LMFDB - Embedded newform 7623.2.a.ce.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7623,2,Mod(1,7623)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [3,2,0,8,-1,0,-3,6,0,4,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
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			$-0.363328$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.36333 q^{2} -0.141336 q^{4} -3.14134 q^{5} -1.00000 q^{7} -2.91934 q^{8} -4.28267 q^{10} -4.77801 q^{13} -1.36333 q^{14} -3.69735 q^{16} +4.77801 q^{17} -7.00933 q^{19} +0.443984 q^{20} -5.14134 q^{23} +4.86799 q^{25} -6.51399 q^{26} +0.141336 q^{28} -7.00933 q^{29} -3.63667 q^{31} +0.797984 q^{32} +6.51399 q^{34} +3.14134 q^{35} -9.86799 q^{37} -9.55602 q^{38} +9.17064 q^{40} +3.22199 q^{41} +4.28267 q^{43} -7.00933 q^{46} +0.778008 q^{47} +1.00000 q^{49} +6.63667 q^{50} +0.675305 q^{52} -2.28267 q^{53} +2.91934 q^{56} -9.55602 q^{58} +0.363328 q^{59} +3.22199 q^{61} -4.95798 q^{62} +8.48262 q^{64} +15.0093 q^{65} -6.59465 q^{67} -0.675305 q^{68} +4.28267 q^{70} +15.1600 q^{71} -3.22199 q^{73} -13.4533 q^{74} +0.990671 q^{76} -3.71733 q^{79} +11.6146 q^{80} +4.39263 q^{82} -1.55602 q^{83} -15.0093 q^{85} +5.83869 q^{86} +5.58532 q^{89} +4.77801 q^{91} +0.726656 q^{92} +1.06068 q^{94} +22.0187 q^{95} +6.15066 q^{97} +1.36333 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 8 q^{4} - q^{5} - 3 q^{7} + 6 q^{8} + 4 q^{10} - 8 q^{13} - 2 q^{14} + 10 q^{16} + 8 q^{17} + 14 q^{20} - 7 q^{23} + 2 q^{25} + 12 q^{26} - 8 q^{28} - 13 q^{31} + 34 q^{32} - 12 q^{34}+ \cdots + 2 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 8 * q^4 - q^5 - 3 * q^7 + 6 * q^8 + 4 * q^10 - 8 * q^13 - 2 * q^14 + 10 * q^16 + 8 * q^17 + 14 * q^20 - 7 * q^23 + 2 * q^25 + 12 * q^26 - 8 * q^28 - 13 * q^31 + 34 * q^32 - 12 * q^34 + q^35 - 17 * q^37 - 16 * q^38 + 36 * q^40 + 16 * q^41 - 4 * q^43 - 4 * q^47 + 3 * q^49 + 22 * q^50 + 10 * q^53 - 6 * q^56 - 16 * q^58 - q^59 + 16 * q^61 + 4 * q^62 + 34 * q^64 + 24 * q^65 - 3 * q^67 - 4 * q^70 - 5 * q^71 - 16 * q^73 - 32 * q^74 + 24 * q^76 - 28 * q^79 + 56 * q^80 + 28 * q^82 + 8 * q^83 - 24 * q^85 - 12 * q^86 + 21 * q^89 + 8 * q^91 - 2 * q^92 - 20 * q^94 + 24 * q^95 - 11 * q^97 + 2 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.36333	0.964019	0.482009	−	0.876166i	$-0.339907\pi$
$2$	1.36333	0.964019	0.482009	+	0.876166i	$0.339907\pi$
$3$	0	0
$3$	0	0
$4$	−0.141336	−0.0706681
$5$	−3.14134	−1.40485	−0.702424	−	0.711759i	$-0.747899\pi$
$5$	−3.14134	−1.40485	−0.702424	+	0.711759i	$0.747899\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.91934	−1.03214
$9$	0	0
$10$	−4.28267	−1.35430
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.77801	−1.32518	−0.662590	−	0.748982i	$-0.730543\pi$
$13$	−4.77801	−1.32518	−0.662590	+	0.748982i	$0.730543\pi$
$14$	−1.36333	−0.364365
$15$	0	0
$16$	−3.69735	−0.924338
$17$	4.77801	1.15884	0.579419	−	0.815030i	$-0.303280\pi$
$17$	4.77801	1.15884	0.579419	+	0.815030i	$0.303280\pi$
$18$	0	0
$19$	−7.00933	−1.60805	−0.804025	−	0.594595i	$-0.797312\pi$
$19$	−7.00933	−1.60805	−0.804025	+	0.594595i	$0.797312\pi$
$20$	0.443984	0.0992779
$21$	0	0
$22$	0	0
$23$	−5.14134	−1.07204	−0.536021	−	0.844204i	$-0.680073\pi$
$23$	−5.14134	−1.07204	−0.536021	+	0.844204i	$0.680073\pi$
$24$	0	0
$25$	4.86799	0.973599
$26$	−6.51399	−1.27750
$27$	0	0
$28$	0.141336	0.0267100
$29$	−7.00933	−1.30160	−0.650800	−	0.759249i	$-0.725566\pi$
$29$	−7.00933	−1.30160	−0.650800	+	0.759249i	$0.725566\pi$
$30$	0	0
$31$	−3.63667	−0.653166	−0.326583	−	0.945169i	$-0.605897\pi$
$31$	−3.63667	−0.653166	−0.326583	+	0.945169i	$0.605897\pi$
$32$	0.797984	0.141065
$33$	0	0
$34$	6.51399	1.11714
$35$	3.14134	0.530983
$36$	0	0
$37$	−9.86799	−1.62229	−0.811144	−	0.584846i	$-0.801155\pi$
$37$	−9.86799	−1.62229	−0.811144	+	0.584846i	$0.801155\pi$
$38$	−9.55602	−1.55019
$39$	0	0
$40$	9.17064	1.45001
$41$	3.22199	0.503191	0.251595	−	0.967833i	$-0.419045\pi$
$41$	3.22199	0.503191	0.251595	+	0.967833i	$0.419045\pi$
$42$	0	0
$43$	4.28267	0.653101	0.326551	−	0.945180i	$-0.394114\pi$
$43$	4.28267	0.653101	0.326551	+	0.945180i	$0.394114\pi$
$44$	0	0
$45$	0	0
$46$	−7.00933	−1.03347
$47$	0.778008	0.113484	0.0567421	−	0.998389i	$-0.481929\pi$
$47$	0.778008	0.113484	0.0567421	+	0.998389i	$0.481929\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	6.63667	0.938567
$51$	0	0
$52$	0.675305	0.0936480
$53$	−2.28267	−0.313549	−0.156775	−	0.987634i	$-0.550110\pi$
$53$	−2.28267	−0.313549	−0.156775	+	0.987634i	$0.550110\pi$
$54$	0	0
$55$	0	0
$56$	2.91934	0.390114
$57$	0	0
$58$	−9.55602	−1.25477
$59$	0.363328	0.0473013	0.0236507	−	0.999720i	$-0.492471\pi$
$59$	0.363328	0.0473013	0.0236507	+	0.999720i	$0.492471\pi$
$60$	0	0
$61$	3.22199	0.412534	0.206267	−	0.978496i	$-0.433868\pi$
$61$	3.22199	0.412534	0.206267	+	0.978496i	$0.433868\pi$
$62$	−4.95798	−0.629664
$63$	0	0
$64$	8.48262	1.06033
$65$	15.0093	1.86168
$66$	0	0
$67$	−6.59465	−0.805665	−0.402832	−	0.915274i	$-0.631974\pi$
$67$	−6.59465	−0.805665	−0.402832	+	0.915274i	$0.631974\pi$
$68$	−0.675305	−0.0818928
$69$	0	0
$70$	4.28267	0.511877
$71$	15.1600	1.79916	0.899580	−	0.436756i	$-0.143873\pi$
$71$	15.1600	1.79916	0.899580	+	0.436756i	$0.143873\pi$
$72$	0	0
$73$	−3.22199	−0.377106	−0.188553	−	0.982063i	$-0.560380\pi$
$73$	−3.22199	−0.377106	−0.188553	+	0.982063i	$0.560380\pi$
$74$	−13.4533	−1.56392
$75$	0	0
$76$	0.990671	0.113638
$77$	0	0
$78$	0	0
$79$	−3.71733	−0.418232	−0.209116	−	0.977891i	$-0.567059\pi$
$79$	−3.71733	−0.418232	−0.209116	+	0.977891i	$0.567059\pi$
$80$	11.6146	1.29855
$81$	0	0
$82$	4.39263	0.485085
$83$	−1.55602	−0.170795	−0.0853975	−	0.996347i	$-0.527216\pi$
$83$	−1.55602	−0.170795	−0.0853975	+	0.996347i	$0.527216\pi$
$84$	0	0
$85$	−15.0093	−1.62799
$86$	5.83869	0.629602
$87$	0	0
$88$	0	0
$89$	5.58532	0.592043	0.296021	−	0.955181i	$-0.404340\pi$
$89$	5.58532	0.592043	0.296021	+	0.955181i	$0.404340\pi$
$90$	0	0
$91$	4.77801	0.500871
$92$	0.726656	0.0757592
$93$	0	0
$94$	1.06068	0.109401
$95$	22.0187	2.25907
$96$	0	0
$97$	6.15066	0.624505	0.312253	−	0.949999i	$-0.398916\pi$
$97$	6.15066	0.624505	0.312253	+	0.949999i	$0.398916\pi$
$98$	1.36333	0.137717
$99$	0	0
$100$	−0.688023	−0.0688023
$101$	10.7967	1.07431	0.537154	−	0.843484i	$-0.319499\pi$
$101$	10.7967	1.07431	0.537154	+	0.843484i	$0.319499\pi$
$102$	0	0
$103$	−13.2406	−1.30464	−0.652320	−	0.757944i	$-0.726204\pi$
$103$	−13.2406	−1.30464	−0.652320	+	0.757944i	$0.726204\pi$
$104$	13.9486	1.36778
$105$	0	0
$106$	−3.11203	−0.302267
$107$	−14.0187	−1.35523	−0.677617	−	0.735415i	$-0.736987\pi$
$107$	−14.0187	−1.35523	−0.677617	+	0.735415i	$0.736987\pi$
$108$	0	0
$109$	−15.5747	−1.49178	−0.745892	−	0.666067i	$-0.767976\pi$
$109$	−15.5747	−1.49178	−0.745892	+	0.666067i	$0.767976\pi$
$110$	0	0
$111$	0	0
$112$	3.69735	0.349367
$113$	−4.13201	−0.388707	−0.194353	−	0.980932i	$-0.562261\pi$
$113$	−4.13201	−0.388707	−0.194353	+	0.980932i	$0.562261\pi$
$114$	0	0
$115$	16.1507	1.50606
$116$	0.990671	0.0919815
$117$	0	0
$118$	0.495336	0.0455993
$119$	−4.77801	−0.437999
$120$	0	0
$121$	0	0
$122$	4.39263	0.397690
$123$	0	0
$124$	0.513993	0.0461579
$125$	0.414680	0.0370901
$126$	0	0
$127$	22.0187	1.95384	0.976920	−	0.213606i	$-0.0685208\pi$
$127$	22.0187	1.95384	0.976920	+	0.213606i	$0.0685208\pi$
$128$	9.96862	0.881110
$129$	0	0
$130$	20.4626	1.79469
$131$	−15.0093	−1.31137	−0.655686	−	0.755034i	$-0.727620\pi$
$131$	−15.0093	−1.31137	−0.655686	+	0.755034i	$0.727620\pi$
$132$	0	0
$133$	7.00933	0.607786
$134$	−8.99067	−0.776676
$135$	0	0
$136$	−13.9486	−1.19609
$137$	10.6974	0.913936	0.456968	−	0.889483i	$-0.348935\pi$
$137$	10.6974	0.913936	0.456968	+	0.889483i	$0.348935\pi$
$138$	0	0
$139$	4.10270	0.347987	0.173993	−	0.984747i	$-0.444333\pi$
$139$	4.10270	0.347987	0.173993	+	0.984747i	$0.444333\pi$
$140$	−0.443984	−0.0375235
$141$	0	0
$142$	20.6680	1.73442
$143$	0	0
$144$	0	0
$145$	22.0187	1.82855
$146$	−4.39263	−0.363537
$147$	0	0
$148$	1.39470	0.114644
$149$	−14.0187	−1.14845	−0.574227	−	0.818696i	$-0.694697\pi$
$149$	−14.0187	−1.14845	−0.574227	+	0.818696i	$0.694697\pi$
$150$	0	0
$151$	−6.82936	−0.555765	−0.277883	−	0.960615i	$-0.589633\pi$
$151$	−6.82936	−0.555765	−0.277883	+	0.960615i	$0.589633\pi$
$152$	20.4626	1.65974
$153$	0	0
$154$	0	0
$155$	11.4240	0.917598
$156$	0	0
$157$	−23.4427	−1.87093	−0.935464	−	0.353421i	$-0.885018\pi$
$157$	−23.4427	−1.87093	−0.935464	+	0.353421i	$0.885018\pi$
$158$	−5.06794	−0.403183
$159$	0	0
$160$	−2.50674	−0.198175
$161$	5.14134	0.405194
$162$	0	0
$163$	−0.990671	−0.0775954	−0.0387977	−	0.999247i	$-0.512353\pi$
$163$	−0.990671	−0.0775954	−0.0387977	+	0.999247i	$0.512353\pi$
$164$	−0.455384	−0.0355595
$165$	0	0
$166$	−2.12136	−0.164649
$167$	−0.565344	−0.0437477	−0.0218738	−	0.999761i	$-0.506963\pi$
$167$	−0.565344	−0.0437477	−0.0218738	+	0.999761i	$0.506963\pi$
$168$	0	0
$169$	9.82936	0.756105
$170$	−20.4626	−1.56941
$171$	0	0
$172$	−0.605296	−0.0461534
$173$	17.2406	1.31078	0.655391	−	0.755290i	$-0.272504\pi$
$173$	17.2406	1.31078	0.655391	+	0.755290i	$0.272504\pi$
$174$	0	0
$175$	−4.86799	−0.367986
$176$	0	0
$177$	0	0
$178$	7.61462	0.570740
$179$	−11.3213	−0.846194	−0.423097	−	0.906084i	$-0.639057\pi$
$179$	−11.3213	−0.846194	−0.423097	+	0.906084i	$0.639057\pi$
$180$	0	0
$181$	14.8773	1.10582	0.552911	−	0.833240i	$-0.313517\pi$
$181$	14.8773	1.10582	0.552911	+	0.833240i	$0.313517\pi$
$182$	6.51399	0.482849
$183$	0	0
$184$	15.0093	1.10650
$185$	30.9987	2.27907
$186$	0	0
$187$	0	0
$188$	−0.109961	−0.00801970
$189$	0	0
$190$	30.0187	2.17778
$191$	−3.84934	−0.278528	−0.139264	−	0.990255i	$-0.544474\pi$
$191$	−3.84934	−0.278528	−0.139264	+	0.990255i	$0.544474\pi$
$192$	0	0
$193$	−2.54669	−0.183315	−0.0916573	−	0.995791i	$-0.529216\pi$
$193$	−2.54669	−0.183315	−0.0916573	+	0.995791i	$0.529216\pi$
$194$	8.38538	0.602035
$195$	0	0
$196$	−0.141336	−0.0100954
$197$	10.5467	0.751420	0.375710	−	0.926737i	$-0.377399\pi$
$197$	10.5467	0.751420	0.375710	+	0.926737i	$0.377399\pi$
$198$	0	0
$199$	11.6846	0.828302	0.414151	−	0.910208i	$-0.364079\pi$
$199$	11.6846	0.828302	0.414151	+	0.910208i	$0.364079\pi$
$200$	−14.2113	−1.00489
$201$	0	0
$202$	14.7194	1.03565
$203$	7.00933	0.491958
$204$	0	0
$205$	−10.1214	−0.706906
$206$	−18.0514	−1.25770
$207$	0	0
$208$	17.6660	1.22492
$209$	0	0
$210$	0	0
$211$	11.1120	0.764984	0.382492	−	0.923959i	$-0.375066\pi$
$211$	11.1120	0.764984	0.382492	+	0.923959i	$0.375066\pi$
$212$	0.322624	0.0221579
$213$	0	0
$214$	−19.1120	−1.30647
$215$	−13.4533	−0.917508
$216$	0	0
$217$	3.63667	0.246873
$218$	−21.2334	−1.43811
$219$	0	0
$220$	0	0
$221$	−22.8294	−1.53567
$222$	0	0
$223$	28.0407	1.87774	0.938872	−	0.344266i	$-0.111872\pi$
$223$	28.0407	1.87774	0.938872	+	0.344266i	$0.111872\pi$
$224$	−0.797984	−0.0533175
$225$	0	0
$226$	−5.63328	−0.374720
$227$	23.0093	1.52718	0.763591	−	0.645700i	$-0.223434\pi$
$227$	23.0093	1.52718	0.763591	+	0.645700i	$0.223434\pi$
$228$	0	0
$229$	13.4240	0.887083	0.443542	−	0.896254i	$-0.353722\pi$
$229$	13.4240	0.887083	0.443542	+	0.896254i	$0.353722\pi$
$230$	22.0187	1.45187
$231$	0	0
$232$	20.4626	1.34344
$233$	3.53736	0.231740	0.115870	−	0.993264i	$-0.463034\pi$
$233$	3.53736	0.231740	0.115870	+	0.993264i	$0.463034\pi$
$234$	0	0
$235$	−2.44398	−0.159428
$236$	−0.0513514	−0.00334269
$237$	0	0
$238$	−6.51399	−0.422239
$239$	−22.0187	−1.42427	−0.712134	−	0.702043i	$-0.752271\pi$
$239$	−22.0187	−1.42427	−0.712134	+	0.702043i	$0.752271\pi$
$240$	0	0
$241$	−0.315366	−0.0203145	−0.0101573	−	0.999948i	$-0.503233\pi$
$241$	−0.315366	−0.0203145	−0.0101573	+	0.999948i	$0.503233\pi$
$242$	0	0
$243$	0	0
$244$	−0.455384	−0.0291530
$245$	−3.14134	−0.200693
$246$	0	0
$247$	33.4906	2.13096
$248$	10.6167	0.674161
$249$	0	0
$250$	0.565344	0.0357555
$251$	−18.6460	−1.17693	−0.588463	−	0.808524i	$-0.700267\pi$
$251$	−18.6460	−1.17693	−0.588463	+	0.808524i	$0.700267\pi$
$252$	0	0
$253$	0	0
$254$	30.0187	1.88354
$255$	0	0
$256$	−3.37473	−0.210920
$257$	8.54669	0.533128	0.266564	−	0.963817i	$-0.414112\pi$
$257$	8.54669	0.533128	0.266564	+	0.963817i	$0.414112\pi$
$258$	0	0
$259$	9.86799	0.613167
$260$	−2.12136	−0.131561
$261$	0	0
$262$	−20.4626	−1.26419
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	7.17064	0.440489
$266$	9.55602	0.585917
$267$	0	0
$268$	0.932062	0.0569348
$269$	−7.55602	−0.460698	−0.230349	−	0.973108i	$-0.573987\pi$
$269$	−7.55602	−0.460698	−0.230349	+	0.973108i	$0.573987\pi$
$270$	0	0
$271$	14.0187	0.851573	0.425786	−	0.904824i	$-0.359998\pi$
$271$	14.0187	0.851573	0.425786	+	0.904824i	$0.359998\pi$
$272$	−17.6660	−1.07116
$273$	0	0
$274$	14.5840	0.881052
$275$	0	0
$276$	0	0
$277$	−6.01866	−0.361626	−0.180813	−	0.983517i	$-0.557873\pi$
$277$	−6.01866	−0.361626	−0.180813	+	0.983517i	$0.557873\pi$
$278$	5.59333	0.335466
$279$	0	0
$280$	−9.17064	−0.548051
$281$	20.6680	1.23295	0.616476	−	0.787374i	$-0.288560\pi$
$281$	20.6680	1.23295	0.616476	+	0.787374i	$0.288560\pi$
$282$	0	0
$283$	9.91595	0.589442	0.294721	−	0.955583i	$-0.404773\pi$
$283$	9.91595	0.589442	0.294721	+	0.955583i	$0.404773\pi$
$284$	−2.14265	−0.127143
$285$	0	0
$286$	0	0
$287$	−3.22199	−0.190188
$288$	0	0
$289$	5.82936	0.342903
$290$	30.0187	1.76276
$291$	0	0
$292$	0.455384	0.0266493
$293$	26.7967	1.56548	0.782739	−	0.622350i	$-0.213822\pi$
$293$	26.7967	1.56548	0.782739	+	0.622350i	$0.213822\pi$
$294$	0	0
$295$	−1.14134	−0.0664512
$296$	28.8081	1.67443
$297$	0	0
$298$	−19.1120	−1.10713
$299$	24.5653	1.42065
$300$	0	0
$301$	−4.28267	−0.246849
$302$	−9.31066	−0.535768
$303$	0	0
$304$	25.9160	1.48638
$305$	−10.1214	−0.579547
$306$	0	0
$307$	11.8973	0.679015	0.339507	−	0.940603i	$-0.389740\pi$
$307$	11.8973	0.679015	0.339507	+	0.940603i	$0.389740\pi$
$308$	0	0
$309$	0	0
$310$	15.5747	0.884582
$311$	21.2406	1.20445	0.602223	−	0.798328i	$-0.294282\pi$
$311$	21.2406	1.20445	0.602223	+	0.798328i	$0.294282\pi$
$312$	0	0
$313$	−12.6974	−0.717697	−0.358848	−	0.933396i	$-0.616830\pi$
$313$	−12.6974	−0.717697	−0.358848	+	0.933396i	$0.616830\pi$
$314$	−31.9600	−1.80361
$315$	0	0
$316$	0.525393	0.0295556
$317$	11.8680	0.666573	0.333286	−	0.942826i	$-0.391842\pi$
$317$	11.8680	0.666573	0.333286	+	0.942826i	$0.391842\pi$
$318$	0	0
$319$	0	0
$320$	−26.6468	−1.48960
$321$	0	0
$322$	7.00933	0.390615
$323$	−33.4906	−1.86347
$324$	0	0
$325$	−23.2593	−1.29019
$326$	−1.35061	−0.0748034
$327$	0	0
$328$	−9.40610	−0.519365
$329$	−0.778008	−0.0428930
$330$	0	0
$331$	−11.8867	−0.653349	−0.326675	−	0.945137i	$-0.605928\pi$
$331$	−11.8867	−0.653349	−0.326675	+	0.945137i	$0.605928\pi$
$332$	0.219921	0.0120697
$333$	0	0
$334$	−0.770750	−0.0421736
$335$	20.7160	1.13184
$336$	0	0
$337$	0.990671	0.0539653	0.0269827	−	0.999636i	$-0.491410\pi$
$337$	0.990671	0.0539653	0.0269827	+	0.999636i	$0.491410\pi$
$338$	13.4006	0.728899
$339$	0	0
$340$	2.12136	0.115047
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−12.5026	−0.674095
$345$	0	0
$346$	23.5047	1.26362
$347$	15.1893	0.815404	0.407702	−	0.913115i	$-0.366330\pi$
$347$	15.1893	0.815404	0.407702	+	0.913115i	$0.366330\pi$
$348$	0	0
$349$	−29.7033	−1.58998	−0.794990	−	0.606622i	$-0.792524\pi$
$349$	−29.7033	−1.58998	−0.794990	+	0.606622i	$0.792524\pi$
$350$	−6.63667	−0.354745
$351$	0	0
$352$	0	0
$353$	6.71601	0.357457	0.178729	−	0.983898i	$-0.442802\pi$
$353$	6.71601	0.357457	0.178729	+	0.983898i	$0.442802\pi$
$354$	0	0
$355$	−47.6226	−2.52755
$356$	−0.789407	−0.0418385
$357$	0	0
$358$	−15.4347	−0.815747
$359$	−17.7360	−0.936069	−0.468035	−	0.883710i	$-0.655038\pi$
$359$	−17.7360	−0.936069	−0.468035	+	0.883710i	$0.655038\pi$
$360$	0	0
$361$	30.1307	1.58583
$362$	20.2827	1.06603
$363$	0	0
$364$	−0.675305	−0.0353956
$365$	10.1214	0.529776
$366$	0	0
$367$	−8.09931	−0.422781	−0.211390	−	0.977402i	$-0.567799\pi$
$367$	−8.09931	−0.422781	−0.211390	+	0.977402i	$0.567799\pi$
$368$	19.0093	0.990930
$369$	0	0
$370$	42.2614	2.19706
$371$	2.28267	0.118510
$372$	0	0
$373$	0.565344	0.0292724	0.0146362	−	0.999893i	$-0.495341\pi$
$373$	0.565344	0.0292724	0.0146362	+	0.999893i	$0.495341\pi$
$374$	0	0
$375$	0	0
$376$	−2.27127	−0.117132
$377$	33.4906	1.72486
$378$	0	0
$379$	−22.5360	−1.15760	−0.578799	−	0.815470i	$-0.696479\pi$
$379$	−22.5360	−1.15760	−0.578799	+	0.815470i	$0.696479\pi$
$380$	−3.11203	−0.159644
$381$	0	0
$382$	−5.24791	−0.268506
$383$	−21.3913	−1.09305	−0.546523	−	0.837444i	$-0.684049\pi$
$383$	−21.3913	−1.09305	−0.546523	+	0.837444i	$0.684049\pi$
$384$	0	0
$385$	0	0
$386$	−3.47197	−0.176719
$387$	0	0
$388$	−0.869311	−0.0441326
$389$	−10.6787	−0.541431	−0.270716	−	0.962659i	$-0.587260\pi$
$389$	−10.6787	−0.541431	−0.270716	+	0.962659i	$0.587260\pi$
$390$	0	0
$391$	−24.5653	−1.24232
$392$	−2.91934	−0.147449
$393$	0	0
$394$	14.3786	0.724383
$395$	11.6774	0.587553
$396$	0	0
$397$	−23.9160	−1.20031	−0.600154	−	0.799885i	$-0.704894\pi$
$397$	−23.9160	−1.20031	−0.600154	+	0.799885i	$0.704894\pi$
$398$	15.9300	0.798498
$399$	0	0
$400$	−17.9987	−0.899934
$401$	−11.6587	−0.582209	−0.291104	−	0.956691i	$-0.594023\pi$
$401$	−11.6587	−0.582209	−0.291104	+	0.956691i	$0.594023\pi$
$402$	0	0
$403$	17.3760	0.865563
$404$	−1.52596	−0.0759193
$405$	0	0
$406$	9.55602	0.474257
$407$	0	0
$408$	0	0
$409$	32.8153	1.62261	0.811307	−	0.584621i	$-0.198757\pi$
$409$	32.8153	1.62261	0.811307	+	0.584621i	$0.198757\pi$
$410$	−13.7987	−0.681471
$411$	0	0
$412$	1.87138	0.0921964
$413$	−0.363328	−0.0178782
$414$	0	0
$415$	4.88797	0.239941
$416$	−3.81277	−0.186937
$417$	0	0
$418$	0	0
$419$	−10.7967	−0.527452	−0.263726	−	0.964598i	$-0.584951\pi$
$419$	−10.7967	−0.527452	−0.263726	+	0.964598i	$0.584951\pi$
$420$	0	0
$421$	−1.15198	−0.0561442	−0.0280721	−	0.999606i	$-0.508937\pi$
$421$	−1.15198	−0.0561442	−0.0280721	+	0.999606i	$0.508937\pi$
$422$	15.1493	0.737459
$423$	0	0
$424$	6.66391	0.323628
$425$	23.2593	1.12824
$426$	0	0
$427$	−3.22199	−0.155923
$428$	1.98134	0.0957718
$429$	0	0
$430$	−18.3413	−0.884495
$431$	31.9600	1.53946	0.769731	−	0.638369i	$-0.220391\pi$
$431$	31.9600	1.53946	0.769731	+	0.638369i	$0.220391\pi$
$432$	0	0
$433$	−4.43334	−0.213053	−0.106526	−	0.994310i	$-0.533973\pi$
$433$	−4.43334	−0.213053	−0.106526	+	0.994310i	$0.533973\pi$
$434$	4.95798	0.237991
$435$	0	0
$436$	2.20126	0.105421
$437$	36.0373	1.72390
$438$	0	0
$439$	−27.4720	−1.31117	−0.655583	−	0.755123i	$-0.727577\pi$
$439$	−27.4720	−1.31117	−0.655583	+	0.755123i	$0.727577\pi$
$440$	0	0
$441$	0	0
$442$	−31.1239	−1.48041
$443$	7.79073	0.370149	0.185074	−	0.982725i	$-0.440747\pi$
$443$	7.79073	0.370149	0.185074	+	0.982725i	$0.440747\pi$
$444$	0	0
$445$	−17.5454	−0.831730
$446$	38.2287	1.81018
$447$	0	0
$448$	−8.48262	−0.400766
$449$	−20.5106	−0.967955	−0.483978	−	0.875080i	$-0.660808\pi$
$449$	−20.5106	−0.967955	−0.483978	+	0.875080i	$0.660808\pi$
$450$	0	0
$451$	0	0
$452$	0.584002	0.0274691
$453$	0	0
$454$	31.3693	1.47223
$455$	−15.0093	−0.703648
$456$	0	0
$457$	−8.56534	−0.400670	−0.200335	−	0.979727i	$-0.564203\pi$
$457$	−8.56534	−0.400670	−0.200335	+	0.979727i	$0.564203\pi$
$458$	18.3013	0.855165
$459$	0	0
$460$	−2.28267	−0.106430
$461$	−9.66598	−0.450189	−0.225095	−	0.974337i	$-0.572269\pi$
$461$	−9.66598	−0.450189	−0.225095	+	0.974337i	$0.572269\pi$
$462$	0	0
$463$	11.4240	0.530919	0.265459	−	0.964122i	$-0.414476\pi$
$463$	11.4240	0.530919	0.265459	+	0.964122i	$0.414476\pi$
$464$	25.9160	1.20312
$465$	0	0
$466$	4.82258	0.223402
$467$	−18.2207	−0.843152	−0.421576	−	0.906793i	$-0.638523\pi$
$467$	−18.2207	−0.843152	−0.421576	+	0.906793i	$0.638523\pi$
$468$	0	0
$469$	6.59465	0.304513
$470$	−3.33195	−0.153692
$471$	0	0
$472$	−1.06068	−0.0488218
$473$	0	0
$474$	0	0
$475$	−34.1214	−1.56560
$476$	0.675305	0.0309526
$477$	0	0
$478$	−30.0187	−1.37302
$479$	−3.47197	−0.158638	−0.0793192	−	0.996849i	$-0.525275\pi$
$479$	−3.47197	−0.158638	−0.0793192	+	0.996849i	$0.525275\pi$
$480$	0	0
$481$	47.1493	2.14983
$482$	−0.429948	−0.0195836
$483$	0	0
$484$	0	0
$485$	−19.3213	−0.877335
$486$	0	0
$487$	−27.4613	−1.24439	−0.622196	−	0.782862i	$-0.713759\pi$
$487$	−27.4613	−1.24439	−0.622196	+	0.782862i	$0.713759\pi$
$488$	−9.40610	−0.425794
$489$	0	0
$490$	−4.28267	−0.193471
$491$	6.26401	0.282691	0.141346	−	0.989960i	$-0.454857\pi$
$491$	6.26401	0.282691	0.141346	+	0.989960i	$0.454857\pi$
$492$	0	0
$493$	−33.4906	−1.50834
$494$	45.6587	2.05428
$495$	0	0
$496$	13.4461	0.603746
$497$	−15.1600	−0.680019
$498$	0	0
$499$	−20.6680	−0.925229	−0.462614	−	0.886560i	$-0.653089\pi$
$499$	−20.6680	−0.925229	−0.462614	+	0.886560i	$0.653089\pi$
$500$	−0.0586092	−0.00262108
$501$	0	0
$502$	−25.4206	−1.13458
$503$	−22.0187	−0.981763	−0.490882	−	0.871226i	$-0.663325\pi$
$503$	−22.0187	−0.981763	−0.490882	+	0.871226i	$0.663325\pi$
$504$	0	0
$505$	−33.9160	−1.50924
$506$	0	0
$507$	0	0
$508$	−3.11203	−0.138074
$509$	−21.4240	−0.949602	−0.474801	−	0.880093i	$-0.657480\pi$
$509$	−21.4240	−0.949602	−0.474801	+	0.880093i	$0.657480\pi$
$510$	0	0
$511$	3.22199	0.142533
$512$	−24.5381	−1.08444
$513$	0	0
$514$	11.6519	0.513945
$515$	41.5933	1.83282
$516$	0	0
$517$	0	0
$518$	13.4533	0.591105
$519$	0	0
$520$	−43.8174	−1.92152
$521$	−13.9453	−0.610953	−0.305476	−	0.952200i	$-0.598816\pi$
$521$	−13.9453	−0.610953	−0.305476	+	0.952200i	$0.598816\pi$
$522$	0	0
$523$	−18.4813	−0.808131	−0.404065	−	0.914730i	$-0.632403\pi$
$523$	−18.4813	−0.808131	−0.404065	+	0.914730i	$0.632403\pi$
$524$	2.12136	0.0926721
$525$	0	0
$526$	−21.8133	−0.951103
$527$	−17.3760	−0.756912
$528$	0	0
$529$	3.43334	0.149276
$530$	9.77594	0.424640
$531$	0	0
$532$	−0.990671	−0.0429510
$533$	−15.3947	−0.666819
$534$	0	0
$535$	44.0373	1.90390
$536$	19.2520	0.831562
$537$	0	0
$538$	−10.3013	−0.444122
$539$	0	0
$540$	0	0
$541$	37.4533	1.61024	0.805122	−	0.593109i	$-0.202100\pi$
$541$	37.4533	1.61024	0.805122	+	0.593109i	$0.202100\pi$
$542$	19.1120	0.820932
$543$	0	0
$544$	3.81277	0.163471
$545$	48.9253	2.09573
$546$	0	0
$547$	−17.1307	−0.732455	−0.366228	−	0.930525i	$-0.619351\pi$
$547$	−17.1307	−0.732455	−0.366228	+	0.930525i	$0.619351\pi$
$548$	−1.51192	−0.0645861
$549$	0	0
$550$	0	0
$551$	49.1307	2.09304
$552$	0	0
$553$	3.71733	0.158077
$554$	−8.20541	−0.348614
$555$	0	0
$556$	−0.579860	−0.0245915
$557$	−24.5653	−1.04087	−0.520434	−	0.853902i	$-0.674230\pi$
$557$	−24.5653	−1.04087	−0.520434	+	0.853902i	$0.674230\pi$
$558$	0	0
$559$	−20.4626	−0.865478
$560$	−11.6146	−0.490807
$561$	0	0
$562$	28.1773	1.18859
$563$	15.0093	0.632568	0.316284	−	0.948665i	$-0.397565\pi$
$563$	15.0093	0.632568	0.316284	+	0.948665i	$0.397565\pi$
$564$	0	0
$565$	12.9800	0.546074
$566$	13.5187	0.568233
$567$	0	0
$568$	−44.2572	−1.85699
$569$	−12.1027	−0.507372	−0.253686	−	0.967287i	$-0.581643\pi$
$569$	−12.1027	−0.507372	−0.253686	+	0.967287i	$0.581643\pi$
$570$	0	0
$571$	38.9439	1.62975	0.814877	−	0.579634i	$-0.196805\pi$
$571$	38.9439	1.62975	0.814877	+	0.579634i	$0.196805\pi$
$572$	0	0
$573$	0	0
$574$	−4.39263	−0.183345
$575$	−25.0280	−1.04374
$576$	0	0
$577$	38.1507	1.58823	0.794116	−	0.607766i	$-0.207934\pi$
$577$	38.1507	1.58823	0.794116	+	0.607766i	$0.207934\pi$
$578$	7.94733	0.330565
$579$	0	0
$580$	−3.11203	−0.129220
$581$	1.55602	0.0645544
$582$	0	0
$583$	0	0
$584$	9.40610	0.389227
$585$	0	0
$586$	36.5327	1.50915
$587$	34.7967	1.43621	0.718106	−	0.695934i	$-0.245009\pi$
$587$	34.7967	1.43621	0.718106	+	0.695934i	$0.245009\pi$
$588$	0	0
$589$	25.4906	1.05032
$590$	−1.55602	−0.0640602
$591$	0	0
$592$	36.4854	1.49954
$593$	18.5913	0.763452	0.381726	−	0.924276i	$-0.375330\pi$
$593$	18.5913	0.763452	0.381726	+	0.924276i	$0.375330\pi$
$594$	0	0
$595$	15.0093	0.615322
$596$	1.98134	0.0811590
$597$	0	0
$598$	33.4906	1.36953
$599$	−24.6680	−1.00791	−0.503955	−	0.863730i	$-0.668122\pi$
$599$	−24.6680	−1.00791	−0.503955	+	0.863730i	$0.668122\pi$
$600$	0	0
$601$	−30.1286	−1.22897	−0.614486	−	0.788928i	$-0.710637\pi$
$601$	−30.1286	−1.22897	−0.614486	+	0.788928i	$0.710637\pi$
$602$	−5.83869	−0.237967
$603$	0	0
$604$	0.965235	0.0392749
$605$	0	0
$606$	0	0
$607$	−13.6587	−0.554390	−0.277195	−	0.960814i	$-0.589405\pi$
$607$	−13.6587	−0.554390	−0.277195	+	0.960814i	$0.589405\pi$
$608$	−5.59333	−0.226840
$609$	0	0
$610$	−13.7987	−0.558694
$611$	−3.71733	−0.150387
$612$	0	0
$613$	−45.0280	−1.81866	−0.909332	−	0.416072i	$-0.863406\pi$
$613$	−45.0280	−1.81866	−0.909332	+	0.416072i	$0.863406\pi$
$614$	16.2199	0.654583
$615$	0	0
$616$	0	0
$617$	8.26401	0.332697	0.166348	−	0.986067i	$-0.446802\pi$
$617$	8.26401	0.332697	0.166348	+	0.986067i	$0.446802\pi$
$618$	0	0
$619$	43.0500	1.73033	0.865163	−	0.501490i	$-0.167215\pi$
$619$	43.0500	1.73033	0.865163	+	0.501490i	$0.167215\pi$
$620$	−1.61462	−0.0648449
$621$	0	0
$622$	28.9580	1.16111
$623$	−5.58532	−0.223771
$624$	0	0
$625$	−25.6426	−1.02570
$626$	−17.3107	−0.691873
$627$	0	0
$628$	3.31330	0.132215
$629$	−47.1493	−1.87997
$630$	0	0
$631$	1.03863	0.0413473	0.0206737	−	0.999786i	$-0.493419\pi$
$631$	1.03863	0.0413473	0.0206737	+	0.999786i	$0.493419\pi$
$632$	10.8522	0.431676
$633$	0	0
$634$	16.1800	0.642589
$635$	−69.1680	−2.74485
$636$	0	0
$637$	−4.77801	−0.189312
$638$	0	0
$639$	0	0
$640$	−31.3148	−1.23783
$641$	−33.6413	−1.32875	−0.664376	−	0.747399i	$-0.731302\pi$
$641$	−33.6413	−1.32875	−0.664376	+	0.747399i	$0.731302\pi$
$642$	0	0
$643$	−21.0314	−0.829396	−0.414698	−	0.909959i	$-0.636113\pi$
$643$	−21.0314	−0.829396	−0.414698	+	0.909959i	$0.636113\pi$
$644$	−0.726656	−0.0286343
$645$	0	0
$646$	−45.6587	−1.79642
$647$	−25.2300	−0.991894	−0.495947	−	0.868353i	$-0.665179\pi$
$647$	−25.2300	−0.991894	−0.495947	+	0.868353i	$0.665179\pi$
$648$	0	0
$649$	0	0
$650$	−31.7101	−1.24377
$651$	0	0
$652$	0.140018	0.00548351
$653$	−20.6387	−0.807656	−0.403828	−	0.914835i	$-0.632321\pi$
$653$	−20.6387	−0.807656	−0.403828	+	0.914835i	$0.632321\pi$
$654$	0	0
$655$	47.1493	1.84228
$656$	−11.9128	−0.465118
$657$	0	0
$658$	−1.06068	−0.0413496
$659$	18.5067	0.720920	0.360460	−	0.932775i	$-0.382620\pi$
$659$	18.5067	0.720920	0.360460	+	0.932775i	$0.382620\pi$
$660$	0	0
$661$	16.8001	0.653446	0.326723	−	0.945120i	$-0.394056\pi$
$661$	16.8001	0.653446	0.326723	+	0.945120i	$0.394056\pi$
$662$	−16.2054	−0.629841
$663$	0	0
$664$	4.54255	0.176285
$665$	−22.0187	−0.853847
$666$	0	0
$667$	36.0373	1.39537
$668$	0.0799036	0.00309156
$669$	0	0
$670$	28.2427	1.09111
$671$	0	0
$672$	0	0
$673$	17.5560	0.676735	0.338367	−	0.941014i	$-0.390125\pi$
$673$	17.5560	0.676735	0.338367	+	0.941014i	$0.390125\pi$
$674$	1.35061	0.0520236
$675$	0	0
$676$	−1.38924	−0.0534324
$677$	3.64732	0.140178	0.0700889	−	0.997541i	$-0.477672\pi$
$677$	3.64732	0.140178	0.0700889	+	0.997541i	$0.477672\pi$
$678$	0	0
$679$	−6.15066	−0.236041
$680$	43.8174	1.68032
$681$	0	0
$682$	0	0
$683$	−27.4720	−1.05119	−0.525593	−	0.850736i	$-0.676156\pi$
$683$	−27.4720	−1.05119	−0.525593	+	0.850736i	$0.676156\pi$
$684$	0	0
$685$	−33.6040	−1.28394
$686$	−1.36333	−0.0520521
$687$	0	0
$688$	−15.8345	−0.603686
$689$	10.9066	0.415509
$690$	0	0
$691$	−5.95666	−0.226602	−0.113301	−	0.993561i	$-0.536142\pi$
$691$	−5.95666	−0.226602	−0.113301	+	0.993561i	$0.536142\pi$
$692$	−2.43673	−0.0926304
$693$	0	0
$694$	20.7080	0.786065
$695$	−12.8880	−0.488869
$696$	0	0
$697$	15.3947	0.583116
$698$	−40.4953	−1.53277
$699$	0	0
$700$	0.688023	0.0260048
$701$	−7.36927	−0.278333	−0.139167	−	0.990269i	$-0.544442\pi$
$701$	−7.36927	−0.278333	−0.139167	+	0.990269i	$0.544442\pi$
$702$	0	0
$703$	69.1680	2.60872
$704$	0	0
$705$	0	0
$706$	9.15613	0.344595
$707$	−10.7967	−0.406050
$708$	0	0
$709$	−34.7160	−1.30379	−0.651894	−	0.758310i	$-0.726025\pi$
$709$	−34.7160	−1.30379	−0.651894	+	0.758310i	$0.726025\pi$
$710$	−64.9253	−2.43660
$711$	0	0
$712$	−16.3055	−0.611073
$713$	18.6974	0.700221
$714$	0	0
$715$	0	0
$716$	1.60011	0.0597989
$717$	0	0
$718$	−24.1800	−0.902388
$719$	4.06200	0.151487	0.0757435	−	0.997127i	$-0.475867\pi$
$719$	4.06200	0.151487	0.0757435	+	0.997127i	$0.475867\pi$
$720$	0	0
$721$	13.2406	0.493108
$722$	41.0780	1.52877
$723$	0	0
$724$	−2.10270	−0.0781463
$725$	−34.1214	−1.26724
$726$	0	0
$727$	−30.5433	−1.13279	−0.566394	−	0.824135i	$-0.691662\pi$
$727$	−30.5433	−1.13279	−0.566394	+	0.824135i	$0.691662\pi$
$728$	−13.9486	−0.516971
$729$	0	0
$730$	13.7987	0.510714
$731$	20.4626	0.756838
$732$	0	0
$733$	−30.8340	−1.13888	−0.569440	−	0.822033i	$-0.692840\pi$
$733$	−30.8340	−1.13888	−0.569440	+	0.822033i	$0.692840\pi$
$734$	−11.0420	−0.407568
$735$	0	0
$736$	−4.10270	−0.151228
$737$	0	0
$738$	0	0
$739$	22.2241	0.817525	0.408763	−	0.912641i	$-0.365960\pi$
$739$	22.2241	0.817525	0.408763	+	0.912641i	$0.365960\pi$
$740$	−4.38123	−0.161057
$741$	0	0
$742$	3.11203	0.114246
$743$	31.1493	1.14276	0.571379	−	0.820686i	$-0.306409\pi$
$743$	31.1493	1.14276	0.571379	+	0.820686i	$0.306409\pi$
$744$	0	0
$745$	44.0373	1.61340
$746$	0.770750	0.0282192
$747$	0	0
$748$	0	0
$749$	14.0187	0.512231
$750$	0	0
$751$	11.9894	0.437498	0.218749	−	0.975781i	$-0.429802\pi$
$751$	11.9894	0.437498	0.218749	+	0.975781i	$0.429802\pi$
$752$	−2.87657	−0.104898
$753$	0	0
$754$	45.6587	1.66279
$755$	21.4533	0.780766
$756$	0	0
$757$	−11.4533	−0.416278	−0.208139	−	0.978099i	$-0.566741\pi$
$757$	−11.4533	−0.416278	−0.208139	+	0.978099i	$0.566741\pi$
$758$	−30.7240	−1.11595
$759$	0	0
$760$	−64.2800	−2.33168
$761$	−7.89004	−0.286014	−0.143007	−	0.989722i	$-0.545677\pi$
$761$	−7.89004	−0.286014	−0.143007	+	0.989722i	$0.545677\pi$
$762$	0	0
$763$	15.5747	0.563841
$764$	0.544050	0.0196830
$765$	0	0
$766$	−29.1634	−1.05372
$767$	−1.73599	−0.0626828
$768$	0	0
$769$	−48.8153	−1.76033	−0.880163	−	0.474672i	$-0.842567\pi$
$769$	−48.8153	−1.76033	−0.880163	+	0.474672i	$0.842567\pi$
$770$	0	0
$771$	0	0
$772$	0.359939	0.0129545
$773$	3.19608	0.114955	0.0574774	−	0.998347i	$-0.481694\pi$
$773$	3.19608	0.114955	0.0574774	+	0.998347i	$0.481694\pi$
$774$	0	0
$775$	−17.7033	−0.635921
$776$	−17.9559	−0.644579
$777$	0	0
$778$	−14.5586	−0.521950
$779$	−22.5840	−0.809156
$780$	0	0
$781$	0	0
$782$	−33.4906	−1.19762
$783$	0	0
$784$	−3.69735	−0.132048
$785$	73.6413	2.62837
$786$	0	0
$787$	−42.4813	−1.51429	−0.757147	−	0.653244i	$-0.773408\pi$
$787$	−42.4813	−1.51429	−0.757147	+	0.653244i	$0.773408\pi$
$788$	−1.49063	−0.0531014
$789$	0	0
$790$	15.9201	0.566412
$791$	4.13201	0.146917
$792$	0	0
$793$	−15.3947	−0.546682
$794$	−32.6053	−1.15712
$795$	0	0
$796$	−1.65146	−0.0585345
$797$	28.8587	1.02223	0.511113	−	0.859513i	$-0.329233\pi$
$797$	28.8587	1.02223	0.511113	+	0.859513i	$0.329233\pi$
$798$	0	0
$799$	3.71733	0.131510
$800$	3.88458	0.137341
$801$	0	0
$802$	−15.8947	−0.561260
$803$	0	0
$804$	0	0
$805$	−16.1507	−0.569236
$806$	23.6893	0.834418
$807$	0	0
$808$	−31.5192	−1.10884
$809$	29.4533	1.03552	0.517762	−	0.855525i	$-0.326765\pi$
$809$	29.4533	1.03552	0.517762	+	0.855525i	$0.326765\pi$
$810$	0	0
$811$	−2.48130	−0.0871302	−0.0435651	−	0.999051i	$-0.513872\pi$
$811$	−2.48130	−0.0871302	−0.0435651	+	0.999051i	$0.513872\pi$
$812$	−0.990671	−0.0347657
$813$	0	0
$814$	0	0
$815$	3.11203	0.109010
$816$	0	0
$817$	−30.0187	−1.05022
$818$	44.7381	1.56423
$819$	0	0
$820$	1.43051	0.0499557
$821$	14.0187	0.489255	0.244627	−	0.969617i	$-0.421334\pi$
$821$	14.0187	0.489255	0.244627	+	0.969617i	$0.421334\pi$
$822$	0	0
$823$	−42.7933	−1.49168	−0.745840	−	0.666125i	$-0.767952\pi$
$823$	−42.7933	−1.49168	−0.745840	+	0.666125i	$0.767952\pi$
$824$	38.6540	1.34658
$825$	0	0
$826$	−0.495336	−0.0172349
$827$	−9.17064	−0.318894	−0.159447	−	0.987206i	$-0.550971\pi$
$827$	−9.17064	−0.318894	−0.159447	+	0.987206i	$0.550971\pi$
$828$	0	0
$829$	29.1973	1.01406	0.507032	−	0.861927i	$-0.330743\pi$
$829$	29.1973	1.01406	0.507032	+	0.861927i	$0.330743\pi$
$830$	6.66391	0.231308
$831$	0	0
$832$	−40.5300	−1.40513
$833$	4.77801	0.165548
$834$	0	0
$835$	1.77594	0.0614588
$836$	0	0
$837$	0	0
$838$	−14.7194	−0.508473
$839$	−19.7326	−0.681245	−0.340622	−	0.940200i	$-0.610638\pi$
$839$	−19.7326	−0.681245	−0.340622	+	0.940200i	$0.610638\pi$
$840$	0	0
$841$	20.1307	0.694162
$842$	−1.57053	−0.0541241
$843$	0	0
$844$	−1.57053	−0.0540599
$845$	−30.8773	−1.06221
$846$	0	0
$847$	0	0
$848$	8.43984	0.289825
$849$	0	0
$850$	31.7101	1.08765
$851$	50.7347	1.73916
$852$	0	0
$853$	28.3527	0.970777	0.485389	−	0.874298i	$-0.338678\pi$
$853$	28.3527	0.970777	0.485389	+	0.874298i	$0.338678\pi$
$854$	−4.39263	−0.150313
$855$	0	0
$856$	40.9253	1.39880
$857$	13.2033	0.451017	0.225509	−	0.974241i	$-0.427596\pi$
$857$	13.2033	0.451017	0.225509	+	0.974241i	$0.427596\pi$
$858$	0	0
$859$	−20.8260	−0.710573	−0.355286	−	0.934757i	$-0.615617\pi$
$859$	−20.8260	−0.710573	−0.355286	+	0.934757i	$0.615617\pi$
$860$	1.90144	0.0648385
$861$	0	0
$862$	43.5720	1.48407
$863$	−34.9439	−1.18951	−0.594753	−	0.803909i	$-0.702750\pi$
$863$	−34.9439	−1.18951	−0.594753	+	0.803909i	$0.702750\pi$
$864$	0	0
$865$	−54.1587	−1.84145
$866$	−6.04409	−0.205387
$867$	0	0
$868$	−0.513993	−0.0174461
$869$	0	0
$870$	0	0
$871$	31.5093	1.06765
$872$	45.4678	1.53973
$873$	0	0
$874$	49.1307	1.66187
$875$	−0.414680	−0.0140187
$876$	0	0
$877$	26.4813	0.894210	0.447105	−	0.894481i	$-0.352455\pi$
$877$	26.4813	0.894210	0.447105	+	0.894481i	$0.352455\pi$
$878$	−37.4533	−1.26399
$879$	0	0
$880$	0	0
$881$	−17.2627	−0.581595	−0.290798	−	0.956785i	$-0.593921\pi$
$881$	−17.2627	−0.581595	−0.290798	+	0.956785i	$0.593921\pi$
$882$	0	0
$883$	−44.6027	−1.50100	−0.750499	−	0.660871i	$-0.770187\pi$
$883$	−44.6027	−1.50100	−0.750499	+	0.660871i	$0.770187\pi$
$884$	3.22661	0.108523
$885$	0	0
$886$	10.6213	0.356830
$887$	30.7894	1.03381	0.516904	−	0.856044i	$-0.327085\pi$
$887$	30.7894	1.03381	0.516904	+	0.856044i	$0.327085\pi$
$888$	0	0
$889$	−22.0187	−0.738482
$890$	−23.9201	−0.801803
$891$	0	0
$892$	−3.96316	−0.132697
$893$	−5.45331	−0.182488
$894$	0	0
$895$	35.5640	1.18877
$896$	−9.96862	−0.333028
$897$	0	0
$898$	−27.9627	−0.933127
$899$	25.4906	0.850160
$900$	0	0
$901$	−10.9066	−0.363352
$902$	0	0
$903$	0	0
$904$	12.0628	0.401201
$905$	−46.7347	−1.55351
$906$	0	0
$907$	23.7801	0.789605	0.394802	−	0.918766i	$-0.370813\pi$
$907$	23.7801	0.789605	0.394802	+	0.918766i	$0.370813\pi$
$908$	−3.25205	−0.107923
$909$	0	0
$910$	−20.4626	−0.678330
$911$	−49.0280	−1.62437	−0.812185	−	0.583400i	$-0.801722\pi$
$911$	−49.0280	−1.62437	−0.812185	+	0.583400i	$0.801722\pi$
$912$	0	0
$913$	0	0
$914$	−11.6774	−0.386253
$915$	0	0
$916$	−1.89730	−0.0626885
$917$	15.0093	0.495652
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	−47.1493	−1.55447
$921$	0	0
$922$	−13.1779	−0.433991
$923$	−72.4346	−2.38421
$924$	0	0
$925$	−48.0373	−1.57946
$926$	15.5747	0.511816
$927$	0	0
$928$	−5.59333	−0.183610
$929$	56.5840	1.85646	0.928230	−	0.372006i	$-0.121330\pi$
$929$	56.5840	1.85646	0.928230	+	0.372006i	$0.121330\pi$
$930$	0	0
$931$	−7.00933	−0.229721
$932$	−0.499956	−0.0163766
$933$	0	0
$934$	−24.8408	−0.812814
$935$	0	0
$936$	0	0
$937$	−16.8153	−0.549333	−0.274666	−	0.961540i	$-0.588567\pi$
$937$	−16.8153	−0.549333	−0.274666	+	0.961540i	$0.588567\pi$
$938$	8.99067	0.293556
$939$	0	0
$940$	0.345423	0.0112665
$941$	−7.25931	−0.236647	−0.118323	−	0.992975i	$-0.537752\pi$
$941$	−7.25931	−0.236647	−0.118323	+	0.992975i	$0.537752\pi$
$942$	0	0
$943$	−16.5653	−0.539442
$944$	−1.34335	−0.0437224
$945$	0	0
$946$	0	0
$947$	25.6040	0.832017	0.416009	−	0.909361i	$-0.363429\pi$
$947$	25.6040	0.832017	0.416009	+	0.909361i	$0.363429\pi$
$948$	0	0
$949$	15.3947	0.499733
$950$	−46.5186	−1.50926
$951$	0	0
$952$	13.9486	0.452078
$953$	−7.14935	−0.231590	−0.115795	−	0.993273i	$-0.536942\pi$
$953$	−7.14935	−0.231590	−0.115795	+	0.993273i	$0.536942\pi$
$954$	0	0
$955$	12.0921	0.391290
$956$	3.11203	0.100650
$957$	0	0
$958$	−4.73344	−0.152930
$959$	−10.6974	−0.345436
$960$	0	0
$961$	−17.7746	−0.573375
$962$	64.2800	2.07247
$963$	0	0
$964$	0.0445726	0.00143559
$965$	8.00000	0.257529
$966$	0	0
$967$	15.9600	0.513241	0.256620	−	0.966512i	$-0.417391\pi$
$967$	15.9600	0.513241	0.256620	+	0.966512i	$0.417391\pi$
$968$	0	0
$969$	0	0
$970$	−26.3413	−0.845768
$971$	30.4779	0.978083	0.489041	−	0.872261i	$-0.337347\pi$
$971$	30.4779	0.978083	0.489041	+	0.872261i	$0.337347\pi$
$972$	0	0
$973$	−4.10270	−0.131527
$974$	−37.4388	−1.19962
$975$	0	0
$976$	−11.9128	−0.381321
$977$	−4.13201	−0.132195	−0.0660973	−	0.997813i	$-0.521055\pi$
$977$	−4.13201	−0.132195	−0.0660973	+	0.997813i	$0.521055\pi$
$978$	0	0
$979$	0	0
$980$	0.443984	0.0141826
$981$	0	0
$982$	8.53991	0.272519
$983$	25.0246	0.798161	0.399080	−	0.916916i	$-0.369329\pi$
$983$	25.0246	0.798161	0.399080	+	0.916916i	$0.369329\pi$
$984$	0	0
$985$	−33.1307	−1.05563
$986$	−45.6587	−1.45407
$987$	0	0
$988$	−4.73344	−0.150591
$989$	−22.0187	−0.700153
$990$	0	0
$991$	−27.5747	−0.875938	−0.437969	−	0.898990i	$-0.644302\pi$
$991$	−27.5747	−0.875938	−0.437969	+	0.898990i	$0.644302\pi$
$992$	−2.90201	−0.0921388
$993$	0	0
$994$	−20.6680	−0.655551
$995$	−36.7054	−1.16364
$996$	0	0
$997$	23.6846	0.750100	0.375050	−	0.927005i	$-0.377626\pi$
$997$	23.6846	0.750100	0.375050	+	0.927005i	$0.377626\pi$
$998$	−28.1773	−0.891938
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ce.1.2		3
3.2	odd	2		847.2.a.i.1.2	✓	3
11.10	odd	2		7623.2.a.bz.1.2		3
21.20	even	2		5929.2.a.t.1.2		3
33.2	even	10		847.2.f.t.323.2		12
33.5	odd	10		847.2.f.u.729.2		12
33.8	even	10		847.2.f.t.372.2		12
33.14	odd	10		847.2.f.u.372.2		12
33.17	even	10		847.2.f.t.729.2		12
33.20	odd	10		847.2.f.u.323.2		12
33.26	odd	10		847.2.f.u.148.2		12
33.29	even	10		847.2.f.t.148.2		12
33.32	even	2		847.2.a.j.1.2	yes	3
231.230	odd	2		5929.2.a.y.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.i.1.2	✓	3	3.2	odd	2
847.2.a.j.1.2	yes	3	33.32	even	2
847.2.f.t.148.2		12	33.29	even	10
847.2.f.t.323.2		12	33.2	even	10
847.2.f.t.372.2		12	33.8	even	10
847.2.f.t.729.2		12	33.17	even	10
847.2.f.u.148.2		12	33.26	odd	10
847.2.f.u.323.2		12	33.20	odd	10
847.2.f.u.372.2		12	33.14	odd	10
847.2.f.u.729.2		12	33.5	odd	10
5929.2.a.t.1.2		3	21.20	even	2
5929.2.a.y.1.2		3	231.230	odd	2
7623.2.a.bz.1.2		3	11.10	odd	2
7623.2.a.ce.1.2		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+1.36333 q^{2} -0.141336 q^{4} -3.14134 q^{5} -1.00000 q^{7} -2.91934 q^{8} -4.28267 q^{10} -4.77801 q^{13} -1.36333 q^{14} -3.69735 q^{16} +4.77801 q^{17} -7.00933 q^{19} +0.443984 q^{20} -5.14134 q^{23} +4.86799 q^{25} -6.51399 q^{26} +0.141336 q^{28} -7.00933 q^{29} -3.63667 q^{31} +0.797984 q^{32} +6.51399 q^{34} +3.14134 q^{35} -9.86799 q^{37} -9.55602 q^{38} +9.17064 q^{40} +3.22199 q^{41} +4.28267 q^{43} -7.00933 q^{46} +0.778008 q^{47} +1.00000 q^{49} +6.63667 q^{50} +0.675305 q^{52} -2.28267 q^{53} +2.91934 q^{56} -9.55602 q^{58} +0.363328 q^{59} +3.22199 q^{61} -4.95798 q^{62} +8.48262 q^{64} +15.0093 q^{65} -6.59465 q^{67} -0.675305 q^{68} +4.28267 q^{70} +15.1600 q^{71} -3.22199 q^{73} -13.4533 q^{74} +0.990671 q^{76} -3.71733 q^{79} +11.6146 q^{80} +4.39263 q^{82} -1.55602 q^{83} -15.0093 q^{85} +5.83869 q^{86} +5.58532 q^{89} +4.77801 q^{91} +0.726656 q^{92} +1.06068 q^{94} +22.0187 q^{95} +6.15066 q^{97} +1.36333 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 8 q^{4} - q^{5} - 3 q^{7} + 6 q^{8} + 4 q^{10} - 8 q^{13} - 2 q^{14} + 10 q^{16} + 8 q^{17} + 14 q^{20} - 7 q^{23} + 2 q^{25} + 12 q^{26} - 8 q^{28} - 13 q^{31} + 34 q^{32} - 12 q^{34}+ \cdots + 2 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 8 * q^4 - q^5 - 3 * q^7 + 6 * q^8 + 4 * q^10 - 8 * q^13 - 2 * q^14 + 10 * q^16 + 8 * q^17 + 14 * q^20 - 7 * q^23 + 2 * q^25 + 12 * q^26 - 8 * q^28 - 13 * q^31 + 34 * q^32 - 12 * q^34 + q^35 - 17 * q^37 - 16 * q^38 + 36 * q^40 + 16 * q^41 - 4 * q^43 - 4 * q^47 + 3 * q^49 + 22 * q^50 + 10 * q^53 - 6 * q^56 - 16 * q^58 - q^59 + 16 * q^61 + 4 * q^62 + 34 * q^64 + 24 * q^65 - 3 * q^67 - 4 * q^70 - 5 * q^71 - 16 * q^73 - 32 * q^74 + 24 * q^76 - 28 * q^79 + 56 * q^80 + 28 * q^82 + 8 * q^83 - 24 * q^85 - 12 * q^86 + 21 * q^89 + 8 * q^91 - 2 * q^92 - 20 * q^94 + 24 * q^95 - 11 * q^97 + 2 * q^98

Introduction

L-functions

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