LMFDB - Embedded newform 2304.3.h.f.2177.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(2177,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2304.2177");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.h (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$62.7794529086$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2177.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2304.2177
Dual form			2304.3.h.f.2177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.24264 q^{5} +4.00000 q^{7} +O(q^{10})$	$q-4.24264 q^{5} +4.00000 q^{7} +16.9706 q^{11} +8.00000i q^{13} -12.7279i q^{17} -16.0000i q^{19} +16.9706i q^{23} -7.00000 q^{25} -4.24264 q^{29} +44.0000 q^{31} -16.9706 q^{35} +34.0000i q^{37} -46.6690i q^{41} +40.0000i q^{43} -84.8528i q^{47} -33.0000 q^{49} +38.1838 q^{53} -72.0000 q^{55} +33.9411 q^{59} +50.0000i q^{61} -33.9411i q^{65} +8.00000i q^{67} +50.9117i q^{71} +16.0000 q^{73} +67.8823 q^{77} -76.0000 q^{79} -118.794 q^{83} +54.0000i q^{85} -12.7279i q^{89} +32.0000i q^{91} +67.8823i q^{95} +176.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{7}+O(q^{10}) $ 4 * q + 16 * q^7	$ 4 q + 16 q^{7} - 28 q^{25} + 176 q^{31} - 132 q^{49} - 288 q^{55} + 64 q^{73} - 304 q^{79} + 704 q^{97}+O(q^{100}) $ 4 * q + 16 * q^7 - 28 * q^25 + 176 * q^31 - 132 * q^49 - 288 * q^55 + 64 * q^73 - 304 * q^79 + 704 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	−4.24264	−0.848528	−0.424264	−	0.905539i	$-0.639467\pi$
$5$	−4.24264	−0.848528	−0.424264	+	0.905539i	$0.639467\pi$
$6$	0	0
$7$	4.00000	0.571429	0.285714	−	0.958315i	$-0.407769\pi$
$7$	4.00000	0.571429	0.285714	+	0.958315i	$0.407769\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	16.9706	1.54278	0.771389	−	0.636364i	$-0.219562\pi$
$11$	16.9706	1.54278	0.771389	+	0.636364i	$0.219562\pi$
$12$	0	0
$13$	8.00000i	0.615385i	0.951486	+	0.307692i	$0.0995567\pi$
$13$	8.00000i	0.615385i	−0.951486	+	0.307692i	$0.900443\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 12.7279i	− 0.748701i	−0.927287	−	0.374351i	$-0.877866\pi$
$17$	− 12.7279i	− 0.748701i	0.927287	−	0.374351i	$-0.122134\pi$
$18$	0	0
$19$	− 16.0000i	− 0.842105i	−0.907036	−	0.421053i	$-0.861661\pi$
$19$	− 16.0000i	− 0.842105i	0.907036	−	0.421053i	$-0.138339\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	16.9706i	0.737851i	0.929459	+	0.368925i	$0.120274\pi$
$23$	16.9706i	0.737851i	−0.929459	+	0.368925i	$0.879726\pi$
$24$	0	0
$25$	−7.00000	−0.280000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.24264	−0.146298	−0.0731490	−	0.997321i	$-0.523305\pi$
$29$	−4.24264	−0.146298	−0.0731490	+	0.997321i	$0.523305\pi$
$30$	0	0
$31$	44.0000	1.41935	0.709677	−	0.704527i	$-0.248841\pi$
$31$	44.0000	1.41935	0.709677	+	0.704527i	$0.248841\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−16.9706	−0.484873
$36$	0	0
$37$	34.0000i	0.918919i	0.888199	+	0.459459i	$0.151957\pi$
$37$	34.0000i	0.918919i	−0.888199	+	0.459459i	$0.848043\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 46.6690i	− 1.13827i	−0.822244	−	0.569135i	$-0.807278\pi$
$41$	− 46.6690i	− 1.13827i	0.822244	−	0.569135i	$-0.192722\pi$
$42$	0	0
$43$	40.0000i	0.930233i	0.885250	+	0.465116i	$0.153987\pi$
$43$	40.0000i	0.930233i	−0.885250	+	0.465116i	$0.846013\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 84.8528i	− 1.80538i	−0.430293	−	0.902690i	$-0.641590\pi$
$47$	− 84.8528i	− 1.80538i	0.430293	−	0.902690i	$-0.358410\pi$
$48$	0	0
$49$	−33.0000	−0.673469
$50$	0	0
$51$	0	0
$52$	0	0
$53$	38.1838	0.720448	0.360224	−	0.932866i	$-0.382700\pi$
$53$	38.1838	0.720448	0.360224	+	0.932866i	$0.382700\pi$
$54$	0	0
$55$	−72.0000	−1.30909
$56$	0	0
$57$	0	0
$58$	0	0
$59$	33.9411	0.575273	0.287637	−	0.957740i	$-0.407130\pi$
$59$	33.9411	0.575273	0.287637	+	0.957740i	$0.407130\pi$
$60$	0	0
$61$	50.0000i	0.819672i	0.912159	+	0.409836i	$0.134414\pi$
$61$	50.0000i	0.819672i	−0.912159	+	0.409836i	$0.865586\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 33.9411i	− 0.522171i
$66$	0	0
$67$	8.00000i	0.119403i	0.998216	+	0.0597015i	$0.0190149\pi$
$67$	8.00000i	0.119403i	−0.998216	+	0.0597015i	$0.980985\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	50.9117i	0.717066i	0.933517	+	0.358533i	$0.116723\pi$
$71$	50.9117i	0.717066i	−0.933517	+	0.358533i	$0.883277\pi$
$72$	0	0
$73$	16.0000	0.219178	0.109589	−	0.993977i	$-0.465047\pi$
$73$	16.0000	0.219178	0.109589	+	0.993977i	$0.465047\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	67.8823	0.881588
$78$	0	0
$79$	−76.0000	−0.962025	−0.481013	−	0.876714i	$-0.659731\pi$
$79$	−76.0000	−0.962025	−0.481013	+	0.876714i	$0.659731\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−118.794	−1.43125	−0.715626	−	0.698484i	$-0.753859\pi$
$83$	−118.794	−1.43125	−0.715626	+	0.698484i	$0.753859\pi$
$84$	0	0
$85$	54.0000i	0.635294i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12.7279i	− 0.143010i	−0.997440	−	0.0715052i	$-0.977220\pi$
$89$	− 12.7279i	− 0.143010i	0.997440	−	0.0715052i	$-0.0227802\pi$
$90$	0	0
$91$	32.0000i	0.351648i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	67.8823i	0.714550i
$96$	0	0
$97$	176.000	1.81443	0.907216	−	0.420664i	$-0.138203\pi$
$97$	176.000	1.81443	0.907216	+	0.420664i	$0.138203\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	29.6985	0.294044	0.147022	−	0.989133i	$-0.453031\pi$
$101$	29.6985	0.294044	0.147022	+	0.989133i	$0.453031\pi$
$102$	0	0
$103$	28.0000	0.271845	0.135922	−	0.990719i	$-0.456600\pi$
$103$	28.0000	0.271845	0.135922	+	0.990719i	$0.456600\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	56.0000i	0.513761i	0.966443	+	0.256881i	$0.0826948\pi$
$109$	56.0000i	0.513761i	−0.966443	+	0.256881i	$0.917305\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 156.978i	− 1.38918i	−0.719404	−	0.694592i	$-0.755585\pi$
$113$	− 156.978i	− 1.38918i	0.719404	−	0.694592i	$-0.244415\pi$
$114$	0	0
$115$	− 72.0000i	− 0.626087i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 50.9117i	− 0.427829i
$120$	0	0
$121$	167.000	1.38017
$122$	0	0
$123$	0	0
$124$	0	0
$125$	135.765	1.08612
$126$	0	0
$127$	92.0000	0.724409	0.362205	−	0.932099i	$-0.382024\pi$
$127$	92.0000	0.724409	0.362205	+	0.932099i	$0.382024\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	169.706	1.29546	0.647731	−	0.761869i	$-0.275718\pi$
$131$	169.706	1.29546	0.647731	+	0.761869i	$0.275718\pi$
$132$	0	0
$133$	− 64.0000i	− 0.481203i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 156.978i	− 1.14582i	−0.819617	−	0.572911i	$-0.805814\pi$
$137$	− 156.978i	− 1.14582i	0.819617	−	0.572911i	$-0.194186\pi$
$138$	0	0
$139$	− 152.000i	− 1.09353i	−0.837288	−	0.546763i	$-0.815860\pi$
$139$	− 152.000i	− 1.09353i	0.837288	−	0.546763i	$-0.184140\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	135.765i	0.949402i
$144$	0	0
$145$	18.0000	0.124138
$146$	0	0
$147$	0	0
$148$	0	0
$149$	275.772	1.85082	0.925408	−	0.378972i	$-0.123722\pi$
$149$	275.772	1.85082	0.925408	+	0.378972i	$0.123722\pi$
$150$	0	0
$151$	148.000	0.980132	0.490066	−	0.871685i	$-0.336973\pi$
$151$	148.000	0.980132	0.490066	+	0.871685i	$0.336973\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−186.676	−1.20436
$156$	0	0
$157$	− 82.0000i	− 0.522293i	−0.965299	−	0.261146i	$-0.915899\pi$
$157$	− 82.0000i	− 0.522293i	0.965299	−	0.261146i	$-0.0841006\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	67.8823i	0.421629i
$162$	0	0
$163$	56.0000i	0.343558i	0.985135	+	0.171779i	$0.0549515\pi$
$163$	56.0000i	0.343558i	−0.985135	+	0.171779i	$0.945048\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 33.9411i	− 0.203240i	−0.994823	−	0.101620i	$-0.967597\pi$
$167$	− 33.9411i	− 0.203240i	0.994823	−	0.101620i	$-0.0324026\pi$
$168$	0	0
$169$	105.000	0.621302
$170$	0	0
$171$	0	0
$172$	0	0
$173$	173.948	1.00548	0.502741	−	0.864437i	$-0.332325\pi$
$173$	173.948	1.00548	0.502741	+	0.864437i	$0.332325\pi$
$174$	0	0
$175$	−28.0000	−0.160000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	203.647	1.13769	0.568846	−	0.822444i	$-0.307390\pi$
$179$	203.647	1.13769	0.568846	+	0.822444i	$0.307390\pi$
$180$	0	0
$181$	232.000i	1.28177i	0.767638	+	0.640884i	$0.221432\pi$
$181$	232.000i	1.28177i	−0.767638	+	0.640884i	$0.778568\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 144.250i	− 0.779729i
$186$	0	0
$187$	− 216.000i	− 1.15508i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 33.9411i	− 0.177702i	−0.996045	−	0.0888511i	$-0.971680\pi$
$191$	− 33.9411i	− 0.177702i	0.996045	−	0.0888511i	$-0.0283195\pi$
$192$	0	0
$193$	206.000	1.06736	0.533679	−	0.845687i	$-0.320809\pi$
$193$	206.000	1.06736	0.533679	+	0.845687i	$0.320809\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	165.463	0.839914	0.419957	−	0.907544i	$-0.362045\pi$
$197$	165.463	0.839914	0.419957	+	0.907544i	$0.362045\pi$
$198$	0	0
$199$	−20.0000	−0.100503	−0.0502513	−	0.998737i	$-0.516002\pi$
$199$	−20.0000	−0.100503	−0.0502513	+	0.998737i	$0.516002\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−16.9706	−0.0835988
$204$	0	0
$205$	198.000i	0.965854i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 271.529i	− 1.29918i
$210$	0	0
$211$	296.000i	1.40284i	0.712746	+	0.701422i	$0.247451\pi$
$211$	296.000i	1.40284i	−0.712746	+	0.701422i	$0.752549\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 169.706i	− 0.789328i
$216$	0	0
$217$	176.000	0.811060
$218$	0	0
$219$	0	0
$220$	0	0
$221$	101.823	0.460739
$222$	0	0
$223$	−436.000	−1.95516	−0.977578	−	0.210571i	$-0.932468\pi$
$223$	−436.000	−1.95516	−0.977578	+	0.210571i	$0.932468\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.9706	−0.0747602	−0.0373801	−	0.999301i	$-0.511901\pi$
$227$	−16.9706	−0.0747602	−0.0373801	+	0.999301i	$0.511901\pi$
$228$	0	0
$229$	− 8.00000i	− 0.0349345i	−0.999847	−	0.0174672i	$-0.994440\pi$
$229$	− 8.00000i	− 0.0349345i	0.999847	−	0.0174672i	$-0.00556028\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.7279i	0.0546263i	0.999627	+	0.0273131i	$0.00869512\pi$
$233$	12.7279i	0.0546263i	−0.999627	+	0.0273131i	$0.991305\pi$
$234$	0	0
$235$	360.000i	1.53191i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	135.765i	0.568052i	0.958817	+	0.284026i	$0.0916703\pi$
$239$	135.765i	0.568052i	−0.958817	+	0.284026i	$0.908330\pi$
$240$	0	0
$241$	32.0000	0.132780	0.0663900	−	0.997794i	$-0.478852\pi$
$241$	32.0000	0.132780	0.0663900	+	0.997794i	$0.478852\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	140.007	0.571458
$246$	0	0
$247$	128.000	0.518219
$248$	0	0
$249$	0	0
$250$	0	0
$251$	50.9117	0.202835	0.101418	−	0.994844i	$-0.467662\pi$
$251$	50.9117	0.202835	0.101418	+	0.994844i	$0.467662\pi$
$252$	0	0
$253$	288.000i	1.13834i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 182.434i	− 0.709858i	−0.934893	−	0.354929i	$-0.884505\pi$
$257$	− 182.434i	− 0.709858i	0.934893	−	0.354929i	$-0.115495\pi$
$258$	0	0
$259$	136.000i	0.525097i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 373.352i	− 1.41959i	−0.704408	−	0.709795i	$-0.748787\pi$
$263$	− 373.352i	− 1.41959i	0.704408	−	0.709795i	$-0.251213\pi$
$264$	0	0
$265$	−162.000	−0.611321
$266$	0	0
$267$	0	0
$268$	0	0
$269$	343.654	1.27752	0.638762	−	0.769404i	$-0.279447\pi$
$269$	343.654	1.27752	0.638762	+	0.769404i	$0.279447\pi$
$270$	0	0
$271$	380.000	1.40221	0.701107	−	0.713056i	$-0.252690\pi$
$271$	380.000	1.40221	0.701107	+	0.713056i	$0.252690\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−118.794	−0.431978
$276$	0	0
$277$	328.000i	1.18412i	0.805896	+	0.592058i	$0.201684\pi$
$277$	328.000i	1.18412i	−0.805896	+	0.592058i	$0.798316\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 284.257i	− 1.01159i	−0.862654	−	0.505795i	$-0.831199\pi$
$281$	− 284.257i	− 1.01159i	0.862654	−	0.505795i	$-0.168801\pi$
$282$	0	0
$283$	208.000i	0.734982i	0.930027	+	0.367491i	$0.119783\pi$
$283$	208.000i	0.734982i	−0.930027	+	0.367491i	$0.880217\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 186.676i	− 0.650440i
$288$	0	0
$289$	127.000	0.439446
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−436.992	−1.49144	−0.745720	−	0.666259i	$-0.767894\pi$
$293$	−436.992	−1.49144	−0.745720	+	0.666259i	$0.767894\pi$
$294$	0	0
$295$	−144.000	−0.488136
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−135.765	−0.454062
$300$	0	0
$301$	160.000i	0.531561i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 212.132i	− 0.695515i
$306$	0	0
$307$	− 520.000i	− 1.69381i	−0.531743	−	0.846906i	$-0.678463\pi$
$307$	− 520.000i	− 1.69381i	0.531743	−	0.846906i	$-0.321537\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	373.352i	1.20049i	0.799816	+	0.600245i	$0.204930\pi$
$311$	373.352i	1.20049i	−0.799816	+	0.600245i	$0.795070\pi$
$312$	0	0
$313$	94.0000	0.300319	0.150160	−	0.988662i	$-0.452021\pi$
$313$	94.0000	0.300319	0.150160	+	0.988662i	$0.452021\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−335.169	−1.05731	−0.528657	−	0.848835i	$-0.677304\pi$
$317$	−335.169	−1.05731	−0.528657	+	0.848835i	$0.677304\pi$
$318$	0	0
$319$	−72.0000	−0.225705
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−203.647	−0.630485
$324$	0	0
$325$	− 56.0000i	− 0.172308i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 339.411i	− 1.03165i
$330$	0	0
$331$	− 536.000i	− 1.61934i	−0.586889	−	0.809668i	$-0.699647\pi$
$331$	− 536.000i	− 1.61934i	0.586889	−	0.809668i	$-0.300353\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 33.9411i	− 0.101317i
$336$	0	0
$337$	−208.000	−0.617211	−0.308605	−	0.951190i	$-0.599862\pi$
$337$	−208.000	−0.617211	−0.308605	+	0.951190i	$0.599862\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	746.705	2.18975
$342$	0	0
$343$	−328.000	−0.956268
$344$	0	0
$345$	0	0
$346$	0	0
$347$	288.500	0.831411	0.415705	−	0.909499i	$-0.363535\pi$
$347$	288.500	0.831411	0.415705	+	0.909499i	$0.363535\pi$
$348$	0	0
$349$	− 238.000i	− 0.681948i	−0.940073	−	0.340974i	$-0.889243\pi$
$349$	− 238.000i	− 0.681948i	0.940073	−	0.340974i	$-0.110757\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 224.860i	− 0.636997i	−0.947923	−	0.318499i	$-0.896821\pi$
$353$	− 224.860i	− 0.636997i	0.947923	−	0.318499i	$-0.103179\pi$
$354$	0	0
$355$	− 216.000i	− 0.608451i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	560.029i	1.55997i	0.625799	+	0.779984i	$0.284773\pi$
$359$	560.029i	1.55997i	−0.625799	+	0.779984i	$0.715227\pi$
$360$	0	0
$361$	105.000	0.290859
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−67.8823	−0.185979
$366$	0	0
$367$	284.000	0.773842	0.386921	−	0.922113i	$-0.373539\pi$
$367$	284.000	0.773842	0.386921	+	0.922113i	$0.373539\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	152.735	0.411685
$372$	0	0
$373$	190.000i	0.509383i	0.967022	+	0.254692i	$0.0819740\pi$
$373$	190.000i	0.509383i	−0.967022	+	0.254692i	$0.918026\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 33.9411i	− 0.0900295i
$378$	0	0
$379$	160.000i	0.422164i	0.977468	+	0.211082i	$0.0676986\pi$
$379$	160.000i	0.422164i	−0.977468	+	0.211082i	$0.932301\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 271.529i	− 0.708953i	−0.935065	−	0.354477i	$-0.884659\pi$
$383$	− 271.529i	− 0.708953i	0.935065	−	0.354477i	$-0.115341\pi$
$384$	0	0
$385$	−288.000	−0.748052
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−403.051	−1.03612	−0.518060	−	0.855344i	$-0.673346\pi$
$389$	−403.051	−1.03612	−0.518060	+	0.855344i	$0.673346\pi$
$390$	0	0
$391$	216.000	0.552430
$392$	0	0
$393$	0	0
$394$	0	0
$395$	322.441	0.816306
$396$	0	0
$397$	146.000i	0.367758i	0.982949	+	0.183879i	$0.0588655\pi$
$397$	146.000i	0.367758i	−0.982949	+	0.183879i	$0.941135\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	326.683i	0.814672i	0.913278	+	0.407336i	$0.133542\pi$
$401$	326.683i	0.814672i	−0.913278	+	0.407336i	$0.866458\pi$
$402$	0	0
$403$	352.000i	0.873449i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	576.999i	1.41769i
$408$	0	0
$409$	−368.000	−0.899756	−0.449878	−	0.893090i	$-0.648532\pi$
$409$	−368.000	−0.899756	−0.449878	+	0.893090i	$0.648532\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	135.765	0.328728
$414$	0	0
$415$	504.000	1.21446
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−390.323	−0.931558	−0.465779	−	0.884901i	$-0.654226\pi$
$419$	−390.323	−0.931558	−0.465779	+	0.884901i	$0.654226\pi$
$420$	0	0
$421$	40.0000i	0.0950119i	0.998871	+	0.0475059i	$0.0151273\pi$
$421$	40.0000i	0.0950119i	−0.998871	+	0.0475059i	$0.984873\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	89.0955i	0.209636i
$426$	0	0
$427$	200.000i	0.468384i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 152.735i	− 0.354374i	−0.984177	−	0.177187i	$-0.943300\pi$
$431$	− 152.735i	− 0.354374i	0.984177	−	0.177187i	$-0.0566997\pi$
$432$	0	0
$433$	542.000	1.25173	0.625866	−	0.779931i	$-0.284746\pi$
$433$	542.000	1.25173	0.625866	+	0.779931i	$0.284746\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	271.529	0.621348
$438$	0	0
$439$	4.00000	0.00911162	0.00455581	−	0.999990i	$-0.498550\pi$
$439$	4.00000	0.00911162	0.00455581	+	0.999990i	$0.498550\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	322.441	0.727857	0.363929	−	0.931427i	$-0.381435\pi$
$443$	322.441	0.727857	0.363929	+	0.931427i	$0.381435\pi$
$444$	0	0
$445$	54.0000i	0.121348i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	216.375i	0.481904i	0.970537	+	0.240952i	$0.0774596\pi$
$449$	216.375i	0.481904i	−0.970537	+	0.240952i	$0.922540\pi$
$450$	0	0
$451$	− 792.000i	− 1.75610i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 135.765i	− 0.298384i
$456$	0	0
$457$	400.000	0.875274	0.437637	−	0.899152i	$-0.355816\pi$
$457$	400.000	0.875274	0.437637	+	0.899152i	$0.355816\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	301.227	0.653422	0.326711	−	0.945124i	$-0.394060\pi$
$461$	301.227	0.653422	0.326711	+	0.945124i	$0.394060\pi$
$462$	0	0
$463$	−604.000	−1.30454	−0.652268	−	0.757989i	$-0.726182\pi$
$463$	−604.000	−1.30454	−0.652268	+	0.757989i	$0.726182\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−356.382	−0.763130	−0.381565	−	0.924342i	$-0.624615\pi$
$467$	−356.382	−0.763130	−0.381565	+	0.924342i	$0.624615\pi$
$468$	0	0
$469$	32.0000i	0.0682303i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	678.823i	1.43514i
$474$	0	0
$475$	112.000i	0.235789i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	526.087i	1.09830i	0.835723	+	0.549152i	$0.185049\pi$
$479$	526.087i	1.09830i	−0.835723	+	0.549152i	$0.814951\pi$
$480$	0	0
$481$	−272.000	−0.565489
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−746.705	−1.53960
$486$	0	0
$487$	−596.000	−1.22382	−0.611910	−	0.790928i	$-0.709598\pi$
$487$	−596.000	−1.22382	−0.611910	+	0.790928i	$0.709598\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−271.529	−0.553012	−0.276506	−	0.961012i	$-0.589177\pi$
$491$	−271.529	−0.553012	−0.276506	+	0.961012i	$0.589177\pi$
$492$	0	0
$493$	54.0000i	0.109533i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	203.647i	0.409752i
$498$	0	0
$499$	224.000i	0.448898i	0.974486	+	0.224449i	$0.0720582\pi$
$499$	224.000i	0.448898i	−0.974486	+	0.224449i	$0.927942\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 865.499i	− 1.72067i	−0.509726	−	0.860337i	$-0.670253\pi$
$503$	− 865.499i	− 1.72067i	0.509726	−	0.860337i	$-0.329747\pi$
$504$	0	0
$505$	−126.000	−0.249505
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−479.418	−0.941883	−0.470941	−	0.882164i	$-0.656086\pi$
$509$	−479.418	−0.941883	−0.470941	+	0.882164i	$0.656086\pi$
$510$	0	0
$511$	64.0000	0.125245
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−118.794	−0.230668
$516$	0	0
$517$	− 1440.00i	− 2.78530i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	521.845i	1.00162i	0.865557	+	0.500811i	$0.166965\pi$
$521$	521.845i	1.00162i	−0.865557	+	0.500811i	$0.833035\pi$
$522$	0	0
$523$	736.000i	1.40727i	0.710564	+	0.703633i	$0.248440\pi$
$523$	736.000i	1.40727i	−0.710564	+	0.703633i	$0.751560\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 560.029i	− 1.06267i
$528$	0	0
$529$	241.000	0.455577
$530$	0	0
$531$	0	0
$532$	0	0
$533$	373.352	0.700474
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−560.029	−1.03901
$540$	0	0
$541$	− 808.000i	− 1.49353i	−0.665088	−	0.746765i	$-0.731606\pi$
$541$	− 808.000i	− 1.49353i	0.665088	−	0.746765i	$-0.268394\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 237.588i	− 0.435941i
$546$	0	0
$547$	536.000i	0.979890i	0.871753	+	0.489945i	$0.162983\pi$
$547$	536.000i	0.979890i	−0.871753	+	0.489945i	$0.837017\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	67.8823i	0.123198i
$552$	0	0
$553$	−304.000	−0.549729
$554$	0	0
$555$	0	0
$556$	0	0
$557$	165.463	0.297061	0.148531	−	0.988908i	$-0.452546\pi$
$557$	165.463	0.297061	0.148531	+	0.988908i	$0.452546\pi$
$558$	0	0
$559$	−320.000	−0.572451
$560$	0	0
$561$	0	0
$562$	0	0
$563$	322.441	0.572719	0.286359	−	0.958122i	$-0.407555\pi$
$563$	322.441	0.572719	0.286359	+	0.958122i	$0.407555\pi$
$564$	0	0
$565$	666.000i	1.17876i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 156.978i	− 0.275883i	−0.990440	−	0.137942i	$-0.955951\pi$
$569$	− 156.978i	− 0.275883i	0.990440	−	0.137942i	$-0.0440487\pi$
$570$	0	0
$571$	− 368.000i	− 0.644483i	−0.946657	−	0.322242i	$-0.895564\pi$
$571$	− 368.000i	− 0.644483i	0.946657	−	0.322242i	$-0.104436\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 118.794i	− 0.206598i
$576$	0	0
$577$	−142.000	−0.246101	−0.123050	−	0.992400i	$-0.539268\pi$
$577$	−142.000	−0.246101	−0.123050	+	0.992400i	$0.539268\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−475.176	−0.817858
$582$	0	0
$583$	648.000	1.11149
$584$	0	0
$585$	0	0
$586$	0	0
$587$	373.352	0.636035	0.318017	−	0.948085i	$-0.396983\pi$
$587$	373.352	0.636035	0.318017	+	0.948085i	$0.396983\pi$
$588$	0	0
$589$	− 704.000i	− 1.19525i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1107.33i	1.86733i	0.358142	+	0.933667i	$0.383410\pi$
$593$	1107.33i	1.86733i	−0.358142	+	0.933667i	$0.616590\pi$
$594$	0	0
$595$	216.000i	0.363025i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 797.616i	− 1.33158i	−0.746139	−	0.665790i	$-0.768095\pi$
$599$	− 797.616i	− 1.33158i	0.746139	−	0.665790i	$-0.231905\pi$
$600$	0	0
$601$	−158.000	−0.262895	−0.131448	−	0.991323i	$-0.541963\pi$
$601$	−158.000	−0.262895	−0.131448	+	0.991323i	$0.541963\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−708.521	−1.17111
$606$	0	0
$607$	332.000	0.546952	0.273476	−	0.961879i	$-0.411827\pi$
$607$	332.000	0.546952	0.273476	+	0.961879i	$0.411827\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	678.823	1.11100
$612$	0	0
$613$	− 578.000i	− 0.942904i	−0.881892	−	0.471452i	$-0.843730\pi$
$613$	− 578.000i	− 0.942904i	0.881892	−	0.471452i	$-0.156270\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	55.1543i	0.0893911i	0.999001	+	0.0446956i	$0.0142318\pi$
$617$	55.1543i	0.0893911i	−0.999001	+	0.0446956i	$0.985768\pi$
$618$	0	0
$619$	− 896.000i	− 1.44750i	−0.690064	−	0.723748i	$-0.742418\pi$
$619$	− 896.000i	− 1.44750i	0.690064	−	0.723748i	$-0.257582\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 50.9117i	− 0.0817202i
$624$	0	0
$625$	−401.000	−0.641600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	432.749	0.687996
$630$	0	0
$631$	−20.0000	−0.0316957	−0.0158479	−	0.999874i	$-0.505045\pi$
$631$	−20.0000	−0.0316957	−0.0158479	+	0.999874i	$0.505045\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−390.323	−0.614682
$636$	0	0
$637$	− 264.000i	− 0.414443i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	258.801i	0.403746i	0.979412	+	0.201873i	$0.0647028\pi$
$641$	258.801i	0.403746i	−0.979412	+	0.201873i	$0.935297\pi$
$642$	0	0
$643$	728.000i	1.13219i	0.824339	+	0.566096i	$0.191547\pi$
$643$	728.000i	1.13219i	−0.824339	+	0.566096i	$0.808453\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	458.205i	0.708200i	0.935208	+	0.354100i	$0.115213\pi$
$647$	458.205i	0.708200i	−0.935208	+	0.354100i	$0.884787\pi$
$648$	0	0
$649$	576.000	0.887519
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−301.227	−0.461298	−0.230649	−	0.973037i	$-0.574085\pi$
$653$	−301.227	−0.461298	−0.230649	+	0.973037i	$0.574085\pi$
$654$	0	0
$655$	−720.000	−1.09924
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1052.17	1.59662	0.798312	−	0.602244i	$-0.205727\pi$
$659$	1052.17	1.59662	0.798312	+	0.602244i	$0.205727\pi$
$660$	0	0
$661$	− 62.0000i	− 0.0937973i	−0.998900	−	0.0468986i	$-0.985066\pi$
$661$	− 62.0000i	− 0.0937973i	0.998900	−	0.0468986i	$-0.0149338\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	271.529i	0.408314i
$666$	0	0
$667$	− 72.0000i	− 0.107946i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	848.528i	1.26457i
$672$	0	0
$673$	−670.000	−0.995542	−0.497771	−	0.867308i	$-0.665848\pi$
$673$	−670.000	−0.995542	−0.497771	+	0.867308i	$0.665848\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1294.01	−1.91138	−0.955691	−	0.294372i	$-0.904889\pi$
$677$	−1294.01	−1.91138	−0.955691	+	0.294372i	$0.904889\pi$
$678$	0	0
$679$	704.000	1.03682
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−560.029	−0.819954	−0.409977	−	0.912096i	$-0.634463\pi$
$683$	−560.029	−0.819954	−0.409977	+	0.912096i	$0.634463\pi$
$684$	0	0
$685$	666.000i	0.972263i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	305.470i	0.443353i
$690$	0	0
$691$	− 40.0000i	− 0.0578871i	−0.999581	−	0.0289436i	$-0.990786\pi$
$691$	− 40.0000i	− 0.0578871i	0.999581	−	0.0289436i	$-0.00921431\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	644.881i	0.927887i
$696$	0	0
$697$	−594.000	−0.852224
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−954.594	−1.36176	−0.680880	−	0.732395i	$-0.738403\pi$
$701$	−954.594	−1.36176	−0.680880	+	0.732395i	$0.738403\pi$
$702$	0	0
$703$	544.000	0.773826
$704$	0	0
$705$	0	0
$706$	0	0
$707$	118.794	0.168025
$708$	0	0
$709$	− 968.000i	− 1.36530i	−0.730744	−	0.682652i	$-0.760827\pi$
$709$	− 968.000i	− 1.36530i	0.730744	−	0.682652i	$-0.239173\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	746.705i	1.04727i
$714$	0	0
$715$	− 576.000i	− 0.805594i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1170.97i	1.62861i	0.580439	+	0.814304i	$0.302881\pi$
$719$	1170.97i	1.62861i	−0.580439	+	0.814304i	$0.697119\pi$
$720$	0	0
$721$	112.000	0.155340
$722$	0	0
$723$	0	0
$724$	0	0
$725$	29.6985	0.0409634
$726$	0	0
$727$	508.000	0.698762	0.349381	−	0.936981i	$-0.386392\pi$
$727$	508.000	0.698762	0.349381	+	0.936981i	$0.386392\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	509.117	0.696466
$732$	0	0
$733$	− 1144.00i	− 1.56071i	−0.625337	−	0.780355i	$-0.715039\pi$
$733$	− 1144.00i	− 1.56071i	0.625337	−	0.780355i	$-0.284961\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	135.765i	0.184212i
$738$	0	0
$739$	− 304.000i	− 0.411367i	−0.978619	−	0.205683i	$-0.934058\pi$
$739$	− 304.000i	− 0.411367i	0.978619	−	0.205683i	$-0.0659417\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 848.528i	− 1.14203i	−0.820940	−	0.571015i	$-0.806550\pi$
$743$	− 848.528i	− 1.14203i	0.820940	−	0.571015i	$-0.193450\pi$
$744$	0	0
$745$	−1170.00	−1.57047
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	188.000	0.250333	0.125166	−	0.992136i	$-0.460054\pi$
$751$	188.000	0.250333	0.125166	+	0.992136i	$0.460054\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−627.911	−0.831670
$756$	0	0
$757$	1240.00i	1.63804i	0.573761	+	0.819022i	$0.305484\pi$
$757$	1240.00i	1.63804i	−0.573761	+	0.819022i	$0.694516\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	156.978i	0.206278i	0.994667	+	0.103139i	$0.0328887\pi$
$761$	156.978i	0.206278i	−0.994667	+	0.103139i	$0.967111\pi$
$762$	0	0
$763$	224.000i	0.293578i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	271.529i	0.354014i
$768$	0	0
$769$	−910.000	−1.18336	−0.591678	−	0.806175i	$-0.701534\pi$
$769$	−910.000	−1.18336	−0.591678	+	0.806175i	$0.701534\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1387.34	−1.79475	−0.897376	−	0.441266i	$-0.854529\pi$
$773$	−1387.34	−1.79475	−0.897376	+	0.441266i	$0.854529\pi$
$774$	0	0
$775$	−308.000	−0.397419
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−746.705	−0.958543
$780$	0	0
$781$	864.000i	1.10627i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	347.897i	0.443180i
$786$	0	0
$787$	− 1360.00i	− 1.72808i	−0.503422	−	0.864041i	$-0.667926\pi$
$787$	− 1360.00i	− 1.72808i	0.503422	−	0.864041i	$-0.332074\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 627.911i	− 0.793819i
$792$	0	0
$793$	−400.000	−0.504414
$794$	0	0
$795$	0	0
$796$	0	0
$797$	106.066	0.133082	0.0665408	−	0.997784i	$-0.478804\pi$
$797$	106.066	0.133082	0.0665408	+	0.997784i	$0.478804\pi$
$798$	0	0
$799$	−1080.00	−1.35169
$800$	0	0
$801$	0	0
$802$	0	0
$803$	271.529	0.338143
$804$	0	0
$805$	− 288.000i	− 0.357764i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1107.33i	1.36876i	0.729124	+	0.684381i	$0.239928\pi$
$809$	1107.33i	1.36876i	−0.729124	+	0.684381i	$0.760072\pi$
$810$	0	0
$811$	160.000i	0.197287i	0.995123	+	0.0986436i	$0.0314504\pi$
$811$	160.000i	0.197287i	−0.995123	+	0.0986436i	$0.968550\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 237.588i	− 0.291519i
$816$	0	0
$817$	640.000	0.783354
$818$	0	0
$819$	0	0
$820$	0	0
$821$	436.992	0.532268	0.266134	−	0.963936i	$-0.414254\pi$
$821$	436.992	0.532268	0.266134	+	0.963936i	$0.414254\pi$
$822$	0	0
$823$	−332.000	−0.403402	−0.201701	−	0.979447i	$-0.564647\pi$
$823$	−332.000	−0.403402	−0.201701	+	0.979447i	$0.564647\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	101.823	0.123124	0.0615619	−	0.998103i	$-0.480392\pi$
$827$	101.823	0.123124	0.0615619	+	0.998103i	$0.480392\pi$
$828$	0	0
$829$	632.000i	0.762364i	0.924500	+	0.381182i	$0.124483\pi$
$829$	632.000i	0.762364i	−0.924500	+	0.381182i	$0.875517\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	420.021i	0.504227i
$834$	0	0
$835$	144.000i	0.172455i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 729.734i	− 0.869767i	−0.900487	−	0.434883i	$-0.856790\pi$
$839$	− 729.734i	− 0.869767i	0.900487	−	0.434883i	$-0.143210\pi$
$840$	0	0
$841$	−823.000	−0.978597
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−445.477	−0.527192
$846$	0	0
$847$	668.000	0.788666
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−576.999	−0.678025
$852$	0	0
$853$	− 446.000i	− 0.522860i	−0.965222	−	0.261430i	$-0.915806\pi$
$853$	− 446.000i	− 0.522860i	0.965222	−	0.261430i	$-0.0841941\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	428.507i	0.500008i	0.968245	+	0.250004i	$0.0804319\pi$
$857$	428.507i	0.500008i	−0.968245	+	0.250004i	$0.919568\pi$
$858$	0	0
$859$	− 728.000i	− 0.847497i	−0.905780	−	0.423749i	$-0.860714\pi$
$859$	− 728.000i	− 0.847497i	0.905780	−	0.423749i	$-0.139286\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 916.410i	− 1.06189i	−0.847407	−	0.530945i	$-0.821837\pi$
$863$	− 916.410i	− 1.06189i	0.847407	−	0.530945i	$-0.178163\pi$
$864$	0	0
$865$	−738.000	−0.853179
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1289.76	−1.48419
$870$	0	0
$871$	−64.0000	−0.0734788
$872$	0	0
$873$	0	0
$874$	0	0
$875$	543.058	0.620638
$876$	0	0
$877$	− 910.000i	− 1.03763i	−0.854887	−	0.518814i	$-0.826374\pi$
$877$	− 910.000i	− 1.03763i	0.854887	−	0.518814i	$-0.173626\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 929.138i	− 1.05464i	−0.849667	−	0.527320i	$-0.823197\pi$
$881$	− 929.138i	− 1.05464i	0.849667	−	0.527320i	$-0.176803\pi$
$882$	0	0
$883$	1064.00i	1.20498i	0.798125	+	0.602492i	$0.205825\pi$
$883$	1064.00i	1.20498i	−0.798125	+	0.602492i	$0.794175\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1391.59i	1.56887i	0.620212	+	0.784434i	$0.287047\pi$
$887$	1391.59i	1.56887i	−0.620212	+	0.784434i	$0.712953\pi$
$888$	0	0
$889$	368.000	0.413948
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1357.65	−1.52032
$894$	0	0
$895$	−864.000	−0.965363
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−186.676	−0.207649
$900$	0	0
$901$	− 486.000i	− 0.539401i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 984.293i	− 1.08762i
$906$	0	0
$907$	1768.00i	1.94928i	0.223771	+	0.974642i	$0.428163\pi$
$907$	1768.00i	1.94928i	−0.223771	+	0.974642i	$0.571837\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 237.588i	− 0.260799i	−0.991462	−	0.130399i	$-0.958374\pi$
$911$	− 237.588i	− 0.260799i	0.991462	−	0.130399i	$-0.0416260\pi$
$912$	0	0
$913$	−2016.00	−2.20811
$914$	0	0
$915$	0	0
$916$	0	0
$917$	678.823	0.740264
$918$	0	0
$919$	−380.000	−0.413493	−0.206746	−	0.978395i	$-0.566288\pi$
$919$	−380.000	−0.413493	−0.206746	+	0.978395i	$0.566288\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−407.294	−0.441271
$924$	0	0
$925$	− 238.000i	− 0.257297i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	666.095i	0.717002i	0.933529	+	0.358501i	$0.116712\pi$
$929$	666.095i	0.717002i	−0.933529	+	0.358501i	$0.883288\pi$
$930$	0	0
$931$	528.000i	0.567132i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	916.410i	0.980118i
$936$	0	0
$937$	178.000	0.189968	0.0949840	−	0.995479i	$-0.469720\pi$
$937$	178.000	0.189968	0.0949840	+	0.995479i	$0.469720\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	436.992	0.464391	0.232196	−	0.972669i	$-0.425409\pi$
$941$	436.992	0.464391	0.232196	+	0.972669i	$0.425409\pi$
$942$	0	0
$943$	792.000	0.839873
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1798.88	−1.89956	−0.949778	−	0.312924i	$-0.898691\pi$
$947$	−1798.88	−1.89956	−0.949778	+	0.312924i	$0.898691\pi$
$948$	0	0
$949$	128.000i	0.134879i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1310.98i	− 1.37563i	−0.725886	−	0.687815i	$-0.758570\pi$
$953$	− 1310.98i	− 1.37563i	0.725886	−	0.687815i	$-0.241430\pi$
$954$	0	0
$955$	144.000i	0.150785i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 627.911i	− 0.654756i
$960$	0	0
$961$	975.000	1.01457
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−873.984	−0.905683
$966$	0	0
$967$	−1700.00	−1.75801	−0.879007	−	0.476808i	$-0.841794\pi$
$967$	−1700.00	−1.75801	−0.879007	+	0.476808i	$0.841794\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	458.205	0.471890	0.235945	−	0.971766i	$-0.424181\pi$
$971$	458.205	0.471890	0.235945	+	0.971766i	$0.424181\pi$
$972$	0	0
$973$	− 608.000i	− 0.624872i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	759.433i	0.777311i	0.921383	+	0.388655i	$0.127060\pi$
$977$	759.433i	0.777311i	−0.921383	+	0.388655i	$0.872940\pi$
$978$	0	0
$979$	− 216.000i	− 0.220633i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1052.17i	1.07037i	0.844734	+	0.535186i	$0.179758\pi$
$983$	1052.17i	1.07037i	−0.844734	+	0.535186i	$0.820242\pi$
$984$	0	0
$985$	−702.000	−0.712690
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−678.823	−0.686373
$990$	0	0
$991$	−772.000	−0.779011	−0.389506	−	0.921024i	$-0.627354\pi$
$991$	−772.000	−0.779011	−0.389506	+	0.921024i	$0.627354\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	84.8528	0.0852792
$996$	0	0
$997$	− 194.000i	− 0.194584i	−0.995256	−	0.0972919i	$-0.968982\pi$
$997$	− 194.000i	− 0.194584i	0.995256	−	0.0972919i	$-0.0310180\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.h.f.2177.2		4
3.2	odd	2	inner	2304.3.h.f.2177.4		4
4.3	odd	2		2304.3.h.c.2177.2		4
8.3	odd	2		2304.3.h.c.2177.3		4
8.5	even	2	inner	2304.3.h.f.2177.3		4
12.11	even	2		2304.3.h.c.2177.4		4
16.3	odd	4		576.3.e.f.449.2		2
16.5	even	4		18.3.b.a.17.2	yes	2
16.11	odd	4		144.3.e.b.17.1		2
16.13	even	4		576.3.e.c.449.2		2
24.5	odd	2	inner	2304.3.h.f.2177.1		4
24.11	even	2		2304.3.h.c.2177.1		4
48.5	odd	4		18.3.b.a.17.1	✓	2
48.11	even	4		144.3.e.b.17.2		2
48.29	odd	4		576.3.e.c.449.1		2
48.35	even	4		576.3.e.f.449.1		2
80.27	even	4		3600.3.c.b.449.3		4
80.37	odd	4		450.3.b.b.449.1		4
80.43	even	4		3600.3.c.b.449.1		4
80.53	odd	4		450.3.b.b.449.4		4
80.59	odd	4		3600.3.l.d.1601.1		2
80.69	even	4		450.3.d.f.251.1		2
112.5	odd	12		882.3.s.d.557.2		4
112.37	even	12		882.3.s.b.557.2		4
112.53	even	12		882.3.s.b.863.1		4
112.69	odd	4		882.3.b.a.197.2		2
112.101	odd	12		882.3.s.d.863.1		4
144.5	odd	12		162.3.d.b.107.2		4
144.11	even	12		1296.3.q.f.1025.1		4
144.43	odd	12		1296.3.q.f.1025.2		4
144.59	even	12		1296.3.q.f.593.2		4
144.85	even	12		162.3.d.b.107.1		4
144.101	odd	12		162.3.d.b.53.1		4
144.133	even	12		162.3.d.b.53.2		4
144.139	odd	12		1296.3.q.f.593.1		4
176.21	odd	4		2178.3.c.d.485.1		2
208.5	odd	4		3042.3.d.a.3041.3		4
208.21	odd	4		3042.3.d.a.3041.2		4
208.181	even	4		3042.3.c.e.1691.1		2
240.53	even	4		450.3.b.b.449.2		4
240.59	even	4		3600.3.l.d.1601.2		2
240.107	odd	4		3600.3.c.b.449.4		4
240.149	odd	4		450.3.d.f.251.2		2
240.197	even	4		450.3.b.b.449.3		4
240.203	odd	4		3600.3.c.b.449.2		4
336.5	even	12		882.3.s.d.557.1		4
336.53	odd	12		882.3.s.b.863.2		4
336.101	even	12		882.3.s.d.863.2		4
336.149	odd	12		882.3.s.b.557.1		4
336.293	even	4		882.3.b.a.197.1		2
528.197	even	4		2178.3.c.d.485.2		2
624.5	even	4		3042.3.d.a.3041.1		4
624.389	odd	4		3042.3.c.e.1691.2		2
624.437	even	4		3042.3.d.a.3041.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.3.b.a.17.1	✓	2	48.5	odd	4
18.3.b.a.17.2	yes	2	16.5	even	4
144.3.e.b.17.1		2	16.11	odd	4
144.3.e.b.17.2		2	48.11	even	4
162.3.d.b.53.1		4	144.101	odd	12
162.3.d.b.53.2		4	144.133	even	12
162.3.d.b.107.1		4	144.85	even	12
162.3.d.b.107.2		4	144.5	odd	12
450.3.b.b.449.1		4	80.37	odd	4
450.3.b.b.449.2		4	240.53	even	4
450.3.b.b.449.3		4	240.197	even	4
450.3.b.b.449.4		4	80.53	odd	4
450.3.d.f.251.1		2	80.69	even	4
450.3.d.f.251.2		2	240.149	odd	4
576.3.e.c.449.1		2	48.29	odd	4
576.3.e.c.449.2		2	16.13	even	4
576.3.e.f.449.1		2	48.35	even	4
576.3.e.f.449.2		2	16.3	odd	4
882.3.b.a.197.1		2	336.293	even	4
882.3.b.a.197.2		2	112.69	odd	4
882.3.s.b.557.1		4	336.149	odd	12
882.3.s.b.557.2		4	112.37	even	12
882.3.s.b.863.1		4	112.53	even	12
882.3.s.b.863.2		4	336.53	odd	12
882.3.s.d.557.1		4	336.5	even	12
882.3.s.d.557.2		4	112.5	odd	12
882.3.s.d.863.1		4	112.101	odd	12
882.3.s.d.863.2		4	336.101	even	12
1296.3.q.f.593.1		4	144.139	odd	12
1296.3.q.f.593.2		4	144.59	even	12
1296.3.q.f.1025.1		4	144.11	even	12
1296.3.q.f.1025.2		4	144.43	odd	12
2178.3.c.d.485.1		2	176.21	odd	4
2178.3.c.d.485.2		2	528.197	even	4
2304.3.h.c.2177.1		4	24.11	even	2
2304.3.h.c.2177.2		4	4.3	odd	2
2304.3.h.c.2177.3		4	8.3	odd	2
2304.3.h.c.2177.4		4	12.11	even	2
2304.3.h.f.2177.1		4	24.5	odd	2	inner
2304.3.h.f.2177.2		4	1.1	even	1	trivial
2304.3.h.f.2177.3		4	8.5	even	2	inner
2304.3.h.f.2177.4		4	3.2	odd	2	inner
3042.3.c.e.1691.1		2	208.181	even	4
3042.3.c.e.1691.2		2	624.389	odd	4
3042.3.d.a.3041.1		4	624.5	even	4
3042.3.d.a.3041.2		4	208.21	odd	4
3042.3.d.a.3041.3		4	208.5	odd	4
3042.3.d.a.3041.4		4	624.437	even	4
3600.3.c.b.449.1		4	80.43	even	4
3600.3.c.b.449.2		4	240.203	odd	4
3600.3.c.b.449.3		4	80.27	even	4
3600.3.c.b.449.4		4	240.107	odd	4
3600.3.l.d.1601.1		2	80.59	odd	4
3600.3.l.d.1601.2		2	240.59	even	4

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.24264 q^{5} +4.00000 q^{7} +O(q^{10})\)	\(q-4.24264 q^{5} +4.00000 q^{7} +16.9706 q^{11} +8.00000i q^{13} -12.7279i q^{17} -16.0000i q^{19} +16.9706i q^{23} -7.00000 q^{25} -4.24264 q^{29} +44.0000 q^{31} -16.9706 q^{35} +34.0000i q^{37} -46.6690i q^{41} +40.0000i q^{43} -84.8528i q^{47} -33.0000 q^{49} +38.1838 q^{53} -72.0000 q^{55} +33.9411 q^{59} +50.0000i q^{61} -33.9411i q^{65} +8.00000i q^{67} +50.9117i q^{71} +16.0000 q^{73} +67.8823 q^{77} -76.0000 q^{79} -118.794 q^{83} +54.0000i q^{85} -12.7279i q^{89} +32.0000i q^{91} +67.8823i q^{95} +176.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{7}+O(q^{10}) \) 4 * q + 16 * q^7	\( 4 q + 16 q^{7} - 28 q^{25} + 176 q^{31} - 132 q^{49} - 288 q^{55} + 64 q^{73} - 304 q^{79} + 704 q^{97}+O(q^{100}) \) 4 * q + 16 * q^7 - 28 * q^25 + 176 * q^31 - 132 * q^49 - 288 * q^55 + 64 * q^73 - 304 * q^79 + 704 * q^97

Introduction

L-functions

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