LMFDB - Embedded newform 5225.2.a.p.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,-3,-3,9,0,0,-13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 $ x^9 - 3x^8 - 9x^7 + 29x^6 + 23x^5 - 84x^4 - 23x^3 + 89x^2 + 8x - 27
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.682920$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.682920 q^{2} -2.48419 q^{3} -1.53362 q^{4} +1.69650 q^{6} -0.856948 q^{7} +2.41318 q^{8} +3.17118 q^{9} +1.00000 q^{11} +3.80980 q^{12} -5.50630 q^{13} +0.585228 q^{14} +1.41923 q^{16} +3.50630 q^{17} -2.16566 q^{18} +1.00000 q^{19} +2.12882 q^{21} -0.682920 q^{22} -6.75237 q^{23} -5.99479 q^{24} +3.76036 q^{26} -0.425246 q^{27} +1.31423 q^{28} -0.474680 q^{29} -5.19745 q^{31} -5.79558 q^{32} -2.48419 q^{33} -2.39452 q^{34} -4.86339 q^{36} -0.420904 q^{37} -0.682920 q^{38} +13.6787 q^{39} +8.25141 q^{41} -1.45381 q^{42} +1.53320 q^{43} -1.53362 q^{44} +4.61133 q^{46} -2.48849 q^{47} -3.52563 q^{48} -6.26564 q^{49} -8.71030 q^{51} +8.44457 q^{52} +4.79753 q^{53} +0.290409 q^{54} -2.06797 q^{56} -2.48419 q^{57} +0.324169 q^{58} +1.75917 q^{59} +3.42433 q^{61} +3.54944 q^{62} -2.71754 q^{63} +1.11946 q^{64} +1.69650 q^{66} +5.12424 q^{67} -5.37733 q^{68} +16.7742 q^{69} -12.1937 q^{71} +7.65263 q^{72} +12.7638 q^{73} +0.287444 q^{74} -1.53362 q^{76} -0.856948 q^{77} -9.34144 q^{78} +10.8691 q^{79} -8.45715 q^{81} -5.63505 q^{82} -2.94669 q^{83} -3.26480 q^{84} -1.04705 q^{86} +1.17919 q^{87} +2.41318 q^{88} +3.08736 q^{89} +4.71861 q^{91} +10.3556 q^{92} +12.9114 q^{93} +1.69944 q^{94} +14.3973 q^{96} +8.88210 q^{97} +4.27893 q^{98} +3.17118 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{2} - 3 q^{3} + 9 q^{4} - 13 q^{7} - 9 q^{8} + 12 q^{9} + 9 q^{11} + 5 q^{12} - 5 q^{13} - 2 q^{14} + q^{16} - 13 q^{17} - q^{18} + 9 q^{19} + q^{21} - 3 q^{22} - 8 q^{23} - 7 q^{24} + 8 q^{26}+ \cdots + 12 q^{99}+O(q^{100}) $ 9 * q - 3 * q^2 - 3 * q^3 + 9 * q^4 - 13 * q^7 - 9 * q^8 + 12 * q^9 + 9 * q^11 + 5 * q^12 - 5 * q^13 - 2 * q^14 + q^16 - 13 * q^17 - q^18 + 9 * q^19 + q^21 - 3 * q^22 - 8 * q^23 - 7 * q^24 + 8 * q^26 - 6 * q^27 - 10 * q^28 - 3 * q^29 - 9 * q^31 - 6 * q^32 - 3 * q^33 - 2 * q^34 - 15 * q^36 + 7 * q^37 - 3 * q^38 - 2 * q^39 + 9 * q^41 + 9 * q^42 - 23 * q^43 + 9 * q^44 - 32 * q^46 - 20 * q^47 + 18 * q^48 - 4 * q^49 + 8 * q^51 - 9 * q^52 + 5 * q^53 + 9 * q^54 + 4 * q^56 - 3 * q^57 - 3 * q^58 + 19 * q^59 + q^61 - 18 * q^62 - 24 * q^63 + 23 * q^64 + 10 * q^67 - 9 * q^68 - 28 * q^69 - 24 * q^72 - 12 * q^73 + 5 * q^74 + 9 * q^76 - 13 * q^77 - 19 * q^78 - 21 * q^79 - 3 * q^81 + 14 * q^82 - 47 * q^83 - 11 * q^84 + 12 * q^86 + 2 * q^87 - 9 * q^88 - 2 * q^89 + 2 * q^91 + 19 * q^92 + 2 * q^93 + 26 * q^94 - 55 * q^96 + 32 * q^97 + 12 * q^99

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 $

$2$	−0.682920	−0.482898	−0.241449	−	0.970414i	$-0.577623\pi$
$2$	−0.682920	−0.482898	−0.241449	+	0.970414i	$0.577623\pi$
$3$	−2.48419	−1.43425	−0.717123	−	0.696947i	$-0.754541\pi$
$3$	−2.48419	−1.43425	−0.717123	+	0.696947i	$0.754541\pi$
$4$	−1.53362	−0.766810
$5$	0	0
$5$	0	0
$6$	1.69650	0.692594
$7$	−0.856948	−0.323896	−0.161948	−	0.986799i	$-0.551778\pi$
$7$	−0.856948	−0.323896	−0.161948	+	0.986799i	$0.551778\pi$
$8$	2.41318	0.853188
$9$	3.17118	1.05706
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	3.80980	1.09979
$13$	−5.50630	−1.52717	−0.763586	−	0.645706i	$-0.776563\pi$
$13$	−5.50630	−1.52717	−0.763586	+	0.645706i	$0.776563\pi$
$14$	0.585228	0.156409
$15$	0	0
$16$	1.41923	0.354807
$17$	3.50630	0.850402	0.425201	−	0.905099i	$-0.360203\pi$
$17$	3.50630	0.850402	0.425201	+	0.905099i	$0.360203\pi$
$18$	−2.16566	−0.510452
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.12882	0.464546
$22$	−0.682920	−0.145599
$23$	−6.75237	−1.40797	−0.703984	−	0.710216i	$-0.748597\pi$
$23$	−6.75237	−1.40797	−0.703984	+	0.710216i	$0.748597\pi$
$24$	−5.99479	−1.22368
$25$	0	0
$26$	3.76036	0.737468
$27$	−0.425246	−0.0818387
$28$	1.31423	0.248367
$29$	−0.474680	−0.0881458	−0.0440729	−	0.999028i	$-0.514033\pi$
$29$	−0.474680	−0.0881458	−0.0440729	+	0.999028i	$0.514033\pi$
$30$	0	0
$31$	−5.19745	−0.933489	−0.466745	−	0.884392i	$-0.654573\pi$
$31$	−5.19745	−0.933489	−0.466745	+	0.884392i	$0.654573\pi$
$32$	−5.79558	−1.02452
$33$	−2.48419	−0.432441
$34$	−2.39452	−0.410657
$35$	0	0
$36$	−4.86339	−0.810564
$37$	−0.420904	−0.0691961	−0.0345981	−	0.999401i	$-0.511015\pi$
$37$	−0.420904	−0.0691961	−0.0345981	+	0.999401i	$0.511015\pi$
$38$	−0.682920	−0.110784
$39$	13.6787	2.19034
$40$	0	0
$41$	8.25141	1.28865	0.644327	−	0.764750i	$-0.277138\pi$
$41$	8.25141	1.28865	0.644327	+	0.764750i	$0.277138\pi$
$42$	−1.45381	−0.224328
$43$	1.53320	0.233811	0.116905	−	0.993143i	$-0.462703\pi$
$43$	1.53320	0.233811	0.116905	+	0.993143i	$0.462703\pi$
$44$	−1.53362	−0.231202
$45$	0	0
$46$	4.61133	0.679904
$47$	−2.48849	−0.362984	−0.181492	−	0.983392i	$-0.558093\pi$
$47$	−2.48849	−0.362984	−0.181492	+	0.983392i	$0.558093\pi$
$48$	−3.52563	−0.508881
$49$	−6.26564	−0.895091
$50$	0	0
$51$	−8.71030	−1.21969
$52$	8.44457	1.17105
$53$	4.79753	0.658991	0.329495	−	0.944157i	$-0.393121\pi$
$53$	4.79753	0.658991	0.329495	+	0.944157i	$0.393121\pi$
$54$	0.290409	0.0395197
$55$	0	0
$56$	−2.06797	−0.276344
$57$	−2.48419	−0.329039
$58$	0.324169	0.0425654
$59$	1.75917	0.229025	0.114513	−	0.993422i	$-0.463469\pi$
$59$	1.75917	0.229025	0.114513	+	0.993422i	$0.463469\pi$
$60$	0	0
$61$	3.42433	0.438440	0.219220	−	0.975675i	$-0.429649\pi$
$61$	3.42433	0.438440	0.219220	+	0.975675i	$0.429649\pi$
$62$	3.54944	0.450780
$63$	−2.71754	−0.342378
$64$	1.11946	0.139933
$65$	0	0
$66$	1.69650	0.208825
$67$	5.12424	0.626026	0.313013	−	0.949749i	$-0.398662\pi$
$67$	5.12424	0.626026	0.313013	+	0.949749i	$0.398662\pi$
$68$	−5.37733	−0.652097
$69$	16.7742	2.01937
$70$	0	0
$71$	−12.1937	−1.44713	−0.723563	−	0.690258i	$-0.757497\pi$
$71$	−12.1937	−1.44713	−0.723563	+	0.690258i	$0.757497\pi$
$72$	7.65263	0.901872
$73$	12.7638	1.49389	0.746945	−	0.664886i	$-0.231520\pi$
$73$	12.7638	1.49389	0.746945	+	0.664886i	$0.231520\pi$
$74$	0.287444	0.0334146
$75$	0	0
$76$	−1.53362	−0.175918
$77$	−0.856948	−0.0976583
$78$	−9.34144	−1.05771
$79$	10.8691	1.22287	0.611437	−	0.791293i	$-0.290592\pi$
$79$	10.8691	1.22287	0.611437	+	0.791293i	$0.290592\pi$
$80$	0	0
$81$	−8.45715	−0.939684
$82$	−5.63505	−0.622288
$83$	−2.94669	−0.323441	−0.161721	−	0.986837i	$-0.551704\pi$
$83$	−2.94669	−0.323441	−0.161721	+	0.986837i	$0.551704\pi$
$84$	−3.26480	−0.356219
$85$	0	0
$86$	−1.04705	−0.112907
$87$	1.17919	0.126423
$88$	2.41318	0.257246
$89$	3.08736	0.327259	0.163629	−	0.986522i	$-0.447680\pi$
$89$	3.08736	0.327259	0.163629	+	0.986522i	$0.447680\pi$
$90$	0	0
$91$	4.71861	0.494645
$92$	10.3556	1.07964
$93$	12.9114	1.33885
$94$	1.69944	0.175284
$95$	0	0
$96$	14.3973	1.46942
$97$	8.88210	0.901840	0.450920	−	0.892564i	$-0.351096\pi$
$97$	8.88210	0.901840	0.450920	+	0.892564i	$0.351096\pi$
$98$	4.27893	0.432238
$99$	3.17118	0.318716
$100$	0	0
$101$	−2.09965	−0.208923	−0.104462	−	0.994529i	$-0.533312\pi$
$101$	−2.09965	−0.208923	−0.104462	+	0.994529i	$0.533312\pi$
$102$	5.94844	0.588983
$103$	3.91556	0.385812	0.192906	−	0.981217i	$-0.438209\pi$
$103$	3.91556	0.385812	0.192906	+	0.981217i	$0.438209\pi$
$104$	−13.2877	−1.30297
$105$	0	0
$106$	−3.27633	−0.318225
$107$	18.0439	1.74437	0.872184	−	0.489178i	$-0.162703\pi$
$107$	18.0439	1.74437	0.872184	+	0.489178i	$0.162703\pi$
$108$	0.652166	0.0627547
$109$	3.34558	0.320448	0.160224	−	0.987081i	$-0.448778\pi$
$109$	3.34558	0.320448	0.160224	+	0.987081i	$0.448778\pi$
$110$	0	0
$111$	1.04560	0.0992442
$112$	−1.21621	−0.114921
$113$	−7.26819	−0.683733	−0.341867	−	0.939748i	$-0.611059\pi$
$113$	−7.26819	−0.683733	−0.341867	+	0.939748i	$0.611059\pi$
$114$	1.69650	0.158892
$115$	0	0
$116$	0.727978	0.0675911
$117$	−17.4615	−1.61431
$118$	−1.20138	−0.110596
$119$	−3.00472	−0.275442
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.33854	−0.211722
$123$	−20.4980	−1.84825
$124$	7.97091	0.715809
$125$	0	0
$126$	1.85586	0.165333
$127$	16.9303	1.50232	0.751162	−	0.660118i	$-0.229494\pi$
$127$	16.9303	1.50232	0.751162	+	0.660118i	$0.229494\pi$
$128$	10.8267	0.956951
$129$	−3.80875	−0.335342
$130$	0	0
$131$	5.26619	0.460109	0.230055	−	0.973178i	$-0.426109\pi$
$131$	5.26619	0.460109	0.230055	+	0.973178i	$0.426109\pi$
$132$	3.80980	0.331600
$133$	−0.856948	−0.0743069
$134$	−3.49945	−0.302307
$135$	0	0
$136$	8.46133	0.725553
$137$	20.6078	1.76064	0.880321	−	0.474378i	$-0.157327\pi$
$137$	20.6078	1.76064	0.880321	+	0.474378i	$0.157327\pi$
$138$	−11.4554	−0.975149
$139$	1.26192	0.107034	0.0535171	−	0.998567i	$-0.482957\pi$
$139$	1.26192	0.107034	0.0535171	+	0.998567i	$0.482957\pi$
$140$	0	0
$141$	6.18188	0.520608
$142$	8.32733	0.698814
$143$	−5.50630	−0.460460
$144$	4.50063	0.375053
$145$	0	0
$146$	−8.71666	−0.721396
$147$	15.5650	1.28378
$148$	0.645506	0.0530603
$149$	−11.3117	−0.926692	−0.463346	−	0.886177i	$-0.653351\pi$
$149$	−11.3117	−0.926692	−0.463346	+	0.886177i	$0.653351\pi$
$150$	0	0
$151$	−3.70074	−0.301162	−0.150581	−	0.988598i	$-0.548114\pi$
$151$	−3.70074	−0.301162	−0.150581	+	0.988598i	$0.548114\pi$
$152$	2.41318	0.195735
$153$	11.1191	0.898927
$154$	0.585228	0.0471590
$155$	0	0
$156$	−20.9779	−1.67957
$157$	−2.91493	−0.232637	−0.116318	−	0.993212i	$-0.537109\pi$
$157$	−2.91493	−0.232637	−0.116318	+	0.993212i	$0.537109\pi$
$158$	−7.42276	−0.590523
$159$	−11.9179	−0.945155
$160$	0	0
$161$	5.78644	0.456035
$162$	5.77556	0.453771
$163$	−18.4278	−1.44338	−0.721690	−	0.692216i	$-0.756634\pi$
$163$	−18.4278	−1.44338	−0.721690	+	0.692216i	$0.756634\pi$
$164$	−12.6545	−0.988152
$165$	0	0
$166$	2.01235	0.156189
$167$	−15.1433	−1.17182	−0.585912	−	0.810374i	$-0.699264\pi$
$167$	−15.1433	−1.17182	−0.585912	+	0.810374i	$0.699264\pi$
$168$	5.13723	0.396346
$169$	17.3193	1.33226
$170$	0	0
$171$	3.17118	0.242506
$172$	−2.35135	−0.179288
$173$	0.110615	0.00840989	0.00420495	−	0.999991i	$-0.498662\pi$
$173$	0.110615	0.00840989	0.00420495	+	0.999991i	$0.498662\pi$
$174$	−0.805295	−0.0610493
$175$	0	0
$176$	1.41923	0.106978
$177$	−4.37012	−0.328478
$178$	−2.10842	−0.158033
$179$	−9.42389	−0.704375	−0.352187	−	0.935930i	$-0.614562\pi$
$179$	−9.42389	−0.704375	−0.352187	+	0.935930i	$0.614562\pi$
$180$	0	0
$181$	−12.2876	−0.913334	−0.456667	−	0.889638i	$-0.650957\pi$
$181$	−12.2876	−0.913334	−0.456667	+	0.889638i	$0.650957\pi$
$182$	−3.22244	−0.238863
$183$	−8.50667	−0.628831
$184$	−16.2947	−1.20126
$185$	0	0
$186$	−8.81748	−0.646529
$187$	3.50630	0.256406
$188$	3.81640	0.278340
$189$	0.364414	0.0265072
$190$	0	0
$191$	16.4558	1.19070	0.595351	−	0.803466i	$-0.297013\pi$
$191$	16.4558	1.19070	0.595351	+	0.803466i	$0.297013\pi$
$192$	−2.78096	−0.200698
$193$	−2.79958	−0.201518	−0.100759	−	0.994911i	$-0.532127\pi$
$193$	−2.79958	−0.201518	−0.100759	+	0.994911i	$0.532127\pi$
$194$	−6.06577	−0.435497
$195$	0	0
$196$	9.60911	0.686365
$197$	7.19454	0.512589	0.256295	−	0.966599i	$-0.417498\pi$
$197$	7.19454	0.512589	0.256295	+	0.966599i	$0.417498\pi$
$198$	−2.16566	−0.153907
$199$	−11.9287	−0.845607	−0.422803	−	0.906221i	$-0.638954\pi$
$199$	−11.9287	−0.845607	−0.422803	+	0.906221i	$0.638954\pi$
$200$	0	0
$201$	−12.7296	−0.897875
$202$	1.43390	0.100889
$203$	0.406776	0.0285501
$204$	13.3583	0.935267
$205$	0	0
$206$	−2.67402	−0.186308
$207$	−21.4130	−1.48831
$208$	−7.81470	−0.541852
$209$	1.00000	0.0691714
$210$	0	0
$211$	18.0430	1.24213	0.621065	−	0.783759i	$-0.286700\pi$
$211$	18.0430	1.24213	0.621065	+	0.783759i	$0.286700\pi$
$212$	−7.35758	−0.505321
$213$	30.2914	2.07553
$214$	−12.3225	−0.842351
$215$	0	0
$216$	−1.02620	−0.0698238
$217$	4.45395	0.302354
$218$	−2.28476	−0.154744
$219$	−31.7077	−2.14261
$220$	0	0
$221$	−19.3067	−1.29871
$222$	−0.714064	−0.0479248
$223$	24.5001	1.64065	0.820323	−	0.571901i	$-0.193794\pi$
$223$	24.5001	1.64065	0.820323	+	0.571901i	$0.193794\pi$
$224$	4.96652	0.331839
$225$	0	0
$226$	4.96359	0.330173
$227$	−16.6675	−1.10626	−0.553131	−	0.833094i	$-0.686567\pi$
$227$	−16.6675	−1.10626	−0.553131	+	0.833094i	$0.686567\pi$
$228$	3.80980	0.252310
$229$	14.8344	0.980282	0.490141	−	0.871643i	$-0.336945\pi$
$229$	14.8344	0.980282	0.490141	+	0.871643i	$0.336945\pi$
$230$	0	0
$231$	2.12882	0.140066
$232$	−1.14549	−0.0752050
$233$	−28.8509	−1.89009	−0.945043	−	0.326947i	$-0.893980\pi$
$233$	−28.8509	−1.89009	−0.945043	+	0.326947i	$0.893980\pi$
$234$	11.9248	0.779548
$235$	0	0
$236$	−2.69790	−0.175619
$237$	−27.0010	−1.75390
$238$	2.05198	0.133010
$239$	8.29173	0.536347	0.268174	−	0.963371i	$-0.413580\pi$
$239$	8.29173	0.536347	0.268174	+	0.963371i	$0.413580\pi$
$240$	0	0
$241$	3.48268	0.224339	0.112170	−	0.993689i	$-0.464220\pi$
$241$	3.48268	0.224339	0.112170	+	0.993689i	$0.464220\pi$
$242$	−0.682920	−0.0438998
$243$	22.2849	1.42958
$244$	−5.25162	−0.336200
$245$	0	0
$246$	13.9985	0.892513
$247$	−5.50630	−0.350357
$248$	−12.5424	−0.796442
$249$	7.32013	0.463894
$250$	0	0
$251$	−28.8156	−1.81883	−0.909413	−	0.415895i	$-0.863468\pi$
$251$	−28.8156	−1.81883	−0.909413	+	0.415895i	$0.863468\pi$
$252$	4.16767	0.262539
$253$	−6.75237	−0.424518
$254$	−11.5621	−0.725469
$255$	0	0
$256$	−9.63267	−0.602042
$257$	8.44177	0.526583	0.263292	−	0.964716i	$-0.415192\pi$
$257$	8.44177	0.526583	0.263292	+	0.964716i	$0.415192\pi$
$258$	2.60108	0.161936
$259$	0.360693	0.0224124
$260$	0	0
$261$	−1.50530	−0.0931755
$262$	−3.59639	−0.222186
$263$	−21.4210	−1.32088	−0.660438	−	0.750881i	$-0.729629\pi$
$263$	−21.4210	−1.32088	−0.660438	+	0.750881i	$0.729629\pi$
$264$	−5.99479	−0.368954
$265$	0	0
$266$	0.585228	0.0358826
$267$	−7.66957	−0.469370
$268$	−7.85864	−0.480043
$269$	10.3642	0.631916	0.315958	−	0.948773i	$-0.397674\pi$
$269$	10.3642	0.631916	0.315958	+	0.948773i	$0.397674\pi$
$270$	0	0
$271$	−26.1079	−1.58594	−0.792972	−	0.609258i	$-0.791467\pi$
$271$	−26.1079	−1.58594	−0.792972	+	0.609258i	$0.791467\pi$
$272$	4.97624	0.301729
$273$	−11.7219	−0.709443
$274$	−14.0735	−0.850210
$275$	0	0
$276$	−25.7252	−1.54847
$277$	18.7663	1.12756	0.563779	−	0.825925i	$-0.309347\pi$
$277$	18.7663	1.12756	0.563779	+	0.825925i	$0.309347\pi$
$278$	−0.861788	−0.0516866
$279$	−16.4821	−0.986755
$280$	0	0
$281$	24.6322	1.46944	0.734718	−	0.678373i	$-0.237315\pi$
$281$	24.6322	1.46944	0.734718	+	0.678373i	$0.237315\pi$
$282$	−4.22173	−0.251400
$283$	13.7870	0.819550	0.409775	−	0.912187i	$-0.365607\pi$
$283$	13.7870	0.819550	0.409775	+	0.912187i	$0.365607\pi$
$284$	18.7005	1.10967
$285$	0	0
$286$	3.76036	0.222355
$287$	−7.07103	−0.417390
$288$	−18.3788	−1.08298
$289$	−4.70587	−0.276816
$290$	0	0
$291$	−22.0648	−1.29346
$292$	−19.5748	−1.14553
$293$	3.53033	0.206244	0.103122	−	0.994669i	$-0.467117\pi$
$293$	3.53033	0.206244	0.103122	+	0.994669i	$0.467117\pi$
$294$	−10.6297	−0.619935
$295$	0	0
$296$	−1.01572	−0.0590373
$297$	−0.425246	−0.0246753
$298$	7.72501	0.447498
$299$	37.1806	2.15021
$300$	0	0
$301$	−1.31387	−0.0757304
$302$	2.52731	0.145430
$303$	5.21593	0.299647
$304$	1.41923	0.0813984
$305$	0	0
$306$	−7.59347	−0.434090
$307$	−5.59192	−0.319148	−0.159574	−	0.987186i	$-0.551012\pi$
$307$	−5.59192	−0.319148	−0.159574	+	0.987186i	$0.551012\pi$
$308$	1.31423	0.0748854
$309$	−9.72699	−0.553349
$310$	0	0
$311$	−3.38109	−0.191724	−0.0958621	−	0.995395i	$-0.530561\pi$
$311$	−3.38109	−0.191724	−0.0958621	+	0.995395i	$0.530561\pi$
$312$	33.0091	1.86877
$313$	−13.5676	−0.766886	−0.383443	−	0.923565i	$-0.625262\pi$
$313$	−13.5676	−0.766886	−0.383443	+	0.923565i	$0.625262\pi$
$314$	1.99066	0.112340
$315$	0	0
$316$	−16.6691	−0.937712
$317$	9.02947	0.507145	0.253573	−	0.967316i	$-0.418394\pi$
$317$	9.02947	0.507145	0.253573	+	0.967316i	$0.418394\pi$
$318$	8.13901	0.456413
$319$	−0.474680	−0.0265770
$320$	0	0
$321$	−44.8244	−2.50185
$322$	−3.95167	−0.220218
$323$	3.50630	0.195096
$324$	12.9701	0.720559
$325$	0	0
$326$	12.5847	0.697005
$327$	−8.31104	−0.459602
$328$	19.9121	1.09946
$329$	2.13251	0.117569
$330$	0	0
$331$	−18.6966	−1.02766	−0.513829	−	0.857892i	$-0.671774\pi$
$331$	−18.6966	−1.02766	−0.513829	+	0.857892i	$0.671774\pi$
$332$	4.51910	0.248018
$333$	−1.33476	−0.0731445
$334$	10.3417	0.565871
$335$	0	0
$336$	3.02128	0.164824
$337$	−29.3264	−1.59751	−0.798755	−	0.601657i	$-0.794508\pi$
$337$	−29.3264	−1.59751	−0.798755	+	0.601657i	$0.794508\pi$
$338$	−11.8277	−0.643343
$339$	18.0555	0.980642
$340$	0	0
$341$	−5.19745	−0.281458
$342$	−2.16566	−0.117106
$343$	11.3680	0.613813
$344$	3.69989	0.199485
$345$	0	0
$346$	−0.0755411	−0.00406112
$347$	−11.0048	−0.590768	−0.295384	−	0.955379i	$-0.595448\pi$
$347$	−11.0048	−0.590768	−0.295384	+	0.955379i	$0.595448\pi$
$348$	−1.80843	−0.0969422
$349$	−24.1430	−1.29235	−0.646174	−	0.763190i	$-0.723632\pi$
$349$	−24.1430	−1.29235	−0.646174	+	0.763190i	$0.723632\pi$
$350$	0	0
$351$	2.34153	0.124982
$352$	−5.79558	−0.308906
$353$	−13.1068	−0.697602	−0.348801	−	0.937197i	$-0.613411\pi$
$353$	−13.1068	−0.697602	−0.348801	+	0.937197i	$0.613411\pi$
$354$	2.98444	0.158621
$355$	0	0
$356$	−4.73483	−0.250945
$357$	7.46428	0.395051
$358$	6.43577	0.340141
$359$	−11.2347	−0.592942	−0.296471	−	0.955042i	$-0.595810\pi$
$359$	−11.2347	−0.592942	−0.296471	+	0.955042i	$0.595810\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	8.39149	0.441047
$363$	−2.48419	−0.130386
$364$	−7.23656	−0.379299
$365$	0	0
$366$	5.80938	0.303661
$367$	−0.796321	−0.0415676	−0.0207838	−	0.999784i	$-0.506616\pi$
$367$	−0.796321	−0.0415676	−0.0207838	+	0.999784i	$0.506616\pi$
$368$	−9.58316	−0.499557
$369$	26.1667	1.36218
$370$	0	0
$371$	−4.11123	−0.213445
$372$	−19.8012	−1.02665
$373$	−4.22370	−0.218695	−0.109347	−	0.994004i	$-0.534876\pi$
$373$	−4.22370	−0.218695	−0.109347	+	0.994004i	$0.534876\pi$
$374$	−2.39452	−0.123818
$375$	0	0
$376$	−6.00518	−0.309694
$377$	2.61373	0.134614
$378$	−0.248866	−0.0128003
$379$	−22.5751	−1.15961	−0.579803	−	0.814757i	$-0.696870\pi$
$379$	−22.5751	−1.15961	−0.579803	+	0.814757i	$0.696870\pi$
$380$	0	0
$381$	−42.0581	−2.15470
$382$	−11.2380	−0.574987
$383$	−32.5159	−1.66148	−0.830742	−	0.556658i	$-0.812083\pi$
$383$	−32.5159	−1.66148	−0.830742	+	0.556658i	$0.812083\pi$
$384$	−26.8954	−1.37250
$385$	0	0
$386$	1.91189	0.0973128
$387$	4.86206	0.247152
$388$	−13.6218	−0.691540
$389$	−35.4867	−1.79925	−0.899624	−	0.436665i	$-0.856160\pi$
$389$	−35.4867	−1.79925	−0.899624	+	0.436665i	$0.856160\pi$
$390$	0	0
$391$	−23.6758	−1.19734
$392$	−15.1201	−0.763681
$393$	−13.0822	−0.659910
$394$	−4.91329	−0.247528
$395$	0	0
$396$	−4.86339	−0.244394
$397$	−11.7223	−0.588325	−0.294162	−	0.955755i	$-0.595041\pi$
$397$	−11.7223	−0.588325	−0.294162	+	0.955755i	$0.595041\pi$
$398$	8.14639	0.408341
$399$	2.12882	0.106574
$400$	0	0
$401$	−33.5093	−1.67337	−0.836687	−	0.547681i	$-0.815511\pi$
$401$	−33.5093	−1.67337	−0.836687	+	0.547681i	$0.815511\pi$
$402$	8.69329	0.433582
$403$	28.6187	1.42560
$404$	3.22007	0.160205
$405$	0	0
$406$	−0.277796	−0.0137868
$407$	−0.420904	−0.0208634
$408$	−21.0195	−1.04062
$409$	−6.50358	−0.321581	−0.160791	−	0.986989i	$-0.551404\pi$
$409$	−6.50358	−0.321581	−0.160791	+	0.986989i	$0.551404\pi$
$410$	0	0
$411$	−51.1936	−2.52519
$412$	−6.00499	−0.295844
$413$	−1.50752	−0.0741803
$414$	14.6234	0.718700
$415$	0	0
$416$	31.9122	1.56462
$417$	−3.13483	−0.153513
$418$	−0.682920	−0.0334027
$419$	−7.49790	−0.366296	−0.183148	−	0.983085i	$-0.558629\pi$
$419$	−7.49790	−0.366296	−0.183148	+	0.983085i	$0.558629\pi$
$420$	0	0
$421$	11.5280	0.561840	0.280920	−	0.959731i	$-0.409360\pi$
$421$	11.5280	0.561840	0.280920	+	0.959731i	$0.409360\pi$
$422$	−12.3219	−0.599822
$423$	−7.89146	−0.383696
$424$	11.5773	0.562243
$425$	0	0
$426$	−20.6866	−1.00227
$427$	−2.93447	−0.142009
$428$	−27.6724	−1.33760
$429$	13.6787	0.660412
$430$	0	0
$431$	22.4459	1.08118	0.540590	−	0.841286i	$-0.318201\pi$
$431$	22.4459	1.08118	0.540590	+	0.841286i	$0.318201\pi$
$432$	−0.603522	−0.0290370
$433$	27.4999	1.32156	0.660781	−	0.750579i	$-0.270225\pi$
$433$	27.4999	1.32156	0.660781	+	0.750579i	$0.270225\pi$
$434$	−3.04169	−0.146006
$435$	0	0
$436$	−5.13084	−0.245723
$437$	−6.75237	−0.323010
$438$	21.6538	1.03466
$439$	8.18489	0.390644	0.195322	−	0.980739i	$-0.437425\pi$
$439$	8.18489	0.390644	0.195322	+	0.980739i	$0.437425\pi$
$440$	0	0
$441$	−19.8695	−0.946166
$442$	13.1850	0.627144
$443$	11.1175	0.528209	0.264104	−	0.964494i	$-0.414924\pi$
$443$	11.1175	0.528209	0.264104	+	0.964494i	$0.414924\pi$
$444$	−1.60356	−0.0761015
$445$	0	0
$446$	−16.7316	−0.792264
$447$	28.1004	1.32910
$448$	−0.959323	−0.0453237
$449$	11.1426	0.525854	0.262927	−	0.964816i	$-0.415312\pi$
$449$	11.1426	0.525854	0.262927	+	0.964816i	$0.415312\pi$
$450$	0	0
$451$	8.25141	0.388544
$452$	11.1466	0.524294
$453$	9.19332	0.431940
$454$	11.3826	0.534212
$455$	0	0
$456$	−5.99479	−0.280732
$457$	−18.3241	−0.857164	−0.428582	−	0.903503i	$-0.640987\pi$
$457$	−18.3241	−0.857164	−0.428582	+	0.903503i	$0.640987\pi$
$458$	−10.1307	−0.473376
$459$	−1.49104	−0.0695958
$460$	0	0
$461$	31.6216	1.47276	0.736382	−	0.676566i	$-0.236533\pi$
$461$	31.6216	1.47276	0.736382	+	0.676566i	$0.236533\pi$
$462$	−1.45381	−0.0676376
$463$	7.76077	0.360674	0.180337	−	0.983605i	$-0.442281\pi$
$463$	7.76077	0.360674	0.180337	+	0.983605i	$0.442281\pi$
$464$	−0.673679	−0.0312748
$465$	0	0
$466$	19.7029	0.912718
$467$	−31.2308	−1.44519	−0.722594	−	0.691273i	$-0.757050\pi$
$467$	−31.2308	−1.44519	−0.722594	+	0.691273i	$0.757050\pi$
$468$	26.7793	1.23787
$469$	−4.39121	−0.202767
$470$	0	0
$471$	7.24122	0.333658
$472$	4.24521	0.195401
$473$	1.53320	0.0704966
$474$	18.4395	0.846955
$475$	0	0
$476$	4.60809	0.211212
$477$	15.2138	0.696593
$478$	−5.66259	−0.259001
$479$	−18.2169	−0.832353	−0.416177	−	0.909284i	$-0.636630\pi$
$479$	−18.2169	−0.832353	−0.416177	+	0.909284i	$0.636630\pi$
$480$	0	0
$481$	2.31762	0.105674
$482$	−2.37840	−0.108333
$483$	−14.3746	−0.654066
$484$	−1.53362	−0.0697100
$485$	0	0
$486$	−15.2188	−0.690339
$487$	−39.2194	−1.77720	−0.888600	−	0.458683i	$-0.848321\pi$
$487$	−39.2194	−1.77720	−0.888600	+	0.458683i	$0.848321\pi$
$488$	8.26352	0.374072
$489$	45.7782	2.07016
$490$	0	0
$491$	36.5478	1.64938	0.824688	−	0.565587i	$-0.191350\pi$
$491$	36.5478	1.64938	0.824688	+	0.565587i	$0.191350\pi$
$492$	31.4362	1.41725
$493$	−1.66437	−0.0749594
$494$	3.76036	0.169187
$495$	0	0
$496$	−7.37637	−0.331209
$497$	10.4494	0.468719
$498$	−4.99906	−0.224013
$499$	31.4475	1.40778	0.703892	−	0.710307i	$-0.251444\pi$
$499$	31.4475	1.40778	0.703892	+	0.710307i	$0.251444\pi$
$500$	0	0
$501$	37.6188	1.68068
$502$	19.6788	0.878307
$503$	−20.7020	−0.923057	−0.461528	−	0.887125i	$-0.652699\pi$
$503$	−20.7020	−0.923057	−0.461528	+	0.887125i	$0.652699\pi$
$504$	−6.55791	−0.292113
$505$	0	0
$506$	4.61133	0.204999
$507$	−43.0244	−1.91078
$508$	−25.9647	−1.15200
$509$	23.5196	1.04249	0.521245	−	0.853407i	$-0.325468\pi$
$509$	23.5196	1.04249	0.521245	+	0.853407i	$0.325468\pi$
$510$	0	0
$511$	−10.9379	−0.483865
$512$	−15.0750	−0.666226
$513$	−0.425246	−0.0187751
$514$	−5.76506	−0.254286
$515$	0	0
$516$	5.84118	0.257144
$517$	−2.48849	−0.109444
$518$	−0.246324	−0.0108229
$519$	−0.274788	−0.0120619
$520$	0	0
$521$	28.6887	1.25687	0.628437	−	0.777860i	$-0.283695\pi$
$521$	28.6887	1.25687	0.628437	+	0.777860i	$0.283695\pi$
$522$	1.02800	0.0449942
$523$	−5.70672	−0.249537	−0.124769	−	0.992186i	$-0.539819\pi$
$523$	−5.70672	−0.249537	−0.124769	+	0.992186i	$0.539819\pi$
$524$	−8.07634	−0.352816
$525$	0	0
$526$	14.6288	0.637848
$527$	−18.2238	−0.793842
$528$	−3.52563	−0.153433
$529$	22.5945	0.982371
$530$	0	0
$531$	5.57866	0.242093
$532$	1.31423	0.0569792
$533$	−45.4347	−1.96800
$534$	5.23770	0.226658
$535$	0	0
$536$	12.3657	0.534118
$537$	23.4107	1.01025
$538$	−7.07792	−0.305151
$539$	−6.26564	−0.269880
$540$	0	0
$541$	3.50382	0.150641	0.0753205	−	0.997159i	$-0.476002\pi$
$541$	3.50382	0.150641	0.0753205	+	0.997159i	$0.476002\pi$
$542$	17.8296	0.765849
$543$	30.5248	1.30994
$544$	−20.3210	−0.871257
$545$	0	0
$546$	8.00514	0.342588
$547$	−28.3788	−1.21339	−0.606696	−	0.794934i	$-0.707505\pi$
$547$	−28.3788	−1.21339	−0.606696	+	0.794934i	$0.707505\pi$
$548$	−31.6045	−1.35008
$549$	10.8592	0.463458
$550$	0	0
$551$	−0.474680	−0.0202220
$552$	40.4791	1.72290
$553$	−9.31430	−0.396084
$554$	−12.8159	−0.544495
$555$	0	0
$556$	−1.93530	−0.0820749
$557$	0.440370	0.0186591	0.00932953	−	0.999956i	$-0.497030\pi$
$557$	0.440370	0.0186591	0.00932953	+	0.999956i	$0.497030\pi$
$558$	11.2559	0.476502
$559$	−8.44226	−0.357069
$560$	0	0
$561$	−8.71030	−0.367749
$562$	−16.8219	−0.709587
$563$	−44.5147	−1.87607	−0.938036	−	0.346537i	$-0.887358\pi$
$563$	−44.5147	−1.87607	−0.938036	+	0.346537i	$0.887358\pi$
$564$	−9.48065	−0.399207
$565$	0	0
$566$	−9.41540	−0.395759
$567$	7.24734	0.304360
$568$	−29.4256	−1.23467
$569$	5.32611	0.223282	0.111641	−	0.993749i	$-0.464389\pi$
$569$	5.32611	0.223282	0.111641	+	0.993749i	$0.464389\pi$
$570$	0	0
$571$	11.2850	0.472264	0.236132	−	0.971721i	$-0.424120\pi$
$571$	11.2850	0.472264	0.236132	+	0.971721i	$0.424120\pi$
$572$	8.44457	0.353085
$573$	−40.8793	−1.70776
$574$	4.82895	0.201557
$575$	0	0
$576$	3.55002	0.147918
$577$	−44.9805	−1.87256	−0.936281	−	0.351251i	$-0.885757\pi$
$577$	−44.9805	−1.87256	−0.936281	+	0.351251i	$0.885757\pi$
$578$	3.21374	0.133674
$579$	6.95469	0.289027
$580$	0	0
$581$	2.52516	0.104761
$582$	15.0685	0.624609
$583$	4.79753	0.198693
$584$	30.8014	1.27457
$585$	0	0
$586$	−2.41094	−0.0995948
$587$	12.8765	0.531470	0.265735	−	0.964046i	$-0.414385\pi$
$587$	12.8765	0.531470	0.265735	+	0.964046i	$0.414385\pi$
$588$	−23.8708	−0.984416
$589$	−5.19745	−0.214157
$590$	0	0
$591$	−17.8726	−0.735179
$592$	−0.597359	−0.0245513
$593$	25.1716	1.03367	0.516837	−	0.856084i	$-0.327109\pi$
$593$	25.1716	1.03367	0.516837	+	0.856084i	$0.327109\pi$
$594$	0.290409	0.0119156
$595$	0	0
$596$	17.3479	0.710597
$597$	29.6332	1.21281
$598$	−25.3914	−1.03833
$599$	30.2614	1.23645	0.618224	−	0.786002i	$-0.287853\pi$
$599$	30.2614	1.23645	0.618224	+	0.786002i	$0.287853\pi$
$600$	0	0
$601$	33.1672	1.35292	0.676458	−	0.736481i	$-0.263514\pi$
$601$	33.1672	1.35292	0.676458	+	0.736481i	$0.263514\pi$
$602$	0.897271	0.0365700
$603$	16.2499	0.661747
$604$	5.67552	0.230934
$605$	0	0
$606$	−3.56207	−0.144699
$607$	10.4625	0.424660	0.212330	−	0.977198i	$-0.431895\pi$
$607$	10.4625	0.424660	0.212330	+	0.977198i	$0.431895\pi$
$608$	−5.79558	−0.235042
$609$	−1.01051	−0.0409478
$610$	0	0
$611$	13.7024	0.554339
$612$	−17.0525	−0.689306
$613$	9.24317	0.373328	0.186664	−	0.982424i	$-0.440232\pi$
$613$	9.24317	0.373328	0.186664	+	0.982424i	$0.440232\pi$
$614$	3.81884	0.154116
$615$	0	0
$616$	−2.06797	−0.0833210
$617$	1.95689	0.0787814	0.0393907	−	0.999224i	$-0.487458\pi$
$617$	1.95689	0.0787814	0.0393907	+	0.999224i	$0.487458\pi$
$618$	6.64276	0.267211
$619$	−48.3987	−1.94531	−0.972653	−	0.232263i	$-0.925387\pi$
$619$	−48.3987	−1.94531	−0.972653	+	0.232263i	$0.925387\pi$
$620$	0	0
$621$	2.87142	0.115226
$622$	2.30902	0.0925832
$623$	−2.64570	−0.105998
$624$	19.4132	0.777149
$625$	0	0
$626$	9.26559	0.370327
$627$	−2.48419	−0.0992088
$628$	4.47039	0.178388
$629$	−1.47581	−0.0588445
$630$	0	0
$631$	30.8039	1.22628	0.613141	−	0.789973i	$-0.289906\pi$
$631$	30.8039	1.22628	0.613141	+	0.789973i	$0.289906\pi$
$632$	26.2292	1.04334
$633$	−44.8222	−1.78152
$634$	−6.16641	−0.244899
$635$	0	0
$636$	18.2776	0.724754
$637$	34.5005	1.36696
$638$	0.324169	0.0128340
$639$	−38.6684	−1.52970
$640$	0	0
$641$	−7.12059	−0.281246	−0.140623	−	0.990063i	$-0.544911\pi$
$641$	−7.12059	−0.281246	−0.140623	+	0.990063i	$0.544911\pi$
$642$	30.6115	1.20814
$643$	43.8135	1.72783	0.863917	−	0.503634i	$-0.168004\pi$
$643$	43.8135	1.72783	0.863917	+	0.503634i	$0.168004\pi$
$644$	−8.87419	−0.349692
$645$	0	0
$646$	−2.39452	−0.0942112
$647$	−19.4584	−0.764988	−0.382494	−	0.923958i	$-0.624935\pi$
$647$	−19.4584	−0.764988	−0.382494	+	0.923958i	$0.624935\pi$
$648$	−20.4086	−0.801727
$649$	1.75917	0.0690536
$650$	0	0
$651$	−11.0644	−0.433649
$652$	28.2613	1.10680
$653$	35.4429	1.38699	0.693493	−	0.720463i	$-0.256071\pi$
$653$	35.4429	1.38699	0.693493	+	0.720463i	$0.256071\pi$
$654$	5.67578	0.221940
$655$	0	0
$656$	11.7106	0.457224
$657$	40.4763	1.57913
$658$	−1.45633	−0.0567738
$659$	−7.91185	−0.308202	−0.154101	−	0.988055i	$-0.549248\pi$
$659$	−7.91185	−0.308202	−0.154101	+	0.988055i	$0.549248\pi$
$660$	0	0
$661$	−37.9285	−1.47525	−0.737625	−	0.675211i	$-0.764053\pi$
$661$	−37.9285	−1.47525	−0.737625	+	0.675211i	$0.764053\pi$
$662$	12.7683	0.496254
$663$	47.9615	1.86267
$664$	−7.11090	−0.275956
$665$	0	0
$666$	0.911536	0.0353213
$667$	3.20522	0.124106
$668$	23.2241	0.898567
$669$	−60.8627	−2.35309
$670$	0	0
$671$	3.42433	0.132195
$672$	−12.3377	−0.475939
$673$	5.19834	0.200381	0.100191	−	0.994968i	$-0.468055\pi$
$673$	5.19834	0.200381	0.100191	+	0.994968i	$0.468055\pi$
$674$	20.0276	0.771434
$675$	0	0
$676$	−26.5613	−1.02159
$677$	12.7010	0.488137	0.244069	−	0.969758i	$-0.421518\pi$
$677$	12.7010	0.488137	0.244069	+	0.969758i	$0.421518\pi$
$678$	−12.3305	−0.473550
$679$	−7.61150	−0.292103
$680$	0	0
$681$	41.4052	1.58665
$682$	3.54944	0.135915
$683$	−45.3282	−1.73444	−0.867219	−	0.497928i	$-0.834095\pi$
$683$	−45.3282	−1.73444	−0.867219	+	0.497928i	$0.834095\pi$
$684$	−4.86339	−0.185956
$685$	0	0
$686$	−7.76342	−0.296409
$687$	−36.8513	−1.40597
$688$	2.17596	0.0829578
$689$	−26.4166	−1.00639
$690$	0	0
$691$	30.5149	1.16084	0.580421	−	0.814317i	$-0.302888\pi$
$691$	30.5149	1.16084	0.580421	+	0.814317i	$0.302888\pi$
$692$	−0.169641	−0.00644879
$693$	−2.71754	−0.103231
$694$	7.51539	0.285280
$695$	0	0
$696$	2.84561	0.107862
$697$	28.9319	1.09587
$698$	16.4878	0.624072
$699$	71.6710	2.71085
$700$	0	0
$701$	0.621590	0.0234771	0.0117386	−	0.999931i	$-0.496263\pi$
$701$	0.621590	0.0234771	0.0117386	+	0.999931i	$0.496263\pi$
$702$	−1.59908	−0.0603534
$703$	−0.420904	−0.0158747
$704$	1.11946	0.0421914
$705$	0	0
$706$	8.95087	0.336870
$707$	1.79930	0.0676695
$708$	6.70210	0.251880
$709$	−21.9142	−0.823004	−0.411502	−	0.911409i	$-0.634996\pi$
$709$	−21.9142	−0.823004	−0.411502	+	0.911409i	$0.634996\pi$
$710$	0	0
$711$	34.4680	1.29265
$712$	7.45035	0.279214
$713$	35.0951	1.31432
$714$	−5.09751	−0.190769
$715$	0	0
$716$	14.4527	0.540121
$717$	−20.5982	−0.769254
$718$	7.67238	0.286331
$719$	−43.0880	−1.60691	−0.803455	−	0.595365i	$-0.797007\pi$
$719$	−43.0880	−1.60691	−0.803455	+	0.595365i	$0.797007\pi$
$720$	0	0
$721$	−3.35544	−0.124963
$722$	−0.682920	−0.0254157
$723$	−8.65163	−0.321758
$724$	18.8446	0.700353
$725$	0	0
$726$	1.69650	0.0629631
$727$	−30.8306	−1.14344	−0.571721	−	0.820448i	$-0.693724\pi$
$727$	−30.8306	−1.14344	−0.571721	+	0.820448i	$0.693724\pi$
$728$	11.3869	0.422025
$729$	−29.9883	−1.11068
$730$	0	0
$731$	5.37586	0.198833
$732$	13.0460	0.482194
$733$	17.6688	0.652610	0.326305	−	0.945264i	$-0.394196\pi$
$733$	17.6688	0.652610	0.326305	+	0.945264i	$0.394196\pi$
$734$	0.543824	0.0200729
$735$	0	0
$736$	39.1339	1.44250
$737$	5.12424	0.188754
$738$	−17.8698	−0.657796
$739$	−6.72966	−0.247555	−0.123777	−	0.992310i	$-0.539501\pi$
$739$	−6.72966	−0.247555	−0.123777	+	0.992310i	$0.539501\pi$
$740$	0	0
$741$	13.6787	0.502499
$742$	2.80764	0.103072
$743$	−48.4780	−1.77848	−0.889242	−	0.457437i	$-0.848767\pi$
$743$	−48.4780	−1.77848	−0.889242	+	0.457437i	$0.848767\pi$
$744$	31.1576	1.14229
$745$	0	0
$746$	2.88445	0.105607
$747$	−9.34449	−0.341897
$748$	−5.37733	−0.196615
$749$	−15.4627	−0.564994
$750$	0	0
$751$	−36.2684	−1.32345	−0.661726	−	0.749746i	$-0.730176\pi$
$751$	−36.2684	−1.32345	−0.661726	+	0.749746i	$0.730176\pi$
$752$	−3.53174	−0.128789
$753$	71.5834	2.60864
$754$	−1.78497	−0.0650047
$755$	0	0
$756$	−0.558873	−0.0203260
$757$	5.13939	0.186794	0.0933972	−	0.995629i	$-0.470227\pi$
$757$	5.13939	0.186794	0.0933972	+	0.995629i	$0.470227\pi$
$758$	15.4170	0.559971
$759$	16.7742	0.608863
$760$	0	0
$761$	10.2119	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$761$	10.2119	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$762$	28.7223	1.04050
$763$	−2.86699	−0.103792
$764$	−25.2370	−0.913042
$765$	0	0
$766$	22.2057	0.802326
$767$	−9.68654	−0.349761
$768$	23.9294	0.863476
$769$	−49.2015	−1.77425	−0.887126	−	0.461527i	$-0.847302\pi$
$769$	−49.2015	−1.77425	−0.887126	+	0.461527i	$0.847302\pi$
$770$	0	0
$771$	−20.9709	−0.755249
$772$	4.29350	0.154526
$773$	40.7240	1.46474	0.732371	−	0.680906i	$-0.238414\pi$
$773$	40.7240	1.46474	0.732371	+	0.680906i	$0.238414\pi$
$774$	−3.32040	−0.119349
$775$	0	0
$776$	21.4341	0.769440
$777$	−0.896028	−0.0321448
$778$	24.2346	0.868853
$779$	8.25141	0.295637
$780$	0	0
$781$	−12.1937	−0.436325
$782$	16.1687	0.578192
$783$	0.201856	0.00721374
$784$	−8.89238	−0.317585
$785$	0	0
$786$	8.93410	0.318669
$787$	−7.23417	−0.257870	−0.128935	−	0.991653i	$-0.541156\pi$
$787$	−7.23417	−0.257870	−0.128935	+	0.991653i	$0.541156\pi$
$788$	−11.0337	−0.393059
$789$	53.2138	1.89446
$790$	0	0
$791$	6.22846	0.221459
$792$	7.65263	0.271925
$793$	−18.8554	−0.669574
$794$	8.00539	0.284101
$795$	0	0
$796$	18.2942	0.648420
$797$	−43.9784	−1.55779	−0.778897	−	0.627152i	$-0.784220\pi$
$797$	−43.9784	−1.55779	−0.778897	+	0.627152i	$0.784220\pi$
$798$	−1.45381	−0.0514645
$799$	−8.72540	−0.308682
$800$	0	0
$801$	9.79056	0.345933
$802$	22.8842	0.808068
$803$	12.7638	0.450425
$804$	19.5223	0.688500
$805$	0	0
$806$	−19.5443	−0.688419
$807$	−25.7466	−0.906323
$808$	−5.06685	−0.178251
$809$	11.9798	0.421187	0.210593	−	0.977574i	$-0.432460\pi$
$809$	11.9798	0.421187	0.210593	+	0.977574i	$0.432460\pi$
$810$	0	0
$811$	30.0059	1.05365	0.526825	−	0.849974i	$-0.323382\pi$
$811$	30.0059	1.05365	0.526825	+	0.849974i	$0.323382\pi$
$812$	−0.623840	−0.0218925
$813$	64.8570	2.27463
$814$	0.287444	0.0100749
$815$	0	0
$816$	−12.3619	−0.432753
$817$	1.53320	0.0536399
$818$	4.44143	0.155291
$819$	14.9636	0.522870
$820$	0	0
$821$	26.5123	0.925284	0.462642	−	0.886545i	$-0.346901\pi$
$821$	26.5123	0.925284	0.462642	+	0.886545i	$0.346901\pi$
$822$	34.9611	1.21941
$823$	−13.2244	−0.460972	−0.230486	−	0.973076i	$-0.574032\pi$
$823$	−13.2244	−0.460972	−0.230486	+	0.973076i	$0.574032\pi$
$824$	9.44896	0.329170
$825$	0	0
$826$	1.02952	0.0358215
$827$	−25.4465	−0.884861	−0.442431	−	0.896803i	$-0.645884\pi$
$827$	−25.4465	−0.884861	−0.442431	+	0.896803i	$0.645884\pi$
$828$	32.8394	1.14125
$829$	−28.6999	−0.996789	−0.498395	−	0.866950i	$-0.666077\pi$
$829$	−28.6999	−0.996789	−0.498395	+	0.866950i	$0.666077\pi$
$830$	0	0
$831$	−46.6190	−1.61720
$832$	−6.16410	−0.213702
$833$	−21.9692	−0.761188
$834$	2.14084	0.0741313
$835$	0	0
$836$	−1.53362	−0.0530413
$837$	2.21020	0.0763956
$838$	5.12047	0.176884
$839$	−4.10582	−0.141749	−0.0708743	−	0.997485i	$-0.522579\pi$
$839$	−4.10582	−0.141749	−0.0708743	+	0.997485i	$0.522579\pi$
$840$	0	0
$841$	−28.7747	−0.992230
$842$	−7.87271	−0.271311
$843$	−61.1911	−2.10753
$844$	−27.6711	−0.952478
$845$	0	0
$846$	5.38924	0.185286
$847$	−0.856948	−0.0294451
$848$	6.80879	0.233815
$849$	−34.2494	−1.17544
$850$	0	0
$851$	2.84210	0.0974259
$852$	−46.4555	−1.59154
$853$	51.2689	1.75541	0.877707	−	0.479198i	$-0.159072\pi$
$853$	51.2689	1.75541	0.877707	+	0.479198i	$0.159072\pi$
$854$	2.00401	0.0685758
$855$	0	0
$856$	43.5431	1.48827
$857$	34.2836	1.17110	0.585552	−	0.810635i	$-0.300878\pi$
$857$	34.2836	1.17110	0.585552	+	0.810635i	$0.300878\pi$
$858$	−9.34144	−0.318912
$859$	39.5431	1.34919	0.674596	−	0.738187i	$-0.264318\pi$
$859$	39.5431	1.34919	0.674596	+	0.738187i	$0.264318\pi$
$860$	0	0
$861$	17.5658	0.598639
$862$	−15.3288	−0.522100
$863$	13.1157	0.446462	0.223231	−	0.974766i	$-0.428340\pi$
$863$	13.1157	0.446462	0.223231	+	0.974766i	$0.428340\pi$
$864$	2.46455	0.0838457
$865$	0	0
$866$	−18.7802	−0.638179
$867$	11.6903	0.397022
$868$	−6.83066	−0.231848
$869$	10.8691	0.368711
$870$	0	0
$871$	−28.2156	−0.956050
$872$	8.07349	0.273403
$873$	28.1667	0.953300
$874$	4.61133	0.155981
$875$	0	0
$876$	48.6275	1.64297
$877$	34.1036	1.15160	0.575798	−	0.817592i	$-0.304691\pi$
$877$	34.1036	1.15160	0.575798	+	0.817592i	$0.304691\pi$
$878$	−5.58963	−0.188641
$879$	−8.77000	−0.295805
$880$	0	0
$881$	39.1343	1.31847	0.659234	−	0.751938i	$-0.270881\pi$
$881$	39.1343	1.31847	0.659234	+	0.751938i	$0.270881\pi$
$882$	13.5693	0.456901
$883$	−44.1281	−1.48503	−0.742515	−	0.669829i	$-0.766367\pi$
$883$	−44.1281	−1.48503	−0.742515	+	0.669829i	$0.766367\pi$
$884$	29.6092	0.995864
$885$	0	0
$886$	−7.59238	−0.255071
$887$	48.8255	1.63940	0.819700	−	0.572793i	$-0.194140\pi$
$887$	48.8255	1.63940	0.819700	+	0.572793i	$0.194140\pi$
$888$	2.52323	0.0846740
$889$	−14.5084	−0.486597
$890$	0	0
$891$	−8.45715	−0.283325
$892$	−37.5738	−1.25806
$893$	−2.48849	−0.0832742
$894$	−19.1904	−0.641821
$895$	0	0
$896$	−9.27789	−0.309953
$897$	−92.3635	−3.08393
$898$	−7.60954	−0.253934
$899$	2.46712	0.0822832
$900$	0	0
$901$	16.8216	0.560407
$902$	−5.63505	−0.187627
$903$	3.26391	0.108616
$904$	−17.5394	−0.583353
$905$	0	0
$906$	−6.27830	−0.208583
$907$	25.4676	0.845638	0.422819	−	0.906214i	$-0.361041\pi$
$907$	25.4676	0.845638	0.422819	+	0.906214i	$0.361041\pi$
$908$	25.5617	0.848293
$909$	−6.65838	−0.220845
$910$	0	0
$911$	−23.4140	−0.775740	−0.387870	−	0.921714i	$-0.626789\pi$
$911$	−23.4140	−0.775740	−0.387870	+	0.921714i	$0.626789\pi$
$912$	−3.52563	−0.116745
$913$	−2.94669	−0.0975212
$914$	12.5139	0.413923
$915$	0	0
$916$	−22.7503	−0.751690
$917$	−4.51286	−0.149028
$918$	1.01826	0.0336077
$919$	−18.7511	−0.618542	−0.309271	−	0.950974i	$-0.600085\pi$
$919$	−18.7511	−0.618542	−0.309271	+	0.950974i	$0.600085\pi$
$920$	0	0
$921$	13.8914	0.457736
$922$	−21.5950	−0.711194
$923$	67.1422	2.21001
$924$	−3.26480	−0.107404
$925$	0	0
$926$	−5.29999	−0.174169
$927$	12.4170	0.407827
$928$	2.75105	0.0903075
$929$	40.7701	1.33762	0.668812	−	0.743432i	$-0.266803\pi$
$929$	40.7701	1.33762	0.668812	+	0.743432i	$0.266803\pi$
$930$	0	0
$931$	−6.26564	−0.205348
$932$	44.2463	1.44934
$933$	8.39927	0.274980
$934$	21.3281	0.697878
$935$	0	0
$936$	−42.1377	−1.37731
$937$	6.54322	0.213758	0.106879	−	0.994272i	$-0.465914\pi$
$937$	6.54322	0.213758	0.106879	+	0.994272i	$0.465914\pi$
$938$	2.99885	0.0979159
$939$	33.7044	1.09990
$940$	0	0
$941$	−34.5714	−1.12700	−0.563498	−	0.826117i	$-0.690545\pi$
$941$	−34.5714	−1.12700	−0.563498	+	0.826117i	$0.690545\pi$
$942$	−4.94518	−0.161123
$943$	−55.7166	−1.81438
$944$	2.49667	0.0812597
$945$	0	0
$946$	−1.04705	−0.0340427
$947$	−17.6584	−0.573822	−0.286911	−	0.957957i	$-0.592628\pi$
$947$	−17.6584	−0.573822	−0.286911	+	0.957957i	$0.592628\pi$
$948$	41.4092	1.34491
$949$	−70.2813	−2.28143
$950$	0	0
$951$	−22.4309	−0.727371
$952$	−7.25093	−0.235004
$953$	42.7340	1.38429	0.692145	−	0.721758i	$-0.256666\pi$
$953$	42.7340	1.38429	0.692145	+	0.721758i	$0.256666\pi$
$954$	−10.3898	−0.336383
$955$	0	0
$956$	−12.7164	−0.411276
$957$	1.17919	0.0381179
$958$	12.4407	0.401941
$959$	−17.6598	−0.570265
$960$	0	0
$961$	−3.98652	−0.128597
$962$	−1.58275	−0.0510299
$963$	57.2204	1.84390
$964$	−5.34111	−0.172026
$965$	0	0
$966$	9.81670	0.315847
$967$	3.62541	0.116585	0.0582927	−	0.998300i	$-0.481434\pi$
$967$	3.62541	0.116585	0.0582927	+	0.998300i	$0.481434\pi$
$968$	2.41318	0.0775626
$969$	−8.71030	−0.279815
$970$	0	0
$971$	2.86101	0.0918142	0.0459071	−	0.998946i	$-0.485382\pi$
$971$	2.86101	0.0918142	0.0459071	+	0.998946i	$0.485382\pi$
$972$	−34.1765	−1.09621
$973$	−1.08140	−0.0346680
$974$	26.7837	0.858206
$975$	0	0
$976$	4.85990	0.155562
$977$	−29.4990	−0.943756	−0.471878	−	0.881664i	$-0.656424\pi$
$977$	−29.4990	−0.943756	−0.471878	+	0.881664i	$0.656424\pi$
$978$	−31.2629	−0.999676
$979$	3.08736	0.0986723
$980$	0	0
$981$	10.6094	0.338733
$982$	−24.9592	−0.796480
$983$	−31.1737	−0.994287	−0.497144	−	0.867668i	$-0.665618\pi$
$983$	−31.1737	−0.994287	−0.497144	+	0.867668i	$0.665618\pi$
$984$	−49.4655	−1.57690
$985$	0	0
$986$	1.13663	0.0361977
$987$	−5.29755	−0.168623
$988$	8.44457	0.268657
$989$	−10.3527	−0.329198
$990$	0	0
$991$	−55.9145	−1.77618	−0.888092	−	0.459666i	$-0.847969\pi$
$991$	−55.9145	−1.77618	−0.888092	+	0.459666i	$0.847969\pi$
$992$	30.1222	0.956382
$993$	46.4459	1.47392
$994$	−7.13609	−0.226343
$995$	0	0
$996$	−11.2263	−0.355719
$997$	−42.1768	−1.33575	−0.667877	−	0.744272i	$-0.732797\pi$
$997$	−42.1768	−1.33575	−0.667877	+	0.744272i	$0.732797\pi$
$998$	−21.4762	−0.679816
$999$	0.178988	0.00566292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.p.1.5		9
5.4	even	2		1045.2.a.k.1.5	✓	9
15.14	odd	2		9405.2.a.bh.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.k.1.5	✓	9	5.4	even	2
5225.2.a.p.1.5		9	1.1	even	1	trivial
9405.2.a.bh.1.5		9	15.14	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.682920 q^{2} -2.48419 q^{3} -1.53362 q^{4} +1.69650 q^{6} -0.856948 q^{7} +2.41318 q^{8} +3.17118 q^{9} +1.00000 q^{11} +3.80980 q^{12} -5.50630 q^{13} +0.585228 q^{14} +1.41923 q^{16} +3.50630 q^{17} -2.16566 q^{18} +1.00000 q^{19} +2.12882 q^{21} -0.682920 q^{22} -6.75237 q^{23} -5.99479 q^{24} +3.76036 q^{26} -0.425246 q^{27} +1.31423 q^{28} -0.474680 q^{29} -5.19745 q^{31} -5.79558 q^{32} -2.48419 q^{33} -2.39452 q^{34} -4.86339 q^{36} -0.420904 q^{37} -0.682920 q^{38} +13.6787 q^{39} +8.25141 q^{41} -1.45381 q^{42} +1.53320 q^{43} -1.53362 q^{44} +4.61133 q^{46} -2.48849 q^{47} -3.52563 q^{48} -6.26564 q^{49} -8.71030 q^{51} +8.44457 q^{52} +4.79753 q^{53} +0.290409 q^{54} -2.06797 q^{56} -2.48419 q^{57} +0.324169 q^{58} +1.75917 q^{59} +3.42433 q^{61} +3.54944 q^{62} -2.71754 q^{63} +1.11946 q^{64} +1.69650 q^{66} +5.12424 q^{67} -5.37733 q^{68} +16.7742 q^{69} -12.1937 q^{71} +7.65263 q^{72} +12.7638 q^{73} +0.287444 q^{74} -1.53362 q^{76} -0.856948 q^{77} -9.34144 q^{78} +10.8691 q^{79} -8.45715 q^{81} -5.63505 q^{82} -2.94669 q^{83} -3.26480 q^{84} -1.04705 q^{86} +1.17919 q^{87} +2.41318 q^{88} +3.08736 q^{89} +4.71861 q^{91} +10.3556 q^{92} +12.9114 q^{93} +1.69944 q^{94} +14.3973 q^{96} +8.88210 q^{97} +4.27893 q^{98} +3.17118 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{2} - 3 q^{3} + 9 q^{4} - 13 q^{7} - 9 q^{8} + 12 q^{9} + 9 q^{11} + 5 q^{12} - 5 q^{13} - 2 q^{14} + q^{16} - 13 q^{17} - q^{18} + 9 q^{19} + q^{21} - 3 q^{22} - 8 q^{23} - 7 q^{24} + 8 q^{26}+ \cdots + 12 q^{99}+O(q^{100}) \) 9 * q - 3 * q^2 - 3 * q^3 + 9 * q^4 - 13 * q^7 - 9 * q^8 + 12 * q^9 + 9 * q^11 + 5 * q^12 - 5 * q^13 - 2 * q^14 + q^16 - 13 * q^17 - q^18 + 9 * q^19 + q^21 - 3 * q^22 - 8 * q^23 - 7 * q^24 + 8 * q^26 - 6 * q^27 - 10 * q^28 - 3 * q^29 - 9 * q^31 - 6 * q^32 - 3 * q^33 - 2 * q^34 - 15 * q^36 + 7 * q^37 - 3 * q^38 - 2 * q^39 + 9 * q^41 + 9 * q^42 - 23 * q^43 + 9 * q^44 - 32 * q^46 - 20 * q^47 + 18 * q^48 - 4 * q^49 + 8 * q^51 - 9 * q^52 + 5 * q^53 + 9 * q^54 + 4 * q^56 - 3 * q^57 - 3 * q^58 + 19 * q^59 + q^61 - 18 * q^62 - 24 * q^63 + 23 * q^64 + 10 * q^67 - 9 * q^68 - 28 * q^69 - 24 * q^72 - 12 * q^73 + 5 * q^74 + 9 * q^76 - 13 * q^77 - 19 * q^78 - 21 * q^79 - 3 * q^81 + 14 * q^82 - 47 * q^83 - 11 * q^84 + 12 * q^86 + 2 * q^87 - 9 * q^88 - 2 * q^89 + 2 * q^91 + 19 * q^92 + 2 * q^93 + 26 * q^94 - 55 * q^96 + 32 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more