LMFDB - Embedded newform 1680.2.t.j.1009.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(1009,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1680.1009");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.t (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.5
Root			$1.45161 + 1.45161i$ of defining polynomial
Character	$\chi$	$=$	1680.1009
Dual form			1680.2.t.j.1009.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-0.311108 + 2.21432i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-0.311108 + 2.21432i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +5.05086 q^{11} +3.37778i q^{13} +(-2.21432 - 0.311108i) q^{15} -7.18421i q^{17} +8.23506 q^{19} +1.00000 q^{21} +6.23506i q^{23} +(-4.80642 - 1.37778i) q^{25} -1.00000i q^{27} +2.00000 q^{29} +4.62222 q^{31} +5.05086i q^{33} +(2.21432 + 0.311108i) q^{35} +4.85728i q^{37} -3.37778 q^{39} -3.37778 q^{41} -1.24443i q^{43} +(0.311108 - 2.21432i) q^{45} -1.00000 q^{49} +7.18421 q^{51} +4.62222i q^{53} +(-1.57136 + 11.1842i) q^{55} +8.23506i q^{57} -11.6128 q^{59} +0.488863 q^{61} +1.00000i q^{63} +(-7.47949 - 1.05086i) q^{65} +3.61285i q^{67} -6.23506 q^{69} +10.2953 q^{71} +16.2351i q^{73} +(1.37778 - 4.80642i) q^{75} -5.05086i q^{77} +1.24443 q^{79} +1.00000 q^{81} -11.6128i q^{83} +(15.9081 + 2.23506i) q^{85} +2.00000i q^{87} -6.99063 q^{89} +3.37778 q^{91} +4.62222i q^{93} +(-2.56199 + 18.2351i) q^{95} +8.23506i q^{97} -5.05086 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^5 - 6 * q^9	$ 6 q - 2 q^{5} - 6 q^{9} + 4 q^{11} - 4 q^{19} + 6 q^{21} - 2 q^{25} + 12 q^{29} + 28 q^{31} - 20 q^{39} - 20 q^{41} + 2 q^{45} - 6 q^{49} + 16 q^{51} - 36 q^{55} - 16 q^{59} + 4 q^{61} + 8 q^{65} + 16 q^{69} + 36 q^{71} + 8 q^{75} + 8 q^{79} + 6 q^{81} + 16 q^{85} + 12 q^{89} + 20 q^{91} + 12 q^{95} - 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^5 - 6 * q^9 + 4 * q^11 - 4 * q^19 + 6 * q^21 - 2 * q^25 + 12 * q^29 + 28 * q^31 - 20 * q^39 - 20 * q^41 + 2 * q^45 - 6 * q^49 + 16 * q^51 - 36 * q^55 - 16 * q^59 + 4 * q^61 + 8 * q^65 + 16 * q^69 + 36 * q^71 + 8 * q^75 + 8 * q^79 + 6 * q^81 + 16 * q^85 + 12 * q^89 + 20 * q^91 + 12 * q^95 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$-1$	$1$	$1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−0.311108	+	2.21432i	−0.139132	+	0.990274i
$5$	−0.311108	+	2.21432i	−0.139132	+	0.990274i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	5.05086			1.52289			0.761445	−	0.648229i	$-0.224490\pi$
$11$	5.05086			1.52289			0.761445	+	0.648229i	$0.224490\pi$
$12$		0			0
$13$			3.37778i			0.936829i	0.883509	+	0.468414i	$0.155175\pi$
$13$			3.37778i			0.936829i	−0.883509	+	0.468414i	$0.844825\pi$
$14$		0			0
$15$	−2.21432	−	0.311108i	−0.571735	−	0.0803277i
$16$		0			0
$17$		−	7.18421i		−	1.74243i	−0.490905	−	0.871213i	$-0.663334\pi$
$17$		−	7.18421i		−	1.74243i	0.490905	−	0.871213i	$-0.336666\pi$
$18$		0			0
$19$	8.23506			1.88925			0.944627	−	0.328147i	$-0.106424\pi$
$19$	8.23506			1.88925			0.944627	+	0.328147i	$0.106424\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$			6.23506i			1.30010i	0.759891	+	0.650050i	$0.225252\pi$
$23$			6.23506i			1.30010i	−0.759891	+	0.650050i	$0.774748\pi$
$24$		0			0
$25$	−4.80642	−	1.37778i	−0.961285	−	0.275557i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	4.62222			0.830174			0.415087	−	0.909782i	$-0.363751\pi$
$31$	4.62222			0.830174			0.415087	+	0.909782i	$0.363751\pi$
$32$		0			0
$33$			5.05086i			0.879241i
$34$		0			0
$35$	2.21432	+	0.311108i	0.374288	+	0.0525868i
$36$		0			0
$37$			4.85728i			0.798532i	0.916835	+	0.399266i	$0.130735\pi$
$37$			4.85728i			0.798532i	−0.916835	+	0.399266i	$0.869265\pi$
$38$		0			0
$39$	−3.37778			−0.540878
$40$		0			0
$41$	−3.37778			−0.527521			−0.263761	−	0.964588i	$-0.584963\pi$
$41$	−3.37778			−0.527521			−0.263761	+	0.964588i	$0.584963\pi$
$42$		0			0
$43$		−	1.24443i		−	0.189774i	−0.995488	−	0.0948870i	$-0.969751\pi$
$43$		−	1.24443i		−	0.189774i	0.995488	−	0.0948870i	$-0.0302490\pi$
$44$		0			0
$45$	0.311108	−	2.21432i	0.0463772	−	0.330091i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	7.18421			1.00599
$52$		0			0
$53$			4.62222i			0.634910i	0.948273	+	0.317455i	$0.102828\pi$
$53$			4.62222i			0.634910i	−0.948273	+	0.317455i	$0.897172\pi$
$54$		0			0
$55$	−1.57136	+	11.1842i	−0.211882	+	1.50808i
$56$		0			0
$57$			8.23506i			1.09076i
$58$		0			0
$59$	−11.6128			−1.51186			−0.755932	−	0.654650i	$-0.772816\pi$
$59$	−11.6128			−1.51186			−0.755932	+	0.654650i	$0.772816\pi$
$60$		0			0
$61$	0.488863			0.0625924			0.0312962	−	0.999510i	$-0.490036\pi$
$61$	0.488863			0.0625924			0.0312962	+	0.999510i	$0.490036\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−7.47949	−	1.05086i	−0.927717	−	0.130343i
$66$		0			0
$67$			3.61285i			0.441380i	0.975344	+	0.220690i	$0.0708308\pi$
$67$			3.61285i			0.441380i	−0.975344	+	0.220690i	$0.929169\pi$
$68$		0			0
$69$	−6.23506			−0.750613
$70$		0			0
$71$	10.2953			1.22183			0.610913	−	0.791698i	$-0.290803\pi$
$71$	10.2953			1.22183			0.610913	+	0.791698i	$0.290803\pi$
$72$		0			0
$73$			16.2351i			1.90017i	0.311991	+	0.950085i	$0.399004\pi$
$73$			16.2351i			1.90017i	−0.311991	+	0.950085i	$0.600996\pi$
$74$		0			0
$75$	1.37778	−	4.80642i	0.159093	−	0.554998i
$76$		0			0
$77$		−	5.05086i		−	0.575598i
$78$		0			0
$79$	1.24443			0.140009			0.0700047	−	0.997547i	$-0.477699\pi$
$79$	1.24443			0.140009			0.0700047	+	0.997547i	$0.477699\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	11.6128i		−	1.27468i	−0.770585	−	0.637338i	$-0.780036\pi$
$83$		−	11.6128i		−	1.27468i	0.770585	−	0.637338i	$-0.219964\pi$
$84$		0			0
$85$	15.9081	+	2.23506i	1.72548	+	0.242427i
$86$		0			0
$87$			2.00000i			0.214423i
$88$		0			0
$89$	−6.99063			−0.741006			−0.370503	−	0.928831i	$-0.620815\pi$
$89$	−6.99063			−0.741006			−0.370503	+	0.928831i	$0.620815\pi$
$90$		0			0
$91$	3.37778			0.354088
$92$		0			0
$93$			4.62222i			0.479301i
$94$		0			0
$95$	−2.56199	+	18.2351i	−0.262855	+	1.87088i
$96$		0			0
$97$			8.23506i			0.836144i	0.908414	+	0.418072i	$0.137294\pi$
$97$			8.23506i			0.836144i	−0.908414	+	0.418072i	$0.862706\pi$
$98$		0			0
$99$	−5.05086			−0.507630
$100$		0			0
$101$	−9.47949			−0.943245			−0.471622	−	0.881801i	$-0.656331\pi$
$101$	−9.47949			−0.943245			−0.471622	+	0.881801i	$0.656331\pi$
$102$		0			0
$103$		−	16.8573i		−	1.66100i	−0.557021	−	0.830499i	$-0.688056\pi$
$103$		−	16.8573i		−	1.66100i	0.557021	−	0.830499i	$-0.311944\pi$
$104$		0			0
$105$	−0.311108	+	2.21432i	−0.0303610	+	0.216095i
$106$		0			0
$107$			15.4795i			1.49646i	0.663440	+	0.748230i	$0.269096\pi$
$107$			15.4795i			1.49646i	−0.663440	+	0.748230i	$0.730904\pi$
$108$		0			0
$109$	1.61285			0.154483			0.0772414	−	0.997012i	$-0.475389\pi$
$109$	1.61285			0.154483			0.0772414	+	0.997012i	$0.475389\pi$
$110$		0			0
$111$	−4.85728			−0.461033
$112$		0			0
$113$			1.86665i			0.175599i	0.996138	+	0.0877997i	$0.0279835\pi$
$113$			1.86665i			0.175599i	−0.996138	+	0.0877997i	$0.972016\pi$
$114$		0			0
$115$	−13.8064	−	1.93978i	−1.28746	−	0.180885i
$116$		0			0
$117$		−	3.37778i		−	0.312276i
$118$		0			0
$119$	−7.18421			−0.658575
$120$		0			0
$121$	14.5111			1.31919
$122$		0			0
$123$		−	3.37778i		−	0.304565i
$124$		0			0
$125$	4.54617	−	10.2143i	0.406622	−	0.913597i
$126$		0			0
$127$			12.8573i			1.14090i	0.821333	+	0.570450i	$0.193231\pi$
$127$			12.8573i			1.14090i	−0.821333	+	0.570450i	$0.806769\pi$
$128$		0			0
$129$	1.24443			0.109566
$130$		0			0
$131$	−21.7146			−1.89721			−0.948605	−	0.316463i	$-0.897505\pi$
$131$	−21.7146			−1.89721			−0.948605	+	0.316463i	$0.897505\pi$
$132$		0			0
$133$		−	8.23506i		−	0.714071i
$134$		0			0
$135$	2.21432	+	0.311108i	0.190578	+	0.0267759i
$136$		0			0
$137$		−	9.47949i		−	0.809888i	−0.914342	−	0.404944i	$-0.867291\pi$
$137$		−	9.47949i		−	0.809888i	0.914342	−	0.404944i	$-0.132709\pi$
$138$		0			0
$139$	10.1334			0.859500			0.429750	−	0.902948i	$-0.358602\pi$
$139$	10.1334			0.859500			0.429750	+	0.902948i	$0.358602\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			17.0607i			1.42669i
$144$		0			0
$145$	−0.622216	+	4.42864i	−0.0516722	+	0.367778i
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	−7.24443			−0.593487			−0.296743	−	0.954957i	$-0.595901\pi$
$149$	−7.24443			−0.593487			−0.296743	+	0.954957i	$0.595901\pi$
$150$		0			0
$151$	8.85728			0.720795			0.360398	−	0.932799i	$-0.382641\pi$
$151$	8.85728			0.720795			0.360398	+	0.932799i	$0.382641\pi$
$152$		0			0
$153$			7.18421i			0.580809i
$154$		0			0
$155$	−1.43801	+	10.2351i	−0.115504	+	0.822100i
$156$		0			0
$157$		−	13.4795i		−	1.07578i	−0.843015	−	0.537890i	$-0.819221\pi$
$157$		−	13.4795i		−	1.07578i	0.843015	−	0.537890i	$-0.180779\pi$
$158$		0			0
$159$	−4.62222			−0.366566
$160$		0			0
$161$	6.23506			0.491392
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$	−11.1842	−	1.57136i	−0.870689	−	0.122330i
$166$		0			0
$167$			2.10171i			0.162635i	0.996688	+	0.0813176i	$0.0259128\pi$
$167$			2.10171i			0.162635i	−0.996688	+	0.0813176i	$0.974087\pi$
$168$		0			0
$169$	1.59057			0.122352
$170$		0			0
$171$	−8.23506			−0.629751
$172$		0			0
$173$			11.1842i			0.850320i	0.905118	+	0.425160i	$0.139782\pi$
$173$			11.1842i			0.850320i	−0.905118	+	0.425160i	$0.860218\pi$
$174$		0			0
$175$	−1.37778	+	4.80642i	−0.104151	+	0.363331i
$176$		0			0
$177$		−	11.6128i		−	0.872875i
$178$		0			0
$179$	20.6637			1.54448			0.772239	−	0.635332i	$-0.219137\pi$
$179$	20.6637			1.54448			0.772239	+	0.635332i	$0.219137\pi$
$180$		0			0
$181$	24.9590			1.85519			0.927594	−	0.373591i	$-0.121874\pi$
$181$	24.9590			1.85519			0.927594	+	0.373591i	$0.121874\pi$
$182$		0			0
$183$			0.488863i			0.0361378i
$184$		0			0
$185$	−10.7556	−	1.51114i	−0.790765	−	0.111101i
$186$		0			0
$187$		−	36.2864i		−	2.65352i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	−9.52098			−0.688914			−0.344457	−	0.938802i	$-0.611937\pi$
$191$	−9.52098			−0.688914			−0.344457	+	0.938802i	$0.611937\pi$
$192$		0			0
$193$		−	22.9590i		−	1.65262i	−0.563212	−	0.826312i	$-0.690435\pi$
$193$		−	22.9590i		−	1.65262i	0.563212	−	0.826312i	$-0.309565\pi$
$194$		0			0
$195$	1.05086	−	7.47949i	0.0752533	−	0.535618i
$196$		0			0
$197$			19.8479i			1.41411i	0.707161	+	0.707053i	$0.249976\pi$
$197$			19.8479i			1.41411i	−0.707161	+	0.707053i	$0.750024\pi$
$198$		0			0
$199$	−8.23506			−0.583768			−0.291884	−	0.956454i	$-0.594282\pi$
$199$	−8.23506			−0.583768			−0.291884	+	0.956454i	$0.594282\pi$
$200$		0			0
$201$	−3.61285			−0.254831
$202$		0			0
$203$		−	2.00000i		−	0.140372i
$204$		0			0
$205$	1.05086	−	7.47949i	0.0733949	−	0.522391i
$206$		0			0
$207$		−	6.23506i		−	0.433367i
$208$		0			0
$209$	41.5941			2.87712
$210$		0			0
$211$	11.6128			0.799461			0.399731	−	0.916633i	$-0.369104\pi$
$211$	11.6128			0.799461			0.399731	+	0.916633i	$0.369104\pi$
$212$		0			0
$213$			10.2953i			0.705421i
$214$		0			0
$215$	2.75557	+	0.387152i	0.187928	+	0.0264036i
$216$		0			0
$217$		−	4.62222i		−	0.313776i
$218$		0			0
$219$	−16.2351			−1.09706
$220$		0			0
$221$	24.2667			1.63236
$222$		0			0
$223$		−	2.48886i		−	0.166667i	−0.996522	−	0.0833333i	$-0.973443\pi$
$223$		−	2.48886i		−	0.166667i	0.996522	−	0.0833333i	$-0.0265566\pi$
$224$		0			0
$225$	4.80642	+	1.37778i	0.320428	+	0.0918523i
$226$		0			0
$227$		−	0.857279i		−	0.0568996i	−0.999595	−	0.0284498i	$-0.990943\pi$
$227$		−	0.857279i		−	0.0568996i	0.999595	−	0.0284498i	$-0.00905708\pi$
$228$		0			0
$229$	14.4701			0.956213			0.478106	−	0.878302i	$-0.341323\pi$
$229$	14.4701			0.956213			0.478106	+	0.878302i	$0.341323\pi$
$230$		0			0
$231$	5.05086			0.332322
$232$		0			0
$233$		−	3.96836i		−	0.259976i	−0.991516	−	0.129988i	$-0.958506\pi$
$233$		−	3.96836i		−	0.259976i	0.991516	−	0.129988i	$-0.0414938\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			1.24443i			0.0808345i
$238$		0			0
$239$	1.90813			0.123427			0.0617135	−	0.998094i	$-0.480344\pi$
$239$	1.90813			0.123427			0.0617135	+	0.998094i	$0.480344\pi$
$240$		0			0
$241$	0.755569			0.0486705			0.0243352	−	0.999704i	$-0.492253\pi$
$241$	0.755569			0.0486705			0.0243352	+	0.999704i	$0.492253\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	0.311108	−	2.21432i	0.0198759	−	0.141468i
$246$		0			0
$247$			27.8163i			1.76991i
$248$		0			0
$249$	11.6128			0.735934
$250$		0			0
$251$	9.12399			0.575901			0.287950	−	0.957645i	$-0.407026\pi$
$251$	9.12399			0.575901			0.287950	+	0.957645i	$0.407026\pi$
$252$		0			0
$253$			31.4924i			1.97991i
$254$		0			0
$255$	−2.23506	+	15.9081i	−0.139965	+	0.996206i
$256$		0			0
$257$			1.28592i			0.0802134i	0.999195	+	0.0401067i	$0.0127698\pi$
$257$			1.28592i			0.0802134i	−0.999195	+	0.0401067i	$0.987230\pi$
$258$		0			0
$259$	4.85728			0.301817
$260$		0			0
$261$	−2.00000			−0.123797
$262$		0			0
$263$		−	7.00937i		−	0.432216i	−0.976369	−	0.216108i	$-0.930664\pi$
$263$		−	7.00937i		−	0.432216i	0.976369	−	0.216108i	$-0.0693364\pi$
$264$		0			0
$265$	−10.2351	−	1.43801i	−0.628735	−	0.0883361i
$266$		0			0
$267$		−	6.99063i		−	0.427820i
$268$		0			0
$269$	−23.1941			−1.41417			−0.707083	−	0.707130i	$-0.749989\pi$
$269$	−23.1941			−1.41417			−0.707083	+	0.707130i	$0.749989\pi$
$270$		0			0
$271$	−7.11108			−0.431967			−0.215984	−	0.976397i	$-0.569296\pi$
$271$	−7.11108			−0.431967			−0.215984	+	0.976397i	$0.569296\pi$
$272$		0			0
$273$			3.37778i			0.204433i
$274$		0			0
$275$	−24.2766	−	6.95899i	−1.46393	−	0.419643i
$276$		0			0
$277$		−	7.34614i		−	0.441387i	−0.975343	−	0.220693i	$-0.929168\pi$
$277$		−	7.34614i		−	0.441387i	0.975343	−	0.220693i	$-0.0708320\pi$
$278$		0			0
$279$	−4.62222			−0.276725
$280$		0			0
$281$	−19.9813			−1.19198			−0.595991	−	0.802991i	$-0.703241\pi$
$281$	−19.9813			−1.19198			−0.595991	+	0.802991i	$0.703241\pi$
$282$		0			0
$283$		−	4.85728i		−	0.288735i	−0.989524	−	0.144368i	$-0.953885\pi$
$283$		−	4.85728i		−	0.288735i	0.989524	−	0.144368i	$-0.0461148\pi$
$284$		0			0
$285$	−18.2351	−	2.56199i	−1.08015	−	0.151759i
$286$		0			0
$287$			3.37778i			0.199384i
$288$		0			0
$289$	−34.6128			−2.03605
$290$		0			0
$291$	−8.23506			−0.482748
$292$		0			0
$293$		−	11.1842i		−	0.653388i	−0.945130	−	0.326694i	$-0.894065\pi$
$293$		−	11.1842i		−	0.653388i	0.945130	−	0.326694i	$-0.105935\pi$
$294$		0			0
$295$	3.61285	−	25.7146i	0.210348	−	1.49716i
$296$		0			0
$297$		−	5.05086i		−	0.293080i
$298$		0			0
$299$	−21.0607			−1.21797
$300$		0			0
$301$	−1.24443			−0.0717278
$302$		0			0
$303$		−	9.47949i		−	0.544583i
$304$		0			0
$305$	−0.152089	+	1.08250i	−0.00870859	+	0.0619837i
$306$		0			0
$307$		−	23.3461i		−	1.33243i	−0.745758	−	0.666217i	$-0.767912\pi$
$307$		−	23.3461i		−	1.33243i	0.745758	−	0.666217i	$-0.232088\pi$
$308$		0			0
$309$	16.8573			0.958977
$310$		0			0
$311$	32.8573			1.86317			0.931583	−	0.363530i	$-0.118428\pi$
$311$	32.8573			1.86317			0.931583	+	0.363530i	$0.118428\pi$
$312$		0			0
$313$			28.8256i			1.62932i	0.579938	+	0.814661i	$0.303077\pi$
$313$			28.8256i			1.62932i	−0.579938	+	0.814661i	$0.696923\pi$
$314$		0			0
$315$	−2.21432	−	0.311108i	−0.124763	−	0.0175289i
$316$		0			0
$317$		−	25.3590i		−	1.42431i	−0.702024	−	0.712153i	$-0.747720\pi$
$317$		−	25.3590i		−	1.42431i	0.702024	−	0.712153i	$-0.252280\pi$
$318$		0			0
$319$	10.1017			0.565587
$320$		0			0
$321$	−15.4795			−0.863981
$322$		0			0
$323$		−	59.1624i		−	3.29188i
$324$		0			0
$325$	4.65386	−	16.2351i	0.258150	−	0.900559i
$326$		0			0
$327$			1.61285i			0.0891907i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.89829			−0.104339			−0.0521697	−	0.998638i	$-0.516614\pi$
$331$	−1.89829			−0.104339			−0.0521697	+	0.998638i	$0.516614\pi$
$332$		0			0
$333$		−	4.85728i		−	0.266177i
$334$		0			0
$335$	−8.00000	−	1.12399i	−0.437087	−	0.0614099i
$336$		0			0
$337$		−	23.2257i		−	1.26518i	−0.774485	−	0.632592i	$-0.781991\pi$
$337$		−	23.2257i		−	1.26518i	0.774485	−	0.632592i	$-0.218009\pi$
$338$		0			0
$339$	−1.86665			−0.101382
$340$		0			0
$341$	23.3461			1.26426
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	1.93978	−	13.8064i	0.104434	−	0.743313i
$346$		0			0
$347$			7.47949i			0.401520i	0.979640	+	0.200760i	$0.0643412\pi$
$347$			7.47949i			0.401520i	−0.979640	+	0.200760i	$0.935659\pi$
$348$		0			0
$349$	−29.2257			−1.56442			−0.782208	−	0.623018i	$-0.785906\pi$
$349$	−29.2257			−1.56442			−0.782208	+	0.623018i	$0.785906\pi$
$350$		0			0
$351$	3.37778			0.180293
$352$		0			0
$353$		−	30.4099i		−	1.61856i	−0.587426	−	0.809278i	$-0.699859\pi$
$353$		−	30.4099i		−	1.61856i	0.587426	−	0.809278i	$-0.300141\pi$
$354$		0			0
$355$	−3.20294	+	22.7971i	−0.169995	+	1.20994i
$356$		0			0
$357$		−	7.18421i		−	0.380229i
$358$		0			0
$359$	−16.1936			−0.854664			−0.427332	−	0.904095i	$-0.640546\pi$
$359$	−16.1936			−0.854664			−0.427332	+	0.904095i	$0.640546\pi$
$360$		0			0
$361$	48.8163			2.56928
$362$		0			0
$363$			14.5111i			0.761637i
$364$		0			0
$365$	−35.9496	−	5.05086i	−1.88169	−	0.264374i
$366$		0			0
$367$		−	5.51114i		−	0.287679i	−0.989601	−	0.143840i	$-0.954055\pi$
$367$		−	5.51114i		−	0.287679i	0.989601	−	0.143840i	$-0.0459449\pi$
$368$		0			0
$369$	3.37778			0.175840
$370$		0			0
$371$	4.62222			0.239973
$372$		0			0
$373$			1.63158i			0.0844802i	0.999107	+	0.0422401i	$0.0134494\pi$
$373$			1.63158i			0.0844802i	−0.999107	+	0.0422401i	$0.986551\pi$
$374$		0			0
$375$	10.2143	+	4.54617i	0.527465	+	0.234763i
$376$		0			0
$377$			6.75557i			0.347929i
$378$		0			0
$379$	20.8573			1.07137			0.535683	−	0.844419i	$-0.320054\pi$
$379$	20.8573			1.07137			0.535683	+	0.844419i	$0.320054\pi$
$380$		0			0
$381$	−12.8573			−0.658698
$382$		0			0
$383$		−	20.0830i		−	1.02619i	−0.858331	−	0.513096i	$-0.828498\pi$
$383$		−	20.0830i		−	1.02619i	0.858331	−	0.513096i	$-0.171502\pi$
$384$		0			0
$385$	11.1842	+	1.57136i	0.570000	+	0.0800839i
$386$		0			0
$387$			1.24443i			0.0632580i
$388$		0			0
$389$	−33.2257			−1.68461			−0.842305	−	0.539001i	$-0.818802\pi$
$389$	−33.2257			−1.68461			−0.842305	+	0.539001i	$0.818802\pi$
$390$		0			0
$391$	44.7940			2.26533
$392$		0			0
$393$		−	21.7146i		−	1.09535i
$394$		0			0
$395$	−0.387152	+	2.75557i	−0.0194797	+	0.138648i
$396$		0			0
$397$		−	22.7239i		−	1.14048i	−0.821478	−	0.570241i	$-0.806850\pi$
$397$		−	22.7239i		−	1.14048i	0.821478	−	0.570241i	$-0.193150\pi$
$398$		0			0
$399$	8.23506			0.412269
$400$		0			0
$401$	12.7556			0.636983			0.318491	−	0.947926i	$-0.396824\pi$
$401$	12.7556			0.636983			0.318491	+	0.947926i	$0.396824\pi$
$402$		0			0
$403$			15.6128i			0.777731i
$404$		0			0
$405$	−0.311108	+	2.21432i	−0.0154591	+	0.110030i
$406$		0			0
$407$			24.5334i			1.21608i
$408$		0			0
$409$	−27.4479			−1.35721			−0.678604	−	0.734504i	$-0.737415\pi$
$409$	−27.4479			−1.35721			−0.678604	+	0.734504i	$0.737415\pi$
$410$		0			0
$411$	9.47949			0.467589
$412$		0			0
$413$			11.6128i			0.571431i
$414$		0			0
$415$	25.7146	+	3.61285i	1.26228	+	0.177348i
$416$		0			0
$417$			10.1334i			0.496232i
$418$		0			0
$419$	2.36842			0.115705			0.0578524	−	0.998325i	$-0.481575\pi$
$419$	2.36842			0.115705			0.0578524	+	0.998325i	$0.481575\pi$
$420$		0			0
$421$	39.3274			1.91670			0.958350	−	0.285596i	$-0.0921914\pi$
$421$	39.3274			1.91670			0.958350	+	0.285596i	$0.0921914\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.89829	+	34.5303i	−0.480138	+	1.67497i
$426$		0			0
$427$		−	0.488863i		−	0.0236577i
$428$		0			0
$429$	−17.0607			−0.823698
$430$		0			0
$431$	−8.19358			−0.394671			−0.197335	−	0.980336i	$-0.563229\pi$
$431$	−8.19358			−0.394671			−0.197335	+	0.980336i	$0.563229\pi$
$432$		0			0
$433$		−	14.6035i		−	0.701798i	−0.936413	−	0.350899i	$-0.885876\pi$
$433$		−	14.6035i		−	0.701798i	0.936413	−	0.350899i	$-0.114124\pi$
$434$		0			0
$435$	−4.42864	−	0.622216i	−0.212337	−	0.0298330i
$436$		0			0
$437$			51.3461i			2.45622i
$438$		0			0
$439$	5.27607			0.251813			0.125907	−	0.992042i	$-0.459816\pi$
$439$	5.27607			0.251813			0.125907	+	0.992042i	$0.459816\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	18.5018i		−	0.879046i	−0.898231	−	0.439523i	$-0.855148\pi$
$443$		−	18.5018i		−	0.879046i	0.898231	−	0.439523i	$-0.144852\pi$
$444$		0			0
$445$	2.17484	−	15.4795i	0.103097	−	0.733798i
$446$		0			0
$447$		−	7.24443i		−	0.342650i
$448$		0			0
$449$	−11.7146			−0.552844			−0.276422	−	0.961036i	$-0.589149\pi$
$449$	−11.7146			−0.552844			−0.276422	+	0.961036i	$0.589149\pi$
$450$		0			0
$451$	−17.0607			−0.803357
$452$		0			0
$453$			8.85728i			0.416151i
$454$		0			0
$455$	−1.05086	+	7.47949i	−0.0492648	+	0.350644i
$456$		0			0
$457$			18.9590i			0.886864i	0.896308	+	0.443432i	$0.146239\pi$
$457$			18.9590i			0.886864i	−0.896308	+	0.443432i	$0.853761\pi$
$458$		0			0
$459$	−7.18421			−0.335330
$460$		0			0
$461$	−15.1111			−0.703793			−0.351897	−	0.936039i	$-0.614463\pi$
$461$	−15.1111			−0.703793			−0.351897	+	0.936039i	$0.614463\pi$
$462$		0			0
$463$			22.5718i			1.04900i	0.851410	+	0.524501i	$0.175748\pi$
$463$			22.5718i			1.04900i	−0.851410	+	0.524501i	$0.824252\pi$
$464$		0			0
$465$	−10.2351	−	1.43801i	−0.474640	−	0.0666860i
$466$		0			0
$467$		−	22.3684i		−	1.03509i	−0.855657	−	0.517543i	$-0.826847\pi$
$467$		−	22.3684i		−	1.03509i	0.855657	−	0.517543i	$-0.173153\pi$
$468$		0			0
$469$	3.61285			0.166826
$470$		0			0
$471$	13.4795			0.621102
$472$		0			0
$473$		−	6.28544i		−	0.289005i
$474$		0			0
$475$	−39.5812	−	11.3461i	−1.81611	−	0.520597i
$476$		0			0
$477$		−	4.62222i		−	0.211637i
$478$		0			0
$479$	−32.8573			−1.50129			−0.750644	−	0.660707i	$-0.770256\pi$
$479$	−32.8573			−1.50129			−0.750644	+	0.660707i	$0.770256\pi$
$480$		0			0
$481$	−16.4068			−0.748088
$482$		0			0
$483$			6.23506i			0.283705i
$484$		0			0
$485$	−18.2351	−	2.56199i	−0.828012	−	0.116334i
$486$		0			0
$487$			3.40943i			0.154496i	0.997012	+	0.0772479i	$0.0246133\pi$
$487$			3.40943i			0.154496i	−0.997012	+	0.0772479i	$0.975387\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$	33.7877			1.52482			0.762409	−	0.647096i	$-0.224017\pi$
$491$	33.7877			1.52482			0.762409	+	0.647096i	$0.224017\pi$
$492$		0			0
$493$		−	14.3684i		−	0.647121i
$494$		0			0
$495$	1.57136	−	11.1842i	0.0706274	−	0.502693i
$496$		0			0
$497$		−	10.2953i		−	0.461807i
$498$		0			0
$499$	13.6316			0.610233			0.305117	−	0.952315i	$-0.401305\pi$
$499$	13.6316			0.610233			0.305117	+	0.952315i	$0.401305\pi$
$500$		0			0
$501$	−2.10171			−0.0938975
$502$		0			0
$503$		−	7.34614i		−	0.327548i	−0.986498	−	0.163774i	$-0.947633\pi$
$503$		−	7.34614i		−	0.327548i	0.986498	−	0.163774i	$-0.0523668\pi$
$504$		0			0
$505$	2.94914	−	20.9906i	0.131235	−	0.934071i
$506$		0			0
$507$			1.59057i			0.0706398i
$508$		0			0
$509$	15.9684			0.707785			0.353892	−	0.935286i	$-0.384858\pi$
$509$	15.9684			0.707785			0.353892	+	0.935286i	$0.384858\pi$
$510$		0			0
$511$	16.2351			0.718197
$512$		0			0
$513$		−	8.23506i		−	0.363587i
$514$		0			0
$515$	37.3274	+	5.24443i	1.64484	+	0.231097i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−11.1842			−0.490932
$520$		0			0
$521$	−1.09234			−0.0478564			−0.0239282	−	0.999714i	$-0.507617\pi$
$521$	−1.09234			−0.0478564			−0.0239282	+	0.999714i	$0.507617\pi$
$522$		0			0
$523$		−	17.5111i		−	0.765709i	−0.923809	−	0.382854i	$-0.874941\pi$
$523$		−	17.5111i		−	0.765709i	0.923809	−	0.382854i	$-0.125059\pi$
$524$		0			0
$525$	−4.80642	−	1.37778i	−0.209770	−	0.0601314i
$526$		0			0
$527$		−	33.2070i		−	1.44652i
$528$		0			0
$529$	−15.8760			−0.690262
$530$		0			0
$531$	11.6128			0.503955
$532$		0			0
$533$		−	11.4094i		−	0.494197i
$534$		0			0
$535$	−34.2766	−	4.81579i	−1.48190	−	0.208205i
$536$		0			0
$537$			20.6637i			0.891705i
$538$		0			0
$539$	−5.05086			−0.217556
$540$		0			0
$541$	−15.3274			−0.658977			−0.329488	−	0.944160i	$-0.606876\pi$
$541$	−15.3274			−0.658977			−0.329488	+	0.944160i	$0.606876\pi$
$542$		0			0
$543$			24.9590i			1.07109i
$544$		0			0
$545$	−0.501770	+	3.57136i	−0.0214934	+	0.152980i
$546$		0			0
$547$		−	5.32741i		−	0.227783i	−0.993493	−	0.113892i	$-0.963668\pi$
$547$		−	5.32741i		−	0.227783i	0.993493	−	0.113892i	$-0.0363317\pi$
$548$		0			0
$549$	−0.488863			−0.0208641
$550$		0			0
$551$	16.4701			0.701651
$552$		0			0
$553$		−	1.24443i		−	0.0529186i
$554$		0			0
$555$	1.51114	−	10.7556i	0.0641442	−	0.456548i
$556$		0			0
$557$			0.355509i			0.0150634i	0.999972	+	0.00753171i	$0.00239744\pi$
$557$			0.355509i			0.0150634i	−0.999972	+	0.00753171i	$0.997603\pi$
$558$		0			0
$559$	4.20342			0.177786
$560$		0			0
$561$	36.2864			1.53201
$562$		0			0
$563$		−	16.4701i		−	0.694133i	−0.937841	−	0.347067i	$-0.887178\pi$
$563$		−	16.4701i		−	0.694133i	0.937841	−	0.347067i	$-0.112822\pi$
$564$		0			0
$565$	−4.13335	−	0.580728i	−0.173891	−	0.0244314i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	20.9590			0.878647			0.439323	−	0.898329i	$-0.355218\pi$
$569$	20.9590			0.878647			0.439323	+	0.898329i	$0.355218\pi$
$570$		0			0
$571$	−17.5111			−0.732818			−0.366409	−	0.930454i	$-0.619413\pi$
$571$	−17.5111			−0.732818			−0.366409	+	0.930454i	$0.619413\pi$
$572$		0			0
$573$		−	9.52098i		−	0.397745i
$574$		0			0
$575$	8.59057	−	29.9684i	0.358252	−	1.24977i
$576$		0			0
$577$			0.152089i			0.00633155i	0.999995	+	0.00316577i	$0.00100770\pi$
$577$			0.152089i			0.00633155i	−0.999995	+	0.00316577i	$0.998992\pi$
$578$		0			0
$579$	22.9590			0.954143
$580$		0			0
$581$	−11.6128			−0.481782
$582$		0			0
$583$			23.3461i			0.966898i
$584$		0			0
$585$	7.47949	+	1.05086i	0.309239	+	0.0434475i
$586$		0			0
$587$		−	34.1847i		−	1.41095i	−0.708733	−	0.705476i	$-0.750733\pi$
$587$		−	34.1847i		−	1.41095i	0.708733	−	0.705476i	$-0.249267\pi$
$588$		0			0
$589$	38.0642			1.56841
$590$		0			0
$591$	−19.8479			−0.816434
$592$		0			0
$593$		−	19.3047i		−	0.792747i	−0.918089	−	0.396374i	$-0.870269\pi$
$593$		−	19.3047i		−	0.792747i	0.918089	−	0.396374i	$-0.129731\pi$
$594$		0			0
$595$	2.23506	−	15.9081i	0.0916287	−	0.652170i
$596$		0			0
$597$		−	8.23506i		−	0.337039i
$598$		0			0
$599$	−25.9081			−1.05858			−0.529289	−	0.848442i	$-0.677541\pi$
$599$	−25.9081			−1.05858			−0.529289	+	0.848442i	$0.677541\pi$
$600$		0			0
$601$	−22.7368			−0.927455			−0.463727	−	0.885978i	$-0.653488\pi$
$601$	−22.7368			−0.927455			−0.463727	+	0.885978i	$0.653488\pi$
$602$		0			0
$603$		−	3.61285i		−	0.147127i
$604$		0			0
$605$	−4.51453	+	32.1323i	−0.183542	+	1.30636i
$606$		0			0
$607$			32.9403i			1.33700i	0.743711	+	0.668502i	$0.233064\pi$
$607$			32.9403i			1.33700i	−0.743711	+	0.668502i	$0.766936\pi$
$608$		0			0
$609$	2.00000			0.0810441
$610$		0			0
$611$		0			0
$612$		0			0
$613$			13.1240i			0.530073i	0.964238	+	0.265036i	$0.0853840\pi$
$613$			13.1240i			0.530073i	−0.964238	+	0.265036i	$0.914616\pi$
$614$		0			0
$615$	7.47949	+	1.05086i	0.301602	+	0.0423746i
$616$		0			0
$617$			18.2538i			0.734870i	0.930049	+	0.367435i	$0.119764\pi$
$617$			18.2538i			0.734870i	−0.930049	+	0.367435i	$0.880236\pi$
$618$		0			0
$619$	7.64449			0.307258			0.153629	−	0.988129i	$-0.450904\pi$
$619$	7.64449			0.307258			0.153629	+	0.988129i	$0.450904\pi$
$620$		0			0
$621$	6.23506			0.250204
$622$		0			0
$623$			6.99063i			0.280074i
$624$		0			0
$625$	21.2034	+	13.2444i	0.848137	+	0.529777i
$626$		0			0
$627$			41.5941i			1.66111i
$628$		0			0
$629$	34.8957			1.39138
$630$		0			0
$631$	−28.6735			−1.14148			−0.570738	−	0.821132i	$-0.693343\pi$
$631$	−28.6735			−1.14148			−0.570738	+	0.821132i	$0.693343\pi$
$632$		0			0
$633$			11.6128i			0.461569i
$634$		0			0
$635$	−28.4701	−	4.00000i	−1.12980	−	0.158735i
$636$		0			0
$637$		−	3.37778i		−	0.133833i
$638$		0			0
$639$	−10.2953			−0.407275
$640$		0			0
$641$	15.4479			0.610153			0.305077	−	0.952328i	$-0.401318\pi$
$641$	15.4479			0.610153			0.305077	+	0.952328i	$0.401318\pi$
$642$		0			0
$643$			25.8350i			1.01883i	0.860520	+	0.509417i	$0.170139\pi$
$643$			25.8350i			1.01883i	−0.860520	+	0.509417i	$0.829861\pi$
$644$		0			0
$645$	−0.387152	+	2.75557i	−0.0152441	+	0.108500i
$646$		0			0
$647$			23.6128i			0.928317i	0.885752	+	0.464158i	$0.153643\pi$
$647$			23.6128i			0.928317i	−0.885752	+	0.464158i	$0.846357\pi$
$648$		0			0
$649$	−58.6548			−2.30240
$650$		0			0
$651$	4.62222			0.181159
$652$		0			0
$653$		−	36.7052i		−	1.43639i	−0.695844	−	0.718193i	$-0.744970\pi$
$653$		−	36.7052i		−	1.43639i	0.695844	−	0.718193i	$-0.255030\pi$
$654$		0			0
$655$	6.75557	−	48.0830i	0.263962	−	1.87876i
$656$		0			0
$657$		−	16.2351i		−	0.633390i
$658$		0			0
$659$	37.9911			1.47992			0.739962	−	0.672649i	$-0.234844\pi$
$659$	37.9911			1.47992			0.739962	+	0.672649i	$0.234844\pi$
$660$		0			0
$661$	−49.2257			−1.91466			−0.957329	−	0.289001i	$-0.906677\pi$
$661$	−49.2257			−1.91466			−0.957329	+	0.289001i	$0.906677\pi$
$662$		0			0
$663$			24.2667i			0.942441i
$664$		0			0
$665$	18.2351	+	2.56199i	0.707125	+	0.0993498i
$666$		0			0
$667$			12.4701i			0.482845i
$668$		0			0
$669$	2.48886			0.0962250
$670$		0			0
$671$	2.46917			0.0953214
$672$		0			0
$673$			11.2257i			0.432719i	0.976314	+	0.216359i	$0.0694183\pi$
$673$			11.2257i			0.432719i	−0.976314	+	0.216359i	$0.930582\pi$
$674$		0			0
$675$	−1.37778	+	4.80642i	−0.0530309	+	0.184999i
$676$		0			0
$677$			9.55262i			0.367137i	0.983007	+	0.183569i	$0.0587649\pi$
$677$			9.55262i			0.367137i	−0.983007	+	0.183569i	$0.941235\pi$
$678$		0			0
$679$	8.23506			0.316033
$680$		0			0
$681$	0.857279			0.0328510
$682$		0			0
$683$		−	17.9684i		−	0.687540i	−0.939054	−	0.343770i	$-0.888296\pi$
$683$		−	17.9684i		−	0.687540i	0.939054	−	0.343770i	$-0.111704\pi$
$684$		0			0
$685$	20.9906	+	2.94914i	0.802011	+	0.112681i
$686$		0			0
$687$			14.4701i			0.552070i
$688$		0			0
$689$	−15.6128			−0.594802
$690$		0			0
$691$	−0.355509			−0.0135242			−0.00676211	−	0.999977i	$-0.502152\pi$
$691$	−0.355509			−0.0135242			−0.00676211	+	0.999977i	$0.502152\pi$
$692$		0			0
$693$			5.05086i			0.191866i
$694$		0			0
$695$	−3.15257	+	22.4385i	−0.119584	+	0.851140i
$696$		0			0
$697$			24.2667i			0.919167i
$698$		0			0
$699$	3.96836			0.150097
$700$		0			0
$701$	13.2257			0.499528			0.249764	−	0.968307i	$-0.419647\pi$
$701$	13.2257			0.499528			0.249764	+	0.968307i	$0.419647\pi$
$702$		0			0
$703$			40.0000i			1.50863i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			9.47949i			0.356513i
$708$		0			0
$709$	10.9777			0.412277			0.206139	−	0.978523i	$-0.433910\pi$
$709$	10.9777			0.412277			0.206139	+	0.978523i	$0.433910\pi$
$710$		0			0
$711$	−1.24443			−0.0466698
$712$		0			0
$713$			28.8198i			1.07931i
$714$		0			0
$715$	−37.7778	−	5.30772i	−1.41281	−	0.198497i
$716$		0			0
$717$			1.90813i			0.0712606i
$718$		0			0
$719$	−21.9813			−0.819763			−0.409881	−	0.912139i	$-0.634430\pi$
$719$	−21.9813			−0.819763			−0.409881	+	0.912139i	$0.634430\pi$
$720$		0			0
$721$	−16.8573			−0.627798
$722$		0			0
$723$			0.755569i			0.0280999i
$724$		0			0
$725$	−9.61285	−	2.75557i	−0.357012	−	0.102339i
$726$		0			0
$727$			13.0607i			0.484395i	0.970227	+	0.242197i	$0.0778681\pi$
$727$			13.0607i			0.484395i	−0.970227	+	0.242197i	$0.922132\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−8.94025			−0.330667
$732$		0			0
$733$		−	13.5625i		−	0.500941i	−0.968124	−	0.250471i	$-0.919415\pi$
$733$		−	13.5625i		−	0.500941i	0.968124	−	0.250471i	$-0.0805853\pi$
$734$		0			0
$735$	2.21432	+	0.311108i	0.0816764	+	0.0114754i
$736$		0			0
$737$			18.2480i			0.672173i
$738$		0			0
$739$	10.2854			0.378356			0.189178	−	0.981943i	$-0.439418\pi$
$739$	10.2854			0.378356			0.189178	+	0.981943i	$0.439418\pi$
$740$		0			0
$741$	−27.8163			−1.02186
$742$		0			0
$743$			21.4608i			0.787319i	0.919256	+	0.393659i	$0.128791\pi$
$743$			21.4608i			0.787319i	−0.919256	+	0.393659i	$0.871209\pi$
$744$		0			0
$745$	2.25380	−	16.0415i	0.0825728	−	0.587715i
$746$		0			0
$747$			11.6128i			0.424892i
$748$		0			0
$749$	15.4795			0.565608
$750$		0			0
$751$	−46.0642			−1.68091			−0.840454	−	0.541883i	$-0.817712\pi$
$751$	−46.0642			−1.68091			−0.840454	+	0.541883i	$0.817712\pi$
$752$		0			0
$753$			9.12399i			0.332497i
$754$		0			0
$755$	−2.75557	+	19.6128i	−0.100285	+	0.713785i
$756$		0			0
$757$			27.1052i			0.985157i	0.870268	+	0.492579i	$0.163946\pi$
$757$			27.1052i			0.985157i	−0.870268	+	0.492579i	$0.836054\pi$
$758$		0			0
$759$	−31.4924			−1.14310
$760$		0			0
$761$	39.4608			1.43045			0.715226	−	0.698894i	$-0.246324\pi$
$761$	39.4608			1.43045			0.715226	+	0.698894i	$0.246324\pi$
$762$		0			0
$763$		−	1.61285i		−	0.0583890i
$764$		0			0
$765$	−15.9081	−	2.23506i	−0.575160	−	0.0808089i
$766$		0			0
$767$		−	39.2257i		−	1.41636i
$768$		0			0
$769$	7.51114			0.270859			0.135429	−	0.990787i	$-0.456759\pi$
$769$	7.51114			0.270859			0.135429	+	0.990787i	$0.456759\pi$
$770$		0			0
$771$	−1.28592			−0.0463112
$772$		0			0
$773$		−	1.46965i		−	0.0528596i	−0.999651	−	0.0264298i	$-0.991586\pi$
$773$		−	1.46965i		−	0.0528596i	0.999651	−	0.0264298i	$-0.00841385\pi$
$774$		0			0
$775$	−22.2163	−	6.36842i	−0.798034	−	0.228760i
$776$		0			0
$777$			4.85728i			0.174254i
$778$		0			0
$779$	−27.8163			−0.996621
$780$		0			0
$781$	52.0000			1.86071
$782$		0			0
$783$		−	2.00000i		−	0.0714742i
$784$		0			0
$785$	29.8479	+	4.19358i	1.06532	+	0.149675i
$786$		0			0
$787$			16.2034i			0.577590i	0.957391	+	0.288795i	$0.0932545\pi$
$787$			16.2034i			0.577590i	−0.957391	+	0.288795i	$0.906745\pi$
$788$		0			0
$789$	7.00937			0.249540
$790$		0			0
$791$	1.86665			0.0663703
$792$		0			0
$793$			1.65127i			0.0586384i
$794$		0			0
$795$	1.43801	−	10.2351i	0.0510009	−	0.363000i
$796$		0			0
$797$		−	0.161933i		−	0.00573597i	−0.999996	−	0.00286799i	$-0.999087\pi$
$797$		−	0.161933i		−	0.00573597i	0.999996	−	0.00286799i	$-0.000912909\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	6.99063			0.247002
$802$		0			0
$803$			82.0010i			2.89375i
$804$		0			0
$805$	−1.93978	+	13.8064i	−0.0683682	+	0.486613i
$806$		0			0
$807$		−	23.1941i		−	0.816469i
$808$		0			0
$809$	42.9403			1.50970			0.754849	−	0.655898i	$-0.227710\pi$
$809$	42.9403			1.50970			0.754849	+	0.655898i	$0.227710\pi$
$810$		0			0
$811$	41.2958			1.45009			0.725045	−	0.688701i	$-0.241819\pi$
$811$	41.2958			1.45009			0.725045	+	0.688701i	$0.241819\pi$
$812$		0			0
$813$		−	7.11108i		−	0.249396i
$814$		0			0
$815$	8.85728	+	1.24443i	0.310257	+	0.0435905i
$816$		0			0
$817$		−	10.2480i		−	0.358531i
$818$		0			0
$819$	−3.37778			−0.118029
$820$		0			0
$821$	−43.2070			−1.50793			−0.753967	−	0.656913i	$-0.771862\pi$
$821$	−43.2070			−1.50793			−0.753967	+	0.656913i	$0.771862\pi$
$822$		0			0
$823$			11.1427i			0.388411i	0.980961	+	0.194205i	$0.0622128\pi$
$823$			11.1427i			0.388411i	−0.980961	+	0.194205i	$0.937787\pi$
$824$		0			0
$825$	6.95899	−	24.2766i	0.242281	−	0.845201i
$826$		0			0
$827$			36.9906i			1.28629i	0.765744	+	0.643145i	$0.222371\pi$
$827$			36.9906i			1.28629i	−0.765744	+	0.643145i	$0.777629\pi$
$828$		0			0
$829$	22.6735			0.787485			0.393742	−	0.919221i	$-0.371180\pi$
$829$	22.6735			0.787485			0.393742	+	0.919221i	$0.371180\pi$
$830$		0			0
$831$	7.34614			0.254835
$832$		0			0
$833$			7.18421i			0.248918i
$834$		0			0
$835$	−4.65386	−	0.653858i	−0.161053	−	0.0226277i
$836$		0			0
$837$		−	4.62222i		−	0.159767i
$838$		0			0
$839$	−21.8983			−0.756013			−0.378006	−	0.925803i	$-0.623390\pi$
$839$	−21.8983			−0.756013			−0.378006	+	0.925803i	$0.623390\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		−	19.9813i		−	0.688191i
$844$		0			0
$845$	−0.494840	+	3.52204i	−0.0170230	+	0.121162i
$846$		0			0
$847$		−	14.5111i		−	0.498609i
$848$		0			0
$849$	4.85728			0.166701
$850$		0			0
$851$	−30.2854			−1.03817
$852$		0			0
$853$			52.4010i			1.79418i	0.441851	+	0.897088i	$0.354322\pi$
$853$			52.4010i			1.79418i	−0.441851	+	0.897088i	$0.645678\pi$
$854$		0			0
$855$	2.56199	−	18.2351i	0.0876183	−	0.623626i
$856$		0			0
$857$			24.5116i			0.837301i	0.908147	+	0.418650i	$0.137497\pi$
$857$			24.5116i			0.837301i	−0.908147	+	0.418650i	$0.862503\pi$
$858$		0			0
$859$	57.4795			1.96118			0.980588	−	0.196082i	$-0.0628218\pi$
$859$	57.4795			1.96118			0.980588	+	0.196082i	$0.0628218\pi$
$860$		0			0
$861$	−3.37778			−0.115115
$862$		0			0
$863$		−	21.1941i		−	0.721454i	−0.932671	−	0.360727i	$-0.882529\pi$
$863$		−	21.1941i		−	0.721454i	0.932671	−	0.360727i	$-0.117471\pi$
$864$		0			0
$865$	−24.7654	−	3.47949i	−0.842049	−	0.118306i
$866$		0			0
$867$		−	34.6128i		−	1.17551i
$868$		0			0
$869$	6.28544			0.213219
$870$		0			0
$871$	−12.2034			−0.413497
$872$		0			0
$873$		−	8.23506i		−	0.278715i
$874$		0			0
$875$	−10.2143	−	4.54617i	−0.345307	−	0.153689i
$876$		0			0
$877$		−	8.38715i		−	0.283214i	−0.989923	−	0.141607i	$-0.954773\pi$
$877$		−	8.38715i		−	0.283214i	0.989923	−	0.141607i	$-0.0452269\pi$
$878$		0			0
$879$	11.1842			0.377234
$880$		0			0
$881$	29.1753			0.982941			0.491471	−	0.870894i	$-0.336459\pi$
$881$	29.1753			0.982941			0.491471	+	0.870894i	$0.336459\pi$
$882$		0			0
$883$			19.8796i			0.669000i	0.942396	+	0.334500i	$0.108567\pi$
$883$			19.8796i			0.669000i	−0.942396	+	0.334500i	$0.891433\pi$
$884$		0			0
$885$	25.7146	+	3.61285i	0.864385	+	0.121445i
$886$		0			0
$887$			14.6923i			0.493319i	0.969102	+	0.246659i	$0.0793329\pi$
$887$			14.6923i			0.493319i	−0.969102	+	0.246659i	$0.920667\pi$
$888$		0			0
$889$	12.8573			0.431219
$890$		0			0
$891$	5.05086			0.169210
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−6.42864	+	45.7560i	−0.214886	+	1.52946i
$896$		0			0
$897$		−	21.0607i		−	0.703196i
$898$		0			0
$899$	9.24443			0.308319
$900$		0			0
$901$	33.2070			1.10628
$902$		0			0
$903$		−	1.24443i		−	0.0414121i
$904$		0			0
$905$	−7.76494	+	55.2672i	−0.258115	+	1.83714i
$906$		0			0
$907$		−	19.8796i		−	0.660090i	−0.943965	−	0.330045i	$-0.892936\pi$
$907$		−	19.8796i		−	0.660090i	0.943965	−	0.330045i	$-0.107064\pi$
$908$		0			0
$909$	9.47949			0.314415
$910$		0			0
$911$	53.3372			1.76714			0.883571	−	0.468297i	$-0.155132\pi$
$911$	53.3372			1.76714			0.883571	+	0.468297i	$0.155132\pi$
$912$		0			0
$913$		−	58.6548i		−	1.94119i
$914$		0			0
$915$	−1.08250	−	0.152089i	−0.0357863	−	0.00502791i
$916$		0			0
$917$			21.7146i			0.717078i
$918$		0			0
$919$	−21.1240			−0.696816			−0.348408	−	0.937343i	$-0.613278\pi$
$919$	−21.1240			−0.696816			−0.348408	+	0.937343i	$0.613278\pi$
$920$		0			0
$921$	23.3461			0.769282
$922$		0			0
$923$			34.7753i			1.14464i
$924$		0			0
$925$	6.69228	−	23.3461i	0.220041	−	0.767616i
$926$		0			0
$927$			16.8573i			0.553666i
$928$		0			0
$929$	−2.72393			−0.0893691			−0.0446846	−	0.999001i	$-0.514228\pi$
$929$	−2.72393			−0.0893691			−0.0446846	+	0.999001i	$0.514228\pi$
$930$		0			0
$931$	−8.23506			−0.269893
$932$		0			0
$933$			32.8573i			1.07570i
$934$		0			0
$935$	80.3497	+	11.2890i	2.62772	+	0.369189i
$936$		0			0
$937$			15.3145i			0.500303i	0.968207	+	0.250151i	$0.0804804\pi$
$937$			15.3145i			0.500303i	−0.968207	+	0.250151i	$0.919520\pi$
$938$		0			0
$939$	−28.8256			−0.940689
$940$		0			0
$941$	−30.6035			−0.997645			−0.498822	−	0.866704i	$-0.666234\pi$
$941$	−30.6035			−0.997645			−0.498822	+	0.866704i	$0.666234\pi$
$942$		0			0
$943$		−	21.0607i		−	0.685831i
$944$		0			0
$945$	0.311108	−	2.21432i	0.0101203	−	0.0720318i
$946$		0			0
$947$		−	2.74266i		−	0.0891245i	−0.999007	−	0.0445623i	$-0.985811\pi$
$947$		−	2.74266i		−	0.0891245i	0.999007	−	0.0445623i	$-0.0141893\pi$
$948$		0			0
$949$	−54.8385			−1.78013
$950$		0			0
$951$	25.3590			0.822323
$952$		0			0
$953$		−	47.8479i		−	1.54995i	−0.631994	−	0.774973i	$-0.717763\pi$
$953$		−	47.8479i		−	1.54995i	0.631994	−	0.774973i	$-0.282237\pi$
$954$		0			0
$955$	2.96205	−	21.0825i	0.0958497	−	0.682214i
$956$		0			0
$957$			10.1017i			0.326542i
$958$		0			0
$959$	−9.47949			−0.306109
$960$		0			0
$961$	−9.63512			−0.310810
$962$		0			0
$963$		−	15.4795i		−	0.498820i
$964$		0			0
$965$	50.8385	+	7.14272i	1.63655	+	0.229932i
$966$		0			0
$967$		−	38.7181i		−	1.24509i	−0.782584	−	0.622545i	$-0.786099\pi$
$967$		−	38.7181i		−	1.24509i	0.782584	−	0.622545i	$-0.213901\pi$
$968$		0			0
$969$	59.1624			1.90057
$970$		0			0
$971$	−13.7778			−0.442152			−0.221076	−	0.975257i	$-0.570957\pi$
$971$	−13.7778			−0.442152			−0.221076	+	0.975257i	$0.570957\pi$
$972$		0			0
$973$		−	10.1334i		−	0.324860i
$974$		0			0
$975$	16.2351	+	4.65386i	0.519938	+	0.149043i
$976$		0			0
$977$		−	4.03164i		−	0.128984i	−0.997918	−	0.0644918i	$-0.979457\pi$
$977$		−	4.03164i		−	0.128984i	0.997918	−	0.0644918i	$-0.0205426\pi$
$978$		0			0
$979$	−35.3087			−1.12847
$980$		0			0
$981$	−1.61285			−0.0514943
$982$		0			0
$983$		−	47.4924i		−	1.51477i	−0.652967	−	0.757386i	$-0.726476\pi$
$983$		−	47.4924i		−	1.51477i	0.652967	−	0.757386i	$-0.273524\pi$
$984$		0			0
$985$	−43.9496	−	6.17484i	−1.40035	−	0.196747i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.75911			0.246725
$990$		0			0
$991$	−1.16146			−0.0368949			−0.0184474	−	0.999830i	$-0.505872\pi$
$991$	−1.16146			−0.0368949			−0.0184474	+	0.999830i	$0.505872\pi$
$992$		0			0
$993$		−	1.89829i		−	0.0602404i
$994$		0			0
$995$	2.56199	−	18.2351i	0.0812206	−	0.578090i
$996$		0			0
$997$			18.4572i			0.584546i	0.956335	+	0.292273i	$0.0944116\pi$
$997$			18.4572i			0.584546i	−0.956335	+	0.292273i	$0.905588\pi$
$998$		0			0
$999$	4.85728			0.153678

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.t.j.1009.5		6
3.2	odd	2		5040.2.t.z.1009.3		6
4.3	odd	2		840.2.t.d.169.2	✓	6
5.2	odd	4		8400.2.a.dl.1.3		3
5.3	odd	4		8400.2.a.di.1.3		3
5.4	even	2	inner	1680.2.t.j.1009.2		6
12.11	even	2		2520.2.t.k.1009.3		6
15.14	odd	2		5040.2.t.z.1009.4		6
20.3	even	4		4200.2.a.bp.1.1		3
20.7	even	4		4200.2.a.bn.1.1		3
20.19	odd	2		840.2.t.d.169.5	yes	6
60.59	even	2		2520.2.t.k.1009.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.d.169.2	✓	6	4.3	odd	2
840.2.t.d.169.5	yes	6	20.19	odd	2
1680.2.t.j.1009.2		6	5.4	even	2	inner
1680.2.t.j.1009.5		6	1.1	even	1	trivial
2520.2.t.k.1009.3		6	12.11	even	2
2520.2.t.k.1009.4		6	60.59	even	2
4200.2.a.bn.1.1		3	20.7	even	4
4200.2.a.bp.1.1		3	20.3	even	4
5040.2.t.z.1009.3		6	3.2	odd	2
5040.2.t.z.1009.4		6	15.14	odd	2
8400.2.a.di.1.3		3	5.3	odd	4
8400.2.a.dl.1.3		3	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-0.311108 + 2.21432i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-0.311108 + 2.21432i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +5.05086 q^{11} +3.37778i q^{13} +(-2.21432 - 0.311108i) q^{15} -7.18421i q^{17} +8.23506 q^{19} +1.00000 q^{21} +6.23506i q^{23} +(-4.80642 - 1.37778i) q^{25} -1.00000i q^{27} +2.00000 q^{29} +4.62222 q^{31} +5.05086i q^{33} +(2.21432 + 0.311108i) q^{35} +4.85728i q^{37} -3.37778 q^{39} -3.37778 q^{41} -1.24443i q^{43} +(0.311108 - 2.21432i) q^{45} -1.00000 q^{49} +7.18421 q^{51} +4.62222i q^{53} +(-1.57136 + 11.1842i) q^{55} +8.23506i q^{57} -11.6128 q^{59} +0.488863 q^{61} +1.00000i q^{63} +(-7.47949 - 1.05086i) q^{65} +3.61285i q^{67} -6.23506 q^{69} +10.2953 q^{71} +16.2351i q^{73} +(1.37778 - 4.80642i) q^{75} -5.05086i q^{77} +1.24443 q^{79} +1.00000 q^{81} -11.6128i q^{83} +(15.9081 + 2.23506i) q^{85} +2.00000i q^{87} -6.99063 q^{89} +3.37778 q^{91} +4.62222i q^{93} +(-2.56199 + 18.2351i) q^{95} +8.23506i q^{97} -5.05086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^5 - 6 * q^9	\( 6 q - 2 q^{5} - 6 q^{9} + 4 q^{11} - 4 q^{19} + 6 q^{21} - 2 q^{25} + 12 q^{29} + 28 q^{31} - 20 q^{39} - 20 q^{41} + 2 q^{45} - 6 q^{49} + 16 q^{51} - 36 q^{55} - 16 q^{59} + 4 q^{61} + 8 q^{65} + 16 q^{69} + 36 q^{71} + 8 q^{75} + 8 q^{79} + 6 q^{81} + 16 q^{85} + 12 q^{89} + 20 q^{91} + 12 q^{95} - 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^5 - 6 * q^9 + 4 * q^11 - 4 * q^19 + 6 * q^21 - 2 * q^25 + 12 * q^29 + 28 * q^31 - 20 * q^39 - 20 * q^41 + 2 * q^45 - 6 * q^49 + 16 * q^51 - 36 * q^55 - 16 * q^59 + 4 * q^61 + 8 * q^65 + 16 * q^69 + 36 * q^71 + 8 * q^75 + 8 * q^79 + 6 * q^81 + 16 * q^85 + 12 * q^89 + 20 * q^91 + 12 * q^95 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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