LMFDB - Embedded newform 975.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(1,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, 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("975.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.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:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1821184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} + 13x - 4 $ x^5 - 8x^3 + 13x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.22308$ of defining polynomial
Character	$\chi$	$=$	975.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.22308 q^{2} -1.00000 q^{3} -0.504085 q^{4} -1.22308 q^{6} +4.18388 q^{7} -3.06269 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.22308 q^{2} -1.00000 q^{3} -0.504085 q^{4} -1.22308 q^{6} +4.18388 q^{7} -3.06269 q^{8} +1.00000 q^{9} +1.89812 q^{11} +0.504085 q^{12} -1.00000 q^{13} +5.11720 q^{14} -2.73773 q^{16} -1.73773 q^{17} +1.22308 q^{18} -1.11720 q^{19} -4.18388 q^{21} +2.32154 q^{22} +9.30108 q^{23} +3.06269 q^{24} -1.22308 q^{26} -1.00000 q^{27} -2.10903 q^{28} +5.00817 q^{29} +10.0095 q^{31} +2.77692 q^{32} -1.89812 q^{33} -2.12537 q^{34} -0.504085 q^{36} -11.3011 q^{37} -1.36642 q^{38} +1.00000 q^{39} +8.02349 q^{41} -5.11720 q^{42} +1.00817 q^{43} -0.956813 q^{44} +11.3759 q^{46} +5.63584 q^{47} +2.73773 q^{48} +10.5048 q^{49} +1.73773 q^{51} +0.504085 q^{52} +0.174373 q^{53} -1.22308 q^{54} -12.8139 q^{56} +1.11720 q^{57} +6.12537 q^{58} +8.64402 q^{59} +3.27044 q^{61} +12.2424 q^{62} +4.18388 q^{63} +8.87184 q^{64} -2.32154 q^{66} -11.4543 q^{67} +0.875963 q^{68} -9.30108 q^{69} -5.37357 q^{71} -3.06269 q^{72} -4.01634 q^{73} -13.8221 q^{74} +0.563165 q^{76} +7.94149 q^{77} +1.22308 q^{78} -0.613690 q^{79} +1.00000 q^{81} +9.81334 q^{82} -9.08333 q^{83} +2.10903 q^{84} +1.23307 q^{86} -5.00817 q^{87} -5.81334 q^{88} -1.46831 q^{89} -4.18388 q^{91} -4.68854 q^{92} -10.0095 q^{93} +6.89307 q^{94} -2.77692 q^{96} +3.85359 q^{97} +12.8482 q^{98} +1.89812 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 6 q^{4} - 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 6 * q^4 - 5 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + 6 q^{4} - 5 q^{7} + 5 q^{9} + 5 q^{11} - 6 q^{12} - 5 q^{13} + 12 q^{14} + 5 q^{17} + 8 q^{19} + 5 q^{21} + 8 q^{22} + 7 q^{23} - 5 q^{27} - 14 q^{28} + 8 q^{29} + 12 q^{31} + 20 q^{32} - 5 q^{33} + 20 q^{34} + 6 q^{36} - 17 q^{37} + 24 q^{38} + 5 q^{39} + 5 q^{41} - 12 q^{42} - 12 q^{43} + 18 q^{44} - 12 q^{46} + 10 q^{47} + 22 q^{49} - 5 q^{51} - 6 q^{52} + 13 q^{53} - 8 q^{57} + 8 q^{59} + 13 q^{61} + 40 q^{62} - 5 q^{63} - 16 q^{64} - 8 q^{66} - 28 q^{67} + 14 q^{68} - 7 q^{69} + 5 q^{71} + 14 q^{73} + 12 q^{74} + 35 q^{77} + q^{79} + 5 q^{81} - 16 q^{82} + 6 q^{83} + 14 q^{84} - 8 q^{87} + 36 q^{88} + 19 q^{89} + 5 q^{91} + 10 q^{92} - 12 q^{93} - 12 q^{94} - 20 q^{96} + 13 q^{97} - 28 q^{98} + 5 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 6 * q^4 - 5 * q^7 + 5 * q^9 + 5 * q^11 - 6 * q^12 - 5 * q^13 + 12 * q^14 + 5 * q^17 + 8 * q^19 + 5 * q^21 + 8 * q^22 + 7 * q^23 - 5 * q^27 - 14 * q^28 + 8 * q^29 + 12 * q^31 + 20 * q^32 - 5 * q^33 + 20 * q^34 + 6 * q^36 - 17 * q^37 + 24 * q^38 + 5 * q^39 + 5 * q^41 - 12 * q^42 - 12 * q^43 + 18 * q^44 - 12 * q^46 + 10 * q^47 + 22 * q^49 - 5 * q^51 - 6 * q^52 + 13 * q^53 - 8 * q^57 + 8 * q^59 + 13 * q^61 + 40 * q^62 - 5 * q^63 - 16 * q^64 - 8 * q^66 - 28 * q^67 + 14 * q^68 - 7 * q^69 + 5 * q^71 + 14 * q^73 + 12 * q^74 + 35 * q^77 + q^79 + 5 * q^81 - 16 * q^82 + 6 * q^83 + 14 * q^84 - 8 * q^87 + 36 * q^88 + 19 * q^89 + 5 * q^91 + 10 * q^92 - 12 * q^93 - 12 * q^94 - 20 * q^96 + 13 * q^97 - 28 * q^98 + 5 * 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 $

$2$	1.22308	0.864845	0.432423	−	0.901671i	$-0.357659\pi$
$2$	1.22308	0.864845	0.432423	+	0.901671i	$0.357659\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.504085	−0.252043
$5$	0	0
$5$	0	0
$6$	−1.22308	−0.499319
$7$	4.18388	1.58136	0.790679	−	0.612231i	$-0.209728\pi$
$7$	4.18388	1.58136	0.790679	+	0.612231i	$0.209728\pi$
$8$	−3.06269	−1.08282
$9$	1.00000	0.333333
$10$	0	0
$11$	1.89812	0.572304	0.286152	−	0.958184i	$-0.407624\pi$
$11$	1.89812	0.572304	0.286152	+	0.958184i	$0.407624\pi$
$12$	0.504085	0.145517
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	5.11720	1.36763
$15$	0	0
$16$	−2.73773	−0.684432
$17$	−1.73773	−0.421461	−0.210730	−	0.977544i	$-0.567584\pi$
$17$	−1.73773	−0.421461	−0.210730	+	0.977544i	$0.567584\pi$
$18$	1.22308	0.288282
$19$	−1.11720	−0.256304	−0.128152	−	0.991755i	$-0.540905\pi$
$19$	−1.11720	−0.256304	−0.128152	+	0.991755i	$0.540905\pi$
$20$	0	0
$21$	−4.18388	−0.912997
$22$	2.32154	0.494954
$23$	9.30108	1.93941	0.969705	−	0.244280i	$-0.0785515\pi$
$23$	9.30108	1.93941	0.969705	+	0.244280i	$0.0785515\pi$
$24$	3.06269	0.625168
$25$	0	0
$26$	−1.22308	−0.239865
$27$	−1.00000	−0.192450
$28$	−2.10903	−0.398570
$29$	5.00817	0.929994	0.464997	−	0.885312i	$-0.346055\pi$
$29$	5.00817	0.929994	0.464997	+	0.885312i	$0.346055\pi$
$30$	0	0
$31$	10.0095	1.79776	0.898880	−	0.438194i	$-0.144382\pi$
$31$	10.0095	1.79776	0.898880	+	0.438194i	$0.144382\pi$
$32$	2.77692	0.490895
$33$	−1.89812	−0.330420
$34$	−2.12537	−0.364498
$35$	0	0
$36$	−0.504085	−0.0840142
$37$	−11.3011	−1.85789	−0.928943	−	0.370222i	$-0.879282\pi$
$37$	−11.3011	−1.85789	−0.928943	+	0.370222i	$0.879282\pi$
$38$	−1.36642	−0.221663
$39$	1.00000	0.160128
$40$	0	0
$41$	8.02349	1.25306	0.626529	−	0.779398i	$-0.284475\pi$
$41$	8.02349	1.25306	0.626529	+	0.779398i	$0.284475\pi$
$42$	−5.11720	−0.789601
$43$	1.00817	0.153745	0.0768723	−	0.997041i	$-0.475507\pi$
$43$	1.00817	0.153745	0.0768723	+	0.997041i	$0.475507\pi$
$44$	−0.956813	−0.144245
$45$	0	0
$46$	11.3759	1.67729
$47$	5.63584	0.822072	0.411036	−	0.911619i	$-0.365167\pi$
$47$	5.63584	0.822072	0.411036	+	0.911619i	$0.365167\pi$
$48$	2.73773	0.395157
$49$	10.5048	1.50069
$50$	0	0
$51$	1.73773	0.243331
$52$	0.504085	0.0699040
$53$	0.174373	0.0239520	0.0119760	−	0.999928i	$-0.496188\pi$
$53$	0.174373	0.0239520	0.0119760	+	0.999928i	$0.496188\pi$
$54$	−1.22308	−0.166440
$55$	0	0
$56$	−12.8139	−1.71233
$57$	1.11720	0.147977
$58$	6.12537	0.804301
$59$	8.64402	1.12535	0.562677	−	0.826677i	$-0.309771\pi$
$59$	8.64402	1.12535	0.562677	+	0.826677i	$0.309771\pi$
$60$	0	0
$61$	3.27044	0.418737	0.209369	−	0.977837i	$-0.432859\pi$
$61$	3.27044	0.418737	0.209369	+	0.977837i	$0.432859\pi$
$62$	12.2424	1.55478
$63$	4.18388	0.527119
$64$	8.87184	1.10898
$65$	0	0
$66$	−2.32154	−0.285762
$67$	−11.4543	−1.39937	−0.699684	−	0.714452i	$-0.746676\pi$
$67$	−11.4543	−1.39937	−0.699684	+	0.714452i	$0.746676\pi$
$68$	0.875963	0.106226
$69$	−9.30108	−1.11972
$70$	0	0
$71$	−5.37357	−0.637726	−0.318863	−	0.947801i	$-0.603301\pi$
$71$	−5.37357	−0.637726	−0.318863	+	0.947801i	$0.603301\pi$
$72$	−3.06269	−0.360941
$73$	−4.01634	−0.470077	−0.235039	−	0.971986i	$-0.575522\pi$
$73$	−4.01634	−0.470077	−0.235039	+	0.971986i	$0.575522\pi$
$74$	−13.8221	−1.60678
$75$	0	0
$76$	0.563165	0.0645995
$77$	7.94149	0.905017
$78$	1.22308	0.138486
$79$	−0.613690	−0.0690456	−0.0345228	−	0.999404i	$-0.510991\pi$
$79$	−0.613690	−0.0690456	−0.0345228	+	0.999404i	$0.510991\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	9.81334	1.08370
$83$	−9.08333	−0.997025	−0.498513	−	0.866882i	$-0.666120\pi$
$83$	−9.08333	−0.997025	−0.498513	+	0.866882i	$0.666120\pi$
$84$	2.10903	0.230114
$85$	0	0
$86$	1.23307	0.132965
$87$	−5.00817	−0.536932
$88$	−5.81334	−0.619704
$89$	−1.46831	−0.155640	−0.0778201	−	0.996967i	$-0.524796\pi$
$89$	−1.46831	−0.155640	−0.0778201	+	0.996967i	$0.524796\pi$
$90$	0	0
$91$	−4.18388	−0.438590
$92$	−4.68854	−0.488814
$93$	−10.0095	−1.03794
$94$	6.89307	0.710965
$95$	0	0
$96$	−2.77692	−0.283419
$97$	3.85359	0.391273	0.195637	−	0.980676i	$-0.437323\pi$
$97$	3.85359	0.391273	0.195637	+	0.980676i	$0.437323\pi$
$98$	12.8482	1.29787
$99$	1.89812	0.190768
$100$	0	0
$101$	−2.57152	−0.255876	−0.127938	−	0.991782i	$-0.540836\pi$
$101$	−2.57152	−0.255876	−0.127938	+	0.991782i	$0.540836\pi$
$102$	2.12537	0.210443
$103$	−1.53272	−0.151023	−0.0755115	−	0.997145i	$-0.524059\pi$
$103$	−1.53272	−0.151023	−0.0755115	+	0.997145i	$0.524059\pi$
$104$	3.06269	0.300321
$105$	0	0
$106$	0.213272	0.0207148
$107$	−8.63003	−0.834297	−0.417148	−	0.908838i	$-0.636970\pi$
$107$	−8.63003	−0.834297	−0.417148	+	0.908838i	$0.636970\pi$
$108$	0.504085	0.0485056
$109$	12.0306	1.15233	0.576163	−	0.817335i	$-0.304549\pi$
$109$	12.0306	1.15233	0.576163	+	0.817335i	$0.304549\pi$
$110$	0	0
$111$	11.3011	1.07265
$112$	−11.4543	−1.08233
$113$	−7.93111	−0.746096	−0.373048	−	0.927812i	$-0.621687\pi$
$113$	−7.93111	−0.746096	−0.373048	+	0.927812i	$0.621687\pi$
$114$	1.36642	0.127977
$115$	0	0
$116$	−2.52454	−0.234398
$117$	−1.00000	−0.0924500
$118$	10.5723	0.973258
$119$	−7.27044	−0.666480
$120$	0	0
$121$	−7.39715	−0.672468
$122$	4.00000	0.362143
$123$	−8.02349	−0.723454
$124$	−5.04564	−0.453112
$125$	0	0
$126$	5.11720	0.455877
$127$	−2.35142	−0.208655	−0.104327	−	0.994543i	$-0.533269\pi$
$127$	−2.35142	−0.208655	−0.104327	+	0.994543i	$0.533269\pi$
$128$	5.29709	0.468201
$129$	−1.00817	−0.0887645
$130$	0	0
$131$	−9.46383	−0.826859	−0.413429	−	0.910536i	$-0.635669\pi$
$131$	−9.46383	−0.826859	−0.413429	+	0.910536i	$0.635669\pi$
$132$	0.956813	0.0832799
$133$	−4.67424	−0.405308
$134$	−14.0095	−1.21024
$135$	0	0
$136$	5.32211	0.456368
$137$	4.48953	0.383566	0.191783	−	0.981437i	$-0.438573\pi$
$137$	4.48953	0.383566	0.191783	+	0.981437i	$0.438573\pi$
$138$	−11.3759	−0.968383
$139$	−13.5048	−1.14547	−0.572733	−	0.819742i	$-0.694117\pi$
$139$	−13.5048	−1.14547	−0.572733	+	0.819742i	$0.694117\pi$
$140$	0	0
$141$	−5.63584	−0.474624
$142$	−6.57229	−0.551534
$143$	−1.89812	−0.158729
$144$	−2.73773	−0.228144
$145$	0	0
$146$	−4.91229	−0.406544
$147$	−10.5048	−0.866425
$148$	5.69671	0.468267
$149$	5.44034	0.445690	0.222845	−	0.974854i	$-0.428466\pi$
$149$	5.44034	0.445690	0.222845	+	0.974854i	$0.428466\pi$
$150$	0	0
$151$	−8.55039	−0.695821	−0.347911	−	0.937528i	$-0.613109\pi$
$151$	−8.55039	−0.695821	−0.347911	+	0.937528i	$0.613109\pi$
$152$	3.42164	0.277532
$153$	−1.73773	−0.140487
$154$	9.71305	0.782700
$155$	0	0
$156$	−0.504085	−0.0403591
$157$	−17.1430	−1.36816	−0.684082	−	0.729405i	$-0.739797\pi$
$157$	−17.1430	−1.36816	−0.684082	+	0.729405i	$0.739797\pi$
$158$	−0.750590	−0.0597137
$159$	−0.174373	−0.0138287
$160$	0	0
$161$	38.9146	3.06690
$162$	1.22308	0.0960939
$163$	−18.2145	−1.42667	−0.713336	−	0.700823i	$-0.752816\pi$
$163$	−18.2145	−1.42667	−0.713336	+	0.700823i	$0.752816\pi$
$164$	−4.04452	−0.315824
$165$	0	0
$166$	−11.1096	−0.862273
$167$	7.97296	0.616967	0.308483	−	0.951230i	$-0.400179\pi$
$167$	7.97296	0.616967	0.308483	+	0.951230i	$0.400179\pi$
$168$	12.8139	0.988615
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.11720	−0.0854346
$172$	−0.508204	−0.0387502
$173$	12.9086	0.981426	0.490713	−	0.871321i	$-0.336736\pi$
$173$	12.9086	0.981426	0.490713	+	0.871321i	$0.336736\pi$
$174$	−6.12537	−0.464363
$175$	0	0
$176$	−5.19653	−0.391703
$177$	−8.64402	−0.649724
$178$	−1.79585	−0.134605
$179$	−7.47546	−0.558742	−0.279371	−	0.960183i	$-0.590126\pi$
$179$	−7.47546	−0.558742	−0.279371	+	0.960183i	$0.590126\pi$
$180$	0	0
$181$	−6.16275	−0.458073	−0.229037	−	0.973418i	$-0.573558\pi$
$181$	−6.16275	−0.458073	−0.229037	+	0.973418i	$0.573558\pi$
$182$	−5.11720	−0.379312
$183$	−3.27044	−0.241758
$184$	−28.4863	−2.10004
$185$	0	0
$186$	−12.2424	−0.897655
$187$	−3.29841	−0.241204
$188$	−2.84095	−0.207197
$189$	−4.18388	−0.304332
$190$	0	0
$191$	3.79776	0.274796	0.137398	−	0.990516i	$-0.456126\pi$
$191$	3.79776	0.274796	0.137398	+	0.990516i	$0.456126\pi$
$192$	−8.87184	−0.640270
$193$	−17.3174	−1.24654	−0.623268	−	0.782009i	$-0.714195\pi$
$193$	−17.3174	−1.24654	−0.623268	+	0.782009i	$0.714195\pi$
$194$	4.71324	0.338391
$195$	0	0
$196$	−5.29534	−0.378238
$197$	6.31640	0.450025	0.225012	−	0.974356i	$-0.427758\pi$
$197$	6.31640	0.450025	0.225012	+	0.974356i	$0.427758\pi$
$198$	2.32154	0.164985
$199$	−2.77377	−0.196627	−0.0983135	−	0.995155i	$-0.531345\pi$
$199$	−2.77377	−0.196627	−0.0983135	+	0.995155i	$0.531345\pi$
$200$	0	0
$201$	11.4543	0.807926
$202$	−3.14517	−0.221293
$203$	20.9536	1.47065
$204$	−0.875963	−0.0613297
$205$	0	0
$206$	−1.87463	−0.130611
$207$	9.30108	0.646470
$208$	2.73773	0.189827
$209$	−2.12058	−0.146684
$210$	0	0
$211$	10.7928	0.743005	0.371503	−	0.928432i	$-0.378843\pi$
$211$	10.7928	0.743005	0.371503	+	0.928432i	$0.378843\pi$
$212$	−0.0878990	−0.00603693
$213$	5.37357	0.368191
$214$	−10.5552	−0.721538
$215$	0	0
$216$	3.06269	0.208389
$217$	41.8786	2.84290
$218$	14.7144	0.996584
$219$	4.01634	0.271399
$220$	0	0
$221$	1.73773	0.116892
$222$	13.8221	0.927677
$223$	23.1526	1.55041	0.775205	−	0.631710i	$-0.217647\pi$
$223$	23.1526	1.55041	0.775205	+	0.631710i	$0.217647\pi$
$224$	11.6183	0.776281
$225$	0	0
$226$	−9.70035	−0.645258
$227$	25.0690	1.66389	0.831945	−	0.554858i	$-0.187227\pi$
$227$	25.0690	1.66389	0.831945	+	0.554858i	$0.187227\pi$
$228$	−0.563165	−0.0372965
$229$	0.786917	0.0520009	0.0260005	−	0.999662i	$-0.491723\pi$
$229$	0.786917	0.0520009	0.0260005	+	0.999662i	$0.491723\pi$
$230$	0	0
$231$	−7.94149	−0.522512
$232$	−15.3385	−1.00702
$233$	−1.85627	−0.121608	−0.0608040	−	0.998150i	$-0.519366\pi$
$233$	−1.85627	−0.121608	−0.0608040	+	0.998150i	$0.519366\pi$
$234$	−1.22308	−0.0799550
$235$	0	0
$236$	−4.35732	−0.283637
$237$	0.613690	0.0398635
$238$	−8.89230	−0.576402
$239$	−7.35723	−0.475900	−0.237950	−	0.971277i	$-0.576475\pi$
$239$	−7.35723	−0.475900	−0.237950	+	0.971277i	$0.576475\pi$
$240$	0	0
$241$	−16.0497	−1.03385	−0.516924	−	0.856031i	$-0.672923\pi$
$241$	−16.0497	−1.03385	−0.516924	+	0.856031i	$0.672923\pi$
$242$	−9.04728	−0.581581
$243$	−1.00000	−0.0641500
$244$	−1.64858	−0.105540
$245$	0	0
$246$	−9.81334	−0.625676
$247$	1.11720	0.0710859
$248$	−30.6560	−1.94666
$249$	9.08333	0.575633
$250$	0	0
$251$	16.1185	1.01739	0.508697	−	0.860946i	$-0.330128\pi$
$251$	16.1185	1.01739	0.508697	+	0.860946i	$0.330128\pi$
$252$	−2.10903	−0.132857
$253$	17.6545	1.10993
$254$	−2.87596	−0.180454
$255$	0	0
$256$	−11.2649	−0.704059
$257$	9.78194	0.610180	0.305090	−	0.952323i	$-0.401313\pi$
$257$	9.78194	0.610180	0.305090	+	0.952323i	$0.401313\pi$
$258$	−1.23307	−0.0767675
$259$	−47.2824	−2.93798
$260$	0	0
$261$	5.00817	0.309998
$262$	−11.5750	−0.715105
$263$	3.12518	0.192707	0.0963535	−	0.995347i	$-0.469282\pi$
$263$	3.12518	0.192707	0.0963535	+	0.995347i	$0.469282\pi$
$264$	5.81334	0.357786
$265$	0	0
$266$	−5.71695	−0.350529
$267$	1.46831	0.0898589
$268$	5.77395	0.352700
$269$	15.8444	0.966048	0.483024	−	0.875607i	$-0.339538\pi$
$269$	15.8444	0.966048	0.483024	+	0.875607i	$0.339538\pi$
$270$	0	0
$271$	6.44501	0.391506	0.195753	−	0.980653i	$-0.437285\pi$
$271$	6.44501	0.391506	0.195753	+	0.980653i	$0.437285\pi$
$272$	4.75742	0.288461
$273$	4.18388	0.253220
$274$	5.49103	0.331725
$275$	0	0
$276$	4.68854	0.282217
$277$	13.6676	0.821206	0.410603	−	0.911814i	$-0.365318\pi$
$277$	13.6676	0.821206	0.410603	+	0.911814i	$0.365318\pi$
$278$	−16.5175	−0.990651
$279$	10.0095	0.599253
$280$	0	0
$281$	−21.2966	−1.27045	−0.635224	−	0.772328i	$-0.719092\pi$
$281$	−21.2966	−1.27045	−0.635224	+	0.772328i	$0.719092\pi$
$282$	−6.89307	−0.410476
$283$	−22.1524	−1.31682	−0.658411	−	0.752659i	$-0.728771\pi$
$283$	−22.1524	−1.31682	−0.658411	+	0.752659i	$0.728771\pi$
$284$	2.70874	0.160734
$285$	0	0
$286$	−2.32154	−0.137276
$287$	33.5693	1.98153
$288$	2.77692	0.163632
$289$	−13.9803	−0.822371
$290$	0	0
$291$	−3.85359	−0.225902
$292$	2.02458	0.118479
$293$	21.9677	1.28336	0.641682	−	0.766971i	$-0.278237\pi$
$293$	21.9677	1.28336	0.641682	+	0.766971i	$0.278237\pi$
$294$	−12.8482	−0.749324
$295$	0	0
$296$	34.6117	2.01176
$297$	−1.89812	−0.110140
$298$	6.65395	0.385453
$299$	−9.30108	−0.537895
$300$	0	0
$301$	4.21806	0.243125
$302$	−10.4578	−0.601778
$303$	2.57152	0.147730
$304$	3.05860	0.175422
$305$	0	0
$306$	−2.12537	−0.121499
$307$	−24.5516	−1.40124	−0.700618	−	0.713537i	$-0.747092\pi$
$307$	−24.5516	−1.40124	−0.700618	+	0.713537i	$0.747092\pi$
$308$	−4.00319	−0.228103
$309$	1.53272	0.0871931
$310$	0	0
$311$	−28.3295	−1.60642	−0.803210	−	0.595697i	$-0.796876\pi$
$311$	−28.3295	−1.60642	−0.803210	+	0.595697i	$0.796876\pi$
$312$	−3.06269	−0.173390
$313$	18.1497	1.02588	0.512941	−	0.858424i	$-0.328556\pi$
$313$	18.1497	1.02588	0.512941	+	0.858424i	$0.328556\pi$
$314$	−20.9673	−1.18325
$315$	0	0
$316$	0.309352	0.0174024
$317$	−10.3021	−0.578624	−0.289312	−	0.957235i	$-0.593426\pi$
$317$	−10.3021	−0.578624	−0.289312	+	0.957235i	$0.593426\pi$
$318$	−0.213272	−0.0119597
$319$	9.50609	0.532239
$320$	0	0
$321$	8.63003	0.481681
$322$	47.5955	2.65239
$323$	1.94139	0.108022
$324$	−0.504085	−0.0280047
$325$	0	0
$326$	−22.2777	−1.23385
$327$	−12.0306	−0.665296
$328$	−24.5734	−1.35684
$329$	23.5797	1.29999
$330$	0	0
$331$	33.3002	1.83034	0.915172	−	0.403062i	$-0.132054\pi$
$331$	33.3002	1.83034	0.915172	+	0.403062i	$0.132054\pi$
$332$	4.57877	0.251293
$333$	−11.3011	−0.619295
$334$	9.75154	0.533581
$335$	0	0
$336$	11.4543	0.624885
$337$	13.2437	0.721431	0.360716	−	0.932676i	$-0.382532\pi$
$337$	13.2437	0.721431	0.360716	+	0.932676i	$0.382532\pi$
$338$	1.22308	0.0665266
$339$	7.93111	0.430759
$340$	0	0
$341$	18.9992	1.02887
$342$	−1.36642	−0.0738877
$343$	14.6639	0.791774
$344$	−3.08771	−0.166478
$345$	0	0
$346$	15.7883	0.848782
$347$	−24.0250	−1.28973	−0.644864	−	0.764297i	$-0.723086\pi$
$347$	−24.0250	−1.28973	−0.644864	+	0.764297i	$0.723086\pi$
$348$	2.52454	0.135330
$349$	6.23173	0.333577	0.166789	−	0.985993i	$-0.446660\pi$
$349$	6.23173	0.333577	0.166789	+	0.985993i	$0.446660\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	5.27093	0.280941
$353$	−30.2041	−1.60760	−0.803801	−	0.594898i	$-0.797192\pi$
$353$	−30.2041	−1.60760	−0.803801	+	0.594898i	$0.797192\pi$
$354$	−10.5723	−0.561911
$355$	0	0
$356$	0.740151	0.0392279
$357$	7.27044	0.384793
$358$	−9.14305	−0.483225
$359$	12.8525	0.678329	0.339164	−	0.940727i	$-0.389856\pi$
$359$	12.8525	0.678329	0.339164	+	0.940727i	$0.389856\pi$
$360$	0	0
$361$	−17.7519	−0.934308
$362$	−7.53751	−0.396163
$363$	7.39715	0.388250
$364$	2.10903	0.110543
$365$	0	0
$366$	−4.00000	−0.209083
$367$	−27.3923	−1.42986	−0.714932	−	0.699194i	$-0.753543\pi$
$367$	−27.3923	−1.42986	−0.714932	+	0.699194i	$0.753543\pi$
$368$	−25.4638	−1.32739
$369$	8.02349	0.417686
$370$	0	0
$371$	0.729557	0.0378767
$372$	5.04564	0.261604
$373$	18.9113	0.979191	0.489595	−	0.871950i	$-0.337144\pi$
$373$	18.9113	0.979191	0.489595	+	0.871950i	$0.337144\pi$
$374$	−4.03421	−0.208604
$375$	0	0
$376$	−17.2608	−0.890159
$377$	−5.00817	−0.257934
$378$	−5.11720	−0.263200
$379$	20.7424	1.06546	0.532732	−	0.846284i	$-0.321166\pi$
$379$	20.7424	1.06546	0.532732	+	0.846284i	$0.321166\pi$
$380$	0	0
$381$	2.35142	0.120467
$382$	4.64495	0.237656
$383$	21.0833	1.07731	0.538654	−	0.842527i	$-0.318933\pi$
$383$	21.0833	1.07731	0.538654	+	0.842527i	$0.318933\pi$
$384$	−5.29709	−0.270316
$385$	0	0
$386$	−21.1805	−1.07806
$387$	1.00817	0.0512482
$388$	−1.94254	−0.0986175
$389$	−22.4583	−1.13868	−0.569341	−	0.822101i	$-0.692802\pi$
$389$	−22.4583	−1.13868	−0.569341	+	0.822101i	$0.692802\pi$
$390$	0	0
$391$	−16.1627	−0.817385
$392$	−32.1731	−1.62498
$393$	9.46383	0.477387
$394$	7.72544	0.389202
$395$	0	0
$396$	−0.956813	−0.0480817
$397$	−17.3038	−0.868450	−0.434225	−	0.900804i	$-0.642978\pi$
$397$	−17.3038	−0.868450	−0.434225	+	0.900804i	$0.642978\pi$
$398$	−3.39253	−0.170052
$399$	4.67424	0.234005
$400$	0	0
$401$	−8.01044	−0.400022	−0.200011	−	0.979794i	$-0.564098\pi$
$401$	−8.01044	−0.400022	−0.200011	+	0.979794i	$0.564098\pi$
$402$	14.0095	0.698731
$403$	−10.0095	−0.498609
$404$	1.29627	0.0644917
$405$	0	0
$406$	25.6278	1.27189
$407$	−21.4508	−1.06328
$408$	−5.32211	−0.263484
$409$	26.6885	1.31966	0.659832	−	0.751413i	$-0.270628\pi$
$409$	26.6885	1.31966	0.659832	+	0.751413i	$0.270628\pi$
$410$	0	0
$411$	−4.48953	−0.221452
$412$	0.772619	0.0380642
$413$	36.1655	1.77959
$414$	11.3759	0.559096
$415$	0	0
$416$	−2.77692	−0.136150
$417$	13.5048	0.661335
$418$	−2.59363	−0.126859
$419$	1.83868	0.0898253	0.0449127	−	0.998991i	$-0.485699\pi$
$419$	1.83868	0.0898253	0.0449127	+	0.998991i	$0.485699\pi$
$420$	0	0
$421$	6.16399	0.300415	0.150207	−	0.988655i	$-0.452006\pi$
$421$	6.16399	0.300415	0.150207	+	0.988655i	$0.452006\pi$
$422$	13.2004	0.642585
$423$	5.63584	0.274024
$424$	−0.534051	−0.0259358
$425$	0	0
$426$	6.57229	0.318428
$427$	13.6831	0.662173
$428$	4.35027	0.210278
$429$	1.89812	0.0916420
$430$	0	0
$431$	−31.1292	−1.49944	−0.749720	−	0.661756i	$-0.769812\pi$
$431$	−31.1292	−1.49944	−0.749720	+	0.661756i	$0.769812\pi$
$432$	2.73773	0.131719
$433$	−4.01634	−0.193013	−0.0965065	−	0.995332i	$-0.530767\pi$
$433$	−4.01634	−0.193013	−0.0965065	+	0.995332i	$0.530767\pi$
$434$	51.2207	2.45867
$435$	0	0
$436$	−6.06447	−0.290435
$437$	−10.3912	−0.497078
$438$	4.91229	0.234718
$439$	−20.3808	−0.972723	−0.486362	−	0.873758i	$-0.661676\pi$
$439$	−20.3808	−0.972723	−0.486362	+	0.873758i	$0.661676\pi$
$440$	0	0
$441$	10.5048	0.500231
$442$	2.12537	0.101094
$443$	14.4967	0.688758	0.344379	−	0.938831i	$-0.388090\pi$
$443$	14.4967	0.688758	0.344379	+	0.938831i	$0.388090\pi$
$444$	−5.69671	−0.270354
$445$	0	0
$446$	28.3173	1.34086
$447$	−5.44034	−0.257319
$448$	37.1187	1.75370
$449$	−3.09141	−0.145893	−0.0729464	−	0.997336i	$-0.523240\pi$
$449$	−3.09141	−0.145893	−0.0729464	+	0.997336i	$0.523240\pi$
$450$	0	0
$451$	15.2295	0.717130
$452$	3.99796	0.188048
$453$	8.55039	0.401732
$454$	30.6613	1.43901
$455$	0	0
$456$	−3.42164	−0.160233
$457$	−23.7556	−1.11124	−0.555620	−	0.831436i	$-0.687519\pi$
$457$	−23.7556	−1.11124	−0.555620	+	0.831436i	$0.687519\pi$
$458$	0.962459	0.0449728
$459$	1.73773	0.0811102
$460$	0	0
$461$	38.1201	1.77543	0.887716	−	0.460392i	$-0.152291\pi$
$461$	38.1201	1.77543	0.887716	+	0.460392i	$0.152291\pi$
$462$	−9.71305	−0.451892
$463$	−26.7670	−1.24397	−0.621985	−	0.783029i	$-0.713673\pi$
$463$	−26.7670	−1.24397	−0.621985	+	0.783029i	$0.713673\pi$
$464$	−13.7110	−0.636517
$465$	0	0
$466$	−2.27035	−0.105172
$467$	5.70861	0.264163	0.132082	−	0.991239i	$-0.457834\pi$
$467$	5.70861	0.264163	0.132082	+	0.991239i	$0.457834\pi$
$468$	0.504085	0.0233013
$469$	−47.9235	−2.21290
$470$	0	0
$471$	17.1430	0.789910
$472$	−26.4739	−1.21856
$473$	1.91363	0.0879886
$474$	0.750590	0.0344757
$475$	0	0
$476$	3.66492	0.167981
$477$	0.174373	0.00798401
$478$	−8.99845	−0.411580
$479$	−41.9688	−1.91760	−0.958802	−	0.284074i	$-0.908314\pi$
$479$	−41.9688	−1.91760	−0.958802	+	0.284074i	$0.908314\pi$
$480$	0	0
$481$	11.3011	0.515285
$482$	−19.6299	−0.894119
$483$	−38.9146	−1.77068
$484$	3.72879	0.169491
$485$	0	0
$486$	−1.22308	−0.0554799
$487$	27.1205	1.22895	0.614473	−	0.788938i	$-0.289369\pi$
$487$	27.1205	1.22895	0.614473	+	0.788938i	$0.289369\pi$
$488$	−10.0163	−0.453418
$489$	18.2145	0.823689
$490$	0	0
$491$	−2.50973	−0.113262	−0.0566312	−	0.998395i	$-0.518036\pi$
$491$	−2.50973	−0.113262	−0.0566312	+	0.998395i	$0.518036\pi$
$492$	4.04452	0.182341
$493$	−8.70284	−0.391956
$494$	1.36642	0.0614783
$495$	0	0
$496$	−27.4033	−1.23044
$497$	−22.4824	−1.00847
$498$	11.1096	0.497833
$499$	−17.3400	−0.776244	−0.388122	−	0.921608i	$-0.626876\pi$
$499$	−17.3400	−0.776244	−0.388122	+	0.921608i	$0.626876\pi$
$500$	0	0
$501$	−7.97296	−0.356206
$502$	19.7142	0.879888
$503$	−0.235551	−0.0105027	−0.00525136	−	0.999986i	$-0.501672\pi$
$503$	−0.235551	−0.0105027	−0.00525136	+	0.999986i	$0.501672\pi$
$504$	−12.8139	−0.570777
$505$	0	0
$506$	21.5928	0.959919
$507$	−1.00000	−0.0444116
$508$	1.18532	0.0525899
$509$	30.4076	1.34779	0.673896	−	0.738826i	$-0.264619\pi$
$509$	30.4076	1.34779	0.673896	+	0.738826i	$0.264619\pi$
$510$	0	0
$511$	−16.8039	−0.743360
$512$	−24.3721	−1.07710
$513$	1.11720	0.0493257
$514$	11.9641	0.527712
$515$	0	0
$516$	0.508204	0.0223724
$517$	10.6975	0.470475
$518$	−57.8299	−2.54090
$519$	−12.9086	−0.566627
$520$	0	0
$521$	34.5000	1.51147	0.755735	−	0.654877i	$-0.227280\pi$
$521$	34.5000	1.51147	0.755735	+	0.654877i	$0.227280\pi$
$522$	6.12537	0.268100
$523$	−31.5967	−1.38163	−0.690813	−	0.723034i	$-0.742747\pi$
$523$	−31.5967	−1.38163	−0.690813	+	0.723034i	$0.742747\pi$
$524$	4.77058	0.208404
$525$	0	0
$526$	3.82234	0.166662
$527$	−17.3938	−0.757686
$528$	5.19653	0.226150
$529$	63.5101	2.76131
$530$	0	0
$531$	8.64402	0.375118
$532$	2.35622	0.102155
$533$	−8.02349	−0.347536
$534$	1.79585	0.0777140
$535$	0	0
$536$	35.0810	1.51527
$537$	7.47546	0.322590
$538$	19.3789	0.835482
$539$	19.9394	0.858852
$540$	0	0
$541$	−36.5481	−1.57133	−0.785663	−	0.618655i	$-0.787678\pi$
$541$	−36.5481	−1.57133	−0.785663	+	0.618655i	$0.787678\pi$
$542$	7.88273	0.338592
$543$	6.16275	0.264469
$544$	−4.82554	−0.206893
$545$	0	0
$546$	5.11720	0.218996
$547$	−24.4439	−1.04514	−0.522572	−	0.852595i	$-0.675027\pi$
$547$	−24.4439	−1.04514	−0.522572	+	0.852595i	$0.675027\pi$
$548$	−2.26310	−0.0966750
$549$	3.27044	0.139579
$550$	0	0
$551$	−5.59514	−0.238361
$552$	28.4863	1.21246
$553$	−2.56761	−0.109186
$554$	16.7165	0.710216
$555$	0	0
$556$	6.80759	0.288706
$557$	−30.8005	−1.30506	−0.652530	−	0.757762i	$-0.726293\pi$
$557$	−30.8005	−1.30506	−0.652530	+	0.757762i	$0.726293\pi$
$558$	12.2424	0.518262
$559$	−1.00817	−0.0426411
$560$	0	0
$561$	3.29841	0.139259
$562$	−26.0474	−1.09874
$563$	−2.32559	−0.0980121	−0.0490060	−	0.998798i	$-0.515605\pi$
$563$	−2.32559	−0.0980121	−0.0490060	+	0.998798i	$0.515605\pi$
$564$	2.84095	0.119625
$565$	0	0
$566$	−27.0940	−1.13885
$567$	4.18388	0.175706
$568$	16.4576	0.690544
$569$	27.3891	1.14821	0.574105	−	0.818782i	$-0.305350\pi$
$569$	27.3891	1.14821	0.574105	+	0.818782i	$0.305350\pi$
$570$	0	0
$571$	−5.97175	−0.249910	−0.124955	−	0.992162i	$-0.539879\pi$
$571$	−5.97175	−0.249910	−0.124955	+	0.992162i	$0.539879\pi$
$572$	0.956813	0.0400064
$573$	−3.79776	−0.158654
$574$	41.0578	1.71372
$575$	0	0
$576$	8.87184	0.369660
$577$	7.66635	0.319154	0.159577	−	0.987185i	$-0.448987\pi$
$577$	7.66635	0.319154	0.159577	+	0.987185i	$0.448987\pi$
$578$	−17.0990	−0.711223
$579$	17.3174	0.719688
$580$	0	0
$581$	−38.0036	−1.57665
$582$	−4.71324	−0.195370
$583$	0.330981	0.0137078
$584$	12.3008	0.509010
$585$	0	0
$586$	26.8681	1.10991
$587$	−11.6195	−0.479588	−0.239794	−	0.970824i	$-0.577080\pi$
$587$	−11.6195	−0.479588	−0.239794	+	0.970824i	$0.577080\pi$
$588$	5.29534	0.218376
$589$	−11.1826	−0.460773
$590$	0	0
$591$	−6.31640	−0.259822
$592$	30.9393	1.27160
$593$	−24.3280	−0.999032	−0.499516	−	0.866305i	$-0.666489\pi$
$593$	−24.3280	−0.999032	−0.499516	+	0.866305i	$0.666489\pi$
$594$	−2.32154	−0.0952540
$595$	0	0
$596$	−2.74239	−0.112333
$597$	2.77377	0.113523
$598$	−11.3759	−0.465196
$599$	−0.702208	−0.0286914	−0.0143457	−	0.999897i	$-0.504567\pi$
$599$	−0.702208	−0.0286914	−0.0143457	+	0.999897i	$0.504567\pi$
$600$	0	0
$601$	−4.61254	−0.188150	−0.0940748	−	0.995565i	$-0.529989\pi$
$601$	−4.61254	−0.188150	−0.0940748	+	0.995565i	$0.529989\pi$
$602$	5.15901	0.210266
$603$	−11.4543	−0.466456
$604$	4.31013	0.175377
$605$	0	0
$606$	3.14517	0.127764
$607$	18.0763	0.733693	0.366847	−	0.930281i	$-0.380437\pi$
$607$	18.0763	0.733693	0.366847	+	0.930281i	$0.380437\pi$
$608$	−3.10239	−0.125818
$609$	−20.9536	−0.849082
$610$	0	0
$611$	−5.63584	−0.228002
$612$	0.875963	0.0354087
$613$	34.4468	1.39129	0.695647	−	0.718384i	$-0.255118\pi$
$613$	34.4468	1.39129	0.695647	+	0.718384i	$0.255118\pi$
$614$	−30.0285	−1.21185
$615$	0	0
$616$	−24.3223	−0.979974
$617$	39.0566	1.57236	0.786180	−	0.617997i	$-0.212056\pi$
$617$	39.0566	1.57236	0.786180	+	0.617997i	$0.212056\pi$
$618$	1.87463	0.0754086
$619$	−4.92711	−0.198037	−0.0990186	−	0.995086i	$-0.531570\pi$
$619$	−4.92711	−0.198037	−0.0990186	+	0.995086i	$0.531570\pi$
$620$	0	0
$621$	−9.30108	−0.373240
$622$	−34.6491	−1.38930
$623$	−6.14322	−0.246123
$624$	−2.73773	−0.109597
$625$	0	0
$626$	22.1985	0.887229
$627$	2.12058	0.0846878
$628$	8.64156	0.344836
$629$	19.6382	0.783026
$630$	0	0
$631$	6.90181	0.274757	0.137378	−	0.990519i	$-0.456132\pi$
$631$	6.90181	0.274757	0.137378	+	0.990519i	$0.456132\pi$
$632$	1.87954	0.0747641
$633$	−10.7928	−0.428974
$634$	−12.6003	−0.500420
$635$	0	0
$636$	0.0878990	0.00348542
$637$	−10.5048	−0.416217
$638$	11.6267	0.460304
$639$	−5.37357	−0.212575
$640$	0	0
$641$	12.4366	0.491218	0.245609	−	0.969369i	$-0.421012\pi$
$641$	12.4366	0.491218	0.245609	+	0.969369i	$0.421012\pi$
$642$	10.5552	0.416580
$643$	−23.6920	−0.934322	−0.467161	−	0.884172i	$-0.654723\pi$
$643$	−23.6920	−0.934322	−0.467161	+	0.884172i	$0.654723\pi$
$644$	−19.6163	−0.772990
$645$	0	0
$646$	2.37447	0.0934223
$647$	20.7168	0.814461	0.407230	−	0.913325i	$-0.366495\pi$
$647$	20.7168	0.814461	0.407230	+	0.913325i	$0.366495\pi$
$648$	−3.06269	−0.120314
$649$	16.4074	0.644045
$650$	0	0
$651$	−41.8786	−1.64135
$652$	9.18167	0.359582
$653$	−13.7390	−0.537648	−0.268824	−	0.963189i	$-0.586635\pi$
$653$	−13.7390	−0.537648	−0.268824	+	0.963189i	$0.586635\pi$
$654$	−14.7144	−0.575378
$655$	0	0
$656$	−21.9661	−0.857633
$657$	−4.01634	−0.156692
$658$	28.8398	1.12429
$659$	−19.4786	−0.758780	−0.379390	−	0.925237i	$-0.623866\pi$
$659$	−19.4786	−0.758780	−0.379390	+	0.925237i	$0.623866\pi$
$660$	0	0
$661$	40.7519	1.58506	0.792532	−	0.609831i	$-0.208763\pi$
$661$	40.7519	1.58506	0.792532	+	0.609831i	$0.208763\pi$
$662$	40.7287	1.58297
$663$	−1.73773	−0.0674877
$664$	27.8194	1.07960
$665$	0	0
$666$	−13.8221	−0.535595
$667$	46.5814	1.80364
$668$	−4.01905	−0.155502
$669$	−23.1526	−0.895130
$670$	0	0
$671$	6.20768	0.239645
$672$	−11.6183	−0.448186
$673$	23.4755	0.904912	0.452456	−	0.891787i	$-0.350548\pi$
$673$	23.4755	0.904912	0.452456	+	0.891787i	$0.350548\pi$
$674$	16.1981	0.623927
$675$	0	0
$676$	−0.504085	−0.0193879
$677$	18.6618	0.717232	0.358616	−	0.933485i	$-0.383249\pi$
$677$	18.6618	0.717232	0.358616	+	0.933485i	$0.383249\pi$
$678$	9.70035	0.372540
$679$	16.1230	0.618743
$680$	0	0
$681$	−25.0690	−0.960647
$682$	23.2375	0.889809
$683$	−18.3291	−0.701343	−0.350672	−	0.936498i	$-0.614047\pi$
$683$	−18.3291	−0.701343	−0.350672	+	0.936498i	$0.614047\pi$
$684$	0.563165	0.0215332
$685$	0	0
$686$	17.9350	0.684762
$687$	−0.786917	−0.0300228
$688$	−2.76010	−0.105228
$689$	−0.174373	−0.00664310
$690$	0	0
$691$	26.9911	1.02679	0.513394	−	0.858153i	$-0.328388\pi$
$691$	26.9911	1.02679	0.513394	+	0.858153i	$0.328388\pi$
$692$	−6.50706	−0.247361
$693$	7.94149	0.301672
$694$	−29.3844	−1.11542
$695$	0	0
$696$	15.3385	0.581403
$697$	−13.9426	−0.528115
$698$	7.62188	0.288493
$699$	1.85627	0.0702104
$700$	0	0
$701$	15.2498	0.575976	0.287988	−	0.957634i	$-0.407014\pi$
$701$	15.2498	0.575976	0.287988	+	0.957634i	$0.407014\pi$
$702$	1.22308	0.0461620
$703$	12.6256	0.476183
$704$	16.8398	0.634674
$705$	0	0
$706$	−36.9419	−1.39033
$707$	−10.7589	−0.404632
$708$	4.35732	0.163758
$709$	15.1710	0.569760	0.284880	−	0.958563i	$-0.408046\pi$
$709$	15.1710	0.569760	0.284880	+	0.958563i	$0.408046\pi$
$710$	0	0
$711$	−0.613690	−0.0230152
$712$	4.49696	0.168531
$713$	93.0992	3.48659
$714$	8.89230	0.332786
$715$	0	0
$716$	3.76827	0.140827
$717$	7.35723	0.274761
$718$	15.7196	0.586649
$719$	9.25854	0.345285	0.172643	−	0.984985i	$-0.444769\pi$
$719$	9.25854	0.345285	0.172643	+	0.984985i	$0.444769\pi$
$720$	0	0
$721$	−6.41270	−0.238821
$722$	−21.7119	−0.808032
$723$	16.0497	0.596893
$724$	3.10655	0.115454
$725$	0	0
$726$	9.04728	0.335776
$727$	28.5175	1.05765	0.528827	−	0.848730i	$-0.322632\pi$
$727$	28.5175	1.05765	0.528827	+	0.848730i	$0.322632\pi$
$728$	12.8139	0.474915
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.75193	−0.0647973
$732$	1.64858	0.0609333
$733$	−20.7882	−0.767828	−0.383914	−	0.923369i	$-0.625424\pi$
$733$	−20.7882	−0.767828	−0.383914	+	0.923369i	$0.625424\pi$
$734$	−33.5028	−1.23661
$735$	0	0
$736$	25.8284	0.952047
$737$	−21.7416	−0.800864
$738$	9.81334	0.361234
$739$	−23.9528	−0.881117	−0.440558	−	0.897724i	$-0.645220\pi$
$739$	−23.9528	−0.881117	−0.440558	+	0.897724i	$0.645220\pi$
$740$	0	0
$741$	−1.11720	−0.0410415
$742$	0.892304	0.0327575
$743$	13.0976	0.480505	0.240253	−	0.970710i	$-0.422770\pi$
$743$	13.0976	0.480505	0.240253	+	0.970710i	$0.422770\pi$
$744$	30.6560	1.12390
$745$	0	0
$746$	23.1300	0.846849
$747$	−9.08333	−0.332342
$748$	1.66268	0.0607936
$749$	−36.1070	−1.31932
$750$	0	0
$751$	37.3273	1.36209	0.681046	−	0.732240i	$-0.261525\pi$
$751$	37.3273	1.36209	0.681046	+	0.732240i	$0.261525\pi$
$752$	−15.4294	−0.562653
$753$	−16.1185	−0.587392
$754$	−6.12537	−0.223073
$755$	0	0
$756$	2.10903	0.0767047
$757$	15.6701	0.569539	0.284770	−	0.958596i	$-0.408083\pi$
$757$	15.6701	0.569539	0.284770	+	0.958596i	$0.408083\pi$
$758$	25.3695	0.921461
$759$	−17.6545	−0.640819
$760$	0	0
$761$	−15.9872	−0.579535	−0.289768	−	0.957097i	$-0.593578\pi$
$761$	−15.9872	−0.579535	−0.289768	+	0.957097i	$0.593578\pi$
$762$	2.87596	0.104185
$763$	50.3347	1.82224
$764$	−1.91439	−0.0692604
$765$	0	0
$766$	25.7865	0.931705
$767$	−8.64402	−0.312117
$768$	11.2649	0.406489
$769$	−30.5787	−1.10270	−0.551349	−	0.834275i	$-0.685887\pi$
$769$	−30.5787	−1.10270	−0.551349	+	0.834275i	$0.685887\pi$
$770$	0	0
$771$	−9.78194	−0.352288
$772$	8.72946	0.314180
$773$	−13.3491	−0.480136	−0.240068	−	0.970756i	$-0.577170\pi$
$773$	−13.3491	−0.480136	−0.240068	+	0.970756i	$0.577170\pi$
$774$	1.23307	0.0443218
$775$	0	0
$776$	−11.8024	−0.423680
$777$	47.2824	1.69625
$778$	−27.4682	−0.984784
$779$	−8.96386	−0.321164
$780$	0	0
$781$	−10.1997	−0.364973
$782$	−19.7683	−0.706912
$783$	−5.00817	−0.178977
$784$	−28.7594	−1.02712
$785$	0	0
$786$	11.5750	0.412866
$787$	50.0198	1.78301	0.891506	−	0.453009i	$-0.149649\pi$
$787$	50.0198	1.78301	0.891506	+	0.453009i	$0.149649\pi$
$788$	−3.18400	−0.113425
$789$	−3.12518	−0.111259
$790$	0	0
$791$	−33.1828	−1.17985
$792$	−5.81334	−0.206568
$793$	−3.27044	−0.116137
$794$	−21.1638	−0.751075
$795$	0	0
$796$	1.39821	0.0495584
$797$	−17.6557	−0.625397	−0.312698	−	0.949852i	$-0.601233\pi$
$797$	−17.6557	−0.625397	−0.312698	+	0.949852i	$0.601233\pi$
$798$	5.71695	0.202378
$799$	−9.79356	−0.346471
$800$	0	0
$801$	−1.46831	−0.0518800
$802$	−9.79737	−0.345957
$803$	−7.62349	−0.269027
$804$	−5.77395	−0.203632
$805$	0	0
$806$	−12.2424	−0.431220
$807$	−15.8444	−0.557748
$808$	7.87577	0.277069
$809$	−9.94274	−0.349568	−0.174784	−	0.984607i	$-0.555923\pi$
$809$	−9.94274	−0.349568	−0.174784	+	0.984607i	$0.555923\pi$
$810$	0	0
$811$	−38.3512	−1.34669	−0.673346	−	0.739328i	$-0.735143\pi$
$811$	−38.3512	−1.34669	−0.673346	+	0.739328i	$0.735143\pi$
$812$	−10.5624	−0.370667
$813$	−6.44501	−0.226036
$814$	−26.2359	−0.919569
$815$	0	0
$816$	−4.75742	−0.166543
$817$	−1.12633	−0.0394053
$818$	32.6421	1.14130
$819$	−4.18388	−0.146197
$820$	0	0
$821$	−51.5792	−1.80013	−0.900064	−	0.435758i	$-0.856480\pi$
$821$	−51.5792	−1.80013	−0.900064	+	0.435758i	$0.856480\pi$
$822$	−5.49103	−0.191522
$823$	−39.8188	−1.38800	−0.693998	−	0.719977i	$-0.744152\pi$
$823$	−39.8188	−1.38800	−0.693998	+	0.719977i	$0.744152\pi$
$824$	4.69423	0.163531
$825$	0	0
$826$	44.2332	1.53907
$827$	−16.8916	−0.587377	−0.293689	−	0.955901i	$-0.594883\pi$
$827$	−16.8916	−0.587377	−0.293689	+	0.955901i	$0.594883\pi$
$828$	−4.68854	−0.162938
$829$	19.4332	0.674943	0.337471	−	0.941336i	$-0.390428\pi$
$829$	19.4332	0.674943	0.337471	+	0.941336i	$0.390428\pi$
$830$	0	0
$831$	−13.6676	−0.474124
$832$	−8.87184	−0.307576
$833$	−18.2546	−0.632483
$834$	16.5175	0.571953
$835$	0	0
$836$	1.06895	0.0369705
$837$	−10.0095	−0.345979
$838$	2.24884	0.0776850
$839$	−36.2400	−1.25114	−0.625571	−	0.780167i	$-0.715134\pi$
$839$	−36.2400	−1.25114	−0.625571	+	0.780167i	$0.715134\pi$
$840$	0	0
$841$	−3.91823	−0.135111
$842$	7.53903	0.259812
$843$	21.2966	0.733494
$844$	−5.44048	−0.187269
$845$	0	0
$846$	6.89307	0.236988
$847$	−30.9488	−1.06341
$848$	−0.477387	−0.0163935
$849$	22.1524	0.760267
$850$	0	0
$851$	−105.112	−3.60320
$852$	−2.70874	−0.0927999
$853$	2.44146	0.0835940	0.0417970	−	0.999126i	$-0.486692\pi$
$853$	2.44146	0.0835940	0.0417970	+	0.999126i	$0.486692\pi$
$854$	16.7355	0.572678
$855$	0	0
$856$	26.4311	0.903396
$857$	−20.9217	−0.714672	−0.357336	−	0.933976i	$-0.616315\pi$
$857$	−20.9217	−0.714672	−0.357336	+	0.933976i	$0.616315\pi$
$858$	2.32154	0.0792561
$859$	14.7240	0.502376	0.251188	−	0.967938i	$-0.419179\pi$
$859$	14.7240	0.502376	0.251188	+	0.967938i	$0.419179\pi$
$860$	0	0
$861$	−33.5693	−1.14404
$862$	−38.0733	−1.29678
$863$	−21.1748	−0.720799	−0.360399	−	0.932798i	$-0.617360\pi$
$863$	−21.1748	−0.720799	−0.360399	+	0.932798i	$0.617360\pi$
$864$	−2.77692	−0.0944729
$865$	0	0
$866$	−4.91229	−0.166926
$867$	13.9803	0.474796
$868$	−21.1104	−0.716533
$869$	−1.16486	−0.0395150
$870$	0	0
$871$	11.4543	0.388115
$872$	−36.8461	−1.24777
$873$	3.85359	0.130424
$874$	−12.7092	−0.429896
$875$	0	0
$876$	−2.02458	−0.0684042
$877$	−36.7401	−1.24062	−0.620312	−	0.784355i	$-0.712994\pi$
$877$	−36.7401	−1.24062	−0.620312	+	0.784355i	$0.712994\pi$
$878$	−24.9273	−0.841255
$879$	−21.9677	−0.740951
$880$	0	0
$881$	−16.7074	−0.562886	−0.281443	−	0.959578i	$-0.590813\pi$
$881$	−16.7074	−0.562886	−0.281443	+	0.959578i	$0.590813\pi$
$882$	12.8482	0.432622
$883$	−41.2791	−1.38915	−0.694576	−	0.719420i	$-0.744408\pi$
$883$	−41.2791	−1.38915	−0.694576	+	0.719420i	$0.744408\pi$
$884$	−0.875963	−0.0294618
$885$	0	0
$886$	17.7305	0.595669
$887$	47.5592	1.59688	0.798441	−	0.602073i	$-0.205658\pi$
$887$	47.5592	1.59688	0.798441	+	0.602073i	$0.205658\pi$
$888$	−34.6117	−1.16149
$889$	−9.83805	−0.329958
$890$	0	0
$891$	1.89812	0.0635893
$892$	−11.6709	−0.390769
$893$	−6.29638	−0.210700
$894$	−6.65395	−0.222541
$895$	0	0
$896$	22.1624	0.740394
$897$	9.30108	0.310554
$898$	−3.78103	−0.126175
$899$	50.1293	1.67191
$900$	0	0
$901$	−0.303013	−0.0100948
$902$	18.6269	0.620207
$903$	−4.21806	−0.140368
$904$	24.2905	0.807890
$905$	0	0
$906$	10.4578	0.347436
$907$	−16.1360	−0.535788	−0.267894	−	0.963448i	$-0.586328\pi$
$907$	−16.1360	−0.535788	−0.267894	+	0.963448i	$0.586328\pi$
$908$	−12.6369	−0.419371
$909$	−2.57152	−0.0852921
$910$	0	0
$911$	42.5496	1.40973	0.704866	−	0.709341i	$-0.251007\pi$
$911$	42.5496	1.40973	0.704866	+	0.709341i	$0.251007\pi$
$912$	−3.05860	−0.101280
$913$	−17.2412	−0.570601
$914$	−29.0549	−0.961050
$915$	0	0
$916$	−0.396673	−0.0131065
$917$	−39.5955	−1.30756
$918$	2.12537	0.0701478
$919$	−31.3306	−1.03350	−0.516750	−	0.856136i	$-0.672858\pi$
$919$	−31.3306	−1.03350	−0.516750	+	0.856136i	$0.672858\pi$
$920$	0	0
$921$	24.5516	0.809004
$922$	46.6238	1.53547
$923$	5.37357	0.176873
$924$	4.00319	0.131695
$925$	0	0
$926$	−32.7381	−1.07584
$927$	−1.53272	−0.0503410
$928$	13.9073	0.456530
$929$	−33.9021	−1.11229	−0.556145	−	0.831085i	$-0.687720\pi$
$929$	−33.9021	−1.11229	−0.556145	+	0.831085i	$0.687720\pi$
$930$	0	0
$931$	−11.7360	−0.384633
$932$	0.935716	0.0306504
$933$	28.3295	0.927467
$934$	6.98207	0.228460
$935$	0	0
$936$	3.06269	0.100107
$937$	36.0449	1.17754	0.588768	−	0.808302i	$-0.299613\pi$
$937$	36.0449	1.17754	0.588768	+	0.808302i	$0.299613\pi$
$938$	−58.6141	−1.91382
$939$	−18.1497	−0.592293
$940$	0	0
$941$	46.8764	1.52813	0.764063	−	0.645141i	$-0.223201\pi$
$941$	46.8764	1.52813	0.764063	+	0.645141i	$0.223201\pi$
$942$	20.9673	0.683150
$943$	74.6271	2.43019
$944$	−23.6650	−0.770229
$945$	0	0
$946$	2.34051	0.0760965
$947$	23.2987	0.757107	0.378553	−	0.925579i	$-0.376422\pi$
$947$	23.2987	0.757107	0.378553	+	0.925579i	$0.376422\pi$
$948$	−0.309352	−0.0100473
$949$	4.01634	0.130376
$950$	0	0
$951$	10.3021	0.334069
$952$	22.2671	0.721680
$953$	17.2170	0.557713	0.278857	−	0.960333i	$-0.410045\pi$
$953$	17.2170	0.557713	0.278857	+	0.960333i	$0.410045\pi$
$954$	0.213272	0.00690493
$955$	0	0
$956$	3.70867	0.119947
$957$	−9.50609	−0.307288
$958$	−51.3311	−1.65843
$959$	18.7836	0.606556
$960$	0	0
$961$	69.1902	2.23194
$962$	13.8221	0.445642
$963$	−8.63003	−0.278099
$964$	8.09039	0.260574
$965$	0	0
$966$	−47.5955	−1.53136
$967$	2.31848	0.0745573	0.0372787	−	0.999305i	$-0.488131\pi$
$967$	2.31848	0.0745573	0.0372787	+	0.999305i	$0.488131\pi$
$968$	22.6552	0.728164
$969$	−1.94139	−0.0623665
$970$	0	0
$971$	−16.4953	−0.529358	−0.264679	−	0.964337i	$-0.585266\pi$
$971$	−16.4953	−0.529358	−0.264679	+	0.964337i	$0.585266\pi$
$972$	0.504085	0.0161685
$973$	−56.5027	−1.81139
$974$	33.1704	1.06285
$975$	0	0
$976$	−8.95358	−0.286597
$977$	−30.2337	−0.967263	−0.483631	−	0.875272i	$-0.660682\pi$
$977$	−30.2337	−0.967263	−0.483631	+	0.875272i	$0.660682\pi$
$978$	22.2777	0.712364
$979$	−2.78702	−0.0890735
$980$	0	0
$981$	12.0306	0.384109
$982$	−3.06959	−0.0979545
$983$	−27.7015	−0.883539	−0.441770	−	0.897129i	$-0.645649\pi$
$983$	−27.7015	−0.883539	−0.441770	+	0.897129i	$0.645649\pi$
$984$	24.5734	0.783372
$985$	0	0
$986$	−10.6442	−0.338981
$987$	−23.5797	−0.750550
$988$	−0.563165	−0.0179167
$989$	9.37708	0.298174
$990$	0	0
$991$	−45.3385	−1.44022	−0.720112	−	0.693858i	$-0.755910\pi$
$991$	−45.3385	−1.44022	−0.720112	+	0.693858i	$0.755910\pi$
$992$	27.7956	0.882512
$993$	−33.3002	−1.05675
$994$	−27.4977	−0.872173
$995$	0	0
$996$	−4.57877	−0.145084
$997$	13.2601	0.419950	0.209975	−	0.977707i	$-0.432662\pi$
$997$	13.2601	0.419950	0.209975	+	0.977707i	$0.432662\pi$
$998$	−21.2081	−0.671331
$999$	11.3011	0.357550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.a.r.1.4		5
3.2	odd	2		2925.2.a.bl.1.2		5
5.2	odd	4		195.2.c.b.79.7	yes	10
5.3	odd	4		195.2.c.b.79.4	✓	10
5.4	even	2		975.2.a.s.1.2		5
15.2	even	4		585.2.c.c.469.4		10
15.8	even	4		585.2.c.c.469.7		10
15.14	odd	2		2925.2.a.bm.1.4		5
20.3	even	4		3120.2.l.p.1249.8		10
20.7	even	4		3120.2.l.p.1249.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.c.b.79.4	✓	10	5.3	odd	4
195.2.c.b.79.7	yes	10	5.2	odd	4
585.2.c.c.469.4		10	15.2	even	4
585.2.c.c.469.7		10	15.8	even	4
975.2.a.r.1.4		5	1.1	even	1	trivial
975.2.a.s.1.2		5	5.4	even	2
2925.2.a.bl.1.2		5	3.2	odd	2
2925.2.a.bm.1.4		5	15.14	odd	2
3120.2.l.p.1249.3		10	20.7	even	4
3120.2.l.p.1249.8		10	20.3	even	4

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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

\(f(q)\)	\(=\)	\(q+1.22308 q^{2} -1.00000 q^{3} -0.504085 q^{4} -1.22308 q^{6} +4.18388 q^{7} -3.06269 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.22308 q^{2} -1.00000 q^{3} -0.504085 q^{4} -1.22308 q^{6} +4.18388 q^{7} -3.06269 q^{8} +1.00000 q^{9} +1.89812 q^{11} +0.504085 q^{12} -1.00000 q^{13} +5.11720 q^{14} -2.73773 q^{16} -1.73773 q^{17} +1.22308 q^{18} -1.11720 q^{19} -4.18388 q^{21} +2.32154 q^{22} +9.30108 q^{23} +3.06269 q^{24} -1.22308 q^{26} -1.00000 q^{27} -2.10903 q^{28} +5.00817 q^{29} +10.0095 q^{31} +2.77692 q^{32} -1.89812 q^{33} -2.12537 q^{34} -0.504085 q^{36} -11.3011 q^{37} -1.36642 q^{38} +1.00000 q^{39} +8.02349 q^{41} -5.11720 q^{42} +1.00817 q^{43} -0.956813 q^{44} +11.3759 q^{46} +5.63584 q^{47} +2.73773 q^{48} +10.5048 q^{49} +1.73773 q^{51} +0.504085 q^{52} +0.174373 q^{53} -1.22308 q^{54} -12.8139 q^{56} +1.11720 q^{57} +6.12537 q^{58} +8.64402 q^{59} +3.27044 q^{61} +12.2424 q^{62} +4.18388 q^{63} +8.87184 q^{64} -2.32154 q^{66} -11.4543 q^{67} +0.875963 q^{68} -9.30108 q^{69} -5.37357 q^{71} -3.06269 q^{72} -4.01634 q^{73} -13.8221 q^{74} +0.563165 q^{76} +7.94149 q^{77} +1.22308 q^{78} -0.613690 q^{79} +1.00000 q^{81} +9.81334 q^{82} -9.08333 q^{83} +2.10903 q^{84} +1.23307 q^{86} -5.00817 q^{87} -5.81334 q^{88} -1.46831 q^{89} -4.18388 q^{91} -4.68854 q^{92} -10.0095 q^{93} +6.89307 q^{94} -2.77692 q^{96} +3.85359 q^{97} +12.8482 q^{98} +1.89812 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 6 q^{4} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 6 * q^4 - 5 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + 6 q^{4} - 5 q^{7} + 5 q^{9} + 5 q^{11} - 6 q^{12} - 5 q^{13} + 12 q^{14} + 5 q^{17} + 8 q^{19} + 5 q^{21} + 8 q^{22} + 7 q^{23} - 5 q^{27} - 14 q^{28} + 8 q^{29} + 12 q^{31} + 20 q^{32} - 5 q^{33} + 20 q^{34} + 6 q^{36} - 17 q^{37} + 24 q^{38} + 5 q^{39} + 5 q^{41} - 12 q^{42} - 12 q^{43} + 18 q^{44} - 12 q^{46} + 10 q^{47} + 22 q^{49} - 5 q^{51} - 6 q^{52} + 13 q^{53} - 8 q^{57} + 8 q^{59} + 13 q^{61} + 40 q^{62} - 5 q^{63} - 16 q^{64} - 8 q^{66} - 28 q^{67} + 14 q^{68} - 7 q^{69} + 5 q^{71} + 14 q^{73} + 12 q^{74} + 35 q^{77} + q^{79} + 5 q^{81} - 16 q^{82} + 6 q^{83} + 14 q^{84} - 8 q^{87} + 36 q^{88} + 19 q^{89} + 5 q^{91} + 10 q^{92} - 12 q^{93} - 12 q^{94} - 20 q^{96} + 13 q^{97} - 28 q^{98} + 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 6 * q^4 - 5 * q^7 + 5 * q^9 + 5 * q^11 - 6 * q^12 - 5 * q^13 + 12 * q^14 + 5 * q^17 + 8 * q^19 + 5 * q^21 + 8 * q^22 + 7 * q^23 - 5 * q^27 - 14 * q^28 + 8 * q^29 + 12 * q^31 + 20 * q^32 - 5 * q^33 + 20 * q^34 + 6 * q^36 - 17 * q^37 + 24 * q^38 + 5 * q^39 + 5 * q^41 - 12 * q^42 - 12 * q^43 + 18 * q^44 - 12 * q^46 + 10 * q^47 + 22 * q^49 - 5 * q^51 - 6 * q^52 + 13 * q^53 - 8 * q^57 + 8 * q^59 + 13 * q^61 + 40 * q^62 - 5 * q^63 - 16 * q^64 - 8 * q^66 - 28 * q^67 + 14 * q^68 - 7 * q^69 + 5 * q^71 + 14 * q^73 + 12 * q^74 + 35 * q^77 + q^79 + 5 * q^81 - 16 * q^82 + 6 * q^83 + 14 * q^84 - 8 * q^87 + 36 * q^88 + 19 * q^89 + 5 * q^91 + 10 * q^92 - 12 * q^93 - 12 * q^94 - 20 * q^96 + 13 * q^97 - 28 * q^98 + 5 * q^99

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