LMFDB - Embedded newform 4416.2.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4416,2,Mod(1,4416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4416, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4416.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4416 = 2^{6} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4416.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$35.2619375326$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	4416.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.07838 q^{5} +4.34017 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.07838 q^{5} +4.34017 q^{7} +1.00000 q^{9} +3.41855 q^{11} -2.00000 q^{13} +1.07838 q^{15} -2.34017 q^{17} -0.921622 q^{19} -4.34017 q^{21} -1.00000 q^{23} -3.83710 q^{25} -1.00000 q^{27} -6.68035 q^{29} -8.68035 q^{31} -3.41855 q^{33} -4.68035 q^{35} -7.26180 q^{37} +2.00000 q^{39} +8.83710 q^{41} -7.75872 q^{43} -1.07838 q^{45} -12.6803 q^{47} +11.8371 q^{49} +2.34017 q^{51} +11.9155 q^{53} -3.68649 q^{55} +0.921622 q^{57} +1.84324 q^{59} -4.73820 q^{61} +4.34017 q^{63} +2.15676 q^{65} +7.75872 q^{67} +1.00000 q^{69} +2.52359 q^{71} +2.00000 q^{73} +3.83710 q^{75} +14.8371 q^{77} -12.3402 q^{79} +1.00000 q^{81} +0.894960 q^{83} +2.52359 q^{85} +6.68035 q^{87} -8.49693 q^{89} -8.68035 q^{91} +8.68035 q^{93} +0.993857 q^{95} +8.15676 q^{97} +3.41855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 4 q^{17} - 6 q^{19} - 2 q^{21} - 3 q^{23} + 17 q^{25} - 3 q^{27} + 2 q^{29} - 4 q^{31} + 4 q^{33} + 8 q^{35} - 14 q^{37} + 6 q^{39} - 2 q^{41} + 2 q^{43} - 16 q^{47} + 7 q^{49} - 4 q^{51} + 4 q^{53} - 24 q^{55} + 6 q^{57} + 12 q^{59} - 22 q^{61} + 2 q^{63} - 2 q^{67} + 3 q^{69} - 8 q^{71} + 6 q^{73} - 17 q^{75} + 16 q^{77} - 26 q^{79} + 3 q^{81} + 4 q^{83} - 8 q^{85} - 2 q^{87} - 8 q^{89} - 4 q^{91} + 4 q^{93} - 32 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 4 * q^17 - 6 * q^19 - 2 * q^21 - 3 * q^23 + 17 * q^25 - 3 * q^27 + 2 * q^29 - 4 * q^31 + 4 * q^33 + 8 * q^35 - 14 * q^37 + 6 * q^39 - 2 * q^41 + 2 * q^43 - 16 * q^47 + 7 * q^49 - 4 * q^51 + 4 * q^53 - 24 * q^55 + 6 * q^57 + 12 * q^59 - 22 * q^61 + 2 * q^63 - 2 * q^67 + 3 * q^69 - 8 * q^71 + 6 * q^73 - 17 * q^75 + 16 * q^77 - 26 * q^79 + 3 * q^81 + 4 * q^83 - 8 * q^85 - 2 * q^87 - 8 * q^89 - 4 * q^91 + 4 * q^93 - 32 * q^95 + 18 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.07838	−0.482265	−0.241133	−	0.970492i	$-0.577519\pi$
$5$	−1.07838	−0.482265	−0.241133	+	0.970492i	$0.577519\pi$
$6$	0	0
$7$	4.34017	1.64043	0.820216	−	0.572055i	$-0.193853\pi$
$7$	4.34017	1.64043	0.820216	+	0.572055i	$0.193853\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.41855	1.03073	0.515366	−	0.856970i	$-0.327656\pi$
$11$	3.41855	1.03073	0.515366	+	0.856970i	$0.327656\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	1.07838	0.278436
$16$	0	0
$17$	−2.34017	−0.567575	−0.283788	−	0.958887i	$-0.591591\pi$
$17$	−2.34017	−0.567575	−0.283788	+	0.958887i	$0.591591\pi$
$18$	0	0
$19$	−0.921622	−0.211435	−0.105717	−	0.994396i	$-0.533714\pi$
$19$	−0.921622	−0.211435	−0.105717	+	0.994396i	$0.533714\pi$
$20$	0	0
$21$	−4.34017	−0.947103
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.83710	−0.767420
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.68035	−1.24051	−0.620255	−	0.784401i	$-0.712971\pi$
$29$	−6.68035	−1.24051	−0.620255	+	0.784401i	$0.712971\pi$
$30$	0	0
$31$	−8.68035	−1.55904	−0.779518	−	0.626380i	$-0.784536\pi$
$31$	−8.68035	−1.55904	−0.779518	+	0.626380i	$0.784536\pi$
$32$	0	0
$33$	−3.41855	−0.595093
$34$	0	0
$35$	−4.68035	−0.791123
$36$	0	0
$37$	−7.26180	−1.19383	−0.596916	−	0.802304i	$-0.703607\pi$
$37$	−7.26180	−1.19383	−0.596916	+	0.802304i	$0.703607\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	8.83710	1.38012	0.690062	−	0.723751i	$-0.257583\pi$
$41$	8.83710	1.38012	0.690062	+	0.723751i	$0.257583\pi$
$42$	0	0
$43$	−7.75872	−1.18319	−0.591597	−	0.806234i	$-0.701502\pi$
$43$	−7.75872	−1.18319	−0.591597	+	0.806234i	$0.701502\pi$
$44$	0	0
$45$	−1.07838	−0.160755
$46$	0	0
$47$	−12.6803	−1.84962	−0.924809	−	0.380431i	$-0.875776\pi$
$47$	−12.6803	−1.84962	−0.924809	+	0.380431i	$0.875776\pi$
$48$	0	0
$49$	11.8371	1.69101
$50$	0	0
$51$	2.34017	0.327690
$52$	0	0
$53$	11.9155	1.63672	0.818358	−	0.574708i	$-0.194884\pi$
$53$	11.9155	1.63672	0.818358	+	0.574708i	$0.194884\pi$
$54$	0	0
$55$	−3.68649	−0.497086
$56$	0	0
$57$	0.921622	0.122072
$58$	0	0
$59$	1.84324	0.239970	0.119985	−	0.992776i	$-0.461715\pi$
$59$	1.84324	0.239970	0.119985	+	0.992776i	$0.461715\pi$
$60$	0	0
$61$	−4.73820	−0.606665	−0.303332	−	0.952885i	$-0.598099\pi$
$61$	−4.73820	−0.606665	−0.303332	+	0.952885i	$0.598099\pi$
$62$	0	0
$63$	4.34017	0.546810
$64$	0	0
$65$	2.15676	0.267513
$66$	0	0
$67$	7.75872	0.947879	0.473939	−	0.880557i	$-0.342832\pi$
$67$	7.75872	0.947879	0.473939	+	0.880557i	$0.342832\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	2.52359	0.299495	0.149748	−	0.988724i	$-0.452154\pi$
$71$	2.52359	0.299495	0.149748	+	0.988724i	$0.452154\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	3.83710	0.443070
$76$	0	0
$77$	14.8371	1.69084
$78$	0	0
$79$	−12.3402	−1.38838	−0.694189	−	0.719793i	$-0.744237\pi$
$79$	−12.3402	−1.38838	−0.694189	+	0.719793i	$0.744237\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.894960	0.0982347	0.0491173	−	0.998793i	$-0.484359\pi$
$83$	0.894960	0.0982347	0.0491173	+	0.998793i	$0.484359\pi$
$84$	0	0
$85$	2.52359	0.273722
$86$	0	0
$87$	6.68035	0.716208
$88$	0	0
$89$	−8.49693	−0.900673	−0.450336	−	0.892859i	$-0.648696\pi$
$89$	−8.49693	−0.900673	−0.450336	+	0.892859i	$0.648696\pi$
$90$	0	0
$91$	−8.68035	−0.909948
$92$	0	0
$93$	8.68035	0.900110
$94$	0	0
$95$	0.993857	0.101968
$96$	0	0
$97$	8.15676	0.828193	0.414097	−	0.910233i	$-0.364098\pi$
$97$	8.15676	0.828193	0.414097	+	0.910233i	$0.364098\pi$
$98$	0	0
$99$	3.41855	0.343577
$100$	0	0
$101$	6.31351	0.628218	0.314109	−	0.949387i	$-0.398294\pi$
$101$	6.31351	0.628218	0.314109	+	0.949387i	$0.398294\pi$
$102$	0	0
$103$	−1.81658	−0.178993	−0.0894966	−	0.995987i	$-0.528526\pi$
$103$	−1.81658	−0.178993	−0.0894966	+	0.995987i	$0.528526\pi$
$104$	0	0
$105$	4.68035	0.456755
$106$	0	0
$107$	5.94214	0.574448	0.287224	−	0.957863i	$-0.407268\pi$
$107$	5.94214	0.574448	0.287224	+	0.957863i	$0.407268\pi$
$108$	0	0
$109$	−14.7792	−1.41559	−0.707797	−	0.706416i	$-0.750311\pi$
$109$	−14.7792	−1.41559	−0.707797	+	0.706416i	$0.750311\pi$
$110$	0	0
$111$	7.26180	0.689259
$112$	0	0
$113$	−4.18342	−0.393543	−0.196771	−	0.980449i	$-0.563046\pi$
$113$	−4.18342	−0.393543	−0.196771	+	0.980449i	$0.563046\pi$
$114$	0	0
$115$	1.07838	0.100559
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−10.1568	−0.931068
$120$	0	0
$121$	0.686489	0.0624081
$122$	0	0
$123$	−8.83710	−0.796815
$124$	0	0
$125$	9.52973	0.852365
$126$	0	0
$127$	−20.9939	−1.86290	−0.931452	−	0.363865i	$-0.881457\pi$
$127$	−20.9939	−1.86290	−0.931452	+	0.363865i	$0.881457\pi$
$128$	0	0
$129$	7.75872	0.683118
$130$	0	0
$131$	−10.8371	−0.946842	−0.473421	−	0.880836i	$-0.656981\pi$
$131$	−10.8371	−0.946842	−0.473421	+	0.880836i	$0.656981\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	1.07838	0.0928120
$136$	0	0
$137$	−1.65983	−0.141809	−0.0709043	−	0.997483i	$-0.522588\pi$
$137$	−1.65983	−0.141809	−0.0709043	+	0.997483i	$0.522588\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	12.6803	1.06788
$142$	0	0
$143$	−6.83710	−0.571747
$144$	0	0
$145$	7.20394	0.598254
$146$	0	0
$147$	−11.8371	−0.976308
$148$	0	0
$149$	3.23513	0.265032	0.132516	−	0.991181i	$-0.457694\pi$
$149$	3.23513	0.265032	0.132516	+	0.991181i	$0.457694\pi$
$150$	0	0
$151$	−17.3607	−1.41279	−0.706397	−	0.707816i	$-0.749680\pi$
$151$	−17.3607	−1.41279	−0.706397	+	0.707816i	$0.749680\pi$
$152$	0	0
$153$	−2.34017	−0.189192
$154$	0	0
$155$	9.36069	0.751869
$156$	0	0
$157$	5.78539	0.461724	0.230862	−	0.972986i	$-0.425845\pi$
$157$	5.78539	0.461724	0.230862	+	0.972986i	$0.425845\pi$
$158$	0	0
$159$	−11.9155	−0.944959
$160$	0	0
$161$	−4.34017	−0.342054
$162$	0	0
$163$	−2.15676	−0.168930	−0.0844651	−	0.996426i	$-0.526918\pi$
$163$	−2.15676	−0.168930	−0.0844651	+	0.996426i	$0.526918\pi$
$164$	0	0
$165$	3.68649	0.286993
$166$	0	0
$167$	5.47641	0.423777	0.211889	−	0.977294i	$-0.432039\pi$
$167$	5.47641	0.423777	0.211889	+	0.977294i	$0.432039\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−0.921622	−0.0704782
$172$	0	0
$173$	9.31965	0.708560	0.354280	−	0.935139i	$-0.384726\pi$
$173$	9.31965	0.708560	0.354280	+	0.935139i	$0.384726\pi$
$174$	0	0
$175$	−16.6537	−1.25890
$176$	0	0
$177$	−1.84324	−0.138547
$178$	0	0
$179$	4.36683	0.326393	0.163196	−	0.986594i	$-0.447820\pi$
$179$	4.36683	0.326393	0.163196	+	0.986594i	$0.447820\pi$
$180$	0	0
$181$	3.94214	0.293017	0.146509	−	0.989209i	$-0.453196\pi$
$181$	3.94214	0.293017	0.146509	+	0.989209i	$0.453196\pi$
$182$	0	0
$183$	4.73820	0.350258
$184$	0	0
$185$	7.83096	0.575744
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	−4.34017	−0.315701
$190$	0	0
$191$	15.5174	1.12280	0.561402	−	0.827544i	$-0.310262\pi$
$191$	15.5174	1.12280	0.561402	+	0.827544i	$0.310262\pi$
$192$	0	0
$193$	−16.8371	−1.21196	−0.605981	−	0.795479i	$-0.707219\pi$
$193$	−16.8371	−1.21196	−0.605981	+	0.795479i	$0.707219\pi$
$194$	0	0
$195$	−2.15676	−0.154448
$196$	0	0
$197$	−9.20394	−0.655753	−0.327877	−	0.944721i	$-0.606333\pi$
$197$	−9.20394	−0.655753	−0.327877	+	0.944721i	$0.606333\pi$
$198$	0	0
$199$	26.6947	1.89234	0.946169	−	0.323672i	$-0.104917\pi$
$199$	26.6947	1.89234	0.946169	+	0.323672i	$0.104917\pi$
$200$	0	0
$201$	−7.75872	−0.547258
$202$	0	0
$203$	−28.9939	−2.03497
$204$	0	0
$205$	−9.52973	−0.665585
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−3.15061	−0.217932
$210$	0	0
$211$	−19.5174	−1.34364	−0.671818	−	0.740716i	$-0.734486\pi$
$211$	−19.5174	−1.34364	−0.671818	+	0.740716i	$0.734486\pi$
$212$	0	0
$213$	−2.52359	−0.172914
$214$	0	0
$215$	8.36683	0.570613
$216$	0	0
$217$	−37.6742	−2.55749
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	4.68035	0.314834
$222$	0	0
$223$	29.6742	1.98713	0.993566	−	0.113256i	$-0.0361281\pi$
$223$	29.6742	1.98713	0.993566	+	0.113256i	$0.0361281\pi$
$224$	0	0
$225$	−3.83710	−0.255807
$226$	0	0
$227$	−13.2618	−0.880216	−0.440108	−	0.897945i	$-0.645060\pi$
$227$	−13.2618	−0.880216	−0.440108	+	0.897945i	$0.645060\pi$
$228$	0	0
$229$	11.9421	0.789159	0.394579	−	0.918862i	$-0.370890\pi$
$229$	11.9421	0.789159	0.394579	+	0.918862i	$0.370890\pi$
$230$	0	0
$231$	−14.8371	−0.976210
$232$	0	0
$233$	5.68649	0.372534	0.186267	−	0.982499i	$-0.440361\pi$
$233$	5.68649	0.372534	0.186267	+	0.982499i	$0.440361\pi$
$234$	0	0
$235$	13.6742	0.892007
$236$	0	0
$237$	12.3402	0.801580
$238$	0	0
$239$	−21.6742	−1.40199	−0.700994	−	0.713167i	$-0.747260\pi$
$239$	−21.6742	−1.40199	−0.700994	+	0.713167i	$0.747260\pi$
$240$	0	0
$241$	16.1568	1.04075	0.520374	−	0.853938i	$-0.325793\pi$
$241$	16.1568	1.04075	0.520374	+	0.853938i	$0.325793\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−12.7649	−0.815517
$246$	0	0
$247$	1.84324	0.117283
$248$	0	0
$249$	−0.894960	−0.0567158
$250$	0	0
$251$	−28.0989	−1.77359	−0.886793	−	0.462166i	$-0.847072\pi$
$251$	−28.0989	−1.77359	−0.886793	+	0.462166i	$0.847072\pi$
$252$	0	0
$253$	−3.41855	−0.214922
$254$	0	0
$255$	−2.52359	−0.158033
$256$	0	0
$257$	−0.523590	−0.0326607	−0.0163303	−	0.999867i	$-0.505198\pi$
$257$	−0.523590	−0.0326607	−0.0163303	+	0.999867i	$0.505198\pi$
$258$	0	0
$259$	−31.5174	−1.95840
$260$	0	0
$261$	−6.68035	−0.413503
$262$	0	0
$263$	4.99386	0.307934	0.153967	−	0.988076i	$-0.450795\pi$
$263$	4.99386	0.307934	0.153967	+	0.988076i	$0.450795\pi$
$264$	0	0
$265$	−12.8494	−0.789332
$266$	0	0
$267$	8.49693	0.520004
$268$	0	0
$269$	3.84324	0.234327	0.117163	−	0.993113i	$-0.462620\pi$
$269$	3.84324	0.234327	0.117163	+	0.993113i	$0.462620\pi$
$270$	0	0
$271$	−24.6803	−1.49922	−0.749612	−	0.661877i	$-0.769760\pi$
$271$	−24.6803	−1.49922	−0.749612	+	0.661877i	$0.769760\pi$
$272$	0	0
$273$	8.68035	0.525358
$274$	0	0
$275$	−13.1173	−0.791005
$276$	0	0
$277$	−25.8843	−1.55524	−0.777618	−	0.628737i	$-0.783572\pi$
$277$	−25.8843	−1.55524	−0.777618	+	0.628737i	$0.783572\pi$
$278$	0	0
$279$	−8.68035	−0.519679
$280$	0	0
$281$	29.1773	1.74057	0.870285	−	0.492548i	$-0.163934\pi$
$281$	29.1773	1.74057	0.870285	+	0.492548i	$0.163934\pi$
$282$	0	0
$283$	2.28231	0.135669	0.0678347	−	0.997697i	$-0.478391\pi$
$283$	2.28231	0.135669	0.0678347	+	0.997697i	$0.478391\pi$
$284$	0	0
$285$	−0.993857	−0.0588710
$286$	0	0
$287$	38.3545	2.26400
$288$	0	0
$289$	−11.5236	−0.677858
$290$	0	0
$291$	−8.15676	−0.478157
$292$	0	0
$293$	7.60197	0.444112	0.222056	−	0.975034i	$-0.428723\pi$
$293$	7.60197	0.444112	0.222056	+	0.975034i	$0.428723\pi$
$294$	0	0
$295$	−1.98771	−0.115729
$296$	0	0
$297$	−3.41855	−0.198364
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	−33.6742	−1.94095
$302$	0	0
$303$	−6.31351	−0.362702
$304$	0	0
$305$	5.10957	0.292573
$306$	0	0
$307$	10.8904	0.621549	0.310775	−	0.950484i	$-0.399412\pi$
$307$	10.8904	0.621549	0.310775	+	0.950484i	$0.399412\pi$
$308$	0	0
$309$	1.81658	0.103342
$310$	0	0
$311$	−11.5174	−0.653095	−0.326547	−	0.945181i	$-0.605885\pi$
$311$	−11.5174	−0.653095	−0.326547	+	0.945181i	$0.605885\pi$
$312$	0	0
$313$	−4.21008	−0.237968	−0.118984	−	0.992896i	$-0.537964\pi$
$313$	−4.21008	−0.237968	−0.118984	+	0.992896i	$0.537964\pi$
$314$	0	0
$315$	−4.68035	−0.263708
$316$	0	0
$317$	−13.3197	−0.748106	−0.374053	−	0.927407i	$-0.622032\pi$
$317$	−13.3197	−0.748106	−0.374053	+	0.927407i	$0.622032\pi$
$318$	0	0
$319$	−22.8371	−1.27863
$320$	0	0
$321$	−5.94214	−0.331658
$322$	0	0
$323$	2.15676	0.120005
$324$	0	0
$325$	7.67420	0.425688
$326$	0	0
$327$	14.7792	0.817294
$328$	0	0
$329$	−55.0349	−3.03417
$330$	0	0
$331$	−2.15676	−0.118546	−0.0592730	−	0.998242i	$-0.518878\pi$
$331$	−2.15676	−0.118546	−0.0592730	+	0.998242i	$0.518878\pi$
$332$	0	0
$333$	−7.26180	−0.397944
$334$	0	0
$335$	−8.36683	−0.457129
$336$	0	0
$337$	24.3545	1.32668	0.663338	−	0.748320i	$-0.269139\pi$
$337$	24.3545	1.32668	0.663338	+	0.748320i	$0.269139\pi$
$338$	0	0
$339$	4.18342	0.227212
$340$	0	0
$341$	−29.6742	−1.60695
$342$	0	0
$343$	20.9939	1.13356
$344$	0	0
$345$	−1.07838	−0.0580579
$346$	0	0
$347$	−26.0410	−1.39796	−0.698978	−	0.715143i	$-0.746362\pi$
$347$	−26.0410	−1.39796	−0.698978	+	0.715143i	$0.746362\pi$
$348$	0	0
$349$	0.523590	0.0280272	0.0140136	−	0.999902i	$-0.495539\pi$
$349$	0.523590	0.0280272	0.0140136	+	0.999902i	$0.495539\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	28.0410	1.49247	0.746237	−	0.665680i	$-0.231859\pi$
$353$	28.0410	1.49247	0.746237	+	0.665680i	$0.231859\pi$
$354$	0	0
$355$	−2.72138	−0.144436
$356$	0	0
$357$	10.1568	0.537553
$358$	0	0
$359$	−9.84324	−0.519507	−0.259753	−	0.965675i	$-0.583641\pi$
$359$	−9.84324	−0.519507	−0.259753	+	0.965675i	$0.583641\pi$
$360$	0	0
$361$	−18.1506	−0.955295
$362$	0	0
$363$	−0.686489	−0.0360313
$364$	0	0
$365$	−2.15676	−0.112890
$366$	0	0
$367$	17.3340	0.904829	0.452414	−	0.891808i	$-0.350563\pi$
$367$	17.3340	0.904829	0.452414	+	0.891808i	$0.350563\pi$
$368$	0	0
$369$	8.83710	0.460041
$370$	0	0
$371$	51.7152	2.68492
$372$	0	0
$373$	−23.2618	−1.20445	−0.602225	−	0.798326i	$-0.705719\pi$
$373$	−23.2618	−1.20445	−0.602225	+	0.798326i	$0.705719\pi$
$374$	0	0
$375$	−9.52973	−0.492113
$376$	0	0
$377$	13.3607	0.688111
$378$	0	0
$379$	25.7998	1.32524	0.662622	−	0.748954i	$-0.269443\pi$
$379$	25.7998	1.32524	0.662622	+	0.748954i	$0.269443\pi$
$380$	0	0
$381$	20.9939	1.07555
$382$	0	0
$383$	−6.35455	−0.324702	−0.162351	−	0.986733i	$-0.551908\pi$
$383$	−6.35455	−0.324702	−0.162351	+	0.986733i	$0.551908\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	−7.75872	−0.394398
$388$	0	0
$389$	−3.54864	−0.179923	−0.0899617	−	0.995945i	$-0.528674\pi$
$389$	−3.54864	−0.179923	−0.0899617	+	0.995945i	$0.528674\pi$
$390$	0	0
$391$	2.34017	0.118348
$392$	0	0
$393$	10.8371	0.546659
$394$	0	0
$395$	13.3074	0.669566
$396$	0	0
$397$	10.6270	0.533355	0.266677	−	0.963786i	$-0.414074\pi$
$397$	10.6270	0.533355	0.266677	+	0.963786i	$0.414074\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	36.0677	1.80113	0.900567	−	0.434716i	$-0.143151\pi$
$401$	36.0677	1.80113	0.900567	+	0.434716i	$0.143151\pi$
$402$	0	0
$403$	17.3607	0.864798
$404$	0	0
$405$	−1.07838	−0.0535850
$406$	0	0
$407$	−24.8248	−1.23052
$408$	0	0
$409$	−29.1506	−1.44141	−0.720703	−	0.693244i	$-0.756181\pi$
$409$	−29.1506	−1.44141	−0.720703	+	0.693244i	$0.756181\pi$
$410$	0	0
$411$	1.65983	0.0818732
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	−0.965105	−0.0473752
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−12.7792	−0.624307	−0.312153	−	0.950032i	$-0.601050\pi$
$419$	−12.7792	−0.624307	−0.312153	+	0.950032i	$0.601050\pi$
$420$	0	0
$421$	−28.2557	−1.37710	−0.688548	−	0.725191i	$-0.741752\pi$
$421$	−28.2557	−1.37710	−0.688548	+	0.725191i	$0.741752\pi$
$422$	0	0
$423$	−12.6803	−0.616540
$424$	0	0
$425$	8.97948	0.435569
$426$	0	0
$427$	−20.5646	−0.995192
$428$	0	0
$429$	6.83710	0.330098
$430$	0	0
$431$	37.8720	1.82423	0.912115	−	0.409935i	$-0.134448\pi$
$431$	37.8720	1.82423	0.912115	+	0.409935i	$0.134448\pi$
$432$	0	0
$433$	−12.8371	−0.616912	−0.308456	−	0.951239i	$-0.599812\pi$
$433$	−12.8371	−0.616912	−0.308456	+	0.951239i	$0.599812\pi$
$434$	0	0
$435$	−7.20394	−0.345402
$436$	0	0
$437$	0.921622	0.0440872
$438$	0	0
$439$	−25.3607	−1.21040	−0.605200	−	0.796074i	$-0.706907\pi$
$439$	−25.3607	−1.21040	−0.605200	+	0.796074i	$0.706907\pi$
$440$	0	0
$441$	11.8371	0.563671
$442$	0	0
$443$	−37.9877	−1.80485	−0.902425	−	0.430846i	$-0.858215\pi$
$443$	−37.9877	−1.80485	−0.902425	+	0.430846i	$0.858215\pi$
$444$	0	0
$445$	9.16290	0.434363
$446$	0	0
$447$	−3.23513	−0.153017
$448$	0	0
$449$	−38.1978	−1.80267	−0.901333	−	0.433128i	$-0.857410\pi$
$449$	−38.1978	−1.80267	−0.901333	+	0.433128i	$0.857410\pi$
$450$	0	0
$451$	30.2101	1.42254
$452$	0	0
$453$	17.3607	0.815676
$454$	0	0
$455$	9.36069	0.438836
$456$	0	0
$457$	11.8432	0.554004	0.277002	−	0.960869i	$-0.410659\pi$
$457$	11.8432	0.554004	0.277002	+	0.960869i	$0.410659\pi$
$458$	0	0
$459$	2.34017	0.109230
$460$	0	0
$461$	39.1917	1.82534	0.912669	−	0.408700i	$-0.134018\pi$
$461$	39.1917	1.82534	0.912669	+	0.408700i	$0.134018\pi$
$462$	0	0
$463$	25.3607	1.17861	0.589306	−	0.807910i	$-0.299401\pi$
$463$	25.3607	1.17861	0.589306	+	0.807910i	$0.299401\pi$
$464$	0	0
$465$	−9.36069	−0.434092
$466$	0	0
$467$	13.2618	0.613683	0.306841	−	0.951761i	$-0.400728\pi$
$467$	13.2618	0.613683	0.306841	+	0.951761i	$0.400728\pi$
$468$	0	0
$469$	33.6742	1.55493
$470$	0	0
$471$	−5.78539	−0.266577
$472$	0	0
$473$	−26.5236	−1.21956
$474$	0	0
$475$	3.53636	0.162259
$476$	0	0
$477$	11.9155	0.545572
$478$	0	0
$479$	−11.6865	−0.533969	−0.266985	−	0.963701i	$-0.586027\pi$
$479$	−11.6865	−0.533969	−0.266985	+	0.963701i	$0.586027\pi$
$480$	0	0
$481$	14.5236	0.662219
$482$	0	0
$483$	4.34017	0.197485
$484$	0	0
$485$	−8.79606	−0.399409
$486$	0	0
$487$	−5.04718	−0.228710	−0.114355	−	0.993440i	$-0.536480\pi$
$487$	−5.04718	−0.228710	−0.114355	+	0.993440i	$0.536480\pi$
$488$	0	0
$489$	2.15676	0.0975319
$490$	0	0
$491$	−0.680346	−0.0307036	−0.0153518	−	0.999882i	$-0.504887\pi$
$491$	−0.680346	−0.0307036	−0.0153518	+	0.999882i	$0.504887\pi$
$492$	0	0
$493$	15.6332	0.704082
$494$	0	0
$495$	−3.68649	−0.165695
$496$	0	0
$497$	10.9528	0.491301
$498$	0	0
$499$	25.1917	1.12773	0.563867	−	0.825866i	$-0.309313\pi$
$499$	25.1917	1.12773	0.563867	+	0.825866i	$0.309313\pi$
$500$	0	0
$501$	−5.47641	−0.244668
$502$	0	0
$503$	−19.8843	−0.886596	−0.443298	−	0.896374i	$-0.646192\pi$
$503$	−19.8843	−0.886596	−0.443298	+	0.896374i	$0.646192\pi$
$504$	0	0
$505$	−6.80835	−0.302968
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−33.4017	−1.48051	−0.740253	−	0.672329i	$-0.765294\pi$
$509$	−33.4017	−1.48051	−0.740253	+	0.672329i	$0.765294\pi$
$510$	0	0
$511$	8.68035	0.383996
$512$	0	0
$513$	0.921622	0.0406906
$514$	0	0
$515$	1.95896	0.0863222
$516$	0	0
$517$	−43.3484	−1.90646
$518$	0	0
$519$	−9.31965	−0.409087
$520$	0	0
$521$	27.3874	1.19986	0.599931	−	0.800052i	$-0.295195\pi$
$521$	27.3874	1.19986	0.599931	+	0.800052i	$0.295195\pi$
$522$	0	0
$523$	28.0722	1.22751	0.613757	−	0.789495i	$-0.289658\pi$
$523$	28.0722	1.22751	0.613757	+	0.789495i	$0.289658\pi$
$524$	0	0
$525$	16.6537	0.726826
$526$	0	0
$527$	20.3135	0.884870
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	1.84324	0.0799900
$532$	0	0
$533$	−17.6742	−0.765555
$534$	0	0
$535$	−6.40787	−0.277036
$536$	0	0
$537$	−4.36683	−0.188443
$538$	0	0
$539$	40.4657	1.74298
$540$	0	0
$541$	41.8843	1.80075	0.900373	−	0.435119i	$-0.143294\pi$
$541$	41.8843	1.80075	0.900373	+	0.435119i	$0.143294\pi$
$542$	0	0
$543$	−3.94214	−0.169173
$544$	0	0
$545$	15.9376	0.682692
$546$	0	0
$547$	31.2039	1.33418	0.667092	−	0.744975i	$-0.267539\pi$
$547$	31.2039	1.33418	0.667092	+	0.744975i	$0.267539\pi$
$548$	0	0
$549$	−4.73820	−0.202222
$550$	0	0
$551$	6.15676	0.262287
$552$	0	0
$553$	−53.5585	−2.27754
$554$	0	0
$555$	−7.83096	−0.332406
$556$	0	0
$557$	−6.60811	−0.279995	−0.139997	−	0.990152i	$-0.544709\pi$
$557$	−6.60811	−0.279995	−0.139997	+	0.990152i	$0.544709\pi$
$558$	0	0
$559$	15.5174	0.656318
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	−29.2618	−1.23324	−0.616619	−	0.787262i	$-0.711498\pi$
$563$	−29.2618	−1.23324	−0.616619	+	0.787262i	$0.711498\pi$
$564$	0	0
$565$	4.51130	0.189792
$566$	0	0
$567$	4.34017	0.182270
$568$	0	0
$569$	11.3340	0.475147	0.237574	−	0.971370i	$-0.423648\pi$
$569$	11.3340	0.475147	0.237574	+	0.971370i	$0.423648\pi$
$570$	0	0
$571$	−34.9627	−1.46314	−0.731571	−	0.681765i	$-0.761212\pi$
$571$	−34.9627	−1.46314	−0.731571	+	0.681765i	$0.761212\pi$
$572$	0	0
$573$	−15.5174	−0.648251
$574$	0	0
$575$	3.83710	0.160018
$576$	0	0
$577$	1.15061	0.0479006	0.0239503	−	0.999713i	$-0.492376\pi$
$577$	1.15061	0.0479006	0.0239503	+	0.999713i	$0.492376\pi$
$578$	0	0
$579$	16.8371	0.699726
$580$	0	0
$581$	3.88428	0.161147
$582$	0	0
$583$	40.7337	1.68702
$584$	0	0
$585$	2.15676	0.0891709
$586$	0	0
$587$	−16.5113	−0.681494	−0.340747	−	0.940155i	$-0.610680\pi$
$587$	−16.5113	−0.681494	−0.340747	+	0.940155i	$0.610680\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	9.20394	0.378599
$592$	0	0
$593$	11.1629	0.458405	0.229203	−	0.973379i	$-0.426388\pi$
$593$	11.1629	0.458405	0.229203	+	0.973379i	$0.426388\pi$
$594$	0	0
$595$	10.9528	0.449022
$596$	0	0
$597$	−26.6947	−1.09254
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	41.1506	1.67857	0.839284	−	0.543693i	$-0.182974\pi$
$601$	41.1506	1.67857	0.839284	+	0.543693i	$0.182974\pi$
$602$	0	0
$603$	7.75872	0.315960
$604$	0	0
$605$	−0.740294	−0.0300973
$606$	0	0
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	0	0
$609$	28.9939	1.17489
$610$	0	0
$611$	25.3607	1.02598
$612$	0	0
$613$	−4.73820	−0.191374	−0.0956871	−	0.995411i	$-0.530505\pi$
$613$	−4.73820	−0.191374	−0.0956871	+	0.995411i	$0.530505\pi$
$614$	0	0
$615$	9.52973	0.384276
$616$	0	0
$617$	−34.4846	−1.38830	−0.694150	−	0.719831i	$-0.744219\pi$
$617$	−34.4846	−1.38830	−0.694150	+	0.719831i	$0.744219\pi$
$618$	0	0
$619$	−41.0661	−1.65059	−0.825293	−	0.564705i	$-0.808990\pi$
$619$	−41.0661	−1.65059	−0.825293	+	0.564705i	$0.808990\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−36.8781	−1.47749
$624$	0	0
$625$	8.90885	0.356354
$626$	0	0
$627$	3.15061	0.125823
$628$	0	0
$629$	16.9939	0.677589
$630$	0	0
$631$	−22.8638	−0.910192	−0.455096	−	0.890442i	$-0.650395\pi$
$631$	−22.8638	−0.910192	−0.455096	+	0.890442i	$0.650395\pi$
$632$	0	0
$633$	19.5174	0.775749
$634$	0	0
$635$	22.6393	0.898414
$636$	0	0
$637$	−23.6742	−0.938006
$638$	0	0
$639$	2.52359	0.0998317
$640$	0	0
$641$	34.7070	1.37084	0.685422	−	0.728146i	$-0.259618\pi$
$641$	34.7070	1.37084	0.685422	+	0.728146i	$0.259618\pi$
$642$	0	0
$643$	2.96266	0.116836	0.0584180	−	0.998292i	$-0.481394\pi$
$643$	2.96266	0.116836	0.0584180	+	0.998292i	$0.481394\pi$
$644$	0	0
$645$	−8.36683	−0.329444
$646$	0	0
$647$	−14.4703	−0.568885	−0.284442	−	0.958693i	$-0.591808\pi$
$647$	−14.4703	−0.568885	−0.284442	+	0.958693i	$0.591808\pi$
$648$	0	0
$649$	6.30122	0.247345
$650$	0	0
$651$	37.6742	1.47657
$652$	0	0
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	0	0
$655$	11.6865	0.456629
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	−35.1338	−1.36862	−0.684309	−	0.729192i	$-0.739896\pi$
$659$	−35.1338	−1.36862	−0.684309	+	0.729192i	$0.739896\pi$
$660$	0	0
$661$	42.2967	1.64515	0.822575	−	0.568656i	$-0.192537\pi$
$661$	42.2967	1.64515	0.822575	+	0.568656i	$0.192537\pi$
$662$	0	0
$663$	−4.68035	−0.181770
$664$	0	0
$665$	4.31351	0.167271
$666$	0	0
$667$	6.68035	0.258664
$668$	0	0
$669$	−29.6742	−1.14727
$670$	0	0
$671$	−16.1978	−0.625309
$672$	0	0
$673$	9.15061	0.352730	0.176365	−	0.984325i	$-0.443566\pi$
$673$	9.15061	0.352730	0.176365	+	0.984325i	$0.443566\pi$
$674$	0	0
$675$	3.83710	0.147690
$676$	0	0
$677$	43.0037	1.65277	0.826383	−	0.563108i	$-0.190395\pi$
$677$	43.0037	1.65277	0.826383	+	0.563108i	$0.190395\pi$
$678$	0	0
$679$	35.4017	1.35859
$680$	0	0
$681$	13.2618	0.508193
$682$	0	0
$683$	44.1978	1.69118	0.845591	−	0.533832i	$-0.179248\pi$
$683$	44.1978	1.69118	0.845591	+	0.533832i	$0.179248\pi$
$684$	0	0
$685$	1.78992	0.0683893
$686$	0	0
$687$	−11.9421	−0.455621
$688$	0	0
$689$	−23.8310	−0.907887
$690$	0	0
$691$	22.4703	0.854809	0.427405	−	0.904060i	$-0.359428\pi$
$691$	22.4703	0.854809	0.427405	+	0.904060i	$0.359428\pi$
$692$	0	0
$693$	14.8371	0.563615
$694$	0	0
$695$	−4.31351	−0.163621
$696$	0	0
$697$	−20.6803	−0.783324
$698$	0	0
$699$	−5.68649	−0.215083
$700$	0	0
$701$	24.4801	0.924601	0.462300	−	0.886723i	$-0.347024\pi$
$701$	24.4801	0.924601	0.462300	+	0.886723i	$0.347024\pi$
$702$	0	0
$703$	6.69263	0.252417
$704$	0	0
$705$	−13.6742	−0.515000
$706$	0	0
$707$	27.4017	1.03055
$708$	0	0
$709$	−32.5692	−1.22316	−0.611580	−	0.791182i	$-0.709466\pi$
$709$	−32.5692	−1.22316	−0.611580	+	0.791182i	$0.709466\pi$
$710$	0	0
$711$	−12.3402	−0.462793
$712$	0	0
$713$	8.68035	0.325082
$714$	0	0
$715$	7.37298	0.275734
$716$	0	0
$717$	21.6742	0.809438
$718$	0	0
$719$	7.20394	0.268661	0.134331	−	0.990937i	$-0.457112\pi$
$719$	7.20394	0.268661	0.134331	+	0.990937i	$0.457112\pi$
$720$	0	0
$721$	−7.88428	−0.293626
$722$	0	0
$723$	−16.1568	−0.600876
$724$	0	0
$725$	25.6332	0.951992
$726$	0	0
$727$	16.6537	0.617651	0.308825	−	0.951119i	$-0.400064\pi$
$727$	16.6537	0.617651	0.308825	+	0.951119i	$0.400064\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.1568	0.671552
$732$	0	0
$733$	−46.2434	−1.70804	−0.854019	−	0.520242i	$-0.825842\pi$
$733$	−46.2434	−1.70804	−0.854019	+	0.520242i	$0.825842\pi$
$734$	0	0
$735$	12.7649	0.470839
$736$	0	0
$737$	26.5236	0.977009
$738$	0	0
$739$	−12.7337	−0.468416	−0.234208	−	0.972187i	$-0.575250\pi$
$739$	−12.7337	−0.468416	−0.234208	+	0.972187i	$0.575250\pi$
$740$	0	0
$741$	−1.84324	−0.0677133
$742$	0	0
$743$	20.3668	0.747187	0.373593	−	0.927593i	$-0.378126\pi$
$743$	20.3668	0.747187	0.373593	+	0.927593i	$0.378126\pi$
$744$	0	0
$745$	−3.48870	−0.127816
$746$	0	0
$747$	0.894960	0.0327449
$748$	0	0
$749$	25.7899	0.942343
$750$	0	0
$751$	−3.85762	−0.140767	−0.0703833	−	0.997520i	$-0.522422\pi$
$751$	−3.85762	−0.140767	−0.0703833	+	0.997520i	$0.522422\pi$
$752$	0	0
$753$	28.0989	1.02398
$754$	0	0
$755$	18.7214	0.681341
$756$	0	0
$757$	1.61634	0.0587470	0.0293735	−	0.999569i	$-0.490649\pi$
$757$	1.61634	0.0587470	0.0293735	+	0.999569i	$0.490649\pi$
$758$	0	0
$759$	3.41855	0.124086
$760$	0	0
$761$	20.0410	0.726487	0.363244	−	0.931694i	$-0.381669\pi$
$761$	20.0410	0.726487	0.363244	+	0.931694i	$0.381669\pi$
$762$	0	0
$763$	−64.1445	−2.32219
$764$	0	0
$765$	2.52359	0.0912406
$766$	0	0
$767$	−3.68649	−0.133111
$768$	0	0
$769$	−23.1629	−0.835275	−0.417638	−	0.908614i	$-0.637142\pi$
$769$	−23.1629	−0.835275	−0.417638	+	0.908614i	$0.637142\pi$
$770$	0	0
$771$	0.523590	0.0188566
$772$	0	0
$773$	−41.2762	−1.48460	−0.742300	−	0.670067i	$-0.766265\pi$
$773$	−41.2762	−1.48460	−0.742300	+	0.670067i	$0.766265\pi$
$774$	0	0
$775$	33.3074	1.19644
$776$	0	0
$777$	31.5174	1.13068
$778$	0	0
$779$	−8.14447	−0.291806
$780$	0	0
$781$	8.62702	0.308699
$782$	0	0
$783$	6.68035	0.238736
$784$	0	0
$785$	−6.23883	−0.222673
$786$	0	0
$787$	−13.4329	−0.478832	−0.239416	−	0.970917i	$-0.576956\pi$
$787$	−13.4329	−0.478832	−0.239416	+	0.970917i	$0.576956\pi$
$788$	0	0
$789$	−4.99386	−0.177786
$790$	0	0
$791$	−18.1568	−0.645580
$792$	0	0
$793$	9.47641	0.336517
$794$	0	0
$795$	12.8494	0.455721
$796$	0	0
$797$	5.07838	0.179885	0.0899427	−	0.995947i	$-0.471332\pi$
$797$	5.07838	0.179885	0.0899427	+	0.995947i	$0.471332\pi$
$798$	0	0
$799$	29.6742	1.04980
$800$	0	0
$801$	−8.49693	−0.300224
$802$	0	0
$803$	6.83710	0.241276
$804$	0	0
$805$	4.68035	0.164961
$806$	0	0
$807$	−3.84324	−0.135289
$808$	0	0
$809$	8.35455	0.293730	0.146865	−	0.989157i	$-0.453082\pi$
$809$	8.35455	0.293730	0.146865	+	0.989157i	$0.453082\pi$
$810$	0	0
$811$	−44.8781	−1.57588	−0.787942	−	0.615749i	$-0.788854\pi$
$811$	−44.8781	−1.57588	−0.787942	+	0.615749i	$0.788854\pi$
$812$	0	0
$813$	24.6803	0.865578
$814$	0	0
$815$	2.32580	0.0814691
$816$	0	0
$817$	7.15061	0.250168
$818$	0	0
$819$	−8.68035	−0.303316
$820$	0	0
$821$	−49.3484	−1.72227	−0.861136	−	0.508375i	$-0.830246\pi$
$821$	−49.3484	−1.72227	−0.861136	+	0.508375i	$0.830246\pi$
$822$	0	0
$823$	10.0410	0.350009	0.175004	−	0.984568i	$-0.444006\pi$
$823$	10.0410	0.350009	0.175004	+	0.984568i	$0.444006\pi$
$824$	0	0
$825$	13.1173	0.456687
$826$	0	0
$827$	−54.0866	−1.88078	−0.940388	−	0.340104i	$-0.889538\pi$
$827$	−54.0866	−1.88078	−0.940388	+	0.340104i	$0.889538\pi$
$828$	0	0
$829$	20.8371	0.723702	0.361851	−	0.932236i	$-0.382145\pi$
$829$	20.8371	0.723702	0.361851	+	0.932236i	$0.382145\pi$
$830$	0	0
$831$	25.8843	0.897916
$832$	0	0
$833$	−27.7009	−0.959778
$834$	0	0
$835$	−5.90564	−0.204373
$836$	0	0
$837$	8.68035	0.300037
$838$	0	0
$839$	8.68035	0.299679	0.149839	−	0.988710i	$-0.452124\pi$
$839$	8.68035	0.299679	0.149839	+	0.988710i	$0.452124\pi$
$840$	0	0
$841$	15.6270	0.538863
$842$	0	0
$843$	−29.1773	−1.00492
$844$	0	0
$845$	9.70540	0.333876
$846$	0	0
$847$	2.97948	0.102376
$848$	0	0
$849$	−2.28231	−0.0783288
$850$	0	0
$851$	7.26180	0.248931
$852$	0	0
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	0	0
$855$	0.993857	0.0339892
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	−38.3545	−1.30712
$862$	0	0
$863$	29.8432	1.01588	0.507938	−	0.861394i	$-0.330408\pi$
$863$	29.8432	1.01588	0.507938	+	0.861394i	$0.330408\pi$
$864$	0	0
$865$	−10.0501	−0.341714
$866$	0	0
$867$	11.5236	0.391362
$868$	0	0
$869$	−42.1855	−1.43105
$870$	0	0
$871$	−15.5174	−0.525789
$872$	0	0
$873$	8.15676	0.276064
$874$	0	0
$875$	41.3607	1.39825
$876$	0	0
$877$	−11.4764	−0.387531	−0.193765	−	0.981048i	$-0.562070\pi$
$877$	−11.4764	−0.387531	−0.193765	+	0.981048i	$0.562070\pi$
$878$	0	0
$879$	−7.60197	−0.256408
$880$	0	0
$881$	−26.8227	−0.903681	−0.451840	−	0.892099i	$-0.649232\pi$
$881$	−26.8227	−0.903681	−0.451840	+	0.892099i	$0.649232\pi$
$882$	0	0
$883$	2.15676	0.0725806	0.0362903	−	0.999341i	$-0.488446\pi$
$883$	2.15676	0.0725806	0.0362903	+	0.999341i	$0.488446\pi$
$884$	0	0
$885$	1.98771	0.0668163
$886$	0	0
$887$	29.4140	0.987626	0.493813	−	0.869568i	$-0.335603\pi$
$887$	29.4140	0.987626	0.493813	+	0.869568i	$0.335603\pi$
$888$	0	0
$889$	−91.1170	−3.05597
$890$	0	0
$891$	3.41855	0.114526
$892$	0	0
$893$	11.6865	0.391073
$894$	0	0
$895$	−4.70910	−0.157408
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	57.9877	1.93400
$900$	0	0
$901$	−27.8843	−0.928960
$902$	0	0
$903$	33.6742	1.12061
$904$	0	0
$905$	−4.25112	−0.141312
$906$	0	0
$907$	44.2700	1.46996	0.734981	−	0.678088i	$-0.237191\pi$
$907$	44.2700	1.46996	0.734981	+	0.678088i	$0.237191\pi$
$908$	0	0
$909$	6.31351	0.209406
$910$	0	0
$911$	20.5113	0.679570	0.339785	−	0.940503i	$-0.389646\pi$
$911$	20.5113	0.679570	0.339785	+	0.940503i	$0.389646\pi$
$912$	0	0
$913$	3.05947	0.101254
$914$	0	0
$915$	−5.10957	−0.168917
$916$	0	0
$917$	−47.0349	−1.55323
$918$	0	0
$919$	−2.01438	−0.0664481	−0.0332241	−	0.999448i	$-0.510577\pi$
$919$	−2.01438	−0.0664481	−0.0332241	+	0.999448i	$0.510577\pi$
$920$	0	0
$921$	−10.8904	−0.358852
$922$	0	0
$923$	−5.04718	−0.166130
$924$	0	0
$925$	27.8642	0.916171
$926$	0	0
$927$	−1.81658	−0.0596644
$928$	0	0
$929$	−4.21008	−0.138128	−0.0690641	−	0.997612i	$-0.522001\pi$
$929$	−4.21008	−0.138128	−0.0690641	+	0.997612i	$0.522001\pi$
$930$	0	0
$931$	−10.9093	−0.357539
$932$	0	0
$933$	11.5174	0.377064
$934$	0	0
$935$	8.62702	0.282134
$936$	0	0
$937$	−20.6393	−0.674257	−0.337128	−	0.941459i	$-0.609456\pi$
$937$	−20.6393	−0.674257	−0.337128	+	0.941459i	$0.609456\pi$
$938$	0	0
$939$	4.21008	0.137391
$940$	0	0
$941$	52.5958	1.71457	0.857287	−	0.514838i	$-0.172148\pi$
$941$	52.5958	1.71457	0.857287	+	0.514838i	$0.172148\pi$
$942$	0	0
$943$	−8.83710	−0.287776
$944$	0	0
$945$	4.68035	0.152252
$946$	0	0
$947$	−44.0288	−1.43074	−0.715371	−	0.698745i	$-0.753742\pi$
$947$	−44.0288	−1.43074	−0.715371	+	0.698745i	$0.753742\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	13.3197	0.431919
$952$	0	0
$953$	6.39350	0.207106	0.103553	−	0.994624i	$-0.466979\pi$
$953$	6.39350	0.207106	0.103553	+	0.994624i	$0.466979\pi$
$954$	0	0
$955$	−16.7337	−0.541489
$956$	0	0
$957$	22.8371	0.738219
$958$	0	0
$959$	−7.20394	−0.232627
$960$	0	0
$961$	44.3484	1.43059
$962$	0	0
$963$	5.94214	0.191483
$964$	0	0
$965$	18.1568	0.584487
$966$	0	0
$967$	−39.0882	−1.25699	−0.628496	−	0.777813i	$-0.716329\pi$
$967$	−39.0882	−1.25699	−0.628496	+	0.777813i	$0.716329\pi$
$968$	0	0
$969$	−2.15676	−0.0692850
$970$	0	0
$971$	54.1933	1.73914	0.869572	−	0.493806i	$-0.164395\pi$
$971$	54.1933	1.73914	0.869572	+	0.493806i	$0.164395\pi$
$972$	0	0
$973$	17.3607	0.556558
$974$	0	0
$975$	−7.67420	−0.245771
$976$	0	0
$977$	47.7009	1.52609	0.763043	−	0.646348i	$-0.223704\pi$
$977$	47.7009	1.52609	0.763043	+	0.646348i	$0.223704\pi$
$978$	0	0
$979$	−29.0472	−0.928352
$980$	0	0
$981$	−14.7792	−0.471865
$982$	0	0
$983$	49.5585	1.58067	0.790335	−	0.612675i	$-0.209906\pi$
$983$	49.5585	1.58067	0.790335	+	0.612675i	$0.209906\pi$
$984$	0	0
$985$	9.92532	0.316247
$986$	0	0
$987$	55.0349	1.75178
$988$	0	0
$989$	7.75872	0.246713
$990$	0	0
$991$	−20.2602	−0.643586	−0.321793	−	0.946810i	$-0.604286\pi$
$991$	−20.2602	−0.643586	−0.321793	+	0.946810i	$0.604286\pi$
$992$	0	0
$993$	2.15676	0.0684426
$994$	0	0
$995$	−28.7870	−0.912609
$996$	0	0
$997$	−1.57077	−0.0497468	−0.0248734	−	0.999691i	$-0.507918\pi$
$997$	−1.57077	−0.0497468	−0.0248734	+	0.999691i	$0.507918\pi$
$998$	0	0
$999$	7.26180	0.229753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4416.2.a.bp.1.2		3
4.3	odd	2		4416.2.a.bs.1.2		3
8.3	odd	2		1104.2.a.o.1.2		3
8.5	even	2		552.2.a.g.1.2	✓	3
24.5	odd	2		1656.2.a.n.1.2		3
24.11	even	2		3312.2.a.bf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.a.g.1.2	✓	3	8.5	even	2
1104.2.a.o.1.2		3	8.3	odd	2
1656.2.a.n.1.2		3	24.5	odd	2
3312.2.a.bf.1.2		3	24.11	even	2
4416.2.a.bp.1.2		3	1.1	even	1	trivial
4416.2.a.bs.1.2		3	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4416 = 2^{6} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4416.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.07838 q^{5} +4.34017 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.07838 q^{5} +4.34017 q^{7} +1.00000 q^{9} +3.41855 q^{11} -2.00000 q^{13} +1.07838 q^{15} -2.34017 q^{17} -0.921622 q^{19} -4.34017 q^{21} -1.00000 q^{23} -3.83710 q^{25} -1.00000 q^{27} -6.68035 q^{29} -8.68035 q^{31} -3.41855 q^{33} -4.68035 q^{35} -7.26180 q^{37} +2.00000 q^{39} +8.83710 q^{41} -7.75872 q^{43} -1.07838 q^{45} -12.6803 q^{47} +11.8371 q^{49} +2.34017 q^{51} +11.9155 q^{53} -3.68649 q^{55} +0.921622 q^{57} +1.84324 q^{59} -4.73820 q^{61} +4.34017 q^{63} +2.15676 q^{65} +7.75872 q^{67} +1.00000 q^{69} +2.52359 q^{71} +2.00000 q^{73} +3.83710 q^{75} +14.8371 q^{77} -12.3402 q^{79} +1.00000 q^{81} +0.894960 q^{83} +2.52359 q^{85} +6.68035 q^{87} -8.49693 q^{89} -8.68035 q^{91} +8.68035 q^{93} +0.993857 q^{95} +8.15676 q^{97} +3.41855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 4 q^{17} - 6 q^{19} - 2 q^{21} - 3 q^{23} + 17 q^{25} - 3 q^{27} + 2 q^{29} - 4 q^{31} + 4 q^{33} + 8 q^{35} - 14 q^{37} + 6 q^{39} - 2 q^{41} + 2 q^{43} - 16 q^{47} + 7 q^{49} - 4 q^{51} + 4 q^{53} - 24 q^{55} + 6 q^{57} + 12 q^{59} - 22 q^{61} + 2 q^{63} - 2 q^{67} + 3 q^{69} - 8 q^{71} + 6 q^{73} - 17 q^{75} + 16 q^{77} - 26 q^{79} + 3 q^{81} + 4 q^{83} - 8 q^{85} - 2 q^{87} - 8 q^{89} - 4 q^{91} + 4 q^{93} - 32 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 4 * q^17 - 6 * q^19 - 2 * q^21 - 3 * q^23 + 17 * q^25 - 3 * q^27 + 2 * q^29 - 4 * q^31 + 4 * q^33 + 8 * q^35 - 14 * q^37 + 6 * q^39 - 2 * q^41 + 2 * q^43 - 16 * q^47 + 7 * q^49 - 4 * q^51 + 4 * q^53 - 24 * q^55 + 6 * q^57 + 12 * q^59 - 22 * q^61 + 2 * q^63 - 2 * q^67 + 3 * q^69 - 8 * q^71 + 6 * q^73 - 17 * q^75 + 16 * q^77 - 26 * q^79 + 3 * q^81 + 4 * q^83 - 8 * q^85 - 2 * q^87 - 8 * q^89 - 4 * q^91 + 4 * q^93 - 32 * q^95 + 18 * q^97 - 4 * q^99

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