LMFDB - Embedded newform 5225.2.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.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:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.71250$ 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)

$f(q)$	$=$	\(q+0.779856 q^{2} +2.98063 q^{3} -1.39182 q^{4} +2.32446 q^{6} -1.06736 q^{7} -2.64513 q^{8} +5.88418 q^{9} +1.00000 q^{11} -4.14852 q^{12} +0.0563258 q^{13} -0.832387 q^{14} +0.720827 q^{16} +4.53628 q^{17} +4.58881 q^{18} -1.00000 q^{19} -3.18141 q^{21} +0.779856 q^{22} +1.07949 q^{23} -7.88418 q^{24} +0.0439260 q^{26} +8.59667 q^{27} +1.48558 q^{28} +0.299905 q^{29} +9.18548 q^{31} +5.85241 q^{32} +2.98063 q^{33} +3.53764 q^{34} -8.18974 q^{36} -4.50448 q^{37} -0.779856 q^{38} +0.167887 q^{39} +12.0009 q^{41} -2.48104 q^{42} -10.7260 q^{43} -1.39182 q^{44} +0.841844 q^{46} -2.89630 q^{47} +2.14852 q^{48} -5.86074 q^{49} +13.5210 q^{51} -0.0783957 q^{52} +12.3213 q^{53} +6.70416 q^{54} +2.82331 q^{56} -2.98063 q^{57} +0.233882 q^{58} -1.14582 q^{59} -8.09599 q^{61} +7.16335 q^{62} -6.28054 q^{63} +3.12238 q^{64} +2.32446 q^{66} -11.2733 q^{67} -6.31370 q^{68} +3.21755 q^{69} +13.4948 q^{71} -15.5644 q^{72} +11.1470 q^{73} -3.51284 q^{74} +1.39182 q^{76} -1.06736 q^{77} +0.130927 q^{78} +11.4250 q^{79} +7.97100 q^{81} +9.35899 q^{82} +13.9802 q^{83} +4.42797 q^{84} -8.36475 q^{86} +0.893906 q^{87} -2.64513 q^{88} -0.183185 q^{89} -0.0601200 q^{91} -1.50246 q^{92} +27.3785 q^{93} -2.25870 q^{94} +17.4439 q^{96} -5.66263 q^{97} -4.57053 q^{98} +5.88418 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} - 6 q^{12} - 4 q^{13} - 14 q^{14} + 8 q^{16} + 4 q^{17} + 20 q^{18} - 5 q^{19} + 10 q^{21} - 2 q^{22} - 3 q^{23} - 14 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) $ 5 * q - 2 * q^2 - q^3 + 6 * q^4 - 2 * q^6 - 6 * q^7 - 6 * q^8 + 4 * q^9 + 5 * q^11 - 6 * q^12 - 4 * q^13 - 14 * q^14 + 8 * q^16 + 4 * q^17 + 20 * q^18 - 5 * q^19 + 10 * q^21 - 2 * q^22 - 3 * q^23 - 14 * q^24 - 6 * q^26 + 11 * q^27 + 10 * q^28 + 10 * q^29 + 11 * q^31 - 14 * q^32 - q^33 - 4 * q^34 - 26 * q^36 - q^37 + 2 * q^38 + 2 * q^39 + 2 * q^41 + 16 * q^42 - 20 * q^43 + 6 * q^44 - 4 * q^46 + 20 * q^47 - 4 * q^48 + 3 * q^49 + 24 * q^51 - 6 * q^52 + 14 * q^53 + 16 * q^54 - 38 * q^56 + q^57 + 6 * q^58 + 3 * q^59 - 10 * q^61 + 6 * q^62 - 24 * q^63 - 2 * q^66 - 9 * q^67 - 24 * q^68 - 5 * q^69 + 23 * q^71 + 12 * q^72 + 8 * q^74 - 6 * q^76 - 6 * q^77 + 22 * q^78 + 44 * q^79 + q^81 + 30 * q^82 + 14 * q^83 + 14 * q^84 + 52 * q^86 - 28 * q^87 - 6 * q^88 - 27 * q^89 + 24 * q^91 - 58 * q^92 + 27 * q^93 - 8 * q^94 + 50 * q^96 - 15 * q^97 + 10 * q^98 + 4 * 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 $

$2$	0.779856	0.551441	0.275721	−	0.961238i	$-0.411084\pi$
$2$	0.779856	0.551441	0.275721	+	0.961238i	$0.411084\pi$
$3$	2.98063	1.72087	0.860435	−	0.509561i	$-0.170192\pi$
$3$	2.98063	1.72087	0.860435	+	0.509561i	$0.170192\pi$
$4$	−1.39182	−0.695912
$5$	0	0
$5$	0	0
$6$	2.32446	0.948959
$7$	−1.06736	−0.403424	−0.201712	−	0.979445i	$-0.564651\pi$
$7$	−1.06736	−0.403424	−0.201712	+	0.979445i	$0.564651\pi$
$8$	−2.64513	−0.935196
$9$	5.88418	1.96139
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−4.14852	−1.19757
$13$	0.0563258	0.0156220	0.00781098	−	0.999969i	$-0.497514\pi$
$13$	0.0563258	0.0156220	0.00781098	+	0.999969i	$0.497514\pi$
$14$	−0.832387	−0.222465
$15$	0	0
$16$	0.720827	0.180207
$17$	4.53628	1.10021	0.550104	−	0.835096i	$-0.314588\pi$
$17$	4.53628	1.10021	0.550104	+	0.835096i	$0.314588\pi$
$18$	4.58881	1.08159
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.18141	−0.694241
$22$	0.779856	0.166266
$23$	1.07949	0.225088	0.112544	−	0.993647i	$-0.464100\pi$
$23$	1.07949	0.225088	0.112544	+	0.993647i	$0.464100\pi$
$24$	−7.88418	−1.60935
$25$	0	0
$26$	0.0439260	0.00861460
$27$	8.59667	1.65443
$28$	1.48558	0.280748
$29$	0.299905	0.0556909	0.0278455	−	0.999612i	$-0.491135\pi$
$29$	0.299905	0.0556909	0.0278455	+	0.999612i	$0.491135\pi$
$30$	0	0
$31$	9.18548	1.64976	0.824880	−	0.565307i	$-0.191242\pi$
$31$	9.18548	1.64976	0.824880	+	0.565307i	$0.191242\pi$
$32$	5.85241	1.03457
$33$	2.98063	0.518862
$34$	3.53764	0.606701
$35$	0	0
$36$	−8.18974	−1.36496
$37$	−4.50448	−0.740531	−0.370266	−	0.928926i	$-0.620733\pi$
$37$	−4.50448	−0.740531	−0.370266	+	0.928926i	$0.620733\pi$
$38$	−0.779856	−0.126509
$39$	0.167887	0.0268834
$40$	0	0
$41$	12.0009	1.87423	0.937115	−	0.349020i	$-0.113486\pi$
$41$	12.0009	1.87423	0.937115	+	0.349020i	$0.113486\pi$
$42$	−2.48104	−0.382833
$43$	−10.7260	−1.63570	−0.817851	−	0.575430i	$-0.804835\pi$
$43$	−10.7260	−1.63570	−0.817851	+	0.575430i	$0.804835\pi$
$44$	−1.39182	−0.209826
$45$	0	0
$46$	0.841844	0.124123
$47$	−2.89630	−0.422469	−0.211235	−	0.977435i	$-0.567748\pi$
$47$	−2.89630	−0.422469	−0.211235	+	0.977435i	$0.567748\pi$
$48$	2.14852	0.310112
$49$	−5.86074	−0.837249
$50$	0	0
$51$	13.5210	1.89332
$52$	−0.0783957	−0.0108715
$53$	12.3213	1.69246	0.846230	−	0.532818i	$-0.178867\pi$
$53$	12.3213	1.69246	0.846230	+	0.532818i	$0.178867\pi$
$54$	6.70416	0.912321
$55$	0	0
$56$	2.82331	0.377281
$57$	−2.98063	−0.394795
$58$	0.233882	0.0307103
$59$	−1.14582	−0.149173	−0.0745863	−	0.997215i	$-0.523764\pi$
$59$	−1.14582	−0.149173	−0.0745863	+	0.997215i	$0.523764\pi$
$60$	0	0
$61$	−8.09599	−1.03659	−0.518293	−	0.855203i	$-0.673432\pi$
$61$	−8.09599	−1.03659	−0.518293	+	0.855203i	$0.673432\pi$
$62$	7.16335	0.909746
$63$	−6.28054	−0.791273
$64$	3.12238	0.390298
$65$	0	0
$66$	2.32446	0.286122
$67$	−11.2733	−1.37726	−0.688628	−	0.725115i	$-0.741787\pi$
$67$	−11.2733	−1.37726	−0.688628	+	0.725115i	$0.741787\pi$
$68$	−6.31370	−0.765649
$69$	3.21755	0.387348
$70$	0	0
$71$	13.4948	1.60154	0.800771	−	0.598970i	$-0.204423\pi$
$71$	13.4948	1.60154	0.800771	+	0.598970i	$0.204423\pi$
$72$	−15.5644	−1.83429
$73$	11.1470	1.30466	0.652330	−	0.757935i	$-0.273792\pi$
$73$	11.1470	1.30466	0.652330	+	0.757935i	$0.273792\pi$
$74$	−3.51284	−0.408360
$75$	0	0
$76$	1.39182	0.159653
$77$	−1.06736	−0.121637
$78$	0.130927	0.0148246
$79$	11.4250	1.28541	0.642706	−	0.766113i	$-0.277812\pi$
$79$	11.4250	1.28541	0.642706	+	0.766113i	$0.277812\pi$
$80$	0	0
$81$	7.97100	0.885666
$82$	9.35899	1.03353
$83$	13.9802	1.53452	0.767261	−	0.641335i	$-0.221619\pi$
$83$	13.9802	1.53452	0.767261	+	0.641335i	$0.221619\pi$
$84$	4.42797	0.483131
$85$	0	0
$86$	−8.36475	−0.901994
$87$	0.893906	0.0958368
$88$	−2.64513	−0.281972
$89$	−0.183185	−0.0194176	−0.00970878	−	0.999953i	$-0.503090\pi$
$89$	−0.183185	−0.0194176	−0.00970878	+	0.999953i	$0.503090\pi$
$90$	0	0
$91$	−0.0601200	−0.00630228
$92$	−1.50246	−0.156642
$93$	27.3785	2.83902
$94$	−2.25870	−0.232967
$95$	0	0
$96$	17.4439	1.78036
$97$	−5.66263	−0.574953	−0.287477	−	0.957788i	$-0.592816\pi$
$97$	−5.66263	−0.574953	−0.287477	+	0.957788i	$0.592816\pi$
$98$	−4.57053	−0.461694
$99$	5.88418	0.591382
$100$	0	0
$101$	8.00759	0.796785	0.398392	−	0.917215i	$-0.369568\pi$
$101$	8.00759	0.796785	0.398392	+	0.917215i	$0.369568\pi$
$102$	10.5444	1.04405
$103$	−6.18725	−0.609648	−0.304824	−	0.952409i	$-0.598598\pi$
$103$	−6.18725	−0.609648	−0.304824	+	0.952409i	$0.598598\pi$
$104$	−0.148989	−0.0146096
$105$	0	0
$106$	9.60883	0.933292
$107$	7.29027	0.704777	0.352388	−	0.935854i	$-0.385370\pi$
$107$	7.29027	0.704777	0.352388	+	0.935854i	$0.385370\pi$
$108$	−11.9651	−1.15134
$109$	7.79895	0.747004	0.373502	−	0.927629i	$-0.378157\pi$
$109$	7.79895	0.747004	0.373502	+	0.927629i	$0.378157\pi$
$110$	0	0
$111$	−13.4262	−1.27436
$112$	−0.769382	−0.0726998
$113$	0.430558	0.0405035	0.0202517	−	0.999795i	$-0.493553\pi$
$113$	0.430558	0.0405035	0.0202517	+	0.999795i	$0.493553\pi$
$114$	−2.32446	−0.217706
$115$	0	0
$116$	−0.417415	−0.0387560
$117$	0.331431	0.0306408
$118$	−0.893572	−0.0822600
$119$	−4.84184	−0.443851
$120$	0	0
$121$	1.00000	0.0909091
$122$	−6.31370	−0.571616
$123$	35.7704	3.22531
$124$	−12.7846	−1.14809
$125$	0	0
$126$	−4.89791	−0.436341
$127$	3.13888	0.278531	0.139265	−	0.990255i	$-0.455526\pi$
$127$	3.13888	0.278531	0.139265	+	0.990255i	$0.455526\pi$
$128$	−9.26981	−0.819343
$129$	−31.9703	−2.81483
$130$	0	0
$131$	12.4315	1.08615	0.543075	−	0.839684i	$-0.317260\pi$
$131$	12.4315	1.08615	0.543075	+	0.839684i	$0.317260\pi$
$132$	−4.14852	−0.361082
$133$	1.06736	0.0925519
$134$	−8.79157	−0.759476
$135$	0	0
$136$	−11.9991	−1.02891
$137$	−13.8301	−1.18159	−0.590794	−	0.806822i	$-0.701186\pi$
$137$	−13.8301	−1.18159	−0.590794	+	0.806822i	$0.701186\pi$
$138$	2.50923	0.213600
$139$	15.9968	1.35683	0.678417	−	0.734677i	$-0.262666\pi$
$139$	15.9968	1.35683	0.678417	+	0.734677i	$0.262666\pi$
$140$	0	0
$141$	−8.63281	−0.727014
$142$	10.5240	0.883157
$143$	0.0563258	0.00471020
$144$	4.24147	0.353456
$145$	0	0
$146$	8.69307	0.719443
$147$	−17.4687	−1.44080
$148$	6.26944	0.515345
$149$	11.3620	0.930812	0.465406	−	0.885097i	$-0.345908\pi$
$149$	11.3620	0.930812	0.465406	+	0.885097i	$0.345908\pi$
$150$	0	0
$151$	4.66341	0.379503	0.189751	−	0.981832i	$-0.439232\pi$
$151$	4.66341	0.379503	0.189751	+	0.981832i	$0.439232\pi$
$152$	2.64513	0.214549
$153$	26.6923	2.15794
$154$	−0.832387	−0.0670757
$155$	0	0
$156$	−0.233669	−0.0187085
$157$	9.05206	0.722433	0.361217	−	0.932482i	$-0.382361\pi$
$157$	9.05206	0.722433	0.361217	+	0.932482i	$0.382361\pi$
$158$	8.90984	0.708829
$159$	36.7253	2.91250
$160$	0	0
$161$	−1.15220	−0.0908062
$162$	6.21623	0.488393
$163$	−2.36761	−0.185446	−0.0927229	−	0.995692i	$-0.529557\pi$
$163$	−2.36761	−0.185446	−0.0927229	+	0.995692i	$0.529557\pi$
$164$	−16.7032	−1.30430
$165$	0	0
$166$	10.9025	0.846199
$167$	9.27361	0.717613	0.358807	−	0.933412i	$-0.383184\pi$
$167$	9.27361	0.717613	0.358807	+	0.933412i	$0.383184\pi$
$168$	8.41526	0.649251
$169$	−12.9968	−0.999756
$170$	0	0
$171$	−5.88418	−0.449974
$172$	14.9287	1.13831
$173$	−10.7172	−0.814812	−0.407406	−	0.913247i	$-0.633567\pi$
$173$	−10.7172	−0.814812	−0.407406	+	0.913247i	$0.633567\pi$
$174$	0.697118	0.0528484
$175$	0	0
$176$	0.720827	0.0543344
$177$	−3.41526	−0.256707
$178$	−0.142858	−0.0107076
$179$	−22.9070	−1.71215	−0.856073	−	0.516854i	$-0.827103\pi$
$179$	−22.9070	−1.71215	−0.856073	+	0.516854i	$0.827103\pi$
$180$	0	0
$181$	5.90522	0.438931	0.219466	−	0.975620i	$-0.429569\pi$
$181$	5.90522	0.438931	0.219466	+	0.975620i	$0.429569\pi$
$182$	−0.0468849	−0.00347534
$183$	−24.1312	−1.78383
$184$	−2.85539	−0.210502
$185$	0	0
$186$	21.3513	1.56555
$187$	4.53628	0.331725
$188$	4.03114	0.294001
$189$	−9.17575	−0.667438
$190$	0	0
$191$	6.44628	0.466437	0.233218	−	0.972424i	$-0.425074\pi$
$191$	6.44628	0.466437	0.233218	+	0.972424i	$0.425074\pi$
$192$	9.30668	0.671652
$193$	−1.43606	−0.103370	−0.0516849	−	0.998663i	$-0.516459\pi$
$193$	−1.43606	−0.103370	−0.0516849	+	0.998663i	$0.516459\pi$
$194$	−4.41604	−0.317053
$195$	0	0
$196$	8.15713	0.582652
$197$	4.75399	0.338708	0.169354	−	0.985555i	$-0.445832\pi$
$197$	4.75399	0.338708	0.169354	+	0.985555i	$0.445832\pi$
$198$	4.58881	0.326112
$199$	−2.36002	−0.167298	−0.0836489	−	0.996495i	$-0.526657\pi$
$199$	−2.36002	−0.167298	−0.0836489	+	0.996495i	$0.526657\pi$
$200$	0	0
$201$	−33.6017	−2.37008
$202$	6.24476	0.439380
$203$	−0.320107	−0.0224671
$204$	−18.8188	−1.31758
$205$	0	0
$206$	−4.82517	−0.336185
$207$	6.35189	0.441487
$208$	0.0406011	0.00281518
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−24.5133	−1.68756	−0.843781	−	0.536687i	$-0.819676\pi$
$211$	−24.5133	−1.68756	−0.843781	+	0.536687i	$0.819676\pi$
$212$	−17.1491	−1.17780
$213$	40.2232	2.75605
$214$	5.68536	0.388643
$215$	0	0
$216$	−22.7393	−1.54722
$217$	−9.80422	−0.665554
$218$	6.08205	0.411929
$219$	33.2252	2.24515
$220$	0	0
$221$	0.255509	0.0171874
$222$	−10.4705	−0.702734
$223$	−8.47427	−0.567479	−0.283740	−	0.958901i	$-0.591575\pi$
$223$	−8.47427	−0.567479	−0.283740	+	0.958901i	$0.591575\pi$
$224$	−6.24663	−0.417371
$225$	0	0
$226$	0.335773	0.0223353
$227$	−13.7491	−0.912558	−0.456279	−	0.889837i	$-0.650818\pi$
$227$	−13.7491	−0.912558	−0.456279	+	0.889837i	$0.650818\pi$
$228$	4.14852	0.274742
$229$	−1.99640	−0.131926	−0.0659629	−	0.997822i	$-0.521012\pi$
$229$	−1.99640	−0.131926	−0.0659629	+	0.997822i	$0.521012\pi$
$230$	0	0
$231$	−3.18141	−0.209321
$232$	−0.793288	−0.0520819
$233$	−11.3481	−0.743437	−0.371718	−	0.928346i	$-0.621231\pi$
$233$	−11.3481	−0.743437	−0.371718	+	0.928346i	$0.621231\pi$
$234$	0.258468	0.0168966
$235$	0	0
$236$	1.59478	0.103811
$237$	34.0537	2.21203
$238$	−3.77594	−0.244758
$239$	5.21045	0.337036	0.168518	−	0.985699i	$-0.446102\pi$
$239$	5.21045	0.337036	0.168518	+	0.985699i	$0.446102\pi$
$240$	0	0
$241$	6.60827	0.425676	0.212838	−	0.977087i	$-0.431729\pi$
$241$	6.60827	0.425676	0.212838	+	0.977087i	$0.431729\pi$
$242$	0.779856	0.0501310
$243$	−2.03139	−0.130314
$244$	11.2682	0.721373
$245$	0	0
$246$	27.8957	1.77857
$247$	−0.0563258	−0.00358393
$248$	−24.2968	−1.54285
$249$	41.6697	2.64071
$250$	0	0
$251$	24.0024	1.51502	0.757510	−	0.652823i	$-0.226415\pi$
$251$	24.0024	1.51502	0.757510	+	0.652823i	$0.226415\pi$
$252$	8.74141	0.550657
$253$	1.07949	0.0678667
$254$	2.44788	0.153593
$255$	0	0
$256$	−13.4739	−0.842117
$257$	5.68903	0.354872	0.177436	−	0.984132i	$-0.443220\pi$
$257$	5.68903	0.354872	0.177436	+	0.984132i	$0.443220\pi$
$258$	−24.9322	−1.55221
$259$	4.80790	0.298748
$260$	0	0
$261$	1.76469	0.109232
$262$	9.69481	0.598948
$263$	−13.9857	−0.862395	−0.431197	−	0.902258i	$-0.641909\pi$
$263$	−13.9857	−0.862395	−0.431197	+	0.902258i	$0.641909\pi$
$264$	−7.88418	−0.485237
$265$	0	0
$266$	0.832387	0.0510369
$267$	−0.546007	−0.0334151
$268$	15.6905	0.958450
$269$	15.4020	0.939075	0.469538	−	0.882912i	$-0.344421\pi$
$269$	15.4020	0.939075	0.469538	+	0.882912i	$0.344421\pi$
$270$	0	0
$271$	−25.3911	−1.54240	−0.771199	−	0.636595i	$-0.780342\pi$
$271$	−25.3911	−1.54240	−0.771199	+	0.636595i	$0.780342\pi$
$272$	3.26987	0.198265
$273$	−0.179196	−0.0108454
$274$	−10.7855	−0.651577
$275$	0	0
$276$	−4.47827	−0.269560
$277$	−19.9798	−1.20047	−0.600235	−	0.799824i	$-0.704926\pi$
$277$	−19.9798	−1.20047	−0.600235	+	0.799824i	$0.704926\pi$
$278$	12.4752	0.748214
$279$	54.0490	3.23583
$280$	0	0
$281$	6.18130	0.368745	0.184373	−	0.982856i	$-0.440975\pi$
$281$	6.18130	0.368745	0.184373	+	0.982856i	$0.440975\pi$
$282$	−6.73235	−0.400906
$283$	7.12127	0.423316	0.211658	−	0.977344i	$-0.432114\pi$
$283$	7.12127	0.423316	0.211658	+	0.977344i	$0.432114\pi$
$284$	−18.7825	−1.11453
$285$	0	0
$286$	0.0439260	0.00259740
$287$	−12.8093	−0.756110
$288$	34.4366	2.02920
$289$	3.57781	0.210459
$290$	0	0
$291$	−16.8782	−0.989420
$292$	−15.5147	−0.907929
$293$	−19.1342	−1.11783	−0.558916	−	0.829224i	$-0.688783\pi$
$293$	−19.1342	−1.11783	−0.558916	+	0.829224i	$0.688783\pi$
$294$	−13.6231	−0.794514
$295$	0	0
$296$	11.9149	0.692542
$297$	8.59667	0.498829
$298$	8.86073	0.513288
$299$	0.0608030	0.00351633
$300$	0	0
$301$	11.4485	0.659882
$302$	3.63679	0.209274
$303$	23.8677	1.37116
$304$	−0.720827	−0.0413422
$305$	0	0
$306$	20.8161	1.18998
$307$	−6.82573	−0.389565	−0.194782	−	0.980846i	$-0.562400\pi$
$307$	−6.82573	−0.389565	−0.194782	+	0.980846i	$0.562400\pi$
$308$	1.48558	0.0846487
$309$	−18.4419	−1.04912
$310$	0	0
$311$	−12.8609	−0.729276	−0.364638	−	0.931149i	$-0.618807\pi$
$311$	−12.8609	−0.729276	−0.364638	+	0.931149i	$0.618807\pi$
$312$	−0.444083	−0.0251412
$313$	3.01707	0.170535	0.0852673	−	0.996358i	$-0.472826\pi$
$313$	3.01707	0.170535	0.0852673	+	0.996358i	$0.472826\pi$
$314$	7.05930	0.398380
$315$	0	0
$316$	−15.9016	−0.894534
$317$	−17.7857	−0.998943	−0.499471	−	0.866330i	$-0.666472\pi$
$317$	−17.7857	−0.998943	−0.499471	+	0.866330i	$0.666472\pi$
$318$	28.6404	1.60607
$319$	0.299905	0.0167914
$320$	0	0
$321$	21.7296	1.21283
$322$	−0.898551	−0.0500743
$323$	−4.53628	−0.252405
$324$	−11.0942	−0.616346
$325$	0	0
$326$	−1.84640	−0.102262
$327$	23.2458	1.28550
$328$	−31.7441	−1.75277
$329$	3.09140	0.170434
$330$	0	0
$331$	−26.3860	−1.45030	−0.725152	−	0.688589i	$-0.758230\pi$
$331$	−26.3860	−1.45030	−0.725152	+	0.688589i	$0.758230\pi$
$332$	−19.4579	−1.06789
$333$	−26.5051	−1.45247
$334$	7.23207	0.395722
$335$	0	0
$336$	−2.29325	−0.125107
$337$	13.2024	0.719181	0.359590	−	0.933110i	$-0.382916\pi$
$337$	13.2024	0.719181	0.359590	+	0.933110i	$0.382916\pi$
$338$	−10.1357	−0.551307
$339$	1.28334	0.0697012
$340$	0	0
$341$	9.18548	0.497422
$342$	−4.58881	−0.248134
$343$	13.7271	0.741191
$344$	28.3718	1.52970
$345$	0	0
$346$	−8.35786	−0.449321
$347$	−16.4809	−0.884740	−0.442370	−	0.896833i	$-0.645862\pi$
$347$	−16.4809	−0.884740	−0.442370	+	0.896833i	$0.645862\pi$
$348$	−1.24416	−0.0666940
$349$	−19.0165	−1.01793	−0.508966	−	0.860787i	$-0.669972\pi$
$349$	−19.0165	−1.01793	−0.508966	+	0.860787i	$0.669972\pi$
$350$	0	0
$351$	0.484214	0.0258455
$352$	5.85241	0.311934
$353$	−13.1955	−0.702325	−0.351162	−	0.936315i	$-0.614213\pi$
$353$	−13.1955	−0.702325	−0.351162	+	0.936315i	$0.614213\pi$
$354$	−2.66341	−0.141559
$355$	0	0
$356$	0.254961	0.0135129
$357$	−14.4318	−0.763810
$358$	−17.8641	−0.944148
$359$	−2.88756	−0.152400	−0.0761998	−	0.997093i	$-0.524279\pi$
$359$	−2.88756	−0.152400	−0.0761998	+	0.997093i	$0.524279\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	4.60522	0.242045
$363$	2.98063	0.156443
$364$	0.0836765	0.00438584
$365$	0	0
$366$	−18.8188	−0.983676
$367$	2.10833	0.110054	0.0550269	−	0.998485i	$-0.482476\pi$
$367$	2.10833	0.110054	0.0550269	+	0.998485i	$0.482476\pi$
$368$	0.778123	0.0405624
$369$	70.6156	3.67610
$370$	0	0
$371$	−13.1513	−0.682780
$372$	−38.1061	−1.97571
$373$	−8.07305	−0.418007	−0.209003	−	0.977915i	$-0.567022\pi$
$373$	−8.07305	−0.418007	−0.209003	+	0.977915i	$0.567022\pi$
$374$	3.53764	0.182927
$375$	0	0
$376$	7.66111	0.395091
$377$	0.0168924	0.000870002 0
$378$	−7.15576	−0.368053
$379$	14.3355	0.736365	0.368183	−	0.929754i	$-0.379980\pi$
$379$	14.3355	0.736365	0.368183	+	0.929754i	$0.379980\pi$
$380$	0	0
$381$	9.35586	0.479315
$382$	5.02717	0.257212
$383$	9.79867	0.500689	0.250344	−	0.968157i	$-0.419456\pi$
$383$	9.79867	0.500689	0.250344	+	0.968157i	$0.419456\pi$
$384$	−27.6299	−1.40998
$385$	0	0
$386$	−1.11992	−0.0570024
$387$	−63.1138	−3.20825
$388$	7.88139	0.400117
$389$	−19.9236	−1.01016	−0.505082	−	0.863071i	$-0.668538\pi$
$389$	−19.9236	−1.01016	−0.505082	+	0.863071i	$0.668538\pi$
$390$	0	0
$391$	4.89685	0.247644
$392$	15.5024	0.782992
$393$	37.0539	1.86912
$394$	3.70743	0.186778
$395$	0	0
$396$	−8.18974	−0.411550
$397$	−13.9941	−0.702342	−0.351171	−	0.936311i	$-0.614216\pi$
$397$	−13.9941	−0.702342	−0.351171	+	0.936311i	$0.614216\pi$
$398$	−1.84048	−0.0922549
$399$	3.18141	0.159270
$400$	0	0
$401$	−18.6293	−0.930301	−0.465151	−	0.885232i	$-0.654000\pi$
$401$	−18.6293	−0.930301	−0.465151	+	0.885232i	$0.654000\pi$
$402$	−26.2045	−1.30696
$403$	0.517380	0.0257725
$404$	−11.1452	−0.554492
$405$	0	0
$406$	−0.249637	−0.0123893
$407$	−4.50448	−0.223279
$408$	−35.7648	−1.77062
$409$	−30.8428	−1.52508	−0.762540	−	0.646941i	$-0.776048\pi$
$409$	−30.8428	−1.52508	−0.762540	+	0.646941i	$0.776048\pi$
$410$	0	0
$411$	−41.2226	−2.03336
$412$	8.61157	0.424262
$413$	1.22300	0.0601799
$414$	4.95356	0.243454
$415$	0	0
$416$	0.329642	0.0161620
$417$	47.6807	2.33493
$418$	−0.779856	−0.0381440
$419$	37.9347	1.85323	0.926616	−	0.376009i	$-0.122704\pi$
$419$	37.9347	1.85323	0.926616	+	0.376009i	$0.122704\pi$
$420$	0	0
$421$	−18.4642	−0.899891	−0.449945	−	0.893056i	$-0.648557\pi$
$421$	−18.4642	−0.899891	−0.449945	+	0.893056i	$0.648557\pi$
$422$	−19.1168	−0.930592
$423$	−17.0423	−0.828627
$424$	−32.5915	−1.58278
$425$	0	0
$426$	31.3683	1.51980
$427$	8.64134	0.418184
$428$	−10.1468	−0.490463
$429$	0.167887	0.00810564
$430$	0	0
$431$	33.6742	1.62203	0.811013	−	0.585027i	$-0.198916\pi$
$431$	33.6742	1.62203	0.811013	+	0.585027i	$0.198916\pi$
$432$	6.19671	0.298139
$433$	−11.4041	−0.548047	−0.274024	−	0.961723i	$-0.588355\pi$
$433$	−11.4041	−0.548047	−0.274024	+	0.961723i	$0.588355\pi$
$434$	−7.64588	−0.367014
$435$	0	0
$436$	−10.8548	−0.519849
$437$	−1.07949	−0.0516388
$438$	25.9108	1.23807
$439$	−3.76890	−0.179880	−0.0899399	−	0.995947i	$-0.528667\pi$
$439$	−3.76890	−0.179880	−0.0899399	+	0.995947i	$0.528667\pi$
$440$	0	0
$441$	−34.4856	−1.64217
$442$	0.199261	0.00947786
$443$	−4.51841	−0.214676	−0.107338	−	0.994223i	$-0.534233\pi$
$443$	−4.51841	−0.214676	−0.107338	+	0.994223i	$0.534233\pi$
$444$	18.6869	0.886842
$445$	0	0
$446$	−6.60871	−0.312931
$447$	33.8660	1.60181
$448$	−3.33271	−0.157456
$449$	−37.3611	−1.76318	−0.881590	−	0.472016i	$-0.843526\pi$
$449$	−37.3611	−1.76318	−0.881590	+	0.472016i	$0.843526\pi$
$450$	0	0
$451$	12.0009	0.565102
$452$	−0.599261	−0.0281869
$453$	13.8999	0.653075
$454$	−10.7223	−0.503222
$455$	0	0
$456$	7.88418	0.369210
$457$	−9.37773	−0.438672	−0.219336	−	0.975649i	$-0.570389\pi$
$457$	−9.37773	−0.438672	−0.219336	+	0.975649i	$0.570389\pi$
$458$	−1.55690	−0.0727494
$459$	38.9969	1.82022
$460$	0	0
$461$	31.6785	1.47542	0.737708	−	0.675120i	$-0.235908\pi$
$461$	31.6785	1.47542	0.737708	+	0.675120i	$0.235908\pi$
$462$	−2.48104	−0.115429
$463$	37.1284	1.72550	0.862752	−	0.505628i	$-0.168739\pi$
$463$	37.1284	1.72550	0.862752	+	0.505628i	$0.168739\pi$
$464$	0.216179	0.0100359
$465$	0	0
$466$	−8.84986	−0.409962
$467$	15.9678	0.738902	0.369451	−	0.929250i	$-0.379546\pi$
$467$	15.9678	0.738902	0.369451	+	0.929250i	$0.379546\pi$
$468$	−0.461294	−0.0213233
$469$	12.0327	0.555619
$470$	0	0
$471$	26.9809	1.24321
$472$	3.03084	0.139506
$473$	−10.7260	−0.493183
$474$	26.5570	1.21980
$475$	0	0
$476$	6.73900	0.308882
$477$	72.5006	3.31958
$478$	4.06340	0.185855
$479$	28.1729	1.28725	0.643627	−	0.765340i	$-0.277429\pi$
$479$	28.1729	1.28725	0.643627	+	0.765340i	$0.277429\pi$
$480$	0	0
$481$	−0.253718	−0.0115686
$482$	5.15350	0.234735
$483$	−3.43429	−0.156266
$484$	−1.39182	−0.0632648
$485$	0	0
$486$	−1.58419	−0.0718604
$487$	−24.0613	−1.09032	−0.545161	−	0.838331i	$-0.683532\pi$
$487$	−24.0613	−1.09032	−0.545161	+	0.838331i	$0.683532\pi$
$488$	21.4150	0.969410
$489$	−7.05699	−0.319128
$490$	0	0
$491$	−29.0197	−1.30964	−0.654821	−	0.755784i	$-0.727256\pi$
$491$	−29.0197	−1.30964	−0.654821	+	0.755784i	$0.727256\pi$
$492$	−49.7861	−2.24453
$493$	1.36045	0.0612716
$494$	−0.0439260	−0.00197632
$495$	0	0
$496$	6.62114	0.297298
$497$	−14.4039	−0.646102
$498$	32.4964	1.45620
$499$	−15.0257	−0.672642	−0.336321	−	0.941747i	$-0.609183\pi$
$499$	−15.0257	−0.672642	−0.336321	+	0.941747i	$0.609183\pi$
$500$	0	0
$501$	27.6412	1.23492
$502$	18.7184	0.835445
$503$	28.7785	1.28317	0.641585	−	0.767052i	$-0.278277\pi$
$503$	28.7785	1.28317	0.641585	+	0.767052i	$0.278277\pi$
$504$	16.6129	0.739996
$505$	0	0
$506$	0.841844	0.0374245
$507$	−38.7388	−1.72045
$508$	−4.36878	−0.193833
$509$	−10.4772	−0.464395	−0.232197	−	0.972669i	$-0.574592\pi$
$509$	−10.4772	−0.464395	−0.232197	+	0.972669i	$0.574592\pi$
$510$	0	0
$511$	−11.8979	−0.526332
$512$	8.03194	0.354965
$513$	−8.59667	−0.379552
$514$	4.43662	0.195691
$515$	0	0
$516$	44.4971	1.95888
$517$	−2.89630	−0.127379
$518$	3.74947	0.164742
$519$	−31.9440	−1.40219
$520$	0	0
$521$	−24.7590	−1.08471	−0.542357	−	0.840148i	$-0.682468\pi$
$521$	−24.7590	−1.08471	−0.542357	+	0.840148i	$0.682468\pi$
$522$	1.37621	0.0602349
$523$	−14.8566	−0.649635	−0.324818	−	0.945777i	$-0.605303\pi$
$523$	−14.8566	−0.649635	−0.324818	+	0.945777i	$0.605303\pi$
$524$	−17.3025	−0.755865
$525$	0	0
$526$	−10.9068	−0.475560
$527$	41.6679	1.81508
$528$	2.14852	0.0935023
$529$	−21.8347	−0.949335
$530$	0	0
$531$	−6.74219	−0.292586
$532$	−1.48558	−0.0644080
$533$	0.675962	0.0292792
$534$	−0.425807	−0.0184265
$535$	0	0
$536$	29.8195	1.28801
$537$	−68.2773	−2.94638
$538$	12.0113	0.517845
$539$	−5.86074	−0.252440
$540$	0	0
$541$	3.88960	0.167227	0.0836134	−	0.996498i	$-0.473354\pi$
$541$	3.88960	0.167227	0.0836134	+	0.996498i	$0.473354\pi$
$542$	−19.8014	−0.850541
$543$	17.6013	0.755343
$544$	26.5481	1.13824
$545$	0	0
$546$	−0.139747	−0.00598061
$547$	−17.5180	−0.749015	−0.374507	−	0.927224i	$-0.622188\pi$
$547$	−17.5180	−0.749015	−0.374507	+	0.927224i	$0.622188\pi$
$548$	19.2491	0.822283
$549$	−47.6382	−2.03315
$550$	0	0
$551$	−0.299905	−0.0127764
$552$	−8.51086	−0.362246
$553$	−12.1946	−0.518567
$554$	−15.5814	−0.661988
$555$	0	0
$556$	−22.2648	−0.944237
$557$	−28.9860	−1.22818	−0.614088	−	0.789237i	$-0.710476\pi$
$557$	−28.9860	−1.22818	−0.614088	+	0.789237i	$0.710476\pi$
$558$	42.1504	1.78437
$559$	−0.604152	−0.0255529
$560$	0	0
$561$	13.5210	0.570856
$562$	4.82052	0.203341
$563$	−29.6112	−1.24796	−0.623982	−	0.781438i	$-0.714486\pi$
$563$	−29.6112	−1.24796	−0.623982	+	0.781438i	$0.714486\pi$
$564$	12.0154	0.505938
$565$	0	0
$566$	5.55356	0.233434
$567$	−8.50793	−0.357299
$568$	−35.6957	−1.49776
$569$	−4.05818	−0.170128	−0.0850640	−	0.996375i	$-0.527109\pi$
$569$	−4.05818	−0.170128	−0.0850640	+	0.996375i	$0.527109\pi$
$570$	0	0
$571$	20.3380	0.851117	0.425559	−	0.904931i	$-0.360078\pi$
$571$	20.3380	0.851117	0.425559	+	0.904931i	$0.360078\pi$
$572$	−0.0783957	−0.00327789
$573$	19.2140	0.802677
$574$	−9.98942	−0.416950
$575$	0	0
$576$	18.3726	0.765527
$577$	−24.4768	−1.01898	−0.509492	−	0.860476i	$-0.670167\pi$
$577$	−24.4768	−1.01898	−0.509492	+	0.860476i	$0.670167\pi$
$578$	2.79017	0.116056
$579$	−4.28037	−0.177886
$580$	0	0
$581$	−14.9219	−0.619064
$582$	−13.1626	−0.545607
$583$	12.3213	0.510296
$584$	−29.4854	−1.22011
$585$	0	0
$586$	−14.9219	−0.616419
$587$	3.34628	0.138116	0.0690579	−	0.997613i	$-0.478001\pi$
$587$	3.34628	0.138116	0.0690579	+	0.997613i	$0.478001\pi$
$588$	24.3134	1.00267
$589$	−9.18548	−0.378481
$590$	0	0
$591$	14.1699	0.582872
$592$	−3.24695	−0.133449
$593$	39.2063	1.61001	0.805006	−	0.593267i	$-0.202162\pi$
$593$	39.2063	1.61001	0.805006	+	0.593267i	$0.202162\pi$
$594$	6.70416	0.275075
$595$	0	0
$596$	−15.8139	−0.647764
$597$	−7.03437	−0.287898
$598$	0.0474175	0.00193905
$599$	5.72987	0.234116	0.117058	−	0.993125i	$-0.462654\pi$
$599$	5.72987	0.234116	0.117058	+	0.993125i	$0.462654\pi$
$600$	0	0
$601$	13.9163	0.567656	0.283828	−	0.958875i	$-0.408395\pi$
$601$	13.9163	0.567656	0.283828	+	0.958875i	$0.408395\pi$
$602$	8.92820	0.363886
$603$	−66.3343	−2.70134
$604$	−6.49065	−0.264101
$605$	0	0
$606$	18.6134	0.756116
$607$	−0.156175	−0.00633897	−0.00316948	−	0.999995i	$-0.501009\pi$
$607$	−0.156175	−0.00633897	−0.00316948	+	0.999995i	$0.501009\pi$
$608$	−5.85241	−0.237347
$609$	−0.954120	−0.0386629
$610$	0	0
$611$	−0.163137	−0.00659980
$612$	−37.1509	−1.50174
$613$	−40.1902	−1.62327	−0.811634	−	0.584166i	$-0.801422\pi$
$613$	−40.1902	−1.62327	−0.811634	+	0.584166i	$0.801422\pi$
$614$	−5.32308	−0.214822
$615$	0	0
$616$	2.82331	0.113755
$617$	−12.8606	−0.517747	−0.258874	−	0.965911i	$-0.583351\pi$
$617$	−12.8606	−0.517747	−0.258874	+	0.965911i	$0.583351\pi$
$618$	−14.3820	−0.578531
$619$	−2.03398	−0.0817526	−0.0408763	−	0.999164i	$-0.513015\pi$
$619$	−2.03398	−0.0817526	−0.0408763	+	0.999164i	$0.513015\pi$
$620$	0	0
$621$	9.27999	0.372393
$622$	−10.0297	−0.402153
$623$	0.195524	0.00783352
$624$	0.121017	0.00484456
$625$	0	0
$626$	2.35288	0.0940398
$627$	−2.98063	−0.119035
$628$	−12.5989	−0.502750
$629$	−20.4336	−0.814739
$630$	0	0
$631$	37.6984	1.50075	0.750375	−	0.661012i	$-0.229873\pi$
$631$	37.6984	1.50075	0.750375	+	0.661012i	$0.229873\pi$
$632$	−30.2206	−1.20211
$633$	−73.0651	−2.90408
$634$	−13.8703	−0.550858
$635$	0	0
$636$	−51.1151	−2.02685
$637$	−0.330111	−0.0130795
$638$	0.233882	0.00925950
$639$	79.4060	3.14125
$640$	0	0
$641$	2.84360	0.112315	0.0561576	−	0.998422i	$-0.482115\pi$
$641$	2.84360	0.112315	0.0561576	+	0.998422i	$0.482115\pi$
$642$	16.9460	0.668804
$643$	−30.6389	−1.20828	−0.604140	−	0.796878i	$-0.706483\pi$
$643$	−30.6389	−1.20828	−0.604140	+	0.796878i	$0.706483\pi$
$644$	1.60366	0.0631932
$645$	0	0
$646$	−3.53764	−0.139187
$647$	6.76617	0.266006	0.133003	−	0.991116i	$-0.457538\pi$
$647$	6.76617	0.266006	0.133003	+	0.991116i	$0.457538\pi$
$648$	−21.0844	−0.828272
$649$	−1.14582	−0.0449772
$650$	0	0
$651$	−29.2228	−1.14533
$652$	3.29530	0.129054
$653$	−11.9015	−0.465743	−0.232872	−	0.972508i	$-0.574812\pi$
$653$	−11.9015	−0.465743	−0.232872	+	0.972508i	$0.574812\pi$
$654$	18.1284	0.708876
$655$	0	0
$656$	8.65059	0.337749
$657$	65.5910	2.55895
$658$	2.41085	0.0939845
$659$	−11.2353	−0.437664	−0.218832	−	0.975763i	$-0.570225\pi$
$659$	−11.2353	−0.437664	−0.218832	+	0.975763i	$0.570225\pi$
$660$	0	0
$661$	−22.9273	−0.891768	−0.445884	−	0.895091i	$-0.647111\pi$
$661$	−22.9273	−0.891768	−0.445884	+	0.895091i	$0.647111\pi$
$662$	−20.5772	−0.799757
$663$	0.761580	0.0295773
$664$	−36.9794	−1.43508
$665$	0	0
$666$	−20.6702	−0.800953
$667$	0.323743	0.0125354
$668$	−12.9072	−0.499396
$669$	−25.2587	−0.976558
$670$	0	0
$671$	−8.09599	−0.312542
$672$	−18.6189	−0.718240
$673$	3.09828	0.119430	0.0597150	−	0.998215i	$-0.480981\pi$
$673$	3.09828	0.119430	0.0597150	+	0.998215i	$0.480981\pi$
$674$	10.2960	0.396586
$675$	0	0
$676$	18.0893	0.695743
$677$	−33.3985	−1.28361	−0.641804	−	0.766868i	$-0.721814\pi$
$677$	−33.3985	−1.28361	−0.641804	+	0.766868i	$0.721814\pi$
$678$	1.00082	0.0384361
$679$	6.04407	0.231950
$680$	0	0
$681$	−40.9810	−1.57039
$682$	7.16335	0.274299
$683$	−6.01634	−0.230209	−0.115104	−	0.993353i	$-0.536720\pi$
$683$	−6.01634	−0.230209	−0.115104	+	0.993353i	$0.536720\pi$
$684$	8.18974	0.313143
$685$	0	0
$686$	10.7051	0.408723
$687$	−5.95054	−0.227027
$688$	−7.73160	−0.294765
$689$	0.694007	0.0264396
$690$	0	0
$691$	−15.6730	−0.596227	−0.298114	−	0.954530i	$-0.596357\pi$
$691$	−15.6730	−0.596227	−0.298114	+	0.954530i	$0.596357\pi$
$692$	14.9164	0.567038
$693$	−6.28054	−0.238578
$694$	−12.8527	−0.487882
$695$	0	0
$696$	−2.36450	−0.0896262
$697$	54.4395	2.06204
$698$	−14.8302	−0.561330
$699$	−33.8244	−1.27936
$700$	0	0
$701$	18.0567	0.681992	0.340996	−	0.940065i	$-0.389236\pi$
$701$	18.0567	0.681992	0.340996	+	0.940065i	$0.389236\pi$
$702$	0.377617	0.0142523
$703$	4.50448	0.169890
$704$	3.12238	0.117679
$705$	0	0
$706$	−10.2906	−0.387291
$707$	−8.54699	−0.321442
$708$	4.75344	0.178645
$709$	16.2048	0.608586	0.304293	−	0.952579i	$-0.401580\pi$
$709$	16.2048	0.608586	0.304293	+	0.952579i	$0.401580\pi$
$710$	0	0
$711$	67.2266	2.52120
$712$	0.484549	0.0181592
$713$	9.91560	0.371342
$714$	−11.2547	−0.421196
$715$	0	0
$716$	31.8825	1.19150
$717$	15.5304	0.579995
$718$	−2.25188	−0.0840395
$719$	−27.9403	−1.04200	−0.520998	−	0.853558i	$-0.674440\pi$
$719$	−27.9403	−1.04200	−0.520998	+	0.853558i	$0.674440\pi$
$720$	0	0
$721$	6.60403	0.245947
$722$	0.779856	0.0290232
$723$	19.6968	0.732533
$724$	−8.21903	−0.305458
$725$	0	0
$726$	2.32446	0.0862690
$727$	30.7020	1.13867	0.569337	−	0.822104i	$-0.307200\pi$
$727$	30.7020	1.13867	0.569337	+	0.822104i	$0.307200\pi$
$728$	0.159025	0.00589387
$729$	−29.9678	−1.10992
$730$	0	0
$731$	−48.6562	−1.79961
$732$	33.5864	1.24139
$733$	−44.9333	−1.65965	−0.829826	−	0.558023i	$-0.811560\pi$
$733$	−44.9333	−1.65965	−0.829826	+	0.558023i	$0.811560\pi$
$734$	1.64419	0.0606882
$735$	0	0
$736$	6.31760	0.232870
$737$	−11.2733	−0.415258
$738$	55.0700	2.02715
$739$	−3.19172	−0.117409	−0.0587047	−	0.998275i	$-0.518697\pi$
$739$	−3.19172	−0.117409	−0.0587047	+	0.998275i	$0.518697\pi$
$740$	0	0
$741$	−0.167887	−0.00616747
$742$	−10.2561	−0.376513
$743$	−17.4348	−0.639619	−0.319810	−	0.947482i	$-0.603619\pi$
$743$	−17.4348	−0.639619	−0.319810	+	0.947482i	$0.603619\pi$
$744$	−72.4199	−2.65504
$745$	0	0
$746$	−6.29581	−0.230506
$747$	82.2617	3.00980
$748$	−6.31370	−0.230852
$749$	−7.78135	−0.284324
$750$	0	0
$751$	9.55633	0.348715	0.174358	−	0.984682i	$-0.444215\pi$
$751$	9.55633	0.348715	0.174358	+	0.984682i	$0.444215\pi$
$752$	−2.08773	−0.0761317
$753$	71.5425	2.60715
$754$	0.0131736	0.000479755 0
$755$	0	0
$756$	12.7710	0.464478
$757$	48.5259	1.76370	0.881852	−	0.471527i	$-0.156297\pi$
$757$	48.5259	1.76370	0.881852	+	0.471527i	$0.156297\pi$
$758$	11.1796	0.406062
$759$	3.21755	0.116790
$760$	0	0
$761$	38.6035	1.39937	0.699687	−	0.714449i	$-0.253323\pi$
$761$	38.6035	1.39937	0.699687	+	0.714449i	$0.253323\pi$
$762$	7.29622	0.264314
$763$	−8.32429	−0.301360
$764$	−8.97210	−0.324599
$765$	0	0
$766$	7.64155	0.276101
$767$	−0.0645391	−0.00233037
$768$	−40.1607	−1.44917
$769$	53.2658	1.92081	0.960406	−	0.278603i	$-0.0898712\pi$
$769$	53.2658	1.92081	0.960406	+	0.278603i	$0.0898712\pi$
$770$	0	0
$771$	16.9569	0.610688
$772$	1.99874	0.0719363
$773$	−9.31748	−0.335126	−0.167563	−	0.985861i	$-0.553590\pi$
$773$	−9.31748	−0.335126	−0.167563	+	0.985861i	$0.553590\pi$
$774$	−49.2196	−1.76916
$775$	0	0
$776$	14.9784	0.537694
$777$	14.3306	0.514107
$778$	−15.5375	−0.557047
$779$	−12.0009	−0.429978
$780$	0	0
$781$	13.4948	0.482883
$782$	3.81884	0.136561
$783$	2.57818	0.0921367
$784$	−4.22458	−0.150878
$785$	0	0
$786$	28.8967	1.03071
$787$	15.0616	0.536889	0.268444	−	0.963295i	$-0.413490\pi$
$787$	15.0616	0.536889	0.268444	+	0.963295i	$0.413490\pi$
$788$	−6.61672	−0.235711
$789$	−41.6862	−1.48407
$790$	0	0
$791$	−0.459561	−0.0163401
$792$	−15.5644	−0.553058
$793$	−0.456013	−0.0161935
$794$	−10.9133	−0.387300
$795$	0	0
$796$	3.28474	0.116425
$797$	19.0593	0.675114	0.337557	−	0.941305i	$-0.390399\pi$
$797$	19.0593	0.675114	0.337557	+	0.941305i	$0.390399\pi$
$798$	2.48104	0.0878279
$799$	−13.1384	−0.464804
$800$	0	0
$801$	−1.07789	−0.0380855
$802$	−14.5281	−0.513006
$803$	11.1470	0.393370
$804$	46.7676	1.64937
$805$	0	0
$806$	0.403481	0.0142120
$807$	45.9077	1.61603
$808$	−21.1811	−0.745150
$809$	−15.2273	−0.535363	−0.267681	−	0.963507i	$-0.586257\pi$
$809$	−15.2273	−0.535363	−0.267681	+	0.963507i	$0.586257\pi$
$810$	0	0
$811$	−43.3354	−1.52171	−0.760857	−	0.648920i	$-0.775221\pi$
$811$	−43.3354	−1.52171	−0.760857	+	0.648920i	$0.775221\pi$
$812$	0.445532	0.0156351
$813$	−75.6814	−2.65426
$814$	−3.51284	−0.123125
$815$	0	0
$816$	9.74628	0.341188
$817$	10.7260	0.375256
$818$	−24.0530	−0.840992
$819$	−0.353756	−0.0123612
$820$	0	0
$821$	19.4279	0.678040	0.339020	−	0.940779i	$-0.389905\pi$
$821$	19.4279	0.678040	0.339020	+	0.940779i	$0.389905\pi$
$822$	−32.1477	−1.12128
$823$	−2.35683	−0.0821539	−0.0410769	−	0.999156i	$-0.513079\pi$
$823$	−2.35683	−0.0821539	−0.0410769	+	0.999156i	$0.513079\pi$
$824$	16.3661	0.570141
$825$	0	0
$826$	0.953763	0.0331857
$827$	29.8127	1.03669	0.518344	−	0.855172i	$-0.326549\pi$
$827$	29.8127	1.03669	0.518344	+	0.855172i	$0.326549\pi$
$828$	−8.84072	−0.307236
$829$	52.6510	1.82864	0.914322	−	0.404988i	$-0.132724\pi$
$829$	52.6510	1.82864	0.914322	+	0.404988i	$0.132724\pi$
$830$	0	0
$831$	−59.5524	−2.06585
$832$	0.175871	0.00609722
$833$	−26.5859	−0.921148
$834$	37.1841	1.28758
$835$	0	0
$836$	1.39182	0.0481373
$837$	78.9645	2.72941
$838$	29.5836	1.02195
$839$	29.7892	1.02844	0.514218	−	0.857660i	$-0.328082\pi$
$839$	29.7892	1.02844	0.514218	+	0.857660i	$0.328082\pi$
$840$	0	0
$841$	−28.9101	−0.996899
$842$	−14.3994	−0.496237
$843$	18.4242	0.634563
$844$	34.1182	1.17440
$845$	0	0
$846$	−13.2906	−0.456939
$847$	−1.06736	−0.0366749
$848$	8.88152	0.304992
$849$	21.2259	0.728471
$850$	0	0
$851$	−4.86252	−0.166685
$852$	−55.9836	−1.91797
$853$	35.2393	1.20657	0.603285	−	0.797526i	$-0.293858\pi$
$853$	35.2393	1.20657	0.603285	+	0.797526i	$0.293858\pi$
$854$	6.73900	0.230604
$855$	0	0
$856$	−19.2837	−0.659105
$857$	21.5877	0.737421	0.368711	−	0.929544i	$-0.379799\pi$
$857$	21.5877	0.737421	0.368711	+	0.929544i	$0.379799\pi$
$858$	0.130927	0.00446979
$859$	27.7669	0.947396	0.473698	−	0.880687i	$-0.342919\pi$
$859$	27.7669	0.947396	0.473698	+	0.880687i	$0.342919\pi$
$860$	0	0
$861$	−38.1799	−1.30117
$862$	26.2610	0.894453
$863$	−7.11774	−0.242291	−0.121145	−	0.992635i	$-0.538657\pi$
$863$	−7.11774	−0.242291	−0.121145	+	0.992635i	$0.538657\pi$
$864$	50.3112	1.71162
$865$	0	0
$866$	−8.89357	−0.302216
$867$	10.6641	0.362173
$868$	13.6458	0.463167
$869$	11.4250	0.387566
$870$	0	0
$871$	−0.634980	−0.0215155
$872$	−20.6293	−0.698595
$873$	−33.3199	−1.12771
$874$	−0.841844	−0.0284758
$875$	0	0
$876$	−46.2436	−1.56243
$877$	44.1031	1.48925	0.744627	−	0.667481i	$-0.232627\pi$
$877$	44.1031	1.48925	0.744627	+	0.667481i	$0.232627\pi$
$878$	−2.93920	−0.0991931
$879$	−57.0321	−1.92364
$880$	0	0
$881$	−37.9204	−1.27757	−0.638785	−	0.769385i	$-0.720563\pi$
$881$	−37.9204	−1.27757	−0.638785	+	0.769385i	$0.720563\pi$
$882$	−26.8938	−0.905562
$883$	−4.85252	−0.163300	−0.0816502	−	0.996661i	$-0.526019\pi$
$883$	−4.85252	−0.163300	−0.0816502	+	0.996661i	$0.526019\pi$
$884$	−0.355624	−0.0119609
$885$	0	0
$886$	−3.52371	−0.118381
$887$	−11.2754	−0.378591	−0.189295	−	0.981920i	$-0.560620\pi$
$887$	−11.2754	−0.378591	−0.189295	+	0.981920i	$0.560620\pi$
$888$	35.5141	1.19177
$889$	−3.35032	−0.112366
$890$	0	0
$891$	7.97100	0.267038
$892$	11.7947	0.394916
$893$	2.89630	0.0969210
$894$	26.4106	0.883302
$895$	0	0
$896$	9.89423	0.330543
$897$	0.181231	0.00605114
$898$	−29.1363	−0.972290
$899$	2.75477	0.0918767
$900$	0	0
$901$	55.8928	1.86206
$902$	9.35899	0.311620
$903$	34.1239	1.13557
$904$	−1.13888	−0.0378787
$905$	0	0
$906$	10.8399	0.360133
$907$	−24.0191	−0.797540	−0.398770	−	0.917051i	$-0.630563\pi$
$907$	−24.0191	−0.797540	−0.398770	+	0.917051i	$0.630563\pi$
$908$	19.1363	0.635061
$909$	47.1181	1.56281
$910$	0	0
$911$	7.48540	0.248002	0.124001	−	0.992282i	$-0.460427\pi$
$911$	7.48540	0.248002	0.124001	+	0.992282i	$0.460427\pi$
$912$	−2.14852	−0.0711446
$913$	13.9802	0.462676
$914$	−7.31328	−0.241902
$915$	0	0
$916$	2.77864	0.0918089
$917$	−13.2689	−0.438179
$918$	30.4119	1.00374
$919$	40.0187	1.32009	0.660047	−	0.751224i	$-0.270536\pi$
$919$	40.0187	1.32009	0.660047	+	0.751224i	$0.270536\pi$
$920$	0	0
$921$	−20.3450	−0.670390
$922$	24.7047	0.813605
$923$	0.760108	0.0250193
$924$	4.42797	0.145669
$925$	0	0
$926$	28.9548	0.951514
$927$	−36.4069	−1.19576
$928$	1.75517	0.0576161
$929$	−32.3099	−1.06005	−0.530027	−	0.847981i	$-0.677818\pi$
$929$	−32.3099	−1.06005	−0.530027	+	0.847981i	$0.677818\pi$
$930$	0	0
$931$	5.86074	0.192078
$932$	15.7945	0.517367
$933$	−38.3337	−1.25499
$934$	12.4526	0.407461
$935$	0	0
$936$	−0.876679	−0.0286552
$937$	54.1000	1.76737	0.883684	−	0.468083i	$-0.155055\pi$
$937$	54.1000	1.76737	0.883684	+	0.468083i	$0.155055\pi$
$938$	9.38378	0.306391
$939$	8.99277	0.293468
$940$	0	0
$941$	26.6955	0.870248	0.435124	−	0.900371i	$-0.356705\pi$
$941$	26.6955	0.870248	0.435124	+	0.900371i	$0.356705\pi$
$942$	21.0412	0.685559
$943$	12.9548	0.421868
$944$	−0.825935	−0.0268819
$945$	0	0
$946$	−8.36475	−0.271961
$947$	39.4463	1.28183	0.640916	−	0.767611i	$-0.278555\pi$
$947$	39.4463	1.28183	0.640916	+	0.767611i	$0.278555\pi$
$948$	−47.3968	−1.53938
$949$	0.627865	0.0203814
$950$	0	0
$951$	−53.0126	−1.71905
$952$	12.8073	0.415088
$953$	48.7523	1.57924	0.789621	−	0.613595i	$-0.210277\pi$
$953$	48.7523	1.57924	0.789621	+	0.613595i	$0.210277\pi$
$954$	56.5400	1.83055
$955$	0	0
$956$	−7.25203	−0.234547
$957$	0.893906	0.0288959
$958$	21.9708	0.709845
$959$	14.7618	0.476682
$960$	0	0
$961$	53.3730	1.72171
$962$	−0.197864	−0.00637938
$963$	42.8972	1.38234
$964$	−9.19755	−0.296233
$965$	0	0
$966$	−2.67825	−0.0861713
$967$	1.39478	0.0448532	0.0224266	−	0.999748i	$-0.492861\pi$
$967$	1.39478	0.0448532	0.0224266	+	0.999748i	$0.492861\pi$
$968$	−2.64513	−0.0850178
$969$	−13.5210	−0.434356
$970$	0	0
$971$	−3.11733	−0.100040	−0.0500200	−	0.998748i	$-0.515929\pi$
$971$	−3.11733	−0.100040	−0.0500200	+	0.998748i	$0.515929\pi$
$972$	2.82734	0.0906870
$973$	−17.0744	−0.547380
$974$	−18.7644	−0.601249
$975$	0	0
$976$	−5.83580	−0.186800
$977$	22.3690	0.715647	0.357823	−	0.933789i	$-0.383519\pi$
$977$	22.3690	0.715647	0.357823	+	0.933789i	$0.383519\pi$
$978$	−5.50343	−0.175980
$979$	−0.183185	−0.00585462
$980$	0	0
$981$	45.8904	1.46517
$982$	−22.6312	−0.722191
$983$	−42.2421	−1.34731	−0.673657	−	0.739044i	$-0.735277\pi$
$983$	−42.2421	−1.34731	−0.673657	+	0.739044i	$0.735277\pi$
$984$	−94.6174	−3.01629
$985$	0	0
$986$	1.06096	0.0337877
$987$	9.21433	0.293295
$988$	0.0783957	0.00249410
$989$	−11.5786	−0.368178
$990$	0	0
$991$	−1.69828	−0.0539477	−0.0269738	−	0.999636i	$-0.508587\pi$
$991$	−1.69828	−0.0539477	−0.0269738	+	0.999636i	$0.508587\pi$
$992$	53.7572	1.70679
$993$	−78.6469	−2.49578
$994$	−11.2329	−0.356287
$995$	0	0
$996$	−57.9970	−1.83770
$997$	−18.9376	−0.599759	−0.299879	−	0.953977i	$-0.596946\pi$
$997$	−18.9376	−0.599759	−0.299879	+	0.953977i	$0.596946\pi$
$998$	−11.7179	−0.370923
$999$	−38.7235	−1.22516

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.h.1.4		5
5.4	even	2		209.2.a.c.1.2	✓	5
15.14	odd	2		1881.2.a.k.1.4		5
20.19	odd	2		3344.2.a.t.1.5		5
55.54	odd	2		2299.2.a.n.1.4		5
95.94	odd	2		3971.2.a.h.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.2	✓	5	5.4	even	2
1881.2.a.k.1.4		5	15.14	odd	2
2299.2.a.n.1.4		5	55.54	odd	2
3344.2.a.t.1.5		5	20.19	odd	2
3971.2.a.h.1.4		5	95.94	odd	2
5225.2.a.h.1.4		5	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)

\(f(q)\)	\(=\)	\(q+0.779856 q^{2} +2.98063 q^{3} -1.39182 q^{4} +2.32446 q^{6} -1.06736 q^{7} -2.64513 q^{8} +5.88418 q^{9} +1.00000 q^{11} -4.14852 q^{12} +0.0563258 q^{13} -0.832387 q^{14} +0.720827 q^{16} +4.53628 q^{17} +4.58881 q^{18} -1.00000 q^{19} -3.18141 q^{21} +0.779856 q^{22} +1.07949 q^{23} -7.88418 q^{24} +0.0439260 q^{26} +8.59667 q^{27} +1.48558 q^{28} +0.299905 q^{29} +9.18548 q^{31} +5.85241 q^{32} +2.98063 q^{33} +3.53764 q^{34} -8.18974 q^{36} -4.50448 q^{37} -0.779856 q^{38} +0.167887 q^{39} +12.0009 q^{41} -2.48104 q^{42} -10.7260 q^{43} -1.39182 q^{44} +0.841844 q^{46} -2.89630 q^{47} +2.14852 q^{48} -5.86074 q^{49} +13.5210 q^{51} -0.0783957 q^{52} +12.3213 q^{53} +6.70416 q^{54} +2.82331 q^{56} -2.98063 q^{57} +0.233882 q^{58} -1.14582 q^{59} -8.09599 q^{61} +7.16335 q^{62} -6.28054 q^{63} +3.12238 q^{64} +2.32446 q^{66} -11.2733 q^{67} -6.31370 q^{68} +3.21755 q^{69} +13.4948 q^{71} -15.5644 q^{72} +11.1470 q^{73} -3.51284 q^{74} +1.39182 q^{76} -1.06736 q^{77} +0.130927 q^{78} +11.4250 q^{79} +7.97100 q^{81} +9.35899 q^{82} +13.9802 q^{83} +4.42797 q^{84} -8.36475 q^{86} +0.893906 q^{87} -2.64513 q^{88} -0.183185 q^{89} -0.0601200 q^{91} -1.50246 q^{92} +27.3785 q^{93} -2.25870 q^{94} +17.4439 q^{96} -5.66263 q^{97} -4.57053 q^{98} +5.88418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} - 6 q^{12} - 4 q^{13} - 14 q^{14} + 8 q^{16} + 4 q^{17} + 20 q^{18} - 5 q^{19} + 10 q^{21} - 2 q^{22} - 3 q^{23} - 14 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - q^3 + 6 * q^4 - 2 * q^6 - 6 * q^7 - 6 * q^8 + 4 * q^9 + 5 * q^11 - 6 * q^12 - 4 * q^13 - 14 * q^14 + 8 * q^16 + 4 * q^17 + 20 * q^18 - 5 * q^19 + 10 * q^21 - 2 * q^22 - 3 * q^23 - 14 * q^24 - 6 * q^26 + 11 * q^27 + 10 * q^28 + 10 * q^29 + 11 * q^31 - 14 * q^32 - q^33 - 4 * q^34 - 26 * q^36 - q^37 + 2 * q^38 + 2 * q^39 + 2 * q^41 + 16 * q^42 - 20 * q^43 + 6 * q^44 - 4 * q^46 + 20 * q^47 - 4 * q^48 + 3 * q^49 + 24 * q^51 - 6 * q^52 + 14 * q^53 + 16 * q^54 - 38 * q^56 + q^57 + 6 * q^58 + 3 * q^59 - 10 * q^61 + 6 * q^62 - 24 * q^63 - 2 * q^66 - 9 * q^67 - 24 * q^68 - 5 * q^69 + 23 * q^71 + 12 * q^72 + 8 * q^74 - 6 * q^76 - 6 * q^77 + 22 * q^78 + 44 * q^79 + q^81 + 30 * q^82 + 14 * q^83 + 14 * q^84 + 52 * q^86 - 28 * q^87 - 6 * q^88 - 27 * q^89 + 24 * q^91 - 58 * q^92 + 27 * q^93 - 8 * q^94 + 50 * q^96 - 15 * q^97 + 10 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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