LMFDB - Embedded newform 2025.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(1,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, 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("2025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2025.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:	$16.1697064093$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.11661.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 3 $ x^4 - 2x^3 - 4x^2 + 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.63372$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.63372 q^{2} +4.93650 q^{4} -1.79743 q^{7} -7.73393 q^{8} +O(q^{10})$	$q-2.63372 q^{2} +4.93650 q^{4} -1.79743 q^{7} -7.73393 q^{8} -1.80812 q^{11} +1.97183 q^{13} +4.73393 q^{14} +10.4960 q^{16} -4.80812 q^{17} +2.96467 q^{19} +4.76210 q^{22} +1.73393 q^{23} -5.19325 q^{26} -8.87300 q^{28} +7.36765 q^{29} -2.62303 q^{31} -12.1758 q^{32} +12.6633 q^{34} +11.6351 q^{37} -7.80812 q^{38} +2.46648 q^{41} -7.27814 q^{43} -8.92580 q^{44} -4.56668 q^{46} -6.29208 q^{47} -3.76926 q^{49} +9.73393 q^{52} -1.72540 q^{53} +13.9012 q^{56} -19.4044 q^{58} -11.0260 q^{59} -12.6704 q^{61} +6.90833 q^{62} +11.0756 q^{64} +9.10374 q^{67} -23.7353 q^{68} -1.27460 q^{71} -3.58770 q^{73} -30.6436 q^{74} +14.6351 q^{76} +3.24997 q^{77} +2.11090 q^{79} -6.49602 q^{82} -1.09883 q^{83} +19.1686 q^{86} +13.9839 q^{88} +13.2935 q^{89} -3.54422 q^{91} +8.55953 q^{92} +16.5716 q^{94} -3.83276 q^{97} +9.92718 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8	$ 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8} + q^{11} + 2 q^{13} - 3 q^{14} + 4 q^{16} - 11 q^{17} + 2 q^{19} + 3 q^{22} - 15 q^{23} + 10 q^{26} - 4 q^{28} - q^{29} - 4 q^{31} - 10 q^{32} + 9 q^{34} - q^{37} - 23 q^{38} + 5 q^{41} - 10 q^{43} - 22 q^{44} - 20 q^{47} - 3 q^{49} + 17 q^{52} - 20 q^{53} + 30 q^{56} - 18 q^{58} - 17 q^{59} - 13 q^{61} + 6 q^{62} + 19 q^{64} + 17 q^{67} - 34 q^{68} + 8 q^{71} + 2 q^{73} - 40 q^{74} + 11 q^{76} - 12 q^{77} - 7 q^{79} + 12 q^{82} - 30 q^{83} + 34 q^{86} + 9 q^{88} + 9 q^{89} - 17 q^{91} + 12 q^{92} + 3 q^{94} - 19 q^{97} - 13 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8 + q^11 + 2 * q^13 - 3 * q^14 + 4 * q^16 - 11 * q^17 + 2 * q^19 + 3 * q^22 - 15 * q^23 + 10 * q^26 - 4 * q^28 - q^29 - 4 * q^31 - 10 * q^32 + 9 * q^34 - q^37 - 23 * q^38 + 5 * q^41 - 10 * q^43 - 22 * q^44 - 20 * q^47 - 3 * q^49 + 17 * q^52 - 20 * q^53 + 30 * q^56 - 18 * q^58 - 17 * q^59 - 13 * q^61 + 6 * q^62 + 19 * q^64 + 17 * q^67 - 34 * q^68 + 8 * q^71 + 2 * q^73 - 40 * q^74 + 11 * q^76 - 12 * q^77 - 7 * q^79 + 12 * q^82 - 30 * q^83 + 34 * q^86 + 9 * q^88 + 9 * q^89 - 17 * q^91 + 12 * q^92 + 3 * q^94 - 19 * q^97 - 13 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.63372	−1.86232	−0.931162	−	0.364606i	$-0.881204\pi$
$2$	−2.63372	−1.86232	−0.931162	+	0.364606i	$0.881204\pi$
$3$	0	0
$3$	0	0
$4$	4.93650	2.46825
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.79743	−0.679364	−0.339682	−	0.940540i	$-0.610319\pi$
$7$	−1.79743	−0.679364	−0.339682	+	0.940540i	$0.610319\pi$
$8$	−7.73393	−2.73436
$9$	0	0
$10$	0	0
$11$	−1.80812	−0.545170	−0.272585	−	0.962132i	$-0.587879\pi$
$11$	−1.80812	−0.545170	−0.272585	+	0.962132i	$0.587879\pi$
$12$	0	0
$13$	1.97183	0.546887	0.273443	−	0.961888i	$-0.411837\pi$
$13$	1.97183	0.546887	0.273443	+	0.961888i	$0.411837\pi$
$14$	4.73393	1.26520
$15$	0	0
$16$	10.4960	2.62401
$17$	−4.80812	−1.16614	−0.583071	−	0.812421i	$-0.698149\pi$
$17$	−4.80812	−1.16614	−0.583071	+	0.812421i	$0.698149\pi$
$18$	0	0
$19$	2.96467	0.680142	0.340071	−	0.940400i	$-0.389549\pi$
$19$	2.96467	0.680142	0.340071	+	0.940400i	$0.389549\pi$
$20$	0	0
$21$	0	0
$22$	4.76210	1.01528
$23$	1.73393	0.361549	0.180774	−	0.983525i	$-0.442140\pi$
$23$	1.73393	0.361549	0.180774	+	0.983525i	$0.442140\pi$
$24$	0	0
$25$	0	0
$26$	−5.19325	−1.01848
$27$	0	0
$28$	−8.87300	−1.67684
$29$	7.36765	1.36814	0.684069	−	0.729417i	$-0.260209\pi$
$29$	7.36765	1.36814	0.684069	+	0.729417i	$0.260209\pi$
$30$	0	0
$31$	−2.62303	−0.471110	−0.235555	−	0.971861i	$-0.575691\pi$
$31$	−2.62303	−0.471110	−0.235555	+	0.971861i	$0.575691\pi$
$32$	−12.1758	−2.15239
$33$	0	0
$34$	12.6633	2.17173
$35$	0	0
$36$	0	0
$37$	11.6351	1.91280	0.956399	−	0.292063i	$-0.0943417\pi$
$37$	11.6351	1.91280	0.956399	+	0.292063i	$0.0943417\pi$
$38$	−7.80812	−1.26664
$39$	0	0
$40$	0	0
$41$	2.46648	0.385199	0.192600	−	0.981277i	$-0.438308\pi$
$41$	2.46648	0.385199	0.192600	+	0.981277i	$0.438308\pi$
$42$	0	0
$43$	−7.27814	−1.10991	−0.554953	−	0.831882i	$-0.687264\pi$
$43$	−7.27814	−1.10991	−0.554953	+	0.831882i	$0.687264\pi$
$44$	−8.92580	−1.34562
$45$	0	0
$46$	−4.56668	−0.673321
$47$	−6.29208	−0.917794	−0.458897	−	0.888489i	$-0.651755\pi$
$47$	−6.29208	−0.917794	−0.458897	+	0.888489i	$0.651755\pi$
$48$	0	0
$49$	−3.76926	−0.538465
$50$	0	0
$51$	0	0
$52$	9.73393	1.34985
$53$	−1.72540	−0.237001	−0.118501	−	0.992954i	$-0.537809\pi$
$53$	−1.72540	−0.237001	−0.118501	+	0.992954i	$0.537809\pi$
$54$	0	0
$55$	0	0
$56$	13.9012	1.85762
$57$	0	0
$58$	−19.4044	−2.54792
$59$	−11.0260	−1.43546	−0.717732	−	0.696320i	$-0.754820\pi$
$59$	−11.0260	−1.43546	−0.717732	+	0.696320i	$0.754820\pi$
$60$	0	0
$61$	−12.6704	−1.62228	−0.811141	−	0.584851i	$-0.801153\pi$
$61$	−12.6704	−1.62228	−0.811141	+	0.584851i	$0.801153\pi$
$62$	6.90833	0.877358
$63$	0	0
$64$	11.0756	1.38445
$65$	0	0
$66$	0	0
$67$	9.10374	1.11220	0.556100	−	0.831116i	$-0.312297\pi$
$67$	9.10374	1.11220	0.556100	+	0.831116i	$0.312297\pi$
$68$	−23.7353	−2.87833
$69$	0	0
$70$	0	0
$71$	−1.27460	−0.151268	−0.0756338	−	0.997136i	$-0.524098\pi$
$71$	−1.27460	−0.151268	−0.0756338	+	0.997136i	$0.524098\pi$
$72$	0	0
$73$	−3.58770	−0.419908	−0.209954	−	0.977711i	$-0.567331\pi$
$73$	−3.58770	−0.419908	−0.209954	+	0.977711i	$0.567331\pi$
$74$	−30.6436	−3.56225
$75$	0	0
$76$	14.6351	1.67876
$77$	3.24997	0.370369
$78$	0	0
$79$	2.11090	0.237495	0.118747	−	0.992924i	$-0.462112\pi$
$79$	2.11090	0.237495	0.118747	+	0.992924i	$0.462112\pi$
$80$	0	0
$81$	0	0
$82$	−6.49602	−0.717366
$83$	−1.09883	−0.120612	−0.0603061	−	0.998180i	$-0.519208\pi$
$83$	−1.09883	−0.120612	−0.0603061	+	0.998180i	$0.519208\pi$
$84$	0	0
$85$	0	0
$86$	19.1686	2.06701
$87$	0	0
$88$	13.9839	1.49069
$89$	13.2935	1.40910	0.704552	−	0.709653i	$-0.251148\pi$
$89$	13.2935	1.40910	0.704552	+	0.709653i	$0.251148\pi$
$90$	0	0
$91$	−3.54422	−0.371535
$92$	8.55953	0.892392
$93$	0	0
$94$	16.5716	1.70923
$95$	0	0
$96$	0	0
$97$	−3.83276	−0.389157	−0.194579	−	0.980887i	$-0.562334\pi$
$97$	−3.83276	−0.389157	−0.194579	+	0.980887i	$0.562334\pi$
$98$	9.92718	1.00280
$99$	0	0
$100$	0	0
$101$	−6.55237	−0.651985	−0.325993	−	0.945372i	$-0.605698\pi$
$101$	−6.55237	−0.651985	−0.325993	+	0.945372i	$0.605698\pi$
$102$	0	0
$103$	−8.07557	−0.795710	−0.397855	−	0.917448i	$-0.630245\pi$
$103$	−8.07557	−0.795710	−0.397855	+	0.917448i	$0.630245\pi$
$104$	−15.2500	−1.49538
$105$	0	0
$106$	4.54422	0.441373
$107$	8.97674	0.867814	0.433907	−	0.900958i	$-0.357135\pi$
$107$	8.97674	0.867814	0.433907	+	0.900958i	$0.357135\pi$
$108$	0	0
$109$	−6.34164	−0.607419	−0.303710	−	0.952765i	$-0.598225\pi$
$109$	−6.34164	−0.607419	−0.303710	+	0.952765i	$0.598225\pi$
$110$	0	0
$111$	0	0
$112$	−18.8658	−1.78265
$113$	14.9025	1.40191	0.700957	−	0.713204i	$-0.252757\pi$
$113$	14.9025	1.40191	0.700957	+	0.713204i	$0.252757\pi$
$114$	0	0
$115$	0	0
$116$	36.3704	3.37691
$117$	0	0
$118$	29.0394	2.67330
$119$	8.64225	0.792234
$120$	0	0
$121$	−7.73069	−0.702790
$122$	33.3704	3.02121
$123$	0	0
$124$	−12.9486	−1.16282
$125$	0	0
$126$	0	0
$127$	3.62303	0.321492	0.160746	−	0.986996i	$-0.448610\pi$
$127$	3.62303	0.321492	0.160746	+	0.986996i	$0.448610\pi$
$128$	−4.81844	−0.425894
$129$	0	0
$130$	0	0
$131$	−7.29345	−0.637232	−0.318616	−	0.947884i	$-0.603218\pi$
$131$	−7.29345	−0.637232	−0.318616	+	0.947884i	$0.603218\pi$
$132$	0	0
$133$	−5.32878	−0.462064
$134$	−23.9767	−2.07127
$135$	0	0
$136$	37.1857	3.18865
$137$	7.12621	0.608833	0.304417	−	0.952539i	$-0.401539\pi$
$137$	7.12621	0.608833	0.304417	+	0.952539i	$0.401539\pi$
$138$	0	0
$139$	−14.7107	−1.24774	−0.623871	−	0.781527i	$-0.714441\pi$
$139$	−14.7107	−1.24774	−0.623871	+	0.781527i	$0.714441\pi$
$140$	0	0
$141$	0	0
$142$	3.35695	0.281709
$143$	−3.56531	−0.298146
$144$	0	0
$145$	0	0
$146$	9.44900	0.782005
$147$	0	0
$148$	57.4366	4.72126
$149$	0.565309	0.0463119	0.0231560	−	0.999732i	$-0.492629\pi$
$149$	0.565309	0.0463119	0.0231560	+	0.999732i	$0.492629\pi$
$150$	0	0
$151$	0.153385	0.0124823	0.00624115	−	0.999981i	$-0.498013\pi$
$151$	0.153385	0.0124823	0.00624115	+	0.999981i	$0.498013\pi$
$152$	−22.9285	−1.85975
$153$	0	0
$154$	−8.55953	−0.689746
$155$	0	0
$156$	0	0
$157$	−11.4607	−0.914663	−0.457332	−	0.889296i	$-0.651195\pi$
$157$	−11.4607	−0.914663	−0.457332	+	0.889296i	$0.651195\pi$
$158$	−5.55953	−0.442292
$159$	0	0
$160$	0	0
$161$	−3.11661	−0.245623
$162$	0	0
$163$	−22.0595	−1.72783	−0.863915	−	0.503637i	$-0.831995\pi$
$163$	−22.0595	−1.72783	−0.863915	+	0.503637i	$0.831995\pi$
$164$	12.1758	0.950768
$165$	0	0
$166$	2.89401	0.224619
$167$	−17.0684	−1.32079	−0.660397	−	0.750917i	$-0.729612\pi$
$167$	−17.0684	−1.32079	−0.660397	+	0.750917i	$0.729612\pi$
$168$	0	0
$169$	−9.11189	−0.700915
$170$	0	0
$171$	0	0
$172$	−35.9285	−2.73953
$173$	−11.9447	−0.908135	−0.454067	−	0.890967i	$-0.650028\pi$
$173$	−11.9447	−0.908135	−0.454067	+	0.890967i	$0.650028\pi$
$174$	0	0
$175$	0	0
$176$	−18.9781	−1.43053
$177$	0	0
$178$	−35.0113	−2.62421
$179$	−8.54921	−0.638998	−0.319499	−	0.947587i	$-0.603515\pi$
$179$	−8.54921	−0.638998	−0.319499	+	0.947587i	$0.603515\pi$
$180$	0	0
$181$	−10.5524	−0.784351	−0.392176	−	0.919890i	$-0.628277\pi$
$181$	−10.5524	−0.784351	−0.392176	+	0.919890i	$0.628277\pi$
$182$	9.33449	0.691918
$183$	0	0
$184$	−13.4101	−0.988603
$185$	0	0
$186$	0	0
$187$	8.69368	0.635745
$188$	−31.0608	−2.26534
$189$	0	0
$190$	0	0
$191$	17.3372	1.25448	0.627239	−	0.778827i	$-0.284185\pi$
$191$	17.3372	1.25448	0.627239	+	0.778827i	$0.284185\pi$
$192$	0	0
$193$	1.55953	0.112257	0.0561286	−	0.998424i	$-0.482124\pi$
$193$	1.55953	0.112257	0.0561286	+	0.998424i	$0.482124\pi$
$194$	10.0944	0.724737
$195$	0	0
$196$	−18.6069	−1.32907
$197$	−17.9767	−1.28079	−0.640395	−	0.768046i	$-0.721229\pi$
$197$	−17.9767	−1.28079	−0.640395	+	0.768046i	$0.721229\pi$
$198$	0	0
$199$	11.0225	0.781362	0.390681	−	0.920526i	$-0.372240\pi$
$199$	11.0225	0.781362	0.390681	+	0.920526i	$0.372240\pi$
$200$	0	0
$201$	0	0
$202$	17.2571	1.21421
$203$	−13.2428	−0.929463
$204$	0	0
$205$	0	0
$206$	21.2688	1.48187
$207$	0	0
$208$	20.6964	1.43503
$209$	−5.36049	−0.370793
$210$	0	0
$211$	−23.9285	−1.64731	−0.823655	−	0.567092i	$-0.808068\pi$
$211$	−23.9285	−1.64731	−0.823655	+	0.567092i	$0.808068\pi$
$212$	−8.51742	−0.584979
$213$	0	0
$214$	−23.6423	−1.61615
$215$	0	0
$216$	0	0
$217$	4.71470	0.320055
$218$	16.7021	1.13121
$219$	0	0
$220$	0	0
$221$	−9.48079	−0.637747
$222$	0	0
$223$	21.7107	1.45385	0.726927	−	0.686715i	$-0.240948\pi$
$223$	21.7107	1.45385	0.726927	+	0.686715i	$0.240948\pi$
$224$	21.8851	1.46226
$225$	0	0
$226$	−39.2492	−2.61082
$227$	−14.1002	−0.935863	−0.467932	−	0.883765i	$-0.655001\pi$
$227$	−14.1002	−0.935863	−0.467932	+	0.883765i	$0.655001\pi$
$228$	0	0
$229$	3.67758	0.243021	0.121511	−	0.992590i	$-0.461226\pi$
$229$	3.67758	0.243021	0.121511	+	0.992590i	$0.461226\pi$
$230$	0	0
$231$	0	0
$232$	−56.9809	−3.74098
$233$	5.34164	0.349943	0.174971	−	0.984574i	$-0.444017\pi$
$233$	5.34164	0.349943	0.174971	+	0.984574i	$0.444017\pi$
$234$	0	0
$235$	0	0
$236$	−54.4299	−3.54308
$237$	0	0
$238$	−22.7613	−1.47540
$239$	22.0335	1.42523	0.712613	−	0.701557i	$-0.247512\pi$
$239$	22.0335	1.42523	0.712613	+	0.701557i	$0.247512\pi$
$240$	0	0
$241$	−18.6472	−1.20117	−0.600585	−	0.799561i	$-0.705066\pi$
$241$	−18.6472	−1.20117	−0.600585	+	0.799561i	$0.705066\pi$
$242$	20.3605	1.30882
$243$	0	0
$244$	−62.5475	−4.00420
$245$	0	0
$246$	0	0
$247$	5.84582	0.371961
$248$	20.2863	1.28818
$249$	0	0
$250$	0	0
$251$	−14.6929	−0.927407	−0.463704	−	0.885990i	$-0.653480\pi$
$251$	−14.6929	−0.927407	−0.463704	+	0.885990i	$0.653480\pi$
$252$	0	0
$253$	−3.13515	−0.197105
$254$	−9.54205	−0.598721
$255$	0	0
$256$	−9.46070	−0.591294
$257$	−22.2089	−1.38536	−0.692678	−	0.721247i	$-0.743569\pi$
$257$	−22.2089	−1.38536	−0.692678	+	0.721247i	$0.743569\pi$
$258$	0	0
$259$	−20.9132	−1.29949
$260$	0	0
$261$	0	0
$262$	19.2089	1.18673
$263$	−5.74001	−0.353944	−0.176972	−	0.984216i	$-0.556630\pi$
$263$	−5.74001	−0.353944	−0.176972	+	0.984216i	$0.556630\pi$
$264$	0	0
$265$	0	0
$266$	14.0345	0.860513
$267$	0	0
$268$	44.9406	2.74519
$269$	15.6162	0.952139	0.476070	−	0.879408i	$-0.342061\pi$
$269$	15.6162	0.952139	0.476070	+	0.879408i	$0.342061\pi$
$270$	0	0
$271$	−6.75315	−0.410225	−0.205112	−	0.978738i	$-0.565756\pi$
$271$	−6.75315	−0.410225	−0.205112	+	0.978738i	$0.565756\pi$
$272$	−50.4662	−3.05996
$273$	0	0
$274$	−18.7685	−1.13384
$275$	0	0
$276$	0	0
$277$	30.2966	1.82034	0.910172	−	0.414230i	$-0.135949\pi$
$277$	30.2966	1.82034	0.910172	+	0.414230i	$0.135949\pi$
$278$	38.7438	2.32370
$279$	0	0
$280$	0	0
$281$	−18.6351	−1.11168	−0.555838	−	0.831290i	$-0.687603\pi$
$281$	−18.6351	−1.11168	−0.555838	+	0.831290i	$0.687603\pi$
$282$	0	0
$283$	−5.67366	−0.337264	−0.168632	−	0.985679i	$-0.553935\pi$
$283$	−5.67366	−0.337264	−0.168632	+	0.985679i	$0.553935\pi$
$284$	−6.29208	−0.373366
$285$	0	0
$286$	9.39004	0.555245
$287$	−4.43332	−0.261690
$288$	0	0
$289$	6.11806	0.359886
$290$	0	0
$291$	0	0
$292$	−17.7107	−1.03644
$293$	−18.2773	−1.06777	−0.533887	−	0.845556i	$-0.679269\pi$
$293$	−18.2773	−1.06777	−0.533887	+	0.845556i	$0.679269\pi$
$294$	0	0
$295$	0	0
$296$	−89.9850	−5.23027
$297$	0	0
$298$	−1.48887	−0.0862478
$299$	3.41900	0.197726
$300$	0	0
$301$	13.0819	0.754030
$302$	−0.403974	−0.0232461
$303$	0	0
$304$	31.1173	1.78470
$305$	0	0
$306$	0	0
$307$	−15.5050	−0.884915	−0.442458	−	0.896789i	$-0.645893\pi$
$307$	−15.5050	−0.884915	−0.442458	+	0.896789i	$0.645893\pi$
$308$	16.0435	0.914162
$309$	0	0
$310$	0	0
$311$	−30.4464	−1.72646	−0.863228	−	0.504814i	$-0.831561\pi$
$311$	−30.4464	−1.72646	−0.863228	+	0.504814i	$0.831561\pi$
$312$	0	0
$313$	−6.94936	−0.392801	−0.196401	−	0.980524i	$-0.562925\pi$
$313$	−6.94936	−0.392801	−0.196401	+	0.980524i	$0.562925\pi$
$314$	30.1843	1.70340
$315$	0	0
$316$	10.4205	0.586196
$317$	16.1451	0.906797	0.453398	−	0.891308i	$-0.350212\pi$
$317$	16.1451	0.906797	0.453398	+	0.891308i	$0.350212\pi$
$318$	0	0
$319$	−13.3216	−0.745868
$320$	0	0
$321$	0	0
$322$	8.20828	0.457430
$323$	−14.2545	−0.793142
$324$	0	0
$325$	0	0
$326$	58.0985	3.21778
$327$	0	0
$328$	−19.0756	−1.05327
$329$	11.3096	0.623516
$330$	0	0
$331$	12.6222	0.693781	0.346890	−	0.937906i	$-0.387238\pi$
$331$	12.6222	0.693781	0.346890	+	0.937906i	$0.387238\pi$
$332$	−5.42437	−0.297701
$333$	0	0
$334$	44.9535	2.45975
$335$	0	0
$336$	0	0
$337$	−6.92040	−0.376978	−0.188489	−	0.982075i	$-0.560359\pi$
$337$	−6.92040	−0.376978	−0.188489	+	0.982075i	$0.560359\pi$
$338$	23.9982	1.30533
$339$	0	0
$340$	0	0
$341$	4.74276	0.256835
$342$	0	0
$343$	19.3570	1.04518
$344$	56.2886	3.03488
$345$	0	0
$346$	31.4589	1.69124
$347$	−13.8063	−0.741163	−0.370581	−	0.928800i	$-0.620842\pi$
$347$	−13.8063	−0.741163	−0.370581	+	0.928800i	$0.620842\pi$
$348$	0	0
$349$	6.56768	0.351560	0.175780	−	0.984429i	$-0.443755\pi$
$349$	6.56768	0.351560	0.175780	+	0.984429i	$0.443755\pi$
$350$	0	0
$351$	0	0
$352$	22.0153	1.17342
$353$	3.52499	0.187616	0.0938082	−	0.995590i	$-0.470096\pi$
$353$	3.52499	0.187616	0.0938082	+	0.995590i	$0.470096\pi$
$354$	0	0
$355$	0	0
$356$	65.6231	3.47802
$357$	0	0
$358$	22.5162	1.19002
$359$	−22.9285	−1.21012	−0.605061	−	0.796179i	$-0.706851\pi$
$359$	−22.9285	−1.21012	−0.605061	+	0.796179i	$0.706851\pi$
$360$	0	0
$361$	−10.2107	−0.537407
$362$	27.7920	1.46072
$363$	0	0
$364$	−17.4960	−0.917041
$365$	0	0
$366$	0	0
$367$	4.17931	0.218158	0.109079	−	0.994033i	$-0.465210\pi$
$367$	4.17931	0.218158	0.109079	+	0.994033i	$0.465210\pi$
$368$	18.1993	0.948706
$369$	0	0
$370$	0	0
$371$	3.10127	0.161010
$372$	0	0
$373$	6.84091	0.354209	0.177104	−	0.984192i	$-0.443327\pi$
$373$	6.84091	0.354209	0.177104	+	0.984192i	$0.443327\pi$
$374$	−22.8968	−1.18396
$375$	0	0
$376$	48.6625	2.50958
$377$	14.5277	0.748217
$378$	0	0
$379$	−12.7764	−0.656280	−0.328140	−	0.944629i	$-0.606422\pi$
$379$	−12.7764	−0.656280	−0.328140	+	0.944629i	$0.606422\pi$
$380$	0	0
$381$	0	0
$382$	−45.6615	−2.33624
$383$	−7.53459	−0.385000	−0.192500	−	0.981297i	$-0.561659\pi$
$383$	−7.53459	−0.385000	−0.192500	+	0.981297i	$0.561659\pi$
$384$	0	0
$385$	0	0
$386$	−4.10736	−0.209059
$387$	0	0
$388$	−18.9204	−0.960538
$389$	−5.45175	−0.276415	−0.138207	−	0.990403i	$-0.544134\pi$
$389$	−5.45175	−0.276415	−0.138207	+	0.990403i	$0.544134\pi$
$390$	0	0
$391$	−8.33693	−0.421617
$392$	29.1511	1.47235
$393$	0	0
$394$	47.3458	2.38525
$395$	0	0
$396$	0	0
$397$	−5.64549	−0.283339	−0.141670	−	0.989914i	$-0.545247\pi$
$397$	−5.64549	−0.283339	−0.141670	+	0.989914i	$0.545247\pi$
$398$	−29.0301	−1.45515
$399$	0	0
$400$	0	0
$401$	5.50418	0.274865	0.137433	−	0.990511i	$-0.456115\pi$
$401$	5.50418	0.274865	0.137433	+	0.990511i	$0.456115\pi$
$402$	0	0
$403$	−5.17216	−0.257643
$404$	−32.3458	−1.60926
$405$	0	0
$406$	34.8779	1.73096
$407$	−21.0377	−1.04280
$408$	0	0
$409$	32.8530	1.62448	0.812238	−	0.583327i	$-0.198249\pi$
$409$	32.8530	1.62448	0.812238	+	0.583327i	$0.198249\pi$
$410$	0	0
$411$	0	0
$412$	−39.8650	−1.96401
$413$	19.8184	0.975202
$414$	0	0
$415$	0	0
$416$	−24.0085	−1.17712
$417$	0	0
$418$	14.1181	0.690537
$419$	−22.8591	−1.11674	−0.558369	−	0.829593i	$-0.688573\pi$
$419$	−22.8591	−1.11674	−0.558369	+	0.829593i	$0.688573\pi$
$420$	0	0
$421$	17.9414	0.874411	0.437205	−	0.899362i	$-0.355968\pi$
$421$	17.9414	0.874411	0.437205	+	0.899362i	$0.355968\pi$
$422$	63.0212	3.06782
$423$	0	0
$424$	13.3441	0.648046
$425$	0	0
$426$	0	0
$427$	22.7742	1.10212
$428$	44.3137	2.14198
$429$	0	0
$430$	0	0
$431$	6.18871	0.298100	0.149050	−	0.988830i	$-0.452378\pi$
$431$	6.18871	0.298100	0.149050	+	0.988830i	$0.452378\pi$
$432$	0	0
$433$	−3.11806	−0.149844	−0.0749221	−	0.997189i	$-0.523871\pi$
$433$	−3.11806	−0.149844	−0.0749221	+	0.997189i	$0.523871\pi$
$434$	−12.4172	−0.596045
$435$	0	0
$436$	−31.3055	−1.49926
$437$	5.14052	0.245904
$438$	0	0
$439$	13.5099	0.644792	0.322396	−	0.946605i	$-0.395512\pi$
$439$	13.5099	0.644792	0.322396	+	0.946605i	$0.395512\pi$
$440$	0	0
$441$	0	0
$442$	24.9698	1.18769
$443$	24.2773	1.15345	0.576726	−	0.816938i	$-0.304330\pi$
$443$	24.2773	1.15345	0.576726	+	0.816938i	$0.304330\pi$
$444$	0	0
$445$	0	0
$446$	−57.1799	−2.70755
$447$	0	0
$448$	−19.9075	−0.940543
$449$	24.1437	1.13941	0.569705	−	0.821849i	$-0.307057\pi$
$449$	24.1437	1.13941	0.569705	+	0.821849i	$0.307057\pi$
$450$	0	0
$451$	−4.45970	−0.209999
$452$	73.5664	3.46027
$453$	0	0
$454$	37.1360	1.74288
$455$	0	0
$456$	0	0
$457$	−2.82157	−0.131987	−0.0659937	−	0.997820i	$-0.521022\pi$
$457$	−2.82157	−0.131987	−0.0659937	+	0.997820i	$0.521022\pi$
$458$	−9.68573	−0.452585
$459$	0	0
$460$	0	0
$461$	−21.4572	−0.999363	−0.499681	−	0.866209i	$-0.666550\pi$
$461$	−21.4572	−0.999363	−0.499681	+	0.866209i	$0.666550\pi$
$462$	0	0
$463$	−19.8033	−0.920339	−0.460170	−	0.887831i	$-0.652211\pi$
$463$	−19.8033	−0.920339	−0.460170	+	0.887831i	$0.652211\pi$
$464$	77.3310	3.59000
$465$	0	0
$466$	−14.0684	−0.651707
$467$	−22.7210	−1.05140	−0.525701	−	0.850669i	$-0.676197\pi$
$467$	−22.7210	−1.05140	−0.525701	+	0.850669i	$0.676197\pi$
$468$	0	0
$469$	−16.3633	−0.755588
$470$	0	0
$471$	0	0
$472$	85.2743	3.92507
$473$	13.1598	0.605088
$474$	0	0
$475$	0	0
$476$	42.6625	1.95543
$477$	0	0
$478$	−58.0300	−2.65423
$479$	−21.2880	−0.972672	−0.486336	−	0.873772i	$-0.661667\pi$
$479$	−21.2880	−0.972672	−0.486336	+	0.873772i	$0.661667\pi$
$480$	0	0
$481$	22.9424	1.04608
$482$	49.1115	2.23697
$483$	0	0
$484$	−38.1625	−1.73466
$485$	0	0
$486$	0	0
$487$	−9.58690	−0.434424	−0.217212	−	0.976124i	$-0.569696\pi$
$487$	−9.58690	−0.434424	−0.217212	+	0.976124i	$0.569696\pi$
$488$	97.9921	4.43590
$489$	0	0
$490$	0	0
$491$	37.8443	1.70789	0.853945	−	0.520363i	$-0.174203\pi$
$491$	37.8443	1.70789	0.853945	+	0.520363i	$0.174203\pi$
$492$	0	0
$493$	−35.4246	−1.59544
$494$	−15.3963	−0.692711
$495$	0	0
$496$	−27.5314	−1.23619
$497$	2.29101	0.102766
$498$	0	0
$499$	16.9253	0.757681	0.378840	−	0.925462i	$-0.376323\pi$
$499$	16.9253	0.757681	0.378840	+	0.925462i	$0.376323\pi$
$500$	0	0
$501$	0	0
$502$	38.6970	1.72713
$503$	−40.4168	−1.80210	−0.901048	−	0.433719i	$-0.857201\pi$
$503$	−40.4168	−1.80210	−0.901048	+	0.433719i	$0.857201\pi$
$504$	0	0
$505$	0	0
$506$	8.25713	0.367074
$507$	0	0
$508$	17.8851	0.793522
$509$	41.4067	1.83532	0.917660	−	0.397366i	$-0.130076\pi$
$509$	41.4067	1.83532	0.917660	+	0.397366i	$0.130076\pi$
$510$	0	0
$511$	6.44863	0.285270
$512$	34.5537	1.52707
$513$	0	0
$514$	58.4922	2.57998
$515$	0	0
$516$	0	0
$517$	11.3769	0.500354
$518$	55.0797	2.42006
$519$	0	0
$520$	0	0
$521$	17.0301	0.746103	0.373052	−	0.927811i	$-0.378311\pi$
$521$	17.0301	0.746103	0.373052	+	0.927811i	$0.378311\pi$
$522$	0	0
$523$	9.57651	0.418751	0.209376	−	0.977835i	$-0.432857\pi$
$523$	9.57651	0.418751	0.209376	+	0.977835i	$0.432857\pi$
$524$	−36.0041	−1.57285
$525$	0	0
$526$	15.1176	0.659159
$527$	12.6118	0.549380
$528$	0	0
$529$	−19.9935	−0.869283
$530$	0	0
$531$	0	0
$532$	−26.3055	−1.14049
$533$	4.86347	0.210660
$534$	0	0
$535$	0	0
$536$	−70.4077	−3.04115
$537$	0	0
$538$	−41.1289	−1.77319
$539$	6.81528	0.293555
$540$	0	0
$541$	−0.833751	−0.0358458	−0.0179229	−	0.999839i	$-0.505705\pi$
$541$	−0.833751	−0.0358458	−0.0179229	+	0.999839i	$0.505705\pi$
$542$	17.7859	0.763971
$543$	0	0
$544$	58.5426	2.50999
$545$	0	0
$546$	0	0
$547$	−28.3270	−1.21117	−0.605587	−	0.795779i	$-0.707062\pi$
$547$	−28.3270	−1.21117	−0.605587	+	0.795779i	$0.707062\pi$
$548$	35.1785	1.50275
$549$	0	0
$550$	0	0
$551$	21.8427	0.930529
$552$	0	0
$553$	−3.79419	−0.161345
$554$	−79.7928	−3.39007
$555$	0	0
$556$	−72.6192	−3.07974
$557$	−11.5042	−0.487448	−0.243724	−	0.969845i	$-0.578369\pi$
$557$	−11.5042	−0.487448	−0.243724	+	0.969845i	$0.578369\pi$
$558$	0	0
$559$	−14.3512	−0.606993
$560$	0	0
$561$	0	0
$562$	49.0797	2.07030
$563$	−33.0059	−1.39103	−0.695517	−	0.718510i	$-0.744824\pi$
$563$	−33.0059	−1.39103	−0.695517	+	0.718510i	$0.744824\pi$
$564$	0	0
$565$	0	0
$566$	14.9429	0.628095
$567$	0	0
$568$	9.85769	0.413619
$569$	27.0088	1.13227	0.566135	−	0.824313i	$-0.308438\pi$
$569$	27.0088	1.13227	0.566135	+	0.824313i	$0.308438\pi$
$570$	0	0
$571$	−24.4244	−1.02213	−0.511064	−	0.859543i	$-0.670749\pi$
$571$	−24.4244	−1.02213	−0.511064	+	0.859543i	$0.670749\pi$
$572$	−17.6001	−0.735899
$573$	0	0
$574$	11.6761	0.487352
$575$	0	0
$576$	0	0
$577$	14.7976	0.616033	0.308017	−	0.951381i	$-0.400335\pi$
$577$	14.7976	0.616033	0.308017	+	0.951381i	$0.400335\pi$
$578$	−16.1133	−0.670224
$579$	0	0
$580$	0	0
$581$	1.97507	0.0819396
$582$	0	0
$583$	3.11973	0.129206
$584$	27.7470	1.14818
$585$	0	0
$586$	48.1375	1.98854
$587$	−30.5780	−1.26209	−0.631044	−	0.775747i	$-0.717373\pi$
$587$	−30.5780	−1.26209	−0.631044	+	0.775747i	$0.717373\pi$
$588$	0	0
$589$	−7.77641	−0.320421
$590$	0	0
$591$	0	0
$592$	122.122	5.01919
$593$	5.09990	0.209428	0.104714	−	0.994502i	$-0.466607\pi$
$593$	5.09990	0.209428	0.104714	+	0.994502i	$0.466607\pi$
$594$	0	0
$595$	0	0
$596$	2.79065	0.114309
$597$	0	0
$598$	−9.00471	−0.368230
$599$	0.565309	0.0230979	0.0115490	−	0.999933i	$-0.496324\pi$
$599$	0.565309	0.0230979	0.0115490	+	0.999933i	$0.496324\pi$
$600$	0	0
$601$	−11.0096	−0.449091	−0.224546	−	0.974464i	$-0.572090\pi$
$601$	−11.0096	−0.449091	−0.224546	+	0.974464i	$0.572090\pi$
$602$	−34.4542	−1.40425
$603$	0	0
$604$	0.757185	0.0308094
$605$	0	0
$606$	0	0
$607$	−19.0983	−0.775174	−0.387587	−	0.921833i	$-0.626691\pi$
$607$	−19.0983	−0.775174	−0.387587	+	0.921833i	$0.626691\pi$
$608$	−36.0972	−1.46393
$609$	0	0
$610$	0	0
$611$	−12.4069	−0.501929
$612$	0	0
$613$	9.33918	0.377206	0.188603	−	0.982053i	$-0.439604\pi$
$613$	9.33918	0.377206	0.188603	+	0.982053i	$0.439604\pi$
$614$	40.8358	1.64800
$615$	0	0
$616$	−25.1350	−1.01272
$617$	24.4154	0.982928	0.491464	−	0.870898i	$-0.336462\pi$
$617$	24.4154	0.982928	0.491464	+	0.870898i	$0.336462\pi$
$618$	0	0
$619$	39.4863	1.58709	0.793544	−	0.608513i	$-0.208234\pi$
$619$	39.4863	1.58709	0.793544	+	0.608513i	$0.208234\pi$
$620$	0	0
$621$	0	0
$622$	80.1874	3.21522
$623$	−23.8940	−0.957293
$624$	0	0
$625$	0	0
$626$	18.3027	0.731523
$627$	0	0
$628$	−56.5757	−2.25762
$629$	−55.9430	−2.23059
$630$	0	0
$631$	42.1634	1.67850	0.839249	−	0.543747i	$-0.182995\pi$
$631$	42.1634	1.67850	0.839249	+	0.543747i	$0.182995\pi$
$632$	−16.3255	−0.649395
$633$	0	0
$634$	−42.5216	−1.68875
$635$	0	0
$636$	0	0
$637$	−7.43232	−0.294479
$638$	35.0855	1.38905
$639$	0	0
$640$	0	0
$641$	−35.3155	−1.39488	−0.697438	−	0.716645i	$-0.745677\pi$
$641$	−35.3155	−1.39488	−0.697438	+	0.716645i	$0.745677\pi$
$642$	0	0
$643$	14.1954	0.559813	0.279906	−	0.960027i	$-0.409697\pi$
$643$	14.1954	0.559813	0.279906	+	0.960027i	$0.409697\pi$
$644$	−15.3851	−0.606259
$645$	0	0
$646$	37.5424	1.47709
$647$	−17.4897	−0.687593	−0.343796	−	0.939044i	$-0.611713\pi$
$647$	−17.4897	−0.687593	−0.343796	+	0.939044i	$0.611713\pi$
$648$	0	0
$649$	19.9364	0.782572
$650$	0	0
$651$	0	0
$652$	−108.897	−4.26472
$653$	10.9772	0.429569	0.214785	−	0.976661i	$-0.431095\pi$
$653$	10.9772	0.429569	0.214785	+	0.976661i	$0.431095\pi$
$654$	0	0
$655$	0	0
$656$	25.8882	1.01077
$657$	0	0
$658$	−29.7862	−1.16119
$659$	15.7876	0.614998	0.307499	−	0.951548i	$-0.400508\pi$
$659$	15.7876	0.614998	0.307499	+	0.951548i	$0.400508\pi$
$660$	0	0
$661$	49.8932	1.94062	0.970311	−	0.241862i	$-0.0777581\pi$
$661$	49.8932	1.94062	0.970311	+	0.241862i	$0.0777581\pi$
$662$	−33.2435	−1.29204
$663$	0	0
$664$	8.49827	0.329797
$665$	0	0
$666$	0	0
$667$	12.7750	0.494649
$668$	−84.2582	−3.26005
$669$	0	0
$670$	0	0
$671$	22.9097	0.884419
$672$	0	0
$673$	28.8395	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$673$	28.8395	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$674$	18.2264	0.702055
$675$	0	0
$676$	−44.9809	−1.73003
$677$	−10.4636	−0.402150	−0.201075	−	0.979576i	$-0.564443\pi$
$677$	−10.4636	−0.402150	−0.201075	+	0.979576i	$0.564443\pi$
$678$	0	0
$679$	6.88910	0.264379
$680$	0	0
$681$	0	0
$682$	−12.4911	−0.478309
$683$	16.1875	0.619396	0.309698	−	0.950835i	$-0.399772\pi$
$683$	16.1875	0.619396	0.309698	+	0.950835i	$0.399772\pi$
$684$	0	0
$685$	0	0
$686$	−50.9809	−1.94646
$687$	0	0
$688$	−76.3916	−2.91240
$689$	−3.40218	−0.129613
$690$	0	0
$691$	9.88362	0.375991	0.187995	−	0.982170i	$-0.439801\pi$
$691$	9.88362	0.375991	0.187995	+	0.982170i	$0.439801\pi$
$692$	−58.9648	−2.24150
$693$	0	0
$694$	36.3621	1.38029
$695$	0	0
$696$	0	0
$697$	−11.8591	−0.449197
$698$	−17.2974	−0.654718
$699$	0	0
$700$	0	0
$701$	−43.9692	−1.66069	−0.830346	−	0.557248i	$-0.811857\pi$
$701$	−43.9692	−1.66069	−0.830346	+	0.557248i	$0.811857\pi$
$702$	0	0
$703$	34.4942	1.30097
$704$	−20.0260	−0.754758
$705$	0	0
$706$	−9.28385	−0.349402
$707$	11.7774	0.442935
$708$	0	0
$709$	25.2260	0.947384	0.473692	−	0.880691i	$-0.342921\pi$
$709$	25.2260	0.947384	0.473692	+	0.880691i	$0.342921\pi$
$710$	0	0
$711$	0	0
$712$	−102.811	−3.85299
$713$	−4.54814	−0.170329
$714$	0	0
$715$	0	0
$716$	−42.2032	−1.57721
$717$	0	0
$718$	60.3875	2.25364
$719$	36.8600	1.37465	0.687323	−	0.726352i	$-0.258786\pi$
$719$	36.8600	1.37465	0.687323	+	0.726352i	$0.258786\pi$
$720$	0	0
$721$	14.5153	0.540576
$722$	26.8922	1.00082
$723$	0	0
$724$	−52.0918	−1.93598
$725$	0	0
$726$	0	0
$727$	−38.2451	−1.41843	−0.709217	−	0.704990i	$-0.750951\pi$
$727$	−38.2451	−1.41843	−0.709217	+	0.704990i	$0.750951\pi$
$728$	27.4107	1.01591
$729$	0	0
$730$	0	0
$731$	34.9942	1.29431
$732$	0	0
$733$	39.5832	1.46204	0.731020	−	0.682356i	$-0.239045\pi$
$733$	39.5832	1.46204	0.731020	+	0.682356i	$0.239045\pi$
$734$	−11.0072	−0.406282
$735$	0	0
$736$	−21.1119	−0.778195
$737$	−16.4607	−0.606338
$738$	0	0
$739$	−8.24773	−0.303398	−0.151699	−	0.988427i	$-0.548474\pi$
$739$	−8.24773	−0.303398	−0.151699	+	0.988427i	$0.548474\pi$
$740$	0	0
$741$	0	0
$742$	−8.16790	−0.299853
$743$	−1.30818	−0.0479925	−0.0239963	−	0.999712i	$-0.507639\pi$
$743$	−1.30818	−0.0479925	−0.0239963	+	0.999712i	$0.507639\pi$
$744$	0	0
$745$	0	0
$746$	−18.0171	−0.659651
$747$	0	0
$748$	42.9164	1.56918
$749$	−16.1350	−0.589562
$750$	0	0
$751$	28.4468	1.03804	0.519020	−	0.854762i	$-0.326297\pi$
$751$	28.4468	1.03804	0.519020	+	0.854762i	$0.326297\pi$
$752$	−66.0418	−2.40830
$753$	0	0
$754$	−38.2620	−1.39342
$755$	0	0
$756$	0	0
$757$	−38.2012	−1.38845	−0.694223	−	0.719760i	$-0.744252\pi$
$757$	−38.2012	−1.38845	−0.694223	+	0.719760i	$0.744252\pi$
$758$	33.6495	1.22221
$759$	0	0
$760$	0	0
$761$	−22.1904	−0.804401	−0.402200	−	0.915552i	$-0.631755\pi$
$761$	−22.1904	−0.804401	−0.402200	+	0.915552i	$0.631755\pi$
$762$	0	0
$763$	11.3986	0.412659
$764$	85.5852	3.09637
$765$	0	0
$766$	19.8440	0.716994
$767$	−21.7414	−0.785036
$768$	0	0
$769$	−16.9130	−0.609900	−0.304950	−	0.952368i	$-0.598640\pi$
$769$	−16.9130	−0.609900	−0.304950	+	0.952368i	$0.598640\pi$
$770$	0	0
$771$	0	0
$772$	7.69860	0.277079
$773$	38.6464	1.39001	0.695007	−	0.719003i	$-0.255401\pi$
$773$	38.6464	1.39001	0.695007	+	0.719003i	$0.255401\pi$
$774$	0	0
$775$	0	0
$776$	29.6423	1.06409
$777$	0	0
$778$	14.3584	0.514774
$779$	7.31230	0.261990
$780$	0	0
$781$	2.30464	0.0824665
$782$	21.9572	0.785187
$783$	0	0
$784$	−39.5622	−1.41294
$785$	0	0
$786$	0	0
$787$	10.9147	0.389067	0.194533	−	0.980896i	$-0.437681\pi$
$787$	10.9147	0.389067	0.194533	+	0.980896i	$0.437681\pi$
$788$	−88.7422	−3.16131
$789$	0	0
$790$	0	0
$791$	−26.7862	−0.952409
$792$	0	0
$793$	−24.9839	−0.887204
$794$	14.8687	0.527669
$795$	0	0
$796$	54.4124	1.92860
$797$	2.92550	0.103627	0.0518133	−	0.998657i	$-0.483500\pi$
$797$	2.92550	0.103627	0.0518133	+	0.998657i	$0.483500\pi$
$798$	0	0
$799$	30.2531	1.07028
$800$	0	0
$801$	0	0
$802$	−14.4965	−0.511888
$803$	6.48700	0.228921
$804$	0	0
$805$	0	0
$806$	13.6220	0.479816
$807$	0	0
$808$	50.6755	1.78276
$809$	37.9241	1.33334	0.666671	−	0.745352i	$-0.267719\pi$
$809$	37.9241	1.33334	0.666671	+	0.745352i	$0.267719\pi$
$810$	0	0
$811$	19.5050	0.684912	0.342456	−	0.939534i	$-0.388741\pi$
$811$	19.5050	0.684912	0.342456	+	0.939534i	$0.388741\pi$
$812$	−65.3731	−2.29415
$813$	0	0
$814$	55.4075	1.94203
$815$	0	0
$816$	0	0
$817$	−21.5773	−0.754894
$818$	−86.5257	−3.02530
$819$	0	0
$820$	0	0
$821$	−44.7524	−1.56187	−0.780934	−	0.624613i	$-0.785257\pi$
$821$	−44.7524	−1.56187	−0.780934	+	0.624613i	$0.785257\pi$
$822$	0	0
$823$	−35.8994	−1.25137	−0.625687	−	0.780074i	$-0.715181\pi$
$823$	−35.8994	−1.25137	−0.625687	+	0.780074i	$0.715181\pi$
$824$	62.4559	2.17575
$825$	0	0
$826$	−52.1963	−1.81614
$827$	16.2717	0.565822	0.282911	−	0.959146i	$-0.408700\pi$
$827$	16.2717	0.565822	0.282911	+	0.959146i	$0.408700\pi$
$828$	0	0
$829$	−20.6592	−0.717525	−0.358762	−	0.933429i	$-0.616801\pi$
$829$	−20.6592	−0.717525	−0.358762	+	0.933429i	$0.616801\pi$
$830$	0	0
$831$	0	0
$832$	21.8391	0.757135
$833$	18.1230	0.627926
$834$	0	0
$835$	0	0
$836$	−26.4621	−0.915210
$837$	0	0
$838$	60.2044	2.07973
$839$	12.2367	0.422459	0.211229	−	0.977437i	$-0.432253\pi$
$839$	12.2367	0.422459	0.211229	+	0.977437i	$0.432253\pi$
$840$	0	0
$841$	25.2823	0.871802
$842$	−47.2527	−1.62844
$843$	0	0
$844$	−118.123	−4.06597
$845$	0	0
$846$	0	0
$847$	13.8953	0.477450
$848$	−18.1098	−0.621893
$849$	0	0
$850$	0	0
$851$	20.1744	0.691570
$852$	0	0
$853$	16.4293	0.562528	0.281264	−	0.959630i	$-0.409246\pi$
$853$	16.4293	0.562528	0.281264	+	0.959630i	$0.409246\pi$
$854$	−59.9809	−2.05250
$855$	0	0
$856$	−69.4255	−2.37291
$857$	−49.0897	−1.67687	−0.838436	−	0.545000i	$-0.816530\pi$
$857$	−49.0897	−1.67687	−0.838436	+	0.545000i	$0.816530\pi$
$858$	0	0
$859$	41.0908	1.40200	0.700999	−	0.713162i	$-0.252738\pi$
$859$	41.0908	1.40200	0.700999	+	0.713162i	$0.252738\pi$
$860$	0	0
$861$	0	0
$862$	−16.2994	−0.555158
$863$	−47.8366	−1.62838	−0.814188	−	0.580601i	$-0.802817\pi$
$863$	−47.8366	−1.62838	−0.814188	+	0.580601i	$0.802817\pi$
$864$	0	0
$865$	0	0
$866$	8.21210	0.279058
$867$	0	0
$868$	23.2741	0.789975
$869$	−3.81677	−0.129475
$870$	0	0
$871$	17.9510	0.608247
$872$	49.0458	1.66090
$873$	0	0
$874$	−13.5387	−0.457954
$875$	0	0
$876$	0	0
$877$	−11.3772	−0.384180	−0.192090	−	0.981377i	$-0.561527\pi$
$877$	−11.3772	−0.384180	−0.192090	+	0.981377i	$0.561527\pi$
$878$	−35.5813	−1.20081
$879$	0	0
$880$	0	0
$881$	43.9924	1.48214	0.741071	−	0.671426i	$-0.234318\pi$
$881$	43.9924	1.48214	0.741071	+	0.671426i	$0.234318\pi$
$882$	0	0
$883$	37.6820	1.26810	0.634051	−	0.773291i	$-0.281391\pi$
$883$	37.6820	1.26810	0.634051	+	0.773291i	$0.281391\pi$
$884$	−46.8019	−1.57412
$885$	0	0
$886$	−63.9398	−2.14810
$887$	33.2833	1.11754	0.558771	−	0.829322i	$-0.311273\pi$
$887$	33.2833	1.11754	0.558771	+	0.829322i	$0.311273\pi$
$888$	0	0
$889$	−6.51213	−0.218410
$890$	0	0
$891$	0	0
$892$	107.175	3.58847
$893$	−18.6539	−0.624230
$894$	0	0
$895$	0	0
$896$	8.66080	0.289337
$897$	0	0
$898$	−63.5878	−2.12195
$899$	−19.3255	−0.644543
$900$	0	0
$901$	8.29592	0.276377
$902$	11.7456	0.391086
$903$	0	0
$904$	−115.255	−3.83333
$905$	0	0
$906$	0	0
$907$	−14.0784	−0.467464	−0.233732	−	0.972301i	$-0.575094\pi$
$907$	−14.0784	−0.467464	−0.233732	+	0.972301i	$0.575094\pi$
$908$	−69.6056	−2.30994
$909$	0	0
$910$	0	0
$911$	−7.31522	−0.242364	−0.121182	−	0.992630i	$-0.538668\pi$
$911$	−7.31522	−0.242364	−0.121182	+	0.992630i	$0.538668\pi$
$912$	0	0
$913$	1.98682	0.0657542
$914$	7.43123	0.245803
$915$	0	0
$916$	18.1544	0.599838
$917$	13.1094	0.432912
$918$	0	0
$919$	4.44684	0.146688	0.0733438	−	0.997307i	$-0.476633\pi$
$919$	4.44684	0.146688	0.0733438	+	0.997307i	$0.476633\pi$
$920$	0	0
$921$	0	0
$922$	56.5124	1.86114
$923$	−2.51330	−0.0827262
$924$	0	0
$925$	0	0
$926$	52.1565	1.71397
$927$	0	0
$928$	−89.7068	−2.94477
$929$	−13.1133	−0.430232	−0.215116	−	0.976588i	$-0.569013\pi$
$929$	−13.1133	−0.430232	−0.215116	+	0.976588i	$0.569013\pi$
$930$	0	0
$931$	−11.1746	−0.366233
$932$	26.3690	0.863746
$933$	0	0
$934$	59.8408	1.95805
$935$	0	0
$936$	0	0
$937$	−33.5187	−1.09501	−0.547504	−	0.836803i	$-0.684422\pi$
$937$	−33.5187	−1.09501	−0.547504	+	0.836803i	$0.684422\pi$
$938$	43.0964	1.40715
$939$	0	0
$940$	0	0
$941$	−1.79076	−0.0583772	−0.0291886	−	0.999574i	$-0.509292\pi$
$941$	−1.79076	−0.0583772	−0.0291886	+	0.999574i	$0.509292\pi$
$942$	0	0
$943$	4.27669	0.139268
$944$	−115.729	−3.76667
$945$	0	0
$946$	−34.6592	−1.12687
$947$	52.6350	1.71041	0.855204	−	0.518292i	$-0.173432\pi$
$947$	52.6350	1.71041	0.855204	+	0.518292i	$0.173432\pi$
$948$	0	0
$949$	−7.07432	−0.229642
$950$	0	0
$951$	0	0
$952$	−66.8386	−2.16625
$953$	18.4072	0.596268	0.298134	−	0.954524i	$-0.403636\pi$
$953$	18.4072	0.596268	0.298134	+	0.954524i	$0.403636\pi$
$954$	0	0
$955$	0	0
$956$	108.768	3.51781
$957$	0	0
$958$	56.0666	1.81143
$959$	−12.8088	−0.413619
$960$	0	0
$961$	−24.1197	−0.778056
$962$	−60.4240	−1.94815
$963$	0	0
$964$	−92.0517	−2.96479
$965$	0	0
$966$	0	0
$967$	−6.09579	−0.196027	−0.0980137	−	0.995185i	$-0.531249\pi$
$967$	−6.09579	−0.196027	−0.0980137	+	0.995185i	$0.531249\pi$
$968$	59.7886	1.92168
$969$	0	0
$970$	0	0
$971$	34.1077	1.09457	0.547283	−	0.836947i	$-0.315662\pi$
$971$	34.1077	1.09457	0.547283	+	0.836947i	$0.315662\pi$
$972$	0	0
$973$	26.4414	0.847671
$974$	25.2493	0.809038
$975$	0	0
$976$	−132.989	−4.25688
$977$	−48.7914	−1.56098	−0.780488	−	0.625171i	$-0.785029\pi$
$977$	−48.7914	−1.56098	−0.780488	+	0.625171i	$0.785029\pi$
$978$	0	0
$979$	−24.0362	−0.768201
$980$	0	0
$981$	0	0
$982$	−99.6715	−3.18065
$983$	−33.4172	−1.06584	−0.532922	−	0.846164i	$-0.678906\pi$
$983$	−33.4172	−1.06584	−0.532922	+	0.846164i	$0.678906\pi$
$984$	0	0
$985$	0	0
$986$	93.2985	2.97123
$987$	0	0
$988$	28.8579	0.918092
$989$	−12.6198	−0.401285
$990$	0	0
$991$	−19.3512	−0.614713	−0.307356	−	0.951595i	$-0.599444\pi$
$991$	−19.3512	−0.614713	−0.307356	+	0.951595i	$0.599444\pi$
$992$	31.9374	1.01401
$993$	0	0
$994$	−6.03388	−0.191383
$995$	0	0
$996$	0	0
$997$	39.5442	1.25238	0.626189	−	0.779671i	$-0.284614\pi$
$997$	39.5442	1.25238	0.626189	+	0.779671i	$0.284614\pi$
$998$	−44.5766	−1.41105
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.p.1.1		4
3.2	odd	2		2025.2.a.y.1.4		4
5.2	odd	4		2025.2.b.o.649.1		8
5.3	odd	4		2025.2.b.o.649.8		8
5.4	even	2		2025.2.a.z.1.4		4
9.2	odd	6		225.2.e.c.76.1	✓	8
9.4	even	3		675.2.e.e.451.4		8
9.5	odd	6		225.2.e.c.151.1	yes	8
9.7	even	3		675.2.e.e.226.4		8
15.2	even	4		2025.2.b.n.649.8		8
15.8	even	4		2025.2.b.n.649.1		8
15.14	odd	2		2025.2.a.q.1.1		4
45.2	even	12		225.2.k.c.49.8		16
45.4	even	6		675.2.e.c.451.1		8
45.7	odd	12		675.2.k.c.199.1		16
45.13	odd	12		675.2.k.c.424.1		16
45.14	odd	6		225.2.e.e.151.4	yes	8
45.22	odd	12		675.2.k.c.424.8		16
45.23	even	12		225.2.k.c.124.8		16
45.29	odd	6		225.2.e.e.76.4	yes	8
45.32	even	12		225.2.k.c.124.1		16
45.34	even	6		675.2.e.c.226.1		8
45.38	even	12		225.2.k.c.49.1		16
45.43	odd	12		675.2.k.c.199.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.e.c.76.1	✓	8	9.2	odd	6
225.2.e.c.151.1	yes	8	9.5	odd	6
225.2.e.e.76.4	yes	8	45.29	odd	6
225.2.e.e.151.4	yes	8	45.14	odd	6
225.2.k.c.49.1		16	45.38	even	12
225.2.k.c.49.8		16	45.2	even	12
225.2.k.c.124.1		16	45.32	even	12
225.2.k.c.124.8		16	45.23	even	12
675.2.e.c.226.1		8	45.34	even	6
675.2.e.c.451.1		8	45.4	even	6
675.2.e.e.226.4		8	9.7	even	3
675.2.e.e.451.4		8	9.4	even	3
675.2.k.c.199.1		16	45.7	odd	12
675.2.k.c.199.8		16	45.43	odd	12
675.2.k.c.424.1		16	45.13	odd	12
675.2.k.c.424.8		16	45.22	odd	12
2025.2.a.p.1.1		4	1.1	even	1	trivial
2025.2.a.q.1.1		4	15.14	odd	2
2025.2.a.y.1.4		4	3.2	odd	2
2025.2.a.z.1.4		4	5.4	even	2
2025.2.b.n.649.1		8	15.8	even	4
2025.2.b.n.649.8		8	15.2	even	4
2025.2.b.o.649.1		8	5.2	odd	4
2025.2.b.o.649.8		8	5.3	odd	4

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:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.63372 q^{2} +4.93650 q^{4} -1.79743 q^{7} -7.73393 q^{8} +O(q^{10})\)	\(q-2.63372 q^{2} +4.93650 q^{4} -1.79743 q^{7} -7.73393 q^{8} -1.80812 q^{11} +1.97183 q^{13} +4.73393 q^{14} +10.4960 q^{16} -4.80812 q^{17} +2.96467 q^{19} +4.76210 q^{22} +1.73393 q^{23} -5.19325 q^{26} -8.87300 q^{28} +7.36765 q^{29} -2.62303 q^{31} -12.1758 q^{32} +12.6633 q^{34} +11.6351 q^{37} -7.80812 q^{38} +2.46648 q^{41} -7.27814 q^{43} -8.92580 q^{44} -4.56668 q^{46} -6.29208 q^{47} -3.76926 q^{49} +9.73393 q^{52} -1.72540 q^{53} +13.9012 q^{56} -19.4044 q^{58} -11.0260 q^{59} -12.6704 q^{61} +6.90833 q^{62} +11.0756 q^{64} +9.10374 q^{67} -23.7353 q^{68} -1.27460 q^{71} -3.58770 q^{73} -30.6436 q^{74} +14.6351 q^{76} +3.24997 q^{77} +2.11090 q^{79} -6.49602 q^{82} -1.09883 q^{83} +19.1686 q^{86} +13.9839 q^{88} +13.2935 q^{89} -3.54422 q^{91} +8.55953 q^{92} +16.5716 q^{94} -3.83276 q^{97} +9.92718 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8	\( 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8} + q^{11} + 2 q^{13} - 3 q^{14} + 4 q^{16} - 11 q^{17} + 2 q^{19} + 3 q^{22} - 15 q^{23} + 10 q^{26} - 4 q^{28} - q^{29} - 4 q^{31} - 10 q^{32} + 9 q^{34} - q^{37} - 23 q^{38} + 5 q^{41} - 10 q^{43} - 22 q^{44} - 20 q^{47} - 3 q^{49} + 17 q^{52} - 20 q^{53} + 30 q^{56} - 18 q^{58} - 17 q^{59} - 13 q^{61} + 6 q^{62} + 19 q^{64} + 17 q^{67} - 34 q^{68} + 8 q^{71} + 2 q^{73} - 40 q^{74} + 11 q^{76} - 12 q^{77} - 7 q^{79} + 12 q^{82} - 30 q^{83} + 34 q^{86} + 9 q^{88} + 9 q^{89} - 17 q^{91} + 12 q^{92} + 3 q^{94} - 19 q^{97} - 13 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8 + q^11 + 2 * q^13 - 3 * q^14 + 4 * q^16 - 11 * q^17 + 2 * q^19 + 3 * q^22 - 15 * q^23 + 10 * q^26 - 4 * q^28 - q^29 - 4 * q^31 - 10 * q^32 + 9 * q^34 - q^37 - 23 * q^38 + 5 * q^41 - 10 * q^43 - 22 * q^44 - 20 * q^47 - 3 * q^49 + 17 * q^52 - 20 * q^53 + 30 * q^56 - 18 * q^58 - 17 * q^59 - 13 * q^61 + 6 * q^62 + 19 * q^64 + 17 * q^67 - 34 * q^68 + 8 * q^71 + 2 * q^73 - 40 * q^74 + 11 * q^76 - 12 * q^77 - 7 * q^79 + 12 * q^82 - 30 * q^83 + 34 * q^86 + 9 * q^88 + 9 * q^89 - 17 * q^91 + 12 * q^92 + 3 * q^94 - 19 * q^97 - 13 * q^98

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