LMFDB - Embedded newform 8464.2.a.bh.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [2,0,4,0,0,0,0,0,2,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1058)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+2.00000 q^{3} +1.73205 q^{5} +3.46410 q^{7} +1.00000 q^{9} -3.46410 q^{11} -5.00000 q^{13} +3.46410 q^{15} -6.92820 q^{17} -3.46410 q^{19} +6.92820 q^{21} -2.00000 q^{25} -4.00000 q^{27} -3.00000 q^{29} +8.00000 q^{31} -6.92820 q^{33} +6.00000 q^{35} -10.0000 q^{39} -9.00000 q^{41} -6.92820 q^{43} +1.73205 q^{45} +6.00000 q^{47} +5.00000 q^{49} -13.8564 q^{51} -1.73205 q^{53} -6.00000 q^{55} -6.92820 q^{57} +6.00000 q^{59} +5.19615 q^{61} +3.46410 q^{63} -8.66025 q^{65} +6.92820 q^{67} -6.00000 q^{71} -11.0000 q^{73} -4.00000 q^{75} -12.0000 q^{77} +6.92820 q^{79} -11.0000 q^{81} -12.0000 q^{85} -6.00000 q^{87} -1.73205 q^{89} -17.3205 q^{91} +16.0000 q^{93} -6.00000 q^{95} +12.1244 q^{97} -3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 2 q^{9} - 10 q^{13} - 4 q^{25} - 8 q^{27} - 6 q^{29} + 16 q^{31} + 12 q^{35} - 20 q^{39} - 18 q^{41} + 12 q^{47} + 10 q^{49} - 12 q^{55} + 12 q^{59} - 12 q^{71} - 22 q^{73} - 8 q^{75} - 24 q^{77}+ \cdots - 12 q^{95}+O(q^{100}) $ 2 * q + 4 * q^3 + 2 * q^9 - 10 * q^13 - 4 * q^25 - 8 * q^27 - 6 * q^29 + 16 * q^31 + 12 * q^35 - 20 * q^39 - 18 * q^41 + 12 * q^47 + 10 * q^49 - 12 * q^55 + 12 * q^59 - 12 * q^71 - 22 * q^73 - 8 * q^75 - 24 * q^77 - 22 * q^81 - 24 * q^85 - 12 * q^87 + 32 * q^93 - 12 * q^95

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	3.46410	0.894427
$16$	0	0
$17$	−6.92820	−1.68034	−0.840168	−	0.542326i	$-0.817544\pi$
$17$	−6.92820	−1.68034	−0.840168	+	0.542326i	$0.817544\pi$
$18$	0	0
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	0	0
$21$	6.92820	1.51186
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	−6.92820	−1.20605
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−10.0000	−1.60128
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	−6.92820	−1.05654	−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820	−1.05654	−0.528271	+	0.849076i	$0.677159\pi$
$44$	0	0
$45$	1.73205	0.258199
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	−13.8564	−1.94029
$52$	0	0
$53$	−1.73205	−0.237915	−0.118958	−	0.992899i	$-0.537955\pi$
$53$	−1.73205	−0.237915	−0.118958	+	0.992899i	$0.537955\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	0	0
$57$	−6.92820	−0.917663
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	5.19615	0.665299	0.332650	−	0.943051i	$-0.392057\pi$
$61$	5.19615	0.665299	0.332650	+	0.943051i	$0.392057\pi$
$62$	0	0
$63$	3.46410	0.436436
$64$	0	0
$65$	−8.66025	−1.07417
$66$	0	0
$67$	6.92820	0.846415	0.423207	−	0.906033i	$-0.360904\pi$
$67$	6.92820	0.846415	0.423207	+	0.906033i	$0.360904\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−12.0000	−1.36753
$78$	0	0
$79$	6.92820	0.779484	0.389742	−	0.920924i	$-0.372564\pi$
$79$	6.92820	0.779484	0.389742	+	0.920924i	$0.372564\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−1.73205	−0.183597	−0.0917985	−	0.995778i	$-0.529262\pi$
$89$	−1.73205	−0.183597	−0.0917985	+	0.995778i	$0.529262\pi$
$90$	0	0
$91$	−17.3205	−1.81568
$92$	0	0
$93$	16.0000	1.65912
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	12.1244	1.23104	0.615521	−	0.788121i	$-0.288946\pi$
$97$	12.1244	1.23104	0.615521	+	0.788121i	$0.288946\pi$
$98$	0	0
$99$	−3.46410	−0.348155
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	12.0000	1.17108
$106$	0	0
$107$	3.46410	0.334887	0.167444	−	0.985882i	$-0.446449\pi$
$107$	3.46410	0.334887	0.167444	+	0.985882i	$0.446449\pi$
$108$	0	0
$109$	15.5885	1.49310	0.746552	−	0.665327i	$-0.231708\pi$
$109$	15.5885	1.49310	0.746552	+	0.665327i	$0.231708\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.19615	−0.488813	−0.244406	−	0.969673i	$-0.578593\pi$
$113$	−5.19615	−0.488813	−0.244406	+	0.969673i	$0.578593\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.00000	−0.462250
$118$	0	0
$119$	−24.0000	−2.20008
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−18.0000	−1.62301
$124$	0	0
$125$	−12.1244	−1.08444
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	−13.8564	−1.21999
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	−6.92820	−0.596285
$136$	0	0
$137$	−15.5885	−1.33181	−0.665906	−	0.746036i	$-0.731955\pi$
$137$	−15.5885	−1.33181	−0.665906	+	0.746036i	$0.731955\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	17.3205	1.44841
$144$	0	0
$145$	−5.19615	−0.431517
$146$	0	0
$147$	10.0000	0.824786
$148$	0	0
$149$	5.19615	0.425685	0.212843	−	0.977086i	$-0.431728\pi$
$149$	5.19615	0.425685	0.212843	+	0.977086i	$0.431728\pi$
$150$	0	0
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	0	0
$153$	−6.92820	−0.560112
$154$	0	0
$155$	13.8564	1.11297
$156$	0	0
$157$	−5.19615	−0.414698	−0.207349	−	0.978267i	$-0.566484\pi$
$157$	−5.19615	−0.414698	−0.207349	+	0.978267i	$0.566484\pi$
$158$	0	0
$159$	−3.46410	−0.274721
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	0	0
$165$	−12.0000	−0.934199
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−3.46410	−0.264906
$172$	0	0
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	0	0
$175$	−6.92820	−0.523723
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	13.8564	1.02994	0.514969	−	0.857209i	$-0.327803\pi$
$181$	13.8564	1.02994	0.514969	+	0.857209i	$0.327803\pi$
$182$	0	0
$183$	10.3923	0.768221
$184$	0	0
$185$	0	0
$186$	0	0
$187$	24.0000	1.75505
$188$	0	0
$189$	−13.8564	−1.00791
$190$	0	0
$191$	−6.92820	−0.501307	−0.250654	−	0.968077i	$-0.580646\pi$
$191$	−6.92820	−0.501307	−0.250654	+	0.968077i	$0.580646\pi$
$192$	0	0
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	0	0
$195$	−17.3205	−1.24035
$196$	0	0
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	0	0
$199$	−3.46410	−0.245564	−0.122782	−	0.992434i	$-0.539182\pi$
$199$	−3.46410	−0.245564	−0.122782	+	0.992434i	$0.539182\pi$
$200$	0	0
$201$	13.8564	0.977356
$202$	0	0
$203$	−10.3923	−0.729397
$204$	0	0
$205$	−15.5885	−1.08875
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	−12.0000	−0.818393
$216$	0	0
$217$	27.7128	1.88127
$218$	0	0
$219$	−22.0000	−1.48662
$220$	0	0
$221$	34.6410	2.33021
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	6.92820	0.459841	0.229920	−	0.973209i	$-0.426153\pi$
$227$	6.92820	0.459841	0.229920	+	0.973209i	$0.426153\pi$
$228$	0	0
$229$	13.8564	0.915657	0.457829	−	0.889041i	$-0.348627\pi$
$229$	13.8564	0.915657	0.457829	+	0.889041i	$0.348627\pi$
$230$	0	0
$231$	−24.0000	−1.57908
$232$	0	0
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	0	0
$235$	10.3923	0.677919
$236$	0	0
$237$	13.8564	0.900070
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−25.9808	−1.67357	−0.836784	−	0.547533i	$-0.815567\pi$
$241$	−25.9808	−1.67357	−0.836784	+	0.547533i	$0.815567\pi$
$242$	0	0
$243$	−10.0000	−0.641500
$244$	0	0
$245$	8.66025	0.553283
$246$	0	0
$247$	17.3205	1.10208
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.46410	0.218652	0.109326	−	0.994006i	$-0.465131\pi$
$251$	3.46410	0.218652	0.109326	+	0.994006i	$0.465131\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−24.0000	−1.50294
$256$	0	0
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.00000	−0.185695
$262$	0	0
$263$	−17.3205	−1.06803	−0.534014	−	0.845476i	$-0.679317\pi$
$263$	−17.3205	−1.06803	−0.534014	+	0.845476i	$0.679317\pi$
$264$	0	0
$265$	−3.00000	−0.184289
$266$	0	0
$267$	−3.46410	−0.212000
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	0	0
$273$	−34.6410	−2.09657
$274$	0	0
$275$	6.92820	0.417786
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	6.92820	0.413302	0.206651	−	0.978415i	$-0.433744\pi$
$281$	6.92820	0.413302	0.206651	+	0.978415i	$0.433744\pi$
$282$	0	0
$283$	27.7128	1.64736	0.823678	−	0.567058i	$-0.191918\pi$
$283$	27.7128	1.64736	0.823678	+	0.567058i	$0.191918\pi$
$284$	0	0
$285$	−12.0000	−0.710819
$286$	0	0
$287$	−31.1769	−1.84032
$288$	0	0
$289$	31.0000	1.82353
$290$	0	0
$291$	24.2487	1.42148
$292$	0	0
$293$	32.9090	1.92256	0.961281	−	0.275570i	$-0.0888664\pi$
$293$	32.9090	1.92256	0.961281	+	0.275570i	$0.0888664\pi$
$294$	0	0
$295$	10.3923	0.605063
$296$	0	0
$297$	13.8564	0.804030
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−24.0000	−1.38334
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	9.00000	0.515339
$306$	0	0
$307$	26.0000	1.48390	0.741949	−	0.670456i	$-0.233902\pi$
$307$	26.0000	1.48390	0.741949	+	0.670456i	$0.233902\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	25.9808	1.46852	0.734260	−	0.678869i	$-0.237529\pi$
$313$	25.9808	1.46852	0.734260	+	0.678869i	$0.237529\pi$
$314$	0	0
$315$	6.00000	0.338062
$316$	0	0
$317$	21.0000	1.17948	0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000	1.17948	0.589739	+	0.807594i	$0.299231\pi$
$318$	0	0
$319$	10.3923	0.581857
$320$	0	0
$321$	6.92820	0.386695
$322$	0	0
$323$	24.0000	1.33540
$324$	0	0
$325$	10.0000	0.554700
$326$	0	0
$327$	31.1769	1.72409
$328$	0	0
$329$	20.7846	1.14589
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	1.73205	0.0943508	0.0471754	−	0.998887i	$-0.484978\pi$
$337$	1.73205	0.0943508	0.0471754	+	0.998887i	$0.484978\pi$
$338$	0	0
$339$	−10.3923	−0.564433
$340$	0	0
$341$	−27.7128	−1.50073
$342$	0	0
$343$	−6.92820	−0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	20.0000	1.06752
$352$	0	0
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	0	0
$355$	−10.3923	−0.551566
$356$	0	0
$357$	−48.0000	−2.54043
$358$	0	0
$359$	24.2487	1.27980	0.639899	−	0.768459i	$-0.278976\pi$
$359$	24.2487	1.27980	0.639899	+	0.768459i	$0.278976\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	−19.0526	−0.997257
$366$	0	0
$367$	24.2487	1.26577	0.632886	−	0.774245i	$-0.281870\pi$
$367$	24.2487	1.26577	0.632886	+	0.774245i	$0.281870\pi$
$368$	0	0
$369$	−9.00000	−0.468521
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	−24.2487	−1.25220
$376$	0	0
$377$	15.0000	0.772539
$378$	0	0
$379$	31.1769	1.60145	0.800725	−	0.599032i	$-0.204448\pi$
$379$	31.1769	1.60145	0.800725	+	0.599032i	$0.204448\pi$
$380$	0	0
$381$	−32.0000	−1.63941
$382$	0	0
$383$	−6.92820	−0.354015	−0.177007	−	0.984210i	$-0.556642\pi$
$383$	−6.92820	−0.354015	−0.177007	+	0.984210i	$0.556642\pi$
$384$	0	0
$385$	−20.7846	−1.05928
$386$	0	0
$387$	−6.92820	−0.352180
$388$	0	0
$389$	−13.8564	−0.702548	−0.351274	−	0.936273i	$-0.614251\pi$
$389$	−13.8564	−0.702548	−0.351274	+	0.936273i	$0.614251\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	0	0
$399$	−24.0000	−1.20150
$400$	0	0
$401$	−29.4449	−1.47041	−0.735203	−	0.677847i	$-0.762913\pi$
$401$	−29.4449	−1.47041	−0.735203	+	0.677847i	$0.762913\pi$
$402$	0	0
$403$	−40.0000	−1.99254
$404$	0	0
$405$	−19.0526	−0.946729
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	−31.1769	−1.53784
$412$	0	0
$413$	20.7846	1.02274
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.0000	−0.783523
$418$	0	0
$419$	−13.8564	−0.676930	−0.338465	−	0.940979i	$-0.609908\pi$
$419$	−13.8564	−0.676930	−0.338465	+	0.940979i	$0.609908\pi$
$420$	0	0
$421$	−6.92820	−0.337660	−0.168830	−	0.985645i	$-0.553999\pi$
$421$	−6.92820	−0.337660	−0.168830	+	0.985645i	$0.553999\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	13.8564	0.672134
$426$	0	0
$427$	18.0000	0.871081
$428$	0	0
$429$	34.6410	1.67248
$430$	0	0
$431$	24.2487	1.16802	0.584010	−	0.811747i	$-0.301483\pi$
$431$	24.2487	1.16802	0.584010	+	0.811747i	$0.301483\pi$
$432$	0	0
$433$	29.4449	1.41503	0.707515	−	0.706698i	$-0.249816\pi$
$433$	29.4449	1.41503	0.707515	+	0.706698i	$0.249816\pi$
$434$	0	0
$435$	−10.3923	−0.498273
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	5.00000	0.238095
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	0	0
$447$	10.3923	0.491539
$448$	0	0
$449$	−33.0000	−1.55737	−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000	−1.55737	−0.778683	+	0.627417i	$0.784112\pi$
$450$	0	0
$451$	31.1769	1.46806
$452$	0	0
$453$	−28.0000	−1.31555
$454$	0	0
$455$	−30.0000	−1.40642
$456$	0	0
$457$	−12.1244	−0.567153	−0.283577	−	0.958950i	$-0.591521\pi$
$457$	−12.1244	−0.567153	−0.283577	+	0.958950i	$0.591521\pi$
$458$	0	0
$459$	27.7128	1.29352
$460$	0	0
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	0	0
$463$	28.0000	1.30127	0.650635	−	0.759390i	$-0.274503\pi$
$463$	28.0000	1.30127	0.650635	+	0.759390i	$0.274503\pi$
$464$	0	0
$465$	27.7128	1.28515
$466$	0	0
$467$	20.7846	0.961797	0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846	0.961797	0.480899	+	0.876776i	$0.340311\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	0	0
$471$	−10.3923	−0.478852
$472$	0	0
$473$	24.0000	1.10352
$474$	0	0
$475$	6.92820	0.317888
$476$	0	0
$477$	−1.73205	−0.0793052
$478$	0	0
$479$	−34.6410	−1.58279	−0.791394	−	0.611306i	$-0.790644\pi$
$479$	−34.6410	−1.58279	−0.791394	+	0.611306i	$0.790644\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	21.0000	0.953561
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	20.7846	0.936092
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	−20.7846	−0.932317
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	0	0
$503$	−17.3205	−0.772283	−0.386142	−	0.922440i	$-0.626192\pi$
$503$	−17.3205	−0.772283	−0.386142	+	0.922440i	$0.626192\pi$
$504$	0	0
$505$	5.19615	0.231226
$506$	0	0
$507$	24.0000	1.06588
$508$	0	0
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	−38.1051	−1.68567
$512$	0	0
$513$	13.8564	0.611775
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−20.7846	−0.914106
$518$	0	0
$519$	42.0000	1.84360
$520$	0	0
$521$	−20.7846	−0.910590	−0.455295	−	0.890341i	$-0.650466\pi$
$521$	−20.7846	−0.910590	−0.455295	+	0.890341i	$0.650466\pi$
$522$	0	0
$523$	13.8564	0.605898	0.302949	−	0.953007i	$-0.402029\pi$
$523$	13.8564	0.605898	0.302949	+	0.953007i	$0.402029\pi$
$524$	0	0
$525$	−13.8564	−0.604743
$526$	0	0
$527$	−55.4256	−2.41438
$528$	0	0
$529$	0	0
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	45.0000	1.94917
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	−17.3205	−0.746047
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	0	0
$543$	27.7128	1.18927
$544$	0	0
$545$	27.0000	1.15655
$546$	0	0
$547$	−40.0000	−1.71028	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000	−1.71028	−0.855138	+	0.518400i	$0.826528\pi$
$548$	0	0
$549$	5.19615	0.221766
$550$	0	0
$551$	10.3923	0.442727
$552$	0	0
$553$	24.0000	1.02058
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.3731	−1.54118	−0.770588	−	0.637333i	$-0.780037\pi$
$557$	−36.3731	−1.54118	−0.770588	+	0.637333i	$0.780037\pi$
$558$	0	0
$559$	34.6410	1.46516
$560$	0	0
$561$	48.0000	2.02656
$562$	0	0
$563$	34.6410	1.45994	0.729972	−	0.683477i	$-0.239533\pi$
$563$	34.6410	1.45994	0.729972	+	0.683477i	$0.239533\pi$
$564$	0	0
$565$	−9.00000	−0.378633
$566$	0	0
$567$	−38.1051	−1.60026
$568$	0	0
$569$	29.4449	1.23439	0.617196	−	0.786809i	$-0.288268\pi$
$569$	29.4449	1.23439	0.617196	+	0.786809i	$0.288268\pi$
$570$	0	0
$571$	−31.1769	−1.30471	−0.652357	−	0.757912i	$-0.726220\pi$
$571$	−31.1769	−1.30471	−0.652357	+	0.757912i	$0.726220\pi$
$572$	0	0
$573$	−13.8564	−0.578860
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	0	0
$579$	−38.0000	−1.57923
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	−8.66025	−0.358057
$586$	0	0
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	−27.7128	−1.14189
$590$	0	0
$591$	−30.0000	−1.23404
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−41.5692	−1.70417
$596$	0	0
$597$	−6.92820	−0.283552
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	6.92820	0.282138
$604$	0	0
$605$	1.73205	0.0704179
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	0	0
$609$	−20.7846	−0.842235
$610$	0	0
$611$	−30.0000	−1.21367
$612$	0	0
$613$	−15.5885	−0.629612	−0.314806	−	0.949156i	$-0.601939\pi$
$613$	−15.5885	−0.629612	−0.314806	+	0.949156i	$0.601939\pi$
$614$	0	0
$615$	−31.1769	−1.25717
$616$	0	0
$617$	41.5692	1.67351	0.836757	−	0.547575i	$-0.184449\pi$
$617$	41.5692	1.67351	0.836757	+	0.547575i	$0.184449\pi$
$618$	0	0
$619$	−13.8564	−0.556936	−0.278468	−	0.960446i	$-0.589827\pi$
$619$	−13.8564	−0.556936	−0.278468	+	0.960446i	$0.589827\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	24.0000	0.958468
$628$	0	0
$629$	0	0
$630$	0	0
$631$	13.8564	0.551615	0.275807	−	0.961213i	$-0.411055\pi$
$631$	13.8564	0.551615	0.275807	+	0.961213i	$0.411055\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	−27.7128	−1.09975
$636$	0	0
$637$	−25.0000	−0.990536
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−5.19615	−0.205236	−0.102618	−	0.994721i	$-0.532722\pi$
$641$	−5.19615	−0.205236	−0.102618	+	0.994721i	$0.532722\pi$
$642$	0	0
$643$	−38.1051	−1.50272	−0.751360	−	0.659893i	$-0.770602\pi$
$643$	−38.1051	−1.50272	−0.751360	+	0.659893i	$0.770602\pi$
$644$	0	0
$645$	−24.0000	−0.944999
$646$	0	0
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	0	0
$649$	−20.7846	−0.815867
$650$	0	0
$651$	55.4256	2.17230
$652$	0	0
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	0	0
$655$	−10.3923	−0.406061
$656$	0	0
$657$	−11.0000	−0.429151
$658$	0	0
$659$	−27.7128	−1.07954	−0.539769	−	0.841813i	$-0.681488\pi$
$659$	−27.7128	−1.07954	−0.539769	+	0.841813i	$0.681488\pi$
$660$	0	0
$661$	−29.4449	−1.14527	−0.572636	−	0.819810i	$-0.694079\pi$
$661$	−29.4449	−1.14527	−0.572636	+	0.819810i	$0.694079\pi$
$662$	0	0
$663$	69.2820	2.69069
$664$	0	0
$665$	−20.7846	−0.805993
$666$	0	0
$667$	0	0
$668$	0	0
$669$	20.0000	0.773245
$670$	0	0
$671$	−18.0000	−0.694882
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	0	0
$675$	8.00000	0.307920
$676$	0	0
$677$	−43.3013	−1.66420	−0.832102	−	0.554623i	$-0.812862\pi$
$677$	−43.3013	−1.66420	−0.832102	+	0.554623i	$0.812862\pi$
$678$	0	0
$679$	42.0000	1.61181
$680$	0	0
$681$	13.8564	0.530979
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	−27.0000	−1.03162
$686$	0	0
$687$	27.7128	1.05731
$688$	0	0
$689$	8.66025	0.329929
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	−12.0000	−0.455842
$694$	0	0
$695$	−13.8564	−0.525603
$696$	0	0
$697$	62.3538	2.36182
$698$	0	0
$699$	30.0000	1.13470
$700$	0	0
$701$	13.8564	0.523349	0.261675	−	0.965156i	$-0.415725\pi$
$701$	13.8564	0.523349	0.261675	+	0.965156i	$0.415725\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	20.7846	0.782794
$706$	0	0
$707$	10.3923	0.390843
$708$	0	0
$709$	−1.73205	−0.0650485	−0.0325243	−	0.999471i	$-0.510355\pi$
$709$	−1.73205	−0.0650485	−0.0325243	+	0.999471i	$0.510355\pi$
$710$	0	0
$711$	6.92820	0.259828
$712$	0	0
$713$	0	0
$714$	0	0
$715$	30.0000	1.12194
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−51.9615	−1.93247
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	10.3923	0.385429	0.192715	−	0.981255i	$-0.438271\pi$
$727$	10.3923	0.385429	0.192715	+	0.981255i	$0.438271\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	48.0000	1.77534
$732$	0	0
$733$	−19.0526	−0.703722	−0.351861	−	0.936052i	$-0.614451\pi$
$733$	−19.0526	−0.703722	−0.351861	+	0.936052i	$0.614451\pi$
$734$	0	0
$735$	17.3205	0.638877
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	0	0
$741$	34.6410	1.27257
$742$	0	0
$743$	−38.1051	−1.39794	−0.698971	−	0.715150i	$-0.746358\pi$
$743$	−38.1051	−1.39794	−0.698971	+	0.715150i	$0.746358\pi$
$744$	0	0
$745$	9.00000	0.329734
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	24.2487	0.884848	0.442424	−	0.896806i	$-0.354119\pi$
$751$	24.2487	0.884848	0.442424	+	0.896806i	$0.354119\pi$
$752$	0	0
$753$	6.92820	0.252478
$754$	0	0
$755$	−24.2487	−0.882501
$756$	0	0
$757$	−22.5167	−0.818382	−0.409191	−	0.912449i	$-0.634189\pi$
$757$	−22.5167	−0.818382	−0.409191	+	0.912449i	$0.634189\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	0	0
$763$	54.0000	1.95493
$764$	0	0
$765$	−12.0000	−0.433861
$766$	0	0
$767$	−30.0000	−1.08324
$768$	0	0
$769$	6.92820	0.249837	0.124919	−	0.992167i	$-0.460133\pi$
$769$	6.92820	0.249837	0.124919	+	0.992167i	$0.460133\pi$
$770$	0	0
$771$	−42.0000	−1.51259
$772$	0	0
$773$	34.6410	1.24595	0.622975	−	0.782241i	$-0.285924\pi$
$773$	34.6410	1.24595	0.622975	+	0.782241i	$0.285924\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	0	0
$778$	0	0
$779$	31.1769	1.11703
$780$	0	0
$781$	20.7846	0.743732
$782$	0	0
$783$	12.0000	0.428845
$784$	0	0
$785$	−9.00000	−0.321224
$786$	0	0
$787$	−24.2487	−0.864373	−0.432187	−	0.901784i	$-0.642258\pi$
$787$	−24.2487	−0.864373	−0.432187	+	0.901784i	$0.642258\pi$
$788$	0	0
$789$	−34.6410	−1.23325
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−25.9808	−0.922604
$794$	0	0
$795$	−6.00000	−0.212798
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	−41.5692	−1.47061
$800$	0	0
$801$	−1.73205	−0.0611990
$802$	0	0
$803$	38.1051	1.34470
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.0000	0.422420
$808$	0	0
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	−10.0000	−0.351147	−0.175574	−	0.984466i	$-0.556178\pi$
$811$	−10.0000	−0.351147	−0.175574	+	0.984466i	$0.556178\pi$
$812$	0	0
$813$	28.0000	0.982003
$814$	0	0
$815$	3.46410	0.121342
$816$	0	0
$817$	24.0000	0.839654
$818$	0	0
$819$	−17.3205	−0.605228
$820$	0	0
$821$	3.00000	0.104701	0.0523504	−	0.998629i	$-0.483329\pi$
$821$	3.00000	0.104701	0.0523504	+	0.998629i	$0.483329\pi$
$822$	0	0
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	0	0
$825$	13.8564	0.482418
$826$	0	0
$827$	−45.0333	−1.56596	−0.782981	−	0.622046i	$-0.786302\pi$
$827$	−45.0333	−1.56596	−0.782981	+	0.622046i	$0.786302\pi$
$828$	0	0
$829$	−7.00000	−0.243120	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000	−0.243120	−0.121560	+	0.992584i	$0.538790\pi$
$830$	0	0
$831$	−52.0000	−1.80386
$832$	0	0
$833$	−34.6410	−1.20024
$834$	0	0
$835$	−20.7846	−0.719281
$836$	0	0
$837$	−32.0000	−1.10608
$838$	0	0
$839$	−3.46410	−0.119594	−0.0597970	−	0.998211i	$-0.519045\pi$
$839$	−3.46410	−0.119594	−0.0597970	+	0.998211i	$0.519045\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	13.8564	0.477240
$844$	0	0
$845$	20.7846	0.715012
$846$	0	0
$847$	3.46410	0.119028
$848$	0	0
$849$	55.4256	1.90220
$850$	0	0
$851$	0	0
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	15.0000	0.512390	0.256195	−	0.966625i	$-0.417531\pi$
$857$	15.0000	0.512390	0.256195	+	0.966625i	$0.417531\pi$
$858$	0	0
$859$	−2.00000	−0.0682391	−0.0341196	−	0.999418i	$-0.510863\pi$
$859$	−2.00000	−0.0682391	−0.0341196	+	0.999418i	$0.510863\pi$
$860$	0	0
$861$	−62.3538	−2.12501
$862$	0	0
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	36.3731	1.23672
$866$	0	0
$867$	62.0000	2.10563
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	−34.6410	−1.17377
$872$	0	0
$873$	12.1244	0.410347
$874$	0	0
$875$	−42.0000	−1.41986
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	65.8179	2.21998
$880$	0	0
$881$	−29.4449	−0.992023	−0.496011	−	0.868316i	$-0.665203\pi$
$881$	−29.4449	−0.992023	−0.496011	+	0.868316i	$0.665203\pi$
$882$	0	0
$883$	38.0000	1.27880	0.639401	−	0.768874i	$-0.279182\pi$
$883$	38.0000	1.27880	0.639401	+	0.768874i	$0.279182\pi$
$884$	0	0
$885$	20.7846	0.698667
$886$	0	0
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	−55.4256	−1.85892
$890$	0	0
$891$	38.1051	1.27657
$892$	0	0
$893$	−20.7846	−0.695530
$894$	0	0
$895$	−20.7846	−0.694753
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−24.0000	−0.800445
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	−48.0000	−1.59734
$904$	0	0
$905$	24.0000	0.797787
$906$	0	0
$907$	−31.1769	−1.03521	−0.517606	−	0.855619i	$-0.673177\pi$
$907$	−31.1769	−1.03521	−0.517606	+	0.855619i	$0.673177\pi$
$908$	0	0
$909$	3.00000	0.0995037
$910$	0	0
$911$	20.7846	0.688625	0.344312	−	0.938855i	$-0.388112\pi$
$911$	20.7846	0.688625	0.344312	+	0.938855i	$0.388112\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	18.0000	0.595062
$916$	0	0
$917$	−20.7846	−0.686368
$918$	0	0
$919$	45.0333	1.48551	0.742756	−	0.669562i	$-0.233518\pi$
$919$	45.0333	1.48551	0.742756	+	0.669562i	$0.233518\pi$
$920$	0	0
$921$	52.0000	1.71346
$922$	0	0
$923$	30.0000	0.987462
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−17.3205	−0.567657
$932$	0	0
$933$	36.0000	1.17859
$934$	0	0
$935$	41.5692	1.35946
$936$	0	0
$937$	−20.7846	−0.679004	−0.339502	−	0.940605i	$-0.610258\pi$
$937$	−20.7846	−0.679004	−0.339502	+	0.940605i	$0.610258\pi$
$938$	0	0
$939$	51.9615	1.69570
$940$	0	0
$941$	−6.92820	−0.225853	−0.112926	−	0.993603i	$-0.536022\pi$
$941$	−6.92820	−0.225853	−0.112926	+	0.993603i	$0.536022\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−24.0000	−0.780720
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	55.0000	1.78538
$950$	0	0
$951$	42.0000	1.36194
$952$	0	0
$953$	22.5167	0.729386	0.364693	−	0.931128i	$-0.381174\pi$
$953$	22.5167	0.729386	0.364693	+	0.931128i	$0.381174\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	0	0
$957$	20.7846	0.671871
$958$	0	0
$959$	−54.0000	−1.74375
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	3.46410	0.111629
$964$	0	0
$965$	−32.9090	−1.05938
$966$	0	0
$967$	50.0000	1.60789	0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000	1.60789	0.803946	+	0.594703i	$0.202730\pi$
$968$	0	0
$969$	48.0000	1.54198
$970$	0	0
$971$	3.46410	0.111168	0.0555842	−	0.998454i	$-0.482298\pi$
$971$	3.46410	0.111168	0.0555842	+	0.998454i	$0.482298\pi$
$972$	0	0
$973$	−27.7128	−0.888432
$974$	0	0
$975$	20.0000	0.640513
$976$	0	0
$977$	36.3731	1.16368	0.581839	−	0.813304i	$-0.302333\pi$
$977$	36.3731	1.16368	0.581839	+	0.813304i	$0.302333\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	15.5885	0.497701
$982$	0	0
$983$	6.92820	0.220975	0.110488	−	0.993877i	$-0.464759\pi$
$983$	6.92820	0.220975	0.110488	+	0.993877i	$0.464759\pi$
$984$	0	0
$985$	−25.9808	−0.827816
$986$	0	0
$987$	41.5692	1.32316
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	16.0000	0.507745
$994$	0	0
$995$	−6.00000	−0.190213
$996$	0	0
$997$	5.00000	0.158352	0.0791758	−	0.996861i	$-0.474771\pi$
$997$	5.00000	0.158352	0.0791758	+	0.996861i	$0.474771\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.bh.1.2		2
4.3	odd	2		1058.2.a.g.1.2	yes	2
12.11	even	2		9522.2.a.t.1.1		2
23.22	odd	2	inner	8464.2.a.bh.1.1		2
92.91	even	2		1058.2.a.g.1.1	✓	2
276.275	odd	2		9522.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1058.2.a.g.1.1	✓	2	92.91	even	2
1058.2.a.g.1.2	yes	2	4.3	odd	2
8464.2.a.bh.1.1		2	23.22	odd	2	inner
8464.2.a.bh.1.2		2	1.1	even	1	trivial
9522.2.a.t.1.1		2	12.11	even	2
9522.2.a.t.1.2		2	276.275	odd	2

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Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{3} +1.73205 q^{5} +3.46410 q^{7} +1.00000 q^{9} -3.46410 q^{11} -5.00000 q^{13} +3.46410 q^{15} -6.92820 q^{17} -3.46410 q^{19} +6.92820 q^{21} -2.00000 q^{25} -4.00000 q^{27} -3.00000 q^{29} +8.00000 q^{31} -6.92820 q^{33} +6.00000 q^{35} -10.0000 q^{39} -9.00000 q^{41} -6.92820 q^{43} +1.73205 q^{45} +6.00000 q^{47} +5.00000 q^{49} -13.8564 q^{51} -1.73205 q^{53} -6.00000 q^{55} -6.92820 q^{57} +6.00000 q^{59} +5.19615 q^{61} +3.46410 q^{63} -8.66025 q^{65} +6.92820 q^{67} -6.00000 q^{71} -11.0000 q^{73} -4.00000 q^{75} -12.0000 q^{77} +6.92820 q^{79} -11.0000 q^{81} -12.0000 q^{85} -6.00000 q^{87} -1.73205 q^{89} -17.3205 q^{91} +16.0000 q^{93} -6.00000 q^{95} +12.1244 q^{97} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 2 q^{9} - 10 q^{13} - 4 q^{25} - 8 q^{27} - 6 q^{29} + 16 q^{31} + 12 q^{35} - 20 q^{39} - 18 q^{41} + 12 q^{47} + 10 q^{49} - 12 q^{55} + 12 q^{59} - 12 q^{71} - 22 q^{73} - 8 q^{75} - 24 q^{77}+ \cdots - 12 q^{95}+O(q^{100}) \) 2 * q + 4 * q^3 + 2 * q^9 - 10 * q^13 - 4 * q^25 - 8 * q^27 - 6 * q^29 + 16 * q^31 + 12 * q^35 - 20 * q^39 - 18 * q^41 + 12 * q^47 + 10 * q^49 - 12 * q^55 + 12 * q^59 - 12 * q^71 - 22 * q^73 - 8 * q^75 - 24 * q^77 - 22 * q^81 - 24 * q^85 - 12 * q^87 + 32 * q^93 - 12 * q^95

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