LMFDB - Embedded newform 5265.2.a.ba.1.7

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			1.7
Root			$-1.96921$ 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)

$f(q)$	$=$	$q+1.96921 q^{2} +1.87778 q^{4} +1.00000 q^{5} -3.02828 q^{7} -0.240686 q^{8} +O(q^{10})$	\(q+1.96921 q^{2} +1.87778 q^{4} +1.00000 q^{5} -3.02828 q^{7} -0.240686 q^{8} +1.96921 q^{10} +4.30395 q^{11} -1.00000 q^{13} -5.96330 q^{14} -4.22951 q^{16} +0.303306 q^{17} -6.04909 q^{19} +1.87778 q^{20} +8.47536 q^{22} -2.95049 q^{23} +1.00000 q^{25} -1.96921 q^{26} -5.68642 q^{28} -8.88535 q^{29} +0.0639895 q^{31} -7.84741 q^{32} +0.597272 q^{34} -3.02828 q^{35} +11.1045 q^{37} -11.9119 q^{38} -0.240686 q^{40} -8.19698 q^{41} +2.91072 q^{43} +8.08184 q^{44} -5.81012 q^{46} +6.88864 q^{47} +2.17046 q^{49} +1.96921 q^{50} -1.87778 q^{52} -11.8296 q^{53} +4.30395 q^{55} +0.728863 q^{56} -17.4971 q^{58} -9.15068 q^{59} +1.31406 q^{61} +0.126009 q^{62} -6.99415 q^{64} -1.00000 q^{65} -15.8313 q^{67} +0.569540 q^{68} -5.96330 q^{70} +2.85405 q^{71} -11.2819 q^{73} +21.8670 q^{74} -11.3588 q^{76} -13.0335 q^{77} +10.3596 q^{79} -4.22951 q^{80} -16.1415 q^{82} +7.06927 q^{83} +0.303306 q^{85} +5.73180 q^{86} -1.03590 q^{88} -4.26865 q^{89} +3.02828 q^{91} -5.54036 q^{92} +13.5651 q^{94} -6.04909 q^{95} +5.83707 q^{97} +4.27408 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.96921	1.39244	0.696220	−	0.717829i	$-0.254864\pi$
$2$	1.96921	1.39244	0.696220	+	0.717829i	$0.254864\pi$
$3$	0	0
$3$	0	0
$4$	1.87778	0.938888
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.02828	−1.14458	−0.572290	−	0.820051i	$-0.693945\pi$
$7$	−3.02828	−1.14458	−0.572290	+	0.820051i	$0.693945\pi$
$8$	−0.240686	−0.0850953
$9$	0	0
$10$	1.96921	0.622718
$11$	4.30395	1.29769	0.648844	−	0.760921i	$-0.275253\pi$
$11$	4.30395	1.29769	0.648844	+	0.760921i	$0.275253\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−5.96330	−1.59376
$15$	0	0
$16$	−4.22951	−1.05738
$17$	0.303306	0.0735625	0.0367812	−	0.999323i	$-0.488290\pi$
$17$	0.303306	0.0735625	0.0367812	+	0.999323i	$0.488290\pi$
$18$	0	0
$19$	−6.04909	−1.38776	−0.693878	−	0.720092i	$-0.744099\pi$
$19$	−6.04909	−1.38776	−0.693878	+	0.720092i	$0.744099\pi$
$20$	1.87778	0.419883
$21$	0	0
$22$	8.47536	1.80695
$23$	−2.95049	−0.615220	−0.307610	−	0.951513i	$-0.599529\pi$
$23$	−2.95049	−0.615220	−0.307610	+	0.951513i	$0.599529\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.96921	−0.386193
$27$	0	0
$28$	−5.68642	−1.07463
$29$	−8.88535	−1.64997	−0.824984	−	0.565156i	$-0.808816\pi$
$29$	−8.88535	−1.64997	−0.824984	+	0.565156i	$0.808816\pi$
$30$	0	0
$31$	0.0639895	0.0114929	0.00574643	−	0.999983i	$-0.498171\pi$
$31$	0.0639895	0.0114929	0.00574643	+	0.999983i	$0.498171\pi$
$32$	−7.84741	−1.38724
$33$	0	0
$34$	0.597272	0.102431
$35$	−3.02828	−0.511872
$36$	0	0
$37$	11.1045	1.82556	0.912781	−	0.408450i	$-0.133930\pi$
$37$	11.1045	1.82556	0.912781	+	0.408450i	$0.133930\pi$
$38$	−11.9119	−1.93237
$39$	0	0
$40$	−0.240686	−0.0380558
$41$	−8.19698	−1.28015	−0.640076	−	0.768311i	$-0.721097\pi$
$41$	−8.19698	−1.28015	−0.640076	+	0.768311i	$0.721097\pi$
$42$	0	0
$43$	2.91072	0.443880	0.221940	−	0.975060i	$-0.428761\pi$
$43$	2.91072	0.443880	0.221940	+	0.975060i	$0.428761\pi$
$44$	8.08184	1.21838
$45$	0	0
$46$	−5.81012	−0.856656
$47$	6.88864	1.00481	0.502405	−	0.864632i	$-0.332449\pi$
$47$	6.88864	1.00481	0.502405	+	0.864632i	$0.332449\pi$
$48$	0	0
$49$	2.17046	0.310065
$50$	1.96921	0.278488
$51$	0	0
$52$	−1.87778	−0.260401
$53$	−11.8296	−1.62493	−0.812463	−	0.583012i	$-0.801874\pi$
$53$	−11.8296	−1.62493	−0.812463	+	0.583012i	$0.801874\pi$
$54$	0	0
$55$	4.30395	0.580344
$56$	0.728863	0.0973985
$57$	0	0
$58$	−17.4971	−2.29748
$59$	−9.15068	−1.19132	−0.595658	−	0.803238i	$-0.703109\pi$
$59$	−9.15068	−1.19132	−0.595658	+	0.803238i	$0.703109\pi$
$60$	0	0
$61$	1.31406	0.168248	0.0841241	−	0.996455i	$-0.473191\pi$
$61$	1.31406	0.168248	0.0841241	+	0.996455i	$0.473191\pi$
$62$	0.126009	0.0160031
$63$	0	0
$64$	−6.99415	−0.874269
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−15.8313	−1.93410	−0.967049	−	0.254590i	$-0.918060\pi$
$67$	−15.8313	−1.93410	−0.967049	+	0.254590i	$0.918060\pi$
$68$	0.569540	0.0690669
$69$	0	0
$70$	−5.96330	−0.712751
$71$	2.85405	0.338714	0.169357	−	0.985555i	$-0.445831\pi$
$71$	2.85405	0.338714	0.169357	+	0.985555i	$0.445831\pi$
$72$	0	0
$73$	−11.2819	−1.32045	−0.660223	−	0.751070i	$-0.729538\pi$
$73$	−11.2819	−1.32045	−0.660223	+	0.751070i	$0.729538\pi$
$74$	21.8670	2.54198
$75$	0	0
$76$	−11.3588	−1.30295
$77$	−13.0335	−1.48531
$78$	0	0
$79$	10.3596	1.16554	0.582772	−	0.812635i	$-0.301968\pi$
$79$	10.3596	1.16554	0.582772	+	0.812635i	$0.301968\pi$
$80$	−4.22951	−0.472874
$81$	0	0
$82$	−16.1415	−1.78254
$83$	7.06927	0.775953	0.387977	−	0.921669i	$-0.373174\pi$
$83$	7.06927	0.775953	0.387977	+	0.921669i	$0.373174\pi$
$84$	0	0
$85$	0.303306	0.0328981
$86$	5.73180	0.618076
$87$	0	0
$88$	−1.03590	−0.110427
$89$	−4.26865	−0.452476	−0.226238	−	0.974072i	$-0.572643\pi$
$89$	−4.26865	−0.452476	−0.226238	+	0.974072i	$0.572643\pi$
$90$	0	0
$91$	3.02828	0.317450
$92$	−5.54036	−0.577622
$93$	0	0
$94$	13.5651	1.39914
$95$	−6.04909	−0.620624
$96$	0	0
$97$	5.83707	0.592665	0.296333	−	0.955085i	$-0.404236\pi$
$97$	5.83707	0.592665	0.296333	+	0.955085i	$0.404236\pi$
$98$	4.27408	0.431747
$99$	0	0
$100$	1.87778	0.187778
$101$	−10.6489	−1.05960	−0.529801	−	0.848122i	$-0.677733\pi$
$101$	−10.6489	−1.05960	−0.529801	+	0.848122i	$0.677733\pi$
$102$	0	0
$103$	−17.9164	−1.76536	−0.882679	−	0.469976i	$-0.844262\pi$
$103$	−17.9164	−1.76536	−0.882679	+	0.469976i	$0.844262\pi$
$104$	0.240686	0.0236012
$105$	0	0
$106$	−23.2950	−2.26261
$107$	1.12190	0.108458	0.0542288	−	0.998529i	$-0.482730\pi$
$107$	1.12190	0.108458	0.0542288	+	0.998529i	$0.482730\pi$
$108$	0	0
$109$	14.1268	1.35310	0.676552	−	0.736394i	$-0.263473\pi$
$109$	14.1268	1.35310	0.676552	+	0.736394i	$0.263473\pi$
$110$	8.47536	0.808094
$111$	0	0
$112$	12.8081	1.21025
$113$	0.289617	0.0272449	0.0136224	−	0.999907i	$-0.495664\pi$
$113$	0.289617	0.0272449	0.0136224	+	0.999907i	$0.495664\pi$
$114$	0	0
$115$	−2.95049	−0.275135
$116$	−16.6847	−1.54914
$117$	0	0
$118$	−18.0196	−1.65884
$119$	−0.918494	−0.0841982
$120$	0	0
$121$	7.52395	0.683995
$122$	2.58766	0.234276
$123$	0	0
$124$	0.120158	0.0107905
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.70408	0.417419	0.208710	−	0.977978i	$-0.433074\pi$
$127$	4.70408	0.417419	0.208710	+	0.977978i	$0.433074\pi$
$128$	1.92189	0.169873
$129$	0	0
$130$	−1.96921	−0.172711
$131$	22.0861	1.92967	0.964835	−	0.262855i	$-0.0846640\pi$
$131$	22.0861	1.92967	0.964835	+	0.262855i	$0.0846640\pi$
$132$	0	0
$133$	18.3183	1.58840
$134$	−31.1751	−2.69311
$135$	0	0
$136$	−0.0730014	−0.00625982
$137$	2.16882	0.185295	0.0926473	−	0.995699i	$-0.470467\pi$
$137$	2.16882	0.185295	0.0926473	+	0.995699i	$0.470467\pi$
$138$	0	0
$139$	−1.21287	−0.102874	−0.0514370	−	0.998676i	$-0.516380\pi$
$139$	−1.21287	−0.102874	−0.0514370	+	0.998676i	$0.516380\pi$
$140$	−5.68642	−0.480590
$141$	0	0
$142$	5.62022	0.471639
$143$	−4.30395	−0.359914
$144$	0	0
$145$	−8.88535	−0.737888
$146$	−22.2164	−1.83864
$147$	0	0
$148$	20.8517	1.71400
$149$	−5.87020	−0.480906	−0.240453	−	0.970661i	$-0.577296\pi$
$149$	−5.87020	−0.480906	−0.240453	+	0.970661i	$0.577296\pi$
$150$	0	0
$151$	−9.30445	−0.757186	−0.378593	−	0.925563i	$-0.623592\pi$
$151$	−9.30445	−0.757186	−0.378593	+	0.925563i	$0.623592\pi$
$152$	1.45593	0.118092
$153$	0	0
$154$	−25.6657	−2.06820
$155$	0.0639895	0.00513976
$156$	0	0
$157$	−18.2945	−1.46006	−0.730031	−	0.683414i	$-0.760495\pi$
$157$	−18.2945	−1.46006	−0.730031	+	0.683414i	$0.760495\pi$
$158$	20.4002	1.62295
$159$	0	0
$160$	−7.84741	−0.620392
$161$	8.93490	0.704169
$162$	0	0
$163$	−11.9524	−0.936186	−0.468093	−	0.883679i	$-0.655059\pi$
$163$	−11.9524	−0.936186	−0.468093	+	0.883679i	$0.655059\pi$
$164$	−15.3921	−1.20192
$165$	0	0
$166$	13.9209	1.08047
$167$	−13.3298	−1.03149	−0.515746	−	0.856742i	$-0.672485\pi$
$167$	−13.3298	−1.03149	−0.515746	+	0.856742i	$0.672485\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0.597272	0.0458087
$171$	0	0
$172$	5.46567	0.416753
$173$	16.5734	1.26006	0.630028	−	0.776572i	$-0.283043\pi$
$173$	16.5734	1.26006	0.630028	+	0.776572i	$0.283043\pi$
$174$	0	0
$175$	−3.02828	−0.228916
$176$	−18.2036	−1.37215
$177$	0	0
$178$	−8.40586	−0.630045
$179$	−13.5386	−1.01192	−0.505961	−	0.862556i	$-0.668862\pi$
$179$	−13.5386	−1.01192	−0.505961	+	0.862556i	$0.668862\pi$
$180$	0	0
$181$	−3.59303	−0.267068	−0.133534	−	0.991044i	$-0.542632\pi$
$181$	−3.59303	−0.267068	−0.133534	+	0.991044i	$0.542632\pi$
$182$	5.96330	0.442029
$183$	0	0
$184$	0.710141	0.0523523
$185$	11.1045	0.816416
$186$	0	0
$187$	1.30541	0.0954612
$188$	12.9353	0.943404
$189$	0	0
$190$	−11.9119	−0.864181
$191$	−20.5414	−1.48632	−0.743161	−	0.669112i	$-0.766674\pi$
$191$	−20.5414	−1.48632	−0.743161	+	0.669112i	$0.766674\pi$
$192$	0	0
$193$	−1.52627	−0.109863	−0.0549316	−	0.998490i	$-0.517494\pi$
$193$	−1.52627	−0.109863	−0.0549316	+	0.998490i	$0.517494\pi$
$194$	11.4944	0.825250
$195$	0	0
$196$	4.07563	0.291117
$197$	1.07799	0.0768033	0.0384016	−	0.999262i	$-0.487773\pi$
$197$	1.07799	0.0768033	0.0384016	+	0.999262i	$0.487773\pi$
$198$	0	0
$199$	2.00263	0.141963	0.0709813	−	0.997478i	$-0.477387\pi$
$199$	2.00263	0.141963	0.0709813	+	0.997478i	$0.477387\pi$
$200$	−0.240686	−0.0170191
$201$	0	0
$202$	−20.9698	−1.47543
$203$	26.9073	1.88852
$204$	0	0
$205$	−8.19698	−0.572502
$206$	−35.2812	−2.45815
$207$	0	0
$208$	4.22951	0.293264
$209$	−26.0350	−1.80088
$210$	0	0
$211$	−0.235221	−0.0161933	−0.00809663	−	0.999967i	$-0.502577\pi$
$211$	−0.235221	−0.0161933	−0.00809663	+	0.999967i	$0.502577\pi$
$212$	−22.2134	−1.52562
$213$	0	0
$214$	2.20924	0.151021
$215$	2.91072	0.198509
$216$	0	0
$217$	−0.193778	−0.0131545
$218$	27.8186	1.88412
$219$	0	0
$220$	8.08184	0.544878
$221$	−0.303306	−0.0204026
$222$	0	0
$223$	0.448147	0.0300101	0.0150051	−	0.999887i	$-0.495224\pi$
$223$	0.448147	0.0300101	0.0150051	+	0.999887i	$0.495224\pi$
$224$	23.7641	1.58781
$225$	0	0
$226$	0.570315	0.0379368
$227$	9.79559	0.650156	0.325078	−	0.945687i	$-0.394609\pi$
$227$	9.79559	0.650156	0.325078	+	0.945687i	$0.394609\pi$
$228$	0	0
$229$	2.86231	0.189147	0.0945735	−	0.995518i	$-0.469851\pi$
$229$	2.86231	0.189147	0.0945735	+	0.995518i	$0.469851\pi$
$230$	−5.81012	−0.383108
$231$	0	0
$232$	2.13858	0.140405
$233$	15.5072	1.01591	0.507955	−	0.861384i	$-0.330402\pi$
$233$	15.5072	1.01591	0.507955	+	0.861384i	$0.330402\pi$
$234$	0	0
$235$	6.88864	0.449365
$236$	−17.1829	−1.11851
$237$	0	0
$238$	−1.80870	−0.117241
$239$	7.28109	0.470974	0.235487	−	0.971877i	$-0.424331\pi$
$239$	7.28109	0.470974	0.235487	+	0.971877i	$0.424331\pi$
$240$	0	0
$241$	10.9886	0.707840	0.353920	−	0.935276i	$-0.384848\pi$
$241$	10.9886	0.707840	0.353920	+	0.935276i	$0.384848\pi$
$242$	14.8162	0.952422
$243$	0	0
$244$	2.46751	0.157966
$245$	2.17046	0.138665
$246$	0	0
$247$	6.04909	0.384894
$248$	−0.0154014	−0.000977988 0
$249$	0	0
$250$	1.96921	0.124544
$251$	−9.53838	−0.602057	−0.301029	−	0.953615i	$-0.597330\pi$
$251$	−9.53838	−0.602057	−0.301029	+	0.953615i	$0.597330\pi$
$252$	0	0
$253$	−12.6987	−0.798364
$254$	9.26330	0.581231
$255$	0	0
$256$	17.7729	1.11081
$257$	−0.751099	−0.0468523	−0.0234261	−	0.999726i	$-0.507457\pi$
$257$	−0.751099	−0.0468523	−0.0234261	+	0.999726i	$0.507457\pi$
$258$	0	0
$259$	−33.6274	−2.08950
$260$	−1.87778	−0.116455
$261$	0	0
$262$	43.4921	2.68695
$263$	−20.8375	−1.28490	−0.642449	−	0.766329i	$-0.722081\pi$
$263$	−20.8375	−1.28490	−0.642449	+	0.766329i	$0.722081\pi$
$264$	0	0
$265$	−11.8296	−0.726689
$266$	36.0726	2.21175
$267$	0	0
$268$	−29.7276	−1.81590
$269$	15.5184	0.946176	0.473088	−	0.881015i	$-0.343139\pi$
$269$	15.5184	0.946176	0.473088	+	0.881015i	$0.343139\pi$
$270$	0	0
$271$	0.983023	0.0597144	0.0298572	−	0.999554i	$-0.490495\pi$
$271$	0.983023	0.0597144	0.0298572	+	0.999554i	$0.490495\pi$
$272$	−1.28283	−0.0777833
$273$	0	0
$274$	4.27085	0.258012
$275$	4.30395	0.259538
$276$	0	0
$277$	24.2823	1.45898	0.729492	−	0.683989i	$-0.239756\pi$
$277$	24.2823	1.45898	0.729492	+	0.683989i	$0.239756\pi$
$278$	−2.38838	−0.143246
$279$	0	0
$280$	0.728863	0.0435579
$281$	3.04044	0.181378	0.0906888	−	0.995879i	$-0.471093\pi$
$281$	3.04044	0.181378	0.0906888	+	0.995879i	$0.471093\pi$
$282$	0	0
$283$	−25.2618	−1.50166	−0.750829	−	0.660497i	$-0.770346\pi$
$283$	−25.2618	−1.50166	−0.750829	+	0.660497i	$0.770346\pi$
$284$	5.35927	0.318014
$285$	0	0
$286$	−8.47536	−0.501158
$287$	24.8227	1.46524
$288$	0	0
$289$	−16.9080	−0.994589
$290$	−17.4971	−1.02746
$291$	0	0
$292$	−21.1849	−1.23975
$293$	26.4117	1.54299	0.771495	−	0.636236i	$-0.219509\pi$
$293$	26.4117	1.54299	0.771495	+	0.636236i	$0.219509\pi$
$294$	0	0
$295$	−9.15068	−0.532773
$296$	−2.67269	−0.155347
$297$	0	0
$298$	−11.5596	−0.669632
$299$	2.95049	0.170631
$300$	0	0
$301$	−8.81445	−0.508057
$302$	−18.3224	−1.05434
$303$	0	0
$304$	25.5847	1.46738
$305$	1.31406	0.0752429
$306$	0	0
$307$	−15.3240	−0.874586	−0.437293	−	0.899319i	$-0.644063\pi$
$307$	−15.3240	−0.874586	−0.437293	+	0.899319i	$0.644063\pi$
$308$	−24.4741	−1.39454
$309$	0	0
$310$	0.126009	0.00715680
$311$	−2.91293	−0.165177	−0.0825884	−	0.996584i	$-0.526319\pi$
$311$	−2.91293	−0.165177	−0.0825884	+	0.996584i	$0.526319\pi$
$312$	0	0
$313$	−10.8911	−0.615602	−0.307801	−	0.951451i	$-0.599593\pi$
$313$	−10.8911	−0.615602	−0.307801	+	0.951451i	$0.599593\pi$
$314$	−36.0257	−2.03305
$315$	0	0
$316$	19.4530	1.09432
$317$	8.22471	0.461945	0.230973	−	0.972960i	$-0.425809\pi$
$317$	8.22471	0.461945	0.230973	+	0.972960i	$0.425809\pi$
$318$	0	0
$319$	−38.2421	−2.14115
$320$	−6.99415	−0.390985
$321$	0	0
$322$	17.5947	0.980512
$323$	−1.83472	−0.102087
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−23.5368	−1.30358
$327$	0	0
$328$	1.97290	0.108935
$329$	−20.8607	−1.15009
$330$	0	0
$331$	18.0502	0.992129	0.496064	−	0.868286i	$-0.334778\pi$
$331$	18.0502	0.992129	0.496064	+	0.868286i	$0.334778\pi$
$332$	13.2745	0.728533
$333$	0	0
$334$	−26.2492	−1.43629
$335$	−15.8313	−0.864955
$336$	0	0
$337$	36.6044	1.99397	0.996985	−	0.0775934i	$-0.0247236\pi$
$337$	36.6044	1.99397	0.996985	+	0.0775934i	$0.0247236\pi$
$338$	1.96921	0.107111
$339$	0	0
$340$	0.569540	0.0308876
$341$	0.275407	0.0149141
$342$	0	0
$343$	14.6252	0.789686
$344$	−0.700568	−0.0377721
$345$	0	0
$346$	32.6365	1.75455
$347$	−5.82933	−0.312935	−0.156467	−	0.987683i	$-0.550011\pi$
$347$	−5.82933	−0.312935	−0.156467	+	0.987683i	$0.550011\pi$
$348$	0	0
$349$	3.01712	0.161503	0.0807513	−	0.996734i	$-0.474268\pi$
$349$	3.01712	0.161503	0.0807513	+	0.996734i	$0.474268\pi$
$350$	−5.96330	−0.318752
$351$	0	0
$352$	−33.7748	−1.80020
$353$	9.36184	0.498280	0.249140	−	0.968467i	$-0.419852\pi$
$353$	9.36184	0.498280	0.249140	+	0.968467i	$0.419852\pi$
$354$	0	0
$355$	2.85405	0.151478
$356$	−8.01557	−0.424824
$357$	0	0
$358$	−26.6603	−1.40904
$359$	−5.22577	−0.275806	−0.137903	−	0.990446i	$-0.544036\pi$
$359$	−5.22577	−0.275806	−0.137903	+	0.990446i	$0.544036\pi$
$360$	0	0
$361$	17.5915	0.925868
$362$	−7.07542	−0.371876
$363$	0	0
$364$	5.68642	0.298050
$365$	−11.2819	−0.590521
$366$	0	0
$367$	−4.30163	−0.224543	−0.112272	−	0.993678i	$-0.535813\pi$
$367$	−4.30163	−0.224543	−0.112272	+	0.993678i	$0.535813\pi$
$368$	12.4791	0.650520
$369$	0	0
$370$	21.8670	1.13681
$371$	35.8234	1.85986
$372$	0	0
$373$	26.5300	1.37367	0.686836	−	0.726812i	$-0.258999\pi$
$373$	26.5300	1.37367	0.686836	+	0.726812i	$0.258999\pi$
$374$	2.57063	0.132924
$375$	0	0
$376$	−1.65800	−0.0855047
$377$	8.88535	0.457619
$378$	0	0
$379$	−3.48090	−0.178802	−0.0894008	−	0.995996i	$-0.528495\pi$
$379$	−3.48090	−0.178802	−0.0894008	+	0.995996i	$0.528495\pi$
$380$	−11.3588	−0.582696
$381$	0	0
$382$	−40.4502	−2.06961
$383$	−24.6376	−1.25892	−0.629461	−	0.777032i	$-0.716724\pi$
$383$	−24.6376	−1.25892	−0.629461	+	0.777032i	$0.716724\pi$
$384$	0	0
$385$	−13.0335	−0.664251
$386$	−3.00554	−0.152978
$387$	0	0
$388$	10.9607	0.556446
$389$	18.4191	0.933883	0.466942	−	0.884288i	$-0.345356\pi$
$389$	18.4191	0.933883	0.466942	+	0.884288i	$0.345356\pi$
$390$	0	0
$391$	−0.894901	−0.0452571
$392$	−0.522399	−0.0263851
$393$	0	0
$394$	2.12278	0.106944
$395$	10.3596	0.521248
$396$	0	0
$397$	15.1217	0.758934	0.379467	−	0.925205i	$-0.376107\pi$
$397$	15.1217	0.758934	0.379467	+	0.925205i	$0.376107\pi$
$398$	3.94359	0.197674
$399$	0	0
$400$	−4.22951	−0.211476
$401$	−0.275314	−0.0137485	−0.00687426	−	0.999976i	$-0.502188\pi$
$401$	−0.275314	−0.0137485	−0.00687426	+	0.999976i	$0.502188\pi$
$402$	0	0
$403$	−0.0639895	−0.00318754
$404$	−19.9962	−0.994847
$405$	0	0
$406$	52.9860	2.62965
$407$	47.7930	2.36901
$408$	0	0
$409$	−12.5525	−0.620682	−0.310341	−	0.950625i	$-0.600443\pi$
$409$	−12.5525	−0.620682	−0.310341	+	0.950625i	$0.600443\pi$
$410$	−16.1415	−0.797174
$411$	0	0
$412$	−33.6430	−1.65747
$413$	27.7108	1.36356
$414$	0	0
$415$	7.06927	0.347017
$416$	7.84741	0.384751
$417$	0	0
$418$	−51.2682	−2.50761
$419$	−22.8702	−1.11728	−0.558641	−	0.829409i	$-0.688677\pi$
$419$	−22.8702	−1.11728	−0.558641	+	0.829409i	$0.688677\pi$
$420$	0	0
$421$	−24.9758	−1.21724	−0.608622	−	0.793460i	$-0.708277\pi$
$421$	−24.9758	−1.21724	−0.608622	+	0.793460i	$0.708277\pi$
$422$	−0.463198	−0.0225481
$423$	0	0
$424$	2.84723	0.138274
$425$	0.303306	0.0147125
$426$	0	0
$427$	−3.97934	−0.192574
$428$	2.10667	0.101830
$429$	0	0
$430$	5.73180	0.276412
$431$	34.4612	1.65994	0.829969	−	0.557809i	$-0.188358\pi$
$431$	34.4612	1.65994	0.829969	+	0.557809i	$0.188358\pi$
$432$	0	0
$433$	15.6584	0.752495	0.376247	−	0.926519i	$-0.377214\pi$
$433$	15.6584	0.752495	0.376247	+	0.926519i	$0.377214\pi$
$434$	−0.381589	−0.0183168
$435$	0	0
$436$	26.5270	1.27041
$437$	17.8478	0.853775
$438$	0	0
$439$	−28.9267	−1.38060	−0.690299	−	0.723525i	$-0.742521\pi$
$439$	−28.9267	−1.38060	−0.690299	+	0.723525i	$0.742521\pi$
$440$	−1.03590	−0.0493845
$441$	0	0
$442$	−0.597272	−0.0284093
$443$	25.2012	1.19735	0.598673	−	0.800993i	$-0.295695\pi$
$443$	25.2012	1.19735	0.598673	+	0.800993i	$0.295695\pi$
$444$	0	0
$445$	−4.26865	−0.202353
$446$	0.882494	0.0417873
$447$	0	0
$448$	21.1802	1.00067
$449$	21.5777	1.01832	0.509158	−	0.860673i	$-0.329957\pi$
$449$	21.5777	1.01832	0.509158	+	0.860673i	$0.329957\pi$
$450$	0	0
$451$	−35.2794	−1.66124
$452$	0.543835	0.0255799
$453$	0	0
$454$	19.2895	0.905303
$455$	3.02828	0.141968
$456$	0	0
$457$	0.494576	0.0231353	0.0115676	−	0.999933i	$-0.496318\pi$
$457$	0.494576	0.0231353	0.0115676	+	0.999933i	$0.496318\pi$
$458$	5.63649	0.263376
$459$	0	0
$460$	−5.54036	−0.258320
$461$	−13.6336	−0.634980	−0.317490	−	0.948262i	$-0.602840\pi$
$461$	−13.6336	−0.634980	−0.317490	+	0.948262i	$0.602840\pi$
$462$	0	0
$463$	20.7556	0.964595	0.482297	−	0.876008i	$-0.339803\pi$
$463$	20.7556	0.964595	0.482297	+	0.876008i	$0.339803\pi$
$464$	37.5807	1.74464
$465$	0	0
$466$	30.5368	1.41459
$467$	34.9625	1.61787	0.808935	−	0.587898i	$-0.200044\pi$
$467$	34.9625	1.61787	0.808935	+	0.587898i	$0.200044\pi$
$468$	0	0
$469$	47.9415	2.21373
$470$	13.5651	0.625714
$471$	0	0
$472$	2.20244	0.101375
$473$	12.5276	0.576018
$474$	0	0
$475$	−6.04909	−0.277551
$476$	−1.72472	−0.0790526
$477$	0	0
$478$	14.3380	0.655803
$479$	−30.5754	−1.39702	−0.698512	−	0.715598i	$-0.746154\pi$
$479$	−30.5754	−1.39702	−0.698512	+	0.715598i	$0.746154\pi$
$480$	0	0
$481$	−11.1045	−0.506320
$482$	21.6389	0.985625
$483$	0	0
$484$	14.1283	0.642195
$485$	5.83707	0.265048
$486$	0	0
$487$	38.2764	1.73447	0.867235	−	0.497899i	$-0.165895\pi$
$487$	38.2764	1.73447	0.867235	+	0.497899i	$0.165895\pi$
$488$	−0.316276	−0.0143171
$489$	0	0
$490$	4.27408	0.193083
$491$	−6.60748	−0.298192	−0.149096	−	0.988823i	$-0.547636\pi$
$491$	−6.60748	−0.298192	−0.149096	+	0.988823i	$0.547636\pi$
$492$	0	0
$493$	−2.69498	−0.121376
$494$	11.9119	0.535942
$495$	0	0
$496$	−0.270644	−0.0121523
$497$	−8.64287	−0.387686
$498$	0	0
$499$	−22.0475	−0.986983	−0.493491	−	0.869751i	$-0.664280\pi$
$499$	−22.0475	−0.986983	−0.493491	+	0.869751i	$0.664280\pi$
$500$	1.87778	0.0839767
$501$	0	0
$502$	−18.7831	−0.838329
$503$	16.8779	0.752550	0.376275	−	0.926508i	$-0.377205\pi$
$503$	16.8779	0.752550	0.376275	+	0.926508i	$0.377205\pi$
$504$	0	0
$505$	−10.6489	−0.473868
$506$	−25.0065	−1.11167
$507$	0	0
$508$	8.83320	0.391910
$509$	22.9551	1.01746	0.508732	−	0.860925i	$-0.330114\pi$
$509$	22.9551	1.01746	0.508732	+	0.860925i	$0.330114\pi$
$510$	0	0
$511$	34.1647	1.51136
$512$	31.1547	1.37686
$513$	0	0
$514$	−1.47907	−0.0652390
$515$	−17.9164	−0.789492
$516$	0	0
$517$	29.6483	1.30393
$518$	−66.2192	−2.90951
$519$	0	0
$520$	0.240686	0.0105548
$521$	−10.6723	−0.467562	−0.233781	−	0.972289i	$-0.575110\pi$
$521$	−10.6723	−0.467562	−0.233781	+	0.972289i	$0.575110\pi$
$522$	0	0
$523$	−2.51360	−0.109912	−0.0549559	−	0.998489i	$-0.517502\pi$
$523$	−2.51360	−0.109912	−0.0549559	+	0.998489i	$0.517502\pi$
$524$	41.4727	1.81174
$525$	0	0
$526$	−41.0334	−1.78914
$527$	0.0194084	0.000845442 0
$528$	0	0
$529$	−14.2946	−0.621505
$530$	−23.2950	−1.01187
$531$	0	0
$532$	34.3977	1.49133
$533$	8.19698	0.355051
$534$	0	0
$535$	1.12190	0.0485038
$536$	3.81036	0.164583
$537$	0	0
$538$	30.5590	1.31749
$539$	9.34153	0.402368
$540$	0	0
$541$	−10.6855	−0.459404	−0.229702	−	0.973261i	$-0.573775\pi$
$541$	−10.6855	−0.459404	−0.229702	+	0.973261i	$0.573775\pi$
$542$	1.93578	0.0831487
$543$	0	0
$544$	−2.38016	−0.102049
$545$	14.1268	0.605127
$546$	0	0
$547$	−12.1228	−0.518332	−0.259166	−	0.965833i	$-0.583448\pi$
$547$	−12.1228	−0.518332	−0.259166	+	0.965833i	$0.583448\pi$
$548$	4.07255	0.173971
$549$	0	0
$550$	8.47536	0.361391
$551$	53.7483	2.28975
$552$	0	0
$553$	−31.3717	−1.33406
$554$	47.8170	2.03155
$555$	0	0
$556$	−2.27749	−0.0965871
$557$	−13.7918	−0.584377	−0.292189	−	0.956361i	$-0.594383\pi$
$557$	−13.7918	−0.584377	−0.292189	+	0.956361i	$0.594383\pi$
$558$	0	0
$559$	−2.91072	−0.123110
$560$	12.8081	0.541242
$561$	0	0
$562$	5.98726	0.252557
$563$	10.7491	0.453019	0.226509	−	0.974009i	$-0.427269\pi$
$563$	10.7491	0.453019	0.226509	+	0.974009i	$0.427269\pi$
$564$	0	0
$565$	0.289617	0.0121843
$566$	−49.7457	−2.09097
$567$	0	0
$568$	−0.686931	−0.0288230
$569$	−11.2612	−0.472096	−0.236048	−	0.971741i	$-0.575852\pi$
$569$	−11.2612	−0.472096	−0.236048	+	0.971741i	$0.575852\pi$
$570$	0	0
$571$	25.9816	1.08730	0.543648	−	0.839313i	$-0.317042\pi$
$571$	25.9816	1.08730	0.543648	+	0.839313i	$0.317042\pi$
$572$	−8.08184	−0.337919
$573$	0	0
$574$	48.8811	2.04026
$575$	−2.95049	−0.123044
$576$	0	0
$577$	5.83579	0.242947	0.121474	−	0.992595i	$-0.461238\pi$
$577$	5.83579	0.242947	0.121474	+	0.992595i	$0.461238\pi$
$578$	−33.2954	−1.38490
$579$	0	0
$580$	−16.6847	−0.692794
$581$	−21.4077	−0.888141
$582$	0	0
$583$	−50.9142	−2.10865
$584$	2.71539	0.112364
$585$	0	0
$586$	52.0101	2.14852
$587$	11.4088	0.470892	0.235446	−	0.971887i	$-0.424345\pi$
$587$	11.4088	0.470892	0.235446	+	0.971887i	$0.424345\pi$
$588$	0	0
$589$	−0.387078	−0.0159493
$590$	−18.0196	−0.741854
$591$	0	0
$592$	−46.9664	−1.93031
$593$	21.1795	0.869737	0.434869	−	0.900494i	$-0.356795\pi$
$593$	21.1795	0.869737	0.434869	+	0.900494i	$0.356795\pi$
$594$	0	0
$595$	−0.918494	−0.0376546
$596$	−11.0229	−0.451516
$597$	0	0
$598$	5.81012	0.237594
$599$	14.8324	0.606037	0.303019	−	0.952985i	$-0.402006\pi$
$599$	14.8324	0.606037	0.303019	+	0.952985i	$0.402006\pi$
$600$	0	0
$601$	13.2067	0.538711	0.269355	−	0.963041i	$-0.413189\pi$
$601$	13.2067	0.538711	0.269355	+	0.963041i	$0.413189\pi$
$602$	−17.3575	−0.707438
$603$	0	0
$604$	−17.4717	−0.710912
$605$	7.52395	0.305892
$606$	0	0
$607$	−33.4305	−1.35690	−0.678451	−	0.734646i	$-0.737348\pi$
$607$	−33.4305	−1.35690	−0.678451	+	0.734646i	$0.737348\pi$
$608$	47.4697	1.92515
$609$	0	0
$610$	2.58766	0.104771
$611$	−6.88864	−0.278684
$612$	0	0
$613$	−33.4817	−1.35231	−0.676157	−	0.736758i	$-0.736356\pi$
$613$	−33.4817	−1.35231	−0.676157	+	0.736758i	$0.736356\pi$
$614$	−30.1761	−1.21781
$615$	0	0
$616$	3.13699	0.126393
$617$	38.1133	1.53438	0.767191	−	0.641418i	$-0.221654\pi$
$617$	38.1133	1.53438	0.767191	+	0.641418i	$0.221654\pi$
$618$	0	0
$619$	42.7994	1.72025	0.860126	−	0.510082i	$-0.170385\pi$
$619$	42.7994	1.72025	0.860126	+	0.510082i	$0.170385\pi$
$620$	0.120158	0.00482566
$621$	0	0
$622$	−5.73615	−0.229999
$623$	12.9267	0.517895
$624$	0	0
$625$	1.00000	0.0400000
$626$	−21.4469	−0.857189
$627$	0	0
$628$	−34.3530	−1.37084
$629$	3.36805	0.134293
$630$	0	0
$631$	−34.5751	−1.37641	−0.688206	−	0.725516i	$-0.741601\pi$
$631$	−34.5751	−1.37641	−0.688206	+	0.725516i	$0.741601\pi$
$632$	−2.49341	−0.0991824
$633$	0	0
$634$	16.1961	0.643231
$635$	4.70408	0.186676
$636$	0	0
$637$	−2.17046	−0.0859967
$638$	−75.3066	−2.98142
$639$	0	0
$640$	1.92189	0.0759694
$641$	21.1887	0.836903	0.418452	−	0.908239i	$-0.362573\pi$
$641$	21.1887	0.836903	0.418452	+	0.908239i	$0.362573\pi$
$642$	0	0
$643$	25.9017	1.02146	0.510732	−	0.859740i	$-0.329374\pi$
$643$	25.9017	1.02146	0.510732	+	0.859740i	$0.329374\pi$
$644$	16.7777	0.661135
$645$	0	0
$646$	−3.61295	−0.142150
$647$	31.6341	1.24366	0.621832	−	0.783151i	$-0.286389\pi$
$647$	31.6341	1.24366	0.621832	+	0.783151i	$0.286389\pi$
$648$	0	0
$649$	−39.3840	−1.54596
$650$	−1.96921	−0.0772386
$651$	0	0
$652$	−22.4440	−0.878973
$653$	17.1268	0.670223	0.335112	−	0.942178i	$-0.391226\pi$
$653$	17.1268	0.670223	0.335112	+	0.942178i	$0.391226\pi$
$654$	0	0
$655$	22.0861	0.862975
$656$	34.6692	1.35361
$657$	0	0
$658$	−41.0790	−1.60143
$659$	−38.6509	−1.50562	−0.752812	−	0.658236i	$-0.771303\pi$
$659$	−38.6509	−1.50562	−0.752812	+	0.658236i	$0.771303\pi$
$660$	0	0
$661$	−44.2640	−1.72167	−0.860836	−	0.508883i	$-0.830059\pi$
$661$	−44.2640	−1.72167	−0.860836	+	0.508883i	$0.830059\pi$
$662$	35.5446	1.38148
$663$	0	0
$664$	−1.70147	−0.0660300
$665$	18.3183	0.710354
$666$	0	0
$667$	26.2161	1.01509
$668$	−25.0304	−0.968455
$669$	0	0
$670$	−31.1751	−1.20440
$671$	5.65565	0.218334
$672$	0	0
$673$	−28.4616	−1.09711	−0.548557	−	0.836113i	$-0.684823\pi$
$673$	−28.4616	−1.09711	−0.548557	+	0.836113i	$0.684823\pi$
$674$	72.0817	2.77648
$675$	0	0
$676$	1.87778	0.0722221
$677$	13.6371	0.524117	0.262058	−	0.965052i	$-0.415599\pi$
$677$	13.6371	0.524117	0.262058	+	0.965052i	$0.415599\pi$
$678$	0	0
$679$	−17.6763	−0.678353
$680$	−0.0730014	−0.00279948
$681$	0	0
$682$	0.542334	0.0207670
$683$	−47.3362	−1.81127	−0.905636	−	0.424057i	$-0.860606\pi$
$683$	−47.3362	−1.81127	−0.905636	+	0.424057i	$0.860606\pi$
$684$	0	0
$685$	2.16882	0.0828663
$686$	28.8000	1.09959
$687$	0	0
$688$	−12.3109	−0.469349
$689$	11.8296	0.450674
$690$	0	0
$691$	16.6234	0.632383	0.316191	−	0.948695i	$-0.397596\pi$
$691$	16.6234	0.632383	0.316191	+	0.948695i	$0.397596\pi$
$692$	31.1212	1.18305
$693$	0	0
$694$	−11.4791	−0.435743
$695$	−1.21287	−0.0460066
$696$	0	0
$697$	−2.48619	−0.0941712
$698$	5.94133	0.224883
$699$	0	0
$700$	−5.68642	−0.214927
$701$	−21.7166	−0.820223	−0.410111	−	0.912035i	$-0.634510\pi$
$701$	−21.7166	−0.820223	−0.410111	+	0.912035i	$0.634510\pi$
$702$	0	0
$703$	−67.1719	−2.53343
$704$	−30.1024	−1.13453
$705$	0	0
$706$	18.4354	0.693825
$707$	32.2477	1.21280
$708$	0	0
$709$	−40.2986	−1.51345	−0.756723	−	0.653735i	$-0.773201\pi$
$709$	−40.2986	−1.51345	−0.756723	+	0.653735i	$0.773201\pi$
$710$	5.62022	0.210923
$711$	0	0
$712$	1.02740	0.0385036
$713$	−0.188800	−0.00707063
$714$	0	0
$715$	−4.30395	−0.160958
$716$	−25.4225	−0.950082
$717$	0	0
$718$	−10.2906	−0.384043
$719$	−42.5521	−1.58693	−0.793463	−	0.608619i	$-0.791724\pi$
$719$	−42.5521	−1.58693	−0.793463	+	0.608619i	$0.791724\pi$
$720$	0	0
$721$	54.2559	2.02060
$722$	34.6413	1.28922
$723$	0	0
$724$	−6.74690	−0.250747
$725$	−8.88535	−0.329994
$726$	0	0
$727$	−22.7167	−0.842517	−0.421259	−	0.906941i	$-0.638412\pi$
$727$	−22.7167	−0.842517	−0.421259	+	0.906941i	$0.638412\pi$
$728$	−0.728863	−0.0270135
$729$	0	0
$730$	−22.2164	−0.822265
$731$	0.882837	0.0326529
$732$	0	0
$733$	−28.5083	−1.05298	−0.526490	−	0.850181i	$-0.676492\pi$
$733$	−28.5083	−1.05298	−0.526490	+	0.850181i	$0.676492\pi$
$734$	−8.47080	−0.312663
$735$	0	0
$736$	23.1537	0.853457
$737$	−68.1370	−2.50986
$738$	0	0
$739$	−10.2174	−0.375852	−0.187926	−	0.982183i	$-0.560177\pi$
$739$	−10.2174	−0.375852	−0.187926	+	0.982183i	$0.560177\pi$
$740$	20.8517	0.766523
$741$	0	0
$742$	70.5438	2.58974
$743$	−52.6674	−1.93218	−0.966090	−	0.258205i	$-0.916869\pi$
$743$	−52.6674	−1.93218	−0.966090	+	0.258205i	$0.916869\pi$
$744$	0	0
$745$	−5.87020	−0.215068
$746$	52.2431	1.91276
$747$	0	0
$748$	2.45127	0.0896273
$749$	−3.39741	−0.124139
$750$	0	0
$751$	0.0431881	0.00157596	0.000787978	−	1.00000i	$-0.499749\pi$
$751$	0.0431881	0.00157596	0.000787978		1.00000i	$0.499749\pi$
$752$	−29.1356	−1.06246
$753$	0	0
$754$	17.4971	0.637207
$755$	−9.30445	−0.338624
$756$	0	0
$757$	−20.9410	−0.761113	−0.380557	−	0.924758i	$-0.624268\pi$
$757$	−20.9410	−0.761113	−0.380557	+	0.924758i	$0.624268\pi$
$758$	−6.85460	−0.248970
$759$	0	0
$760$	1.45593	0.0528122
$761$	−15.2220	−0.551797	−0.275899	−	0.961187i	$-0.588975\pi$
$761$	−15.2220	−0.551797	−0.275899	+	0.961187i	$0.588975\pi$
$762$	0	0
$763$	−42.7799	−1.54874
$764$	−38.5721	−1.39549
$765$	0	0
$766$	−48.5165	−1.75297
$767$	9.15068	0.330412
$768$	0	0
$769$	−16.2583	−0.586291	−0.293145	−	0.956068i	$-0.594702\pi$
$769$	−16.2583	−0.586291	−0.293145	+	0.956068i	$0.594702\pi$
$770$	−25.6657	−0.924929
$771$	0	0
$772$	−2.86599	−0.103149
$773$	−14.6825	−0.528093	−0.264047	−	0.964510i	$-0.585057\pi$
$773$	−14.6825	−0.528093	−0.264047	+	0.964510i	$0.585057\pi$
$774$	0	0
$775$	0.0639895	0.00229857
$776$	−1.40490	−0.0504330
$777$	0	0
$778$	36.2709	1.30038
$779$	49.5843	1.77654
$780$	0	0
$781$	12.2837	0.439545
$782$	−1.76224	−0.0630177
$783$	0	0
$784$	−9.17997	−0.327856
$785$	−18.2945	−0.652960
$786$	0	0
$787$	−18.6828	−0.665970	−0.332985	−	0.942932i	$-0.608056\pi$
$787$	−18.6828	−0.665970	−0.332985	+	0.942932i	$0.608056\pi$
$788$	2.02421	0.0721097
$789$	0	0
$790$	20.4002	0.725806
$791$	−0.877040	−0.0311839
$792$	0	0
$793$	−1.31406	−0.0466637
$794$	29.7777	1.05677
$795$	0	0
$796$	3.76049	0.133287
$797$	22.2679	0.788770	0.394385	−	0.918945i	$-0.370958\pi$
$797$	22.2679	0.788770	0.394385	+	0.918945i	$0.370958\pi$
$798$	0	0
$799$	2.08936	0.0739163
$800$	−7.84741	−0.277448
$801$	0	0
$802$	−0.542150	−0.0191440
$803$	−48.5567	−1.71353
$804$	0	0
$805$	8.93490	0.314914
$806$	−0.126009	−0.00443846
$807$	0	0
$808$	2.56303	0.0901671
$809$	10.6028	0.372776	0.186388	−	0.982476i	$-0.440322\pi$
$809$	10.6028	0.372776	0.186388	+	0.982476i	$0.440322\pi$
$810$	0	0
$811$	−29.8215	−1.04718	−0.523588	−	0.851972i	$-0.675407\pi$
$811$	−29.8215	−1.04718	−0.523588	+	0.851972i	$0.675407\pi$
$812$	50.5259	1.77311
$813$	0	0
$814$	94.1143	3.29870
$815$	−11.9524	−0.418675
$816$	0	0
$817$	−17.6072	−0.615997
$818$	−24.7185	−0.864262
$819$	0	0
$820$	−15.3921	−0.537515
$821$	−19.2272	−0.671033	−0.335516	−	0.942034i	$-0.608911\pi$
$821$	−19.2272	−0.671033	−0.335516	+	0.942034i	$0.608911\pi$
$822$	0	0
$823$	12.0589	0.420346	0.210173	−	0.977664i	$-0.432597\pi$
$823$	12.0589	0.420346	0.210173	+	0.977664i	$0.432597\pi$
$824$	4.31223	0.150224
$825$	0	0
$826$	54.5683	1.89867
$827$	−12.5315	−0.435765	−0.217882	−	0.975975i	$-0.569915\pi$
$827$	−12.5315	−0.435765	−0.217882	+	0.975975i	$0.569915\pi$
$828$	0	0
$829$	−44.7564	−1.55445	−0.777227	−	0.629220i	$-0.783374\pi$
$829$	−44.7564	−1.55445	−0.777227	+	0.629220i	$0.783374\pi$
$830$	13.9209	0.483200
$831$	0	0
$832$	6.99415	0.242479
$833$	0.658312	0.0228092
$834$	0	0
$835$	−13.3298	−0.461297
$836$	−48.8878	−1.69082
$837$	0	0
$838$	−45.0361	−1.55575
$839$	19.9128	0.687466	0.343733	−	0.939067i	$-0.388309\pi$
$839$	19.9128	0.687466	0.343733	+	0.939067i	$0.388309\pi$
$840$	0	0
$841$	49.9495	1.72240
$842$	−49.1824	−1.69494
$843$	0	0
$844$	−0.441692	−0.0152037
$845$	1.00000	0.0344010
$846$	0	0
$847$	−22.7846	−0.782888
$848$	50.0336	1.71816
$849$	0	0
$850$	0.597272	0.0204863
$851$	−32.7636	−1.12312
$852$	0	0
$853$	−3.21977	−0.110243	−0.0551215	−	0.998480i	$-0.517555\pi$
$853$	−3.21977	−0.110243	−0.0551215	+	0.998480i	$0.517555\pi$
$854$	−7.83614	−0.268147
$855$	0	0
$856$	−0.270024	−0.00922924
$857$	3.17474	0.108447	0.0542236	−	0.998529i	$-0.482732\pi$
$857$	3.17474	0.108447	0.0542236	+	0.998529i	$0.482732\pi$
$858$	0	0
$859$	25.6653	0.875688	0.437844	−	0.899051i	$-0.355742\pi$
$859$	25.6653	0.875688	0.437844	+	0.899051i	$0.355742\pi$
$860$	5.46567	0.186378
$861$	0	0
$862$	67.8613	2.31136
$863$	40.8808	1.39160	0.695799	−	0.718236i	$-0.255050\pi$
$863$	40.8808	1.39160	0.695799	+	0.718236i	$0.255050\pi$
$864$	0	0
$865$	16.5734	0.563514
$866$	30.8346	1.04780
$867$	0	0
$868$	−0.363871	−0.0123506
$869$	44.5871	1.51251
$870$	0	0
$871$	15.8313	0.536422
$872$	−3.40013	−0.115143
$873$	0	0
$874$	35.1460	1.18883
$875$	−3.02828	−0.102374
$876$	0	0
$877$	49.4950	1.67133	0.835663	−	0.549242i	$-0.185084\pi$
$877$	49.4950	1.67133	0.835663	+	0.549242i	$0.185084\pi$
$878$	−56.9627	−1.92240
$879$	0	0
$880$	−18.2036	−0.613643
$881$	46.0194	1.55043	0.775216	−	0.631696i	$-0.217641\pi$
$881$	46.0194	1.55043	0.775216	+	0.631696i	$0.217641\pi$
$882$	0	0
$883$	−23.7208	−0.798267	−0.399134	−	0.916893i	$-0.630689\pi$
$883$	−23.7208	−0.798267	−0.399134	+	0.916893i	$0.630689\pi$
$884$	−0.569540	−0.0191557
$885$	0	0
$886$	49.6264	1.66723
$887$	−54.3335	−1.82434	−0.912170	−	0.409812i	$-0.865594\pi$
$887$	−54.3335	−1.82434	−0.912170	+	0.409812i	$0.865594\pi$
$888$	0	0
$889$	−14.2452	−0.477770
$890$	−8.40586	−0.281765
$891$	0	0
$892$	0.841519	0.0281761
$893$	−41.6700	−1.39443
$894$	0	0
$895$	−13.5386	−0.452546
$896$	−5.82002	−0.194433
$897$	0	0
$898$	42.4910	1.41794
$899$	−0.568569	−0.0189628
$900$	0	0
$901$	−3.58800	−0.119534
$902$	−69.4723	−2.31318
$903$	0	0
$904$	−0.0697067	−0.00231841
$905$	−3.59303	−0.119436
$906$	0	0
$907$	42.8928	1.42423	0.712116	−	0.702062i	$-0.247737\pi$
$907$	42.8928	1.42423	0.712116	+	0.702062i	$0.247737\pi$
$908$	18.3939	0.610423
$909$	0	0
$910$	5.96330	0.197682
$911$	−27.1213	−0.898568	−0.449284	−	0.893389i	$-0.648321\pi$
$911$	−27.1213	−0.898568	−0.449284	+	0.893389i	$0.648321\pi$
$912$	0	0
$913$	30.4258	1.00695
$914$	0.973922	0.0322145
$915$	0	0
$916$	5.37478	0.177588
$917$	−66.8828	−2.20866
$918$	0	0
$919$	2.27072	0.0749043	0.0374522	−	0.999298i	$-0.488076\pi$
$919$	2.27072	0.0749043	0.0374522	+	0.999298i	$0.488076\pi$
$920$	0.710141	0.0234127
$921$	0	0
$922$	−26.8474	−0.884171
$923$	−2.85405	−0.0939424
$924$	0	0
$925$	11.1045	0.365112
$926$	40.8721	1.34314
$927$	0	0
$928$	69.7270	2.28890
$929$	−0.471876	−0.0154818	−0.00774088	−	0.999970i	$-0.502464\pi$
$929$	−0.471876	−0.0154818	−0.00774088	+	0.999970i	$0.502464\pi$
$930$	0	0
$931$	−13.1293	−0.430295
$932$	29.1190	0.953825
$933$	0	0
$934$	68.8483	2.25279
$935$	1.30541	0.0426915
$936$	0	0
$937$	44.6044	1.45716	0.728581	−	0.684960i	$-0.240180\pi$
$937$	44.6044	1.45716	0.728581	+	0.684960i	$0.240180\pi$
$938$	94.4067	3.08249
$939$	0	0
$940$	12.9353	0.421903
$941$	−22.5025	−0.733562	−0.366781	−	0.930307i	$-0.619540\pi$
$941$	−22.5025	−0.733562	−0.366781	+	0.930307i	$0.619540\pi$
$942$	0	0
$943$	24.1851	0.787575
$944$	38.7029	1.25967
$945$	0	0
$946$	24.6694	0.802070
$947$	14.2679	0.463645	0.231822	−	0.972758i	$-0.425531\pi$
$947$	14.2679	0.463645	0.231822	+	0.972758i	$0.425531\pi$
$948$	0	0
$949$	11.2819	0.366226
$950$	−11.9119	−0.386473
$951$	0	0
$952$	0.221068	0.00716487
$953$	49.4934	1.60325	0.801624	−	0.597829i	$-0.203970\pi$
$953$	49.4934	1.60325	0.801624	+	0.597829i	$0.203970\pi$
$954$	0	0
$955$	−20.5414	−0.664704
$956$	13.6722	0.442192
$957$	0	0
$958$	−60.2093	−1.94527
$959$	−6.56778	−0.212085
$960$	0	0
$961$	−30.9959	−0.999868
$962$	−21.8670	−0.705020
$963$	0	0
$964$	20.6342	0.664583
$965$	−1.52627	−0.0491324
$966$	0	0
$967$	29.3646	0.944301	0.472151	−	0.881518i	$-0.343478\pi$
$967$	29.3646	0.944301	0.472151	+	0.881518i	$0.343478\pi$
$968$	−1.81091	−0.0582048
$969$	0	0
$970$	11.4944	0.369063
$971$	7.22467	0.231851	0.115925	−	0.993258i	$-0.463017\pi$
$971$	7.22467	0.231851	0.115925	+	0.993258i	$0.463017\pi$
$972$	0	0
$973$	3.67289	0.117748
$974$	75.3742	2.41514
$975$	0	0
$976$	−5.55784	−0.177902
$977$	40.7149	1.30258	0.651292	−	0.758827i	$-0.274227\pi$
$977$	40.7149	1.30258	0.651292	+	0.758827i	$0.274227\pi$
$978$	0	0
$979$	−18.3720	−0.587173
$980$	4.07563	0.130191
$981$	0	0
$982$	−13.0115	−0.415214
$983$	34.5286	1.10129	0.550646	−	0.834739i	$-0.314381\pi$
$983$	34.5286	1.10129	0.550646	+	0.834739i	$0.314381\pi$
$984$	0	0
$985$	1.07799	0.0343475
$986$	−5.30697	−0.169008
$987$	0	0
$988$	11.3588	0.361373
$989$	−8.58804	−0.273084
$990$	0	0
$991$	−14.9325	−0.474346	−0.237173	−	0.971467i	$-0.576221\pi$
$991$	−14.9325	−0.474346	−0.237173	+	0.971467i	$0.576221\pi$
$992$	−0.502152	−0.0159433
$993$	0	0
$994$	−17.0196	−0.539829
$995$	2.00263	0.0634876
$996$	0	0
$997$	19.4908	0.617279	0.308640	−	0.951179i	$-0.400126\pi$
$997$	19.4908	0.617279	0.308640	+	0.951179i	$0.400126\pi$
$998$	−43.4161	−1.37431
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.ba.1.7		8
3.2	odd	2		5265.2.a.bf.1.2		8
9.2	odd	6		585.2.i.e.391.7	yes	16
9.4	even	3		1755.2.i.f.586.2		16
9.5	odd	6		585.2.i.e.196.7	✓	16
9.7	even	3		1755.2.i.f.1171.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.e.196.7	✓	16	9.5	odd	6
585.2.i.e.391.7	yes	16	9.2	odd	6
1755.2.i.f.586.2		16	9.4	even	3
1755.2.i.f.1171.2		16	9.7	even	3
5265.2.a.ba.1.7		8	1.1	even	1	trivial
5265.2.a.bf.1.2		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)

\(f(q)\)	\(=\)	\(q+1.96921 q^{2} +1.87778 q^{4} +1.00000 q^{5} -3.02828 q^{7} -0.240686 q^{8} +O(q^{10})\)	\(q+1.96921 q^{2} +1.87778 q^{4} +1.00000 q^{5} -3.02828 q^{7} -0.240686 q^{8} +1.96921 q^{10} +4.30395 q^{11} -1.00000 q^{13} -5.96330 q^{14} -4.22951 q^{16} +0.303306 q^{17} -6.04909 q^{19} +1.87778 q^{20} +8.47536 q^{22} -2.95049 q^{23} +1.00000 q^{25} -1.96921 q^{26} -5.68642 q^{28} -8.88535 q^{29} +0.0639895 q^{31} -7.84741 q^{32} +0.597272 q^{34} -3.02828 q^{35} +11.1045 q^{37} -11.9119 q^{38} -0.240686 q^{40} -8.19698 q^{41} +2.91072 q^{43} +8.08184 q^{44} -5.81012 q^{46} +6.88864 q^{47} +2.17046 q^{49} +1.96921 q^{50} -1.87778 q^{52} -11.8296 q^{53} +4.30395 q^{55} +0.728863 q^{56} -17.4971 q^{58} -9.15068 q^{59} +1.31406 q^{61} +0.126009 q^{62} -6.99415 q^{64} -1.00000 q^{65} -15.8313 q^{67} +0.569540 q^{68} -5.96330 q^{70} +2.85405 q^{71} -11.2819 q^{73} +21.8670 q^{74} -11.3588 q^{76} -13.0335 q^{77} +10.3596 q^{79} -4.22951 q^{80} -16.1415 q^{82} +7.06927 q^{83} +0.303306 q^{85} +5.73180 q^{86} -1.03590 q^{88} -4.26865 q^{89} +3.02828 q^{91} -5.54036 q^{92} +13.5651 q^{94} -6.04909 q^{95} +5.83707 q^{97} +4.27408 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8}+O(q^{10}) \) 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

Introduction

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