LMFDB - Embedded newform 6036.2.a.f.1.10

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:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 12 x^{12} + 112 x^{11} + 7 x^{10} - 710 x^{9} + 281 x^{8} + 1850 x^{7} - 830 x^{6} + \cdots + 1 $ x^14 - 6x^13 - 12x^12 + 112x^11 + 7x^10 - 710x^9 + 281x^8 + 1850x^7 - 830x^6 - 1918x^5 + 623x^4 + 480x^3 - 29x^2 - 17*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-0.362365$ of defining polynomial
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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.362365 q^{5} +0.748260 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.362365 q^{5} +0.748260 q^{7} +1.00000 q^{9} -1.78145 q^{11} +4.77943 q^{13} -0.362365 q^{15} -1.20002 q^{17} +1.70695 q^{19} -0.748260 q^{21} +0.559632 q^{23} -4.86869 q^{25} -1.00000 q^{27} -7.80059 q^{29} -5.19570 q^{31} +1.78145 q^{33} +0.271143 q^{35} -0.872692 q^{37} -4.77943 q^{39} +5.27553 q^{41} -10.1729 q^{43} +0.362365 q^{45} +5.67478 q^{47} -6.44011 q^{49} +1.20002 q^{51} -1.88353 q^{53} -0.645535 q^{55} -1.70695 q^{57} +4.92700 q^{59} -8.26261 q^{61} +0.748260 q^{63} +1.73190 q^{65} -0.345994 q^{67} -0.559632 q^{69} +13.9606 q^{71} -11.7647 q^{73} +4.86869 q^{75} -1.33299 q^{77} +11.4594 q^{79} +1.00000 q^{81} +11.9599 q^{83} -0.434845 q^{85} +7.80059 q^{87} -15.0708 q^{89} +3.57626 q^{91} +5.19570 q^{93} +0.618540 q^{95} -15.5504 q^{97} -1.78145 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9} + q^{11} - q^{13} + 6 q^{15} - 6 q^{17} + q^{19} + 7 q^{21} + 10 q^{23} - 10 q^{25} - 14 q^{27} - 6 q^{29} - 5 q^{31} - q^{33} + 17 q^{35} - 12 q^{37} + q^{39} - 21 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{47} - 19 q^{49} + 6 q^{51} - q^{53} - q^{57} + 14 q^{59} - 19 q^{61} - 7 q^{63} + 17 q^{65} + 17 q^{67} - 10 q^{69} - 13 q^{71} - 12 q^{73} + 10 q^{75} - 9 q^{77} - 8 q^{79} + 14 q^{81} + 11 q^{83} - 17 q^{85} + 6 q^{87} - 9 q^{89} - 5 q^{91} + 5 q^{93} + 8 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9 + q^11 - q^13 + 6 * q^15 - 6 * q^17 + q^19 + 7 * q^21 + 10 * q^23 - 10 * q^25 - 14 * q^27 - 6 * q^29 - 5 * q^31 - q^33 + 17 * q^35 - 12 * q^37 + q^39 - 21 * q^41 + 8 * q^43 - 6 * q^45 + 18 * q^47 - 19 * q^49 + 6 * q^51 - q^53 - q^57 + 14 * q^59 - 19 * q^61 - 7 * q^63 + 17 * q^65 + 17 * q^67 - 10 * q^69 - 13 * q^71 - 12 * q^73 + 10 * q^75 - 9 * q^77 - 8 * q^79 + 14 * q^81 + 11 * q^83 - 17 * q^85 + 6 * q^87 - 9 * q^89 - 5 * q^91 + 5 * q^93 + 8 * q^95 - 25 * q^97 + 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
$5$	0.362365	0.162054	0.0810272	−	0.996712i	$-0.474180\pi$
$5$	0.362365	0.162054	0.0810272	+	0.996712i	$0.474180\pi$
$6$	0	0
$7$	0.748260	0.282816	0.141408	−	0.989951i	$-0.454837\pi$
$7$	0.748260	0.282816	0.141408	+	0.989951i	$0.454837\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.78145	−0.537128	−0.268564	−	0.963262i	$-0.586549\pi$
$11$	−1.78145	−0.537128	−0.268564	+	0.963262i	$0.586549\pi$
$12$	0	0
$13$	4.77943	1.32558	0.662788	−	0.748807i	$-0.269373\pi$
$13$	4.77943	1.32558	0.662788	+	0.748807i	$0.269373\pi$
$14$	0	0
$15$	−0.362365	−0.0935622
$16$	0	0
$17$	−1.20002	−0.291047	−0.145524	−	0.989355i	$-0.546487\pi$
$17$	−1.20002	−0.291047	−0.145524	+	0.989355i	$0.546487\pi$
$18$	0	0
$19$	1.70695	0.391602	0.195801	−	0.980644i	$-0.437269\pi$
$19$	1.70695	0.391602	0.195801	+	0.980644i	$0.437269\pi$
$20$	0	0
$21$	−0.748260	−0.163284
$22$	0	0
$23$	0.559632	0.116691	0.0583457	−	0.998296i	$-0.481417\pi$
$23$	0.559632	0.116691	0.0583457	+	0.998296i	$0.481417\pi$
$24$	0	0
$25$	−4.86869	−0.973738
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.80059	−1.44853	−0.724267	−	0.689520i	$-0.757822\pi$
$29$	−7.80059	−1.44853	−0.724267	+	0.689520i	$0.757822\pi$
$30$	0	0
$31$	−5.19570	−0.933176	−0.466588	−	0.884475i	$-0.654517\pi$
$31$	−5.19570	−0.933176	−0.466588	+	0.884475i	$0.654517\pi$
$32$	0	0
$33$	1.78145	0.310111
$34$	0	0
$35$	0.271143	0.0458315
$36$	0	0
$37$	−0.872692	−0.143470	−0.0717348	−	0.997424i	$-0.522854\pi$
$37$	−0.872692	−0.143470	−0.0717348	+	0.997424i	$0.522854\pi$
$38$	0	0
$39$	−4.77943	−0.765321
$40$	0	0
$41$	5.27553	0.823900	0.411950	−	0.911206i	$-0.364848\pi$
$41$	5.27553	0.823900	0.411950	+	0.911206i	$0.364848\pi$
$42$	0	0
$43$	−10.1729	−1.55136	−0.775679	−	0.631128i	$-0.782592\pi$
$43$	−10.1729	−1.55136	−0.775679	+	0.631128i	$0.782592\pi$
$44$	0	0
$45$	0.362365	0.0540182
$46$	0	0
$47$	5.67478	0.827752	0.413876	−	0.910333i	$-0.364175\pi$
$47$	5.67478	0.827752	0.413876	+	0.910333i	$0.364175\pi$
$48$	0	0
$49$	−6.44011	−0.920015
$50$	0	0
$51$	1.20002	0.168036
$52$	0	0
$53$	−1.88353	−0.258723	−0.129361	−	0.991598i	$-0.541293\pi$
$53$	−1.88353	−0.258723	−0.129361	+	0.991598i	$0.541293\pi$
$54$	0	0
$55$	−0.645535	−0.0870439
$56$	0	0
$57$	−1.70695	−0.226091
$58$	0	0
$59$	4.92700	0.641441	0.320721	−	0.947174i	$-0.396075\pi$
$59$	4.92700	0.641441	0.320721	+	0.947174i	$0.396075\pi$
$60$	0	0
$61$	−8.26261	−1.05792	−0.528959	−	0.848647i	$-0.677418\pi$
$61$	−8.26261	−1.05792	−0.528959	+	0.848647i	$0.677418\pi$
$62$	0	0
$63$	0.748260	0.0942719
$64$	0	0
$65$	1.73190	0.214815
$66$	0	0
$67$	−0.345994	−0.0422699	−0.0211349	−	0.999777i	$-0.506728\pi$
$67$	−0.345994	−0.0422699	−0.0211349	+	0.999777i	$0.506728\pi$
$68$	0	0
$69$	−0.559632	−0.0673718
$70$	0	0
$71$	13.9606	1.65681	0.828407	−	0.560127i	$-0.189248\pi$
$71$	13.9606	1.65681	0.828407	+	0.560127i	$0.189248\pi$
$72$	0	0
$73$	−11.7647	−1.37695	−0.688476	−	0.725259i	$-0.741720\pi$
$73$	−11.7647	−1.37695	−0.688476	+	0.725259i	$0.741720\pi$
$74$	0	0
$75$	4.86869	0.562188
$76$	0	0
$77$	−1.33299	−0.151908
$78$	0	0
$79$	11.4594	1.28928	0.644642	−	0.764485i	$-0.277007\pi$
$79$	11.4594	1.28928	0.644642	+	0.764485i	$0.277007\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.9599	1.31277	0.656387	−	0.754425i	$-0.272084\pi$
$83$	11.9599	1.31277	0.656387	+	0.754425i	$0.272084\pi$
$84$	0	0
$85$	−0.434845	−0.0471655
$86$	0	0
$87$	7.80059	0.836311
$88$	0	0
$89$	−15.0708	−1.59750	−0.798749	−	0.601665i	$-0.794504\pi$
$89$	−15.0708	−1.59750	−0.798749	+	0.601665i	$0.794504\pi$
$90$	0	0
$91$	3.57626	0.374893
$92$	0	0
$93$	5.19570	0.538769
$94$	0	0
$95$	0.618540	0.0634608
$96$	0	0
$97$	−15.5504	−1.57890	−0.789452	−	0.613812i	$-0.789635\pi$
$97$	−15.5504	−1.57890	−0.789452	+	0.613812i	$0.789635\pi$
$98$	0	0
$99$	−1.78145	−0.179043
$100$	0	0
$101$	9.79122	0.974263	0.487131	−	0.873329i	$-0.338043\pi$
$101$	9.79122	0.974263	0.487131	+	0.873329i	$0.338043\pi$
$102$	0	0
$103$	−3.18928	−0.314249	−0.157125	−	0.987579i	$-0.550222\pi$
$103$	−3.18928	−0.314249	−0.157125	+	0.987579i	$0.550222\pi$
$104$	0	0
$105$	−0.271143	−0.0264608
$106$	0	0
$107$	−9.97877	−0.964684	−0.482342	−	0.875983i	$-0.660214\pi$
$107$	−9.97877	−0.964684	−0.482342	+	0.875983i	$0.660214\pi$
$108$	0	0
$109$	5.90307	0.565411	0.282706	−	0.959207i	$-0.408768\pi$
$109$	5.90307	0.565411	0.282706	+	0.959207i	$0.408768\pi$
$110$	0	0
$111$	0.872692	0.0828322
$112$	0	0
$113$	8.73259	0.821493	0.410746	−	0.911750i	$-0.365268\pi$
$113$	8.73259	0.821493	0.410746	+	0.911750i	$0.365268\pi$
$114$	0	0
$115$	0.202791	0.0189104
$116$	0	0
$117$	4.77943	0.441859
$118$	0	0
$119$	−0.897926	−0.0823128
$120$	0	0
$121$	−7.82643	−0.711494
$122$	0	0
$123$	−5.27553	−0.475679
$124$	0	0
$125$	−3.57607	−0.319853
$126$	0	0
$127$	−7.71435	−0.684538	−0.342269	−	0.939602i	$-0.611195\pi$
$127$	−7.71435	−0.684538	−0.342269	+	0.939602i	$0.611195\pi$
$128$	0	0
$129$	10.1729	0.895676
$130$	0	0
$131$	4.22997	0.369575	0.184787	−	0.982779i	$-0.440840\pi$
$131$	4.22997	0.369575	0.184787	+	0.982779i	$0.440840\pi$
$132$	0	0
$133$	1.27724	0.110751
$134$	0	0
$135$	−0.362365	−0.0311874
$136$	0	0
$137$	7.38241	0.630722	0.315361	−	0.948972i	$-0.397874\pi$
$137$	7.38241	0.630722	0.315361	+	0.948972i	$0.397874\pi$
$138$	0	0
$139$	10.6511	0.903418	0.451709	−	0.892165i	$-0.350814\pi$
$139$	10.6511	0.903418	0.451709	+	0.892165i	$0.350814\pi$
$140$	0	0
$141$	−5.67478	−0.477903
$142$	0	0
$143$	−8.51432	−0.712003
$144$	0	0
$145$	−2.82666	−0.234741
$146$	0	0
$147$	6.44011	0.531171
$148$	0	0
$149$	−13.0050	−1.06541	−0.532706	−	0.846300i	$-0.678825\pi$
$149$	−13.0050	−1.06541	−0.532706	+	0.846300i	$0.678825\pi$
$150$	0	0
$151$	−18.6881	−1.52082	−0.760408	−	0.649446i	$-0.775001\pi$
$151$	−18.6881	−1.52082	−0.760408	+	0.649446i	$0.775001\pi$
$152$	0	0
$153$	−1.20002	−0.0970158
$154$	0	0
$155$	−1.88274	−0.151225
$156$	0	0
$157$	17.6466	1.40835	0.704175	−	0.710026i	$-0.251317\pi$
$157$	17.6466	1.40835	0.704175	+	0.710026i	$0.251317\pi$
$158$	0	0
$159$	1.88353	0.149374
$160$	0	0
$161$	0.418750	0.0330021
$162$	0	0
$163$	−15.4521	−1.21030	−0.605152	−	0.796110i	$-0.706888\pi$
$163$	−15.4521	−1.21030	−0.605152	+	0.796110i	$0.706888\pi$
$164$	0	0
$165$	0.645535	0.0502548
$166$	0	0
$167$	10.5874	0.819278	0.409639	−	0.912248i	$-0.365655\pi$
$167$	10.5874	0.819278	0.409639	+	0.912248i	$0.365655\pi$
$168$	0	0
$169$	9.84296	0.757150
$170$	0	0
$171$	1.70695	0.130534
$172$	0	0
$173$	−22.2287	−1.69001	−0.845007	−	0.534755i	$-0.820404\pi$
$173$	−22.2287	−1.69001	−0.845007	+	0.534755i	$0.820404\pi$
$174$	0	0
$175$	−3.64305	−0.275388
$176$	0	0
$177$	−4.92700	−0.370336
$178$	0	0
$179$	−4.78638	−0.357751	−0.178876	−	0.983872i	$-0.557246\pi$
$179$	−4.78638	−0.357751	−0.178876	+	0.983872i	$0.557246\pi$
$180$	0	0
$181$	3.86910	0.287588	0.143794	−	0.989608i	$-0.454070\pi$
$181$	3.86910	0.287588	0.143794	+	0.989608i	$0.454070\pi$
$182$	0	0
$183$	8.26261	0.610790
$184$	0	0
$185$	−0.316233	−0.0232499
$186$	0	0
$187$	2.13778	0.156330
$188$	0	0
$189$	−0.748260	−0.0544279
$190$	0	0
$191$	−5.86319	−0.424246	−0.212123	−	0.977243i	$-0.568038\pi$
$191$	−5.86319	−0.424246	−0.212123	+	0.977243i	$0.568038\pi$
$192$	0	0
$193$	2.83984	0.204416	0.102208	−	0.994763i	$-0.467409\pi$
$193$	2.83984	0.204416	0.102208	+	0.994763i	$0.467409\pi$
$194$	0	0
$195$	−1.73190	−0.124024
$196$	0	0
$197$	−7.98428	−0.568856	−0.284428	−	0.958697i	$-0.591804\pi$
$197$	−7.98428	−0.568856	−0.284428	+	0.958697i	$0.591804\pi$
$198$	0	0
$199$	0.277867	0.0196975	0.00984874	−	0.999951i	$-0.496865\pi$
$199$	0.277867	0.0196975	0.00984874	+	0.999951i	$0.496865\pi$
$200$	0	0
$201$	0.345994	0.0244045
$202$	0	0
$203$	−5.83687	−0.409668
$204$	0	0
$205$	1.91167	0.133517
$206$	0	0
$207$	0.559632	0.0388971
$208$	0	0
$209$	−3.04085	−0.210340
$210$	0	0
$211$	−17.4352	−1.20029	−0.600143	−	0.799893i	$-0.704890\pi$
$211$	−17.4352	−1.20029	−0.600143	+	0.799893i	$0.704890\pi$
$212$	0	0
$213$	−13.9606	−0.956562
$214$	0	0
$215$	−3.68631	−0.251404
$216$	0	0
$217$	−3.88773	−0.263917
$218$	0	0
$219$	11.7647	0.794983
$220$	0	0
$221$	−5.73541	−0.385805
$222$	0	0
$223$	−10.1614	−0.680461	−0.340230	−	0.940342i	$-0.610505\pi$
$223$	−10.1614	−0.680461	−0.340230	+	0.940342i	$0.610505\pi$
$224$	0	0
$225$	−4.86869	−0.324579
$226$	0	0
$227$	4.98533	0.330888	0.165444	−	0.986219i	$-0.447094\pi$
$227$	4.98533	0.330888	0.165444	+	0.986219i	$0.447094\pi$
$228$	0	0
$229$	−5.63725	−0.372520	−0.186260	−	0.982501i	$-0.559637\pi$
$229$	−5.63725	−0.372520	−0.186260	+	0.982501i	$0.559637\pi$
$230$	0	0
$231$	1.33299	0.0877042
$232$	0	0
$233$	−1.07165	−0.0702058	−0.0351029	−	0.999384i	$-0.511176\pi$
$233$	−1.07165	−0.0702058	−0.0351029	+	0.999384i	$0.511176\pi$
$234$	0	0
$235$	2.05634	0.134141
$236$	0	0
$237$	−11.4594	−0.744368
$238$	0	0
$239$	−20.3357	−1.31541	−0.657703	−	0.753277i	$-0.728472\pi$
$239$	−20.3357	−1.31541	−0.657703	+	0.753277i	$0.728472\pi$
$240$	0	0
$241$	−1.21543	−0.0782927	−0.0391463	−	0.999233i	$-0.512464\pi$
$241$	−1.21543	−0.0782927	−0.0391463	+	0.999233i	$0.512464\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.33367	−0.149093
$246$	0	0
$247$	8.15826	0.519098
$248$	0	0
$249$	−11.9599	−0.757930
$250$	0	0
$251$	−4.45275	−0.281055	−0.140528	−	0.990077i	$-0.544880\pi$
$251$	−4.45275	−0.281055	−0.140528	+	0.990077i	$0.544880\pi$
$252$	0	0
$253$	−0.996957	−0.0626782
$254$	0	0
$255$	0.434845	0.0272310
$256$	0	0
$257$	−20.7639	−1.29522	−0.647608	−	0.761974i	$-0.724230\pi$
$257$	−20.7639	−1.29522	−0.647608	+	0.761974i	$0.724230\pi$
$258$	0	0
$259$	−0.653000	−0.0405755
$260$	0	0
$261$	−7.80059	−0.482844
$262$	0	0
$263$	12.6893	0.782457	0.391228	−	0.920294i	$-0.372050\pi$
$263$	12.6893	0.782457	0.391228	+	0.920294i	$0.372050\pi$
$264$	0	0
$265$	−0.682525	−0.0419272
$266$	0	0
$267$	15.0708	0.922316
$268$	0	0
$269$	5.74993	0.350579	0.175290	−	0.984517i	$-0.443914\pi$
$269$	5.74993	0.350579	0.175290	+	0.984517i	$0.443914\pi$
$270$	0	0
$271$	−2.37825	−0.144468	−0.0722342	−	0.997388i	$-0.523013\pi$
$271$	−2.37825	−0.144468	−0.0722342	+	0.997388i	$0.523013\pi$
$272$	0	0
$273$	−3.57626	−0.216445
$274$	0	0
$275$	8.67334	0.523022
$276$	0	0
$277$	−3.16071	−0.189909	−0.0949544	−	0.995482i	$-0.530271\pi$
$277$	−3.16071	−0.189909	−0.0949544	+	0.995482i	$0.530271\pi$
$278$	0	0
$279$	−5.19570	−0.311059
$280$	0	0
$281$	−0.886688	−0.0528954	−0.0264477	−	0.999650i	$-0.508420\pi$
$281$	−0.886688	−0.0528954	−0.0264477	+	0.999650i	$0.508420\pi$
$282$	0	0
$283$	22.7832	1.35432	0.677159	−	0.735837i	$-0.263211\pi$
$283$	22.7832	1.35432	0.677159	+	0.735837i	$0.263211\pi$
$284$	0	0
$285$	−0.618540	−0.0366391
$286$	0	0
$287$	3.94747	0.233012
$288$	0	0
$289$	−15.5600	−0.915291
$290$	0	0
$291$	15.5504	0.911581
$292$	0	0
$293$	11.2177	0.655342	0.327671	−	0.944792i	$-0.393736\pi$
$293$	11.2177	0.655342	0.327671	+	0.944792i	$0.393736\pi$
$294$	0	0
$295$	1.78537	0.103948
$296$	0	0
$297$	1.78145	0.103370
$298$	0	0
$299$	2.67472	0.154683
$300$	0	0
$301$	−7.61199	−0.438748
$302$	0	0
$303$	−9.79122	−0.562491
$304$	0	0
$305$	−2.99408	−0.171440
$306$	0	0
$307$	−4.18794	−0.239018	−0.119509	−	0.992833i	$-0.538132\pi$
$307$	−4.18794	−0.239018	−0.119509	+	0.992833i	$0.538132\pi$
$308$	0	0
$309$	3.18928	0.181432
$310$	0	0
$311$	19.7539	1.12014	0.560070	−	0.828446i	$-0.310774\pi$
$311$	19.7539	1.12014	0.560070	+	0.828446i	$0.310774\pi$
$312$	0	0
$313$	−31.0441	−1.75472	−0.877358	−	0.479837i	$-0.840696\pi$
$313$	−31.0441	−1.75472	−0.877358	+	0.479837i	$0.840696\pi$
$314$	0	0
$315$	0.271143	0.0152772
$316$	0	0
$317$	−23.4958	−1.31965	−0.659827	−	0.751417i	$-0.729371\pi$
$317$	−23.4958	−1.31965	−0.659827	+	0.751417i	$0.729371\pi$
$318$	0	0
$319$	13.8964	0.778047
$320$	0	0
$321$	9.97877	0.556960
$322$	0	0
$323$	−2.04838	−0.113975
$324$	0	0
$325$	−23.2696	−1.29076
$326$	0	0
$327$	−5.90307	−0.326440
$328$	0	0
$329$	4.24621	0.234101
$330$	0	0
$331$	28.5747	1.57060	0.785302	−	0.619112i	$-0.212507\pi$
$331$	28.5747	1.57060	0.785302	+	0.619112i	$0.212507\pi$
$332$	0	0
$333$	−0.872692	−0.0478232
$334$	0	0
$335$	−0.125376	−0.00685003
$336$	0	0
$337$	−7.24100	−0.394442	−0.197221	−	0.980359i	$-0.563192\pi$
$337$	−7.24100	−0.394442	−0.197221	+	0.980359i	$0.563192\pi$
$338$	0	0
$339$	−8.73259	−0.474289
$340$	0	0
$341$	9.25589	0.501234
$342$	0	0
$343$	−10.0567	−0.543010
$344$	0	0
$345$	−0.202791	−0.0109179
$346$	0	0
$347$	10.2583	0.550694	0.275347	−	0.961345i	$-0.411207\pi$
$347$	10.2583	0.550694	0.275347	+	0.961345i	$0.411207\pi$
$348$	0	0
$349$	29.1523	1.56049	0.780243	−	0.625477i	$-0.215096\pi$
$349$	29.1523	1.56049	0.780243	+	0.625477i	$0.215096\pi$
$350$	0	0
$351$	−4.77943	−0.255107
$352$	0	0
$353$	1.18265	0.0629459	0.0314730	−	0.999505i	$-0.489980\pi$
$353$	1.18265	0.0629459	0.0314730	+	0.999505i	$0.489980\pi$
$354$	0	0
$355$	5.05882	0.268494
$356$	0	0
$357$	0.897926	0.0475233
$358$	0	0
$359$	−16.4324	−0.867267	−0.433634	−	0.901089i	$-0.642769\pi$
$359$	−16.4324	−0.867267	−0.433634	+	0.901089i	$0.642769\pi$
$360$	0	0
$361$	−16.0863	−0.846648
$362$	0	0
$363$	7.82643	0.410781
$364$	0	0
$365$	−4.26311	−0.223141
$366$	0	0
$367$	9.72449	0.507614	0.253807	−	0.967255i	$-0.418317\pi$
$367$	9.72449	0.507614	0.253807	+	0.967255i	$0.418317\pi$
$368$	0	0
$369$	5.27553	0.274633
$370$	0	0
$371$	−1.40937	−0.0731709
$372$	0	0
$373$	−28.7004	−1.48605	−0.743024	−	0.669265i	$-0.766609\pi$
$373$	−28.7004	−1.48605	−0.743024	+	0.669265i	$0.766609\pi$
$374$	0	0
$375$	3.57607	0.184667
$376$	0	0
$377$	−37.2824	−1.92014
$378$	0	0
$379$	−31.5213	−1.61914	−0.809569	−	0.587024i	$-0.800299\pi$
$379$	−31.5213	−1.61914	−0.809569	+	0.587024i	$0.800299\pi$
$380$	0	0
$381$	7.71435	0.395218
$382$	0	0
$383$	−5.69648	−0.291077	−0.145538	−	0.989353i	$-0.546491\pi$
$383$	−5.69648	−0.291077	−0.145538	+	0.989353i	$0.546491\pi$
$384$	0	0
$385$	−0.483028	−0.0246174
$386$	0	0
$387$	−10.1729	−0.517119
$388$	0	0
$389$	11.5892	0.587594	0.293797	−	0.955868i	$-0.405081\pi$
$389$	11.5892	0.587594	0.293797	+	0.955868i	$0.405081\pi$
$390$	0	0
$391$	−0.671569	−0.0339627
$392$	0	0
$393$	−4.22997	−0.213374
$394$	0	0
$395$	4.15248	0.208934
$396$	0	0
$397$	−21.4887	−1.07849	−0.539243	−	0.842150i	$-0.681290\pi$
$397$	−21.4887	−1.07849	−0.539243	+	0.842150i	$0.681290\pi$
$398$	0	0
$399$	−1.27724	−0.0639422
$400$	0	0
$401$	28.7406	1.43523	0.717617	−	0.696438i	$-0.245233\pi$
$401$	28.7406	1.43523	0.717617	+	0.696438i	$0.245233\pi$
$402$	0	0
$403$	−24.8325	−1.23699
$404$	0	0
$405$	0.362365	0.0180061
$406$	0	0
$407$	1.55466	0.0770615
$408$	0	0
$409$	28.6332	1.41582	0.707912	−	0.706301i	$-0.249638\pi$
$409$	28.6332	1.41582	0.707912	+	0.706301i	$0.249638\pi$
$410$	0	0
$411$	−7.38241	−0.364148
$412$	0	0
$413$	3.68668	0.181410
$414$	0	0
$415$	4.33386	0.212741
$416$	0	0
$417$	−10.6511	−0.521589
$418$	0	0
$419$	−22.8541	−1.11650	−0.558249	−	0.829674i	$-0.688526\pi$
$419$	−22.8541	−1.11650	−0.558249	+	0.829674i	$0.688526\pi$
$420$	0	0
$421$	4.17087	0.203276	0.101638	−	0.994821i	$-0.467592\pi$
$421$	4.17087	0.203276	0.101638	+	0.994821i	$0.467592\pi$
$422$	0	0
$423$	5.67478	0.275917
$424$	0	0
$425$	5.84252	0.283404
$426$	0	0
$427$	−6.18258	−0.299196
$428$	0	0
$429$	8.51432	0.411075
$430$	0	0
$431$	−20.1432	−0.970264	−0.485132	−	0.874441i	$-0.661228\pi$
$431$	−20.1432	−0.970264	−0.485132	+	0.874441i	$0.661228\pi$
$432$	0	0
$433$	13.6327	0.655146	0.327573	−	0.944826i	$-0.393769\pi$
$433$	13.6327	0.655146	0.327573	+	0.944826i	$0.393769\pi$
$434$	0	0
$435$	2.82666	0.135528
$436$	0	0
$437$	0.955266	0.0456966
$438$	0	0
$439$	18.6521	0.890216	0.445108	−	0.895477i	$-0.353165\pi$
$439$	18.6521	0.890216	0.445108	+	0.895477i	$0.353165\pi$
$440$	0	0
$441$	−6.44011	−0.306672
$442$	0	0
$443$	−15.6207	−0.742161	−0.371080	−	0.928601i	$-0.621013\pi$
$443$	−15.6207	−0.742161	−0.371080	+	0.928601i	$0.621013\pi$
$444$	0	0
$445$	−5.46111	−0.258882
$446$	0	0
$447$	13.0050	0.615116
$448$	0	0
$449$	15.6569	0.738893	0.369446	−	0.929252i	$-0.379547\pi$
$449$	15.6569	0.738893	0.369446	+	0.929252i	$0.379547\pi$
$450$	0	0
$451$	−9.39810	−0.442539
$452$	0	0
$453$	18.6881	0.878043
$454$	0	0
$455$	1.29591	0.0607532
$456$	0	0
$457$	−37.6789	−1.76254	−0.881271	−	0.472611i	$-0.843312\pi$
$457$	−37.6789	−1.76254	−0.881271	+	0.472611i	$0.843312\pi$
$458$	0	0
$459$	1.20002	0.0560121
$460$	0	0
$461$	−27.0992	−1.26214	−0.631068	−	0.775728i	$-0.717383\pi$
$461$	−27.0992	−1.26214	−0.631068	+	0.775728i	$0.717383\pi$
$462$	0	0
$463$	5.63480	0.261871	0.130936	−	0.991391i	$-0.458202\pi$
$463$	5.63480	0.261871	0.130936	+	0.991391i	$0.458202\pi$
$464$	0	0
$465$	1.88274	0.0873100
$466$	0	0
$467$	−37.9821	−1.75760	−0.878802	−	0.477186i	$-0.841657\pi$
$467$	−37.9821	−1.75760	−0.878802	+	0.477186i	$0.841657\pi$
$468$	0	0
$469$	−0.258893	−0.0119546
$470$	0	0
$471$	−17.6466	−0.813111
$472$	0	0
$473$	18.1226	0.833277
$474$	0	0
$475$	−8.31063	−0.381318
$476$	0	0
$477$	−1.88353	−0.0862410
$478$	0	0
$479$	37.0893	1.69465	0.847327	−	0.531072i	$-0.178211\pi$
$479$	37.0893	1.69465	0.847327	+	0.531072i	$0.178211\pi$
$480$	0	0
$481$	−4.17097	−0.190180
$482$	0	0
$483$	−0.418750	−0.0190538
$484$	0	0
$485$	−5.63492	−0.255869
$486$	0	0
$487$	−4.61916	−0.209314	−0.104657	−	0.994508i	$-0.533374\pi$
$487$	−4.61916	−0.209314	−0.104657	+	0.994508i	$0.533374\pi$
$488$	0	0
$489$	15.4521	0.698770
$490$	0	0
$491$	−23.8573	−1.07667	−0.538333	−	0.842732i	$-0.680946\pi$
$491$	−23.8573	−1.07667	−0.538333	+	0.842732i	$0.680946\pi$
$492$	0	0
$493$	9.36086	0.421592
$494$	0	0
$495$	−0.645535	−0.0290146
$496$	0	0
$497$	10.4461	0.468573
$498$	0	0
$499$	−21.9740	−0.983693	−0.491847	−	0.870682i	$-0.663678\pi$
$499$	−21.9740	−0.983693	−0.491847	+	0.870682i	$0.663678\pi$
$500$	0	0
$501$	−10.5874	−0.473010
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	3.54799	0.157884
$506$	0	0
$507$	−9.84296	−0.437141
$508$	0	0
$509$	−18.9967	−0.842013	−0.421007	−	0.907058i	$-0.638323\pi$
$509$	−18.9967	−0.842013	−0.421007	+	0.907058i	$0.638323\pi$
$510$	0	0
$511$	−8.80303	−0.389423
$512$	0	0
$513$	−1.70695	−0.0753638
$514$	0	0
$515$	−1.15568	−0.0509255
$516$	0	0
$517$	−10.1093	−0.444609
$518$	0	0
$519$	22.2287	0.975730
$520$	0	0
$521$	17.8359	0.781404	0.390702	−	0.920517i	$-0.372232\pi$
$521$	17.8359	0.781404	0.390702	+	0.920517i	$0.372232\pi$
$522$	0	0
$523$	−6.83897	−0.299047	−0.149524	−	0.988758i	$-0.547774\pi$
$523$	−6.83897	−0.299047	−0.149524	+	0.988758i	$0.547774\pi$
$524$	0	0
$525$	3.64305	0.158996
$526$	0	0
$527$	6.23494	0.271598
$528$	0	0
$529$	−22.6868	−0.986383
$530$	0	0
$531$	4.92700	0.213814
$532$	0	0
$533$	25.2140	1.09214
$534$	0	0
$535$	−3.61595	−0.156331
$536$	0	0
$537$	4.78638	0.206548
$538$	0	0
$539$	11.4727	0.494166
$540$	0	0
$541$	26.1506	1.12430	0.562150	−	0.827035i	$-0.309974\pi$
$541$	26.1506	1.12430	0.562150	+	0.827035i	$0.309974\pi$
$542$	0	0
$543$	−3.86910	−0.166039
$544$	0	0
$545$	2.13906	0.0916274
$546$	0	0
$547$	−8.70868	−0.372356	−0.186178	−	0.982516i	$-0.559610\pi$
$547$	−8.70868	−0.372356	−0.186178	+	0.982516i	$0.559610\pi$
$548$	0	0
$549$	−8.26261	−0.352640
$550$	0	0
$551$	−13.3152	−0.567248
$552$	0	0
$553$	8.57461	0.364630
$554$	0	0
$555$	0.316233	0.0134233
$556$	0	0
$557$	12.4633	0.528087	0.264044	−	0.964511i	$-0.414944\pi$
$557$	12.4633	0.528087	0.264044	+	0.964511i	$0.414944\pi$
$558$	0	0
$559$	−48.6208	−2.05644
$560$	0	0
$561$	−2.13778	−0.0902570
$562$	0	0
$563$	3.97990	0.167733	0.0838664	−	0.996477i	$-0.473273\pi$
$563$	3.97990	0.167733	0.0838664	+	0.996477i	$0.473273\pi$
$564$	0	0
$565$	3.16438	0.133127
$566$	0	0
$567$	0.748260	0.0314240
$568$	0	0
$569$	−37.2488	−1.56155	−0.780775	−	0.624813i	$-0.785175\pi$
$569$	−37.2488	−1.56155	−0.780775	+	0.624813i	$0.785175\pi$
$570$	0	0
$571$	−32.6117	−1.36476	−0.682378	−	0.730999i	$-0.739054\pi$
$571$	−32.6117	−1.36476	−0.682378	+	0.730999i	$0.739054\pi$
$572$	0	0
$573$	5.86319	0.244938
$574$	0	0
$575$	−2.72468	−0.113627
$576$	0	0
$577$	−6.47122	−0.269400	−0.134700	−	0.990886i	$-0.543007\pi$
$577$	−6.47122	−0.269400	−0.134700	+	0.990886i	$0.543007\pi$
$578$	0	0
$579$	−2.83984	−0.118020
$580$	0	0
$581$	8.94914	0.371273
$582$	0	0
$583$	3.35542	0.138967
$584$	0	0
$585$	1.73190	0.0716051
$586$	0	0
$587$	20.7280	0.855537	0.427769	−	0.903888i	$-0.359300\pi$
$587$	20.7280	0.855537	0.427769	+	0.903888i	$0.359300\pi$
$588$	0	0
$589$	−8.86882	−0.365433
$590$	0	0
$591$	7.98428	0.328429
$592$	0	0
$593$	−29.5092	−1.21180	−0.605898	−	0.795542i	$-0.707186\pi$
$593$	−29.5092	−1.21180	−0.605898	+	0.795542i	$0.707186\pi$
$594$	0	0
$595$	−0.325377	−0.0133391
$596$	0	0
$597$	−0.277867	−0.0113723
$598$	0	0
$599$	22.4486	0.917225	0.458612	−	0.888636i	$-0.348347\pi$
$599$	22.4486	0.917225	0.458612	+	0.888636i	$0.348347\pi$
$600$	0	0
$601$	−23.7897	−0.970403	−0.485202	−	0.874402i	$-0.661254\pi$
$601$	−23.7897	−0.970403	−0.485202	+	0.874402i	$0.661254\pi$
$602$	0	0
$603$	−0.345994	−0.0140900
$604$	0	0
$605$	−2.83602	−0.115301
$606$	0	0
$607$	21.5235	0.873611	0.436805	−	0.899556i	$-0.356110\pi$
$607$	21.5235	0.873611	0.436805	+	0.899556i	$0.356110\pi$
$608$	0	0
$609$	5.83687	0.236522
$610$	0	0
$611$	27.1222	1.09725
$612$	0	0
$613$	17.1181	0.691393	0.345697	−	0.938346i	$-0.387643\pi$
$613$	17.1181	0.691393	0.345697	+	0.938346i	$0.387643\pi$
$614$	0	0
$615$	−1.91167	−0.0770859
$616$	0	0
$617$	−3.82238	−0.153883	−0.0769417	−	0.997036i	$-0.524516\pi$
$617$	−3.82238	−0.153883	−0.0769417	+	0.997036i	$0.524516\pi$
$618$	0	0
$619$	10.2615	0.412444	0.206222	−	0.978505i	$-0.433883\pi$
$619$	10.2615	0.412444	0.206222	+	0.978505i	$0.433883\pi$
$620$	0	0
$621$	−0.559632	−0.0224573
$622$	0	0
$623$	−11.2768	−0.451797
$624$	0	0
$625$	23.0476	0.921905
$626$	0	0
$627$	3.04085	0.121440
$628$	0	0
$629$	1.04725	0.0417565
$630$	0	0
$631$	−15.7928	−0.628703	−0.314352	−	0.949307i	$-0.601787\pi$
$631$	−15.7928	−0.628703	−0.314352	+	0.949307i	$0.601787\pi$
$632$	0	0
$633$	17.4352	0.692985
$634$	0	0
$635$	−2.79541	−0.110932
$636$	0	0
$637$	−30.7800	−1.21955
$638$	0	0
$639$	13.9606	0.552271
$640$	0	0
$641$	−45.2752	−1.78826	−0.894131	−	0.447805i	$-0.852206\pi$
$641$	−45.2752	−1.78826	−0.894131	+	0.447805i	$0.852206\pi$
$642$	0	0
$643$	34.3268	1.35371	0.676857	−	0.736114i	$-0.263341\pi$
$643$	34.3268	1.35371	0.676857	+	0.736114i	$0.263341\pi$
$644$	0	0
$645$	3.68631	0.145148
$646$	0	0
$647$	32.2191	1.26666	0.633332	−	0.773880i	$-0.281687\pi$
$647$	32.2191	1.26666	0.633332	+	0.773880i	$0.281687\pi$
$648$	0	0
$649$	−8.77722	−0.344536
$650$	0	0
$651$	3.88773	0.152372
$652$	0	0
$653$	30.4055	1.18986	0.594929	−	0.803778i	$-0.297180\pi$
$653$	30.4055	1.18986	0.594929	+	0.803778i	$0.297180\pi$
$654$	0	0
$655$	1.53279	0.0598912
$656$	0	0
$657$	−11.7647	−0.458984
$658$	0	0
$659$	7.55614	0.294345	0.147173	−	0.989111i	$-0.452983\pi$
$659$	7.55614	0.294345	0.147173	+	0.989111i	$0.452983\pi$
$660$	0	0
$661$	24.4677	0.951684	0.475842	−	0.879531i	$-0.342143\pi$
$661$	24.4677	0.951684	0.475842	+	0.879531i	$0.342143\pi$
$662$	0	0
$663$	5.73541	0.222745
$664$	0	0
$665$	0.462828	0.0179477
$666$	0	0
$667$	−4.36546	−0.169031
$668$	0	0
$669$	10.1614	0.392864
$670$	0	0
$671$	14.7194	0.568237
$672$	0	0
$673$	−21.1099	−0.813727	−0.406863	−	0.913489i	$-0.633378\pi$
$673$	−21.1099	−0.813727	−0.406863	+	0.913489i	$0.633378\pi$
$674$	0	0
$675$	4.86869	0.187396
$676$	0	0
$677$	5.33301	0.204964	0.102482	−	0.994735i	$-0.467322\pi$
$677$	5.33301	0.204964	0.102482	+	0.994735i	$0.467322\pi$
$678$	0	0
$679$	−11.6357	−0.446539
$680$	0	0
$681$	−4.98533	−0.191038
$682$	0	0
$683$	−13.0010	−0.497469	−0.248734	−	0.968572i	$-0.580015\pi$
$683$	−13.0010	−0.497469	−0.248734	+	0.968572i	$0.580015\pi$
$684$	0	0
$685$	2.67513	0.102211
$686$	0	0
$687$	5.63725	0.215074
$688$	0	0
$689$	−9.00221	−0.342957
$690$	0	0
$691$	−33.2944	−1.26658	−0.633289	−	0.773916i	$-0.718295\pi$
$691$	−33.2944	−1.26658	−0.633289	+	0.773916i	$0.718295\pi$
$692$	0	0
$693$	−1.33299	−0.0506360
$694$	0	0
$695$	3.85960	0.146403
$696$	0	0
$697$	−6.33074	−0.239794
$698$	0	0
$699$	1.07165	0.0405334
$700$	0	0
$701$	19.5720	0.739222	0.369611	−	0.929187i	$-0.379491\pi$
$701$	19.5720	0.739222	0.369611	+	0.929187i	$0.379491\pi$
$702$	0	0
$703$	−1.48964	−0.0561830
$704$	0	0
$705$	−2.05634	−0.0774463
$706$	0	0
$707$	7.32637	0.275537
$708$	0	0
$709$	9.78270	0.367397	0.183699	−	0.982983i	$-0.441193\pi$
$709$	9.78270	0.367397	0.183699	+	0.982983i	$0.441193\pi$
$710$	0	0
$711$	11.4594	0.429761
$712$	0	0
$713$	−2.90768	−0.108894
$714$	0	0
$715$	−3.08529	−0.115383
$716$	0	0
$717$	20.3357	0.759450
$718$	0	0
$719$	30.4332	1.13497	0.567484	−	0.823384i	$-0.307917\pi$
$719$	30.4332	1.13497	0.567484	+	0.823384i	$0.307917\pi$
$720$	0	0
$721$	−2.38641	−0.0888746
$722$	0	0
$723$	1.21543	0.0452023
$724$	0	0
$725$	37.9787	1.41049
$726$	0	0
$727$	−1.81336	−0.0672540	−0.0336270	−	0.999434i	$-0.510706\pi$
$727$	−1.81336	−0.0672540	−0.0336270	+	0.999434i	$0.510706\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.2077	0.451519
$732$	0	0
$733$	−37.1759	−1.37312	−0.686562	−	0.727071i	$-0.740881\pi$
$733$	−37.1759	−1.37312	−0.686562	+	0.727071i	$0.740881\pi$
$734$	0	0
$735$	2.33367	0.0860787
$736$	0	0
$737$	0.616371	0.0227043
$738$	0	0
$739$	25.5022	0.938115	0.469057	−	0.883168i	$-0.344594\pi$
$739$	25.5022	0.938115	0.469057	+	0.883168i	$0.344594\pi$
$740$	0	0
$741$	−8.15826	−0.299701
$742$	0	0
$743$	−24.3967	−0.895028	−0.447514	−	0.894277i	$-0.647690\pi$
$743$	−24.3967	−0.895028	−0.447514	+	0.894277i	$0.647690\pi$
$744$	0	0
$745$	−4.71256	−0.172655
$746$	0	0
$747$	11.9599	0.437591
$748$	0	0
$749$	−7.46671	−0.272828
$750$	0	0
$751$	−40.4706	−1.47679	−0.738396	−	0.674368i	$-0.764416\pi$
$751$	−40.4706	−1.47679	−0.738396	+	0.674368i	$0.764416\pi$
$752$	0	0
$753$	4.45275	0.162267
$754$	0	0
$755$	−6.77191	−0.246455
$756$	0	0
$757$	11.4028	0.414442	0.207221	−	0.978294i	$-0.433558\pi$
$757$	11.4028	0.414442	0.207221	+	0.978294i	$0.433558\pi$
$758$	0	0
$759$	0.996957	0.0361873
$760$	0	0
$761$	−18.9423	−0.686658	−0.343329	−	0.939215i	$-0.611555\pi$
$761$	−18.9423	−0.686658	−0.343329	+	0.939215i	$0.611555\pi$
$762$	0	0
$763$	4.41703	0.159907
$764$	0	0
$765$	−0.434845	−0.0157218
$766$	0	0
$767$	23.5483	0.850279
$768$	0	0
$769$	−25.0048	−0.901696	−0.450848	−	0.892601i	$-0.648878\pi$
$769$	−25.0048	−0.901696	−0.450848	+	0.892601i	$0.648878\pi$
$770$	0	0
$771$	20.7639	0.747793
$772$	0	0
$773$	−46.2900	−1.66494	−0.832468	−	0.554073i	$-0.813073\pi$
$773$	−46.2900	−1.66494	−0.832468	+	0.554073i	$0.813073\pi$
$774$	0	0
$775$	25.2963	0.908669
$776$	0	0
$777$	0.653000	0.0234262
$778$	0	0
$779$	9.00509	0.322641
$780$	0	0
$781$	−24.8701	−0.889921
$782$	0	0
$783$	7.80059	0.278770
$784$	0	0
$785$	6.39450	0.228229
$786$	0	0
$787$	26.7578	0.953813	0.476907	−	0.878954i	$-0.341758\pi$
$787$	26.7578	0.953813	0.476907	+	0.878954i	$0.341758\pi$
$788$	0	0
$789$	−12.6893	−0.451751
$790$	0	0
$791$	6.53424	0.232331
$792$	0	0
$793$	−39.4906	−1.40235
$794$	0	0
$795$	0.682525	0.0242067
$796$	0	0
$797$	25.1696	0.891555	0.445777	−	0.895144i	$-0.352927\pi$
$797$	25.1696	0.891555	0.445777	+	0.895144i	$0.352927\pi$
$798$	0	0
$799$	−6.80985	−0.240915
$800$	0	0
$801$	−15.0708	−0.532499
$802$	0	0
$803$	20.9582	0.739599
$804$	0	0
$805$	0.151740	0.00534814
$806$	0	0
$807$	−5.74993	−0.202407
$808$	0	0
$809$	40.4113	1.42078	0.710392	−	0.703806i	$-0.248518\pi$
$809$	40.4113	1.42078	0.710392	+	0.703806i	$0.248518\pi$
$810$	0	0
$811$	−12.5959	−0.442303	−0.221151	−	0.975240i	$-0.570981\pi$
$811$	−12.5959	−0.442303	−0.221151	+	0.975240i	$0.570981\pi$
$812$	0	0
$813$	2.37825	0.0834089
$814$	0	0
$815$	−5.59931	−0.196135
$816$	0	0
$817$	−17.3647	−0.607514
$818$	0	0
$819$	3.57626	0.124964
$820$	0	0
$821$	−43.0135	−1.50118	−0.750590	−	0.660768i	$-0.770231\pi$
$821$	−43.0135	−1.50118	−0.750590	+	0.660768i	$0.770231\pi$
$822$	0	0
$823$	−14.2143	−0.495480	−0.247740	−	0.968827i	$-0.579688\pi$
$823$	−14.2143	−0.495480	−0.247740	+	0.968827i	$0.579688\pi$
$824$	0	0
$825$	−8.67334	−0.301967
$826$	0	0
$827$	2.16318	0.0752210	0.0376105	−	0.999292i	$-0.488025\pi$
$827$	2.16318	0.0752210	0.0376105	+	0.999292i	$0.488025\pi$
$828$	0	0
$829$	30.1683	1.04779	0.523893	−	0.851784i	$-0.324479\pi$
$829$	30.1683	1.04779	0.523893	+	0.851784i	$0.324479\pi$
$830$	0	0
$831$	3.16071	0.109644
$832$	0	0
$833$	7.72825	0.267768
$834$	0	0
$835$	3.83650	0.132768
$836$	0	0
$837$	5.19570	0.179590
$838$	0	0
$839$	−3.68864	−0.127346	−0.0636729	−	0.997971i	$-0.520281\pi$
$839$	−3.68864	−0.127346	−0.0636729	+	0.997971i	$0.520281\pi$
$840$	0	0
$841$	31.8492	1.09825
$842$	0	0
$843$	0.886688	0.0305392
$844$	0	0
$845$	3.56674	0.122700
$846$	0	0
$847$	−5.85620	−0.201222
$848$	0	0
$849$	−22.7832	−0.781916
$850$	0	0
$851$	−0.488386	−0.0167417
$852$	0	0
$853$	36.5674	1.25204	0.626022	−	0.779805i	$-0.284682\pi$
$853$	36.5674	1.25204	0.626022	+	0.779805i	$0.284682\pi$
$854$	0	0
$855$	0.618540	0.0211536
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	32.1413	1.09665	0.548323	−	0.836267i	$-0.315267\pi$
$859$	32.1413	1.09665	0.548323	+	0.836267i	$0.315267\pi$
$860$	0	0
$861$	−3.94747	−0.134529
$862$	0	0
$863$	−27.4806	−0.935450	−0.467725	−	0.883874i	$-0.654926\pi$
$863$	−27.4806	−0.935450	−0.467725	+	0.883874i	$0.654926\pi$
$864$	0	0
$865$	−8.05489	−0.273874
$866$	0	0
$867$	15.5600	0.528444
$868$	0	0
$869$	−20.4144	−0.692510
$870$	0	0
$871$	−1.65365	−0.0560319
$872$	0	0
$873$	−15.5504	−0.526302
$874$	0	0
$875$	−2.67583	−0.0904594
$876$	0	0
$877$	6.31724	0.213318	0.106659	−	0.994296i	$-0.465985\pi$
$877$	6.31724	0.213318	0.106659	+	0.994296i	$0.465985\pi$
$878$	0	0
$879$	−11.2177	−0.378362
$880$	0	0
$881$	40.9209	1.37866	0.689330	−	0.724447i	$-0.257905\pi$
$881$	40.9209	1.37866	0.689330	+	0.724447i	$0.257905\pi$
$882$	0	0
$883$	−10.2184	−0.343878	−0.171939	−	0.985108i	$-0.555003\pi$
$883$	−10.2184	−0.343878	−0.171939	+	0.985108i	$0.555003\pi$
$884$	0	0
$885$	−1.78537	−0.0600147
$886$	0	0
$887$	17.4688	0.586544	0.293272	−	0.956029i	$-0.405256\pi$
$887$	17.4688	0.586544	0.293272	+	0.956029i	$0.405256\pi$
$888$	0	0
$889$	−5.77234	−0.193598
$890$	0	0
$891$	−1.78145	−0.0596809
$892$	0	0
$893$	9.68659	0.324149
$894$	0	0
$895$	−1.73442	−0.0579752
$896$	0	0
$897$	−2.67472	−0.0893064
$898$	0	0
$899$	40.5295	1.35174
$900$	0	0
$901$	2.26027	0.0753006
$902$	0	0
$903$	7.61199	0.253311
$904$	0	0
$905$	1.40202	0.0466049
$906$	0	0
$907$	22.2872	0.740033	0.370017	−	0.929025i	$-0.379352\pi$
$907$	22.2872	0.740033	0.370017	+	0.929025i	$0.379352\pi$
$908$	0	0
$909$	9.79122	0.324754
$910$	0	0
$911$	2.96849	0.0983505	0.0491752	−	0.998790i	$-0.484341\pi$
$911$	2.96849	0.0983505	0.0491752	+	0.998790i	$0.484341\pi$
$912$	0	0
$913$	−21.3060	−0.705127
$914$	0	0
$915$	2.99408	0.0989812
$916$	0	0
$917$	3.16512	0.104521
$918$	0	0
$919$	19.7060	0.650041	0.325021	−	0.945707i	$-0.394629\pi$
$919$	19.7060	0.650041	0.325021	+	0.945707i	$0.394629\pi$
$920$	0	0
$921$	4.18794	0.137997
$922$	0	0
$923$	66.7235	2.19623
$924$	0	0
$925$	4.24887	0.139702
$926$	0	0
$927$	−3.18928	−0.104750
$928$	0	0
$929$	29.4625	0.966634	0.483317	−	0.875446i	$-0.339432\pi$
$929$	29.4625	0.966634	0.483317	+	0.875446i	$0.339432\pi$
$930$	0	0
$931$	−10.9930	−0.360280
$932$	0	0
$933$	−19.7539	−0.646713
$934$	0	0
$935$	0.774655	0.0253339
$936$	0	0
$937$	−8.93193	−0.291794	−0.145897	−	0.989300i	$-0.546607\pi$
$937$	−8.93193	−0.291794	−0.145897	+	0.989300i	$0.546607\pi$
$938$	0	0
$939$	31.0441	1.01309
$940$	0	0
$941$	−9.73715	−0.317422	−0.158711	−	0.987325i	$-0.550734\pi$
$941$	−9.73715	−0.317422	−0.158711	+	0.987325i	$0.550734\pi$
$942$	0	0
$943$	2.95236	0.0961420
$944$	0	0
$945$	−0.271143	−0.00882028
$946$	0	0
$947$	9.34828	0.303778	0.151889	−	0.988398i	$-0.451464\pi$
$947$	9.34828	0.303778	0.151889	+	0.988398i	$0.451464\pi$
$948$	0	0
$949$	−56.2285	−1.82525
$950$	0	0
$951$	23.4958	0.761903
$952$	0	0
$953$	41.9950	1.36035	0.680175	−	0.733049i	$-0.261904\pi$
$953$	41.9950	1.36035	0.680175	+	0.733049i	$0.261904\pi$
$954$	0	0
$955$	−2.12461	−0.0687509
$956$	0	0
$957$	−13.8964	−0.449206
$958$	0	0
$959$	5.52396	0.178378
$960$	0	0
$961$	−4.00468	−0.129183
$962$	0	0
$963$	−9.97877	−0.321561
$964$	0	0
$965$	1.02906	0.0331265
$966$	0	0
$967$	−40.4991	−1.30236	−0.651181	−	0.758922i	$-0.725726\pi$
$967$	−40.4991	−1.30236	−0.651181	+	0.758922i	$0.725726\pi$
$968$	0	0
$969$	2.04838	0.0658033
$970$	0	0
$971$	−0.799125	−0.0256452	−0.0128226	−	0.999918i	$-0.504082\pi$
$971$	−0.799125	−0.0256452	−0.0128226	+	0.999918i	$0.504082\pi$
$972$	0	0
$973$	7.96982	0.255501
$974$	0	0
$975$	23.2696	0.745223
$976$	0	0
$977$	−37.3966	−1.19642	−0.598211	−	0.801339i	$-0.704122\pi$
$977$	−37.3966	−1.19642	−0.598211	+	0.801339i	$0.704122\pi$
$978$	0	0
$979$	26.8478	0.858060
$980$	0	0
$981$	5.90307	0.188470
$982$	0	0
$983$	44.9145	1.43255	0.716274	−	0.697819i	$-0.245846\pi$
$983$	44.9145	1.43255	0.716274	+	0.697819i	$0.245846\pi$
$984$	0	0
$985$	−2.89322	−0.0921857
$986$	0	0
$987$	−4.24621	−0.135158
$988$	0	0
$989$	−5.69310	−0.181030
$990$	0	0
$991$	7.79965	0.247764	0.123882	−	0.992297i	$-0.460466\pi$
$991$	7.79965	0.247764	0.123882	+	0.992297i	$0.460466\pi$
$992$	0	0
$993$	−28.5747	−0.906789
$994$	0	0
$995$	0.100689	0.00319206
$996$	0	0
$997$	21.0040	0.665203	0.332601	−	0.943067i	$-0.392074\pi$
$997$	21.0040	0.665203	0.332601	+	0.943067i	$0.392074\pi$
$998$	0	0
$999$	0.872692	0.0276107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.f.1.10	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.f.1.10	✓	14	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.362365 q^{5} +0.748260 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.362365 q^{5} +0.748260 q^{7} +1.00000 q^{9} -1.78145 q^{11} +4.77943 q^{13} -0.362365 q^{15} -1.20002 q^{17} +1.70695 q^{19} -0.748260 q^{21} +0.559632 q^{23} -4.86869 q^{25} -1.00000 q^{27} -7.80059 q^{29} -5.19570 q^{31} +1.78145 q^{33} +0.271143 q^{35} -0.872692 q^{37} -4.77943 q^{39} +5.27553 q^{41} -10.1729 q^{43} +0.362365 q^{45} +5.67478 q^{47} -6.44011 q^{49} +1.20002 q^{51} -1.88353 q^{53} -0.645535 q^{55} -1.70695 q^{57} +4.92700 q^{59} -8.26261 q^{61} +0.748260 q^{63} +1.73190 q^{65} -0.345994 q^{67} -0.559632 q^{69} +13.9606 q^{71} -11.7647 q^{73} +4.86869 q^{75} -1.33299 q^{77} +11.4594 q^{79} +1.00000 q^{81} +11.9599 q^{83} -0.434845 q^{85} +7.80059 q^{87} -15.0708 q^{89} +3.57626 q^{91} +5.19570 q^{93} +0.618540 q^{95} -15.5504 q^{97} -1.78145 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9} + q^{11} - q^{13} + 6 q^{15} - 6 q^{17} + q^{19} + 7 q^{21} + 10 q^{23} - 10 q^{25} - 14 q^{27} - 6 q^{29} - 5 q^{31} - q^{33} + 17 q^{35} - 12 q^{37} + q^{39} - 21 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{47} - 19 q^{49} + 6 q^{51} - q^{53} - q^{57} + 14 q^{59} - 19 q^{61} - 7 q^{63} + 17 q^{65} + 17 q^{67} - 10 q^{69} - 13 q^{71} - 12 q^{73} + 10 q^{75} - 9 q^{77} - 8 q^{79} + 14 q^{81} + 11 q^{83} - 17 q^{85} + 6 q^{87} - 9 q^{89} - 5 q^{91} + 5 q^{93} + 8 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9 + q^11 - q^13 + 6 * q^15 - 6 * q^17 + q^19 + 7 * q^21 + 10 * q^23 - 10 * q^25 - 14 * q^27 - 6 * q^29 - 5 * q^31 - q^33 + 17 * q^35 - 12 * q^37 + q^39 - 21 * q^41 + 8 * q^43 - 6 * q^45 + 18 * q^47 - 19 * q^49 + 6 * q^51 - q^53 - q^57 + 14 * q^59 - 19 * q^61 - 7 * q^63 + 17 * q^65 + 17 * q^67 - 10 * q^69 - 13 * q^71 - 12 * q^73 + 10 * q^75 - 9 * q^77 - 8 * q^79 + 14 * q^81 + 11 * q^83 - 17 * q^85 + 6 * q^87 - 9 * q^89 - 5 * q^91 + 5 * q^93 + 8 * q^95 - 25 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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