LMFDB - Embedded newform 7623.2.a.cy.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 $ x^10 - 19x^8 - x^7 + 124x^6 + 6x^5 - 316x^4 + 17x^3 + 253x^2 - 70*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.33330$ 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)

$f(q)$	$=$	$q-1.33330 q^{2} -0.222305 q^{4} +0.873210 q^{5} +1.00000 q^{7} +2.96300 q^{8} +O(q^{10})$	$q-1.33330 q^{2} -0.222305 q^{4} +0.873210 q^{5} +1.00000 q^{7} +2.96300 q^{8} -1.16425 q^{10} -0.0395440 q^{13} -1.33330 q^{14} -3.50597 q^{16} +2.04633 q^{17} +5.69421 q^{19} -0.194119 q^{20} -5.97870 q^{23} -4.23750 q^{25} +0.0527241 q^{26} -0.222305 q^{28} -4.67254 q^{29} +7.47891 q^{31} -1.25149 q^{32} -2.72838 q^{34} +0.873210 q^{35} +11.5704 q^{37} -7.59210 q^{38} +2.58733 q^{40} -6.14882 q^{41} +1.79689 q^{43} +7.97142 q^{46} -6.04773 q^{47} +1.00000 q^{49} +5.64987 q^{50} +0.00879083 q^{52} +0.124401 q^{53} +2.96300 q^{56} +6.22991 q^{58} +14.3460 q^{59} -7.55944 q^{61} -9.97165 q^{62} +8.68056 q^{64} -0.0345302 q^{65} +4.85938 q^{67} -0.454910 q^{68} -1.16425 q^{70} +11.2503 q^{71} -0.612954 q^{73} -15.4269 q^{74} -1.26585 q^{76} +0.155227 q^{79} -3.06145 q^{80} +8.19824 q^{82} +8.91599 q^{83} +1.78688 q^{85} -2.39580 q^{86} -5.17689 q^{89} -0.0395440 q^{91} +1.32910 q^{92} +8.06345 q^{94} +4.97224 q^{95} +7.94222 q^{97} -1.33330 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8}+O(q^{10}) $ 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8	$ 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8} - 6 q^{10} + 6 q^{13} + 38 q^{16} - 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} + 18 q^{28} + 14 q^{29} + 26 q^{31} + 41 q^{32} + 21 q^{34} - 5 q^{35} + 24 q^{37} - 8 q^{38} - 5 q^{40} - 19 q^{41} - 6 q^{43} - q^{46} - 15 q^{47} + 10 q^{49} + q^{50} - 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} - 11 q^{62} + 53 q^{64} + 29 q^{65} + 38 q^{67} - 87 q^{68} - 6 q^{70} - 26 q^{71} - q^{73} + 39 q^{74} - 2 q^{76} + 5 q^{79} - 6 q^{80} + 5 q^{82} - 6 q^{83} - q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} + 42 q^{95} + 24 q^{97}+O(q^{100}) $ 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8 - 6 * q^10 + 6 * q^13 + 38 * q^16 - 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 + 18 * q^28 + 14 * q^29 + 26 * q^31 + 41 * q^32 + 21 * q^34 - 5 * q^35 + 24 * q^37 - 8 * q^38 - 5 * q^40 - 19 * q^41 - 6 * q^43 - q^46 - 15 * q^47 + 10 * q^49 + q^50 - 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 - 11 * q^62 + 53 * q^64 + 29 * q^65 + 38 * q^67 - 87 * q^68 - 6 * q^70 - 26 * q^71 - q^73 + 39 * q^74 - 2 * q^76 + 5 * q^79 - 6 * q^80 + 5 * q^82 - 6 * q^83 - q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 + 42 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.33330	−0.942787	−0.471394	−	0.881923i	$-0.656249\pi$
$2$	−1.33330	−0.942787	−0.471394	+	0.881923i	$0.656249\pi$
$3$	0	0
$3$	0	0
$4$	−0.222305	−0.111152
$5$	0.873210	0.390512	0.195256	−	0.980752i	$-0.437446\pi$
$5$	0.873210	0.390512	0.195256	+	0.980752i	$0.437446\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.96300	1.04758
$9$	0	0
$10$	−1.16425	−0.368169
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.0395440	−0.0109675	−0.00548377	−	0.999985i	$-0.501746\pi$
$13$	−0.0395440	−0.0109675	−0.00548377	+	0.999985i	$0.501746\pi$
$14$	−1.33330	−0.356340
$15$	0	0
$16$	−3.50597	−0.876493
$17$	2.04633	0.496308	0.248154	−	0.968721i	$-0.420176\pi$
$17$	2.04633	0.496308	0.248154	+	0.968721i	$0.420176\pi$
$18$	0	0
$19$	5.69421	1.30634	0.653170	−	0.757211i	$-0.273439\pi$
$19$	5.69421	1.30634	0.653170	+	0.757211i	$0.273439\pi$
$20$	−0.194119	−0.0434063
$21$	0	0
$22$	0	0
$23$	−5.97870	−1.24665	−0.623323	−	0.781965i	$-0.714218\pi$
$23$	−5.97870	−1.24665	−0.623323	+	0.781965i	$0.714218\pi$
$24$	0	0
$25$	−4.23750	−0.847501
$26$	0.0527241	0.0103401
$27$	0	0
$28$	−0.222305	−0.0420117
$29$	−4.67254	−0.867669	−0.433834	−	0.900993i	$-0.642840\pi$
$29$	−4.67254	−0.867669	−0.433834	+	0.900993i	$0.642840\pi$
$30$	0	0
$31$	7.47891	1.34325	0.671626	−	0.740890i	$-0.265596\pi$
$31$	7.47891	1.34325	0.671626	+	0.740890i	$0.265596\pi$
$32$	−1.25149	−0.221234
$33$	0	0
$34$	−2.72838	−0.467913
$35$	0.873210	0.147600
$36$	0	0
$37$	11.5704	1.90217	0.951085	−	0.308930i	$-0.0999708\pi$
$37$	11.5704	1.90217	0.951085	+	0.308930i	$0.0999708\pi$
$38$	−7.59210	−1.23160
$39$	0	0
$40$	2.58733	0.409092
$41$	−6.14882	−0.960285	−0.480142	−	0.877191i	$-0.659415\pi$
$41$	−6.14882	−0.960285	−0.480142	+	0.877191i	$0.659415\pi$
$42$	0	0
$43$	1.79689	0.274024	0.137012	−	0.990569i	$-0.456250\pi$
$43$	1.79689	0.274024	0.137012	+	0.990569i	$0.456250\pi$
$44$	0	0
$45$	0	0
$46$	7.97142	1.17532
$47$	−6.04773	−0.882152	−0.441076	−	0.897470i	$-0.645403\pi$
$47$	−6.04773	−0.882152	−0.441076	+	0.897470i	$0.645403\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.64987	0.799013
$51$	0	0
$52$	0.00879083	0.00121907
$53$	0.124401	0.0170878	0.00854390	−	0.999964i	$-0.497280\pi$
$53$	0.124401	0.0170878	0.00854390	+	0.999964i	$0.497280\pi$
$54$	0	0
$55$	0	0
$56$	2.96300	0.395948
$57$	0	0
$58$	6.22991	0.818027
$59$	14.3460	1.86769	0.933845	−	0.357679i	$-0.116432\pi$
$59$	14.3460	1.86769	0.933845	+	0.357679i	$0.116432\pi$
$60$	0	0
$61$	−7.55944	−0.967887	−0.483944	−	0.875099i	$-0.660796\pi$
$61$	−7.55944	−0.967887	−0.483944	+	0.875099i	$0.660796\pi$
$62$	−9.97165	−1.26640
$63$	0	0
$64$	8.68056	1.08507
$65$	−0.0345302	−0.00428295
$66$	0	0
$67$	4.85938	0.593668	0.296834	−	0.954929i	$-0.404069\pi$
$67$	4.85938	0.593668	0.296834	+	0.954929i	$0.404069\pi$
$68$	−0.454910	−0.0551659
$69$	0	0
$70$	−1.16425	−0.139155
$71$	11.2503	1.33516	0.667582	−	0.744536i	$-0.267329\pi$
$71$	11.2503	1.33516	0.667582	+	0.744536i	$0.267329\pi$
$72$	0	0
$73$	−0.612954	−0.0717409	−0.0358704	−	0.999356i	$-0.511420\pi$
$73$	−0.612954	−0.0717409	−0.0358704	+	0.999356i	$0.511420\pi$
$74$	−15.4269	−1.79334
$75$	0	0
$76$	−1.26585	−0.145203
$77$	0	0
$78$	0	0
$79$	0.155227	0.0174644	0.00873220	−	0.999962i	$-0.497220\pi$
$79$	0.155227	0.0174644	0.00873220	+	0.999962i	$0.497220\pi$
$80$	−3.06145	−0.342281
$81$	0	0
$82$	8.19824	0.905344
$83$	8.91599	0.978657	0.489328	−	0.872100i	$-0.337242\pi$
$83$	8.91599	0.978657	0.489328	+	0.872100i	$0.337242\pi$
$84$	0	0
$85$	1.78688	0.193814
$86$	−2.39580	−0.258346
$87$	0	0
$88$	0	0
$89$	−5.17689	−0.548749	−0.274374	−	0.961623i	$-0.588471\pi$
$89$	−5.17689	−0.548749	−0.274374	+	0.961623i	$0.588471\pi$
$90$	0	0
$91$	−0.0395440	−0.00414534
$92$	1.32910	0.138568
$93$	0	0
$94$	8.06345	0.831682
$95$	4.97224	0.510141
$96$	0	0
$97$	7.94222	0.806410	0.403205	−	0.915110i	$-0.367896\pi$
$97$	7.94222	0.806410	0.403205	+	0.915110i	$0.367896\pi$
$98$	−1.33330	−0.134684
$99$	0	0
$100$	0.942018	0.0942018
$101$	−9.09560	−0.905046	−0.452523	−	0.891753i	$-0.649476\pi$
$101$	−9.09560	−0.905046	−0.452523	+	0.891753i	$0.649476\pi$
$102$	0	0
$103$	3.43750	0.338707	0.169354	−	0.985555i	$-0.445832\pi$
$103$	3.43750	0.338707	0.169354	+	0.985555i	$0.445832\pi$
$104$	−0.117169	−0.0114894
$105$	0	0
$106$	−0.165864	−0.0161102
$107$	3.29161	0.318212	0.159106	−	0.987262i	$-0.449139\pi$
$107$	3.29161	0.318212	0.159106	+	0.987262i	$0.449139\pi$
$108$	0	0
$109$	−9.99510	−0.957357	−0.478678	−	0.877990i	$-0.658884\pi$
$109$	−9.99510	−0.957357	−0.478678	+	0.877990i	$0.658884\pi$
$110$	0	0
$111$	0	0
$112$	−3.50597	−0.331283
$113$	−4.40261	−0.414163	−0.207081	−	0.978324i	$-0.566397\pi$
$113$	−4.40261	−0.414163	−0.207081	+	0.978324i	$0.566397\pi$
$114$	0	0
$115$	−5.22067	−0.486830
$116$	1.03873	0.0964435
$117$	0	0
$118$	−19.1275	−1.76083
$119$	2.04633	0.187587
$120$	0	0
$121$	0	0
$122$	10.0790	0.912512
$123$	0	0
$124$	−1.66260	−0.149306
$125$	−8.06628	−0.721470
$126$	0	0
$127$	−12.9823	−1.15200	−0.575998	−	0.817451i	$-0.695386\pi$
$127$	−12.9823	−1.15200	−0.575998	+	0.817451i	$0.695386\pi$
$128$	−9.07082	−0.801755
$129$	0	0
$130$	0.0460393	0.00403791
$131$	18.1255	1.58363	0.791817	−	0.610758i	$-0.209135\pi$
$131$	18.1255	1.58363	0.791817	+	0.610758i	$0.209135\pi$
$132$	0	0
$133$	5.69421	0.493750
$134$	−6.47902	−0.559702
$135$	0	0
$136$	6.06329	0.519923
$137$	−5.20699	−0.444863	−0.222431	−	0.974948i	$-0.571399\pi$
$137$	−5.20699	−0.444863	−0.222431	+	0.974948i	$0.571399\pi$
$138$	0	0
$139$	18.1482	1.53931	0.769657	−	0.638458i	$-0.220427\pi$
$139$	18.1482	1.53931	0.769657	+	0.638458i	$0.220427\pi$
$140$	−0.194119	−0.0164061
$141$	0	0
$142$	−15.0000	−1.25878
$143$	0	0
$144$	0	0
$145$	−4.08011	−0.338835
$146$	0.817253	0.0676364
$147$	0	0
$148$	−2.57217	−0.211431
$149$	11.0495	0.905212	0.452606	−	0.891711i	$-0.350494\pi$
$149$	11.0495	0.905212	0.452606	+	0.891711i	$0.350494\pi$
$150$	0	0
$151$	−15.9314	−1.29648	−0.648240	−	0.761436i	$-0.724495\pi$
$151$	−15.9314	−1.29648	−0.648240	+	0.761436i	$0.724495\pi$
$152$	16.8720	1.36850
$153$	0	0
$154$	0	0
$155$	6.53066	0.524555
$156$	0	0
$157$	16.0035	1.27722	0.638610	−	0.769531i	$-0.279510\pi$
$157$	16.0035	1.27722	0.638610	+	0.769531i	$0.279510\pi$
$158$	−0.206964	−0.0164652
$159$	0	0
$160$	−1.09281	−0.0863945
$161$	−5.97870	−0.471188
$162$	0	0
$163$	22.0034	1.72344	0.861721	−	0.507383i	$-0.169387\pi$
$163$	22.0034	1.72344	0.861721	+	0.507383i	$0.169387\pi$
$164$	1.36691	0.106738
$165$	0	0
$166$	−11.8877	−0.922665
$167$	−4.72101	−0.365323	−0.182661	−	0.983176i	$-0.558471\pi$
$167$	−4.72101	−0.365323	−0.182661	+	0.983176i	$0.558471\pi$
$168$	0	0
$169$	−12.9984	−0.999880
$170$	−2.38245	−0.182725
$171$	0	0
$172$	−0.399458	−0.0304584
$173$	−9.47572	−0.720426	−0.360213	−	0.932870i	$-0.617296\pi$
$173$	−9.47572	−0.720426	−0.360213	+	0.932870i	$0.617296\pi$
$174$	0	0
$175$	−4.23750	−0.320325
$176$	0	0
$177$	0	0
$178$	6.90235	0.517353
$179$	−17.7185	−1.32434	−0.662170	−	0.749353i	$-0.730364\pi$
$179$	−17.7185	−1.32434	−0.662170	+	0.749353i	$0.730364\pi$
$180$	0	0
$181$	16.2742	1.20965	0.604826	−	0.796358i	$-0.293243\pi$
$181$	16.2742	1.20965	0.604826	+	0.796358i	$0.293243\pi$
$182$	0.0527241	0.00390817
$183$	0	0
$184$	−17.7149	−1.30596
$185$	10.1034	0.742819
$186$	0	0
$187$	0	0
$188$	1.34444	0.0980534
$189$	0	0
$190$	−6.62950	−0.480954
$191$	−6.44604	−0.466419	−0.233210	−	0.972426i	$-0.574923\pi$
$191$	−6.44604	−0.466419	−0.233210	+	0.972426i	$0.574923\pi$
$192$	0	0
$193$	1.81074	0.130340	0.0651701	−	0.997874i	$-0.479241\pi$
$193$	1.81074	0.130340	0.0651701	+	0.997874i	$0.479241\pi$
$194$	−10.5894	−0.760273
$195$	0	0
$196$	−0.222305	−0.0158789
$197$	−6.85772	−0.488592	−0.244296	−	0.969701i	$-0.578557\pi$
$197$	−6.85772	−0.488592	−0.244296	+	0.969701i	$0.578557\pi$
$198$	0	0
$199$	−5.07400	−0.359687	−0.179843	−	0.983695i	$-0.557559\pi$
$199$	−5.07400	−0.359687	−0.179843	+	0.983695i	$0.557559\pi$
$200$	−12.5557	−0.887825
$201$	0	0
$202$	12.1272	0.853266
$203$	−4.67254	−0.327948
$204$	0	0
$205$	−5.36922	−0.375002
$206$	−4.58323	−0.319329
$207$	0	0
$208$	0.138640	0.00961296
$209$	0	0
$210$	0	0
$211$	−8.10688	−0.558101	−0.279050	−	0.960276i	$-0.590020\pi$
$211$	−8.10688	−0.558101	−0.279050	+	0.960276i	$0.590020\pi$
$212$	−0.0276550	−0.00189935
$213$	0	0
$214$	−4.38871	−0.300006
$215$	1.56907	0.107009
$216$	0	0
$217$	7.47891	0.507702
$218$	13.3265	0.902583
$219$	0	0
$220$	0	0
$221$	−0.0809201	−0.00544328
$222$	0	0
$223$	−16.2050	−1.08517	−0.542583	−	0.840002i	$-0.682554\pi$
$223$	−16.2050	−1.08517	−0.542583	+	0.840002i	$0.682554\pi$
$224$	−1.25149	−0.0836187
$225$	0	0
$226$	5.87001	0.390467
$227$	23.6742	1.57131	0.785655	−	0.618665i	$-0.212326\pi$
$227$	23.6742	1.57131	0.785655	+	0.618665i	$0.212326\pi$
$228$	0	0
$229$	10.1567	0.671176	0.335588	−	0.942009i	$-0.391065\pi$
$229$	10.1567	0.671176	0.335588	+	0.942009i	$0.391065\pi$
$230$	6.96073	0.458977
$231$	0	0
$232$	−13.8448	−0.908952
$233$	23.5158	1.54057	0.770287	−	0.637697i	$-0.220113\pi$
$233$	23.5158	1.54057	0.770287	+	0.637697i	$0.220113\pi$
$234$	0	0
$235$	−5.28094	−0.344491
$236$	−3.18919	−0.207598
$237$	0	0
$238$	−2.72838	−0.176854
$239$	7.82554	0.506192	0.253096	−	0.967441i	$-0.418551\pi$
$239$	7.82554	0.506192	0.253096	+	0.967441i	$0.418551\pi$
$240$	0	0
$241$	6.86139	0.441981	0.220991	−	0.975276i	$-0.429071\pi$
$241$	6.86139	0.441981	0.220991	+	0.975276i	$0.429071\pi$
$242$	0	0
$243$	0	0
$244$	1.68050	0.107583
$245$	0.873210	0.0557874
$246$	0	0
$247$	−0.225172	−0.0143273
$248$	22.1600	1.40716
$249$	0	0
$250$	10.7548	0.680193
$251$	2.56709	0.162033	0.0810166	−	0.996713i	$-0.474183\pi$
$251$	2.56709	0.162033	0.0810166	+	0.996713i	$0.474183\pi$
$252$	0	0
$253$	0	0
$254$	17.3094	1.08609
$255$	0	0
$256$	−5.26696	−0.329185
$257$	19.5915	1.22208	0.611042	−	0.791598i	$-0.290750\pi$
$257$	19.5915	1.22208	0.611042	+	0.791598i	$0.290750\pi$
$258$	0	0
$259$	11.5704	0.718953
$260$	0.00767624	0.000476060 0
$261$	0	0
$262$	−24.1668	−1.49303
$263$	−26.5306	−1.63595	−0.817974	−	0.575255i	$-0.804903\pi$
$263$	−26.5306	−1.63595	−0.817974	+	0.575255i	$0.804903\pi$
$264$	0	0
$265$	0.108628	0.00667298
$266$	−7.59210	−0.465501
$267$	0	0
$268$	−1.08026	−0.0659876
$269$	17.1248	1.04412	0.522058	−	0.852910i	$-0.325164\pi$
$269$	17.1248	1.04412	0.522058	+	0.852910i	$0.325164\pi$
$270$	0	0
$271$	−5.29554	−0.321681	−0.160840	−	0.986980i	$-0.551420\pi$
$271$	−5.29554	−0.321681	−0.160840	+	0.986980i	$0.551420\pi$
$272$	−7.17438	−0.435010
$273$	0	0
$274$	6.94249	0.419411
$275$	0	0
$276$	0	0
$277$	−2.45004	−0.147209	−0.0736044	−	0.997288i	$-0.523450\pi$
$277$	−2.45004	−0.147209	−0.0736044	+	0.997288i	$0.523450\pi$
$278$	−24.1971	−1.45125
$279$	0	0
$280$	2.58733	0.154622
$281$	0.301746	0.0180006	0.00900031	−	0.999959i	$-0.497135\pi$
$281$	0.301746	0.0180006	0.00900031	+	0.999959i	$0.497135\pi$
$282$	0	0
$283$	−24.0094	−1.42721	−0.713605	−	0.700548i	$-0.752939\pi$
$283$	−24.0094	−1.42721	−0.713605	+	0.700548i	$0.752939\pi$
$284$	−2.50100	−0.148407
$285$	0	0
$286$	0	0
$287$	−6.14882	−0.362954
$288$	0	0
$289$	−12.8125	−0.753678
$290$	5.44002	0.319449
$291$	0	0
$292$	0.136263	0.00797418
$293$	−1.96403	−0.114740	−0.0573700	−	0.998353i	$-0.518271\pi$
$293$	−1.96403	−0.114740	−0.0573700	+	0.998353i	$0.518271\pi$
$294$	0	0
$295$	12.5271	0.729354
$296$	34.2833	1.99268
$297$	0	0
$298$	−14.7324	−0.853422
$299$	0.236422	0.0136726
$300$	0	0
$301$	1.79689	0.103571
$302$	21.2414	1.22230
$303$	0	0
$304$	−19.9637	−1.14500
$305$	−6.60098	−0.377971
$306$	0	0
$307$	−20.0180	−1.14249	−0.571243	−	0.820781i	$-0.693538\pi$
$307$	−20.0180	−1.14249	−0.571243	+	0.820781i	$0.693538\pi$
$308$	0	0
$309$	0	0
$310$	−8.70735	−0.494544
$311$	−17.7041	−1.00391	−0.501954	−	0.864894i	$-0.667385\pi$
$311$	−17.7041	−1.00391	−0.501954	+	0.864894i	$0.667385\pi$
$312$	0	0
$313$	28.3315	1.60139	0.800697	−	0.599070i	$-0.204463\pi$
$313$	28.3315	1.60139	0.800697	+	0.599070i	$0.204463\pi$
$314$	−21.3375	−1.20415
$315$	0	0
$316$	−0.0345077	−0.00194121
$317$	21.9554	1.23314	0.616570	−	0.787300i	$-0.288522\pi$
$317$	21.9554	1.23314	0.616570	+	0.787300i	$0.288522\pi$
$318$	0	0
$319$	0	0
$320$	7.57995	0.423732
$321$	0	0
$322$	7.97142	0.444230
$323$	11.6522	0.648347
$324$	0	0
$325$	0.167568	0.00929499
$326$	−29.3372	−1.62484
$327$	0	0
$328$	−18.2190	−1.00598
$329$	−6.04773	−0.333422
$330$	0	0
$331$	23.1465	1.27225	0.636123	−	0.771587i	$-0.280537\pi$
$331$	23.1465	1.27225	0.636123	+	0.771587i	$0.280537\pi$
$332$	−1.98207	−0.108780
$333$	0	0
$334$	6.29454	0.344422
$335$	4.24326	0.231834
$336$	0	0
$337$	−35.0111	−1.90717	−0.953587	−	0.301117i	$-0.902640\pi$
$337$	−35.0111	−1.90717	−0.953587	+	0.301117i	$0.902640\pi$
$338$	17.3308	0.942674
$339$	0	0
$340$	−0.397232	−0.0215429
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.32420	0.287062
$345$	0	0
$346$	12.6340	0.679208
$347$	−3.75958	−0.201825	−0.100912	−	0.994895i	$-0.532176\pi$
$347$	−3.75958	−0.201825	−0.100912	+	0.994895i	$0.532176\pi$
$348$	0	0
$349$	12.1740	0.651658	0.325829	−	0.945429i	$-0.394357\pi$
$349$	12.1740	0.651658	0.325829	+	0.945429i	$0.394357\pi$
$350$	5.64987	0.301998
$351$	0	0
$352$	0	0
$353$	5.79059	0.308202	0.154101	−	0.988055i	$-0.450752\pi$
$353$	5.79059	0.308202	0.154101	+	0.988055i	$0.450752\pi$
$354$	0	0
$355$	9.82387	0.521397
$356$	1.15085	0.0609948
$357$	0	0
$358$	23.6241	1.24857
$359$	5.76086	0.304046	0.152023	−	0.988377i	$-0.451421\pi$
$359$	5.76086	0.304046	0.152023	+	0.988377i	$0.451421\pi$
$360$	0	0
$361$	13.4240	0.706525
$362$	−21.6984	−1.14044
$363$	0	0
$364$	0.00879083	0.000460765 0
$365$	−0.535238	−0.0280156
$366$	0	0
$367$	14.2580	0.744263	0.372132	−	0.928180i	$-0.378627\pi$
$367$	14.2580	0.744263	0.372132	+	0.928180i	$0.378627\pi$
$368$	20.9612	1.09268
$369$	0	0
$370$	−13.4709	−0.700320
$371$	0.124401	0.00645858
$372$	0	0
$373$	−32.2223	−1.66841	−0.834203	−	0.551458i	$-0.814072\pi$
$373$	−32.2223	−1.66841	−0.834203	+	0.551458i	$0.814072\pi$
$374$	0	0
$375$	0	0
$376$	−17.9195	−0.924125
$377$	0.184771	0.00951619
$378$	0	0
$379$	−28.5774	−1.46792	−0.733960	−	0.679193i	$-0.762330\pi$
$379$	−28.5774	−1.46792	−0.733960	+	0.679193i	$0.762330\pi$
$380$	−1.10535	−0.0567035
$381$	0	0
$382$	8.59452	0.439734
$383$	20.1613	1.03019	0.515097	−	0.857132i	$-0.327756\pi$
$383$	20.1613	1.03019	0.515097	+	0.857132i	$0.327756\pi$
$384$	0	0
$385$	0	0
$386$	−2.41427	−0.122883
$387$	0	0
$388$	−1.76559	−0.0896345
$389$	16.5608	0.839663	0.419832	−	0.907602i	$-0.362089\pi$
$389$	16.5608	0.839663	0.419832	+	0.907602i	$0.362089\pi$
$390$	0	0
$391$	−12.2344	−0.618721
$392$	2.96300	0.149654
$393$	0	0
$394$	9.14342	0.460639
$395$	0.135546	0.00682005
$396$	0	0
$397$	30.9920	1.55545	0.777723	−	0.628607i	$-0.216375\pi$
$397$	30.9920	1.55545	0.777723	+	0.628607i	$0.216375\pi$
$398$	6.76518	0.339108
$399$	0	0
$400$	14.8566	0.742828
$401$	−3.48264	−0.173915	−0.0869574	−	0.996212i	$-0.527714\pi$
$401$	−3.48264	−0.173915	−0.0869574	+	0.996212i	$0.527714\pi$
$402$	0	0
$403$	−0.295746	−0.0147322
$404$	2.02200	0.100598
$405$	0	0
$406$	6.22991	0.309185
$407$	0	0
$408$	0	0
$409$	26.5883	1.31471	0.657353	−	0.753583i	$-0.271676\pi$
$409$	26.5883	1.31471	0.657353	+	0.753583i	$0.271676\pi$
$410$	7.15879	0.353547
$411$	0	0
$412$	−0.764174	−0.0376481
$413$	14.3460	0.705920
$414$	0	0
$415$	7.78553	0.382177
$416$	0.0494889	0.00242639
$417$	0	0
$418$	0	0
$419$	−29.1491	−1.42402	−0.712012	−	0.702167i	$-0.752216\pi$
$419$	−29.1491	−1.42402	−0.712012	+	0.702167i	$0.752216\pi$
$420$	0	0
$421$	−19.9909	−0.974297	−0.487148	−	0.873319i	$-0.661963\pi$
$421$	−19.9909	−0.974297	−0.487148	+	0.873319i	$0.661963\pi$
$422$	10.8089	0.526170
$423$	0	0
$424$	0.368601	0.0179008
$425$	−8.67134	−0.420622
$426$	0	0
$427$	−7.55944	−0.365827
$428$	−0.731740	−0.0353700
$429$	0	0
$430$	−2.09204	−0.100887
$431$	16.9285	0.815417	0.407709	−	0.913112i	$-0.366328\pi$
$431$	16.9285	0.815417	0.407709	+	0.913112i	$0.366328\pi$
$432$	0	0
$433$	−8.55148	−0.410958	−0.205479	−	0.978662i	$-0.565875\pi$
$433$	−8.55148	−0.410958	−0.205479	+	0.978662i	$0.565875\pi$
$434$	−9.97165	−0.478654
$435$	0	0
$436$	2.22196	0.106413
$437$	−34.0440	−1.62854
$438$	0	0
$439$	33.9408	1.61990	0.809952	−	0.586496i	$-0.199493\pi$
$439$	33.9408	1.61990	0.809952	+	0.586496i	$0.199493\pi$
$440$	0	0
$441$	0	0
$442$	0.107891	0.00513185
$443$	35.1601	1.67051	0.835254	−	0.549864i	$-0.185320\pi$
$443$	35.1601	1.67051	0.835254	+	0.549864i	$0.185320\pi$
$444$	0	0
$445$	−4.52051	−0.214293
$446$	21.6061	1.02308
$447$	0	0
$448$	8.68056	0.410118
$449$	1.92800	0.0909879	0.0454939	−	0.998965i	$-0.485514\pi$
$449$	1.92800	0.0909879	0.0454939	+	0.998965i	$0.485514\pi$
$450$	0	0
$451$	0	0
$452$	0.978723	0.0460352
$453$	0	0
$454$	−31.5648	−1.48141
$455$	−0.0345302	−0.00161880
$456$	0	0
$457$	7.65166	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$457$	7.65166	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$458$	−13.5420	−0.632776
$459$	0	0
$460$	1.16058	0.0541123
$461$	40.7525	1.89803	0.949017	−	0.315224i	$-0.102080\pi$
$461$	40.7525	1.89803	0.949017	+	0.315224i	$0.102080\pi$
$462$	0	0
$463$	−8.37407	−0.389176	−0.194588	−	0.980885i	$-0.562337\pi$
$463$	−8.37407	−0.389176	−0.194588	+	0.980885i	$0.562337\pi$
$464$	16.3818	0.760505
$465$	0	0
$466$	−31.3537	−1.45243
$467$	−9.49614	−0.439429	−0.219714	−	0.975564i	$-0.570513\pi$
$467$	−9.49614	−0.439429	−0.219714	+	0.975564i	$0.570513\pi$
$468$	0	0
$469$	4.85938	0.224385
$470$	7.04109	0.324781
$471$	0	0
$472$	42.5072	1.95655
$473$	0	0
$474$	0	0
$475$	−24.1292	−1.10712
$476$	−0.454910	−0.0208507
$477$	0	0
$478$	−10.4338	−0.477232
$479$	−36.2983	−1.65851	−0.829256	−	0.558869i	$-0.811235\pi$
$479$	−36.2983	−1.65851	−0.829256	+	0.558869i	$0.811235\pi$
$480$	0	0
$481$	−0.457542	−0.0208621
$482$	−9.14831	−0.416694
$483$	0	0
$484$	0	0
$485$	6.93523	0.314912
$486$	0	0
$487$	5.36736	0.243218	0.121609	−	0.992578i	$-0.461195\pi$
$487$	5.36736	0.243218	0.121609	+	0.992578i	$0.461195\pi$
$488$	−22.3987	−1.01394
$489$	0	0
$490$	−1.16425	−0.0525956
$491$	23.0614	1.04075	0.520373	−	0.853939i	$-0.325793\pi$
$491$	23.0614	1.04075	0.520373	+	0.853939i	$0.325793\pi$
$492$	0	0
$493$	−9.56156	−0.430631
$494$	0.300222	0.0135076
$495$	0	0
$496$	−26.2208	−1.17735
$497$	11.2503	0.504644
$498$	0	0
$499$	33.1392	1.48351	0.741756	−	0.670670i	$-0.233993\pi$
$499$	33.1392	1.48351	0.741756	+	0.670670i	$0.233993\pi$
$500$	1.79318	0.0801932
$501$	0	0
$502$	−3.42270	−0.152763
$503$	−18.1118	−0.807563	−0.403782	−	0.914855i	$-0.632304\pi$
$503$	−18.1118	−0.807563	−0.403782	+	0.914855i	$0.632304\pi$
$504$	0	0
$505$	−7.94237	−0.353431
$506$	0	0
$507$	0	0
$508$	2.88604	0.128047
$509$	15.5919	0.691100	0.345550	−	0.938400i	$-0.387692\pi$
$509$	15.5919	0.691100	0.345550	+	0.938400i	$0.387692\pi$
$510$	0	0
$511$	−0.612954	−0.0271155
$512$	25.1641	1.11211
$513$	0	0
$514$	−26.1214	−1.15217
$515$	3.00166	0.132269
$516$	0	0
$517$	0	0
$518$	−15.4269	−0.677819
$519$	0	0
$520$	−0.102313	−0.00448673
$521$	15.4692	0.677716	0.338858	−	0.940838i	$-0.389959\pi$
$521$	15.4692	0.677716	0.338858	+	0.940838i	$0.389959\pi$
$522$	0	0
$523$	17.3464	0.758504	0.379252	−	0.925293i	$-0.376181\pi$
$523$	17.3464	0.758504	0.379252	+	0.925293i	$0.376181\pi$
$524$	−4.02940	−0.176025
$525$	0	0
$526$	35.3733	1.54235
$527$	15.3043	0.666667
$528$	0	0
$529$	12.7449	0.554126
$530$	−0.144834	−0.00629120
$531$	0	0
$532$	−1.26585	−0.0548816
$533$	0.243149	0.0105320
$534$	0	0
$535$	2.87427	0.124265
$536$	14.3984	0.621914
$537$	0	0
$538$	−22.8325	−0.984380
$539$	0	0
$540$	0	0
$541$	−30.2945	−1.30246	−0.651232	−	0.758879i	$-0.725748\pi$
$541$	−30.2945	−1.30246	−0.651232	+	0.758879i	$0.725748\pi$
$542$	7.06055	0.303277
$543$	0	0
$544$	−2.56096	−0.109800
$545$	−8.72782	−0.373859
$546$	0	0
$547$	30.4490	1.30191	0.650954	−	0.759118i	$-0.274369\pi$
$547$	30.4490	1.30191	0.650954	+	0.759118i	$0.274369\pi$
$548$	1.15754	0.0494476
$549$	0	0
$550$	0	0
$551$	−26.6064	−1.13347
$552$	0	0
$553$	0.155227	0.00660092
$554$	3.26665	0.138787
$555$	0	0
$556$	−4.03445	−0.171099
$557$	7.17456	0.303996	0.151998	−	0.988381i	$-0.451429\pi$
$557$	7.17456	0.303996	0.151998	+	0.988381i	$0.451429\pi$
$558$	0	0
$559$	−0.0710563	−0.00300536
$560$	−3.06145	−0.129370
$561$	0	0
$562$	−0.402318	−0.0169708
$563$	9.01364	0.379880	0.189940	−	0.981796i	$-0.439171\pi$
$563$	9.01364	0.379880	0.189940	+	0.981796i	$0.439171\pi$
$564$	0	0
$565$	−3.84441	−0.161735
$566$	32.0118	1.34556
$567$	0	0
$568$	33.3347	1.39869
$569$	7.45999	0.312739	0.156370	−	0.987699i	$-0.450021\pi$
$569$	7.45999	0.312739	0.156370	+	0.987699i	$0.450021\pi$
$570$	0	0
$571$	37.5921	1.57318	0.786590	−	0.617475i	$-0.211844\pi$
$571$	37.5921	1.57318	0.786590	+	0.617475i	$0.211844\pi$
$572$	0	0
$573$	0	0
$574$	8.19824	0.342188
$575$	25.3348	1.05653
$576$	0	0
$577$	−12.1230	−0.504688	−0.252344	−	0.967638i	$-0.581201\pi$
$577$	−12.1230	−0.504688	−0.252344	+	0.967638i	$0.581201\pi$
$578$	17.0830	0.710558
$579$	0	0
$580$	0.907029	0.0376623
$581$	8.91599	0.369897
$582$	0	0
$583$	0	0
$584$	−1.81619	−0.0751543
$585$	0	0
$586$	2.61865	0.108175
$587$	1.28073	0.0528615	0.0264307	−	0.999651i	$-0.491586\pi$
$587$	1.28073	0.0528615	0.0264307	+	0.999651i	$0.491586\pi$
$588$	0	0
$589$	42.5865	1.75474
$590$	−16.7024	−0.687626
$591$	0	0
$592$	−40.5656	−1.66724
$593$	−24.7324	−1.01564	−0.507819	−	0.861464i	$-0.669548\pi$
$593$	−24.7324	−1.01564	−0.507819	+	0.861464i	$0.669548\pi$
$594$	0	0
$595$	1.78688	0.0732548
$596$	−2.45636	−0.100617
$597$	0	0
$598$	−0.315222	−0.0128904
$599$	27.9003	1.13998	0.569988	−	0.821653i	$-0.306948\pi$
$599$	27.9003	1.13998	0.569988	+	0.821653i	$0.306948\pi$
$600$	0	0
$601$	34.4256	1.40425	0.702124	−	0.712055i	$-0.252235\pi$
$601$	34.4256	1.40425	0.702124	+	0.712055i	$0.252235\pi$
$602$	−2.39580	−0.0976456
$603$	0	0
$604$	3.54163	0.144107
$605$	0	0
$606$	0	0
$607$	26.4243	1.07253	0.536264	−	0.844050i	$-0.319835\pi$
$607$	26.4243	1.07253	0.536264	+	0.844050i	$0.319835\pi$
$608$	−7.12624	−0.289007
$609$	0	0
$610$	8.80111	0.356346
$611$	0.239152	0.00967503
$612$	0	0
$613$	39.6007	1.59946	0.799729	−	0.600361i	$-0.204976\pi$
$613$	39.6007	1.59946	0.799729	+	0.600361i	$0.204976\pi$
$614$	26.6900	1.07712
$615$	0	0
$616$	0	0
$617$	−17.0640	−0.686971	−0.343486	−	0.939158i	$-0.611608\pi$
$617$	−17.0640	−0.686971	−0.343486	+	0.939158i	$0.611608\pi$
$618$	0	0
$619$	29.2807	1.17689	0.588445	−	0.808538i	$-0.299741\pi$
$619$	29.2807	1.17689	0.588445	+	0.808538i	$0.299741\pi$
$620$	−1.45180	−0.0583056
$621$	0	0
$622$	23.6049	0.946471
$623$	−5.17689	−0.207408
$624$	0	0
$625$	14.1440	0.565758
$626$	−37.7745	−1.50977
$627$	0	0
$628$	−3.55766	−0.141966
$629$	23.6770	0.944062
$630$	0	0
$631$	−40.2944	−1.60410	−0.802048	−	0.597260i	$-0.796256\pi$
$631$	−40.2944	−1.60410	−0.802048	+	0.597260i	$0.796256\pi$
$632$	0.459938	0.0182954
$633$	0	0
$634$	−29.2732	−1.16259
$635$	−11.3363	−0.449867
$636$	0	0
$637$	−0.0395440	−0.00156679
$638$	0	0
$639$	0	0
$640$	−7.92074	−0.313095
$641$	12.5572	0.495980	0.247990	−	0.968763i	$-0.420230\pi$
$641$	12.5572	0.495980	0.247990	+	0.968763i	$0.420230\pi$
$642$	0	0
$643$	−4.13705	−0.163149	−0.0815747	−	0.996667i	$-0.525995\pi$
$643$	−4.13705	−0.163149	−0.0815747	+	0.996667i	$0.525995\pi$
$644$	1.32910	0.0523737
$645$	0	0
$646$	−15.5359	−0.611254
$647$	37.6088	1.47856	0.739278	−	0.673400i	$-0.235167\pi$
$647$	37.6088	1.47856	0.739278	+	0.673400i	$0.235167\pi$
$648$	0	0
$649$	0	0
$650$	−0.223419	−0.00876320
$651$	0	0
$652$	−4.89147	−0.191565
$653$	−6.02137	−0.235634	−0.117817	−	0.993035i	$-0.537590\pi$
$653$	−6.02137	−0.235634	−0.117817	+	0.993035i	$0.537590\pi$
$654$	0	0
$655$	15.8274	0.618428
$656$	21.5576	0.841682
$657$	0	0
$658$	8.06345	0.314346
$659$	−38.6192	−1.50439	−0.752195	−	0.658940i	$-0.771005\pi$
$659$	−38.6192	−1.50439	−0.752195	+	0.658940i	$0.771005\pi$
$660$	0	0
$661$	43.0833	1.67575	0.837874	−	0.545864i	$-0.183798\pi$
$661$	43.0833	1.67575	0.837874	+	0.545864i	$0.183798\pi$
$662$	−30.8613	−1.19946
$663$	0	0
$664$	26.4181	1.02522
$665$	4.97224	0.192815
$666$	0	0
$667$	27.9357	1.08168
$668$	1.04950	0.0406066
$669$	0	0
$670$	−5.65755	−0.218570
$671$	0	0
$672$	0	0
$673$	21.0171	0.810149	0.405075	−	0.914284i	$-0.367246\pi$
$673$	21.0171	0.810149	0.405075	+	0.914284i	$0.367246\pi$
$674$	46.6803	1.79806
$675$	0	0
$676$	2.88962	0.111139
$677$	−23.8886	−0.918115	−0.459057	−	0.888407i	$-0.651813\pi$
$677$	−23.8886	−0.918115	−0.459057	+	0.888407i	$0.651813\pi$
$678$	0	0
$679$	7.94222	0.304794
$680$	5.29453	0.203036
$681$	0	0
$682$	0	0
$683$	13.5672	0.519134	0.259567	−	0.965725i	$-0.416420\pi$
$683$	13.5672	0.519134	0.259567	+	0.965725i	$0.416420\pi$
$684$	0	0
$685$	−4.54679	−0.173724
$686$	−1.33330	−0.0509057
$687$	0	0
$688$	−6.29985	−0.240180
$689$	−0.00491932	−0.000187411 0
$690$	0	0
$691$	−4.91171	−0.186850	−0.0934251	−	0.995626i	$-0.529782\pi$
$691$	−4.91171	−0.186850	−0.0934251	+	0.995626i	$0.529782\pi$
$692$	2.10650	0.0800771
$693$	0	0
$694$	5.01266	0.190278
$695$	15.8472	0.601120
$696$	0	0
$697$	−12.5825	−0.476597
$698$	−16.2316	−0.614375
$699$	0	0
$700$	0.942018	0.0356049
$701$	36.9242	1.39461	0.697304	−	0.716775i	$-0.254383\pi$
$701$	36.9242	1.39461	0.697304	+	0.716775i	$0.254383\pi$
$702$	0	0
$703$	65.8845	2.48488
$704$	0	0
$705$	0	0
$706$	−7.72061	−0.290569
$707$	−9.09560	−0.342075
$708$	0	0
$709$	0.489376	0.0183789	0.00918945	−	0.999958i	$-0.497075\pi$
$709$	0.489376	0.0183789	0.00918945	+	0.999958i	$0.497075\pi$
$710$	−13.0982	−0.491566
$711$	0	0
$712$	−15.3391	−0.574858
$713$	−44.7142	−1.67456
$714$	0	0
$715$	0	0
$716$	3.93890	0.147204
$717$	0	0
$718$	−7.68096	−0.286651
$719$	−10.1392	−0.378127	−0.189064	−	0.981965i	$-0.560545\pi$
$719$	−10.1392	−0.378127	−0.189064	+	0.981965i	$0.560545\pi$
$720$	0	0
$721$	3.43750	0.128019
$722$	−17.8982	−0.666103
$723$	0	0
$724$	−3.61784	−0.134456
$725$	19.7999	0.735350
$726$	0	0
$727$	−10.5834	−0.392516	−0.196258	−	0.980552i	$-0.562879\pi$
$727$	−10.5834	−0.392516	−0.196258	+	0.980552i	$0.562879\pi$
$728$	−0.117169	−0.00434257
$729$	0	0
$730$	0.713634	0.0264128
$731$	3.67704	0.136000
$732$	0	0
$733$	−12.5752	−0.464475	−0.232238	−	0.972659i	$-0.574605\pi$
$733$	−12.5752	−0.464475	−0.232238	+	0.972659i	$0.574605\pi$
$734$	−19.0103	−0.701682
$735$	0	0
$736$	7.48229	0.275801
$737$	0	0
$738$	0	0
$739$	1.39740	0.0514043	0.0257022	−	0.999670i	$-0.491818\pi$
$739$	1.39740	0.0514043	0.0257022	+	0.999670i	$0.491818\pi$
$740$	−2.24604	−0.0825662
$741$	0	0
$742$	−0.165864	−0.00608907
$743$	24.1170	0.884767	0.442383	−	0.896826i	$-0.354133\pi$
$743$	24.1170	0.884767	0.442383	+	0.896826i	$0.354133\pi$
$744$	0	0
$745$	9.64856	0.353496
$746$	42.9620	1.57295
$747$	0	0
$748$	0	0
$749$	3.29161	0.120273
$750$	0	0
$751$	−19.2248	−0.701523	−0.350762	−	0.936465i	$-0.614077\pi$
$751$	−19.2248	−0.701523	−0.350762	+	0.936465i	$0.614077\pi$
$752$	21.2032	0.773200
$753$	0	0
$754$	−0.246355	−0.00897174
$755$	−13.9115	−0.506290
$756$	0	0
$757$	−42.4814	−1.54401	−0.772006	−	0.635615i	$-0.780747\pi$
$757$	−42.4814	−1.54401	−0.772006	+	0.635615i	$0.780747\pi$
$758$	38.1023	1.38394
$759$	0	0
$760$	14.7328	0.534414
$761$	14.7018	0.532939	0.266470	−	0.963843i	$-0.414143\pi$
$761$	14.7018	0.532939	0.266470	+	0.963843i	$0.414143\pi$
$762$	0	0
$763$	−9.99510	−0.361847
$764$	1.43299	0.0518437
$765$	0	0
$766$	−26.8811	−0.971254
$767$	−0.567298	−0.0204839
$768$	0	0
$769$	45.4931	1.64052	0.820262	−	0.571988i	$-0.193828\pi$
$769$	45.4931	1.64052	0.820262	+	0.571988i	$0.193828\pi$
$770$	0	0
$771$	0	0
$772$	−0.402537	−0.0144876
$773$	−6.05381	−0.217741	−0.108870	−	0.994056i	$-0.534723\pi$
$773$	−6.05381	−0.217741	−0.108870	+	0.994056i	$0.534723\pi$
$774$	0	0
$775$	−31.6919	−1.13841
$776$	23.5328	0.844779
$777$	0	0
$778$	−22.0805	−0.791624
$779$	−35.0127	−1.25446
$780$	0	0
$781$	0	0
$782$	16.3122	0.583322
$783$	0	0
$784$	−3.50597	−0.125213
$785$	13.9744	0.498769
$786$	0	0
$787$	−54.7630	−1.95209	−0.976046	−	0.217566i	$-0.930188\pi$
$787$	−54.7630	−1.95209	−0.976046	+	0.217566i	$0.930188\pi$
$788$	1.52451	0.0543083
$789$	0	0
$790$	−0.180723	−0.00642985
$791$	−4.40261	−0.156539
$792$	0	0
$793$	0.298931	0.0106153
$794$	−41.3218	−1.46645
$795$	0	0
$796$	1.12798	0.0399801
$797$	43.2922	1.53349	0.766744	−	0.641953i	$-0.221876\pi$
$797$	43.2922	1.53349	0.766744	+	0.641953i	$0.221876\pi$
$798$	0	0
$799$	−12.3757	−0.437819
$800$	5.30319	0.187496
$801$	0	0
$802$	4.64342	0.163965
$803$	0	0
$804$	0	0
$805$	−5.22067	−0.184004
$806$	0.394319	0.0138893
$807$	0	0
$808$	−26.9503	−0.948108
$809$	−15.5173	−0.545559	−0.272779	−	0.962077i	$-0.587943\pi$
$809$	−15.5173	−0.545559	−0.272779	+	0.962077i	$0.587943\pi$
$810$	0	0
$811$	−41.4421	−1.45523	−0.727615	−	0.685986i	$-0.759371\pi$
$811$	−41.4421	−1.45523	−0.727615	+	0.685986i	$0.759371\pi$
$812$	1.03873	0.0364522
$813$	0	0
$814$	0	0
$815$	19.2136	0.673024
$816$	0	0
$817$	10.2319	0.357968
$818$	−35.4502	−1.23949
$819$	0	0
$820$	1.19360	0.0416824
$821$	−40.0922	−1.39923	−0.699613	−	0.714522i	$-0.746644\pi$
$821$	−40.0922	−1.39923	−0.699613	+	0.714522i	$0.746644\pi$
$822$	0	0
$823$	4.23033	0.147460	0.0737301	−	0.997278i	$-0.476510\pi$
$823$	4.23033	0.147460	0.0737301	+	0.997278i	$0.476510\pi$
$824$	10.1853	0.354823
$825$	0	0
$826$	−19.1275	−0.665532
$827$	47.6889	1.65830	0.829152	−	0.559023i	$-0.188824\pi$
$827$	47.6889	1.65830	0.829152	+	0.559023i	$0.188824\pi$
$828$	0	0
$829$	4.68523	0.162725	0.0813624	−	0.996685i	$-0.474073\pi$
$829$	4.68523	0.162725	0.0813624	+	0.996685i	$0.474073\pi$
$830$	−10.3805	−0.360311
$831$	0	0
$832$	−0.343264	−0.0119005
$833$	2.04633	0.0709012
$834$	0	0
$835$	−4.12244	−0.142663
$836$	0	0
$837$	0	0
$838$	38.8645	1.34255
$839$	6.93707	0.239494	0.119747	−	0.992804i	$-0.461792\pi$
$839$	6.93707	0.239494	0.119747	+	0.992804i	$0.461792\pi$
$840$	0	0
$841$	−7.16738	−0.247151
$842$	26.6539	0.918555
$843$	0	0
$844$	1.80220	0.0620343
$845$	−11.3504	−0.390465
$846$	0	0
$847$	0	0
$848$	−0.436146	−0.0149773
$849$	0	0
$850$	11.5615	0.396557
$851$	−69.1763	−2.37133
$852$	0	0
$853$	6.23883	0.213613	0.106807	−	0.994280i	$-0.465937\pi$
$853$	6.23883	0.213613	0.106807	+	0.994280i	$0.465937\pi$
$854$	10.0790	0.344897
$855$	0	0
$856$	9.75304	0.333352
$857$	26.5684	0.907559	0.453779	−	0.891114i	$-0.350075\pi$
$857$	26.5684	0.907559	0.453779	+	0.891114i	$0.350075\pi$
$858$	0	0
$859$	44.6475	1.52335	0.761677	−	0.647957i	$-0.224377\pi$
$859$	44.6475	1.52335	0.761677	+	0.647957i	$0.224377\pi$
$860$	−0.348811	−0.0118944
$861$	0	0
$862$	−22.5708	−0.768765
$863$	21.2309	0.722707	0.361353	−	0.932429i	$-0.382315\pi$
$863$	21.2309	0.722707	0.361353	+	0.932429i	$0.382315\pi$
$864$	0	0
$865$	−8.27430	−0.281335
$866$	11.4017	0.387446
$867$	0	0
$868$	−1.66260	−0.0564323
$869$	0	0
$870$	0	0
$871$	−0.192159	−0.00651107
$872$	−29.6155	−1.00291
$873$	0	0
$874$	45.3909	1.53537
$875$	−8.06628	−0.272690
$876$	0	0
$877$	−9.63512	−0.325355	−0.162677	−	0.986679i	$-0.552013\pi$
$877$	−9.63512	−0.325355	−0.162677	+	0.986679i	$0.552013\pi$
$878$	−45.2533	−1.52722
$879$	0	0
$880$	0	0
$881$	38.9952	1.31378	0.656891	−	0.753985i	$-0.271871\pi$
$881$	38.9952	1.31378	0.656891	+	0.753985i	$0.271871\pi$
$882$	0	0
$883$	14.8304	0.499081	0.249541	−	0.968364i	$-0.419720\pi$
$883$	14.8304	0.499081	0.249541	+	0.968364i	$0.419720\pi$
$884$	0.0179889	0.000605034 0
$885$	0	0
$886$	−46.8791	−1.57493
$887$	−17.6091	−0.591254	−0.295627	−	0.955303i	$-0.595529\pi$
$887$	−17.6091	−0.591254	−0.295627	+	0.955303i	$0.595529\pi$
$888$	0	0
$889$	−12.9823	−0.435413
$890$	6.02721	0.202032
$891$	0	0
$892$	3.60245	0.120619
$893$	−34.4370	−1.15239
$894$	0	0
$895$	−15.4720	−0.517170
$896$	−9.07082	−0.303035
$897$	0	0
$898$	−2.57060	−0.0857822
$899$	−34.9455	−1.16550
$900$	0	0
$901$	0.254566	0.00848081
$902$	0	0
$903$	0	0
$904$	−13.0450	−0.433869
$905$	14.2108	0.472383
$906$	0	0
$907$	46.0126	1.52782	0.763911	−	0.645322i	$-0.223277\pi$
$907$	46.0126	1.52782	0.763911	+	0.645322i	$0.223277\pi$
$908$	−5.26289	−0.174655
$909$	0	0
$910$	0.0460393	0.00152619
$911$	−19.5000	−0.646064	−0.323032	−	0.946388i	$-0.604702\pi$
$911$	−19.5000	−0.646064	−0.323032	+	0.946388i	$0.604702\pi$
$912$	0	0
$913$	0	0
$914$	−10.2020	−0.337451
$915$	0	0
$916$	−2.25789	−0.0746029
$917$	18.1255	0.598558
$918$	0	0
$919$	−11.7377	−0.387191	−0.193596	−	0.981081i	$-0.562015\pi$
$919$	−11.7377	−0.387191	−0.193596	+	0.981081i	$0.562015\pi$
$920$	−15.4689	−0.509993
$921$	0	0
$922$	−54.3354	−1.78944
$923$	−0.444882	−0.0146435
$924$	0	0
$925$	−49.0298	−1.61209
$926$	11.1652	0.366910
$927$	0	0
$928$	5.84764	0.191958
$929$	−35.6903	−1.17096	−0.585480	−	0.810687i	$-0.699094\pi$
$929$	−35.6903	−1.17096	−0.585480	+	0.810687i	$0.699094\pi$
$930$	0	0
$931$	5.69421	0.186620
$932$	−5.22769	−0.171239
$933$	0	0
$934$	12.6612	0.414288
$935$	0	0
$936$	0	0
$937$	−35.3541	−1.15497	−0.577485	−	0.816402i	$-0.695966\pi$
$937$	−35.3541	−1.15497	−0.577485	+	0.816402i	$0.695966\pi$
$938$	−6.47902	−0.211548
$939$	0	0
$940$	1.17398	0.0382910
$941$	−30.5431	−0.995678	−0.497839	−	0.867270i	$-0.665873\pi$
$941$	−30.5431	−0.995678	−0.497839	+	0.867270i	$0.665873\pi$
$942$	0	0
$943$	36.7620	1.19713
$944$	−50.2966	−1.63702
$945$	0	0
$946$	0	0
$947$	−38.9527	−1.26579	−0.632897	−	0.774236i	$-0.718134\pi$
$947$	−38.9527	−1.26579	−0.632897	+	0.774236i	$0.718134\pi$
$948$	0	0
$949$	0.0242387	0.000786821 0
$950$	32.1715	1.04378
$951$	0	0
$952$	6.06329	0.196512
$953$	35.2846	1.14298	0.571490	−	0.820609i	$-0.306366\pi$
$953$	35.2846	1.14298	0.571490	+	0.820609i	$0.306366\pi$
$954$	0	0
$955$	−5.62875	−0.182142
$956$	−1.73966	−0.0562645
$957$	0	0
$958$	48.3966	1.56362
$959$	−5.20699	−0.168142
$960$	0	0
$961$	24.9341	0.804326
$962$	0.610042	0.0196685
$963$	0	0
$964$	−1.52532	−0.0491273
$965$	1.58116	0.0508993
$966$	0	0
$967$	−31.4791	−1.01230	−0.506150	−	0.862445i	$-0.668932\pi$
$967$	−31.4791	−1.01230	−0.506150	+	0.862445i	$0.668932\pi$
$968$	0	0
$969$	0	0
$970$	−9.24675	−0.296895
$971$	34.4771	1.10642	0.553211	−	0.833041i	$-0.313402\pi$
$971$	34.4771	1.10642	0.553211	+	0.833041i	$0.313402\pi$
$972$	0	0
$973$	18.1482	0.581806
$974$	−7.15631	−0.229303
$975$	0	0
$976$	26.5032	0.848346
$977$	−5.51932	−0.176579	−0.0882894	−	0.996095i	$-0.528140\pi$
$977$	−5.51932	−0.176579	−0.0882894	+	0.996095i	$0.528140\pi$
$978$	0	0
$979$	0	0
$980$	−0.194119	−0.00620090
$981$	0	0
$982$	−30.7478	−0.981202
$983$	33.0076	1.05278	0.526390	−	0.850243i	$-0.323545\pi$
$983$	33.0076	1.05278	0.526390	+	0.850243i	$0.323545\pi$
$984$	0	0
$985$	−5.98823	−0.190801
$986$	12.7485	0.405993
$987$	0	0
$988$	0.0500568	0.00159252
$989$	−10.7431	−0.341610
$990$	0	0
$991$	−40.9399	−1.30050	−0.650249	−	0.759721i	$-0.725335\pi$
$991$	−40.9399	−1.30050	−0.650249	+	0.759721i	$0.725335\pi$
$992$	−9.35978	−0.297173
$993$	0	0
$994$	−15.0000	−0.475772
$995$	−4.43067	−0.140462
$996$	0	0
$997$	37.8795	1.19966	0.599828	−	0.800129i	$-0.295236\pi$
$997$	37.8795	1.19966	0.599828	+	0.800129i	$0.295236\pi$
$998$	−44.1845	−1.39864
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cy.1.4		10
3.2	odd	2		2541.2.a.br.1.7		10
11.2	odd	10		693.2.m.j.631.2		20
11.6	odd	10		693.2.m.j.190.2		20
11.10	odd	2		7623.2.a.cx.1.7		10
33.2	even	10		231.2.j.g.169.4	✓	20
33.17	even	10		231.2.j.g.190.4	yes	20
33.32	even	2		2541.2.a.bq.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.g.169.4	✓	20	33.2	even	10
231.2.j.g.190.4	yes	20	33.17	even	10
693.2.m.j.190.2		20	11.6	odd	10
693.2.m.j.631.2		20	11.2	odd	10
2541.2.a.bq.1.4		10	33.32	even	2
2541.2.a.br.1.7		10	3.2	odd	2
7623.2.a.cx.1.7		10	11.10	odd	2
7623.2.a.cy.1.4		10	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)

\(f(q)\)	\(=\)	\(q-1.33330 q^{2} -0.222305 q^{4} +0.873210 q^{5} +1.00000 q^{7} +2.96300 q^{8} +O(q^{10})\)	\(q-1.33330 q^{2} -0.222305 q^{4} +0.873210 q^{5} +1.00000 q^{7} +2.96300 q^{8} -1.16425 q^{10} -0.0395440 q^{13} -1.33330 q^{14} -3.50597 q^{16} +2.04633 q^{17} +5.69421 q^{19} -0.194119 q^{20} -5.97870 q^{23} -4.23750 q^{25} +0.0527241 q^{26} -0.222305 q^{28} -4.67254 q^{29} +7.47891 q^{31} -1.25149 q^{32} -2.72838 q^{34} +0.873210 q^{35} +11.5704 q^{37} -7.59210 q^{38} +2.58733 q^{40} -6.14882 q^{41} +1.79689 q^{43} +7.97142 q^{46} -6.04773 q^{47} +1.00000 q^{49} +5.64987 q^{50} +0.00879083 q^{52} +0.124401 q^{53} +2.96300 q^{56} +6.22991 q^{58} +14.3460 q^{59} -7.55944 q^{61} -9.97165 q^{62} +8.68056 q^{64} -0.0345302 q^{65} +4.85938 q^{67} -0.454910 q^{68} -1.16425 q^{70} +11.2503 q^{71} -0.612954 q^{73} -15.4269 q^{74} -1.26585 q^{76} +0.155227 q^{79} -3.06145 q^{80} +8.19824 q^{82} +8.91599 q^{83} +1.78688 q^{85} -2.39580 q^{86} -5.17689 q^{89} -0.0395440 q^{91} +1.32910 q^{92} +8.06345 q^{94} +4.97224 q^{95} +7.94222 q^{97} -1.33330 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8}+O(q^{10}) \) 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8	\( 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8} - 6 q^{10} + 6 q^{13} + 38 q^{16} - 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} + 18 q^{28} + 14 q^{29} + 26 q^{31} + 41 q^{32} + 21 q^{34} - 5 q^{35} + 24 q^{37} - 8 q^{38} - 5 q^{40} - 19 q^{41} - 6 q^{43} - q^{46} - 15 q^{47} + 10 q^{49} + q^{50} - 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} - 11 q^{62} + 53 q^{64} + 29 q^{65} + 38 q^{67} - 87 q^{68} - 6 q^{70} - 26 q^{71} - q^{73} + 39 q^{74} - 2 q^{76} + 5 q^{79} - 6 q^{80} + 5 q^{82} - 6 q^{83} - q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} + 42 q^{95} + 24 q^{97}+O(q^{100}) \) 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8 - 6 * q^10 + 6 * q^13 + 38 * q^16 - 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 + 18 * q^28 + 14 * q^29 + 26 * q^31 + 41 * q^32 + 21 * q^34 - 5 * q^35 + 24 * q^37 - 8 * q^38 - 5 * q^40 - 19 * q^41 - 6 * q^43 - q^46 - 15 * q^47 + 10 * q^49 + q^50 - 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 - 11 * q^62 + 53 * q^64 + 29 * q^65 + 38 * q^67 - 87 * q^68 - 6 * q^70 - 26 * q^71 - q^73 + 39 * q^74 - 2 * q^76 + 5 * q^79 - 6 * q^80 + 5 * q^82 - 6 * q^83 - q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 + 42 * q^95 + 24 * q^97

Introduction

L-functions

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