LMFDB - Embedded newform 6025.2.a.q.1.2

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			1.2
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)

$f(q)$	$=$	$q-2.73687 q^{2} -1.37229 q^{3} +5.49044 q^{4} +3.75578 q^{6} -0.639768 q^{7} -9.55288 q^{8} -1.11681 q^{9} +O(q^{10})$	\(q-2.73687 q^{2} -1.37229 q^{3} +5.49044 q^{4} +3.75578 q^{6} -0.639768 q^{7} -9.55288 q^{8} -1.11681 q^{9} -1.22132 q^{11} -7.53449 q^{12} -4.43214 q^{13} +1.75096 q^{14} +15.1641 q^{16} +3.84662 q^{17} +3.05657 q^{18} -1.09889 q^{19} +0.877948 q^{21} +3.34260 q^{22} +0.641133 q^{23} +13.1093 q^{24} +12.1302 q^{26} +5.64947 q^{27} -3.51261 q^{28} +4.61275 q^{29} +6.14522 q^{31} -22.3963 q^{32} +1.67601 q^{33} -10.5277 q^{34} -6.13180 q^{36} -6.41169 q^{37} +3.00750 q^{38} +6.08219 q^{39} +10.5205 q^{41} -2.40283 q^{42} +0.924563 q^{43} -6.70561 q^{44} -1.75470 q^{46} +3.26969 q^{47} -20.8095 q^{48} -6.59070 q^{49} -5.27869 q^{51} -24.3344 q^{52} -10.0158 q^{53} -15.4619 q^{54} +6.11162 q^{56} +1.50799 q^{57} -12.6245 q^{58} -4.76465 q^{59} -1.50404 q^{61} -16.8187 q^{62} +0.714501 q^{63} +30.9675 q^{64} -4.58703 q^{66} -6.85863 q^{67} +21.1197 q^{68} -0.879822 q^{69} -6.73643 q^{71} +10.6688 q^{72} +10.0399 q^{73} +17.5480 q^{74} -6.03337 q^{76} +0.781364 q^{77} -16.6462 q^{78} -10.2451 q^{79} -4.40229 q^{81} -28.7932 q^{82} -16.6773 q^{83} +4.82032 q^{84} -2.53041 q^{86} -6.33004 q^{87} +11.6672 q^{88} -2.77942 q^{89} +2.83554 q^{91} +3.52010 q^{92} -8.43304 q^{93} -8.94872 q^{94} +30.7343 q^{96} +2.14152 q^{97} +18.0379 q^{98} +1.36399 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.73687	−1.93526	−0.967629	−	0.252378i	$-0.918787\pi$
$2$	−2.73687	−1.93526	−0.967629	+	0.252378i	$0.918787\pi$
$3$	−1.37229	−0.792293	−0.396147	−	0.918187i	$-0.629653\pi$
$3$	−1.37229	−0.792293	−0.396147	+	0.918187i	$0.629653\pi$
$4$	5.49044	2.74522
$5$	0	0
$5$	0	0
$6$	3.75578	1.53329
$7$	−0.639768	−0.241809	−0.120905	−	0.992664i	$-0.538580\pi$
$7$	−0.639768	−0.241809	−0.120905	+	0.992664i	$0.538580\pi$
$8$	−9.55288	−3.37745
$9$	−1.11681	−0.372271
$10$	0	0
$11$	−1.22132	−0.368243	−0.184122	−	0.982903i	$-0.558944\pi$
$11$	−1.22132	−0.368243	−0.184122	+	0.982903i	$0.558944\pi$
$12$	−7.53449	−2.17502
$13$	−4.43214	−1.22925	−0.614627	−	0.788818i	$-0.710694\pi$
$13$	−4.43214	−1.22925	−0.614627	+	0.788818i	$0.710694\pi$
$14$	1.75096	0.467963
$15$	0	0
$16$	15.1641	3.79102
$17$	3.84662	0.932943	0.466471	−	0.884536i	$-0.345525\pi$
$17$	3.84662	0.932943	0.466471	+	0.884536i	$0.345525\pi$
$18$	3.05657	0.720441
$19$	−1.09889	−0.252102	−0.126051	−	0.992024i	$-0.540230\pi$
$19$	−1.09889	−0.252102	−0.126051	+	0.992024i	$0.540230\pi$
$20$	0	0
$21$	0.877948	0.191584
$22$	3.34260	0.712645
$23$	0.641133	0.133686	0.0668428	−	0.997764i	$-0.478707\pi$
$23$	0.641133	0.133686	0.0668428	+	0.997764i	$0.478707\pi$
$24$	13.1093	2.67593
$25$	0	0
$26$	12.1302	2.37892
$27$	5.64947	1.08724
$28$	−3.51261	−0.663820
$29$	4.61275	0.856566	0.428283	−	0.903645i	$-0.359119\pi$
$29$	4.61275	0.856566	0.428283	+	0.903645i	$0.359119\pi$
$30$	0	0
$31$	6.14522	1.10371	0.551857	−	0.833938i	$-0.313919\pi$
$31$	6.14522	1.10371	0.551857	+	0.833938i	$0.313919\pi$
$32$	−22.3963	−3.95914
$33$	1.67601	0.291757
$34$	−10.5277	−1.80548
$35$	0	0
$36$	−6.13180	−1.02197
$37$	−6.41169	−1.05408	−0.527038	−	0.849842i	$-0.676697\pi$
$37$	−6.41169	−1.05408	−0.527038	+	0.849842i	$0.676697\pi$
$38$	3.00750	0.487882
$39$	6.08219	0.973930
$40$	0	0
$41$	10.5205	1.64302	0.821512	−	0.570191i	$-0.193131\pi$
$41$	10.5205	1.64302	0.821512	+	0.570191i	$0.193131\pi$
$42$	−2.40283	−0.370764
$43$	0.924563	0.140995	0.0704973	−	0.997512i	$-0.477541\pi$
$43$	0.924563	0.140995	0.0704973	+	0.997512i	$0.477541\pi$
$44$	−6.70561	−1.01091
$45$	0	0
$46$	−1.75470	−0.258716
$47$	3.26969	0.476934	0.238467	−	0.971151i	$-0.423355\pi$
$47$	3.26969	0.476934	0.238467	+	0.971151i	$0.423355\pi$
$48$	−20.8095	−3.00360
$49$	−6.59070	−0.941528
$50$	0	0
$51$	−5.27869	−0.739164
$52$	−24.3344	−3.37458
$53$	−10.0158	−1.37578	−0.687890	−	0.725815i	$-0.741463\pi$
$53$	−10.0158	−1.37578	−0.687890	+	0.725815i	$0.741463\pi$
$54$	−15.4619	−2.10409
$55$	0	0
$56$	6.11162	0.816700
$57$	1.50799	0.199739
$58$	−12.6245	−1.65768
$59$	−4.76465	−0.620305	−0.310152	−	0.950687i	$-0.600380\pi$
$59$	−4.76465	−0.620305	−0.310152	+	0.950687i	$0.600380\pi$
$60$	0	0
$61$	−1.50404	−0.192572	−0.0962862	−	0.995354i	$-0.530696\pi$
$61$	−1.50404	−0.192572	−0.0962862	+	0.995354i	$0.530696\pi$
$62$	−16.8187	−2.13597
$63$	0.714501	0.0900187
$64$	30.9675	3.87094
$65$	0	0
$66$	−4.58703	−0.564624
$67$	−6.85863	−0.837915	−0.418958	−	0.908006i	$-0.637604\pi$
$67$	−6.85863	−0.837915	−0.418958	+	0.908006i	$0.637604\pi$
$68$	21.1197	2.56113
$69$	−0.879822	−0.105918
$70$	0	0
$71$	−6.73643	−0.799467	−0.399733	−	0.916631i	$-0.630897\pi$
$71$	−6.73643	−0.799467	−0.399733	+	0.916631i	$0.630897\pi$
$72$	10.6688	1.25733
$73$	10.0399	1.17508	0.587539	−	0.809196i	$-0.300097\pi$
$73$	10.0399	1.17508	0.587539	+	0.809196i	$0.300097\pi$
$74$	17.5480	2.03991
$75$	0	0
$76$	−6.03337	−0.692075
$77$	0.781364	0.0890447
$78$	−16.6462	−1.88481
$79$	−10.2451	−1.15266	−0.576332	−	0.817215i	$-0.695517\pi$
$79$	−10.2451	−1.15266	−0.576332	+	0.817215i	$0.695517\pi$
$80$	0	0
$81$	−4.40229	−0.489143
$82$	−28.7932	−3.17968
$83$	−16.6773	−1.83057	−0.915284	−	0.402810i	$-0.868034\pi$
$83$	−16.6773	−1.83057	−0.915284	+	0.402810i	$0.868034\pi$
$84$	4.82032	0.525940
$85$	0	0
$86$	−2.53041	−0.272861
$87$	−6.33004	−0.678651
$88$	11.6672	1.24372
$89$	−2.77942	−0.294618	−0.147309	−	0.989091i	$-0.547061\pi$
$89$	−2.77942	−0.294618	−0.147309	+	0.989091i	$0.547061\pi$
$90$	0	0
$91$	2.83554	0.297245
$92$	3.52010	0.366996
$93$	−8.43304	−0.874466
$94$	−8.94872	−0.922990
$95$	0	0
$96$	30.7343	3.13680
$97$	2.14152	0.217438	0.108719	−	0.994073i	$-0.465325\pi$
$97$	2.14152	0.217438	0.108719	+	0.994073i	$0.465325\pi$
$98$	18.0379	1.82210
$99$	1.36399	0.137086
$100$	0	0
$101$	−13.4987	−1.34317	−0.671584	−	0.740928i	$-0.734386\pi$
$101$	−13.4987	−1.34317	−0.671584	+	0.740928i	$0.734386\pi$
$102$	14.4471	1.43047
$103$	2.92184	0.287898	0.143949	−	0.989585i	$-0.454020\pi$
$103$	2.92184	0.287898	0.143949	+	0.989585i	$0.454020\pi$
$104$	42.3397	4.15175
$105$	0	0
$106$	27.4120	2.66249
$107$	12.1577	1.17533	0.587666	−	0.809104i	$-0.300047\pi$
$107$	12.1577	1.17533	0.587666	+	0.809104i	$0.300047\pi$
$108$	31.0181	2.98472
$109$	−11.5960	−1.11070	−0.555349	−	0.831617i	$-0.687415\pi$
$109$	−11.5960	−1.11070	−0.555349	+	0.831617i	$0.687415\pi$
$110$	0	0
$111$	8.79872	0.835137
$112$	−9.70148	−0.916704
$113$	5.69763	0.535988	0.267994	−	0.963421i	$-0.413639\pi$
$113$	5.69763	0.535988	0.267994	+	0.963421i	$0.413639\pi$
$114$	−4.12718	−0.386545
$115$	0	0
$116$	25.3260	2.35146
$117$	4.94988	0.457616
$118$	13.0402	1.20045
$119$	−2.46094	−0.225594
$120$	0	0
$121$	−9.50837	−0.864397
$122$	4.11635	0.372677
$123$	−14.4372	−1.30176
$124$	33.7400	3.02994
$125$	0	0
$126$	−1.95550	−0.174209
$127$	−9.31889	−0.826917	−0.413459	−	0.910523i	$-0.635679\pi$
$127$	−9.31889	−0.826917	−0.413459	+	0.910523i	$0.635679\pi$
$128$	−39.9614	−3.53213
$129$	−1.26877	−0.111709
$130$	0	0
$131$	11.8918	1.03899	0.519497	−	0.854473i	$-0.326120\pi$
$131$	11.8918	1.03899	0.519497	+	0.854473i	$0.326120\pi$
$132$	9.20206	0.800936
$133$	0.703032	0.0609606
$134$	18.7712	1.62158
$135$	0	0
$136$	−36.7463	−3.15097
$137$	15.7177	1.34286	0.671429	−	0.741069i	$-0.265681\pi$
$137$	15.7177	1.34286	0.671429	+	0.741069i	$0.265681\pi$
$138$	2.40796	0.204979
$139$	−14.4202	−1.22311	−0.611553	−	0.791204i	$-0.709455\pi$
$139$	−14.4202	−1.22311	−0.611553	+	0.791204i	$0.709455\pi$
$140$	0	0
$141$	−4.48697	−0.377871
$142$	18.4367	1.54717
$143$	5.41308	0.452665
$144$	−16.9354	−1.41129
$145$	0	0
$146$	−27.4778	−2.27408
$147$	9.04436	0.745967
$148$	−35.2030	−2.89367
$149$	3.44283	0.282048	0.141024	−	0.990006i	$-0.454961\pi$
$149$	3.44283	0.282048	0.141024	+	0.990006i	$0.454961\pi$
$150$	0	0
$151$	2.93942	0.239206	0.119603	−	0.992822i	$-0.461838\pi$
$151$	2.93942	0.239206	0.119603	+	0.992822i	$0.461838\pi$
$152$	10.4975	0.851461
$153$	−4.29596	−0.347308
$154$	−2.13849	−0.172324
$155$	0	0
$156$	33.3939	2.67365
$157$	20.1810	1.61062	0.805309	−	0.592855i	$-0.201999\pi$
$157$	20.1810	1.61062	0.805309	+	0.592855i	$0.201999\pi$
$158$	28.0395	2.23070
$159$	13.7446	1.09002
$160$	0	0
$161$	−0.410176	−0.0323264
$162$	12.0485	0.946617
$163$	−14.1236	−1.10624	−0.553122	−	0.833100i	$-0.686564\pi$
$163$	−14.1236	−1.10624	−0.553122	+	0.833100i	$0.686564\pi$
$164$	57.7621	4.51047
$165$	0	0
$166$	45.6435	3.54262
$167$	23.7195	1.83547	0.917736	−	0.397192i	$-0.130015\pi$
$167$	23.7195	1.83547	0.917736	+	0.397192i	$0.130015\pi$
$168$	−8.38693	−0.647066
$169$	6.64387	0.511067
$170$	0	0
$171$	1.22725	0.0938502
$172$	5.07626	0.387061
$173$	−7.91840	−0.602025	−0.301012	−	0.953620i	$-0.597325\pi$
$173$	−7.91840	−0.602025	−0.301012	+	0.953620i	$0.597325\pi$
$174$	17.3245	1.31337
$175$	0	0
$176$	−18.5202	−1.39602
$177$	6.53850	0.491464
$178$	7.60690	0.570161
$179$	−4.82792	−0.360856	−0.180428	−	0.983588i	$-0.557748\pi$
$179$	−4.82792	−0.360856	−0.180428	+	0.983588i	$0.557748\pi$
$180$	0	0
$181$	−22.6631	−1.68454	−0.842268	−	0.539059i	$-0.818780\pi$
$181$	−22.6631	−1.68454	−0.842268	+	0.539059i	$0.818780\pi$
$182$	−7.76050	−0.575246
$183$	2.06398	0.152574
$184$	−6.12467	−0.451516
$185$	0	0
$186$	23.0801	1.69232
$187$	−4.69797	−0.343550
$188$	17.9521	1.30929
$189$	−3.61435	−0.262905
$190$	0	0
$191$	−9.98358	−0.722387	−0.361193	−	0.932491i	$-0.617631\pi$
$191$	−9.98358	−0.722387	−0.361193	+	0.932491i	$0.617631\pi$
$192$	−42.4965	−3.06692
$193$	−14.6595	−1.05521	−0.527607	−	0.849489i	$-0.676911\pi$
$193$	−14.6595	−1.05521	−0.527607	+	0.849489i	$0.676911\pi$
$194$	−5.86104	−0.420798
$195$	0	0
$196$	−36.1858	−2.58470
$197$	21.5295	1.53392	0.766958	−	0.641697i	$-0.221769\pi$
$197$	21.5295	1.53392	0.766958	+	0.641697i	$0.221769\pi$
$198$	−3.73307	−0.265297
$199$	−7.92980	−0.562129	−0.281064	−	0.959689i	$-0.590687\pi$
$199$	−7.92980	−0.562129	−0.281064	+	0.959689i	$0.590687\pi$
$200$	0	0
$201$	9.41205	0.663875
$202$	36.9441	2.59938
$203$	−2.95109	−0.207126
$204$	−28.9823	−2.02917
$205$	0	0
$206$	−7.99670	−0.557156
$207$	−0.716026	−0.0497673
$208$	−67.2093	−4.66013
$209$	1.34210	0.0928347
$210$	0	0
$211$	5.35265	0.368491	0.184246	−	0.982880i	$-0.441016\pi$
$211$	5.35265	0.368491	0.184246	+	0.982880i	$0.441016\pi$
$212$	−54.9913	−3.77682
$213$	9.24435	0.633412
$214$	−33.2741	−2.27457
$215$	0	0
$216$	−53.9687	−3.67211
$217$	−3.93152	−0.266889
$218$	31.7368	2.14949
$219$	−13.7776	−0.931007
$220$	0	0
$221$	−17.0488	−1.14682
$222$	−24.0809	−1.61621
$223$	28.5194	1.90980	0.954902	−	0.296922i	$-0.0959601\pi$
$223$	28.5194	1.90980	0.954902	+	0.296922i	$0.0959601\pi$
$224$	14.3284	0.957358
$225$	0	0
$226$	−15.5937	−1.03727
$227$	−6.47292	−0.429623	−0.214811	−	0.976656i	$-0.568914\pi$
$227$	−6.47292	−0.429623	−0.214811	+	0.976656i	$0.568914\pi$
$228$	8.27955	0.548326
$229$	−9.36212	−0.618666	−0.309333	−	0.950954i	$-0.600106\pi$
$229$	−9.36212	−0.618666	−0.309333	+	0.950954i	$0.600106\pi$
$230$	0	0
$231$	−1.07226	−0.0705495
$232$	−44.0650	−2.89301
$233$	9.69257	0.634981	0.317491	−	0.948261i	$-0.397160\pi$
$233$	9.69257	0.634981	0.317491	+	0.948261i	$0.397160\pi$
$234$	−13.5472	−0.885605
$235$	0	0
$236$	−26.1601	−1.70287
$237$	14.0593	0.913248
$238$	6.73528	0.436583
$239$	1.83905	0.118958	0.0594791	−	0.998230i	$-0.481056\pi$
$239$	1.83905	0.118958	0.0594791	+	0.998230i	$0.481056\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	26.0231	1.67283
$243$	−10.9072	−0.699697
$244$	−8.25784	−0.528654
$245$	0	0
$246$	39.5127	2.51924
$247$	4.87042	0.309897
$248$	−58.7046	−3.72774
$249$	22.8861	1.45035
$250$	0	0
$251$	19.1051	1.20590	0.602950	−	0.797779i	$-0.293992\pi$
$251$	19.1051	1.20590	0.602950	+	0.797779i	$0.293992\pi$
$252$	3.92293	0.247121
$253$	−0.783032	−0.0492288
$254$	25.5046	1.60030
$255$	0	0
$256$	47.4341	2.96463
$257$	−4.92468	−0.307193	−0.153596	−	0.988134i	$-0.549086\pi$
$257$	−4.92468	−0.307193	−0.153596	+	0.988134i	$0.549086\pi$
$258$	3.47246	0.216186
$259$	4.10199	0.254886
$260$	0	0
$261$	−5.15158	−0.318875
$262$	−32.5463	−2.01072
$263$	−22.4117	−1.38197	−0.690983	−	0.722871i	$-0.742822\pi$
$263$	−22.4117	−1.38197	−0.690983	+	0.722871i	$0.742822\pi$
$264$	−16.0108	−0.985394
$265$	0	0
$266$	−1.92410	−0.117974
$267$	3.81418	0.233424
$268$	−37.6569	−2.30026
$269$	32.2758	1.96789	0.983945	−	0.178473i	$-0.0571158\pi$
$269$	32.2758	1.96789	0.983945	+	0.178473i	$0.0571158\pi$
$270$	0	0
$271$	18.7714	1.14028	0.570142	−	0.821546i	$-0.306888\pi$
$271$	18.7714	1.14028	0.570142	+	0.821546i	$0.306888\pi$
$272$	58.3304	3.53680
$273$	−3.89119	−0.235506
$274$	−43.0174	−2.59878
$275$	0	0
$276$	−4.83061	−0.290769
$277$	−1.61695	−0.0971531	−0.0485765	−	0.998819i	$-0.515468\pi$
$277$	−1.61695	−0.0971531	−0.0485765	+	0.998819i	$0.515468\pi$
$278$	39.4662	2.36702
$279$	−6.86307	−0.410881
$280$	0	0
$281$	−1.13217	−0.0675396	−0.0337698	−	0.999430i	$-0.510751\pi$
$281$	−1.13217	−0.0675396	−0.0337698	+	0.999430i	$0.510751\pi$
$282$	12.2803	0.731278
$283$	−12.5855	−0.748129	−0.374064	−	0.927403i	$-0.622036\pi$
$283$	−12.5855	−0.748129	−0.374064	+	0.927403i	$0.622036\pi$
$284$	−36.9860	−2.19471
$285$	0	0
$286$	−14.8149	−0.876023
$287$	−6.73067	−0.397299
$288$	25.0125	1.47387
$289$	−2.20350	−0.129618
$290$	0	0
$291$	−2.93878	−0.172275
$292$	55.1234	3.22585
$293$	6.09708	0.356195	0.178098	−	0.984013i	$-0.443006\pi$
$293$	6.09708	0.356195	0.178098	+	0.984013i	$0.443006\pi$
$294$	−24.7532	−1.44364
$295$	0	0
$296$	61.2501	3.56009
$297$	−6.89984	−0.400369
$298$	−9.42257	−0.545835
$299$	−2.84159	−0.164334
$300$	0	0
$301$	−0.591505	−0.0340938
$302$	−8.04480	−0.462926
$303$	18.5241	1.06418
$304$	−16.6636	−0.955722
$305$	0	0
$306$	11.7575	0.672130
$307$	−9.29750	−0.530636	−0.265318	−	0.964161i	$-0.585477\pi$
$307$	−9.29750	−0.530636	−0.265318	+	0.964161i	$0.585477\pi$
$308$	4.29003	0.244447
$309$	−4.00962	−0.228100
$310$	0	0
$311$	20.3604	1.15453	0.577267	−	0.816556i	$-0.304119\pi$
$311$	20.3604	1.15453	0.577267	+	0.816556i	$0.304119\pi$
$312$	−58.1024	−3.28940
$313$	−18.4109	−1.04065	−0.520323	−	0.853970i	$-0.674188\pi$
$313$	−18.4109	−1.04065	−0.520323	+	0.853970i	$0.674188\pi$
$314$	−55.2327	−3.11696
$315$	0	0
$316$	−56.2502	−3.16432
$317$	−25.6936	−1.44309	−0.721547	−	0.692366i	$-0.756569\pi$
$317$	−25.6936	−1.44309	−0.721547	+	0.692366i	$0.756569\pi$
$318$	−37.6173	−2.10947
$319$	−5.63366	−0.315425
$320$	0	0
$321$	−16.6840	−0.931208
$322$	1.12260	0.0625599
$323$	−4.22700	−0.235196
$324$	−24.1705	−1.34281
$325$	0	0
$326$	38.6544	2.14087
$327$	15.9131	0.879999
$328$	−100.501	−5.54924
$329$	−2.09184	−0.115327
$330$	0	0
$331$	1.75061	0.0962225	0.0481112	−	0.998842i	$-0.484680\pi$
$331$	1.75061	0.0962225	0.0481112	+	0.998842i	$0.484680\pi$
$332$	−91.5655	−5.02531
$333$	7.16067	0.392402
$334$	−64.9172	−3.55211
$335$	0	0
$336$	13.3133	0.726298
$337$	−11.5741	−0.630480	−0.315240	−	0.949012i	$-0.602085\pi$
$337$	−11.5741	−0.630480	−0.315240	+	0.949012i	$0.602085\pi$
$338$	−18.1834	−0.989046
$339$	−7.81881	−0.424660
$340$	0	0
$341$	−7.50531	−0.406435
$342$	−3.35882	−0.181624
$343$	8.69489	0.469480
$344$	−8.83223	−0.476202
$345$	0	0
$346$	21.6716	1.16507
$347$	−30.6495	−1.64535	−0.822677	−	0.568509i	$-0.807520\pi$
$347$	−30.6495	−1.64535	−0.822677	+	0.568509i	$0.807520\pi$
$348$	−34.7547	−1.86305
$349$	34.9615	1.87144	0.935722	−	0.352739i	$-0.114750\pi$
$349$	34.9615	1.87144	0.935722	+	0.352739i	$0.114750\pi$
$350$	0	0
$351$	−25.0393	−1.33650
$352$	27.3531	1.45793
$353$	16.3277	0.869038	0.434519	−	0.900663i	$-0.356918\pi$
$353$	16.3277	0.869038	0.434519	+	0.900663i	$0.356918\pi$
$354$	−17.8950	−0.951108
$355$	0	0
$356$	−15.2602	−0.808791
$357$	3.37713	0.178737
$358$	13.2134	0.698349
$359$	30.1153	1.58943	0.794713	−	0.606986i	$-0.207621\pi$
$359$	30.1153	1.58943	0.794713	+	0.606986i	$0.207621\pi$
$360$	0	0
$361$	−17.7924	−0.936445
$362$	62.0259	3.26001
$363$	13.0483	0.684856
$364$	15.5684	0.816004
$365$	0	0
$366$	−5.64884	−0.295270
$367$	30.3655	1.58506	0.792532	−	0.609831i	$-0.208762\pi$
$367$	30.3655	1.58506	0.792532	+	0.609831i	$0.208762\pi$
$368$	9.72219	0.506804
$369$	−11.7494	−0.611651
$370$	0	0
$371$	6.40780	0.332677
$372$	−46.3011	−2.40060
$373$	4.18160	0.216515	0.108258	−	0.994123i	$-0.465473\pi$
$373$	4.18160	0.216515	0.108258	+	0.994123i	$0.465473\pi$
$374$	12.8577	0.664857
$375$	0	0
$376$	−31.2350	−1.61082
$377$	−20.4443	−1.05294
$378$	9.89199	0.508789
$379$	−18.3382	−0.941968	−0.470984	−	0.882142i	$-0.656101\pi$
$379$	−18.3382	−0.941968	−0.470984	+	0.882142i	$0.656101\pi$
$380$	0	0
$381$	12.7882	0.655161
$382$	27.3237	1.39800
$383$	13.6114	0.695510	0.347755	−	0.937585i	$-0.386944\pi$
$383$	13.6114	0.695510	0.347755	+	0.937585i	$0.386944\pi$
$384$	54.8388	2.79848
$385$	0	0
$386$	40.1211	2.04211
$387$	−1.03256	−0.0524882
$388$	11.7579	0.596915
$389$	16.4625	0.834681	0.417340	−	0.908750i	$-0.362962\pi$
$389$	16.4625	0.834681	0.417340	+	0.908750i	$0.362962\pi$
$390$	0	0
$391$	2.46620	0.124721
$392$	62.9601	3.17997
$393$	−16.3191	−0.823187
$394$	−58.9235	−2.96852
$395$	0	0
$396$	7.48892	0.376332
$397$	−14.3066	−0.718026	−0.359013	−	0.933333i	$-0.616887\pi$
$397$	−14.3066	−0.718026	−0.359013	+	0.933333i	$0.616887\pi$
$398$	21.7028	1.08786
$399$	−0.964765	−0.0482987
$400$	0	0
$401$	30.6772	1.53195	0.765973	−	0.642873i	$-0.222258\pi$
$401$	30.6772	1.53195	0.765973	+	0.642873i	$0.222258\pi$
$402$	−25.7595	−1.28477
$403$	−27.2365	−1.35675
$404$	−74.1137	−3.68729
$405$	0	0
$406$	8.07673	0.400842
$407$	7.83076	0.388156
$408$	50.4267	2.49649
$409$	−18.5846	−0.918947	−0.459473	−	0.888191i	$-0.651962\pi$
$409$	−18.5846	−0.918947	−0.459473	+	0.888191i	$0.651962\pi$
$410$	0	0
$411$	−21.5693	−1.06394
$412$	16.0422	0.790343
$413$	3.04827	0.149996
$414$	1.95967	0.0963125
$415$	0	0
$416$	99.2635	4.86679
$417$	19.7887	0.969058
$418$	−3.67314	−0.179659
$419$	19.8470	0.969587	0.484794	−	0.874629i	$-0.338895\pi$
$419$	19.8470	0.969587	0.484794	+	0.874629i	$0.338895\pi$
$420$	0	0
$421$	−27.6111	−1.34568	−0.672840	−	0.739788i	$-0.734926\pi$
$421$	−27.6111	−1.34568	−0.672840	+	0.739788i	$0.734926\pi$
$422$	−14.6495	−0.713126
$423$	−3.65164	−0.177549
$424$	95.6800	4.64663
$425$	0	0
$426$	−25.3005	−1.22582
$427$	0.962235	0.0465658
$428$	66.7513	3.22655
$429$	−7.42833	−0.358643
$430$	0	0
$431$	−9.85411	−0.474656	−0.237328	−	0.971430i	$-0.576272\pi$
$431$	−9.85411	−0.474656	−0.237328	+	0.971430i	$0.576272\pi$
$432$	85.6690	4.12175
$433$	30.3606	1.45904	0.729519	−	0.683960i	$-0.239744\pi$
$433$	30.3606	1.45904	0.729519	+	0.683960i	$0.239744\pi$
$434$	10.7600	0.516498
$435$	0	0
$436$	−63.6673	−3.04911
$437$	−0.704532	−0.0337023
$438$	37.7076	1.80174
$439$	6.26733	0.299123	0.149562	−	0.988752i	$-0.452214\pi$
$439$	6.26733	0.299123	0.149562	+	0.988752i	$0.452214\pi$
$440$	0	0
$441$	7.36058	0.350504
$442$	46.6602	2.21940
$443$	16.9876	0.807105	0.403552	−	0.914957i	$-0.367775\pi$
$443$	16.9876	0.807105	0.403552	+	0.914957i	$0.367775\pi$
$444$	48.3089	2.29264
$445$	0	0
$446$	−78.0539	−3.69596
$447$	−4.72457	−0.223464
$448$	−19.8120	−0.936030
$449$	35.2723	1.66460	0.832300	−	0.554325i	$-0.187023\pi$
$449$	35.2723	1.66460	0.832300	+	0.554325i	$0.187023\pi$
$450$	0	0
$451$	−12.8489	−0.605033
$452$	31.2825	1.47141
$453$	−4.03374	−0.189522
$454$	17.7155	0.831431
$455$	0	0
$456$	−14.4057	−0.674607
$457$	5.84116	0.273238	0.136619	−	0.990624i	$-0.456376\pi$
$457$	5.84116	0.273238	0.136619	+	0.990624i	$0.456376\pi$
$458$	25.6229	1.19728
$459$	21.7314	1.01433
$460$	0	0
$461$	19.6890	0.917008	0.458504	−	0.888692i	$-0.348385\pi$
$461$	19.6890	0.917008	0.458504	+	0.888692i	$0.348385\pi$
$462$	2.93463	0.136531
$463$	39.8545	1.85220	0.926099	−	0.377281i	$-0.123141\pi$
$463$	39.8545	1.85220	0.926099	+	0.377281i	$0.123141\pi$
$464$	69.9480	3.24726
$465$	0	0
$466$	−26.5273	−1.22885
$467$	−10.3436	−0.478647	−0.239323	−	0.970940i	$-0.576926\pi$
$467$	−10.3436	−0.478647	−0.239323	+	0.970940i	$0.576926\pi$
$468$	27.1770	1.25626
$469$	4.38793	0.202616
$470$	0	0
$471$	−27.6942	−1.27608
$472$	45.5161	2.09505
$473$	−1.12919	−0.0519203
$474$	−38.4784	−1.76737
$475$	0	0
$476$	−13.5117	−0.619306
$477$	11.1858	0.512163
$478$	−5.03324	−0.230215
$479$	−42.5400	−1.94370	−0.971851	−	0.235595i	$-0.924296\pi$
$479$	−42.5400	−1.94370	−0.971851	+	0.235595i	$0.924296\pi$
$480$	0	0
$481$	28.4175	1.29573
$482$	2.73687	0.124661
$483$	0.562882	0.0256120
$484$	−52.2051	−2.37296
$485$	0	0
$486$	29.8515	1.35409
$487$	33.5960	1.52238	0.761190	−	0.648529i	$-0.224616\pi$
$487$	33.5960	1.52238	0.761190	+	0.648529i	$0.224616\pi$
$488$	14.3679	0.650404
$489$	19.3817	0.876470
$490$	0	0
$491$	21.5477	0.972433	0.486217	−	0.873838i	$-0.338377\pi$
$491$	21.5477	0.972433	0.486217	+	0.873838i	$0.338377\pi$
$492$	−79.2665	−3.57361
$493$	17.7435	0.799127
$494$	−13.3297	−0.599731
$495$	0	0
$496$	93.1866	4.18420
$497$	4.30975	0.193319
$498$	−62.6362	−2.80679
$499$	12.0650	0.540102	0.270051	−	0.962846i	$-0.412959\pi$
$499$	12.0650	0.540102	0.270051	+	0.962846i	$0.412959\pi$
$500$	0	0
$501$	−32.5501	−1.45423
$502$	−52.2880	−2.33373
$503$	20.6590	0.921140	0.460570	−	0.887623i	$-0.347645\pi$
$503$	20.6590	0.921140	0.460570	+	0.887623i	$0.347645\pi$
$504$	−6.82554	−0.304034
$505$	0	0
$506$	2.14305	0.0952704
$507$	−9.11733	−0.404915
$508$	−51.1648	−2.27007
$509$	−20.5056	−0.908896	−0.454448	−	0.890773i	$-0.650164\pi$
$509$	−20.5056	−0.908896	−0.454448	+	0.890773i	$0.650164\pi$
$510$	0	0
$511$	−6.42319	−0.284145
$512$	−49.8980	−2.20520
$513$	−6.20812	−0.274095
$514$	13.4782	0.594497
$515$	0	0
$516$	−6.96611	−0.306666
$517$	−3.99336	−0.175628
$518$	−11.2266	−0.493269
$519$	10.8664	0.476980
$520$	0	0
$521$	−1.75347	−0.0768208	−0.0384104	−	0.999262i	$-0.512229\pi$
$521$	−1.75347	−0.0768208	−0.0384104	+	0.999262i	$0.512229\pi$
$522$	14.0992	0.617105
$523$	2.65988	0.116309	0.0581543	−	0.998308i	$-0.481478\pi$
$523$	2.65988	0.116309	0.0581543	+	0.998308i	$0.481478\pi$
$524$	65.2913	2.85227
$525$	0	0
$526$	61.3379	2.67446
$527$	23.6384	1.02970
$528$	25.4152	1.10605
$529$	−22.5889	−0.982128
$530$	0	0
$531$	5.32123	0.230922
$532$	3.85995	0.167350
$533$	−46.6283	−2.01970
$534$	−10.4389	−0.451735
$535$	0	0
$536$	65.5196	2.83002
$537$	6.62532	0.285904
$538$	−88.3345	−3.80837
$539$	8.04938	0.346711
$540$	0	0
$541$	15.6618	0.673353	0.336676	−	0.941620i	$-0.390697\pi$
$541$	15.6618	0.673353	0.336676	+	0.941620i	$0.390697\pi$
$542$	−51.3750	−2.20674
$543$	31.1004	1.33465
$544$	−86.1501	−3.69365
$545$	0	0
$546$	10.6497	0.455764
$547$	−1.45170	−0.0620703	−0.0310352	−	0.999518i	$-0.509880\pi$
$547$	−1.45170	−0.0620703	−0.0310352	+	0.999518i	$0.509880\pi$
$548$	86.2974	3.68644
$549$	1.67973	0.0716892
$550$	0	0
$551$	−5.06888	−0.215942
$552$	8.40483	0.357733
$553$	6.55449	0.278725
$554$	4.42537	0.188016
$555$	0	0
$556$	−79.1732	−3.35769
$557$	−29.5512	−1.25212	−0.626062	−	0.779773i	$-0.715334\pi$
$557$	−29.5512	−1.25212	−0.626062	+	0.779773i	$0.715334\pi$
$558$	18.7833	0.795161
$559$	−4.09779	−0.173318
$560$	0	0
$561$	6.44699	0.272192
$562$	3.09860	0.130707
$563$	11.5106	0.485113	0.242556	−	0.970137i	$-0.422014\pi$
$563$	11.5106	0.485113	0.242556	+	0.970137i	$0.422014\pi$
$564$	−24.6355	−1.03734
$565$	0	0
$566$	34.4448	1.44782
$567$	2.81644	0.118279
$568$	64.3522	2.70016
$569$	27.3601	1.14700	0.573498	−	0.819207i	$-0.305586\pi$
$569$	27.3601	1.14700	0.573498	+	0.819207i	$0.305586\pi$
$570$	0	0
$571$	32.4788	1.35919	0.679597	−	0.733586i	$-0.262155\pi$
$571$	32.4788	1.35919	0.679597	+	0.733586i	$0.262155\pi$
$572$	29.7202	1.24266
$573$	13.7004	0.572342
$574$	18.4209	0.768876
$575$	0	0
$576$	−34.5850	−1.44104
$577$	16.9082	0.703899	0.351950	−	0.936019i	$-0.385519\pi$
$577$	16.9082	0.703899	0.351950	+	0.936019i	$0.385519\pi$
$578$	6.03069	0.250844
$579$	20.1171	0.836039
$580$	0	0
$581$	10.6696	0.442648
$582$	8.04306	0.333396
$583$	12.2326	0.506622
$584$	−95.9097	−3.96877
$585$	0	0
$586$	−16.6869	−0.689329
$587$	27.7908	1.14705	0.573525	−	0.819188i	$-0.305576\pi$
$587$	27.7908	1.14705	0.573525	+	0.819188i	$0.305576\pi$
$588$	49.6576	2.04784
$589$	−6.75290	−0.278248
$590$	0	0
$591$	−29.5448	−1.21531
$592$	−97.2274	−3.99602
$593$	−44.4704	−1.82618	−0.913091	−	0.407756i	$-0.866311\pi$
$593$	−44.4704	−1.82618	−0.913091	+	0.407756i	$0.866311\pi$
$594$	18.8839	0.774818
$595$	0	0
$596$	18.9027	0.774283
$597$	10.8820	0.445371
$598$	7.77706	0.318028
$599$	−6.46648	−0.264213	−0.132107	−	0.991236i	$-0.542174\pi$
$599$	−6.46648	−0.264213	−0.132107	+	0.991236i	$0.542174\pi$
$600$	0	0
$601$	−35.0556	−1.42995	−0.714973	−	0.699152i	$-0.753561\pi$
$601$	−35.0556	−1.42995	−0.714973	+	0.699152i	$0.753561\pi$
$602$	1.61887	0.0659803
$603$	7.65981	0.311932
$604$	16.1387	0.656675
$605$	0	0
$606$	−50.6981	−2.05947
$607$	−32.7458	−1.32911	−0.664555	−	0.747239i	$-0.731379\pi$
$607$	−32.7458	−1.32911	−0.664555	+	0.747239i	$0.731379\pi$
$608$	24.6110	0.998107
$609$	4.04975	0.164104
$610$	0	0
$611$	−14.4917	−0.586273
$612$	−23.5867	−0.953437
$613$	4.14429	0.167386	0.0836931	−	0.996492i	$-0.473328\pi$
$613$	4.14429	0.167386	0.0836931	+	0.996492i	$0.473328\pi$
$614$	25.4460	1.02692
$615$	0	0
$616$	−7.46427	−0.300744
$617$	−0.192219	−0.00773844	−0.00386922	−	0.999993i	$-0.501232\pi$
$617$	−0.192219	−0.00773844	−0.00386922	+	0.999993i	$0.501232\pi$
$618$	10.9738	0.441431
$619$	47.0896	1.89269	0.946345	−	0.323159i	$-0.104745\pi$
$619$	47.0896	1.89269	0.946345	+	0.323159i	$0.104745\pi$
$620$	0	0
$621$	3.62206	0.145348
$622$	−55.7238	−2.23432
$623$	1.77818	0.0712414
$624$	92.2308	3.69219
$625$	0	0
$626$	50.3882	2.01392
$627$	−1.84175	−0.0735523
$628$	110.803	4.42150
$629$	−24.6634	−0.983393
$630$	0	0
$631$	31.8829	1.26924	0.634618	−	0.772826i	$-0.281157\pi$
$631$	31.8829	1.26924	0.634618	+	0.772826i	$0.281157\pi$
$632$	97.8702	3.89307
$633$	−7.34539	−0.291953
$634$	70.3198	2.79276
$635$	0	0
$636$	75.4642	2.99235
$637$	29.2109	1.15738
$638$	15.4186	0.610428
$639$	7.52333	0.297618
$640$	0	0
$641$	−0.973151	−0.0384372	−0.0192186	−	0.999815i	$-0.506118\pi$
$641$	−0.973151	−0.0384372	−0.0192186	+	0.999815i	$0.506118\pi$
$642$	45.6618	1.80213
$643$	24.1374	0.951884	0.475942	−	0.879477i	$-0.342107\pi$
$643$	24.1374	0.951884	0.475942	+	0.879477i	$0.342107\pi$
$644$	−2.25205	−0.0887432
$645$	0	0
$646$	11.5687	0.455166
$647$	−42.2322	−1.66032	−0.830160	−	0.557525i	$-0.811751\pi$
$647$	−42.2322	−1.66032	−0.830160	+	0.557525i	$0.811751\pi$
$648$	42.0545	1.65206
$649$	5.81919	0.228423
$650$	0	0
$651$	5.39519	0.211454
$652$	−77.5447	−3.03688
$653$	−16.1285	−0.631159	−0.315579	−	0.948899i	$-0.602199\pi$
$653$	−16.1285	−0.631159	−0.315579	+	0.948899i	$0.602199\pi$
$654$	−43.5522	−1.70302
$655$	0	0
$656$	159.533	6.22873
$657$	−11.2127	−0.437448
$658$	5.72510	0.223188
$659$	−0.880038	−0.0342814	−0.0171407	−	0.999853i	$-0.505456\pi$
$659$	−0.880038	−0.0342814	−0.0171407	+	0.999853i	$0.505456\pi$
$660$	0	0
$661$	−18.9767	−0.738106	−0.369053	−	0.929408i	$-0.620318\pi$
$661$	−18.9767	−0.738106	−0.369053	+	0.929408i	$0.620318\pi$
$662$	−4.79120	−0.186215
$663$	23.3959	0.908621
$664$	159.316	6.18265
$665$	0	0
$666$	−19.5978	−0.759399
$667$	2.95739	0.114510
$668$	130.231	5.03877
$669$	−39.1370	−1.51312
$670$	0	0
$671$	1.83692	0.0709135
$672$	−19.6628	−0.758508
$673$	−23.7405	−0.915127	−0.457564	−	0.889177i	$-0.651278\pi$
$673$	−23.7405	−0.915127	−0.457564	+	0.889177i	$0.651278\pi$
$674$	31.6767	1.22014
$675$	0	0
$676$	36.4778	1.40299
$677$	20.9370	0.804675	0.402337	−	0.915492i	$-0.368198\pi$
$677$	20.9370	0.804675	0.402337	+	0.915492i	$0.368198\pi$
$678$	21.3991	0.821826
$679$	−1.37007	−0.0525785
$680$	0	0
$681$	8.88274	0.340387
$682$	20.5410	0.786557
$683$	−10.3808	−0.397210	−0.198605	−	0.980080i	$-0.563641\pi$
$683$	−10.3808	−0.397210	−0.198605	+	0.980080i	$0.563641\pi$
$684$	6.73815	0.257640
$685$	0	0
$686$	−23.7968	−0.908564
$687$	12.8476	0.490165
$688$	14.0201	0.534513
$689$	44.3916	1.69118
$690$	0	0
$691$	4.26464	0.162235	0.0811173	−	0.996705i	$-0.474151\pi$
$691$	4.26464	0.162235	0.0811173	+	0.996705i	$0.474151\pi$
$692$	−43.4755	−1.65269
$693$	−0.872638	−0.0331488
$694$	83.8837	3.18418
$695$	0	0
$696$	60.4701	2.29211
$697$	40.4683	1.53285
$698$	−95.6849	−3.62173
$699$	−13.3010	−0.503091
$700$	0	0
$701$	−31.1234	−1.17551	−0.587757	−	0.809037i	$-0.699989\pi$
$701$	−31.1234	−1.17551	−0.587757	+	0.809037i	$0.699989\pi$
$702$	68.5291	2.58646
$703$	7.04572	0.265734
$704$	−37.8214	−1.42545
$705$	0	0
$706$	−44.6868	−1.68181
$707$	8.63601	0.324791
$708$	35.8992	1.34918
$709$	11.9842	0.450075	0.225038	−	0.974350i	$-0.427749\pi$
$709$	11.9842	0.450075	0.225038	+	0.974350i	$0.427749\pi$
$710$	0	0
$711$	11.4419	0.429104
$712$	26.5515	0.995058
$713$	3.93991	0.147551
$714$	−9.24277	−0.345902
$715$	0	0
$716$	−26.5074	−0.990629
$717$	−2.52371	−0.0942499
$718$	−82.4216	−3.07595
$719$	38.2754	1.42743	0.713716	−	0.700435i	$-0.247011\pi$
$719$	38.2754	1.42743	0.713716	+	0.700435i	$0.247011\pi$
$720$	0	0
$721$	−1.86930	−0.0696164
$722$	48.6956	1.81226
$723$	1.37229	0.0510361
$724$	−124.431	−4.62442
$725$	0	0
$726$	−35.7113	−1.32537
$727$	8.54679	0.316983	0.158491	−	0.987360i	$-0.449337\pi$
$727$	8.54679	0.316983	0.158491	+	0.987360i	$0.449337\pi$
$728$	−27.0876	−1.00393
$729$	28.1747	1.04351
$730$	0	0
$731$	3.55644	0.131540
$732$	11.3322	0.418849
$733$	−3.50210	−0.129353	−0.0646765	−	0.997906i	$-0.520602\pi$
$733$	−3.50210	−0.129353	−0.0646765	+	0.997906i	$0.520602\pi$
$734$	−83.1062	−3.06751
$735$	0	0
$736$	−14.3590	−0.529280
$737$	8.37661	0.308557
$738$	32.1566	1.18370
$739$	9.75976	0.359018	0.179509	−	0.983756i	$-0.442549\pi$
$739$	9.75976	0.359018	0.179509	+	0.983756i	$0.442549\pi$
$740$	0	0
$741$	−6.68364	−0.245529
$742$	−17.5373	−0.643815
$743$	17.3646	0.637045	0.318523	−	0.947915i	$-0.396813\pi$
$743$	17.3646	0.637045	0.318523	+	0.947915i	$0.396813\pi$
$744$	80.5598	2.95347
$745$	0	0
$746$	−11.4445	−0.419013
$747$	18.6254	0.681468
$748$	−25.7939	−0.943120
$749$	−7.77812	−0.284206
$750$	0	0
$751$	36.2816	1.32393	0.661967	−	0.749533i	$-0.269722\pi$
$751$	36.2816	1.32393	0.661967	+	0.749533i	$0.269722\pi$
$752$	49.5819	1.80806
$753$	−26.2177	−0.955427
$754$	55.9535	2.03771
$755$	0	0
$756$	−19.8444	−0.721733
$757$	−17.4512	−0.634274	−0.317137	−	0.948380i	$-0.602721\pi$
$757$	−17.4512	−0.634274	−0.317137	+	0.948380i	$0.602721\pi$
$758$	50.1891	1.82295
$759$	1.07455	0.0390036
$760$	0	0
$761$	−30.8147	−1.11703	−0.558516	−	0.829494i	$-0.688629\pi$
$761$	−30.8147	−1.11703	−0.558516	+	0.829494i	$0.688629\pi$
$762$	−34.9997	−1.26791
$763$	7.41877	0.268577
$764$	−54.8143	−1.98311
$765$	0	0
$766$	−37.2526	−1.34599
$767$	21.1176	0.762513
$768$	−65.0935	−2.34886
$769$	47.8280	1.72472	0.862361	−	0.506295i	$-0.168985\pi$
$769$	47.8280	1.72472	0.862361	+	0.506295i	$0.168985\pi$
$770$	0	0
$771$	6.75810	0.243387
$772$	−80.4871	−2.89679
$773$	−31.6875	−1.13972	−0.569861	−	0.821741i	$-0.693003\pi$
$773$	−31.6875	−1.13972	−0.569861	+	0.821741i	$0.693003\pi$
$774$	2.82599	0.101578
$775$	0	0
$776$	−20.4576	−0.734386
$777$	−5.62914	−0.201944
$778$	−45.0556	−1.61532
$779$	−11.5608	−0.414209
$780$	0	0
$781$	8.22736	0.294398
$782$	−6.74965	−0.241367
$783$	26.0596	0.931294
$784$	−99.9418	−3.56935
$785$	0	0
$786$	44.6631	1.59308
$787$	44.2245	1.57643	0.788216	−	0.615398i	$-0.211005\pi$
$787$	44.2245	1.57643	0.788216	+	0.615398i	$0.211005\pi$
$788$	118.207	4.21094
$789$	30.7554	1.09492
$790$	0	0
$791$	−3.64516	−0.129607
$792$	−13.0300	−0.463003
$793$	6.66611	0.236721
$794$	39.1551	1.38956
$795$	0	0
$796$	−43.5381	−1.54317
$797$	33.1040	1.17260	0.586302	−	0.810092i	$-0.300583\pi$
$797$	33.1040	1.17260	0.586302	+	0.810092i	$0.300583\pi$
$798$	2.64043	0.0934703
$799$	12.5773	0.444952
$800$	0	0
$801$	3.10409	0.109678
$802$	−83.9594	−2.96471
$803$	−12.2619	−0.432715
$804$	51.6763	1.82248
$805$	0	0
$806$	74.5427	2.62565
$807$	−44.2918	−1.55915
$808$	128.951	4.53649
$809$	−41.3689	−1.45445	−0.727227	−	0.686398i	$-0.759191\pi$
$809$	−41.3689	−1.45445	−0.727227	+	0.686398i	$0.759191\pi$
$810$	0	0
$811$	10.6961	0.375592	0.187796	−	0.982208i	$-0.439866\pi$
$811$	10.6961	0.375592	0.187796	+	0.982208i	$0.439866\pi$
$812$	−16.2028	−0.568606
$813$	−25.7599	−0.903440
$814$	−21.4317	−0.751182
$815$	0	0
$816$	−80.0464	−2.80218
$817$	−1.01599	−0.0355450
$818$	50.8635	1.77840
$819$	−3.16677	−0.110656
$820$	0	0
$821$	2.86174	0.0998753	0.0499376	−	0.998752i	$-0.484098\pi$
$821$	2.86174	0.0998753	0.0499376	+	0.998752i	$0.484098\pi$
$822$	59.0324	2.05899
$823$	−14.4643	−0.504195	−0.252098	−	0.967702i	$-0.581120\pi$
$823$	−14.4643	−0.504195	−0.252098	+	0.967702i	$0.581120\pi$
$824$	−27.9120	−0.972361
$825$	0	0
$826$	−8.34271	−0.290280
$827$	11.2853	0.392430	0.196215	−	0.980561i	$-0.437135\pi$
$827$	11.2853	0.392430	0.196215	+	0.980561i	$0.437135\pi$
$828$	−3.93130	−0.136622
$829$	19.9553	0.693077	0.346538	−	0.938036i	$-0.387357\pi$
$829$	19.9553	0.693077	0.346538	+	0.938036i	$0.387357\pi$
$830$	0	0
$831$	2.21893	0.0769737
$832$	−137.252	−4.75837
$833$	−25.3519	−0.878392
$834$	−54.1591	−1.87538
$835$	0	0
$836$	7.36870	0.254852
$837$	34.7173	1.20000
$838$	−54.3185	−1.87640
$839$	4.03287	0.139230	0.0696152	−	0.997574i	$-0.477823\pi$
$839$	4.03287	0.139230	0.0696152	+	0.997574i	$0.477823\pi$
$840$	0	0
$841$	−7.72255	−0.266295
$842$	75.5678	2.60424
$843$	1.55367	0.0535112
$844$	29.3884	1.01159
$845$	0	0
$846$	9.99405	0.343602
$847$	6.08314	0.209019
$848$	−151.881	−5.21561
$849$	17.2709	0.592737
$850$	0	0
$851$	−4.11075	−0.140915
$852$	50.7555	1.73886
$853$	46.7085	1.59927	0.799634	−	0.600487i	$-0.205027\pi$
$853$	46.7085	1.59927	0.799634	+	0.600487i	$0.205027\pi$
$854$	−2.63351	−0.0901169
$855$	0	0
$856$	−116.141	−3.96963
$857$	4.23605	0.144701	0.0723503	−	0.997379i	$-0.476950\pi$
$857$	4.23605	0.144701	0.0723503	+	0.997379i	$0.476950\pi$
$858$	20.3304	0.694067
$859$	8.33367	0.284341	0.142171	−	0.989842i	$-0.454592\pi$
$859$	8.33367	0.284341	0.142171	+	0.989842i	$0.454592\pi$
$860$	0	0
$861$	9.23645	0.314777
$862$	26.9694	0.918581
$863$	−22.4573	−0.764455	−0.382227	−	0.924068i	$-0.624843\pi$
$863$	−22.4573	−0.764455	−0.382227	+	0.924068i	$0.624843\pi$
$864$	−126.527	−4.30454
$865$	0	0
$866$	−83.0930	−2.82362
$867$	3.02385	0.102695
$868$	−21.5858	−0.732668
$869$	12.5126	0.424461
$870$	0	0
$871$	30.3984	1.03001
$872$	110.775	3.75133
$873$	−2.39167	−0.0809459
$874$	1.92821	0.0652227
$875$	0	0
$876$	−75.6454	−2.55582
$877$	34.0560	1.14999	0.574994	−	0.818157i	$-0.305004\pi$
$877$	34.0560	1.14999	0.574994	+	0.818157i	$0.305004\pi$
$878$	−17.1528	−0.578880
$879$	−8.36697	−0.282211
$880$	0	0
$881$	47.7500	1.60874	0.804369	−	0.594130i	$-0.202503\pi$
$881$	47.7500	1.60874	0.804369	+	0.594130i	$0.202503\pi$
$882$	−20.1449	−0.678315
$883$	−49.5142	−1.66629	−0.833143	−	0.553057i	$-0.813461\pi$
$883$	−49.5142	−1.66629	−0.833143	+	0.553057i	$0.813461\pi$
$884$	−93.6053	−3.14829
$885$	0	0
$886$	−46.4928	−1.56196
$887$	−48.9165	−1.64245	−0.821227	−	0.570602i	$-0.806710\pi$
$887$	−48.9165	−1.64245	−0.821227	+	0.570602i	$0.806710\pi$
$888$	−84.0531	−2.82064
$889$	5.96192	0.199956
$890$	0	0
$891$	5.37662	0.180124
$892$	156.584	5.24283
$893$	−3.59302	−0.120236
$894$	12.9305	0.432461
$895$	0	0
$896$	25.5660	0.854101
$897$	3.89950	0.130200
$898$	−96.5355	−3.22143
$899$	28.3464	0.945405
$900$	0	0
$901$	−38.5271	−1.28352
$902$	35.1658	1.17089
$903$	0.811718	0.0270123
$904$	−54.4287	−1.81027
$905$	0	0
$906$	11.0398	0.366773
$907$	0.496627	0.0164902	0.00824511	−	0.999966i	$-0.497375\pi$
$907$	0.496627	0.0164902	0.00824511	+	0.999966i	$0.497375\pi$
$908$	−35.5392	−1.17941
$909$	15.0755	0.500023
$910$	0	0
$911$	24.9005	0.824989	0.412495	−	0.910960i	$-0.364658\pi$
$911$	24.9005	0.824989	0.412495	+	0.910960i	$0.364658\pi$
$912$	22.8673	0.757212
$913$	20.3683	0.674094
$914$	−15.9865	−0.528786
$915$	0	0
$916$	−51.4022	−1.69838
$917$	−7.60800	−0.251238
$918$	−59.4759	−1.96300
$919$	33.4552	1.10358	0.551792	−	0.833982i	$-0.313944\pi$
$919$	33.4552	1.10358	0.551792	+	0.833982i	$0.313944\pi$
$920$	0	0
$921$	12.7589	0.420420
$922$	−53.8862	−1.77465
$923$	29.8568	0.982748
$924$	−5.88718	−0.193674
$925$	0	0
$926$	−109.077	−3.58448
$927$	−3.26316	−0.107176
$928$	−103.308	−3.39127
$929$	−22.7514	−0.746450	−0.373225	−	0.927741i	$-0.621748\pi$
$929$	−22.7514	−0.746450	−0.373225	+	0.927741i	$0.621748\pi$
$930$	0	0
$931$	7.24242	0.237361
$932$	53.2165	1.74316
$933$	−27.9404	−0.914729
$934$	28.3092	0.926304
$935$	0	0
$936$	−47.2855	−1.54558
$937$	−14.8408	−0.484826	−0.242413	−	0.970173i	$-0.577939\pi$
$937$	−14.8408	−0.484826	−0.242413	+	0.970173i	$0.577939\pi$
$938$	−12.0092	−0.392114
$939$	25.2651	0.824497
$940$	0	0
$941$	−31.7427	−1.03478	−0.517391	−	0.855749i	$-0.673097\pi$
$941$	−31.7427	−1.03478	−0.517391	+	0.855749i	$0.673097\pi$
$942$	75.7954	2.46955
$943$	6.74504	0.219649
$944$	−72.2515	−2.35159
$945$	0	0
$946$	3.09045	0.100479
$947$	−60.8621	−1.97775	−0.988876	−	0.148743i	$-0.952477\pi$
$947$	−60.8621	−1.97775	−0.988876	+	0.148743i	$0.952477\pi$
$948$	77.1917	2.50707
$949$	−44.4981	−1.44447
$950$	0	0
$951$	35.2591	1.14335
$952$	23.5091	0.761934
$953$	−17.7123	−0.573759	−0.286879	−	0.957967i	$-0.592618\pi$
$953$	−17.7123	−0.573759	−0.286879	+	0.957967i	$0.592618\pi$
$954$	−30.6141	−0.991168
$955$	0	0
$956$	10.0972	0.326567
$957$	7.73103	0.249909
$958$	116.426	3.76157
$959$	−10.0557	−0.324716
$960$	0	0
$961$	6.76378	0.218186
$962$	−77.7750	−2.50757
$963$	−13.5779	−0.437542
$964$	−5.49044	−0.176835
$965$	0	0
$966$	−1.54053	−0.0495658
$967$	16.8430	0.541634	0.270817	−	0.962631i	$-0.412706\pi$
$967$	16.8430	0.541634	0.270817	+	0.962631i	$0.412706\pi$
$968$	90.8322	2.91946
$969$	5.80068	0.186345
$970$	0	0
$971$	−6.13593	−0.196911	−0.0984556	−	0.995141i	$-0.531390\pi$
$971$	−6.13593	−0.196911	−0.0984556	+	0.995141i	$0.531390\pi$
$972$	−59.8853	−1.92082
$973$	9.22557	0.295758
$974$	−91.9477	−2.94620
$975$	0	0
$976$	−22.8073	−0.730045
$977$	29.8640	0.955433	0.477716	−	0.878514i	$-0.341465\pi$
$977$	29.8640	0.955433	0.477716	+	0.878514i	$0.341465\pi$
$978$	−53.0451	−1.69619
$979$	3.39457	0.108491
$980$	0	0
$981$	12.9506	0.413481
$982$	−58.9731	−1.88191
$983$	33.2178	1.05948	0.529742	−	0.848159i	$-0.322289\pi$
$983$	33.2178	1.05948	0.529742	+	0.848159i	$0.322289\pi$
$984$	137.917	4.39662
$985$	0	0
$986$	−48.5616	−1.54652
$987$	2.87062	0.0913729
$988$	26.7407	0.850736
$989$	0.592768	0.0188489
$990$	0	0
$991$	−10.2899	−0.326870	−0.163435	−	0.986554i	$-0.552257\pi$
$991$	−10.2899	−0.326870	−0.163435	+	0.986554i	$0.552257\pi$
$992$	−137.630	−4.36976
$993$	−2.40236	−0.0762364
$994$	−11.7952	−0.374121
$995$	0	0
$996$	125.655	3.98152
$997$	−54.3710	−1.72195	−0.860974	−	0.508649i	$-0.830145\pi$
$997$	−54.3710	−1.72195	−0.860974	+	0.508649i	$0.830145\pi$
$998$	−33.0202	−1.04524
$999$	−36.2227	−1.14604

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.q.1.2		66
5.2	odd	4		1205.2.b.d.724.2	✓	66
5.3	odd	4		1205.2.b.d.724.65	yes	66
5.4	even	2	inner	6025.2.a.q.1.65		66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.d.724.2	✓	66	5.2	odd	4
1205.2.b.d.724.65	yes	66	5.3	odd	4
6025.2.a.q.1.2		66	1.1	even	1	trivial
6025.2.a.q.1.65		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)

\(f(q)\)	\(=\)	\(q-2.73687 q^{2} -1.37229 q^{3} +5.49044 q^{4} +3.75578 q^{6} -0.639768 q^{7} -9.55288 q^{8} -1.11681 q^{9} +O(q^{10})\)	\(q-2.73687 q^{2} -1.37229 q^{3} +5.49044 q^{4} +3.75578 q^{6} -0.639768 q^{7} -9.55288 q^{8} -1.11681 q^{9} -1.22132 q^{11} -7.53449 q^{12} -4.43214 q^{13} +1.75096 q^{14} +15.1641 q^{16} +3.84662 q^{17} +3.05657 q^{18} -1.09889 q^{19} +0.877948 q^{21} +3.34260 q^{22} +0.641133 q^{23} +13.1093 q^{24} +12.1302 q^{26} +5.64947 q^{27} -3.51261 q^{28} +4.61275 q^{29} +6.14522 q^{31} -22.3963 q^{32} +1.67601 q^{33} -10.5277 q^{34} -6.13180 q^{36} -6.41169 q^{37} +3.00750 q^{38} +6.08219 q^{39} +10.5205 q^{41} -2.40283 q^{42} +0.924563 q^{43} -6.70561 q^{44} -1.75470 q^{46} +3.26969 q^{47} -20.8095 q^{48} -6.59070 q^{49} -5.27869 q^{51} -24.3344 q^{52} -10.0158 q^{53} -15.4619 q^{54} +6.11162 q^{56} +1.50799 q^{57} -12.6245 q^{58} -4.76465 q^{59} -1.50404 q^{61} -16.8187 q^{62} +0.714501 q^{63} +30.9675 q^{64} -4.58703 q^{66} -6.85863 q^{67} +21.1197 q^{68} -0.879822 q^{69} -6.73643 q^{71} +10.6688 q^{72} +10.0399 q^{73} +17.5480 q^{74} -6.03337 q^{76} +0.781364 q^{77} -16.6462 q^{78} -10.2451 q^{79} -4.40229 q^{81} -28.7932 q^{82} -16.6773 q^{83} +4.82032 q^{84} -2.53041 q^{86} -6.33004 q^{87} +11.6672 q^{88} -2.77942 q^{89} +2.83554 q^{91} +3.52010 q^{92} -8.43304 q^{93} -8.94872 q^{94} +30.7343 q^{96} +2.14152 q^{97} +18.0379 q^{98} +1.36399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) \) 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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