LMFDB - Embedded newform 8036.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + \cdots + 3381 $ x^15 - x^14 - 37x^13 + 39x^12 + 537x^11 - 616x^10 - 3853x^9 + 4929x^8 + 13971x^7 - 20311x^6 - 22309x^5 + 38415x^4 + 8429x^3 - 22584x^2 - 399*x + 3381
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.24213$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.24213 q^{3} -1.49679 q^{5} +7.51143 q^{9} +O(q^{10})$	$q-3.24213 q^{3} -1.49679 q^{5} +7.51143 q^{9} -2.60103 q^{11} -2.15963 q^{13} +4.85278 q^{15} -7.81895 q^{17} +4.98834 q^{19} -5.00028 q^{23} -2.75963 q^{25} -14.6267 q^{27} +0.404401 q^{29} -1.13325 q^{31} +8.43288 q^{33} -1.67690 q^{37} +7.00181 q^{39} +1.00000 q^{41} -0.0986445 q^{43} -11.2430 q^{45} -6.72337 q^{47} +25.3501 q^{51} +13.0309 q^{53} +3.89319 q^{55} -16.1729 q^{57} -15.1985 q^{59} +1.87235 q^{61} +3.23251 q^{65} -4.60485 q^{67} +16.2116 q^{69} -1.21035 q^{71} -3.16046 q^{73} +8.94708 q^{75} +12.5713 q^{79} +24.8873 q^{81} -12.0343 q^{83} +11.7033 q^{85} -1.31112 q^{87} -12.3299 q^{89} +3.67414 q^{93} -7.46649 q^{95} -16.3259 q^{97} -19.5374 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * q^99

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	0
$2$	0	0
$3$	−3.24213	−1.87185	−0.935923	−	0.352204i	$-0.885432\pi$
$3$	−3.24213	−1.87185	−0.935923	+	0.352204i	$0.885432\pi$
$4$	0	0
$5$	−1.49679	−0.669384	−0.334692	−	0.942328i	$-0.608632\pi$
$5$	−1.49679	−0.669384	−0.334692	+	0.942328i	$0.608632\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.51143	2.50381
$10$	0	0
$11$	−2.60103	−0.784240	−0.392120	−	0.919914i	$-0.628258\pi$
$11$	−2.60103	−0.784240	−0.392120	+	0.919914i	$0.628258\pi$
$12$	0	0
$13$	−2.15963	−0.598974	−0.299487	−	0.954100i	$-0.596815\pi$
$13$	−2.15963	−0.598974	−0.299487	+	0.954100i	$0.596815\pi$
$14$	0	0
$15$	4.85278	1.25298
$16$	0	0
$17$	−7.81895	−1.89637	−0.948187	−	0.317712i	$-0.897085\pi$
$17$	−7.81895	−1.89637	−0.948187	+	0.317712i	$0.897085\pi$
$18$	0	0
$19$	4.98834	1.14440	0.572202	−	0.820113i	$-0.306089\pi$
$19$	4.98834	1.14440	0.572202	+	0.820113i	$0.306089\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.00028	−1.04263	−0.521315	−	0.853364i	$-0.674558\pi$
$23$	−5.00028	−1.04263	−0.521315	+	0.853364i	$0.674558\pi$
$24$	0	0
$25$	−2.75963	−0.551926
$26$	0	0
$27$	−14.6267	−2.81490
$28$	0	0
$29$	0.404401	0.0750953	0.0375476	−	0.999295i	$-0.488045\pi$
$29$	0.404401	0.0750953	0.0375476	+	0.999295i	$0.488045\pi$
$30$	0	0
$31$	−1.13325	−0.203537	−0.101769	−	0.994808i	$-0.532450\pi$
$31$	−1.13325	−0.203537	−0.101769	+	0.994808i	$0.532450\pi$
$32$	0	0
$33$	8.43288	1.46798
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.67690	−0.275680	−0.137840	−	0.990454i	$-0.544016\pi$
$37$	−1.67690	−0.275680	−0.137840	+	0.990454i	$0.544016\pi$
$38$	0	0
$39$	7.00181	1.12119
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−0.0986445	−0.0150431	−0.00752157	−	0.999972i	$-0.502394\pi$
$43$	−0.0986445	−0.0150431	−0.00752157	+	0.999972i	$0.502394\pi$
$44$	0	0
$45$	−11.2430	−1.67601
$46$	0	0
$47$	−6.72337	−0.980704	−0.490352	−	0.871524i	$-0.663132\pi$
$47$	−6.72337	−0.980704	−0.490352	+	0.871524i	$0.663132\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	25.3501	3.54972
$52$	0	0
$53$	13.0309	1.78993	0.894964	−	0.446138i	$-0.147201\pi$
$53$	13.0309	1.78993	0.894964	+	0.446138i	$0.147201\pi$
$54$	0	0
$55$	3.89319	0.524957
$56$	0	0
$57$	−16.1729	−2.14215
$58$	0	0
$59$	−15.1985	−1.97868	−0.989341	−	0.145618i	$-0.953483\pi$
$59$	−15.1985	−1.97868	−0.989341	+	0.145618i	$0.953483\pi$
$60$	0	0
$61$	1.87235	0.239730	0.119865	−	0.992790i	$-0.461754\pi$
$61$	1.87235	0.239730	0.119865	+	0.992790i	$0.461754\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.23251	0.400943
$66$	0	0
$67$	−4.60485	−0.562572	−0.281286	−	0.959624i	$-0.590761\pi$
$67$	−4.60485	−0.562572	−0.281286	+	0.959624i	$0.590761\pi$
$68$	0	0
$69$	16.2116	1.95164
$70$	0	0
$71$	−1.21035	−0.143642	−0.0718212	−	0.997418i	$-0.522881\pi$
$71$	−1.21035	−0.143642	−0.0718212	+	0.997418i	$0.522881\pi$
$72$	0	0
$73$	−3.16046	−0.369904	−0.184952	−	0.982748i	$-0.559213\pi$
$73$	−3.16046	−0.369904	−0.184952	+	0.982748i	$0.559213\pi$
$74$	0	0
$75$	8.94708	1.03312
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.5713	1.41439	0.707193	−	0.707020i	$-0.249961\pi$
$79$	12.5713	1.41439	0.707193	+	0.707020i	$0.249961\pi$
$80$	0	0
$81$	24.8873	2.76525
$82$	0	0
$83$	−12.0343	−1.32094	−0.660469	−	0.750853i	$-0.729643\pi$
$83$	−12.0343	−1.32094	−0.660469	+	0.750853i	$0.729643\pi$
$84$	0	0
$85$	11.7033	1.26940
$86$	0	0
$87$	−1.31112	−0.140567
$88$	0	0
$89$	−12.3299	−1.30696	−0.653481	−	0.756943i	$-0.726692\pi$
$89$	−12.3299	−1.30696	−0.653481	+	0.756943i	$0.726692\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.67414	0.380990
$94$	0	0
$95$	−7.46649	−0.766045
$96$	0	0
$97$	−16.3259	−1.65764	−0.828822	−	0.559512i	$-0.810988\pi$
$97$	−16.3259	−1.65764	−0.828822	+	0.559512i	$0.810988\pi$
$98$	0	0
$99$	−19.5374	−1.96359
$100$	0	0
$101$	−13.1826	−1.31172	−0.655859	−	0.754883i	$-0.727694\pi$
$101$	−13.1826	−1.31172	−0.655859	+	0.754883i	$0.727694\pi$
$102$	0	0
$103$	2.41497	0.237954	0.118977	−	0.992897i	$-0.462039\pi$
$103$	2.41497	0.237954	0.118977	+	0.992897i	$0.462039\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.9079	−1.73122	−0.865609	−	0.500721i	$-0.833068\pi$
$107$	−17.9079	−1.73122	−0.865609	+	0.500721i	$0.833068\pi$
$108$	0	0
$109$	11.0933	1.06255	0.531273	−	0.847201i	$-0.321714\pi$
$109$	11.0933	1.06255	0.531273	+	0.847201i	$0.321714\pi$
$110$	0	0
$111$	5.43673	0.516031
$112$	0	0
$113$	−4.48247	−0.421675	−0.210838	−	0.977521i	$-0.567619\pi$
$113$	−4.48247	−0.421675	−0.210838	+	0.977521i	$0.567619\pi$
$114$	0	0
$115$	7.48436	0.697920
$116$	0	0
$117$	−16.2219	−1.49972
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.23465	−0.384968
$122$	0	0
$123$	−3.24213	−0.292333
$124$	0	0
$125$	11.6145	1.03883
$126$	0	0
$127$	−13.7744	−1.22228	−0.611142	−	0.791521i	$-0.709290\pi$
$127$	−13.7744	−1.22228	−0.611142	+	0.791521i	$0.709290\pi$
$128$	0	0
$129$	0.319818	0.0281584
$130$	0	0
$131$	3.79644	0.331697	0.165848	−	0.986151i	$-0.446964\pi$
$131$	3.79644	0.331697	0.165848	+	0.986151i	$0.446964\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	21.8930	1.88425
$136$	0	0
$137$	−0.00898688	−0.000767801 0	−0.000383900	−	1.00000i	$-0.500122\pi$
$137$	−0.00898688	−0.000767801 0	−0.000383900		1.00000i	$0.500122\pi$
$138$	0	0
$139$	1.05225	0.0892507	0.0446254	−	0.999004i	$-0.485791\pi$
$139$	1.05225	0.0892507	0.0446254	+	0.999004i	$0.485791\pi$
$140$	0	0
$141$	21.7981	1.83573
$142$	0	0
$143$	5.61726	0.469739
$144$	0	0
$145$	−0.605302	−0.0502676
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.58959	−0.457917	−0.228959	−	0.973436i	$-0.573532\pi$
$149$	−5.58959	−0.457917	−0.228959	+	0.973436i	$0.573532\pi$
$150$	0	0
$151$	−20.8788	−1.69909	−0.849546	−	0.527515i	$-0.823124\pi$
$151$	−20.8788	−1.69909	−0.849546	+	0.527515i	$0.823124\pi$
$152$	0	0
$153$	−58.7315	−4.74816
$154$	0	0
$155$	1.69623	0.136244
$156$	0	0
$157$	13.7208	1.09504	0.547521	−	0.836792i	$-0.315572\pi$
$157$	13.7208	1.09504	0.547521	+	0.836792i	$0.315572\pi$
$158$	0	0
$159$	−42.2478	−3.35047
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−24.0930	−1.88711	−0.943556	−	0.331214i	$-0.892542\pi$
$163$	−24.0930	−1.88711	−0.943556	+	0.331214i	$0.892542\pi$
$164$	0	0
$165$	−12.6222	−0.982639
$166$	0	0
$167$	−5.46141	−0.422616	−0.211308	−	0.977419i	$-0.567772\pi$
$167$	−5.46141	−0.422616	−0.211308	+	0.977419i	$0.567772\pi$
$168$	0	0
$169$	−8.33600	−0.641230
$170$	0	0
$171$	37.4696	2.86537
$172$	0	0
$173$	16.9973	1.29228	0.646139	−	0.763220i	$-0.276383\pi$
$173$	16.9973	1.29228	0.646139	+	0.763220i	$0.276383\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	49.2757	3.70379
$178$	0	0
$179$	−9.11221	−0.681078	−0.340539	−	0.940230i	$-0.610610\pi$
$179$	−9.11221	−0.681078	−0.340539	+	0.940230i	$0.610610\pi$
$180$	0	0
$181$	4.21950	0.313633	0.156816	−	0.987628i	$-0.449877\pi$
$181$	4.21950	0.313633	0.156816	+	0.987628i	$0.449877\pi$
$182$	0	0
$183$	−6.07040	−0.448737
$184$	0	0
$185$	2.50996	0.184536
$186$	0	0
$187$	20.3373	1.48721
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.16197	−0.518222	−0.259111	−	0.965848i	$-0.583430\pi$
$191$	−7.16197	−0.518222	−0.259111	+	0.965848i	$0.583430\pi$
$192$	0	0
$193$	−1.15715	−0.0832934	−0.0416467	−	0.999132i	$-0.513260\pi$
$193$	−1.15715	−0.0832934	−0.0416467	+	0.999132i	$0.513260\pi$
$194$	0	0
$195$	−10.4802	−0.750504
$196$	0	0
$197$	−2.30106	−0.163944	−0.0819720	−	0.996635i	$-0.526122\pi$
$197$	−2.30106	−0.163944	−0.0819720	+	0.996635i	$0.526122\pi$
$198$	0	0
$199$	−17.1511	−1.21581	−0.607905	−	0.794010i	$-0.707990\pi$
$199$	−17.1511	−1.21581	−0.607905	+	0.794010i	$0.707990\pi$
$200$	0	0
$201$	14.9295	1.05305
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.49679	−0.104540
$206$	0	0
$207$	−37.5592	−2.61055
$208$	0	0
$209$	−12.9748	−0.897487
$210$	0	0
$211$	6.24067	0.429625	0.214813	−	0.976655i	$-0.431086\pi$
$211$	6.24067	0.429625	0.214813	+	0.976655i	$0.431086\pi$
$212$	0	0
$213$	3.92412	0.268877
$214$	0	0
$215$	0.147650	0.0100696
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.2466	0.692403
$220$	0	0
$221$	16.8860	1.13588
$222$	0	0
$223$	5.45026	0.364976	0.182488	−	0.983208i	$-0.441585\pi$
$223$	5.45026	0.364976	0.182488	+	0.983208i	$0.441585\pi$
$224$	0	0
$225$	−20.7287	−1.38192
$226$	0	0
$227$	11.3106	0.750709	0.375354	−	0.926881i	$-0.377521\pi$
$227$	11.3106	0.750709	0.375354	+	0.926881i	$0.377521\pi$
$228$	0	0
$229$	6.96496	0.460257	0.230129	−	0.973160i	$-0.426085\pi$
$229$	6.96496	0.460257	0.230129	+	0.973160i	$0.426085\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.40261	0.550474	0.275237	−	0.961376i	$-0.411244\pi$
$233$	8.40261	0.550474	0.275237	+	0.961376i	$0.411244\pi$
$234$	0	0
$235$	10.0635	0.656467
$236$	0	0
$237$	−40.7580	−2.64751
$238$	0	0
$239$	−8.49866	−0.549732	−0.274866	−	0.961482i	$-0.588634\pi$
$239$	−8.49866	−0.549732	−0.274866	+	0.961482i	$0.588634\pi$
$240$	0	0
$241$	−2.87685	−0.185314	−0.0926572	−	0.995698i	$-0.529536\pi$
$241$	−2.87685	−0.185314	−0.0926572	+	0.995698i	$0.529536\pi$
$242$	0	0
$243$	−36.8079	−2.36123
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.7730	−0.685468
$248$	0	0
$249$	39.0169	2.47259
$250$	0	0
$251$	−10.3806	−0.655217	−0.327608	−	0.944814i	$-0.606243\pi$
$251$	−10.3806	−0.655217	−0.327608	+	0.944814i	$0.606243\pi$
$252$	0	0
$253$	13.0059	0.817672
$254$	0	0
$255$	−37.9437	−2.37613
$256$	0	0
$257$	−15.1372	−0.944233	−0.472117	−	0.881536i	$-0.656510\pi$
$257$	−15.1372	−0.944233	−0.472117	+	0.881536i	$0.656510\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.03763	0.188024
$262$	0	0
$263$	−21.5130	−1.32655	−0.663276	−	0.748375i	$-0.730834\pi$
$263$	−21.5130	−1.32655	−0.663276	+	0.748375i	$0.730834\pi$
$264$	0	0
$265$	−19.5044	−1.19815
$266$	0	0
$267$	39.9750	2.44643
$268$	0	0
$269$	20.8515	1.27134	0.635668	−	0.771962i	$-0.280725\pi$
$269$	20.8515	1.27134	0.635668	+	0.771962i	$0.280725\pi$
$270$	0	0
$271$	31.0705	1.88740	0.943700	−	0.330801i	$-0.107319\pi$
$271$	31.0705	1.88740	0.943700	+	0.330801i	$0.107319\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.17787	0.432842
$276$	0	0
$277$	−15.1088	−0.907801	−0.453901	−	0.891052i	$-0.649968\pi$
$277$	−15.1088	−0.907801	−0.453901	+	0.891052i	$0.649968\pi$
$278$	0	0
$279$	−8.51230	−0.509618
$280$	0	0
$281$	−2.60682	−0.155510	−0.0777548	−	0.996973i	$-0.524775\pi$
$281$	−2.60682	−0.155510	−0.0777548	+	0.996973i	$0.524775\pi$
$282$	0	0
$283$	−9.81175	−0.583248	−0.291624	−	0.956533i	$-0.594196\pi$
$283$	−9.81175	−0.583248	−0.291624	+	0.956533i	$0.594196\pi$
$284$	0	0
$285$	24.2073	1.43392
$286$	0	0
$287$	0	0
$288$	0	0
$289$	44.1360	2.59624
$290$	0	0
$291$	52.9307	3.10286
$292$	0	0
$293$	13.7070	0.800769	0.400384	−	0.916347i	$-0.368877\pi$
$293$	13.7070	0.800769	0.400384	+	0.916347i	$0.368877\pi$
$294$	0	0
$295$	22.7490	1.32450
$296$	0	0
$297$	38.0443	2.20756
$298$	0	0
$299$	10.7988	0.624508
$300$	0	0
$301$	0	0
$302$	0	0
$303$	42.7398	2.45534
$304$	0	0
$305$	−2.80251	−0.160471
$306$	0	0
$307$	−21.2774	−1.21436	−0.607182	−	0.794563i	$-0.707700\pi$
$307$	−21.2774	−1.21436	−0.607182	+	0.794563i	$0.707700\pi$
$308$	0	0
$309$	−7.82965	−0.445413
$310$	0	0
$311$	22.8754	1.29714	0.648571	−	0.761154i	$-0.275367\pi$
$311$	22.8754	1.29714	0.648571	+	0.761154i	$0.275367\pi$
$312$	0	0
$313$	33.8192	1.91157	0.955787	−	0.294060i	$-0.0950066\pi$
$313$	33.8192	1.91157	0.955787	+	0.294060i	$0.0950066\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.8336	0.833138	0.416569	−	0.909104i	$-0.363232\pi$
$317$	14.8336	0.833138	0.416569	+	0.909104i	$0.363232\pi$
$318$	0	0
$319$	−1.05186	−0.0588927
$320$	0	0
$321$	58.0597	3.24057
$322$	0	0
$323$	−39.0036	−2.17022
$324$	0	0
$325$	5.95978	0.330589
$326$	0	0
$327$	−35.9660	−1.98892
$328$	0	0
$329$	0	0
$330$	0	0
$331$	29.3364	1.61247	0.806236	−	0.591594i	$-0.201501\pi$
$331$	29.3364	1.61247	0.806236	+	0.591594i	$0.201501\pi$
$332$	0	0
$333$	−12.5959	−0.690251
$334$	0	0
$335$	6.89248	0.376576
$336$	0	0
$337$	−25.8091	−1.40591	−0.702957	−	0.711232i	$-0.748137\pi$
$337$	−25.8091	−1.40591	−0.702957	+	0.711232i	$0.748137\pi$
$338$	0	0
$339$	14.5328	0.789311
$340$	0	0
$341$	2.94761	0.159622
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−24.2653	−1.30640
$346$	0	0
$347$	3.80579	0.204305	0.102153	−	0.994769i	$-0.467427\pi$
$347$	3.80579	0.204305	0.102153	+	0.994769i	$0.467427\pi$
$348$	0	0
$349$	−0.239149	−0.0128013	−0.00640067	−	0.999980i	$-0.502037\pi$
$349$	−0.239149	−0.0128013	−0.00640067	+	0.999980i	$0.502037\pi$
$350$	0	0
$351$	31.5882	1.68605
$352$	0	0
$353$	−4.72219	−0.251337	−0.125668	−	0.992072i	$-0.540108\pi$
$353$	−4.72219	−0.251337	−0.125668	+	0.992072i	$0.540108\pi$
$354$	0	0
$355$	1.81164	0.0961519
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.2266	−1.64807	−0.824037	−	0.566535i	$-0.808283\pi$
$359$	−31.2266	−1.64807	−0.824037	+	0.566535i	$0.808283\pi$
$360$	0	0
$361$	5.88355	0.309660
$362$	0	0
$363$	13.7293	0.720602
$364$	0	0
$365$	4.73053	0.247607
$366$	0	0
$367$	26.0787	1.36130	0.680649	−	0.732610i	$-0.261698\pi$
$367$	26.0787	1.36130	0.680649	+	0.732610i	$0.261698\pi$
$368$	0	0
$369$	7.51143	0.391029
$370$	0	0
$371$	0	0
$372$	0	0
$373$	8.91964	0.461841	0.230921	−	0.972973i	$-0.425826\pi$
$373$	8.91964	0.461841	0.230921	+	0.972973i	$0.425826\pi$
$374$	0	0
$375$	−37.6558	−1.94454
$376$	0	0
$377$	−0.873356	−0.0449801
$378$	0	0
$379$	10.9678	0.563380	0.281690	−	0.959506i	$-0.409105\pi$
$379$	10.9678	0.563380	0.281690	+	0.959506i	$0.409105\pi$
$380$	0	0
$381$	44.6586	2.28793
$382$	0	0
$383$	−21.5001	−1.09860	−0.549302	−	0.835624i	$-0.685106\pi$
$383$	−21.5001	−1.09860	−0.549302	+	0.835624i	$0.685106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.740961	−0.0376652
$388$	0	0
$389$	−16.3029	−0.826589	−0.413295	−	0.910597i	$-0.635622\pi$
$389$	−16.3029	−0.826589	−0.413295	+	0.910597i	$0.635622\pi$
$390$	0	0
$391$	39.0969	1.97722
$392$	0	0
$393$	−12.3086	−0.620885
$394$	0	0
$395$	−18.8166	−0.946767
$396$	0	0
$397$	−21.9023	−1.09925	−0.549623	−	0.835413i	$-0.685229\pi$
$397$	−21.9023	−1.09925	−0.549623	+	0.835413i	$0.685229\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.9606	0.597284	0.298642	−	0.954365i	$-0.403466\pi$
$401$	11.9606	0.597284	0.298642	+	0.954365i	$0.403466\pi$
$402$	0	0
$403$	2.44739	0.121913
$404$	0	0
$405$	−37.2509	−1.85101
$406$	0	0
$407$	4.36166	0.216199
$408$	0	0
$409$	−30.0777	−1.48725	−0.743624	−	0.668598i	$-0.766895\pi$
$409$	−30.0777	−1.48725	−0.743624	+	0.668598i	$0.766895\pi$
$410$	0	0
$411$	0.0291367	0.00143721
$412$	0	0
$413$	0	0
$414$	0	0
$415$	18.0128	0.884215
$416$	0	0
$417$	−3.41154	−0.167064
$418$	0	0
$419$	12.5534	0.613273	0.306636	−	0.951827i	$-0.400796\pi$
$419$	12.5534	0.613273	0.306636	+	0.951827i	$0.400796\pi$
$420$	0	0
$421$	30.0470	1.46440	0.732200	−	0.681090i	$-0.238494\pi$
$421$	30.0470	1.46440	0.732200	+	0.681090i	$0.238494\pi$
$422$	0	0
$423$	−50.5021	−2.45550
$424$	0	0
$425$	21.5774	1.04666
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−18.2119	−0.879279
$430$	0	0
$431$	−24.3805	−1.17437	−0.587185	−	0.809453i	$-0.699764\pi$
$431$	−24.3805	−1.17437	−0.587185	+	0.809453i	$0.699764\pi$
$432$	0	0
$433$	11.9120	0.572455	0.286228	−	0.958162i	$-0.407599\pi$
$433$	11.9120	0.572455	0.286228	+	0.958162i	$0.407599\pi$
$434$	0	0
$435$	1.96247	0.0940932
$436$	0	0
$437$	−24.9431	−1.19319
$438$	0	0
$439$	−11.1595	−0.532614	−0.266307	−	0.963888i	$-0.585804\pi$
$439$	−11.1595	−0.532614	−0.266307	+	0.963888i	$0.585804\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.6578	−0.791434	−0.395717	−	0.918372i	$-0.629504\pi$
$443$	−16.6578	−0.791434	−0.395717	+	0.918372i	$0.629504\pi$
$444$	0	0
$445$	18.4552	0.874859
$446$	0	0
$447$	18.1222	0.857151
$448$	0	0
$449$	32.0634	1.51317	0.756583	−	0.653897i	$-0.226867\pi$
$449$	32.0634	1.51317	0.756583	+	0.653897i	$0.226867\pi$
$450$	0	0
$451$	−2.60103	−0.122478
$452$	0	0
$453$	67.6918	3.18044
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.51609	0.304810	0.152405	−	0.988318i	$-0.451298\pi$
$457$	6.51609	0.304810	0.152405	+	0.988318i	$0.451298\pi$
$458$	0	0
$459$	114.365	5.33810
$460$	0	0
$461$	11.0378	0.514082	0.257041	−	0.966400i	$-0.417252\pi$
$461$	11.0378	0.514082	0.257041	+	0.966400i	$0.417252\pi$
$462$	0	0
$463$	25.7136	1.19501	0.597507	−	0.801864i	$-0.296158\pi$
$463$	25.7136	1.19501	0.597507	+	0.801864i	$0.296158\pi$
$464$	0	0
$465$	−5.49940	−0.255029
$466$	0	0
$467$	−21.9608	−1.01623	−0.508113	−	0.861291i	$-0.669657\pi$
$467$	−21.9608	−1.01623	−0.508113	+	0.861291i	$0.669657\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−44.4848	−2.04975
$472$	0	0
$473$	0.256577	0.0117974
$474$	0	0
$475$	−13.7660	−0.631626
$476$	0	0
$477$	97.8805	4.48164
$478$	0	0
$479$	15.4932	0.707901	0.353951	−	0.935264i	$-0.384838\pi$
$479$	15.4932	0.707901	0.353951	+	0.935264i	$0.384838\pi$
$480$	0	0
$481$	3.62148	0.165125
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.4364	1.10960
$486$	0	0
$487$	34.0797	1.54430	0.772150	−	0.635440i	$-0.219181\pi$
$487$	34.0797	1.54430	0.772150	+	0.635440i	$0.219181\pi$
$488$	0	0
$489$	78.1128	3.53238
$490$	0	0
$491$	−0.00976244	−0.000440573 0	−0.000220286	−	1.00000i	$-0.500070\pi$
$491$	−0.00976244	−0.000440573 0	−0.000220286		1.00000i	$0.500070\pi$
$492$	0	0
$493$	−3.16199	−0.142409
$494$	0	0
$495$	29.2434	1.31439
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−21.9496	−0.982599	−0.491300	−	0.870991i	$-0.663478\pi$
$499$	−21.9496	−0.982599	−0.491300	+	0.870991i	$0.663478\pi$
$500$	0	0
$501$	17.7066	0.791073
$502$	0	0
$503$	−20.0786	−0.895260	−0.447630	−	0.894219i	$-0.647732\pi$
$503$	−20.0786	−0.895260	−0.447630	+	0.894219i	$0.647732\pi$
$504$	0	0
$505$	19.7316	0.878043
$506$	0	0
$507$	27.0264	1.20029
$508$	0	0
$509$	1.84080	0.0815922	0.0407961	−	0.999167i	$-0.487011\pi$
$509$	1.84080	0.0815922	0.0407961	+	0.999167i	$0.487011\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−72.9627	−3.22138
$514$	0	0
$515$	−3.61469	−0.159282
$516$	0	0
$517$	17.4877	0.769107
$518$	0	0
$519$	−55.1074	−2.41895
$520$	0	0
$521$	−5.94737	−0.260559	−0.130279	−	0.991477i	$-0.541587\pi$
$521$	−5.94737	−0.260559	−0.130279	+	0.991477i	$0.541587\pi$
$522$	0	0
$523$	9.87253	0.431696	0.215848	−	0.976427i	$-0.430748\pi$
$523$	9.87253	0.431696	0.215848	+	0.976427i	$0.430748\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.86080	0.385982
$528$	0	0
$529$	2.00280	0.0870785
$530$	0	0
$531$	−114.163	−4.95424
$532$	0	0
$533$	−2.15963	−0.0935440
$534$	0	0
$535$	26.8043	1.15885
$536$	0	0
$537$	29.5430	1.27487
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.60499	−0.154991	−0.0774954	−	0.996993i	$-0.524692\pi$
$541$	−3.60499	−0.154991	−0.0774954	+	0.996993i	$0.524692\pi$
$542$	0	0
$543$	−13.6802	−0.587072
$544$	0	0
$545$	−16.6043	−0.711251
$546$	0	0
$547$	38.4142	1.64247	0.821235	−	0.570590i	$-0.193285\pi$
$547$	38.4142	1.64247	0.821235	+	0.570590i	$0.193285\pi$
$548$	0	0
$549$	14.0640	0.600237
$550$	0	0
$551$	2.01729	0.0859394
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−8.13763	−0.345423
$556$	0	0
$557$	41.9327	1.77674	0.888372	−	0.459123i	$-0.151836\pi$
$557$	41.9327	1.77674	0.888372	+	0.459123i	$0.151836\pi$
$558$	0	0
$559$	0.213036	0.00901044
$560$	0	0
$561$	−65.9363	−2.78383
$562$	0	0
$563$	−13.0546	−0.550186	−0.275093	−	0.961418i	$-0.588709\pi$
$563$	−13.0546	−0.550186	−0.275093	+	0.961418i	$0.588709\pi$
$564$	0	0
$565$	6.70930	0.282262
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.77130	0.409634	0.204817	−	0.978800i	$-0.434340\pi$
$569$	9.77130	0.409634	0.204817	+	0.978800i	$0.434340\pi$
$570$	0	0
$571$	−30.9066	−1.29340	−0.646701	−	0.762744i	$-0.723852\pi$
$571$	−30.9066	−1.29340	−0.646701	+	0.762744i	$0.723852\pi$
$572$	0	0
$573$	23.2201	0.970033
$574$	0	0
$575$	13.7989	0.575455
$576$	0	0
$577$	32.3793	1.34797	0.673983	−	0.738747i	$-0.264582\pi$
$577$	32.3793	1.34797	0.673983	+	0.738747i	$0.264582\pi$
$578$	0	0
$579$	3.75163	0.155912
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−33.8937	−1.40373
$584$	0	0
$585$	24.2807	1.00389
$586$	0	0
$587$	40.3400	1.66501	0.832506	−	0.554016i	$-0.186905\pi$
$587$	40.3400	1.66501	0.832506	+	0.554016i	$0.186905\pi$
$588$	0	0
$589$	−5.65302	−0.232929
$590$	0	0
$591$	7.46035	0.306878
$592$	0	0
$593$	7.24357	0.297458	0.148729	−	0.988878i	$-0.452482\pi$
$593$	7.24357	0.297458	0.148729	+	0.988878i	$0.452482\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	55.6062	2.27581
$598$	0	0
$599$	−32.0113	−1.30794	−0.653972	−	0.756519i	$-0.726899\pi$
$599$	−32.0113	−1.30794	−0.653972	+	0.756519i	$0.726899\pi$
$600$	0	0
$601$	30.6709	1.25109	0.625546	−	0.780187i	$-0.284876\pi$
$601$	30.6709	1.25109	0.625546	+	0.780187i	$0.284876\pi$
$602$	0	0
$603$	−34.5890	−1.40857
$604$	0	0
$605$	6.33837	0.257692
$606$	0	0
$607$	−40.0906	−1.62723	−0.813615	−	0.581405i	$-0.802503\pi$
$607$	−40.0906	−1.62723	−0.813615	+	0.581405i	$0.802503\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	14.5200	0.587416
$612$	0	0
$613$	−11.3707	−0.459258	−0.229629	−	0.973278i	$-0.573751\pi$
$613$	−11.3707	−0.459258	−0.229629	+	0.973278i	$0.573751\pi$
$614$	0	0
$615$	4.85278	0.195683
$616$	0	0
$617$	−5.51261	−0.221929	−0.110965	−	0.993824i	$-0.535394\pi$
$617$	−5.51261	−0.221929	−0.110965	+	0.993824i	$0.535394\pi$
$618$	0	0
$619$	21.2191	0.852869	0.426434	−	0.904519i	$-0.359770\pi$
$619$	21.2191	0.852869	0.426434	+	0.904519i	$0.359770\pi$
$620$	0	0
$621$	73.1374	2.93490
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−3.58631	−0.143453
$626$	0	0
$627$	42.0661	1.67996
$628$	0	0
$629$	13.1116	0.522793
$630$	0	0
$631$	−18.1401	−0.722147	−0.361073	−	0.932537i	$-0.617590\pi$
$631$	−18.1401	−0.722147	−0.361073	+	0.932537i	$0.617590\pi$
$632$	0	0
$633$	−20.2331	−0.804193
$634$	0	0
$635$	20.6174	0.818177
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−9.09148	−0.359653
$640$	0	0
$641$	−9.53870	−0.376756	−0.188378	−	0.982097i	$-0.560323\pi$
$641$	−9.53870	−0.376756	−0.188378	+	0.982097i	$0.560323\pi$
$642$	0	0
$643$	28.3896	1.11958	0.559788	−	0.828636i	$-0.310883\pi$
$643$	28.3896	1.11958	0.559788	+	0.828636i	$0.310883\pi$
$644$	0	0
$645$	−0.478700	−0.0188488
$646$	0	0
$647$	47.6272	1.87242	0.936208	−	0.351445i	$-0.114310\pi$
$647$	47.6272	1.87242	0.936208	+	0.351445i	$0.114310\pi$
$648$	0	0
$649$	39.5318	1.55176
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.5006	1.11532	0.557658	−	0.830071i	$-0.311700\pi$
$653$	28.5006	1.11532	0.557658	+	0.830071i	$0.311700\pi$
$654$	0	0
$655$	−5.68246	−0.222032
$656$	0	0
$657$	−23.7396	−0.926168
$658$	0	0
$659$	42.8168	1.66791	0.833954	−	0.551835i	$-0.186072\pi$
$659$	42.8168	1.66791	0.833954	+	0.551835i	$0.186072\pi$
$660$	0	0
$661$	32.7509	1.27386	0.636931	−	0.770921i	$-0.280204\pi$
$661$	32.7509	1.27386	0.636931	+	0.770921i	$0.280204\pi$
$662$	0	0
$663$	−54.7468	−2.12619
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.02212	−0.0782966
$668$	0	0
$669$	−17.6705	−0.683180
$670$	0	0
$671$	−4.87003	−0.188005
$672$	0	0
$673$	−12.7103	−0.489944	−0.244972	−	0.969530i	$-0.578779\pi$
$673$	−12.7103	−0.489944	−0.244972	+	0.969530i	$0.578779\pi$
$674$	0	0
$675$	40.3641	1.55362
$676$	0	0
$677$	−2.17361	−0.0835388	−0.0417694	−	0.999127i	$-0.513299\pi$
$677$	−2.17361	−0.0835388	−0.0417694	+	0.999127i	$0.513299\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−36.6704	−1.40521
$682$	0	0
$683$	−11.5806	−0.443119	−0.221559	−	0.975147i	$-0.571115\pi$
$683$	−11.5806	−0.443119	−0.221559	+	0.975147i	$0.571115\pi$
$684$	0	0
$685$	0.0134514	0.000513953 0
$686$	0	0
$687$	−22.5813	−0.861531
$688$	0	0
$689$	−28.1419	−1.07212
$690$	0	0
$691$	−38.9363	−1.48121	−0.740603	−	0.671943i	$-0.765460\pi$
$691$	−38.9363	−1.48121	−0.740603	+	0.671943i	$0.765460\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.57500	−0.0597430
$696$	0	0
$697$	−7.81895	−0.296164
$698$	0	0
$699$	−27.2424	−1.03040
$700$	0	0
$701$	−28.2623	−1.06745	−0.533726	−	0.845658i	$-0.679209\pi$
$701$	−28.2623	−1.06745	−0.533726	+	0.845658i	$0.679209\pi$
$702$	0	0
$703$	−8.36494	−0.315490
$704$	0	0
$705$	−32.6271	−1.22881
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10.6038	−0.398233	−0.199116	−	0.979976i	$-0.563807\pi$
$709$	−10.6038	−0.398233	−0.199116	+	0.979976i	$0.563807\pi$
$710$	0	0
$711$	94.4287	3.54135
$712$	0	0
$713$	5.66655	0.212214
$714$	0	0
$715$	−8.40784	−0.314435
$716$	0	0
$717$	27.5538	1.02901
$718$	0	0
$719$	33.3423	1.24346	0.621729	−	0.783232i	$-0.286430\pi$
$719$	33.3423	1.24346	0.621729	+	0.783232i	$0.286430\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	9.32714	0.346880
$724$	0	0
$725$	−1.11600	−0.0414470
$726$	0	0
$727$	−53.5627	−1.98653	−0.993265	−	0.115861i	$-0.963037\pi$
$727$	−53.5627	−1.98653	−0.993265	+	0.115861i	$0.963037\pi$
$728$	0	0
$729$	44.6743	1.65460
$730$	0	0
$731$	0.771296	0.0285274
$732$	0	0
$733$	−3.09276	−0.114234	−0.0571169	−	0.998367i	$-0.518191\pi$
$733$	−3.09276	−0.114234	−0.0571169	+	0.998367i	$0.518191\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.9773	0.441191
$738$	0	0
$739$	7.80935	0.287271	0.143636	−	0.989631i	$-0.454121\pi$
$739$	7.80935	0.287271	0.143636	+	0.989631i	$0.454121\pi$
$740$	0	0
$741$	34.9274	1.28309
$742$	0	0
$743$	16.2197	0.595042	0.297521	−	0.954715i	$-0.403840\pi$
$743$	16.2197	0.595042	0.297521	+	0.954715i	$0.403840\pi$
$744$	0	0
$745$	8.36643	0.306522
$746$	0	0
$747$	−90.3950	−3.30738
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−20.5274	−0.749056	−0.374528	−	0.927216i	$-0.622195\pi$
$751$	−20.5274	−0.749056	−0.374528	+	0.927216i	$0.622195\pi$
$752$	0	0
$753$	33.6552	1.22646
$754$	0	0
$755$	31.2511	1.13734
$756$	0	0
$757$	−7.72214	−0.280666	−0.140333	−	0.990104i	$-0.544817\pi$
$757$	−7.72214	−0.280666	−0.140333	+	0.990104i	$0.544817\pi$
$758$	0	0
$759$	−42.1668	−1.53056
$760$	0	0
$761$	−20.6779	−0.749574	−0.374787	−	0.927111i	$-0.622284\pi$
$761$	−20.6779	−0.749574	−0.374787	+	0.927111i	$0.622284\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	87.9085	3.17834
$766$	0	0
$767$	32.8232	1.18518
$768$	0	0
$769$	17.9006	0.645511	0.322756	−	0.946482i	$-0.395391\pi$
$769$	17.9006	0.645511	0.322756	+	0.946482i	$0.395391\pi$
$770$	0	0
$771$	49.0769	1.76746
$772$	0	0
$773$	3.94378	0.141848	0.0709240	−	0.997482i	$-0.477405\pi$
$773$	3.94378	0.141848	0.0709240	+	0.997482i	$0.477405\pi$
$774$	0	0
$775$	3.12734	0.112337
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.98834	0.178726
$780$	0	0
$781$	3.14816	0.112650
$782$	0	0
$783$	−5.91503	−0.211386
$784$	0	0
$785$	−20.5372	−0.733003
$786$	0	0
$787$	−15.1175	−0.538882	−0.269441	−	0.963017i	$-0.586839\pi$
$787$	−15.1175	−0.538882	−0.269441	+	0.963017i	$0.586839\pi$
$788$	0	0
$789$	69.7482	2.48310
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.04358	−0.143592
$794$	0	0
$795$	63.2360	2.24275
$796$	0	0
$797$	51.2221	1.81438	0.907190	−	0.420721i	$-0.138223\pi$
$797$	51.2221	1.81438	0.907190	+	0.420721i	$0.138223\pi$
$798$	0	0
$799$	52.5697	1.85978
$800$	0	0
$801$	−92.6148	−3.27238
$802$	0	0
$803$	8.22044	0.290093
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−67.6033	−2.37975
$808$	0	0
$809$	−29.2576	−1.02864	−0.514322	−	0.857597i	$-0.671956\pi$
$809$	−29.2576	−1.02864	−0.514322	+	0.857597i	$0.671956\pi$
$810$	0	0
$811$	42.6431	1.49740	0.748702	−	0.662907i	$-0.230678\pi$
$811$	42.6431	1.49740	0.748702	+	0.662907i	$0.230678\pi$
$812$	0	0
$813$	−100.735	−3.53292
$814$	0	0
$815$	36.0621	1.26320
$816$	0	0
$817$	−0.492072	−0.0172154
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14.3946	−0.502376	−0.251188	−	0.967938i	$-0.580821\pi$
$821$	−14.3946	−0.502376	−0.251188	+	0.967938i	$0.580821\pi$
$822$	0	0
$823$	−49.0112	−1.70842	−0.854211	−	0.519927i	$-0.825959\pi$
$823$	−49.0112	−1.70842	−0.854211	+	0.519927i	$0.825959\pi$
$824$	0	0
$825$	−23.2716	−0.810214
$826$	0	0
$827$	18.0400	0.627312	0.313656	−	0.949537i	$-0.398446\pi$
$827$	18.0400	0.627312	0.313656	+	0.949537i	$0.398446\pi$
$828$	0	0
$829$	12.9888	0.451120	0.225560	−	0.974229i	$-0.427579\pi$
$829$	12.9888	0.451120	0.225560	+	0.974229i	$0.427579\pi$
$830$	0	0
$831$	48.9848	1.69926
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8.17457	0.282892
$836$	0	0
$837$	16.5756	0.572937
$838$	0	0
$839$	20.9007	0.721571	0.360785	−	0.932649i	$-0.382509\pi$
$839$	20.9007	0.721571	0.360785	+	0.932649i	$0.382509\pi$
$840$	0	0
$841$	−28.8365	−0.994361
$842$	0	0
$843$	8.45164	0.291090
$844$	0	0
$845$	12.4772	0.429229
$846$	0	0
$847$	0	0
$848$	0	0
$849$	31.8110	1.09175
$850$	0	0
$851$	8.38496	0.287433
$852$	0	0
$853$	−25.5850	−0.876013	−0.438006	−	0.898972i	$-0.644315\pi$
$853$	−25.5850	−0.876013	−0.438006	+	0.898972i	$0.644315\pi$
$854$	0	0
$855$	−56.0840	−1.91803
$856$	0	0
$857$	23.8589	0.815005	0.407503	−	0.913204i	$-0.366400\pi$
$857$	23.8589	0.815005	0.407503	+	0.913204i	$0.366400\pi$
$858$	0	0
$859$	30.2417	1.03183	0.515917	−	0.856639i	$-0.327451\pi$
$859$	30.2417	1.03183	0.515917	+	0.856639i	$0.327451\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33.1988	1.13010	0.565049	−	0.825057i	$-0.308857\pi$
$863$	33.1988	1.13010	0.565049	+	0.825057i	$0.308857\pi$
$864$	0	0
$865$	−25.4413	−0.865030
$866$	0	0
$867$	−143.095	−4.85975
$868$	0	0
$869$	−32.6984	−1.10922
$870$	0	0
$871$	9.94477	0.336966
$872$	0	0
$873$	−122.631	−4.15042
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−22.9618	−0.775364	−0.387682	−	0.921793i	$-0.626724\pi$
$877$	−22.9618	−0.775364	−0.387682	+	0.921793i	$0.626724\pi$
$878$	0	0
$879$	−44.4398	−1.49892
$880$	0	0
$881$	0.0996818	0.00335837	0.00167918	−	0.999999i	$-0.499465\pi$
$881$	0.0996818	0.00335837	0.00167918	+	0.999999i	$0.499465\pi$
$882$	0	0
$883$	42.2637	1.42229	0.711143	−	0.703048i	$-0.248178\pi$
$883$	42.2637	1.42229	0.711143	+	0.703048i	$0.248178\pi$
$884$	0	0
$885$	−73.7552	−2.47926
$886$	0	0
$887$	−43.0159	−1.44433	−0.722167	−	0.691719i	$-0.756854\pi$
$887$	−43.0159	−1.44433	−0.722167	+	0.691719i	$0.756854\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−64.7325	−2.16862
$892$	0	0
$893$	−33.5385	−1.12232
$894$	0	0
$895$	13.6390	0.455903
$896$	0	0
$897$	−35.0110	−1.16898
$898$	0	0
$899$	−0.458285	−0.0152847
$900$	0	0
$901$	−101.888	−3.39437
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.31569	−0.209941
$906$	0	0
$907$	12.7608	0.423715	0.211858	−	0.977301i	$-0.432049\pi$
$907$	12.7608	0.423715	0.211858	+	0.977301i	$0.432049\pi$
$908$	0	0
$909$	−99.0202	−3.28429
$910$	0	0
$911$	−22.4015	−0.742196	−0.371098	−	0.928594i	$-0.621019\pi$
$911$	−22.4015	−0.742196	−0.371098	+	0.928594i	$0.621019\pi$
$912$	0	0
$913$	31.3016	1.03593
$914$	0	0
$915$	9.08610	0.300377
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−47.4888	−1.56651	−0.783256	−	0.621700i	$-0.786442\pi$
$919$	−47.4888	−1.56651	−0.783256	+	0.621700i	$0.786442\pi$
$920$	0	0
$921$	68.9841	2.27310
$922$	0	0
$923$	2.61391	0.0860380
$924$	0	0
$925$	4.62762	0.152155
$926$	0	0
$927$	18.1399	0.595791
$928$	0	0
$929$	−19.9053	−0.653072	−0.326536	−	0.945185i	$-0.605881\pi$
$929$	−19.9053	−0.653072	−0.326536	+	0.945185i	$0.605881\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−74.1650	−2.42805
$934$	0	0
$935$	−30.4406	−0.995515
$936$	0	0
$937$	−25.0489	−0.818310	−0.409155	−	0.912465i	$-0.634177\pi$
$937$	−25.0489	−0.818310	−0.409155	+	0.912465i	$0.634177\pi$
$938$	0	0
$939$	−109.646	−3.57817
$940$	0	0
$941$	−12.7460	−0.415509	−0.207755	−	0.978181i	$-0.566616\pi$
$941$	−12.7460	−0.415509	−0.207755	+	0.978181i	$0.566616\pi$
$942$	0	0
$943$	−5.00028	−0.162832
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.6762	0.964347	0.482174	−	0.876076i	$-0.339847\pi$
$947$	29.6762	0.964347	0.482174	+	0.876076i	$0.339847\pi$
$948$	0	0
$949$	6.82542	0.221563
$950$	0	0
$951$	−48.0925	−1.55951
$952$	0	0
$953$	29.3545	0.950887	0.475443	−	0.879746i	$-0.342288\pi$
$953$	29.3545	0.950887	0.475443	+	0.879746i	$0.342288\pi$
$954$	0	0
$955$	10.7200	0.346890
$956$	0	0
$957$	3.41026	0.110238
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.7158	−0.958573
$962$	0	0
$963$	−134.514	−4.33464
$964$	0	0
$965$	1.73201	0.0557552
$966$	0	0
$967$	−0.156677	−0.00503841	−0.00251920	−	0.999997i	$-0.500802\pi$
$967$	−0.156677	−0.00503841	−0.00251920	+	0.999997i	$0.500802\pi$
$968$	0	0
$969$	126.455	4.06232
$970$	0	0
$971$	−28.8865	−0.927011	−0.463505	−	0.886094i	$-0.653409\pi$
$971$	−28.8865	−0.927011	−0.463505	+	0.886094i	$0.653409\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−19.3224	−0.618812
$976$	0	0
$977$	−37.9126	−1.21293	−0.606466	−	0.795109i	$-0.707414\pi$
$977$	−37.9126	−1.21293	−0.606466	+	0.795109i	$0.707414\pi$
$978$	0	0
$979$	32.0703	1.02497
$980$	0	0
$981$	83.3266	2.66041
$982$	0	0
$983$	−15.8649	−0.506012	−0.253006	−	0.967465i	$-0.581419\pi$
$983$	−15.8649	−0.506012	−0.253006	+	0.967465i	$0.581419\pi$
$984$	0	0
$985$	3.44420	0.109741
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.493250	0.0156844
$990$	0	0
$991$	43.8398	1.39262	0.696308	−	0.717743i	$-0.254825\pi$
$991$	43.8398	1.39262	0.696308	+	0.717743i	$0.254825\pi$
$992$	0	0
$993$	−95.1124	−3.01830
$994$	0	0
$995$	25.6716	0.813843
$996$	0	0
$997$	−21.9847	−0.696262	−0.348131	−	0.937446i	$-0.613184\pi$
$997$	−21.9847	−0.696262	−0.348131	+	0.937446i	$0.613184\pi$
$998$	0	0
$999$	24.5274	0.776013

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.r.1.1		15
7.3	odd	6		1148.2.i.e.821.1	yes	30
7.5	odd	6		1148.2.i.e.165.1	✓	30
7.6	odd	2		8036.2.a.q.1.15		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.e.165.1	✓	30	7.5	odd	6
1148.2.i.e.821.1	yes	30	7.3	odd	6
8036.2.a.q.1.15		15	7.6	odd	2
8036.2.a.r.1.1		15	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-3.24213 q^{3} -1.49679 q^{5} +7.51143 q^{9} +O(q^{10})\)	\(q-3.24213 q^{3} -1.49679 q^{5} +7.51143 q^{9} -2.60103 q^{11} -2.15963 q^{13} +4.85278 q^{15} -7.81895 q^{17} +4.98834 q^{19} -5.00028 q^{23} -2.75963 q^{25} -14.6267 q^{27} +0.404401 q^{29} -1.13325 q^{31} +8.43288 q^{33} -1.67690 q^{37} +7.00181 q^{39} +1.00000 q^{41} -0.0986445 q^{43} -11.2430 q^{45} -6.72337 q^{47} +25.3501 q^{51} +13.0309 q^{53} +3.89319 q^{55} -16.1729 q^{57} -15.1985 q^{59} +1.87235 q^{61} +3.23251 q^{65} -4.60485 q^{67} +16.2116 q^{69} -1.21035 q^{71} -3.16046 q^{73} +8.94708 q^{75} +12.5713 q^{79} +24.8873 q^{81} -12.0343 q^{83} +11.7033 q^{85} -1.31112 q^{87} -12.3299 q^{89} +3.67414 q^{93} -7.46649 q^{95} -16.3259 q^{97} -19.5374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * q^99

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