LMFDB - Embedded newform 3360.2.j.a.1009.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3360,2,Mod(1009,3360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("3360.1009");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.j (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:	$26.8297350792$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3360.1009
Dual form			3360.2.j.a.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +1.00000 q^{9} -6.00000 q^{13} +(2.00000 - 1.00000i) q^{15} -2.00000i q^{17} -4.00000i q^{19} +1.00000i q^{21} +4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -1.00000 q^{27} -6.00000i q^{29} +8.00000 q^{31} +(1.00000 + 2.00000i) q^{35} -2.00000 q^{37} +6.00000 q^{39} -8.00000 q^{41} +6.00000 q^{43} +(-2.00000 + 1.00000i) q^{45} +2.00000i q^{47} -1.00000 q^{49} +2.00000i q^{51} -6.00000 q^{53} +4.00000i q^{57} +6.00000i q^{59} +10.0000i q^{61} -1.00000i q^{63} +(12.0000 - 6.00000i) q^{65} +2.00000 q^{67} -4.00000i q^{69} +8.00000 q^{71} +6.00000i q^{73} +(-3.00000 + 4.00000i) q^{75} -10.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +(2.00000 + 4.00000i) q^{85} +6.00000i q^{87} +6.00000i q^{91} -8.00000 q^{93} +(4.00000 + 8.00000i) q^{95} -2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} - 12 q^{13} + 4 q^{15} + 6 q^{25} - 2 q^{27} + 16 q^{31} + 2 q^{35} - 4 q^{37} + 12 q^{39} - 16 q^{41} + 12 q^{43} - 4 q^{45} - 2 q^{49} - 12 q^{53} + 24 q^{65} + 4 q^{67} + 16 q^{71} - 6 q^{75} - 20 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} - 16 q^{93} + 8 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 - 12 * q^13 + 4 * q^15 + 6 * q^25 - 2 * q^27 + 16 * q^31 + 2 * q^35 - 4 * q^37 + 12 * q^39 - 16 * q^41 + 12 * q^43 - 4 * q^45 - 2 * q^49 - 12 * q^53 + 24 * q^65 + 4 * q^67 + 16 * q^71 - 6 * q^75 - 20 * q^79 + 2 * q^81 - 8 * q^83 + 4 * q^85 - 16 * q^93 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−6.00000			−1.66410			−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000			−1.66410			−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$	2.00000	−	1.00000i	0.516398	−	0.258199i
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.00000	+	2.00000i	0.169031	+	0.338062i
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$	6.00000			0.960769
$40$		0			0
$41$	−8.00000			−1.24939			−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000			−1.24939			−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$	6.00000			0.914991			0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000			0.914991			0.457496	+	0.889212i	$0.348747\pi$
$44$		0			0
$45$	−2.00000	+	1.00000i	−0.298142	+	0.149071i
$46$		0			0
$47$			2.00000i			0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$			2.00000i			0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$			2.00000i			0.280056i
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			4.00000i			0.529813i
$58$		0			0
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$			10.0000i			1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$			10.0000i			1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	12.0000	−	6.00000i	1.48842	−	0.744208i
$66$		0			0
$67$	2.00000			0.244339			0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000			0.244339			0.122169	+	0.992509i	$0.461015\pi$
$68$		0			0
$69$		−	4.00000i		−	0.481543i
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$	−3.00000	+	4.00000i	−0.346410	+	0.461880i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−10.0000			−1.12509			−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000			−1.12509			−0.562544	+	0.826767i	$0.690177\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$		0			0
$85$	2.00000	+	4.00000i	0.216930	+	0.433861i
$86$		0			0
$87$			6.00000i			0.643268i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$			6.00000i			0.628971i
$92$		0			0
$93$	−8.00000			−0.829561
$94$		0			0
$95$	4.00000	+	8.00000i	0.410391	+	0.820783i
$96$		0			0
$97$		−	2.00000i		−	0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$		−	2.00000i		−	0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			10.0000i			0.995037i	0.867453	+	0.497519i	$0.165755\pi$
$101$			10.0000i			0.995037i	−0.867453	+	0.497519i	$0.834245\pi$
$102$		0			0
$103$		−	16.0000i		−	1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$		−	16.0000i		−	1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$		0			0
$105$	−1.00000	−	2.00000i	−0.0975900	−	0.195180i
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$		−	16.0000i		−	1.53252i	−0.642529	−	0.766261i	$-0.722115\pi$
$109$		−	16.0000i		−	1.53252i	0.642529	−	0.766261i	$-0.277885\pi$
$110$		0			0
$111$	2.00000			0.189832
$112$		0			0
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$	−4.00000	−	8.00000i	−0.373002	−	0.746004i
$116$		0			0
$117$	−6.00000			−0.554700
$118$		0			0
$119$	−2.00000			−0.183340
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	8.00000			0.721336
$124$		0			0
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$		−	8.00000i		−	0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$		−	8.00000i		−	0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$		0			0
$129$	−6.00000			−0.528271
$130$		0			0
$131$			10.0000i			0.873704i	0.899533	+	0.436852i	$0.143907\pi$
$131$			10.0000i			0.873704i	−0.899533	+	0.436852i	$0.856093\pi$
$132$		0			0
$133$	−4.00000			−0.346844
$134$		0			0
$135$	2.00000	−	1.00000i	0.172133	−	0.0860663i
$136$		0			0
$137$			18.0000i			1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$			18.0000i			1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$		0			0
$139$			16.0000i			1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$			16.0000i			1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$		0			0
$141$		−	2.00000i		−	0.168430i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	6.00000	+	12.0000i	0.498273	+	0.996546i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$			14.0000i			1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$			14.0000i			1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$		0			0
$153$		−	2.00000i		−	0.161690i
$154$		0			0
$155$	−16.0000	+	8.00000i	−1.28515	+	0.642575i
$156$		0			0
$157$	−22.0000			−1.75579			−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000			−1.75579			−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$		0			0
$161$	4.00000			0.315244
$162$		0			0
$163$	6.00000			0.469956			0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000			0.469956			0.234978	+	0.972001i	$0.424498\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			22.0000i			1.70241i	0.524832	+	0.851206i	$0.324128\pi$
$167$			22.0000i			1.70241i	−0.524832	+	0.851206i	$0.675872\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$		0			0
$171$		−	4.00000i		−	0.305888i
$172$		0			0
$173$	24.0000			1.82469			0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000			1.82469			0.912343	+	0.409426i	$0.134271\pi$
$174$		0			0
$175$	−4.00000	−	3.00000i	−0.302372	−	0.226779i
$176$		0			0
$177$		−	6.00000i		−	0.450988i
$178$		0			0
$179$			16.0000i			1.19590i	0.801535	+	0.597948i	$0.204017\pi$
$179$			16.0000i			1.19590i	−0.801535	+	0.597948i	$0.795983\pi$
$180$		0			0
$181$		−	10.0000i		−	0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$		−	10.0000i		−	0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$		0			0
$183$		−	10.0000i		−	0.739221i
$184$		0			0
$185$	4.00000	−	2.00000i	0.294086	−	0.147043i
$186$		0			0
$187$		0			0
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$		0			0
$195$	−12.0000	+	6.00000i	−0.859338	+	0.429669i
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$	−2.00000			−0.141069
$202$		0			0
$203$	−6.00000			−0.421117
$204$		0			0
$205$	16.0000	−	8.00000i	1.11749	−	0.558744i
$206$		0			0
$207$			4.00000i			0.278019i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			20.0000i			1.37686i	0.725304	+	0.688428i	$0.241699\pi$
$211$			20.0000i			1.37686i	−0.725304	+	0.688428i	$0.758301\pi$
$212$		0			0
$213$	−8.00000			−0.548151
$214$		0			0
$215$	−12.0000	+	6.00000i	−0.818393	+	0.409197i
$216$		0			0
$217$		−	8.00000i		−	0.543075i
$218$		0			0
$219$		−	6.00000i		−	0.405442i
$220$		0			0
$221$			12.0000i			0.807207i
$222$		0			0
$223$			4.00000i			0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$			4.00000i			0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$		0			0
$225$	3.00000	−	4.00000i	0.200000	−	0.266667i
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$			14.0000i			0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$			14.0000i			0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			6.00000i			0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$			6.00000i			0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$		0			0
$235$	−2.00000	−	4.00000i	−0.130466	−	0.260931i
$236$		0			0
$237$	10.0000			0.649570
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	2.00000	−	1.00000i	0.127775	−	0.0638877i
$246$		0			0
$247$			24.0000i			1.52708i
$248$		0			0
$249$	4.00000			0.253490
$250$		0			0
$251$		−	10.0000i		−	0.631194i	−0.948893	−	0.315597i	$-0.897795\pi$
$251$		−	10.0000i		−	0.631194i	0.948893	−	0.315597i	$-0.102205\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−2.00000	−	4.00000i	−0.125245	−	0.250490i
$256$		0			0
$257$			18.0000i			1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$			18.0000i			1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$		0			0
$259$			2.00000i			0.124274i
$260$		0			0
$261$		−	6.00000i		−	0.371391i
$262$		0			0
$263$			24.0000i			1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$			24.0000i			1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$		0			0
$265$	12.0000	−	6.00000i	0.737154	−	0.368577i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			14.0000i			0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$			14.0000i			0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$		0			0
$271$	28.0000			1.70088			0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000			1.70088			0.850439	+	0.526073i	$0.176336\pi$
$272$		0			0
$273$		−	6.00000i		−	0.363137i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	18.0000			1.08152			0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000			1.08152			0.540758	+	0.841178i	$0.318138\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$	−4.00000	−	8.00000i	−0.236940	−	0.473879i
$286$		0			0
$287$			8.00000i			0.472225i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$			2.00000i			0.117242i
$292$		0			0
$293$	−16.0000			−0.934730			−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000			−0.934730			−0.467365	+	0.884064i	$0.654797\pi$
$294$		0			0
$295$	−6.00000	−	12.0000i	−0.349334	−	0.698667i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	24.0000i		−	1.38796i
$300$		0			0
$301$		−	6.00000i		−	0.345834i
$302$		0			0
$303$		−	10.0000i		−	0.574485i
$304$		0			0
$305$	−10.0000	−	20.0000i	−0.572598	−	1.14520i
$306$		0			0
$307$	−28.0000			−1.59804			−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000			−1.59804			−0.799022	+	0.601302i	$0.794649\pi$
$308$		0			0
$309$			16.0000i			0.910208i
$310$		0			0
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$		−	14.0000i		−	0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$		−	14.0000i		−	0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$		0			0
$315$	1.00000	+	2.00000i	0.0563436	+	0.112687i
$316$		0			0
$317$	18.0000			1.01098			0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000			1.01098			0.505490	+	0.862832i	$0.331312\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$	−8.00000			−0.445132
$324$		0			0
$325$	−18.0000	+	24.0000i	−0.998460	+	1.33128i
$326$		0			0
$327$			16.0000i			0.884802i
$328$		0			0
$329$	2.00000			0.110264
$330$		0			0
$331$		−	20.0000i		−	1.09930i	−0.835395	−	0.549650i	$-0.814761\pi$
$331$		−	20.0000i		−	1.09930i	0.835395	−	0.549650i	$-0.185239\pi$
$332$		0			0
$333$	−2.00000			−0.109599
$334$		0			0
$335$	−4.00000	+	2.00000i	−0.218543	+	0.109272i
$336$		0			0
$337$			8.00000i			0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$			8.00000i			0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$		0			0
$339$		−	6.00000i		−	0.325875i
$340$		0			0
$341$		0			0
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	4.00000	+	8.00000i	0.215353	+	0.430706i
$346$		0			0
$347$	12.0000			0.644194			0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000			0.644194			0.322097	+	0.946707i	$0.395612\pi$
$348$		0			0
$349$		−	6.00000i		−	0.321173i	−0.987022	−	0.160586i	$-0.948662\pi$
$349$		−	6.00000i		−	0.321173i	0.987022	−	0.160586i	$-0.0513385\pi$
$350$		0			0
$351$	6.00000			0.320256
$352$		0			0
$353$		−	14.0000i		−	0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$		−	14.0000i		−	0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$		0			0
$355$	−16.0000	+	8.00000i	−0.849192	+	0.424596i
$356$		0			0
$357$	2.00000			0.105851
$358$		0			0
$359$	20.0000			1.05556			0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000			1.05556			0.527780	+	0.849381i	$0.323025\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	−11.0000			−0.577350
$364$		0			0
$365$	−6.00000	−	12.0000i	−0.314054	−	0.628109i
$366$		0			0
$367$		−	28.0000i		−	1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$		−	28.0000i		−	1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$		0			0
$369$	−8.00000			−0.416463
$370$		0			0
$371$			6.00000i			0.311504i
$372$		0			0
$373$	−26.0000			−1.34623			−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000			−1.34623			−0.673114	+	0.739538i	$0.735044\pi$
$374$		0			0
$375$	2.00000	−	11.0000i	0.103280	−	0.568038i
$376$		0			0
$377$			36.0000i			1.85409i
$378$		0			0
$379$		−	4.00000i		−	0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$		−	4.00000i		−	0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$		0			0
$381$			8.00000i			0.409852i
$382$		0			0
$383$		−	26.0000i		−	1.32854i	−0.747494	−	0.664269i	$-0.768743\pi$
$383$		−	26.0000i		−	1.32854i	0.747494	−	0.664269i	$-0.231257\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	6.00000			0.304997
$388$		0			0
$389$		−	26.0000i		−	1.31825i	−0.752032	−	0.659126i	$-0.770926\pi$
$389$		−	26.0000i		−	1.31825i	0.752032	−	0.659126i	$-0.229074\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$		−	10.0000i		−	0.504433i
$394$		0			0
$395$	20.0000	−	10.0000i	1.00631	−	0.503155i
$396$		0			0
$397$	38.0000			1.90717			0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000			1.90717			0.953583	+	0.301131i	$0.0973643\pi$
$398$		0			0
$399$	4.00000			0.200250
$400$		0			0
$401$	2.00000			0.0998752			0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000			0.0998752			0.0499376	+	0.998752i	$0.484098\pi$
$402$		0			0
$403$	−48.0000			−2.39105
$404$		0			0
$405$	−2.00000	+	1.00000i	−0.0993808	+	0.0496904i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$		−	18.0000i		−	0.887875i
$412$		0			0
$413$	6.00000			0.295241
$414$		0			0
$415$	8.00000	−	4.00000i	0.392705	−	0.196352i
$416$		0			0
$417$		−	16.0000i		−	0.783523i
$418$		0			0
$419$		−	34.0000i		−	1.66101i	−0.557012	−	0.830504i	$-0.688052\pi$
$419$		−	34.0000i		−	1.66101i	0.557012	−	0.830504i	$-0.311948\pi$
$420$		0			0
$421$			20.0000i			0.974740i	0.873195	+	0.487370i	$0.162044\pi$
$421$			20.0000i			0.974740i	−0.873195	+	0.487370i	$0.837956\pi$
$422$		0			0
$423$			2.00000i			0.0972433i
$424$		0			0
$425$	−8.00000	−	6.00000i	−0.388057	−	0.291043i
$426$		0			0
$427$	10.0000			0.483934
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.0000			−0.578020			−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000			−0.578020			−0.289010	+	0.957326i	$0.593326\pi$
$432$		0			0
$433$			6.00000i			0.288342i	0.989553	+	0.144171i	$0.0460515\pi$
$433$			6.00000i			0.288342i	−0.989553	+	0.144171i	$0.953949\pi$
$434$		0			0
$435$	−6.00000	−	12.0000i	−0.287678	−	0.575356i
$436$		0			0
$437$	16.0000			0.765384
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	36.0000			1.71041			0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000			1.71041			0.855206	+	0.518289i	$0.173431\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	14.0000i		−	0.662177i
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	2.00000			0.0939682
$454$		0			0
$455$	−6.00000	−	12.0000i	−0.281284	−	0.562569i
$456$		0			0
$457$		−	12.0000i		−	0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$457$		−	12.0000i		−	0.561336i	0.959805	−	0.280668i	$-0.0905560\pi$
$458$		0			0
$459$			2.00000i			0.0933520i
$460$		0			0
$461$		−	30.0000i		−	1.39724i	−0.715493	−	0.698620i	$-0.753798\pi$
$461$		−	30.0000i		−	1.39724i	0.715493	−	0.698620i	$-0.246202\pi$
$462$		0			0
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$		0			0
$465$	16.0000	−	8.00000i	0.741982	−	0.370991i
$466$		0			0
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$		−	2.00000i		−	0.0923514i
$470$		0			0
$471$	22.0000			1.01371
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−16.0000	−	12.0000i	−0.734130	−	0.550598i
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	12.0000			0.547153
$482$		0			0
$483$	−4.00000			−0.182006
$484$		0			0
$485$	2.00000	+	4.00000i	0.0908153	+	0.181631i
$486$		0			0
$487$			32.0000i			1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$			32.0000i			1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$		0			0
$489$	−6.00000			−0.271329
$490$		0			0
$491$			20.0000i			0.902587i	0.892375	+	0.451294i	$0.149037\pi$
$491$			20.0000i			0.902587i	−0.892375	+	0.451294i	$0.850963\pi$
$492$		0			0
$493$	−12.0000			−0.540453
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	8.00000i		−	0.358849i
$498$		0			0
$499$		−	4.00000i		−	0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$		−	4.00000i		−	0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$		0			0
$501$		−	22.0000i		−	0.982888i
$502$		0			0
$503$		−	6.00000i		−	0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$		−	6.00000i		−	0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$		0			0
$505$	−10.0000	−	20.0000i	−0.444994	−	0.889988i
$506$		0			0
$507$	−23.0000			−1.02147
$508$		0			0
$509$			34.0000i			1.50702i	0.657434	+	0.753512i	$0.271642\pi$
$509$			34.0000i			1.50702i	−0.657434	+	0.753512i	$0.728358\pi$
$510$		0			0
$511$	6.00000			0.265424
$512$		0			0
$513$			4.00000i			0.176604i
$514$		0			0
$515$	16.0000	+	32.0000i	0.705044	+	1.41009i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−24.0000			−1.05348
$520$		0			0
$521$	−28.0000			−1.22670			−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000			−1.22670			−0.613351	+	0.789810i	$0.710179\pi$
$522$		0			0
$523$	16.0000			0.699631			0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000			0.699631			0.349816	+	0.936819i	$0.386244\pi$
$524$		0			0
$525$	4.00000	+	3.00000i	0.174574	+	0.130931i
$526$		0			0
$527$		−	16.0000i		−	0.696971i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$			6.00000i			0.260378i
$532$		0			0
$533$	48.0000			2.07911
$534$		0			0
$535$	−24.0000	+	12.0000i	−1.03761	+	0.518805i
$536$		0			0
$537$		−	16.0000i		−	0.690451i
$538$		0			0
$539$		0			0
$540$		0			0
$541$			20.0000i			0.859867i	0.902861	+	0.429934i	$0.141463\pi$
$541$			20.0000i			0.859867i	−0.902861	+	0.429934i	$0.858537\pi$
$542$		0			0
$543$			10.0000i			0.429141i
$544$		0			0
$545$	16.0000	+	32.0000i	0.685365	+	1.37073i
$546$		0			0
$547$	2.00000			0.0855138			0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000			0.0855138			0.0427569	+	0.999086i	$0.486386\pi$
$548$		0			0
$549$			10.0000i			0.426790i
$550$		0			0
$551$	−24.0000			−1.02243
$552$		0			0
$553$			10.0000i			0.425243i
$554$		0			0
$555$	−4.00000	+	2.00000i	−0.169791	+	0.0848953i
$556$		0			0
$557$	−2.00000			−0.0847427			−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000			−0.0847427			−0.0423714	+	0.999102i	$0.513491\pi$
$558$		0			0
$559$	−36.0000			−1.52264
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.00000			−0.168580			−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000			−0.168580			−0.0842900	+	0.996441i	$0.526862\pi$
$564$		0			0
$565$	−6.00000	−	12.0000i	−0.252422	−	0.504844i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−30.0000			−1.25767			−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000			−1.25767			−0.628833	+	0.777541i	$0.716467\pi$
$570$		0			0
$571$		−	20.0000i		−	0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$		−	20.0000i		−	0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$		0			0
$573$	−8.00000			−0.334205
$574$		0			0
$575$	16.0000	+	12.0000i	0.667246	+	0.500435i
$576$		0			0
$577$			38.0000i			1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$			38.0000i			1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$		0			0
$579$		−	16.0000i		−	0.664937i
$580$		0			0
$581$			4.00000i			0.165948i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	12.0000	−	6.00000i	0.496139	−	0.248069i
$586$		0			0
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		0			0
$589$		−	32.0000i		−	1.31854i
$590$		0			0
$591$	−18.0000			−0.740421
$592$		0			0
$593$		−	14.0000i		−	0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$		−	14.0000i		−	0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$		0			0
$595$	4.00000	−	2.00000i	0.163984	−	0.0819920i
$596$		0			0
$597$	20.0000			0.818546
$598$		0			0
$599$	−40.0000			−1.63436			−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000			−1.63436			−0.817178	+	0.576386i	$0.804463\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$	2.00000			0.0814463
$604$		0			0
$605$	−22.0000	+	11.0000i	−0.894427	+	0.447214i
$606$		0			0
$607$		−	8.00000i		−	0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$		−	8.00000i		−	0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$		0			0
$609$	6.00000			0.243132
$610$		0			0
$611$		−	12.0000i		−	0.485468i
$612$		0			0
$613$	−26.0000			−1.05013			−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000			−1.05013			−0.525065	+	0.851062i	$0.675959\pi$
$614$		0			0
$615$	−16.0000	+	8.00000i	−0.645182	+	0.322591i
$616$		0			0
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$		0			0
$619$		−	44.0000i		−	1.76851i	−0.467005	−	0.884255i	$-0.654667\pi$
$619$		−	44.0000i		−	1.76851i	0.467005	−	0.884255i	$-0.345333\pi$
$620$		0			0
$621$		−	4.00000i		−	0.160514i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			4.00000i			0.159490i
$630$		0			0
$631$	18.0000			0.716569			0.358284	−	0.933613i	$-0.383362\pi$
$631$	18.0000			0.716569			0.358284	+	0.933613i	$0.383362\pi$
$632$		0			0
$633$		−	20.0000i		−	0.794929i
$634$		0			0
$635$	8.00000	+	16.0000i	0.317470	+	0.634941i
$636$		0			0
$637$	6.00000			0.237729
$638$		0			0
$639$	8.00000			0.316475
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$	16.0000			0.630978			0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000			0.630978			0.315489	+	0.948929i	$0.397831\pi$
$644$		0			0
$645$	12.0000	−	6.00000i	0.472500	−	0.236250i
$646$		0			0
$647$			42.0000i			1.65119i	0.564263	+	0.825595i	$0.309160\pi$
$647$			42.0000i			1.65119i	−0.564263	+	0.825595i	$0.690840\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$			8.00000i			0.313545i
$652$		0			0
$653$	14.0000			0.547862			0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000			0.547862			0.273931	+	0.961749i	$0.411676\pi$
$654$		0			0
$655$	−10.0000	−	20.0000i	−0.390732	−	0.781465i
$656$		0			0
$657$			6.00000i			0.234082i
$658$		0			0
$659$			36.0000i			1.40236i	0.712984	+	0.701180i	$0.247343\pi$
$659$			36.0000i			1.40236i	−0.712984	+	0.701180i	$0.752657\pi$
$660$		0			0
$661$			30.0000i			1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$			30.0000i			1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$		0			0
$663$		−	12.0000i		−	0.466041i
$664$		0			0
$665$	8.00000	−	4.00000i	0.310227	−	0.155113i
$666$		0			0
$667$	24.0000			0.929284
$668$		0			0
$669$		−	4.00000i		−	0.154649i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	44.0000i		−	1.69608i	−0.529936	−	0.848038i	$-0.677784\pi$
$673$		−	44.0000i		−	1.69608i	0.529936	−	0.848038i	$-0.322216\pi$
$674$		0			0
$675$	−3.00000	+	4.00000i	−0.115470	+	0.153960i
$676$		0			0
$677$	−32.0000			−1.22986			−0.614930	−	0.788582i	$-0.710816\pi$
$677$	−32.0000			−1.22986			−0.614930	+	0.788582i	$0.710816\pi$
$678$		0			0
$679$	−2.00000			−0.0767530
$680$		0			0
$681$	−12.0000			−0.459841
$682$		0			0
$683$	−44.0000			−1.68361			−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000			−1.68361			−0.841807	+	0.539779i	$0.818508\pi$
$684$		0			0
$685$	−18.0000	−	36.0000i	−0.687745	−	1.37549i
$686$		0			0
$687$		−	14.0000i		−	0.534133i
$688$		0			0
$689$	36.0000			1.37149
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−16.0000	−	32.0000i	−0.606915	−	1.21383i
$696$		0			0
$697$			16.0000i			0.606043i
$698$		0			0
$699$		−	6.00000i		−	0.226941i
$700$		0			0
$701$		−	50.0000i		−	1.88847i	−0.329267	−	0.944237i	$-0.606802\pi$
$701$		−	50.0000i		−	1.88847i	0.329267	−	0.944237i	$-0.393198\pi$
$702$		0			0
$703$			8.00000i			0.301726i
$704$		0			0
$705$	2.00000	+	4.00000i	0.0753244	+	0.150649i
$706$		0			0
$707$	10.0000			0.376089
$708$		0			0
$709$		−	16.0000i		−	0.600893i	−0.953799	−	0.300446i	$-0.902864\pi$
$709$		−	16.0000i		−	0.600893i	0.953799	−	0.300446i	$-0.0971356\pi$
$710$		0			0
$711$	−10.0000			−0.375029
$712$		0			0
$713$			32.0000i			1.19841i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−16.0000			−0.595871
$722$		0			0
$723$	18.0000			0.669427
$724$		0			0
$725$	−24.0000	−	18.0000i	−0.891338	−	0.668503i
$726$		0			0
$727$			52.0000i			1.92857i	0.264861	+	0.964287i	$0.414674\pi$
$727$			52.0000i			1.92857i	−0.264861	+	0.964287i	$0.585326\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	12.0000i		−	0.443836i
$732$		0			0
$733$	14.0000			0.517102			0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000			0.517102			0.258551	+	0.965998i	$0.416755\pi$
$734$		0			0
$735$	−2.00000	+	1.00000i	−0.0737711	+	0.0368856i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	4.00000i		−	0.147142i	−0.997290	−	0.0735712i	$-0.976560\pi$
$739$		−	4.00000i		−	0.147142i	0.997290	−	0.0735712i	$-0.0234396\pi$
$740$		0			0
$741$		−	24.0000i		−	0.881662i
$742$		0			0
$743$			24.0000i			0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$			24.0000i			0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$		0			0
$745$	−14.0000	−	28.0000i	−0.512920	−	1.02584i
$746$		0			0
$747$	−4.00000			−0.146352
$748$		0			0
$749$		−	12.0000i		−	0.438470i
$750$		0			0
$751$	38.0000			1.38664			0.693320	−	0.720630i	$-0.256147\pi$
$751$	38.0000			1.38664			0.693320	+	0.720630i	$0.256147\pi$
$752$		0			0
$753$			10.0000i			0.364420i
$754$		0			0
$755$	4.00000	−	2.00000i	0.145575	−	0.0727875i
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−8.00000			−0.290000			−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000			−0.290000			−0.145000	+	0.989432i	$0.546318\pi$
$762$		0			0
$763$	−16.0000			−0.579239
$764$		0			0
$765$	2.00000	+	4.00000i	0.0723102	+	0.144620i
$766$		0			0
$767$		−	36.0000i		−	1.29988i
$768$		0			0
$769$	−50.0000			−1.80305			−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000			−1.80305			−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		−	18.0000i		−	0.648254i
$772$		0			0
$773$	24.0000			0.863220			0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000			0.863220			0.431610	+	0.902060i	$0.357946\pi$
$774$		0			0
$775$	24.0000	−	32.0000i	0.862105	−	1.14947i
$776$		0			0
$777$		−	2.00000i		−	0.0717496i
$778$		0			0
$779$			32.0000i			1.14652i
$780$		0			0
$781$		0			0
$782$		0			0
$783$			6.00000i			0.214423i
$784$		0			0
$785$	44.0000	−	22.0000i	1.57043	−	0.785214i
$786$		0			0
$787$	52.0000			1.85360			0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000			1.85360			0.926800	+	0.375555i	$0.122548\pi$
$788$		0			0
$789$		−	24.0000i		−	0.854423i
$790$		0			0
$791$	6.00000			0.213335
$792$		0			0
$793$		−	60.0000i		−	2.13066i
$794$		0			0
$795$	−12.0000	+	6.00000i	−0.425596	+	0.212798i
$796$		0			0
$797$	8.00000			0.283375			0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000			0.283375			0.141687	+	0.989911i	$0.454747\pi$
$798$		0			0
$799$	4.00000			0.141510
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−8.00000	+	4.00000i	−0.281963	+	0.140981i
$806$		0			0
$807$		−	14.0000i		−	0.492823i
$808$		0			0
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$			20.0000i			0.702295i	0.936320	+	0.351147i	$0.114208\pi$
$811$			20.0000i			0.702295i	−0.936320	+	0.351147i	$0.885792\pi$
$812$		0			0
$813$	−28.0000			−0.982003
$814$		0			0
$815$	−12.0000	+	6.00000i	−0.420342	+	0.210171i
$816$		0			0
$817$		−	24.0000i		−	0.839654i
$818$		0			0
$819$			6.00000i			0.209657i
$820$		0			0
$821$			10.0000i			0.349002i	0.984657	+	0.174501i	$0.0558313\pi$
$821$			10.0000i			0.349002i	−0.984657	+	0.174501i	$0.944169\pi$
$822$		0			0
$823$		−	16.0000i		−	0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$		−	16.0000i		−	0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	52.0000			1.80822			0.904109	−	0.427303i	$-0.140536\pi$
$827$	52.0000			1.80822			0.904109	+	0.427303i	$0.140536\pi$
$828$		0			0
$829$		−	46.0000i		−	1.59765i	−0.601566	−	0.798823i	$-0.705456\pi$
$829$		−	46.0000i		−	1.59765i	0.601566	−	0.798823i	$-0.294544\pi$
$830$		0			0
$831$	−18.0000			−0.624413
$832$		0			0
$833$			2.00000i			0.0692959i
$834$		0			0
$835$	−22.0000	−	44.0000i	−0.761341	−	1.52268i
$836$		0			0
$837$	−8.00000			−0.276520
$838$		0			0
$839$	−40.0000			−1.38095			−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000			−1.38095			−0.690477	+	0.723355i	$0.742599\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$	18.0000			0.619953
$844$		0			0
$845$	−46.0000	+	23.0000i	−1.58245	+	0.791224i
$846$		0			0
$847$		−	11.0000i		−	0.377964i
$848$		0			0
$849$	4.00000			0.137280
$850$		0			0
$851$		−	8.00000i		−	0.274236i
$852$		0			0
$853$	−26.0000			−0.890223			−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000			−0.890223			−0.445112	+	0.895475i	$0.646836\pi$
$854$		0			0
$855$	4.00000	+	8.00000i	0.136797	+	0.273594i
$856$		0			0
$857$			18.0000i			0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$			18.0000i			0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$		0			0
$859$			36.0000i			1.22830i	0.789188	+	0.614152i	$0.210502\pi$
$859$			36.0000i			1.22830i	−0.789188	+	0.614152i	$0.789498\pi$
$860$		0			0
$861$		−	8.00000i		−	0.272639i
$862$		0			0
$863$		−	36.0000i		−	1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$		−	36.0000i		−	1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$		0			0
$865$	−48.0000	+	24.0000i	−1.63205	+	0.816024i
$866$		0			0
$867$	−13.0000			−0.441503
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$		−	2.00000i		−	0.0676897i
$874$		0			0
$875$	11.0000	+	2.00000i	0.371868	+	0.0676123i
$876$		0			0
$877$	18.0000			0.607817			0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000			0.607817			0.303908	+	0.952701i	$0.401708\pi$
$878$		0			0
$879$	16.0000			0.539667
$880$		0			0
$881$	−28.0000			−0.943344			−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000			−0.943344			−0.471672	+	0.881774i	$0.656349\pi$
$882$		0			0
$883$	26.0000			0.874970			0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000			0.874970			0.437485	+	0.899226i	$0.355869\pi$
$884$		0			0
$885$	6.00000	+	12.0000i	0.201688	+	0.403376i
$886$		0			0
$887$			42.0000i			1.41022i	0.709097	+	0.705111i	$0.249103\pi$
$887$			42.0000i			1.41022i	−0.709097	+	0.705111i	$0.750897\pi$
$888$		0			0
$889$	−8.00000			−0.268311
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.00000			0.267710
$894$		0			0
$895$	−16.0000	−	32.0000i	−0.534821	−	1.06964i
$896$		0			0
$897$			24.0000i			0.801337i
$898$		0			0
$899$		−	48.0000i		−	1.60089i
$900$		0			0
$901$			12.0000i			0.399778i
$902$		0			0
$903$			6.00000i			0.199667i
$904$		0			0
$905$	10.0000	+	20.0000i	0.332411	+	0.664822i
$906$		0			0
$907$	−18.0000			−0.597680			−0.298840	−	0.954303i	$-0.596600\pi$
$907$	−18.0000			−0.597680			−0.298840	+	0.954303i	$0.596600\pi$
$908$		0			0
$909$			10.0000i			0.331679i
$910$		0			0
$911$	28.0000			0.927681			0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000			0.927681			0.463841	+	0.885919i	$0.346471\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	10.0000	+	20.0000i	0.330590	+	0.661180i
$916$		0			0
$917$	10.0000			0.330229
$918$		0			0
$919$	−50.0000			−1.64935			−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000			−1.64935			−0.824674	+	0.565608i	$0.808641\pi$
$920$		0			0
$921$	28.0000			0.922631
$922$		0			0
$923$	−48.0000			−1.57994
$924$		0			0
$925$	−6.00000	+	8.00000i	−0.197279	+	0.263038i
$926$		0			0
$927$		−	16.0000i		−	0.525509i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$			4.00000i			0.131095i
$932$		0			0
$933$	−8.00000			−0.261908
$934$		0			0
$935$		0			0
$936$		0			0
$937$			38.0000i			1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$			38.0000i			1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$		0			0
$939$			14.0000i			0.456873i
$940$		0			0
$941$		−	50.0000i		−	1.62995i	−0.579494	−	0.814977i	$-0.696750\pi$
$941$		−	50.0000i		−	1.62995i	0.579494	−	0.814977i	$-0.303250\pi$
$942$		0			0
$943$		−	32.0000i		−	1.04206i
$944$		0			0
$945$	−1.00000	−	2.00000i	−0.0325300	−	0.0650600i
$946$		0			0
$947$	12.0000			0.389948			0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000			0.389948			0.194974	+	0.980808i	$0.437538\pi$
$948$		0			0
$949$		−	36.0000i		−	1.16861i
$950$		0			0
$951$	−18.0000			−0.583690
$952$		0			0
$953$		−	34.0000i		−	1.10137i	−0.834714	−	0.550684i	$-0.814367\pi$
$953$		−	34.0000i		−	1.10137i	0.834714	−	0.550684i	$-0.185633\pi$
$954$		0			0
$955$	−16.0000	+	8.00000i	−0.517748	+	0.258874i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	18.0000			0.581250
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	12.0000			0.386695
$964$		0			0
$965$	−16.0000	−	32.0000i	−0.515058	−	1.03012i
$966$		0			0
$967$		−	8.00000i		−	0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$		−	8.00000i		−	0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$		0			0
$969$	8.00000			0.256997
$970$		0			0
$971$		−	10.0000i		−	0.320915i	−0.987043	−	0.160458i	$-0.948703\pi$
$971$		−	10.0000i		−	0.320915i	0.987043	−	0.160458i	$-0.0512970\pi$
$972$		0			0
$973$	16.0000			0.512936
$974$		0			0
$975$	18.0000	−	24.0000i	0.576461	−	0.768615i
$976$		0			0
$977$			18.0000i			0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$			18.0000i			0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	16.0000i		−	0.510841i
$982$		0			0
$983$			14.0000i			0.446531i	0.974758	+	0.223265i	$0.0716716\pi$
$983$			14.0000i			0.446531i	−0.974758	+	0.223265i	$0.928328\pi$
$984$		0			0
$985$	−36.0000	+	18.0000i	−1.14706	+	0.573528i
$986$		0			0
$987$	−2.00000			−0.0636607
$988$		0			0
$989$			24.0000i			0.763156i
$990$		0			0
$991$	−22.0000			−0.698853			−0.349427	−	0.936964i	$-0.613624\pi$
$991$	−22.0000			−0.698853			−0.349427	+	0.936964i	$0.613624\pi$
$992$		0			0
$993$			20.0000i			0.634681i
$994$		0			0
$995$	40.0000	−	20.0000i	1.26809	−	0.634043i
$996$		0			0
$997$	−2.00000			−0.0633406			−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000			−0.0633406			−0.0316703	+	0.999498i	$0.510083\pi$
$998$		0			0
$999$	2.00000			0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.j.a.1009.2		2
4.3	odd	2		840.2.j.b.589.1	✓	2
5.4	even	2		3360.2.j.d.1009.2		2
8.3	odd	2		840.2.j.c.589.1	yes	2
8.5	even	2		3360.2.j.d.1009.1		2
20.19	odd	2		840.2.j.c.589.2	yes	2
40.19	odd	2		840.2.j.b.589.2	yes	2
40.29	even	2	inner	3360.2.j.a.1009.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.j.b.589.1	✓	2	4.3	odd	2
840.2.j.b.589.2	yes	2	40.19	odd	2
840.2.j.c.589.1	yes	2	8.3	odd	2
840.2.j.c.589.2	yes	2	20.19	odd	2
3360.2.j.a.1009.1		2	40.29	even	2	inner
3360.2.j.a.1009.2		2	1.1	even	1	trivial
3360.2.j.d.1009.1		2	8.5	even	2
3360.2.j.d.1009.2		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +1.00000 q^{9} -6.00000 q^{13} +(2.00000 - 1.00000i) q^{15} -2.00000i q^{17} -4.00000i q^{19} +1.00000i q^{21} +4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -1.00000 q^{27} -6.00000i q^{29} +8.00000 q^{31} +(1.00000 + 2.00000i) q^{35} -2.00000 q^{37} +6.00000 q^{39} -8.00000 q^{41} +6.00000 q^{43} +(-2.00000 + 1.00000i) q^{45} +2.00000i q^{47} -1.00000 q^{49} +2.00000i q^{51} -6.00000 q^{53} +4.00000i q^{57} +6.00000i q^{59} +10.0000i q^{61} -1.00000i q^{63} +(12.0000 - 6.00000i) q^{65} +2.00000 q^{67} -4.00000i q^{69} +8.00000 q^{71} +6.00000i q^{73} +(-3.00000 + 4.00000i) q^{75} -10.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +(2.00000 + 4.00000i) q^{85} +6.00000i q^{87} +6.00000i q^{91} -8.00000 q^{93} +(4.00000 + 8.00000i) q^{95} -2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} - 12 q^{13} + 4 q^{15} + 6 q^{25} - 2 q^{27} + 16 q^{31} + 2 q^{35} - 4 q^{37} + 12 q^{39} - 16 q^{41} + 12 q^{43} - 4 q^{45} - 2 q^{49} - 12 q^{53} + 24 q^{65} + 4 q^{67} + 16 q^{71} - 6 q^{75} - 20 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} - 16 q^{93} + 8 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 - 12 * q^13 + 4 * q^15 + 6 * q^25 - 2 * q^27 + 16 * q^31 + 2 * q^35 - 4 * q^37 + 12 * q^39 - 16 * q^41 + 12 * q^43 - 4 * q^45 - 2 * q^49 - 12 * q^53 + 24 * q^65 + 4 * q^67 + 16 * q^71 - 6 * q^75 - 20 * q^79 + 2 * q^81 - 8 * q^83 + 4 * q^85 - 16 * q^93 + 8 * q^95

Introduction

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