LMFDB - Embedded newform 5040.2.t.y.1009.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(1009,5040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5040, 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("5040.1009");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.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:	$40.2446026187$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.2
Root			$1.32001 + 1.32001i$ of defining polynomial
Character	$\chi$	$=$	5040.1009
Dual form			5040.2.t.y.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.32001 + 1.80487i) q^{5} +1.00000i q^{7} +O(q^{10})$	$q+(-1.32001 + 1.80487i) q^{5} +1.00000i q^{7} +2.48486 q^{11} +4.15516i q^{13} -5.76491i q^{17} -1.60975 q^{19} -7.28005i q^{23} +(-1.51514 - 4.76491i) q^{25} +1.45459 q^{29} +2.24977 q^{31} +(-1.80487 - 1.32001i) q^{35} -6.00000i q^{37} -11.2800 q^{41} -5.28005i q^{43} +3.45459i q^{47} -1.00000 q^{49} -9.21949i q^{53} +(-3.28005 + 4.48486i) q^{55} +5.92007 q^{59} +5.35998 q^{61} +(-7.49954 - 5.48486i) q^{65} +7.52982i q^{67} -4.24977 q^{71} -7.28005i q^{73} +2.48486i q^{77} +16.9844 q^{79} +10.1093i q^{83} +(10.4049 + 7.60975i) q^{85} -11.4693 q^{89} -4.15516 q^{91} +(2.12489 - 2.90539i) q^{95} -2.73463i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 14 q^{11} + 8 q^{19} - 10 q^{25} + 6 q^{29} - 20 q^{31} - 2 q^{35} - 36 q^{41} - 6 q^{49} + 12 q^{55} - 12 q^{59} + 48 q^{61} + 22 q^{65} + 8 q^{71} + 34 q^{79} + 14 q^{85} - 10 q^{91} - 4 q^{95}+O(q^{100}) $ 6 * q + 14 * q^11 + 8 * q^19 - 10 * q^25 + 6 * q^29 - 20 * q^31 - 2 * q^35 - 36 * q^41 - 6 * q^49 + 12 * q^55 - 12 * q^59 + 48 * q^61 + 22 * q^65 + 8 * q^71 + 34 * q^79 + 14 * q^85 - 10 * q^91 - 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times$.

$n$	$2017$	$2801$	$3151$	$3601$	$3781$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.32001	+	1.80487i	−0.590327	+	0.807164i
$5$	−1.32001	+	1.80487i	−0.590327	+	0.807164i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.48486			0.749214			0.374607	−	0.927184i	$-0.377778\pi$
$11$	2.48486			0.749214			0.374607	+	0.927184i	$0.377778\pi$
$12$		0			0
$13$			4.15516i			1.15243i	0.817297	+	0.576217i	$0.195472\pi$
$13$			4.15516i			1.15243i	−0.817297	+	0.576217i	$0.804528\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.76491i		−	1.39820i	−0.715026	−	0.699098i	$-0.753585\pi$
$17$		−	5.76491i		−	1.39820i	0.715026	−	0.699098i	$-0.246415\pi$
$18$		0			0
$19$	−1.60975			−0.369301			−0.184651	−	0.982804i	$-0.559115\pi$
$19$	−1.60975			−0.369301			−0.184651	+	0.982804i	$0.559115\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.28005i		−	1.51799i	−0.651094	−	0.758997i	$-0.725690\pi$
$23$		−	7.28005i		−	1.51799i	0.651094	−	0.758997i	$-0.274310\pi$
$24$		0			0
$25$	−1.51514	−	4.76491i	−0.303028	−	0.952982i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.45459			0.270110			0.135055	−	0.990838i	$-0.456879\pi$
$29$	1.45459			0.270110			0.135055	+	0.990838i	$0.456879\pi$
$30$		0			0
$31$	2.24977			0.404071			0.202035	−	0.979378i	$-0.435244\pi$
$31$	2.24977			0.404071			0.202035	+	0.979378i	$0.435244\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.80487	−	1.32001i	−0.305079	−	0.223123i
$36$		0			0
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11.2800			−1.76165			−0.880824	−	0.473444i	$-0.843010\pi$
$41$	−11.2800			−1.76165			−0.880824	+	0.473444i	$0.843010\pi$
$42$		0			0
$43$		−	5.28005i		−	0.805200i	−0.915376	−	0.402600i	$-0.868107\pi$
$43$		−	5.28005i		−	0.805200i	0.915376	−	0.402600i	$-0.131893\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.45459i			0.503903i	0.967740	+	0.251952i	$0.0810724\pi$
$47$			3.45459i			0.503903i	−0.967740	+	0.251952i	$0.918928\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.21949i		−	1.26639i	−0.773990	−	0.633197i	$-0.781742\pi$
$53$		−	9.21949i		−	1.26639i	0.773990	−	0.633197i	$-0.218258\pi$
$54$		0			0
$55$	−3.28005	+	4.48486i	−0.442281	+	0.604739i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.92007			0.770728			0.385364	−	0.922765i	$-0.374076\pi$
$59$	5.92007			0.770728			0.385364	+	0.922765i	$0.374076\pi$
$60$		0			0
$61$	5.35998			0.686275			0.343137	−	0.939285i	$-0.388510\pi$
$61$	5.35998			0.686275			0.343137	+	0.939285i	$0.388510\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.49954	−	5.48486i	−0.930204	−	0.680313i
$66$		0			0
$67$			7.52982i			0.919914i	0.887941	+	0.459957i	$0.152135\pi$
$67$			7.52982i			0.919914i	−0.887941	+	0.459957i	$0.847865\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.24977			−0.504355			−0.252178	−	0.967681i	$-0.581147\pi$
$71$	−4.24977			−0.504355			−0.252178	+	0.967681i	$0.581147\pi$
$72$		0			0
$73$		−	7.28005i		−	0.852065i	−0.904708	−	0.426033i	$-0.859911\pi$
$73$		−	7.28005i		−	0.852065i	0.904708	−	0.426033i	$-0.140089\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.48486i			0.283176i
$78$		0			0
$79$	16.9844			1.91089			0.955447	−	0.295162i	$-0.0953735\pi$
$79$	16.9844			1.91089			0.955447	+	0.295162i	$0.0953735\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			10.1093i			1.10964i	0.831971	+	0.554819i	$0.187213\pi$
$83$			10.1093i			1.10964i	−0.831971	+	0.554819i	$0.812787\pi$
$84$		0			0
$85$	10.4049	+	7.60975i	1.12857	+	0.825393i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.4693			−1.21574			−0.607870	−	0.794037i	$-0.707976\pi$
$89$	−11.4693			−1.21574			−0.607870	+	0.794037i	$0.707976\pi$
$90$		0			0
$91$	−4.15516			−0.435579
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.12489	−	2.90539i	0.218009	−	0.298087i
$96$		0			0
$97$		−	2.73463i		−	0.277660i	−0.990316	−	0.138830i	$-0.955666\pi$
$97$		−	2.73463i		−	0.277660i	0.990316	−	0.138830i	$-0.0443341\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.57947			0.455674			0.227837	−	0.973699i	$-0.426835\pi$
$101$	4.57947			0.455674			0.227837	+	0.973699i	$0.426835\pi$
$102$		0			0
$103$		−	2.48486i		−	0.244841i	−0.992478	−	0.122420i	$-0.960934\pi$
$103$		−	2.48486i		−	0.244841i	0.992478	−	0.122420i	$-0.0390656\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	9.95413			0.953433			0.476716	−	0.879057i	$-0.341827\pi$
$109$	9.95413			0.953433			0.476716	+	0.879057i	$0.341827\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	18.4995i		−	1.74029i	−0.492795	−	0.870145i	$-0.664025\pi$
$113$		−	18.4995i		−	1.74029i	0.492795	−	0.870145i	$-0.335975\pi$
$114$		0			0
$115$	13.1396	+	9.60975i	1.22527	+	0.896114i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.76491			0.528468
$120$		0			0
$121$	−4.82546			−0.438678
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.6001	+	3.55510i	0.948098	+	0.317978i
$126$		0			0
$127$			7.15894i			0.635253i	0.948216	+	0.317627i	$0.102886\pi$
$127$			7.15894i			0.635253i	−0.948216	+	0.317627i	$0.897114\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.85952			0.686689			0.343345	−	0.939209i	$-0.388440\pi$
$131$	7.85952			0.686689			0.343345	+	0.939209i	$0.388440\pi$
$132$		0			0
$133$		−	1.60975i		−	0.139583i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			10.5601i			0.902210i	0.892471	+	0.451105i	$0.148970\pi$
$137$			10.5601i			0.902210i	−0.892471	+	0.451105i	$0.851030\pi$
$138$		0			0
$139$	9.79897			0.831137			0.415569	−	0.909562i	$-0.363583\pi$
$139$	9.79897			0.831137			0.415569	+	0.909562i	$0.363583\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			10.3250i			0.863420i
$144$		0			0
$145$	−1.92007	+	2.62534i	−0.159453	+	0.218023i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.49954			0.204770			0.102385	−	0.994745i	$-0.467353\pi$
$149$	2.49954			0.204770			0.102385	+	0.994745i	$0.467353\pi$
$150$		0			0
$151$	3.76491			0.306384			0.153192	−	0.988196i	$-0.451045\pi$
$151$	3.76491			0.306384			0.153192	+	0.988196i	$0.451045\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.96972	+	4.06055i	−0.238534	+	0.326151i
$156$		0			0
$157$		−	5.67030i		−	0.452539i	−0.974065	−	0.226270i	$-0.927347\pi$
$157$		−	5.67030i		−	0.452539i	0.974065	−	0.226270i	$-0.0726530\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.28005			0.573748
$162$		0			0
$163$		−	10.2498i		−	0.802824i	−0.915898	−	0.401412i	$-0.868520\pi$
$163$		−	10.2498i		−	0.802824i	0.915898	−	0.401412i	$-0.131480\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2.95504i			0.228668i	0.993442	+	0.114334i	$0.0364734\pi$
$167$			2.95504i			0.228668i	−0.993442	+	0.114334i	$0.963527\pi$
$168$		0			0
$169$	−4.26537			−0.328105
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	22.4049i		−	1.70342i	−0.524017	−	0.851708i	$-0.675567\pi$
$173$		−	22.4049i		−	1.70342i	0.524017	−	0.851708i	$-0.324433\pi$
$174$		0			0
$175$	4.76491	−	1.51514i	0.360193	−	0.114534i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−14.5601			−1.08827			−0.544136	−	0.838997i	$-0.683143\pi$
$179$	−14.5601			−1.08827			−0.544136	+	0.838997i	$0.683143\pi$
$180$		0			0
$181$	−9.67030			−0.718788			−0.359394	−	0.933186i	$-0.617017\pi$
$181$	−9.67030			−0.718788			−0.359394	+	0.933186i	$0.617017\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.8292	+	7.92007i	0.796182	+	0.582295i
$186$		0			0
$187$		−	14.3250i		−	1.04755i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.01468			0.579922			0.289961	−	0.957038i	$-0.406358\pi$
$191$	8.01468			0.579922			0.289961	+	0.957038i	$0.406358\pi$
$192$		0			0
$193$			3.46927i			0.249723i	0.992174	+	0.124862i	$0.0398487\pi$
$193$			3.46927i			0.249723i	−0.992174	+	0.124862i	$0.960151\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	16.2791i		−	1.15984i	−0.814673	−	0.579920i	$-0.803084\pi$
$197$		−	16.2791i		−	1.15984i	0.814673	−	0.579920i	$-0.196916\pi$
$198$		0			0
$199$	26.8704			1.90479			0.952397	−	0.304862i	$-0.0986102\pi$
$199$	26.8704			1.90479			0.952397	+	0.304862i	$0.0986102\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			1.45459i			0.102092i
$204$		0			0
$205$	14.8898	−	20.3591i	1.03995	−	1.42194i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	−17.2342			−1.18645			−0.593225	−	0.805037i	$-0.702145\pi$
$211$	−17.2342			−1.18645			−0.593225	+	0.805037i	$0.702145\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	9.52982	+	6.96972i	0.649928	+	0.475331i
$216$		0			0
$217$			2.24977i			0.152724i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	23.9541			1.61133
$222$		0			0
$223$			0.235091i			0.0157429i	0.999969	+	0.00787143i	$0.00250558\pi$
$223$			0.235091i			0.0157429i	−0.999969	+	0.00787143i	$0.997494\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			0.564792i			0.0374865i	0.999824	+	0.0187433i	$0.00596652\pi$
$227$			0.564792i			0.0374865i	−0.999824	+	0.0187433i	$0.994033\pi$
$228$		0			0
$229$	−7.11021			−0.469856			−0.234928	−	0.972013i	$-0.575485\pi$
$229$	−7.11021			−0.469856			−0.234928	+	0.972013i	$0.575485\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	8.49954i		−	0.556823i	−0.960462	−	0.278412i	$-0.910192\pi$
$233$		−	8.49954i		−	0.556823i	0.960462	−	0.278412i	$-0.0898080\pi$
$234$		0			0
$235$	−6.23509	−	4.56009i	−0.406732	−	0.297468i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.26537			0.469958			0.234979	−	0.972001i	$-0.424498\pi$
$239$	7.26537			0.469958			0.234979	+	0.972001i	$0.424498\pi$
$240$		0			0
$241$	7.28005			0.468949			0.234475	−	0.972122i	$-0.424663\pi$
$241$	7.28005			0.468949			0.234475	+	0.972122i	$0.424663\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.32001	−	1.80487i	0.0843325	−	0.115309i
$246$		0			0
$247$		−	6.68876i		−	0.425596i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	22.9192			1.44664			0.723322	−	0.690511i	$-0.242614\pi$
$251$	22.9192			1.44664			0.723322	+	0.690511i	$0.242614\pi$
$252$		0			0
$253$		−	18.0899i		−	1.13730i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			0.719953i			0.0449094i	0.999748	+	0.0224547i	$0.00714816\pi$
$257$			0.719953i			0.0449094i	−0.999748	+	0.0224547i	$0.992852\pi$
$258$		0			0
$259$	6.00000			0.372822
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	18.5601i		−	1.14446i	−0.820092	−	0.572232i	$-0.806078\pi$
$263$		−	18.5601i		−	1.14446i	0.820092	−	0.572232i	$-0.193922\pi$
$264$		0			0
$265$	16.6400	+	12.1698i	1.02219	+	0.747587i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−22.1698			−1.35172			−0.675860	−	0.737030i	$-0.736227\pi$
$269$	−22.1698			−1.35172			−0.675860	+	0.737030i	$0.736227\pi$
$270$		0			0
$271$	7.87890			0.478609			0.239304	−	0.970945i	$-0.423081\pi$
$271$	7.87890			0.478609			0.239304	+	0.970945i	$0.423081\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.76491	−	11.8401i	−0.227033	−	0.713987i
$276$		0			0
$277$			2.78051i			0.167064i	0.996505	+	0.0835322i	$0.0266201\pi$
$277$			2.78051i			0.167064i	−0.996505	+	0.0835322i	$0.973380\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	23.7044			1.41408			0.707042	−	0.707172i	$-0.250029\pi$
$281$	23.7044			1.41408			0.707042	+	0.707172i	$0.250029\pi$
$282$		0			0
$283$		−	26.8439i		−	1.59571i	−0.602852	−	0.797853i	$-0.705969\pi$
$283$		−	26.8439i		−	1.59571i	0.602852	−	0.797853i	$-0.294031\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	11.2800i		−	0.665840i
$288$		0			0
$289$	−16.2342			−0.954951
$290$		0			0
$291$		0			0
$292$		0			0
$293$			9.93475i			0.580394i	0.956967	+	0.290197i	$0.0937209\pi$
$293$			9.93475i			0.580394i	−0.956967	+	0.290197i	$0.906279\pi$
$294$		0			0
$295$	−7.81456	+	10.6850i	−0.454981	+	0.622104i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	30.2498			1.74939
$300$		0			0
$301$	5.28005			0.304337
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.07523	+	9.67408i	−0.405127	+	0.553936i
$306$		0			0
$307$		−	32.0946i		−	1.83174i	−0.401479	−	0.915868i	$-0.631504\pi$
$307$		−	32.0946i		−	1.83174i	0.401479	−	0.915868i	$-0.368496\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.0606			−0.570482			−0.285241	−	0.958456i	$-0.592074\pi$
$311$	−10.0606			−0.570482			−0.285241	+	0.958456i	$0.592074\pi$
$312$		0			0
$313$			20.8245i			1.17707i	0.808471	+	0.588536i	$0.200296\pi$
$313$			20.8245i			1.17707i	−0.808471	+	0.588536i	$0.799704\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			15.7502i			0.884621i	0.896862	+	0.442311i	$0.145841\pi$
$317$			15.7502i			0.884621i	−0.896862	+	0.442311i	$0.854159\pi$
$318$		0			0
$319$	3.61445			0.202370
$320$		0			0
$321$		0			0
$322$		0			0
$323$			9.28005i			0.516356i
$324$		0			0
$325$	19.7990	−	6.29564i	1.09825	−	0.349219i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.45459			−0.190457
$330$		0			0
$331$	25.7796			1.41697			0.708487	−	0.705724i	$-0.249378\pi$
$331$	25.7796			1.41697			0.708487	+	0.705724i	$0.249378\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−13.5904	−	9.93945i	−0.742521	−	0.543050i
$336$		0			0
$337$		−	0.0605522i		−	0.00329849i	−0.999999	−	0.00164924i	$-0.999475\pi$
$337$		−	0.0605522i		−	0.00329849i	0.999999	−	0.00164924i	$-0.000524971\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5.59037			0.302736
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			20.9310i			1.12363i	0.827262	+	0.561817i	$0.189897\pi$
$347$			20.9310i			1.12363i	−0.827262	+	0.561817i	$0.810103\pi$
$348$		0			0
$349$	−5.48108			−0.293396			−0.146698	−	0.989181i	$-0.546864\pi$
$349$	−5.48108			−0.293396			−0.146698	+	0.989181i	$0.546864\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			26.9239i			1.43301i	0.697581	+	0.716506i	$0.254260\pi$
$353$			26.9239i			1.43301i	−0.697581	+	0.716506i	$0.745740\pi$
$354$		0			0
$355$	5.60975	−	7.67030i	0.297734	−	0.407097i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.06055			−0.319864			−0.159932	−	0.987128i	$-0.551127\pi$
$359$	−6.06055			−0.319864			−0.159932	+	0.987128i	$0.551127\pi$
$360$		0			0
$361$	−16.4087			−0.863616
$362$		0			0
$363$		0			0
$364$		0			0
$365$	13.1396	+	9.60975i	0.687756	+	0.502997i
$366$		0			0
$367$			30.7034i			1.60271i	0.598191	+	0.801353i	$0.295886\pi$
$367$			30.7034i			1.60271i	−0.598191	+	0.801353i	$0.704114\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.21949			0.478652
$372$		0			0
$373$		−	10.8099i		−	0.559714i	−0.960042	−	0.279857i	$-0.909713\pi$
$373$		−	10.8099i		−	0.559714i	0.960042	−	0.279857i	$-0.0902870\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6.04404i			0.311284i
$378$		0			0
$379$	−9.28005			−0.476684			−0.238342	−	0.971181i	$-0.576604\pi$
$379$	−9.28005			−0.476684			−0.238342	+	0.971181i	$0.576604\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	5.28005i		−	0.269798i	−0.990859	−	0.134899i	$-0.956929\pi$
$383$		−	5.28005i		−	0.269798i	0.990859	−	0.134899i	$-0.0430710\pi$
$384$		0			0
$385$	−4.48486	−	3.28005i	−0.228570	−	0.167167i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.48395			0.278047			0.139024	−	0.990289i	$-0.455604\pi$
$389$	5.48395			0.278047			0.139024	+	0.990289i	$0.455604\pi$
$390$		0			0
$391$	−41.9688			−2.12245
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−22.4196	+	30.6547i	−1.12805	+	1.54241i
$396$		0			0
$397$		−	6.40493i		−	0.321454i	−0.986999	−	0.160727i	$-0.948616\pi$
$397$		−	6.40493i		−	0.321454i	0.986999	−	0.160727i	$-0.0513839\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.20482			0.0601656			0.0300828	−	0.999547i	$-0.490423\pi$
$401$	1.20482			0.0601656			0.0300828	+	0.999547i	$0.490423\pi$
$402$		0			0
$403$			9.34816i			0.465665i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	14.9092i		−	0.739020i
$408$		0			0
$409$	2.31032			0.114238			0.0571191	−	0.998367i	$-0.481809\pi$
$409$	2.31032			0.114238			0.0571191	+	0.998367i	$0.481809\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			5.92007i			0.291308i
$414$		0			0
$415$	−18.2460	−	13.3444i	−0.895660	−	0.655050i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	19.7384			0.964285			0.482142	−	0.876093i	$-0.339859\pi$
$419$	19.7384			0.964285			0.482142	+	0.876093i	$0.339859\pi$
$420$		0			0
$421$	30.1433			1.46910			0.734548	−	0.678556i	$-0.237394\pi$
$421$	30.1433			1.46910			0.734548	+	0.678556i	$0.237394\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−27.4693	+	8.73463i	−1.33246	+	0.423692i
$426$		0			0
$427$			5.35998i			0.259387i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.61353			−0.222226			−0.111113	−	0.993808i	$-0.535442\pi$
$431$	−4.61353			−0.222226			−0.111113	+	0.993808i	$0.535442\pi$
$432$		0			0
$433$		−	11.4399i		−	0.549767i	−0.961478	−	0.274883i	$-0.911361\pi$
$433$		−	11.4399i		−	0.549767i	0.961478	−	0.274883i	$-0.0886393\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			11.7190i			0.560598i
$438$		0			0
$439$	12.0294			0.574130			0.287065	−	0.957911i	$-0.407320\pi$
$439$	12.0294			0.574130			0.287065	+	0.957911i	$0.407320\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	14.4390i		−	0.686017i	−0.939332	−	0.343009i	$-0.888554\pi$
$443$		−	14.4390i		−	0.686017i	0.939332	−	0.343009i	$-0.111446\pi$
$444$		0			0
$445$	15.1396	−	20.7006i	0.717684	−	0.981301i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	19.2342			0.907717			0.453858	−	0.891074i	$-0.350047\pi$
$449$	19.2342			0.907717			0.453858	+	0.891074i	$0.350047\pi$
$450$		0			0
$451$	−28.0294			−1.31985
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.48486	−	7.49954i	0.257134	−	0.351584i
$456$		0			0
$457$		−	34.8780i		−	1.63152i	−0.578388	−	0.815762i	$-0.696318\pi$
$457$		−	34.8780i		−	1.63152i	0.578388	−	0.815762i	$-0.303682\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.2607			0.710760			0.355380	−	0.934722i	$-0.384351\pi$
$461$	15.2607			0.710760			0.355380	+	0.934722i	$0.384351\pi$
$462$		0			0
$463$		−	24.9991i		−	1.16181i	−0.813973	−	0.580903i	$-0.802700\pi$
$463$		−	24.9991i		−	1.16181i	0.813973	−	0.580903i	$-0.197300\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	5.06433i		−	0.234349i	−0.993111	−	0.117175i	$-0.962616\pi$
$467$		−	5.06433i		−	0.234349i	0.993111	−	0.117175i	$-0.0373837\pi$
$468$		0			0
$469$	−7.52982			−0.347695
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	13.1202i		−	0.603267i
$474$		0			0
$475$	2.43899	+	7.67030i	0.111909	+	0.351937i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−41.8089			−1.91030			−0.955150	−	0.296123i	$-0.904306\pi$
$479$	−41.8089			−1.91030			−0.955150	+	0.296123i	$0.904306\pi$
$480$		0			0
$481$	24.9310			1.13675
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.93567	+	3.60975i	0.224117	+	0.163910i
$486$		0			0
$487$			21.3406i			0.967035i	0.875335	+	0.483517i	$0.160641\pi$
$487$			21.3406i			0.967035i	−0.875335	+	0.483517i	$0.839359\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8.35620			0.377110			0.188555	−	0.982063i	$-0.439620\pi$
$491$	8.35620			0.377110			0.188555	+	0.982063i	$0.439620\pi$
$492$		0			0
$493$		−	8.38555i		−	0.377666i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	4.24977i		−	0.190628i
$498$		0			0
$499$	−38.8539			−1.73934			−0.869670	−	0.493634i	$-0.835668\pi$
$499$	−38.8539			−1.73934			−0.869670	+	0.493634i	$0.835668\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	17.7044i		−	0.789398i	−0.918810	−	0.394699i	$-0.870849\pi$
$503$		−	17.7044i		−	0.789398i	0.918810	−	0.394699i	$-0.129151\pi$
$504$		0			0
$505$	−6.04496	+	8.26537i	−0.268997	+	0.367804i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.11021			0.315154			0.157577	−	0.987507i	$-0.449632\pi$
$509$	7.11021			0.315154			0.157577	+	0.987507i	$0.449632\pi$
$510$		0			0
$511$	7.28005			0.322050
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.48486	+	3.28005i	0.197627	+	0.144536i
$516$		0			0
$517$			8.58417i			0.377531i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−18.1892			−0.796884			−0.398442	−	0.917194i	$-0.630449\pi$
$521$	−18.1892			−0.796884			−0.398442	+	0.917194i	$0.630449\pi$
$522$		0			0
$523$			13.9882i			0.611661i	0.952086	+	0.305830i	$0.0989340\pi$
$523$			13.9882i			0.611661i	−0.952086	+	0.305830i	$0.901066\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	12.9697i		−	0.564970i
$528$		0			0
$529$	−29.9991			−1.30431
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	46.8704i		−	2.03018i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.48486			−0.107031
$540$		0			0
$541$	−20.8245			−0.895317			−0.447659	−	0.894205i	$-0.647742\pi$
$541$	−20.8245			−0.895317			−0.447659	+	0.894205i	$0.647742\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−13.1396	+	17.9659i	−0.562837	+	0.769576i
$546$		0			0
$547$		−	3.09839i		−	0.132478i	−0.997804	−	0.0662388i	$-0.978900\pi$
$547$		−	3.09839i		−	0.132478i	0.997804	−	0.0662388i	$-0.0210999\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.34152			−0.0997519
$552$		0			0
$553$			16.9844i			0.722250i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			18.8099i			0.797000i	0.917168	+	0.398500i	$0.130469\pi$
$557$			18.8099i			0.797000i	−0.917168	+	0.398500i	$0.869531\pi$
$558$		0			0
$559$	21.9394			0.927940
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	28.9503i		−	1.22011i	−0.792358	−	0.610056i	$-0.791147\pi$
$563$		−	28.9503i		−	1.22011i	0.792358	−	0.610056i	$-0.208853\pi$
$564$		0			0
$565$	33.3893	+	24.4196i	1.40470	+	1.02734i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−29.9688			−1.25636			−0.628179	−	0.778069i	$-0.716199\pi$
$569$	−29.9688			−1.25636			−0.628179	+	0.778069i	$0.716199\pi$
$570$		0			0
$571$	−0.280964			−0.0117580			−0.00587898	−	0.999983i	$-0.501871\pi$
$571$	−0.280964			−0.0117580			−0.00587898	+	0.999983i	$0.501871\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−34.6888	+	11.0303i	−1.44662	+	0.459994i
$576$		0			0
$577$			25.9541i			1.08048i	0.841510	+	0.540242i	$0.181667\pi$
$577$			25.9541i			1.08048i	−0.841510	+	0.540242i	$0.818333\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.1093			−0.419404
$582$		0			0
$583$		−	22.9092i		−	0.948801i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	22.2304i		−	0.917547i	−0.888553	−	0.458773i	$-0.848289\pi$
$587$		−	22.2304i		−	0.917547i	0.888553	−	0.458773i	$-0.151711\pi$
$588$		0			0
$589$	−3.62156			−0.149224
$590$		0			0
$591$		0			0
$592$		0			0
$593$			35.0743i			1.44033i	0.693803	+	0.720165i	$0.255934\pi$
$593$			35.0743i			1.44033i	−0.693803	+	0.720165i	$0.744066\pi$
$594$		0			0
$595$	−7.60975	+	10.4049i	−0.311969	+	0.426561i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	21.7262			0.887707			0.443853	−	0.896099i	$-0.353611\pi$
$599$	21.7262			0.887707			0.443853	+	0.896099i	$0.353611\pi$
$600$		0			0
$601$	−23.9688			−0.977708			−0.488854	−	0.872366i	$-0.662585\pi$
$601$	−23.9688			−0.977708			−0.488854	+	0.872366i	$0.662585\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.36967	−	8.70935i	0.258964	−	0.354085i
$606$		0			0
$607$		−	43.3241i		−	1.75847i	−0.476388	−	0.879235i	$-0.658054\pi$
$607$		−	43.3241i		−	1.75847i	0.476388	−	0.879235i	$-0.341946\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−14.3544			−0.580715
$612$		0			0
$613$			8.90917i			0.359838i	0.983681	+	0.179919i	$0.0575836\pi$
$613$			8.90917i			0.359838i	−0.983681	+	0.179919i	$0.942416\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	27.2413i		−	1.09669i	−0.836251	−	0.548347i	$-0.815257\pi$
$617$		−	27.2413i		−	1.09669i	0.836251	−	0.548347i	$-0.184743\pi$
$618$		0			0
$619$	24.1405			0.970288			0.485144	−	0.874434i	$-0.338767\pi$
$619$	24.1405			0.970288			0.485144	+	0.874434i	$0.338767\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	11.4693i		−	0.459506i
$624$		0			0
$625$	−20.4087	+	14.4390i	−0.816349	+	0.577560i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−34.5895			−1.37917
$630$		0			0
$631$	−8.01468			−0.319059			−0.159530	−	0.987193i	$-0.550998\pi$
$631$	−8.01468			−0.319059			−0.159530	+	0.987193i	$0.550998\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.9210	−	9.44989i	−0.512754	−	0.375007i
$636$		0			0
$637$		−	4.15516i		−	0.164633i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	49.1495			1.94129			0.970645	−	0.240516i	$-0.0773166\pi$
$641$	49.1495			1.94129			0.970645	+	0.240516i	$0.0773166\pi$
$642$		0			0
$643$			7.24599i			0.285754i	0.989740	+	0.142877i	$0.0456353\pi$
$643$			7.24599i			0.285754i	−0.989740	+	0.142877i	$0.954365\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			30.2791i			1.19040i	0.803579	+	0.595198i	$0.202926\pi$
$647$			30.2791i			1.19040i	−0.803579	+	0.595198i	$0.797074\pi$
$648$		0			0
$649$	14.7106			0.577440
$650$		0			0
$651$		0			0
$652$		0			0
$653$			7.96881i			0.311844i	0.987769	+	0.155922i	$0.0498348\pi$
$653$			7.96881i			0.311844i	−0.987769	+	0.155922i	$0.950165\pi$
$654$		0			0
$655$	−10.3747	+	14.1854i	−0.405371	+	0.554271i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−20.9239			−0.815078			−0.407539	−	0.913188i	$-0.633613\pi$
$659$	−20.9239			−0.815078			−0.407539	+	0.913188i	$0.633613\pi$
$660$		0			0
$661$	46.1992			1.79694			0.898470	−	0.439034i	$-0.144679\pi$
$661$	46.1992			1.79694			0.898470	+	0.439034i	$0.144679\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.90539	+	2.12489i	0.112666	+	0.0823995i
$666$		0			0
$667$		−	10.5895i		−	0.410025i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.3188			0.514167
$672$		0			0
$673$		−	40.3784i		−	1.55647i	−0.627970	−	0.778237i	$-0.716114\pi$
$673$		−	40.3784i		−	1.55647i	0.627970	−	0.778237i	$-0.283886\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			44.3737i			1.70542i	0.522383	+	0.852711i	$0.325043\pi$
$677$			44.3737i			1.70542i	−0.522383	+	0.852711i	$0.674957\pi$
$678$		0			0
$679$	2.73463			0.104946
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	17.9394i		−	0.686434i	−0.939256	−	0.343217i	$-0.888483\pi$
$683$		−	17.9394i		−	0.686434i	0.939256	−	0.343217i	$-0.111517\pi$
$684$		0			0
$685$	−19.0596	−	13.9394i	−0.728231	−	0.532599i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	38.3085			1.45944
$690$		0			0
$691$	12.7905			0.486573			0.243287	−	0.969954i	$-0.421774\pi$
$691$	12.7905			0.486573			0.243287	+	0.969954i	$0.421774\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12.9348	+	17.6859i	−0.490643	+	0.670864i
$696$		0			0
$697$			65.0284i			2.46313i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	19.4234			0.733611			0.366806	−	0.930298i	$-0.380451\pi$
$701$	19.4234			0.733611			0.366806	+	0.930298i	$0.380451\pi$
$702$		0			0
$703$			9.65848i			0.364277i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			4.57947i			0.172229i
$708$		0			0
$709$	13.4839			0.506400			0.253200	−	0.967414i	$-0.418517\pi$
$709$	13.4839			0.506400			0.253200	+	0.967414i	$0.418517\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	16.3784i		−	0.613377i
$714$		0			0
$715$	−18.6353	−	13.6291i	−0.696922	−	0.509700i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.3700			−0.573203			−0.286601	−	0.958050i	$-0.592526\pi$
$719$	−15.3700			−0.573203			−0.286601	+	0.958050i	$0.592526\pi$
$720$		0			0
$721$	2.48486			0.0925411
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.20390	−	6.93097i	−0.0818507	−	0.257410i
$726$		0			0
$727$			13.1589i			0.488038i	0.969770	+	0.244019i	$0.0784660\pi$
$727$			13.1589i			0.488038i	−0.969770	+	0.244019i	$0.921534\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−30.4390			−1.12583
$732$		0			0
$733$		−	10.0265i		−	0.370337i	−0.982707	−	0.185169i	$-0.940717\pi$
$733$		−	10.0265i		−	0.370337i	0.982707	−	0.185169i	$-0.0592831\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			18.7106i			0.689212i
$738$		0			0
$739$	19.2266			0.707262			0.353631	−	0.935385i	$-0.384947\pi$
$739$	19.2266			0.707262			0.353631	+	0.935385i	$0.384947\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	5.21949i		−	0.191485i	−0.995406	−	0.0957423i	$-0.969478\pi$
$743$		−	5.21949i		−	0.191485i	0.995406	−	0.0957423i	$-0.0305225\pi$
$744$		0			0
$745$	−3.29942	+	4.51136i	−0.120882	+	0.165283i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−29.8548			−1.08942			−0.544709	−	0.838625i	$-0.683360\pi$
$751$	−29.8548			−1.08942			−0.544709	+	0.838625i	$0.683360\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4.96972	+	6.79518i	−0.180867	+	0.247302i
$756$		0			0
$757$			18.0294i			0.655288i	0.944801	+	0.327644i	$0.106255\pi$
$757$			18.0294i			0.655288i	−0.944801	+	0.327644i	$0.893745\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−8.22041			−0.297990			−0.148995	−	0.988838i	$-0.547604\pi$
$761$	−8.22041			−0.297990			−0.148995	+	0.988838i	$0.547604\pi$
$762$		0			0
$763$			9.95413i			0.360364i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			24.5988i			0.888213i
$768$		0			0
$769$	−50.6888			−1.82788			−0.913942	−	0.405845i	$-0.866977\pi$
$769$	−50.6888			−1.82788			−0.913942	+	0.405845i	$0.866977\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	6.99622i		−	0.251637i	−0.992053	−	0.125818i	$-0.959844\pi$
$773$		−	6.99622i		−	0.251637i	0.992053	−	0.125818i	$-0.0401556\pi$
$774$		0			0
$775$	−3.40871	−	10.7200i	−0.122445	−	0.385072i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	18.1580			0.650579
$780$		0			0
$781$	−10.5601			−0.377870
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10.2342	+	7.48486i	0.365273	+	0.267146i
$786$		0			0
$787$		−	14.3444i		−	0.511322i	−0.966767	−	0.255661i	$-0.917707\pi$
$787$		−	14.3444i		−	0.511322i	0.966767	−	0.255661i	$-0.0822931\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	18.4995			0.657768
$792$		0			0
$793$			22.2716i			0.790887i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	13.9348i		−	0.493594i	−0.969067	−	0.246797i	$-0.920622\pi$
$797$		−	13.9348i		−	0.493594i	0.969067	−	0.246797i	$-0.0793781\pi$
$798$		0			0
$799$	19.9154			0.704555
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	18.0899i		−	0.638379i
$804$		0			0
$805$	−9.60975	+	13.1396i	−0.338699	+	0.463109i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5.58325			−0.196297			−0.0981483	−	0.995172i	$-0.531292\pi$
$809$	−5.58325			−0.196297			−0.0981483	+	0.995172i	$0.531292\pi$
$810$		0			0
$811$	6.57947			0.231036			0.115518	−	0.993305i	$-0.463147\pi$
$811$	6.57947			0.231036			0.115518	+	0.993305i	$0.463147\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18.4995	+	13.5298i	0.648011	+	0.473929i
$816$		0			0
$817$			8.49954i			0.297361i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16.3250			−0.569747			−0.284873	−	0.958565i	$-0.591952\pi$
$821$	−16.3250			−0.569747			−0.284873	+	0.958565i	$0.591952\pi$
$822$		0			0
$823$		−	12.3491i		−	0.430462i	−0.976563	−	0.215231i	$-0.930950\pi$
$823$		−	12.3491i		−	0.430462i	0.976563	−	0.215231i	$-0.0690504\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	6.40963i		−	0.222885i	−0.993771	−	0.111442i	$-0.964453\pi$
$827$		−	6.40963i		−	0.222885i	0.993771	−	0.111442i	$-0.0355470\pi$
$828$		0			0
$829$	26.1698			0.908916			0.454458	−	0.890768i	$-0.349833\pi$
$829$	26.1698			0.908916			0.454458	+	0.890768i	$0.349833\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			5.76491i			0.199742i
$834$		0			0
$835$	−5.33348	−	3.90069i	−0.184573	−	0.134989i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.12958			0.0389975			0.0194988	−	0.999810i	$-0.493793\pi$
$839$	1.12958			0.0389975			0.0194988	+	0.999810i	$0.493793\pi$
$840$		0			0
$841$	−26.8842			−0.927041
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.63033	−	7.69845i	0.193689	−	0.264835i
$846$		0			0
$847$		−	4.82546i		−	0.165805i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−43.6803			−1.49734
$852$		0			0
$853$			6.29095i			0.215398i	0.994184	+	0.107699i	$0.0343483\pi$
$853$			6.29095i			0.215398i	−0.994184	+	0.107699i	$0.965652\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	25.0596i		−	0.856021i	−0.903774	−	0.428010i	$-0.859215\pi$
$857$		−	25.0596i		−	0.856021i	0.903774	−	0.428010i	$-0.140785\pi$
$858$		0			0
$859$	53.7896			1.83528			0.917638	−	0.397417i	$-0.130093\pi$
$859$	53.7896			1.83528			0.917638	+	0.397417i	$0.130093\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	58.0899i		−	1.97740i	−0.149896	−	0.988702i	$-0.547894\pi$
$863$		−	58.0899i		−	1.97740i	0.149896	−	0.988702i	$-0.452106\pi$
$864$		0			0
$865$	40.4381	+	29.5748i	1.37494	+	1.00557i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	42.2039			1.43167
$870$		0			0
$871$	−31.2876			−1.06014
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.55510	+	10.6001i	−0.120184	+	0.358347i
$876$		0			0
$877$			24.0899i			0.813459i	0.913549	+	0.406729i	$0.133331\pi$
$877$			24.0899i			0.813459i	−0.913549	+	0.406729i	$0.866669\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−40.9679			−1.38024			−0.690122	−	0.723693i	$-0.742443\pi$
$881$	−40.9679			−1.38024			−0.690122	+	0.723693i	$0.742443\pi$
$882$		0			0
$883$			43.8014i			1.47403i	0.675874	+	0.737017i	$0.263766\pi$
$883$			43.8014i			1.47403i	−0.675874	+	0.737017i	$0.736234\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1.77959i			0.0597527i	0.999554	+	0.0298764i	$0.00951136\pi$
$887$			1.77959i			0.0597527i	−0.999554	+	0.0298764i	$0.990489\pi$
$888$		0			0
$889$	−7.15894			−0.240103
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	5.56101i		−	0.186092i
$894$		0			0
$895$	19.2195	−	26.2791i	0.642437	−	0.878414i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.27248			0.109143
$900$		0			0
$901$	−53.1495			−1.77067
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.7649	−	17.4537i	0.424320	−	0.580180i
$906$		0			0
$907$			53.0284i			1.76078i	0.474250	+	0.880390i	$0.342719\pi$
$907$			53.0284i			1.76078i	−0.474250	+	0.880390i	$0.657281\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	28.1798			0.933639			0.466820	−	0.884353i	$-0.345400\pi$
$911$	28.1798			0.933639			0.466820	+	0.884353i	$0.345400\pi$
$912$		0			0
$913$			25.1202i			0.831357i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			7.85952i			0.259544i
$918$		0			0
$919$	38.4243			1.26750			0.633751	−	0.773538i	$-0.281515\pi$
$919$	38.4243			1.26750			0.633751	+	0.773538i	$0.281515\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	17.6585i		−	0.581236i
$924$		0			0
$925$	−28.5895	+	9.09083i	−0.940015	+	0.298905i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−36.6576			−1.20270			−0.601348	−	0.798987i	$-0.705369\pi$
$929$	−36.6576			−1.20270			−0.601348	+	0.798987i	$0.705369\pi$
$930$		0			0
$931$	1.60975			0.0527573
$932$		0			0
$933$		0			0
$934$		0			0
$935$	25.8548	+	18.9092i	0.845543	+	0.618396i
$936$		0			0
$937$		−	5.83302i		−	0.190557i	−0.995451	−	0.0952783i	$-0.969626\pi$
$937$		−	5.83302i		−	0.190557i	0.995451	−	0.0952783i	$-0.0303741\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	2.23796			0.0729553			0.0364776	−	0.999334i	$-0.488386\pi$
$941$	2.23796			0.0729553			0.0364776	+	0.999334i	$0.488386\pi$
$942$		0			0
$943$			82.1193i			2.67417i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			17.7502i			0.576805i	0.957509	+	0.288402i	$0.0931241\pi$
$947$			17.7502i			0.576805i	−0.957509	+	0.288402i	$0.906876\pi$
$948$		0			0
$949$	30.2498			0.981949
$950$		0			0
$951$		0			0
$952$		0			0
$953$			32.0294i			1.03753i	0.854916	+	0.518766i	$0.173609\pi$
$953$			32.0294i			1.03753i	−0.854916	+	0.518766i	$0.826391\pi$
$954$		0			0
$955$	−10.5795	+	14.4655i	−0.342344	+	0.468092i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.5601			−0.341003
$960$		0			0
$961$	−25.9385			−0.836727
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.26159	−	4.57947i	−0.201568	−	0.147418i
$966$		0			0
$967$			43.4305i			1.39663i	0.715790	+	0.698316i	$0.246067\pi$
$967$			43.4305i			1.39663i	−0.715790	+	0.698316i	$0.753933\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.79897			−0.0577316			−0.0288658	−	0.999583i	$-0.509190\pi$
$971$	−1.79897			−0.0577316			−0.0288658	+	0.999583i	$0.509190\pi$
$972$		0			0
$973$			9.79897i			0.314140i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	22.4390i		−	0.717887i	−0.933359	−	0.358943i	$-0.883137\pi$
$977$		−	22.4390i		−	0.717887i	0.933359	−	0.358943i	$-0.116863\pi$
$978$		0			0
$979$	−28.4995			−0.910849
$980$		0			0
$981$		0			0
$982$		0			0
$983$			37.6950i			1.20228i	0.799143	+	0.601141i	$0.205287\pi$
$983$			37.6950i			1.20228i	−0.799143	+	0.601141i	$0.794713\pi$
$984$		0			0
$985$	29.3818	+	21.4886i	0.936181	+	0.684685i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−38.4390			−1.22229
$990$		0			0
$991$	−13.5686			−0.431020			−0.215510	−	0.976502i	$-0.569141\pi$
$991$	−13.5686			−0.431020			−0.215510	+	0.976502i	$0.569141\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−35.4693	+	48.4977i	−1.12445	+	1.53748i
$996$		0			0
$997$		−	14.2838i		−	0.452373i	−0.974084	−	0.226187i	$-0.927374\pi$
$997$		−	14.2838i		−	0.452373i	0.974084	−	0.226187i	$-0.0726259\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.t.y.1009.2		6
3.2	odd	2		560.2.g.f.449.6		6
4.3	odd	2		2520.2.t.g.1009.2		6
5.4	even	2	inner	5040.2.t.y.1009.1		6
12.11	even	2		280.2.g.b.169.1	✓	6
15.2	even	4		2800.2.a.br.1.3		3
15.8	even	4		2800.2.a.bq.1.1		3
15.14	odd	2		560.2.g.f.449.1		6
20.19	odd	2		2520.2.t.g.1009.1		6
24.5	odd	2		2240.2.g.m.449.1		6
24.11	even	2		2240.2.g.l.449.6		6
60.23	odd	4		1400.2.a.t.1.3		3
60.47	odd	4		1400.2.a.s.1.1		3
60.59	even	2		280.2.g.b.169.6	yes	6
84.83	odd	2		1960.2.g.c.1569.6		6
120.29	odd	2		2240.2.g.m.449.6		6
120.59	even	2		2240.2.g.l.449.1		6
420.83	even	4		9800.2.a.cd.1.1		3
420.167	even	4		9800.2.a.cg.1.3		3
420.419	odd	2		1960.2.g.c.1569.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.g.b.169.1	✓	6	12.11	even	2
280.2.g.b.169.6	yes	6	60.59	even	2
560.2.g.f.449.1		6	15.14	odd	2
560.2.g.f.449.6		6	3.2	odd	2
1400.2.a.s.1.1		3	60.47	odd	4
1400.2.a.t.1.3		3	60.23	odd	4
1960.2.g.c.1569.1		6	420.419	odd	2
1960.2.g.c.1569.6		6	84.83	odd	2
2240.2.g.l.449.1		6	120.59	even	2
2240.2.g.l.449.6		6	24.11	even	2
2240.2.g.m.449.1		6	24.5	odd	2
2240.2.g.m.449.6		6	120.29	odd	2
2520.2.t.g.1009.1		6	20.19	odd	2
2520.2.t.g.1009.2		6	4.3	odd	2
2800.2.a.bq.1.1		3	15.8	even	4
2800.2.a.br.1.3		3	15.2	even	4
5040.2.t.y.1009.1		6	5.4	even	2	inner
5040.2.t.y.1009.2		6	1.1	even	1	trivial
9800.2.a.cd.1.1		3	420.83	even	4
9800.2.a.cg.1.3		3	420.167	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.32001 + 1.80487i) q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+(-1.32001 + 1.80487i) q^{5} +1.00000i q^{7} +2.48486 q^{11} +4.15516i q^{13} -5.76491i q^{17} -1.60975 q^{19} -7.28005i q^{23} +(-1.51514 - 4.76491i) q^{25} +1.45459 q^{29} +2.24977 q^{31} +(-1.80487 - 1.32001i) q^{35} -6.00000i q^{37} -11.2800 q^{41} -5.28005i q^{43} +3.45459i q^{47} -1.00000 q^{49} -9.21949i q^{53} +(-3.28005 + 4.48486i) q^{55} +5.92007 q^{59} +5.35998 q^{61} +(-7.49954 - 5.48486i) q^{65} +7.52982i q^{67} -4.24977 q^{71} -7.28005i q^{73} +2.48486i q^{77} +16.9844 q^{79} +10.1093i q^{83} +(10.4049 + 7.60975i) q^{85} -11.4693 q^{89} -4.15516 q^{91} +(2.12489 - 2.90539i) q^{95} -2.73463i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 14 q^{11} + 8 q^{19} - 10 q^{25} + 6 q^{29} - 20 q^{31} - 2 q^{35} - 36 q^{41} - 6 q^{49} + 12 q^{55} - 12 q^{59} + 48 q^{61} + 22 q^{65} + 8 q^{71} + 34 q^{79} + 14 q^{85} - 10 q^{91} - 4 q^{95}+O(q^{100}) \) 6 * q + 14 * q^11 + 8 * q^19 - 10 * q^25 + 6 * q^29 - 20 * q^31 - 2 * q^35 - 36 * q^41 - 6 * q^49 + 12 * q^55 - 12 * q^59 + 48 * q^61 + 22 * q^65 + 8 * q^71 + 34 * q^79 + 14 * q^85 - 10 * q^91 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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