LMFDB - Embedded newform 5265.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5265,2,Mod(1,5265)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5265, 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("5265.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5265 = 3^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5265.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:	$42.0412366642$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 31x^{5} - x^{4} - 70x^{3} + 66x^{2} - 19x + 1 $ x^8 - 3x^7 - 8x^6 + 31x^5 - x^4 - 70x^3 + 66x^2 - 19x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.44829$ of defining polynomial
Character	$\chi$	$=$	5265.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44829 q^{2} +3.99411 q^{4} +1.00000 q^{5} -3.93452 q^{7} -4.88214 q^{8} +O(q^{10})$	\(q-2.44829 q^{2} +3.99411 q^{4} +1.00000 q^{5} -3.93452 q^{7} -4.88214 q^{8} -2.44829 q^{10} -4.67286 q^{11} -1.00000 q^{13} +9.63283 q^{14} +3.96467 q^{16} +7.96946 q^{17} -0.120966 q^{19} +3.99411 q^{20} +11.4405 q^{22} -0.328157 q^{23} +1.00000 q^{25} +2.44829 q^{26} -15.7149 q^{28} +0.272925 q^{29} -6.57612 q^{31} +0.0576370 q^{32} -19.5115 q^{34} -3.93452 q^{35} -7.43198 q^{37} +0.296159 q^{38} -4.88214 q^{40} -0.0910132 q^{41} +7.79669 q^{43} -18.6639 q^{44} +0.803421 q^{46} +11.5760 q^{47} +8.48046 q^{49} -2.44829 q^{50} -3.99411 q^{52} +11.1782 q^{53} -4.67286 q^{55} +19.2089 q^{56} -0.668199 q^{58} -5.55560 q^{59} +13.2654 q^{61} +16.1002 q^{62} -8.07045 q^{64} -1.00000 q^{65} +6.82138 q^{67} +31.8309 q^{68} +9.63283 q^{70} +6.32442 q^{71} -7.37048 q^{73} +18.1956 q^{74} -0.483150 q^{76} +18.3855 q^{77} -11.9175 q^{79} +3.96467 q^{80} +0.222826 q^{82} -1.54855 q^{83} +7.96946 q^{85} -19.0885 q^{86} +22.8136 q^{88} +3.59499 q^{89} +3.93452 q^{91} -1.31069 q^{92} -28.3412 q^{94} -0.120966 q^{95} +5.27957 q^{97} -20.7626 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8}+O(q^{10}) $ 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8	$ 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8} - 3 q^{10} - 6 q^{11} - 8 q^{13} - 10 q^{14} + 11 q^{16} + 2 q^{17} - 10 q^{19} + 9 q^{20} + 3 q^{22} - 6 q^{23} + 8 q^{25} + 3 q^{26} - 34 q^{28} - 14 q^{29} - 31 q^{31} - q^{32} - 7 q^{34} - 11 q^{35} + q^{37} - 9 q^{38} + 6 q^{40} + 12 q^{41} + 15 q^{43} - 16 q^{44} - 32 q^{46} + 18 q^{47} + 17 q^{49} - 3 q^{50} - 9 q^{52} - 2 q^{53} - 6 q^{55} - 16 q^{56} - 42 q^{58} - 24 q^{59} - 9 q^{61} + 20 q^{62} - 30 q^{64} - 8 q^{65} - 18 q^{67} + 14 q^{68} - 10 q^{70} - 10 q^{71} + 6 q^{73} + 37 q^{74} - 53 q^{76} + 34 q^{77} - 3 q^{79} + 11 q^{80} - 34 q^{82} + 10 q^{83} + 2 q^{85} - 60 q^{86} - 14 q^{88} + 13 q^{89} + 11 q^{91} - 5 q^{92} + 17 q^{94} - 10 q^{95} - 34 q^{97} + 30 q^{98}+O(q^{100}) $ 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8 - 3 * q^10 - 6 * q^11 - 8 * q^13 - 10 * q^14 + 11 * q^16 + 2 * q^17 - 10 * q^19 + 9 * q^20 + 3 * q^22 - 6 * q^23 + 8 * q^25 + 3 * q^26 - 34 * q^28 - 14 * q^29 - 31 * q^31 - q^32 - 7 * q^34 - 11 * q^35 + q^37 - 9 * q^38 + 6 * q^40 + 12 * q^41 + 15 * q^43 - 16 * q^44 - 32 * q^46 + 18 * q^47 + 17 * q^49 - 3 * q^50 - 9 * q^52 - 2 * q^53 - 6 * q^55 - 16 * q^56 - 42 * q^58 - 24 * q^59 - 9 * q^61 + 20 * q^62 - 30 * q^64 - 8 * q^65 - 18 * q^67 + 14 * q^68 - 10 * q^70 - 10 * q^71 + 6 * q^73 + 37 * q^74 - 53 * q^76 + 34 * q^77 - 3 * q^79 + 11 * q^80 - 34 * q^82 + 10 * q^83 + 2 * q^85 - 60 * q^86 - 14 * q^88 + 13 * q^89 + 11 * q^91 - 5 * q^92 + 17 * q^94 - 10 * q^95 - 34 * q^97 + 30 * 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 $

$2$	−2.44829	−1.73120	−0.865600	−	0.500736i	$-0.833063\pi$
$2$	−2.44829	−1.73120	−0.865600	+	0.500736i	$0.833063\pi$
$3$	0	0
$3$	0	0
$4$	3.99411	1.99705
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.93452	−1.48711	−0.743555	−	0.668675i	$-0.766862\pi$
$7$	−3.93452	−1.48711	−0.743555	+	0.668675i	$0.766862\pi$
$8$	−4.88214	−1.72610
$9$	0	0
$10$	−2.44829	−0.774216
$11$	−4.67286	−1.40892	−0.704461	−	0.709743i	$-0.748811\pi$
$11$	−4.67286	−1.40892	−0.704461	+	0.709743i	$0.748811\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	9.63283	2.57448
$15$	0	0
$16$	3.96467	0.991167
$17$	7.96946	1.93288	0.966439	−	0.256897i	$-0.0827001\pi$
$17$	7.96946	1.93288	0.966439	+	0.256897i	$0.0827001\pi$
$18$	0	0
$19$	−0.120966	−0.0277515	−0.0138757	−	0.999904i	$-0.504417\pi$
$19$	−0.120966	−0.0277515	−0.0138757	+	0.999904i	$0.504417\pi$
$20$	3.99411	0.893109
$21$	0	0
$22$	11.4405	2.43912
$23$	−0.328157	−0.0684254	−0.0342127	−	0.999415i	$-0.510892\pi$
$23$	−0.328157	−0.0684254	−0.0342127	+	0.999415i	$0.510892\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.44829	0.480148
$27$	0	0
$28$	−15.7149	−2.96984
$29$	0.272925	0.0506809	0.0253405	−	0.999679i	$-0.491933\pi$
$29$	0.272925	0.0506809	0.0253405	+	0.999679i	$0.491933\pi$
$30$	0	0
$31$	−6.57612	−1.18111	−0.590553	−	0.806999i	$-0.701090\pi$
$31$	−6.57612	−1.18111	−0.590553	+	0.806999i	$0.701090\pi$
$32$	0.0576370	0.0101889
$33$	0	0
$34$	−19.5115	−3.34620
$35$	−3.93452	−0.665055
$36$	0	0
$37$	−7.43198	−1.22181	−0.610905	−	0.791704i	$-0.709194\pi$
$37$	−7.43198	−1.22181	−0.610905	+	0.791704i	$0.709194\pi$
$38$	0.296159	0.0480433
$39$	0	0
$40$	−4.88214	−0.771934
$41$	−0.0910132	−0.0142139	−0.00710694	−	0.999975i	$-0.502262\pi$
$41$	−0.0910132	−0.0142139	−0.00710694	+	0.999975i	$0.502262\pi$
$42$	0	0
$43$	7.79669	1.18898	0.594492	−	0.804101i	$-0.297353\pi$
$43$	7.79669	1.18898	0.594492	+	0.804101i	$0.297353\pi$
$44$	−18.6639	−2.81369
$45$	0	0
$46$	0.803421	0.118458
$47$	11.5760	1.68853	0.844263	−	0.535929i	$-0.180039\pi$
$47$	11.5760	1.68853	0.844263	+	0.535929i	$0.180039\pi$
$48$	0	0
$49$	8.48046	1.21149
$50$	−2.44829	−0.346240
$51$	0	0
$52$	−3.99411	−0.553883
$53$	11.1782	1.53544	0.767721	−	0.640784i	$-0.221391\pi$
$53$	11.1782	1.53544	0.767721	+	0.640784i	$0.221391\pi$
$54$	0	0
$55$	−4.67286	−0.630089
$56$	19.2089	2.56690
$57$	0	0
$58$	−0.668199	−0.0877388
$59$	−5.55560	−0.723277	−0.361639	−	0.932318i	$-0.617783\pi$
$59$	−5.55560	−0.723277	−0.361639	+	0.932318i	$0.617783\pi$
$60$	0	0
$61$	13.2654	1.69846	0.849230	−	0.528023i	$-0.177067\pi$
$61$	13.2654	1.69846	0.849230	+	0.528023i	$0.177067\pi$
$62$	16.1002	2.04473
$63$	0	0
$64$	−8.07045	−1.00881
$65$	−1.00000	−0.124035
$66$	0	0
$67$	6.82138	0.833365	0.416682	−	0.909052i	$-0.363193\pi$
$67$	6.82138	0.833365	0.416682	+	0.909052i	$0.363193\pi$
$68$	31.8309	3.86006
$69$	0	0
$70$	9.63283	1.15134
$71$	6.32442	0.750570	0.375285	−	0.926909i	$-0.377545\pi$
$71$	6.32442	0.750570	0.375285	+	0.926909i	$0.377545\pi$
$72$	0	0
$73$	−7.37048	−0.862649	−0.431324	−	0.902197i	$-0.641954\pi$
$73$	−7.37048	−0.862649	−0.431324	+	0.902197i	$0.641954\pi$
$74$	18.1956	2.11520
$75$	0	0
$76$	−0.483150	−0.0554211
$77$	18.3855	2.09522
$78$	0	0
$79$	−11.9175	−1.34083	−0.670413	−	0.741988i	$-0.733883\pi$
$79$	−11.9175	−1.34083	−0.670413	+	0.741988i	$0.733883\pi$
$80$	3.96467	0.443264
$81$	0	0
$82$	0.222826	0.0246071
$83$	−1.54855	−0.169976	−0.0849878	−	0.996382i	$-0.527085\pi$
$83$	−1.54855	−0.169976	−0.0849878	+	0.996382i	$0.527085\pi$
$84$	0	0
$85$	7.96946	0.864409
$86$	−19.0885	−2.05837
$87$	0	0
$88$	22.8136	2.43194
$89$	3.59499	0.381068	0.190534	−	0.981681i	$-0.438978\pi$
$89$	3.59499	0.381068	0.190534	+	0.981681i	$0.438978\pi$
$90$	0	0
$91$	3.93452	0.412450
$92$	−1.31069	−0.136649
$93$	0	0
$94$	−28.3412	−2.92318
$95$	−0.120966	−0.0124108
$96$	0	0
$97$	5.27957	0.536059	0.268030	−	0.963411i	$-0.413627\pi$
$97$	5.27957	0.536059	0.268030	+	0.963411i	$0.413627\pi$
$98$	−20.7626	−2.09734
$99$	0	0
$100$	3.99411	0.399411
$101$	0.357091	0.0355319	0.0177659	−	0.999842i	$-0.494345\pi$
$101$	0.357091	0.0355319	0.0177659	+	0.999842i	$0.494345\pi$
$102$	0	0
$103$	−4.61245	−0.454478	−0.227239	−	0.973839i	$-0.572970\pi$
$103$	−4.61245	−0.454478	−0.227239	+	0.973839i	$0.572970\pi$
$104$	4.88214	0.478733
$105$	0	0
$106$	−27.3674	−2.65816
$107$	−1.48003	−0.143080	−0.0715399	−	0.997438i	$-0.522791\pi$
$107$	−1.48003	−0.143080	−0.0715399	+	0.997438i	$0.522791\pi$
$108$	0	0
$109$	−14.6820	−1.40628	−0.703140	−	0.711051i	$-0.748219\pi$
$109$	−14.6820	−1.40628	−0.703140	+	0.711051i	$0.748219\pi$
$110$	11.4405	1.09081
$111$	0	0
$112$	−15.5991	−1.47397
$113$	−7.07380	−0.665447	−0.332724	−	0.943024i	$-0.607968\pi$
$113$	−7.07380	−0.665447	−0.332724	+	0.943024i	$0.607968\pi$
$114$	0	0
$115$	−0.328157	−0.0306008
$116$	1.09009	0.101212
$117$	0	0
$118$	13.6017	1.25214
$119$	−31.3560	−2.87440
$120$	0	0
$121$	10.8357	0.985060
$122$	−32.4775	−2.94037
$123$	0	0
$124$	−26.2657	−2.35873
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.23715	0.109779	0.0548896	−	0.998492i	$-0.482519\pi$
$127$	1.23715	0.109779	0.0548896	+	0.998492i	$0.482519\pi$
$128$	19.6435	1.73626
$129$	0	0
$130$	2.44829	0.214729
$131$	−12.7528	−1.11421	−0.557107	−	0.830441i	$-0.688089\pi$
$131$	−12.7528	−1.11421	−0.557107	+	0.830441i	$0.688089\pi$
$132$	0	0
$133$	0.475943	0.0412695
$134$	−16.7007	−1.44272
$135$	0	0
$136$	−38.9080	−3.33634
$137$	−7.32899	−0.626158	−0.313079	−	0.949727i	$-0.601360\pi$
$137$	−7.32899	−0.626158	−0.313079	+	0.949727i	$0.601360\pi$
$138$	0	0
$139$	−16.8547	−1.42960	−0.714799	−	0.699329i	$-0.753482\pi$
$139$	−16.8547	−1.42960	−0.714799	+	0.699329i	$0.753482\pi$
$140$	−15.7149	−1.32815
$141$	0	0
$142$	−15.4840	−1.29939
$143$	4.67286	0.390765
$144$	0	0
$145$	0.272925	0.0226652
$146$	18.0450	1.49342
$147$	0	0
$148$	−29.6841	−2.44002
$149$	8.22079	0.673473	0.336737	−	0.941599i	$-0.390677\pi$
$149$	8.22079	0.673473	0.336737	+	0.941599i	$0.390677\pi$
$150$	0	0
$151$	15.4012	1.25333	0.626665	−	0.779289i	$-0.284419\pi$
$151$	15.4012	1.25333	0.626665	+	0.779289i	$0.284419\pi$
$152$	0.590572	0.0479017
$153$	0	0
$154$	−45.0129	−3.62725
$155$	−6.57612	−0.528207
$156$	0	0
$157$	−20.1799	−1.61053	−0.805264	−	0.592916i	$-0.797977\pi$
$157$	−20.1799	−1.61053	−0.805264	+	0.592916i	$0.797977\pi$
$158$	29.1775	2.32124
$159$	0	0
$160$	0.0576370	0.00455661
$161$	1.29114	0.101756
$162$	0	0
$163$	8.27259	0.647959	0.323980	−	0.946064i	$-0.394979\pi$
$163$	8.27259	0.647959	0.323980	+	0.946064i	$0.394979\pi$
$164$	−0.363516	−0.0283859
$165$	0	0
$166$	3.79130	0.294262
$167$	13.6325	1.05492	0.527458	−	0.849581i	$-0.323145\pi$
$167$	13.6325	1.05492	0.527458	+	0.849581i	$0.323145\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−19.5115	−1.49646
$171$	0	0
$172$	31.1408	2.37446
$173$	−3.46478	−0.263422	−0.131711	−	0.991288i	$-0.542047\pi$
$173$	−3.46478	−0.263422	−0.131711	+	0.991288i	$0.542047\pi$
$174$	0	0
$175$	−3.93452	−0.297422
$176$	−18.5264	−1.39648
$177$	0	0
$178$	−8.80156	−0.659705
$179$	6.80739	0.508808	0.254404	−	0.967098i	$-0.418121\pi$
$179$	6.80739	0.508808	0.254404	+	0.967098i	$0.418121\pi$
$180$	0	0
$181$	−8.23721	−0.612267	−0.306133	−	0.951989i	$-0.599035\pi$
$181$	−8.23721	−0.612267	−0.306133	+	0.951989i	$0.599035\pi$
$182$	−9.63283	−0.714033
$183$	0	0
$184$	1.60211	0.118109
$185$	−7.43198	−0.546410
$186$	0	0
$187$	−37.2402	−2.72327
$188$	46.2356	3.37208
$189$	0	0
$190$	0.296159	0.0214856
$191$	−17.7269	−1.28267	−0.641337	−	0.767260i	$-0.721620\pi$
$191$	−17.7269	−1.28267	−0.641337	+	0.767260i	$0.721620\pi$
$192$	0	0
$193$	11.6033	0.835227	0.417613	−	0.908625i	$-0.362867\pi$
$193$	11.6033	0.835227	0.417613	+	0.908625i	$0.362867\pi$
$194$	−12.9259	−0.928026
$195$	0	0
$196$	33.8718	2.41942
$197$	−2.00369	−0.142757	−0.0713783	−	0.997449i	$-0.522740\pi$
$197$	−2.00369	−0.142757	−0.0713783	+	0.997449i	$0.522740\pi$
$198$	0	0
$199$	11.2572	0.798004	0.399002	−	0.916950i	$-0.369357\pi$
$199$	11.2572	0.798004	0.399002	+	0.916950i	$0.369357\pi$
$200$	−4.88214	−0.345220
$201$	0	0
$202$	−0.874260	−0.0615127
$203$	−1.07383	−0.0753681
$204$	0	0
$205$	−0.0910132	−0.00635664
$206$	11.2926	0.786793
$207$	0	0
$208$	−3.96467	−0.274900
$209$	0.565257	0.0390996
$210$	0	0
$211$	−4.86410	−0.334859	−0.167429	−	0.985884i	$-0.553547\pi$
$211$	−4.86410	−0.334859	−0.167429	+	0.985884i	$0.553547\pi$
$212$	44.6469	3.06636
$213$	0	0
$214$	3.62353	0.247700
$215$	7.79669	0.531730
$216$	0	0
$217$	25.8739	1.75643
$218$	35.9457	2.43455
$219$	0	0
$220$	−18.6639	−1.25832
$221$	−7.96946	−0.536084
$222$	0	0
$223$	−14.0824	−0.943025	−0.471513	−	0.881859i	$-0.656292\pi$
$223$	−14.0824	−0.943025	−0.471513	+	0.881859i	$0.656292\pi$
$224$	−0.226774	−0.0151520
$225$	0	0
$226$	17.3187	1.15202
$227$	7.96332	0.528544	0.264272	−	0.964448i	$-0.414868\pi$
$227$	7.96332	0.528544	0.264272	+	0.964448i	$0.414868\pi$
$228$	0	0
$229$	13.6285	0.900597	0.450298	−	0.892878i	$-0.351318\pi$
$229$	13.6285	0.900597	0.450298	+	0.892878i	$0.351318\pi$
$230$	0.803421	0.0529760
$231$	0	0
$232$	−1.33246	−0.0874802
$233$	−6.76769	−0.443366	−0.221683	−	0.975119i	$-0.571155\pi$
$233$	−6.76769	−0.443366	−0.221683	+	0.975119i	$0.571155\pi$
$234$	0	0
$235$	11.5760	0.755132
$236$	−22.1896	−1.44442
$237$	0	0
$238$	76.7685	4.97616
$239$	−25.4676	−1.64736	−0.823681	−	0.567054i	$-0.808083\pi$
$239$	−25.4676	−1.64736	−0.823681	+	0.567054i	$0.808083\pi$
$240$	0	0
$241$	−20.1625	−1.29878	−0.649392	−	0.760454i	$-0.724976\pi$
$241$	−20.1625	−1.29878	−0.649392	+	0.760454i	$0.724976\pi$
$242$	−26.5288	−1.70534
$243$	0	0
$244$	52.9834	3.39191
$245$	8.48046	0.541797
$246$	0	0
$247$	0.120966	0.00769687
$248$	32.1056	2.03870
$249$	0	0
$250$	−2.44829	−0.154843
$251$	−22.3167	−1.40862	−0.704308	−	0.709894i	$-0.748743\pi$
$251$	−22.3167	−1.40862	−0.704308	+	0.709894i	$0.748743\pi$
$252$	0	0
$253$	1.53343	0.0964060
$254$	−3.02890	−0.190050
$255$	0	0
$256$	−31.9520	−1.99700
$257$	3.45471	0.215499	0.107750	−	0.994178i	$-0.465636\pi$
$257$	3.45471	0.215499	0.107750	+	0.994178i	$0.465636\pi$
$258$	0	0
$259$	29.2413	1.81696
$260$	−3.99411	−0.247704
$261$	0	0
$262$	31.2224	1.92893
$263$	−14.9744	−0.923359	−0.461680	−	0.887047i	$-0.652753\pi$
$263$	−14.9744	−0.923359	−0.461680	+	0.887047i	$0.652753\pi$
$264$	0	0
$265$	11.1782	0.686671
$266$	−1.16524	−0.0714457
$267$	0	0
$268$	27.2453	1.66427
$269$	−10.5996	−0.646269	−0.323135	−	0.946353i	$-0.604737\pi$
$269$	−10.5996	−0.646269	−0.323135	+	0.946353i	$0.604737\pi$
$270$	0	0
$271$	−9.73751	−0.591512	−0.295756	−	0.955264i	$-0.595571\pi$
$271$	−9.73751	−0.591512	−0.295756	+	0.955264i	$0.595571\pi$
$272$	31.5963	1.91581
$273$	0	0
$274$	17.9435	1.08400
$275$	−4.67286	−0.281784
$276$	0	0
$277$	−11.0416	−0.663425	−0.331713	−	0.943381i	$-0.607626\pi$
$277$	−11.0416	−0.663425	−0.331713	+	0.943381i	$0.607626\pi$
$278$	41.2652	2.47492
$279$	0	0
$280$	19.2089	1.14795
$281$	21.3065	1.27104	0.635519	−	0.772085i	$-0.280786\pi$
$281$	21.3065	1.27104	0.635519	+	0.772085i	$0.280786\pi$
$282$	0	0
$283$	−3.27051	−0.194412	−0.0972060	−	0.995264i	$-0.530991\pi$
$283$	−3.27051	−0.194412	−0.0972060	+	0.995264i	$0.530991\pi$
$284$	25.2604	1.49893
$285$	0	0
$286$	−11.4405	−0.676492
$287$	0.358093	0.0211376
$288$	0	0
$289$	46.5123	2.73602
$290$	−0.668199	−0.0392380
$291$	0	0
$292$	−29.4385	−1.72276
$293$	28.9968	1.69401	0.847004	−	0.531586i	$-0.178404\pi$
$293$	28.9968	1.69401	0.847004	+	0.531586i	$0.178404\pi$
$294$	0	0
$295$	−5.55560	−0.323459
$296$	36.2840	2.10896
$297$	0	0
$298$	−20.1268	−1.16592
$299$	0.328157	0.0189778
$300$	0	0
$301$	−30.6762	−1.76815
$302$	−37.7065	−2.16976
$303$	0	0
$304$	−0.479590	−0.0275063
$305$	13.2654	0.759574
$306$	0	0
$307$	7.67485	0.438027	0.219013	−	0.975722i	$-0.429716\pi$
$307$	7.67485	0.438027	0.219013	+	0.975722i	$0.429716\pi$
$308$	73.4336	4.18427
$309$	0	0
$310$	16.1002	0.914431
$311$	6.58949	0.373656	0.186828	−	0.982393i	$-0.440179\pi$
$311$	6.58949	0.373656	0.186828	+	0.982393i	$0.440179\pi$
$312$	0	0
$313$	−8.13303	−0.459706	−0.229853	−	0.973225i	$-0.573825\pi$
$313$	−8.13303	−0.459706	−0.229853	+	0.973225i	$0.573825\pi$
$314$	49.4061	2.78815
$315$	0	0
$316$	−47.5999	−2.67770
$317$	−23.3461	−1.31125	−0.655624	−	0.755088i	$-0.727594\pi$
$317$	−23.3461	−1.31125	−0.655624	+	0.755088i	$0.727594\pi$
$318$	0	0
$319$	−1.27534	−0.0714054
$320$	−8.07045	−0.451152
$321$	0	0
$322$	−3.16108	−0.176160
$323$	−0.964032	−0.0536402
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−20.2537	−1.12175
$327$	0	0
$328$	0.444339	0.0245345
$329$	−45.5458	−2.51102
$330$	0	0
$331$	−4.48044	−0.246268	−0.123134	−	0.992390i	$-0.539294\pi$
$331$	−4.48044	−0.246268	−0.123134	+	0.992390i	$0.539294\pi$
$332$	−6.18508	−0.339450
$333$	0	0
$334$	−33.3763	−1.82627
$335$	6.82138	0.372692
$336$	0	0
$337$	−4.64674	−0.253124	−0.126562	−	0.991959i	$-0.540394\pi$
$337$	−4.64674	−0.253124	−0.126562	+	0.991959i	$0.540394\pi$
$338$	−2.44829	−0.133169
$339$	0	0
$340$	31.8309	1.72627
$341$	30.7293	1.66409
$342$	0	0
$343$	−5.82489	−0.314514
$344$	−38.0646	−2.05230
$345$	0	0
$346$	8.48277	0.456037
$347$	4.42523	0.237559	0.118779	−	0.992921i	$-0.462102\pi$
$347$	4.42523	0.237559	0.118779	+	0.992921i	$0.462102\pi$
$348$	0	0
$349$	−0.0135151	−0.000723446 0	−0.000361723	−	1.00000i	$-0.500115\pi$
$349$	−0.0135151	−0.000723446 0	−0.000361723		1.00000i	$0.500115\pi$
$350$	9.63283	0.514897
$351$	0	0
$352$	−0.269330	−0.0143553
$353$	−29.7072	−1.58116	−0.790578	−	0.612361i	$-0.790220\pi$
$353$	−29.7072	−1.58116	−0.790578	+	0.612361i	$0.790220\pi$
$354$	0	0
$355$	6.32442	0.335665
$356$	14.3588	0.761013
$357$	0	0
$358$	−16.6664	−0.880849
$359$	−35.9145	−1.89550	−0.947748	−	0.319020i	$-0.896646\pi$
$359$	−35.9145	−1.89550	−0.947748	+	0.319020i	$0.896646\pi$
$360$	0	0
$361$	−18.9854	−0.999230
$362$	20.1670	1.05996
$363$	0	0
$364$	15.7149	0.823684
$365$	−7.37048	−0.385788
$366$	0	0
$367$	2.95972	0.154496	0.0772481	−	0.997012i	$-0.475387\pi$
$367$	2.95972	0.154496	0.0772481	+	0.997012i	$0.475387\pi$
$368$	−1.30103	−0.0678210
$369$	0	0
$370$	18.1956	0.945945
$371$	−43.9808	−2.28337
$372$	0	0
$373$	4.56441	0.236336	0.118168	−	0.992994i	$-0.462298\pi$
$373$	4.56441	0.236336	0.118168	+	0.992994i	$0.462298\pi$
$374$	91.1747	4.71453
$375$	0	0
$376$	−56.5154	−2.91456
$377$	−0.272925	−0.0140564
$378$	0	0
$379$	−22.9852	−1.18067	−0.590336	−	0.807158i	$-0.701005\pi$
$379$	−22.9852	−1.18067	−0.590336	+	0.807158i	$0.701005\pi$
$380$	−0.483150	−0.0247851
$381$	0	0
$382$	43.4005	2.22056
$383$	−9.20542	−0.470375	−0.235187	−	0.971950i	$-0.575570\pi$
$383$	−9.20542	−0.470375	−0.235187	+	0.971950i	$0.575570\pi$
$384$	0	0
$385$	18.3855	0.937011
$386$	−28.4083	−1.44594
$387$	0	0
$388$	21.0872	1.07054
$389$	26.0241	1.31948	0.659738	−	0.751495i	$-0.270667\pi$
$389$	26.0241	1.31948	0.659738	+	0.751495i	$0.270667\pi$
$390$	0	0
$391$	−2.61523	−0.132258
$392$	−41.4028	−2.09116
$393$	0	0
$394$	4.90560	0.247140
$395$	−11.9175	−0.599636
$396$	0	0
$397$	−32.0420	−1.60814	−0.804071	−	0.594533i	$-0.797337\pi$
$397$	−32.0420	−1.60814	−0.804071	+	0.594533i	$0.797337\pi$
$398$	−27.5609	−1.38150
$399$	0	0
$400$	3.96467	0.198233
$401$	−0.824590	−0.0411780	−0.0205890	−	0.999788i	$-0.506554\pi$
$401$	−0.824590	−0.0411780	−0.0205890	+	0.999788i	$0.506554\pi$
$402$	0	0
$403$	6.57612	0.327580
$404$	1.42626	0.0709590
$405$	0	0
$406$	2.62904	0.130477
$407$	34.7286	1.72143
$408$	0	0
$409$	28.2192	1.39535	0.697675	−	0.716415i	$-0.254218\pi$
$409$	28.2192	1.39535	0.697675	+	0.716415i	$0.254218\pi$
$410$	0.222826	0.0110046
$411$	0	0
$412$	−18.4226	−0.907617
$413$	21.8586	1.07559
$414$	0	0
$415$	−1.54855	−0.0760154
$416$	−0.0576370	−0.00282589
$417$	0	0
$418$	−1.38391	−0.0676893
$419$	−24.8224	−1.21265	−0.606327	−	0.795216i	$-0.707358\pi$
$419$	−24.8224	−1.21265	−0.606327	+	0.795216i	$0.707358\pi$
$420$	0	0
$421$	32.8065	1.59889	0.799444	−	0.600740i	$-0.205127\pi$
$421$	32.8065	1.59889	0.799444	+	0.600740i	$0.205127\pi$
$422$	11.9087	0.579707
$423$	0	0
$424$	−54.5735	−2.65032
$425$	7.96946	0.386576
$426$	0	0
$427$	−52.1930	−2.52579
$428$	−5.91139	−0.285738
$429$	0	0
$430$	−19.0885	−0.920531
$431$	−4.17377	−0.201043	−0.100522	−	0.994935i	$-0.532051\pi$
$431$	−4.17377	−0.201043	−0.100522	+	0.994935i	$0.532051\pi$
$432$	0	0
$433$	20.3832	0.979555	0.489777	−	0.871848i	$-0.337078\pi$
$433$	20.3832	0.979555	0.489777	+	0.871848i	$0.337078\pi$
$434$	−63.3467	−3.04074
$435$	0	0
$436$	−58.6415	−2.80842
$437$	0.0396957	0.00189890
$438$	0	0
$439$	18.2228	0.869727	0.434864	−	0.900496i	$-0.356797\pi$
$439$	18.2228	0.869727	0.434864	+	0.900496i	$0.356797\pi$
$440$	22.8136	1.08760
$441$	0	0
$442$	19.5115	0.928068
$443$	−6.95230	−0.330314	−0.165157	−	0.986267i	$-0.552813\pi$
$443$	−6.95230	−0.330314	−0.165157	+	0.986267i	$0.552813\pi$
$444$	0	0
$445$	3.59499	0.170419
$446$	34.4777	1.63256
$447$	0	0
$448$	31.7534	1.50021
$449$	−13.6819	−0.645687	−0.322843	−	0.946452i	$-0.604639\pi$
$449$	−13.6819	−0.645687	−0.322843	+	0.946452i	$0.604639\pi$
$450$	0	0
$451$	0.425292	0.0200262
$452$	−28.2535	−1.32893
$453$	0	0
$454$	−19.4965	−0.915016
$455$	3.93452	0.184453
$456$	0	0
$457$	16.8387	0.787680	0.393840	−	0.919179i	$-0.371146\pi$
$457$	16.8387	0.787680	0.393840	+	0.919179i	$0.371146\pi$
$458$	−33.3665	−1.55911
$459$	0	0
$460$	−1.31069	−0.0611113
$461$	39.3709	1.83369	0.916844	−	0.399247i	$-0.130728\pi$
$461$	39.3709	1.83369	0.916844	+	0.399247i	$0.130728\pi$
$462$	0	0
$463$	−2.88341	−0.134003	−0.0670016	−	0.997753i	$-0.521343\pi$
$463$	−2.88341	−0.134003	−0.0670016	+	0.997753i	$0.521343\pi$
$464$	1.08206	0.0502333
$465$	0	0
$466$	16.5692	0.767555
$467$	−25.1278	−1.16278	−0.581388	−	0.813626i	$-0.697490\pi$
$467$	−25.1278	−1.16278	−0.581388	+	0.813626i	$0.697490\pi$
$468$	0	0
$469$	−26.8389	−1.23930
$470$	−28.3412	−1.30728
$471$	0	0
$472$	27.1232	1.24845
$473$	−36.4329	−1.67519
$474$	0	0
$475$	−0.120966	−0.00555029
$476$	−125.239	−5.74033
$477$	0	0
$478$	62.3520	2.85191
$479$	−21.4658	−0.980798	−0.490399	−	0.871498i	$-0.663149\pi$
$479$	−21.4658	−0.980798	−0.490399	+	0.871498i	$0.663149\pi$
$480$	0	0
$481$	7.43198	0.338869
$482$	49.3637	2.24845
$483$	0	0
$484$	43.2788	1.96722
$485$	5.27957	0.239733
$486$	0	0
$487$	−32.0627	−1.45290	−0.726450	−	0.687220i	$-0.758831\pi$
$487$	−32.0627	−1.45290	−0.726450	+	0.687220i	$0.758831\pi$
$488$	−64.7635	−2.93171
$489$	0	0
$490$	−20.7626	−0.937958
$491$	9.73999	0.439560	0.219780	−	0.975549i	$-0.429466\pi$
$491$	9.73999	0.439560	0.219780	+	0.975549i	$0.429466\pi$
$492$	0	0
$493$	2.17507	0.0979600
$494$	−0.296159	−0.0133248
$495$	0	0
$496$	−26.0721	−1.17067
$497$	−24.8836	−1.11618
$498$	0	0
$499$	31.7717	1.42230	0.711149	−	0.703042i	$-0.248175\pi$
$499$	31.7717	1.42230	0.711149	+	0.703042i	$0.248175\pi$
$500$	3.99411	0.178622
$501$	0	0
$502$	54.6376	2.43860
$503$	7.30099	0.325535	0.162768	−	0.986664i	$-0.447958\pi$
$503$	7.30099	0.325535	0.162768	+	0.986664i	$0.447958\pi$
$504$	0	0
$505$	0.357091	0.0158903
$506$	−3.75428	−0.166898
$507$	0	0
$508$	4.94130	0.219235
$509$	10.7912	0.478312	0.239156	−	0.970981i	$-0.423129\pi$
$509$	10.7912	0.478312	0.239156	+	0.970981i	$0.423129\pi$
$510$	0	0
$511$	28.9993	1.28285
$512$	38.9407	1.72095
$513$	0	0
$514$	−8.45813	−0.373072
$515$	−4.61245	−0.203249
$516$	0	0
$517$	−54.0928	−2.37900
$518$	−71.5910	−3.14553
$519$	0	0
$520$	4.88214	0.214096
$521$	3.58465	0.157046	0.0785231	−	0.996912i	$-0.474980\pi$
$521$	3.58465	0.157046	0.0785231	+	0.996912i	$0.474980\pi$
$522$	0	0
$523$	4.70071	0.205548	0.102774	−	0.994705i	$-0.467228\pi$
$523$	4.70071	0.205548	0.102774	+	0.994705i	$0.467228\pi$
$524$	−50.9359	−2.22514
$525$	0	0
$526$	36.6615	1.59852
$527$	−52.4081	−2.28293
$528$	0	0
$529$	−22.8923	−0.995318
$530$	−27.3674	−1.18876
$531$	0	0
$532$	1.90097	0.0824173
$533$	0.0910132	0.00394222
$534$	0	0
$535$	−1.48003	−0.0639872
$536$	−33.3030	−1.43847
$537$	0	0
$538$	25.9509	1.11882
$539$	−39.6280	−1.70690
$540$	0	0
$541$	31.2736	1.34456	0.672278	−	0.740299i	$-0.265316\pi$
$541$	31.2736	1.34456	0.672278	+	0.740299i	$0.265316\pi$
$542$	23.8402	1.02403
$543$	0	0
$544$	0.459336	0.0196939
$545$	−14.6820	−0.628908
$546$	0	0
$547$	−3.45324	−0.147650	−0.0738249	−	0.997271i	$-0.523521\pi$
$547$	−3.45324	−0.147650	−0.0738249	+	0.997271i	$0.523521\pi$
$548$	−29.2728	−1.25047
$549$	0	0
$550$	11.4405	0.487825
$551$	−0.0330146	−0.00140647
$552$	0	0
$553$	46.8898	1.99396
$554$	27.0330	1.14852
$555$	0	0
$556$	−67.3195	−2.85498
$557$	−12.5254	−0.530720	−0.265360	−	0.964149i	$-0.585491\pi$
$557$	−12.5254	−0.530720	−0.265360	+	0.964149i	$0.585491\pi$
$558$	0	0
$559$	−7.79669	−0.329765
$560$	−15.5991	−0.659181
$561$	0	0
$562$	−52.1643	−2.20042
$563$	26.4317	1.11396	0.556981	−	0.830525i	$-0.311960\pi$
$563$	26.4317	1.11396	0.556981	+	0.830525i	$0.311960\pi$
$564$	0	0
$565$	−7.07380	−0.297597
$566$	8.00716	0.336566
$567$	0	0
$568$	−30.8767	−1.29556
$569$	−22.3724	−0.937899	−0.468949	−	0.883225i	$-0.655367\pi$
$569$	−22.3724	−0.937899	−0.468949	+	0.883225i	$0.655367\pi$
$570$	0	0
$571$	−14.9412	−0.625271	−0.312636	−	0.949873i	$-0.601212\pi$
$571$	−14.9412	−0.625271	−0.312636	+	0.949873i	$0.601212\pi$
$572$	18.6639	0.780377
$573$	0	0
$574$	−0.876715	−0.0365934
$575$	−0.328157	−0.0136851
$576$	0	0
$577$	−2.65841	−0.110671	−0.0553355	−	0.998468i	$-0.517623\pi$
$577$	−2.65841	−0.110671	−0.0553355	+	0.998468i	$0.517623\pi$
$578$	−113.875	−4.73659
$579$	0	0
$580$	1.09009	0.0452636
$581$	6.09281	0.252772
$582$	0	0
$583$	−52.2342	−2.16332
$584$	35.9837	1.48902
$585$	0	0
$586$	−70.9924	−2.93267
$587$	11.4982	0.474581	0.237291	−	0.971439i	$-0.423741\pi$
$587$	11.4982	0.474581	0.237291	+	0.971439i	$0.423741\pi$
$588$	0	0
$589$	0.795486	0.0327774
$590$	13.6017	0.559973
$591$	0	0
$592$	−29.4653	−1.21102
$593$	22.0373	0.904964	0.452482	−	0.891774i	$-0.350539\pi$
$593$	22.0373	0.904964	0.452482	+	0.891774i	$0.350539\pi$
$594$	0	0
$595$	−31.3560	−1.28547
$596$	32.8347	1.34496
$597$	0	0
$598$	−0.803421	−0.0328543
$599$	−24.1227	−0.985626	−0.492813	−	0.870135i	$-0.664031\pi$
$599$	−24.1227	−0.985626	−0.492813	+	0.870135i	$0.664031\pi$
$600$	0	0
$601$	0.342838	0.0139846	0.00699232	−	0.999976i	$-0.497774\pi$
$601$	0.342838	0.0139846	0.00699232	+	0.999976i	$0.497774\pi$
$602$	75.1042	3.06102
$603$	0	0
$604$	61.5139	2.50297
$605$	10.8357	0.440532
$606$	0	0
$607$	−37.0885	−1.50538	−0.752688	−	0.658377i	$-0.771243\pi$
$607$	−37.0885	−1.50538	−0.752688	+	0.658377i	$0.771243\pi$
$608$	−0.00697211	−0.000282756 0
$609$	0	0
$610$	−32.4775	−1.31497
$611$	−11.5760	−0.468313
$612$	0	0
$613$	38.1878	1.54239	0.771194	−	0.636600i	$-0.219660\pi$
$613$	38.1878	1.54239	0.771194	+	0.636600i	$0.219660\pi$
$614$	−18.7902	−0.758312
$615$	0	0
$616$	−89.7605	−3.61656
$617$	12.9173	0.520030	0.260015	−	0.965605i	$-0.416273\pi$
$617$	12.9173	0.520030	0.260015	+	0.965605i	$0.416273\pi$
$618$	0	0
$619$	−6.49730	−0.261149	−0.130574	−	0.991439i	$-0.541682\pi$
$619$	−6.49730	−0.261149	−0.130574	+	0.991439i	$0.541682\pi$
$620$	−26.2657	−1.05486
$621$	0	0
$622$	−16.1330	−0.646873
$623$	−14.1446	−0.566690
$624$	0	0
$625$	1.00000	0.0400000
$626$	19.9120	0.795843
$627$	0	0
$628$	−80.6005	−3.21631
$629$	−59.2288	−2.36161
$630$	0	0
$631$	−37.1832	−1.48024	−0.740119	−	0.672476i	$-0.765231\pi$
$631$	−37.1832	−1.48024	−0.740119	+	0.672476i	$0.765231\pi$
$632$	58.1831	2.31440
$633$	0	0
$634$	57.1580	2.27003
$635$	1.23715	0.0490948
$636$	0	0
$637$	−8.48046	−0.336008
$638$	3.12240	0.123617
$639$	0	0
$640$	19.6435	0.776478
$641$	−41.5824	−1.64241	−0.821203	−	0.570636i	$-0.806697\pi$
$641$	−41.5824	−1.64241	−0.821203	+	0.570636i	$0.806697\pi$
$642$	0	0
$643$	−14.4993	−0.571796	−0.285898	−	0.958260i	$-0.592292\pi$
$643$	−14.4993	−0.571796	−0.285898	+	0.958260i	$0.592292\pi$
$644$	5.15695	0.203212
$645$	0	0
$646$	2.36023	0.0928619
$647$	4.64226	0.182506	0.0912531	−	0.995828i	$-0.470913\pi$
$647$	4.64226	0.182506	0.0912531	+	0.995828i	$0.470913\pi$
$648$	0	0
$649$	25.9606	1.01904
$650$	2.44829	0.0960297
$651$	0	0
$652$	33.0416	1.29401
$653$	−10.5742	−0.413801	−0.206901	−	0.978362i	$-0.566338\pi$
$653$	−10.5742	−0.413801	−0.206901	+	0.978362i	$0.566338\pi$
$654$	0	0
$655$	−12.7528	−0.498292
$656$	−0.360837	−0.0140883
$657$	0	0
$658$	111.509	4.34708
$659$	31.3027	1.21938	0.609690	−	0.792640i	$-0.291294\pi$
$659$	31.3027	1.21938	0.609690	+	0.792640i	$0.291294\pi$
$660$	0	0
$661$	−13.0407	−0.507224	−0.253612	−	0.967306i	$-0.581619\pi$
$661$	−13.0407	−0.507224	−0.253612	+	0.967306i	$0.581619\pi$
$662$	10.9694	0.426338
$663$	0	0
$664$	7.56025	0.293395
$665$	0.475943	0.0184563
$666$	0	0
$667$	−0.0895622	−0.00346786
$668$	54.4497	2.10672
$669$	0	0
$670$	−16.7007	−0.645204
$671$	−61.9874	−2.39300
$672$	0	0
$673$	−12.2791	−0.473323	−0.236661	−	0.971592i	$-0.576053\pi$
$673$	−12.2791	−0.473323	−0.236661	+	0.971592i	$0.576053\pi$
$674$	11.3766	0.438209
$675$	0	0
$676$	3.99411	0.153619
$677$	−36.2842	−1.39452	−0.697258	−	0.716820i	$-0.745597\pi$
$677$	−36.2842	−1.39452	−0.697258	+	0.716820i	$0.745597\pi$
$678$	0	0
$679$	−20.7726	−0.797179
$680$	−38.9080	−1.49205
$681$	0	0
$682$	−75.2342	−2.88086
$683$	25.2755	0.967141	0.483571	−	0.875305i	$-0.339340\pi$
$683$	25.2755	0.967141	0.483571	+	0.875305i	$0.339340\pi$
$684$	0	0
$685$	−7.32899	−0.280026
$686$	14.2610	0.544487
$687$	0	0
$688$	30.9113	1.17848
$689$	−11.1782	−0.425855
$690$	0	0
$691$	−48.6756	−1.85171	−0.925854	−	0.377881i	$-0.876653\pi$
$691$	−48.6756	−1.85171	−0.925854	+	0.377881i	$0.876653\pi$
$692$	−13.8387	−0.526068
$693$	0	0
$694$	−10.8342	−0.411262
$695$	−16.8547	−0.639336
$696$	0	0
$697$	−0.725326	−0.0274737
$698$	0.0330888	0.00125243
$699$	0	0
$700$	−15.7149	−0.593967
$701$	−22.9449	−0.866615	−0.433308	−	0.901246i	$-0.642654\pi$
$701$	−22.9449	−0.866615	−0.433308	+	0.901246i	$0.642654\pi$
$702$	0	0
$703$	0.899015	0.0339070
$704$	37.7121	1.42133
$705$	0	0
$706$	72.7318	2.73730
$707$	−1.40498	−0.0528397
$708$	0	0
$709$	−22.0143	−0.826765	−0.413382	−	0.910558i	$-0.635653\pi$
$709$	−22.0143	−0.826765	−0.413382	+	0.910558i	$0.635653\pi$
$710$	−15.4840	−0.581104
$711$	0	0
$712$	−17.5512	−0.657761
$713$	2.15800	0.0808176
$714$	0	0
$715$	4.67286	0.174755
$716$	27.1894	1.01612
$717$	0	0
$718$	87.9290	3.28148
$719$	−35.7966	−1.33499	−0.667494	−	0.744615i	$-0.732633\pi$
$719$	−35.7966	−1.33499	−0.667494	+	0.744615i	$0.732633\pi$
$720$	0	0
$721$	18.1478	0.675859
$722$	46.4816	1.72987
$723$	0	0
$724$	−32.9003	−1.22273
$725$	0.272925	0.0101362
$726$	0	0
$727$	−31.1374	−1.15482	−0.577410	−	0.816454i	$-0.695937\pi$
$727$	−31.1374	−1.15482	−0.577410	+	0.816454i	$0.695937\pi$
$728$	−19.2089	−0.711929
$729$	0	0
$730$	18.0450	0.667877
$731$	62.1354	2.29816
$732$	0	0
$733$	−23.3211	−0.861384	−0.430692	−	0.902499i	$-0.641730\pi$
$733$	−23.3211	−0.861384	−0.430692	+	0.902499i	$0.641730\pi$
$734$	−7.24624	−0.267464
$735$	0	0
$736$	−0.0189140	−0.000697178 0
$737$	−31.8754	−1.17415
$738$	0	0
$739$	−23.4244	−0.861682	−0.430841	−	0.902428i	$-0.641783\pi$
$739$	−23.4244	−0.861682	−0.430841	+	0.902428i	$0.641783\pi$
$740$	−29.6841	−1.09121
$741$	0	0
$742$	107.678	3.95297
$743$	−39.6720	−1.45543	−0.727713	−	0.685882i	$-0.759416\pi$
$743$	−39.6720	−1.45543	−0.727713	+	0.685882i	$0.759416\pi$
$744$	0	0
$745$	8.22079	0.301186
$746$	−11.1750	−0.409145
$747$	0	0
$748$	−148.741	−5.43852
$749$	5.82320	0.212775
$750$	0	0
$751$	−21.1673	−0.772405	−0.386203	−	0.922414i	$-0.626213\pi$
$751$	−21.1673	−0.772405	−0.386203	+	0.922414i	$0.626213\pi$
$752$	45.8948	1.67361
$753$	0	0
$754$	0.668199	0.0243344
$755$	15.4012	0.560506
$756$	0	0
$757$	−6.75801	−0.245624	−0.122812	−	0.992430i	$-0.539191\pi$
$757$	−6.75801	−0.245624	−0.122812	+	0.992430i	$0.539191\pi$
$758$	56.2744	2.04398
$759$	0	0
$760$	0.590572	0.0214223
$761$	38.3625	1.39064	0.695320	−	0.718700i	$-0.255263\pi$
$761$	38.3625	1.39064	0.695320	+	0.718700i	$0.255263\pi$
$762$	0	0
$763$	57.7667	2.09129
$764$	−70.8031	−2.56157
$765$	0	0
$766$	22.5375	0.814313
$767$	5.55560	0.200601
$768$	0	0
$769$	−17.0162	−0.613621	−0.306811	−	0.951771i	$-0.599262\pi$
$769$	−17.0162	−0.613621	−0.306811	+	0.951771i	$0.599262\pi$
$770$	−45.0129	−1.62215
$771$	0	0
$772$	46.3450	1.66799
$773$	−21.8081	−0.784382	−0.392191	−	0.919884i	$-0.628283\pi$
$773$	−21.8081	−0.784382	−0.392191	+	0.919884i	$0.628283\pi$
$774$	0	0
$775$	−6.57612	−0.236221
$776$	−25.7756	−0.925291
$777$	0	0
$778$	−63.7146	−2.28428
$779$	0.0110095	0.000394456 0
$780$	0	0
$781$	−29.5531	−1.05749
$782$	6.40283	0.228965
$783$	0	0
$784$	33.6222	1.20079
$785$	−20.1799	−0.720250
$786$	0	0
$787$	−4.82600	−0.172028	−0.0860141	−	0.996294i	$-0.527413\pi$
$787$	−4.82600	−0.172028	−0.0860141	+	0.996294i	$0.527413\pi$
$788$	−8.00293	−0.285093
$789$	0	0
$790$	29.1775	1.03809
$791$	27.8320	0.989592
$792$	0	0
$793$	−13.2654	−0.471068
$794$	78.4480	2.78401
$795$	0	0
$796$	44.9626	1.59366
$797$	−25.1459	−0.890713	−0.445357	−	0.895353i	$-0.646923\pi$
$797$	−25.1459	−0.890713	−0.445357	+	0.895353i	$0.646923\pi$
$798$	0	0
$799$	92.2541	3.26371
$800$	0.0576370	0.00203778
$801$	0	0
$802$	2.01883	0.0712874
$803$	34.4412	1.21540
$804$	0	0
$805$	1.29114	0.0455067
$806$	−16.1002	−0.567106
$807$	0	0
$808$	−1.74337	−0.0613315
$809$	51.2155	1.80064	0.900320	−	0.435228i	$-0.143332\pi$
$809$	51.2155	1.80064	0.900320	+	0.435228i	$0.143332\pi$
$810$	0	0
$811$	25.7297	0.903491	0.451745	−	0.892147i	$-0.350802\pi$
$811$	25.7297	0.903491	0.451745	+	0.892147i	$0.350802\pi$
$812$	−4.28899	−0.150514
$813$	0	0
$814$	−85.0256	−2.98015
$815$	8.27259	0.289776
$816$	0	0
$817$	−0.943133	−0.0329961
$818$	−69.0887	−2.41563
$819$	0	0
$820$	−0.363516	−0.0126945
$821$	−21.2596	−0.741966	−0.370983	−	0.928640i	$-0.620979\pi$
$821$	−21.2596	−0.741966	−0.370983	+	0.928640i	$0.620979\pi$
$822$	0	0
$823$	−12.3955	−0.432080	−0.216040	−	0.976385i	$-0.569314\pi$
$823$	−12.3955	−0.432080	−0.216040	+	0.976385i	$0.569314\pi$
$824$	22.5186	0.784474
$825$	0	0
$826$	−53.5162	−1.86207
$827$	41.1108	1.42956	0.714781	−	0.699349i	$-0.246527\pi$
$827$	41.1108	1.42956	0.714781	+	0.699349i	$0.246527\pi$
$828$	0	0
$829$	−28.6599	−0.995400	−0.497700	−	0.867349i	$-0.665822\pi$
$829$	−28.6599	−0.995400	−0.497700	+	0.867349i	$0.665822\pi$
$830$	3.79130	0.131598
$831$	0	0
$832$	8.07045	0.279793
$833$	67.5846	2.34167
$834$	0	0
$835$	13.6325	0.471773
$836$	2.25770	0.0780840
$837$	0	0
$838$	60.7723	2.09934
$839$	34.5390	1.19242	0.596209	−	0.802829i	$-0.296673\pi$
$839$	34.5390	1.19242	0.596209	+	0.802829i	$0.296673\pi$
$840$	0	0
$841$	−28.9255	−0.997431
$842$	−80.3196	−2.76800
$843$	0	0
$844$	−19.4277	−0.668730
$845$	1.00000	0.0344010
$846$	0	0
$847$	−42.6331	−1.46489
$848$	44.3178	1.52188
$849$	0	0
$850$	−19.5115	−0.669239
$851$	2.43885	0.0836028
$852$	0	0
$853$	−28.4337	−0.973550	−0.486775	−	0.873527i	$-0.661827\pi$
$853$	−28.4337	−0.973550	−0.486775	+	0.873527i	$0.661827\pi$
$854$	127.783	4.37266
$855$	0	0
$856$	7.22571	0.246970
$857$	−21.7514	−0.743013	−0.371507	−	0.928430i	$-0.621159\pi$
$857$	−21.7514	−0.743013	−0.371507	+	0.928430i	$0.621159\pi$
$858$	0	0
$859$	−3.86541	−0.131886	−0.0659431	−	0.997823i	$-0.521006\pi$
$859$	−3.86541	−0.131886	−0.0659431	+	0.997823i	$0.521006\pi$
$860$	31.1408	1.06189
$861$	0	0
$862$	10.2186	0.348046
$863$	−10.1359	−0.345030	−0.172515	−	0.985007i	$-0.555189\pi$
$863$	−10.1359	−0.345030	−0.172515	+	0.985007i	$0.555189\pi$
$864$	0	0
$865$	−3.46478	−0.117806
$866$	−49.9039	−1.69580
$867$	0	0
$868$	103.343	3.50769
$869$	55.6890	1.88912
$870$	0	0
$871$	−6.82138	−0.231134
$872$	71.6796	2.42738
$873$	0	0
$874$	−0.0971865	−0.00328738
$875$	−3.93452	−0.133011
$876$	0	0
$877$	19.8811	0.671338	0.335669	−	0.941980i	$-0.391038\pi$
$877$	19.8811	0.671338	0.335669	+	0.941980i	$0.391038\pi$
$878$	−44.6146	−1.50567
$879$	0	0
$880$	−18.5264	−0.624524
$881$	21.4750	0.723511	0.361755	−	0.932273i	$-0.382178\pi$
$881$	21.4750	0.723511	0.361755	+	0.932273i	$0.382178\pi$
$882$	0	0
$883$	−32.0639	−1.07904	−0.539519	−	0.841974i	$-0.681394\pi$
$883$	−32.0639	−1.07904	−0.539519	+	0.841974i	$0.681394\pi$
$884$	−31.8309	−1.07059
$885$	0	0
$886$	17.0212	0.571839
$887$	7.55516	0.253677	0.126839	−	0.991923i	$-0.459517\pi$
$887$	7.55516	0.253677	0.126839	+	0.991923i	$0.459517\pi$
$888$	0	0
$889$	−4.86759	−0.163254
$890$	−8.80156	−0.295029
$891$	0	0
$892$	−56.2465	−1.88327
$893$	−1.40029	−0.0468591
$894$	0	0
$895$	6.80739	0.227546
$896$	−77.2878	−2.58200
$897$	0	0
$898$	33.4971	1.11781
$899$	−1.79479	−0.0598595
$900$	0	0
$901$	89.0841	2.96782
$902$	−1.04124	−0.0346694
$903$	0	0
$904$	34.5353	1.14863
$905$	−8.23721	−0.273814
$906$	0	0
$907$	39.0898	1.29795	0.648977	−	0.760808i	$-0.275197\pi$
$907$	39.0898	1.29795	0.648977	+	0.760808i	$0.275197\pi$
$908$	31.8064	1.05553
$909$	0	0
$910$	−9.63283	−0.319325
$911$	35.8503	1.18777	0.593887	−	0.804549i	$-0.297593\pi$
$911$	35.8503	1.18777	0.593887	+	0.804549i	$0.297593\pi$
$912$	0	0
$913$	7.23617	0.239482
$914$	−41.2259	−1.36363
$915$	0	0
$916$	54.4337	1.79854
$917$	50.1760	1.65696
$918$	0	0
$919$	−20.3071	−0.669869	−0.334934	−	0.942241i	$-0.608714\pi$
$919$	−20.3071	−0.669869	−0.334934	+	0.942241i	$0.608714\pi$
$920$	1.60211	0.0528199
$921$	0	0
$922$	−96.3913	−3.17448
$923$	−6.32442	−0.208171
$924$	0	0
$925$	−7.43198	−0.244362
$926$	7.05940	0.231986
$927$	0	0
$928$	0.0157306	0.000516382 0
$929$	3.81213	0.125072	0.0625359	−	0.998043i	$-0.480081\pi$
$929$	3.81213	0.125072	0.0625359	+	0.998043i	$0.480081\pi$
$930$	0	0
$931$	−1.02585	−0.0336207
$932$	−27.0309	−0.885426
$933$	0	0
$934$	61.5201	2.01300
$935$	−37.2402	−1.21788
$936$	0	0
$937$	−0.408946	−0.0133597	−0.00667983	−	0.999978i	$-0.502126\pi$
$937$	−0.408946	−0.0133597	−0.00667983	+	0.999978i	$0.502126\pi$
$938$	65.7093	2.14548
$939$	0	0
$940$	46.2356	1.50804
$941$	12.5875	0.410340	0.205170	−	0.978726i	$-0.434225\pi$
$941$	12.5875	0.410340	0.205170	+	0.978726i	$0.434225\pi$
$942$	0	0
$943$	0.0298666	0.000972590 0
$944$	−22.0261	−0.716889
$945$	0	0
$946$	89.1981	2.90008
$947$	35.8294	1.16430	0.582149	−	0.813082i	$-0.302212\pi$
$947$	35.8294	1.16430	0.582149	+	0.813082i	$0.302212\pi$
$948$	0	0
$949$	7.37048	0.239256
$950$	0.296159	0.00960867
$951$	0	0
$952$	153.084	4.96150
$953$	−6.87652	−0.222752	−0.111376	−	0.993778i	$-0.535526\pi$
$953$	−6.87652	−0.222752	−0.111376	+	0.993778i	$0.535526\pi$
$954$	0	0
$955$	−17.7269	−0.573629
$956$	−101.720	−3.28987
$957$	0	0
$958$	52.5545	1.69796
$959$	28.8361	0.931165
$960$	0	0
$961$	12.2453	0.395011
$962$	−18.1956	−0.586650
$963$	0	0
$964$	−80.5313	−2.59374
$965$	11.6033	0.373525
$966$	0	0
$967$	−16.0530	−0.516230	−0.258115	−	0.966114i	$-0.583101\pi$
$967$	−16.0530	−0.516230	−0.258115	+	0.966114i	$0.583101\pi$
$968$	−52.9012	−1.70031
$969$	0	0
$970$	−12.9259	−0.415026
$971$	29.7306	0.954100	0.477050	−	0.878876i	$-0.341706\pi$
$971$	29.7306	0.954100	0.477050	+	0.878876i	$0.341706\pi$
$972$	0	0
$973$	66.3153	2.12597
$974$	78.4986	2.51526
$975$	0	0
$976$	52.5929	1.68346
$977$	−25.2997	−0.809408	−0.404704	−	0.914448i	$-0.632626\pi$
$977$	−25.2997	−0.809408	−0.404704	+	0.914448i	$0.632626\pi$
$978$	0	0
$979$	−16.7989	−0.536895
$980$	33.8718	1.08200
$981$	0	0
$982$	−23.8463	−0.760966
$983$	26.3608	0.840778	0.420389	−	0.907344i	$-0.361894\pi$
$983$	26.3608	0.840778	0.420389	+	0.907344i	$0.361894\pi$
$984$	0	0
$985$	−2.00369	−0.0638427
$986$	−5.32518	−0.169588
$987$	0	0
$988$	0.483150	0.0153711
$989$	−2.55854	−0.0813567
$990$	0	0
$991$	−47.5111	−1.50924	−0.754621	−	0.656161i	$-0.772179\pi$
$991$	−47.5111	−1.50924	−0.754621	+	0.656161i	$0.772179\pi$
$992$	−0.379028	−0.0120341
$993$	0	0
$994$	60.9221	1.93233
$995$	11.2572	0.356878
$996$	0	0
$997$	−38.7083	−1.22591	−0.612953	−	0.790119i	$-0.710018\pi$
$997$	−38.7083	−1.22591	−0.612953	+	0.790119i	$0.710018\pi$
$998$	−77.7863	−2.46228
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.ba.1.1		8
3.2	odd	2		5265.2.a.bf.1.8		8
9.2	odd	6		585.2.i.e.391.1	yes	16
9.4	even	3		1755.2.i.f.586.8		16
9.5	odd	6		585.2.i.e.196.1	✓	16
9.7	even	3		1755.2.i.f.1171.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.e.196.1	✓	16	9.5	odd	6
585.2.i.e.391.1	yes	16	9.2	odd	6
1755.2.i.f.586.8		16	9.4	even	3
1755.2.i.f.1171.8		16	9.7	even	3
5265.2.a.ba.1.1		8	1.1	even	1	trivial
5265.2.a.bf.1.8		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44829 q^{2} +3.99411 q^{4} +1.00000 q^{5} -3.93452 q^{7} -4.88214 q^{8} +O(q^{10})\)	\(q-2.44829 q^{2} +3.99411 q^{4} +1.00000 q^{5} -3.93452 q^{7} -4.88214 q^{8} -2.44829 q^{10} -4.67286 q^{11} -1.00000 q^{13} +9.63283 q^{14} +3.96467 q^{16} +7.96946 q^{17} -0.120966 q^{19} +3.99411 q^{20} +11.4405 q^{22} -0.328157 q^{23} +1.00000 q^{25} +2.44829 q^{26} -15.7149 q^{28} +0.272925 q^{29} -6.57612 q^{31} +0.0576370 q^{32} -19.5115 q^{34} -3.93452 q^{35} -7.43198 q^{37} +0.296159 q^{38} -4.88214 q^{40} -0.0910132 q^{41} +7.79669 q^{43} -18.6639 q^{44} +0.803421 q^{46} +11.5760 q^{47} +8.48046 q^{49} -2.44829 q^{50} -3.99411 q^{52} +11.1782 q^{53} -4.67286 q^{55} +19.2089 q^{56} -0.668199 q^{58} -5.55560 q^{59} +13.2654 q^{61} +16.1002 q^{62} -8.07045 q^{64} -1.00000 q^{65} +6.82138 q^{67} +31.8309 q^{68} +9.63283 q^{70} +6.32442 q^{71} -7.37048 q^{73} +18.1956 q^{74} -0.483150 q^{76} +18.3855 q^{77} -11.9175 q^{79} +3.96467 q^{80} +0.222826 q^{82} -1.54855 q^{83} +7.96946 q^{85} -19.0885 q^{86} +22.8136 q^{88} +3.59499 q^{89} +3.93452 q^{91} -1.31069 q^{92} -28.3412 q^{94} -0.120966 q^{95} +5.27957 q^{97} -20.7626 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8}+O(q^{10}) \) 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8	\( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8} - 3 q^{10} - 6 q^{11} - 8 q^{13} - 10 q^{14} + 11 q^{16} + 2 q^{17} - 10 q^{19} + 9 q^{20} + 3 q^{22} - 6 q^{23} + 8 q^{25} + 3 q^{26} - 34 q^{28} - 14 q^{29} - 31 q^{31} - q^{32} - 7 q^{34} - 11 q^{35} + q^{37} - 9 q^{38} + 6 q^{40} + 12 q^{41} + 15 q^{43} - 16 q^{44} - 32 q^{46} + 18 q^{47} + 17 q^{49} - 3 q^{50} - 9 q^{52} - 2 q^{53} - 6 q^{55} - 16 q^{56} - 42 q^{58} - 24 q^{59} - 9 q^{61} + 20 q^{62} - 30 q^{64} - 8 q^{65} - 18 q^{67} + 14 q^{68} - 10 q^{70} - 10 q^{71} + 6 q^{73} + 37 q^{74} - 53 q^{76} + 34 q^{77} - 3 q^{79} + 11 q^{80} - 34 q^{82} + 10 q^{83} + 2 q^{85} - 60 q^{86} - 14 q^{88} + 13 q^{89} + 11 q^{91} - 5 q^{92} + 17 q^{94} - 10 q^{95} - 34 q^{97} + 30 q^{98}+O(q^{100}) \) 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8 - 3 * q^10 - 6 * q^11 - 8 * q^13 - 10 * q^14 + 11 * q^16 + 2 * q^17 - 10 * q^19 + 9 * q^20 + 3 * q^22 - 6 * q^23 + 8 * q^25 + 3 * q^26 - 34 * q^28 - 14 * q^29 - 31 * q^31 - q^32 - 7 * q^34 - 11 * q^35 + q^37 - 9 * q^38 + 6 * q^40 + 12 * q^41 + 15 * q^43 - 16 * q^44 - 32 * q^46 + 18 * q^47 + 17 * q^49 - 3 * q^50 - 9 * q^52 - 2 * q^53 - 6 * q^55 - 16 * q^56 - 42 * q^58 - 24 * q^59 - 9 * q^61 + 20 * q^62 - 30 * q^64 - 8 * q^65 - 18 * q^67 + 14 * q^68 - 10 * q^70 - 10 * q^71 + 6 * q^73 + 37 * q^74 - 53 * q^76 + 34 * q^77 - 3 * q^79 + 11 * q^80 - 34 * q^82 + 10 * q^83 + 2 * q^85 - 60 * q^86 - 14 * q^88 + 13 * q^89 + 11 * q^91 - 5 * q^92 + 17 * q^94 - 10 * q^95 - 34 * q^97 + 30 * q^98

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