LMFDB - Embedded newform 6025.2.a.o.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.568441 q^{2} -1.77259 q^{3} -1.67687 q^{4} +1.00761 q^{6} -2.89075 q^{7} +2.09009 q^{8} +0.142065 q^{9} +O(q^{10})$	\(q-0.568441 q^{2} -1.77259 q^{3} -1.67687 q^{4} +1.00761 q^{6} -2.89075 q^{7} +2.09009 q^{8} +0.142065 q^{9} -2.23192 q^{11} +2.97241 q^{12} +4.54908 q^{13} +1.64322 q^{14} +2.16566 q^{16} +1.59105 q^{17} -0.0807554 q^{18} +7.63341 q^{19} +5.12410 q^{21} +1.26872 q^{22} +5.02583 q^{23} -3.70486 q^{24} -2.58588 q^{26} +5.06594 q^{27} +4.84742 q^{28} +4.68915 q^{29} -0.922532 q^{31} -5.41122 q^{32} +3.95628 q^{33} -0.904415 q^{34} -0.238225 q^{36} -4.28380 q^{37} -4.33914 q^{38} -8.06364 q^{39} -4.67537 q^{41} -2.91275 q^{42} -2.70462 q^{43} +3.74266 q^{44} -2.85689 q^{46} +6.94778 q^{47} -3.83882 q^{48} +1.35641 q^{49} -2.82027 q^{51} -7.62824 q^{52} -10.7509 q^{53} -2.87969 q^{54} -6.04191 q^{56} -13.5309 q^{57} -2.66550 q^{58} -11.9479 q^{59} -15.0001 q^{61} +0.524405 q^{62} -0.410673 q^{63} -1.25536 q^{64} -2.24891 q^{66} +2.34500 q^{67} -2.66798 q^{68} -8.90873 q^{69} -11.1530 q^{71} +0.296927 q^{72} +5.97498 q^{73} +2.43509 q^{74} -12.8003 q^{76} +6.45193 q^{77} +4.58370 q^{78} +11.3464 q^{79} -9.40601 q^{81} +2.65767 q^{82} +17.5096 q^{83} -8.59247 q^{84} +1.53742 q^{86} -8.31192 q^{87} -4.66491 q^{88} +8.70909 q^{89} -13.1502 q^{91} -8.42769 q^{92} +1.63527 q^{93} -3.94940 q^{94} +9.59186 q^{96} +10.4327 q^{97} -0.771041 q^{98} -0.317078 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

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 $

$2$	−0.568441	−0.401948	−0.200974	−	0.979597i	$-0.564411\pi$
$2$	−0.568441	−0.401948	−0.200974	+	0.979597i	$0.564411\pi$
$3$	−1.77259	−1.02340	−0.511702	−	0.859163i	$-0.670985\pi$
$3$	−1.77259	−1.02340	−0.511702	+	0.859163i	$0.670985\pi$
$4$	−1.67687	−0.838437
$5$	0	0
$5$	0	0
$6$	1.00761	0.411355
$7$	−2.89075	−1.09260	−0.546300	−	0.837590i	$-0.683964\pi$
$7$	−2.89075	−1.09260	−0.546300	+	0.837590i	$0.683964\pi$
$8$	2.09009	0.738957
$9$	0.142065	0.0473549
$10$	0	0
$11$	−2.23192	−0.672950	−0.336475	−	0.941692i	$-0.609235\pi$
$11$	−2.23192	−0.672950	−0.336475	+	0.941692i	$0.609235\pi$
$12$	2.97241	0.858060
$13$	4.54908	1.26169	0.630844	−	0.775910i	$-0.282709\pi$
$13$	4.54908	1.26169	0.630844	+	0.775910i	$0.282709\pi$
$14$	1.64322	0.439169
$15$	0	0
$16$	2.16566	0.541415
$17$	1.59105	0.385885	0.192943	−	0.981210i	$-0.438197\pi$
$17$	1.59105	0.385885	0.192943	+	0.981210i	$0.438197\pi$
$18$	−0.0807554	−0.0190342
$19$	7.63341	1.75123	0.875613	−	0.483014i	$-0.160458\pi$
$19$	7.63341	1.75123	0.875613	+	0.483014i	$0.160458\pi$
$20$	0	0
$21$	5.12410	1.11817
$22$	1.26872	0.270491
$23$	5.02583	1.04796	0.523979	−	0.851731i	$-0.324447\pi$
$23$	5.02583	1.04796	0.523979	+	0.851731i	$0.324447\pi$
$24$	−3.70486	−0.756251
$25$	0	0
$26$	−2.58588	−0.507133
$27$	5.06594	0.974940
$28$	4.84742	0.916076
$29$	4.68915	0.870753	0.435376	−	0.900248i	$-0.356615\pi$
$29$	4.68915	0.870753	0.435376	+	0.900248i	$0.356615\pi$
$30$	0	0
$31$	−0.922532	−0.165692	−0.0828458	−	0.996562i	$-0.526401\pi$
$31$	−0.922532	−0.165692	−0.0828458	+	0.996562i	$0.526401\pi$
$32$	−5.41122	−0.956578
$33$	3.95628	0.688700
$34$	−0.904415	−0.155106
$35$	0	0
$36$	−0.238225	−0.0397041
$37$	−4.28380	−0.704253	−0.352126	−	0.935952i	$-0.614541\pi$
$37$	−4.28380	−0.704253	−0.352126	+	0.935952i	$0.614541\pi$
$38$	−4.33914	−0.703902
$39$	−8.06364	−1.29122
$40$	0	0
$41$	−4.67537	−0.730171	−0.365085	−	0.930974i	$-0.618960\pi$
$41$	−4.67537	−0.730171	−0.365085	+	0.930974i	$0.618960\pi$
$42$	−2.91275	−0.449447
$43$	−2.70462	−0.412450	−0.206225	−	0.978505i	$-0.566118\pi$
$43$	−2.70462	−0.412450	−0.206225	+	0.978505i	$0.566118\pi$
$44$	3.74266	0.564227
$45$	0	0
$46$	−2.85689	−0.421225
$47$	6.94778	1.01344	0.506719	−	0.862111i	$-0.330858\pi$
$47$	6.94778	1.01344	0.506719	+	0.862111i	$0.330858\pi$
$48$	−3.83882	−0.554086
$49$	1.35641	0.193773
$50$	0	0
$51$	−2.82027	−0.394916
$52$	−7.62824	−1.05785
$53$	−10.7509	−1.47675	−0.738375	−	0.674391i	$-0.764406\pi$
$53$	−10.7509	−1.47675	−0.738375	+	0.674391i	$0.764406\pi$
$54$	−2.87969	−0.391876
$55$	0	0
$56$	−6.04191	−0.807384
$57$	−13.5309	−1.79221
$58$	−2.66550	−0.349998
$59$	−11.9479	−1.55549	−0.777743	−	0.628582i	$-0.783636\pi$
$59$	−11.9479	−1.55549	−0.777743	+	0.628582i	$0.783636\pi$
$60$	0	0
$61$	−15.0001	−1.92057	−0.960283	−	0.279027i	$-0.909988\pi$
$61$	−15.0001	−1.92057	−0.960283	+	0.279027i	$0.909988\pi$
$62$	0.524405	0.0665995
$63$	−0.410673	−0.0517399
$64$	−1.25536	−0.156920
$65$	0	0
$66$	−2.24891	−0.276822
$67$	2.34500	0.286488	0.143244	−	0.989687i	$-0.454247\pi$
$67$	2.34500	0.286488	0.143244	+	0.989687i	$0.454247\pi$
$68$	−2.66798	−0.323541
$69$	−8.90873	−1.07248
$70$	0	0
$71$	−11.1530	−1.32362	−0.661809	−	0.749672i	$-0.730211\pi$
$71$	−11.1530	−1.32362	−0.661809	+	0.749672i	$0.730211\pi$
$72$	0.296927	0.0349932
$73$	5.97498	0.699318	0.349659	−	0.936877i	$-0.386297\pi$
$73$	5.97498	0.699318	0.349659	+	0.936877i	$0.386297\pi$
$74$	2.43509	0.283073
$75$	0	0
$76$	−12.8003	−1.46829
$77$	6.45193	0.735265
$78$	4.58370	0.519002
$79$	11.3464	1.27657	0.638286	−	0.769800i	$-0.279644\pi$
$79$	11.3464	1.27657	0.638286	+	0.769800i	$0.279644\pi$
$80$	0	0
$81$	−9.40601	−1.04511
$82$	2.65767	0.293491
$83$	17.5096	1.92192	0.960962	−	0.276681i	$-0.0892344\pi$
$83$	17.5096	1.92192	0.960962	+	0.276681i	$0.0892344\pi$
$84$	−8.59247	−0.937516
$85$	0	0
$86$	1.53742	0.165784
$87$	−8.31192	−0.891132
$88$	−4.66491	−0.497281
$89$	8.70909	0.923162	0.461581	−	0.887098i	$-0.347282\pi$
$89$	8.70909	0.923162	0.461581	+	0.887098i	$0.347282\pi$
$90$	0	0
$91$	−13.1502	−1.37852
$92$	−8.42769	−0.878648
$93$	1.63527	0.169569
$94$	−3.94940	−0.407350
$95$	0	0
$96$	9.59186	0.978965
$97$	10.4327	1.05928	0.529638	−	0.848224i	$-0.322328\pi$
$97$	10.4327	1.05928	0.529638	+	0.848224i	$0.322328\pi$
$98$	−0.771041	−0.0778869
$99$	−0.317078	−0.0318675
$100$	0	0
$101$	4.20380	0.418294	0.209147	−	0.977884i	$-0.432931\pi$
$101$	4.20380	0.418294	0.209147	+	0.977884i	$0.432931\pi$
$102$	1.60315	0.158736
$103$	3.69485	0.364065	0.182032	−	0.983293i	$-0.441732\pi$
$103$	3.69485	0.364065	0.182032	+	0.983293i	$0.441732\pi$
$104$	9.50797	0.932333
$105$	0	0
$106$	6.11125	0.593577
$107$	9.36949	0.905783	0.452891	−	0.891566i	$-0.350393\pi$
$107$	9.36949	0.905783	0.452891	+	0.891566i	$0.350393\pi$
$108$	−8.49495	−0.817427
$109$	0.556728	0.0533249	0.0266624	−	0.999644i	$-0.491512\pi$
$109$	0.556728	0.0533249	0.0266624	+	0.999644i	$0.491512\pi$
$110$	0	0
$111$	7.59341	0.720735
$112$	−6.26037	−0.591550
$113$	2.81136	0.264470	0.132235	−	0.991218i	$-0.457785\pi$
$113$	2.81136	0.264470	0.132235	+	0.991218i	$0.457785\pi$
$114$	7.69151	0.720376
$115$	0	0
$116$	−7.86312	−0.730072
$117$	0.646264	0.0597471
$118$	6.79169	0.625226
$119$	−4.59931	−0.421618
$120$	0	0
$121$	−6.01851	−0.547138
$122$	8.52667	0.771969
$123$	8.28751	0.747259
$124$	1.54697	0.138922
$125$	0	0
$126$	0.233443	0.0207968
$127$	−17.5308	−1.55561	−0.777805	−	0.628506i	$-0.783667\pi$
$127$	−17.5308	−1.55561	−0.777805	+	0.628506i	$0.783667\pi$
$128$	11.5360	1.01965
$129$	4.79417	0.422103
$130$	0	0
$131$	4.77092	0.416837	0.208418	−	0.978040i	$-0.433168\pi$
$131$	4.77092	0.416837	0.208418	+	0.978040i	$0.433168\pi$
$132$	−6.63419	−0.577432
$133$	−22.0663	−1.91339
$134$	−1.33300	−0.115153
$135$	0	0
$136$	3.32542	0.285153
$137$	13.4927	1.15276	0.576378	−	0.817183i	$-0.304465\pi$
$137$	13.4927	1.15276	0.576378	+	0.817183i	$0.304465\pi$
$138$	5.06408	0.431084
$139$	−6.90059	−0.585301	−0.292650	−	0.956220i	$-0.594537\pi$
$139$	−6.90059	−0.585301	−0.292650	+	0.956220i	$0.594537\pi$
$140$	0	0
$141$	−12.3155	−1.03716
$142$	6.33983	0.532026
$143$	−10.1532	−0.849053
$144$	0.307664	0.0256386
$145$	0	0
$146$	−3.39642	−0.281090
$147$	−2.40436	−0.198308
$148$	7.18340	0.590472
$149$	1.31147	0.107439	0.0537197	−	0.998556i	$-0.482892\pi$
$149$	1.31147	0.107439	0.0537197	+	0.998556i	$0.482892\pi$
$150$	0	0
$151$	−23.6949	−1.92826	−0.964132	−	0.265424i	$-0.914488\pi$
$151$	−23.6949	−1.92826	−0.964132	+	0.265424i	$0.914488\pi$
$152$	15.9545	1.29408
$153$	0.226031	0.0182736
$154$	−3.66754	−0.295539
$155$	0	0
$156$	13.5217	1.08260
$157$	0.931398	0.0743337	0.0371668	−	0.999309i	$-0.488167\pi$
$157$	0.931398	0.0743337	0.0371668	+	0.999309i	$0.488167\pi$
$158$	−6.44977	−0.513116
$159$	19.0569	1.51131
$160$	0	0
$161$	−14.5284	−1.14500
$162$	5.34676	0.420081
$163$	14.5382	1.13872	0.569360	−	0.822088i	$-0.307191\pi$
$163$	14.5382	1.13872	0.569360	+	0.822088i	$0.307191\pi$
$164$	7.84002	0.612202
$165$	0	0
$166$	−9.95315	−0.772514
$167$	6.80462	0.526557	0.263279	−	0.964720i	$-0.415196\pi$
$167$	6.80462	0.526557	0.263279	+	0.964720i	$0.415196\pi$
$168$	10.7098	0.826280
$169$	7.69413	0.591856
$170$	0	0
$171$	1.08444	0.0829291
$172$	4.53531	0.345814
$173$	11.3260	0.861102	0.430551	−	0.902566i	$-0.358319\pi$
$173$	11.3260	0.861102	0.430551	+	0.902566i	$0.358319\pi$
$174$	4.72484	0.358189
$175$	0	0
$176$	−4.83359	−0.364345
$177$	21.1787	1.59189
$178$	−4.95060	−0.371063
$179$	−20.2148	−1.51093	−0.755464	−	0.655191i	$-0.772588\pi$
$179$	−20.2148	−1.51093	−0.755464	+	0.655191i	$0.772588\pi$
$180$	0	0
$181$	16.3792	1.21745	0.608727	−	0.793380i	$-0.291681\pi$
$181$	16.3792	1.21745	0.608727	+	0.793380i	$0.291681\pi$
$182$	7.47513	0.554094
$183$	26.5890	1.96551
$184$	10.5044	0.774396
$185$	0	0
$186$	−0.929554	−0.0681582
$187$	−3.55109	−0.259682
$188$	−11.6506	−0.849704
$189$	−14.6443	−1.06522
$190$	0	0
$191$	−11.4017	−0.825000	−0.412500	−	0.910958i	$-0.635344\pi$
$191$	−11.4017	−0.825000	−0.412500	+	0.910958i	$0.635344\pi$
$192$	2.22523	0.160592
$193$	−24.8394	−1.78798	−0.893989	−	0.448090i	$-0.852104\pi$
$193$	−24.8394	−1.78798	−0.893989	+	0.448090i	$0.852104\pi$
$194$	−5.93035	−0.425775
$195$	0	0
$196$	−2.27454	−0.162467
$197$	17.6572	1.25802	0.629010	−	0.777397i	$-0.283460\pi$
$197$	17.6572	1.25802	0.629010	+	0.777397i	$0.283460\pi$
$198$	0.180240	0.0128091
$199$	21.3701	1.51488	0.757441	−	0.652903i	$-0.226449\pi$
$199$	21.3701	1.51488	0.757441	+	0.652903i	$0.226449\pi$
$200$	0	0
$201$	−4.15672	−0.293193
$202$	−2.38961	−0.168133
$203$	−13.5551	−0.951384
$204$	4.72923	0.331113
$205$	0	0
$206$	−2.10031	−0.146335
$207$	0.713993	0.0496260
$208$	9.85176	0.683097
$209$	−17.0372	−1.17849
$210$	0	0
$211$	9.78312	0.673498	0.336749	−	0.941595i	$-0.390673\pi$
$211$	9.78312	0.673498	0.336749	+	0.941595i	$0.390673\pi$
$212$	18.0279	1.23816
$213$	19.7697	1.35460
$214$	−5.32600	−0.364078
$215$	0	0
$216$	10.5882	0.720439
$217$	2.66681	0.181035
$218$	−0.316467	−0.0214338
$219$	−10.5912	−0.715684
$220$	0	0
$221$	7.23779	0.486867
$222$	−4.31641	−0.289698
$223$	6.19052	0.414548	0.207274	−	0.978283i	$-0.433541\pi$
$223$	6.19052	0.414548	0.207274	+	0.978283i	$0.433541\pi$
$224$	15.6425	1.04516
$225$	0	0
$226$	−1.59809	−0.106303
$227$	−28.1701	−1.86972	−0.934859	−	0.355019i	$-0.884474\pi$
$227$	−28.1701	−1.86972	−0.934859	+	0.355019i	$0.884474\pi$
$228$	22.6896	1.50266
$229$	20.2029	1.33504	0.667522	−	0.744590i	$-0.267355\pi$
$229$	20.2029	1.33504	0.667522	+	0.744590i	$0.267355\pi$
$230$	0	0
$231$	−11.4366	−0.752473
$232$	9.80072	0.643449
$233$	−25.1668	−1.64873	−0.824366	−	0.566058i	$-0.808468\pi$
$233$	−25.1668	−1.64873	−0.824366	+	0.566058i	$0.808468\pi$
$234$	−0.367363	−0.0240152
$235$	0	0
$236$	20.0352	1.30418
$237$	−20.1125	−1.30645
$238$	2.61444	0.169469
$239$	−4.50795	−0.291595	−0.145797	−	0.989314i	$-0.546575\pi$
$239$	−4.50795	−0.291595	−0.145797	+	0.989314i	$0.546575\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	3.42117	0.219921
$243$	1.47516	0.0946314
$244$	25.1533	1.61027
$245$	0	0
$246$	−4.71096	−0.300360
$247$	34.7250	2.20950
$248$	−1.92817	−0.122439
$249$	−31.0372	−1.96690
$250$	0	0
$251$	16.7571	1.05770	0.528849	−	0.848716i	$-0.322624\pi$
$251$	16.7571	1.05770	0.528849	+	0.848716i	$0.322624\pi$
$252$	0.688647	0.0433807
$253$	−11.2173	−0.705224
$254$	9.96524	0.625275
$255$	0	0
$256$	−4.04684	−0.252927
$257$	−28.9757	−1.80746	−0.903728	−	0.428107i	$-0.859180\pi$
$257$	−28.9757	−1.80746	−0.903728	+	0.428107i	$0.859180\pi$
$258$	−2.72520	−0.169664
$259$	12.3834	0.769466
$260$	0	0
$261$	0.666162	0.0412344
$262$	−2.71198	−0.167547
$263$	7.48205	0.461363	0.230682	−	0.973029i	$-0.425904\pi$
$263$	7.48205	0.461363	0.230682	+	0.973029i	$0.425904\pi$
$264$	8.26897	0.508920
$265$	0	0
$266$	12.5434	0.769083
$267$	−15.4376	−0.944767
$268$	−3.93228	−0.240202
$269$	−24.3384	−1.48394	−0.741968	−	0.670435i	$-0.766108\pi$
$269$	−24.3384	−1.48394	−0.741968	+	0.670435i	$0.766108\pi$
$270$	0	0
$271$	16.2435	0.986725	0.493363	−	0.869824i	$-0.335768\pi$
$271$	16.2435	0.986725	0.493363	+	0.869824i	$0.335768\pi$
$272$	3.44566	0.208924
$273$	23.3099	1.41078
$274$	−7.66979	−0.463349
$275$	0	0
$276$	14.9388	0.899211
$277$	−0.333460	−0.0200356	−0.0100178	−	0.999950i	$-0.503189\pi$
$277$	−0.333460	−0.0200356	−0.0100178	+	0.999950i	$0.503189\pi$
$278$	3.92258	0.235261
$279$	−0.131059	−0.00784631
$280$	0	0
$281$	−6.13116	−0.365754	−0.182877	−	0.983136i	$-0.558541\pi$
$281$	−6.13116	−0.365754	−0.182877	+	0.983136i	$0.558541\pi$
$282$	7.00066	0.416883
$283$	5.83443	0.346821	0.173410	−	0.984850i	$-0.444521\pi$
$283$	5.83443	0.346821	0.173410	+	0.984850i	$0.444521\pi$
$284$	18.7022	1.10977
$285$	0	0
$286$	5.77150	0.341276
$287$	13.5153	0.797784
$288$	−0.768743	−0.0452986
$289$	−14.4686	−0.851093
$290$	0	0
$291$	−18.4928	−1.08407
$292$	−10.0193	−0.586334
$293$	19.8347	1.15876	0.579378	−	0.815059i	$-0.303295\pi$
$293$	19.8347	1.15876	0.579378	+	0.815059i	$0.303295\pi$
$294$	1.36674	0.0797097
$295$	0	0
$296$	−8.95352	−0.520413
$297$	−11.3068	−0.656087
$298$	−0.745491	−0.0431851
$299$	22.8629	1.32220
$300$	0	0
$301$	7.81836	0.450643
$302$	13.4691	0.775062
$303$	−7.45160	−0.428083
$304$	16.5314	0.948139
$305$	0	0
$306$	−0.128485	−0.00734503
$307$	−4.34370	−0.247908	−0.123954	−	0.992288i	$-0.539558\pi$
$307$	−4.34370	−0.247908	−0.123954	+	0.992288i	$0.539558\pi$
$308$	−10.8191	−0.616474
$309$	−6.54945	−0.372585
$310$	0	0
$311$	28.6558	1.62492	0.812461	−	0.583016i	$-0.198127\pi$
$311$	28.6558	1.62492	0.812461	+	0.583016i	$0.198127\pi$
$312$	−16.8537	−0.954153
$313$	1.92456	0.108783	0.0543914	−	0.998520i	$-0.482678\pi$
$313$	1.92456	0.108783	0.0543914	+	0.998520i	$0.482678\pi$
$314$	−0.529445	−0.0298783
$315$	0	0
$316$	−19.0265	−1.07033
$317$	−17.8922	−1.00493	−0.502464	−	0.864598i	$-0.667573\pi$
$317$	−17.8922	−1.00493	−0.502464	+	0.864598i	$0.667573\pi$
$318$	−10.8327	−0.607469
$319$	−10.4658	−0.585974
$320$	0	0
$321$	−16.6082	−0.926982
$322$	8.25854	0.460231
$323$	12.1451	0.675772
$324$	15.7727	0.876261
$325$	0	0
$326$	−8.26411	−0.457707
$327$	−0.986849	−0.0545729
$328$	−9.77193	−0.539565
$329$	−20.0843	−1.10728
$330$	0	0
$331$	−13.9659	−0.767633	−0.383817	−	0.923409i	$-0.625391\pi$
$331$	−13.9659	−0.767633	−0.383817	+	0.923409i	$0.625391\pi$
$332$	−29.3613	−1.61141
$333$	−0.608577	−0.0333498
$334$	−3.86802	−0.211649
$335$	0	0
$336$	11.0971	0.605394
$337$	9.30104	0.506660	0.253330	−	0.967380i	$-0.418474\pi$
$337$	9.30104	0.506660	0.253330	+	0.967380i	$0.418474\pi$
$338$	−4.37366	−0.237896
$339$	−4.98338	−0.270660
$340$	0	0
$341$	2.05902	0.111502
$342$	−0.616439	−0.0333332
$343$	16.3142	0.880883
$344$	−5.65288	−0.304783
$345$	0	0
$346$	−6.43818	−0.346119
$347$	13.4461	0.721826	0.360913	−	0.932599i	$-0.382465\pi$
$347$	13.4461	0.721826	0.360913	+	0.932599i	$0.382465\pi$
$348$	13.9381	0.747158
$349$	−18.3481	−0.982149	−0.491074	−	0.871118i	$-0.663396\pi$
$349$	−18.3481	−0.982149	−0.491074	+	0.871118i	$0.663396\pi$
$350$	0	0
$351$	23.0454	1.23007
$352$	12.0774	0.643730
$353$	15.1926	0.808619	0.404310	−	0.914622i	$-0.367512\pi$
$353$	15.1926	0.808619	0.404310	+	0.914622i	$0.367512\pi$
$354$	−12.0389	−0.639858
$355$	0	0
$356$	−14.6041	−0.774013
$357$	8.15267	0.431485
$358$	11.4909	0.607315
$359$	19.4551	1.02680	0.513400	−	0.858150i	$-0.328386\pi$
$359$	19.4551	1.02680	0.513400	+	0.858150i	$0.328386\pi$
$360$	0	0
$361$	39.2690	2.06679
$362$	−9.31058	−0.489353
$363$	10.6683	0.559943
$364$	22.0513	1.15580
$365$	0	0
$366$	−15.1143	−0.790035
$367$	−11.7135	−0.611438	−0.305719	−	0.952122i	$-0.598897\pi$
$367$	−11.7135	−0.611438	−0.305719	+	0.952122i	$0.598897\pi$
$368$	10.8842	0.567380
$369$	−0.664205	−0.0345771
$370$	0	0
$371$	31.0781	1.61350
$372$	−2.74214	−0.142173
$373$	24.8893	1.28872	0.644358	−	0.764724i	$-0.277124\pi$
$373$	24.8893	1.28872	0.644358	+	0.764724i	$0.277124\pi$
$374$	2.01859	0.104379
$375$	0	0
$376$	14.5215	0.748887
$377$	21.3313	1.09862
$378$	8.32444	0.428163
$379$	30.6521	1.57449	0.787247	−	0.616638i	$-0.211506\pi$
$379$	30.6521	1.57449	0.787247	+	0.616638i	$0.211506\pi$
$380$	0	0
$381$	31.0749	1.59202
$382$	6.48121	0.331608
$383$	10.7802	0.550840	0.275420	−	0.961324i	$-0.411183\pi$
$383$	10.7802	0.550840	0.275420	+	0.961324i	$0.411183\pi$
$384$	−20.4486	−1.04352
$385$	0	0
$386$	14.1197	0.718675
$387$	−0.384231	−0.0195315
$388$	−17.4943	−0.888137
$389$	10.6262	0.538769	0.269385	−	0.963033i	$-0.413180\pi$
$389$	10.6262	0.538769	0.269385	+	0.963033i	$0.413180\pi$
$390$	0	0
$391$	7.99633	0.404392
$392$	2.83502	0.143190
$393$	−8.45686	−0.426592
$394$	−10.0371	−0.505659
$395$	0	0
$396$	0.531699	0.0267189
$397$	−27.1704	−1.36364	−0.681821	−	0.731519i	$-0.738812\pi$
$397$	−27.1704	−1.36364	−0.681821	+	0.731519i	$0.738812\pi$
$398$	−12.1476	−0.608905
$399$	39.1144	1.95817
$400$	0	0
$401$	−3.92491	−0.196001	−0.0980003	−	0.995186i	$-0.531245\pi$
$401$	−3.92491	−0.196001	−0.0980003	+	0.995186i	$0.531245\pi$
$402$	2.36285	0.117848
$403$	−4.19667	−0.209051
$404$	−7.04925	−0.350713
$405$	0	0
$406$	7.70529	0.382407
$407$	9.56112	0.473927
$408$	−5.89460	−0.291826
$409$	−16.1199	−0.797080	−0.398540	−	0.917151i	$-0.630483\pi$
$409$	−16.1199	−0.797080	−0.398540	+	0.917151i	$0.630483\pi$
$410$	0	0
$411$	−23.9169	−1.17974
$412$	−6.19581	−0.305246
$413$	34.5384	1.69952
$414$	−0.405863	−0.0199471
$415$	0	0
$416$	−24.6161	−1.20690
$417$	12.2319	0.598999
$418$	9.68464	0.473691
$419$	9.67629	0.472718	0.236359	−	0.971666i	$-0.424046\pi$
$419$	9.67629	0.472718	0.236359	+	0.971666i	$0.424046\pi$
$420$	0	0
$421$	−28.4866	−1.38835	−0.694177	−	0.719805i	$-0.744231\pi$
$421$	−28.4866	−1.38835	−0.694177	+	0.719805i	$0.744231\pi$
$422$	−5.56113	−0.270711
$423$	0.987034	0.0479912
$424$	−22.4703	−1.09125
$425$	0	0
$426$	−11.2379	−0.544478
$427$	43.3615	2.09841
$428$	−15.7115	−0.759442
$429$	17.9974	0.868924
$430$	0	0
$431$	−2.47201	−0.119072	−0.0595362	−	0.998226i	$-0.518962\pi$
$431$	−2.47201	−0.119072	−0.0595362	+	0.998226i	$0.518962\pi$
$432$	10.9711	0.527847
$433$	−31.2957	−1.50398	−0.751988	−	0.659177i	$-0.770905\pi$
$433$	−31.2957	−1.50398	−0.751988	+	0.659177i	$0.770905\pi$
$434$	−1.51592	−0.0727666
$435$	0	0
$436$	−0.933563	−0.0447096
$437$	38.3643	1.83521
$438$	6.02045	0.287668
$439$	27.0431	1.29070	0.645349	−	0.763888i	$-0.276712\pi$
$439$	27.0431	1.29070	0.645349	+	0.763888i	$0.276712\pi$
$440$	0	0
$441$	0.192698	0.00917612
$442$	−4.11426	−0.195695
$443$	18.2060	0.864995	0.432497	−	0.901635i	$-0.357632\pi$
$443$	18.2060	0.864995	0.432497	+	0.901635i	$0.357632\pi$
$444$	−12.7332	−0.604291
$445$	0	0
$446$	−3.51895	−0.166627
$447$	−2.32469	−0.109954
$448$	3.62892	0.171451
$449$	−20.1640	−0.951599	−0.475799	−	0.879554i	$-0.657841\pi$
$449$	−20.1640	−0.951599	−0.475799	+	0.879554i	$0.657841\pi$
$450$	0	0
$451$	10.4351	0.491369
$452$	−4.71430	−0.221742
$453$	42.0013	1.97339
$454$	16.0131	0.751530
$455$	0	0
$456$	−28.2807	−1.32437
$457$	−33.2854	−1.55703	−0.778513	−	0.627628i	$-0.784026\pi$
$457$	−33.2854	−1.55703	−0.778513	+	0.627628i	$0.784026\pi$
$458$	−11.4841	−0.536619
$459$	8.06014	0.376215
$460$	0	0
$461$	−23.9462	−1.11529	−0.557644	−	0.830080i	$-0.688294\pi$
$461$	−23.9462	−1.11529	−0.557644	+	0.830080i	$0.688294\pi$
$462$	6.50103	0.302455
$463$	21.6766	1.00740	0.503698	−	0.863880i	$-0.331972\pi$
$463$	21.6766	1.00740	0.503698	+	0.863880i	$0.331972\pi$
$464$	10.1551	0.471439
$465$	0	0
$466$	14.3058	0.662705
$467$	−18.9628	−0.877492	−0.438746	−	0.898611i	$-0.644577\pi$
$467$	−18.9628	−0.877492	−0.438746	+	0.898611i	$0.644577\pi$
$468$	−1.08370	−0.0500942
$469$	−6.77881	−0.313016
$470$	0	0
$471$	−1.65098	−0.0760733
$472$	−24.9722	−1.14944
$473$	6.03650	0.277559
$474$	11.4328	0.525125
$475$	0	0
$476$	7.71247	0.353500
$477$	−1.52732	−0.0699313
$478$	2.56250	0.117206
$479$	6.74368	0.308126	0.154063	−	0.988061i	$-0.450764\pi$
$479$	6.74368	0.308126	0.154063	+	0.988061i	$0.450764\pi$
$480$	0	0
$481$	−19.4874	−0.888547
$482$	0.568441	0.0258918
$483$	25.7529	1.17180
$484$	10.0923	0.458741
$485$	0	0
$486$	−0.838540	−0.0380369
$487$	−21.8176	−0.988650	−0.494325	−	0.869277i	$-0.664585\pi$
$487$	−21.8176	−0.988650	−0.494325	+	0.869277i	$0.664585\pi$
$488$	−31.3515	−1.41922
$489$	−25.7702	−1.16537
$490$	0	0
$491$	−3.83334	−0.172996	−0.0864981	−	0.996252i	$-0.527568\pi$
$491$	−3.83334	−0.172996	−0.0864981	+	0.996252i	$0.527568\pi$
$492$	−13.8971	−0.626530
$493$	7.46065	0.336011
$494$	−19.7391	−0.888105
$495$	0	0
$496$	−1.99789	−0.0897079
$497$	32.2405	1.44618
$498$	17.6428	0.790594
$499$	−32.4901	−1.45446	−0.727228	−	0.686396i	$-0.759192\pi$
$499$	−32.4901	−1.45446	−0.727228	+	0.686396i	$0.759192\pi$
$500$	0	0
$501$	−12.0618	−0.538881
$502$	−9.52541	−0.425140
$503$	12.5298	0.558674	0.279337	−	0.960193i	$-0.409885\pi$
$503$	12.5298	0.558674	0.279337	+	0.960193i	$0.409885\pi$
$504$	−0.858342	−0.0382336
$505$	0	0
$506$	6.37636	0.283464
$507$	−13.6385	−0.605708
$508$	29.3970	1.30428
$509$	−14.4691	−0.641331	−0.320666	−	0.947192i	$-0.603907\pi$
$509$	−14.4691	−0.641331	−0.320666	+	0.947192i	$0.603907\pi$
$510$	0	0
$511$	−17.2721	−0.764074
$512$	−20.7717	−0.917988
$513$	38.6704	1.70734
$514$	16.4710	0.726504
$515$	0	0
$516$	−8.03922	−0.353907
$517$	−15.5069	−0.681994
$518$	−7.03922	−0.309286
$519$	−20.0764	−0.881255
$520$	0	0
$521$	9.37255	0.410619	0.205309	−	0.978697i	$-0.434180\pi$
$521$	9.37255	0.410619	0.205309	+	0.978697i	$0.434180\pi$
$522$	−0.378674	−0.0165741
$523$	41.1945	1.80131	0.900656	−	0.434533i	$-0.143087\pi$
$523$	41.1945	1.80131	0.900656	+	0.434533i	$0.143087\pi$
$524$	−8.00023	−0.349492
$525$	0	0
$526$	−4.25310	−0.185444
$527$	−1.46779	−0.0639380
$528$	8.56795	0.372872
$529$	2.25900	0.0982175
$530$	0	0
$531$	−1.69738	−0.0736599
$532$	37.0024	1.60426
$533$	−21.2686	−0.921247
$534$	8.77537	0.379748
$535$	0	0
$536$	4.90126	0.211702
$537$	35.8325	1.54629
$538$	13.8349	0.596466
$539$	−3.02741	−0.130400
$540$	0	0
$541$	22.2675	0.957355	0.478678	−	0.877991i	$-0.341116\pi$
$541$	22.2675	0.957355	0.478678	+	0.877991i	$0.341116\pi$
$542$	−9.23350	−0.396613
$543$	−29.0335	−1.24595
$544$	−8.60950	−0.369129
$545$	0	0
$546$	−13.2503	−0.567061
$547$	−4.53586	−0.193939	−0.0969696	−	0.995287i	$-0.530915\pi$
$547$	−4.53586	−0.193939	−0.0969696	+	0.995287i	$0.530915\pi$
$548$	−22.6255	−0.966514
$549$	−2.13098	−0.0909482
$550$	0	0
$551$	35.7942	1.52488
$552$	−18.6200	−0.792520
$553$	−32.7996	−1.39478
$554$	0.189552	0.00805330
$555$	0	0
$556$	11.5714	0.490738
$557$	−16.1310	−0.683492	−0.341746	−	0.939792i	$-0.611018\pi$
$557$	−16.1310	−0.683492	−0.341746	+	0.939792i	$0.611018\pi$
$558$	0.0744994	0.00315381
$559$	−12.3035	−0.520384
$560$	0	0
$561$	6.29462	0.265759
$562$	3.48520	0.147014
$563$	−7.39442	−0.311638	−0.155819	−	0.987786i	$-0.549802\pi$
$563$	−7.39442	−0.311638	−0.155819	+	0.987786i	$0.549802\pi$
$564$	20.6516	0.869591
$565$	0	0
$566$	−3.31653	−0.139404
$567$	27.1904	1.14189
$568$	−23.3107	−0.978097
$569$	−40.7574	−1.70864	−0.854319	−	0.519750i	$-0.826025\pi$
$569$	−40.7574	−1.70864	−0.854319	+	0.519750i	$0.826025\pi$
$570$	0	0
$571$	17.3835	0.727475	0.363737	−	0.931502i	$-0.381501\pi$
$571$	17.3835	0.727475	0.363737	+	0.931502i	$0.381501\pi$
$572$	17.0257	0.711878
$573$	20.2106	0.844308
$574$	−7.68266	−0.320668
$575$	0	0
$576$	−0.178342	−0.00743092
$577$	37.2698	1.55156	0.775781	−	0.631002i	$-0.217356\pi$
$577$	37.2698	1.55156	0.775781	+	0.631002i	$0.217356\pi$
$578$	8.22453	0.342095
$579$	44.0299	1.82982
$580$	0	0
$581$	−50.6157	−2.09989
$582$	10.5121	0.435739
$583$	23.9952	0.993779
$584$	12.4882	0.516766
$585$	0	0
$586$	−11.2749	−0.465760
$587$	4.79844	0.198053	0.0990263	−	0.995085i	$-0.468427\pi$
$587$	4.79844	0.198053	0.0990263	+	0.995085i	$0.468427\pi$
$588$	4.03181	0.166269
$589$	−7.04207	−0.290163
$590$	0	0
$591$	−31.2989	−1.28746
$592$	−9.27726	−0.381293
$593$	40.0832	1.64602	0.823009	−	0.568028i	$-0.192293\pi$
$593$	40.0832	1.64602	0.823009	+	0.568028i	$0.192293\pi$
$594$	6.42724	0.263713
$595$	0	0
$596$	−2.19916	−0.0900813
$597$	−37.8803	−1.55034
$598$	−12.9962	−0.531455
$599$	−2.01706	−0.0824147	−0.0412073	−	0.999151i	$-0.513120\pi$
$599$	−2.01706	−0.0824147	−0.0412073	+	0.999151i	$0.513120\pi$
$600$	0	0
$601$	45.3262	1.84889	0.924446	−	0.381313i	$-0.124528\pi$
$601$	45.3262	1.84889	0.924446	+	0.381313i	$0.124528\pi$
$602$	−4.44428	−0.181135
$603$	0.333142	0.0135666
$604$	39.7334	1.61673
$605$	0	0
$606$	4.23579	0.172067
$607$	3.87231	0.157172	0.0785861	−	0.996907i	$-0.474959\pi$
$607$	3.87231	0.157172	0.0785861	+	0.996907i	$0.474959\pi$
$608$	−41.3061	−1.67518
$609$	24.0277	0.973650
$610$	0	0
$611$	31.6060	1.27864
$612$	−0.379026	−0.0153212
$613$	33.2275	1.34205	0.671023	−	0.741436i	$-0.265855\pi$
$613$	33.2275	1.34205	0.671023	+	0.741436i	$0.265855\pi$
$614$	2.46914	0.0996463
$615$	0	0
$616$	13.4851	0.543329
$617$	10.8343	0.436171	0.218086	−	0.975930i	$-0.430019\pi$
$617$	10.8343	0.436171	0.218086	+	0.975930i	$0.430019\pi$
$618$	3.72298	0.149760
$619$	24.8457	0.998634	0.499317	−	0.866419i	$-0.333584\pi$
$619$	24.8457	0.998634	0.499317	+	0.866419i	$0.333584\pi$
$620$	0	0
$621$	25.4606	1.02170
$622$	−16.2891	−0.653135
$623$	−25.1758	−1.00865
$624$	−17.4631	−0.699083
$625$	0	0
$626$	−1.09400	−0.0437251
$627$	30.1999	1.20607
$628$	−1.56184	−0.0623241
$629$	−6.81572	−0.271761
$630$	0	0
$631$	0.100786	0.00401224	0.00200612	−	0.999998i	$-0.499361\pi$
$631$	0.100786	0.00401224	0.00200612	+	0.999998i	$0.499361\pi$
$632$	23.7150	0.943331
$633$	−17.3414	−0.689260
$634$	10.1707	0.403929
$635$	0	0
$636$	−31.9560	−1.26714
$637$	6.17043	0.244482
$638$	5.94920	0.235531
$639$	−1.58445	−0.0626798
$640$	0	0
$641$	34.2836	1.35412	0.677061	−	0.735927i	$-0.263253\pi$
$641$	34.2836	1.35412	0.677061	+	0.735927i	$0.263253\pi$
$642$	9.44080	0.372599
$643$	17.7413	0.699647	0.349823	−	0.936816i	$-0.386242\pi$
$643$	17.7413	0.699647	0.349823	+	0.936816i	$0.386242\pi$
$644$	24.3623	0.960010
$645$	0	0
$646$	−6.90378	−0.271625
$647$	15.7198	0.618008	0.309004	−	0.951061i	$-0.400004\pi$
$647$	15.7198	0.618008	0.309004	+	0.951061i	$0.400004\pi$
$648$	−19.6594	−0.772293
$649$	26.6669	1.04677
$650$	0	0
$651$	−4.72715	−0.185271
$652$	−24.3788	−0.954746
$653$	−19.8665	−0.777437	−0.388718	−	0.921357i	$-0.627082\pi$
$653$	−19.8665	−0.777437	−0.388718	+	0.921357i	$0.627082\pi$
$654$	0.560965	0.0219355
$655$	0	0
$656$	−10.1253	−0.395325
$657$	0.848833	0.0331161
$658$	11.4167	0.445070
$659$	21.2040	0.825992	0.412996	−	0.910733i	$-0.364482\pi$
$659$	21.2040	0.825992	0.412996	+	0.910733i	$0.364482\pi$
$660$	0	0
$661$	−22.4937	−0.874903	−0.437451	−	0.899242i	$-0.644119\pi$
$661$	−22.4937	−0.874903	−0.437451	+	0.899242i	$0.644119\pi$
$662$	7.93877	0.308549
$663$	−12.8296	−0.498261
$664$	36.5965	1.42022
$665$	0	0
$666$	0.345940	0.0134049
$667$	23.5669	0.912513
$668$	−11.4105	−0.441485
$669$	−10.9732	−0.424250
$670$	0	0
$671$	33.4791	1.29245
$672$	−27.7276	−1.06962
$673$	−47.4421	−1.82876	−0.914379	−	0.404858i	$-0.867321\pi$
$673$	−47.4421	−1.82876	−0.914379	+	0.404858i	$0.867321\pi$
$674$	−5.28709	−0.203651
$675$	0	0
$676$	−12.9021	−0.496234
$677$	6.72870	0.258605	0.129302	−	0.991605i	$-0.458726\pi$
$677$	6.72870	0.258605	0.129302	+	0.991605i	$0.458726\pi$
$678$	2.83276	0.108791
$679$	−30.1582	−1.15737
$680$	0	0
$681$	49.9340	1.91348
$682$	−1.17043	−0.0448182
$683$	−24.3017	−0.929877	−0.464939	−	0.885343i	$-0.653924\pi$
$683$	−24.3017	−0.929877	−0.464939	+	0.885343i	$0.653924\pi$
$684$	−1.81847	−0.0695308
$685$	0	0
$686$	−9.27364	−0.354069
$687$	−35.8114	−1.36629
$688$	−5.85728	−0.223307
$689$	−48.9067	−1.86320
$690$	0	0
$691$	19.1318	0.727808	0.363904	−	0.931436i	$-0.381444\pi$
$691$	19.1318	0.727808	0.363904	+	0.931436i	$0.381444\pi$
$692$	−18.9923	−0.721980
$693$	0.916591	0.0348184
$694$	−7.64333	−0.290137
$695$	0	0
$696$	−17.3726	−0.658508
$697$	−7.43873	−0.281762
$698$	10.4298	0.394773
$699$	44.6103	1.68732
$700$	0	0
$701$	16.3145	0.616190	0.308095	−	0.951356i	$-0.400308\pi$
$701$	16.3145	0.616190	0.308095	+	0.951356i	$0.400308\pi$
$702$	−13.0999	−0.494425
$703$	−32.7000	−1.23331
$704$	2.80187	0.105599
$705$	0	0
$706$	−8.63608	−0.325023
$707$	−12.1521	−0.457027
$708$	−35.5141	−1.33470
$709$	21.7993	0.818691	0.409346	−	0.912379i	$-0.365757\pi$
$709$	21.7993	0.818691	0.409346	+	0.912379i	$0.365757\pi$
$710$	0	0
$711$	1.61192	0.0604519
$712$	18.2027	0.682177
$713$	−4.63649	−0.173638
$714$	−4.63431	−0.173435
$715$	0	0
$716$	33.8977	1.26682
$717$	7.99073	0.298419
$718$	−11.0591	−0.412720
$719$	38.1750	1.42369	0.711844	−	0.702337i	$-0.247860\pi$
$719$	38.1750	1.42369	0.711844	+	0.702337i	$0.247860\pi$
$720$	0	0
$721$	−10.6809	−0.397777
$722$	−22.3221	−0.830743
$723$	1.77259	0.0659232
$724$	−27.4658	−1.02076
$725$	0	0
$726$	−6.06432	−0.225068
$727$	−14.7414	−0.546727	−0.273363	−	0.961911i	$-0.588136\pi$
$727$	−14.7414	−0.546727	−0.273363	+	0.961911i	$0.588136\pi$
$728$	−27.4851	−1.01867
$729$	25.6032	0.948266
$730$	0	0
$731$	−4.30317	−0.159159
$732$	−44.5864	−1.64796
$733$	6.40298	0.236499	0.118250	−	0.992984i	$-0.462272\pi$
$733$	6.40298	0.236499	0.118250	+	0.992984i	$0.462272\pi$
$734$	6.65842	0.245767
$735$	0	0
$736$	−27.1959	−1.00245
$737$	−5.23387	−0.192792
$738$	0.377561	0.0138982
$739$	28.7087	1.05607	0.528034	−	0.849223i	$-0.322929\pi$
$739$	28.7087	1.05607	0.528034	+	0.849223i	$0.322929\pi$
$740$	0	0
$741$	−61.5531	−2.26121
$742$	−17.6661	−0.648542
$743$	11.4084	0.418535	0.209268	−	0.977858i	$-0.432892\pi$
$743$	11.4084	0.418535	0.209268	+	0.977858i	$0.432892\pi$
$744$	3.41785	0.125305
$745$	0	0
$746$	−14.1481	−0.517998
$747$	2.48749	0.0910125
$748$	5.95474	0.217727
$749$	−27.0848	−0.989658
$750$	0	0
$751$	−22.3682	−0.816228	−0.408114	−	0.912931i	$-0.633813\pi$
$751$	−22.3682	−0.816228	−0.408114	+	0.912931i	$0.633813\pi$
$752$	15.0465	0.548690
$753$	−29.7034	−1.08245
$754$	−12.1256	−0.441588
$755$	0	0
$756$	24.5567	0.893120
$757$	15.8277	0.575266	0.287633	−	0.957741i	$-0.407132\pi$
$757$	15.8277	0.575266	0.287633	+	0.957741i	$0.407132\pi$
$758$	−17.4239	−0.632865
$759$	19.8836	0.721729
$760$	0	0
$761$	9.62041	0.348739	0.174370	−	0.984680i	$-0.444211\pi$
$761$	9.62041	0.348739	0.174370	+	0.984680i	$0.444211\pi$
$762$	−17.6643	−0.639909
$763$	−1.60936	−0.0582627
$764$	19.1193	0.691711
$765$	0	0
$766$	−6.12788	−0.221409
$767$	−54.3521	−1.96254
$768$	7.17337	0.258847
$769$	14.2096	0.512413	0.256206	−	0.966622i	$-0.417527\pi$
$769$	14.2096	0.512413	0.256206	+	0.966622i	$0.417527\pi$
$770$	0	0
$771$	51.3620	1.84976
$772$	41.6525	1.49911
$773$	22.8239	0.820919	0.410460	−	0.911879i	$-0.365368\pi$
$773$	22.8239	0.820919	0.410460	+	0.911879i	$0.365368\pi$
$774$	0.218412	0.00785067
$775$	0	0
$776$	21.8052	0.782760
$777$	−21.9506	−0.787474
$778$	−6.04036	−0.216557
$779$	−35.6891	−1.27869
$780$	0	0
$781$	24.8927	0.890730
$782$	−4.54544	−0.162545
$783$	23.7549	0.848932
$784$	2.93753	0.104912
$785$	0	0
$786$	4.80723	0.171468
$787$	−30.6863	−1.09385	−0.546924	−	0.837182i	$-0.684202\pi$
$787$	−30.6863	−1.09385	−0.546924	+	0.837182i	$0.684202\pi$
$788$	−29.6088	−1.05477
$789$	−13.2626	−0.472161
$790$	0	0
$791$	−8.12693	−0.288960
$792$	−0.662719	−0.0235487
$793$	−68.2367	−2.42316
$794$	15.4448	0.548114
$795$	0	0
$796$	−35.8349	−1.27013
$797$	5.95905	0.211080	0.105540	−	0.994415i	$-0.466343\pi$
$797$	5.95905	0.211080	0.105540	+	0.994415i	$0.466343\pi$
$798$	−22.2342	−0.787082
$799$	11.0542	0.391071
$800$	0	0
$801$	1.23725	0.0437162
$802$	2.23108	0.0787821
$803$	−13.3357	−0.470606
$804$	6.97031	0.245824
$805$	0	0
$806$	2.38556	0.0840278
$807$	43.1419	1.51867
$808$	8.78630	0.309101
$809$	42.7677	1.50363	0.751816	−	0.659373i	$-0.229178\pi$
$809$	42.7677	1.50363	0.751816	+	0.659373i	$0.229178\pi$
$810$	0	0
$811$	−26.8066	−0.941308	−0.470654	−	0.882318i	$-0.655982\pi$
$811$	−26.8066	−0.941308	−0.470654	+	0.882318i	$0.655982\pi$
$812$	22.7303	0.797676
$813$	−28.7931	−1.00982
$814$	−5.43493	−0.190494
$815$	0	0
$816$	−6.10774	−0.213814
$817$	−20.6455	−0.722293
$818$	9.16324	0.320385
$819$	−1.86818	−0.0652796
$820$	0	0
$821$	−51.5956	−1.80070	−0.900350	−	0.435167i	$-0.856689\pi$
$821$	−51.5956	−1.80070	−0.900350	+	0.435167i	$0.856689\pi$
$822$	13.5954	0.474193
$823$	32.3384	1.12725	0.563623	−	0.826032i	$-0.309407\pi$
$823$	32.3384	1.12725	0.563623	+	0.826032i	$0.309407\pi$
$824$	7.72256	0.269028
$825$	0	0
$826$	−19.6330	−0.683121
$827$	28.0372	0.974949	0.487474	−	0.873137i	$-0.337918\pi$
$827$	28.0372	0.974949	0.487474	+	0.873137i	$0.337918\pi$
$828$	−1.19728	−0.0416083
$829$	−16.6055	−0.576733	−0.288366	−	0.957520i	$-0.593112\pi$
$829$	−16.6055	−0.576733	−0.288366	+	0.957520i	$0.593112\pi$
$830$	0	0
$831$	0.591086	0.0205046
$832$	−5.71073	−0.197984
$833$	2.15812	0.0747743
$834$	−6.95311	−0.240767
$835$	0	0
$836$	28.5693	0.988088
$837$	−4.67349	−0.161540
$838$	−5.50040	−0.190008
$839$	−28.7535	−0.992682	−0.496341	−	0.868128i	$-0.665323\pi$
$839$	−28.7535	−0.992682	−0.496341	+	0.868128i	$0.665323\pi$
$840$	0	0
$841$	−7.01189	−0.241789
$842$	16.1930	0.558047
$843$	10.8680	0.374314
$844$	−16.4051	−0.564686
$845$	0	0
$846$	−0.561071	−0.0192900
$847$	17.3980	0.597802
$848$	−23.2828	−0.799534
$849$	−10.3420	−0.354938
$850$	0	0
$851$	−21.5297	−0.738028
$852$	−33.1513	−1.13574
$853$	6.62356	0.226786	0.113393	−	0.993550i	$-0.463828\pi$
$853$	6.62356	0.226786	0.113393	+	0.993550i	$0.463828\pi$
$854$	−24.6484	−0.843452
$855$	0	0
$856$	19.5830	0.669335
$857$	−29.4169	−1.00486	−0.502431	−	0.864618i	$-0.667561\pi$
$857$	−29.4169	−1.00486	−0.502431	+	0.864618i	$0.667561\pi$
$858$	−10.2305	−0.349263
$859$	29.7654	1.01558	0.507792	−	0.861480i	$-0.330462\pi$
$859$	29.7654	1.01558	0.507792	+	0.861480i	$0.330462\pi$
$860$	0	0
$861$	−23.9571	−0.816455
$862$	1.40519	0.0478610
$863$	−33.6211	−1.14448	−0.572238	−	0.820088i	$-0.693925\pi$
$863$	−33.6211	−1.14448	−0.572238	+	0.820088i	$0.693925\pi$
$864$	−27.4129	−0.932606
$865$	0	0
$866$	17.7898	0.604521
$867$	25.6468	0.871011
$868$	−4.47190	−0.151786
$869$	−25.3243	−0.859069
$870$	0	0
$871$	10.6676	0.361458
$872$	1.16361	0.0394048
$873$	1.48211	0.0501619
$874$	−21.8078	−0.737660
$875$	0	0
$876$	17.7601	0.600057
$877$	46.2046	1.56022	0.780109	−	0.625644i	$-0.215164\pi$
$877$	46.2046	1.56022	0.780109	+	0.625644i	$0.215164\pi$
$878$	−15.3724	−0.518794
$879$	−35.1588	−1.18588
$880$	0	0
$881$	33.6218	1.13275	0.566373	−	0.824149i	$-0.308346\pi$
$881$	33.6218	1.13275	0.566373	+	0.824149i	$0.308346\pi$
$882$	−0.109538	−0.00368833
$883$	0.110180	0.00370785	0.00185393	−	0.999998i	$-0.499410\pi$
$883$	0.110180	0.00370785	0.00185393	+	0.999998i	$0.499410\pi$
$884$	−12.1369	−0.408207
$885$	0	0
$886$	−10.3491	−0.347683
$887$	1.27515	0.0428154	0.0214077	−	0.999771i	$-0.493185\pi$
$887$	1.27515	0.0428154	0.0214077	+	0.999771i	$0.493185\pi$
$888$	15.8709	0.532592
$889$	50.6772	1.69966
$890$	0	0
$891$	20.9935	0.703309
$892$	−10.3807	−0.347573
$893$	53.0353	1.77476
$894$	1.32145	0.0441958
$895$	0	0
$896$	−33.3478	−1.11407
$897$	−40.5265	−1.35314
$898$	11.4621	0.382494
$899$	−4.32589	−0.144277
$900$	0	0
$901$	−17.1052	−0.569856
$902$	−5.93172	−0.197505
$903$	−13.8587	−0.461190
$904$	5.87598	0.195432
$905$	0	0
$906$	−23.8752	−0.793202
$907$	−1.55443	−0.0516140	−0.0258070	−	0.999667i	$-0.508216\pi$
$907$	−1.55443	−0.0516140	−0.0258070	+	0.999667i	$0.508216\pi$
$908$	47.2378	1.56764
$909$	0.597211	0.0198083
$910$	0	0
$911$	5.05866	0.167601	0.0838005	−	0.996483i	$-0.473294\pi$
$911$	5.05866	0.167601	0.0838005	+	0.996483i	$0.473294\pi$
$912$	−29.3033	−0.970329
$913$	−39.0800	−1.29336
$914$	18.9208	0.625844
$915$	0	0
$916$	−33.8777	−1.11935
$917$	−13.7915	−0.455436
$918$	−4.58171	−0.151219
$919$	4.98229	0.164351	0.0821753	−	0.996618i	$-0.473813\pi$
$919$	4.98229	0.164351	0.0821753	+	0.996618i	$0.473813\pi$
$920$	0	0
$921$	7.69959	0.253710
$922$	13.6120	0.448288
$923$	−50.7359	−1.66999
$924$	19.1777	0.630902
$925$	0	0
$926$	−12.3219	−0.404921
$927$	0.524908	0.0172402
$928$	−25.3740	−0.832943
$929$	−4.88467	−0.160261	−0.0801304	−	0.996784i	$-0.525534\pi$
$929$	−4.88467	−0.160261	−0.0801304	+	0.996784i	$0.525534\pi$
$930$	0	0
$931$	10.3541	0.339341
$932$	42.2016	1.38236
$933$	−50.7949	−1.66295
$934$	10.7792	0.352707
$935$	0	0
$936$	1.35075	0.0441505
$937$	59.1046	1.93086	0.965431	−	0.260659i	$-0.0839397\pi$
$937$	59.1046	1.93086	0.965431	+	0.260659i	$0.0839397\pi$
$938$	3.85335	0.125816
$939$	−3.41146	−0.111329
$940$	0	0
$941$	37.0677	1.20837	0.604187	−	0.796843i	$-0.293498\pi$
$941$	37.0677	1.20837	0.604187	+	0.796843i	$0.293498\pi$
$942$	0.938487	0.0305776
$943$	−23.4976	−0.765189
$944$	−25.8751	−0.842164
$945$	0	0
$946$	−3.43139	−0.111564
$947$	44.4539	1.44456	0.722279	−	0.691602i	$-0.243095\pi$
$947$	44.4539	1.44456	0.722279	+	0.691602i	$0.243095\pi$
$948$	33.7262	1.09537
$949$	27.1806	0.882321
$950$	0	0
$951$	31.7155	1.02845
$952$	−9.61295	−0.311558
$953$	−24.6524	−0.798569	−0.399285	−	0.916827i	$-0.630741\pi$
$953$	−24.6524	−0.798569	−0.399285	+	0.916827i	$0.630741\pi$
$954$	0.868193	0.0281088
$955$	0	0
$956$	7.55926	0.244484
$957$	18.5516	0.599687
$958$	−3.83338	−0.123851
$959$	−39.0039	−1.25950
$960$	0	0
$961$	−30.1489	−0.972546
$962$	11.0774	0.357150
$963$	1.33107	0.0428933
$964$	1.67687	0.0540085
$965$	0	0
$966$	−14.6390	−0.471002
$967$	−5.83670	−0.187696	−0.0938478	−	0.995587i	$-0.529917\pi$
$967$	−5.83670	−0.187696	−0.0938478	+	0.995587i	$0.529917\pi$
$968$	−12.5792	−0.404311
$969$	−21.5283	−0.691587
$970$	0	0
$971$	39.7591	1.27593	0.637965	−	0.770066i	$-0.279777\pi$
$971$	39.7591	1.27593	0.637965	+	0.770066i	$0.279777\pi$
$972$	−2.47365	−0.0793425
$973$	19.9479	0.639499
$974$	12.4020	0.397386
$975$	0	0
$976$	−32.4851	−1.03982
$977$	−6.42778	−0.205643	−0.102821	−	0.994700i	$-0.532787\pi$
$977$	−6.42778	−0.205643	−0.102821	+	0.994700i	$0.532787\pi$
$978$	14.6489	0.468419
$979$	−19.4380	−0.621242
$980$	0	0
$981$	0.0790914	0.00252519
$982$	2.17903	0.0695355
$983$	−8.61131	−0.274658	−0.137329	−	0.990525i	$-0.543852\pi$
$983$	−8.61131	−0.274658	−0.137329	+	0.990525i	$0.543852\pi$
$984$	17.3216	0.552192
$985$	0	0
$986$	−4.24094	−0.135059
$987$	35.6011	1.13320
$988$	−58.2295	−1.85253
$989$	−13.5930	−0.432231
$990$	0	0
$991$	50.7845	1.61322	0.806612	−	0.591082i	$-0.201299\pi$
$991$	50.7845	1.61322	0.806612	+	0.591082i	$0.201299\pi$
$992$	4.99203	0.158497
$993$	24.7557	0.785599
$994$	−18.3268	−0.581292
$995$	0	0
$996$	52.0455	1.64913
$997$	21.8466	0.691888	0.345944	−	0.938255i	$-0.387559\pi$
$997$	21.8466	0.691888	0.345944	+	0.938255i	$0.387559\pi$
$998$	18.4687	0.584616
$999$	−21.7015	−0.686605

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.o.1.16	yes	40
5.4	even	2		6025.2.a.l.1.25	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.25	✓	40	5.4	even	2
6025.2.a.o.1.16	yes	40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.568441 q^{2} -1.77259 q^{3} -1.67687 q^{4} +1.00761 q^{6} -2.89075 q^{7} +2.09009 q^{8} +0.142065 q^{9} +O(q^{10})\)	\(q-0.568441 q^{2} -1.77259 q^{3} -1.67687 q^{4} +1.00761 q^{6} -2.89075 q^{7} +2.09009 q^{8} +0.142065 q^{9} -2.23192 q^{11} +2.97241 q^{12} +4.54908 q^{13} +1.64322 q^{14} +2.16566 q^{16} +1.59105 q^{17} -0.0807554 q^{18} +7.63341 q^{19} +5.12410 q^{21} +1.26872 q^{22} +5.02583 q^{23} -3.70486 q^{24} -2.58588 q^{26} +5.06594 q^{27} +4.84742 q^{28} +4.68915 q^{29} -0.922532 q^{31} -5.41122 q^{32} +3.95628 q^{33} -0.904415 q^{34} -0.238225 q^{36} -4.28380 q^{37} -4.33914 q^{38} -8.06364 q^{39} -4.67537 q^{41} -2.91275 q^{42} -2.70462 q^{43} +3.74266 q^{44} -2.85689 q^{46} +6.94778 q^{47} -3.83882 q^{48} +1.35641 q^{49} -2.82027 q^{51} -7.62824 q^{52} -10.7509 q^{53} -2.87969 q^{54} -6.04191 q^{56} -13.5309 q^{57} -2.66550 q^{58} -11.9479 q^{59} -15.0001 q^{61} +0.524405 q^{62} -0.410673 q^{63} -1.25536 q^{64} -2.24891 q^{66} +2.34500 q^{67} -2.66798 q^{68} -8.90873 q^{69} -11.1530 q^{71} +0.296927 q^{72} +5.97498 q^{73} +2.43509 q^{74} -12.8003 q^{76} +6.45193 q^{77} +4.58370 q^{78} +11.3464 q^{79} -9.40601 q^{81} +2.65767 q^{82} +17.5096 q^{83} -8.59247 q^{84} +1.53742 q^{86} -8.31192 q^{87} -4.66491 q^{88} +8.70909 q^{89} -13.1502 q^{91} -8.42769 q^{92} +1.63527 q^{93} -3.94940 q^{94} +9.59186 q^{96} +10.4327 q^{97} -0.771041 q^{98} -0.317078 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

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Database

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