LMFDB - Embedded newform 6024.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$0$
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} - 29 x^{12} + 210 x^{11} + 280 x^{10} - 2796 x^{9} - 863 x^{8} + 17652 x^{7} + \cdots - 3364 $ x^14 - 6x^13 - 29x^12 + 210x^11 + 280x^10 - 2796x^9 - 863x^8 + 17652x^7 - 1300x^6 - 53458x^5 + 10892x^4 + 67298x^3 - 13761x^2 - 22140*x - 3364
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.30404$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.30404 q^{5} -0.450333 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.30404 q^{5} -0.450333 q^{7} +1.00000 q^{9} -3.25032 q^{11} -0.857958 q^{13} +3.30404 q^{15} -0.105230 q^{17} -7.07933 q^{19} +0.450333 q^{21} -5.24654 q^{23} +5.91670 q^{25} -1.00000 q^{27} +1.37602 q^{29} -1.41438 q^{31} +3.25032 q^{33} +1.48792 q^{35} -7.72697 q^{37} +0.857958 q^{39} +7.02839 q^{41} -6.52856 q^{43} -3.30404 q^{45} -2.73518 q^{47} -6.79720 q^{49} +0.105230 q^{51} -3.85852 q^{53} +10.7392 q^{55} +7.07933 q^{57} -1.73811 q^{59} -9.85050 q^{61} -0.450333 q^{63} +2.83473 q^{65} -9.84986 q^{67} +5.24654 q^{69} +12.2757 q^{71} -2.94578 q^{73} -5.91670 q^{75} +1.46373 q^{77} -1.59014 q^{79} +1.00000 q^{81} +1.64530 q^{83} +0.347684 q^{85} -1.37602 q^{87} -1.73887 q^{89} +0.386366 q^{91} +1.41438 q^{93} +23.3904 q^{95} -7.69389 q^{97} -3.25032 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * 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
$5$	−3.30404	−1.47761	−0.738806	−	0.673918i	$-0.764610\pi$
$5$	−3.30404	−1.47761	−0.738806	+	0.673918i	$0.764610\pi$
$6$	0	0
$7$	−0.450333	−0.170210	−0.0851049	−	0.996372i	$-0.527123\pi$
$7$	−0.450333	−0.170210	−0.0851049	+	0.996372i	$0.527123\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.25032	−0.980008	−0.490004	−	0.871720i	$-0.663005\pi$
$11$	−3.25032	−0.980008	−0.490004	+	0.871720i	$0.663005\pi$
$12$	0	0
$13$	−0.857958	−0.237955	−0.118977	−	0.992897i	$-0.537962\pi$
$13$	−0.857958	−0.237955	−0.118977	+	0.992897i	$0.537962\pi$
$14$	0	0
$15$	3.30404	0.853100
$16$	0	0
$17$	−0.105230	−0.0255220	−0.0127610	−	0.999919i	$-0.504062\pi$
$17$	−0.105230	−0.0255220	−0.0127610	+	0.999919i	$0.504062\pi$
$18$	0	0
$19$	−7.07933	−1.62411	−0.812055	−	0.583581i	$-0.801651\pi$
$19$	−7.07933	−1.62411	−0.812055	+	0.583581i	$0.801651\pi$
$20$	0	0
$21$	0.450333	0.0982706
$22$	0	0
$23$	−5.24654	−1.09398	−0.546990	−	0.837139i	$-0.684226\pi$
$23$	−5.24654	−1.09398	−0.546990	+	0.837139i	$0.684226\pi$
$24$	0	0
$25$	5.91670	1.18334
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.37602	0.255521	0.127760	−	0.991805i	$-0.459221\pi$
$29$	1.37602	0.255521	0.127760	+	0.991805i	$0.459221\pi$
$30$	0	0
$31$	−1.41438	−0.254029	−0.127015	−	0.991901i	$-0.540540\pi$
$31$	−1.41438	−0.254029	−0.127015	+	0.991901i	$0.540540\pi$
$32$	0	0
$33$	3.25032	0.565808
$34$	0	0
$35$	1.48792	0.251504
$36$	0	0
$37$	−7.72697	−1.27031	−0.635153	−	0.772386i	$-0.719063\pi$
$37$	−7.72697	−1.27031	−0.635153	+	0.772386i	$0.719063\pi$
$38$	0	0
$39$	0.857958	0.137383
$40$	0	0
$41$	7.02839	1.09765	0.548825	−	0.835937i	$-0.315075\pi$
$41$	7.02839	1.09765	0.548825	+	0.835937i	$0.315075\pi$
$42$	0	0
$43$	−6.52856	−0.995596	−0.497798	−	0.867293i	$-0.665858\pi$
$43$	−6.52856	−0.995596	−0.497798	+	0.867293i	$0.665858\pi$
$44$	0	0
$45$	−3.30404	−0.492538
$46$	0	0
$47$	−2.73518	−0.398968	−0.199484	−	0.979901i	$-0.563927\pi$
$47$	−2.73518	−0.398968	−0.199484	+	0.979901i	$0.563927\pi$
$48$	0	0
$49$	−6.79720	−0.971029
$50$	0	0
$51$	0.105230	0.0147351
$52$	0	0
$53$	−3.85852	−0.530009	−0.265004	−	0.964247i	$-0.585373\pi$
$53$	−3.85852	−0.530009	−0.265004	+	0.964247i	$0.585373\pi$
$54$	0	0
$55$	10.7392	1.44807
$56$	0	0
$57$	7.07933	0.937680
$58$	0	0
$59$	−1.73811	−0.226283	−0.113141	−	0.993579i	$-0.536091\pi$
$59$	−1.73811	−0.226283	−0.113141	+	0.993579i	$0.536091\pi$
$60$	0	0
$61$	−9.85050	−1.26123	−0.630614	−	0.776097i	$-0.717197\pi$
$61$	−9.85050	−1.26123	−0.630614	+	0.776097i	$0.717197\pi$
$62$	0	0
$63$	−0.450333	−0.0567366
$64$	0	0
$65$	2.83473	0.351605
$66$	0	0
$67$	−9.84986	−1.20335	−0.601676	−	0.798740i	$-0.705500\pi$
$67$	−9.84986	−1.20335	−0.601676	+	0.798740i	$0.705500\pi$
$68$	0	0
$69$	5.24654	0.631609
$70$	0	0
$71$	12.2757	1.45686	0.728430	−	0.685120i	$-0.240250\pi$
$71$	12.2757	1.45686	0.728430	+	0.685120i	$0.240250\pi$
$72$	0	0
$73$	−2.94578	−0.344778	−0.172389	−	0.985029i	$-0.555149\pi$
$73$	−2.94578	−0.344778	−0.172389	+	0.985029i	$0.555149\pi$
$74$	0	0
$75$	−5.91670	−0.683202
$76$	0	0
$77$	1.46373	0.166807
$78$	0	0
$79$	−1.59014	−0.178905	−0.0894524	−	0.995991i	$-0.528512\pi$
$79$	−1.59014	−0.178905	−0.0894524	+	0.995991i	$0.528512\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.64530	0.180595	0.0902975	−	0.995915i	$-0.471218\pi$
$83$	1.64530	0.180595	0.0902975	+	0.995915i	$0.471218\pi$
$84$	0	0
$85$	0.347684	0.0377116
$86$	0	0
$87$	−1.37602	−0.147525
$88$	0	0
$89$	−1.73887	−0.184320	−0.0921601	−	0.995744i	$-0.529377\pi$
$89$	−1.73887	−0.184320	−0.0921601	+	0.995744i	$0.529377\pi$
$90$	0	0
$91$	0.386366	0.0405022
$92$	0	0
$93$	1.41438	0.146664
$94$	0	0
$95$	23.3904	2.39981
$96$	0	0
$97$	−7.69389	−0.781196	−0.390598	−	0.920561i	$-0.627732\pi$
$97$	−7.69389	−0.781196	−0.390598	+	0.920561i	$0.627732\pi$
$98$	0	0
$99$	−3.25032	−0.326669
$100$	0	0
$101$	8.47436	0.843231	0.421615	−	0.906775i	$-0.361463\pi$
$101$	8.47436	0.843231	0.421615	+	0.906775i	$0.361463\pi$
$102$	0	0
$103$	−10.5577	−1.04028	−0.520141	−	0.854080i	$-0.674121\pi$
$103$	−10.5577	−1.04028	−0.520141	+	0.854080i	$0.674121\pi$
$104$	0	0
$105$	−1.48792	−0.145206
$106$	0	0
$107$	−8.39020	−0.811111	−0.405555	−	0.914070i	$-0.632922\pi$
$107$	−8.39020	−0.811111	−0.405555	+	0.914070i	$0.632922\pi$
$108$	0	0
$109$	−4.72074	−0.452165	−0.226083	−	0.974108i	$-0.572592\pi$
$109$	−4.72074	−0.452165	−0.226083	+	0.974108i	$0.572592\pi$
$110$	0	0
$111$	7.72697	0.733411
$112$	0	0
$113$	−2.41657	−0.227331	−0.113666	−	0.993519i	$-0.536259\pi$
$113$	−2.41657	−0.227331	−0.113666	+	0.993519i	$0.536259\pi$
$114$	0	0
$115$	17.3348	1.61648
$116$	0	0
$117$	−0.857958	−0.0793182
$118$	0	0
$119$	0.0473884	0.00434409
$120$	0	0
$121$	−0.435418	−0.0395835
$122$	0	0
$123$	−7.02839	−0.633729
$124$	0	0
$125$	−3.02881	−0.270905
$126$	0	0
$127$	16.9207	1.50147	0.750735	−	0.660603i	$-0.229699\pi$
$127$	16.9207	1.50147	0.750735	+	0.660603i	$0.229699\pi$
$128$	0	0
$129$	6.52856	0.574808
$130$	0	0
$131$	−10.0042	−0.874073	−0.437036	−	0.899444i	$-0.643972\pi$
$131$	−10.0042	−0.874073	−0.437036	+	0.899444i	$0.643972\pi$
$132$	0	0
$133$	3.18805	0.276439
$134$	0	0
$135$	3.30404	0.284367
$136$	0	0
$137$	5.15265	0.440220	0.220110	−	0.975475i	$-0.429358\pi$
$137$	5.15265	0.440220	0.220110	+	0.975475i	$0.429358\pi$
$138$	0	0
$139$	12.5932	1.06814	0.534072	−	0.845439i	$-0.320661\pi$
$139$	12.5932	1.06814	0.534072	+	0.845439i	$0.320661\pi$
$140$	0	0
$141$	2.73518	0.230344
$142$	0	0
$143$	2.78864	0.233198
$144$	0	0
$145$	−4.54643	−0.377561
$146$	0	0
$147$	6.79720	0.560624
$148$	0	0
$149$	8.06303	0.660549	0.330274	−	0.943885i	$-0.392859\pi$
$149$	8.06303	0.660549	0.330274	+	0.943885i	$0.392859\pi$
$150$	0	0
$151$	−10.2888	−0.837292	−0.418646	−	0.908150i	$-0.637495\pi$
$151$	−10.2888	−0.837292	−0.418646	+	0.908150i	$0.637495\pi$
$152$	0	0
$153$	−0.105230	−0.00850732
$154$	0	0
$155$	4.67316	0.375357
$156$	0	0
$157$	1.55957	0.124467	0.0622335	−	0.998062i	$-0.480178\pi$
$157$	1.55957	0.124467	0.0622335	+	0.998062i	$0.480178\pi$
$158$	0	0
$159$	3.85852	0.306001
$160$	0	0
$161$	2.36269	0.186206
$162$	0	0
$163$	−4.45342	−0.348819	−0.174409	−	0.984673i	$-0.555802\pi$
$163$	−4.45342	−0.348819	−0.174409	+	0.984673i	$0.555802\pi$
$164$	0	0
$165$	−10.7392	−0.836045
$166$	0	0
$167$	−3.46303	−0.267977	−0.133989	−	0.990983i	$-0.542779\pi$
$167$	−3.46303	−0.267977	−0.133989	+	0.990983i	$0.542779\pi$
$168$	0	0
$169$	−12.2639	−0.943378
$170$	0	0
$171$	−7.07933	−0.541370
$172$	0	0
$173$	−4.38865	−0.333663	−0.166831	−	0.985985i	$-0.553354\pi$
$173$	−4.38865	−0.333663	−0.166831	+	0.985985i	$0.553354\pi$
$174$	0	0
$175$	−2.66448	−0.201416
$176$	0	0
$177$	1.73811	0.130644
$178$	0	0
$179$	−5.67834	−0.424419	−0.212210	−	0.977224i	$-0.568066\pi$
$179$	−5.67834	−0.424419	−0.212210	+	0.977224i	$0.568066\pi$
$180$	0	0
$181$	8.41598	0.625555	0.312777	−	0.949826i	$-0.398741\pi$
$181$	8.41598	0.625555	0.312777	+	0.949826i	$0.398741\pi$
$182$	0	0
$183$	9.85050	0.728170
$184$	0	0
$185$	25.5302	1.87702
$186$	0	0
$187$	0.342030	0.0250117
$188$	0	0
$189$	0.450333	0.0327569
$190$	0	0
$191$	4.95642	0.358634	0.179317	−	0.983791i	$-0.442611\pi$
$191$	4.95642	0.358634	0.179317	+	0.983791i	$0.442611\pi$
$192$	0	0
$193$	3.97943	0.286445	0.143223	−	0.989690i	$-0.454253\pi$
$193$	3.97943	0.286445	0.143223	+	0.989690i	$0.454253\pi$
$194$	0	0
$195$	−2.83473	−0.202999
$196$	0	0
$197$	15.9071	1.13333	0.566665	−	0.823948i	$-0.308233\pi$
$197$	15.9071	1.13333	0.566665	+	0.823948i	$0.308233\pi$
$198$	0	0
$199$	−6.47868	−0.459261	−0.229631	−	0.973278i	$-0.573752\pi$
$199$	−6.47868	−0.459261	−0.229631	+	0.973278i	$0.573752\pi$
$200$	0	0
$201$	9.84986	0.694756
$202$	0	0
$203$	−0.619667	−0.0434921
$204$	0	0
$205$	−23.2221	−1.62190
$206$	0	0
$207$	−5.24654	−0.364660
$208$	0	0
$209$	23.0101	1.59164
$210$	0	0
$211$	−5.12881	−0.353082	−0.176541	−	0.984293i	$-0.556491\pi$
$211$	−5.12881	−0.353082	−0.176541	+	0.984293i	$0.556491\pi$
$212$	0	0
$213$	−12.2757	−0.841119
$214$	0	0
$215$	21.5706	1.47111
$216$	0	0
$217$	0.636940	0.0432383
$218$	0	0
$219$	2.94578	0.199058
$220$	0	0
$221$	0.0902827	0.00607307
$222$	0	0
$223$	−9.47092	−0.634219	−0.317110	−	0.948389i	$-0.602712\pi$
$223$	−9.47092	−0.634219	−0.317110	+	0.948389i	$0.602712\pi$
$224$	0	0
$225$	5.91670	0.394447
$226$	0	0
$227$	5.67241	0.376491	0.188246	−	0.982122i	$-0.439720\pi$
$227$	5.67241	0.376491	0.188246	+	0.982122i	$0.439720\pi$
$228$	0	0
$229$	16.6636	1.10116	0.550580	−	0.834782i	$-0.314407\pi$
$229$	16.6636	1.10116	0.550580	+	0.834782i	$0.314407\pi$
$230$	0	0
$231$	−1.46373	−0.0963061
$232$	0	0
$233$	8.39455	0.549945	0.274973	−	0.961452i	$-0.411331\pi$
$233$	8.39455	0.549945	0.274973	+	0.961452i	$0.411331\pi$
$234$	0	0
$235$	9.03717	0.589520
$236$	0	0
$237$	1.59014	0.103291
$238$	0	0
$239$	−18.1156	−1.17180	−0.585902	−	0.810382i	$-0.699260\pi$
$239$	−18.1156	−1.17180	−0.585902	+	0.810382i	$0.699260\pi$
$240$	0	0
$241$	10.8126	0.696502	0.348251	−	0.937401i	$-0.386776\pi$
$241$	10.8126	0.696502	0.348251	+	0.937401i	$0.386776\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	22.4582	1.43480
$246$	0	0
$247$	6.07377	0.386465
$248$	0	0
$249$	−1.64530	−0.104267
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	17.0529	1.07211
$254$	0	0
$255$	−0.347684	−0.0217728
$256$	0	0
$257$	0.696300	0.0434340	0.0217170	−	0.999764i	$-0.493087\pi$
$257$	0.696300	0.0434340	0.0217170	+	0.999764i	$0.493087\pi$
$258$	0	0
$259$	3.47971	0.216218
$260$	0	0
$261$	1.37602	0.0851736
$262$	0	0
$263$	−11.6918	−0.720945	−0.360473	−	0.932770i	$-0.617385\pi$
$263$	−11.6918	−0.720945	−0.360473	+	0.932770i	$0.617385\pi$
$264$	0	0
$265$	12.7487	0.783148
$266$	0	0
$267$	1.73887	0.106417
$268$	0	0
$269$	26.3471	1.60641	0.803207	−	0.595700i	$-0.203125\pi$
$269$	26.3471	1.60641	0.803207	+	0.595700i	$0.203125\pi$
$270$	0	0
$271$	3.70896	0.225303	0.112652	−	0.993635i	$-0.464066\pi$
$271$	3.70896	0.225303	0.112652	+	0.993635i	$0.464066\pi$
$272$	0	0
$273$	−0.386366	−0.0233840
$274$	0	0
$275$	−19.2312	−1.15968
$276$	0	0
$277$	10.3422	0.621404	0.310702	−	0.950507i	$-0.399436\pi$
$277$	10.3422	0.621404	0.310702	+	0.950507i	$0.399436\pi$
$278$	0	0
$279$	−1.41438	−0.0846765
$280$	0	0
$281$	2.70871	0.161588	0.0807940	−	0.996731i	$-0.474254\pi$
$281$	2.70871	0.161588	0.0807940	+	0.996731i	$0.474254\pi$
$282$	0	0
$283$	13.3829	0.795529	0.397764	−	0.917488i	$-0.369786\pi$
$283$	13.3829	0.795529	0.397764	+	0.917488i	$0.369786\pi$
$284$	0	0
$285$	−23.3904	−1.38553
$286$	0	0
$287$	−3.16511	−0.186831
$288$	0	0
$289$	−16.9889	−0.999349
$290$	0	0
$291$	7.69389	0.451024
$292$	0	0
$293$	−0.785711	−0.0459017	−0.0229509	−	0.999737i	$-0.507306\pi$
$293$	−0.785711	−0.0459017	−0.0229509	+	0.999737i	$0.507306\pi$
$294$	0	0
$295$	5.74279	0.334358
$296$	0	0
$297$	3.25032	0.188603
$298$	0	0
$299$	4.50131	0.260318
$300$	0	0
$301$	2.94002	0.169460
$302$	0	0
$303$	−8.47436	−0.486839
$304$	0	0
$305$	32.5465	1.86361
$306$	0	0
$307$	31.0052	1.76956	0.884780	−	0.466010i	$-0.154309\pi$
$307$	31.0052	1.76956	0.884780	+	0.466010i	$0.154309\pi$
$308$	0	0
$309$	10.5577	0.600607
$310$	0	0
$311$	11.7456	0.666033	0.333017	−	0.942921i	$-0.391933\pi$
$311$	11.7456	0.666033	0.333017	+	0.942921i	$0.391933\pi$
$312$	0	0
$313$	16.7919	0.949137	0.474569	−	0.880219i	$-0.342604\pi$
$313$	16.7919	0.949137	0.474569	+	0.880219i	$0.342604\pi$
$314$	0	0
$315$	1.48792	0.0838347
$316$	0	0
$317$	3.16039	0.177505	0.0887526	−	0.996054i	$-0.471712\pi$
$317$	3.16039	0.177505	0.0887526	+	0.996054i	$0.471712\pi$
$318$	0	0
$319$	−4.47251	−0.250412
$320$	0	0
$321$	8.39020	0.468295
$322$	0	0
$323$	0.744956	0.0414505
$324$	0	0
$325$	−5.07628	−0.281581
$326$	0	0
$327$	4.72074	0.261058
$328$	0	0
$329$	1.23174	0.0679082
$330$	0	0
$331$	1.51201	0.0831073	0.0415537	−	0.999136i	$-0.486769\pi$
$331$	1.51201	0.0831073	0.0415537	+	0.999136i	$0.486769\pi$
$332$	0	0
$333$	−7.72697	−0.423435
$334$	0	0
$335$	32.5444	1.77809
$336$	0	0
$337$	18.8170	1.02503	0.512514	−	0.858679i	$-0.328714\pi$
$337$	18.8170	1.02503	0.512514	+	0.858679i	$0.328714\pi$
$338$	0	0
$339$	2.41657	0.131250
$340$	0	0
$341$	4.59718	0.248951
$342$	0	0
$343$	6.21333	0.335488
$344$	0	0
$345$	−17.3348	−0.933274
$346$	0	0
$347$	−25.9052	−1.39066	−0.695332	−	0.718688i	$-0.744743\pi$
$347$	−25.9052	−1.39066	−0.695332	+	0.718688i	$0.744743\pi$
$348$	0	0
$349$	1.22206	0.0654154	0.0327077	−	0.999465i	$-0.489587\pi$
$349$	1.22206	0.0654154	0.0327077	+	0.999465i	$0.489587\pi$
$350$	0	0
$351$	0.857958	0.0457944
$352$	0	0
$353$	29.0816	1.54786	0.773929	−	0.633273i	$-0.218289\pi$
$353$	29.0816	1.54786	0.773929	+	0.633273i	$0.218289\pi$
$354$	0	0
$355$	−40.5595	−2.15268
$356$	0	0
$357$	−0.0473884	−0.00250806
$358$	0	0
$359$	11.7059	0.617814	0.308907	−	0.951092i	$-0.400037\pi$
$359$	11.7059	0.617814	0.308907	+	0.951092i	$0.400037\pi$
$360$	0	0
$361$	31.1169	1.63773
$362$	0	0
$363$	0.435418	0.0228535
$364$	0	0
$365$	9.73300	0.509448
$366$	0	0
$367$	−23.3277	−1.21769	−0.608847	−	0.793288i	$-0.708368\pi$
$367$	−23.3277	−1.21769	−0.608847	+	0.793288i	$0.708368\pi$
$368$	0	0
$369$	7.02839	0.365883
$370$	0	0
$371$	1.73762	0.0902127
$372$	0	0
$373$	6.61503	0.342513	0.171257	−	0.985226i	$-0.445217\pi$
$373$	6.61503	0.342513	0.171257	+	0.985226i	$0.445217\pi$
$374$	0	0
$375$	3.02881	0.156407
$376$	0	0
$377$	−1.18057	−0.0608024
$378$	0	0
$379$	−20.9546	−1.07637	−0.538183	−	0.842828i	$-0.680889\pi$
$379$	−20.9546	−1.07637	−0.538183	+	0.842828i	$0.680889\pi$
$380$	0	0
$381$	−16.9207	−0.866874
$382$	0	0
$383$	19.4279	0.992719	0.496359	−	0.868117i	$-0.334670\pi$
$383$	19.4279	0.992719	0.496359	+	0.868117i	$0.334670\pi$
$384$	0	0
$385$	−4.83621	−0.246476
$386$	0	0
$387$	−6.52856	−0.331865
$388$	0	0
$389$	−36.8292	−1.86731	−0.933657	−	0.358167i	$-0.883402\pi$
$389$	−36.8292	−1.86731	−0.933657	+	0.358167i	$0.883402\pi$
$390$	0	0
$391$	0.552092	0.0279205
$392$	0	0
$393$	10.0042	0.504646
$394$	0	0
$395$	5.25390	0.264352
$396$	0	0
$397$	−32.9158	−1.65199	−0.825997	−	0.563674i	$-0.809387\pi$
$397$	−32.9158	−1.65199	−0.825997	+	0.563674i	$0.809387\pi$
$398$	0	0
$399$	−3.18805	−0.159602
$400$	0	0
$401$	−23.2777	−1.16243	−0.581215	−	0.813750i	$-0.697423\pi$
$401$	−23.2777	−1.16243	−0.581215	+	0.813750i	$0.697423\pi$
$402$	0	0
$403$	1.21348	0.0604475
$404$	0	0
$405$	−3.30404	−0.164179
$406$	0	0
$407$	25.1151	1.24491
$408$	0	0
$409$	10.9722	0.542541	0.271270	−	0.962503i	$-0.412556\pi$
$409$	10.9722	0.542541	0.271270	+	0.962503i	$0.412556\pi$
$410$	0	0
$411$	−5.15265	−0.254161
$412$	0	0
$413$	0.782728	0.0385155
$414$	0	0
$415$	−5.43614	−0.266850
$416$	0	0
$417$	−12.5932	−0.616693
$418$	0	0
$419$	−10.1855	−0.497592	−0.248796	−	0.968556i	$-0.580035\pi$
$419$	−10.1855	−0.497592	−0.248796	+	0.968556i	$0.580035\pi$
$420$	0	0
$421$	5.43426	0.264850	0.132425	−	0.991193i	$-0.457724\pi$
$421$	5.43426	0.264850	0.132425	+	0.991193i	$0.457724\pi$
$422$	0	0
$423$	−2.73518	−0.132989
$424$	0	0
$425$	−0.622613	−0.0302012
$426$	0	0
$427$	4.43600	0.214673
$428$	0	0
$429$	−2.78864	−0.134637
$430$	0	0
$431$	24.4285	1.17668	0.588340	−	0.808614i	$-0.299782\pi$
$431$	24.4285	1.17668	0.588340	+	0.808614i	$0.299782\pi$
$432$	0	0
$433$	10.1844	0.489433	0.244717	−	0.969595i	$-0.421305\pi$
$433$	10.1844	0.489433	0.244717	+	0.969595i	$0.421305\pi$
$434$	0	0
$435$	4.54643	0.217985
$436$	0	0
$437$	37.1420	1.77674
$438$	0	0
$439$	18.3878	0.877600	0.438800	−	0.898585i	$-0.355404\pi$
$439$	18.3878	0.877600	0.438800	+	0.898585i	$0.355404\pi$
$440$	0	0
$441$	−6.79720	−0.323676
$442$	0	0
$443$	−22.5296	−1.07041	−0.535207	−	0.844721i	$-0.679766\pi$
$443$	−22.5296	−1.07041	−0.535207	+	0.844721i	$0.679766\pi$
$444$	0	0
$445$	5.74531	0.272354
$446$	0	0
$447$	−8.06303	−0.381368
$448$	0	0
$449$	16.7474	0.790357	0.395178	−	0.918604i	$-0.370683\pi$
$449$	16.7474	0.790357	0.395178	+	0.918604i	$0.370683\pi$
$450$	0	0
$451$	−22.8445	−1.07571
$452$	0	0
$453$	10.2888	0.483411
$454$	0	0
$455$	−1.27657	−0.0598466
$456$	0	0
$457$	−34.4961	−1.61366	−0.806830	−	0.590784i	$-0.798819\pi$
$457$	−34.4961	−1.61366	−0.806830	+	0.590784i	$0.798819\pi$
$458$	0	0
$459$	0.105230	0.00491170
$460$	0	0
$461$	35.4899	1.65293	0.826465	−	0.562988i	$-0.190348\pi$
$461$	35.4899	1.65293	0.826465	+	0.562988i	$0.190348\pi$
$462$	0	0
$463$	28.9981	1.34766	0.673829	−	0.738887i	$-0.264648\pi$
$463$	28.9981	1.34766	0.673829	+	0.738887i	$0.264648\pi$
$464$	0	0
$465$	−4.67316	−0.216713
$466$	0	0
$467$	−9.55160	−0.441995	−0.220998	−	0.975274i	$-0.570931\pi$
$467$	−9.55160	−0.441995	−0.220998	+	0.975274i	$0.570931\pi$
$468$	0	0
$469$	4.43571	0.204822
$470$	0	0
$471$	−1.55957	−0.0718610
$472$	0	0
$473$	21.2199	0.975692
$474$	0	0
$475$	−41.8863	−1.92187
$476$	0	0
$477$	−3.85852	−0.176670
$478$	0	0
$479$	−30.7438	−1.40472	−0.702360	−	0.711822i	$-0.747870\pi$
$479$	−30.7438	−1.40472	−0.702360	+	0.711822i	$0.747870\pi$
$480$	0	0
$481$	6.62941	0.302275
$482$	0	0
$483$	−2.36269	−0.107506
$484$	0	0
$485$	25.4209	1.15431
$486$	0	0
$487$	−20.6232	−0.934527	−0.467263	−	0.884118i	$-0.654760\pi$
$487$	−20.6232	−0.934527	−0.467263	+	0.884118i	$0.654760\pi$
$488$	0	0
$489$	4.45342	0.201391
$490$	0	0
$491$	−20.9367	−0.944859	−0.472430	−	0.881368i	$-0.656623\pi$
$491$	−20.9367	−0.944859	−0.472430	+	0.881368i	$0.656623\pi$
$492$	0	0
$493$	−0.144798	−0.00652139
$494$	0	0
$495$	10.7392	0.482691
$496$	0	0
$497$	−5.52816	−0.247972
$498$	0	0
$499$	7.07890	0.316895	0.158448	−	0.987367i	$-0.449351\pi$
$499$	7.07890	0.316895	0.158448	+	0.987367i	$0.449351\pi$
$500$	0	0
$501$	3.46303	0.154717
$502$	0	0
$503$	−18.3438	−0.817911	−0.408956	−	0.912554i	$-0.634107\pi$
$503$	−18.3438	−0.817911	−0.408956	+	0.912554i	$0.634107\pi$
$504$	0	0
$505$	−27.9997	−1.24597
$506$	0	0
$507$	12.2639	0.544659
$508$	0	0
$509$	−20.0594	−0.889117	−0.444559	−	0.895750i	$-0.646639\pi$
$509$	−20.0594	−0.889117	−0.444559	+	0.895750i	$0.646639\pi$
$510$	0	0
$511$	1.32658	0.0586846
$512$	0	0
$513$	7.07933	0.312560
$514$	0	0
$515$	34.8831	1.53713
$516$	0	0
$517$	8.89023	0.390992
$518$	0	0
$519$	4.38865	0.192640
$520$	0	0
$521$	−14.1988	−0.622060	−0.311030	−	0.950400i	$-0.600674\pi$
$521$	−14.1988	−0.622060	−0.311030	+	0.950400i	$0.600674\pi$
$522$	0	0
$523$	3.66810	0.160395	0.0801973	−	0.996779i	$-0.474445\pi$
$523$	3.66810	0.160395	0.0801973	+	0.996779i	$0.474445\pi$
$524$	0	0
$525$	2.66448	0.116288
$526$	0	0
$527$	0.148834	0.00648333
$528$	0	0
$529$	4.52620	0.196791
$530$	0	0
$531$	−1.73811	−0.0754275
$532$	0	0
$533$	−6.03006	−0.261191
$534$	0	0
$535$	27.7216	1.19851
$536$	0	0
$537$	5.67834	0.245039
$538$	0	0
$539$	22.0931	0.951616
$540$	0	0
$541$	−1.06729	−0.0458865	−0.0229432	−	0.999737i	$-0.507304\pi$
$541$	−1.06729	−0.0458865	−0.0229432	+	0.999737i	$0.507304\pi$
$542$	0	0
$543$	−8.41598	−0.361164
$544$	0	0
$545$	15.5975	0.668125
$546$	0	0
$547$	20.8597	0.891896	0.445948	−	0.895059i	$-0.352867\pi$
$547$	20.8597	0.891896	0.445948	+	0.895059i	$0.352867\pi$
$548$	0	0
$549$	−9.85050	−0.420409
$550$	0	0
$551$	−9.74130	−0.414994
$552$	0	0
$553$	0.716093	0.0304514
$554$	0	0
$555$	−25.5302	−1.08370
$556$	0	0
$557$	15.1423	0.641601	0.320801	−	0.947147i	$-0.396048\pi$
$557$	15.1423	0.641601	0.320801	+	0.947147i	$0.396048\pi$
$558$	0	0
$559$	5.60123	0.236907
$560$	0	0
$561$	−0.342030	−0.0144405
$562$	0	0
$563$	−13.4809	−0.568152	−0.284076	−	0.958802i	$-0.591687\pi$
$563$	−13.4809	−0.568152	−0.284076	+	0.958802i	$0.591687\pi$
$564$	0	0
$565$	7.98444	0.335908
$566$	0	0
$567$	−0.450333	−0.0189122
$568$	0	0
$569$	2.00420	0.0840202	0.0420101	−	0.999117i	$-0.486624\pi$
$569$	2.00420	0.0840202	0.0420101	+	0.999117i	$0.486624\pi$
$570$	0	0
$571$	−18.7771	−0.785795	−0.392898	−	0.919582i	$-0.628527\pi$
$571$	−18.7771	−0.785795	−0.392898	+	0.919582i	$0.628527\pi$
$572$	0	0
$573$	−4.95642	−0.207057
$574$	0	0
$575$	−31.0422	−1.29455
$576$	0	0
$577$	16.0846	0.669609	0.334804	−	0.942288i	$-0.391330\pi$
$577$	16.0846	0.669609	0.334804	+	0.942288i	$0.391330\pi$
$578$	0	0
$579$	−3.97943	−0.165379
$580$	0	0
$581$	−0.740932	−0.0307390
$582$	0	0
$583$	12.5414	0.519413
$584$	0	0
$585$	2.83473	0.117202
$586$	0	0
$587$	15.2952	0.631301	0.315651	−	0.948875i	$-0.397777\pi$
$587$	15.2952	0.631301	0.315651	+	0.948875i	$0.397777\pi$
$588$	0	0
$589$	10.0128	0.412572
$590$	0	0
$591$	−15.9071	−0.654329
$592$	0	0
$593$	−0.690447	−0.0283533	−0.0141766	−	0.999900i	$-0.504513\pi$
$593$	−0.690447	−0.0283533	−0.0141766	+	0.999900i	$0.504513\pi$
$594$	0	0
$595$	−0.156573	−0.00641888
$596$	0	0
$597$	6.47868	0.265155
$598$	0	0
$599$	−9.92271	−0.405431	−0.202715	−	0.979238i	$-0.564977\pi$
$599$	−9.92271	−0.405431	−0.202715	+	0.979238i	$0.564977\pi$
$600$	0	0
$601$	−16.4114	−0.669434	−0.334717	−	0.942319i	$-0.608641\pi$
$601$	−16.4114	−0.669434	−0.334717	+	0.942319i	$0.608641\pi$
$602$	0	0
$603$	−9.84986	−0.401117
$604$	0	0
$605$	1.43864	0.0584890
$606$	0	0
$607$	30.5945	1.24179	0.620897	−	0.783892i	$-0.286769\pi$
$607$	30.5945	1.24179	0.620897	+	0.783892i	$0.286769\pi$
$608$	0	0
$609$	0.619667	0.0251102
$610$	0	0
$611$	2.34667	0.0949362
$612$	0	0
$613$	8.98923	0.363071	0.181536	−	0.983384i	$-0.441893\pi$
$613$	8.98923	0.363071	0.181536	+	0.983384i	$0.441893\pi$
$614$	0	0
$615$	23.2221	0.936406
$616$	0	0
$617$	−1.51748	−0.0610916	−0.0305458	−	0.999533i	$-0.509725\pi$
$617$	−1.51748	−0.0610916	−0.0305458	+	0.999533i	$0.509725\pi$
$618$	0	0
$619$	−16.8356	−0.676680	−0.338340	−	0.941024i	$-0.609865\pi$
$619$	−16.8356	−0.676680	−0.338340	+	0.941024i	$0.609865\pi$
$620$	0	0
$621$	5.24654	0.210536
$622$	0	0
$623$	0.783071	0.0313731
$624$	0	0
$625$	−19.5762	−0.783047
$626$	0	0
$627$	−23.0101	−0.918934
$628$	0	0
$629$	0.813107	0.0324207
$630$	0	0
$631$	28.0165	1.11532	0.557660	−	0.830069i	$-0.311699\pi$
$631$	28.0165	1.11532	0.557660	+	0.830069i	$0.311699\pi$
$632$	0	0
$633$	5.12881	0.203852
$634$	0	0
$635$	−55.9068	−2.21859
$636$	0	0
$637$	5.83171	0.231061
$638$	0	0
$639$	12.2757	0.485620
$640$	0	0
$641$	−3.57139	−0.141061	−0.0705307	−	0.997510i	$-0.522469\pi$
$641$	−3.57139	−0.141061	−0.0705307	+	0.997510i	$0.522469\pi$
$642$	0	0
$643$	−27.2953	−1.07642	−0.538212	−	0.842810i	$-0.680900\pi$
$643$	−27.2953	−1.07642	−0.538212	+	0.842810i	$0.680900\pi$
$644$	0	0
$645$	−21.5706	−0.849343
$646$	0	0
$647$	−31.3648	−1.23308	−0.616540	−	0.787324i	$-0.711466\pi$
$647$	−31.3648	−1.23308	−0.616540	+	0.787324i	$0.711466\pi$
$648$	0	0
$649$	5.64941	0.221759
$650$	0	0
$651$	−0.636940	−0.0249636
$652$	0	0
$653$	33.8394	1.32424	0.662120	−	0.749398i	$-0.269657\pi$
$653$	33.8394	1.32424	0.662120	+	0.749398i	$0.269657\pi$
$654$	0	0
$655$	33.0544	1.29154
$656$	0	0
$657$	−2.94578	−0.114926
$658$	0	0
$659$	−4.90671	−0.191138	−0.0955692	−	0.995423i	$-0.530467\pi$
$659$	−4.90671	−0.191138	−0.0955692	+	0.995423i	$0.530467\pi$
$660$	0	0
$661$	13.6075	0.529271	0.264636	−	0.964348i	$-0.414748\pi$
$661$	13.6075	0.529271	0.264636	+	0.964348i	$0.414748\pi$
$662$	0	0
$663$	−0.0902827	−0.00350629
$664$	0	0
$665$	−10.5335	−0.408470
$666$	0	0
$667$	−7.21935	−0.279534
$668$	0	0
$669$	9.47092	0.366167
$670$	0	0
$671$	32.0173	1.23601
$672$	0	0
$673$	−12.8792	−0.496456	−0.248228	−	0.968702i	$-0.579848\pi$
$673$	−12.8792	−0.496456	−0.248228	+	0.968702i	$0.579848\pi$
$674$	0	0
$675$	−5.91670	−0.227734
$676$	0	0
$677$	−14.2331	−0.547021	−0.273510	−	0.961869i	$-0.588185\pi$
$677$	−14.2331	−0.547021	−0.273510	+	0.961869i	$0.588185\pi$
$678$	0	0
$679$	3.46481	0.132967
$680$	0	0
$681$	−5.67241	−0.217367
$682$	0	0
$683$	−36.6718	−1.40321	−0.701604	−	0.712567i	$-0.747532\pi$
$683$	−36.6718	−1.40321	−0.701604	+	0.712567i	$0.747532\pi$
$684$	0	0
$685$	−17.0246	−0.650475
$686$	0	0
$687$	−16.6636	−0.635755
$688$	0	0
$689$	3.31045	0.126118
$690$	0	0
$691$	−13.7349	−0.522499	−0.261250	−	0.965271i	$-0.584135\pi$
$691$	−13.7349	−0.522499	−0.261250	+	0.965271i	$0.584135\pi$
$692$	0	0
$693$	1.46373	0.0556023
$694$	0	0
$695$	−41.6086	−1.57830
$696$	0	0
$697$	−0.739596	−0.0280142
$698$	0	0
$699$	−8.39455	−0.317511
$700$	0	0
$701$	42.9944	1.62388	0.811938	−	0.583744i	$-0.198413\pi$
$701$	42.9944	1.62388	0.811938	+	0.583744i	$0.198413\pi$
$702$	0	0
$703$	54.7018	2.06312
$704$	0	0
$705$	−9.03717	−0.340359
$706$	0	0
$707$	−3.81628	−0.143526
$708$	0	0
$709$	2.52235	0.0947287	0.0473644	−	0.998878i	$-0.484918\pi$
$709$	2.52235	0.0947287	0.0473644	+	0.998878i	$0.484918\pi$
$710$	0	0
$711$	−1.59014	−0.0596350
$712$	0	0
$713$	7.42058	0.277903
$714$	0	0
$715$	−9.21378	−0.344576
$716$	0	0
$717$	18.1156	0.676541
$718$	0	0
$719$	2.87343	0.107161	0.0535803	−	0.998564i	$-0.482937\pi$
$719$	2.87343	0.107161	0.0535803	+	0.998564i	$0.482937\pi$
$720$	0	0
$721$	4.75448	0.177066
$722$	0	0
$723$	−10.8126	−0.402126
$724$	0	0
$725$	8.14150	0.302368
$726$	0	0
$727$	−32.3349	−1.19924	−0.599618	−	0.800286i	$-0.704681\pi$
$727$	−32.3349	−1.19924	−0.599618	+	0.800286i	$0.704681\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.686999	0.0254096
$732$	0	0
$733$	−22.9801	−0.848790	−0.424395	−	0.905477i	$-0.639513\pi$
$733$	−22.9801	−0.848790	−0.424395	+	0.905477i	$0.639513\pi$
$734$	0	0
$735$	−22.4582	−0.828385
$736$	0	0
$737$	32.0152	1.17930
$738$	0	0
$739$	33.9047	1.24720	0.623602	−	0.781742i	$-0.285668\pi$
$739$	33.9047	1.24720	0.623602	+	0.781742i	$0.285668\pi$
$740$	0	0
$741$	−6.07377	−0.223125
$742$	0	0
$743$	−25.1804	−0.923780	−0.461890	−	0.886937i	$-0.652829\pi$
$743$	−25.1804	−0.923780	−0.461890	+	0.886937i	$0.652829\pi$
$744$	0	0
$745$	−26.6406	−0.976036
$746$	0	0
$747$	1.64530	0.0601983
$748$	0	0
$749$	3.77838	0.138059
$750$	0	0
$751$	23.2355	0.847877	0.423938	−	0.905691i	$-0.360647\pi$
$751$	23.2355	0.847877	0.423938	+	0.905691i	$0.360647\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	33.9947	1.23719
$756$	0	0
$757$	1.40226	0.0509661	0.0254830	−	0.999675i	$-0.491888\pi$
$757$	1.40226	0.0509661	0.0254830	+	0.999675i	$0.491888\pi$
$758$	0	0
$759$	−17.0529	−0.618983
$760$	0	0
$761$	14.4778	0.524819	0.262409	−	0.964957i	$-0.415483\pi$
$761$	14.4778	0.524819	0.262409	+	0.964957i	$0.415483\pi$
$762$	0	0
$763$	2.12590	0.0769629
$764$	0	0
$765$	0.347684	0.0125705
$766$	0	0
$767$	1.49123	0.0538450
$768$	0	0
$769$	−17.9681	−0.647946	−0.323973	−	0.946066i	$-0.605019\pi$
$769$	−17.9681	−0.647946	−0.323973	+	0.946066i	$0.605019\pi$
$770$	0	0
$771$	−0.696300	−0.0250766
$772$	0	0
$773$	28.2064	1.01451	0.507256	−	0.861795i	$-0.330660\pi$
$773$	28.2064	1.01451	0.507256	+	0.861795i	$0.330660\pi$
$774$	0	0
$775$	−8.36844	−0.300603
$776$	0	0
$777$	−3.47971	−0.124834
$778$	0	0
$779$	−49.7563	−1.78270
$780$	0	0
$781$	−39.9000	−1.42774
$782$	0	0
$783$	−1.37602	−0.0491750
$784$	0	0
$785$	−5.15288	−0.183914
$786$	0	0
$787$	−33.5079	−1.19443	−0.597214	−	0.802082i	$-0.703726\pi$
$787$	−33.5079	−1.19443	−0.597214	+	0.802082i	$0.703726\pi$
$788$	0	0
$789$	11.6918	0.416238
$790$	0	0
$791$	1.08826	0.0386940
$792$	0	0
$793$	8.45132	0.300115
$794$	0	0
$795$	−12.7487	−0.452151
$796$	0	0
$797$	−48.0816	−1.70314	−0.851568	−	0.524244i	$-0.824348\pi$
$797$	−48.0816	−1.70314	−0.851568	+	0.524244i	$0.824348\pi$
$798$	0	0
$799$	0.287823	0.0101824
$800$	0	0
$801$	−1.73887	−0.0614400
$802$	0	0
$803$	9.57474	0.337885
$804$	0	0
$805$	−7.80643	−0.275140
$806$	0	0
$807$	−26.3471	−0.927464
$808$	0	0
$809$	−32.9791	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$809$	−32.9791	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$810$	0	0
$811$	−22.2980	−0.782988	−0.391494	−	0.920181i	$-0.628042\pi$
$811$	−22.2980	−0.782988	−0.391494	+	0.920181i	$0.628042\pi$
$812$	0	0
$813$	−3.70896	−0.130079
$814$	0	0
$815$	14.7143	0.515419
$816$	0	0
$817$	46.2178	1.61696
$818$	0	0
$819$	0.386366	0.0135007
$820$	0	0
$821$	−13.6366	−0.475920	−0.237960	−	0.971275i	$-0.576479\pi$
$821$	−13.6366	−0.475920	−0.237960	+	0.971275i	$0.576479\pi$
$822$	0	0
$823$	−33.2144	−1.15778	−0.578891	−	0.815405i	$-0.696514\pi$
$823$	−33.2144	−1.15778	−0.578891	+	0.815405i	$0.696514\pi$
$824$	0	0
$825$	19.2312	0.669543
$826$	0	0
$827$	−38.0272	−1.32234	−0.661168	−	0.750238i	$-0.729939\pi$
$827$	−38.0272	−1.32234	−0.661168	+	0.750238i	$0.729939\pi$
$828$	0	0
$829$	−13.5122	−0.469299	−0.234650	−	0.972080i	$-0.575394\pi$
$829$	−13.5122	−0.469299	−0.234650	+	0.972080i	$0.575394\pi$
$830$	0	0
$831$	−10.3422	−0.358768
$832$	0	0
$833$	0.715268	0.0247826
$834$	0	0
$835$	11.4420	0.395967
$836$	0	0
$837$	1.41438	0.0488880
$838$	0	0
$839$	−37.4989	−1.29461	−0.647303	−	0.762233i	$-0.724103\pi$
$839$	−37.4989	−1.29461	−0.647303	+	0.762233i	$0.724103\pi$
$840$	0	0
$841$	−27.1066	−0.934709
$842$	0	0
$843$	−2.70871	−0.0932929
$844$	0	0
$845$	40.5205	1.39395
$846$	0	0
$847$	0.196083	0.00673749
$848$	0	0
$849$	−13.3829	−0.459299
$850$	0	0
$851$	40.5399	1.38969
$852$	0	0
$853$	52.9076	1.81152	0.905762	−	0.423788i	$-0.139300\pi$
$853$	52.9076	1.81152	0.905762	+	0.423788i	$0.139300\pi$
$854$	0	0
$855$	23.3904	0.799935
$856$	0	0
$857$	−26.9673	−0.921184	−0.460592	−	0.887612i	$-0.652363\pi$
$857$	−26.9673	−0.921184	−0.460592	+	0.887612i	$0.652363\pi$
$858$	0	0
$859$	−11.1169	−0.379303	−0.189652	−	0.981851i	$-0.560736\pi$
$859$	−11.1169	−0.379303	−0.189652	+	0.981851i	$0.560736\pi$
$860$	0	0
$861$	3.16511	0.107867
$862$	0	0
$863$	−27.7619	−0.945026	−0.472513	−	0.881324i	$-0.656653\pi$
$863$	−27.7619	−0.945026	−0.472513	+	0.881324i	$0.656653\pi$
$864$	0	0
$865$	14.5003	0.493025
$866$	0	0
$867$	16.9889	0.576974
$868$	0	0
$869$	5.16847	0.175328
$870$	0	0
$871$	8.45077	0.286343
$872$	0	0
$873$	−7.69389	−0.260399
$874$	0	0
$875$	1.36397	0.0461107
$876$	0	0
$877$	45.7512	1.54491	0.772454	−	0.635071i	$-0.219029\pi$
$877$	45.7512	1.54491	0.772454	+	0.635071i	$0.219029\pi$
$878$	0	0
$879$	0.785711	0.0265014
$880$	0	0
$881$	8.59002	0.289405	0.144703	−	0.989475i	$-0.453777\pi$
$881$	8.59002	0.289405	0.144703	+	0.989475i	$0.453777\pi$
$882$	0	0
$883$	−48.5638	−1.63430	−0.817151	−	0.576424i	$-0.804448\pi$
$883$	−48.5638	−1.63430	−0.817151	+	0.576424i	$0.804448\pi$
$884$	0	0
$885$	−5.74279	−0.193042
$886$	0	0
$887$	−9.28523	−0.311768	−0.155884	−	0.987775i	$-0.549823\pi$
$887$	−9.28523	−0.311768	−0.155884	+	0.987775i	$0.549823\pi$
$888$	0	0
$889$	−7.61995	−0.255565
$890$	0	0
$891$	−3.25032	−0.108890
$892$	0	0
$893$	19.3633	0.647967
$894$	0	0
$895$	18.7615	0.627127
$896$	0	0
$897$	−4.50131	−0.150294
$898$	0	0
$899$	−1.94621	−0.0649098
$900$	0	0
$901$	0.406031	0.0135269
$902$	0	0
$903$	−2.94002	−0.0978378
$904$	0	0
$905$	−27.8068	−0.924328
$906$	0	0
$907$	−8.70614	−0.289083	−0.144541	−	0.989499i	$-0.546171\pi$
$907$	−8.70614	−0.289083	−0.144541	+	0.989499i	$0.546171\pi$
$908$	0	0
$909$	8.47436	0.281077
$910$	0	0
$911$	3.97126	0.131574	0.0657869	−	0.997834i	$-0.479044\pi$
$911$	3.97126	0.131574	0.0657869	+	0.997834i	$0.479044\pi$
$912$	0	0
$913$	−5.34775	−0.176985
$914$	0	0
$915$	−32.5465	−1.07595
$916$	0	0
$917$	4.50523	0.148776
$918$	0	0
$919$	18.5349	0.611409	0.305705	−	0.952126i	$-0.401108\pi$
$919$	18.5349	0.611409	0.305705	+	0.952126i	$0.401108\pi$
$920$	0	0
$921$	−31.0052	−1.02166
$922$	0	0
$923$	−10.5321	−0.346667
$924$	0	0
$925$	−45.7181	−1.50320
$926$	0	0
$927$	−10.5577	−0.346761
$928$	0	0
$929$	5.12940	0.168290	0.0841451	−	0.996454i	$-0.473184\pi$
$929$	5.12940	0.168290	0.0841451	+	0.996454i	$0.473184\pi$
$930$	0	0
$931$	48.1196	1.57706
$932$	0	0
$933$	−11.7456	−0.384535
$934$	0	0
$935$	−1.13008	−0.0369577
$936$	0	0
$937$	−5.65734	−0.184817	−0.0924086	−	0.995721i	$-0.529457\pi$
$937$	−5.65734	−0.184817	−0.0924086	+	0.995721i	$0.529457\pi$
$938$	0	0
$939$	−16.7919	−0.547985
$940$	0	0
$941$	−23.3420	−0.760927	−0.380463	−	0.924796i	$-0.624236\pi$
$941$	−23.3420	−0.760927	−0.380463	+	0.924796i	$0.624236\pi$
$942$	0	0
$943$	−36.8747	−1.20081
$944$	0	0
$945$	−1.48792	−0.0484020
$946$	0	0
$947$	36.5585	1.18799	0.593996	−	0.804468i	$-0.297550\pi$
$947$	36.5585	1.18799	0.593996	+	0.804468i	$0.297550\pi$
$948$	0	0
$949$	2.52736	0.0820415
$950$	0	0
$951$	−3.16039	−0.102483
$952$	0	0
$953$	17.6006	0.570139	0.285069	−	0.958507i	$-0.407983\pi$
$953$	17.6006	0.570139	0.285069	+	0.958507i	$0.407983\pi$
$954$	0	0
$955$	−16.3762	−0.529922
$956$	0	0
$957$	4.47251	0.144576
$958$	0	0
$959$	−2.32041	−0.0749298
$960$	0	0
$961$	−28.9995	−0.935469
$962$	0	0
$963$	−8.39020	−0.270370
$964$	0	0
$965$	−13.1482	−0.423255
$966$	0	0
$967$	−52.0191	−1.67282	−0.836411	−	0.548102i	$-0.815351\pi$
$967$	−52.0191	−1.67282	−0.836411	+	0.548102i	$0.815351\pi$
$968$	0	0
$969$	−0.744956	−0.0239314
$970$	0	0
$971$	5.76789	0.185100	0.0925502	−	0.995708i	$-0.470498\pi$
$971$	5.76789	0.185100	0.0925502	+	0.995708i	$0.470498\pi$
$972$	0	0
$973$	−5.67115	−0.181809
$974$	0	0
$975$	5.07628	0.162571
$976$	0	0
$977$	−26.3573	−0.843244	−0.421622	−	0.906772i	$-0.638539\pi$
$977$	−26.3573	−0.843244	−0.421622	+	0.906772i	$0.638539\pi$
$978$	0	0
$979$	5.65189	0.180635
$980$	0	0
$981$	−4.72074	−0.150722
$982$	0	0
$983$	31.5739	1.00705	0.503526	−	0.863980i	$-0.332036\pi$
$983$	31.5739	1.00705	0.503526	+	0.863980i	$0.332036\pi$
$984$	0	0
$985$	−52.5576	−1.67462
$986$	0	0
$987$	−1.23174	−0.0392068
$988$	0	0
$989$	34.2524	1.08916
$990$	0	0
$991$	−3.36975	−0.107044	−0.0535218	−	0.998567i	$-0.517045\pi$
$991$	−3.36975	−0.107044	−0.0535218	+	0.998567i	$0.517045\pi$
$992$	0	0
$993$	−1.51201	−0.0479820
$994$	0	0
$995$	21.4058	0.678610
$996$	0	0
$997$	−4.81712	−0.152560	−0.0762799	−	0.997086i	$-0.524304\pi$
$997$	−4.81712	−0.152560	−0.0762799	+	0.997086i	$0.524304\pi$
$998$	0	0
$999$	7.72697	0.244470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.p.1.1	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.p.1.1	✓	14	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.30404 q^{5} -0.450333 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.30404 q^{5} -0.450333 q^{7} +1.00000 q^{9} -3.25032 q^{11} -0.857958 q^{13} +3.30404 q^{15} -0.105230 q^{17} -7.07933 q^{19} +0.450333 q^{21} -5.24654 q^{23} +5.91670 q^{25} -1.00000 q^{27} +1.37602 q^{29} -1.41438 q^{31} +3.25032 q^{33} +1.48792 q^{35} -7.72697 q^{37} +0.857958 q^{39} +7.02839 q^{41} -6.52856 q^{43} -3.30404 q^{45} -2.73518 q^{47} -6.79720 q^{49} +0.105230 q^{51} -3.85852 q^{53} +10.7392 q^{55} +7.07933 q^{57} -1.73811 q^{59} -9.85050 q^{61} -0.450333 q^{63} +2.83473 q^{65} -9.84986 q^{67} +5.24654 q^{69} +12.2757 q^{71} -2.94578 q^{73} -5.91670 q^{75} +1.46373 q^{77} -1.59014 q^{79} +1.00000 q^{81} +1.64530 q^{83} +0.347684 q^{85} -1.37602 q^{87} -1.73887 q^{89} +0.386366 q^{91} +1.41438 q^{93} +23.3904 q^{95} -7.69389 q^{97} -3.25032 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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