LMFDB - Embedded newform 8464.2.a.ch.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8464,2,Mod(1,8464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8464, 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("8464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.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:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 $ x^15 - x^14 - 30x^13 + 28x^12 + 354x^11 - 302x^10 - 2111x^9 + 1596x^8 + 6777x^7 - 4292x^6 - 11506x^5 + 5297x^4 + 9480x^3 - 1825x^2 - 3136*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-0.851979$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.851979 q^{3} -3.06283 q^{5} +1.27761 q^{7} -2.27413 q^{9} +O(q^{10})$	$q+0.851979 q^{3} -3.06283 q^{5} +1.27761 q^{7} -2.27413 q^{9} +0.214716 q^{11} -0.936210 q^{13} -2.60947 q^{15} +2.53327 q^{17} -4.57466 q^{19} +1.08850 q^{21} +4.38095 q^{25} -4.49345 q^{27} +2.09356 q^{29} -7.27444 q^{31} +0.182934 q^{33} -3.91312 q^{35} +10.3124 q^{37} -0.797632 q^{39} +5.32900 q^{41} +9.65778 q^{43} +6.96529 q^{45} -5.35449 q^{47} -5.36770 q^{49} +2.15829 q^{51} -14.3811 q^{53} -0.657639 q^{55} -3.89752 q^{57} -4.11009 q^{59} -7.12221 q^{61} -2.90546 q^{63} +2.86746 q^{65} +4.00341 q^{67} -7.07859 q^{71} -3.87820 q^{73} +3.73248 q^{75} +0.274324 q^{77} +13.5325 q^{79} +2.99407 q^{81} +10.4771 q^{83} -7.75898 q^{85} +1.78367 q^{87} +4.82358 q^{89} -1.19611 q^{91} -6.19767 q^{93} +14.0114 q^{95} -11.6193 q^{97} -0.488293 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * 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
$3$	0.851979	0.491890	0.245945	−	0.969284i	$-0.420902\pi$
$3$	0.851979	0.491890	0.245945	+	0.969284i	$0.420902\pi$
$4$	0	0
$5$	−3.06283	−1.36974	−0.684870	−	0.728665i	$-0.740141\pi$
$5$	−3.06283	−1.36974	−0.684870	+	0.728665i	$0.740141\pi$
$6$	0	0
$7$	1.27761	0.482892	0.241446	−	0.970414i	$-0.422378\pi$
$7$	1.27761	0.482892	0.241446	+	0.970414i	$0.422378\pi$
$8$	0	0
$9$	−2.27413	−0.758044
$10$	0	0
$11$	0.214716	0.0647393	0.0323697	−	0.999476i	$-0.489695\pi$
$11$	0.214716	0.0647393	0.0323697	+	0.999476i	$0.489695\pi$
$12$	0	0
$13$	−0.936210	−0.259658	−0.129829	−	0.991536i	$-0.541443\pi$
$13$	−0.936210	−0.259658	−0.129829	+	0.991536i	$0.541443\pi$
$14$	0	0
$15$	−2.60947	−0.673762
$16$	0	0
$17$	2.53327	0.614408	0.307204	−	0.951644i	$-0.400607\pi$
$17$	2.53327	0.614408	0.307204	+	0.951644i	$0.400607\pi$
$18$	0	0
$19$	−4.57466	−1.04950	−0.524750	−	0.851257i	$-0.675841\pi$
$19$	−4.57466	−1.04950	−0.524750	+	0.851257i	$0.675841\pi$
$20$	0	0
$21$	1.08850	0.237530
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	4.38095	0.876190
$26$	0	0
$27$	−4.49345	−0.864765
$28$	0	0
$29$	2.09356	0.388764	0.194382	−	0.980926i	$-0.437730\pi$
$29$	2.09356	0.388764	0.194382	+	0.980926i	$0.437730\pi$
$30$	0	0
$31$	−7.27444	−1.30653	−0.653264	−	0.757131i	$-0.726601\pi$
$31$	−7.27444	−1.30653	−0.653264	+	0.757131i	$0.726601\pi$
$32$	0	0
$33$	0.182934	0.0318447
$34$	0	0
$35$	−3.91312	−0.661437
$36$	0	0
$37$	10.3124	1.69535	0.847675	−	0.530516i	$-0.178002\pi$
$37$	10.3124	1.69535	0.847675	+	0.530516i	$0.178002\pi$
$38$	0	0
$39$	−0.797632	−0.127723
$40$	0	0
$41$	5.32900	0.832250	0.416125	−	0.909307i	$-0.363388\pi$
$41$	5.32900	0.832250	0.416125	+	0.909307i	$0.363388\pi$
$42$	0	0
$43$	9.65778	1.47280	0.736399	−	0.676548i	$-0.236525\pi$
$43$	9.65778	1.47280	0.736399	+	0.676548i	$0.236525\pi$
$44$	0	0
$45$	6.96529	1.03832
$46$	0	0
$47$	−5.35449	−0.781033	−0.390516	−	0.920596i	$-0.627703\pi$
$47$	−5.35449	−0.781033	−0.390516	+	0.920596i	$0.627703\pi$
$48$	0	0
$49$	−5.36770	−0.766815
$50$	0	0
$51$	2.15829	0.302221
$52$	0	0
$53$	−14.3811	−1.97540	−0.987698	−	0.156372i	$-0.950020\pi$
$53$	−14.3811	−1.97540	−0.987698	+	0.156372i	$0.950020\pi$
$54$	0	0
$55$	−0.657639	−0.0886761
$56$	0	0
$57$	−3.89752	−0.516239
$58$	0	0
$59$	−4.11009	−0.535089	−0.267544	−	0.963546i	$-0.586212\pi$
$59$	−4.11009	−0.535089	−0.267544	+	0.963546i	$0.586212\pi$
$60$	0	0
$61$	−7.12221	−0.911906	−0.455953	−	0.890004i	$-0.650701\pi$
$61$	−7.12221	−0.911906	−0.455953	+	0.890004i	$0.650701\pi$
$62$	0	0
$63$	−2.90546	−0.366054
$64$	0	0
$65$	2.86746	0.355664
$66$	0	0
$67$	4.00341	0.489095	0.244547	−	0.969637i	$-0.421361\pi$
$67$	4.00341	0.489095	0.244547	+	0.969637i	$0.421361\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.07859	−0.840074	−0.420037	−	0.907507i	$-0.637983\pi$
$71$	−7.07859	−0.840074	−0.420037	+	0.907507i	$0.637983\pi$
$72$	0	0
$73$	−3.87820	−0.453909	−0.226955	−	0.973905i	$-0.572877\pi$
$73$	−3.87820	−0.453909	−0.226955	+	0.973905i	$0.572877\pi$
$74$	0	0
$75$	3.73248	0.430989
$76$	0	0
$77$	0.274324	0.0312621
$78$	0	0
$79$	13.5325	1.52253	0.761263	−	0.648443i	$-0.224580\pi$
$79$	13.5325	1.52253	0.761263	+	0.648443i	$0.224580\pi$
$80$	0	0
$81$	2.99407	0.332674
$82$	0	0
$83$	10.4771	1.15001	0.575003	−	0.818151i	$-0.305001\pi$
$83$	10.4771	1.15001	0.575003	+	0.818151i	$0.305001\pi$
$84$	0	0
$85$	−7.75898	−0.841579
$86$	0	0
$87$	1.78367	0.191229
$88$	0	0
$89$	4.82358	0.511298	0.255649	−	0.966770i	$-0.417711\pi$
$89$	4.82358	0.511298	0.255649	+	0.966770i	$0.417711\pi$
$90$	0	0
$91$	−1.19611	−0.125387
$92$	0	0
$93$	−6.19767	−0.642668
$94$	0	0
$95$	14.0114	1.43754
$96$	0	0
$97$	−11.6193	−1.17976	−0.589882	−	0.807489i	$-0.700826\pi$
$97$	−11.6193	−1.17976	−0.589882	+	0.807489i	$0.700826\pi$
$98$	0	0
$99$	−0.488293	−0.0490752
$100$	0	0
$101$	16.6907	1.66079	0.830395	−	0.557176i	$-0.188115\pi$
$101$	16.6907	1.66079	0.830395	+	0.557176i	$0.188115\pi$
$102$	0	0
$103$	2.00502	0.197561	0.0987803	−	0.995109i	$-0.468506\pi$
$103$	2.00502	0.197561	0.0987803	+	0.995109i	$0.468506\pi$
$104$	0	0
$105$	−3.33389	−0.325355
$106$	0	0
$107$	−3.29818	−0.318847	−0.159423	−	0.987210i	$-0.550963\pi$
$107$	−3.29818	−0.318847	−0.159423	+	0.987210i	$0.550963\pi$
$108$	0	0
$109$	0.480766	0.0460491	0.0230245	−	0.999735i	$-0.492670\pi$
$109$	0.480766	0.0460491	0.0230245	+	0.999735i	$0.492670\pi$
$110$	0	0
$111$	8.78596	0.833926
$112$	0	0
$113$	−2.97077	−0.279467	−0.139733	−	0.990189i	$-0.544625\pi$
$113$	−2.97077	−0.279467	−0.139733	+	0.990189i	$0.544625\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.12907	0.196832
$118$	0	0
$119$	3.23654	0.296693
$120$	0	0
$121$	−10.9539	−0.995809
$122$	0	0
$123$	4.54020	0.409376
$124$	0	0
$125$	1.89605	0.169588
$126$	0	0
$127$	−14.5367	−1.28992	−0.644961	−	0.764215i	$-0.723127\pi$
$127$	−14.5367	−1.28992	−0.644961	+	0.764215i	$0.723127\pi$
$128$	0	0
$129$	8.22823	0.724455
$130$	0	0
$131$	18.7906	1.64174	0.820871	−	0.571114i	$-0.193489\pi$
$131$	18.7906	1.64174	0.820871	+	0.571114i	$0.193489\pi$
$132$	0	0
$133$	−5.84465	−0.506795
$134$	0	0
$135$	13.7627	1.18450
$136$	0	0
$137$	20.9794	1.79239	0.896197	−	0.443657i	$-0.146319\pi$
$137$	20.9794	1.79239	0.896197	+	0.443657i	$0.146319\pi$
$138$	0	0
$139$	13.7231	1.16398	0.581991	−	0.813195i	$-0.302274\pi$
$139$	13.7231	1.16398	0.581991	+	0.813195i	$0.302274\pi$
$140$	0	0
$141$	−4.56191	−0.384182
$142$	0	0
$143$	−0.201019	−0.0168101
$144$	0	0
$145$	−6.41222	−0.532506
$146$	0	0
$147$	−4.57317	−0.377189
$148$	0	0
$149$	9.25678	0.758345	0.379173	−	0.925326i	$-0.376209\pi$
$149$	9.25678	0.758345	0.379173	+	0.925326i	$0.376209\pi$
$150$	0	0
$151$	−8.06770	−0.656540	−0.328270	−	0.944584i	$-0.606466\pi$
$151$	−8.06770	−0.656540	−0.328270	+	0.944584i	$0.606466\pi$
$152$	0	0
$153$	−5.76098	−0.465748
$154$	0	0
$155$	22.2804	1.78960
$156$	0	0
$157$	−2.90079	−0.231508	−0.115754	−	0.993278i	$-0.536928\pi$
$157$	−2.90079	−0.231508	−0.115754	+	0.993278i	$0.536928\pi$
$158$	0	0
$159$	−12.2524	−0.971678
$160$	0	0
$161$	0	0
$162$	0	0
$163$	22.6992	1.77794	0.888970	−	0.457966i	$-0.151422\pi$
$163$	22.6992	1.77794	0.888970	+	0.457966i	$0.151422\pi$
$164$	0	0
$165$	−0.560295	−0.0436189
$166$	0	0
$167$	20.2588	1.56767	0.783837	−	0.620967i	$-0.213260\pi$
$167$	20.2588	1.56767	0.783837	+	0.620967i	$0.213260\pi$
$168$	0	0
$169$	−12.1235	−0.932578
$170$	0	0
$171$	10.4034	0.795567
$172$	0	0
$173$	8.44263	0.641881	0.320941	−	0.947099i	$-0.396001\pi$
$173$	8.44263	0.641881	0.320941	+	0.947099i	$0.396001\pi$
$174$	0	0
$175$	5.59716	0.423105
$176$	0	0
$177$	−3.50171	−0.263205
$178$	0	0
$179$	−17.9559	−1.34208	−0.671042	−	0.741420i	$-0.734153\pi$
$179$	−17.9559	−1.34208	−0.671042	+	0.741420i	$0.734153\pi$
$180$	0	0
$181$	4.40113	0.327134	0.163567	−	0.986532i	$-0.447700\pi$
$181$	4.40113	0.327134	0.163567	+	0.986532i	$0.447700\pi$
$182$	0	0
$183$	−6.06798	−0.448558
$184$	0	0
$185$	−31.5852	−2.32219
$186$	0	0
$187$	0.543933	0.0397763
$188$	0	0
$189$	−5.74089	−0.417588
$190$	0	0
$191$	15.3773	1.11266	0.556332	−	0.830960i	$-0.312208\pi$
$191$	15.3773	1.11266	0.556332	+	0.830960i	$0.312208\pi$
$192$	0	0
$193$	12.8811	0.927203	0.463602	−	0.886044i	$-0.346557\pi$
$193$	12.8811	0.927203	0.463602	+	0.886044i	$0.346557\pi$
$194$	0	0
$195$	2.44301	0.174948
$196$	0	0
$197$	2.61390	0.186233	0.0931165	−	0.995655i	$-0.470317\pi$
$197$	2.61390	0.186233	0.0931165	+	0.995655i	$0.470317\pi$
$198$	0	0
$199$	18.1986	1.29007	0.645033	−	0.764155i	$-0.276844\pi$
$199$	18.1986	1.29007	0.645033	+	0.764155i	$0.276844\pi$
$200$	0	0
$201$	3.41082	0.240581
$202$	0	0
$203$	2.67476	0.187731
$204$	0	0
$205$	−16.3218	−1.13997
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.982253	−0.0679439
$210$	0	0
$211$	−6.95046	−0.478489	−0.239245	−	0.970959i	$-0.576900\pi$
$211$	−6.95046	−0.478489	−0.239245	+	0.970959i	$0.576900\pi$
$212$	0	0
$213$	−6.03081	−0.413224
$214$	0	0
$215$	−29.5802	−2.01735
$216$	0	0
$217$	−9.29391	−0.630912
$218$	0	0
$219$	−3.30415	−0.223274
$220$	0	0
$221$	−2.37167	−0.159536
$222$	0	0
$223$	−13.9435	−0.933723	−0.466861	−	0.884331i	$-0.654615\pi$
$223$	−13.9435	−0.933723	−0.466861	+	0.884331i	$0.654615\pi$
$224$	0	0
$225$	−9.96285	−0.664190
$226$	0	0
$227$	10.7339	0.712432	0.356216	−	0.934404i	$-0.384067\pi$
$227$	10.7339	0.712432	0.356216	+	0.934404i	$0.384067\pi$
$228$	0	0
$229$	5.31107	0.350965	0.175483	−	0.984483i	$-0.443851\pi$
$229$	5.31107	0.350965	0.175483	+	0.984483i	$0.443851\pi$
$230$	0	0
$231$	0.233718	0.0153775
$232$	0	0
$233$	−5.92970	−0.388468	−0.194234	−	0.980955i	$-0.562222\pi$
$233$	−5.92970	−0.388468	−0.194234	+	0.980955i	$0.562222\pi$
$234$	0	0
$235$	16.3999	1.06981
$236$	0	0
$237$	11.5294	0.748916
$238$	0	0
$239$	−19.1959	−1.24168	−0.620838	−	0.783939i	$-0.713208\pi$
$239$	−19.1959	−1.24168	−0.620838	+	0.783939i	$0.713208\pi$
$240$	0	0
$241$	12.0896	0.778759	0.389380	−	0.921077i	$-0.372689\pi$
$241$	12.0896	0.778759	0.389380	+	0.921077i	$0.372689\pi$
$242$	0	0
$243$	16.0312	1.02840
$244$	0	0
$245$	16.4404	1.05034
$246$	0	0
$247$	4.28285	0.272511
$248$	0	0
$249$	8.92624	0.565677
$250$	0	0
$251$	−2.05454	−0.129681	−0.0648407	−	0.997896i	$-0.520654\pi$
$251$	−2.05454	−0.129681	−0.0648407	+	0.997896i	$0.520654\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−6.61049	−0.413965
$256$	0	0
$257$	10.9886	0.685450	0.342725	−	0.939436i	$-0.388650\pi$
$257$	10.9886	0.685450	0.342725	+	0.939436i	$0.388650\pi$
$258$	0	0
$259$	13.1753	0.818671
$260$	0	0
$261$	−4.76103	−0.294700
$262$	0	0
$263$	18.1706	1.12045	0.560225	−	0.828341i	$-0.310715\pi$
$263$	18.1706	1.12045	0.560225	+	0.828341i	$0.310715\pi$
$264$	0	0
$265$	44.0469	2.70578
$266$	0	0
$267$	4.10959	0.251503
$268$	0	0
$269$	−7.64841	−0.466332	−0.233166	−	0.972437i	$-0.574909\pi$
$269$	−7.64841	−0.466332	−0.233166	+	0.972437i	$0.574909\pi$
$270$	0	0
$271$	−26.2717	−1.59589	−0.797946	−	0.602729i	$-0.794080\pi$
$271$	−26.2717	−1.59589	−0.797946	+	0.602729i	$0.794080\pi$
$272$	0	0
$273$	−1.01906	−0.0616766
$274$	0	0
$275$	0.940660	0.0567239
$276$	0	0
$277$	−6.45543	−0.387869	−0.193935	−	0.981014i	$-0.562125\pi$
$277$	−6.45543	−0.387869	−0.193935	+	0.981014i	$0.562125\pi$
$278$	0	0
$279$	16.5430	0.990405
$280$	0	0
$281$	−18.8358	−1.12365	−0.561824	−	0.827257i	$-0.689900\pi$
$281$	−18.8358	−1.12365	−0.561824	+	0.827257i	$0.689900\pi$
$282$	0	0
$283$	8.54282	0.507818	0.253909	−	0.967228i	$-0.418284\pi$
$283$	8.54282	0.507818	0.253909	+	0.967228i	$0.418284\pi$
$284$	0	0
$285$	11.9374	0.707113
$286$	0	0
$287$	6.80840	0.401887
$288$	0	0
$289$	−10.5826	−0.622503
$290$	0	0
$291$	−9.89943	−0.580315
$292$	0	0
$293$	−12.0959	−0.706652	−0.353326	−	0.935500i	$-0.614949\pi$
$293$	−12.0959	−0.706652	−0.353326	+	0.935500i	$0.614949\pi$
$294$	0	0
$295$	12.5885	0.732933
$296$	0	0
$297$	−0.964816	−0.0559843
$298$	0	0
$299$	0	0
$300$	0	0
$301$	12.3389	0.711203
$302$	0	0
$303$	14.2201	0.816926
$304$	0	0
$305$	21.8142	1.24907
$306$	0	0
$307$	25.6874	1.46606	0.733028	−	0.680198i	$-0.238106\pi$
$307$	25.6874	1.46606	0.733028	+	0.680198i	$0.238106\pi$
$308$	0	0
$309$	1.70824	0.0971782
$310$	0	0
$311$	30.6251	1.73659	0.868294	−	0.496050i	$-0.165217\pi$
$311$	30.6251	1.73659	0.868294	+	0.496050i	$0.165217\pi$
$312$	0	0
$313$	26.6190	1.50459	0.752297	−	0.658825i	$-0.228946\pi$
$313$	26.6190	1.50459	0.752297	+	0.658825i	$0.228946\pi$
$314$	0	0
$315$	8.89894	0.501399
$316$	0	0
$317$	−3.26665	−0.183473	−0.0917366	−	0.995783i	$-0.529242\pi$
$317$	−3.26665	−0.183473	−0.0917366	+	0.995783i	$0.529242\pi$
$318$	0	0
$319$	0.449521	0.0251683
$320$	0	0
$321$	−2.80998	−0.156838
$322$	0	0
$323$	−11.5888	−0.644821
$324$	0	0
$325$	−4.10149	−0.227510
$326$	0	0
$327$	0.409603	0.0226511
$328$	0	0
$329$	−6.84097	−0.377155
$330$	0	0
$331$	28.0044	1.53926	0.769631	−	0.638489i	$-0.220440\pi$
$331$	28.0044	1.53926	0.769631	+	0.638489i	$0.220440\pi$
$332$	0	0
$333$	−23.4518	−1.28515
$334$	0	0
$335$	−12.2618	−0.669933
$336$	0	0
$337$	13.0357	0.710102	0.355051	−	0.934847i	$-0.384464\pi$
$337$	13.0357	0.710102	0.355051	+	0.934847i	$0.384464\pi$
$338$	0	0
$339$	−2.53104	−0.137467
$340$	0	0
$341$	−1.56194	−0.0845837
$342$	0	0
$343$	−15.8011	−0.853181
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0648	1.61396	0.806981	−	0.590578i	$-0.201100\pi$
$347$	30.0648	1.61396	0.806981	+	0.590578i	$0.201100\pi$
$348$	0	0
$349$	10.6937	0.572420	0.286210	−	0.958167i	$-0.407605\pi$
$349$	10.6937	0.572420	0.286210	+	0.958167i	$0.407605\pi$
$350$	0	0
$351$	4.20681	0.224543
$352$	0	0
$353$	32.6627	1.73846	0.869231	−	0.494407i	$-0.164615\pi$
$353$	32.6627	1.73846	0.869231	+	0.494407i	$0.164615\pi$
$354$	0	0
$355$	21.6805	1.15068
$356$	0	0
$357$	2.75746	0.145940
$358$	0	0
$359$	−13.8867	−0.732912	−0.366456	−	0.930435i	$-0.619429\pi$
$359$	−13.8867	−0.732912	−0.366456	+	0.930435i	$0.619429\pi$
$360$	0	0
$361$	1.92753	0.101449
$362$	0	0
$363$	−9.33249	−0.489829
$364$	0	0
$365$	11.8783	0.621738
$366$	0	0
$367$	−8.70583	−0.454441	−0.227220	−	0.973843i	$-0.572964\pi$
$367$	−8.70583	−0.454441	−0.227220	+	0.973843i	$0.572964\pi$
$368$	0	0
$369$	−12.1188	−0.630882
$370$	0	0
$371$	−18.3735	−0.953904
$372$	0	0
$373$	0.926810	0.0479884	0.0239942	−	0.999712i	$-0.492362\pi$
$373$	0.926810	0.0479884	0.0239942	+	0.999712i	$0.492362\pi$
$374$	0	0
$375$	1.61540	0.0834187
$376$	0	0
$377$	−1.96001	−0.100946
$378$	0	0
$379$	32.2342	1.65576	0.827879	−	0.560907i	$-0.189547\pi$
$379$	32.2342	1.65576	0.827879	+	0.560907i	$0.189547\pi$
$380$	0	0
$381$	−12.3850	−0.634501
$382$	0	0
$383$	9.96947	0.509416	0.254708	−	0.967018i	$-0.418021\pi$
$383$	9.96947	0.509416	0.254708	+	0.967018i	$0.418021\pi$
$384$	0	0
$385$	−0.840209	−0.0428210
$386$	0	0
$387$	−21.9631	−1.11645
$388$	0	0
$389$	−33.1532	−1.68094	−0.840468	−	0.541861i	$-0.817720\pi$
$389$	−33.1532	−1.68094	−0.840468	+	0.541861i	$0.817720\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	16.0092	0.807557
$394$	0	0
$395$	−41.4478	−2.08547
$396$	0	0
$397$	17.1002	0.858235	0.429117	−	0.903249i	$-0.358825\pi$
$397$	17.1002	0.858235	0.429117	+	0.903249i	$0.358825\pi$
$398$	0	0
$399$	−4.97952	−0.249288
$400$	0	0
$401$	21.7662	1.08695	0.543476	−	0.839425i	$-0.317108\pi$
$401$	21.7662	1.08695	0.543476	+	0.839425i	$0.317108\pi$
$402$	0	0
$403$	6.81040	0.339250
$404$	0	0
$405$	−9.17033	−0.455678
$406$	0	0
$407$	2.21424	0.109756
$408$	0	0
$409$	−33.8617	−1.67435	−0.837176	−	0.546934i	$-0.815795\pi$
$409$	−33.8617	−1.67435	−0.837176	+	0.546934i	$0.815795\pi$
$410$	0	0
$411$	17.8740	0.881661
$412$	0	0
$413$	−5.25111	−0.258390
$414$	0	0
$415$	−32.0895	−1.57521
$416$	0	0
$417$	11.6918	0.572552
$418$	0	0
$419$	2.38219	0.116378	0.0581888	−	0.998306i	$-0.481467\pi$
$419$	2.38219	0.116378	0.0581888	+	0.998306i	$0.481467\pi$
$420$	0	0
$421$	−25.6412	−1.24968	−0.624839	−	0.780754i	$-0.714835\pi$
$421$	−25.6412	−1.24968	−0.624839	+	0.780754i	$0.714835\pi$
$422$	0	0
$423$	12.1768	0.592057
$424$	0	0
$425$	11.0981	0.538338
$426$	0	0
$427$	−9.09943	−0.440352
$428$	0	0
$429$	−0.171264	−0.00826872
$430$	0	0
$431$	0.0332953	0.00160378	0.000801890	−	1.00000i	$-0.499745\pi$
$431$	0.0332953	0.00160378	0.000801890		1.00000i	$0.499745\pi$
$432$	0	0
$433$	1.19610	0.0574809	0.0287404	−	0.999587i	$-0.490850\pi$
$433$	1.19610	0.0574809	0.0287404	+	0.999587i	$0.490850\pi$
$434$	0	0
$435$	−5.46308	−0.261935
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−15.4367	−0.736754	−0.368377	−	0.929677i	$-0.620086\pi$
$439$	−15.4367	−0.736754	−0.368377	+	0.929677i	$0.620086\pi$
$440$	0	0
$441$	12.2069	0.581279
$442$	0	0
$443$	−7.08067	−0.336413	−0.168206	−	0.985752i	$-0.553798\pi$
$443$	−7.08067	−0.336413	−0.168206	+	0.985752i	$0.553798\pi$
$444$	0	0
$445$	−14.7738	−0.700346
$446$	0	0
$447$	7.88659	0.373023
$448$	0	0
$449$	−9.68948	−0.457275	−0.228637	−	0.973512i	$-0.573427\pi$
$449$	−9.68948	−0.457275	−0.228637	+	0.973512i	$0.573427\pi$
$450$	0	0
$451$	1.14422	0.0538793
$452$	0	0
$453$	−6.87352	−0.322946
$454$	0	0
$455$	3.66350	0.171747
$456$	0	0
$457$	−14.0201	−0.655832	−0.327916	−	0.944707i	$-0.606346\pi$
$457$	−14.0201	−0.655832	−0.327916	+	0.944707i	$0.606346\pi$
$458$	0	0
$459$	−11.3831	−0.531318
$460$	0	0
$461$	−0.854133	−0.0397809	−0.0198905	−	0.999802i	$-0.506332\pi$
$461$	−0.854133	−0.0397809	−0.0198905	+	0.999802i	$0.506332\pi$
$462$	0	0
$463$	6.95729	0.323333	0.161666	−	0.986845i	$-0.448313\pi$
$463$	6.95729	0.323333	0.161666	+	0.986845i	$0.448313\pi$
$464$	0	0
$465$	18.9824	0.880289
$466$	0	0
$467$	28.8360	1.33437	0.667185	−	0.744892i	$-0.267499\pi$
$467$	28.8360	1.33437	0.667185	+	0.744892i	$0.267499\pi$
$468$	0	0
$469$	5.11481	0.236180
$470$	0	0
$471$	−2.47141	−0.113877
$472$	0	0
$473$	2.07368	0.0953479
$474$	0	0
$475$	−20.0414	−0.919561
$476$	0	0
$477$	32.7045	1.49744
$478$	0	0
$479$	−3.66277	−0.167356	−0.0836781	−	0.996493i	$-0.526667\pi$
$479$	−3.66277	−0.167356	−0.0836781	+	0.996493i	$0.526667\pi$
$480$	0	0
$481$	−9.65458	−0.440211
$482$	0	0
$483$	0	0
$484$	0	0
$485$	35.5881	1.61597
$486$	0	0
$487$	31.0523	1.40711	0.703556	−	0.710640i	$-0.251594\pi$
$487$	31.0523	1.40711	0.703556	+	0.710640i	$0.251594\pi$
$488$	0	0
$489$	19.3393	0.874551
$490$	0	0
$491$	19.2978	0.870897	0.435449	−	0.900214i	$-0.356590\pi$
$491$	19.2978	0.870897	0.435449	+	0.900214i	$0.356590\pi$
$492$	0	0
$493$	5.30354	0.238860
$494$	0	0
$495$	1.49556	0.0672204
$496$	0	0
$497$	−9.04370	−0.405665
$498$	0	0
$499$	−29.9353	−1.34009	−0.670044	−	0.742322i	$-0.733725\pi$
$499$	−29.9353	−1.34009	−0.670044	+	0.742322i	$0.733725\pi$
$500$	0	0
$501$	17.2601	0.771123
$502$	0	0
$503$	−4.60802	−0.205461	−0.102731	−	0.994709i	$-0.532758\pi$
$503$	−4.60802	−0.205461	−0.102731	+	0.994709i	$0.532758\pi$
$504$	0	0
$505$	−51.1209	−2.27485
$506$	0	0
$507$	−10.3290	−0.458726
$508$	0	0
$509$	4.08685	0.181146	0.0905732	−	0.995890i	$-0.471130\pi$
$509$	4.08685	0.181146	0.0905732	+	0.995890i	$0.471130\pi$
$510$	0	0
$511$	−4.95484	−0.219189
$512$	0	0
$513$	20.5560	0.907570
$514$	0	0
$515$	−6.14105	−0.270607
$516$	0	0
$517$	−1.14969	−0.0505635
$518$	0	0
$519$	7.19295	0.315735
$520$	0	0
$521$	−4.46372	−0.195559	−0.0977795	−	0.995208i	$-0.531174\pi$
$521$	−4.46372	−0.195559	−0.0977795	+	0.995208i	$0.531174\pi$
$522$	0	0
$523$	14.4676	0.632622	0.316311	−	0.948655i	$-0.397556\pi$
$523$	14.4676	0.632622	0.316311	+	0.948655i	$0.397556\pi$
$524$	0	0
$525$	4.76866	0.208121
$526$	0	0
$527$	−18.4281	−0.802740
$528$	0	0
$529$	0	0
$530$	0	0
$531$	9.34690	0.405621
$532$	0	0
$533$	−4.98907	−0.216100
$534$	0	0
$535$	10.1018	0.436737
$536$	0	0
$537$	−15.2980	−0.660158
$538$	0	0
$539$	−1.15253	−0.0496431
$540$	0	0
$541$	3.23509	0.139088	0.0695438	−	0.997579i	$-0.477846\pi$
$541$	3.23509	0.139088	0.0695438	+	0.997579i	$0.477846\pi$
$542$	0	0
$543$	3.74967	0.160914
$544$	0	0
$545$	−1.47251	−0.0630753
$546$	0	0
$547$	−3.25602	−0.139217	−0.0696087	−	0.997574i	$-0.522175\pi$
$547$	−3.25602	−0.139217	−0.0696087	+	0.997574i	$0.522175\pi$
$548$	0	0
$549$	16.1968	0.691265
$550$	0	0
$551$	−9.57732	−0.408008
$552$	0	0
$553$	17.2893	0.735216
$554$	0	0
$555$	−26.9099	−1.14226
$556$	0	0
$557$	10.8424	0.459409	0.229704	−	0.973260i	$-0.426224\pi$
$557$	10.8424	0.459409	0.229704	+	0.973260i	$0.426224\pi$
$558$	0	0
$559$	−9.04171	−0.382424
$560$	0	0
$561$	0.463420	0.0195656
$562$	0	0
$563$	5.18464	0.218506	0.109253	−	0.994014i	$-0.465154\pi$
$563$	5.18464	0.218506	0.109253	+	0.994014i	$0.465154\pi$
$564$	0	0
$565$	9.09898	0.382797
$566$	0	0
$567$	3.82526	0.160646
$568$	0	0
$569$	27.1955	1.14010	0.570048	−	0.821612i	$-0.306925\pi$
$569$	27.1955	1.14010	0.570048	+	0.821612i	$0.306925\pi$
$570$	0	0
$571$	−0.394683	−0.0165170	−0.00825848	−	0.999966i	$-0.502629\pi$
$571$	−0.394683	−0.0165170	−0.00825848	+	0.999966i	$0.502629\pi$
$572$	0	0
$573$	13.1012	0.547309
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5.90886	−0.245989	−0.122995	−	0.992407i	$-0.539250\pi$
$577$	−5.90886	−0.245989	−0.122995	+	0.992407i	$0.539250\pi$
$578$	0	0
$579$	10.9744	0.456082
$580$	0	0
$581$	13.3856	0.555329
$582$	0	0
$583$	−3.08785	−0.127886
$584$	0	0
$585$	−6.52097	−0.269609
$586$	0	0
$587$	35.4408	1.46280	0.731400	−	0.681949i	$-0.238867\pi$
$587$	35.4408	1.46280	0.731400	+	0.681949i	$0.238867\pi$
$588$	0	0
$589$	33.2781	1.37120
$590$	0	0
$591$	2.22699	0.0916062
$592$	0	0
$593$	30.4720	1.25134	0.625668	−	0.780089i	$-0.284826\pi$
$593$	30.4720	1.25134	0.625668	+	0.780089i	$0.284826\pi$
$594$	0	0
$595$	−9.91297	−0.406392
$596$	0	0
$597$	15.5048	0.634571
$598$	0	0
$599$	−22.6521	−0.925539	−0.462770	−	0.886479i	$-0.653144\pi$
$599$	−22.6521	−0.925539	−0.462770	+	0.886479i	$0.653144\pi$
$600$	0	0
$601$	−14.7340	−0.601012	−0.300506	−	0.953780i	$-0.597155\pi$
$601$	−14.7340	−0.601012	−0.300506	+	0.953780i	$0.597155\pi$
$602$	0	0
$603$	−9.10429	−0.370755
$604$	0	0
$605$	33.5500	1.36400
$606$	0	0
$607$	40.1133	1.62815	0.814075	−	0.580760i	$-0.197245\pi$
$607$	40.1133	1.62815	0.814075	+	0.580760i	$0.197245\pi$
$608$	0	0
$609$	2.27884	0.0923432
$610$	0	0
$611$	5.01293	0.202801
$612$	0	0
$613$	−27.3131	−1.10317	−0.551584	−	0.834119i	$-0.685976\pi$
$613$	−27.3131	−1.10317	−0.551584	+	0.834119i	$0.685976\pi$
$614$	0	0
$615$	−13.9059	−0.560739
$616$	0	0
$617$	11.8387	0.476609	0.238305	−	0.971190i	$-0.423408\pi$
$617$	11.8387	0.476609	0.238305	+	0.971190i	$0.423408\pi$
$618$	0	0
$619$	10.0634	0.404483	0.202242	−	0.979336i	$-0.435177\pi$
$619$	10.0634	0.404483	0.202242	+	0.979336i	$0.435177\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.16267	0.246902
$624$	0	0
$625$	−27.7120	−1.10848
$626$	0	0
$627$	−0.836859	−0.0334209
$628$	0	0
$629$	26.1241	1.04164
$630$	0	0
$631$	−28.1840	−1.12199	−0.560993	−	0.827821i	$-0.689580\pi$
$631$	−28.1840	−1.12199	−0.560993	+	0.827821i	$0.689580\pi$
$632$	0	0
$633$	−5.92165	−0.235364
$634$	0	0
$635$	44.5235	1.76686
$636$	0	0
$637$	5.02530	0.199110
$638$	0	0
$639$	16.0976	0.636813
$640$	0	0
$641$	32.2777	1.27489	0.637446	−	0.770495i	$-0.279991\pi$
$641$	32.2777	1.27489	0.637446	+	0.770495i	$0.279991\pi$
$642$	0	0
$643$	37.3247	1.47194	0.735971	−	0.677013i	$-0.236726\pi$
$643$	37.3247	1.47194	0.735971	+	0.677013i	$0.236726\pi$
$644$	0	0
$645$	−25.2017	−0.992315
$646$	0	0
$647$	25.0401	0.984429	0.492214	−	0.870474i	$-0.336188\pi$
$647$	25.0401	0.984429	0.492214	+	0.870474i	$0.336188\pi$
$648$	0	0
$649$	−0.882503	−0.0346413
$650$	0	0
$651$	−7.91822	−0.310340
$652$	0	0
$653$	−13.3196	−0.521235	−0.260617	−	0.965442i	$-0.583926\pi$
$653$	−13.3196	−0.521235	−0.260617	+	0.965442i	$0.583926\pi$
$654$	0	0
$655$	−57.5524	−2.24876
$656$	0	0
$657$	8.81955	0.344083
$658$	0	0
$659$	−20.4750	−0.797594	−0.398797	−	0.917039i	$-0.630572\pi$
$659$	−20.4750	−0.797594	−0.398797	+	0.917039i	$0.630572\pi$
$660$	0	0
$661$	−20.7350	−0.806497	−0.403249	−	0.915090i	$-0.632119\pi$
$661$	−20.7350	−0.806497	−0.403249	+	0.915090i	$0.632119\pi$
$662$	0	0
$663$	−2.02061	−0.0784742
$664$	0	0
$665$	17.9012	0.694178
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−11.8795	−0.459289
$670$	0	0
$671$	−1.52925	−0.0590362
$672$	0	0
$673$	−11.8152	−0.455441	−0.227720	−	0.973727i	$-0.573127\pi$
$673$	−11.8152	−0.455441	−0.227720	+	0.973727i	$0.573127\pi$
$674$	0	0
$675$	−19.6856	−0.757698
$676$	0	0
$677$	27.5736	1.05974	0.529871	−	0.848078i	$-0.322240\pi$
$677$	27.5736	1.05974	0.529871	+	0.848078i	$0.322240\pi$
$678$	0	0
$679$	−14.8450	−0.569699
$680$	0	0
$681$	9.14504	0.350439
$682$	0	0
$683$	9.35498	0.357958	0.178979	−	0.983853i	$-0.442721\pi$
$683$	9.35498	0.357958	0.178979	+	0.983853i	$0.442721\pi$
$684$	0	0
$685$	−64.2565	−2.45511
$686$	0	0
$687$	4.52492	0.172636
$688$	0	0
$689$	13.4637	0.512927
$690$	0	0
$691$	25.0171	0.951696	0.475848	−	0.879528i	$-0.342141\pi$
$691$	25.0171	0.951696	0.475848	+	0.879528i	$0.342141\pi$
$692$	0	0
$693$	−0.623849	−0.0236981
$694$	0	0
$695$	−42.0317	−1.59435
$696$	0	0
$697$	13.4998	0.511341
$698$	0	0
$699$	−5.05198	−0.191084
$700$	0	0
$701$	−1.65840	−0.0626368	−0.0313184	−	0.999509i	$-0.509971\pi$
$701$	−1.65840	−0.0626368	−0.0313184	+	0.999509i	$0.509971\pi$
$702$	0	0
$703$	−47.1758	−1.77927
$704$	0	0
$705$	13.9724	0.526230
$706$	0	0
$707$	21.3243	0.801982
$708$	0	0
$709$	−35.3779	−1.32865	−0.664323	−	0.747446i	$-0.731280\pi$
$709$	−35.3779	−1.32865	−0.664323	+	0.747446i	$0.731280\pi$
$710$	0	0
$711$	−30.7747	−1.15414
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0.615689	0.0230255
$716$	0	0
$717$	−16.3545	−0.610769
$718$	0	0
$719$	1.71569	0.0639845	0.0319922	−	0.999488i	$-0.489815\pi$
$719$	1.71569	0.0639845	0.0319922	+	0.999488i	$0.489815\pi$
$720$	0	0
$721$	2.56164	0.0954005
$722$	0	0
$723$	10.3001	0.383064
$724$	0	0
$725$	9.17177	0.340631
$726$	0	0
$727$	18.4919	0.685826	0.342913	−	0.939367i	$-0.388586\pi$
$727$	18.4919	0.685826	0.342913	+	0.939367i	$0.388586\pi$
$728$	0	0
$729$	4.67607	0.173188
$730$	0	0
$731$	24.4657	0.904898
$732$	0	0
$733$	−20.2545	−0.748117	−0.374058	−	0.927405i	$-0.622034\pi$
$733$	−20.2545	−0.748117	−0.374058	+	0.927405i	$0.622034\pi$
$734$	0	0
$735$	14.0069	0.516651
$736$	0	0
$737$	0.859597	0.0316637
$738$	0	0
$739$	−47.1141	−1.73312	−0.866560	−	0.499073i	$-0.833674\pi$
$739$	−47.1141	−1.73312	−0.866560	+	0.499073i	$0.833674\pi$
$740$	0	0
$741$	3.64890	0.134046
$742$	0	0
$743$	−45.2537	−1.66020	−0.830099	−	0.557616i	$-0.811716\pi$
$743$	−45.2537	−1.66020	−0.830099	+	0.557616i	$0.811716\pi$
$744$	0	0
$745$	−28.3520	−1.03874
$746$	0	0
$747$	−23.8262	−0.871756
$748$	0	0
$749$	−4.21379	−0.153969
$750$	0	0
$751$	40.6924	1.48489	0.742444	−	0.669909i	$-0.233667\pi$
$751$	40.6924	1.48489	0.742444	+	0.669909i	$0.233667\pi$
$752$	0	0
$753$	−1.75043	−0.0637890
$754$	0	0
$755$	24.7100	0.899290
$756$	0	0
$757$	−14.4685	−0.525867	−0.262934	−	0.964814i	$-0.584690\pi$
$757$	−14.4685	−0.525867	−0.262934	+	0.964814i	$0.584690\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32.9896	−1.19587	−0.597935	−	0.801544i	$-0.704012\pi$
$761$	−32.9896	−1.19587	−0.597935	+	0.801544i	$0.704012\pi$
$762$	0	0
$763$	0.614233	0.0222367
$764$	0	0
$765$	17.6449	0.637954
$766$	0	0
$767$	3.84791	0.138940
$768$	0	0
$769$	49.6899	1.79187	0.895933	−	0.444190i	$-0.146508\pi$
$769$	49.6899	1.79187	0.895933	+	0.444190i	$0.146508\pi$
$770$	0	0
$771$	9.36205	0.337166
$772$	0	0
$773$	−13.6146	−0.489682	−0.244841	−	0.969563i	$-0.578736\pi$
$773$	−13.6146	−0.489682	−0.244841	+	0.969563i	$0.578736\pi$
$774$	0	0
$775$	−31.8689	−1.14477
$776$	0	0
$777$	11.2251	0.402697
$778$	0	0
$779$	−24.3784	−0.873446
$780$	0	0
$781$	−1.51989	−0.0543858
$782$	0	0
$783$	−9.40730	−0.336189
$784$	0	0
$785$	8.88463	0.317106
$786$	0	0
$787$	2.50969	0.0894607	0.0447304	−	0.998999i	$-0.485757\pi$
$787$	2.50969	0.0894607	0.0447304	+	0.998999i	$0.485757\pi$
$788$	0	0
$789$	15.4810	0.551138
$790$	0	0
$791$	−3.79550	−0.134952
$792$	0	0
$793$	6.66789	0.236784
$794$	0	0
$795$	37.5271	1.33095
$796$	0	0
$797$	−21.6351	−0.766353	−0.383177	−	0.923675i	$-0.625170\pi$
$797$	−21.6351	−0.766353	−0.383177	+	0.923675i	$0.625170\pi$
$798$	0	0
$799$	−13.5644	−0.479872
$800$	0	0
$801$	−10.9694	−0.387586
$802$	0	0
$803$	−0.832713	−0.0293858
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.51629	−0.229384
$808$	0	0
$809$	−38.6204	−1.35782	−0.678910	−	0.734222i	$-0.737547\pi$
$809$	−38.6204	−1.35782	−0.678910	+	0.734222i	$0.737547\pi$
$810$	0	0
$811$	27.4312	0.963241	0.481620	−	0.876380i	$-0.340048\pi$
$811$	27.4312	0.963241	0.481620	+	0.876380i	$0.340048\pi$
$812$	0	0
$813$	−22.3829	−0.785004
$814$	0	0
$815$	−69.5239	−2.43532
$816$	0	0
$817$	−44.1811	−1.54570
$818$	0	0
$819$	2.72012	0.0950487
$820$	0	0
$821$	18.5908	0.648825	0.324412	−	0.945916i	$-0.394833\pi$
$821$	18.5908	0.648825	0.324412	+	0.945916i	$0.394833\pi$
$822$	0	0
$823$	−37.3220	−1.30096	−0.650481	−	0.759523i	$-0.725433\pi$
$823$	−37.3220	−1.30096	−0.650481	+	0.759523i	$0.725433\pi$
$824$	0	0
$825$	0.801423	0.0279020
$826$	0	0
$827$	19.9983	0.695409	0.347704	−	0.937604i	$-0.386961\pi$
$827$	19.9983	0.695409	0.347704	+	0.937604i	$0.386961\pi$
$828$	0	0
$829$	31.7857	1.10396	0.551982	−	0.833856i	$-0.313872\pi$
$829$	31.7857	1.10396	0.551982	+	0.833856i	$0.313872\pi$
$830$	0	0
$831$	−5.49990	−0.190789
$832$	0	0
$833$	−13.5978	−0.471137
$834$	0	0
$835$	−62.0493	−2.14731
$836$	0	0
$837$	32.6873	1.12984
$838$	0	0
$839$	−21.9600	−0.758143	−0.379071	−	0.925367i	$-0.623756\pi$
$839$	−21.9600	−0.758143	−0.379071	+	0.925367i	$0.623756\pi$
$840$	0	0
$841$	−24.6170	−0.848863
$842$	0	0
$843$	−16.0477	−0.552711
$844$	0	0
$845$	37.1323	1.27739
$846$	0	0
$847$	−13.9948	−0.480868
$848$	0	0
$849$	7.27830	0.249791
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−52.2768	−1.78992	−0.894962	−	0.446143i	$-0.852797\pi$
$853$	−52.2768	−1.78992	−0.894962	+	0.446143i	$0.852797\pi$
$854$	0	0
$855$	−31.8638	−1.08972
$856$	0	0
$857$	−19.4336	−0.663839	−0.331919	−	0.943308i	$-0.607696\pi$
$857$	−19.4336	−0.663839	−0.331919	+	0.943308i	$0.607696\pi$
$858$	0	0
$859$	13.8885	0.473869	0.236935	−	0.971526i	$-0.423857\pi$
$859$	13.8885	0.473869	0.236935	+	0.971526i	$0.423857\pi$
$860$	0	0
$861$	5.80062	0.197684
$862$	0	0
$863$	21.0088	0.715147	0.357574	−	0.933885i	$-0.383604\pi$
$863$	21.0088	0.715147	0.357574	+	0.933885i	$0.383604\pi$
$864$	0	0
$865$	−25.8584	−0.879211
$866$	0	0
$867$	−9.01612	−0.306203
$868$	0	0
$869$	2.90565	0.0985673
$870$	0	0
$871$	−3.74804	−0.126997
$872$	0	0
$873$	26.4239	0.894313
$874$	0	0
$875$	2.42242	0.0818928
$876$	0	0
$877$	28.9817	0.978643	0.489321	−	0.872103i	$-0.337244\pi$
$877$	28.9817	0.978643	0.489321	+	0.872103i	$0.337244\pi$
$878$	0	0
$879$	−10.3055	−0.347595
$880$	0	0
$881$	8.42478	0.283838	0.141919	−	0.989878i	$-0.454673\pi$
$881$	8.42478	0.283838	0.141919	+	0.989878i	$0.454673\pi$
$882$	0	0
$883$	9.48055	0.319046	0.159523	−	0.987194i	$-0.449004\pi$
$883$	9.48055	0.319046	0.159523	+	0.987194i	$0.449004\pi$
$884$	0	0
$885$	10.7252	0.360523
$886$	0	0
$887$	0.0531606	0.00178496	0.000892479	−	1.00000i	$-0.499716\pi$
$887$	0.0531606	0.00178496	0.000892479		1.00000i	$0.499716\pi$
$888$	0	0
$889$	−18.5723	−0.622894
$890$	0	0
$891$	0.642875	0.0215371
$892$	0	0
$893$	24.4950	0.819693
$894$	0	0
$895$	54.9958	1.83831
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−15.2295	−0.507931
$900$	0	0
$901$	−36.4312	−1.21370
$902$	0	0
$903$	10.5125	0.349834
$904$	0	0
$905$	−13.4799	−0.448088
$906$	0	0
$907$	11.6406	0.386521	0.193260	−	0.981147i	$-0.438094\pi$
$907$	11.6406	0.386521	0.193260	+	0.981147i	$0.438094\pi$
$908$	0	0
$909$	−37.9569	−1.25895
$910$	0	0
$911$	22.1617	0.734251	0.367126	−	0.930171i	$-0.380342\pi$
$911$	22.1617	0.734251	0.367126	+	0.930171i	$0.380342\pi$
$912$	0	0
$913$	2.24959	0.0744507
$914$	0	0
$915$	18.5852	0.614408
$916$	0	0
$917$	24.0071	0.792784
$918$	0	0
$919$	6.37804	0.210392	0.105196	−	0.994451i	$-0.466453\pi$
$919$	6.37804	0.210392	0.105196	+	0.994451i	$0.466453\pi$
$920$	0	0
$921$	21.8851	0.721139
$922$	0	0
$923$	6.62705	0.218132
$924$	0	0
$925$	45.1781	1.48545
$926$	0	0
$927$	−4.55968	−0.149760
$928$	0	0
$929$	8.34940	0.273935	0.136967	−	0.990576i	$-0.456264\pi$
$929$	8.34940	0.273935	0.136967	+	0.990576i	$0.456264\pi$
$930$	0	0
$931$	24.5554	0.804772
$932$	0	0
$933$	26.0919	0.854211
$934$	0	0
$935$	−1.66598	−0.0544833
$936$	0	0
$937$	−35.5078	−1.15999	−0.579995	−	0.814620i	$-0.696946\pi$
$937$	−35.5078	−1.15999	−0.579995	+	0.814620i	$0.696946\pi$
$938$	0	0
$939$	22.6788	0.740095
$940$	0	0
$941$	41.5764	1.35535	0.677676	−	0.735361i	$-0.262987\pi$
$941$	41.5764	1.35535	0.677676	+	0.735361i	$0.262987\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	17.5834	0.571988
$946$	0	0
$947$	−14.3776	−0.467208	−0.233604	−	0.972332i	$-0.575052\pi$
$947$	−14.3776	−0.467208	−0.233604	+	0.972332i	$0.575052\pi$
$948$	0	0
$949$	3.63081	0.117861
$950$	0	0
$951$	−2.78311	−0.0902487
$952$	0	0
$953$	−13.5070	−0.437534	−0.218767	−	0.975777i	$-0.570203\pi$
$953$	−13.5070	−0.437534	−0.218767	+	0.975777i	$0.570203\pi$
$954$	0	0
$955$	−47.0982	−1.52406
$956$	0	0
$957$	0.382982	0.0123801
$958$	0	0
$959$	26.8036	0.865533
$960$	0	0
$961$	21.9174	0.707013
$962$	0	0
$963$	7.50049	0.241700
$964$	0	0
$965$	−39.4527	−1.27003
$966$	0	0
$967$	−20.6391	−0.663709	−0.331855	−	0.943331i	$-0.607674\pi$
$967$	−20.6391	−0.663709	−0.331855	+	0.943331i	$0.607674\pi$
$968$	0	0
$969$	−9.87345	−0.317181
$970$	0	0
$971$	−41.9342	−1.34573	−0.672866	−	0.739765i	$-0.734937\pi$
$971$	−41.9342	−1.34573	−0.672866	+	0.739765i	$0.734937\pi$
$972$	0	0
$973$	17.5329	0.562078
$974$	0	0
$975$	−3.49438	−0.111910
$976$	0	0
$977$	12.4018	0.396767	0.198384	−	0.980124i	$-0.436431\pi$
$977$	12.4018	0.396767	0.198384	+	0.980124i	$0.436431\pi$
$978$	0	0
$979$	1.03570	0.0331011
$980$	0	0
$981$	−1.09333	−0.0349072
$982$	0	0
$983$	−37.1960	−1.18637	−0.593185	−	0.805066i	$-0.702130\pi$
$983$	−37.1960	−1.18637	−0.593185	+	0.805066i	$0.702130\pi$
$984$	0	0
$985$	−8.00595	−0.255091
$986$	0	0
$987$	−5.82836	−0.185519
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−19.2652	−0.611980	−0.305990	−	0.952035i	$-0.598987\pi$
$991$	−19.2652	−0.611980	−0.305990	+	0.952035i	$0.598987\pi$
$992$	0	0
$993$	23.8592	0.757148
$994$	0	0
$995$	−55.7394	−1.76706
$996$	0	0
$997$	−17.2126	−0.545129	−0.272565	−	0.962137i	$-0.587872\pi$
$997$	−17.2126	−0.545129	−0.272565	+	0.962137i	$0.587872\pi$
$998$	0	0
$999$	−46.3383	−1.46608

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.ch.1.10		15
4.3	odd	2		4232.2.a.ba.1.6		15
23.11	odd	22		368.2.m.e.305.2		30
23.21	odd	22		368.2.m.e.257.2		30
23.22	odd	2		8464.2.a.cg.1.10		15
92.11	even	22		184.2.i.b.121.2	yes	30
92.67	even	22		184.2.i.b.73.2	✓	30
92.91	even	2		4232.2.a.bb.1.6		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.73.2	✓	30	92.67	even	22
184.2.i.b.121.2	yes	30	92.11	even	22
368.2.m.e.257.2		30	23.21	odd	22
368.2.m.e.305.2		30	23.11	odd	22
4232.2.a.ba.1.6		15	4.3	odd	2
4232.2.a.bb.1.6		15	92.91	even	2
8464.2.a.cg.1.10		15	23.22	odd	2
8464.2.a.ch.1.10		15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+0.851979 q^{3} -3.06283 q^{5} +1.27761 q^{7} -2.27413 q^{9} +O(q^{10})\)	\(q+0.851979 q^{3} -3.06283 q^{5} +1.27761 q^{7} -2.27413 q^{9} +0.214716 q^{11} -0.936210 q^{13} -2.60947 q^{15} +2.53327 q^{17} -4.57466 q^{19} +1.08850 q^{21} +4.38095 q^{25} -4.49345 q^{27} +2.09356 q^{29} -7.27444 q^{31} +0.182934 q^{33} -3.91312 q^{35} +10.3124 q^{37} -0.797632 q^{39} +5.32900 q^{41} +9.65778 q^{43} +6.96529 q^{45} -5.35449 q^{47} -5.36770 q^{49} +2.15829 q^{51} -14.3811 q^{53} -0.657639 q^{55} -3.89752 q^{57} -4.11009 q^{59} -7.12221 q^{61} -2.90546 q^{63} +2.86746 q^{65} +4.00341 q^{67} -7.07859 q^{71} -3.87820 q^{73} +3.73248 q^{75} +0.274324 q^{77} +13.5325 q^{79} +2.99407 q^{81} +10.4771 q^{83} -7.75898 q^{85} +1.78367 q^{87} +4.82358 q^{89} -1.19611 q^{91} -6.19767 q^{93} +14.0114 q^{95} -11.6193 q^{97} -0.488293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * q^99

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