LMFDB - Embedded newform 8037.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 15 x^{10} + 14 x^{9} + 84 x^{8} - 76 x^{7} - 213 x^{6} + 196 x^{5} + 225 x^{4} + \cdots - 17 $ x^12 - x^11 - 15x^10 + 14x^9 + 84x^8 - 76x^7 - 213x^6 + 196x^5 + 225x^4 - 229x^3 - 49x^2 + 83x - 17
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.75952$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.75952 q^{2} +1.09589 q^{4} +1.58069 q^{5} +4.03503 q^{7} +1.59079 q^{8} +O(q^{10})$	\(q-1.75952 q^{2} +1.09589 q^{4} +1.58069 q^{5} +4.03503 q^{7} +1.59079 q^{8} -2.78124 q^{10} +3.16192 q^{11} -6.50974 q^{13} -7.09970 q^{14} -4.99080 q^{16} -2.47968 q^{17} +1.00000 q^{19} +1.73227 q^{20} -5.56344 q^{22} -2.42270 q^{23} -2.50143 q^{25} +11.4540 q^{26} +4.42197 q^{28} +7.50776 q^{29} -0.413833 q^{31} +5.59982 q^{32} +4.36304 q^{34} +6.37813 q^{35} -3.76092 q^{37} -1.75952 q^{38} +2.51454 q^{40} +2.25256 q^{41} -12.1666 q^{43} +3.46512 q^{44} +4.26277 q^{46} -1.00000 q^{47} +9.28149 q^{49} +4.40130 q^{50} -7.13398 q^{52} -10.0758 q^{53} +4.99801 q^{55} +6.41889 q^{56} -13.2100 q^{58} -0.751872 q^{59} -8.83736 q^{61} +0.728145 q^{62} +0.128647 q^{64} -10.2899 q^{65} +13.5517 q^{67} -2.71747 q^{68} -11.2224 q^{70} -1.91541 q^{71} -15.3157 q^{73} +6.61740 q^{74} +1.09589 q^{76} +12.7584 q^{77} +11.1424 q^{79} -7.88890 q^{80} -3.96342 q^{82} -5.54491 q^{83} -3.91960 q^{85} +21.4073 q^{86} +5.02995 q^{88} +11.2544 q^{89} -26.2670 q^{91} -2.65502 q^{92} +1.75952 q^{94} +1.58069 q^{95} -0.608257 q^{97} -16.3309 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) $ 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7	$ 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7} + q^{10} + 4 q^{11} - 17 q^{13} - 3 q^{14} - 19 q^{16} + 6 q^{17} + 12 q^{19} - 5 q^{20} - 8 q^{22} + 13 q^{23} - 7 q^{25} + 19 q^{26} - 29 q^{28} + 2 q^{29} - 14 q^{31} + 21 q^{32} - 6 q^{34} + 3 q^{35} - 2 q^{37} + q^{38} + 8 q^{40} - 8 q^{41} - 42 q^{43} - 24 q^{44} - 9 q^{46} - 12 q^{47} - 5 q^{49} + 33 q^{50} - 26 q^{52} - 3 q^{53} - 12 q^{55} - 7 q^{56} - 16 q^{58} - 8 q^{59} - 6 q^{61} + 24 q^{62} - 22 q^{64} - 22 q^{65} - 29 q^{67} + 30 q^{68} - 34 q^{70} + 7 q^{71} - 48 q^{73} - 25 q^{74} + 7 q^{76} + 18 q^{77} - 11 q^{79} - 3 q^{80} + 28 q^{82} + 57 q^{83} - 7 q^{85} - 9 q^{86} - 11 q^{88} + 2 q^{89} - 4 q^{91} + 13 q^{92} - q^{94} + 7 q^{95} - 14 q^{97} - 58 q^{98}+O(q^{100}) $ 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7 + q^10 + 4 * q^11 - 17 * q^13 - 3 * q^14 - 19 * q^16 + 6 * q^17 + 12 * q^19 - 5 * q^20 - 8 * q^22 + 13 * q^23 - 7 * q^25 + 19 * q^26 - 29 * q^28 + 2 * q^29 - 14 * q^31 + 21 * q^32 - 6 * q^34 + 3 * q^35 - 2 * q^37 + q^38 + 8 * q^40 - 8 * q^41 - 42 * q^43 - 24 * q^44 - 9 * q^46 - 12 * q^47 - 5 * q^49 + 33 * q^50 - 26 * q^52 - 3 * q^53 - 12 * q^55 - 7 * q^56 - 16 * q^58 - 8 * q^59 - 6 * q^61 + 24 * q^62 - 22 * q^64 - 22 * q^65 - 29 * q^67 + 30 * q^68 - 34 * q^70 + 7 * q^71 - 48 * q^73 - 25 * q^74 + 7 * q^76 + 18 * q^77 - 11 * q^79 - 3 * q^80 + 28 * q^82 + 57 * q^83 - 7 * q^85 - 9 * q^86 - 11 * q^88 + 2 * q^89 - 4 * q^91 + 13 * q^92 - q^94 + 7 * q^95 - 14 * q^97 - 58 * 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 $

$2$	−1.75952	−1.24417	−0.622083	−	0.782952i	$-0.713713\pi$
$2$	−1.75952	−1.24417	−0.622083	+	0.782952i	$0.713713\pi$
$3$	0	0
$3$	0	0
$4$	1.09589	0.547947
$5$	1.58069	0.706905	0.353453	−	0.935452i	$-0.385008\pi$
$5$	1.58069	0.706905	0.353453	+	0.935452i	$0.385008\pi$
$6$	0	0
$7$	4.03503	1.52510	0.762550	−	0.646930i	$-0.223947\pi$
$7$	4.03503	1.52510	0.762550	+	0.646930i	$0.223947\pi$
$8$	1.59079	0.562429
$9$	0	0
$10$	−2.78124	−0.879507
$11$	3.16192	0.953354	0.476677	−	0.879078i	$-0.341841\pi$
$11$	3.16192	0.953354	0.476677	+	0.879078i	$0.341841\pi$
$12$	0	0
$13$	−6.50974	−1.80548	−0.902739	−	0.430190i	$-0.858447\pi$
$13$	−6.50974	−1.80548	−0.902739	+	0.430190i	$0.858447\pi$
$14$	−7.09970	−1.89748
$15$	0	0
$16$	−4.99080	−1.24770
$17$	−2.47968	−0.601411	−0.300705	−	0.953717i	$-0.597222\pi$
$17$	−2.47968	−0.601411	−0.300705	+	0.953717i	$0.597222\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.73227	0.387346
$21$	0	0
$22$	−5.56344	−1.18613
$23$	−2.42270	−0.505167	−0.252583	−	0.967575i	$-0.581280\pi$
$23$	−2.42270	−0.505167	−0.252583	+	0.967575i	$0.581280\pi$
$24$	0	0
$25$	−2.50143	−0.500285
$26$	11.4540	2.24631
$27$	0	0
$28$	4.42197	0.835673
$29$	7.50776	1.39416	0.697078	−	0.716995i	$-0.254483\pi$
$29$	7.50776	1.39416	0.697078	+	0.716995i	$0.254483\pi$
$30$	0	0
$31$	−0.413833	−0.0743265	−0.0371633	−	0.999309i	$-0.511832\pi$
$31$	−0.413833	−0.0743265	−0.0371633	+	0.999309i	$0.511832\pi$
$32$	5.59982	0.989917
$33$	0	0
$34$	4.36304	0.748254
$35$	6.37813	1.07810
$36$	0	0
$37$	−3.76092	−0.618292	−0.309146	−	0.951015i	$-0.600043\pi$
$37$	−3.76092	−0.618292	−0.309146	+	0.951015i	$0.600043\pi$
$38$	−1.75952	−0.285431
$39$	0	0
$40$	2.51454	0.397584
$41$	2.25256	0.351791	0.175896	−	0.984409i	$-0.443718\pi$
$41$	2.25256	0.351791	0.175896	+	0.984409i	$0.443718\pi$
$42$	0	0
$43$	−12.1666	−1.85539	−0.927695	−	0.373340i	$-0.878213\pi$
$43$	−12.1666	−1.85539	−0.927695	+	0.373340i	$0.878213\pi$
$44$	3.46512	0.522387
$45$	0	0
$46$	4.26277	0.628511
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	9.28149	1.32593
$50$	4.40130	0.622437
$51$	0	0
$52$	−7.13398	−0.989305
$53$	−10.0758	−1.38402	−0.692008	−	0.721890i	$-0.743274\pi$
$53$	−10.0758	−1.38402	−0.692008	+	0.721890i	$0.743274\pi$
$54$	0	0
$55$	4.99801	0.673931
$56$	6.41889	0.857760
$57$	0	0
$58$	−13.2100	−1.73456
$59$	−0.751872	−0.0978854	−0.0489427	−	0.998802i	$-0.515585\pi$
$59$	−0.751872	−0.0978854	−0.0489427	+	0.998802i	$0.515585\pi$
$60$	0	0
$61$	−8.83736	−1.13151	−0.565754	−	0.824574i	$-0.691415\pi$
$61$	−8.83736	−1.13151	−0.565754	+	0.824574i	$0.691415\pi$
$62$	0.728145	0.0924745
$63$	0	0
$64$	0.128647	0.0160809
$65$	−10.2899	−1.27630
$66$	0	0
$67$	13.5517	1.65560	0.827799	−	0.561024i	$-0.189593\pi$
$67$	13.5517	1.65560	0.827799	+	0.561024i	$0.189593\pi$
$68$	−2.71747	−0.329541
$69$	0	0
$70$	−11.2224	−1.34133
$71$	−1.91541	−0.227318	−0.113659	−	0.993520i	$-0.536257\pi$
$71$	−1.91541	−0.227318	−0.113659	+	0.993520i	$0.536257\pi$
$72$	0	0
$73$	−15.3157	−1.79257	−0.896285	−	0.443479i	$-0.853744\pi$
$73$	−15.3157	−1.79257	−0.896285	+	0.443479i	$0.853744\pi$
$74$	6.61740	0.769257
$75$	0	0
$76$	1.09589	0.125708
$77$	12.7584	1.45396
$78$	0	0
$79$	11.1424	1.25362	0.626808	−	0.779174i	$-0.284361\pi$
$79$	11.1424	1.25362	0.626808	+	0.779174i	$0.284361\pi$
$80$	−7.88890	−0.882006
$81$	0	0
$82$	−3.96342	−0.437686
$83$	−5.54491	−0.608633	−0.304316	−	0.952571i	$-0.598428\pi$
$83$	−5.54491	−0.608633	−0.304316	+	0.952571i	$0.598428\pi$
$84$	0	0
$85$	−3.91960	−0.425140
$86$	21.4073	2.30841
$87$	0	0
$88$	5.02995	0.536194
$89$	11.2544	1.19297	0.596484	−	0.802625i	$-0.296564\pi$
$89$	11.2544	1.19297	0.596484	+	0.802625i	$0.296564\pi$
$90$	0	0
$91$	−26.2670	−2.75353
$92$	−2.65502	−0.276805
$93$	0	0
$94$	1.75952	0.181480
$95$	1.58069	0.162175
$96$	0	0
$97$	−0.608257	−0.0617592	−0.0308796	−	0.999523i	$-0.509831\pi$
$97$	−0.608257	−0.0617592	−0.0308796	+	0.999523i	$0.509831\pi$
$98$	−16.3309	−1.64967
$99$	0	0
$100$	−2.74130	−0.274130
$101$	−6.69224	−0.665903	−0.332952	−	0.942944i	$-0.608045\pi$
$101$	−6.69224	−0.665903	−0.332952	+	0.942944i	$0.608045\pi$
$102$	0	0
$103$	0.286403	0.0282202	0.0141101	−	0.999900i	$-0.495508\pi$
$103$	0.286403	0.0282202	0.0141101	+	0.999900i	$0.495508\pi$
$104$	−10.3556	−1.01545
$105$	0	0
$106$	17.7285	1.72194
$107$	−16.9656	−1.64012	−0.820061	−	0.572276i	$-0.806061\pi$
$107$	−16.9656	−1.64012	−0.820061	+	0.572276i	$0.806061\pi$
$108$	0	0
$109$	−5.10657	−0.489120	−0.244560	−	0.969634i	$-0.578644\pi$
$109$	−5.10657	−0.489120	−0.244560	+	0.969634i	$0.578644\pi$
$110$	−8.79407	−0.838481
$111$	0	0
$112$	−20.1381	−1.90287
$113$	−21.0917	−1.98414	−0.992069	−	0.125693i	$-0.959885\pi$
$113$	−21.0917	−1.98414	−0.992069	+	0.125693i	$0.959885\pi$
$114$	0	0
$115$	−3.82953	−0.357105
$116$	8.22771	0.763923
$117$	0	0
$118$	1.32293	0.121786
$119$	−10.0056	−0.917211
$120$	0	0
$121$	−1.00227	−0.0911159
$122$	15.5495	1.40778
$123$	0	0
$124$	−0.453516	−0.0407270
$125$	−11.8574	−1.06056
$126$	0	0
$127$	−7.86461	−0.697871	−0.348936	−	0.937147i	$-0.613457\pi$
$127$	−7.86461	−0.697871	−0.348936	+	0.937147i	$0.613457\pi$
$128$	−11.4260	−1.00992
$129$	0	0
$130$	18.1052	1.58793
$131$	5.97582	0.522109	0.261055	−	0.965324i	$-0.415930\pi$
$131$	5.97582	0.522109	0.261055	+	0.965324i	$0.415930\pi$
$132$	0	0
$133$	4.03503	0.349882
$134$	−23.8443	−2.05984
$135$	0	0
$136$	−3.94465	−0.338251
$137$	−12.9545	−1.10678	−0.553390	−	0.832922i	$-0.686666\pi$
$137$	−12.9545	−1.10678	−0.553390	+	0.832922i	$0.686666\pi$
$138$	0	0
$139$	−17.2954	−1.46698	−0.733488	−	0.679702i	$-0.762109\pi$
$139$	−17.2954	−1.46698	−0.733488	+	0.679702i	$0.762109\pi$
$140$	6.98975	0.590741
$141$	0	0
$142$	3.37020	0.282821
$143$	−20.5833	−1.72126
$144$	0	0
$145$	11.8674	0.985536
$146$	26.9482	2.23025
$147$	0	0
$148$	−4.12157	−0.338791
$149$	−2.04248	−0.167326	−0.0836632	−	0.996494i	$-0.526662\pi$
$149$	−2.04248	−0.167326	−0.0836632	+	0.996494i	$0.526662\pi$
$150$	0	0
$151$	16.2264	1.32048	0.660241	−	0.751053i	$-0.270454\pi$
$151$	16.2264	1.32048	0.660241	+	0.751053i	$0.270454\pi$
$152$	1.59079	0.129030
$153$	0	0
$154$	−22.4487	−1.80897
$155$	−0.654140	−0.0525418
$156$	0	0
$157$	−16.8324	−1.34337	−0.671687	−	0.740835i	$-0.734430\pi$
$157$	−16.8324	−1.34337	−0.671687	+	0.740835i	$0.734430\pi$
$158$	−19.6052	−1.55970
$159$	0	0
$160$	8.85156	0.699777
$161$	−9.77566	−0.770430
$162$	0	0
$163$	−6.01140	−0.470849	−0.235425	−	0.971893i	$-0.575648\pi$
$163$	−6.01140	−0.470849	−0.235425	+	0.971893i	$0.575648\pi$
$164$	2.46857	0.192763
$165$	0	0
$166$	9.75635	0.757240
$167$	11.2262	0.868713	0.434357	−	0.900741i	$-0.356976\pi$
$167$	11.2262	0.868713	0.434357	+	0.900741i	$0.356976\pi$
$168$	0	0
$169$	29.3767	2.25975
$170$	6.89660	0.528945
$171$	0	0
$172$	−13.3333	−1.01665
$173$	−10.7553	−0.817710	−0.408855	−	0.912599i	$-0.634072\pi$
$173$	−10.7553	−0.817710	−0.408855	+	0.912599i	$0.634072\pi$
$174$	0	0
$175$	−10.0933	−0.762985
$176$	−15.7805	−1.18950
$177$	0	0
$178$	−19.8024	−1.48425
$179$	26.6881	1.99476	0.997381	−	0.0723244i	$-0.0230417\pi$
$179$	26.6881	1.99476	0.997381	+	0.0723244i	$0.0230417\pi$
$180$	0	0
$181$	−19.1977	−1.42695	−0.713477	−	0.700679i	$-0.752881\pi$
$181$	−19.1977	−1.42695	−0.713477	+	0.700679i	$0.752881\pi$
$182$	46.2172	3.42585
$183$	0	0
$184$	−3.85400	−0.284121
$185$	−5.94484	−0.437073
$186$	0	0
$187$	−7.84055	−0.573358
$188$	−1.09589	−0.0799262
$189$	0	0
$190$	−2.78124	−0.201773
$191$	10.4495	0.756099	0.378049	−	0.925785i	$-0.376595\pi$
$191$	10.4495	0.756099	0.378049	+	0.925785i	$0.376595\pi$
$192$	0	0
$193$	3.56266	0.256446	0.128223	−	0.991745i	$-0.459073\pi$
$193$	3.56266	0.256446	0.128223	+	0.991745i	$0.459073\pi$
$194$	1.07024	0.0768386
$195$	0	0
$196$	10.1715	0.726537
$197$	1.86396	0.132801	0.0664007	−	0.997793i	$-0.478848\pi$
$197$	1.86396	0.132801	0.0664007	+	0.997793i	$0.478848\pi$
$198$	0	0
$199$	−9.32555	−0.661071	−0.330535	−	0.943794i	$-0.607229\pi$
$199$	−9.32555	−0.661071	−0.330535	+	0.943794i	$0.607229\pi$
$200$	−3.97924	−0.281375
$201$	0	0
$202$	11.7751	0.828493
$203$	30.2941	2.12623
$204$	0	0
$205$	3.56060	0.248683
$206$	−0.503931	−0.0351105
$207$	0	0
$208$	32.4888	2.25270
$209$	3.16192	0.218714
$210$	0	0
$211$	20.9360	1.44129	0.720645	−	0.693304i	$-0.243846\pi$
$211$	20.9360	1.44129	0.720645	+	0.693304i	$0.243846\pi$
$212$	−11.0420	−0.758367
$213$	0	0
$214$	29.8512	2.04058
$215$	−19.2316	−1.31158
$216$	0	0
$217$	−1.66983	−0.113355
$218$	8.98508	0.608547
$219$	0	0
$220$	5.47728	0.369278
$221$	16.1421	1.08583
$222$	0	0
$223$	−11.1596	−0.747303	−0.373652	−	0.927569i	$-0.621894\pi$
$223$	−11.1596	−0.747303	−0.373652	+	0.927569i	$0.621894\pi$
$224$	22.5954	1.50972
$225$	0	0
$226$	37.1111	2.46860
$227$	16.6184	1.10300	0.551501	−	0.834174i	$-0.314055\pi$
$227$	16.6184	1.10300	0.551501	+	0.834174i	$0.314055\pi$
$228$	0	0
$229$	−1.66755	−0.110195	−0.0550974	−	0.998481i	$-0.517547\pi$
$229$	−1.66755	−0.110195	−0.0550974	+	0.998481i	$0.517547\pi$
$230$	6.73811	0.444298
$231$	0	0
$232$	11.9433	0.784114
$233$	11.6038	0.760190	0.380095	−	0.924947i	$-0.375891\pi$
$233$	11.6038	0.760190	0.380095	+	0.924947i	$0.375891\pi$
$234$	0	0
$235$	−1.58069	−0.103113
$236$	−0.823972	−0.0536360
$237$	0	0
$238$	17.6050	1.14116
$239$	1.73674	0.112340	0.0561701	−	0.998421i	$-0.482111\pi$
$239$	1.73674	0.112340	0.0561701	+	0.998421i	$0.482111\pi$
$240$	0	0
$241$	−18.3018	−1.17893	−0.589463	−	0.807795i	$-0.700661\pi$
$241$	−18.3018	−1.17893	−0.589463	+	0.807795i	$0.700661\pi$
$242$	1.76352	0.113363
$243$	0	0
$244$	−9.68481	−0.620006
$245$	14.6711	0.937305
$246$	0	0
$247$	−6.50974	−0.414205
$248$	−0.658321	−0.0418034
$249$	0	0
$250$	20.8633	1.31951
$251$	9.28589	0.586120	0.293060	−	0.956094i	$-0.405326\pi$
$251$	9.28589	0.586120	0.293060	+	0.956094i	$0.405326\pi$
$252$	0	0
$253$	−7.66036	−0.481603
$254$	13.8379	0.868267
$255$	0	0
$256$	19.8469	1.24043
$257$	−20.7151	−1.29217	−0.646085	−	0.763265i	$-0.723595\pi$
$257$	−20.7151	−1.29217	−0.646085	+	0.763265i	$0.723595\pi$
$258$	0	0
$259$	−15.1754	−0.942956
$260$	−11.2766	−0.699345
$261$	0	0
$262$	−10.5145	−0.649590
$263$	15.4295	0.951421	0.475711	−	0.879602i	$-0.342191\pi$
$263$	15.4295	0.951421	0.475711	+	0.879602i	$0.342191\pi$
$264$	0	0
$265$	−15.9267	−0.978368
$266$	−7.09970	−0.435311
$267$	0	0
$268$	14.8512	0.907180
$269$	−16.0149	−0.976449	−0.488224	−	0.872718i	$-0.662355\pi$
$269$	−16.0149	−0.976449	−0.488224	+	0.872718i	$0.662355\pi$
$270$	0	0
$271$	10.8086	0.656576	0.328288	−	0.944578i	$-0.393528\pi$
$271$	10.8086	0.656576	0.328288	+	0.944578i	$0.393528\pi$
$272$	12.3756	0.750381
$273$	0	0
$274$	22.7937	1.37702
$275$	−7.90930	−0.476949
$276$	0	0
$277$	−8.08670	−0.485883	−0.242941	−	0.970041i	$-0.578112\pi$
$277$	−8.08670	−0.485883	−0.242941	+	0.970041i	$0.578112\pi$
$278$	30.4315	1.82516
$279$	0	0
$280$	10.1463	0.606355
$281$	18.9715	1.13174	0.565872	−	0.824493i	$-0.308540\pi$
$281$	18.9715	1.13174	0.565872	+	0.824493i	$0.308540\pi$
$282$	0	0
$283$	−4.30053	−0.255640	−0.127820	−	0.991797i	$-0.540798\pi$
$283$	−4.30053	−0.255640	−0.127820	+	0.991797i	$0.540798\pi$
$284$	−2.09909	−0.124558
$285$	0	0
$286$	36.2166	2.14153
$287$	9.08917	0.536517
$288$	0	0
$289$	−10.8512	−0.638305
$290$	−20.8809	−1.22617
$291$	0	0
$292$	−16.7844	−0.982232
$293$	−2.09388	−0.122326	−0.0611630	−	0.998128i	$-0.519481\pi$
$293$	−2.09388	−0.122326	−0.0611630	+	0.998128i	$0.519481\pi$
$294$	0	0
$295$	−1.18847	−0.0691957
$296$	−5.98283	−0.347745
$297$	0	0
$298$	3.59377	0.208182
$299$	15.7711	0.912067
$300$	0	0
$301$	−49.0926	−2.82965
$302$	−28.5505	−1.64290
$303$	0	0
$304$	−4.99080	−0.286242
$305$	−13.9691	−0.799869
$306$	0	0
$307$	4.77060	0.272273	0.136136	−	0.990690i	$-0.456531\pi$
$307$	4.77060	0.272273	0.136136	+	0.990690i	$0.456531\pi$
$308$	13.9819	0.796692
$309$	0	0
$310$	1.15097	0.0653707
$311$	9.82619	0.557192	0.278596	−	0.960408i	$-0.410131\pi$
$311$	9.82619	0.557192	0.278596	+	0.960408i	$0.410131\pi$
$312$	0	0
$313$	−26.8883	−1.51982	−0.759910	−	0.650029i	$-0.774757\pi$
$313$	−26.8883	−1.51982	−0.759910	+	0.650029i	$0.774757\pi$
$314$	29.6169	1.67138
$315$	0	0
$316$	12.2109	0.686915
$317$	28.3283	1.59107	0.795537	−	0.605904i	$-0.207189\pi$
$317$	28.3283	1.59107	0.795537	+	0.605904i	$0.207189\pi$
$318$	0	0
$319$	23.7389	1.32912
$320$	0.203351	0.0113676
$321$	0	0
$322$	17.2004	0.958542
$323$	−2.47968	−0.137973
$324$	0	0
$325$	16.2836	0.903254
$326$	10.5772	0.585814
$327$	0	0
$328$	3.58335	0.197858
$329$	−4.03503	−0.222459
$330$	0	0
$331$	−2.97501	−0.163522	−0.0817608	−	0.996652i	$-0.526054\pi$
$331$	−2.97501	−0.163522	−0.0817608	+	0.996652i	$0.526054\pi$
$332$	−6.07663	−0.333498
$333$	0	0
$334$	−19.7528	−1.08082
$335$	21.4209	1.17035
$336$	0	0
$337$	30.4448	1.65843	0.829217	−	0.558927i	$-0.188787\pi$
$337$	30.4448	1.65843	0.829217	+	0.558927i	$0.188787\pi$
$338$	−51.6888	−2.81150
$339$	0	0
$340$	−4.29546	−0.232954
$341$	−1.30850	−0.0708595
$342$	0	0
$343$	9.20590	0.497072
$344$	−19.3545	−1.04352
$345$	0	0
$346$	18.9241	1.01737
$347$	1.04912	0.0563198	0.0281599	−	0.999603i	$-0.491035\pi$
$347$	1.04912	0.0563198	0.0281599	+	0.999603i	$0.491035\pi$
$348$	0	0
$349$	1.68451	0.0901696	0.0450848	−	0.998983i	$-0.485644\pi$
$349$	1.68451	0.0901696	0.0450848	+	0.998983i	$0.485644\pi$
$350$	17.7594	0.949279
$351$	0	0
$352$	17.7062	0.943742
$353$	18.2693	0.972379	0.486189	−	0.873853i	$-0.338387\pi$
$353$	18.2693	0.972379	0.486189	+	0.873853i	$0.338387\pi$
$354$	0	0
$355$	−3.02767	−0.160692
$356$	12.3337	0.653683
$357$	0	0
$358$	−46.9581	−2.48181
$359$	21.1769	1.11768	0.558838	−	0.829277i	$-0.311247\pi$
$359$	21.1769	1.11768	0.558838	+	0.829277i	$0.311247\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	33.7786	1.77537
$363$	0	0
$364$	−28.7858	−1.50879
$365$	−24.2094	−1.26718
$366$	0	0
$367$	−0.143772	−0.00750485	−0.00375243	−	0.999993i	$-0.501194\pi$
$367$	−0.143772	−0.00750485	−0.00375243	+	0.999993i	$0.501194\pi$
$368$	12.0912	0.630297
$369$	0	0
$370$	10.4600	0.543792
$371$	−40.6562	−2.11076
$372$	0	0
$373$	9.11508	0.471961	0.235980	−	0.971758i	$-0.424170\pi$
$373$	9.11508	0.471961	0.235980	+	0.971758i	$0.424170\pi$
$374$	13.7956	0.713351
$375$	0	0
$376$	−1.59079	−0.0820387
$377$	−48.8736	−2.51712
$378$	0	0
$379$	−20.5258	−1.05434	−0.527170	−	0.849760i	$-0.676747\pi$
$379$	−20.5258	−1.05434	−0.527170	+	0.849760i	$0.676747\pi$
$380$	1.73227	0.0888633
$381$	0	0
$382$	−18.3860	−0.940712
$383$	18.7172	0.956403	0.478201	−	0.878250i	$-0.341289\pi$
$383$	18.7172	0.956403	0.478201	+	0.878250i	$0.341289\pi$
$384$	0	0
$385$	20.1671	1.02781
$386$	−6.26855	−0.319061
$387$	0	0
$388$	−0.666585	−0.0338407
$389$	−11.0375	−0.559623	−0.279811	−	0.960055i	$-0.590272\pi$
$389$	−11.0375	−0.559623	−0.279811	+	0.960055i	$0.590272\pi$
$390$	0	0
$391$	6.00751	0.303813
$392$	14.7649	0.745740
$393$	0	0
$394$	−3.27966	−0.165227
$395$	17.6126	0.886187
$396$	0	0
$397$	33.7861	1.69568	0.847839	−	0.530254i	$-0.177903\pi$
$397$	33.7861	1.69568	0.847839	+	0.530254i	$0.177903\pi$
$398$	16.4084	0.822481
$399$	0	0
$400$	12.4841	0.624206
$401$	−22.5126	−1.12422	−0.562112	−	0.827061i	$-0.690011\pi$
$401$	−22.5126	−1.12422	−0.562112	+	0.827061i	$0.690011\pi$
$402$	0	0
$403$	2.69394	0.134195
$404$	−7.33398	−0.364879
$405$	0	0
$406$	−53.3029	−2.64538
$407$	−11.8917	−0.589451
$408$	0	0
$409$	26.7536	1.32288	0.661440	−	0.749998i	$-0.269945\pi$
$409$	26.7536	1.32288	0.661440	+	0.749998i	$0.269945\pi$
$410$	−6.26493	−0.309403
$411$	0	0
$412$	0.313868	0.0154631
$413$	−3.03383	−0.149285
$414$	0	0
$415$	−8.76477	−0.430246
$416$	−36.4534	−1.78727
$417$	0	0
$418$	−5.56344	−0.272117
$419$	11.0233	0.538524	0.269262	−	0.963067i	$-0.413220\pi$
$419$	11.0233	0.538524	0.269262	+	0.963067i	$0.413220\pi$
$420$	0	0
$421$	−4.33452	−0.211252	−0.105626	−	0.994406i	$-0.533685\pi$
$421$	−4.33452	−0.211252	−0.105626	+	0.994406i	$0.533685\pi$
$422$	−36.8371	−1.79320
$423$	0	0
$424$	−16.0285	−0.778411
$425$	6.20274	0.300877
$426$	0	0
$427$	−35.6591	−1.72566
$428$	−18.5924	−0.898700
$429$	0	0
$430$	33.8383	1.63183
$431$	4.08139	0.196594	0.0982968	−	0.995157i	$-0.468661\pi$
$431$	4.08139	0.196594	0.0982968	+	0.995157i	$0.468661\pi$
$432$	0	0
$433$	16.9477	0.814454	0.407227	−	0.913327i	$-0.366496\pi$
$433$	16.9477	0.814454	0.407227	+	0.913327i	$0.366496\pi$
$434$	2.93809	0.141033
$435$	0	0
$436$	−5.59625	−0.268012
$437$	−2.42270	−0.115893
$438$	0	0
$439$	−40.9613	−1.95497	−0.977487	−	0.210995i	$-0.932330\pi$
$439$	−40.9613	−1.95497	−0.977487	+	0.210995i	$0.932330\pi$
$440$	7.95077	0.379038
$441$	0	0
$442$	−28.4022	−1.35096
$443$	−18.6203	−0.884678	−0.442339	−	0.896848i	$-0.645851\pi$
$443$	−18.6203	−0.884678	−0.442339	+	0.896848i	$0.645851\pi$
$444$	0	0
$445$	17.7898	0.843316
$446$	19.6355	0.929769
$447$	0	0
$448$	0.519095	0.0245249
$449$	21.0655	0.994143	0.497071	−	0.867710i	$-0.334409\pi$
$449$	21.0655	0.994143	0.497071	+	0.867710i	$0.334409\pi$
$450$	0	0
$451$	7.12242	0.335382
$452$	−23.1142	−1.08720
$453$	0	0
$454$	−29.2403	−1.37232
$455$	−41.5200	−1.94649
$456$	0	0
$457$	−11.4136	−0.533905	−0.266953	−	0.963710i	$-0.586017\pi$
$457$	−11.4136	−0.533905	−0.266953	+	0.963710i	$0.586017\pi$
$458$	2.93408	0.137100
$459$	0	0
$460$	−4.19675	−0.195675
$461$	−20.3304	−0.946882	−0.473441	−	0.880825i	$-0.656988\pi$
$461$	−20.3304	−0.946882	−0.473441	+	0.880825i	$0.656988\pi$
$462$	0	0
$463$	39.0906	1.81670	0.908348	−	0.418215i	$-0.137344\pi$
$463$	39.0906	1.81670	0.908348	+	0.418215i	$0.137344\pi$
$464$	−37.4698	−1.73949
$465$	0	0
$466$	−20.4171	−0.945802
$467$	−15.3743	−0.711438	−0.355719	−	0.934593i	$-0.615764\pi$
$467$	−15.3743	−0.711438	−0.355719	+	0.934593i	$0.615764\pi$
$468$	0	0
$469$	54.6814	2.52495
$470$	2.78124	0.128289
$471$	0	0
$472$	−1.19607	−0.0550536
$473$	−38.4698	−1.76884
$474$	0	0
$475$	−2.50143	−0.114773
$476$	−10.9651	−0.502583
$477$	0	0
$478$	−3.05582	−0.139770
$479$	−14.0880	−0.643696	−0.321848	−	0.946791i	$-0.604304\pi$
$479$	−14.0880	−0.643696	−0.321848	+	0.946791i	$0.604304\pi$
$480$	0	0
$481$	24.4826	1.11631
$482$	32.2024	1.46678
$483$	0	0
$484$	−1.09839	−0.0499266
$485$	−0.961465	−0.0436579
$486$	0	0
$487$	−1.10681	−0.0501544	−0.0250772	−	0.999686i	$-0.507983\pi$
$487$	−1.10681	−0.0501544	−0.0250772	+	0.999686i	$0.507983\pi$
$488$	−14.0584	−0.636393
$489$	0	0
$490$	−25.8141	−1.16616
$491$	−35.0436	−1.58149	−0.790747	−	0.612143i	$-0.790307\pi$
$491$	−35.0436	−1.58149	−0.790747	+	0.612143i	$0.790307\pi$
$492$	0	0
$493$	−18.6169	−0.838461
$494$	11.4540	0.515339
$495$	0	0
$496$	2.06536	0.0927373
$497$	−7.72875	−0.346682
$498$	0	0
$499$	11.4723	0.513570	0.256785	−	0.966469i	$-0.417337\pi$
$499$	11.4723	0.513570	0.256785	+	0.966469i	$0.417337\pi$
$500$	−12.9945	−0.581130
$501$	0	0
$502$	−16.3387	−0.729231
$503$	−25.8686	−1.15342	−0.576712	−	0.816947i	$-0.695665\pi$
$503$	−25.8686	−1.15342	−0.576712	+	0.816947i	$0.695665\pi$
$504$	0	0
$505$	−10.5783	−0.470730
$506$	13.4785	0.599194
$507$	0	0
$508$	−8.61877	−0.382396
$509$	−20.0616	−0.889216	−0.444608	−	0.895725i	$-0.646657\pi$
$509$	−20.0616	−0.889216	−0.444608	+	0.895725i	$0.646657\pi$
$510$	0	0
$511$	−61.7994	−2.73385
$512$	−12.0690	−0.533377
$513$	0	0
$514$	36.4485	1.60767
$515$	0.452714	0.0199490
$516$	0	0
$517$	−3.16192	−0.139061
$518$	26.7014	1.17319
$519$	0	0
$520$	−16.3690	−0.717829
$521$	7.28664	0.319234	0.159617	−	0.987179i	$-0.448974\pi$
$521$	7.28664	0.319234	0.159617	+	0.987179i	$0.448974\pi$
$522$	0	0
$523$	−29.3445	−1.28315	−0.641573	−	0.767062i	$-0.721718\pi$
$523$	−29.3445	−1.28315	−0.641573	+	0.767062i	$0.721718\pi$
$524$	6.54886	0.286088
$525$	0	0
$526$	−27.1484	−1.18372
$527$	1.02617	0.0447008
$528$	0	0
$529$	−17.1305	−0.744806
$530$	28.0232	1.21725
$531$	0	0
$532$	4.42197	0.191717
$533$	−14.6636	−0.635151
$534$	0	0
$535$	−26.8172	−1.15941
$536$	21.5578	0.931157
$537$	0	0
$538$	28.1785	1.21486
$539$	29.3473	1.26408
$540$	0	0
$541$	−44.6089	−1.91788	−0.958942	−	0.283601i	$-0.908471\pi$
$541$	−44.6089	−1.91788	−0.958942	+	0.283601i	$0.908471\pi$
$542$	−19.0179	−0.816889
$543$	0	0
$544$	−13.8858	−0.595347
$545$	−8.07189	−0.345762
$546$	0	0
$547$	9.17944	0.392485	0.196242	−	0.980555i	$-0.437126\pi$
$547$	9.17944	0.392485	0.196242	+	0.980555i	$0.437126\pi$
$548$	−14.1968	−0.606457
$549$	0	0
$550$	13.9165	0.593403
$551$	7.50776	0.319841
$552$	0	0
$553$	44.9599	1.91189
$554$	14.2287	0.604518
$555$	0	0
$556$	−18.9539	−0.803825
$557$	18.1128	0.767466	0.383733	−	0.923444i	$-0.374638\pi$
$557$	18.1128	0.767466	0.383733	+	0.923444i	$0.374638\pi$
$558$	0	0
$559$	79.2014	3.34986
$560$	−31.8320	−1.34515
$561$	0	0
$562$	−33.3806	−1.40808
$563$	7.45643	0.314251	0.157126	−	0.987579i	$-0.449777\pi$
$563$	7.45643	0.314251	0.157126	+	0.987579i	$0.449777\pi$
$564$	0	0
$565$	−33.3394	−1.40260
$566$	7.56685	0.318059
$567$	0	0
$568$	−3.04702	−0.127850
$569$	−10.1899	−0.427182	−0.213591	−	0.976923i	$-0.568516\pi$
$569$	−10.1899	−0.427182	−0.213591	+	0.976923i	$0.568516\pi$
$570$	0	0
$571$	34.3066	1.43569	0.717843	−	0.696205i	$-0.245130\pi$
$571$	34.3066	1.43569	0.717843	+	0.696205i	$0.245130\pi$
$572$	−22.5571	−0.943158
$573$	0	0
$574$	−15.9925	−0.667515
$575$	6.06019	0.252728
$576$	0	0
$577$	−37.3132	−1.55337	−0.776685	−	0.629889i	$-0.783100\pi$
$577$	−37.3132	−1.55337	−0.776685	+	0.629889i	$0.783100\pi$
$578$	19.0928	0.794157
$579$	0	0
$580$	13.0054	0.540021
$581$	−22.3739	−0.928225
$582$	0	0
$583$	−31.8588	−1.31946
$584$	−24.3641	−1.00819
$585$	0	0
$586$	3.68422	0.152194
$587$	7.52886	0.310749	0.155375	−	0.987856i	$-0.450342\pi$
$587$	7.52886	0.310749	0.155375	+	0.987856i	$0.450342\pi$
$588$	0	0
$589$	−0.413833	−0.0170517
$590$	2.09114	0.0860909
$591$	0	0
$592$	18.7700	0.771443
$593$	5.05221	0.207469	0.103735	−	0.994605i	$-0.466921\pi$
$593$	5.05221	0.207469	0.103735	+	0.994605i	$0.466921\pi$
$594$	0	0
$595$	−15.8157	−0.648381
$596$	−2.23834	−0.0916859
$597$	0	0
$598$	−27.7495	−1.13476
$599$	36.6651	1.49809	0.749047	−	0.662517i	$-0.230512\pi$
$599$	36.6651	1.49809	0.749047	+	0.662517i	$0.230512\pi$
$600$	0	0
$601$	20.7571	0.846698	0.423349	−	0.905967i	$-0.360854\pi$
$601$	20.7571	0.846698	0.423349	+	0.905967i	$0.360854\pi$
$602$	86.3792	3.52056
$603$	0	0
$604$	17.7824	0.723554
$605$	−1.58428	−0.0644103
$606$	0	0
$607$	19.6646	0.798163	0.399082	−	0.916915i	$-0.369329\pi$
$607$	19.6646	0.798163	0.399082	+	0.916915i	$0.369329\pi$
$608$	5.59982	0.227103
$609$	0	0
$610$	24.5789	0.995169
$611$	6.50974	0.263356
$612$	0	0
$613$	−47.7222	−1.92748	−0.963741	−	0.266840i	$-0.914021\pi$
$613$	−47.7222	−1.92748	−0.963741	+	0.266840i	$0.914021\pi$
$614$	−8.39395	−0.338752
$615$	0	0
$616$	20.2960	0.817749
$617$	2.44027	0.0982418	0.0491209	−	0.998793i	$-0.484358\pi$
$617$	2.44027	0.0982418	0.0491209	+	0.998793i	$0.484358\pi$
$618$	0	0
$619$	5.00583	0.201201	0.100601	−	0.994927i	$-0.467924\pi$
$619$	5.00583	0.201201	0.100601	+	0.994927i	$0.467924\pi$
$620$	−0.716868	−0.0287901
$621$	0	0
$622$	−17.2893	−0.693239
$623$	45.4121	1.81940
$624$	0	0
$625$	−6.23574	−0.249429
$626$	47.3105	1.89091
$627$	0	0
$628$	−18.4466	−0.736098
$629$	9.32588	0.371847
$630$	0	0
$631$	−14.8947	−0.592950	−0.296475	−	0.955041i	$-0.595811\pi$
$631$	−14.8947	−0.592950	−0.296475	+	0.955041i	$0.595811\pi$
$632$	17.7252	0.705070
$633$	0	0
$634$	−49.8440	−1.97956
$635$	−12.4315	−0.493329
$636$	0	0
$637$	−60.4201	−2.39393
$638$	−41.7690	−1.65365
$639$	0	0
$640$	−18.0609	−0.713921
$641$	−33.5843	−1.32650	−0.663250	−	0.748398i	$-0.730824\pi$
$641$	−33.5843	−1.32650	−0.663250	+	0.748398i	$0.730824\pi$
$642$	0	0
$643$	−7.46920	−0.294556	−0.147278	−	0.989095i	$-0.547051\pi$
$643$	−7.46920	−0.294556	−0.147278	+	0.989095i	$0.547051\pi$
$644$	−10.7131	−0.422154
$645$	0	0
$646$	4.36304	0.171661
$647$	4.59074	0.180481	0.0902403	−	0.995920i	$-0.471236\pi$
$647$	4.59074	0.180481	0.0902403	+	0.995920i	$0.471236\pi$
$648$	0	0
$649$	−2.37736	−0.0933195
$650$	−28.6513	−1.12380
$651$	0	0
$652$	−6.58785	−0.258000
$653$	47.1219	1.84402	0.922011	−	0.387164i	$-0.126545\pi$
$653$	47.1219	1.84402	0.922011	+	0.387164i	$0.126545\pi$
$654$	0	0
$655$	9.44590	0.369082
$656$	−11.2421	−0.438930
$657$	0	0
$658$	7.09970	0.276775
$659$	0.981040	0.0382159	0.0191079	−	0.999817i	$-0.493917\pi$
$659$	0.981040	0.0382159	0.0191079	+	0.999817i	$0.493917\pi$
$660$	0	0
$661$	−31.2637	−1.21602	−0.608009	−	0.793930i	$-0.708032\pi$
$661$	−31.2637	−1.21602	−0.608009	+	0.793930i	$0.708032\pi$
$662$	5.23458	0.203448
$663$	0	0
$664$	−8.82078	−0.342313
$665$	6.37813	0.247333
$666$	0	0
$667$	−18.1890	−0.704282
$668$	12.3028	0.476008
$669$	0	0
$670$	−37.6905	−1.45611
$671$	−27.9430	−1.07873
$672$	0	0
$673$	−20.4137	−0.786889	−0.393444	−	0.919348i	$-0.628717\pi$
$673$	−20.4137	−0.786889	−0.393444	+	0.919348i	$0.628717\pi$
$674$	−53.5681	−2.06337
$675$	0	0
$676$	32.1937	1.23822
$677$	4.61318	0.177299	0.0886495	−	0.996063i	$-0.471745\pi$
$677$	4.61318	0.177299	0.0886495	+	0.996063i	$0.471745\pi$
$678$	0	0
$679$	−2.45434	−0.0941888
$680$	−6.23526	−0.239111
$681$	0	0
$682$	2.30233	0.0881609
$683$	26.1179	0.999373	0.499686	−	0.866206i	$-0.333449\pi$
$683$	26.1179	0.999373	0.499686	+	0.866206i	$0.333449\pi$
$684$	0	0
$685$	−20.4771	−0.782389
$686$	−16.1979	−0.618439
$687$	0	0
$688$	60.7211	2.31497
$689$	65.5908	2.49881
$690$	0	0
$691$	33.3929	1.27033	0.635163	−	0.772378i	$-0.280933\pi$
$691$	33.3929	1.27033	0.635163	+	0.772378i	$0.280933\pi$
$692$	−11.7867	−0.448061
$693$	0	0
$694$	−1.84595	−0.0700712
$695$	−27.3386	−1.03701
$696$	0	0
$697$	−5.58564	−0.211571
$698$	−2.96392	−0.112186
$699$	0	0
$700$	−11.0612	−0.418075
$701$	−30.2059	−1.14086	−0.570430	−	0.821346i	$-0.693224\pi$
$701$	−30.2059	−1.14086	−0.570430	+	0.821346i	$0.693224\pi$
$702$	0	0
$703$	−3.76092	−0.141846
$704$	0.406771	0.0153308
$705$	0	0
$706$	−32.1452	−1.20980
$707$	−27.0034	−1.01557
$708$	0	0
$709$	−14.4748	−0.543613	−0.271807	−	0.962352i	$-0.587621\pi$
$709$	−14.4748	−0.543613	−0.271807	+	0.962352i	$0.587621\pi$
$710$	5.32723	0.199927
$711$	0	0
$712$	17.9035	0.670960
$713$	1.00259	0.0375473
$714$	0	0
$715$	−32.5357	−1.21677
$716$	29.2473	1.09302
$717$	0	0
$718$	−37.2612	−1.39057
$719$	15.4055	0.574527	0.287264	−	0.957852i	$-0.407254\pi$
$719$	15.4055	0.574527	0.287264	+	0.957852i	$0.407254\pi$
$720$	0	0
$721$	1.15565	0.0430385
$722$	−1.75952	−0.0654824
$723$	0	0
$724$	−21.0386	−0.781895
$725$	−18.7801	−0.697476
$726$	0	0
$727$	−8.39144	−0.311221	−0.155611	−	0.987818i	$-0.549735\pi$
$727$	−8.39144	−0.311221	−0.155611	+	0.987818i	$0.549735\pi$
$728$	−41.7853	−1.54867
$729$	0	0
$730$	42.5968	1.57658
$731$	30.1693	1.11585
$732$	0	0
$733$	39.4244	1.45617	0.728087	−	0.685485i	$-0.240410\pi$
$733$	39.4244	1.45617	0.728087	+	0.685485i	$0.240410\pi$
$734$	0.252970	0.00933728
$735$	0	0
$736$	−13.5667	−0.500073
$737$	42.8492	1.57837
$738$	0	0
$739$	14.0127	0.515466	0.257733	−	0.966216i	$-0.417024\pi$
$739$	14.0127	0.515466	0.257733	+	0.966216i	$0.417024\pi$
$740$	−6.51491	−0.239493
$741$	0	0
$742$	71.5351	2.62614
$743$	−18.8212	−0.690484	−0.345242	−	0.938514i	$-0.612203\pi$
$743$	−18.8212	−0.690484	−0.345242	+	0.938514i	$0.612203\pi$
$744$	0	0
$745$	−3.22852	−0.118284
$746$	−16.0381	−0.587197
$747$	0	0
$748$	−8.59240	−0.314169
$749$	−68.4566	−2.50135
$750$	0	0
$751$	−20.8358	−0.760309	−0.380155	−	0.924923i	$-0.624129\pi$
$751$	−20.8358	−0.760309	−0.380155	+	0.924923i	$0.624129\pi$
$752$	4.99080	0.181996
$753$	0	0
$754$	85.9938	3.13171
$755$	25.6488	0.933456
$756$	0	0
$757$	−13.3900	−0.486669	−0.243334	−	0.969942i	$-0.578241\pi$
$757$	−13.3900	−0.486669	−0.243334	+	0.969942i	$0.578241\pi$
$758$	36.1154	1.31177
$759$	0	0
$760$	2.51454	0.0912120
$761$	−45.6069	−1.65325	−0.826624	−	0.562754i	$-0.809742\pi$
$761$	−45.6069	−1.65325	−0.826624	+	0.562754i	$0.809742\pi$
$762$	0	0
$763$	−20.6052	−0.745957
$764$	11.4515	0.414302
$765$	0	0
$766$	−32.9331	−1.18992
$767$	4.89449	0.176730
$768$	0	0
$769$	23.4053	0.844016	0.422008	−	0.906592i	$-0.361325\pi$
$769$	23.4053	0.844016	0.422008	+	0.906592i	$0.361325\pi$
$770$	−35.4843	−1.27877
$771$	0	0
$772$	3.90429	0.140519
$773$	−37.9415	−1.36466	−0.682330	−	0.731044i	$-0.739033\pi$
$773$	−37.9415	−1.36466	−0.682330	+	0.731044i	$0.739033\pi$
$774$	0	0
$775$	1.03517	0.0371845
$776$	−0.967609	−0.0347351
$777$	0	0
$778$	19.4206	0.696263
$779$	2.25256	0.0807065
$780$	0	0
$781$	−6.05638	−0.216714
$782$	−10.5703	−0.377993
$783$	0	0
$784$	−46.3221	−1.65436
$785$	−26.6068	−0.949638
$786$	0	0
$787$	9.27000	0.330440	0.165220	−	0.986257i	$-0.447167\pi$
$787$	9.27000	0.330440	0.165220	+	0.986257i	$0.447167\pi$
$788$	2.04270	0.0727681
$789$	0	0
$790$	−30.9897	−1.10256
$791$	−85.1056	−3.02601
$792$	0	0
$793$	57.5289	2.04291
$794$	−59.4472	−2.10970
$795$	0	0
$796$	−10.2198	−0.362231
$797$	16.5685	0.586887	0.293443	−	0.955976i	$-0.405199\pi$
$797$	16.5685	0.586887	0.293443	+	0.955976i	$0.405199\pi$
$798$	0	0
$799$	2.47968	0.0877248
$800$	−14.0075	−0.495241
$801$	0	0
$802$	39.6112	1.39872
$803$	−48.4271	−1.70895
$804$	0	0
$805$	−15.4523	−0.544621
$806$	−4.74003	−0.166961
$807$	0	0
$808$	−10.6460	−0.374523
$809$	−30.4565	−1.07080	−0.535398	−	0.844600i	$-0.679838\pi$
$809$	−30.4565	−1.07080	−0.535398	+	0.844600i	$0.679838\pi$
$810$	0	0
$811$	40.5454	1.42374	0.711871	−	0.702310i	$-0.247848\pi$
$811$	40.5454	1.42374	0.711871	+	0.702310i	$0.247848\pi$
$812$	33.1991	1.16506
$813$	0	0
$814$	20.9237	0.733374
$815$	−9.50215	−0.332846
$816$	0	0
$817$	−12.1666	−0.425656
$818$	−47.0734	−1.64588
$819$	0	0
$820$	3.90204	0.136265
$821$	16.6967	0.582717	0.291359	−	0.956614i	$-0.405893\pi$
$821$	16.6967	0.582717	0.291359	+	0.956614i	$0.405893\pi$
$822$	0	0
$823$	9.69485	0.337941	0.168971	−	0.985621i	$-0.445956\pi$
$823$	9.69485	0.337941	0.168971	+	0.985621i	$0.445956\pi$
$824$	0.455607	0.0158718
$825$	0	0
$826$	5.33807	0.185735
$827$	−9.59520	−0.333658	−0.166829	−	0.985986i	$-0.553353\pi$
$827$	−9.59520	−0.333658	−0.166829	+	0.985986i	$0.553353\pi$
$828$	0	0
$829$	18.6077	0.646272	0.323136	−	0.946352i	$-0.395263\pi$
$829$	18.6077	0.646272	0.323136	+	0.946352i	$0.395263\pi$
$830$	15.4217	0.535296
$831$	0	0
$832$	−0.837458	−0.0290336
$833$	−23.0151	−0.797427
$834$	0	0
$835$	17.7452	0.614098
$836$	3.46512	0.119844
$837$	0	0
$838$	−19.3957	−0.670013
$839$	−2.37366	−0.0819478	−0.0409739	−	0.999160i	$-0.513046\pi$
$839$	−2.37366	−0.0819478	−0.0409739	+	0.999160i	$0.513046\pi$
$840$	0	0
$841$	27.3665	0.943672
$842$	7.62666	0.262832
$843$	0	0
$844$	22.9436	0.789750
$845$	46.4354	1.59743
$846$	0	0
$847$	−4.04421	−0.138961
$848$	50.2863	1.72684
$849$	0	0
$850$	−10.9138	−0.374341
$851$	9.11157	0.312340
$852$	0	0
$853$	2.53130	0.0866702	0.0433351	−	0.999061i	$-0.486202\pi$
$853$	2.53130	0.0866702	0.0433351	+	0.999061i	$0.486202\pi$
$854$	62.7426	2.14701
$855$	0	0
$856$	−26.9886	−0.922452
$857$	12.9401	0.442027	0.221013	−	0.975271i	$-0.429064\pi$
$857$	12.9401	0.442027	0.221013	+	0.975271i	$0.429064\pi$
$858$	0	0
$859$	−3.18924	−0.108815	−0.0544077	−	0.998519i	$-0.517327\pi$
$859$	−3.18924	−0.108815	−0.0544077	+	0.998519i	$0.517327\pi$
$860$	−21.0758	−0.718678
$861$	0	0
$862$	−7.18127	−0.244595
$863$	27.9988	0.953091	0.476546	−	0.879150i	$-0.341889\pi$
$863$	27.9988	0.953091	0.476546	+	0.879150i	$0.341889\pi$
$864$	0	0
$865$	−17.0008	−0.578043
$866$	−29.8197	−1.01332
$867$	0	0
$868$	−1.82995	−0.0621127
$869$	35.2313	1.19514
$870$	0	0
$871$	−88.2178	−2.98915
$872$	−8.12347	−0.275095
$873$	0	0
$874$	4.26277	0.144190
$875$	−47.8451	−1.61746
$876$	0	0
$877$	45.2828	1.52909	0.764546	−	0.644569i	$-0.222963\pi$
$877$	45.2828	1.52909	0.764546	+	0.644569i	$0.222963\pi$
$878$	72.0720	2.43231
$879$	0	0
$880$	−24.9441	−0.840864
$881$	2.75149	0.0927001	0.0463501	−	0.998925i	$-0.485241\pi$
$881$	2.75149	0.0927001	0.0463501	+	0.998925i	$0.485241\pi$
$882$	0	0
$883$	23.1772	0.779974	0.389987	−	0.920820i	$-0.372480\pi$
$883$	23.1772	0.779974	0.389987	+	0.920820i	$0.372480\pi$
$884$	17.6900	0.594979
$885$	0	0
$886$	32.7627	1.10069
$887$	6.06502	0.203643	0.101822	−	0.994803i	$-0.467533\pi$
$887$	6.06502	0.203643	0.101822	+	0.994803i	$0.467533\pi$
$888$	0	0
$889$	−31.7340	−1.06432
$890$	−31.3014	−1.04922
$891$	0	0
$892$	−12.2297	−0.409482
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	42.1856	1.41011
$896$	−46.1042	−1.54023
$897$	0	0
$898$	−37.0651	−1.23688
$899$	−3.10696	−0.103623
$900$	0	0
$901$	24.9847	0.832363
$902$	−12.5320	−0.417270
$903$	0	0
$904$	−33.5524	−1.11594
$905$	−30.3456	−1.00872
$906$	0	0
$907$	−15.9820	−0.530674	−0.265337	−	0.964156i	$-0.585483\pi$
$907$	−15.9820	−0.530674	−0.265337	+	0.964156i	$0.585483\pi$
$908$	18.2120	0.604386
$909$	0	0
$910$	73.0550	2.42175
$911$	−28.9890	−0.960447	−0.480224	−	0.877146i	$-0.659445\pi$
$911$	−28.9890	−0.960447	−0.480224	+	0.877146i	$0.659445\pi$
$912$	0	0
$913$	−17.5325	−0.580243
$914$	20.0824	0.664266
$915$	0	0
$916$	−1.82746	−0.0603808
$917$	24.1126	0.796269
$918$	0	0
$919$	26.0635	0.859755	0.429878	−	0.902887i	$-0.358557\pi$
$919$	26.0635	0.859755	0.429878	+	0.902887i	$0.358557\pi$
$920$	−6.09197	−0.200846
$921$	0	0
$922$	35.7717	1.17808
$923$	12.4688	0.410417
$924$	0	0
$925$	9.40767	0.309322
$926$	−68.7806	−2.26027
$927$	0	0
$928$	42.0421	1.38010
$929$	6.76015	0.221793	0.110897	−	0.993832i	$-0.464628\pi$
$929$	6.76015	0.221793	0.110897	+	0.993832i	$0.464628\pi$
$930$	0	0
$931$	9.28149	0.304189
$932$	12.7165	0.416544
$933$	0	0
$934$	27.0513	0.885146
$935$	−12.3935	−0.405309
$936$	0	0
$937$	−46.0645	−1.50486	−0.752430	−	0.658672i	$-0.771118\pi$
$937$	−46.0645	−1.50486	−0.752430	+	0.658672i	$0.771118\pi$
$938$	−96.2127	−3.14146
$939$	0	0
$940$	−1.73227	−0.0565003
$941$	−37.6341	−1.22684	−0.613418	−	0.789758i	$-0.710206\pi$
$941$	−37.6341	−1.22684	−0.613418	+	0.789758i	$0.710206\pi$
$942$	0	0
$943$	−5.45728	−0.177713
$944$	3.75245	0.122132
$945$	0	0
$946$	67.6882	2.20073
$947$	49.5046	1.60868	0.804341	−	0.594168i	$-0.202518\pi$
$947$	49.5046	1.60868	0.804341	+	0.594168i	$0.202518\pi$
$948$	0	0
$949$	99.7014	3.23644
$950$	4.40130	0.142797
$951$	0	0
$952$	−15.9168	−0.515866
$953$	−8.39637	−0.271985	−0.135993	−	0.990710i	$-0.543422\pi$
$953$	−8.39637	−0.271985	−0.135993	+	0.990710i	$0.543422\pi$
$954$	0	0
$955$	16.5174	0.534490
$956$	1.90328	0.0615565
$957$	0	0
$958$	24.7880	0.800864
$959$	−52.2720	−1.68795
$960$	0	0
$961$	−30.8287	−0.994476
$962$	−43.0775	−1.38888
$963$	0	0
$964$	−20.0569	−0.645988
$965$	5.63145	0.181283
$966$	0	0
$967$	−9.23790	−0.297071	−0.148535	−	0.988907i	$-0.547456\pi$
$967$	−9.23790	−0.297071	−0.148535	+	0.988907i	$0.547456\pi$
$968$	−1.59441	−0.0512462
$969$	0	0
$970$	1.69171	0.0543176
$971$	51.7463	1.66062	0.830308	−	0.557304i	$-0.188164\pi$
$971$	51.7463	1.66062	0.830308	+	0.557304i	$0.188164\pi$
$972$	0	0
$973$	−69.7875	−2.23728
$974$	1.94745	0.0624004
$975$	0	0
$976$	44.1056	1.41178
$977$	−23.4606	−0.750573	−0.375286	−	0.926909i	$-0.622456\pi$
$977$	−23.4606	−0.750573	−0.375286	+	0.926909i	$0.622456\pi$
$978$	0	0
$979$	35.5856	1.13732
$980$	16.0780	0.513593
$981$	0	0
$982$	61.6597	1.96764
$983$	8.83607	0.281827	0.140913	−	0.990022i	$-0.454996\pi$
$983$	8.83607	0.281827	0.140913	+	0.990022i	$0.454996\pi$
$984$	0	0
$985$	2.94633	0.0938780
$986$	32.7566	1.04318
$987$	0	0
$988$	−7.13398	−0.226962
$989$	29.4760	0.937281
$990$	0	0
$991$	52.6061	1.67109	0.835545	−	0.549422i	$-0.185152\pi$
$991$	52.6061	1.67109	0.835545	+	0.549422i	$0.185152\pi$
$992$	−2.31739	−0.0735771
$993$	0	0
$994$	13.5989	0.431330
$995$	−14.7408	−0.467314
$996$	0	0
$997$	−42.5678	−1.34814	−0.674068	−	0.738670i	$-0.735454\pi$
$997$	−42.5678	−1.34814	−0.674068	+	0.738670i	$0.735454\pi$
$998$	−20.1857	−0.638966
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.m.1.3		12
3.2	odd	2		893.2.a.a.1.10	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.a.1.10	✓	12	3.2	odd	2
8037.2.a.m.1.3		12	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-1.75952 q^{2} +1.09589 q^{4} +1.58069 q^{5} +4.03503 q^{7} +1.59079 q^{8} +O(q^{10})\)	\(q-1.75952 q^{2} +1.09589 q^{4} +1.58069 q^{5} +4.03503 q^{7} +1.59079 q^{8} -2.78124 q^{10} +3.16192 q^{11} -6.50974 q^{13} -7.09970 q^{14} -4.99080 q^{16} -2.47968 q^{17} +1.00000 q^{19} +1.73227 q^{20} -5.56344 q^{22} -2.42270 q^{23} -2.50143 q^{25} +11.4540 q^{26} +4.42197 q^{28} +7.50776 q^{29} -0.413833 q^{31} +5.59982 q^{32} +4.36304 q^{34} +6.37813 q^{35} -3.76092 q^{37} -1.75952 q^{38} +2.51454 q^{40} +2.25256 q^{41} -12.1666 q^{43} +3.46512 q^{44} +4.26277 q^{46} -1.00000 q^{47} +9.28149 q^{49} +4.40130 q^{50} -7.13398 q^{52} -10.0758 q^{53} +4.99801 q^{55} +6.41889 q^{56} -13.2100 q^{58} -0.751872 q^{59} -8.83736 q^{61} +0.728145 q^{62} +0.128647 q^{64} -10.2899 q^{65} +13.5517 q^{67} -2.71747 q^{68} -11.2224 q^{70} -1.91541 q^{71} -15.3157 q^{73} +6.61740 q^{74} +1.09589 q^{76} +12.7584 q^{77} +11.1424 q^{79} -7.88890 q^{80} -3.96342 q^{82} -5.54491 q^{83} -3.91960 q^{85} +21.4073 q^{86} +5.02995 q^{88} +11.2544 q^{89} -26.2670 q^{91} -2.65502 q^{92} +1.75952 q^{94} +1.58069 q^{95} -0.608257 q^{97} -16.3309 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) \) 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7} + q^{10} + 4 q^{11} - 17 q^{13} - 3 q^{14} - 19 q^{16} + 6 q^{17} + 12 q^{19} - 5 q^{20} - 8 q^{22} + 13 q^{23} - 7 q^{25} + 19 q^{26} - 29 q^{28} + 2 q^{29} - 14 q^{31} + 21 q^{32} - 6 q^{34} + 3 q^{35} - 2 q^{37} + q^{38} + 8 q^{40} - 8 q^{41} - 42 q^{43} - 24 q^{44} - 9 q^{46} - 12 q^{47} - 5 q^{49} + 33 q^{50} - 26 q^{52} - 3 q^{53} - 12 q^{55} - 7 q^{56} - 16 q^{58} - 8 q^{59} - 6 q^{61} + 24 q^{62} - 22 q^{64} - 22 q^{65} - 29 q^{67} + 30 q^{68} - 34 q^{70} + 7 q^{71} - 48 q^{73} - 25 q^{74} + 7 q^{76} + 18 q^{77} - 11 q^{79} - 3 q^{80} + 28 q^{82} + 57 q^{83} - 7 q^{85} - 9 q^{86} - 11 q^{88} + 2 q^{89} - 4 q^{91} + 13 q^{92} - q^{94} + 7 q^{95} - 14 q^{97} - 58 q^{98}+O(q^{100}) \) 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7 + q^10 + 4 * q^11 - 17 * q^13 - 3 * q^14 - 19 * q^16 + 6 * q^17 + 12 * q^19 - 5 * q^20 - 8 * q^22 + 13 * q^23 - 7 * q^25 + 19 * q^26 - 29 * q^28 + 2 * q^29 - 14 * q^31 + 21 * q^32 - 6 * q^34 + 3 * q^35 - 2 * q^37 + q^38 + 8 * q^40 - 8 * q^41 - 42 * q^43 - 24 * q^44 - 9 * q^46 - 12 * q^47 - 5 * q^49 + 33 * q^50 - 26 * q^52 - 3 * q^53 - 12 * q^55 - 7 * q^56 - 16 * q^58 - 8 * q^59 - 6 * q^61 + 24 * q^62 - 22 * q^64 - 22 * q^65 - 29 * q^67 + 30 * q^68 - 34 * q^70 + 7 * q^71 - 48 * q^73 - 25 * q^74 + 7 * q^76 + 18 * q^77 - 11 * q^79 - 3 * q^80 + 28 * q^82 + 57 * q^83 - 7 * q^85 - 9 * q^86 - 11 * q^88 + 2 * q^89 - 4 * q^91 + 13 * q^92 - q^94 + 7 * q^95 - 14 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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