LMFDB - Embedded newform 5265.2.a.bb.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} - x^{7} - 10x^{6} + 7x^{5} + 33x^{4} - 14x^{3} - 38x^{2} + 7x + 9 $ x^8 - x^7 - 10x^6 + 7x^5 + 33x^4 - 14x^3 - 38x^2 + 7x + 9
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			$2.42368$ 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)

$f(q)$	$=$	$q-2.42368 q^{2} +3.87423 q^{4} +1.00000 q^{5} -1.81135 q^{7} -4.54253 q^{8} +O(q^{10})$	\(q-2.42368 q^{2} +3.87423 q^{4} +1.00000 q^{5} -1.81135 q^{7} -4.54253 q^{8} -2.42368 q^{10} +0.734130 q^{11} +1.00000 q^{13} +4.39013 q^{14} +3.26118 q^{16} +4.77113 q^{17} -4.07121 q^{19} +3.87423 q^{20} -1.77930 q^{22} +0.673190 q^{23} +1.00000 q^{25} -2.42368 q^{26} -7.01758 q^{28} -2.50666 q^{29} -6.55662 q^{31} +1.18099 q^{32} -11.5637 q^{34} -1.81135 q^{35} +6.08491 q^{37} +9.86731 q^{38} -4.54253 q^{40} +3.12077 q^{41} -8.92060 q^{43} +2.84418 q^{44} -1.63160 q^{46} +3.57839 q^{47} -3.71901 q^{49} -2.42368 q^{50} +3.87423 q^{52} -12.8213 q^{53} +0.734130 q^{55} +8.22810 q^{56} +6.07534 q^{58} +13.3288 q^{59} -4.84499 q^{61} +15.8911 q^{62} -9.38471 q^{64} +1.00000 q^{65} -2.87305 q^{67} +18.4844 q^{68} +4.39013 q^{70} -9.86873 q^{71} +11.8648 q^{73} -14.7479 q^{74} -15.7728 q^{76} -1.32976 q^{77} -4.11201 q^{79} +3.26118 q^{80} -7.56375 q^{82} +12.7150 q^{83} +4.77113 q^{85} +21.6207 q^{86} -3.33480 q^{88} +5.47463 q^{89} -1.81135 q^{91} +2.60809 q^{92} -8.67286 q^{94} -4.07121 q^{95} -1.70276 q^{97} +9.01370 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 5 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 8 * q - q^2 + 5 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8	$ 8 q - q^{2} + 5 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8} - q^{10} - 9 q^{11} + 8 q^{13} + 3 q^{14} - 13 q^{16} + 6 q^{17} - 11 q^{19} + 5 q^{20} - 4 q^{22} - 3 q^{23} + 8 q^{25} - q^{26} - 13 q^{28} - 8 q^{29} - 18 q^{31} - 3 q^{32} - 9 q^{34} - 6 q^{35} - 18 q^{37} + 8 q^{38} - 6 q^{40} + 17 q^{41} - 17 q^{43} + 5 q^{44} + 3 q^{46} - 11 q^{47} - 16 q^{49} - q^{50} + 5 q^{52} + 10 q^{53} - 9 q^{55} + q^{56} - 10 q^{58} - 7 q^{59} - 21 q^{61} - 29 q^{62} - 10 q^{64} + 8 q^{65} - 13 q^{67} + 16 q^{68} + 3 q^{70} - 34 q^{71} - 16 q^{73} + 4 q^{74} - 2 q^{76} - 18 q^{77} - 37 q^{79} - 13 q^{80} + q^{82} - 3 q^{83} + 6 q^{85} + 2 q^{86} - 19 q^{88} + 14 q^{89} - 6 q^{91} + 14 q^{92} - 44 q^{94} - 11 q^{95} - 17 q^{97} + 45 q^{98}+O(q^{100}) $ 8 * q - q^2 + 5 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8 - q^10 - 9 * q^11 + 8 * q^13 + 3 * q^14 - 13 * q^16 + 6 * q^17 - 11 * q^19 + 5 * q^20 - 4 * q^22 - 3 * q^23 + 8 * q^25 - q^26 - 13 * q^28 - 8 * q^29 - 18 * q^31 - 3 * q^32 - 9 * q^34 - 6 * q^35 - 18 * q^37 + 8 * q^38 - 6 * q^40 + 17 * q^41 - 17 * q^43 + 5 * q^44 + 3 * q^46 - 11 * q^47 - 16 * q^49 - q^50 + 5 * q^52 + 10 * q^53 - 9 * q^55 + q^56 - 10 * q^58 - 7 * q^59 - 21 * q^61 - 29 * q^62 - 10 * q^64 + 8 * q^65 - 13 * q^67 + 16 * q^68 + 3 * q^70 - 34 * q^71 - 16 * q^73 + 4 * q^74 - 2 * q^76 - 18 * q^77 - 37 * q^79 - 13 * q^80 + q^82 - 3 * q^83 + 6 * q^85 + 2 * q^86 - 19 * q^88 + 14 * q^89 - 6 * q^91 + 14 * q^92 - 44 * q^94 - 11 * q^95 - 17 * q^97 + 45 * 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 $

$2$	−2.42368	−1.71380	−0.856900	−	0.515482i	$-0.827613\pi$
$2$	−2.42368	−1.71380	−0.856900	+	0.515482i	$0.827613\pi$
$3$	0	0
$3$	0	0
$4$	3.87423	1.93711
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.81135	−0.684626	−0.342313	−	0.939586i	$-0.611210\pi$
$7$	−1.81135	−0.684626	−0.342313	+	0.939586i	$0.611210\pi$
$8$	−4.54253	−1.60603
$9$	0	0
$10$	−2.42368	−0.766435
$11$	0.734130	0.221348	0.110674	−	0.993857i	$-0.464699\pi$
$11$	0.734130	0.221348	0.110674	+	0.993857i	$0.464699\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	4.39013	1.17331
$15$	0	0
$16$	3.26118	0.815296
$17$	4.77113	1.15717	0.578584	−	0.815623i	$-0.303605\pi$
$17$	4.77113	1.15717	0.578584	+	0.815623i	$0.303605\pi$
$18$	0	0
$19$	−4.07121	−0.934000	−0.467000	−	0.884257i	$-0.654665\pi$
$19$	−4.07121	−0.934000	−0.467000	+	0.884257i	$0.654665\pi$
$20$	3.87423	0.866304
$21$	0	0
$22$	−1.77930	−0.379347
$23$	0.673190	0.140370	0.0701849	−	0.997534i	$-0.477641\pi$
$23$	0.673190	0.140370	0.0701849	+	0.997534i	$0.477641\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.42368	−0.475323
$27$	0	0
$28$	−7.01758	−1.32620
$29$	−2.50666	−0.465475	−0.232737	−	0.972540i	$-0.574768\pi$
$29$	−2.50666	−0.465475	−0.232737	+	0.972540i	$0.574768\pi$
$30$	0	0
$31$	−6.55662	−1.17760	−0.588802	−	0.808278i	$-0.700400\pi$
$31$	−6.55662	−1.17760	−0.588802	+	0.808278i	$0.700400\pi$
$32$	1.18099	0.208772
$33$	0	0
$34$	−11.5637	−1.98316
$35$	−1.81135	−0.306174
$36$	0	0
$37$	6.08491	1.00035	0.500176	−	0.865924i	$-0.333269\pi$
$37$	6.08491	1.00035	0.500176	+	0.865924i	$0.333269\pi$
$38$	9.86731	1.60069
$39$	0	0
$40$	−4.54253	−0.718237
$41$	3.12077	0.487383	0.243691	−	0.969853i	$-0.421642\pi$
$41$	3.12077	0.487383	0.243691	+	0.969853i	$0.421642\pi$
$42$	0	0
$43$	−8.92060	−1.36038	−0.680189	−	0.733037i	$-0.738102\pi$
$43$	−8.92060	−1.36038	−0.680189	+	0.733037i	$0.738102\pi$
$44$	2.84418	0.428777
$45$	0	0
$46$	−1.63160	−0.240566
$47$	3.57839	0.521961	0.260981	−	0.965344i	$-0.415954\pi$
$47$	3.57839	0.521961	0.260981	+	0.965344i	$0.415954\pi$
$48$	0	0
$49$	−3.71901	−0.531288
$50$	−2.42368	−0.342760
$51$	0	0
$52$	3.87423	0.537259
$53$	−12.8213	−1.76114	−0.880570	−	0.473917i	$-0.842840\pi$
$53$	−12.8213	−1.76114	−0.880570	+	0.473917i	$0.842840\pi$
$54$	0	0
$55$	0.734130	0.0989900
$56$	8.22810	1.09953
$57$	0	0
$58$	6.07534	0.797731
$59$	13.3288	1.73527	0.867633	−	0.497205i	$-0.165640\pi$
$59$	13.3288	1.73527	0.867633	+	0.497205i	$0.165640\pi$
$60$	0	0
$61$	−4.84499	−0.620338	−0.310169	−	0.950681i	$-0.600386\pi$
$61$	−4.84499	−0.620338	−0.310169	+	0.950681i	$0.600386\pi$
$62$	15.8911	2.01818
$63$	0	0
$64$	−9.38471	−1.17309
$65$	1.00000	0.124035
$66$	0	0
$67$	−2.87305	−0.350998	−0.175499	−	0.984480i	$-0.556154\pi$
$67$	−2.87305	−0.350998	−0.175499	+	0.984480i	$0.556154\pi$
$68$	18.4844	2.24157
$69$	0	0
$70$	4.39013	0.524721
$71$	−9.86873	−1.17120	−0.585601	−	0.810599i	$-0.699141\pi$
$71$	−9.86873	−1.17120	−0.585601	+	0.810599i	$0.699141\pi$
$72$	0	0
$73$	11.8648	1.38866	0.694332	−	0.719655i	$-0.255700\pi$
$73$	11.8648	1.38866	0.694332	+	0.719655i	$0.255700\pi$
$74$	−14.7479	−1.71441
$75$	0	0
$76$	−15.7728	−1.80926
$77$	−1.32976	−0.151541
$78$	0	0
$79$	−4.11201	−0.462637	−0.231319	−	0.972878i	$-0.574304\pi$
$79$	−4.11201	−0.462637	−0.231319	+	0.972878i	$0.574304\pi$
$80$	3.26118	0.364611
$81$	0	0
$82$	−7.56375	−0.835277
$83$	12.7150	1.39565	0.697826	−	0.716267i	$-0.254151\pi$
$83$	12.7150	1.39565	0.697826	+	0.716267i	$0.254151\pi$
$84$	0	0
$85$	4.77113	0.517501
$86$	21.6207	2.33142
$87$	0	0
$88$	−3.33480	−0.355491
$89$	5.47463	0.580309	0.290155	−	0.956980i	$-0.406293\pi$
$89$	5.47463	0.580309	0.290155	+	0.956980i	$0.406293\pi$
$90$	0	0
$91$	−1.81135	−0.189881
$92$	2.60809	0.271912
$93$	0	0
$94$	−8.67286	−0.894538
$95$	−4.07121	−0.417697
$96$	0	0
$97$	−1.70276	−0.172889	−0.0864444	−	0.996257i	$-0.527550\pi$
$97$	−1.70276	−0.172889	−0.0864444	+	0.996257i	$0.527550\pi$
$98$	9.01370	0.910521
$99$	0	0
$100$	3.87423	0.387423
$101$	−15.2915	−1.52156	−0.760781	−	0.649009i	$-0.775184\pi$
$101$	−15.2915	−1.52156	−0.760781	+	0.649009i	$0.775184\pi$
$102$	0	0
$103$	−2.36751	−0.233278	−0.116639	−	0.993174i	$-0.537212\pi$
$103$	−2.36751	−0.233278	−0.116639	+	0.993174i	$0.537212\pi$
$104$	−4.54253	−0.445432
$105$	0	0
$106$	31.0747	3.01824
$107$	1.75602	0.169761	0.0848806	−	0.996391i	$-0.472949\pi$
$107$	1.75602	0.169761	0.0848806	+	0.996391i	$0.472949\pi$
$108$	0	0
$109$	7.37960	0.706838	0.353419	−	0.935465i	$-0.385019\pi$
$109$	7.37960	0.706838	0.353419	+	0.935465i	$0.385019\pi$
$110$	−1.77930	−0.169649
$111$	0	0
$112$	−5.90714	−0.558172
$113$	4.46249	0.419796	0.209898	−	0.977723i	$-0.432687\pi$
$113$	4.46249	0.419796	0.209898	+	0.977723i	$0.432687\pi$
$114$	0	0
$115$	0.673190	0.0627753
$116$	−9.71136	−0.901677
$117$	0	0
$118$	−32.3048	−2.97390
$119$	−8.64218	−0.792227
$120$	0	0
$121$	−10.4611	−0.951005
$122$	11.7427	1.06314
$123$	0	0
$124$	−25.4018	−2.28115
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.28055	−0.379837	−0.189919	−	0.981800i	$-0.560822\pi$
$127$	−4.28055	−0.379837	−0.189919	+	0.981800i	$0.560822\pi$
$128$	20.3836	1.80167
$129$	0	0
$130$	−2.42368	−0.212571
$131$	7.24187	0.632725	0.316363	−	0.948638i	$-0.397538\pi$
$131$	7.24187	0.632725	0.316363	+	0.948638i	$0.397538\pi$
$132$	0	0
$133$	7.37438	0.639440
$134$	6.96335	0.601542
$135$	0	0
$136$	−21.6730	−1.85844
$137$	−9.68085	−0.827091	−0.413546	−	0.910483i	$-0.635710\pi$
$137$	−9.68085	−0.827091	−0.413546	+	0.910483i	$0.635710\pi$
$138$	0	0
$139$	−11.7899	−1.00000	−0.500001	−	0.866025i	$-0.666667\pi$
$139$	−11.7899	−1.00000	−0.500001	+	0.866025i	$0.666667\pi$
$140$	−7.01758	−0.593094
$141$	0	0
$142$	23.9186	2.00721
$143$	0.734130	0.0613910
$144$	0	0
$145$	−2.50666	−0.208167
$146$	−28.7564	−2.37989
$147$	0	0
$148$	23.5743	1.93780
$149$	−18.9480	−1.55228	−0.776142	−	0.630558i	$-0.782826\pi$
$149$	−18.9480	−1.55228	−0.776142	+	0.630558i	$0.782826\pi$
$150$	0	0
$151$	−18.2346	−1.48391	−0.741954	−	0.670450i	$-0.766101\pi$
$151$	−18.2346	−1.48391	−0.741954	+	0.670450i	$0.766101\pi$
$152$	18.4936	1.50003
$153$	0	0
$154$	3.22293	0.259711
$155$	−6.55662	−0.526640
$156$	0	0
$157$	−8.73613	−0.697219	−0.348610	−	0.937268i	$-0.613346\pi$
$157$	−8.73613	−0.697219	−0.348610	+	0.937268i	$0.613346\pi$
$158$	9.96620	0.792868
$159$	0	0
$160$	1.18099	0.0933656
$161$	−1.21938	−0.0961008
$162$	0	0
$163$	3.41141	0.267202	0.133601	−	0.991035i	$-0.457346\pi$
$163$	3.41141	0.267202	0.133601	+	0.991035i	$0.457346\pi$
$164$	12.0906	0.944116
$165$	0	0
$166$	−30.8171	−2.39187
$167$	−19.5468	−1.51257	−0.756287	−	0.654240i	$-0.772989\pi$
$167$	−19.5468	−1.51257	−0.756287	+	0.654240i	$0.772989\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−11.5637	−0.886895
$171$	0	0
$172$	−34.5604	−2.63521
$173$	18.7294	1.42397	0.711984	−	0.702196i	$-0.247797\pi$
$173$	18.7294	1.42397	0.711984	+	0.702196i	$0.247797\pi$
$174$	0	0
$175$	−1.81135	−0.136925
$176$	2.39413	0.180464
$177$	0	0
$178$	−13.2687	−0.994535
$179$	−19.0809	−1.42617	−0.713087	−	0.701076i	$-0.752704\pi$
$179$	−19.0809	−1.42617	−0.713087	+	0.701076i	$0.752704\pi$
$180$	0	0
$181$	8.14795	0.605632	0.302816	−	0.953049i	$-0.402073\pi$
$181$	8.14795	0.605632	0.302816	+	0.953049i	$0.402073\pi$
$182$	4.39013	0.325418
$183$	0	0
$184$	−3.05798	−0.225438
$185$	6.08491	0.447371
$186$	0	0
$187$	3.50263	0.256137
$188$	13.8635	1.01110
$189$	0	0
$190$	9.86731	0.715850
$191$	−9.70009	−0.701874	−0.350937	−	0.936399i	$-0.614137\pi$
$191$	−9.70009	−0.701874	−0.350937	+	0.936399i	$0.614137\pi$
$192$	0	0
$193$	14.8986	1.07243	0.536214	−	0.844082i	$-0.319854\pi$
$193$	14.8986	1.07243	0.536214	+	0.844082i	$0.319854\pi$
$194$	4.12694	0.296297
$195$	0	0
$196$	−14.4083	−1.02916
$197$	17.3196	1.23397	0.616984	−	0.786976i	$-0.288354\pi$
$197$	17.3196	1.23397	0.616984	+	0.786976i	$0.288354\pi$
$198$	0	0
$199$	0.684429	0.0485179	0.0242589	−	0.999706i	$-0.492277\pi$
$199$	0.684429	0.0485179	0.0242589	+	0.999706i	$0.492277\pi$
$200$	−4.54253	−0.321205
$201$	0	0
$202$	37.0617	2.60765
$203$	4.54043	0.318676
$204$	0	0
$205$	3.12077	0.217964
$206$	5.73809	0.399792
$207$	0	0
$208$	3.26118	0.226122
$209$	−2.98880	−0.206739
$210$	0	0
$211$	8.10044	0.557657	0.278829	−	0.960341i	$-0.410054\pi$
$211$	8.10044	0.557657	0.278829	+	0.960341i	$0.410054\pi$
$212$	−49.6726	−3.41153
$213$	0	0
$214$	−4.25604	−0.290937
$215$	−8.92060	−0.608380
$216$	0	0
$217$	11.8763	0.806217
$218$	−17.8858	−1.21138
$219$	0	0
$220$	2.84418	0.191755
$221$	4.77113	0.320941
$222$	0	0
$223$	6.45699	0.432392	0.216196	−	0.976350i	$-0.430635\pi$
$223$	6.45699	0.432392	0.216196	+	0.976350i	$0.430635\pi$
$224$	−2.13919	−0.142931
$225$	0	0
$226$	−10.8157	−0.719447
$227$	−21.0494	−1.39710	−0.698548	−	0.715563i	$-0.746170\pi$
$227$	−21.0494	−1.39710	−0.698548	+	0.715563i	$0.746170\pi$
$228$	0	0
$229$	−10.3981	−0.687128	−0.343564	−	0.939129i	$-0.611634\pi$
$229$	−10.3981	−0.687128	−0.343564	+	0.939129i	$0.611634\pi$
$230$	−1.63160	−0.107584
$231$	0	0
$232$	11.3866	0.747564
$233$	28.4639	1.86473	0.932365	−	0.361517i	$-0.117741\pi$
$233$	28.4639	1.86473	0.932365	+	0.361517i	$0.117741\pi$
$234$	0	0
$235$	3.57839	0.233428
$236$	51.6389	3.36141
$237$	0	0
$238$	20.9459	1.35772
$239$	−12.0398	−0.778791	−0.389395	−	0.921071i	$-0.627316\pi$
$239$	−12.0398	−0.778791	−0.389395	+	0.921071i	$0.627316\pi$
$240$	0	0
$241$	25.6541	1.65253	0.826263	−	0.563284i	$-0.190462\pi$
$241$	25.6541	1.65253	0.826263	+	0.563284i	$0.190462\pi$
$242$	25.3543	1.62983
$243$	0	0
$244$	−18.7706	−1.20166
$245$	−3.71901	−0.237599
$246$	0	0
$247$	−4.07121	−0.259045
$248$	29.7836	1.89126
$249$	0	0
$250$	−2.42368	−0.153287
$251$	17.0690	1.07738	0.538692	−	0.842503i	$-0.318919\pi$
$251$	17.0690	1.07738	0.538692	+	0.842503i	$0.318919\pi$
$252$	0	0
$253$	0.494209	0.0310706
$254$	10.3747	0.650965
$255$	0	0
$256$	−30.6338	−1.91461
$257$	−10.5843	−0.660232	−0.330116	−	0.943940i	$-0.607088\pi$
$257$	−10.5843	−0.660232	−0.330116	+	0.943940i	$0.607088\pi$
$258$	0	0
$259$	−11.0219	−0.684867
$260$	3.87423	0.240269
$261$	0	0
$262$	−17.5520	−1.08437
$263$	−1.29606	−0.0799185	−0.0399593	−	0.999201i	$-0.512723\pi$
$263$	−1.29606	−0.0799185	−0.0399593	+	0.999201i	$0.512723\pi$
$264$	0	0
$265$	−12.8213	−0.787606
$266$	−17.8731	−1.09587
$267$	0	0
$268$	−11.1308	−0.679924
$269$	0.133747	0.00815471	0.00407736	−	0.999992i	$-0.498702\pi$
$269$	0.133747	0.00815471	0.00407736	+	0.999992i	$0.498702\pi$
$270$	0	0
$271$	20.2516	1.23020	0.615098	−	0.788451i	$-0.289116\pi$
$271$	20.2516	1.23020	0.615098	+	0.788451i	$0.289116\pi$
$272$	15.5595	0.943434
$273$	0	0
$274$	23.4633	1.41747
$275$	0.734130	0.0442697
$276$	0	0
$277$	−23.8416	−1.43250	−0.716252	−	0.697841i	$-0.754144\pi$
$277$	−23.8416	−1.43250	−0.716252	+	0.697841i	$0.754144\pi$
$278$	28.5749	1.71381
$279$	0	0
$280$	8.22810	0.491723
$281$	24.3388	1.45193	0.725966	−	0.687731i	$-0.241393\pi$
$281$	24.3388	1.45193	0.725966	+	0.687731i	$0.241393\pi$
$282$	0	0
$283$	−16.1986	−0.962905	−0.481453	−	0.876472i	$-0.659891\pi$
$283$	−16.1986	−0.962905	−0.481453	+	0.876472i	$0.659891\pi$
$284$	−38.2337	−2.26875
$285$	0	0
$286$	−1.77930	−0.105212
$287$	−5.65281	−0.333675
$288$	0	0
$289$	5.76366	0.339039
$290$	6.07534	0.356756
$291$	0	0
$292$	45.9668	2.69000
$293$	−21.5945	−1.26156	−0.630782	−	0.775960i	$-0.717266\pi$
$293$	−21.5945	−1.26156	−0.630782	+	0.775960i	$0.717266\pi$
$294$	0	0
$295$	13.3288	0.776035
$296$	−27.6409	−1.60659
$297$	0	0
$298$	45.9240	2.66031
$299$	0.673190	0.0389316
$300$	0	0
$301$	16.1583	0.931350
$302$	44.1948	2.54312
$303$	0	0
$304$	−13.2770	−0.761486
$305$	−4.84499	−0.277424
$306$	0	0
$307$	3.90718	0.222994	0.111497	−	0.993765i	$-0.464435\pi$
$307$	3.90718	0.222994	0.111497	+	0.993765i	$0.464435\pi$
$308$	−5.15181	−0.293552
$309$	0	0
$310$	15.8911	0.902556
$311$	6.79303	0.385197	0.192599	−	0.981278i	$-0.438308\pi$
$311$	6.79303	0.385197	0.192599	+	0.981278i	$0.438308\pi$
$312$	0	0
$313$	23.2536	1.31437	0.657186	−	0.753728i	$-0.271746\pi$
$313$	23.2536	1.31437	0.657186	+	0.753728i	$0.271746\pi$
$314$	21.1736	1.19490
$315$	0	0
$316$	−15.9309	−0.896181
$317$	18.8837	1.06061	0.530306	−	0.847806i	$-0.322077\pi$
$317$	18.8837	1.06061	0.530306	+	0.847806i	$0.322077\pi$
$318$	0	0
$319$	−1.84021	−0.103032
$320$	−9.38471	−0.524621
$321$	0	0
$322$	2.95539	0.164698
$323$	−19.4243	−1.08079
$324$	0	0
$325$	1.00000	0.0554700
$326$	−8.26817	−0.457932
$327$	0	0
$328$	−14.1762	−0.782749
$329$	−6.48171	−0.357348
$330$	0	0
$331$	−25.0316	−1.37586	−0.687930	−	0.725777i	$-0.741480\pi$
$331$	−25.0316	−1.37586	−0.687930	+	0.725777i	$0.741480\pi$
$332$	49.2608	2.70354
$333$	0	0
$334$	47.3751	2.59225
$335$	−2.87305	−0.156971
$336$	0	0
$337$	−21.4662	−1.16934	−0.584669	−	0.811272i	$-0.698776\pi$
$337$	−21.4662	−1.16934	−0.584669	+	0.811272i	$0.698776\pi$
$338$	−2.42368	−0.131831
$339$	0	0
$340$	18.4844	1.00246
$341$	−4.81341	−0.260661
$342$	0	0
$343$	19.4159	1.04836
$344$	40.5221	2.18480
$345$	0	0
$346$	−45.3940	−2.44040
$347$	30.8255	1.65480	0.827400	−	0.561613i	$-0.189819\pi$
$347$	30.8255	1.65480	0.827400	+	0.561613i	$0.189819\pi$
$348$	0	0
$349$	25.7243	1.37699	0.688494	−	0.725242i	$-0.258272\pi$
$349$	25.7243	1.37699	0.688494	+	0.725242i	$0.258272\pi$
$350$	4.39013	0.234662
$351$	0	0
$352$	0.867001	0.0462113
$353$	9.44972	0.502958	0.251479	−	0.967863i	$-0.419083\pi$
$353$	9.44972	0.502958	0.251479	+	0.967863i	$0.419083\pi$
$354$	0	0
$355$	−9.86873	−0.523778
$356$	21.2099	1.12413
$357$	0	0
$358$	46.2460	2.44418
$359$	−26.4303	−1.39494	−0.697469	−	0.716615i	$-0.745691\pi$
$359$	−26.4303	−1.39494	−0.697469	+	0.716615i	$0.745691\pi$
$360$	0	0
$361$	−2.42525	−0.127645
$362$	−19.7480	−1.03793
$363$	0	0
$364$	−7.01758	−0.367821
$365$	11.8648	0.621030
$366$	0	0
$367$	−30.5131	−1.59277	−0.796387	−	0.604788i	$-0.793258\pi$
$367$	−30.5131	−1.59277	−0.796387	+	0.604788i	$0.793258\pi$
$368$	2.19540	0.114443
$369$	0	0
$370$	−14.7479	−0.766705
$371$	23.2238	1.20572
$372$	0	0
$373$	6.16928	0.319433	0.159717	−	0.987163i	$-0.448942\pi$
$373$	6.16928	0.319433	0.159717	+	0.987163i	$0.448942\pi$
$374$	−8.48925	−0.438968
$375$	0	0
$376$	−16.2549	−0.838283
$377$	−2.50666	−0.129099
$378$	0	0
$379$	−28.0041	−1.43847	−0.719237	−	0.694765i	$-0.755508\pi$
$379$	−28.0041	−1.43847	−0.719237	+	0.694765i	$0.755508\pi$
$380$	−15.7728	−0.809127
$381$	0	0
$382$	23.5099	1.20287
$383$	−2.61921	−0.133835	−0.0669177	−	0.997758i	$-0.521317\pi$
$383$	−2.61921	−0.133835	−0.0669177	+	0.997758i	$0.521317\pi$
$384$	0	0
$385$	−1.32976	−0.0677711
$386$	−36.1095	−1.83793
$387$	0	0
$388$	−6.59687	−0.334905
$389$	−33.0715	−1.67679	−0.838397	−	0.545061i	$-0.816507\pi$
$389$	−33.0715	−1.67679	−0.838397	+	0.545061i	$0.816507\pi$
$390$	0	0
$391$	3.21188	0.162432
$392$	16.8937	0.853262
$393$	0	0
$394$	−41.9771	−2.11477
$395$	−4.11201	−0.206898
$396$	0	0
$397$	−23.3740	−1.17311	−0.586553	−	0.809911i	$-0.699516\pi$
$397$	−23.3740	−1.17311	−0.586553	+	0.809911i	$0.699516\pi$
$398$	−1.65884	−0.0831499
$399$	0	0
$400$	3.26118	0.163059
$401$	−25.1126	−1.25406	−0.627032	−	0.778994i	$-0.715730\pi$
$401$	−25.1126	−1.25406	−0.627032	+	0.778994i	$0.715730\pi$
$402$	0	0
$403$	−6.55662	−0.326608
$404$	−59.2428	−2.94744
$405$	0	0
$406$	−11.0046	−0.546147
$407$	4.46711	0.221426
$408$	0	0
$409$	−10.2602	−0.507335	−0.253668	−	0.967291i	$-0.581637\pi$
$409$	−10.2602	−0.507335	−0.253668	+	0.967291i	$0.581637\pi$
$410$	−7.56375	−0.373547
$411$	0	0
$412$	−9.17228	−0.451886
$413$	−24.1432	−1.18801
$414$	0	0
$415$	12.7150	0.624154
$416$	1.18099	0.0579029
$417$	0	0
$418$	7.24389	0.354310
$419$	11.8757	0.580165	0.290082	−	0.957002i	$-0.406317\pi$
$419$	11.8757	0.580165	0.290082	+	0.957002i	$0.406317\pi$
$420$	0	0
$421$	14.8379	0.723153	0.361577	−	0.932342i	$-0.382239\pi$
$421$	14.8379	0.723153	0.361577	+	0.932342i	$0.382239\pi$
$422$	−19.6329	−0.955714
$423$	0	0
$424$	58.2411	2.82844
$425$	4.77113	0.231434
$426$	0	0
$427$	8.77598	0.424699
$428$	6.80323	0.328847
$429$	0	0
$430$	21.6207	1.04264
$431$	−38.6608	−1.86222	−0.931112	−	0.364733i	$-0.881160\pi$
$431$	−38.6608	−1.86222	−0.931112	+	0.364733i	$0.881160\pi$
$432$	0	0
$433$	10.3454	0.497169	0.248585	−	0.968610i	$-0.420035\pi$
$433$	10.3454	0.497169	0.248585	+	0.968610i	$0.420035\pi$
$434$	−28.7844	−1.38170
$435$	0	0
$436$	28.5903	1.36922
$437$	−2.74070	−0.131105
$438$	0	0
$439$	35.3504	1.68718	0.843592	−	0.536984i	$-0.180436\pi$
$439$	35.3504	1.68718	0.843592	+	0.536984i	$0.180436\pi$
$440$	−3.33480	−0.158981
$441$	0	0
$442$	−11.5637	−0.550029
$443$	−18.9131	−0.898589	−0.449294	−	0.893384i	$-0.648325\pi$
$443$	−18.9131	−0.898589	−0.449294	+	0.893384i	$0.648325\pi$
$444$	0	0
$445$	5.47463	0.259522
$446$	−15.6497	−0.741034
$447$	0	0
$448$	16.9990	0.803127
$449$	−36.0165	−1.69972	−0.849862	−	0.527006i	$-0.823314\pi$
$449$	−36.0165	−1.69972	−0.849862	+	0.527006i	$0.823314\pi$
$450$	0	0
$451$	2.29105	0.107881
$452$	17.2887	0.813193
$453$	0	0
$454$	51.0170	2.39435
$455$	−1.81135	−0.0849174
$456$	0	0
$457$	−35.0066	−1.63754	−0.818771	−	0.574120i	$-0.805344\pi$
$457$	−35.0066	−1.63754	−0.818771	+	0.574120i	$0.805344\pi$
$458$	25.2017	1.17760
$459$	0	0
$460$	2.60809	0.121603
$461$	8.51720	0.396686	0.198343	−	0.980133i	$-0.436444\pi$
$461$	8.51720	0.396686	0.198343	+	0.980133i	$0.436444\pi$
$462$	0	0
$463$	−20.0189	−0.930356	−0.465178	−	0.885217i	$-0.654010\pi$
$463$	−20.0189	−0.930356	−0.465178	+	0.885217i	$0.654010\pi$
$464$	−8.17467	−0.379499
$465$	0	0
$466$	−68.9873	−3.19578
$467$	−27.4282	−1.26922	−0.634612	−	0.772831i	$-0.718840\pi$
$467$	−27.4282	−1.26922	−0.634612	+	0.772831i	$0.718840\pi$
$468$	0	0
$469$	5.20409	0.240303
$470$	−8.67286	−0.400049
$471$	0	0
$472$	−60.5466	−2.78688
$473$	−6.54887	−0.301117
$474$	0	0
$475$	−4.07121	−0.186800
$476$	−33.4818	−1.53463
$477$	0	0
$478$	29.1807	1.33469
$479$	2.67605	0.122272	0.0611359	−	0.998129i	$-0.480528\pi$
$479$	2.67605	0.122272	0.0611359	+	0.998129i	$0.480528\pi$
$480$	0	0
$481$	6.08491	0.277448
$482$	−62.1774	−2.83210
$483$	0	0
$484$	−40.5285	−1.84220
$485$	−1.70276	−0.0773182
$486$	0	0
$487$	−28.0976	−1.27323	−0.636613	−	0.771184i	$-0.719665\pi$
$487$	−28.0976	−1.27323	−0.636613	+	0.771184i	$0.719665\pi$
$488$	22.0085	0.996279
$489$	0	0
$490$	9.01370	0.407198
$491$	7.55666	0.341027	0.170514	−	0.985355i	$-0.445457\pi$
$491$	7.55666	0.341027	0.170514	+	0.985355i	$0.445457\pi$
$492$	0	0
$493$	−11.9596	−0.538632
$494$	9.86731	0.443951
$495$	0	0
$496$	−21.3823	−0.960095
$497$	17.8757	0.801835
$498$	0	0
$499$	17.3776	0.777928	0.388964	−	0.921253i	$-0.372833\pi$
$499$	17.3776	0.777928	0.388964	+	0.921253i	$0.372833\pi$
$500$	3.87423	0.173261
$501$	0	0
$502$	−41.3697	−1.84642
$503$	−12.5517	−0.559651	−0.279826	−	0.960051i	$-0.590277\pi$
$503$	−12.5517	−0.559651	−0.279826	+	0.960051i	$0.590277\pi$
$504$	0	0
$505$	−15.2915	−0.680463
$506$	−1.19780	−0.0532489
$507$	0	0
$508$	−16.5838	−0.735788
$509$	19.8053	0.877853	0.438927	−	0.898523i	$-0.355359\pi$
$509$	19.8053	0.877853	0.438927	+	0.898523i	$0.355359\pi$
$510$	0	0
$511$	−21.4912	−0.950715
$512$	33.4795	1.47960
$513$	0	0
$514$	25.6530	1.13151
$515$	−2.36751	−0.104325
$516$	0	0
$517$	2.62700	0.115535
$518$	26.7135	1.17373
$519$	0	0
$520$	−4.54253	−0.199203
$521$	27.6044	1.20937	0.604685	−	0.796464i	$-0.293299\pi$
$521$	27.6044	1.20937	0.604685	+	0.796464i	$0.293299\pi$
$522$	0	0
$523$	7.57067	0.331042	0.165521	−	0.986206i	$-0.447069\pi$
$523$	7.57067	0.331042	0.165521	+	0.986206i	$0.447069\pi$
$524$	28.0567	1.22566
$525$	0	0
$526$	3.14124	0.136964
$527$	−31.2825	−1.36269
$528$	0	0
$529$	−22.5468	−0.980296
$530$	31.0747	1.34980
$531$	0	0
$532$	28.5700	1.23867
$533$	3.12077	0.135176
$534$	0	0
$535$	1.75602	0.0759195
$536$	13.0509	0.563713
$537$	0	0
$538$	−0.324160	−0.0139756
$539$	−2.73024	−0.117600
$540$	0	0
$541$	−18.6196	−0.800520	−0.400260	−	0.916402i	$-0.631080\pi$
$541$	−18.6196	−0.800520	−0.400260	+	0.916402i	$0.631080\pi$
$542$	−49.0834	−2.10831
$543$	0	0
$544$	5.63466	0.241584
$545$	7.37960	0.316107
$546$	0	0
$547$	21.4930	0.918975	0.459487	−	0.888184i	$-0.348033\pi$
$547$	21.4930	0.918975	0.459487	+	0.888184i	$0.348033\pi$
$548$	−37.5058	−1.60217
$549$	0	0
$550$	−1.77930	−0.0758694
$551$	10.2051	0.434753
$552$	0	0
$553$	7.44829	0.316733
$554$	57.7845	2.45503
$555$	0	0
$556$	−45.6766	−1.93712
$557$	−19.4675	−0.824865	−0.412432	−	0.910988i	$-0.635321\pi$
$557$	−19.4675	−0.824865	−0.412432	+	0.910988i	$0.635321\pi$
$558$	0	0
$559$	−8.92060	−0.377301
$560$	−5.90714	−0.249622
$561$	0	0
$562$	−58.9895	−2.48832
$563$	−29.6229	−1.24846	−0.624228	−	0.781242i	$-0.714586\pi$
$563$	−29.6229	−1.24846	−0.624228	+	0.781242i	$0.714586\pi$
$564$	0	0
$565$	4.46249	0.187739
$566$	39.2602	1.65023
$567$	0	0
$568$	44.8290	1.88098
$569$	−38.3644	−1.60832	−0.804160	−	0.594413i	$-0.797384\pi$
$569$	−38.3644	−1.60832	−0.804160	+	0.594413i	$0.797384\pi$
$570$	0	0
$571$	19.4454	0.813764	0.406882	−	0.913481i	$-0.366616\pi$
$571$	19.4454	0.813764	0.406882	+	0.913481i	$0.366616\pi$
$572$	2.84418	0.118921
$573$	0	0
$574$	13.7006	0.571852
$575$	0.673190	0.0280740
$576$	0	0
$577$	−12.7513	−0.530842	−0.265421	−	0.964133i	$-0.585511\pi$
$577$	−12.7513	−0.530842	−0.265421	+	0.964133i	$0.585511\pi$
$578$	−13.9693	−0.581045
$579$	0	0
$580$	−9.71136	−0.403242
$581$	−23.0313	−0.955499
$582$	0	0
$583$	−9.41249	−0.389825
$584$	−53.8960	−2.23023
$585$	0	0
$586$	52.3382	2.16207
$587$	6.92768	0.285936	0.142968	−	0.989727i	$-0.454335\pi$
$587$	6.92768	0.285936	0.142968	+	0.989727i	$0.454335\pi$
$588$	0	0
$589$	26.6934	1.09988
$590$	−32.3048	−1.32997
$591$	0	0
$592$	19.8440	0.815583
$593$	−5.71698	−0.234768	−0.117384	−	0.993087i	$-0.537451\pi$
$593$	−5.71698	−0.234768	−0.117384	+	0.993087i	$0.537451\pi$
$594$	0	0
$595$	−8.64218	−0.354295
$596$	−73.4090	−3.00695
$597$	0	0
$598$	−1.63160	−0.0667210
$599$	−16.8127	−0.686948	−0.343474	−	0.939162i	$-0.611604\pi$
$599$	−16.8127	−0.686948	−0.343474	+	0.939162i	$0.611604\pi$
$600$	0	0
$601$	−26.6098	−1.08544	−0.542718	−	0.839915i	$-0.682605\pi$
$601$	−26.6098	−1.08544	−0.542718	+	0.839915i	$0.682605\pi$
$602$	−39.1626	−1.59615
$603$	0	0
$604$	−70.6449	−2.87450
$605$	−10.4611	−0.425302
$606$	0	0
$607$	−38.4768	−1.56173	−0.780863	−	0.624703i	$-0.785220\pi$
$607$	−38.4768	−1.56173	−0.780863	+	0.624703i	$0.785220\pi$
$608$	−4.80807	−0.194993
$609$	0	0
$610$	11.7427	0.475449
$611$	3.57839	0.144766
$612$	0	0
$613$	23.9205	0.966142	0.483071	−	0.875581i	$-0.339521\pi$
$613$	23.9205	0.966142	0.483071	+	0.875581i	$0.339521\pi$
$614$	−9.46975	−0.382168
$615$	0	0
$616$	6.04049	0.243378
$617$	−41.4685	−1.66946	−0.834730	−	0.550660i	$-0.814376\pi$
$617$	−41.4685	−1.66946	−0.834730	+	0.550660i	$0.814376\pi$
$618$	0	0
$619$	45.3650	1.82337	0.911687	−	0.410885i	$-0.134780\pi$
$619$	45.3650	1.82337	0.911687	+	0.410885i	$0.134780\pi$
$620$	−25.4018	−1.02016
$621$	0	0
$622$	−16.4641	−0.660152
$623$	−9.91646	−0.397295
$624$	0	0
$625$	1.00000	0.0400000
$626$	−56.3594	−2.25257
$627$	0	0
$628$	−33.8458	−1.35059
$629$	29.0319	1.15758
$630$	0	0
$631$	−36.5598	−1.45542	−0.727712	−	0.685883i	$-0.759416\pi$
$631$	−36.5598	−1.45542	−0.727712	+	0.685883i	$0.759416\pi$
$632$	18.6789	0.743008
$633$	0	0
$634$	−45.7680	−1.81768
$635$	−4.28055	−0.169868
$636$	0	0
$637$	−3.71901	−0.147353
$638$	4.46008	0.176576
$639$	0	0
$640$	20.3836	0.805731
$641$	−10.2837	−0.406181	−0.203090	−	0.979160i	$-0.565099\pi$
$641$	−10.2837	−0.406181	−0.203090	+	0.979160i	$0.565099\pi$
$642$	0	0
$643$	−20.5168	−0.809104	−0.404552	−	0.914515i	$-0.632573\pi$
$643$	−20.5168	−0.809104	−0.404552	+	0.914515i	$0.632573\pi$
$644$	−4.72416	−0.186158
$645$	0	0
$646$	47.0782	1.85227
$647$	5.54925	0.218164	0.109082	−	0.994033i	$-0.465209\pi$
$647$	5.54925	0.218164	0.109082	+	0.994033i	$0.465209\pi$
$648$	0	0
$649$	9.78509	0.384098
$650$	−2.42368	−0.0950646
$651$	0	0
$652$	13.2166	0.517601
$653$	−4.88907	−0.191324	−0.0956620	−	0.995414i	$-0.530497\pi$
$653$	−4.88907	−0.191324	−0.0956620	+	0.995414i	$0.530497\pi$
$654$	0	0
$655$	7.24187	0.282963
$656$	10.1774	0.397361
$657$	0	0
$658$	15.7096	0.612423
$659$	−37.8287	−1.47360	−0.736798	−	0.676113i	$-0.763663\pi$
$659$	−37.8287	−1.47360	−0.736798	+	0.676113i	$0.763663\pi$
$660$	0	0
$661$	12.4724	0.485120	0.242560	−	0.970136i	$-0.422013\pi$
$661$	12.4724	0.485120	0.242560	+	0.970136i	$0.422013\pi$
$662$	60.6686	2.35795
$663$	0	0
$664$	−57.7582	−2.24145
$665$	7.37438	0.285966
$666$	0	0
$667$	−1.68746	−0.0653386
$668$	−75.7286	−2.93003
$669$	0	0
$670$	6.96335	0.269018
$671$	−3.55685	−0.137311
$672$	0	0
$673$	−41.0640	−1.58290	−0.791451	−	0.611232i	$-0.790674\pi$
$673$	−41.0640	−1.58290	−0.791451	+	0.611232i	$0.790674\pi$
$674$	52.0272	2.00401
$675$	0	0
$676$	3.87423	0.149009
$677$	13.0645	0.502110	0.251055	−	0.967973i	$-0.419222\pi$
$677$	13.0645	0.502110	0.251055	+	0.967973i	$0.419222\pi$
$678$	0	0
$679$	3.08429	0.118364
$680$	−21.6730	−0.831121
$681$	0	0
$682$	11.6662	0.446720
$683$	38.8228	1.48551	0.742756	−	0.669562i	$-0.233518\pi$
$683$	38.8228	1.48551	0.742756	+	0.669562i	$0.233518\pi$
$684$	0	0
$685$	−9.68085	−0.369886
$686$	−47.0579	−1.79668
$687$	0	0
$688$	−29.0917	−1.10911
$689$	−12.8213	−0.488452
$690$	0	0
$691$	−35.5986	−1.35424	−0.677118	−	0.735875i	$-0.736771\pi$
$691$	−35.5986	−1.35424	−0.677118	+	0.735875i	$0.736771\pi$
$692$	72.5618	2.75839
$693$	0	0
$694$	−74.7112	−2.83600
$695$	−11.7899	−0.447215
$696$	0	0
$697$	14.8896	0.563984
$698$	−62.3474	−2.35988
$699$	0	0
$700$	−7.01758	−0.265240
$701$	7.02815	0.265450	0.132725	−	0.991153i	$-0.457627\pi$
$701$	7.02815	0.265450	0.132725	+	0.991153i	$0.457627\pi$
$702$	0	0
$703$	−24.7729	−0.934329
$704$	−6.88959	−0.259661
$705$	0	0
$706$	−22.9031	−0.861969
$707$	27.6983	1.04170
$708$	0	0
$709$	20.8603	0.783426	0.391713	−	0.920087i	$-0.371883\pi$
$709$	20.8603	0.783426	0.391713	+	0.920087i	$0.371883\pi$
$710$	23.9186	0.897650
$711$	0	0
$712$	−24.8686	−0.931992
$713$	−4.41385	−0.165300
$714$	0	0
$715$	0.734130	0.0274549
$716$	−73.9238	−2.76266
$717$	0	0
$718$	64.0586	2.39065
$719$	−23.9744	−0.894093	−0.447046	−	0.894511i	$-0.647524\pi$
$719$	−23.9744	−0.894093	−0.447046	+	0.894511i	$0.647524\pi$
$720$	0	0
$721$	4.28839	0.159708
$722$	5.87803	0.218758
$723$	0	0
$724$	31.5670	1.17318
$725$	−2.50666	−0.0930949
$726$	0	0
$727$	19.3935	0.719264	0.359632	−	0.933094i	$-0.382902\pi$
$727$	19.3935	0.719264	0.359632	+	0.933094i	$0.382902\pi$
$728$	8.22810	0.304954
$729$	0	0
$730$	−28.7564	−1.06432
$731$	−42.5613	−1.57419
$732$	0	0
$733$	25.8255	0.953886	0.476943	−	0.878934i	$-0.341745\pi$
$733$	25.8255	0.953886	0.476943	+	0.878934i	$0.341745\pi$
$734$	73.9541	2.72970
$735$	0	0
$736$	0.795032	0.0293053
$737$	−2.10919	−0.0776929
$738$	0	0
$739$	2.55601	0.0940243	0.0470121	−	0.998894i	$-0.485030\pi$
$739$	2.55601	0.0940243	0.0470121	+	0.998894i	$0.485030\pi$
$740$	23.5743	0.866609
$741$	0	0
$742$	−56.2871	−2.06637
$743$	41.9652	1.53955	0.769777	−	0.638313i	$-0.220368\pi$
$743$	41.9652	1.53955	0.769777	+	0.638313i	$0.220368\pi$
$744$	0	0
$745$	−18.9480	−0.694202
$746$	−14.9524	−0.547445
$747$	0	0
$748$	13.5700	0.496167
$749$	−3.18077	−0.116223
$750$	0	0
$751$	24.1350	0.880698	0.440349	−	0.897827i	$-0.354855\pi$
$751$	24.1350	0.880698	0.440349	+	0.897827i	$0.354855\pi$
$752$	11.6698	0.425553
$753$	0	0
$754$	6.07534	0.221251
$755$	−18.2346	−0.663624
$756$	0	0
$757$	−46.3001	−1.68281	−0.841403	−	0.540409i	$-0.818270\pi$
$757$	−46.3001	−1.68281	−0.841403	+	0.540409i	$0.818270\pi$
$758$	67.8730	2.46526
$759$	0	0
$760$	18.4936	0.670833
$761$	−28.8878	−1.04718	−0.523591	−	0.851970i	$-0.675408\pi$
$761$	−28.8878	−1.04718	−0.523591	+	0.851970i	$0.675408\pi$
$762$	0	0
$763$	−13.3670	−0.483919
$764$	−37.5804	−1.35961
$765$	0	0
$766$	6.34813	0.229367
$767$	13.3288	0.481276
$768$	0	0
$769$	13.3812	0.482539	0.241270	−	0.970458i	$-0.422436\pi$
$769$	13.3812	0.482539	0.241270	+	0.970458i	$0.422436\pi$
$770$	3.22293	0.116146
$771$	0	0
$772$	57.7207	2.07741
$773$	23.0723	0.829854	0.414927	−	0.909855i	$-0.363807\pi$
$773$	23.0723	0.829854	0.414927	+	0.909855i	$0.363807\pi$
$774$	0	0
$775$	−6.55662	−0.235521
$776$	7.73482	0.277664
$777$	0	0
$778$	80.1548	2.87369
$779$	−12.7053	−0.455215
$780$	0	0
$781$	−7.24492	−0.259244
$782$	−7.78456	−0.278375
$783$	0	0
$784$	−12.1284	−0.433157
$785$	−8.73613	−0.311806
$786$	0	0
$787$	−19.0160	−0.677846	−0.338923	−	0.940814i	$-0.610063\pi$
$787$	−19.0160	−0.677846	−0.338923	+	0.940814i	$0.610063\pi$
$788$	67.0999	2.39033
$789$	0	0
$790$	9.96620	0.354581
$791$	−8.08314	−0.287403
$792$	0	0
$793$	−4.84499	−0.172051
$794$	56.6511	2.01047
$795$	0	0
$796$	2.65163	0.0939846
$797$	0.300380	0.0106400	0.00532001	−	0.999986i	$-0.498307\pi$
$797$	0.300380	0.0106400	0.00532001	+	0.999986i	$0.498307\pi$
$798$	0	0
$799$	17.0729	0.603997
$800$	1.18099	0.0417544
$801$	0	0
$802$	60.8649	2.14922
$803$	8.71027	0.307379
$804$	0	0
$805$	−1.21938	−0.0429776
$806$	15.8911	0.559742
$807$	0	0
$808$	69.4621	2.44367
$809$	−25.7408	−0.904999	−0.452500	−	0.891765i	$-0.649468\pi$
$809$	−25.7408	−0.904999	−0.452500	+	0.891765i	$0.649468\pi$
$810$	0	0
$811$	−40.1066	−1.40833	−0.704167	−	0.710035i	$-0.748679\pi$
$811$	−40.1066	−1.40833	−0.704167	+	0.710035i	$0.748679\pi$
$812$	17.5907	0.617311
$813$	0	0
$814$	−10.8268	−0.379481
$815$	3.41141	0.119497
$816$	0	0
$817$	36.3176	1.27059
$818$	24.8675	0.869472
$819$	0	0
$820$	12.0906	0.422221
$821$	19.9053	0.694699	0.347349	−	0.937736i	$-0.387082\pi$
$821$	19.9053	0.694699	0.347349	+	0.937736i	$0.387082\pi$
$822$	0	0
$823$	−24.2386	−0.844904	−0.422452	−	0.906385i	$-0.638831\pi$
$823$	−24.2386	−0.844904	−0.422452	+	0.906385i	$0.638831\pi$
$824$	10.7545	0.374650
$825$	0	0
$826$	58.5153	2.03601
$827$	16.5356	0.574999	0.287500	−	0.957781i	$-0.407176\pi$
$827$	16.5356	0.574999	0.287500	+	0.957781i	$0.407176\pi$
$828$	0	0
$829$	−47.6635	−1.65542	−0.827711	−	0.561154i	$-0.810358\pi$
$829$	−47.6635	−1.65542	−0.827711	+	0.561154i	$0.810358\pi$
$830$	−30.8171	−1.06968
$831$	0	0
$832$	−9.38471	−0.325356
$833$	−17.7439	−0.614789
$834$	0	0
$835$	−19.5468	−0.676444
$836$	−11.5793	−0.400478
$837$	0	0
$838$	−28.7828	−0.994287
$839$	52.5712	1.81496	0.907480	−	0.420094i	$-0.138003\pi$
$839$	52.5712	1.81496	0.907480	+	0.420094i	$0.138003\pi$
$840$	0	0
$841$	−22.7167	−0.783333
$842$	−35.9622	−1.23934
$843$	0	0
$844$	31.3829	1.08025
$845$	1.00000	0.0344010
$846$	0	0
$847$	18.9486	0.651082
$848$	−41.8126	−1.43585
$849$	0	0
$850$	−11.5637	−0.396631
$851$	4.09630	0.140419
$852$	0	0
$853$	−14.9290	−0.511160	−0.255580	−	0.966788i	$-0.582266\pi$
$853$	−14.9290	−0.511160	−0.255580	+	0.966788i	$0.582266\pi$
$854$	−21.2702	−0.727850
$855$	0	0
$856$	−7.97679	−0.272641
$857$	36.8715	1.25951	0.629754	−	0.776795i	$-0.283156\pi$
$857$	36.8715	1.25951	0.629754	+	0.776795i	$0.283156\pi$
$858$	0	0
$859$	−8.74605	−0.298411	−0.149206	−	0.988806i	$-0.547672\pi$
$859$	−8.74605	−0.298411	−0.149206	+	0.988806i	$0.547672\pi$
$860$	−34.5604	−1.17850
$861$	0	0
$862$	93.7014	3.19148
$863$	−32.1874	−1.09567	−0.547836	−	0.836586i	$-0.684548\pi$
$863$	−32.1874	−1.09567	−0.547836	+	0.836586i	$0.684548\pi$
$864$	0	0
$865$	18.7294	0.636818
$866$	−25.0740	−0.852049
$867$	0	0
$868$	46.0116	1.56173
$869$	−3.01875	−0.102404
$870$	0	0
$871$	−2.87305	−0.0973495
$872$	−33.5220	−1.13520
$873$	0	0
$874$	6.64258	0.224688
$875$	−1.81135	−0.0612348
$876$	0	0
$877$	−37.7977	−1.27634	−0.638169	−	0.769896i	$-0.720308\pi$
$877$	−37.7977	−1.27634	−0.638169	+	0.769896i	$0.720308\pi$
$878$	−85.6782	−2.89150
$879$	0	0
$880$	2.39413	0.0807061
$881$	4.76707	0.160607	0.0803033	−	0.996770i	$-0.474411\pi$
$881$	4.76707	0.160607	0.0803033	+	0.996770i	$0.474411\pi$
$882$	0	0
$883$	24.7892	0.834221	0.417111	−	0.908856i	$-0.363043\pi$
$883$	24.7892	0.834221	0.417111	+	0.908856i	$0.363043\pi$
$884$	18.4844	0.621699
$885$	0	0
$886$	45.8393	1.54000
$887$	6.43243	0.215980	0.107990	−	0.994152i	$-0.465559\pi$
$887$	6.43243	0.215980	0.107990	+	0.994152i	$0.465559\pi$
$888$	0	0
$889$	7.75357	0.260046
$890$	−13.2687	−0.444769
$891$	0	0
$892$	25.0158	0.837592
$893$	−14.5684	−0.487512
$894$	0	0
$895$	−19.0809	−0.637804
$896$	−36.9217	−1.23347
$897$	0	0
$898$	87.2925	2.91299
$899$	16.4352	0.548144
$900$	0	0
$901$	−61.1720	−2.03794
$902$	−5.55277	−0.184887
$903$	0	0
$904$	−20.2710	−0.674204
$905$	8.14795	0.270847
$906$	0	0
$907$	−21.9638	−0.729295	−0.364647	−	0.931146i	$-0.618810\pi$
$907$	−21.9638	−0.729295	−0.364647	+	0.931146i	$0.618810\pi$
$908$	−81.5501	−2.70634
$909$	0	0
$910$	4.39013	0.145531
$911$	48.8550	1.61864	0.809319	−	0.587369i	$-0.199836\pi$
$911$	48.8550	1.61864	0.809319	+	0.587369i	$0.199836\pi$
$912$	0	0
$913$	9.33445	0.308925
$914$	84.8449	2.80642
$915$	0	0
$916$	−40.2847	−1.33104
$917$	−13.1176	−0.433180
$918$	0	0
$919$	−34.7323	−1.14571	−0.572856	−	0.819656i	$-0.694165\pi$
$919$	−34.7323	−1.14571	−0.572856	+	0.819656i	$0.694165\pi$
$920$	−3.05798	−0.100819
$921$	0	0
$922$	−20.6430	−0.679840
$923$	−9.86873	−0.324833
$924$	0	0
$925$	6.08491	0.200071
$926$	48.5193	1.59444
$927$	0	0
$928$	−2.96034	−0.0971780
$929$	−11.2434	−0.368884	−0.184442	−	0.982843i	$-0.559048\pi$
$929$	−11.2434	−0.368884	−0.184442	+	0.982843i	$0.559048\pi$
$930$	0	0
$931$	15.1409	0.496223
$932$	110.276	3.61220
$933$	0	0
$934$	66.4771	2.17520
$935$	3.50263	0.114548
$936$	0	0
$937$	−35.1034	−1.14678	−0.573388	−	0.819284i	$-0.694371\pi$
$937$	−35.1034	−1.14678	−0.573388	+	0.819284i	$0.694371\pi$
$938$	−12.6131	−0.411831
$939$	0	0
$940$	13.8635	0.452177
$941$	18.4392	0.601100	0.300550	−	0.953766i	$-0.402830\pi$
$941$	18.4392	0.601100	0.300550	+	0.953766i	$0.402830\pi$
$942$	0	0
$943$	2.10087	0.0684138
$944$	43.4678	1.41475
$945$	0	0
$946$	15.8724	0.516055
$947$	6.82313	0.221722	0.110861	−	0.993836i	$-0.464639\pi$
$947$	6.82313	0.221722	0.110861	+	0.993836i	$0.464639\pi$
$948$	0	0
$949$	11.8648	0.385146
$950$	9.86731	0.320138
$951$	0	0
$952$	39.2573	1.27234
$953$	31.3321	1.01495	0.507474	−	0.861667i	$-0.330579\pi$
$953$	31.3321	1.01495	0.507474	+	0.861667i	$0.330579\pi$
$954$	0	0
$955$	−9.70009	−0.313888
$956$	−46.6450	−1.50861
$957$	0	0
$958$	−6.48589	−0.209550
$959$	17.5354	0.566248
$960$	0	0
$961$	11.9892	0.386749
$962$	−14.7479	−0.475490
$963$	0	0
$964$	99.3898	3.20113
$965$	14.8986	0.479604
$966$	0	0
$967$	−52.6569	−1.69333	−0.846666	−	0.532125i	$-0.821394\pi$
$967$	−52.6569	−1.69333	−0.846666	+	0.532125i	$0.821394\pi$
$968$	47.5196	1.52734
$969$	0	0
$970$	4.12694	0.132508
$971$	16.7981	0.539076	0.269538	−	0.962990i	$-0.413129\pi$
$971$	16.7981	0.539076	0.269538	+	0.962990i	$0.413129\pi$
$972$	0	0
$973$	21.3556	0.684628
$974$	68.0997	2.18205
$975$	0	0
$976$	−15.8004	−0.505759
$977$	−43.9857	−1.40723	−0.703613	−	0.710583i	$-0.748431\pi$
$977$	−43.9857	−1.40723	−0.703613	+	0.710583i	$0.748431\pi$
$978$	0	0
$979$	4.01909	0.128451
$980$	−14.4083	−0.460256
$981$	0	0
$982$	−18.3149	−0.584453
$983$	12.4167	0.396030	0.198015	−	0.980199i	$-0.436551\pi$
$983$	12.4167	0.396030	0.198015	+	0.980199i	$0.436551\pi$
$984$	0	0
$985$	17.3196	0.551847
$986$	28.9862	0.923109
$987$	0	0
$988$	−15.7728	−0.501799
$989$	−6.00526	−0.190956
$990$	0	0
$991$	1.42630	0.0453080	0.0226540	−	0.999743i	$-0.492788\pi$
$991$	1.42630	0.0453080	0.0226540	+	0.999743i	$0.492788\pi$
$992$	−7.74331	−0.245850
$993$	0	0
$994$	−43.3250	−1.37419
$995$	0.684429	0.0216978
$996$	0	0
$997$	−29.8110	−0.944123	−0.472062	−	0.881566i	$-0.656490\pi$
$997$	−29.8110	−0.944123	−0.472062	+	0.881566i	$0.656490\pi$
$998$	−42.1177	−1.33321
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bb.1.1		8
3.2	odd	2		5265.2.a.be.1.8		8
9.2	odd	6		1755.2.i.e.1171.1		16
9.4	even	3		585.2.i.f.196.8	✓	16
9.5	odd	6		1755.2.i.e.586.1		16
9.7	even	3		585.2.i.f.391.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.f.196.8	✓	16	9.4	even	3
585.2.i.f.391.8	yes	16	9.7	even	3
1755.2.i.e.586.1		16	9.5	odd	6
1755.2.i.e.1171.1		16	9.2	odd	6
5265.2.a.bb.1.1		8	1.1	even	1	trivial
5265.2.a.be.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)

\(f(q)\)	\(=\)	\(q-2.42368 q^{2} +3.87423 q^{4} +1.00000 q^{5} -1.81135 q^{7} -4.54253 q^{8} +O(q^{10})\)	\(q-2.42368 q^{2} +3.87423 q^{4} +1.00000 q^{5} -1.81135 q^{7} -4.54253 q^{8} -2.42368 q^{10} +0.734130 q^{11} +1.00000 q^{13} +4.39013 q^{14} +3.26118 q^{16} +4.77113 q^{17} -4.07121 q^{19} +3.87423 q^{20} -1.77930 q^{22} +0.673190 q^{23} +1.00000 q^{25} -2.42368 q^{26} -7.01758 q^{28} -2.50666 q^{29} -6.55662 q^{31} +1.18099 q^{32} -11.5637 q^{34} -1.81135 q^{35} +6.08491 q^{37} +9.86731 q^{38} -4.54253 q^{40} +3.12077 q^{41} -8.92060 q^{43} +2.84418 q^{44} -1.63160 q^{46} +3.57839 q^{47} -3.71901 q^{49} -2.42368 q^{50} +3.87423 q^{52} -12.8213 q^{53} +0.734130 q^{55} +8.22810 q^{56} +6.07534 q^{58} +13.3288 q^{59} -4.84499 q^{61} +15.8911 q^{62} -9.38471 q^{64} +1.00000 q^{65} -2.87305 q^{67} +18.4844 q^{68} +4.39013 q^{70} -9.86873 q^{71} +11.8648 q^{73} -14.7479 q^{74} -15.7728 q^{76} -1.32976 q^{77} -4.11201 q^{79} +3.26118 q^{80} -7.56375 q^{82} +12.7150 q^{83} +4.77113 q^{85} +21.6207 q^{86} -3.33480 q^{88} +5.47463 q^{89} -1.81135 q^{91} +2.60809 q^{92} -8.67286 q^{94} -4.07121 q^{95} -1.70276 q^{97} +9.01370 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 5 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 8 * q - q^2 + 5 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8	\( 8 q - q^{2} + 5 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8} - q^{10} - 9 q^{11} + 8 q^{13} + 3 q^{14} - 13 q^{16} + 6 q^{17} - 11 q^{19} + 5 q^{20} - 4 q^{22} - 3 q^{23} + 8 q^{25} - q^{26} - 13 q^{28} - 8 q^{29} - 18 q^{31} - 3 q^{32} - 9 q^{34} - 6 q^{35} - 18 q^{37} + 8 q^{38} - 6 q^{40} + 17 q^{41} - 17 q^{43} + 5 q^{44} + 3 q^{46} - 11 q^{47} - 16 q^{49} - q^{50} + 5 q^{52} + 10 q^{53} - 9 q^{55} + q^{56} - 10 q^{58} - 7 q^{59} - 21 q^{61} - 29 q^{62} - 10 q^{64} + 8 q^{65} - 13 q^{67} + 16 q^{68} + 3 q^{70} - 34 q^{71} - 16 q^{73} + 4 q^{74} - 2 q^{76} - 18 q^{77} - 37 q^{79} - 13 q^{80} + q^{82} - 3 q^{83} + 6 q^{85} + 2 q^{86} - 19 q^{88} + 14 q^{89} - 6 q^{91} + 14 q^{92} - 44 q^{94} - 11 q^{95} - 17 q^{97} + 45 q^{98}+O(q^{100}) \) 8 * q - q^2 + 5 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8 - q^10 - 9 * q^11 + 8 * q^13 + 3 * q^14 - 13 * q^16 + 6 * q^17 - 11 * q^19 + 5 * q^20 - 4 * q^22 - 3 * q^23 + 8 * q^25 - q^26 - 13 * q^28 - 8 * q^29 - 18 * q^31 - 3 * q^32 - 9 * q^34 - 6 * q^35 - 18 * q^37 + 8 * q^38 - 6 * q^40 + 17 * q^41 - 17 * q^43 + 5 * q^44 + 3 * q^46 - 11 * q^47 - 16 * q^49 - q^50 + 5 * q^52 + 10 * q^53 - 9 * q^55 + q^56 - 10 * q^58 - 7 * q^59 - 21 * q^61 - 29 * q^62 - 10 * q^64 + 8 * q^65 - 13 * q^67 + 16 * q^68 + 3 * q^70 - 34 * q^71 - 16 * q^73 + 4 * q^74 - 2 * q^76 - 18 * q^77 - 37 * q^79 - 13 * q^80 + q^82 - 3 * q^83 + 6 * q^85 + 2 * q^86 - 19 * q^88 + 14 * q^89 - 6 * q^91 + 14 * q^92 - 44 * q^94 - 11 * q^95 - 17 * q^97 + 45 * q^98

Introduction

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