LMFDB - Embedded newform 7623.2.a.bk.1.1

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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-2.23607 q^{2} +3.00000 q^{4} +0.618034 q^{5} +1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	$q-2.23607 q^{2} +3.00000 q^{4} +0.618034 q^{5} +1.00000 q^{7} -2.23607 q^{8} -1.38197 q^{10} +3.23607 q^{13} -2.23607 q^{14} -1.00000 q^{16} -4.85410 q^{17} +2.85410 q^{19} +1.85410 q^{20} -4.38197 q^{23} -4.61803 q^{25} -7.23607 q^{26} +3.00000 q^{28} +6.00000 q^{29} -3.09017 q^{31} +6.70820 q^{32} +10.8541 q^{34} +0.618034 q^{35} +4.61803 q^{37} -6.38197 q^{38} -1.38197 q^{40} -7.38197 q^{41} +9.70820 q^{43} +9.79837 q^{46} -4.47214 q^{47} +1.00000 q^{49} +10.3262 q^{50} +9.70820 q^{52} -6.76393 q^{53} -2.23607 q^{56} -13.4164 q^{58} -3.23607 q^{59} +6.90983 q^{62} -13.0000 q^{64} +2.00000 q^{65} -14.5623 q^{68} -1.38197 q^{70} +4.94427 q^{71} +13.7082 q^{73} -10.3262 q^{74} +8.56231 q^{76} -10.0000 q^{79} -0.618034 q^{80} +16.5066 q^{82} -16.4721 q^{83} -3.00000 q^{85} -21.7082 q^{86} -5.61803 q^{89} +3.23607 q^{91} -13.1459 q^{92} +10.0000 q^{94} +1.76393 q^{95} -6.00000 q^{97} -2.23607 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} - q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 6 * q^4 - q^5 + 2 * q^7	$ 2 q + 6 q^{4} - q^{5} + 2 q^{7} - 5 q^{10} + 2 q^{13} - 2 q^{16} - 3 q^{17} - q^{19} - 3 q^{20} - 11 q^{23} - 7 q^{25} - 10 q^{26} + 6 q^{28} + 12 q^{29} + 5 q^{31} + 15 q^{34} - q^{35} + 7 q^{37} - 15 q^{38} - 5 q^{40} - 17 q^{41} + 6 q^{43} - 5 q^{46} + 2 q^{49} + 5 q^{50} + 6 q^{52} - 18 q^{53} - 2 q^{59} + 25 q^{62} - 26 q^{64} + 4 q^{65} - 9 q^{68} - 5 q^{70} - 8 q^{71} + 14 q^{73} - 5 q^{74} - 3 q^{76} - 20 q^{79} + q^{80} - 5 q^{82} - 24 q^{83} - 6 q^{85} - 30 q^{86} - 9 q^{89} + 2 q^{91} - 33 q^{92} + 20 q^{94} + 8 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q + 6 * q^4 - q^5 + 2 * q^7 - 5 * q^10 + 2 * q^13 - 2 * q^16 - 3 * q^17 - q^19 - 3 * q^20 - 11 * q^23 - 7 * q^25 - 10 * q^26 + 6 * q^28 + 12 * q^29 + 5 * q^31 + 15 * q^34 - q^35 + 7 * q^37 - 15 * q^38 - 5 * q^40 - 17 * q^41 + 6 * q^43 - 5 * q^46 + 2 * q^49 + 5 * q^50 + 6 * q^52 - 18 * q^53 - 2 * q^59 + 25 * q^62 - 26 * q^64 + 4 * q^65 - 9 * q^68 - 5 * q^70 - 8 * q^71 + 14 * q^73 - 5 * q^74 - 3 * q^76 - 20 * q^79 + q^80 - 5 * q^82 - 24 * q^83 - 6 * q^85 - 30 * q^86 - 9 * q^89 + 2 * q^91 - 33 * q^92 + 20 * q^94 + 8 * q^95 - 12 * 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 $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	0.618034	0.276393	0.138197	−	0.990405i	$-0.455869\pi$
$5$	0.618034	0.276393	0.138197	+	0.990405i	$0.455869\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	−1.38197	−0.437016
$11$	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	$0.351865\pi$
$14$	−2.23607	−0.597614
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−4.85410	−1.17729	−0.588646	−	0.808391i	$-0.700339\pi$
$17$	−4.85410	−1.17729	−0.588646	+	0.808391i	$0.700339\pi$
$18$	0	0
$19$	2.85410	0.654776	0.327388	−	0.944890i	$-0.393832\pi$
$19$	2.85410	0.654776	0.327388	+	0.944890i	$0.393832\pi$
$20$	1.85410	0.414590
$21$	0	0
$22$	0	0
$23$	−4.38197	−0.913703	−0.456852	−	0.889543i	$-0.651023\pi$
$23$	−4.38197	−0.913703	−0.456852	+	0.889543i	$0.651023\pi$
$24$	0	0
$25$	−4.61803	−0.923607
$26$	−7.23607	−1.41911
$27$	0	0
$28$	3.00000	0.566947
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−3.09017	−0.555011	−0.277505	−	0.960724i	$-0.589508\pi$
$31$	−3.09017	−0.555011	−0.277505	+	0.960724i	$0.589508\pi$
$32$	6.70820	1.18585
$33$	0	0
$34$	10.8541	1.86146
$35$	0.618034	0.104467
$36$	0	0
$37$	4.61803	0.759200	0.379600	−	0.925151i	$-0.376062\pi$
$37$	4.61803	0.759200	0.379600	+	0.925151i	$0.376062\pi$
$38$	−6.38197	−1.03529
$39$	0	0
$40$	−1.38197	−0.218508
$41$	−7.38197	−1.15287	−0.576435	−	0.817143i	$-0.695556\pi$
$41$	−7.38197	−1.15287	−0.576435	+	0.817143i	$0.695556\pi$
$42$	0	0
$43$	9.70820	1.48049	0.740244	−	0.672339i	$-0.234710\pi$
$43$	9.70820	1.48049	0.740244	+	0.672339i	$0.234710\pi$
$44$	0	0
$45$	0	0
$46$	9.79837	1.44469
$47$	−4.47214	−0.652328	−0.326164	−	0.945313i	$-0.605756\pi$
$47$	−4.47214	−0.652328	−0.326164	+	0.945313i	$0.605756\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	10.3262	1.46035
$51$	0	0
$52$	9.70820	1.34629
$53$	−6.76393	−0.929098	−0.464549	−	0.885548i	$-0.653783\pi$
$53$	−6.76393	−0.929098	−0.464549	+	0.885548i	$0.653783\pi$
$54$	0	0
$55$	0	0
$56$	−2.23607	−0.298807
$57$	0	0
$58$	−13.4164	−1.76166
$59$	−3.23607	−0.421300	−0.210650	−	0.977562i	$-0.567558\pi$
$59$	−3.23607	−0.421300	−0.210650	+	0.977562i	$0.567558\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	6.90983	0.877549
$63$	0	0
$64$	−13.0000	−1.62500
$65$	2.00000	0.248069
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−14.5623	−1.76594
$69$	0	0
$70$	−1.38197	−0.165177
$71$	4.94427	0.586777	0.293389	−	0.955993i	$-0.405217\pi$
$71$	4.94427	0.586777	0.293389	+	0.955993i	$0.405217\pi$
$72$	0	0
$73$	13.7082	1.60442	0.802212	−	0.597039i	$-0.203656\pi$
$73$	13.7082	1.60442	0.802212	+	0.597039i	$0.203656\pi$
$74$	−10.3262	−1.20040
$75$	0	0
$76$	8.56231	0.982164
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	−0.618034	−0.0690983
$81$	0	0
$82$	16.5066	1.82285
$83$	−16.4721	−1.80805	−0.904026	−	0.427478i	$-0.859402\pi$
$83$	−16.4721	−1.80805	−0.904026	+	0.427478i	$0.859402\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	−21.7082	−2.34086
$87$	0	0
$88$	0	0
$89$	−5.61803	−0.595510	−0.297755	−	0.954642i	$-0.596238\pi$
$89$	−5.61803	−0.595510	−0.297755	+	0.954642i	$0.596238\pi$
$90$	0	0
$91$	3.23607	0.339232
$92$	−13.1459	−1.37055
$93$	0	0
$94$	10.0000	1.03142
$95$	1.76393	0.180976
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	−2.23607	−0.225877
$99$	0	0
$100$	−13.8541	−1.38541
$101$	6.79837	0.676463	0.338232	−	0.941063i	$-0.390171\pi$
$101$	6.79837	0.676463	0.338232	+	0.941063i	$0.390171\pi$
$102$	0	0
$103$	1.32624	0.130678	0.0653391	−	0.997863i	$-0.479187\pi$
$103$	1.32624	0.130678	0.0653391	+	0.997863i	$0.479187\pi$
$104$	−7.23607	−0.709555
$105$	0	0
$106$	15.1246	1.46903
$107$	−15.7984	−1.52729	−0.763643	−	0.645638i	$-0.776591\pi$
$107$	−15.7984	−1.52729	−0.763643	+	0.645638i	$0.776591\pi$
$108$	0	0
$109$	−17.5623	−1.68216	−0.841082	−	0.540908i	$-0.818081\pi$
$109$	−17.5623	−1.68216	−0.841082	+	0.540908i	$0.818081\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	5.23607	0.492568	0.246284	−	0.969198i	$-0.420790\pi$
$113$	5.23607	0.492568	0.246284	+	0.969198i	$0.420790\pi$
$114$	0	0
$115$	−2.70820	−0.252541
$116$	18.0000	1.67126
$117$	0	0
$118$	7.23607	0.666134
$119$	−4.85410	−0.444975
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	−9.27051	−0.832516
$125$	−5.94427	−0.531672
$126$	0	0
$127$	0.291796	0.0258927	0.0129464	−	0.999916i	$-0.495879\pi$
$127$	0.291796	0.0258927	0.0129464	+	0.999916i	$0.495879\pi$
$128$	15.6525	1.38350
$129$	0	0
$130$	−4.47214	−0.392232
$131$	16.6525	1.45493	0.727467	−	0.686143i	$-0.240698\pi$
$131$	16.6525	1.45493	0.727467	+	0.686143i	$0.240698\pi$
$132$	0	0
$133$	2.85410	0.247482
$134$	0	0
$135$	0	0
$136$	10.8541	0.930732
$137$	−8.18034	−0.698894	−0.349447	−	0.936956i	$-0.613630\pi$
$137$	−8.18034	−0.698894	−0.349447	+	0.936956i	$0.613630\pi$
$138$	0	0
$139$	13.1459	1.11502	0.557510	−	0.830170i	$-0.311757\pi$
$139$	13.1459	1.11502	0.557510	+	0.830170i	$0.311757\pi$
$140$	1.85410	0.156700
$141$	0	0
$142$	−11.0557	−0.927776
$143$	0	0
$144$	0	0
$145$	3.70820	0.307950
$146$	−30.6525	−2.53682
$147$	0	0
$148$	13.8541	1.13880
$149$	4.94427	0.405051	0.202525	−	0.979277i	$-0.435085\pi$
$149$	4.94427	0.405051	0.202525	+	0.979277i	$0.435085\pi$
$150$	0	0
$151$	−15.5279	−1.26364	−0.631820	−	0.775115i	$-0.717692\pi$
$151$	−15.5279	−1.26364	−0.631820	+	0.775115i	$0.717692\pi$
$152$	−6.38197	−0.517646
$153$	0	0
$154$	0	0
$155$	−1.90983	−0.153401
$156$	0	0
$157$	−23.1246	−1.84554	−0.922772	−	0.385345i	$-0.874082\pi$
$157$	−23.1246	−1.84554	−0.922772	+	0.385345i	$0.874082\pi$
$158$	22.3607	1.77892
$159$	0	0
$160$	4.14590	0.327762
$161$	−4.38197	−0.345347
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	−22.1459	−1.72930
$165$	0	0
$166$	36.8328	2.85878
$167$	21.2361	1.64330	0.821648	−	0.569995i	$-0.193055\pi$
$167$	21.2361	1.64330	0.821648	+	0.569995i	$0.193055\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	6.70820	0.514496
$171$	0	0
$172$	29.1246	2.22073
$173$	13.6180	1.03536	0.517680	−	0.855574i	$-0.326796\pi$
$173$	13.6180	1.03536	0.517680	+	0.855574i	$0.326796\pi$
$174$	0	0
$175$	−4.61803	−0.349091
$176$	0	0
$177$	0	0
$178$	12.5623	0.941585
$179$	14.7984	1.10608	0.553041	−	0.833154i	$-0.313467\pi$
$179$	14.7984	1.10608	0.553041	+	0.833154i	$0.313467\pi$
$180$	0	0
$181$	−19.8885	−1.47830	−0.739152	−	0.673539i	$-0.764773\pi$
$181$	−19.8885	−1.47830	−0.739152	+	0.673539i	$0.764773\pi$
$182$	−7.23607	−0.536373
$183$	0	0
$184$	9.79837	0.722346
$185$	2.85410	0.209838
$186$	0	0
$187$	0	0
$188$	−13.4164	−0.978492
$189$	0	0
$190$	−3.94427	−0.286148
$191$	−4.90983	−0.355263	−0.177631	−	0.984097i	$-0.556843\pi$
$191$	−4.90983	−0.355263	−0.177631	+	0.984097i	$0.556843\pi$
$192$	0	0
$193$	−21.8541	−1.57309	−0.786546	−	0.617531i	$-0.788133\pi$
$193$	−21.8541	−1.57309	−0.786546	+	0.617531i	$0.788133\pi$
$194$	13.4164	0.963242
$195$	0	0
$196$	3.00000	0.214286
$197$	−22.4721	−1.60107	−0.800537	−	0.599284i	$-0.795452\pi$
$197$	−22.4721	−1.60107	−0.800537	+	0.599284i	$0.795452\pi$
$198$	0	0
$199$	13.1459	0.931888	0.465944	−	0.884814i	$-0.345715\pi$
$199$	13.1459	0.931888	0.465944	+	0.884814i	$0.345715\pi$
$200$	10.3262	0.730175
$201$	0	0
$202$	−15.2016	−1.06958
$203$	6.00000	0.421117
$204$	0	0
$205$	−4.56231	−0.318645
$206$	−2.96556	−0.206620
$207$	0	0
$208$	−3.23607	−0.224381
$209$	0	0
$210$	0	0
$211$	−18.3607	−1.26400	−0.632001	−	0.774968i	$-0.717766\pi$
$211$	−18.3607	−1.26400	−0.632001	+	0.774968i	$0.717766\pi$
$212$	−20.2918	−1.39365
$213$	0	0
$214$	35.3262	2.41485
$215$	6.00000	0.409197
$216$	0	0
$217$	−3.09017	−0.209774
$218$	39.2705	2.65973
$219$	0	0
$220$	0	0
$221$	−15.7082	−1.05665
$222$	0	0
$223$	19.6180	1.31372	0.656860	−	0.754012i	$-0.271884\pi$
$223$	19.6180	1.31372	0.656860	+	0.754012i	$0.271884\pi$
$224$	6.70820	0.448211
$225$	0	0
$226$	−11.7082	−0.778818
$227$	−12.4721	−0.827805	−0.413902	−	0.910321i	$-0.635835\pi$
$227$	−12.4721	−0.827805	−0.413902	+	0.910321i	$0.635835\pi$
$228$	0	0
$229$	16.1803	1.06923	0.534613	−	0.845097i	$-0.320457\pi$
$229$	16.1803	1.06923	0.534613	+	0.845097i	$0.320457\pi$
$230$	6.05573	0.399303
$231$	0	0
$232$	−13.4164	−0.880830
$233$	15.7082	1.02908	0.514539	−	0.857467i	$-0.327963\pi$
$233$	15.7082	1.02908	0.514539	+	0.857467i	$0.327963\pi$
$234$	0	0
$235$	−2.76393	−0.180299
$236$	−9.70820	−0.631950
$237$	0	0
$238$	10.8541	0.703567
$239$	19.0344	1.23124	0.615618	−	0.788045i	$-0.288907\pi$
$239$	19.0344	1.23124	0.615618	+	0.788045i	$0.288907\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.618034	0.0394847
$246$	0	0
$247$	9.23607	0.587677
$248$	6.90983	0.438775
$249$	0	0
$250$	13.2918	0.840647
$251$	−27.7082	−1.74893	−0.874463	−	0.485092i	$-0.838786\pi$
$251$	−27.7082	−1.74893	−0.874463	+	0.485092i	$0.838786\pi$
$252$	0	0
$253$	0	0
$254$	−0.652476	−0.0409400
$255$	0	0
$256$	−9.00000	−0.562500
$257$	−24.4508	−1.52520	−0.762601	−	0.646869i	$-0.776078\pi$
$257$	−24.4508	−1.52520	−0.762601	+	0.646869i	$0.776078\pi$
$258$	0	0
$259$	4.61803	0.286951
$260$	6.00000	0.372104
$261$	0	0
$262$	−37.2361	−2.30045
$263$	−10.8541	−0.669293	−0.334646	−	0.942344i	$-0.608617\pi$
$263$	−10.8541	−0.669293	−0.334646	+	0.942344i	$0.608617\pi$
$264$	0	0
$265$	−4.18034	−0.256796
$266$	−6.38197	−0.391303
$267$	0	0
$268$	0	0
$269$	13.4164	0.818013	0.409006	−	0.912532i	$-0.365875\pi$
$269$	13.4164	0.818013	0.409006	+	0.912532i	$0.365875\pi$
$270$	0	0
$271$	−22.5066	−1.36718	−0.683589	−	0.729868i	$-0.739582\pi$
$271$	−22.5066	−1.36718	−0.683589	+	0.729868i	$0.739582\pi$
$272$	4.85410	0.294323
$273$	0	0
$274$	18.2918	1.10505
$275$	0	0
$276$	0	0
$277$	−0.0901699	−0.00541779	−0.00270889	−	0.999996i	$-0.500862\pi$
$277$	−0.0901699	−0.00541779	−0.00270889	+	0.999996i	$0.500862\pi$
$278$	−29.3951	−1.76300
$279$	0	0
$280$	−1.38197	−0.0825883
$281$	26.9443	1.60736	0.803680	−	0.595061i	$-0.202872\pi$
$281$	26.9443	1.60736	0.803680	+	0.595061i	$0.202872\pi$
$282$	0	0
$283$	−17.2705	−1.02663	−0.513313	−	0.858202i	$-0.671582\pi$
$283$	−17.2705	−1.02663	−0.513313	+	0.858202i	$0.671582\pi$
$284$	14.8328	0.880166
$285$	0	0
$286$	0	0
$287$	−7.38197	−0.435744
$288$	0	0
$289$	6.56231	0.386018
$290$	−8.29180	−0.486911
$291$	0	0
$292$	41.1246	2.40664
$293$	18.2148	1.06412	0.532059	−	0.846707i	$-0.321418\pi$
$293$	18.2148	1.06412	0.532059	+	0.846707i	$0.321418\pi$
$294$	0	0
$295$	−2.00000	−0.116445
$296$	−10.3262	−0.600200
$297$	0	0
$298$	−11.0557	−0.640441
$299$	−14.1803	−0.820070
$300$	0	0
$301$	9.70820	0.559572
$302$	34.7214	1.99799
$303$	0	0
$304$	−2.85410	−0.163694
$305$	0	0
$306$	0	0
$307$	−17.5623	−1.00233	−0.501167	−	0.865351i	$-0.667096\pi$
$307$	−17.5623	−1.00233	−0.501167	+	0.865351i	$0.667096\pi$
$308$	0	0
$309$	0	0
$310$	4.27051	0.242549
$311$	−20.6525	−1.17109	−0.585547	−	0.810638i	$-0.699120\pi$
$311$	−20.6525	−1.17109	−0.585547	+	0.810638i	$0.699120\pi$
$312$	0	0
$313$	−7.81966	−0.441993	−0.220997	−	0.975275i	$-0.570931\pi$
$313$	−7.81966	−0.441993	−0.220997	+	0.975275i	$0.570931\pi$
$314$	51.7082	2.91806
$315$	0	0
$316$	−30.0000	−1.68763
$317$	−6.76393	−0.379900	−0.189950	−	0.981794i	$-0.560833\pi$
$317$	−6.76393	−0.379900	−0.189950	+	0.981794i	$0.560833\pi$
$318$	0	0
$319$	0	0
$320$	−8.03444	−0.449139
$321$	0	0
$322$	9.79837	0.546042
$323$	−13.8541	−0.770863
$324$	0	0
$325$	−14.9443	−0.828959
$326$	22.3607	1.23844
$327$	0	0
$328$	16.5066	0.911423
$329$	−4.47214	−0.246557
$330$	0	0
$331$	17.4164	0.957292	0.478646	−	0.878008i	$-0.341128\pi$
$331$	17.4164	0.957292	0.478646	+	0.878008i	$0.341128\pi$
$332$	−49.4164	−2.71208
$333$	0	0
$334$	−47.4853	−2.59828
$335$	0	0
$336$	0	0
$337$	27.0902	1.47570	0.737848	−	0.674967i	$-0.235842\pi$
$337$	27.0902	1.47570	0.737848	+	0.674967i	$0.235842\pi$
$338$	5.65248	0.307454
$339$	0	0
$340$	−9.00000	−0.488094
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−21.7082	−1.17043
$345$	0	0
$346$	−30.4508	−1.63705
$347$	14.5066	0.778754	0.389377	−	0.921078i	$-0.372690\pi$
$347$	14.5066	0.778754	0.389377	+	0.921078i	$0.372690\pi$
$348$	0	0
$349$	−21.7082	−1.16201	−0.581007	−	0.813899i	$-0.697341\pi$
$349$	−21.7082	−1.16201	−0.581007	+	0.813899i	$0.697341\pi$
$350$	10.3262	0.551961
$351$	0	0
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	3.05573	0.162181
$356$	−16.8541	−0.893266
$357$	0	0
$358$	−33.0902	−1.74887
$359$	−6.43769	−0.339768	−0.169884	−	0.985464i	$-0.554339\pi$
$359$	−6.43769	−0.339768	−0.169884	+	0.985464i	$0.554339\pi$
$360$	0	0
$361$	−10.8541	−0.571269
$362$	44.4721	2.33740
$363$	0	0
$364$	9.70820	0.508848
$365$	8.47214	0.443452
$366$	0	0
$367$	2.72949	0.142478	0.0712391	−	0.997459i	$-0.477305\pi$
$367$	2.72949	0.142478	0.0712391	+	0.997459i	$0.477305\pi$
$368$	4.38197	0.228426
$369$	0	0
$370$	−6.38197	−0.331783
$371$	−6.76393	−0.351166
$372$	0	0
$373$	16.3262	0.845341	0.422670	−	0.906284i	$-0.361093\pi$
$373$	16.3262	0.845341	0.422670	+	0.906284i	$0.361093\pi$
$374$	0	0
$375$	0	0
$376$	10.0000	0.515711
$377$	19.4164	0.999996
$378$	0	0
$379$	21.5279	1.10581	0.552906	−	0.833244i	$-0.313519\pi$
$379$	21.5279	1.10581	0.552906	+	0.833244i	$0.313519\pi$
$380$	5.29180	0.271463
$381$	0	0
$382$	10.9787	0.561720
$383$	20.8328	1.06451	0.532254	−	0.846585i	$-0.321345\pi$
$383$	20.8328	1.06451	0.532254	+	0.846585i	$0.321345\pi$
$384$	0	0
$385$	0	0
$386$	48.8673	2.48728
$387$	0	0
$388$	−18.0000	−0.913812
$389$	−15.2361	−0.772499	−0.386250	−	0.922394i	$-0.626230\pi$
$389$	−15.2361	−0.772499	−0.386250	+	0.922394i	$0.626230\pi$
$390$	0	0
$391$	21.2705	1.07570
$392$	−2.23607	−0.112938
$393$	0	0
$394$	50.2492	2.53152
$395$	−6.18034	−0.310967
$396$	0	0
$397$	30.5410	1.53281	0.766405	−	0.642358i	$-0.222044\pi$
$397$	30.5410	1.53281	0.766405	+	0.642358i	$0.222044\pi$
$398$	−29.3951	−1.47344
$399$	0	0
$400$	4.61803	0.230902
$401$	−7.41641	−0.370358	−0.185179	−	0.982705i	$-0.559286\pi$
$401$	−7.41641	−0.370358	−0.185179	+	0.982705i	$0.559286\pi$
$402$	0	0
$403$	−10.0000	−0.498135
$404$	20.3951	1.01470
$405$	0	0
$406$	−13.4164	−0.665845
$407$	0	0
$408$	0	0
$409$	−22.1803	−1.09675	−0.548374	−	0.836233i	$-0.684753\pi$
$409$	−22.1803	−1.09675	−0.548374	+	0.836233i	$0.684753\pi$
$410$	10.2016	0.503822
$411$	0	0
$412$	3.97871	0.196017
$413$	−3.23607	−0.159236
$414$	0	0
$415$	−10.1803	−0.499733
$416$	21.7082	1.06433
$417$	0	0
$418$	0	0
$419$	3.23607	0.158092	0.0790461	−	0.996871i	$-0.474813\pi$
$419$	3.23607	0.158092	0.0790461	+	0.996871i	$0.474813\pi$
$420$	0	0
$421$	9.79837	0.477544	0.238772	−	0.971076i	$-0.423255\pi$
$421$	9.79837	0.477544	0.238772	+	0.971076i	$0.423255\pi$
$422$	41.0557	1.99856
$423$	0	0
$424$	15.1246	0.734516
$425$	22.4164	1.08736
$426$	0	0
$427$	0	0
$428$	−47.3951	−2.29093
$429$	0	0
$430$	−13.4164	−0.646997
$431$	−15.0902	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$431$	−15.0902	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$432$	0	0
$433$	−31.2361	−1.50111	−0.750555	−	0.660808i	$-0.770214\pi$
$433$	−31.2361	−1.50111	−0.750555	+	0.660808i	$0.770214\pi$
$434$	6.90983	0.331682
$435$	0	0
$436$	−52.6869	−2.52325
$437$	−12.5066	−0.598271
$438$	0	0
$439$	1.85410	0.0884915	0.0442457	−	0.999021i	$-0.485912\pi$
$439$	1.85410	0.0884915	0.0442457	+	0.999021i	$0.485912\pi$
$440$	0	0
$441$	0	0
$442$	35.1246	1.67071
$443$	−20.3820	−0.968376	−0.484188	−	0.874964i	$-0.660885\pi$
$443$	−20.3820	−0.968376	−0.484188	+	0.874964i	$0.660885\pi$
$444$	0	0
$445$	−3.47214	−0.164595
$446$	−43.8673	−2.07717
$447$	0	0
$448$	−13.0000	−0.614192
$449$	41.8885	1.97684	0.988421	−	0.151734i	$-0.0484858\pi$
$449$	41.8885	1.97684	0.988421	+	0.151734i	$0.0484858\pi$
$450$	0	0
$451$	0	0
$452$	15.7082	0.738852
$453$	0	0
$454$	27.8885	1.30887
$455$	2.00000	0.0937614
$456$	0	0
$457$	21.4164	1.00182	0.500909	−	0.865500i	$-0.332999\pi$
$457$	21.4164	1.00182	0.500909	+	0.865500i	$0.332999\pi$
$458$	−36.1803	−1.69060
$459$	0	0
$460$	−8.12461	−0.378812
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−23.4164	−1.08825	−0.544126	−	0.839003i	$-0.683139\pi$
$463$	−23.4164	−1.08825	−0.544126	+	0.839003i	$0.683139\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−35.1246	−1.62712
$467$	−26.1803	−1.21148	−0.605741	−	0.795662i	$-0.707123\pi$
$467$	−26.1803	−1.21148	−0.605741	+	0.795662i	$0.707123\pi$
$468$	0	0
$469$	0	0
$470$	6.18034	0.285078
$471$	0	0
$472$	7.23607	0.333067
$473$	0	0
$474$	0	0
$475$	−13.1803	−0.604755
$476$	−14.5623	−0.667462
$477$	0	0
$478$	−42.5623	−1.94675
$479$	0.944272	0.0431449	0.0215724	−	0.999767i	$-0.493133\pi$
$479$	0.944272	0.0431449	0.0215724	+	0.999767i	$0.493133\pi$
$480$	0	0
$481$	14.9443	0.681400
$482$	−26.8328	−1.22220
$483$	0	0
$484$	0	0
$485$	−3.70820	−0.168381
$486$	0	0
$487$	27.7082	1.25558	0.627789	−	0.778383i	$-0.283960\pi$
$487$	27.7082	1.25558	0.627789	+	0.778383i	$0.283960\pi$
$488$	0	0
$489$	0	0
$490$	−1.38197	−0.0624309
$491$	−1.50658	−0.0679909	−0.0339955	−	0.999422i	$-0.510823\pi$
$491$	−1.50658	−0.0679909	−0.0339955	+	0.999422i	$0.510823\pi$
$492$	0	0
$493$	−29.1246	−1.31171
$494$	−20.6525	−0.929199
$495$	0	0
$496$	3.09017	0.138753
$497$	4.94427	0.221781
$498$	0	0
$499$	21.1246	0.945668	0.472834	−	0.881152i	$-0.343231\pi$
$499$	21.1246	0.945668	0.472834	+	0.881152i	$0.343231\pi$
$500$	−17.8328	−0.797508
$501$	0	0
$502$	61.9574	2.76530
$503$	14.4721	0.645281	0.322640	−	0.946522i	$-0.395430\pi$
$503$	14.4721	0.645281	0.322640	+	0.946522i	$0.395430\pi$
$504$	0	0
$505$	4.20163	0.186970
$506$	0	0
$507$	0	0
$508$	0.875388	0.0388391
$509$	27.0902	1.20075	0.600375	−	0.799718i	$-0.295018\pi$
$509$	27.0902	1.20075	0.600375	+	0.799718i	$0.295018\pi$
$510$	0	0
$511$	13.7082	0.606415
$512$	−11.1803	−0.494106
$513$	0	0
$514$	54.6738	2.41156
$515$	0.819660	0.0361185
$516$	0	0
$517$	0	0
$518$	−10.3262	−0.453709
$519$	0	0
$520$	−4.47214	−0.196116
$521$	12.3262	0.540022	0.270011	−	0.962857i	$-0.412973\pi$
$521$	12.3262	0.540022	0.270011	+	0.962857i	$0.412973\pi$
$522$	0	0
$523$	13.1459	0.574830	0.287415	−	0.957806i	$-0.407204\pi$
$523$	13.1459	0.574830	0.287415	+	0.957806i	$0.407204\pi$
$524$	49.9574	2.18240
$525$	0	0
$526$	24.2705	1.05824
$527$	15.0000	0.653410
$528$	0	0
$529$	−3.79837	−0.165147
$530$	9.34752	0.406031
$531$	0	0
$532$	8.56231	0.371223
$533$	−23.8885	−1.03473
$534$	0	0
$535$	−9.76393	−0.422132
$536$	0	0
$537$	0	0
$538$	−30.0000	−1.29339
$539$	0	0
$540$	0	0
$541$	−16.3262	−0.701920	−0.350960	−	0.936390i	$-0.614145\pi$
$541$	−16.3262	−0.701920	−0.350960	+	0.936390i	$0.614145\pi$
$542$	50.3262	2.16170
$543$	0	0
$544$	−32.5623	−1.39610
$545$	−10.8541	−0.464939
$546$	0	0
$547$	−36.4721	−1.55944	−0.779718	−	0.626131i	$-0.784638\pi$
$547$	−36.4721	−1.55944	−0.779718	+	0.626131i	$0.784638\pi$
$548$	−24.5410	−1.04834
$549$	0	0
$550$	0	0
$551$	17.1246	0.729533
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0.201626	0.00856627
$555$	0	0
$556$	39.4377	1.67253
$557$	−22.3607	−0.947452	−0.473726	−	0.880672i	$-0.657091\pi$
$557$	−22.3607	−0.947452	−0.473726	+	0.880672i	$0.657091\pi$
$558$	0	0
$559$	31.4164	1.32877
$560$	−0.618034	−0.0261167
$561$	0	0
$562$	−60.2492	−2.54146
$563$	9.52786	0.401552	0.200776	−	0.979637i	$-0.435654\pi$
$563$	9.52786	0.401552	0.200776	+	0.979637i	$0.435654\pi$
$564$	0	0
$565$	3.23607	0.136142
$566$	38.6180	1.62324
$567$	0	0
$568$	−11.0557	−0.463888
$569$	−11.5967	−0.486161	−0.243080	−	0.970006i	$-0.578158\pi$
$569$	−11.5967	−0.486161	−0.243080	+	0.970006i	$0.578158\pi$
$570$	0	0
$571$	15.3475	0.642274	0.321137	−	0.947033i	$-0.395935\pi$
$571$	15.3475	0.642274	0.321137	+	0.947033i	$0.395935\pi$
$572$	0	0
$573$	0	0
$574$	16.5066	0.688971
$575$	20.2361	0.843902
$576$	0	0
$577$	−35.7082	−1.48655	−0.743276	−	0.668985i	$-0.766729\pi$
$577$	−35.7082	−1.48655	−0.743276	+	0.668985i	$0.766729\pi$
$578$	−14.6738	−0.610348
$579$	0	0
$580$	11.1246	0.461924
$581$	−16.4721	−0.683379
$582$	0	0
$583$	0	0
$584$	−30.6525	−1.26841
$585$	0	0
$586$	−40.7295	−1.68252
$587$	1.41641	0.0584614	0.0292307	−	0.999573i	$-0.490694\pi$
$587$	1.41641	0.0584614	0.0292307	+	0.999573i	$0.490694\pi$
$588$	0	0
$589$	−8.81966	−0.363408
$590$	4.47214	0.184115
$591$	0	0
$592$	−4.61803	−0.189800
$593$	−22.5066	−0.924234	−0.462117	−	0.886819i	$-0.652910\pi$
$593$	−22.5066	−0.924234	−0.462117	+	0.886819i	$0.652910\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	14.8328	0.607576
$597$	0	0
$598$	31.7082	1.29664
$599$	23.6180	0.965007	0.482503	−	0.875894i	$-0.339728\pi$
$599$	23.6180	0.965007	0.482503	+	0.875894i	$0.339728\pi$
$600$	0	0
$601$	3.52786	0.143905	0.0719523	−	0.997408i	$-0.477077\pi$
$601$	3.52786	0.143905	0.0719523	+	0.997408i	$0.477077\pi$
$602$	−21.7082	−0.884760
$603$	0	0
$604$	−46.5836	−1.89546
$605$	0	0
$606$	0	0
$607$	−11.9787	−0.486201	−0.243100	−	0.970001i	$-0.578164\pi$
$607$	−11.9787	−0.486201	−0.243100	+	0.970001i	$0.578164\pi$
$608$	19.1459	0.776469
$609$	0	0
$610$	0	0
$611$	−14.4721	−0.585480
$612$	0	0
$613$	−21.9787	−0.887712	−0.443856	−	0.896098i	$-0.646390\pi$
$613$	−21.9787	−0.887712	−0.443856	+	0.896098i	$0.646390\pi$
$614$	39.2705	1.58483
$615$	0	0
$616$	0	0
$617$	14.0689	0.566392	0.283196	−	0.959062i	$-0.408605\pi$
$617$	14.0689	0.566392	0.283196	+	0.959062i	$0.408605\pi$
$618$	0	0
$619$	−36.6869	−1.47457	−0.737286	−	0.675581i	$-0.763893\pi$
$619$	−36.6869	−1.47457	−0.737286	+	0.675581i	$0.763893\pi$
$620$	−5.72949	−0.230102
$621$	0	0
$622$	46.1803	1.85166
$623$	−5.61803	−0.225082
$624$	0	0
$625$	19.4164	0.776656
$626$	17.4853	0.698853
$627$	0	0
$628$	−69.3738	−2.76832
$629$	−22.4164	−0.893801
$630$	0	0
$631$	−45.3050	−1.80356	−0.901781	−	0.432194i	$-0.857740\pi$
$631$	−45.3050	−1.80356	−0.901781	+	0.432194i	$0.857740\pi$
$632$	22.3607	0.889460
$633$	0	0
$634$	15.1246	0.600675
$635$	0.180340	0.00715657
$636$	0	0
$637$	3.23607	0.128218
$638$	0	0
$639$	0	0
$640$	9.67376	0.382389
$641$	−12.7639	−0.504145	−0.252073	−	0.967708i	$-0.581112\pi$
$641$	−12.7639	−0.504145	−0.252073	+	0.967708i	$0.581112\pi$
$642$	0	0
$643$	10.2705	0.405029	0.202515	−	0.979279i	$-0.435089\pi$
$643$	10.2705	0.405029	0.202515	+	0.979279i	$0.435089\pi$
$644$	−13.1459	−0.518021
$645$	0	0
$646$	30.9787	1.21884
$647$	−36.4721	−1.43387	−0.716934	−	0.697141i	$-0.754455\pi$
$647$	−36.4721	−1.43387	−0.716934	+	0.697141i	$0.754455\pi$
$648$	0	0
$649$	0	0
$650$	33.4164	1.31070
$651$	0	0
$652$	−30.0000	−1.17489
$653$	−16.3607	−0.640243	−0.320121	−	0.947377i	$-0.603724\pi$
$653$	−16.3607	−0.640243	−0.320121	+	0.947377i	$0.603724\pi$
$654$	0	0
$655$	10.2918	0.402134
$656$	7.38197	0.288217
$657$	0	0
$658$	10.0000	0.389841
$659$	25.8541	1.00713	0.503566	−	0.863957i	$-0.332021\pi$
$659$	25.8541	1.00713	0.503566	+	0.863957i	$0.332021\pi$
$660$	0	0
$661$	−32.5410	−1.26570	−0.632849	−	0.774275i	$-0.718115\pi$
$661$	−32.5410	−1.26570	−0.632849	+	0.774275i	$0.718115\pi$
$662$	−38.9443	−1.51361
$663$	0	0
$664$	36.8328	1.42939
$665$	1.76393	0.0684023
$666$	0	0
$667$	−26.2918	−1.01802
$668$	63.7082	2.46494
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−44.8328	−1.72818	−0.864089	−	0.503339i	$-0.832105\pi$
$673$	−44.8328	−1.72818	−0.864089	+	0.503339i	$0.832105\pi$
$674$	−60.5755	−2.33328
$675$	0	0
$676$	−7.58359	−0.291677
$677$	−34.3607	−1.32059	−0.660294	−	0.751007i	$-0.729568\pi$
$677$	−34.3607	−1.32059	−0.660294	+	0.751007i	$0.729568\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	6.70820	0.257248
$681$	0	0
$682$	0	0
$683$	−26.7984	−1.02541	−0.512706	−	0.858564i	$-0.671357\pi$
$683$	−26.7984	−1.02541	−0.512706	+	0.858564i	$0.671357\pi$
$684$	0	0
$685$	−5.05573	−0.193169
$686$	−2.23607	−0.0853735
$687$	0	0
$688$	−9.70820	−0.370122
$689$	−21.8885	−0.833887
$690$	0	0
$691$	48.7426	1.85426	0.927129	−	0.374743i	$-0.122269\pi$
$691$	48.7426	1.85426	0.927129	+	0.374743i	$0.122269\pi$
$692$	40.8541	1.55304
$693$	0	0
$694$	−32.4377	−1.23132
$695$	8.12461	0.308184
$696$	0	0
$697$	35.8328	1.35726
$698$	48.5410	1.83730
$699$	0	0
$700$	−13.8541	−0.523636
$701$	−39.5967	−1.49555	−0.747774	−	0.663953i	$-0.768877\pi$
$701$	−39.5967	−1.49555	−0.747774	+	0.663953i	$0.768877\pi$
$702$	0	0
$703$	13.1803	0.497106
$704$	0	0
$705$	0	0
$706$	−40.2492	−1.51480
$707$	6.79837	0.255679
$708$	0	0
$709$	−8.03444	−0.301740	−0.150870	−	0.988554i	$-0.548207\pi$
$709$	−8.03444	−0.301740	−0.150870	+	0.988554i	$0.548207\pi$
$710$	−6.83282	−0.256431
$711$	0	0
$712$	12.5623	0.470792
$713$	13.5410	0.507115
$714$	0	0
$715$	0	0
$716$	44.3951	1.65912
$717$	0	0
$718$	14.3951	0.537221
$719$	−39.0132	−1.45495	−0.727473	−	0.686137i	$-0.759305\pi$
$719$	−39.0132	−1.45495	−0.727473	+	0.686137i	$0.759305\pi$
$720$	0	0
$721$	1.32624	0.0493917
$722$	24.2705	0.903255
$723$	0	0
$724$	−59.6656	−2.21746
$725$	−27.7082	−1.02906
$726$	0	0
$727$	−18.8541	−0.699260	−0.349630	−	0.936888i	$-0.613693\pi$
$727$	−18.8541	−0.699260	−0.349630	+	0.936888i	$0.613693\pi$
$728$	−7.23607	−0.268187
$729$	0	0
$730$	−18.9443	−0.701159
$731$	−47.1246	−1.74297
$732$	0	0
$733$	−18.0689	−0.667389	−0.333695	−	0.942681i	$-0.608295\pi$
$733$	−18.0689	−0.667389	−0.333695	+	0.942681i	$0.608295\pi$
$734$	−6.10333	−0.225278
$735$	0	0
$736$	−29.3951	−1.08352
$737$	0	0
$738$	0	0
$739$	41.1246	1.51279	0.756397	−	0.654113i	$-0.226958\pi$
$739$	41.1246	1.51279	0.756397	+	0.654113i	$0.226958\pi$
$740$	8.56231	0.314757
$741$	0	0
$742$	15.1246	0.555242
$743$	16.3262	0.598952	0.299476	−	0.954104i	$-0.403188\pi$
$743$	16.3262	0.598952	0.299476	+	0.954104i	$0.403188\pi$
$744$	0	0
$745$	3.05573	0.111953
$746$	−36.5066	−1.33660
$747$	0	0
$748$	0	0
$749$	−15.7984	−0.577260
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	4.47214	0.163082
$753$	0	0
$754$	−43.4164	−1.58113
$755$	−9.59675	−0.349261
$756$	0	0
$757$	10.5623	0.383894	0.191947	−	0.981405i	$-0.438520\pi$
$757$	10.5623	0.383894	0.191947	+	0.981405i	$0.438520\pi$
$758$	−48.1378	−1.74844
$759$	0	0
$760$	−3.94427	−0.143074
$761$	−1.41641	−0.0513447	−0.0256724	−	0.999670i	$-0.508173\pi$
$761$	−1.41641	−0.0513447	−0.0256724	+	0.999670i	$0.508173\pi$
$762$	0	0
$763$	−17.5623	−0.635798
$764$	−14.7295	−0.532894
$765$	0	0
$766$	−46.5836	−1.68313
$767$	−10.4721	−0.378127
$768$	0	0
$769$	1.05573	0.0380705	0.0190353	−	0.999819i	$-0.493941\pi$
$769$	1.05573	0.0380705	0.0190353	+	0.999819i	$0.493941\pi$
$770$	0	0
$771$	0	0
$772$	−65.5623	−2.35964
$773$	−8.47214	−0.304722	−0.152361	−	0.988325i	$-0.548688\pi$
$773$	−8.47214	−0.304722	−0.152361	+	0.988325i	$0.548688\pi$
$774$	0	0
$775$	14.2705	0.512612
$776$	13.4164	0.481621
$777$	0	0
$778$	34.0689	1.22143
$779$	−21.0689	−0.754871
$780$	0	0
$781$	0	0
$782$	−47.5623	−1.70082
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	−14.2918	−0.510096
$786$	0	0
$787$	−25.8541	−0.921599	−0.460800	−	0.887504i	$-0.652437\pi$
$787$	−25.8541	−0.921599	−0.460800	+	0.887504i	$0.652437\pi$
$788$	−67.4164	−2.40161
$789$	0	0
$790$	13.8197	0.491681
$791$	5.23607	0.186173
$792$	0	0
$793$	0	0
$794$	−68.2918	−2.42359
$795$	0	0
$796$	39.4377	1.39783
$797$	−41.5623	−1.47221	−0.736106	−	0.676866i	$-0.763338\pi$
$797$	−41.5623	−1.47221	−0.736106	+	0.676866i	$0.763338\pi$
$798$	0	0
$799$	21.7082	0.767981
$800$	−30.9787	−1.09526
$801$	0	0
$802$	16.5836	0.585587
$803$	0	0
$804$	0	0
$805$	−2.70820	−0.0954516
$806$	22.3607	0.787621
$807$	0	0
$808$	−15.2016	−0.534791
$809$	−7.23607	−0.254407	−0.127203	−	0.991877i	$-0.540600\pi$
$809$	−7.23607	−0.254407	−0.127203	+	0.991877i	$0.540600\pi$
$810$	0	0
$811$	8.58359	0.301411	0.150705	−	0.988579i	$-0.451846\pi$
$811$	8.58359	0.301411	0.150705	+	0.988579i	$0.451846\pi$
$812$	18.0000	0.631676
$813$	0	0
$814$	0	0
$815$	−6.18034	−0.216488
$816$	0	0
$817$	27.7082	0.969387
$818$	49.5967	1.73411
$819$	0	0
$820$	−13.6869	−0.477968
$821$	10.9443	0.381958	0.190979	−	0.981594i	$-0.438834\pi$
$821$	10.9443	0.381958	0.190979	+	0.981594i	$0.438834\pi$
$822$	0	0
$823$	4.83282	0.168461	0.0842307	−	0.996446i	$-0.473157\pi$
$823$	4.83282	0.168461	0.0842307	+	0.996446i	$0.473157\pi$
$824$	−2.96556	−0.103310
$825$	0	0
$826$	7.23607	0.251775
$827$	−29.8673	−1.03859	−0.519293	−	0.854596i	$-0.673805\pi$
$827$	−29.8673	−1.03859	−0.519293	+	0.854596i	$0.673805\pi$
$828$	0	0
$829$	−30.2492	−1.05060	−0.525299	−	0.850917i	$-0.676047\pi$
$829$	−30.2492	−1.05060	−0.525299	+	0.850917i	$0.676047\pi$
$830$	22.7639	0.790148
$831$	0	0
$832$	−42.0689	−1.45848
$833$	−4.85410	−0.168185
$834$	0	0
$835$	13.1246	0.454196
$836$	0	0
$837$	0	0
$838$	−7.23607	−0.249966
$839$	40.2492	1.38956	0.694779	−	0.719224i	$-0.255502\pi$
$839$	40.2492	1.38956	0.694779	+	0.719224i	$0.255502\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−21.9098	−0.755063
$843$	0	0
$844$	−55.0820	−1.89600
$845$	−1.56231	−0.0537450
$846$	0	0
$847$	0	0
$848$	6.76393	0.232274
$849$	0	0
$850$	−50.1246	−1.71926
$851$	−20.2361	−0.693683
$852$	0	0
$853$	−41.7082	−1.42806	−0.714031	−	0.700114i	$-0.753132\pi$
$853$	−41.7082	−1.42806	−0.714031	+	0.700114i	$0.753132\pi$
$854$	0	0
$855$	0	0
$856$	35.3262	1.20743
$857$	34.3607	1.17374	0.586869	−	0.809682i	$-0.300360\pi$
$857$	34.3607	1.17374	0.586869	+	0.809682i	$0.300360\pi$
$858$	0	0
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	18.0000	0.613795
$861$	0	0
$862$	33.7426	1.14928
$863$	12.3262	0.419590	0.209795	−	0.977745i	$-0.432720\pi$
$863$	12.3262	0.419590	0.209795	+	0.977745i	$0.432720\pi$
$864$	0	0
$865$	8.41641	0.286166
$866$	69.8460	2.37346
$867$	0	0
$868$	−9.27051	−0.314662
$869$	0	0
$870$	0	0
$871$	0	0
$872$	39.2705	1.32987
$873$	0	0
$874$	27.9656	0.945949
$875$	−5.94427	−0.200953
$876$	0	0
$877$	12.4721	0.421154	0.210577	−	0.977577i	$-0.432466\pi$
$877$	12.4721	0.421154	0.210577	+	0.977577i	$0.432466\pi$
$878$	−4.14590	−0.139917
$879$	0	0
$880$	0	0
$881$	−11.3262	−0.381591	−0.190795	−	0.981630i	$-0.561107\pi$
$881$	−11.3262	−0.381591	−0.190795	+	0.981630i	$0.561107\pi$
$882$	0	0
$883$	19.5967	0.659483	0.329742	−	0.944071i	$-0.393038\pi$
$883$	19.5967	0.659483	0.329742	+	0.944071i	$0.393038\pi$
$884$	−47.1246	−1.58497
$885$	0	0
$886$	45.5755	1.53114
$887$	−27.5967	−0.926608	−0.463304	−	0.886199i	$-0.653336\pi$
$887$	−27.5967	−0.926608	−0.463304	+	0.886199i	$0.653336\pi$
$888$	0	0
$889$	0.291796	0.00978653
$890$	7.76393	0.260248
$891$	0	0
$892$	58.8541	1.97058
$893$	−12.7639	−0.427129
$894$	0	0
$895$	9.14590	0.305714
$896$	15.6525	0.522913
$897$	0	0
$898$	−93.6656	−3.12566
$899$	−18.5410	−0.618378
$900$	0	0
$901$	32.8328	1.09382
$902$	0	0
$903$	0	0
$904$	−11.7082	−0.389409
$905$	−12.2918	−0.408593
$906$	0	0
$907$	16.2918	0.540960	0.270480	−	0.962726i	$-0.412818\pi$
$907$	16.2918	0.540960	0.270480	+	0.962726i	$0.412818\pi$
$908$	−37.4164	−1.24171
$909$	0	0
$910$	−4.47214	−0.148250
$911$	0.944272	0.0312851	0.0156426	−	0.999878i	$-0.495021\pi$
$911$	0.944272	0.0312851	0.0156426	+	0.999878i	$0.495021\pi$
$912$	0	0
$913$	0	0
$914$	−47.8885	−1.58401
$915$	0	0
$916$	48.5410	1.60384
$917$	16.6525	0.549913
$918$	0	0
$919$	−30.4721	−1.00518	−0.502592	−	0.864524i	$-0.667620\pi$
$919$	−30.4721	−1.00518	−0.502592	+	0.864524i	$0.667620\pi$
$920$	6.05573	0.199651
$921$	0	0
$922$	−13.4164	−0.441846
$923$	16.0000	0.526646
$924$	0	0
$925$	−21.3262	−0.701202
$926$	52.3607	1.72068
$927$	0	0
$928$	40.2492	1.32125
$929$	−23.7295	−0.778539	−0.389270	−	0.921124i	$-0.627273\pi$
$929$	−23.7295	−0.778539	−0.389270	+	0.921124i	$0.627273\pi$
$930$	0	0
$931$	2.85410	0.0935394
$932$	47.1246	1.54362
$933$	0	0
$934$	58.5410	1.91552
$935$	0	0
$936$	0	0
$937$	34.0689	1.11298	0.556491	−	0.830854i	$-0.312147\pi$
$937$	34.0689	1.11298	0.556491	+	0.830854i	$0.312147\pi$
$938$	0	0
$939$	0	0
$940$	−8.29180	−0.270449
$941$	0.493422	0.0160851	0.00804255	−	0.999968i	$-0.497440\pi$
$941$	0.493422	0.0160851	0.00804255	+	0.999968i	$0.497440\pi$
$942$	0	0
$943$	32.3475	1.05338
$944$	3.23607	0.105325
$945$	0	0
$946$	0	0
$947$	−13.8541	−0.450198	−0.225099	−	0.974336i	$-0.572271\pi$
$947$	−13.8541	−0.450198	−0.225099	+	0.974336i	$0.572271\pi$
$948$	0	0
$949$	44.3607	1.44001
$950$	29.4721	0.956202
$951$	0	0
$952$	10.8541	0.351783
$953$	−11.8885	−0.385108	−0.192554	−	0.981286i	$-0.561677\pi$
$953$	−11.8885	−0.385108	−0.192554	+	0.981286i	$0.561677\pi$
$954$	0	0
$955$	−3.03444	−0.0981922
$956$	57.1033	1.84685
$957$	0	0
$958$	−2.11146	−0.0682181
$959$	−8.18034	−0.264157
$960$	0	0
$961$	−21.4508	−0.691963
$962$	−33.4164	−1.07739
$963$	0	0
$964$	36.0000	1.15948
$965$	−13.5066	−0.434792
$966$	0	0
$967$	−3.63932	−0.117033	−0.0585163	−	0.998286i	$-0.518637\pi$
$967$	−3.63932	−0.117033	−0.0585163	+	0.998286i	$0.518637\pi$
$968$	0	0
$969$	0	0
$970$	8.29180	0.266234
$971$	−54.5410	−1.75030	−0.875152	−	0.483848i	$-0.839239\pi$
$971$	−54.5410	−1.75030	−0.875152	+	0.483848i	$0.839239\pi$
$972$	0	0
$973$	13.1459	0.421438
$974$	−61.9574	−1.98524
$975$	0	0
$976$	0	0
$977$	13.6393	0.436361	0.218180	−	0.975908i	$-0.429988\pi$
$977$	13.6393	0.436361	0.218180	+	0.975908i	$0.429988\pi$
$978$	0	0
$979$	0	0
$980$	1.85410	0.0592271
$981$	0	0
$982$	3.36881	0.107503
$983$	10.9443	0.349068	0.174534	−	0.984651i	$-0.444158\pi$
$983$	10.9443	0.349068	0.174534	+	0.984651i	$0.444158\pi$
$984$	0	0
$985$	−13.8885	−0.442526
$986$	65.1246	2.07399
$987$	0	0
$988$	27.7082	0.881515
$989$	−42.5410	−1.35273
$990$	0	0
$991$	−33.2361	−1.05578	−0.527889	−	0.849313i	$-0.677016\pi$
$991$	−33.2361	−1.05578	−0.527889	+	0.849313i	$0.677016\pi$
$992$	−20.7295	−0.658162
$993$	0	0
$994$	−11.0557	−0.350666
$995$	8.12461	0.257568
$996$	0	0
$997$	−44.4296	−1.40710	−0.703549	−	0.710647i	$-0.748403\pi$
$997$	−44.4296	−1.40710	−0.703549	+	0.710647i	$0.748403\pi$
$998$	−47.2361	−1.49523
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.bk.1.1		2
3.2	odd	2		2541.2.a.w.1.2		2
11.3	even	5		693.2.m.a.64.1		4
11.4	even	5		693.2.m.a.379.1		4
11.10	odd	2		7623.2.a.bj.1.2		2
33.14	odd	10		231.2.j.e.64.1	✓	4
33.26	odd	10		231.2.j.e.148.1	yes	4
33.32	even	2		2541.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.e.64.1	✓	4	33.14	odd	10
231.2.j.e.148.1	yes	4	33.26	odd	10
693.2.m.a.64.1		4	11.3	even	5
693.2.m.a.379.1		4	11.4	even	5
2541.2.a.v.1.1		2	33.32	even	2
2541.2.a.w.1.2		2	3.2	odd	2
7623.2.a.bj.1.2		2	11.10	odd	2
7623.2.a.bk.1.1		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)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} +3.00000 q^{4} +0.618034 q^{5} +1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q-2.23607 q^{2} +3.00000 q^{4} +0.618034 q^{5} +1.00000 q^{7} -2.23607 q^{8} -1.38197 q^{10} +3.23607 q^{13} -2.23607 q^{14} -1.00000 q^{16} -4.85410 q^{17} +2.85410 q^{19} +1.85410 q^{20} -4.38197 q^{23} -4.61803 q^{25} -7.23607 q^{26} +3.00000 q^{28} +6.00000 q^{29} -3.09017 q^{31} +6.70820 q^{32} +10.8541 q^{34} +0.618034 q^{35} +4.61803 q^{37} -6.38197 q^{38} -1.38197 q^{40} -7.38197 q^{41} +9.70820 q^{43} +9.79837 q^{46} -4.47214 q^{47} +1.00000 q^{49} +10.3262 q^{50} +9.70820 q^{52} -6.76393 q^{53} -2.23607 q^{56} -13.4164 q^{58} -3.23607 q^{59} +6.90983 q^{62} -13.0000 q^{64} +2.00000 q^{65} -14.5623 q^{68} -1.38197 q^{70} +4.94427 q^{71} +13.7082 q^{73} -10.3262 q^{74} +8.56231 q^{76} -10.0000 q^{79} -0.618034 q^{80} +16.5066 q^{82} -16.4721 q^{83} -3.00000 q^{85} -21.7082 q^{86} -5.61803 q^{89} +3.23607 q^{91} -13.1459 q^{92} +10.0000 q^{94} +1.76393 q^{95} -6.00000 q^{97} -2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} - q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 6 * q^4 - q^5 + 2 * q^7	\( 2 q + 6 q^{4} - q^{5} + 2 q^{7} - 5 q^{10} + 2 q^{13} - 2 q^{16} - 3 q^{17} - q^{19} - 3 q^{20} - 11 q^{23} - 7 q^{25} - 10 q^{26} + 6 q^{28} + 12 q^{29} + 5 q^{31} + 15 q^{34} - q^{35} + 7 q^{37} - 15 q^{38} - 5 q^{40} - 17 q^{41} + 6 q^{43} - 5 q^{46} + 2 q^{49} + 5 q^{50} + 6 q^{52} - 18 q^{53} - 2 q^{59} + 25 q^{62} - 26 q^{64} + 4 q^{65} - 9 q^{68} - 5 q^{70} - 8 q^{71} + 14 q^{73} - 5 q^{74} - 3 q^{76} - 20 q^{79} + q^{80} - 5 q^{82} - 24 q^{83} - 6 q^{85} - 30 q^{86} - 9 q^{89} + 2 q^{91} - 33 q^{92} + 20 q^{94} + 8 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q + 6 * q^4 - q^5 + 2 * q^7 - 5 * q^10 + 2 * q^13 - 2 * q^16 - 3 * q^17 - q^19 - 3 * q^20 - 11 * q^23 - 7 * q^25 - 10 * q^26 + 6 * q^28 + 12 * q^29 + 5 * q^31 + 15 * q^34 - q^35 + 7 * q^37 - 15 * q^38 - 5 * q^40 - 17 * q^41 + 6 * q^43 - 5 * q^46 + 2 * q^49 + 5 * q^50 + 6 * q^52 - 18 * q^53 - 2 * q^59 + 25 * q^62 - 26 * q^64 + 4 * q^65 - 9 * q^68 - 5 * q^70 - 8 * q^71 + 14 * q^73 - 5 * q^74 - 3 * q^76 - 20 * q^79 + q^80 - 5 * q^82 - 24 * q^83 - 6 * q^85 - 30 * q^86 - 9 * q^89 + 2 * q^91 - 33 * q^92 + 20 * q^94 + 8 * q^95 - 12 * q^97

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