LMFDB - Embedded newform 5225.2.a.y.1.15

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 = [15,5,4,17,0,-1,21] 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:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 $ x^15 - 5x^14 - 11x^13 + 87x^12 - 4x^11 - 545x^10 + 431x^9 + 1480x^8 - 1763x^7 - 1609x^6 + 2516x^5 + 391x^4 - 1081x^3 - 45x^2 + 153x + 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$2.66335$ 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)

$f(q)$	$=$	\(q+2.66335 q^{2} -1.48323 q^{3} +5.09344 q^{4} -3.95037 q^{6} +2.27906 q^{7} +8.23893 q^{8} -0.800022 q^{9} +1.00000 q^{11} -7.55476 q^{12} +1.52624 q^{13} +6.06993 q^{14} +11.7563 q^{16} +3.39286 q^{17} -2.13074 q^{18} +1.00000 q^{19} -3.38037 q^{21} +2.66335 q^{22} -4.55963 q^{23} -12.2203 q^{24} +4.06493 q^{26} +5.63632 q^{27} +11.6082 q^{28} +9.48668 q^{29} +0.999697 q^{31} +14.8333 q^{32} -1.48323 q^{33} +9.03639 q^{34} -4.07487 q^{36} -7.35542 q^{37} +2.66335 q^{38} -2.26377 q^{39} -10.9835 q^{41} -9.00311 q^{42} +10.7429 q^{43} +5.09344 q^{44} -12.1439 q^{46} -12.5964 q^{47} -17.4373 q^{48} -1.80590 q^{49} -5.03241 q^{51} +7.77384 q^{52} +6.26507 q^{53} +15.0115 q^{54} +18.7770 q^{56} -1.48323 q^{57} +25.2664 q^{58} +12.8815 q^{59} +9.11146 q^{61} +2.66254 q^{62} -1.82330 q^{63} +15.9937 q^{64} -3.95037 q^{66} +1.97376 q^{67} +17.2814 q^{68} +6.76299 q^{69} -9.85507 q^{71} -6.59133 q^{72} +5.69357 q^{73} -19.5901 q^{74} +5.09344 q^{76} +2.27906 q^{77} -6.02923 q^{78} +13.5023 q^{79} -5.95990 q^{81} -29.2528 q^{82} -9.93334 q^{83} -17.2177 q^{84} +28.6122 q^{86} -14.0709 q^{87} +8.23893 q^{88} -3.48710 q^{89} +3.47840 q^{91} -23.2242 q^{92} -1.48278 q^{93} -33.5486 q^{94} -22.0012 q^{96} +6.80514 q^{97} -4.80975 q^{98} -0.800022 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) $ 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 21 * q^7 + 9 * q^8 + 15 * q^9 + 15 * q^11 + 11 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 17 * q^17 + 22 * q^18 + 15 * q^19 + 6 * q^21 + 5 * q^22 + 26 * q^23 + q^24 + 3 * q^26 + q^27 + 46 * q^28 + 9 * q^29 + 14 * q^31 + 18 * q^32 + 4 * q^33 - 13 * q^34 + 12 * q^36 + 9 * q^37 + 5 * q^38 - 22 * q^39 + 4 * q^41 - 6 * q^42 + 28 * q^43 + 17 * q^44 + 27 * q^46 + 14 * q^47 - 4 * q^48 + 32 * q^49 - 40 * q^51 + 14 * q^52 + 3 * q^53 - 39 * q^54 + 34 * q^56 + 4 * q^57 + 26 * q^58 + q^59 + 2 * q^61 - 3 * q^62 + 45 * q^63 + 5 * q^64 - q^66 + 37 * q^67 + 26 * q^68 - 7 * q^69 - 7 * q^71 + 16 * q^72 + 42 * q^73 - 43 * q^74 + 17 * q^76 + 21 * q^77 - 64 * q^78 - 10 * q^79 + 31 * q^81 + 22 * q^82 + 14 * q^83 - 32 * q^84 + 37 * q^86 + 29 * q^87 + 9 * q^88 + 15 * q^89 - 22 * q^91 + 26 * q^92 - 18 * q^93 - 44 * q^94 + 71 * q^96 + 8 * q^97 - 10 * q^98 + 15 * 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 $

$2$	2.66335	1.88327	0.941637	−	0.336629i	$-0.109287\pi$
$2$	2.66335	1.88327	0.941637	+	0.336629i	$0.109287\pi$
$3$	−1.48323	−0.856344	−0.428172	−	0.903697i	$-0.640842\pi$
$3$	−1.48323	−0.856344	−0.428172	+	0.903697i	$0.640842\pi$
$4$	5.09344	2.54672
$5$	0	0
$5$	0	0
$6$	−3.95037	−1.61273
$7$	2.27906	0.861402	0.430701	−	0.902495i	$-0.358266\pi$
$7$	2.27906	0.861402	0.430701	+	0.902495i	$0.358266\pi$
$8$	8.23893	2.91290
$9$	−0.800022	−0.266674
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−7.55476	−2.18087
$13$	1.52624	0.423304	0.211652	−	0.977345i	$-0.432116\pi$
$13$	1.52624	0.423304	0.211652	+	0.977345i	$0.432116\pi$
$14$	6.06993	1.62226
$15$	0	0
$16$	11.7563	2.93907
$17$	3.39286	0.822891	0.411445	−	0.911434i	$-0.365024\pi$
$17$	3.39286	0.822891	0.411445	+	0.911434i	$0.365024\pi$
$18$	−2.13074	−0.502221
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.38037	−0.737657
$22$	2.66335	0.567829
$23$	−4.55963	−0.950749	−0.475375	−	0.879783i	$-0.657687\pi$
$23$	−4.55963	−0.950749	−0.475375	+	0.879783i	$0.657687\pi$
$24$	−12.2203	−2.49445
$25$	0	0
$26$	4.06493	0.797198
$27$	5.63632	1.08471
$28$	11.6082	2.19375
$29$	9.48668	1.76163	0.880816	−	0.473459i	$-0.156995\pi$
$29$	9.48668	1.76163	0.880816	+	0.473459i	$0.156995\pi$
$30$	0	0
$31$	0.999697	0.179551	0.0897754	−	0.995962i	$-0.471385\pi$
$31$	0.999697	0.179551	0.0897754	+	0.995962i	$0.471385\pi$
$32$	14.8333	2.62218
$33$	−1.48323	−0.258198
$34$	9.03639	1.54973
$35$	0	0
$36$	−4.07487	−0.679145
$37$	−7.35542	−1.20922	−0.604611	−	0.796521i	$-0.706672\pi$
$37$	−7.35542	−1.20922	−0.604611	+	0.796521i	$0.706672\pi$
$38$	2.66335	0.432053
$39$	−2.26377	−0.362494
$40$	0	0
$41$	−10.9835	−1.71533	−0.857664	−	0.514210i	$-0.828085\pi$
$41$	−10.9835	−1.71533	−0.857664	+	0.514210i	$0.828085\pi$
$42$	−9.00311	−1.38921
$43$	10.7429	1.63828	0.819140	−	0.573593i	$-0.194451\pi$
$43$	10.7429	1.63828	0.819140	+	0.573593i	$0.194451\pi$
$44$	5.09344	0.767866
$45$	0	0
$46$	−12.1439	−1.79052
$47$	−12.5964	−1.83737	−0.918685	−	0.394991i	$-0.870748\pi$
$47$	−12.5964	−1.83737	−0.918685	+	0.394991i	$0.870748\pi$
$48$	−17.4373	−2.51686
$49$	−1.80590	−0.257986
$50$	0	0
$51$	−5.03241	−0.704678
$52$	7.77384	1.07804
$53$	6.26507	0.860573	0.430287	−	0.902692i	$-0.358413\pi$
$53$	6.26507	0.860573	0.430287	+	0.902692i	$0.358413\pi$
$54$	15.0115	2.04281
$55$	0	0
$56$	18.7770	2.50918
$57$	−1.48323	−0.196459
$58$	25.2664	3.31764
$59$	12.8815	1.67703	0.838516	−	0.544877i	$-0.183424\pi$
$59$	12.8815	1.67703	0.838516	+	0.544877i	$0.183424\pi$
$60$	0	0
$61$	9.11146	1.16660	0.583302	−	0.812256i	$-0.301761\pi$
$61$	9.11146	1.16660	0.583302	+	0.812256i	$0.301761\pi$
$62$	2.66254	0.338144
$63$	−1.82330	−0.229714
$64$	15.9937	1.99921
$65$	0	0
$66$	−3.95037	−0.486257
$67$	1.97376	0.241133	0.120566	−	0.992705i	$-0.461529\pi$
$67$	1.97376	0.241133	0.120566	+	0.992705i	$0.461529\pi$
$68$	17.2814	2.09567
$69$	6.76299	0.814169
$70$	0	0
$71$	−9.85507	−1.16958	−0.584791	−	0.811184i	$-0.698823\pi$
$71$	−9.85507	−1.16958	−0.584791	+	0.811184i	$0.698823\pi$
$72$	−6.59133	−0.776796
$73$	5.69357	0.666382	0.333191	−	0.942859i	$-0.391875\pi$
$73$	5.69357	0.666382	0.333191	+	0.942859i	$0.391875\pi$
$74$	−19.5901	−2.27730
$75$	0	0
$76$	5.09344	0.584258
$77$	2.27906	0.259723
$78$	−6.02923	−0.682676
$79$	13.5023	1.51913	0.759564	−	0.650433i	$-0.225412\pi$
$79$	13.5023	1.51913	0.759564	+	0.650433i	$0.225412\pi$
$80$	0	0
$81$	−5.95990	−0.662211
$82$	−29.2528	−3.23043
$83$	−9.93334	−1.09033	−0.545163	−	0.838330i	$-0.683532\pi$
$83$	−9.93334	−1.09033	−0.545163	+	0.838330i	$0.683532\pi$
$84$	−17.2177	−1.87861
$85$	0	0
$86$	28.6122	3.08533
$87$	−14.0709	−1.50856
$88$	8.23893	0.878273
$89$	−3.48710	−0.369632	−0.184816	−	0.982773i	$-0.559169\pi$
$89$	−3.48710	−0.369632	−0.184816	+	0.982773i	$0.559169\pi$
$90$	0	0
$91$	3.47840	0.364635
$92$	−23.2242	−2.42129
$93$	−1.48278	−0.153757
$94$	−33.5486	−3.46027
$95$	0	0
$96$	−22.0012	−2.24549
$97$	6.80514	0.690957	0.345478	−	0.938427i	$-0.387717\pi$
$97$	6.80514	0.690957	0.345478	+	0.938427i	$0.387717\pi$
$98$	−4.80975	−0.485859
$99$	−0.800022	−0.0804053
$100$	0	0
$101$	1.97228	0.196249	0.0981245	−	0.995174i	$-0.468716\pi$
$101$	1.97228	0.196249	0.0981245	+	0.995174i	$0.468716\pi$
$102$	−13.4031	−1.32710
$103$	−0.301074	−0.0296657	−0.0148329	−	0.999890i	$-0.504722\pi$
$103$	−0.301074	−0.0296657	−0.0148329	+	0.999890i	$0.504722\pi$
$104$	12.5746	1.23304
$105$	0	0
$106$	16.6861	1.62070
$107$	5.51379	0.533038	0.266519	−	0.963830i	$-0.414127\pi$
$107$	5.51379	0.533038	0.266519	+	0.963830i	$0.414127\pi$
$108$	28.7083	2.76245
$109$	−0.0672977	−0.00644595	−0.00322297	−	0.999995i	$-0.501026\pi$
$109$	−0.0672977	−0.00644595	−0.00322297	+	0.999995i	$0.501026\pi$
$110$	0	0
$111$	10.9098	1.03551
$112$	26.7933	2.53172
$113$	6.65021	0.625599	0.312800	−	0.949819i	$-0.398733\pi$
$113$	6.65021	0.625599	0.312800	+	0.949819i	$0.398733\pi$
$114$	−3.95037	−0.369986
$115$	0	0
$116$	48.3199	4.48639
$117$	−1.22103	−0.112884
$118$	34.3080	3.15831
$119$	7.73253	0.708840
$120$	0	0
$121$	1.00000	0.0909091
$122$	24.2670	2.19703
$123$	16.2910	1.46891
$124$	5.09190	0.457266
$125$	0	0
$126$	−4.85608	−0.432614
$127$	12.5167	1.11068	0.555339	−	0.831624i	$-0.312589\pi$
$127$	12.5167	1.11068	0.555339	+	0.831624i	$0.312589\pi$
$128$	12.9302	1.14288
$129$	−15.9342	−1.40293
$130$	0	0
$131$	−7.11523	−0.621660	−0.310830	−	0.950465i	$-0.600607\pi$
$131$	−7.11523	−0.621660	−0.310830	+	0.950465i	$0.600607\pi$
$132$	−7.55476	−0.657558
$133$	2.27906	0.197619
$134$	5.25681	0.454119
$135$	0	0
$136$	27.9536	2.39700
$137$	−7.95408	−0.679563	−0.339781	−	0.940504i	$-0.610353\pi$
$137$	−7.95408	−0.679563	−0.339781	+	0.940504i	$0.610353\pi$
$138$	18.0122	1.53330
$139$	−8.97715	−0.761432	−0.380716	−	0.924692i	$-0.624322\pi$
$139$	−8.97715	−0.761432	−0.380716	+	0.924692i	$0.624322\pi$
$140$	0	0
$141$	18.6833	1.57342
$142$	−26.2475	−2.20264
$143$	1.52624	0.127631
$144$	−9.40530	−0.783775
$145$	0	0
$146$	15.1640	1.25498
$147$	2.67857	0.220925
$148$	−37.4644	−3.07956
$149$	−16.2455	−1.33089	−0.665444	−	0.746448i	$-0.731758\pi$
$149$	−16.2455	−1.33089	−0.665444	+	0.746448i	$0.731758\pi$
$150$	0	0
$151$	−16.7242	−1.36100	−0.680499	−	0.732749i	$-0.738237\pi$
$151$	−16.7242	−1.36100	−0.680499	+	0.732749i	$0.738237\pi$
$152$	8.23893	0.668266
$153$	−2.71437	−0.219444
$154$	6.06993	0.489129
$155$	0	0
$156$	−11.5304	−0.923172
$157$	15.0972	1.20489	0.602444	−	0.798161i	$-0.294194\pi$
$157$	15.0972	1.20489	0.602444	+	0.798161i	$0.294194\pi$
$158$	35.9614	2.86093
$159$	−9.29255	−0.736947
$160$	0	0
$161$	−10.3917	−0.818978
$162$	−15.8733	−1.24712
$163$	23.8250	1.86612	0.933060	−	0.359721i	$-0.117128\pi$
$163$	23.8250	1.86612	0.933060	+	0.359721i	$0.117128\pi$
$164$	−55.9437	−4.36847
$165$	0	0
$166$	−26.4560	−2.05338
$167$	3.48426	0.269620	0.134810	−	0.990871i	$-0.456958\pi$
$167$	3.48426	0.269620	0.134810	+	0.990871i	$0.456958\pi$
$168$	−27.8506	−2.14872
$169$	−10.6706	−0.820814
$170$	0	0
$171$	−0.800022	−0.0611792
$172$	54.7185	4.17225
$173$	8.59439	0.653420	0.326710	−	0.945125i	$-0.394060\pi$
$173$	8.59439	0.653420	0.326710	+	0.945125i	$0.394060\pi$
$174$	−37.4759	−2.84104
$175$	0	0
$176$	11.7563	0.886164
$177$	−19.1063	−1.43612
$178$	−9.28737	−0.696118
$179$	16.8791	1.26160	0.630802	−	0.775944i	$-0.282726\pi$
$179$	16.8791	1.26160	0.630802	+	0.775944i	$0.282726\pi$
$180$	0	0
$181$	−12.8785	−0.957253	−0.478626	−	0.878019i	$-0.658865\pi$
$181$	−12.8785	−0.957253	−0.478626	+	0.878019i	$0.658865\pi$
$182$	9.26419	0.686708
$183$	−13.5144	−0.999014
$184$	−37.5665	−2.76944
$185$	0	0
$186$	−3.94917	−0.289567
$187$	3.39286	0.248111
$188$	−64.1589	−4.67927
$189$	12.8455	0.934371
$190$	0	0
$191$	−9.90809	−0.716924	−0.358462	−	0.933544i	$-0.616699\pi$
$191$	−9.90809	−0.716924	−0.358462	+	0.933544i	$0.616699\pi$
$192$	−23.7223	−1.71201
$193$	9.46456	0.681274	0.340637	−	0.940195i	$-0.389357\pi$
$193$	9.46456	0.681274	0.340637	+	0.940195i	$0.389357\pi$
$194$	18.1245	1.30126
$195$	0	0
$196$	−9.19826	−0.657019
$197$	−2.58989	−0.184522	−0.0922610	−	0.995735i	$-0.529409\pi$
$197$	−2.58989	−0.184522	−0.0922610	+	0.995735i	$0.529409\pi$
$198$	−2.13074	−0.151425
$199$	−7.62690	−0.540656	−0.270328	−	0.962768i	$-0.587132\pi$
$199$	−7.62690	−0.540656	−0.270328	+	0.962768i	$0.587132\pi$
$200$	0	0
$201$	−2.92754	−0.206493
$202$	5.25287	0.369591
$203$	21.6207	1.51747
$204$	−25.6323	−1.79462
$205$	0	0
$206$	−0.801867	−0.0558687
$207$	3.64781	0.253540
$208$	17.9430	1.24412
$209$	1.00000	0.0691714
$210$	0	0
$211$	−24.0159	−1.65332	−0.826662	−	0.562700i	$-0.809763\pi$
$211$	−24.0159	−1.65332	−0.826662	+	0.562700i	$0.809763\pi$
$212$	31.9108	2.19164
$213$	14.6173	1.00156
$214$	14.6852	1.00386
$215$	0	0
$216$	46.4372	3.15965
$217$	2.27837	0.154666
$218$	−0.179237	−0.0121395
$219$	−8.44489	−0.570653
$220$	0	0
$221$	5.17834	0.348333
$222$	29.0566	1.95015
$223$	−5.88063	−0.393796	−0.196898	−	0.980424i	$-0.563087\pi$
$223$	−5.88063	−0.393796	−0.196898	+	0.980424i	$0.563087\pi$
$224$	33.8059	2.25875
$225$	0	0
$226$	17.7119	1.17818
$227$	−15.1534	−1.00577	−0.502883	−	0.864355i	$-0.667727\pi$
$227$	−15.1534	−1.00577	−0.502883	+	0.864355i	$0.667727\pi$
$228$	−7.55476	−0.500326
$229$	7.95745	0.525843	0.262921	−	0.964817i	$-0.415314\pi$
$229$	7.95745	0.525843	0.262921	+	0.964817i	$0.415314\pi$
$230$	0	0
$231$	−3.38037	−0.222412
$232$	78.1601	5.13146
$233$	−1.24861	−0.0817991	−0.0408996	−	0.999163i	$-0.513022\pi$
$233$	−1.24861	−0.0817991	−0.0408996	+	0.999163i	$0.513022\pi$
$234$	−3.25203	−0.212592
$235$	0	0
$236$	65.6114	4.27094
$237$	−20.0270	−1.30090
$238$	20.5945	1.33494
$239$	17.1689	1.11056	0.555282	−	0.831662i	$-0.312610\pi$
$239$	17.1689	1.11056	0.555282	+	0.831662i	$0.312610\pi$
$240$	0	0
$241$	−8.48920	−0.546837	−0.273419	−	0.961895i	$-0.588154\pi$
$241$	−8.48920	−0.546837	−0.273419	+	0.961895i	$0.588154\pi$
$242$	2.66335	0.171207
$243$	−8.06904	−0.517629
$244$	46.4087	2.97102
$245$	0	0
$246$	43.3887	2.76636
$247$	1.52624	0.0971126
$248$	8.23644	0.523014
$249$	14.7334	0.933694
$250$	0	0
$251$	−8.61021	−0.543472	−0.271736	−	0.962372i	$-0.587598\pi$
$251$	−8.61021	−0.543472	−0.271736	+	0.962372i	$0.587598\pi$
$252$	−9.28686	−0.585017
$253$	−4.55963	−0.286662
$254$	33.3364	2.09171
$255$	0	0
$256$	2.45035	0.153147
$257$	−13.8864	−0.866213	−0.433106	−	0.901343i	$-0.642583\pi$
$257$	−13.8864	−0.866213	−0.433106	+	0.901343i	$0.642583\pi$
$258$	−42.4385	−2.64211
$259$	−16.7634	−1.04163
$260$	0	0
$261$	−7.58955	−0.469782
$262$	−18.9504	−1.17076
$263$	6.17658	0.380864	0.190432	−	0.981700i	$-0.439011\pi$
$263$	6.17658	0.380864	0.190432	+	0.981700i	$0.439011\pi$
$264$	−12.2203	−0.752104
$265$	0	0
$266$	6.06993	0.372171
$267$	5.17218	0.316532
$268$	10.0532	0.614098
$269$	−17.9268	−1.09301	−0.546507	−	0.837454i	$-0.684043\pi$
$269$	−17.9268	−1.09301	−0.546507	+	0.837454i	$0.684043\pi$
$270$	0	0
$271$	−19.1484	−1.16318	−0.581592	−	0.813481i	$-0.697570\pi$
$271$	−19.1484	−1.16318	−0.581592	+	0.813481i	$0.697570\pi$
$272$	39.8875	2.41854
$273$	−5.15927	−0.312253
$274$	−21.1845	−1.27980
$275$	0	0
$276$	34.4469	2.07346
$277$	12.7314	0.764955	0.382477	−	0.923965i	$-0.375071\pi$
$277$	12.7314	0.764955	0.382477	+	0.923965i	$0.375071\pi$
$278$	−23.9093	−1.43398
$279$	−0.799780	−0.0478816
$280$	0	0
$281$	12.6290	0.753381	0.376690	−	0.926339i	$-0.377062\pi$
$281$	12.6290	0.753381	0.376690	+	0.926339i	$0.377062\pi$
$282$	49.7603	2.96318
$283$	−12.3980	−0.736982	−0.368491	−	0.929631i	$-0.620125\pi$
$283$	−12.3980	−0.736982	−0.368491	+	0.929631i	$0.620125\pi$
$284$	−50.1962	−2.97860
$285$	0	0
$286$	4.06493	0.240364
$287$	−25.0319	−1.47759
$288$	−11.8670	−0.699267
$289$	−5.48847	−0.322851
$290$	0	0
$291$	−10.0936	−0.591697
$292$	28.9999	1.69709
$293$	−11.3319	−0.662016	−0.331008	−	0.943628i	$-0.607389\pi$
$293$	−11.3319	−0.662016	−0.331008	+	0.943628i	$0.607389\pi$
$294$	7.13398	0.416062
$295$	0	0
$296$	−60.6008	−3.52235
$297$	5.63632	0.327052
$298$	−43.2676	−2.50643
$299$	−6.95911	−0.402456
$300$	0	0
$301$	24.4837	1.41122
$302$	−44.5425	−2.56313
$303$	−2.92535	−0.168057
$304$	11.7563	0.674270
$305$	0	0
$306$	−7.22932	−0.413273
$307$	8.22400	0.469368	0.234684	−	0.972072i	$-0.424594\pi$
$307$	8.22400	0.469368	0.234684	+	0.972072i	$0.424594\pi$
$308$	11.6082	0.661441
$309$	0.446563	0.0254041
$310$	0	0
$311$	2.48566	0.140949	0.0704745	−	0.997514i	$-0.477549\pi$
$311$	2.48566	0.140949	0.0704745	+	0.997514i	$0.477549\pi$
$312$	−18.6511	−1.05591
$313$	21.7727	1.23067	0.615333	−	0.788267i	$-0.289022\pi$
$313$	21.7727	1.23067	0.615333	+	0.788267i	$0.289022\pi$
$314$	40.2092	2.26914
$315$	0	0
$316$	68.7732	3.86880
$317$	4.23879	0.238074	0.119037	−	0.992890i	$-0.462019\pi$
$317$	4.23879	0.238074	0.119037	+	0.992890i	$0.462019\pi$
$318$	−24.7493	−1.38787
$319$	9.48668	0.531152
$320$	0	0
$321$	−8.17823	−0.456464
$322$	−27.6767	−1.54236
$323$	3.39286	0.188784
$324$	−30.3564	−1.68647
$325$	0	0
$326$	63.4544	3.51442
$327$	0.0998181	0.00551995
$328$	−90.4920	−4.99658
$329$	−28.7078	−1.58271
$330$	0	0
$331$	27.9785	1.53784	0.768919	−	0.639346i	$-0.220795\pi$
$331$	27.9785	1.53784	0.768919	+	0.639346i	$0.220795\pi$
$332$	−50.5949	−2.77676
$333$	5.88450	0.322468
$334$	9.27981	0.507768
$335$	0	0
$336$	−39.7406	−2.16803
$337$	−21.4572	−1.16885	−0.584423	−	0.811449i	$-0.698679\pi$
$337$	−21.4572	−1.16885	−0.584423	+	0.811449i	$0.698679\pi$
$338$	−28.4195	−1.54582
$339$	−9.86381	−0.535728
$340$	0	0
$341$	0.999697	0.0541366
$342$	−2.13074	−0.115217
$343$	−20.0691	−1.08363
$344$	88.5102	4.77215
$345$	0	0
$346$	22.8899	1.23057
$347$	31.2123	1.67556	0.837782	−	0.546005i	$-0.183852\pi$
$347$	31.2123	1.67556	0.837782	+	0.546005i	$0.183852\pi$
$348$	−71.6696	−3.84189
$349$	−23.4707	−1.25636	−0.628180	−	0.778068i	$-0.716200\pi$
$349$	−23.4707	−1.25636	−0.628180	+	0.778068i	$0.716200\pi$
$350$	0	0
$351$	8.60239	0.459162
$352$	14.8333	0.790616
$353$	13.5458	0.720971	0.360485	−	0.932765i	$-0.382611\pi$
$353$	13.5458	0.720971	0.360485	+	0.932765i	$0.382611\pi$
$354$	−50.8868	−2.70460
$355$	0	0
$356$	−17.7613	−0.941349
$357$	−11.4691	−0.607011
$358$	44.9550	2.37595
$359$	16.2248	0.856313	0.428156	−	0.903705i	$-0.359163\pi$
$359$	16.2248	0.856313	0.428156	+	0.903705i	$0.359163\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−34.3000	−1.80277
$363$	−1.48323	−0.0778495
$364$	17.7170	0.928624
$365$	0	0
$366$	−35.9937	−1.88142
$367$	5.01745	0.261909	0.130954	−	0.991388i	$-0.458196\pi$
$367$	5.01745	0.261909	0.130954	+	0.991388i	$0.458196\pi$
$368$	−53.6044	−2.79432
$369$	8.78702	0.457434
$370$	0	0
$371$	14.2784	0.741300
$372$	−7.55247	−0.391577
$373$	−33.2251	−1.72033	−0.860166	−	0.510015i	$-0.829640\pi$
$373$	−33.2251	−1.72033	−0.860166	+	0.510015i	$0.829640\pi$
$374$	9.03639	0.467261
$375$	0	0
$376$	−103.781	−5.35208
$377$	14.4790	0.745706
$378$	34.2120	1.75968
$379$	−13.0037	−0.667957	−0.333978	−	0.942581i	$-0.608391\pi$
$379$	−13.0037	−0.667957	−0.333978	+	0.942581i	$0.608391\pi$
$380$	0	0
$381$	−18.5652	−0.951122
$382$	−26.3887	−1.35016
$383$	−31.0556	−1.58687	−0.793435	−	0.608655i	$-0.791709\pi$
$383$	−31.0556	−1.58687	−0.793435	+	0.608655i	$0.791709\pi$
$384$	−19.1785	−0.978698
$385$	0	0
$386$	25.2074	1.28303
$387$	−8.59458	−0.436887
$388$	34.6616	1.75968
$389$	28.0459	1.42199	0.710993	−	0.703199i	$-0.248246\pi$
$389$	28.0459	1.42199	0.710993	+	0.703199i	$0.248246\pi$
$390$	0	0
$391$	−15.4702	−0.782363
$392$	−14.8787	−0.751488
$393$	10.5535	0.532355
$394$	−6.89779	−0.347505
$395$	0	0
$396$	−4.07487	−0.204770
$397$	31.5695	1.58443	0.792214	−	0.610243i	$-0.208928\pi$
$397$	31.5695	1.58443	0.792214	+	0.610243i	$0.208928\pi$
$398$	−20.3131	−1.01820
$399$	−3.38037	−0.169230
$400$	0	0
$401$	−28.1565	−1.40607	−0.703034	−	0.711156i	$-0.748172\pi$
$401$	−28.1565	−1.40607	−0.703034	+	0.711156i	$0.748172\pi$
$402$	−7.79707	−0.388882
$403$	1.52578	0.0760046
$404$	10.0457	0.499792
$405$	0	0
$406$	57.5835	2.85782
$407$	−7.35542	−0.364594
$408$	−41.4617	−2.05266
$409$	−15.1780	−0.750503	−0.375251	−	0.926923i	$-0.622444\pi$
$409$	−15.1780	−0.750503	−0.375251	+	0.926923i	$0.622444\pi$
$410$	0	0
$411$	11.7977	0.581940
$412$	−1.53351	−0.0755504
$413$	29.3577	1.44460
$414$	9.71540	0.477486
$415$	0	0
$416$	22.6392	1.10998
$417$	13.3152	0.652048
$418$	2.66335	0.130269
$419$	−35.4606	−1.73237	−0.866183	−	0.499728i	$-0.833433\pi$
$419$	−35.4606	−1.73237	−0.866183	+	0.499728i	$0.833433\pi$
$420$	0	0
$421$	11.9716	0.583458	0.291729	−	0.956501i	$-0.405769\pi$
$421$	11.9716	0.583458	0.291729	+	0.956501i	$0.405769\pi$
$422$	−63.9628	−3.11366
$423$	10.0774	0.489979
$424$	51.6175	2.50677
$425$	0	0
$426$	38.9312	1.88622
$427$	20.7655	1.00491
$428$	28.0842	1.35750
$429$	−2.26377	−0.109296
$430$	0	0
$431$	14.4099	0.694099	0.347050	−	0.937847i	$-0.387184\pi$
$431$	14.4099	0.694099	0.347050	+	0.937847i	$0.387184\pi$
$432$	66.2622	3.18804
$433$	−26.9997	−1.29752	−0.648761	−	0.760992i	$-0.724713\pi$
$433$	−26.9997	−1.29752	−0.648761	+	0.760992i	$0.724713\pi$
$434$	6.06809	0.291278
$435$	0	0
$436$	−0.342777	−0.0164160
$437$	−4.55963	−0.218117
$438$	−22.4917	−1.07470
$439$	−14.7306	−0.703055	−0.351527	−	0.936178i	$-0.614338\pi$
$439$	−14.7306	−0.703055	−0.351527	+	0.936178i	$0.614338\pi$
$440$	0	0
$441$	1.44476	0.0687982
$442$	13.7917	0.656006
$443$	3.56318	0.169292	0.0846460	−	0.996411i	$-0.473024\pi$
$443$	3.56318	0.169292	0.0846460	+	0.996411i	$0.473024\pi$
$444$	55.5684	2.63716
$445$	0	0
$446$	−15.6622	−0.741626
$447$	24.0959	1.13970
$448$	36.4505	1.72212
$449$	−2.98503	−0.140872	−0.0704362	−	0.997516i	$-0.522439\pi$
$449$	−2.98503	−0.140872	−0.0704362	+	0.997516i	$0.522439\pi$
$450$	0	0
$451$	−10.9835	−0.517191
$452$	33.8725	1.59323
$453$	24.8059	1.16548
$454$	−40.3588	−1.89413
$455$	0	0
$456$	−12.2203	−0.572266
$457$	−7.09332	−0.331811	−0.165906	−	0.986142i	$-0.553055\pi$
$457$	−7.09332	−0.331811	−0.165906	+	0.986142i	$0.553055\pi$
$458$	21.1935	0.990306
$459$	19.1233	0.892597
$460$	0	0
$461$	−25.7684	−1.20015	−0.600077	−	0.799943i	$-0.704863\pi$
$461$	−25.7684	−1.20015	−0.600077	+	0.799943i	$0.704863\pi$
$462$	−9.00311	−0.418863
$463$	−35.3532	−1.64300	−0.821502	−	0.570206i	$-0.806863\pi$
$463$	−35.3532	−1.64300	−0.821502	+	0.570206i	$0.806863\pi$
$464$	111.528	5.17756
$465$	0	0
$466$	−3.32549	−0.154050
$467$	−31.6989	−1.46685	−0.733425	−	0.679770i	$-0.762079\pi$
$467$	−31.6989	−1.46685	−0.733425	+	0.679770i	$0.762079\pi$
$468$	−6.21925	−0.287485
$469$	4.49830	0.207712
$470$	0	0
$471$	−22.3927	−1.03180
$472$	106.130	4.88503
$473$	10.7429	0.493960
$474$	−53.3391	−2.44995
$475$	0	0
$476$	39.3852	1.80522
$477$	−5.01219	−0.229493
$478$	45.7268	2.09150
$479$	−1.37837	−0.0629795	−0.0314898	−	0.999504i	$-0.510025\pi$
$479$	−1.37837	−0.0629795	−0.0314898	+	0.999504i	$0.510025\pi$
$480$	0	0
$481$	−11.2262	−0.511869
$482$	−22.6097	−1.02984
$483$	15.4132	0.701327
$484$	5.09344	0.231520
$485$	0	0
$486$	−21.4907	−0.974837
$487$	29.4556	1.33476	0.667380	−	0.744717i	$-0.267416\pi$
$487$	29.4556	1.33476	0.667380	+	0.744717i	$0.267416\pi$
$488$	75.0688	3.39820
$489$	−35.3380	−1.59804
$490$	0	0
$491$	27.7324	1.25155	0.625773	−	0.780005i	$-0.284784\pi$
$491$	27.7324	1.25155	0.625773	+	0.780005i	$0.284784\pi$
$492$	82.9774	3.74091
$493$	32.1870	1.44963
$494$	4.06493	0.182890
$495$	0	0
$496$	11.7527	0.527713
$497$	−22.4602	−1.00748
$498$	39.2404	1.75840
$499$	−6.97004	−0.312022	−0.156011	−	0.987755i	$-0.549863\pi$
$499$	−6.97004	−0.312022	−0.156011	+	0.987755i	$0.549863\pi$
$500$	0	0
$501$	−5.16796	−0.230888
$502$	−22.9320	−1.02351
$503$	29.1479	1.29964	0.649820	−	0.760088i	$-0.274844\pi$
$503$	29.1479	1.29964	0.649820	+	0.760088i	$0.274844\pi$
$504$	−15.0220	−0.669134
$505$	0	0
$506$	−12.1439	−0.539863
$507$	15.8269	0.702899
$508$	63.7531	2.82859
$509$	−8.38114	−0.371488	−0.185744	−	0.982598i	$-0.559469\pi$
$509$	−8.38114	−0.371488	−0.185744	+	0.982598i	$0.559469\pi$
$510$	0	0
$511$	12.9760	0.574023
$512$	−19.3343	−0.854462
$513$	5.63632	0.248849
$514$	−36.9845	−1.63132
$515$	0	0
$516$	−81.1602	−3.57288
$517$	−12.5964	−0.553988
$518$	−44.6469	−1.96167
$519$	−12.7475	−0.559552
$520$	0	0
$521$	−34.6749	−1.51914	−0.759568	−	0.650428i	$-0.774590\pi$
$521$	−34.6749	−1.51914	−0.759568	+	0.650428i	$0.774590\pi$
$522$	−20.2137	−0.884728
$523$	12.2052	0.533697	0.266848	−	0.963739i	$-0.414018\pi$
$523$	12.2052	0.533697	0.266848	+	0.963739i	$0.414018\pi$
$524$	−36.2410	−1.58320
$525$	0	0
$526$	16.4504	0.717272
$527$	3.39184	0.147751
$528$	−17.4373	−0.758862
$529$	−2.20975	−0.0960760
$530$	0	0
$531$	−10.3055	−0.447221
$532$	11.6082	0.503281
$533$	−16.7634	−0.726105
$534$	13.7753	0.596117
$535$	0	0
$536$	16.2616	0.702396
$537$	−25.0356	−1.08037
$538$	−47.7453	−2.05845
$539$	−1.80590	−0.0777857
$540$	0	0
$541$	−15.8649	−0.682085	−0.341043	−	0.940048i	$-0.610780\pi$
$541$	−15.8649	−0.682085	−0.341043	+	0.940048i	$0.610780\pi$
$542$	−50.9990	−2.19059
$543$	19.1018	0.819738
$544$	50.3273	2.15777
$545$	0	0
$546$	−13.7410	−0.588058
$547$	−21.2077	−0.906775	−0.453387	−	0.891314i	$-0.649785\pi$
$547$	−21.2077	−0.906775	−0.453387	+	0.891314i	$0.649785\pi$
$548$	−40.5136	−1.73066
$549$	−7.28938	−0.311103
$550$	0	0
$551$	9.48668	0.404146
$552$	55.7199	2.37159
$553$	30.7725	1.30858
$554$	33.9082	1.44062
$555$	0	0
$556$	−45.7246	−1.93916
$557$	12.9761	0.549814	0.274907	−	0.961471i	$-0.411353\pi$
$557$	12.9761	0.549814	0.274907	+	0.961471i	$0.411353\pi$
$558$	−2.13010	−0.0901741
$559$	16.3963	0.693491
$560$	0	0
$561$	−5.03241	−0.212468
$562$	33.6354	1.41882
$563$	12.2246	0.515207	0.257603	−	0.966251i	$-0.417067\pi$
$563$	12.2246	0.515207	0.257603	+	0.966251i	$0.417067\pi$
$564$	95.1626	4.00707
$565$	0	0
$566$	−33.0201	−1.38794
$567$	−13.5829	−0.570430
$568$	−81.1952	−3.40688
$569$	−14.1468	−0.593067	−0.296533	−	0.955022i	$-0.595831\pi$
$569$	−14.1468	−0.593067	−0.296533	+	0.955022i	$0.595831\pi$
$570$	0	0
$571$	−30.4417	−1.27395	−0.636974	−	0.770886i	$-0.719814\pi$
$571$	−30.4417	−1.27395	−0.636974	+	0.770886i	$0.719814\pi$
$572$	7.77384	0.325041
$573$	14.6960	0.613934
$574$	−66.6688	−2.78270
$575$	0	0
$576$	−12.7953	−0.533137
$577$	−23.4791	−0.977446	−0.488723	−	0.872439i	$-0.662537\pi$
$577$	−23.4791	−0.977446	−0.488723	+	0.872439i	$0.662537\pi$
$578$	−14.6177	−0.608017
$579$	−14.0381	−0.583405
$580$	0	0
$581$	−22.6386	−0.939209
$582$	−26.8828	−1.11433
$583$	6.26507	0.259473
$584$	46.9090	1.94111
$585$	0	0
$586$	−30.1808	−1.24676
$587$	29.7356	1.22732	0.613659	−	0.789571i	$-0.289697\pi$
$587$	29.7356	1.22732	0.613659	+	0.789571i	$0.289697\pi$
$588$	13.6432	0.562634
$589$	0.999697	0.0411918
$590$	0	0
$591$	3.84141	0.158014
$592$	−86.4724	−3.55399
$593$	−34.8478	−1.43103	−0.715513	−	0.698599i	$-0.753807\pi$
$593$	−34.8478	−1.43103	−0.715513	+	0.698599i	$0.753807\pi$
$594$	15.0115	0.615929
$595$	0	0
$596$	−82.7458	−3.38940
$597$	11.3125	0.462988
$598$	−18.5346	−0.757935
$599$	10.1488	0.414669	0.207335	−	0.978270i	$-0.433521\pi$
$599$	10.1488	0.414669	0.207335	+	0.978270i	$0.433521\pi$
$600$	0	0
$601$	−14.8810	−0.607008	−0.303504	−	0.952830i	$-0.598157\pi$
$601$	−14.8810	−0.607008	−0.303504	+	0.952830i	$0.598157\pi$
$602$	65.2088	2.65771
$603$	−1.57905	−0.0643039
$604$	−85.1839	−3.46608
$605$	0	0
$606$	−7.79123	−0.316497
$607$	11.1790	0.453743	0.226871	−	0.973925i	$-0.427150\pi$
$607$	11.1790	0.453743	0.226871	+	0.973925i	$0.427150\pi$
$608$	14.8333	0.601569
$609$	−32.0685	−1.29948
$610$	0	0
$611$	−19.2251	−0.777766
$612$	−13.8255	−0.558862
$613$	−41.2298	−1.66526	−0.832628	−	0.553833i	$-0.813165\pi$
$613$	−41.2298	−1.66526	−0.832628	+	0.553833i	$0.813165\pi$
$614$	21.9034	0.883949
$615$	0	0
$616$	18.7770	0.756547
$617$	11.2150	0.451499	0.225750	−	0.974185i	$-0.427517\pi$
$617$	11.2150	0.451499	0.225750	+	0.974185i	$0.427517\pi$
$618$	1.18935	0.0478429
$619$	−3.80803	−0.153058	−0.0765289	−	0.997067i	$-0.524384\pi$
$619$	−3.80803	−0.153058	−0.0765289	+	0.997067i	$0.524384\pi$
$620$	0	0
$621$	−25.6995	−1.03129
$622$	6.62019	0.265445
$623$	−7.94729	−0.318402
$624$	−26.6136	−1.06540
$625$	0	0
$626$	57.9884	2.31768
$627$	−1.48323	−0.0592346
$628$	76.8968	3.06852
$629$	−24.9559	−0.995058
$630$	0	0
$631$	11.6663	0.464429	0.232215	−	0.972665i	$-0.425403\pi$
$631$	11.6663	0.464429	0.232215	+	0.972665i	$0.425403\pi$
$632$	111.245	4.42507
$633$	35.6212	1.41581
$634$	11.2894	0.448359
$635$	0	0
$636$	−47.3311	−1.87680
$637$	−2.75625	−0.109207
$638$	25.2664	1.00030
$639$	7.88427	0.311897
$640$	0	0
$641$	−13.4013	−0.529321	−0.264661	−	0.964342i	$-0.585260\pi$
$641$	−13.4013	−0.529321	−0.264661	+	0.964342i	$0.585260\pi$
$642$	−21.7815	−0.859647
$643$	−9.60060	−0.378611	−0.189305	−	0.981918i	$-0.560624\pi$
$643$	−9.60060	−0.378611	−0.189305	+	0.981918i	$0.560624\pi$
$644$	−52.9293	−2.08571
$645$	0	0
$646$	9.03639	0.355532
$647$	25.0402	0.984432	0.492216	−	0.870473i	$-0.336187\pi$
$647$	25.0402	0.984432	0.492216	+	0.870473i	$0.336187\pi$
$648$	−49.1032	−1.92896
$649$	12.8815	0.505644
$650$	0	0
$651$	−3.37934	−0.132447
$652$	121.351	4.75249
$653$	48.7909	1.90934	0.954669	−	0.297671i	$-0.0962098\pi$
$653$	48.7909	1.90934	0.954669	+	0.297671i	$0.0962098\pi$
$654$	0.265851	0.0103956
$655$	0	0
$656$	−129.125	−5.04148
$657$	−4.55499	−0.177707
$658$	−76.4591	−2.98069
$659$	−11.6452	−0.453631	−0.226816	−	0.973938i	$-0.572832\pi$
$659$	−11.6452	−0.453631	−0.226816	+	0.973938i	$0.572832\pi$
$660$	0	0
$661$	18.1320	0.705252	0.352626	−	0.935764i	$-0.385289\pi$
$661$	18.1320	0.705252	0.352626	+	0.935764i	$0.385289\pi$
$662$	74.5166	2.89617
$663$	−7.68068	−0.298293
$664$	−81.8401	−3.17601
$665$	0	0
$666$	15.6725	0.607297
$667$	−43.2558	−1.67487
$668$	17.7469	0.686647
$669$	8.72234	0.337225
$670$	0	0
$671$	9.11146	0.351744
$672$	−50.1420	−1.93427
$673$	11.1544	0.429970	0.214985	−	0.976617i	$-0.431030\pi$
$673$	11.1544	0.429970	0.214985	+	0.976617i	$0.431030\pi$
$674$	−57.1480	−2.20126
$675$	0	0
$676$	−54.3500	−2.09038
$677$	−42.5574	−1.63562	−0.817808	−	0.575492i	$-0.804811\pi$
$677$	−42.5574	−1.63562	−0.817808	+	0.575492i	$0.804811\pi$
$678$	−26.2708	−1.00892
$679$	15.5093	0.595192
$680$	0	0
$681$	22.4760	0.861282
$682$	2.66254	0.101954
$683$	−37.0003	−1.41578	−0.707888	−	0.706325i	$-0.750352\pi$
$683$	−37.0003	−1.41578	−0.707888	+	0.706325i	$0.750352\pi$
$684$	−4.07487	−0.155807
$685$	0	0
$686$	−53.4512	−2.04078
$687$	−11.8027	−0.450303
$688$	126.297	4.81503
$689$	9.56202	0.364284
$690$	0	0
$691$	30.0210	1.14205	0.571026	−	0.820932i	$-0.306546\pi$
$691$	30.0210	1.14205	0.571026	+	0.820932i	$0.306546\pi$
$692$	43.7751	1.66408
$693$	−1.82330	−0.0692613
$694$	83.1294	3.15555
$695$	0	0
$696$	−115.930	−4.39430
$697$	−37.2654	−1.41153
$698$	−62.5109	−2.36607
$699$	1.85198	0.0700482
$700$	0	0
$701$	−30.8213	−1.16410	−0.582052	−	0.813151i	$-0.697750\pi$
$701$	−30.8213	−1.16410	−0.582052	+	0.813151i	$0.697750\pi$
$702$	22.9112	0.864728
$703$	−7.35542	−0.277415
$704$	15.9937	0.602784
$705$	0	0
$706$	36.0773	1.35779
$707$	4.49493	0.169049
$708$	−97.3169	−3.65739
$709$	23.6622	0.888651	0.444325	−	0.895865i	$-0.353443\pi$
$709$	23.6622	0.888651	0.444325	+	0.895865i	$0.353443\pi$
$710$	0	0
$711$	−10.8021	−0.405112
$712$	−28.7300	−1.07670
$713$	−4.55825	−0.170708
$714$	−30.5464	−1.14317
$715$	0	0
$716$	85.9728	3.21296
$717$	−25.4655	−0.951025
$718$	43.2124	1.61267
$719$	8.20483	0.305989	0.152994	−	0.988227i	$-0.451108\pi$
$719$	8.20483	0.305989	0.152994	+	0.988227i	$0.451108\pi$
$720$	0	0
$721$	−0.686165	−0.0255541
$722$	2.66335	0.0991197
$723$	12.5914	0.468281
$724$	−65.5960	−2.43786
$725$	0	0
$726$	−3.95037	−0.146612
$727$	−36.9180	−1.36921	−0.684606	−	0.728913i	$-0.740026\pi$
$727$	−36.9180	−1.36921	−0.684606	+	0.728913i	$0.740026\pi$
$728$	28.6583	1.06215
$729$	29.8479	1.10548
$730$	0	0
$731$	36.4493	1.34813
$732$	−68.8349	−2.54421
$733$	−15.9166	−0.587891	−0.293946	−	0.955822i	$-0.594968\pi$
$733$	−15.9166	−0.587891	−0.293946	+	0.955822i	$0.594968\pi$
$734$	13.3632	0.493246
$735$	0	0
$736$	−67.6343	−2.49303
$737$	1.97376	0.0727042
$738$	23.4029	0.861473
$739$	−52.6814	−1.93792	−0.968959	−	0.247223i	$-0.920482\pi$
$739$	−52.6814	−1.93792	−0.968959	+	0.247223i	$0.920482\pi$
$740$	0	0
$741$	−2.26377	−0.0831618
$742$	38.0285	1.39607
$743$	−19.3770	−0.710874	−0.355437	−	0.934700i	$-0.615668\pi$
$743$	−19.3770	−0.710874	−0.355437	+	0.934700i	$0.615668\pi$
$744$	−12.2165	−0.447880
$745$	0	0
$746$	−88.4902	−3.23986
$747$	7.94689	0.290762
$748$	17.2814	0.631869
$749$	12.5662	0.459160
$750$	0	0
$751$	−20.9486	−0.764424	−0.382212	−	0.924075i	$-0.624838\pi$
$751$	−20.9486	−0.764424	−0.382212	+	0.924075i	$0.624838\pi$
$752$	−148.087	−5.40016
$753$	12.7709	0.465399
$754$	38.5626	1.40437
$755$	0	0
$756$	65.4277	2.37958
$757$	−4.85815	−0.176572	−0.0882862	−	0.996095i	$-0.528139\pi$
$757$	−4.85815	−0.176572	−0.0882862	+	0.996095i	$0.528139\pi$
$758$	−34.6335	−1.25795
$759$	6.76299	0.245481
$760$	0	0
$761$	−45.7286	−1.65766	−0.828831	−	0.559500i	$-0.810993\pi$
$761$	−45.7286	−1.65766	−0.828831	+	0.559500i	$0.810993\pi$
$762$	−49.4456	−1.79122
$763$	−0.153375	−0.00555255
$764$	−50.4663	−1.82581
$765$	0	0
$766$	−82.7121	−2.98851
$767$	19.6604	0.709894
$768$	−3.63443	−0.131146
$769$	42.0970	1.51806	0.759029	−	0.651057i	$-0.225674\pi$
$769$	42.0970	1.51806	0.759029	+	0.651057i	$0.225674\pi$
$770$	0	0
$771$	20.5968	0.741777
$772$	48.2072	1.73501
$773$	0.276331	0.00993893	0.00496947	−	0.999988i	$-0.498418\pi$
$773$	0.276331	0.00993893	0.00496947	+	0.999988i	$0.498418\pi$
$774$	−22.8904	−0.822778
$775$	0	0
$776$	56.0671	2.01269
$777$	24.8640	0.891992
$778$	74.6962	2.67799
$779$	−10.9835	−0.393523
$780$	0	0
$781$	−9.85507	−0.352642
$782$	−41.2026	−1.47340
$783$	53.4699	1.91086
$784$	−21.2307	−0.758240
$785$	0	0
$786$	28.1078	1.00257
$787$	−30.0118	−1.06980	−0.534902	−	0.844914i	$-0.679651\pi$
$787$	−30.0118	−1.06980	−0.534902	+	0.844914i	$0.679651\pi$
$788$	−13.1915	−0.469926
$789$	−9.16131	−0.326151
$790$	0	0
$791$	15.1562	0.538893
$792$	−6.59133	−0.234213
$793$	13.9063	0.493828
$794$	84.0807	2.98391
$795$	0	0
$796$	−38.8472	−1.37690
$797$	−53.5284	−1.89607	−0.948037	−	0.318160i	$-0.896935\pi$
$797$	−53.5284	−1.89607	−0.948037	+	0.318160i	$0.896935\pi$
$798$	−9.00311	−0.318707
$799$	−42.7378	−1.51195
$800$	0	0
$801$	2.78976	0.0985712
$802$	−74.9906	−2.64801
$803$	5.69357	0.200922
$804$	−14.9113	−0.525879
$805$	0	0
$806$	4.06369	0.143138
$807$	26.5896	0.935997
$808$	16.2495	0.571654
$809$	32.1617	1.13074	0.565372	−	0.824836i	$-0.308733\pi$
$809$	32.1617	1.13074	0.565372	+	0.824836i	$0.308733\pi$
$810$	0	0
$811$	−11.7382	−0.412185	−0.206093	−	0.978533i	$-0.566075\pi$
$811$	−11.7382	−0.412185	−0.206093	+	0.978533i	$0.566075\pi$
$812$	110.124	3.86458
$813$	28.4015	0.996085
$814$	−19.5901	−0.686631
$815$	0	0
$816$	−59.1624	−2.07110
$817$	10.7429	0.375847
$818$	−40.4243	−1.41340
$819$	−2.78280	−0.0972387
$820$	0	0
$821$	15.5521	0.542772	0.271386	−	0.962471i	$-0.412518\pi$
$821$	15.5521	0.542772	0.271386	+	0.962471i	$0.412518\pi$
$822$	31.4215	1.09595
$823$	49.9472	1.74105	0.870524	−	0.492125i	$-0.163780\pi$
$823$	49.9472	1.74105	0.870524	+	0.492125i	$0.163780\pi$
$824$	−2.48053	−0.0864134
$825$	0	0
$826$	78.1900	2.72058
$827$	−22.9363	−0.797575	−0.398787	−	0.917043i	$-0.630569\pi$
$827$	−22.9363	−0.797575	−0.398787	+	0.917043i	$0.630569\pi$
$828$	18.5799	0.645697
$829$	−37.1605	−1.29064	−0.645318	−	0.763914i	$-0.723275\pi$
$829$	−37.1605	−1.29064	−0.645318	+	0.763914i	$0.723275\pi$
$830$	0	0
$831$	−18.8836	−0.655065
$832$	24.4102	0.846273
$833$	−6.12718	−0.212294
$834$	35.4631	1.22799
$835$	0	0
$836$	5.09344	0.176160
$837$	5.63461	0.194761
$838$	−94.4441	−3.26252
$839$	−10.8879	−0.375893	−0.187946	−	0.982179i	$-0.560183\pi$
$839$	−10.8879	−0.375893	−0.187946	+	0.982179i	$0.560183\pi$
$840$	0	0
$841$	60.9970	2.10335
$842$	31.8845	1.09881
$843$	−18.7317	−0.645154
$844$	−122.324	−4.21055
$845$	0	0
$846$	26.8396	0.922765
$847$	2.27906	0.0783093
$848$	73.6540	2.52929
$849$	18.3890	0.631110
$850$	0	0
$851$	33.5380	1.14967
$852$	74.4527	2.55071
$853$	46.7100	1.59932	0.799659	−	0.600454i	$-0.205013\pi$
$853$	46.7100	1.59932	0.799659	+	0.600454i	$0.205013\pi$
$854$	55.3059	1.89253
$855$	0	0
$856$	45.4277	1.55269
$857$	−35.0245	−1.19642	−0.598208	−	0.801341i	$-0.704120\pi$
$857$	−35.0245	−1.19642	−0.598208	+	0.801341i	$0.704120\pi$
$858$	−6.02923	−0.205834
$859$	15.6843	0.535141	0.267571	−	0.963538i	$-0.413779\pi$
$859$	15.6843	0.535141	0.267571	+	0.963538i	$0.413779\pi$
$860$	0	0
$861$	37.1282	1.26532
$862$	38.3786	1.30718
$863$	−21.9764	−0.748084	−0.374042	−	0.927412i	$-0.622028\pi$
$863$	−21.9764	−0.748084	−0.374042	+	0.927412i	$0.622028\pi$
$864$	83.6050	2.84430
$865$	0	0
$866$	−71.9097	−2.44359
$867$	8.14067	0.276472
$868$	11.6047	0.393890
$869$	13.5023	0.458034
$870$	0	0
$871$	3.01243	0.102072
$872$	−0.554461	−0.0187764
$873$	−5.44426	−0.184260
$874$	−12.1439	−0.410774
$875$	0	0
$876$	−43.0136	−1.45329
$877$	24.7003	0.834069	0.417035	−	0.908891i	$-0.363069\pi$
$877$	24.7003	0.834069	0.417035	+	0.908891i	$0.363069\pi$
$878$	−39.2329	−1.32404
$879$	16.8078	0.566914
$880$	0	0
$881$	−33.6606	−1.13405	−0.567027	−	0.823699i	$-0.691906\pi$
$881$	−33.6606	−1.13405	−0.567027	+	0.823699i	$0.691906\pi$
$882$	3.84791	0.129566
$883$	32.9203	1.10786	0.553929	−	0.832564i	$-0.313128\pi$
$883$	32.9203	1.10786	0.553929	+	0.832564i	$0.313128\pi$
$884$	26.3756	0.887107
$885$	0	0
$886$	9.49001	0.318823
$887$	41.6748	1.39930	0.699651	−	0.714485i	$-0.253339\pi$
$887$	41.6748	1.39930	0.699651	+	0.714485i	$0.253339\pi$
$888$	89.8850	3.01634
$889$	28.5263	0.956740
$890$	0	0
$891$	−5.95990	−0.199664
$892$	−29.9527	−1.00289
$893$	−12.5964	−0.421522
$894$	64.1759	2.14636
$895$	0	0
$896$	29.4686	0.984478
$897$	10.3220	0.344641
$898$	−7.95019	−0.265301
$899$	9.48380	0.316302
$900$	0	0
$901$	21.2565	0.708158
$902$	−29.2528	−0.974013
$903$	−36.3151	−1.20849
$904$	54.7907	1.82231
$905$	0	0
$906$	66.0668	2.19492
$907$	33.7269	1.11988	0.559941	−	0.828532i	$-0.310824\pi$
$907$	33.7269	1.11988	0.559941	+	0.828532i	$0.310824\pi$
$908$	−77.1830	−2.56141
$909$	−1.57787	−0.0523345
$910$	0	0
$911$	−27.4759	−0.910318	−0.455159	−	0.890410i	$-0.650418\pi$
$911$	−27.4759	−0.910318	−0.455159	+	0.890410i	$0.650418\pi$
$912$	−17.4373	−0.577407
$913$	−9.93334	−0.328746
$914$	−18.8920	−0.624892
$915$	0	0
$916$	40.5308	1.33918
$917$	−16.2160	−0.535500
$918$	50.9320	1.68101
$919$	6.84795	0.225893	0.112947	−	0.993601i	$-0.463971\pi$
$919$	6.84795	0.225893	0.112947	+	0.993601i	$0.463971\pi$
$920$	0	0
$921$	−12.1981	−0.401941
$922$	−68.6303	−2.26022
$923$	−15.0412	−0.495088
$924$	−17.2177	−0.566422
$925$	0	0
$926$	−94.1581	−3.09423
$927$	0.240866	0.00791108
$928$	140.719	4.61931
$929$	9.06304	0.297349	0.148674	−	0.988886i	$-0.452499\pi$
$929$	9.06304	0.297349	0.148674	+	0.988886i	$0.452499\pi$
$930$	0	0
$931$	−1.80590	−0.0591861
$932$	−6.35972	−0.208320
$933$	−3.68681	−0.120701
$934$	−84.4253	−2.76248
$935$	0	0
$936$	−10.0600	−0.328821
$937$	31.5974	1.03224	0.516122	−	0.856515i	$-0.327375\pi$
$937$	31.5974	1.03224	0.516122	+	0.856515i	$0.327375\pi$
$938$	11.9806	0.391179
$939$	−32.2940	−1.05387
$940$	0	0
$941$	−5.58295	−0.181999	−0.0909995	−	0.995851i	$-0.529006\pi$
$941$	−5.58295	−0.181999	−0.0909995	+	0.995851i	$0.529006\pi$
$942$	−59.6395	−1.94316
$943$	50.0805	1.63085
$944$	151.439	4.92892
$945$	0	0
$946$	28.6122	0.930262
$947$	26.8235	0.871646	0.435823	−	0.900032i	$-0.356457\pi$
$947$	26.8235	0.871646	0.435823	+	0.900032i	$0.356457\pi$
$948$	−102.007	−3.31302
$949$	8.68978	0.282082
$950$	0	0
$951$	−6.28712	−0.203874
$952$	63.7078	2.06478
$953$	−15.6816	−0.507976	−0.253988	−	0.967207i	$-0.581742\pi$
$953$	−15.6816	−0.507976	−0.253988	+	0.967207i	$0.581742\pi$
$954$	−13.3492	−0.432198
$955$	0	0
$956$	87.4489	2.82830
$957$	−14.0709	−0.454849
$958$	−3.67110	−0.118608
$959$	−18.1278	−0.585377
$960$	0	0
$961$	−30.0006	−0.967761
$962$	−29.8992	−0.963990
$963$	−4.41115	−0.142147
$964$	−43.2393	−1.39264
$965$	0	0
$966$	41.0509	1.32079
$967$	−13.6161	−0.437863	−0.218931	−	0.975740i	$-0.570257\pi$
$967$	−13.6161	−0.437863	−0.218931	+	0.975740i	$0.570257\pi$
$968$	8.23893	0.264809
$969$	−5.03241	−0.161664
$970$	0	0
$971$	59.1158	1.89712	0.948558	−	0.316604i	$-0.102543\pi$
$971$	59.1158	1.89712	0.948558	+	0.316604i	$0.102543\pi$
$972$	−41.0992	−1.31826
$973$	−20.4594	−0.655899
$974$	78.4506	2.51372
$975$	0	0
$976$	107.117	3.42873
$977$	22.4291	0.717571	0.358785	−	0.933420i	$-0.383191\pi$
$977$	22.4291	0.717571	0.358785	+	0.933420i	$0.383191\pi$
$978$	−94.1177	−3.00955
$979$	−3.48710	−0.111448
$980$	0	0
$981$	0.0538396	0.00171897
$982$	73.8612	2.35700
$983$	−7.13855	−0.227684	−0.113842	−	0.993499i	$-0.536316\pi$
$983$	−7.13855	−0.227684	−0.113842	+	0.993499i	$0.536316\pi$
$984$	134.221	4.27880
$985$	0	0
$986$	85.7253	2.73005
$987$	42.5804	1.35535
$988$	7.77384	0.247319
$989$	−48.9838	−1.55759
$990$	0	0
$991$	−18.7708	−0.596274	−0.298137	−	0.954523i	$-0.596365\pi$
$991$	−18.7708	−0.596274	−0.298137	+	0.954523i	$0.596365\pi$
$992$	14.8288	0.470814
$993$	−41.4986	−1.31692
$994$	−59.8196	−1.89736
$995$	0	0
$996$	75.0440	2.37786
$997$	31.9203	1.01093	0.505463	−	0.862848i	$-0.331322\pi$
$997$	31.9203	1.01093	0.505463	+	0.862848i	$0.331322\pi$
$998$	−18.5637	−0.587623
$999$	−41.4574	−1.31166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.y.1.15	yes	15
5.4	even	2		5225.2.a.r.1.1	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.r.1.1	✓	15	5.4	even	2
5225.2.a.y.1.15	yes	15	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.66335 q^{2} -1.48323 q^{3} +5.09344 q^{4} -3.95037 q^{6} +2.27906 q^{7} +8.23893 q^{8} -0.800022 q^{9} +1.00000 q^{11} -7.55476 q^{12} +1.52624 q^{13} +6.06993 q^{14} +11.7563 q^{16} +3.39286 q^{17} -2.13074 q^{18} +1.00000 q^{19} -3.38037 q^{21} +2.66335 q^{22} -4.55963 q^{23} -12.2203 q^{24} +4.06493 q^{26} +5.63632 q^{27} +11.6082 q^{28} +9.48668 q^{29} +0.999697 q^{31} +14.8333 q^{32} -1.48323 q^{33} +9.03639 q^{34} -4.07487 q^{36} -7.35542 q^{37} +2.66335 q^{38} -2.26377 q^{39} -10.9835 q^{41} -9.00311 q^{42} +10.7429 q^{43} +5.09344 q^{44} -12.1439 q^{46} -12.5964 q^{47} -17.4373 q^{48} -1.80590 q^{49} -5.03241 q^{51} +7.77384 q^{52} +6.26507 q^{53} +15.0115 q^{54} +18.7770 q^{56} -1.48323 q^{57} +25.2664 q^{58} +12.8815 q^{59} +9.11146 q^{61} +2.66254 q^{62} -1.82330 q^{63} +15.9937 q^{64} -3.95037 q^{66} +1.97376 q^{67} +17.2814 q^{68} +6.76299 q^{69} -9.85507 q^{71} -6.59133 q^{72} +5.69357 q^{73} -19.5901 q^{74} +5.09344 q^{76} +2.27906 q^{77} -6.02923 q^{78} +13.5023 q^{79} -5.95990 q^{81} -29.2528 q^{82} -9.93334 q^{83} -17.2177 q^{84} +28.6122 q^{86} -14.0709 q^{87} +8.23893 q^{88} -3.48710 q^{89} +3.47840 q^{91} -23.2242 q^{92} -1.48278 q^{93} -33.5486 q^{94} -22.0012 q^{96} +6.80514 q^{97} -4.80975 q^{98} -0.800022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) \) 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 21 * q^7 + 9 * q^8 + 15 * q^9 + 15 * q^11 + 11 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 17 * q^17 + 22 * q^18 + 15 * q^19 + 6 * q^21 + 5 * q^22 + 26 * q^23 + q^24 + 3 * q^26 + q^27 + 46 * q^28 + 9 * q^29 + 14 * q^31 + 18 * q^32 + 4 * q^33 - 13 * q^34 + 12 * q^36 + 9 * q^37 + 5 * q^38 - 22 * q^39 + 4 * q^41 - 6 * q^42 + 28 * q^43 + 17 * q^44 + 27 * q^46 + 14 * q^47 - 4 * q^48 + 32 * q^49 - 40 * q^51 + 14 * q^52 + 3 * q^53 - 39 * q^54 + 34 * q^56 + 4 * q^57 + 26 * q^58 + q^59 + 2 * q^61 - 3 * q^62 + 45 * q^63 + 5 * q^64 - q^66 + 37 * q^67 + 26 * q^68 - 7 * q^69 - 7 * q^71 + 16 * q^72 + 42 * q^73 - 43 * q^74 + 17 * q^76 + 21 * q^77 - 64 * q^78 - 10 * q^79 + 31 * q^81 + 22 * q^82 + 14 * q^83 - 32 * q^84 + 37 * q^86 + 29 * q^87 + 9 * q^88 + 15 * q^89 - 22 * q^91 + 26 * q^92 - 18 * q^93 - 44 * q^94 + 71 * q^96 + 8 * q^97 - 10 * q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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