LMFDB - Embedded newform 6036.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.35282 q^{5} +4.52941 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.35282 q^{5} +4.52941 q^{7} +1.00000 q^{9} -1.05764 q^{11} +3.27256 q^{13} +4.35282 q^{15} -0.821600 q^{17} +6.20538 q^{19} -4.52941 q^{21} +4.58343 q^{23} +13.9470 q^{25} -1.00000 q^{27} -2.39594 q^{29} +3.22680 q^{31} +1.05764 q^{33} -19.7157 q^{35} -7.35998 q^{37} -3.27256 q^{39} -4.28190 q^{41} +4.67415 q^{43} -4.35282 q^{45} +9.26046 q^{47} +13.5155 q^{49} +0.821600 q^{51} -2.53746 q^{53} +4.60373 q^{55} -6.20538 q^{57} -10.3806 q^{59} -5.03856 q^{61} +4.52941 q^{63} -14.2448 q^{65} -3.07201 q^{67} -4.58343 q^{69} +4.37291 q^{71} -8.07671 q^{73} -13.9470 q^{75} -4.79050 q^{77} -2.23645 q^{79} +1.00000 q^{81} +3.91592 q^{83} +3.57628 q^{85} +2.39594 q^{87} -0.496745 q^{89} +14.8227 q^{91} -3.22680 q^{93} -27.0109 q^{95} -2.38609 q^{97} -1.05764 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−4.35282	−1.94664	−0.973320	−	0.229453i	$-0.926306\pi$
$5$	−4.35282	−1.94664	−0.973320	+	0.229453i	$0.926306\pi$
$6$	0	0
$7$	4.52941	1.71195	0.855977	−	0.517013i	$-0.172956\pi$
$7$	4.52941	1.71195	0.855977	+	0.517013i	$0.172956\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.05764	−0.318892	−0.159446	−	0.987207i	$-0.550971\pi$
$11$	−1.05764	−0.318892	−0.159446	+	0.987207i	$0.550971\pi$
$12$	0	0
$13$	3.27256	0.907644	0.453822	−	0.891092i	$-0.350060\pi$
$13$	3.27256	0.907644	0.453822	+	0.891092i	$0.350060\pi$
$14$	0	0
$15$	4.35282	1.12389
$16$	0	0
$17$	−0.821600	−0.199267	−0.0996337	−	0.995024i	$-0.531767\pi$
$17$	−0.821600	−0.199267	−0.0996337	+	0.995024i	$0.531767\pi$
$18$	0	0
$19$	6.20538	1.42361	0.711806	−	0.702376i	$-0.247877\pi$
$19$	6.20538	1.42361	0.711806	+	0.702376i	$0.247877\pi$
$20$	0	0
$21$	−4.52941	−0.988398
$22$	0	0
$23$	4.58343	0.955711	0.477855	−	0.878439i	$-0.341414\pi$
$23$	4.58343	0.955711	0.477855	+	0.878439i	$0.341414\pi$
$24$	0	0
$25$	13.9470	2.78941
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.39594	−0.444915	−0.222457	−	0.974942i	$-0.571408\pi$
$29$	−2.39594	−0.444915	−0.222457	+	0.974942i	$0.571408\pi$
$30$	0	0
$31$	3.22680	0.579550	0.289775	−	0.957095i	$-0.406420\pi$
$31$	3.22680	0.579550	0.289775	+	0.957095i	$0.406420\pi$
$32$	0	0
$33$	1.05764	0.184112
$34$	0	0
$35$	−19.7157	−3.33256
$36$	0	0
$37$	−7.35998	−1.20997	−0.604987	−	0.796235i	$-0.706822\pi$
$37$	−7.35998	−1.20997	−0.604987	+	0.796235i	$0.706822\pi$
$38$	0	0
$39$	−3.27256	−0.524029
$40$	0	0
$41$	−4.28190	−0.668721	−0.334360	−	0.942445i	$-0.608520\pi$
$41$	−4.28190	−0.668721	−0.334360	+	0.942445i	$0.608520\pi$
$42$	0	0
$43$	4.67415	0.712801	0.356401	−	0.934333i	$-0.384004\pi$
$43$	4.67415	0.712801	0.356401	+	0.934333i	$0.384004\pi$
$44$	0	0
$45$	−4.35282	−0.648880
$46$	0	0
$47$	9.26046	1.35078	0.675389	−	0.737462i	$-0.263976\pi$
$47$	9.26046	1.35078	0.675389	+	0.737462i	$0.263976\pi$
$48$	0	0
$49$	13.5155	1.93079
$50$	0	0
$51$	0.821600	0.115047
$52$	0	0
$53$	−2.53746	−0.348547	−0.174273	−	0.984697i	$-0.555758\pi$
$53$	−2.53746	−0.348547	−0.174273	+	0.984697i	$0.555758\pi$
$54$	0	0
$55$	4.60373	0.620767
$56$	0	0
$57$	−6.20538	−0.821923
$58$	0	0
$59$	−10.3806	−1.35144	−0.675720	−	0.737158i	$-0.736167\pi$
$59$	−10.3806	−1.35144	−0.675720	+	0.737158i	$0.736167\pi$
$60$	0	0
$61$	−5.03856	−0.645121	−0.322561	−	0.946549i	$-0.604544\pi$
$61$	−5.03856	−0.645121	−0.322561	+	0.946549i	$0.604544\pi$
$62$	0	0
$63$	4.52941	0.570652
$64$	0	0
$65$	−14.2448	−1.76686
$66$	0	0
$67$	−3.07201	−0.375306	−0.187653	−	0.982235i	$-0.560088\pi$
$67$	−3.07201	−0.375306	−0.187653	+	0.982235i	$0.560088\pi$
$68$	0	0
$69$	−4.58343	−0.551780
$70$	0	0
$71$	4.37291	0.518969	0.259484	−	0.965747i	$-0.416447\pi$
$71$	4.37291	0.518969	0.259484	+	0.965747i	$0.416447\pi$
$72$	0	0
$73$	−8.07671	−0.945308	−0.472654	−	0.881248i	$-0.656704\pi$
$73$	−8.07671	−0.945308	−0.472654	+	0.881248i	$0.656704\pi$
$74$	0	0
$75$	−13.9470	−1.61046
$76$	0	0
$77$	−4.79050	−0.545928
$78$	0	0
$79$	−2.23645	−0.251621	−0.125810	−	0.992054i	$-0.540153\pi$
$79$	−2.23645	−0.251621	−0.125810	+	0.992054i	$0.540153\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.91592	0.429828	0.214914	−	0.976633i	$-0.431053\pi$
$83$	3.91592	0.429828	0.214914	+	0.976633i	$0.431053\pi$
$84$	0	0
$85$	3.57628	0.387902
$86$	0	0
$87$	2.39594	0.256872
$88$	0	0
$89$	−0.496745	−0.0526549	−0.0263274	−	0.999653i	$-0.508381\pi$
$89$	−0.496745	−0.0526549	−0.0263274	+	0.999653i	$0.508381\pi$
$90$	0	0
$91$	14.8227	1.55385
$92$	0	0
$93$	−3.22680	−0.334603
$94$	0	0
$95$	−27.0109	−2.77126
$96$	0	0
$97$	−2.38609	−0.242271	−0.121135	−	0.992636i	$-0.538654\pi$
$97$	−2.38609	−0.242271	−0.121135	+	0.992636i	$0.538654\pi$
$98$	0	0
$99$	−1.05764	−0.106297
$100$	0	0
$101$	16.5226	1.64406	0.822031	−	0.569442i	$-0.192841\pi$
$101$	16.5226	1.64406	0.822031	+	0.569442i	$0.192841\pi$
$102$	0	0
$103$	5.81697	0.573163	0.286582	−	0.958056i	$-0.407481\pi$
$103$	5.81697	0.573163	0.286582	+	0.958056i	$0.407481\pi$
$104$	0	0
$105$	19.7157	1.92405
$106$	0	0
$107$	16.3751	1.58304	0.791521	−	0.611143i	$-0.209290\pi$
$107$	16.3751	1.58304	0.791521	+	0.611143i	$0.209290\pi$
$108$	0	0
$109$	−17.1123	−1.63906	−0.819532	−	0.573034i	$-0.805766\pi$
$109$	−17.1123	−1.63906	−0.819532	+	0.573034i	$0.805766\pi$
$110$	0	0
$111$	7.35998	0.698579
$112$	0	0
$113$	−3.36438	−0.316495	−0.158247	−	0.987400i	$-0.550584\pi$
$113$	−3.36438	−0.316495	−0.158247	+	0.987400i	$0.550584\pi$
$114$	0	0
$115$	−19.9508	−1.86042
$116$	0	0
$117$	3.27256	0.302548
$118$	0	0
$119$	−3.72136	−0.341137
$120$	0	0
$121$	−9.88139	−0.898308
$122$	0	0
$123$	4.28190	0.386086
$124$	0	0
$125$	−38.9448	−3.48333
$126$	0	0
$127$	−4.73957	−0.420569	−0.210284	−	0.977640i	$-0.567439\pi$
$127$	−4.73957	−0.420569	−0.210284	+	0.977640i	$0.567439\pi$
$128$	0	0
$129$	−4.67415	−0.411536
$130$	0	0
$131$	2.04201	0.178411	0.0892057	−	0.996013i	$-0.471567\pi$
$131$	2.04201	0.178411	0.0892057	+	0.996013i	$0.471567\pi$
$132$	0	0
$133$	28.1067	2.43716
$134$	0	0
$135$	4.35282	0.374631
$136$	0	0
$137$	10.7240	0.916215	0.458107	−	0.888897i	$-0.348528\pi$
$137$	10.7240	0.916215	0.458107	+	0.888897i	$0.348528\pi$
$138$	0	0
$139$	19.6787	1.66912	0.834562	−	0.550913i	$-0.185721\pi$
$139$	19.6787	1.66912	0.834562	+	0.550913i	$0.185721\pi$
$140$	0	0
$141$	−9.26046	−0.779872
$142$	0	0
$143$	−3.46120	−0.289440
$144$	0	0
$145$	10.4291	0.866089
$146$	0	0
$147$	−13.5155	−1.11474
$148$	0	0
$149$	13.8512	1.13473	0.567365	−	0.823466i	$-0.307963\pi$
$149$	13.8512	1.13473	0.567365	+	0.823466i	$0.307963\pi$
$150$	0	0
$151$	9.97448	0.811712	0.405856	−	0.913937i	$-0.366973\pi$
$151$	9.97448	0.811712	0.405856	+	0.913937i	$0.366973\pi$
$152$	0	0
$153$	−0.821600	−0.0664224
$154$	0	0
$155$	−14.0457	−1.12817
$156$	0	0
$157$	−15.3424	−1.22445	−0.612227	−	0.790682i	$-0.709726\pi$
$157$	−15.3424	−1.22445	−0.612227	+	0.790682i	$0.709726\pi$
$158$	0	0
$159$	2.53746	0.201234
$160$	0	0
$161$	20.7602	1.63613
$162$	0	0
$163$	8.68771	0.680474	0.340237	−	0.940340i	$-0.389493\pi$
$163$	8.68771	0.680474	0.340237	+	0.940340i	$0.389493\pi$
$164$	0	0
$165$	−4.60373	−0.358400
$166$	0	0
$167$	19.4450	1.50470	0.752350	−	0.658764i	$-0.228920\pi$
$167$	19.4450	1.50470	0.752350	+	0.658764i	$0.228920\pi$
$168$	0	0
$169$	−2.29037	−0.176182
$170$	0	0
$171$	6.20538	0.474537
$172$	0	0
$173$	−10.2254	−0.777421	−0.388710	−	0.921360i	$-0.627079\pi$
$173$	−10.2254	−0.777421	−0.388710	+	0.921360i	$0.627079\pi$
$174$	0	0
$175$	63.1718	4.77534
$176$	0	0
$177$	10.3806	0.780254
$178$	0	0
$179$	1.21223	0.0906064	0.0453032	−	0.998973i	$-0.485575\pi$
$179$	1.21223	0.0906064	0.0453032	+	0.998973i	$0.485575\pi$
$180$	0	0
$181$	−3.65026	−0.271322	−0.135661	−	0.990755i	$-0.543316\pi$
$181$	−3.65026	−0.271322	−0.135661	+	0.990755i	$0.543316\pi$
$182$	0	0
$183$	5.03856	0.372461
$184$	0	0
$185$	32.0367	2.35538
$186$	0	0
$187$	0.868961	0.0635447
$188$	0	0
$189$	−4.52941	−0.329466
$190$	0	0
$191$	4.31692	0.312361	0.156181	−	0.987729i	$-0.450082\pi$
$191$	4.31692	0.312361	0.156181	+	0.987729i	$0.450082\pi$
$192$	0	0
$193$	6.29825	0.453358	0.226679	−	0.973969i	$-0.427213\pi$
$193$	6.29825	0.453358	0.226679	+	0.973969i	$0.427213\pi$
$194$	0	0
$195$	14.2448	1.02009
$196$	0	0
$197$	−27.8337	−1.98307	−0.991535	−	0.129839i	$-0.958554\pi$
$197$	−27.8337	−1.98307	−0.991535	+	0.129839i	$0.958554\pi$
$198$	0	0
$199$	−3.11661	−0.220931	−0.110465	−	0.993880i	$-0.535234\pi$
$199$	−3.11661	−0.220931	−0.110465	+	0.993880i	$0.535234\pi$
$200$	0	0
$201$	3.07201	0.216683
$202$	0	0
$203$	−10.8522	−0.761674
$204$	0	0
$205$	18.6383	1.30176
$206$	0	0
$207$	4.58343	0.318570
$208$	0	0
$209$	−6.56309	−0.453978
$210$	0	0
$211$	−9.81779	−0.675885	−0.337942	−	0.941167i	$-0.609731\pi$
$211$	−9.81779	−0.675885	−0.337942	+	0.941167i	$0.609731\pi$
$212$	0	0
$213$	−4.37291	−0.299627
$214$	0	0
$215$	−20.3457	−1.38757
$216$	0	0
$217$	14.6155	0.992163
$218$	0	0
$219$	8.07671	0.545774
$220$	0	0
$221$	−2.68873	−0.180864
$222$	0	0
$223$	2.96394	0.198480	0.0992399	−	0.995064i	$-0.468359\pi$
$223$	2.96394	0.198480	0.0992399	+	0.995064i	$0.468359\pi$
$224$	0	0
$225$	13.9470	0.929802
$226$	0	0
$227$	22.2796	1.47875	0.739375	−	0.673294i	$-0.235121\pi$
$227$	22.2796	1.47875	0.739375	+	0.673294i	$0.235121\pi$
$228$	0	0
$229$	−6.42292	−0.424438	−0.212219	−	0.977222i	$-0.568069\pi$
$229$	−6.42292	−0.424438	−0.212219	+	0.977222i	$0.568069\pi$
$230$	0	0
$231$	4.79050	0.315192
$232$	0	0
$233$	−2.07609	−0.136009	−0.0680045	−	0.997685i	$-0.521663\pi$
$233$	−2.07609	−0.136009	−0.0680045	+	0.997685i	$0.521663\pi$
$234$	0	0
$235$	−40.3091	−2.62948
$236$	0	0
$237$	2.23645	0.145273
$238$	0	0
$239$	10.8471	0.701637	0.350819	−	0.936443i	$-0.385903\pi$
$239$	10.8471	0.701637	0.350819	+	0.936443i	$0.385903\pi$
$240$	0	0
$241$	−19.1121	−1.23112	−0.615560	−	0.788090i	$-0.711070\pi$
$241$	−19.1121	−1.23112	−0.615560	+	0.788090i	$0.711070\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−58.8306	−3.75855
$246$	0	0
$247$	20.3075	1.29213
$248$	0	0
$249$	−3.91592	−0.248161
$250$	0	0
$251$	−0.653466	−0.0412464	−0.0206232	−	0.999787i	$-0.506565\pi$
$251$	−0.653466	−0.0412464	−0.0206232	+	0.999787i	$0.506565\pi$
$252$	0	0
$253$	−4.84763	−0.304768
$254$	0	0
$255$	−3.57628	−0.223955
$256$	0	0
$257$	19.3408	1.20644	0.603222	−	0.797573i	$-0.293883\pi$
$257$	19.3408	1.20644	0.603222	+	0.797573i	$0.293883\pi$
$258$	0	0
$259$	−33.3364	−2.07142
$260$	0	0
$261$	−2.39594	−0.148305
$262$	0	0
$263$	6.11362	0.376982	0.188491	−	0.982075i	$-0.439640\pi$
$263$	6.11362	0.376982	0.188491	+	0.982075i	$0.439640\pi$
$264$	0	0
$265$	11.0451	0.678495
$266$	0	0
$267$	0.496745	0.0304003
$268$	0	0
$269$	16.9230	1.03181	0.515906	−	0.856645i	$-0.327456\pi$
$269$	16.9230	1.03181	0.515906	+	0.856645i	$0.327456\pi$
$270$	0	0
$271$	15.2149	0.924241	0.462121	−	0.886817i	$-0.347089\pi$
$271$	15.2149	0.924241	0.462121	+	0.886817i	$0.347089\pi$
$272$	0	0
$273$	−14.8227	−0.897113
$274$	0	0
$275$	−14.7510	−0.889518
$276$	0	0
$277$	28.5273	1.71404	0.857020	−	0.515282i	$-0.172313\pi$
$277$	28.5273	1.71404	0.857020	+	0.515282i	$0.172313\pi$
$278$	0	0
$279$	3.22680	0.193183
$280$	0	0
$281$	−12.3250	−0.735248	−0.367624	−	0.929974i	$-0.619829\pi$
$281$	−12.3250	−0.735248	−0.367624	+	0.929974i	$0.619829\pi$
$282$	0	0
$283$	−17.3636	−1.03216	−0.516080	−	0.856540i	$-0.672609\pi$
$283$	−17.3636	−1.03216	−0.516080	+	0.856540i	$0.672609\pi$
$284$	0	0
$285$	27.0109	1.59999
$286$	0	0
$287$	−19.3945	−1.14482
$288$	0	0
$289$	−16.3250	−0.960293
$290$	0	0
$291$	2.38609	0.139875
$292$	0	0
$293$	−14.7636	−0.862496	−0.431248	−	0.902233i	$-0.641927\pi$
$293$	−14.7636	−0.862496	−0.431248	+	0.902233i	$0.641927\pi$
$294$	0	0
$295$	45.1849	2.63077
$296$	0	0
$297$	1.05764	0.0613707
$298$	0	0
$299$	14.9995	0.867445
$300$	0	0
$301$	21.1711	1.22028
$302$	0	0
$303$	−16.5226	−0.949200
$304$	0	0
$305$	21.9319	1.25582
$306$	0	0
$307$	12.2740	0.700511	0.350256	−	0.936654i	$-0.386095\pi$
$307$	12.2740	0.700511	0.350256	+	0.936654i	$0.386095\pi$
$308$	0	0
$309$	−5.81697	−0.330916
$310$	0	0
$311$	−9.78212	−0.554693	−0.277347	−	0.960770i	$-0.589455\pi$
$311$	−9.78212	−0.554693	−0.277347	+	0.960770i	$0.589455\pi$
$312$	0	0
$313$	22.2407	1.25712	0.628560	−	0.777761i	$-0.283645\pi$
$313$	22.2407	1.25712	0.628560	+	0.777761i	$0.283645\pi$
$314$	0	0
$315$	−19.7157	−1.11085
$316$	0	0
$317$	29.8559	1.67688	0.838438	−	0.544997i	$-0.183469\pi$
$317$	29.8559	1.67688	0.838438	+	0.544997i	$0.183469\pi$
$318$	0	0
$319$	2.53405	0.141880
$320$	0	0
$321$	−16.3751	−0.913969
$322$	0	0
$323$	−5.09834	−0.283679
$324$	0	0
$325$	45.6424	2.53179
$326$	0	0
$327$	17.1123	0.946314
$328$	0	0
$329$	41.9444	2.31247
$330$	0	0
$331$	20.4854	1.12598	0.562990	−	0.826464i	$-0.309651\pi$
$331$	20.4854	1.12598	0.562990	+	0.826464i	$0.309651\pi$
$332$	0	0
$333$	−7.35998	−0.403325
$334$	0	0
$335$	13.3719	0.730585
$336$	0	0
$337$	−18.0879	−0.985313	−0.492656	−	0.870224i	$-0.663974\pi$
$337$	−18.0879	−0.985313	−0.492656	+	0.870224i	$0.663974\pi$
$338$	0	0
$339$	3.36438	0.182728
$340$	0	0
$341$	−3.41280	−0.184814
$342$	0	0
$343$	29.5115	1.59347
$344$	0	0
$345$	19.9508	1.07412
$346$	0	0
$347$	16.0628	0.862295	0.431148	−	0.902281i	$-0.358109\pi$
$347$	16.0628	0.862295	0.431148	+	0.902281i	$0.358109\pi$
$348$	0	0
$349$	−18.2232	−0.975464	−0.487732	−	0.872993i	$-0.662176\pi$
$349$	−18.2232	−0.975464	−0.487732	+	0.872993i	$0.662176\pi$
$350$	0	0
$351$	−3.27256	−0.174676
$352$	0	0
$353$	35.9672	1.91434	0.957172	−	0.289521i	$-0.0934961\pi$
$353$	35.9672	1.91434	0.957172	+	0.289521i	$0.0934961\pi$
$354$	0	0
$355$	−19.0345	−1.01025
$356$	0	0
$357$	3.72136	0.196955
$358$	0	0
$359$	−3.46377	−0.182811	−0.0914055	−	0.995814i	$-0.529136\pi$
$359$	−3.46377	−0.182811	−0.0914055	+	0.995814i	$0.529136\pi$
$360$	0	0
$361$	19.5068	1.02667
$362$	0	0
$363$	9.88139	0.518638
$364$	0	0
$365$	35.1565	1.84017
$366$	0	0
$367$	33.1443	1.73012	0.865058	−	0.501672i	$-0.167281\pi$
$367$	33.1443	1.73012	0.865058	+	0.501672i	$0.167281\pi$
$368$	0	0
$369$	−4.28190	−0.222907
$370$	0	0
$371$	−11.4932	−0.596697
$372$	0	0
$373$	−3.95958	−0.205020	−0.102510	−	0.994732i	$-0.532687\pi$
$373$	−3.95958	−0.205020	−0.102510	+	0.994732i	$0.532687\pi$
$374$	0	0
$375$	38.9448	2.01110
$376$	0	0
$377$	−7.84085	−0.403824
$378$	0	0
$379$	−20.6399	−1.06020	−0.530100	−	0.847935i	$-0.677846\pi$
$379$	−20.6399	−1.06020	−0.530100	+	0.847935i	$0.677846\pi$
$380$	0	0
$381$	4.73957	0.242815
$382$	0	0
$383$	−29.2361	−1.49389	−0.746947	−	0.664883i	$-0.768481\pi$
$383$	−29.2361	−1.49389	−0.746947	+	0.664883i	$0.768481\pi$
$384$	0	0
$385$	20.8522	1.06273
$386$	0	0
$387$	4.67415	0.237600
$388$	0	0
$389$	33.7283	1.71009	0.855046	−	0.518552i	$-0.173529\pi$
$389$	33.7283	1.71009	0.855046	+	0.518552i	$0.173529\pi$
$390$	0	0
$391$	−3.76575	−0.190442
$392$	0	0
$393$	−2.04201	−0.103006
$394$	0	0
$395$	9.73488	0.489815
$396$	0	0
$397$	36.0650	1.81005	0.905024	−	0.425360i	$-0.139853\pi$
$397$	36.0650	1.81005	0.905024	+	0.425360i	$0.139853\pi$
$398$	0	0
$399$	−28.1067	−1.40710
$400$	0	0
$401$	−3.05637	−0.152628	−0.0763138	−	0.997084i	$-0.524315\pi$
$401$	−3.05637	−0.152628	−0.0763138	+	0.997084i	$0.524315\pi$
$402$	0	0
$403$	10.5599	0.526025
$404$	0	0
$405$	−4.35282	−0.216293
$406$	0	0
$407$	7.78424	0.385851
$408$	0	0
$409$	22.6294	1.11895	0.559477	−	0.828846i	$-0.311002\pi$
$409$	22.6294	1.11895	0.559477	+	0.828846i	$0.311002\pi$
$410$	0	0
$411$	−10.7240	−0.528977
$412$	0	0
$413$	−47.0180	−2.31360
$414$	0	0
$415$	−17.0453	−0.836720
$416$	0	0
$417$	−19.6787	−0.963670
$418$	0	0
$419$	25.4245	1.24207	0.621035	−	0.783783i	$-0.286712\pi$
$419$	25.4245	1.24207	0.621035	+	0.783783i	$0.286712\pi$
$420$	0	0
$421$	12.5592	0.612098	0.306049	−	0.952016i	$-0.400993\pi$
$421$	12.5592	0.612098	0.306049	+	0.952016i	$0.400993\pi$
$422$	0	0
$423$	9.26046	0.450259
$424$	0	0
$425$	−11.4589	−0.555837
$426$	0	0
$427$	−22.8217	−1.10442
$428$	0	0
$429$	3.46120	0.167108
$430$	0	0
$431$	36.1168	1.73969	0.869844	−	0.493327i	$-0.164220\pi$
$431$	36.1168	1.73969	0.869844	+	0.493327i	$0.164220\pi$
$432$	0	0
$433$	32.4490	1.55940	0.779699	−	0.626155i	$-0.215372\pi$
$433$	32.4490	1.55940	0.779699	+	0.626155i	$0.215372\pi$
$434$	0	0
$435$	−10.4291	−0.500037
$436$	0	0
$437$	28.4419	1.36056
$438$	0	0
$439$	−31.8930	−1.52217	−0.761086	−	0.648651i	$-0.775333\pi$
$439$	−31.8930	−1.52217	−0.761086	+	0.648651i	$0.775333\pi$
$440$	0	0
$441$	13.5155	0.643597
$442$	0	0
$443$	17.7156	0.841696	0.420848	−	0.907131i	$-0.361733\pi$
$443$	17.7156	0.841696	0.420848	+	0.907131i	$0.361733\pi$
$444$	0	0
$445$	2.16224	0.102500
$446$	0	0
$447$	−13.8512	−0.655137
$448$	0	0
$449$	−7.64492	−0.360786	−0.180393	−	0.983595i	$-0.557737\pi$
$449$	−7.64492	−0.360786	−0.180393	+	0.983595i	$0.557737\pi$
$450$	0	0
$451$	4.52873	0.213249
$452$	0	0
$453$	−9.97448	−0.468642
$454$	0	0
$455$	−64.5207	−3.02478
$456$	0	0
$457$	−40.5453	−1.89663	−0.948315	−	0.317331i	$-0.897213\pi$
$457$	−40.5453	−1.89663	−0.948315	+	0.317331i	$0.897213\pi$
$458$	0	0
$459$	0.821600	0.0383490
$460$	0	0
$461$	32.7071	1.52332	0.761661	−	0.647976i	$-0.224384\pi$
$461$	32.7071	1.52332	0.761661	+	0.647976i	$0.224384\pi$
$462$	0	0
$463$	−5.86978	−0.272792	−0.136396	−	0.990654i	$-0.543552\pi$
$463$	−5.86978	−0.272792	−0.136396	+	0.990654i	$0.543552\pi$
$464$	0	0
$465$	14.0457	0.651352
$466$	0	0
$467$	−22.2518	−1.02969	−0.514846	−	0.857282i	$-0.672151\pi$
$467$	−22.2518	−1.02969	−0.514846	+	0.857282i	$0.672151\pi$
$468$	0	0
$469$	−13.9144	−0.642506
$470$	0	0
$471$	15.3424	0.706939
$472$	0	0
$473$	−4.94359	−0.227306
$474$	0	0
$475$	86.5466	3.97103
$476$	0	0
$477$	−2.53746	−0.116182
$478$	0	0
$479$	−32.7526	−1.49651	−0.748253	−	0.663413i	$-0.769107\pi$
$479$	−32.7526	−1.49651	−0.748253	+	0.663413i	$0.769107\pi$
$480$	0	0
$481$	−24.0860	−1.09823
$482$	0	0
$483$	−20.7602	−0.944622
$484$	0	0
$485$	10.3862	0.471614
$486$	0	0
$487$	10.8538	0.491834	0.245917	−	0.969291i	$-0.420911\pi$
$487$	10.8538	0.491834	0.245917	+	0.969291i	$0.420911\pi$
$488$	0	0
$489$	−8.68771	−0.392872
$490$	0	0
$491$	−9.73311	−0.439249	−0.219625	−	0.975584i	$-0.570483\pi$
$491$	−9.73311	−0.439249	−0.219625	+	0.975584i	$0.570483\pi$
$492$	0	0
$493$	1.96850	0.0886570
$494$	0	0
$495$	4.60373	0.206922
$496$	0	0
$497$	19.8067	0.888451
$498$	0	0
$499$	5.53749	0.247892	0.123946	−	0.992289i	$-0.460445\pi$
$499$	5.53749	0.247892	0.123946	+	0.992289i	$0.460445\pi$
$500$	0	0
$501$	−19.4450	−0.868739
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−71.9200	−3.20040
$506$	0	0
$507$	2.29037	0.101719
$508$	0	0
$509$	30.5738	1.35516	0.677580	−	0.735449i	$-0.263029\pi$
$509$	30.5738	1.35516	0.677580	+	0.735449i	$0.263029\pi$
$510$	0	0
$511$	−36.5827	−1.61832
$512$	0	0
$513$	−6.20538	−0.273974
$514$	0	0
$515$	−25.3202	−1.11574
$516$	0	0
$517$	−9.79427	−0.430752
$518$	0	0
$519$	10.2254	0.448844
$520$	0	0
$521$	8.96937	0.392955	0.196478	−	0.980508i	$-0.437050\pi$
$521$	8.96937	0.392955	0.196478	+	0.980508i	$0.437050\pi$
$522$	0	0
$523$	−15.3234	−0.670045	−0.335022	−	0.942210i	$-0.608744\pi$
$523$	−15.3234	−0.670045	−0.335022	+	0.942210i	$0.608744\pi$
$524$	0	0
$525$	−63.1718	−2.75704
$526$	0	0
$527$	−2.65114	−0.115485
$528$	0	0
$529$	−1.99219	−0.0866171
$530$	0	0
$531$	−10.3806	−0.450480
$532$	0	0
$533$	−14.0128	−0.606960
$534$	0	0
$535$	−71.2779	−3.08161
$536$	0	0
$537$	−1.21223	−0.0523116
$538$	0	0
$539$	−14.2946	−0.615713
$540$	0	0
$541$	22.6037	0.971808	0.485904	−	0.874012i	$-0.338491\pi$
$541$	22.6037	0.971808	0.485904	+	0.874012i	$0.338491\pi$
$542$	0	0
$543$	3.65026	0.156648
$544$	0	0
$545$	74.4868	3.19067
$546$	0	0
$547$	−5.13096	−0.219384	−0.109692	−	0.993966i	$-0.534986\pi$
$547$	−5.13096	−0.219384	−0.109692	+	0.993966i	$0.534986\pi$
$548$	0	0
$549$	−5.03856	−0.215040
$550$	0	0
$551$	−14.8677	−0.633386
$552$	0	0
$553$	−10.1298	−0.430763
$554$	0	0
$555$	−32.0367	−1.35988
$556$	0	0
$557$	−39.1488	−1.65879	−0.829394	−	0.558664i	$-0.811314\pi$
$557$	−39.1488	−1.65879	−0.829394	+	0.558664i	$0.811314\pi$
$558$	0	0
$559$	15.2964	0.646970
$560$	0	0
$561$	−0.868961	−0.0366875
$562$	0	0
$563$	26.5625	1.11947	0.559737	−	0.828670i	$-0.310902\pi$
$563$	26.5625	1.11947	0.559737	+	0.828670i	$0.310902\pi$
$564$	0	0
$565$	14.6445	0.616101
$566$	0	0
$567$	4.52941	0.190217
$568$	0	0
$569$	−0.364054	−0.0152619	−0.00763097	−	0.999971i	$-0.502429\pi$
$569$	−0.364054	−0.0152619	−0.00763097	+	0.999971i	$0.502429\pi$
$570$	0	0
$571$	22.2371	0.930592	0.465296	−	0.885155i	$-0.345948\pi$
$571$	22.2371	0.930592	0.465296	+	0.885155i	$0.345948\pi$
$572$	0	0
$573$	−4.31692	−0.180342
$574$	0	0
$575$	63.9252	2.66586
$576$	0	0
$577$	13.1985	0.549459	0.274730	−	0.961521i	$-0.411412\pi$
$577$	13.1985	0.549459	0.274730	+	0.961521i	$0.411412\pi$
$578$	0	0
$579$	−6.29825	−0.261747
$580$	0	0
$581$	17.7368	0.735846
$582$	0	0
$583$	2.68373	0.111149
$584$	0	0
$585$	−14.2448	−0.588952
$586$	0	0
$587$	−3.71450	−0.153314	−0.0766569	−	0.997058i	$-0.524425\pi$
$587$	−3.71450	−0.153314	−0.0766569	+	0.997058i	$0.524425\pi$
$588$	0	0
$589$	20.0235	0.825054
$590$	0	0
$591$	27.8337	1.14493
$592$	0	0
$593$	12.5057	0.513547	0.256774	−	0.966472i	$-0.417340\pi$
$593$	12.5057	0.513547	0.256774	+	0.966472i	$0.417340\pi$
$594$	0	0
$595$	16.1984	0.664070
$596$	0	0
$597$	3.11661	0.127555
$598$	0	0
$599$	−17.6122	−0.719616	−0.359808	−	0.933026i	$-0.617158\pi$
$599$	−17.6122	−0.719616	−0.359808	+	0.933026i	$0.617158\pi$
$600$	0	0
$601$	−39.9569	−1.62988	−0.814939	−	0.579547i	$-0.803230\pi$
$601$	−39.9569	−1.62988	−0.814939	+	0.579547i	$0.803230\pi$
$602$	0	0
$603$	−3.07201	−0.125102
$604$	0	0
$605$	43.0119	1.74868
$606$	0	0
$607$	33.8689	1.37470	0.687348	−	0.726328i	$-0.258775\pi$
$607$	33.8689	1.37470	0.687348	+	0.726328i	$0.258775\pi$
$608$	0	0
$609$	10.8522	0.439753
$610$	0	0
$611$	30.3054	1.22603
$612$	0	0
$613$	−2.27108	−0.0917280	−0.0458640	−	0.998948i	$-0.514604\pi$
$613$	−2.27108	−0.0917280	−0.0458640	+	0.998948i	$0.514604\pi$
$614$	0	0
$615$	−18.6383	−0.751570
$616$	0	0
$617$	−34.7961	−1.40084	−0.700420	−	0.713731i	$-0.747004\pi$
$617$	−34.7961	−1.40084	−0.700420	+	0.713731i	$0.747004\pi$
$618$	0	0
$619$	2.01074	0.0808186	0.0404093	−	0.999183i	$-0.487134\pi$
$619$	2.01074	0.0808186	0.0404093	+	0.999183i	$0.487134\pi$
$620$	0	0
$621$	−4.58343	−0.183927
$622$	0	0
$623$	−2.24996	−0.0901428
$624$	0	0
$625$	99.7844	3.99138
$626$	0	0
$627$	6.56309	0.262104
$628$	0	0
$629$	6.04696	0.241108
$630$	0	0
$631$	−19.0081	−0.756699	−0.378350	−	0.925663i	$-0.623508\pi$
$631$	−19.0081	−0.756699	−0.378350	+	0.925663i	$0.623508\pi$
$632$	0	0
$633$	9.81779	0.390222
$634$	0	0
$635$	20.6305	0.818696
$636$	0	0
$637$	44.2303	1.75247
$638$	0	0
$639$	4.37291	0.172990
$640$	0	0
$641$	−12.4540	−0.491902	−0.245951	−	0.969282i	$-0.579100\pi$
$641$	−12.4540	−0.491902	−0.245951	+	0.969282i	$0.579100\pi$
$642$	0	0
$643$	−0.106778	−0.00421092	−0.00210546	−	0.999998i	$-0.500670\pi$
$643$	−0.106778	−0.00421092	−0.00210546	+	0.999998i	$0.500670\pi$
$644$	0	0
$645$	20.3457	0.801112
$646$	0	0
$647$	21.7232	0.854026	0.427013	−	0.904246i	$-0.359566\pi$
$647$	21.7232	0.854026	0.427013	+	0.904246i	$0.359566\pi$
$648$	0	0
$649$	10.9790	0.430963
$650$	0	0
$651$	−14.6155	−0.572826
$652$	0	0
$653$	25.5520	0.999926	0.499963	−	0.866047i	$-0.333347\pi$
$653$	25.5520	0.999926	0.499963	+	0.866047i	$0.333347\pi$
$654$	0	0
$655$	−8.88850	−0.347303
$656$	0	0
$657$	−8.07671	−0.315103
$658$	0	0
$659$	−32.5153	−1.26662	−0.633308	−	0.773900i	$-0.718303\pi$
$659$	−32.5153	−1.26662	−0.633308	+	0.773900i	$0.718303\pi$
$660$	0	0
$661$	−23.7138	−0.922361	−0.461181	−	0.887306i	$-0.652574\pi$
$661$	−23.7138	−0.922361	−0.461181	+	0.887306i	$0.652574\pi$
$662$	0	0
$663$	2.68873	0.104422
$664$	0	0
$665$	−122.343	−4.74427
$666$	0	0
$667$	−10.9816	−0.425210
$668$	0	0
$669$	−2.96394	−0.114592
$670$	0	0
$671$	5.32900	0.205724
$672$	0	0
$673$	6.66566	0.256942	0.128471	−	0.991713i	$-0.458993\pi$
$673$	6.66566	0.256942	0.128471	+	0.991713i	$0.458993\pi$
$674$	0	0
$675$	−13.9470	−0.536821
$676$	0	0
$677$	−45.7883	−1.75979	−0.879893	−	0.475172i	$-0.842386\pi$
$677$	−45.7883	−1.75979	−0.879893	+	0.475172i	$0.842386\pi$
$678$	0	0
$679$	−10.8076	−0.414757
$680$	0	0
$681$	−22.2796	−0.853756
$682$	0	0
$683$	10.2127	0.390780	0.195390	−	0.980726i	$-0.437403\pi$
$683$	10.2127	0.390780	0.195390	+	0.980726i	$0.437403\pi$
$684$	0	0
$685$	−46.6797	−1.78354
$686$	0	0
$687$	6.42292	0.245050
$688$	0	0
$689$	−8.30398	−0.316357
$690$	0	0
$691$	−36.7659	−1.39864	−0.699321	−	0.714808i	$-0.746514\pi$
$691$	−36.7659	−1.39864	−0.699321	+	0.714808i	$0.746514\pi$
$692$	0	0
$693$	−4.79050	−0.181976
$694$	0	0
$695$	−85.6578	−3.24918
$696$	0	0
$697$	3.51801	0.133254
$698$	0	0
$699$	2.07609	0.0785249
$700$	0	0
$701$	6.07273	0.229364	0.114682	−	0.993402i	$-0.463415\pi$
$701$	6.07273	0.229364	0.114682	+	0.993402i	$0.463415\pi$
$702$	0	0
$703$	−45.6715	−1.72253
$704$	0	0
$705$	40.3091	1.51813
$706$	0	0
$707$	74.8377	2.81456
$708$	0	0
$709$	−28.2925	−1.06255	−0.531274	−	0.847200i	$-0.678286\pi$
$709$	−28.2925	−1.06255	−0.531274	+	0.847200i	$0.678286\pi$
$710$	0	0
$711$	−2.23645	−0.0838736
$712$	0	0
$713$	14.7898	0.553882
$714$	0	0
$715$	15.0660	0.563436
$716$	0	0
$717$	−10.8471	−0.405091
$718$	0	0
$719$	15.3739	0.573349	0.286675	−	0.958028i	$-0.407450\pi$
$719$	15.3739	0.573349	0.286675	+	0.958028i	$0.407450\pi$
$720$	0	0
$721$	26.3474	0.981230
$722$	0	0
$723$	19.1121	0.710788
$724$	0	0
$725$	−33.4162	−1.24105
$726$	0	0
$727$	6.36541	0.236080	0.118040	−	0.993009i	$-0.462339\pi$
$727$	6.36541	0.236080	0.118040	+	0.993009i	$0.462339\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.84028	−0.142038
$732$	0	0
$733$	14.9431	0.551936	0.275968	−	0.961167i	$-0.411002\pi$
$733$	14.9431	0.551936	0.275968	+	0.961167i	$0.411002\pi$
$734$	0	0
$735$	58.8306	2.17000
$736$	0	0
$737$	3.24909	0.119682
$738$	0	0
$739$	−20.2443	−0.744700	−0.372350	−	0.928092i	$-0.621448\pi$
$739$	−20.2443	−0.744700	−0.372350	+	0.928092i	$0.621448\pi$
$740$	0	0
$741$	−20.3075	−0.746013
$742$	0	0
$743$	14.2351	0.522236	0.261118	−	0.965307i	$-0.415909\pi$
$743$	14.2351	0.522236	0.261118	+	0.965307i	$0.415909\pi$
$744$	0	0
$745$	−60.2915	−2.20891
$746$	0	0
$747$	3.91592	0.143276
$748$	0	0
$749$	74.1695	2.71010
$750$	0	0
$751$	−27.9012	−1.01813	−0.509065	−	0.860728i	$-0.670009\pi$
$751$	−27.9012	−1.01813	−0.509065	+	0.860728i	$0.670009\pi$
$752$	0	0
$753$	0.653466	0.0238136
$754$	0	0
$755$	−43.4171	−1.58011
$756$	0	0
$757$	−32.4604	−1.17979	−0.589897	−	0.807479i	$-0.700832\pi$
$757$	−32.4604	−1.17979	−0.589897	+	0.807479i	$0.700832\pi$
$758$	0	0
$759$	4.84763	0.175958
$760$	0	0
$761$	12.9118	0.468054	0.234027	−	0.972230i	$-0.424810\pi$
$761$	12.9118	0.468054	0.234027	+	0.972230i	$0.424810\pi$
$762$	0	0
$763$	−77.5087	−2.80600
$764$	0	0
$765$	3.57628	0.129301
$766$	0	0
$767$	−33.9711	−1.22663
$768$	0	0
$769$	34.6314	1.24884	0.624419	−	0.781089i	$-0.285336\pi$
$769$	34.6314	1.24884	0.624419	+	0.781089i	$0.285336\pi$
$770$	0	0
$771$	−19.3408	−0.696541
$772$	0	0
$773$	9.40139	0.338145	0.169072	−	0.985604i	$-0.445923\pi$
$773$	9.40139	0.338145	0.169072	+	0.985604i	$0.445923\pi$
$774$	0	0
$775$	45.0042	1.61660
$776$	0	0
$777$	33.3364	1.19594
$778$	0	0
$779$	−26.5708	−0.951999
$780$	0	0
$781$	−4.62498	−0.165495
$782$	0	0
$783$	2.39594	0.0856239
$784$	0	0
$785$	66.7825	2.38357
$786$	0	0
$787$	−4.10286	−0.146251	−0.0731256	−	0.997323i	$-0.523297\pi$
$787$	−4.10286	−0.146251	−0.0731256	+	0.997323i	$0.523297\pi$
$788$	0	0
$789$	−6.11362	−0.217651
$790$	0	0
$791$	−15.2387	−0.541824
$792$	0	0
$793$	−16.4890	−0.585540
$794$	0	0
$795$	−11.0451	−0.391729
$796$	0	0
$797$	−7.29586	−0.258432	−0.129216	−	0.991616i	$-0.541246\pi$
$797$	−7.29586	−0.258432	−0.129216	+	0.991616i	$0.541246\pi$
$798$	0	0
$799$	−7.60840	−0.269166
$800$	0	0
$801$	−0.496745	−0.0175516
$802$	0	0
$803$	8.54229	0.301451
$804$	0	0
$805$	−90.3654	−3.18496
$806$	0	0
$807$	−16.9230	−0.595717
$808$	0	0
$809$	14.7709	0.519316	0.259658	−	0.965701i	$-0.416390\pi$
$809$	14.7709	0.519316	0.259658	+	0.965701i	$0.416390\pi$
$810$	0	0
$811$	−45.2152	−1.58772	−0.793860	−	0.608101i	$-0.791932\pi$
$811$	−45.2152	−1.58772	−0.793860	+	0.608101i	$0.791932\pi$
$812$	0	0
$813$	−15.2149	−0.533611
$814$	0	0
$815$	−37.8160	−1.32464
$816$	0	0
$817$	29.0049	1.01475
$818$	0	0
$819$	14.8227	0.517949
$820$	0	0
$821$	−41.8618	−1.46099	−0.730494	−	0.682919i	$-0.760710\pi$
$821$	−41.8618	−1.46099	−0.730494	+	0.682919i	$0.760710\pi$
$822$	0	0
$823$	44.4038	1.54782	0.773910	−	0.633296i	$-0.218298\pi$
$823$	44.4038	1.54782	0.773910	+	0.633296i	$0.218298\pi$
$824$	0	0
$825$	14.7510	0.513564
$826$	0	0
$827$	−11.8056	−0.410522	−0.205261	−	0.978707i	$-0.565804\pi$
$827$	−11.8056	−0.410522	−0.205261	+	0.978707i	$0.565804\pi$
$828$	0	0
$829$	28.9232	1.00454	0.502272	−	0.864710i	$-0.332498\pi$
$829$	28.9232	1.00454	0.502272	+	0.864710i	$0.332498\pi$
$830$	0	0
$831$	−28.5273	−0.989602
$832$	0	0
$833$	−11.1044	−0.384743
$834$	0	0
$835$	−84.6406	−2.92911
$836$	0	0
$837$	−3.22680	−0.111534
$838$	0	0
$839$	−2.73283	−0.0943477	−0.0471738	−	0.998887i	$-0.515021\pi$
$839$	−2.73283	−0.0943477	−0.0471738	+	0.998887i	$0.515021\pi$
$840$	0	0
$841$	−23.2595	−0.802051
$842$	0	0
$843$	12.3250	0.424496
$844$	0	0
$845$	9.96957	0.342964
$846$	0	0
$847$	−44.7568	−1.53786
$848$	0	0
$849$	17.3636	0.595918
$850$	0	0
$851$	−33.7339	−1.15638
$852$	0	0
$853$	−38.8217	−1.32923	−0.664615	−	0.747186i	$-0.731404\pi$
$853$	−38.8217	−1.32923	−0.664615	+	0.747186i	$0.731404\pi$
$854$	0	0
$855$	−27.0109	−0.923753
$856$	0	0
$857$	−21.5394	−0.735773	−0.367887	−	0.929871i	$-0.619919\pi$
$857$	−21.5394	−0.735773	−0.367887	+	0.929871i	$0.619919\pi$
$858$	0	0
$859$	37.4784	1.27875	0.639373	−	0.768897i	$-0.279194\pi$
$859$	37.4784	1.27875	0.639373	+	0.768897i	$0.279194\pi$
$860$	0	0
$861$	19.3945	0.660962
$862$	0	0
$863$	−40.0938	−1.36481	−0.682404	−	0.730975i	$-0.739066\pi$
$863$	−40.0938	−1.36481	−0.682404	+	0.730975i	$0.739066\pi$
$864$	0	0
$865$	44.5092	1.51336
$866$	0	0
$867$	16.3250	0.554425
$868$	0	0
$869$	2.36537	0.0802398
$870$	0	0
$871$	−10.0533	−0.340644
$872$	0	0
$873$	−2.38609	−0.0807569
$874$	0	0
$875$	−176.397	−5.96330
$876$	0	0
$877$	−16.4417	−0.555196	−0.277598	−	0.960697i	$-0.589538\pi$
$877$	−16.4417	−0.555196	−0.277598	+	0.960697i	$0.589538\pi$
$878$	0	0
$879$	14.7636	0.497962
$880$	0	0
$881$	46.2546	1.55836	0.779179	−	0.626801i	$-0.215636\pi$
$881$	46.2546	1.55836	0.779179	+	0.626801i	$0.215636\pi$
$882$	0	0
$883$	17.1895	0.578474	0.289237	−	0.957258i	$-0.406598\pi$
$883$	17.1895	0.578474	0.289237	+	0.957258i	$0.406598\pi$
$884$	0	0
$885$	−45.1849	−1.51887
$886$	0	0
$887$	−28.4684	−0.955874	−0.477937	−	0.878394i	$-0.658615\pi$
$887$	−28.4684	−0.955874	−0.477937	+	0.878394i	$0.658615\pi$
$888$	0	0
$889$	−21.4674	−0.719995
$890$	0	0
$891$	−1.05764	−0.0354324
$892$	0	0
$893$	57.4647	1.92298
$894$	0	0
$895$	−5.27662	−0.176378
$896$	0	0
$897$	−14.9995	−0.500820
$898$	0	0
$899$	−7.73121	−0.257850
$900$	0	0
$901$	2.08478	0.0694540
$902$	0	0
$903$	−21.1711	−0.704531
$904$	0	0
$905$	15.8889	0.528166
$906$	0	0
$907$	12.7422	0.423098	0.211549	−	0.977367i	$-0.432149\pi$
$907$	12.7422	0.423098	0.211549	+	0.977367i	$0.432149\pi$
$908$	0	0
$909$	16.5226	0.548021
$910$	0	0
$911$	8.19637	0.271558	0.135779	−	0.990739i	$-0.456646\pi$
$911$	8.19637	0.271558	0.135779	+	0.990739i	$0.456646\pi$
$912$	0	0
$913$	−4.14165	−0.137069
$914$	0	0
$915$	−21.9319	−0.725047
$916$	0	0
$917$	9.24910	0.305432
$918$	0	0
$919$	48.1024	1.58675	0.793377	−	0.608731i	$-0.208321\pi$
$919$	48.1024	1.58675	0.793377	+	0.608731i	$0.208321\pi$
$920$	0	0
$921$	−12.2740	−0.404440
$922$	0	0
$923$	14.3106	0.471039
$924$	0	0
$925$	−102.650	−3.37511
$926$	0	0
$927$	5.81697	0.191054
$928$	0	0
$929$	−10.8455	−0.355830	−0.177915	−	0.984046i	$-0.556935\pi$
$929$	−10.8455	−0.355830	−0.177915	+	0.984046i	$0.556935\pi$
$930$	0	0
$931$	83.8690	2.74870
$932$	0	0
$933$	9.78212	0.320252
$934$	0	0
$935$	−3.78243	−0.123699
$936$	0	0
$937$	40.7170	1.33017	0.665083	−	0.746770i	$-0.268396\pi$
$937$	40.7170	1.33017	0.665083	+	0.746770i	$0.268396\pi$
$938$	0	0
$939$	−22.2407	−0.725799
$940$	0	0
$941$	−21.2682	−0.693325	−0.346663	−	0.937990i	$-0.612685\pi$
$941$	−21.2682	−0.693325	−0.346663	+	0.937990i	$0.612685\pi$
$942$	0	0
$943$	−19.6258	−0.639104
$944$	0	0
$945$	19.7157	0.641351
$946$	0	0
$947$	−21.9072	−0.711889	−0.355945	−	0.934507i	$-0.615841\pi$
$947$	−21.9072	−0.711889	−0.355945	+	0.934507i	$0.615841\pi$
$948$	0	0
$949$	−26.4315	−0.858003
$950$	0	0
$951$	−29.8559	−0.968145
$952$	0	0
$953$	−9.13776	−0.296001	−0.148001	−	0.988987i	$-0.547284\pi$
$953$	−9.13776	−0.296001	−0.148001	+	0.988987i	$0.547284\pi$
$954$	0	0
$955$	−18.7908	−0.608055
$956$	0	0
$957$	−2.53405	−0.0819143
$958$	0	0
$959$	48.5734	1.56852
$960$	0	0
$961$	−20.5878	−0.664122
$962$	0	0
$963$	16.3751	0.527680
$964$	0	0
$965$	−27.4152	−0.882525
$966$	0	0
$967$	59.0700	1.89956	0.949782	−	0.312912i	$-0.101304\pi$
$967$	59.0700	1.89956	0.949782	+	0.312912i	$0.101304\pi$
$968$	0	0
$969$	5.09834	0.163782
$970$	0	0
$971$	28.4647	0.913476	0.456738	−	0.889601i	$-0.349018\pi$
$971$	28.4647	0.913476	0.456738	+	0.889601i	$0.349018\pi$
$972$	0	0
$973$	89.1328	2.85747
$974$	0	0
$975$	−45.6424	−1.46173
$976$	0	0
$977$	38.9169	1.24506	0.622531	−	0.782595i	$-0.286104\pi$
$977$	38.9169	1.24506	0.622531	+	0.782595i	$0.286104\pi$
$978$	0	0
$979$	0.525380	0.0167912
$980$	0	0
$981$	−17.1123	−0.546354
$982$	0	0
$983$	−15.0208	−0.479089	−0.239544	−	0.970885i	$-0.576998\pi$
$983$	−15.0208	−0.479089	−0.239544	+	0.970885i	$0.576998\pi$
$984$	0	0
$985$	121.155	3.86032
$986$	0	0
$987$	−41.9444	−1.33511
$988$	0	0
$989$	21.4236	0.681232
$990$	0	0
$991$	12.2252	0.388346	0.194173	−	0.980967i	$-0.437798\pi$
$991$	12.2252	0.388346	0.194173	+	0.980967i	$0.437798\pi$
$992$	0	0
$993$	−20.4854	−0.650085
$994$	0	0
$995$	13.5661	0.430073
$996$	0	0
$997$	25.9183	0.820841	0.410421	−	0.911896i	$-0.365382\pi$
$997$	25.9183	0.820841	0.410421	+	0.911896i	$0.365382\pi$
$998$	0	0
$999$	7.35998	0.232860

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.1	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.1	✓	26	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.35282 q^{5} +4.52941 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.35282 q^{5} +4.52941 q^{7} +1.00000 q^{9} -1.05764 q^{11} +3.27256 q^{13} +4.35282 q^{15} -0.821600 q^{17} +6.20538 q^{19} -4.52941 q^{21} +4.58343 q^{23} +13.9470 q^{25} -1.00000 q^{27} -2.39594 q^{29} +3.22680 q^{31} +1.05764 q^{33} -19.7157 q^{35} -7.35998 q^{37} -3.27256 q^{39} -4.28190 q^{41} +4.67415 q^{43} -4.35282 q^{45} +9.26046 q^{47} +13.5155 q^{49} +0.821600 q^{51} -2.53746 q^{53} +4.60373 q^{55} -6.20538 q^{57} -10.3806 q^{59} -5.03856 q^{61} +4.52941 q^{63} -14.2448 q^{65} -3.07201 q^{67} -4.58343 q^{69} +4.37291 q^{71} -8.07671 q^{73} -13.9470 q^{75} -4.79050 q^{77} -2.23645 q^{79} +1.00000 q^{81} +3.91592 q^{83} +3.57628 q^{85} +2.39594 q^{87} -0.496745 q^{89} +14.8227 q^{91} -3.22680 q^{93} -27.0109 q^{95} -2.38609 q^{97} -1.05764 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more