LMFDB - Embedded newform 7569.2.a.bp.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7569, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7569.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.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:	$60.4387692899$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 15x^{10} + 78x^{8} - 169x^{6} + 148x^{4} - 36x^{2} + 1 $ x^12 - 15x^10 + 78x^8 - 169x^6 + 148x^4 - 36*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.178275$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.16308 q^{2} -0.647237 q^{4} -1.89937 q^{5} +1.52574 q^{7} +3.07896 q^{8} +O(q^{10})$	\(q-1.16308 q^{2} -0.647237 q^{4} -1.89937 q^{5} +1.52574 q^{7} +3.07896 q^{8} +2.20913 q^{10} +0.794830 q^{11} +6.44341 q^{13} -1.77456 q^{14} -2.28661 q^{16} -4.03114 q^{17} +5.97758 q^{19} +1.22934 q^{20} -0.924454 q^{22} -5.82280 q^{23} -1.39239 q^{25} -7.49423 q^{26} -0.987516 q^{28} -0.631076 q^{31} -3.49840 q^{32} +4.68855 q^{34} -2.89795 q^{35} -2.53226 q^{37} -6.95242 q^{38} -5.84808 q^{40} +2.49005 q^{41} -8.19675 q^{43} -0.514444 q^{44} +6.77241 q^{46} +6.75222 q^{47} -4.67211 q^{49} +1.61947 q^{50} -4.17041 q^{52} +0.689457 q^{53} -1.50968 q^{55} +4.69769 q^{56} -2.67206 q^{59} -8.18512 q^{61} +0.733994 q^{62} +8.64215 q^{64} -12.2384 q^{65} +3.95370 q^{67} +2.60910 q^{68} +3.37055 q^{70} +3.12799 q^{71} -0.915217 q^{73} +2.94523 q^{74} -3.86891 q^{76} +1.21271 q^{77} -12.8832 q^{79} +4.34312 q^{80} -2.89614 q^{82} -12.6241 q^{83} +7.65662 q^{85} +9.53350 q^{86} +2.44725 q^{88} -1.83979 q^{89} +9.83098 q^{91} +3.76873 q^{92} -7.85340 q^{94} -11.3536 q^{95} -0.120628 q^{97} +5.43406 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{4} - 8 q^{5} + 10 q^{7}+O(q^{10}) $ 12 * q + 8 * q^4 - 8 * q^5 + 10 * q^7	$ 12 q + 8 q^{4} - 8 q^{5} + 10 q^{7} + 12 q^{13} + 16 q^{16} - 24 q^{20} - 38 q^{22} - 30 q^{23} - 8 q^{25} - 12 q^{28} + 6 q^{34} - 44 q^{35} + 6 q^{49} - 6 q^{52} - 32 q^{53} - 44 q^{59} + 16 q^{62} - 2 q^{64} - 8 q^{65} + 30 q^{67} - 70 q^{71} - 56 q^{74} - 34 q^{80} - 62 q^{82} - 82 q^{83} - 44 q^{86} - 66 q^{88} - 32 q^{91} + 22 q^{92} + 10 q^{94}+O(q^{100}) $ 12 * q + 8 * q^4 - 8 * q^5 + 10 * q^7 + 12 * q^13 + 16 * q^16 - 24 * q^20 - 38 * q^22 - 30 * q^23 - 8 * q^25 - 12 * q^28 + 6 * q^34 - 44 * q^35 + 6 * q^49 - 6 * q^52 - 32 * q^53 - 44 * q^59 + 16 * q^62 - 2 * q^64 - 8 * q^65 + 30 * q^67 - 70 * q^71 - 56 * q^74 - 34 * q^80 - 62 * q^82 - 82 * q^83 - 44 * q^86 - 66 * q^88 - 32 * q^91 + 22 * q^92 + 10 * q^94

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 $

$2$	−1.16308	−0.822424	−0.411212	−	0.911540i	$-0.634894\pi$
$2$	−1.16308	−0.822424	−0.411212	+	0.911540i	$0.634894\pi$
$3$	0	0
$3$	0	0
$4$	−0.647237	−0.323618
$5$	−1.89937	−0.849424	−0.424712	−	0.905329i	$-0.639625\pi$
$5$	−1.89937	−0.849424	−0.424712	+	0.905329i	$0.639625\pi$
$6$	0	0
$7$	1.52574	0.576676	0.288338	−	0.957529i	$-0.406897\pi$
$7$	1.52574	0.576676	0.288338	+	0.957529i	$0.406897\pi$
$8$	3.07896	1.08858
$9$	0	0
$10$	2.20913	0.698587
$11$	0.794830	0.239650	0.119825	−	0.992795i	$-0.461767\pi$
$11$	0.794830	0.239650	0.119825	+	0.992795i	$0.461767\pi$
$12$	0	0
$13$	6.44341	1.78708	0.893540	−	0.448983i	$-0.148214\pi$
$13$	6.44341	1.78708	0.893540	+	0.448983i	$0.148214\pi$
$14$	−1.77456	−0.474272
$15$	0	0
$16$	−2.28661	−0.571653
$17$	−4.03114	−0.977695	−0.488847	−	0.872369i	$-0.662583\pi$
$17$	−4.03114	−0.977695	−0.488847	+	0.872369i	$0.662583\pi$
$18$	0	0
$19$	5.97758	1.37135	0.685675	−	0.727908i	$-0.259507\pi$
$19$	5.97758	1.37135	0.685675	+	0.727908i	$0.259507\pi$
$20$	1.22934	0.274889
$21$	0	0
$22$	−0.924454	−0.197094
$23$	−5.82280	−1.21414	−0.607069	−	0.794649i	$-0.707655\pi$
$23$	−5.82280	−1.21414	−0.607069	+	0.794649i	$0.707655\pi$
$24$	0	0
$25$	−1.39239	−0.278479
$26$	−7.49423	−1.46974
$27$	0	0
$28$	−0.987516	−0.186623
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−0.631076	−0.113345	−0.0566723	−	0.998393i	$-0.518049\pi$
$31$	−0.631076	−0.113345	−0.0566723	+	0.998393i	$0.518049\pi$
$32$	−3.49840	−0.618435
$33$	0	0
$34$	4.68855	0.804080
$35$	−2.89795	−0.489842
$36$	0	0
$37$	−2.53226	−0.416301	−0.208151	−	0.978097i	$-0.566744\pi$
$37$	−2.53226	−0.416301	−0.208151	+	0.978097i	$0.566744\pi$
$38$	−6.95242	−1.12783
$39$	0	0
$40$	−5.84808	−0.924662
$41$	2.49005	0.388881	0.194441	−	0.980914i	$-0.437711\pi$
$41$	2.49005	0.388881	0.194441	+	0.980914i	$0.437711\pi$
$42$	0	0
$43$	−8.19675	−1.24999	−0.624996	−	0.780628i	$-0.714900\pi$
$43$	−8.19675	−1.24999	−0.624996	+	0.780628i	$0.714900\pi$
$44$	−0.514444	−0.0775553
$45$	0	0
$46$	6.77241	0.998537
$47$	6.75222	0.984913	0.492456	−	0.870337i	$-0.336099\pi$
$47$	6.75222	0.984913	0.492456	+	0.870337i	$0.336099\pi$
$48$	0	0
$49$	−4.67211	−0.667445
$50$	1.61947	0.229028
$51$	0	0
$52$	−4.17041	−0.578332
$53$	0.689457	0.0947042	0.0473521	−	0.998878i	$-0.484922\pi$
$53$	0.689457	0.0947042	0.0473521	+	0.998878i	$0.484922\pi$
$54$	0	0
$55$	−1.50968	−0.203565
$56$	4.69769	0.627755
$57$	0	0
$58$	0	0
$59$	−2.67206	−0.347873	−0.173936	−	0.984757i	$-0.555649\pi$
$59$	−2.67206	−0.347873	−0.173936	+	0.984757i	$0.555649\pi$
$60$	0	0
$61$	−8.18512	−1.04800	−0.523999	−	0.851719i	$-0.675560\pi$
$61$	−8.18512	−1.04800	−0.523999	+	0.851719i	$0.675560\pi$
$62$	0.733994	0.0932173
$63$	0	0
$64$	8.64215	1.08027
$65$	−12.2384	−1.51799
$66$	0	0
$67$	3.95370	0.483021	0.241511	−	0.970398i	$-0.422357\pi$
$67$	3.95370	0.483021	0.241511	+	0.970398i	$0.422357\pi$
$68$	2.60910	0.316400
$69$	0	0
$70$	3.37055	0.402858
$71$	3.12799	0.371224	0.185612	−	0.982623i	$-0.440573\pi$
$71$	3.12799	0.371224	0.185612	+	0.982623i	$0.440573\pi$
$72$	0	0
$73$	−0.915217	−0.107118	−0.0535590	−	0.998565i	$-0.517057\pi$
$73$	−0.915217	−0.107118	−0.0535590	+	0.998565i	$0.517057\pi$
$74$	2.94523	0.342376
$75$	0	0
$76$	−3.86891	−0.443794
$77$	1.21271	0.138201
$78$	0	0
$79$	−12.8832	−1.44947	−0.724737	−	0.689026i	$-0.758039\pi$
$79$	−12.8832	−1.44947	−0.724737	+	0.689026i	$0.758039\pi$
$80$	4.34312	0.485576
$81$	0	0
$82$	−2.89614	−0.319825
$83$	−12.6241	−1.38568	−0.692839	−	0.721092i	$-0.743640\pi$
$83$	−12.6241	−1.38568	−0.692839	+	0.721092i	$0.743640\pi$
$84$	0	0
$85$	7.65662	0.830478
$86$	9.53350	1.02802
$87$	0	0
$88$	2.44725	0.260878
$89$	−1.83979	−0.195017	−0.0975085	−	0.995235i	$-0.531087\pi$
$89$	−1.83979	−0.195017	−0.0975085	+	0.995235i	$0.531087\pi$
$90$	0	0
$91$	9.83098	1.03057
$92$	3.76873	0.392918
$93$	0	0
$94$	−7.85340	−0.810016
$95$	−11.3536	−1.16486
$96$	0	0
$97$	−0.120628	−0.0122479	−0.00612394	−	0.999981i	$-0.501949\pi$
$97$	−0.120628	−0.0122479	−0.00612394	+	0.999981i	$0.501949\pi$
$98$	5.43406	0.548923
$99$	0	0
$100$	0.901209	0.0901209
$101$	8.87014	0.882611	0.441306	−	0.897357i	$-0.354515\pi$
$101$	8.87014	0.882611	0.441306	+	0.897357i	$0.354515\pi$
$102$	0	0
$103$	6.58536	0.648874	0.324437	−	0.945907i	$-0.394825\pi$
$103$	6.58536	0.648874	0.324437	+	0.945907i	$0.394825\pi$
$104$	19.8390	1.94537
$105$	0	0
$106$	−0.801896	−0.0778870
$107$	−9.75138	−0.942701	−0.471351	−	0.881946i	$-0.656233\pi$
$107$	−9.75138	−0.942701	−0.471351	+	0.881946i	$0.656233\pi$
$108$	0	0
$109$	−2.57634	−0.246769	−0.123384	−	0.992359i	$-0.539375\pi$
$109$	−2.57634	−0.246769	−0.123384	+	0.992359i	$0.539375\pi$
$110$	1.75588	0.167417
$111$	0	0
$112$	−3.48878	−0.329658
$113$	5.30400	0.498959	0.249479	−	0.968380i	$-0.419741\pi$
$113$	5.30400	0.498959	0.249479	+	0.968380i	$0.419741\pi$
$114$	0	0
$115$	11.0597	1.03132
$116$	0	0
$117$	0	0
$118$	3.10783	0.286099
$119$	−6.15048	−0.563813
$120$	0	0
$121$	−10.3682	−0.942568
$122$	9.51998	0.861898
$123$	0	0
$124$	0.408456	0.0366804
$125$	12.1415	1.08597
$126$	0	0
$127$	12.7856	1.13454	0.567269	−	0.823533i	$-0.308000\pi$
$127$	12.7856	1.13454	0.567269	+	0.823533i	$0.308000\pi$
$128$	−3.05475	−0.270004
$129$	0	0
$130$	14.2343	1.24843
$131$	17.2193	1.50446	0.752228	−	0.658903i	$-0.228979\pi$
$131$	17.2193	1.50446	0.752228	+	0.658903i	$0.228979\pi$
$132$	0	0
$133$	9.12023	0.790824
$134$	−4.59848	−0.397249
$135$	0	0
$136$	−12.4117	−1.06430
$137$	−2.68136	−0.229084	−0.114542	−	0.993418i	$-0.536540\pi$
$137$	−2.68136	−0.229084	−0.114542	+	0.993418i	$0.536540\pi$
$138$	0	0
$139$	−13.7310	−1.16465	−0.582325	−	0.812956i	$-0.697857\pi$
$139$	−13.7310	−1.16465	−0.582325	+	0.812956i	$0.697857\pi$
$140$	1.87566	0.158522
$141$	0	0
$142$	−3.63811	−0.305303
$143$	5.12142	0.428275
$144$	0	0
$145$	0	0
$146$	1.06447	0.0880965
$147$	0	0
$148$	1.63897	0.134723
$149$	−0.576892	−0.0472609	−0.0236304	−	0.999721i	$-0.507523\pi$
$149$	−0.576892	−0.0472609	−0.0236304	+	0.999721i	$0.507523\pi$
$150$	0	0
$151$	−22.1168	−1.79984	−0.899921	−	0.436053i	$-0.856376\pi$
$151$	−22.1168	−1.79984	−0.899921	+	0.436053i	$0.856376\pi$
$152$	18.4047	1.49282
$153$	0	0
$154$	−1.41048	−0.113660
$155$	1.19865	0.0962776
$156$	0	0
$157$	1.53417	0.122440	0.0612199	−	0.998124i	$-0.480501\pi$
$157$	1.53417	0.122440	0.0612199	+	0.998124i	$0.480501\pi$
$158$	14.9842	1.19208
$159$	0	0
$160$	6.64475	0.525313
$161$	−8.88409	−0.700164
$162$	0	0
$163$	15.4394	1.20931	0.604653	−	0.796489i	$-0.293312\pi$
$163$	15.4394	1.20931	0.604653	+	0.796489i	$0.293312\pi$
$164$	−1.61166	−0.125849
$165$	0	0
$166$	14.6829	1.13962
$167$	9.01466	0.697576	0.348788	−	0.937202i	$-0.386593\pi$
$167$	9.01466	0.697576	0.348788	+	0.937202i	$0.386593\pi$
$168$	0	0
$169$	28.5176	2.19366
$170$	−8.90529	−0.683005
$171$	0	0
$172$	5.30524	0.404521
$173$	9.60089	0.729942	0.364971	−	0.931019i	$-0.381079\pi$
$173$	9.60089	0.729942	0.364971	+	0.931019i	$0.381079\pi$
$174$	0	0
$175$	−2.12443	−0.160592
$176$	−1.81747	−0.136997
$177$	0	0
$178$	2.13983	0.160387
$179$	−24.4563	−1.82795	−0.913976	−	0.405768i	$-0.867004\pi$
$179$	−24.4563	−1.82795	−0.913976	+	0.405768i	$0.867004\pi$
$180$	0	0
$181$	−5.85747	−0.435383	−0.217691	−	0.976018i	$-0.569853\pi$
$181$	−5.85747	−0.435383	−0.217691	+	0.976018i	$0.569853\pi$
$182$	−11.4342	−0.847563
$183$	0	0
$184$	−17.9282	−1.32168
$185$	4.80970	0.353616
$186$	0	0
$187$	−3.20407	−0.234305
$188$	−4.37029	−0.318736
$189$	0	0
$190$	13.2052	0.958007
$191$	18.9202	1.36902	0.684511	−	0.729003i	$-0.260016\pi$
$191$	18.9202	1.36902	0.684511	+	0.729003i	$0.260016\pi$
$192$	0	0
$193$	3.53259	0.254281	0.127141	−	0.991885i	$-0.459420\pi$
$193$	3.53259	0.254281	0.127141	+	0.991885i	$0.459420\pi$
$194$	0.140300	0.0100729
$195$	0	0
$196$	3.02396	0.215997
$197$	−22.0854	−1.57352	−0.786759	−	0.617260i	$-0.788243\pi$
$197$	−22.0854	−1.57352	−0.786759	+	0.617260i	$0.788243\pi$
$198$	0	0
$199$	8.38466	0.594373	0.297187	−	0.954819i	$-0.403952\pi$
$199$	8.38466	0.594373	0.297187	+	0.954819i	$0.403952\pi$
$200$	−4.28712	−0.303145
$201$	0	0
$202$	−10.3167	−0.725881
$203$	0	0
$204$	0	0
$205$	−4.72954	−0.330325
$206$	−7.65932	−0.533650
$207$	0	0
$208$	−14.7336	−1.02159
$209$	4.75116	0.328645
$210$	0	0
$211$	3.73405	0.257062	0.128531	−	0.991705i	$-0.458974\pi$
$211$	3.73405	0.257062	0.128531	+	0.991705i	$0.458974\pi$
$212$	−0.446242	−0.0306480
$213$	0	0
$214$	11.3417	0.775300
$215$	15.5687	1.06177
$216$	0	0
$217$	−0.962859	−0.0653631
$218$	2.99650	0.202949
$219$	0	0
$220$	0.977118	0.0658773
$221$	−25.9743	−1.74722
$222$	0	0
$223$	4.27748	0.286441	0.143220	−	0.989691i	$-0.454254\pi$
$223$	4.27748	0.286441	0.143220	+	0.989691i	$0.454254\pi$
$224$	−5.33765	−0.356636
$225$	0	0
$226$	−6.16900	−0.410356
$227$	−4.28954	−0.284707	−0.142353	−	0.989816i	$-0.545467\pi$
$227$	−4.28954	−0.284707	−0.142353	+	0.989816i	$0.545467\pi$
$228$	0	0
$229$	7.39575	0.488725	0.244362	−	0.969684i	$-0.421421\pi$
$229$	7.39575	0.488725	0.244362	+	0.969684i	$0.421421\pi$
$230$	−12.8633	−0.848181
$231$	0	0
$232$	0	0
$233$	−11.6776	−0.765023	−0.382511	−	0.923951i	$-0.624941\pi$
$233$	−11.6776	−0.765023	−0.382511	+	0.923951i	$0.624941\pi$
$234$	0	0
$235$	−12.8250	−0.836608
$236$	1.72946	0.112578
$237$	0	0
$238$	7.15352	0.463694
$239$	2.64677	0.171205	0.0856026	−	0.996329i	$-0.472718\pi$
$239$	2.64677	0.171205	0.0856026	+	0.996329i	$0.472718\pi$
$240$	0	0
$241$	−26.7417	−1.72259	−0.861293	−	0.508109i	$-0.830345\pi$
$241$	−26.7417	−1.72259	−0.861293	+	0.508109i	$0.830345\pi$
$242$	12.0591	0.775190
$243$	0	0
$244$	5.29771	0.339151
$245$	8.87407	0.566944
$246$	0	0
$247$	38.5160	2.45071
$248$	−1.94306	−0.123384
$249$	0	0
$250$	−14.1216	−0.893129
$251$	7.66160	0.483596	0.241798	−	0.970327i	$-0.422263\pi$
$251$	7.66160	0.483596	0.241798	+	0.970327i	$0.422263\pi$
$252$	0	0
$253$	−4.62814	−0.290969
$254$	−14.8707	−0.933072
$255$	0	0
$256$	−13.7314	−0.858210
$257$	11.8853	0.741384	0.370692	−	0.928756i	$-0.379120\pi$
$257$	11.8853	0.741384	0.370692	+	0.928756i	$0.379120\pi$
$258$	0	0
$259$	−3.86358	−0.240071
$260$	7.92116	0.491249
$261$	0	0
$262$	−20.0275	−1.23730
$263$	−15.8491	−0.977297	−0.488648	−	0.872481i	$-0.662510\pi$
$263$	−15.8491	−0.977297	−0.488648	+	0.872481i	$0.662510\pi$
$264$	0	0
$265$	−1.30953	−0.0804440
$266$	−10.6076	−0.650393
$267$	0	0
$268$	−2.55898	−0.156315
$269$	1.03867	0.0633289	0.0316644	−	0.999499i	$-0.489919\pi$
$269$	1.03867	0.0633289	0.0316644	+	0.999499i	$0.489919\pi$
$270$	0	0
$271$	12.3202	0.748400	0.374200	−	0.927348i	$-0.377917\pi$
$271$	12.3202	0.748400	0.374200	+	0.927348i	$0.377917\pi$
$272$	9.21765	0.558902
$273$	0	0
$274$	3.11865	0.188404
$275$	−1.10672	−0.0667376
$276$	0	0
$277$	−12.2808	−0.737883	−0.368941	−	0.929453i	$-0.620280\pi$
$277$	−12.2808	−0.737883	−0.368941	+	0.929453i	$0.620280\pi$
$278$	15.9703	0.957836
$279$	0	0
$280$	−8.92265	−0.533231
$281$	−0.656198	−0.0391455	−0.0195727	−	0.999808i	$-0.506231\pi$
$281$	−0.656198	−0.0391455	−0.0195727	+	0.999808i	$0.506231\pi$
$282$	0	0
$283$	3.02486	0.179810	0.0899048	−	0.995950i	$-0.471344\pi$
$283$	3.02486	0.179810	0.0899048	+	0.995950i	$0.471344\pi$
$284$	−2.02455	−0.120135
$285$	0	0
$286$	−5.95664	−0.352223
$287$	3.79918	0.224258
$288$	0	0
$289$	−0.749913	−0.0441125
$290$	0	0
$291$	0	0
$292$	0.592362	0.0346654
$293$	−15.5801	−0.910198	−0.455099	−	0.890441i	$-0.650396\pi$
$293$	−15.5801	−0.910198	−0.455099	+	0.890441i	$0.650396\pi$
$294$	0	0
$295$	5.07523	0.295491
$296$	−7.79673	−0.453176
$297$	0	0
$298$	0.670974	0.0388685
$299$	−37.5187	−2.16976
$300$	0	0
$301$	−12.5061	−0.720841
$302$	25.7237	1.48023
$303$	0	0
$304$	−13.6684	−0.783936
$305$	15.5466	0.890194
$306$	0	0
$307$	12.6496	0.721952	0.360976	−	0.932575i	$-0.382444\pi$
$307$	12.6496	0.721952	0.360976	+	0.932575i	$0.382444\pi$
$308$	−0.784908	−0.0447243
$309$	0	0
$310$	−1.39413	−0.0791810
$311$	16.5615	0.939116	0.469558	−	0.882902i	$-0.344413\pi$
$311$	16.5615	0.939116	0.469558	+	0.882902i	$0.344413\pi$
$312$	0	0
$313$	−3.52683	−0.199348	−0.0996740	−	0.995020i	$-0.531780\pi$
$313$	−3.52683	−0.199348	−0.0996740	+	0.995020i	$0.531780\pi$
$314$	−1.78436	−0.100697
$315$	0	0
$316$	8.33848	0.469076
$317$	5.33178	0.299463	0.149731	−	0.988727i	$-0.452159\pi$
$317$	5.33178	0.299463	0.149731	+	0.988727i	$0.452159\pi$
$318$	0	0
$319$	0	0
$320$	−16.4146	−0.917606
$321$	0	0
$322$	10.3329	0.575832
$323$	−24.0964	−1.34076
$324$	0	0
$325$	−8.97177	−0.497664
$326$	−17.9573	−0.994563
$327$	0	0
$328$	7.66677	0.423327
$329$	10.3021	0.567975
$330$	0	0
$331$	−0.600782	−0.0330220	−0.0165110	−	0.999864i	$-0.505256\pi$
$331$	−0.600782	−0.0330220	−0.0165110	+	0.999864i	$0.505256\pi$
$332$	8.17080	0.448431
$333$	0	0
$334$	−10.4848	−0.573703
$335$	−7.50954	−0.410290
$336$	0	0
$337$	−24.1449	−1.31526	−0.657628	−	0.753342i	$-0.728440\pi$
$337$	−24.1449	−1.31526	−0.657628	+	0.753342i	$0.728440\pi$
$338$	−33.1683	−1.80412
$339$	0	0
$340$	−4.95565	−0.268758
$341$	−0.501598	−0.0271631
$342$	0	0
$343$	−17.8086	−0.961575
$344$	−25.2374	−1.36071
$345$	0	0
$346$	−11.1666	−0.600322
$347$	−16.7193	−0.897540	−0.448770	−	0.893647i	$-0.648138\pi$
$347$	−16.7193	−0.897540	−0.448770	+	0.893647i	$0.648138\pi$
$348$	0	0
$349$	10.7789	0.576983	0.288491	−	0.957482i	$-0.406846\pi$
$349$	10.7789	0.576983	0.288491	+	0.957482i	$0.406846\pi$
$350$	2.47089	0.132075
$351$	0	0
$352$	−2.78063	−0.148208
$353$	−10.4429	−0.555821	−0.277911	−	0.960607i	$-0.589642\pi$
$353$	−10.4429	−0.555821	−0.277911	+	0.960607i	$0.589642\pi$
$354$	0	0
$355$	−5.94120	−0.315326
$356$	1.19078	0.0631111
$357$	0	0
$358$	28.4448	1.50335
$359$	3.58689	0.189309	0.0946543	−	0.995510i	$-0.469825\pi$
$359$	3.58689	0.189309	0.0946543	+	0.995510i	$0.469825\pi$
$360$	0	0
$361$	16.7314	0.880601
$362$	6.81273	0.358069
$363$	0	0
$364$	−6.36297	−0.333510
$365$	1.73834	0.0909887
$366$	0	0
$367$	−9.22595	−0.481590	−0.240795	−	0.970576i	$-0.577408\pi$
$367$	−9.22595	−0.481590	−0.240795	+	0.970576i	$0.577408\pi$
$368$	13.3145	0.694066
$369$	0	0
$370$	−5.59409	−0.290823
$371$	1.05193	0.0546136
$372$	0	0
$373$	−21.5039	−1.11343	−0.556715	−	0.830704i	$-0.687938\pi$
$373$	−21.5039	−1.11343	−0.556715	+	0.830704i	$0.687938\pi$
$374$	3.72660	0.192698
$375$	0	0
$376$	20.7898	1.07215
$377$	0	0
$378$	0	0
$379$	−18.9907	−0.975487	−0.487743	−	0.872987i	$-0.662180\pi$
$379$	−18.9907	−0.975487	−0.487743	+	0.872987i	$0.662180\pi$
$380$	7.34849	0.376969
$381$	0	0
$382$	−22.0058	−1.12592
$383$	−29.2661	−1.49543	−0.747714	−	0.664021i	$-0.768848\pi$
$383$	−29.2661	−1.49543	−0.747714	+	0.664021i	$0.768848\pi$
$384$	0	0
$385$	−2.30338	−0.117391
$386$	−4.10870	−0.209127
$387$	0	0
$388$	0.0780746	0.00396364
$389$	−35.4572	−1.79775	−0.898876	−	0.438203i	$-0.855615\pi$
$389$	−35.4572	−1.79775	−0.898876	+	0.438203i	$0.855615\pi$
$390$	0	0
$391$	23.4725	1.18706
$392$	−14.3852	−0.726564
$393$	0	0
$394$	25.6871	1.29410
$395$	24.4700	1.23122
$396$	0	0
$397$	16.1019	0.808129	0.404064	−	0.914731i	$-0.367597\pi$
$397$	16.1019	0.808129	0.404064	+	0.914731i	$0.367597\pi$
$398$	−9.75206	−0.488827
$399$	0	0
$400$	3.18386	0.159193
$401$	24.1448	1.20574	0.602868	−	0.797841i	$-0.294025\pi$
$401$	24.1448	1.20574	0.602868	+	0.797841i	$0.294025\pi$
$402$	0	0
$403$	−4.06628	−0.202556
$404$	−5.74108	−0.285629
$405$	0	0
$406$	0	0
$407$	−2.01272	−0.0997668
$408$	0	0
$409$	−16.4675	−0.814264	−0.407132	−	0.913369i	$-0.633471\pi$
$409$	−16.4675	−0.814264	−0.407132	+	0.913369i	$0.633471\pi$
$410$	5.50084	0.271667
$411$	0	0
$412$	−4.26229	−0.209988
$413$	−4.07687	−0.200610
$414$	0	0
$415$	23.9779	1.17703
$416$	−22.5416	−1.10519
$417$	0	0
$418$	−5.52599	−0.270285
$419$	−15.5115	−0.757789	−0.378894	−	0.925440i	$-0.623696\pi$
$419$	−15.5115	−0.757789	−0.378894	+	0.925440i	$0.623696\pi$
$420$	0	0
$421$	−12.2515	−0.597101	−0.298551	−	0.954394i	$-0.596503\pi$
$421$	−12.2515	−0.597101	−0.298551	+	0.954394i	$0.596503\pi$
$422$	−4.34301	−0.211414
$423$	0	0
$424$	2.12281	0.103093
$425$	5.61294	0.272267
$426$	0	0
$427$	−12.4884	−0.604355
$428$	6.31145	0.305075
$429$	0	0
$430$	−18.1076	−0.873228
$431$	−21.3021	−1.02608	−0.513042	−	0.858364i	$-0.671481\pi$
$431$	−21.3021	−1.02608	−0.513042	+	0.858364i	$0.671481\pi$
$432$	0	0
$433$	23.7180	1.13981	0.569906	−	0.821710i	$-0.306980\pi$
$433$	23.7180	1.13981	0.569906	+	0.821710i	$0.306980\pi$
$434$	1.11988	0.0537562
$435$	0	0
$436$	1.66750	0.0798589
$437$	−34.8062	−1.66501
$438$	0	0
$439$	−20.1785	−0.963067	−0.481533	−	0.876428i	$-0.659920\pi$
$439$	−20.1785	−0.963067	−0.481533	+	0.876428i	$0.659920\pi$
$440$	−4.64823	−0.221596
$441$	0	0
$442$	30.2103	1.43696
$443$	−18.0735	−0.858697	−0.429348	−	0.903139i	$-0.641257\pi$
$443$	−18.0735	−0.858697	−0.429348	+	0.903139i	$0.641257\pi$
$444$	0	0
$445$	3.49444	0.165652
$446$	−4.97506	−0.235576
$447$	0	0
$448$	13.1857	0.622965
$449$	30.9126	1.45886	0.729429	−	0.684057i	$-0.239786\pi$
$449$	30.9126	1.45886	0.729429	+	0.684057i	$0.239786\pi$
$450$	0	0
$451$	1.97917	0.0931955
$452$	−3.43295	−0.161472
$453$	0	0
$454$	4.98909	0.234150
$455$	−18.6727	−0.875388
$456$	0	0
$457$	−25.0580	−1.17216	−0.586082	−	0.810252i	$-0.699330\pi$
$457$	−25.0580	−1.17216	−0.586082	+	0.810252i	$0.699330\pi$
$458$	−8.60187	−0.401939
$459$	0	0
$460$	−7.15822	−0.333754
$461$	−34.9121	−1.62602	−0.813010	−	0.582250i	$-0.802173\pi$
$461$	−34.9121	−1.62602	−0.813010	+	0.582250i	$0.802173\pi$
$462$	0	0
$463$	13.9627	0.648901	0.324450	−	0.945903i	$-0.394821\pi$
$463$	13.9627	0.648901	0.324450	+	0.945903i	$0.394821\pi$
$464$	0	0
$465$	0	0
$466$	13.5820	0.629173
$467$	−39.8109	−1.84223	−0.921115	−	0.389290i	$-0.872720\pi$
$467$	−39.8109	−1.84223	−0.921115	+	0.389290i	$0.872720\pi$
$468$	0	0
$469$	6.03232	0.278547
$470$	14.9165	0.688047
$471$	0	0
$472$	−8.22716	−0.378686
$473$	−6.51503	−0.299561
$474$	0	0
$475$	−8.32314	−0.381892
$476$	3.98081	0.182460
$477$	0	0
$478$	−3.07841	−0.140803
$479$	−12.1225	−0.553891	−0.276945	−	0.960886i	$-0.589322\pi$
$479$	−12.1225	−0.553891	−0.276945	+	0.960886i	$0.589322\pi$
$480$	0	0
$481$	−16.3164	−0.743964
$482$	31.1029	1.41670
$483$	0	0
$484$	6.71071	0.305032
$485$	0.229116	0.0104036
$486$	0	0
$487$	12.8270	0.581247	0.290624	−	0.956837i	$-0.406137\pi$
$487$	12.8270	0.581247	0.290624	+	0.956837i	$0.406137\pi$
$488$	−25.2016	−1.14082
$489$	0	0
$490$	−10.3213	−0.466268
$491$	24.1103	1.08808	0.544041	−	0.839058i	$-0.316893\pi$
$491$	24.1103	1.08808	0.544041	+	0.839058i	$0.316893\pi$
$492$	0	0
$493$	0	0
$494$	−44.7973	−2.01553
$495$	0	0
$496$	1.44303	0.0647937
$497$	4.77250	0.214076
$498$	0	0
$499$	18.4126	0.824259	0.412130	−	0.911125i	$-0.364785\pi$
$499$	18.4126	0.824259	0.412130	+	0.911125i	$0.364785\pi$
$500$	−7.85844	−0.351440
$501$	0	0
$502$	−8.91108	−0.397721
$503$	−10.9102	−0.486463	−0.243232	−	0.969968i	$-0.578208\pi$
$503$	−10.9102	−0.486463	−0.243232	+	0.969968i	$0.578208\pi$
$504$	0	0
$505$	−16.8477	−0.749711
$506$	5.38291	0.239300
$507$	0	0
$508$	−8.27531	−0.367157
$509$	24.2119	1.07318	0.536588	−	0.843844i	$-0.319713\pi$
$509$	24.2119	1.07318	0.536588	+	0.843844i	$0.319713\pi$
$510$	0	0
$511$	−1.39638	−0.0617724
$512$	22.0802	0.975817
$513$	0	0
$514$	−13.8236	−0.609732
$515$	−12.5080	−0.551170
$516$	0	0
$517$	5.36687	0.236035
$518$	4.49366	0.197440
$519$	0	0
$520$	−37.6816	−1.65245
$521$	41.7398	1.82865	0.914327	−	0.404976i	$-0.132720\pi$
$521$	41.7398	1.82865	0.914327	+	0.404976i	$0.132720\pi$
$522$	0	0
$523$	−18.9831	−0.830074	−0.415037	−	0.909804i	$-0.636231\pi$
$523$	−18.9831	−0.830074	−0.415037	+	0.909804i	$0.636231\pi$
$524$	−11.1450	−0.486870
$525$	0	0
$526$	18.4338	0.803753
$527$	2.54396	0.110816
$528$	0	0
$529$	10.9050	0.474132
$530$	1.52310	0.0661591
$531$	0	0
$532$	−5.90295	−0.255925
$533$	16.0444	0.694962
$534$	0	0
$535$	18.5215	0.800753
$536$	12.1733	0.525805
$537$	0	0
$538$	−1.20806	−0.0520832
$539$	−3.71354	−0.159953
$540$	0	0
$541$	3.36327	0.144598	0.0722992	−	0.997383i	$-0.476966\pi$
$541$	3.36327	0.144598	0.0722992	+	0.997383i	$0.476966\pi$
$542$	−14.3294	−0.615502
$543$	0	0
$544$	14.1025	0.604641
$545$	4.89343	0.209611
$546$	0	0
$547$	−17.0391	−0.728540	−0.364270	−	0.931293i	$-0.618681\pi$
$547$	−17.0391	−0.728540	−0.364270	+	0.931293i	$0.618681\pi$
$548$	1.73548	0.0741358
$549$	0	0
$550$	1.28720	0.0548866
$551$	0	0
$552$	0	0
$553$	−19.6564	−0.835876
$554$	14.2836	0.606852
$555$	0	0
$556$	8.88722	0.376902
$557$	4.93067	0.208919	0.104460	−	0.994529i	$-0.466689\pi$
$557$	4.93067	0.208919	0.104460	+	0.994529i	$0.466689\pi$
$558$	0	0
$559$	−52.8150	−2.23384
$560$	6.62648	0.280020
$561$	0	0
$562$	0.763213	0.0321942
$563$	−0.638084	−0.0268920	−0.0134460	−	0.999910i	$-0.504280\pi$
$563$	−0.638084	−0.0268920	−0.0134460	+	0.999910i	$0.504280\pi$
$564$	0	0
$565$	−10.0743	−0.423828
$566$	−3.51817	−0.147880
$567$	0	0
$568$	9.63094	0.404105
$569$	−31.5258	−1.32163	−0.660815	−	0.750549i	$-0.729789\pi$
$569$	−31.5258	−1.32163	−0.660815	+	0.750549i	$0.729789\pi$
$570$	0	0
$571$	−12.7801	−0.534832	−0.267416	−	0.963581i	$-0.586170\pi$
$571$	−12.7801	−0.534832	−0.267416	+	0.963581i	$0.586170\pi$
$572$	−3.31477	−0.138598
$573$	0	0
$574$	−4.41876	−0.184436
$575$	8.10764	0.338112
$576$	0	0
$577$	12.3175	0.512786	0.256393	−	0.966573i	$-0.417466\pi$
$577$	12.3175	0.512786	0.256393	+	0.966573i	$0.417466\pi$
$578$	0.872212	0.0362792
$579$	0	0
$580$	0	0
$581$	−19.2612	−0.799087
$582$	0	0
$583$	0.548001	0.0226959
$584$	−2.81792	−0.116606
$585$	0	0
$586$	18.1209	0.748569
$587$	27.6342	1.14059	0.570293	−	0.821441i	$-0.306830\pi$
$587$	27.6342	1.14059	0.570293	+	0.821441i	$0.306830\pi$
$588$	0	0
$589$	−3.77230	−0.155435
$590$	−5.90292	−0.243019
$591$	0	0
$592$	5.79030	0.237980
$593$	−2.27226	−0.0933105	−0.0466552	−	0.998911i	$-0.514856\pi$
$593$	−2.27226	−0.0933105	−0.0466552	+	0.998911i	$0.514856\pi$
$594$	0	0
$595$	11.6820	0.478916
$596$	0.373386	0.0152945
$597$	0	0
$598$	43.6374	1.78447
$599$	−36.1311	−1.47628	−0.738139	−	0.674649i	$-0.764295\pi$
$599$	−36.1311	−1.47628	−0.738139	+	0.674649i	$0.764295\pi$
$600$	0	0
$601$	37.8045	1.54208	0.771040	−	0.636787i	$-0.219737\pi$
$601$	37.8045	1.54208	0.771040	+	0.636787i	$0.219737\pi$
$602$	14.5457	0.592837
$603$	0	0
$604$	14.3148	0.582462
$605$	19.6931	0.800640
$606$	0	0
$607$	12.5466	0.509252	0.254626	−	0.967040i	$-0.418048\pi$
$607$	12.5466	0.509252	0.254626	+	0.967040i	$0.418048\pi$
$608$	−20.9119	−0.848091
$609$	0	0
$610$	−18.0820	−0.732117
$611$	43.5073	1.76012
$612$	0	0
$613$	−36.6463	−1.48013	−0.740065	−	0.672535i	$-0.765205\pi$
$613$	−36.6463	−1.48013	−0.740065	+	0.672535i	$0.765205\pi$
$614$	−14.7126	−0.593751
$615$	0	0
$616$	3.73387	0.150442
$617$	−40.2748	−1.62140	−0.810701	−	0.585460i	$-0.800914\pi$
$617$	−40.2748	−1.62140	−0.810701	+	0.585460i	$0.800914\pi$
$618$	0	0
$619$	−4.55867	−0.183228	−0.0916142	−	0.995795i	$-0.529203\pi$
$619$	−4.55867	−0.183228	−0.0916142	+	0.995795i	$0.529203\pi$
$620$	−0.775808	−0.0311572
$621$	0	0
$622$	−19.2624	−0.772351
$623$	−2.80704	−0.112462
$624$	0	0
$625$	−16.0993	−0.643971
$626$	4.10199	0.163949
$627$	0	0
$628$	−0.992968	−0.0396237
$629$	10.2079	0.407016
$630$	0	0
$631$	36.4307	1.45028	0.725142	−	0.688599i	$-0.241774\pi$
$631$	36.4307	1.45028	0.725142	+	0.688599i	$0.241774\pi$
$632$	−39.6668	−1.57786
$633$	0	0
$634$	−6.20130	−0.246285
$635$	−24.2846	−0.963704
$636$	0	0
$637$	−30.1044	−1.19278
$638$	0	0
$639$	0	0
$640$	5.80210	0.229348
$641$	−8.71576	−0.344252	−0.172126	−	0.985075i	$-0.555064\pi$
$641$	−8.71576	−0.344252	−0.172126	+	0.985075i	$0.555064\pi$
$642$	0	0
$643$	23.5712	0.929559	0.464780	−	0.885426i	$-0.346134\pi$
$643$	23.5712	0.929559	0.464780	+	0.885426i	$0.346134\pi$
$644$	5.75011	0.226586
$645$	0	0
$646$	28.0262	1.10268
$647$	16.7106	0.656963	0.328481	−	0.944510i	$-0.393463\pi$
$647$	16.7106	0.656963	0.328481	+	0.944510i	$0.393463\pi$
$648$	0	0
$649$	−2.12384	−0.0833678
$650$	10.4349	0.409291
$651$	0	0
$652$	−9.99294	−0.391354
$653$	19.3825	0.758496	0.379248	−	0.925295i	$-0.376183\pi$
$653$	19.3825	0.758496	0.379248	+	0.925295i	$0.376183\pi$
$654$	0	0
$655$	−32.7058	−1.27792
$656$	−5.69379	−0.222305
$657$	0	0
$658$	−11.9822	−0.467117
$659$	−41.2814	−1.60809	−0.804047	−	0.594566i	$-0.797324\pi$
$659$	−41.2814	−1.60809	−0.804047	+	0.594566i	$0.797324\pi$
$660$	0	0
$661$	3.24439	0.126192	0.0630960	−	0.998007i	$-0.479903\pi$
$661$	3.24439	0.126192	0.0630960	+	0.998007i	$0.479903\pi$
$662$	0.698760	0.0271581
$663$	0	0
$664$	−38.8692	−1.50842
$665$	−17.3227	−0.671745
$666$	0	0
$667$	0	0
$668$	−5.83462	−0.225748
$669$	0	0
$670$	8.73422	0.337432
$671$	−6.50578	−0.251153
$672$	0	0
$673$	25.7451	0.992399	0.496199	−	0.868209i	$-0.334728\pi$
$673$	25.7451	0.992399	0.496199	+	0.868209i	$0.334728\pi$
$674$	28.0825	1.08170
$675$	0	0
$676$	−18.4576	−0.709908
$677$	41.5377	1.59642	0.798211	−	0.602378i	$-0.205780\pi$
$677$	41.5377	1.59642	0.798211	+	0.602378i	$0.205780\pi$
$678$	0	0
$679$	−0.184046	−0.00706305
$680$	23.5744	0.904038
$681$	0	0
$682$	0.583401	0.0223396
$683$	−48.2204	−1.84510	−0.922551	−	0.385876i	$-0.873899\pi$
$683$	−48.2204	−1.84510	−0.922551	+	0.385876i	$0.873899\pi$
$684$	0	0
$685$	5.09290	0.194590
$686$	20.7129	0.790823
$687$	0	0
$688$	18.7428	0.714562
$689$	4.44245	0.169244
$690$	0	0
$691$	−4.13218	−0.157196	−0.0785978	−	0.996906i	$-0.525044\pi$
$691$	−4.13218	−0.157196	−0.0785978	+	0.996906i	$0.525044\pi$
$692$	−6.21405	−0.236223
$693$	0	0
$694$	19.4460	0.738159
$695$	26.0803	0.989281
$696$	0	0
$697$	−10.0378	−0.380207
$698$	−12.5368	−0.474525
$699$	0	0
$700$	1.37501	0.0519706
$701$	15.3801	0.580898	0.290449	−	0.956890i	$-0.406195\pi$
$701$	15.3801	0.580898	0.290449	+	0.956890i	$0.406195\pi$
$702$	0	0
$703$	−15.1368	−0.570895
$704$	6.86904	0.258887
$705$	0	0
$706$	12.1460	0.457121
$707$	13.5335	0.508981
$708$	0	0
$709$	−41.6729	−1.56506	−0.782529	−	0.622614i	$-0.786071\pi$
$709$	−41.6729	−1.56506	−0.782529	+	0.622614i	$0.786071\pi$
$710$	6.91011	0.259332
$711$	0	0
$712$	−5.66463	−0.212291
$713$	3.67463	0.137616
$714$	0	0
$715$	−9.72747	−0.363787
$716$	15.8290	0.591559
$717$	0	0
$718$	−4.17185	−0.155692
$719$	46.5973	1.73779	0.868893	−	0.495001i	$-0.164832\pi$
$719$	46.5973	1.73779	0.868893	+	0.495001i	$0.164832\pi$
$720$	0	0
$721$	10.0475	0.374190
$722$	−19.4600	−0.724227
$723$	0	0
$724$	3.79117	0.140898
$725$	0	0
$726$	0	0
$727$	−5.83560	−0.216431	−0.108215	−	0.994127i	$-0.534514\pi$
$727$	−5.83560	−0.216431	−0.108215	+	0.994127i	$0.534514\pi$
$728$	30.2692	1.12185
$729$	0	0
$730$	−2.02183	−0.0748313
$731$	33.0422	1.22211
$732$	0	0
$733$	−8.38662	−0.309767	−0.154883	−	0.987933i	$-0.549500\pi$
$733$	−8.38662	−0.309767	−0.154883	+	0.987933i	$0.549500\pi$
$734$	10.7305	0.396072
$735$	0	0
$736$	20.3705	0.750865
$737$	3.14252	0.115756
$738$	0	0
$739$	39.6588	1.45887	0.729436	−	0.684049i	$-0.239783\pi$
$739$	39.6588	1.45887	0.729436	+	0.684049i	$0.239783\pi$
$740$	−3.11302	−0.114437
$741$	0	0
$742$	−1.22349	−0.0449156
$743$	−19.1090	−0.701042	−0.350521	−	0.936555i	$-0.613995\pi$
$743$	−19.1090	−0.701042	−0.350521	+	0.936555i	$0.613995\pi$
$744$	0	0
$745$	1.09573	0.0401445
$746$	25.0108	0.915711
$747$	0	0
$748$	2.07379	0.0758254
$749$	−14.8781	−0.543633
$750$	0	0
$751$	−32.0784	−1.17056	−0.585278	−	0.810832i	$-0.699015\pi$
$751$	−32.0784	−1.17056	−0.585278	+	0.810832i	$0.699015\pi$
$752$	−15.4397	−0.563028
$753$	0	0
$754$	0	0
$755$	42.0080	1.52883
$756$	0	0
$757$	11.4842	0.417402	0.208701	−	0.977979i	$-0.433076\pi$
$757$	11.4842	0.417402	0.208701	+	0.977979i	$0.433076\pi$
$758$	22.0878	0.802264
$759$	0	0
$760$	−34.9573	−1.26804
$761$	−29.5856	−1.07248	−0.536239	−	0.844066i	$-0.680155\pi$
$761$	−29.5856	−1.07248	−0.536239	+	0.844066i	$0.680155\pi$
$762$	0	0
$763$	−3.93083	−0.142306
$764$	−12.2459	−0.443040
$765$	0	0
$766$	34.0389	1.22988
$767$	−17.2172	−0.621677
$768$	0	0
$769$	−50.6990	−1.82825	−0.914127	−	0.405428i	$-0.867122\pi$
$769$	−50.6990	−1.82825	−0.914127	+	0.405428i	$0.867122\pi$
$770$	2.67902	0.0965451
$771$	0	0
$772$	−2.28642	−0.0822901
$773$	35.0782	1.26167	0.630837	−	0.775916i	$-0.282712\pi$
$773$	35.0782	1.26167	0.630837	+	0.775916i	$0.282712\pi$
$774$	0	0
$775$	0.878707	0.0315641
$776$	−0.371407	−0.0133327
$777$	0	0
$778$	41.2397	1.47851
$779$	14.8845	0.533292
$780$	0	0
$781$	2.48622	0.0889639
$782$	−27.3005	−0.976264
$783$	0	0
$784$	10.6833	0.381547
$785$	−2.91395	−0.104003
$786$	0	0
$787$	−26.0072	−0.927056	−0.463528	−	0.886082i	$-0.653417\pi$
$787$	−26.0072	−0.927056	−0.463528	+	0.886082i	$0.653417\pi$
$788$	14.2945	0.509219
$789$	0	0
$790$	−28.4606	−1.01258
$791$	8.09254	0.287738
$792$	0	0
$793$	−52.7401	−1.87286
$794$	−18.7278	−0.664625
$795$	0	0
$796$	−5.42686	−0.192350
$797$	−36.6478	−1.29813	−0.649066	−	0.760732i	$-0.724840\pi$
$797$	−36.6478	−1.29813	−0.649066	+	0.760732i	$0.724840\pi$
$798$	0	0
$799$	−27.2191	−0.962944
$800$	4.87115	0.172221
$801$	0	0
$802$	−28.0824	−0.991626
$803$	−0.727443	−0.0256709
$804$	0	0
$805$	16.8742	0.594736
$806$	4.72943	0.166587
$807$	0	0
$808$	27.3108	0.960790
$809$	49.5564	1.74231	0.871155	−	0.491009i	$-0.163372\pi$
$809$	49.5564	1.74231	0.871155	+	0.491009i	$0.163372\pi$
$810$	0	0
$811$	23.2411	0.816106	0.408053	−	0.912958i	$-0.366208\pi$
$811$	23.2411	0.816106	0.408053	+	0.912958i	$0.366208\pi$
$812$	0	0
$813$	0	0
$814$	2.34096	0.0820506
$815$	−29.3251	−1.02721
$816$	0	0
$817$	−48.9967	−1.71418
$818$	19.1530	0.669670
$819$	0	0
$820$	3.06113	0.106899
$821$	−8.86769	−0.309485	−0.154742	−	0.987955i	$-0.549455\pi$
$821$	−8.86769	−0.309485	−0.154742	+	0.987955i	$0.549455\pi$
$822$	0	0
$823$	50.5714	1.76281	0.881404	−	0.472364i	$-0.156599\pi$
$823$	50.5714	1.76281	0.881404	+	0.472364i	$0.156599\pi$
$824$	20.2760	0.706349
$825$	0	0
$826$	4.74174	0.164986
$827$	−35.7629	−1.24360	−0.621798	−	0.783177i	$-0.713598\pi$
$827$	−35.7629	−1.24360	−0.621798	+	0.783177i	$0.713598\pi$
$828$	0	0
$829$	−36.7260	−1.27555	−0.637774	−	0.770224i	$-0.720145\pi$
$829$	−36.7260	−1.27555	−0.637774	+	0.770224i	$0.720145\pi$
$830$	−27.8883	−0.968017
$831$	0	0
$832$	55.6849	1.93053
$833$	18.8339	0.652558
$834$	0	0
$835$	−17.1222	−0.592538
$836$	−3.07513	−0.106355
$837$	0	0
$838$	18.0412	0.623224
$839$	−19.9596	−0.689083	−0.344541	−	0.938771i	$-0.611966\pi$
$839$	−19.9596	−0.689083	−0.344541	+	0.938771i	$0.611966\pi$
$840$	0	0
$841$	0	0
$842$	14.2495	0.491071
$843$	0	0
$844$	−2.41681	−0.0831901
$845$	−54.1654	−1.86335
$846$	0	0
$847$	−15.8193	−0.543556
$848$	−1.57652	−0.0541379
$849$	0	0
$850$	−6.52831	−0.223919
$851$	14.7449	0.505448
$852$	0	0
$853$	48.9528	1.67611	0.838056	−	0.545584i	$-0.183692\pi$
$853$	48.9528	1.67611	0.838056	+	0.545584i	$0.183692\pi$
$854$	14.5250	0.497036
$855$	0	0
$856$	−30.0241	−1.02620
$857$	−47.5088	−1.62287	−0.811435	−	0.584443i	$-0.801313\pi$
$857$	−47.5088	−1.62287	−0.811435	+	0.584443i	$0.801313\pi$
$858$	0	0
$859$	10.6069	0.361904	0.180952	−	0.983492i	$-0.442082\pi$
$859$	10.6069	0.361904	0.180952	+	0.983492i	$0.442082\pi$
$860$	−10.0766	−0.343610
$861$	0	0
$862$	24.7761	0.843876
$863$	−19.6910	−0.670291	−0.335145	−	0.942166i	$-0.608785\pi$
$863$	−19.6910	−0.670291	−0.335145	+	0.942166i	$0.608785\pi$
$864$	0	0
$865$	−18.2356	−0.620030
$866$	−27.5860	−0.937409
$867$	0	0
$868$	0.623198	0.0211527
$869$	−10.2400	−0.347367
$870$	0	0
$871$	25.4753	0.863198
$872$	−7.93245	−0.268627
$873$	0	0
$874$	40.4826	1.36934
$875$	18.5248	0.626253
$876$	0	0
$877$	7.69046	0.259688	0.129844	−	0.991534i	$-0.458552\pi$
$877$	7.69046	0.259688	0.129844	+	0.991534i	$0.458552\pi$
$878$	23.4693	0.792049
$879$	0	0
$880$	3.45204	0.116368
$881$	27.5871	0.929434	0.464717	−	0.885459i	$-0.346156\pi$
$881$	27.5871	0.929434	0.464717	+	0.885459i	$0.346156\pi$
$882$	0	0
$883$	−28.4582	−0.957696	−0.478848	−	0.877898i	$-0.658945\pi$
$883$	−28.4582	−0.957696	−0.478848	+	0.877898i	$0.658945\pi$
$884$	16.8115	0.565433
$885$	0	0
$886$	21.0210	0.706213
$887$	22.5316	0.756536	0.378268	−	0.925696i	$-0.376520\pi$
$887$	22.5316	0.756536	0.378268	+	0.925696i	$0.376520\pi$
$888$	0	0
$889$	19.5075	0.654261
$890$	−4.06432	−0.136236
$891$	0	0
$892$	−2.76854	−0.0926976
$893$	40.3619	1.35066
$894$	0	0
$895$	46.4516	1.55271
$896$	−4.66076	−0.155705
$897$	0	0
$898$	−35.9540	−1.19980
$899$	0	0
$900$	0	0
$901$	−2.77930	−0.0925918
$902$	−2.30194	−0.0766463
$903$	0	0
$904$	16.3308	0.543155
$905$	11.1255	0.369824
$906$	0	0
$907$	41.3501	1.37301	0.686504	−	0.727126i	$-0.259145\pi$
$907$	41.3501	1.37301	0.686504	+	0.727126i	$0.259145\pi$
$908$	2.77635	0.0921363
$909$	0	0
$910$	21.7179	0.719940
$911$	8.27165	0.274052	0.137026	−	0.990567i	$-0.456246\pi$
$911$	8.27165	0.274052	0.137026	+	0.990567i	$0.456246\pi$
$912$	0	0
$913$	−10.0340	−0.332078
$914$	29.1446	0.964016
$915$	0	0
$916$	−4.78680	−0.158160
$917$	26.2722	0.867583
$918$	0	0
$919$	−27.7281	−0.914665	−0.457332	−	0.889296i	$-0.651195\pi$
$919$	−27.7281	−0.914665	−0.457332	+	0.889296i	$0.651195\pi$
$920$	34.0522	1.12267
$921$	0	0
$922$	40.6057	1.33728
$923$	20.1549	0.663407
$924$	0	0
$925$	3.52591	0.115931
$926$	−16.2398	−0.533672
$927$	0	0
$928$	0	0
$929$	29.3190	0.961924	0.480962	−	0.876742i	$-0.340288\pi$
$929$	29.3190	0.961924	0.480962	+	0.876742i	$0.340288\pi$
$930$	0	0
$931$	−27.9279	−0.915301
$932$	7.55815	0.247576
$933$	0	0
$934$	46.3034	1.51509
$935$	6.08572	0.199024
$936$	0	0
$937$	3.74169	0.122236	0.0611179	−	0.998131i	$-0.480533\pi$
$937$	3.74169	0.122236	0.0611179	+	0.998131i	$0.480533\pi$
$938$	−7.01610	−0.229084
$939$	0	0
$940$	8.30079	0.270742
$941$	−1.33740	−0.0435982	−0.0217991	−	0.999762i	$-0.506939\pi$
$941$	−1.33740	−0.0435982	−0.0217991	+	0.999762i	$0.506939\pi$
$942$	0	0
$943$	−14.4991	−0.472156
$944$	6.10996	0.198862
$945$	0	0
$946$	7.57752	0.246366
$947$	5.89635	0.191606	0.0958029	−	0.995400i	$-0.469458\pi$
$947$	5.89635	0.191606	0.0958029	+	0.995400i	$0.469458\pi$
$948$	0	0
$949$	−5.89712	−0.191429
$950$	9.68051	0.314077
$951$	0	0
$952$	−18.9371	−0.613753
$953$	12.7197	0.412031	0.206016	−	0.978549i	$-0.433950\pi$
$953$	12.7197	0.412031	0.206016	+	0.978549i	$0.433950\pi$
$954$	0	0
$955$	−35.9365	−1.16288
$956$	−1.71309	−0.0554052
$957$	0	0
$958$	14.0995	0.455533
$959$	−4.09106	−0.132107
$960$	0	0
$961$	−30.6017	−0.987153
$962$	18.9774	0.611854
$963$	0	0
$964$	17.3082	0.557460
$965$	−6.70970	−0.215993
$966$	0	0
$967$	−6.93437	−0.222994	−0.111497	−	0.993765i	$-0.535565\pi$
$967$	−6.93437	−0.222994	−0.111497	+	0.993765i	$0.535565\pi$
$968$	−31.9234	−1.02606
$969$	0	0
$970$	−0.266481	−0.00855620
$971$	4.84664	0.155536	0.0777680	−	0.996971i	$-0.475221\pi$
$971$	4.84664	0.155536	0.0777680	+	0.996971i	$0.475221\pi$
$972$	0	0
$973$	−20.9500	−0.671625
$974$	−14.9189	−0.478032
$975$	0	0
$976$	18.7162	0.599090
$977$	3.15904	0.101067	0.0505334	−	0.998722i	$-0.483908\pi$
$977$	3.15904	0.101067	0.0505334	+	0.998722i	$0.483908\pi$
$978$	0	0
$979$	−1.46232	−0.0467359
$980$	−5.74363	−0.183473
$981$	0	0
$982$	−28.0423	−0.894865
$983$	40.6134	1.29536	0.647682	−	0.761910i	$-0.275738\pi$
$983$	40.6134	1.29536	0.647682	+	0.761910i	$0.275738\pi$
$984$	0	0
$985$	41.9483	1.33658
$986$	0	0
$987$	0	0
$988$	−24.9290	−0.793096
$989$	47.7281	1.51766
$990$	0	0
$991$	23.6067	0.749891	0.374946	−	0.927047i	$-0.377661\pi$
$991$	23.6067	0.749891	0.374946	+	0.927047i	$0.377661\pi$
$992$	2.20775	0.0700962
$993$	0	0
$994$	−5.55081	−0.176061
$995$	−15.9256	−0.504875
$996$	0	0
$997$	−43.0436	−1.36320	−0.681602	−	0.731723i	$-0.738717\pi$
$997$	−43.0436	−1.36320	−0.681602	+	0.731723i	$0.738717\pi$
$998$	−21.4153	−0.677891
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bp.1.4		12
3.2	odd	2		841.2.a.k.1.9		12
29.18	odd	28		261.2.o.a.208.2		12
29.21	odd	28		261.2.o.a.64.2		12
29.28	even	2	inner	7569.2.a.bp.1.9		12
87.2	even	28		841.2.e.a.236.2		12
87.5	odd	14		841.2.d.l.605.3		24
87.8	even	28		841.2.e.i.267.2		12
87.11	even	28		841.2.e.i.63.2		12
87.14	even	28		841.2.e.h.196.1		12
87.17	even	4		841.2.b.e.840.4		12
87.20	odd	14		841.2.d.m.574.3		24
87.23	odd	14		841.2.d.l.645.2		24
87.26	even	28		841.2.e.e.270.1		12
87.32	even	28		841.2.e.f.270.2		12
87.35	odd	14		841.2.d.l.645.3		24
87.38	odd	14		841.2.d.m.574.2		24
87.41	even	4		841.2.b.e.840.9		12
87.44	even	28		841.2.e.a.196.2		12
87.47	even	28		29.2.e.a.5.1	✓	12
87.50	even	28		29.2.e.a.6.1	yes	12
87.53	odd	14		841.2.d.l.605.2		24
87.56	even	28		841.2.e.h.236.1		12
87.62	odd	14		841.2.d.k.190.3		24
87.65	odd	14		841.2.d.k.571.2		24
87.68	even	28		841.2.e.f.651.2		12
87.71	odd	14		841.2.d.m.778.2		24
87.74	odd	14		841.2.d.m.778.3		24
87.77	even	28		841.2.e.e.651.1		12
87.80	odd	14		841.2.d.k.571.3		24
87.83	odd	14		841.2.d.k.190.2		24
87.86	odd	2		841.2.a.k.1.4		12
348.47	odd	28		464.2.y.d.353.1		12
348.311	odd	28		464.2.y.d.209.1		12
435.47	odd	28		725.2.p.a.324.2		24
435.134	even	28		725.2.q.a.701.2		12
435.137	odd	28		725.2.p.a.499.3		24
435.224	even	28		725.2.q.a.151.2		12
435.308	odd	28		725.2.p.a.324.3		24
435.398	odd	28		725.2.p.a.499.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.e.a.5.1	✓	12	87.47	even	28
29.2.e.a.6.1	yes	12	87.50	even	28
261.2.o.a.64.2		12	29.21	odd	28
261.2.o.a.208.2		12	29.18	odd	28
464.2.y.d.209.1		12	348.311	odd	28
464.2.y.d.353.1		12	348.47	odd	28
725.2.p.a.324.2		24	435.47	odd	28
725.2.p.a.324.3		24	435.308	odd	28
725.2.p.a.499.2		24	435.398	odd	28
725.2.p.a.499.3		24	435.137	odd	28
725.2.q.a.151.2		12	435.224	even	28
725.2.q.a.701.2		12	435.134	even	28
841.2.a.k.1.4		12	87.86	odd	2
841.2.a.k.1.9		12	3.2	odd	2
841.2.b.e.840.4		12	87.17	even	4
841.2.b.e.840.9		12	87.41	even	4
841.2.d.k.190.2		24	87.83	odd	14
841.2.d.k.190.3		24	87.62	odd	14
841.2.d.k.571.2		24	87.65	odd	14
841.2.d.k.571.3		24	87.80	odd	14
841.2.d.l.605.2		24	87.53	odd	14
841.2.d.l.605.3		24	87.5	odd	14
841.2.d.l.645.2		24	87.23	odd	14
841.2.d.l.645.3		24	87.35	odd	14
841.2.d.m.574.2		24	87.38	odd	14
841.2.d.m.574.3		24	87.20	odd	14
841.2.d.m.778.2		24	87.71	odd	14
841.2.d.m.778.3		24	87.74	odd	14
841.2.e.a.196.2		12	87.44	even	28
841.2.e.a.236.2		12	87.2	even	28
841.2.e.e.270.1		12	87.26	even	28
841.2.e.e.651.1		12	87.77	even	28
841.2.e.f.270.2		12	87.32	even	28
841.2.e.f.651.2		12	87.68	even	28
841.2.e.h.196.1		12	87.14	even	28
841.2.e.h.236.1		12	87.56	even	28
841.2.e.i.63.2		12	87.11	even	28
841.2.e.i.267.2		12	87.8	even	28
7569.2.a.bp.1.4		12	1.1	even	1	trivial
7569.2.a.bp.1.9		12	29.28	even	2	inner

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:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q-1.16308 q^{2} -0.647237 q^{4} -1.89937 q^{5} +1.52574 q^{7} +3.07896 q^{8} +O(q^{10})\)	\(q-1.16308 q^{2} -0.647237 q^{4} -1.89937 q^{5} +1.52574 q^{7} +3.07896 q^{8} +2.20913 q^{10} +0.794830 q^{11} +6.44341 q^{13} -1.77456 q^{14} -2.28661 q^{16} -4.03114 q^{17} +5.97758 q^{19} +1.22934 q^{20} -0.924454 q^{22} -5.82280 q^{23} -1.39239 q^{25} -7.49423 q^{26} -0.987516 q^{28} -0.631076 q^{31} -3.49840 q^{32} +4.68855 q^{34} -2.89795 q^{35} -2.53226 q^{37} -6.95242 q^{38} -5.84808 q^{40} +2.49005 q^{41} -8.19675 q^{43} -0.514444 q^{44} +6.77241 q^{46} +6.75222 q^{47} -4.67211 q^{49} +1.61947 q^{50} -4.17041 q^{52} +0.689457 q^{53} -1.50968 q^{55} +4.69769 q^{56} -2.67206 q^{59} -8.18512 q^{61} +0.733994 q^{62} +8.64215 q^{64} -12.2384 q^{65} +3.95370 q^{67} +2.60910 q^{68} +3.37055 q^{70} +3.12799 q^{71} -0.915217 q^{73} +2.94523 q^{74} -3.86891 q^{76} +1.21271 q^{77} -12.8832 q^{79} +4.34312 q^{80} -2.89614 q^{82} -12.6241 q^{83} +7.65662 q^{85} +9.53350 q^{86} +2.44725 q^{88} -1.83979 q^{89} +9.83098 q^{91} +3.76873 q^{92} -7.85340 q^{94} -11.3536 q^{95} -0.120628 q^{97} +5.43406 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{4} - 8 q^{5} + 10 q^{7}+O(q^{10}) \) 12 * q + 8 * q^4 - 8 * q^5 + 10 * q^7	\( 12 q + 8 q^{4} - 8 q^{5} + 10 q^{7} + 12 q^{13} + 16 q^{16} - 24 q^{20} - 38 q^{22} - 30 q^{23} - 8 q^{25} - 12 q^{28} + 6 q^{34} - 44 q^{35} + 6 q^{49} - 6 q^{52} - 32 q^{53} - 44 q^{59} + 16 q^{62} - 2 q^{64} - 8 q^{65} + 30 q^{67} - 70 q^{71} - 56 q^{74} - 34 q^{80} - 62 q^{82} - 82 q^{83} - 44 q^{86} - 66 q^{88} - 32 q^{91} + 22 q^{92} + 10 q^{94}+O(q^{100}) \) 12 * q + 8 * q^4 - 8 * q^5 + 10 * q^7 + 12 * q^13 + 16 * q^16 - 24 * q^20 - 38 * q^22 - 30 * q^23 - 8 * q^25 - 12 * q^28 + 6 * q^34 - 44 * q^35 + 6 * q^49 - 6 * q^52 - 32 * q^53 - 44 * q^59 + 16 * q^62 - 2 * q^64 - 8 * q^65 + 30 * q^67 - 70 * q^71 - 56 * q^74 - 34 * q^80 - 62 * q^82 - 82 * q^83 - 44 * q^86 - 66 * q^88 - 32 * q^91 + 22 * q^92 + 10 * q^94

Introduction

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