LMFDB - Embedded newform 6025.2.a.k.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:	$0$
Dimension:	$25$
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)

$f(q)$	$=$	$q+0.430350 q^{2} -3.17127 q^{3} -1.81480 q^{4} -1.36476 q^{6} -2.65145 q^{7} -1.64170 q^{8} +7.05698 q^{9} +O(q^{10})$	\(q+0.430350 q^{2} -3.17127 q^{3} -1.81480 q^{4} -1.36476 q^{6} -2.65145 q^{7} -1.64170 q^{8} +7.05698 q^{9} +3.22530 q^{11} +5.75522 q^{12} +4.47301 q^{13} -1.14105 q^{14} +2.92309 q^{16} +4.30952 q^{17} +3.03697 q^{18} +7.19031 q^{19} +8.40848 q^{21} +1.38801 q^{22} +0.762968 q^{23} +5.20627 q^{24} +1.92496 q^{26} -12.8658 q^{27} +4.81185 q^{28} -6.90304 q^{29} +9.63351 q^{31} +4.54135 q^{32} -10.2283 q^{33} +1.85460 q^{34} -12.8070 q^{36} -0.950857 q^{37} +3.09435 q^{38} -14.1851 q^{39} +9.26787 q^{41} +3.61859 q^{42} +4.55503 q^{43} -5.85328 q^{44} +0.328343 q^{46} +10.2429 q^{47} -9.26993 q^{48} +0.0301952 q^{49} -13.6667 q^{51} -8.11761 q^{52} -11.2168 q^{53} -5.53678 q^{54} +4.35288 q^{56} -22.8024 q^{57} -2.97072 q^{58} +4.09917 q^{59} -0.708973 q^{61} +4.14578 q^{62} -18.7112 q^{63} -3.89182 q^{64} -4.40176 q^{66} -1.19242 q^{67} -7.82091 q^{68} -2.41958 q^{69} +15.1227 q^{71} -11.5854 q^{72} +1.32162 q^{73} -0.409201 q^{74} -13.0490 q^{76} -8.55174 q^{77} -6.10457 q^{78} -10.4590 q^{79} +19.6300 q^{81} +3.98843 q^{82} -4.67370 q^{83} -15.2597 q^{84} +1.96026 q^{86} +21.8914 q^{87} -5.29498 q^{88} +4.03267 q^{89} -11.8600 q^{91} -1.38463 q^{92} -30.5505 q^{93} +4.40801 q^{94} -14.4019 q^{96} -4.97671 q^{97} +0.0129945 q^{98} +22.7609 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.430350	0.304303	0.152152	−	0.988357i	$-0.451380\pi$
$2$	0.430350	0.304303	0.152152	+	0.988357i	$0.451380\pi$
$3$	−3.17127	−1.83094	−0.915468	−	0.402391i	$-0.868179\pi$
$3$	−3.17127	−1.83094	−0.915468	+	0.402391i	$0.868179\pi$
$4$	−1.81480	−0.907400
$5$	0	0
$5$	0	0
$6$	−1.36476	−0.557160
$7$	−2.65145	−1.00215	−0.501077	−	0.865403i	$-0.667063\pi$
$7$	−2.65145	−1.00215	−0.501077	+	0.865403i	$0.667063\pi$
$8$	−1.64170	−0.580428
$9$	7.05698	2.35233
$10$	0	0
$11$	3.22530	0.972466	0.486233	−	0.873829i	$-0.338371\pi$
$11$	3.22530	0.972466	0.486233	+	0.873829i	$0.338371\pi$
$12$	5.75522	1.66139
$13$	4.47301	1.24059	0.620294	−	0.784369i	$-0.287013\pi$
$13$	4.47301	1.24059	0.620294	+	0.784369i	$0.287013\pi$
$14$	−1.14105	−0.304959
$15$	0	0
$16$	2.92309	0.730773
$17$	4.30952	1.04521	0.522606	−	0.852575i	$-0.324960\pi$
$17$	4.30952	1.04521	0.522606	+	0.852575i	$0.324960\pi$
$18$	3.03697	0.715820
$19$	7.19031	1.64957	0.824785	−	0.565447i	$-0.191296\pi$
$19$	7.19031	1.64957	0.824785	+	0.565447i	$0.191296\pi$
$20$	0	0
$21$	8.40848	1.83488
$22$	1.38801	0.295925
$23$	0.762968	0.159090	0.0795449	−	0.996831i	$-0.474653\pi$
$23$	0.762968	0.159090	0.0795449	+	0.996831i	$0.474653\pi$
$24$	5.20627	1.06273
$25$	0	0
$26$	1.92496	0.377515
$27$	−12.8658	−2.47602
$28$	4.81185	0.909354
$29$	−6.90304	−1.28186	−0.640931	−	0.767598i	$-0.721452\pi$
$29$	−6.90304	−1.28186	−0.640931	+	0.767598i	$0.721452\pi$
$30$	0	0
$31$	9.63351	1.73023	0.865114	−	0.501574i	$-0.167246\pi$
$31$	9.63351	1.73023	0.865114	+	0.501574i	$0.167246\pi$
$32$	4.54135	0.802805
$33$	−10.2283	−1.78052
$34$	1.85460	0.318061
$35$	0	0
$36$	−12.8070	−2.13450
$37$	−0.950857	−0.156320	−0.0781600	−	0.996941i	$-0.524904\pi$
$37$	−0.950857	−0.156320	−0.0781600	+	0.996941i	$0.524904\pi$
$38$	3.09435	0.501969
$39$	−14.1851	−2.27144
$40$	0	0
$41$	9.26787	1.44740	0.723699	−	0.690115i	$-0.242440\pi$
$41$	9.26787	1.44740	0.723699	+	0.690115i	$0.242440\pi$
$42$	3.61859	0.558360
$43$	4.55503	0.694636	0.347318	−	0.937747i	$-0.387093\pi$
$43$	4.55503	0.694636	0.347318	+	0.937747i	$0.387093\pi$
$44$	−5.85328	−0.882415
$45$	0	0
$46$	0.328343	0.0484116
$47$	10.2429	1.49407	0.747037	−	0.664783i	$-0.231476\pi$
$47$	10.2429	1.49407	0.747037	+	0.664783i	$0.231476\pi$
$48$	−9.26993	−1.33800
$49$	0.0301952	0.00431359
$50$	0	0
$51$	−13.6667	−1.91371
$52$	−8.11761	−1.12571
$53$	−11.2168	−1.54074	−0.770370	−	0.637597i	$-0.779929\pi$
$53$	−11.2168	−1.54074	−0.770370	+	0.637597i	$0.779929\pi$
$54$	−5.53678	−0.753461
$55$	0	0
$56$	4.35288	0.581678
$57$	−22.8024	−3.02026
$58$	−2.97072	−0.390075
$59$	4.09917	0.533667	0.266834	−	0.963743i	$-0.414023\pi$
$59$	4.09917	0.533667	0.266834	+	0.963743i	$0.414023\pi$
$60$	0	0
$61$	−0.708973	−0.0907747	−0.0453874	−	0.998969i	$-0.514452\pi$
$61$	−0.708973	−0.0907747	−0.0453874	+	0.998969i	$0.514452\pi$
$62$	4.14578	0.526514
$63$	−18.7112	−2.35739
$64$	−3.89182	−0.486477
$65$	0	0
$66$	−4.40176	−0.541819
$67$	−1.19242	−0.145677	−0.0728384	−	0.997344i	$-0.523206\pi$
$67$	−1.19242	−0.145677	−0.0728384	+	0.997344i	$0.523206\pi$
$68$	−7.82091	−0.948424
$69$	−2.41958	−0.291283
$70$	0	0
$71$	15.1227	1.79473	0.897366	−	0.441287i	$-0.145478\pi$
$71$	15.1227	1.79473	0.897366	+	0.441287i	$0.145478\pi$
$72$	−11.5854	−1.36536
$73$	1.32162	0.154684	0.0773419	−	0.997005i	$-0.475357\pi$
$73$	1.32162	0.154684	0.0773419	+	0.997005i	$0.475357\pi$
$74$	−0.409201	−0.0475687
$75$	0	0
$76$	−13.0490	−1.49682
$77$	−8.55174	−0.974561
$78$	−6.10457	−0.691206
$79$	−10.4590	−1.17673	−0.588363	−	0.808597i	$-0.700227\pi$
$79$	−10.4590	−1.17673	−0.588363	+	0.808597i	$0.700227\pi$
$80$	0	0
$81$	19.6300	2.18111
$82$	3.98843	0.440448
$83$	−4.67370	−0.513006	−0.256503	−	0.966543i	$-0.582570\pi$
$83$	−4.67370	−0.513006	−0.256503	+	0.966543i	$0.582570\pi$
$84$	−15.2597	−1.66497
$85$	0	0
$86$	1.96026	0.211380
$87$	21.8914	2.34701
$88$	−5.29498	−0.564446
$89$	4.03267	0.427462	0.213731	−	0.976893i	$-0.431438\pi$
$89$	4.03267	0.427462	0.213731	+	0.976893i	$0.431438\pi$
$90$	0	0
$91$	−11.8600	−1.24326
$92$	−1.38463	−0.144358
$93$	−30.5505	−3.16794
$94$	4.40801	0.454652
$95$	0	0
$96$	−14.4019	−1.46988
$97$	−4.97671	−0.505308	−0.252654	−	0.967557i	$-0.581303\pi$
$97$	−4.97671	−0.505308	−0.252654	+	0.967557i	$0.581303\pi$
$98$	0.0129945	0.00131264
$99$	22.7609	2.28756
$100$	0	0
$101$	−10.9010	−1.08469	−0.542343	−	0.840157i	$-0.682463\pi$
$101$	−10.9010	−1.08469	−0.542343	+	0.840157i	$0.682463\pi$
$102$	−5.88144	−0.582350
$103$	7.97420	0.785721	0.392861	−	0.919598i	$-0.371485\pi$
$103$	7.97420	0.785721	0.392861	+	0.919598i	$0.371485\pi$
$104$	−7.34332	−0.720072
$105$	0	0
$106$	−4.82713	−0.468852
$107$	7.24246	0.700155	0.350077	−	0.936721i	$-0.386155\pi$
$107$	7.24246	0.700155	0.350077	+	0.936721i	$0.386155\pi$
$108$	23.3488	2.24674
$109$	13.3590	1.27956	0.639780	−	0.768558i	$-0.279026\pi$
$109$	13.3590	1.27956	0.639780	+	0.768558i	$0.279026\pi$
$110$	0	0
$111$	3.01543	0.286212
$112$	−7.75044	−0.732348
$113$	14.1201	1.32831	0.664153	−	0.747597i	$-0.268792\pi$
$113$	14.1201	1.32831	0.664153	+	0.747597i	$0.268792\pi$
$114$	−9.81302	−0.919074
$115$	0	0
$116$	12.5276	1.16316
$117$	31.5659	2.91827
$118$	1.76408	0.162397
$119$	−11.4265	−1.04746
$120$	0	0
$121$	−0.597409	−0.0543099
$122$	−0.305107	−0.0276230
$123$	−29.3910	−2.65009
$124$	−17.4829	−1.57001
$125$	0	0
$126$	−8.05237	−0.717362
$127$	6.08519	0.539973	0.269986	−	0.962864i	$-0.412981\pi$
$127$	6.08519	0.539973	0.269986	+	0.962864i	$0.412981\pi$
$128$	−10.7575	−0.950841
$129$	−14.4452	−1.27183
$130$	0	0
$131$	−14.1515	−1.23643	−0.618213	−	0.786010i	$-0.712143\pi$
$131$	−14.1515	−1.23643	−0.618213	+	0.786010i	$0.712143\pi$
$132$	18.5624	1.61565
$133$	−19.0647	−1.65312
$134$	−0.513156	−0.0443299
$135$	0	0
$136$	−7.07492	−0.606670
$137$	3.99258	0.341109	0.170554	−	0.985348i	$-0.445444\pi$
$137$	3.99258	0.341109	0.170554	+	0.985348i	$0.445444\pi$
$138$	−1.04127	−0.0886384
$139$	10.1276	0.859009	0.429504	−	0.903065i	$-0.358688\pi$
$139$	10.1276	0.859009	0.429504	+	0.903065i	$0.358688\pi$
$140$	0	0
$141$	−32.4829	−2.73555
$142$	6.50804	0.546143
$143$	14.4268	1.20643
$144$	20.6282	1.71902
$145$	0	0
$146$	0.568758	0.0470708
$147$	−0.0957571	−0.00789791
$148$	1.72561	0.141845
$149$	−16.1675	−1.32450	−0.662248	−	0.749284i	$-0.730398\pi$
$149$	−16.1675	−1.32450	−0.662248	+	0.749284i	$0.730398\pi$
$150$	0	0
$151$	4.48166	0.364713	0.182356	−	0.983233i	$-0.441628\pi$
$151$	4.48166	0.364713	0.182356	+	0.983233i	$0.441628\pi$
$152$	−11.8043	−0.957456
$153$	30.4122	2.45868
$154$	−3.68024	−0.296562
$155$	0	0
$156$	25.7432	2.06110
$157$	−1.05468	−0.0841722	−0.0420861	−	0.999114i	$-0.513400\pi$
$157$	−1.05468	−0.0841722	−0.0420861	+	0.999114i	$0.513400\pi$
$158$	−4.50101	−0.358082
$159$	35.5714	2.82100
$160$	0	0
$161$	−2.02297	−0.159433
$162$	8.44776	0.663718
$163$	3.07665	0.240982	0.120491	−	0.992714i	$-0.461553\pi$
$163$	3.07665	0.240982	0.120491	+	0.992714i	$0.461553\pi$
$164$	−16.8193	−1.31337
$165$	0	0
$166$	−2.01133	−0.156109
$167$	21.1310	1.63517	0.817584	−	0.575810i	$-0.195313\pi$
$167$	21.1310	1.63517	0.817584	+	0.575810i	$0.195313\pi$
$168$	−13.8042	−1.06502
$169$	7.00778	0.539060
$170$	0	0
$171$	50.7418	3.88032
$172$	−8.26646	−0.630312
$173$	13.7386	1.04452	0.522262	−	0.852785i	$-0.325088\pi$
$173$	13.7386	1.04452	0.522262	+	0.852785i	$0.325088\pi$
$174$	9.42097	0.714202
$175$	0	0
$176$	9.42787	0.710652
$177$	−12.9996	−0.977110
$178$	1.73546	0.130078
$179$	−0.267118	−0.0199653	−0.00998265	−	0.999950i	$-0.503178\pi$
$179$	−0.267118	−0.0199653	−0.00998265	+	0.999950i	$0.503178\pi$
$180$	0	0
$181$	8.18703	0.608537	0.304269	−	0.952586i	$-0.401588\pi$
$181$	8.18703	0.608537	0.304269	+	0.952586i	$0.401588\pi$
$182$	−5.10393	−0.378329
$183$	2.24835	0.166203
$184$	−1.25256	−0.0923402
$185$	0	0
$186$	−13.1474	−0.964014
$187$	13.8995	1.01643
$188$	−18.5887	−1.35572
$189$	34.1130	2.48135
$190$	0	0
$191$	7.64694	0.553313	0.276657	−	0.960969i	$-0.410774\pi$
$191$	7.64694	0.553313	0.276657	+	0.960969i	$0.410774\pi$
$192$	12.3420	0.890709
$193$	−23.9739	−1.72568	−0.862838	−	0.505480i	$-0.831315\pi$
$193$	−23.9739	−1.72568	−0.862838	+	0.505480i	$0.831315\pi$
$194$	−2.14173	−0.153767
$195$	0	0
$196$	−0.0547981	−0.00391415
$197$	20.3066	1.44679	0.723393	−	0.690436i	$-0.242581\pi$
$197$	20.3066	1.44679	0.723393	+	0.690436i	$0.242581\pi$
$198$	9.79515	0.696111
$199$	−24.0978	−1.70825	−0.854124	−	0.520069i	$-0.825906\pi$
$199$	−24.0978	−1.70825	−0.854124	+	0.520069i	$0.825906\pi$
$200$	0	0
$201$	3.78148	0.266725
$202$	−4.69123	−0.330074
$203$	18.3031	1.28462
$204$	24.8022	1.73650
$205$	0	0
$206$	3.43170	0.239098
$207$	5.38425	0.374231
$208$	13.0750	0.906589
$209$	23.1909	1.60415
$210$	0	0
$211$	1.64638	0.113342	0.0566709	−	0.998393i	$-0.481951\pi$
$211$	1.64638	0.113342	0.0566709	+	0.998393i	$0.481951\pi$
$212$	20.3562	1.39807
$213$	−47.9582	−3.28604
$214$	3.11679	0.213059
$215$	0	0
$216$	21.1217	1.43715
$217$	−25.5428	−1.73396
$218$	5.74904	0.389374
$219$	−4.19121	−0.283216
$220$	0	0
$221$	19.2765	1.29668
$222$	1.29769	0.0870952
$223$	−3.79259	−0.253971	−0.126985	−	0.991905i	$-0.540530\pi$
$223$	−3.79259	−0.253971	−0.126985	+	0.991905i	$0.540530\pi$
$224$	−12.0412	−0.804534
$225$	0	0
$226$	6.07657	0.404208
$227$	3.02970	0.201088	0.100544	−	0.994933i	$-0.467942\pi$
$227$	3.02970	0.201088	0.100544	+	0.994933i	$0.467942\pi$
$228$	41.3818	2.74058
$229$	−26.4127	−1.74540	−0.872701	−	0.488255i	$-0.837634\pi$
$229$	−26.4127	−1.74540	−0.872701	+	0.488255i	$0.837634\pi$
$230$	0	0
$231$	27.1199	1.78436
$232$	11.3327	0.744029
$233$	−17.7428	−1.16237	−0.581185	−	0.813771i	$-0.697411\pi$
$233$	−17.7428	−1.16237	−0.581185	+	0.813771i	$0.697411\pi$
$234$	13.5844	0.888038
$235$	0	0
$236$	−7.43918	−0.484249
$237$	33.1682	2.15451
$238$	−4.91738	−0.318746
$239$	−18.4188	−1.19141	−0.595706	−	0.803202i	$-0.703128\pi$
$239$	−18.4188	−1.19141	−0.595706	+	0.803202i	$0.703128\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−0.257095	−0.0165267
$243$	−23.6547	−1.51745
$244$	1.28664	0.0823689
$245$	0	0
$246$	−12.6484	−0.806432
$247$	32.1623	2.04644
$248$	−15.8153	−1.00427
$249$	14.8216	0.939280
$250$	0	0
$251$	−3.32450	−0.209840	−0.104920	−	0.994481i	$-0.533459\pi$
$251$	−3.32450	−0.209840	−0.104920	+	0.994481i	$0.533459\pi$
$252$	33.9571	2.13910
$253$	2.46080	0.154709
$254$	2.61876	0.164315
$255$	0	0
$256$	3.15413	0.197133
$257$	10.0638	0.627762	0.313881	−	0.949462i	$-0.398371\pi$
$257$	10.0638	0.627762	0.313881	+	0.949462i	$0.398371\pi$
$258$	−6.21651	−0.387023
$259$	2.52115	0.156657
$260$	0	0
$261$	−48.7146	−3.01536
$262$	−6.09012	−0.376249
$263$	11.1303	0.686322	0.343161	−	0.939277i	$-0.388502\pi$
$263$	11.1303	0.686322	0.343161	+	0.939277i	$0.388502\pi$
$264$	16.7918	1.03346
$265$	0	0
$266$	−8.20451	−0.503051
$267$	−12.7887	−0.782655
$268$	2.16400	0.132187
$269$	13.7203	0.836542	0.418271	−	0.908322i	$-0.362636\pi$
$269$	13.7203	0.836542	0.418271	+	0.908322i	$0.362636\pi$
$270$	0	0
$271$	16.6731	1.01282	0.506410	−	0.862293i	$-0.330972\pi$
$271$	16.6731	1.01282	0.506410	+	0.862293i	$0.330972\pi$
$272$	12.5971	0.763813
$273$	37.6112	2.27633
$274$	1.71820	0.103801
$275$	0	0
$276$	4.39105	0.264310
$277$	18.0619	1.08524	0.542618	−	0.839980i	$-0.317433\pi$
$277$	18.0619	1.08524	0.542618	+	0.839980i	$0.317433\pi$
$278$	4.35840	0.261399
$279$	67.9834	4.07006
$280$	0	0
$281$	−24.6368	−1.46971	−0.734855	−	0.678224i	$-0.762750\pi$
$281$	−24.6368	−1.46971	−0.734855	+	0.678224i	$0.762750\pi$
$282$	−13.9790	−0.832438
$283$	0.420384	0.0249892	0.0124946	−	0.999922i	$-0.496023\pi$
$283$	0.420384	0.0249892	0.0124946	+	0.999922i	$0.496023\pi$
$284$	−27.4446	−1.62854
$285$	0	0
$286$	6.20857	0.367121
$287$	−24.5733	−1.45052
$288$	32.0482	1.88846
$289$	1.57193	0.0924667
$290$	0	0
$291$	15.7825	0.925187
$292$	−2.39847	−0.140360
$293$	15.5110	0.906161	0.453080	−	0.891470i	$-0.350325\pi$
$293$	15.5110	0.906161	0.453080	+	0.891470i	$0.350325\pi$
$294$	−0.0412090	−0.00240336
$295$	0	0
$296$	1.56102	0.0907325
$297$	−41.4961	−2.40785
$298$	−6.95770	−0.403049
$299$	3.41276	0.197365
$300$	0	0
$301$	−12.0774	−0.696132
$302$	1.92868	0.110983
$303$	34.5699	1.98599
$304$	21.0179	1.20546
$305$	0	0
$306$	13.0879	0.748183
$307$	6.46407	0.368924	0.184462	−	0.982840i	$-0.440946\pi$
$307$	6.46407	0.368924	0.184462	+	0.982840i	$0.440946\pi$
$308$	15.5197	0.884316
$309$	−25.2884	−1.43861
$310$	0	0
$311$	−20.3178	−1.15211	−0.576057	−	0.817409i	$-0.695409\pi$
$311$	−20.3178	−1.15211	−0.576057	+	0.817409i	$0.695409\pi$
$312$	23.2877	1.31841
$313$	−7.26538	−0.410664	−0.205332	−	0.978692i	$-0.565827\pi$
$313$	−7.26538	−0.410664	−0.205332	+	0.978692i	$0.565827\pi$
$314$	−0.453879	−0.0256139
$315$	0	0
$316$	18.9809	1.06776
$317$	−16.0182	−0.899670	−0.449835	−	0.893112i	$-0.648517\pi$
$317$	−16.0182	−0.899670	−0.449835	+	0.893112i	$0.648517\pi$
$318$	15.3081	0.858438
$319$	−22.2644	−1.24657
$320$	0	0
$321$	−22.9678	−1.28194
$322$	−0.870586	−0.0485159
$323$	30.9867	1.72415
$324$	−35.6245	−1.97914
$325$	0	0
$326$	1.32404	0.0733315
$327$	−42.3650	−2.34279
$328$	−15.2150	−0.840111
$329$	−27.1584	−1.49729
$330$	0	0
$331$	−35.2128	−1.93547	−0.967736	−	0.251966i	$-0.918923\pi$
$331$	−35.2128	−1.93547	−0.967736	+	0.251966i	$0.918923\pi$
$332$	8.48183	0.465501
$333$	−6.71018	−0.367715
$334$	9.09373	0.497587
$335$	0	0
$336$	24.5788	1.34088
$337$	−32.0461	−1.74566	−0.872831	−	0.488022i	$-0.837719\pi$
$337$	−32.0461	−1.74566	−0.872831	+	0.488022i	$0.837719\pi$
$338$	3.01580	0.164038
$339$	−44.7786	−2.43204
$340$	0	0
$341$	31.0710	1.68259
$342$	21.8367	1.18080
$343$	18.4801	0.997832
$344$	−7.47798	−0.403186
$345$	0	0
$346$	5.91239	0.317852
$347$	−7.92879	−0.425640	−0.212820	−	0.977091i	$-0.568265\pi$
$347$	−7.92879	−0.425640	−0.212820	+	0.977091i	$0.568265\pi$
$348$	−39.7285	−2.12967
$349$	37.0106	1.98113	0.990567	−	0.137028i	$-0.0437551\pi$
$349$	37.0106	1.98113	0.990567	+	0.137028i	$0.0437551\pi$
$350$	0	0
$351$	−57.5487	−3.07172
$352$	14.6472	0.780700
$353$	4.27939	0.227769	0.113884	−	0.993494i	$-0.463671\pi$
$353$	4.27939	0.227769	0.113884	+	0.993494i	$0.463671\pi$
$354$	−5.59438	−0.297338
$355$	0	0
$356$	−7.31848	−0.387879
$357$	36.2365	1.91784
$358$	−0.114954	−0.00607551
$359$	−20.8295	−1.09934	−0.549669	−	0.835383i	$-0.685246\pi$
$359$	−20.8295	−1.09934	−0.549669	+	0.835383i	$0.685246\pi$
$360$	0	0
$361$	32.7005	1.72108
$362$	3.52329	0.185180
$363$	1.89455	0.0994379
$364$	21.5234	1.12813
$365$	0	0
$366$	0.967576	0.0505760
$367$	18.9891	0.991225	0.495613	−	0.868544i	$-0.334944\pi$
$367$	18.9891	0.991225	0.495613	+	0.868544i	$0.334944\pi$
$368$	2.23023	0.116259
$369$	65.4032	3.40475
$370$	0	0
$371$	29.7407	1.54406
$372$	55.4430	2.87459
$373$	−21.5360	−1.11509	−0.557547	−	0.830146i	$-0.688257\pi$
$373$	−21.5360	−1.11509	−0.557547	+	0.830146i	$0.688257\pi$
$374$	5.98165	0.309304
$375$	0	0
$376$	−16.8157	−0.867202
$377$	−30.8773	−1.59026
$378$	14.6805	0.755084
$379$	32.0961	1.64867	0.824333	−	0.566105i	$-0.191550\pi$
$379$	32.0961	1.64867	0.824333	+	0.566105i	$0.191550\pi$
$380$	0	0
$381$	−19.2978	−0.988656
$382$	3.29086	0.168375
$383$	−22.9341	−1.17188	−0.585940	−	0.810355i	$-0.699274\pi$
$383$	−22.9341	−1.17188	−0.585940	+	0.810355i	$0.699274\pi$
$384$	34.1151	1.74093
$385$	0	0
$386$	−10.3171	−0.525129
$387$	32.1447	1.63401
$388$	9.03173	0.458516
$389$	−18.2497	−0.925295	−0.462648	−	0.886542i	$-0.653101\pi$
$389$	−18.2497	−0.925295	−0.462648	+	0.886542i	$0.653101\pi$
$390$	0	0
$391$	3.28802	0.166283
$392$	−0.0495713	−0.00250373
$393$	44.8784	2.26382
$394$	8.73895	0.440262
$395$	0	0
$396$	−41.3065	−2.07573
$397$	−32.8923	−1.65082	−0.825408	−	0.564536i	$-0.809055\pi$
$397$	−32.8923	−1.65082	−0.825408	+	0.564536i	$0.809055\pi$
$398$	−10.3705	−0.519826
$399$	60.4595	3.02676
$400$	0	0
$401$	1.69101	0.0844453	0.0422226	−	0.999108i	$-0.486556\pi$
$401$	1.69101	0.0844453	0.0422226	+	0.999108i	$0.486556\pi$
$402$	1.62736	0.0811653
$403$	43.0907	2.14650
$404$	19.7831	0.984244
$405$	0	0
$406$	7.87673	0.390915
$407$	−3.06680	−0.152016
$408$	22.4365	1.11077
$409$	−2.15359	−0.106488	−0.0532441	−	0.998582i	$-0.516956\pi$
$409$	−2.15359	−0.106488	−0.0532441	+	0.998582i	$0.516956\pi$
$410$	0	0
$411$	−12.6616	−0.624548
$412$	−14.4716	−0.712963
$413$	−10.8688	−0.534817
$414$	2.31711	0.113880
$415$	0	0
$416$	20.3135	0.995950
$417$	−32.1173	−1.57279
$418$	9.98021	0.488148
$419$	8.52025	0.416241	0.208121	−	0.978103i	$-0.433265\pi$
$419$	8.52025	0.416241	0.208121	+	0.978103i	$0.433265\pi$
$420$	0	0
$421$	25.7534	1.25514	0.627572	−	0.778558i	$-0.284049\pi$
$421$	25.7534	1.25514	0.627572	+	0.778558i	$0.284049\pi$
$422$	0.708521	0.0344903
$423$	72.2836	3.51455
$424$	18.4145	0.894289
$425$	0	0
$426$	−20.6388	−0.999952
$427$	1.87981	0.0909703
$428$	−13.1436	−0.635320
$429$	−45.7514	−2.20890
$430$	0	0
$431$	15.0277	0.723860	0.361930	−	0.932205i	$-0.382118\pi$
$431$	15.0277	0.723860	0.361930	+	0.932205i	$0.382118\pi$
$432$	−37.6079	−1.80941
$433$	23.5613	1.13228	0.566141	−	0.824308i	$-0.308436\pi$
$433$	23.5613	1.13228	0.566141	+	0.824308i	$0.308436\pi$
$434$	−10.9923	−0.527649
$435$	0	0
$436$	−24.2439	−1.16107
$437$	5.48597	0.262430
$438$	−1.80369	−0.0861836
$439$	26.8410	1.28105	0.640525	−	0.767937i	$-0.278717\pi$
$439$	26.8410	1.28105	0.640525	+	0.767937i	$0.278717\pi$
$440$	0	0
$441$	0.213086	0.0101470
$442$	8.29564	0.394583
$443$	21.2196	1.00817	0.504086	−	0.863653i	$-0.331829\pi$
$443$	21.2196	1.00817	0.504086	+	0.863653i	$0.331829\pi$
$444$	−5.47240	−0.259708
$445$	0	0
$446$	−1.63214	−0.0772841
$447$	51.2717	2.42507
$448$	10.3190	0.487525
$449$	−0.288969	−0.0136373	−0.00681864	−	0.999977i	$-0.502170\pi$
$449$	−0.288969	−0.0136373	−0.00681864	+	0.999977i	$0.502170\pi$
$450$	0	0
$451$	29.8917	1.40755
$452$	−25.6251	−1.20530
$453$	−14.2126	−0.667765
$454$	1.30383	0.0611918
$455$	0	0
$456$	37.4347	1.75304
$457$	1.65657	0.0774911	0.0387456	−	0.999249i	$-0.487664\pi$
$457$	1.65657	0.0774911	0.0387456	+	0.999249i	$0.487664\pi$
$458$	−11.3667	−0.531132
$459$	−55.4453	−2.58796
$460$	0	0
$461$	−25.9563	−1.20890	−0.604452	−	0.796642i	$-0.706608\pi$
$461$	−25.9563	−1.20890	−0.604452	+	0.796642i	$0.706608\pi$
$462$	11.6710	0.542986
$463$	22.6743	1.05376	0.526881	−	0.849939i	$-0.323361\pi$
$463$	22.6743	1.05376	0.526881	+	0.849939i	$0.323361\pi$
$464$	−20.1782	−0.936751
$465$	0	0
$466$	−7.63562	−0.353713
$467$	−21.8000	−1.00878	−0.504392	−	0.863475i	$-0.668283\pi$
$467$	−21.8000	−1.00878	−0.504392	+	0.863475i	$0.668283\pi$
$468$	−57.2858	−2.64803
$469$	3.16163	0.145991
$470$	0	0
$471$	3.34466	0.154114
$472$	−6.72961	−0.309755
$473$	14.6914	0.675509
$474$	14.2739	0.655624
$475$	0	0
$476$	20.7368	0.950468
$477$	−79.1564	−3.62432
$478$	−7.92652	−0.362551
$479$	3.81278	0.174211	0.0871053	−	0.996199i	$-0.472238\pi$
$479$	3.81278	0.174211	0.0871053	+	0.996199i	$0.472238\pi$
$480$	0	0
$481$	−4.25319	−0.193929
$482$	0.430350	0.0196019
$483$	6.41540	0.291911
$484$	1.08418	0.0492808
$485$	0	0
$486$	−10.1798	−0.461765
$487$	−4.70939	−0.213403	−0.106701	−	0.994291i	$-0.534029\pi$
$487$	−4.70939	−0.213403	−0.106701	+	0.994291i	$0.534029\pi$
$488$	1.16392	0.0526882
$489$	−9.75690	−0.441222
$490$	0	0
$491$	−18.7992	−0.848396	−0.424198	−	0.905570i	$-0.639444\pi$
$491$	−18.7992	−0.848396	−0.424198	+	0.905570i	$0.639444\pi$
$492$	53.3387	2.40469
$493$	−29.7488	−1.33982
$494$	13.8410	0.622737
$495$	0	0
$496$	28.1596	1.26441
$497$	−40.0971	−1.79860
$498$	6.37847	0.285826
$499$	−4.59120	−0.205530	−0.102765	−	0.994706i	$-0.532769\pi$
$499$	−4.59120	−0.205530	−0.102765	+	0.994706i	$0.532769\pi$
$500$	0	0
$501$	−67.0123	−2.99389
$502$	−1.43070	−0.0638551
$503$	15.2949	0.681964	0.340982	−	0.940070i	$-0.389240\pi$
$503$	15.2949	0.681964	0.340982	+	0.940070i	$0.389240\pi$
$504$	30.7182	1.36830
$505$	0	0
$506$	1.05901	0.0470786
$507$	−22.2236	−0.986984
$508$	−11.0434	−0.489971
$509$	−23.8935	−1.05906	−0.529530	−	0.848291i	$-0.677632\pi$
$509$	−23.8935	−1.05906	−0.529530	+	0.848291i	$0.677632\pi$
$510$	0	0
$511$	−3.50421	−0.155017
$512$	22.8725	1.01083
$513$	−92.5089	−4.08437
$514$	4.33095	0.191030
$515$	0	0
$516$	26.2152	1.15406
$517$	33.0363	1.45294
$518$	1.08498	0.0476712
$519$	−43.5687	−1.91246
$520$	0	0
$521$	8.74634	0.383184	0.191592	−	0.981475i	$-0.438635\pi$
$521$	8.74634	0.383184	0.191592	+	0.981475i	$0.438635\pi$
$522$	−20.9643	−0.917583
$523$	−12.4068	−0.542510	−0.271255	−	0.962508i	$-0.587439\pi$
$523$	−12.4068	−0.542510	−0.271255	+	0.962508i	$0.587439\pi$
$524$	25.6822	1.12193
$525$	0	0
$526$	4.78991	0.208850
$527$	41.5158	1.80845
$528$	−29.8983	−1.30116
$529$	−22.4179	−0.974690
$530$	0	0
$531$	28.9278	1.25536
$532$	34.5987	1.50004
$533$	41.4553	1.79563
$534$	−5.50361	−0.238164
$535$	0	0
$536$	1.95759	0.0845549
$537$	0.847103	0.0365552
$538$	5.90453	0.254562
$539$	0.0973886	0.00419482
$540$	0	0
$541$	36.8470	1.58418	0.792089	−	0.610406i	$-0.208994\pi$
$541$	36.8470	1.58418	0.792089	+	0.610406i	$0.208994\pi$
$542$	7.17527	0.308204
$543$	−25.9633	−1.11419
$544$	19.5710	0.839100
$545$	0	0
$546$	16.1860	0.692695
$547$	−28.4146	−1.21492	−0.607459	−	0.794351i	$-0.707811\pi$
$547$	−28.4146	−1.21492	−0.607459	+	0.794351i	$0.707811\pi$
$548$	−7.24573	−0.309522
$549$	−5.00321	−0.213532
$550$	0	0
$551$	−49.6350	−2.11452
$552$	3.97222	0.169069
$553$	27.7314	1.17926
$554$	7.77294	0.330241
$555$	0	0
$556$	−18.3795	−0.779464
$557$	−44.0534	−1.86660	−0.933302	−	0.359093i	$-0.883086\pi$
$557$	−44.0534	−1.86660	−0.933302	+	0.359093i	$0.883086\pi$
$558$	29.2567	1.23853
$559$	20.3747	0.861757
$560$	0	0
$561$	−44.0791	−1.86102
$562$	−10.6025	−0.447237
$563$	28.1045	1.18446	0.592232	−	0.805767i	$-0.298247\pi$
$563$	28.1045	1.18446	0.592232	+	0.805767i	$0.298247\pi$
$564$	58.9499	2.48224
$565$	0	0
$566$	0.180912	0.00760430
$567$	−52.0479	−2.18581
$568$	−24.8269	−1.04171
$569$	−7.44561	−0.312136	−0.156068	−	0.987746i	$-0.549882\pi$
$569$	−7.44561	−0.312136	−0.156068	+	0.987746i	$0.549882\pi$
$570$	0	0
$571$	−37.9663	−1.58884	−0.794419	−	0.607370i	$-0.792225\pi$
$571$	−37.9663	−1.58884	−0.794419	+	0.607370i	$0.792225\pi$
$572$	−26.1818	−1.09471
$573$	−24.2505	−1.01308
$574$	−10.5751	−0.441397
$575$	0	0
$576$	−27.4645	−1.14435
$577$	32.4537	1.35106	0.675532	−	0.737330i	$-0.263914\pi$
$577$	32.4537	1.35106	0.675532	+	0.737330i	$0.263914\pi$
$578$	0.676482	0.0281379
$579$	76.0277	3.15960
$580$	0	0
$581$	12.3921	0.514111
$582$	6.79200	0.281537
$583$	−36.1775	−1.49832
$584$	−2.16970	−0.0897828
$585$	0	0
$586$	6.67514	0.275748
$587$	−24.8595	−1.02606	−0.513032	−	0.858370i	$-0.671478\pi$
$587$	−24.8595	−1.02606	−0.513032	+	0.858370i	$0.671478\pi$
$588$	0.173780	0.00716656
$589$	69.2679	2.85413
$590$	0	0
$591$	−64.3979	−2.64897
$592$	−2.77944	−0.114234
$593$	−32.6081	−1.33905	−0.669527	−	0.742788i	$-0.733503\pi$
$593$	−32.6081	−1.33905	−0.669527	+	0.742788i	$0.733503\pi$
$594$	−17.8578	−0.732715
$595$	0	0
$596$	29.3408	1.20185
$597$	76.4207	3.12769
$598$	1.46868	0.0600588
$599$	33.5509	1.37085	0.685426	−	0.728143i	$-0.259616\pi$
$599$	33.5509	1.37085	0.685426	+	0.728143i	$0.259616\pi$
$600$	0	0
$601$	−42.0371	−1.71473	−0.857364	−	0.514710i	$-0.827900\pi$
$601$	−42.0371	−1.71473	−0.857364	+	0.514710i	$0.827900\pi$
$602$	−5.19752	−0.211835
$603$	−8.41485	−0.342679
$604$	−8.13332	−0.330940
$605$	0	0
$606$	14.8772	0.604343
$607$	−48.0145	−1.94885	−0.974425	−	0.224714i	$-0.927855\pi$
$607$	−48.0145	−1.94885	−0.974425	+	0.224714i	$0.927855\pi$
$608$	32.6537	1.32428
$609$	−58.0441	−2.35206
$610$	0	0
$611$	45.8163	1.85353
$612$	−55.1919	−2.23100
$613$	11.4330	0.461775	0.230887	−	0.972980i	$-0.425837\pi$
$613$	11.4330	0.461775	0.230887	+	0.972980i	$0.425837\pi$
$614$	2.78181	0.112265
$615$	0	0
$616$	14.0394	0.565662
$617$	23.7126	0.954635	0.477317	−	0.878731i	$-0.341609\pi$
$617$	23.7126	0.954635	0.477317	+	0.878731i	$0.341609\pi$
$618$	−10.8828	−0.437772
$619$	−18.9962	−0.763521	−0.381760	−	0.924261i	$-0.624682\pi$
$619$	−18.9962	−0.763521	−0.381760	+	0.924261i	$0.624682\pi$
$620$	0	0
$621$	−9.81618	−0.393910
$622$	−8.74374	−0.350592
$623$	−10.6924	−0.428383
$624$	−41.4644	−1.65991
$625$	0	0
$626$	−3.12666	−0.124966
$627$	−73.5448	−2.93710
$628$	1.91402	0.0763778
$629$	−4.09774	−0.163387
$630$	0	0
$631$	16.6365	0.662290	0.331145	−	0.943580i	$-0.392565\pi$
$631$	16.6365	0.662290	0.331145	+	0.943580i	$0.392565\pi$
$632$	17.1705	0.683004
$633$	−5.22114	−0.207522
$634$	−6.89342	−0.273773
$635$	0	0
$636$	−64.5550	−2.55977
$637$	0.135063	0.00535139
$638$	−9.58148	−0.379335
$639$	106.720	4.22179
$640$	0	0
$641$	7.81030	0.308488	0.154244	−	0.988033i	$-0.450706\pi$
$641$	7.81030	0.308488	0.154244	+	0.988033i	$0.450706\pi$
$642$	−9.88419	−0.390098
$643$	−4.73776	−0.186839	−0.0934195	−	0.995627i	$-0.529780\pi$
$643$	−4.73776	−0.186839	−0.0934195	+	0.995627i	$0.529780\pi$
$644$	3.67129	0.144669
$645$	0	0
$646$	13.3351	0.524664
$647$	−13.8303	−0.543727	−0.271863	−	0.962336i	$-0.587640\pi$
$647$	−13.8303	−0.543727	−0.271863	+	0.962336i	$0.587640\pi$
$648$	−32.2265	−1.26598
$649$	13.2211	0.518973
$650$	0	0
$651$	81.0031	3.17476
$652$	−5.58350	−0.218667
$653$	−19.9066	−0.779007	−0.389504	−	0.921025i	$-0.627353\pi$
$653$	−19.9066	−0.779007	−0.389504	+	0.921025i	$0.627353\pi$
$654$	−18.2318	−0.712919
$655$	0	0
$656$	27.0909	1.05772
$657$	9.32663	0.363867
$658$	−11.6876	−0.455631
$659$	3.71186	0.144593	0.0722967	−	0.997383i	$-0.476967\pi$
$659$	3.71186	0.144593	0.0722967	+	0.997383i	$0.476967\pi$
$660$	0	0
$661$	14.0605	0.546889	0.273445	−	0.961888i	$-0.411837\pi$
$661$	14.0605	0.546889	0.273445	+	0.961888i	$0.411837\pi$
$662$	−15.1538	−0.588971
$663$	−61.1310	−2.37413
$664$	7.67281	0.297763
$665$	0	0
$666$	−2.88772	−0.111897
$667$	−5.26680	−0.203931
$668$	−38.3486	−1.48375
$669$	12.0273	0.465004
$670$	0	0
$671$	−2.28666	−0.0882753
$672$	38.1858	1.47305
$673$	−11.0280	−0.425098	−0.212549	−	0.977150i	$-0.568177\pi$
$673$	−11.0280	−0.425098	−0.212549	+	0.977150i	$0.568177\pi$
$674$	−13.7910	−0.531211
$675$	0	0
$676$	−12.7177	−0.489143
$677$	11.3937	0.437896	0.218948	−	0.975736i	$-0.429737\pi$
$677$	11.3937	0.437896	0.218948	+	0.975736i	$0.429737\pi$
$678$	−19.2705	−0.740078
$679$	13.1955	0.506397
$680$	0	0
$681$	−9.60801	−0.368180
$682$	13.3714	0.512017
$683$	15.9236	0.609298	0.304649	−	0.952465i	$-0.401461\pi$
$683$	15.9236	0.609298	0.304649	+	0.952465i	$0.401461\pi$
$684$	−92.0862	−3.52100
$685$	0	0
$686$	7.95291	0.303643
$687$	83.7620	3.19572
$688$	13.3148	0.507621
$689$	−50.1726	−1.91142
$690$	0	0
$691$	−2.93703	−0.111730	−0.0558649	−	0.998438i	$-0.517792\pi$
$691$	−2.93703	−0.111730	−0.0558649	+	0.998438i	$0.517792\pi$
$692$	−24.9327	−0.947800
$693$	−60.3494	−2.29248
$694$	−3.41215	−0.129524
$695$	0	0
$696$	−35.9391	−1.36227
$697$	39.9401	1.51284
$698$	15.9275	0.602866
$699$	56.2673	2.12823
$700$	0	0
$701$	29.3291	1.10774	0.553872	−	0.832602i	$-0.313150\pi$
$701$	29.3291	1.10774	0.553872	+	0.832602i	$0.313150\pi$
$702$	−24.7661	−0.934735
$703$	−6.83696	−0.257861
$704$	−12.5523	−0.473083
$705$	0	0
$706$	1.84163	0.0693108
$707$	28.9034	1.08702
$708$	23.5917	0.886629
$709$	20.8469	0.782923	0.391461	−	0.920195i	$-0.371970\pi$
$709$	20.8469	0.782923	0.391461	+	0.920195i	$0.371970\pi$
$710$	0	0
$711$	−73.8087	−2.76804
$712$	−6.62042	−0.248111
$713$	7.35006	0.275262
$714$	15.5944	0.583604
$715$	0	0
$716$	0.484765	0.0181165
$717$	58.4110	2.18140
$718$	−8.96396	−0.334532
$719$	13.6549	0.509242	0.254621	−	0.967041i	$-0.418049\pi$
$719$	13.6549	0.509242	0.254621	+	0.967041i	$0.418049\pi$
$720$	0	0
$721$	−21.1432	−0.787414
$722$	14.0727	0.523730
$723$	−3.17127	−0.117941
$724$	−14.8578	−0.552186
$725$	0	0
$726$	0.815318	0.0302593
$727$	13.0999	0.485848	0.242924	−	0.970045i	$-0.421893\pi$
$727$	13.0999	0.485848	0.242924	+	0.970045i	$0.421893\pi$
$728$	19.4705	0.721624
$729$	16.1255	0.597242
$730$	0	0
$731$	19.6300	0.726041
$732$	−4.08030	−0.150812
$733$	49.4747	1.82739	0.913695	−	0.406401i	$-0.133216\pi$
$733$	49.4747	1.82739	0.913695	+	0.406401i	$0.133216\pi$
$734$	8.17197	0.301633
$735$	0	0
$736$	3.46490	0.127718
$737$	−3.84591	−0.141666
$738$	28.1462	1.03608
$739$	3.81156	0.140210	0.0701052	−	0.997540i	$-0.477667\pi$
$739$	3.81156	0.140210	0.0701052	+	0.997540i	$0.477667\pi$
$740$	0	0
$741$	−101.995	−3.74689
$742$	12.7989	0.469862
$743$	30.7779	1.12913	0.564566	−	0.825388i	$-0.309043\pi$
$743$	30.7779	1.12913	0.564566	+	0.825388i	$0.309043\pi$
$744$	50.1547	1.83876
$745$	0	0
$746$	−9.26803	−0.339327
$747$	−32.9822	−1.20676
$748$	−25.2248	−0.922310
$749$	−19.2030	−0.701663
$750$	0	0
$751$	45.2907	1.65268	0.826341	−	0.563170i	$-0.190418\pi$
$751$	45.2907	1.65268	0.826341	+	0.563170i	$0.190418\pi$
$752$	29.9408	1.09183
$753$	10.5429	0.384204
$754$	−13.2881	−0.483923
$755$	0	0
$756$	−61.9082	−2.25158
$757$	45.6537	1.65931	0.829655	−	0.558276i	$-0.188537\pi$
$757$	45.6537	1.65931	0.829655	+	0.558276i	$0.188537\pi$
$758$	13.8125	0.501694
$759$	−7.80388	−0.283263
$760$	0	0
$761$	17.4199	0.631472	0.315736	−	0.948847i	$-0.397749\pi$
$761$	17.4199	0.631472	0.315736	+	0.948847i	$0.397749\pi$
$762$	−8.30480	−0.300851
$763$	−35.4207	−1.28232
$764$	−13.8777	−0.502076
$765$	0	0
$766$	−9.86970	−0.356607
$767$	18.3356	0.662061
$768$	−10.0026	−0.360938
$769$	−14.1180	−0.509110	−0.254555	−	0.967058i	$-0.581929\pi$
$769$	−14.1180	−0.509110	−0.254555	+	0.967058i	$0.581929\pi$
$770$	0	0
$771$	−31.9150	−1.14939
$772$	43.5077	1.56588
$773$	−25.4809	−0.916485	−0.458242	−	0.888827i	$-0.651521\pi$
$773$	−25.4809	−0.916485	−0.458242	+	0.888827i	$0.651521\pi$
$774$	13.8335	0.497234
$775$	0	0
$776$	8.17025	0.293295
$777$	−7.99526	−0.286828
$778$	−7.85375	−0.281570
$779$	66.6388	2.38758
$780$	0	0
$781$	48.7753	1.74532
$782$	1.41500	0.0506003
$783$	88.8130	3.17392
$784$	0.0882633	0.00315226
$785$	0	0
$786$	19.3134	0.688887
$787$	−39.1637	−1.39604	−0.698018	−	0.716080i	$-0.745935\pi$
$787$	−39.1637	−1.39604	−0.698018	+	0.716080i	$0.745935\pi$
$788$	−36.8524	−1.31281
$789$	−35.2971	−1.25661
$790$	0	0
$791$	−37.4387	−1.33117
$792$	−37.3665	−1.32776
$793$	−3.17124	−0.112614
$794$	−14.1552	−0.502349
$795$	0	0
$796$	43.7327	1.55006
$797$	−46.6723	−1.65322	−0.826610	−	0.562776i	$-0.809733\pi$
$797$	−46.6723	−1.65322	−0.826610	+	0.562776i	$0.809733\pi$
$798$	26.0187	0.921054
$799$	44.1417	1.56162
$800$	0	0
$801$	28.4584	1.00553
$802$	0.727728	0.0256970
$803$	4.26262	0.150425
$804$	−6.86262	−0.242026
$805$	0	0
$806$	18.5441	0.653188
$807$	−43.5108	−1.53165
$808$	17.8961	0.629582
$809$	−25.4789	−0.895789	−0.447895	−	0.894086i	$-0.647826\pi$
$809$	−25.4789	−0.895789	−0.447895	+	0.894086i	$0.647826\pi$
$810$	0	0
$811$	55.7675	1.95826	0.979131	−	0.203230i	$-0.0651441\pi$
$811$	55.7675	1.95826	0.979131	+	0.203230i	$0.0651441\pi$
$812$	−33.2164	−1.16567
$813$	−52.8750	−1.85441
$814$	−1.31980	−0.0462589
$815$	0	0
$816$	−39.9489	−1.39849
$817$	32.7521	1.14585
$818$	−0.926797	−0.0324047
$819$	−83.6954	−2.92456
$820$	0	0
$821$	24.8303	0.866585	0.433293	−	0.901253i	$-0.357352\pi$
$821$	24.8303	0.866585	0.433293	+	0.901253i	$0.357352\pi$
$822$	−5.44890	−0.190052
$823$	−32.1344	−1.12013	−0.560067	−	0.828447i	$-0.689225\pi$
$823$	−32.1344	−1.12013	−0.560067	+	0.828447i	$0.689225\pi$
$824$	−13.0912	−0.456055
$825$	0	0
$826$	−4.67737	−0.162747
$827$	6.76763	0.235334	0.117667	−	0.993053i	$-0.462459\pi$
$827$	6.76763	0.235334	0.117667	+	0.993053i	$0.462459\pi$
$828$	−9.77133	−0.339577
$829$	−18.6501	−0.647745	−0.323873	−	0.946101i	$-0.604985\pi$
$829$	−18.6501	−0.647745	−0.323873	+	0.946101i	$0.604985\pi$
$830$	0	0
$831$	−57.2793	−1.98700
$832$	−17.4081	−0.603518
$833$	0.130127	0.00450862
$834$	−13.8217	−0.478605
$835$	0	0
$836$	−42.0869	−1.45561
$837$	−123.943	−4.28408
$838$	3.66669	0.126664
$839$	29.4412	1.01642	0.508212	−	0.861232i	$-0.330307\pi$
$839$	29.4412	1.01642	0.508212	+	0.861232i	$0.330307\pi$
$840$	0	0
$841$	18.6520	0.643171
$842$	11.0830	0.381945
$843$	78.1301	2.69094
$844$	−2.98786	−0.102846
$845$	0	0
$846$	31.1072	1.06949
$847$	1.58400	0.0544269
$848$	−32.7876	−1.12593
$849$	−1.33315	−0.0457537
$850$	0	0
$851$	−0.725474	−0.0248689
$852$	87.0344	2.98175
$853$	−32.2758	−1.10510	−0.552552	−	0.833479i	$-0.686346\pi$
$853$	−32.2758	−1.10510	−0.552552	+	0.833479i	$0.686346\pi$
$854$	0.808975	0.0276826
$855$	0	0
$856$	−11.8899	−0.406389
$857$	43.0908	1.47195	0.735977	−	0.677007i	$-0.236723\pi$
$857$	43.0908	1.47195	0.735977	+	0.677007i	$0.236723\pi$
$858$	−19.6891	−0.672174
$859$	12.1974	0.416170	0.208085	−	0.978111i	$-0.433277\pi$
$859$	12.1974	0.416170	0.208085	+	0.978111i	$0.433277\pi$
$860$	0	0
$861$	77.9287	2.65580
$862$	6.46718	0.220273
$863$	3.79622	0.129225	0.0646124	−	0.997910i	$-0.479419\pi$
$863$	3.79622	0.129225	0.0646124	+	0.997910i	$0.479419\pi$
$864$	−58.4280	−1.98776
$865$	0	0
$866$	10.1396	0.344557
$867$	−4.98503	−0.169301
$868$	46.3550	1.57339
$869$	−33.7334	−1.14433
$870$	0	0
$871$	−5.33369	−0.180725
$872$	−21.9314	−0.742692
$873$	−35.1205	−1.18865
$874$	2.36089	0.0798582
$875$	0	0
$876$	7.60621	0.256990
$877$	−15.2867	−0.516195	−0.258097	−	0.966119i	$-0.583096\pi$
$877$	−15.2867	−0.516195	−0.258097	+	0.966119i	$0.583096\pi$
$878$	11.5510	0.389828
$879$	−49.1895	−1.65912
$880$	0	0
$881$	11.7503	0.395876	0.197938	−	0.980215i	$-0.436575\pi$
$881$	11.7503	0.395876	0.197938	+	0.980215i	$0.436575\pi$
$882$	0.0917017	0.00308776
$883$	−32.8028	−1.10390	−0.551951	−	0.833877i	$-0.686116\pi$
$883$	−32.8028	−1.10390	−0.551951	+	0.833877i	$0.686116\pi$
$884$	−34.9830	−1.17660
$885$	0	0
$886$	9.13184	0.306790
$887$	1.27137	0.0426884	0.0213442	−	0.999772i	$-0.493205\pi$
$887$	1.27137	0.0426884	0.0213442	+	0.999772i	$0.493205\pi$
$888$	−4.95042	−0.166125
$889$	−16.1346	−0.541136
$890$	0	0
$891$	63.3126	2.12105
$892$	6.88279	0.230453
$893$	73.6493	2.46458
$894$	22.0648	0.737956
$895$	0	0
$896$	28.5231	0.952890
$897$	−10.8228	−0.361363
$898$	−0.124358	−0.00414987
$899$	−66.5005	−2.21792
$900$	0	0
$901$	−48.3388	−1.61040
$902$	12.8639	0.428321
$903$	38.3009	1.27457
$904$	−23.1809	−0.770985
$905$	0	0
$906$	−6.11638	−0.203203
$907$	15.4194	0.511991	0.255996	−	0.966678i	$-0.417597\pi$
$907$	15.4194	0.511991	0.255996	+	0.966678i	$0.417597\pi$
$908$	−5.49830	−0.182467
$909$	−76.9278	−2.55153
$910$	0	0
$911$	−25.7711	−0.853833	−0.426917	−	0.904291i	$-0.640400\pi$
$911$	−25.7711	−0.853833	−0.426917	+	0.904291i	$0.640400\pi$
$912$	−66.6536	−2.20712
$913$	−15.0741	−0.498881
$914$	0.712905	0.0235808
$915$	0	0
$916$	47.9338	1.58378
$917$	37.5221	1.23909
$918$	−23.8609	−0.787526
$919$	−21.6172	−0.713087	−0.356543	−	0.934279i	$-0.616045\pi$
$919$	−21.6172	−0.713087	−0.356543	+	0.934279i	$0.616045\pi$
$920$	0	0
$921$	−20.4993	−0.675476
$922$	−11.1703	−0.367873
$923$	67.6438	2.22652
$924$	−49.2172	−1.61913
$925$	0	0
$926$	9.75787	0.320663
$927$	56.2737	1.84827
$928$	−31.3491	−1.02909
$929$	51.5207	1.69034	0.845170	−	0.534498i	$-0.179499\pi$
$929$	51.5207	1.69034	0.845170	+	0.534498i	$0.179499\pi$
$930$	0	0
$931$	0.217112	0.00711557
$932$	32.1996	1.05473
$933$	64.4332	2.10945
$934$	−9.38162	−0.306976
$935$	0	0
$936$	−51.8217	−1.69384
$937$	0.906215	0.0296048	0.0148024	−	0.999890i	$-0.495288\pi$
$937$	0.906215	0.0296048	0.0148024	+	0.999890i	$0.495288\pi$
$938$	1.36061	0.0444255
$939$	23.0405	0.751899
$940$	0	0
$941$	1.37157	0.0447118	0.0223559	−	0.999750i	$-0.492883\pi$
$941$	1.37157	0.0447118	0.0223559	+	0.999750i	$0.492883\pi$
$942$	1.43938	0.0468974
$943$	7.07109	0.230266
$944$	11.9823	0.389990
$945$	0	0
$946$	6.32242	0.205560
$947$	−4.65593	−0.151297	−0.0756487	−	0.997135i	$-0.524103\pi$
$947$	−4.65593	−0.151297	−0.0756487	+	0.997135i	$0.524103\pi$
$948$	−60.1937	−1.95500
$949$	5.91161	0.191899
$950$	0	0
$951$	50.7980	1.64724
$952$	18.7588	0.607977
$953$	−11.1834	−0.362265	−0.181133	−	0.983459i	$-0.557976\pi$
$953$	−11.1834	−0.362265	−0.181133	+	0.983459i	$0.557976\pi$
$954$	−34.0649	−1.10289
$955$	0	0
$956$	33.4264	1.08109
$957$	70.6065	2.28239
$958$	1.64083	0.0530128
$959$	−10.5861	−0.341844
$960$	0	0
$961$	61.8045	1.99369
$962$	−1.83036	−0.0590132
$963$	51.1099	1.64699
$964$	−1.81480	−0.0584507
$965$	0	0
$966$	2.76087	0.0888294
$967$	16.7080	0.537293	0.268646	−	0.963239i	$-0.413424\pi$
$967$	16.7080	0.537293	0.268646	+	0.963239i	$0.413424\pi$
$968$	0.980765	0.0315230
$969$	−98.2674	−3.15681
$970$	0	0
$971$	50.1260	1.60862	0.804310	−	0.594210i	$-0.202535\pi$
$971$	50.1260	1.60862	0.804310	+	0.594210i	$0.202535\pi$
$972$	42.9285	1.37693
$973$	−26.8527	−0.860860
$974$	−2.02668	−0.0649391
$975$	0	0
$976$	−2.07240	−0.0663358
$977$	−26.3812	−0.844008	−0.422004	−	0.906594i	$-0.638673\pi$
$977$	−26.3812	−0.844008	−0.422004	+	0.906594i	$0.638673\pi$
$978$	−4.19888	−0.134265
$979$	13.0066	0.415692
$980$	0	0
$981$	94.2741	3.00994
$982$	−8.09023	−0.258170
$983$	53.0104	1.69077	0.845384	−	0.534159i	$-0.179372\pi$
$983$	53.0104	1.69077	0.845384	+	0.534159i	$0.179372\pi$
$984$	48.2511	1.53819
$985$	0	0
$986$	−12.8024	−0.407711
$987$	86.1268	2.74145
$988$	−58.3681	−1.85694
$989$	3.47534	0.110509
$990$	0	0
$991$	−49.2563	−1.56468	−0.782339	−	0.622853i	$-0.785974\pi$
$991$	−49.2563	−1.56468	−0.782339	+	0.622853i	$0.785974\pi$
$992$	43.7491	1.38904
$993$	111.670	3.54373
$994$	−17.2558	−0.547319
$995$	0	0
$996$	−26.8982	−0.852303
$997$	−38.0068	−1.20369	−0.601843	−	0.798614i	$-0.705567\pi$
$997$	−38.0068	−1.20369	−0.601843	+	0.798614i	$0.705567\pi$
$998$	−1.97582	−0.0625435
$999$	12.2335	0.387051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.k.1.14		25
5.4	even	2		1205.2.a.d.1.12	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.d.1.12	✓	25	5.4	even	2
6025.2.a.k.1.14		25	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.430350 q^{2} -3.17127 q^{3} -1.81480 q^{4} -1.36476 q^{6} -2.65145 q^{7} -1.64170 q^{8} +7.05698 q^{9} +O(q^{10})\)	\(q+0.430350 q^{2} -3.17127 q^{3} -1.81480 q^{4} -1.36476 q^{6} -2.65145 q^{7} -1.64170 q^{8} +7.05698 q^{9} +3.22530 q^{11} +5.75522 q^{12} +4.47301 q^{13} -1.14105 q^{14} +2.92309 q^{16} +4.30952 q^{17} +3.03697 q^{18} +7.19031 q^{19} +8.40848 q^{21} +1.38801 q^{22} +0.762968 q^{23} +5.20627 q^{24} +1.92496 q^{26} -12.8658 q^{27} +4.81185 q^{28} -6.90304 q^{29} +9.63351 q^{31} +4.54135 q^{32} -10.2283 q^{33} +1.85460 q^{34} -12.8070 q^{36} -0.950857 q^{37} +3.09435 q^{38} -14.1851 q^{39} +9.26787 q^{41} +3.61859 q^{42} +4.55503 q^{43} -5.85328 q^{44} +0.328343 q^{46} +10.2429 q^{47} -9.26993 q^{48} +0.0301952 q^{49} -13.6667 q^{51} -8.11761 q^{52} -11.2168 q^{53} -5.53678 q^{54} +4.35288 q^{56} -22.8024 q^{57} -2.97072 q^{58} +4.09917 q^{59} -0.708973 q^{61} +4.14578 q^{62} -18.7112 q^{63} -3.89182 q^{64} -4.40176 q^{66} -1.19242 q^{67} -7.82091 q^{68} -2.41958 q^{69} +15.1227 q^{71} -11.5854 q^{72} +1.32162 q^{73} -0.409201 q^{74} -13.0490 q^{76} -8.55174 q^{77} -6.10457 q^{78} -10.4590 q^{79} +19.6300 q^{81} +3.98843 q^{82} -4.67370 q^{83} -15.2597 q^{84} +1.96026 q^{86} +21.8914 q^{87} -5.29498 q^{88} +4.03267 q^{89} -11.8600 q^{91} -1.38463 q^{92} -30.5505 q^{93} +4.40801 q^{94} -14.4019 q^{96} -4.97671 q^{97} +0.0129945 q^{98} +22.7609 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more