LMFDB - Embedded newform 775.2.a.l.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [775,2,Mod(1,775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.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:	$6.18840615665$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 16x^{8} + 88x^{6} - 183x^{4} + 92x^{2} - 4 $ x^10 - 16x^8 + 88x^6 - 183x^4 + 92x^2 - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.805123$ of defining polynomial
Character	$\chi$	$=$	775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.805123 q^{2} +0.681921 q^{3} -1.35178 q^{4} +0.549030 q^{6} +0.986124 q^{7} -2.69859 q^{8} -2.53498 q^{9} +O(q^{10})$	\(q+0.805123 q^{2} +0.681921 q^{3} -1.35178 q^{4} +0.549030 q^{6} +0.986124 q^{7} -2.69859 q^{8} -2.53498 q^{9} +2.38926 q^{11} -0.921805 q^{12} +4.76974 q^{13} +0.793951 q^{14} +0.530857 q^{16} +3.84142 q^{17} -2.04097 q^{18} +8.19613 q^{19} +0.672458 q^{21} +1.92364 q^{22} +4.45634 q^{23} -1.84023 q^{24} +3.84023 q^{26} -3.77442 q^{27} -1.33302 q^{28} +3.45097 q^{29} -1.00000 q^{31} +5.82459 q^{32} +1.62928 q^{33} +3.09281 q^{34} +3.42673 q^{36} -10.8488 q^{37} +6.59889 q^{38} +3.25258 q^{39} +0.896680 q^{41} +0.541412 q^{42} -4.76322 q^{43} -3.22974 q^{44} +3.58790 q^{46} -7.25383 q^{47} +0.362002 q^{48} -6.02756 q^{49} +2.61954 q^{51} -6.44763 q^{52} -5.40370 q^{53} -3.03887 q^{54} -2.66115 q^{56} +5.58911 q^{57} +2.77845 q^{58} +12.3331 q^{59} +10.0083 q^{61} -0.805123 q^{62} -2.49981 q^{63} +3.62779 q^{64} +1.31177 q^{66} +2.55109 q^{67} -5.19274 q^{68} +3.03887 q^{69} -1.98949 q^{71} +6.84089 q^{72} -9.23260 q^{73} -8.73465 q^{74} -11.0793 q^{76} +2.35610 q^{77} +2.61873 q^{78} -3.09281 q^{79} +5.03110 q^{81} +0.721938 q^{82} -0.367875 q^{83} -0.909014 q^{84} -3.83498 q^{86} +2.35329 q^{87} -6.44763 q^{88} +4.54903 q^{89} +4.70355 q^{91} -6.02398 q^{92} -0.681921 q^{93} -5.84023 q^{94} +3.97191 q^{96} +16.1212 q^{97} -4.85293 q^{98} -6.05673 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 12 q^{4} + 8 q^{6} + 22 q^{9}+O(q^{10}) $ 10 * q + 12 * q^4 + 8 * q^6 + 22 * q^9	$ 10 q + 12 q^{4} + 8 q^{6} + 22 q^{9} + 16 q^{11} - 6 q^{14} + 8 q^{16} - 4 q^{19} + 20 q^{21} - 8 q^{24} + 28 q^{26} + 32 q^{29} - 10 q^{31} - 28 q^{34} + 44 q^{36} - 16 q^{39} + 36 q^{41} + 52 q^{44} + 8 q^{46} + 22 q^{49} + 20 q^{51} + 12 q^{56} + 12 q^{59} - 70 q^{64} + 12 q^{71} + 28 q^{74} - 30 q^{76} + 28 q^{79} - 14 q^{81} - 40 q^{84} + 36 q^{86} + 48 q^{89} - 4 q^{91} - 48 q^{94} - 60 q^{96} + 64 q^{99}+O(q^{100}) $ 10 * q + 12 * q^4 + 8 * q^6 + 22 * q^9 + 16 * q^11 - 6 * q^14 + 8 * q^16 - 4 * q^19 + 20 * q^21 - 8 * q^24 + 28 * q^26 + 32 * q^29 - 10 * q^31 - 28 * q^34 + 44 * q^36 - 16 * q^39 + 36 * q^41 + 52 * q^44 + 8 * q^46 + 22 * q^49 + 20 * q^51 + 12 * q^56 + 12 * q^59 - 70 * q^64 + 12 * q^71 + 28 * q^74 - 30 * q^76 + 28 * q^79 - 14 * q^81 - 40 * q^84 + 36 * q^86 + 48 * q^89 - 4 * q^91 - 48 * q^94 - 60 * q^96 + 64 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.805123	0.569308	0.284654	−	0.958630i	$-0.408121\pi$
$2$	0.805123	0.569308	0.284654	+	0.958630i	$0.408121\pi$
$3$	0.681921	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$3$	0.681921	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$4$	−1.35178	−0.675889
$5$	0	0
$5$	0	0
$6$	0.549030	0.224141
$7$	0.986124	0.372720	0.186360	−	0.982482i	$-0.440331\pi$
$7$	0.986124	0.372720	0.186360	+	0.982482i	$0.440331\pi$
$8$	−2.69859	−0.954096
$9$	−2.53498	−0.844995
$10$	0	0
$11$	2.38926	0.720388	0.360194	−	0.932877i	$-0.382711\pi$
$11$	2.38926	0.720388	0.360194	+	0.932877i	$0.382711\pi$
$12$	−0.921805	−0.266102
$13$	4.76974	1.32289	0.661444	−	0.749995i	$-0.269944\pi$
$13$	4.76974	1.32289	0.661444	+	0.749995i	$0.269944\pi$
$14$	0.793951	0.212192
$15$	0	0
$16$	0.530857	0.132714
$17$	3.84142	0.931680	0.465840	−	0.884869i	$-0.345752\pi$
$17$	3.84142	0.931680	0.465840	+	0.884869i	$0.345752\pi$
$18$	−2.04097	−0.481062
$19$	8.19613	1.88032	0.940161	−	0.340732i	$-0.110675\pi$
$19$	8.19613	1.88032	0.940161	+	0.340732i	$0.110675\pi$
$20$	0	0
$21$	0.672458	0.146742
$22$	1.92364	0.410122
$23$	4.45634	0.929211	0.464606	−	0.885518i	$-0.346196\pi$
$23$	4.45634	0.929211	0.464606	+	0.885518i	$0.346196\pi$
$24$	−1.84023	−0.375635
$25$	0	0
$26$	3.84023	0.753130
$27$	−3.77442	−0.726388
$28$	−1.33302	−0.251917
$29$	3.45097	0.640829	0.320415	−	0.947277i	$-0.396178\pi$
$29$	3.45097	0.640829	0.320415	+	0.947277i	$0.396178\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	5.82459	1.02965
$33$	1.62928	0.283622
$34$	3.09281	0.530413
$35$	0	0
$36$	3.42673	0.571122
$37$	−10.8488	−1.78354	−0.891770	−	0.452490i	$-0.850536\pi$
$37$	−10.8488	−1.78354	−0.891770	+	0.452490i	$0.850536\pi$
$38$	6.59889	1.07048
$39$	3.25258	0.520830
$40$	0	0
$41$	0.896680	0.140038	0.0700190	−	0.997546i	$-0.477694\pi$
$41$	0.896680	0.140038	0.0700190	+	0.997546i	$0.477694\pi$
$42$	0.541412	0.0835416
$43$	−4.76322	−0.726384	−0.363192	−	0.931714i	$-0.618313\pi$
$43$	−4.76322	−0.726384	−0.363192	+	0.931714i	$0.618313\pi$
$44$	−3.22974	−0.486902
$45$	0	0
$46$	3.58790	0.529007
$47$	−7.25383	−1.05808	−0.529040	−	0.848597i	$-0.677448\pi$
$47$	−7.25383	−1.05808	−0.529040	+	0.848597i	$0.677448\pi$
$48$	0.362002	0.0522506
$49$	−6.02756	−0.861080
$50$	0	0
$51$	2.61954	0.366809
$52$	−6.44763	−0.894125
$53$	−5.40370	−0.742256	−0.371128	−	0.928582i	$-0.621029\pi$
$53$	−5.40370	−0.742256	−0.371128	+	0.928582i	$0.621029\pi$
$54$	−3.03887	−0.413538
$55$	0	0
$56$	−2.66115	−0.355611
$57$	5.58911	0.740296
$58$	2.77845	0.364829
$59$	12.3331	1.60563	0.802814	−	0.596230i	$-0.203335\pi$
$59$	12.3331	1.60563	0.802814	+	0.596230i	$0.203335\pi$
$60$	0	0
$61$	10.0083	1.28143	0.640713	−	0.767781i	$-0.278639\pi$
$61$	10.0083	1.28143	0.640713	+	0.767781i	$0.278639\pi$
$62$	−0.805123	−0.102251
$63$	−2.49981	−0.314946
$64$	3.62779	0.453474
$65$	0	0
$66$	1.31177	0.161468
$67$	2.55109	0.311665	0.155833	−	0.987783i	$-0.450194\pi$
$67$	2.55109	0.311665	0.155833	+	0.987783i	$0.450194\pi$
$68$	−5.19274	−0.629712
$69$	3.03887	0.365837
$70$	0	0
$71$	−1.98949	−0.236109	−0.118055	−	0.993007i	$-0.537666\pi$
$71$	−1.98949	−0.236109	−0.118055	+	0.993007i	$0.537666\pi$
$72$	6.84089	0.806206
$73$	−9.23260	−1.08059	−0.540297	−	0.841474i	$-0.681688\pi$
$73$	−9.23260	−1.08059	−0.540297	+	0.841474i	$0.681688\pi$
$74$	−8.73465	−1.01538
$75$	0	0
$76$	−11.0793	−1.27089
$77$	2.35610	0.268503
$78$	2.61873	0.296513
$79$	−3.09281	−0.347968	−0.173984	−	0.984748i	$-0.555664\pi$
$79$	−3.09281	−0.347968	−0.173984	+	0.984748i	$0.555664\pi$
$80$	0	0
$81$	5.03110	0.559011
$82$	0.721938	0.0797247
$83$	−0.367875	−0.0403796	−0.0201898	−	0.999796i	$-0.506427\pi$
$83$	−0.367875	−0.0403796	−0.0201898	+	0.999796i	$0.506427\pi$
$84$	−0.909014	−0.0991816
$85$	0	0
$86$	−3.83498	−0.413536
$87$	2.35329	0.252299
$88$	−6.44763	−0.687319
$89$	4.54903	0.482196	0.241098	−	0.970501i	$-0.422492\pi$
$89$	4.54903	0.482196	0.241098	+	0.970501i	$0.422492\pi$
$90$	0	0
$91$	4.70355	0.493067
$92$	−6.02398	−0.628044
$93$	−0.681921	−0.0707119
$94$	−5.84023	−0.602373
$95$	0	0
$96$	3.97191	0.405381
$97$	16.1212	1.63686	0.818432	−	0.574604i	$-0.194844\pi$
$97$	16.1212	1.63686	0.818432	+	0.574604i	$0.194844\pi$
$98$	−4.85293	−0.490219
$99$	−6.05673	−0.608724
$100$	0	0
$101$	2.59064	0.257778	0.128889	−	0.991659i	$-0.458859\pi$
$101$	2.59064	0.257778	0.128889	+	0.991659i	$0.458859\pi$
$102$	2.10905	0.208827
$103$	10.7910	1.06327	0.531637	−	0.846973i	$-0.321577\pi$
$103$	10.7910	1.06327	0.531637	+	0.846973i	$0.321577\pi$
$104$	−12.8716	−1.26216
$105$	0	0
$106$	−4.35064	−0.422572
$107$	−5.68887	−0.549964	−0.274982	−	0.961449i	$-0.588672\pi$
$107$	−5.68887	−0.549964	−0.274982	+	0.961449i	$0.588672\pi$
$108$	5.10218	0.490957
$109$	−3.24025	−0.310360	−0.155180	−	0.987886i	$-0.549596\pi$
$109$	−3.24025	−0.310360	−0.155180	+	0.987886i	$0.549596\pi$
$110$	0	0
$111$	−7.39805	−0.702192
$112$	0.523491	0.0494652
$113$	−17.2196	−1.61989	−0.809943	−	0.586508i	$-0.800502\pi$
$113$	−17.2196	−1.61989	−0.809943	+	0.586508i	$0.800502\pi$
$114$	4.49992	0.421456
$115$	0	0
$116$	−4.66494	−0.433129
$117$	−12.0912	−1.11783
$118$	9.92963	0.914096
$119$	3.78811	0.347256
$120$	0	0
$121$	−5.29146	−0.481041
$122$	8.05787	0.729525
$123$	0.611465	0.0551339
$124$	1.35178	0.121393
$125$	0	0
$126$	−2.01265	−0.179301
$127$	−10.2339	−0.908113	−0.454057	−	0.890973i	$-0.650024\pi$
$127$	−10.2339	−0.908113	−0.454057	+	0.890973i	$0.650024\pi$
$128$	−8.72836	−0.771485
$129$	−3.24814	−0.285983
$130$	0	0
$131$	2.43692	0.212915	0.106458	−	0.994317i	$-0.466049\pi$
$131$	2.43692	0.212915	0.106458	+	0.994317i	$0.466049\pi$
$132$	−2.20243	−0.191697
$133$	8.08240	0.700833
$134$	2.05394	0.177433
$135$	0	0
$136$	−10.3664	−0.888913
$137$	−5.47070	−0.467393	−0.233697	−	0.972310i	$-0.575082\pi$
$137$	−5.47070	−0.467393	−0.233697	+	0.972310i	$0.575082\pi$
$138$	2.44666	0.208274
$139$	−4.72640	−0.400888	−0.200444	−	0.979705i	$-0.564238\pi$
$139$	−4.72640	−0.400888	−0.200444	+	0.979705i	$0.564238\pi$
$140$	0	0
$141$	−4.94654	−0.416574
$142$	−1.60178	−0.134419
$143$	11.3961	0.952992
$144$	−1.34571	−0.112143
$145$	0	0
$146$	−7.43338	−0.615191
$147$	−4.11032	−0.339013
$148$	14.6652	1.20547
$149$	4.47852	0.366895	0.183447	−	0.983030i	$-0.441274\pi$
$149$	4.47852	0.366895	0.183447	+	0.983030i	$0.441274\pi$
$150$	0	0
$151$	−21.0647	−1.71422	−0.857111	−	0.515131i	$-0.827743\pi$
$151$	−21.0647	−1.71422	−0.857111	+	0.515131i	$0.827743\pi$
$152$	−22.1180	−1.79401
$153$	−9.73793	−0.787265
$154$	1.89695	0.152861
$155$	0	0
$156$	−4.39677	−0.352023
$157$	8.30695	0.662967	0.331483	−	0.943461i	$-0.392451\pi$
$157$	8.30695	0.662967	0.331483	+	0.943461i	$0.392451\pi$
$158$	−2.49009	−0.198101
$159$	−3.68490	−0.292231
$160$	0	0
$161$	4.39450	0.346336
$162$	4.05065	0.318249
$163$	6.32231	0.495202	0.247601	−	0.968862i	$-0.420358\pi$
$163$	6.32231	0.495202	0.247601	+	0.968862i	$0.420358\pi$
$164$	−1.21211	−0.0946501
$165$	0	0
$166$	−0.296185	−0.0229884
$167$	20.8304	1.61190	0.805951	−	0.591982i	$-0.201655\pi$
$167$	20.8304	1.61190	0.805951	+	0.591982i	$0.201655\pi$
$168$	−1.81469	−0.140006
$169$	9.75042	0.750032
$170$	0	0
$171$	−20.7771	−1.58886
$172$	6.43881	0.490955
$173$	7.92324	0.602392	0.301196	−	0.953562i	$-0.402614\pi$
$173$	7.92324	0.602392	0.301196	+	0.953562i	$0.402614\pi$
$174$	1.89469	0.143636
$175$	0	0
$176$	1.26835	0.0956057
$177$	8.41017	0.632147
$178$	3.66253	0.274518
$179$	16.7347	1.25081	0.625403	−	0.780302i	$-0.284934\pi$
$179$	16.7347	1.25081	0.625403	+	0.780302i	$0.284934\pi$
$180$	0	0
$181$	7.24734	0.538690	0.269345	−	0.963044i	$-0.413193\pi$
$181$	7.24734	0.538690	0.269345	+	0.963044i	$0.413193\pi$
$182$	3.78694	0.280707
$183$	6.82484	0.504506
$184$	−12.0258	−0.886557
$185$	0	0
$186$	−0.549030	−0.0402568
$187$	9.17812	0.671171
$188$	9.80557	0.715145
$189$	−3.72205	−0.270739
$190$	0	0
$191$	4.20218	0.304059	0.152030	−	0.988376i	$-0.451419\pi$
$191$	4.20218	0.304059	0.152030	+	0.988376i	$0.451419\pi$
$192$	2.47387	0.178536
$193$	−23.8041	−1.71345	−0.856727	−	0.515770i	$-0.827506\pi$
$193$	−23.8041	−1.71345	−0.856727	+	0.515770i	$0.827506\pi$
$194$	12.9796	0.931879
$195$	0	0
$196$	8.14792	0.581994
$197$	−8.15443	−0.580979	−0.290490	−	0.956878i	$-0.593818\pi$
$197$	−8.15443	−0.580979	−0.290490	+	0.956878i	$0.593818\pi$
$198$	−4.87641	−0.346551
$199$	−11.5827	−0.821072	−0.410536	−	0.911844i	$-0.634658\pi$
$199$	−11.5827	−0.821072	−0.410536	+	0.911844i	$0.634658\pi$
$200$	0	0
$201$	1.73964	0.122705
$202$	2.08578	0.146755
$203$	3.40308	0.238850
$204$	−3.54104	−0.247922
$205$	0	0
$206$	8.68812	0.605330
$207$	−11.2968	−0.785179
$208$	2.53205	0.175566
$209$	19.5827	1.35456
$210$	0	0
$211$	−10.5498	−0.726281	−0.363141	−	0.931734i	$-0.618296\pi$
$211$	−10.5498	−0.726281	−0.363141	+	0.931734i	$0.618296\pi$
$212$	7.30460	0.501682
$213$	−1.35668	−0.0929579
$214$	−4.58024	−0.313099
$215$	0	0
$216$	10.1856	0.693044
$217$	−0.986124	−0.0669425
$218$	−2.60880	−0.176690
$219$	−6.29590	−0.425438
$220$	0	0
$221$	18.3226	1.23251
$222$	−5.95634	−0.399763
$223$	−6.93392	−0.464329	−0.232165	−	0.972677i	$-0.574581\pi$
$223$	−6.93392	−0.464329	−0.232165	+	0.972677i	$0.574581\pi$
$224$	5.74377	0.383772
$225$	0	0
$226$	−13.8639	−0.922214
$227$	−5.13174	−0.340606	−0.170303	−	0.985392i	$-0.554475\pi$
$227$	−5.13174	−0.340606	−0.170303	+	0.985392i	$0.554475\pi$
$228$	−7.55523	−0.500358
$229$	−6.49167	−0.428981	−0.214491	−	0.976726i	$-0.568809\pi$
$229$	−6.49167	−0.428981	−0.214491	+	0.976726i	$0.568809\pi$
$230$	0	0
$231$	1.60668	0.105711
$232$	−9.31276	−0.611413
$233$	6.80912	0.446080	0.223040	−	0.974809i	$-0.428402\pi$
$233$	6.80912	0.446080	0.223040	+	0.974809i	$0.428402\pi$
$234$	−9.73491	−0.636391
$235$	0	0
$236$	−16.6716	−1.08523
$237$	−2.10905	−0.136998
$238$	3.04989	0.197695
$239$	−25.2638	−1.63418	−0.817091	−	0.576509i	$-0.804415\pi$
$239$	−25.2638	−1.63418	−0.817091	+	0.576509i	$0.804415\pi$
$240$	0	0
$241$	23.5440	1.51661	0.758303	−	0.651903i	$-0.226029\pi$
$241$	23.5440	1.51661	0.758303	+	0.651903i	$0.226029\pi$
$242$	−4.26027	−0.273861
$243$	14.7541	0.946474
$244$	−13.5289	−0.866101
$245$	0	0
$246$	0.492304	0.0313882
$247$	39.0934	2.48745
$248$	2.69859	0.171361
$249$	−0.250862	−0.0158977
$250$	0	0
$251$	−13.6711	−0.862913	−0.431456	−	0.902134i	$-0.642000\pi$
$251$	−13.6711	−0.862913	−0.431456	+	0.902134i	$0.642000\pi$
$252$	3.37918	0.212869
$253$	10.6473	0.669393
$254$	−8.23956	−0.516996
$255$	0	0
$256$	−14.2830	−0.892687
$257$	−23.7946	−1.48427	−0.742133	−	0.670252i	$-0.766186\pi$
$257$	−23.7946	−1.48427	−0.742133	+	0.670252i	$0.766186\pi$
$258$	−2.61515	−0.162812
$259$	−10.6983	−0.664760
$260$	0	0
$261$	−8.74815	−0.541497
$262$	1.96202	0.121214
$263$	−22.4886	−1.38670	−0.693352	−	0.720599i	$-0.743867\pi$
$263$	−22.4886	−1.38670	−0.693352	+	0.720599i	$0.743867\pi$
$264$	−4.39677	−0.270603
$265$	0	0
$266$	6.50732	0.398990
$267$	3.10208	0.189844
$268$	−3.44850	−0.210651
$269$	24.3015	1.48169	0.740845	−	0.671676i	$-0.234425\pi$
$269$	24.3015	1.48169	0.740845	+	0.671676i	$0.234425\pi$
$270$	0	0
$271$	−7.73030	−0.469582	−0.234791	−	0.972046i	$-0.575441\pi$
$271$	−7.73030	−0.469582	−0.234791	+	0.972046i	$0.575441\pi$
$272$	2.03924	0.123647
$273$	3.20745	0.194124
$274$	−4.40458	−0.266091
$275$	0	0
$276$	−4.10788	−0.247265
$277$	1.98758	0.119422	0.0597111	−	0.998216i	$-0.480982\pi$
$277$	1.98758	0.119422	0.0597111	+	0.998216i	$0.480982\pi$
$278$	−3.80533	−0.228229
$279$	2.53498	0.151766
$280$	0	0
$281$	−11.3024	−0.674247	−0.337124	−	0.941460i	$-0.609454\pi$
$281$	−11.3024	−0.674247	−0.337124	+	0.941460i	$0.609454\pi$
$282$	−3.98257	−0.237159
$283$	−25.7441	−1.53033	−0.765163	−	0.643836i	$-0.777342\pi$
$283$	−25.7441	−1.53033	−0.765163	+	0.643836i	$0.777342\pi$
$284$	2.68935	0.159584
$285$	0	0
$286$	9.17528	0.542546
$287$	0.884238	0.0521949
$288$	−14.7652	−0.870050
$289$	−2.24353	−0.131972
$290$	0	0
$291$	10.9934	0.644445
$292$	12.4804	0.730361
$293$	8.05723	0.470708	0.235354	−	0.971910i	$-0.424375\pi$
$293$	8.05723	0.470708	0.235354	+	0.971910i	$0.424375\pi$
$294$	−3.30931	−0.193003
$295$	0	0
$296$	29.2766	1.70167
$297$	−9.01806	−0.523281
$298$	3.60576	0.208876
$299$	21.2556	1.22924
$300$	0	0
$301$	−4.69713	−0.270738
$302$	−16.9597	−0.975920
$303$	1.76661	0.101489
$304$	4.35097	0.249545
$305$	0	0
$306$	−7.84023	−0.448196
$307$	−9.40600	−0.536829	−0.268414	−	0.963304i	$-0.586500\pi$
$307$	−9.40600	−0.536829	−0.268414	+	0.963304i	$0.586500\pi$
$308$	−3.18493	−0.181478
$309$	7.35864	0.418618
$310$	0	0
$311$	−2.60801	−0.147887	−0.0739434	−	0.997262i	$-0.523558\pi$
$311$	−2.60801	−0.147887	−0.0739434	+	0.997262i	$0.523558\pi$
$312$	−8.77740	−0.496922
$313$	−3.58901	−0.202863	−0.101431	−	0.994843i	$-0.532342\pi$
$313$	−3.58901	−0.202863	−0.101431	+	0.994843i	$0.532342\pi$
$314$	6.68812	0.377432
$315$	0	0
$316$	4.18079	0.235188
$317$	1.11411	0.0625749	0.0312875	−	0.999510i	$-0.490039\pi$
$317$	1.11411	0.0625749	0.0312875	+	0.999510i	$0.490039\pi$
$318$	−2.96679	−0.166370
$319$	8.24525	0.461645
$320$	0	0
$321$	−3.87936	−0.216525
$322$	3.53812	0.197171
$323$	31.4847	1.75186
$324$	−6.80092	−0.377829
$325$	0	0
$326$	5.09024	0.281922
$327$	−2.20959	−0.122191
$328$	−2.41977	−0.133610
$329$	−7.15318	−0.394368
$330$	0	0
$331$	9.85555	0.541710	0.270855	−	0.962620i	$-0.412694\pi$
$331$	9.85555	0.541710	0.270855	+	0.962620i	$0.412694\pi$
$332$	0.497286	0.0272921
$333$	27.5016	1.50708
$334$	16.7710	0.917668
$335$	0	0
$336$	0.356979	0.0194748
$337$	−11.3127	−0.616244	−0.308122	−	0.951347i	$-0.599700\pi$
$337$	−11.3127	−0.616244	−0.308122	+	0.951347i	$0.599700\pi$
$338$	7.85028	0.426999
$339$	−11.7424	−0.637761
$340$	0	0
$341$	−2.38926	−0.129385
$342$	−16.7281	−0.904551
$343$	−12.8468	−0.693661
$344$	12.8540	0.693041
$345$	0	0
$346$	6.37918	0.342947
$347$	8.93172	0.479480	0.239740	−	0.970837i	$-0.422938\pi$
$347$	8.93172	0.479480	0.239740	+	0.970837i	$0.422938\pi$
$348$	−3.18112	−0.170526
$349$	−7.21243	−0.386073	−0.193036	−	0.981192i	$-0.561834\pi$
$349$	−7.21243	−0.386073	−0.193036	+	0.981192i	$0.561834\pi$
$350$	0	0
$351$	−18.0030	−0.960929
$352$	13.9164	0.741748
$353$	27.8602	1.48285	0.741424	−	0.671037i	$-0.234151\pi$
$353$	27.8602	1.48285	0.741424	+	0.671037i	$0.234151\pi$
$354$	6.77122	0.359886
$355$	0	0
$356$	−6.14928	−0.325911
$357$	2.58319	0.136717
$358$	13.4734	0.712094
$359$	25.6468	1.35359	0.676794	−	0.736173i	$-0.263369\pi$
$359$	25.6468	1.35359	0.676794	+	0.736173i	$0.263369\pi$
$360$	0	0
$361$	48.1766	2.53561
$362$	5.83499	0.306681
$363$	−3.60835	−0.189389
$364$	−6.35816	−0.333258
$365$	0	0
$366$	5.49483	0.287219
$367$	−15.7335	−0.821284	−0.410642	−	0.911797i	$-0.634695\pi$
$367$	−15.7335	−0.821284	−0.410642	+	0.911797i	$0.634695\pi$
$368$	2.36568	0.123320
$369$	−2.27307	−0.118331
$370$	0	0
$371$	−5.32872	−0.276653
$372$	0.921805	0.0477934
$373$	−24.6582	−1.27675	−0.638377	−	0.769724i	$-0.720394\pi$
$373$	−24.6582	−1.27675	−0.638377	+	0.769724i	$0.720394\pi$
$374$	7.38952	0.382103
$375$	0	0
$376$	19.5751	1.00951
$377$	16.4602	0.847745
$378$	−2.99670	−0.154134
$379$	−3.25703	−0.167302	−0.0836512	−	0.996495i	$-0.526658\pi$
$379$	−3.25703	−0.167302	−0.0836512	+	0.996495i	$0.526658\pi$
$380$	0	0
$381$	−6.97872	−0.357531
$382$	3.38327	0.173103
$383$	−35.3352	−1.80555	−0.902773	−	0.430118i	$-0.858472\pi$
$383$	−35.3352	−1.80555	−0.902773	+	0.430118i	$0.858472\pi$
$384$	−5.95205	−0.303739
$385$	0	0
$386$	−19.1652	−0.975483
$387$	12.0747	0.613791
$388$	−21.7923	−1.10634
$389$	11.0770	0.561628	0.280814	−	0.959762i	$-0.409396\pi$
$389$	11.0770	0.561628	0.280814	+	0.959762i	$0.409396\pi$
$390$	0	0
$391$	17.1187	0.865728
$392$	16.2659	0.821553
$393$	1.66179	0.0838262
$394$	−6.56532	−0.330756
$395$	0	0
$396$	8.18734	0.411430
$397$	−6.46616	−0.324527	−0.162264	−	0.986747i	$-0.551879\pi$
$397$	−6.46616	−0.324527	−0.162264	+	0.986747i	$0.551879\pi$
$398$	−9.32546	−0.467443
$399$	5.51156	0.275923
$400$	0	0
$401$	35.9280	1.79416	0.897081	−	0.441867i	$-0.145684\pi$
$401$	35.9280	1.79416	0.897081	+	0.441867i	$0.145684\pi$
$402$	1.40062	0.0698568
$403$	−4.76974	−0.237598
$404$	−3.50196	−0.174229
$405$	0	0
$406$	2.73990	0.135979
$407$	−25.9207	−1.28484
$408$	−7.06907	−0.349971
$409$	0.324797	0.0160602	0.00803009	−	0.999968i	$-0.497444\pi$
$409$	0.324797	0.0160602	0.00803009	+	0.999968i	$0.497444\pi$
$410$	0	0
$411$	−3.73058	−0.184016
$412$	−14.5871	−0.718654
$413$	12.1619	0.598449
$414$	−9.09527	−0.447008
$415$	0	0
$416$	27.7818	1.36211
$417$	−3.22303	−0.157832
$418$	15.7664	0.771162
$419$	16.3226	0.797409	0.398705	−	0.917079i	$-0.369460\pi$
$419$	16.3226	0.797409	0.398705	+	0.917079i	$0.369460\pi$
$420$	0	0
$421$	−14.1523	−0.689739	−0.344870	−	0.938651i	$-0.612077\pi$
$421$	−14.1523	−0.689739	−0.344870	+	0.938651i	$0.612077\pi$
$422$	−8.49392	−0.413477
$423$	18.3884	0.894072
$424$	14.5824	0.708184
$425$	0	0
$426$	−1.09229	−0.0529216
$427$	9.86938	0.477613
$428$	7.69008	0.371714
$429$	7.77126	0.375200
$430$	0	0
$431$	1.09361	0.0526775	0.0263388	−	0.999653i	$-0.491615\pi$
$431$	1.09361	0.0526775	0.0263388	+	0.999653i	$0.491615\pi$
$432$	−2.00368	−0.0964020
$433$	13.3170	0.639972	0.319986	−	0.947422i	$-0.396322\pi$
$433$	13.3170	0.639972	0.319986	+	0.947422i	$0.396322\pi$
$434$	−0.793951	−0.0381109
$435$	0	0
$436$	4.38010	0.209769
$437$	36.5248	1.74722
$438$	−5.06897	−0.242205
$439$	−7.68345	−0.366711	−0.183355	−	0.983047i	$-0.558696\pi$
$439$	−7.68345	−0.366711	−0.183355	+	0.983047i	$0.558696\pi$
$440$	0	0
$441$	15.2798	0.727608
$442$	14.7519	0.701676
$443$	−21.5788	−1.02524	−0.512619	−	0.858616i	$-0.671325\pi$
$443$	−21.5788	−1.02524	−0.512619	+	0.858616i	$0.671325\pi$
$444$	10.0005	0.474604
$445$	0	0
$446$	−5.58265	−0.264346
$447$	3.05399	0.144449
$448$	3.57746	0.169019
$449$	−19.4236	−0.916657	−0.458329	−	0.888783i	$-0.651552\pi$
$449$	−19.4236	−0.916657	−0.458329	+	0.888783i	$0.651552\pi$
$450$	0	0
$451$	2.14240	0.100882
$452$	23.2771	1.09486
$453$	−14.3645	−0.674902
$454$	−4.13168	−0.193909
$455$	0	0
$456$	−15.0827	−0.706314
$457$	−1.97506	−0.0923895	−0.0461948	−	0.998932i	$-0.514709\pi$
$457$	−1.97506	−0.0923895	−0.0461948	+	0.998932i	$0.514709\pi$
$458$	−5.22659	−0.244222
$459$	−14.4991	−0.676761
$460$	0	0
$461$	16.5678	0.771640	0.385820	−	0.922574i	$-0.373919\pi$
$461$	16.5678	0.771640	0.385820	+	0.922574i	$0.373919\pi$
$462$	1.29357	0.0601824
$463$	−14.4094	−0.669660	−0.334830	−	0.942279i	$-0.608679\pi$
$463$	−14.4094	−0.669660	−0.334830	+	0.942279i	$0.608679\pi$
$464$	1.83197	0.0850472
$465$	0	0
$466$	5.48218	0.253957
$467$	−3.51711	−0.162752	−0.0813761	−	0.996683i	$-0.525931\pi$
$467$	−3.51711	−0.162752	−0.0813761	+	0.996683i	$0.525931\pi$
$468$	16.3446	0.755531
$469$	2.51569	0.116164
$470$	0	0
$471$	5.66468	0.261015
$472$	−33.2819	−1.53192
$473$	−11.3806	−0.523278
$474$	−1.69805	−0.0779938
$475$	0	0
$476$	−5.12068	−0.234706
$477$	13.6983	0.627202
$478$	−20.3405	−0.930353
$479$	−27.5796	−1.26014	−0.630072	−	0.776537i	$-0.716975\pi$
$479$	−27.5796	−1.26014	−0.630072	+	0.776537i	$0.716975\pi$
$480$	0	0
$481$	−51.7462	−2.35942
$482$	18.9558	0.863415
$483$	2.99670	0.136355
$484$	7.15287	0.325131
$485$	0	0
$486$	11.8788	0.538835
$487$	−18.6282	−0.844126	−0.422063	−	0.906567i	$-0.638694\pi$
$487$	−18.6282	−0.844126	−0.422063	+	0.906567i	$0.638694\pi$
$488$	−27.0082	−1.22260
$489$	4.31132	0.194964
$490$	0	0
$491$	36.9962	1.66961	0.834807	−	0.550542i	$-0.185579\pi$
$491$	36.9962	1.66961	0.834807	+	0.550542i	$0.185579\pi$
$492$	−0.826565	−0.0372644
$493$	13.2566	0.597048
$494$	31.4750	1.41613
$495$	0	0
$496$	−0.530857	−0.0238362
$497$	−1.96188	−0.0880026
$498$	−0.201975	−0.00905070
$499$	−21.0625	−0.942885	−0.471442	−	0.881897i	$-0.656267\pi$
$499$	−21.0625	−0.942885	−0.471442	+	0.881897i	$0.656267\pi$
$500$	0	0
$501$	14.2047	0.634617
$502$	−11.0069	−0.491263
$503$	21.4540	0.956585	0.478292	−	0.878201i	$-0.341256\pi$
$503$	21.4540	0.956585	0.478292	+	0.878201i	$0.341256\pi$
$504$	6.74596	0.300489
$505$	0	0
$506$	8.57241	0.381090
$507$	6.64901	0.295293
$508$	13.8340	0.613784
$509$	31.9575	1.41649	0.708246	−	0.705965i	$-0.249487\pi$
$509$	31.9575	1.41649	0.708246	+	0.705965i	$0.249487\pi$
$510$	0	0
$511$	−9.10449	−0.402759
$512$	5.95716	0.263272
$513$	−30.9356	−1.36584
$514$	−19.1576	−0.845005
$515$	0	0
$516$	4.39076	0.193292
$517$	−17.3313	−0.762228
$518$	−8.61345	−0.378453
$519$	5.40302	0.237166
$520$	0	0
$521$	18.1636	0.795761	0.397880	−	0.917437i	$-0.369746\pi$
$521$	18.1636	0.795761	0.397880	+	0.917437i	$0.369746\pi$
$522$	−7.04334	−0.308279
$523$	31.5577	1.37992	0.689962	−	0.723846i	$-0.257627\pi$
$523$	31.5577	1.37992	0.689962	+	0.723846i	$0.257627\pi$
$524$	−3.29418	−0.143907
$525$	0	0
$526$	−18.1061	−0.789461
$527$	−3.84142	−0.167335
$528$	0.864917	0.0376407
$529$	−3.14102	−0.136566
$530$	0	0
$531$	−31.2641	−1.35675
$532$	−10.9256	−0.473685
$533$	4.27693	0.185255
$534$	2.49755	0.108080
$535$	0	0
$536$	−6.88435	−0.297358
$537$	11.4117	0.492451
$538$	19.5657	0.843538
$539$	−14.4014	−0.620311
$540$	0	0
$541$	7.20266	0.309667	0.154833	−	0.987941i	$-0.450516\pi$
$541$	7.20266	0.309667	0.154833	+	0.987941i	$0.450516\pi$
$542$	−6.22384	−0.267337
$543$	4.94211	0.212086
$544$	22.3747	0.959306
$545$	0	0
$546$	2.58239	0.110516
$547$	27.7349	1.18586	0.592929	−	0.805255i	$-0.297972\pi$
$547$	27.7349	1.18586	0.592929	+	0.805255i	$0.297972\pi$
$548$	7.39517	0.315906
$549$	−25.3708	−1.08280
$550$	0	0
$551$	28.2846	1.20496
$552$	−8.20068	−0.349044
$553$	−3.04989	−0.129695
$554$	1.60025	0.0679879
$555$	0	0
$556$	6.38904	0.270956
$557$	−19.8242	−0.839977	−0.419989	−	0.907529i	$-0.637966\pi$
$557$	−19.8242	−0.839977	−0.419989	+	0.907529i	$0.637966\pi$
$558$	2.04097	0.0864013
$559$	−22.7193	−0.960925
$560$	0	0
$561$	6.25875	0.264245
$562$	−9.09985	−0.383854
$563$	−14.1312	−0.595561	−0.297780	−	0.954634i	$-0.596246\pi$
$563$	−14.1312	−0.595561	−0.297780	+	0.954634i	$0.596246\pi$
$564$	6.68662	0.281558
$565$	0	0
$566$	−20.7271	−0.871227
$567$	4.96128	0.208354
$568$	5.36883	0.225271
$569$	35.0298	1.46852	0.734262	−	0.678866i	$-0.237528\pi$
$569$	35.0298	1.46852	0.734262	+	0.678866i	$0.237528\pi$
$570$	0	0
$571$	−14.5385	−0.608419	−0.304209	−	0.952605i	$-0.598392\pi$
$571$	−14.5385	−0.608419	−0.304209	+	0.952605i	$0.598392\pi$
$572$	−15.4050	−0.644117
$573$	2.86556	0.119710
$574$	0.711920	0.0297150
$575$	0	0
$576$	−9.19640	−0.383183
$577$	28.2453	1.17587	0.587934	−	0.808909i	$-0.299941\pi$
$577$	28.2453	1.17587	0.587934	+	0.808909i	$0.299941\pi$
$578$	−1.80632	−0.0751328
$579$	−16.2325	−0.674599
$580$	0	0
$581$	−0.362771	−0.0150503
$582$	8.85104	0.366887
$583$	−12.9108	−0.534712
$584$	24.9150	1.03099
$585$	0	0
$586$	6.48706	0.267978
$587$	−35.5176	−1.46597	−0.732983	−	0.680247i	$-0.761873\pi$
$587$	−35.5176	−1.46597	−0.732983	+	0.680247i	$0.761873\pi$
$588$	5.55624	0.229135
$589$	−8.19613	−0.337716
$590$	0	0
$591$	−5.56068	−0.228736
$592$	−5.75919	−0.236701
$593$	42.1657	1.73154	0.865768	−	0.500445i	$-0.166830\pi$
$593$	42.1657	1.73154	0.865768	+	0.500445i	$0.166830\pi$
$594$	−7.26064	−0.297908
$595$	0	0
$596$	−6.05396	−0.247980
$597$	−7.89845	−0.323262
$598$	17.1134	0.699817
$599$	−11.0920	−0.453207	−0.226603	−	0.973987i	$-0.572762\pi$
$599$	−11.0920	−0.453207	−0.226603	+	0.973987i	$0.572762\pi$
$600$	0	0
$601$	−32.7506	−1.33593	−0.667963	−	0.744194i	$-0.732834\pi$
$601$	−32.7506	−1.33593	−0.667963	+	0.744194i	$0.732834\pi$
$602$	−3.78176	−0.154133
$603$	−6.46697	−0.263355
$604$	28.4748	1.15862
$605$	0	0
$606$	1.42234	0.0577785
$607$	19.5174	0.792188	0.396094	−	0.918210i	$-0.370366\pi$
$607$	19.5174	0.792188	0.396094	+	0.918210i	$0.370366\pi$
$608$	47.7391	1.93608
$609$	2.32063	0.0940368
$610$	0	0
$611$	−34.5989	−1.39972
$612$	13.1635	0.532103
$613$	25.6007	1.03400	0.517002	−	0.855984i	$-0.327048\pi$
$613$	25.6007	1.03400	0.517002	+	0.855984i	$0.327048\pi$
$614$	−7.57298	−0.305621
$615$	0	0
$616$	−6.35816	−0.256178
$617$	19.9855	0.804587	0.402294	−	0.915511i	$-0.368213\pi$
$617$	19.9855	0.804587	0.402294	+	0.915511i	$0.368213\pi$
$618$	5.92461	0.238323
$619$	−43.6616	−1.75491	−0.877453	−	0.479662i	$-0.840759\pi$
$619$	−43.6616	−1.75491	−0.877453	+	0.479662i	$0.840759\pi$
$620$	0	0
$621$	−16.8201	−0.674968
$622$	−2.09977	−0.0841930
$623$	4.48591	0.179724
$624$	1.72666	0.0691216
$625$	0	0
$626$	−2.88960	−0.115491
$627$	13.3538	0.533300
$628$	−11.2292	−0.448092
$629$	−41.6749	−1.66169
$630$	0	0
$631$	−47.5079	−1.89126	−0.945630	−	0.325244i	$-0.894554\pi$
$631$	−47.5079	−1.89126	−0.945630	+	0.325244i	$0.894554\pi$
$632$	8.34623	0.331995
$633$	−7.19416	−0.285942
$634$	0.896999	0.0356244
$635$	0	0
$636$	4.98116	0.197516
$637$	−28.7499	−1.13911
$638$	6.63844	0.262818
$639$	5.04333	0.199511
$640$	0	0
$641$	9.60774	0.379483	0.189741	−	0.981834i	$-0.439235\pi$
$641$	9.60774	0.379483	0.189741	+	0.981834i	$0.439235\pi$
$642$	−3.12336	−0.123269
$643$	−3.86327	−0.152352	−0.0761762	−	0.997094i	$-0.524271\pi$
$643$	−3.86327	−0.152352	−0.0761762	+	0.997094i	$0.524271\pi$
$644$	−5.94039	−0.234084
$645$	0	0
$646$	25.3491	0.997346
$647$	−7.35942	−0.289329	−0.144664	−	0.989481i	$-0.546210\pi$
$647$	−7.35942	−0.289329	−0.144664	+	0.989481i	$0.546210\pi$
$648$	−13.5769	−0.533350
$649$	29.4668	1.15667
$650$	0	0
$651$	−0.672458	−0.0263557
$652$	−8.54636	−0.334701
$653$	−4.71091	−0.184352	−0.0921761	−	0.995743i	$-0.529382\pi$
$653$	−4.71091	−0.184352	−0.0921761	+	0.995743i	$0.529382\pi$
$654$	−1.77899	−0.0695642
$655$	0	0
$656$	0.476009	0.0185850
$657$	23.4045	0.913096
$658$	−5.75919	−0.224516
$659$	−38.2904	−1.49158	−0.745791	−	0.666180i	$-0.767928\pi$
$659$	−38.2904	−1.49158	−0.745791	+	0.666180i	$0.767928\pi$
$660$	0	0
$661$	−40.8986	−1.59077	−0.795386	−	0.606104i	$-0.792732\pi$
$661$	−40.8986	−1.59077	−0.795386	+	0.606104i	$0.792732\pi$
$662$	7.93493	0.308400
$663$	12.4945	0.485247
$664$	0.992745	0.0385260
$665$	0	0
$666$	22.1422	0.857993
$667$	15.3787	0.595466
$668$	−28.1580	−1.08947
$669$	−4.72838	−0.182810
$670$	0	0
$671$	23.9123	0.923123
$672$	3.91679	0.151094
$673$	16.2428	0.626116	0.313058	−	0.949734i	$-0.398647\pi$
$673$	16.2428	0.626116	0.313058	+	0.949734i	$0.398647\pi$
$674$	−9.10814	−0.350832
$675$	0	0
$676$	−13.1804	−0.506938
$677$	−25.6007	−0.983917	−0.491958	−	0.870619i	$-0.663719\pi$
$677$	−25.6007	−0.983917	−0.491958	+	0.870619i	$0.663719\pi$
$678$	−9.45409	−0.363082
$679$	15.8975	0.610091
$680$	0	0
$681$	−3.49944	−0.134099
$682$	−1.92364	−0.0736601
$683$	8.82902	0.337833	0.168917	−	0.985630i	$-0.445973\pi$
$683$	8.82902	0.337833	0.168917	+	0.985630i	$0.445973\pi$
$684$	28.0860	1.07389
$685$	0	0
$686$	−10.3432	−0.394907
$687$	−4.42680	−0.168893
$688$	−2.52859	−0.0964016
$689$	−25.7743	−0.981921
$690$	0	0
$691$	−24.8206	−0.944222	−0.472111	−	0.881539i	$-0.656508\pi$
$691$	−24.8206	−0.944222	−0.472111	+	0.881539i	$0.656508\pi$
$692$	−10.7105	−0.407150
$693$	−5.97268	−0.226883
$694$	7.19113	0.272972
$695$	0	0
$696$	−6.35056	−0.240718
$697$	3.44452	0.130471
$698$	−5.80689	−0.219794
$699$	4.64328	0.175625
$700$	0	0
$701$	−27.7129	−1.04670	−0.523351	−	0.852117i	$-0.675318\pi$
$701$	−27.7129	−1.04670	−0.523351	+	0.852117i	$0.675318\pi$
$702$	−14.4946	−0.547064
$703$	−88.9185	−3.35363
$704$	8.66773	0.326677
$705$	0	0
$706$	22.4309	0.844197
$707$	2.55469	0.0960789
$708$	−11.3687	−0.427261
$709$	5.61375	0.210829	0.105414	−	0.994428i	$-0.466383\pi$
$709$	5.61375	0.210829	0.105414	+	0.994428i	$0.466383\pi$
$710$	0	0
$711$	7.84023	0.294031
$712$	−12.2760	−0.460062
$713$	−4.45634	−0.166891
$714$	2.07979	0.0778341
$715$	0	0
$716$	−22.6215	−0.845406
$717$	−17.2279	−0.643389
$718$	20.6488	0.770608
$719$	24.7322	0.922355	0.461178	−	0.887308i	$-0.347427\pi$
$719$	24.7322	0.922355	0.461178	+	0.887308i	$0.347427\pi$
$720$	0	0
$721$	10.6413	0.396303
$722$	38.7880	1.44354
$723$	16.0552	0.597098
$724$	−9.79678	−0.364095
$725$	0	0
$726$	−2.90517	−0.107821
$727$	−0.618123	−0.0229249	−0.0114625	−	0.999934i	$-0.503649\pi$
$727$	−0.618123	−0.0229249	−0.0114625	+	0.999934i	$0.503649\pi$
$728$	−12.6930	−0.470433
$729$	−5.03218	−0.186377
$730$	0	0
$731$	−18.2975	−0.676758
$732$	−9.22566	−0.340990
$733$	−2.11941	−0.0782824	−0.0391412	−	0.999234i	$-0.512462\pi$
$733$	−2.11941	−0.0782824	−0.0391412	+	0.999234i	$0.512462\pi$
$734$	−12.6674	−0.467563
$735$	0	0
$736$	25.9564	0.956764
$737$	6.09520	0.224520
$738$	−1.83010	−0.0673669
$739$	19.4330	0.714853	0.357426	−	0.933941i	$-0.383654\pi$
$739$	19.4330	0.714853	0.357426	+	0.933941i	$0.383654\pi$
$740$	0	0
$741$	26.6586	0.979328
$742$	−4.29027	−0.157501
$743$	3.94330	0.144666	0.0723328	−	0.997381i	$-0.476956\pi$
$743$	3.94330	0.144666	0.0723328	+	0.997381i	$0.476956\pi$
$744$	1.84023	0.0674660
$745$	0	0
$746$	−19.8529	−0.726866
$747$	0.932558	0.0341205
$748$	−12.4068	−0.453637
$749$	−5.60993	−0.204982
$750$	0	0
$751$	33.1090	1.20817	0.604083	−	0.796921i	$-0.293539\pi$
$751$	33.1090	1.20817	0.604083	+	0.796921i	$0.293539\pi$
$752$	−3.85075	−0.140422
$753$	−9.32262	−0.339735
$754$	13.2525	0.482628
$755$	0	0
$756$	5.03138	0.182989
$757$	−32.3374	−1.17532	−0.587661	−	0.809107i	$-0.699951\pi$
$757$	−32.3374	−1.17532	−0.587661	+	0.809107i	$0.699951\pi$
$758$	−2.62231	−0.0952466
$759$	7.26064	0.263545
$760$	0	0
$761$	49.0776	1.77906	0.889531	−	0.456876i	$-0.151032\pi$
$761$	49.0776	1.77906	0.889531	+	0.456876i	$0.151032\pi$
$762$	−5.61873	−0.203545
$763$	−3.19529	−0.115677
$764$	−5.68042	−0.205510
$765$	0	0
$766$	−28.4492	−1.02791
$767$	58.8255	2.12407
$768$	−9.73987	−0.351457
$769$	−1.22675	−0.0442377	−0.0221188	−	0.999755i	$-0.507041\pi$
$769$	−1.22675	−0.0442377	−0.0221188	+	0.999755i	$0.507041\pi$
$770$	0	0
$771$	−16.2260	−0.584366
$772$	32.1778	1.15810
$773$	−6.45534	−0.232182	−0.116091	−	0.993239i	$-0.537036\pi$
$773$	−6.45534	−0.232182	−0.116091	+	0.993239i	$0.537036\pi$
$774$	9.72161	0.349436
$775$	0	0
$776$	−43.5046	−1.56173
$777$	−7.29540	−0.261721
$778$	8.91838	0.319739
$779$	7.34931	0.263316
$780$	0	0
$781$	−4.75340	−0.170090
$782$	13.7826	0.492865
$783$	−13.0254	−0.465490
$784$	−3.19977	−0.114278
$785$	0	0
$786$	1.33794	0.0477229
$787$	−52.4233	−1.86869	−0.934344	−	0.356372i	$-0.884014\pi$
$787$	−52.4233	−1.86869	−0.934344	+	0.356372i	$0.884014\pi$
$788$	11.0230	0.392677
$789$	−15.3354	−0.545955
$790$	0	0
$791$	−16.9807	−0.603764
$792$	16.3446	0.580781
$793$	47.7368	1.69518
$794$	−5.20605	−0.184756
$795$	0	0
$796$	15.6572	0.554954
$797$	39.8463	1.41143	0.705715	−	0.708496i	$-0.250626\pi$
$797$	39.8463	1.41143	0.705715	+	0.708496i	$0.250626\pi$
$798$	4.43748	0.157085
$799$	−27.8650	−0.985792
$800$	0	0
$801$	−11.5317	−0.407453
$802$	28.9265	1.02143
$803$	−22.0590	−0.778447
$804$	−2.35161	−0.0829348
$805$	0	0
$806$	−3.84023	−0.135266
$807$	16.5717	0.583352
$808$	−6.99107	−0.245945
$809$	30.8217	1.08363	0.541817	−	0.840496i	$-0.317736\pi$
$809$	30.8217	1.08363	0.541817	+	0.840496i	$0.317736\pi$
$810$	0	0
$811$	18.8829	0.663069	0.331534	−	0.943443i	$-0.392434\pi$
$811$	18.8829	0.663069	0.331534	+	0.943443i	$0.392434\pi$
$812$	−4.60021	−0.161436
$813$	−5.27145	−0.184878
$814$	−20.8693	−0.731469
$815$	0	0
$816$	1.39060	0.0486808
$817$	−39.0400	−1.36584
$818$	0.261501	0.00914319
$819$	−11.9234	−0.416639
$820$	0	0
$821$	−5.23982	−0.182871	−0.0914355	−	0.995811i	$-0.529146\pi$
$821$	−5.23982	−0.182871	−0.0914355	+	0.995811i	$0.529146\pi$
$822$	−3.00358	−0.104762
$823$	32.1940	1.12221	0.561106	−	0.827744i	$-0.310376\pi$
$823$	32.1940	1.12221	0.561106	+	0.827744i	$0.310376\pi$
$824$	−29.1206	−1.01447
$825$	0	0
$826$	9.79184	0.340702
$827$	23.3522	0.812035	0.406018	−	0.913865i	$-0.366917\pi$
$827$	23.3522	0.812035	0.406018	+	0.913865i	$0.366917\pi$
$828$	15.2707	0.530693
$829$	−13.9086	−0.483067	−0.241533	−	0.970393i	$-0.577650\pi$
$829$	−13.9086	−0.483067	−0.241533	+	0.970393i	$0.577650\pi$
$830$	0	0
$831$	1.35537	0.0470173
$832$	17.3036	0.599896
$833$	−23.1544	−0.802251
$834$	−2.59493	−0.0898552
$835$	0	0
$836$	−26.4714	−0.915532
$837$	3.77442	0.130463
$838$	13.1417	0.453971
$839$	−47.0380	−1.62393	−0.811965	−	0.583706i	$-0.801602\pi$
$839$	−47.0380	−1.62393	−0.811965	+	0.583706i	$0.801602\pi$
$840$	0	0
$841$	−17.0908	−0.589338
$842$	−11.3943	−0.392674
$843$	−7.70737	−0.265456
$844$	14.2610	0.490885
$845$	0	0
$846$	14.8049	0.509002
$847$	−5.21803	−0.179294
$848$	−2.86859	−0.0985079
$849$	−17.5554	−0.602500
$850$	0	0
$851$	−48.3462	−1.65728
$852$	1.83392	0.0628292
$853$	−3.48847	−0.119443	−0.0597215	−	0.998215i	$-0.519021\pi$
$853$	−3.48847	−0.119443	−0.0597215	+	0.998215i	$0.519021\pi$
$854$	7.94606	0.271909
$855$	0	0
$856$	15.3519	0.524718
$857$	−34.5380	−1.17980	−0.589898	−	0.807478i	$-0.700832\pi$
$857$	−34.5380	−1.17980	−0.589898	+	0.807478i	$0.700832\pi$
$858$	6.25682	0.213604
$859$	26.7027	0.911084	0.455542	−	0.890214i	$-0.349445\pi$
$859$	26.7027	0.911084	0.455542	+	0.890214i	$0.349445\pi$
$860$	0	0
$861$	0.602980	0.0205495
$862$	0.880493	0.0299897
$863$	40.9566	1.39418	0.697089	−	0.716985i	$-0.254478\pi$
$863$	40.9566	1.39418	0.697089	+	0.716985i	$0.254478\pi$
$864$	−21.9844	−0.747926
$865$	0	0
$866$	10.7218	0.364341
$867$	−1.52991	−0.0519584
$868$	1.33302	0.0452457
$869$	−7.38952	−0.250672
$870$	0	0
$871$	12.1680	0.412298
$872$	8.74412	0.296113
$873$	−40.8671	−1.38314
$874$	29.4069	0.994704
$875$	0	0
$876$	8.51066	0.287549
$877$	−28.3662	−0.957858	−0.478929	−	0.877854i	$-0.658975\pi$
$877$	−28.3662	−0.957858	−0.478929	+	0.877854i	$0.658975\pi$
$878$	−6.18612	−0.208771
$879$	5.49439	0.185321
$880$	0	0
$881$	41.3066	1.39166	0.695828	−	0.718209i	$-0.255038\pi$
$881$	41.3066	1.39166	0.695828	+	0.718209i	$0.255038\pi$
$882$	12.3021	0.414233
$883$	15.6274	0.525904	0.262952	−	0.964809i	$-0.415304\pi$
$883$	15.6274	0.525904	0.262952	+	0.964809i	$0.415304\pi$
$884$	−24.7680	−0.833038
$885$	0	0
$886$	−17.3736	−0.583676
$887$	25.2885	0.849104	0.424552	−	0.905403i	$-0.360432\pi$
$887$	25.2885	0.849104	0.424552	+	0.905403i	$0.360432\pi$
$888$	19.9643	0.669959
$889$	−10.0919	−0.338472
$890$	0	0
$891$	12.0206	0.402704
$892$	9.37311	0.313835
$893$	−59.4534	−1.98953
$894$	2.45884	0.0822359
$895$	0	0
$896$	−8.60724	−0.287548
$897$	14.4946	0.483962
$898$	−15.6384	−0.521860
$899$	−3.45097	−0.115096
$900$	0	0
$901$	−20.7579	−0.691545
$902$	1.72489	0.0574327
$903$	−3.20307	−0.106591
$904$	46.4688	1.54553
$905$	0	0
$906$	−11.5652	−0.384227
$907$	−34.3017	−1.13897	−0.569484	−	0.822002i	$-0.692857\pi$
$907$	−34.3017	−1.13897	−0.569484	+	0.822002i	$0.692857\pi$
$908$	6.93697	0.230212
$909$	−6.56722	−0.217821
$910$	0	0
$911$	−45.3919	−1.50390	−0.751950	−	0.659220i	$-0.770887\pi$
$911$	−45.3919	−1.50390	−0.751950	+	0.659220i	$0.770887\pi$
$912$	2.96702	0.0982478
$913$	−0.878948	−0.0290889
$914$	−1.59017	−0.0525981
$915$	0	0
$916$	8.77529	0.289944
$917$	2.40311	0.0793577
$918$	−11.6736	−0.385285
$919$	59.8589	1.97456	0.987282	−	0.158979i	$-0.0508201\pi$
$919$	59.8589	1.97456	0.987282	+	0.158979i	$0.0508201\pi$
$920$	0	0
$921$	−6.41415	−0.211353
$922$	13.3391	0.439300
$923$	−9.48936	−0.312346
$924$	−2.17187	−0.0714492
$925$	0	0
$926$	−11.6013	−0.381242
$927$	−27.3551	−0.898460
$928$	20.1005	0.659831
$929$	41.3564	1.35686	0.678430	−	0.734665i	$-0.262661\pi$
$929$	41.3564	1.35686	0.678430	+	0.734665i	$0.262661\pi$
$930$	0	0
$931$	−49.4027	−1.61911
$932$	−9.20442	−0.301501
$933$	−1.77846	−0.0582241
$934$	−2.83170	−0.0926561
$935$	0	0
$936$	32.6293	1.06652
$937$	34.5380	1.12831	0.564154	−	0.825670i	$-0.309203\pi$
$937$	34.5380	1.12831	0.564154	+	0.825670i	$0.309203\pi$
$938$	2.02544	0.0661329
$939$	−2.44742	−0.0798686
$940$	0	0
$941$	−44.6687	−1.45616	−0.728079	−	0.685493i	$-0.759587\pi$
$941$	−44.6687	−1.45616	−0.728079	+	0.685493i	$0.759587\pi$
$942$	4.56077	0.148598
$943$	3.99591	0.130125
$944$	6.54709	0.213090
$945$	0	0
$946$	−9.16274	−0.297906
$947$	−46.0665	−1.49696	−0.748480	−	0.663158i	$-0.769216\pi$
$947$	−46.0665	−1.49696	−0.748480	+	0.663158i	$0.769216\pi$
$948$	2.85097	0.0925952
$949$	−44.0371	−1.42950
$950$	0	0
$951$	0.759738	0.0246362
$952$	−10.2226	−0.331315
$953$	−18.6857	−0.605288	−0.302644	−	0.953104i	$-0.597869\pi$
$953$	−18.6857	−0.605288	−0.302644	+	0.953104i	$0.597869\pi$
$954$	11.0288	0.357071
$955$	0	0
$956$	34.1511	1.10453
$957$	5.62261	0.181753
$958$	−22.2050	−0.717410
$959$	−5.39479	−0.174207
$960$	0	0
$961$	1.00000	0.0322581
$962$	−41.6620	−1.34324
$963$	14.4212	0.464716
$964$	−31.8263	−1.02506
$965$	0	0
$966$	2.41271	0.0776278
$967$	−43.2252	−1.39003	−0.695014	−	0.718996i	$-0.744602\pi$
$967$	−43.2252	−1.39003	−0.695014	+	0.718996i	$0.744602\pi$
$968$	14.2795	0.458960
$969$	21.4701	0.689719
$970$	0	0
$971$	29.6317	0.950927	0.475464	−	0.879735i	$-0.342280\pi$
$971$	29.6317	0.950927	0.475464	+	0.879735i	$0.342280\pi$
$972$	−19.9442	−0.639711
$973$	−4.66081	−0.149419
$974$	−14.9980	−0.480567
$975$	0	0
$976$	5.31295	0.170063
$977$	−39.1485	−1.25247	−0.626236	−	0.779633i	$-0.715405\pi$
$977$	−39.1485	−1.25247	−0.626236	+	0.779633i	$0.715405\pi$
$978$	3.47114	0.110995
$979$	10.8688	0.347368
$980$	0	0
$981$	8.21398	0.262252
$982$	29.7865	0.950525
$983$	6.47513	0.206525	0.103262	−	0.994654i	$-0.467072\pi$
$983$	6.47513	0.206525	0.103262	+	0.994654i	$0.467072\pi$
$984$	−1.65009	−0.0526031
$985$	0	0
$986$	10.6732	0.339904
$987$	−4.87790	−0.155265
$988$	−52.8456	−1.68124
$989$	−21.2265	−0.674965
$990$	0	0
$991$	41.5628	1.32029	0.660143	−	0.751140i	$-0.270496\pi$
$991$	41.5628	1.32029	0.660143	+	0.751140i	$0.270496\pi$
$992$	−5.82459	−0.184931
$993$	6.72071	0.213275
$994$	−1.57956	−0.0501005
$995$	0	0
$996$	0.339109	0.0107451
$997$	−21.2037	−0.671529	−0.335764	−	0.941946i	$-0.608995\pi$
$997$	−21.2037	−0.671529	−0.335764	+	0.941946i	$0.608995\pi$
$998$	−16.9579	−0.536792
$999$	40.9481	1.29554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.a.l.1.7		10
3.2	odd	2		6975.2.a.ch.1.4		10
5.2	odd	4		155.2.b.b.94.7	yes	10
5.3	odd	4		155.2.b.b.94.4	✓	10
5.4	even	2	inner	775.2.a.l.1.4		10
15.2	even	4		1395.2.c.e.559.4		10
15.8	even	4		1395.2.c.e.559.7		10
15.14	odd	2		6975.2.a.ch.1.7		10
20.3	even	4		2480.2.d.g.1489.5		10
20.7	even	4		2480.2.d.g.1489.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.b.b.94.4	✓	10	5.3	odd	4
155.2.b.b.94.7	yes	10	5.2	odd	4
775.2.a.l.1.4		10	5.4	even	2	inner
775.2.a.l.1.7		10	1.1	even	1	trivial
1395.2.c.e.559.4		10	15.2	even	4
1395.2.c.e.559.7		10	15.8	even	4
2480.2.d.g.1489.5		10	20.3	even	4
2480.2.d.g.1489.6		10	20.7	even	4
6975.2.a.ch.1.4		10	3.2	odd	2
6975.2.a.ch.1.7		10	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.a (trivial)

\(f(q)\)	\(=\)	\(q+0.805123 q^{2} +0.681921 q^{3} -1.35178 q^{4} +0.549030 q^{6} +0.986124 q^{7} -2.69859 q^{8} -2.53498 q^{9} +O(q^{10})\)	\(q+0.805123 q^{2} +0.681921 q^{3} -1.35178 q^{4} +0.549030 q^{6} +0.986124 q^{7} -2.69859 q^{8} -2.53498 q^{9} +2.38926 q^{11} -0.921805 q^{12} +4.76974 q^{13} +0.793951 q^{14} +0.530857 q^{16} +3.84142 q^{17} -2.04097 q^{18} +8.19613 q^{19} +0.672458 q^{21} +1.92364 q^{22} +4.45634 q^{23} -1.84023 q^{24} +3.84023 q^{26} -3.77442 q^{27} -1.33302 q^{28} +3.45097 q^{29} -1.00000 q^{31} +5.82459 q^{32} +1.62928 q^{33} +3.09281 q^{34} +3.42673 q^{36} -10.8488 q^{37} +6.59889 q^{38} +3.25258 q^{39} +0.896680 q^{41} +0.541412 q^{42} -4.76322 q^{43} -3.22974 q^{44} +3.58790 q^{46} -7.25383 q^{47} +0.362002 q^{48} -6.02756 q^{49} +2.61954 q^{51} -6.44763 q^{52} -5.40370 q^{53} -3.03887 q^{54} -2.66115 q^{56} +5.58911 q^{57} +2.77845 q^{58} +12.3331 q^{59} +10.0083 q^{61} -0.805123 q^{62} -2.49981 q^{63} +3.62779 q^{64} +1.31177 q^{66} +2.55109 q^{67} -5.19274 q^{68} +3.03887 q^{69} -1.98949 q^{71} +6.84089 q^{72} -9.23260 q^{73} -8.73465 q^{74} -11.0793 q^{76} +2.35610 q^{77} +2.61873 q^{78} -3.09281 q^{79} +5.03110 q^{81} +0.721938 q^{82} -0.367875 q^{83} -0.909014 q^{84} -3.83498 q^{86} +2.35329 q^{87} -6.44763 q^{88} +4.54903 q^{89} +4.70355 q^{91} -6.02398 q^{92} -0.681921 q^{93} -5.84023 q^{94} +3.97191 q^{96} +16.1212 q^{97} -4.85293 q^{98} -6.05673 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 12 q^{4} + 8 q^{6} + 22 q^{9}+O(q^{10}) \) 10 * q + 12 * q^4 + 8 * q^6 + 22 * q^9	\( 10 q + 12 q^{4} + 8 q^{6} + 22 q^{9} + 16 q^{11} - 6 q^{14} + 8 q^{16} - 4 q^{19} + 20 q^{21} - 8 q^{24} + 28 q^{26} + 32 q^{29} - 10 q^{31} - 28 q^{34} + 44 q^{36} - 16 q^{39} + 36 q^{41} + 52 q^{44} + 8 q^{46} + 22 q^{49} + 20 q^{51} + 12 q^{56} + 12 q^{59} - 70 q^{64} + 12 q^{71} + 28 q^{74} - 30 q^{76} + 28 q^{79} - 14 q^{81} - 40 q^{84} + 36 q^{86} + 48 q^{89} - 4 q^{91} - 48 q^{94} - 60 q^{96} + 64 q^{99}+O(q^{100}) \) 10 * q + 12 * q^4 + 8 * q^6 + 22 * q^9 + 16 * q^11 - 6 * q^14 + 8 * q^16 - 4 * q^19 + 20 * q^21 - 8 * q^24 + 28 * q^26 + 32 * q^29 - 10 * q^31 - 28 * q^34 + 44 * q^36 - 16 * q^39 + 36 * q^41 + 52 * q^44 + 8 * q^46 + 22 * q^49 + 20 * q^51 + 12 * q^56 + 12 * q^59 - 70 * q^64 + 12 * q^71 + 28 * q^74 - 30 * q^76 + 28 * q^79 - 14 * q^81 - 40 * q^84 + 36 * q^86 + 48 * q^89 - 4 * q^91 - 48 * q^94 - 60 * q^96 + 64 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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