LMFDB - Embedded newform 6633.2.a.w.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6633,2,Mod(1,6633)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6633, 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("6633.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6633 = 3^{2} \cdot 11 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6633.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:	$52.9647716607$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - x^{16} - 26 x^{15} + 25 x^{14} + 272 x^{13} - 244 x^{12} - 1472 x^{11} + 1186 x^{10} + 4406 x^{9} + \cdots - 2 $ x^17 - x^16 - 26x^15 + 25x^14 + 272x^13 - 244x^12 - 1472x^11 + 1186x^10 + 4406x^9 - 3042x^8 - 7180x^7 + 4055x^6 + 5665x^5 - 2655x^4 - 1418x^3 + 810x^2 - 76*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 737)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.484318$ of defining polynomial
Character	$\chi$	$=$	6633.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.484318 q^{2} -1.76544 q^{4} -3.54277 q^{5} -0.730966 q^{7} +1.82367 q^{8} +O(q^{10})$	\(q-0.484318 q^{2} -1.76544 q^{4} -3.54277 q^{5} -0.730966 q^{7} +1.82367 q^{8} +1.71583 q^{10} -1.00000 q^{11} +5.91392 q^{13} +0.354020 q^{14} +2.64764 q^{16} -4.21395 q^{17} +0.168612 q^{19} +6.25454 q^{20} +0.484318 q^{22} -4.61992 q^{23} +7.55123 q^{25} -2.86422 q^{26} +1.29047 q^{28} +6.17507 q^{29} +7.65946 q^{31} -4.92963 q^{32} +2.04089 q^{34} +2.58965 q^{35} -9.78458 q^{37} -0.0816618 q^{38} -6.46084 q^{40} -2.91240 q^{41} +1.61544 q^{43} +1.76544 q^{44} +2.23751 q^{46} -8.51279 q^{47} -6.46569 q^{49} -3.65720 q^{50} -10.4407 q^{52} +10.6095 q^{53} +3.54277 q^{55} -1.33304 q^{56} -2.99070 q^{58} -7.46564 q^{59} -6.10978 q^{61} -3.70961 q^{62} -2.90776 q^{64} -20.9517 q^{65} -1.00000 q^{67} +7.43946 q^{68} -1.25421 q^{70} -13.2783 q^{71} -13.6988 q^{73} +4.73885 q^{74} -0.297674 q^{76} +0.730966 q^{77} +3.43703 q^{79} -9.37998 q^{80} +1.41053 q^{82} -1.41858 q^{83} +14.9291 q^{85} -0.782387 q^{86} -1.82367 q^{88} +2.82870 q^{89} -4.32288 q^{91} +8.15618 q^{92} +4.12290 q^{94} -0.597354 q^{95} +2.90243 q^{97} +3.13145 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) $ 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7	$ 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7} + 10 q^{10} - 17 q^{11} + q^{13} + 11 q^{14} + 19 q^{16} - 2 q^{17} + 13 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 33 q^{25} - 12 q^{26} + 44 q^{28} + 5 q^{29} + 16 q^{31} + 24 q^{32} + 4 q^{34} + 2 q^{35} + 29 q^{37} + 19 q^{38} + 31 q^{40} + 6 q^{41} + 19 q^{43} - 19 q^{44} - 33 q^{46} - 40 q^{47} + 23 q^{49} + 3 q^{50} - 28 q^{52} - 15 q^{53} + 10 q^{55} + 38 q^{56} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 3 q^{62} - 4 q^{64} + 30 q^{65} - 17 q^{67} + 13 q^{68} + 71 q^{70} - 2 q^{71} + 41 q^{73} + 13 q^{74} + 21 q^{76} - 20 q^{77} + 41 q^{79} + 23 q^{80} - 8 q^{82} - 2 q^{83} - 36 q^{85} + 54 q^{86} - q^{89} + 16 q^{91} - 36 q^{92} + 12 q^{94} + 31 q^{95} + 3 q^{97} + 64 q^{98}+O(q^{100}) $ 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7 + 10 * q^10 - 17 * q^11 + q^13 + 11 * q^14 + 19 * q^16 - 2 * q^17 + 13 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 33 * q^25 - 12 * q^26 + 44 * q^28 + 5 * q^29 + 16 * q^31 + 24 * q^32 + 4 * q^34 + 2 * q^35 + 29 * q^37 + 19 * q^38 + 31 * q^40 + 6 * q^41 + 19 * q^43 - 19 * q^44 - 33 * q^46 - 40 * q^47 + 23 * q^49 + 3 * q^50 - 28 * q^52 - 15 * q^53 + 10 * q^55 + 38 * q^56 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 3 * q^62 - 4 * q^64 + 30 * q^65 - 17 * q^67 + 13 * q^68 + 71 * q^70 - 2 * q^71 + 41 * q^73 + 13 * q^74 + 21 * q^76 - 20 * q^77 + 41 * q^79 + 23 * q^80 - 8 * q^82 - 2 * q^83 - 36 * q^85 + 54 * q^86 - q^89 + 16 * q^91 - 36 * q^92 + 12 * q^94 + 31 * q^95 + 3 * q^97 + 64 * 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 $

$2$	−0.484318	−0.342464	−0.171232	−	0.985231i	$-0.554775\pi$
$2$	−0.484318	−0.342464	−0.171232	+	0.985231i	$0.554775\pi$
$3$	0	0
$3$	0	0
$4$	−1.76544	−0.882718
$5$	−3.54277	−1.58438	−0.792188	−	0.610277i	$-0.791058\pi$
$5$	−3.54277	−1.58438	−0.792188	+	0.610277i	$0.791058\pi$
$6$	0	0
$7$	−0.730966	−0.276279	−0.138140	−	0.990413i	$-0.544112\pi$
$7$	−0.730966	−0.276279	−0.138140	+	0.990413i	$0.544112\pi$
$8$	1.82367	0.644764
$9$	0	0
$10$	1.71583	0.542592
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.91392	1.64023	0.820114	−	0.572201i	$-0.193910\pi$
$13$	5.91392	1.64023	0.820114	+	0.572201i	$0.193910\pi$
$14$	0.354020	0.0946158
$15$	0	0
$16$	2.64764	0.661909
$17$	−4.21395	−1.02203	−0.511016	−	0.859571i	$-0.670731\pi$
$17$	−4.21395	−1.02203	−0.511016	+	0.859571i	$0.670731\pi$
$18$	0	0
$19$	0.168612	0.0386822	0.0193411	−	0.999813i	$-0.493843\pi$
$19$	0.168612	0.0386822	0.0193411	+	0.999813i	$0.493843\pi$
$20$	6.25454	1.39856
$21$	0	0
$22$	0.484318	0.103257
$23$	−4.61992	−0.963321	−0.481660	−	0.876358i	$-0.659966\pi$
$23$	−4.61992	−0.963321	−0.481660	+	0.876358i	$0.659966\pi$
$24$	0	0
$25$	7.55123	1.51025
$26$	−2.86422	−0.561720
$27$	0	0
$28$	1.29047	0.243877
$29$	6.17507	1.14668	0.573341	−	0.819317i	$-0.305647\pi$
$29$	6.17507	1.14668	0.573341	+	0.819317i	$0.305647\pi$
$30$	0	0
$31$	7.65946	1.37568	0.687839	−	0.725863i	$-0.258559\pi$
$31$	7.65946	1.37568	0.687839	+	0.725863i	$0.258559\pi$
$32$	−4.92963	−0.871444
$33$	0	0
$34$	2.04089	0.350010
$35$	2.58965	0.437730
$36$	0	0
$37$	−9.78458	−1.60858	−0.804288	−	0.594240i	$-0.797453\pi$
$37$	−9.78458	−1.60858	−0.804288	+	0.594240i	$0.797453\pi$
$38$	−0.0816618	−0.0132473
$39$	0	0
$40$	−6.46084	−1.02155
$41$	−2.91240	−0.454840	−0.227420	−	0.973797i	$-0.573029\pi$
$41$	−2.91240	−0.454840	−0.227420	+	0.973797i	$0.573029\pi$
$42$	0	0
$43$	1.61544	0.246353	0.123176	−	0.992385i	$-0.460692\pi$
$43$	1.61544	0.246353	0.123176	+	0.992385i	$0.460692\pi$
$44$	1.76544	0.266150
$45$	0	0
$46$	2.23751	0.329903
$47$	−8.51279	−1.24172	−0.620859	−	0.783922i	$-0.713216\pi$
$47$	−8.51279	−1.24172	−0.620859	+	0.783922i	$0.713216\pi$
$48$	0	0
$49$	−6.46569	−0.923670
$50$	−3.65720	−0.517206
$51$	0	0
$52$	−10.4407	−1.44786
$53$	10.6095	1.45733	0.728667	−	0.684869i	$-0.240140\pi$
$53$	10.6095	1.45733	0.728667	+	0.684869i	$0.240140\pi$
$54$	0	0
$55$	3.54277	0.477707
$56$	−1.33304	−0.178135
$57$	0	0
$58$	−2.99070	−0.392698
$59$	−7.46564	−0.971943	−0.485972	−	0.873975i	$-0.661534\pi$
$59$	−7.46564	−0.971943	−0.485972	+	0.873975i	$0.661534\pi$
$60$	0	0
$61$	−6.10978	−0.782277	−0.391139	−	0.920332i	$-0.627919\pi$
$61$	−6.10978	−0.782277	−0.391139	+	0.920332i	$0.627919\pi$
$62$	−3.70961	−0.471121
$63$	0	0
$64$	−2.90776	−0.363471
$65$	−20.9517	−2.59874
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	7.43946	0.902167
$69$	0	0
$70$	−1.25421	−0.149907
$71$	−13.2783	−1.57585	−0.787925	−	0.615772i	$-0.788844\pi$
$71$	−13.2783	−1.57585	−0.787925	+	0.615772i	$0.788844\pi$
$72$	0	0
$73$	−13.6988	−1.60332	−0.801661	−	0.597778i	$-0.796050\pi$
$73$	−13.6988	−1.60332	−0.801661	+	0.597778i	$0.796050\pi$
$74$	4.73885	0.550880
$75$	0	0
$76$	−0.297674	−0.0341455
$77$	0.730966	0.0833013
$78$	0	0
$79$	3.43703	0.386697	0.193348	−	0.981130i	$-0.438065\pi$
$79$	3.43703	0.386697	0.193348	+	0.981130i	$0.438065\pi$
$80$	−9.37998	−1.04871
$81$	0	0
$82$	1.41053	0.155767
$83$	−1.41858	−0.155709	−0.0778546	−	0.996965i	$-0.524807\pi$
$83$	−1.41858	−0.155709	−0.0778546	+	0.996965i	$0.524807\pi$
$84$	0	0
$85$	14.9291	1.61928
$86$	−0.782387	−0.0843670
$87$	0	0
$88$	−1.82367	−0.194404
$89$	2.82870	0.299842	0.149921	−	0.988698i	$-0.452098\pi$
$89$	2.82870	0.299842	0.149921	+	0.988698i	$0.452098\pi$
$90$	0	0
$91$	−4.32288	−0.453161
$92$	8.15618	0.850341
$93$	0	0
$94$	4.12290	0.425244
$95$	−0.597354	−0.0612872
$96$	0	0
$97$	2.90243	0.294697	0.147348	−	0.989085i	$-0.452926\pi$
$97$	2.90243	0.294697	0.147348	+	0.989085i	$0.452926\pi$
$98$	3.13145	0.316324
$99$	0	0
$100$	−13.3312	−1.33312
$101$	7.31446	0.727816	0.363908	−	0.931435i	$-0.381442\pi$
$101$	7.31446	0.727816	0.363908	+	0.931435i	$0.381442\pi$
$102$	0	0
$103$	1.32283	0.130342	0.0651710	−	0.997874i	$-0.479241\pi$
$103$	1.32283	0.130342	0.0651710	+	0.997874i	$0.479241\pi$
$104$	10.7850	1.05756
$105$	0	0
$106$	−5.13839	−0.499085
$107$	15.1012	1.45989	0.729944	−	0.683507i	$-0.239546\pi$
$107$	15.1012	1.45989	0.729944	+	0.683507i	$0.239546\pi$
$108$	0	0
$109$	−5.14666	−0.492961	−0.246480	−	0.969148i	$-0.579274\pi$
$109$	−5.14666	−0.492961	−0.246480	+	0.969148i	$0.579274\pi$
$110$	−1.71583	−0.163598
$111$	0	0
$112$	−1.93533	−0.182872
$113$	9.70384	0.912861	0.456430	−	0.889759i	$-0.349128\pi$
$113$	9.70384	0.912861	0.456430	+	0.889759i	$0.349128\pi$
$114$	0	0
$115$	16.3673	1.52626
$116$	−10.9017	−1.01220
$117$	0	0
$118$	3.61574	0.332856
$119$	3.08025	0.282366
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.95908	0.267902
$123$	0	0
$124$	−13.5223	−1.21434
$125$	−9.03844	−0.808423
$126$	0	0
$127$	−15.2083	−1.34951	−0.674757	−	0.738040i	$-0.735752\pi$
$127$	−15.2083	−1.34951	−0.674757	+	0.738040i	$0.735752\pi$
$128$	11.2676	0.995920
$129$	0	0
$130$	10.1473	0.889975
$131$	12.1031	1.05746	0.528729	−	0.848791i	$-0.322669\pi$
$131$	12.1031	1.05746	0.528729	+	0.848791i	$0.322669\pi$
$132$	0	0
$133$	−0.123250	−0.0106871
$134$	0.484318	0.0418387
$135$	0	0
$136$	−7.68485	−0.658970
$137$	−5.30742	−0.453443	−0.226722	−	0.973960i	$-0.572801\pi$
$137$	−5.30742	−0.453443	−0.226722	+	0.973960i	$0.572801\pi$
$138$	0	0
$139$	−7.00710	−0.594334	−0.297167	−	0.954826i	$-0.596042\pi$
$139$	−7.00710	−0.594334	−0.297167	+	0.954826i	$0.596042\pi$
$140$	−4.57186	−0.386392
$141$	0	0
$142$	6.43094	0.539672
$143$	−5.91392	−0.494547
$144$	0	0
$145$	−21.8769	−1.81677
$146$	6.63457	0.549081
$147$	0	0
$148$	17.2741	1.41992
$149$	−11.5103	−0.942957	−0.471478	−	0.881878i	$-0.656279\pi$
$149$	−11.5103	−0.942957	−0.471478	+	0.881878i	$0.656279\pi$
$150$	0	0
$151$	−15.9876	−1.30105	−0.650524	−	0.759485i	$-0.725451\pi$
$151$	−15.9876	−1.30105	−0.650524	+	0.759485i	$0.725451\pi$
$152$	0.307492	0.0249409
$153$	0	0
$154$	−0.354020	−0.0285277
$155$	−27.1357	−2.17959
$156$	0	0
$157$	3.83587	0.306136	0.153068	−	0.988216i	$-0.451085\pi$
$157$	3.83587	0.306136	0.153068	+	0.988216i	$0.451085\pi$
$158$	−1.66462	−0.132430
$159$	0	0
$160$	17.4646	1.38070
$161$	3.37701	0.266145
$162$	0	0
$163$	10.2941	0.806293	0.403146	−	0.915136i	$-0.367917\pi$
$163$	10.2941	0.806293	0.403146	+	0.915136i	$0.367917\pi$
$164$	5.14165	0.401496
$165$	0	0
$166$	0.687043	0.0533249
$167$	13.6114	1.05328	0.526641	−	0.850088i	$-0.323451\pi$
$167$	13.6114	1.05328	0.526641	+	0.850088i	$0.323451\pi$
$168$	0	0
$169$	21.9745	1.69035
$170$	−7.23041	−0.554547
$171$	0	0
$172$	−2.85196	−0.217460
$173$	4.57323	0.347697	0.173848	−	0.984772i	$-0.444380\pi$
$173$	4.57323	0.347697	0.173848	+	0.984772i	$0.444380\pi$
$174$	0	0
$175$	−5.51970	−0.417250
$176$	−2.64764	−0.199573
$177$	0	0
$178$	−1.36999	−0.102685
$179$	−0.302067	−0.0225775	−0.0112888	−	0.999936i	$-0.503593\pi$
$179$	−0.302067	−0.0225775	−0.0112888	+	0.999936i	$0.503593\pi$
$180$	0	0
$181$	−11.3794	−0.845825	−0.422912	−	0.906171i	$-0.638992\pi$
$181$	−11.3794	−0.845825	−0.422912	+	0.906171i	$0.638992\pi$
$182$	2.09365	0.155191
$183$	0	0
$184$	−8.42521	−0.621115
$185$	34.6645	2.54859
$186$	0	0
$187$	4.21395	0.308155
$188$	15.0288	1.09609
$189$	0	0
$190$	0.289309	0.0209887
$191$	−3.03842	−0.219852	−0.109926	−	0.993940i	$-0.535061\pi$
$191$	−3.03842	−0.219852	−0.109926	+	0.993940i	$0.535061\pi$
$192$	0	0
$193$	6.36026	0.457821	0.228911	−	0.973447i	$-0.426484\pi$
$193$	6.36026	0.457821	0.228911	+	0.973447i	$0.426484\pi$
$194$	−1.40570	−0.100923
$195$	0	0
$196$	11.4148	0.815340
$197$	8.66708	0.617504	0.308752	−	0.951143i	$-0.400089\pi$
$197$	8.66708	0.617504	0.308752	+	0.951143i	$0.400089\pi$
$198$	0	0
$199$	8.42039	0.596905	0.298453	−	0.954424i	$-0.403529\pi$
$199$	8.42039	0.596905	0.298453	+	0.954424i	$0.403529\pi$
$200$	13.7709	0.973753
$201$	0	0
$202$	−3.54252	−0.249251
$203$	−4.51377	−0.316804
$204$	0	0
$205$	10.3180	0.720638
$206$	−0.640668	−0.0446375
$207$	0	0
$208$	15.6579	1.08568
$209$	−0.168612	−0.0116631
$210$	0	0
$211$	26.3438	1.81359	0.906793	−	0.421577i	$-0.138523\pi$
$211$	26.3438	1.81359	0.906793	+	0.421577i	$0.138523\pi$
$212$	−18.7305	−1.28641
$213$	0	0
$214$	−7.31378	−0.499960
$215$	−5.72314	−0.390315
$216$	0	0
$217$	−5.59880	−0.380072
$218$	2.49262	0.168822
$219$	0	0
$220$	−6.25454	−0.421681
$221$	−24.9210	−1.67637
$222$	0	0
$223$	−4.34416	−0.290906	−0.145453	−	0.989365i	$-0.546464\pi$
$223$	−4.34416	−0.290906	−0.145453	+	0.989365i	$0.546464\pi$
$224$	3.60340	0.240762
$225$	0	0
$226$	−4.69974	−0.312622
$227$	−12.8788	−0.854798	−0.427399	−	0.904063i	$-0.640570\pi$
$227$	−12.8788	−0.854798	−0.427399	+	0.904063i	$0.640570\pi$
$228$	0	0
$229$	−11.7427	−0.775980	−0.387990	−	0.921664i	$-0.626831\pi$
$229$	−11.7427	−0.775980	−0.387990	+	0.921664i	$0.626831\pi$
$230$	−7.92699	−0.522691
$231$	0	0
$232$	11.2613	0.739339
$233$	−19.7223	−1.29205	−0.646027	−	0.763315i	$-0.723571\pi$
$233$	−19.7223	−1.29205	−0.646027	+	0.763315i	$0.723571\pi$
$234$	0	0
$235$	30.1589	1.96735
$236$	13.1801	0.857952
$237$	0	0
$238$	−1.49182	−0.0967005
$239$	1.28368	0.0830344	0.0415172	−	0.999138i	$-0.486781\pi$
$239$	1.28368	0.0830344	0.0415172	+	0.999138i	$0.486781\pi$
$240$	0	0
$241$	−3.66187	−0.235882	−0.117941	−	0.993021i	$-0.537629\pi$
$241$	−3.66187	−0.235882	−0.117941	+	0.993021i	$0.537629\pi$
$242$	−0.484318	−0.0311331
$243$	0	0
$244$	10.7864	0.690530
$245$	22.9065	1.46344
$246$	0	0
$247$	0.997158	0.0634476
$248$	13.9683	0.886988
$249$	0	0
$250$	4.37748	0.276856
$251$	10.4416	0.659066	0.329533	−	0.944144i	$-0.393109\pi$
$251$	10.4416	0.659066	0.329533	+	0.944144i	$0.393109\pi$
$252$	0	0
$253$	4.61992	0.290452
$254$	7.36563	0.462161
$255$	0	0
$256$	0.358454	0.0224033
$257$	−14.4427	−0.900910	−0.450455	−	0.892799i	$-0.648738\pi$
$257$	−14.4427	−0.900910	−0.450455	+	0.892799i	$0.648738\pi$
$258$	0	0
$259$	7.15220	0.444416
$260$	36.9889	2.29395
$261$	0	0
$262$	−5.86177	−0.362141
$263$	29.6930	1.83095	0.915474	−	0.402377i	$-0.131816\pi$
$263$	29.6930	1.83095	0.915474	+	0.402377i	$0.131816\pi$
$264$	0	0
$265$	−37.5872	−2.30896
$266$	0.0596920	0.00365995
$267$	0	0
$268$	1.76544	0.107841
$269$	31.3154	1.90934	0.954668	−	0.297672i	$-0.0962103\pi$
$269$	31.3154	1.90934	0.954668	+	0.297672i	$0.0962103\pi$
$270$	0	0
$271$	−0.561536	−0.0341109	−0.0170554	−	0.999855i	$-0.505429\pi$
$271$	−0.561536	−0.0341109	−0.0170554	+	0.999855i	$0.505429\pi$
$272$	−11.1570	−0.676493
$273$	0	0
$274$	2.57048	0.155288
$275$	−7.55123	−0.455357
$276$	0	0
$277$	−21.5577	−1.29527	−0.647637	−	0.761949i	$-0.724243\pi$
$277$	−21.5577	−1.29527	−0.647637	+	0.761949i	$0.724243\pi$
$278$	3.39366	0.203538
$279$	0	0
$280$	4.72266	0.282233
$281$	−9.02729	−0.538523	−0.269261	−	0.963067i	$-0.586780\pi$
$281$	−9.02729	−0.538523	−0.269261	+	0.963067i	$0.586780\pi$
$282$	0	0
$283$	6.20713	0.368976	0.184488	−	0.982835i	$-0.440937\pi$
$283$	6.20713	0.368976	0.184488	+	0.982835i	$0.440937\pi$
$284$	23.4421	1.39103
$285$	0	0
$286$	2.86422	0.169365
$287$	2.12886	0.125663
$288$	0	0
$289$	0.757372	0.0445513
$290$	10.5954	0.622181
$291$	0	0
$292$	24.1844	1.41528
$293$	1.85972	0.108646	0.0543231	−	0.998523i	$-0.482700\pi$
$293$	1.85972	0.108646	0.0543231	+	0.998523i	$0.482700\pi$
$294$	0	0
$295$	26.4491	1.53992
$296$	−17.8438	−1.03715
$297$	0	0
$298$	5.57462	0.322929
$299$	−27.3219	−1.58007
$300$	0	0
$301$	−1.18083	−0.0680621
$302$	7.74306	0.445563
$303$	0	0
$304$	0.446423	0.0256041
$305$	21.6456	1.23942
$306$	0	0
$307$	−24.5053	−1.39859	−0.699295	−	0.714834i	$-0.746502\pi$
$307$	−24.5053	−1.39859	−0.699295	+	0.714834i	$0.746502\pi$
$308$	−1.29047	−0.0735316
$309$	0	0
$310$	13.1423	0.746433
$311$	−6.46510	−0.366602	−0.183301	−	0.983057i	$-0.558678\pi$
$311$	−6.46510	−0.366602	−0.183301	+	0.983057i	$0.558678\pi$
$312$	0	0
$313$	34.6259	1.95717	0.978586	−	0.205837i	$-0.0659918\pi$
$313$	34.6259	1.95717	0.978586	+	0.205837i	$0.0659918\pi$
$314$	−1.85778	−0.104841
$315$	0	0
$316$	−6.06787	−0.341344
$317$	−25.7860	−1.44829	−0.724143	−	0.689650i	$-0.757765\pi$
$317$	−25.7860	−1.44829	−0.724143	+	0.689650i	$0.757765\pi$
$318$	0	0
$319$	−6.17507	−0.345737
$320$	10.3015	0.575874
$321$	0	0
$322$	−1.63555	−0.0911454
$323$	−0.710522	−0.0395345
$324$	0	0
$325$	44.6574	2.47715
$326$	−4.98560	−0.276127
$327$	0	0
$328$	−5.31125	−0.293265
$329$	6.22256	0.343061
$330$	0	0
$331$	14.2125	0.781189	0.390595	−	0.920563i	$-0.372269\pi$
$331$	14.2125	0.781189	0.390595	+	0.920563i	$0.372269\pi$
$332$	2.50441	0.137447
$333$	0	0
$334$	−6.59224	−0.360711
$335$	3.54277	0.193562
$336$	0	0
$337$	28.3462	1.54412	0.772059	−	0.635551i	$-0.219227\pi$
$337$	28.3462	1.54412	0.772059	+	0.635551i	$0.219227\pi$
$338$	−10.6426	−0.578883
$339$	0	0
$340$	−26.3563	−1.42937
$341$	−7.65946	−0.414783
$342$	0	0
$343$	9.84296	0.531470
$344$	2.94603	0.158839
$345$	0	0
$346$	−2.21490	−0.119074
$347$	24.0576	1.29148	0.645740	−	0.763557i	$-0.276549\pi$
$347$	24.0576	1.29148	0.645740	+	0.763557i	$0.276549\pi$
$348$	0	0
$349$	20.8930	1.11838	0.559189	−	0.829040i	$-0.311113\pi$
$349$	20.8930	1.11838	0.559189	+	0.829040i	$0.311113\pi$
$350$	2.67329	0.142893
$351$	0	0
$352$	4.92963	0.262750
$353$	17.2540	0.918336	0.459168	−	0.888349i	$-0.348148\pi$
$353$	17.2540	0.918336	0.459168	+	0.888349i	$0.348148\pi$
$354$	0	0
$355$	47.0421	2.49674
$356$	−4.99390	−0.264676
$357$	0	0
$358$	0.146296	0.00773200
$359$	−23.9204	−1.26247	−0.631236	−	0.775591i	$-0.717452\pi$
$359$	−23.9204	−1.26247	−0.631236	+	0.775591i	$0.717452\pi$
$360$	0	0
$361$	−18.9716	−0.998504
$362$	5.51125	0.289665
$363$	0	0
$364$	7.63177	0.400013
$365$	48.5317	2.54027
$366$	0	0
$367$	13.2067	0.689382	0.344691	−	0.938716i	$-0.387984\pi$
$367$	13.2067	0.689382	0.344691	+	0.938716i	$0.387984\pi$
$368$	−12.2319	−0.637631
$369$	0	0
$370$	−16.7887	−0.872801
$371$	−7.75522	−0.402631
$372$	0	0
$373$	11.4746	0.594132	0.297066	−	0.954857i	$-0.403992\pi$
$373$	11.4746	0.594132	0.297066	+	0.954857i	$0.403992\pi$
$374$	−2.04089	−0.105532
$375$	0	0
$376$	−15.5245	−0.800615
$377$	36.5189	1.88082
$378$	0	0
$379$	30.0655	1.54436	0.772180	−	0.635404i	$-0.219166\pi$
$379$	30.0655	1.54436	0.772180	+	0.635404i	$0.219166\pi$
$380$	1.05459	0.0540993
$381$	0	0
$382$	1.47156	0.0752915
$383$	37.9318	1.93822	0.969111	−	0.246623i	$-0.0793210\pi$
$383$	37.9318	1.93822	0.969111	+	0.246623i	$0.0793210\pi$
$384$	0	0
$385$	−2.58965	−0.131981
$386$	−3.08039	−0.156788
$387$	0	0
$388$	−5.12405	−0.260134
$389$	−25.6996	−1.30302	−0.651511	−	0.758639i	$-0.725865\pi$
$389$	−25.6996	−1.30302	−0.651511	+	0.758639i	$0.725865\pi$
$390$	0	0
$391$	19.4681	0.984545
$392$	−11.7913	−0.595549
$393$	0	0
$394$	−4.19762	−0.211473
$395$	−12.1766	−0.612673
$396$	0	0
$397$	18.3801	0.922469	0.461234	−	0.887278i	$-0.347407\pi$
$397$	18.3801	0.922469	0.461234	+	0.887278i	$0.347407\pi$
$398$	−4.07814	−0.204419
$399$	0	0
$400$	19.9929	0.999647
$401$	−17.3042	−0.864129	−0.432064	−	0.901843i	$-0.642215\pi$
$401$	−17.3042	−0.864129	−0.432064	+	0.901843i	$0.642215\pi$
$402$	0	0
$403$	45.2974	2.25643
$404$	−12.9132	−0.642456
$405$	0	0
$406$	2.18610	0.108494
$407$	9.78458	0.485004
$408$	0	0
$409$	−2.52942	−0.125072	−0.0625359	−	0.998043i	$-0.519919\pi$
$409$	−2.52942	−0.125072	−0.0625359	+	0.998043i	$0.519919\pi$
$410$	−4.99717	−0.246793
$411$	0	0
$412$	−2.33537	−0.115055
$413$	5.45713	0.268528
$414$	0	0
$415$	5.02570	0.246702
$416$	−29.1535	−1.42937
$417$	0	0
$418$	0.0816618	0.00399421
$419$	27.7685	1.35658	0.678291	−	0.734793i	$-0.262721\pi$
$419$	27.7685	1.35658	0.678291	+	0.734793i	$0.262721\pi$
$420$	0	0
$421$	−19.8369	−0.966792	−0.483396	−	0.875402i	$-0.660597\pi$
$421$	−19.8369	−0.966792	−0.483396	+	0.875402i	$0.660597\pi$
$422$	−12.7588	−0.621088
$423$	0	0
$424$	19.3483	0.939636
$425$	−31.8205	−1.54352
$426$	0	0
$427$	4.46604	0.216127
$428$	−26.6602	−1.28867
$429$	0	0
$430$	2.77182	0.133669
$431$	−31.0209	−1.49422	−0.747112	−	0.664699i	$-0.768560\pi$
$431$	−31.0209	−1.49422	−0.747112	+	0.664699i	$0.768560\pi$
$432$	0	0
$433$	33.5320	1.61145	0.805723	−	0.592293i	$-0.201777\pi$
$433$	33.5320	1.61145	0.805723	+	0.592293i	$0.201777\pi$
$434$	2.71160	0.130161
$435$	0	0
$436$	9.08611	0.435146
$437$	−0.778974	−0.0372634
$438$	0	0
$439$	−29.6763	−1.41637	−0.708186	−	0.706026i	$-0.750486\pi$
$439$	−29.6763	−1.41637	−0.708186	+	0.706026i	$0.750486\pi$
$440$	6.46084	0.308008
$441$	0	0
$442$	12.0697	0.574096
$443$	−3.73608	−0.177507	−0.0887534	−	0.996054i	$-0.528288\pi$
$443$	−3.73608	−0.177507	−0.0887534	+	0.996054i	$0.528288\pi$
$444$	0	0
$445$	−10.0215	−0.475062
$446$	2.10395	0.0996250
$447$	0	0
$448$	2.12548	0.100419
$449$	17.9398	0.846631	0.423316	−	0.905982i	$-0.360866\pi$
$449$	17.9398	0.846631	0.423316	+	0.905982i	$0.360866\pi$
$450$	0	0
$451$	2.91240	0.137139
$452$	−17.1315	−0.805799
$453$	0	0
$454$	6.23745	0.292738
$455$	15.3150	0.717977
$456$	0	0
$457$	−27.3544	−1.27959	−0.639793	−	0.768547i	$-0.720980\pi$
$457$	−27.3544	−1.27959	−0.639793	+	0.768547i	$0.720980\pi$
$458$	5.68720	0.265746
$459$	0	0
$460$	−28.8955	−1.34726
$461$	13.5464	0.630919	0.315459	−	0.948939i	$-0.397841\pi$
$461$	13.5464	0.630919	0.315459	+	0.948939i	$0.397841\pi$
$462$	0	0
$463$	16.2620	0.755761	0.377881	−	0.925854i	$-0.376653\pi$
$463$	16.2620	0.755761	0.377881	+	0.925854i	$0.376653\pi$
$464$	16.3493	0.758999
$465$	0	0
$466$	9.55189	0.442483
$467$	17.3313	0.801998	0.400999	−	0.916078i	$-0.368663\pi$
$467$	17.3313	0.801998	0.400999	+	0.916078i	$0.368663\pi$
$468$	0	0
$469$	0.730966	0.0337529
$470$	−14.6065	−0.673747
$471$	0	0
$472$	−13.6148	−0.626674
$473$	−1.61544	−0.0742781
$474$	0	0
$475$	1.27323	0.0584197
$476$	−5.43799	−0.249250
$477$	0	0
$478$	−0.621710	−0.0284363
$479$	−5.86243	−0.267861	−0.133931	−	0.990991i	$-0.542760\pi$
$479$	−5.86243	−0.267861	−0.133931	+	0.990991i	$0.542760\pi$
$480$	0	0
$481$	−57.8653	−2.63843
$482$	1.77351	0.0807811
$483$	0	0
$484$	−1.76544	−0.0802471
$485$	−10.2826	−0.466911
$486$	0	0
$487$	−4.40785	−0.199739	−0.0998695	−	0.995001i	$-0.531843\pi$
$487$	−4.40785	−0.199739	−0.0998695	+	0.995001i	$0.531843\pi$
$488$	−11.1422	−0.504384
$489$	0	0
$490$	−11.0940	−0.501176
$491$	22.8238	1.03002	0.515011	−	0.857184i	$-0.327788\pi$
$491$	22.8238	1.03002	0.515011	+	0.857184i	$0.327788\pi$
$492$	0	0
$493$	−26.0214	−1.17195
$494$	−0.482941	−0.0217286
$495$	0	0
$496$	20.2795	0.910575
$497$	9.70602	0.435374
$498$	0	0
$499$	28.0390	1.25520	0.627600	−	0.778536i	$-0.284038\pi$
$499$	28.0390	1.25520	0.627600	+	0.778536i	$0.284038\pi$
$500$	15.9568	0.713610
$501$	0	0
$502$	−5.05704	−0.225707
$503$	−13.5372	−0.603594	−0.301797	−	0.953372i	$-0.597586\pi$
$503$	−13.5372	−0.603594	−0.301797	+	0.953372i	$0.597586\pi$
$504$	0	0
$505$	−25.9135	−1.15313
$506$	−2.23751	−0.0994695
$507$	0	0
$508$	26.8492	1.19124
$509$	−6.15050	−0.272616	−0.136308	−	0.990667i	$-0.543524\pi$
$509$	−6.15050	−0.272616	−0.136308	+	0.990667i	$0.543524\pi$
$510$	0	0
$511$	10.0134	0.442965
$512$	−22.7087	−1.00359
$513$	0	0
$514$	6.99485	0.308530
$515$	−4.68647	−0.206511
$516$	0	0
$517$	8.51279	0.374392
$518$	−3.46394	−0.152197
$519$	0	0
$520$	−38.2089	−1.67557
$521$	13.4524	0.589360	0.294680	−	0.955596i	$-0.404787\pi$
$521$	13.4524	0.589360	0.294680	+	0.955596i	$0.404787\pi$
$522$	0	0
$523$	−36.0827	−1.57779	−0.788893	−	0.614530i	$-0.789346\pi$
$523$	−36.0827	−1.57779	−0.788893	+	0.614530i	$0.789346\pi$
$524$	−21.3673	−0.933436
$525$	0	0
$526$	−14.3808	−0.627035
$527$	−32.2766	−1.40599
$528$	0	0
$529$	−1.65630	−0.0720132
$530$	18.2042	0.790738
$531$	0	0
$532$	0.217589	0.00943369
$533$	−17.2237	−0.746041
$534$	0	0
$535$	−53.5001	−2.31301
$536$	−1.82367	−0.0787705
$537$	0	0
$538$	−15.1666	−0.653880
$539$	6.46569	0.278497
$540$	0	0
$541$	30.2824	1.30194	0.650971	−	0.759102i	$-0.274362\pi$
$541$	30.2824	1.30194	0.650971	+	0.759102i	$0.274362\pi$
$542$	0.271962	0.0116818
$543$	0	0
$544$	20.7732	0.890645
$545$	18.2335	0.781035
$546$	0	0
$547$	−38.4642	−1.64461	−0.822305	−	0.569047i	$-0.807312\pi$
$547$	−38.4642	−1.64461	−0.822305	+	0.569047i	$0.807312\pi$
$548$	9.36991	0.400263
$549$	0	0
$550$	3.65720	0.155943
$551$	1.04119	0.0443562
$552$	0	0
$553$	−2.51236	−0.106836
$554$	10.4408	0.443585
$555$	0	0
$556$	12.3706	0.524629
$557$	34.5049	1.46202	0.731009	−	0.682367i	$-0.239050\pi$
$557$	34.5049	1.46202	0.731009	+	0.682367i	$0.239050\pi$
$558$	0	0
$559$	9.55360	0.404074
$560$	6.85644	0.289738
$561$	0	0
$562$	4.37208	0.184425
$563$	40.7463	1.71725	0.858626	−	0.512603i	$-0.171319\pi$
$563$	40.7463	1.71725	0.858626	+	0.512603i	$0.171319\pi$
$564$	0	0
$565$	−34.3785	−1.44631
$566$	−3.00622	−0.126361
$567$	0	0
$568$	−24.2153	−1.01605
$569$	2.17648	0.0912429	0.0456215	−	0.998959i	$-0.485473\pi$
$569$	2.17648	0.0912429	0.0456215	+	0.998959i	$0.485473\pi$
$570$	0	0
$571$	16.4020	0.686404	0.343202	−	0.939262i	$-0.388488\pi$
$571$	16.4020	0.686404	0.343202	+	0.939262i	$0.388488\pi$
$572$	10.4407	0.436546
$573$	0	0
$574$	−1.03105	−0.0430351
$575$	−34.8861	−1.45485
$576$	0	0
$577$	41.4773	1.72672	0.863362	−	0.504584i	$-0.168354\pi$
$577$	41.4773	1.72672	0.863362	+	0.504584i	$0.168354\pi$
$578$	−0.366809	−0.0152572
$579$	0	0
$580$	38.6222	1.60370
$581$	1.03693	0.0430192
$582$	0	0
$583$	−10.6095	−0.439403
$584$	−24.9821	−1.03376
$585$	0	0
$586$	−0.900696	−0.0372074
$587$	24.0553	0.992869	0.496435	−	0.868074i	$-0.334642\pi$
$587$	24.0553	0.992869	0.496435	+	0.868074i	$0.334642\pi$
$588$	0	0
$589$	1.29148	0.0532143
$590$	−12.8097	−0.527369
$591$	0	0
$592$	−25.9060	−1.06473
$593$	−30.9995	−1.27300	−0.636498	−	0.771279i	$-0.719617\pi$
$593$	−30.9995	−1.27300	−0.636498	+	0.771279i	$0.719617\pi$
$594$	0	0
$595$	−10.9126	−0.447375
$596$	20.3206	0.832365
$597$	0	0
$598$	13.2325	0.541116
$599$	24.7255	1.01026	0.505128	−	0.863045i	$-0.331445\pi$
$599$	24.7255	1.01026	0.505128	+	0.863045i	$0.331445\pi$
$600$	0	0
$601$	−12.6843	−0.517405	−0.258703	−	0.965957i	$-0.583295\pi$
$601$	−12.6843	−0.517405	−0.258703	+	0.965957i	$0.583295\pi$
$602$	0.571899	0.0233089
$603$	0	0
$604$	28.2250	1.14846
$605$	−3.54277	−0.144034
$606$	0	0
$607$	32.4377	1.31661	0.658304	−	0.752752i	$-0.271274\pi$
$607$	32.4377	1.31661	0.658304	+	0.752752i	$0.271274\pi$
$608$	−0.831195	−0.0337094
$609$	0	0
$610$	−10.4833	−0.424458
$611$	−50.3440	−2.03670
$612$	0	0
$613$	−24.0359	−0.970802	−0.485401	−	0.874292i	$-0.661326\pi$
$613$	−24.0359	−0.970802	−0.485401	+	0.874292i	$0.661326\pi$
$614$	11.8683	0.478967
$615$	0	0
$616$	1.33304	0.0537097
$617$	2.56216	0.103148	0.0515742	−	0.998669i	$-0.483576\pi$
$617$	2.56216	0.103148	0.0515742	+	0.998669i	$0.483576\pi$
$618$	0	0
$619$	−10.2973	−0.413885	−0.206943	−	0.978353i	$-0.566351\pi$
$619$	−10.2973	−0.413885	−0.206943	+	0.978353i	$0.566351\pi$
$620$	47.9064	1.92397
$621$	0	0
$622$	3.13116	0.125548
$623$	−2.06769	−0.0828401
$624$	0	0
$625$	−5.73503	−0.229401
$626$	−16.7699	−0.670262
$627$	0	0
$628$	−6.77199	−0.270232
$629$	41.2317	1.64402
$630$	0	0
$631$	41.1107	1.63659	0.818296	−	0.574798i	$-0.194919\pi$
$631$	41.1107	1.63659	0.818296	+	0.574798i	$0.194919\pi$
$632$	6.26801	0.249328
$633$	0	0
$634$	12.4886	0.495987
$635$	53.8794	2.13814
$636$	0	0
$637$	−38.2376	−1.51503
$638$	2.99070	0.118403
$639$	0	0
$640$	−39.9184	−1.57791
$641$	48.2660	1.90639	0.953197	−	0.302350i	$-0.0977712\pi$
$641$	48.2660	1.90639	0.953197	+	0.302350i	$0.0977712\pi$
$642$	0	0
$643$	5.01096	0.197613	0.0988065	−	0.995107i	$-0.468498\pi$
$643$	5.01096	0.197613	0.0988065	+	0.995107i	$0.468498\pi$
$644$	−5.96189	−0.234931
$645$	0	0
$646$	0.344119	0.0135392
$647$	−43.6747	−1.71703	−0.858514	−	0.512790i	$-0.828612\pi$
$647$	−43.6747	−1.71703	−0.858514	+	0.512790i	$0.828612\pi$
$648$	0	0
$649$	7.46564	0.293052
$650$	−21.6284	−0.848335
$651$	0	0
$652$	−18.1735	−0.711729
$653$	−6.30539	−0.246749	−0.123374	−	0.992360i	$-0.539372\pi$
$653$	−6.30539	−0.246749	−0.123374	+	0.992360i	$0.539372\pi$
$654$	0	0
$655$	−42.8787	−1.67541
$656$	−7.71097	−0.301063
$657$	0	0
$658$	−3.01370	−0.117486
$659$	−36.9811	−1.44058	−0.720290	−	0.693673i	$-0.755991\pi$
$659$	−36.9811	−1.44058	−0.720290	+	0.693673i	$0.755991\pi$
$660$	0	0
$661$	21.2536	0.826671	0.413335	−	0.910579i	$-0.364364\pi$
$661$	21.2536	0.826671	0.413335	+	0.910579i	$0.364364\pi$
$662$	−6.88336	−0.267530
$663$	0	0
$664$	−2.58702	−0.100396
$665$	0.436645	0.0169324
$666$	0	0
$667$	−28.5283	−1.10462
$668$	−24.0301	−0.929751
$669$	0	0
$670$	−1.71583	−0.0662882
$671$	6.10978	0.235866
$672$	0	0
$673$	−5.47856	−0.211183	−0.105591	−	0.994410i	$-0.533674\pi$
$673$	−5.47856	−0.211183	−0.105591	+	0.994410i	$0.533674\pi$
$674$	−13.7286	−0.528805
$675$	0	0
$676$	−38.7946	−1.49210
$677$	−31.5575	−1.21285	−0.606426	−	0.795140i	$-0.707398\pi$
$677$	−31.5575	−1.21285	−0.606426	+	0.795140i	$0.707398\pi$
$678$	0	0
$679$	−2.12158	−0.0814186
$680$	27.2257	1.04406
$681$	0	0
$682$	3.70961	0.142048
$683$	−27.5428	−1.05389	−0.526947	−	0.849898i	$-0.676663\pi$
$683$	−27.5428	−1.05389	−0.526947	+	0.849898i	$0.676663\pi$
$684$	0	0
$685$	18.8030	0.718424
$686$	−4.76712	−0.182010
$687$	0	0
$688$	4.27710	0.163063
$689$	62.7441	2.39036
$690$	0	0
$691$	16.7413	0.636867	0.318434	−	0.947945i	$-0.396843\pi$
$691$	16.7413	0.636867	0.318434	+	0.947945i	$0.396843\pi$
$692$	−8.07375	−0.306918
$693$	0	0
$694$	−11.6515	−0.442286
$695$	24.8245	0.941649
$696$	0	0
$697$	12.2727	0.464862
$698$	−10.1189	−0.383005
$699$	0	0
$700$	9.74467	0.368314
$701$	28.1391	1.06280	0.531400	−	0.847121i	$-0.321666\pi$
$701$	28.1391	1.06280	0.531400	+	0.847121i	$0.321666\pi$
$702$	0	0
$703$	−1.64980	−0.0622233
$704$	2.90776	0.109591
$705$	0	0
$706$	−8.35640	−0.314497
$707$	−5.34662	−0.201080
$708$	0	0
$709$	−4.94252	−0.185620	−0.0928100	−	0.995684i	$-0.529585\pi$
$709$	−4.94252	−0.185620	−0.0928100	+	0.995684i	$0.529585\pi$
$710$	−22.7834	−0.855044
$711$	0	0
$712$	5.15862	0.193327
$713$	−35.3861	−1.32522
$714$	0	0
$715$	20.9517	0.783549
$716$	0.533280	0.0199296
$717$	0	0
$718$	11.5851	0.432352
$719$	14.6252	0.545426	0.272713	−	0.962095i	$-0.412079\pi$
$719$	14.6252	0.545426	0.272713	+	0.962095i	$0.412079\pi$
$720$	0	0
$721$	−0.966941	−0.0360108
$722$	9.18827	0.341952
$723$	0	0
$724$	20.0896	0.746625
$725$	46.6294	1.73177
$726$	0	0
$727$	9.59018	0.355680	0.177840	−	0.984059i	$-0.443089\pi$
$727$	9.59018	0.355680	0.177840	+	0.984059i	$0.443089\pi$
$728$	−7.88349	−0.292182
$729$	0	0
$730$	−23.5048	−0.869951
$731$	−6.80739	−0.251780
$732$	0	0
$733$	27.7809	1.02611	0.513056	−	0.858355i	$-0.328513\pi$
$733$	27.7809	1.02611	0.513056	+	0.858355i	$0.328513\pi$
$734$	−6.39622	−0.236089
$735$	0	0
$736$	22.7745	0.839480
$737$	1.00000	0.0368355
$738$	0	0
$739$	−46.2390	−1.70093	−0.850465	−	0.526032i	$-0.823679\pi$
$739$	−46.2390	−1.70093	−0.850465	+	0.526032i	$0.823679\pi$
$740$	−61.1980	−2.24968
$741$	0	0
$742$	3.75599	0.137887
$743$	8.63575	0.316815	0.158407	−	0.987374i	$-0.449364\pi$
$743$	8.63575	0.316815	0.158407	+	0.987374i	$0.449364\pi$
$744$	0	0
$745$	40.7782	1.49400
$746$	−5.55735	−0.203469
$747$	0	0
$748$	−7.43946	−0.272014
$749$	−11.0385	−0.403337
$750$	0	0
$751$	−8.34168	−0.304392	−0.152196	−	0.988350i	$-0.548635\pi$
$751$	−8.34168	−0.304392	−0.152196	+	0.988350i	$0.548635\pi$
$752$	−22.5388	−0.821905
$753$	0	0
$754$	−17.6867	−0.644113
$755$	56.6403	2.06135
$756$	0	0
$757$	11.4502	0.416164	0.208082	−	0.978111i	$-0.433278\pi$
$757$	11.4502	0.416164	0.208082	+	0.978111i	$0.433278\pi$
$758$	−14.5612	−0.528888
$759$	0	0
$760$	−1.08937	−0.0395158
$761$	−40.0996	−1.45361	−0.726805	−	0.686844i	$-0.758996\pi$
$761$	−40.0996	−1.45361	−0.726805	+	0.686844i	$0.758996\pi$
$762$	0	0
$763$	3.76204	0.136195
$764$	5.36413	0.194067
$765$	0	0
$766$	−18.3710	−0.663772
$767$	−44.1512	−1.59421
$768$	0	0
$769$	25.1415	0.906626	0.453313	−	0.891351i	$-0.350242\pi$
$769$	25.1415	0.906626	0.453313	+	0.891351i	$0.350242\pi$
$770$	1.25421	0.0451987
$771$	0	0
$772$	−11.2286	−0.404127
$773$	−50.8150	−1.82769	−0.913845	−	0.406063i	$-0.866901\pi$
$773$	−50.8150	−1.82769	−0.913845	+	0.406063i	$0.866901\pi$
$774$	0	0
$775$	57.8384	2.07761
$776$	5.29306	0.190010
$777$	0	0
$778$	12.4468	0.446239
$779$	−0.491065	−0.0175942
$780$	0	0
$781$	13.2783	0.475136
$782$	−9.42876	−0.337172
$783$	0	0
$784$	−17.1188	−0.611386
$785$	−13.5896	−0.485034
$786$	0	0
$787$	1.86474	0.0664707	0.0332354	−	0.999448i	$-0.489419\pi$
$787$	1.86474	0.0664707	0.0332354	+	0.999448i	$0.489419\pi$
$788$	−15.3012	−0.545082
$789$	0	0
$790$	5.89736	0.209819
$791$	−7.09318	−0.252204
$792$	0	0
$793$	−36.1328	−1.28311
$794$	−8.90179	−0.315913
$795$	0	0
$796$	−14.8657	−0.526899
$797$	21.4736	0.760634	0.380317	−	0.924856i	$-0.375815\pi$
$797$	21.4736	0.760634	0.380317	+	0.924856i	$0.375815\pi$
$798$	0	0
$799$	35.8725	1.26908
$800$	−37.2248	−1.31610
$801$	0	0
$802$	8.38072	0.295933
$803$	13.6988	0.483420
$804$	0	0
$805$	−11.9640	−0.421675
$806$	−21.9384	−0.772746
$807$	0	0
$808$	13.3391	0.469269
$809$	−0.305286	−0.0107333	−0.00536665	−	0.999986i	$-0.501708\pi$
$809$	−0.305286	−0.0107333	−0.00536665	+	0.999986i	$0.501708\pi$
$810$	0	0
$811$	−26.2074	−0.920268	−0.460134	−	0.887849i	$-0.652199\pi$
$811$	−26.2074	−0.920268	−0.460134	+	0.887849i	$0.652199\pi$
$812$	7.96877	0.279649
$813$	0	0
$814$	−4.73885	−0.166097
$815$	−36.4695	−1.27747
$816$	0	0
$817$	0.272383	0.00952947
$818$	1.22504	0.0428326
$819$	0	0
$820$	−18.2157	−0.636120
$821$	53.1835	1.85612	0.928059	−	0.372433i	$-0.121476\pi$
$821$	53.1835	1.85612	0.928059	+	0.372433i	$0.121476\pi$
$822$	0	0
$823$	−24.6875	−0.860551	−0.430275	−	0.902698i	$-0.641584\pi$
$823$	−24.6875	−0.860551	−0.430275	+	0.902698i	$0.641584\pi$
$824$	2.41240	0.0840398
$825$	0	0
$826$	−2.64298	−0.0919612
$827$	−5.93040	−0.206220	−0.103110	−	0.994670i	$-0.532879\pi$
$827$	−5.93040	−0.206220	−0.103110	+	0.994670i	$0.532879\pi$
$828$	0	0
$829$	−46.4177	−1.61215	−0.806077	−	0.591811i	$-0.798413\pi$
$829$	−46.4177	−1.61215	−0.806077	+	0.591811i	$0.798413\pi$
$830$	−2.43404	−0.0844866
$831$	0	0
$832$	−17.1963	−0.596174
$833$	27.2461	0.944021
$834$	0	0
$835$	−48.2221	−1.66879
$836$	0.297674	0.0102953
$837$	0	0
$838$	−13.4488	−0.464581
$839$	14.1077	0.487050	0.243525	−	0.969895i	$-0.421696\pi$
$839$	14.1077	0.487050	0.243525	+	0.969895i	$0.421696\pi$
$840$	0	0
$841$	9.13147	0.314878
$842$	9.60737	0.331092
$843$	0	0
$844$	−46.5084	−1.60088
$845$	−77.8506	−2.67814
$846$	0	0
$847$	−0.730966	−0.0251163
$848$	28.0902	0.964623
$849$	0	0
$850$	15.4112	0.528601
$851$	45.2040	1.54957
$852$	0	0
$853$	−44.4038	−1.52036	−0.760178	−	0.649715i	$-0.774888\pi$
$853$	−44.4038	−1.52036	−0.760178	+	0.649715i	$0.774888\pi$
$854$	−2.16298	−0.0740158
$855$	0	0
$856$	27.5396	0.941284
$857$	35.9889	1.22936	0.614680	−	0.788777i	$-0.289285\pi$
$857$	35.9889	1.22936	0.614680	+	0.788777i	$0.289285\pi$
$858$	0	0
$859$	−36.9757	−1.26160	−0.630798	−	0.775947i	$-0.717272\pi$
$859$	−36.9757	−1.26160	−0.630798	+	0.775947i	$0.717272\pi$
$860$	10.1038	0.344538
$861$	0	0
$862$	15.0240	0.511718
$863$	23.5787	0.802628	0.401314	−	0.915941i	$-0.368554\pi$
$863$	23.5787	0.802628	0.401314	+	0.915941i	$0.368554\pi$
$864$	0	0
$865$	−16.2019	−0.550882
$866$	−16.2401	−0.551863
$867$	0	0
$868$	9.88433	0.335496
$869$	−3.43703	−0.116593
$870$	0	0
$871$	−5.91392	−0.200386
$872$	−9.38581	−0.317843
$873$	0	0
$874$	0.377271	0.0127614
$875$	6.60680	0.223350
$876$	0	0
$877$	34.8587	1.17709	0.588547	−	0.808463i	$-0.299700\pi$
$877$	34.8587	1.17709	0.588547	+	0.808463i	$0.299700\pi$
$878$	14.3728	0.485057
$879$	0	0
$880$	9.37998	0.316199
$881$	32.5187	1.09558	0.547792	−	0.836615i	$-0.315469\pi$
$881$	32.5187	1.09558	0.547792	+	0.836615i	$0.315469\pi$
$882$	0	0
$883$	18.8622	0.634765	0.317382	−	0.948298i	$-0.397196\pi$
$883$	18.8622	0.634765	0.317382	+	0.948298i	$0.397196\pi$
$884$	43.9964	1.47976
$885$	0	0
$886$	1.80945	0.0607898
$887$	−19.7831	−0.664252	−0.332126	−	0.943235i	$-0.607766\pi$
$887$	−19.7831	−0.664252	−0.332126	+	0.943235i	$0.607766\pi$
$888$	0	0
$889$	11.1167	0.372843
$890$	4.85357	0.162692
$891$	0	0
$892$	7.66933	0.256788
$893$	−1.43536	−0.0480324
$894$	0	0
$895$	1.07015	0.0357713
$896$	−8.23620	−0.275152
$897$	0	0
$898$	−8.68856	−0.289941
$899$	47.2977	1.57747
$900$	0	0
$901$	−44.7081	−1.48944
$902$	−1.41053	−0.0469654
$903$	0	0
$904$	17.6966	0.588580
$905$	40.3147	1.34010
$906$	0	0
$907$	3.13028	0.103939	0.0519696	−	0.998649i	$-0.483450\pi$
$907$	3.13028	0.103939	0.0519696	+	0.998649i	$0.483450\pi$
$908$	22.7368	0.754546
$909$	0	0
$910$	−7.41731	−0.245882
$911$	9.70729	0.321617	0.160808	−	0.986986i	$-0.448590\pi$
$911$	9.70729	0.321617	0.160808	+	0.986986i	$0.448590\pi$
$912$	0	0
$913$	1.41858	0.0469481
$914$	13.2482	0.438213
$915$	0	0
$916$	20.7310	0.684972
$917$	−8.84699	−0.292153
$918$	0	0
$919$	54.9215	1.81169	0.905847	−	0.423606i	$-0.139236\pi$
$919$	54.9215	1.81169	0.905847	+	0.423606i	$0.139236\pi$
$920$	29.8486	0.984079
$921$	0	0
$922$	−6.56076	−0.216067
$923$	−78.5271	−2.58475
$924$	0	0
$925$	−73.8857	−2.42935
$926$	−7.87600	−0.258821
$927$	0	0
$928$	−30.4408	−0.999269
$929$	−4.07418	−0.133670	−0.0668348	−	0.997764i	$-0.521290\pi$
$929$	−4.07418	−0.133670	−0.0668348	+	0.997764i	$0.521290\pi$
$930$	0	0
$931$	−1.09019	−0.0357296
$932$	34.8186	1.14052
$933$	0	0
$934$	−8.39387	−0.274656
$935$	−14.9291	−0.488233
$936$	0	0
$937$	−41.8992	−1.36879	−0.684394	−	0.729113i	$-0.739933\pi$
$937$	−41.8992	−1.36879	−0.684394	+	0.729113i	$0.739933\pi$
$938$	−0.354020	−0.0115592
$939$	0	0
$940$	−53.2436	−1.73661
$941$	−30.0825	−0.980661	−0.490331	−	0.871537i	$-0.663124\pi$
$941$	−30.0825	−0.980661	−0.490331	+	0.871537i	$0.663124\pi$
$942$	0	0
$943$	13.4551	0.438157
$944$	−19.7663	−0.643338
$945$	0	0
$946$	0.782387	0.0254376
$947$	−19.1756	−0.623124	−0.311562	−	0.950226i	$-0.600852\pi$
$947$	−19.1756	−0.623124	−0.311562	+	0.950226i	$0.600852\pi$
$948$	0	0
$949$	−81.0137	−2.62981
$950$	−0.616647	−0.0200067
$951$	0	0
$952$	5.61736	0.182060
$953$	7.20564	0.233414	0.116707	−	0.993166i	$-0.462766\pi$
$953$	7.20564	0.233414	0.116707	+	0.993166i	$0.462766\pi$
$954$	0	0
$955$	10.7644	0.348328
$956$	−2.26626	−0.0732960
$957$	0	0
$958$	2.83928	0.0917330
$959$	3.87954	0.125277
$960$	0	0
$961$	27.6673	0.892493
$962$	28.0252	0.903568
$963$	0	0
$964$	6.46480	0.208217
$965$	−22.5329	−0.725361
$966$	0	0
$967$	47.0846	1.51414	0.757070	−	0.653334i	$-0.226630\pi$
$967$	47.0846	1.51414	0.757070	+	0.653334i	$0.226630\pi$
$968$	1.82367	0.0586149
$969$	0	0
$970$	4.98007	0.159900
$971$	−45.5554	−1.46194	−0.730971	−	0.682409i	$-0.760932\pi$
$971$	−45.5554	−1.46194	−0.730971	+	0.682409i	$0.760932\pi$
$972$	0	0
$973$	5.12195	0.164202
$974$	2.13480	0.0684035
$975$	0	0
$976$	−16.1765	−0.517797
$977$	28.0005	0.895816	0.447908	−	0.894080i	$-0.352169\pi$
$977$	28.0005	0.895816	0.447908	+	0.894080i	$0.352169\pi$
$978$	0	0
$979$	−2.82870	−0.0904058
$980$	−40.4399	−1.29181
$981$	0	0
$982$	−11.0540	−0.352746
$983$	16.3410	0.521196	0.260598	−	0.965447i	$-0.416080\pi$
$983$	16.3410	0.521196	0.260598	+	0.965447i	$0.416080\pi$
$984$	0	0
$985$	−30.7055	−0.978358
$986$	12.6026	0.401350
$987$	0	0
$988$	−1.76042	−0.0560064
$989$	−7.46322	−0.237317
$990$	0	0
$991$	−19.9773	−0.634601	−0.317300	−	0.948325i	$-0.602776\pi$
$991$	−19.9773	−0.634601	−0.317300	+	0.948325i	$0.602776\pi$
$992$	−37.7583	−1.19883
$993$	0	0
$994$	−4.70080	−0.149100
$995$	−29.8315	−0.945723
$996$	0	0
$997$	40.2636	1.27516	0.637580	−	0.770384i	$-0.279936\pi$
$997$	40.2636	1.27516	0.637580	+	0.770384i	$0.279936\pi$
$998$	−13.5798	−0.429861
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6633.2.a.w.1.7		17
3.2	odd	2		737.2.a.f.1.11	✓	17
33.32	even	2		8107.2.a.o.1.7		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
737.2.a.f.1.11	✓	17	3.2	odd	2
6633.2.a.w.1.7		17	1.1	even	1	trivial
8107.2.a.o.1.7		17	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6633 = 3^{2} \cdot 11 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6633.a (trivial)

\(f(q)\)	\(=\)	\(q-0.484318 q^{2} -1.76544 q^{4} -3.54277 q^{5} -0.730966 q^{7} +1.82367 q^{8} +O(q^{10})\)	\(q-0.484318 q^{2} -1.76544 q^{4} -3.54277 q^{5} -0.730966 q^{7} +1.82367 q^{8} +1.71583 q^{10} -1.00000 q^{11} +5.91392 q^{13} +0.354020 q^{14} +2.64764 q^{16} -4.21395 q^{17} +0.168612 q^{19} +6.25454 q^{20} +0.484318 q^{22} -4.61992 q^{23} +7.55123 q^{25} -2.86422 q^{26} +1.29047 q^{28} +6.17507 q^{29} +7.65946 q^{31} -4.92963 q^{32} +2.04089 q^{34} +2.58965 q^{35} -9.78458 q^{37} -0.0816618 q^{38} -6.46084 q^{40} -2.91240 q^{41} +1.61544 q^{43} +1.76544 q^{44} +2.23751 q^{46} -8.51279 q^{47} -6.46569 q^{49} -3.65720 q^{50} -10.4407 q^{52} +10.6095 q^{53} +3.54277 q^{55} -1.33304 q^{56} -2.99070 q^{58} -7.46564 q^{59} -6.10978 q^{61} -3.70961 q^{62} -2.90776 q^{64} -20.9517 q^{65} -1.00000 q^{67} +7.43946 q^{68} -1.25421 q^{70} -13.2783 q^{71} -13.6988 q^{73} +4.73885 q^{74} -0.297674 q^{76} +0.730966 q^{77} +3.43703 q^{79} -9.37998 q^{80} +1.41053 q^{82} -1.41858 q^{83} +14.9291 q^{85} -0.782387 q^{86} -1.82367 q^{88} +2.82870 q^{89} -4.32288 q^{91} +8.15618 q^{92} +4.12290 q^{94} -0.597354 q^{95} +2.90243 q^{97} +3.13145 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) \) 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7} + 10 q^{10} - 17 q^{11} + q^{13} + 11 q^{14} + 19 q^{16} - 2 q^{17} + 13 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 33 q^{25} - 12 q^{26} + 44 q^{28} + 5 q^{29} + 16 q^{31} + 24 q^{32} + 4 q^{34} + 2 q^{35} + 29 q^{37} + 19 q^{38} + 31 q^{40} + 6 q^{41} + 19 q^{43} - 19 q^{44} - 33 q^{46} - 40 q^{47} + 23 q^{49} + 3 q^{50} - 28 q^{52} - 15 q^{53} + 10 q^{55} + 38 q^{56} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 3 q^{62} - 4 q^{64} + 30 q^{65} - 17 q^{67} + 13 q^{68} + 71 q^{70} - 2 q^{71} + 41 q^{73} + 13 q^{74} + 21 q^{76} - 20 q^{77} + 41 q^{79} + 23 q^{80} - 8 q^{82} - 2 q^{83} - 36 q^{85} + 54 q^{86} - q^{89} + 16 q^{91} - 36 q^{92} + 12 q^{94} + 31 q^{95} + 3 q^{97} + 64 q^{98}+O(q^{100}) \) 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7 + 10 * q^10 - 17 * q^11 + q^13 + 11 * q^14 + 19 * q^16 - 2 * q^17 + 13 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 33 * q^25 - 12 * q^26 + 44 * q^28 + 5 * q^29 + 16 * q^31 + 24 * q^32 + 4 * q^34 + 2 * q^35 + 29 * q^37 + 19 * q^38 + 31 * q^40 + 6 * q^41 + 19 * q^43 - 19 * q^44 - 33 * q^46 - 40 * q^47 + 23 * q^49 + 3 * q^50 - 28 * q^52 - 15 * q^53 + 10 * q^55 + 38 * q^56 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 3 * q^62 - 4 * q^64 + 30 * q^65 - 17 * q^67 + 13 * q^68 + 71 * q^70 - 2 * q^71 + 41 * q^73 + 13 * q^74 + 21 * q^76 - 20 * q^77 + 41 * q^79 + 23 * q^80 - 8 * q^82 - 2 * q^83 - 36 * q^85 + 54 * q^86 - q^89 + 16 * q^91 - 36 * q^92 + 12 * q^94 + 31 * q^95 + 3 * q^97 + 64 * q^98

Introduction

L-functions

Modular forms

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Database

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