LMFDB - Embedded newform 6728.2.a.z.1.12

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			1.12
Root			$-2.76658$ 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)

$f(q)$	$=$	$q+2.76658 q^{3} +0.385552 q^{5} +1.24280 q^{7} +4.65398 q^{9} +O(q^{10})$	$q+2.76658 q^{3} +0.385552 q^{5} +1.24280 q^{7} +4.65398 q^{9} -4.31882 q^{11} -4.39038 q^{13} +1.06666 q^{15} -3.23759 q^{17} +3.79169 q^{19} +3.43831 q^{21} -4.69267 q^{23} -4.85135 q^{25} +4.57587 q^{27} -5.38296 q^{31} -11.9484 q^{33} +0.479165 q^{35} -0.973790 q^{37} -12.1464 q^{39} -6.85312 q^{41} -12.6267 q^{43} +1.79435 q^{45} -3.62993 q^{47} -5.45544 q^{49} -8.95707 q^{51} +8.91633 q^{53} -1.66513 q^{55} +10.4900 q^{57} -0.479557 q^{59} -5.21253 q^{61} +5.78398 q^{63} -1.69272 q^{65} -1.40456 q^{67} -12.9827 q^{69} -9.03766 q^{71} +8.96979 q^{73} -13.4217 q^{75} -5.36744 q^{77} +13.4972 q^{79} -1.30242 q^{81} +17.0677 q^{83} -1.24826 q^{85} +5.42584 q^{89} -5.45638 q^{91} -14.8924 q^{93} +1.46189 q^{95} +14.7374 q^{97} -20.0997 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.76658	1.59729	0.798644	−	0.601804i	$-0.205551\pi$
$3$	2.76658	1.59729	0.798644	+	0.601804i	$0.205551\pi$
$4$	0	0
$5$	0.385552	0.172424	0.0862120	−	0.996277i	$-0.472524\pi$
$5$	0.385552	0.172424	0.0862120	+	0.996277i	$0.472524\pi$
$6$	0	0
$7$	1.24280	0.469735	0.234868	−	0.972027i	$-0.424534\pi$
$7$	1.24280	0.469735	0.234868	+	0.972027i	$0.424534\pi$
$8$	0	0
$9$	4.65398	1.55133
$10$	0	0
$11$	−4.31882	−1.30217	−0.651087	−	0.759003i	$-0.725687\pi$
$11$	−4.31882	−1.30217	−0.651087	+	0.759003i	$0.725687\pi$
$12$	0	0
$13$	−4.39038	−1.21767	−0.608836	−	0.793296i	$-0.708363\pi$
$13$	−4.39038	−1.21767	−0.608836	+	0.793296i	$0.708363\pi$
$14$	0	0
$15$	1.06666	0.275411
$16$	0	0
$17$	−3.23759	−0.785232	−0.392616	−	0.919703i	$-0.628430\pi$
$17$	−3.23759	−0.785232	−0.392616	+	0.919703i	$0.628430\pi$
$18$	0	0
$19$	3.79169	0.869874	0.434937	−	0.900461i	$-0.356771\pi$
$19$	3.79169	0.869874	0.434937	+	0.900461i	$0.356771\pi$
$20$	0	0
$21$	3.43831	0.750302
$22$	0	0
$23$	−4.69267	−0.978490	−0.489245	−	0.872146i	$-0.662728\pi$
$23$	−4.69267	−0.978490	−0.489245	+	0.872146i	$0.662728\pi$
$24$	0	0
$25$	−4.85135	−0.970270
$26$	0	0
$27$	4.57587	0.880626
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−5.38296	−0.966809	−0.483404	−	0.875397i	$-0.660600\pi$
$31$	−5.38296	−0.966809	−0.483404	+	0.875397i	$0.660600\pi$
$32$	0	0
$33$	−11.9484	−2.07995
$34$	0	0
$35$	0.479165	0.0809936
$36$	0	0
$37$	−0.973790	−0.160090	−0.0800451	−	0.996791i	$-0.525506\pi$
$37$	−0.973790	−0.160090	−0.0800451	+	0.996791i	$0.525506\pi$
$38$	0	0
$39$	−12.1464	−1.94497
$40$	0	0
$41$	−6.85312	−1.07028	−0.535139	−	0.844764i	$-0.679741\pi$
$41$	−6.85312	−1.07028	−0.535139	+	0.844764i	$0.679741\pi$
$42$	0	0
$43$	−12.6267	−1.92556	−0.962779	−	0.270288i	$-0.912881\pi$
$43$	−12.6267	−1.92556	−0.962779	+	0.270288i	$0.912881\pi$
$44$	0	0
$45$	1.79435	0.267486
$46$	0	0
$47$	−3.62993	−0.529480	−0.264740	−	0.964320i	$-0.585286\pi$
$47$	−3.62993	−0.529480	−0.264740	+	0.964320i	$0.585286\pi$
$48$	0	0
$49$	−5.45544	−0.779349
$50$	0	0
$51$	−8.95707	−1.25424
$52$	0	0
$53$	8.91633	1.22475	0.612376	−	0.790566i	$-0.290214\pi$
$53$	8.91633	1.22475	0.612376	+	0.790566i	$0.290214\pi$
$54$	0	0
$55$	−1.66513	−0.224526
$56$	0	0
$57$	10.4900	1.38944
$58$	0	0
$59$	−0.479557	−0.0624330	−0.0312165	−	0.999513i	$-0.509938\pi$
$59$	−0.479557	−0.0624330	−0.0312165	+	0.999513i	$0.509938\pi$
$60$	0	0
$61$	−5.21253	−0.667396	−0.333698	−	0.942680i	$-0.608297\pi$
$61$	−5.21253	−0.667396	−0.333698	+	0.942680i	$0.608297\pi$
$62$	0	0
$63$	5.78398	0.728712
$64$	0	0
$65$	−1.69272	−0.209956
$66$	0	0
$67$	−1.40456	−0.171595	−0.0857973	−	0.996313i	$-0.527344\pi$
$67$	−1.40456	−0.171595	−0.0857973	+	0.996313i	$0.527344\pi$
$68$	0	0
$69$	−12.9827	−1.56293
$70$	0	0
$71$	−9.03766	−1.07257	−0.536287	−	0.844036i	$-0.680173\pi$
$71$	−9.03766	−1.07257	−0.536287	+	0.844036i	$0.680173\pi$
$72$	0	0
$73$	8.96979	1.04983	0.524917	−	0.851153i	$-0.324096\pi$
$73$	8.96979	1.04983	0.524917	+	0.851153i	$0.324096\pi$
$74$	0	0
$75$	−13.4217	−1.54980
$76$	0	0
$77$	−5.36744	−0.611677
$78$	0	0
$79$	13.4972	1.51856	0.759279	−	0.650765i	$-0.225552\pi$
$79$	13.4972	1.51856	0.759279	+	0.650765i	$0.225552\pi$
$80$	0	0
$81$	−1.30242	−0.144713
$82$	0	0
$83$	17.0677	1.87342	0.936711	−	0.350104i	$-0.113854\pi$
$83$	17.0677	1.87342	0.936711	+	0.350104i	$0.113854\pi$
$84$	0	0
$85$	−1.24826	−0.135393
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.42584	0.575138	0.287569	−	0.957760i	$-0.407153\pi$
$89$	5.42584	0.575138	0.287569	+	0.957760i	$0.407153\pi$
$90$	0	0
$91$	−5.45638	−0.571984
$92$	0	0
$93$	−14.8924	−1.54427
$94$	0	0
$95$	1.46189	0.149987
$96$	0	0
$97$	14.7374	1.49636	0.748178	−	0.663498i	$-0.230929\pi$
$97$	14.7374	1.49636	0.748178	+	0.663498i	$0.230929\pi$
$98$	0	0
$99$	−20.0997	−2.02010
$100$	0	0
$101$	−6.13471	−0.610426	−0.305213	−	0.952284i	$-0.598728\pi$
$101$	−6.13471	−0.610426	−0.305213	+	0.952284i	$0.598728\pi$
$102$	0	0
$103$	19.5073	1.92211	0.961055	−	0.276358i	$-0.0891274\pi$
$103$	19.5073	1.92211	0.961055	+	0.276358i	$0.0891274\pi$
$104$	0	0
$105$	1.32565	0.129370
$106$	0	0
$107$	7.19282	0.695357	0.347678	−	0.937614i	$-0.386970\pi$
$107$	7.19282	0.695357	0.347678	+	0.937614i	$0.386970\pi$
$108$	0	0
$109$	−17.5948	−1.68528	−0.842639	−	0.538479i	$-0.818999\pi$
$109$	−17.5948	−1.68528	−0.842639	+	0.538479i	$0.818999\pi$
$110$	0	0
$111$	−2.69407	−0.255710
$112$	0	0
$113$	16.1455	1.51884	0.759422	−	0.650598i	$-0.225482\pi$
$113$	16.1455	1.51884	0.759422	+	0.650598i	$0.225482\pi$
$114$	0	0
$115$	−1.80927	−0.168715
$116$	0	0
$117$	−20.4327	−1.88901
$118$	0	0
$119$	−4.02369	−0.368851
$120$	0	0
$121$	7.65222	0.695657
$122$	0	0
$123$	−18.9597	−1.70954
$124$	0	0
$125$	−3.79821	−0.339722
$126$	0	0
$127$	−6.26860	−0.556249	−0.278124	−	0.960545i	$-0.589713\pi$
$127$	−6.26860	−0.556249	−0.278124	+	0.960545i	$0.589713\pi$
$128$	0	0
$129$	−34.9329	−3.07567
$130$	0	0
$131$	−6.64950	−0.580970	−0.290485	−	0.956880i	$-0.593817\pi$
$131$	−6.64950	−0.580970	−0.290485	+	0.956880i	$0.593817\pi$
$132$	0	0
$133$	4.71232	0.408610
$134$	0	0
$135$	1.76423	0.151841
$136$	0	0
$137$	1.63349	0.139559	0.0697794	−	0.997562i	$-0.477770\pi$
$137$	1.63349	0.139559	0.0697794	+	0.997562i	$0.477770\pi$
$138$	0	0
$139$	1.58496	0.134434	0.0672171	−	0.997738i	$-0.478588\pi$
$139$	1.58496	0.134434	0.0672171	+	0.997738i	$0.478588\pi$
$140$	0	0
$141$	−10.0425	−0.845732
$142$	0	0
$143$	18.9613	1.58562
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−15.0929	−1.24484
$148$	0	0
$149$	5.29865	0.434082	0.217041	−	0.976162i	$-0.430359\pi$
$149$	5.29865	0.434082	0.217041	+	0.976162i	$0.430359\pi$
$150$	0	0
$151$	−0.791921	−0.0644456	−0.0322228	−	0.999481i	$-0.510259\pi$
$151$	−0.791921	−0.0644456	−0.0322228	+	0.999481i	$0.510259\pi$
$152$	0	0
$153$	−15.0677	−1.21815
$154$	0	0
$155$	−2.07541	−0.166701
$156$	0	0
$157$	−3.72631	−0.297392	−0.148696	−	0.988883i	$-0.547508\pi$
$157$	−3.72631	−0.297392	−0.148696	+	0.988883i	$0.547508\pi$
$158$	0	0
$159$	24.6678	1.95628
$160$	0	0
$161$	−5.83207	−0.459631
$162$	0	0
$163$	0.517363	0.0405230	0.0202615	−	0.999795i	$-0.493550\pi$
$163$	0.517363	0.0405230	0.0202615	+	0.999795i	$0.493550\pi$
$164$	0	0
$165$	−4.60672	−0.358633
$166$	0	0
$167$	−25.3820	−1.96412	−0.982058	−	0.188580i	$-0.939611\pi$
$167$	−25.3820	−1.96412	−0.982058	+	0.188580i	$0.939611\pi$
$168$	0	0
$169$	6.27545	0.482727
$170$	0	0
$171$	17.6465	1.34946
$172$	0	0
$173$	13.0345	0.990996	0.495498	−	0.868609i	$-0.334986\pi$
$173$	13.0345	0.990996	0.495498	+	0.868609i	$0.334986\pi$
$174$	0	0
$175$	−6.02927	−0.455770
$176$	0	0
$177$	−1.32673	−0.0997235
$178$	0	0
$179$	−6.87518	−0.513875	−0.256938	−	0.966428i	$-0.582714\pi$
$179$	−6.87518	−0.513875	−0.256938	+	0.966428i	$0.582714\pi$
$180$	0	0
$181$	−21.8047	−1.62073	−0.810367	−	0.585923i	$-0.800732\pi$
$181$	−21.8047	−1.62073	−0.810367	+	0.585923i	$0.800732\pi$
$182$	0	0
$183$	−14.4209	−1.06602
$184$	0	0
$185$	−0.375447	−0.0276034
$186$	0	0
$187$	13.9826	1.02251
$188$	0	0
$189$	5.68690	0.413661
$190$	0	0
$191$	−12.6291	−0.913809	−0.456905	−	0.889516i	$-0.651042\pi$
$191$	−12.6291	−0.913809	−0.456905	+	0.889516i	$0.651042\pi$
$192$	0	0
$193$	−8.25947	−0.594530	−0.297265	−	0.954795i	$-0.596074\pi$
$193$	−8.25947	−0.594530	−0.297265	+	0.954795i	$0.596074\pi$
$194$	0	0
$195$	−4.68305	−0.335360
$196$	0	0
$197$	25.5070	1.81730	0.908650	−	0.417558i	$-0.137114\pi$
$197$	25.5070	1.81730	0.908650	+	0.417558i	$0.137114\pi$
$198$	0	0
$199$	2.14123	0.151788	0.0758940	−	0.997116i	$-0.475819\pi$
$199$	2.14123	0.151788	0.0758940	+	0.997116i	$0.475819\pi$
$200$	0	0
$201$	−3.88584	−0.274086
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.64223	−0.184542
$206$	0	0
$207$	−21.8396	−1.51796
$208$	0	0
$209$	−16.3756	−1.13273
$210$	0	0
$211$	−16.3088	−1.12274	−0.561370	−	0.827565i	$-0.689726\pi$
$211$	−16.3088	−1.12274	−0.561370	+	0.827565i	$0.689726\pi$
$212$	0	0
$213$	−25.0034	−1.71321
$214$	0	0
$215$	−4.86826	−0.332013
$216$	0	0
$217$	−6.68996	−0.454144
$218$	0	0
$219$	24.8157	1.67689
$220$	0	0
$221$	14.2143	0.956155
$222$	0	0
$223$	−7.39156	−0.494975	−0.247488	−	0.968891i	$-0.579605\pi$
$223$	−7.39156	−0.494975	−0.247488	+	0.968891i	$0.579605\pi$
$224$	0	0
$225$	−22.5781	−1.50521
$226$	0	0
$227$	27.8613	1.84922	0.924609	−	0.380917i	$-0.124392\pi$
$227$	27.8613	1.84922	0.924609	+	0.380917i	$0.124392\pi$
$228$	0	0
$229$	10.9409	0.722996	0.361498	−	0.932373i	$-0.382265\pi$
$229$	10.9409	0.722996	0.361498	+	0.932373i	$0.382265\pi$
$230$	0	0
$231$	−14.8495	−0.977023
$232$	0	0
$233$	22.6821	1.48596	0.742978	−	0.669316i	$-0.233413\pi$
$233$	22.6821	1.48596	0.742978	+	0.669316i	$0.233413\pi$
$234$	0	0
$235$	−1.39953	−0.0912951
$236$	0	0
$237$	37.3412	2.42557
$238$	0	0
$239$	19.0986	1.23538	0.617691	−	0.786421i	$-0.288068\pi$
$239$	19.0986	1.23538	0.617691	+	0.786421i	$0.288068\pi$
$240$	0	0
$241$	18.6433	1.20092	0.600460	−	0.799655i	$-0.294984\pi$
$241$	18.6433	1.20092	0.600460	+	0.799655i	$0.294984\pi$
$242$	0	0
$243$	−17.3309	−1.11177
$244$	0	0
$245$	−2.10336	−0.134378
$246$	0	0
$247$	−16.6470	−1.05922
$248$	0	0
$249$	47.2192	2.99239
$250$	0	0
$251$	−1.95482	−0.123387	−0.0616935	−	0.998095i	$-0.519650\pi$
$251$	−1.95482	−0.123387	−0.0616935	+	0.998095i	$0.519650\pi$
$252$	0	0
$253$	20.2668	1.27416
$254$	0	0
$255$	−3.45341	−0.216261
$256$	0	0
$257$	16.6609	1.03928	0.519640	−	0.854385i	$-0.326066\pi$
$257$	16.6609	1.03928	0.519640	+	0.854385i	$0.326066\pi$
$258$	0	0
$259$	−1.21023	−0.0752000
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.89626	−0.301916	−0.150958	−	0.988540i	$-0.548236\pi$
$263$	−4.89626	−0.301916	−0.150958	+	0.988540i	$0.548236\pi$
$264$	0	0
$265$	3.43771	0.211177
$266$	0	0
$267$	15.0110	0.918661
$268$	0	0
$269$	10.3898	0.633478	0.316739	−	0.948513i	$-0.397412\pi$
$269$	10.3898	0.633478	0.316739	+	0.948513i	$0.397412\pi$
$270$	0	0
$271$	−3.89105	−0.236365	−0.118182	−	0.992992i	$-0.537707\pi$
$271$	−3.89105	−0.236365	−0.118182	+	0.992992i	$0.537707\pi$
$272$	0	0
$273$	−15.0955	−0.913622
$274$	0	0
$275$	20.9521	1.26346
$276$	0	0
$277$	−16.7403	−1.00583	−0.502913	−	0.864337i	$-0.667738\pi$
$277$	−16.7403	−1.00583	−0.502913	+	0.864337i	$0.667738\pi$
$278$	0	0
$279$	−25.0522	−1.49984
$280$	0	0
$281$	−2.61645	−0.156085	−0.0780423	−	0.996950i	$-0.524867\pi$
$281$	−2.61645	−0.156085	−0.0780423	+	0.996950i	$0.524867\pi$
$282$	0	0
$283$	23.0703	1.37139	0.685694	−	0.727890i	$-0.259499\pi$
$283$	23.0703	1.37139	0.685694	+	0.727890i	$0.259499\pi$
$284$	0	0
$285$	4.04445	0.239572
$286$	0	0
$287$	−8.51708	−0.502747
$288$	0	0
$289$	−6.51799	−0.383411
$290$	0	0
$291$	40.7722	2.39011
$292$	0	0
$293$	−2.73397	−0.159720	−0.0798600	−	0.996806i	$-0.525447\pi$
$293$	−2.73397	−0.159720	−0.0798600	+	0.996806i	$0.525447\pi$
$294$	0	0
$295$	−0.184894	−0.0107650
$296$	0	0
$297$	−19.7624	−1.14673
$298$	0	0
$299$	20.6026	1.19148
$300$	0	0
$301$	−15.6925	−0.904502
$302$	0	0
$303$	−16.9722	−0.975026
$304$	0	0
$305$	−2.00970	−0.115075
$306$	0	0
$307$	−15.3288	−0.874859	−0.437429	−	0.899253i	$-0.644111\pi$
$307$	−15.3288	−0.874859	−0.437429	+	0.899253i	$0.644111\pi$
$308$	0	0
$309$	53.9685	3.07016
$310$	0	0
$311$	−12.1432	−0.688580	−0.344290	−	0.938863i	$-0.611880\pi$
$311$	−12.1432	−0.688580	−0.344290	+	0.938863i	$0.611880\pi$
$312$	0	0
$313$	4.25978	0.240777	0.120388	−	0.992727i	$-0.461586\pi$
$313$	4.25978	0.240777	0.120388	+	0.992727i	$0.461586\pi$
$314$	0	0
$315$	2.23002	0.125648
$316$	0	0
$317$	−21.0794	−1.18394	−0.591969	−	0.805961i	$-0.701649\pi$
$317$	−21.0794	−1.18394	−0.591969	+	0.805961i	$0.701649\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	19.8995	1.11068
$322$	0	0
$323$	−12.2760	−0.683052
$324$	0	0
$325$	21.2993	1.18147
$326$	0	0
$327$	−48.6775	−2.69187
$328$	0	0
$329$	−4.51129	−0.248715
$330$	0	0
$331$	4.72121	0.259501	0.129751	−	0.991547i	$-0.458582\pi$
$331$	4.72121	0.259501	0.129751	+	0.991547i	$0.458582\pi$
$332$	0	0
$333$	−4.53200	−0.248352
$334$	0	0
$335$	−0.541531	−0.0295870
$336$	0	0
$337$	−21.2092	−1.15534	−0.577670	−	0.816270i	$-0.696038\pi$
$337$	−21.2092	−1.15534	−0.577670	+	0.816270i	$0.696038\pi$
$338$	0	0
$339$	44.6680	2.42603
$340$	0	0
$341$	23.2481	1.25895
$342$	0	0
$343$	−15.4797	−0.835823
$344$	0	0
$345$	−5.00549	−0.269487
$346$	0	0
$347$	−13.9042	−0.746416	−0.373208	−	0.927748i	$-0.621742\pi$
$347$	−13.9042	−0.746416	−0.373208	+	0.927748i	$0.621742\pi$
$348$	0	0
$349$	−17.1560	−0.918339	−0.459170	−	0.888349i	$-0.651853\pi$
$349$	−17.1560	−0.918339	−0.459170	+	0.888349i	$0.651853\pi$
$350$	0	0
$351$	−20.0898	−1.07231
$352$	0	0
$353$	21.8583	1.16340	0.581700	−	0.813403i	$-0.302388\pi$
$353$	21.8583	1.16340	0.581700	+	0.813403i	$0.302388\pi$
$354$	0	0
$355$	−3.48449	−0.184937
$356$	0	0
$357$	−11.1319	−0.589161
$358$	0	0
$359$	−13.4982	−0.712410	−0.356205	−	0.934408i	$-0.615929\pi$
$359$	−13.4982	−0.712410	−0.356205	+	0.934408i	$0.615929\pi$
$360$	0	0
$361$	−4.62308	−0.243320
$362$	0	0
$363$	21.1705	1.11116
$364$	0	0
$365$	3.45832	0.181017
$366$	0	0
$367$	7.03127	0.367029	0.183515	−	0.983017i	$-0.441253\pi$
$367$	7.03127	0.367029	0.183515	+	0.983017i	$0.441253\pi$
$368$	0	0
$369$	−31.8943	−1.66035
$370$	0	0
$371$	11.0812	0.575309
$372$	0	0
$373$	−21.9880	−1.13850	−0.569248	−	0.822166i	$-0.692766\pi$
$373$	−21.9880	−1.13850	−0.569248	+	0.822166i	$0.692766\pi$
$374$	0	0
$375$	−10.5080	−0.542633
$376$	0	0
$377$	0	0
$378$	0	0
$379$	12.4001	0.636951	0.318476	−	0.947931i	$-0.396829\pi$
$379$	12.4001	0.636951	0.318476	+	0.947931i	$0.396829\pi$
$380$	0	0
$381$	−17.3426	−0.888489
$382$	0	0
$383$	7.44987	0.380671	0.190335	−	0.981719i	$-0.439042\pi$
$383$	7.44987	0.380671	0.190335	+	0.981719i	$0.439042\pi$
$384$	0	0
$385$	−2.06943	−0.105468
$386$	0	0
$387$	−58.7646	−2.98717
$388$	0	0
$389$	20.3307	1.03081	0.515405	−	0.856947i	$-0.327642\pi$
$389$	20.3307	1.03081	0.515405	+	0.856947i	$0.327642\pi$
$390$	0	0
$391$	15.1930	0.768341
$392$	0	0
$393$	−18.3964	−0.927976
$394$	0	0
$395$	5.20388	0.261836
$396$	0	0
$397$	−36.3085	−1.82227	−0.911136	−	0.412105i	$-0.864794\pi$
$397$	−36.3085	−1.82227	−0.911136	+	0.412105i	$0.864794\pi$
$398$	0	0
$399$	13.0370	0.652668
$400$	0	0
$401$	−12.7715	−0.637776	−0.318888	−	0.947792i	$-0.603309\pi$
$401$	−12.7715	−0.637776	−0.318888	+	0.947792i	$0.603309\pi$
$402$	0	0
$403$	23.6333	1.17726
$404$	0	0
$405$	−0.502149	−0.0249520
$406$	0	0
$407$	4.20563	0.208465
$408$	0	0
$409$	1.62828	0.0805131	0.0402565	−	0.999189i	$-0.487182\pi$
$409$	1.62828	0.0805131	0.0402565	+	0.999189i	$0.487182\pi$
$410$	0	0
$411$	4.51919	0.222915
$412$	0	0
$413$	−0.595995	−0.0293270
$414$	0	0
$415$	6.58048	0.323023
$416$	0	0
$417$	4.38491	0.214730
$418$	0	0
$419$	19.3604	0.945820	0.472910	−	0.881111i	$-0.343204\pi$
$419$	19.3604	0.945820	0.472910	+	0.881111i	$0.343204\pi$
$420$	0	0
$421$	−26.2766	−1.28065	−0.640323	−	0.768106i	$-0.721199\pi$
$421$	−26.2766	−1.28065	−0.640323	+	0.768106i	$0.721199\pi$
$422$	0	0
$423$	−16.8936	−0.821396
$424$	0	0
$425$	15.7067	0.761887
$426$	0	0
$427$	−6.47814	−0.313499
$428$	0	0
$429$	52.4579	2.53269
$430$	0	0
$431$	−16.4821	−0.793914	−0.396957	−	0.917837i	$-0.629934\pi$
$431$	−16.4821	−0.793914	−0.396957	+	0.917837i	$0.629934\pi$
$432$	0	0
$433$	−23.2609	−1.11785	−0.558923	−	0.829220i	$-0.688785\pi$
$433$	−23.2609	−1.11785	−0.558923	+	0.829220i	$0.688785\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17.7932	−0.851163
$438$	0	0
$439$	29.9242	1.42820	0.714101	−	0.700042i	$-0.246835\pi$
$439$	29.9242	1.42820	0.714101	+	0.700042i	$0.246835\pi$
$440$	0	0
$441$	−25.3895	−1.20902
$442$	0	0
$443$	14.5560	0.691579	0.345789	−	0.938312i	$-0.387611\pi$
$443$	14.5560	0.691579	0.345789	+	0.938312i	$0.387611\pi$
$444$	0	0
$445$	2.09194	0.0991676
$446$	0	0
$447$	14.6592	0.693354
$448$	0	0
$449$	−15.7482	−0.743205	−0.371603	−	0.928392i	$-0.621192\pi$
$449$	−15.7482	−0.743205	−0.371603	+	0.928392i	$0.621192\pi$
$450$	0	0
$451$	29.5974	1.39369
$452$	0	0
$453$	−2.19091	−0.102938
$454$	0	0
$455$	−2.10372	−0.0986237
$456$	0	0
$457$	23.8968	1.11785	0.558923	−	0.829220i	$-0.311215\pi$
$457$	23.8968	1.11785	0.558923	+	0.829220i	$0.311215\pi$
$458$	0	0
$459$	−14.8148	−0.691496
$460$	0	0
$461$	−4.41898	−0.205812	−0.102906	−	0.994691i	$-0.532814\pi$
$461$	−4.41898	−0.205812	−0.102906	+	0.994691i	$0.532814\pi$
$462$	0	0
$463$	−28.6300	−1.33055	−0.665275	−	0.746599i	$-0.731686\pi$
$463$	−28.6300	−1.33055	−0.665275	+	0.746599i	$0.731686\pi$
$464$	0	0
$465$	−5.74180	−0.266269
$466$	0	0
$467$	−25.9113	−1.19903	−0.599515	−	0.800364i	$-0.704640\pi$
$467$	−25.9113	−1.19903	−0.599515	+	0.800364i	$0.704640\pi$
$468$	0	0
$469$	−1.74559	−0.0806040
$470$	0	0
$471$	−10.3091	−0.475020
$472$	0	0
$473$	54.5326	2.50741
$474$	0	0
$475$	−18.3948	−0.844012
$476$	0	0
$477$	41.4964	1.89999
$478$	0	0
$479$	−6.72845	−0.307431	−0.153715	−	0.988115i	$-0.549124\pi$
$479$	−6.72845	−0.307431	−0.153715	+	0.988115i	$0.549124\pi$
$480$	0	0
$481$	4.27531	0.194937
$482$	0	0
$483$	−16.1349	−0.734163
$484$	0	0
$485$	5.68203	0.258008
$486$	0	0
$487$	24.7539	1.12171	0.560853	−	0.827915i	$-0.310473\pi$
$487$	24.7539	1.12171	0.560853	+	0.827915i	$0.310473\pi$
$488$	0	0
$489$	1.43133	0.0647269
$490$	0	0
$491$	−15.3672	−0.693511	−0.346756	−	0.937955i	$-0.612717\pi$
$491$	−15.3672	−0.693511	−0.346756	+	0.937955i	$0.612717\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−7.74948	−0.348313
$496$	0	0
$497$	−11.2320	−0.503825
$498$	0	0
$499$	−13.0979	−0.586343	−0.293172	−	0.956060i	$-0.594711\pi$
$499$	−13.0979	−0.586343	−0.293172	+	0.956060i	$0.594711\pi$
$500$	0	0
$501$	−70.2213	−3.13726
$502$	0	0
$503$	−9.06338	−0.404116	−0.202058	−	0.979374i	$-0.564763\pi$
$503$	−9.06338	−0.404116	−0.202058	+	0.979374i	$0.564763\pi$
$504$	0	0
$505$	−2.36525	−0.105252
$506$	0	0
$507$	17.3616	0.771054
$508$	0	0
$509$	−4.85193	−0.215058	−0.107529	−	0.994202i	$-0.534294\pi$
$509$	−4.85193	−0.215058	−0.107529	+	0.994202i	$0.534294\pi$
$510$	0	0
$511$	11.1477	0.493144
$512$	0	0
$513$	17.3503	0.766034
$514$	0	0
$515$	7.52107	0.331418
$516$	0	0
$517$	15.6770	0.689475
$518$	0	0
$519$	36.0611	1.58290
$520$	0	0
$521$	−25.5713	−1.12030	−0.560150	−	0.828391i	$-0.689257\pi$
$521$	−25.5713	−1.12030	−0.560150	+	0.828391i	$0.689257\pi$
$522$	0	0
$523$	13.8459	0.605439	0.302719	−	0.953080i	$-0.402105\pi$
$523$	13.8459	0.605439	0.302719	+	0.953080i	$0.402105\pi$
$524$	0	0
$525$	−16.6805	−0.727995
$526$	0	0
$527$	17.4278	0.759169
$528$	0	0
$529$	−0.978809	−0.0425569
$530$	0	0
$531$	−2.23185	−0.0968540
$532$	0	0
$533$	30.0878	1.30325
$534$	0	0
$535$	2.77321	0.119896
$536$	0	0
$537$	−19.0208	−0.820807
$538$	0	0
$539$	23.5611	1.01485
$540$	0	0
$541$	−28.6908	−1.23352	−0.616758	−	0.787153i	$-0.711554\pi$
$541$	−28.6908	−1.23352	−0.616758	+	0.787153i	$0.711554\pi$
$542$	0	0
$543$	−60.3246	−2.58878
$544$	0	0
$545$	−6.78371	−0.290582
$546$	0	0
$547$	12.6855	0.542394	0.271197	−	0.962524i	$-0.412581\pi$
$547$	12.6855	0.542394	0.271197	+	0.962524i	$0.412581\pi$
$548$	0	0
$549$	−24.2590	−1.03535
$550$	0	0
$551$	0	0
$552$	0	0
$553$	16.7744	0.713320
$554$	0	0
$555$	−1.03870	−0.0440905
$556$	0	0
$557$	7.41564	0.314211	0.157105	−	0.987582i	$-0.449784\pi$
$557$	7.41564	0.314211	0.157105	+	0.987582i	$0.449784\pi$
$558$	0	0
$559$	55.4362	2.34470
$560$	0	0
$561$	38.6840	1.63324
$562$	0	0
$563$	21.1847	0.892828	0.446414	−	0.894826i	$-0.352701\pi$
$563$	21.1847	0.892828	0.446414	+	0.894826i	$0.352701\pi$
$564$	0	0
$565$	6.22494	0.261885
$566$	0	0
$567$	−1.61865	−0.0679767
$568$	0	0
$569$	8.08511	0.338945	0.169473	−	0.985535i	$-0.445794\pi$
$569$	8.08511	0.338945	0.169473	+	0.985535i	$0.445794\pi$
$570$	0	0
$571$	−7.19732	−0.301199	−0.150599	−	0.988595i	$-0.548120\pi$
$571$	−7.19732	−0.301199	−0.150599	+	0.988595i	$0.548120\pi$
$572$	0	0
$573$	−34.9394	−1.45962
$574$	0	0
$575$	22.7658	0.949400
$576$	0	0
$577$	−19.9643	−0.831125	−0.415563	−	0.909565i	$-0.636415\pi$
$577$	−19.9643	−0.831125	−0.415563	+	0.909565i	$0.636415\pi$
$578$	0	0
$579$	−22.8505	−0.949635
$580$	0	0
$581$	21.2118	0.880012
$582$	0	0
$583$	−38.5081	−1.59484
$584$	0	0
$585$	−7.87788	−0.325710
$586$	0	0
$587$	0.751113	0.0310018	0.0155009	−	0.999880i	$-0.495066\pi$
$587$	0.751113	0.0310018	0.0155009	+	0.999880i	$0.495066\pi$
$588$	0	0
$589$	−20.4105	−0.841001
$590$	0	0
$591$	70.5673	2.90275
$592$	0	0
$593$	−43.3160	−1.77878	−0.889388	−	0.457153i	$-0.848869\pi$
$593$	−43.3160	−1.77878	−0.889388	+	0.457153i	$0.848869\pi$
$594$	0	0
$595$	−1.55134	−0.0635987
$596$	0	0
$597$	5.92390	0.242449
$598$	0	0
$599$	−4.95432	−0.202428	−0.101214	−	0.994865i	$-0.532273\pi$
$599$	−4.95432	−0.202428	−0.101214	+	0.994865i	$0.532273\pi$
$600$	0	0
$601$	−46.2865	−1.88806	−0.944032	−	0.329853i	$-0.893001\pi$
$601$	−46.2865	−1.88806	−0.944032	+	0.329853i	$0.893001\pi$
$602$	0	0
$603$	−6.53680	−0.266199
$604$	0	0
$605$	2.95033	0.119948
$606$	0	0
$607$	−26.5455	−1.07745	−0.538724	−	0.842483i	$-0.681093\pi$
$607$	−26.5455	−1.07745	−0.538724	+	0.842483i	$0.681093\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	15.9368	0.644733
$612$	0	0
$613$	42.9356	1.73415	0.867077	−	0.498174i	$-0.165996\pi$
$613$	42.9356	1.73415	0.867077	+	0.498174i	$0.165996\pi$
$614$	0	0
$615$	−7.30996	−0.294766
$616$	0	0
$617$	−25.1939	−1.01427	−0.507134	−	0.861867i	$-0.669295\pi$
$617$	−25.1939	−1.01427	−0.507134	+	0.861867i	$0.669295\pi$
$618$	0	0
$619$	−5.25356	−0.211159	−0.105579	−	0.994411i	$-0.533670\pi$
$619$	−5.25356	−0.211159	−0.105579	+	0.994411i	$0.533670\pi$
$620$	0	0
$621$	−21.4731	−0.861684
$622$	0	0
$623$	6.74325	0.270162
$624$	0	0
$625$	22.7923	0.911694
$626$	0	0
$627$	−45.3046	−1.80929
$628$	0	0
$629$	3.15274	0.125708
$630$	0	0
$631$	−7.01590	−0.279299	−0.139649	−	0.990201i	$-0.544598\pi$
$631$	−7.01590	−0.279299	−0.139649	+	0.990201i	$0.544598\pi$
$632$	0	0
$633$	−45.1195	−1.79334
$634$	0	0
$635$	−2.41687	−0.0959106
$636$	0	0
$637$	23.9515	0.948992
$638$	0	0
$639$	−42.0611	−1.66391
$640$	0	0
$641$	23.6285	0.933270	0.466635	−	0.884450i	$-0.345466\pi$
$641$	23.6285	0.933270	0.466635	+	0.884450i	$0.345466\pi$
$642$	0	0
$643$	12.1904	0.480741	0.240371	−	0.970681i	$-0.422731\pi$
$643$	12.1904	0.480741	0.240371	+	0.970681i	$0.422731\pi$
$644$	0	0
$645$	−13.4684	−0.530319
$646$	0	0
$647$	−20.9951	−0.825404	−0.412702	−	0.910866i	$-0.635415\pi$
$647$	−20.9951	−0.825404	−0.412702	+	0.910866i	$0.635415\pi$
$648$	0	0
$649$	2.07112	0.0812987
$650$	0	0
$651$	−18.5083	−0.725398
$652$	0	0
$653$	−33.1162	−1.29594	−0.647968	−	0.761668i	$-0.724381\pi$
$653$	−33.1162	−1.29594	−0.647968	+	0.761668i	$0.724381\pi$
$654$	0	0
$655$	−2.56373	−0.100173
$656$	0	0
$657$	41.7452	1.62864
$658$	0	0
$659$	28.1676	1.09726	0.548628	−	0.836067i	$-0.315150\pi$
$659$	28.1676	1.09726	0.548628	+	0.836067i	$0.315150\pi$
$660$	0	0
$661$	35.0204	1.36214	0.681068	−	0.732220i	$-0.261516\pi$
$661$	35.0204	1.36214	0.681068	+	0.732220i	$0.261516\pi$
$662$	0	0
$663$	39.3250	1.52725
$664$	0	0
$665$	1.81684	0.0704542
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−20.4493	−0.790618
$670$	0	0
$671$	22.5120	0.869065
$672$	0	0
$673$	−14.5817	−0.562083	−0.281042	−	0.959696i	$-0.590680\pi$
$673$	−14.5817	−0.562083	−0.281042	+	0.959696i	$0.590680\pi$
$674$	0	0
$675$	−22.1991	−0.854445
$676$	0	0
$677$	11.5726	0.444770	0.222385	−	0.974959i	$-0.428616\pi$
$677$	11.5726	0.444770	0.222385	+	0.974959i	$0.428616\pi$
$678$	0	0
$679$	18.3157	0.702891
$680$	0	0
$681$	77.0806	2.95373
$682$	0	0
$683$	−26.6866	−1.02114	−0.510568	−	0.859837i	$-0.670565\pi$
$683$	−26.6866	−1.02114	−0.510568	+	0.859837i	$0.670565\pi$
$684$	0	0
$685$	0.629796	0.0240633
$686$	0	0
$687$	30.2689	1.15483
$688$	0	0
$689$	−39.1461	−1.49135
$690$	0	0
$691$	16.7908	0.638750	0.319375	−	0.947628i	$-0.396527\pi$
$691$	16.7908	0.638750	0.319375	+	0.947628i	$0.396527\pi$
$692$	0	0
$693$	−24.9800	−0.948910
$694$	0	0
$695$	0.611083	0.0231797
$696$	0	0
$697$	22.1876	0.840416
$698$	0	0
$699$	62.7520	2.37350
$700$	0	0
$701$	−24.6200	−0.929886	−0.464943	−	0.885341i	$-0.653925\pi$
$701$	−24.6200	−0.929886	−0.464943	+	0.885341i	$0.653925\pi$
$702$	0	0
$703$	−3.69231	−0.139258
$704$	0	0
$705$	−3.87191	−0.145824
$706$	0	0
$707$	−7.62423	−0.286739
$708$	0	0
$709$	26.7715	1.00543	0.502713	−	0.864454i	$-0.332335\pi$
$709$	26.7715	1.00543	0.502713	+	0.864454i	$0.332335\pi$
$710$	0	0
$711$	62.8159	2.35578
$712$	0	0
$713$	25.2605	0.946013
$714$	0	0
$715$	7.31055	0.273399
$716$	0	0
$717$	52.8377	1.97326
$718$	0	0
$719$	−9.76861	−0.364308	−0.182154	−	0.983270i	$-0.558307\pi$
$719$	−9.76861	−0.364308	−0.182154	+	0.983270i	$0.558307\pi$
$720$	0	0
$721$	24.2437	0.902882
$722$	0	0
$723$	51.5782	1.91821
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.54797	0.0944989	0.0472495	−	0.998883i	$-0.484954\pi$
$727$	2.54797	0.0944989	0.0472495	+	0.998883i	$0.484954\pi$
$728$	0	0
$729$	−44.0400	−1.63111
$730$	0	0
$731$	40.8802	1.51201
$732$	0	0
$733$	−40.0607	−1.47968	−0.739838	−	0.672785i	$-0.765098\pi$
$733$	−40.0607	−1.47968	−0.739838	+	0.672785i	$0.765098\pi$
$734$	0	0
$735$	−5.81911	−0.214641
$736$	0	0
$737$	6.06605	0.223446
$738$	0	0
$739$	−9.97429	−0.366910	−0.183455	−	0.983028i	$-0.558728\pi$
$739$	−9.97429	−0.366910	−0.183455	+	0.983028i	$0.558728\pi$
$740$	0	0
$741$	−46.0552	−1.69188
$742$	0	0
$743$	−3.33734	−0.122435	−0.0612176	−	0.998124i	$-0.519498\pi$
$743$	−3.33734	−0.122435	−0.0612176	+	0.998124i	$0.519498\pi$
$744$	0	0
$745$	2.04290	0.0748462
$746$	0	0
$747$	79.4326	2.90629
$748$	0	0
$749$	8.93926	0.326633
$750$	0	0
$751$	36.1700	1.31986	0.659931	−	0.751326i	$-0.270585\pi$
$751$	36.1700	1.31986	0.659931	+	0.751326i	$0.270585\pi$
$752$	0	0
$753$	−5.40817	−0.197085
$754$	0	0
$755$	−0.305327	−0.0111120
$756$	0	0
$757$	−40.8867	−1.48605	−0.743026	−	0.669263i	$-0.766610\pi$
$757$	−40.8867	−1.48605	−0.743026	+	0.669263i	$0.766610\pi$
$758$	0	0
$759$	56.0698	2.03521
$760$	0	0
$761$	14.0405	0.508969	0.254484	−	0.967077i	$-0.418094\pi$
$761$	14.0405	0.508969	0.254484	+	0.967077i	$0.418094\pi$
$762$	0	0
$763$	−21.8669	−0.791634
$764$	0	0
$765$	−5.80938	−0.210038
$766$	0	0
$767$	2.10544	0.0760230
$768$	0	0
$769$	−29.6420	−1.06892	−0.534459	−	0.845194i	$-0.679485\pi$
$769$	−29.6420	−1.06892	−0.534459	+	0.845194i	$0.679485\pi$
$770$	0	0
$771$	46.0938	1.66003
$772$	0	0
$773$	−39.0544	−1.40469	−0.702344	−	0.711838i	$-0.747863\pi$
$773$	−39.0544	−1.40469	−0.702344	+	0.711838i	$0.747863\pi$
$774$	0	0
$775$	26.1146	0.938065
$776$	0	0
$777$	−3.34820	−0.120116
$778$	0	0
$779$	−25.9849	−0.931007
$780$	0	0
$781$	39.0321	1.39668
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.43668	−0.0512775
$786$	0	0
$787$	34.1022	1.21561	0.607806	−	0.794086i	$-0.292050\pi$
$787$	34.1022	1.21561	0.607806	+	0.794086i	$0.292050\pi$
$788$	0	0
$789$	−13.5459	−0.482247
$790$	0	0
$791$	20.0657	0.713455
$792$	0	0
$793$	22.8850	0.812670
$794$	0	0
$795$	9.51071	0.337310
$796$	0	0
$797$	−27.8622	−0.986929	−0.493464	−	0.869766i	$-0.664270\pi$
$797$	−27.8622	−0.986929	−0.493464	+	0.869766i	$0.664270\pi$
$798$	0	0
$799$	11.7522	0.415764
$800$	0	0
$801$	25.2518	0.892227
$802$	0	0
$803$	−38.7389	−1.36707
$804$	0	0
$805$	−2.24856	−0.0792515
$806$	0	0
$807$	28.7443	1.01185
$808$	0	0
$809$	−18.3118	−0.643808	−0.321904	−	0.946772i	$-0.604323\pi$
$809$	−18.3118	−0.643808	−0.321904	+	0.946772i	$0.604323\pi$
$810$	0	0
$811$	48.7533	1.71196	0.855980	−	0.517009i	$-0.172955\pi$
$811$	48.7533	1.71196	0.855980	+	0.517009i	$0.172955\pi$
$812$	0	0
$813$	−10.7649	−0.377542
$814$	0	0
$815$	0.199470	0.00698714
$816$	0	0
$817$	−47.8767	−1.67499
$818$	0	0
$819$	−25.3939	−0.887333
$820$	0	0
$821$	−14.5534	−0.507917	−0.253959	−	0.967215i	$-0.581733\pi$
$821$	−14.5534	−0.507917	−0.253959	+	0.967215i	$0.581733\pi$
$822$	0	0
$823$	−14.3136	−0.498942	−0.249471	−	0.968382i	$-0.580257\pi$
$823$	−14.3136	−0.498942	−0.249471	+	0.968382i	$0.580257\pi$
$824$	0	0
$825$	57.9658	2.01811
$826$	0	0
$827$	−15.9061	−0.553108	−0.276554	−	0.960998i	$-0.589192\pi$
$827$	−15.9061	−0.553108	−0.276554	+	0.960998i	$0.589192\pi$
$828$	0	0
$829$	−35.5681	−1.23533	−0.617666	−	0.786441i	$-0.711922\pi$
$829$	−35.5681	−1.23533	−0.617666	+	0.786441i	$0.711922\pi$
$830$	0	0
$831$	−46.3134	−1.60659
$832$	0	0
$833$	17.6625	0.611969
$834$	0	0
$835$	−9.78606	−0.338661
$836$	0	0
$837$	−24.6317	−0.851397
$838$	0	0
$839$	−12.9857	−0.448317	−0.224159	−	0.974553i	$-0.571963\pi$
$839$	−12.9857	−0.448317	−0.224159	+	0.974553i	$0.571963\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	−7.23864	−0.249312
$844$	0	0
$845$	2.41951	0.0832338
$846$	0	0
$847$	9.51020	0.326774
$848$	0	0
$849$	63.8259	2.19050
$850$	0	0
$851$	4.56968	0.156647
$852$	0	0
$853$	−3.28244	−0.112389	−0.0561943	−	0.998420i	$-0.517897\pi$
$853$	−3.28244	−0.112389	−0.0561943	+	0.998420i	$0.517897\pi$
$854$	0	0
$855$	6.80362	0.232679
$856$	0	0
$857$	2.76222	0.0943555	0.0471778	−	0.998887i	$-0.484977\pi$
$857$	2.76222	0.0943555	0.0471778	+	0.998887i	$0.484977\pi$
$858$	0	0
$859$	21.7727	0.742874	0.371437	−	0.928458i	$-0.378865\pi$
$859$	21.7727	0.742874	0.371437	+	0.928458i	$0.378865\pi$
$860$	0	0
$861$	−23.5632	−0.803032
$862$	0	0
$863$	22.5370	0.767167	0.383584	−	0.923506i	$-0.374690\pi$
$863$	22.5370	0.767167	0.383584	+	0.923506i	$0.374690\pi$
$864$	0	0
$865$	5.02548	0.170871
$866$	0	0
$867$	−18.0326	−0.612418
$868$	0	0
$869$	−58.2922	−1.97743
$870$	0	0
$871$	6.16656	0.208946
$872$	0	0
$873$	68.5875	2.32134
$874$	0	0
$875$	−4.72042	−0.159579
$876$	0	0
$877$	13.5310	0.456908	0.228454	−	0.973555i	$-0.426633\pi$
$877$	13.5310	0.456908	0.228454	+	0.973555i	$0.426633\pi$
$878$	0	0
$879$	−7.56375	−0.255119
$880$	0	0
$881$	−46.1728	−1.55560	−0.777801	−	0.628511i	$-0.783665\pi$
$881$	−46.1728	−1.55560	−0.777801	+	0.628511i	$0.783665\pi$
$882$	0	0
$883$	−49.3616	−1.66115	−0.830575	−	0.556906i	$-0.811988\pi$
$883$	−49.3616	−1.66115	−0.830575	+	0.556906i	$0.811988\pi$
$884$	0	0
$885$	−0.511525	−0.0171947
$886$	0	0
$887$	−11.5191	−0.386772	−0.193386	−	0.981123i	$-0.561947\pi$
$887$	−11.5191	−0.386772	−0.193386	+	0.981123i	$0.561947\pi$
$888$	0	0
$889$	−7.79063	−0.261290
$890$	0	0
$891$	5.62490	0.188441
$892$	0	0
$893$	−13.7636	−0.460581
$894$	0	0
$895$	−2.65074	−0.0886045
$896$	0	0
$897$	56.9989	1.90314
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−28.8675	−0.961715
$902$	0	0
$903$	−43.4147	−1.44475
$904$	0	0
$905$	−8.40686	−0.279453
$906$	0	0
$907$	32.8988	1.09239	0.546194	−	0.837659i	$-0.316076\pi$
$907$	32.8988	1.09239	0.546194	+	0.837659i	$0.316076\pi$
$908$	0	0
$909$	−28.5508	−0.946971
$910$	0	0
$911$	6.81813	0.225895	0.112947	−	0.993601i	$-0.463971\pi$
$911$	6.81813	0.225895	0.112947	+	0.993601i	$0.463971\pi$
$912$	0	0
$913$	−73.7123	−2.43952
$914$	0	0
$915$	−5.56000	−0.183808
$916$	0	0
$917$	−8.26402	−0.272902
$918$	0	0
$919$	54.0063	1.78150	0.890752	−	0.454489i	$-0.150178\pi$
$919$	54.0063	1.78150	0.890752	+	0.454489i	$0.150178\pi$
$920$	0	0
$921$	−42.4083	−1.39740
$922$	0	0
$923$	39.6788	1.30604
$924$	0	0
$925$	4.72420	0.155331
$926$	0	0
$927$	90.7865	2.98182
$928$	0	0
$929$	36.8689	1.20963	0.604816	−	0.796366i	$-0.293247\pi$
$929$	36.8689	1.20963	0.604816	+	0.796366i	$0.293247\pi$
$930$	0	0
$931$	−20.6854	−0.677935
$932$	0	0
$933$	−33.5953	−1.09986
$934$	0	0
$935$	5.39101	0.176305
$936$	0	0
$937$	22.3469	0.730043	0.365021	−	0.930999i	$-0.381062\pi$
$937$	22.3469	0.730043	0.365021	+	0.930999i	$0.381062\pi$
$938$	0	0
$939$	11.7850	0.384590
$940$	0	0
$941$	−19.7291	−0.643150	−0.321575	−	0.946884i	$-0.604212\pi$
$941$	−19.7291	−0.643150	−0.321575	+	0.946884i	$0.604212\pi$
$942$	0	0
$943$	32.1595	1.04726
$944$	0	0
$945$	2.19259	0.0713251
$946$	0	0
$947$	59.1800	1.92309	0.961545	−	0.274646i	$-0.0885606\pi$
$947$	59.1800	1.92309	0.961545	+	0.274646i	$0.0885606\pi$
$948$	0	0
$949$	−39.3808	−1.27835
$950$	0	0
$951$	−58.3179	−1.89109
$952$	0	0
$953$	22.5820	0.731503	0.365751	−	0.930713i	$-0.380812\pi$
$953$	22.5820	0.731503	0.365751	+	0.930713i	$0.380812\pi$
$954$	0	0
$955$	−4.86917	−0.157563
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.03011	0.0655556
$960$	0	0
$961$	−2.02371	−0.0652811
$962$	0	0
$963$	33.4753	1.07872
$964$	0	0
$965$	−3.18445	−0.102511
$966$	0	0
$967$	8.00751	0.257504	0.128752	−	0.991677i	$-0.458903\pi$
$967$	8.00751	0.257504	0.128752	+	0.991677i	$0.458903\pi$
$968$	0	0
$969$	−33.9624	−1.09103
$970$	0	0
$971$	14.5194	0.465950	0.232975	−	0.972483i	$-0.425154\pi$
$971$	14.5194	0.465950	0.232975	+	0.972483i	$0.425154\pi$
$972$	0	0
$973$	1.96979	0.0631485
$974$	0	0
$975$	58.9262	1.88715
$976$	0	0
$977$	−21.6607	−0.692987	−0.346493	−	0.938052i	$-0.612628\pi$
$977$	−21.6607	−0.692987	−0.346493	+	0.938052i	$0.612628\pi$
$978$	0	0
$979$	−23.4332	−0.748930
$980$	0	0
$981$	−81.8859	−2.61442
$982$	0	0
$983$	9.05114	0.288687	0.144343	−	0.989528i	$-0.453893\pi$
$983$	9.05114	0.288687	0.144343	+	0.989528i	$0.453893\pi$
$984$	0	0
$985$	9.83428	0.313346
$986$	0	0
$987$	−12.4808	−0.397270
$988$	0	0
$989$	59.2531	1.88414
$990$	0	0
$991$	−58.5410	−1.85962	−0.929809	−	0.368043i	$-0.880028\pi$
$991$	−58.5410	−1.85962	−0.929809	+	0.368043i	$0.880028\pi$
$992$	0	0
$993$	13.0616	0.414498
$994$	0	0
$995$	0.825556	0.0261719
$996$	0	0
$997$	7.52081	0.238186	0.119093	−	0.992883i	$-0.462001\pi$
$997$	7.52081	0.238186	0.119093	+	0.992883i	$0.462001\pi$
$998$	0	0
$999$	−4.45594	−0.140980

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6728.2.a.z.1.12		12
29.23	even	7		232.2.m.d.65.1	yes	24
29.24	even	7		232.2.m.d.25.1	✓	24
29.28	even	2		6728.2.a.bb.1.1		12
116.23	odd	14		464.2.u.i.65.4		24
116.111	odd	14		464.2.u.i.257.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.m.d.25.1	✓	24	29.24	even	7
232.2.m.d.65.1	yes	24	29.23	even	7
464.2.u.i.65.4		24	116.23	odd	14
464.2.u.i.257.4		24	116.111	odd	14
6728.2.a.z.1.12		12	1.1	even	1	trivial
6728.2.a.bb.1.1		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)

\(f(q)\)	\(=\)	\(q+2.76658 q^{3} +0.385552 q^{5} +1.24280 q^{7} +4.65398 q^{9} +O(q^{10})\)	\(q+2.76658 q^{3} +0.385552 q^{5} +1.24280 q^{7} +4.65398 q^{9} -4.31882 q^{11} -4.39038 q^{13} +1.06666 q^{15} -3.23759 q^{17} +3.79169 q^{19} +3.43831 q^{21} -4.69267 q^{23} -4.85135 q^{25} +4.57587 q^{27} -5.38296 q^{31} -11.9484 q^{33} +0.479165 q^{35} -0.973790 q^{37} -12.1464 q^{39} -6.85312 q^{41} -12.6267 q^{43} +1.79435 q^{45} -3.62993 q^{47} -5.45544 q^{49} -8.95707 q^{51} +8.91633 q^{53} -1.66513 q^{55} +10.4900 q^{57} -0.479557 q^{59} -5.21253 q^{61} +5.78398 q^{63} -1.69272 q^{65} -1.40456 q^{67} -12.9827 q^{69} -9.03766 q^{71} +8.96979 q^{73} -13.4217 q^{75} -5.36744 q^{77} +13.4972 q^{79} -1.30242 q^{81} +17.0677 q^{83} -1.24826 q^{85} +5.42584 q^{89} -5.45638 q^{91} -14.8924 q^{93} +1.46189 q^{95} +14.7374 q^{97} -20.0997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9}+O(q^{10}) \) 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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