LMFDB - Embedded newform 7623.2.a.t.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 847)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.47214 q^{8} +O(q^{10})$	$q-0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.47214 q^{8} +0.381966 q^{10} +1.23607 q^{13} +0.381966 q^{14} +3.14590 q^{16} +3.09017 q^{17} +1.76393 q^{19} +1.85410 q^{20} -5.09017 q^{23} -4.00000 q^{25} -0.472136 q^{26} +1.85410 q^{28} -4.61803 q^{29} -4.23607 q^{31} -4.14590 q^{32} -1.18034 q^{34} +1.00000 q^{35} +6.47214 q^{37} -0.673762 q^{38} -1.47214 q^{40} -11.1803 q^{41} +12.5623 q^{43} +1.94427 q^{46} +6.61803 q^{47} +1.00000 q^{49} +1.52786 q^{50} -2.29180 q^{52} +2.38197 q^{53} -1.47214 q^{56} +1.76393 q^{58} -11.0902 q^{59} +7.61803 q^{61} +1.61803 q^{62} -4.70820 q^{64} -1.23607 q^{65} -8.32624 q^{67} -5.72949 q^{68} -0.381966 q^{70} +16.0902 q^{71} +14.2361 q^{73} -2.47214 q^{74} -3.27051 q^{76} -6.38197 q^{79} -3.14590 q^{80} +4.27051 q^{82} +2.70820 q^{83} -3.09017 q^{85} -4.79837 q^{86} -6.85410 q^{89} -1.23607 q^{91} +9.43769 q^{92} -2.52786 q^{94} -1.76393 q^{95} +7.00000 q^{97} -0.381966 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 3 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8	$ 2 q - 3 q^{2} + 3 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8} + 3 q^{10} - 2 q^{13} + 3 q^{14} + 13 q^{16} - 5 q^{17} + 8 q^{19} - 3 q^{20} + q^{23} - 8 q^{25} + 8 q^{26} - 3 q^{28} - 7 q^{29} - 4 q^{31} - 15 q^{32} + 20 q^{34} + 2 q^{35} + 4 q^{37} - 17 q^{38} + 6 q^{40} + 5 q^{43} - 14 q^{46} + 11 q^{47} + 2 q^{49} + 12 q^{50} - 18 q^{52} + 7 q^{53} + 6 q^{56} + 8 q^{58} - 11 q^{59} + 13 q^{61} + q^{62} + 4 q^{64} + 2 q^{65} - q^{67} - 45 q^{68} - 3 q^{70} + 21 q^{71} + 24 q^{73} + 4 q^{74} + 27 q^{76} - 15 q^{79} - 13 q^{80} - 25 q^{82} - 8 q^{83} + 5 q^{85} + 15 q^{86} - 7 q^{89} + 2 q^{91} + 39 q^{92} - 14 q^{94} - 8 q^{95} + 14 q^{97} - 3 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8 + 3 * q^10 - 2 * q^13 + 3 * q^14 + 13 * q^16 - 5 * q^17 + 8 * q^19 - 3 * q^20 + q^23 - 8 * q^25 + 8 * q^26 - 3 * q^28 - 7 * q^29 - 4 * q^31 - 15 * q^32 + 20 * q^34 + 2 * q^35 + 4 * q^37 - 17 * q^38 + 6 * q^40 + 5 * q^43 - 14 * q^46 + 11 * q^47 + 2 * q^49 + 12 * q^50 - 18 * q^52 + 7 * q^53 + 6 * q^56 + 8 * q^58 - 11 * q^59 + 13 * q^61 + q^62 + 4 * q^64 + 2 * q^65 - q^67 - 45 * q^68 - 3 * q^70 + 21 * q^71 + 24 * q^73 + 4 * q^74 + 27 * q^76 - 15 * q^79 - 13 * q^80 - 25 * q^82 - 8 * q^83 + 5 * q^85 + 15 * q^86 - 7 * q^89 + 2 * q^91 + 39 * q^92 - 14 * q^94 - 8 * q^95 + 14 * q^97 - 3 * 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 $

$2$	−0.381966	−0.270091	−0.135045	−	0.990839i	$-0.543118\pi$
$2$	−0.381966	−0.270091	−0.135045	+	0.990839i	$0.543118\pi$
$3$	0	0
$3$	0	0
$4$	−1.85410	−0.927051
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.47214	0.520479
$9$	0	0
$10$	0.381966	0.120788
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0.381966	0.102085
$15$	0	0
$16$	3.14590	0.786475
$17$	3.09017	0.749476	0.374738	−	0.927131i	$-0.377733\pi$
$17$	3.09017	0.749476	0.374738	+	0.927131i	$0.377733\pi$
$18$	0	0
$19$	1.76393	0.404674	0.202337	−	0.979316i	$-0.435146\pi$
$19$	1.76393	0.404674	0.202337	+	0.979316i	$0.435146\pi$
$20$	1.85410	0.414590
$21$	0	0
$22$	0	0
$23$	−5.09017	−1.06137	−0.530687	−	0.847568i	$-0.678066\pi$
$23$	−5.09017	−1.06137	−0.530687	+	0.847568i	$0.678066\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−0.472136	−0.0925935
$27$	0	0
$28$	1.85410	0.350392
$29$	−4.61803	−0.857547	−0.428774	−	0.903412i	$-0.641054\pi$
$29$	−4.61803	−0.857547	−0.428774	+	0.903412i	$0.641054\pi$
$30$	0	0
$31$	−4.23607	−0.760820	−0.380410	−	0.924818i	$-0.624217\pi$
$31$	−4.23607	−0.760820	−0.380410	+	0.924818i	$0.624217\pi$
$32$	−4.14590	−0.732898
$33$	0	0
$34$	−1.18034	−0.202427
$35$	1.00000	0.169031
$36$	0	0
$37$	6.47214	1.06401	0.532006	−	0.846740i	$-0.321438\pi$
$37$	6.47214	1.06401	0.532006	+	0.846740i	$0.321438\pi$
$38$	−0.673762	−0.109299
$39$	0	0
$40$	−1.47214	−0.232765
$41$	−11.1803	−1.74608	−0.873038	−	0.487652i	$-0.837853\pi$
$41$	−11.1803	−1.74608	−0.873038	+	0.487652i	$0.837853\pi$
$42$	0	0
$43$	12.5623	1.91573	0.957867	−	0.287213i	$-0.0927286\pi$
$43$	12.5623	1.91573	0.957867	+	0.287213i	$0.0927286\pi$
$44$	0	0
$45$	0	0
$46$	1.94427	0.286667
$47$	6.61803	0.965339	0.482670	−	0.875802i	$-0.339667\pi$
$47$	6.61803	0.965339	0.482670	+	0.875802i	$0.339667\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.52786	0.216073
$51$	0	0
$52$	−2.29180	−0.317815
$53$	2.38197	0.327188	0.163594	−	0.986528i	$-0.447691\pi$
$53$	2.38197	0.327188	0.163594	+	0.986528i	$0.447691\pi$
$54$	0	0
$55$	0	0
$56$	−1.47214	−0.196722
$57$	0	0
$58$	1.76393	0.231616
$59$	−11.0902	−1.44382	−0.721909	−	0.691988i	$-0.756735\pi$
$59$	−11.0902	−1.44382	−0.721909	+	0.691988i	$0.756735\pi$
$60$	0	0
$61$	7.61803	0.975389	0.487695	−	0.873014i	$-0.337838\pi$
$61$	7.61803	0.975389	0.487695	+	0.873014i	$0.337838\pi$
$62$	1.61803	0.205491
$63$	0	0
$64$	−4.70820	−0.588525
$65$	−1.23607	−0.153315
$66$	0	0
$67$	−8.32624	−1.01721	−0.508606	−	0.860999i	$-0.669839\pi$
$67$	−8.32624	−1.01721	−0.508606	+	0.860999i	$0.669839\pi$
$68$	−5.72949	−0.694803
$69$	0	0
$70$	−0.381966	−0.0456537
$71$	16.0902	1.90955	0.954776	−	0.297326i	$-0.0960949\pi$
$71$	16.0902	1.90955	0.954776	+	0.297326i	$0.0960949\pi$
$72$	0	0
$73$	14.2361	1.66621	0.833103	−	0.553118i	$-0.186562\pi$
$73$	14.2361	1.66621	0.833103	+	0.553118i	$0.186562\pi$
$74$	−2.47214	−0.287380
$75$	0	0
$76$	−3.27051	−0.375153
$77$	0	0
$78$	0	0
$79$	−6.38197	−0.718027	−0.359014	−	0.933332i	$-0.616887\pi$
$79$	−6.38197	−0.718027	−0.359014	+	0.933332i	$0.616887\pi$
$80$	−3.14590	−0.351722
$81$	0	0
$82$	4.27051	0.471599
$83$	2.70820	0.297264	0.148632	−	0.988893i	$-0.452513\pi$
$83$	2.70820	0.297264	0.148632	+	0.988893i	$0.452513\pi$
$84$	0	0
$85$	−3.09017	−0.335176
$86$	−4.79837	−0.517422
$87$	0	0
$88$	0	0
$89$	−6.85410	−0.726533	−0.363267	−	0.931685i	$-0.618339\pi$
$89$	−6.85410	−0.726533	−0.363267	+	0.931685i	$0.618339\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	9.43769	0.983948
$93$	0	0
$94$	−2.52786	−0.260729
$95$	−1.76393	−0.180976
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	−0.381966	−0.0385844
$99$	0	0
$100$	7.41641	0.741641
$101$	−14.7984	−1.47249	−0.736247	−	0.676713i	$-0.763404\pi$
$101$	−14.7984	−1.47249	−0.736247	+	0.676713i	$0.763404\pi$
$102$	0	0
$103$	8.85410	0.872421	0.436210	−	0.899845i	$-0.356320\pi$
$103$	8.85410	0.872421	0.436210	+	0.899845i	$0.356320\pi$
$104$	1.81966	0.178432
$105$	0	0
$106$	−0.909830	−0.0883705
$107$	2.70820	0.261812	0.130906	−	0.991395i	$-0.458211\pi$
$107$	2.70820	0.261812	0.130906	+	0.991395i	$0.458211\pi$
$108$	0	0
$109$	1.52786	0.146343	0.0731714	−	0.997319i	$-0.476688\pi$
$109$	1.52786	0.146343	0.0731714	+	0.997319i	$0.476688\pi$
$110$	0	0
$111$	0	0
$112$	−3.14590	−0.297259
$113$	−0.145898	−0.0137249	−0.00686247	−	0.999976i	$-0.502184\pi$
$113$	−0.145898	−0.0137249	−0.00686247	+	0.999976i	$0.502184\pi$
$114$	0	0
$115$	5.09017	0.474661
$116$	8.56231	0.794990
$117$	0	0
$118$	4.23607	0.389962
$119$	−3.09017	−0.283275
$120$	0	0
$121$	0	0
$122$	−2.90983	−0.263444
$123$	0	0
$124$	7.85410	0.705319
$125$	9.00000	0.804984
$126$	0	0
$127$	−2.94427	−0.261262	−0.130631	−	0.991431i	$-0.541700\pi$
$127$	−2.94427	−0.261262	−0.130631	+	0.991431i	$0.541700\pi$
$128$	10.0902	0.891853
$129$	0	0
$130$	0.472136	0.0414091
$131$	18.9443	1.65517	0.827584	−	0.561341i	$-0.189715\pi$
$131$	18.9443	1.65517	0.827584	+	0.561341i	$0.189715\pi$
$132$	0	0
$133$	−1.76393	−0.152952
$134$	3.18034	0.274740
$135$	0	0
$136$	4.54915	0.390086
$137$	15.3262	1.30941	0.654704	−	0.755885i	$-0.272793\pi$
$137$	15.3262	1.30941	0.654704	+	0.755885i	$0.272793\pi$
$138$	0	0
$139$	−11.9443	−1.01310	−0.506550	−	0.862211i	$-0.669079\pi$
$139$	−11.9443	−1.01310	−0.506550	+	0.862211i	$0.669079\pi$
$140$	−1.85410	−0.156700
$141$	0	0
$142$	−6.14590	−0.515752
$143$	0	0
$144$	0	0
$145$	4.61803	0.383507
$146$	−5.43769	−0.450027
$147$	0	0
$148$	−12.0000	−0.986394
$149$	−8.85410	−0.725356	−0.362678	−	0.931914i	$-0.618138\pi$
$149$	−8.85410	−0.725356	−0.362678	+	0.931914i	$0.618138\pi$
$150$	0	0
$151$	−0.0557281	−0.00453509	−0.00226754	−	0.999997i	$-0.500722\pi$
$151$	−0.0557281	−0.00453509	−0.00226754	+	0.999997i	$0.500722\pi$
$152$	2.59675	0.210624
$153$	0	0
$154$	0	0
$155$	4.23607	0.340249
$156$	0	0
$157$	19.8885	1.58728	0.793639	−	0.608389i	$-0.208184\pi$
$157$	19.8885	1.58728	0.793639	+	0.608389i	$0.208184\pi$
$158$	2.43769	0.193933
$159$	0	0
$160$	4.14590	0.327762
$161$	5.09017	0.401162
$162$	0	0
$163$	−8.70820	−0.682079	−0.341040	−	0.940049i	$-0.610779\pi$
$163$	−8.70820	−0.682079	−0.341040	+	0.940049i	$0.610779\pi$
$164$	20.7295	1.61870
$165$	0	0
$166$	−1.03444	−0.0802883
$167$	6.47214	0.500829	0.250414	−	0.968139i	$-0.419433\pi$
$167$	6.47214	0.500829	0.250414	+	0.968139i	$0.419433\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	1.18034	0.0905279
$171$	0	0
$172$	−23.2918	−1.77598
$173$	−15.3820	−1.16947	−0.584735	−	0.811225i	$-0.698801\pi$
$173$	−15.3820	−1.16947	−0.584735	+	0.811225i	$0.698801\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	2.61803	0.196230
$179$	3.76393	0.281329	0.140665	−	0.990057i	$-0.455076\pi$
$179$	3.76393	0.281329	0.140665	+	0.990057i	$0.455076\pi$
$180$	0	0
$181$	−7.41641	−0.551257	−0.275629	−	0.961264i	$-0.588886\pi$
$181$	−7.41641	−0.551257	−0.275629	+	0.961264i	$0.588886\pi$
$182$	0.472136	0.0349970
$183$	0	0
$184$	−7.49342	−0.552422
$185$	−6.47214	−0.475841
$186$	0	0
$187$	0	0
$188$	−12.2705	−0.894919
$189$	0	0
$190$	0.673762	0.0488798
$191$	−15.7639	−1.14064	−0.570319	−	0.821423i	$-0.693180\pi$
$191$	−15.7639	−1.14064	−0.570319	+	0.821423i	$0.693180\pi$
$192$	0	0
$193$	15.3262	1.10321	0.551603	−	0.834107i	$-0.314016\pi$
$193$	15.3262	1.10321	0.551603	+	0.834107i	$0.314016\pi$
$194$	−2.67376	−0.191965
$195$	0	0
$196$	−1.85410	−0.132436
$197$	−4.29180	−0.305778	−0.152889	−	0.988243i	$-0.548858\pi$
$197$	−4.29180	−0.305778	−0.152889	+	0.988243i	$0.548858\pi$
$198$	0	0
$199$	−8.23607	−0.583839	−0.291920	−	0.956443i	$-0.594294\pi$
$199$	−8.23607	−0.583839	−0.291920	+	0.956443i	$0.594294\pi$
$200$	−5.88854	−0.416383
$201$	0	0
$202$	5.65248	0.397707
$203$	4.61803	0.324122
$204$	0	0
$205$	11.1803	0.780869
$206$	−3.38197	−0.235633
$207$	0	0
$208$	3.88854	0.269622
$209$	0	0
$210$	0	0
$211$	−15.6180	−1.07519	−0.537595	−	0.843203i	$-0.680667\pi$
$211$	−15.6180	−1.07519	−0.537595	+	0.843203i	$0.680667\pi$
$212$	−4.41641	−0.303320
$213$	0	0
$214$	−1.03444	−0.0707130
$215$	−12.5623	−0.856742
$216$	0	0
$217$	4.23607	0.287563
$218$	−0.583592	−0.0395258
$219$	0	0
$220$	0	0
$221$	3.81966	0.256938
$222$	0	0
$223$	2.03444	0.136236	0.0681182	−	0.997677i	$-0.478301\pi$
$223$	2.03444	0.136236	0.0681182	+	0.997677i	$0.478301\pi$
$224$	4.14590	0.277009
$225$	0	0
$226$	0.0557281	0.00370698
$227$	−16.0344	−1.06424	−0.532122	−	0.846668i	$-0.678605\pi$
$227$	−16.0344	−1.06424	−0.532122	+	0.846668i	$0.678605\pi$
$228$	0	0
$229$	6.76393	0.446973	0.223487	−	0.974707i	$-0.428256\pi$
$229$	6.76393	0.446973	0.223487	+	0.974707i	$0.428256\pi$
$230$	−1.94427	−0.128201
$231$	0	0
$232$	−6.79837	−0.446335
$233$	−29.4164	−1.92713	−0.963566	−	0.267469i	$-0.913813\pi$
$233$	−29.4164	−1.92713	−0.963566	+	0.267469i	$0.913813\pi$
$234$	0	0
$235$	−6.61803	−0.431713
$236$	20.5623	1.33849
$237$	0	0
$238$	1.18034	0.0765101
$239$	−17.0902	−1.10547	−0.552736	−	0.833357i	$-0.686416\pi$
$239$	−17.0902	−1.10547	−0.552736	+	0.833357i	$0.686416\pi$
$240$	0	0
$241$	−16.2705	−1.04808	−0.524038	−	0.851695i	$-0.675575\pi$
$241$	−16.2705	−1.04808	−0.524038	+	0.851695i	$0.675575\pi$
$242$	0	0
$243$	0	0
$244$	−14.1246	−0.904236
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	2.18034	0.138732
$248$	−6.23607	−0.395991
$249$	0	0
$250$	−3.43769	−0.217419
$251$	−23.0000	−1.45175	−0.725874	−	0.687828i	$-0.758564\pi$
$251$	−23.0000	−1.45175	−0.725874	+	0.687828i	$0.758564\pi$
$252$	0	0
$253$	0	0
$254$	1.12461	0.0705644
$255$	0	0
$256$	5.56231	0.347644
$257$	8.56231	0.534102	0.267051	−	0.963682i	$-0.413951\pi$
$257$	8.56231	0.534102	0.267051	+	0.963682i	$0.413951\pi$
$258$	0	0
$259$	−6.47214	−0.402159
$260$	2.29180	0.142131
$261$	0	0
$262$	−7.23607	−0.447046
$263$	−17.1246	−1.05595	−0.527974	−	0.849260i	$-0.677048\pi$
$263$	−17.1246	−1.05595	−0.527974	+	0.849260i	$0.677048\pi$
$264$	0	0
$265$	−2.38197	−0.146323
$266$	0.673762	0.0413110
$267$	0	0
$268$	15.4377	0.943007
$269$	−16.8541	−1.02761	−0.513806	−	0.857906i	$-0.671765\pi$
$269$	−16.8541	−1.02761	−0.513806	+	0.857906i	$0.671765\pi$
$270$	0	0
$271$	4.79837	0.291480	0.145740	−	0.989323i	$-0.453444\pi$
$271$	4.79837	0.291480	0.145740	+	0.989323i	$0.453444\pi$
$272$	9.72136	0.589444
$273$	0	0
$274$	−5.85410	−0.353659
$275$	0	0
$276$	0	0
$277$	−23.0344	−1.38401	−0.692003	−	0.721895i	$-0.743271\pi$
$277$	−23.0344	−1.38401	−0.692003	+	0.721895i	$0.743271\pi$
$278$	4.56231	0.273629
$279$	0	0
$280$	1.47214	0.0879770
$281$	−25.1803	−1.50213	−0.751067	−	0.660226i	$-0.770460\pi$
$281$	−25.1803	−1.50213	−0.751067	+	0.660226i	$0.770460\pi$
$282$	0	0
$283$	−11.9443	−0.710013	−0.355007	−	0.934864i	$-0.615521\pi$
$283$	−11.9443	−0.710013	−0.355007	+	0.934864i	$0.615521\pi$
$284$	−29.8328	−1.77025
$285$	0	0
$286$	0	0
$287$	11.1803	0.659955
$288$	0	0
$289$	−7.45085	−0.438285
$290$	−1.76393	−0.103582
$291$	0	0
$292$	−26.3951	−1.54466
$293$	−11.0000	−0.642627	−0.321313	−	0.946973i	$-0.604124\pi$
$293$	−11.0000	−0.642627	−0.321313	+	0.946973i	$0.604124\pi$
$294$	0	0
$295$	11.0902	0.645695
$296$	9.52786	0.553796
$297$	0	0
$298$	3.38197	0.195912
$299$	−6.29180	−0.363864
$300$	0	0
$301$	−12.5623	−0.724079
$302$	0.0212862	0.00122489
$303$	0	0
$304$	5.54915	0.318266
$305$	−7.61803	−0.436207
$306$	0	0
$307$	23.1803	1.32297	0.661486	−	0.749958i	$-0.269926\pi$
$307$	23.1803	1.32297	0.661486	+	0.749958i	$0.269926\pi$
$308$	0	0
$309$	0	0
$310$	−1.61803	−0.0918982
$311$	−9.00000	−0.510343	−0.255172	−	0.966896i	$-0.582132\pi$
$311$	−9.00000	−0.510343	−0.255172	+	0.966896i	$0.582132\pi$
$312$	0	0
$313$	18.3262	1.03586	0.517930	−	0.855423i	$-0.326703\pi$
$313$	18.3262	1.03586	0.517930	+	0.855423i	$0.326703\pi$
$314$	−7.59675	−0.428709
$315$	0	0
$316$	11.8328	0.665648
$317$	4.43769	0.249246	0.124623	−	0.992204i	$-0.460228\pi$
$317$	4.43769	0.249246	0.124623	+	0.992204i	$0.460228\pi$
$318$	0	0
$319$	0	0
$320$	4.70820	0.263197
$321$	0	0
$322$	−1.94427	−0.108350
$323$	5.45085	0.303293
$324$	0	0
$325$	−4.94427	−0.274259
$326$	3.32624	0.184223
$327$	0	0
$328$	−16.4590	−0.908795
$329$	−6.61803	−0.364864
$330$	0	0
$331$	6.18034	0.339702	0.169851	−	0.985470i	$-0.445671\pi$
$331$	6.18034	0.339702	0.169851	+	0.985470i	$0.445671\pi$
$332$	−5.02129	−0.275579
$333$	0	0
$334$	−2.47214	−0.135269
$335$	8.32624	0.454911
$336$	0	0
$337$	−25.9443	−1.41327	−0.706637	−	0.707576i	$-0.749789\pi$
$337$	−25.9443	−1.41327	−0.706637	+	0.707576i	$0.749789\pi$
$338$	4.38197	0.238348
$339$	0	0
$340$	5.72949	0.310725
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	18.4934	0.997099
$345$	0	0
$346$	5.87539	0.315863
$347$	2.34752	0.126022	0.0630108	−	0.998013i	$-0.479930\pi$
$347$	2.34752	0.126022	0.0630108	+	0.998013i	$0.479930\pi$
$348$	0	0
$349$	−32.7984	−1.75566	−0.877828	−	0.478975i	$-0.841008\pi$
$349$	−32.7984	−1.75566	−0.877828	+	0.478975i	$0.841008\pi$
$350$	−1.52786	−0.0816678
$351$	0	0
$352$	0	0
$353$	−24.0902	−1.28219	−0.641095	−	0.767461i	$-0.721520\pi$
$353$	−24.0902	−1.28219	−0.641095	+	0.767461i	$0.721520\pi$
$354$	0	0
$355$	−16.0902	−0.853978
$356$	12.7082	0.673533
$357$	0	0
$358$	−1.43769	−0.0759845
$359$	−15.6180	−0.824288	−0.412144	−	0.911119i	$-0.635220\pi$
$359$	−15.6180	−0.824288	−0.412144	+	0.911119i	$0.635220\pi$
$360$	0	0
$361$	−15.8885	−0.836239
$362$	2.83282	0.148889
$363$	0	0
$364$	2.29180	0.120123
$365$	−14.2361	−0.745150
$366$	0	0
$367$	32.8328	1.71386	0.856930	−	0.515434i	$-0.172369\pi$
$367$	32.8328	1.71386	0.856930	+	0.515434i	$0.172369\pi$
$368$	−16.0132	−0.834743
$369$	0	0
$370$	2.47214	0.128520
$371$	−2.38197	−0.123666
$372$	0	0
$373$	15.4377	0.799334	0.399667	−	0.916661i	$-0.369126\pi$
$373$	15.4377	0.799334	0.399667	+	0.916661i	$0.369126\pi$
$374$	0	0
$375$	0	0
$376$	9.74265	0.502439
$377$	−5.70820	−0.293987
$378$	0	0
$379$	32.0689	1.64727	0.823634	−	0.567122i	$-0.191943\pi$
$379$	32.0689	1.64727	0.823634	+	0.567122i	$0.191943\pi$
$380$	3.27051	0.167774
$381$	0	0
$382$	6.02129	0.308076
$383$	34.5967	1.76781	0.883906	−	0.467665i	$-0.154905\pi$
$383$	34.5967	1.76781	0.883906	+	0.467665i	$0.154905\pi$
$384$	0	0
$385$	0	0
$386$	−5.85410	−0.297966
$387$	0	0
$388$	−12.9787	−0.658894
$389$	34.1803	1.73301	0.866506	−	0.499167i	$-0.166360\pi$
$389$	34.1803	1.73301	0.866506	+	0.499167i	$0.166360\pi$
$390$	0	0
$391$	−15.7295	−0.795475
$392$	1.47214	0.0743541
$393$	0	0
$394$	1.63932	0.0825878
$395$	6.38197	0.321112
$396$	0	0
$397$	−0.819660	−0.0411376	−0.0205688	−	0.999788i	$-0.506548\pi$
$397$	−0.819660	−0.0411376	−0.0205688	+	0.999788i	$0.506548\pi$
$398$	3.14590	0.157690
$399$	0	0
$400$	−12.5836	−0.629180
$401$	17.5279	0.875300	0.437650	−	0.899145i	$-0.355811\pi$
$401$	17.5279	0.875300	0.437650	+	0.899145i	$0.355811\pi$
$402$	0	0
$403$	−5.23607	−0.260827
$404$	27.4377	1.36508
$405$	0	0
$406$	−1.76393	−0.0875425
$407$	0	0
$408$	0	0
$409$	17.7082	0.875614	0.437807	−	0.899069i	$-0.355755\pi$
$409$	17.7082	0.875614	0.437807	+	0.899069i	$0.355755\pi$
$410$	−4.27051	−0.210905
$411$	0	0
$412$	−16.4164	−0.808778
$413$	11.0902	0.545712
$414$	0	0
$415$	−2.70820	−0.132941
$416$	−5.12461	−0.251255
$417$	0	0
$418$	0	0
$419$	−29.8885	−1.46015	−0.730075	−	0.683367i	$-0.760515\pi$
$419$	−29.8885	−1.46015	−0.730075	+	0.683367i	$0.760515\pi$
$420$	0	0
$421$	0.763932	0.0372318	0.0186159	−	0.999827i	$-0.494074\pi$
$421$	0.763932	0.0372318	0.0186159	+	0.999827i	$0.494074\pi$
$422$	5.96556	0.290399
$423$	0	0
$424$	3.50658	0.170294
$425$	−12.3607	−0.599581
$426$	0	0
$427$	−7.61803	−0.368663
$428$	−5.02129	−0.242713
$429$	0	0
$430$	4.79837	0.231398
$431$	−13.8541	−0.667329	−0.333664	−	0.942692i	$-0.608285\pi$
$431$	−13.8541	−0.667329	−0.333664	+	0.942692i	$0.608285\pi$
$432$	0	0
$433$	−8.96556	−0.430857	−0.215429	−	0.976520i	$-0.569115\pi$
$433$	−8.96556	−0.430857	−0.215429	+	0.976520i	$0.569115\pi$
$434$	−1.61803	−0.0776681
$435$	0	0
$436$	−2.83282	−0.135667
$437$	−8.97871	−0.429510
$438$	0	0
$439$	−3.38197	−0.161412	−0.0807062	−	0.996738i	$-0.525718\pi$
$439$	−3.38197	−0.161412	−0.0807062	+	0.996738i	$0.525718\pi$
$440$	0	0
$441$	0	0
$442$	−1.45898	−0.0693966
$443$	−3.09017	−0.146818	−0.0734092	−	0.997302i	$-0.523388\pi$
$443$	−3.09017	−0.146818	−0.0734092	+	0.997302i	$0.523388\pi$
$444$	0	0
$445$	6.85410	0.324916
$446$	−0.777088	−0.0367962
$447$	0	0
$448$	4.70820	0.222442
$449$	−12.3262	−0.581711	−0.290856	−	0.956767i	$-0.593940\pi$
$449$	−12.3262	−0.581711	−0.290856	+	0.956767i	$0.593940\pi$
$450$	0	0
$451$	0	0
$452$	0.270510	0.0127237
$453$	0	0
$454$	6.12461	0.287442
$455$	1.23607	0.0579478
$456$	0	0
$457$	0.145898	0.00682482	0.00341241	−	0.999994i	$-0.498914\pi$
$457$	0.145898	0.00682482	0.00341241	+	0.999994i	$0.498914\pi$
$458$	−2.58359	−0.120723
$459$	0	0
$460$	−9.43769	−0.440035
$461$	−18.5066	−0.861937	−0.430969	−	0.902367i	$-0.641828\pi$
$461$	−18.5066	−0.861937	−0.430969	+	0.902367i	$0.641828\pi$
$462$	0	0
$463$	22.2361	1.03340	0.516699	−	0.856167i	$-0.327161\pi$
$463$	22.2361	1.03340	0.516699	+	0.856167i	$0.327161\pi$
$464$	−14.5279	−0.674439
$465$	0	0
$466$	11.2361	0.520501
$467$	4.50658	0.208540	0.104270	−	0.994549i	$-0.466749\pi$
$467$	4.50658	0.208540	0.104270	+	0.994549i	$0.466749\pi$
$468$	0	0
$469$	8.32624	0.384470
$470$	2.52786	0.116602
$471$	0	0
$472$	−16.3262	−0.751476
$473$	0	0
$474$	0	0
$475$	−7.05573	−0.323739
$476$	5.72949	0.262611
$477$	0	0
$478$	6.52786	0.298578
$479$	21.7639	0.994419	0.497210	−	0.867630i	$-0.334358\pi$
$479$	21.7639	0.994419	0.497210	+	0.867630i	$0.334358\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	6.21478	0.283076
$483$	0	0
$484$	0	0
$485$	−7.00000	−0.317854
$486$	0	0
$487$	−32.5967	−1.47710	−0.738550	−	0.674199i	$-0.764489\pi$
$487$	−32.5967	−1.47710	−0.738550	+	0.674199i	$0.764489\pi$
$488$	11.2148	0.507669
$489$	0	0
$490$	0.381966	0.0172555
$491$	22.1459	0.999430	0.499715	−	0.866190i	$-0.333438\pi$
$491$	22.1459	0.999430	0.499715	+	0.866190i	$0.333438\pi$
$492$	0	0
$493$	−14.2705	−0.642711
$494$	−0.832816	−0.0374702
$495$	0	0
$496$	−13.3262	−0.598366
$497$	−16.0902	−0.721743
$498$	0	0
$499$	−10.0902	−0.451698	−0.225849	−	0.974162i	$-0.572516\pi$
$499$	−10.0902	−0.451698	−0.225849	+	0.974162i	$0.572516\pi$
$500$	−16.6869	−0.746262
$501$	0	0
$502$	8.78522	0.392103
$503$	−34.8328	−1.55312	−0.776559	−	0.630044i	$-0.783037\pi$
$503$	−34.8328	−1.55312	−0.776559	+	0.630044i	$0.783037\pi$
$504$	0	0
$505$	14.7984	0.658519
$506$	0	0
$507$	0	0
$508$	5.45898	0.242203
$509$	10.7426	0.476159	0.238080	−	0.971246i	$-0.423482\pi$
$509$	10.7426	0.476159	0.238080	+	0.971246i	$0.423482\pi$
$510$	0	0
$511$	−14.2361	−0.629767
$512$	−22.3050	−0.985749
$513$	0	0
$514$	−3.27051	−0.144256
$515$	−8.85410	−0.390158
$516$	0	0
$517$	0	0
$518$	2.47214	0.108619
$519$	0	0
$520$	−1.81966	−0.0797974
$521$	12.2361	0.536072	0.268036	−	0.963409i	$-0.413625\pi$
$521$	12.2361	0.536072	0.268036	+	0.963409i	$0.413625\pi$
$522$	0	0
$523$	−16.5623	−0.724219	−0.362110	−	0.932136i	$-0.617943\pi$
$523$	−16.5623	−0.724219	−0.362110	+	0.932136i	$0.617943\pi$
$524$	−35.1246	−1.53443
$525$	0	0
$526$	6.54102	0.285202
$527$	−13.0902	−0.570217
$528$	0	0
$529$	2.90983	0.126514
$530$	0.909830	0.0395205
$531$	0	0
$532$	3.27051	0.141795
$533$	−13.8197	−0.598596
$534$	0	0
$535$	−2.70820	−0.117086
$536$	−12.2574	−0.529437
$537$	0	0
$538$	6.43769	0.277549
$539$	0	0
$540$	0	0
$541$	−0.708204	−0.0304481	−0.0152240	−	0.999884i	$-0.504846\pi$
$541$	−0.708204	−0.0304481	−0.0152240	+	0.999884i	$0.504846\pi$
$542$	−1.83282	−0.0787262
$543$	0	0
$544$	−12.8115	−0.549290
$545$	−1.52786	−0.0654465
$546$	0	0
$547$	1.72949	0.0739477	0.0369738	−	0.999316i	$-0.488228\pi$
$547$	1.72949	0.0739477	0.0369738	+	0.999316i	$0.488228\pi$
$548$	−28.4164	−1.21389
$549$	0	0
$550$	0	0
$551$	−8.14590	−0.347027
$552$	0	0
$553$	6.38197	0.271389
$554$	8.79837	0.373807
$555$	0	0
$556$	22.1459	0.939195
$557$	−9.76393	−0.413711	−0.206856	−	0.978371i	$-0.566323\pi$
$557$	−9.76393	−0.413711	−0.206856	+	0.978371i	$0.566323\pi$
$558$	0	0
$559$	15.5279	0.656759
$560$	3.14590	0.132938
$561$	0	0
$562$	9.61803	0.405712
$563$	−31.9787	−1.34774	−0.673871	−	0.738849i	$-0.735370\pi$
$563$	−31.9787	−1.34774	−0.673871	+	0.738849i	$0.735370\pi$
$564$	0	0
$565$	0.145898	0.00613798
$566$	4.56231	0.191768
$567$	0	0
$568$	23.6869	0.993881
$569$	37.3050	1.56390	0.781952	−	0.623338i	$-0.214224\pi$
$569$	37.3050	1.56390	0.781952	+	0.623338i	$0.214224\pi$
$570$	0	0
$571$	−34.3050	−1.43562	−0.717809	−	0.696240i	$-0.754855\pi$
$571$	−34.3050	−1.43562	−0.717809	+	0.696240i	$0.754855\pi$
$572$	0	0
$573$	0	0
$574$	−4.27051	−0.178248
$575$	20.3607	0.849099
$576$	0	0
$577$	9.65248	0.401838	0.200919	−	0.979608i	$-0.435607\pi$
$577$	9.65248	0.401838	0.200919	+	0.979608i	$0.435607\pi$
$578$	2.84597	0.118377
$579$	0	0
$580$	−8.56231	−0.355530
$581$	−2.70820	−0.112355
$582$	0	0
$583$	0	0
$584$	20.9574	0.867225
$585$	0	0
$586$	4.20163	0.173568
$587$	−1.00000	−0.0412744	−0.0206372	−	0.999787i	$-0.506569\pi$
$587$	−1.00000	−0.0412744	−0.0206372	+	0.999787i	$0.506569\pi$
$588$	0	0
$589$	−7.47214	−0.307884
$590$	−4.23607	−0.174396
$591$	0	0
$592$	20.3607	0.836819
$593$	11.1246	0.456833	0.228417	−	0.973564i	$-0.426645\pi$
$593$	11.1246	0.456833	0.228417	+	0.973564i	$0.426645\pi$
$594$	0	0
$595$	3.09017	0.126685
$596$	16.4164	0.672442
$597$	0	0
$598$	2.40325	0.0982763
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−13.7082	−0.559169	−0.279585	−	0.960121i	$-0.590197\pi$
$601$	−13.7082	−0.559169	−0.279585	+	0.960121i	$0.590197\pi$
$602$	4.79837	0.195567
$603$	0	0
$604$	0.103326	0.00420426
$605$	0	0
$606$	0	0
$607$	−20.2016	−0.819959	−0.409979	−	0.912095i	$-0.634464\pi$
$607$	−20.2016	−0.819959	−0.409979	+	0.912095i	$0.634464\pi$
$608$	−7.31308	−0.296585
$609$	0	0
$610$	2.90983	0.117816
$611$	8.18034	0.330941
$612$	0	0
$613$	0.888544	0.0358879	0.0179440	−	0.999839i	$-0.494288\pi$
$613$	0.888544	0.0358879	0.0179440	+	0.999839i	$0.494288\pi$
$614$	−8.85410	−0.357322
$615$	0	0
$616$	0	0
$617$	−9.41641	−0.379090	−0.189545	−	0.981872i	$-0.560701\pi$
$617$	−9.41641	−0.379090	−0.189545	+	0.981872i	$0.560701\pi$
$618$	0	0
$619$	−39.7082	−1.59601	−0.798004	−	0.602653i	$-0.794111\pi$
$619$	−39.7082	−1.59601	−0.798004	+	0.602653i	$0.794111\pi$
$620$	−7.85410	−0.315428
$621$	0	0
$622$	3.43769	0.137839
$623$	6.85410	0.274604
$624$	0	0
$625$	11.0000	0.440000
$626$	−7.00000	−0.279776
$627$	0	0
$628$	−36.8754	−1.47149
$629$	20.0000	0.797452
$630$	0	0
$631$	17.8328	0.709913	0.354957	−	0.934883i	$-0.384496\pi$
$631$	17.8328	0.709913	0.354957	+	0.934883i	$0.384496\pi$
$632$	−9.39512	−0.373718
$633$	0	0
$634$	−1.69505	−0.0673190
$635$	2.94427	0.116840
$636$	0	0
$637$	1.23607	0.0489748
$638$	0	0
$639$	0	0
$640$	−10.0902	−0.398849
$641$	−31.6525	−1.25020	−0.625099	−	0.780546i	$-0.714941\pi$
$641$	−31.6525	−1.25020	−0.625099	+	0.780546i	$0.714941\pi$
$642$	0	0
$643$	1.58359	0.0624508	0.0312254	−	0.999512i	$-0.490059\pi$
$643$	1.58359	0.0624508	0.0312254	+	0.999512i	$0.490059\pi$
$644$	−9.43769	−0.371897
$645$	0	0
$646$	−2.08204	−0.0819167
$647$	−6.18034	−0.242974	−0.121487	−	0.992593i	$-0.538766\pi$
$647$	−6.18034	−0.242974	−0.121487	+	0.992593i	$0.538766\pi$
$648$	0	0
$649$	0	0
$650$	1.88854	0.0740748
$651$	0	0
$652$	16.1459	0.632322
$653$	−18.2361	−0.713632	−0.356816	−	0.934175i	$-0.616138\pi$
$653$	−18.2361	−0.713632	−0.356816	+	0.934175i	$0.616138\pi$
$654$	0	0
$655$	−18.9443	−0.740214
$656$	−35.1722	−1.37324
$657$	0	0
$658$	2.52786	0.0985464
$659$	31.4721	1.22598	0.612990	−	0.790091i	$-0.289967\pi$
$659$	31.4721	1.22598	0.612990	+	0.790091i	$0.289967\pi$
$660$	0	0
$661$	−34.5623	−1.34432	−0.672159	−	0.740407i	$-0.734633\pi$
$661$	−34.5623	−1.34432	−0.672159	+	0.740407i	$0.734633\pi$
$662$	−2.36068	−0.0917504
$663$	0	0
$664$	3.98684	0.154720
$665$	1.76393	0.0684023
$666$	0	0
$667$	23.5066	0.910178
$668$	−12.0000	−0.464294
$669$	0	0
$670$	−3.18034	−0.122867
$671$	0	0
$672$	0	0
$673$	6.58359	0.253779	0.126889	−	0.991917i	$-0.459501\pi$
$673$	6.58359	0.253779	0.126889	+	0.991917i	$0.459501\pi$
$674$	9.90983	0.381712
$675$	0	0
$676$	21.2705	0.818097
$677$	−36.2705	−1.39399	−0.696994	−	0.717077i	$-0.745480\pi$
$677$	−36.2705	−1.39399	−0.696994	+	0.717077i	$0.745480\pi$
$678$	0	0
$679$	−7.00000	−0.268635
$680$	−4.54915	−0.174452
$681$	0	0
$682$	0	0
$683$	−0.493422	−0.0188803	−0.00944014	−	0.999955i	$-0.503005\pi$
$683$	−0.493422	−0.0188803	−0.00944014	+	0.999955i	$0.503005\pi$
$684$	0	0
$685$	−15.3262	−0.585585
$686$	0.381966	0.0145835
$687$	0	0
$688$	39.5197	1.50668
$689$	2.94427	0.112168
$690$	0	0
$691$	18.5410	0.705334	0.352667	−	0.935749i	$-0.385275\pi$
$691$	18.5410	0.705334	0.352667	+	0.935749i	$0.385275\pi$
$692$	28.5197	1.08416
$693$	0	0
$694$	−0.896674	−0.0340373
$695$	11.9443	0.453072
$696$	0	0
$697$	−34.5492	−1.30864
$698$	12.5279	0.474187
$699$	0	0
$700$	−7.41641	−0.280314
$701$	18.5066	0.698984	0.349492	−	0.936939i	$-0.386354\pi$
$701$	18.5066	0.698984	0.349492	+	0.936939i	$0.386354\pi$
$702$	0	0
$703$	11.4164	0.430578
$704$	0	0
$705$	0	0
$706$	9.20163	0.346308
$707$	14.7984	0.556550
$708$	0	0
$709$	−7.03444	−0.264184	−0.132092	−	0.991237i	$-0.542169\pi$
$709$	−7.03444	−0.264184	−0.132092	+	0.991237i	$0.542169\pi$
$710$	6.14590	0.230651
$711$	0	0
$712$	−10.0902	−0.378145
$713$	21.5623	0.807515
$714$	0	0
$715$	0	0
$716$	−6.97871	−0.260807
$717$	0	0
$718$	5.96556	0.222633
$719$	2.88854	0.107725	0.0538623	−	0.998548i	$-0.482847\pi$
$719$	2.88854	0.107725	0.0538623	+	0.998548i	$0.482847\pi$
$720$	0	0
$721$	−8.85410	−0.329744
$722$	6.06888	0.225860
$723$	0	0
$724$	13.7508	0.511044
$725$	18.4721	0.686038
$726$	0	0
$727$	−12.4508	−0.461776	−0.230888	−	0.972980i	$-0.574163\pi$
$727$	−12.4508	−0.461776	−0.230888	+	0.972980i	$0.574163\pi$
$728$	−1.81966	−0.0674411
$729$	0	0
$730$	5.43769	0.201258
$731$	38.8197	1.43580
$732$	0	0
$733$	−52.9443	−1.95554	−0.977771	−	0.209677i	$-0.932759\pi$
$733$	−52.9443	−1.95554	−0.977771	+	0.209677i	$0.932759\pi$
$734$	−12.5410	−0.462897
$735$	0	0
$736$	21.1033	0.777879
$737$	0	0
$738$	0	0
$739$	11.3820	0.418692	0.209346	−	0.977842i	$-0.432866\pi$
$739$	11.3820	0.418692	0.209346	+	0.977842i	$0.432866\pi$
$740$	12.0000	0.441129
$741$	0	0
$742$	0.909830	0.0334009
$743$	−14.9098	−0.546989	−0.273494	−	0.961874i	$-0.588179\pi$
$743$	−14.9098	−0.546989	−0.273494	+	0.961874i	$0.588179\pi$
$744$	0	0
$745$	8.85410	0.324389
$746$	−5.89667	−0.215893
$747$	0	0
$748$	0	0
$749$	−2.70820	−0.0989556
$750$	0	0
$751$	41.3050	1.50724	0.753620	−	0.657311i	$-0.228306\pi$
$751$	41.3050	1.50724	0.753620	+	0.657311i	$0.228306\pi$
$752$	20.8197	0.759215
$753$	0	0
$754$	2.18034	0.0794033
$755$	0.0557281	0.00202815
$756$	0	0
$757$	−14.7639	−0.536604	−0.268302	−	0.963335i	$-0.586463\pi$
$757$	−14.7639	−0.536604	−0.268302	+	0.963335i	$0.586463\pi$
$758$	−12.2492	−0.444912
$759$	0	0
$760$	−2.59675	−0.0941939
$761$	15.3050	0.554804	0.277402	−	0.960754i	$-0.410527\pi$
$761$	15.3050	0.554804	0.277402	+	0.960754i	$0.410527\pi$
$762$	0	0
$763$	−1.52786	−0.0553124
$764$	29.2279	1.05743
$765$	0	0
$766$	−13.2148	−0.477469
$767$	−13.7082	−0.494975
$768$	0	0
$769$	−48.5623	−1.75120	−0.875601	−	0.483035i	$-0.839534\pi$
$769$	−48.5623	−1.75120	−0.875601	+	0.483035i	$0.839534\pi$
$770$	0	0
$771$	0	0
$772$	−28.4164	−1.02273
$773$	−40.2361	−1.44719	−0.723595	−	0.690224i	$-0.757512\pi$
$773$	−40.2361	−1.44719	−0.723595	+	0.690224i	$0.757512\pi$
$774$	0	0
$775$	16.9443	0.608656
$776$	10.3050	0.369926
$777$	0	0
$778$	−13.0557	−0.468071
$779$	−19.7214	−0.706591
$780$	0	0
$781$	0	0
$782$	6.00813	0.214850
$783$	0	0
$784$	3.14590	0.112354
$785$	−19.8885	−0.709853
$786$	0	0
$787$	−20.5623	−0.732967	−0.366484	−	0.930425i	$-0.619438\pi$
$787$	−20.5623	−0.732967	−0.366484	+	0.930425i	$0.619438\pi$
$788$	7.95743	0.283472
$789$	0	0
$790$	−2.43769	−0.0867293
$791$	0.145898	0.00518754
$792$	0	0
$793$	9.41641	0.334386
$794$	0.313082	0.0111109
$795$	0	0
$796$	15.2705	0.541249
$797$	−32.2361	−1.14186	−0.570930	−	0.820999i	$-0.693417\pi$
$797$	−32.2361	−1.14186	−0.570930	+	0.820999i	$0.693417\pi$
$798$	0	0
$799$	20.4508	0.723499
$800$	16.5836	0.586319
$801$	0	0
$802$	−6.69505	−0.236410
$803$	0	0
$804$	0	0
$805$	−5.09017	−0.179405
$806$	2.00000	0.0704470
$807$	0	0
$808$	−21.7852	−0.766401
$809$	32.5066	1.14287	0.571435	−	0.820647i	$-0.306387\pi$
$809$	32.5066	1.14287	0.571435	+	0.820647i	$0.306387\pi$
$810$	0	0
$811$	−1.16718	−0.0409854	−0.0204927	−	0.999790i	$-0.506523\pi$
$811$	−1.16718	−0.0409854	−0.0204927	+	0.999790i	$0.506523\pi$
$812$	−8.56231	−0.300478
$813$	0	0
$814$	0	0
$815$	8.70820	0.305035
$816$	0	0
$817$	22.1591	0.775247
$818$	−6.76393	−0.236495
$819$	0	0
$820$	−20.7295	−0.723905
$821$	56.6656	1.97764	0.988822	−	0.149100i	$-0.0476377\pi$
$821$	56.6656	1.97764	0.988822	+	0.149100i	$0.0476377\pi$
$822$	0	0
$823$	−31.8885	−1.11156	−0.555782	−	0.831328i	$-0.687581\pi$
$823$	−31.8885	−1.11156	−0.555782	+	0.831328i	$0.687581\pi$
$824$	13.0344	0.454076
$825$	0	0
$826$	−4.23607	−0.147392
$827$	−19.5967	−0.681446	−0.340723	−	0.940164i	$-0.610672\pi$
$827$	−19.5967	−0.681446	−0.340723	+	0.940164i	$0.610672\pi$
$828$	0	0
$829$	−47.9787	−1.66637	−0.833185	−	0.552995i	$-0.813485\pi$
$829$	−47.9787	−1.66637	−0.833185	+	0.552995i	$0.813485\pi$
$830$	1.03444	0.0359060
$831$	0	0
$832$	−5.81966	−0.201760
$833$	3.09017	0.107068
$834$	0	0
$835$	−6.47214	−0.223978
$836$	0	0
$837$	0	0
$838$	11.4164	0.394373
$839$	−30.5967	−1.05632	−0.528159	−	0.849146i	$-0.677117\pi$
$839$	−30.5967	−1.05632	−0.528159	+	0.849146i	$0.677117\pi$
$840$	0	0
$841$	−7.67376	−0.264612
$842$	−0.291796	−0.0100560
$843$	0	0
$844$	28.9574	0.996756
$845$	11.4721	0.394653
$846$	0	0
$847$	0	0
$848$	7.49342	0.257325
$849$	0	0
$850$	4.72136	0.161941
$851$	−32.9443	−1.12932
$852$	0	0
$853$	34.9098	1.19529	0.597645	−	0.801761i	$-0.296103\pi$
$853$	34.9098	1.19529	0.597645	+	0.801761i	$0.296103\pi$
$854$	2.90983	0.0995723
$855$	0	0
$856$	3.98684	0.136268
$857$	−15.0902	−0.515470	−0.257735	−	0.966216i	$-0.582976\pi$
$857$	−15.0902	−0.515470	−0.257735	+	0.966216i	$0.582976\pi$
$858$	0	0
$859$	21.5623	0.735696	0.367848	−	0.929886i	$-0.380095\pi$
$859$	21.5623	0.735696	0.367848	+	0.929886i	$0.380095\pi$
$860$	23.2918	0.794244
$861$	0	0
$862$	5.29180	0.180239
$863$	47.9443	1.63204	0.816021	−	0.578022i	$-0.196175\pi$
$863$	47.9443	1.63204	0.816021	+	0.578022i	$0.196175\pi$
$864$	0	0
$865$	15.3820	0.523003
$866$	3.42454	0.116371
$867$	0	0
$868$	−7.85410	−0.266586
$869$	0	0
$870$	0	0
$871$	−10.2918	−0.348724
$872$	2.24922	0.0761683
$873$	0	0
$874$	3.42956	0.116007
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	1.29180	0.0435960
$879$	0	0
$880$	0	0
$881$	−14.1459	−0.476587	−0.238294	−	0.971193i	$-0.576588\pi$
$881$	−14.1459	−0.476587	−0.238294	+	0.971193i	$0.576588\pi$
$882$	0	0
$883$	31.6312	1.06447	0.532237	−	0.846595i	$-0.321351\pi$
$883$	31.6312	1.06447	0.532237	+	0.846595i	$0.321351\pi$
$884$	−7.08204	−0.238195
$885$	0	0
$886$	1.18034	0.0396543
$887$	57.2148	1.92108	0.960542	−	0.278134i	$-0.0897161\pi$
$887$	57.2148	1.92108	0.960542	+	0.278134i	$0.0897161\pi$
$888$	0	0
$889$	2.94427	0.0987477
$890$	−2.61803	−0.0877567
$891$	0	0
$892$	−3.77206	−0.126298
$893$	11.6738	0.390648
$894$	0	0
$895$	−3.76393	−0.125814
$896$	−10.0902	−0.337089
$897$	0	0
$898$	4.70820	0.157115
$899$	19.5623	0.652439
$900$	0	0
$901$	7.36068	0.245220
$902$	0	0
$903$	0	0
$904$	−0.214782	−0.00714353
$905$	7.41641	0.246530
$906$	0	0
$907$	−15.7639	−0.523433	−0.261716	−	0.965145i	$-0.584289\pi$
$907$	−15.7639	−0.523433	−0.261716	+	0.965145i	$0.584289\pi$
$908$	29.7295	0.986608
$909$	0	0
$910$	−0.472136	−0.0156512
$911$	45.1803	1.49689	0.748446	−	0.663196i	$-0.230800\pi$
$911$	45.1803	1.49689	0.748446	+	0.663196i	$0.230800\pi$
$912$	0	0
$913$	0	0
$914$	−0.0557281	−0.00184332
$915$	0	0
$916$	−12.5410	−0.414367
$917$	−18.9443	−0.625595
$918$	0	0
$919$	14.8885	0.491128	0.245564	−	0.969380i	$-0.421027\pi$
$919$	14.8885	0.491128	0.245564	+	0.969380i	$0.421027\pi$
$920$	7.49342	0.247051
$921$	0	0
$922$	7.06888	0.232801
$923$	19.8885	0.654639
$924$	0	0
$925$	−25.8885	−0.851210
$926$	−8.49342	−0.279111
$927$	0	0
$928$	19.1459	0.628495
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	0	0
$931$	1.76393	0.0578105
$932$	54.5410	1.78655
$933$	0	0
$934$	−1.72136	−0.0563246
$935$	0	0
$936$	0	0
$937$	51.7214	1.68966	0.844832	−	0.535032i	$-0.179701\pi$
$937$	51.7214	1.68966	0.844832	+	0.535032i	$0.179701\pi$
$938$	−3.18034	−0.103842
$939$	0	0
$940$	12.2705	0.400220
$941$	27.0689	0.882420	0.441210	−	0.897404i	$-0.354549\pi$
$941$	27.0689	0.882420	0.441210	+	0.897404i	$0.354549\pi$
$942$	0	0
$943$	56.9098	1.85324
$944$	−34.8885	−1.13553
$945$	0	0
$946$	0	0
$947$	−6.29180	−0.204456	−0.102228	−	0.994761i	$-0.532597\pi$
$947$	−6.29180	−0.204456	−0.102228	+	0.994761i	$0.532597\pi$
$948$	0	0
$949$	17.5967	0.571215
$950$	2.69505	0.0874389
$951$	0	0
$952$	−4.54915	−0.147439
$953$	−6.23607	−0.202006	−0.101003	−	0.994886i	$-0.532205\pi$
$953$	−6.23607	−0.202006	−0.101003	+	0.994886i	$0.532205\pi$
$954$	0	0
$955$	15.7639	0.510109
$956$	31.6869	1.02483
$957$	0	0
$958$	−8.31308	−0.268583
$959$	−15.3262	−0.494910
$960$	0	0
$961$	−13.0557	−0.421153
$962$	−3.05573	−0.0985206
$963$	0	0
$964$	30.1672	0.971620
$965$	−15.3262	−0.493369
$966$	0	0
$967$	−27.4508	−0.882760	−0.441380	−	0.897320i	$-0.645511\pi$
$967$	−27.4508	−0.882760	−0.441380	+	0.897320i	$0.645511\pi$
$968$	0	0
$969$	0	0
$970$	2.67376	0.0858493
$971$	28.2492	0.906561	0.453280	−	0.891368i	$-0.350254\pi$
$971$	28.2492	0.906561	0.453280	+	0.891368i	$0.350254\pi$
$972$	0	0
$973$	11.9443	0.382916
$974$	12.4508	0.398951
$975$	0	0
$976$	23.9656	0.767119
$977$	10.8197	0.346152	0.173076	−	0.984909i	$-0.444629\pi$
$977$	10.8197	0.346152	0.173076	+	0.984909i	$0.444629\pi$
$978$	0	0
$979$	0	0
$980$	1.85410	0.0592271
$981$	0	0
$982$	−8.45898	−0.269937
$983$	12.3820	0.394923	0.197462	−	0.980311i	$-0.436730\pi$
$983$	12.3820	0.394923	0.197462	+	0.980311i	$0.436730\pi$
$984$	0	0
$985$	4.29180	0.136748
$986$	5.45085	0.173590
$987$	0	0
$988$	−4.04257	−0.128611
$989$	−63.9443	−2.03331
$990$	0	0
$991$	−0.729490	−0.0231730	−0.0115865	−	0.999933i	$-0.503688\pi$
$991$	−0.729490	−0.0231730	−0.0115865	+	0.999933i	$0.503688\pi$
$992$	17.5623	0.557604
$993$	0	0
$994$	6.14590	0.194936
$995$	8.23607	0.261101
$996$	0	0
$997$	55.8673	1.76933	0.884667	−	0.466224i	$-0.154386\pi$
$997$	55.8673	1.76933	0.884667	+	0.466224i	$0.154386\pi$
$998$	3.85410	0.121999
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.t.1.2		2
3.2	odd	2		847.2.a.h.1.1	yes	2
11.10	odd	2		7623.2.a.bx.1.1		2
21.20	even	2		5929.2.a.s.1.1		2
33.2	even	10		847.2.f.d.323.1		4
33.5	odd	10		847.2.f.j.729.1		4
33.8	even	10		847.2.f.l.372.1		4
33.14	odd	10		847.2.f.c.372.1		4
33.17	even	10		847.2.f.d.729.1		4
33.20	odd	10		847.2.f.j.323.1		4
33.26	odd	10		847.2.f.c.148.1		4
33.29	even	10		847.2.f.l.148.1		4
33.32	even	2		847.2.a.d.1.2	✓	2
231.230	odd	2		5929.2.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.d.1.2	✓	2	33.32	even	2
847.2.a.h.1.1	yes	2	3.2	odd	2
847.2.f.c.148.1		4	33.26	odd	10
847.2.f.c.372.1		4	33.14	odd	10
847.2.f.d.323.1		4	33.2	even	10
847.2.f.d.729.1		4	33.17	even	10
847.2.f.j.323.1		4	33.20	odd	10
847.2.f.j.729.1		4	33.5	odd	10
847.2.f.l.148.1		4	33.29	even	10
847.2.f.l.372.1		4	33.8	even	10
5929.2.a.i.1.2		2	231.230	odd	2
5929.2.a.s.1.1		2	21.20	even	2
7623.2.a.t.1.2		2	1.1	even	1	trivial
7623.2.a.bx.1.1		2	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)

\(f(q)\)	\(=\)	\(q-0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.47214 q^{8} +O(q^{10})\)	\(q-0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.47214 q^{8} +0.381966 q^{10} +1.23607 q^{13} +0.381966 q^{14} +3.14590 q^{16} +3.09017 q^{17} +1.76393 q^{19} +1.85410 q^{20} -5.09017 q^{23} -4.00000 q^{25} -0.472136 q^{26} +1.85410 q^{28} -4.61803 q^{29} -4.23607 q^{31} -4.14590 q^{32} -1.18034 q^{34} +1.00000 q^{35} +6.47214 q^{37} -0.673762 q^{38} -1.47214 q^{40} -11.1803 q^{41} +12.5623 q^{43} +1.94427 q^{46} +6.61803 q^{47} +1.00000 q^{49} +1.52786 q^{50} -2.29180 q^{52} +2.38197 q^{53} -1.47214 q^{56} +1.76393 q^{58} -11.0902 q^{59} +7.61803 q^{61} +1.61803 q^{62} -4.70820 q^{64} -1.23607 q^{65} -8.32624 q^{67} -5.72949 q^{68} -0.381966 q^{70} +16.0902 q^{71} +14.2361 q^{73} -2.47214 q^{74} -3.27051 q^{76} -6.38197 q^{79} -3.14590 q^{80} +4.27051 q^{82} +2.70820 q^{83} -3.09017 q^{85} -4.79837 q^{86} -6.85410 q^{89} -1.23607 q^{91} +9.43769 q^{92} -2.52786 q^{94} -1.76393 q^{95} +7.00000 q^{97} -0.381966 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 3 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8	\( 2 q - 3 q^{2} + 3 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8} + 3 q^{10} - 2 q^{13} + 3 q^{14} + 13 q^{16} - 5 q^{17} + 8 q^{19} - 3 q^{20} + q^{23} - 8 q^{25} + 8 q^{26} - 3 q^{28} - 7 q^{29} - 4 q^{31} - 15 q^{32} + 20 q^{34} + 2 q^{35} + 4 q^{37} - 17 q^{38} + 6 q^{40} + 5 q^{43} - 14 q^{46} + 11 q^{47} + 2 q^{49} + 12 q^{50} - 18 q^{52} + 7 q^{53} + 6 q^{56} + 8 q^{58} - 11 q^{59} + 13 q^{61} + q^{62} + 4 q^{64} + 2 q^{65} - q^{67} - 45 q^{68} - 3 q^{70} + 21 q^{71} + 24 q^{73} + 4 q^{74} + 27 q^{76} - 15 q^{79} - 13 q^{80} - 25 q^{82} - 8 q^{83} + 5 q^{85} + 15 q^{86} - 7 q^{89} + 2 q^{91} + 39 q^{92} - 14 q^{94} - 8 q^{95} + 14 q^{97} - 3 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8 + 3 * q^10 - 2 * q^13 + 3 * q^14 + 13 * q^16 - 5 * q^17 + 8 * q^19 - 3 * q^20 + q^23 - 8 * q^25 + 8 * q^26 - 3 * q^28 - 7 * q^29 - 4 * q^31 - 15 * q^32 + 20 * q^34 + 2 * q^35 + 4 * q^37 - 17 * q^38 + 6 * q^40 + 5 * q^43 - 14 * q^46 + 11 * q^47 + 2 * q^49 + 12 * q^50 - 18 * q^52 + 7 * q^53 + 6 * q^56 + 8 * q^58 - 11 * q^59 + 13 * q^61 + q^62 + 4 * q^64 + 2 * q^65 - q^67 - 45 * q^68 - 3 * q^70 + 21 * q^71 + 24 * q^73 + 4 * q^74 + 27 * q^76 - 15 * q^79 - 13 * q^80 - 25 * q^82 - 8 * q^83 + 5 * q^85 + 15 * q^86 - 7 * q^89 + 2 * q^91 + 39 * q^92 - 14 * q^94 - 8 * q^95 + 14 * q^97 - 3 * q^98

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