LMFDB - Embedded newform 2025.2.a.i.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ 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)

$f(q)$	$=$	$q-2.30278 q^{2} +3.30278 q^{4} +4.30278 q^{7} -3.00000 q^{8} +O(q^{10})$	$q-2.30278 q^{2} +3.30278 q^{4} +4.30278 q^{7} -3.00000 q^{8} -5.30278 q^{11} -0.302776 q^{13} -9.90833 q^{14} +0.302776 q^{16} -2.30278 q^{17} -7.21110 q^{19} +12.2111 q^{22} +6.90833 q^{23} +0.697224 q^{26} +14.2111 q^{28} +1.39445 q^{29} +8.21110 q^{31} +5.30278 q^{32} +5.30278 q^{34} -1.90833 q^{37} +16.6056 q^{38} +5.30278 q^{41} +5.69722 q^{43} -17.5139 q^{44} -15.9083 q^{46} -6.21110 q^{47} +11.5139 q^{49} -1.00000 q^{52} +5.51388 q^{53} -12.9083 q^{56} -3.21110 q^{58} +3.69722 q^{59} -0.302776 q^{61} -18.9083 q^{62} -12.8167 q^{64} -5.60555 q^{67} -7.60555 q^{68} +4.39445 q^{71} +9.60555 q^{73} +4.39445 q^{74} -23.8167 q^{76} -22.8167 q^{77} -2.60555 q^{79} -12.2111 q^{82} +9.00000 q^{83} -13.1194 q^{86} +15.9083 q^{88} -12.0000 q^{89} -1.30278 q^{91} +22.8167 q^{92} +14.3028 q^{94} +2.00000 q^{97} -26.5139 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - q^2 + 3 * q^4 + 5 * q^7 - 6 * q^8	$ 2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8} - 7 q^{11} + 3 q^{13} - 9 q^{14} - 3 q^{16} - q^{17} + 10 q^{22} + 3 q^{23} + 5 q^{26} + 14 q^{28} + 10 q^{29} + 2 q^{31} + 7 q^{32} + 7 q^{34} + 7 q^{37} + 26 q^{38} + 7 q^{41} + 15 q^{43} - 17 q^{44} - 21 q^{46} + 2 q^{47} + 5 q^{49} - 2 q^{52} - 7 q^{53} - 15 q^{56} + 8 q^{58} + 11 q^{59} + 3 q^{61} - 27 q^{62} - 4 q^{64} - 4 q^{67} - 8 q^{68} + 16 q^{71} + 12 q^{73} + 16 q^{74} - 26 q^{76} - 24 q^{77} + 2 q^{79} - 10 q^{82} + 18 q^{83} - q^{86} + 21 q^{88} - 24 q^{89} + q^{91} + 24 q^{92} + 25 q^{94} + 4 q^{97} - 35 q^{98}+O(q^{100}) $ 2 * q - q^2 + 3 * q^4 + 5 * q^7 - 6 * q^8 - 7 * q^11 + 3 * q^13 - 9 * q^14 - 3 * q^16 - q^17 + 10 * q^22 + 3 * q^23 + 5 * q^26 + 14 * q^28 + 10 * q^29 + 2 * q^31 + 7 * q^32 + 7 * q^34 + 7 * q^37 + 26 * q^38 + 7 * q^41 + 15 * q^43 - 17 * q^44 - 21 * q^46 + 2 * q^47 + 5 * q^49 - 2 * q^52 - 7 * q^53 - 15 * q^56 + 8 * q^58 + 11 * q^59 + 3 * q^61 - 27 * q^62 - 4 * q^64 - 4 * q^67 - 8 * q^68 + 16 * q^71 + 12 * q^73 + 16 * q^74 - 26 * q^76 - 24 * q^77 + 2 * q^79 - 10 * q^82 + 18 * q^83 - q^86 + 21 * q^88 - 24 * q^89 + q^91 + 24 * q^92 + 25 * q^94 + 4 * q^97 - 35 * 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 $

$2$	−2.30278	−1.62831	−0.814154	−	0.580649i	$-0.802799\pi$
$2$	−2.30278	−1.62831	−0.814154	+	0.580649i	$0.802799\pi$
$3$	0	0
$3$	0	0
$4$	3.30278	1.65139
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.30278	1.62630	0.813148	−	0.582057i	$-0.197752\pi$
$7$	4.30278	1.62630	0.813148	+	0.582057i	$0.197752\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	0	0
$11$	−5.30278	−1.59885	−0.799424	−	0.600768i	$-0.794862\pi$
$11$	−5.30278	−1.59885	−0.799424	+	0.600768i	$0.794862\pi$
$12$	0	0
$13$	−0.302776	−0.0839749	−0.0419874	−	0.999118i	$-0.513369\pi$
$13$	−0.302776	−0.0839749	−0.0419874	+	0.999118i	$0.513369\pi$
$14$	−9.90833	−2.64811
$15$	0	0
$16$	0.302776	0.0756939
$17$	−2.30278	−0.558505	−0.279253	−	0.960218i	$-0.590087\pi$
$17$	−2.30278	−0.558505	−0.279253	+	0.960218i	$0.590087\pi$
$18$	0	0
$19$	−7.21110	−1.65434	−0.827170	−	0.561951i	$-0.810051\pi$
$19$	−7.21110	−1.65434	−0.827170	+	0.561951i	$0.810051\pi$
$20$	0	0
$21$	0	0
$22$	12.2111	2.60342
$23$	6.90833	1.44049	0.720243	−	0.693722i	$-0.244030\pi$
$23$	6.90833	1.44049	0.720243	+	0.693722i	$0.244030\pi$
$24$	0	0
$25$	0	0
$26$	0.697224	0.136737
$27$	0	0
$28$	14.2111	2.68565
$29$	1.39445	0.258943	0.129471	−	0.991583i	$-0.458672\pi$
$29$	1.39445	0.258943	0.129471	+	0.991583i	$0.458672\pi$
$30$	0	0
$31$	8.21110	1.47476	0.737379	−	0.675479i	$-0.236063\pi$
$31$	8.21110	1.47476	0.737379	+	0.675479i	$0.236063\pi$
$32$	5.30278	0.937407
$33$	0	0
$34$	5.30278	0.909419
$35$	0	0
$36$	0	0
$37$	−1.90833	−0.313727	−0.156864	−	0.987620i	$-0.550138\pi$
$37$	−1.90833	−0.313727	−0.156864	+	0.987620i	$0.550138\pi$
$38$	16.6056	2.69378
$39$	0	0
$40$	0	0
$41$	5.30278	0.828154	0.414077	−	0.910242i	$-0.364104\pi$
$41$	5.30278	0.828154	0.414077	+	0.910242i	$0.364104\pi$
$42$	0	0
$43$	5.69722	0.868819	0.434409	−	0.900716i	$-0.356957\pi$
$43$	5.69722	0.868819	0.434409	+	0.900716i	$0.356957\pi$
$44$	−17.5139	−2.64032
$45$	0	0
$46$	−15.9083	−2.34555
$47$	−6.21110	−0.905982	−0.452991	−	0.891515i	$-0.649643\pi$
$47$	−6.21110	−0.905982	−0.452991	+	0.891515i	$0.649643\pi$
$48$	0	0
$49$	11.5139	1.64484
$50$	0	0
$51$	0	0
$52$	−1.00000	−0.138675
$53$	5.51388	0.757389	0.378695	−	0.925522i	$-0.376373\pi$
$53$	5.51388	0.757389	0.378695	+	0.925522i	$0.376373\pi$
$54$	0	0
$55$	0	0
$56$	−12.9083	−1.72495
$57$	0	0
$58$	−3.21110	−0.421638
$59$	3.69722	0.481338	0.240669	−	0.970607i	$-0.422633\pi$
$59$	3.69722	0.481338	0.240669	+	0.970607i	$0.422633\pi$
$60$	0	0
$61$	−0.302776	−0.0387664	−0.0193832	−	0.999812i	$-0.506170\pi$
$61$	−0.302776	−0.0387664	−0.0193832	+	0.999812i	$0.506170\pi$
$62$	−18.9083	−2.40136
$63$	0	0
$64$	−12.8167	−1.60208
$65$	0	0
$66$	0	0
$67$	−5.60555	−0.684827	−0.342414	−	0.939549i	$-0.611244\pi$
$67$	−5.60555	−0.684827	−0.342414	+	0.939549i	$0.611244\pi$
$68$	−7.60555	−0.922309
$69$	0	0
$70$	0	0
$71$	4.39445	0.521525	0.260763	−	0.965403i	$-0.416026\pi$
$71$	4.39445	0.521525	0.260763	+	0.965403i	$0.416026\pi$
$72$	0	0
$73$	9.60555	1.12424	0.562122	−	0.827054i	$-0.309985\pi$
$73$	9.60555	1.12424	0.562122	+	0.827054i	$0.309985\pi$
$74$	4.39445	0.510844
$75$	0	0
$76$	−23.8167	−2.73196
$77$	−22.8167	−2.60020
$78$	0	0
$79$	−2.60555	−0.293147	−0.146574	−	0.989200i	$-0.546825\pi$
$79$	−2.60555	−0.293147	−0.146574	+	0.989200i	$0.546825\pi$
$80$	0	0
$81$	0	0
$82$	−12.2111	−1.34849
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	0	0
$86$	−13.1194	−1.41470
$87$	0	0
$88$	15.9083	1.69583
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	−1.30278	−0.136568
$92$	22.8167	2.37880
$93$	0	0
$94$	14.3028	1.47522
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−26.5139	−2.67831
$99$	0	0
$100$	0	0
$101$	14.7250	1.46519	0.732595	−	0.680665i	$-0.238309\pi$
$101$	14.7250	1.46519	0.732595	+	0.680665i	$0.238309\pi$
$102$	0	0
$103$	17.9083	1.76456	0.882280	−	0.470725i	$-0.156008\pi$
$103$	17.9083	1.76456	0.882280	+	0.470725i	$0.156008\pi$
$104$	0.908327	0.0890688
$105$	0	0
$106$	−12.6972	−1.23326
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	0	0
$109$	11.9083	1.14061	0.570305	−	0.821433i	$-0.306825\pi$
$109$	11.9083	1.14061	0.570305	+	0.821433i	$0.306825\pi$
$110$	0	0
$111$	0	0
$112$	1.30278	0.123101
$113$	−11.3028	−1.06328	−0.531638	−	0.846972i	$-0.678423\pi$
$113$	−11.3028	−1.06328	−0.531638	+	0.846972i	$0.678423\pi$
$114$	0	0
$115$	0	0
$116$	4.60555	0.427615
$117$	0	0
$118$	−8.51388	−0.783766
$119$	−9.90833	−0.908295
$120$	0	0
$121$	17.1194	1.55631
$122$	0.697224	0.0631237
$123$	0	0
$124$	27.1194	2.43540
$125$	0	0
$126$	0	0
$127$	3.60555	0.319941	0.159970	−	0.987122i	$-0.448860\pi$
$127$	3.60555	0.319941	0.159970	+	0.987122i	$0.448860\pi$
$128$	18.9083	1.67128
$129$	0	0
$130$	0	0
$131$	1.81665	0.158722	0.0793609	−	0.996846i	$-0.474712\pi$
$131$	1.81665	0.158722	0.0793609	+	0.996846i	$0.474712\pi$
$132$	0	0
$133$	−31.0278	−2.69045
$134$	12.9083	1.11511
$135$	0	0
$136$	6.90833	0.592384
$137$	12.9083	1.10283	0.551416	−	0.834230i	$-0.314088\pi$
$137$	12.9083	1.10283	0.551416	+	0.834230i	$0.314088\pi$
$138$	0	0
$139$	−7.90833	−0.670776	−0.335388	−	0.942080i	$-0.608867\pi$
$139$	−7.90833	−0.670776	−0.335388	+	0.942080i	$0.608867\pi$
$140$	0	0
$141$	0	0
$142$	−10.1194	−0.849204
$143$	1.60555	0.134263
$144$	0	0
$145$	0	0
$146$	−22.1194	−1.83062
$147$	0	0
$148$	−6.30278	−0.518085
$149$	6.21110	0.508833	0.254417	−	0.967095i	$-0.418117\pi$
$149$	6.21110	0.508833	0.254417	+	0.967095i	$0.418117\pi$
$150$	0	0
$151$	−23.1194	−1.88143	−0.940716	−	0.339195i	$-0.889846\pi$
$151$	−23.1194	−1.88143	−0.940716	+	0.339195i	$0.889846\pi$
$152$	21.6333	1.75469
$153$	0	0
$154$	52.5416	4.23393
$155$	0	0
$156$	0	0
$157$	3.18335	0.254059	0.127029	−	0.991899i	$-0.459456\pi$
$157$	3.18335	0.254059	0.127029	+	0.991899i	$0.459456\pi$
$158$	6.00000	0.477334
$159$	0	0
$160$	0	0
$161$	29.7250	2.34266
$162$	0	0
$163$	−2.81665	−0.220617	−0.110309	−	0.993897i	$-0.535184\pi$
$163$	−2.81665	−0.220617	−0.110309	+	0.993897i	$0.535184\pi$
$164$	17.5139	1.36760
$165$	0	0
$166$	−20.7250	−1.60857
$167$	0.908327	0.0702884	0.0351442	−	0.999382i	$-0.488811\pi$
$167$	0.908327	0.0702884	0.0351442	+	0.999382i	$0.488811\pi$
$168$	0	0
$169$	−12.9083	−0.992948
$170$	0	0
$171$	0	0
$172$	18.8167	1.43476
$173$	−7.81665	−0.594289	−0.297145	−	0.954832i	$-0.596034\pi$
$173$	−7.81665	−0.594289	−0.297145	+	0.954832i	$0.596034\pi$
$174$	0	0
$175$	0	0
$176$	−1.60555	−0.121023
$177$	0	0
$178$	27.6333	2.07120
$179$	15.4222	1.15271	0.576355	−	0.817200i	$-0.304475\pi$
$179$	15.4222	1.15271	0.576355	+	0.817200i	$0.304475\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	3.00000	0.222375
$183$	0	0
$184$	−20.7250	−1.52787
$185$	0	0
$186$	0	0
$187$	12.2111	0.892964
$188$	−20.5139	−1.49613
$189$	0	0
$190$	0	0
$191$	9.21110	0.666492	0.333246	−	0.942840i	$-0.391856\pi$
$191$	9.21110	0.666492	0.333246	+	0.942840i	$0.391856\pi$
$192$	0	0
$193$	6.81665	0.490673	0.245337	−	0.969438i	$-0.421101\pi$
$193$	6.81665	0.490673	0.245337	+	0.969438i	$0.421101\pi$
$194$	−4.60555	−0.330659
$195$	0	0
$196$	38.0278	2.71627
$197$	2.09167	0.149026	0.0745128	−	0.997220i	$-0.476260\pi$
$197$	2.09167	0.149026	0.0745128	+	0.997220i	$0.476260\pi$
$198$	0	0
$199$	−7.48612	−0.530677	−0.265339	−	0.964155i	$-0.585484\pi$
$199$	−7.48612	−0.530677	−0.265339	+	0.964155i	$0.585484\pi$
$200$	0	0
$201$	0	0
$202$	−33.9083	−2.38578
$203$	6.00000	0.421117
$204$	0	0
$205$	0	0
$206$	−41.2389	−2.87325
$207$	0	0
$208$	−0.0916731	−0.00635638
$209$	38.2389	2.64504
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	18.2111	1.25074
$213$	0	0
$214$	−20.7250	−1.41673
$215$	0	0
$216$	0	0
$217$	35.3305	2.39839
$218$	−27.4222	−1.85727
$219$	0	0
$220$	0	0
$221$	0.697224	0.0469004
$222$	0	0
$223$	17.0000	1.13840	0.569202	−	0.822198i	$-0.307252\pi$
$223$	17.0000	1.13840	0.569202	+	0.822198i	$0.307252\pi$
$224$	22.8167	1.52450
$225$	0	0
$226$	26.0278	1.73134
$227$	−25.6056	−1.69950	−0.849750	−	0.527186i	$-0.823247\pi$
$227$	−25.6056	−1.69950	−0.849750	+	0.527186i	$0.823247\pi$
$228$	0	0
$229$	−19.4222	−1.28346	−0.641728	−	0.766933i	$-0.721782\pi$
$229$	−19.4222	−1.28346	−0.641728	+	0.766933i	$0.721782\pi$
$230$	0	0
$231$	0	0
$232$	−4.18335	−0.274650
$233$	7.18335	0.470597	0.235298	−	0.971923i	$-0.424393\pi$
$233$	7.18335	0.470597	0.235298	+	0.971923i	$0.424393\pi$
$234$	0	0
$235$	0	0
$236$	12.2111	0.794875
$237$	0	0
$238$	22.8167	1.47898
$239$	24.4222	1.57974	0.789871	−	0.613274i	$-0.210148\pi$
$239$	24.4222	1.57974	0.789871	+	0.613274i	$0.210148\pi$
$240$	0	0
$241$	4.78890	0.308480	0.154240	−	0.988033i	$-0.450707\pi$
$241$	4.78890	0.308480	0.154240	+	0.988033i	$0.450707\pi$
$242$	−39.4222	−2.53416
$243$	0	0
$244$	−1.00000	−0.0640184
$245$	0	0
$246$	0	0
$247$	2.18335	0.138923
$248$	−24.6333	−1.56422
$249$	0	0
$250$	0	0
$251$	26.7250	1.68687	0.843433	−	0.537235i	$-0.180531\pi$
$251$	26.7250	1.68687	0.843433	+	0.537235i	$0.180531\pi$
$252$	0	0
$253$	−36.6333	−2.30312
$254$	−8.30278	−0.520962
$255$	0	0
$256$	−17.9083	−1.11927
$257$	21.6333	1.34945	0.674724	−	0.738070i	$-0.264263\pi$
$257$	21.6333	1.34945	0.674724	+	0.738070i	$0.264263\pi$
$258$	0	0
$259$	−8.21110	−0.510213
$260$	0	0
$261$	0	0
$262$	−4.18335	−0.258448
$263$	11.0278	0.680001	0.340000	−	0.940425i	$-0.389573\pi$
$263$	11.0278	0.680001	0.340000	+	0.940425i	$0.389573\pi$
$264$	0	0
$265$	0	0
$266$	71.4500	4.38088
$267$	0	0
$268$	−18.5139	−1.13092
$269$	−17.3028	−1.05497	−0.527484	−	0.849565i	$-0.676865\pi$
$269$	−17.3028	−1.05497	−0.527484	+	0.849565i	$0.676865\pi$
$270$	0	0
$271$	9.60555	0.583496	0.291748	−	0.956495i	$-0.405763\pi$
$271$	9.60555	0.583496	0.291748	+	0.956495i	$0.405763\pi$
$272$	−0.697224	−0.0422754
$273$	0	0
$274$	−29.7250	−1.79575
$275$	0	0
$276$	0	0
$277$	29.2111	1.75513	0.877563	−	0.479462i	$-0.159168\pi$
$277$	29.2111	1.75513	0.877563	+	0.479462i	$0.159168\pi$
$278$	18.2111	1.09223
$279$	0	0
$280$	0	0
$281$	−4.81665	−0.287337	−0.143669	−	0.989626i	$-0.545890\pi$
$281$	−4.81665	−0.287337	−0.143669	+	0.989626i	$0.545890\pi$
$282$	0	0
$283$	15.1194	0.898757	0.449378	−	0.893342i	$-0.351646\pi$
$283$	15.1194	0.898757	0.449378	+	0.893342i	$0.351646\pi$
$284$	14.5139	0.861240
$285$	0	0
$286$	−3.69722	−0.218621
$287$	22.8167	1.34682
$288$	0	0
$289$	−11.6972	−0.688072
$290$	0	0
$291$	0	0
$292$	31.7250	1.85656
$293$	−1.18335	−0.0691318	−0.0345659	−	0.999402i	$-0.511005\pi$
$293$	−1.18335	−0.0691318	−0.0345659	+	0.999402i	$0.511005\pi$
$294$	0	0
$295$	0	0
$296$	5.72498	0.332758
$297$	0	0
$298$	−14.3028	−0.828538
$299$	−2.09167	−0.120965
$300$	0	0
$301$	24.5139	1.41296
$302$	53.2389	3.06355
$303$	0	0
$304$	−2.18335	−0.125223
$305$	0	0
$306$	0	0
$307$	4.78890	0.273317	0.136658	−	0.990618i	$-0.456364\pi$
$307$	4.78890	0.273317	0.136658	+	0.990618i	$0.456364\pi$
$308$	−75.3583	−4.29394
$309$	0	0
$310$	0	0
$311$	−21.0000	−1.19080	−0.595400	−	0.803429i	$-0.703007\pi$
$311$	−21.0000	−1.19080	−0.595400	+	0.803429i	$0.703007\pi$
$312$	0	0
$313$	10.7250	0.606212	0.303106	−	0.952957i	$-0.401976\pi$
$313$	10.7250	0.606212	0.303106	+	0.952957i	$0.401976\pi$
$314$	−7.33053	−0.413686
$315$	0	0
$316$	−8.60555	−0.484100
$317$	−21.4222	−1.20319	−0.601595	−	0.798801i	$-0.705468\pi$
$317$	−21.4222	−1.20319	−0.601595	+	0.798801i	$0.705468\pi$
$318$	0	0
$319$	−7.39445	−0.414010
$320$	0	0
$321$	0	0
$322$	−68.4500	−3.81457
$323$	16.6056	0.923958
$324$	0	0
$325$	0	0
$326$	6.48612	0.359233
$327$	0	0
$328$	−15.9083	−0.878390
$329$	−26.7250	−1.47340
$330$	0	0
$331$	−16.4222	−0.902646	−0.451323	−	0.892361i	$-0.649048\pi$
$331$	−16.4222	−0.902646	−0.451323	+	0.892361i	$0.649048\pi$
$332$	29.7250	1.63137
$333$	0	0
$334$	−2.09167	−0.114451
$335$	0	0
$336$	0	0
$337$	−11.3944	−0.620695	−0.310348	−	0.950623i	$-0.600445\pi$
$337$	−11.3944	−0.620695	−0.310348	+	0.950623i	$0.600445\pi$
$338$	29.7250	1.61683
$339$	0	0
$340$	0	0
$341$	−43.5416	−2.35791
$342$	0	0
$343$	19.4222	1.04870
$344$	−17.0917	−0.921521
$345$	0	0
$346$	18.0000	0.967686
$347$	−26.9361	−1.44600	−0.723002	−	0.690846i	$-0.757238\pi$
$347$	−26.9361	−1.44600	−0.723002	+	0.690846i	$0.757238\pi$
$348$	0	0
$349$	−5.39445	−0.288758	−0.144379	−	0.989522i	$-0.546118\pi$
$349$	−5.39445	−0.288758	−0.144379	+	0.989522i	$0.546118\pi$
$350$	0	0
$351$	0	0
$352$	−28.1194	−1.49877
$353$	17.7250	0.943406	0.471703	−	0.881758i	$-0.343640\pi$
$353$	17.7250	0.943406	0.471703	+	0.881758i	$0.343640\pi$
$354$	0	0
$355$	0	0
$356$	−39.6333	−2.10056
$357$	0	0
$358$	−35.5139	−1.87697
$359$	9.90833	0.522941	0.261471	−	0.965211i	$-0.415793\pi$
$359$	9.90833	0.522941	0.261471	+	0.965211i	$0.415793\pi$
$360$	0	0
$361$	33.0000	1.73684
$362$	16.1194	0.847218
$363$	0	0
$364$	−4.30278	−0.225527
$365$	0	0
$366$	0	0
$367$	13.3028	0.694399	0.347200	−	0.937791i	$-0.387133\pi$
$367$	13.3028	0.694399	0.347200	+	0.937791i	$0.387133\pi$
$368$	2.09167	0.109036
$369$	0	0
$370$	0	0
$371$	23.7250	1.23174
$372$	0	0
$373$	−8.39445	−0.434648	−0.217324	−	0.976100i	$-0.569733\pi$
$373$	−8.39445	−0.434648	−0.217324	+	0.976100i	$0.569733\pi$
$374$	−28.1194	−1.45402
$375$	0	0
$376$	18.6333	0.960939
$377$	−0.422205	−0.0217447
$378$	0	0
$379$	5.69722	0.292647	0.146323	−	0.989237i	$-0.453256\pi$
$379$	5.69722	0.292647	0.146323	+	0.989237i	$0.453256\pi$
$380$	0	0
$381$	0	0
$382$	−21.2111	−1.08525
$383$	−5.72498	−0.292533	−0.146266	−	0.989245i	$-0.546726\pi$
$383$	−5.72498	−0.292533	−0.146266	+	0.989245i	$0.546726\pi$
$384$	0	0
$385$	0	0
$386$	−15.6972	−0.798968
$387$	0	0
$388$	6.60555	0.335346
$389$	1.18335	0.0599980	0.0299990	−	0.999550i	$-0.490450\pi$
$389$	1.18335	0.0599980	0.0299990	+	0.999550i	$0.490450\pi$
$390$	0	0
$391$	−15.9083	−0.804519
$392$	−34.5416	−1.74462
$393$	0	0
$394$	−4.81665	−0.242660
$395$	0	0
$396$	0	0
$397$	−14.3944	−0.722437	−0.361218	−	0.932481i	$-0.617639\pi$
$397$	−14.3944	−0.722437	−0.361218	+	0.932481i	$0.617639\pi$
$398$	17.2389	0.864106
$399$	0	0
$400$	0	0
$401$	−4.18335	−0.208906	−0.104453	−	0.994530i	$-0.533309\pi$
$401$	−4.18335	−0.208906	−0.104453	+	0.994530i	$0.533309\pi$
$402$	0	0
$403$	−2.48612	−0.123843
$404$	48.6333	2.41960
$405$	0	0
$406$	−13.8167	−0.685709
$407$	10.1194	0.501601
$408$	0	0
$409$	30.5416	1.51019	0.755093	−	0.655617i	$-0.227592\pi$
$409$	30.5416	1.51019	0.755093	+	0.655617i	$0.227592\pi$
$410$	0	0
$411$	0	0
$412$	59.1472	2.91397
$413$	15.9083	0.782798
$414$	0	0
$415$	0	0
$416$	−1.60555	−0.0787186
$417$	0	0
$418$	−88.0555	−4.30694
$419$	35.3028	1.72465	0.862327	−	0.506352i	$-0.169006\pi$
$419$	35.3028	1.72465	0.862327	+	0.506352i	$0.169006\pi$
$420$	0	0
$421$	12.6056	0.614357	0.307178	−	0.951652i	$-0.400615\pi$
$421$	12.6056	0.614357	0.307178	+	0.951652i	$0.400615\pi$
$422$	−11.5139	−0.560487
$423$	0	0
$424$	−16.5416	−0.803333
$425$	0	0
$426$	0	0
$427$	−1.30278	−0.0630457
$428$	29.7250	1.43681
$429$	0	0
$430$	0	0
$431$	10.8167	0.521020	0.260510	−	0.965471i	$-0.416109\pi$
$431$	10.8167	0.521020	0.260510	+	0.965471i	$0.416109\pi$
$432$	0	0
$433$	16.2389	0.780390	0.390195	−	0.920732i	$-0.372408\pi$
$433$	16.2389	0.780390	0.390195	+	0.920732i	$0.372408\pi$
$434$	−81.3583	−3.90532
$435$	0	0
$436$	39.3305	1.88359
$437$	−49.8167	−2.38305
$438$	0	0
$439$	−13.4861	−0.643657	−0.321829	−	0.946798i	$-0.604298\pi$
$439$	−13.4861	−0.643657	−0.321829	+	0.946798i	$0.604298\pi$
$440$	0	0
$441$	0	0
$442$	−1.60555	−0.0763683
$443$	−29.0917	−1.38219	−0.691094	−	0.722765i	$-0.742871\pi$
$443$	−29.0917	−1.38219	−0.691094	+	0.722765i	$0.742871\pi$
$444$	0	0
$445$	0	0
$446$	−39.1472	−1.85367
$447$	0	0
$448$	−55.1472	−2.60546
$449$	1.88057	0.0887496	0.0443748	−	0.999015i	$-0.485870\pi$
$449$	1.88057	0.0887496	0.0443748	+	0.999015i	$0.485870\pi$
$450$	0	0
$451$	−28.1194	−1.32409
$452$	−37.3305	−1.75588
$453$	0	0
$454$	58.9638	2.76731
$455$	0	0
$456$	0	0
$457$	−3.57779	−0.167362	−0.0836811	−	0.996493i	$-0.526668\pi$
$457$	−3.57779	−0.167362	−0.0836811	+	0.996493i	$0.526668\pi$
$458$	44.7250	2.08986
$459$	0	0
$460$	0	0
$461$	−23.7250	−1.10498	−0.552491	−	0.833519i	$-0.686323\pi$
$461$	−23.7250	−1.10498	−0.552491	+	0.833519i	$0.686323\pi$
$462$	0	0
$463$	−17.3305	−0.805418	−0.402709	−	0.915328i	$-0.631931\pi$
$463$	−17.3305	−0.805418	−0.402709	+	0.915328i	$0.631931\pi$
$464$	0.422205	0.0196004
$465$	0	0
$466$	−16.5416	−0.766276
$467$	5.57779	0.258110	0.129055	−	0.991637i	$-0.458806\pi$
$467$	5.57779	0.258110	0.129055	+	0.991637i	$0.458806\pi$
$468$	0	0
$469$	−24.1194	−1.11373
$470$	0	0
$471$	0	0
$472$	−11.0917	−0.510536
$473$	−30.2111	−1.38911
$474$	0	0
$475$	0	0
$476$	−32.7250	−1.49995
$477$	0	0
$478$	−56.2389	−2.57231
$479$	−33.9083	−1.54931	−0.774656	−	0.632383i	$-0.782077\pi$
$479$	−33.9083	−1.54931	−0.774656	+	0.632383i	$0.782077\pi$
$480$	0	0
$481$	0.577795	0.0263452
$482$	−11.0278	−0.502301
$483$	0	0
$484$	56.5416	2.57007
$485$	0	0
$486$	0	0
$487$	−7.63331	−0.345898	−0.172949	−	0.984931i	$-0.555330\pi$
$487$	−7.63331	−0.345898	−0.172949	+	0.984931i	$0.555330\pi$
$488$	0.908327	0.0411180
$489$	0	0
$490$	0	0
$491$	−23.2389	−1.04876	−0.524378	−	0.851486i	$-0.675702\pi$
$491$	−23.2389	−1.04876	−0.524378	+	0.851486i	$0.675702\pi$
$492$	0	0
$493$	−3.21110	−0.144621
$494$	−5.02776	−0.226209
$495$	0	0
$496$	2.48612	0.111630
$497$	18.9083	0.848154
$498$	0	0
$499$	33.3305	1.49208	0.746040	−	0.665901i	$-0.231953\pi$
$499$	33.3305	1.49208	0.746040	+	0.665901i	$0.231953\pi$
$500$	0	0
$501$	0	0
$502$	−61.5416	−2.74674
$503$	−25.3305	−1.12943	−0.564716	−	0.825285i	$-0.691014\pi$
$503$	−25.3305	−1.12943	−0.564716	+	0.825285i	$0.691014\pi$
$504$	0	0
$505$	0	0
$506$	84.3583	3.75018
$507$	0	0
$508$	11.9083	0.528347
$509$	5.78890	0.256588	0.128294	−	0.991736i	$-0.459050\pi$
$509$	5.78890	0.256588	0.128294	+	0.991736i	$0.459050\pi$
$510$	0	0
$511$	41.3305	1.82836
$512$	3.42221	0.151242
$513$	0	0
$514$	−49.8167	−2.19732
$515$	0	0
$516$	0	0
$517$	32.9361	1.44853
$518$	18.9083	0.830784
$519$	0	0
$520$	0	0
$521$	41.0278	1.79746	0.898729	−	0.438504i	$-0.144491\pi$
$521$	41.0278	1.79746	0.898729	+	0.438504i	$0.144491\pi$
$522$	0	0
$523$	37.2389	1.62834	0.814171	−	0.580625i	$-0.197192\pi$
$523$	37.2389	1.62834	0.814171	+	0.580625i	$0.197192\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−25.3944	−1.10725
$527$	−18.9083	−0.823660
$528$	0	0
$529$	24.7250	1.07500
$530$	0	0
$531$	0	0
$532$	−102.478	−4.44297
$533$	−1.60555	−0.0695441
$534$	0	0
$535$	0	0
$536$	16.8167	0.726369
$537$	0	0
$538$	39.8444	1.71781
$539$	−61.0555	−2.62985
$540$	0	0
$541$	−20.3305	−0.874078	−0.437039	−	0.899443i	$-0.643973\pi$
$541$	−20.3305	−0.874078	−0.437039	+	0.899443i	$0.643973\pi$
$542$	−22.1194	−0.950111
$543$	0	0
$544$	−12.2111	−0.523547
$545$	0	0
$546$	0	0
$547$	−39.6611	−1.69578	−0.847892	−	0.530168i	$-0.822129\pi$
$547$	−39.6611	−1.69578	−0.847892	+	0.530168i	$0.822129\pi$
$548$	42.6333	1.82120
$549$	0	0
$550$	0	0
$551$	−10.0555	−0.428379
$552$	0	0
$553$	−11.2111	−0.476745
$554$	−67.2666	−2.85788
$555$	0	0
$556$	−26.1194	−1.10771
$557$	33.6333	1.42509	0.712544	−	0.701627i	$-0.247543\pi$
$557$	33.6333	1.42509	0.712544	+	0.701627i	$0.247543\pi$
$558$	0	0
$559$	−1.72498	−0.0729589
$560$	0	0
$561$	0	0
$562$	11.0917	0.467874
$563$	36.6333	1.54391	0.771955	−	0.635677i	$-0.219279\pi$
$563$	36.6333	1.54391	0.771955	+	0.635677i	$0.219279\pi$
$564$	0	0
$565$	0	0
$566$	−34.8167	−1.46345
$567$	0	0
$568$	−13.1833	−0.553161
$569$	−19.8167	−0.830757	−0.415379	−	0.909649i	$-0.636351\pi$
$569$	−19.8167	−0.830757	−0.415379	+	0.909649i	$0.636351\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	5.30278	0.221720
$573$	0	0
$574$	−52.5416	−2.19305
$575$	0	0
$576$	0	0
$577$	−10.6333	−0.442670	−0.221335	−	0.975198i	$-0.571041\pi$
$577$	−10.6333	−0.442670	−0.221335	+	0.975198i	$0.571041\pi$
$578$	26.9361	1.12039
$579$	0	0
$580$	0	0
$581$	38.7250	1.60658
$582$	0	0
$583$	−29.2389	−1.21095
$584$	−28.8167	−1.19244
$585$	0	0
$586$	2.72498	0.112568
$587$	4.81665	0.198805	0.0994023	−	0.995047i	$-0.468307\pi$
$587$	4.81665	0.198805	0.0994023	+	0.995047i	$0.468307\pi$
$588$	0	0
$589$	−59.2111	−2.43975
$590$	0	0
$591$	0	0
$592$	−0.577795	−0.0237472
$593$	37.1194	1.52431	0.762156	−	0.647393i	$-0.224141\pi$
$593$	37.1194	1.52431	0.762156	+	0.647393i	$0.224141\pi$
$594$	0	0
$595$	0	0
$596$	20.5139	0.840281
$597$	0	0
$598$	4.81665	0.196968
$599$	−19.3305	−0.789824	−0.394912	−	0.918719i	$-0.629225\pi$
$599$	−19.3305	−0.789824	−0.394912	+	0.918719i	$0.629225\pi$
$600$	0	0
$601$	17.9083	0.730496	0.365248	−	0.930910i	$-0.380984\pi$
$601$	17.9083	0.730496	0.365248	+	0.930910i	$0.380984\pi$
$602$	−56.4500	−2.30073
$603$	0	0
$604$	−76.3583	−3.10697
$605$	0	0
$606$	0	0
$607$	28.2389	1.14618	0.573090	−	0.819492i	$-0.305745\pi$
$607$	28.2389	1.14618	0.573090	+	0.819492i	$0.305745\pi$
$608$	−38.2389	−1.55079
$609$	0	0
$610$	0	0
$611$	1.88057	0.0760797
$612$	0	0
$613$	−17.3944	−0.702555	−0.351278	−	0.936271i	$-0.614253\pi$
$613$	−17.3944	−0.702555	−0.351278	+	0.936271i	$0.614253\pi$
$614$	−11.0278	−0.445044
$615$	0	0
$616$	68.4500	2.75793
$617$	−4.81665	−0.193911	−0.0969556	−	0.995289i	$-0.530910\pi$
$617$	−4.81665	−0.193911	−0.0969556	+	0.995289i	$0.530910\pi$
$618$	0	0
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	0	0
$621$	0	0
$622$	48.3583	1.93899
$623$	−51.6333	−2.06864
$624$	0	0
$625$	0	0
$626$	−24.6972	−0.987100
$627$	0	0
$628$	10.5139	0.419549
$629$	4.39445	0.175218
$630$	0	0
$631$	−3.30278	−0.131481	−0.0657407	−	0.997837i	$-0.520941\pi$
$631$	−3.30278	−0.131481	−0.0657407	+	0.997837i	$0.520941\pi$
$632$	7.81665	0.310930
$633$	0	0
$634$	49.3305	1.95917
$635$	0	0
$636$	0	0
$637$	−3.48612	−0.138125
$638$	17.0278	0.674135
$639$	0	0
$640$	0	0
$641$	−32.0917	−1.26754	−0.633772	−	0.773520i	$-0.718494\pi$
$641$	−32.0917	−1.26754	−0.633772	+	0.773520i	$0.718494\pi$
$642$	0	0
$643$	5.48612	0.216352	0.108176	−	0.994132i	$-0.465499\pi$
$643$	5.48612	0.216352	0.108176	+	0.994132i	$0.465499\pi$
$644$	98.1749	3.86863
$645$	0	0
$646$	−38.2389	−1.50449
$647$	−33.8444	−1.33056	−0.665281	−	0.746593i	$-0.731688\pi$
$647$	−33.8444	−1.33056	−0.665281	+	0.746593i	$0.731688\pi$
$648$	0	0
$649$	−19.6056	−0.769585
$650$	0	0
$651$	0	0
$652$	−9.30278	−0.364325
$653$	1.11943	0.0438067	0.0219033	−	0.999760i	$-0.493027\pi$
$653$	1.11943	0.0438067	0.0219033	+	0.999760i	$0.493027\pi$
$654$	0	0
$655$	0	0
$656$	1.60555	0.0626862
$657$	0	0
$658$	61.5416	2.39914
$659$	−22.1194	−0.861651	−0.430825	−	0.902435i	$-0.641777\pi$
$659$	−22.1194	−0.861651	−0.430825	+	0.902435i	$0.641777\pi$
$660$	0	0
$661$	−10.2111	−0.397166	−0.198583	−	0.980084i	$-0.563634\pi$
$661$	−10.2111	−0.397166	−0.198583	+	0.980084i	$0.563634\pi$
$662$	37.8167	1.46979
$663$	0	0
$664$	−27.0000	−1.04780
$665$	0	0
$666$	0	0
$667$	9.63331	0.373003
$668$	3.00000	0.116073
$669$	0	0
$670$	0	0
$671$	1.60555	0.0619816
$672$	0	0
$673$	16.0278	0.617825	0.308912	−	0.951091i	$-0.400035\pi$
$673$	16.0278	0.617825	0.308912	+	0.951091i	$0.400035\pi$
$674$	26.2389	1.01068
$675$	0	0
$676$	−42.6333	−1.63974
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	0	0
$679$	8.60555	0.330251
$680$	0	0
$681$	0	0
$682$	100.267	3.83941
$683$	−32.7250	−1.25219	−0.626093	−	0.779748i	$-0.715347\pi$
$683$	−32.7250	−1.25219	−0.626093	+	0.779748i	$0.715347\pi$
$684$	0	0
$685$	0	0
$686$	−44.7250	−1.70761
$687$	0	0
$688$	1.72498	0.0657643
$689$	−1.66947	−0.0636017
$690$	0	0
$691$	33.0555	1.25749	0.628745	−	0.777611i	$-0.283569\pi$
$691$	33.0555	1.25749	0.628745	+	0.777611i	$0.283569\pi$
$692$	−25.8167	−0.981402
$693$	0	0
$694$	62.0278	2.35454
$695$	0	0
$696$	0	0
$697$	−12.2111	−0.462528
$698$	12.4222	0.470187
$699$	0	0
$700$	0	0
$701$	−50.2389	−1.89750	−0.948748	−	0.316034i	$-0.897649\pi$
$701$	−50.2389	−1.89750	−0.948748	+	0.316034i	$0.897649\pi$
$702$	0	0
$703$	13.7611	0.519011
$704$	67.9638	2.56148
$705$	0	0
$706$	−40.8167	−1.53616
$707$	63.3583	2.38283
$708$	0	0
$709$	−28.2111	−1.05949	−0.529745	−	0.848157i	$-0.677712\pi$
$709$	−28.2111	−1.05949	−0.529745	+	0.848157i	$0.677712\pi$
$710$	0	0
$711$	0	0
$712$	36.0000	1.34916
$713$	56.7250	2.12437
$714$	0	0
$715$	0	0
$716$	50.9361	1.90357
$717$	0	0
$718$	−22.8167	−0.851510
$719$	−24.9083	−0.928924	−0.464462	−	0.885593i	$-0.653752\pi$
$719$	−24.9083	−0.928924	−0.464462	+	0.885593i	$0.653752\pi$
$720$	0	0
$721$	77.0555	2.86970
$722$	−75.9916	−2.82811
$723$	0	0
$724$	−23.1194	−0.859227
$725$	0	0
$726$	0	0
$727$	11.9083	0.441655	0.220828	−	0.975313i	$-0.429124\pi$
$727$	11.9083	0.441655	0.220828	+	0.975313i	$0.429124\pi$
$728$	3.90833	0.144852
$729$	0	0
$730$	0	0
$731$	−13.1194	−0.485240
$732$	0	0
$733$	−24.7250	−0.913238	−0.456619	−	0.889662i	$-0.650940\pi$
$733$	−24.7250	−0.913238	−0.456619	+	0.889662i	$0.650940\pi$
$734$	−30.6333	−1.13070
$735$	0	0
$736$	36.6333	1.35032
$737$	29.7250	1.09493
$738$	0	0
$739$	33.3305	1.22608	0.613042	−	0.790051i	$-0.289946\pi$
$739$	33.3305	1.22608	0.613042	+	0.790051i	$0.289946\pi$
$740$	0	0
$741$	0	0
$742$	−54.6333	−2.00565
$743$	6.97224	0.255787	0.127893	−	0.991788i	$-0.459178\pi$
$743$	6.97224	0.255787	0.127893	+	0.991788i	$0.459178\pi$
$744$	0	0
$745$	0	0
$746$	19.3305	0.707741
$747$	0	0
$748$	40.3305	1.47463
$749$	38.7250	1.41498
$750$	0	0
$751$	34.5139	1.25943	0.629715	−	0.776827i	$-0.283172\pi$
$751$	34.5139	1.25943	0.629715	+	0.776827i	$0.283172\pi$
$752$	−1.88057	−0.0685774
$753$	0	0
$754$	0.972244	0.0354070
$755$	0	0
$756$	0	0
$757$	−20.1194	−0.731253	−0.365627	−	0.930762i	$-0.619145\pi$
$757$	−20.1194	−0.731253	−0.365627	+	0.930762i	$0.619145\pi$
$758$	−13.1194	−0.476519
$759$	0	0
$760$	0	0
$761$	−36.3583	−1.31799	−0.658993	−	0.752149i	$-0.729018\pi$
$761$	−36.3583	−1.31799	−0.658993	+	0.752149i	$0.729018\pi$
$762$	0	0
$763$	51.2389	1.85497
$764$	30.4222	1.10064
$765$	0	0
$766$	13.1833	0.476334
$767$	−1.11943	−0.0404203
$768$	0	0
$769$	−25.3583	−0.914443	−0.457222	−	0.889353i	$-0.651155\pi$
$769$	−25.3583	−0.914443	−0.457222	+	0.889353i	$0.651155\pi$
$770$	0	0
$771$	0	0
$772$	22.5139	0.810292
$773$	−44.4500	−1.59875	−0.799377	−	0.600830i	$-0.794837\pi$
$773$	−44.4500	−1.59875	−0.799377	+	0.600830i	$0.794837\pi$
$774$	0	0
$775$	0	0
$776$	−6.00000	−0.215387
$777$	0	0
$778$	−2.72498	−0.0976953
$779$	−38.2389	−1.37005
$780$	0	0
$781$	−23.3028	−0.833839
$782$	36.6333	1.31000
$783$	0	0
$784$	3.48612	0.124504
$785$	0	0
$786$	0	0
$787$	46.5139	1.65804	0.829020	−	0.559218i	$-0.188899\pi$
$787$	46.5139	1.65804	0.829020	+	0.559218i	$0.188899\pi$
$788$	6.90833	0.246099
$789$	0	0
$790$	0	0
$791$	−48.6333	−1.72920
$792$	0	0
$793$	0.0916731	0.00325541
$794$	33.1472	1.17635
$795$	0	0
$796$	−24.7250	−0.876354
$797$	19.1194	0.677245	0.338622	−	0.940922i	$-0.390039\pi$
$797$	19.1194	0.677245	0.338622	+	0.940922i	$0.390039\pi$
$798$	0	0
$799$	14.3028	0.505996
$800$	0	0
$801$	0	0
$802$	9.63331	0.340164
$803$	−50.9361	−1.79750
$804$	0	0
$805$	0	0
$806$	5.72498	0.201654
$807$	0	0
$808$	−44.1749	−1.55407
$809$	32.0278	1.12604	0.563018	−	0.826445i	$-0.309640\pi$
$809$	32.0278	1.12604	0.563018	+	0.826445i	$0.309640\pi$
$810$	0	0
$811$	16.9361	0.594706	0.297353	−	0.954768i	$-0.403896\pi$
$811$	16.9361	0.594706	0.297353	+	0.954768i	$0.403896\pi$
$812$	19.8167	0.695428
$813$	0	0
$814$	−23.3028	−0.816762
$815$	0	0
$816$	0	0
$817$	−41.0833	−1.43732
$818$	−70.3305	−2.45905
$819$	0	0
$820$	0	0
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	0	0
$823$	50.6333	1.76497	0.882483	−	0.470344i	$-0.155870\pi$
$823$	50.6333	1.76497	0.882483	+	0.470344i	$0.155870\pi$
$824$	−53.7250	−1.87160
$825$	0	0
$826$	−36.6333	−1.27464
$827$	49.9638	1.73741	0.868706	−	0.495327i	$-0.164952\pi$
$827$	49.9638	1.73741	0.868706	+	0.495327i	$0.164952\pi$
$828$	0	0
$829$	−12.7889	−0.444177	−0.222088	−	0.975027i	$-0.571287\pi$
$829$	−12.7889	−0.444177	−0.222088	+	0.975027i	$0.571287\pi$
$830$	0	0
$831$	0	0
$832$	3.88057	0.134535
$833$	−26.5139	−0.918651
$834$	0	0
$835$	0	0
$836$	126.294	4.36798
$837$	0	0
$838$	−81.2944	−2.80827
$839$	47.0278	1.62358	0.811789	−	0.583951i	$-0.198494\pi$
$839$	47.0278	1.62358	0.811789	+	0.583951i	$0.198494\pi$
$840$	0	0
$841$	−27.0555	−0.932949
$842$	−29.0278	−1.00036
$843$	0	0
$844$	16.5139	0.568431
$845$	0	0
$846$	0	0
$847$	73.6611	2.53102
$848$	1.66947	0.0573298
$849$	0	0
$850$	0	0
$851$	−13.1833	−0.451919
$852$	0	0
$853$	43.6611	1.49493	0.747463	−	0.664303i	$-0.231272\pi$
$853$	43.6611	1.49493	0.747463	+	0.664303i	$0.231272\pi$
$854$	3.00000	0.102658
$855$	0	0
$856$	−27.0000	−0.922841
$857$	42.1472	1.43972	0.719860	−	0.694119i	$-0.244206\pi$
$857$	42.1472	1.43972	0.719860	+	0.694119i	$0.244206\pi$
$858$	0	0
$859$	−27.6611	−0.943783	−0.471892	−	0.881657i	$-0.656429\pi$
$859$	−27.6611	−0.943783	−0.471892	+	0.881657i	$0.656429\pi$
$860$	0	0
$861$	0	0
$862$	−24.9083	−0.848381
$863$	22.7527	0.774512	0.387256	−	0.921972i	$-0.373423\pi$
$863$	22.7527	0.774512	0.387256	+	0.921972i	$0.373423\pi$
$864$	0	0
$865$	0	0
$866$	−37.3944	−1.27072
$867$	0	0
$868$	116.689	3.96068
$869$	13.8167	0.468698
$870$	0	0
$871$	1.69722	0.0575083
$872$	−35.7250	−1.20980
$873$	0	0
$874$	114.717	3.88035
$875$	0	0
$876$	0	0
$877$	24.1194	0.814455	0.407228	−	0.913327i	$-0.366496\pi$
$877$	24.1194	0.814455	0.407228	+	0.913327i	$0.366496\pi$
$878$	31.0555	1.04807
$879$	0	0
$880$	0	0
$881$	31.6056	1.06482	0.532409	−	0.846487i	$-0.321287\pi$
$881$	31.6056	1.06482	0.532409	+	0.846487i	$0.321287\pi$
$882$	0	0
$883$	−35.8167	−1.20533	−0.602663	−	0.797996i	$-0.705894\pi$
$883$	−35.8167	−1.20533	−0.602663	+	0.797996i	$0.705894\pi$
$884$	2.30278	0.0774507
$885$	0	0
$886$	66.9916	2.25063
$887$	31.5416	1.05906	0.529532	−	0.848290i	$-0.322367\pi$
$887$	31.5416	1.05906	0.529532	+	0.848290i	$0.322367\pi$
$888$	0	0
$889$	15.5139	0.520319
$890$	0	0
$891$	0	0
$892$	56.1472	1.87995
$893$	44.7889	1.49880
$894$	0	0
$895$	0	0
$896$	81.3583	2.71799
$897$	0	0
$898$	−4.33053	−0.144512
$899$	11.4500	0.381878
$900$	0	0
$901$	−12.6972	−0.423006
$902$	64.7527	2.15603
$903$	0	0
$904$	33.9083	1.12777
$905$	0	0
$906$	0	0
$907$	27.7527	0.921515	0.460757	−	0.887526i	$-0.347578\pi$
$907$	27.7527	0.921515	0.460757	+	0.887526i	$0.347578\pi$
$908$	−84.5694	−2.80653
$909$	0	0
$910$	0	0
$911$	45.4222	1.50490	0.752452	−	0.658647i	$-0.228871\pi$
$911$	45.4222	1.50490	0.752452	+	0.658647i	$0.228871\pi$
$912$	0	0
$913$	−47.7250	−1.57947
$914$	8.23886	0.272517
$915$	0	0
$916$	−64.1472	−2.11948
$917$	7.81665	0.258129
$918$	0	0
$919$	39.3305	1.29739	0.648697	−	0.761047i	$-0.275314\pi$
$919$	39.3305	1.29739	0.648697	+	0.761047i	$0.275314\pi$
$920$	0	0
$921$	0	0
$922$	54.6333	1.79925
$923$	−1.33053	−0.0437950
$924$	0	0
$925$	0	0
$926$	39.9083	1.31147
$927$	0	0
$928$	7.39445	0.242735
$929$	−56.8722	−1.86592	−0.932958	−	0.359986i	$-0.882781\pi$
$929$	−56.8722	−1.86592	−0.932958	+	0.359986i	$0.882781\pi$
$930$	0	0
$931$	−83.0278	−2.72112
$932$	23.7250	0.777138
$933$	0	0
$934$	−12.8444	−0.420282
$935$	0	0
$936$	0	0
$937$	−49.4222	−1.61455	−0.807277	−	0.590173i	$-0.799059\pi$
$937$	−49.4222	−1.61455	−0.807277	+	0.590173i	$0.799059\pi$
$938$	55.5416	1.81350
$939$	0	0
$940$	0	0
$941$	18.4861	0.602630	0.301315	−	0.953525i	$-0.402574\pi$
$941$	18.4861	0.602630	0.301315	+	0.953525i	$0.402574\pi$
$942$	0	0
$943$	36.6333	1.19294
$944$	1.11943	0.0364343
$945$	0	0
$946$	69.5694	2.26190
$947$	−34.8167	−1.13139	−0.565695	−	0.824615i	$-0.691392\pi$
$947$	−34.8167	−1.13139	−0.565695	+	0.824615i	$0.691392\pi$
$948$	0	0
$949$	−2.90833	−0.0944083
$950$	0	0
$951$	0	0
$952$	29.7250	0.963392
$953$	12.4222	0.402395	0.201197	−	0.979551i	$-0.435517\pi$
$953$	12.4222	0.402395	0.201197	+	0.979551i	$0.435517\pi$
$954$	0	0
$955$	0	0
$956$	80.6611	2.60877
$957$	0	0
$958$	78.0833	2.52276
$959$	55.5416	1.79353
$960$	0	0
$961$	36.4222	1.17491
$962$	−1.33053	−0.0428981
$963$	0	0
$964$	15.8167	0.509420
$965$	0	0
$966$	0	0
$967$	−2.81665	−0.0905775	−0.0452887	−	0.998974i	$-0.514421\pi$
$967$	−2.81665	−0.0905775	−0.0452887	+	0.998974i	$0.514421\pi$
$968$	−51.3583	−1.65072
$969$	0	0
$970$	0	0
$971$	−10.5416	−0.338297	−0.169149	−	0.985591i	$-0.554102\pi$
$971$	−10.5416	−0.338297	−0.169149	+	0.985591i	$0.554102\pi$
$972$	0	0
$973$	−34.0278	−1.09088
$974$	17.5778	0.563229
$975$	0	0
$976$	−0.0916731	−0.00293438
$977$	−16.1194	−0.515706	−0.257853	−	0.966184i	$-0.583015\pi$
$977$	−16.1194	−0.515706	−0.257853	+	0.966184i	$0.583015\pi$
$978$	0	0
$979$	63.6333	2.03373
$980$	0	0
$981$	0	0
$982$	53.5139	1.70770
$983$	−12.6333	−0.402940	−0.201470	−	0.979495i	$-0.564572\pi$
$983$	−12.6333	−0.402940	−0.201470	+	0.979495i	$0.564572\pi$
$984$	0	0
$985$	0	0
$986$	7.39445	0.235487
$987$	0	0
$988$	7.21110	0.229416
$989$	39.3583	1.25152
$990$	0	0
$991$	−21.0917	−0.669999	−0.335000	−	0.942218i	$-0.608736\pi$
$991$	−21.0917	−0.669999	−0.335000	+	0.942218i	$0.608736\pi$
$992$	43.5416	1.38245
$993$	0	0
$994$	−43.5416	−1.38106
$995$	0	0
$996$	0	0
$997$	−30.2389	−0.957674	−0.478837	−	0.877904i	$-0.658942\pi$
$997$	−30.2389	−0.957674	−0.478837	+	0.877904i	$0.658942\pi$
$998$	−76.7527	−2.42957
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.i.1.1	yes	2
3.2	odd	2		2025.2.a.l.1.2	yes	2
5.2	odd	4		2025.2.b.i.649.1		4
5.3	odd	4		2025.2.b.i.649.4		4
5.4	even	2		2025.2.a.k.1.2	yes	2
15.2	even	4		2025.2.b.j.649.4		4
15.8	even	4		2025.2.b.j.649.1		4
15.14	odd	2		2025.2.a.h.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2025.2.a.h.1.1	✓	2	15.14	odd	2
2025.2.a.i.1.1	yes	2	1.1	even	1	trivial
2025.2.a.k.1.2	yes	2	5.4	even	2
2025.2.a.l.1.2	yes	2	3.2	odd	2
2025.2.b.i.649.1		4	5.2	odd	4
2025.2.b.i.649.4		4	5.3	odd	4
2025.2.b.j.649.1		4	15.8	even	4
2025.2.b.j.649.4		4	15.2	even	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)

\(f(q)\)	\(=\)	\(q-2.30278 q^{2} +3.30278 q^{4} +4.30278 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q-2.30278 q^{2} +3.30278 q^{4} +4.30278 q^{7} -3.00000 q^{8} -5.30278 q^{11} -0.302776 q^{13} -9.90833 q^{14} +0.302776 q^{16} -2.30278 q^{17} -7.21110 q^{19} +12.2111 q^{22} +6.90833 q^{23} +0.697224 q^{26} +14.2111 q^{28} +1.39445 q^{29} +8.21110 q^{31} +5.30278 q^{32} +5.30278 q^{34} -1.90833 q^{37} +16.6056 q^{38} +5.30278 q^{41} +5.69722 q^{43} -17.5139 q^{44} -15.9083 q^{46} -6.21110 q^{47} +11.5139 q^{49} -1.00000 q^{52} +5.51388 q^{53} -12.9083 q^{56} -3.21110 q^{58} +3.69722 q^{59} -0.302776 q^{61} -18.9083 q^{62} -12.8167 q^{64} -5.60555 q^{67} -7.60555 q^{68} +4.39445 q^{71} +9.60555 q^{73} +4.39445 q^{74} -23.8167 q^{76} -22.8167 q^{77} -2.60555 q^{79} -12.2111 q^{82} +9.00000 q^{83} -13.1194 q^{86} +15.9083 q^{88} -12.0000 q^{89} -1.30278 q^{91} +22.8167 q^{92} +14.3028 q^{94} +2.00000 q^{97} -26.5139 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - q^2 + 3 * q^4 + 5 * q^7 - 6 * q^8	\( 2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8} - 7 q^{11} + 3 q^{13} - 9 q^{14} - 3 q^{16} - q^{17} + 10 q^{22} + 3 q^{23} + 5 q^{26} + 14 q^{28} + 10 q^{29} + 2 q^{31} + 7 q^{32} + 7 q^{34} + 7 q^{37} + 26 q^{38} + 7 q^{41} + 15 q^{43} - 17 q^{44} - 21 q^{46} + 2 q^{47} + 5 q^{49} - 2 q^{52} - 7 q^{53} - 15 q^{56} + 8 q^{58} + 11 q^{59} + 3 q^{61} - 27 q^{62} - 4 q^{64} - 4 q^{67} - 8 q^{68} + 16 q^{71} + 12 q^{73} + 16 q^{74} - 26 q^{76} - 24 q^{77} + 2 q^{79} - 10 q^{82} + 18 q^{83} - q^{86} + 21 q^{88} - 24 q^{89} + q^{91} + 24 q^{92} + 25 q^{94} + 4 q^{97} - 35 q^{98}+O(q^{100}) \) 2 * q - q^2 + 3 * q^4 + 5 * q^7 - 6 * q^8 - 7 * q^11 + 3 * q^13 - 9 * q^14 - 3 * q^16 - q^17 + 10 * q^22 + 3 * q^23 + 5 * q^26 + 14 * q^28 + 10 * q^29 + 2 * q^31 + 7 * q^32 + 7 * q^34 + 7 * q^37 + 26 * q^38 + 7 * q^41 + 15 * q^43 - 17 * q^44 - 21 * q^46 + 2 * q^47 + 5 * q^49 - 2 * q^52 - 7 * q^53 - 15 * q^56 + 8 * q^58 + 11 * q^59 + 3 * q^61 - 27 * q^62 - 4 * q^64 - 4 * q^67 - 8 * q^68 + 16 * q^71 + 12 * q^73 + 16 * q^74 - 26 * q^76 - 24 * q^77 + 2 * q^79 - 10 * q^82 + 18 * q^83 - q^86 + 21 * q^88 - 24 * q^89 + q^91 + 24 * q^92 + 25 * q^94 + 4 * q^97 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more