LMFDB - Embedded newform 7168.2.a.bb.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7168,2,Mod(1,7168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7168, 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("7168.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.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:	$57.2367681689$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.9433055232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 6x^{6} + 32x^{5} + 9x^{4} - 76x^{3} - 4x^{2} + 48x - 8 $ x^8 - 4x^7 - 6x^6 + 32x^5 + 9x^4 - 76x^3 - 4x^2 + 48*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1792)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.177920$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.67251 q^{3} +2.65618 q^{5} +1.00000 q^{7} -0.202696 q^{9} +O(q^{10})$	$q+1.67251 q^{3} +2.65618 q^{5} +1.00000 q^{7} -0.202696 q^{9} -0.826484 q^{11} -5.57281 q^{13} +4.44250 q^{15} -1.74896 q^{17} -5.93216 q^{19} +1.67251 q^{21} +3.04150 q^{23} +2.05530 q^{25} -5.35655 q^{27} -6.26943 q^{29} -7.90794 q^{31} -1.38231 q^{33} +2.65618 q^{35} +8.30514 q^{37} -9.32061 q^{39} +1.38922 q^{41} -2.45934 q^{43} -0.538396 q^{45} -1.80017 q^{47} +1.00000 q^{49} -2.92516 q^{51} -13.7698 q^{53} -2.19529 q^{55} -9.92162 q^{57} -6.70340 q^{59} +4.38770 q^{61} -0.202696 q^{63} -14.8024 q^{65} +6.80390 q^{67} +5.08696 q^{69} +1.11625 q^{71} +11.2521 q^{73} +3.43752 q^{75} -0.826484 q^{77} -7.61158 q^{79} -8.35083 q^{81} +15.8207 q^{83} -4.64556 q^{85} -10.4857 q^{87} -0.428825 q^{89} -5.57281 q^{91} -13.2261 q^{93} -15.7569 q^{95} -19.2163 q^{97} +0.167525 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) $ 8 * q - 8 * q^5 + 8 * q^7 + 12 * q^9	$ 8 q - 8 q^{5} + 8 q^{7} + 12 q^{9} - 12 q^{11} - 20 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 12 q^{25} + 12 q^{27} - 8 q^{29} - 4 q^{31} + 8 q^{33} - 8 q^{35} - 8 q^{37} - 16 q^{39} - 12 q^{41} + 4 q^{43} - 52 q^{45} + 20 q^{47} + 8 q^{49} - 32 q^{51} - 40 q^{53} + 24 q^{55} - 4 q^{57} - 4 q^{59} + 8 q^{61} + 12 q^{63} + 36 q^{65} - 28 q^{67} - 4 q^{69} - 16 q^{71} + 16 q^{73} + 28 q^{75} - 12 q^{77} + 20 q^{81} + 8 q^{83} - 16 q^{85} + 20 q^{87} + 16 q^{89} - 20 q^{91} - 16 q^{93} - 40 q^{95} - 36 q^{97} + 4 q^{99}+O(q^{100}) $ 8 * q - 8 * q^5 + 8 * q^7 + 12 * q^9 - 12 * q^11 - 20 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 12 * q^25 + 12 * q^27 - 8 * q^29 - 4 * q^31 + 8 * q^33 - 8 * q^35 - 8 * q^37 - 16 * q^39 - 12 * q^41 + 4 * q^43 - 52 * q^45 + 20 * q^47 + 8 * q^49 - 32 * q^51 - 40 * q^53 + 24 * q^55 - 4 * q^57 - 4 * q^59 + 8 * q^61 + 12 * q^63 + 36 * q^65 - 28 * q^67 - 4 * q^69 - 16 * q^71 + 16 * q^73 + 28 * q^75 - 12 * q^77 + 20 * q^81 + 8 * q^83 - 16 * q^85 + 20 * q^87 + 16 * q^89 - 20 * q^91 - 16 * q^93 - 40 * q^95 - 36 * q^97 + 4 * 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
$3$	1.67251	0.965627	0.482813	−	0.875723i	$-0.339615\pi$
$3$	1.67251	0.965627	0.482813	+	0.875723i	$0.339615\pi$
$4$	0	0
$5$	2.65618	1.18788	0.593940	−	0.804509i	$-0.297572\pi$
$5$	2.65618	1.18788	0.593940	+	0.804509i	$0.297572\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.202696	−0.0675652
$10$	0	0
$11$	−0.826484	−0.249194	−0.124597	−	0.992207i	$-0.539764\pi$
$11$	−0.826484	−0.249194	−0.124597	+	0.992207i	$0.539764\pi$
$12$	0	0
$13$	−5.57281	−1.54562	−0.772810	−	0.634637i	$-0.781150\pi$
$13$	−5.57281	−1.54562	−0.772810	+	0.634637i	$0.781150\pi$
$14$	0	0
$15$	4.44250	1.14705
$16$	0	0
$17$	−1.74896	−0.424185	−0.212093	−	0.977250i	$-0.568028\pi$
$17$	−1.74896	−0.424185	−0.212093	+	0.977250i	$0.568028\pi$
$18$	0	0
$19$	−5.93216	−1.36093	−0.680465	−	0.732781i	$-0.738222\pi$
$19$	−5.93216	−1.36093	−0.680465	+	0.732781i	$0.738222\pi$
$20$	0	0
$21$	1.67251	0.364973
$22$	0	0
$23$	3.04150	0.634198	0.317099	−	0.948393i	$-0.397291\pi$
$23$	3.04150	0.634198	0.317099	+	0.948393i	$0.397291\pi$
$24$	0	0
$25$	2.05530	0.411060
$26$	0	0
$27$	−5.35655	−1.03087
$28$	0	0
$29$	−6.26943	−1.16420	−0.582102	−	0.813116i	$-0.697770\pi$
$29$	−6.26943	−1.16420	−0.582102	+	0.813116i	$0.697770\pi$
$30$	0	0
$31$	−7.90794	−1.42031	−0.710154	−	0.704046i	$-0.751375\pi$
$31$	−7.90794	−1.42031	−0.710154	+	0.704046i	$0.751375\pi$
$32$	0	0
$33$	−1.38231	−0.240629
$34$	0	0
$35$	2.65618	0.448977
$36$	0	0
$37$	8.30514	1.36536	0.682678	−	0.730719i	$-0.260815\pi$
$37$	8.30514	1.36536	0.682678	+	0.730719i	$0.260815\pi$
$38$	0	0
$39$	−9.32061	−1.49249
$40$	0	0
$41$	1.38922	0.216960	0.108480	−	0.994099i	$-0.465402\pi$
$41$	1.38922	0.216960	0.108480	+	0.994099i	$0.465402\pi$
$42$	0	0
$43$	−2.45934	−0.375046	−0.187523	−	0.982260i	$-0.560046\pi$
$43$	−2.45934	−0.375046	−0.187523	+	0.982260i	$0.560046\pi$
$44$	0	0
$45$	−0.538396	−0.0802594
$46$	0	0
$47$	−1.80017	−0.262582	−0.131291	−	0.991344i	$-0.541912\pi$
$47$	−1.80017	−0.262582	−0.131291	+	0.991344i	$0.541912\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.92516	−0.409604
$52$	0	0
$53$	−13.7698	−1.89143	−0.945717	−	0.324992i	$-0.894639\pi$
$53$	−13.7698	−1.89143	−0.945717	+	0.324992i	$0.894639\pi$
$54$	0	0
$55$	−2.19529	−0.296013
$56$	0	0
$57$	−9.92162	−1.31415
$58$	0	0
$59$	−6.70340	−0.872708	−0.436354	−	0.899775i	$-0.643731\pi$
$59$	−6.70340	−0.872708	−0.436354	+	0.899775i	$0.643731\pi$
$60$	0	0
$61$	4.38770	0.561787	0.280893	−	0.959739i	$-0.409369\pi$
$61$	4.38770	0.561787	0.280893	+	0.959739i	$0.409369\pi$
$62$	0	0
$63$	−0.202696	−0.0255372
$64$	0	0
$65$	−14.8024	−1.83601
$66$	0	0
$67$	6.80390	0.831228	0.415614	−	0.909541i	$-0.363567\pi$
$67$	6.80390	0.831228	0.415614	+	0.909541i	$0.363567\pi$
$68$	0	0
$69$	5.08696	0.612398
$70$	0	0
$71$	1.11625	0.132475	0.0662375	−	0.997804i	$-0.478900\pi$
$71$	1.11625	0.132475	0.0662375	+	0.997804i	$0.478900\pi$
$72$	0	0
$73$	11.2521	1.31696	0.658478	−	0.752600i	$-0.271201\pi$
$73$	11.2521	1.31696	0.658478	+	0.752600i	$0.271201\pi$
$74$	0	0
$75$	3.43752	0.396931
$76$	0	0
$77$	−0.826484	−0.0941866
$78$	0	0
$79$	−7.61158	−0.856370	−0.428185	−	0.903691i	$-0.640847\pi$
$79$	−7.61158	−0.856370	−0.428185	+	0.903691i	$0.640847\pi$
$80$	0	0
$81$	−8.35083	−0.927870
$82$	0	0
$83$	15.8207	1.73655	0.868273	−	0.496086i	$-0.165230\pi$
$83$	15.8207	1.73655	0.868273	+	0.496086i	$0.165230\pi$
$84$	0	0
$85$	−4.64556	−0.503881
$86$	0	0
$87$	−10.4857	−1.12419
$88$	0	0
$89$	−0.428825	−0.0454554	−0.0227277	−	0.999742i	$-0.507235\pi$
$89$	−0.428825	−0.0454554	−0.0227277	+	0.999742i	$0.507235\pi$
$90$	0	0
$91$	−5.57281	−0.584190
$92$	0	0
$93$	−13.2261	−1.37149
$94$	0	0
$95$	−15.7569	−1.61662
$96$	0	0
$97$	−19.2163	−1.95111	−0.975557	−	0.219745i	$-0.929478\pi$
$97$	−19.2163	−1.95111	−0.975557	+	0.219745i	$0.929478\pi$
$98$	0	0
$99$	0.167525	0.0168369
$100$	0	0
$101$	−6.89495	−0.686073	−0.343037	−	0.939322i	$-0.611456\pi$
$101$	−6.89495	−0.686073	−0.343037	+	0.939322i	$0.611456\pi$
$102$	0	0
$103$	−6.09849	−0.600902	−0.300451	−	0.953797i	$-0.597137\pi$
$103$	−6.09849	−0.600902	−0.300451	+	0.953797i	$0.597137\pi$
$104$	0	0
$105$	4.44250	0.433544
$106$	0	0
$107$	7.35376	0.710915	0.355457	−	0.934692i	$-0.384325\pi$
$107$	7.35376	0.710915	0.355457	+	0.934692i	$0.384325\pi$
$108$	0	0
$109$	−10.8902	−1.04309	−0.521547	−	0.853222i	$-0.674645\pi$
$109$	−10.8902	−1.04309	−0.521547	+	0.853222i	$0.674645\pi$
$110$	0	0
$111$	13.8905	1.31842
$112$	0	0
$113$	14.2646	1.34190	0.670952	−	0.741500i	$-0.265885\pi$
$113$	14.2646	1.34190	0.670952	+	0.741500i	$0.265885\pi$
$114$	0	0
$115$	8.07879	0.753351
$116$	0	0
$117$	1.12958	0.104430
$118$	0	0
$119$	−1.74896	−0.160327
$120$	0	0
$121$	−10.3169	−0.937902
$122$	0	0
$123$	2.32349	0.209502
$124$	0	0
$125$	−7.82165	−0.699590
$126$	0	0
$127$	8.98310	0.797121	0.398561	−	0.917142i	$-0.369510\pi$
$127$	8.98310	0.797121	0.398561	+	0.917142i	$0.369510\pi$
$128$	0	0
$129$	−4.11328	−0.362154
$130$	0	0
$131$	18.0378	1.57597	0.787985	−	0.615694i	$-0.211124\pi$
$131$	18.0378	1.57597	0.787985	+	0.615694i	$0.211124\pi$
$132$	0	0
$133$	−5.93216	−0.514383
$134$	0	0
$135$	−14.2280	−1.22455
$136$	0	0
$137$	−11.7927	−1.00751	−0.503757	−	0.863845i	$-0.668049\pi$
$137$	−11.7927	−1.00751	−0.503757	+	0.863845i	$0.668049\pi$
$138$	0	0
$139$	0.741071	0.0628568	0.0314284	−	0.999506i	$-0.489994\pi$
$139$	0.741071	0.0628568	0.0314284	+	0.999506i	$0.489994\pi$
$140$	0	0
$141$	−3.01081	−0.253556
$142$	0	0
$143$	4.60584	0.385160
$144$	0	0
$145$	−16.6527	−1.38294
$146$	0	0
$147$	1.67251	0.137947
$148$	0	0
$149$	−2.81229	−0.230392	−0.115196	−	0.993343i	$-0.536750\pi$
$149$	−2.81229	−0.230392	−0.115196	+	0.993343i	$0.536750\pi$
$150$	0	0
$151$	−15.1887	−1.23604	−0.618020	−	0.786162i	$-0.712065\pi$
$151$	−15.1887	−1.23604	−0.618020	+	0.786162i	$0.712065\pi$
$152$	0	0
$153$	0.354507	0.0286602
$154$	0	0
$155$	−21.0049	−1.68716
$156$	0	0
$157$	11.1953	0.893484	0.446742	−	0.894663i	$-0.352584\pi$
$157$	11.1953	0.893484	0.446742	+	0.894663i	$0.352584\pi$
$158$	0	0
$159$	−23.0303	−1.82642
$160$	0	0
$161$	3.04150	0.239704
$162$	0	0
$163$	−8.94026	−0.700255	−0.350128	−	0.936702i	$-0.613862\pi$
$163$	−8.94026	−0.700255	−0.350128	+	0.936702i	$0.613862\pi$
$164$	0	0
$165$	−3.67166	−0.285838
$166$	0	0
$167$	22.5263	1.74314	0.871570	−	0.490271i	$-0.163102\pi$
$167$	22.5263	1.74314	0.871570	+	0.490271i	$0.163102\pi$
$168$	0	0
$169$	18.0563	1.38894
$170$	0	0
$171$	1.20242	0.0919515
$172$	0	0
$173$	−0.0149780	−0.00113876	−0.000569379	−	1.00000i	$-0.500181\pi$
$173$	−0.0149780	−0.00113876	−0.000569379		1.00000i	$0.500181\pi$
$174$	0	0
$175$	2.05530	0.155366
$176$	0	0
$177$	−11.2115	−0.842710
$178$	0	0
$179$	−20.3038	−1.51757	−0.758787	−	0.651338i	$-0.774208\pi$
$179$	−20.3038	−1.51757	−0.758787	+	0.651338i	$0.774208\pi$
$180$	0	0
$181$	−12.2375	−0.909605	−0.454803	−	0.890592i	$-0.650290\pi$
$181$	−12.2375	−0.909605	−0.454803	+	0.890592i	$0.650290\pi$
$182$	0	0
$183$	7.33848	0.542476
$184$	0	0
$185$	22.0600	1.62188
$186$	0	0
$187$	1.44549	0.105705
$188$	0	0
$189$	−5.35655	−0.389632
$190$	0	0
$191$	17.2085	1.24517	0.622583	−	0.782554i	$-0.286083\pi$
$191$	17.2085	1.24517	0.622583	+	0.782554i	$0.286083\pi$
$192$	0	0
$193$	7.00982	0.504578	0.252289	−	0.967652i	$-0.418817\pi$
$193$	7.00982	0.504578	0.252289	+	0.967652i	$0.418817\pi$
$194$	0	0
$195$	−24.7572	−1.77290
$196$	0	0
$197$	−21.8036	−1.55345	−0.776723	−	0.629843i	$-0.783119\pi$
$197$	−21.8036	−1.55345	−0.776723	+	0.629843i	$0.783119\pi$
$198$	0	0
$199$	15.3483	1.08801	0.544005	−	0.839082i	$-0.316907\pi$
$199$	15.3483	1.08801	0.544005	+	0.839082i	$0.316907\pi$
$200$	0	0
$201$	11.3796	0.802656
$202$	0	0
$203$	−6.26943	−0.440028
$204$	0	0
$205$	3.69002	0.257722
$206$	0	0
$207$	−0.616500	−0.0428497
$208$	0	0
$209$	4.90283	0.339136
$210$	0	0
$211$	−6.19572	−0.426531	−0.213266	−	0.976994i	$-0.568410\pi$
$211$	−6.19572	−0.426531	−0.213266	+	0.976994i	$0.568410\pi$
$212$	0	0
$213$	1.86695	0.127921
$214$	0	0
$215$	−6.53246	−0.445510
$216$	0	0
$217$	−7.90794	−0.536826
$218$	0	0
$219$	18.8193	1.27169
$220$	0	0
$221$	9.74663	0.655629
$222$	0	0
$223$	−0.528935	−0.0354201	−0.0177101	−	0.999843i	$-0.505638\pi$
$223$	−0.528935	−0.0354201	−0.0177101	+	0.999843i	$0.505638\pi$
$224$	0	0
$225$	−0.416601	−0.0277734
$226$	0	0
$227$	25.2769	1.67769	0.838845	−	0.544371i	$-0.183232\pi$
$227$	25.2769	1.67769	0.838845	+	0.544371i	$0.183232\pi$
$228$	0	0
$229$	−12.3496	−0.816084	−0.408042	−	0.912963i	$-0.633788\pi$
$229$	−12.3496	−0.816084	−0.408042	+	0.912963i	$0.633788\pi$
$230$	0	0
$231$	−1.38231	−0.0909491
$232$	0	0
$233$	26.9485	1.76545	0.882727	−	0.469885i	$-0.155705\pi$
$233$	26.9485	1.76545	0.882727	+	0.469885i	$0.155705\pi$
$234$	0	0
$235$	−4.78158	−0.311916
$236$	0	0
$237$	−12.7305	−0.826934
$238$	0	0
$239$	−19.8050	−1.28108	−0.640539	−	0.767926i	$-0.721289\pi$
$239$	−19.8050	−1.28108	−0.640539	+	0.767926i	$0.721289\pi$
$240$	0	0
$241$	−3.73993	−0.240910	−0.120455	−	0.992719i	$-0.538435\pi$
$241$	−3.73993	−0.240910	−0.120455	+	0.992719i	$0.538435\pi$
$242$	0	0
$243$	2.10278	0.134894
$244$	0	0
$245$	2.65618	0.169697
$246$	0	0
$247$	33.0588	2.10348
$248$	0	0
$249$	26.4603	1.67686
$250$	0	0
$251$	14.3509	0.905818	0.452909	−	0.891557i	$-0.350386\pi$
$251$	14.3509	0.905818	0.452909	+	0.891557i	$0.350386\pi$
$252$	0	0
$253$	−2.51376	−0.158039
$254$	0	0
$255$	−7.76976	−0.486561
$256$	0	0
$257$	12.6111	0.786657	0.393328	−	0.919398i	$-0.371324\pi$
$257$	12.6111	0.786657	0.393328	+	0.919398i	$0.371324\pi$
$258$	0	0
$259$	8.30514	0.516056
$260$	0	0
$261$	1.27079	0.0786597
$262$	0	0
$263$	−7.01176	−0.432364	−0.216182	−	0.976353i	$-0.569360\pi$
$263$	−7.01176	−0.432364	−0.216182	+	0.976353i	$0.569360\pi$
$264$	0	0
$265$	−36.5752	−2.24680
$266$	0	0
$267$	−0.717217	−0.0438929
$268$	0	0
$269$	−5.54805	−0.338271	−0.169135	−	0.985593i	$-0.554098\pi$
$269$	−5.54805	−0.338271	−0.169135	+	0.985593i	$0.554098\pi$
$270$	0	0
$271$	−23.5746	−1.43205	−0.716026	−	0.698073i	$-0.754041\pi$
$271$	−23.5746	−1.43205	−0.716026	+	0.698073i	$0.754041\pi$
$272$	0	0
$273$	−9.32061	−0.564109
$274$	0	0
$275$	−1.69868	−0.102434
$276$	0	0
$277$	−11.1677	−0.671003	−0.335502	−	0.942040i	$-0.608906\pi$
$277$	−11.1677	−0.671003	−0.335502	+	0.942040i	$0.608906\pi$
$278$	0	0
$279$	1.60290	0.0959634
$280$	0	0
$281$	33.1753	1.97907	0.989536	−	0.144288i	$-0.0460891\pi$
$281$	33.1753	1.97907	0.989536	+	0.144288i	$0.0460891\pi$
$282$	0	0
$283$	−2.91674	−0.173382	−0.0866912	−	0.996235i	$-0.527629\pi$
$283$	−2.91674	−0.173382	−0.0866912	+	0.996235i	$0.527629\pi$
$284$	0	0
$285$	−26.3536	−1.56105
$286$	0	0
$287$	1.38922	0.0820030
$288$	0	0
$289$	−13.9411	−0.820067
$290$	0	0
$291$	−32.1395	−1.88405
$292$	0	0
$293$	7.53003	0.439909	0.219954	−	0.975510i	$-0.429409\pi$
$293$	7.53003	0.439909	0.219954	+	0.975510i	$0.429409\pi$
$294$	0	0
$295$	−17.8054	−1.03667
$296$	0	0
$297$	4.42711	0.256887
$298$	0	0
$299$	−16.9497	−0.980229
$300$	0	0
$301$	−2.45934	−0.141754
$302$	0	0
$303$	−11.5319	−0.662491
$304$	0	0
$305$	11.6545	0.667336
$306$	0	0
$307$	14.0353	0.801037	0.400518	−	0.916289i	$-0.368830\pi$
$307$	14.0353	0.801037	0.400518	+	0.916289i	$0.368830\pi$
$308$	0	0
$309$	−10.1998	−0.580247
$310$	0	0
$311$	9.78126	0.554644	0.277322	−	0.960777i	$-0.410553\pi$
$311$	9.78126	0.554644	0.277322	+	0.960777i	$0.410553\pi$
$312$	0	0
$313$	5.68720	0.321460	0.160730	−	0.986998i	$-0.448615\pi$
$313$	5.68720	0.321460	0.160730	+	0.986998i	$0.448615\pi$
$314$	0	0
$315$	−0.538396	−0.0303352
$316$	0	0
$317$	−30.9235	−1.73683	−0.868417	−	0.495834i	$-0.834863\pi$
$317$	−30.9235	−1.73683	−0.868417	+	0.495834i	$0.834863\pi$
$318$	0	0
$319$	5.18159	0.290113
$320$	0	0
$321$	12.2993	0.686478
$322$	0	0
$323$	10.3751	0.577286
$324$	0	0
$325$	−11.4538	−0.635343
$326$	0	0
$327$	−18.2141	−1.00724
$328$	0	0
$329$	−1.80017	−0.0992466
$330$	0	0
$331$	12.3877	0.680890	0.340445	−	0.940265i	$-0.389422\pi$
$331$	12.3877	0.680890	0.340445	+	0.940265i	$0.389422\pi$
$332$	0	0
$333$	−1.68342	−0.0922506
$334$	0	0
$335$	18.0724	0.987400
$336$	0	0
$337$	−19.1758	−1.04457	−0.522286	−	0.852771i	$-0.674921\pi$
$337$	−19.1758	−1.04457	−0.522286	+	0.852771i	$0.674921\pi$
$338$	0	0
$339$	23.8578	1.29578
$340$	0	0
$341$	6.53579	0.353933
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	13.5119	0.727456
$346$	0	0
$347$	9.00913	0.483635	0.241818	−	0.970322i	$-0.422256\pi$
$347$	9.00913	0.483635	0.241818	+	0.970322i	$0.422256\pi$
$348$	0	0
$349$	19.9165	1.06610	0.533052	−	0.846082i	$-0.321045\pi$
$349$	19.9165	1.06610	0.533052	+	0.846082i	$0.321045\pi$
$350$	0	0
$351$	29.8511	1.59333
$352$	0	0
$353$	12.8957	0.686371	0.343185	−	0.939268i	$-0.388494\pi$
$353$	12.8957	0.686371	0.343185	+	0.939268i	$0.388494\pi$
$354$	0	0
$355$	2.96498	0.157365
$356$	0	0
$357$	−2.92516	−0.154816
$358$	0	0
$359$	33.5832	1.77245	0.886227	−	0.463252i	$-0.153318\pi$
$359$	33.5832	1.77245	0.886227	+	0.463252i	$0.153318\pi$
$360$	0	0
$361$	16.1905	0.852130
$362$	0	0
$363$	−17.2552	−0.905663
$364$	0	0
$365$	29.8875	1.56439
$366$	0	0
$367$	−7.74580	−0.404328	−0.202164	−	0.979352i	$-0.564797\pi$
$367$	−7.74580	−0.404328	−0.202164	+	0.979352i	$0.564797\pi$
$368$	0	0
$369$	−0.281589	−0.0146589
$370$	0	0
$371$	−13.7698	−0.714895
$372$	0	0
$373$	2.61073	0.135178	0.0675892	−	0.997713i	$-0.478469\pi$
$373$	2.61073	0.135178	0.0675892	+	0.997713i	$0.478469\pi$
$374$	0	0
$375$	−13.0818	−0.675543
$376$	0	0
$377$	34.9384	1.79942
$378$	0	0
$379$	34.5705	1.77577	0.887884	−	0.460068i	$-0.152175\pi$
$379$	34.5705	1.77577	0.887884	+	0.460068i	$0.152175\pi$
$380$	0	0
$381$	15.0244	0.769722
$382$	0	0
$383$	22.2480	1.13682	0.568411	−	0.822745i	$-0.307559\pi$
$383$	22.2480	1.13682	0.568411	+	0.822745i	$0.307559\pi$
$384$	0	0
$385$	−2.19529	−0.111882
$386$	0	0
$387$	0.498498	0.0253401
$388$	0	0
$389$	−9.80992	−0.497383	−0.248691	−	0.968583i	$-0.580000\pi$
$389$	−9.80992	−0.497383	−0.248691	+	0.968583i	$0.580000\pi$
$390$	0	0
$391$	−5.31947	−0.269017
$392$	0	0
$393$	30.1685	1.52180
$394$	0	0
$395$	−20.2177	−1.01727
$396$	0	0
$397$	−23.1991	−1.16433	−0.582164	−	0.813072i	$-0.697794\pi$
$397$	−23.1991	−1.16433	−0.582164	+	0.813072i	$0.697794\pi$
$398$	0	0
$399$	−9.92162	−0.496702
$400$	0	0
$401$	22.8150	1.13932	0.569662	−	0.821879i	$-0.307074\pi$
$401$	22.8150	1.13932	0.569662	+	0.821879i	$0.307074\pi$
$402$	0	0
$403$	44.0695	2.19526
$404$	0	0
$405$	−22.1813	−1.10220
$406$	0	0
$407$	−6.86407	−0.340239
$408$	0	0
$409$	13.9196	0.688281	0.344140	−	0.938918i	$-0.388170\pi$
$409$	13.9196	0.688281	0.344140	+	0.938918i	$0.388170\pi$
$410$	0	0
$411$	−19.7234	−0.972883
$412$	0	0
$413$	−6.70340	−0.329853
$414$	0	0
$415$	42.0226	2.06281
$416$	0	0
$417$	1.23945	0.0606962
$418$	0	0
$419$	−35.6151	−1.73991	−0.869955	−	0.493131i	$-0.835852\pi$
$419$	−35.6151	−1.73991	−0.869955	+	0.493131i	$0.835852\pi$
$420$	0	0
$421$	−22.7369	−1.10813	−0.554064	−	0.832474i	$-0.686924\pi$
$421$	−22.7369	−1.10813	−0.554064	+	0.832474i	$0.686924\pi$
$422$	0	0
$423$	0.364887	0.0177414
$424$	0	0
$425$	−3.59464	−0.174366
$426$	0	0
$427$	4.38770	0.212335
$428$	0	0
$429$	7.70334	0.371921
$430$	0	0
$431$	0.695976	0.0335240	0.0167620	−	0.999860i	$-0.494664\pi$
$431$	0.695976	0.0335240	0.0167620	+	0.999860i	$0.494664\pi$
$432$	0	0
$433$	−26.4982	−1.27342	−0.636711	−	0.771103i	$-0.719705\pi$
$433$	−26.4982	−1.27342	−0.636711	+	0.771103i	$0.719705\pi$
$434$	0	0
$435$	−27.8520	−1.33540
$436$	0	0
$437$	−18.0427	−0.863099
$438$	0	0
$439$	−5.34131	−0.254927	−0.127463	−	0.991843i	$-0.540684\pi$
$439$	−5.34131	−0.254927	−0.127463	+	0.991843i	$0.540684\pi$
$440$	0	0
$441$	−0.202696	−0.00965217
$442$	0	0
$443$	25.5034	1.21171	0.605853	−	0.795577i	$-0.292832\pi$
$443$	25.5034	1.21171	0.605853	+	0.795577i	$0.292832\pi$
$444$	0	0
$445$	−1.13904	−0.0539956
$446$	0	0
$447$	−4.70360	−0.222473
$448$	0	0
$449$	5.57561	0.263129	0.131565	−	0.991308i	$-0.458000\pi$
$449$	5.57561	0.263129	0.131565	+	0.991308i	$0.458000\pi$
$450$	0	0
$451$	−1.14817	−0.0540651
$452$	0	0
$453$	−25.4034	−1.19355
$454$	0	0
$455$	−14.8024	−0.693948
$456$	0	0
$457$	−11.8678	−0.555151	−0.277576	−	0.960704i	$-0.589531\pi$
$457$	−11.8678	−0.555151	−0.277576	+	0.960704i	$0.589531\pi$
$458$	0	0
$459$	9.36840	0.437279
$460$	0	0
$461$	−35.6344	−1.65966	−0.829830	−	0.558017i	$-0.811562\pi$
$461$	−35.6344	−1.65966	−0.829830	+	0.558017i	$0.811562\pi$
$462$	0	0
$463$	31.1785	1.44899	0.724494	−	0.689281i	$-0.242074\pi$
$463$	31.1785	1.44899	0.724494	+	0.689281i	$0.242074\pi$
$464$	0	0
$465$	−35.1310	−1.62916
$466$	0	0
$467$	−6.87676	−0.318219	−0.159109	−	0.987261i	$-0.550862\pi$
$467$	−6.87676	−0.318219	−0.159109	+	0.987261i	$0.550862\pi$
$468$	0	0
$469$	6.80390	0.314175
$470$	0	0
$471$	18.7243	0.862772
$472$	0	0
$473$	2.03261	0.0934594
$474$	0	0
$475$	−12.1924	−0.559424
$476$	0	0
$477$	2.79109	0.127795
$478$	0	0
$479$	−13.2313	−0.604555	−0.302277	−	0.953220i	$-0.597747\pi$
$479$	−13.2313	−0.604555	−0.302277	+	0.953220i	$0.597747\pi$
$480$	0	0
$481$	−46.2830	−2.11032
$482$	0	0
$483$	5.08696	0.231465
$484$	0	0
$485$	−51.0419	−2.31769
$486$	0	0
$487$	27.6451	1.25272	0.626360	−	0.779534i	$-0.284544\pi$
$487$	27.6451	1.25272	0.626360	+	0.779534i	$0.284544\pi$
$488$	0	0
$489$	−14.9527	−0.676185
$490$	0	0
$491$	−7.46915	−0.337078	−0.168539	−	0.985695i	$-0.553905\pi$
$491$	−7.46915	−0.337078	−0.168539	+	0.985695i	$0.553905\pi$
$492$	0	0
$493$	10.9650	0.493838
$494$	0	0
$495$	0.444976	0.0200002
$496$	0	0
$497$	1.11625	0.0500709
$498$	0	0
$499$	−40.7824	−1.82567	−0.912836	−	0.408327i	$-0.866112\pi$
$499$	−40.7824	−1.82567	−0.912836	+	0.408327i	$0.866112\pi$
$500$	0	0
$501$	37.6756	1.68322
$502$	0	0
$503$	−16.8595	−0.751729	−0.375864	−	0.926675i	$-0.622654\pi$
$503$	−16.8595	−0.751729	−0.375864	+	0.926675i	$0.622654\pi$
$504$	0	0
$505$	−18.3142	−0.814973
$506$	0	0
$507$	30.1993	1.34120
$508$	0	0
$509$	−30.2500	−1.34081	−0.670403	−	0.741997i	$-0.733879\pi$
$509$	−30.2500	−1.34081	−0.670403	+	0.741997i	$0.733879\pi$
$510$	0	0
$511$	11.2521	0.497762
$512$	0	0
$513$	31.7759	1.40294
$514$	0	0
$515$	−16.1987	−0.713799
$516$	0	0
$517$	1.48781	0.0654339
$518$	0	0
$519$	−0.0250509	−0.00109961
$520$	0	0
$521$	−29.9153	−1.31061	−0.655307	−	0.755363i	$-0.727461\pi$
$521$	−29.9153	−1.31061	−0.655307	+	0.755363i	$0.727461\pi$
$522$	0	0
$523$	17.8490	0.780484	0.390242	−	0.920712i	$-0.372391\pi$
$523$	17.8490	0.780484	0.390242	+	0.920712i	$0.372391\pi$
$524$	0	0
$525$	3.43752	0.150026
$526$	0	0
$527$	13.8307	0.602473
$528$	0	0
$529$	−13.7492	−0.597793
$530$	0	0
$531$	1.35875	0.0589647
$532$	0	0
$533$	−7.74186	−0.335337
$534$	0	0
$535$	19.5329	0.844482
$536$	0	0
$537$	−33.9583	−1.46541
$538$	0	0
$539$	−0.826484	−0.0355992
$540$	0	0
$541$	9.87716	0.424652	0.212326	−	0.977199i	$-0.431896\pi$
$541$	9.87716	0.424652	0.212326	+	0.977199i	$0.431896\pi$
$542$	0	0
$543$	−20.4674	−0.878339
$544$	0	0
$545$	−28.9264	−1.23907
$546$	0	0
$547$	−26.9678	−1.15306	−0.576530	−	0.817076i	$-0.695594\pi$
$547$	−26.9678	−1.15306	−0.576530	+	0.817076i	$0.695594\pi$
$548$	0	0
$549$	−0.889367	−0.0379572
$550$	0	0
$551$	37.1912	1.58440
$552$	0	0
$553$	−7.61158	−0.323677
$554$	0	0
$555$	36.8956	1.56613
$556$	0	0
$557$	−15.1355	−0.641310	−0.320655	−	0.947196i	$-0.603903\pi$
$557$	−15.1355	−0.641310	−0.320655	+	0.947196i	$0.603903\pi$
$558$	0	0
$559$	13.7055	0.579679
$560$	0	0
$561$	2.41760	0.102071
$562$	0	0
$563$	9.23929	0.389390	0.194695	−	0.980864i	$-0.437628\pi$
$563$	9.23929	0.389390	0.194695	+	0.980864i	$0.437628\pi$
$564$	0	0
$565$	37.8895	1.59402
$566$	0	0
$567$	−8.35083	−0.350702
$568$	0	0
$569$	−36.7286	−1.53974	−0.769871	−	0.638199i	$-0.779680\pi$
$569$	−36.7286	−1.53974	−0.769871	+	0.638199i	$0.779680\pi$
$570$	0	0
$571$	−33.6452	−1.40801	−0.704005	−	0.710195i	$-0.748607\pi$
$571$	−33.6452	−1.40801	−0.704005	+	0.710195i	$0.748607\pi$
$572$	0	0
$573$	28.7815	1.20236
$574$	0	0
$575$	6.25121	0.260694
$576$	0	0
$577$	−5.87217	−0.244462	−0.122231	−	0.992502i	$-0.539005\pi$
$577$	−5.87217	−0.244462	−0.122231	+	0.992502i	$0.539005\pi$
$578$	0	0
$579$	11.7240	0.487234
$580$	0	0
$581$	15.8207	0.656353
$582$	0	0
$583$	11.3806	0.471335
$584$	0	0
$585$	3.00038	0.124051
$586$	0	0
$587$	15.5578	0.642138	0.321069	−	0.947056i	$-0.395958\pi$
$587$	15.5578	0.642138	0.321069	+	0.947056i	$0.395958\pi$
$588$	0	0
$589$	46.9111	1.93294
$590$	0	0
$591$	−36.4669	−1.50005
$592$	0	0
$593$	−26.8167	−1.10123	−0.550616	−	0.834759i	$-0.685607\pi$
$593$	−26.8167	−1.10123	−0.550616	+	0.834759i	$0.685607\pi$
$594$	0	0
$595$	−4.64556	−0.190449
$596$	0	0
$597$	25.6702	1.05061
$598$	0	0
$599$	−44.9307	−1.83582	−0.917909	−	0.396791i	$-0.870124\pi$
$599$	−44.9307	−1.83582	−0.917909	+	0.396791i	$0.870124\pi$
$600$	0	0
$601$	40.2092	1.64017	0.820083	−	0.572244i	$-0.193927\pi$
$601$	40.2092	1.64017	0.820083	+	0.572244i	$0.193927\pi$
$602$	0	0
$603$	−1.37912	−0.0561621
$604$	0	0
$605$	−27.4036	−1.11412
$606$	0	0
$607$	20.7271	0.841286	0.420643	−	0.907226i	$-0.361805\pi$
$607$	20.7271	0.841286	0.420643	+	0.907226i	$0.361805\pi$
$608$	0	0
$609$	−10.4857	−0.424903
$610$	0	0
$611$	10.0320	0.405852
$612$	0	0
$613$	−7.68740	−0.310491	−0.155246	−	0.987876i	$-0.549617\pi$
$613$	−7.68740	−0.310491	−0.155246	+	0.987876i	$0.549617\pi$
$614$	0	0
$615$	6.17161	0.248863
$616$	0	0
$617$	24.8297	0.999607	0.499803	−	0.866139i	$-0.333406\pi$
$617$	24.8297	0.999607	0.499803	+	0.866139i	$0.333406\pi$
$618$	0	0
$619$	18.6927	0.751324	0.375662	−	0.926757i	$-0.377415\pi$
$619$	18.6927	0.751324	0.375662	+	0.926757i	$0.377415\pi$
$620$	0	0
$621$	−16.2920	−0.653775
$622$	0	0
$623$	−0.428825	−0.0171805
$624$	0	0
$625$	−31.0522	−1.24209
$626$	0	0
$627$	8.20006	0.327479
$628$	0	0
$629$	−14.5254	−0.579164
$630$	0	0
$631$	30.4138	1.21075	0.605377	−	0.795939i	$-0.293022\pi$
$631$	30.4138	1.21075	0.605377	+	0.795939i	$0.293022\pi$
$632$	0	0
$633$	−10.3624	−0.411870
$634$	0	0
$635$	23.8608	0.946885
$636$	0	0
$637$	−5.57281	−0.220803
$638$	0	0
$639$	−0.226260	−0.00895071
$640$	0	0
$641$	−41.7486	−1.64897	−0.824485	−	0.565884i	$-0.808535\pi$
$641$	−41.7486	−1.64897	−0.824485	+	0.565884i	$0.808535\pi$
$642$	0	0
$643$	−21.7983	−0.859640	−0.429820	−	0.902915i	$-0.641423\pi$
$643$	−21.7983	−0.859640	−0.429820	+	0.902915i	$0.641423\pi$
$644$	0	0
$645$	−10.9256	−0.430196
$646$	0	0
$647$	8.85923	0.348292	0.174146	−	0.984720i	$-0.444283\pi$
$647$	8.85923	0.348292	0.174146	+	0.984720i	$0.444283\pi$
$648$	0	0
$649$	5.54026	0.217474
$650$	0	0
$651$	−13.2261	−0.518373
$652$	0	0
$653$	3.43172	0.134293	0.0671467	−	0.997743i	$-0.478610\pi$
$653$	3.43172	0.134293	0.0671467	+	0.997743i	$0.478610\pi$
$654$	0	0
$655$	47.9117	1.87206
$656$	0	0
$657$	−2.28075	−0.0889804
$658$	0	0
$659$	−32.9576	−1.28384	−0.641922	−	0.766770i	$-0.721863\pi$
$659$	−32.9576	−1.28384	−0.641922	+	0.766770i	$0.721863\pi$
$660$	0	0
$661$	−32.8710	−1.27853	−0.639266	−	0.768985i	$-0.720762\pi$
$661$	−32.8710	−1.27853	−0.639266	+	0.768985i	$0.720762\pi$
$662$	0	0
$663$	16.3014	0.633093
$664$	0	0
$665$	−15.7569	−0.611026
$666$	0	0
$667$	−19.0685	−0.738335
$668$	0	0
$669$	−0.884651	−0.0342026
$670$	0	0
$671$	−3.62636	−0.139994
$672$	0	0
$673$	46.7430	1.80181	0.900906	−	0.434014i	$-0.142903\pi$
$673$	46.7430	1.80181	0.900906	+	0.434014i	$0.142903\pi$
$674$	0	0
$675$	−11.0093	−0.423750
$676$	0	0
$677$	42.5577	1.63562	0.817812	−	0.575485i	$-0.195187\pi$
$677$	42.5577	1.63562	0.817812	+	0.575485i	$0.195187\pi$
$678$	0	0
$679$	−19.2163	−0.737452
$680$	0	0
$681$	42.2760	1.62002
$682$	0	0
$683$	−36.0188	−1.37822	−0.689110	−	0.724657i	$-0.741998\pi$
$683$	−36.0188	−1.37822	−0.689110	+	0.724657i	$0.741998\pi$
$684$	0	0
$685$	−31.3234	−1.19681
$686$	0	0
$687$	−20.6549	−0.788033
$688$	0	0
$689$	76.7368	2.92344
$690$	0	0
$691$	21.9821	0.836240	0.418120	−	0.908392i	$-0.362689\pi$
$691$	21.9821	0.836240	0.418120	+	0.908392i	$0.362689\pi$
$692$	0	0
$693$	0.167525	0.00636374
$694$	0	0
$695$	1.96842	0.0746664
$696$	0	0
$697$	−2.42969	−0.0920311
$698$	0	0
$699$	45.0718	1.70477
$700$	0	0
$701$	7.91840	0.299074	0.149537	−	0.988756i	$-0.452222\pi$
$701$	7.91840	0.299074	0.149537	+	0.988756i	$0.452222\pi$
$702$	0	0
$703$	−49.2674	−1.85815
$704$	0	0
$705$	−7.99726	−0.301194
$706$	0	0
$707$	−6.89495	−0.259311
$708$	0	0
$709$	−0.969949	−0.0364272	−0.0182136	−	0.999834i	$-0.505798\pi$
$709$	−0.969949	−0.0364272	−0.0182136	+	0.999834i	$0.505798\pi$
$710$	0	0
$711$	1.54283	0.0578608
$712$	0	0
$713$	−24.0520	−0.900756
$714$	0	0
$715$	12.2340	0.457524
$716$	0	0
$717$	−33.1241	−1.23704
$718$	0	0
$719$	−10.0440	−0.374579	−0.187289	−	0.982305i	$-0.559970\pi$
$719$	−10.0440	−0.374579	−0.187289	+	0.982305i	$0.559970\pi$
$720$	0	0
$721$	−6.09849	−0.227119
$722$	0	0
$723$	−6.25509	−0.232629
$724$	0	0
$725$	−12.8856	−0.478558
$726$	0	0
$727$	−5.53235	−0.205184	−0.102592	−	0.994724i	$-0.532714\pi$
$727$	−5.53235	−0.205184	−0.102592	+	0.994724i	$0.532714\pi$
$728$	0	0
$729$	28.5694	1.05813
$730$	0	0
$731$	4.30129	0.159089
$732$	0	0
$733$	−5.63576	−0.208162	−0.104081	−	0.994569i	$-0.533190\pi$
$733$	−5.63576	−0.208162	−0.104081	+	0.994569i	$0.533190\pi$
$734$	0	0
$735$	4.44250	0.163864
$736$	0	0
$737$	−5.62331	−0.207137
$738$	0	0
$739$	3.10659	0.114278	0.0571388	−	0.998366i	$-0.481802\pi$
$739$	3.10659	0.114278	0.0571388	+	0.998366i	$0.481802\pi$
$740$	0	0
$741$	55.2913	2.03118
$742$	0	0
$743$	−22.5404	−0.826927	−0.413464	−	0.910521i	$-0.635681\pi$
$743$	−22.5404	−0.826927	−0.413464	+	0.910521i	$0.635681\pi$
$744$	0	0
$745$	−7.46996	−0.273678
$746$	0	0
$747$	−3.20678	−0.117330
$748$	0	0
$749$	7.35376	0.268701
$750$	0	0
$751$	14.8752	0.542805	0.271403	−	0.962466i	$-0.412513\pi$
$751$	14.8752	0.542805	0.271403	+	0.962466i	$0.412513\pi$
$752$	0	0
$753$	24.0020	0.874682
$754$	0	0
$755$	−40.3440	−1.46827
$756$	0	0
$757$	−1.38945	−0.0505003	−0.0252502	−	0.999681i	$-0.508038\pi$
$757$	−1.38945	−0.0505003	−0.0252502	+	0.999681i	$0.508038\pi$
$758$	0	0
$759$	−4.20429	−0.152606
$760$	0	0
$761$	−19.9186	−0.722049	−0.361024	−	0.932556i	$-0.617573\pi$
$761$	−19.9186	−0.722049	−0.361024	+	0.932556i	$0.617573\pi$
$762$	0	0
$763$	−10.8902	−0.394253
$764$	0	0
$765$	0.941634	0.0340448
$766$	0	0
$767$	37.3568	1.34888
$768$	0	0
$769$	−9.78665	−0.352915	−0.176458	−	0.984308i	$-0.556464\pi$
$769$	−9.78665	−0.352915	−0.176458	+	0.984308i	$0.556464\pi$
$770$	0	0
$771$	21.0922	0.759617
$772$	0	0
$773$	27.8576	1.00197	0.500985	−	0.865456i	$-0.332971\pi$
$773$	27.8576	1.00197	0.500985	+	0.865456i	$0.332971\pi$
$774$	0	0
$775$	−16.2532	−0.583832
$776$	0	0
$777$	13.8905	0.498318
$778$	0	0
$779$	−8.24107	−0.295267
$780$	0	0
$781$	−0.922567	−0.0330121
$782$	0	0
$783$	33.5825	1.20014
$784$	0	0
$785$	29.7368	1.06135
$786$	0	0
$787$	43.7329	1.55891	0.779454	−	0.626459i	$-0.215496\pi$
$787$	43.7329	1.55891	0.779454	+	0.626459i	$0.215496\pi$
$788$	0	0
$789$	−11.7273	−0.417502
$790$	0	0
$791$	14.2646	0.507192
$792$	0	0
$793$	−24.4518	−0.868309
$794$	0	0
$795$	−61.1726	−2.16957
$796$	0	0
$797$	−45.9626	−1.62808	−0.814040	−	0.580809i	$-0.802736\pi$
$797$	−45.9626	−1.62808	−0.814040	+	0.580809i	$0.802736\pi$
$798$	0	0
$799$	3.14843	0.111383
$800$	0	0
$801$	0.0869210	0.00307120
$802$	0	0
$803$	−9.29966	−0.328178
$804$	0	0
$805$	8.07879	0.284740
$806$	0	0
$807$	−9.27920	−0.326643
$808$	0	0
$809$	−20.8417	−0.732756	−0.366378	−	0.930466i	$-0.619402\pi$
$809$	−20.8417	−0.732756	−0.366378	+	0.930466i	$0.619402\pi$
$810$	0	0
$811$	−2.73444	−0.0960193	−0.0480096	−	0.998847i	$-0.515288\pi$
$811$	−2.73444	−0.0960193	−0.0480096	+	0.998847i	$0.515288\pi$
$812$	0	0
$813$	−39.4288	−1.38283
$814$	0	0
$815$	−23.7469	−0.831819
$816$	0	0
$817$	14.5892	0.510411
$818$	0	0
$819$	1.12958	0.0394709
$820$	0	0
$821$	9.56765	0.333913	0.166957	−	0.985964i	$-0.446606\pi$
$821$	9.56765	0.333913	0.166957	+	0.985964i	$0.446606\pi$
$822$	0	0
$823$	35.9651	1.25366	0.626832	−	0.779154i	$-0.284351\pi$
$823$	35.9651	1.25366	0.626832	+	0.779154i	$0.284351\pi$
$824$	0	0
$825$	−2.84106	−0.0989130
$826$	0	0
$827$	−32.7916	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$827$	−32.7916	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$828$	0	0
$829$	−26.0624	−0.905185	−0.452593	−	0.891717i	$-0.649501\pi$
$829$	−26.0624	−0.905185	−0.452593	+	0.891717i	$0.649501\pi$
$830$	0	0
$831$	−18.6782	−0.647938
$832$	0	0
$833$	−1.74896	−0.0605979
$834$	0	0
$835$	59.8340	2.07064
$836$	0	0
$837$	42.3593	1.46415
$838$	0	0
$839$	34.0165	1.17438	0.587191	−	0.809449i	$-0.300234\pi$
$839$	34.0165	1.17438	0.587191	+	0.809449i	$0.300234\pi$
$840$	0	0
$841$	10.3058	0.355371
$842$	0	0
$843$	55.4861	1.91104
$844$	0	0
$845$	47.9607	1.64990
$846$	0	0
$847$	−10.3169	−0.354494
$848$	0	0
$849$	−4.87829	−0.167423
$850$	0	0
$851$	25.2601	0.865906
$852$	0	0
$853$	15.5570	0.532663	0.266331	−	0.963882i	$-0.414188\pi$
$853$	15.5570	0.532663	0.266331	+	0.963882i	$0.414188\pi$
$854$	0	0
$855$	3.19385	0.109227
$856$	0	0
$857$	−0.996249	−0.0340312	−0.0170156	−	0.999855i	$-0.505416\pi$
$857$	−0.996249	−0.0340312	−0.0170156	+	0.999855i	$0.505416\pi$
$858$	0	0
$859$	−33.3053	−1.13636	−0.568181	−	0.822903i	$-0.692353\pi$
$859$	−33.3053	−1.13636	−0.568181	+	0.822903i	$0.692353\pi$
$860$	0	0
$861$	2.32349	0.0791843
$862$	0	0
$863$	−45.1680	−1.53754	−0.768768	−	0.639528i	$-0.779130\pi$
$863$	−45.1680	−1.53754	−0.768768	+	0.639528i	$0.779130\pi$
$864$	0	0
$865$	−0.0397843	−0.00135271
$866$	0	0
$867$	−23.3168	−0.791879
$868$	0	0
$869$	6.29086	0.213403
$870$	0	0
$871$	−37.9168	−1.28476
$872$	0	0
$873$	3.89505	0.131827
$874$	0	0
$875$	−7.82165	−0.264420
$876$	0	0
$877$	−6.46529	−0.218317	−0.109159	−	0.994024i	$-0.534816\pi$
$877$	−6.46529	−0.218317	−0.109159	+	0.994024i	$0.534816\pi$
$878$	0	0
$879$	12.5941	0.424788
$880$	0	0
$881$	−29.5811	−0.996613	−0.498306	−	0.867001i	$-0.666045\pi$
$881$	−29.5811	−0.996613	−0.498306	+	0.867001i	$0.666045\pi$
$882$	0	0
$883$	−29.6304	−0.997143	−0.498572	−	0.866849i	$-0.666142\pi$
$883$	−29.6304	−0.997143	−0.498572	+	0.866849i	$0.666142\pi$
$884$	0	0
$885$	−29.7799	−1.00104
$886$	0	0
$887$	33.7805	1.13424	0.567120	−	0.823635i	$-0.308058\pi$
$887$	33.7805	1.13424	0.567120	+	0.823635i	$0.308058\pi$
$888$	0	0
$889$	8.98310	0.301284
$890$	0	0
$891$	6.90183	0.231220
$892$	0	0
$893$	10.6789	0.357355
$894$	0	0
$895$	−53.9305	−1.80270
$896$	0	0
$897$	−28.3487	−0.946535
$898$	0	0
$899$	49.5783	1.65353
$900$	0	0
$901$	24.0829	0.802318
$902$	0	0
$903$	−4.11328	−0.136881
$904$	0	0
$905$	−32.5050	−1.08050
$906$	0	0
$907$	−53.6507	−1.78144	−0.890721	−	0.454551i	$-0.849800\pi$
$907$	−53.6507	−1.78144	−0.890721	+	0.454551i	$0.849800\pi$
$908$	0	0
$909$	1.39758	0.0463547
$910$	0	0
$911$	−16.3562	−0.541906	−0.270953	−	0.962593i	$-0.587339\pi$
$911$	−16.3562	−0.541906	−0.270953	+	0.962593i	$0.587339\pi$
$912$	0	0
$913$	−13.0756	−0.432738
$914$	0	0
$915$	19.4924	0.644397
$916$	0	0
$917$	18.0378	0.595661
$918$	0	0
$919$	6.74372	0.222455	0.111227	−	0.993795i	$-0.464522\pi$
$919$	6.74372	0.222455	0.111227	+	0.993795i	$0.464522\pi$
$920$	0	0
$921$	23.4742	0.773503
$922$	0	0
$923$	−6.22068	−0.204756
$924$	0	0
$925$	17.0696	0.561244
$926$	0	0
$927$	1.23614	0.0406000
$928$	0	0
$929$	−3.61776	−0.118695	−0.0593475	−	0.998237i	$-0.518902\pi$
$929$	−3.61776	−0.118695	−0.0593475	+	0.998237i	$0.518902\pi$
$930$	0	0
$931$	−5.93216	−0.194419
$932$	0	0
$933$	16.3593	0.535579
$934$	0	0
$935$	3.83948	0.125564
$936$	0	0
$937$	−2.79455	−0.0912940	−0.0456470	−	0.998958i	$-0.514535\pi$
$937$	−2.79455	−0.0912940	−0.0456470	+	0.998958i	$0.514535\pi$
$938$	0	0
$939$	9.51193	0.310410
$940$	0	0
$941$	−24.5879	−0.801541	−0.400771	−	0.916178i	$-0.631258\pi$
$941$	−24.5879	−0.801541	−0.400771	+	0.916178i	$0.631258\pi$
$942$	0	0
$943$	4.22532	0.137595
$944$	0	0
$945$	−14.2280	−0.462836
$946$	0	0
$947$	−42.2713	−1.37363	−0.686816	−	0.726832i	$-0.740992\pi$
$947$	−42.2713	−1.37363	−0.686816	+	0.726832i	$0.740992\pi$
$948$	0	0
$949$	−62.7057	−2.03551
$950$	0	0
$951$	−51.7199	−1.67713
$952$	0	0
$953$	−26.9944	−0.874434	−0.437217	−	0.899356i	$-0.644036\pi$
$953$	−26.9944	−0.874434	−0.437217	+	0.899356i	$0.644036\pi$
$954$	0	0
$955$	45.7090	1.47911
$956$	0	0
$957$	8.66628	0.280141
$958$	0	0
$959$	−11.7927	−0.380805
$960$	0	0
$961$	31.5355	1.01727
$962$	0	0
$963$	−1.49058	−0.0480331
$964$	0	0
$965$	18.6194	0.599379
$966$	0	0
$967$	3.64431	0.117193	0.0585966	−	0.998282i	$-0.481337\pi$
$967$	3.64431	0.117193	0.0585966	+	0.998282i	$0.481337\pi$
$968$	0	0
$969$	17.3525	0.557443
$970$	0	0
$971$	14.6356	0.469680	0.234840	−	0.972034i	$-0.424543\pi$
$971$	14.6356	0.469680	0.234840	+	0.972034i	$0.424543\pi$
$972$	0	0
$973$	0.741071	0.0237576
$974$	0	0
$975$	−19.1567	−0.613504
$976$	0	0
$977$	11.7791	0.376846	0.188423	−	0.982088i	$-0.439662\pi$
$977$	11.7791	0.376846	0.188423	+	0.982088i	$0.439662\pi$
$978$	0	0
$979$	0.354418	0.0113272
$980$	0	0
$981$	2.20740	0.0704769
$982$	0	0
$983$	3.39640	0.108328	0.0541642	−	0.998532i	$-0.482751\pi$
$983$	3.39640	0.108328	0.0541642	+	0.998532i	$0.482751\pi$
$984$	0	0
$985$	−57.9145	−1.84531
$986$	0	0
$987$	−3.01081	−0.0958352
$988$	0	0
$989$	−7.48010	−0.237853
$990$	0	0
$991$	24.0326	0.763421	0.381710	−	0.924282i	$-0.375335\pi$
$991$	24.0326	0.763421	0.381710	+	0.924282i	$0.375335\pi$
$992$	0	0
$993$	20.7186	0.657485
$994$	0	0
$995$	40.7678	1.29243
$996$	0	0
$997$	−2.41840	−0.0765913	−0.0382957	−	0.999266i	$-0.512193\pi$
$997$	−2.41840	−0.0765913	−0.0382957	+	0.999266i	$0.512193\pi$
$998$	0	0
$999$	−44.4869	−1.40750

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.bb.1.6		8
4.3	odd	2		7168.2.a.ba.1.3		8
8.3	odd	2		7168.2.a.be.1.6		8
8.5	even	2		7168.2.a.bf.1.3		8
32.3	odd	8		1792.2.m.g.1345.3	yes	16
32.5	even	8		1792.2.m.h.449.3	yes	16
32.11	odd	8		1792.2.m.g.449.3	yes	16
32.13	even	8		1792.2.m.h.1345.3	yes	16
32.19	odd	8		1792.2.m.f.1345.6	yes	16
32.21	even	8		1792.2.m.e.449.6	✓	16
32.27	odd	8		1792.2.m.f.449.6	yes	16
32.29	even	8		1792.2.m.e.1345.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1792.2.m.e.449.6	✓	16	32.21	even	8
1792.2.m.e.1345.6	yes	16	32.29	even	8
1792.2.m.f.449.6	yes	16	32.27	odd	8
1792.2.m.f.1345.6	yes	16	32.19	odd	8
1792.2.m.g.449.3	yes	16	32.11	odd	8
1792.2.m.g.1345.3	yes	16	32.3	odd	8
1792.2.m.h.449.3	yes	16	32.5	even	8
1792.2.m.h.1345.3	yes	16	32.13	even	8
7168.2.a.ba.1.3		8	4.3	odd	2
7168.2.a.bb.1.6		8	1.1	even	1	trivial
7168.2.a.be.1.6		8	8.3	odd	2
7168.2.a.bf.1.3		8	8.5	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:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q+1.67251 q^{3} +2.65618 q^{5} +1.00000 q^{7} -0.202696 q^{9} +O(q^{10})\)	\(q+1.67251 q^{3} +2.65618 q^{5} +1.00000 q^{7} -0.202696 q^{9} -0.826484 q^{11} -5.57281 q^{13} +4.44250 q^{15} -1.74896 q^{17} -5.93216 q^{19} +1.67251 q^{21} +3.04150 q^{23} +2.05530 q^{25} -5.35655 q^{27} -6.26943 q^{29} -7.90794 q^{31} -1.38231 q^{33} +2.65618 q^{35} +8.30514 q^{37} -9.32061 q^{39} +1.38922 q^{41} -2.45934 q^{43} -0.538396 q^{45} -1.80017 q^{47} +1.00000 q^{49} -2.92516 q^{51} -13.7698 q^{53} -2.19529 q^{55} -9.92162 q^{57} -6.70340 q^{59} +4.38770 q^{61} -0.202696 q^{63} -14.8024 q^{65} +6.80390 q^{67} +5.08696 q^{69} +1.11625 q^{71} +11.2521 q^{73} +3.43752 q^{75} -0.826484 q^{77} -7.61158 q^{79} -8.35083 q^{81} +15.8207 q^{83} -4.64556 q^{85} -10.4857 q^{87} -0.428825 q^{89} -5.57281 q^{91} -13.2261 q^{93} -15.7569 q^{95} -19.2163 q^{97} +0.167525 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 8 * q - 8 * q^5 + 8 * q^7 + 12 * q^9	\( 8 q - 8 q^{5} + 8 q^{7} + 12 q^{9} - 12 q^{11} - 20 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 12 q^{25} + 12 q^{27} - 8 q^{29} - 4 q^{31} + 8 q^{33} - 8 q^{35} - 8 q^{37} - 16 q^{39} - 12 q^{41} + 4 q^{43} - 52 q^{45} + 20 q^{47} + 8 q^{49} - 32 q^{51} - 40 q^{53} + 24 q^{55} - 4 q^{57} - 4 q^{59} + 8 q^{61} + 12 q^{63} + 36 q^{65} - 28 q^{67} - 4 q^{69} - 16 q^{71} + 16 q^{73} + 28 q^{75} - 12 q^{77} + 20 q^{81} + 8 q^{83} - 16 q^{85} + 20 q^{87} + 16 q^{89} - 20 q^{91} - 16 q^{93} - 40 q^{95} - 36 q^{97} + 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^5 + 8 * q^7 + 12 * q^9 - 12 * q^11 - 20 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 12 * q^25 + 12 * q^27 - 8 * q^29 - 4 * q^31 + 8 * q^33 - 8 * q^35 - 8 * q^37 - 16 * q^39 - 12 * q^41 + 4 * q^43 - 52 * q^45 + 20 * q^47 + 8 * q^49 - 32 * q^51 - 40 * q^53 + 24 * q^55 - 4 * q^57 - 4 * q^59 + 8 * q^61 + 12 * q^63 + 36 * q^65 - 28 * q^67 - 4 * q^69 - 16 * q^71 + 16 * q^73 + 28 * q^75 - 12 * q^77 + 20 * q^81 + 8 * q^83 - 16 * q^85 + 20 * q^87 + 16 * q^89 - 20 * q^91 - 16 * q^93 - 40 * q^95 - 36 * q^97 + 4 * q^99

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