LMFDB - Embedded newform 6025.2.a.q.1.13

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:	$66$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	$q-1.90752 q^{2} +0.479938 q^{3} +1.63863 q^{4} -0.915491 q^{6} +1.00580 q^{7} +0.689327 q^{8} -2.76966 q^{9} +O(q^{10})$	\(q-1.90752 q^{2} +0.479938 q^{3} +1.63863 q^{4} -0.915491 q^{6} +1.00580 q^{7} +0.689327 q^{8} -2.76966 q^{9} -2.65217 q^{11} +0.786439 q^{12} -6.61306 q^{13} -1.91859 q^{14} -4.59216 q^{16} +2.85120 q^{17} +5.28318 q^{18} -0.253184 q^{19} +0.482723 q^{21} +5.05906 q^{22} -6.53547 q^{23} +0.330834 q^{24} +12.6145 q^{26} -2.76908 q^{27} +1.64813 q^{28} +4.50076 q^{29} +1.25952 q^{31} +7.38097 q^{32} -1.27288 q^{33} -5.43871 q^{34} -4.53844 q^{36} +4.79681 q^{37} +0.482954 q^{38} -3.17386 q^{39} -9.09759 q^{41} -0.920803 q^{42} -2.76330 q^{43} -4.34591 q^{44} +12.4665 q^{46} -11.4879 q^{47} -2.20395 q^{48} -5.98836 q^{49} +1.36840 q^{51} -10.8363 q^{52} -7.13433 q^{53} +5.28207 q^{54} +0.693326 q^{56} -0.121513 q^{57} -8.58528 q^{58} +0.910403 q^{59} +9.96827 q^{61} -2.40255 q^{62} -2.78573 q^{63} -4.89502 q^{64} +2.42804 q^{66} -1.66535 q^{67} +4.67205 q^{68} -3.13662 q^{69} +12.0656 q^{71} -1.90920 q^{72} -8.27365 q^{73} -9.14999 q^{74} -0.414874 q^{76} -2.66756 q^{77} +6.05419 q^{78} -3.95246 q^{79} +6.97999 q^{81} +17.3538 q^{82} +5.70936 q^{83} +0.791002 q^{84} +5.27105 q^{86} +2.16009 q^{87} -1.82821 q^{88} +10.0007 q^{89} -6.65143 q^{91} -10.7092 q^{92} +0.604491 q^{93} +21.9134 q^{94} +3.54241 q^{96} -3.40346 q^{97} +11.4229 q^{98} +7.34560 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * 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 $

$2$	−1.90752	−1.34882	−0.674410	−	0.738357i	$-0.735602\pi$
$2$	−1.90752	−1.34882	−0.674410	+	0.738357i	$0.735602\pi$
$3$	0.479938	0.277092	0.138546	−	0.990356i	$-0.455757\pi$
$3$	0.479938	0.277092	0.138546	+	0.990356i	$0.455757\pi$
$4$	1.63863	0.819313
$5$	0	0
$5$	0	0
$6$	−0.915491	−0.373748
$7$	1.00580	0.380157	0.190079	−	0.981769i	$-0.439126\pi$
$7$	1.00580	0.380157	0.190079	+	0.981769i	$0.439126\pi$
$8$	0.689327	0.243714
$9$	−2.76966	−0.923220
$10$	0	0
$11$	−2.65217	−0.799659	−0.399829	−	0.916590i	$-0.630931\pi$
$11$	−2.65217	−0.799659	−0.399829	+	0.916590i	$0.630931\pi$
$12$	0.786439	0.227025
$13$	−6.61306	−1.83413	−0.917066	−	0.398736i	$-0.869449\pi$
$13$	−6.61306	−1.83413	−0.917066	+	0.398736i	$0.869449\pi$
$14$	−1.91859	−0.512764
$15$	0	0
$16$	−4.59216	−1.14804
$17$	2.85120	0.691517	0.345759	−	0.938323i	$-0.387622\pi$
$17$	2.85120	0.691517	0.345759	+	0.938323i	$0.387622\pi$
$18$	5.28318	1.24526
$19$	−0.253184	−0.0580845	−0.0290422	−	0.999578i	$-0.509246\pi$
$19$	−0.253184	−0.0580845	−0.0290422	+	0.999578i	$0.509246\pi$
$20$	0	0
$21$	0.482723	0.105339
$22$	5.05906	1.07859
$23$	−6.53547	−1.36274	−0.681370	−	0.731939i	$-0.738616\pi$
$23$	−6.53547	−1.36274	−0.681370	+	0.731939i	$0.738616\pi$
$24$	0.330834	0.0675313
$25$	0	0
$26$	12.6145	2.47391
$27$	−2.76908	−0.532910
$28$	1.64813	0.311468
$29$	4.50076	0.835770	0.417885	−	0.908500i	$-0.362771\pi$
$29$	4.50076	0.835770	0.417885	+	0.908500i	$0.362771\pi$
$30$	0	0
$31$	1.25952	0.226216	0.113108	−	0.993583i	$-0.463919\pi$
$31$	1.25952	0.226216	0.113108	+	0.993583i	$0.463919\pi$
$32$	7.38097	1.30478
$33$	−1.27288	−0.221579
$34$	−5.43871	−0.932732
$35$	0	0
$36$	−4.53844	−0.756406
$37$	4.79681	0.788590	0.394295	−	0.918984i	$-0.370989\pi$
$37$	4.79681	0.788590	0.394295	+	0.918984i	$0.370989\pi$
$38$	0.482954	0.0783454
$39$	−3.17386	−0.508224
$40$	0	0
$41$	−9.09759	−1.42080	−0.710402	−	0.703796i	$-0.751487\pi$
$41$	−9.09759	−1.42080	−0.710402	+	0.703796i	$0.751487\pi$
$42$	−0.920803	−0.142083
$43$	−2.76330	−0.421400	−0.210700	−	0.977551i	$-0.567574\pi$
$43$	−2.76330	−0.421400	−0.210700	+	0.977551i	$0.567574\pi$
$44$	−4.34591	−0.655171
$45$	0	0
$46$	12.4665	1.83809
$47$	−11.4879	−1.67569	−0.837843	−	0.545912i	$-0.816183\pi$
$47$	−11.4879	−1.67569	−0.837843	+	0.545912i	$0.816183\pi$
$48$	−2.20395	−0.318113
$49$	−5.98836	−0.855480
$50$	0	0
$51$	1.36840	0.191614
$52$	−10.8363	−1.50273
$53$	−7.13433	−0.979976	−0.489988	−	0.871729i	$-0.662999\pi$
$53$	−7.13433	−0.979976	−0.489988	+	0.871729i	$0.662999\pi$
$54$	5.28207	0.718799
$55$	0	0
$56$	0.693326	0.0926496
$57$	−0.121513	−0.0160948
$58$	−8.58528	−1.12730
$59$	0.910403	0.118524	0.0592622	−	0.998242i	$-0.481125\pi$
$59$	0.910403	0.118524	0.0592622	+	0.998242i	$0.481125\pi$
$60$	0	0
$61$	9.96827	1.27631	0.638153	−	0.769910i	$-0.279699\pi$
$61$	9.96827	1.27631	0.638153	+	0.769910i	$0.279699\pi$
$62$	−2.40255	−0.305125
$63$	−2.78573	−0.350969
$64$	−4.89502	−0.611878
$65$	0	0
$66$	2.42804	0.298870
$67$	−1.66535	−0.203455	−0.101727	−	0.994812i	$-0.532437\pi$
$67$	−1.66535	−0.203455	−0.101727	+	0.994812i	$0.532437\pi$
$68$	4.67205	0.566569
$69$	−3.13662	−0.377605
$70$	0	0
$71$	12.0656	1.43192	0.715959	−	0.698142i	$-0.245990\pi$
$71$	12.0656	1.43192	0.715959	+	0.698142i	$0.245990\pi$
$72$	−1.90920	−0.225001
$73$	−8.27365	−0.968357	−0.484179	−	0.874969i	$-0.660882\pi$
$73$	−8.27365	−0.968357	−0.484179	+	0.874969i	$0.660882\pi$
$74$	−9.14999	−1.06367
$75$	0	0
$76$	−0.414874	−0.0475894
$77$	−2.66756	−0.303996
$78$	6.05419	0.685502
$79$	−3.95246	−0.444687	−0.222343	−	0.974968i	$-0.571371\pi$
$79$	−3.95246	−0.444687	−0.222343	+	0.974968i	$0.571371\pi$
$80$	0	0
$81$	6.97999	0.775555
$82$	17.3538	1.91641
$83$	5.70936	0.626683	0.313342	−	0.949640i	$-0.398551\pi$
$83$	5.70936	0.626683	0.313342	+	0.949640i	$0.398551\pi$
$84$	0.791002	0.0863054
$85$	0	0
$86$	5.27105	0.568392
$87$	2.16009	0.231586
$88$	−1.82821	−0.194888
$89$	10.0007	1.06007	0.530037	−	0.847974i	$-0.322178\pi$
$89$	10.0007	1.06007	0.530037	+	0.847974i	$0.322178\pi$
$90$	0	0
$91$	−6.65143	−0.697259
$92$	−10.7092	−1.11651
$93$	0.604491	0.0626828
$94$	21.9134	2.26020
$95$	0	0
$96$	3.54241	0.361546
$97$	−3.40346	−0.345569	−0.172784	−	0.984960i	$-0.555276\pi$
$97$	−3.40346	−0.345569	−0.172784	+	0.984960i	$0.555276\pi$
$98$	11.4229	1.15389
$99$	7.34560	0.738261
$100$	0	0
$101$	−6.19893	−0.616817	−0.308408	−	0.951254i	$-0.599796\pi$
$101$	−6.19893	−0.616817	−0.308408	+	0.951254i	$0.599796\pi$
$102$	−2.61025	−0.258453
$103$	−2.75922	−0.271874	−0.135937	−	0.990717i	$-0.543404\pi$
$103$	−2.75922	−0.271874	−0.135937	+	0.990717i	$0.543404\pi$
$104$	−4.55856	−0.447003
$105$	0	0
$106$	13.6089	1.32181
$107$	14.5269	1.40437	0.702186	−	0.711993i	$-0.252207\pi$
$107$	14.5269	1.40437	0.702186	+	0.711993i	$0.252207\pi$
$108$	−4.53749	−0.436620
$109$	18.6386	1.78525	0.892627	−	0.450795i	$-0.148860\pi$
$109$	18.6386	1.78525	0.892627	+	0.450795i	$0.148860\pi$
$110$	0	0
$111$	2.30217	0.218512
$112$	−4.61880	−0.436436
$113$	−1.16295	−0.109401	−0.0547004	−	0.998503i	$-0.517420\pi$
$113$	−1.16295	−0.109401	−0.0547004	+	0.998503i	$0.517420\pi$
$114$	0.231788	0.0217089
$115$	0	0
$116$	7.37507	0.684758
$117$	18.3159	1.69331
$118$	−1.73661	−0.159868
$119$	2.86774	0.262886
$120$	0	0
$121$	−3.96601	−0.360546
$122$	−19.0147	−1.72151
$123$	−4.36628	−0.393694
$124$	2.06388	0.185342
$125$	0	0
$126$	5.31383	0.473394
$127$	21.7276	1.92801	0.964006	−	0.265879i	$-0.0856622\pi$
$127$	21.7276	1.92801	0.964006	+	0.265879i	$0.0856622\pi$
$128$	−5.42460	−0.479471
$129$	−1.32621	−0.116767
$130$	0	0
$131$	13.8845	1.21309	0.606546	−	0.795048i	$-0.292555\pi$
$131$	13.8845	1.21309	0.606546	+	0.795048i	$0.292555\pi$
$132$	−2.08577	−0.181543
$133$	−0.254653	−0.0220812
$134$	3.17669	0.274424
$135$	0	0
$136$	1.96541	0.168532
$137$	−8.63725	−0.737930	−0.368965	−	0.929443i	$-0.620288\pi$
$137$	−8.63725	−0.737930	−0.368965	+	0.929443i	$0.620288\pi$
$138$	5.98317	0.509321
$139$	14.4160	1.22275	0.611373	−	0.791342i	$-0.290617\pi$
$139$	14.4160	1.22275	0.611373	+	0.791342i	$0.290617\pi$
$140$	0	0
$141$	−5.51349	−0.464320
$142$	−23.0153	−1.93140
$143$	17.5389	1.46668
$144$	12.7187	1.05989
$145$	0	0
$146$	15.7821	1.30614
$147$	−2.87404	−0.237047
$148$	7.86017	0.646102
$149$	3.40386	0.278855	0.139427	−	0.990232i	$-0.455474\pi$
$149$	3.40386	0.278855	0.139427	+	0.990232i	$0.455474\pi$
$150$	0	0
$151$	2.83680	0.230855	0.115428	−	0.993316i	$-0.463176\pi$
$151$	2.83680	0.230855	0.115428	+	0.993316i	$0.463176\pi$
$152$	−0.174527	−0.0141560
$153$	−7.89685	−0.638422
$154$	5.08841	0.410036
$155$	0	0
$156$	−5.20077	−0.416395
$157$	−15.5542	−1.24136	−0.620679	−	0.784065i	$-0.713143\pi$
$157$	−15.5542	−1.24136	−0.620679	+	0.784065i	$0.713143\pi$
$158$	7.53939	0.599802
$159$	−3.42404	−0.271544
$160$	0	0
$161$	−6.57339	−0.518056
$162$	−13.3145	−1.04608
$163$	−8.35344	−0.654292	−0.327146	−	0.944974i	$-0.606087\pi$
$163$	−8.35344	−0.654292	−0.327146	+	0.944974i	$0.606087\pi$
$164$	−14.9075	−1.16408
$165$	0	0
$166$	−10.8907	−0.845283
$167$	−12.7098	−0.983510	−0.491755	−	0.870733i	$-0.663645\pi$
$167$	−12.7098	−0.983510	−0.491755	+	0.870733i	$0.663645\pi$
$168$	0.332754	0.0256725
$169$	30.7325	2.36404
$170$	0	0
$171$	0.701234	0.0536247
$172$	−4.52802	−0.345258
$173$	−24.9121	−1.89403	−0.947014	−	0.321191i	$-0.895917\pi$
$173$	−24.9121	−1.89403	−0.947014	+	0.321191i	$0.895917\pi$
$174$	−4.12041	−0.312367
$175$	0	0
$176$	12.1792	0.918039
$177$	0.436937	0.0328422
$178$	−19.0766	−1.42985
$179$	15.0137	1.12218	0.561090	−	0.827755i	$-0.310382\pi$
$179$	15.0137	1.12218	0.561090	+	0.827755i	$0.310382\pi$
$180$	0	0
$181$	15.8939	1.18138	0.590691	−	0.806898i	$-0.298855\pi$
$181$	15.8939	1.18138	0.590691	+	0.806898i	$0.298855\pi$
$182$	12.6877	0.940476
$183$	4.78415	0.353655
$184$	−4.50508	−0.332119
$185$	0	0
$186$	−1.15308	−0.0845477
$187$	−7.56186	−0.552978
$188$	−18.8244	−1.37291
$189$	−2.78515	−0.202590
$190$	0	0
$191$	−9.24070	−0.668634	−0.334317	−	0.942461i	$-0.608506\pi$
$191$	−9.24070	−0.668634	−0.334317	+	0.942461i	$0.608506\pi$
$192$	−2.34931	−0.169547
$193$	−6.53030	−0.470061	−0.235031	−	0.971988i	$-0.575519\pi$
$193$	−6.53030	−0.470061	−0.235031	+	0.971988i	$0.575519\pi$
$194$	6.49216	0.466110
$195$	0	0
$196$	−9.81269	−0.700906
$197$	22.1308	1.57675	0.788377	−	0.615193i	$-0.210922\pi$
$197$	22.1308	1.57675	0.788377	+	0.615193i	$0.210922\pi$
$198$	−14.0119	−0.995780
$199$	−20.7253	−1.46918	−0.734590	−	0.678511i	$-0.762625\pi$
$199$	−20.7253	−1.46918	−0.734590	+	0.678511i	$0.762625\pi$
$200$	0	0
$201$	−0.799265	−0.0563758
$202$	11.8246	0.831974
$203$	4.52688	0.317724
$204$	2.24230	0.156992
$205$	0	0
$206$	5.26326	0.366709
$207$	18.1010	1.25811
$208$	30.3682	2.10566
$209$	0.671487	0.0464477
$210$	0	0
$211$	−28.6898	−1.97509	−0.987543	−	0.157347i	$-0.949706\pi$
$211$	−28.6898	−1.97509	−0.987543	+	0.157347i	$0.949706\pi$
$212$	−11.6905	−0.802907
$213$	5.79072	0.396774
$214$	−27.7104	−1.89424
$215$	0	0
$216$	−1.90880	−0.129877
$217$	1.26683	0.0859977
$218$	−35.5535	−2.40799
$219$	−3.97084	−0.268325
$220$	0	0
$221$	−18.8551	−1.26833
$222$	−4.39143	−0.294734
$223$	−12.2269	−0.818777	−0.409389	−	0.912360i	$-0.634258\pi$
$223$	−12.2269	−0.818777	−0.409389	+	0.912360i	$0.634258\pi$
$224$	7.42379	0.496023
$225$	0	0
$226$	2.21834	0.147562
$227$	1.27160	0.0843993	0.0421997	−	0.999109i	$-0.486563\pi$
$227$	1.27160	0.0843993	0.0421997	+	0.999109i	$0.486563\pi$
$228$	−0.199114	−0.0131867
$229$	−9.48292	−0.626649	−0.313325	−	0.949646i	$-0.601443\pi$
$229$	−9.48292	−0.626649	−0.313325	+	0.949646i	$0.601443\pi$
$230$	0	0
$231$	−1.28026	−0.0842350
$232$	3.10250	0.203689
$233$	14.1289	0.925614	0.462807	−	0.886459i	$-0.346842\pi$
$233$	14.1289	0.925614	0.462807	+	0.886459i	$0.346842\pi$
$234$	−34.9379	−2.28396
$235$	0	0
$236$	1.49181	0.0971086
$237$	−1.89694	−0.123219
$238$	−5.47027	−0.354585
$239$	11.9323	0.771833	0.385917	−	0.922534i	$-0.373885\pi$
$239$	11.9323	0.771833	0.385917	+	0.922534i	$0.373885\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	7.56523	0.486312
$243$	11.6572	0.747810
$244$	16.3343	1.04569
$245$	0	0
$246$	8.32876	0.531022
$247$	1.67432	0.106535
$248$	0.868219	0.0551320
$249$	2.74014	0.173649
$250$	0	0
$251$	19.4681	1.22882	0.614408	−	0.788988i	$-0.289395\pi$
$251$	19.4681	1.22882	0.614408	+	0.788988i	$0.289395\pi$
$252$	−4.56477	−0.287553
$253$	17.3332	1.08973
$254$	−41.4458	−2.60054
$255$	0	0
$256$	20.1376	1.25860
$257$	5.80178	0.361905	0.180953	−	0.983492i	$-0.442082\pi$
$257$	5.80178	0.361905	0.180953	+	0.983492i	$0.442082\pi$
$258$	2.52978	0.157497
$259$	4.82464	0.299788
$260$	0	0
$261$	−12.4656	−0.771600
$262$	−26.4849	−1.63624
$263$	11.4944	0.708773	0.354387	−	0.935099i	$-0.384690\pi$
$263$	11.4944	0.708773	0.354387	+	0.935099i	$0.384690\pi$
$264$	−0.877428	−0.0540020
$265$	0	0
$266$	0.485756	0.0297836
$267$	4.79973	0.293739
$268$	−2.72889	−0.166693
$269$	−21.8743	−1.33370	−0.666850	−	0.745192i	$-0.732358\pi$
$269$	−21.8743	−1.33370	−0.666850	+	0.745192i	$0.732358\pi$
$270$	0	0
$271$	−10.7172	−0.651022	−0.325511	−	0.945538i	$-0.605536\pi$
$271$	−10.7172	−0.651022	−0.325511	+	0.945538i	$0.605536\pi$
$272$	−13.0932	−0.793889
$273$	−3.19227	−0.193205
$274$	16.4757	0.995335
$275$	0	0
$276$	−5.13975	−0.309377
$277$	1.88442	0.113224	0.0566118	−	0.998396i	$-0.481970\pi$
$277$	1.88442	0.113224	0.0566118	+	0.998396i	$0.481970\pi$
$278$	−27.4987	−1.64926
$279$	−3.48843	−0.208847
$280$	0	0
$281$	8.29198	0.494658	0.247329	−	0.968932i	$-0.420447\pi$
$281$	8.29198	0.494658	0.247329	+	0.968932i	$0.420447\pi$
$282$	10.5171	0.626283
$283$	2.76723	0.164495	0.0822475	−	0.996612i	$-0.473790\pi$
$283$	2.76723	0.164495	0.0822475	+	0.996612i	$0.473790\pi$
$284$	19.7709	1.17319
$285$	0	0
$286$	−33.4558	−1.97829
$287$	−9.15037	−0.540129
$288$	−20.4428	−1.20460
$289$	−8.87066	−0.521804
$290$	0	0
$291$	−1.63345	−0.0957545
$292$	−13.5574	−0.793388
$293$	8.07780	0.471910	0.235955	−	0.971764i	$-0.424178\pi$
$293$	8.07780	0.471910	0.235955	+	0.971764i	$0.424178\pi$
$294$	5.48229	0.319734
$295$	0	0
$296$	3.30657	0.192190
$297$	7.34406	0.426146
$298$	−6.49292	−0.376125
$299$	43.2194	2.49944
$300$	0	0
$301$	−2.77934	−0.160198
$302$	−5.41125	−0.311382
$303$	−2.97510	−0.170915
$304$	1.16266	0.0666832
$305$	0	0
$306$	15.0634	0.861117
$307$	−2.64845	−0.151155	−0.0755775	−	0.997140i	$-0.524080\pi$
$307$	−2.64845	−0.151155	−0.0755775	+	0.997140i	$0.524080\pi$
$308$	−4.37113	−0.249068
$309$	−1.32425	−0.0753342
$310$	0	0
$311$	13.7863	0.781751	0.390876	−	0.920443i	$-0.372172\pi$
$311$	13.7863	0.781751	0.390876	+	0.920443i	$0.372172\pi$
$312$	−2.18783	−0.123861
$313$	1.62006	0.0915713	0.0457857	−	0.998951i	$-0.485421\pi$
$313$	1.62006	0.0915713	0.0457857	+	0.998951i	$0.485421\pi$
$314$	29.6699	1.67437
$315$	0	0
$316$	−6.47661	−0.364338
$317$	26.7969	1.50507	0.752533	−	0.658555i	$-0.228832\pi$
$317$	26.7969	1.50507	0.752533	+	0.658555i	$0.228832\pi$
$318$	6.53142	0.366264
$319$	−11.9368	−0.668331
$320$	0	0
$321$	6.97203	0.389141
$322$	12.5389	0.698764
$323$	−0.721879	−0.0401664
$324$	11.4376	0.635422
$325$	0	0
$326$	15.9343	0.882521
$327$	8.94538	0.494681
$328$	−6.27121	−0.346270
$329$	−11.5546	−0.637024
$330$	0	0
$331$	5.26858	0.289587	0.144794	−	0.989462i	$-0.453748\pi$
$331$	5.26858	0.289587	0.144794	+	0.989462i	$0.453748\pi$
$332$	9.35550	0.513450
$333$	−13.2855	−0.728042
$334$	24.2441	1.32658
$335$	0	0
$336$	−2.21674	−0.120933
$337$	10.5091	0.572466	0.286233	−	0.958160i	$-0.407597\pi$
$337$	10.5091	0.572466	0.286233	+	0.958160i	$0.407597\pi$
$338$	−58.6228	−3.18866
$339$	−0.558142	−0.0303141
$340$	0	0
$341$	−3.34045	−0.180896
$342$	−1.33762	−0.0723301
$343$	−13.0637	−0.705375
$344$	−1.90482	−0.102701
$345$	0	0
$346$	47.5202	2.55470
$347$	14.4461	0.775505	0.387753	−	0.921763i	$-0.373251\pi$
$347$	14.4461	0.775505	0.387753	+	0.921763i	$0.373251\pi$
$348$	3.53958	0.189741
$349$	17.5773	0.940890	0.470445	−	0.882429i	$-0.344093\pi$
$349$	17.5773	0.940890	0.470445	+	0.882429i	$0.344093\pi$
$350$	0	0
$351$	18.3121	0.977427
$352$	−19.5756	−1.04338
$353$	34.8145	1.85299	0.926494	−	0.376310i	$-0.122807\pi$
$353$	34.8145	1.85299	0.926494	+	0.376310i	$0.122807\pi$
$354$	−0.833466	−0.0442982
$355$	0	0
$356$	16.3875	0.868533
$357$	1.37634	0.0728436
$358$	−28.6390	−1.51362
$359$	−23.2399	−1.22655	−0.613276	−	0.789868i	$-0.710149\pi$
$359$	−23.2399	−1.22655	−0.613276	+	0.789868i	$0.710149\pi$
$360$	0	0
$361$	−18.9359	−0.996626
$362$	−30.3178	−1.59347
$363$	−1.90344	−0.0999046
$364$	−10.8992	−0.571273
$365$	0	0
$366$	−9.12586	−0.477016
$367$	−11.6918	−0.610305	−0.305152	−	0.952304i	$-0.598707\pi$
$367$	−11.6918	−0.610305	−0.305152	+	0.952304i	$0.598707\pi$
$368$	30.0119	1.56448
$369$	25.1972	1.31171
$370$	0	0
$371$	−7.17573	−0.372545
$372$	0.990534	0.0513568
$373$	24.1077	1.24825	0.624125	−	0.781324i	$-0.285455\pi$
$373$	24.1077	1.24825	0.624125	+	0.781324i	$0.285455\pi$
$374$	14.4244	0.745867
$375$	0	0
$376$	−7.91893	−0.408388
$377$	−29.7638	−1.53291
$378$	5.31272	0.273257
$379$	2.09554	0.107641	0.0538204	−	0.998551i	$-0.482860\pi$
$379$	2.09554	0.107641	0.0538204	+	0.998551i	$0.482860\pi$
$380$	0	0
$381$	10.4279	0.534238
$382$	17.6268	0.901866
$383$	−27.1281	−1.38618	−0.693091	−	0.720851i	$-0.743751\pi$
$383$	−27.1281	−1.38618	−0.693091	+	0.720851i	$0.743751\pi$
$384$	−2.60347	−0.132858
$385$	0	0
$386$	12.4567	0.634028
$387$	7.65341	0.389045
$388$	−5.57700	−0.283129
$389$	6.17362	0.313015	0.156507	−	0.987677i	$-0.449976\pi$
$389$	6.17362	0.313015	0.156507	+	0.987677i	$0.449976\pi$
$390$	0	0
$391$	−18.6339	−0.942358
$392$	−4.12794	−0.208492
$393$	6.66369	0.336139
$394$	−42.2149	−2.12675
$395$	0	0
$396$	12.0367	0.604867
$397$	1.29252	0.0648699	0.0324350	−	0.999474i	$-0.489674\pi$
$397$	1.29252	0.0648699	0.0324350	+	0.999474i	$0.489674\pi$
$398$	39.5339	1.98166
$399$	−0.122218	−0.00611855
$400$	0	0
$401$	−5.57947	−0.278625	−0.139313	−	0.990248i	$-0.544489\pi$
$401$	−5.57947	−0.278625	−0.139313	+	0.990248i	$0.544489\pi$
$402$	1.52461	0.0760408
$403$	−8.32926	−0.414910
$404$	−10.1577	−0.505366
$405$	0	0
$406$	−8.63510	−0.428553
$407$	−12.7219	−0.630603
$408$	0.943274	0.0466990
$409$	−9.74745	−0.481980	−0.240990	−	0.970528i	$-0.577472\pi$
$409$	−9.74745	−0.481980	−0.240990	+	0.970528i	$0.577472\pi$
$410$	0	0
$411$	−4.14535	−0.204475
$412$	−4.52133	−0.222750
$413$	0.915685	0.0450579
$414$	−34.5280	−1.69696
$415$	0	0
$416$	−48.8108	−2.39314
$417$	6.91877	0.338814
$418$	−1.28087	−0.0626496
$419$	29.9222	1.46179	0.730896	−	0.682488i	$-0.239102\pi$
$419$	29.9222	1.46179	0.730896	+	0.682488i	$0.239102\pi$
$420$	0	0
$421$	−2.49935	−0.121811	−0.0609054	−	0.998144i	$-0.519399\pi$
$421$	−2.49935	−0.121811	−0.0609054	+	0.998144i	$0.519399\pi$
$422$	54.7263	2.66404
$423$	31.8176	1.54703
$424$	−4.91789	−0.238834
$425$	0	0
$426$	−11.0459	−0.535176
$427$	10.0261	0.485197
$428$	23.8042	1.15062
$429$	8.41760	0.406406
$430$	0	0
$431$	36.0583	1.73687	0.868435	−	0.495804i	$-0.165126\pi$
$431$	36.0583	1.73687	0.868435	+	0.495804i	$0.165126\pi$
$432$	12.7160	0.611801
$433$	13.8568	0.665915	0.332957	−	0.942942i	$-0.391953\pi$
$433$	13.8568	0.665915	0.332957	+	0.942942i	$0.391953\pi$
$434$	−2.41649	−0.115995
$435$	0	0
$436$	30.5417	1.46268
$437$	1.65468	0.0791540
$438$	7.57445	0.361921
$439$	−3.61937	−0.172743	−0.0863715	−	0.996263i	$-0.527527\pi$
$439$	−3.61937	−0.172743	−0.0863715	+	0.996263i	$0.527527\pi$
$440$	0	0
$441$	16.5857	0.789796
$442$	35.9665	1.71075
$443$	−39.0785	−1.85668	−0.928339	−	0.371736i	$-0.878763\pi$
$443$	−39.0785	−1.85668	−0.928339	+	0.371736i	$0.878763\pi$
$444$	3.77240	0.179030
$445$	0	0
$446$	23.3231	1.10438
$447$	1.63364	0.0772686
$448$	−4.92342	−0.232610
$449$	17.8522	0.842497	0.421249	−	0.906945i	$-0.361592\pi$
$449$	17.8522	0.842497	0.421249	+	0.906945i	$0.361592\pi$
$450$	0	0
$451$	24.1283	1.13616
$452$	−1.90563	−0.0896335
$453$	1.36149	0.0639683
$454$	−2.42561	−0.113839
$455$	0	0
$456$	−0.0837620	−0.00392252
$457$	−10.0148	−0.468472	−0.234236	−	0.972180i	$-0.575259\pi$
$457$	−10.0148	−0.468472	−0.234236	+	0.972180i	$0.575259\pi$
$458$	18.0889	0.845236
$459$	−7.89520	−0.368516
$460$	0	0
$461$	4.11951	0.191865	0.0959323	−	0.995388i	$-0.469417\pi$
$461$	4.11951	0.191865	0.0959323	+	0.995388i	$0.469417\pi$
$462$	2.44212	0.113618
$463$	14.8939	0.692179	0.346089	−	0.938202i	$-0.387509\pi$
$463$	14.8939	0.692179	0.346089	+	0.938202i	$0.387509\pi$
$464$	−20.6682	−0.959497
$465$	0	0
$466$	−26.9511	−1.24849
$467$	−39.4611	−1.82604	−0.913020	−	0.407915i	$-0.866256\pi$
$467$	−39.4611	−1.82604	−0.913020	+	0.407915i	$0.866256\pi$
$468$	30.0129	1.38735
$469$	−1.67501	−0.0773449
$470$	0	0
$471$	−7.46504	−0.343971
$472$	0.627565	0.0288860
$473$	7.32874	0.336976
$474$	3.61844	0.166201
$475$	0	0
$476$	4.69916	0.215386
$477$	19.7597	0.904733
$478$	−22.7610	−1.04106
$479$	−35.3230	−1.61395	−0.806974	−	0.590587i	$-0.798896\pi$
$479$	−35.3230	−1.61395	−0.806974	+	0.590587i	$0.798896\pi$
$480$	0	0
$481$	−31.7215	−1.44638
$482$	1.90752	0.0868851
$483$	−3.15482	−0.143549
$484$	−6.49881	−0.295400
$485$	0	0
$486$	−22.2363	−1.00866
$487$	22.1282	1.00272	0.501362	−	0.865238i	$-0.332833\pi$
$487$	22.1282	1.00272	0.501362	+	0.865238i	$0.332833\pi$
$488$	6.87139	0.311053
$489$	−4.00913	−0.181299
$490$	0	0
$491$	28.9403	1.30606	0.653029	−	0.757333i	$-0.273498\pi$
$491$	28.9403	1.30606	0.653029	+	0.757333i	$0.273498\pi$
$492$	−7.15470	−0.322559
$493$	12.8326	0.577950
$494$	−3.19380	−0.143696
$495$	0	0
$496$	−5.78390	−0.259705
$497$	12.1356	0.544354
$498$	−5.22687	−0.234221
$499$	−21.7983	−0.975827	−0.487913	−	0.872892i	$-0.662242\pi$
$499$	−21.7983	−0.975827	−0.487913	+	0.872892i	$0.662242\pi$
$500$	0	0
$501$	−6.09990	−0.272523
$502$	−37.1358	−1.65745
$503$	25.0641	1.11755	0.558777	−	0.829318i	$-0.311271\pi$
$503$	25.0641	1.11755	0.558777	+	0.829318i	$0.311271\pi$
$504$	−1.92028	−0.0855360
$505$	0	0
$506$	−33.0633	−1.46984
$507$	14.7497	0.655058
$508$	35.6034	1.57965
$509$	31.9583	1.41653	0.708263	−	0.705949i	$-0.249479\pi$
$509$	31.9583	1.41653	0.708263	+	0.705949i	$0.249479\pi$
$510$	0	0
$511$	−8.32166	−0.368128
$512$	−27.5636	−1.21815
$513$	0.701088	0.0309538
$514$	−11.0670	−0.488145
$515$	0	0
$516$	−2.17317	−0.0956685
$517$	30.4679	1.33998
$518$	−9.20308	−0.404360
$519$	−11.9562	−0.524821
$520$	0	0
$521$	40.8603	1.79012	0.895061	−	0.445945i	$-0.147132\pi$
$521$	40.8603	1.79012	0.895061	+	0.445945i	$0.147132\pi$
$522$	23.7783	1.04075
$523$	15.4225	0.674380	0.337190	−	0.941437i	$-0.390524\pi$
$523$	15.4225	0.674380	0.337190	+	0.941437i	$0.390524\pi$
$524$	22.7515	0.993902
$525$	0	0
$526$	−21.9257	−0.956007
$527$	3.59114	0.156432
$528$	5.84525	0.254382
$529$	19.7124	0.857060
$530$	0	0
$531$	−2.52151	−0.109424
$532$	−0.417282	−0.0180915
$533$	60.1629	2.60594
$534$	−9.15557	−0.396200
$535$	0	0
$536$	−1.14797	−0.0495848
$537$	7.20567	0.310948
$538$	41.7256	1.79892
$539$	15.8821	0.684092
$540$	0	0
$541$	−32.8616	−1.41283	−0.706415	−	0.707798i	$-0.749689\pi$
$541$	−32.8616	−1.41283	−0.706415	+	0.707798i	$0.749689\pi$
$542$	20.4432	0.878111
$543$	7.62807	0.327352
$544$	21.0446	0.902280
$545$	0	0
$546$	6.08932	0.260599
$547$	−24.1636	−1.03316	−0.516580	−	0.856239i	$-0.672795\pi$
$547$	−24.1636	−1.03316	−0.516580	+	0.856239i	$0.672795\pi$
$548$	−14.1532	−0.604596
$549$	−27.6087	−1.17831
$550$	0	0
$551$	−1.13952	−0.0485453
$552$	−2.16216	−0.0920276
$553$	−3.97539	−0.169051
$554$	−3.59456	−0.152718
$555$	0	0
$556$	23.6224	1.00181
$557$	−11.7775	−0.499029	−0.249515	−	0.968371i	$-0.580271\pi$
$557$	−11.7775	−0.499029	−0.249515	+	0.968371i	$0.580271\pi$
$558$	6.65425	0.281697
$559$	18.2739	0.772903
$560$	0	0
$561$	−3.62922	−0.153226
$562$	−15.8171	−0.667204
$563$	16.8164	0.708726	0.354363	−	0.935108i	$-0.384698\pi$
$563$	16.8164	0.708726	0.354363	+	0.935108i	$0.384698\pi$
$564$	−9.03455	−0.380423
$565$	0	0
$566$	−5.27855	−0.221874
$567$	7.02049	0.294833
$568$	8.31711	0.348978
$569$	11.0090	0.461523	0.230761	−	0.973010i	$-0.425878\pi$
$569$	11.0090	0.461523	0.230761	+	0.973010i	$0.425878\pi$
$570$	0	0
$571$	−7.19134	−0.300948	−0.150474	−	0.988614i	$-0.548080\pi$
$571$	−7.19134	−0.300948	−0.150474	+	0.988614i	$0.548080\pi$
$572$	28.7398	1.20167
$573$	−4.43497	−0.185273
$574$	17.4545	0.728537
$575$	0	0
$576$	13.5575	0.564898
$577$	−19.1617	−0.797713	−0.398856	−	0.917013i	$-0.630593\pi$
$577$	−19.1617	−0.797713	−0.398856	+	0.917013i	$0.630593\pi$
$578$	16.9210	0.703819
$579$	−3.13414	−0.130250
$580$	0	0
$581$	5.74248	0.238238
$582$	3.11584	0.129156
$583$	18.9214	0.783646
$584$	−5.70325	−0.236002
$585$	0	0
$586$	−15.4086	−0.636522
$587$	−38.0506	−1.57052	−0.785258	−	0.619168i	$-0.787470\pi$
$587$	−38.0506	−1.57052	−0.785258	+	0.619168i	$0.787470\pi$
$588$	−4.70948	−0.194216
$589$	−0.318890	−0.0131396
$590$	0	0
$591$	10.6214	0.436906
$592$	−22.0277	−0.905332
$593$	−4.74158	−0.194713	−0.0973566	−	0.995250i	$-0.531039\pi$
$593$	−4.74158	−0.194713	−0.0973566	+	0.995250i	$0.531039\pi$
$594$	−14.0089	−0.574794
$595$	0	0
$596$	5.57765	0.228470
$597$	−9.94688	−0.407099
$598$	−82.4419	−3.37130
$599$	19.6047	0.801025	0.400512	−	0.916291i	$-0.368832\pi$
$599$	19.6047	0.801025	0.400512	+	0.916291i	$0.368832\pi$
$600$	0	0
$601$	6.03922	0.246345	0.123172	−	0.992385i	$-0.460693\pi$
$601$	6.03922	0.246345	0.123172	+	0.992385i	$0.460693\pi$
$602$	5.30163	0.216078
$603$	4.61245	0.187834
$604$	4.64845	0.189143
$605$	0	0
$606$	5.67506	0.230534
$607$	−9.29720	−0.377362	−0.188681	−	0.982038i	$-0.560421\pi$
$607$	−9.29720	−0.377362	−0.188681	+	0.982038i	$0.560421\pi$
$608$	−1.86875	−0.0757876
$609$	2.17262	0.0880390
$610$	0	0
$611$	75.9703	3.07343
$612$	−12.9400	−0.523068
$613$	34.1747	1.38030	0.690152	−	0.723664i	$-0.257544\pi$
$613$	34.1747	1.38030	0.690152	+	0.723664i	$0.257544\pi$
$614$	5.05196	0.203881
$615$	0	0
$616$	−1.83882	−0.0740881
$617$	26.3076	1.05910	0.529551	−	0.848278i	$-0.322360\pi$
$617$	26.3076	1.05910	0.529551	+	0.848278i	$0.322360\pi$
$618$	2.52604	0.101612
$619$	−11.1210	−0.446989	−0.223495	−	0.974705i	$-0.571746\pi$
$619$	−11.1210	−0.446989	−0.223495	+	0.974705i	$0.571746\pi$
$620$	0	0
$621$	18.0972	0.726217
$622$	−26.2977	−1.05444
$623$	10.0588	0.402995
$624$	14.5749	0.583461
$625$	0	0
$626$	−3.09030	−0.123513
$627$	0.322272	0.0128703
$628$	−25.4875	−1.01706
$629$	13.6766	0.545324
$630$	0	0
$631$	39.5300	1.57366	0.786832	−	0.617168i	$-0.211720\pi$
$631$	39.5300	1.57366	0.786832	+	0.617168i	$0.211720\pi$
$632$	−2.72454	−0.108376
$633$	−13.7693	−0.547282
$634$	−51.1156	−2.03006
$635$	0	0
$636$	−5.61072	−0.222480
$637$	39.6014	1.56906
$638$	22.7696	0.901458
$639$	−33.4175	−1.32197
$640$	0	0
$641$	15.0420	0.594122	0.297061	−	0.954858i	$-0.403993\pi$
$641$	15.0420	0.594122	0.297061	+	0.954858i	$0.403993\pi$
$642$	−13.2993	−0.524881
$643$	−46.2988	−1.82585	−0.912923	−	0.408133i	$-0.866180\pi$
$643$	−46.2988	−1.82585	−0.912923	+	0.408133i	$0.866180\pi$
$644$	−10.7713	−0.424450
$645$	0	0
$646$	1.37700	0.0541772
$647$	−4.95546	−0.194819	−0.0974096	−	0.995244i	$-0.531056\pi$
$647$	−4.95546	−0.194819	−0.0974096	+	0.995244i	$0.531056\pi$
$648$	4.81150	0.189013
$649$	−2.41454	−0.0947790
$650$	0	0
$651$	0.607998	0.0238293
$652$	−13.6882	−0.536070
$653$	−9.58503	−0.375091	−0.187546	−	0.982256i	$-0.560053\pi$
$653$	−9.58503	−0.375091	−0.187546	+	0.982256i	$0.560053\pi$
$654$	−17.0635	−0.667235
$655$	0	0
$656$	41.7775	1.63114
$657$	22.9152	0.894007
$658$	22.0406	0.859231
$659$	13.2928	0.517812	0.258906	−	0.965903i	$-0.416638\pi$
$659$	13.2928	0.517812	0.258906	+	0.965903i	$0.416638\pi$
$660$	0	0
$661$	−47.7090	−1.85566	−0.927832	−	0.372998i	$-0.878330\pi$
$661$	−47.7090	−1.85566	−0.927832	+	0.372998i	$0.878330\pi$
$662$	−10.0499	−0.390601
$663$	−9.04930	−0.351446
$664$	3.93561	0.152731
$665$	0	0
$666$	25.3424	0.981997
$667$	−29.4146	−1.13894
$668$	−20.8265	−0.805803
$669$	−5.86818	−0.226877
$670$	0	0
$671$	−26.4375	−1.02061
$672$	3.56296	0.137444
$673$	0.733363	0.0282691	0.0141345	−	0.999900i	$-0.495501\pi$
$673$	0.733363	0.0282691	0.0141345	+	0.999900i	$0.495501\pi$
$674$	−20.0463	−0.772154
$675$	0	0
$676$	50.3591	1.93689
$677$	2.20028	0.0845637	0.0422818	−	0.999106i	$-0.486537\pi$
$677$	2.20028	0.0845637	0.0422818	+	0.999106i	$0.486537\pi$
$678$	1.06467	0.0408883
$679$	−3.42321	−0.131371
$680$	0	0
$681$	0.610291	0.0233864
$682$	6.37197	0.243995
$683$	29.3717	1.12388	0.561938	−	0.827179i	$-0.310056\pi$
$683$	29.3717	1.12388	0.561938	+	0.827179i	$0.310056\pi$
$684$	1.14906	0.0439354
$685$	0	0
$686$	24.9193	0.951423
$687$	−4.55122	−0.173640
$688$	12.6895	0.483783
$689$	47.1797	1.79741
$690$	0	0
$691$	35.9500	1.36760	0.683801	−	0.729669i	$-0.260326\pi$
$691$	35.9500	1.36760	0.683801	+	0.729669i	$0.260326\pi$
$692$	−40.8216	−1.55180
$693$	7.38822	0.280655
$694$	−27.5561	−1.04602
$695$	0	0
$696$	1.48901	0.0564406
$697$	−25.9390	−0.982511
$698$	−33.5290	−1.26909
$699$	6.78099	0.256481
$700$	0	0
$701$	−4.13295	−0.156099	−0.0780496	−	0.996949i	$-0.524869\pi$
$701$	−4.13295	−0.156099	−0.0780496	+	0.996949i	$0.524869\pi$
$702$	−34.9306	−1.31837
$703$	−1.21448	−0.0458048
$704$	12.9824	0.489293
$705$	0	0
$706$	−66.4093	−2.49935
$707$	−6.23490	−0.234487
$708$	0.715977	0.0269081
$709$	−9.76219	−0.366627	−0.183313	−	0.983055i	$-0.558682\pi$
$709$	−9.76219	−0.366627	−0.183313	+	0.983055i	$0.558682\pi$
$710$	0	0
$711$	10.9470	0.410543
$712$	6.89377	0.258355
$713$	−8.23154	−0.308274
$714$	−2.62539	−0.0982528
$715$	0	0
$716$	24.6019	0.919417
$717$	5.72675	0.213869
$718$	44.3305	1.65440
$719$	17.2120	0.641898	0.320949	−	0.947097i	$-0.395998\pi$
$719$	17.2120	0.641898	0.320949	+	0.947097i	$0.395998\pi$
$720$	0	0
$721$	−2.77523	−0.103355
$722$	36.1206	1.34427
$723$	−0.479938	−0.0178491
$724$	26.0441	0.967921
$725$	0	0
$726$	3.63084	0.134753
$727$	−14.9143	−0.553142	−0.276571	−	0.960993i	$-0.589198\pi$
$727$	−14.9143	−0.553142	−0.276571	+	0.960993i	$0.589198\pi$
$728$	−4.58501	−0.169932
$729$	−15.3452	−0.568342
$730$	0	0
$731$	−7.87873	−0.291405
$732$	7.83944	0.289754
$733$	−6.95822	−0.257008	−0.128504	−	0.991709i	$-0.541017\pi$
$733$	−6.95822	−0.257008	−0.128504	+	0.991709i	$0.541017\pi$
$734$	22.3022	0.823191
$735$	0	0
$736$	−48.2381	−1.77808
$737$	4.41679	0.162694
$738$	−48.0642	−1.76927
$739$	21.2966	0.783410	0.391705	−	0.920091i	$-0.371885\pi$
$739$	21.2966	0.783410	0.391705	+	0.920091i	$0.371885\pi$
$740$	0	0
$741$	0.803571	0.0295199
$742$	13.6878	0.502496
$743$	32.9629	1.20929	0.604645	−	0.796495i	$-0.293315\pi$
$743$	32.9629	1.20929	0.604645	+	0.796495i	$0.293315\pi$
$744$	0.416692	0.0152767
$745$	0	0
$746$	−45.9859	−1.68366
$747$	−15.8130	−0.578566
$748$	−12.3911	−0.453062
$749$	14.6112	0.533883
$750$	0	0
$751$	24.7608	0.903535	0.451768	−	0.892136i	$-0.350794\pi$
$751$	24.7608	0.903535	0.451768	+	0.892136i	$0.350794\pi$
$752$	52.7543	1.92375
$753$	9.34349	0.340496
$754$	56.7750	2.06762
$755$	0	0
$756$	−4.56381	−0.165984
$757$	−32.5617	−1.18348	−0.591738	−	0.806131i	$-0.701558\pi$
$757$	−32.5617	−1.18348	−0.591738	+	0.806131i	$0.701558\pi$
$758$	−3.99729	−0.145188
$759$	8.31885	0.301955
$760$	0	0
$761$	−18.8210	−0.682261	−0.341131	−	0.940016i	$-0.610810\pi$
$761$	−18.8210	−0.682261	−0.341131	+	0.940016i	$0.610810\pi$
$762$	−19.8914	−0.720590
$763$	18.7468	0.678678
$764$	−15.1421	−0.547820
$765$	0	0
$766$	51.7473	1.86971
$767$	−6.02055	−0.217389
$768$	9.66478	0.348748
$769$	−31.8266	−1.14770	−0.573849	−	0.818961i	$-0.694550\pi$
$769$	−31.8266	−1.14770	−0.573849	+	0.818961i	$0.694550\pi$
$770$	0	0
$771$	2.78450	0.100281
$772$	−10.7007	−0.385127
$773$	19.2083	0.690873	0.345437	−	0.938442i	$-0.387731\pi$
$773$	19.2083	0.690873	0.345437	+	0.938442i	$0.387731\pi$
$774$	−14.5990	−0.524751
$775$	0	0
$776$	−2.34610	−0.0842199
$777$	2.31553	0.0830691
$778$	−11.7763	−0.422200
$779$	2.30337	0.0825267
$780$	0	0
$781$	−31.9999	−1.14505
$782$	35.5446	1.27107
$783$	−12.4630	−0.445390
$784$	27.4995	0.982125
$785$	0	0
$786$	−12.7111	−0.453390
$787$	−10.0024	−0.356547	−0.178273	−	0.983981i	$-0.557051\pi$
$787$	−10.0024	−0.356547	−0.178273	+	0.983981i	$0.557051\pi$
$788$	36.2641	1.29185
$789$	5.51659	0.196396
$790$	0	0
$791$	−1.16969	−0.0415895
$792$	5.06352	0.179924
$793$	−65.9207	−2.34091
$794$	−2.46551	−0.0874978
$795$	0	0
$796$	−33.9611	−1.20372
$797$	16.7212	0.592294	0.296147	−	0.955142i	$-0.404298\pi$
$797$	16.7212	0.592294	0.296147	+	0.955142i	$0.404298\pi$
$798$	0.233133	0.00825281
$799$	−32.7543	−1.15877
$800$	0	0
$801$	−27.6986	−0.978682
$802$	10.6429	0.375815
$803$	21.9431	0.774355
$804$	−1.30970	−0.0461894
$805$	0	0
$806$	15.8882	0.559639
$807$	−10.4983	−0.369558
$808$	−4.27309	−0.150327
$809$	3.97269	0.139672	0.0698361	−	0.997558i	$-0.477752\pi$
$809$	3.97269	0.139672	0.0698361	+	0.997558i	$0.477752\pi$
$810$	0	0
$811$	−5.72091	−0.200888	−0.100444	−	0.994943i	$-0.532026\pi$
$811$	−5.72091	−0.200888	−0.100444	+	0.994943i	$0.532026\pi$
$812$	7.41786	0.260316
$813$	−5.14358	−0.180393
$814$	24.2673	0.850569
$815$	0	0
$816$	−6.28390	−0.219981
$817$	0.699625	0.0244768
$818$	18.5934	0.650104
$819$	18.4222	0.643723
$820$	0	0
$821$	42.9239	1.49805	0.749027	−	0.662539i	$-0.230521\pi$
$821$	42.9239	1.49805	0.749027	+	0.662539i	$0.230521\pi$
$822$	7.90733	0.275800
$823$	4.87252	0.169845	0.0849226	−	0.996388i	$-0.472936\pi$
$823$	4.87252	0.169845	0.0849226	+	0.996388i	$0.472936\pi$
$824$	−1.90200	−0.0662594
$825$	0	0
$826$	−1.74669	−0.0607750
$827$	−10.8538	−0.377422	−0.188711	−	0.982033i	$-0.560431\pi$
$827$	−10.8538	−0.377422	−0.188711	+	0.982033i	$0.560431\pi$
$828$	29.6608	1.03078
$829$	1.39402	0.0484162	0.0242081	−	0.999707i	$-0.492294\pi$
$829$	1.39402	0.0484162	0.0242081	+	0.999707i	$0.492294\pi$
$830$	0	0
$831$	0.904403	0.0313734
$832$	32.3711	1.12226
$833$	−17.0740	−0.591579
$834$	−13.1977	−0.456999
$835$	0	0
$836$	1.10032	0.0380552
$837$	−3.48771	−0.120553
$838$	−57.0771	−1.97169
$839$	−37.5022	−1.29472	−0.647360	−	0.762184i	$-0.724127\pi$
$839$	−37.5022	−1.29472	−0.647360	+	0.762184i	$0.724127\pi$
$840$	0	0
$841$	−8.74315	−0.301488
$842$	4.76755	0.164301
$843$	3.97964	0.137066
$844$	−47.0119	−1.61821
$845$	0	0
$846$	−60.6927	−2.08666
$847$	−3.98902	−0.137064
$848$	32.7620	1.12505
$849$	1.32810	0.0455803
$850$	0	0
$851$	−31.3494	−1.07464
$852$	9.48883	0.325082
$853$	41.1049	1.40741	0.703703	−	0.710495i	$-0.251529\pi$
$853$	41.1049	1.40741	0.703703	+	0.710495i	$0.251529\pi$
$854$	−19.1250	−0.654443
$855$	0	0
$856$	10.0138	0.342265
$857$	47.3654	1.61797	0.808986	−	0.587828i	$-0.200017\pi$
$857$	47.3654	1.61797	0.808986	+	0.587828i	$0.200017\pi$
$858$	−16.0567	−0.548168
$859$	−24.4805	−0.835265	−0.417632	−	0.908616i	$-0.637140\pi$
$859$	−24.4805	−0.835265	−0.417632	+	0.908616i	$0.637140\pi$
$860$	0	0
$861$	−4.39161	−0.149666
$862$	−68.7819	−2.34272
$863$	−31.7921	−1.08222	−0.541108	−	0.840953i	$-0.681995\pi$
$863$	−31.7921	−1.08222	−0.541108	+	0.840953i	$0.681995\pi$
$864$	−20.4385	−0.695332
$865$	0	0
$866$	−26.4321	−0.898198
$867$	−4.25737	−0.144588
$868$	2.07585	0.0704591
$869$	10.4826	0.355597
$870$	0	0
$871$	11.0131	0.373163
$872$	12.8481	0.435091
$873$	9.42642	0.319036
$874$	−3.15633	−0.106764
$875$	0	0
$876$	−6.50672	−0.219842
$877$	21.5378	0.727280	0.363640	−	0.931540i	$-0.381534\pi$
$877$	21.5378	0.727280	0.363640	+	0.931540i	$0.381534\pi$
$878$	6.90401	0.232999
$879$	3.87685	0.130763
$880$	0	0
$881$	−44.0697	−1.48475	−0.742374	−	0.669986i	$-0.766300\pi$
$881$	−44.0697	−1.48475	−0.742374	+	0.669986i	$0.766300\pi$
$882$	−31.6376	−1.06529
$883$	1.31082	0.0441126	0.0220563	−	0.999757i	$-0.492979\pi$
$883$	1.31082	0.0441126	0.0220563	+	0.999757i	$0.492979\pi$
$884$	−30.8965	−1.03916
$885$	0	0
$886$	74.5430	2.50432
$887$	31.2402	1.04894	0.524472	−	0.851428i	$-0.324263\pi$
$887$	31.2402	1.04894	0.524472	+	0.851428i	$0.324263\pi$
$888$	1.58695	0.0532545
$889$	21.8537	0.732949
$890$	0	0
$891$	−18.5121	−0.620179
$892$	−20.0354	−0.670835
$893$	2.90856	0.0973313
$894$	−3.11620	−0.104221
$895$	0	0
$896$	−5.45607	−0.182274
$897$	20.7427	0.692577
$898$	−34.0534	−1.13638
$899$	5.66879	0.189065
$900$	0	0
$901$	−20.3414	−0.677670
$902$	−46.0252	−1.53247
$903$	−1.33391	−0.0443897
$904$	−0.801650	−0.0266625
$905$	0	0
$906$	−2.59706	−0.0862817
$907$	33.5648	1.11450	0.557251	−	0.830344i	$-0.311856\pi$
$907$	33.5648	1.11450	0.557251	+	0.830344i	$0.311856\pi$
$908$	2.08368	0.0691495
$909$	17.1689	0.569457
$910$	0	0
$911$	45.8399	1.51874	0.759372	−	0.650657i	$-0.225506\pi$
$911$	45.8399	1.51874	0.759372	+	0.650657i	$0.225506\pi$
$912$	0.558006	0.0184774
$913$	−15.1422	−0.501133
$914$	19.1034	0.631884
$915$	0	0
$916$	−15.5390	−0.513422
$917$	13.9650	0.461166
$918$	15.0602	0.497062
$919$	19.7885	0.652764	0.326382	−	0.945238i	$-0.394170\pi$
$919$	19.7885	0.652764	0.326382	+	0.945238i	$0.394170\pi$
$920$	0	0
$921$	−1.27109	−0.0418839
$922$	−7.85804	−0.258791
$923$	−79.7902	−2.62633
$924$	−2.09787	−0.0690149
$925$	0	0
$926$	−28.4104	−0.933624
$927$	7.64209	0.250999
$928$	33.2200	1.09050
$929$	28.4786	0.934352	0.467176	−	0.884164i	$-0.345272\pi$
$929$	28.4786	0.934352	0.467176	+	0.884164i	$0.345272\pi$
$930$	0	0
$931$	1.51616	0.0496901
$932$	23.1520	0.758368
$933$	6.61659	0.216617
$934$	75.2727	2.46300
$935$	0	0
$936$	12.6257	0.412682
$937$	39.8519	1.30191	0.650953	−	0.759118i	$-0.274370\pi$
$937$	39.8519	1.30191	0.650953	+	0.759118i	$0.274370\pi$
$938$	3.19512	0.104324
$939$	0.777530	0.0253737
$940$	0	0
$941$	28.5053	0.929248	0.464624	−	0.885508i	$-0.346190\pi$
$941$	28.5053	0.929248	0.464624	+	0.885508i	$0.346190\pi$
$942$	14.2397	0.463955
$943$	59.4570	1.93619
$944$	−4.18071	−0.136071
$945$	0	0
$946$	−13.9797	−0.454520
$947$	−4.92424	−0.160016	−0.0800081	−	0.996794i	$-0.525495\pi$
$947$	−4.92424	−0.160016	−0.0800081	+	0.996794i	$0.525495\pi$
$948$	−3.10837	−0.100955
$949$	54.7141	1.77610
$950$	0	0
$951$	12.8609	0.417042
$952$	1.97681	0.0640688
$953$	−9.29302	−0.301030	−0.150515	−	0.988608i	$-0.548093\pi$
$953$	−9.29302	−0.301030	−0.150515	+	0.988608i	$0.548093\pi$
$954$	−37.6919	−1.22032
$955$	0	0
$956$	19.5525	0.632373
$957$	−5.72891	−0.185189
$958$	67.3793	2.17692
$959$	−8.68737	−0.280530
$960$	0	0
$961$	−29.4136	−0.948826
$962$	60.5094	1.95090
$963$	−40.2347	−1.29654
$964$	−1.63863	−0.0527766
$965$	0	0
$966$	6.01788	0.193622
$967$	28.3165	0.910598	0.455299	−	0.890339i	$-0.349532\pi$
$967$	28.3165	0.910598	0.455299	+	0.890339i	$0.349532\pi$
$968$	−2.73388	−0.0878701
$969$	−0.346457	−0.0111298
$970$	0	0
$971$	−2.96567	−0.0951728	−0.0475864	−	0.998867i	$-0.515153\pi$
$971$	−2.96567	−0.0951728	−0.0475864	+	0.998867i	$0.515153\pi$
$972$	19.1018	0.612691
$973$	14.4996	0.464836
$974$	−42.2099	−1.35249
$975$	0	0
$976$	−45.7758	−1.46525
$977$	45.3827	1.45192	0.725960	−	0.687737i	$-0.241396\pi$
$977$	45.3827	1.45192	0.725960	+	0.687737i	$0.241396\pi$
$978$	7.64750	0.244540
$979$	−26.5236	−0.847698
$980$	0	0
$981$	−51.6226	−1.64818
$982$	−55.2042	−1.76164
$983$	27.3131	0.871154	0.435577	−	0.900152i	$-0.356544\pi$
$983$	27.3131	0.871154	0.435577	+	0.900152i	$0.356544\pi$
$984$	−3.00979	−0.0959487
$985$	0	0
$986$	−24.4784	−0.779550
$987$	−5.54548	−0.176515
$988$	2.74359	0.0872852
$989$	18.0595	0.574258
$990$	0	0
$991$	33.6795	1.06987	0.534933	−	0.844895i	$-0.320337\pi$
$991$	33.6795	1.06987	0.534933	+	0.844895i	$0.320337\pi$
$992$	9.29646	0.295163
$993$	2.52859	0.0802424
$994$	−23.1488	−0.734236
$995$	0	0
$996$	4.49006	0.142273
$997$	−4.10550	−0.130023	−0.0650113	−	0.997885i	$-0.520708\pi$
$997$	−4.10550	−0.130023	−0.0650113	+	0.997885i	$0.520708\pi$
$998$	41.5807	1.31621
$999$	−13.2827	−0.420247

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.q.1.13		66
5.2	odd	4		1205.2.b.d.724.13	✓	66
5.3	odd	4		1205.2.b.d.724.54	yes	66
5.4	even	2	inner	6025.2.a.q.1.54		66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.d.724.13	✓	66	5.2	odd	4
1205.2.b.d.724.54	yes	66	5.3	odd	4
6025.2.a.q.1.13		66	1.1	even	1	trivial
6025.2.a.q.1.54		66	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.90752 q^{2} +0.479938 q^{3} +1.63863 q^{4} -0.915491 q^{6} +1.00580 q^{7} +0.689327 q^{8} -2.76966 q^{9} +O(q^{10})\)	\(q-1.90752 q^{2} +0.479938 q^{3} +1.63863 q^{4} -0.915491 q^{6} +1.00580 q^{7} +0.689327 q^{8} -2.76966 q^{9} -2.65217 q^{11} +0.786439 q^{12} -6.61306 q^{13} -1.91859 q^{14} -4.59216 q^{16} +2.85120 q^{17} +5.28318 q^{18} -0.253184 q^{19} +0.482723 q^{21} +5.05906 q^{22} -6.53547 q^{23} +0.330834 q^{24} +12.6145 q^{26} -2.76908 q^{27} +1.64813 q^{28} +4.50076 q^{29} +1.25952 q^{31} +7.38097 q^{32} -1.27288 q^{33} -5.43871 q^{34} -4.53844 q^{36} +4.79681 q^{37} +0.482954 q^{38} -3.17386 q^{39} -9.09759 q^{41} -0.920803 q^{42} -2.76330 q^{43} -4.34591 q^{44} +12.4665 q^{46} -11.4879 q^{47} -2.20395 q^{48} -5.98836 q^{49} +1.36840 q^{51} -10.8363 q^{52} -7.13433 q^{53} +5.28207 q^{54} +0.693326 q^{56} -0.121513 q^{57} -8.58528 q^{58} +0.910403 q^{59} +9.96827 q^{61} -2.40255 q^{62} -2.78573 q^{63} -4.89502 q^{64} +2.42804 q^{66} -1.66535 q^{67} +4.67205 q^{68} -3.13662 q^{69} +12.0656 q^{71} -1.90920 q^{72} -8.27365 q^{73} -9.14999 q^{74} -0.414874 q^{76} -2.66756 q^{77} +6.05419 q^{78} -3.95246 q^{79} +6.97999 q^{81} +17.3538 q^{82} +5.70936 q^{83} +0.791002 q^{84} +5.27105 q^{86} +2.16009 q^{87} -1.82821 q^{88} +10.0007 q^{89} -6.65143 q^{91} -10.7092 q^{92} +0.604491 q^{93} +21.9134 q^{94} +3.54241 q^{96} -3.40346 q^{97} +11.4229 q^{98} +7.34560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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