LMFDB - Embedded newform 6728.2.a.z.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6728,2,Mod(1,6728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6728, 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("6728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6728 = 2^{3} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6728.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:	$53.7233504799$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 18 x^{10} + 83 x^{9} + 83 x^{8} - 577 x^{7} + 121 x^{6} + 1416 x^{5} - 1289 x^{4} + \cdots + 64 $ x^12 - 4x^11 - 18x^10 + 83x^9 + 83x^8 - 577x^7 + 121x^6 + 1416x^5 - 1289x^4 - 465x^3 + 1056x^2 - 464*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.25326$ of defining polynomial
Character	$\chi$	$=$	6728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.25326 q^{3} -1.00069 q^{5} +0.939808 q^{7} +2.07720 q^{9} +O(q^{10})$	$q+2.25326 q^{3} -1.00069 q^{5} +0.939808 q^{7} +2.07720 q^{9} -1.56038 q^{11} +2.80542 q^{13} -2.25482 q^{15} -2.32349 q^{17} -1.86480 q^{19} +2.11764 q^{21} +0.0629887 q^{23} -3.99862 q^{25} -2.07931 q^{27} -8.13324 q^{31} -3.51596 q^{33} -0.940457 q^{35} -9.87033 q^{37} +6.32136 q^{39} -0.928123 q^{41} +5.51441 q^{43} -2.07864 q^{45} -4.88234 q^{47} -6.11676 q^{49} -5.23544 q^{51} +0.420878 q^{53} +1.56146 q^{55} -4.20190 q^{57} -9.07031 q^{59} -2.48734 q^{61} +1.95217 q^{63} -2.80736 q^{65} +5.80734 q^{67} +0.141930 q^{69} +15.2867 q^{71} -8.84034 q^{73} -9.00995 q^{75} -1.46646 q^{77} +3.28129 q^{79} -10.9168 q^{81} -14.0598 q^{83} +2.32509 q^{85} +7.82720 q^{89} +2.63656 q^{91} -18.3263 q^{93} +1.86609 q^{95} +18.6597 q^{97} -3.24123 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9	$ 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9} - 3 q^{11} - 3 q^{13} + 3 q^{15} - 8 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} + 12 q^{25} - 7 q^{27} - 29 q^{31} - 46 q^{33} + 17 q^{35} - 38 q^{37} + 10 q^{39} - 11 q^{41} - 9 q^{43} - 54 q^{45} - 34 q^{47} + 33 q^{49} + 17 q^{51} - 15 q^{53} - 2 q^{55} - q^{57} + 57 q^{59} - 37 q^{61} + 9 q^{63} - 59 q^{65} + 33 q^{67} + 21 q^{69} - 21 q^{71} - 13 q^{73} - 13 q^{75} + 3 q^{77} - 32 q^{79} + 36 q^{81} + 48 q^{83} - 17 q^{85} - 20 q^{89} - 2 q^{91} - 37 q^{93} - 7 q^{97} + 16 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9 - 3 * q^11 - 3 * q^13 + 3 * q^15 - 8 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 + 12 * q^25 - 7 * q^27 - 29 * q^31 - 46 * q^33 + 17 * q^35 - 38 * q^37 + 10 * q^39 - 11 * q^41 - 9 * q^43 - 54 * q^45 - 34 * q^47 + 33 * q^49 + 17 * q^51 - 15 * q^53 - 2 * q^55 - q^57 + 57 * q^59 - 37 * q^61 + 9 * q^63 - 59 * q^65 + 33 * q^67 + 21 * q^69 - 21 * q^71 - 13 * q^73 - 13 * q^75 + 3 * q^77 - 32 * q^79 + 36 * q^81 + 48 * q^83 - 17 * q^85 - 20 * q^89 - 2 * q^91 - 37 * q^93 - 7 * q^97 + 16 * 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
$3$	2.25326	1.30092	0.650462	−	0.759539i	$-0.274576\pi$
$3$	2.25326	1.30092	0.650462	+	0.759539i	$0.274576\pi$
$4$	0	0
$5$	−1.00069	−0.447522	−0.223761	−	0.974644i	$-0.571834\pi$
$5$	−1.00069	−0.447522	−0.223761	+	0.974644i	$0.571834\pi$
$6$	0	0
$7$	0.939808	0.355214	0.177607	−	0.984101i	$-0.443164\pi$
$7$	0.939808	0.355214	0.177607	+	0.984101i	$0.443164\pi$
$8$	0	0
$9$	2.07720	0.692401
$10$	0	0
$11$	−1.56038	−0.470473	−0.235237	−	0.971938i	$-0.575587\pi$
$11$	−1.56038	−0.470473	−0.235237	+	0.971938i	$0.575587\pi$
$12$	0	0
$13$	2.80542	0.778085	0.389042	−	0.921220i	$-0.372806\pi$
$13$	2.80542	0.778085	0.389042	+	0.921220i	$0.372806\pi$
$14$	0	0
$15$	−2.25482	−0.582192
$16$	0	0
$17$	−2.32349	−0.563530	−0.281765	−	0.959484i	$-0.590920\pi$
$17$	−2.32349	−0.563530	−0.281765	+	0.959484i	$0.590920\pi$
$18$	0	0
$19$	−1.86480	−0.427815	−0.213908	−	0.976854i	$-0.568619\pi$
$19$	−1.86480	−0.427815	−0.213908	+	0.976854i	$0.568619\pi$
$20$	0	0
$21$	2.11764	0.462106
$22$	0	0
$23$	0.0629887	0.0131341	0.00656703	−	0.999978i	$-0.497910\pi$
$23$	0.0629887	0.0131341	0.00656703	+	0.999978i	$0.497910\pi$
$24$	0	0
$25$	−3.99862	−0.799724
$26$	0	0
$27$	−2.07931	−0.400163
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−8.13324	−1.46077	−0.730387	−	0.683034i	$-0.760660\pi$
$31$	−8.13324	−1.46077	−0.730387	+	0.683034i	$0.760660\pi$
$32$	0	0
$33$	−3.51596	−0.612049
$34$	0	0
$35$	−0.940457	−0.158966
$36$	0	0
$37$	−9.87033	−1.62267	−0.811336	−	0.584580i	$-0.801259\pi$
$37$	−9.87033	−1.62267	−0.811336	+	0.584580i	$0.801259\pi$
$38$	0	0
$39$	6.32136	1.01223
$40$	0	0
$41$	−0.928123	−0.144949	−0.0724743	−	0.997370i	$-0.523090\pi$
$41$	−0.928123	−0.144949	−0.0724743	+	0.997370i	$0.523090\pi$
$42$	0	0
$43$	5.51441	0.840939	0.420470	−	0.907307i	$-0.361865\pi$
$43$	5.51441	0.840939	0.420470	+	0.907307i	$0.361865\pi$
$44$	0	0
$45$	−2.07864	−0.309865
$46$	0	0
$47$	−4.88234	−0.712163	−0.356081	−	0.934455i	$-0.615887\pi$
$47$	−4.88234	−0.712163	−0.356081	+	0.934455i	$0.615887\pi$
$48$	0	0
$49$	−6.11676	−0.873823
$50$	0	0
$51$	−5.23544	−0.733109
$52$	0	0
$53$	0.420878	0.0578120	0.0289060	−	0.999582i	$-0.490798\pi$
$53$	0.420878	0.0578120	0.0289060	+	0.999582i	$0.490798\pi$
$54$	0	0
$55$	1.56146	0.210547
$56$	0	0
$57$	−4.20190	−0.556555
$58$	0	0
$59$	−9.07031	−1.18085	−0.590427	−	0.807091i	$-0.701041\pi$
$59$	−9.07031	−1.18085	−0.590427	+	0.807091i	$0.701041\pi$
$60$	0	0
$61$	−2.48734	−0.318471	−0.159235	−	0.987241i	$-0.550903\pi$
$61$	−2.48734	−0.318471	−0.159235	+	0.987241i	$0.550903\pi$
$62$	0	0
$63$	1.95217	0.245951
$64$	0	0
$65$	−2.80736	−0.348210
$66$	0	0
$67$	5.80734	0.709479	0.354740	−	0.934965i	$-0.384570\pi$
$67$	5.80734	0.709479	0.354740	+	0.934965i	$0.384570\pi$
$68$	0	0
$69$	0.141930	0.0170864
$70$	0	0
$71$	15.2867	1.81420	0.907101	−	0.420914i	$-0.138290\pi$
$71$	15.2867	1.81420	0.907101	+	0.420914i	$0.138290\pi$
$72$	0	0
$73$	−8.84034	−1.03468	−0.517342	−	0.855779i	$-0.673078\pi$
$73$	−8.84034	−1.03468	−0.517342	+	0.855779i	$0.673078\pi$
$74$	0	0
$75$	−9.00995	−1.04038
$76$	0	0
$77$	−1.46646	−0.167119
$78$	0	0
$79$	3.28129	0.369174	0.184587	−	0.982816i	$-0.440905\pi$
$79$	3.28129	0.369174	0.184587	+	0.982816i	$0.440905\pi$
$80$	0	0
$81$	−10.9168	−1.21298
$82$	0	0
$83$	−14.0598	−1.54327	−0.771633	−	0.636068i	$-0.780560\pi$
$83$	−14.0598	−1.54327	−0.771633	+	0.636068i	$0.780560\pi$
$84$	0	0
$85$	2.32509	0.252192
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.82720	0.829682	0.414841	−	0.909894i	$-0.363837\pi$
$89$	7.82720	0.829682	0.414841	+	0.909894i	$0.363837\pi$
$90$	0	0
$91$	2.63656	0.276387
$92$	0	0
$93$	−18.3263	−1.90035
$94$	0	0
$95$	1.86609	0.191457
$96$	0	0
$97$	18.6597	1.89461	0.947305	−	0.320334i	$-0.103795\pi$
$97$	18.6597	1.89461	0.947305	+	0.320334i	$0.103795\pi$
$98$	0	0
$99$	−3.24123	−0.325756
$100$	0	0
$101$	0.173714	0.0172852	0.00864261	−	0.999963i	$-0.497249\pi$
$101$	0.173714	0.0172852	0.00864261	+	0.999963i	$0.497249\pi$
$102$	0	0
$103$	−12.2182	−1.20389	−0.601945	−	0.798537i	$-0.705607\pi$
$103$	−12.2182	−1.20389	−0.601945	+	0.798537i	$0.705607\pi$
$104$	0	0
$105$	−2.11910	−0.206803
$106$	0	0
$107$	6.63689	0.641613	0.320806	−	0.947145i	$-0.396046\pi$
$107$	6.63689	0.641613	0.320806	+	0.947145i	$0.396046\pi$
$108$	0	0
$109$	12.9525	1.24063	0.620313	−	0.784354i	$-0.287006\pi$
$109$	12.9525	1.24063	0.620313	+	0.784354i	$0.287006\pi$
$110$	0	0
$111$	−22.2405	−2.11097
$112$	0	0
$113$	5.55194	0.522283	0.261141	−	0.965301i	$-0.415901\pi$
$113$	5.55194	0.522283	0.261141	+	0.965301i	$0.415901\pi$
$114$	0	0
$115$	−0.0630322	−0.00587778
$116$	0	0
$117$	5.82743	0.538747
$118$	0	0
$119$	−2.18364	−0.200174
$120$	0	0
$121$	−8.56520	−0.778655
$122$	0	0
$123$	−2.09131	−0.188567
$124$	0	0
$125$	9.00483	0.805416
$126$	0	0
$127$	−13.7965	−1.22424	−0.612122	−	0.790764i	$-0.709684\pi$
$127$	−13.7965	−1.22424	−0.612122	+	0.790764i	$0.709684\pi$
$128$	0	0
$129$	12.4254	1.09400
$130$	0	0
$131$	−14.2456	−1.24465	−0.622323	−	0.782761i	$-0.713811\pi$
$131$	−14.2456	−1.24465	−0.622323	+	0.782761i	$0.713811\pi$
$132$	0	0
$133$	−1.75256	−0.151966
$134$	0	0
$135$	2.08074	0.179082
$136$	0	0
$137$	15.6573	1.33769	0.668846	−	0.743401i	$-0.266788\pi$
$137$	15.6573	1.33769	0.668846	+	0.743401i	$0.266788\pi$
$138$	0	0
$139$	13.4531	1.14108	0.570538	−	0.821271i	$-0.306735\pi$
$139$	13.4531	1.14108	0.570538	+	0.821271i	$0.306735\pi$
$140$	0	0
$141$	−11.0012	−0.926469
$142$	0	0
$143$	−4.37754	−0.366068
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−13.7827	−1.13678
$148$	0	0
$149$	−12.6026	−1.03244	−0.516222	−	0.856455i	$-0.672662\pi$
$149$	−12.6026	−1.03244	−0.516222	+	0.856455i	$0.672662\pi$
$150$	0	0
$151$	7.91925	0.644460	0.322230	−	0.946661i	$-0.395568\pi$
$151$	7.91925	0.644460	0.322230	+	0.946661i	$0.395568\pi$
$152$	0	0
$153$	−4.82636	−0.390188
$154$	0	0
$155$	8.13885	0.653728
$156$	0	0
$157$	2.41258	0.192545	0.0962724	−	0.995355i	$-0.469308\pi$
$157$	2.41258	0.192545	0.0962724	+	0.995355i	$0.469308\pi$
$158$	0	0
$159$	0.948349	0.0752090
$160$	0	0
$161$	0.0591973	0.00466540
$162$	0	0
$163$	−8.76026	−0.686157	−0.343078	−	0.939307i	$-0.611470\pi$
$163$	−8.76026	−0.686157	−0.343078	+	0.939307i	$0.611470\pi$
$164$	0	0
$165$	3.51838	0.273906
$166$	0	0
$167$	1.02842	0.0795816	0.0397908	−	0.999208i	$-0.487331\pi$
$167$	1.02842	0.0795816	0.0397908	+	0.999208i	$0.487331\pi$
$168$	0	0
$169$	−5.12959	−0.394584
$170$	0	0
$171$	−3.87357	−0.296220
$172$	0	0
$173$	−19.7374	−1.50061	−0.750303	−	0.661094i	$-0.770093\pi$
$173$	−19.7374	−1.50061	−0.750303	+	0.661094i	$0.770093\pi$
$174$	0	0
$175$	−3.75794	−0.284073
$176$	0	0
$177$	−20.4378	−1.53620
$178$	0	0
$179$	19.4502	1.45377	0.726887	−	0.686757i	$-0.240966\pi$
$179$	19.4502	1.45377	0.726887	+	0.686757i	$0.240966\pi$
$180$	0	0
$181$	−9.49312	−0.705618	−0.352809	−	0.935695i	$-0.614774\pi$
$181$	−9.49312	−0.705618	−0.352809	+	0.935695i	$0.614774\pi$
$182$	0	0
$183$	−5.60463	−0.414306
$184$	0	0
$185$	9.87713	0.726181
$186$	0	0
$187$	3.62554	0.265126
$188$	0	0
$189$	−1.95415	−0.142144
$190$	0	0
$191$	3.24135	0.234536	0.117268	−	0.993100i	$-0.462586\pi$
$191$	3.24135	0.234536	0.117268	+	0.993100i	$0.462586\pi$
$192$	0	0
$193$	7.37377	0.530776	0.265388	−	0.964142i	$-0.414500\pi$
$193$	7.37377	0.530776	0.265388	+	0.964142i	$0.414500\pi$
$194$	0	0
$195$	−6.32572	−0.452995
$196$	0	0
$197$	−5.71344	−0.407066	−0.203533	−	0.979068i	$-0.565242\pi$
$197$	−5.71344	−0.407066	−0.203533	+	0.979068i	$0.565242\pi$
$198$	0	0
$199$	−6.97827	−0.494676	−0.247338	−	0.968929i	$-0.579556\pi$
$199$	−6.97827	−0.494676	−0.247338	+	0.968929i	$0.579556\pi$
$200$	0	0
$201$	13.0855	0.922978
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.928764	0.0648677
$206$	0	0
$207$	0.130840	0.00909403
$208$	0	0
$209$	2.90981	0.201276
$210$	0	0
$211$	−10.5469	−0.726076	−0.363038	−	0.931774i	$-0.618261\pi$
$211$	−10.5469	−0.726076	−0.363038	+	0.931774i	$0.618261\pi$
$212$	0	0
$213$	34.4451	2.36014
$214$	0	0
$215$	−5.51821	−0.376339
$216$	0	0
$217$	−7.64369	−0.518887
$218$	0	0
$219$	−19.9196	−1.34604
$220$	0	0
$221$	−6.51838	−0.438474
$222$	0	0
$223$	2.65281	0.177646	0.0888228	−	0.996047i	$-0.471690\pi$
$223$	2.65281	0.177646	0.0888228	+	0.996047i	$0.471690\pi$
$224$	0	0
$225$	−8.30594	−0.553729
$226$	0	0
$227$	−12.0050	−0.796797	−0.398399	−	0.917212i	$-0.630434\pi$
$227$	−12.0050	−0.796797	−0.398399	+	0.917212i	$0.630434\pi$
$228$	0	0
$229$	−22.2898	−1.47295	−0.736474	−	0.676466i	$-0.763511\pi$
$229$	−22.2898	−1.47295	−0.736474	+	0.676466i	$0.763511\pi$
$230$	0	0
$231$	−3.30433	−0.217409
$232$	0	0
$233$	22.9976	1.50662	0.753311	−	0.657665i	$-0.228456\pi$
$233$	22.9976	1.50662	0.753311	+	0.657665i	$0.228456\pi$
$234$	0	0
$235$	4.88571	0.318709
$236$	0	0
$237$	7.39362	0.480267
$238$	0	0
$239$	−10.7951	−0.698279	−0.349139	−	0.937071i	$-0.613526\pi$
$239$	−10.7951	−0.698279	−0.349139	+	0.937071i	$0.613526\pi$
$240$	0	0
$241$	16.5887	1.06857	0.534286	−	0.845304i	$-0.320581\pi$
$241$	16.5887	1.06857	0.534286	+	0.845304i	$0.320581\pi$
$242$	0	0
$243$	−18.3606	−1.17783
$244$	0	0
$245$	6.12098	0.391055
$246$	0	0
$247$	−5.23157	−0.332877
$248$	0	0
$249$	−31.6805	−2.00767
$250$	0	0
$251$	18.0624	1.14009	0.570044	−	0.821614i	$-0.306926\pi$
$251$	18.0624	1.14009	0.570044	+	0.821614i	$0.306926\pi$
$252$	0	0
$253$	−0.0982866	−0.00617922
$254$	0	0
$255$	5.23905	0.328082
$256$	0	0
$257$	3.52774	0.220054	0.110027	−	0.993929i	$-0.464906\pi$
$257$	3.52774	0.220054	0.110027	+	0.993929i	$0.464906\pi$
$258$	0	0
$259$	−9.27621	−0.576396
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.5193	−1.20361	−0.601806	−	0.798642i	$-0.705552\pi$
$263$	−19.5193	−1.20361	−0.601806	+	0.798642i	$0.705552\pi$
$264$	0	0
$265$	−0.421168	−0.0258721
$266$	0	0
$267$	17.6368	1.07935
$268$	0	0
$269$	−14.4854	−0.883191	−0.441596	−	0.897214i	$-0.645587\pi$
$269$	−14.4854	−0.883191	−0.441596	+	0.897214i	$0.645587\pi$
$270$	0	0
$271$	−24.0715	−1.46224	−0.731120	−	0.682249i	$-0.761002\pi$
$271$	−24.0715	−1.46224	−0.731120	+	0.682249i	$0.761002\pi$
$272$	0	0
$273$	5.94087	0.359558
$274$	0	0
$275$	6.23938	0.376249
$276$	0	0
$277$	6.35892	0.382071	0.191035	−	0.981583i	$-0.438815\pi$
$277$	6.35892	0.382071	0.191035	+	0.981583i	$0.438815\pi$
$278$	0	0
$279$	−16.8944	−1.01144
$280$	0	0
$281$	25.7439	1.53575	0.767877	−	0.640598i	$-0.221313\pi$
$281$	25.7439	1.53575	0.767877	+	0.640598i	$0.221313\pi$
$282$	0	0
$283$	−0.738208	−0.0438819	−0.0219410	−	0.999759i	$-0.506985\pi$
$283$	−0.738208	−0.0438819	−0.0219410	+	0.999759i	$0.506985\pi$
$284$	0	0
$285$	4.20479	0.249071
$286$	0	0
$287$	−0.872258	−0.0514878
$288$	0	0
$289$	−11.6014	−0.682434
$290$	0	0
$291$	42.0453	2.46474
$292$	0	0
$293$	−23.2639	−1.35909	−0.679544	−	0.733634i	$-0.737823\pi$
$293$	−23.2639	−1.35909	−0.679544	+	0.733634i	$0.737823\pi$
$294$	0	0
$295$	9.07657	0.528458
$296$	0	0
$297$	3.24452	0.188266
$298$	0	0
$299$	0.176710	0.0102194
$300$	0	0
$301$	5.18249	0.298714
$302$	0	0
$303$	0.391424	0.0224867
$304$	0	0
$305$	2.48905	0.142523
$306$	0	0
$307$	−29.2633	−1.67014	−0.835071	−	0.550142i	$-0.814574\pi$
$307$	−29.2633	−1.67014	−0.835071	+	0.550142i	$0.814574\pi$
$308$	0	0
$309$	−27.5307	−1.56617
$310$	0	0
$311$	−3.81149	−0.216130	−0.108065	−	0.994144i	$-0.534465\pi$
$311$	−3.81149	−0.216130	−0.108065	+	0.994144i	$0.534465\pi$
$312$	0	0
$313$	−14.0647	−0.794985	−0.397493	−	0.917605i	$-0.630120\pi$
$313$	−14.0647	−0.794985	−0.397493	+	0.917605i	$0.630120\pi$
$314$	0	0
$315$	−1.95352	−0.110068
$316$	0	0
$317$	5.13069	0.288168	0.144084	−	0.989565i	$-0.453976\pi$
$317$	5.13069	0.288168	0.144084	+	0.989565i	$0.453976\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	14.9547	0.834689
$322$	0	0
$323$	4.33286	0.241087
$324$	0	0
$325$	−11.2178	−0.622253
$326$	0	0
$327$	29.1854	1.61396
$328$	0	0
$329$	−4.58847	−0.252970
$330$	0	0
$331$	18.2958	1.00563	0.502815	−	0.864394i	$-0.332298\pi$
$331$	18.2958	1.00563	0.502815	+	0.864394i	$0.332298\pi$
$332$	0	0
$333$	−20.5027	−1.12354
$334$	0	0
$335$	−5.81134	−0.317508
$336$	0	0
$337$	−12.3075	−0.670431	−0.335215	−	0.942142i	$-0.608809\pi$
$337$	−12.3075	−0.670431	−0.335215	+	0.942142i	$0.608809\pi$
$338$	0	0
$339$	12.5100	0.679450
$340$	0	0
$341$	12.6910	0.687255
$342$	0	0
$343$	−12.3272	−0.665608
$344$	0	0
$345$	−0.142028	−0.00764654
$346$	0	0
$347$	35.8923	1.92680	0.963400	−	0.268067i	$-0.0863848\pi$
$347$	35.8923	1.92680	0.963400	+	0.268067i	$0.0863848\pi$
$348$	0	0
$349$	−1.36783	−0.0732182	−0.0366091	−	0.999330i	$-0.511656\pi$
$349$	−1.36783	−0.0732182	−0.0366091	+	0.999330i	$0.511656\pi$
$350$	0	0
$351$	−5.83334	−0.311361
$352$	0	0
$353$	−25.5257	−1.35859	−0.679297	−	0.733863i	$-0.737715\pi$
$353$	−25.5257	−1.35859	−0.679297	+	0.733863i	$0.737715\pi$
$354$	0	0
$355$	−15.2973	−0.811895
$356$	0	0
$357$	−4.92031	−0.260411
$358$	0	0
$359$	4.87655	0.257375	0.128687	−	0.991685i	$-0.458924\pi$
$359$	4.87655	0.257375	0.128687	+	0.991685i	$0.458924\pi$
$360$	0	0
$361$	−15.5225	−0.816974
$362$	0	0
$363$	−19.2997	−1.01297
$364$	0	0
$365$	8.84643	0.463044
$366$	0	0
$367$	3.91496	0.204359	0.102180	−	0.994766i	$-0.467418\pi$
$367$	3.91496	0.204359	0.102180	+	0.994766i	$0.467418\pi$
$368$	0	0
$369$	−1.92790	−0.100362
$370$	0	0
$371$	0.395544	0.0205356
$372$	0	0
$373$	38.1002	1.97275	0.986376	−	0.164506i	$-0.0526031\pi$
$373$	38.1002	1.97275	0.986376	+	0.164506i	$0.0526031\pi$
$374$	0	0
$375$	20.2903	1.04778
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−24.9396	−1.28106	−0.640531	−	0.767932i	$-0.721286\pi$
$379$	−24.9396	−1.28106	−0.640531	+	0.767932i	$0.721286\pi$
$380$	0	0
$381$	−31.0872	−1.59265
$382$	0	0
$383$	18.5652	0.948636	0.474318	−	0.880354i	$-0.342695\pi$
$383$	18.5652	0.948636	0.474318	+	0.880354i	$0.342695\pi$
$384$	0	0
$385$	1.46747	0.0747893
$386$	0	0
$387$	11.4545	0.582267
$388$	0	0
$389$	30.5463	1.54876	0.774380	−	0.632721i	$-0.218062\pi$
$389$	30.5463	1.54876	0.774380	+	0.632721i	$0.218062\pi$
$390$	0	0
$391$	−0.146354	−0.00740143
$392$	0	0
$393$	−32.0992	−1.61919
$394$	0	0
$395$	−3.28356	−0.165214
$396$	0	0
$397$	30.8272	1.54717	0.773587	−	0.633690i	$-0.218460\pi$
$397$	30.8272	1.54717	0.773587	+	0.633690i	$0.218460\pi$
$398$	0	0
$399$	−3.94898	−0.197696
$400$	0	0
$401$	−13.1933	−0.658844	−0.329422	−	0.944183i	$-0.606854\pi$
$401$	−13.1933	−0.658844	−0.329422	+	0.944183i	$0.606854\pi$
$402$	0	0
$403$	−22.8172	−1.13661
$404$	0	0
$405$	10.9244	0.542836
$406$	0	0
$407$	15.4015	0.763423
$408$	0	0
$409$	−24.3232	−1.20271	−0.601353	−	0.798983i	$-0.705372\pi$
$409$	−24.3232	−1.20271	−0.601353	+	0.798983i	$0.705372\pi$
$410$	0	0
$411$	35.2800	1.74023
$412$	0	0
$413$	−8.52436	−0.419456
$414$	0	0
$415$	14.0695	0.690646
$416$	0	0
$417$	30.3133	1.48445
$418$	0	0
$419$	7.06723	0.345257	0.172628	−	0.984987i	$-0.444774\pi$
$419$	7.06723	0.345257	0.172628	+	0.984987i	$0.444774\pi$
$420$	0	0
$421$	2.91890	0.142259	0.0711293	−	0.997467i	$-0.477340\pi$
$421$	2.91890	0.142259	0.0711293	+	0.997467i	$0.477340\pi$
$422$	0	0
$423$	−10.1416	−0.493102
$424$	0	0
$425$	9.29076	0.450668
$426$	0	0
$427$	−2.33762	−0.113125
$428$	0	0
$429$	−9.86375	−0.476226
$430$	0	0
$431$	14.9656	0.720868	0.360434	−	0.932785i	$-0.382629\pi$
$431$	14.9656	0.720868	0.360434	+	0.932785i	$0.382629\pi$
$432$	0	0
$433$	−19.6398	−0.943830	−0.471915	−	0.881644i	$-0.656437\pi$
$433$	−19.6398	−0.943830	−0.471915	+	0.881644i	$0.656437\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.117462	−0.00561895
$438$	0	0
$439$	−32.3522	−1.54409	−0.772044	−	0.635569i	$-0.780766\pi$
$439$	−32.3522	−1.54409	−0.772044	+	0.635569i	$0.780766\pi$
$440$	0	0
$441$	−12.7057	−0.605036
$442$	0	0
$443$	6.06833	0.288315	0.144157	−	0.989555i	$-0.453953\pi$
$443$	6.06833	0.288315	0.144157	+	0.989555i	$0.453953\pi$
$444$	0	0
$445$	−7.83260	−0.371301
$446$	0	0
$447$	−28.3970	−1.34313
$448$	0	0
$449$	−10.7686	−0.508204	−0.254102	−	0.967177i	$-0.581780\pi$
$449$	−10.7686	−0.508204	−0.254102	+	0.967177i	$0.581780\pi$
$450$	0	0
$451$	1.44823	0.0681944
$452$	0	0
$453$	17.8442	0.838393
$454$	0	0
$455$	−2.63838	−0.123689
$456$	0	0
$457$	−5.62238	−0.263004	−0.131502	−	0.991316i	$-0.541980\pi$
$457$	−5.62238	−0.263004	−0.131502	+	0.991316i	$0.541980\pi$
$458$	0	0
$459$	4.83125	0.225504
$460$	0	0
$461$	−31.5582	−1.46981	−0.734906	−	0.678169i	$-0.762774\pi$
$461$	−31.5582	−1.46981	−0.734906	+	0.678169i	$0.762774\pi$
$462$	0	0
$463$	−16.4251	−0.763340	−0.381670	−	0.924299i	$-0.624651\pi$
$463$	−16.4251	−0.763340	−0.381670	+	0.924299i	$0.624651\pi$
$464$	0	0
$465$	18.3390	0.850450
$466$	0	0
$467$	−21.1043	−0.976591	−0.488296	−	0.872678i	$-0.662381\pi$
$467$	−21.1043	−0.976591	−0.488296	+	0.872678i	$0.662381\pi$
$468$	0	0
$469$	5.45779	0.252017
$470$	0	0
$471$	5.43618	0.250486
$472$	0	0
$473$	−8.60459	−0.395639
$474$	0	0
$475$	7.45664	0.342134
$476$	0	0
$477$	0.874248	0.0400291
$478$	0	0
$479$	26.1648	1.19550	0.597750	−	0.801683i	$-0.296062\pi$
$479$	26.1648	1.19550	0.597750	+	0.801683i	$0.296062\pi$
$480$	0	0
$481$	−27.6905	−1.26258
$482$	0	0
$483$	0.133387	0.00606933
$484$	0	0
$485$	−18.6726	−0.847880
$486$	0	0
$487$	−4.17922	−0.189378	−0.0946892	−	0.995507i	$-0.530186\pi$
$487$	−4.17922	−0.189378	−0.0946892	+	0.995507i	$0.530186\pi$
$488$	0	0
$489$	−19.7392	−0.892637
$490$	0	0
$491$	19.5416	0.881898	0.440949	−	0.897532i	$-0.354642\pi$
$491$	19.5416	0.881898	0.440949	+	0.897532i	$0.354642\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	3.24347	0.145783
$496$	0	0
$497$	14.3666	0.644430
$498$	0	0
$499$	27.0667	1.21167	0.605836	−	0.795590i	$-0.292839\pi$
$499$	27.0667	1.21167	0.605836	+	0.795590i	$0.292839\pi$
$500$	0	0
$501$	2.31730	0.103530
$502$	0	0
$503$	−7.27995	−0.324597	−0.162299	−	0.986742i	$-0.551891\pi$
$503$	−7.27995	−0.324597	−0.162299	+	0.986742i	$0.551891\pi$
$504$	0	0
$505$	−0.173834	−0.00773552
$506$	0	0
$507$	−11.5583	−0.513323
$508$	0	0
$509$	−25.6164	−1.13543	−0.567714	−	0.823226i	$-0.692172\pi$
$509$	−25.6164	−1.13543	−0.567714	+	0.823226i	$0.692172\pi$
$510$	0	0
$511$	−8.30822	−0.367534
$512$	0	0
$513$	3.87750	0.171196
$514$	0	0
$515$	12.2266	0.538767
$516$	0	0
$517$	7.61833	0.335054
$518$	0	0
$519$	−44.4736	−1.95217
$520$	0	0
$521$	15.4273	0.675884	0.337942	−	0.941167i	$-0.390269\pi$
$521$	15.4273	0.675884	0.337942	+	0.941167i	$0.390269\pi$
$522$	0	0
$523$	11.7860	0.515364	0.257682	−	0.966230i	$-0.417041\pi$
$523$	11.7860	0.515364	0.257682	+	0.966230i	$0.417041\pi$
$524$	0	0
$525$	−8.46763	−0.369558
$526$	0	0
$527$	18.8975	0.823189
$528$	0	0
$529$	−22.9960	−0.999827
$530$	0	0
$531$	−18.8409	−0.817624
$532$	0	0
$533$	−2.60378	−0.112782
$534$	0	0
$535$	−6.64147	−0.287136
$536$	0	0
$537$	43.8264	1.89125
$538$	0	0
$539$	9.54449	0.411110
$540$	0	0
$541$	−12.1581	−0.522718	−0.261359	−	0.965242i	$-0.584171\pi$
$541$	−12.1581	−0.522718	−0.261359	+	0.965242i	$0.584171\pi$
$542$	0	0
$543$	−21.3905	−0.917955
$544$	0	0
$545$	−12.9614	−0.555207
$546$	0	0
$547$	27.7493	1.18647	0.593237	−	0.805028i	$-0.297850\pi$
$547$	27.7493	1.18647	0.593237	+	0.805028i	$0.297850\pi$
$548$	0	0
$549$	−5.16670	−0.220509
$550$	0	0
$551$	0	0
$552$	0	0
$553$	3.08379	0.131136
$554$	0	0
$555$	22.2558	0.944706
$556$	0	0
$557$	−12.3962	−0.525242	−0.262621	−	0.964899i	$-0.584587\pi$
$557$	−12.3962	−0.525242	−0.262621	+	0.964899i	$0.584587\pi$
$558$	0	0
$559$	15.4703	0.654322
$560$	0	0
$561$	8.16930	0.344908
$562$	0	0
$563$	23.0470	0.971314	0.485657	−	0.874149i	$-0.338580\pi$
$563$	23.0470	0.971314	0.485657	+	0.874149i	$0.338580\pi$
$564$	0	0
$565$	−5.55577	−0.233733
$566$	0	0
$567$	−10.2597	−0.430868
$568$	0	0
$569$	36.2379	1.51917	0.759586	−	0.650407i	$-0.225402\pi$
$569$	36.2379	1.51917	0.759586	+	0.650407i	$0.225402\pi$
$570$	0	0
$571$	−8.74070	−0.365787	−0.182893	−	0.983133i	$-0.558546\pi$
$571$	−8.74070	−0.365787	−0.182893	+	0.983133i	$0.558546\pi$
$572$	0	0
$573$	7.30361	0.305113
$574$	0	0
$575$	−0.251868	−0.0105036
$576$	0	0
$577$	40.0327	1.66658	0.833291	−	0.552835i	$-0.186454\pi$
$577$	40.0327	1.66658	0.833291	+	0.552835i	$0.186454\pi$
$578$	0	0
$579$	16.6151	0.690498
$580$	0	0
$581$	−13.2135	−0.548190
$582$	0	0
$583$	−0.656730	−0.0271990
$584$	0	0
$585$	−5.83145	−0.241101
$586$	0	0
$587$	41.3657	1.70734	0.853672	−	0.520811i	$-0.174370\pi$
$587$	41.3657	1.70734	0.853672	+	0.520811i	$0.174370\pi$
$588$	0	0
$589$	15.1669	0.624941
$590$	0	0
$591$	−12.8739	−0.529561
$592$	0	0
$593$	11.5802	0.475544	0.237772	−	0.971321i	$-0.423583\pi$
$593$	11.5802	0.475544	0.237772	+	0.971321i	$0.423583\pi$
$594$	0	0
$595$	2.18514	0.0895822
$596$	0	0
$597$	−15.7239	−0.643536
$598$	0	0
$599$	43.2871	1.76866	0.884331	−	0.466860i	$-0.154615\pi$
$599$	43.2871	1.76866	0.884331	+	0.466860i	$0.154615\pi$
$600$	0	0
$601$	−0.184905	−0.00754245	−0.00377123	−	0.999993i	$-0.501200\pi$
$601$	−0.184905	−0.00754245	−0.00377123	+	0.999993i	$0.501200\pi$
$602$	0	0
$603$	12.0630	0.491244
$604$	0	0
$605$	8.57111	0.348465
$606$	0	0
$607$	−10.6681	−0.433005	−0.216502	−	0.976282i	$-0.569465\pi$
$607$	−10.6681	−0.433005	−0.216502	+	0.976282i	$0.569465\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13.6970	−0.554123
$612$	0	0
$613$	−33.9109	−1.36965	−0.684823	−	0.728709i	$-0.740121\pi$
$613$	−33.9109	−1.36965	−0.684823	+	0.728709i	$0.740121\pi$
$614$	0	0
$615$	2.09275	0.0843878
$616$	0	0
$617$	37.7768	1.52084	0.760418	−	0.649434i	$-0.224994\pi$
$617$	37.7768	1.52084	0.760418	+	0.649434i	$0.224994\pi$
$618$	0	0
$619$	−14.0955	−0.566548	−0.283274	−	0.959039i	$-0.591421\pi$
$619$	−14.0955	−0.566548	−0.283274	+	0.959039i	$0.591421\pi$
$620$	0	0
$621$	−0.130973	−0.00525576
$622$	0	0
$623$	7.35607	0.294715
$624$	0	0
$625$	10.9821	0.439282
$626$	0	0
$627$	6.55657	0.261844
$628$	0	0
$629$	22.9336	0.914423
$630$	0	0
$631$	36.1342	1.43848	0.719240	−	0.694761i	$-0.244490\pi$
$631$	36.1342	1.43848	0.719240	+	0.694761i	$0.244490\pi$
$632$	0	0
$633$	−23.7649	−0.944569
$634$	0	0
$635$	13.8060	0.547876
$636$	0	0
$637$	−17.1601	−0.679908
$638$	0	0
$639$	31.7536	1.25615
$640$	0	0
$641$	−34.3641	−1.35730	−0.678650	−	0.734461i	$-0.737435\pi$
$641$	−34.3641	−1.35730	−0.678650	+	0.734461i	$0.737435\pi$
$642$	0	0
$643$	12.3189	0.485810	0.242905	−	0.970050i	$-0.421900\pi$
$643$	12.3189	0.485810	0.242905	+	0.970050i	$0.421900\pi$
$644$	0	0
$645$	−12.4340	−0.489588
$646$	0	0
$647$	36.6380	1.44039	0.720193	−	0.693773i	$-0.244053\pi$
$647$	36.6380	1.44039	0.720193	+	0.693773i	$0.244053\pi$
$648$	0	0
$649$	14.1532	0.555560
$650$	0	0
$651$	−17.2233	−0.675033
$652$	0	0
$653$	−34.5088	−1.35043	−0.675217	−	0.737619i	$-0.735950\pi$
$653$	−34.5088	−1.35043	−0.675217	+	0.737619i	$0.735950\pi$
$654$	0	0
$655$	14.2554	0.557006
$656$	0	0
$657$	−18.3632	−0.716415
$658$	0	0
$659$	37.7229	1.46948	0.734739	−	0.678350i	$-0.237305\pi$
$659$	37.7229	1.46948	0.734739	+	0.678350i	$0.237305\pi$
$660$	0	0
$661$	28.7991	1.12016	0.560078	−	0.828440i	$-0.310771\pi$
$661$	28.7991	1.12016	0.560078	+	0.828440i	$0.310771\pi$
$662$	0	0
$663$	−14.6876	−0.570421
$664$	0	0
$665$	1.75377	0.0680082
$666$	0	0
$667$	0	0
$668$	0	0
$669$	5.97749	0.231103
$670$	0	0
$671$	3.88120	0.149832
$672$	0	0
$673$	37.9968	1.46467	0.732334	−	0.680946i	$-0.238431\pi$
$673$	37.9968	1.46467	0.732334	+	0.680946i	$0.238431\pi$
$674$	0	0
$675$	8.31436	0.320020
$676$	0	0
$677$	−12.7871	−0.491448	−0.245724	−	0.969340i	$-0.579026\pi$
$677$	−12.7871	−0.491448	−0.245724	+	0.969340i	$0.579026\pi$
$678$	0	0
$679$	17.5366	0.672992
$680$	0	0
$681$	−27.0504	−1.03657
$682$	0	0
$683$	27.9785	1.07057	0.535284	−	0.844672i	$-0.320205\pi$
$683$	27.9785	1.07057	0.535284	+	0.844672i	$0.320205\pi$
$684$	0	0
$685$	−15.6681	−0.598647
$686$	0	0
$687$	−50.2247	−1.91619
$688$	0	0
$689$	1.18074	0.0449826
$690$	0	0
$691$	23.2413	0.884140	0.442070	−	0.896981i	$-0.354244\pi$
$691$	23.2413	0.884140	0.442070	+	0.896981i	$0.354244\pi$
$692$	0	0
$693$	−3.04614	−0.115713
$694$	0	0
$695$	−13.4624	−0.510656
$696$	0	0
$697$	2.15649	0.0816828
$698$	0	0
$699$	51.8196	1.96000
$700$	0	0
$701$	2.35147	0.0888137	0.0444068	−	0.999014i	$-0.485860\pi$
$701$	2.35147	0.0888137	0.0444068	+	0.999014i	$0.485860\pi$
$702$	0	0
$703$	18.4062	0.694204
$704$	0	0
$705$	11.0088	0.414615
$706$	0	0
$707$	0.163258	0.00613995
$708$	0	0
$709$	−21.9932	−0.825971	−0.412985	−	0.910738i	$-0.635514\pi$
$709$	−21.9932	−0.825971	−0.412985	+	0.910738i	$0.635514\pi$
$710$	0	0
$711$	6.81591	0.255617
$712$	0	0
$713$	−0.512303	−0.0191859
$714$	0	0
$715$	4.38056	0.163824
$716$	0	0
$717$	−24.3243	−0.908407
$718$	0	0
$719$	15.6423	0.583360	0.291680	−	0.956516i	$-0.405786\pi$
$719$	15.6423	0.583360	0.291680	+	0.956516i	$0.405786\pi$
$720$	0	0
$721$	−11.4827	−0.427639
$722$	0	0
$723$	37.3787	1.39013
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−38.0405	−1.41084	−0.705422	−	0.708788i	$-0.749242\pi$
$727$	−38.0405	−1.41084	−0.705422	+	0.708788i	$0.749242\pi$
$728$	0	0
$729$	−8.62079	−0.319288
$730$	0	0
$731$	−12.8127	−0.473894
$732$	0	0
$733$	−48.8086	−1.80279	−0.901393	−	0.433002i	$-0.857454\pi$
$733$	−48.8086	−1.80279	−0.901393	+	0.433002i	$0.857454\pi$
$734$	0	0
$735$	13.7922	0.508732
$736$	0	0
$737$	−9.06167	−0.333791
$738$	0	0
$739$	−12.1920	−0.448491	−0.224245	−	0.974533i	$-0.571992\pi$
$739$	−12.1920	−0.448491	−0.224245	+	0.974533i	$0.571992\pi$
$740$	0	0
$741$	−11.7881	−0.433047
$742$	0	0
$743$	−27.7012	−1.01626	−0.508129	−	0.861281i	$-0.669663\pi$
$743$	−27.7012	−1.01626	−0.508129	+	0.861281i	$0.669663\pi$
$744$	0	0
$745$	12.6113	0.462042
$746$	0	0
$747$	−29.2051	−1.06856
$748$	0	0
$749$	6.23741	0.227910
$750$	0	0
$751$	0.438088	0.0159861	0.00799304	−	0.999968i	$-0.497456\pi$
$751$	0.438088	0.0159861	0.00799304	+	0.999968i	$0.497456\pi$
$752$	0	0
$753$	40.6994	1.48317
$754$	0	0
$755$	−7.92472	−0.288410
$756$	0	0
$757$	9.40826	0.341949	0.170975	−	0.985275i	$-0.445308\pi$
$757$	9.40826	0.341949	0.170975	+	0.985275i	$0.445308\pi$
$758$	0	0
$759$	−0.221466	−0.00803869
$760$	0	0
$761$	−45.9926	−1.66723	−0.833616	−	0.552345i	$-0.813733\pi$
$761$	−45.9926	−1.66723	−0.833616	+	0.552345i	$0.813733\pi$
$762$	0	0
$763$	12.1729	0.440688
$764$	0	0
$765$	4.82969	0.174618
$766$	0	0
$767$	−25.4461	−0.918805
$768$	0	0
$769$	−3.50959	−0.126559	−0.0632795	−	0.997996i	$-0.520156\pi$
$769$	−3.50959	−0.126559	−0.0632795	+	0.997996i	$0.520156\pi$
$770$	0	0
$771$	7.94894	0.286274
$772$	0	0
$773$	−5.32469	−0.191516	−0.0957578	−	0.995405i	$-0.530527\pi$
$773$	−5.32469	−0.191516	−0.0957578	+	0.995405i	$0.530527\pi$
$774$	0	0
$775$	32.5217	1.16822
$776$	0	0
$777$	−20.9018	−0.749847
$778$	0	0
$779$	1.73077	0.0620112
$780$	0	0
$781$	−23.8532	−0.853533
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2.41424	−0.0861680
$786$	0	0
$787$	18.6753	0.665701	0.332851	−	0.942980i	$-0.391990\pi$
$787$	18.6753	0.665701	0.332851	+	0.942980i	$0.391990\pi$
$788$	0	0
$789$	−43.9822	−1.56581
$790$	0	0
$791$	5.21776	0.185522
$792$	0	0
$793$	−6.97803	−0.247797
$794$	0	0
$795$	−0.949003	−0.0336577
$796$	0	0
$797$	23.2448	0.823372	0.411686	−	0.911326i	$-0.364940\pi$
$797$	23.2448	0.823372	0.411686	+	0.911326i	$0.364940\pi$
$798$	0	0
$799$	11.3441	0.401325
$800$	0	0
$801$	16.2587	0.574472
$802$	0	0
$803$	13.7943	0.486791
$804$	0	0
$805$	−0.0592382	−0.00208787
$806$	0	0
$807$	−32.6395	−1.14896
$808$	0	0
$809$	30.0014	1.05479	0.527397	−	0.849619i	$-0.323168\pi$
$809$	30.0014	1.05479	0.527397	+	0.849619i	$0.323168\pi$
$810$	0	0
$811$	−16.8432	−0.591445	−0.295723	−	0.955274i	$-0.595560\pi$
$811$	−16.8432	−0.591445	−0.295723	+	0.955274i	$0.595560\pi$
$812$	0	0
$813$	−54.2395	−1.90226
$814$	0	0
$815$	8.76630	0.307070
$816$	0	0
$817$	−10.2833	−0.359767
$818$	0	0
$819$	5.47667	0.191370
$820$	0	0
$821$	−50.0949	−1.74833	−0.874163	−	0.485633i	$-0.838589\pi$
$821$	−50.0949	−1.74833	−0.874163	+	0.485633i	$0.838589\pi$
$822$	0	0
$823$	19.1188	0.666439	0.333220	−	0.942849i	$-0.391865\pi$
$823$	19.1188	0.666439	0.333220	+	0.942849i	$0.391865\pi$
$824$	0	0
$825$	14.0590	0.489471
$826$	0	0
$827$	16.6670	0.579567	0.289784	−	0.957092i	$-0.406417\pi$
$827$	16.6670	0.579567	0.289784	+	0.957092i	$0.406417\pi$
$828$	0	0
$829$	33.7770	1.17312	0.586562	−	0.809904i	$-0.300481\pi$
$829$	33.7770	1.17312	0.586562	+	0.809904i	$0.300481\pi$
$830$	0	0
$831$	14.3283	0.497044
$832$	0	0
$833$	14.2122	0.492425
$834$	0	0
$835$	−1.02913	−0.0356145
$836$	0	0
$837$	16.9115	0.584547
$838$	0	0
$839$	−28.3298	−0.978052	−0.489026	−	0.872269i	$-0.662648\pi$
$839$	−28.3298	−0.978052	−0.489026	+	0.872269i	$0.662648\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	58.0079	1.99790
$844$	0	0
$845$	5.13313	0.176585
$846$	0	0
$847$	−8.04965	−0.276589
$848$	0	0
$849$	−1.66338	−0.0570870
$850$	0	0
$851$	−0.621719	−0.0213123
$852$	0	0
$853$	−39.1546	−1.34063	−0.670314	−	0.742077i	$-0.733841\pi$
$853$	−39.1546	−1.34063	−0.670314	+	0.742077i	$0.733841\pi$
$854$	0	0
$855$	3.87625	0.132565
$856$	0	0
$857$	15.9931	0.546314	0.273157	−	0.961970i	$-0.411932\pi$
$857$	15.9931	0.546314	0.273157	+	0.961970i	$0.411932\pi$
$858$	0	0
$859$	54.4035	1.85622	0.928112	−	0.372302i	$-0.121431\pi$
$859$	54.4035	1.85622	0.928112	+	0.372302i	$0.121431\pi$
$860$	0	0
$861$	−1.96543	−0.0669816
$862$	0	0
$863$	17.2418	0.586917	0.293459	−	0.955972i	$-0.405194\pi$
$863$	17.2418	0.586917	0.293459	+	0.955972i	$0.405194\pi$
$864$	0	0
$865$	19.7510	0.671555
$866$	0	0
$867$	−26.1410	−0.887795
$868$	0	0
$869$	−5.12008	−0.173687
$870$	0	0
$871$	16.2920	0.552035
$872$	0	0
$873$	38.7601	1.31183
$874$	0	0
$875$	8.46281	0.286095
$876$	0	0
$877$	−14.5423	−0.491058	−0.245529	−	0.969389i	$-0.578962\pi$
$877$	−14.5423	−0.491058	−0.245529	+	0.969389i	$0.578962\pi$
$878$	0	0
$879$	−52.4196	−1.76807
$880$	0	0
$881$	−26.9280	−0.907227	−0.453614	−	0.891198i	$-0.649865\pi$
$881$	−26.9280	−0.907227	−0.453614	+	0.891198i	$0.649865\pi$
$882$	0	0
$883$	6.33037	0.213034	0.106517	−	0.994311i	$-0.466030\pi$
$883$	6.33037	0.213034	0.106517	+	0.994311i	$0.466030\pi$
$884$	0	0
$885$	20.4519	0.687484
$886$	0	0
$887$	−24.3954	−0.819119	−0.409559	−	0.912283i	$-0.634318\pi$
$887$	−24.3954	−0.819119	−0.409559	+	0.912283i	$0.634318\pi$
$888$	0	0
$889$	−12.9661	−0.434869
$890$	0	0
$891$	17.0344	0.570676
$892$	0	0
$893$	9.10461	0.304674
$894$	0	0
$895$	−19.4636	−0.650596
$896$	0	0
$897$	0.398175	0.0132947
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−0.977906	−0.0325788
$902$	0	0
$903$	11.6775	0.388603
$904$	0	0
$905$	9.49967	0.315780
$906$	0	0
$907$	−49.8436	−1.65503	−0.827515	−	0.561444i	$-0.810246\pi$
$907$	−49.8436	−1.65503	−0.827515	+	0.561444i	$0.810246\pi$
$908$	0	0
$909$	0.360840	0.0119683
$910$	0	0
$911$	4.86062	0.161039	0.0805197	−	0.996753i	$-0.474342\pi$
$911$	4.86062	0.161039	0.0805197	+	0.996753i	$0.474342\pi$
$912$	0	0
$913$	21.9387	0.726066
$914$	0	0
$915$	5.60849	0.185411
$916$	0	0
$917$	−13.3882	−0.442116
$918$	0	0
$919$	−45.2031	−1.49111	−0.745557	−	0.666442i	$-0.767816\pi$
$919$	−45.2031	−1.49111	−0.745557	+	0.666442i	$0.767816\pi$
$920$	0	0
$921$	−65.9379	−2.17273
$922$	0	0
$923$	42.8858	1.41160
$924$	0	0
$925$	39.4677	1.29769
$926$	0	0
$927$	−25.3796	−0.833575
$928$	0	0
$929$	−57.1434	−1.87481	−0.937407	−	0.348236i	$-0.886781\pi$
$929$	−57.1434	−1.87481	−0.937407	+	0.348236i	$0.886781\pi$
$930$	0	0
$931$	11.4066	0.373835
$932$	0	0
$933$	−8.58831	−0.281169
$934$	0	0
$935$	−3.62804	−0.118650
$936$	0	0
$937$	49.6478	1.62192	0.810961	−	0.585100i	$-0.198945\pi$
$937$	49.6478	1.62192	0.810961	+	0.585100i	$0.198945\pi$
$938$	0	0
$939$	−31.6916	−1.03421
$940$	0	0
$941$	33.4755	1.09127	0.545636	−	0.838022i	$-0.316288\pi$
$941$	33.4755	1.09127	0.545636	+	0.838022i	$0.316288\pi$
$942$	0	0
$943$	−0.0584613	−0.00190376
$944$	0	0
$945$	1.95550	0.0636124
$946$	0	0
$947$	−5.96513	−0.193841	−0.0969203	−	0.995292i	$-0.530899\pi$
$947$	−5.96513	−0.193841	−0.0969203	+	0.995292i	$0.530899\pi$
$948$	0	0
$949$	−24.8009	−0.805071
$950$	0	0
$951$	11.5608	0.374885
$952$	0	0
$953$	−24.5231	−0.794381	−0.397191	−	0.917736i	$-0.630015\pi$
$953$	−24.5231	−0.794381	−0.397191	+	0.917736i	$0.630015\pi$
$954$	0	0
$955$	−3.24358	−0.104960
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.7149	0.475167
$960$	0	0
$961$	35.1496	1.13386
$962$	0	0
$963$	13.7862	0.444253
$964$	0	0
$965$	−7.37886	−0.237534
$966$	0	0
$967$	22.4233	0.721084	0.360542	−	0.932743i	$-0.382592\pi$
$967$	22.4233	0.721084	0.360542	+	0.932743i	$0.382592\pi$
$968$	0	0
$969$	9.76307	0.313635
$970$	0	0
$971$	23.3556	0.749517	0.374758	−	0.927123i	$-0.377726\pi$
$971$	23.3556	0.749517	0.374758	+	0.927123i	$0.377726\pi$
$972$	0	0
$973$	12.6433	0.405326
$974$	0	0
$975$	−25.2767	−0.809503
$976$	0	0
$977$	−2.84886	−0.0911430	−0.0455715	−	0.998961i	$-0.514511\pi$
$977$	−2.84886	−0.0911430	−0.0455715	+	0.998961i	$0.514511\pi$
$978$	0	0
$979$	−12.2134	−0.390343
$980$	0	0
$981$	26.9050	0.859010
$982$	0	0
$983$	−24.8988	−0.794150	−0.397075	−	0.917786i	$-0.629975\pi$
$983$	−24.8988	−0.794150	−0.397075	+	0.917786i	$0.629975\pi$
$984$	0	0
$985$	5.71738	0.182171
$986$	0	0
$987$	−10.3390	−0.329095
$988$	0	0
$989$	0.347346	0.0110449
$990$	0	0
$991$	−53.9625	−1.71418	−0.857088	−	0.515171i	$-0.827729\pi$
$991$	−53.9625	−1.71418	−0.857088	+	0.515171i	$0.827729\pi$
$992$	0	0
$993$	41.2254	1.30825
$994$	0	0
$995$	6.98308	0.221379
$996$	0	0
$997$	2.82060	0.0893292	0.0446646	−	0.999002i	$-0.485778\pi$
$997$	2.82060	0.0893292	0.0446646	+	0.999002i	$0.485778\pi$
$998$	0	0
$999$	20.5234	0.649333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6728.2.a.z.1.10		12
29.7	even	7		232.2.m.d.49.4	✓	24
29.25	even	7		232.2.m.d.161.4	yes	24
29.28	even	2		6728.2.a.bb.1.3		12
116.7	odd	14		464.2.u.i.49.1		24
116.83	odd	14		464.2.u.i.161.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.m.d.49.4	✓	24	29.7	even	7
232.2.m.d.161.4	yes	24	29.25	even	7
464.2.u.i.49.1		24	116.7	odd	14
464.2.u.i.161.1		24	116.83	odd	14
6728.2.a.z.1.10		12	1.1	even	1	trivial
6728.2.a.bb.1.3		12	29.28	even	2

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:	\( N \)	\(=\)	\( 6728 = 2^{3} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6728.a (trivial)

\(f(q)\)	\(=\)	\(q+2.25326 q^{3} -1.00069 q^{5} +0.939808 q^{7} +2.07720 q^{9} +O(q^{10})\)	\(q+2.25326 q^{3} -1.00069 q^{5} +0.939808 q^{7} +2.07720 q^{9} -1.56038 q^{11} +2.80542 q^{13} -2.25482 q^{15} -2.32349 q^{17} -1.86480 q^{19} +2.11764 q^{21} +0.0629887 q^{23} -3.99862 q^{25} -2.07931 q^{27} -8.13324 q^{31} -3.51596 q^{33} -0.940457 q^{35} -9.87033 q^{37} +6.32136 q^{39} -0.928123 q^{41} +5.51441 q^{43} -2.07864 q^{45} -4.88234 q^{47} -6.11676 q^{49} -5.23544 q^{51} +0.420878 q^{53} +1.56146 q^{55} -4.20190 q^{57} -9.07031 q^{59} -2.48734 q^{61} +1.95217 q^{63} -2.80736 q^{65} +5.80734 q^{67} +0.141930 q^{69} +15.2867 q^{71} -8.84034 q^{73} -9.00995 q^{75} -1.46646 q^{77} +3.28129 q^{79} -10.9168 q^{81} -14.0598 q^{83} +2.32509 q^{85} +7.82720 q^{89} +2.63656 q^{91} -18.3263 q^{93} +1.86609 q^{95} +18.6597 q^{97} -3.24123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9	\( 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9} - 3 q^{11} - 3 q^{13} + 3 q^{15} - 8 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} + 12 q^{25} - 7 q^{27} - 29 q^{31} - 46 q^{33} + 17 q^{35} - 38 q^{37} + 10 q^{39} - 11 q^{41} - 9 q^{43} - 54 q^{45} - 34 q^{47} + 33 q^{49} + 17 q^{51} - 15 q^{53} - 2 q^{55} - q^{57} + 57 q^{59} - 37 q^{61} + 9 q^{63} - 59 q^{65} + 33 q^{67} + 21 q^{69} - 21 q^{71} - 13 q^{73} - 13 q^{75} + 3 q^{77} - 32 q^{79} + 36 q^{81} + 48 q^{83} - 17 q^{85} - 20 q^{89} - 2 q^{91} - 37 q^{93} - 7 q^{97} + 16 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9 - 3 * q^11 - 3 * q^13 + 3 * q^15 - 8 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 + 12 * q^25 - 7 * q^27 - 29 * q^31 - 46 * q^33 + 17 * q^35 - 38 * q^37 + 10 * q^39 - 11 * q^41 - 9 * q^43 - 54 * q^45 - 34 * q^47 + 33 * q^49 + 17 * q^51 - 15 * q^53 - 2 * q^55 - q^57 + 57 * q^59 - 37 * q^61 + 9 * q^63 - 59 * q^65 + 33 * q^67 + 21 * q^69 - 21 * q^71 - 13 * q^73 - 13 * q^75 + 3 * q^77 - 32 * q^79 + 36 * q^81 + 48 * q^83 - 17 * q^85 - 20 * q^89 - 2 * q^91 - 37 * q^93 - 7 * q^97 + 16 * q^99

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