LMFDB - Embedded newform 1152.4.d.m.577.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,4,Mod(577,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1152.577");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1152.d (of order $2$, degree $1$, 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:	$67.9702003266$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.4
Root			$-1.32288 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1152.577
Dual form			1152.4.d.m.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.00000i q^{5} +21.1660 q^{7} +O(q^{10})$	$q+6.00000i q^{5} +21.1660 q^{7} +42.3320i q^{11} +20.0000i q^{13} +8.00000 q^{17} +84.6640i q^{19} -169.328 q^{23} +89.0000 q^{25} +46.0000i q^{29} -21.1660 q^{31} +126.996i q^{35} +164.000i q^{37} -312.000 q^{41} -423.320i q^{43} +169.328 q^{47} +105.000 q^{49} +266.000i q^{53} -253.992 q^{55} -253.992i q^{59} -132.000i q^{61} -120.000 q^{65} +507.984i q^{67} +677.312 q^{71} -246.000 q^{73} +896.000i q^{77} +232.826 q^{79} -973.636i q^{83} +48.0000i q^{85} -1392.00 q^{89} +423.320i q^{91} -507.984 q^{95} -302.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 32 q^{17} + 356 q^{25} - 1248 q^{41} + 420 q^{49} - 480 q^{65} - 984 q^{73} - 5568 q^{89} - 1208 q^{97}+O(q^{100}) $ 4 * q + 32 * q^17 + 356 * q^25 - 1248 * q^41 + 420 * q^49 - 480 * q^65 - 984 * q^73 - 5568 * q^89 - 1208 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	6.00000i	0.536656i	0.963328	+	0.268328i	$0.0864711\pi$
$5$	6.00000i	0.536656i	−0.963328	+	0.268328i	$0.913529\pi$
$6$	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
$11$	42.3320i	1.16033i	0.814500	+	0.580163i	$0.197011\pi$
$11$	42.3320i	1.16033i	−0.814500	+	0.580163i	$0.802989\pi$
$12$	0	0
$13$	20.0000i	0.426692i	0.976977	+	0.213346i	$0.0684362\pi$
$13$	20.0000i	0.426692i	−0.976977	+	0.213346i	$0.931564\pi$
$14$	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
$19$	84.6640i	1.02228i	0.859498	+	0.511139i	$0.170776\pi$
$19$	84.6640i	1.02228i	−0.859498	+	0.511139i	$0.829224\pi$
$20$	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
$29$	46.0000i	0.294551i	0.989095	+	0.147276i	$0.0470504\pi$
$29$	46.0000i	0.294551i	−0.989095	+	0.147276i	$0.952950\pi$
$30$	0	0
$31$	−21.1660	−0.122630	−0.0613150	−	0.998118i	$-0.519529\pi$
$31$	−21.1660	−0.122630	−0.0613150	+	0.998118i	$0.519529\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	126.996i	0.613322i
$36$	0	0
$37$	164.000i	0.728687i	0.931265	+	0.364344i	$0.118707\pi$
$37$	164.000i	0.728687i	−0.931265	+	0.364344i	$0.881293\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−312.000	−1.18844	−0.594222	−	0.804301i	$-0.702540\pi$
$41$	−312.000	−1.18844	−0.594222	+	0.804301i	$0.702540\pi$
$42$	0	0
$43$	− 423.320i	− 1.50130i	−0.660702	−	0.750648i	$-0.729741\pi$
$43$	− 423.320i	− 1.50130i	0.660702	−	0.750648i	$-0.270259\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	169.328	0.525511	0.262756	−	0.964862i	$-0.415369\pi$
$47$	169.328	0.525511	0.262756	+	0.964862i	$0.415369\pi$
$48$	0	0
$49$	105.000	0.306122
$50$	0	0
$51$	0	0
$52$	0	0
$53$	266.000i	0.689395i	0.938714	+	0.344697i	$0.112018\pi$
$53$	266.000i	0.689395i	−0.938714	+	0.344697i	$0.887982\pi$
$54$	0	0
$55$	−253.992	−0.622696
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 253.992i	− 0.560457i	−0.959933	−	0.280228i	$-0.909590\pi$
$59$	− 253.992i	− 0.560457i	0.959933	−	0.280228i	$-0.0904102\pi$
$60$	0	0
$61$	− 132.000i	− 0.277063i	−0.990358	−	0.138532i	$-0.955762\pi$
$61$	− 132.000i	− 0.277063i	0.990358	−	0.138532i	$-0.0442383\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−120.000	−0.228987
$66$	0	0
$67$	507.984i	0.926271i	0.886287	+	0.463135i	$0.153276\pi$
$67$	507.984i	0.926271i	−0.886287	+	0.463135i	$0.846724\pi$
$68$	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	$-0.563187\pi$
$73$	−246.000	−0.394413	−0.197206	+	0.980362i	$0.563187\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	896.000i	1.32609i
$78$	0	0
$79$	232.826	0.331582	0.165791	−	0.986161i	$-0.446982\pi$
$79$	232.826	0.331582	0.165791	+	0.986161i	$0.446982\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 973.636i	− 1.28760i	−0.765195	−	0.643798i	$-0.777358\pi$
$83$	− 973.636i	− 1.28760i	0.765195	−	0.643798i	$-0.222642\pi$
$84$	0	0
$85$	48.0000i	0.0612510i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1392.00	−1.65788	−0.828942	−	0.559334i	$-0.811057\pi$
$89$	−1392.00	−1.65788	−0.828942	+	0.559334i	$0.811057\pi$
$90$	0	0
$91$	423.320i	0.487649i
$92$	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
$101$	1466.00i	1.44428i	0.691746	+	0.722141i	$0.256842\pi$
$101$	1466.00i	1.44428i	−0.691746	+	0.722141i	$0.743158\pi$
$102$	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
$107$	423.320i	0.382466i	0.981545	+	0.191233i	$0.0612487\pi$
$107$	423.320i	0.382466i	−0.981545	+	0.191233i	$0.938751\pi$
$108$	0	0
$109$	− 1564.00i	− 1.37435i	−0.726492	−	0.687174i	$-0.758851\pi$
$109$	− 1564.00i	− 1.37435i	0.726492	−	0.687174i	$-0.241149\pi$
$110$	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
$115$	− 1015.97i	− 0.823822i
$116$	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
$125$	1284.00i	0.918756i
$126$	0	0
$127$	−994.802	−0.695074	−0.347537	−	0.937666i	$-0.612982\pi$
$127$	−994.802	−0.695074	−0.347537	+	0.937666i	$0.612982\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2539.92i	1.69400i	0.531593	+	0.847000i	$0.321594\pi$
$131$	2539.92i	1.69400i	−0.531593	+	0.847000i	$0.678406\pi$
$132$	0	0
$133$	1792.00i	1.16832i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1992.00	−1.24225	−0.621124	−	0.783712i	$-0.713324\pi$
$137$	−1992.00	−1.24225	−0.621124	+	0.783712i	$0.713324\pi$
$138$	0	0
$139$	1185.30i	0.723277i	0.932318	+	0.361639i	$0.117783\pi$
$139$	1185.30i	0.723277i	−0.932318	+	0.361639i	$0.882217\pi$
$140$	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
$149$	1974.00i	1.08534i	0.839944	+	0.542672i	$0.182587\pi$
$149$	1974.00i	1.08534i	−0.839944	+	0.542672i	$0.817413\pi$
$150$	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
$155$	− 126.996i	− 0.0658101i
$156$	0	0
$157$	− 1796.00i	− 0.912971i	−0.889731	−	0.456485i	$-0.849108\pi$
$157$	− 1796.00i	− 0.912971i	0.889731	−	0.456485i	$-0.150892\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3584.00	−1.75440
$162$	0	0
$163$	− 2624.59i	− 1.26119i	−0.776114	−	0.630593i	$-0.782812\pi$
$163$	− 2624.59i	− 1.26119i	0.776114	−	0.630593i	$-0.217188\pi$
$164$	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
$173$	4290.00i	1.88533i	0.333736	+	0.942667i	$0.391691\pi$
$173$	4290.00i	1.88533i	−0.333736	+	0.942667i	$0.608309\pi$
$174$	0	0
$175$	1883.77	0.813714
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1608.62i	− 0.671696i	−0.941916	−	0.335848i	$-0.890977\pi$
$179$	− 1608.62i	− 0.671696i	0.941916	−	0.335848i	$-0.109023\pi$
$180$	0	0
$181$	3196.00i	1.31247i	0.754557	+	0.656234i	$0.227852\pi$
$181$	3196.00i	1.31247i	−0.754557	+	0.656234i	$0.772148\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−984.000	−0.391055
$186$	0	0
$187$	338.656i	0.132433i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3217.23	1.21880	0.609400	−	0.792863i	$-0.291410\pi$
$191$	3217.23	1.21880	0.609400	+	0.792863i	$0.291410\pi$
$192$	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
$197$	3834.00i	1.38661i	0.720647	+	0.693303i	$0.243845\pi$
$197$	3834.00i	1.38661i	−0.720647	+	0.693303i	$0.756155\pi$
$198$	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
$203$	973.636i	0.336630i
$204$	0	0
$205$	− 1872.00i	− 0.637786i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3584.00	−1.18617
$210$	0	0
$211$	2878.58i	0.939192i	0.882882	+	0.469596i	$0.155600\pi$
$211$	2878.58i	0.939192i	−0.882882	+	0.469596i	$0.844400\pi$
$212$	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
$221$	160.000i	0.0487003i
$222$	0	0
$223$	−910.138	−0.273307	−0.136653	−	0.990619i	$-0.543635\pi$
$223$	−910.138	−0.273307	−0.136653	+	0.990619i	$0.543635\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3259.57i	− 0.953062i	−0.879158	−	0.476531i	$-0.841894\pi$
$227$	− 3259.57i	− 0.953062i	0.879158	−	0.476531i	$-0.158106\pi$
$228$	0	0
$229$	5260.00i	1.51786i	0.651171	+	0.758931i	$0.274278\pi$
$229$	5260.00i	1.51786i	−0.651171	+	0.758931i	$0.725722\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1936.00	−0.544342	−0.272171	−	0.962249i	$-0.587742\pi$
$233$	−1936.00	−0.544342	−0.272171	+	0.962249i	$0.587742\pi$
$234$	0	0
$235$	1015.97i	0.282019i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1523.95	0.412453	0.206227	−	0.978504i	$-0.433882\pi$
$239$	1523.95	0.412453	0.206227	+	0.978504i	$0.433882\pi$
$240$	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
$245$	630.000i	0.164283i
$246$	0	0
$247$	−1693.28	−0.436198
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 2666.92i	− 0.670655i	−0.942102	−	0.335327i	$-0.891153\pi$
$251$	− 2666.92i	− 0.670655i	0.942102	−	0.335327i	$-0.108847\pi$
$252$	0	0
$253$	− 7168.00i	− 1.78122i
$254$	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
$259$	3471.23i	0.832786i
$260$	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
$269$	4350.00i	0.985963i	0.870040	+	0.492982i	$0.164093\pi$
$269$	4350.00i	0.985963i	−0.870040	+	0.492982i	$0.835907\pi$
$270$	0	0
$271$	−1079.47	−0.241967	−0.120983	−	0.992655i	$-0.538605\pi$
$271$	−1079.47	−0.241967	−0.120983	+	0.992655i	$0.538605\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3767.55i	0.826152i
$276$	0	0
$277$	7532.00i	1.63377i	0.576801	+	0.816885i	$0.304301\pi$
$277$	7532.00i	1.63377i	−0.576801	+	0.816885i	$0.695699\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1824.00	−0.387227	−0.193613	−	0.981078i	$-0.562021\pi$
$281$	−1824.00	−0.387227	−0.193613	+	0.981078i	$0.562021\pi$
$282$	0	0
$283$	− 8805.06i	− 1.84949i	−0.380584	−	0.924746i	$-0.624277\pi$
$283$	− 8805.06i	− 1.84949i	0.380584	−	0.924746i	$-0.375723\pi$
$284$	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
$293$	858.000i	0.171075i	0.996335	+	0.0855374i	$0.0272607\pi$
$293$	858.000i	0.171075i	−0.996335	+	0.0855374i	$0.972739\pi$
$294$	0	0
$295$	1523.95	0.300773
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 3386.56i	− 0.655016i
$300$	0	0
$301$	− 8960.00i	− 1.71577i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	792.000	0.148688
$306$	0	0
$307$	− 1523.95i	− 0.283311i	−0.989916	−	0.141656i	$-0.954757\pi$
$307$	− 1523.95i	− 0.283311i	0.989916	−	0.141656i	$-0.0452426\pi$
$308$	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	$-0.503392\pi$
$313$	−118.000	−0.0213091	−0.0106546	+	0.999943i	$0.503392\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4594.00i	− 0.813958i	−0.913437	−	0.406979i	$-0.866582\pi$
$317$	− 4594.00i	− 0.813958i	0.913437	−	0.406979i	$-0.133418\pi$
$318$	0	0
$319$	−1947.27	−0.341775
$320$	0	0
$321$	0	0
$322$	0	0
$323$	677.312i	0.116677i
$324$	0	0
$325$	1780.00i	0.303805i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3584.00	0.600585
$330$	0	0
$331$	− 3725.22i	− 0.618600i	−0.950965	−	0.309300i	$-0.899905\pi$
$331$	− 3725.22i	− 0.618600i	0.950965	−	0.309300i	$-0.100095\pi$
$332$	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
$341$	− 896.000i	− 0.142291i
$342$	0	0
$343$	−5037.51	−0.793003
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11302.6i	1.74858i	0.485402	+	0.874291i	$0.338673\pi$
$347$	11302.6i	1.74858i	−0.485402	+	0.874291i	$0.661327\pi$
$348$	0	0
$349$	− 6388.00i	− 0.979776i	−0.871785	−	0.489888i	$-0.837038\pi$
$349$	− 6388.00i	− 0.979776i	0.871785	−	0.489888i	$-0.162962\pi$
$350$	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
$355$	4063.87i	0.607572i
$356$	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
$365$	− 1476.00i	− 0.211664i
$366$	0	0
$367$	5397.33	0.767680	0.383840	−	0.923400i	$-0.374601\pi$
$367$	5397.33	0.767680	0.383840	+	0.923400i	$0.374601\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5630.16i	0.787879i
$372$	0	0
$373$	8276.00i	1.14883i	0.818563	+	0.574417i	$0.194771\pi$
$373$	8276.00i	1.14883i	−0.818563	+	0.574417i	$0.805229\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−920.000	−0.125683
$378$	0	0
$379$	13800.2i	1.87037i	0.354158	+	0.935186i	$0.384767\pi$
$379$	13800.2i	1.87037i	−0.354158	+	0.935186i	$0.615233\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1862.61	0.248498	0.124249	−	0.992251i	$-0.460348\pi$
$383$	1862.61	0.248498	0.124249	+	0.992251i	$0.460348\pi$
$384$	0	0
$385$	−5376.00	−0.711653
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 6170.00i	− 0.804194i	−0.915597	−	0.402097i	$-0.868281\pi$
$389$	− 6170.00i	− 0.804194i	0.915597	−	0.402097i	$-0.131719\pi$
$390$	0	0
$391$	−1354.62	−0.175208
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1396.96i	0.177946i
$396$	0	0
$397$	− 8644.00i	− 1.09277i	−0.837534	−	0.546385i	$-0.816003\pi$
$397$	− 8644.00i	− 1.09277i	0.837534	−	0.546385i	$-0.183997\pi$
$398$	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
$403$	− 423.320i	− 0.0523253i
$404$	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	$-0.516825\pi$
$409$	−874.000	−0.105664	−0.0528319	+	0.998603i	$0.516825\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5376.00i	− 0.640522i
$414$	0	0
$415$	5841.82	0.690997
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 14265.9i	− 1.66333i	−0.555279	−	0.831664i	$-0.687389\pi$
$419$	− 14265.9i	− 1.66333i	0.555279	−	0.831664i	$-0.312611\pi$
$420$	0	0
$421$	11564.0i	1.33871i	0.742945	+	0.669353i	$0.233428\pi$
$421$	11564.0i	1.33871i	−0.742945	+	0.669353i	$0.766572\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	712.000	0.0812637
$426$	0	0
$427$	− 2793.91i	− 0.316644i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7619.76	0.851580	0.425790	−	0.904822i	$-0.359996\pi$
$431$	7619.76	0.851580	0.425790	+	0.904822i	$0.359996\pi$
$432$	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
$437$	− 14336.0i	− 1.56930i
$438$	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
$443$	− 9016.72i	− 0.967037i	−0.875334	−	0.483518i	$-0.839359\pi$
$443$	− 9016.72i	− 0.967037i	0.875334	−	0.483518i	$-0.160641\pi$
$444$	0	0
$445$	− 8352.00i	− 0.889714i
$446$	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
$451$	− 13207.6i	− 1.37898i
$452$	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	$-0.338791\pi$
$457$	9478.00	0.970158	0.485079	+	0.874470i	$0.338791\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 12606.0i	− 1.27358i	−0.771038	−	0.636790i	$-0.780262\pi$
$461$	− 12606.0i	− 1.27358i	0.771038	−	0.636790i	$-0.219738\pi$
$462$	0	0
$463$	−14371.7	−1.44257	−0.721286	−	0.692638i	$-0.756448\pi$
$463$	−14371.7	−1.44257	−0.721286	+	0.692638i	$0.756448\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4698.85i	0.465604i	0.972524	+	0.232802i	$0.0747894\pi$
$467$	4698.85i	0.465604i	−0.972524	+	0.232802i	$0.925211\pi$
$468$	0	0
$469$	10752.0i	1.05860i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	17920.0	1.74199
$474$	0	0
$475$	7535.10i	0.727861i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10837.0	1.03373	0.516863	−	0.856068i	$-0.327100\pi$
$479$	10837.0	1.03373	0.516863	+	0.856068i	$0.327100\pi$
$480$	0	0
$481$	−3280.00	−0.310925
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1812.00i	− 0.169647i
$486$	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
$491$	− 13715.6i	− 1.26064i	−0.776335	−	0.630321i	$-0.782923\pi$
$491$	− 13715.6i	− 1.26064i	0.776335	−	0.630321i	$-0.217077\pi$
$492$	0	0
$493$	368.000i	0.0336184i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14336.0	1.29388
$498$	0	0
$499$	− 6773.12i	− 0.607629i	−0.952731	−	0.303814i	$-0.901740\pi$
$499$	− 6773.12i	− 0.607629i	0.952731	−	0.303814i	$-0.0982602\pi$
$500$	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
$509$	− 13458.0i	− 1.17194i	−0.810334	−	0.585968i	$-0.800714\pi$
$509$	− 13458.0i	− 1.17194i	0.810334	−	0.585968i	$-0.199286\pi$
$510$	0	0
$511$	−5206.84	−0.450757
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7492.77i	0.641108i
$516$	0	0
$517$	7168.00i	0.609765i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5352.00	0.450049	0.225024	−	0.974353i	$-0.427754\pi$
$521$	5352.00	0.450049	0.225024	+	0.974353i	$0.427754\pi$
$522$	0	0
$523$	6011.15i	0.502580i	0.967912	+	0.251290i	$0.0808547\pi$
$523$	6011.15i	0.502580i	−0.967912	+	0.251290i	$0.919145\pi$
$524$	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
$533$	− 6240.00i	− 0.507100i
$534$	0	0
$535$	−2539.92	−0.205253
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4444.86i	0.355202i
$540$	0	0
$541$	− 2396.00i	− 0.190411i	−0.995458	−	0.0952053i	$-0.969649\pi$
$541$	− 2396.00i	− 0.190411i	0.995458	−	0.0952053i	$-0.0303507\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9384.00	0.737553
$546$	0	0
$547$	1269.96i	0.0992680i	0.998767	+	0.0496340i	$0.0158055\pi$
$547$	1269.96i	0.0992680i	−0.998767	+	0.0496340i	$0.984195\pi$
$548$	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
$557$	− 254.000i	− 0.0193219i	−0.999953	−	0.00966097i	$-0.996925\pi$
$557$	− 254.000i	− 0.0193219i	0.999953	−	0.00966097i	$-0.00307523\pi$
$558$	0	0
$559$	8466.40	0.640592
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 10032.7i	− 0.751026i	−0.926817	−	0.375513i	$-0.877467\pi$
$563$	− 10032.7i	− 0.751026i	0.926817	−	0.375513i	$-0.122533\pi$
$564$	0	0
$565$	− 9312.00i	− 0.693378i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9320.00	0.686669	0.343335	−	0.939213i	$-0.388444\pi$
$569$	9320.00	0.686669	0.343335	+	0.939213i	$0.388444\pi$
$570$	0	0
$571$	− 13376.9i	− 0.980397i	−0.871611	−	0.490198i	$-0.836924\pi$
$571$	− 13376.9i	− 0.980397i	0.871611	−	0.490198i	$-0.163076\pi$
$572$	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
$581$	− 20608.0i	− 1.47154i
$582$	0	0
$583$	−11260.3	−0.799922
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26076.5i	1.83355i	0.399405	+	0.916775i	$0.369217\pi$
$587$	26076.5i	1.83355i	−0.399405	+	0.916775i	$0.630783\pi$
$588$	0	0
$589$	− 1792.00i	− 0.125362i
$590$	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
$595$	1015.97i	0.0700011i
$596$	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	$-0.613639\pi$
$601$	−10298.0	−0.698942	−0.349471	+	0.936947i	$0.613639\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2766.00i	− 0.185874i
$606$	0	0
$607$	23219.1	1.55261	0.776305	−	0.630357i	$-0.217091\pi$
$607$	23219.1	1.55261	0.776305	+	0.630357i	$0.217091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3386.56i	0.224232i
$612$	0	0
$613$	5652.00i	0.372402i	0.982512	+	0.186201i	$0.0596175\pi$
$613$	5652.00i	0.372402i	−0.982512	+	0.186201i	$0.940383\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26848.0	1.75180	0.875899	−	0.482494i	$-0.160269\pi$
$617$	26848.0	1.75180	0.875899	+	0.482494i	$0.160269\pi$
$618$	0	0
$619$	3725.22i	0.241889i	0.992659	+	0.120944i	$0.0385923\pi$
$619$	3725.22i	0.241889i	−0.992659	+	0.120944i	$0.961408\pi$
$620$	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
$629$	1312.00i	0.0831683i
$630$	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
$635$	− 5968.81i	− 0.373016i
$636$	0	0
$637$	2100.00i	0.130620i
$638$	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
$643$	24637.2i	1.51104i	0.655126	+	0.755519i	$0.272615\pi$
$643$	24637.2i	1.51104i	−0.655126	+	0.755519i	$0.727385\pi$
$644$	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
$653$	− 29054.0i	− 1.74115i	−0.492036	−	0.870575i	$-0.663747\pi$
$653$	− 29054.0i	− 1.74115i	0.492036	−	0.870575i	$-0.336253\pi$
$654$	0	0
$655$	−15239.5	−0.909096
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 3047.91i	− 0.180166i	−0.995934	−	0.0900831i	$-0.971287\pi$
$659$	− 3047.91i	− 0.180166i	0.995934	−	0.0900831i	$-0.0287133\pi$
$660$	0	0
$661$	− 9628.00i	− 0.566544i	−0.959040	−	0.283272i	$-0.908580\pi$
$661$	− 9628.00i	− 0.566544i	0.959040	−	0.283272i	$-0.0914200\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−10752.0	−0.626984
$666$	0	0
$667$	− 7789.09i	− 0.452166i
$668$	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
$677$	− 4922.00i	− 0.279421i	−0.990192	−	0.139710i	$-0.955383\pi$
$677$	− 4922.00i	− 0.279421i	0.990192	−	0.139710i	$-0.0446171\pi$
$678$	0	0
$679$	−6392.14	−0.361278
$680$	0	0
$681$	0	0
$682$	0	0
$683$	42.3320i	0.00237158i	0.999999	+	0.00118579i	$0.000377449\pi$
$683$	42.3320i	0.00237158i	−0.999999	+	0.00118579i	$0.999623\pi$
$684$	0	0
$685$	− 11952.0i	− 0.666661i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5320.00	−0.294159
$690$	0	0
$691$	18372.1i	1.01144i	0.862697	+	0.505722i	$0.168774\pi$
$691$	18372.1i	1.01144i	−0.862697	+	0.505722i	$0.831226\pi$
$692$	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
$701$	19310.0i	1.04041i	0.854041	+	0.520206i	$0.174145\pi$
$701$	19310.0i	1.04041i	−0.854041	+	0.520206i	$0.825855\pi$
$702$	0	0
$703$	−13884.9	−0.744920
$704$	0	0
$705$	0	0
$706$	0	0
$707$	31029.4i	1.65061i
$708$	0	0
$709$	− 18820.0i	− 0.996897i	−0.866919	−	0.498448i	$-0.833903\pi$
$709$	− 18820.0i	− 0.996897i	0.866919	−	0.498448i	$-0.166097\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3584.00	0.188249
$714$	0	0
$715$	− 5079.84i	− 0.265700i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	27431.1	1.42282	0.711411	−	0.702776i	$-0.248056\pi$
$719$	27431.1	1.42282	0.711411	+	0.702776i	$0.248056\pi$
$720$	0	0
$721$	26432.0	1.36530
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4094.00i	0.209720i
$726$	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
$731$	− 3386.56i	− 0.171350i
$732$	0	0
$733$	16468.0i	0.829822i	0.909862	+	0.414911i	$0.136187\pi$
$733$	16468.0i	0.829822i	−0.909862	+	0.414911i	$0.863813\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21504.0	−1.07478
$738$	0	0
$739$	− 15070.2i	− 0.750157i	−0.926993	−	0.375079i	$-0.877616\pi$
$739$	− 15070.2i	− 0.750157i	0.926993	−	0.375079i	$-0.122384\pi$
$740$	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
$749$	8960.00i	0.437105i
$750$	0	0
$751$	1756.78	0.0853605	0.0426803	−	0.999089i	$-0.486410\pi$
$751$	1756.78	0.0853605	0.0426803	+	0.999089i	$0.486410\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 18668.4i	− 0.899885i
$756$	0	0
$757$	− 1572.00i	− 0.0754760i	−0.999288	−	0.0377380i	$-0.987985\pi$
$757$	− 1572.00i	− 0.0754760i	0.999288	−	0.0377380i	$-0.0120152\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17816.0	0.848659	0.424329	−	0.905508i	$-0.360510\pi$
$761$	17816.0	0.848659	0.424329	+	0.905508i	$0.360510\pi$
$762$	0	0
$763$	− 33103.6i	− 1.57068i
$764$	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
$773$	13978.0i	0.650393i	0.945646	+	0.325196i	$0.105430\pi$
$773$	13978.0i	0.650393i	−0.945646	+	0.325196i	$0.894570\pi$
$774$	0	0
$775$	−1883.77	−0.0873125
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 26415.2i	− 1.21492i
$780$	0	0
$781$	28672.0i	1.31366i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10776.0	0.489952
$786$	0	0
$787$	− 1777.94i	− 0.0805297i	−0.999189	−	0.0402649i	$-0.987180\pi$
$787$	− 1777.94i	− 0.0805297i	0.999189	−	0.0402649i	$-0.0128202\pi$
$788$	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
$797$	9650.00i	0.428884i	0.976737	+	0.214442i	$0.0687933\pi$
$797$	9650.00i	0.428884i	−0.976737	+	0.214442i	$0.931207\pi$
$798$	0	0
$799$	1354.62	0.0599789
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 10413.7i	− 0.457647i
$804$	0	0
$805$	− 21504.0i	− 0.941511i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38024.0	−1.65248	−0.826238	−	0.563322i	$-0.809523\pi$
$809$	−38024.0	−1.65248	−0.826238	+	0.563322i	$0.809523\pi$
$810$	0	0
$811$	− 931.304i	− 0.0403237i	−0.999797	−	0.0201619i	$-0.993582\pi$
$811$	− 931.304i	− 0.0403237i	0.999797	−	0.0201619i	$-0.00641815\pi$
$812$	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
$821$	6858.00i	0.291530i	0.989319	+	0.145765i	$0.0465643\pi$
$821$	6858.00i	0.291530i	−0.989319	+	0.145765i	$0.953436\pi$
$822$	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
$827$	− 8212.41i	− 0.345313i	−0.984982	−	0.172656i	$-0.944765\pi$
$827$	− 8212.41i	− 0.345313i	0.984982	−	0.172656i	$-0.0552350\pi$
$828$	0	0
$829$	25124.0i	1.05258i	0.850304	+	0.526292i	$0.176418\pi$
$829$	25124.0i	1.05258i	−0.850304	+	0.526292i	$0.823582\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	840.000	0.0349391
$834$	0	0
$835$	− 21335.3i	− 0.884239i
$836$	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
$845$	10782.0i	0.438949i
$846$	0	0
$847$	−9757.53	−0.395836
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 27769.8i	− 1.11861i
$852$	0	0
$853$	− 32812.0i	− 1.31707i	−0.752550	−	0.658535i	$-0.771176\pi$
$853$	− 32812.0i	− 1.31707i	0.752550	−	0.658535i	$-0.228824\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21944.0	−0.874671	−0.437335	−	0.899299i	$-0.644078\pi$
$857$	−21944.0	−0.874671	−0.437335	+	0.899299i	$0.644078\pi$
$858$	0	0
$859$	48173.8i	1.91347i	0.290962	+	0.956735i	$0.406025\pi$
$859$	48173.8i	1.91347i	−0.290962	+	0.956735i	$0.593975\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12361.0	0.487569	0.243784	−	0.969829i	$-0.421611\pi$
$863$	12361.0	0.487569	0.243784	+	0.969829i	$0.421611\pi$
$864$	0	0
$865$	−25740.0	−1.01178
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9856.00i	0.384743i
$870$	0	0
$871$	−10159.7	−0.395233
$872$	0	0
$873$	0	0
$874$	0	0
$875$	27177.2i	1.05001i
$876$	0	0
$877$	33212.0i	1.27878i	0.768883	+	0.639390i	$0.220813\pi$
$877$	33212.0i	1.27878i	−0.768883	+	0.639390i	$0.779187\pi$
$878$	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
$883$	16678.8i	0.635659i	0.948148	+	0.317829i	$0.102954\pi$
$883$	16678.8i	0.635659i	−0.948148	+	0.317829i	$0.897046\pi$
$884$	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
$893$	14336.0i	0.537218i
$894$	0	0
$895$	9651.70	0.360470
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 973.636i	− 0.0361208i
$900$	0	0
$901$	2128.00i	0.0786836i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19176.0	−0.704345
$906$	0	0
$907$	13292.3i	0.486617i	0.969949	+	0.243309i	$0.0782328\pi$
$907$	13292.3i	0.486617i	−0.969949	+	0.243309i	$0.921767\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2539.92	−0.0923725	−0.0461862	−	0.998933i	$-0.514707\pi$
$911$	−2539.92	−0.0923725	−0.0461862	+	0.998933i	$0.514707\pi$
$912$	0	0
$913$	41216.0	1.49403
$914$	0	0
$915$	0	0
$916$	0	0
$917$	53760.0i	1.93600i
$918$	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
$923$	13546.2i	0.483077i
$924$	0	0
$925$	14596.0i	0.518825i
$926$	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
$931$	8889.72i	0.312942i
$932$	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	$-0.377556\pi$
$937$	21526.0	0.750506	0.375253	+	0.926923i	$0.377556\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5218.00i	0.180767i	0.995907	+	0.0903836i	$0.0288093\pi$
$941$	5218.00i	0.180767i	−0.995907	+	0.0903836i	$0.971191\pi$
$942$	0	0
$943$	52830.4	1.82438
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29717.1i	1.01972i	0.860257	+	0.509860i	$0.170303\pi$
$947$	29717.1i	1.01972i	−0.860257	+	0.509860i	$0.829697\pi$
$948$	0	0
$949$	− 4920.00i	− 0.168293i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−48888.0	−1.66174	−0.830870	−	0.556467i	$-0.812157\pi$
$953$	−48888.0	−1.66174	−0.830870	+	0.556467i	$0.812157\pi$
$954$	0	0
$955$	19303.4i	0.654077i
$956$	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
$965$	2772.00i	0.0924703i
$966$	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
$971$	12403.3i	0.409928i	0.978769	+	0.204964i	$0.0657078\pi$
$971$	12403.3i	0.409928i	−0.978769	+	0.204964i	$0.934292\pi$
$972$	0	0
$973$	25088.0i	0.826603i
$974$	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
$979$	− 58926.2i	− 1.92369i
$980$	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
$989$	71680.0i	2.30464i
$990$	0	0
$991$	−10773.5	−0.345340	−0.172670	−	0.984980i	$-0.555239\pi$
$991$	−10773.5	−0.345340	−0.172670	+	0.984980i	$0.555239\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	22732.3i	0.724284i
$996$	0	0
$997$	− 62044.0i	− 1.97087i	−0.170065	−	0.985433i	$-0.554398\pi$
$997$	− 62044.0i	− 1.97087i	0.170065	−	0.985433i	$-0.445602\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.d.k.577.1	✓	4	12.11	even	2
1152.4.d.k.577.2	yes	4	3.2	odd	2
1152.4.d.k.577.3	yes	4	24.11	even	2
1152.4.d.k.577.4	yes	4	24.5	odd	2
1152.4.d.m.577.1	yes	4	8.3	odd	2	inner
1152.4.d.m.577.2	yes	4	8.5	even	2	inner
1152.4.d.m.577.3	yes	4	4.3	odd	2	inner
1152.4.d.m.577.4	yes	4	1.1	even	1	trivial
2304.4.a.q.1.1		2	48.29	odd	4
2304.4.a.q.1.2		2	48.35	even	4
2304.4.a.r.1.1		2	16.5	even	4
2304.4.a.r.1.2		2	16.11	odd	4
2304.4.a.br.1.1		2	16.13	even	4
2304.4.a.br.1.2		2	16.3	odd	4
2304.4.a.bs.1.1		2	48.5	odd	4
2304.4.a.bs.1.2		2	48.11	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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1152.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.00000i q^{5} +21.1660 q^{7} +O(q^{10})\)	\(q+6.00000i q^{5} +21.1660 q^{7} +42.3320i q^{11} +20.0000i q^{13} +8.00000 q^{17} +84.6640i q^{19} -169.328 q^{23} +89.0000 q^{25} +46.0000i q^{29} -21.1660 q^{31} +126.996i q^{35} +164.000i q^{37} -312.000 q^{41} -423.320i q^{43} +169.328 q^{47} +105.000 q^{49} +266.000i q^{53} -253.992 q^{55} -253.992i q^{59} -132.000i q^{61} -120.000 q^{65} +507.984i q^{67} +677.312 q^{71} -246.000 q^{73} +896.000i q^{77} +232.826 q^{79} -973.636i q^{83} +48.0000i q^{85} -1392.00 q^{89} +423.320i q^{91} -507.984 q^{95} -302.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 32 q^{17} + 356 q^{25} - 1248 q^{41} + 420 q^{49} - 480 q^{65} - 984 q^{73} - 5568 q^{89} - 1208 q^{97}+O(q^{100}) \) 4 * q + 32 * q^17 + 356 * q^25 - 1248 * q^41 + 420 * q^49 - 480 * q^65 - 984 * q^73 - 5568 * q^89 - 1208 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more