LMFDB - Embedded newform 7623.2.a.bx.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:	$0$
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			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} +7.47214 q^{8} +O(q^{10})$	$q+2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} +7.47214 q^{8} -2.61803 q^{10} +3.23607 q^{13} +2.61803 q^{14} +9.85410 q^{16} +8.09017 q^{17} -6.23607 q^{19} -4.85410 q^{20} +6.09017 q^{23} -4.00000 q^{25} +8.47214 q^{26} +4.85410 q^{28} +2.38197 q^{29} +0.236068 q^{31} +10.8541 q^{32} +21.1803 q^{34} -1.00000 q^{35} -2.47214 q^{37} -16.3262 q^{38} -7.47214 q^{40} -11.1803 q^{41} +7.56231 q^{43} +15.9443 q^{46} +4.38197 q^{47} +1.00000 q^{49} -10.4721 q^{50} +15.7082 q^{52} +4.61803 q^{53} +7.47214 q^{56} +6.23607 q^{58} +0.0901699 q^{59} -5.38197 q^{61} +0.618034 q^{62} +8.70820 q^{64} -3.23607 q^{65} +7.32624 q^{67} +39.2705 q^{68} -2.61803 q^{70} +4.90983 q^{71} -9.76393 q^{73} -6.47214 q^{74} -30.2705 q^{76} +8.61803 q^{79} -9.85410 q^{80} -29.2705 q^{82} +10.7082 q^{83} -8.09017 q^{85} +19.7984 q^{86} -0.145898 q^{89} +3.23607 q^{91} +29.5623 q^{92} +11.4721 q^{94} +6.23607 q^{95} +7.00000 q^{97} +2.61803 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.61803	1.85123	0.925615	−	0.378467i	$-0.123549\pi$
$2$	2.61803	1.85123	0.925615	+	0.378467i	$0.123549\pi$
$3$	0	0
$3$	0	0
$4$	4.85410	2.42705
$5$	−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
$8$	7.47214	2.64180
$9$	0	0
$10$	−2.61803	−0.827895
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.23607	0.897524	0.448762	−	0.893651i	$-0.351865\pi$
$13$	3.23607	0.897524	0.448762	+	0.893651i	$0.351865\pi$
$14$	2.61803	0.699699
$15$	0	0
$16$	9.85410	2.46353
$17$	8.09017	1.96215	0.981077	−	0.193617i	$-0.0620219\pi$
$17$	8.09017	1.96215	0.981077	+	0.193617i	$0.0620219\pi$
$18$	0	0
$19$	−6.23607	−1.43065	−0.715326	−	0.698791i	$-0.753722\pi$
$19$	−6.23607	−1.43065	−0.715326	+	0.698791i	$0.753722\pi$
$20$	−4.85410	−1.08541
$21$	0	0
$22$	0	0
$23$	6.09017	1.26989	0.634944	−	0.772558i	$-0.281023\pi$
$23$	6.09017	1.26989	0.634944	+	0.772558i	$0.281023\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	8.47214	1.66152
$27$	0	0
$28$	4.85410	0.917339
$29$	2.38197	0.442320	0.221160	−	0.975238i	$-0.429016\pi$
$29$	2.38197	0.442320	0.221160	+	0.975238i	$0.429016\pi$
$30$	0	0
$31$	0.236068	0.0423991	0.0211995	−	0.999775i	$-0.493251\pi$
$31$	0.236068	0.0423991	0.0211995	+	0.999775i	$0.493251\pi$
$32$	10.8541	1.91875
$33$	0	0
$34$	21.1803	3.63240
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−2.47214	−0.406417	−0.203208	−	0.979136i	$-0.565137\pi$
$37$	−2.47214	−0.406417	−0.203208	+	0.979136i	$0.565137\pi$
$38$	−16.3262	−2.64847
$39$	0	0
$40$	−7.47214	−1.18145
$41$	−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
$43$	7.56231	1.15324	0.576620	−	0.817012i	$-0.304371\pi$
$43$	7.56231	1.15324	0.576620	+	0.817012i	$0.304371\pi$
$44$	0	0
$45$	0	0
$46$	15.9443	2.35085
$47$	4.38197	0.639175	0.319588	−	0.947557i	$-0.396456\pi$
$47$	4.38197	0.639175	0.319588	+	0.947557i	$0.396456\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−10.4721	−1.48098
$51$	0	0
$52$	15.7082	2.17834
$53$	4.61803	0.634336	0.317168	−	0.948369i	$-0.397268\pi$
$53$	4.61803	0.634336	0.317168	+	0.948369i	$0.397268\pi$
$54$	0	0
$55$	0	0
$56$	7.47214	0.998506
$57$	0	0
$58$	6.23607	0.818836
$59$	0.0901699	0.0117391	0.00586956	−	0.999983i	$-0.498132\pi$
$59$	0.0901699	0.0117391	0.00586956	+	0.999983i	$0.498132\pi$
$60$	0	0
$61$	−5.38197	−0.689090	−0.344545	−	0.938770i	$-0.611967\pi$
$61$	−5.38197	−0.689090	−0.344545	+	0.938770i	$0.611967\pi$
$62$	0.618034	0.0784904
$63$	0	0
$64$	8.70820	1.08853
$65$	−3.23607	−0.401385
$66$	0	0
$67$	7.32624	0.895042	0.447521	−	0.894273i	$-0.352307\pi$
$67$	7.32624	0.895042	0.447521	+	0.894273i	$0.352307\pi$
$68$	39.2705	4.76225
$69$	0	0
$70$	−2.61803	−0.312915
$71$	4.90983	0.582690	0.291345	−	0.956618i	$-0.405897\pi$
$71$	4.90983	0.582690	0.291345	+	0.956618i	$0.405897\pi$
$72$	0	0
$73$	−9.76393	−1.14278	−0.571391	−	0.820678i	$-0.693596\pi$
$73$	−9.76393	−1.14278	−0.571391	+	0.820678i	$0.693596\pi$
$74$	−6.47214	−0.752371
$75$	0	0
$76$	−30.2705	−3.47227
$77$	0	0
$78$	0	0
$79$	8.61803	0.969605	0.484802	−	0.874624i	$-0.338892\pi$
$79$	8.61803	0.969605	0.484802	+	0.874624i	$0.338892\pi$
$80$	−9.85410	−1.10172
$81$	0	0
$82$	−29.2705	−3.23239
$83$	10.7082	1.17538	0.587689	−	0.809087i	$-0.300038\pi$
$83$	10.7082	1.17538	0.587689	+	0.809087i	$0.300038\pi$
$84$	0	0
$85$	−8.09017	−0.877502
$86$	19.7984	2.13491
$87$	0	0
$88$	0	0
$89$	−0.145898	−0.0154652	−0.00773258	−	0.999970i	$-0.502461\pi$
$89$	−0.145898	−0.0154652	−0.00773258	+	0.999970i	$0.502461\pi$
$90$	0	0
$91$	3.23607	0.339232
$92$	29.5623	3.08208
$93$	0	0
$94$	11.4721	1.18326
$95$	6.23607	0.639807
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	2.61803	0.264461
$99$	0	0
$100$	−19.4164	−1.94164
$101$	−9.79837	−0.974975	−0.487487	−	0.873130i	$-0.662086\pi$
$101$	−9.79837	−0.974975	−0.487487	+	0.873130i	$0.662086\pi$
$102$	0	0
$103$	2.14590	0.211442	0.105721	−	0.994396i	$-0.466285\pi$
$103$	2.14590	0.211442	0.105721	+	0.994396i	$0.466285\pi$
$104$	24.1803	2.37108
$105$	0	0
$106$	12.0902	1.17430
$107$	10.7082	1.03520	0.517601	−	0.855622i	$-0.326825\pi$
$107$	10.7082	1.03520	0.517601	+	0.855622i	$0.326825\pi$
$108$	0	0
$109$	−10.4721	−1.00305	−0.501524	−	0.865144i	$-0.667227\pi$
$109$	−10.4721	−1.00305	−0.501524	+	0.865144i	$0.667227\pi$
$110$	0	0
$111$	0	0
$112$	9.85410	0.931125
$113$	−6.85410	−0.644780	−0.322390	−	0.946607i	$-0.604486\pi$
$113$	−6.85410	−0.644780	−0.322390	+	0.946607i	$0.604486\pi$
$114$	0	0
$115$	−6.09017	−0.567911
$116$	11.5623	1.07353
$117$	0	0
$118$	0.236068	0.0217318
$119$	8.09017	0.741625
$120$	0	0
$121$	0	0
$122$	−14.0902	−1.27566
$123$	0	0
$124$	1.14590	0.102905
$125$	9.00000	0.804984
$126$	0	0
$127$	−14.9443	−1.32609	−0.663045	−	0.748580i	$-0.730736\pi$
$127$	−14.9443	−1.32609	−0.663045	+	0.748580i	$0.730736\pi$
$128$	1.09017	0.0963583
$129$	0	0
$130$	−8.47214	−0.743055
$131$	−1.05573	−0.0922394	−0.0461197	−	0.998936i	$-0.514686\pi$
$131$	−1.05573	−0.0922394	−0.0461197	+	0.998936i	$0.514686\pi$
$132$	0	0
$133$	−6.23607	−0.540736
$134$	19.1803	1.65693
$135$	0	0
$136$	60.4508	5.18362
$137$	−0.326238	−0.0278724	−0.0139362	−	0.999903i	$-0.504436\pi$
$137$	−0.326238	−0.0278724	−0.0139362	+	0.999903i	$0.504436\pi$
$138$	0	0
$139$	−5.94427	−0.504187	−0.252093	−	0.967703i	$-0.581119\pi$
$139$	−5.94427	−0.504187	−0.252093	+	0.967703i	$0.581119\pi$
$140$	−4.85410	−0.410246
$141$	0	0
$142$	12.8541	1.07869
$143$	0	0
$144$	0	0
$145$	−2.38197	−0.197812
$146$	−25.5623	−2.11555
$147$	0	0
$148$	−12.0000	−0.986394
$149$	2.14590	0.175799	0.0878994	−	0.996129i	$-0.471985\pi$
$149$	2.14590	0.175799	0.0878994	+	0.996129i	$0.471985\pi$
$150$	0	0
$151$	17.9443	1.46028	0.730142	−	0.683295i	$-0.239454\pi$
$151$	17.9443	1.46028	0.730142	+	0.683295i	$0.239454\pi$
$152$	−46.5967	−3.77950
$153$	0	0
$154$	0	0
$155$	−0.236068	−0.0189614
$156$	0	0
$157$	−15.8885	−1.26804	−0.634022	−	0.773315i	$-0.718597\pi$
$157$	−15.8885	−1.26804	−0.634022	+	0.773315i	$0.718597\pi$
$158$	22.5623	1.79496
$159$	0	0
$160$	−10.8541	−0.858092
$161$	6.09017	0.479973
$162$	0	0
$163$	4.70820	0.368775	0.184387	−	0.982854i	$-0.440970\pi$
$163$	4.70820	0.368775	0.184387	+	0.982854i	$0.440970\pi$
$164$	−54.2705	−4.23781
$165$	0	0
$166$	28.0344	2.17589
$167$	2.47214	0.191300	0.0956498	−	0.995415i	$-0.469507\pi$
$167$	2.47214	0.191300	0.0956498	+	0.995415i	$0.469507\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	−21.1803	−1.62446
$171$	0	0
$172$	36.7082	2.79897
$173$	17.6180	1.33947	0.669737	−	0.742598i	$-0.266407\pi$
$173$	17.6180	1.33947	0.669737	+	0.742598i	$0.266407\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	0	0
$178$	−0.381966	−0.0286296
$179$	8.23607	0.615593	0.307796	−	0.951452i	$-0.400408\pi$
$179$	8.23607	0.615593	0.307796	+	0.951452i	$0.400408\pi$
$180$	0	0
$181$	19.4164	1.44321	0.721605	−	0.692305i	$-0.243405\pi$
$181$	19.4164	1.44321	0.721605	+	0.692305i	$0.243405\pi$
$182$	8.47214	0.627996
$183$	0	0
$184$	45.5066	3.35479
$185$	2.47214	0.181755
$186$	0	0
$187$	0	0
$188$	21.2705	1.55131
$189$	0	0
$190$	16.3262	1.18443
$191$	−20.2361	−1.46423	−0.732115	−	0.681181i	$-0.761467\pi$
$191$	−20.2361	−1.46423	−0.732115	+	0.681181i	$0.761467\pi$
$192$	0	0
$193$	0.326238	0.0234831	0.0117416	−	0.999931i	$-0.496262\pi$
$193$	0.326238	0.0234831	0.0117416	+	0.999931i	$0.496262\pi$
$194$	18.3262	1.31575
$195$	0	0
$196$	4.85410	0.346722
$197$	17.7082	1.26166	0.630829	−	0.775922i	$-0.282715\pi$
$197$	17.7082	1.26166	0.630829	+	0.775922i	$0.282715\pi$
$198$	0	0
$199$	−3.76393	−0.266818	−0.133409	−	0.991061i	$-0.542592\pi$
$199$	−3.76393	−0.266818	−0.133409	+	0.991061i	$0.542592\pi$
$200$	−29.8885	−2.11344
$201$	0	0
$202$	−25.6525	−1.80490
$203$	2.38197	0.167181
$204$	0	0
$205$	11.1803	0.780869
$206$	5.61803	0.391427
$207$	0	0
$208$	31.8885	2.21107
$209$	0	0
$210$	0	0
$211$	13.3820	0.921253	0.460626	−	0.887594i	$-0.347625\pi$
$211$	13.3820	0.921253	0.460626	+	0.887594i	$0.347625\pi$
$212$	22.4164	1.53957
$213$	0	0
$214$	28.0344	1.91639
$215$	−7.56231	−0.515745
$216$	0	0
$217$	0.236068	0.0160253
$218$	−27.4164	−1.85687
$219$	0	0
$220$	0	0
$221$	26.1803	1.76108
$222$	0	0
$223$	−27.0344	−1.81036	−0.905180	−	0.425028i	$-0.860264\pi$
$223$	−27.0344	−1.81036	−0.905180	+	0.425028i	$0.860264\pi$
$224$	10.8541	0.725220
$225$	0	0
$226$	−17.9443	−1.19364
$227$	−13.0344	−0.865126	−0.432563	−	0.901604i	$-0.642391\pi$
$227$	−13.0344	−0.865126	−0.432563	+	0.901604i	$0.642391\pi$
$228$	0	0
$229$	11.2361	0.742500	0.371250	−	0.928533i	$-0.378929\pi$
$229$	11.2361	0.742500	0.371250	+	0.928533i	$0.378929\pi$
$230$	−15.9443	−1.05133
$231$	0	0
$232$	17.7984	1.16852
$233$	2.58359	0.169257	0.0846284	−	0.996413i	$-0.473030\pi$
$233$	2.58359	0.169257	0.0846284	+	0.996413i	$0.473030\pi$
$234$	0	0
$235$	−4.38197	−0.285848
$236$	0.437694	0.0284915
$237$	0	0
$238$	21.1803	1.37292
$239$	5.90983	0.382275	0.191138	−	0.981563i	$-0.438782\pi$
$239$	5.90983	0.382275	0.191138	+	0.981563i	$0.438782\pi$
$240$	0	0
$241$	−17.2705	−1.11249	−0.556246	−	0.831018i	$-0.687759\pi$
$241$	−17.2705	−1.11249	−0.556246	+	0.831018i	$0.687759\pi$
$242$	0	0
$243$	0	0
$244$	−26.1246	−1.67246
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−20.1803	−1.28404
$248$	1.76393	0.112010
$249$	0	0
$250$	23.5623	1.49021
$251$	−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
$254$	−39.1246	−2.45490
$255$	0	0
$256$	−14.5623	−0.910144
$257$	−11.5623	−0.721237	−0.360618	−	0.932713i	$-0.617434\pi$
$257$	−11.5623	−0.721237	−0.360618	+	0.932713i	$0.617434\pi$
$258$	0	0
$259$	−2.47214	−0.153611
$260$	−15.7082	−0.974181
$261$	0	0
$262$	−2.76393	−0.170756
$263$	−23.1246	−1.42592	−0.712962	−	0.701202i	$-0.752647\pi$
$263$	−23.1246	−1.42592	−0.712962	+	0.701202i	$0.752647\pi$
$264$	0	0
$265$	−4.61803	−0.283684
$266$	−16.3262	−1.00103
$267$	0	0
$268$	35.5623	2.17231
$269$	−10.1459	−0.618606	−0.309303	−	0.950963i	$-0.600096\pi$
$269$	−10.1459	−0.618606	−0.309303	+	0.950963i	$0.600096\pi$
$270$	0	0
$271$	19.7984	1.20267	0.601333	−	0.798999i	$-0.294637\pi$
$271$	19.7984	1.20267	0.601333	+	0.798999i	$0.294637\pi$
$272$	79.7214	4.83382
$273$	0	0
$274$	−0.854102	−0.0515982
$275$	0	0
$276$	0	0
$277$	−6.03444	−0.362574	−0.181287	−	0.983430i	$-0.558026\pi$
$277$	−6.03444	−0.362574	−0.181287	+	0.983430i	$0.558026\pi$
$278$	−15.5623	−0.933365
$279$	0	0
$280$	−7.47214	−0.446546
$281$	2.81966	0.168207	0.0841034	−	0.996457i	$-0.473197\pi$
$281$	2.81966	0.168207	0.0841034	+	0.996457i	$0.473197\pi$
$282$	0	0
$283$	−5.94427	−0.353350	−0.176675	−	0.984269i	$-0.556534\pi$
$283$	−5.94427	−0.353350	−0.176675	+	0.984269i	$0.556534\pi$
$284$	23.8328	1.41422
$285$	0	0
$286$	0	0
$287$	−11.1803	−0.659955
$288$	0	0
$289$	48.4508	2.85005
$290$	−6.23607	−0.366195
$291$	0	0
$292$	−47.3951	−2.77359
$293$	11.0000	0.642627	0.321313	−	0.946973i	$-0.395876\pi$
$293$	11.0000	0.642627	0.321313	+	0.946973i	$0.395876\pi$
$294$	0	0
$295$	−0.0901699	−0.00524990
$296$	−18.4721	−1.07367
$297$	0	0
$298$	5.61803	0.325444
$299$	19.7082	1.13975
$300$	0	0
$301$	7.56231	0.435884
$302$	46.9787	2.70332
$303$	0	0
$304$	−61.4508	−3.52445
$305$	5.38197	0.308170
$306$	0	0
$307$	−0.819660	−0.0467805	−0.0233902	−	0.999726i	$-0.507446\pi$
$307$	−0.819660	−0.0467805	−0.0233902	+	0.999726i	$0.507446\pi$
$308$	0	0
$309$	0	0
$310$	−0.618034	−0.0351020
$311$	−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
$313$	2.67376	0.151130	0.0755650	−	0.997141i	$-0.475924\pi$
$313$	2.67376	0.151130	0.0755650	+	0.997141i	$0.475924\pi$
$314$	−41.5967	−2.34744
$315$	0	0
$316$	41.8328	2.35328
$317$	24.5623	1.37956	0.689778	−	0.724021i	$-0.257708\pi$
$317$	24.5623	1.37956	0.689778	+	0.724021i	$0.257708\pi$
$318$	0	0
$319$	0	0
$320$	−8.70820	−0.486803
$321$	0	0
$322$	15.9443	0.888540
$323$	−50.4508	−2.80716
$324$	0	0
$325$	−12.9443	−0.718019
$326$	12.3262	0.682687
$327$	0	0
$328$	−83.5410	−4.61278
$329$	4.38197	0.241586
$330$	0	0
$331$	−16.1803	−0.889352	−0.444676	−	0.895692i	$-0.646681\pi$
$331$	−16.1803	−0.889352	−0.444676	+	0.895692i	$0.646681\pi$
$332$	51.9787	2.85270
$333$	0	0
$334$	6.47214	0.354140
$335$	−7.32624	−0.400275
$336$	0	0
$337$	8.05573	0.438823	0.219412	−	0.975632i	$-0.429586\pi$
$337$	8.05573	0.438823	0.219412	+	0.975632i	$0.429586\pi$
$338$	−6.61803	−0.359974
$339$	0	0
$340$	−39.2705	−2.12974
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	56.5066	3.04663
$345$	0	0
$346$	46.1246	2.47967
$347$	−33.6525	−1.80656	−0.903280	−	0.429052i	$-0.858848\pi$
$347$	−33.6525	−1.80656	−0.903280	+	0.429052i	$0.858848\pi$
$348$	0	0
$349$	8.20163	0.439023	0.219511	−	0.975610i	$-0.429554\pi$
$349$	8.20163	0.439023	0.219511	+	0.975610i	$0.429554\pi$
$350$	−10.4721	−0.559759
$351$	0	0
$352$	0	0
$353$	−12.9098	−0.687121	−0.343560	−	0.939131i	$-0.611633\pi$
$353$	−12.9098	−0.687121	−0.343560	+	0.939131i	$0.611633\pi$
$354$	0	0
$355$	−4.90983	−0.260587
$356$	−0.708204	−0.0375347
$357$	0	0
$358$	21.5623	1.13960
$359$	13.3820	0.706273	0.353137	−	0.935572i	$-0.385115\pi$
$359$	13.3820	0.706273	0.353137	+	0.935572i	$0.385115\pi$
$360$	0	0
$361$	19.8885	1.04677
$362$	50.8328	2.67171
$363$	0	0
$364$	15.7082	0.823334
$365$	9.76393	0.511068
$366$	0	0
$367$	−20.8328	−1.08746	−0.543732	−	0.839259i	$-0.682989\pi$
$367$	−20.8328	−1.08746	−0.543732	+	0.839259i	$0.682989\pi$
$368$	60.0132	3.12840
$369$	0	0
$370$	6.47214	0.336470
$371$	4.61803	0.239756
$372$	0	0
$373$	−35.5623	−1.84135	−0.920673	−	0.390334i	$-0.872359\pi$
$373$	−35.5623	−1.84135	−0.920673	+	0.390334i	$0.872359\pi$
$374$	0	0
$375$	0	0
$376$	32.7426	1.68857
$377$	7.70820	0.396993
$378$	0	0
$379$	−26.0689	−1.33907	−0.669534	−	0.742781i	$-0.733506\pi$
$379$	−26.0689	−1.33907	−0.669534	+	0.742781i	$0.733506\pi$
$380$	30.2705	1.55284
$381$	0	0
$382$	−52.9787	−2.71063
$383$	−14.5967	−0.745859	−0.372929	−	0.927860i	$-0.621647\pi$
$383$	−14.5967	−0.745859	−0.372929	+	0.927860i	$0.621647\pi$
$384$	0	0
$385$	0	0
$386$	0.854102	0.0434726
$387$	0	0
$388$	33.9787	1.72501
$389$	11.8197	0.599281	0.299640	−	0.954052i	$-0.403133\pi$
$389$	11.8197	0.599281	0.299640	+	0.954052i	$0.403133\pi$
$390$	0	0
$391$	49.2705	2.49172
$392$	7.47214	0.377400
$393$	0	0
$394$	46.3607	2.33562
$395$	−8.61803	−0.433620
$396$	0	0
$397$	−23.1803	−1.16339	−0.581694	−	0.813408i	$-0.697610\pi$
$397$	−23.1803	−1.16339	−0.581694	+	0.813408i	$0.697610\pi$
$398$	−9.85410	−0.493941
$399$	0	0
$400$	−39.4164	−1.97082
$401$	26.4721	1.32196	0.660978	−	0.750406i	$-0.270142\pi$
$401$	26.4721	1.32196	0.660978	+	0.750406i	$0.270142\pi$
$402$	0	0
$403$	0.763932	0.0380542
$404$	−47.5623	−2.36631
$405$	0	0
$406$	6.23607	0.309491
$407$	0	0
$408$	0	0
$409$	−4.29180	−0.212216	−0.106108	−	0.994355i	$-0.533839\pi$
$409$	−4.29180	−0.212216	−0.106108	+	0.994355i	$0.533839\pi$
$410$	29.2705	1.44557
$411$	0	0
$412$	10.4164	0.513180
$413$	0.0901699	0.00443697
$414$	0	0
$415$	−10.7082	−0.525645
$416$	35.1246	1.72213
$417$	0	0
$418$	0	0
$419$	5.88854	0.287674	0.143837	−	0.989601i	$-0.454056\pi$
$419$	5.88854	0.287674	0.143837	+	0.989601i	$0.454056\pi$
$420$	0	0
$421$	5.23607	0.255190	0.127595	−	0.991826i	$-0.459274\pi$
$421$	5.23607	0.255190	0.127595	+	0.991826i	$0.459274\pi$
$422$	35.0344	1.70545
$423$	0	0
$424$	34.5066	1.67579
$425$	−32.3607	−1.56972
$426$	0	0
$427$	−5.38197	−0.260452
$428$	51.9787	2.51249
$429$	0	0
$430$	−19.7984	−0.954762
$431$	7.14590	0.344206	0.172103	−	0.985079i	$-0.444944\pi$
$431$	7.14590	0.344206	0.172103	+	0.985079i	$0.444944\pi$
$432$	0	0
$433$	−38.0344	−1.82782	−0.913909	−	0.405918i	$-0.866952\pi$
$433$	−38.0344	−1.82782	−0.913909	+	0.405918i	$0.866952\pi$
$434$	0.618034	0.0296666
$435$	0	0
$436$	−50.8328	−2.43445
$437$	−37.9787	−1.81677
$438$	0	0
$439$	5.61803	0.268134	0.134067	−	0.990972i	$-0.457196\pi$
$439$	5.61803	0.268134	0.134067	+	0.990972i	$0.457196\pi$
$440$	0	0
$441$	0	0
$442$	68.5410	3.26016
$443$	8.09017	0.384376	0.192188	−	0.981358i	$-0.438442\pi$
$443$	8.09017	0.384376	0.192188	+	0.981358i	$0.438442\pi$
$444$	0	0
$445$	0.145898	0.00691623
$446$	−70.7771	−3.35139
$447$	0	0
$448$	8.70820	0.411424
$449$	3.32624	0.156975	0.0784874	−	0.996915i	$-0.474991\pi$
$449$	3.32624	0.156975	0.0784874	+	0.996915i	$0.474991\pi$
$450$	0	0
$451$	0	0
$452$	−33.2705	−1.56491
$453$	0	0
$454$	−34.1246	−1.60155
$455$	−3.23607	−0.151709
$456$	0	0
$457$	−6.85410	−0.320621	−0.160311	−	0.987067i	$-0.551250\pi$
$457$	−6.85410	−0.320621	−0.160311	+	0.987067i	$0.551250\pi$
$458$	29.4164	1.37454
$459$	0	0
$460$	−29.5623	−1.37835
$461$	−19.5066	−0.908512	−0.454256	−	0.890871i	$-0.650095\pi$
$461$	−19.5066	−0.908512	−0.454256	+	0.890871i	$0.650095\pi$
$462$	0	0
$463$	17.7639	0.825560	0.412780	−	0.910831i	$-0.364558\pi$
$463$	17.7639	0.825560	0.412780	+	0.910831i	$0.364558\pi$
$464$	23.4721	1.08967
$465$	0	0
$466$	6.76393	0.313333
$467$	−33.5066	−1.55050	−0.775250	−	0.631655i	$-0.782376\pi$
$467$	−33.5066	−1.55050	−0.775250	+	0.631655i	$0.782376\pi$
$468$	0	0
$469$	7.32624	0.338294
$470$	−11.4721	−0.529170
$471$	0	0
$472$	0.673762	0.0310124
$473$	0	0
$474$	0	0
$475$	24.9443	1.14452
$476$	39.2705	1.79996
$477$	0	0
$478$	15.4721	0.707679
$479$	−26.2361	−1.19876	−0.599378	−	0.800466i	$-0.704585\pi$
$479$	−26.2361	−1.19876	−0.599378	+	0.800466i	$0.704585\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	−45.2148	−2.05948
$483$	0	0
$484$	0	0
$485$	−7.00000	−0.317854
$486$	0	0
$487$	16.5967	0.752070	0.376035	−	0.926605i	$-0.377287\pi$
$487$	16.5967	0.752070	0.376035	+	0.926605i	$0.377287\pi$
$488$	−40.2148	−1.82044
$489$	0	0
$490$	−2.61803	−0.118271
$491$	−28.8541	−1.30217	−0.651084	−	0.759006i	$-0.725685\pi$
$491$	−28.8541	−1.30217	−0.651084	+	0.759006i	$0.725685\pi$
$492$	0	0
$493$	19.2705	0.867900
$494$	−52.8328	−2.37706
$495$	0	0
$496$	2.32624	0.104451
$497$	4.90983	0.220236
$498$	0	0
$499$	1.09017	0.0488027	0.0244014	−	0.999702i	$-0.492232\pi$
$499$	1.09017	0.0488027	0.0244014	+	0.999702i	$0.492232\pi$
$500$	43.6869	1.95374
$501$	0	0
$502$	−60.2148	−2.68752
$503$	−18.8328	−0.839714	−0.419857	−	0.907590i	$-0.637920\pi$
$503$	−18.8328	−0.839714	−0.419857	+	0.907590i	$0.637920\pi$
$504$	0	0
$505$	9.79837	0.436022
$506$	0	0
$507$	0	0
$508$	−72.5410	−3.21849
$509$	−31.7426	−1.40697	−0.703484	−	0.710711i	$-0.748373\pi$
$509$	−31.7426	−1.40697	−0.703484	+	0.710711i	$0.748373\pi$
$510$	0	0
$511$	−9.76393	−0.431931
$512$	−40.3050	−1.78124
$513$	0	0
$514$	−30.2705	−1.33517
$515$	−2.14590	−0.0945596
$516$	0	0
$517$	0	0
$518$	−6.47214	−0.284369
$519$	0	0
$520$	−24.1803	−1.06038
$521$	7.76393	0.340144	0.170072	−	0.985432i	$-0.445600\pi$
$521$	7.76393	0.340144	0.170072	+	0.985432i	$0.445600\pi$
$522$	0	0
$523$	−3.56231	−0.155769	−0.0778844	−	0.996962i	$-0.524817\pi$
$523$	−3.56231	−0.155769	−0.0778844	+	0.996962i	$0.524817\pi$
$524$	−5.12461	−0.223870
$525$	0	0
$526$	−60.5410	−2.63971
$527$	1.90983	0.0831935
$528$	0	0
$529$	14.0902	0.612616
$530$	−12.0902	−0.525163
$531$	0	0
$532$	−30.2705	−1.31239
$533$	−36.1803	−1.56714
$534$	0	0
$535$	−10.7082	−0.462956
$536$	54.7426	2.36452
$537$	0	0
$538$	−26.5623	−1.14518
$539$	0	0
$540$	0	0
$541$	−12.7082	−0.546368	−0.273184	−	0.961962i	$-0.588077\pi$
$541$	−12.7082	−0.546368	−0.273184	+	0.961962i	$0.588077\pi$
$542$	51.8328	2.22641
$543$	0	0
$544$	87.8115	3.76489
$545$	10.4721	0.448577
$546$	0	0
$547$	−35.2705	−1.50806	−0.754029	−	0.656841i	$-0.771892\pi$
$547$	−35.2705	−1.50806	−0.754029	+	0.656841i	$0.771892\pi$
$548$	−1.58359	−0.0676477
$549$	0	0
$550$	0	0
$551$	−14.8541	−0.632806
$552$	0	0
$553$	8.61803	0.366476
$554$	−15.7984	−0.671209
$555$	0	0
$556$	−28.8541	−1.22369
$557$	14.2361	0.603202	0.301601	−	0.953434i	$-0.402479\pi$
$557$	14.2361	0.603202	0.301601	+	0.953434i	$0.402479\pi$
$558$	0	0
$559$	24.4721	1.03506
$560$	−9.85410	−0.416412
$561$	0	0
$562$	7.38197	0.311389
$563$	−14.9787	−0.631278	−0.315639	−	0.948879i	$-0.602219\pi$
$563$	−14.9787	−0.631278	−0.315639	+	0.948879i	$0.602219\pi$
$564$	0	0
$565$	6.85410	0.288354
$566$	−15.5623	−0.654133
$567$	0	0
$568$	36.6869	1.53935
$569$	25.3050	1.06084	0.530419	−	0.847735i	$-0.322034\pi$
$569$	25.3050	1.06084	0.530419	+	0.847735i	$0.322034\pi$
$570$	0	0
$571$	−28.3050	−1.18453	−0.592263	−	0.805745i	$-0.701765\pi$
$571$	−28.3050	−1.18453	−0.592263	+	0.805745i	$0.701765\pi$
$572$	0	0
$573$	0	0
$574$	−29.2705	−1.22173
$575$	−24.3607	−1.01591
$576$	0	0
$577$	−21.6525	−0.901404	−0.450702	−	0.892674i	$-0.648826\pi$
$577$	−21.6525	−0.901404	−0.450702	+	0.892674i	$0.648826\pi$
$578$	126.846	5.27610
$579$	0	0
$580$	−11.5623	−0.480099
$581$	10.7082	0.444251
$582$	0	0
$583$	0	0
$584$	−72.9574	−3.01900
$585$	0	0
$586$	28.7984	1.18965
$587$	−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
$589$	−1.47214	−0.0606583
$590$	−0.236068	−0.00971876
$591$	0	0
$592$	−24.3607	−1.00122
$593$	29.1246	1.19600	0.598002	−	0.801494i	$-0.295961\pi$
$593$	29.1246	1.19600	0.598002	+	0.801494i	$0.295961\pi$
$594$	0	0
$595$	−8.09017	−0.331665
$596$	10.4164	0.426673
$597$	0	0
$598$	51.5967	2.10995
$599$	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
$601$	0.291796	0.0119026	0.00595130	−	0.999982i	$-0.498106\pi$
$601$	0.291796	0.0119026	0.00595130	+	0.999982i	$0.498106\pi$
$602$	19.7984	0.806921
$603$	0	0
$604$	87.1033	3.54418
$605$	0	0
$606$	0	0
$607$	44.7984	1.81831	0.909155	−	0.416458i	$-0.136729\pi$
$607$	44.7984	1.81831	0.909155	+	0.416458i	$0.136729\pi$
$608$	−67.6869	−2.74507
$609$	0	0
$610$	14.0902	0.570494
$611$	14.1803	0.573675
$612$	0	0
$613$	34.8885	1.40914	0.704568	−	0.709637i	$-0.251141\pi$
$613$	34.8885	1.40914	0.704568	+	0.709637i	$0.251141\pi$
$614$	−2.14590	−0.0866014
$615$	0	0
$616$	0	0
$617$	17.4164	0.701158	0.350579	−	0.936533i	$-0.385985\pi$
$617$	17.4164	0.701158	0.350579	+	0.936533i	$0.385985\pi$
$618$	0	0
$619$	−26.2918	−1.05676	−0.528378	−	0.849009i	$-0.677200\pi$
$619$	−26.2918	−1.05676	−0.528378	+	0.849009i	$0.677200\pi$
$620$	−1.14590	−0.0460204
$621$	0	0
$622$	−23.5623	−0.944762
$623$	−0.145898	−0.00584528
$624$	0	0
$625$	11.0000	0.440000
$626$	7.00000	0.279776
$627$	0	0
$628$	−77.1246	−3.07761
$629$	−20.0000	−0.797452
$630$	0	0
$631$	−35.8328	−1.42648	−0.713241	−	0.700919i	$-0.752773\pi$
$631$	−35.8328	−1.42648	−0.713241	+	0.700919i	$0.752773\pi$
$632$	64.3951	2.56150
$633$	0	0
$634$	64.3050	2.55388
$635$	14.9443	0.593045
$636$	0	0
$637$	3.23607	0.128218
$638$	0	0
$639$	0	0
$640$	−1.09017	−0.0430928
$641$	−0.347524	−0.0137264	−0.00686319	−	0.999976i	$-0.502185\pi$
$641$	−0.347524	−0.0137264	−0.00686319	+	0.999976i	$0.502185\pi$
$642$	0	0
$643$	28.4164	1.12063	0.560317	−	0.828278i	$-0.310679\pi$
$643$	28.4164	1.12063	0.560317	+	0.828278i	$0.310679\pi$
$644$	29.5623	1.16492
$645$	0	0
$646$	−132.082	−5.19670
$647$	16.1803	0.636115	0.318057	−	0.948071i	$-0.396970\pi$
$647$	16.1803	0.636115	0.318057	+	0.948071i	$0.396970\pi$
$648$	0	0
$649$	0	0
$650$	−33.8885	−1.32922
$651$	0	0
$652$	22.8541	0.895036
$653$	−13.7639	−0.538624	−0.269312	−	0.963053i	$-0.586796\pi$
$653$	−13.7639	−0.538624	−0.269312	+	0.963053i	$0.586796\pi$
$654$	0	0
$655$	1.05573	0.0412507
$656$	−110.172	−4.30150
$657$	0	0
$658$	11.4721	0.447230
$659$	−22.5279	−0.877561	−0.438780	−	0.898594i	$-0.644589\pi$
$659$	−22.5279	−0.877561	−0.438780	+	0.898594i	$0.644589\pi$
$660$	0	0
$661$	−14.4377	−0.561561	−0.280781	−	0.959772i	$-0.590593\pi$
$661$	−14.4377	−0.561561	−0.280781	+	0.959772i	$0.590593\pi$
$662$	−42.3607	−1.64639
$663$	0	0
$664$	80.0132	3.10511
$665$	6.23607	0.241824
$666$	0	0
$667$	14.5066	0.561697
$668$	12.0000	0.464294
$669$	0	0
$670$	−19.1803	−0.741001
$671$	0	0
$672$	0	0
$673$	−33.4164	−1.28811	−0.644054	−	0.764980i	$-0.722749\pi$
$673$	−33.4164	−1.28811	−0.644054	+	0.764980i	$0.722749\pi$
$674$	21.0902	0.812363
$675$	0	0
$676$	−12.2705	−0.471943
$677$	2.72949	0.104903	0.0524514	−	0.998623i	$-0.483297\pi$
$677$	2.72949	0.104903	0.0524514	+	0.998623i	$0.483297\pi$
$678$	0	0
$679$	7.00000	0.268635
$680$	−60.4508	−2.31818
$681$	0	0
$682$	0	0
$683$	−38.5066	−1.47341	−0.736707	−	0.676213i	$-0.763620\pi$
$683$	−38.5066	−1.47341	−0.736707	+	0.676213i	$0.763620\pi$
$684$	0	0
$685$	0.326238	0.0124649
$686$	2.61803	0.0999570
$687$	0	0
$688$	74.5197	2.84104
$689$	14.9443	0.569331
$690$	0	0
$691$	−48.5410	−1.84659	−0.923294	−	0.384095i	$-0.874514\pi$
$691$	−48.5410	−1.84659	−0.923294	+	0.384095i	$0.874514\pi$
$692$	85.5197	3.25097
$693$	0	0
$694$	−88.1033	−3.34436
$695$	5.94427	0.225479
$696$	0	0
$697$	−90.4508	−3.42607
$698$	21.4721	0.812732
$699$	0	0
$700$	−19.4164	−0.733871
$701$	19.5066	0.736753	0.368377	−	0.929677i	$-0.379914\pi$
$701$	19.5066	0.736753	0.368377	+	0.929677i	$0.379914\pi$
$702$	0	0
$703$	15.4164	0.581441
$704$	0	0
$705$	0	0
$706$	−33.7984	−1.27202
$707$	−9.79837	−0.368506
$708$	0	0
$709$	22.0344	0.827521	0.413760	−	0.910386i	$-0.364215\pi$
$709$	22.0344	0.827521	0.413760	+	0.910386i	$0.364215\pi$
$710$	−12.8541	−0.482406
$711$	0	0
$712$	−1.09017	−0.0408558
$713$	1.43769	0.0538421
$714$	0	0
$715$	0	0
$716$	39.9787	1.49407
$717$	0	0
$718$	35.0344	1.30747
$719$	−32.8885	−1.22654	−0.613268	−	0.789875i	$-0.710145\pi$
$719$	−32.8885	−1.22654	−0.613268	+	0.789875i	$0.710145\pi$
$720$	0	0
$721$	2.14590	0.0799174
$722$	52.0689	1.93780
$723$	0	0
$724$	94.2492	3.50274
$725$	−9.52786	−0.353856
$726$	0	0
$727$	43.4508	1.61150	0.805751	−	0.592254i	$-0.201762\pi$
$727$	43.4508	1.61150	0.805751	+	0.592254i	$0.201762\pi$
$728$	24.1803	0.896183
$729$	0	0
$730$	25.5623	0.946103
$731$	61.1803	2.26284
$732$	0	0
$733$	35.0557	1.29481	0.647406	−	0.762145i	$-0.275854\pi$
$733$	35.0557	1.29481	0.647406	+	0.762145i	$0.275854\pi$
$734$	−54.5410	−2.01315
$735$	0	0
$736$	66.1033	2.43660
$737$	0	0
$738$	0	0
$739$	−13.6180	−0.500947	−0.250474	−	0.968123i	$-0.580586\pi$
$739$	−13.6180	−0.500947	−0.250474	+	0.968123i	$0.580586\pi$
$740$	12.0000	0.441129
$741$	0	0
$742$	12.0902	0.443844
$743$	26.0902	0.957156	0.478578	−	0.878045i	$-0.341152\pi$
$743$	26.0902	0.957156	0.478578	+	0.878045i	$0.341152\pi$
$744$	0	0
$745$	−2.14590	−0.0786196
$746$	−93.1033	−3.40875
$747$	0	0
$748$	0	0
$749$	10.7082	0.391269
$750$	0	0
$751$	−21.3050	−0.777429	−0.388714	−	0.921358i	$-0.627081\pi$
$751$	−21.3050	−0.777429	−0.388714	+	0.921358i	$0.627081\pi$
$752$	43.1803	1.57462
$753$	0	0
$754$	20.1803	0.734925
$755$	−17.9443	−0.653059
$756$	0	0
$757$	−19.2361	−0.699147	−0.349573	−	0.936909i	$-0.613673\pi$
$757$	−19.2361	−0.699147	−0.349573	+	0.936909i	$0.613673\pi$
$758$	−68.2492	−2.47892
$759$	0	0
$760$	46.5967	1.69024
$761$	47.3050	1.71480	0.857402	−	0.514648i	$-0.172077\pi$
$761$	47.3050	1.71480	0.857402	+	0.514648i	$0.172077\pi$
$762$	0	0
$763$	−10.4721	−0.379117
$764$	−98.2279	−3.55376
$765$	0	0
$766$	−38.2148	−1.38076
$767$	0.291796	0.0105361
$768$	0	0
$769$	28.4377	1.02549	0.512745	−	0.858541i	$-0.328629\pi$
$769$	28.4377	1.02549	0.512745	+	0.858541i	$0.328629\pi$
$770$	0	0
$771$	0	0
$772$	1.58359	0.0569947
$773$	−35.7639	−1.28634	−0.643170	−	0.765724i	$-0.722381\pi$
$773$	−35.7639	−1.28634	−0.643170	+	0.765724i	$0.722381\pi$
$774$	0	0
$775$	−0.944272	−0.0339192
$776$	52.3050	1.87764
$777$	0	0
$778$	30.9443	1.10941
$779$	69.7214	2.49803
$780$	0	0
$781$	0	0
$782$	128.992	4.61274
$783$	0	0
$784$	9.85410	0.351932
$785$	15.8885	0.567086
$786$	0	0
$787$	0.437694	0.0156021	0.00780105	−	0.999970i	$-0.497517\pi$
$787$	0.437694	0.0156021	0.00780105	+	0.999970i	$0.497517\pi$
$788$	85.9574	3.06211
$789$	0	0
$790$	−22.5623	−0.802731
$791$	−6.85410	−0.243704
$792$	0	0
$793$	−17.4164	−0.618475
$794$	−60.6869	−2.15370
$795$	0	0
$796$	−18.2705	−0.647581
$797$	−27.7639	−0.983449	−0.491724	−	0.870751i	$-0.663633\pi$
$797$	−27.7639	−0.983449	−0.491724	+	0.870751i	$0.663633\pi$
$798$	0	0
$799$	35.4508	1.25416
$800$	−43.4164	−1.53500
$801$	0	0
$802$	69.3050	2.44724
$803$	0	0
$804$	0	0
$805$	−6.09017	−0.214650
$806$	2.00000	0.0704470
$807$	0	0
$808$	−73.2148	−2.57569
$809$	5.50658	0.193601	0.0968005	−	0.995304i	$-0.469139\pi$
$809$	5.50658	0.193601	0.0968005	+	0.995304i	$0.469139\pi$
$810$	0	0
$811$	54.8328	1.92544	0.962720	−	0.270499i	$-0.0871886\pi$
$811$	54.8328	1.92544	0.962720	+	0.270499i	$0.0871886\pi$
$812$	11.5623	0.405757
$813$	0	0
$814$	0	0
$815$	−4.70820	−0.164921
$816$	0	0
$817$	−47.1591	−1.64989
$818$	−11.2361	−0.392860
$819$	0	0
$820$	54.2705	1.89521
$821$	50.6656	1.76824	0.884121	−	0.467257i	$-0.154758\pi$
$821$	50.6656	1.76824	0.884121	+	0.467257i	$0.154758\pi$
$822$	0	0
$823$	3.88854	0.135546	0.0677731	−	0.997701i	$-0.478411\pi$
$823$	3.88854	0.135546	0.0677731	+	0.997701i	$0.478411\pi$
$824$	16.0344	0.558586
$825$	0	0
$826$	0.236068	0.00821386
$827$	−29.5967	−1.02918	−0.514590	−	0.857436i	$-0.672056\pi$
$827$	−29.5967	−1.02918	−0.514590	+	0.857436i	$0.672056\pi$
$828$	0	0
$829$	−1.02129	−0.0354707	−0.0177354	−	0.999843i	$-0.505646\pi$
$829$	−1.02129	−0.0354707	−0.0177354	+	0.999843i	$0.505646\pi$
$830$	−28.0344	−0.973090
$831$	0	0
$832$	28.1803	0.976978
$833$	8.09017	0.280308
$834$	0	0
$835$	−2.47214	−0.0855518
$836$	0	0
$837$	0	0
$838$	15.4164	0.532551
$839$	18.5967	0.642031	0.321016	−	0.947074i	$-0.395976\pi$
$839$	18.5967	0.642031	0.321016	+	0.947074i	$0.395976\pi$
$840$	0	0
$841$	−23.3262	−0.804353
$842$	13.7082	0.472416
$843$	0	0
$844$	64.9574	2.23593
$845$	2.52786	0.0869612
$846$	0	0
$847$	0	0
$848$	45.5066	1.56270
$849$	0	0
$850$	−84.7214	−2.90592
$851$	−15.0557	−0.516104
$852$	0	0
$853$	−46.0902	−1.57810	−0.789049	−	0.614331i	$-0.789426\pi$
$853$	−46.0902	−1.57810	−0.789049	+	0.614331i	$0.789426\pi$
$854$	−14.0902	−0.482156
$855$	0	0
$856$	80.0132	2.73479
$857$	3.90983	0.133557	0.0667786	−	0.997768i	$-0.478728\pi$
$857$	3.90983	0.133557	0.0667786	+	0.997768i	$0.478728\pi$
$858$	0	0
$859$	1.43769	0.0490535	0.0245267	−	0.999699i	$-0.492192\pi$
$859$	1.43769	0.0490535	0.0245267	+	0.999699i	$0.492192\pi$
$860$	−36.7082	−1.25174
$861$	0	0
$862$	18.7082	0.637204
$863$	30.0557	1.02311	0.511554	−	0.859251i	$-0.329070\pi$
$863$	30.0557	1.02311	0.511554	+	0.859251i	$0.329070\pi$
$864$	0	0
$865$	−17.6180	−0.599031
$866$	−99.5755	−3.38371
$867$	0	0
$868$	1.14590	0.0388943
$869$	0	0
$870$	0	0
$871$	23.7082	0.803322
$872$	−78.2492	−2.64985
$873$	0	0
$874$	−99.4296	−3.36326
$875$	9.00000	0.304256
$876$	0	0
$877$	13.0000	0.438979	0.219489	−	0.975615i	$-0.429561\pi$
$877$	13.0000	0.438979	0.219489	+	0.975615i	$0.429561\pi$
$878$	14.7082	0.496378
$879$	0	0
$880$	0	0
$881$	−20.8541	−0.702593	−0.351296	−	0.936264i	$-0.614259\pi$
$881$	−20.8541	−0.702593	−0.351296	+	0.936264i	$0.614259\pi$
$882$	0	0
$883$	−46.6312	−1.56926	−0.784632	−	0.619961i	$-0.787148\pi$
$883$	−46.6312	−1.56926	−0.784632	+	0.619961i	$0.787148\pi$
$884$	127.082	4.27423
$885$	0	0
$886$	21.1803	0.711567
$887$	−5.78522	−0.194249	−0.0971243	−	0.995272i	$-0.530964\pi$
$887$	−5.78522	−0.194249	−0.0971243	+	0.995272i	$0.530964\pi$
$888$	0	0
$889$	−14.9443	−0.501215
$890$	0.381966	0.0128035
$891$	0	0
$892$	−131.228	−4.39384
$893$	−27.3262	−0.914438
$894$	0	0
$895$	−8.23607	−0.275301
$896$	1.09017	0.0364200
$897$	0	0
$898$	8.70820	0.290597
$899$	0.562306	0.0187540
$900$	0	0
$901$	37.3607	1.24466
$902$	0	0
$903$	0	0
$904$	−51.2148	−1.70338
$905$	−19.4164	−0.645423
$906$	0	0
$907$	−20.2361	−0.671928	−0.335964	−	0.941875i	$-0.609062\pi$
$907$	−20.2361	−0.671928	−0.335964	+	0.941875i	$0.609062\pi$
$908$	−63.2705	−2.09971
$909$	0	0
$910$	−8.47214	−0.280849
$911$	22.8197	0.756049	0.378025	−	0.925796i	$-0.376603\pi$
$911$	22.8197	0.756049	0.378025	+	0.925796i	$0.376603\pi$
$912$	0	0
$913$	0	0
$914$	−17.9443	−0.593544
$915$	0	0
$916$	54.5410	1.80209
$917$	−1.05573	−0.0348632
$918$	0	0
$919$	20.8885	0.689049	0.344525	−	0.938777i	$-0.388040\pi$
$919$	20.8885	0.689049	0.344525	+	0.938777i	$0.388040\pi$
$920$	−45.5066	−1.50031
$921$	0	0
$922$	−51.0689	−1.68186
$923$	15.8885	0.522978
$924$	0	0
$925$	9.88854	0.325133
$926$	46.5066	1.52830
$927$	0	0
$928$	25.8541	0.848702
$929$	−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
$931$	−6.23607	−0.204379
$932$	12.5410	0.410795
$933$	0	0
$934$	−87.7214	−2.87033
$935$	0	0
$936$	0	0
$937$	37.7214	1.23230	0.616152	−	0.787628i	$-0.288691\pi$
$937$	37.7214	1.23230	0.616152	+	0.787628i	$0.288691\pi$
$938$	19.1803	0.626260
$939$	0	0
$940$	−21.2705	−0.693768
$941$	31.0689	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$941$	31.0689	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$942$	0	0
$943$	−68.0902	−2.21732
$944$	0.888544	0.0289196
$945$	0	0
$946$	0	0
$947$	−19.7082	−0.640431	−0.320215	−	0.947345i	$-0.603755\pi$
$947$	−19.7082	−0.640431	−0.320215	+	0.947345i	$0.603755\pi$
$948$	0	0
$949$	−31.5967	−1.02567
$950$	65.3050	2.11877
$951$	0	0
$952$	60.4508	1.95922
$953$	1.76393	0.0571394	0.0285697	−	0.999592i	$-0.490905\pi$
$953$	1.76393	0.0571394	0.0285697	+	0.999592i	$0.490905\pi$
$954$	0	0
$955$	20.2361	0.654824
$956$	28.6869	0.927801
$957$	0	0
$958$	−68.6869	−2.21917
$959$	−0.326238	−0.0105348
$960$	0	0
$961$	−30.9443	−0.998202
$962$	−20.9443	−0.675270
$963$	0	0
$964$	−83.8328	−2.70007
$965$	−0.326238	−0.0105020
$966$	0	0
$967$	−28.4508	−0.914918	−0.457459	−	0.889231i	$-0.651240\pi$
$967$	−28.4508	−0.914918	−0.457459	+	0.889231i	$0.651240\pi$
$968$	0	0
$969$	0	0
$970$	−18.3262	−0.588420
$971$	−52.2492	−1.67676	−0.838379	−	0.545088i	$-0.816496\pi$
$971$	−52.2492	−1.67676	−0.838379	+	0.545088i	$0.816496\pi$
$972$	0	0
$973$	−5.94427	−0.190565
$974$	43.4508	1.39226
$975$	0	0
$976$	−53.0344	−1.69759
$977$	33.1803	1.06153	0.530767	−	0.847518i	$-0.321904\pi$
$977$	33.1803	1.06153	0.530767	+	0.847518i	$0.321904\pi$
$978$	0	0
$979$	0	0
$980$	−4.85410	−0.155059
$981$	0	0
$982$	−75.5410	−2.41061
$983$	14.6180	0.466243	0.233121	−	0.972448i	$-0.425106\pi$
$983$	14.6180	0.466243	0.233121	+	0.972448i	$0.425106\pi$
$984$	0	0
$985$	−17.7082	−0.564230
$986$	50.4508	1.60668
$987$	0	0
$988$	−97.9574	−3.11644
$989$	46.0557	1.46449
$990$	0	0
$991$	−34.2705	−1.08864	−0.544319	−	0.838878i	$-0.683212\pi$
$991$	−34.2705	−1.08864	−0.544319	+	0.838878i	$0.683212\pi$
$992$	2.56231	0.0813533
$993$	0	0
$994$	12.8541	0.407707
$995$	3.76393	0.119325
$996$	0	0
$997$	26.8673	0.850895	0.425447	−	0.904983i	$-0.360117\pi$
$997$	26.8673	0.850895	0.425447	+	0.904983i	$0.360117\pi$
$998$	2.85410	0.0903450
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} +7.47214 q^{8} +O(q^{10})\)	\(q+2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} +7.47214 q^{8} -2.61803 q^{10} +3.23607 q^{13} +2.61803 q^{14} +9.85410 q^{16} +8.09017 q^{17} -6.23607 q^{19} -4.85410 q^{20} +6.09017 q^{23} -4.00000 q^{25} +8.47214 q^{26} +4.85410 q^{28} +2.38197 q^{29} +0.236068 q^{31} +10.8541 q^{32} +21.1803 q^{34} -1.00000 q^{35} -2.47214 q^{37} -16.3262 q^{38} -7.47214 q^{40} -11.1803 q^{41} +7.56231 q^{43} +15.9443 q^{46} +4.38197 q^{47} +1.00000 q^{49} -10.4721 q^{50} +15.7082 q^{52} +4.61803 q^{53} +7.47214 q^{56} +6.23607 q^{58} +0.0901699 q^{59} -5.38197 q^{61} +0.618034 q^{62} +8.70820 q^{64} -3.23607 q^{65} +7.32624 q^{67} +39.2705 q^{68} -2.61803 q^{70} +4.90983 q^{71} -9.76393 q^{73} -6.47214 q^{74} -30.2705 q^{76} +8.61803 q^{79} -9.85410 q^{80} -29.2705 q^{82} +10.7082 q^{83} -8.09017 q^{85} +19.7984 q^{86} -0.145898 q^{89} +3.23607 q^{91} +29.5623 q^{92} +11.4721 q^{94} +6.23607 q^{95} +7.00000 q^{97} +2.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) 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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