LMFDB - Embedded newform 8464.2.a.ch.1.3

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			1.3
Root			$2.49141$ 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)

$f(q)$	$=$	$q-2.49141 q^{3} -2.70623 q^{5} -3.48198 q^{7} +3.20713 q^{9} +O(q^{10})$	$q-2.49141 q^{3} -2.70623 q^{5} -3.48198 q^{7} +3.20713 q^{9} +4.95599 q^{11} -1.54369 q^{13} +6.74234 q^{15} +1.95771 q^{17} +4.64029 q^{19} +8.67504 q^{21} +2.32370 q^{25} -0.516035 q^{27} +0.0794550 q^{29} +6.97280 q^{31} -12.3474 q^{33} +9.42305 q^{35} -11.5259 q^{37} +3.84596 q^{39} +8.96232 q^{41} +9.79502 q^{43} -8.67923 q^{45} -7.58145 q^{47} +5.12418 q^{49} -4.87745 q^{51} -9.11026 q^{53} -13.4121 q^{55} -11.5609 q^{57} -8.19215 q^{59} +8.12359 q^{61} -11.1671 q^{63} +4.17758 q^{65} +3.24911 q^{67} +4.39028 q^{71} -14.8871 q^{73} -5.78929 q^{75} -17.2567 q^{77} -5.71432 q^{79} -8.33572 q^{81} +0.464051 q^{83} -5.29801 q^{85} -0.197955 q^{87} +1.93417 q^{89} +5.37509 q^{91} -17.3721 q^{93} -12.5577 q^{95} +1.69399 q^{97} +15.8945 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.49141	−1.43842	−0.719208	−	0.694795i	$-0.755495\pi$
$3$	−2.49141	−1.43842	−0.719208	+	0.694795i	$0.755495\pi$
$4$	0	0
$5$	−2.70623	−1.21026	−0.605132	−	0.796125i	$-0.706880\pi$
$5$	−2.70623	−1.21026	−0.605132	+	0.796125i	$0.706880\pi$
$6$	0	0
$7$	−3.48198	−1.31606	−0.658032	−	0.752990i	$-0.728611\pi$
$7$	−3.48198	−1.31606	−0.658032	+	0.752990i	$0.728611\pi$
$8$	0	0
$9$	3.20713	1.06904
$10$	0	0
$11$	4.95599	1.49429	0.747144	−	0.664662i	$-0.231425\pi$
$11$	4.95599	1.49429	0.747144	+	0.664662i	$0.231425\pi$
$12$	0	0
$13$	−1.54369	−0.428142	−0.214071	−	0.976818i	$-0.568672\pi$
$13$	−1.54369	−0.428142	−0.214071	+	0.976818i	$0.568672\pi$
$14$	0	0
$15$	6.74234	1.74086
$16$	0	0
$17$	1.95771	0.474813	0.237407	−	0.971410i	$-0.423703\pi$
$17$	1.95771	0.474813	0.237407	+	0.971410i	$0.423703\pi$
$18$	0	0
$19$	4.64029	1.06455	0.532277	−	0.846570i	$-0.321336\pi$
$19$	4.64029	1.06455	0.532277	+	0.846570i	$0.321336\pi$
$20$	0	0
$21$	8.67504	1.89305
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	2.32370	0.464740
$26$	0	0
$27$	−0.516035	−0.0993111
$28$	0	0
$29$	0.0794550	0.0147544	0.00737722	−	0.999973i	$-0.497652\pi$
$29$	0.0794550	0.0147544	0.00737722	+	0.999973i	$0.497652\pi$
$30$	0	0
$31$	6.97280	1.25235	0.626176	−	0.779682i	$-0.284619\pi$
$31$	6.97280	1.25235	0.626176	+	0.779682i	$0.284619\pi$
$32$	0	0
$33$	−12.3474	−2.14941
$34$	0	0
$35$	9.42305	1.59279
$36$	0	0
$37$	−11.5259	−1.89484	−0.947420	−	0.319993i	$-0.896319\pi$
$37$	−11.5259	−1.89484	−0.947420	+	0.319993i	$0.896319\pi$
$38$	0	0
$39$	3.84596	0.615846
$40$	0	0
$41$	8.96232	1.39968	0.699840	−	0.714300i	$-0.253255\pi$
$41$	8.96232	1.39968	0.699840	+	0.714300i	$0.253255\pi$
$42$	0	0
$43$	9.79502	1.49373	0.746863	−	0.664977i	$-0.231559\pi$
$43$	9.79502	1.49373	0.746863	+	0.664977i	$0.231559\pi$
$44$	0	0
$45$	−8.67923	−1.29382
$46$	0	0
$47$	−7.58145	−1.10587	−0.552934	−	0.833225i	$-0.686492\pi$
$47$	−7.58145	−1.10587	−0.552934	+	0.833225i	$0.686492\pi$
$48$	0	0
$49$	5.12418	0.732026
$50$	0	0
$51$	−4.87745	−0.682979
$52$	0	0
$53$	−9.11026	−1.25139	−0.625695	−	0.780068i	$-0.715185\pi$
$53$	−9.11026	−1.25139	−0.625695	+	0.780068i	$0.715185\pi$
$54$	0	0
$55$	−13.4121	−1.80848
$56$	0	0
$57$	−11.5609	−1.53127
$58$	0	0
$59$	−8.19215	−1.06653	−0.533263	−	0.845949i	$-0.679035\pi$
$59$	−8.19215	−1.06653	−0.533263	+	0.845949i	$0.679035\pi$
$60$	0	0
$61$	8.12359	1.04012	0.520060	−	0.854130i	$-0.325910\pi$
$61$	8.12359	1.04012	0.520060	+	0.854130i	$0.325910\pi$
$62$	0	0
$63$	−11.1671	−1.40693
$64$	0	0
$65$	4.17758	0.518165
$66$	0	0
$67$	3.24911	0.396942	0.198471	−	0.980107i	$-0.436403\pi$
$67$	3.24911	0.396942	0.198471	+	0.980107i	$0.436403\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.39028	0.521030	0.260515	−	0.965470i	$-0.416108\pi$
$71$	4.39028	0.521030	0.260515	+	0.965470i	$0.416108\pi$
$72$	0	0
$73$	−14.8871	−1.74240	−0.871201	−	0.490927i	$-0.836658\pi$
$73$	−14.8871	−1.74240	−0.871201	+	0.490927i	$0.836658\pi$
$74$	0	0
$75$	−5.78929	−0.668489
$76$	0	0
$77$	−17.2567	−1.96658
$78$	0	0
$79$	−5.71432	−0.642911	−0.321456	−	0.946925i	$-0.604172\pi$
$79$	−5.71432	−0.642911	−0.321456	+	0.946925i	$0.604172\pi$
$80$	0	0
$81$	−8.33572	−0.926191
$82$	0	0
$83$	0.464051	0.0509363	0.0254681	−	0.999676i	$-0.491892\pi$
$83$	0.464051	0.0509363	0.0254681	+	0.999676i	$0.491892\pi$
$84$	0	0
$85$	−5.29801	−0.574650
$86$	0	0
$87$	−0.197955	−0.0212230
$88$	0	0
$89$	1.93417	0.205022	0.102511	−	0.994732i	$-0.467312\pi$
$89$	1.93417	0.205022	0.102511	+	0.994732i	$0.467312\pi$
$90$	0	0
$91$	5.37509	0.563462
$92$	0	0
$93$	−17.3721	−1.80140
$94$	0	0
$95$	−12.5577	−1.28839
$96$	0	0
$97$	1.69399	0.171999	0.0859993	−	0.996295i	$-0.472592\pi$
$97$	1.69399	0.171999	0.0859993	+	0.996295i	$0.472592\pi$
$98$	0	0
$99$	15.8945	1.59746
$100$	0	0
$101$	3.90709	0.388770	0.194385	−	0.980925i	$-0.437729\pi$
$101$	3.90709	0.388770	0.194385	+	0.980925i	$0.437729\pi$
$102$	0	0
$103$	−4.92082	−0.484863	−0.242431	−	0.970169i	$-0.577945\pi$
$103$	−4.92082	−0.484863	−0.242431	+	0.970169i	$0.577945\pi$
$104$	0	0
$105$	−23.4767	−2.29109
$106$	0	0
$107$	11.6698	1.12816	0.564081	−	0.825719i	$-0.309231\pi$
$107$	11.6698	1.12816	0.564081	+	0.825719i	$0.309231\pi$
$108$	0	0
$109$	0.915561	0.0876949	0.0438474	−	0.999038i	$-0.486038\pi$
$109$	0.915561	0.0876949	0.0438474	+	0.999038i	$0.486038\pi$
$110$	0	0
$111$	28.7156	2.72557
$112$	0	0
$113$	−16.8474	−1.58487	−0.792434	−	0.609958i	$-0.791186\pi$
$113$	−16.8474	−1.58487	−0.792434	+	0.609958i	$0.791186\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.95080	−0.457702
$118$	0	0
$119$	−6.81669	−0.624885
$120$	0	0
$121$	13.5619	1.23290
$122$	0	0
$123$	−22.3288	−2.01332
$124$	0	0
$125$	7.24270	0.647806
$126$	0	0
$127$	0.966172	0.0857339	0.0428670	−	0.999081i	$-0.486351\pi$
$127$	0.966172	0.0857339	0.0428670	+	0.999081i	$0.486351\pi$
$128$	0	0
$129$	−24.4034	−2.14860
$130$	0	0
$131$	−0.870410	−0.0760480	−0.0380240	−	0.999277i	$-0.512106\pi$
$131$	−0.870410	−0.0760480	−0.0380240	+	0.999277i	$0.512106\pi$
$132$	0	0
$133$	−16.1574	−1.40102
$134$	0	0
$135$	1.39651	0.120193
$136$	0	0
$137$	3.08371	0.263459	0.131730	−	0.991286i	$-0.457947\pi$
$137$	3.08371	0.263459	0.131730	+	0.991286i	$0.457947\pi$
$138$	0	0
$139$	−4.17897	−0.354456	−0.177228	−	0.984170i	$-0.556713\pi$
$139$	−4.17897	−0.354456	−0.177228	+	0.984170i	$0.556713\pi$
$140$	0	0
$141$	18.8885	1.59070
$142$	0	0
$143$	−7.65050	−0.639767
$144$	0	0
$145$	−0.215024	−0.0178568
$146$	0	0
$147$	−12.7664	−1.05296
$148$	0	0
$149$	11.1308	0.911873	0.455937	−	0.890012i	$-0.349304\pi$
$149$	11.1308	0.911873	0.455937	+	0.890012i	$0.349304\pi$
$150$	0	0
$151$	−1.14409	−0.0931046	−0.0465523	−	0.998916i	$-0.514823\pi$
$151$	−1.14409	−0.0931046	−0.0465523	+	0.998916i	$0.514823\pi$
$152$	0	0
$153$	6.27861	0.507595
$154$	0	0
$155$	−18.8700	−1.51568
$156$	0	0
$157$	0.947415	0.0756119	0.0378060	−	0.999285i	$-0.487963\pi$
$157$	0.947415	0.0756119	0.0378060	+	0.999285i	$0.487963\pi$
$158$	0	0
$159$	22.6974	1.80002
$160$	0	0
$161$	0	0
$162$	0	0
$163$	22.9131	1.79469	0.897347	−	0.441326i	$-0.145492\pi$
$163$	22.9131	1.79469	0.897347	+	0.441326i	$0.145492\pi$
$164$	0	0
$165$	33.4150	2.60135
$166$	0	0
$167$	−8.85465	−0.685193	−0.342597	−	0.939483i	$-0.611306\pi$
$167$	−8.85465	−0.685193	−0.342597	+	0.939483i	$0.611306\pi$
$168$	0	0
$169$	−10.6170	−0.816695
$170$	0	0
$171$	14.8820	1.13805
$172$	0	0
$173$	−15.2019	−1.15578	−0.577889	−	0.816115i	$-0.696123\pi$
$173$	−15.2019	−1.15578	−0.577889	+	0.816115i	$0.696123\pi$
$174$	0	0
$175$	−8.09107	−0.611627
$176$	0	0
$177$	20.4100	1.53411
$178$	0	0
$179$	−22.3205	−1.66832	−0.834158	−	0.551526i	$-0.814046\pi$
$179$	−22.3205	−1.66832	−0.834158	+	0.551526i	$0.814046\pi$
$180$	0	0
$181$	13.3445	0.991890	0.495945	−	0.868354i	$-0.334822\pi$
$181$	13.3445	0.991890	0.495945	+	0.868354i	$0.334822\pi$
$182$	0	0
$183$	−20.2392	−1.49612
$184$	0	0
$185$	31.1917	2.29326
$186$	0	0
$187$	9.70237	0.709508
$188$	0	0
$189$	1.79682	0.130700
$190$	0	0
$191$	−11.5669	−0.836954	−0.418477	−	0.908227i	$-0.637436\pi$
$191$	−11.5669	−0.836954	−0.418477	+	0.908227i	$0.637436\pi$
$192$	0	0
$193$	−14.2980	−1.02919	−0.514596	−	0.857433i	$-0.672058\pi$
$193$	−14.2980	−1.02919	−0.514596	+	0.857433i	$0.672058\pi$
$194$	0	0
$195$	−10.4081	−0.745337
$196$	0	0
$197$	−14.1980	−1.01157	−0.505784	−	0.862660i	$-0.668797\pi$
$197$	−14.1980	−1.01157	−0.505784	+	0.862660i	$0.668797\pi$
$198$	0	0
$199$	−6.65497	−0.471758	−0.235879	−	0.971782i	$-0.575797\pi$
$199$	−6.65497	−0.471758	−0.235879	+	0.971782i	$0.575797\pi$
$200$	0	0
$201$	−8.09486	−0.570967
$202$	0	0
$203$	−0.276661	−0.0194178
$204$	0	0
$205$	−24.2541	−1.69398
$206$	0	0
$207$	0	0
$208$	0	0
$209$	22.9972	1.59075
$210$	0	0
$211$	16.5370	1.13846	0.569228	−	0.822180i	$-0.307242\pi$
$211$	16.5370	1.13846	0.569228	+	0.822180i	$0.307242\pi$
$212$	0	0
$213$	−10.9380	−0.749458
$214$	0	0
$215$	−26.5076	−1.80780
$216$	0	0
$217$	−24.2791	−1.64818
$218$	0	0
$219$	37.0898	2.50630
$220$	0	0
$221$	−3.02209	−0.203287
$222$	0	0
$223$	−4.01926	−0.269150	−0.134575	−	0.990903i	$-0.542967\pi$
$223$	−4.01926	−0.269150	−0.134575	+	0.990903i	$0.542967\pi$
$224$	0	0
$225$	7.45239	0.496826
$226$	0	0
$227$	13.6425	0.905485	0.452743	−	0.891641i	$-0.350446\pi$
$227$	13.6425	0.905485	0.452743	+	0.891641i	$0.350446\pi$
$228$	0	0
$229$	−2.39057	−0.157973	−0.0789867	−	0.996876i	$-0.525168\pi$
$229$	−2.39057	−0.157973	−0.0789867	+	0.996876i	$0.525168\pi$
$230$	0	0
$231$	42.9934	2.82876
$232$	0	0
$233$	1.72583	0.113063	0.0565313	−	0.998401i	$-0.481996\pi$
$233$	1.72583	0.113063	0.0565313	+	0.998401i	$0.481996\pi$
$234$	0	0
$235$	20.5172	1.33839
$236$	0	0
$237$	14.2367	0.924774
$238$	0	0
$239$	10.2434	0.662594	0.331297	−	0.943527i	$-0.392514\pi$
$239$	10.2434	0.662594	0.331297	+	0.943527i	$0.392514\pi$
$240$	0	0
$241$	5.72212	0.368594	0.184297	−	0.982871i	$-0.440999\pi$
$241$	5.72212	0.368594	0.184297	+	0.982871i	$0.440999\pi$
$242$	0	0
$243$	22.3158	1.43156
$244$	0	0
$245$	−13.8672	−0.885945
$246$	0	0
$247$	−7.16315	−0.455781
$248$	0	0
$249$	−1.15614	−0.0732676
$250$	0	0
$251$	5.68716	0.358970	0.179485	−	0.983761i	$-0.442557\pi$
$251$	5.68716	0.358970	0.179485	+	0.983761i	$0.442557\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	13.1995	0.826586
$256$	0	0
$257$	18.1195	1.13026	0.565131	−	0.825001i	$-0.308826\pi$
$257$	18.1195	1.13026	0.565131	+	0.825001i	$0.308826\pi$
$258$	0	0
$259$	40.1328	2.49373
$260$	0	0
$261$	0.254822	0.0157731
$262$	0	0
$263$	25.3662	1.56415	0.782074	−	0.623186i	$-0.214162\pi$
$263$	25.3662	1.56415	0.782074	+	0.623186i	$0.214162\pi$
$264$	0	0
$265$	24.6545	1.51451
$266$	0	0
$267$	−4.81881	−0.294907
$268$	0	0
$269$	2.64267	0.161126	0.0805631	−	0.996750i	$-0.474328\pi$
$269$	2.64267	0.161126	0.0805631	+	0.996750i	$0.474328\pi$
$270$	0	0
$271$	−8.46475	−0.514197	−0.257098	−	0.966385i	$-0.582766\pi$
$271$	−8.46475	−0.514197	−0.257098	+	0.966385i	$0.582766\pi$
$272$	0	0
$273$	−13.3915	−0.810493
$274$	0	0
$275$	11.5162	0.694455
$276$	0	0
$277$	−14.2482	−0.856089	−0.428044	−	0.903758i	$-0.640797\pi$
$277$	−14.2482	−0.856089	−0.428044	+	0.903758i	$0.640797\pi$
$278$	0	0
$279$	22.3626	1.33882
$280$	0	0
$281$	−3.74537	−0.223430	−0.111715	−	0.993740i	$-0.535634\pi$
$281$	−3.74537	−0.223430	−0.111715	+	0.993740i	$0.535634\pi$
$282$	0	0
$283$	−10.1494	−0.603319	−0.301659	−	0.953416i	$-0.597541\pi$
$283$	−10.1494	−0.603319	−0.301659	+	0.953416i	$0.597541\pi$
$284$	0	0
$285$	31.2864	1.85325
$286$	0	0
$287$	−31.2066	−1.84207
$288$	0	0
$289$	−13.1674	−0.774552
$290$	0	0
$291$	−4.22042	−0.247405
$292$	0	0
$293$	24.7698	1.44707	0.723533	−	0.690290i	$-0.242517\pi$
$293$	24.7698	1.44707	0.723533	+	0.690290i	$0.242517\pi$
$294$	0	0
$295$	22.1699	1.29078
$296$	0	0
$297$	−2.55747	−0.148399
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−34.1061	−1.96584
$302$	0	0
$303$	−9.73417	−0.559214
$304$	0	0
$305$	−21.9843	−1.25882
$306$	0	0
$307$	30.2985	1.72923	0.864614	−	0.502436i	$-0.167563\pi$
$307$	30.2985	1.72923	0.864614	+	0.502436i	$0.167563\pi$
$308$	0	0
$309$	12.2598	0.697434
$310$	0	0
$311$	−19.1345	−1.08502	−0.542510	−	0.840049i	$-0.682526\pi$
$311$	−19.1345	−1.08502	−0.542510	+	0.840049i	$0.682526\pi$
$312$	0	0
$313$	33.6423	1.90157	0.950787	−	0.309845i	$-0.100277\pi$
$313$	33.6423	1.90157	0.950787	+	0.309845i	$0.100277\pi$
$314$	0	0
$315$	30.2209	1.70275
$316$	0	0
$317$	1.34405	0.0754894	0.0377447	−	0.999287i	$-0.487983\pi$
$317$	1.34405	0.0754894	0.0377447	+	0.999287i	$0.487983\pi$
$318$	0	0
$319$	0.393779	0.0220474
$320$	0	0
$321$	−29.0743	−1.62277
$322$	0	0
$323$	9.08432	0.505465
$324$	0	0
$325$	−3.58706	−0.198975
$326$	0	0
$327$	−2.28104	−0.126142
$328$	0	0
$329$	26.3985	1.45539
$330$	0	0
$331$	−0.00328224	−0.000180408 0	−9.02040e−5	−	1.00000i	$-0.500029\pi$
$331$	−0.00328224	−0.000180408 0	−9.02040e−5		1.00000i	$0.500029\pi$
$332$	0	0
$333$	−36.9649	−2.02566
$334$	0	0
$335$	−8.79284	−0.480404
$336$	0	0
$337$	7.07638	0.385475	0.192738	−	0.981250i	$-0.438263\pi$
$337$	7.07638	0.385475	0.192738	+	0.981250i	$0.438263\pi$
$338$	0	0
$339$	41.9737	2.27970
$340$	0	0
$341$	34.5571	1.87137
$342$	0	0
$343$	6.53156	0.352671
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.0623	−0.969635	−0.484818	−	0.874615i	$-0.661114\pi$
$347$	−18.0623	−0.969635	−0.484818	+	0.874615i	$0.661114\pi$
$348$	0	0
$349$	14.0442	0.751768	0.375884	−	0.926667i	$-0.377339\pi$
$349$	14.0442	0.751768	0.375884	+	0.926667i	$0.377339\pi$
$350$	0	0
$351$	0.796597	0.0425192
$352$	0	0
$353$	−7.42191	−0.395028	−0.197514	−	0.980300i	$-0.563287\pi$
$353$	−7.42191	−0.395028	−0.197514	+	0.980300i	$0.563287\pi$
$354$	0	0
$355$	−11.8811	−0.630584
$356$	0	0
$357$	16.9832	0.898845
$358$	0	0
$359$	6.77100	0.357360	0.178680	−	0.983907i	$-0.442817\pi$
$359$	6.77100	0.357360	0.178680	+	0.983907i	$0.442817\pi$
$360$	0	0
$361$	2.53227	0.133277
$362$	0	0
$363$	−33.7881	−1.77342
$364$	0	0
$365$	40.2879	2.10877
$366$	0	0
$367$	2.23720	0.116781	0.0583905	−	0.998294i	$-0.481403\pi$
$367$	2.23720	0.116781	0.0583905	+	0.998294i	$0.481403\pi$
$368$	0	0
$369$	28.7433	1.49632
$370$	0	0
$371$	31.7217	1.64691
$372$	0	0
$373$	15.8241	0.819340	0.409670	−	0.912234i	$-0.365644\pi$
$373$	15.8241	0.819340	0.409670	+	0.912234i	$0.365644\pi$
$374$	0	0
$375$	−18.0445	−0.931815
$376$	0	0
$377$	−0.122654	−0.00631699
$378$	0	0
$379$	−9.80855	−0.503832	−0.251916	−	0.967749i	$-0.581061\pi$
$379$	−9.80855	−0.503832	−0.251916	+	0.967749i	$0.581061\pi$
$380$	0	0
$381$	−2.40713	−0.123321
$382$	0	0
$383$	18.5503	0.947877	0.473938	−	0.880558i	$-0.342832\pi$
$383$	18.5503	0.947877	0.473938	+	0.880558i	$0.342832\pi$
$384$	0	0
$385$	46.7006	2.38008
$386$	0	0
$387$	31.4139	1.59686
$388$	0	0
$389$	−0.417705	−0.0211785	−0.0105892	−	0.999944i	$-0.503371\pi$
$389$	−0.417705	−0.0211785	−0.0105892	+	0.999944i	$0.503371\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	2.16855	0.109389
$394$	0	0
$395$	15.4643	0.778092
$396$	0	0
$397$	29.5139	1.48126	0.740629	−	0.671914i	$-0.234528\pi$
$397$	29.5139	1.48126	0.740629	+	0.671914i	$0.234528\pi$
$398$	0	0
$399$	40.2547	2.01525
$400$	0	0
$401$	−9.93783	−0.496271	−0.248136	−	0.968725i	$-0.579818\pi$
$401$	−9.93783	−0.496271	−0.248136	+	0.968725i	$0.579818\pi$
$402$	0	0
$403$	−10.7638	−0.536184
$404$	0	0
$405$	22.5584	1.12094
$406$	0	0
$407$	−57.1221	−2.83144
$408$	0	0
$409$	−24.1687	−1.19506	−0.597532	−	0.801845i	$-0.703852\pi$
$409$	−24.1687	−1.19506	−0.597532	+	0.801845i	$0.703852\pi$
$410$	0	0
$411$	−7.68279	−0.378964
$412$	0	0
$413$	28.5249	1.40362
$414$	0	0
$415$	−1.25583	−0.0616463
$416$	0	0
$417$	10.4115	0.509855
$418$	0	0
$419$	−14.4914	−0.707953	−0.353976	−	0.935254i	$-0.615171\pi$
$419$	−14.4914	−0.707953	−0.353976	+	0.935254i	$0.615171\pi$
$420$	0	0
$421$	5.50298	0.268199	0.134099	−	0.990968i	$-0.457186\pi$
$421$	5.50298	0.268199	0.134099	+	0.990968i	$0.457186\pi$
$422$	0	0
$423$	−24.3147	−1.18222
$424$	0	0
$425$	4.54912	0.220665
$426$	0	0
$427$	−28.2862	−1.36886
$428$	0	0
$429$	19.0605	0.920252
$430$	0	0
$431$	−9.26770	−0.446409	−0.223205	−	0.974772i	$-0.571652\pi$
$431$	−9.26770	−0.446409	−0.223205	+	0.974772i	$0.571652\pi$
$432$	0	0
$433$	−30.7522	−1.47786	−0.738929	−	0.673783i	$-0.764668\pi$
$433$	−30.7522	−1.47786	−0.738929	+	0.673783i	$0.764668\pi$
$434$	0	0
$435$	0.535713	0.0256855
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−11.1705	−0.533138	−0.266569	−	0.963816i	$-0.585890\pi$
$439$	−11.1705	−0.533138	−0.266569	+	0.963816i	$0.585890\pi$
$440$	0	0
$441$	16.4339	0.782566
$442$	0	0
$443$	−31.2320	−1.48388	−0.741940	−	0.670467i	$-0.766094\pi$
$443$	−31.2320	−1.48388	−0.741940	+	0.670467i	$0.766094\pi$
$444$	0	0
$445$	−5.23432	−0.248130
$446$	0	0
$447$	−27.7315	−1.31165
$448$	0	0
$449$	−13.0032	−0.613659	−0.306829	−	0.951765i	$-0.599268\pi$
$449$	−13.0032	−0.613659	−0.306829	+	0.951765i	$0.599268\pi$
$450$	0	0
$451$	44.4172	2.09152
$452$	0	0
$453$	2.85039	0.133923
$454$	0	0
$455$	−14.5462	−0.681938
$456$	0	0
$457$	−12.1396	−0.567867	−0.283933	−	0.958844i	$-0.591639\pi$
$457$	−12.1396	−0.567867	−0.283933	+	0.958844i	$0.591639\pi$
$458$	0	0
$459$	−1.01025	−0.0471542
$460$	0	0
$461$	−25.5360	−1.18933	−0.594664	−	0.803974i	$-0.702715\pi$
$461$	−25.5360	−1.18933	−0.594664	+	0.803974i	$0.702715\pi$
$462$	0	0
$463$	33.0113	1.53417	0.767084	−	0.641547i	$-0.221707\pi$
$463$	33.0113	1.53417	0.767084	+	0.641547i	$0.221707\pi$
$464$	0	0
$465$	47.0130	2.18017
$466$	0	0
$467$	24.8261	1.14881	0.574407	−	0.818570i	$-0.305233\pi$
$467$	24.8261	1.14881	0.574407	+	0.818570i	$0.305233\pi$
$468$	0	0
$469$	−11.3133	−0.522401
$470$	0	0
$471$	−2.36040	−0.108761
$472$	0	0
$473$	48.5441	2.23206
$474$	0	0
$475$	10.7826	0.494741
$476$	0	0
$477$	−29.2178	−1.33779
$478$	0	0
$479$	−9.96883	−0.455488	−0.227744	−	0.973721i	$-0.573135\pi$
$479$	−9.96883	−0.455488	−0.227744	+	0.973721i	$0.573135\pi$
$480$	0	0
$481$	17.7923	0.811260
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.58433	−0.208164
$486$	0	0
$487$	−34.6281	−1.56915	−0.784574	−	0.620035i	$-0.787118\pi$
$487$	−34.6281	−1.56915	−0.784574	+	0.620035i	$0.787118\pi$
$488$	0	0
$489$	−57.0860	−2.58152
$490$	0	0
$491$	−3.69003	−0.166529	−0.0832644	−	0.996527i	$-0.526535\pi$
$491$	−3.69003	−0.166529	−0.0832644	+	0.996527i	$0.526535\pi$
$492$	0	0
$493$	0.155550	0.00700560
$494$	0	0
$495$	−43.0142	−1.93334
$496$	0	0
$497$	−15.2868	−0.685709
$498$	0	0
$499$	28.5977	1.28021	0.640104	−	0.768289i	$-0.278891\pi$
$499$	28.5977	1.28021	0.640104	+	0.768289i	$0.278891\pi$
$500$	0	0
$501$	22.0606	0.985593
$502$	0	0
$503$	−5.63401	−0.251208	−0.125604	−	0.992080i	$-0.540087\pi$
$503$	−5.63401	−0.251208	−0.125604	+	0.992080i	$0.540087\pi$
$504$	0	0
$505$	−10.5735	−0.470515
$506$	0	0
$507$	26.4514	1.17475
$508$	0	0
$509$	21.8183	0.967081	0.483540	−	0.875322i	$-0.339351\pi$
$509$	21.8183	0.967081	0.483540	+	0.875322i	$0.339351\pi$
$510$	0	0
$511$	51.8365	2.29311
$512$	0	0
$513$	−2.39455	−0.105722
$514$	0	0
$515$	13.3169	0.586812
$516$	0	0
$517$	−37.5736	−1.65248
$518$	0	0
$519$	37.8742	1.66249
$520$	0	0
$521$	23.2023	1.01651	0.508255	−	0.861206i	$-0.330291\pi$
$521$	23.2023	1.01651	0.508255	+	0.861206i	$0.330291\pi$
$522$	0	0
$523$	−0.387775	−0.0169562	−0.00847810	−	0.999964i	$-0.502699\pi$
$523$	−0.387775	−0.0169562	−0.00847810	+	0.999964i	$0.502699\pi$
$524$	0	0
$525$	20.1582	0.879775
$526$	0	0
$527$	13.6507	0.594633
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−26.2732	−1.14016
$532$	0	0
$533$	−13.8350	−0.599261
$534$	0	0
$535$	−31.5812	−1.36537
$536$	0	0
$537$	55.6096	2.39973
$538$	0	0
$539$	25.3954	1.09386
$540$	0	0
$541$	−31.8593	−1.36974	−0.684868	−	0.728667i	$-0.740140\pi$
$541$	−31.8593	−1.36974	−0.684868	+	0.728667i	$0.740140\pi$
$542$	0	0
$543$	−33.2467	−1.42675
$544$	0	0
$545$	−2.47772	−0.106134
$546$	0	0
$547$	22.7035	0.970731	0.485366	−	0.874311i	$-0.338687\pi$
$547$	22.7035	0.970731	0.485366	+	0.874311i	$0.338687\pi$
$548$	0	0
$549$	26.0534	1.11193
$550$	0	0
$551$	0.368694	0.0157069
$552$	0	0
$553$	19.8971	0.846112
$554$	0	0
$555$	−77.7112	−3.29866
$556$	0	0
$557$	22.7772	0.965100	0.482550	−	0.875868i	$-0.339711\pi$
$557$	22.7772	0.965100	0.482550	+	0.875868i	$0.339711\pi$
$558$	0	0
$559$	−15.1205	−0.639527
$560$	0	0
$561$	−24.1726	−1.02057
$562$	0	0
$563$	13.8785	0.584908	0.292454	−	0.956280i	$-0.405528\pi$
$563$	13.8785	0.584908	0.292454	+	0.956280i	$0.405528\pi$
$564$	0	0
$565$	45.5929	1.91811
$566$	0	0
$567$	29.0248	1.21893
$568$	0	0
$569$	−11.4125	−0.478436	−0.239218	−	0.970966i	$-0.576891\pi$
$569$	−11.4125	−0.478436	−0.239218	+	0.970966i	$0.576891\pi$
$570$	0	0
$571$	−33.3948	−1.39753	−0.698764	−	0.715353i	$-0.746266\pi$
$571$	−33.3948	−1.39753	−0.698764	+	0.715353i	$0.746266\pi$
$572$	0	0
$573$	28.8180	1.20389
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.5237	0.771150	0.385575	−	0.922676i	$-0.374003\pi$
$577$	18.5237	0.771150	0.385575	+	0.922676i	$0.374003\pi$
$578$	0	0
$579$	35.6221	1.48041
$580$	0	0
$581$	−1.61582	−0.0670354
$582$	0	0
$583$	−45.1504	−1.86994
$584$	0	0
$585$	13.3980	0.553940
$586$	0	0
$587$	10.9092	0.450270	0.225135	−	0.974328i	$-0.427718\pi$
$587$	10.9092	0.450270	0.225135	+	0.974328i	$0.427718\pi$
$588$	0	0
$589$	32.3558	1.33320
$590$	0	0
$591$	35.3731	1.45506
$592$	0	0
$593$	−9.31065	−0.382342	−0.191171	−	0.981557i	$-0.561229\pi$
$593$	−9.31065	−0.382342	−0.191171	+	0.981557i	$0.561229\pi$
$594$	0	0
$595$	18.4476	0.756276
$596$	0	0
$597$	16.5803	0.678585
$598$	0	0
$599$	12.8667	0.525718	0.262859	−	0.964834i	$-0.415335\pi$
$599$	12.8667	0.525718	0.262859	+	0.964834i	$0.415335\pi$
$600$	0	0
$601$	40.8201	1.66509	0.832544	−	0.553959i	$-0.186884\pi$
$601$	40.8201	1.66509	0.832544	+	0.553959i	$0.186884\pi$
$602$	0	0
$603$	10.4203	0.424347
$604$	0	0
$605$	−36.7015	−1.49213
$606$	0	0
$607$	35.1187	1.42542	0.712711	−	0.701458i	$-0.247467\pi$
$607$	35.1187	1.42542	0.712711	+	0.701458i	$0.247467\pi$
$608$	0	0
$609$	0.689276	0.0279309
$610$	0	0
$611$	11.7034	0.473468
$612$	0	0
$613$	46.5823	1.88144	0.940720	−	0.339184i	$-0.110151\pi$
$613$	46.5823	1.88144	0.940720	+	0.339184i	$0.110151\pi$
$614$	0	0
$615$	60.4270	2.43665
$616$	0	0
$617$	−6.78904	−0.273317	−0.136658	−	0.990618i	$-0.543636\pi$
$617$	−6.78904	−0.273317	−0.136658	+	0.990618i	$0.543636\pi$
$618$	0	0
$619$	−22.8754	−0.919441	−0.459721	−	0.888064i	$-0.652050\pi$
$619$	−22.8754	−0.919441	−0.459721	+	0.888064i	$0.652050\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.73474	−0.269822
$624$	0	0
$625$	−31.2189	−1.24876
$626$	0	0
$627$	−57.2955	−2.28816
$628$	0	0
$629$	−22.5642	−0.899695
$630$	0	0
$631$	−1.25957	−0.0501429	−0.0250714	−	0.999686i	$-0.507981\pi$
$631$	−1.25957	−0.0501429	−0.0250714	+	0.999686i	$0.507981\pi$
$632$	0	0
$633$	−41.2005	−1.63757
$634$	0	0
$635$	−2.61469	−0.103761
$636$	0	0
$637$	−7.91013	−0.313411
$638$	0	0
$639$	14.0802	0.557003
$640$	0	0
$641$	−10.2259	−0.403900	−0.201950	−	0.979396i	$-0.564728\pi$
$641$	−10.2259	−0.403900	−0.201950	+	0.979396i	$0.564728\pi$
$642$	0	0
$643$	10.1651	0.400871	0.200435	−	0.979707i	$-0.435764\pi$
$643$	10.1651	0.400871	0.200435	+	0.979707i	$0.435764\pi$
$644$	0	0
$645$	66.0414	2.60038
$646$	0	0
$647$	6.09090	0.239458	0.119729	−	0.992807i	$-0.461797\pi$
$647$	6.09090	0.239458	0.119729	+	0.992807i	$0.461797\pi$
$648$	0	0
$649$	−40.6002	−1.59370
$650$	0	0
$651$	60.4893	2.37076
$652$	0	0
$653$	30.9964	1.21298	0.606492	−	0.795090i	$-0.292576\pi$
$653$	30.9964	1.21298	0.606492	+	0.795090i	$0.292576\pi$
$654$	0	0
$655$	2.35553	0.0920382
$656$	0	0
$657$	−47.7448	−1.86270
$658$	0	0
$659$	−8.63446	−0.336351	−0.168175	−	0.985757i	$-0.553788\pi$
$659$	−8.63446	−0.336351	−0.168175	+	0.985757i	$0.553788\pi$
$660$	0	0
$661$	−11.4037	−0.443554	−0.221777	−	0.975097i	$-0.571186\pi$
$661$	−11.4037	−0.443554	−0.221777	+	0.975097i	$0.571186\pi$
$662$	0	0
$663$	7.52926	0.292412
$664$	0	0
$665$	43.7257	1.69561
$666$	0	0
$667$	0	0
$668$	0	0
$669$	10.0136	0.387150
$670$	0	0
$671$	40.2604	1.55424
$672$	0	0
$673$	−20.5972	−0.793962	−0.396981	−	0.917827i	$-0.629942\pi$
$673$	−20.5972	−0.793962	−0.396981	+	0.917827i	$0.629942\pi$
$674$	0	0
$675$	−1.19911	−0.0461538
$676$	0	0
$677$	25.7244	0.988669	0.494334	−	0.869272i	$-0.335412\pi$
$677$	25.7244	0.988669	0.494334	+	0.869272i	$0.335412\pi$
$678$	0	0
$679$	−5.89843	−0.226361
$680$	0	0
$681$	−33.9891	−1.30246
$682$	0	0
$683$	−29.0338	−1.11095	−0.555473	−	0.831535i	$-0.687463\pi$
$683$	−29.0338	−1.11095	−0.555473	+	0.831535i	$0.687463\pi$
$684$	0	0
$685$	−8.34524	−0.318855
$686$	0	0
$687$	5.95589	0.227232
$688$	0	0
$689$	14.0634	0.535773
$690$	0	0
$691$	−22.9857	−0.874419	−0.437210	−	0.899360i	$-0.644033\pi$
$691$	−22.9857	−0.874419	−0.437210	+	0.899360i	$0.644033\pi$
$692$	0	0
$693$	−55.3443	−2.10236
$694$	0	0
$695$	11.3093	0.428985
$696$	0	0
$697$	17.5456	0.664587
$698$	0	0
$699$	−4.29974	−0.162631
$700$	0	0
$701$	49.4016	1.86587	0.932936	−	0.360041i	$-0.117237\pi$
$701$	49.4016	1.86587	0.932936	+	0.360041i	$0.117237\pi$
$702$	0	0
$703$	−53.4833	−2.01716
$704$	0	0
$705$	−51.1167	−1.92517
$706$	0	0
$707$	−13.6044	−0.511647
$708$	0	0
$709$	42.2797	1.58785	0.793925	−	0.608016i	$-0.208034\pi$
$709$	42.2797	1.58785	0.793925	+	0.608016i	$0.208034\pi$
$710$	0	0
$711$	−18.3265	−0.687299
$712$	0	0
$713$	0	0
$714$	0	0
$715$	20.7040	0.774287
$716$	0	0
$717$	−25.5206	−0.953086
$718$	0	0
$719$	−13.0023	−0.484905	−0.242452	−	0.970163i	$-0.577952\pi$
$719$	−13.0023	−0.484905	−0.242452	+	0.970163i	$0.577952\pi$
$720$	0	0
$721$	17.1342	0.638110
$722$	0	0
$723$	−14.2561	−0.530192
$724$	0	0
$725$	0.184630	0.00685697
$726$	0	0
$727$	47.0987	1.74679	0.873397	−	0.487009i	$-0.161912\pi$
$727$	47.0987	1.74679	0.873397	+	0.487009i	$0.161912\pi$
$728$	0	0
$729$	−30.5907	−1.13299
$730$	0	0
$731$	19.1758	0.709242
$732$	0	0
$733$	−16.6346	−0.614415	−0.307207	−	0.951643i	$-0.599394\pi$
$733$	−16.6346	−0.614415	−0.307207	+	0.951643i	$0.599394\pi$
$734$	0	0
$735$	34.5490	1.27436
$736$	0	0
$737$	16.1025	0.593145
$738$	0	0
$739$	−5.11672	−0.188222	−0.0941109	−	0.995562i	$-0.530001\pi$
$739$	−5.11672	−0.188222	−0.0941109	+	0.995562i	$0.530001\pi$
$740$	0	0
$741$	17.8464	0.655602
$742$	0	0
$743$	−3.92013	−0.143816	−0.0719078	−	0.997411i	$-0.522909\pi$
$743$	−3.92013	−0.143816	−0.0719078	+	0.997411i	$0.522909\pi$
$744$	0	0
$745$	−30.1226	−1.10361
$746$	0	0
$747$	1.48827	0.0544530
$748$	0	0
$749$	−40.6340	−1.48473
$750$	0	0
$751$	−6.79454	−0.247936	−0.123968	−	0.992286i	$-0.539562\pi$
$751$	−6.79454	−0.247936	−0.123968	+	0.992286i	$0.539562\pi$
$752$	0	0
$753$	−14.1690	−0.516349
$754$	0	0
$755$	3.09617	0.112681
$756$	0	0
$757$	−30.2155	−1.09820	−0.549101	−	0.835756i	$-0.685030\pi$
$757$	−30.2155	−1.09820	−0.549101	+	0.835756i	$0.685030\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.4942	0.416665	0.208333	−	0.978058i	$-0.433196\pi$
$761$	11.4942	0.416665	0.208333	+	0.978058i	$0.433196\pi$
$762$	0	0
$763$	−3.18797	−0.115412
$764$	0	0
$765$	−16.9914	−0.614325
$766$	0	0
$767$	12.6461	0.456625
$768$	0	0
$769$	−51.9295	−1.87262	−0.936312	−	0.351169i	$-0.885784\pi$
$769$	−51.9295	−1.87262	−0.936312	+	0.351169i	$0.885784\pi$
$770$	0	0
$771$	−45.1430	−1.62579
$772$	0	0
$773$	−5.32916	−0.191676	−0.0958382	−	0.995397i	$-0.530553\pi$
$773$	−5.32916	−0.191676	−0.0958382	+	0.995397i	$0.530553\pi$
$774$	0	0
$775$	16.2027	0.582017
$776$	0	0
$777$	−99.9873	−3.58702
$778$	0	0
$779$	41.5878	1.49004
$780$	0	0
$781$	21.7582	0.778568
$782$	0	0
$783$	−0.0410016	−0.00146528
$784$	0	0
$785$	−2.56392	−0.0915104
$786$	0	0
$787$	39.8059	1.41893	0.709464	−	0.704742i	$-0.248937\pi$
$787$	39.8059	1.41893	0.709464	+	0.704742i	$0.248937\pi$
$788$	0	0
$789$	−63.1977	−2.24990
$790$	0	0
$791$	58.6622	2.08579
$792$	0	0
$793$	−12.5403	−0.445319
$794$	0	0
$795$	−61.4245	−2.17850
$796$	0	0
$797$	−14.1927	−0.502732	−0.251366	−	0.967892i	$-0.580880\pi$
$797$	−14.1927	−0.502732	−0.251366	+	0.967892i	$0.580880\pi$
$798$	0	0
$799$	−14.8422	−0.525081
$800$	0	0
$801$	6.20313	0.219177
$802$	0	0
$803$	−73.7803	−2.60365
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.58397	−0.231767
$808$	0	0
$809$	−50.1648	−1.76370	−0.881850	−	0.471529i	$-0.843702\pi$
$809$	−50.1648	−1.76370	−0.881850	+	0.471529i	$0.843702\pi$
$810$	0	0
$811$	−35.3222	−1.24033	−0.620165	−	0.784471i	$-0.712934\pi$
$811$	−35.3222	−1.24033	−0.620165	+	0.784471i	$0.712934\pi$
$812$	0	0
$813$	21.0892	0.739629
$814$	0	0
$815$	−62.0082	−2.17205
$816$	0	0
$817$	45.4517	1.59015
$818$	0	0
$819$	17.2386	0.602365
$820$	0	0
$821$	33.8318	1.18074	0.590369	−	0.807134i	$-0.298982\pi$
$821$	33.8318	1.18074	0.590369	+	0.807134i	$0.298982\pi$
$822$	0	0
$823$	1.31116	0.0457041	0.0228521	−	0.999739i	$-0.492725\pi$
$823$	1.31116	0.0457041	0.0228521	+	0.999739i	$0.492725\pi$
$824$	0	0
$825$	−28.6917	−0.998915
$826$	0	0
$827$	−9.91522	−0.344786	−0.172393	−	0.985028i	$-0.555150\pi$
$827$	−9.91522	−0.344786	−0.172393	+	0.985028i	$0.555150\pi$
$828$	0	0
$829$	39.5179	1.37251	0.686256	−	0.727360i	$-0.259253\pi$
$829$	39.5179	1.37251	0.686256	+	0.727360i	$0.259253\pi$
$830$	0	0
$831$	35.4980	1.23141
$832$	0	0
$833$	10.0316	0.347576
$834$	0	0
$835$	23.9627	0.829265
$836$	0	0
$837$	−3.59821	−0.124372
$838$	0	0
$839$	27.1891	0.938672	0.469336	−	0.883020i	$-0.344493\pi$
$839$	27.1891	0.938672	0.469336	+	0.883020i	$0.344493\pi$
$840$	0	0
$841$	−28.9937	−0.999782
$842$	0	0
$843$	9.33125	0.321385
$844$	0	0
$845$	28.7322	0.988416
$846$	0	0
$847$	−47.2221	−1.62257
$848$	0	0
$849$	25.2863	0.867824
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−30.1830	−1.03344	−0.516722	−	0.856153i	$-0.672848\pi$
$853$	−30.1830	−1.03344	−0.516722	+	0.856153i	$0.672848\pi$
$854$	0	0
$855$	−40.2741	−1.37735
$856$	0	0
$857$	17.2197	0.588213	0.294106	−	0.955773i	$-0.404978\pi$
$857$	17.2197	0.588213	0.294106	+	0.955773i	$0.404978\pi$
$858$	0	0
$859$	11.8510	0.404352	0.202176	−	0.979349i	$-0.435199\pi$
$859$	11.8510	0.404352	0.202176	+	0.979349i	$0.435199\pi$
$860$	0	0
$861$	77.7485	2.64966
$862$	0	0
$863$	36.5134	1.24293	0.621465	−	0.783442i	$-0.286538\pi$
$863$	36.5134	1.24293	0.621465	+	0.783442i	$0.286538\pi$
$864$	0	0
$865$	41.1399	1.39880
$866$	0	0
$867$	32.8054	1.11413
$868$	0	0
$869$	−28.3201	−0.960694
$870$	0	0
$871$	−5.01561	−0.169947
$872$	0	0
$873$	5.43284	0.183874
$874$	0	0
$875$	−25.2189	−0.852555
$876$	0	0
$877$	−16.3949	−0.553617	−0.276808	−	0.960925i	$-0.589277\pi$
$877$	−16.3949	−0.553617	−0.276808	+	0.960925i	$0.589277\pi$
$878$	0	0
$879$	−61.7117	−2.08148
$880$	0	0
$881$	9.42130	0.317412	0.158706	−	0.987326i	$-0.449268\pi$
$881$	9.42130	0.317412	0.158706	+	0.987326i	$0.449268\pi$
$882$	0	0
$883$	−48.9746	−1.64813	−0.824063	−	0.566498i	$-0.808298\pi$
$883$	−48.9746	−1.64813	−0.824063	+	0.566498i	$0.808298\pi$
$884$	0	0
$885$	−55.2342	−1.85668
$886$	0	0
$887$	15.3259	0.514593	0.257297	−	0.966333i	$-0.417168\pi$
$887$	15.3259	0.514593	0.257297	+	0.966333i	$0.417168\pi$
$888$	0	0
$889$	−3.36419	−0.112831
$890$	0	0
$891$	−41.3118	−1.38400
$892$	0	0
$893$	−35.1801	−1.17726
$894$	0	0
$895$	60.4046	2.01910
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.554024	0.0184777
$900$	0	0
$901$	−17.8352	−0.594177
$902$	0	0
$903$	84.9722	2.82770
$904$	0	0
$905$	−36.1134	−1.20045
$906$	0	0
$907$	−29.4280	−0.977140	−0.488570	−	0.872525i	$-0.662481\pi$
$907$	−29.4280	−0.977140	−0.488570	+	0.872525i	$0.662481\pi$
$908$	0	0
$909$	12.5305	0.415612
$910$	0	0
$911$	−6.37521	−0.211220	−0.105610	−	0.994408i	$-0.533680\pi$
$911$	−6.37521	−0.211220	−0.105610	+	0.994408i	$0.533680\pi$
$912$	0	0
$913$	2.29984	0.0761134
$914$	0	0
$915$	54.7720	1.81071
$916$	0	0
$917$	3.03075	0.100084
$918$	0	0
$919$	−21.7958	−0.718978	−0.359489	−	0.933149i	$-0.617049\pi$
$919$	−21.7958	−0.718978	−0.359489	+	0.933149i	$0.617049\pi$
$920$	0	0
$921$	−75.4861	−2.48735
$922$	0	0
$923$	−6.77721	−0.223075
$924$	0	0
$925$	−26.7826	−0.880607
$926$	0	0
$927$	−15.7817	−0.518338
$928$	0	0
$929$	19.8298	0.650596	0.325298	−	0.945612i	$-0.394535\pi$
$929$	19.8298	0.650596	0.325298	+	0.945612i	$0.394535\pi$
$930$	0	0
$931$	23.7777	0.779282
$932$	0	0
$933$	47.6720	1.56071
$934$	0	0
$935$	−26.2569	−0.858692
$936$	0	0
$937$	−50.8286	−1.66050	−0.830249	−	0.557393i	$-0.811802\pi$
$937$	−50.8286	−1.66050	−0.830249	+	0.557393i	$0.811802\pi$
$938$	0	0
$939$	−83.8167	−2.73526
$940$	0	0
$941$	39.1326	1.27569	0.637843	−	0.770167i	$-0.279827\pi$
$941$	39.1326	1.27569	0.637843	+	0.770167i	$0.279827\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−4.86263	−0.158181
$946$	0	0
$947$	49.5559	1.61035	0.805174	−	0.593038i	$-0.202072\pi$
$947$	49.5559	1.61035	0.805174	+	0.593038i	$0.202072\pi$
$948$	0	0
$949$	22.9810	0.745995
$950$	0	0
$951$	−3.34858	−0.108585
$952$	0	0
$953$	33.8397	1.09618	0.548088	−	0.836421i	$-0.315356\pi$
$953$	33.8397	1.09618	0.548088	+	0.836421i	$0.315356\pi$
$954$	0	0
$955$	31.3028	1.01294
$956$	0	0
$957$	−0.981064	−0.0317133
$958$	0	0
$959$	−10.7374	−0.346729
$960$	0	0
$961$	17.6199	0.568384
$962$	0	0
$963$	37.4265	1.20605
$964$	0	0
$965$	38.6937	1.24559
$966$	0	0
$967$	−3.69582	−0.118850	−0.0594248	−	0.998233i	$-0.518927\pi$
$967$	−3.69582	−0.118850	−0.0594248	+	0.998233i	$0.518927\pi$
$968$	0	0
$969$	−22.6328	−0.727069
$970$	0	0
$971$	62.0824	1.99232	0.996160	−	0.0875519i	$-0.0279044\pi$
$971$	62.0824	1.99232	0.996160	+	0.0875519i	$0.0279044\pi$
$972$	0	0
$973$	14.5511	0.466487
$974$	0	0
$975$	8.93685	0.286208
$976$	0	0
$977$	−2.36003	−0.0755040	−0.0377520	−	0.999287i	$-0.512020\pi$
$977$	−2.36003	−0.0755040	−0.0377520	+	0.999287i	$0.512020\pi$
$978$	0	0
$979$	9.58573	0.306361
$980$	0	0
$981$	2.93632	0.0937495
$982$	0	0
$983$	44.1013	1.40661	0.703307	−	0.710887i	$-0.251706\pi$
$983$	44.1013	1.40661	0.703307	+	0.710887i	$0.251706\pi$
$984$	0	0
$985$	38.4232	1.22427
$986$	0	0
$987$	−65.7694	−2.09346
$988$	0	0
$989$	0	0
$990$	0	0
$991$	14.9250	0.474109	0.237055	−	0.971496i	$-0.423818\pi$
$991$	14.9250	0.474109	0.237055	+	0.971496i	$0.423818\pi$
$992$	0	0
$993$	0.00817740	0.000259502 0
$994$	0	0
$995$	18.0099	0.570952
$996$	0	0
$997$	−13.3432	−0.422584	−0.211292	−	0.977423i	$-0.567767\pi$
$997$	−13.3432	−0.422584	−0.211292	+	0.977423i	$0.567767\pi$
$998$	0	0
$999$	5.94775	0.188179

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.ch.1.3		15
4.3	odd	2		4232.2.a.ba.1.13		15
23.15	odd	22		368.2.m.e.225.3		30
23.20	odd	22		368.2.m.e.193.3		30
23.22	odd	2		8464.2.a.cg.1.3		15
92.15	even	22		184.2.i.b.41.1	yes	30
92.43	even	22		184.2.i.b.9.1	✓	30
92.91	even	2		4232.2.a.bb.1.13		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.9.1	✓	30	92.43	even	22
184.2.i.b.41.1	yes	30	92.15	even	22
368.2.m.e.193.3		30	23.20	odd	22
368.2.m.e.225.3		30	23.15	odd	22
4232.2.a.ba.1.13		15	4.3	odd	2
4232.2.a.bb.1.13		15	92.91	even	2
8464.2.a.cg.1.3		15	23.22	odd	2
8464.2.a.ch.1.3		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)

\(f(q)\)	\(=\)	\(q-2.49141 q^{3} -2.70623 q^{5} -3.48198 q^{7} +3.20713 q^{9} +O(q^{10})\)	\(q-2.49141 q^{3} -2.70623 q^{5} -3.48198 q^{7} +3.20713 q^{9} +4.95599 q^{11} -1.54369 q^{13} +6.74234 q^{15} +1.95771 q^{17} +4.64029 q^{19} +8.67504 q^{21} +2.32370 q^{25} -0.516035 q^{27} +0.0794550 q^{29} +6.97280 q^{31} -12.3474 q^{33} +9.42305 q^{35} -11.5259 q^{37} +3.84596 q^{39} +8.96232 q^{41} +9.79502 q^{43} -8.67923 q^{45} -7.58145 q^{47} +5.12418 q^{49} -4.87745 q^{51} -9.11026 q^{53} -13.4121 q^{55} -11.5609 q^{57} -8.19215 q^{59} +8.12359 q^{61} -11.1671 q^{63} +4.17758 q^{65} +3.24911 q^{67} +4.39028 q^{71} -14.8871 q^{73} -5.78929 q^{75} -17.2567 q^{77} -5.71432 q^{79} -8.33572 q^{81} +0.464051 q^{83} -5.29801 q^{85} -0.197955 q^{87} +1.93417 q^{89} +5.37509 q^{91} -17.3721 q^{93} -12.5577 q^{95} +1.69399 q^{97} +15.8945 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) \) 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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