LMFDB - Embedded newform 2325.2.a.q.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2325,2,Mod(1,2325)] 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("2325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2325 = 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,-3,1,0,1,-2,-3,3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.5652184699$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	2325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.48119 q^{2} -1.00000 q^{3} +0.193937 q^{4} -1.48119 q^{6} -2.48119 q^{7} -2.67513 q^{8} +1.00000 q^{9} +2.00000 q^{11} -0.193937 q^{12} +5.83146 q^{13} -3.67513 q^{14} -4.35026 q^{16} -3.76845 q^{17} +1.48119 q^{18} +2.48119 q^{21} +2.96239 q^{22} -0.806063 q^{23} +2.67513 q^{24} +8.63752 q^{26} -1.00000 q^{27} -0.481194 q^{28} +4.06300 q^{29} +1.00000 q^{31} -1.09332 q^{32} -2.00000 q^{33} -5.58181 q^{34} +0.193937 q^{36} -12.1441 q^{37} -5.83146 q^{39} +9.92478 q^{41} +3.67513 q^{42} +10.3127 q^{43} +0.387873 q^{44} -1.19394 q^{46} +1.58181 q^{47} +4.35026 q^{48} -0.843675 q^{49} +3.76845 q^{51} +1.13093 q^{52} +7.11871 q^{53} -1.48119 q^{54} +6.63752 q^{56} +6.01810 q^{58} +9.41327 q^{59} +6.31265 q^{61} +1.48119 q^{62} -2.48119 q^{63} +7.08110 q^{64} -2.96239 q^{66} +14.4060 q^{67} -0.730841 q^{68} +0.806063 q^{69} +12.2496 q^{71} -2.67513 q^{72} -1.25694 q^{73} -17.9878 q^{74} -4.96239 q^{77} -8.63752 q^{78} +12.8568 q^{79} +1.00000 q^{81} +14.7005 q^{82} +0.0303172 q^{83} +0.481194 q^{84} +15.2750 q^{86} -4.06300 q^{87} -5.35026 q^{88} -13.7259 q^{89} -14.4690 q^{91} -0.156325 q^{92} -1.00000 q^{93} +2.34297 q^{94} +1.09332 q^{96} +12.0000 q^{97} -1.24965 q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - 3 q^{3} + q^{4} + q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9} + 6 q^{11} - q^{12} + 2 q^{13} - 6 q^{14} - 3 q^{16} - q^{18} + 2 q^{21} - 2 q^{22} - 2 q^{23} + 3 q^{24} + 10 q^{26} - 3 q^{27}+ \cdots + 6 q^{99}+O(q^{100}) $ 3 * q - q^2 - 3 * q^3 + q^4 + q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9 + 6 * q^11 - q^12 + 2 * q^13 - 6 * q^14 - 3 * q^16 - q^18 + 2 * q^21 - 2 * q^22 - 2 * q^23 + 3 * q^24 + 10 * q^26 - 3 * q^27 + 4 * q^28 + 8 * q^29 + 3 * q^31 + 3 * q^32 - 6 * q^33 - 18 * q^34 + q^36 - 2 * q^39 + 8 * q^41 + 6 * q^42 + 10 * q^43 + 2 * q^44 - 4 * q^46 + 6 * q^47 + 3 * q^48 - 13 * q^49 + 8 * q^52 + q^54 + 4 * q^56 - 10 * q^58 + 14 * q^59 - 2 * q^61 - q^62 - 2 * q^63 - 11 * q^64 + 2 * q^66 + 16 * q^67 + 20 * q^68 + 2 * q^69 + 20 * q^71 - 3 * q^72 - 28 * q^74 - 4 * q^77 - 10 * q^78 + 8 * q^79 + 3 * q^81 + 24 * q^82 - 2 * q^83 - 4 * q^84 + 14 * q^86 - 8 * q^87 - 6 * q^88 - 6 * q^89 - 12 * q^91 + 10 * q^92 - 3 * q^93 - 16 * q^94 - 3 * q^96 + 36 * q^97 + 13 * q^98 + 6 * 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 $

$2$	1.48119	1.04736	0.523681	−	0.851914i	$-0.324558\pi$
$2$	1.48119	1.04736	0.523681	+	0.851914i	$0.324558\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.193937	0.0969683
$5$	0	0
$5$	0	0
$6$	−1.48119	−0.604695
$7$	−2.48119	−0.937803	−0.468902	−	0.883250i	$-0.655350\pi$
$7$	−2.48119	−0.937803	−0.468902	+	0.883250i	$0.655350\pi$
$8$	−2.67513	−0.945802
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−0.193937	−0.0559847
$13$	5.83146	1.61735	0.808677	−	0.588252i	$-0.200184\pi$
$13$	5.83146	1.61735	0.808677	+	0.588252i	$0.200184\pi$
$14$	−3.67513	−0.982220
$15$	0	0
$16$	−4.35026	−1.08757
$17$	−3.76845	−0.913984	−0.456992	−	0.889471i	$-0.651073\pi$
$17$	−3.76845	−0.913984	−0.456992	+	0.889471i	$0.651073\pi$
$18$	1.48119	0.349121
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	2.48119	0.541441
$22$	2.96239	0.631583
$23$	−0.806063	−0.168076	−0.0840379	−	0.996463i	$-0.526782\pi$
$23$	−0.806063	−0.168076	−0.0840379	+	0.996463i	$0.526782\pi$
$24$	2.67513	0.546059
$25$	0	0
$26$	8.63752	1.69396
$27$	−1.00000	−0.192450
$28$	−0.481194	−0.0909372
$29$	4.06300	0.754481	0.377240	−	0.926115i	$-0.376873\pi$
$29$	4.06300	0.754481	0.377240	+	0.926115i	$0.376873\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.09332	−0.193274
$33$	−2.00000	−0.348155
$34$	−5.58181	−0.957272
$35$	0	0
$36$	0.193937	0.0323228
$37$	−12.1441	−1.99648	−0.998239	−	0.0593136i	$-0.981109\pi$
$37$	−12.1441	−1.99648	−0.998239	+	0.0593136i	$0.981109\pi$
$38$	0	0
$39$	−5.83146	−0.933780
$40$	0	0
$41$	9.92478	1.54999	0.774995	−	0.631967i	$-0.217752\pi$
$41$	9.92478	1.54999	0.774995	+	0.631967i	$0.217752\pi$
$42$	3.67513	0.567085
$43$	10.3127	1.57266	0.786332	−	0.617804i	$-0.211977\pi$
$43$	10.3127	1.57266	0.786332	+	0.617804i	$0.211977\pi$
$44$	0.387873	0.0584741
$45$	0	0
$46$	−1.19394	−0.176036
$47$	1.58181	0.230731	0.115365	−	0.993323i	$-0.463196\pi$
$47$	1.58181	0.230731	0.115365	+	0.993323i	$0.463196\pi$
$48$	4.35026	0.627906
$49$	−0.843675	−0.120525
$50$	0	0
$51$	3.76845	0.527689
$52$	1.13093	0.156832
$53$	7.11871	0.977831	0.488915	−	0.872331i	$-0.337393\pi$
$53$	7.11871	0.977831	0.488915	+	0.872331i	$0.337393\pi$
$54$	−1.48119	−0.201565
$55$	0	0
$56$	6.63752	0.886976
$57$	0	0
$58$	6.01810	0.790215
$59$	9.41327	1.22550	0.612751	−	0.790276i	$-0.290063\pi$
$59$	9.41327	1.22550	0.612751	+	0.790276i	$0.290063\pi$
$60$	0	0
$61$	6.31265	0.808252	0.404126	−	0.914703i	$-0.367576\pi$
$61$	6.31265	0.808252	0.404126	+	0.914703i	$0.367576\pi$
$62$	1.48119	0.188112
$63$	−2.48119	−0.312601
$64$	7.08110	0.885138
$65$	0	0
$66$	−2.96239	−0.364645
$67$	14.4060	1.75997	0.879985	−	0.475002i	$-0.157553\pi$
$67$	14.4060	1.75997	0.879985	+	0.475002i	$0.157553\pi$
$68$	−0.730841	−0.0886274
$69$	0.806063	0.0970386
$70$	0	0
$71$	12.2496	1.45377	0.726883	−	0.686762i	$-0.240968\pi$
$71$	12.2496	1.45377	0.726883	+	0.686762i	$0.240968\pi$
$72$	−2.67513	−0.315267
$73$	−1.25694	−0.147114	−0.0735569	−	0.997291i	$-0.523435\pi$
$73$	−1.25694	−0.147114	−0.0735569	+	0.997291i	$0.523435\pi$
$74$	−17.9878	−2.09104
$75$	0	0
$76$	0	0
$77$	−4.96239	−0.565517
$78$	−8.63752	−0.978006
$79$	12.8568	1.44651	0.723254	−	0.690582i	$-0.242645\pi$
$79$	12.8568	1.44651	0.723254	+	0.690582i	$0.242645\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	14.7005	1.62340
$83$	0.0303172	0.00332774	0.00166387	−	0.999999i	$-0.499470\pi$
$83$	0.0303172	0.00332774	0.00166387	+	0.999999i	$0.499470\pi$
$84$	0.481194	0.0525026
$85$	0	0
$86$	15.2750	1.64715
$87$	−4.06300	−0.435600
$88$	−5.35026	−0.570340
$89$	−13.7259	−1.45494	−0.727472	−	0.686137i	$-0.759305\pi$
$89$	−13.7259	−1.45494	−0.727472	+	0.686137i	$0.759305\pi$
$90$	0	0
$91$	−14.4690	−1.51676
$92$	−0.156325	−0.0162980
$93$	−1.00000	−0.103695
$94$	2.34297	0.241659
$95$	0	0
$96$	1.09332	0.111587
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	−1.24965	−0.126233
$99$	2.00000	0.201008
$100$	0	0
$101$	−10.2374	−1.01866	−0.509331	−	0.860571i	$-0.670107\pi$
$101$	−10.2374	−1.01866	−0.509331	+	0.860571i	$0.670107\pi$
$102$	5.58181	0.552682
$103$	13.7562	1.35544	0.677721	−	0.735319i	$-0.262968\pi$
$103$	13.7562	1.35544	0.677721	+	0.735319i	$0.262968\pi$
$104$	−15.5999	−1.52970
$105$	0	0
$106$	10.5442	1.02414
$107$	4.15633	0.401807	0.200904	−	0.979611i	$-0.435612\pi$
$107$	4.15633	0.401807	0.200904	+	0.979611i	$0.435612\pi$
$108$	−0.193937	−0.0186616
$109$	−13.8945	−1.33085	−0.665424	−	0.746466i	$-0.731749\pi$
$109$	−13.8945	−1.33085	−0.665424	+	0.746466i	$0.731749\pi$
$110$	0	0
$111$	12.1441	1.15267
$112$	10.7938	1.01992
$113$	8.57452	0.806623	0.403311	−	0.915063i	$-0.367859\pi$
$113$	8.57452	0.806623	0.403311	+	0.915063i	$0.367859\pi$
$114$	0	0
$115$	0	0
$116$	0.787965	0.0731607
$117$	5.83146	0.539118
$118$	13.9429	1.28355
$119$	9.35026	0.857137
$120$	0	0
$121$	−7.00000	−0.636364
$122$	9.35026	0.846533
$123$	−9.92478	−0.894887
$124$	0.193937	0.0174160
$125$	0	0
$126$	−3.67513	−0.327407
$127$	−7.66291	−0.679973	−0.339987	−	0.940430i	$-0.610423\pi$
$127$	−7.66291	−0.679973	−0.339987	+	0.940430i	$0.610423\pi$
$128$	12.6751	1.12033
$129$	−10.3127	−0.907978
$130$	0	0
$131$	−6.37565	−0.557044	−0.278522	−	0.960430i	$-0.589844\pi$
$131$	−6.37565	−0.557044	−0.278522	+	0.960430i	$0.589844\pi$
$132$	−0.387873	−0.0337600
$133$	0	0
$134$	21.3380	1.84333
$135$	0	0
$136$	10.0811	0.864447
$137$	−2.93207	−0.250504	−0.125252	−	0.992125i	$-0.539974\pi$
$137$	−2.93207	−0.250504	−0.125252	+	0.992125i	$0.539974\pi$
$138$	1.19394	0.101635
$139$	−1.73813	−0.147427	−0.0737133	−	0.997279i	$-0.523485\pi$
$139$	−1.73813	−0.147427	−0.0737133	+	0.997279i	$0.523485\pi$
$140$	0	0
$141$	−1.58181	−0.133212
$142$	18.1441	1.52262
$143$	11.6629	0.975302
$144$	−4.35026	−0.362522
$145$	0	0
$146$	−1.86177	−0.154081
$147$	0.843675	0.0695851
$148$	−2.35519	−0.193595
$149$	−15.8496	−1.29845	−0.649223	−	0.760598i	$-0.724906\pi$
$149$	−15.8496	−1.29845	−0.649223	+	0.760598i	$0.724906\pi$
$150$	0	0
$151$	15.8945	1.29347	0.646736	−	0.762714i	$-0.276133\pi$
$151$	15.8945	1.29347	0.646736	+	0.762714i	$0.276133\pi$
$152$	0	0
$153$	−3.76845	−0.304661
$154$	−7.35026	−0.592301
$155$	0	0
$156$	−1.13093	−0.0905471
$157$	0.463096	0.0369591	0.0184795	−	0.999829i	$-0.494117\pi$
$157$	0.463096	0.0369591	0.0184795	+	0.999829i	$0.494117\pi$
$158$	19.0435	1.51502
$159$	−7.11871	−0.564551
$160$	0	0
$161$	2.00000	0.157622
$162$	1.48119	0.116374
$163$	−14.5320	−1.13823	−0.569116	−	0.822257i	$-0.692715\pi$
$163$	−14.5320	−1.13823	−0.569116	+	0.822257i	$0.692715\pi$
$164$	1.92478	0.150300
$165$	0	0
$166$	0.0449056	0.00348535
$167$	3.87399	0.299779	0.149889	−	0.988703i	$-0.452108\pi$
$167$	3.87399	0.299779	0.149889	+	0.988703i	$0.452108\pi$
$168$	−6.63752	−0.512096
$169$	21.0059	1.61584
$170$	0	0
$171$	0	0
$172$	2.00000	0.152499
$173$	−12.8265	−0.975183	−0.487592	−	0.873072i	$-0.662124\pi$
$173$	−12.8265	−0.975183	−0.487592	+	0.873072i	$0.662124\pi$
$174$	−6.01810	−0.456231
$175$	0	0
$176$	−8.70052	−0.655827
$177$	−9.41327	−0.707544
$178$	−20.3307	−1.52385
$179$	−2.23743	−0.167233	−0.0836166	−	0.996498i	$-0.526647\pi$
$179$	−2.23743	−0.167233	−0.0836166	+	0.996498i	$0.526647\pi$
$180$	0	0
$181$	−20.2374	−1.50424	−0.752118	−	0.659028i	$-0.770968\pi$
$181$	−20.2374	−1.50424	−0.752118	+	0.659028i	$0.770968\pi$
$182$	−21.4314	−1.58860
$183$	−6.31265	−0.466645
$184$	2.15633	0.158966
$185$	0	0
$186$	−1.48119	−0.108606
$187$	−7.53690	−0.551153
$188$	0.306771	0.0223736
$189$	2.48119	0.180480
$190$	0	0
$191$	−9.78655	−0.708130	−0.354065	−	0.935221i	$-0.615201\pi$
$191$	−9.78655	−0.708130	−0.354065	+	0.935221i	$0.615201\pi$
$192$	−7.08110	−0.511035
$193$	−4.83638	−0.348130	−0.174065	−	0.984734i	$-0.555690\pi$
$193$	−4.83638	−0.348130	−0.174065	+	0.984734i	$0.555690\pi$
$194$	17.7743	1.27612
$195$	0	0
$196$	−0.163619	−0.0116871
$197$	10.0059	0.712889	0.356445	−	0.934316i	$-0.383989\pi$
$197$	10.0059	0.712889	0.356445	+	0.934316i	$0.383989\pi$
$198$	2.96239	0.210528
$199$	−3.05808	−0.216782	−0.108391	−	0.994108i	$-0.534570\pi$
$199$	−3.05808	−0.216782	−0.108391	+	0.994108i	$0.534570\pi$
$200$	0	0
$201$	−14.4060	−1.01612
$202$	−15.1636	−1.06691
$203$	−10.0811	−0.707555
$204$	0.730841	0.0511691
$205$	0	0
$206$	20.3757	1.41964
$207$	−0.806063	−0.0560253
$208$	−25.3684	−1.75898
$209$	0	0
$210$	0	0
$211$	−8.49929	−0.585115	−0.292558	−	0.956248i	$-0.594506\pi$
$211$	−8.49929	−0.585115	−0.292558	+	0.956248i	$0.594506\pi$
$212$	1.38058	0.0948185
$213$	−12.2496	−0.839332
$214$	6.15633	0.420838
$215$	0	0
$216$	2.67513	0.182020
$217$	−2.48119	−0.168434
$218$	−20.5804	−1.39388
$219$	1.25694	0.0849362
$220$	0	0
$221$	−21.9756	−1.47824
$222$	17.9878	1.20726
$223$	−21.7235	−1.45472	−0.727358	−	0.686258i	$-0.759252\pi$
$223$	−21.7235	−1.45472	−0.727358	+	0.686258i	$0.759252\pi$
$224$	2.71274	0.181253
$225$	0	0
$226$	12.7005	0.844826
$227$	14.0205	0.930571	0.465286	−	0.885161i	$-0.345952\pi$
$227$	14.0205	0.930571	0.465286	+	0.885161i	$0.345952\pi$
$228$	0	0
$229$	−7.02302	−0.464094	−0.232047	−	0.972705i	$-0.574542\pi$
$229$	−7.02302	−0.464094	−0.232047	+	0.972705i	$0.574542\pi$
$230$	0	0
$231$	4.96239	0.326501
$232$	−10.8691	−0.713589
$233$	10.9321	0.716184	0.358092	−	0.933686i	$-0.383427\pi$
$233$	10.9321	0.716184	0.358092	+	0.933686i	$0.383427\pi$
$234$	8.63752	0.564652
$235$	0	0
$236$	1.82558	0.118835
$237$	−12.8568	−0.835142
$238$	13.8496	0.897733
$239$	7.42548	0.480315	0.240157	−	0.970734i	$-0.422801\pi$
$239$	7.42548	0.480315	0.240157	+	0.970734i	$0.422801\pi$
$240$	0	0
$241$	−10.4387	−0.672413	−0.336207	−	0.941788i	$-0.609144\pi$
$241$	−10.4387	−0.672413	−0.336207	+	0.941788i	$0.609144\pi$
$242$	−10.3684	−0.666503
$243$	−1.00000	−0.0641500
$244$	1.22425	0.0783748
$245$	0	0
$246$	−14.7005	−0.937271
$247$	0	0
$248$	−2.67513	−0.169871
$249$	−0.0303172	−0.00192127
$250$	0	0
$251$	10.7612	0.679238	0.339619	−	0.940563i	$-0.389702\pi$
$251$	10.7612	0.679238	0.339619	+	0.940563i	$0.389702\pi$
$252$	−0.481194	−0.0303124
$253$	−1.61213	−0.101354
$254$	−11.3503	−0.712179
$255$	0	0
$256$	4.61213	0.288258
$257$	21.9452	1.36891	0.684453	−	0.729057i	$-0.260041\pi$
$257$	21.9452	1.36891	0.684453	+	0.729057i	$0.260041\pi$
$258$	−15.2750	−0.950982
$259$	30.1319	1.87230
$260$	0	0
$261$	4.06300	0.251494
$262$	−9.44358	−0.583427
$263$	−5.73813	−0.353829	−0.176914	−	0.984226i	$-0.556612\pi$
$263$	−5.73813	−0.353829	−0.176914	+	0.984226i	$0.556612\pi$
$264$	5.35026	0.329286
$265$	0	0
$266$	0	0
$267$	13.7259	0.840012
$268$	2.79384	0.170661
$269$	13.3625	0.814725	0.407362	−	0.913267i	$-0.366449\pi$
$269$	13.3625	0.814725	0.407362	+	0.913267i	$0.366449\pi$
$270$	0	0
$271$	3.03761	0.184522	0.0922609	−	0.995735i	$-0.470591\pi$
$271$	3.03761	0.184522	0.0922609	+	0.995735i	$0.470591\pi$
$272$	16.3938	0.994017
$273$	14.4690	0.875702
$274$	−4.34297	−0.262368
$275$	0	0
$276$	0.156325	0.00940967
$277$	16.1441	0.970005	0.485003	−	0.874513i	$-0.338819\pi$
$277$	16.1441	0.970005	0.485003	+	0.874513i	$0.338819\pi$
$278$	−2.57452	−0.154409
$279$	1.00000	0.0598684
$280$	0	0
$281$	28.2130	1.68305	0.841523	−	0.540221i	$-0.181660\pi$
$281$	28.2130	1.68305	0.841523	+	0.540221i	$0.181660\pi$
$282$	−2.34297	−0.139522
$283$	13.0557	0.776081	0.388041	−	0.921642i	$-0.373152\pi$
$283$	13.0557	0.776081	0.388041	+	0.921642i	$0.373152\pi$
$284$	2.37565	0.140969
$285$	0	0
$286$	17.2750	1.02149
$287$	−24.6253	−1.45359
$288$	−1.09332	−0.0644246
$289$	−2.79877	−0.164633
$290$	0	0
$291$	−12.0000	−0.703452
$292$	−0.243767	−0.0142654
$293$	−2.00588	−0.117185	−0.0585924	−	0.998282i	$-0.518661\pi$
$293$	−2.00588	−0.117185	−0.0585924	+	0.998282i	$0.518661\pi$
$294$	1.24965	0.0728809
$295$	0	0
$296$	32.4871	1.88827
$297$	−2.00000	−0.116052
$298$	−23.4763	−1.35994
$299$	−4.70052	−0.271838
$300$	0	0
$301$	−25.5877	−1.47485
$302$	23.5428	1.35473
$303$	10.2374	0.588125
$304$	0	0
$305$	0	0
$306$	−5.58181	−0.319091
$307$	−0.743059	−0.0424086	−0.0212043	−	0.999775i	$-0.506750\pi$
$307$	−0.743059	−0.0424086	−0.0212043	+	0.999775i	$0.506750\pi$
$308$	−0.962389	−0.0548372
$309$	−13.7562	−0.782565
$310$	0	0
$311$	19.1368	1.08515	0.542575	−	0.840008i	$-0.317450\pi$
$311$	19.1368	1.08515	0.542575	+	0.840008i	$0.317450\pi$
$312$	15.5999	0.883171
$313$	11.3176	0.639707	0.319854	−	0.947467i	$-0.396366\pi$
$313$	11.3176	0.639707	0.319854	+	0.947467i	$0.396366\pi$
$314$	0.685935	0.0387096
$315$	0	0
$316$	2.49341	0.140265
$317$	−30.3185	−1.70286	−0.851429	−	0.524470i	$-0.824264\pi$
$317$	−30.3185	−1.70286	−0.851429	+	0.524470i	$0.824264\pi$
$318$	−10.5442	−0.591289
$319$	8.12601	0.454969
$320$	0	0
$321$	−4.15633	−0.231983
$322$	2.96239	0.165087
$323$	0	0
$324$	0.193937	0.0107743
$325$	0	0
$326$	−21.5247	−1.19214
$327$	13.8945	0.768365
$328$	−26.5501	−1.46598
$329$	−3.92478	−0.216380
$330$	0	0
$331$	−6.68006	−0.367169	−0.183585	−	0.983004i	$-0.558770\pi$
$331$	−6.68006	−0.367169	−0.183585	+	0.983004i	$0.558770\pi$
$332$	0.00587961	0.000322685 0
$333$	−12.1441	−0.665493
$334$	5.73813	0.313977
$335$	0	0
$336$	−10.7938	−0.588853
$337$	−14.2193	−0.774576	−0.387288	−	0.921959i	$-0.626588\pi$
$337$	−14.2193	−0.774576	−0.387288	+	0.921959i	$0.626588\pi$
$338$	31.1138	1.69237
$339$	−8.57452	−0.465704
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	19.4617	1.05083
$344$	−27.5877	−1.48743
$345$	0	0
$346$	−18.9986	−1.02137
$347$	−17.5271	−0.940902	−0.470451	−	0.882426i	$-0.655909\pi$
$347$	−17.5271	−0.940902	−0.470451	+	0.882426i	$0.655909\pi$
$348$	−0.787965	−0.0422394
$349$	−14.7308	−0.788524	−0.394262	−	0.918998i	$-0.629000\pi$
$349$	−14.7308	−0.788524	−0.394262	+	0.918998i	$0.629000\pi$
$350$	0	0
$351$	−5.83146	−0.311260
$352$	−2.18664	−0.116548
$353$	30.9321	1.64635	0.823174	−	0.567789i	$-0.192201\pi$
$353$	30.9321	1.64635	0.823174	+	0.567789i	$0.192201\pi$
$354$	−13.9429	−0.741055
$355$	0	0
$356$	−2.66196	−0.141083
$357$	−9.35026	−0.494868
$358$	−3.31406	−0.175154
$359$	−3.02539	−0.159674	−0.0798371	−	0.996808i	$-0.525440\pi$
$359$	−3.02539	−0.159674	−0.0798371	+	0.996808i	$0.525440\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−29.9756	−1.57548
$363$	7.00000	0.367405
$364$	−2.80606	−0.147078
$365$	0	0
$366$	−9.35026	−0.488746
$367$	31.3258	1.63519	0.817597	−	0.575790i	$-0.195306\pi$
$367$	31.3258	1.63519	0.817597	+	0.575790i	$0.195306\pi$
$368$	3.50659	0.182793
$369$	9.92478	0.516663
$370$	0	0
$371$	−17.6629	−0.917013
$372$	−0.193937	−0.0100551
$373$	33.4518	1.73207	0.866035	−	0.499983i	$-0.166660\pi$
$373$	33.4518	1.73207	0.866035	+	0.499983i	$0.166660\pi$
$374$	−11.1636	−0.577257
$375$	0	0
$376$	−4.23155	−0.218225
$377$	23.6932	1.22026
$378$	3.67513	0.189028
$379$	−1.40105	−0.0719669	−0.0359835	−	0.999352i	$-0.511456\pi$
$379$	−1.40105	−0.0719669	−0.0359835	+	0.999352i	$0.511456\pi$
$380$	0	0
$381$	7.66291	0.392583
$382$	−14.4958	−0.741669
$383$	32.1319	1.64186	0.820931	−	0.571027i	$-0.193455\pi$
$383$	32.1319	1.64186	0.820931	+	0.571027i	$0.193455\pi$
$384$	−12.6751	−0.646825
$385$	0	0
$386$	−7.16362	−0.364619
$387$	10.3127	0.524221
$388$	2.32724	0.118148
$389$	25.7621	1.30619	0.653095	−	0.757276i	$-0.273470\pi$
$389$	25.7621	1.30619	0.653095	+	0.757276i	$0.273470\pi$
$390$	0	0
$391$	3.03761	0.153619
$392$	2.25694	0.113993
$393$	6.37565	0.321609
$394$	14.8207	0.746654
$395$	0	0
$396$	0.387873	0.0194914
$397$	−4.38787	−0.220221	−0.110111	−	0.993919i	$-0.535120\pi$
$397$	−4.38787	−0.220221	−0.110111	+	0.993919i	$0.535120\pi$
$398$	−4.52961	−0.227049
$399$	0	0
$400$	0	0
$401$	−17.3479	−0.866312	−0.433156	−	0.901319i	$-0.642600\pi$
$401$	−17.3479	−0.866312	−0.433156	+	0.901319i	$0.642600\pi$
$402$	−21.3380	−1.06424
$403$	5.83146	0.290486
$404$	−1.98541	−0.0987779
$405$	0	0
$406$	−14.9321	−0.741066
$407$	−24.2882	−1.20392
$408$	−10.0811	−0.499089
$409$	14.3127	0.707715	0.353858	−	0.935299i	$-0.384870\pi$
$409$	14.3127	0.707715	0.353858	+	0.935299i	$0.384870\pi$
$410$	0	0
$411$	2.93207	0.144628
$412$	2.66784	0.131435
$413$	−23.3561	−1.14928
$414$	−1.19394	−0.0586788
$415$	0	0
$416$	−6.37565	−0.312592
$417$	1.73813	0.0851168
$418$	0	0
$419$	−8.31028	−0.405984	−0.202992	−	0.979180i	$-0.565067\pi$
$419$	−8.31028	−0.405984	−0.202992	+	0.979180i	$0.565067\pi$
$420$	0	0
$421$	0.00587961	0.000286555 0	0.000143277	−	1.00000i	$-0.499954\pi$
$421$	0.00587961	0.000286555 0	0.000143277		1.00000i	$0.499954\pi$
$422$	−12.5891	−0.612828
$423$	1.58181	0.0769102
$424$	−19.0435	−0.924834
$425$	0	0
$426$	−18.1441	−0.879085
$427$	−15.6629	−0.757981
$428$	0.806063	0.0389625
$429$	−11.6629	−0.563091
$430$	0	0
$431$	−1.86177	−0.0896785	−0.0448392	−	0.998994i	$-0.514278\pi$
$431$	−1.86177	−0.0896785	−0.0448392	+	0.998994i	$0.514278\pi$
$432$	4.35026	0.209302
$433$	−23.6566	−1.13686	−0.568431	−	0.822731i	$-0.692449\pi$
$433$	−23.6566	−1.13686	−0.568431	+	0.822731i	$0.692449\pi$
$434$	−3.67513	−0.176412
$435$	0	0
$436$	−2.69464	−0.129050
$437$	0	0
$438$	1.86177	0.0889590
$439$	1.67276	0.0798365	0.0399183	−	0.999203i	$-0.487290\pi$
$439$	1.67276	0.0798365	0.0399183	+	0.999203i	$0.487290\pi$
$440$	0	0
$441$	−0.843675	−0.0401750
$442$	−32.5501	−1.54825
$443$	10.2315	0.486116	0.243058	−	0.970012i	$-0.421850\pi$
$443$	10.2315	0.486116	0.243058	+	0.970012i	$0.421850\pi$
$444$	2.35519	0.111772
$445$	0	0
$446$	−32.1768	−1.52362
$447$	15.8496	0.749658
$448$	−17.5696	−0.830085
$449$	11.9224	0.562653	0.281327	−	0.959612i	$-0.409226\pi$
$449$	11.9224	0.562653	0.281327	+	0.959612i	$0.409226\pi$
$450$	0	0
$451$	19.8496	0.934679
$452$	1.66291	0.0782168
$453$	−15.8945	−0.746787
$454$	20.7670	0.974645
$455$	0	0
$456$	0	0
$457$	−9.16713	−0.428820	−0.214410	−	0.976744i	$-0.568783\pi$
$457$	−9.16713	−0.428820	−0.214410	+	0.976744i	$0.568783\pi$
$458$	−10.4025	−0.486075
$459$	3.76845	0.175896
$460$	0	0
$461$	27.2628	1.26976	0.634878	−	0.772612i	$-0.281050\pi$
$461$	27.2628	1.26976	0.634878	+	0.772612i	$0.281050\pi$
$462$	7.35026	0.341965
$463$	−6.77575	−0.314896	−0.157448	−	0.987527i	$-0.550327\pi$
$463$	−6.77575	−0.314896	−0.157448	+	0.987527i	$0.550327\pi$
$464$	−17.6751	−0.820547
$465$	0	0
$466$	16.1925	0.750104
$467$	−0.655618	−0.0303384	−0.0151692	−	0.999885i	$-0.504829\pi$
$467$	−0.655618	−0.0303384	−0.0151692	+	0.999885i	$0.504829\pi$
$468$	1.13093	0.0522774
$469$	−35.7440	−1.65051
$470$	0	0
$471$	−0.463096	−0.0213383
$472$	−25.1817	−1.15908
$473$	20.6253	0.948352
$474$	−19.0435	−0.874697
$475$	0	0
$476$	1.81336	0.0831151
$477$	7.11871	0.325944
$478$	10.9986	0.503064
$479$	8.31028	0.379706	0.189853	−	0.981812i	$-0.439199\pi$
$479$	8.31028	0.379706	0.189853	+	0.981812i	$0.439199\pi$
$480$	0	0
$481$	−70.8178	−3.22901
$482$	−15.4617	−0.704260
$483$	−2.00000	−0.0910032
$484$	−1.35756	−0.0617071
$485$	0	0
$486$	−1.48119	−0.0671883
$487$	−9.17347	−0.415690	−0.207845	−	0.978162i	$-0.566645\pi$
$487$	−9.17347	−0.415690	−0.207845	+	0.978162i	$0.566645\pi$
$488$	−16.8872	−0.764446
$489$	14.5320	0.657159
$490$	0	0
$491$	30.4241	1.37302	0.686510	−	0.727121i	$-0.259142\pi$
$491$	30.4241	1.37302	0.686510	+	0.727121i	$0.259142\pi$
$492$	−1.92478	−0.0867757
$493$	−15.3112	−0.689583
$494$	0	0
$495$	0	0
$496$	−4.35026	−0.195333
$497$	−30.3938	−1.36335
$498$	−0.0449056	−0.00201227
$499$	−25.1490	−1.12583	−0.562913	−	0.826516i	$-0.690319\pi$
$499$	−25.1490	−1.12583	−0.562913	+	0.826516i	$0.690319\pi$
$500$	0	0
$501$	−3.87399	−0.173077
$502$	15.9394	0.711409
$503$	−39.2955	−1.75210	−0.876050	−	0.482220i	$-0.839831\pi$
$503$	−39.2955	−1.75210	−0.876050	+	0.482220i	$0.839831\pi$
$504$	6.63752	0.295659
$505$	0	0
$506$	−2.38787	−0.106154
$507$	−21.0059	−0.932904
$508$	−1.48612	−0.0659359
$509$	26.9356	1.19390	0.596949	−	0.802279i	$-0.296379\pi$
$509$	26.9356	1.19390	0.596949	+	0.802279i	$0.296379\pi$
$510$	0	0
$511$	3.11871	0.137964
$512$	−18.5188	−0.818423
$513$	0	0
$514$	32.5052	1.43374
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	3.16362	0.139136
$518$	44.6312	1.96098
$519$	12.8265	0.563022
$520$	0	0
$521$	9.25060	0.405276	0.202638	−	0.979254i	$-0.435048\pi$
$521$	9.25060	0.405276	0.202638	+	0.979254i	$0.435048\pi$
$522$	6.01810	0.263405
$523$	5.79877	0.253562	0.126781	−	0.991931i	$-0.459535\pi$
$523$	5.79877	0.253562	0.126781	+	0.991931i	$0.459535\pi$
$524$	−1.23647	−0.0540156
$525$	0	0
$526$	−8.49929	−0.370587
$527$	−3.76845	−0.164156
$528$	8.70052	0.378642
$529$	−22.3503	−0.971751
$530$	0	0
$531$	9.41327	0.408501
$532$	0	0
$533$	57.8759	2.50688
$534$	20.3307	0.879798
$535$	0	0
$536$	−38.5379	−1.66458
$537$	2.23743	0.0965521
$538$	19.7924	0.853312
$539$	−1.68735	−0.0726793
$540$	0	0
$541$	−12.3674	−0.531716	−0.265858	−	0.964012i	$-0.585655\pi$
$541$	−12.3674	−0.531716	−0.265858	+	0.964012i	$0.585655\pi$
$542$	4.49929	0.193261
$543$	20.2374	0.868471
$544$	4.12013	0.176649
$545$	0	0
$546$	21.4314	0.917178
$547$	−35.7826	−1.52995	−0.764976	−	0.644058i	$-0.777249\pi$
$547$	−35.7826	−1.52995	−0.764976	+	0.644058i	$0.777249\pi$
$548$	−0.568636	−0.0242909
$549$	6.31265	0.269417
$550$	0	0
$551$	0	0
$552$	−2.15633	−0.0917793
$553$	−31.9003	−1.35654
$554$	23.9126	1.01595
$555$	0	0
$556$	−0.337088	−0.0142957
$557$	12.2433	0.518766	0.259383	−	0.965775i	$-0.416481\pi$
$557$	12.2433	0.518766	0.259383	+	0.965775i	$0.416481\pi$
$558$	1.48119	0.0627040
$559$	60.1378	2.54356
$560$	0	0
$561$	7.53690	0.318208
$562$	41.7889	1.76276
$563$	−42.0322	−1.77145	−0.885724	−	0.464213i	$-0.846337\pi$
$563$	−42.0322	−1.77145	−0.885724	+	0.464213i	$0.846337\pi$
$564$	−0.306771	−0.0129174
$565$	0	0
$566$	19.3380	0.812839
$567$	−2.48119	−0.104200
$568$	−32.7694	−1.37497
$569$	−32.8505	−1.37716	−0.688582	−	0.725158i	$-0.741767\pi$
$569$	−32.8505	−1.37716	−0.688582	+	0.725158i	$0.741767\pi$
$570$	0	0
$571$	36.8119	1.54053	0.770266	−	0.637723i	$-0.220123\pi$
$571$	36.8119	1.54053	0.770266	+	0.637723i	$0.220123\pi$
$572$	2.26187	0.0945733
$573$	9.78655	0.408839
$574$	−36.4749	−1.52243
$575$	0	0
$576$	7.08110	0.295046
$577$	16.8119	0.699890	0.349945	−	0.936770i	$-0.386200\pi$
$577$	16.8119	0.699890	0.349945	+	0.936770i	$0.386200\pi$
$578$	−4.14552	−0.172431
$579$	4.83638	0.200993
$580$	0	0
$581$	−0.0752228	−0.00312077
$582$	−17.7743	−0.736770
$583$	14.2374	0.589654
$584$	3.36248	0.139140
$585$	0	0
$586$	−2.97110	−0.122735
$587$	20.9624	0.865210	0.432605	−	0.901583i	$-0.357594\pi$
$587$	20.9624	0.865210	0.432605	+	0.901583i	$0.357594\pi$
$588$	0.163619	0.00674755
$589$	0	0
$590$	0	0
$591$	−10.0059	−0.411587
$592$	52.8300	2.17130
$593$	−22.6907	−0.931794	−0.465897	−	0.884839i	$-0.654268\pi$
$593$	−22.6907	−0.931794	−0.465897	+	0.884839i	$0.654268\pi$
$594$	−2.96239	−0.121548
$595$	0	0
$596$	−3.07381	−0.125908
$597$	3.05808	0.125159
$598$	−6.96239	−0.284713
$599$	−24.7245	−1.01022	−0.505108	−	0.863056i	$-0.668547\pi$
$599$	−24.7245	−1.01022	−0.505108	+	0.863056i	$0.668547\pi$
$600$	0	0
$601$	−0.412311	−0.0168185	−0.00840925	−	0.999965i	$-0.502677\pi$
$601$	−0.412311	−0.0168185	−0.00840925	+	0.999965i	$0.502677\pi$
$602$	−37.9003	−1.54470
$603$	14.4060	0.586657
$604$	3.08252	0.125426
$605$	0	0
$606$	15.1636	0.615980
$607$	−34.7186	−1.40919	−0.704593	−	0.709612i	$-0.748870\pi$
$607$	−34.7186	−1.40919	−0.704593	+	0.709612i	$0.748870\pi$
$608$	0	0
$609$	10.0811	0.408507
$610$	0	0
$611$	9.22425	0.373173
$612$	−0.730841	−0.0295425
$613$	28.5682	1.15386	0.576929	−	0.816794i	$-0.304251\pi$
$613$	28.5682	1.15386	0.576929	+	0.816794i	$0.304251\pi$
$614$	−1.10062	−0.0444172
$615$	0	0
$616$	13.2750	0.534867
$617$	28.5501	1.14938	0.574691	−	0.818370i	$-0.305122\pi$
$617$	28.5501	1.14938	0.574691	+	0.818370i	$0.305122\pi$
$618$	−20.3757	−0.819629
$619$	4.23155	0.170080	0.0850401	−	0.996378i	$-0.472898\pi$
$619$	4.23155	0.170080	0.0850401	+	0.996378i	$0.472898\pi$
$620$	0	0
$621$	0.806063	0.0323462
$622$	28.3453	1.13654
$623$	34.0567	1.36445
$624$	25.3684	1.01555
$625$	0	0
$626$	16.7635	0.670005
$627$	0	0
$628$	0.0898112	0.00358386
$629$	45.7645	1.82475
$630$	0	0
$631$	−2.17679	−0.0866568	−0.0433284	−	0.999061i	$-0.513796\pi$
$631$	−2.17679	−0.0866568	−0.0433284	+	0.999061i	$0.513796\pi$
$632$	−34.3938	−1.36811
$633$	8.49929	0.337817
$634$	−44.9076	−1.78351
$635$	0	0
$636$	−1.38058	−0.0547435
$637$	−4.91985	−0.194932
$638$	12.0362	0.476518
$639$	12.2496	0.484589
$640$	0	0
$641$	−44.3268	−1.75080	−0.875401	−	0.483397i	$-0.839403\pi$
$641$	−44.3268	−1.75080	−0.875401	+	0.483397i	$0.839403\pi$
$642$	−6.15633	−0.242971
$643$	34.8265	1.37342	0.686712	−	0.726929i	$-0.259053\pi$
$643$	34.8265	1.37342	0.686712	+	0.726929i	$0.259053\pi$
$644$	0.387873	0.0152843
$645$	0	0
$646$	0	0
$647$	28.4142	1.11708	0.558539	−	0.829478i	$-0.311362\pi$
$647$	28.4142	1.11708	0.558539	+	0.829478i	$0.311362\pi$
$648$	−2.67513	−0.105089
$649$	18.8265	0.739006
$650$	0	0
$651$	2.48119	0.0972457
$652$	−2.81828	−0.110372
$653$	28.4544	1.11351	0.556753	−	0.830678i	$-0.312047\pi$
$653$	28.4544	1.11351	0.556753	+	0.830678i	$0.312047\pi$
$654$	20.5804	0.804757
$655$	0	0
$656$	−43.1754	−1.68572
$657$	−1.25694	−0.0490379
$658$	−5.81336	−0.226628
$659$	19.3479	0.753687	0.376843	−	0.926277i	$-0.377009\pi$
$659$	19.3479	0.753687	0.376843	+	0.926277i	$0.377009\pi$
$660$	0	0
$661$	−20.1721	−0.784602	−0.392301	−	0.919837i	$-0.628321\pi$
$661$	−20.1721	−0.784602	−0.392301	+	0.919837i	$0.628321\pi$
$662$	−9.89446	−0.384559
$663$	21.9756	0.853460
$664$	−0.0811024	−0.00314738
$665$	0	0
$666$	−17.9878	−0.697012
$667$	−3.27504	−0.126810
$668$	0.751309	0.0290690
$669$	21.7235	0.839881
$670$	0	0
$671$	12.6253	0.487394
$672$	−2.71274	−0.104646
$673$	−46.7040	−1.80031	−0.900154	−	0.435572i	$-0.856546\pi$
$673$	−46.7040	−1.80031	−0.900154	+	0.435572i	$0.856546\pi$
$674$	−21.0616	−0.811262
$675$	0	0
$676$	4.07381	0.156685
$677$	−44.4299	−1.70758	−0.853791	−	0.520616i	$-0.825702\pi$
$677$	−44.4299	−1.70758	−0.853791	+	0.520616i	$0.825702\pi$
$678$	−12.7005	−0.487761
$679$	−29.7743	−1.14263
$680$	0	0
$681$	−14.0205	−0.537266
$682$	2.96239	0.113436
$683$	−3.21837	−0.123148	−0.0615738	−	0.998103i	$-0.519612\pi$
$683$	−3.21837	−0.123148	−0.0615738	+	0.998103i	$0.519612\pi$
$684$	0	0
$685$	0	0
$686$	28.8265	1.10060
$687$	7.02302	0.267945
$688$	−44.8627	−1.71038
$689$	41.5125	1.58150
$690$	0	0
$691$	−32.9741	−1.25440	−0.627198	−	0.778860i	$-0.715798\pi$
$691$	−32.9741	−1.25440	−0.627198	+	0.778860i	$0.715798\pi$
$692$	−2.48753	−0.0945618
$693$	−4.96239	−0.188506
$694$	−25.9610	−0.985465
$695$	0	0
$696$	10.8691	0.411991
$697$	−37.4010	−1.41667
$698$	−21.8192	−0.825870
$699$	−10.9321	−0.413489
$700$	0	0
$701$	33.3357	1.25907	0.629536	−	0.776972i	$-0.283245\pi$
$701$	33.3357	1.25907	0.629536	+	0.776972i	$0.283245\pi$
$702$	−8.63752	−0.326002
$703$	0	0
$704$	14.1622	0.533758
$705$	0	0
$706$	45.8164	1.72432
$707$	25.4010	0.955305
$708$	−1.82558	−0.0686094
$709$	−34.8773	−1.30985	−0.654923	−	0.755696i	$-0.727299\pi$
$709$	−34.8773	−1.30985	−0.654923	+	0.755696i	$0.727299\pi$
$710$	0	0
$711$	12.8568	0.482169
$712$	36.7186	1.37609
$713$	−0.806063	−0.0301873
$714$	−13.8496	−0.518307
$715$	0	0
$716$	−0.433919	−0.0162163
$717$	−7.42548	−0.277310
$718$	−4.48119	−0.167237
$719$	1.37470	0.0512676	0.0256338	−	0.999671i	$-0.491840\pi$
$719$	1.37470	0.0512676	0.0256338	+	0.999671i	$0.491840\pi$
$720$	0	0
$721$	−34.1319	−1.27114
$722$	−28.1427	−1.04736
$723$	10.4387	0.388218
$724$	−3.92478	−0.145863
$725$	0	0
$726$	10.3684	0.384806
$727$	−46.3209	−1.71795	−0.858974	−	0.512020i	$-0.828897\pi$
$727$	−46.3209	−1.71795	−0.858974	+	0.512020i	$0.828897\pi$
$728$	38.7064	1.43455
$729$	1.00000	0.0370370
$730$	0	0
$731$	−38.8627	−1.43739
$732$	−1.22425	−0.0452497
$733$	29.0738	1.07387	0.536933	−	0.843625i	$-0.319583\pi$
$733$	29.0738	1.07387	0.536933	+	0.843625i	$0.319583\pi$
$734$	46.3996	1.71264
$735$	0	0
$736$	0.881286	0.0324846
$737$	28.8119	1.06130
$738$	14.7005	0.541134
$739$	18.7454	0.689562	0.344781	−	0.938683i	$-0.387953\pi$
$739$	18.7454	0.689562	0.344781	+	0.938683i	$0.387953\pi$
$740$	0	0
$741$	0	0
$742$	−26.1622	−0.960445
$743$	−51.8858	−1.90350	−0.951752	−	0.306869i	$-0.900719\pi$
$743$	−51.8858	−1.90350	−0.951752	+	0.306869i	$0.900719\pi$
$744$	2.67513	0.0980751
$745$	0	0
$746$	49.5487	1.81411
$747$	0.0303172	0.00110925
$748$	−1.46168	−0.0534444
$749$	−10.3127	−0.376816
$750$	0	0
$751$	47.9149	1.74844	0.874220	−	0.485530i	$-0.161373\pi$
$751$	47.9149	1.74844	0.874220	+	0.485530i	$0.161373\pi$
$752$	−6.88129	−0.250935
$753$	−10.7612	−0.392158
$754$	35.0943	1.27806
$755$	0	0
$756$	0.481194	0.0175009
$757$	−27.6302	−1.00424	−0.502119	−	0.864799i	$-0.667446\pi$
$757$	−27.6302	−1.00424	−0.502119	+	0.864799i	$0.667446\pi$
$758$	−2.07522	−0.0753755
$759$	1.61213	0.0585165
$760$	0	0
$761$	11.0108	0.399141	0.199571	−	0.979883i	$-0.436045\pi$
$761$	11.0108	0.399141	0.199571	+	0.979883i	$0.436045\pi$
$762$	11.3503	0.411177
$763$	34.4749	1.24807
$764$	−1.89797	−0.0686661
$765$	0	0
$766$	47.5936	1.71963
$767$	54.8930	1.98207
$768$	−4.61213	−0.166426
$769$	−1.52118	−0.0548550	−0.0274275	−	0.999624i	$-0.508732\pi$
$769$	−1.52118	−0.0548550	−0.0274275	+	0.999624i	$0.508732\pi$
$770$	0	0
$771$	−21.9452	−0.790339
$772$	−0.937951	−0.0337576
$773$	47.5085	1.70876	0.854381	−	0.519647i	$-0.173937\pi$
$773$	47.5085	1.70876	0.854381	+	0.519647i	$0.173937\pi$
$774$	15.2750	0.549050
$775$	0	0
$776$	−32.1016	−1.15238
$777$	−30.1319	−1.08098
$778$	38.1587	1.36806
$779$	0	0
$780$	0	0
$781$	24.4993	0.876654
$782$	4.49929	0.160894
$783$	−4.06300	−0.145200
$784$	3.67021	0.131079
$785$	0	0
$786$	9.44358	0.336841
$787$	27.2144	0.970089	0.485044	−	0.874490i	$-0.338803\pi$
$787$	27.2144	0.970089	0.485044	+	0.874490i	$0.338803\pi$
$788$	1.94051	0.0691277
$789$	5.73813	0.204283
$790$	0	0
$791$	−21.2750	−0.756453
$792$	−5.35026	−0.190113
$793$	36.8119	1.30723
$794$	−6.49929	−0.230651
$795$	0	0
$796$	−0.593073	−0.0210209
$797$	1.94525	0.0689041	0.0344521	−	0.999406i	$-0.489031\pi$
$797$	1.94525	0.0689041	0.0344521	+	0.999406i	$0.489031\pi$
$798$	0	0
$799$	−5.96097	−0.210884
$800$	0	0
$801$	−13.7259	−0.484981
$802$	−25.6956	−0.907343
$803$	−2.51388	−0.0887129
$804$	−2.79384	−0.0985313
$805$	0	0
$806$	8.63752	0.304244
$807$	−13.3625	−0.470382
$808$	27.3865	0.963452
$809$	9.81573	0.345103	0.172551	−	0.985001i	$-0.444799\pi$
$809$	9.81573	0.345103	0.172551	+	0.985001i	$0.444799\pi$
$810$	0	0
$811$	3.71179	0.130338	0.0651692	−	0.997874i	$-0.479241\pi$
$811$	3.71179	0.130338	0.0651692	+	0.997874i	$0.479241\pi$
$812$	−1.95509	−0.0686104
$813$	−3.03761	−0.106534
$814$	−35.9756	−1.26094
$815$	0	0
$816$	−16.3938	−0.573896
$817$	0	0
$818$	21.1998	0.741234
$819$	−14.4690	−0.505587
$820$	0	0
$821$	30.9403	1.07982	0.539912	−	0.841721i	$-0.318457\pi$
$821$	30.9403	1.07982	0.539912	+	0.841721i	$0.318457\pi$
$822$	4.34297	0.151478
$823$	−29.4471	−1.02646	−0.513231	−	0.858251i	$-0.671551\pi$
$823$	−29.4471	−1.02646	−0.513231	+	0.858251i	$0.671551\pi$
$824$	−36.7997	−1.28198
$825$	0	0
$826$	−34.5950	−1.20371
$827$	−9.11871	−0.317089	−0.158544	−	0.987352i	$-0.550680\pi$
$827$	−9.11871	−0.317089	−0.158544	+	0.987352i	$0.550680\pi$
$828$	−0.156325	−0.00543268
$829$	1.19982	0.0416713	0.0208357	−	0.999783i	$-0.493367\pi$
$829$	1.19982	0.0416713	0.0208357	+	0.999783i	$0.493367\pi$
$830$	0	0
$831$	−16.1441	−0.560033
$832$	41.2931	1.43158
$833$	3.17935	0.110158
$834$	2.57452	0.0891482
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	−12.3091	−0.425212
$839$	20.7753	0.717243	0.358621	−	0.933483i	$-0.383247\pi$
$839$	20.7753	0.717243	0.358621	+	0.933483i	$0.383247\pi$
$840$	0	0
$841$	−12.4920	−0.430759
$842$	0.00870884	0.000300126 0
$843$	−28.2130	−0.971707
$844$	−1.64832	−0.0567376
$845$	0	0
$846$	2.34297	0.0805529
$847$	17.3684	0.596784
$848$	−30.9683	−1.06345
$849$	−13.0557	−0.448071
$850$	0	0
$851$	9.78892	0.335560
$852$	−2.37565	−0.0813886
$853$	−41.0757	−1.40641	−0.703203	−	0.710989i	$-0.748247\pi$
$853$	−41.0757	−1.40641	−0.703203	+	0.710989i	$0.748247\pi$
$854$	−23.1998	−0.793881
$855$	0	0
$856$	−11.1187	−0.380030
$857$	27.3766	0.935167	0.467584	−	0.883949i	$-0.345125\pi$
$857$	27.3766	0.935167	0.467584	+	0.883949i	$0.345125\pi$
$858$	−17.2750	−0.589760
$859$	−32.1016	−1.09529	−0.547646	−	0.836710i	$-0.684476\pi$
$859$	−32.1016	−1.09529	−0.547646	+	0.836710i	$0.684476\pi$
$860$	0	0
$861$	24.6253	0.839228
$862$	−2.75765	−0.0939259
$863$	−55.2966	−1.88232	−0.941160	−	0.337962i	$-0.890263\pi$
$863$	−55.2966	−1.88232	−0.941160	+	0.337962i	$0.890263\pi$
$864$	1.09332	0.0371955
$865$	0	0
$866$	−35.0400	−1.19071
$867$	2.79877	0.0950512
$868$	−0.481194	−0.0163328
$869$	25.7137	0.872277
$870$	0	0
$871$	84.0078	2.84650
$872$	37.1695	1.25872
$873$	12.0000	0.406138
$874$	0	0
$875$	0	0
$876$	0.243767	0.00823611
$877$	−51.1392	−1.72685	−0.863424	−	0.504479i	$-0.831684\pi$
$877$	−51.1392	−1.72685	−0.863424	+	0.504479i	$0.831684\pi$
$878$	2.47768	0.0836178
$879$	2.00588	0.0676566
$880$	0	0
$881$	39.1632	1.31944	0.659720	−	0.751511i	$-0.270675\pi$
$881$	39.1632	1.31944	0.659720	+	0.751511i	$0.270675\pi$
$882$	−1.24965	−0.0420778
$883$	−40.6155	−1.36682	−0.683409	−	0.730035i	$-0.739504\pi$
$883$	−40.6155	−1.36682	−0.683409	+	0.730035i	$0.739504\pi$
$884$	−4.26187	−0.143342
$885$	0	0
$886$	15.1549	0.509139
$887$	6.16617	0.207040	0.103520	−	0.994627i	$-0.466989\pi$
$887$	6.16617	0.207040	0.103520	+	0.994627i	$0.466989\pi$
$888$	−32.4871	−1.09019
$889$	19.0132	0.637681
$890$	0	0
$891$	2.00000	0.0670025
$892$	−4.21299	−0.141061
$893$	0	0
$894$	23.4763	0.785164
$895$	0	0
$896$	−31.4495	−1.05065
$897$	4.70052	0.156946
$898$	17.6594	0.589302
$899$	4.06300	0.135509
$900$	0	0
$901$	−26.8265	−0.893721
$902$	29.4010	0.978948
$903$	25.5877	0.851505
$904$	−22.9380	−0.762905
$905$	0	0
$906$	−23.5428	−0.782157
$907$	21.4944	0.713709	0.356854	−	0.934160i	$-0.383849\pi$
$907$	21.4944	0.713709	0.356854	+	0.934160i	$0.383849\pi$
$908$	2.71908	0.0902359
$909$	−10.2374	−0.339554
$910$	0	0
$911$	17.4471	0.578048	0.289024	−	0.957322i	$-0.406669\pi$
$911$	17.4471	0.578048	0.289024	+	0.957322i	$0.406669\pi$
$912$	0	0
$913$	0.0606343	0.00200670
$914$	−13.5783	−0.449130
$915$	0	0
$916$	−1.36202	−0.0450024
$917$	15.8192	0.522397
$918$	5.58181	0.184227
$919$	32.5355	1.07325	0.536623	−	0.843822i	$-0.319700\pi$
$919$	32.5355	1.07325	0.536623	+	0.843822i	$0.319700\pi$
$920$	0	0
$921$	0.743059	0.0244846
$922$	40.3815	1.32989
$923$	71.4333	2.35125
$924$	0.962389	0.0316603
$925$	0	0
$926$	−10.0362	−0.329810
$927$	13.7562	0.451814
$928$	−4.44217	−0.145821
$929$	45.4641	1.49163	0.745814	−	0.666155i	$-0.232061\pi$
$929$	45.4641	1.49163	0.745814	+	0.666155i	$0.232061\pi$
$930$	0	0
$931$	0	0
$932$	2.12013	0.0694471
$933$	−19.1368	−0.626511
$934$	−0.971097	−0.0317753
$935$	0	0
$936$	−15.5999	−0.509899
$937$	56.2031	1.83608	0.918038	−	0.396492i	$-0.129772\pi$
$937$	56.2031	1.83608	0.918038	+	0.396492i	$0.129772\pi$
$938$	−52.9438	−1.72868
$939$	−11.3176	−0.369335
$940$	0	0
$941$	13.1251	0.427864	0.213932	−	0.976849i	$-0.431373\pi$
$941$	13.1251	0.427864	0.213932	+	0.976849i	$0.431373\pi$
$942$	−0.685935	−0.0223490
$943$	−8.00000	−0.260516
$944$	−40.9502	−1.33281
$945$	0	0
$946$	30.5501	0.993269
$947$	55.1939	1.79356	0.896781	−	0.442475i	$-0.145899\pi$
$947$	55.1939	1.79356	0.896781	+	0.442475i	$0.145899\pi$
$948$	−2.49341	−0.0809823
$949$	−7.32979	−0.237935
$950$	0	0
$951$	30.3185	0.983146
$952$	−25.0132	−0.810682
$953$	−10.4690	−0.339123	−0.169562	−	0.985520i	$-0.554235\pi$
$953$	−10.4690	−0.339123	−0.169562	+	0.985520i	$0.554235\pi$
$954$	10.5442	0.341381
$955$	0	0
$956$	1.44007	0.0465753
$957$	−8.12601	−0.262677
$958$	12.3091	0.397690
$959$	7.27504	0.234923
$960$	0	0
$961$	1.00000	0.0322581
$962$	−104.895	−3.38195
$963$	4.15633	0.133936
$964$	−2.02444	−0.0652027
$965$	0	0
$966$	−2.96239	−0.0953133
$967$	7.37661	0.237216	0.118608	−	0.992941i	$-0.462157\pi$
$967$	7.37661	0.237216	0.118608	+	0.992941i	$0.462157\pi$
$968$	18.7259	0.601874
$969$	0	0
$970$	0	0
$971$	30.4020	0.975647	0.487823	−	0.872942i	$-0.337791\pi$
$971$	30.4020	0.975647	0.487823	+	0.872942i	$0.337791\pi$
$972$	−0.193937	−0.00622052
$973$	4.31265	0.138257
$974$	−13.5877	−0.435378
$975$	0	0
$976$	−27.4617	−0.879027
$977$	−12.8510	−0.411139	−0.205569	−	0.978643i	$-0.565905\pi$
$977$	−12.8510	−0.411139	−0.205569	+	0.978643i	$0.565905\pi$
$978$	21.5247	0.688284
$979$	−27.4518	−0.877364
$980$	0	0
$981$	−13.8945	−0.443616
$982$	45.0640	1.43805
$983$	−30.1162	−0.960556	−0.480278	−	0.877116i	$-0.659464\pi$
$983$	−30.1162	−0.960556	−0.480278	+	0.877116i	$0.659464\pi$
$984$	26.5501	0.846386
$985$	0	0
$986$	−22.6789	−0.722244
$987$	3.92478	0.124927
$988$	0	0
$989$	−8.31265	−0.264327
$990$	0	0
$991$	−27.9873	−0.889047	−0.444523	−	0.895767i	$-0.646627\pi$
$991$	−27.9873	−0.889047	−0.444523	+	0.895767i	$0.646627\pi$
$992$	−1.09332	−0.0347130
$993$	6.68006	0.211985
$994$	−45.0191	−1.42792
$995$	0	0
$996$	−0.00587961	−0.000186302 0
$997$	22.0968	0.699814	0.349907	−	0.936785i	$-0.386213\pi$
$997$	22.0968	0.699814	0.349907	+	0.936785i	$0.386213\pi$
$998$	−37.2506	−1.17915
$999$	12.1441	0.384223

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.a.q.1.3		3
3.2	odd	2		6975.2.a.be.1.1		3
5.2	odd	4		2325.2.c.o.1024.5		6
5.3	odd	4		2325.2.c.o.1024.2		6
5.4	even	2		465.2.a.f.1.1	✓	3
15.14	odd	2		1395.2.a.i.1.3		3
20.19	odd	2		7440.2.a.bp.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.f.1.1	✓	3	5.4	even	2
1395.2.a.i.1.3		3	15.14	odd	2
2325.2.a.q.1.3		3	1.1	even	1	trivial
2325.2.c.o.1024.2		6	5.3	odd	4
2325.2.c.o.1024.5		6	5.2	odd	4
6975.2.a.be.1.1		3	3.2	odd	2
7440.2.a.bp.1.1		3	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2325.a (trivial)

\(f(q)\)	\(=\)	\(q+1.48119 q^{2} -1.00000 q^{3} +0.193937 q^{4} -1.48119 q^{6} -2.48119 q^{7} -2.67513 q^{8} +1.00000 q^{9} +2.00000 q^{11} -0.193937 q^{12} +5.83146 q^{13} -3.67513 q^{14} -4.35026 q^{16} -3.76845 q^{17} +1.48119 q^{18} +2.48119 q^{21} +2.96239 q^{22} -0.806063 q^{23} +2.67513 q^{24} +8.63752 q^{26} -1.00000 q^{27} -0.481194 q^{28} +4.06300 q^{29} +1.00000 q^{31} -1.09332 q^{32} -2.00000 q^{33} -5.58181 q^{34} +0.193937 q^{36} -12.1441 q^{37} -5.83146 q^{39} +9.92478 q^{41} +3.67513 q^{42} +10.3127 q^{43} +0.387873 q^{44} -1.19394 q^{46} +1.58181 q^{47} +4.35026 q^{48} -0.843675 q^{49} +3.76845 q^{51} +1.13093 q^{52} +7.11871 q^{53} -1.48119 q^{54} +6.63752 q^{56} +6.01810 q^{58} +9.41327 q^{59} +6.31265 q^{61} +1.48119 q^{62} -2.48119 q^{63} +7.08110 q^{64} -2.96239 q^{66} +14.4060 q^{67} -0.730841 q^{68} +0.806063 q^{69} +12.2496 q^{71} -2.67513 q^{72} -1.25694 q^{73} -17.9878 q^{74} -4.96239 q^{77} -8.63752 q^{78} +12.8568 q^{79} +1.00000 q^{81} +14.7005 q^{82} +0.0303172 q^{83} +0.481194 q^{84} +15.2750 q^{86} -4.06300 q^{87} -5.35026 q^{88} -13.7259 q^{89} -14.4690 q^{91} -0.156325 q^{92} -1.00000 q^{93} +2.34297 q^{94} +1.09332 q^{96} +12.0000 q^{97} -1.24965 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 3 q^{3} + q^{4} + q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9} + 6 q^{11} - q^{12} + 2 q^{13} - 6 q^{14} - 3 q^{16} - q^{18} + 2 q^{21} - 2 q^{22} - 2 q^{23} + 3 q^{24} + 10 q^{26} - 3 q^{27}+ \cdots + 6 q^{99}+O(q^{100}) \) 3 * q - q^2 - 3 * q^3 + q^4 + q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9 + 6 * q^11 - q^12 + 2 * q^13 - 6 * q^14 - 3 * q^16 - q^18 + 2 * q^21 - 2 * q^22 - 2 * q^23 + 3 * q^24 + 10 * q^26 - 3 * q^27 + 4 * q^28 + 8 * q^29 + 3 * q^31 + 3 * q^32 - 6 * q^33 - 18 * q^34 + q^36 - 2 * q^39 + 8 * q^41 + 6 * q^42 + 10 * q^43 + 2 * q^44 - 4 * q^46 + 6 * q^47 + 3 * q^48 - 13 * q^49 + 8 * q^52 + q^54 + 4 * q^56 - 10 * q^58 + 14 * q^59 - 2 * q^61 - q^62 - 2 * q^63 - 11 * q^64 + 2 * q^66 + 16 * q^67 + 20 * q^68 + 2 * q^69 + 20 * q^71 - 3 * q^72 - 28 * q^74 - 4 * q^77 - 10 * q^78 + 8 * q^79 + 3 * q^81 + 24 * q^82 - 2 * q^83 - 4 * q^84 + 14 * q^86 - 8 * q^87 - 6 * q^88 - 6 * q^89 - 12 * q^91 + 10 * q^92 - 3 * q^93 - 16 * q^94 - 3 * q^96 + 36 * q^97 + 13 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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