	0.987330i	$0.550724\pi$
$140$	0	0
$141$	0	0
$142$	−2.00000	−0.167836
$143$	0	0
$144$	0	0
$145$	10.2583	0.851909
$146$	22.2621	1.84242
$147$	0	0
$148$	0	0
$149$	23.3109	1.90971	0.954853	−	0.297079i	$-0.0960126\pi$
$149$	23.3109	1.90971	0.954853	+	0.297079i	$0.0960126\pi$
$150$	0	0
$151$	7.51669	0.611699	0.305850	−	0.952080i	$-0.401060\pi$
$151$	7.51669	0.611699	0.305850	+	0.952080i	$0.401060\pi$
$152$	16.2399	1.31723
$153$	0	0
$154$	0	0
$155$	14.5075	1.16527
$156$	0	0
$157$	−12.9666	−1.03485	−0.517425	−	0.855729i	$-0.673109\pi$
$157$	−12.9666	−1.03485	−0.517425	+	0.855729i	$0.673109\pi$
$158$	19.0683	1.51699
$159$	0	0
$160$	0	0
$161$	3.87729	0.305573
$162$	0	0
$163$	−24.4499	−1.91507	−0.957534	−	0.288321i	$-0.906903\pi$
$163$	−24.4499	−1.91507	−0.957534	+	0.288321i	$0.906903\pi$
$164$	0	0
$165$	0	0
$166$	15.7417	1.22179
$167$	−8.66796	−0.670747	−0.335373	−	0.942085i	$-0.608862\pi$
$167$	−8.66796	−0.670747	−0.335373	+	0.942085i	$0.608862\pi$
$168$	0	0
$169$	19.9666	1.53589
$170$	20.4825	1.57094
$171$	0	0
$172$	0	0
$173$	5.10883	0.388417	0.194208	−	0.980960i	$-0.437786\pi$
$173$	5.10883	0.388417	0.194208	+	0.980960i	$0.437786\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	8.25834	0.618989
$179$	14.1421	1.05703	0.528516	−	0.848923i	$-0.322748\pi$
$179$	14.1421	1.05703	0.528516	+	0.848923i	$0.322748\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	−8.11993	−0.601889
$183$	0	0
$184$	10.9666	0.808471
$185$	9.21598	0.677573
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−21.4833	−1.55856
$191$	−1.41421	−0.102329	−0.0511645	−	0.998690i	$-0.516293\pi$
$191$	−1.41421	−0.102329	−0.0511645	+	0.998690i	$0.516293\pi$
$192$	0	0
$193$	16.4833	1.18649	0.593247	−	0.805020i	$-0.297846\pi$
$193$	16.4833	1.18649	0.593247	+	0.805020i	$0.297846\pi$
$194$	−10.2177	−0.733585
$195$	0	0
$196$	0	0
$197$	22.2621	1.58611	0.793053	−	0.609152i	$-0.208490\pi$
$197$	22.2621	1.58611	0.793053	+	0.609152i	$0.208490\pi$
$198$	0	0
$199$	2.25834	0.160090	0.0800448	−	0.996791i	$-0.474494\pi$
$199$	2.25834	0.160090	0.0800448	+	0.996791i	$0.474494\pi$
$200$	−5.65685	−0.400000
$201$	0	0
$202$	−12.2583	−0.862494
$203$	−3.87729	−0.272132
$204$	0	0
$205$	−14.9666	−1.04531
$206$	−18.7029	−1.30310
$207$	0	0
$208$	−22.9666	−1.59245
$209$	0	0
$210$	0	0
$211$	9.51669	0.655156	0.327578	−	0.944824i	$-0.393768\pi$
$211$	9.51669	0.655156	0.327578	+	0.944824i	$0.393768\pi$
$212$	0	0
$213$	0	0
$214$	−13.4833	−0.921700
$215$	2.64575	0.180439
$216$	0	0
$217$	−5.48331	−0.372232
$218$	−23.3109	−1.57882
$219$	0	0
$220$	0	0
$221$	31.4309	2.11427
$222$	0	0
$223$	17.4833	1.17077	0.585385	−	0.810756i	$-0.300943\pi$
$223$	17.4833	1.17077	0.585385	+	0.810756i	$0.300943\pi$
$224$	0	0
$225$	0	0
$226$	7.48331	0.497783
$227$	27.7362	1.84092	0.920460	−	0.390838i	$-0.127815\pi$
$227$	27.7362	1.84092	0.920460	+	0.390838i	$0.127815\pi$
$228$	0	0
$229$	13.4833	0.891003	0.445501	−	0.895281i	$-0.353025\pi$
$229$	13.4833	0.891003	0.445501	+	0.895281i	$0.353025\pi$
$230$	−14.5075	−0.956595
$231$	0	0
$232$	−10.9666	−0.719995
$233$	6.34036	0.415371	0.207686	−	0.978196i	$-0.433407\pi$
$233$	6.34036	0.415371	0.207686	+	0.978196i	$0.433407\pi$
$234$	0	0
$235$	−14.4833	−0.944787
$236$	0	0
$237$	0	0
$238$	−7.74166	−0.501817
$239$	−8.11993	−0.525235	−0.262617	−	0.964900i	$-0.584586\pi$
$239$	−8.11993	−0.525235	−0.262617	+	0.964900i	$0.584586\pi$
$240$	0	0
$241$	−3.22497	−0.207739	−0.103869	−	0.994591i	$-0.533122\pi$
$241$	−3.22497	−0.207739	−0.103869	+	0.994591i	$0.533122\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.64575	0.169031
$246$	0	0
$247$	−32.9666	−2.09762
$248$	−15.5092	−0.984832
$249$	0	0
$250$	−11.2250	−0.709930
$251$	−2.09772	−0.132407	−0.0662036	−	0.997806i	$-0.521089\pi$
$251$	−2.09772	−0.132407	−0.0662036	+	0.997806i	$0.521089\pi$
$252$	0	0
$253$	0	0
$254$	23.3109	1.46266
$255$	0	0
$256$	0	0
$257$	0.548027	0.0341850	0.0170925	−	0.999854i	$-0.494559\pi$
$257$	0.548027	0.0341850	0.0170925	+	0.999854i	$0.494559\pi$
$258$	0	0
$259$	−3.48331	−0.216443
$260$	0	0
$261$	0	0
$262$	12.2583	0.757323
$263$	−20.1171	−1.24048	−0.620238	−	0.784413i	$-0.712964\pi$
$263$	−20.1171	−1.24048	−0.620238	+	0.784413i	$0.712964\pi$
$264$	0	0
$265$	35.4833	2.17972
$266$	8.11993	0.497865
$267$	0	0
$268$	0	0
$269$	−24.3598	−1.48524	−0.742621	−	0.669712i	$-0.766418\pi$
$269$	−24.3598	−1.48524	−0.742621	+	0.669712i	$0.766418\pi$
$270$	0	0
$271$	8.96663	0.544684	0.272342	−	0.962201i	$-0.412202\pi$
$271$	8.96663	0.544684	0.272342	+	0.962201i	$0.412202\pi$
$272$	−21.8967	−1.32768
$273$	0	0
$274$	12.4499	0.752129
$275$	0	0
$276$	0	0
$277$	19.9666	1.19968	0.599839	−	0.800121i	$-0.295231\pi$
$277$	19.9666	1.19968	0.599839	+	0.800121i	$0.295231\pi$
$278$	−5.29150	−0.317363
$279$	0	0
$280$	7.48331	0.447214
$281$	−16.2399	−0.968789	−0.484394	−	0.874850i	$-0.660960\pi$
$281$	−16.2399	−0.968789	−0.484394	+	0.874850i	$0.660960\pi$
$282$	0	0
$283$	−8.96663	−0.533011	−0.266505	−	0.963833i	$-0.585869\pi$
$283$	−8.96663	−0.533011	−0.266505	+	0.963833i	$0.585869\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	12.9666	0.762743
$290$	14.5075	0.851909
$291$	0	0
$292$	0	0
$293$	−28.1016	−1.64171	−0.820856	−	0.571135i	$-0.806503\pi$
$293$	−28.1016	−1.64171	−0.820856	+	0.571135i	$0.806503\pi$
$294$	0	0
$295$	21.9666	1.27895
$296$	−9.85230	−0.572653
$297$	0	0
$298$	32.9666	1.90971
$299$	−22.2621	−1.28745
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	10.6302	0.611699
$303$	0	0
$304$	22.9666	1.31723
$305$	−4.60799	−0.263853
$306$	0	0
$307$	6.70829	0.382862	0.191431	−	0.981506i	$-0.438687\pi$
$307$	6.70829	0.382862	0.191431	+	0.981506i	$0.438687\pi$
$308$	0	0
$309$	0	0
$310$	20.5167	1.16527
$311$	15.3265	0.869085	0.434542	−	0.900651i	$-0.356910\pi$
$311$	15.3265	0.869085	0.434542	+	0.900651i	$0.356910\pi$
$312$	0	0
$313$	−4.51669	−0.255298	−0.127649	−	0.991819i	$-0.540743\pi$
$313$	−4.51669	−0.255298	−0.127649	+	0.991819i	$0.540743\pi$
$314$	−18.3376	−1.03485
$315$	0	0
$316$	0	0
$317$	9.53414	0.535491	0.267745	−	0.963490i	$-0.413721\pi$
$317$	9.53414	0.535491	0.267745	+	0.963490i	$0.413721\pi$
$318$	0	0
$319$	0	0
$320$	21.1660	1.18322
$321$	0	0
$322$	5.48331	0.305573
$323$	−31.4309	−1.74886
$324$	0	0
$325$	11.4833	0.636980
$326$	−34.5774	−1.91507
$327$	0	0
$328$	16.0000	0.883452
$329$	5.47418	0.301801
$330$	0	0
$331$	20.4833	1.12586	0.562932	−	0.826503i	$-0.309673\pi$
$331$	20.4833	1.12586	0.562932	+	0.826503i	$0.309673\pi$
$332$	0	0
$333$	0	0
$334$	−12.2583	−0.670747
$335$	−17.1532	−0.937182
$336$	0	0
$337$	−22.9666	−1.25107	−0.625536	−	0.780195i	$-0.715120\pi$
$337$	−22.9666	−1.25107	−0.625536	+	0.780195i	$0.715120\pi$
$338$	28.2371	1.53589
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−2.82843	−0.152499
$345$	0	0
$346$	7.22497	0.388417
$347$	33.1632	1.78030	0.890148	−	0.455672i	$-0.150601\pi$
$347$	33.1632	1.78030	0.890148	+	0.455672i	$0.150601\pi$
$348$	0	0
$349$	33.4833	1.79232	0.896160	−	0.443730i	$-0.146345\pi$
$349$	33.4833	1.79232	0.896160	+	0.443730i	$0.146345\pi$
$350$	−2.82843	−0.151186
$351$	0	0
$352$	0	0
$353$	−22.9928	−1.22378	−0.611891	−	0.790942i	$-0.709591\pi$
$353$	−22.9928	−1.22378	−0.611891	+	0.790942i	$0.709591\pi$
$354$	0	0
$355$	−3.74166	−0.198587
$356$	0	0
$357$	0	0
$358$	20.0000	1.05703
$359$	−14.0949	−0.743903	−0.371951	−	0.928252i	$-0.621311\pi$
$359$	−14.0949	−0.743903	−0.371951	+	0.928252i	$0.621311\pi$
$360$	0	0
$361$	13.9666	0.735086
$362$	0	0
$363$	0	0
$364$	0	0
$365$	41.6485	2.17998
$366$	0	0
$367$	17.2250	0.899136	0.449568	−	0.893246i	$-0.351578\pi$
$367$	17.2250	0.899136	0.449568	+	0.893246i	$0.351578\pi$
$368$	15.5092	0.808471
$369$	0	0
$370$	13.0334	0.677573
$371$	−13.4114	−0.696287
$372$	0	0
$373$	19.4499	1.00708	0.503540	−	0.863972i	$-0.332031\pi$
$373$	19.4499	1.00708	0.503540	+	0.863972i	$0.332031\pi$
$374$	0	0
$375$	0	0
$376$	15.4833	0.798491
$377$	22.2621	1.14655
$378$	0	0
$379$	−34.4833	−1.77129	−0.885644	−	0.464364i	$-0.846283\pi$
$379$	−34.4833	−1.77129	−0.885644	+	0.464364i	$0.846283\pi$
$380$	0	0
$381$	0	0
$382$	−2.00000	−0.102329
$383$	−21.3487	−1.09087	−0.545433	−	0.838154i	$-0.683635\pi$
$383$	−21.3487	−1.09087	−0.545433	+	0.838154i	$0.683635\pi$
$384$	0	0
$385$	0	0
$386$	23.3109	1.18649
$387$	0	0
$388$	0	0
$389$	−33.5758	−1.70236	−0.851180	−	0.524874i	$-0.824112\pi$
$389$	−33.5758	−1.70236	−0.851180	+	0.524874i	$0.824112\pi$
$390$	0	0
$391$	−21.2250	−1.07339
$392$	−2.82843	−0.142857
$393$	0	0
$394$	31.4833	1.58611
$395$	35.6735	1.79493
$396$	0	0
$397$	1.48331	0.0744454	0.0372227	−	0.999307i	$-0.488149\pi$
$397$	1.48331	0.0744454	0.0372227	+	0.999307i	$0.488149\pi$
$398$	3.19378	0.160090
$399$	0	0
$400$	−8.00000	−0.400000
$401$	8.11993	0.405490	0.202745	−	0.979232i	$-0.435014\pi$
$401$	8.11993	0.405490	0.202745	+	0.979232i	$0.435014\pi$
$402$	0	0
$403$	31.4833	1.56830
$404$	0	0
$405$	0	0
$406$	−5.48331	−0.272132
$407$	0	0
$408$	0	0
$409$	−9.22497	−0.456146	−0.228073	−	0.973644i	$-0.573242\pi$
$409$	−9.22497	−0.456146	−0.228073	+	0.973644i	$0.573242\pi$
$410$	−21.1660	−1.04531
$411$	0	0
$412$	0	0
$413$	−8.30261	−0.408545
$414$	0	0
$415$	29.4499	1.44564
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−29.4686	−1.43964	−0.719818	−	0.694163i	$-0.755775\pi$
$419$	−29.4686	−1.43964	−0.719818	+	0.694163i	$0.755775\pi$
$420$	0	0
$421$	−7.51669	−0.366341	−0.183170	−	0.983081i	$-0.558636\pi$
$421$	−7.51669	−0.366341	−0.183170	+	0.983081i	$0.558636\pi$
$422$	13.4586	0.655156
$423$	0	0
$424$	−37.9333	−1.84220
$425$	10.9484	0.531073
$426$	0	0
$427$	1.74166	0.0842847
$428$	0	0
$429$	0	0
$430$	3.74166	0.180439
$431$	−15.8745	−0.764648	−0.382324	−	0.924028i	$-0.624876\pi$
$431$	−15.8745	−0.764648	−0.382324	+	0.924028i	$0.624876\pi$
$432$	0	0
$433$	−1.74166	−0.0836987	−0.0418494	−	0.999124i	$-0.513325\pi$
$433$	−1.74166	−0.0836987	−0.0418494	+	0.999124i	$0.513325\pi$
$434$	−7.75458	−0.372232
$435$	0	0
$436$	0	0
$437$	22.2621	1.06494
$438$	0	0
$439$	19.4833	0.929888	0.464944	−	0.885340i	$-0.346074\pi$
$439$	19.4833	0.929888	0.464944	+	0.885340i	$0.346074\pi$
$440$	0	0
$441$	0	0
$442$	44.4499	2.11427
$443$	−27.2354	−1.29399	−0.646997	−	0.762493i	$-0.723975\pi$
$443$	−27.2354	−1.29399	−0.646997	+	0.762493i	$0.723975\pi$
$444$	0	0
$445$	15.4499	0.732398
$446$	24.7251	1.17077
$447$	0	0
$448$	−8.00000	−0.377964
$449$	10.2177	0.482201	0.241100	−	0.970500i	$-0.422492\pi$
$449$	10.2177	0.482201	0.241100	+	0.970500i	$0.422492\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	39.2250	1.84092
$455$	−15.1910	−0.712165
$456$	0	0
$457$	23.0000	1.07589	0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000	1.07589	0.537947	+	0.842978i	$0.319200\pi$
$458$	19.0683	0.891003
$459$	0	0
$460$	0	0
$461$	7.20655	0.335643	0.167821	−	0.985817i	$-0.446327\pi$
$461$	7.20655	0.335643	0.167821	+	0.985817i	$0.446327\pi$
$462$	0	0
$463$	−17.4833	−0.812519	−0.406259	−	0.913758i	$-0.633167\pi$
$463$	−17.4833	−0.812519	−0.406259	+	0.913758i	$0.633167\pi$
$464$	−15.5092	−0.719995
$465$	0	0
$466$	8.96663	0.415371
$467$	20.8007	0.962540	0.481270	−	0.876572i	$-0.340176\pi$
$467$	20.8007	0.962540	0.481270	+	0.876572i	$0.340176\pi$
$468$	0	0
$469$	6.48331	0.299372
$470$	−20.4825	−0.944787
$471$	0	0
$472$	−23.4833	−1.08091
$473$	0	0
$474$	0	0
$475$	−11.4833	−0.526891
$476$	0	0
$477$	0	0
$478$	−11.4833	−0.525235
$479$	23.1754	1.05891	0.529457	−	0.848337i	$-0.322396\pi$
$479$	23.1754	1.05891	0.529457	+	0.848337i	$0.322396\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	−4.56080	−0.207739
$483$	0	0
$484$	0	0
$485$	−19.1155	−0.867989
$486$	0	0
$487$	5.51669	0.249985	0.124992	−	0.992158i	$-0.460109\pi$
$487$	5.51669	0.249985	0.124992	+	0.992158i	$0.460109\pi$
$488$	4.92615	0.222996
$489$	0	0
$490$	3.74166	0.169031
$491$	−29.6985	−1.34027	−0.670137	−	0.742237i	$-0.733765\pi$
$491$	−29.6985	−1.34027	−0.670137	+	0.742237i	$0.733765\pi$
$492$	0	0
$493$	21.2250	0.955925
$494$	−46.6219	−2.09762
$495$	0	0
$496$	−21.9333	−0.984832
$497$	1.41421	0.0634361
$498$	0	0
$499$	−17.9666	−0.804297	−0.402148	−	0.915574i	$-0.631736\pi$
$499$	−17.9666	−0.804297	−0.402148	+	0.915574i	$0.631736\pi$
$500$	0	0
$501$	0	0
$502$	−2.96663	−0.132407
$503$	−34.3948	−1.53359	−0.766793	−	0.641894i	$-0.778149\pi$
$503$	−34.3948	−1.53359	−0.766793	+	0.641894i	$0.778149\pi$
$504$	0	0
$505$	−22.9333	−1.02052
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11.8617	−0.525762	−0.262881	−	0.964828i	$-0.584673\pi$
$509$	−11.8617	−0.525762	−0.262881	+	0.964828i	$0.584673\pi$
$510$	0	0
$511$	−15.7417	−0.696370
$512$	−22.6274	−1.00000
$513$	0	0
$514$	0.775028	0.0341850
$515$	−34.9900	−1.54184
$516$	0	0
$517$	0	0
$518$	−4.92615	−0.216443
$519$	0	0
$520$	−42.9666	−1.88421
$521$	−7.20655	−0.315725	−0.157862	−	0.987461i	$-0.550460\pi$
$521$	−7.20655	−0.315725	−0.157862	+	0.987461i	$0.550460\pi$
$522$	0	0
$523$	−7.22497	−0.315926	−0.157963	−	0.987445i	$-0.550493\pi$
$523$	−7.22497	−0.315926	−0.157963	+	0.987445i	$0.550493\pi$
$524$	0	0
$525$	0	0
$526$	−28.4499	−1.24048
$527$	30.0166	1.30755
$528$	0	0
$529$	−7.96663	−0.346375
$530$	50.1810	2.17972
$531$	0	0
$532$	0	0
$533$	−32.4797	−1.40685
$534$	0	0
$535$	−25.2250	−1.09057
$536$	18.3376	0.792063
$537$	0	0
$538$	−34.4499	−1.48524
$539$	0	0
$540$	0	0
$541$	15.0000	0.644900	0.322450	−	0.946586i	$-0.395494\pi$
$541$	15.0000	0.644900	0.322450	+	0.946586i	$0.395494\pi$
$542$	12.6807	0.544684
$543$	0	0
$544$	0	0
$545$	−43.6108	−1.86808
$546$	0	0
$547$	−10.0000	−0.427569	−0.213785	−	0.976881i	$-0.568579\pi$
$547$	−10.0000	−0.427569	−0.213785	+	0.976881i	$0.568579\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−22.2621	−0.948396
$552$	0	0
$553$	−13.4833	−0.573369
$554$	28.2371	1.19968
$555$	0	0
$556$	0	0
$557$	−46.9872	−1.99091	−0.995456	−	0.0952236i	$-0.969643\pi$
$557$	−46.9872	−1.99091	−0.995456	+	0.0952236i	$0.969643\pi$
$558$	0	0
$559$	5.74166	0.242846
$560$	10.5830	0.447214
$561$	0	0
$562$	−22.9666	−0.968789
$563$	−13.2288	−0.557526	−0.278763	−	0.960360i	$-0.589924\pi$
$563$	−13.2288	−0.557526	−0.278763	+	0.960360i	$0.589924\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	−12.6807	−0.533011
$567$	0	0
$568$	4.00000	0.167836
$569$	18.3848	0.770730	0.385365	−	0.922764i	$-0.374076\pi$
$569$	18.3848	0.770730	0.385365	+	0.922764i	$0.374076\pi$
$570$	0	0
$571$	26.9666	1.12852	0.564259	−	0.825598i	$-0.309162\pi$
$571$	26.9666	1.12852	0.564259	+	0.825598i	$0.309162\pi$
$572$	0	0
$573$	0	0
$574$	8.00000	0.333914
$575$	−7.75458	−0.323388
$576$	0	0
$577$	−18.7083	−0.778836	−0.389418	−	0.921061i	$-0.627324\pi$
$577$	−18.7083	−0.778836	−0.389418	+	0.921061i	$0.627324\pi$
$578$	18.3376	0.762743
$579$	0	0
$580$	0	0
$581$	−11.1310	−0.461793
$582$	0	0
$583$	0	0
$584$	−44.5241	−1.84242
$585$	0	0
$586$	−39.7417	−1.64171
$587$	1.54970	0.0639628	0.0319814	−	0.999488i	$-0.489818\pi$
$587$	1.54970	0.0639628	0.0319814	+	0.999488i	$0.489818\pi$
$588$	0	0
$589$	−31.4833	−1.29725
$590$	31.0655	1.27895
$591$	0	0
$592$	−13.9333	−0.572653
$593$	−39.6863	−1.62972	−0.814860	−	0.579658i	$-0.803186\pi$
$593$	−39.6863	−1.62972	−0.814860	+	0.579658i	$0.803186\pi$
$594$	0	0
$595$	−14.4833	−0.593758
$596$	0	0
$597$	0	0
$598$	−31.4833	−1.28745
$599$	29.6513	1.21152	0.605759	−	0.795648i	$-0.292869\pi$
$599$	29.6513	1.21152	0.605759	+	0.795648i	$0.292869\pi$
$600$	0	0
$601$	37.2250	1.51844	0.759219	−	0.650835i	$-0.225581\pi$
$601$	37.2250	1.51844	0.759219	+	0.650835i	$0.225581\pi$
$602$	−1.41421	−0.0576390
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	35.7417	1.45071	0.725355	−	0.688375i	$-0.241676\pi$
$607$	35.7417	1.45071	0.725355	+	0.688375i	$0.241676\pi$
$608$	0	0
$609$	0	0
$610$	−6.51669	−0.263853
$611$	−31.4309	−1.27156
$612$	0	0
$613$	26.4833	1.06965	0.534826	−	0.844963i	$-0.320377\pi$
$613$	26.4833	1.06965	0.534826	+	0.844963i	$0.320377\pi$
$614$	9.48695	0.382862
$615$	0	0
$616$	0	0
$617$	−2.09772	−0.0844512	−0.0422256	−	0.999108i	$-0.513445\pi$
$617$	−2.09772	−0.0844512	−0.0422256	+	0.999108i	$0.513445\pi$
$618$	0	0
$619$	34.4499	1.38466	0.692330	−	0.721581i	$-0.256584\pi$
$619$	34.4499	1.38466	0.692330	+	0.721581i	$0.256584\pi$
$620$	0	0
$621$	0	0
$622$	21.6749	0.869085
$623$	−5.83953	−0.233956
$624$	0	0
$625$	−31.0000	−1.24000
$626$	−6.38756	−0.255298
$627$	0	0
$628$	0	0
$629$	19.0683	0.760302
$630$	0	0
$631$	−25.4499	−1.01315	−0.506573	−	0.862197i	$-0.669088\pi$
$631$	−25.4499	−1.01315	−0.506573	+	0.862197i	$0.669088\pi$
$632$	−38.1366	−1.51699
$633$	0	0
$634$	13.4833	0.535491
$635$	43.6108	1.73064
$636$	0	0
$637$	5.74166	0.227493
$638$	0	0
$639$	0	0
$640$	29.9333	1.18322
$641$	−11.2665	−0.445001	−0.222500	−	0.974933i	$-0.571422\pi$
$641$	−11.2665	−0.445001	−0.222500	+	0.974933i	$0.571422\pi$
$642$	0	0
$643$	−40.4499	−1.59519	−0.797595	−	0.603193i	$-0.793895\pi$
$643$	−40.4499	−1.59519	−0.797595	+	0.603193i	$0.793895\pi$
$644$	0	0
$645$	0	0
$646$	−44.4499	−1.74886
$647$	47.9006	1.88317	0.941583	−	0.336781i	$-0.109338\pi$
$647$	47.9006	1.88317	0.941583	+	0.336781i	$0.109338\pi$
$648$	0	0
$649$	0	0
$650$	16.2399	0.636980
$651$	0	0
$652$	0	0
$653$	27.8717	1.09070	0.545352	−	0.838207i	$-0.316396\pi$
$653$	27.8717	1.09070	0.545352	+	0.838207i	$0.316396\pi$
$654$	0	0
$655$	22.9333	0.896077
$656$	22.6274	0.883452
$657$	0	0
$658$	7.74166	0.301801
$659$	−32.5269	−1.26707	−0.633534	−	0.773715i	$-0.718396\pi$
$659$	−32.5269	−1.26707	−0.633534	+	0.773715i	$0.718396\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	28.9678	1.12586
$663$	0	0
$664$	−31.4833	−1.22179
$665$	15.1910	0.589082
$666$	0	0
$667$	−15.0334	−0.582094
$668$	0	0
$669$	0	0
$670$	−24.2583	−0.937182
$671$	0	0
$672$	0	0
$673$	−5.44994	−0.210080	−0.105040	−	0.994468i	$-0.533497\pi$
$673$	−5.44994	−0.210080	−0.105040	+	0.994468i	$0.533497\pi$
$674$	−32.4797	−1.25107
$675$	0	0
$676$	0	0
$677$	15.3265	0.589044	0.294522	−	0.955645i	$-0.404839\pi$
$677$	15.3265	0.589044	0.294522	+	0.955645i	$0.404839\pi$
$678$	0	0
$679$	7.22497	0.277269
$680$	−40.9650	−1.57094
$681$	0	0
$682$	0	0
$683$	−14.1421	−0.541134	−0.270567	−	0.962701i	$-0.587211\pi$
$683$	−14.1421	−0.541134	−0.270567	+	0.962701i	$0.587211\pi$
$684$	0	0
$685$	23.2917	0.889931
$686$	−1.41421	−0.0539949
$687$	0	0
$688$	−4.00000	−0.152499
$689$	77.0038	2.93361
$690$	0	0
$691$	−10.4499	−0.397535	−0.198767	−	0.980047i	$-0.563694\pi$
$691$	−10.4499	−0.397535	−0.198767	+	0.980047i	$0.563694\pi$
$692$	0	0
$693$	0	0
$694$	46.8999	1.78030
$695$	−9.89949	−0.375509
$696$	0	0
$697$	−30.9666	−1.17294
$698$	47.3526	1.79232
$699$	0	0
$700$	0	0
$701$	4.97334	0.187841	0.0939203	−	0.995580i	$-0.470060\pi$
$701$	4.97334	0.187841	0.0939203	+	0.995580i	$0.470060\pi$
$702$	0	0
$703$	−20.0000	−0.754314
$704$	0	0
$705$	0	0
$706$	−32.5167	−1.22378
$707$	8.66796	0.325992
$708$	0	0
$709$	40.4166	1.51788	0.758938	−	0.651163i	$-0.225718\pi$
$709$	40.4166	1.51788	0.758938	+	0.651163i	$0.225718\pi$
$710$	−5.29150	−0.198587
$711$	0	0
$712$	−16.5167	−0.618989
$713$	−21.2604	−0.796208
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−19.9333	−0.743903
$719$	−2.82843	−0.105483	−0.0527413	−	0.998608i	$-0.516796\pi$
$719$	−2.82843	−0.105483	−0.0527413	+	0.998608i	$0.516796\pi$
$720$	0	0
$721$	13.2250	0.492524
$722$	19.7518	0.735086
$723$	0	0
$724$	0	0
$725$	7.75458	0.287998
$726$	0	0
$727$	10.1916	0.377986	0.188993	−	0.981978i	$-0.439478\pi$
$727$	10.1916	0.377986	0.188993	+	0.981978i	$0.439478\pi$
$728$	16.2399	0.601889
$729$	0	0
$730$	58.8999	2.17998
$731$	5.47418	0.202470
$732$	0	0
$733$	−1.29171	−0.0477105	−0.0238553	−	0.999715i	$-0.507594\pi$
$733$	−1.29171	−0.0477105	−0.0238553	+	0.999715i	$0.507594\pi$
$734$	24.3598	0.899136
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−27.9333	−1.02754	−0.513771	−	0.857927i	$-0.671752\pi$
$739$	−27.9333	−1.02754	−0.513771	+	0.857927i	$0.671752\pi$
$740$	0	0
$741$	0	0
$742$	−18.9666	−0.696287
$743$	−32.1616	−1.17989	−0.589947	−	0.807442i	$-0.700851\pi$
$743$	−32.1616	−1.17989	−0.589947	+	0.807442i	$0.700851\pi$
$744$	0	0
$745$	61.6749	2.25959
$746$	27.5064	1.00708
$747$	0	0
$748$	0	0
$749$	9.53414	0.348370
$750$	0	0
$751$	−31.9666	−1.16648	−0.583239	−	0.812300i	$-0.698215\pi$
$751$	−31.9666	−1.16648	−0.583239	+	0.812300i	$0.698215\pi$
$752$	21.8967	0.798491
$753$	0	0
$754$	31.4833	1.14655
$755$	19.8873	0.723772
$756$	0	0
$757$	−24.4833	−0.889861	−0.444931	−	0.895565i	$-0.646772\pi$
$757$	−24.4833	−0.889861	−0.444931	+	0.895565i	$0.646772\pi$
$758$	−48.7668	−1.77129
$759$	0	0
$760$	42.9666	1.55856
$761$	−41.5130	−1.50485	−0.752423	−	0.658680i	$-0.771115\pi$
$761$	−41.5130	−1.50485	−0.752423	+	0.658680i	$0.771115\pi$
$762$	0	0
$763$	16.4833	0.596736
$764$	0	0
$765$	0	0
$766$	−30.1916	−1.09087
$767$	47.6707	1.72129
$768$	0	0
$769$	3.55006	0.128018	0.0640091	−	0.997949i	$-0.479611\pi$
$769$	3.55006	0.128018	0.0640091	+	0.997949i	$0.479611\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−46.8045	−1.68344	−0.841721	−	0.539913i	$-0.818457\pi$
$773$	−46.8045	−1.68344	−0.841721	+	0.539913i	$0.818457\pi$
$774$	0	0
$775$	10.9666	0.393933
$776$	20.4353	0.733585
$777$	0	0
$778$	−47.4833	−1.70236
$779$	32.4797	1.16371
$780$	0	0
$781$	0	0
$782$	−30.0166	−1.07339
$783$	0	0
$784$	−4.00000	−0.142857
$785$	−34.3065	−1.22445
$786$	0	0
$787$	7.55006	0.269130	0.134565	−	0.990905i	$-0.457036\pi$
$787$	7.55006	0.269130	0.134565	+	0.990905i	$0.457036\pi$
$788$	0	0
$789$	0	0
$790$	50.4499	1.79493
$791$	−5.29150	−0.188144
$792$	0	0
$793$	−10.0000	−0.355110
$794$	2.09772	0.0744454
$795$	0	0
$796$	0	0
$797$	−25.5441	−0.904820	−0.452410	−	0.891810i	$-0.649436\pi$
$797$	−25.5441	−0.904820	−0.452410	+	0.891810i	$0.649436\pi$
$798$	0	0
$799$	−29.9666	−1.06014
$800$	0	0
$801$	0	0
$802$	11.4833	0.405490
$803$	0	0
$804$	0	0
$805$	10.2583	0.361559
$806$	44.5241	1.56830
$807$	0	0
$808$	24.5167	0.862494
$809$	−10.9955	−0.386583	−0.193291	−	0.981141i	$-0.561916\pi$
$809$	−10.9955	−0.386583	−0.193291	+	0.981141i	$0.561916\pi$
$810$	0	0
$811$	26.7750	0.940198	0.470099	−	0.882614i	$-0.344218\pi$
$811$	26.7750	0.940198	0.470099	+	0.882614i	$0.344218\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−64.6885	−2.26594
$816$	0	0
$817$	−5.74166	−0.200875
$818$	−13.0461	−0.456146
$819$	0	0
$820$	0	0
$821$	33.9411	1.18455	0.592277	−	0.805735i	$-0.298229\pi$
$821$	33.9411	1.18455	0.592277	+	0.805735i	$0.298229\pi$
$822$	0	0
$823$	−55.9333	−1.94971	−0.974855	−	0.222838i	$-0.928468\pi$
$823$	−55.9333	−1.94971	−0.974855	+	0.222838i	$0.928468\pi$
$824$	37.4059	1.30310
$825$	0	0
$826$	−11.7417	−0.408545
$827$	−21.4842	−0.747078	−0.373539	−	0.927615i	$-0.621856\pi$
$827$	−21.4842	−0.747078	−0.373539	+	0.927615i	$0.621856\pi$
$828$	0	0
$829$	4.77503	0.165844	0.0829218	−	0.996556i	$-0.473575\pi$
$829$	4.77503	0.165844	0.0829218	+	0.996556i	$0.473575\pi$
$830$	41.6485	1.44564
$831$	0	0
$832$	45.9333	1.59245
$833$	5.47418	0.189669
$834$	0	0
$835$	−22.9333	−0.793638
$836$	0	0
$837$	0	0
$838$	−41.6749	−1.43964
$839$	−22.0794	−0.762265	−0.381133	−	0.924520i	$-0.624466\pi$
$839$	−22.0794	−0.762265	−0.381133	+	0.924520i	$0.624466\pi$
$840$	0	0
$841$	−13.9666	−0.481608
$842$	−10.6302	−0.366341
$843$	0	0
$844$	0	0
$845$	52.8267	1.81729
$846$	0	0
$847$	0	0
$848$	−53.6457	−1.84220
$849$	0	0
$850$	15.4833	0.531073
$851$	−13.5058	−0.462973
$852$	0	0
$853$	−38.9666	−1.33419	−0.667096	−	0.744972i	$-0.732463\pi$
$853$	−38.9666	−1.33419	−0.667096	+	0.744972i	$0.732463\pi$
$854$	2.46308	0.0842847
$855$	0	0
$856$	26.9666	0.921700
$857$	−12.8634	−0.439406	−0.219703	−	0.975567i	$-0.570509\pi$
$857$	−12.8634	−0.439406	−0.219703	+	0.975567i	$0.570509\pi$
$858$	0	0
$859$	−30.4499	−1.03894	−0.519469	−	0.854489i	$-0.673870\pi$
$859$	−30.4499	−1.03894	−0.519469	+	0.854489i	$0.673870\pi$
$860$	0	0
$861$	0	0
$862$	−22.4499	−0.764648
$863$	18.7029	0.636655	0.318328	−	0.947981i	$-0.396879\pi$
$863$	18.7029	0.636655	0.318328	+	0.947981i	$0.396879\pi$
$864$	0	0
$865$	13.5167	0.459581
$866$	−2.46308	−0.0836987
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−37.2250	−1.26132
$872$	46.6219	1.57882
$873$	0	0
$874$	31.4833	1.06494
$875$	7.93725	0.268328
$876$	0	0
$877$	−48.8999	−1.65123	−0.825616	−	0.564232i	$-0.809172\pi$
$877$	−48.8999	−1.65123	−0.825616	+	0.564232i	$0.809172\pi$
$878$	27.5536	0.929888
$879$	0	0
$880$	0	0
$881$	11.6791	0.393478	0.196739	−	0.980456i	$-0.436965\pi$
$881$	11.6791	0.393478	0.196739	+	0.980456i	$0.436965\pi$
$882$	0	0
$883$	52.4166	1.76396	0.881979	−	0.471289i	$-0.156211\pi$
$883$	52.4166	1.76396	0.881979	+	0.471289i	$0.156211\pi$
$884$	0	0
$885$	0	0
$886$	−38.5167	−1.29399
$887$	−29.4686	−0.989459	−0.494730	−	0.869047i	$-0.664733\pi$
$887$	−29.4686	−0.989459	−0.494730	+	0.869047i	$0.664733\pi$
$888$	0	0
$889$	−16.4833	−0.552833
$890$	21.8495	0.732398
$891$	0	0
$892$	0	0
$893$	31.4309	1.05179
$894$	0	0
$895$	37.4166	1.25070
$896$	−11.3137	−0.377964
$897$	0	0
$898$	14.4499	0.482201
$899$	21.2604	0.709074
$900$	0	0
$901$	73.4166	2.44586
$902$	0	0
$903$	0	0
$904$	−14.9666	−0.497783
$905$	0	0
$906$	0	0
$907$	−1.48331	−0.0492527	−0.0246263	−	0.999697i	$-0.507840\pi$
$907$	−1.48331	−0.0492527	−0.0246263	+	0.999697i	$0.507840\pi$
$908$	0	0
$909$	0	0
$910$	−21.4833	−0.712165
$911$	−38.8673	−1.28773	−0.643865	−	0.765139i	$-0.722670\pi$
$911$	−38.8673	−1.28773	−0.643865	+	0.765139i	$0.722670\pi$
$912$	0	0
$913$	0	0
$914$	32.5269	1.07589
$915$	0	0
$916$	0	0
$917$	−8.66796	−0.286241
$918$	0	0
$919$	27.0334	0.891749	0.445874	−	0.895096i	$-0.352893\pi$
$919$	27.0334	0.891749	0.445874	+	0.895096i	$0.352893\pi$
$920$	29.0150	0.956595
$921$	0	0
$922$	10.1916	0.335643
$923$	−8.11993	−0.267271
$924$	0	0
$925$	6.96663	0.229061
$926$	−24.7251	−0.812519
$927$	0	0
$928$	0	0
$929$	−16.4225	−0.538806	−0.269403	−	0.963028i	$-0.586826\pi$
$929$	−16.4225	−0.538806	−0.269403	+	0.963028i	$0.586826\pi$
$930$	0	0
$931$	−5.74166	−0.188175
$932$	0	0
$933$	0	0
$934$	29.4166	0.962540
$935$	0	0
$936$	0	0
$937$	−35.7417	−1.16763	−0.583815	−	0.811887i	$-0.698440\pi$
$937$	−35.7417	−1.16763	−0.583815	+	0.811887i	$0.698440\pi$
$938$	9.16879	0.299372
$939$	0	0
$940$	0	0
$941$	56.8395	1.85292	0.926458	−	0.376399i	$-0.122838\pi$
$941$	56.8395	1.85292	0.926458	+	0.376399i	$0.122838\pi$
$942$	0	0
$943$	21.9333	0.714245
$944$	−33.2104	−1.08091
$945$	0	0
$946$	0	0
$947$	−55.1071	−1.79074	−0.895371	−	0.445322i	$-0.853089\pi$
$947$	−55.1071	−1.79074	−0.895371	+	0.445322i	$0.853089\pi$
$948$	0	0
$949$	90.3832	2.93396
$950$	−16.2399	−0.526891
$951$	0	0
$952$	15.4833	0.501817
$953$	−50.5463	−1.63736	−0.818678	−	0.574253i	$-0.805293\pi$
$953$	−50.5463	−1.63736	−0.818678	+	0.574253i	$0.805293\pi$
$954$	0	0
$955$	−3.74166	−0.121077
$956$	0	0
$957$	0	0
$958$	32.7750	1.05891
$959$	−8.80344	−0.284278
$960$	0	0
$961$	−0.933259	−0.0301051
$962$	28.2843	0.911922
$963$	0	0
$964$	0	0
$965$	43.6108	1.40388
$966$	0	0
$967$	−30.9333	−0.994747	−0.497373	−	0.867537i	$-0.665702\pi$
$967$	−30.9333	−0.994747	−0.497373	+	0.867537i	$0.665702\pi$
$968$	0	0
$969$	0	0
$970$	−27.0334	−0.867989
$971$	−15.3265	−0.491850	−0.245925	−	0.969289i	$-0.579092\pi$
$971$	−15.3265	−0.491850	−0.245925	+	0.969289i	$0.579092\pi$
$972$	0	0
$973$	3.74166	0.119952
$974$	7.80177	0.249985
$975$	0	0
$976$	6.96663	0.222996
$977$	19.7518	0.631916	0.315958	−	0.948773i	$-0.397674\pi$
$977$	19.7518	0.631916	0.315958	+	0.948773i	$0.397674\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−42.0000	−1.34027
$983$	2.82843	0.0902128	0.0451064	−	0.998982i	$-0.485637\pi$
$983$	2.82843	0.0902128	0.0451064	+	0.998982i	$0.485637\pi$
$984$	0	0
$985$	58.8999	1.87671
$986$	30.0166	0.955925
$987$	0	0
$988$	0	0
$989$	−3.87729	−0.123291
$990$	0	0
$991$	0.449944	0.0142930	0.00714648	−	0.999974i	$-0.497725\pi$
$991$	0.449944	0.0142930	0.00714648	+	0.999974i	$0.497725\pi$
$992$	0	0
$993$	0	0
$994$	2.00000	0.0634361
$995$	5.97501	0.189421
$996$	0	0
$997$	−10.9666	−0.347317	−0.173658	−	0.984806i	$-0.555559\pi$
$997$	−10.9666	−0.347317	−0.173658	+	0.984806i	$0.555559\pi$
$998$	−25.4087	−0.804297
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cj.1.4	yes	4
3.2	odd	2	inner	7623.2.a.cj.1.1	✓	4
11.10	odd	2		7623.2.a.ck.1.2	yes	4
33.32	even	2		7623.2.a.ck.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7623.2.a.cj.1.1	✓	4	3.2	odd	2	inner
7623.2.a.cj.1.4	yes	4	1.1	even	1	trivial
7623.2.a.ck.1.2	yes	4	11.10	odd	2
7623.2.a.ck.1.3	yes	4	33.32	even	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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.64575 q^{5} -1.00000 q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.64575 q^{5} -1.00000 q^{7} -2.82843 q^{8} +3.74166 q^{10} +5.74166 q^{13} -1.41421 q^{14} -4.00000 q^{16} +5.47418 q^{17} -5.74166 q^{19} -3.87729 q^{23} +2.00000 q^{25} +8.11993 q^{26} +3.87729 q^{29} +5.48331 q^{31} +7.74166 q^{34} -2.64575 q^{35} +3.48331 q^{37} -8.11993 q^{38} -7.48331 q^{40} -5.65685 q^{41} +1.00000 q^{43} -5.48331 q^{46} -5.47418 q^{47} +1.00000 q^{49} +2.82843 q^{50} +13.4114 q^{53} +2.82843 q^{56} +5.48331 q^{58} +8.30261 q^{59} -1.74166 q^{61} +7.75458 q^{62} +8.00000 q^{64} +15.1910 q^{65} -6.48331 q^{67} -3.74166 q^{70} -1.41421 q^{71} +15.7417 q^{73} +4.92615 q^{74} +13.4833 q^{79} -10.5830 q^{80} -8.00000 q^{82} +11.1310 q^{83} +14.4833 q^{85} +1.41421 q^{86} +5.83953 q^{89} -5.74166 q^{91} -7.74166 q^{94} -15.1910 q^{95} -7.22497 q^{97} +1.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^7	\( 4 q - 4 q^{7} + 8 q^{13} - 16 q^{16} - 8 q^{19} + 8 q^{25} - 8 q^{31} + 16 q^{34} - 16 q^{37} + 4 q^{43} + 8 q^{46} + 4 q^{49} - 8 q^{58} + 8 q^{61} + 32 q^{64} + 4 q^{67} + 48 q^{73} + 24 q^{79} - 32 q^{82} + 28 q^{85} - 8 q^{91} - 16 q^{94} + 16 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 + 8 * q^13 - 16 * q^16 - 8 * q^19 + 8 * q^25 - 8 * q^31 + 16 * q^34 - 16 * q^37 + 4 * q^43 + 8 * q^46 + 4 * q^49 - 8 * q^58 + 8 * q^61 + 32 * q^64 + 4 * q^67 + 48 * q^73 + 24 * q^79 - 32 * q^82 + 28 * q^85 - 8 * q^91 - 16 * q^94 + 16 * q^97

Introduction

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