LMFDB - Embedded newform 4205.2.a.y.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4205,2,Mod(1,4205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4205, 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("4205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4205 = 5 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4205.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:	$33.5770940499$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	4205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.34022 q^{2} +1.23909 q^{3} -0.203815 q^{4} +1.00000 q^{5} +1.66065 q^{6} -3.86391 q^{7} -2.95359 q^{8} -1.46465 q^{9} +1.34022 q^{10} +5.03570 q^{11} -0.252546 q^{12} +3.21526 q^{13} -5.17848 q^{14} +1.23909 q^{15} -3.55083 q^{16} -5.75831 q^{17} -1.96295 q^{18} +4.82162 q^{19} -0.203815 q^{20} -4.78774 q^{21} +6.74894 q^{22} -3.49187 q^{23} -3.65977 q^{24} +1.00000 q^{25} +4.30915 q^{26} -5.53211 q^{27} +0.787523 q^{28} +1.66065 q^{30} -3.10558 q^{31} +1.14830 q^{32} +6.23970 q^{33} -7.71740 q^{34} -3.86391 q^{35} +0.298518 q^{36} +2.71406 q^{37} +6.46203 q^{38} +3.98400 q^{39} -2.95359 q^{40} -12.5230 q^{41} -6.41661 q^{42} -8.64635 q^{43} -1.02635 q^{44} -1.46465 q^{45} -4.67987 q^{46} +0.222881 q^{47} -4.39980 q^{48} +7.92979 q^{49} +1.34022 q^{50} -7.13508 q^{51} -0.655318 q^{52} -6.68898 q^{53} -7.41424 q^{54} +5.03570 q^{55} +11.4124 q^{56} +5.97443 q^{57} -4.63118 q^{59} -0.252546 q^{60} -9.46013 q^{61} -4.16216 q^{62} +5.65928 q^{63} +8.64063 q^{64} +3.21526 q^{65} +8.36256 q^{66} -12.9846 q^{67} +1.17363 q^{68} -4.32675 q^{69} -5.17848 q^{70} +7.50217 q^{71} +4.32598 q^{72} +3.53453 q^{73} +3.63743 q^{74} +1.23909 q^{75} -0.982719 q^{76} -19.4575 q^{77} +5.33943 q^{78} -4.45093 q^{79} -3.55083 q^{80} -2.46084 q^{81} -16.7836 q^{82} +8.43040 q^{83} +0.975813 q^{84} -5.75831 q^{85} -11.5880 q^{86} -14.8734 q^{88} +9.68882 q^{89} -1.96295 q^{90} -12.4235 q^{91} +0.711695 q^{92} -3.84810 q^{93} +0.298709 q^{94} +4.82162 q^{95} +1.42285 q^{96} -15.6823 q^{97} +10.6276 q^{98} -7.37555 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 14 q^{4} + 24 q^{5} - 18 q^{6} - 38 q^{7} + 18 q^{9} - 34 q^{13} - 10 q^{16} + 14 q^{20} - 10 q^{22} - 44 q^{23} - 4 q^{24} + 24 q^{25} - 44 q^{28} - 18 q^{30} - 58 q^{33} + 8 q^{34} - 38 q^{35}+ \cdots + 22 q^{96}+O(q^{100}) $ 24 * q + 14 * q^4 + 24 * q^5 - 18 * q^6 - 38 * q^7 + 18 * q^9 - 34 * q^13 - 10 * q^16 + 14 * q^20 - 10 * q^22 - 44 * q^23 - 4 * q^24 + 24 * q^25 - 44 * q^28 - 18 * q^30 - 58 * q^33 + 8 * q^34 - 38 * q^35 - 28 * q^36 - 44 * q^38 + 2 * q^42 + 18 * q^45 + 26 * q^49 - 48 * q^51 - 4 * q^52 - 28 * q^53 - 40 * q^54 - 32 * q^57 - 66 * q^59 - 2 * q^62 - 60 * q^63 - 12 * q^64 - 34 * q^65 - 144 * q^67 - 4 * q^71 + 12 * q^74 + 60 * q^78 - 10 * q^80 + 76 * q^81 + 4 * q^82 - 114 * q^83 - 30 * q^86 - 104 * q^88 - 32 * q^91 - 134 * q^92 - 32 * q^93 - 66 * q^94 + 22 * q^96

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 $

$2$	1.34022	0.947677	0.473839	−	0.880612i	$-0.342868\pi$
$2$	1.34022	0.947677	0.473839	+	0.880612i	$0.342868\pi$
$3$	1.23909	0.715390	0.357695	−	0.933838i	$-0.383563\pi$
$3$	1.23909	0.715390	0.357695	+	0.933838i	$0.383563\pi$
$4$	−0.203815	−0.101907
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.66065	0.677959
$7$	−3.86391	−1.46042	−0.730210	−	0.683223i	$-0.760578\pi$
$7$	−3.86391	−1.46042	−0.730210	+	0.683223i	$0.760578\pi$
$8$	−2.95359	−1.04425
$9$	−1.46465	−0.488217
$10$	1.34022	0.423814
$11$	5.03570	1.51832	0.759161	−	0.650903i	$-0.225610\pi$
$11$	5.03570	1.51832	0.759161	+	0.650903i	$0.225610\pi$
$12$	−0.252546	−0.0729036
$13$	3.21526	0.891752	0.445876	−	0.895095i	$-0.352892\pi$
$13$	3.21526	0.891752	0.445876	+	0.895095i	$0.352892\pi$
$14$	−5.17848	−1.38401
$15$	1.23909	0.319932
$16$	−3.55083	−0.887707
$17$	−5.75831	−1.39660	−0.698298	−	0.715807i	$-0.746059\pi$
$17$	−5.75831	−1.39660	−0.698298	+	0.715807i	$0.746059\pi$
$18$	−1.96295	−0.462672
$19$	4.82162	1.10616	0.553078	−	0.833129i	$-0.313453\pi$
$19$	4.82162	1.10616	0.553078	+	0.833129i	$0.313453\pi$
$20$	−0.203815	−0.0455744
$21$	−4.78774	−1.04477
$22$	6.74894	1.43888
$23$	−3.49187	−0.728105	−0.364053	−	0.931378i	$-0.618607\pi$
$23$	−3.49187	−0.728105	−0.364053	+	0.931378i	$0.618607\pi$
$24$	−3.65977	−0.747048
$25$	1.00000	0.200000
$26$	4.30915	0.845094
$27$	−5.53211	−1.06466
$28$	0.787523	0.148828
$29$	0	0
$29$	0	0
$30$	1.66065	0.303192
$31$	−3.10558	−0.557779	−0.278890	−	0.960323i	$-0.589966\pi$
$31$	−3.10558	−0.557779	−0.278890	+	0.960323i	$0.589966\pi$
$32$	1.14830	0.202993
$33$	6.23970	1.08619
$34$	−7.71740	−1.32352
$35$	−3.86391	−0.653120
$36$	0.298518	0.0497530
$37$	2.71406	0.446189	0.223094	−	0.974797i	$-0.428384\pi$
$37$	2.71406	0.446189	0.223094	+	0.974797i	$0.428384\pi$
$38$	6.46203	1.04828
$39$	3.98400	0.637951
$40$	−2.95359	−0.467004
$41$	−12.5230	−1.95576	−0.977882	−	0.209157i	$-0.932928\pi$
$41$	−12.5230	−1.95576	−0.977882	+	0.209157i	$0.932928\pi$
$42$	−6.41661	−0.990105
$43$	−8.64635	−1.31856	−0.659278	−	0.751899i	$-0.729138\pi$
$43$	−8.64635	−1.31856	−0.659278	+	0.751899i	$0.729138\pi$
$44$	−1.02635	−0.154728
$45$	−1.46465	−0.218337
$46$	−4.67987	−0.690009
$47$	0.222881	0.0325105	0.0162553	−	0.999868i	$-0.494826\pi$
$47$	0.222881	0.0325105	0.0162553	+	0.999868i	$0.494826\pi$
$48$	−4.39980	−0.635057
$49$	7.92979	1.13283
$50$	1.34022	0.189535
$51$	−7.13508	−0.999111
$52$	−0.655318	−0.0908762
$53$	−6.68898	−0.918802	−0.459401	−	0.888229i	$-0.651936\pi$
$53$	−6.68898	−0.918802	−0.459401	+	0.888229i	$0.651936\pi$
$54$	−7.41424	−1.00895
$55$	5.03570	0.679014
$56$	11.4124	1.52505
$57$	5.97443	0.791333
$58$	0	0
$59$	−4.63118	−0.602928	−0.301464	−	0.953478i	$-0.597475\pi$
$59$	−4.63118	−0.602928	−0.301464	+	0.953478i	$0.597475\pi$
$60$	−0.252546	−0.0326035
$61$	−9.46013	−1.21125	−0.605623	−	0.795752i	$-0.707076\pi$
$61$	−9.46013	−1.21125	−0.605623	+	0.795752i	$0.707076\pi$
$62$	−4.16216	−0.528595
$63$	5.65928	0.713002
$64$	8.64063	1.08008
$65$	3.21526	0.398804
$66$	8.36256	1.02936
$67$	−12.9846	−1.58632	−0.793159	−	0.609014i	$-0.791565\pi$
$67$	−12.9846	−1.58632	−0.793159	+	0.609014i	$0.791565\pi$
$68$	1.17363	0.142324
$69$	−4.32675	−0.520879
$70$	−5.17848	−0.618947
$71$	7.50217	0.890344	0.445172	−	0.895445i	$-0.353142\pi$
$71$	7.50217	0.890344	0.445172	+	0.895445i	$0.353142\pi$
$72$	4.32598	0.509822
$73$	3.53453	0.413685	0.206843	−	0.978374i	$-0.433681\pi$
$73$	3.53453	0.413685	0.206843	+	0.978374i	$0.433681\pi$
$74$	3.63743	0.422843
$75$	1.23909	0.143078
$76$	−0.982719	−0.112726
$77$	−19.4575	−2.21739
$78$	5.33943	0.604571
$79$	−4.45093	−0.500769	−0.250385	−	0.968146i	$-0.580557\pi$
$79$	−4.45093	−0.500769	−0.250385	+	0.968146i	$0.580557\pi$
$80$	−3.55083	−0.396995
$81$	−2.46084	−0.273427
$82$	−16.7836	−1.85343
$83$	8.43040	0.925357	0.462678	−	0.886526i	$-0.346889\pi$
$83$	8.43040	0.925357	0.462678	+	0.886526i	$0.346889\pi$
$84$	0.975813	0.106470
$85$	−5.75831	−0.624577
$86$	−11.5880	−1.24957
$87$	0	0
$88$	−14.8734	−1.58551
$89$	9.68882	1.02701	0.513506	−	0.858086i	$-0.328346\pi$
$89$	9.68882	1.02701	0.513506	+	0.858086i	$0.328346\pi$
$90$	−1.96295	−0.206913
$91$	−12.4235	−1.30233
$92$	0.711695	0.0741994
$93$	−3.84810	−0.399030
$94$	0.298709	0.0308095
$95$	4.82162	0.494688
$96$	1.42285	0.145219
$97$	−15.6823	−1.59230	−0.796149	−	0.605101i	$-0.793133\pi$
$97$	−15.6823	−1.59230	−0.796149	+	0.605101i	$0.793133\pi$
$98$	10.6276	1.07355
$99$	−7.37555	−0.741271
$100$	−0.203815	−0.0203815
$101$	−4.23137	−0.421037	−0.210519	−	0.977590i	$-0.567515\pi$
$101$	−4.23137	−0.421037	−0.210519	+	0.977590i	$0.567515\pi$
$102$	−9.56256	−0.946835
$103$	−1.93739	−0.190897	−0.0954484	−	0.995434i	$-0.530429\pi$
$103$	−1.93739	−0.190897	−0.0954484	+	0.995434i	$0.530429\pi$
$104$	−9.49657	−0.931215
$105$	−4.78774	−0.467235
$106$	−8.96470	−0.870728
$107$	−5.34791	−0.517002	−0.258501	−	0.966011i	$-0.583228\pi$
$107$	−5.34791	−0.517002	−0.258501	+	0.966011i	$0.583228\pi$
$108$	1.12753	0.108496
$109$	−3.65169	−0.349769	−0.174884	−	0.984589i	$-0.555955\pi$
$109$	−3.65169	−0.349769	−0.174884	+	0.984589i	$0.555955\pi$
$110$	6.74894	0.643486
$111$	3.36297	0.319199
$112$	13.7201	1.29643
$113$	9.16662	0.862323	0.431162	−	0.902275i	$-0.358104\pi$
$113$	9.16662	0.862323	0.431162	+	0.902275i	$0.358104\pi$
$114$	8.00705	0.749929
$115$	−3.49187	−0.325618
$116$	0	0
$117$	−4.70923	−0.435369
$118$	−6.20679	−0.571381
$119$	22.2496	2.03962
$120$	−3.65977	−0.334090
$121$	14.3583	1.30530
$122$	−12.6786	−1.14787
$123$	−15.5171	−1.39913
$124$	0.632964	0.0568419
$125$	1.00000	0.0894427
$126$	7.58467	0.675696
$127$	−6.23834	−0.553563	−0.276781	−	0.960933i	$-0.589268\pi$
$127$	−6.23834	−0.553563	−0.276781	+	0.960933i	$0.589268\pi$
$128$	9.28373	0.820574
$129$	−10.7136	−0.943282
$130$	4.30915	0.377937
$131$	0.654087	0.0571478	0.0285739	−	0.999592i	$-0.490903\pi$
$131$	0.654087	0.0571478	0.0285739	+	0.999592i	$0.490903\pi$
$132$	−1.27174	−0.110691
$133$	−18.6303	−1.61545
$134$	−17.4022	−1.50332
$135$	−5.53211	−0.476128
$136$	17.0077	1.45840
$137$	−2.60785	−0.222804	−0.111402	−	0.993775i	$-0.535534\pi$
$137$	−2.60785	−0.222804	−0.111402	+	0.993775i	$0.535534\pi$
$138$	−5.79879	−0.493625
$139$	5.86910	0.497810	0.248905	−	0.968528i	$-0.419929\pi$
$139$	5.86910	0.497810	0.248905	+	0.968528i	$0.419929\pi$
$140$	0.787523	0.0665578
$141$	0.276170	0.0232577
$142$	10.0546	0.843759
$143$	16.1911	1.35397
$144$	5.20073	0.433394
$145$	0	0
$146$	4.73704	0.392040
$147$	9.82574	0.810413
$148$	−0.553166	−0.0454700
$149$	10.4004	0.852037	0.426019	−	0.904714i	$-0.359916\pi$
$149$	10.4004	0.852037	0.426019	+	0.904714i	$0.359916\pi$
$150$	1.66065	0.135592
$151$	−3.48009	−0.283206	−0.141603	−	0.989924i	$-0.545226\pi$
$151$	−3.48009	−0.283206	−0.141603	+	0.989924i	$0.545226\pi$
$152$	−14.2411	−1.15511
$153$	8.43392	0.681842
$154$	−26.0773	−2.10137
$155$	−3.10558	−0.249446
$156$	−0.811999	−0.0650120
$157$	9.31575	0.743478	0.371739	−	0.928337i	$-0.378762\pi$
$157$	9.31575	0.743478	0.371739	+	0.928337i	$0.378762\pi$
$158$	−5.96522	−0.474567
$159$	−8.28826	−0.657302
$160$	1.14830	0.0907811
$161$	13.4923	1.06334
$162$	−3.29807	−0.259121
$163$	0.416983	0.0326606	0.0163303	−	0.999867i	$-0.494802\pi$
$163$	0.416983	0.0326606	0.0163303	+	0.999867i	$0.494802\pi$
$164$	2.55238	0.199307
$165$	6.23970	0.485760
$166$	11.2986	0.876940
$167$	−11.8458	−0.916656	−0.458328	−	0.888783i	$-0.651552\pi$
$167$	−11.8458	−0.916656	−0.458328	+	0.888783i	$0.651552\pi$
$168$	14.1410	1.09100
$169$	−2.66211	−0.204778
$170$	−7.71740	−0.591897
$171$	−7.06200	−0.540044
$172$	1.76226	0.134371
$173$	−5.01268	−0.381107	−0.190553	−	0.981677i	$-0.561028\pi$
$173$	−5.01268	−0.381107	−0.190553	+	0.981677i	$0.561028\pi$
$174$	0	0
$175$	−3.86391	−0.292084
$176$	−17.8809	−1.34783
$177$	−5.73846	−0.431329
$178$	12.9851	0.973276
$179$	−4.57941	−0.342281	−0.171140	−	0.985247i	$-0.554745\pi$
$179$	−4.57941	−0.342281	−0.171140	+	0.985247i	$0.554745\pi$
$180$	0.298518	0.0222502
$181$	1.38162	0.102695	0.0513475	−	0.998681i	$-0.483648\pi$
$181$	1.38162	0.102695	0.0513475	+	0.998681i	$0.483648\pi$
$182$	−16.6502	−1.23419
$183$	−11.7220	−0.866513
$184$	10.3136	0.760326
$185$	2.71406	0.199542
$186$	−5.15730	−0.378151
$187$	−28.9971	−2.12048
$188$	−0.0454265	−0.00331307
$189$	21.3756	1.55484
$190$	6.46203	0.468805
$191$	−25.0514	−1.81265	−0.906327	−	0.422577i	$-0.861126\pi$
$191$	−25.0514	−1.81265	−0.906327	+	0.422577i	$0.861126\pi$
$192$	10.7065	0.772678
$193$	−11.5666	−0.832585	−0.416293	−	0.909231i	$-0.636671\pi$
$193$	−11.5666	−0.832585	−0.416293	+	0.909231i	$0.636671\pi$
$194$	−21.0177	−1.50898
$195$	3.98400	0.285300
$196$	−1.61621	−0.115444
$197$	−22.1833	−1.58049	−0.790246	−	0.612790i	$-0.790047\pi$
$197$	−22.1833	−1.58049	−0.790246	+	0.612790i	$0.790047\pi$
$198$	−9.88485	−0.702485
$199$	20.5710	1.45824	0.729120	−	0.684386i	$-0.239930\pi$
$199$	20.5710	1.45824	0.729120	+	0.684386i	$0.239930\pi$
$200$	−2.95359	−0.208851
$201$	−16.0891	−1.13484
$202$	−5.67096	−0.399008
$203$	0	0
$204$	1.45424	0.101817
$205$	−12.5230	−0.874644
$206$	−2.59653	−0.180909
$207$	5.11437	0.355473
$208$	−11.4168	−0.791615
$209$	24.2803	1.67950
$210$	−6.41661	−0.442788
$211$	4.37352	0.301086	0.150543	−	0.988603i	$-0.451898\pi$
$211$	4.37352	0.301086	0.150543	+	0.988603i	$0.451898\pi$
$212$	1.36331	0.0936329
$213$	9.29588	0.636943
$214$	−7.16736	−0.489951
$215$	−8.64635	−0.589676
$216$	16.3396	1.11177
$217$	11.9997	0.814592
$218$	−4.89407	−0.331468
$219$	4.37961	0.295946
$220$	−1.02635	−0.0691966
$221$	−18.5145	−1.24542
$222$	4.50711	0.302498
$223$	−0.505255	−0.0338344	−0.0169172	−	0.999857i	$-0.505385\pi$
$223$	−0.505255	−0.0338344	−0.0169172	+	0.999857i	$0.505385\pi$
$224$	−4.43693	−0.296455
$225$	−1.46465	−0.0976434
$226$	12.2853	0.817204
$227$	−2.44549	−0.162313	−0.0811564	−	0.996701i	$-0.525861\pi$
$227$	−2.44549	−0.162313	−0.0811564	+	0.996701i	$0.525861\pi$
$228$	−1.21768	−0.0806428
$229$	11.0213	0.728310	0.364155	−	0.931338i	$-0.381358\pi$
$229$	11.0213	0.728310	0.364155	+	0.931338i	$0.381358\pi$
$230$	−4.67987	−0.308581
$231$	−24.1096	−1.58630
$232$	0	0
$233$	11.0552	0.724252	0.362126	−	0.932129i	$-0.382051\pi$
$233$	11.0552	0.724252	0.362126	+	0.932129i	$0.382051\pi$
$234$	−6.31140	−0.412589
$235$	0.222881	0.0145392
$236$	0.943904	0.0614429
$237$	−5.51511	−0.358245
$238$	29.8193	1.93290
$239$	12.6393	0.817571	0.408786	−	0.912630i	$-0.365952\pi$
$239$	12.6393	0.817571	0.408786	+	0.912630i	$0.365952\pi$
$240$	−4.39980	−0.284006
$241$	20.5451	1.32343	0.661713	−	0.749757i	$-0.269830\pi$
$241$	20.5451	1.32343	0.661713	+	0.749757i	$0.269830\pi$
$242$	19.2433	1.23700
$243$	13.5471	0.869049
$244$	1.92812	0.123435
$245$	7.92979	0.506616
$246$	−20.7964	−1.32593
$247$	15.5028	0.986417
$248$	9.17263	0.582463
$249$	10.4460	0.661991
$250$	1.34022	0.0847628
$251$	6.95375	0.438917	0.219458	−	0.975622i	$-0.429571\pi$
$251$	6.95375	0.438917	0.219458	+	0.975622i	$0.429571\pi$
$252$	−1.15345	−0.0726603
$253$	−17.5840	−1.10550
$254$	−8.36073	−0.524599
$255$	−7.13508	−0.446816
$256$	−4.83903	−0.302440
$257$	−20.6193	−1.28620	−0.643099	−	0.765783i	$-0.722352\pi$
$257$	−20.6193	−1.28620	−0.643099	+	0.765783i	$0.722352\pi$
$258$	−14.3586	−0.893927
$259$	−10.4869	−0.651623
$260$	−0.655318	−0.0406411
$261$	0	0
$262$	0.876619	0.0541577
$263$	21.5161	1.32674	0.663370	−	0.748292i	$-0.269126\pi$
$263$	21.5161	1.32674	0.663370	+	0.748292i	$0.269126\pi$
$264$	−18.4295	−1.13426
$265$	−6.68898	−0.410901
$266$	−24.9687	−1.53093
$267$	12.0053	0.734714
$268$	2.64645	0.161658
$269$	21.7280	1.32478	0.662390	−	0.749159i	$-0.269542\pi$
$269$	21.7280	1.32478	0.662390	+	0.749159i	$0.269542\pi$
$270$	−7.41424	−0.451216
$271$	−3.70503	−0.225064	−0.112532	−	0.993648i	$-0.535896\pi$
$271$	−3.70503	−0.225064	−0.112532	+	0.993648i	$0.535896\pi$
$272$	20.4468	1.23977
$273$	−15.3938	−0.931676
$274$	−3.49509	−0.211146
$275$	5.03570	0.303664
$276$	0.881856	0.0530815
$277$	2.66495	0.160121	0.0800606	−	0.996790i	$-0.474489\pi$
$277$	2.66495	0.160121	0.0800606	+	0.996790i	$0.474489\pi$
$278$	7.86587	0.471764
$279$	4.54860	0.272317
$280$	11.4124	0.682022
$281$	0.265091	0.0158140	0.00790701	−	0.999969i	$-0.497483\pi$
$281$	0.265091	0.0158140	0.00790701	+	0.999969i	$0.497483\pi$
$282$	0.370128	0.0220408
$283$	7.65663	0.455139	0.227570	−	0.973762i	$-0.426922\pi$
$283$	7.65663	0.455139	0.227570	+	0.973762i	$0.426922\pi$
$284$	−1.52906	−0.0907328
$285$	5.97443	0.353895
$286$	21.6996	1.28312
$287$	48.3877	2.85624
$288$	−1.68186	−0.0991045
$289$	16.1582	0.950480
$290$	0	0
$291$	−19.4318	−1.13911
$292$	−0.720390	−0.0421576
$293$	−5.42694	−0.317045	−0.158523	−	0.987355i	$-0.550673\pi$
$293$	−5.42694	−0.317045	−0.158523	+	0.987355i	$0.550673\pi$
$294$	13.1686	0.768010
$295$	−4.63118	−0.269638
$296$	−8.01623	−0.465934
$297$	−27.8581	−1.61649
$298$	13.9389	0.807456
$299$	−11.2273	−0.649289
$300$	−0.252546	−0.0145807
$301$	33.4087	1.92565
$302$	−4.66408	−0.268388
$303$	−5.24306	−0.301206
$304$	−17.1208	−0.981943
$305$	−9.46013	−0.541686
$306$	11.3033	0.646166
$307$	23.3143	1.33062	0.665309	−	0.746568i	$-0.268300\pi$
$307$	23.3143	1.33062	0.665309	+	0.746568i	$0.268300\pi$
$308$	3.96573	0.225968
$309$	−2.40061	−0.136566
$310$	−4.16216	−0.236395
$311$	29.8056	1.69012	0.845060	−	0.534671i	$-0.179564\pi$
$311$	29.8056	1.69012	0.845060	+	0.534671i	$0.179564\pi$
$312$	−11.7671	−0.666182
$313$	−18.6046	−1.05160	−0.525798	−	0.850609i	$-0.676233\pi$
$313$	−18.6046	−1.05160	−0.525798	+	0.850609i	$0.676233\pi$
$314$	12.4851	0.704577
$315$	5.65928	0.318864
$316$	0.907167	0.0510321
$317$	3.40919	0.191479	0.0957396	−	0.995406i	$-0.469478\pi$
$317$	3.40919	0.191479	0.0957396	+	0.995406i	$0.469478\pi$
$318$	−11.1081	−0.622910
$319$	0	0
$320$	8.64063	0.483026
$321$	−6.62655	−0.369858
$322$	18.0826	1.00770
$323$	−27.7644	−1.54485
$324$	0.501557	0.0278643
$325$	3.21526	0.178350
$326$	0.558848	0.0309517
$327$	−4.52478	−0.250221
$328$	36.9879	2.04231
$329$	−0.861192	−0.0474791
$330$	8.36256	0.460344
$331$	17.8368	0.980399	0.490200	−	0.871610i	$-0.336924\pi$
$331$	17.8368	0.980399	0.490200	+	0.871610i	$0.336924\pi$
$332$	−1.71824	−0.0943008
$333$	−3.97515	−0.217837
$334$	−15.8760	−0.868695
$335$	−12.9846	−0.709423
$336$	17.0004	0.927450
$337$	9.69957	0.528369	0.264185	−	0.964472i	$-0.414897\pi$
$337$	9.69957	0.528369	0.264185	+	0.964472i	$0.414897\pi$
$338$	−3.56781	−0.194063
$339$	11.3583	0.616898
$340$	1.17363	0.0636490
$341$	−15.6388	−0.846888
$342$	−9.46462	−0.511788
$343$	−3.59262	−0.193983
$344$	25.5378	1.37691
$345$	−4.32675	−0.232944
$346$	−6.71808	−0.361166
$347$	27.8087	1.49285	0.746425	−	0.665469i	$-0.231769\pi$
$347$	27.8087	1.49285	0.746425	+	0.665469i	$0.231769\pi$
$348$	0	0
$349$	−31.0380	−1.66142	−0.830712	−	0.556703i	$-0.812066\pi$
$349$	−31.0380	−1.66142	−0.830712	+	0.556703i	$0.812066\pi$
$350$	−5.17848	−0.276801
$351$	−17.7872	−0.949409
$352$	5.78250	0.308208
$353$	5.96567	0.317520	0.158760	−	0.987317i	$-0.449250\pi$
$353$	5.96567	0.317520	0.158760	+	0.987317i	$0.449250\pi$
$354$	−7.69079	−0.408761
$355$	7.50217	0.398174
$356$	−1.97473	−0.104660
$357$	27.5693	1.45912
$358$	−6.13740	−0.324372
$359$	13.9053	0.733894	0.366947	−	0.930242i	$-0.380403\pi$
$359$	13.9053	0.733894	0.366947	+	0.930242i	$0.380403\pi$
$360$	4.32598	0.227999
$361$	4.24805	0.223582
$362$	1.85167	0.0973216
$363$	17.7913	0.933799
$364$	2.53209	0.132718
$365$	3.53453	0.185006
$366$	−15.7100	−0.821175
$367$	17.4703	0.911942	0.455971	−	0.889995i	$-0.349292\pi$
$367$	17.4703	0.911942	0.455971	+	0.889995i	$0.349292\pi$
$368$	12.3990	0.646344
$369$	18.3418	0.954838
$370$	3.63743	0.189101
$371$	25.8456	1.34184
$372$	0.784301	0.0406641
$373$	1.44817	0.0749835	0.0374917	−	0.999297i	$-0.488063\pi$
$373$	1.44817	0.0749835	0.0374917	+	0.999297i	$0.488063\pi$
$374$	−38.8625	−2.00953
$375$	1.23909	0.0639864
$376$	−0.658300	−0.0339492
$377$	0	0
$378$	28.6479	1.47349
$379$	4.09290	0.210238	0.105119	−	0.994460i	$-0.466478\pi$
$379$	4.09290	0.210238	0.105119	+	0.994460i	$0.466478\pi$
$380$	−0.982719	−0.0504124
$381$	−7.72987	−0.396013
$382$	−33.5743	−1.71781
$383$	0.300571	0.0153585	0.00767924	−	0.999971i	$-0.497556\pi$
$383$	0.300571	0.0153585	0.00767924	+	0.999971i	$0.497556\pi$
$384$	11.5034	0.587030
$385$	−19.4575	−0.991646
$386$	−15.5018	−0.789022
$387$	12.6639	0.643742
$388$	3.19629	0.162267
$389$	−17.2457	−0.874393	−0.437197	−	0.899366i	$-0.644029\pi$
$389$	−17.2457	−0.874393	−0.437197	+	0.899366i	$0.644029\pi$
$390$	5.33943	0.270373
$391$	20.1073	1.01687
$392$	−23.4214	−1.18296
$393$	0.810473	0.0408830
$394$	−29.7304	−1.49780
$395$	−4.45093	−0.223951
$396$	1.50325	0.0755410
$397$	−27.2974	−1.37002	−0.685008	−	0.728536i	$-0.740201\pi$
$397$	−27.2974	−1.37002	−0.685008	+	0.728536i	$0.740201\pi$
$398$	27.5696	1.38194
$399$	−23.0847	−1.15568
$400$	−3.55083	−0.177541
$401$	−8.97403	−0.448142	−0.224071	−	0.974573i	$-0.571935\pi$
$401$	−8.97403	−0.448142	−0.224071	+	0.974573i	$0.571935\pi$
$402$	−21.5629	−1.07546
$403$	−9.98525	−0.497401
$404$	0.862417	0.0429069
$405$	−2.46084	−0.122280
$406$	0	0
$407$	13.6672	0.677458
$408$	21.0741	1.04332
$409$	15.8156	0.782030	0.391015	−	0.920384i	$-0.372124\pi$
$409$	15.8156	0.782030	0.391015	+	0.920384i	$0.372124\pi$
$410$	−16.7836	−0.828881
$411$	−3.23137	−0.159392
$412$	0.394869	0.0194538
$413$	17.8945	0.880529
$414$	6.85437	0.336874
$415$	8.43040	0.413832
$416$	3.69208	0.181019
$417$	7.27235	0.356129
$418$	32.5409	1.59163
$419$	−39.1991	−1.91500	−0.957500	−	0.288434i	$-0.906865\pi$
$419$	−39.1991	−1.91500	−0.957500	+	0.288434i	$0.906865\pi$
$420$	0.975813	0.0476148
$421$	39.5568	1.92788	0.963941	−	0.266115i	$-0.0857402\pi$
$421$	39.5568	1.92788	0.963941	+	0.266115i	$0.0857402\pi$
$422$	5.86148	0.285332
$423$	−0.326443	−0.0158722
$424$	19.7565	0.959462
$425$	−5.75831	−0.279319
$426$	12.4585	0.603617
$427$	36.5531	1.76893
$428$	1.08998	0.0526864
$429$	20.0622	0.968614
$430$	−11.5880	−0.558823
$431$	−3.11697	−0.150139	−0.0750697	−	0.997178i	$-0.523918\pi$
$431$	−3.11697	−0.150139	−0.0750697	+	0.997178i	$0.523918\pi$
$432$	19.6436	0.945103
$433$	−2.09593	−0.100724	−0.0503621	−	0.998731i	$-0.516038\pi$
$433$	−2.09593	−0.100724	−0.0503621	+	0.998731i	$0.516038\pi$
$434$	16.0822	0.771970
$435$	0	0
$436$	0.744270	0.0356441
$437$	−16.8365	−0.805398
$438$	5.86963	0.280462
$439$	3.10731	0.148304	0.0741519	−	0.997247i	$-0.476375\pi$
$439$	3.10731	0.148304	0.0741519	+	0.997247i	$0.476375\pi$
$440$	−14.8734	−0.709062
$441$	−11.6144	−0.553066
$442$	−24.8134	−1.18025
$443$	11.3167	0.537672	0.268836	−	0.963186i	$-0.413361\pi$
$443$	11.3167	0.537672	0.268836	+	0.963186i	$0.413361\pi$
$444$	−0.685423	−0.0325288
$445$	9.68882	0.459294
$446$	−0.677152	−0.0320641
$447$	12.8871	0.609539
$448$	−33.3866	−1.57737
$449$	35.1123	1.65705	0.828526	−	0.559951i	$-0.189180\pi$
$449$	35.1123	1.65705	0.828526	+	0.559951i	$0.189180\pi$
$450$	−1.96295	−0.0925345
$451$	−63.0621	−2.96948
$452$	−1.86830	−0.0878772
$453$	−4.31215	−0.202603
$454$	−3.27749	−0.153820
$455$	−12.4235	−0.582421
$456$	−17.6460	−0.826352
$457$	33.2957	1.55751	0.778754	−	0.627330i	$-0.215852\pi$
$457$	33.2957	1.55751	0.778754	+	0.627330i	$0.215852\pi$
$458$	14.7710	0.690203
$459$	31.8556	1.48689
$460$	0.711695	0.0331830
$461$	−31.9796	−1.48944	−0.744719	−	0.667379i	$-0.767416\pi$
$461$	−31.9796	−1.48944	−0.744719	+	0.667379i	$0.767416\pi$
$462$	−32.3122	−1.50330
$463$	−21.7185	−1.00934	−0.504672	−	0.863311i	$-0.668387\pi$
$463$	−21.7185	−1.00934	−0.504672	+	0.863311i	$0.668387\pi$
$464$	0	0
$465$	−3.84810	−0.178451
$466$	14.8164	0.686357
$467$	−17.1624	−0.794179	−0.397089	−	0.917780i	$-0.629980\pi$
$467$	−17.1624	−0.794179	−0.397089	+	0.917780i	$0.629980\pi$
$468$	0.959812	0.0443673
$469$	50.1712	2.31669
$470$	0.298709	0.0137784
$471$	11.5431	0.531877
$472$	13.6786	0.629610
$473$	−43.5405	−2.00199
$474$	−7.39146	−0.339501
$475$	4.82162	0.221231
$476$	−4.53480	−0.207852
$477$	9.79703	0.448575
$478$	16.9395	0.774794
$479$	−26.1468	−1.19468	−0.597339	−	0.801989i	$-0.703775\pi$
$479$	−26.1468	−1.19468	−0.597339	+	0.801989i	$0.703775\pi$
$480$	1.42285	0.0649439
$481$	8.72640	0.397890
$482$	27.5349	1.25418
$483$	16.7182	0.760702
$484$	−2.92644	−0.133020
$485$	−15.6823	−0.712097
$486$	18.1561	0.823578
$487$	−23.0580	−1.04486	−0.522429	−	0.852683i	$-0.674974\pi$
$487$	−23.0580	−1.04486	−0.522429	+	0.852683i	$0.674974\pi$
$488$	27.9414	1.26485
$489$	0.516680	0.0233651
$490$	10.6276	0.480108
$491$	−2.13628	−0.0964091	−0.0482046	−	0.998837i	$-0.515350\pi$
$491$	−2.13628	−0.0964091	−0.0482046	+	0.998837i	$0.515350\pi$
$492$	3.16263	0.142582
$493$	0	0
$494$	20.7771	0.934806
$495$	−7.37555	−0.331506
$496$	11.0274	0.495145
$497$	−28.9877	−1.30028
$498$	14.0000	0.627354
$499$	9.64391	0.431721	0.215860	−	0.976424i	$-0.430744\pi$
$499$	9.64391	0.431721	0.215860	+	0.976424i	$0.430744\pi$
$500$	−0.203815	−0.00911488
$501$	−14.6780	−0.655767
$502$	9.31954	0.415951
$503$	−33.7545	−1.50504	−0.752519	−	0.658571i	$-0.771161\pi$
$503$	−33.7545	−1.50504	−0.752519	+	0.658571i	$0.771161\pi$
$504$	−16.7152	−0.744554
$505$	−4.23137	−0.188294
$506$	−23.5664	−1.04766
$507$	−3.29860	−0.146496
$508$	1.27147	0.0564122
$509$	22.8073	1.01092	0.505459	−	0.862851i	$-0.331323\pi$
$509$	22.8073	1.01092	0.505459	+	0.862851i	$0.331323\pi$
$510$	−9.56256	−0.423437
$511$	−13.6571	−0.604154
$512$	−25.0528	−1.10719
$513$	−26.6738	−1.17768
$514$	−27.6344	−1.21890
$515$	−1.93739	−0.0853717
$516$	2.18360	0.0961275
$517$	1.12236	0.0493615
$518$	−14.0547	−0.617528
$519$	−6.21117	−0.272640
$520$	−9.49657	−0.416452
$521$	−34.9903	−1.53295	−0.766476	−	0.642272i	$-0.777992\pi$
$521$	−34.9903	−1.53295	−0.766476	+	0.642272i	$0.777992\pi$
$522$	0	0
$523$	−21.9987	−0.961936	−0.480968	−	0.876738i	$-0.659715\pi$
$523$	−21.9987	−0.961936	−0.480968	+	0.876738i	$0.659715\pi$
$524$	−0.133313	−0.00582379
$525$	−4.78774	−0.208954
$526$	28.8363	1.25732
$527$	17.8829	0.778992
$528$	−22.1561	−0.964221
$529$	−10.8068	−0.469863
$530$	−8.96470	−0.389402
$531$	6.78306	0.294360
$532$	3.79714	0.164627
$533$	−40.2647	−1.74406
$534$	16.0898	0.696272
$535$	−5.34791	−0.231210
$536$	38.3512	1.65652
$537$	−5.67430	−0.244864
$538$	29.1203	1.25546
$539$	39.9321	1.72000
$540$	1.12753	0.0485211
$541$	38.0683	1.63669	0.818343	−	0.574730i	$-0.194893\pi$
$541$	38.0683	1.63669	0.818343	+	0.574730i	$0.194893\pi$
$542$	−4.96555	−0.213288
$543$	1.71195	0.0734669
$544$	−6.61227	−0.283499
$545$	−3.65169	−0.156421
$546$	−20.6311	−0.882928
$547$	−36.8179	−1.57422	−0.787110	−	0.616813i	$-0.788424\pi$
$547$	−36.8179	−1.57422	−0.787110	+	0.616813i	$0.788424\pi$
$548$	0.531519	0.0227054
$549$	13.8558	0.591351
$550$	6.74894	0.287776
$551$	0	0
$552$	12.7794	0.543929
$553$	17.1980	0.731333
$554$	3.57161	0.151743
$555$	3.36297	0.142750
$556$	−1.19621	−0.0507306
$557$	16.1095	0.682581	0.341290	−	0.939958i	$-0.389136\pi$
$557$	16.1095	0.682581	0.341290	+	0.939958i	$0.389136\pi$
$558$	6.09611	0.258069
$559$	−27.8003	−1.17583
$560$	13.7201	0.579779
$561$	−35.9301	−1.51697
$562$	0.355280	0.0149866
$563$	2.46470	0.103875	0.0519374	−	0.998650i	$-0.483460\pi$
$563$	2.46470	0.103875	0.0519374	+	0.998650i	$0.483460\pi$
$564$	−0.0562876	−0.00237014
$565$	9.16662	0.385643
$566$	10.2616	0.431325
$567$	9.50847	0.399318
$568$	−22.1584	−0.929745
$569$	−25.1103	−1.05268	−0.526340	−	0.850274i	$-0.676436\pi$
$569$	−25.1103	−1.05268	−0.526340	+	0.850274i	$0.676436\pi$
$570$	8.00705	0.335378
$571$	−36.9106	−1.54466	−0.772330	−	0.635222i	$-0.780909\pi$
$571$	−36.9106	−1.54466	−0.772330	+	0.635222i	$0.780909\pi$
$572$	−3.29999	−0.137979
$573$	−31.0410	−1.29675
$574$	64.8501	2.70679
$575$	−3.49187	−0.145621
$576$	−12.6555	−0.527313
$577$	20.4809	0.852630	0.426315	−	0.904575i	$-0.359812\pi$
$577$	20.4809	0.852630	0.426315	+	0.904575i	$0.359812\pi$
$578$	21.6555	0.900748
$579$	−14.3321	−0.595623
$580$	0	0
$581$	−32.5743	−1.35141
$582$	−26.0429	−1.07951
$583$	−33.6837	−1.39504
$584$	−10.4396	−0.431992
$585$	−4.70923	−0.194703
$586$	−7.27328	−0.300457
$587$	1.31085	0.0541046	0.0270523	−	0.999634i	$-0.491388\pi$
$587$	1.31085	0.0541046	0.0270523	+	0.999634i	$0.491388\pi$
$588$	−2.00263	−0.0825872
$589$	−14.9740	−0.616991
$590$	−6.20679	−0.255530
$591$	−27.4871	−1.13067
$592$	−9.63716	−0.396085
$593$	7.19903	0.295629	0.147814	−	0.989015i	$-0.452776\pi$
$593$	7.19903	0.295629	0.147814	+	0.989015i	$0.452776\pi$
$594$	−37.3359	−1.53191
$595$	22.2496	0.912144
$596$	−2.11977	−0.0868290
$597$	25.4894	1.04321
$598$	−15.0470	−0.615317
$599$	−40.1490	−1.64044	−0.820221	−	0.572046i	$-0.806150\pi$
$599$	−40.1490	−1.64044	−0.820221	+	0.572046i	$0.806150\pi$
$600$	−3.65977	−0.149410
$601$	−25.8016	−1.05247	−0.526234	−	0.850340i	$-0.676397\pi$
$601$	−25.8016	−1.05247	−0.526234	+	0.850340i	$0.676397\pi$
$602$	44.7750	1.82489
$603$	19.0179	0.774468
$604$	0.709295	0.0288608
$605$	14.3583	0.583748
$606$	−7.02684	−0.285446
$607$	−23.2336	−0.943022	−0.471511	−	0.881860i	$-0.656291\pi$
$607$	−23.2336	−0.943022	−0.471511	+	0.881860i	$0.656291\pi$
$608$	5.53667	0.224542
$609$	0	0
$610$	−12.6786	−0.513343
$611$	0.716620	0.0289914
$612$	−1.71896	−0.0694848
$613$	43.4106	1.75334	0.876669	−	0.481095i	$-0.159761\pi$
$613$	43.4106	1.75334	0.876669	+	0.481095i	$0.159761\pi$
$614$	31.2463	1.26100
$615$	−15.5171	−0.625712
$616$	57.4695	2.31551
$617$	−2.92095	−0.117593	−0.0587966	−	0.998270i	$-0.518726\pi$
$617$	−2.92095	−0.117593	−0.0587966	+	0.998270i	$0.518726\pi$
$618$	−3.21734	−0.129420
$619$	6.78634	0.272766	0.136383	−	0.990656i	$-0.456452\pi$
$619$	6.78634	0.272766	0.136383	+	0.990656i	$0.456452\pi$
$620$	0.632964	0.0254205
$621$	19.3174	0.775181
$622$	39.9460	1.60169
$623$	−37.4367	−1.49987
$624$	−14.1465	−0.566314
$625$	1.00000	0.0400000
$626$	−24.9343	−0.996574
$627$	30.0855	1.20150
$628$	−1.89869	−0.0757660
$629$	−15.6284	−0.623145
$630$	7.58467	0.302180
$631$	−13.0844	−0.520881	−0.260440	−	0.965490i	$-0.583868\pi$
$631$	−13.0844	−0.520881	−0.260440	+	0.965490i	$0.583868\pi$
$632$	13.1462	0.522929
$633$	5.41920	0.215394
$634$	4.56906	0.181461
$635$	−6.23834	−0.247561
$636$	1.68927	0.0669840
$637$	25.4963	1.01020
$638$	0	0
$639$	−10.9881	−0.434681
$640$	9.28373	0.366972
$641$	−15.3430	−0.606012	−0.303006	−	0.952989i	$-0.597990\pi$
$641$	−15.3430	−0.606012	−0.303006	+	0.952989i	$0.597990\pi$
$642$	−8.88102	−0.350506
$643$	−14.6890	−0.579277	−0.289638	−	0.957136i	$-0.593535\pi$
$643$	−14.6890	−0.579277	−0.289638	+	0.957136i	$0.593535\pi$
$644$	−2.74993	−0.108362
$645$	−10.7136	−0.421848
$646$	−37.2104	−1.46402
$647$	7.17088	0.281916	0.140958	−	0.990016i	$-0.454982\pi$
$647$	7.17088	0.281916	0.140958	+	0.990016i	$0.454982\pi$
$648$	7.26833	0.285527
$649$	−23.3212	−0.915439
$650$	4.30915	0.169019
$651$	14.8687	0.582751
$652$	−0.0849873	−0.00332836
$653$	8.03005	0.314240	0.157120	−	0.987580i	$-0.449779\pi$
$653$	8.03005	0.314240	0.157120	+	0.987580i	$0.449779\pi$
$654$	−6.06420	−0.237129
$655$	0.654087	0.0255573
$656$	44.4670	1.73615
$657$	−5.17685	−0.201968
$658$	−1.15419	−0.0449948
$659$	−37.4528	−1.45896	−0.729478	−	0.684005i	$-0.760237\pi$
$659$	−37.4528	−1.45896	−0.729478	+	0.684005i	$0.760237\pi$
$660$	−1.27174	−0.0495026
$661$	−23.7604	−0.924171	−0.462085	−	0.886835i	$-0.652899\pi$
$661$	−23.7604	−0.924171	−0.462085	+	0.886835i	$0.652899\pi$
$662$	23.9052	0.929102
$663$	−22.9411	−0.890959
$664$	−24.9000	−0.966306
$665$	−18.6303	−0.722452
$666$	−5.32757	−0.206439
$667$	0	0
$668$	2.41435	0.0934142
$669$	−0.626058	−0.0242048
$670$	−17.4022	−0.672305
$671$	−47.6384	−1.83906
$672$	−5.49776	−0.212081
$673$	−26.7661	−1.03176	−0.515879	−	0.856661i	$-0.672535\pi$
$673$	−26.7661	−1.03176	−0.515879	+	0.856661i	$0.672535\pi$
$674$	12.9995	0.500724
$675$	−5.53211	−0.212931
$676$	0.542578	0.0208684
$677$	−7.77181	−0.298695	−0.149347	−	0.988785i	$-0.547717\pi$
$677$	−7.77181	−0.298695	−0.149347	+	0.988785i	$0.547717\pi$
$678$	15.2226	0.584620
$679$	60.5950	2.32542
$680$	17.0077	0.652216
$681$	−3.03019	−0.116117
$682$	−20.9594	−0.802577
$683$	−11.3325	−0.433627	−0.216814	−	0.976213i	$-0.569566\pi$
$683$	−11.3325	−0.433627	−0.216814	+	0.976213i	$0.569566\pi$
$684$	1.43934	0.0550346
$685$	−2.60785	−0.0996409
$686$	−4.81489	−0.183834
$687$	13.6564	0.521026
$688$	30.7017	1.17049
$689$	−21.5068	−0.819344
$690$	−5.79879	−0.220756
$691$	19.8571	0.755399	0.377700	−	0.925928i	$-0.376715\pi$
$691$	19.8571	0.755399	0.377700	+	0.925928i	$0.376715\pi$
$692$	1.02166	0.0388376
$693$	28.4984	1.08257
$694$	37.2698	1.41474
$695$	5.86910	0.222628
$696$	0	0
$697$	72.1114	2.73141
$698$	−41.5976	−1.57449
$699$	13.6984	0.518123
$700$	0.787523	0.0297656
$701$	31.6059	1.19374	0.596869	−	0.802339i	$-0.296411\pi$
$701$	31.6059	1.19374	0.596869	+	0.802339i	$0.296411\pi$
$702$	−23.8387	−0.899734
$703$	13.0862	0.493554
$704$	43.5117	1.63991
$705$	0.276170	0.0104012
$706$	7.99529	0.300907
$707$	16.3496	0.614891
$708$	1.16958	0.0439556
$709$	39.2218	1.47301	0.736503	−	0.676434i	$-0.236476\pi$
$709$	39.2218	1.47301	0.736503	+	0.676434i	$0.236476\pi$
$710$	10.0546	0.377341
$711$	6.51906	0.244484
$712$	−28.6168	−1.07246
$713$	10.8443	0.406122
$714$	36.9489	1.38278
$715$	16.1911	0.605512
$716$	0.933352	0.0348810
$717$	15.6613	0.584882
$718$	18.6361	0.695495
$719$	−1.67354	−0.0624125	−0.0312063	−	0.999513i	$-0.509935\pi$
$719$	−1.67354	−0.0624125	−0.0312063	+	0.999513i	$0.509935\pi$
$720$	5.20073	0.193820
$721$	7.48590	0.278790
$722$	5.69332	0.211883
$723$	25.4573	0.946766
$724$	−0.281595	−0.0104654
$725$	0	0
$726$	23.8442	0.884940
$727$	38.5023	1.42797	0.713986	−	0.700160i	$-0.246888\pi$
$727$	38.5023	1.42797	0.713986	+	0.700160i	$0.246888\pi$
$728$	36.6939	1.35997
$729$	24.1687	0.895136
$730$	4.73704	0.175326
$731$	49.7884	1.84149
$732$	2.38911	0.0883042
$733$	11.1932	0.413429	0.206715	−	0.978401i	$-0.433723\pi$
$733$	11.1932	0.413429	0.206715	+	0.978401i	$0.433723\pi$
$734$	23.4140	0.864227
$735$	9.82574	0.362428
$736$	−4.00971	−0.147800
$737$	−65.3865	−2.40854
$738$	24.5821	0.904878
$739$	−45.2165	−1.66331	−0.831657	−	0.555289i	$-0.812608\pi$
$739$	−45.2165	−1.66331	−0.831657	+	0.555289i	$0.812608\pi$
$740$	−0.553166	−0.0203348
$741$	19.2094	0.705673
$742$	34.6388	1.27163
$743$	23.9297	0.877895	0.438947	−	0.898513i	$-0.355351\pi$
$743$	23.9297	0.877895	0.438947	+	0.898513i	$0.355351\pi$
$744$	11.3657	0.416688
$745$	10.4004	0.381043
$746$	1.94087	0.0710601
$747$	−12.3476	−0.451775
$748$	5.91005	0.216093
$749$	20.6638	0.755040
$750$	1.66065	0.0606385
$751$	−43.9608	−1.60415	−0.802077	−	0.597221i	$-0.796272\pi$
$751$	−43.9608	−1.60415	−0.802077	+	0.597221i	$0.796272\pi$
$752$	−0.791413	−0.0288599
$753$	8.61633	0.313997
$754$	0	0
$755$	−3.48009	−0.126653
$756$	−4.35666	−0.158450
$757$	42.1923	1.53351	0.766753	−	0.641942i	$-0.221871\pi$
$757$	42.1923	1.53351	0.766753	+	0.641942i	$0.221871\pi$
$758$	5.48538	0.199238
$759$	−21.7882	−0.790862
$760$	−14.2411	−0.516579
$761$	14.5122	0.526066	0.263033	−	0.964787i	$-0.415277\pi$
$761$	14.5122	0.526066	0.263033	+	0.964787i	$0.415277\pi$
$762$	−10.3597	−0.375293
$763$	14.1098	0.510809
$764$	5.10585	0.184723
$765$	8.43392	0.304929
$766$	0.402831	0.0145549
$767$	−14.8904	−0.537663
$768$	−5.99601	−0.216362
$769$	−11.9771	−0.431907	−0.215953	−	0.976404i	$-0.569286\pi$
$769$	−11.9771	−0.431907	−0.215953	+	0.976404i	$0.569286\pi$
$770$	−26.0773	−0.939760
$771$	−25.5492	−0.920133
$772$	2.35746	0.0848467
$773$	20.5104	0.737709	0.368854	−	0.929487i	$-0.379750\pi$
$773$	20.5104	0.737709	0.368854	+	0.929487i	$0.379750\pi$
$774$	16.9724	0.610059
$775$	−3.10558	−0.111556
$776$	46.3192	1.66276
$777$	−12.9942	−0.466164
$778$	−23.1130	−0.828643
$779$	−60.3812	−2.16338
$780$	−0.811999	−0.0290742
$781$	37.7787	1.35183
$782$	26.9481	0.963663
$783$	0	0
$784$	−28.1573	−1.00562
$785$	9.31575	0.332493
$786$	1.08621	0.0387439
$787$	−11.3492	−0.404555	−0.202278	−	0.979328i	$-0.564834\pi$
$787$	−11.3492	−0.404555	−0.202278	+	0.979328i	$0.564834\pi$
$788$	4.52128	0.161064
$789$	26.6604	0.949136
$790$	−5.96522	−0.212233
$791$	−35.4190	−1.25935
$792$	21.7844	0.774074
$793$	−30.4168	−1.08013
$794$	−36.5844	−1.29833
$795$	−8.28826	−0.293954
$796$	−4.19268	−0.148606
$797$	4.45253	0.157717	0.0788584	−	0.996886i	$-0.474872\pi$
$797$	4.45253	0.157717	0.0788584	+	0.996886i	$0.474872\pi$
$798$	−30.9385	−1.09521
$799$	−1.28342	−0.0454041
$800$	1.14830	0.0405985
$801$	−14.1907	−0.501405
$802$	−12.0272	−0.424694
$803$	17.7988	0.628107
$804$	3.27920	0.115648
$805$	13.4923	0.475540
$806$	−13.3824	−0.471376
$807$	26.9230	0.947734
$808$	12.4978	0.439669
$809$	−2.94380	−0.103499	−0.0517493	−	0.998660i	$-0.516480\pi$
$809$	−2.94380	−0.103499	−0.0517493	+	0.998660i	$0.516480\pi$
$810$	−3.29807	−0.115882
$811$	52.6587	1.84910	0.924548	−	0.381066i	$-0.124443\pi$
$811$	52.6587	1.84910	0.924548	+	0.381066i	$0.124443\pi$
$812$	0	0
$813$	−4.59087	−0.161009
$814$	18.3170	0.642011
$815$	0.416983	0.0146063
$816$	25.3354	0.886918
$817$	−41.6894	−1.45853
$818$	21.1963	0.741113
$819$	18.1960	0.635821
$820$	2.55238	0.0891328
$821$	−1.62029	−0.0565483	−0.0282742	−	0.999600i	$-0.509001\pi$
$821$	−1.62029	−0.0565483	−0.0282742	+	0.999600i	$0.509001\pi$
$822$	−4.33074	−0.151052
$823$	−37.5426	−1.30865	−0.654326	−	0.756213i	$-0.727047\pi$
$823$	−37.5426	−1.30865	−0.654326	+	0.756213i	$0.727047\pi$
$824$	5.72227	0.199345
$825$	6.23970	0.217238
$826$	23.9825	0.834457
$827$	8.71039	0.302890	0.151445	−	0.988466i	$-0.451607\pi$
$827$	8.71039	0.302890	0.151445	+	0.988466i	$0.451607\pi$
$828$	−1.04239	−0.0362254
$829$	1.94099	0.0674135	0.0337067	−	0.999432i	$-0.489269\pi$
$829$	1.94099	0.0674135	0.0337067	+	0.999432i	$0.489269\pi$
$830$	11.2986	0.392179
$831$	3.30212	0.114549
$832$	27.7819	0.963163
$833$	−45.6622	−1.58210
$834$	9.74654	0.337495
$835$	−11.8458	−0.409941
$836$	−4.94868	−0.171154
$837$	17.1804	0.593843
$838$	−52.5353	−1.81480
$839$	36.1898	1.24941	0.624705	−	0.780861i	$-0.285219\pi$
$839$	36.1898	1.24941	0.624705	+	0.780861i	$0.285219\pi$
$840$	14.1410	0.487912
$841$	0	0
$842$	53.0148	1.82701
$843$	0.328473	0.0113132
$844$	−0.891390	−0.0306829
$845$	−2.66211	−0.0915794
$846$	−0.437505	−0.0150417
$847$	−55.4792	−1.90629
$848$	23.7514	0.815628
$849$	9.48726	0.325602
$850$	−7.71740	−0.264704
$851$	−9.47714	−0.324872
$852$	−1.89464	−0.0649093
$853$	0.842759	0.0288555	0.0144278	−	0.999896i	$-0.495407\pi$
$853$	0.842759	0.0288555	0.0144278	+	0.999896i	$0.495407\pi$
$854$	48.9891	1.67637
$855$	−7.06200	−0.241515
$856$	15.7955	0.539881
$857$	−18.7743	−0.641317	−0.320658	−	0.947195i	$-0.603904\pi$
$857$	−18.7743	−0.641317	−0.320658	+	0.947195i	$0.603904\pi$
$858$	26.8878	0.917934
$859$	−32.2105	−1.09901	−0.549504	−	0.835491i	$-0.685183\pi$
$859$	−32.2105	−1.09901	−0.549504	+	0.835491i	$0.685183\pi$
$860$	1.76226	0.0600924
$861$	59.9568	2.04332
$862$	−4.17742	−0.142284
$863$	47.1066	1.60353	0.801763	−	0.597642i	$-0.203896\pi$
$863$	47.1066	1.60353	0.801763	+	0.597642i	$0.203896\pi$
$864$	−6.35252	−0.216117
$865$	−5.01268	−0.170436
$866$	−2.80901	−0.0954540
$867$	20.0214	0.679964
$868$	−2.44572	−0.0830130
$869$	−22.4136	−0.760328
$870$	0	0
$871$	−41.7488	−1.41460
$872$	10.7856	0.365247
$873$	22.9691	0.777387
$874$	−22.5646	−0.763257
$875$	−3.86391	−0.130624
$876$	−0.892630	−0.0301592
$877$	−31.8345	−1.07498	−0.537488	−	0.843272i	$-0.680627\pi$
$877$	−31.8345	−1.07498	−0.537488	+	0.843272i	$0.680627\pi$
$878$	4.16448	0.140544
$879$	−6.72448	−0.226811
$880$	−17.8809	−0.602766
$881$	32.5720	1.09738	0.548689	−	0.836027i	$-0.315127\pi$
$881$	32.5720	1.09738	0.548689	+	0.836027i	$0.315127\pi$
$882$	−15.5658	−0.524128
$883$	54.9398	1.84887	0.924435	−	0.381339i	$-0.124537\pi$
$883$	54.9398	1.84887	0.924435	+	0.381339i	$0.124537\pi$
$884$	3.77353	0.126917
$885$	−5.73846	−0.192896
$886$	15.1668	0.509539
$887$	29.8115	1.00097	0.500486	−	0.865745i	$-0.333155\pi$
$887$	29.8115	1.00097	0.500486	+	0.865745i	$0.333155\pi$
$888$	−9.93284	−0.333324
$889$	24.1044	0.808434
$890$	12.9851	0.435262
$891$	−12.3921	−0.415150
$892$	0.102979	0.00344798
$893$	1.07465	0.0359617
$894$	17.2715	0.577646
$895$	−4.57941	−0.153073
$896$	−35.8715	−1.19838
$897$	−13.9116	−0.464495
$898$	47.0581	1.57035
$899$	0	0
$900$	0.298518	0.00995060
$901$	38.5173	1.28320
$902$	−84.5170	−2.81411
$903$	41.3965	1.37759
$904$	−27.0745	−0.900484
$905$	1.38162	0.0459266
$906$	−5.77922	−0.192002
$907$	−31.3088	−1.03959	−0.519796	−	0.854290i	$-0.673992\pi$
$907$	−31.3088	−1.03959	−0.519796	+	0.854290i	$0.673992\pi$
$908$	0.498427	0.0165409
$909$	6.19749	0.205558
$910$	−16.6502	−0.551947
$911$	−30.4463	−1.00873	−0.504365	−	0.863490i	$-0.668273\pi$
$911$	−30.4463	−1.00873	−0.504365	+	0.863490i	$0.668273\pi$
$912$	−21.2142	−0.702472
$913$	42.4530	1.40499
$914$	44.6235	1.47601
$915$	−11.7220	−0.387516
$916$	−2.24631	−0.0742203
$917$	−2.52733	−0.0834598
$918$	42.6935	1.40910
$919$	−38.2199	−1.26076	−0.630379	−	0.776287i	$-0.717101\pi$
$919$	−38.2199	−1.26076	−0.630379	+	0.776287i	$0.717101\pi$
$920$	10.3136	0.340028
$921$	28.8886	0.951911
$922$	−42.8596	−1.41151
$923$	24.1214	0.793967
$924$	4.91390	0.161656
$925$	2.71406	0.0892377
$926$	−29.1075	−0.956533
$927$	2.83760	0.0931991
$928$	0	0
$929$	7.40994	0.243112	0.121556	−	0.992585i	$-0.461212\pi$
$929$	7.40994	0.243112	0.121556	+	0.992585i	$0.461212\pi$
$930$	−5.15730	−0.169114
$931$	38.2345	1.25308
$932$	−2.25322	−0.0738067
$933$	36.9319	1.20910
$934$	−23.0013	−0.752625
$935$	−28.9971	−0.948308
$936$	13.9092	0.454635
$937$	−37.5661	−1.22723	−0.613615	−	0.789605i	$-0.710285\pi$
$937$	−37.5661	−1.22723	−0.613615	+	0.789605i	$0.710285\pi$
$938$	67.2404	2.19548
$939$	−23.0529	−0.752302
$940$	−0.0454265	−0.00148165
$941$	43.2981	1.41148	0.705738	−	0.708473i	$-0.250615\pi$
$941$	43.2981	1.41148	0.705738	+	0.708473i	$0.250615\pi$
$942$	15.4702	0.504047
$943$	43.7287	1.42400
$944$	16.4445	0.535224
$945$	21.3756	0.695348
$946$	−58.3537	−1.89724
$947$	14.5615	0.473185	0.236592	−	0.971609i	$-0.423969\pi$
$947$	14.5615	0.473185	0.236592	+	0.971609i	$0.423969\pi$
$948$	1.12406	0.0365079
$949$	11.3644	0.368905
$950$	6.46203	0.209656
$951$	4.22430	0.136982
$952$	−65.7162	−2.12988
$953$	−8.18056	−0.264994	−0.132497	−	0.991183i	$-0.542300\pi$
$953$	−8.18056	−0.264994	−0.132497	+	0.991183i	$0.542300\pi$
$954$	13.1302	0.425104
$955$	−25.0514	−0.810644
$956$	−2.57609	−0.0833166
$957$	0	0
$958$	−35.0424	−1.13217
$959$	10.0765	0.325387
$960$	10.7065	0.345552
$961$	−21.3554	−0.688882
$962$	11.6953	0.377071
$963$	7.83282	0.252409
$964$	−4.18740	−0.134867
$965$	−11.5666	−0.372343
$966$	22.4060	0.720900
$967$	−21.7758	−0.700262	−0.350131	−	0.936701i	$-0.613863\pi$
$967$	−21.7758	−0.700262	−0.350131	+	0.936701i	$0.613863\pi$
$968$	−42.4086	−1.36306
$969$	−34.4027	−1.10517
$970$	−21.0177	−0.674838
$971$	−25.0348	−0.803404	−0.401702	−	0.915770i	$-0.631581\pi$
$971$	−25.0348	−0.803404	−0.401702	+	0.915770i	$0.631581\pi$
$972$	−2.76111	−0.0885626
$973$	−22.6777	−0.727012
$974$	−30.9027	−0.990187
$975$	3.98400	0.127590
$976$	33.5913	1.07523
$977$	25.3391	0.810671	0.405335	−	0.914168i	$-0.367155\pi$
$977$	25.3391	0.810671	0.405335	+	0.914168i	$0.367155\pi$
$978$	0.692464	0.0221425
$979$	48.7900	1.55933
$980$	−1.61621	−0.0516279
$981$	5.34846	0.170763
$982$	−2.86309	−0.0913647
$983$	0.704210	0.0224608	0.0112304	−	0.999937i	$-0.496425\pi$
$983$	0.704210	0.0224608	0.0112304	+	0.999937i	$0.496425\pi$
$984$	45.8313	1.46105
$985$	−22.1833	−0.706817
$986$	0	0
$987$	−1.06710	−0.0339660
$988$	−3.15970	−0.100523
$989$	30.1919	0.960047
$990$	−9.88485	−0.314161
$991$	−27.1594	−0.862746	−0.431373	−	0.902174i	$-0.641971\pi$
$991$	−27.1594	−0.862746	−0.431373	+	0.902174i	$0.641971\pi$
$992$	−3.56614	−0.113225
$993$	22.1014	0.701368
$994$	−38.8499	−1.23224
$995$	20.5710	0.652145
$996$	−2.12906	−0.0674618
$997$	−5.42615	−0.171848	−0.0859239	−	0.996302i	$-0.527384\pi$
$997$	−5.42615	−0.171848	−0.0859239	+	0.996302i	$0.527384\pi$
$998$	12.9250	0.409132
$999$	−15.0145	−0.475037

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4205.2.a.y.1.19		24
29.3	odd	28		145.2.m.a.96.3	yes	24
29.10	odd	28		145.2.m.a.71.3	✓	24
29.28	even	2	inner	4205.2.a.y.1.6		24
145.3	even	28		725.2.p.b.299.7		48
145.32	even	28		725.2.p.b.299.2		48
145.39	odd	28		725.2.q.b.651.2		24
145.68	even	28		725.2.p.b.274.2		48
145.97	even	28		725.2.p.b.274.7		48
145.119	odd	28		725.2.q.b.676.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.m.a.71.3	✓	24	29.10	odd	28
145.2.m.a.96.3	yes	24	29.3	odd	28
725.2.p.b.274.2		48	145.68	even	28
725.2.p.b.274.7		48	145.97	even	28
725.2.p.b.299.2		48	145.32	even	28
725.2.p.b.299.7		48	145.3	even	28
725.2.q.b.651.2		24	145.39	odd	28
725.2.q.b.676.2		24	145.119	odd	28
4205.2.a.y.1.6		24	29.28	even	2	inner
4205.2.a.y.1.19		24	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:	\( N \)	\(=\)	\( 4205 = 5 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4205.a (trivial)

\(f(q)\)	\(=\)	\(q+1.34022 q^{2} +1.23909 q^{3} -0.203815 q^{4} +1.00000 q^{5} +1.66065 q^{6} -3.86391 q^{7} -2.95359 q^{8} -1.46465 q^{9} +1.34022 q^{10} +5.03570 q^{11} -0.252546 q^{12} +3.21526 q^{13} -5.17848 q^{14} +1.23909 q^{15} -3.55083 q^{16} -5.75831 q^{17} -1.96295 q^{18} +4.82162 q^{19} -0.203815 q^{20} -4.78774 q^{21} +6.74894 q^{22} -3.49187 q^{23} -3.65977 q^{24} +1.00000 q^{25} +4.30915 q^{26} -5.53211 q^{27} +0.787523 q^{28} +1.66065 q^{30} -3.10558 q^{31} +1.14830 q^{32} +6.23970 q^{33} -7.71740 q^{34} -3.86391 q^{35} +0.298518 q^{36} +2.71406 q^{37} +6.46203 q^{38} +3.98400 q^{39} -2.95359 q^{40} -12.5230 q^{41} -6.41661 q^{42} -8.64635 q^{43} -1.02635 q^{44} -1.46465 q^{45} -4.67987 q^{46} +0.222881 q^{47} -4.39980 q^{48} +7.92979 q^{49} +1.34022 q^{50} -7.13508 q^{51} -0.655318 q^{52} -6.68898 q^{53} -7.41424 q^{54} +5.03570 q^{55} +11.4124 q^{56} +5.97443 q^{57} -4.63118 q^{59} -0.252546 q^{60} -9.46013 q^{61} -4.16216 q^{62} +5.65928 q^{63} +8.64063 q^{64} +3.21526 q^{65} +8.36256 q^{66} -12.9846 q^{67} +1.17363 q^{68} -4.32675 q^{69} -5.17848 q^{70} +7.50217 q^{71} +4.32598 q^{72} +3.53453 q^{73} +3.63743 q^{74} +1.23909 q^{75} -0.982719 q^{76} -19.4575 q^{77} +5.33943 q^{78} -4.45093 q^{79} -3.55083 q^{80} -2.46084 q^{81} -16.7836 q^{82} +8.43040 q^{83} +0.975813 q^{84} -5.75831 q^{85} -11.5880 q^{86} -14.8734 q^{88} +9.68882 q^{89} -1.96295 q^{90} -12.4235 q^{91} +0.711695 q^{92} -3.84810 q^{93} +0.298709 q^{94} +4.82162 q^{95} +1.42285 q^{96} -15.6823 q^{97} +10.6276 q^{98} -7.37555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 14 q^{4} + 24 q^{5} - 18 q^{6} - 38 q^{7} + 18 q^{9} - 34 q^{13} - 10 q^{16} + 14 q^{20} - 10 q^{22} - 44 q^{23} - 4 q^{24} + 24 q^{25} - 44 q^{28} - 18 q^{30} - 58 q^{33} + 8 q^{34} - 38 q^{35}+ \cdots + 22 q^{96}+O(q^{100}) \) 24 * q + 14 * q^4 + 24 * q^5 - 18 * q^6 - 38 * q^7 + 18 * q^9 - 34 * q^13 - 10 * q^16 + 14 * q^20 - 10 * q^22 - 44 * q^23 - 4 * q^24 + 24 * q^25 - 44 * q^28 - 18 * q^30 - 58 * q^33 + 8 * q^34 - 38 * q^35 - 28 * q^36 - 44 * q^38 + 2 * q^42 + 18 * q^45 + 26 * q^49 - 48 * q^51 - 4 * q^52 - 28 * q^53 - 40 * q^54 - 32 * q^57 - 66 * q^59 - 2 * q^62 - 60 * q^63 - 12 * q^64 - 34 * q^65 - 144 * q^67 - 4 * q^71 + 12 * q^74 + 60 * q^78 - 10 * q^80 + 76 * q^81 + 4 * q^82 - 114 * q^83 - 30 * q^86 - 104 * q^88 - 32 * q^91 - 134 * q^92 - 32 * q^93 - 66 * q^94 + 22 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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