LMFDB - Embedded newform 2304.4.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2304,4,Mod(1,2304)] 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("2304.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2304.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-12,0,0,0,0,0,0,0,40,0,0,0,16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$135.940400653$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	2304.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.00000 q^{5} +21.1660 q^{7} +42.3320 q^{11} +20.0000 q^{13} +8.00000 q^{17} -84.6640 q^{19} -169.328 q^{23} -89.0000 q^{25} +46.0000 q^{29} +21.1660 q^{31} -126.996 q^{35} -164.000 q^{37} +312.000 q^{41} -423.320 q^{43} -169.328 q^{47} +105.000 q^{49} -266.000 q^{53} -253.992 q^{55} -253.992 q^{59} -132.000 q^{61} -120.000 q^{65} -507.984 q^{67} +677.312 q^{71} +246.000 q^{73} +896.000 q^{77} -232.826 q^{79} +973.636 q^{83} -48.0000 q^{85} +1392.00 q^{89} +423.320 q^{91} +507.984 q^{95} -302.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{5} + 40 q^{13} + 16 q^{17} - 178 q^{25} + 92 q^{29} - 328 q^{37} + 624 q^{41} + 210 q^{49} - 532 q^{53} - 264 q^{61} - 240 q^{65} + 492 q^{73} + 1792 q^{77} - 96 q^{85} + 2784 q^{89} - 604 q^{97}+O(q^{100}) $ 2 * q - 12 * q^5 + 40 * q^13 + 16 * q^17 - 178 * q^25 + 92 * q^29 - 328 * q^37 + 624 * q^41 + 210 * q^49 - 532 * q^53 - 264 * q^61 - 240 * q^65 + 492 * q^73 + 1792 * q^77 - 96 * q^85 + 2784 * q^89 - 604 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	−6.00000	−0.536656	−0.268328	−	0.963328i	$-0.586471\pi$
$5$	−6.00000	−0.536656	−0.268328	+	0.963328i	$0.586471\pi$
$6$	0	0
$7$	21.1660	1.14286	0.571429	−	0.820652i	$-0.306389\pi$
$7$	21.1660	1.14286	0.571429	+	0.820652i	$0.306389\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	42.3320	1.16033	0.580163	−	0.814500i	$-0.302989\pi$
$11$	42.3320	1.16033	0.580163	+	0.814500i	$0.302989\pi$
$12$	0	0
$13$	20.0000	0.426692	0.213346	−	0.976977i	$-0.431564\pi$
$13$	20.0000	0.426692	0.213346	+	0.976977i	$0.431564\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	8.00000	0.114134	0.0570672	−	0.998370i	$-0.481825\pi$
$17$	8.00000	0.114134	0.0570672	+	0.998370i	$0.481825\pi$
$18$	0	0
$19$	−84.6640	−1.02228	−0.511139	−	0.859498i	$-0.670776\pi$
$19$	−84.6640	−1.02228	−0.511139	+	0.859498i	$0.670776\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−169.328	−1.53510	−0.767551	−	0.640988i	$-0.778525\pi$
$23$	−169.328	−1.53510	−0.767551	+	0.640988i	$0.778525\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	46.0000	0.294551	0.147276	−	0.989095i	$-0.452950\pi$
$29$	46.0000	0.294551	0.147276	+	0.989095i	$0.452950\pi$
$30$	0	0
$31$	21.1660	0.122630	0.0613150	−	0.998118i	$-0.480471\pi$
$31$	21.1660	0.122630	0.0613150	+	0.998118i	$0.480471\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−126.996	−0.613322
$36$	0	0
$37$	−164.000	−0.728687	−0.364344	−	0.931265i	$-0.618707\pi$
$37$	−164.000	−0.728687	−0.364344	+	0.931265i	$0.618707\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	312.000	1.18844	0.594222	−	0.804301i	$-0.297460\pi$
$41$	312.000	1.18844	0.594222	+	0.804301i	$0.297460\pi$
$42$	0	0
$43$	−423.320	−1.50130	−0.750648	−	0.660702i	$-0.770259\pi$
$43$	−423.320	−1.50130	−0.750648	+	0.660702i	$0.770259\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−169.328	−0.525511	−0.262756	−	0.964862i	$-0.584631\pi$
$47$	−169.328	−0.525511	−0.262756	+	0.964862i	$0.584631\pi$
$48$	0	0
$49$	105.000	0.306122
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−266.000	−0.689395	−0.344697	−	0.938714i	$-0.612018\pi$
$53$	−266.000	−0.689395	−0.344697	+	0.938714i	$0.612018\pi$
$54$	0	0
$55$	−253.992	−0.622696
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−253.992	−0.560457	−0.280228	−	0.959933i	$-0.590410\pi$
$59$	−253.992	−0.560457	−0.280228	+	0.959933i	$0.590410\pi$
$60$	0	0
$61$	−132.000	−0.277063	−0.138532	−	0.990358i	$-0.544238\pi$
$61$	−132.000	−0.277063	−0.138532	+	0.990358i	$0.544238\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−120.000	−0.228987
$66$	0	0
$67$	−507.984	−0.926271	−0.463135	−	0.886287i	$-0.653276\pi$
$67$	−507.984	−0.926271	−0.463135	+	0.886287i	$0.653276\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	677.312	1.13214	0.566072	−	0.824356i	$-0.308463\pi$
$71$	677.312	1.13214	0.566072	+	0.824356i	$0.308463\pi$
$72$	0	0
$73$	246.000	0.394413	0.197206	−	0.980362i	$-0.436813\pi$
$73$	246.000	0.394413	0.197206	+	0.980362i	$0.436813\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	896.000	1.32609
$78$	0	0
$79$	−232.826	−0.331582	−0.165791	−	0.986161i	$-0.553018\pi$
$79$	−232.826	−0.331582	−0.165791	+	0.986161i	$0.553018\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	973.636	1.28760	0.643798	−	0.765195i	$-0.277358\pi$
$83$	973.636	1.28760	0.643798	+	0.765195i	$0.277358\pi$
$84$	0	0
$85$	−48.0000	−0.0612510
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1392.00	1.65788	0.828942	−	0.559334i	$-0.188943\pi$
$89$	1392.00	1.65788	0.828942	+	0.559334i	$0.188943\pi$
$90$	0	0
$91$	423.320	0.487649
$92$	0	0
$93$	0	0
$94$	0	0
$95$	507.984	0.548611
$96$	0	0
$97$	−302.000	−0.316118	−0.158059	−	0.987430i	$-0.550524\pi$
$97$	−302.000	−0.316118	−0.158059	+	0.987430i	$0.550524\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1466.00	−1.44428	−0.722141	−	0.691746i	$-0.756842\pi$
$101$	−1466.00	−1.44428	−0.722141	+	0.691746i	$0.756842\pi$
$102$	0	0
$103$	1248.79	1.19463	0.597317	−	0.802005i	$-0.296233\pi$
$103$	1248.79	1.19463	0.597317	+	0.802005i	$0.296233\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	423.320	0.382466	0.191233	−	0.981545i	$-0.438751\pi$
$107$	423.320	0.382466	0.191233	+	0.981545i	$0.438751\pi$
$108$	0	0
$109$	−1564.00	−1.37435	−0.687174	−	0.726492i	$-0.741149\pi$
$109$	−1564.00	−1.37435	−0.687174	+	0.726492i	$0.741149\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1552.00	−1.29203	−0.646017	−	0.763323i	$-0.723567\pi$
$113$	−1552.00	−1.29203	−0.646017	+	0.763323i	$0.723567\pi$
$114$	0	0
$115$	1015.97	0.823822
$116$	0	0
$117$	0	0
$118$	0	0
$119$	169.328	0.130439
$120$	0	0
$121$	461.000	0.346356
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1284.00	0.918756
$126$	0	0
$127$	994.802	0.695074	0.347537	−	0.937666i	$-0.387018\pi$
$127$	994.802	0.695074	0.347537	+	0.937666i	$0.387018\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2539.92	−1.69400	−0.847000	−	0.531593i	$-0.821594\pi$
$131$	−2539.92	−1.69400	−0.847000	+	0.531593i	$0.821594\pi$
$132$	0	0
$133$	−1792.00	−1.16832
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1992.00	1.24225	0.621124	−	0.783712i	$-0.286676\pi$
$137$	1992.00	1.24225	0.621124	+	0.783712i	$0.286676\pi$
$138$	0	0
$139$	1185.30	0.723277	0.361639	−	0.932318i	$-0.382217\pi$
$139$	1185.30	0.723277	0.361639	+	0.932318i	$0.382217\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	846.640	0.495102
$144$	0	0
$145$	−276.000	−0.158073
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1974.00	−1.08534	−0.542672	−	0.839944i	$-0.682587\pi$
$149$	−1974.00	−1.08534	−0.542672	+	0.839944i	$0.682587\pi$
$150$	0	0
$151$	−3111.40	−1.67684	−0.838419	−	0.545027i	$-0.816519\pi$
$151$	−3111.40	−1.67684	−0.838419	+	0.545027i	$0.816519\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−126.996	−0.0658101
$156$	0	0
$157$	−1796.00	−0.912971	−0.456485	−	0.889731i	$-0.650892\pi$
$157$	−1796.00	−0.912971	−0.456485	+	0.889731i	$0.650892\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3584.00	−1.75440
$162$	0	0
$163$	2624.59	1.26119	0.630593	−	0.776114i	$-0.282812\pi$
$163$	2624.59	1.26119	0.630593	+	0.776114i	$0.282812\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3555.89	−1.64768	−0.823841	−	0.566820i	$-0.808173\pi$
$167$	−3555.89	−1.64768	−0.823841	+	0.566820i	$0.808173\pi$
$168$	0	0
$169$	−1797.00	−0.817934
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4290.00	1.88533	0.942667	−	0.333736i	$-0.108309\pi$
$173$	4290.00	1.88533	0.942667	+	0.333736i	$0.108309\pi$
$174$	0	0
$175$	−1883.77	−0.813714
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1608.62	0.671696	0.335848	−	0.941916i	$-0.390977\pi$
$179$	1608.62	0.671696	0.335848	+	0.941916i	$0.390977\pi$
$180$	0	0
$181$	−3196.00	−1.31247	−0.656234	−	0.754557i	$-0.727852\pi$
$181$	−3196.00	−1.31247	−0.656234	+	0.754557i	$0.727852\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	984.000	0.391055
$186$	0	0
$187$	338.656	0.132433
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3217.23	−1.21880	−0.609400	−	0.792863i	$-0.708590\pi$
$191$	−3217.23	−1.21880	−0.609400	+	0.792863i	$0.708590\pi$
$192$	0	0
$193$	462.000	0.172308	0.0861541	−	0.996282i	$-0.472542\pi$
$193$	462.000	0.172308	0.0861541	+	0.996282i	$0.472542\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3834.00	−1.38661	−0.693303	−	0.720647i	$-0.743845\pi$
$197$	−3834.00	−1.38661	−0.693303	+	0.720647i	$0.743845\pi$
$198$	0	0
$199$	3788.72	1.34962	0.674811	−	0.737990i	$-0.264225\pi$
$199$	3788.72	1.34962	0.674811	+	0.737990i	$0.264225\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	973.636	0.336630
$204$	0	0
$205$	−1872.00	−0.637786
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3584.00	−1.18617
$210$	0	0
$211$	−2878.58	−0.939192	−0.469596	−	0.882882i	$-0.655600\pi$
$211$	−2878.58	−0.939192	−0.469596	+	0.882882i	$0.655600\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2539.92	0.805680
$216$	0	0
$217$	448.000	0.140148
$218$	0	0
$219$	0	0
$220$	0	0
$221$	160.000	0.0487003
$222$	0	0
$223$	910.138	0.273307	0.136653	−	0.990619i	$-0.456365\pi$
$223$	910.138	0.273307	0.136653	+	0.990619i	$0.456365\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3259.57	0.953062	0.476531	−	0.879158i	$-0.341894\pi$
$227$	3259.57	0.953062	0.476531	+	0.879158i	$0.341894\pi$
$228$	0	0
$229$	−5260.00	−1.51786	−0.758931	−	0.651171i	$-0.774278\pi$
$229$	−5260.00	−1.51786	−0.758931	+	0.651171i	$0.774278\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1936.00	0.544342	0.272171	−	0.962249i	$-0.412258\pi$
$233$	1936.00	0.544342	0.272171	+	0.962249i	$0.412258\pi$
$234$	0	0
$235$	1015.97	0.282019
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1523.95	−0.412453	−0.206227	−	0.978504i	$-0.566118\pi$
$239$	−1523.95	−0.412453	−0.206227	+	0.978504i	$0.566118\pi$
$240$	0	0
$241$	3202.00	0.855846	0.427923	−	0.903815i	$-0.359245\pi$
$241$	3202.00	0.855846	0.427923	+	0.903815i	$0.359245\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−630.000	−0.164283
$246$	0	0
$247$	−1693.28	−0.436198
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2666.92	−0.670655	−0.335327	−	0.942102i	$-0.608847\pi$
$251$	−2666.92	−0.670655	−0.335327	+	0.942102i	$0.608847\pi$
$252$	0	0
$253$	−7168.00	−1.78122
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3104.00	−0.753394	−0.376697	−	0.926337i	$-0.622940\pi$
$257$	−3104.00	−0.753394	−0.376697	+	0.926337i	$0.622940\pi$
$258$	0	0
$259$	−3471.23	−0.832786
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4741.19	1.11161	0.555806	−	0.831312i	$-0.312410\pi$
$263$	4741.19	1.11161	0.555806	+	0.831312i	$0.312410\pi$
$264$	0	0
$265$	1596.00	0.369968
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4350.00	0.985963	0.492982	−	0.870040i	$-0.335907\pi$
$269$	4350.00	0.985963	0.492982	+	0.870040i	$0.335907\pi$
$270$	0	0
$271$	1079.47	0.241967	0.120983	−	0.992655i	$-0.461395\pi$
$271$	1079.47	0.241967	0.120983	+	0.992655i	$0.461395\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3767.55	−0.826152
$276$	0	0
$277$	−7532.00	−1.63377	−0.816885	−	0.576801i	$-0.804301\pi$
$277$	−7532.00	−1.63377	−0.816885	+	0.576801i	$0.804301\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1824.00	0.387227	0.193613	−	0.981078i	$-0.437979\pi$
$281$	1824.00	0.387227	0.193613	+	0.981078i	$0.437979\pi$
$282$	0	0
$283$	−8805.06	−1.84949	−0.924746	−	0.380584i	$-0.875723\pi$
$283$	−8805.06	−1.84949	−0.924746	+	0.380584i	$0.875723\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6603.80	1.35822
$288$	0	0
$289$	−4849.00	−0.986973
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−858.000	−0.171075	−0.0855374	−	0.996335i	$-0.527261\pi$
$293$	−858.000	−0.171075	−0.0855374	+	0.996335i	$0.527261\pi$
$294$	0	0
$295$	1523.95	0.300773
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3386.56	−0.655016
$300$	0	0
$301$	−8960.00	−1.71577
$302$	0	0
$303$	0	0
$304$	0	0
$305$	792.000	0.148688
$306$	0	0
$307$	1523.95	0.283311	0.141656	−	0.989916i	$-0.454757\pi$
$307$	1523.95	0.283311	0.141656	+	0.989916i	$0.454757\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4402.53	−0.802716	−0.401358	−	0.915921i	$-0.631462\pi$
$311$	−4402.53	−0.802716	−0.401358	+	0.915921i	$0.631462\pi$
$312$	0	0
$313$	118.000	0.0213091	0.0106546	−	0.999943i	$-0.496608\pi$
$313$	118.000	0.0213091	0.0106546	+	0.999943i	$0.496608\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4594.00	−0.813958	−0.406979	−	0.913437i	$-0.633418\pi$
$317$	−4594.00	−0.813958	−0.406979	+	0.913437i	$0.633418\pi$
$318$	0	0
$319$	1947.27	0.341775
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−677.312	−0.116677
$324$	0	0
$325$	−1780.00	−0.303805
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3584.00	−0.600585
$330$	0	0
$331$	−3725.22	−0.618600	−0.309300	−	0.950965i	$-0.600095\pi$
$331$	−3725.22	−0.618600	−0.309300	+	0.950965i	$0.600095\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3047.91	0.497089
$336$	0	0
$337$	1166.00	0.188475	0.0942375	−	0.995550i	$-0.469959\pi$
$337$	1166.00	0.188475	0.0942375	+	0.995550i	$0.469959\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	896.000	0.142291
$342$	0	0
$343$	−5037.51	−0.793003
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11302.6	1.74858	0.874291	−	0.485402i	$-0.161327\pi$
$347$	11302.6	1.74858	0.874291	+	0.485402i	$0.161327\pi$
$348$	0	0
$349$	−6388.00	−0.979776	−0.489888	−	0.871785i	$-0.662962\pi$
$349$	−6388.00	−0.979776	−0.489888	+	0.871785i	$0.662962\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8432.00	1.27136	0.635680	−	0.771953i	$-0.280720\pi$
$353$	8432.00	1.27136	0.635680	+	0.771953i	$0.280720\pi$
$354$	0	0
$355$	−4063.87	−0.607572
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2031.94	0.298723	0.149361	−	0.988783i	$-0.452278\pi$
$359$	2031.94	0.298723	0.149361	+	0.988783i	$0.452278\pi$
$360$	0	0
$361$	309.000	0.0450503
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1476.00	−0.211664
$366$	0	0
$367$	−5397.33	−0.767680	−0.383840	−	0.923400i	$-0.625399\pi$
$367$	−5397.33	−0.767680	−0.383840	+	0.923400i	$0.625399\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5630.16	−0.787879
$372$	0	0
$373$	−8276.00	−1.14883	−0.574417	−	0.818563i	$-0.694771\pi$
$373$	−8276.00	−1.14883	−0.574417	+	0.818563i	$0.694771\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	920.000	0.125683
$378$	0	0
$379$	13800.2	1.87037	0.935186	−	0.354158i	$-0.115233\pi$
$379$	13800.2	1.87037	0.935186	+	0.354158i	$0.115233\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1862.61	−0.248498	−0.124249	−	0.992251i	$-0.539652\pi$
$383$	−1862.61	−0.248498	−0.124249	+	0.992251i	$0.539652\pi$
$384$	0	0
$385$	−5376.00	−0.711653
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6170.00	0.804194	0.402097	−	0.915597i	$-0.368281\pi$
$389$	6170.00	0.804194	0.402097	+	0.915597i	$0.368281\pi$
$390$	0	0
$391$	−1354.62	−0.175208
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1396.96	0.177946
$396$	0	0
$397$	−8644.00	−1.09277	−0.546385	−	0.837534i	$-0.683997\pi$
$397$	−8644.00	−1.09277	−0.546385	+	0.837534i	$0.683997\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8152.00	1.01519	0.507595	−	0.861596i	$-0.330535\pi$
$401$	8152.00	1.01519	0.507595	+	0.861596i	$0.330535\pi$
$402$	0	0
$403$	423.320	0.0523253
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6942.45	−0.845515
$408$	0	0
$409$	874.000	0.105664	0.0528319	−	0.998603i	$-0.483175\pi$
$409$	874.000	0.105664	0.0528319	+	0.998603i	$0.483175\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5376.00	−0.640522
$414$	0	0
$415$	−5841.82	−0.690997
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14265.9	1.66333	0.831664	−	0.555279i	$-0.187389\pi$
$419$	14265.9	1.66333	0.831664	+	0.555279i	$0.187389\pi$
$420$	0	0
$421$	−11564.0	−1.33871	−0.669353	−	0.742945i	$-0.733428\pi$
$421$	−11564.0	−1.33871	−0.669353	+	0.742945i	$0.733428\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−712.000	−0.0812637
$426$	0	0
$427$	−2793.91	−0.316644
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7619.76	−0.851580	−0.425790	−	0.904822i	$-0.640004\pi$
$431$	−7619.76	−0.851580	−0.425790	+	0.904822i	$0.640004\pi$
$432$	0	0
$433$	7550.00	0.837944	0.418972	−	0.907999i	$-0.362391\pi$
$433$	7550.00	0.837944	0.418972	+	0.907999i	$0.362391\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14336.0	1.56930
$438$	0	0
$439$	17335.0	1.88463	0.942315	−	0.334727i	$-0.108644\pi$
$439$	17335.0	1.88463	0.942315	+	0.334727i	$0.108644\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9016.72	−0.967037	−0.483518	−	0.875334i	$-0.660641\pi$
$443$	−9016.72	−0.967037	−0.483518	+	0.875334i	$0.660641\pi$
$444$	0	0
$445$	−8352.00	−0.889714
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6568.00	0.690341	0.345170	−	0.938540i	$-0.387821\pi$
$449$	6568.00	0.690341	0.345170	+	0.938540i	$0.387821\pi$
$450$	0	0
$451$	13207.6	1.37898
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2539.92	−0.261700
$456$	0	0
$457$	−9478.00	−0.970158	−0.485079	−	0.874470i	$-0.661209\pi$
$457$	−9478.00	−0.970158	−0.485079	+	0.874470i	$0.661209\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12606.0	−1.27358	−0.636790	−	0.771038i	$-0.719738\pi$
$461$	−12606.0	−1.27358	−0.636790	+	0.771038i	$0.719738\pi$
$462$	0	0
$463$	14371.7	1.44257	0.721286	−	0.692638i	$-0.243552\pi$
$463$	14371.7	1.44257	0.721286	+	0.692638i	$0.243552\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4698.85	−0.465604	−0.232802	−	0.972524i	$-0.574789\pi$
$467$	−4698.85	−0.465604	−0.232802	+	0.972524i	$0.574789\pi$
$468$	0	0
$469$	−10752.0	−1.05860
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−17920.0	−1.74199
$474$	0	0
$475$	7535.10	0.727861
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10837.0	−1.03373	−0.516863	−	0.856068i	$-0.672900\pi$
$479$	−10837.0	−1.03373	−0.516863	+	0.856068i	$0.672900\pi$
$480$	0	0
$481$	−3280.00	−0.310925
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1812.00	0.169647
$486$	0	0
$487$	−13355.8	−1.24272	−0.621362	−	0.783523i	$-0.713420\pi$
$487$	−13355.8	−1.24272	−0.621362	+	0.783523i	$0.713420\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13715.6	−1.26064	−0.630321	−	0.776335i	$-0.717077\pi$
$491$	−13715.6	−1.26064	−0.630321	+	0.776335i	$0.717077\pi$
$492$	0	0
$493$	368.000	0.0336184
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14336.0	1.29388
$498$	0	0
$499$	6773.12	0.607629	0.303814	−	0.952731i	$-0.401740\pi$
$499$	6773.12	0.607629	0.303814	+	0.952731i	$0.401740\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11175.7	0.990652	0.495326	−	0.868707i	$-0.335049\pi$
$503$	11175.7	0.990652	0.495326	+	0.868707i	$0.335049\pi$
$504$	0	0
$505$	8796.00	0.775083
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13458.0	−1.17194	−0.585968	−	0.810334i	$-0.699286\pi$
$509$	−13458.0	−1.17194	−0.585968	+	0.810334i	$0.699286\pi$
$510$	0	0
$511$	5206.84	0.450757
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7492.77	−0.641108
$516$	0	0
$517$	−7168.00	−0.609765
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5352.00	−0.450049	−0.225024	−	0.974353i	$-0.572246\pi$
$521$	−5352.00	−0.450049	−0.225024	+	0.974353i	$0.572246\pi$
$522$	0	0
$523$	6011.15	0.502580	0.251290	−	0.967912i	$-0.419145\pi$
$523$	6011.15	0.502580	0.251290	+	0.967912i	$0.419145\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	169.328	0.0139963
$528$	0	0
$529$	16505.0	1.35654
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6240.00	0.507100
$534$	0	0
$535$	−2539.92	−0.205253
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4444.86	0.355202
$540$	0	0
$541$	−2396.00	−0.190411	−0.0952053	−	0.995458i	$-0.530351\pi$
$541$	−2396.00	−0.190411	−0.0952053	+	0.995458i	$0.530351\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9384.00	0.737553
$546$	0	0
$547$	−1269.96	−0.0992680	−0.0496340	−	0.998767i	$-0.515805\pi$
$547$	−1269.96	−0.0992680	−0.0496340	+	0.998767i	$0.515805\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3894.55	−0.301113
$552$	0	0
$553$	−4928.00	−0.378951
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−254.000	−0.0193219	−0.00966097	−	0.999953i	$-0.503075\pi$
$557$	−254.000	−0.0193219	−0.00966097	+	0.999953i	$0.503075\pi$
$558$	0	0
$559$	−8466.40	−0.640592
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10032.7	0.751026	0.375513	−	0.926817i	$-0.377467\pi$
$563$	10032.7	0.751026	0.375513	+	0.926817i	$0.377467\pi$
$564$	0	0
$565$	9312.00	0.693378
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9320.00	−0.686669	−0.343335	−	0.939213i	$-0.611556\pi$
$569$	−9320.00	−0.686669	−0.343335	+	0.939213i	$0.611556\pi$
$570$	0	0
$571$	−13376.9	−0.980397	−0.490198	−	0.871611i	$-0.663076\pi$
$571$	−13376.9	−0.980397	−0.490198	+	0.871611i	$0.663076\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15070.2	1.09299
$576$	0	0
$577$	−13758.0	−0.992640	−0.496320	−	0.868140i	$-0.665316\pi$
$577$	−13758.0	−0.992640	−0.496320	+	0.868140i	$0.665316\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	20608.0	1.47154
$582$	0	0
$583$	−11260.3	−0.799922
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26076.5	1.83355	0.916775	−	0.399405i	$-0.130783\pi$
$587$	26076.5	1.83355	0.916775	+	0.399405i	$0.130783\pi$
$588$	0	0
$589$	−1792.00	−0.125362
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4080.00	−0.282539	−0.141269	−	0.989971i	$-0.545118\pi$
$593$	−4080.00	−0.282539	−0.141269	+	0.989971i	$0.545118\pi$
$594$	0	0
$595$	−1015.97	−0.0700011
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15747.5	1.07417	0.537083	−	0.843529i	$-0.319526\pi$
$599$	15747.5	1.07417	0.537083	+	0.843529i	$0.319526\pi$
$600$	0	0
$601$	10298.0	0.698942	0.349471	−	0.936947i	$-0.386361\pi$
$601$	10298.0	0.698942	0.349471	+	0.936947i	$0.386361\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2766.00	−0.185874
$606$	0	0
$607$	−23219.1	−1.55261	−0.776305	−	0.630357i	$-0.782909\pi$
$607$	−23219.1	−1.55261	−0.776305	+	0.630357i	$0.782909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3386.56	−0.224232
$612$	0	0
$613$	−5652.00	−0.372402	−0.186201	−	0.982512i	$-0.559617\pi$
$613$	−5652.00	−0.372402	−0.186201	+	0.982512i	$0.559617\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26848.0	−1.75180	−0.875899	−	0.482494i	$-0.839731\pi$
$617$	−26848.0	−1.75180	−0.875899	+	0.482494i	$0.839731\pi$
$618$	0	0
$619$	3725.22	0.241889	0.120944	−	0.992659i	$-0.461408\pi$
$619$	3725.22	0.241889	0.120944	+	0.992659i	$0.461408\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	29463.1	1.89472
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1312.00	−0.0831683
$630$	0	0
$631$	27452.3	1.73195	0.865974	−	0.500089i	$-0.166699\pi$
$631$	27452.3	1.73195	0.865974	+	0.500089i	$0.166699\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5968.81	−0.373016
$636$	0	0
$637$	2100.00	0.130620
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6024.00	0.371191	0.185596	−	0.982626i	$-0.440579\pi$
$641$	6024.00	0.371191	0.185596	+	0.982626i	$0.440579\pi$
$642$	0	0
$643$	−24637.2	−1.51104	−0.755519	−	0.655126i	$-0.772615\pi$
$643$	−24637.2	−1.51104	−0.755519	+	0.655126i	$0.772615\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20827.4	−1.26555	−0.632773	−	0.774338i	$-0.718083\pi$
$647$	−20827.4	−1.26555	−0.632773	+	0.774338i	$0.718083\pi$
$648$	0	0
$649$	−10752.0	−0.650313
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−29054.0	−1.74115	−0.870575	−	0.492036i	$-0.836253\pi$
$653$	−29054.0	−1.74115	−0.870575	+	0.492036i	$0.836253\pi$
$654$	0	0
$655$	15239.5	0.909096
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3047.91	0.180166	0.0900831	−	0.995934i	$-0.471287\pi$
$659$	3047.91	0.180166	0.0900831	+	0.995934i	$0.471287\pi$
$660$	0	0
$661$	9628.00	0.566544	0.283272	−	0.959040i	$-0.408580\pi$
$661$	9628.00	0.566544	0.283272	+	0.959040i	$0.408580\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10752.0	0.626984
$666$	0	0
$667$	−7789.09	−0.452166
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5587.83	−0.321484
$672$	0	0
$673$	−28690.0	−1.64327	−0.821633	−	0.570017i	$-0.806937\pi$
$673$	−28690.0	−1.64327	−0.821633	+	0.570017i	$0.806937\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4922.00	0.279421	0.139710	−	0.990192i	$-0.455383\pi$
$677$	4922.00	0.279421	0.139710	+	0.990192i	$0.455383\pi$
$678$	0	0
$679$	−6392.14	−0.361278
$680$	0	0
$681$	0	0
$682$	0	0
$683$	42.3320	0.00237158	0.00118579	−	0.999999i	$-0.499623\pi$
$683$	42.3320	0.00237158	0.00118579	+	0.999999i	$0.499623\pi$
$684$	0	0
$685$	−11952.0	−0.666661
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5320.00	−0.294159
$690$	0	0
$691$	−18372.1	−1.01144	−0.505722	−	0.862697i	$-0.668774\pi$
$691$	−18372.1	−1.01144	−0.505722	+	0.862697i	$0.668774\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7111.78	−0.388151
$696$	0	0
$697$	2496.00	0.135642
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19310.0	1.04041	0.520206	−	0.854041i	$-0.325855\pi$
$701$	19310.0	1.04041	0.520206	+	0.854041i	$0.325855\pi$
$702$	0	0
$703$	13884.9	0.744920
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−31029.4	−1.65061
$708$	0	0
$709$	18820.0	0.996897	0.498448	−	0.866919i	$-0.333903\pi$
$709$	18820.0	0.996897	0.498448	+	0.866919i	$0.333903\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3584.00	−0.188249
$714$	0	0
$715$	−5079.84	−0.265700
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27431.1	−1.42282	−0.711411	−	0.702776i	$-0.751944\pi$
$719$	−27431.1	−1.42282	−0.711411	+	0.702776i	$0.751944\pi$
$720$	0	0
$721$	26432.0	1.36530
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4094.00	−0.209720
$726$	0	0
$727$	−10561.8	−0.538813	−0.269406	−	0.963027i	$-0.586827\pi$
$727$	−10561.8	−0.538813	−0.269406	+	0.963027i	$0.586827\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3386.56	−0.171350
$732$	0	0
$733$	16468.0	0.829822	0.414911	−	0.909862i	$-0.363813\pi$
$733$	16468.0	0.829822	0.414911	+	0.909862i	$0.363813\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21504.0	−1.07478
$738$	0	0
$739$	15070.2	0.750157	0.375079	−	0.926993i	$-0.377616\pi$
$739$	15070.2	0.750157	0.375079	+	0.926993i	$0.377616\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	34373.6	1.69723	0.848617	−	0.529007i	$-0.177436\pi$
$743$	34373.6	1.69723	0.848617	+	0.529007i	$0.177436\pi$
$744$	0	0
$745$	11844.0	0.582457
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8960.00	0.437105
$750$	0	0
$751$	−1756.78	−0.0853605	−0.0426803	−	0.999089i	$-0.513590\pi$
$751$	−1756.78	−0.0853605	−0.0426803	+	0.999089i	$0.513590\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18668.4	0.899885
$756$	0	0
$757$	1572.00	0.0754760	0.0377380	−	0.999288i	$-0.487985\pi$
$757$	1572.00	0.0754760	0.0377380	+	0.999288i	$0.487985\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−17816.0	−0.848659	−0.424329	−	0.905508i	$-0.639490\pi$
$761$	−17816.0	−0.848659	−0.424329	+	0.905508i	$0.639490\pi$
$762$	0	0
$763$	−33103.6	−1.57068
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5079.84	−0.239143
$768$	0	0
$769$	−21746.0	−1.01974	−0.509870	−	0.860251i	$-0.670307\pi$
$769$	−21746.0	−1.01974	−0.509870	+	0.860251i	$0.670307\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13978.0	−0.650393	−0.325196	−	0.945646i	$-0.605430\pi$
$773$	−13978.0	−0.650393	−0.325196	+	0.945646i	$0.605430\pi$
$774$	0	0
$775$	−1883.77	−0.0873125
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−26415.2	−1.21492
$780$	0	0
$781$	28672.0	1.31366
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10776.0	0.489952
$786$	0	0
$787$	1777.94	0.0805297	0.0402649	−	0.999189i	$-0.487180\pi$
$787$	1777.94	0.0805297	0.0402649	+	0.999189i	$0.487180\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−32849.6	−1.47661
$792$	0	0
$793$	−2640.00	−0.118221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9650.00	0.428884	0.214442	−	0.976737i	$-0.431207\pi$
$797$	9650.00	0.428884	0.214442	+	0.976737i	$0.431207\pi$
$798$	0	0
$799$	−1354.62	−0.0599789
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10413.7	0.457647
$804$	0	0
$805$	21504.0	0.941511
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38024.0	1.65248	0.826238	−	0.563322i	$-0.190477\pi$
$809$	38024.0	1.65248	0.826238	+	0.563322i	$0.190477\pi$
$810$	0	0
$811$	−931.304	−0.0403237	−0.0201619	−	0.999797i	$-0.506418\pi$
$811$	−931.304	−0.0403237	−0.0201619	+	0.999797i	$0.506418\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−15747.5	−0.676824
$816$	0	0
$817$	35840.0	1.53474
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6858.00	−0.291530	−0.145765	−	0.989319i	$-0.546564\pi$
$821$	−6858.00	−0.291530	−0.145765	+	0.989319i	$0.546564\pi$
$822$	0	0
$823$	18816.6	0.796968	0.398484	−	0.917175i	$-0.369536\pi$
$823$	18816.6	0.796968	0.398484	+	0.917175i	$0.369536\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8212.41	−0.345313	−0.172656	−	0.984982i	$-0.555235\pi$
$827$	−8212.41	−0.345313	−0.172656	+	0.984982i	$0.555235\pi$
$828$	0	0
$829$	25124.0	1.05258	0.526292	−	0.850304i	$-0.323582\pi$
$829$	25124.0	1.05258	0.526292	+	0.850304i	$0.323582\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	840.000	0.0349391
$834$	0	0
$835$	21335.3	0.884239
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6095.81	0.250835	0.125418	−	0.992104i	$-0.459973\pi$
$839$	6095.81	0.250835	0.125418	+	0.992104i	$0.459973\pi$
$840$	0	0
$841$	−22273.0	−0.913240
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10782.0	0.438949
$846$	0	0
$847$	9757.53	0.395836
$848$	0	0
$849$	0	0
$850$	0	0
$851$	27769.8	1.11861
$852$	0	0
$853$	32812.0	1.31707	0.658535	−	0.752550i	$-0.271176\pi$
$853$	32812.0	1.31707	0.658535	+	0.752550i	$0.271176\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21944.0	0.874671	0.437335	−	0.899299i	$-0.355922\pi$
$857$	21944.0	0.874671	0.437335	+	0.899299i	$0.355922\pi$
$858$	0	0
$859$	48173.8	1.91347	0.956735	−	0.290962i	$-0.0939752\pi$
$859$	48173.8	1.91347	0.956735	+	0.290962i	$0.0939752\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12361.0	−0.487569	−0.243784	−	0.969829i	$-0.578389\pi$
$863$	−12361.0	−0.487569	−0.243784	+	0.969829i	$0.578389\pi$
$864$	0	0
$865$	−25740.0	−1.01178
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9856.00	−0.384743
$870$	0	0
$871$	−10159.7	−0.395233
$872$	0	0
$873$	0	0
$874$	0	0
$875$	27177.2	1.05001
$876$	0	0
$877$	33212.0	1.27878	0.639390	−	0.768883i	$-0.279187\pi$
$877$	33212.0	1.27878	0.639390	+	0.768883i	$0.279187\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−34336.0	−1.31306	−0.656532	−	0.754298i	$-0.727977\pi$
$881$	−34336.0	−1.31306	−0.656532	+	0.754298i	$0.727977\pi$
$882$	0	0
$883$	−16678.8	−0.635659	−0.317829	−	0.948148i	$-0.602954\pi$
$883$	−16678.8	−0.635659	−0.317829	+	0.948148i	$0.602954\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5079.84	−0.192294	−0.0961468	−	0.995367i	$-0.530652\pi$
$887$	−5079.84	−0.192294	−0.0961468	+	0.995367i	$0.530652\pi$
$888$	0	0
$889$	21056.0	0.794371
$890$	0	0
$891$	0	0
$892$	0	0
$893$	14336.0	0.537218
$894$	0	0
$895$	−9651.70	−0.360470
$896$	0	0
$897$	0	0
$898$	0	0
$899$	973.636	0.0361208
$900$	0	0
$901$	−2128.00	−0.0786836
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19176.0	0.704345
$906$	0	0
$907$	13292.3	0.486617	0.243309	−	0.969949i	$-0.421767\pi$
$907$	13292.3	0.486617	0.243309	+	0.969949i	$0.421767\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2539.92	0.0923725	0.0461862	−	0.998933i	$-0.485293\pi$
$911$	2539.92	0.0923725	0.0461862	+	0.998933i	$0.485293\pi$
$912$	0	0
$913$	41216.0	1.49403
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−53760.0	−1.93600
$918$	0	0
$919$	−35368.4	−1.26953	−0.634764	−	0.772706i	$-0.718903\pi$
$919$	−35368.4	−1.26953	−0.634764	+	0.772706i	$0.718903\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13546.2	0.483077
$924$	0	0
$925$	14596.0	0.518825
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−17576.0	−0.620721	−0.310361	−	0.950619i	$-0.600450\pi$
$929$	−17576.0	−0.620721	−0.310361	+	0.950619i	$0.600450\pi$
$930$	0	0
$931$	−8889.72	−0.312942
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2031.94	−0.0710711
$936$	0	0
$937$	−21526.0	−0.750506	−0.375253	−	0.926923i	$-0.622444\pi$
$937$	−21526.0	−0.750506	−0.375253	+	0.926923i	$0.622444\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5218.00	0.180767	0.0903836	−	0.995907i	$-0.471191\pi$
$941$	5218.00	0.180767	0.0903836	+	0.995907i	$0.471191\pi$
$942$	0	0
$943$	−52830.4	−1.82438
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29717.1	−1.01972	−0.509860	−	0.860257i	$-0.670303\pi$
$947$	−29717.1	−1.01972	−0.509860	+	0.860257i	$0.670303\pi$
$948$	0	0
$949$	4920.00	0.168293
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48888.0	1.66174	0.830870	−	0.556467i	$-0.187843\pi$
$953$	48888.0	1.66174	0.830870	+	0.556467i	$0.187843\pi$
$954$	0	0
$955$	19303.4	0.654077
$956$	0	0
$957$	0	0
$958$	0	0
$959$	42162.7	1.41971
$960$	0	0
$961$	−29343.0	−0.984962
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2772.00	−0.0924703
$966$	0	0
$967$	42988.2	1.42958	0.714791	−	0.699338i	$-0.246522\pi$
$967$	42988.2	1.42958	0.714791	+	0.699338i	$0.246522\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12403.3	0.409928	0.204964	−	0.978769i	$-0.434292\pi$
$971$	12403.3	0.409928	0.204964	+	0.978769i	$0.434292\pi$
$972$	0	0
$973$	25088.0	0.826603
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2040.00	0.0668018	0.0334009	−	0.999442i	$-0.489366\pi$
$977$	2040.00	0.0668018	0.0334009	+	0.999442i	$0.489366\pi$
$978$	0	0
$979$	58926.2	1.92369
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20658.0	0.670284	0.335142	−	0.942168i	$-0.391216\pi$
$983$	20658.0	0.670284	0.335142	+	0.942168i	$0.391216\pi$
$984$	0	0
$985$	23004.0	0.744130
$986$	0	0
$987$	0	0
$988$	0	0
$989$	71680.0	2.30464
$990$	0	0
$991$	10773.5	0.345340	0.172670	−	0.984980i	$-0.444761\pi$
$991$	10773.5	0.345340	0.172670	+	0.984980i	$0.444761\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−22732.3	−0.724284
$996$	0	0
$997$	62044.0	1.97087	0.985433	−	0.170065i	$-0.0543977\pi$
$997$	62044.0	1.97087	0.985433	+	0.170065i	$0.0543977\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.4.a.r.1.2		2
3.2	odd	2		2304.4.a.bs.1.2		2
4.3	odd	2	inner	2304.4.a.r.1.1		2
8.3	odd	2		2304.4.a.br.1.1		2
8.5	even	2		2304.4.a.br.1.2		2
12.11	even	2		2304.4.a.bs.1.1		2
16.3	odd	4		1152.4.d.m.577.4	yes	4
16.5	even	4		1152.4.d.m.577.1	yes	4
16.11	odd	4		1152.4.d.m.577.2	yes	4
16.13	even	4		1152.4.d.m.577.3	yes	4
24.5	odd	2		2304.4.a.q.1.2		2
24.11	even	2		2304.4.a.q.1.1		2
48.5	odd	4		1152.4.d.k.577.3	yes	4
48.11	even	4		1152.4.d.k.577.4	yes	4
48.29	odd	4		1152.4.d.k.577.1	✓	4
48.35	even	4		1152.4.d.k.577.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.d.k.577.1	✓	4	48.29	odd	4
1152.4.d.k.577.2	yes	4	48.35	even	4
1152.4.d.k.577.3	yes	4	48.5	odd	4
1152.4.d.k.577.4	yes	4	48.11	even	4
1152.4.d.m.577.1	yes	4	16.5	even	4
1152.4.d.m.577.2	yes	4	16.11	odd	4
1152.4.d.m.577.3	yes	4	16.13	even	4
1152.4.d.m.577.4	yes	4	16.3	odd	4
2304.4.a.q.1.1		2	24.11	even	2
2304.4.a.q.1.2		2	24.5	odd	2
2304.4.a.r.1.1		2	4.3	odd	2	inner
2304.4.a.r.1.2		2	1.1	even	1	trivial
2304.4.a.br.1.1		2	8.3	odd	2
2304.4.a.br.1.2		2	8.5	even	2
2304.4.a.bs.1.1		2	12.11	even	2
2304.4.a.bs.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2304.a (trivial)

\(f(q)\)	\(=\)	\(q-6.00000 q^{5} +21.1660 q^{7} +42.3320 q^{11} +20.0000 q^{13} +8.00000 q^{17} -84.6640 q^{19} -169.328 q^{23} -89.0000 q^{25} +46.0000 q^{29} +21.1660 q^{31} -126.996 q^{35} -164.000 q^{37} +312.000 q^{41} -423.320 q^{43} -169.328 q^{47} +105.000 q^{49} -266.000 q^{53} -253.992 q^{55} -253.992 q^{59} -132.000 q^{61} -120.000 q^{65} -507.984 q^{67} +677.312 q^{71} +246.000 q^{73} +896.000 q^{77} -232.826 q^{79} +973.636 q^{83} -48.0000 q^{85} +1392.00 q^{89} +423.320 q^{91} +507.984 q^{95} -302.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{5} + 40 q^{13} + 16 q^{17} - 178 q^{25} + 92 q^{29} - 328 q^{37} + 624 q^{41} + 210 q^{49} - 532 q^{53} - 264 q^{61} - 240 q^{65} + 492 q^{73} + 1792 q^{77} - 96 q^{85} + 2784 q^{89} - 604 q^{97}+O(q^{100}) \) 2 * q - 12 * q^5 + 40 * q^13 + 16 * q^17 - 178 * q^25 + 92 * q^29 - 328 * q^37 + 624 * q^41 + 210 * q^49 - 532 * q^53 - 264 * q^61 - 240 * q^65 + 492 * q^73 + 1792 * q^77 - 96 * q^85 + 2784 * q^89 - 604 * q^97

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