LMFDB - Embedded newform 5265.2.a.bf.1.5

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			1.5
Root			$0.655066$ 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)

$f(q)$	$=$	$q+0.655066 q^{2} -1.57089 q^{4} -1.00000 q^{5} -0.777184 q^{7} -2.33917 q^{8} +O(q^{10})$	\(q+0.655066 q^{2} -1.57089 q^{4} -1.00000 q^{5} -0.777184 q^{7} -2.33917 q^{8} -0.655066 q^{10} -4.06797 q^{11} -1.00000 q^{13} -0.509106 q^{14} +1.60947 q^{16} +5.29154 q^{17} +6.65042 q^{19} +1.57089 q^{20} -2.66479 q^{22} +2.30567 q^{23} +1.00000 q^{25} -0.655066 q^{26} +1.22087 q^{28} +6.38016 q^{29} -8.94517 q^{31} +5.73264 q^{32} +3.46630 q^{34} +0.777184 q^{35} -2.80592 q^{37} +4.35646 q^{38} +2.33917 q^{40} +7.60986 q^{41} +8.84366 q^{43} +6.39034 q^{44} +1.51036 q^{46} -4.08782 q^{47} -6.39599 q^{49} +0.655066 q^{50} +1.57089 q^{52} -4.78025 q^{53} +4.06797 q^{55} +1.81796 q^{56} +4.17942 q^{58} -4.74099 q^{59} -14.3660 q^{61} -5.85967 q^{62} +0.536314 q^{64} +1.00000 q^{65} +10.7871 q^{67} -8.31242 q^{68} +0.509106 q^{70} +0.307776 q^{71} -7.50636 q^{73} -1.83806 q^{74} -10.4471 q^{76} +3.16156 q^{77} +7.61897 q^{79} -1.60947 q^{80} +4.98496 q^{82} -12.5297 q^{83} -5.29154 q^{85} +5.79318 q^{86} +9.51567 q^{88} -11.7743 q^{89} +0.777184 q^{91} -3.62195 q^{92} -2.67779 q^{94} -6.65042 q^{95} -6.08536 q^{97} -4.18979 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.655066	0.463201	0.231601	−	0.972811i	$-0.425604\pi$
$2$	0.655066	0.463201	0.231601	+	0.972811i	$0.425604\pi$
$3$	0	0
$3$	0	0
$4$	−1.57089	−0.785445
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.777184	−0.293748	−0.146874	−	0.989155i	$-0.546921\pi$
$7$	−0.777184	−0.293748	−0.146874	+	0.989155i	$0.546921\pi$
$8$	−2.33917	−0.827020
$9$	0	0
$10$	−0.655066	−0.207150
$11$	−4.06797	−1.22654	−0.613270	−	0.789873i	$-0.710146\pi$
$11$	−4.06797	−1.22654	−0.613270	+	0.789873i	$0.710146\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.509106	−0.136064
$15$	0	0
$16$	1.60947	0.402368
$17$	5.29154	1.28339	0.641693	−	0.766961i	$-0.278232\pi$
$17$	5.29154	1.28339	0.641693	+	0.766961i	$0.278232\pi$
$18$	0	0
$19$	6.65042	1.52571	0.762855	−	0.646569i	$-0.223797\pi$
$19$	6.65042	1.52571	0.762855	+	0.646569i	$0.223797\pi$
$20$	1.57089	0.351261
$21$	0	0
$22$	−2.66479	−0.568135
$23$	2.30567	0.480765	0.240383	−	0.970678i	$-0.422727\pi$
$23$	2.30567	0.480765	0.240383	+	0.970678i	$0.422727\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.655066	−0.128469
$27$	0	0
$28$	1.22087	0.230723
$29$	6.38016	1.18477	0.592383	−	0.805656i	$-0.298187\pi$
$29$	6.38016	1.18477	0.592383	+	0.805656i	$0.298187\pi$
$30$	0	0
$31$	−8.94517	−1.60660	−0.803300	−	0.595574i	$-0.796925\pi$
$31$	−8.94517	−1.60660	−0.803300	+	0.595574i	$0.796925\pi$
$32$	5.73264	1.01340
$33$	0	0
$34$	3.46630	0.594466
$35$	0.777184	0.131368
$36$	0	0
$37$	−2.80592	−0.461291	−0.230645	−	0.973038i	$-0.574084\pi$
$37$	−2.80592	−0.461291	−0.230645	+	0.973038i	$0.574084\pi$
$38$	4.35646	0.706711
$39$	0	0
$40$	2.33917	0.369855
$41$	7.60986	1.18846	0.594231	−	0.804295i	$-0.297457\pi$
$41$	7.60986	1.18846	0.594231	+	0.804295i	$0.297457\pi$
$42$	0	0
$43$	8.84366	1.34865	0.674323	−	0.738437i	$-0.264436\pi$
$43$	8.84366	1.34865	0.674323	+	0.738437i	$0.264436\pi$
$44$	6.39034	0.963379
$45$	0	0
$46$	1.51036	0.222691
$47$	−4.08782	−0.596270	−0.298135	−	0.954524i	$-0.596365\pi$
$47$	−4.08782	−0.596270	−0.298135	+	0.954524i	$0.596365\pi$
$48$	0	0
$49$	−6.39599	−0.913712
$50$	0.655066	0.0926403
$51$	0	0
$52$	1.57089	0.217843
$53$	−4.78025	−0.656618	−0.328309	−	0.944570i	$-0.606479\pi$
$53$	−4.78025	−0.656618	−0.328309	+	0.944570i	$0.606479\pi$
$54$	0	0
$55$	4.06797	0.548525
$56$	1.81796	0.242935
$57$	0	0
$58$	4.17942	0.548785
$59$	−4.74099	−0.617224	−0.308612	−	0.951188i	$-0.599864\pi$
$59$	−4.74099	−0.617224	−0.308612	+	0.951188i	$0.599864\pi$
$60$	0	0
$61$	−14.3660	−1.83938	−0.919688	−	0.392651i	$-0.871558\pi$
$61$	−14.3660	−1.83938	−0.919688	+	0.392651i	$0.871558\pi$
$62$	−5.85967	−0.744179
$63$	0	0
$64$	0.536314	0.0670393
$65$	1.00000	0.124035
$66$	0	0
$67$	10.7871	1.31785	0.658925	−	0.752209i	$-0.271011\pi$
$67$	10.7871	1.31785	0.658925	+	0.752209i	$0.271011\pi$
$68$	−8.31242	−1.00803
$69$	0	0
$70$	0.509106	0.0608498
$71$	0.307776	0.0365262	0.0182631	−	0.999833i	$-0.494186\pi$
$71$	0.307776	0.0365262	0.0182631	+	0.999833i	$0.494186\pi$
$72$	0	0
$73$	−7.50636	−0.878553	−0.439277	−	0.898352i	$-0.644765\pi$
$73$	−7.50636	−0.878553	−0.439277	+	0.898352i	$0.644765\pi$
$74$	−1.83806	−0.213670
$75$	0	0
$76$	−10.4471	−1.19836
$77$	3.16156	0.360293
$78$	0	0
$79$	7.61897	0.857201	0.428601	−	0.903494i	$-0.359007\pi$
$79$	7.61897	0.857201	0.428601	+	0.903494i	$0.359007\pi$
$80$	−1.60947	−0.179944
$81$	0	0
$82$	4.98496	0.550497
$83$	−12.5297	−1.37531	−0.687656	−	0.726037i	$-0.741360\pi$
$83$	−12.5297	−1.37531	−0.687656	+	0.726037i	$0.741360\pi$
$84$	0	0
$85$	−5.29154	−0.573948
$86$	5.79318	0.624694
$87$	0	0
$88$	9.51567	1.01437
$89$	−11.7743	−1.24807	−0.624037	−	0.781394i	$-0.714509\pi$
$89$	−11.7743	−1.24807	−0.624037	+	0.781394i	$0.714509\pi$
$90$	0	0
$91$	0.777184	0.0814710
$92$	−3.62195	−0.377615
$93$	0	0
$94$	−2.67779	−0.276193
$95$	−6.65042	−0.682318
$96$	0	0
$97$	−6.08536	−0.617874	−0.308937	−	0.951082i	$-0.599973\pi$
$97$	−6.08536	−0.617874	−0.308937	+	0.951082i	$0.599973\pi$
$98$	−4.18979	−0.423233
$99$	0	0
$100$	−1.57089	−0.157089
$101$	9.35209	0.930568	0.465284	−	0.885162i	$-0.345952\pi$
$101$	9.35209	0.930568	0.465284	+	0.885162i	$0.345952\pi$
$102$	0	0
$103$	−2.21467	−0.218218	−0.109109	−	0.994030i	$-0.534800\pi$
$103$	−2.21467	−0.218218	−0.109109	+	0.994030i	$0.534800\pi$
$104$	2.33917	0.229374
$105$	0	0
$106$	−3.13138	−0.304146
$107$	−15.4985	−1.49830	−0.749148	−	0.662403i	$-0.769537\pi$
$107$	−15.4985	−1.49830	−0.749148	+	0.662403i	$0.769537\pi$
$108$	0	0
$109$	−5.88278	−0.563468	−0.281734	−	0.959493i	$-0.590910\pi$
$109$	−5.88278	−0.563468	−0.281734	+	0.959493i	$0.590910\pi$
$110$	2.66479	0.254078
$111$	0	0
$112$	−1.25085	−0.118195
$113$	10.5758	0.994888	0.497444	−	0.867496i	$-0.334272\pi$
$113$	10.5758	0.994888	0.497444	+	0.867496i	$0.334272\pi$
$114$	0	0
$115$	−2.30567	−0.215005
$116$	−10.0225	−0.930568
$117$	0	0
$118$	−3.10566	−0.285899
$119$	−4.11250	−0.376992
$120$	0	0
$121$	5.54841	0.504401
$122$	−9.41066	−0.852001
$123$	0	0
$124$	14.0519	1.26190
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−11.4873	−1.01934	−0.509668	−	0.860371i	$-0.670232\pi$
$127$	−11.4873	−1.01934	−0.509668	+	0.860371i	$0.670232\pi$
$128$	−11.1140	−0.982345
$129$	0	0
$130$	0.655066	0.0574530
$131$	12.5793	1.09906	0.549531	−	0.835473i	$-0.314806\pi$
$131$	12.5793	1.09906	0.549531	+	0.835473i	$0.314806\pi$
$132$	0	0
$133$	−5.16860	−0.448174
$134$	7.06624	0.610430
$135$	0	0
$136$	−12.3778	−1.06139
$137$	−1.81204	−0.154813	−0.0774067	−	0.997000i	$-0.524664\pi$
$137$	−1.81204	−0.154813	−0.0774067	+	0.997000i	$0.524664\pi$
$138$	0	0
$139$	−20.4105	−1.73119	−0.865597	−	0.500741i	$-0.833061\pi$
$139$	−20.4105	−1.73119	−0.865597	+	0.500741i	$0.833061\pi$
$140$	−1.22087	−0.103182
$141$	0	0
$142$	0.201613	0.0169190
$143$	4.06797	0.340181
$144$	0	0
$145$	−6.38016	−0.529843
$146$	−4.91716	−0.406947
$147$	0	0
$148$	4.40779	0.362318
$149$	−0.464119	−0.0380221	−0.0190111	−	0.999819i	$-0.506052\pi$
$149$	−0.464119	−0.0380221	−0.0190111	+	0.999819i	$0.506052\pi$
$150$	0	0
$151$	−9.96476	−0.810921	−0.405460	−	0.914113i	$-0.632889\pi$
$151$	−9.96476	−0.810921	−0.405460	+	0.914113i	$0.632889\pi$
$152$	−15.5564	−1.26179
$153$	0	0
$154$	2.07103	0.166888
$155$	8.94517	0.718494
$156$	0	0
$157$	−9.31971	−0.743794	−0.371897	−	0.928274i	$-0.621292\pi$
$157$	−9.31971	−0.743794	−0.371897	+	0.928274i	$0.621292\pi$
$158$	4.99093	0.397057
$159$	0	0
$160$	−5.73264	−0.453205
$161$	−1.79193	−0.141224
$162$	0	0
$163$	8.93727	0.700021	0.350010	−	0.936746i	$-0.386178\pi$
$163$	8.93727	0.700021	0.350010	+	0.936746i	$0.386178\pi$
$164$	−11.9543	−0.933470
$165$	0	0
$166$	−8.20776	−0.637046
$167$	−16.3450	−1.26482	−0.632409	−	0.774635i	$-0.717934\pi$
$167$	−16.3450	−1.26482	−0.632409	+	0.774635i	$0.717934\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−3.46630	−0.265853
$171$	0	0
$172$	−13.8924	−1.05929
$173$	11.0008	0.836374	0.418187	−	0.908361i	$-0.362666\pi$
$173$	11.0008	0.836374	0.418187	+	0.908361i	$0.362666\pi$
$174$	0	0
$175$	−0.777184	−0.0587496
$176$	−6.54728	−0.493520
$177$	0	0
$178$	−7.71295	−0.578110
$179$	−2.87227	−0.214683	−0.107342	−	0.994222i	$-0.534234\pi$
$179$	−2.87227	−0.214683	−0.107342	+	0.994222i	$0.534234\pi$
$180$	0	0
$181$	−10.8459	−0.806169	−0.403084	−	0.915163i	$-0.632062\pi$
$181$	−10.8459	−0.806169	−0.403084	+	0.915163i	$0.632062\pi$
$182$	0.509106	0.0377375
$183$	0	0
$184$	−5.39335	−0.397603
$185$	2.80592	0.206296
$186$	0	0
$187$	−21.5258	−1.57413
$188$	6.42151	0.468337
$189$	0	0
$190$	−4.35646	−0.316051
$191$	13.8954	1.00543	0.502716	−	0.864452i	$-0.332334\pi$
$191$	13.8954	1.00543	0.502716	+	0.864452i	$0.332334\pi$
$192$	0	0
$193$	21.7868	1.56825	0.784123	−	0.620606i	$-0.213113\pi$
$193$	21.7868	1.56825	0.784123	+	0.620606i	$0.213113\pi$
$194$	−3.98631	−0.286200
$195$	0	0
$196$	10.0474	0.717670
$197$	−3.29282	−0.234604	−0.117302	−	0.993096i	$-0.537425\pi$
$197$	−3.29282	−0.234604	−0.117302	+	0.993096i	$0.537425\pi$
$198$	0	0
$199$	−19.7792	−1.40211	−0.701055	−	0.713107i	$-0.747287\pi$
$199$	−19.7792	−1.40211	−0.701055	+	0.713107i	$0.747287\pi$
$200$	−2.33917	−0.165404
$201$	0	0
$202$	6.12623	0.431040
$203$	−4.95856	−0.348022
$204$	0	0
$205$	−7.60986	−0.531496
$206$	−1.45075	−0.101079
$207$	0	0
$208$	−1.60947	−0.111597
$209$	−27.0537	−1.87134
$210$	0	0
$211$	−2.42809	−0.167157	−0.0835783	−	0.996501i	$-0.526635\pi$
$211$	−2.42809	−0.167157	−0.0835783	+	0.996501i	$0.526635\pi$
$212$	7.50925	0.515737
$213$	0	0
$214$	−10.1525	−0.694013
$215$	−8.84366	−0.603132
$216$	0	0
$217$	6.95204	0.471935
$218$	−3.85361	−0.260999
$219$	0	0
$220$	−6.39034	−0.430836
$221$	−5.29154	−0.355947
$222$	0	0
$223$	−9.45726	−0.633305	−0.316652	−	0.948542i	$-0.602559\pi$
$223$	−9.45726	−0.633305	−0.316652	+	0.948542i	$0.602559\pi$
$224$	−4.45532	−0.297683
$225$	0	0
$226$	6.92785	0.460834
$227$	24.0821	1.59839	0.799193	−	0.601074i	$-0.205260\pi$
$227$	24.0821	1.59839	0.799193	+	0.601074i	$0.205260\pi$
$228$	0	0
$229$	9.97401	0.659101	0.329550	−	0.944138i	$-0.393103\pi$
$229$	9.97401	0.659101	0.329550	+	0.944138i	$0.393103\pi$
$230$	−1.51036	−0.0995905
$231$	0	0
$232$	−14.9243	−0.979825
$233$	18.2730	1.19710	0.598552	−	0.801084i	$-0.295743\pi$
$233$	18.2730	1.19710	0.598552	+	0.801084i	$0.295743\pi$
$234$	0	0
$235$	4.08782	0.266660
$236$	7.44756	0.484795
$237$	0	0
$238$	−2.69396	−0.174623
$239$	−26.5440	−1.71699	−0.858495	−	0.512822i	$-0.828600\pi$
$239$	−26.5440	−1.71699	−0.858495	+	0.512822i	$0.828600\pi$
$240$	0	0
$241$	14.4187	0.928791	0.464396	−	0.885628i	$-0.346272\pi$
$241$	14.4187	0.928791	0.464396	+	0.885628i	$0.346272\pi$
$242$	3.63457	0.233639
$243$	0	0
$244$	22.5674	1.44473
$245$	6.39599	0.408625
$246$	0	0
$247$	−6.65042	−0.423156
$248$	20.9242	1.32869
$249$	0	0
$250$	−0.655066	−0.0414300
$251$	13.7541	0.868148	0.434074	−	0.900877i	$-0.357076\pi$
$251$	13.7541	0.868148	0.434074	+	0.900877i	$0.357076\pi$
$252$	0	0
$253$	−9.37940	−0.589678
$254$	−7.52496	−0.472158
$255$	0	0
$256$	−8.35300	−0.522063
$257$	−4.91358	−0.306500	−0.153250	−	0.988187i	$-0.548974\pi$
$257$	−4.91358	−0.306500	−0.153250	+	0.988187i	$0.548974\pi$
$258$	0	0
$259$	2.18072	0.135503
$260$	−1.57089	−0.0974224
$261$	0	0
$262$	8.24029	0.509087
$263$	26.0723	1.60769	0.803843	−	0.594841i	$-0.202785\pi$
$263$	26.0723	1.60769	0.803843	+	0.594841i	$0.202785\pi$
$264$	0	0
$265$	4.78025	0.293649
$266$	−3.38577	−0.207595
$267$	0	0
$268$	−16.9453	−1.03510
$269$	29.7277	1.81253	0.906263	−	0.422713i	$-0.138922\pi$
$269$	29.7277	1.81253	0.906263	+	0.422713i	$0.138922\pi$
$270$	0	0
$271$	14.2980	0.868540	0.434270	−	0.900783i	$-0.357006\pi$
$271$	14.2980	0.868540	0.434270	+	0.900783i	$0.357006\pi$
$272$	8.51658	0.516393
$273$	0	0
$274$	−1.18701	−0.0717098
$275$	−4.06797	−0.245308
$276$	0	0
$277$	1.87861	0.112875	0.0564373	−	0.998406i	$-0.482026\pi$
$277$	1.87861	0.112875	0.0564373	+	0.998406i	$0.482026\pi$
$278$	−13.3702	−0.801892
$279$	0	0
$280$	−1.81796	−0.108644
$281$	3.57185	0.213079	0.106539	−	0.994308i	$-0.466023\pi$
$281$	3.57185	0.213079	0.106539	+	0.994308i	$0.466023\pi$
$282$	0	0
$283$	−13.7483	−0.817249	−0.408625	−	0.912703i	$-0.633991\pi$
$283$	−13.7483	−0.817249	−0.408625	+	0.912703i	$0.633991\pi$
$284$	−0.483481	−0.0286893
$285$	0	0
$286$	2.66479	0.157572
$287$	−5.91426	−0.349108
$288$	0	0
$289$	11.0004	0.647081
$290$	−4.17942	−0.245424
$291$	0	0
$292$	11.7917	0.690055
$293$	−7.49807	−0.438042	−0.219021	−	0.975720i	$-0.570286\pi$
$293$	−7.49807	−0.438042	−0.219021	+	0.975720i	$0.570286\pi$
$294$	0	0
$295$	4.74099	0.276031
$296$	6.56352	0.381497
$297$	0	0
$298$	−0.304028	−0.0176119
$299$	−2.30567	−0.133340
$300$	0	0
$301$	−6.87315	−0.396162
$302$	−6.52757	−0.375619
$303$	0	0
$304$	10.7037	0.613897
$305$	14.3660	0.822594
$306$	0	0
$307$	−18.4113	−1.05079	−0.525395	−	0.850858i	$-0.676082\pi$
$307$	−18.4113	−1.05079	−0.525395	+	0.850858i	$0.676082\pi$
$308$	−4.96646	−0.282991
$309$	0	0
$310$	5.85967	0.332807
$311$	17.2648	0.978996	0.489498	−	0.872004i	$-0.337180\pi$
$311$	17.2648	0.978996	0.489498	+	0.872004i	$0.337180\pi$
$312$	0	0
$313$	−9.06061	−0.512136	−0.256068	−	0.966659i	$-0.582427\pi$
$313$	−9.06061	−0.512136	−0.256068	+	0.966659i	$0.582427\pi$
$314$	−6.10502	−0.344526
$315$	0	0
$316$	−11.9686	−0.673284
$317$	8.03887	0.451508	0.225754	−	0.974184i	$-0.427515\pi$
$317$	8.03887	0.451508	0.225754	+	0.974184i	$0.427515\pi$
$318$	0	0
$319$	−25.9543	−1.45316
$320$	−0.536314	−0.0299809
$321$	0	0
$322$	−1.17383	−0.0654150
$323$	35.1909	1.95808
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	5.85450	0.324251
$327$	0	0
$328$	−17.8007	−0.982881
$329$	3.17699	0.175153
$330$	0	0
$331$	−29.8664	−1.64161	−0.820803	−	0.571212i	$-0.806473\pi$
$331$	−29.8664	−1.64161	−0.820803	+	0.571212i	$0.806473\pi$
$332$	19.6827	1.08023
$333$	0	0
$334$	−10.7071	−0.585865
$335$	−10.7871	−0.589361
$336$	0	0
$337$	24.2087	1.31873	0.659366	−	0.751822i	$-0.270825\pi$
$337$	24.2087	1.31873	0.659366	+	0.751822i	$0.270825\pi$
$338$	0.655066	0.0356309
$339$	0	0
$340$	8.31242	0.450804
$341$	36.3887	1.97056
$342$	0	0
$343$	10.4111	0.562149
$344$	−20.6868	−1.11536
$345$	0	0
$346$	7.20624	0.387410
$347$	−22.9143	−1.23011	−0.615053	−	0.788486i	$-0.710865\pi$
$347$	−22.9143	−1.23011	−0.615053	+	0.788486i	$0.710865\pi$
$348$	0	0
$349$	−12.3575	−0.661481	−0.330740	−	0.943722i	$-0.607298\pi$
$349$	−12.3575	−0.661481	−0.330740	+	0.943722i	$0.607298\pi$
$350$	−0.509106	−0.0272129
$351$	0	0
$352$	−23.3202	−1.24297
$353$	−27.0380	−1.43909	−0.719544	−	0.694447i	$-0.755649\pi$
$353$	−27.0380	−1.43909	−0.719544	+	0.694447i	$0.755649\pi$
$354$	0	0
$355$	−0.307776	−0.0163350
$356$	18.4961	0.980294
$357$	0	0
$358$	−1.88152	−0.0994417
$359$	−28.4894	−1.50361	−0.751806	−	0.659384i	$-0.770817\pi$
$359$	−28.4894	−1.50361	−0.751806	+	0.659384i	$0.770817\pi$
$360$	0	0
$361$	25.2280	1.32779
$362$	−7.10477	−0.373418
$363$	0	0
$364$	−1.22087	−0.0639909
$365$	7.50636	0.392901
$366$	0	0
$367$	−30.3201	−1.58269	−0.791347	−	0.611367i	$-0.790620\pi$
$367$	−30.3201	−1.58269	−0.791347	+	0.611367i	$0.790620\pi$
$368$	3.71091	0.193444
$369$	0	0
$370$	1.83806	0.0955563
$371$	3.71514	0.192880
$372$	0	0
$373$	−9.33620	−0.483410	−0.241705	−	0.970350i	$-0.577707\pi$
$373$	−9.33620	−0.483410	−0.241705	+	0.970350i	$0.577707\pi$
$374$	−14.1008	−0.729137
$375$	0	0
$376$	9.56209	0.493127
$377$	−6.38016	−0.328595
$378$	0	0
$379$	−22.5156	−1.15655	−0.578274	−	0.815843i	$-0.696274\pi$
$379$	−22.5156	−1.15655	−0.578274	+	0.815843i	$0.696274\pi$
$380$	10.4471	0.535923
$381$	0	0
$382$	9.10237	0.465718
$383$	−3.03380	−0.155020	−0.0775100	−	0.996992i	$-0.524697\pi$
$383$	−3.03380	−0.155020	−0.0775100	+	0.996992i	$0.524697\pi$
$384$	0	0
$385$	−3.16156	−0.161128
$386$	14.2718	0.726413
$387$	0	0
$388$	9.55942	0.485306
$389$	−18.2987	−0.927782	−0.463891	−	0.885892i	$-0.653547\pi$
$389$	−18.2987	−0.927782	−0.463891	+	0.885892i	$0.653547\pi$
$390$	0	0
$391$	12.2005	0.617008
$392$	14.9613	0.755658
$393$	0	0
$394$	−2.15701	−0.108669
$395$	−7.61897	−0.383352
$396$	0	0
$397$	−29.4753	−1.47932	−0.739661	−	0.672980i	$-0.765014\pi$
$397$	−29.4753	−1.47932	−0.739661	+	0.672980i	$0.765014\pi$
$398$	−12.9567	−0.649459
$399$	0	0
$400$	1.60947	0.0804735
$401$	−27.7675	−1.38664	−0.693322	−	0.720628i	$-0.743854\pi$
$401$	−27.7675	−1.38664	−0.693322	+	0.720628i	$0.743854\pi$
$402$	0	0
$403$	8.94517	0.445591
$404$	−14.6911	−0.730909
$405$	0	0
$406$	−3.24818	−0.161204
$407$	11.4144	0.565792
$408$	0	0
$409$	−9.80833	−0.484991	−0.242495	−	0.970153i	$-0.577966\pi$
$409$	−9.80833	−0.484991	−0.242495	+	0.970153i	$0.577966\pi$
$410$	−4.98496	−0.246190
$411$	0	0
$412$	3.47900	0.171398
$413$	3.68462	0.181308
$414$	0	0
$415$	12.5297	0.615058
$416$	−5.73264	−0.281066
$417$	0	0
$418$	−17.7220	−0.866809
$419$	14.4829	0.707535	0.353768	−	0.935333i	$-0.384900\pi$
$419$	14.4829	0.707535	0.353768	+	0.935333i	$0.384900\pi$
$420$	0	0
$421$	−35.8224	−1.74587	−0.872937	−	0.487832i	$-0.837788\pi$
$421$	−35.8224	−1.74587	−0.872937	+	0.487832i	$0.837788\pi$
$422$	−1.59056	−0.0774272
$423$	0	0
$424$	11.1818	0.543037
$425$	5.29154	0.256677
$426$	0	0
$427$	11.1650	0.540313
$428$	24.3464	1.17683
$429$	0	0
$430$	−5.79318	−0.279372
$431$	7.34712	0.353898	0.176949	−	0.984220i	$-0.443377\pi$
$431$	7.34712	0.353898	0.176949	+	0.984220i	$0.443377\pi$
$432$	0	0
$433$	−6.55302	−0.314918	−0.157459	−	0.987526i	$-0.550330\pi$
$433$	−6.55302	−0.314918	−0.157459	+	0.987526i	$0.550330\pi$
$434$	4.55404	0.218601
$435$	0	0
$436$	9.24119	0.442573
$437$	15.3337	0.733509
$438$	0	0
$439$	−1.71109	−0.0816657	−0.0408329	−	0.999166i	$-0.513001\pi$
$439$	−1.71109	−0.0816657	−0.0408329	+	0.999166i	$0.513001\pi$
$440$	−9.51567	−0.453642
$441$	0	0
$442$	−3.46630	−0.164875
$443$	−22.0916	−1.04960	−0.524801	−	0.851225i	$-0.675860\pi$
$443$	−22.0916	−1.04960	−0.524801	+	0.851225i	$0.675860\pi$
$444$	0	0
$445$	11.7743	0.558156
$446$	−6.19512	−0.293348
$447$	0	0
$448$	−0.416815	−0.0196926
$449$	35.4082	1.67102	0.835508	−	0.549479i	$-0.185174\pi$
$449$	35.4082	1.67102	0.835508	+	0.549479i	$0.185174\pi$
$450$	0	0
$451$	−30.9567	−1.45770
$452$	−16.6134	−0.781430
$453$	0	0
$454$	15.7754	0.740375
$455$	−0.777184	−0.0364349
$456$	0	0
$457$	−2.53430	−0.118550	−0.0592748	−	0.998242i	$-0.518879\pi$
$457$	−2.53430	−0.118550	−0.0592748	+	0.998242i	$0.518879\pi$
$458$	6.53363	0.305296
$459$	0	0
$460$	3.62195	0.168874
$461$	−2.19744	−0.102345	−0.0511724	−	0.998690i	$-0.516296\pi$
$461$	−2.19744	−0.102345	−0.0511724	+	0.998690i	$0.516296\pi$
$462$	0	0
$463$	−25.4537	−1.18294	−0.591468	−	0.806329i	$-0.701451\pi$
$463$	−25.4537	−1.18294	−0.591468	+	0.806329i	$0.701451\pi$
$464$	10.2687	0.476712
$465$	0	0
$466$	11.9700	0.554500
$467$	−15.4769	−0.716187	−0.358093	−	0.933686i	$-0.616573\pi$
$467$	−15.4769	−0.716187	−0.358093	+	0.933686i	$0.616573\pi$
$468$	0	0
$469$	−8.38353	−0.387116
$470$	2.67779	0.123517
$471$	0	0
$472$	11.0900	0.510457
$473$	−35.9758	−1.65417
$474$	0	0
$475$	6.65042	0.305142
$476$	6.46028	0.296106
$477$	0	0
$478$	−17.3881	−0.795312
$479$	−9.62232	−0.439655	−0.219828	−	0.975539i	$-0.570549\pi$
$479$	−9.62232	−0.439655	−0.219828	+	0.975539i	$0.570549\pi$
$480$	0	0
$481$	2.80592	0.127939
$482$	9.44521	0.430217
$483$	0	0
$484$	−8.71593	−0.396179
$485$	6.08536	0.276322
$486$	0	0
$487$	−20.0963	−0.910649	−0.455325	−	0.890325i	$-0.650477\pi$
$487$	−20.0963	−0.910649	−0.455325	+	0.890325i	$0.650477\pi$
$488$	33.6044	1.52120
$489$	0	0
$490$	4.18979	0.189275
$491$	2.96672	0.133886	0.0669430	−	0.997757i	$-0.478675\pi$
$491$	2.96672	0.133886	0.0669430	+	0.997757i	$0.478675\pi$
$492$	0	0
$493$	33.7609	1.52051
$494$	−4.35646	−0.196006
$495$	0	0
$496$	−14.3970	−0.646444
$497$	−0.239198	−0.0107295
$498$	0	0
$499$	13.9441	0.624224	0.312112	−	0.950045i	$-0.398964\pi$
$499$	13.9441	0.624224	0.312112	+	0.950045i	$0.398964\pi$
$500$	1.57089	0.0702523
$501$	0	0
$502$	9.00981	0.402127
$503$	−25.1391	−1.12090	−0.560448	−	0.828190i	$-0.689371\pi$
$503$	−25.1391	−1.12090	−0.560448	+	0.828190i	$0.689371\pi$
$504$	0	0
$505$	−9.35209	−0.416162
$506$	−6.14412	−0.273140
$507$	0	0
$508$	18.0453	0.800632
$509$	−2.08157	−0.0922639	−0.0461320	−	0.998935i	$-0.514689\pi$
$509$	−2.08157	−0.0922639	−0.0461320	+	0.998935i	$0.514689\pi$
$510$	0	0
$511$	5.83382	0.258073
$512$	16.7562	0.740525
$513$	0	0
$514$	−3.21871	−0.141971
$515$	2.21467	0.0975899
$516$	0	0
$517$	16.6291	0.731349
$518$	1.42851	0.0627652
$519$	0	0
$520$	−2.33917	−0.102579
$521$	−16.7112	−0.732130	−0.366065	−	0.930589i	$-0.619295\pi$
$521$	−16.7112	−0.732130	−0.366065	+	0.930589i	$0.619295\pi$
$522$	0	0
$523$	39.1071	1.71003	0.855016	−	0.518602i	$-0.173547\pi$
$523$	39.1071	1.71003	0.855016	+	0.518602i	$0.173547\pi$
$524$	−19.7607	−0.863252
$525$	0	0
$526$	17.0791	0.744683
$527$	−47.3337	−2.06189
$528$	0	0
$529$	−17.6839	−0.768865
$530$	3.13138	0.136018
$531$	0	0
$532$	8.11929	0.352016
$533$	−7.60986	−0.329620
$534$	0	0
$535$	15.4985	0.670058
$536$	−25.2328	−1.08989
$537$	0	0
$538$	19.4736	0.839565
$539$	26.0187	1.12070
$540$	0	0
$541$	4.87732	0.209692	0.104846	−	0.994488i	$-0.466565\pi$
$541$	4.87732	0.209692	0.104846	+	0.994488i	$0.466565\pi$
$542$	9.36611	0.402309
$543$	0	0
$544$	30.3345	1.30058
$545$	5.88278	0.251991
$546$	0	0
$547$	−14.9480	−0.639128	−0.319564	−	0.947565i	$-0.603537\pi$
$547$	−14.9480	−0.639128	−0.319564	+	0.947565i	$0.603537\pi$
$548$	2.84652	0.121597
$549$	0	0
$550$	−2.66479	−0.113627
$551$	42.4307	1.80761
$552$	0	0
$553$	−5.92134	−0.251801
$554$	1.23061	0.0522836
$555$	0	0
$556$	32.0626	1.35976
$557$	−23.5505	−0.997866	−0.498933	−	0.866641i	$-0.666275\pi$
$557$	−23.5505	−0.997866	−0.498933	+	0.866641i	$0.666275\pi$
$558$	0	0
$559$	−8.84366	−0.374047
$560$	1.25085	0.0528583
$561$	0	0
$562$	2.33980	0.0986983
$563$	−0.293153	−0.0123549	−0.00617747	−	0.999981i	$-0.501966\pi$
$563$	−0.293153	−0.0123549	−0.00617747	+	0.999981i	$0.501966\pi$
$564$	0	0
$565$	−10.5758	−0.444928
$566$	−9.00601	−0.378551
$567$	0	0
$568$	−0.719938	−0.0302079
$569$	33.8631	1.41961	0.709807	−	0.704396i	$-0.248782\pi$
$569$	33.8631	1.41961	0.709807	+	0.704396i	$0.248782\pi$
$570$	0	0
$571$	−22.0309	−0.921963	−0.460981	−	0.887410i	$-0.652503\pi$
$571$	−22.0309	−0.921963	−0.460981	+	0.887410i	$0.652503\pi$
$572$	−6.39034	−0.267193
$573$	0	0
$574$	−3.87423	−0.161707
$575$	2.30567	0.0961531
$576$	0	0
$577$	23.7846	0.990165	0.495083	−	0.868846i	$-0.335138\pi$
$577$	23.7846	0.990165	0.495083	+	0.868846i	$0.335138\pi$
$578$	7.20597	0.299729
$579$	0	0
$580$	10.0225	0.416163
$581$	9.73786	0.403995
$582$	0	0
$583$	19.4459	0.805369
$584$	17.5586	0.726581
$585$	0	0
$586$	−4.91173	−0.202902
$587$	11.6973	0.482798	0.241399	−	0.970426i	$-0.422394\pi$
$587$	11.6973	0.482798	0.241399	+	0.970426i	$0.422394\pi$
$588$	0	0
$589$	−59.4891	−2.45121
$590$	3.10566	0.127858
$591$	0	0
$592$	−4.51605	−0.185609
$593$	14.5999	0.599545	0.299773	−	0.954011i	$-0.403089\pi$
$593$	14.5999	0.599545	0.299773	+	0.954011i	$0.403089\pi$
$594$	0	0
$595$	4.11250	0.168596
$596$	0.729080	0.0298643
$597$	0	0
$598$	−1.51036	−0.0617634
$599$	17.4884	0.714557	0.357278	−	0.933998i	$-0.383705\pi$
$599$	17.4884	0.714557	0.357278	+	0.933998i	$0.383705\pi$
$600$	0	0
$601$	−44.7939	−1.82718	−0.913590	−	0.406636i	$-0.866702\pi$
$601$	−44.7939	−1.82718	−0.913590	+	0.406636i	$0.866702\pi$
$602$	−4.50236	−0.183503
$603$	0	0
$604$	15.6535	0.636933
$605$	−5.54841	−0.225575
$606$	0	0
$607$	−42.3787	−1.72010	−0.860050	−	0.510209i	$-0.829568\pi$
$607$	−42.3787	−1.72010	−0.860050	+	0.510209i	$0.829568\pi$
$608$	38.1245	1.54615
$609$	0	0
$610$	9.41066	0.381026
$611$	4.08782	0.165375
$612$	0	0
$613$	−6.73566	−0.272051	−0.136025	−	0.990705i	$-0.543433\pi$
$613$	−6.73566	−0.272051	−0.136025	+	0.990705i	$0.543433\pi$
$614$	−12.0606	−0.486727
$615$	0	0
$616$	−7.39542	−0.297970
$617$	14.7847	0.595211	0.297606	−	0.954689i	$-0.403812\pi$
$617$	14.7847	0.595211	0.297606	+	0.954689i	$0.403812\pi$
$618$	0	0
$619$	27.3795	1.10048	0.550239	−	0.835008i	$-0.314537\pi$
$619$	27.3795	1.10048	0.550239	+	0.835008i	$0.314537\pi$
$620$	−14.0519	−0.564337
$621$	0	0
$622$	11.3096	0.453472
$623$	9.15080	0.366619
$624$	0	0
$625$	1.00000	0.0400000
$626$	−5.93529	−0.237222
$627$	0	0
$628$	14.6402	0.584209
$629$	−14.8476	−0.592014
$630$	0	0
$631$	19.2458	0.766162	0.383081	−	0.923715i	$-0.374863\pi$
$631$	19.2458	0.766162	0.383081	+	0.923715i	$0.374863\pi$
$632$	−17.8220	−0.708923
$633$	0	0
$634$	5.26599	0.209139
$635$	11.4873	0.455861
$636$	0	0
$637$	6.39599	0.253418
$638$	−17.0018	−0.673107
$639$	0	0
$640$	11.1140	0.439318
$641$	−27.9161	−1.10262	−0.551310	−	0.834300i	$-0.685872\pi$
$641$	−27.9161	−1.10262	−0.551310	+	0.834300i	$0.685872\pi$
$642$	0	0
$643$	3.24764	0.128074	0.0640372	−	0.997948i	$-0.479602\pi$
$643$	3.24764	0.128074	0.0640372	+	0.997948i	$0.479602\pi$
$644$	2.81492	0.110923
$645$	0	0
$646$	23.0524	0.906983
$647$	−1.47061	−0.0578155	−0.0289077	−	0.999582i	$-0.509203\pi$
$647$	−1.47061	−0.0578155	−0.0289077	+	0.999582i	$0.509203\pi$
$648$	0	0
$649$	19.2862	0.757050
$650$	−0.655066	−0.0256938
$651$	0	0
$652$	−14.0395	−0.549828
$653$	−2.39978	−0.0939107	−0.0469554	−	0.998897i	$-0.514952\pi$
$653$	−2.39978	−0.0939107	−0.0469554	+	0.998897i	$0.514952\pi$
$654$	0	0
$655$	−12.5793	−0.491515
$656$	12.2479	0.478198
$657$	0	0
$658$	2.08113	0.0811311
$659$	−3.50745	−0.136631	−0.0683154	−	0.997664i	$-0.521762\pi$
$659$	−3.50745	−0.136631	−0.0683154	+	0.997664i	$0.521762\pi$
$660$	0	0
$661$	−20.2946	−0.789369	−0.394685	−	0.918817i	$-0.629146\pi$
$661$	−20.2946	−0.789369	−0.394685	+	0.918817i	$0.629146\pi$
$662$	−19.5644	−0.760394
$663$	0	0
$664$	29.3090	1.13741
$665$	5.16860	0.200430
$666$	0	0
$667$	14.7105	0.569595
$668$	25.6763	0.993444
$669$	0	0
$670$	−7.06624	−0.272993
$671$	58.4404	2.25607
$672$	0	0
$673$	9.89110	0.381274	0.190637	−	0.981661i	$-0.438945\pi$
$673$	9.89110	0.381274	0.190637	+	0.981661i	$0.438945\pi$
$674$	15.8583	0.610838
$675$	0	0
$676$	−1.57089	−0.0604188
$677$	8.03421	0.308780	0.154390	−	0.988010i	$-0.450659\pi$
$677$	8.03421	0.308780	0.154390	+	0.988010i	$0.450659\pi$
$678$	0	0
$679$	4.72944	0.181499
$680$	12.3778	0.474667
$681$	0	0
$682$	23.8370	0.912766
$683$	0.652224	0.0249567	0.0124783	−	0.999922i	$-0.496028\pi$
$683$	0.652224	0.0249567	0.0124783	+	0.999922i	$0.496028\pi$
$684$	0	0
$685$	1.81204	0.0692347
$686$	6.81998	0.260388
$687$	0	0
$688$	14.2336	0.542651
$689$	4.78025	0.182113
$690$	0	0
$691$	41.1289	1.56462	0.782308	−	0.622892i	$-0.214042\pi$
$691$	41.1289	1.56462	0.782308	+	0.622892i	$0.214042\pi$
$692$	−17.2810	−0.656926
$693$	0	0
$694$	−15.0104	−0.569786
$695$	20.4105	0.774214
$696$	0	0
$697$	40.2679	1.52526
$698$	−8.09496	−0.306399
$699$	0	0
$700$	1.22087	0.0461445
$701$	28.9154	1.09212	0.546060	−	0.837746i	$-0.316127\pi$
$701$	28.9154	1.09212	0.546060	+	0.837746i	$0.316127\pi$
$702$	0	0
$703$	−18.6606	−0.703796
$704$	−2.18171	−0.0822264
$705$	0	0
$706$	−17.7117	−0.666588
$707$	−7.26829	−0.273352
$708$	0	0
$709$	−38.2858	−1.43785	−0.718927	−	0.695086i	$-0.755366\pi$
$709$	−38.2858	−1.43785	−0.718927	+	0.695086i	$0.755366\pi$
$710$	−0.201613	−0.00756641
$711$	0	0
$712$	27.5421	1.03218
$713$	−20.6246	−0.772398
$714$	0	0
$715$	−4.06797	−0.152134
$716$	4.51202	0.168622
$717$	0	0
$718$	−18.6624	−0.696475
$719$	−31.0939	−1.15961	−0.579803	−	0.814757i	$-0.696870\pi$
$719$	−31.0939	−1.15961	−0.579803	+	0.814757i	$0.696870\pi$
$720$	0	0
$721$	1.72120	0.0641009
$722$	16.5260	0.615035
$723$	0	0
$724$	17.0377	0.633201
$725$	6.38016	0.236953
$726$	0	0
$727$	−23.3504	−0.866017	−0.433008	−	0.901390i	$-0.642548\pi$
$727$	−23.3504	−0.866017	−0.433008	+	0.901390i	$0.642548\pi$
$728$	−1.81796	−0.0673782
$729$	0	0
$730$	4.91716	0.181992
$731$	46.7966	1.73083
$732$	0	0
$733$	1.11626	0.0412301	0.0206150	−	0.999787i	$-0.493438\pi$
$733$	1.11626	0.0412301	0.0206150	+	0.999787i	$0.493438\pi$
$734$	−19.8616	−0.733106
$735$	0	0
$736$	13.2176	0.487206
$737$	−43.8815	−1.61640
$738$	0	0
$739$	1.73962	0.0639930	0.0319965	−	0.999488i	$-0.489813\pi$
$739$	1.73962	0.0639930	0.0319965	+	0.999488i	$0.489813\pi$
$740$	−4.40779	−0.162034
$741$	0	0
$742$	2.43366	0.0893424
$743$	−27.2753	−1.00063	−0.500317	−	0.865842i	$-0.666783\pi$
$743$	−27.2753	−1.00063	−0.500317	+	0.865842i	$0.666783\pi$
$744$	0	0
$745$	0.464119	0.0170040
$746$	−6.11582	−0.223916
$747$	0	0
$748$	33.8147	1.23639
$749$	12.0452	0.440121
$750$	0	0
$751$	32.6647	1.19195	0.595977	−	0.803002i	$-0.296765\pi$
$751$	32.6647	1.19195	0.595977	+	0.803002i	$0.296765\pi$
$752$	−6.57923	−0.239920
$753$	0	0
$754$	−4.17942	−0.152206
$755$	9.96476	0.362655
$756$	0	0
$757$	43.4319	1.57856	0.789279	−	0.614035i	$-0.210455\pi$
$757$	43.4319	1.57856	0.789279	+	0.614035i	$0.210455\pi$
$758$	−14.7492	−0.535715
$759$	0	0
$760$	15.5564	0.564291
$761$	49.4890	1.79397	0.896987	−	0.442058i	$-0.145751\pi$
$761$	49.4890	1.79397	0.896987	+	0.442058i	$0.145751\pi$
$762$	0	0
$763$	4.57200	0.165518
$764$	−21.8281	−0.789711
$765$	0	0
$766$	−1.98734	−0.0718055
$767$	4.74099	0.171187
$768$	0	0
$769$	−9.04054	−0.326010	−0.163005	−	0.986625i	$-0.552119\pi$
$769$	−9.04054	−0.326010	−0.163005	+	0.986625i	$0.552119\pi$
$770$	−2.07103	−0.0746348
$771$	0	0
$772$	−34.2246	−1.23177
$773$	−8.64817	−0.311053	−0.155527	−	0.987832i	$-0.549707\pi$
$773$	−8.64817	−0.311053	−0.155527	+	0.987832i	$0.549707\pi$
$774$	0	0
$775$	−8.94517	−0.321320
$776$	14.2347	0.510995
$777$	0	0
$778$	−11.9869	−0.429750
$779$	50.6088	1.81325
$780$	0	0
$781$	−1.25202	−0.0448009
$782$	7.99215	0.285799
$783$	0	0
$784$	−10.2942	−0.367648
$785$	9.31971	0.332635
$786$	0	0
$787$	18.3542	0.654258	0.327129	−	0.944980i	$-0.393919\pi$
$787$	18.3542	0.654258	0.327129	+	0.944980i	$0.393919\pi$
$788$	5.17266	0.184268
$789$	0	0
$790$	−4.99093	−0.177569
$791$	−8.21935	−0.292246
$792$	0	0
$793$	14.3660	0.510151
$794$	−19.3082	−0.685224
$795$	0	0
$796$	31.0709	1.10128
$797$	14.2529	0.504862	0.252431	−	0.967615i	$-0.418770\pi$
$797$	14.2529	0.504862	0.252431	+	0.967615i	$0.418770\pi$
$798$	0	0
$799$	−21.6309	−0.765245
$800$	5.73264	0.202679
$801$	0	0
$802$	−18.1896	−0.642295
$803$	30.5357	1.07758
$804$	0	0
$805$	1.79193	0.0631572
$806$	5.85967	0.206398
$807$	0	0
$808$	−21.8761	−0.769598
$809$	−2.15221	−0.0756676	−0.0378338	−	0.999284i	$-0.512046\pi$
$809$	−2.15221	−0.0756676	−0.0378338	+	0.999284i	$0.512046\pi$
$810$	0	0
$811$	−30.3748	−1.06660	−0.533302	−	0.845925i	$-0.679049\pi$
$811$	−30.3748	−1.06660	−0.533302	+	0.845925i	$0.679049\pi$
$812$	7.78934	0.273352
$813$	0	0
$814$	7.47719	0.262075
$815$	−8.93727	−0.313059
$816$	0	0
$817$	58.8140	2.05764
$818$	−6.42510	−0.224648
$819$	0	0
$820$	11.9543	0.417461
$821$	−11.9606	−0.417428	−0.208714	−	0.977977i	$-0.566928\pi$
$821$	−11.9606	−0.417428	−0.208714	+	0.977977i	$0.566928\pi$
$822$	0	0
$823$	3.88386	0.135383	0.0676914	−	0.997706i	$-0.478437\pi$
$823$	3.88386	0.135383	0.0676914	+	0.997706i	$0.478437\pi$
$824$	5.18047	0.180470
$825$	0	0
$826$	2.41367	0.0839822
$827$	−30.4261	−1.05802	−0.529010	−	0.848615i	$-0.677437\pi$
$827$	−30.4261	−1.05802	−0.529010	+	0.848615i	$0.677437\pi$
$828$	0	0
$829$	−23.8577	−0.828613	−0.414307	−	0.910137i	$-0.635976\pi$
$829$	−23.8577	−0.828613	−0.414307	+	0.910137i	$0.635976\pi$
$830$	8.20776	0.284896
$831$	0	0
$832$	−0.536314	−0.0185934
$833$	−33.8446	−1.17265
$834$	0	0
$835$	16.3450	0.565644
$836$	42.4984	1.46984
$837$	0	0
$838$	9.48724	0.327731
$839$	−19.0333	−0.657103	−0.328552	−	0.944486i	$-0.606561\pi$
$839$	−19.0333	−0.657103	−0.328552	+	0.944486i	$0.606561\pi$
$840$	0	0
$841$	11.7064	0.403671
$842$	−23.4660	−0.808691
$843$	0	0
$844$	3.81426	0.131292
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−4.31213	−0.148167
$848$	−7.69368	−0.264202
$849$	0	0
$850$	3.46630	0.118893
$851$	−6.46953	−0.221773
$852$	0	0
$853$	21.0527	0.720832	0.360416	−	0.932792i	$-0.382635\pi$
$853$	21.0527	0.720832	0.360416	+	0.932792i	$0.382635\pi$
$854$	7.31381	0.250273
$855$	0	0
$856$	36.2536	1.23912
$857$	−9.88134	−0.337540	−0.168770	−	0.985655i	$-0.553980\pi$
$857$	−9.88134	−0.337540	−0.168770	+	0.985655i	$0.553980\pi$
$858$	0	0
$859$	−9.82391	−0.335187	−0.167594	−	0.985856i	$-0.553600\pi$
$859$	−9.82391	−0.335187	−0.167594	+	0.985856i	$0.553600\pi$
$860$	13.8924	0.473727
$861$	0	0
$862$	4.81285	0.163926
$863$	46.5764	1.58548	0.792739	−	0.609561i	$-0.208654\pi$
$863$	46.5764	1.58548	0.792739	+	0.609561i	$0.208654\pi$
$864$	0	0
$865$	−11.0008	−0.374038
$866$	−4.29266	−0.145870
$867$	0	0
$868$	−10.9209	−0.370679
$869$	−30.9938	−1.05139
$870$	0	0
$871$	−10.7871	−0.365506
$872$	13.7608	0.465999
$873$	0	0
$874$	10.0446	0.339762
$875$	0.777184	0.0262736
$876$	0	0
$877$	−55.8285	−1.88520	−0.942598	−	0.333931i	$-0.891625\pi$
$877$	−55.8285	−1.88520	−0.942598	+	0.333931i	$0.891625\pi$
$878$	−1.12087	−0.0378277
$879$	0	0
$880$	6.54728	0.220709
$881$	1.22939	0.0414192	0.0207096	−	0.999786i	$-0.493407\pi$
$881$	1.22939	0.0414192	0.0207096	+	0.999786i	$0.493407\pi$
$882$	0	0
$883$	−5.89957	−0.198536	−0.0992681	−	0.995061i	$-0.531650\pi$
$883$	−5.89957	−0.198536	−0.0992681	+	0.995061i	$0.531650\pi$
$884$	8.31242	0.279577
$885$	0	0
$886$	−14.4714	−0.486177
$887$	−42.4663	−1.42588	−0.712939	−	0.701226i	$-0.752636\pi$
$887$	−42.4663	−1.42588	−0.712939	+	0.701226i	$0.752636\pi$
$888$	0	0
$889$	8.92777	0.299428
$890$	7.71295	0.258539
$891$	0	0
$892$	14.8563	0.497426
$893$	−27.1857	−0.909735
$894$	0	0
$895$	2.87227	0.0960094
$896$	8.63759	0.288562
$897$	0	0
$898$	23.1947	0.774016
$899$	−57.0716	−1.90345
$900$	0	0
$901$	−25.2949	−0.842695
$902$	−20.2787	−0.675206
$903$	0	0
$904$	−24.7386	−0.822793
$905$	10.8459	0.360530
$906$	0	0
$907$	−11.1772	−0.371134	−0.185567	−	0.982632i	$-0.559412\pi$
$907$	−11.1772	−0.371134	−0.185567	+	0.982632i	$0.559412\pi$
$908$	−37.8304	−1.25544
$909$	0	0
$910$	−0.509106	−0.0168767
$911$	−28.0289	−0.928638	−0.464319	−	0.885668i	$-0.653701\pi$
$911$	−28.0289	−0.928638	−0.464319	+	0.885668i	$0.653701\pi$
$912$	0	0
$913$	50.9704	1.68687
$914$	−1.66013	−0.0549123
$915$	0	0
$916$	−15.6681	−0.517687
$917$	−9.77645	−0.322847
$918$	0	0
$919$	43.1860	1.42457	0.712287	−	0.701888i	$-0.247659\pi$
$919$	43.1860	1.42457	0.712287	+	0.701888i	$0.247659\pi$
$920$	5.39335	0.177813
$921$	0	0
$922$	−1.43946	−0.0474062
$923$	−0.307776	−0.0101306
$924$	0	0
$925$	−2.80592	−0.0922582
$926$	−16.6739	−0.547937
$927$	0	0
$928$	36.5752	1.20064
$929$	−18.3836	−0.603146	−0.301573	−	0.953443i	$-0.597512\pi$
$929$	−18.3836	−0.603146	−0.301573	+	0.953443i	$0.597512\pi$
$930$	0	0
$931$	−42.5360	−1.39406
$932$	−28.7049	−0.940259
$933$	0	0
$934$	−10.1384	−0.331739
$935$	21.5258	0.703970
$936$	0	0
$937$	−13.2414	−0.432579	−0.216289	−	0.976329i	$-0.569395\pi$
$937$	−13.2414	−0.432579	−0.216289	+	0.976329i	$0.569395\pi$
$938$	−5.49176	−0.179312
$939$	0	0
$940$	−6.42151	−0.209447
$941$	15.3309	0.499774	0.249887	−	0.968275i	$-0.419606\pi$
$941$	15.3309	0.499774	0.249887	+	0.968275i	$0.419606\pi$
$942$	0	0
$943$	17.5458	0.571371
$944$	−7.63048	−0.248351
$945$	0	0
$946$	−23.5665	−0.766213
$947$	44.0956	1.43291	0.716457	−	0.697631i	$-0.245763\pi$
$947$	44.0956	1.43291	0.716457	+	0.697631i	$0.245763\pi$
$948$	0	0
$949$	7.50636	0.243667
$950$	4.35646	0.141342
$951$	0	0
$952$	9.61982	0.311780
$953$	−23.1623	−0.750300	−0.375150	−	0.926964i	$-0.622409\pi$
$953$	−23.1623	−0.750300	−0.375150	+	0.926964i	$0.622409\pi$
$954$	0	0
$955$	−13.8954	−0.449643
$956$	41.6977	1.34860
$957$	0	0
$958$	−6.30325	−0.203649
$959$	1.40829	0.0454761
$960$	0	0
$961$	49.0161	1.58116
$962$	1.83806	0.0592615
$963$	0	0
$964$	−22.6502	−0.729514
$965$	−21.7868	−0.701341
$966$	0	0
$967$	−28.8356	−0.927289	−0.463644	−	0.886021i	$-0.653458\pi$
$967$	−28.8356	−0.927289	−0.463644	+	0.886021i	$0.653458\pi$
$968$	−12.9786	−0.417150
$969$	0	0
$970$	3.98631	0.127993
$971$	−1.72410	−0.0553290	−0.0276645	−	0.999617i	$-0.508807\pi$
$971$	−1.72410	−0.0553290	−0.0276645	+	0.999617i	$0.508807\pi$
$972$	0	0
$973$	15.8627	0.508535
$974$	−13.1644	−0.421814
$975$	0	0
$976$	−23.1216	−0.740105
$977$	3.18384	0.101860	0.0509301	−	0.998702i	$-0.483781\pi$
$977$	3.18384	0.101860	0.0509301	+	0.998702i	$0.483781\pi$
$978$	0	0
$979$	47.8976	1.53081
$980$	−10.0474	−0.320952
$981$	0	0
$982$	1.94339	0.0620162
$983$	11.7189	0.373776	0.186888	−	0.982381i	$-0.440160\pi$
$983$	11.7189	0.373776	0.186888	+	0.982381i	$0.440160\pi$
$984$	0	0
$985$	3.29282	0.104918
$986$	22.1156	0.704304
$987$	0	0
$988$	10.4471	0.332365
$989$	20.3906	0.648382
$990$	0	0
$991$	45.8521	1.45654	0.728271	−	0.685290i	$-0.240324\pi$
$991$	45.8521	1.45654	0.728271	+	0.685290i	$0.240324\pi$
$992$	−51.2795	−1.62812
$993$	0	0
$994$	−0.156690	−0.00496992
$995$	19.7792	0.627043
$996$	0	0
$997$	−24.1678	−0.765402	−0.382701	−	0.923872i	$-0.625006\pi$
$997$	−24.1678	−0.765402	−0.382701	+	0.923872i	$0.625006\pi$
$998$	9.13431	0.289141
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bf.1.5		8
3.2	odd	2		5265.2.a.ba.1.4		8
9.2	odd	6		1755.2.i.f.1171.5		16
9.4	even	3		585.2.i.e.196.4	✓	16
9.5	odd	6		1755.2.i.f.586.5		16
9.7	even	3		585.2.i.e.391.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.e.196.4	✓	16	9.4	even	3
585.2.i.e.391.4	yes	16	9.7	even	3
1755.2.i.f.586.5		16	9.5	odd	6
1755.2.i.f.1171.5		16	9.2	odd	6
5265.2.a.ba.1.4		8	3.2	odd	2
5265.2.a.bf.1.5		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.655066 q^{2} -1.57089 q^{4} -1.00000 q^{5} -0.777184 q^{7} -2.33917 q^{8} +O(q^{10})\)	\(q+0.655066 q^{2} -1.57089 q^{4} -1.00000 q^{5} -0.777184 q^{7} -2.33917 q^{8} -0.655066 q^{10} -4.06797 q^{11} -1.00000 q^{13} -0.509106 q^{14} +1.60947 q^{16} +5.29154 q^{17} +6.65042 q^{19} +1.57089 q^{20} -2.66479 q^{22} +2.30567 q^{23} +1.00000 q^{25} -0.655066 q^{26} +1.22087 q^{28} +6.38016 q^{29} -8.94517 q^{31} +5.73264 q^{32} +3.46630 q^{34} +0.777184 q^{35} -2.80592 q^{37} +4.35646 q^{38} +2.33917 q^{40} +7.60986 q^{41} +8.84366 q^{43} +6.39034 q^{44} +1.51036 q^{46} -4.08782 q^{47} -6.39599 q^{49} +0.655066 q^{50} +1.57089 q^{52} -4.78025 q^{53} +4.06797 q^{55} +1.81796 q^{56} +4.17942 q^{58} -4.74099 q^{59} -14.3660 q^{61} -5.85967 q^{62} +0.536314 q^{64} +1.00000 q^{65} +10.7871 q^{67} -8.31242 q^{68} +0.509106 q^{70} +0.307776 q^{71} -7.50636 q^{73} -1.83806 q^{74} -10.4471 q^{76} +3.16156 q^{77} +7.61897 q^{79} -1.60947 q^{80} +4.98496 q^{82} -12.5297 q^{83} -5.29154 q^{85} +5.79318 q^{86} +9.51567 q^{88} -11.7743 q^{89} +0.777184 q^{91} -3.62195 q^{92} -2.67779 q^{94} -6.65042 q^{95} -6.08536 q^{97} -4.18979 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} + 9 q^{4} - 8 q^{5} - 11 q^{7} - 6 q^{8}+O(q^{10}) \) 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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