LMFDB - Embedded newform 360.4.f.a.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,4,Mod(289,360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(360, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("360.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	360.f (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:	$21.2406876021$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	360.289
Dual form			360.4.f.a.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.00000 + 10.0000i) q^{5} -4.00000i q^{7} +O(q^{10})$	$q+(-5.00000 + 10.0000i) q^{5} -4.00000i q^{7} +28.0000 q^{11} -16.0000i q^{13} +108.000i q^{17} -32.0000 q^{19} +28.0000i q^{23} +(-75.0000 - 100.000i) q^{25} -238.000 q^{29} -180.000 q^{31} +(40.0000 + 20.0000i) q^{35} +40.0000i q^{37} -422.000 q^{41} +276.000i q^{43} +60.0000i q^{47} +327.000 q^{49} -220.000i q^{53} +(-140.000 + 280.000i) q^{55} -804.000 q^{59} -358.000 q^{61} +(160.000 + 80.0000i) q^{65} +884.000i q^{67} +64.0000 q^{71} -152.000i q^{73} -112.000i q^{77} +932.000 q^{79} +1292.00i q^{83} +(-1080.00 - 540.000i) q^{85} -1146.00 q^{89} -64.0000 q^{91} +(160.000 - 320.000i) q^{95} -824.000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{5}+O(q^{10}) $ 2 * q - 10 * q^5	$ 2 q - 10 q^{5} + 56 q^{11} - 64 q^{19} - 150 q^{25} - 476 q^{29} - 360 q^{31} + 80 q^{35} - 844 q^{41} + 654 q^{49} - 280 q^{55} - 1608 q^{59} - 716 q^{61} + 320 q^{65} + 128 q^{71} + 1864 q^{79} - 2160 q^{85} - 2292 q^{89} - 128 q^{91} + 320 q^{95}+O(q^{100}) $ 2 * q - 10 * q^5 + 56 * q^11 - 64 * q^19 - 150 * q^25 - 476 * q^29 - 360 * q^31 + 80 * q^35 - 844 * q^41 + 654 * q^49 - 280 * q^55 - 1608 * q^59 - 716 * q^61 + 320 * q^65 + 128 * q^71 + 1864 * q^79 - 2160 * q^85 - 2292 * q^89 - 128 * q^91 + 320 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$.

$n$	$181$	$217$	$271$	$281$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	−5.00000	+	10.0000i	−0.447214	+	0.894427i
$5$	−5.00000	+	10.0000i	−0.447214	+	0.894427i
$6$		0			0
$7$		−	4.00000i		−	0.215980i	−0.994152	−	0.107990i	$-0.965559\pi$
$7$		−	4.00000i		−	0.215980i	0.994152	−	0.107990i	$-0.0344414\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	28.0000			0.767483			0.383742	−	0.923440i	$-0.374635\pi$
$11$	28.0000			0.767483			0.383742	+	0.923440i	$0.374635\pi$
$12$		0			0
$13$		−	16.0000i		−	0.341354i	−0.985327	−	0.170677i	$-0.945405\pi$
$13$		−	16.0000i		−	0.341354i	0.985327	−	0.170677i	$-0.0545955\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			108.000i			1.54081i	0.637552	+	0.770407i	$0.279947\pi$
$17$			108.000i			1.54081i	−0.637552	+	0.770407i	$0.720053\pi$
$18$		0			0
$19$	−32.0000			−0.386384			−0.193192	−	0.981161i	$-0.561884\pi$
$19$	−32.0000			−0.386384			−0.193192	+	0.981161i	$0.561884\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			28.0000i			0.253844i	0.991913	+	0.126922i	$0.0405097\pi$
$23$			28.0000i			0.253844i	−0.991913	+	0.126922i	$0.959490\pi$
$24$		0			0
$25$	−75.0000	−	100.000i	−0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−238.000			−1.52398			−0.761991	−	0.647587i	$-0.775778\pi$
$29$	−238.000			−1.52398			−0.761991	+	0.647587i	$0.775778\pi$
$30$		0			0
$31$	−180.000			−1.04287			−0.521435	−	0.853291i	$-0.674603\pi$
$31$	−180.000			−1.04287			−0.521435	+	0.853291i	$0.674603\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	40.0000	+	20.0000i	0.193178	+	0.0965891i
$36$		0			0
$37$			40.0000i			0.177729i	0.996044	+	0.0888643i	$0.0283238\pi$
$37$			40.0000i			0.177729i	−0.996044	+	0.0888643i	$0.971676\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−422.000			−1.60745			−0.803724	−	0.595003i	$-0.797151\pi$
$41$	−422.000			−1.60745			−0.803724	+	0.595003i	$0.797151\pi$
$42$		0			0
$43$			276.000i			0.978828i	0.872052	+	0.489414i	$0.162789\pi$
$43$			276.000i			0.978828i	−0.872052	+	0.489414i	$0.837211\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			60.0000i			0.186211i	0.995656	+	0.0931053i	$0.0296793\pi$
$47$			60.0000i			0.186211i	−0.995656	+	0.0931053i	$0.970321\pi$
$48$		0			0
$49$	327.000			0.953353
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	220.000i		−	0.570176i	−0.958501	−	0.285088i	$-0.907977\pi$
$53$		−	220.000i		−	0.570176i	0.958501	−	0.285088i	$-0.0920228\pi$
$54$		0			0
$55$	−140.000	+	280.000i	−0.343229	+	0.686458i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−804.000			−1.77410			−0.887050	−	0.461674i	$-0.847249\pi$
$59$	−804.000			−1.77410			−0.887050	+	0.461674i	$0.847249\pi$
$60$		0			0
$61$	−358.000			−0.751430			−0.375715	−	0.926735i	$-0.622603\pi$
$61$	−358.000			−0.751430			−0.375715	+	0.926735i	$0.622603\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	160.000	+	80.0000i	0.305316	+	0.152658i
$66$		0			0
$67$			884.000i			1.61191i	0.591979	+	0.805954i	$0.298347\pi$
$67$			884.000i			1.61191i	−0.591979	+	0.805954i	$0.701653\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	64.0000			0.106978			0.0534888	−	0.998568i	$-0.482966\pi$
$71$	64.0000			0.106978			0.0534888	+	0.998568i	$0.482966\pi$
$72$		0			0
$73$		−	152.000i		−	0.243702i	−0.992548	−	0.121851i	$-0.961117\pi$
$73$		−	152.000i		−	0.243702i	0.992548	−	0.121851i	$-0.0388830\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	112.000i		−	0.165761i
$78$		0			0
$79$	932.000			1.32732			0.663659	−	0.748035i	$-0.269002\pi$
$79$	932.000			1.32732			0.663659	+	0.748035i	$0.269002\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1292.00i			1.70862i	0.519764	+	0.854310i	$0.326020\pi$
$83$			1292.00i			1.70862i	−0.519764	+	0.854310i	$0.673980\pi$
$84$		0			0
$85$	−1080.00	−	540.000i	−1.37815	−	0.689073i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1146.00			−1.36490			−0.682448	−	0.730934i	$-0.739085\pi$
$89$	−1146.00			−1.36490			−0.682448	+	0.730934i	$0.739085\pi$
$90$		0			0
$91$	−64.0000			−0.0737255
$92$		0			0
$93$		0			0
$94$		0			0
$95$	160.000	−	320.000i	0.172796	−	0.345593i
$96$		0			0
$97$		−	824.000i		−	0.862521i	−0.902227	−	0.431260i	$-0.858069\pi$
$97$		−	824.000i		−	0.862521i	0.902227	−	0.431260i	$-0.141931\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1290.00			1.27089			0.635445	−	0.772147i	$-0.280817\pi$
$101$	1290.00			1.27089			0.635445	+	0.772147i	$0.280817\pi$
$102$		0			0
$103$		−	1604.00i		−	1.53444i	−0.641387	−	0.767218i	$-0.721641\pi$
$103$		−	1604.00i		−	1.53444i	0.641387	−	0.767218i	$-0.278359\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			892.000i			0.805915i	0.915219	+	0.402957i	$0.132018\pi$
$107$			892.000i			0.805915i	−0.915219	+	0.402957i	$0.867982\pi$
$108$		0			0
$109$	−966.000			−0.848863			−0.424431	−	0.905460i	$-0.639526\pi$
$109$	−966.000			−0.848863			−0.424431	+	0.905460i	$0.639526\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1124.00i		−	0.935726i	−0.883801	−	0.467863i	$-0.845024\pi$
$113$		−	1124.00i		−	0.935726i	0.883801	−	0.467863i	$-0.154976\pi$
$114$		0			0
$115$	−280.000	−	140.000i	−0.227045	−	0.113522i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	432.000			0.332785
$120$		0			0
$121$	−547.000			−0.410969
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1375.00	−	250.000i	0.983870	−	0.178885i
$126$		0			0
$127$			1884.00i			1.31636i	0.752860	+	0.658181i	$0.228674\pi$
$127$			1884.00i			1.31636i	−0.752860	+	0.658181i	$0.771326\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	588.000			0.392166			0.196083	−	0.980587i	$-0.437178\pi$
$131$	588.000			0.392166			0.196083	+	0.980587i	$0.437178\pi$
$132$		0			0
$133$			128.000i			0.0834512i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			1060.00i			0.661036i	0.943800	+	0.330518i	$0.107223\pi$
$137$			1060.00i			0.661036i	−0.943800	+	0.330518i	$0.892777\pi$
$138$		0			0
$139$	2864.00			1.74764			0.873818	−	0.486254i	$-0.161637\pi$
$139$	2864.00			1.74764			0.873818	+	0.486254i	$0.161637\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	448.000i		−	0.261984i
$144$		0			0
$145$	1190.00	−	2380.00i	0.681546	−	1.36309i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	342.000			0.188038			0.0940192	−	0.995570i	$-0.470028\pi$
$149$	342.000			0.188038			0.0940192	+	0.995570i	$0.470028\pi$
$150$		0			0
$151$	1636.00			0.881694			0.440847	−	0.897582i	$-0.354678\pi$
$151$	1636.00			0.881694			0.440847	+	0.897582i	$0.354678\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	900.000	−	1800.00i	0.466385	−	0.932771i
$156$		0			0
$157$			2072.00i			1.05327i	0.850091	+	0.526636i	$0.176547\pi$
$157$			2072.00i			1.05327i	−0.850091	+	0.526636i	$0.823453\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	112.000			0.0548251
$162$		0			0
$163$			772.000i			0.370968i	0.982647	+	0.185484i	$0.0593852\pi$
$163$			772.000i			0.370968i	−0.982647	+	0.185484i	$0.940615\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1044.00i		−	0.483755i	−0.970307	−	0.241878i	$-0.922237\pi$
$167$		−	1044.00i		−	0.483755i	0.970307	−	0.241878i	$-0.0777633\pi$
$168$		0			0
$169$	1941.00			0.883477
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	4404.00i		−	1.93543i	−0.252041	−	0.967717i	$-0.581102\pi$
$173$		−	4404.00i		−	1.93543i	0.252041	−	0.967717i	$-0.418898\pi$
$174$		0			0
$175$	−400.000	+	300.000i	−0.172784	+	0.129588i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3452.00			1.44142			0.720711	−	0.693235i	$-0.243815\pi$
$179$	3452.00			1.44142			0.720711	+	0.693235i	$0.243815\pi$
$180$		0			0
$181$	526.000			0.216007			0.108004	−	0.994151i	$-0.465554\pi$
$181$	526.000			0.216007			0.108004	+	0.994151i	$0.465554\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−400.000	−	200.000i	−0.158965	−	0.0794827i
$186$		0			0
$187$			3024.00i			1.18255i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−72.0000			−0.0272761			−0.0136381	−	0.999907i	$-0.504341\pi$
$191$	−72.0000			−0.0272761			−0.0136381	+	0.999907i	$0.504341\pi$
$192$		0			0
$193$			208.000i			0.0775760i	0.999247	+	0.0387880i	$0.0123497\pi$
$193$			208.000i			0.0775760i	−0.999247	+	0.0387880i	$0.987650\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			372.000i			0.134538i	0.997735	+	0.0672688i	$0.0214285\pi$
$197$			372.000i			0.134538i	−0.997735	+	0.0672688i	$0.978571\pi$
$198$		0			0
$199$	−4348.00			−1.54885			−0.774426	−	0.632665i	$-0.781961\pi$
$199$	−4348.00			−1.54885			−0.774426	+	0.632665i	$0.781961\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			952.000i			0.329149i
$204$		0			0
$205$	2110.00	−	4220.00i	0.718872	−	1.43774i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−896.000			−0.296544
$210$		0			0
$211$	−416.000			−0.135728			−0.0678640	−	0.997695i	$-0.521618\pi$
$211$	−416.000			−0.135728			−0.0678640	+	0.997695i	$0.521618\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2760.00	−	1380.00i	−0.875490	−	0.437745i
$216$		0			0
$217$			720.000i			0.225239i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1728.00			0.525963
$222$		0			0
$223$		−	5748.00i		−	1.72607i	−0.505141	−	0.863037i	$-0.668559\pi$
$223$		−	5748.00i		−	1.72607i	0.505141	−	0.863037i	$-0.331441\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			1148.00i			0.335663i	0.985816	+	0.167831i	$0.0536764\pi$
$227$			1148.00i			0.335663i	−0.985816	+	0.167831i	$0.946324\pi$
$228$		0			0
$229$	−3234.00			−0.933226			−0.466613	−	0.884462i	$-0.654526\pi$
$229$	−3234.00			−0.933226			−0.466613	+	0.884462i	$0.654526\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			228.000i			0.0641063i	0.999486	+	0.0320532i	$0.0102046\pi$
$233$			228.000i			0.0641063i	−0.999486	+	0.0320532i	$0.989795\pi$
$234$		0			0
$235$	−600.000	−	300.000i	−0.166552	−	0.0832759i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4760.00			1.28828			0.644140	−	0.764908i	$-0.277216\pi$
$239$	4760.00			1.28828			0.644140	+	0.764908i	$0.277216\pi$
$240$		0			0
$241$	3230.00			0.863330			0.431665	−	0.902034i	$-0.357926\pi$
$241$	3230.00			0.863330			0.431665	+	0.902034i	$0.357926\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1635.00	+	3270.00i	−0.426352	+	0.852705i
$246$		0			0
$247$			512.000i			0.131894i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1708.00			−0.429514			−0.214757	−	0.976668i	$-0.568896\pi$
$251$	−1708.00			−0.429514			−0.214757	+	0.976668i	$0.568896\pi$
$252$		0			0
$253$			784.000i			0.194821i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			6372.00i			1.54659i	0.634044	+	0.773297i	$0.281394\pi$
$257$			6372.00i			1.54659i	−0.634044	+	0.773297i	$0.718606\pi$
$258$		0			0
$259$	160.000			0.0383858
$260$		0			0
$261$		0			0
$262$		0			0
$263$			3036.00i			0.711817i	0.934521	+	0.355908i	$0.115828\pi$
$263$			3036.00i			0.711817i	−0.934521	+	0.355908i	$0.884172\pi$
$264$		0			0
$265$	2200.00	+	1100.00i	0.509981	+	0.254990i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−114.000			−0.0258390			−0.0129195	−	0.999917i	$-0.504113\pi$
$269$	−114.000			−0.0258390			−0.0129195	+	0.999917i	$0.504113\pi$
$270$		0			0
$271$	−5236.00			−1.17367			−0.586835	−	0.809707i	$-0.699626\pi$
$271$	−5236.00			−1.17367			−0.586835	+	0.809707i	$0.699626\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2100.00	−	2800.00i	−0.460490	−	0.613987i
$276$		0			0
$277$			5712.00i			1.23899i	0.785000	+	0.619496i	$0.212663\pi$
$277$			5712.00i			1.23899i	−0.785000	+	0.619496i	$0.787337\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3222.00			−0.684016			−0.342008	−	0.939697i	$-0.611107\pi$
$281$	−3222.00			−0.684016			−0.342008	+	0.939697i	$0.611107\pi$
$282$		0			0
$283$		−	4620.00i		−	0.970426i	−0.874396	−	0.485213i	$-0.838742\pi$
$283$		−	4620.00i		−	0.970426i	0.874396	−	0.485213i	$-0.161258\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			1688.00i			0.347176i
$288$		0			0
$289$	−6751.00			−1.37411
$290$		0			0
$291$		0			0
$292$		0			0
$293$			5404.00i			1.07749i	0.842468	+	0.538746i	$0.181102\pi$
$293$			5404.00i			1.07749i	−0.842468	+	0.538746i	$0.818898\pi$
$294$		0			0
$295$	4020.00	−	8040.00i	0.793402	−	1.58680i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	448.000			0.0866505
$300$		0			0
$301$	1104.00			0.211407
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1790.00	−	3580.00i	0.336050	−	0.672099i
$306$		0			0
$307$			9700.00i			1.80328i	0.432483	+	0.901642i	$0.357638\pi$
$307$			9700.00i			1.80328i	−0.432483	+	0.901642i	$0.642362\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9672.00			−1.76350			−0.881750	−	0.471716i	$-0.843635\pi$
$311$	−9672.00			−1.76350			−0.881750	+	0.471716i	$0.843635\pi$
$312$		0			0
$313$			4048.00i			0.731011i	0.930809	+	0.365506i	$0.119104\pi$
$313$			4048.00i			0.731011i	−0.930809	+	0.365506i	$0.880896\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	84.0000i		−	0.0148830i	−0.999972	−	0.00744150i	$-0.997631\pi$
$317$		−	84.0000i		−	0.0148830i	0.999972	−	0.00744150i	$-0.00236872\pi$
$318$		0			0
$319$	−6664.00			−1.16963
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	3456.00i		−	0.595347i
$324$		0			0
$325$	−1600.00	+	1200.00i	−0.273083	+	0.204812i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	240.000			0.0402177
$330$		0			0
$331$	5416.00			0.899366			0.449683	−	0.893188i	$-0.351537\pi$
$331$	5416.00			0.899366			0.449683	+	0.893188i	$0.351537\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8840.00	−	4420.00i	−1.44173	−	0.720867i
$336$		0			0
$337$			8216.00i			1.32805i	0.747709	+	0.664027i	$0.231154\pi$
$337$			8216.00i			1.32805i	−0.747709	+	0.664027i	$0.768846\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5040.00			−0.800385
$342$		0			0
$343$		−	2680.00i		−	0.421885i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			3836.00i			0.593450i	0.954963	+	0.296725i	$0.0958945\pi$
$347$			3836.00i			0.593450i	−0.954963	+	0.296725i	$0.904105\pi$
$348$		0			0
$349$	2038.00			0.312583			0.156292	−	0.987711i	$-0.450046\pi$
$349$	2038.00			0.312583			0.156292	+	0.987711i	$0.450046\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	5292.00i		−	0.797917i	−0.916969	−	0.398959i	$-0.869372\pi$
$353$		−	5292.00i		−	0.797917i	0.916969	−	0.398959i	$-0.130628\pi$
$354$		0			0
$355$	−320.000	+	640.000i	−0.0478418	+	0.0956836i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3896.00			−0.572766			−0.286383	−	0.958115i	$-0.592453\pi$
$359$	−3896.00			−0.572766			−0.286383	+	0.958115i	$0.592453\pi$
$360$		0			0
$361$	−5835.00			−0.850707
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1520.00	+	760.000i	0.217974	+	0.108987i
$366$		0			0
$367$		−	7652.00i		−	1.08837i	−0.838966	−	0.544184i	$-0.816839\pi$
$367$		−	7652.00i		−	1.08837i	0.838966	−	0.544184i	$-0.183161\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−880.000			−0.123146
$372$		0			0
$373$		−	1576.00i		−	0.218773i	−0.993999	−	0.109386i	$-0.965111\pi$
$373$		−	1576.00i		−	0.218773i	0.993999	−	0.109386i	$-0.0348886\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3808.00i			0.520217i
$378$		0			0
$379$	5416.00			0.734040			0.367020	−	0.930213i	$-0.380378\pi$
$379$	5416.00			0.734040			0.367020	+	0.930213i	$0.380378\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	8292.00i		−	1.10627i	−0.833092	−	0.553135i	$-0.813431\pi$
$383$		−	8292.00i		−	1.10627i	0.833092	−	0.553135i	$-0.186569\pi$
$384$		0			0
$385$	1120.00	+	560.000i	0.148261	+	0.0741305i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9642.00			−1.25673			−0.628366	−	0.777918i	$-0.716276\pi$
$389$	−9642.00			−1.25673			−0.628366	+	0.777918i	$0.716276\pi$
$390$		0			0
$391$	−3024.00			−0.391126
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4660.00	+	9320.00i	−0.593595	+	1.18719i
$396$		0			0
$397$			13032.0i			1.64750i	0.566954	+	0.823750i	$0.308122\pi$
$397$			13032.0i			1.64750i	−0.566954	+	0.823750i	$0.691878\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13358.0			1.66351			0.831754	−	0.555144i	$-0.187337\pi$
$401$	13358.0			1.66351			0.831754	+	0.555144i	$0.187337\pi$
$402$		0			0
$403$			2880.00i			0.355988i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			1120.00i			0.136404i
$408$		0			0
$409$	−6410.00			−0.774949			−0.387474	−	0.921880i	$-0.626652\pi$
$409$	−6410.00			−0.774949			−0.387474	+	0.921880i	$0.626652\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			3216.00i			0.383170i
$414$		0			0
$415$	−12920.0	−	6460.00i	−1.52824	−	0.764118i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7644.00			0.891250			0.445625	−	0.895220i	$-0.352981\pi$
$419$	7644.00			0.891250			0.445625	+	0.895220i	$0.352981\pi$
$420$		0			0
$421$	14674.0			1.69873			0.849367	−	0.527803i	$-0.176984\pi$
$421$	14674.0			1.69873			0.849367	+	0.527803i	$0.176984\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	10800.0	−	8100.00i	1.23265	−	0.924489i
$426$		0			0
$427$			1432.00i			0.162294i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9704.00			1.08451			0.542257	−	0.840213i	$-0.317570\pi$
$431$	9704.00			1.08451			0.542257	+	0.840213i	$0.317570\pi$
$432$		0			0
$433$			1296.00i			0.143838i	0.997410	+	0.0719189i	$0.0229123\pi$
$433$			1296.00i			0.143838i	−0.997410	+	0.0719189i	$0.977088\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	896.000i		−	0.0980812i
$438$		0			0
$439$	15684.0			1.70514			0.852570	−	0.522613i	$-0.175043\pi$
$439$	15684.0			1.70514			0.852570	+	0.522613i	$0.175043\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			5772.00i			0.619043i	0.950892	+	0.309521i	$0.100169\pi$
$443$			5772.00i			0.619043i	−0.950892	+	0.309521i	$0.899831\pi$
$444$		0			0
$445$	5730.00	−	11460.0i	0.610400	−	1.22080i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4782.00			−0.502620			−0.251310	−	0.967907i	$-0.580861\pi$
$449$	−4782.00			−0.502620			−0.251310	+	0.967907i	$0.580861\pi$
$450$		0			0
$451$	−11816.0			−1.23369
$452$		0			0
$453$		0			0
$454$		0			0
$455$	320.000	−	640.000i	0.0329711	−	0.0659421i
$456$		0			0
$457$		−	15000.0i		−	1.53538i	−0.640819	−	0.767692i	$-0.721405\pi$
$457$		−	15000.0i		−	1.53538i	0.640819	−	0.767692i	$-0.278595\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3762.00			0.380073			0.190037	−	0.981777i	$-0.439139\pi$
$461$	3762.00			0.380073			0.190037	+	0.981777i	$0.439139\pi$
$462$		0			0
$463$			5036.00i			0.505492i	0.967533	+	0.252746i	$0.0813337\pi$
$463$			5036.00i			0.505492i	−0.967533	+	0.252746i	$0.918666\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			2268.00i			0.224733i	0.993667	+	0.112367i	$0.0358431\pi$
$467$			2268.00i			0.224733i	−0.993667	+	0.112367i	$0.964157\pi$
$468$		0			0
$469$	3536.00			0.348139
$470$		0			0
$471$		0			0
$472$		0			0
$473$			7728.00i			0.751234i
$474$		0			0
$475$	2400.00	+	3200.00i	0.231831	+	0.309108i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	16208.0			1.54606			0.773030	−	0.634370i	$-0.218740\pi$
$479$	16208.0			1.54606			0.773030	+	0.634370i	$0.218740\pi$
$480$		0			0
$481$	640.000			0.0606684
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8240.00	+	4120.00i	0.771462	+	0.385731i
$486$		0			0
$487$		−	11572.0i		−	1.07675i	−0.842705	−	0.538375i	$-0.819038\pi$
$487$		−	11572.0i		−	1.07675i	0.842705	−	0.538375i	$-0.180962\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5636.00			−0.518023			−0.259011	−	0.965874i	$-0.583397\pi$
$491$	−5636.00			−0.518023			−0.259011	+	0.965874i	$0.583397\pi$
$492$		0			0
$493$		−	25704.0i		−	2.34817i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	256.000i		−	0.0231050i
$498$		0			0
$499$	5560.00			0.498797			0.249399	−	0.968401i	$-0.419767\pi$
$499$	5560.00			0.498797			0.249399	+	0.968401i	$0.419767\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	15172.0i		−	1.34490i	−0.740141	−	0.672451i	$-0.765241\pi$
$503$		−	15172.0i		−	1.34490i	0.740141	−	0.672451i	$-0.234759\pi$
$504$		0			0
$505$	−6450.00	+	12900.0i	−0.568359	+	1.13672i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−17342.0			−1.51016			−0.755079	−	0.655634i	$-0.772402\pi$
$509$	−17342.0			−1.51016			−0.755079	+	0.655634i	$0.772402\pi$
$510$		0			0
$511$	−608.000			−0.0526347
$512$		0			0
$513$		0			0
$514$		0			0
$515$	16040.0	+	8020.00i	1.37244	+	0.686220i
$516$		0			0
$517$			1680.00i			0.142914i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4886.00			0.410863			0.205431	−	0.978672i	$-0.434140\pi$
$521$	4886.00			0.410863			0.205431	+	0.978672i	$0.434140\pi$
$522$		0			0
$523$			18548.0i			1.55076i	0.631495	+	0.775380i	$0.282442\pi$
$523$			18548.0i			1.55076i	−0.631495	+	0.775380i	$0.717558\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	19440.0i		−	1.60687i
$528$		0			0
$529$	11383.0			0.935563
$530$		0			0
$531$		0			0
$532$		0			0
$533$			6752.00i			0.548708i
$534$		0			0
$535$	−8920.00	−	4460.00i	−0.720832	−	0.360416i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9156.00			0.731682
$540$		0			0
$541$	−15770.0			−1.25324			−0.626622	−	0.779323i	$-0.715563\pi$
$541$	−15770.0			−1.25324			−0.626622	+	0.779323i	$0.715563\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4830.00	−	9660.00i	0.379623	−	0.759246i
$546$		0			0
$547$			7700.00i			0.601880i	0.953643	+	0.300940i	$0.0973004\pi$
$547$			7700.00i			0.601880i	−0.953643	+	0.300940i	$0.902700\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7616.00			0.588843
$552$		0			0
$553$		−	3728.00i		−	0.286674i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	19236.0i		−	1.46330i	−0.681683	−	0.731648i	$-0.738752\pi$
$557$		−	19236.0i		−	1.46330i	0.681683	−	0.731648i	$-0.261248\pi$
$558$		0			0
$559$	4416.00			0.334127
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	8388.00i		−	0.627908i	−0.949438	−	0.313954i	$-0.898346\pi$
$563$		−	8388.00i		−	0.627908i	0.949438	−	0.313954i	$-0.101654\pi$
$564$		0			0
$565$	11240.0	+	5620.00i	0.836939	+	0.418469i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−16758.0			−1.23468			−0.617339	−	0.786697i	$-0.711789\pi$
$569$	−16758.0			−1.23468			−0.617339	+	0.786697i	$0.711789\pi$
$570$		0			0
$571$	8056.00			0.590426			0.295213	−	0.955432i	$-0.404609\pi$
$571$	8056.00			0.590426			0.295213	+	0.955432i	$0.404609\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2800.00	−	2100.00i	0.203075	−	0.152306i
$576$		0			0
$577$		−	5728.00i		−	0.413275i	−0.978418	−	0.206638i	$-0.933748\pi$
$577$		−	5728.00i		−	0.413275i	0.978418	−	0.206638i	$-0.0662521\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5168.00			0.369027
$582$		0			0
$583$		−	6160.00i		−	0.437601i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			12172.0i			0.855864i	0.903811	+	0.427932i	$0.140758\pi$
$587$			12172.0i			0.855864i	−0.903811	+	0.427932i	$0.859242\pi$
$588$		0			0
$589$	5760.00			0.402948
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	10708.0i		−	0.741526i	−0.928728	−	0.370763i	$-0.879096\pi$
$593$		−	10708.0i		−	0.741526i	0.928728	−	0.370763i	$-0.120904\pi$
$594$		0			0
$595$	−2160.00	+	4320.00i	−0.148826	+	0.297652i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−9416.00			−0.642283			−0.321141	−	0.947031i	$-0.604066\pi$
$599$	−9416.00			−0.642283			−0.321141	+	0.947031i	$0.604066\pi$
$600$		0			0
$601$	9270.00			0.629170			0.314585	−	0.949229i	$-0.398135\pi$
$601$	9270.00			0.629170			0.314585	+	0.949229i	$0.398135\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2735.00	−	5470.00i	0.183791	−	0.367582i
$606$		0			0
$607$			7996.00i			0.534675i	0.963603	+	0.267337i	$0.0861438\pi$
$607$			7996.00i			0.534675i	−0.963603	+	0.267337i	$0.913856\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	960.000			0.0635637
$612$		0			0
$613$		−	232.000i		−	0.0152861i	−0.999971	−	0.00764306i	$-0.997567\pi$
$613$		−	232.000i		−	0.0152861i	0.999971	−	0.00764306i	$-0.00243289\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			3740.00i			0.244030i	0.992528	+	0.122015i	$0.0389357\pi$
$617$			3740.00i			0.244030i	−0.992528	+	0.122015i	$0.961064\pi$
$618$		0			0
$619$	26000.0			1.68825			0.844126	−	0.536145i	$-0.180120\pi$
$619$	26000.0			1.68825			0.844126	+	0.536145i	$0.180120\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			4584.00i			0.294790i
$624$		0			0
$625$	−4375.00	+	15000.0i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4320.00			−0.273847
$630$		0			0
$631$	11660.0			0.735622			0.367811	−	0.929901i	$-0.380107\pi$
$631$	11660.0			0.735622			0.367811	+	0.929901i	$0.380107\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−18840.0	−	9420.00i	−1.17739	−	0.588695i
$636$		0			0
$637$		−	5232.00i		−	0.325431i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7602.00			0.468426			0.234213	−	0.972185i	$-0.424749\pi$
$641$	7602.00			0.468426			0.234213	+	0.972185i	$0.424749\pi$
$642$		0			0
$643$			29268.0i			1.79505i	0.440963	+	0.897525i	$0.354637\pi$
$643$			29268.0i			1.79505i	−0.440963	+	0.897525i	$0.645363\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			17836.0i			1.08378i	0.840449	+	0.541890i	$0.182291\pi$
$647$			17836.0i			1.08378i	−0.840449	+	0.541890i	$0.817709\pi$
$648$		0			0
$649$	−22512.0			−1.36159
$650$		0			0
$651$		0			0
$652$		0			0
$653$			19188.0i			1.14990i	0.818189	+	0.574950i	$0.194978\pi$
$653$			19188.0i			1.14990i	−0.818189	+	0.574950i	$0.805022\pi$
$654$		0			0
$655$	−2940.00	+	5880.00i	−0.175382	+	0.350764i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	13860.0			0.819285			0.409643	−	0.912246i	$-0.365653\pi$
$659$	13860.0			0.819285			0.409643	+	0.912246i	$0.365653\pi$
$660$		0			0
$661$	−16558.0			−0.974329			−0.487165	−	0.873310i	$-0.661969\pi$
$661$	−16558.0			−0.974329			−0.487165	+	0.873310i	$0.661969\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1280.00	−	640.000i	−0.0746410	−	0.0373205i
$666$		0			0
$667$		−	6664.00i		−	0.386853i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−10024.0			−0.576710
$672$		0			0
$673$		−	4640.00i		−	0.265764i	−0.991132	−	0.132882i	$-0.957577\pi$
$673$		−	4640.00i		−	0.265764i	0.991132	−	0.132882i	$-0.0424231\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	34260.0i		−	1.94493i	−0.233045	−	0.972466i	$-0.574869\pi$
$677$		−	34260.0i		−	1.94493i	0.233045	−	0.972466i	$-0.425131\pi$
$678$		0			0
$679$	−3296.00			−0.186287
$680$		0			0
$681$		0			0
$682$		0			0
$683$			19420.0i			1.08797i	0.839094	+	0.543987i	$0.183086\pi$
$683$			19420.0i			1.08797i	−0.839094	+	0.543987i	$0.816914\pi$
$684$		0			0
$685$	−10600.0	−	5300.00i	−0.591248	−	0.295624i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3520.00			−0.194632
$690$		0			0
$691$	4608.00			0.253685			0.126843	−	0.991923i	$-0.459516\pi$
$691$	4608.00			0.253685			0.126843	+	0.991923i	$0.459516\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−14320.0	+	28640.0i	−0.781566	+	1.56313i
$696$		0			0
$697$		−	45576.0i		−	2.47678i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	2318.00			0.124893			0.0624463	−	0.998048i	$-0.480110\pi$
$701$	2318.00			0.124893			0.0624463	+	0.998048i	$0.480110\pi$
$702$		0			0
$703$		−	1280.00i		−	0.0686716i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	5160.00i		−	0.274486i
$708$		0			0
$709$	−16834.0			−0.891698			−0.445849	−	0.895108i	$-0.647098\pi$
$709$	−16834.0			−0.891698			−0.445849	+	0.895108i	$0.647098\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	5040.00i		−	0.264726i
$714$		0			0
$715$	4480.00	+	2240.00i	0.234325	+	0.117163i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7400.00			−0.383830			−0.191915	−	0.981412i	$-0.561470\pi$
$719$	−7400.00			−0.383830			−0.191915	+	0.981412i	$0.561470\pi$
$720$		0			0
$721$	−6416.00			−0.331407
$722$		0			0
$723$		0			0
$724$		0			0
$725$	17850.0	+	23800.0i	0.914389	+	1.21919i
$726$		0			0
$727$		−	20340.0i		−	1.03765i	−0.854882	−	0.518823i	$-0.826370\pi$
$727$		−	20340.0i		−	1.03765i	0.854882	−	0.518823i	$-0.173630\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−29808.0			−1.50819
$732$		0			0
$733$			4896.00i			0.246709i	0.992363	+	0.123355i	$0.0393653\pi$
$733$			4896.00i			0.246709i	−0.992363	+	0.123355i	$0.960635\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			24752.0i			1.23711i
$738$		0			0
$739$	−26040.0			−1.29621			−0.648103	−	0.761552i	$-0.724438\pi$
$739$	−26040.0			−1.29621			−0.648103	+	0.761552i	$0.724438\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			6780.00i			0.334770i	0.985892	+	0.167385i	$0.0535323\pi$
$743$			6780.00i			0.334770i	−0.985892	+	0.167385i	$0.946468\pi$
$744$		0			0
$745$	−1710.00	+	3420.00i	−0.0840934	+	0.168187i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3568.00			0.174061
$750$		0			0
$751$	−20692.0			−1.00541			−0.502704	−	0.864458i	$-0.667662\pi$
$751$	−20692.0			−1.00541			−0.502704	+	0.864458i	$0.667662\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8180.00	+	16360.0i	−0.394306	+	0.788611i
$756$		0			0
$757$			10816.0i			0.519305i	0.965702	+	0.259653i	$0.0836081\pi$
$757$			10816.0i			0.519305i	−0.965702	+	0.259653i	$0.916392\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13978.0			−0.665837			−0.332919	−	0.942956i	$-0.608033\pi$
$761$	−13978.0			−0.665837			−0.332919	+	0.942956i	$0.608033\pi$
$762$		0			0
$763$			3864.00i			0.183337i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			12864.0i			0.605596i
$768$		0			0
$769$	−2926.00			−0.137210			−0.0686048	−	0.997644i	$-0.521855\pi$
$769$	−2926.00			−0.137210			−0.0686048	+	0.997644i	$0.521855\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	13916.0i		−	0.647508i	−0.946141	−	0.323754i	$-0.895055\pi$
$773$		−	13916.0i		−	0.647508i	0.946141	−	0.323754i	$-0.104945\pi$
$774$		0			0
$775$	13500.0	+	18000.0i	0.625722	+	0.834296i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13504.0			0.621092
$780$		0			0
$781$	1792.00			0.0821035
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−20720.0	−	10360.0i	−0.942075	−	0.471037i
$786$		0			0
$787$		−	29996.0i		−	1.35863i	−0.733847	−	0.679315i	$-0.762277\pi$
$787$		−	29996.0i		−	1.35863i	0.733847	−	0.679315i	$-0.237723\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4496.00			−0.202098
$792$		0			0
$793$			5728.00i			0.256503i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	8940.00i		−	0.397329i	−0.980068	−	0.198664i	$-0.936340\pi$
$797$		−	8940.00i		−	0.397329i	0.980068	−	0.198664i	$-0.0636604\pi$
$798$		0			0
$799$	−6480.00			−0.286916
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	4256.00i		−	0.187037i
$804$		0			0
$805$	−560.000	+	1120.00i	−0.0245185	+	0.0490370i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−10698.0			−0.464922			−0.232461	−	0.972606i	$-0.574678\pi$
$809$	−10698.0			−0.464922			−0.232461	+	0.972606i	$0.574678\pi$
$810$		0			0
$811$	−6408.00			−0.277454			−0.138727	−	0.990331i	$-0.544301\pi$
$811$	−6408.00			−0.277454			−0.138727	+	0.990331i	$0.544301\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7720.00	−	3860.00i	−0.331803	−	0.165902i
$816$		0			0
$817$		−	8832.00i		−	0.378204i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−23130.0			−0.983243			−0.491622	−	0.870809i	$-0.663596\pi$
$821$	−23130.0			−0.983243			−0.491622	+	0.870809i	$0.663596\pi$
$822$		0			0
$823$			11852.0i			0.501986i	0.967989	+	0.250993i	$0.0807572\pi$
$823$			11852.0i			0.501986i	−0.967989	+	0.250993i	$0.919243\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	32628.0i		−	1.37193i	−0.727634	−	0.685965i	$-0.759380\pi$
$827$		−	32628.0i		−	1.37193i	0.727634	−	0.685965i	$-0.240620\pi$
$828$		0			0
$829$	−36694.0			−1.53732			−0.768658	−	0.639660i	$-0.779075\pi$
$829$	−36694.0			−1.53732			−0.768658	+	0.639660i	$0.779075\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			35316.0i			1.46894i
$834$		0			0
$835$	10440.0	+	5220.00i	0.432684	+	0.216342i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1704.00			−0.0701175			−0.0350588	−	0.999385i	$-0.511162\pi$
$839$	−1704.00			−0.0701175			−0.0350588	+	0.999385i	$0.511162\pi$
$840$		0			0
$841$	32255.0			1.32252
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9705.00	+	19410.0i	−0.395103	+	0.790206i
$846$		0			0
$847$			2188.00i			0.0887610i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1120.00			−0.0451153
$852$		0			0
$853$			31880.0i			1.27966i	0.768516	+	0.639830i	$0.220995\pi$
$853$			31880.0i			1.27966i	−0.768516	+	0.639830i	$0.779005\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	7972.00i		−	0.317758i	−0.987298	−	0.158879i	$-0.949212\pi$
$857$		−	7972.00i		−	0.317758i	0.987298	−	0.158879i	$-0.0507879\pi$
$858$		0			0
$859$	6008.00			0.238638			0.119319	−	0.992856i	$-0.461929\pi$
$859$	6008.00			0.238638			0.119319	+	0.992856i	$0.461929\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1716.00i		−	0.0676863i	−0.999427	−	0.0338432i	$-0.989225\pi$
$863$		−	1716.00i		−	0.0676863i	0.999427	−	0.0338432i	$-0.0107747\pi$
$864$		0			0
$865$	44040.0	+	22020.0i	1.73110	+	0.865552i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	26096.0			1.01870
$870$		0			0
$871$	14144.0			0.550231
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1000.00	−	5500.00i	−0.0386356	−	0.212496i
$876$		0			0
$877$			9032.00i			0.347764i	0.984767	+	0.173882i	$0.0556311\pi$
$877$			9032.00i			0.347764i	−0.984767	+	0.173882i	$0.944369\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−27838.0			−1.06457			−0.532285	−	0.846565i	$-0.678666\pi$
$881$	−27838.0			−1.06457			−0.532285	+	0.846565i	$0.678666\pi$
$882$		0			0
$883$		−	4316.00i		−	0.164490i	−0.996612	−	0.0822452i	$-0.973791\pi$
$883$		−	4316.00i		−	0.164490i	0.996612	−	0.0822452i	$-0.0262091\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	43524.0i		−	1.64757i	−0.566904	−	0.823784i	$-0.691859\pi$
$887$		−	43524.0i		−	1.64757i	0.566904	−	0.823784i	$-0.308141\pi$
$888$		0			0
$889$	7536.00			0.284307
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	1920.00i		−	0.0719489i
$894$		0			0
$895$	−17260.0	+	34520.0i	−0.644624	+	1.28925i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	42840.0			1.58931
$900$		0			0
$901$	23760.0			0.878535
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2630.00	+	5260.00i	−0.0966013	+	0.193203i
$906$		0			0
$907$		−	10556.0i		−	0.386446i	−0.981155	−	0.193223i	$-0.938106\pi$
$907$		−	10556.0i		−	0.386446i	0.981155	−	0.193223i	$-0.0618940\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	47472.0			1.72647			0.863237	−	0.504799i	$-0.168433\pi$
$911$	47472.0			1.72647			0.863237	+	0.504799i	$0.168433\pi$
$912$		0			0
$913$			36176.0i			1.31134i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	2352.00i		−	0.0847000i
$918$		0			0
$919$	−11964.0			−0.429441			−0.214720	−	0.976676i	$-0.568884\pi$
$919$	−11964.0			−0.429441			−0.214720	+	0.976676i	$0.568884\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	1024.00i		−	0.0365172i
$924$		0			0
$925$	4000.00	−	3000.00i	0.142183	−	0.106637i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15214.0			−0.537304			−0.268652	−	0.963237i	$-0.586578\pi$
$929$	−15214.0			−0.537304			−0.268652	+	0.963237i	$0.586578\pi$
$930$		0			0
$931$	−10464.0			−0.368361
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−30240.0	−	15120.0i	−1.05770	−	0.528852i
$936$		0			0
$937$		−	39712.0i		−	1.38456i	−0.721628	−	0.692281i	$-0.756606\pi$
$937$		−	39712.0i		−	1.38456i	0.721628	−	0.692281i	$-0.243394\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	36034.0			1.24833			0.624163	−	0.781294i	$-0.285440\pi$
$941$	36034.0			1.24833			0.624163	+	0.781294i	$0.285440\pi$
$942$		0			0
$943$		−	11816.0i		−	0.408040i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	2532.00i		−	0.0868838i	−0.999056	−	0.0434419i	$-0.986168\pi$
$947$		−	2532.00i		−	0.0868838i	0.999056	−	0.0434419i	$-0.0138323\pi$
$948$		0			0
$949$	−2432.00			−0.0831887
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	55284.0i		−	1.87914i	−0.342351	−	0.939572i	$-0.611223\pi$
$953$		−	55284.0i		−	1.87914i	0.342351	−	0.939572i	$-0.388777\pi$
$954$		0			0
$955$	360.000	−	720.000i	0.0121982	−	0.0243965i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4240.00			0.142770
$960$		0			0
$961$	2609.00			0.0875768
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2080.00	−	1040.00i	−0.0693861	−	0.0346930i
$966$		0			0
$967$		−	4372.00i		−	0.145392i	−0.997354	−	0.0726960i	$-0.976840\pi$
$967$		−	4372.00i		−	0.145392i	0.997354	−	0.0726960i	$-0.0231603\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24300.0			−0.803114			−0.401557	−	0.915834i	$-0.631531\pi$
$971$	−24300.0			−0.803114			−0.401557	+	0.915834i	$0.631531\pi$
$972$		0			0
$973$		−	11456.0i		−	0.377454i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			46204.0i			1.51300i	0.653996	+	0.756498i	$0.273091\pi$
$977$			46204.0i			1.51300i	−0.653996	+	0.756498i	$0.726909\pi$
$978$		0			0
$979$	−32088.0			−1.04754
$980$		0			0
$981$		0			0
$982$		0			0
$983$			25468.0i			0.826351i	0.910651	+	0.413176i	$0.135580\pi$
$983$			25468.0i			0.826351i	−0.910651	+	0.413176i	$0.864420\pi$
$984$		0			0
$985$	−3720.00	−	1860.00i	−0.120334	−	0.0601670i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7728.00			−0.248469
$990$		0			0
$991$	11668.0			0.374012			0.187006	−	0.982359i	$-0.440122\pi$
$991$	11668.0			0.374012			0.187006	+	0.982359i	$0.440122\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	21740.0	−	43480.0i	0.692668	−	1.38534i
$996$		0			0
$997$			7224.00i			0.229475i	0.993396	+	0.114737i	$0.0366027\pi$
$997$			7224.00i			0.229475i	−0.993396	+	0.114737i	$0.963397\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.4.f.a.289.2		2
3.2	odd	2		120.4.f.c.49.1	✓	2
4.3	odd	2		720.4.f.b.289.2		2
5.2	odd	4		1800.4.a.u.1.1		1
5.3	odd	4		1800.4.a.o.1.1		1
5.4	even	2	inner	360.4.f.a.289.1		2
12.11	even	2		240.4.f.e.49.2		2
15.2	even	4		600.4.a.f.1.1		1
15.8	even	4		600.4.a.k.1.1		1
15.14	odd	2		120.4.f.c.49.2	yes	2
20.19	odd	2		720.4.f.b.289.1		2
24.5	odd	2		960.4.f.b.769.2		2
24.11	even	2		960.4.f.a.769.1		2
60.23	odd	4		1200.4.a.l.1.1		1
60.47	odd	4		1200.4.a.z.1.1		1
60.59	even	2		240.4.f.e.49.1		2
120.29	odd	2		960.4.f.b.769.1		2
120.59	even	2		960.4.f.a.769.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.f.c.49.1	✓	2	3.2	odd	2
120.4.f.c.49.2	yes	2	15.14	odd	2
240.4.f.e.49.1		2	60.59	even	2
240.4.f.e.49.2		2	12.11	even	2
360.4.f.a.289.1		2	5.4	even	2	inner
360.4.f.a.289.2		2	1.1	even	1	trivial
600.4.a.f.1.1		1	15.2	even	4
600.4.a.k.1.1		1	15.8	even	4
720.4.f.b.289.1		2	20.19	odd	2
720.4.f.b.289.2		2	4.3	odd	2
960.4.f.a.769.1		2	24.11	even	2
960.4.f.a.769.2		2	120.59	even	2
960.4.f.b.769.1		2	120.29	odd	2
960.4.f.b.769.2		2	24.5	odd	2
1200.4.a.l.1.1		1	60.23	odd	4
1200.4.a.z.1.1		1	60.47	odd	4
1800.4.a.o.1.1		1	5.3	odd	4
1800.4.a.u.1.1		1	5.2	odd	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:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	360.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.00000 + 10.0000i) q^{5} -4.00000i q^{7} +O(q^{10})\)	\(q+(-5.00000 + 10.0000i) q^{5} -4.00000i q^{7} +28.0000 q^{11} -16.0000i q^{13} +108.000i q^{17} -32.0000 q^{19} +28.0000i q^{23} +(-75.0000 - 100.000i) q^{25} -238.000 q^{29} -180.000 q^{31} +(40.0000 + 20.0000i) q^{35} +40.0000i q^{37} -422.000 q^{41} +276.000i q^{43} +60.0000i q^{47} +327.000 q^{49} -220.000i q^{53} +(-140.000 + 280.000i) q^{55} -804.000 q^{59} -358.000 q^{61} +(160.000 + 80.0000i) q^{65} +884.000i q^{67} +64.0000 q^{71} -152.000i q^{73} -112.000i q^{77} +932.000 q^{79} +1292.00i q^{83} +(-1080.00 - 540.000i) q^{85} -1146.00 q^{89} -64.0000 q^{91} +(160.000 - 320.000i) q^{95} -824.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5}+O(q^{10}) \) 2 * q - 10 * q^5	\( 2 q - 10 q^{5} + 56 q^{11} - 64 q^{19} - 150 q^{25} - 476 q^{29} - 360 q^{31} + 80 q^{35} - 844 q^{41} + 654 q^{49} - 280 q^{55} - 1608 q^{59} - 716 q^{61} + 320 q^{65} + 128 q^{71} + 1864 q^{79} - 2160 q^{85} - 2292 q^{89} - 128 q^{91} + 320 q^{95}+O(q^{100}) \) 2 * q - 10 * q^5 + 56 * q^11 - 64 * q^19 - 150 * q^25 - 476 * q^29 - 360 * q^31 + 80 * q^35 - 844 * q^41 + 654 * q^49 - 280 * q^55 - 1608 * q^59 - 716 * q^61 + 320 * q^65 + 128 * q^71 + 1864 * q^79 - 2160 * q^85 - 2292 * q^89 - 128 * q^91 + 320 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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