LMFDB - Embedded newform 5040.2.t.a.1009.1

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:	$2$
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 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5040.1009
Dual form			5040.2.t.a.1009.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{11} +1.00000i q^{13} +3.00000i q^{17} -4.00000 q^{19} -2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000 q^{29} +6.00000 q^{31} +(1.00000 - 2.00000i) q^{35} +2.00000i q^{37} +10.0000 q^{41} +9.00000i q^{47} -1.00000 q^{49} -14.0000i q^{53} +(2.00000 + 1.00000i) q^{55} -6.00000 q^{59} -4.00000 q^{61} +(1.00000 - 2.00000i) q^{65} -10.0000i q^{67} -16.0000 q^{71} -10.0000i q^{73} -1.00000i q^{77} -11.0000 q^{79} -4.00000i q^{83} +(3.00000 - 6.00000i) q^{85} +12.0000 q^{89} -1.00000 q^{91} +(8.00000 + 4.00000i) q^{95} -19.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 2 q^{11} - 8 q^{19} + 6 q^{25} - 2 q^{29} + 12 q^{31} + 2 q^{35} + 20 q^{41} - 2 q^{49} + 4 q^{55} - 12 q^{59} - 8 q^{61} + 2 q^{65} - 32 q^{71} - 22 q^{79} + 6 q^{85} + 24 q^{89} - 2 q^{91}+ \cdots + 16 q^{95}+O(q^{100}) $ 2 * q - 4 * q^5 - 2 * q^11 - 8 * q^19 + 6 * q^25 - 2 * q^29 + 12 * q^31 + 2 * q^35 + 20 * q^41 - 2 * q^49 + 4 * q^55 - 12 * q^59 - 8 * q^61 + 2 * q^65 - 32 * q^71 - 22 * q^79 + 6 * q^85 + 24 * q^89 - 2 * q^91 + 16 * q^95

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
$5$	−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
$9$		0			0
$10$		0			0
$11$	−1.00000			−0.301511			−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000			−0.301511			−0.150756	+	0.988571i	$0.548171\pi$
$12$		0			0
$13$			1.00000i			0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$			1.00000i			0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.00000i			0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$			3.00000i			0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.00000i		−	0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$		−	2.00000i		−	0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.00000			−0.185695			−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000			−0.185695			−0.0928477	+	0.995680i	$0.529597\pi$
$30$		0			0
$31$	6.00000			1.07763			0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000			1.07763			0.538816	+	0.842424i	$0.318872\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.00000	−	2.00000i	0.169031	−	0.338062i
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.0000			1.56174			0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000			1.56174			0.780869	+	0.624695i	$0.214777\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			9.00000i			1.31278i	0.754420	+	0.656392i	$0.227918\pi$
$47$			9.00000i			1.31278i	−0.754420	+	0.656392i	$0.772082\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	14.0000i		−	1.92305i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$		−	14.0000i		−	1.92305i	0.274721	−	0.961524i	$-0.411414\pi$
$54$		0			0
$55$	2.00000	+	1.00000i	0.269680	+	0.134840i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000			−0.781133			−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000			−0.781133			−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$	−4.00000			−0.512148			−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000			−0.512148			−0.256074	+	0.966657i	$0.582429\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.00000	−	2.00000i	0.124035	−	0.248069i
$66$		0			0
$67$		−	10.0000i		−	1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$		−	10.0000i		−	1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−16.0000			−1.89885			−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000			−1.89885			−0.949425	+	0.313993i	$0.898333\pi$
$72$		0			0
$73$		−	10.0000i		−	1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$		−	10.0000i		−	1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	1.00000i		−	0.113961i
$78$		0			0
$79$	−11.0000			−1.23760			−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000			−1.23760			−0.618798	+	0.785550i	$0.712380\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$		0			0
$85$	3.00000	−	6.00000i	0.325396	−	0.650791i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.0000			1.27200			0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$		0			0
$91$	−1.00000			−0.104828
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.00000	+	4.00000i	0.820783	+	0.410391i
$96$		0			0
$97$		−	19.0000i		−	1.92916i	−0.263795	−	0.964579i	$-0.584974\pi$
$97$		−	19.0000i		−	1.92916i	0.263795	−	0.964579i	$-0.415026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$			1.00000i			0.0985329i	0.998786	+	0.0492665i	$0.0156884\pi$
$103$			1.00000i			0.0985329i	−0.998786	+	0.0492665i	$0.984312\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			16.0000i			1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$107$			16.0000i			1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$108$		0			0
$109$	15.0000			1.43674			0.718370	−	0.695662i	$-0.244889\pi$
$109$	15.0000			1.43674			0.718370	+	0.695662i	$0.244889\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			14.0000i			1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$			14.0000i			1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$	−2.00000	+	4.00000i	−0.186501	+	0.373002i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.00000			−0.275010
$120$		0			0
$121$	−10.0000			−0.909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$126$		0			0
$127$		−	18.0000i		−	1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$127$		−	18.0000i		−	1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.00000			0.524222			0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000			0.524222			0.262111	+	0.965038i	$0.415581\pi$
$132$		0			0
$133$		−	4.00000i		−	0.346844i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.00000i			0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$			8.00000i			0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$		0			0
$139$	−14.0000			−1.18746			−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000			−1.18746			−0.593732	+	0.804663i	$0.702346\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	1.00000i		−	0.0836242i
$144$		0			0
$145$	2.00000	+	1.00000i	0.166091	+	0.0830455i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$	−13.0000			−1.05792			−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000			−1.05792			−0.528962	+	0.848645i	$0.677419\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.0000	−	6.00000i	−0.963863	−	0.481932i
$156$		0			0
$157$		−	14.0000i		−	1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$		−	14.0000i		−	1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.00000			0.157622
$162$		0			0
$163$			10.0000i			0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$			10.0000i			0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	15.0000i		−	1.16073i	−0.814355	−	0.580367i	$-0.802909\pi$
$167$		−	15.0000i		−	1.16073i	0.814355	−	0.580367i	$-0.197091\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	15.0000i		−	1.14043i	−0.821496	−	0.570214i	$-0.806860\pi$
$173$		−	15.0000i		−	1.14043i	0.821496	−	0.570214i	$-0.193140\pi$
$174$		0			0
$175$	−4.00000	+	3.00000i	−0.302372	+	0.226779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.00000	−	4.00000i	0.147043	−	0.294086i
$186$		0			0
$187$		−	3.00000i		−	0.219382i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.00000			0.217072			0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000			0.217072			0.108536	+	0.994092i	$0.465384\pi$
$192$		0			0
$193$		−	4.00000i		−	0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$		−	4.00000i		−	0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$		0			0
$199$	10.0000			0.708881			0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000			0.708881			0.354441	+	0.935079i	$0.384671\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	1.00000i		−	0.0701862i
$204$		0			0
$205$	−20.0000	−	10.0000i	−1.39686	−	0.698430i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	−1.00000			−0.0688428			−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000			−0.0688428			−0.0344214	+	0.999407i	$0.510959\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			6.00000i			0.407307i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.00000			−0.201802
$222$		0			0
$223$		−	15.0000i		−	1.00447i	−0.864730	−	0.502237i	$-0.832510\pi$
$223$		−	15.0000i		−	1.00447i	0.864730	−	0.502237i	$-0.167490\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	7.00000i		−	0.464606i	−0.972643	−	0.232303i	$-0.925374\pi$
$227$		−	7.00000i		−	0.464606i	0.972643	−	0.232303i	$-0.0746261\pi$
$228$		0			0
$229$	−18.0000			−1.18947			−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000			−1.18947			−0.594737	+	0.803921i	$0.702744\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$	9.00000	−	18.0000i	0.587095	−	1.17419i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−17.0000			−1.09964			−0.549819	−	0.835284i	$-0.685303\pi$
$239$	−17.0000			−1.09964			−0.549819	+	0.835284i	$0.685303\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.00000	+	1.00000i	0.127775	+	0.0638877i
$246$		0			0
$247$		−	4.00000i		−	0.254514i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.0000			0.631194			0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000			0.631194			0.315597	+	0.948893i	$0.397795\pi$
$252$		0			0
$253$			2.00000i			0.125739i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	18.0000i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	18.0000i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$		0			0
$259$	−2.00000			−0.124274
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$	−14.0000	+	28.0000i	−0.860013	+	1.72003i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.00000			−0.365826			−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000			−0.365826			−0.182913	+	0.983129i	$0.558553\pi$
$270$		0			0
$271$	4.00000			0.242983			0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000			0.242983			0.121491	+	0.992592i	$0.461232\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.00000	−	4.00000i	−0.180907	−	0.241209i
$276$		0			0
$277$		−	18.0000i		−	1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$		−	18.0000i		−	1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−15.0000			−0.894825			−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000			−0.894825			−0.447412	+	0.894328i	$0.647654\pi$
$282$		0			0
$283$		−	21.0000i		−	1.24832i	−0.781296	−	0.624160i	$-0.785441\pi$
$283$		−	21.0000i		−	1.24832i	0.781296	−	0.624160i	$-0.214559\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			10.0000i			0.590281i
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	31.0000i		−	1.81104i	−0.424304	−	0.905520i	$-0.639481\pi$
$293$		−	31.0000i		−	1.81104i	0.424304	−	0.905520i	$-0.360519\pi$
$294$		0			0
$295$	12.0000	+	6.00000i	0.698667	+	0.349334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.00000			0.115663
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.00000	+	4.00000i	0.458079	+	0.229039i
$306$		0			0
$307$			25.0000i			1.42683i	0.700744	+	0.713413i	$0.252851\pi$
$307$			25.0000i			1.42683i	−0.700744	+	0.713413i	$0.747149\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	7.00000i		−	0.395663i	−0.980236	−	0.197832i	$-0.936610\pi$
$313$		−	7.00000i		−	0.395663i	0.980236	−	0.197832i	$-0.0633900\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	12.0000i		−	0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$		−	12.0000i		−	0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$		0			0
$319$	1.00000			0.0559893
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$	−4.00000	+	3.00000i	−0.221880	+	0.166410i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−9.00000			−0.496186
$330$		0			0
$331$	−28.0000			−1.53902			−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000			−1.53902			−0.769510	+	0.638635i	$0.779499\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.0000	+	20.0000i	−0.546358	+	1.09272i
$336$		0			0
$337$			34.0000i			1.85210i	0.377403	+	0.926049i	$0.376817\pi$
$337$			34.0000i			1.85210i	−0.377403	+	0.926049i	$0.623183\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.00000			−0.324918
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	22.0000i		−	1.18102i	−0.807030	−	0.590511i	$-0.798926\pi$
$347$		−	22.0000i		−	1.18102i	0.807030	−	0.590511i	$-0.201074\pi$
$348$		0			0
$349$	−8.00000			−0.428230			−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000			−0.428230			−0.214115	+	0.976808i	$0.568687\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			9.00000i			0.479022i	0.970894	+	0.239511i	$0.0769871\pi$
$353$			9.00000i			0.479022i	−0.970894	+	0.239511i	$0.923013\pi$
$354$		0			0
$355$	32.0000	+	16.0000i	1.69838	+	0.849192i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.00000			−0.211112			−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000			−0.211112			−0.105556	+	0.994413i	$0.533662\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.0000	+	20.0000i	−0.523424	+	1.04685i
$366$		0			0
$367$		−	17.0000i		−	0.887393i	−0.896177	−	0.443696i	$-0.853667\pi$
$367$		−	17.0000i		−	0.887393i	0.896177	−	0.443696i	$-0.146333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	14.0000			0.726844
$372$		0			0
$373$		−	20.0000i		−	1.03556i	−0.855514	−	0.517780i	$-0.826758\pi$
$373$		−	20.0000i		−	1.03556i	0.855514	−	0.517780i	$-0.173242\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.00000i		−	0.0515026i
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	24.0000i		−	1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$		−	24.0000i		−	1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$		0			0
$385$	−1.00000	+	2.00000i	−0.0509647	+	0.101929i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.00000			0.456318			0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000			0.456318			0.228159	+	0.973624i	$0.426729\pi$
$390$		0			0
$391$	6.00000			0.303433
$392$		0			0
$393$		0			0
$394$		0			0
$395$	22.0000	+	11.0000i	1.10694	+	0.553470i
$396$		0			0
$397$		−	15.0000i		−	0.752828i	−0.926451	−	0.376414i	$-0.877157\pi$
$397$		−	15.0000i		−	0.752828i	0.926451	−	0.376414i	$-0.122843\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−13.0000			−0.649189			−0.324595	−	0.945853i	$-0.605228\pi$
$401$	−13.0000			−0.649189			−0.324595	+	0.945853i	$0.605228\pi$
$402$		0			0
$403$			6.00000i			0.298881i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2.00000i		−	0.0991363i
$408$		0			0
$409$	28.0000			1.38451			0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000			1.38451			0.692255	+	0.721653i	$0.256617\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	6.00000i		−	0.295241i
$414$		0			0
$415$	−4.00000	+	8.00000i	−0.196352	+	0.392705i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	30.0000			1.46560			0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000			1.46560			0.732798	+	0.680446i	$0.238214\pi$
$420$		0			0
$421$	1.00000			0.0487370			0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000			0.0487370			0.0243685	+	0.999703i	$0.492242\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.0000	+	9.00000i	−0.582086	+	0.436564i
$426$		0			0
$427$		−	4.00000i		−	0.193574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	31.0000			1.49322			0.746609	−	0.665263i	$-0.231681\pi$
$431$	31.0000			1.49322			0.746609	+	0.665263i	$0.231681\pi$
$432$		0			0
$433$		−	6.00000i		−	0.288342i	−0.989553	−	0.144171i	$-0.953949\pi$
$433$		−	6.00000i		−	0.288342i	0.989553	−	0.144171i	$-0.0460515\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			8.00000i			0.382692i
$438$		0			0
$439$	−30.0000			−1.43182			−0.715911	−	0.698192i	$-0.753988\pi$
$439$	−30.0000			−1.43182			−0.715911	+	0.698192i	$0.753988\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−24.0000	−	12.0000i	−1.13771	−	0.568855i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	27.0000			1.27421			0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000			1.27421			0.637104	+	0.770778i	$0.280132\pi$
$450$		0			0
$451$	−10.0000			−0.470882
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.00000	+	1.00000i	0.0937614	+	0.0468807i
$456$		0			0
$457$			18.0000i			0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		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
$465$		0			0
$466$		0			0
$467$			27.0000i			1.24941i	0.780860	+	0.624705i	$0.214781\pi$
$467$			27.0000i			1.24941i	−0.780860	+	0.624705i	$0.785219\pi$
$468$		0			0
$469$	10.0000			0.461757
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−12.0000	−	16.0000i	−0.550598	−	0.734130i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.00000			−0.0913823			−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000			−0.0913823			−0.0456912	+	0.998956i	$0.514549\pi$
$480$		0			0
$481$	−2.00000			−0.0911922
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−19.0000	+	38.0000i	−0.862746	+	1.72549i
$486$		0			0
$487$			38.0000i			1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$			38.0000i			1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.00000			0.225647			0.112823	−	0.993615i	$-0.464011\pi$
$491$	5.00000			0.225647			0.112823	+	0.993615i	$0.464011\pi$
$492$		0			0
$493$		−	3.00000i		−	0.135113i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	16.0000i		−	0.717698i
$498$		0			0
$499$	23.0000			1.02962			0.514811	−	0.857304i	$-0.327862\pi$
$499$	23.0000			1.02962			0.514811	+	0.857304i	$0.327862\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			29.0000i			1.29305i	0.762894	+	0.646523i	$0.223778\pi$
$503$			29.0000i			1.29305i	−0.762894	+	0.646523i	$0.776222\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.0000			0.443242			0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000			0.443242			0.221621	+	0.975133i	$0.428865\pi$
$510$		0			0
$511$	10.0000			0.442374
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.00000	−	2.00000i	0.0440653	−	0.0881305i
$516$		0			0
$517$		−	9.00000i		−	0.395820i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8.00000			0.350486			0.175243	−	0.984525i	$-0.443929\pi$
$521$	8.00000			0.350486			0.175243	+	0.984525i	$0.443929\pi$
$522$		0			0
$523$		−	28.0000i		−	1.22435i	−0.790721	−	0.612177i	$-0.790294\pi$
$523$		−	28.0000i		−	1.22435i	0.790721	−	0.612177i	$-0.209706\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			18.0000i			0.784092i
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$		0			0
$532$		0			0
$533$			10.0000i			0.433148i
$534$		0			0
$535$	16.0000	−	32.0000i	0.691740	−	1.38348i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.00000			0.0430730
$540$		0			0
$541$	−41.0000			−1.76273			−0.881364	−	0.472438i	$-0.843374\pi$
$541$	−41.0000			−1.76273			−0.881364	+	0.472438i	$0.843374\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−30.0000	−	15.0000i	−1.28506	−	0.642529i
$546$		0			0
$547$		−	4.00000i		−	0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$		−	4.00000i		−	0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4.00000			0.170406
$552$		0			0
$553$		−	11.0000i		−	0.467768i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			36.0000i			1.52537i	0.646771	+	0.762684i	$0.276119\pi$
$557$			36.0000i			1.52537i	−0.646771	+	0.762684i	$0.723881\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$		0			0
$565$	14.0000	−	28.0000i	0.588984	−	1.17797i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	44.0000			1.84134			0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000			1.84134			0.920671	+	0.390339i	$0.127642\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.00000	−	6.00000i	0.333623	−	0.250217i
$576$		0			0
$577$			31.0000i			1.29055i	0.763952	+	0.645273i	$0.223257\pi$
$577$			31.0000i			1.29055i	−0.763952	+	0.645273i	$0.776743\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.00000			0.165948
$582$		0			0
$583$			14.0000i			0.579821i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	20.0000i		−	0.825488i	−0.910847	−	0.412744i	$-0.864570\pi$
$587$		−	20.0000i		−	0.825488i	0.910847	−	0.412744i	$-0.135430\pi$
$588$		0			0
$589$	−24.0000			−0.988903
$590$		0			0
$591$		0			0
$592$		0			0
$593$			19.0000i			0.780236i	0.920765	+	0.390118i	$0.127566\pi$
$593$			19.0000i			0.780236i	−0.920765	+	0.390118i	$0.872434\pi$
$594$		0			0
$595$	6.00000	+	3.00000i	0.245976	+	0.122988i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−7.00000			−0.286012			−0.143006	−	0.989722i	$-0.545677\pi$
$599$	−7.00000			−0.286012			−0.143006	+	0.989722i	$0.545677\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	20.0000	+	10.0000i	0.813116	+	0.406558i
$606$		0			0
$607$			1.00000i			0.0405887i	0.999794	+	0.0202944i	$0.00646034\pi$
$607$			1.00000i			0.0405887i	−0.999794	+	0.0202944i	$0.993540\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.00000			−0.364101
$612$		0			0
$613$			4.00000i			0.161558i	0.996732	+	0.0807792i	$0.0257409\pi$
$613$			4.00000i			0.161558i	−0.996732	+	0.0807792i	$0.974259\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	2.00000i		−	0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$		−	2.00000i		−	0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$		0			0
$619$	−38.0000			−1.52735			−0.763674	−	0.645601i	$-0.776607\pi$
$619$	−38.0000			−1.52735			−0.763674	+	0.645601i	$0.776607\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			12.0000i			0.480770i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.00000			−0.239236
$630$		0			0
$631$	13.0000			0.517522			0.258761	−	0.965941i	$-0.416686\pi$
$631$	13.0000			0.517522			0.258761	+	0.965941i	$0.416686\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−18.0000	+	36.0000i	−0.714308	+	1.42862i
$636$		0			0
$637$		−	1.00000i		−	0.0396214i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.0000			0.710957			0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000			0.710957			0.355479	+	0.934684i	$0.384318\pi$
$642$		0			0
$643$		−	15.0000i		−	0.591542i	−0.955259	−	0.295771i	$-0.904423\pi$
$643$		−	15.0000i		−	0.591542i	0.955259	−	0.295771i	$-0.0955766\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	32.0000i		−	1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$647$		−	32.0000i		−	1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$648$		0			0
$649$	6.00000			0.235521
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	24.0000i		−	0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$		−	24.0000i		−	0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$		0			0
$655$	−12.0000	−	6.00000i	−0.468879	−	0.234439i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	21.0000			0.818044			0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000			0.818044			0.409022	+	0.912525i	$0.365870\pi$
$660$		0			0
$661$	−28.0000			−1.08907			−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000			−1.08907			−0.544537	+	0.838737i	$0.683295\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.00000	+	8.00000i	−0.155113	+	0.310227i
$666$		0			0
$667$			2.00000i			0.0774403i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.00000			0.154418
$672$		0			0
$673$			20.0000i			0.770943i	0.922720	+	0.385472i	$0.125961\pi$
$673$			20.0000i			0.770943i	−0.922720	+	0.385472i	$0.874039\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	3.00000i		−	0.115299i	−0.998337	−	0.0576497i	$-0.981639\pi$
$677$		−	3.00000i		−	0.115299i	0.998337	−	0.0576497i	$-0.0183606\pi$
$678$		0			0
$679$	19.0000			0.729153
$680$		0			0
$681$		0			0
$682$		0			0
$683$			4.00000i			0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$			4.00000i			0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$		0			0
$685$	8.00000	−	16.0000i	0.305664	−	0.611329i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	14.0000			0.533358
$690$		0			0
$691$	48.0000			1.82601			0.913003	−	0.407953i	$-0.133757\pi$
$691$	48.0000			1.82601			0.913003	+	0.407953i	$0.133757\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	28.0000	+	14.0000i	1.06210	+	0.531050i
$696$		0			0
$697$			30.0000i			1.13633i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	41.0000			1.54855			0.774274	−	0.632850i	$-0.218115\pi$
$701$	41.0000			1.54855			0.774274	+	0.632850i	$0.218115\pi$
$702$		0			0
$703$		−	8.00000i		−	0.301726i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	17.0000			0.638448			0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000			0.638448			0.319224	+	0.947679i	$0.396578\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	12.0000i		−	0.449404i
$714$		0			0
$715$	−1.00000	+	2.00000i	−0.0373979	+	0.0747958i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	50.0000			1.86469			0.932343	−	0.361576i	$-0.117761\pi$
$719$	50.0000			1.86469			0.932343	+	0.361576i	$0.117761\pi$
$720$		0			0
$721$	−1.00000			−0.0372419
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.00000	−	4.00000i	−0.111417	−	0.148556i
$726$		0			0
$727$			36.0000i			1.33517i	0.744535	+	0.667583i	$0.232671\pi$
$727$			36.0000i			1.33517i	−0.744535	+	0.667583i	$0.767329\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$			9.00000i			0.332423i	0.986090	+	0.166211i	$0.0531534\pi$
$733$			9.00000i			0.332423i	−0.986090	+	0.166211i	$0.946847\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			10.0000i			0.368355i
$738$		0			0
$739$	15.0000			0.551784			0.275892	−	0.961189i	$-0.411027\pi$
$739$	15.0000			0.551784			0.275892	+	0.961189i	$0.411027\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	42.0000i		−	1.54083i	−0.637542	−	0.770415i	$-0.720049\pi$
$743$		−	42.0000i		−	1.54083i	0.637542	−	0.770415i	$-0.279951\pi$
$744$		0			0
$745$	12.0000	+	6.00000i	0.439646	+	0.219823i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−16.0000			−0.584627
$750$		0			0
$751$	−17.0000			−0.620339			−0.310169	−	0.950681i	$-0.600386\pi$
$751$	−17.0000			−0.620339			−0.310169	+	0.950681i	$0.600386\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	26.0000	+	13.0000i	0.946237	+	0.473118i
$756$		0			0
$757$			8.00000i			0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$			8.00000i			0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$			15.0000i			0.543036i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	6.00000i		−	0.216647i
$768$		0			0
$769$	8.00000			0.288487			0.144244	−	0.989542i	$-0.453925\pi$
$769$	8.00000			0.288487			0.144244	+	0.989542i	$0.453925\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	13.0000i		−	0.467578i	−0.972287	−	0.233789i	$-0.924888\pi$
$773$		−	13.0000i		−	0.467578i	0.972287	−	0.233789i	$-0.0751124\pi$
$774$		0			0
$775$	18.0000	+	24.0000i	0.646579	+	0.862105i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−40.0000			−1.43315
$780$		0			0
$781$	16.0000			0.572525
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−14.0000	+	28.0000i	−0.499681	+	0.999363i
$786$		0			0
$787$			31.0000i			1.10503i	0.833503	+	0.552515i	$0.186332\pi$
$787$			31.0000i			1.10503i	−0.833503	+	0.552515i	$0.813668\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14.0000			−0.497783
$792$		0			0
$793$		−	4.00000i		−	0.142044i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			27.0000i			0.956389i	0.878254	+	0.478195i	$0.158709\pi$
$797$			27.0000i			0.956389i	−0.878254	+	0.478195i	$0.841291\pi$
$798$		0			0
$799$	−27.0000			−0.955191
$800$		0			0
$801$		0			0
$802$		0			0
$803$			10.0000i			0.352892i
$804$		0			0
$805$	−4.00000	−	2.00000i	−0.140981	−	0.0704907i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−51.0000			−1.79306			−0.896532	−	0.442978i	$-0.853922\pi$
$809$	−51.0000			−1.79306			−0.896532	+	0.442978i	$0.853922\pi$
$810$		0			0
$811$	42.0000			1.47482			0.737410	−	0.675446i	$-0.236049\pi$
$811$	42.0000			1.47482			0.737410	+	0.675446i	$0.236049\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.0000	−	20.0000i	0.350285	−	0.700569i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−45.0000			−1.57051			−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000			−1.57051			−0.785255	+	0.619172i	$0.787468\pi$
$822$		0			0
$823$		−	10.0000i		−	0.348578i	−0.984695	−	0.174289i	$-0.944237\pi$
$823$		−	10.0000i		−	0.348578i	0.984695	−	0.174289i	$-0.0557627\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	42.0000i		−	1.46048i	−0.683189	−	0.730242i	$-0.739408\pi$
$827$		−	42.0000i		−	1.46048i	0.683189	−	0.730242i	$-0.260592\pi$
$828$		0			0
$829$	50.0000			1.73657			0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000			1.73657			0.868286	+	0.496064i	$0.165222\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	3.00000i		−	0.103944i
$834$		0			0
$835$	−15.0000	+	30.0000i	−0.519096	+	1.03819i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−54.0000			−1.86429			−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000			−1.86429			−0.932144	+	0.362089i	$0.882064\pi$
$840$		0			0
$841$	−28.0000			−0.965517
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−24.0000	−	12.0000i	−0.825625	−	0.412813i
$846$		0			0
$847$		−	10.0000i		−	0.343604i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.00000			0.137118
$852$		0			0
$853$		−	14.0000i		−	0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$		−	14.0000i		−	0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			10.0000i			0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$			10.0000i			0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$		0			0
$859$	−8.00000			−0.272956			−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000			−0.272956			−0.136478	+	0.990643i	$0.543578\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	6.00000i		−	0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$863$		−	6.00000i		−	0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$864$		0			0
$865$	−15.0000	+	30.0000i	−0.510015	+	1.02003i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	11.0000			0.373149
$870$		0			0
$871$	10.0000			0.338837
$872$		0			0
$873$		0			0
$874$		0			0
$875$	11.0000	−	2.00000i	0.371868	−	0.0676123i
$876$		0			0
$877$			52.0000i			1.75592i	0.478738	+	0.877958i	$0.341094\pi$
$877$			52.0000i			1.75592i	−0.478738	+	0.877958i	$0.658906\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−16.0000			−0.539054			−0.269527	−	0.962993i	$-0.586867\pi$
$881$	−16.0000			−0.539054			−0.269527	+	0.962993i	$0.586867\pi$
$882$		0			0
$883$			16.0000i			0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$			16.0000i			0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			12.0000i			0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$			12.0000i			0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$		0			0
$889$	18.0000			0.603701
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	36.0000i		−	1.20469i
$894$		0			0
$895$	40.0000	+	20.0000i	1.33705	+	0.668526i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−6.00000			−0.200111
$900$		0			0
$901$	42.0000			1.39922
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.00000	+	2.00000i	0.132964	+	0.0664822i
$906$		0			0
$907$		−	14.0000i		−	0.464862i	−0.972613	−	0.232431i	$-0.925332\pi$
$907$		−	14.0000i		−	0.464862i	0.972613	−	0.232431i	$-0.0746680\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$			4.00000i			0.132381i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			6.00000i			0.198137i
$918$		0			0
$919$	−27.0000			−0.890648			−0.445324	−	0.895370i	$-0.646911\pi$
$919$	−27.0000			−0.890648			−0.445324	+	0.895370i	$0.646911\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	16.0000i		−	0.526646i
$924$		0			0
$925$	−8.00000	+	6.00000i	−0.263038	+	0.197279i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.0000			0.459325			0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000			0.459325			0.229663	+	0.973270i	$0.426238\pi$
$930$		0			0
$931$	4.00000			0.131095
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.00000	+	6.00000i	−0.0981105	+	0.196221i
$936$		0			0
$937$			1.00000i			0.0326686i	0.999867	+	0.0163343i	$0.00519960\pi$
$937$			1.00000i			0.0326686i	−0.999867	+	0.0163343i	$0.994800\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	28.0000			0.912774			0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000			0.912774			0.456387	+	0.889781i	$0.349143\pi$
$942$		0			0
$943$		−	20.0000i		−	0.651290i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			30.0000i			0.974869i	0.873160	+	0.487435i	$0.162067\pi$
$947$			30.0000i			0.974869i	−0.873160	+	0.487435i	$0.837933\pi$
$948$		0			0
$949$	10.0000			0.324614
$950$		0			0
$951$		0			0
$952$		0			0
$953$			30.0000i			0.971795i	0.874016	+	0.485898i	$0.161507\pi$
$953$			30.0000i			0.971795i	−0.874016	+	0.485898i	$0.838493\pi$
$954$		0			0
$955$	−6.00000	−	3.00000i	−0.194155	−	0.0970777i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.00000			−0.258333
$960$		0			0
$961$	5.00000			0.161290
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.00000	+	8.00000i	−0.128765	+	0.257529i
$966$		0			0
$967$			32.0000i			1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$			32.0000i			1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.0000			−0.577647			−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000			−0.577647			−0.288824	+	0.957382i	$0.593264\pi$
$972$		0			0
$973$		−	14.0000i		−	0.448819i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	40.0000i		−	1.27971i	−0.768494	−	0.639857i	$-0.778994\pi$
$977$		−	40.0000i		−	1.27971i	0.768494	−	0.639857i	$-0.221006\pi$
$978$		0			0
$979$	−12.0000			−0.383522
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1.00000i			0.0318950i	0.999873	+	0.0159475i	$0.00507647\pi$
$983$			1.00000i			0.0318950i	−0.999873	+	0.0159475i	$0.994924\pi$
$984$		0			0
$985$	6.00000	−	12.0000i	0.191176	−	0.382352i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−40.0000			−1.27064			−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000			−1.27064			−0.635321	+	0.772248i	$0.719132\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.0000	−	10.0000i	−0.634043	−	0.317021i
$996$		0			0
$997$		−	27.0000i		−	0.855099i	−0.903992	−	0.427549i	$-0.859377\pi$
$997$		−	27.0000i		−	0.855099i	0.903992	−	0.427549i	$-0.140623\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.t.a.1009.1		2
3.2	odd	2		560.2.g.d.449.2		2
4.3	odd	2		2520.2.t.a.1009.1		2
5.4	even	2	inner	5040.2.t.a.1009.2		2
12.11	even	2		280.2.g.a.169.1	✓	2
15.2	even	4		2800.2.a.u.1.1		1
15.8	even	4		2800.2.a.k.1.1		1
15.14	odd	2		560.2.g.d.449.1		2
20.19	odd	2		2520.2.t.a.1009.2		2
24.5	odd	2		2240.2.g.a.449.1		2
24.11	even	2		2240.2.g.b.449.2		2
60.23	odd	4		1400.2.a.j.1.1		1
60.47	odd	4		1400.2.a.d.1.1		1
60.59	even	2		280.2.g.a.169.2	yes	2
84.83	odd	2		1960.2.g.a.1569.2		2
120.29	odd	2		2240.2.g.a.449.2		2
120.59	even	2		2240.2.g.b.449.1		2
420.83	even	4		9800.2.a.p.1.1		1
420.167	even	4		9800.2.a.bb.1.1		1
420.419	odd	2		1960.2.g.a.1569.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.g.a.169.1	✓	2	12.11	even	2
280.2.g.a.169.2	yes	2	60.59	even	2
560.2.g.d.449.1		2	15.14	odd	2
560.2.g.d.449.2		2	3.2	odd	2
1400.2.a.d.1.1		1	60.47	odd	4
1400.2.a.j.1.1		1	60.23	odd	4
1960.2.g.a.1569.1		2	420.419	odd	2
1960.2.g.a.1569.2		2	84.83	odd	2
2240.2.g.a.449.1		2	24.5	odd	2
2240.2.g.a.449.2		2	120.29	odd	2
2240.2.g.b.449.1		2	120.59	even	2
2240.2.g.b.449.2		2	24.11	even	2
2520.2.t.a.1009.1		2	4.3	odd	2
2520.2.t.a.1009.2		2	20.19	odd	2
2800.2.a.k.1.1		1	15.8	even	4
2800.2.a.u.1.1		1	15.2	even	4
5040.2.t.a.1009.1		2	1.1	even	1	trivial
5040.2.t.a.1009.2		2	5.4	even	2	inner
9800.2.a.p.1.1		1	420.83	even	4
9800.2.a.bb.1.1		1	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)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{11} +1.00000i q^{13} +3.00000i q^{17} -4.00000 q^{19} -2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000 q^{29} +6.00000 q^{31} +(1.00000 - 2.00000i) q^{35} +2.00000i q^{37} +10.0000 q^{41} +9.00000i q^{47} -1.00000 q^{49} -14.0000i q^{53} +(2.00000 + 1.00000i) q^{55} -6.00000 q^{59} -4.00000 q^{61} +(1.00000 - 2.00000i) q^{65} -10.0000i q^{67} -16.0000 q^{71} -10.0000i q^{73} -1.00000i q^{77} -11.0000 q^{79} -4.00000i q^{83} +(3.00000 - 6.00000i) q^{85} +12.0000 q^{89} -1.00000 q^{91} +(8.00000 + 4.00000i) q^{95} -19.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 2 q^{11} - 8 q^{19} + 6 q^{25} - 2 q^{29} + 12 q^{31} + 2 q^{35} + 20 q^{41} - 2 q^{49} + 4 q^{55} - 12 q^{59} - 8 q^{61} + 2 q^{65} - 32 q^{71} - 22 q^{79} + 6 q^{85} + 24 q^{89} - 2 q^{91}+ \cdots + 16 q^{95}+O(q^{100}) \) 2 * q - 4 * q^5 - 2 * q^11 - 8 * q^19 + 6 * q^25 - 2 * q^29 + 12 * q^31 + 2 * q^35 + 20 * q^41 - 2 * q^49 + 4 * q^55 - 12 * q^59 - 8 * q^61 + 2 * q^65 - 32 * q^71 - 22 * q^79 + 6 * q^85 + 24 * q^89 - 2 * q^91 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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