LMFDB - Embedded newform 945.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(1,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, 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("945.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.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:	$7.54586299101$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	945.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278 q^{2} +3.30278 q^{4} -1.00000 q^{5} +1.00000 q^{7} -3.00000 q^{8} +O(q^{10})$	\(q-2.30278 q^{2} +3.30278 q^{4} -1.00000 q^{5} +1.00000 q^{7} -3.00000 q^{8} +2.30278 q^{10} -3.00000 q^{11} +1.30278 q^{13} -2.30278 q^{14} +0.302776 q^{16} +2.30278 q^{17} -3.30278 q^{19} -3.30278 q^{20} +6.90833 q^{22} -0.697224 q^{23} +1.00000 q^{25} -3.00000 q^{26} +3.30278 q^{28} -5.30278 q^{29} -2.39445 q^{31} +5.30278 q^{32} -5.30278 q^{34} -1.00000 q^{35} +3.60555 q^{37} +7.60555 q^{38} +3.00000 q^{40} +6.90833 q^{41} -4.21110 q^{43} -9.90833 q^{44} +1.60555 q^{46} -7.60555 q^{47} +1.00000 q^{49} -2.30278 q^{50} +4.30278 q^{52} +5.51388 q^{53} +3.00000 q^{55} -3.00000 q^{56} +12.2111 q^{58} -6.21110 q^{59} -11.1194 q^{61} +5.51388 q^{62} -12.8167 q^{64} -1.30278 q^{65} +13.5139 q^{67} +7.60555 q^{68} +2.30278 q^{70} -2.09167 q^{71} -10.2111 q^{73} -8.30278 q^{74} -10.9083 q^{76} -3.00000 q^{77} -16.9083 q^{79} -0.302776 q^{80} -15.9083 q^{82} +12.2111 q^{83} -2.30278 q^{85} +9.69722 q^{86} +9.00000 q^{88} -10.8167 q^{89} +1.30278 q^{91} -2.30278 q^{92} +17.5139 q^{94} +3.30278 q^{95} -9.51388 q^{97} -2.30278 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 - 6 * q^8	$ 2 q - q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8} + q^{10} - 6 q^{11} - q^{13} - q^{14} - 3 q^{16} + q^{17} - 3 q^{19} - 3 q^{20} + 3 q^{22} - 5 q^{23} + 2 q^{25} - 6 q^{26} + 3 q^{28} - 7 q^{29} - 12 q^{31} + 7 q^{32} - 7 q^{34} - 2 q^{35} + 8 q^{38} + 6 q^{40} + 3 q^{41} + 6 q^{43} - 9 q^{44} - 4 q^{46} - 8 q^{47} + 2 q^{49} - q^{50} + 5 q^{52} - 7 q^{53} + 6 q^{55} - 6 q^{56} + 10 q^{58} + 2 q^{59} + 3 q^{61} - 7 q^{62} - 4 q^{64} + q^{65} + 9 q^{67} + 8 q^{68} + q^{70} - 15 q^{71} - 6 q^{73} - 13 q^{74} - 11 q^{76} - 6 q^{77} - 23 q^{79} + 3 q^{80} - 21 q^{82} + 10 q^{83} - q^{85} + 23 q^{86} + 18 q^{88} - q^{91} - q^{92} + 17 q^{94} + 3 q^{95} - q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 - 6 * q^8 + q^10 - 6 * q^11 - q^13 - q^14 - 3 * q^16 + q^17 - 3 * q^19 - 3 * q^20 + 3 * q^22 - 5 * q^23 + 2 * q^25 - 6 * q^26 + 3 * q^28 - 7 * q^29 - 12 * q^31 + 7 * q^32 - 7 * q^34 - 2 * q^35 + 8 * q^38 + 6 * q^40 + 3 * q^41 + 6 * q^43 - 9 * q^44 - 4 * q^46 - 8 * q^47 + 2 * q^49 - q^50 + 5 * q^52 - 7 * q^53 + 6 * q^55 - 6 * q^56 + 10 * q^58 + 2 * q^59 + 3 * q^61 - 7 * q^62 - 4 * q^64 + q^65 + 9 * q^67 + 8 * q^68 + q^70 - 15 * q^71 - 6 * q^73 - 13 * q^74 - 11 * q^76 - 6 * q^77 - 23 * q^79 + 3 * q^80 - 21 * q^82 + 10 * q^83 - q^85 + 23 * q^86 + 18 * q^88 - q^91 - q^92 + 17 * q^94 + 3 * q^95 - q^97 - q^98

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 $

$2$	−2.30278	−1.62831	−0.814154	−	0.580649i	$-0.802799\pi$
$2$	−2.30278	−1.62831	−0.814154	+	0.580649i	$0.802799\pi$
$3$	0	0
$3$	0	0
$4$	3.30278	1.65139
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.00000	−1.06066
$9$	0	0
$10$	2.30278	0.728202
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	1.30278	0.361325	0.180662	−	0.983545i	$-0.442176\pi$
$13$	1.30278	0.361325	0.180662	+	0.983545i	$0.442176\pi$
$14$	−2.30278	−0.615443
$15$	0	0
$16$	0.302776	0.0756939
$17$	2.30278	0.558505	0.279253	−	0.960218i	$-0.409913\pi$
$17$	2.30278	0.558505	0.279253	+	0.960218i	$0.409913\pi$
$18$	0	0
$19$	−3.30278	−0.757709	−0.378854	−	0.925456i	$-0.623682\pi$
$19$	−3.30278	−0.757709	−0.378854	+	0.925456i	$0.623682\pi$
$20$	−3.30278	−0.738523
$21$	0	0
$22$	6.90833	1.47286
$23$	−0.697224	−0.145381	−0.0726907	−	0.997355i	$-0.523159\pi$
$23$	−0.697224	−0.145381	−0.0726907	+	0.997355i	$0.523159\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.00000	−0.588348
$27$	0	0
$28$	3.30278	0.624166
$29$	−5.30278	−0.984701	−0.492350	−	0.870397i	$-0.663862\pi$
$29$	−5.30278	−0.984701	−0.492350	+	0.870397i	$0.663862\pi$
$30$	0	0
$31$	−2.39445	−0.430056	−0.215028	−	0.976608i	$-0.568984\pi$
$31$	−2.39445	−0.430056	−0.215028	+	0.976608i	$0.568984\pi$
$32$	5.30278	0.937407
$33$	0	0
$34$	−5.30278	−0.909419
$35$	−1.00000	−0.169031
$36$	0	0
$37$	3.60555	0.592749	0.296374	−	0.955072i	$-0.404222\pi$
$37$	3.60555	0.592749	0.296374	+	0.955072i	$0.404222\pi$
$38$	7.60555	1.23378
$39$	0	0
$40$	3.00000	0.474342
$41$	6.90833	1.07890	0.539450	−	0.842018i	$-0.318632\pi$
$41$	6.90833	1.07890	0.539450	+	0.842018i	$0.318632\pi$
$42$	0	0
$43$	−4.21110	−0.642187	−0.321094	−	0.947047i	$-0.604050\pi$
$43$	−4.21110	−0.642187	−0.321094	+	0.947047i	$0.604050\pi$
$44$	−9.90833	−1.49374
$45$	0	0
$46$	1.60555	0.236726
$47$	−7.60555	−1.10938	−0.554692	−	0.832056i	$-0.687164\pi$
$47$	−7.60555	−1.10938	−0.554692	+	0.832056i	$0.687164\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.30278	−0.325662
$51$	0	0
$52$	4.30278	0.596688
$53$	5.51388	0.757389	0.378695	−	0.925522i	$-0.376373\pi$
$53$	5.51388	0.757389	0.378695	+	0.925522i	$0.376373\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	−3.00000	−0.400892
$57$	0	0
$58$	12.2111	1.60340
$59$	−6.21110	−0.808617	−0.404308	−	0.914623i	$-0.632488\pi$
$59$	−6.21110	−0.808617	−0.404308	+	0.914623i	$0.632488\pi$
$60$	0	0
$61$	−11.1194	−1.42370	−0.711849	−	0.702333i	$-0.752142\pi$
$61$	−11.1194	−1.42370	−0.711849	+	0.702333i	$0.752142\pi$
$62$	5.51388	0.700263
$63$	0	0
$64$	−12.8167	−1.60208
$65$	−1.30278	−0.161589
$66$	0	0
$67$	13.5139	1.65098	0.825491	−	0.564415i	$-0.190898\pi$
$67$	13.5139	1.65098	0.825491	+	0.564415i	$0.190898\pi$
$68$	7.60555	0.922309
$69$	0	0
$70$	2.30278	0.275234
$71$	−2.09167	−0.248236	−0.124118	−	0.992267i	$-0.539610\pi$
$71$	−2.09167	−0.248236	−0.124118	+	0.992267i	$0.539610\pi$
$72$	0	0
$73$	−10.2111	−1.19512	−0.597560	−	0.801825i	$-0.703863\pi$
$73$	−10.2111	−1.19512	−0.597560	+	0.801825i	$0.703863\pi$
$74$	−8.30278	−0.965178
$75$	0	0
$76$	−10.9083	−1.25127
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−16.9083	−1.90234	−0.951168	−	0.308675i	$-0.900115\pi$
$79$	−16.9083	−1.90234	−0.951168	+	0.308675i	$0.900115\pi$
$80$	−0.302776	−0.0338513
$81$	0	0
$82$	−15.9083	−1.75678
$83$	12.2111	1.34034	0.670171	−	0.742206i	$-0.266221\pi$
$83$	12.2111	1.34034	0.670171	+	0.742206i	$0.266221\pi$
$84$	0	0
$85$	−2.30278	−0.249771
$86$	9.69722	1.04568
$87$	0	0
$88$	9.00000	0.959403
$89$	−10.8167	−1.14656	−0.573282	−	0.819358i	$-0.694330\pi$
$89$	−10.8167	−1.14656	−0.573282	+	0.819358i	$0.694330\pi$
$90$	0	0
$91$	1.30278	0.136568
$92$	−2.30278	−0.240081
$93$	0	0
$94$	17.5139	1.80642
$95$	3.30278	0.338858
$96$	0	0
$97$	−9.51388	−0.965988	−0.482994	−	0.875624i	$-0.660451\pi$
$97$	−9.51388	−0.965988	−0.482994	+	0.875624i	$0.660451\pi$
$98$	−2.30278	−0.232615
$99$	0	0
$100$	3.30278	0.330278
$101$	9.21110	0.916539	0.458269	−	0.888813i	$-0.348469\pi$
$101$	9.21110	0.916539	0.458269	+	0.888813i	$0.348469\pi$
$102$	0	0
$103$	−15.7250	−1.54943	−0.774714	−	0.632312i	$-0.782106\pi$
$103$	−15.7250	−1.54943	−0.774714	+	0.632312i	$0.782106\pi$
$104$	−3.90833	−0.383243
$105$	0	0
$106$	−12.6972	−1.23326
$107$	−15.0000	−1.45010	−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000	−1.45010	−0.725052	+	0.688694i	$0.758184\pi$
$108$	0	0
$109$	−1.69722	−0.162565	−0.0812823	−	0.996691i	$-0.525902\pi$
$109$	−1.69722	−0.162565	−0.0812823	+	0.996691i	$0.525902\pi$
$110$	−6.90833	−0.658683
$111$	0	0
$112$	0.302776	0.0286096
$113$	−15.9083	−1.49653	−0.748265	−	0.663400i	$-0.769113\pi$
$113$	−15.9083	−1.49653	−0.748265	+	0.663400i	$0.769113\pi$
$114$	0	0
$115$	0.697224	0.0650165
$116$	−17.5139	−1.62612
$117$	0	0
$118$	14.3028	1.31668
$119$	2.30278	0.211095
$120$	0	0
$121$	−2.00000	−0.181818
$122$	25.6056	2.31822
$123$	0	0
$124$	−7.90833	−0.710189
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−18.7250	−1.66157	−0.830787	−	0.556591i	$-0.812109\pi$
$127$	−18.7250	−1.66157	−0.830787	+	0.556591i	$0.812109\pi$
$128$	18.9083	1.67128
$129$	0	0
$130$	3.00000	0.263117
$131$	8.30278	0.725417	0.362708	−	0.931903i	$-0.381852\pi$
$131$	8.30278	0.725417	0.362708	+	0.931903i	$0.381852\pi$
$132$	0	0
$133$	−3.30278	−0.286387
$134$	−31.1194	−2.68831
$135$	0	0
$136$	−6.90833	−0.592384
$137$	−18.4222	−1.57392	−0.786958	−	0.617007i	$-0.788345\pi$
$137$	−18.4222	−1.57392	−0.786958	+	0.617007i	$0.788345\pi$
$138$	0	0
$139$	−5.60555	−0.475457	−0.237728	−	0.971332i	$-0.576403\pi$
$139$	−5.60555	−0.475457	−0.237728	+	0.971332i	$0.576403\pi$
$140$	−3.30278	−0.279135
$141$	0	0
$142$	4.81665	0.404205
$143$	−3.90833	−0.326831
$144$	0	0
$145$	5.30278	0.440372
$146$	23.5139	1.94602
$147$	0	0
$148$	11.9083	0.978858
$149$	−5.51388	−0.451715	−0.225857	−	0.974160i	$-0.572518\pi$
$149$	−5.51388	−0.451715	−0.225857	+	0.974160i	$0.572518\pi$
$150$	0	0
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	9.90833	0.803671
$153$	0	0
$154$	6.90833	0.556689
$155$	2.39445	0.192327
$156$	0	0
$157$	6.60555	0.527180	0.263590	−	0.964635i	$-0.415093\pi$
$157$	6.60555	0.527180	0.263590	+	0.964635i	$0.415093\pi$
$158$	38.9361	3.09759
$159$	0	0
$160$	−5.30278	−0.419221
$161$	−0.697224	−0.0549490
$162$	0	0
$163$	23.2111	1.81803	0.909017	−	0.416759i	$-0.136834\pi$
$163$	23.2111	1.81803	0.909017	+	0.416759i	$0.136834\pi$
$164$	22.8167	1.78168
$165$	0	0
$166$	−28.1194	−2.18249
$167$	−19.6056	−1.51712	−0.758562	−	0.651601i	$-0.774098\pi$
$167$	−19.6056	−1.51712	−0.758562	+	0.651601i	$0.774098\pi$
$168$	0	0
$169$	−11.3028	−0.869444
$170$	5.30278	0.406704
$171$	0	0
$172$	−13.9083	−1.06050
$173$	16.8167	1.27855	0.639273	−	0.768980i	$-0.279235\pi$
$173$	16.8167	1.27855	0.639273	+	0.768980i	$0.279235\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−0.908327	−0.0684677
$177$	0	0
$178$	24.9083	1.86696
$179$	−16.8167	−1.25694	−0.628468	−	0.777836i	$-0.716318\pi$
$179$	−16.8167	−1.25694	−0.628468	+	0.777836i	$0.716318\pi$
$180$	0	0
$181$	17.4222	1.29498	0.647491	−	0.762073i	$-0.275818\pi$
$181$	17.4222	1.29498	0.647491	+	0.762073i	$0.275818\pi$
$182$	−3.00000	−0.222375
$183$	0	0
$184$	2.09167	0.154200
$185$	−3.60555	−0.265085
$186$	0	0
$187$	−6.90833	−0.505187
$188$	−25.1194	−1.83202
$189$	0	0
$190$	−7.60555	−0.551765
$191$	6.69722	0.484594	0.242297	−	0.970202i	$-0.422099\pi$
$191$	6.69722	0.484594	0.242297	+	0.970202i	$0.422099\pi$
$192$	0	0
$193$	6.11943	0.440486	0.220243	−	0.975445i	$-0.429315\pi$
$193$	6.11943	0.440486	0.220243	+	0.975445i	$0.429315\pi$
$194$	21.9083	1.57293
$195$	0	0
$196$	3.30278	0.235913
$197$	22.8167	1.62562	0.812810	−	0.582529i	$-0.197937\pi$
$197$	22.8167	1.62562	0.812810	+	0.582529i	$0.197937\pi$
$198$	0	0
$199$	22.7250	1.61093	0.805466	−	0.592643i	$-0.201915\pi$
$199$	22.7250	1.61093	0.805466	+	0.592643i	$0.201915\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	−21.2111	−1.49241
$203$	−5.30278	−0.372182
$204$	0	0
$205$	−6.90833	−0.482498
$206$	36.2111	2.52295
$207$	0	0
$208$	0.394449	0.0273501
$209$	9.90833	0.685373
$210$	0	0
$211$	−11.3944	−0.784426	−0.392213	−	0.919874i	$-0.628290\pi$
$211$	−11.3944	−0.784426	−0.392213	+	0.919874i	$0.628290\pi$
$212$	18.2111	1.25074
$213$	0	0
$214$	34.5416	2.36122
$215$	4.21110	0.287195
$216$	0	0
$217$	−2.39445	−0.162546
$218$	3.90833	0.264705
$219$	0	0
$220$	9.90833	0.668019
$221$	3.00000	0.201802
$222$	0	0
$223$	20.2111	1.35344	0.676718	−	0.736243i	$-0.263402\pi$
$223$	20.2111	1.35344	0.676718	+	0.736243i	$0.263402\pi$
$224$	5.30278	0.354307
$225$	0	0
$226$	36.6333	2.43681
$227$	6.48612	0.430499	0.215250	−	0.976559i	$-0.430944\pi$
$227$	6.48612	0.430499	0.215250	+	0.976559i	$0.430944\pi$
$228$	0	0
$229$	1.78890	0.118214	0.0591068	−	0.998252i	$-0.481175\pi$
$229$	1.78890	0.118214	0.0591068	+	0.998252i	$0.481175\pi$
$230$	−1.60555	−0.105867
$231$	0	0
$232$	15.9083	1.04443
$233$	9.90833	0.649116	0.324558	−	0.945866i	$-0.394784\pi$
$233$	9.90833	0.649116	0.324558	+	0.945866i	$0.394784\pi$
$234$	0	0
$235$	7.60555	0.496131
$236$	−20.5139	−1.33534
$237$	0	0
$238$	−5.30278	−0.343728
$239$	−5.78890	−0.374453	−0.187226	−	0.982317i	$-0.559950\pi$
$239$	−5.78890	−0.374453	−0.187226	+	0.982317i	$0.559950\pi$
$240$	0	0
$241$	0.119429	0.00769313	0.00384656	−	0.999993i	$-0.498776\pi$
$241$	0.119429	0.00769313	0.00384656	+	0.999993i	$0.498776\pi$
$242$	4.60555	0.296056
$243$	0	0
$244$	−36.7250	−2.35108
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−4.30278	−0.273779
$248$	7.18335	0.456143
$249$	0	0
$250$	2.30278	0.145640
$251$	−4.18335	−0.264050	−0.132025	−	0.991246i	$-0.542148\pi$
$251$	−4.18335	−0.264050	−0.132025	+	0.991246i	$0.542148\pi$
$252$	0	0
$253$	2.09167	0.131502
$254$	43.1194	2.70555
$255$	0	0
$256$	−17.9083	−1.11927
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	3.60555	0.224038
$260$	−4.30278	−0.266847
$261$	0	0
$262$	−19.1194	−1.18120
$263$	−12.4861	−0.769927	−0.384964	−	0.922932i	$-0.625786\pi$
$263$	−12.4861	−0.769927	−0.384964	+	0.922932i	$0.625786\pi$
$264$	0	0
$265$	−5.51388	−0.338715
$266$	7.60555	0.466326
$267$	0	0
$268$	44.6333	2.72641
$269$	−9.42221	−0.574482	−0.287241	−	0.957858i	$-0.592738\pi$
$269$	−9.42221	−0.574482	−0.287241	+	0.957858i	$0.592738\pi$
$270$	0	0
$271$	12.1194	0.736203	0.368101	−	0.929786i	$-0.380008\pi$
$271$	12.1194	0.736203	0.368101	+	0.929786i	$0.380008\pi$
$272$	0.697224	0.0422754
$273$	0	0
$274$	42.4222	2.56282
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−3.09167	−0.185761	−0.0928803	−	0.995677i	$-0.529607\pi$
$277$	−3.09167	−0.185761	−0.0928803	+	0.995677i	$0.529607\pi$
$278$	12.9083	0.774190
$279$	0	0
$280$	3.00000	0.179284
$281$	20.7250	1.23635	0.618174	−	0.786041i	$-0.287873\pi$
$281$	20.7250	1.23635	0.618174	+	0.786041i	$0.287873\pi$
$282$	0	0
$283$	2.90833	0.172882	0.0864410	−	0.996257i	$-0.472451\pi$
$283$	2.90833	0.172882	0.0864410	+	0.996257i	$0.472451\pi$
$284$	−6.90833	−0.409934
$285$	0	0
$286$	9.00000	0.532181
$287$	6.90833	0.407786
$288$	0	0
$289$	−11.6972	−0.688072
$290$	−12.2111	−0.717061
$291$	0	0
$292$	−33.7250	−1.97361
$293$	−20.2389	−1.18237	−0.591183	−	0.806537i	$-0.701339\pi$
$293$	−20.2389	−1.18237	−0.591183	+	0.806537i	$0.701339\pi$
$294$	0	0
$295$	6.21110	0.361624
$296$	−10.8167	−0.628705
$297$	0	0
$298$	12.6972	0.735530
$299$	−0.908327	−0.0525299
$300$	0	0
$301$	−4.21110	−0.242724
$302$	2.30278	0.132510
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	11.1194	0.636697
$306$	0	0
$307$	32.8444	1.87453	0.937265	−	0.348618i	$-0.113349\pi$
$307$	32.8444	1.87453	0.937265	+	0.348618i	$0.113349\pi$
$308$	−9.90833	−0.564579
$309$	0	0
$310$	−5.51388	−0.313167
$311$	−14.0917	−0.799065	−0.399533	−	0.916719i	$-0.630828\pi$
$311$	−14.0917	−0.799065	−0.399533	+	0.916719i	$0.630828\pi$
$312$	0	0
$313$	−14.8167	−0.837487	−0.418743	−	0.908105i	$-0.637529\pi$
$313$	−14.8167	−0.837487	−0.418743	+	0.908105i	$0.637529\pi$
$314$	−15.2111	−0.858412
$315$	0	0
$316$	−55.8444	−3.14149
$317$	−17.2389	−0.968231	−0.484115	−	0.875004i	$-0.660859\pi$
$317$	−17.2389	−0.968231	−0.484115	+	0.875004i	$0.660859\pi$
$318$	0	0
$319$	15.9083	0.890695
$320$	12.8167	0.716473
$321$	0	0
$322$	1.60555	0.0894739
$323$	−7.60555	−0.423184
$324$	0	0
$325$	1.30278	0.0722650
$326$	−53.4500	−2.96032
$327$	0	0
$328$	−20.7250	−1.14435
$329$	−7.60555	−0.419308
$330$	0	0
$331$	13.3028	0.731187	0.365593	−	0.930775i	$-0.380866\pi$
$331$	13.3028	0.731187	0.365593	+	0.930775i	$0.380866\pi$
$332$	40.3305	2.21343
$333$	0	0
$334$	45.1472	2.47034
$335$	−13.5139	−0.738342
$336$	0	0
$337$	8.69722	0.473768	0.236884	−	0.971538i	$-0.423874\pi$
$337$	8.69722	0.473768	0.236884	+	0.971538i	$0.423874\pi$
$338$	26.0278	1.41572
$339$	0	0
$340$	−7.60555	−0.412469
$341$	7.18335	0.389000
$342$	0	0
$343$	1.00000	0.0539949
$344$	12.6333	0.681142
$345$	0	0
$346$	−38.7250	−2.08187
$347$	−19.3944	−1.04115	−0.520574	−	0.853816i	$-0.674282\pi$
$347$	−19.3944	−1.04115	−0.520574	+	0.853816i	$0.674282\pi$
$348$	0	0
$349$	26.6333	1.42565	0.712824	−	0.701343i	$-0.247416\pi$
$349$	26.6333	1.42565	0.712824	+	0.701343i	$0.247416\pi$
$350$	−2.30278	−0.123089
$351$	0	0
$352$	−15.9083	−0.847917
$353$	−23.7250	−1.26275	−0.631377	−	0.775476i	$-0.717510\pi$
$353$	−23.7250	−1.26275	−0.631377	+	0.775476i	$0.717510\pi$
$354$	0	0
$355$	2.09167	0.111014
$356$	−35.7250	−1.89342
$357$	0	0
$358$	38.7250	2.04668
$359$	−1.18335	−0.0624546	−0.0312273	−	0.999512i	$-0.509942\pi$
$359$	−1.18335	−0.0624546	−0.0312273	+	0.999512i	$0.509942\pi$
$360$	0	0
$361$	−8.09167	−0.425878
$362$	−40.1194	−2.10863
$363$	0	0
$364$	4.30278	0.225527
$365$	10.2111	0.534474
$366$	0	0
$367$	−6.30278	−0.329002	−0.164501	−	0.986377i	$-0.552601\pi$
$367$	−6.30278	−0.329002	−0.164501	+	0.986377i	$0.552601\pi$
$368$	−0.211103	−0.0110045
$369$	0	0
$370$	8.30278	0.431641
$371$	5.51388	0.286266
$372$	0	0
$373$	−1.90833	−0.0988094	−0.0494047	−	0.998779i	$-0.515732\pi$
$373$	−1.90833	−0.0988094	−0.0494047	+	0.998779i	$0.515732\pi$
$374$	15.9083	0.822600
$375$	0	0
$376$	22.8167	1.17668
$377$	−6.90833	−0.355797
$378$	0	0
$379$	−25.6333	−1.31669	−0.658347	−	0.752714i	$-0.728744\pi$
$379$	−25.6333	−1.31669	−0.658347	+	0.752714i	$0.728744\pi$
$380$	10.9083	0.559585
$381$	0	0
$382$	−15.4222	−0.789069
$383$	−0.211103	−0.0107868	−0.00539342	−	0.999985i	$-0.501717\pi$
$383$	−0.211103	−0.0107868	−0.00539342	+	0.999985i	$0.501717\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	−14.0917	−0.717247
$387$	0	0
$388$	−31.4222	−1.59522
$389$	8.72498	0.442374	0.221187	−	0.975231i	$-0.429007\pi$
$389$	8.72498	0.442374	0.221187	+	0.975231i	$0.429007\pi$
$390$	0	0
$391$	−1.60555	−0.0811962
$392$	−3.00000	−0.151523
$393$	0	0
$394$	−52.5416	−2.64701
$395$	16.9083	0.850750
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−52.3305	−2.62309
$399$	0	0
$400$	0.302776	0.0151388
$401$	5.72498	0.285892	0.142946	−	0.989730i	$-0.454342\pi$
$401$	5.72498	0.285892	0.142946	+	0.989730i	$0.454342\pi$
$402$	0	0
$403$	−3.11943	−0.155390
$404$	30.4222	1.51356
$405$	0	0
$406$	12.2111	0.606027
$407$	−10.8167	−0.536162
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	15.9083	0.785656
$411$	0	0
$412$	−51.9361	−2.55871
$413$	−6.21110	−0.305628
$414$	0	0
$415$	−12.2111	−0.599419
$416$	6.90833	0.338709
$417$	0	0
$418$	−22.8167	−1.11600
$419$	−13.1833	−0.644049	−0.322024	−	0.946731i	$-0.604363\pi$
$419$	−13.1833	−0.644049	−0.322024	+	0.946731i	$0.604363\pi$
$420$	0	0
$421$	2.69722	0.131455	0.0657273	−	0.997838i	$-0.479063\pi$
$421$	2.69722	0.131455	0.0657273	+	0.997838i	$0.479063\pi$
$422$	26.2389	1.27729
$423$	0	0
$424$	−16.5416	−0.803333
$425$	2.30278	0.111701
$426$	0	0
$427$	−11.1194	−0.538107
$428$	−49.5416	−2.39469
$429$	0	0
$430$	−9.69722	−0.467642
$431$	−35.9361	−1.73098	−0.865490	−	0.500926i	$-0.832993\pi$
$431$	−35.9361	−1.73098	−0.865490	+	0.500926i	$0.832993\pi$
$432$	0	0
$433$	15.3305	0.736738	0.368369	−	0.929680i	$-0.379916\pi$
$433$	15.3305	0.736738	0.368369	+	0.929680i	$0.379916\pi$
$434$	5.51388	0.264675
$435$	0	0
$436$	−5.60555	−0.268457
$437$	2.30278	0.110157
$438$	0	0
$439$	−26.3944	−1.25974	−0.629869	−	0.776701i	$-0.716891\pi$
$439$	−26.3944	−1.25974	−0.629869	+	0.776701i	$0.716891\pi$
$440$	−9.00000	−0.429058
$441$	0	0
$442$	−6.90833	−0.328596
$443$	0.275019	0.0130666	0.00653328	−	0.999979i	$-0.497920\pi$
$443$	0.275019	0.0130666	0.00653328	+	0.999979i	$0.497920\pi$
$444$	0	0
$445$	10.8167	0.512759
$446$	−46.5416	−2.20381
$447$	0	0
$448$	−12.8167	−0.605530
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−20.7250	−0.975901
$452$	−52.5416	−2.47135
$453$	0	0
$454$	−14.9361	−0.700985
$455$	−1.30278	−0.0610751
$456$	0	0
$457$	22.3028	1.04328	0.521640	−	0.853166i	$-0.325320\pi$
$457$	22.3028	1.04328	0.521640	+	0.853166i	$0.325320\pi$
$458$	−4.11943	−0.192488
$459$	0	0
$460$	2.30278	0.107367
$461$	26.5139	1.23487	0.617437	−	0.786620i	$-0.288171\pi$
$461$	26.5139	1.23487	0.617437	+	0.786620i	$0.288171\pi$
$462$	0	0
$463$	10.9361	0.508243	0.254121	−	0.967172i	$-0.418214\pi$
$463$	10.9361	0.508243	0.254121	+	0.967172i	$0.418214\pi$
$464$	−1.60555	−0.0745358
$465$	0	0
$466$	−22.8167	−1.05696
$467$	−9.21110	−0.426239	−0.213119	−	0.977026i	$-0.568362\pi$
$467$	−9.21110	−0.426239	−0.213119	+	0.977026i	$0.568362\pi$
$468$	0	0
$469$	13.5139	0.624013
$470$	−17.5139	−0.807855
$471$	0	0
$472$	18.6333	0.857668
$473$	12.6333	0.580880
$474$	0	0
$475$	−3.30278	−0.151542
$476$	7.60555	0.348600
$477$	0	0
$478$	13.3305	0.609724
$479$	11.9361	0.545374	0.272687	−	0.962103i	$-0.412088\pi$
$479$	11.9361	0.545374	0.272687	+	0.962103i	$0.412088\pi$
$480$	0	0
$481$	4.69722	0.214175
$482$	−0.275019	−0.0125268
$483$	0	0
$484$	−6.60555	−0.300252
$485$	9.51388	0.432003
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	33.3583	1.51006
$489$	0	0
$490$	2.30278	0.104029
$491$	20.7250	0.935305	0.467653	−	0.883912i	$-0.345100\pi$
$491$	20.7250	0.935305	0.467653	+	0.883912i	$0.345100\pi$
$492$	0	0
$493$	−12.2111	−0.549960
$494$	9.90833	0.445797
$495$	0	0
$496$	−0.724981	−0.0325526
$497$	−2.09167	−0.0938244
$498$	0	0
$499$	−40.6333	−1.81900	−0.909498	−	0.415708i	$-0.863534\pi$
$499$	−40.6333	−1.81900	−0.909498	+	0.415708i	$0.863534\pi$
$500$	−3.30278	−0.147705
$501$	0	0
$502$	9.63331	0.429956
$503$	−19.1194	−0.852493	−0.426247	−	0.904607i	$-0.640164\pi$
$503$	−19.1194	−0.852493	−0.426247	+	0.904607i	$0.640164\pi$
$504$	0	0
$505$	−9.21110	−0.409889
$506$	−4.81665	−0.214126
$507$	0	0
$508$	−61.8444	−2.74390
$509$	23.2389	1.03004	0.515022	−	0.857177i	$-0.327784\pi$
$509$	23.2389	1.03004	0.515022	+	0.857177i	$0.327784\pi$
$510$	0	0
$511$	−10.2111	−0.451713
$512$	3.42221	0.151242
$513$	0	0
$514$	−27.6333	−1.21885
$515$	15.7250	0.692925
$516$	0	0
$517$	22.8167	1.00348
$518$	−8.30278	−0.364803
$519$	0	0
$520$	3.90833	0.171391
$521$	43.6056	1.91039	0.955197	−	0.295971i	$-0.0956432\pi$
$521$	43.6056	1.91039	0.955197	+	0.295971i	$0.0956432\pi$
$522$	0	0
$523$	13.7250	0.600152	0.300076	−	0.953915i	$-0.402988\pi$
$523$	13.7250	0.600152	0.300076	+	0.953915i	$0.402988\pi$
$524$	27.4222	1.19794
$525$	0	0
$526$	28.7527	1.25368
$527$	−5.51388	−0.240188
$528$	0	0
$529$	−22.5139	−0.978864
$530$	12.6972	0.551532
$531$	0	0
$532$	−10.9083	−0.472936
$533$	9.00000	0.389833
$534$	0	0
$535$	15.0000	0.648507
$536$	−40.5416	−1.75113
$537$	0	0
$538$	21.6972	0.935434
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−15.5139	−0.666994	−0.333497	−	0.942751i	$-0.608229\pi$
$541$	−15.5139	−0.666994	−0.333497	+	0.942751i	$0.608229\pi$
$542$	−27.9083	−1.19877
$543$	0	0
$544$	12.2111	0.523547
$545$	1.69722	0.0727011
$546$	0	0
$547$	38.2111	1.63379	0.816894	−	0.576787i	$-0.195694\pi$
$547$	38.2111	1.63379	0.816894	+	0.576787i	$0.195694\pi$
$548$	−60.8444	−2.59914
$549$	0	0
$550$	6.90833	0.294572
$551$	17.5139	0.746116
$552$	0	0
$553$	−16.9083	−0.719015
$554$	7.11943	0.302476
$555$	0	0
$556$	−18.5139	−0.785163
$557$	15.4861	0.656168	0.328084	−	0.944649i	$-0.393597\pi$
$557$	15.4861	0.656168	0.328084	+	0.944649i	$0.393597\pi$
$558$	0	0
$559$	−5.48612	−0.232038
$560$	−0.302776	−0.0127946
$561$	0	0
$562$	−47.7250	−2.01316
$563$	−35.9361	−1.51453	−0.757263	−	0.653110i	$-0.773464\pi$
$563$	−35.9361	−1.51453	−0.757263	+	0.653110i	$0.773464\pi$
$564$	0	0
$565$	15.9083	0.669268
$566$	−6.69722	−0.281505
$567$	0	0
$568$	6.27502	0.263294
$569$	30.6333	1.28422	0.642108	−	0.766615i	$-0.278060\pi$
$569$	30.6333	1.28422	0.642108	+	0.766615i	$0.278060\pi$
$570$	0	0
$571$	19.9361	0.834299	0.417150	−	0.908838i	$-0.363029\pi$
$571$	19.9361	0.834299	0.417150	+	0.908838i	$0.363029\pi$
$572$	−12.9083	−0.539724
$573$	0	0
$574$	−15.9083	−0.664001
$575$	−0.697224	−0.0290763
$576$	0	0
$577$	39.6056	1.64880	0.824400	−	0.566007i	$-0.191513\pi$
$577$	39.6056	1.64880	0.824400	+	0.566007i	$0.191513\pi$
$578$	26.9361	1.12039
$579$	0	0
$580$	17.5139	0.727224
$581$	12.2111	0.506602
$582$	0	0
$583$	−16.5416	−0.685085
$584$	30.6333	1.26762
$585$	0	0
$586$	46.6056	1.92526
$587$	2.09167	0.0863326	0.0431663	−	0.999068i	$-0.486255\pi$
$587$	2.09167	0.0863326	0.0431663	+	0.999068i	$0.486255\pi$
$588$	0	0
$589$	7.90833	0.325857
$590$	−14.3028	−0.588836
$591$	0	0
$592$	1.09167	0.0448675
$593$	34.8167	1.42975	0.714874	−	0.699253i	$-0.246484\pi$
$593$	34.8167	1.42975	0.714874	+	0.699253i	$0.246484\pi$
$594$	0	0
$595$	−2.30278	−0.0944046
$596$	−18.2111	−0.745956
$597$	0	0
$598$	2.09167	0.0855349
$599$	−36.4222	−1.48817	−0.744085	−	0.668084i	$-0.767114\pi$
$599$	−36.4222	−1.48817	−0.744085	+	0.668084i	$0.767114\pi$
$600$	0	0
$601$	45.1194	1.84046	0.920230	−	0.391378i	$-0.128002\pi$
$601$	45.1194	1.84046	0.920230	+	0.391378i	$0.128002\pi$
$602$	9.69722	0.395229
$603$	0	0
$604$	−3.30278	−0.134388
$605$	2.00000	0.0813116
$606$	0	0
$607$	14.6333	0.593948	0.296974	−	0.954886i	$-0.404023\pi$
$607$	14.6333	0.593948	0.296974	+	0.954886i	$0.404023\pi$
$608$	−17.5139	−0.710282
$609$	0	0
$610$	−25.6056	−1.03674
$611$	−9.90833	−0.400848
$612$	0	0
$613$	−27.2389	−1.10017	−0.550084	−	0.835110i	$-0.685404\pi$
$613$	−27.2389	−1.10017	−0.550084	+	0.835110i	$0.685404\pi$
$614$	−75.6333	−3.05231
$615$	0	0
$616$	9.00000	0.362620
$617$	−42.9083	−1.72742	−0.863712	−	0.503986i	$-0.831866\pi$
$617$	−42.9083	−1.72742	−0.863712	+	0.503986i	$0.831866\pi$
$618$	0	0
$619$	16.0278	0.644210	0.322105	−	0.946704i	$-0.395610\pi$
$619$	16.0278	0.644210	0.322105	+	0.946704i	$0.395610\pi$
$620$	7.90833	0.317606
$621$	0	0
$622$	32.4500	1.30112
$623$	−10.8167	−0.433360
$624$	0	0
$625$	1.00000	0.0400000
$626$	34.1194	1.36369
$627$	0	0
$628$	21.8167	0.870579
$629$	8.30278	0.331053
$630$	0	0
$631$	−30.7889	−1.22569	−0.612843	−	0.790204i	$-0.709974\pi$
$631$	−30.7889	−1.22569	−0.612843	+	0.790204i	$0.709974\pi$
$632$	50.7250	2.01773
$633$	0	0
$634$	39.6972	1.57658
$635$	18.7250	0.743078
$636$	0	0
$637$	1.30278	0.0516179
$638$	−36.6333	−1.45033
$639$	0	0
$640$	−18.9083	−0.747417
$641$	−5.72498	−0.226123	−0.113062	−	0.993588i	$-0.536066\pi$
$641$	−5.72498	−0.226123	−0.113062	+	0.993588i	$0.536066\pi$
$642$	0	0
$643$	−13.4861	−0.531841	−0.265920	−	0.963995i	$-0.585676\pi$
$643$	−13.4861	−0.531841	−0.265920	+	0.963995i	$0.585676\pi$
$644$	−2.30278	−0.0907421
$645$	0	0
$646$	17.5139	0.689074
$647$	−30.0000	−1.17942	−0.589711	−	0.807614i	$-0.700758\pi$
$647$	−30.0000	−1.17942	−0.589711	+	0.807614i	$0.700758\pi$
$648$	0	0
$649$	18.6333	0.731421
$650$	−3.00000	−0.117670
$651$	0	0
$652$	76.6611	3.00228
$653$	−4.88057	−0.190991	−0.0954957	−	0.995430i	$-0.530444\pi$
$653$	−4.88057	−0.190991	−0.0954957	+	0.995430i	$0.530444\pi$
$654$	0	0
$655$	−8.30278	−0.324416
$656$	2.09167	0.0816661
$657$	0	0
$658$	17.5139	0.682762
$659$	22.1833	0.864140	0.432070	−	0.901840i	$-0.357783\pi$
$659$	22.1833	0.864140	0.432070	+	0.901840i	$0.357783\pi$
$660$	0	0
$661$	7.72498	0.300467	0.150233	−	0.988651i	$-0.451997\pi$
$661$	7.72498	0.300467	0.150233	+	0.988651i	$0.451997\pi$
$662$	−30.6333	−1.19060
$663$	0	0
$664$	−36.6333	−1.42165
$665$	3.30278	0.128076
$666$	0	0
$667$	3.69722	0.143157
$668$	−64.7527	−2.50536
$669$	0	0
$670$	31.1194	1.20225
$671$	33.3583	1.28778
$672$	0	0
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	−20.0278	−0.771440
$675$	0	0
$676$	−37.3305	−1.43579
$677$	17.7889	0.683683	0.341841	−	0.939758i	$-0.388949\pi$
$677$	17.7889	0.683683	0.341841	+	0.939758i	$0.388949\pi$
$678$	0	0
$679$	−9.51388	−0.365109
$680$	6.90833	0.264922
$681$	0	0
$682$	−16.5416	−0.633412
$683$	45.3583	1.73559	0.867793	−	0.496925i	$-0.165538\pi$
$683$	45.3583	1.73559	0.867793	+	0.496925i	$0.165538\pi$
$684$	0	0
$685$	18.4222	0.703876
$686$	−2.30278	−0.0879204
$687$	0	0
$688$	−1.27502	−0.0486097
$689$	7.18335	0.273664
$690$	0	0
$691$	−7.42221	−0.282354	−0.141177	−	0.989984i	$-0.545089\pi$
$691$	−7.42221	−0.282354	−0.141177	+	0.989984i	$0.545089\pi$
$692$	55.5416	2.11138
$693$	0	0
$694$	44.6611	1.69531
$695$	5.60555	0.212631
$696$	0	0
$697$	15.9083	0.602571
$698$	−61.3305	−2.32139
$699$	0	0
$700$	3.30278	0.124833
$701$	−26.0278	−0.983055	−0.491527	−	0.870862i	$-0.663561\pi$
$701$	−26.0278	−0.983055	−0.491527	+	0.870862i	$0.663561\pi$
$702$	0	0
$703$	−11.9083	−0.449131
$704$	38.4500	1.44914
$705$	0	0
$706$	54.6333	2.05615
$707$	9.21110	0.346419
$708$	0	0
$709$	−45.0278	−1.69105	−0.845526	−	0.533934i	$-0.820713\pi$
$709$	−45.0278	−1.69105	−0.845526	+	0.533934i	$0.820713\pi$
$710$	−4.81665	−0.180766
$711$	0	0
$712$	32.4500	1.21611
$713$	1.66947	0.0625221
$714$	0	0
$715$	3.90833	0.146163
$716$	−55.5416	−2.07569
$717$	0	0
$718$	2.72498	0.101695
$719$	−19.3305	−0.720907	−0.360454	−	0.932777i	$-0.617378\pi$
$719$	−19.3305	−0.720907	−0.360454	+	0.932777i	$0.617378\pi$
$720$	0	0
$721$	−15.7250	−0.585629
$722$	18.6333	0.693460
$723$	0	0
$724$	57.5416	2.13852
$725$	−5.30278	−0.196940
$726$	0	0
$727$	−19.4861	−0.722700	−0.361350	−	0.932430i	$-0.617684\pi$
$727$	−19.4861	−0.722700	−0.361350	+	0.932430i	$0.617684\pi$
$728$	−3.90833	−0.144852
$729$	0	0
$730$	−23.5139	−0.870288
$731$	−9.69722	−0.358665
$732$	0	0
$733$	12.5416	0.463236	0.231618	−	0.972807i	$-0.425598\pi$
$733$	12.5416	0.463236	0.231618	+	0.972807i	$0.425598\pi$
$734$	14.5139	0.535717
$735$	0	0
$736$	−3.69722	−0.136281
$737$	−40.5416	−1.49337
$738$	0	0
$739$	−20.1833	−0.742456	−0.371228	−	0.928542i	$-0.621063\pi$
$739$	−20.1833	−0.742456	−0.371228	+	0.928542i	$0.621063\pi$
$740$	−11.9083	−0.437759
$741$	0	0
$742$	−12.6972	−0.466130
$743$	−34.5416	−1.26721	−0.633605	−	0.773657i	$-0.718425\pi$
$743$	−34.5416	−1.26721	−0.633605	+	0.773657i	$0.718425\pi$
$744$	0	0
$745$	5.51388	0.202013
$746$	4.39445	0.160892
$747$	0	0
$748$	−22.8167	−0.834259
$749$	−15.0000	−0.548088
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	−2.30278	−0.0839736
$753$	0	0
$754$	15.9083	0.579347
$755$	1.00000	0.0363937
$756$	0	0
$757$	−14.5416	−0.528525	−0.264262	−	0.964451i	$-0.585128\pi$
$757$	−14.5416	−0.528525	−0.264262	+	0.964451i	$0.585128\pi$
$758$	59.0278	2.14398
$759$	0	0
$760$	−9.90833	−0.359413
$761$	−13.5416	−0.490884	−0.245442	−	0.969411i	$-0.578933\pi$
$761$	−13.5416	−0.490884	−0.245442	+	0.969411i	$0.578933\pi$
$762$	0	0
$763$	−1.69722	−0.0614436
$764$	22.1194	0.800253
$765$	0	0
$766$	0.486122	0.0175643
$767$	−8.09167	−0.292173
$768$	0	0
$769$	−26.8167	−0.967033	−0.483517	−	0.875335i	$-0.660641\pi$
$769$	−26.8167	−0.967033	−0.483517	+	0.875335i	$0.660641\pi$
$770$	−6.90833	−0.248959
$771$	0	0
$772$	20.2111	0.727413
$773$	−1.54163	−0.0554487	−0.0277244	−	0.999616i	$-0.508826\pi$
$773$	−1.54163	−0.0554487	−0.0277244	+	0.999616i	$0.508826\pi$
$774$	0	0
$775$	−2.39445	−0.0860111
$776$	28.5416	1.02458
$777$	0	0
$778$	−20.0917	−0.720321
$779$	−22.8167	−0.817491
$780$	0	0
$781$	6.27502	0.224538
$782$	3.69722	0.132212
$783$	0	0
$784$	0.302776	0.0108134
$785$	−6.60555	−0.235762
$786$	0	0
$787$	−3.02776	−0.107928	−0.0539639	−	0.998543i	$-0.517186\pi$
$787$	−3.02776	−0.107928	−0.0539639	+	0.998543i	$0.517186\pi$
$788$	75.3583	2.68453
$789$	0	0
$790$	−38.9361	−1.38528
$791$	−15.9083	−0.565635
$792$	0	0
$793$	−14.4861	−0.514417
$794$	−4.60555	−0.163445
$795$	0	0
$796$	75.0555	2.66027
$797$	42.9083	1.51989	0.759945	−	0.649987i	$-0.225226\pi$
$797$	42.9083	1.51989	0.759945	+	0.649987i	$0.225226\pi$
$798$	0	0
$799$	−17.5139	−0.619596
$800$	5.30278	0.187481
$801$	0	0
$802$	−13.1833	−0.465520
$803$	30.6333	1.08103
$804$	0	0
$805$	0.697224	0.0245739
$806$	7.18335	0.253023
$807$	0	0
$808$	−27.6333	−0.972136
$809$	19.1833	0.674451	0.337225	−	0.941424i	$-0.390512\pi$
$809$	19.1833	0.674451	0.337225	+	0.941424i	$0.390512\pi$
$810$	0	0
$811$	−24.5139	−0.860799	−0.430399	−	0.902639i	$-0.641627\pi$
$811$	−24.5139	−0.860799	−0.430399	+	0.902639i	$0.641627\pi$
$812$	−17.5139	−0.614617
$813$	0	0
$814$	24.9083	0.873036
$815$	−23.2111	−0.813049
$816$	0	0
$817$	13.9083	0.486591
$818$	−32.2389	−1.12721
$819$	0	0
$820$	−22.8167	−0.796792
$821$	−10.8167	−0.377504	−0.188752	−	0.982025i	$-0.560444\pi$
$821$	−10.8167	−0.377504	−0.188752	+	0.982025i	$0.560444\pi$
$822$	0	0
$823$	32.6333	1.13753	0.568763	−	0.822502i	$-0.307422\pi$
$823$	32.6333	1.13753	0.568763	+	0.822502i	$0.307422\pi$
$824$	47.1749	1.64342
$825$	0	0
$826$	14.3028	0.497657
$827$	−0.633308	−0.0220223	−0.0110111	−	0.999939i	$-0.503505\pi$
$827$	−0.633308	−0.0220223	−0.0110111	+	0.999939i	$0.503505\pi$
$828$	0	0
$829$	−23.0555	−0.800751	−0.400376	−	0.916351i	$-0.631120\pi$
$829$	−23.0555	−0.800751	−0.400376	+	0.916351i	$0.631120\pi$
$830$	28.1194	0.976040
$831$	0	0
$832$	−16.6972	−0.578872
$833$	2.30278	0.0797864
$834$	0	0
$835$	19.6056	0.678478
$836$	32.7250	1.13182
$837$	0	0
$838$	30.3583	1.04871
$839$	11.0917	0.382927	0.191464	−	0.981500i	$-0.438677\pi$
$839$	11.0917	0.382927	0.191464	+	0.981500i	$0.438677\pi$
$840$	0	0
$841$	−0.880571	−0.0303645
$842$	−6.21110	−0.214049
$843$	0	0
$844$	−37.6333	−1.29539
$845$	11.3028	0.388827
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	1.66947	0.0573298
$849$	0	0
$850$	−5.30278	−0.181884
$851$	−2.51388	−0.0861746
$852$	0	0
$853$	29.8444	1.02185	0.510927	−	0.859624i	$-0.329302\pi$
$853$	29.8444	1.02185	0.510927	+	0.859624i	$0.329302\pi$
$854$	25.6056	0.876204
$855$	0	0
$856$	45.0000	1.53807
$857$	34.2666	1.17053	0.585263	−	0.810844i	$-0.300991\pi$
$857$	34.2666	1.17053	0.585263	+	0.810844i	$0.300991\pi$
$858$	0	0
$859$	27.8167	0.949092	0.474546	−	0.880231i	$-0.342612\pi$
$859$	27.8167	0.949092	0.474546	+	0.880231i	$0.342612\pi$
$860$	13.9083	0.474270
$861$	0	0
$862$	82.7527	2.81857
$863$	31.2666	1.06433	0.532164	−	0.846641i	$-0.321379\pi$
$863$	31.2666	1.06433	0.532164	+	0.846641i	$0.321379\pi$
$864$	0	0
$865$	−16.8167	−0.571783
$866$	−35.3028	−1.19964
$867$	0	0
$868$	−7.90833	−0.268426
$869$	50.7250	1.72073
$870$	0	0
$871$	17.6056	0.596541
$872$	5.09167	0.172426
$873$	0	0
$874$	−5.30278	−0.179369
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−22.6333	−0.764272	−0.382136	−	0.924106i	$-0.624811\pi$
$877$	−22.6333	−0.764272	−0.382136	+	0.924106i	$0.624811\pi$
$878$	60.7805	2.05124
$879$	0	0
$880$	0.908327	0.0306197
$881$	45.6333	1.53743	0.768713	−	0.639594i	$-0.220898\pi$
$881$	45.6333	1.53743	0.768713	+	0.639594i	$0.220898\pi$
$882$	0	0
$883$	−6.02776	−0.202850	−0.101425	−	0.994843i	$-0.532340\pi$
$883$	−6.02776	−0.202850	−0.101425	+	0.994843i	$0.532340\pi$
$884$	9.90833	0.333253
$885$	0	0
$886$	−0.633308	−0.0212764
$887$	22.6056	0.759020	0.379510	−	0.925188i	$-0.376093\pi$
$887$	22.6056	0.759020	0.379510	+	0.925188i	$0.376093\pi$
$888$	0	0
$889$	−18.7250	−0.628016
$890$	−24.9083	−0.834929
$891$	0	0
$892$	66.7527	2.23505
$893$	25.1194	0.840590
$894$	0	0
$895$	16.8167	0.562119
$896$	18.9083	0.631683
$897$	0	0
$898$	−41.4500	−1.38320
$899$	12.6972	0.423476
$900$	0	0
$901$	12.6972	0.423006
$902$	47.7250	1.58907
$903$	0	0
$904$	47.7250	1.58731
$905$	−17.4222	−0.579134
$906$	0	0
$907$	−52.5694	−1.74554	−0.872769	−	0.488133i	$-0.837678\pi$
$907$	−52.5694	−1.74554	−0.872769	+	0.488133i	$0.837678\pi$
$908$	21.4222	0.710921
$909$	0	0
$910$	3.00000	0.0994490
$911$	27.6333	0.915532	0.457766	−	0.889073i	$-0.348650\pi$
$911$	27.6333	0.915532	0.457766	+	0.889073i	$0.348650\pi$
$912$	0	0
$913$	−36.6333	−1.21239
$914$	−51.3583	−1.69878
$915$	0	0
$916$	5.90833	0.195217
$917$	8.30278	0.274182
$918$	0	0
$919$	2.48612	0.0820096	0.0410048	−	0.999159i	$-0.486944\pi$
$919$	2.48612	0.0820096	0.0410048	+	0.999159i	$0.486944\pi$
$920$	−2.09167	−0.0689604
$921$	0	0
$922$	−61.0555	−2.01076
$923$	−2.72498	−0.0896938
$924$	0	0
$925$	3.60555	0.118550
$926$	−25.1833	−0.827576
$927$	0	0
$928$	−28.1194	−0.923065
$929$	−39.5694	−1.29823	−0.649115	−	0.760690i	$-0.724861\pi$
$929$	−39.5694	−1.29823	−0.649115	+	0.760690i	$0.724861\pi$
$930$	0	0
$931$	−3.30278	−0.108244
$932$	32.7250	1.07194
$933$	0	0
$934$	21.2111	0.694048
$935$	6.90833	0.225926
$936$	0	0
$937$	44.4222	1.45121	0.725605	−	0.688111i	$-0.241560\pi$
$937$	44.4222	1.45121	0.725605	+	0.688111i	$0.241560\pi$
$938$	−31.1194	−1.01609
$939$	0	0
$940$	25.1194	0.819305
$941$	34.2666	1.11706	0.558530	−	0.829484i	$-0.311366\pi$
$941$	34.2666	1.11706	0.558530	+	0.829484i	$0.311366\pi$
$942$	0	0
$943$	−4.81665	−0.156852
$944$	−1.88057	−0.0612074
$945$	0	0
$946$	−29.0917	−0.945852
$947$	−1.33053	−0.0432365	−0.0216182	−	0.999766i	$-0.506882\pi$
$947$	−1.33053	−0.0432365	−0.0216182	+	0.999766i	$0.506882\pi$
$948$	0	0
$949$	−13.3028	−0.431826
$950$	7.60555	0.246757
$951$	0	0
$952$	−6.90833	−0.223900
$953$	51.4222	1.66573	0.832864	−	0.553477i	$-0.186699\pi$
$953$	51.4222	1.66573	0.832864	+	0.553477i	$0.186699\pi$
$954$	0	0
$955$	−6.69722	−0.216717
$956$	−19.1194	−0.618367
$957$	0	0
$958$	−27.4861	−0.888036
$959$	−18.4222	−0.594884
$960$	0	0
$961$	−25.2666	−0.815052
$962$	−10.8167	−0.348743
$963$	0	0
$964$	0.394449	0.0127043
$965$	−6.11943	−0.196991
$966$	0	0
$967$	−25.8444	−0.831100	−0.415550	−	0.909570i	$-0.636411\pi$
$967$	−25.8444	−0.831100	−0.415550	+	0.909570i	$0.636411\pi$
$968$	6.00000	0.192847
$969$	0	0
$970$	−21.9083	−0.703434
$971$	5.57779	0.179000	0.0895000	−	0.995987i	$-0.471473\pi$
$971$	5.57779	0.179000	0.0895000	+	0.995987i	$0.471473\pi$
$972$	0	0
$973$	−5.60555	−0.179706
$974$	−87.5055	−2.80386
$975$	0	0
$976$	−3.36669	−0.107765
$977$	−0.422205	−0.0135075	−0.00675377	−	0.999977i	$-0.502150\pi$
$977$	−0.422205	−0.0135075	−0.00675377	+	0.999977i	$0.502150\pi$
$978$	0	0
$979$	32.4500	1.03711
$980$	−3.30278	−0.105503
$981$	0	0
$982$	−47.7250	−1.52297
$983$	−20.5139	−0.654291	−0.327146	−	0.944974i	$-0.606087\pi$
$983$	−20.5139	−0.654291	−0.327146	+	0.944974i	$0.606087\pi$
$984$	0	0
$985$	−22.8167	−0.726999
$986$	28.1194	0.895505
$987$	0	0
$988$	−14.2111	−0.452115
$989$	2.93608	0.0933620
$990$	0	0
$991$	−59.0555	−1.87596	−0.937980	−	0.346689i	$-0.887306\pi$
$991$	−59.0555	−1.87596	−0.937980	+	0.346689i	$0.887306\pi$
$992$	−12.6972	−0.403137
$993$	0	0
$994$	4.81665	0.152775
$995$	−22.7250	−0.720430
$996$	0	0
$997$	−14.8167	−0.469248	−0.234624	−	0.972086i	$-0.575386\pi$
$997$	−14.8167	−0.469248	−0.234624	+	0.972086i	$0.575386\pi$
$998$	93.5694	2.96189
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.f.1.1	✓	2
3.2	odd	2		945.2.a.j.1.2	yes	2
5.4	even	2		4725.2.a.bf.1.2		2
7.6	odd	2		6615.2.a.q.1.1		2
15.14	odd	2		4725.2.a.z.1.1		2
21.20	even	2		6615.2.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.f.1.1	✓	2	1.1	even	1	trivial
945.2.a.j.1.2	yes	2	3.2	odd	2
4725.2.a.z.1.1		2	15.14	odd	2
4725.2.a.bf.1.2		2	5.4	even	2
6615.2.a.q.1.1		2	7.6	odd	2
6615.2.a.u.1.2		2	21.20	even	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:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30278 q^{2} +3.30278 q^{4} -1.00000 q^{5} +1.00000 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q-2.30278 q^{2} +3.30278 q^{4} -1.00000 q^{5} +1.00000 q^{7} -3.00000 q^{8} +2.30278 q^{10} -3.00000 q^{11} +1.30278 q^{13} -2.30278 q^{14} +0.302776 q^{16} +2.30278 q^{17} -3.30278 q^{19} -3.30278 q^{20} +6.90833 q^{22} -0.697224 q^{23} +1.00000 q^{25} -3.00000 q^{26} +3.30278 q^{28} -5.30278 q^{29} -2.39445 q^{31} +5.30278 q^{32} -5.30278 q^{34} -1.00000 q^{35} +3.60555 q^{37} +7.60555 q^{38} +3.00000 q^{40} +6.90833 q^{41} -4.21110 q^{43} -9.90833 q^{44} +1.60555 q^{46} -7.60555 q^{47} +1.00000 q^{49} -2.30278 q^{50} +4.30278 q^{52} +5.51388 q^{53} +3.00000 q^{55} -3.00000 q^{56} +12.2111 q^{58} -6.21110 q^{59} -11.1194 q^{61} +5.51388 q^{62} -12.8167 q^{64} -1.30278 q^{65} +13.5139 q^{67} +7.60555 q^{68} +2.30278 q^{70} -2.09167 q^{71} -10.2111 q^{73} -8.30278 q^{74} -10.9083 q^{76} -3.00000 q^{77} -16.9083 q^{79} -0.302776 q^{80} -15.9083 q^{82} +12.2111 q^{83} -2.30278 q^{85} +9.69722 q^{86} +9.00000 q^{88} -10.8167 q^{89} +1.30278 q^{91} -2.30278 q^{92} +17.5139 q^{94} +3.30278 q^{95} -9.51388 q^{97} -2.30278 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 - 6 * q^8	\( 2 q - q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} - 6 q^{8} + q^{10} - 6 q^{11} - q^{13} - q^{14} - 3 q^{16} + q^{17} - 3 q^{19} - 3 q^{20} + 3 q^{22} - 5 q^{23} + 2 q^{25} - 6 q^{26} + 3 q^{28} - 7 q^{29} - 12 q^{31} + 7 q^{32} - 7 q^{34} - 2 q^{35} + 8 q^{38} + 6 q^{40} + 3 q^{41} + 6 q^{43} - 9 q^{44} - 4 q^{46} - 8 q^{47} + 2 q^{49} - q^{50} + 5 q^{52} - 7 q^{53} + 6 q^{55} - 6 q^{56} + 10 q^{58} + 2 q^{59} + 3 q^{61} - 7 q^{62} - 4 q^{64} + q^{65} + 9 q^{67} + 8 q^{68} + q^{70} - 15 q^{71} - 6 q^{73} - 13 q^{74} - 11 q^{76} - 6 q^{77} - 23 q^{79} + 3 q^{80} - 21 q^{82} + 10 q^{83} - q^{85} + 23 q^{86} + 18 q^{88} - q^{91} - q^{92} + 17 q^{94} + 3 q^{95} - q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 - 6 * q^8 + q^10 - 6 * q^11 - q^13 - q^14 - 3 * q^16 + q^17 - 3 * q^19 - 3 * q^20 + 3 * q^22 - 5 * q^23 + 2 * q^25 - 6 * q^26 + 3 * q^28 - 7 * q^29 - 12 * q^31 + 7 * q^32 - 7 * q^34 - 2 * q^35 + 8 * q^38 + 6 * q^40 + 3 * q^41 + 6 * q^43 - 9 * q^44 - 4 * q^46 - 8 * q^47 + 2 * q^49 - q^50 + 5 * q^52 - 7 * q^53 + 6 * q^55 - 6 * q^56 + 10 * q^58 + 2 * q^59 + 3 * q^61 - 7 * q^62 - 4 * q^64 + q^65 + 9 * q^67 + 8 * q^68 + q^70 - 15 * q^71 - 6 * q^73 - 13 * q^74 - 11 * q^76 - 6 * q^77 - 23 * q^79 + 3 * q^80 - 21 * q^82 + 10 * q^83 - q^85 + 23 * q^86 + 18 * q^88 - q^91 - q^92 + 17 * q^94 + 3 * q^95 - q^97 - q^98

Introduction

L-functions

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