LMFDB - Embedded newform 7168.2.a.be.1.4

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:	$0$
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			1.4
Root			$-1.21236$ 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)

$f(q)$	$=$	$q-0.463900 q^{3} -1.98268 q^{5} -1.00000 q^{7} -2.78480 q^{9} +O(q^{10})$	$q-0.463900 q^{3} -1.98268 q^{5} -1.00000 q^{7} -2.78480 q^{9} -0.628034 q^{11} -3.19033 q^{13} +0.919765 q^{15} -4.85578 q^{17} -0.161226 q^{19} +0.463900 q^{21} -3.20529 q^{23} -1.06898 q^{25} +2.68357 q^{27} -1.41010 q^{29} +5.34435 q^{31} +0.291345 q^{33} +1.98268 q^{35} -2.87754 q^{37} +1.47999 q^{39} -9.57673 q^{41} -9.71222 q^{43} +5.52136 q^{45} -9.70703 q^{47} +1.00000 q^{49} +2.25259 q^{51} +10.8106 q^{53} +1.24519 q^{55} +0.0747927 q^{57} +2.12508 q^{59} -2.46295 q^{61} +2.78480 q^{63} +6.32541 q^{65} -12.6783 q^{67} +1.48693 q^{69} -7.18356 q^{71} -9.04029 q^{73} +0.495897 q^{75} +0.628034 q^{77} -9.58806 q^{79} +7.10949 q^{81} +7.49880 q^{83} +9.62746 q^{85} +0.654145 q^{87} -2.49938 q^{89} +3.19033 q^{91} -2.47924 q^{93} +0.319660 q^{95} +5.89073 q^{97} +1.74895 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.463900	−0.267833	−0.133916	−	0.990993i	$-0.542755\pi$
$3$	−0.463900	−0.267833	−0.133916	+	0.990993i	$0.542755\pi$
$4$	0	0
$5$	−1.98268	−0.886682	−0.443341	−	0.896353i	$-0.646207\pi$
$5$	−1.98268	−0.886682	−0.443341	+	0.896353i	$0.646207\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.78480	−0.928266
$10$	0	0
$11$	−0.628034	−0.189359	−0.0946797	−	0.995508i	$-0.530183\pi$
$11$	−0.628034	−0.189359	−0.0946797	+	0.995508i	$0.530183\pi$
$12$	0	0
$13$	−3.19033	−0.884839	−0.442419	−	0.896808i	$-0.645880\pi$
$13$	−3.19033	−0.884839	−0.442419	+	0.896808i	$0.645880\pi$
$14$	0	0
$15$	0.919765	0.237482
$16$	0	0
$17$	−4.85578	−1.17770	−0.588849	−	0.808243i	$-0.700419\pi$
$17$	−4.85578	−1.17770	−0.588849	+	0.808243i	$0.700419\pi$
$18$	0	0
$19$	−0.161226	−0.0369878	−0.0184939	−	0.999829i	$-0.505887\pi$
$19$	−0.161226	−0.0369878	−0.0184939	+	0.999829i	$0.505887\pi$
$20$	0	0
$21$	0.463900	0.101231
$22$	0	0
$23$	−3.20529	−0.668350	−0.334175	−	0.942511i	$-0.608458\pi$
$23$	−3.20529	−0.668350	−0.334175	+	0.942511i	$0.608458\pi$
$24$	0	0
$25$	−1.06898	−0.213795
$26$	0	0
$27$	2.68357	0.516452
$28$	0	0
$29$	−1.41010	−0.261849	−0.130924	−	0.991392i	$-0.541795\pi$
$29$	−1.41010	−0.261849	−0.130924	+	0.991392i	$0.541795\pi$
$30$	0	0
$31$	5.34435	0.959874	0.479937	−	0.877303i	$-0.340660\pi$
$31$	5.34435	0.959874	0.479937	+	0.877303i	$0.340660\pi$
$32$	0	0
$33$	0.291345	0.0507167
$34$	0	0
$35$	1.98268	0.335134
$36$	0	0
$37$	−2.87754	−0.473064	−0.236532	−	0.971624i	$-0.576011\pi$
$37$	−2.87754	−0.473064	−0.236532	+	0.971624i	$0.576011\pi$
$38$	0	0
$39$	1.47999	0.236989
$40$	0	0
$41$	−9.57673	−1.49563	−0.747817	−	0.663905i	$-0.768898\pi$
$41$	−9.57673	−1.49563	−0.747817	+	0.663905i	$0.768898\pi$
$42$	0	0
$43$	−9.71222	−1.48110	−0.740550	−	0.672001i	$-0.765435\pi$
$43$	−9.71222	−1.48110	−0.740550	+	0.672001i	$0.765435\pi$
$44$	0	0
$45$	5.52136	0.823076
$46$	0	0
$47$	−9.70703	−1.41592	−0.707958	−	0.706255i	$-0.750383\pi$
$47$	−9.70703	−1.41592	−0.707958	+	0.706255i	$0.750383\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.25259	0.315426
$52$	0	0
$53$	10.8106	1.48495	0.742476	−	0.669872i	$-0.233651\pi$
$53$	10.8106	1.48495	0.742476	+	0.669872i	$0.233651\pi$
$54$	0	0
$55$	1.24519	0.167902
$56$	0	0
$57$	0.0747927	0.00990653
$58$	0	0
$59$	2.12508	0.276662	0.138331	−	0.990386i	$-0.455826\pi$
$59$	2.12508	0.276662	0.138331	+	0.990386i	$0.455826\pi$
$60$	0	0
$61$	−2.46295	−0.315348	−0.157674	−	0.987491i	$-0.550399\pi$
$61$	−2.46295	−0.315348	−0.157674	+	0.987491i	$0.550399\pi$
$62$	0	0
$63$	2.78480	0.350851
$64$	0	0
$65$	6.32541	0.784571
$66$	0	0
$67$	−12.6783	−1.54890	−0.774450	−	0.632635i	$-0.781973\pi$
$67$	−12.6783	−1.54890	−0.774450	+	0.632635i	$0.781973\pi$
$68$	0	0
$69$	1.48693	0.179006
$70$	0	0
$71$	−7.18356	−0.852532	−0.426266	−	0.904598i	$-0.640171\pi$
$71$	−7.18356	−0.852532	−0.426266	+	0.904598i	$0.640171\pi$
$72$	0	0
$73$	−9.04029	−1.05809	−0.529043	−	0.848595i	$-0.677449\pi$
$73$	−9.04029	−1.05809	−0.529043	+	0.848595i	$0.677449\pi$
$74$	0	0
$75$	0.495897	0.0572613
$76$	0	0
$77$	0.628034	0.0715712
$78$	0	0
$79$	−9.58806	−1.07874	−0.539370	−	0.842069i	$-0.681338\pi$
$79$	−9.58806	−1.07874	−0.539370	+	0.842069i	$0.681338\pi$
$80$	0	0
$81$	7.10949	0.789943
$82$	0	0
$83$	7.49880	0.823100	0.411550	−	0.911387i	$-0.364988\pi$
$83$	7.49880	0.823100	0.411550	+	0.911387i	$0.364988\pi$
$84$	0	0
$85$	9.62746	1.04424
$86$	0	0
$87$	0.654145	0.0701317
$88$	0	0
$89$	−2.49938	−0.264934	−0.132467	−	0.991187i	$-0.542290\pi$
$89$	−2.49938	−0.264934	−0.132467	+	0.991187i	$0.542290\pi$
$90$	0	0
$91$	3.19033	0.334438
$92$	0	0
$93$	−2.47924	−0.257086
$94$	0	0
$95$	0.319660	0.0327964
$96$	0	0
$97$	5.89073	0.598113	0.299057	−	0.954235i	$-0.403328\pi$
$97$	5.89073	0.598113	0.299057	+	0.954235i	$0.403328\pi$
$98$	0	0
$99$	1.74895	0.175776
$100$	0	0
$101$	9.90655	0.985738	0.492869	−	0.870103i	$-0.335948\pi$
$101$	9.90655	0.985738	0.492869	+	0.870103i	$0.335948\pi$
$102$	0	0
$103$	8.10887	0.798991	0.399496	−	0.916735i	$-0.369185\pi$
$103$	8.10887	0.798991	0.399496	+	0.916735i	$0.369185\pi$
$104$	0	0
$105$	−0.919765	−0.0897599
$106$	0	0
$107$	3.17303	0.306749	0.153374	−	0.988168i	$-0.450986\pi$
$107$	3.17303	0.306749	0.153374	+	0.988168i	$0.450986\pi$
$108$	0	0
$109$	−0.423434	−0.0405576	−0.0202788	−	0.999794i	$-0.506455\pi$
$109$	−0.423434	−0.0405576	−0.0202788	+	0.999794i	$0.506455\pi$
$110$	0	0
$111$	1.33489	0.126702
$112$	0	0
$113$	−4.67432	−0.439723	−0.219862	−	0.975531i	$-0.570561\pi$
$113$	−4.67432	−0.439723	−0.219862	+	0.975531i	$0.570561\pi$
$114$	0	0
$115$	6.35507	0.592614
$116$	0	0
$117$	8.88442	0.821365
$118$	0	0
$119$	4.85578	0.445128
$120$	0	0
$121$	−10.6056	−0.964143
$122$	0	0
$123$	4.44264	0.400579
$124$	0	0
$125$	12.0328	1.07625
$126$	0	0
$127$	13.7602	1.22102	0.610509	−	0.792009i	$-0.290965\pi$
$127$	13.7602	1.22102	0.610509	+	0.792009i	$0.290965\pi$
$128$	0	0
$129$	4.50550	0.396687
$130$	0	0
$131$	−9.78426	−0.854855	−0.427427	−	0.904050i	$-0.640580\pi$
$131$	−9.78426	−0.854855	−0.427427	+	0.904050i	$0.640580\pi$
$132$	0	0
$133$	0.161226	0.0139801
$134$	0	0
$135$	−5.32066	−0.457929
$136$	0	0
$137$	−7.10937	−0.607395	−0.303697	−	0.952769i	$-0.598221\pi$
$137$	−7.10937	−0.607395	−0.303697	+	0.952769i	$0.598221\pi$
$138$	0	0
$139$	−15.2977	−1.29754	−0.648768	−	0.760987i	$-0.724715\pi$
$139$	−15.2977	−1.29754	−0.648768	+	0.760987i	$0.724715\pi$
$140$	0	0
$141$	4.50309	0.379228
$142$	0	0
$143$	2.00364	0.167553
$144$	0	0
$145$	2.79578	0.232177
$146$	0	0
$147$	−0.463900	−0.0382618
$148$	0	0
$149$	21.6738	1.77559	0.887794	−	0.460241i	$-0.152237\pi$
$149$	21.6738	1.77559	0.887794	+	0.460241i	$0.152237\pi$
$150$	0	0
$151$	−21.5070	−1.75022	−0.875109	−	0.483925i	$-0.839211\pi$
$151$	−21.5070	−1.75022	−0.875109	+	0.483925i	$0.839211\pi$
$152$	0	0
$153$	13.5224	1.09322
$154$	0	0
$155$	−10.5961	−0.851103
$156$	0	0
$157$	10.9266	0.872036	0.436018	−	0.899938i	$-0.356388\pi$
$157$	10.9266	0.872036	0.436018	+	0.899938i	$0.356388\pi$
$158$	0	0
$159$	−5.01504	−0.397719
$160$	0	0
$161$	3.20529	0.252612
$162$	0	0
$163$	23.9477	1.87573	0.937864	−	0.347003i	$-0.112801\pi$
$163$	23.9477	1.87573	0.937864	+	0.347003i	$0.112801\pi$
$164$	0	0
$165$	−0.577644	−0.0449695
$166$	0	0
$167$	8.57678	0.663691	0.331846	−	0.943334i	$-0.392329\pi$
$167$	8.57678	0.663691	0.331846	+	0.943334i	$0.392329\pi$
$168$	0	0
$169$	−2.82179	−0.217061
$170$	0	0
$171$	0.448981	0.0343345
$172$	0	0
$173$	7.69377	0.584947	0.292473	−	0.956274i	$-0.405522\pi$
$173$	7.69377	0.584947	0.292473	+	0.956274i	$0.405522\pi$
$174$	0	0
$175$	1.06898	0.0808069
$176$	0	0
$177$	−0.985825	−0.0740991
$178$	0	0
$179$	11.7768	0.880242	0.440121	−	0.897939i	$-0.354936\pi$
$179$	11.7768	0.880242	0.440121	+	0.897939i	$0.354936\pi$
$180$	0	0
$181$	−8.82678	−0.656090	−0.328045	−	0.944662i	$-0.606390\pi$
$181$	−8.82678	−0.656090	−0.328045	+	0.944662i	$0.606390\pi$
$182$	0	0
$183$	1.14256	0.0844605
$184$	0	0
$185$	5.70524	0.419457
$186$	0	0
$187$	3.04960	0.223008
$188$	0	0
$189$	−2.68357	−0.195201
$190$	0	0
$191$	−1.57521	−0.113979	−0.0569893	−	0.998375i	$-0.518150\pi$
$191$	−1.57521	−0.113979	−0.0569893	+	0.998375i	$0.518150\pi$
$192$	0	0
$193$	0.649369	0.0467426	0.0233713	−	0.999727i	$-0.492560\pi$
$193$	0.649369	0.0467426	0.0233713	+	0.999727i	$0.492560\pi$
$194$	0	0
$195$	−2.93436	−0.210134
$196$	0	0
$197$	1.65273	0.117752	0.0588760	−	0.998265i	$-0.481248\pi$
$197$	1.65273	0.117752	0.0588760	+	0.998265i	$0.481248\pi$
$198$	0	0
$199$	4.09013	0.289942	0.144971	−	0.989436i	$-0.453691\pi$
$199$	4.09013	0.289942	0.144971	+	0.989436i	$0.453691\pi$
$200$	0	0
$201$	5.88146	0.414846
$202$	0	0
$203$	1.41010	0.0989696
$204$	0	0
$205$	18.9876	1.32615
$206$	0	0
$207$	8.92609	0.620406
$208$	0	0
$209$	0.101255	0.00700398
$210$	0	0
$211$	−26.9295	−1.85391	−0.926953	−	0.375177i	$-0.877582\pi$
$211$	−26.9295	−1.85391	−0.926953	+	0.375177i	$0.877582\pi$
$212$	0	0
$213$	3.33245	0.228336
$214$	0	0
$215$	19.2562	1.31326
$216$	0	0
$217$	−5.34435	−0.362798
$218$	0	0
$219$	4.19379	0.283390
$220$	0	0
$221$	15.4915	1.04207
$222$	0	0
$223$	−26.0121	−1.74190	−0.870950	−	0.491372i	$-0.836496\pi$
$223$	−26.0121	−1.74190	−0.870950	+	0.491372i	$0.836496\pi$
$224$	0	0
$225$	2.97688	0.198459
$226$	0	0
$227$	−0.256838	−0.0170470	−0.00852348	−	0.999964i	$-0.502713\pi$
$227$	−0.256838	−0.0170470	−0.00852348	+	0.999964i	$0.502713\pi$
$228$	0	0
$229$	0.961093	0.0635108	0.0317554	−	0.999496i	$-0.489890\pi$
$229$	0.961093	0.0635108	0.0317554	+	0.999496i	$0.489890\pi$
$230$	0	0
$231$	−0.291345	−0.0191691
$232$	0	0
$233$	19.6676	1.28847	0.644235	−	0.764828i	$-0.277176\pi$
$233$	19.6676	1.28847	0.644235	+	0.764828i	$0.277176\pi$
$234$	0	0
$235$	19.2459	1.25547
$236$	0	0
$237$	4.44790	0.288922
$238$	0	0
$239$	−9.44846	−0.611170	−0.305585	−	0.952165i	$-0.598852\pi$
$239$	−9.44846	−0.611170	−0.305585	+	0.952165i	$0.598852\pi$
$240$	0	0
$241$	22.8233	1.47018	0.735088	−	0.677971i	$-0.237141\pi$
$241$	22.8233	1.47018	0.735088	+	0.677971i	$0.237141\pi$
$242$	0	0
$243$	−11.3488	−0.728025
$244$	0	0
$245$	−1.98268	−0.126669
$246$	0	0
$247$	0.514364	0.0327282
$248$	0	0
$249$	−3.47869	−0.220453
$250$	0	0
$251$	−17.3482	−1.09501	−0.547504	−	0.836803i	$-0.684422\pi$
$251$	−17.3482	−1.09501	−0.547504	+	0.836803i	$0.684422\pi$
$252$	0	0
$253$	2.01303	0.126558
$254$	0	0
$255$	−4.46618	−0.279683
$256$	0	0
$257$	7.71213	0.481070	0.240535	−	0.970641i	$-0.422677\pi$
$257$	7.71213	0.481070	0.240535	+	0.970641i	$0.422677\pi$
$258$	0	0
$259$	2.87754	0.178801
$260$	0	0
$261$	3.92684	0.243065
$262$	0	0
$263$	−9.24972	−0.570362	−0.285181	−	0.958474i	$-0.592054\pi$
$263$	−9.24972	−0.570362	−0.285181	+	0.958474i	$0.592054\pi$
$264$	0	0
$265$	−21.4340	−1.31668
$266$	0	0
$267$	1.15946	0.0709578
$268$	0	0
$269$	−11.8311	−0.721356	−0.360678	−	0.932690i	$-0.617455\pi$
$269$	−11.8311	−0.721356	−0.360678	+	0.932690i	$0.617455\pi$
$270$	0	0
$271$	−1.15740	−0.0703071	−0.0351535	−	0.999382i	$-0.511192\pi$
$271$	−1.15740	−0.0703071	−0.0351535	+	0.999382i	$0.511192\pi$
$272$	0	0
$273$	−1.47999	−0.0895733
$274$	0	0
$275$	0.671353	0.0404841
$276$	0	0
$277$	−23.6068	−1.41839	−0.709197	−	0.705011i	$-0.750942\pi$
$277$	−23.6068	−1.41839	−0.709197	+	0.705011i	$0.750942\pi$
$278$	0	0
$279$	−14.8829	−0.891018
$280$	0	0
$281$	−29.3656	−1.75180	−0.875902	−	0.482490i	$-0.839732\pi$
$281$	−29.3656	−1.75180	−0.875902	+	0.482490i	$0.839732\pi$
$282$	0	0
$283$	18.4557	1.09708	0.548539	−	0.836125i	$-0.315184\pi$
$283$	18.4557	1.09708	0.548539	+	0.836125i	$0.315184\pi$
$284$	0	0
$285$	−0.148290	−0.00878394
$286$	0	0
$287$	9.57673	0.565296
$288$	0	0
$289$	6.57857	0.386975
$290$	0	0
$291$	−2.73271	−0.160194
$292$	0	0
$293$	27.4972	1.60641	0.803203	−	0.595705i	$-0.203127\pi$
$293$	27.4972	1.60641	0.803203	+	0.595705i	$0.203127\pi$
$294$	0	0
$295$	−4.21336	−0.245311
$296$	0	0
$297$	−1.68537	−0.0977952
$298$	0	0
$299$	10.2259	0.591382
$300$	0	0
$301$	9.71222	0.559803
$302$	0	0
$303$	−4.59564	−0.264013
$304$	0	0
$305$	4.88324	0.279613
$306$	0	0
$307$	27.6567	1.57845	0.789226	−	0.614103i	$-0.210482\pi$
$307$	27.6567	1.57845	0.789226	+	0.614103i	$0.210482\pi$
$308$	0	0
$309$	−3.76170	−0.213996
$310$	0	0
$311$	24.5927	1.39453	0.697263	−	0.716815i	$-0.254401\pi$
$311$	24.5927	1.39453	0.697263	+	0.716815i	$0.254401\pi$
$312$	0	0
$313$	−18.0125	−1.01812	−0.509062	−	0.860730i	$-0.670008\pi$
$313$	−18.0125	−1.01812	−0.509062	+	0.860730i	$0.670008\pi$
$314$	0	0
$315$	−5.52136	−0.311094
$316$	0	0
$317$	13.7247	0.770856	0.385428	−	0.922738i	$-0.374054\pi$
$317$	13.7247	0.770856	0.385428	+	0.922738i	$0.374054\pi$
$318$	0	0
$319$	0.885591	0.0495836
$320$	0	0
$321$	−1.47197	−0.0821573
$322$	0	0
$323$	0.782877	0.0435604
$324$	0	0
$325$	3.41038	0.189174
$326$	0	0
$327$	0.196431	0.0108627
$328$	0	0
$329$	9.70703	0.535166
$330$	0	0
$331$	−2.12614	−0.116863	−0.0584317	−	0.998291i	$-0.518610\pi$
$331$	−2.12614	−0.116863	−0.0584317	+	0.998291i	$0.518610\pi$
$332$	0	0
$333$	8.01335	0.439129
$334$	0	0
$335$	25.1370	1.37338
$336$	0	0
$337$	17.7244	0.965512	0.482756	−	0.875755i	$-0.339636\pi$
$337$	17.7244	0.965512	0.482756	+	0.875755i	$0.339636\pi$
$338$	0	0
$339$	2.16842	0.117772
$340$	0	0
$341$	−3.35644	−0.181761
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.94812	−0.158721
$346$	0	0
$347$	6.95932	0.373596	0.186798	−	0.982398i	$-0.440189\pi$
$347$	6.95932	0.373596	0.186798	+	0.982398i	$0.440189\pi$
$348$	0	0
$349$	35.2104	1.88477	0.942386	−	0.334527i	$-0.108577\pi$
$349$	35.2104	1.88477	0.942386	+	0.334527i	$0.108577\pi$
$350$	0	0
$351$	−8.56146	−0.456977
$352$	0	0
$353$	−5.71306	−0.304075	−0.152038	−	0.988375i	$-0.548584\pi$
$353$	−5.71306	−0.304075	−0.152038	+	0.988375i	$0.548584\pi$
$354$	0	0
$355$	14.2427	0.755925
$356$	0	0
$357$	−2.25259	−0.119220
$358$	0	0
$359$	−25.2978	−1.33516	−0.667582	−	0.744536i	$-0.732671\pi$
$359$	−25.2978	−1.33516	−0.667582	+	0.744536i	$0.732671\pi$
$360$	0	0
$361$	−18.9740	−0.998632
$362$	0	0
$363$	4.91992	0.258229
$364$	0	0
$365$	17.9240	0.938186
$366$	0	0
$367$	13.9944	0.730499	0.365250	−	0.930910i	$-0.380984\pi$
$367$	13.9944	0.730499	0.365250	+	0.930910i	$0.380984\pi$
$368$	0	0
$369$	26.6692	1.38835
$370$	0	0
$371$	−10.8106	−0.561259
$372$	0	0
$373$	−9.11095	−0.471747	−0.235874	−	0.971784i	$-0.575795\pi$
$373$	−9.11095	−0.471747	−0.235874	+	0.971784i	$0.575795\pi$
$374$	0	0
$375$	−5.58203	−0.288255
$376$	0	0
$377$	4.49869	0.231694
$378$	0	0
$379$	−28.2898	−1.45315	−0.726576	−	0.687086i	$-0.758889\pi$
$379$	−28.2898	−1.45315	−0.726576	+	0.687086i	$0.758889\pi$
$380$	0	0
$381$	−6.38334	−0.327029
$382$	0	0
$383$	8.23256	0.420664	0.210332	−	0.977630i	$-0.432545\pi$
$383$	8.23256	0.420664	0.210332	+	0.977630i	$0.432545\pi$
$384$	0	0
$385$	−1.24519	−0.0634609
$386$	0	0
$387$	27.0466	1.37485
$388$	0	0
$389$	32.2729	1.63630	0.818150	−	0.575004i	$-0.195000\pi$
$389$	32.2729	1.63630	0.818150	+	0.575004i	$0.195000\pi$
$390$	0	0
$391$	15.5642	0.787115
$392$	0	0
$393$	4.53891	0.228958
$394$	0	0
$395$	19.0101	0.956500
$396$	0	0
$397$	28.6434	1.43757	0.718786	−	0.695231i	$-0.244698\pi$
$397$	28.6434	1.43757	0.718786	+	0.695231i	$0.244698\pi$
$398$	0	0
$399$	−0.0747927	−0.00374432
$400$	0	0
$401$	20.9844	1.04791	0.523956	−	0.851745i	$-0.324456\pi$
$401$	20.9844	1.04791	0.523956	+	0.851745i	$0.324456\pi$
$402$	0	0
$403$	−17.0502	−0.849333
$404$	0	0
$405$	−14.0958	−0.700428
$406$	0	0
$407$	1.80719	0.0895791
$408$	0	0
$409$	−22.9268	−1.13366	−0.566828	−	0.823836i	$-0.691830\pi$
$409$	−22.9268	−1.13366	−0.566828	+	0.823836i	$0.691830\pi$
$410$	0	0
$411$	3.29804	0.162680
$412$	0	0
$413$	−2.12508	−0.104568
$414$	0	0
$415$	−14.8677	−0.729828
$416$	0	0
$417$	7.09661	0.347522
$418$	0	0
$419$	−39.4377	−1.92666	−0.963329	−	0.268323i	$-0.913531\pi$
$419$	−39.4377	−1.92666	−0.963329	+	0.268323i	$0.913531\pi$
$420$	0	0
$421$	5.55909	0.270933	0.135467	−	0.990782i	$-0.456747\pi$
$421$	5.55909	0.270933	0.135467	+	0.990782i	$0.456747\pi$
$422$	0	0
$423$	27.0321	1.31435
$424$	0	0
$425$	5.19071	0.251786
$426$	0	0
$427$	2.46295	0.119190
$428$	0	0
$429$	−0.929487	−0.0448761
$430$	0	0
$431$	30.8494	1.48596	0.742981	−	0.669313i	$-0.233411\pi$
$431$	30.8494	1.48596	0.742981	+	0.669313i	$0.233411\pi$
$432$	0	0
$433$	22.5480	1.08359	0.541794	−	0.840511i	$-0.317745\pi$
$433$	22.5480	1.08359	0.541794	+	0.840511i	$0.317745\pi$
$434$	0	0
$435$	−1.29696	−0.0621845
$436$	0	0
$437$	0.516776	0.0247208
$438$	0	0
$439$	20.4983	0.978329	0.489164	−	0.872192i	$-0.337302\pi$
$439$	20.4983	0.978329	0.489164	+	0.872192i	$0.337302\pi$
$440$	0	0
$441$	−2.78480	−0.132609
$442$	0	0
$443$	−22.4578	−1.06700	−0.533502	−	0.845799i	$-0.679124\pi$
$443$	−22.4578	−1.06700	−0.533502	+	0.845799i	$0.679124\pi$
$444$	0	0
$445$	4.95547	0.234912
$446$	0	0
$447$	−10.0545	−0.475560
$448$	0	0
$449$	28.3273	1.33685	0.668423	−	0.743781i	$-0.266969\pi$
$449$	28.3273	1.33685	0.668423	+	0.743781i	$0.266969\pi$
$450$	0	0
$451$	6.01451	0.283212
$452$	0	0
$453$	9.97711	0.468766
$454$	0	0
$455$	−6.32541	−0.296540
$456$	0	0
$457$	0.979734	0.0458300	0.0229150	−	0.999737i	$-0.492705\pi$
$457$	0.979734	0.0458300	0.0229150	+	0.999737i	$0.492705\pi$
$458$	0	0
$459$	−13.0308	−0.608226
$460$	0	0
$461$	−36.4648	−1.69834	−0.849169	−	0.528122i	$-0.822896\pi$
$461$	−36.4648	−1.69834	−0.849169	+	0.528122i	$0.822896\pi$
$462$	0	0
$463$	−1.56380	−0.0726760	−0.0363380	−	0.999340i	$-0.511569\pi$
$463$	−1.56380	−0.0726760	−0.0363380	+	0.999340i	$0.511569\pi$
$464$	0	0
$465$	4.91555	0.227953
$466$	0	0
$467$	−40.5676	−1.87724	−0.938622	−	0.344947i	$-0.887897\pi$
$467$	−40.5676	−1.87724	−0.938622	+	0.344947i	$0.887897\pi$
$468$	0	0
$469$	12.6783	0.585429
$470$	0	0
$471$	−5.06884	−0.233560
$472$	0	0
$473$	6.09961	0.280460
$474$	0	0
$475$	0.172346	0.00790780
$476$	0	0
$477$	−30.1054	−1.37843
$478$	0	0
$479$	−25.8040	−1.17902	−0.589508	−	0.807762i	$-0.700678\pi$
$479$	−25.8040	−1.17902	−0.589508	+	0.807762i	$0.700678\pi$
$480$	0	0
$481$	9.18029	0.418585
$482$	0	0
$483$	−1.48693	−0.0676579
$484$	0	0
$485$	−11.6794	−0.530336
$486$	0	0
$487$	−2.56961	−0.116440	−0.0582201	−	0.998304i	$-0.518543\pi$
$487$	−2.56961	−0.116440	−0.0582201	+	0.998304i	$0.518543\pi$
$488$	0	0
$489$	−11.1093	−0.502381
$490$	0	0
$491$	−2.98579	−0.134747	−0.0673734	−	0.997728i	$-0.521462\pi$
$491$	−2.98579	−0.134747	−0.0673734	+	0.997728i	$0.521462\pi$
$492$	0	0
$493$	6.84713	0.308379
$494$	0	0
$495$	−3.46761	−0.155857
$496$	0	0
$497$	7.18356	0.322227
$498$	0	0
$499$	−7.50113	−0.335797	−0.167898	−	0.985804i	$-0.553698\pi$
$499$	−7.50113	−0.335797	−0.167898	+	0.985804i	$0.553698\pi$
$500$	0	0
$501$	−3.97877	−0.177758
$502$	0	0
$503$	−37.3432	−1.66505	−0.832525	−	0.553987i	$-0.813106\pi$
$503$	−37.3432	−1.66505	−0.832525	+	0.553987i	$0.813106\pi$
$504$	0	0
$505$	−19.6415	−0.874036
$506$	0	0
$507$	1.30903	0.0581359
$508$	0	0
$509$	30.7313	1.36214	0.681070	−	0.732218i	$-0.261515\pi$
$509$	30.7313	1.36214	0.681070	+	0.732218i	$0.261515\pi$
$510$	0	0
$511$	9.04029	0.399919
$512$	0	0
$513$	−0.432660	−0.0191024
$514$	0	0
$515$	−16.0773	−0.708451
$516$	0	0
$517$	6.09635	0.268117
$518$	0	0
$519$	−3.56914	−0.156668
$520$	0	0
$521$	4.96517	0.217528	0.108764	−	0.994068i	$-0.465311\pi$
$521$	4.96517	0.217528	0.108764	+	0.994068i	$0.465311\pi$
$522$	0	0
$523$	−1.30523	−0.0570737	−0.0285369	−	0.999593i	$-0.509085\pi$
$523$	−1.30523	−0.0570737	−0.0285369	+	0.999593i	$0.509085\pi$
$524$	0	0
$525$	−0.495897	−0.0216427
$526$	0	0
$527$	−25.9510	−1.13044
$528$	0	0
$529$	−12.7261	−0.553309
$530$	0	0
$531$	−5.91792	−0.256816
$532$	0	0
$533$	30.5529	1.32339
$534$	0	0
$535$	−6.29111	−0.271989
$536$	0	0
$537$	−5.46327	−0.235757
$538$	0	0
$539$	−0.628034	−0.0270514
$540$	0	0
$541$	−28.4295	−1.22228	−0.611139	−	0.791523i	$-0.709288\pi$
$541$	−28.4295	−1.22228	−0.611139	+	0.791523i	$0.709288\pi$
$542$	0	0
$543$	4.09474	0.175722
$544$	0	0
$545$	0.839535	0.0359617
$546$	0	0
$547$	23.6760	1.01231	0.506156	−	0.862442i	$-0.331066\pi$
$547$	23.6760	1.01231	0.506156	+	0.862442i	$0.331066\pi$
$548$	0	0
$549$	6.85880	0.292727
$550$	0	0
$551$	0.227345	0.00968521
$552$	0	0
$553$	9.58806	0.407726
$554$	0	0
$555$	−2.64666	−0.112344
$556$	0	0
$557$	−31.3625	−1.32887	−0.664436	−	0.747345i	$-0.731328\pi$
$557$	−31.3625	−1.32887	−0.664436	+	0.747345i	$0.731328\pi$
$558$	0	0
$559$	30.9852	1.31053
$560$	0	0
$561$	−1.41471	−0.0597290
$562$	0	0
$563$	34.3186	1.44636	0.723178	−	0.690662i	$-0.242681\pi$
$563$	34.3186	1.44636	0.723178	+	0.690662i	$0.242681\pi$
$564$	0	0
$565$	9.26769	0.389895
$566$	0	0
$567$	−7.10949	−0.298570
$568$	0	0
$569$	−42.8086	−1.79463	−0.897315	−	0.441392i	$-0.854485\pi$
$569$	−42.8086	−1.79463	−0.897315	+	0.441392i	$0.854485\pi$
$570$	0	0
$571$	−9.75677	−0.408308	−0.204154	−	0.978939i	$-0.565444\pi$
$571$	−9.75677	−0.408308	−0.204154	+	0.978939i	$0.565444\pi$
$572$	0	0
$573$	0.730742	0.0305272
$574$	0	0
$575$	3.42638	0.142890
$576$	0	0
$577$	−2.38332	−0.0992189	−0.0496094	−	0.998769i	$-0.515798\pi$
$577$	−2.38332	−0.0992189	−0.0496094	+	0.998769i	$0.515798\pi$
$578$	0	0
$579$	−0.301242	−0.0125192
$580$	0	0
$581$	−7.49880	−0.311103
$582$	0	0
$583$	−6.78944	−0.281190
$584$	0	0
$585$	−17.6150	−0.728290
$586$	0	0
$587$	3.83243	0.158181	0.0790906	−	0.996867i	$-0.474798\pi$
$587$	3.83243	0.158181	0.0790906	+	0.996867i	$0.474798\pi$
$588$	0	0
$589$	−0.861648	−0.0355036
$590$	0	0
$591$	−0.766700	−0.0315378
$592$	0	0
$593$	−29.3573	−1.20556	−0.602780	−	0.797907i	$-0.705940\pi$
$593$	−29.3573	−1.20556	−0.602780	+	0.797907i	$0.705940\pi$
$594$	0	0
$595$	−9.62746	−0.394687
$596$	0	0
$597$	−1.89741	−0.0776558
$598$	0	0
$599$	40.3716	1.64954	0.824770	−	0.565468i	$-0.191304\pi$
$599$	40.3716	1.64954	0.824770	+	0.565468i	$0.191304\pi$
$600$	0	0
$601$	30.9931	1.26424	0.632118	−	0.774872i	$-0.282186\pi$
$601$	30.9931	1.26424	0.632118	+	0.774872i	$0.282186\pi$
$602$	0	0
$603$	35.3065	1.43779
$604$	0	0
$605$	21.0275	0.854888
$606$	0	0
$607$	9.81876	0.398531	0.199265	−	0.979946i	$-0.436144\pi$
$607$	9.81876	0.398531	0.199265	+	0.979946i	$0.436144\pi$
$608$	0	0
$609$	−0.654145	−0.0265073
$610$	0	0
$611$	30.9686	1.25286
$612$	0	0
$613$	7.14953	0.288767	0.144383	−	0.989522i	$-0.453880\pi$
$613$	7.14953	0.288767	0.144383	+	0.989522i	$0.453880\pi$
$614$	0	0
$615$	−8.80834	−0.355187
$616$	0	0
$617$	34.8190	1.40176	0.700881	−	0.713279i	$-0.252791\pi$
$617$	34.8190	1.40176	0.700881	+	0.713279i	$0.252791\pi$
$618$	0	0
$619$	48.3268	1.94242	0.971209	−	0.238231i	$-0.0765675\pi$
$619$	48.3268	1.94242	0.971209	+	0.238231i	$0.0765675\pi$
$620$	0	0
$621$	−8.60161	−0.345171
$622$	0	0
$623$	2.49938	0.100135
$624$	0	0
$625$	−18.5124	−0.740497
$626$	0	0
$627$	−0.0469724	−0.00187590
$628$	0	0
$629$	13.9727	0.557127
$630$	0	0
$631$	1.37799	0.0548569	0.0274285	−	0.999624i	$-0.491268\pi$
$631$	1.37799	0.0548569	0.0274285	+	0.999624i	$0.491268\pi$
$632$	0	0
$633$	12.4926	0.496537
$634$	0	0
$635$	−27.2821	−1.08266
$636$	0	0
$637$	−3.19033	−0.126406
$638$	0	0
$639$	20.0048	0.791376
$640$	0	0
$641$	−1.24423	−0.0491440	−0.0245720	−	0.999698i	$-0.507822\pi$
$641$	−1.24423	−0.0491440	−0.0245720	+	0.999698i	$0.507822\pi$
$642$	0	0
$643$	−35.4647	−1.39859	−0.699296	−	0.714832i	$-0.746503\pi$
$643$	−35.4647	−1.39859	−0.699296	+	0.714832i	$0.746503\pi$
$644$	0	0
$645$	−8.93297	−0.351735
$646$	0	0
$647$	28.0343	1.10214	0.551071	−	0.834458i	$-0.314219\pi$
$647$	28.0343	1.10214	0.551071	+	0.834458i	$0.314219\pi$
$648$	0	0
$649$	−1.33462	−0.0523886
$650$	0	0
$651$	2.47924	0.0971692
$652$	0	0
$653$	26.2484	1.02718	0.513589	−	0.858036i	$-0.328316\pi$
$653$	26.2484	1.02718	0.513589	+	0.858036i	$0.328316\pi$
$654$	0	0
$655$	19.3991	0.757984
$656$	0	0
$657$	25.1754	0.982185
$658$	0	0
$659$	4.13485	0.161071	0.0805354	−	0.996752i	$-0.474337\pi$
$659$	4.13485	0.161071	0.0805354	+	0.996752i	$0.474337\pi$
$660$	0	0
$661$	−1.60137	−0.0622859	−0.0311430	−	0.999515i	$-0.509915\pi$
$661$	−1.60137	−0.0622859	−0.0311430	+	0.999515i	$0.509915\pi$
$662$	0	0
$663$	−7.18652	−0.279101
$664$	0	0
$665$	−0.319660	−0.0123959
$666$	0	0
$667$	4.51978	0.175007
$668$	0	0
$669$	12.0670	0.466538
$670$	0	0
$671$	1.54681	0.0597141
$672$	0	0
$673$	−50.3603	−1.94125	−0.970623	−	0.240607i	$-0.922654\pi$
$673$	−50.3603	−1.94125	−0.970623	+	0.240607i	$0.922654\pi$
$674$	0	0
$675$	−2.86867	−0.110415
$676$	0	0
$677$	10.4285	0.400799	0.200399	−	0.979714i	$-0.435776\pi$
$677$	10.4285	0.400799	0.200399	+	0.979714i	$0.435776\pi$
$678$	0	0
$679$	−5.89073	−0.226065
$680$	0	0
$681$	0.119147	0.00456573
$682$	0	0
$683$	−33.0316	−1.26392	−0.631960	−	0.775001i	$-0.717749\pi$
$683$	−33.0316	−1.26392	−0.631960	+	0.775001i	$0.717749\pi$
$684$	0	0
$685$	14.0956	0.538566
$686$	0	0
$687$	−0.445851	−0.0170103
$688$	0	0
$689$	−34.4895	−1.31394
$690$	0	0
$691$	17.1546	0.652593	0.326296	−	0.945267i	$-0.394199\pi$
$691$	17.1546	0.652593	0.326296	+	0.945267i	$0.394199\pi$
$692$	0	0
$693$	−1.74895	−0.0664371
$694$	0	0
$695$	30.3305	1.15050
$696$	0	0
$697$	46.5025	1.76141
$698$	0	0
$699$	−9.12382	−0.345094
$700$	0	0
$701$	6.24526	0.235880	0.117940	−	0.993021i	$-0.462371\pi$
$701$	6.24526	0.235880	0.117940	+	0.993021i	$0.462371\pi$
$702$	0	0
$703$	0.463933	0.0174976
$704$	0	0
$705$	−8.92819	−0.336255
$706$	0	0
$707$	−9.90655	−0.372574
$708$	0	0
$709$	−3.83146	−0.143893	−0.0719467	−	0.997408i	$-0.522921\pi$
$709$	−3.83146	−0.143893	−0.0719467	+	0.997408i	$0.522921\pi$
$710$	0	0
$711$	26.7008	1.00136
$712$	0	0
$713$	−17.1302	−0.641531
$714$	0	0
$715$	−3.97258	−0.148566
$716$	0	0
$717$	4.38314	0.163691
$718$	0	0
$719$	−8.99485	−0.335451	−0.167726	−	0.985834i	$-0.553642\pi$
$719$	−8.99485	−0.335451	−0.167726	+	0.985834i	$0.553642\pi$
$720$	0	0
$721$	−8.10887	−0.301990
$722$	0	0
$723$	−10.5877	−0.393761
$724$	0	0
$725$	1.50736	0.0559820
$726$	0	0
$727$	−33.6636	−1.24851	−0.624256	−	0.781219i	$-0.714598\pi$
$727$	−33.6636	−1.24851	−0.624256	+	0.781219i	$0.714598\pi$
$728$	0	0
$729$	−16.0638	−0.594954
$730$	0	0
$731$	47.1604	1.74429
$732$	0	0
$733$	−41.0889	−1.51765	−0.758827	−	0.651293i	$-0.774227\pi$
$733$	−41.0889	−1.51765	−0.758827	+	0.651293i	$0.774227\pi$
$734$	0	0
$735$	0.919765	0.0339261
$736$	0	0
$737$	7.96241	0.293299
$738$	0	0
$739$	−7.96250	−0.292905	−0.146453	−	0.989218i	$-0.546786\pi$
$739$	−7.96250	−0.292905	−0.146453	+	0.989218i	$0.546786\pi$
$740$	0	0
$741$	−0.238613	−0.00876568
$742$	0	0
$743$	−27.7688	−1.01874	−0.509369	−	0.860548i	$-0.670121\pi$
$743$	−27.7688	−1.01874	−0.509369	+	0.860548i	$0.670121\pi$
$744$	0	0
$745$	−42.9723	−1.57438
$746$	0	0
$747$	−20.8826	−0.764056
$748$	0	0
$749$	−3.17303	−0.115940
$750$	0	0
$751$	−45.5138	−1.66082	−0.830410	−	0.557153i	$-0.811894\pi$
$751$	−45.5138	−1.66082	−0.830410	+	0.557153i	$0.811894\pi$
$752$	0	0
$753$	8.04781	0.293279
$754$	0	0
$755$	42.6416	1.55189
$756$	0	0
$757$	29.3644	1.06727	0.533633	−	0.845716i	$-0.320826\pi$
$757$	29.3644	1.06727	0.533633	+	0.845716i	$0.320826\pi$
$758$	0	0
$759$	−0.933846	−0.0338965
$760$	0	0
$761$	3.64393	0.132092	0.0660461	−	0.997817i	$-0.478962\pi$
$761$	3.64393	0.132092	0.0660461	+	0.997817i	$0.478962\pi$
$762$	0	0
$763$	0.423434	0.0153293
$764$	0	0
$765$	−26.8105	−0.969336
$766$	0	0
$767$	−6.77971	−0.244801
$768$	0	0
$769$	−6.23588	−0.224872	−0.112436	−	0.993659i	$-0.535865\pi$
$769$	−6.23588	−0.224872	−0.112436	+	0.993659i	$0.535865\pi$
$770$	0	0
$771$	−3.57766	−0.128846
$772$	0	0
$773$	2.11122	0.0759352	0.0379676	−	0.999279i	$-0.487912\pi$
$773$	2.11122	0.0759352	0.0379676	+	0.999279i	$0.487912\pi$
$774$	0	0
$775$	−5.71298	−0.205216
$776$	0	0
$777$	−1.33489	−0.0478888
$778$	0	0
$779$	1.54402	0.0553201
$780$	0	0
$781$	4.51152	0.161435
$782$	0	0
$783$	−3.78410	−0.135233
$784$	0	0
$785$	−21.6639	−0.773219
$786$	0	0
$787$	12.4016	0.442071	0.221036	−	0.975266i	$-0.429056\pi$
$787$	12.4016	0.442071	0.221036	+	0.975266i	$0.429056\pi$
$788$	0	0
$789$	4.29094	0.152762
$790$	0	0
$791$	4.67432	0.166200
$792$	0	0
$793$	7.85761	0.279032
$794$	0	0
$795$	9.94323	0.352650
$796$	0	0
$797$	32.1695	1.13950	0.569751	−	0.821817i	$-0.307040\pi$
$797$	32.1695	1.13950	0.569751	+	0.821817i	$0.307040\pi$
$798$	0	0
$799$	47.1352	1.66752
$800$	0	0
$801$	6.96026	0.245929
$802$	0	0
$803$	5.67762	0.200359
$804$	0	0
$805$	−6.35507	−0.223987
$806$	0	0
$807$	5.48846	0.193203
$808$	0	0
$809$	−2.69445	−0.0947319	−0.0473660	−	0.998878i	$-0.515083\pi$
$809$	−2.69445	−0.0947319	−0.0473660	+	0.998878i	$0.515083\pi$
$810$	0	0
$811$	−43.1775	−1.51617	−0.758084	−	0.652157i	$-0.773864\pi$
$811$	−43.1775	−1.51617	−0.758084	+	0.652157i	$0.773864\pi$
$812$	0	0
$813$	0.536918	0.0188305
$814$	0	0
$815$	−47.4806	−1.66317
$816$	0	0
$817$	1.56586	0.0547826
$818$	0	0
$819$	−8.88442	−0.310447
$820$	0	0
$821$	21.8510	0.762606	0.381303	−	0.924450i	$-0.375475\pi$
$821$	21.8510	0.762606	0.381303	+	0.924450i	$0.375475\pi$
$822$	0	0
$823$	−22.4104	−0.781178	−0.390589	−	0.920565i	$-0.627729\pi$
$823$	−22.4104	−0.781178	−0.390589	+	0.920565i	$0.627729\pi$
$824$	0	0
$825$	−0.311441	−0.0108430
$826$	0	0
$827$	−22.5736	−0.784963	−0.392481	−	0.919760i	$-0.628383\pi$
$827$	−22.5736	−0.784963	−0.392481	+	0.919760i	$0.628383\pi$
$828$	0	0
$829$	−9.73194	−0.338004	−0.169002	−	0.985616i	$-0.554054\pi$
$829$	−9.73194	−0.338004	−0.169002	+	0.985616i	$0.554054\pi$
$830$	0	0
$831$	10.9512	0.379892
$832$	0	0
$833$	−4.85578	−0.168243
$834$	0	0
$835$	−17.0050	−0.588483
$836$	0	0
$837$	14.3419	0.495729
$838$	0	0
$839$	−24.1568	−0.833986	−0.416993	−	0.908910i	$-0.636916\pi$
$839$	−24.1568	−0.833986	−0.416993	+	0.908910i	$0.636916\pi$
$840$	0	0
$841$	−27.0116	−0.931435
$842$	0	0
$843$	13.6227	0.469190
$844$	0	0
$845$	5.59470	0.192464
$846$	0	0
$847$	10.6056	0.364412
$848$	0	0
$849$	−8.56160	−0.293833
$850$	0	0
$851$	9.22334	0.316172
$852$	0	0
$853$	−30.3658	−1.03971	−0.519853	−	0.854256i	$-0.674013\pi$
$853$	−30.3658	−1.03971	−0.519853	+	0.854256i	$0.674013\pi$
$854$	0	0
$855$	−0.890187	−0.0304438
$856$	0	0
$857$	3.52721	0.120487	0.0602436	−	0.998184i	$-0.480812\pi$
$857$	3.52721	0.120487	0.0602436	+	0.998184i	$0.480812\pi$
$858$	0	0
$859$	55.0973	1.87989	0.939947	−	0.341319i	$-0.110874\pi$
$859$	55.0973	1.87989	0.939947	+	0.341319i	$0.110874\pi$
$860$	0	0
$861$	−4.44264	−0.151405
$862$	0	0
$863$	18.9798	0.646081	0.323040	−	0.946385i	$-0.395295\pi$
$863$	18.9798	0.646081	0.323040	+	0.946385i	$0.395295\pi$
$864$	0	0
$865$	−15.2543	−0.518662
$866$	0	0
$867$	−3.05180	−0.103645
$868$	0	0
$869$	6.02163	0.204270
$870$	0	0
$871$	40.4480	1.37053
$872$	0	0
$873$	−16.4045	−0.555208
$874$	0	0
$875$	−12.0328	−0.406784
$876$	0	0
$877$	9.25719	0.312593	0.156297	−	0.987710i	$-0.450044\pi$
$877$	9.25719	0.312593	0.156297	+	0.987710i	$0.450044\pi$
$878$	0	0
$879$	−12.7560	−0.430248
$880$	0	0
$881$	−26.2370	−0.883946	−0.441973	−	0.897028i	$-0.645721\pi$
$881$	−26.2370	−0.883946	−0.441973	+	0.897028i	$0.645721\pi$
$882$	0	0
$883$	−27.3028	−0.918814	−0.459407	−	0.888226i	$-0.651938\pi$
$883$	−27.3028	−0.918814	−0.459407	+	0.888226i	$0.651938\pi$
$884$	0	0
$885$	1.95458	0.0657024
$886$	0	0
$887$	3.40903	0.114464	0.0572321	−	0.998361i	$-0.481773\pi$
$887$	3.40903	0.114464	0.0572321	+	0.998361i	$0.481773\pi$
$888$	0	0
$889$	−13.7602	−0.461502
$890$	0	0
$891$	−4.46500	−0.149583
$892$	0	0
$893$	1.56502	0.0523715
$894$	0	0
$895$	−23.3497	−0.780494
$896$	0	0
$897$	−4.74381	−0.158391
$898$	0	0
$899$	−7.53607	−0.251342
$900$	0	0
$901$	−52.4940	−1.74883
$902$	0	0
$903$	−4.50550	−0.149934
$904$	0	0
$905$	17.5007	0.581743
$906$	0	0
$907$	−25.3278	−0.840997	−0.420499	−	0.907293i	$-0.638145\pi$
$907$	−25.3278	−0.840997	−0.420499	+	0.907293i	$0.638145\pi$
$908$	0	0
$909$	−27.5877	−0.915027
$910$	0	0
$911$	−23.6991	−0.785186	−0.392593	−	0.919712i	$-0.628422\pi$
$911$	−23.6991	−0.785186	−0.392593	+	0.919712i	$0.628422\pi$
$912$	0	0
$913$	−4.70950	−0.155862
$914$	0	0
$915$	−2.26533	−0.0748896
$916$	0	0
$917$	9.78426	0.323105
$918$	0	0
$919$	29.7662	0.981897	0.490949	−	0.871189i	$-0.336650\pi$
$919$	29.7662	0.981897	0.490949	+	0.871189i	$0.336650\pi$
$920$	0	0
$921$	−12.8299	−0.422761
$922$	0	0
$923$	22.9179	0.754353
$924$	0	0
$925$	3.07601	0.101139
$926$	0	0
$927$	−22.5816	−0.741676
$928$	0	0
$929$	46.6355	1.53006	0.765031	−	0.643993i	$-0.222723\pi$
$929$	46.6355	1.53006	0.765031	+	0.643993i	$0.222723\pi$
$930$	0	0
$931$	−0.161226	−0.00528397
$932$	0	0
$933$	−11.4086	−0.373500
$934$	0	0
$935$	−6.04638	−0.197738
$936$	0	0
$937$	−13.5471	−0.442564	−0.221282	−	0.975210i	$-0.571024\pi$
$937$	−13.5471	−0.442564	−0.221282	+	0.975210i	$0.571024\pi$
$938$	0	0
$939$	8.35597	0.272687
$940$	0	0
$941$	−19.2235	−0.626668	−0.313334	−	0.949643i	$-0.601446\pi$
$941$	−19.2235	−0.626668	−0.313334	+	0.949643i	$0.601446\pi$
$942$	0	0
$943$	30.6962	0.999606
$944$	0	0
$945$	5.32066	0.173081
$946$	0	0
$947$	40.6982	1.32251	0.661257	−	0.750159i	$-0.270023\pi$
$947$	40.6982	1.32251	0.661257	+	0.750159i	$0.270023\pi$
$948$	0	0
$949$	28.8415	0.936236
$950$	0	0
$951$	−6.36689	−0.206460
$952$	0	0
$953$	30.3026	0.981598	0.490799	−	0.871273i	$-0.336705\pi$
$953$	30.3026	0.981598	0.490799	+	0.871273i	$0.336705\pi$
$954$	0	0
$955$	3.12315	0.101063
$956$	0	0
$957$	−0.410826	−0.0132801
$958$	0	0
$959$	7.10937	0.229574
$960$	0	0
$961$	−2.43792	−0.0786425
$962$	0	0
$963$	−8.83625	−0.284744
$964$	0	0
$965$	−1.28749	−0.0414458
$966$	0	0
$967$	−18.8733	−0.606925	−0.303462	−	0.952843i	$-0.598143\pi$
$967$	−18.8733	−0.606925	−0.303462	+	0.952843i	$0.598143\pi$
$968$	0	0
$969$	−0.363176	−0.0116669
$970$	0	0
$971$	40.4012	1.29654	0.648269	−	0.761412i	$-0.275493\pi$
$971$	40.4012	1.29654	0.648269	+	0.761412i	$0.275493\pi$
$972$	0	0
$973$	15.2977	0.490422
$974$	0	0
$975$	−1.58208	−0.0506670
$976$	0	0
$977$	−16.4828	−0.527331	−0.263666	−	0.964614i	$-0.584932\pi$
$977$	−16.4828	−0.527331	−0.263666	+	0.964614i	$0.584932\pi$
$978$	0	0
$979$	1.56970	0.0501677
$980$	0	0
$981$	1.17918	0.0376482
$982$	0	0
$983$	33.4039	1.06542	0.532710	−	0.846298i	$-0.321174\pi$
$983$	33.4039	1.06542	0.532710	+	0.846298i	$0.321174\pi$
$984$	0	0
$985$	−3.27683	−0.104408
$986$	0	0
$987$	−4.50309	−0.143335
$988$	0	0
$989$	31.1305	0.989893
$990$	0	0
$991$	25.3648	0.805740	0.402870	−	0.915257i	$-0.368013\pi$
$991$	25.3648	0.805740	0.402870	+	0.915257i	$0.368013\pi$
$992$	0	0
$993$	0.986318	0.0312998
$994$	0	0
$995$	−8.10943	−0.257086
$996$	0	0
$997$	−15.7551	−0.498968	−0.249484	−	0.968379i	$-0.580261\pi$
$997$	−15.7551	−0.498968	−0.249484	+	0.968379i	$0.580261\pi$
$998$	0	0
$999$	−7.72206	−0.244315

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.be.1.4		8
4.3	odd	2		7168.2.a.bf.1.5		8
8.3	odd	2		7168.2.a.bb.1.4		8
8.5	even	2		7168.2.a.ba.1.5		8
32.3	odd	8		1792.2.m.e.1345.5	yes	16
32.5	even	8		1792.2.m.f.449.5	yes	16
32.11	odd	8		1792.2.m.e.449.5	✓	16
32.13	even	8		1792.2.m.f.1345.5	yes	16
32.19	odd	8		1792.2.m.h.1345.4	yes	16
32.21	even	8		1792.2.m.g.449.4	yes	16
32.27	odd	8		1792.2.m.h.449.4	yes	16
32.29	even	8		1792.2.m.g.1345.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1792.2.m.e.449.5	✓	16	32.11	odd	8
1792.2.m.e.1345.5	yes	16	32.3	odd	8
1792.2.m.f.449.5	yes	16	32.5	even	8
1792.2.m.f.1345.5	yes	16	32.13	even	8
1792.2.m.g.449.4	yes	16	32.21	even	8
1792.2.m.g.1345.4	yes	16	32.29	even	8
1792.2.m.h.449.4	yes	16	32.27	odd	8
1792.2.m.h.1345.4	yes	16	32.19	odd	8
7168.2.a.ba.1.5		8	8.5	even	2
7168.2.a.bb.1.4		8	8.3	odd	2
7168.2.a.be.1.4		8	1.1	even	1	trivial
7168.2.a.bf.1.5		8	4.3	odd	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)

\(f(q)\)	\(=\)	\(q-0.463900 q^{3} -1.98268 q^{5} -1.00000 q^{7} -2.78480 q^{9} +O(q^{10})\)	\(q-0.463900 q^{3} -1.98268 q^{5} -1.00000 q^{7} -2.78480 q^{9} -0.628034 q^{11} -3.19033 q^{13} +0.919765 q^{15} -4.85578 q^{17} -0.161226 q^{19} +0.463900 q^{21} -3.20529 q^{23} -1.06898 q^{25} +2.68357 q^{27} -1.41010 q^{29} +5.34435 q^{31} +0.291345 q^{33} +1.98268 q^{35} -2.87754 q^{37} +1.47999 q^{39} -9.57673 q^{41} -9.71222 q^{43} +5.52136 q^{45} -9.70703 q^{47} +1.00000 q^{49} +2.25259 q^{51} +10.8106 q^{53} +1.24519 q^{55} +0.0747927 q^{57} +2.12508 q^{59} -2.46295 q^{61} +2.78480 q^{63} +6.32541 q^{65} -12.6783 q^{67} +1.48693 q^{69} -7.18356 q^{71} -9.04029 q^{73} +0.495897 q^{75} +0.628034 q^{77} -9.58806 q^{79} +7.10949 q^{81} +7.49880 q^{83} +9.62746 q^{85} +0.654145 q^{87} -2.49938 q^{89} +3.19033 q^{91} -2.47924 q^{93} +0.319660 q^{95} +5.89073 q^{97} +1.74895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) 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

Introduction

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