LMFDB - Embedded newform 2025.2.a.i.1.2

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			1.2
Root			$-1.30278$ 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)

$f(q)$	$=$	$q+1.30278 q^{2} -0.302776 q^{4} +0.697224 q^{7} -3.00000 q^{8} +O(q^{10})$	$q+1.30278 q^{2} -0.302776 q^{4} +0.697224 q^{7} -3.00000 q^{8} -1.69722 q^{11} +3.30278 q^{13} +0.908327 q^{14} -3.30278 q^{16} +1.30278 q^{17} +7.21110 q^{19} -2.21110 q^{22} -3.90833 q^{23} +4.30278 q^{26} -0.211103 q^{28} +8.60555 q^{29} -6.21110 q^{31} +1.69722 q^{32} +1.69722 q^{34} +8.90833 q^{37} +9.39445 q^{38} +1.69722 q^{41} +9.30278 q^{43} +0.513878 q^{44} -5.09167 q^{46} +8.21110 q^{47} -6.51388 q^{49} -1.00000 q^{52} -12.5139 q^{53} -2.09167 q^{56} +11.2111 q^{58} +7.30278 q^{59} +3.30278 q^{61} -8.09167 q^{62} +8.81665 q^{64} +1.60555 q^{67} -0.394449 q^{68} +11.6056 q^{71} +2.39445 q^{73} +11.6056 q^{74} -2.18335 q^{76} -1.18335 q^{77} +4.60555 q^{79} +2.21110 q^{82} +9.00000 q^{83} +12.1194 q^{86} +5.09167 q^{88} -12.0000 q^{89} +2.30278 q^{91} +1.18335 q^{92} +10.6972 q^{94} +2.00000 q^{97} -8.48612 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.30278	0.921201	0.460601	−	0.887607i	$-0.347634\pi$
$2$	1.30278	0.921201	0.460601	+	0.887607i	$0.347634\pi$
$3$	0	0
$3$	0	0
$4$	−0.302776	−0.151388
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.697224	0.263526	0.131763	−	0.991281i	$-0.457936\pi$
$7$	0.697224	0.263526	0.131763	+	0.991281i	$0.457936\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	0	0
$11$	−1.69722	−0.511732	−0.255866	−	0.966712i	$-0.582361\pi$
$11$	−1.69722	−0.511732	−0.255866	+	0.966712i	$0.582361\pi$
$12$	0	0
$13$	3.30278	0.916025	0.458013	−	0.888946i	$-0.348561\pi$
$13$	3.30278	0.916025	0.458013	+	0.888946i	$0.348561\pi$
$14$	0.908327	0.242761
$15$	0	0
$16$	−3.30278	−0.825694
$17$	1.30278	0.315970	0.157985	−	0.987442i	$-0.449500\pi$
$17$	1.30278	0.315970	0.157985	+	0.987442i	$0.449500\pi$
$18$	0	0
$19$	7.21110	1.65434	0.827170	−	0.561951i	$-0.189949\pi$
$19$	7.21110	1.65434	0.827170	+	0.561951i	$0.189949\pi$
$20$	0	0
$21$	0	0
$22$	−2.21110	−0.471409
$23$	−3.90833	−0.814942	−0.407471	−	0.913218i	$-0.633589\pi$
$23$	−3.90833	−0.814942	−0.407471	+	0.913218i	$0.633589\pi$
$24$	0	0
$25$	0	0
$26$	4.30278	0.843844
$27$	0	0
$28$	−0.211103	−0.0398946
$29$	8.60555	1.59801	0.799005	−	0.601324i	$-0.205360\pi$
$29$	8.60555	1.59801	0.799005	+	0.601324i	$0.205360\pi$
$30$	0	0
$31$	−6.21110	−1.11555	−0.557773	−	0.829993i	$-0.688344\pi$
$31$	−6.21110	−1.11555	−0.557773	+	0.829993i	$0.688344\pi$
$32$	1.69722	0.300030
$33$	0	0
$34$	1.69722	0.291072
$35$	0	0
$36$	0	0
$37$	8.90833	1.46452	0.732260	−	0.681025i	$-0.238466\pi$
$37$	8.90833	1.46452	0.732260	+	0.681025i	$0.238466\pi$
$38$	9.39445	1.52398
$39$	0	0
$40$	0	0
$41$	1.69722	0.265062	0.132531	−	0.991179i	$-0.457690\pi$
$41$	1.69722	0.265062	0.132531	+	0.991179i	$0.457690\pi$
$42$	0	0
$43$	9.30278	1.41866	0.709330	−	0.704877i	$-0.248998\pi$
$43$	9.30278	1.41866	0.709330	+	0.704877i	$0.248998\pi$
$44$	0.513878	0.0774701
$45$	0	0
$46$	−5.09167	−0.750726
$47$	8.21110	1.19771	0.598856	−	0.800857i	$-0.295622\pi$
$47$	8.21110	1.19771	0.598856	+	0.800857i	$0.295622\pi$
$48$	0	0
$49$	−6.51388	−0.930554
$50$	0	0
$51$	0	0
$52$	−1.00000	−0.138675
$53$	−12.5139	−1.71891	−0.859457	−	0.511209i	$-0.829198\pi$
$53$	−12.5139	−1.71891	−0.859457	+	0.511209i	$0.829198\pi$
$54$	0	0
$55$	0	0
$56$	−2.09167	−0.279512
$57$	0	0
$58$	11.2111	1.47209
$59$	7.30278	0.950740	0.475370	−	0.879786i	$-0.342314\pi$
$59$	7.30278	0.950740	0.475370	+	0.879786i	$0.342314\pi$
$60$	0	0
$61$	3.30278	0.422877	0.211439	−	0.977391i	$-0.432185\pi$
$61$	3.30278	0.422877	0.211439	+	0.977391i	$0.432185\pi$
$62$	−8.09167	−1.02764
$63$	0	0
$64$	8.81665	1.10208
$65$	0	0
$66$	0	0
$67$	1.60555	0.196149	0.0980747	−	0.995179i	$-0.468732\pi$
$67$	1.60555	0.196149	0.0980747	+	0.995179i	$0.468732\pi$
$68$	−0.394449	−0.0478339
$69$	0	0
$70$	0	0
$71$	11.6056	1.37733	0.688663	−	0.725082i	$-0.258198\pi$
$71$	11.6056	1.37733	0.688663	+	0.725082i	$0.258198\pi$
$72$	0	0
$73$	2.39445	0.280249	0.140125	−	0.990134i	$-0.455250\pi$
$73$	2.39445	0.280249	0.140125	+	0.990134i	$0.455250\pi$
$74$	11.6056	1.34912
$75$	0	0
$76$	−2.18335	−0.250447
$77$	−1.18335	−0.134855
$78$	0	0
$79$	4.60555	0.518165	0.259083	−	0.965855i	$-0.416580\pi$
$79$	4.60555	0.518165	0.259083	+	0.965855i	$0.416580\pi$
$80$	0	0
$81$	0	0
$82$	2.21110	0.244175
$83$	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
$86$	12.1194	1.30687
$87$	0	0
$88$	5.09167	0.542774
$89$	−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
$91$	2.30278	0.241396
$92$	1.18335	0.123372
$93$	0	0
$94$	10.6972	1.10333
$95$	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	$0.467625\pi$
$98$	−8.48612	−0.857228
$99$	0	0
$100$	0	0
$101$	−17.7250	−1.76370	−0.881851	−	0.471529i	$-0.843702\pi$
$101$	−17.7250	−1.76370	−0.881851	+	0.471529i	$0.843702\pi$
$102$	0	0
$103$	7.09167	0.698763	0.349382	−	0.936981i	$-0.386392\pi$
$103$	7.09167	0.698763	0.349382	+	0.936981i	$0.386392\pi$
$104$	−9.90833	−0.971591
$105$	0	0
$106$	−16.3028	−1.58347
$107$	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
$109$	1.09167	0.104563	0.0522817	−	0.998632i	$-0.483351\pi$
$109$	1.09167	0.104563	0.0522817	+	0.998632i	$0.483351\pi$
$110$	0	0
$111$	0	0
$112$	−2.30278	−0.217592
$113$	−7.69722	−0.724094	−0.362047	−	0.932160i	$-0.617922\pi$
$113$	−7.69722	−0.724094	−0.362047	+	0.932160i	$0.617922\pi$
$114$	0	0
$115$	0	0
$116$	−2.60555	−0.241919
$117$	0	0
$118$	9.51388	0.875823
$119$	0.908327	0.0832662
$120$	0	0
$121$	−8.11943	−0.738130
$122$	4.30278	0.389555
$123$	0	0
$124$	1.88057	0.168880
$125$	0	0
$126$	0	0
$127$	−3.60555	−0.319941	−0.159970	−	0.987122i	$-0.551140\pi$
$127$	−3.60555	−0.319941	−0.159970	+	0.987122i	$0.551140\pi$
$128$	8.09167	0.715210
$129$	0	0
$130$	0	0
$131$	−19.8167	−1.73139	−0.865695	−	0.500573i	$-0.833123\pi$
$131$	−19.8167	−1.73139	−0.865695	+	0.500573i	$0.833123\pi$
$132$	0	0
$133$	5.02776	0.435962
$134$	2.09167	0.180693
$135$	0	0
$136$	−3.90833	−0.335136
$137$	2.09167	0.178704	0.0893518	−	0.996000i	$-0.471520\pi$
$137$	2.09167	0.178704	0.0893518	+	0.996000i	$0.471520\pi$
$138$	0	0
$139$	2.90833	0.246681	0.123341	−	0.992364i	$-0.460639\pi$
$139$	2.90833	0.246681	0.123341	+	0.992364i	$0.460639\pi$
$140$	0	0
$141$	0	0
$142$	15.1194	1.26879
$143$	−5.60555	−0.468760
$144$	0	0
$145$	0	0
$146$	3.11943	0.258166
$147$	0	0
$148$	−2.69722	−0.221710
$149$	−8.21110	−0.672680	−0.336340	−	0.941741i	$-0.609189\pi$
$149$	−8.21110	−0.672680	−0.336340	+	0.941741i	$0.609189\pi$
$150$	0	0
$151$	2.11943	0.172477	0.0862384	−	0.996275i	$-0.472515\pi$
$151$	2.11943	0.172477	0.0862384	+	0.996275i	$0.472515\pi$
$152$	−21.6333	−1.75469
$153$	0	0
$154$	−1.54163	−0.124228
$155$	0	0
$156$	0	0
$157$	24.8167	1.98058	0.990292	−	0.139001i	$-0.0443890\pi$
$157$	24.8167	1.98058	0.990292	+	0.139001i	$0.0443890\pi$
$158$	6.00000	0.477334
$159$	0	0
$160$	0	0
$161$	−2.72498	−0.214759
$162$	0	0
$163$	18.8167	1.47383	0.736917	−	0.675983i	$-0.236281\pi$
$163$	18.8167	1.47383	0.736917	+	0.675983i	$0.236281\pi$
$164$	−0.513878	−0.0401271
$165$	0	0
$166$	11.7250	0.910035
$167$	−9.90833	−0.766729	−0.383365	−	0.923597i	$-0.625235\pi$
$167$	−9.90833	−0.766729	−0.383365	+	0.923597i	$0.625235\pi$
$168$	0	0
$169$	−2.09167	−0.160898
$170$	0	0
$171$	0	0
$172$	−2.81665	−0.214768
$173$	13.8167	1.05046	0.525230	−	0.850960i	$-0.323979\pi$
$173$	13.8167	1.05046	0.525230	+	0.850960i	$0.323979\pi$
$174$	0	0
$175$	0	0
$176$	5.60555	0.422534
$177$	0	0
$178$	−15.6333	−1.17177
$179$	−13.4222	−1.00322	−0.501611	−	0.865093i	$-0.667259\pi$
$179$	−13.4222	−1.00322	−0.501611	+	0.865093i	$0.667259\pi$
$180$	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
$184$	11.7250	0.864377
$185$	0	0
$186$	0	0
$187$	−2.21110	−0.161692
$188$	−2.48612	−0.181319
$189$	0	0
$190$	0	0
$191$	−5.21110	−0.377062	−0.188531	−	0.982067i	$-0.560373\pi$
$191$	−5.21110	−0.377062	−0.188531	+	0.982067i	$0.560373\pi$
$192$	0	0
$193$	−14.8167	−1.06653	−0.533263	−	0.845949i	$-0.679034\pi$
$193$	−14.8167	−1.06653	−0.533263	+	0.845949i	$0.679034\pi$
$194$	2.60555	0.187068
$195$	0	0
$196$	1.97224	0.140875
$197$	12.9083	0.919680	0.459840	−	0.888002i	$-0.347907\pi$
$197$	12.9083	0.919680	0.459840	+	0.888002i	$0.347907\pi$
$198$	0	0
$199$	−25.5139	−1.80863	−0.904315	−	0.426865i	$-0.859618\pi$
$199$	−25.5139	−1.80863	−0.904315	+	0.426865i	$0.859618\pi$
$200$	0	0
$201$	0	0
$202$	−23.0917	−1.62472
$203$	6.00000	0.421117
$204$	0	0
$205$	0	0
$206$	9.23886	0.643702
$207$	0	0
$208$	−10.9083	−0.756356
$209$	−12.2389	−0.846580
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	3.78890	0.260223
$213$	0	0
$214$	11.7250	0.801503
$215$	0	0
$216$	0	0
$217$	−4.33053	−0.293976
$218$	1.42221	0.0963239
$219$	0	0
$220$	0	0
$221$	4.30278	0.289436
$222$	0	0
$223$	17.0000	1.13840	0.569202	−	0.822198i	$-0.307252\pi$
$223$	17.0000	1.13840	0.569202	+	0.822198i	$0.307252\pi$
$224$	1.18335	0.0790656
$225$	0	0
$226$	−10.0278	−0.667036
$227$	−18.3944	−1.22088	−0.610441	−	0.792062i	$-0.709008\pi$
$227$	−18.3944	−1.22088	−0.610441	+	0.792062i	$0.709008\pi$
$228$	0	0
$229$	9.42221	0.622637	0.311318	−	0.950306i	$-0.399229\pi$
$229$	9.42221	0.622637	0.311318	+	0.950306i	$0.399229\pi$
$230$	0	0
$231$	0	0
$232$	−25.8167	−1.69495
$233$	28.8167	1.88784	0.943921	−	0.330172i	$-0.107107\pi$
$233$	28.8167	1.88784	0.943921	+	0.330172i	$0.107107\pi$
$234$	0	0
$235$	0	0
$236$	−2.21110	−0.143931
$237$	0	0
$238$	1.18335	0.0767049
$239$	−4.42221	−0.286049	−0.143024	−	0.989719i	$-0.545683\pi$
$239$	−4.42221	−0.286049	−0.143024	+	0.989719i	$0.545683\pi$
$240$	0	0
$241$	19.2111	1.23750	0.618748	−	0.785590i	$-0.287640\pi$
$241$	19.2111	1.23750	0.618748	+	0.785590i	$0.287640\pi$
$242$	−10.5778	−0.679966
$243$	0	0
$244$	−1.00000	−0.0640184
$245$	0	0
$246$	0	0
$247$	23.8167	1.51542
$248$	18.6333	1.18322
$249$	0	0
$250$	0	0
$251$	−5.72498	−0.361358	−0.180679	−	0.983542i	$-0.557829\pi$
$251$	−5.72498	−0.361358	−0.180679	+	0.983542i	$0.557829\pi$
$252$	0	0
$253$	6.63331	0.417032
$254$	−4.69722	−0.294730
$255$	0	0
$256$	−7.09167	−0.443230
$257$	−21.6333	−1.34945	−0.674724	−	0.738070i	$-0.735737\pi$
$257$	−21.6333	−1.34945	−0.674724	+	0.738070i	$0.735737\pi$
$258$	0	0
$259$	6.21110	0.385939
$260$	0	0
$261$	0	0
$262$	−25.8167	−1.59496
$263$	−25.0278	−1.54328	−0.771639	−	0.636061i	$-0.780563\pi$
$263$	−25.0278	−1.54328	−0.771639	+	0.636061i	$0.780563\pi$
$264$	0	0
$265$	0	0
$266$	6.55004	0.401609
$267$	0	0
$268$	−0.486122	−0.0296946
$269$	−13.6972	−0.835135	−0.417567	−	0.908646i	$-0.637117\pi$
$269$	−13.6972	−0.835135	−0.417567	+	0.908646i	$0.637117\pi$
$270$	0	0
$271$	2.39445	0.145452	0.0727262	−	0.997352i	$-0.476830\pi$
$271$	2.39445	0.145452	0.0727262	+	0.997352i	$0.476830\pi$
$272$	−4.30278	−0.260894
$273$	0	0
$274$	2.72498	0.164622
$275$	0	0
$276$	0	0
$277$	14.7889	0.888579	0.444289	−	0.895883i	$-0.353456\pi$
$277$	14.7889	0.888579	0.444289	+	0.895883i	$0.353456\pi$
$278$	3.78890	0.227243
$279$	0	0
$280$	0	0
$281$	16.8167	1.00320	0.501599	−	0.865100i	$-0.332745\pi$
$281$	16.8167	1.00320	0.501599	+	0.865100i	$0.332745\pi$
$282$	0	0
$283$	−10.1194	−0.601538	−0.300769	−	0.953697i	$-0.597243\pi$
$283$	−10.1194	−0.601538	−0.300769	+	0.953697i	$0.597243\pi$
$284$	−3.51388	−0.208510
$285$	0	0
$286$	−7.30278	−0.431822
$287$	1.18335	0.0698507
$288$	0	0
$289$	−15.3028	−0.900163
$290$	0	0
$291$	0	0
$292$	−0.724981	−0.0424263
$293$	−22.8167	−1.33296	−0.666482	−	0.745522i	$-0.732200\pi$
$293$	−22.8167	−1.33296	−0.666482	+	0.745522i	$0.732200\pi$
$294$	0	0
$295$	0	0
$296$	−26.7250	−1.55336
$297$	0	0
$298$	−10.6972	−0.619674
$299$	−12.9083	−0.746508
$300$	0	0
$301$	6.48612	0.373854
$302$	2.76114	0.158886
$303$	0	0
$304$	−23.8167	−1.36598
$305$	0	0
$306$	0	0
$307$	19.2111	1.09644	0.548218	−	0.836336i	$-0.315306\pi$
$307$	19.2111	1.09644	0.548218	+	0.836336i	$0.315306\pi$
$308$	0.358288	0.0204154
$309$	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
$313$	−21.7250	−1.22797	−0.613984	−	0.789318i	$-0.710434\pi$
$313$	−21.7250	−1.22797	−0.613984	+	0.789318i	$0.710434\pi$
$314$	32.3305	1.82452
$315$	0	0
$316$	−1.39445	−0.0784439
$317$	7.42221	0.416873	0.208436	−	0.978036i	$-0.433163\pi$
$317$	7.42221	0.416873	0.208436	+	0.978036i	$0.433163\pi$
$318$	0	0
$319$	−14.6056	−0.817754
$320$	0	0
$321$	0	0
$322$	−3.55004	−0.197836
$323$	9.39445	0.522721
$324$	0	0
$325$	0	0
$326$	24.5139	1.35770
$327$	0	0
$328$	−5.09167	−0.281141
$329$	5.72498	0.315628
$330$	0	0
$331$	12.4222	0.682786	0.341393	−	0.939921i	$-0.389101\pi$
$331$	12.4222	0.682786	0.341393	+	0.939921i	$0.389101\pi$
$332$	−2.72498	−0.149553
$333$	0	0
$334$	−12.9083	−0.706312
$335$	0	0
$336$	0	0
$337$	−18.6056	−1.01351	−0.506754	−	0.862090i	$-0.669155\pi$
$337$	−18.6056	−1.01351	−0.506754	+	0.862090i	$0.669155\pi$
$338$	−2.72498	−0.148219
$339$	0	0
$340$	0	0
$341$	10.5416	0.570862
$342$	0	0
$343$	−9.42221	−0.508751
$344$	−27.9083	−1.50472
$345$	0	0
$346$	18.0000	0.967686
$347$	19.9361	1.07023	0.535113	−	0.844781i	$-0.320269\pi$
$347$	19.9361	1.07023	0.535113	+	0.844781i	$0.320269\pi$
$348$	0	0
$349$	−12.6056	−0.674760	−0.337380	−	0.941369i	$-0.609541\pi$
$349$	−12.6056	−0.674760	−0.337380	+	0.941369i	$0.609541\pi$
$350$	0	0
$351$	0	0
$352$	−2.88057	−0.153535
$353$	−14.7250	−0.783732	−0.391866	−	0.920022i	$-0.628170\pi$
$353$	−14.7250	−0.783732	−0.391866	+	0.920022i	$0.628170\pi$
$354$	0	0
$355$	0	0
$356$	3.63331	0.192565
$357$	0	0
$358$	−17.4861	−0.924170
$359$	−0.908327	−0.0479397	−0.0239698	−	0.999713i	$-0.507631\pi$
$359$	−0.908327	−0.0479397	−0.0239698	+	0.999713i	$0.507631\pi$
$360$	0	0
$361$	33.0000	1.73684
$362$	−9.11943	−0.479307
$363$	0	0
$364$	−0.697224	−0.0365445
$365$	0	0
$366$	0	0
$367$	9.69722	0.506191	0.253095	−	0.967441i	$-0.418551\pi$
$367$	9.69722	0.506191	0.253095	+	0.967441i	$0.418551\pi$
$368$	12.9083	0.672893
$369$	0	0
$370$	0	0
$371$	−8.72498	−0.452978
$372$	0	0
$373$	−15.6056	−0.808025	−0.404012	−	0.914754i	$-0.632385\pi$
$373$	−15.6056	−0.808025	−0.404012	+	0.914754i	$0.632385\pi$
$374$	−2.88057	−0.148951
$375$	0	0
$376$	−24.6333	−1.27037
$377$	28.4222	1.46382
$378$	0	0
$379$	9.30278	0.477851	0.238926	−	0.971038i	$-0.423205\pi$
$379$	9.30278	0.477851	0.238926	+	0.971038i	$0.423205\pi$
$380$	0	0
$381$	0	0
$382$	−6.78890	−0.347350
$383$	26.7250	1.36558	0.682791	−	0.730613i	$-0.260766\pi$
$383$	26.7250	1.36558	0.682791	+	0.730613i	$0.260766\pi$
$384$	0	0
$385$	0	0
$386$	−19.3028	−0.982485
$387$	0	0
$388$	−0.605551	−0.0307422
$389$	22.8167	1.15685	0.578425	−	0.815735i	$-0.303667\pi$
$389$	22.8167	1.15685	0.578425	+	0.815735i	$0.303667\pi$
$390$	0	0
$391$	−5.09167	−0.257497
$392$	19.5416	0.987002
$393$	0	0
$394$	16.8167	0.847211
$395$	0	0
$396$	0	0
$397$	−21.6056	−1.08435	−0.542176	−	0.840265i	$-0.682399\pi$
$397$	−21.6056	−1.08435	−0.542176	+	0.840265i	$0.682399\pi$
$398$	−33.2389	−1.66611
$399$	0	0
$400$	0	0
$401$	−25.8167	−1.28922	−0.644611	−	0.764511i	$-0.722981\pi$
$401$	−25.8167	−1.28922	−0.644611	+	0.764511i	$0.722981\pi$
$402$	0	0
$403$	−20.5139	−1.02187
$404$	5.36669	0.267003
$405$	0	0
$406$	7.81665	0.387934
$407$	−15.1194	−0.749442
$408$	0	0
$409$	−23.5416	−1.16406	−0.582029	−	0.813168i	$-0.697741\pi$
$409$	−23.5416	−1.16406	−0.582029	+	0.813168i	$0.697741\pi$
$410$	0	0
$411$	0	0
$412$	−2.14719	−0.105784
$413$	5.09167	0.250545
$414$	0	0
$415$	0	0
$416$	5.60555	0.274835
$417$	0	0
$418$	−15.9445	−0.779870
$419$	31.6972	1.54851	0.774255	−	0.632873i	$-0.218125\pi$
$419$	31.6972	1.54851	0.774255	+	0.632873i	$0.218125\pi$
$420$	0	0
$421$	5.39445	0.262909	0.131455	−	0.991322i	$-0.458035\pi$
$421$	5.39445	0.262909	0.131455	+	0.991322i	$0.458035\pi$
$422$	6.51388	0.317091
$423$	0	0
$424$	37.5416	1.82318
$425$	0	0
$426$	0	0
$427$	2.30278	0.111439
$428$	−2.72498	−0.131717
$429$	0	0
$430$	0	0
$431$	−10.8167	−0.521020	−0.260510	−	0.965471i	$-0.583891\pi$
$431$	−10.8167	−0.521020	−0.260510	+	0.965471i	$0.583891\pi$
$432$	0	0
$433$	−34.2389	−1.64541	−0.822707	−	0.568465i	$-0.807537\pi$
$433$	−34.2389	−1.64541	−0.822707	+	0.568465i	$0.807537\pi$
$434$	−5.64171	−0.270811
$435$	0	0
$436$	−0.330532	−0.0158296
$437$	−28.1833	−1.34819
$438$	0	0
$439$	−31.5139	−1.50408	−0.752038	−	0.659120i	$-0.770929\pi$
$439$	−31.5139	−1.50408	−0.752038	+	0.659120i	$0.770929\pi$
$440$	0	0
$441$	0	0
$442$	5.60555	0.266629
$443$	−39.9083	−1.89610	−0.948051	−	0.318119i	$-0.896949\pi$
$443$	−39.9083	−1.89610	−0.948051	+	0.318119i	$0.896949\pi$
$444$	0	0
$445$	0	0
$446$	22.1472	1.04870
$447$	0	0
$448$	6.14719	0.290427
$449$	27.1194	1.27985	0.639923	−	0.768439i	$-0.278966\pi$
$449$	27.1194	1.27985	0.639923	+	0.768439i	$0.278966\pi$
$450$	0	0
$451$	−2.88057	−0.135641
$452$	2.33053	0.109619
$453$	0	0
$454$	−23.9638	−1.12468
$455$	0	0
$456$	0	0
$457$	−32.4222	−1.51665	−0.758323	−	0.651879i	$-0.773981\pi$
$457$	−32.4222	−1.51665	−0.758323	+	0.651879i	$0.773981\pi$
$458$	12.2750	0.573574
$459$	0	0
$460$	0	0
$461$	8.72498	0.406363	0.203181	−	0.979141i	$-0.434872\pi$
$461$	8.72498	0.406363	0.203181	+	0.979141i	$0.434872\pi$
$462$	0	0
$463$	22.3305	1.03779	0.518894	−	0.854839i	$-0.326344\pi$
$463$	22.3305	1.03779	0.518894	+	0.854839i	$0.326344\pi$
$464$	−28.4222	−1.31947
$465$	0	0
$466$	37.5416	1.73908
$467$	34.4222	1.59287	0.796435	−	0.604724i	$-0.206717\pi$
$467$	34.4222	1.59287	0.796435	+	0.604724i	$0.206717\pi$
$468$	0	0
$469$	1.11943	0.0516904
$470$	0	0
$471$	0	0
$472$	−21.9083	−1.00841
$473$	−15.7889	−0.725974
$474$	0	0
$475$	0	0
$476$	−0.275019	−0.0126055
$477$	0	0
$478$	−5.76114	−0.263508
$479$	−23.0917	−1.05509	−0.527543	−	0.849528i	$-0.676887\pi$
$479$	−23.0917	−1.05509	−0.527543	+	0.849528i	$0.676887\pi$
$480$	0	0
$481$	29.4222	1.34154
$482$	25.0278	1.13998
$483$	0	0
$484$	2.45837	0.111744
$485$	0	0
$486$	0	0
$487$	35.6333	1.61470	0.807350	−	0.590073i	$-0.200901\pi$
$487$	35.6333	1.61470	0.807350	+	0.590073i	$0.200901\pi$
$488$	−9.90833	−0.448529
$489$	0	0
$490$	0	0
$491$	27.2389	1.22927	0.614636	−	0.788811i	$-0.289303\pi$
$491$	27.2389	1.22927	0.614636	+	0.788811i	$0.289303\pi$
$492$	0	0
$493$	11.2111	0.504923
$494$	31.0278	1.39600
$495$	0	0
$496$	20.5139	0.921100
$497$	8.09167	0.362961
$498$	0	0
$499$	−6.33053	−0.283394	−0.141697	−	0.989910i	$-0.545256\pi$
$499$	−6.33053	−0.283394	−0.141697	+	0.989910i	$0.545256\pi$
$500$	0	0
$501$	0	0
$502$	−7.45837	−0.332883
$503$	14.3305	0.638967	0.319483	−	0.947592i	$-0.396491\pi$
$503$	14.3305	0.638967	0.319483	+	0.947592i	$0.396491\pi$
$504$	0	0
$505$	0	0
$506$	8.64171	0.384171
$507$	0	0
$508$	1.09167	0.0484352
$509$	20.2111	0.895841	0.447921	−	0.894073i	$-0.352165\pi$
$509$	20.2111	0.895841	0.447921	+	0.894073i	$0.352165\pi$
$510$	0	0
$511$	1.66947	0.0738529
$512$	−25.4222	−1.12351
$513$	0	0
$514$	−28.1833	−1.24311
$515$	0	0
$516$	0	0
$517$	−13.9361	−0.612908
$518$	8.09167	0.355528
$519$	0	0
$520$	0	0
$521$	4.97224	0.217838	0.108919	−	0.994051i	$-0.465261\pi$
$521$	4.97224	0.217838	0.108919	+	0.994051i	$0.465261\pi$
$522$	0	0
$523$	−13.2389	−0.578895	−0.289447	−	0.957194i	$-0.593472\pi$
$523$	−13.2389	−0.578895	−0.289447	+	0.957194i	$0.593472\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−32.6056	−1.42167
$527$	−8.09167	−0.352479
$528$	0	0
$529$	−7.72498	−0.335869
$530$	0	0
$531$	0	0
$532$	−1.52228	−0.0659993
$533$	5.60555	0.242803
$534$	0	0
$535$	0	0
$536$	−4.81665	−0.208048
$537$	0	0
$538$	−17.8444	−0.769327
$539$	11.0555	0.476195
$540$	0	0
$541$	19.3305	0.831084	0.415542	−	0.909574i	$-0.363592\pi$
$541$	19.3305	0.831084	0.415542	+	0.909574i	$0.363592\pi$
$542$	3.11943	0.133991
$543$	0	0
$544$	2.21110	0.0948002
$545$	0	0
$546$	0	0
$547$	39.6611	1.69578	0.847892	−	0.530168i	$-0.177871\pi$
$547$	39.6611	1.69578	0.847892	+	0.530168i	$0.177871\pi$
$548$	−0.633308	−0.0270536
$549$	0	0
$550$	0	0
$551$	62.0555	2.64365
$552$	0	0
$553$	3.21110	0.136550
$554$	19.2666	0.818560
$555$	0	0
$556$	−0.880571	−0.0373445
$557$	−9.63331	−0.408176	−0.204088	−	0.978953i	$-0.565423\pi$
$557$	−9.63331	−0.408176	−0.204088	+	0.978953i	$0.565423\pi$
$558$	0	0
$559$	30.7250	1.29953
$560$	0	0
$561$	0	0
$562$	21.9083	0.924147
$563$	−6.63331	−0.279561	−0.139780	−	0.990183i	$-0.544640\pi$
$563$	−6.63331	−0.279561	−0.139780	+	0.990183i	$0.544640\pi$
$564$	0	0
$565$	0	0
$566$	−13.1833	−0.554137
$567$	0	0
$568$	−34.8167	−1.46087
$569$	1.81665	0.0761581	0.0380790	−	0.999275i	$-0.487876\pi$
$569$	1.81665	0.0761581	0.0380790	+	0.999275i	$0.487876\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	1.69722	0.0709645
$573$	0	0
$574$	1.54163	0.0643466
$575$	0	0
$576$	0	0
$577$	32.6333	1.35854	0.679271	−	0.733887i	$-0.262296\pi$
$577$	32.6333	1.35854	0.679271	+	0.733887i	$0.262296\pi$
$578$	−19.9361	−0.829232
$579$	0	0
$580$	0	0
$581$	6.27502	0.260332
$582$	0	0
$583$	21.2389	0.879624
$584$	−7.18335	−0.297249
$585$	0	0
$586$	−29.7250	−1.22793
$587$	−16.8167	−0.694098	−0.347049	−	0.937847i	$-0.612816\pi$
$587$	−16.8167	−0.694098	−0.347049	+	0.937847i	$0.612816\pi$
$588$	0	0
$589$	−44.7889	−1.84549
$590$	0	0
$591$	0	0
$592$	−29.4222	−1.20925
$593$	11.8806	0.487877	0.243938	−	0.969791i	$-0.421561\pi$
$593$	11.8806	0.487877	0.243938	+	0.969791i	$0.421561\pi$
$594$	0	0
$595$	0	0
$596$	2.48612	0.101836
$597$	0	0
$598$	−16.8167	−0.687684
$599$	20.3305	0.830683	0.415342	−	0.909666i	$-0.363662\pi$
$599$	20.3305	0.830683	0.415342	+	0.909666i	$0.363662\pi$
$600$	0	0
$601$	7.09167	0.289275	0.144638	−	0.989485i	$-0.453798\pi$
$601$	7.09167	0.289275	0.144638	+	0.989485i	$0.453798\pi$
$602$	8.44996	0.344395
$603$	0	0
$604$	−0.641712	−0.0261109
$605$	0	0
$606$	0	0
$607$	−22.2389	−0.902647	−0.451324	−	0.892360i	$-0.649048\pi$
$607$	−22.2389	−0.902647	−0.451324	+	0.892360i	$0.649048\pi$
$608$	12.2389	0.496351
$609$	0	0
$610$	0	0
$611$	27.1194	1.09713
$612$	0	0
$613$	−24.6056	−0.993809	−0.496904	−	0.867805i	$-0.665530\pi$
$613$	−24.6056	−0.993809	−0.496904	+	0.867805i	$0.665530\pi$
$614$	25.0278	1.01004
$615$	0	0
$616$	3.55004	0.143035
$617$	16.8167	0.677013	0.338506	−	0.940964i	$-0.390078\pi$
$617$	16.8167	0.677013	0.338506	+	0.940964i	$0.390078\pi$
$618$	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
$622$	−27.3583	−1.09697
$623$	−8.36669	−0.335204
$624$	0	0
$625$	0	0
$626$	−28.3028	−1.13121
$627$	0	0
$628$	−7.51388	−0.299836
$629$	11.6056	0.462744
$630$	0	0
$631$	0.302776	0.0120533	0.00602665	−	0.999982i	$-0.498082\pi$
$631$	0.302776	0.0120533	0.00602665	+	0.999982i	$0.498082\pi$
$632$	−13.8167	−0.549597
$633$	0	0
$634$	9.66947	0.384024
$635$	0	0
$636$	0	0
$637$	−21.5139	−0.852411
$638$	−19.0278	−0.753316
$639$	0	0
$640$	0	0
$641$	−42.9083	−1.69478	−0.847389	−	0.530973i	$-0.821826\pi$
$641$	−42.9083	−1.69478	−0.847389	+	0.530973i	$0.821826\pi$
$642$	0	0
$643$	23.5139	0.927297	0.463648	−	0.886019i	$-0.346540\pi$
$643$	23.5139	0.927297	0.463648	+	0.886019i	$0.346540\pi$
$644$	0.825058	0.0325118
$645$	0	0
$646$	12.2389	0.481531
$647$	23.8444	0.937420	0.468710	−	0.883352i	$-0.344719\pi$
$647$	23.8444	0.937420	0.468710	+	0.883352i	$0.344719\pi$
$648$	0	0
$649$	−12.3944	−0.486525
$650$	0	0
$651$	0	0
$652$	−5.69722	−0.223121
$653$	−24.1194	−0.943866	−0.471933	−	0.881634i	$-0.656444\pi$
$653$	−24.1194	−0.943866	−0.471933	+	0.881634i	$0.656444\pi$
$654$	0	0
$655$	0	0
$656$	−5.60555	−0.218860
$657$	0	0
$658$	7.45837	0.290757
$659$	3.11943	0.121516	0.0607579	−	0.998153i	$-0.480648\pi$
$659$	3.11943	0.121516	0.0607579	+	0.998153i	$0.480648\pi$
$660$	0	0
$661$	4.21110	0.163793	0.0818965	−	0.996641i	$-0.473902\pi$
$661$	4.21110	0.163793	0.0818965	+	0.996641i	$0.473902\pi$
$662$	16.1833	0.628984
$663$	0	0
$664$	−27.0000	−1.04780
$665$	0	0
$666$	0	0
$667$	−33.6333	−1.30229
$668$	3.00000	0.116073
$669$	0	0
$670$	0	0
$671$	−5.60555	−0.216400
$672$	0	0
$673$	−20.0278	−0.772013	−0.386007	−	0.922496i	$-0.626146\pi$
$673$	−20.0278	−0.772013	−0.386007	+	0.922496i	$0.626146\pi$
$674$	−24.2389	−0.933646
$675$	0	0
$676$	0.633308	0.0243580
$677$	−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
$679$	1.39445	0.0535140
$680$	0	0
$681$	0	0
$682$	13.7334	0.525878
$683$	−0.275019	−0.0105233	−0.00526166	−	0.999986i	$-0.501675\pi$
$683$	−0.275019	−0.0105233	−0.00526166	+	0.999986i	$0.501675\pi$
$684$	0	0
$685$	0	0
$686$	−12.2750	−0.468662
$687$	0	0
$688$	−30.7250	−1.17138
$689$	−41.3305	−1.57457
$690$	0	0
$691$	−39.0555	−1.48574	−0.742871	−	0.669435i	$-0.766536\pi$
$691$	−39.0555	−1.48574	−0.742871	+	0.669435i	$0.766536\pi$
$692$	−4.18335	−0.159027
$693$	0	0
$694$	25.9722	0.985893
$695$	0	0
$696$	0	0
$697$	2.21110	0.0837515
$698$	−16.4222	−0.621590
$699$	0	0
$700$	0	0
$701$	0.238859	0.00902158	0.00451079	−	0.999990i	$-0.498564\pi$
$701$	0.238859	0.00902158	0.00451079	+	0.999990i	$0.498564\pi$
$702$	0	0
$703$	64.2389	2.42281
$704$	−14.9638	−0.563971
$705$	0	0
$706$	−19.1833	−0.721975
$707$	−12.3583	−0.464781
$708$	0	0
$709$	−13.7889	−0.517853	−0.258926	−	0.965897i	$-0.583369\pi$
$709$	−13.7889	−0.517853	−0.258926	+	0.965897i	$0.583369\pi$
$710$	0	0
$711$	0	0
$712$	36.0000	1.34916
$713$	24.2750	0.909107
$714$	0	0
$715$	0	0
$716$	4.06392	0.151876
$717$	0	0
$718$	−1.18335	−0.0441621
$719$	−14.0917	−0.525531	−0.262765	−	0.964860i	$-0.584635\pi$
$719$	−14.0917	−0.525531	−0.262765	+	0.964860i	$0.584635\pi$
$720$	0	0
$721$	4.94449	0.184142
$722$	42.9916	1.59998
$723$	0	0
$724$	2.11943	0.0787680
$725$	0	0
$726$	0	0
$727$	1.09167	0.0404879	0.0202440	−	0.999795i	$-0.493556\pi$
$727$	1.09167	0.0404879	0.0202440	+	0.999795i	$0.493556\pi$
$728$	−6.90833	−0.256040
$729$	0	0
$730$	0	0
$731$	12.1194	0.448253
$732$	0	0
$733$	7.72498	0.285329	0.142664	−	0.989771i	$-0.454433\pi$
$733$	7.72498	0.285329	0.142664	+	0.989771i	$0.454433\pi$
$734$	12.6333	0.466304
$735$	0	0
$736$	−6.63331	−0.244507
$737$	−2.72498	−0.100376
$738$	0	0
$739$	−6.33053	−0.232872	−0.116436	−	0.993198i	$-0.537147\pi$
$739$	−6.33053	−0.232872	−0.116436	+	0.993198i	$0.537147\pi$
$740$	0	0
$741$	0	0
$742$	−11.3667	−0.417284
$743$	43.0278	1.57854	0.789268	−	0.614049i	$-0.210460\pi$
$743$	43.0278	1.57854	0.789268	+	0.614049i	$0.210460\pi$
$744$	0	0
$745$	0	0
$746$	−20.3305	−0.744354
$747$	0	0
$748$	0.669468	0.0244782
$749$	6.27502	0.229284
$750$	0	0
$751$	16.4861	0.601587	0.300794	−	0.953689i	$-0.402748\pi$
$751$	16.4861	0.601587	0.300794	+	0.953689i	$0.402748\pi$
$752$	−27.1194	−0.988944
$753$	0	0
$754$	37.0278	1.34847
$755$	0	0
$756$	0	0
$757$	5.11943	0.186069	0.0930344	−	0.995663i	$-0.470343\pi$
$757$	5.11943	0.186069	0.0930344	+	0.995663i	$0.470343\pi$
$758$	12.1194	0.440198
$759$	0	0
$760$	0	0
$761$	39.3583	1.42674	0.713368	−	0.700789i	$-0.247169\pi$
$761$	39.3583	1.42674	0.713368	+	0.700789i	$0.247169\pi$
$762$	0	0
$763$	0.761141	0.0275552
$764$	1.57779	0.0570826
$765$	0	0
$766$	34.8167	1.25798
$767$	24.1194	0.870902
$768$	0	0
$769$	50.3583	1.81597	0.907983	−	0.419007i	$-0.137622\pi$
$769$	50.3583	1.81597	0.907983	+	0.419007i	$0.137622\pi$
$770$	0	0
$771$	0	0
$772$	4.48612	0.161459
$773$	20.4500	0.735534	0.367767	−	0.929918i	$-0.380122\pi$
$773$	20.4500	0.735534	0.367767	+	0.929918i	$0.380122\pi$
$774$	0	0
$775$	0	0
$776$	−6.00000	−0.215387
$777$	0	0
$778$	29.7250	1.06569
$779$	12.2389	0.438503
$780$	0	0
$781$	−19.6972	−0.704822
$782$	−6.63331	−0.237207
$783$	0	0
$784$	21.5139	0.768353
$785$	0	0
$786$	0	0
$787$	28.4861	1.01542	0.507710	−	0.861528i	$-0.330492\pi$
$787$	28.4861	1.01542	0.507710	+	0.861528i	$0.330492\pi$
$788$	−3.90833	−0.139228
$789$	0	0
$790$	0	0
$791$	−5.36669	−0.190818
$792$	0	0
$793$	10.9083	0.387366
$794$	−28.1472	−0.998906
$795$	0	0
$796$	7.72498	0.273805
$797$	−6.11943	−0.216761	−0.108381	−	0.994109i	$-0.534567\pi$
$797$	−6.11943	−0.216761	−0.108381	+	0.994109i	$0.534567\pi$
$798$	0	0
$799$	10.6972	0.378441
$800$	0	0
$801$	0	0
$802$	−33.6333	−1.18763
$803$	−4.06392	−0.143413
$804$	0	0
$805$	0	0
$806$	−26.7250	−0.941347
$807$	0	0
$808$	53.1749	1.87069
$809$	−4.02776	−0.141608	−0.0708042	−	0.997490i	$-0.522557\pi$
$809$	−4.02776	−0.141608	−0.0708042	+	0.997490i	$0.522557\pi$
$810$	0	0
$811$	−29.9361	−1.05120	−0.525599	−	0.850732i	$-0.676159\pi$
$811$	−29.9361	−1.05120	−0.525599	+	0.850732i	$0.676159\pi$
$812$	−1.81665	−0.0637521
$813$	0	0
$814$	−19.6972	−0.690387
$815$	0	0
$816$	0	0
$817$	67.0833	2.34695
$818$	−30.6695	−1.07233
$819$	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
$823$	7.36669	0.256787	0.128393	−	0.991723i	$-0.459018\pi$
$823$	7.36669	0.256787	0.128393	+	0.991723i	$0.459018\pi$
$824$	−21.2750	−0.741150
$825$	0	0
$826$	6.63331	0.230802
$827$	−32.9638	−1.14627	−0.573133	−	0.819463i	$-0.694272\pi$
$827$	−32.9638	−1.14627	−0.573133	+	0.819463i	$0.694272\pi$
$828$	0	0
$829$	−27.2111	−0.945081	−0.472540	−	0.881309i	$-0.656663\pi$
$829$	−27.2111	−0.945081	−0.472540	+	0.881309i	$0.656663\pi$
$830$	0	0
$831$	0	0
$832$	29.1194	1.00953
$833$	−8.48612	−0.294027
$834$	0	0
$835$	0	0
$836$	3.70563	0.128162
$837$	0	0
$838$	41.2944	1.42649
$839$	10.9722	0.378804	0.189402	−	0.981900i	$-0.439345\pi$
$839$	10.9722	0.378804	0.189402	+	0.981900i	$0.439345\pi$
$840$	0	0
$841$	45.0555	1.55364
$842$	7.02776	0.242192
$843$	0	0
$844$	−1.51388	−0.0521098
$845$	0	0
$846$	0	0
$847$	−5.66106	−0.194516
$848$	41.3305	1.41930
$849$	0	0
$850$	0	0
$851$	−34.8167	−1.19350
$852$	0	0
$853$	−35.6611	−1.22101	−0.610506	−	0.792012i	$-0.709034\pi$
$853$	−35.6611	−1.22101	−0.610506	+	0.792012i	$0.709034\pi$
$854$	3.00000	0.102658
$855$	0	0
$856$	−27.0000	−0.922841
$857$	−19.1472	−0.654055	−0.327028	−	0.945015i	$-0.606047\pi$
$857$	−19.1472	−0.654055	−0.327028	+	0.945015i	$0.606047\pi$
$858$	0	0
$859$	51.6611	1.76265	0.881326	−	0.472508i	$-0.156651\pi$
$859$	51.6611	1.76265	0.881326	+	0.472508i	$0.156651\pi$
$860$	0	0
$861$	0	0
$862$	−14.0917	−0.479964
$863$	−45.7527	−1.55744	−0.778721	−	0.627371i	$-0.784131\pi$
$863$	−45.7527	−1.55744	−0.778721	+	0.627371i	$0.784131\pi$
$864$	0	0
$865$	0	0
$866$	−44.6056	−1.51576
$867$	0	0
$868$	1.31118	0.0445043
$869$	−7.81665	−0.265162
$870$	0	0
$871$	5.30278	0.179678
$872$	−3.27502	−0.110906
$873$	0	0
$874$	−36.7166	−1.24196
$875$	0	0
$876$	0	0
$877$	−1.11943	−0.0378004	−0.0189002	−	0.999821i	$-0.506016\pi$
$877$	−1.11943	−0.0378004	−0.0189002	+	0.999821i	$0.506016\pi$
$878$	−41.0555	−1.38556
$879$	0	0
$880$	0	0
$881$	24.3944	0.821870	0.410935	−	0.911665i	$-0.365202\pi$
$881$	24.3944	0.821870	0.410935	+	0.911665i	$0.365202\pi$
$882$	0	0
$883$	−14.1833	−0.477308	−0.238654	−	0.971105i	$-0.576706\pi$
$883$	−14.1833	−0.477308	−0.238654	+	0.971105i	$0.576706\pi$
$884$	−1.30278	−0.0438171
$885$	0	0
$886$	−51.9916	−1.74669
$887$	−22.5416	−0.756874	−0.378437	−	0.925627i	$-0.623538\pi$
$887$	−22.5416	−0.756874	−0.378437	+	0.925627i	$0.623538\pi$
$888$	0	0
$889$	−2.51388	−0.0843128
$890$	0	0
$891$	0	0
$892$	−5.14719	−0.172341
$893$	59.2111	1.98142
$894$	0	0
$895$	0	0
$896$	5.64171	0.188476
$897$	0	0
$898$	35.3305	1.17900
$899$	−53.4500	−1.78266
$900$	0	0
$901$	−16.3028	−0.543124
$902$	−3.75274	−0.124952
$903$	0	0
$904$	23.0917	0.768018
$905$	0	0
$906$	0	0
$907$	−40.7527	−1.35317	−0.676586	−	0.736363i	$-0.736541\pi$
$907$	−40.7527	−1.35317	−0.676586	+	0.736363i	$0.736541\pi$
$908$	5.56939	0.184827
$909$	0	0
$910$	0	0
$911$	16.5778	0.549247	0.274623	−	0.961552i	$-0.411447\pi$
$911$	16.5778	0.549247	0.274623	+	0.961552i	$0.411447\pi$
$912$	0	0
$913$	−15.2750	−0.505529
$914$	−42.2389	−1.39714
$915$	0	0
$916$	−2.85281	−0.0942596
$917$	−13.8167	−0.456266
$918$	0	0
$919$	−0.330532	−0.0109032	−0.00545162	−	0.999985i	$-0.501735\pi$
$919$	−0.330532	−0.0109032	−0.00545162	+	0.999985i	$0.501735\pi$
$920$	0	0
$921$	0	0
$922$	11.3667	0.374342
$923$	38.3305	1.26166
$924$	0	0
$925$	0	0
$926$	29.0917	0.956012
$927$	0	0
$928$	14.6056	0.479451
$929$	36.8722	1.20974	0.604868	−	0.796326i	$-0.293226\pi$
$929$	36.8722	1.20974	0.604868	+	0.796326i	$0.293226\pi$
$930$	0	0
$931$	−46.9722	−1.53945
$932$	−8.72498	−0.285796
$933$	0	0
$934$	44.8444	1.46735
$935$	0	0
$936$	0	0
$937$	−20.5778	−0.672247	−0.336124	−	0.941818i	$-0.609116\pi$
$937$	−20.5778	−0.672247	−0.336124	+	0.941818i	$0.609116\pi$
$938$	1.45837	0.0476173
$939$	0	0
$940$	0	0
$941$	36.5139	1.19032	0.595159	−	0.803608i	$-0.297089\pi$
$941$	36.5139	1.19032	0.595159	+	0.803608i	$0.297089\pi$
$942$	0	0
$943$	−6.63331	−0.216010
$944$	−24.1194	−0.785021
$945$	0	0
$946$	−20.5694	−0.668769
$947$	−13.1833	−0.428401	−0.214201	−	0.976790i	$-0.568715\pi$
$947$	−13.1833	−0.428401	−0.214201	+	0.976790i	$0.568715\pi$
$948$	0	0
$949$	7.90833	0.256715
$950$	0	0
$951$	0	0
$952$	−2.72498	−0.0883171
$953$	−16.4222	−0.531967	−0.265984	−	0.963978i	$-0.585697\pi$
$953$	−16.4222	−0.531967	−0.265984	+	0.963978i	$0.585697\pi$
$954$	0	0
$955$	0	0
$956$	1.33894	0.0433043
$957$	0	0
$958$	−30.0833	−0.971946
$959$	1.45837	0.0470931
$960$	0	0
$961$	7.57779	0.244445
$962$	38.3305	1.23583
$963$	0	0
$964$	−5.81665	−0.187342
$965$	0	0
$966$	0	0
$967$	18.8167	0.605103	0.302551	−	0.953133i	$-0.402162\pi$
$967$	18.8167	0.605103	0.302551	+	0.953133i	$0.402162\pi$
$968$	24.3583	0.782905
$969$	0	0
$970$	0	0
$971$	43.5416	1.39732	0.698659	−	0.715455i	$-0.253781\pi$
$971$	43.5416	1.39732	0.698659	+	0.715455i	$0.253781\pi$
$972$	0	0
$973$	2.02776	0.0650069
$974$	46.4222	1.48746
$975$	0	0
$976$	−10.9083	−0.349167
$977$	9.11943	0.291756	0.145878	−	0.989303i	$-0.453399\pi$
$977$	9.11943	0.291756	0.145878	+	0.989303i	$0.453399\pi$
$978$	0	0
$979$	20.3667	0.650922
$980$	0	0
$981$	0	0
$982$	35.4861	1.13241
$983$	30.6333	0.977051	0.488525	−	0.872550i	$-0.337535\pi$
$983$	30.6333	0.977051	0.488525	+	0.872550i	$0.337535\pi$
$984$	0	0
$985$	0	0
$986$	14.6056	0.465136
$987$	0	0
$988$	−7.21110	−0.229416
$989$	−36.3583	−1.15613
$990$	0	0
$991$	−31.9083	−1.01360	−0.506801	−	0.862063i	$-0.669172\pi$
$991$	−31.9083	−1.01360	−0.506801	+	0.862063i	$0.669172\pi$
$992$	−10.5416	−0.334697
$993$	0	0
$994$	10.5416	0.334360
$995$	0	0
$996$	0	0
$997$	20.2389	0.640971	0.320486	−	0.947253i	$-0.396154\pi$
$997$	20.2389	0.640971	0.320486	+	0.947253i	$0.396154\pi$
$998$	−8.24726	−0.261063
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.i.1.2	yes	2
3.2	odd	2		2025.2.a.l.1.1	yes	2
5.2	odd	4		2025.2.b.i.649.3		4
5.3	odd	4		2025.2.b.i.649.2		4
5.4	even	2		2025.2.a.k.1.1	yes	2
15.2	even	4		2025.2.b.j.649.2		4
15.8	even	4		2025.2.b.j.649.3		4
15.14	odd	2		2025.2.a.h.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2025.2.a.h.1.2	✓	2	15.14	odd	2
2025.2.a.i.1.2	yes	2	1.1	even	1	trivial
2025.2.a.k.1.1	yes	2	5.4	even	2
2025.2.a.l.1.1	yes	2	3.2	odd	2
2025.2.b.i.649.2		4	5.3	odd	4
2025.2.b.i.649.3		4	5.2	odd	4
2025.2.b.j.649.2		4	15.2	even	4
2025.2.b.j.649.3		4	15.8	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)

\(f(q)\)	\(=\)	\(q+1.30278 q^{2} -0.302776 q^{4} +0.697224 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q+1.30278 q^{2} -0.302776 q^{4} +0.697224 q^{7} -3.00000 q^{8} -1.69722 q^{11} +3.30278 q^{13} +0.908327 q^{14} -3.30278 q^{16} +1.30278 q^{17} +7.21110 q^{19} -2.21110 q^{22} -3.90833 q^{23} +4.30278 q^{26} -0.211103 q^{28} +8.60555 q^{29} -6.21110 q^{31} +1.69722 q^{32} +1.69722 q^{34} +8.90833 q^{37} +9.39445 q^{38} +1.69722 q^{41} +9.30278 q^{43} +0.513878 q^{44} -5.09167 q^{46} +8.21110 q^{47} -6.51388 q^{49} -1.00000 q^{52} -12.5139 q^{53} -2.09167 q^{56} +11.2111 q^{58} +7.30278 q^{59} +3.30278 q^{61} -8.09167 q^{62} +8.81665 q^{64} +1.60555 q^{67} -0.394449 q^{68} +11.6056 q^{71} +2.39445 q^{73} +11.6056 q^{74} -2.18335 q^{76} -1.18335 q^{77} +4.60555 q^{79} +2.21110 q^{82} +9.00000 q^{83} +12.1194 q^{86} +5.09167 q^{88} -12.0000 q^{89} +2.30278 q^{91} +1.18335 q^{92} +10.6972 q^{94} +2.00000 q^{97} -8.48612 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8}+O(q^{10}) \) 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

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