LMFDB - Embedded newform 6025.2.a.p.1.14

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:	$1$
Dimension:	$46$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q-1.30874 q^{2} -1.88426 q^{3} -0.287190 q^{4} +2.46601 q^{6} +1.28779 q^{7} +2.99335 q^{8} +0.550432 q^{9} +O(q^{10})$	\(q-1.30874 q^{2} -1.88426 q^{3} -0.287190 q^{4} +2.46601 q^{6} +1.28779 q^{7} +2.99335 q^{8} +0.550432 q^{9} +0.698979 q^{11} +0.541141 q^{12} -5.53802 q^{13} -1.68539 q^{14} -3.34314 q^{16} +3.43199 q^{17} -0.720375 q^{18} -3.98407 q^{19} -2.42653 q^{21} -0.914784 q^{22} +6.27054 q^{23} -5.64024 q^{24} +7.24785 q^{26} +4.61562 q^{27} -0.369841 q^{28} +6.15912 q^{29} -0.212499 q^{31} -1.61138 q^{32} -1.31706 q^{33} -4.49160 q^{34} -0.158079 q^{36} -5.91052 q^{37} +5.21413 q^{38} +10.4351 q^{39} -9.65382 q^{41} +3.17571 q^{42} -7.78957 q^{43} -0.200740 q^{44} -8.20652 q^{46} +3.86035 q^{47} +6.29934 q^{48} -5.34159 q^{49} -6.46676 q^{51} +1.59047 q^{52} +10.2480 q^{53} -6.04066 q^{54} +3.85481 q^{56} +7.50702 q^{57} -8.06070 q^{58} -9.54071 q^{59} +8.82214 q^{61} +0.278107 q^{62} +0.708842 q^{63} +8.79516 q^{64} +1.72369 q^{66} +7.38392 q^{67} -0.985635 q^{68} -11.8153 q^{69} -6.00088 q^{71} +1.64763 q^{72} -2.96121 q^{73} +7.73535 q^{74} +1.14419 q^{76} +0.900139 q^{77} -13.6568 q^{78} -6.80832 q^{79} -10.3483 q^{81} +12.6344 q^{82} +16.1202 q^{83} +0.696877 q^{84} +10.1946 q^{86} -11.6054 q^{87} +2.09228 q^{88} +1.74849 q^{89} -7.13182 q^{91} -1.80084 q^{92} +0.400404 q^{93} -5.05221 q^{94} +3.03625 q^{96} +0.714461 q^{97} +6.99077 q^{98} +0.384741 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * 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 $

$2$	−1.30874	−0.925421	−0.462711	−	0.886509i	$-0.653123\pi$
$2$	−1.30874	−0.925421	−0.462711	+	0.886509i	$0.653123\pi$
$3$	−1.88426	−1.08788	−0.543939	−	0.839125i	$-0.683068\pi$
$3$	−1.88426	−1.08788	−0.543939	+	0.839125i	$0.683068\pi$
$4$	−0.287190	−0.143595
$5$	0	0
$5$	0	0
$6$	2.46601	1.00675
$7$	1.28779	0.486739	0.243370	−	0.969934i	$-0.421747\pi$
$7$	1.28779	0.486739	0.243370	+	0.969934i	$0.421747\pi$
$8$	2.99335	1.05831
$9$	0.550432	0.183477
$10$	0	0
$11$	0.698979	0.210750	0.105375	−	0.994433i	$-0.466396\pi$
$11$	0.698979	0.210750	0.105375	+	0.994433i	$0.466396\pi$
$12$	0.541141	0.156214
$13$	−5.53802	−1.53597	−0.767985	−	0.640468i	$-0.778741\pi$
$13$	−5.53802	−1.53597	−0.767985	+	0.640468i	$0.778741\pi$
$14$	−1.68539	−0.450439
$15$	0	0
$16$	−3.34314	−0.835785
$17$	3.43199	0.832380	0.416190	−	0.909278i	$-0.363365\pi$
$17$	3.43199	0.832380	0.416190	+	0.909278i	$0.363365\pi$
$18$	−0.720375	−0.169794
$19$	−3.98407	−0.914009	−0.457004	−	0.889464i	$-0.651078\pi$
$19$	−3.98407	−0.914009	−0.457004	+	0.889464i	$0.651078\pi$
$20$	0	0
$21$	−2.42653	−0.529513
$22$	−0.914784	−0.195033
$23$	6.27054	1.30750	0.653749	−	0.756712i	$-0.273195\pi$
$23$	6.27054	1.30750	0.653749	+	0.756712i	$0.273195\pi$
$24$	−5.64024	−1.15131
$25$	0	0
$26$	7.24785	1.42142
$27$	4.61562	0.888276
$28$	−0.369841	−0.0698935
$29$	6.15912	1.14372	0.571859	−	0.820352i	$-0.306222\pi$
$29$	6.15912	1.14372	0.571859	+	0.820352i	$0.306222\pi$
$30$	0	0
$31$	−0.212499	−0.0381660	−0.0190830	−	0.999818i	$-0.506075\pi$
$31$	−0.212499	−0.0381660	−0.0190830	+	0.999818i	$0.506075\pi$
$32$	−1.61138	−0.284854
$33$	−1.31706	−0.229270
$34$	−4.49160	−0.770302
$35$	0	0
$36$	−0.158079	−0.0263465
$37$	−5.91052	−0.971683	−0.485841	−	0.874047i	$-0.661487\pi$
$37$	−5.91052	−0.971683	−0.485841	+	0.874047i	$0.661487\pi$
$38$	5.21413	0.845843
$39$	10.4351	1.67095
$40$	0	0
$41$	−9.65382	−1.50767	−0.753837	−	0.657062i	$-0.771799\pi$
$41$	−9.65382	−1.50767	−0.753837	+	0.657062i	$0.771799\pi$
$42$	3.17571	0.490023
$43$	−7.78957	−1.18790	−0.593949	−	0.804502i	$-0.702432\pi$
$43$	−7.78957	−1.18790	−0.593949	+	0.804502i	$0.702432\pi$
$44$	−0.200740	−0.0302627
$45$	0	0
$46$	−8.20652	−1.20999
$47$	3.86035	0.563090	0.281545	−	0.959548i	$-0.409153\pi$
$47$	3.86035	0.563090	0.281545	+	0.959548i	$0.409153\pi$
$48$	6.29934	0.909232
$49$	−5.34159	−0.763085
$50$	0	0
$51$	−6.46676	−0.905527
$52$	1.59047	0.220558
$53$	10.2480	1.40767	0.703837	−	0.710361i	$-0.251468\pi$
$53$	10.2480	1.40767	0.703837	+	0.710361i	$0.251468\pi$
$54$	−6.04066	−0.822030
$55$	0	0
$56$	3.85481	0.515120
$57$	7.50702	0.994329
$58$	−8.06070	−1.05842
$59$	−9.54071	−1.24209	−0.621047	−	0.783773i	$-0.713293\pi$
$59$	−9.54071	−1.24209	−0.621047	+	0.783773i	$0.713293\pi$
$60$	0	0
$61$	8.82214	1.12956	0.564779	−	0.825242i	$-0.308961\pi$
$61$	8.82214	1.12956	0.564779	+	0.825242i	$0.308961\pi$
$62$	0.278107	0.0353196
$63$	0.708842	0.0893057
$64$	8.79516	1.09940
$65$	0	0
$66$	1.72369	0.212172
$67$	7.38392	0.902089	0.451045	−	0.892501i	$-0.351052\pi$
$67$	7.38392	0.902089	0.451045	+	0.892501i	$0.351052\pi$
$68$	−0.985635	−0.119526
$69$	−11.8153	−1.42240
$70$	0	0
$71$	−6.00088	−0.712174	−0.356087	−	0.934453i	$-0.615889\pi$
$71$	−6.00088	−0.712174	−0.356087	+	0.934453i	$0.615889\pi$
$72$	1.64763	0.194176
$73$	−2.96121	−0.346584	−0.173292	−	0.984870i	$-0.555440\pi$
$73$	−2.96121	−0.346584	−0.173292	+	0.984870i	$0.555440\pi$
$74$	7.73535	0.899216
$75$	0	0
$76$	1.14419	0.131247
$77$	0.900139	0.102580
$78$	−13.6568	−1.54633
$79$	−6.80832	−0.765996	−0.382998	−	0.923749i	$-0.625108\pi$
$79$	−6.80832	−0.765996	−0.382998	+	0.923749i	$0.625108\pi$
$80$	0	0
$81$	−10.3483	−1.14981
$82$	12.6344	1.39523
$83$	16.1202	1.76942	0.884709	−	0.466144i	$-0.154357\pi$
$83$	16.1202	1.76942	0.884709	+	0.466144i	$0.154357\pi$
$84$	0.696877	0.0760355
$85$	0	0
$86$	10.1946	1.09931
$87$	−11.6054	−1.24423
$88$	2.09228	0.223038
$89$	1.74849	0.185340	0.0926698	−	0.995697i	$-0.470460\pi$
$89$	1.74849	0.185340	0.0926698	+	0.995697i	$0.470460\pi$
$90$	0	0
$91$	−7.13182	−0.747617
$92$	−1.80084	−0.187750
$93$	0.400404	0.0415199
$94$	−5.05221	−0.521096
$95$	0	0
$96$	3.03625	0.309886
$97$	0.714461	0.0725425	0.0362713	−	0.999342i	$-0.488452\pi$
$97$	0.714461	0.0725425	0.0362713	+	0.999342i	$0.488452\pi$
$98$	6.99077	0.706175
$99$	0.384741	0.0386679
$100$	0	0
$101$	−2.31355	−0.230207	−0.115104	−	0.993353i	$-0.536720\pi$
$101$	−2.31355	−0.230207	−0.115104	+	0.993353i	$0.536720\pi$
$102$	8.46333	0.837994
$103$	−2.90751	−0.286485	−0.143243	−	0.989688i	$-0.545753\pi$
$103$	−2.90751	−0.286485	−0.143243	+	0.989688i	$0.545753\pi$
$104$	−16.5772	−1.62553
$105$	0	0
$106$	−13.4120	−1.30269
$107$	14.8579	1.43637	0.718186	−	0.695851i	$-0.244973\pi$
$107$	14.8579	1.43637	0.718186	+	0.695851i	$0.244973\pi$
$108$	−1.32556	−0.127552
$109$	−11.2125	−1.07396	−0.536980	−	0.843595i	$-0.680435\pi$
$109$	−11.2125	−1.07396	−0.536980	+	0.843595i	$0.680435\pi$
$110$	0	0
$111$	11.1369	1.05707
$112$	−4.30527	−0.406810
$113$	15.7753	1.48402	0.742009	−	0.670390i	$-0.233873\pi$
$113$	15.7753	1.48402	0.742009	+	0.670390i	$0.233873\pi$
$114$	−9.82477	−0.920174
$115$	0	0
$116$	−1.76884	−0.164233
$117$	−3.04831	−0.281816
$118$	12.4863	1.14946
$119$	4.41969	0.405152
$120$	0	0
$121$	−10.5114	−0.955584
$122$	−11.5459	−1.04532
$123$	18.1903	1.64016
$124$	0.0610277	0.00548045
$125$	0	0
$126$	−0.927693	−0.0826454
$127$	−12.8024	−1.13603	−0.568016	−	0.823018i	$-0.692289\pi$
$127$	−12.8024	−1.13603	−0.568016	+	0.823018i	$0.692289\pi$
$128$	−8.28786	−0.732550
$129$	14.6776	1.29229
$130$	0	0
$131$	−1.22476	−0.107008	−0.0535039	−	0.998568i	$-0.517039\pi$
$131$	−1.22476	−0.107008	−0.0535039	+	0.998568i	$0.517039\pi$
$132$	0.378246	0.0329221
$133$	−5.13065	−0.444884
$134$	−9.66366	−0.834813
$135$	0	0
$136$	10.2731	0.880914
$137$	−2.83614	−0.242308	−0.121154	−	0.992634i	$-0.538659\pi$
$137$	−2.83614	−0.242308	−0.121154	+	0.992634i	$0.538659\pi$
$138$	15.4632	1.31632
$139$	2.03279	0.172419	0.0862095	−	0.996277i	$-0.472525\pi$
$139$	2.03279	0.172419	0.0862095	+	0.996277i	$0.472525\pi$
$140$	0	0
$141$	−7.27390	−0.612573
$142$	7.85362	0.659061
$143$	−3.87096	−0.323706
$144$	−1.84017	−0.153348
$145$	0	0
$146$	3.87547	0.320736
$147$	10.0649	0.830143
$148$	1.69744	0.139529
$149$	21.0260	1.72252	0.861259	−	0.508166i	$-0.169676\pi$
$149$	21.0260	1.72252	0.861259	+	0.508166i	$0.169676\pi$
$150$	0	0
$151$	5.32656	0.433469	0.216735	−	0.976231i	$-0.430459\pi$
$151$	5.32656	0.433469	0.216735	+	0.976231i	$0.430459\pi$
$152$	−11.9257	−0.967302
$153$	1.88908	0.152723
$154$	−1.17805	−0.0949300
$155$	0	0
$156$	−2.99685	−0.239940
$157$	−16.8020	−1.34094	−0.670471	−	0.741936i	$-0.733908\pi$
$157$	−16.8020	−1.34094	−0.670471	+	0.741936i	$0.733908\pi$
$158$	8.91035	0.708869
$159$	−19.3099	−1.53138
$160$	0	0
$161$	8.07514	0.636410
$162$	13.5433	1.06406
$163$	−6.28719	−0.492450	−0.246225	−	0.969213i	$-0.579190\pi$
$163$	−6.28719	−0.492450	−0.246225	+	0.969213i	$0.579190\pi$
$164$	2.77249	0.216495
$165$	0	0
$166$	−21.0972	−1.63746
$167$	15.4202	1.19325	0.596626	−	0.802519i	$-0.296508\pi$
$167$	15.4202	1.19325	0.596626	+	0.802519i	$0.296508\pi$
$168$	−7.26345	−0.560387
$169$	17.6697	1.35921
$170$	0	0
$171$	−2.19296	−0.167700
$172$	2.23709	0.170577
$173$	3.72835	0.283461	0.141731	−	0.989905i	$-0.454733\pi$
$173$	3.72835	0.283461	0.141731	+	0.989905i	$0.454733\pi$
$174$	15.1885	1.15143
$175$	0	0
$176$	−2.33678	−0.176142
$177$	17.9772	1.35125
$178$	−2.28833	−0.171517
$179$	10.9282	0.816811	0.408405	−	0.912801i	$-0.366085\pi$
$179$	10.9282	0.816811	0.408405	+	0.912801i	$0.366085\pi$
$180$	0	0
$181$	12.6140	0.937590	0.468795	−	0.883307i	$-0.344688\pi$
$181$	12.6140	0.937590	0.468795	+	0.883307i	$0.344688\pi$
$182$	9.33372	0.691861
$183$	−16.6232	−1.22882
$184$	18.7699	1.38373
$185$	0	0
$186$	−0.524026	−0.0384234
$187$	2.39889	0.175424
$188$	−1.10866	−0.0808571
$189$	5.94396	0.432359
$190$	0	0
$191$	14.4657	1.04670	0.523349	−	0.852118i	$-0.324682\pi$
$191$	14.4657	1.04670	0.523349	+	0.852118i	$0.324682\pi$
$192$	−16.5724	−1.19601
$193$	8.80554	0.633837	0.316918	−	0.948453i	$-0.397352\pi$
$193$	8.80554	0.633837	0.316918	+	0.948453i	$0.397352\pi$
$194$	−0.935046	−0.0671324
$195$	0	0
$196$	1.53405	0.109575
$197$	−20.3391	−1.44910	−0.724552	−	0.689220i	$-0.757953\pi$
$197$	−20.3391	−1.44910	−0.724552	+	0.689220i	$0.757953\pi$
$198$	−0.503527	−0.0357841
$199$	18.6435	1.32160	0.660800	−	0.750562i	$-0.270217\pi$
$199$	18.6435	1.32160	0.660800	+	0.750562i	$0.270217\pi$
$200$	0	0
$201$	−13.9132	−0.981363
$202$	3.02785	0.213039
$203$	7.93166	0.556693
$204$	1.85719	0.130029
$205$	0	0
$206$	3.80518	0.265120
$207$	3.45151	0.239896
$208$	18.5144	1.28374
$209$	−2.78478	−0.192627
$210$	0	0
$211$	−9.60455	−0.661205	−0.330602	−	0.943770i	$-0.607252\pi$
$211$	−9.60455	−0.661205	−0.330602	+	0.943770i	$0.607252\pi$
$212$	−2.94314	−0.202135
$213$	11.3072	0.774758
$214$	−19.4452	−1.32925
$215$	0	0
$216$	13.8161	0.940070
$217$	−0.273655	−0.0185769
$218$	14.6743	0.993866
$219$	5.57970	0.377041
$220$	0	0
$221$	−19.0064	−1.27851
$222$	−14.5754	−0.978237
$223$	21.8910	1.46593	0.732967	−	0.680265i	$-0.238135\pi$
$223$	21.8910	1.46593	0.732967	+	0.680265i	$0.238135\pi$
$224$	−2.07512	−0.138650
$225$	0	0
$226$	−20.6459	−1.37334
$227$	25.5764	1.69757	0.848783	−	0.528742i	$-0.177336\pi$
$227$	25.5764	1.69757	0.848783	+	0.528742i	$0.177336\pi$
$228$	−2.15595	−0.142781
$229$	17.6417	1.16580	0.582899	−	0.812544i	$-0.301918\pi$
$229$	17.6417	1.16580	0.582899	+	0.812544i	$0.301918\pi$
$230$	0	0
$231$	−1.69609	−0.111595
$232$	18.4364	1.21041
$233$	9.74499	0.638416	0.319208	−	0.947685i	$-0.396583\pi$
$233$	9.74499	0.638416	0.319208	+	0.947685i	$0.396583\pi$
$234$	3.98945	0.260799
$235$	0	0
$236$	2.74000	0.178359
$237$	12.8286	0.833310
$238$	−5.78424	−0.374936
$239$	−29.1323	−1.88441	−0.942205	−	0.335037i	$-0.891251\pi$
$239$	−29.1323	−1.88441	−0.942205	+	0.335037i	$0.891251\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	13.7568	0.884318
$243$	5.65206	0.362580
$244$	−2.53363	−0.162199
$245$	0	0
$246$	−23.8064	−1.51784
$247$	22.0639	1.40389
$248$	−0.636084	−0.0403913
$249$	−30.3746	−1.92491
$250$	0	0
$251$	17.9810	1.13495	0.567474	−	0.823391i	$-0.307921\pi$
$251$	17.9810	1.13495	0.567474	+	0.823391i	$0.307921\pi$
$252$	−0.203573	−0.0128239
$253$	4.38297	0.275555
$254$	16.7551	1.05131
$255$	0	0
$256$	−6.74365	−0.421478
$257$	−20.0744	−1.25221	−0.626105	−	0.779739i	$-0.715352\pi$
$257$	−20.0744	−1.25221	−0.626105	+	0.779739i	$0.715352\pi$
$258$	−19.2092	−1.19591
$259$	−7.61151	−0.472956
$260$	0	0
$261$	3.39018	0.209847
$262$	1.60290	0.0990273
$263$	−15.6319	−0.963906	−0.481953	−	0.876197i	$-0.660072\pi$
$263$	−15.6319	−0.963906	−0.481953	+	0.876197i	$0.660072\pi$
$264$	−3.94241	−0.242638
$265$	0	0
$266$	6.71471	0.411705
$267$	−3.29461	−0.201627
$268$	−2.12059	−0.129536
$269$	22.2853	1.35876	0.679379	−	0.733787i	$-0.262249\pi$
$269$	22.2853	1.35876	0.679379	+	0.733787i	$0.262249\pi$
$270$	0	0
$271$	−19.6867	−1.19588	−0.597939	−	0.801541i	$-0.704014\pi$
$271$	−19.6867	−1.19588	−0.597939	+	0.801541i	$0.704014\pi$
$272$	−11.4736	−0.695691
$273$	13.4382	0.813316
$274$	3.71178	0.224237
$275$	0	0
$276$	3.39325	0.204249
$277$	−19.8218	−1.19097	−0.595487	−	0.803365i	$-0.703041\pi$
$277$	−19.8218	−1.19097	−0.595487	+	0.803365i	$0.703041\pi$
$278$	−2.66040	−0.159560
$279$	−0.116966	−0.00700260
$280$	0	0
$281$	−25.1818	−1.50222	−0.751111	−	0.660176i	$-0.770482\pi$
$281$	−25.1818	−1.50222	−0.751111	+	0.660176i	$0.770482\pi$
$282$	9.51967	0.566888
$283$	1.69555	0.100790	0.0503950	−	0.998729i	$-0.483952\pi$
$283$	1.69555	0.100790	0.0503950	+	0.998729i	$0.483952\pi$
$284$	1.72340	0.102265
$285$	0	0
$286$	5.06609	0.299564
$287$	−12.4321	−0.733844
$288$	−0.886955	−0.0522643
$289$	−5.22144	−0.307144
$290$	0	0
$291$	−1.34623	−0.0789174
$292$	0.850433	0.0497678
$293$	21.2874	1.24362	0.621811	−	0.783167i	$-0.286397\pi$
$293$	21.2874	1.24362	0.621811	+	0.783167i	$0.286397\pi$
$294$	−13.1724	−0.768232
$295$	0	0
$296$	−17.6922	−1.02834
$297$	3.22622	0.187204
$298$	−27.5177	−1.59405
$299$	−34.7264	−2.00828
$300$	0	0
$301$	−10.0313	−0.578197
$302$	−6.97110	−0.401142
$303$	4.35933	0.250437
$304$	13.3193	0.763915
$305$	0	0
$306$	−2.47232	−0.141333
$307$	−17.5022	−0.998901	−0.499450	−	0.866342i	$-0.666465\pi$
$307$	−17.5022	−0.998901	−0.499450	+	0.866342i	$0.666465\pi$
$308$	−0.258511	−0.0147300
$309$	5.47850	0.311661
$310$	0	0
$311$	−27.1108	−1.53732	−0.768658	−	0.639661i	$-0.779075\pi$
$311$	−27.1108	−1.53732	−0.768658	+	0.639661i	$0.779075\pi$
$312$	31.2358	1.76838
$313$	−5.43613	−0.307268	−0.153634	−	0.988128i	$-0.549098\pi$
$313$	−5.43613	−0.307268	−0.153634	+	0.988128i	$0.549098\pi$
$314$	21.9894	1.24094
$315$	0	0
$316$	1.95528	0.109993
$317$	1.24021	0.0696573	0.0348287	−	0.999393i	$-0.488911\pi$
$317$	1.24021	0.0696573	0.0348287	+	0.999393i	$0.488911\pi$
$318$	25.2718	1.41717
$319$	4.30509	0.241039
$320$	0	0
$321$	−27.9962	−1.56260
$322$	−10.5683	−0.588948
$323$	−13.6733	−0.760802
$324$	2.97194	0.165108
$325$	0	0
$326$	8.22831	0.455724
$327$	21.1272	1.16834
$328$	−28.8972	−1.59558
$329$	4.97133	0.274078
$330$	0	0
$331$	−13.6324	−0.749307	−0.374654	−	0.927165i	$-0.622238\pi$
$331$	−13.6324	−0.749307	−0.374654	+	0.927165i	$0.622238\pi$
$332$	−4.62956	−0.254080
$333$	−3.25334	−0.178282
$334$	−20.1811	−1.10426
$335$	0	0
$336$	8.11224	0.442559
$337$	−9.45300	−0.514938	−0.257469	−	0.966287i	$-0.582888\pi$
$337$	−9.45300	−0.514938	−0.257469	+	0.966287i	$0.582888\pi$
$338$	−23.1251	−1.25784
$339$	−29.7248	−1.61443
$340$	0	0
$341$	−0.148532	−0.00804348
$342$	2.87002	0.155193
$343$	−15.8934	−0.858163
$344$	−23.3169	−1.25716
$345$	0	0
$346$	−4.87945	−0.262321
$347$	−25.8641	−1.38846	−0.694230	−	0.719754i	$-0.744255\pi$
$347$	−25.8641	−1.38846	−0.694230	+	0.719754i	$0.744255\pi$
$348$	3.33295	0.178665
$349$	10.7675	0.576372	0.288186	−	0.957575i	$-0.406948\pi$
$349$	10.7675	0.576372	0.288186	+	0.957575i	$0.406948\pi$
$350$	0	0
$351$	−25.5614	−1.36437
$352$	−1.12632	−0.0600330
$353$	−29.7801	−1.58504	−0.792518	−	0.609848i	$-0.791230\pi$
$353$	−29.7801	−1.58504	−0.792518	+	0.609848i	$0.791230\pi$
$354$	−23.5275	−1.25047
$355$	0	0
$356$	−0.502150	−0.0266139
$357$	−8.32784	−0.440756
$358$	−14.3022	−0.755894
$359$	9.88958	0.521952	0.260976	−	0.965345i	$-0.415956\pi$
$359$	9.88958	0.521952	0.260976	+	0.965345i	$0.415956\pi$
$360$	0	0
$361$	−3.12718	−0.164588
$362$	−16.5085	−0.867666
$363$	19.8063	1.03956
$364$	2.04819	0.107354
$365$	0	0
$366$	21.7555	1.13718
$367$	24.3156	1.26926	0.634631	−	0.772815i	$-0.281152\pi$
$367$	24.3156	1.26926	0.634631	+	0.772815i	$0.281152\pi$
$368$	−20.9633	−1.09279
$369$	−5.31378	−0.276624
$370$	0	0
$371$	13.1973	0.685171
$372$	−0.114992	−0.00596206
$373$	−6.98503	−0.361671	−0.180836	−	0.983513i	$-0.557880\pi$
$373$	−6.98503	−0.361671	−0.180836	+	0.983513i	$0.557880\pi$
$374$	−3.13953	−0.162341
$375$	0	0
$376$	11.5554	0.595923
$377$	−34.1093	−1.75672
$378$	−7.77911	−0.400114
$379$	24.8142	1.27462	0.637309	−	0.770608i	$-0.280047\pi$
$379$	24.8142	1.27462	0.637309	+	0.770608i	$0.280047\pi$
$380$	0	0
$381$	24.1231	1.23586
$382$	−18.9318	−0.968637
$383$	−7.88965	−0.403142	−0.201571	−	0.979474i	$-0.564605\pi$
$383$	−7.88965	−0.403142	−0.201571	+	0.979474i	$0.564605\pi$
$384$	15.6165	0.796924
$385$	0	0
$386$	−11.5242	−0.586566
$387$	−4.28763	−0.217953
$388$	−0.205186	−0.0104168
$389$	−18.8488	−0.955670	−0.477835	−	0.878450i	$-0.658578\pi$
$389$	−18.8488	−0.955670	−0.477835	+	0.878450i	$0.658578\pi$
$390$	0	0
$391$	21.5204	1.08833
$392$	−15.9892	−0.807578
$393$	2.30776	0.116411
$394$	26.6187	1.34103
$395$	0	0
$396$	−0.110494	−0.00555252
$397$	0.758306	0.0380583	0.0190292	−	0.999819i	$-0.493942\pi$
$397$	0.758306	0.0380583	0.0190292	+	0.999819i	$0.493942\pi$
$398$	−24.3995	−1.22304
$399$	9.66748	0.483979
$400$	0	0
$401$	−34.8303	−1.73934	−0.869672	−	0.493629i	$-0.835670\pi$
$401$	−34.8303	−1.73934	−0.869672	+	0.493629i	$0.835670\pi$
$402$	18.2088	0.908174
$403$	1.17682	0.0586218
$404$	0.664431	0.0330567
$405$	0	0
$406$	−10.3805	−0.515176
$407$	−4.13132	−0.204782
$408$	−19.3572	−0.958326
$409$	17.3471	0.857760	0.428880	−	0.903361i	$-0.358908\pi$
$409$	17.3471	0.857760	0.428880	+	0.903361i	$0.358908\pi$
$410$	0	0
$411$	5.34402	0.263601
$412$	0.835008	0.0411379
$413$	−12.2864	−0.604576
$414$	−4.51714	−0.222005
$415$	0	0
$416$	8.92384	0.437527
$417$	−3.83031	−0.187571
$418$	3.64456	0.178261
$419$	−22.5895	−1.10357	−0.551786	−	0.833986i	$-0.686053\pi$
$419$	−22.5895	−1.10357	−0.551786	+	0.833986i	$0.686053\pi$
$420$	0	0
$421$	−27.8005	−1.35491	−0.677457	−	0.735563i	$-0.736918\pi$
$421$	−27.8005	−1.35491	−0.677457	+	0.735563i	$0.736918\pi$
$422$	12.5699	0.611893
$423$	2.12486	0.103314
$424$	30.6759	1.48975
$425$	0	0
$426$	−14.7982	−0.716977
$427$	11.3611	0.549801
$428$	−4.26706	−0.206256
$429$	7.29389	0.352152
$430$	0	0
$431$	−17.6040	−0.847957	−0.423978	−	0.905672i	$-0.639367\pi$
$431$	−17.6040	−0.847957	−0.423978	+	0.905672i	$0.639367\pi$
$432$	−15.4307	−0.742408
$433$	−7.38130	−0.354723	−0.177361	−	0.984146i	$-0.556756\pi$
$433$	−7.38130	−0.354723	−0.177361	+	0.984146i	$0.556756\pi$
$434$	0.358144	0.0171915
$435$	0	0
$436$	3.22012	0.154216
$437$	−24.9823	−1.19506
$438$	−7.30239	−0.348922
$439$	2.17048	0.103591	0.0517956	−	0.998658i	$-0.483506\pi$
$439$	2.17048	0.103591	0.0517956	+	0.998658i	$0.483506\pi$
$440$	0	0
$441$	−2.94019	−0.140009
$442$	24.8745	1.18316
$443$	12.7026	0.603518	0.301759	−	0.953384i	$-0.402426\pi$
$443$	12.7026	0.603518	0.301759	+	0.953384i	$0.402426\pi$
$444$	−3.19842	−0.151790
$445$	0	0
$446$	−28.6498	−1.35661
$447$	−39.6185	−1.87389
$448$	11.3263	0.535119
$449$	−4.30823	−0.203318	−0.101659	−	0.994819i	$-0.532415\pi$
$449$	−4.30823	−0.203318	−0.101659	+	0.994819i	$0.532415\pi$
$450$	0	0
$451$	−6.74781	−0.317742
$452$	−4.53052	−0.213098
$453$	−10.0366	−0.471561
$454$	−33.4730	−1.57096
$455$	0	0
$456$	22.4711	1.05231
$457$	−3.68356	−0.172310	−0.0861548	−	0.996282i	$-0.527458\pi$
$457$	−3.68356	−0.172310	−0.0861548	+	0.996282i	$0.527458\pi$
$458$	−23.0885	−1.07886
$459$	15.8408	0.739384
$460$	0	0
$461$	37.3221	1.73826	0.869131	−	0.494582i	$-0.164679\pi$
$461$	37.3221	1.73826	0.869131	+	0.494582i	$0.164679\pi$
$462$	2.21975	0.103272
$463$	−12.4345	−0.577880	−0.288940	−	0.957347i	$-0.593303\pi$
$463$	−12.4345	−0.577880	−0.288940	+	0.957347i	$0.593303\pi$
$464$	−20.5908	−0.955903
$465$	0	0
$466$	−12.7537	−0.590804
$467$	3.71254	0.171796	0.0858980	−	0.996304i	$-0.472624\pi$
$467$	3.71254	0.171796	0.0858980	+	0.996304i	$0.472624\pi$
$468$	0.875444	0.0404674
$469$	9.50895	0.439082
$470$	0	0
$471$	31.6592	1.45878
$472$	−28.5586	−1.31452
$473$	−5.44475	−0.250350
$474$	−16.7894	−0.771163
$475$	0	0
$476$	−1.26929	−0.0581779
$477$	5.64085	0.258277
$478$	38.1267	1.74387
$479$	−32.8371	−1.50036	−0.750182	−	0.661231i	$-0.770034\pi$
$479$	−32.8371	−1.50036	−0.750182	+	0.661231i	$0.770034\pi$
$480$	0	0
$481$	32.7326	1.49248
$482$	−1.30874	−0.0596116
$483$	−15.2157	−0.692337
$484$	3.01878	0.137217
$485$	0	0
$486$	−7.39710	−0.335539
$487$	−27.1109	−1.22851	−0.614256	−	0.789107i	$-0.710544\pi$
$487$	−27.1109	−1.22851	−0.614256	+	0.789107i	$0.710544\pi$
$488$	26.4077	1.19542
$489$	11.8467	0.535726
$490$	0	0
$491$	−6.70831	−0.302742	−0.151371	−	0.988477i	$-0.548369\pi$
$491$	−6.70831	−0.302742	−0.151371	+	0.988477i	$0.548369\pi$
$492$	−5.22408	−0.235520
$493$	21.1380	0.952009
$494$	−28.8759	−1.29919
$495$	0	0
$496$	0.710415	0.0318986
$497$	−7.72789	−0.346643
$498$	39.7525	1.78135
$499$	−33.9775	−1.52104	−0.760520	−	0.649314i	$-0.775056\pi$
$499$	−33.9775	−1.52104	−0.760520	+	0.649314i	$0.775056\pi$
$500$	0	0
$501$	−29.0557	−1.29811
$502$	−23.5325	−1.05031
$503$	−6.74907	−0.300926	−0.150463	−	0.988616i	$-0.548076\pi$
$503$	−6.74907	−0.300926	−0.150463	+	0.988616i	$0.548076\pi$
$504$	2.12181	0.0945129
$505$	0	0
$506$	−5.73618	−0.255005
$507$	−33.2942	−1.47865
$508$	3.67674	0.163129
$509$	−5.20927	−0.230897	−0.115448	−	0.993313i	$-0.536831\pi$
$509$	−5.20927	−0.230897	−0.115448	+	0.993313i	$0.536831\pi$
$510$	0	0
$511$	−3.81343	−0.168696
$512$	25.4014	1.12259
$513$	−18.3890	−0.811892
$514$	26.2723	1.15882
$515$	0	0
$516$	−4.21526	−0.185566
$517$	2.69830	0.118671
$518$	9.96152	0.437684
$519$	−7.02518	−0.308371
$520$	0	0
$521$	4.39792	0.192676	0.0963382	−	0.995349i	$-0.469287\pi$
$521$	4.39792	0.192676	0.0963382	+	0.995349i	$0.469287\pi$
$522$	−4.43687	−0.194197
$523$	−28.7186	−1.25578	−0.627889	−	0.778303i	$-0.716081\pi$
$523$	−28.7186	−1.25578	−0.627889	+	0.778303i	$0.716081\pi$
$524$	0.351739	0.0153658
$525$	0	0
$526$	20.4582	0.892019
$527$	−0.729295	−0.0317686
$528$	4.40311	0.191621
$529$	16.3196	0.709549
$530$	0	0
$531$	−5.25152	−0.227896
$532$	1.47347	0.0638832
$533$	53.4631	2.31574
$534$	4.31180	0.186590
$535$	0	0
$536$	22.1026	0.954688
$537$	−20.5915	−0.888590
$538$	−29.1657	−1.25742
$539$	−3.73366	−0.160820
$540$	0	0
$541$	−11.5157	−0.495099	−0.247549	−	0.968875i	$-0.579625\pi$
$541$	−11.5157	−0.495099	−0.247549	+	0.968875i	$0.579625\pi$
$542$	25.7648	1.10669
$543$	−23.7680	−1.01998
$544$	−5.53023	−0.237107
$545$	0	0
$546$	−17.5871	−0.752660
$547$	1.46564	0.0626662	0.0313331	−	0.999509i	$-0.490025\pi$
$547$	1.46564	0.0626662	0.0313331	+	0.999509i	$0.490025\pi$
$548$	0.814512	0.0347942
$549$	4.85599	0.207249
$550$	0	0
$551$	−24.5384	−1.04537
$552$	−35.3673	−1.50533
$553$	−8.76770	−0.372840
$554$	25.9416	1.10215
$555$	0	0
$556$	−0.583798	−0.0247586
$557$	9.02751	0.382508	0.191254	−	0.981541i	$-0.438745\pi$
$557$	9.02751	0.382508	0.191254	+	0.981541i	$0.438745\pi$
$558$	0.153079	0.00648035
$559$	43.1388	1.82458
$560$	0	0
$561$	−4.52013	−0.190840
$562$	32.9566	1.39019
$563$	−1.03855	−0.0437698	−0.0218849	−	0.999760i	$-0.506967\pi$
$563$	−1.03855	−0.0437698	−0.0218849	+	0.999760i	$0.506967\pi$
$564$	2.08900	0.0879626
$565$	0	0
$566$	−2.21904	−0.0932732
$567$	−13.3265	−0.559660
$568$	−17.9627	−0.753699
$569$	−16.4700	−0.690458	−0.345229	−	0.938519i	$-0.612199\pi$
$569$	−16.4700	−0.690458	−0.345229	+	0.938519i	$0.612199\pi$
$570$	0	0
$571$	34.2521	1.43341	0.716703	−	0.697378i	$-0.245650\pi$
$571$	34.2521	1.43341	0.716703	+	0.697378i	$0.245650\pi$
$572$	1.11170	0.0464826
$573$	−27.2570	−1.13868
$574$	16.2704	0.679115
$575$	0	0
$576$	4.84114	0.201714
$577$	−3.72767	−0.155185	−0.0775924	−	0.996985i	$-0.524723\pi$
$577$	−3.72767	−0.155185	−0.0775924	+	0.996985i	$0.524723\pi$
$578$	6.83353	0.284237
$579$	−16.5919	−0.689537
$580$	0	0
$581$	20.7594	0.861245
$582$	1.76187	0.0730318
$583$	7.16315	0.296667
$584$	−8.86394	−0.366792
$585$	0	0
$586$	−27.8597	−1.15088
$587$	−29.8715	−1.23293	−0.616464	−	0.787383i	$-0.711435\pi$
$587$	−29.8715	−1.23293	−0.616464	+	0.787383i	$0.711435\pi$
$588$	−2.89056	−0.119205
$589$	0.846612	0.0348840
$590$	0	0
$591$	38.3242	1.57645
$592$	19.7597	0.812118
$593$	−37.2358	−1.52909	−0.764546	−	0.644569i	$-0.777037\pi$
$593$	−37.2358	−1.52909	−0.764546	+	0.644569i	$0.777037\pi$
$594$	−4.22229	−0.173243
$595$	0	0
$596$	−6.03847	−0.247345
$597$	−35.1291	−1.43774
$598$	45.4479	1.85850
$599$	−14.2870	−0.583753	−0.291877	−	0.956456i	$-0.594280\pi$
$599$	−14.2870	−0.583753	−0.291877	+	0.956456i	$0.594280\pi$
$600$	0	0
$601$	−5.96440	−0.243293	−0.121647	−	0.992573i	$-0.538817\pi$
$601$	−5.96440	−0.243293	−0.121647	+	0.992573i	$0.538817\pi$
$602$	13.1285	0.535076
$603$	4.06435	0.165513
$604$	−1.52974	−0.0622441
$605$	0	0
$606$	−5.70525	−0.231760
$607$	2.52516	0.102493	0.0512466	−	0.998686i	$-0.483681\pi$
$607$	2.52516	0.102493	0.0512466	+	0.998686i	$0.483681\pi$
$608$	6.41984	0.260359
$609$	−14.9453	−0.605614
$610$	0	0
$611$	−21.3787	−0.864890
$612$	−0.542525	−0.0219303
$613$	9.04094	0.365160	0.182580	−	0.983191i	$-0.441555\pi$
$613$	9.04094	0.365160	0.182580	+	0.983191i	$0.441555\pi$
$614$	22.9058	0.924404
$615$	0	0
$616$	2.69443	0.108562
$617$	−38.5569	−1.55224	−0.776120	−	0.630585i	$-0.782815\pi$
$617$	−38.5569	−1.55224	−0.776120	+	0.630585i	$0.782815\pi$
$618$	−7.16995	−0.288418
$619$	−18.0419	−0.725166	−0.362583	−	0.931951i	$-0.618105\pi$
$619$	−18.0419	−0.725166	−0.362583	+	0.931951i	$0.618105\pi$
$620$	0	0
$621$	28.9424	1.16142
$622$	35.4811	1.42266
$623$	2.25169	0.0902121
$624$	−34.8859	−1.39655
$625$	0	0
$626$	7.11449	0.284352
$627$	5.24725	0.209555
$628$	4.82536	0.192553
$629$	−20.2848	−0.808809
$630$	0	0
$631$	−4.00131	−0.159290	−0.0796448	−	0.996823i	$-0.525379\pi$
$631$	−4.00131	−0.159290	−0.0796448	+	0.996823i	$0.525379\pi$
$632$	−20.3797	−0.810659
$633$	18.0975	0.719310
$634$	−1.62312	−0.0644624
$635$	0	0
$636$	5.54563	0.219899
$637$	29.5818	1.17208
$638$	−5.63426	−0.223062
$639$	−3.30308	−0.130668
$640$	0	0
$641$	48.4959	1.91547	0.957736	−	0.287650i	$-0.0928741\pi$
$641$	48.4959	1.91547	0.957736	+	0.287650i	$0.0928741\pi$
$642$	36.6399	1.44606
$643$	−24.8486	−0.979935	−0.489967	−	0.871741i	$-0.662991\pi$
$643$	−24.8486	−0.979935	−0.489967	+	0.871741i	$0.662991\pi$
$644$	−2.31910	−0.0913855
$645$	0	0
$646$	17.8948	0.704063
$647$	−43.2159	−1.69899	−0.849496	−	0.527595i	$-0.823094\pi$
$647$	−43.2159	−1.69899	−0.849496	+	0.527595i	$0.823094\pi$
$648$	−30.9761	−1.21686
$649$	−6.66875	−0.261771
$650$	0	0
$651$	0.515636	0.0202094
$652$	1.80562	0.0707135
$653$	−1.40884	−0.0551323	−0.0275662	−	0.999620i	$-0.508776\pi$
$653$	−1.40884	−0.0551323	−0.0275662	+	0.999620i	$0.508776\pi$
$654$	−27.6501	−1.08120
$655$	0	0
$656$	32.2741	1.26009
$657$	−1.62995	−0.0635904
$658$	−6.50619	−0.253638
$659$	−4.34247	−0.169158	−0.0845792	−	0.996417i	$-0.526955\pi$
$659$	−4.34247	−0.169158	−0.0845792	+	0.996417i	$0.526955\pi$
$660$	0	0
$661$	26.3595	1.02527	0.512633	−	0.858608i	$-0.328670\pi$
$661$	26.3595	1.02527	0.512633	+	0.858608i	$0.328670\pi$
$662$	17.8414	0.693425
$663$	35.8130	1.39086
$664$	48.2532	1.87259
$665$	0	0
$666$	4.25779	0.164986
$667$	38.6210	1.49541
$668$	−4.42854	−0.171345
$669$	−41.2484	−1.59476
$670$	0	0
$671$	6.16648	0.238054
$672$	3.91006	0.150834
$673$	−1.88857	−0.0727989	−0.0363995	−	0.999337i	$-0.511589\pi$
$673$	−1.88857	−0.0727989	−0.0363995	+	0.999337i	$0.511589\pi$
$674$	12.3716	0.476534
$675$	0	0
$676$	−5.07456	−0.195175
$677$	40.6860	1.56369	0.781845	−	0.623473i	$-0.214279\pi$
$677$	40.6860	1.56369	0.781845	+	0.623473i	$0.214279\pi$
$678$	38.9021	1.49403
$679$	0.920077	0.0353093
$680$	0	0
$681$	−48.1926	−1.84674
$682$	0.194391	0.00744361
$683$	−13.1902	−0.504710	−0.252355	−	0.967635i	$-0.581205\pi$
$683$	−13.1902	−0.504710	−0.252355	+	0.967635i	$0.581205\pi$
$684$	0.629798	0.0240809
$685$	0	0
$686$	20.8004	0.794162
$687$	−33.2416	−1.26825
$688$	26.0416	0.992828
$689$	−56.7538	−2.16215
$690$	0	0
$691$	−21.4420	−0.815692	−0.407846	−	0.913051i	$-0.633720\pi$
$691$	−21.4420	−0.815692	−0.407846	+	0.913051i	$0.633720\pi$
$692$	−1.07075	−0.0407037
$693$	0.495466	0.0188212
$694$	33.8495	1.28491
$695$	0	0
$696$	−34.7389	−1.31677
$697$	−33.1318	−1.25496
$698$	−14.0919	−0.533387
$699$	−18.3621	−0.694518
$700$	0	0
$701$	15.8950	0.600348	0.300174	−	0.953885i	$-0.402955\pi$
$701$	15.8950	0.600348	0.300174	+	0.953885i	$0.402955\pi$
$702$	33.4533	1.26261
$703$	23.5479	0.888126
$704$	6.14763	0.231698
$705$	0	0
$706$	38.9746	1.46683
$707$	−2.97937	−0.112051
$708$	−5.16287	−0.194033
$709$	−38.8392	−1.45864	−0.729318	−	0.684175i	$-0.760162\pi$
$709$	−38.8392	−1.45864	−0.729318	+	0.684175i	$0.760162\pi$
$710$	0	0
$711$	−3.74752	−0.140543
$712$	5.23384	0.196146
$713$	−1.33248	−0.0499019
$714$	10.8990	0.407885
$715$	0	0
$716$	−3.13847	−0.117290
$717$	54.8927	2.05001
$718$	−12.9429	−0.483025
$719$	−1.15150	−0.0429436	−0.0214718	−	0.999769i	$-0.506835\pi$
$719$	−1.15150	−0.0429436	−0.0214718	+	0.999769i	$0.506835\pi$
$720$	0	0
$721$	−3.74426	−0.139444
$722$	4.09267	0.152314
$723$	−1.88426	−0.0700764
$724$	−3.62262	−0.134634
$725$	0	0
$726$	−25.9213	−0.962030
$727$	−2.07741	−0.0770469	−0.0385234	−	0.999258i	$-0.512265\pi$
$727$	−2.07741	−0.0770469	−0.0385234	+	0.999258i	$0.512265\pi$
$728$	−21.3480	−0.791209
$729$	20.3950	0.755371
$730$	0	0
$731$	−26.7337	−0.988783
$732$	4.77402	0.176453
$733$	−0.247310	−0.00913459	−0.00456729	−	0.999990i	$-0.501454\pi$
$733$	−0.247310	−0.00913459	−0.00456729	+	0.999990i	$0.501454\pi$
$734$	−31.8228	−1.17460
$735$	0	0
$736$	−10.1042	−0.372446
$737$	5.16120	0.190115
$738$	6.95437	0.255994
$739$	−41.8038	−1.53778	−0.768889	−	0.639382i	$-0.779190\pi$
$739$	−41.8038	−1.53778	−0.768889	+	0.639382i	$0.779190\pi$
$740$	0	0
$741$	−41.5740	−1.52726
$742$	−17.2719	−0.634072
$743$	−16.1617	−0.592916	−0.296458	−	0.955046i	$-0.595805\pi$
$743$	−16.1617	−0.592916	−0.296458	+	0.955046i	$0.595805\pi$
$744$	1.19855	0.0439408
$745$	0	0
$746$	9.14162	0.334698
$747$	8.87306	0.324648
$748$	−0.688938	−0.0251901
$749$	19.1339	0.699139
$750$	0	0
$751$	−8.92639	−0.325729	−0.162864	−	0.986648i	$-0.552073\pi$
$751$	−8.92639	−0.325729	−0.162864	+	0.986648i	$0.552073\pi$
$752$	−12.9057	−0.470622
$753$	−33.8808	−1.23468
$754$	44.6403	1.62570
$755$	0	0
$756$	−1.70705	−0.0620847
$757$	−2.30487	−0.0837720	−0.0418860	−	0.999122i	$-0.513337\pi$
$757$	−2.30487	−0.0837720	−0.0418860	+	0.999122i	$0.513337\pi$
$758$	−32.4754	−1.17956
$759$	−8.25865	−0.299770
$760$	0	0
$761$	−5.45934	−0.197901	−0.0989505	−	0.995092i	$-0.531549\pi$
$761$	−5.45934	−0.197901	−0.0989505	+	0.995092i	$0.531549\pi$
$762$	−31.5709	−1.14369
$763$	−14.4393	−0.522739
$764$	−4.15440	−0.150301
$765$	0	0
$766$	10.3255	0.373076
$767$	52.8366	1.90782
$768$	12.7068	0.458516
$769$	20.6287	0.743890	0.371945	−	0.928255i	$-0.378691\pi$
$769$	20.6287	0.743890	0.371945	+	0.928255i	$0.378691\pi$
$770$	0	0
$771$	37.8254	1.36225
$772$	−2.52887	−0.0910159
$773$	36.9986	1.33075	0.665374	−	0.746510i	$-0.268272\pi$
$773$	36.9986	1.33075	0.665374	+	0.746510i	$0.268272\pi$
$774$	5.61141	0.201698
$775$	0	0
$776$	2.13863	0.0767723
$777$	14.3421	0.514519
$778$	24.6682	0.884398
$779$	38.4615	1.37803
$780$	0	0
$781$	−4.19449	−0.150091
$782$	−28.1647	−1.00717
$783$	28.4281	1.01594
$784$	17.8577	0.637775
$785$	0	0
$786$	−3.02027	−0.107730
$787$	9.55601	0.340635	0.170317	−	0.985389i	$-0.445521\pi$
$787$	9.55601	0.340635	0.170317	+	0.985389i	$0.445521\pi$
$788$	5.84121	0.208084
$789$	29.4546	1.04861
$790$	0	0
$791$	20.3153	0.722330
$792$	1.15166	0.0409225
$793$	−48.8572	−1.73497
$794$	−0.992428	−0.0352200
$795$	0	0
$796$	−5.35423	−0.189776
$797$	31.5410	1.11724	0.558620	−	0.829424i	$-0.311331\pi$
$797$	31.5410	1.11724	0.558620	+	0.829424i	$0.311331\pi$
$798$	−12.6523	−0.447885
$799$	13.2487	0.468705
$800$	0	0
$801$	0.962426	0.0340056
$802$	45.5840	1.60963
$803$	−2.06983	−0.0730426
$804$	3.99574	0.140919
$805$	0	0
$806$	−1.54016	−0.0542499
$807$	−41.9913	−1.47816
$808$	−6.92527	−0.243630
$809$	26.1016	0.917685	0.458842	−	0.888518i	$-0.348264\pi$
$809$	26.1016	0.917685	0.458842	+	0.888518i	$0.348264\pi$
$810$	0	0
$811$	−34.3946	−1.20776	−0.603880	−	0.797076i	$-0.706379\pi$
$811$	−34.3946	−1.20776	−0.603880	+	0.797076i	$0.706379\pi$
$812$	−2.27790	−0.0799385
$813$	37.0948	1.30097
$814$	5.40684	0.189510
$815$	0	0
$816$	21.6193	0.756826
$817$	31.0342	1.08575
$818$	−22.7029	−0.793790
$819$	−3.92558	−0.137171
$820$	0	0
$821$	52.9922	1.84944	0.924720	−	0.380647i	$-0.124299\pi$
$821$	52.9922	1.84944	0.924720	+	0.380647i	$0.124299\pi$
$822$	−6.99395	−0.243942
$823$	−20.4286	−0.712095	−0.356047	−	0.934468i	$-0.615876\pi$
$823$	−20.4286	−0.712095	−0.356047	+	0.934468i	$0.615876\pi$
$824$	−8.70318	−0.303189
$825$	0	0
$826$	16.0798	0.559488
$827$	40.8210	1.41949	0.709743	−	0.704461i	$-0.248811\pi$
$827$	40.8210	1.41949	0.709743	+	0.704461i	$0.248811\pi$
$828$	−0.991240	−0.0344480
$829$	7.89986	0.274374	0.137187	−	0.990545i	$-0.456194\pi$
$829$	7.89986	0.274374	0.137187	+	0.990545i	$0.456194\pi$
$830$	0	0
$831$	37.3494	1.29563
$832$	−48.7078	−1.68864
$833$	−18.3323	−0.635176
$834$	5.01289	0.173582
$835$	0	0
$836$	0.799762	0.0276604
$837$	−0.980816	−0.0339019
$838$	29.5639	1.02127
$839$	45.1697	1.55943	0.779716	−	0.626133i	$-0.215363\pi$
$839$	45.1697	1.55943	0.779716	+	0.626133i	$0.215363\pi$
$840$	0	0
$841$	8.93470	0.308093
$842$	36.3837	1.25387
$843$	47.4491	1.63423
$844$	2.75834	0.0949458
$845$	0	0
$846$	−2.78090	−0.0956093
$847$	−13.5365	−0.465121
$848$	−34.2606	−1.17651
$849$	−3.19486	−0.109647
$850$	0	0
$851$	−37.0621	−1.27047
$852$	−3.24732	−0.111252
$853$	33.8378	1.15859	0.579293	−	0.815120i	$-0.303329\pi$
$853$	33.8378	1.15859	0.579293	+	0.815120i	$0.303329\pi$
$854$	−14.8687	−0.508797
$855$	0	0
$856$	44.4750	1.52012
$857$	22.0350	0.752701	0.376351	−	0.926477i	$-0.377179\pi$
$857$	22.0350	0.752701	0.376351	+	0.926477i	$0.377179\pi$
$858$	−9.54583	−0.325889
$859$	−43.0953	−1.47039	−0.735197	−	0.677853i	$-0.762910\pi$
$859$	−43.0953	−1.47039	−0.735197	+	0.677853i	$0.762910\pi$
$860$	0	0
$861$	23.4253	0.798333
$862$	23.0392	0.784717
$863$	−41.5592	−1.41469	−0.707347	−	0.706867i	$-0.750108\pi$
$863$	−41.5592	−1.41469	−0.707347	+	0.706867i	$0.750108\pi$
$864$	−7.43751	−0.253029
$865$	0	0
$866$	9.66022	0.328268
$867$	9.83855	0.334135
$868$	0.0785910	0.00266755
$869$	−4.75887	−0.161434
$870$	0	0
$871$	−40.8923	−1.38558
$872$	−33.5628	−1.13658
$873$	0.393263	0.0133099
$874$	32.6954	1.10594
$875$	0	0
$876$	−1.60244	−0.0541413
$877$	41.5051	1.40153	0.700763	−	0.713394i	$-0.252843\pi$
$877$	41.5051	1.40153	0.700763	+	0.713394i	$0.252843\pi$
$878$	−2.84060	−0.0958655
$879$	−40.1110	−1.35291
$880$	0	0
$881$	5.22392	0.175998	0.0879991	−	0.996121i	$-0.471953\pi$
$881$	5.22392	0.175998	0.0879991	+	0.996121i	$0.471953\pi$
$882$	3.84795	0.129567
$883$	−31.9624	−1.07562	−0.537810	−	0.843066i	$-0.680748\pi$
$883$	−31.9624	−1.07562	−0.537810	+	0.843066i	$0.680748\pi$
$884$	5.45847	0.183588
$885$	0	0
$886$	−16.6244	−0.558509
$887$	26.7513	0.898222	0.449111	−	0.893476i	$-0.351741\pi$
$887$	26.7513	0.898222	0.449111	+	0.893476i	$0.351741\pi$
$888$	33.3367	1.11871
$889$	−16.4869	−0.552952
$890$	0	0
$891$	−7.23326	−0.242323
$892$	−6.28690	−0.210501
$893$	−15.3799	−0.514669
$894$	51.8504	1.73414
$895$	0	0
$896$	−10.6730	−0.356561
$897$	65.4335	2.18476
$898$	5.63837	0.188155
$899$	−1.30881	−0.0436512
$900$	0	0
$901$	35.1711	1.17172
$902$	8.83116	0.294045
$903$	18.9017	0.629008
$904$	47.2210	1.57055
$905$	0	0
$906$	13.1354	0.436393
$907$	−23.1619	−0.769078	−0.384539	−	0.923109i	$-0.625640\pi$
$907$	−23.1619	−0.769078	−0.384539	+	0.923109i	$0.625640\pi$
$908$	−7.34530	−0.243762
$909$	−1.27346	−0.0422378
$910$	0	0
$911$	49.1366	1.62797	0.813984	−	0.580888i	$-0.197294\pi$
$911$	49.1366	1.62797	0.813984	+	0.580888i	$0.197294\pi$
$912$	−25.0970	−0.831046
$913$	11.2676	0.372905
$914$	4.82083	0.159459
$915$	0	0
$916$	−5.06654	−0.167403
$917$	−1.57724	−0.0520849
$918$	−20.7315	−0.684241
$919$	−52.6144	−1.73559	−0.867794	−	0.496923i	$-0.834463\pi$
$919$	−52.6144	−1.73559	−0.867794	+	0.496923i	$0.834463\pi$
$920$	0	0
$921$	32.9786	1.08668
$922$	−48.8450	−1.60862
$923$	33.2330	1.09388
$924$	0.487102	0.0160245
$925$	0	0
$926$	16.2736	0.534782
$927$	−1.60039	−0.0525636
$928$	−9.92466	−0.325793
$929$	11.0619	0.362930	0.181465	−	0.983397i	$-0.441916\pi$
$929$	11.0619	0.362930	0.181465	+	0.983397i	$0.441916\pi$
$930$	0	0
$931$	21.2813	0.697466
$932$	−2.79867	−0.0916735
$933$	51.0839	1.67241
$934$	−4.85877	−0.158984
$935$	0	0
$936$	−9.12463	−0.298248
$937$	47.7232	1.55905	0.779525	−	0.626371i	$-0.215461\pi$
$937$	47.7232	1.55905	0.779525	+	0.626371i	$0.215461\pi$
$938$	−12.4448	−0.406336
$939$	10.2431	0.334270
$940$	0	0
$941$	−7.26848	−0.236946	−0.118473	−	0.992957i	$-0.537800\pi$
$941$	−7.26848	−0.236946	−0.118473	+	0.992957i	$0.537800\pi$
$942$	−41.4338	−1.34999
$943$	−60.5346	−1.97128
$944$	31.8959	1.03812
$945$	0	0
$946$	7.12577	0.231679
$947$	−42.4508	−1.37947	−0.689734	−	0.724063i	$-0.742272\pi$
$947$	−42.4508	−1.37947	−0.689734	+	0.724063i	$0.742272\pi$
$948$	−3.68426	−0.119659
$949$	16.3993	0.532343
$950$	0	0
$951$	−2.33688	−0.0757787
$952$	13.2297	0.428776
$953$	−31.8336	−1.03119	−0.515596	−	0.856832i	$-0.672429\pi$
$953$	−31.8336	−1.03119	−0.515596	+	0.856832i	$0.672429\pi$
$954$	−7.38242	−0.239015
$955$	0	0
$956$	8.36651	0.270592
$957$	−8.11190	−0.262221
$958$	42.9753	1.38847
$959$	−3.65236	−0.117941
$960$	0	0
$961$	−30.9548	−0.998543
$962$	−42.8385	−1.38117
$963$	8.17830	0.263542
$964$	−0.287190	−0.00924978
$965$	0	0
$966$	19.9134	0.640703
$967$	1.98161	0.0637242	0.0318621	−	0.999492i	$-0.489856\pi$
$967$	1.98161	0.0637242	0.0318621	+	0.999492i	$0.489856\pi$
$968$	−31.4643	−1.01130
$969$	25.7640	0.827660
$970$	0	0
$971$	−24.3073	−0.780057	−0.390029	−	0.920803i	$-0.627535\pi$
$971$	−24.3073	−0.780057	−0.390029	+	0.920803i	$0.627535\pi$
$972$	−1.62322	−0.0520647
$973$	2.61781	0.0839232
$974$	35.4812	1.13689
$975$	0	0
$976$	−29.4936	−0.944068
$977$	−22.5696	−0.722064	−0.361032	−	0.932553i	$-0.617576\pi$
$977$	−22.5696	−0.722064	−0.361032	+	0.932553i	$0.617576\pi$
$978$	−15.5043	−0.495772
$979$	1.22216	0.0390603
$980$	0	0
$981$	−6.17171	−0.197048
$982$	8.77946	0.280164
$983$	−27.4009	−0.873953	−0.436976	−	0.899473i	$-0.643951\pi$
$983$	−27.4009	−0.873953	−0.436976	+	0.899473i	$0.643951\pi$
$984$	54.4499	1.73580
$985$	0	0
$986$	−27.6643	−0.881009
$987$	−9.36727	−0.298164
$988$	−6.33653	−0.201592
$989$	−48.8448	−1.55317
$990$	0	0
$991$	−39.0711	−1.24114	−0.620568	−	0.784153i	$-0.713098\pi$
$991$	−39.0711	−1.24114	−0.620568	+	0.784153i	$0.713098\pi$
$992$	0.342416	0.0108717
$993$	25.6871	0.815154
$994$	10.1138	0.320791
$995$	0	0
$996$	8.72329	0.276408
$997$	55.0824	1.74448	0.872239	−	0.489080i	$-0.162667\pi$
$997$	55.0824	1.74448	0.872239	+	0.489080i	$0.162667\pi$
$998$	44.4678	1.40760
$999$	−27.2807	−0.863123

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.14		46
5.2	odd	4		1205.2.b.c.724.14	✓	46
5.3	odd	4		1205.2.b.c.724.33	yes	46
5.4	even	2	inner	6025.2.a.p.1.33		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.14	✓	46	5.2	odd	4
1205.2.b.c.724.33	yes	46	5.3	odd	4
6025.2.a.p.1.14		46	1.1	even	1	trivial
6025.2.a.p.1.33		46	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)

\(f(q)\)	\(=\)	\(q-1.30874 q^{2} -1.88426 q^{3} -0.287190 q^{4} +2.46601 q^{6} +1.28779 q^{7} +2.99335 q^{8} +0.550432 q^{9} +O(q^{10})\)	\(q-1.30874 q^{2} -1.88426 q^{3} -0.287190 q^{4} +2.46601 q^{6} +1.28779 q^{7} +2.99335 q^{8} +0.550432 q^{9} +0.698979 q^{11} +0.541141 q^{12} -5.53802 q^{13} -1.68539 q^{14} -3.34314 q^{16} +3.43199 q^{17} -0.720375 q^{18} -3.98407 q^{19} -2.42653 q^{21} -0.914784 q^{22} +6.27054 q^{23} -5.64024 q^{24} +7.24785 q^{26} +4.61562 q^{27} -0.369841 q^{28} +6.15912 q^{29} -0.212499 q^{31} -1.61138 q^{32} -1.31706 q^{33} -4.49160 q^{34} -0.158079 q^{36} -5.91052 q^{37} +5.21413 q^{38} +10.4351 q^{39} -9.65382 q^{41} +3.17571 q^{42} -7.78957 q^{43} -0.200740 q^{44} -8.20652 q^{46} +3.86035 q^{47} +6.29934 q^{48} -5.34159 q^{49} -6.46676 q^{51} +1.59047 q^{52} +10.2480 q^{53} -6.04066 q^{54} +3.85481 q^{56} +7.50702 q^{57} -8.06070 q^{58} -9.54071 q^{59} +8.82214 q^{61} +0.278107 q^{62} +0.708842 q^{63} +8.79516 q^{64} +1.72369 q^{66} +7.38392 q^{67} -0.985635 q^{68} -11.8153 q^{69} -6.00088 q^{71} +1.64763 q^{72} -2.96121 q^{73} +7.73535 q^{74} +1.14419 q^{76} +0.900139 q^{77} -13.6568 q^{78} -6.80832 q^{79} -10.3483 q^{81} +12.6344 q^{82} +16.1202 q^{83} +0.696877 q^{84} +10.1946 q^{86} -11.6054 q^{87} +2.09228 q^{88} +1.74849 q^{89} -7.13182 q^{91} -1.80084 q^{92} +0.400404 q^{93} -5.05221 q^{94} +3.03625 q^{96} +0.714461 q^{97} +6.99077 q^{98} +0.384741 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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