LMFDB - Embedded newform 8037.2.a.m.1.12

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			1.12
Root			$2.46212$ 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)

$f(q)$	$=$	$q+2.46212 q^{2} +4.06203 q^{4} +1.39858 q^{5} -2.95389 q^{7} +5.07697 q^{8} +O(q^{10})$	\(q+2.46212 q^{2} +4.06203 q^{4} +1.39858 q^{5} -2.95389 q^{7} +5.07697 q^{8} +3.44347 q^{10} -5.47412 q^{11} +0.289994 q^{13} -7.27284 q^{14} +4.37604 q^{16} -0.545585 q^{17} +1.00000 q^{19} +5.68107 q^{20} -13.4779 q^{22} +2.62789 q^{23} -3.04398 q^{25} +0.714001 q^{26} -11.9988 q^{28} -10.4891 q^{29} +2.38456 q^{31} +0.620398 q^{32} -1.34329 q^{34} -4.13125 q^{35} -2.38151 q^{37} +2.46212 q^{38} +7.10054 q^{40} +4.74360 q^{41} -4.18576 q^{43} -22.2361 q^{44} +6.47017 q^{46} -1.00000 q^{47} +1.72549 q^{49} -7.49464 q^{50} +1.17797 q^{52} -11.1376 q^{53} -7.65600 q^{55} -14.9968 q^{56} -25.8254 q^{58} -6.78274 q^{59} +5.06585 q^{61} +5.87107 q^{62} -7.22459 q^{64} +0.405580 q^{65} +1.98930 q^{67} -2.21618 q^{68} -10.1716 q^{70} +9.03520 q^{71} -10.6215 q^{73} -5.86355 q^{74} +4.06203 q^{76} +16.1700 q^{77} -17.1441 q^{79} +6.12024 q^{80} +11.6793 q^{82} +11.4200 q^{83} -0.763043 q^{85} -10.3058 q^{86} -27.7920 q^{88} +13.6473 q^{89} -0.856613 q^{91} +10.6746 q^{92} -2.46212 q^{94} +1.39858 q^{95} +12.8747 q^{97} +4.24837 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.46212	1.74098	0.870491	−	0.492185i	$-0.163802\pi$
$2$	2.46212	1.74098	0.870491	+	0.492185i	$0.163802\pi$
$3$	0	0
$3$	0	0
$4$	4.06203	2.03102
$5$	1.39858	0.625463	0.312732	−	0.949841i	$-0.398756\pi$
$5$	1.39858	0.625463	0.312732	+	0.949841i	$0.398756\pi$
$6$	0	0
$7$	−2.95389	−1.11647	−0.558234	−	0.829684i	$-0.688521\pi$
$7$	−2.95389	−1.11647	−0.558234	+	0.829684i	$0.688521\pi$
$8$	5.07697	1.79498
$9$	0	0
$10$	3.44347	1.08892
$11$	−5.47412	−1.65051	−0.825255	−	0.564760i	$-0.808969\pi$
$11$	−5.47412	−1.65051	−0.825255	+	0.564760i	$0.808969\pi$
$12$	0	0
$13$	0.289994	0.0804300	0.0402150	−	0.999191i	$-0.487196\pi$
$13$	0.289994	0.0804300	0.0402150	+	0.999191i	$0.487196\pi$
$14$	−7.27284	−1.94375
$15$	0	0
$16$	4.37604	1.09401
$17$	−0.545585	−0.132324	−0.0661618	−	0.997809i	$-0.521075\pi$
$17$	−0.545585	−0.132324	−0.0661618	+	0.997809i	$0.521075\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	5.68107	1.27033
$21$	0	0
$22$	−13.4779	−2.87351
$23$	2.62789	0.547952	0.273976	−	0.961736i	$-0.411661\pi$
$23$	2.62789	0.547952	0.273976	+	0.961736i	$0.411661\pi$
$24$	0	0
$25$	−3.04398	−0.608795
$26$	0.714001	0.140027
$27$	0	0
$28$	−11.9988	−2.26756
$29$	−10.4891	−1.94778	−0.973889	−	0.227023i	$-0.927101\pi$
$29$	−10.4891	−1.94778	−0.973889	+	0.227023i	$0.927101\pi$
$30$	0	0
$31$	2.38456	0.428279	0.214140	−	0.976803i	$-0.431305\pi$
$31$	2.38456	0.428279	0.214140	+	0.976803i	$0.431305\pi$
$32$	0.620398	0.109672
$33$	0	0
$34$	−1.34329	−0.230373
$35$	−4.13125	−0.698309
$36$	0	0
$37$	−2.38151	−0.391517	−0.195759	−	0.980652i	$-0.562717\pi$
$37$	−2.38151	−0.391517	−0.195759	+	0.980652i	$0.562717\pi$
$38$	2.46212	0.399409
$39$	0	0
$40$	7.10054	1.12269
$41$	4.74360	0.740826	0.370413	−	0.928867i	$-0.379216\pi$
$41$	4.74360	0.740826	0.370413	+	0.928867i	$0.379216\pi$
$42$	0	0
$43$	−4.18576	−0.638322	−0.319161	−	0.947700i	$-0.603401\pi$
$43$	−4.18576	−0.638322	−0.319161	+	0.947700i	$0.603401\pi$
$44$	−22.2361	−3.35221
$45$	0	0
$46$	6.47017	0.953975
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	1.72549	0.246499
$50$	−7.49464	−1.05990
$51$	0	0
$52$	1.17797	0.163355
$53$	−11.1376	−1.52987	−0.764935	−	0.644107i	$-0.777229\pi$
$53$	−11.1376	−1.52987	−0.764935	+	0.644107i	$0.777229\pi$
$54$	0	0
$55$	−7.65600	−1.03233
$56$	−14.9968	−2.00404
$57$	0	0
$58$	−25.8254	−3.39105
$59$	−6.78274	−0.883038	−0.441519	−	0.897252i	$-0.645560\pi$
$59$	−6.78274	−0.883038	−0.441519	+	0.897252i	$0.645560\pi$
$60$	0	0
$61$	5.06585	0.648616	0.324308	−	0.945952i	$-0.394869\pi$
$61$	5.06585	0.648616	0.324308	+	0.945952i	$0.394869\pi$
$62$	5.87107	0.745626
$63$	0	0
$64$	−7.22459	−0.903074
$65$	0.405580	0.0503060
$66$	0	0
$67$	1.98930	0.243032	0.121516	−	0.992589i	$-0.461224\pi$
$67$	1.98930	0.243032	0.121516	+	0.992589i	$0.461224\pi$
$68$	−2.21618	−0.268752
$69$	0	0
$70$	−10.1716	−1.21574
$71$	9.03520	1.07228	0.536141	−	0.844129i	$-0.319882\pi$
$71$	9.03520	1.07228	0.536141	+	0.844129i	$0.319882\pi$
$72$	0	0
$73$	−10.6215	−1.24315	−0.621574	−	0.783356i	$-0.713506\pi$
$73$	−10.6215	−1.24315	−0.621574	+	0.783356i	$0.713506\pi$
$74$	−5.86355	−0.681624
$75$	0	0
$76$	4.06203	0.465947
$77$	16.1700	1.84274
$78$	0	0
$79$	−17.1441	−1.92886	−0.964429	−	0.264343i	$-0.914845\pi$
$79$	−17.1441	−1.92886	−0.964429	+	0.264343i	$0.914845\pi$
$80$	6.12024	0.684264
$81$	0	0
$82$	11.6793	1.28976
$83$	11.4200	1.25351	0.626756	−	0.779216i	$-0.284382\pi$
$83$	11.4200	1.25351	0.626756	+	0.779216i	$0.284382\pi$
$84$	0	0
$85$	−0.763043	−0.0827636
$86$	−10.3058	−1.11131
$87$	0	0
$88$	−27.7920	−2.96263
$89$	13.6473	1.44661	0.723305	−	0.690528i	$-0.242622\pi$
$89$	13.6473	1.44661	0.723305	+	0.690528i	$0.242622\pi$
$90$	0	0
$91$	−0.856613	−0.0897974
$92$	10.6746	1.11290
$93$	0	0
$94$	−2.46212	−0.253948
$95$	1.39858	0.143491
$96$	0	0
$97$	12.8747	1.30723	0.653614	−	0.756828i	$-0.273252\pi$
$97$	12.8747	1.30723	0.653614	+	0.756828i	$0.273252\pi$
$98$	4.24837	0.429150
$99$	0	0
$100$	−12.3647	−1.23647
$101$	15.2053	1.51298	0.756492	−	0.654003i	$-0.226912\pi$
$101$	15.2053	1.51298	0.756492	+	0.654003i	$0.226912\pi$
$102$	0	0
$103$	−17.7809	−1.75200	−0.876001	−	0.482309i	$-0.839798\pi$
$103$	−17.7809	−1.75200	−0.876001	+	0.482309i	$0.839798\pi$
$104$	1.47229	0.144370
$105$	0	0
$106$	−27.4222	−2.66348
$107$	4.78423	0.462509	0.231255	−	0.972893i	$-0.425717\pi$
$107$	4.78423	0.462509	0.231255	+	0.972893i	$0.425717\pi$
$108$	0	0
$109$	−5.35742	−0.513148	−0.256574	−	0.966525i	$-0.582594\pi$
$109$	−5.35742	−0.513148	−0.256574	+	0.966525i	$0.582594\pi$
$110$	−18.8500	−1.79727
$111$	0	0
$112$	−12.9264	−1.22143
$113$	4.48887	0.422277	0.211139	−	0.977456i	$-0.432283\pi$
$113$	4.48887	0.422277	0.211139	+	0.977456i	$0.432283\pi$
$114$	0	0
$115$	3.67531	0.342724
$116$	−42.6071	−3.95597
$117$	0	0
$118$	−16.6999	−1.53735
$119$	1.61160	0.147735
$120$	0	0
$121$	18.9660	1.72419
$122$	12.4727	1.12923
$123$	0	0
$124$	9.68615	0.869842
$125$	−11.2501	−1.00624
$126$	0	0
$127$	10.1627	0.901797	0.450899	−	0.892575i	$-0.351104\pi$
$127$	10.1627	0.901797	0.450899	+	0.892575i	$0.351104\pi$
$128$	−19.0286	−1.68191
$129$	0	0
$130$	0.998587	0.0875818
$131$	−1.26066	−0.110145	−0.0550723	−	0.998482i	$-0.517539\pi$
$131$	−1.26066	−0.110145	−0.0550723	+	0.998482i	$0.517539\pi$
$132$	0	0
$133$	−2.95389	−0.256135
$134$	4.89790	0.423114
$135$	0	0
$136$	−2.76992	−0.237518
$137$	−7.27463	−0.621513	−0.310757	−	0.950489i	$-0.600582\pi$
$137$	−7.27463	−0.621513	−0.310757	+	0.950489i	$0.600582\pi$
$138$	0	0
$139$	−16.3111	−1.38349	−0.691746	−	0.722141i	$-0.743158\pi$
$139$	−16.3111	−1.38349	−0.691746	+	0.722141i	$0.743158\pi$
$140$	−16.7813	−1.41828
$141$	0	0
$142$	22.2457	1.86682
$143$	−1.58747	−0.132751
$144$	0	0
$145$	−14.6698	−1.21826
$146$	−26.1513	−2.16430
$147$	0	0
$148$	−9.67375	−0.795178
$149$	−13.9927	−1.14633	−0.573164	−	0.819440i	$-0.694284\pi$
$149$	−13.9927	−1.14633	−0.573164	+	0.819440i	$0.694284\pi$
$150$	0	0
$151$	−3.65341	−0.297310	−0.148655	−	0.988889i	$-0.547494\pi$
$151$	−3.65341	−0.297310	−0.148655	+	0.988889i	$0.547494\pi$
$152$	5.07697	0.411797
$153$	0	0
$154$	39.8124	3.20818
$155$	3.33499	0.267873
$156$	0	0
$157$	−4.92003	−0.392661	−0.196331	−	0.980538i	$-0.562903\pi$
$157$	−4.92003	−0.392661	−0.196331	+	0.980538i	$0.562903\pi$
$158$	−42.2107	−3.35811
$159$	0	0
$160$	0.867676	0.0685958
$161$	−7.76250	−0.611771
$162$	0	0
$163$	−20.6955	−1.62099	−0.810497	−	0.585743i	$-0.800803\pi$
$163$	−20.6955	−1.62099	−0.810497	+	0.585743i	$0.800803\pi$
$164$	19.2687	1.50463
$165$	0	0
$166$	28.1175	2.18234
$167$	−0.175903	−0.0136118	−0.00680590	−	0.999977i	$-0.502166\pi$
$167$	−0.175903	−0.0136118	−0.00680590	+	0.999977i	$0.502166\pi$
$168$	0	0
$169$	−12.9159	−0.993531
$170$	−1.87870	−0.144090
$171$	0	0
$172$	−17.0027	−1.29644
$173$	−3.45179	−0.262435	−0.131217	−	0.991354i	$-0.541889\pi$
$173$	−3.45179	−0.262435	−0.131217	+	0.991354i	$0.541889\pi$
$174$	0	0
$175$	8.99159	0.679700
$176$	−23.9550	−1.80568
$177$	0	0
$178$	33.6013	2.51852
$179$	−13.9574	−1.04323	−0.521613	−	0.853182i	$-0.674670\pi$
$179$	−13.9574	−1.04323	−0.521613	+	0.853182i	$0.674670\pi$
$180$	0	0
$181$	5.46163	0.405960	0.202980	−	0.979183i	$-0.434937\pi$
$181$	5.46163	0.405960	0.202980	+	0.979183i	$0.434937\pi$
$182$	−2.10908	−0.156336
$183$	0	0
$184$	13.3417	0.983563
$185$	−3.33072	−0.244880
$186$	0	0
$187$	2.98660	0.218402
$188$	−4.06203	−0.296254
$189$	0	0
$190$	3.44347	0.249815
$191$	1.26908	0.0918277	0.0459138	−	0.998945i	$-0.485380\pi$
$191$	1.26908	0.0918277	0.0459138	+	0.998945i	$0.485380\pi$
$192$	0	0
$193$	10.6390	0.765811	0.382906	−	0.923788i	$-0.374923\pi$
$193$	10.6390	0.765811	0.382906	+	0.923788i	$0.374923\pi$
$194$	31.6991	2.27586
$195$	0	0
$196$	7.00900	0.500643
$197$	16.1425	1.15011	0.575054	−	0.818115i	$-0.304981\pi$
$197$	16.1425	1.15011	0.575054	+	0.818115i	$0.304981\pi$
$198$	0	0
$199$	4.60061	0.326128	0.163064	−	0.986615i	$-0.447862\pi$
$199$	4.60061	0.326128	0.163064	+	0.986615i	$0.447862\pi$
$200$	−15.4542	−1.09278
$201$	0	0
$202$	37.4373	2.63408
$203$	30.9837	2.17463
$204$	0	0
$205$	6.63430	0.463360
$206$	−43.7787	−3.05020
$207$	0	0
$208$	1.26903	0.0879912
$209$	−5.47412	−0.378653
$210$	0	0
$211$	−2.53645	−0.174617	−0.0873083	−	0.996181i	$-0.527827\pi$
$211$	−2.53645	−0.174617	−0.0873083	+	0.996181i	$0.527827\pi$
$212$	−45.2414	−3.10719
$213$	0	0
$214$	11.7794	0.805220
$215$	−5.85411	−0.399247
$216$	0	0
$217$	−7.04373	−0.478160
$218$	−13.1906	−0.893381
$219$	0	0
$220$	−31.0989	−2.09669
$221$	−0.158216	−0.0106428
$222$	0	0
$223$	11.4976	0.769934	0.384967	−	0.922930i	$-0.374213\pi$
$223$	11.4976	0.769934	0.384967	+	0.922930i	$0.374213\pi$
$224$	−1.83259	−0.122445
$225$	0	0
$226$	11.0521	0.735177
$227$	8.53050	0.566189	0.283094	−	0.959092i	$-0.408639\pi$
$227$	8.53050	0.566189	0.283094	+	0.959092i	$0.408639\pi$
$228$	0	0
$229$	12.5390	0.828600	0.414300	−	0.910140i	$-0.364026\pi$
$229$	12.5390	0.828600	0.414300	+	0.910140i	$0.364026\pi$
$230$	9.04905	0.596676
$231$	0	0
$232$	−53.2529	−3.49622
$233$	0.930826	0.0609804	0.0304902	−	0.999535i	$-0.490293\pi$
$233$	0.930826	0.0609804	0.0304902	+	0.999535i	$0.490293\pi$
$234$	0	0
$235$	−1.39858	−0.0912332
$236$	−27.5517	−1.79346
$237$	0	0
$238$	3.96795	0.257204
$239$	−10.4527	−0.676130	−0.338065	−	0.941123i	$-0.609772\pi$
$239$	−10.4527	−0.676130	−0.338065	+	0.941123i	$0.609772\pi$
$240$	0	0
$241$	18.6685	1.20255	0.601273	−	0.799044i	$-0.294660\pi$
$241$	18.6685	1.20255	0.601273	+	0.799044i	$0.294660\pi$
$242$	46.6967	3.00178
$243$	0	0
$244$	20.5777	1.31735
$245$	2.41324	0.154176
$246$	0	0
$247$	0.289994	0.0184519
$248$	12.1063	0.768753
$249$	0	0
$250$	−27.6992	−1.75185
$251$	11.1469	0.703585	0.351793	−	0.936078i	$-0.385572\pi$
$251$	11.1469	0.703585	0.351793	+	0.936078i	$0.385572\pi$
$252$	0	0
$253$	−14.3854	−0.904401
$254$	25.0219	1.57001
$255$	0	0
$256$	−32.4015	−2.02509
$257$	4.41395	0.275335	0.137667	−	0.990479i	$-0.456040\pi$
$257$	4.41395	0.275335	0.137667	+	0.990479i	$0.456040\pi$
$258$	0	0
$259$	7.03472	0.437116
$260$	1.64748	0.102172
$261$	0	0
$262$	−3.10390	−0.191760
$263$	23.5355	1.45126	0.725630	−	0.688085i	$-0.241548\pi$
$263$	23.5355	1.45126	0.725630	+	0.688085i	$0.241548\pi$
$264$	0	0
$265$	−15.5769	−0.956878
$266$	−7.27284	−0.445926
$267$	0	0
$268$	8.08062	0.493602
$269$	15.4799	0.943828	0.471914	−	0.881645i	$-0.343563\pi$
$269$	15.4799	0.943828	0.471914	+	0.881645i	$0.343563\pi$
$270$	0	0
$271$	−17.3427	−1.05350	−0.526748	−	0.850021i	$-0.676589\pi$
$271$	−17.3427	−1.05350	−0.526748	+	0.850021i	$0.676589\pi$
$272$	−2.38750	−0.144764
$273$	0	0
$274$	−17.9110	−1.08204
$275$	16.6631	1.00482
$276$	0	0
$277$	24.2686	1.45816	0.729078	−	0.684430i	$-0.239949\pi$
$277$	24.2686	1.45816	0.729078	+	0.684430i	$0.239949\pi$
$278$	−40.1600	−2.40864
$279$	0	0
$280$	−20.9743	−1.25345
$281$	−8.88311	−0.529922	−0.264961	−	0.964259i	$-0.585359\pi$
$281$	−8.88311	−0.529922	−0.264961	+	0.964259i	$0.585359\pi$
$282$	0	0
$283$	−23.8941	−1.42036	−0.710179	−	0.704021i	$-0.751386\pi$
$283$	−23.8941	−1.42036	−0.710179	+	0.704021i	$0.751386\pi$
$284$	36.7013	2.17782
$285$	0	0
$286$	−3.90853	−0.231116
$287$	−14.0121	−0.827108
$288$	0	0
$289$	−16.7023	−0.982490
$290$	−36.1189	−2.12098
$291$	0	0
$292$	−43.1447	−2.52485
$293$	7.65834	0.447405	0.223703	−	0.974657i	$-0.428186\pi$
$293$	7.65834	0.447405	0.223703	+	0.974657i	$0.428186\pi$
$294$	0	0
$295$	−9.48620	−0.552308
$296$	−12.0908	−0.702765
$297$	0	0
$298$	−34.4518	−1.99574
$299$	0.762073	0.0440718
$300$	0	0
$301$	12.3643	0.712666
$302$	−8.99512	−0.517611
$303$	0	0
$304$	4.37604	0.250983
$305$	7.08500	0.405686
$306$	0	0
$307$	−13.4815	−0.769432	−0.384716	−	0.923035i	$-0.625701\pi$
$307$	−13.4815	−0.769432	−0.384716	+	0.923035i	$0.625701\pi$
$308$	65.6830	3.74264
$309$	0	0
$310$	8.21115	0.466362
$311$	24.7963	1.40607	0.703034	−	0.711156i	$-0.251828\pi$
$311$	24.7963	1.40607	0.703034	+	0.711156i	$0.251828\pi$
$312$	0	0
$313$	7.20490	0.407245	0.203622	−	0.979049i	$-0.434728\pi$
$313$	7.20490	0.407245	0.203622	+	0.979049i	$0.434728\pi$
$314$	−12.1137	−0.683616
$315$	0	0
$316$	−69.6397	−3.91754
$317$	−17.6208	−0.989681	−0.494841	−	0.868984i	$-0.664774\pi$
$317$	−17.6208	−0.989681	−0.494841	+	0.868984i	$0.664774\pi$
$318$	0	0
$319$	57.4187	3.21483
$320$	−10.1042	−0.564840
$321$	0	0
$322$	−19.1122	−1.06508
$323$	−0.545585	−0.0303571
$324$	0	0
$325$	−0.882736	−0.0489654
$326$	−50.9547	−2.82212
$327$	0	0
$328$	24.0831	1.32977
$329$	2.95389	0.162853
$330$	0	0
$331$	−8.19469	−0.450421	−0.225210	−	0.974310i	$-0.572307\pi$
$331$	−8.19469	−0.450421	−0.225210	+	0.974310i	$0.572307\pi$
$332$	46.3886	2.54590
$333$	0	0
$334$	−0.433095	−0.0236979
$335$	2.78220	0.152008
$336$	0	0
$337$	−9.20281	−0.501309	−0.250655	−	0.968077i	$-0.580646\pi$
$337$	−9.20281	−0.501309	−0.250655	+	0.968077i	$0.580646\pi$
$338$	−31.8005	−1.72972
$339$	0	0
$340$	−3.09951	−0.168094
$341$	−13.0534	−0.706880
$342$	0	0
$343$	15.5803	0.841259
$344$	−21.2510	−1.14578
$345$	0	0
$346$	−8.49872	−0.456894
$347$	−6.24475	−0.335236	−0.167618	−	0.985852i	$-0.553608\pi$
$347$	−6.24475	−0.335236	−0.167618	+	0.985852i	$0.553608\pi$
$348$	0	0
$349$	26.1261	1.39850	0.699249	−	0.714878i	$-0.253518\pi$
$349$	26.1261	1.39850	0.699249	+	0.714878i	$0.253518\pi$
$350$	22.1384	1.18335
$351$	0	0
$352$	−3.39614	−0.181015
$353$	6.87528	0.365934	0.182967	−	0.983119i	$-0.441430\pi$
$353$	6.87528	0.365934	0.182967	+	0.983119i	$0.441430\pi$
$354$	0	0
$355$	12.6364	0.670673
$356$	55.4358	2.93809
$357$	0	0
$358$	−34.3648	−1.81624
$359$	−29.0488	−1.53314	−0.766569	−	0.642162i	$-0.778037\pi$
$359$	−29.0488	−1.53314	−0.766569	+	0.642162i	$0.778037\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	13.4472	0.706768
$363$	0	0
$364$	−3.47959	−0.182380
$365$	−14.8549	−0.777543
$366$	0	0
$367$	13.3554	0.697149	0.348574	−	0.937281i	$-0.386666\pi$
$367$	13.3554	0.697149	0.348574	+	0.937281i	$0.386666\pi$
$368$	11.4997	0.599466
$369$	0	0
$370$	−8.20064	−0.426331
$371$	32.8994	1.70805
$372$	0	0
$373$	28.7744	1.48988	0.744941	−	0.667131i	$-0.232478\pi$
$373$	28.7744	1.48988	0.744941	+	0.667131i	$0.232478\pi$
$374$	7.35336	0.380233
$375$	0	0
$376$	−5.07697	−0.261825
$377$	−3.04178	−0.156660
$378$	0	0
$379$	−31.2791	−1.60670	−0.803350	−	0.595508i	$-0.796951\pi$
$379$	−31.2791	−1.60670	−0.803350	+	0.595508i	$0.796951\pi$
$380$	5.68107	0.291433
$381$	0	0
$382$	3.12464	0.159870
$383$	29.9726	1.53153	0.765764	−	0.643121i	$-0.222361\pi$
$383$	29.9726	1.53153	0.765764	+	0.643121i	$0.222361\pi$
$384$	0	0
$385$	22.6150	1.15257
$386$	26.1945	1.33326
$387$	0	0
$388$	52.2974	2.65500
$389$	−6.36583	−0.322761	−0.161380	−	0.986892i	$-0.551595\pi$
$389$	−6.36583	−0.322761	−0.161380	+	0.986892i	$0.551595\pi$
$390$	0	0
$391$	−1.43373	−0.0725071
$392$	8.76027	0.442460
$393$	0	0
$394$	39.7449	2.00232
$395$	−23.9773	−1.20643
$396$	0	0
$397$	−3.01076	−0.151106	−0.0755529	−	0.997142i	$-0.524072\pi$
$397$	−3.01076	−0.151106	−0.0755529	+	0.997142i	$0.524072\pi$
$398$	11.3272	0.567783
$399$	0	0
$400$	−13.3206	−0.666029
$401$	12.4808	0.623263	0.311631	−	0.950203i	$-0.399125\pi$
$401$	12.4808	0.623263	0.311631	+	0.950203i	$0.399125\pi$
$402$	0	0
$403$	0.691508	0.0344465
$404$	61.7644	3.07289
$405$	0	0
$406$	76.2856	3.78599
$407$	13.0367	0.646203
$408$	0	0
$409$	0.595057	0.0294237	0.0147118	−	0.999892i	$-0.495317\pi$
$409$	0.595057	0.0294237	0.0147118	+	0.999892i	$0.495317\pi$
$410$	16.3344	0.806701
$411$	0	0
$412$	−72.2265	−3.55835
$413$	20.0355	0.985883
$414$	0	0
$415$	15.9718	0.784026
$416$	0.179912	0.00882092
$417$	0	0
$418$	−13.4779	−0.659228
$419$	39.0123	1.90587	0.952937	−	0.303167i	$-0.0980440\pi$
$419$	39.0123	1.90587	0.952937	+	0.303167i	$0.0980440\pi$
$420$	0	0
$421$	−18.4127	−0.897382	−0.448691	−	0.893687i	$-0.648110\pi$
$421$	−18.4127	−0.897382	−0.448691	+	0.893687i	$0.648110\pi$
$422$	−6.24505	−0.304004
$423$	0	0
$424$	−56.5454	−2.74609
$425$	1.66075	0.0805581
$426$	0	0
$427$	−14.9640	−0.724159
$428$	19.4337	0.939364
$429$	0	0
$430$	−14.4135	−0.695082
$431$	−15.7936	−0.760753	−0.380377	−	0.924832i	$-0.624206\pi$
$431$	−15.7936	−0.760753	−0.380377	+	0.924832i	$0.624206\pi$
$432$	0	0
$433$	−19.3944	−0.932036	−0.466018	−	0.884775i	$-0.654312\pi$
$433$	−19.3944	−0.932036	−0.466018	+	0.884775i	$0.654312\pi$
$434$	−17.3425	−0.832467
$435$	0	0
$436$	−21.7620	−1.04221
$437$	2.62789	0.125709
$438$	0	0
$439$	7.50801	0.358338	0.179169	−	0.983818i	$-0.442659\pi$
$439$	7.50801	0.358338	0.179169	+	0.983818i	$0.442659\pi$
$440$	−38.8693	−1.85302
$441$	0	0
$442$	−0.389548	−0.0185289
$443$	−13.3177	−0.632745	−0.316373	−	0.948635i	$-0.602465\pi$
$443$	−13.3177	−0.632745	−0.316373	+	0.948635i	$0.602465\pi$
$444$	0	0
$445$	19.0868	0.904802
$446$	28.3084	1.34044
$447$	0	0
$448$	21.3407	1.00825
$449$	−4.28384	−0.202167	−0.101083	−	0.994878i	$-0.532231\pi$
$449$	−4.28384	−0.202167	−0.101083	+	0.994878i	$0.532231\pi$
$450$	0	0
$451$	−25.9671	−1.22274
$452$	18.2339	0.857652
$453$	0	0
$454$	21.0031	0.985724
$455$	−1.19804	−0.0561650
$456$	0	0
$457$	8.52369	0.398721	0.199361	−	0.979926i	$-0.436113\pi$
$457$	8.52369	0.398721	0.199361	+	0.979926i	$0.436113\pi$
$458$	30.8725	1.44258
$459$	0	0
$460$	14.9292	0.696078
$461$	−24.1410	−1.12436	−0.562179	−	0.827015i	$-0.690037\pi$
$461$	−24.1410	−1.12436	−0.562179	+	0.827015i	$0.690037\pi$
$462$	0	0
$463$	−11.0461	−0.513354	−0.256677	−	0.966497i	$-0.582628\pi$
$463$	−11.0461	−0.513354	−0.256677	+	0.966497i	$0.582628\pi$
$464$	−45.9008	−2.13089
$465$	0	0
$466$	2.29180	0.106166
$467$	−25.2315	−1.16758	−0.583788	−	0.811906i	$-0.698430\pi$
$467$	−25.2315	−1.16758	−0.583788	+	0.811906i	$0.698430\pi$
$468$	0	0
$469$	−5.87619	−0.271337
$470$	−3.44347	−0.158835
$471$	0	0
$472$	−34.4358	−1.58504
$473$	22.9134	1.05356
$474$	0	0
$475$	−3.04398	−0.139667
$476$	6.54637	0.300052
$477$	0	0
$478$	−25.7358	−1.17713
$479$	12.5860	0.575069	0.287535	−	0.957770i	$-0.407164\pi$
$479$	12.5860	0.575069	0.287535	+	0.957770i	$0.407164\pi$
$480$	0	0
$481$	−0.690623	−0.0314897
$482$	45.9642	2.09361
$483$	0	0
$484$	77.0407	3.50185
$485$	18.0063	0.817623
$486$	0	0
$487$	−37.1775	−1.68467	−0.842336	−	0.538952i	$-0.818820\pi$
$487$	−37.1775	−1.68467	−0.842336	+	0.538952i	$0.818820\pi$
$488$	25.7192	1.16425
$489$	0	0
$490$	5.94168	0.268418
$491$	16.3103	0.736072	0.368036	−	0.929812i	$-0.380030\pi$
$491$	16.3103	0.736072	0.368036	+	0.929812i	$0.380030\pi$
$492$	0	0
$493$	5.72270	0.257737
$494$	0.714001	0.0321244
$495$	0	0
$496$	10.4349	0.468542
$497$	−26.6890	−1.19717
$498$	0	0
$499$	18.8093	0.842020	0.421010	−	0.907056i	$-0.361676\pi$
$499$	18.8093	0.842020	0.421010	+	0.907056i	$0.361676\pi$
$500$	−45.6984	−2.04370
$501$	0	0
$502$	27.4450	1.22493
$503$	15.8989	0.708898	0.354449	−	0.935075i	$-0.384668\pi$
$503$	15.8989	0.708898	0.354449	+	0.935075i	$0.384668\pi$
$504$	0	0
$505$	21.2658	0.946316
$506$	−35.4185	−1.57455
$507$	0	0
$508$	41.2814	1.83156
$509$	33.2937	1.47572	0.737860	−	0.674954i	$-0.235837\pi$
$509$	33.2937	1.47572	0.737860	+	0.674954i	$0.235837\pi$
$510$	0	0
$511$	31.3747	1.38793
$512$	−41.7192	−1.84374
$513$	0	0
$514$	10.8677	0.479353
$515$	−24.8680	−1.09581
$516$	0	0
$517$	5.47412	0.240752
$518$	17.3203	0.761011
$519$	0	0
$520$	2.05912	0.0902983
$521$	6.32817	0.277242	0.138621	−	0.990345i	$-0.455733\pi$
$521$	6.32817	0.277242	0.138621	+	0.990345i	$0.455733\pi$
$522$	0	0
$523$	−39.2603	−1.71673	−0.858367	−	0.513036i	$-0.828521\pi$
$523$	−39.2603	−1.71673	−0.858367	+	0.513036i	$0.828521\pi$
$524$	−5.12085	−0.223706
$525$	0	0
$526$	57.9472	2.52662
$527$	−1.30098	−0.0566715
$528$	0	0
$529$	−16.0942	−0.699748
$530$	−38.3521	−1.66591
$531$	0	0
$532$	−11.9988	−0.520215
$533$	1.37562	0.0595846
$534$	0	0
$535$	6.69113	0.289283
$536$	10.0996	0.436238
$537$	0	0
$538$	38.1134	1.64319
$539$	−9.44556	−0.406849
$540$	0	0
$541$	−9.82530	−0.422423	−0.211211	−	0.977440i	$-0.567741\pi$
$541$	−9.82530	−0.422423	−0.211211	+	0.977440i	$0.567741\pi$
$542$	−42.6999	−1.83412
$543$	0	0
$544$	−0.338480	−0.0145122
$545$	−7.49278	−0.320955
$546$	0	0
$547$	20.8977	0.893521	0.446760	−	0.894654i	$-0.352578\pi$
$547$	20.8977	0.893521	0.446760	+	0.894654i	$0.352578\pi$
$548$	−29.5498	−1.26230
$549$	0	0
$550$	41.0266	1.74938
$551$	−10.4891	−0.446851
$552$	0	0
$553$	50.6417	2.15351
$554$	59.7521	2.53862
$555$	0	0
$556$	−66.2564	−2.80990
$557$	−35.5496	−1.50628	−0.753142	−	0.657858i	$-0.771463\pi$
$557$	−35.5496	−1.50628	−0.753142	+	0.657858i	$0.771463\pi$
$558$	0	0
$559$	−1.21385	−0.0513402
$560$	−18.0785	−0.763958
$561$	0	0
$562$	−21.8713	−0.922584
$563$	−42.0913	−1.77394	−0.886969	−	0.461829i	$-0.847193\pi$
$563$	−42.0913	−1.77394	−0.886969	+	0.461829i	$0.847193\pi$
$564$	0	0
$565$	6.27804	0.264119
$566$	−58.8302	−2.47282
$567$	0	0
$568$	45.8714	1.92472
$569$	20.7605	0.870324	0.435162	−	0.900352i	$-0.356691\pi$
$569$	20.7605	0.870324	0.435162	+	0.900352i	$0.356691\pi$
$570$	0	0
$571$	20.1214	0.842055	0.421028	−	0.907048i	$-0.361670\pi$
$571$	20.1214	0.842055	0.421028	+	0.907048i	$0.361670\pi$
$572$	−6.44834	−0.269618
$573$	0	0
$574$	−34.4995	−1.43998
$575$	−7.99923	−0.333591
$576$	0	0
$577$	−35.6855	−1.48561	−0.742803	−	0.669510i	$-0.766504\pi$
$577$	−35.6855	−1.48561	−0.742803	+	0.669510i	$0.766504\pi$
$578$	−41.1232	−1.71050
$579$	0	0
$580$	−59.5894	−2.47432
$581$	−33.7336	−1.39951
$582$	0	0
$583$	60.9688	2.52507
$584$	−53.9248	−2.23142
$585$	0	0
$586$	18.8558	0.778924
$587$	−24.3613	−1.00550	−0.502750	−	0.864432i	$-0.667678\pi$
$587$	−24.3613	−1.00550	−0.502750	+	0.864432i	$0.667678\pi$
$588$	0	0
$589$	2.38456	0.0982540
$590$	−23.3562	−0.961558
$591$	0	0
$592$	−10.4216	−0.428324
$593$	36.5685	1.50169	0.750845	−	0.660479i	$-0.229647\pi$
$593$	36.5685	1.50169	0.750845	+	0.660479i	$0.229647\pi$
$594$	0	0
$595$	2.25395	0.0924029
$596$	−56.8389	−2.32821
$597$	0	0
$598$	1.87631	0.0767282
$599$	−7.92858	−0.323953	−0.161976	−	0.986795i	$-0.551787\pi$
$599$	−7.92858	−0.323953	−0.161976	+	0.986795i	$0.551787\pi$
$600$	0	0
$601$	8.16527	0.333068	0.166534	−	0.986036i	$-0.446742\pi$
$601$	8.16527	0.333068	0.166534	+	0.986036i	$0.446742\pi$
$602$	30.4424	1.24074
$603$	0	0
$604$	−14.8403	−0.603841
$605$	26.5255	1.07842
$606$	0	0
$607$	4.91150	0.199352	0.0996758	−	0.995020i	$-0.468219\pi$
$607$	4.91150	0.199352	0.0996758	+	0.995020i	$0.468219\pi$
$608$	0.620398	0.0251605
$609$	0	0
$610$	17.4441	0.706291
$611$	−0.289994	−0.0117319
$612$	0	0
$613$	−42.8452	−1.73050	−0.865252	−	0.501338i	$-0.832841\pi$
$613$	−42.8452	−1.73050	−0.865252	+	0.501338i	$0.832841\pi$
$614$	−33.1931	−1.33957
$615$	0	0
$616$	82.0945	3.30768
$617$	37.1902	1.49722	0.748611	−	0.663010i	$-0.230721\pi$
$617$	37.1902	1.49722	0.748611	+	0.663010i	$0.230721\pi$
$618$	0	0
$619$	−16.2179	−0.651851	−0.325926	−	0.945395i	$-0.605676\pi$
$619$	−16.2179	−0.651851	−0.325926	+	0.945395i	$0.605676\pi$
$620$	13.5468	0.544054
$621$	0	0
$622$	61.0514	2.44794
$623$	−40.3127	−1.61509
$624$	0	0
$625$	−0.514317	−0.0205727
$626$	17.7393	0.709006
$627$	0	0
$628$	−19.9853	−0.797502
$629$	1.29931	0.0518070
$630$	0	0
$631$	−12.4043	−0.493805	−0.246903	−	0.969040i	$-0.579413\pi$
$631$	−12.4043	−0.493805	−0.246903	+	0.969040i	$0.579413\pi$
$632$	−87.0399	−3.46226
$633$	0	0
$634$	−43.3845	−1.72302
$635$	14.2134	0.564041
$636$	0	0
$637$	0.500383	0.0198259
$638$	141.372	5.59696
$639$	0	0
$640$	−26.6130	−1.05197
$641$	42.3599	1.67312	0.836558	−	0.547878i	$-0.184564\pi$
$641$	42.3599	1.67312	0.836558	+	0.547878i	$0.184564\pi$
$642$	0	0
$643$	−39.0338	−1.53934	−0.769672	−	0.638440i	$-0.779580\pi$
$643$	−39.0338	−1.53934	−0.769672	+	0.638440i	$0.779580\pi$
$644$	−31.5315	−1.24252
$645$	0	0
$646$	−1.34329	−0.0528512
$647$	24.2084	0.951729	0.475864	−	0.879519i	$-0.342135\pi$
$647$	24.2084	0.951729	0.475864	+	0.879519i	$0.342135\pi$
$648$	0	0
$649$	37.1296	1.45746
$650$	−2.17340	−0.0852479
$651$	0	0
$652$	−84.0657	−3.29226
$653$	13.7139	0.536666	0.268333	−	0.963326i	$-0.413527\pi$
$653$	13.7139	0.536666	0.268333	+	0.963326i	$0.413527\pi$
$654$	0	0
$655$	−1.76314	−0.0688915
$656$	20.7582	0.810472
$657$	0	0
$658$	7.27284	0.283525
$659$	24.4731	0.953338	0.476669	−	0.879083i	$-0.341844\pi$
$659$	24.4731	0.953338	0.476669	+	0.879083i	$0.341844\pi$
$660$	0	0
$661$	−23.5662	−0.916620	−0.458310	−	0.888793i	$-0.651545\pi$
$661$	−23.5662	−0.916620	−0.458310	+	0.888793i	$0.651545\pi$
$662$	−20.1763	−0.784174
$663$	0	0
$664$	57.9792	2.25003
$665$	−4.13125	−0.160203
$666$	0	0
$667$	−27.5642	−1.06729
$668$	−0.714525	−0.0276458
$669$	0	0
$670$	6.85010	0.264643
$671$	−27.7311	−1.07055
$672$	0	0
$673$	25.8628	0.996938	0.498469	−	0.866907i	$-0.333896\pi$
$673$	25.8628	0.996938	0.498469	+	0.866907i	$0.333896\pi$
$674$	−22.6584	−0.872770
$675$	0	0
$676$	−52.4648	−2.01788
$677$	−26.4064	−1.01488	−0.507441	−	0.861687i	$-0.669408\pi$
$677$	−26.4064	−1.01488	−0.507441	+	0.861687i	$0.669408\pi$
$678$	0	0
$679$	−38.0305	−1.45948
$680$	−3.87395	−0.148559
$681$	0	0
$682$	−32.1390	−1.23066
$683$	−14.1518	−0.541502	−0.270751	−	0.962649i	$-0.587272\pi$
$683$	−14.1518	−0.541502	−0.270751	+	0.962649i	$0.587272\pi$
$684$	0	0
$685$	−10.1741	−0.388734
$686$	38.3607	1.46462
$687$	0	0
$688$	−18.3171	−0.698331
$689$	−3.22985	−0.123047
$690$	0	0
$691$	17.1420	0.652114	0.326057	−	0.945350i	$-0.394280\pi$
$691$	17.1420	0.652114	0.326057	+	0.945350i	$0.394280\pi$
$692$	−14.0213	−0.533009
$693$	0	0
$694$	−15.3753	−0.583640
$695$	−22.8124	−0.865324
$696$	0	0
$697$	−2.58804	−0.0980288
$698$	64.3256	2.43476
$699$	0	0
$700$	36.5241	1.38048
$701$	7.87825	0.297558	0.148779	−	0.988871i	$-0.452466\pi$
$701$	7.87825	0.297558	0.148779	+	0.988871i	$0.452466\pi$
$702$	0	0
$703$	−2.38151	−0.0898202
$704$	39.5483	1.49053
$705$	0	0
$706$	16.9278	0.637084
$707$	−44.9148	−1.68920
$708$	0	0
$709$	1.51595	0.0569325	0.0284663	−	0.999595i	$-0.490938\pi$
$709$	1.51595	0.0569325	0.0284663	+	0.999595i	$0.490938\pi$
$710$	31.1124	1.16763
$711$	0	0
$712$	69.2869	2.59664
$713$	6.26635	0.234677
$714$	0	0
$715$	−2.22020	−0.0830306
$716$	−56.6955	−2.11881
$717$	0	0
$718$	−71.5216	−2.66916
$719$	11.2427	0.419281	0.209640	−	0.977779i	$-0.432771\pi$
$719$	11.2427	0.419281	0.209640	+	0.977779i	$0.432771\pi$
$720$	0	0
$721$	52.5228	1.95605
$722$	2.46212	0.0916306
$723$	0	0
$724$	22.1853	0.824511
$725$	31.9286	1.18580
$726$	0	0
$727$	12.4911	0.463269	0.231635	−	0.972803i	$-0.425593\pi$
$727$	12.4911	0.463269	0.231635	+	0.972803i	$0.425593\pi$
$728$	−4.34900	−0.161185
$729$	0	0
$730$	−36.5746	−1.35369
$731$	2.28369	0.0844651
$732$	0	0
$733$	37.1249	1.37124	0.685619	−	0.727960i	$-0.259531\pi$
$733$	37.1249	1.37124	0.685619	+	0.727960i	$0.259531\pi$
$734$	32.8827	1.21372
$735$	0	0
$736$	1.63034	0.0600950
$737$	−10.8897	−0.401127
$738$	0	0
$739$	11.0931	0.408067	0.204034	−	0.978964i	$-0.434595\pi$
$739$	11.0931	0.408067	0.204034	+	0.978964i	$0.434595\pi$
$740$	−13.5295	−0.497355
$741$	0	0
$742$	81.0022	2.97368
$743$	−47.3872	−1.73847	−0.869234	−	0.494401i	$-0.835387\pi$
$743$	−47.3872	−1.73847	−0.869234	+	0.494401i	$0.835387\pi$
$744$	0	0
$745$	−19.5699	−0.716987
$746$	70.8460	2.59386
$747$	0	0
$748$	12.1317	0.443577
$749$	−14.1321	−0.516376
$750$	0	0
$751$	−50.0518	−1.82642	−0.913209	−	0.407492i	$-0.866403\pi$
$751$	−50.0518	−1.82642	−0.913209	+	0.407492i	$0.866403\pi$
$752$	−4.37604	−0.159578
$753$	0	0
$754$	−7.48923	−0.272742
$755$	−5.10958	−0.185957
$756$	0	0
$757$	26.4016	0.959583	0.479792	−	0.877382i	$-0.340712\pi$
$757$	26.4016	0.959583	0.479792	+	0.877382i	$0.340712\pi$
$758$	−77.0129	−2.79723
$759$	0	0
$760$	7.10054	0.257564
$761$	−42.8399	−1.55295	−0.776473	−	0.630151i	$-0.782993\pi$
$761$	−42.8399	−1.55295	−0.776473	+	0.630151i	$0.782993\pi$
$762$	0	0
$763$	15.8253	0.572913
$764$	5.15506	0.186504
$765$	0	0
$766$	73.7961	2.66636
$767$	−1.96696	−0.0710227
$768$	0	0
$769$	−17.8667	−0.644290	−0.322145	−	0.946690i	$-0.604404\pi$
$769$	−17.8667	−0.644290	−0.322145	+	0.946690i	$0.604404\pi$
$770$	55.6808	2.00660
$771$	0	0
$772$	43.2159	1.55537
$773$	−17.9819	−0.646763	−0.323382	−	0.946269i	$-0.604820\pi$
$773$	−17.9819	−0.646763	−0.323382	+	0.946269i	$0.604820\pi$
$774$	0	0
$775$	−7.25854	−0.260734
$776$	65.3645	2.34645
$777$	0	0
$778$	−15.6734	−0.561920
$779$	4.74360	0.169957
$780$	0	0
$781$	−49.4598	−1.76981
$782$	−3.53003	−0.126233
$783$	0	0
$784$	7.55082	0.269672
$785$	−6.88105	−0.245595
$786$	0	0
$787$	−28.4875	−1.01547	−0.507735	−	0.861513i	$-0.669517\pi$
$787$	−28.4875	−1.01547	−0.507735	+	0.861513i	$0.669517\pi$
$788$	65.5715	2.33589
$789$	0	0
$790$	−59.0350	−2.10037
$791$	−13.2597	−0.471459
$792$	0	0
$793$	1.46907	0.0521682
$794$	−7.41286	−0.263073
$795$	0	0
$796$	18.6878	0.662372
$797$	29.8610	1.05773	0.528865	−	0.848706i	$-0.322618\pi$
$797$	29.8610	1.05773	0.528865	+	0.848706i	$0.322618\pi$
$798$	0	0
$799$	0.545585	0.0193014
$800$	−1.88848	−0.0667678
$801$	0	0
$802$	30.7293	1.08509
$803$	58.1432	2.05183
$804$	0	0
$805$	−10.8565	−0.382640
$806$	1.70258	0.0599707
$807$	0	0
$808$	77.1968	2.71578
$809$	32.2904	1.13527	0.567636	−	0.823280i	$-0.307858\pi$
$809$	32.2904	1.13527	0.567636	+	0.823280i	$0.307858\pi$
$810$	0	0
$811$	−15.8359	−0.556075	−0.278038	−	0.960570i	$-0.589684\pi$
$811$	−15.8359	−0.556075	−0.278038	+	0.960570i	$0.589684\pi$
$812$	125.857	4.41671
$813$	0	0
$814$	32.0978	1.12503
$815$	−28.9442	−1.01387
$816$	0	0
$817$	−4.18576	−0.146441
$818$	1.46510	0.0512261
$819$	0	0
$820$	26.9487	0.941091
$821$	−3.41950	−0.119341	−0.0596706	−	0.998218i	$-0.519005\pi$
$821$	−3.41950	−0.119341	−0.0596706	+	0.998218i	$0.519005\pi$
$822$	0	0
$823$	−49.7774	−1.73513	−0.867566	−	0.497322i	$-0.834317\pi$
$823$	−49.7774	−1.73513	−0.867566	+	0.497322i	$0.834317\pi$
$824$	−90.2730	−3.14481
$825$	0	0
$826$	49.3298	1.71640
$827$	−1.33419	−0.0463943	−0.0231972	−	0.999731i	$-0.507385\pi$
$827$	−1.33419	−0.0463943	−0.0231972	+	0.999731i	$0.507385\pi$
$828$	0	0
$829$	−20.4432	−0.710021	−0.355010	−	0.934862i	$-0.615523\pi$
$829$	−20.4432	−0.710021	−0.355010	+	0.934862i	$0.615523\pi$
$830$	39.3245	1.36497
$831$	0	0
$832$	−2.09509	−0.0726342
$833$	−0.941402	−0.0326176
$834$	0	0
$835$	−0.246015	−0.00851369
$836$	−22.2361	−0.769051
$837$	0	0
$838$	96.0529	3.31809
$839$	11.2839	0.389563	0.194781	−	0.980847i	$-0.437600\pi$
$839$	11.2839	0.389563	0.194781	+	0.980847i	$0.437600\pi$
$840$	0	0
$841$	81.0214	2.79384
$842$	−45.3343	−1.56232
$843$	0	0
$844$	−10.3031	−0.354649
$845$	−18.0639	−0.621417
$846$	0	0
$847$	−56.0237	−1.92500
$848$	−48.7387	−1.67369
$849$	0	0
$850$	4.08896	0.140250
$851$	−6.25833	−0.214533
$852$	0	0
$853$	4.32511	0.148089	0.0740445	−	0.997255i	$-0.476409\pi$
$853$	4.32511	0.148089	0.0740445	+	0.997255i	$0.476409\pi$
$854$	−36.8432	−1.26075
$855$	0	0
$856$	24.2894	0.830195
$857$	−19.2260	−0.656747	−0.328373	−	0.944548i	$-0.606500\pi$
$857$	−19.2260	−0.656747	−0.328373	+	0.944548i	$0.606500\pi$
$858$	0	0
$859$	−39.1295	−1.33508	−0.667541	−	0.744573i	$-0.732653\pi$
$859$	−39.1295	−1.33508	−0.667541	+	0.744573i	$0.732653\pi$
$860$	−23.7796	−0.810878
$861$	0	0
$862$	−38.8859	−1.32446
$863$	−33.8039	−1.15070	−0.575349	−	0.817908i	$-0.695134\pi$
$863$	−33.8039	−1.15070	−0.575349	+	0.817908i	$0.695134\pi$
$864$	0	0
$865$	−4.82760	−0.164143
$866$	−47.7514	−1.62266
$867$	0	0
$868$	−28.6119	−0.971150
$869$	93.8487	3.18360
$870$	0	0
$871$	0.576887	0.0195471
$872$	−27.1995	−0.921091
$873$	0	0
$874$	6.47017	0.218857
$875$	33.2317	1.12344
$876$	0	0
$877$	57.9053	1.95532	0.977661	−	0.210189i	$-0.0674078\pi$
$877$	57.9053	1.95532	0.977661	+	0.210189i	$0.0674078\pi$
$878$	18.4856	0.623859
$879$	0	0
$880$	−33.5030	−1.12938
$881$	−6.02466	−0.202976	−0.101488	−	0.994837i	$-0.532360\pi$
$881$	−6.02466	−0.202976	−0.101488	+	0.994837i	$0.532360\pi$
$882$	0	0
$883$	−15.5841	−0.524448	−0.262224	−	0.965007i	$-0.584456\pi$
$883$	−15.5841	−0.524448	−0.262224	+	0.965007i	$0.584456\pi$
$884$	−0.642680	−0.0216157
$885$	0	0
$886$	−32.7899	−1.10160
$887$	41.2670	1.38561	0.692806	−	0.721124i	$-0.256374\pi$
$887$	41.2670	1.38561	0.692806	+	0.721124i	$0.256374\pi$
$888$	0	0
$889$	−30.0196	−1.00683
$890$	46.9940	1.57524
$891$	0	0
$892$	46.7035	1.56375
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−19.5206	−0.652500
$896$	56.2085	1.87779
$897$	0	0
$898$	−10.5473	−0.351969
$899$	−25.0119	−0.834193
$900$	0	0
$901$	6.07652	0.202438
$902$	−63.9340	−2.12877
$903$	0	0
$904$	22.7899	0.757980
$905$	7.63852	0.253913
$906$	0	0
$907$	14.8782	0.494024	0.247012	−	0.969012i	$-0.420551\pi$
$907$	14.8782	0.494024	0.247012	+	0.969012i	$0.420551\pi$
$908$	34.6511	1.14994
$909$	0	0
$910$	−2.94972	−0.0977822
$911$	−47.2026	−1.56389	−0.781946	−	0.623347i	$-0.785773\pi$
$911$	−47.2026	−1.56389	−0.781946	+	0.623347i	$0.785773\pi$
$912$	0	0
$913$	−62.5147	−2.06894
$914$	20.9863	0.694166
$915$	0	0
$916$	50.9338	1.68290
$917$	3.72387	0.122973
$918$	0	0
$919$	−24.1979	−0.798215	−0.399108	−	0.916904i	$-0.630680\pi$
$919$	−24.1979	−0.798215	−0.399108	+	0.916904i	$0.630680\pi$
$920$	18.6594	0.615183
$921$	0	0
$922$	−59.4380	−1.95749
$923$	2.62016	0.0862435
$924$	0	0
$925$	7.24925	0.238354
$926$	−27.1967	−0.893740
$927$	0	0
$928$	−6.50743	−0.213617
$929$	39.2845	1.28888	0.644441	−	0.764654i	$-0.277090\pi$
$929$	39.2845	1.28888	0.644441	+	0.764654i	$0.277090\pi$
$930$	0	0
$931$	1.72549	0.0565507
$932$	3.78105	0.123852
$933$	0	0
$934$	−62.1231	−2.03273
$935$	4.17699	0.136602
$936$	0	0
$937$	21.7453	0.710389	0.355195	−	0.934792i	$-0.384415\pi$
$937$	21.7453	0.710389	0.355195	+	0.934792i	$0.384415\pi$
$938$	−14.4679	−0.472393
$939$	0	0
$940$	−5.68107	−0.185296
$941$	7.54566	0.245981	0.122991	−	0.992408i	$-0.460751\pi$
$941$	7.54566	0.245981	0.122991	+	0.992408i	$0.460751\pi$
$942$	0	0
$943$	12.4656	0.405937
$944$	−29.6816	−0.966053
$945$	0	0
$946$	56.4154	1.83422
$947$	−33.5957	−1.09171	−0.545857	−	0.837878i	$-0.683796\pi$
$947$	−33.5957	−1.09171	−0.545857	+	0.837878i	$0.683796\pi$
$948$	0	0
$949$	−3.08016	−0.0999863
$950$	−7.49464	−0.243158
$951$	0	0
$952$	8.18204	0.265181
$953$	38.7699	1.25588	0.627941	−	0.778261i	$-0.283898\pi$
$953$	38.7699	1.25588	0.627941	+	0.778261i	$0.283898\pi$
$954$	0	0
$955$	1.77491	0.0574349
$956$	−42.4592	−1.37323
$957$	0	0
$958$	30.9883	1.00118
$959$	21.4885	0.693899
$960$	0	0
$961$	−25.3139	−0.816577
$962$	−1.70040	−0.0548230
$963$	0	0
$964$	75.8322	2.44239
$965$	14.8795	0.478987
$966$	0	0
$967$	30.7994	0.990443	0.495222	−	0.868767i	$-0.335087\pi$
$967$	30.7994	0.990443	0.495222	+	0.868767i	$0.335087\pi$
$968$	96.2900	3.09488
$969$	0	0
$970$	44.3336	1.42347
$971$	46.4695	1.49128	0.745638	−	0.666351i	$-0.232145\pi$
$971$	46.4695	1.49128	0.745638	+	0.666351i	$0.232145\pi$
$972$	0	0
$973$	48.1814	1.54462
$974$	−91.5354	−2.93298
$975$	0	0
$976$	22.1684	0.709593
$977$	−6.48483	−0.207468	−0.103734	−	0.994605i	$-0.533079\pi$
$977$	−6.48483	−0.207468	−0.103734	+	0.994605i	$0.533079\pi$
$978$	0	0
$979$	−74.7070	−2.38765
$980$	9.80264	0.313134
$981$	0	0
$982$	40.1578	1.28149
$983$	6.97104	0.222342	0.111171	−	0.993801i	$-0.464540\pi$
$983$	6.97104	0.222342	0.111171	+	0.993801i	$0.464540\pi$
$984$	0	0
$985$	22.5766	0.719351
$986$	14.0900	0.448716
$987$	0	0
$988$	1.17797	0.0374761
$989$	−10.9997	−0.349770
$990$	0	0
$991$	4.83394	0.153555	0.0767776	−	0.997048i	$-0.475537\pi$
$991$	4.83394	0.153555	0.0767776	+	0.997048i	$0.475537\pi$
$992$	1.47938	0.0469702
$993$	0	0
$994$	−65.7116	−2.08424
$995$	6.43431	0.203981
$996$	0	0
$997$	−23.4485	−0.742621	−0.371310	−	0.928509i	$-0.621091\pi$
$997$	−23.4485	−0.742621	−0.371310	+	0.928509i	$0.621091\pi$
$998$	46.3108	1.46594
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.m.1.12		12
3.2	odd	2		893.2.a.a.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.a.1.1	✓	12	3.2	odd	2
8037.2.a.m.1.12		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)

\(f(q)\)	\(=\)	\(q+2.46212 q^{2} +4.06203 q^{4} +1.39858 q^{5} -2.95389 q^{7} +5.07697 q^{8} +O(q^{10})\)	\(q+2.46212 q^{2} +4.06203 q^{4} +1.39858 q^{5} -2.95389 q^{7} +5.07697 q^{8} +3.44347 q^{10} -5.47412 q^{11} +0.289994 q^{13} -7.27284 q^{14} +4.37604 q^{16} -0.545585 q^{17} +1.00000 q^{19} +5.68107 q^{20} -13.4779 q^{22} +2.62789 q^{23} -3.04398 q^{25} +0.714001 q^{26} -11.9988 q^{28} -10.4891 q^{29} +2.38456 q^{31} +0.620398 q^{32} -1.34329 q^{34} -4.13125 q^{35} -2.38151 q^{37} +2.46212 q^{38} +7.10054 q^{40} +4.74360 q^{41} -4.18576 q^{43} -22.2361 q^{44} +6.47017 q^{46} -1.00000 q^{47} +1.72549 q^{49} -7.49464 q^{50} +1.17797 q^{52} -11.1376 q^{53} -7.65600 q^{55} -14.9968 q^{56} -25.8254 q^{58} -6.78274 q^{59} +5.06585 q^{61} +5.87107 q^{62} -7.22459 q^{64} +0.405580 q^{65} +1.98930 q^{67} -2.21618 q^{68} -10.1716 q^{70} +9.03520 q^{71} -10.6215 q^{73} -5.86355 q^{74} +4.06203 q^{76} +16.1700 q^{77} -17.1441 q^{79} +6.12024 q^{80} +11.6793 q^{82} +11.4200 q^{83} -0.763043 q^{85} -10.3058 q^{86} -27.7920 q^{88} +13.6473 q^{89} -0.856613 q^{91} +10.6746 q^{92} -2.46212 q^{94} +1.39858 q^{95} +12.8747 q^{97} +4.24837 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) \) 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

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