LMFDB - Embedded newform 2040.2.m.b.409.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2040,2,Mod(409,2040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2040.409"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2040.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-4,0,0,0,-2,0,-4,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$16.2894820123$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			409.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2040.409
Dual form			2040.2.m.b.409.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-2.00000 + 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} -4.00000i q^{13} +(1.00000 + 2.00000i) q^{15} -1.00000i q^{17} -4.00000 q^{19} -4.00000 q^{21} +8.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} -4.00000 q^{31} +2.00000i q^{33} +(4.00000 + 8.00000i) q^{35} +2.00000i q^{37} -4.00000 q^{39} +12.0000 q^{41} +2.00000i q^{43} +(2.00000 - 1.00000i) q^{45} +8.00000i q^{47} -9.00000 q^{49} -1.00000 q^{51} -10.0000i q^{53} +(4.00000 - 2.00000i) q^{55} +4.00000i q^{57} -8.00000 q^{59} -6.00000 q^{61} +4.00000i q^{63} +(4.00000 + 8.00000i) q^{65} +2.00000i q^{67} +8.00000 q^{69} -6.00000 q^{71} +10.0000i q^{73} +(-4.00000 - 3.00000i) q^{75} +8.00000i q^{77} +1.00000 q^{81} +12.0000i q^{83} +(1.00000 + 2.00000i) q^{85} -2.00000 q^{89} -16.0000 q^{91} +4.00000i q^{93} +(8.00000 - 4.00000i) q^{95} -2.00000i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 2 q^{9} - 4 q^{11} + 2 q^{15} - 8 q^{19} - 8 q^{21} + 6 q^{25} - 8 q^{31} + 8 q^{35} - 8 q^{39} + 24 q^{41} + 4 q^{45} - 18 q^{49} - 2 q^{51} + 8 q^{55} - 16 q^{59} - 12 q^{61} + 8 q^{65}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q - 4 * q^5 - 2 * q^9 - 4 * q^11 + 2 * q^15 - 8 * q^19 - 8 * q^21 + 6 * q^25 - 8 * q^31 + 8 * q^35 - 8 * q^39 + 24 * q^41 + 4 * q^45 - 18 * q^49 - 2 * q^51 + 8 * q^55 - 16 * q^59 - 12 * q^61 + 8 * q^65 + 16 * q^69 - 12 * q^71 - 8 * q^75 + 2 * q^81 + 2 * q^85 - 4 * q^89 - 32 * q^91 + 16 * q^95 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2040\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1021$	$1361$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		0			0
$15$	1.00000	+	2.00000i	0.258199	+	0.516398i
$16$		0			0
$17$		−	1.00000i		−	0.242536i
$17$		−	1.00000i		−	0.242536i
$18$		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
$21$	−4.00000			−0.872872
$22$		0			0
$23$			8.00000i			1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$			8.00000i			1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$			2.00000i			0.348155i
$34$		0			0
$35$	4.00000	+	8.00000i	0.676123	+	1.35225i
$36$		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
$39$	−4.00000			−0.640513
$40$		0			0
$41$	12.0000			1.87409			0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000			1.87409			0.937043	+	0.349215i	$0.113552\pi$
$42$		0			0
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$		0			0
$45$	2.00000	−	1.00000i	0.298142	−	0.149071i
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$	−1.00000			−0.140028
$52$		0			0
$53$		−	10.0000i		−	1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$		−	10.0000i		−	1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$		0			0
$55$	4.00000	−	2.00000i	0.539360	−	0.269680i
$56$		0			0
$57$			4.00000i			0.529813i
$58$		0			0
$59$	−8.00000			−1.04151			−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000			−1.04151			−0.520756	+	0.853706i	$0.674350\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$			4.00000i			0.503953i
$64$		0			0
$65$	4.00000	+	8.00000i	0.496139	+	0.992278i
$66$		0			0
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$		0			0
$69$	8.00000			0.963087
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$			10.0000i			1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$			10.0000i			1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$		0			0
$75$	−4.00000	−	3.00000i	−0.461880	−	0.346410i
$76$		0			0
$77$			8.00000i			0.911685i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		0			0
$85$	1.00000	+	2.00000i	0.108465	+	0.216930i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$		0			0
$91$	−16.0000			−1.67726
$92$		0			0
$93$			4.00000i			0.414781i
$94$		0			0
$95$	8.00000	−	4.00000i	0.820783	−	0.410391i
$96$		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
$99$	2.00000			0.201008
$100$		0			0
$101$	−14.0000			−1.39305			−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000			−1.39305			−0.696526	+	0.717532i	$0.745272\pi$
$102$		0			0
$103$			18.0000i			1.77359i	0.462160	+	0.886796i	$0.347074\pi$
$103$			18.0000i			1.77359i	−0.462160	+	0.886796i	$0.652926\pi$
$104$		0			0
$105$	8.00000	−	4.00000i	0.780720	−	0.390360i
$106$		0			0
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$	2.00000			0.189832
$112$		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
$115$	−8.00000	−	16.0000i	−0.746004	−	1.49201i
$116$		0			0
$117$			4.00000i			0.369800i
$118$		0			0
$119$	−4.00000			−0.366679
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		−	12.0000i		−	1.08200i
$124$		0			0
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$		−	10.0000i		−	0.887357i	−0.896186	−	0.443678i	$-0.853673\pi$
$127$		−	10.0000i		−	0.887357i	0.896186	−	0.443678i	$-0.146327\pi$
$128$		0			0
$129$	2.00000			0.176090
$130$		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
$133$			16.0000i			1.38738i
$134$		0			0
$135$	−1.00000	−	2.00000i	−0.0860663	−	0.172133i
$136$		0			0
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$	8.00000			0.673722
$142$		0			0
$143$			8.00000i			0.668994i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			9.00000i			0.742307i
$148$		0			0
$149$	2.00000			0.163846			0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000			0.163846			0.0819232	+	0.996639i	$0.473894\pi$
$150$		0			0
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$		0			0
$153$			1.00000i			0.0808452i
$154$		0			0
$155$	8.00000	−	4.00000i	0.642575	−	0.321288i
$156$		0			0
$157$			12.0000i			0.957704i	0.877896	+	0.478852i	$0.158947\pi$
$157$			12.0000i			0.957704i	−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$	−10.0000			−0.793052
$160$		0			0
$161$	32.0000			2.52195
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$	−2.00000	−	4.00000i	−0.155700	−	0.311400i
$166$		0			0
$167$		−	8.00000i		−	0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$		−	8.00000i		−	0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	4.00000			0.305888
$172$		0			0
$173$		−	10.0000i		−	0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$		−	10.0000i		−	0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$		0			0
$175$	−16.0000	−	12.0000i	−1.20949	−	0.907115i
$176$		0			0
$177$			8.00000i			0.601317i
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	18.0000			1.33793			0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000			1.33793			0.668965	+	0.743294i	$0.266738\pi$
$182$		0			0
$183$			6.00000i			0.443533i
$184$		0			0
$185$	−2.00000	−	4.00000i	−0.147043	−	0.294086i
$186$		0			0
$187$			2.00000i			0.146254i
$188$		0			0
$189$	4.00000			0.290957
$190$		0			0
$191$	−16.0000			−1.15772			−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000			−1.15772			−0.578860	+	0.815427i	$0.696502\pi$
$192$		0			0
$193$			14.0000i			1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$			14.0000i			1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$		0			0
$195$	8.00000	−	4.00000i	0.572892	−	0.286446i
$196$		0			0
$197$		−	26.0000i		−	1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$		−	26.0000i		−	1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$		0			0
$199$	24.0000			1.70131			0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000			1.70131			0.850657	+	0.525720i	$0.176204\pi$
$200$		0			0
$201$	2.00000			0.141069
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−24.0000	+	12.0000i	−1.67623	+	0.838116i
$206$		0			0
$207$		−	8.00000i		−	0.556038i
$208$		0			0
$209$	8.00000			0.553372
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		0			0
$213$			6.00000i			0.411113i
$214$		0			0
$215$	−2.00000	−	4.00000i	−0.136399	−	0.272798i
$216$		0			0
$217$			16.0000i			1.08615i
$218$		0			0
$219$	10.0000			0.675737
$220$		0			0
$221$	−4.00000			−0.269069
$222$		0			0
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$		0			0
$225$	−3.00000	+	4.00000i	−0.200000	+	0.266667i
$226$		0			0
$227$			20.0000i			1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$			20.0000i			1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$		0			0
$229$	−10.0000			−0.660819			−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000			−0.660819			−0.330409	+	0.943838i	$0.607187\pi$
$230$		0			0
$231$	8.00000			0.526361
$232$		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
$235$	−8.00000	−	16.0000i	−0.521862	−	1.04372i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	18.0000	−	9.00000i	1.14998	−	0.574989i
$246$		0			0
$247$			16.0000i			1.01806i
$248$		0			0
$249$	12.0000			0.760469
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$		−	16.0000i		−	1.00591i
$254$		0			0
$255$	2.00000	−	1.00000i	0.125245	−	0.0626224i
$256$		0			0
$257$		−	14.0000i		−	0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$		−	14.0000i		−	0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$		0			0
$261$		0			0
$262$		0			0
$263$			16.0000i			0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$			16.0000i			0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$		0			0
$265$	10.0000	+	20.0000i	0.614295	+	1.22859i
$266$		0			0
$267$			2.00000i			0.122398i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−24.0000			−1.45790			−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000			−1.45790			−0.728948	+	0.684569i	$0.759990\pi$
$272$		0			0
$273$			16.0000i			0.968364i
$274$		0			0
$275$	−6.00000	+	8.00000i	−0.361814	+	0.482418i
$276$		0			0
$277$		−	26.0000i		−	1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$		−	26.0000i		−	1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$		0			0
$279$	4.00000			0.239474
$280$		0			0
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$		−	16.0000i		−	0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$		−	16.0000i		−	0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$		0			0
$285$	−4.00000	−	8.00000i	−0.236940	−	0.473879i
$286$		0			0
$287$		−	48.0000i		−	2.83335i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	−2.00000			−0.117242
$292$		0			0
$293$			10.0000i			0.584206i	0.956387	+	0.292103i	$0.0943550\pi$
$293$			10.0000i			0.584206i	−0.956387	+	0.292103i	$0.905645\pi$
$294$		0			0
$295$	16.0000	−	8.00000i	0.931556	−	0.465778i
$296$		0			0
$297$		−	2.00000i		−	0.116052i
$298$		0			0
$299$	32.0000			1.85061
$300$		0			0
$301$	8.00000			0.461112
$302$		0			0
$303$			14.0000i			0.804279i
$304$		0			0
$305$	12.0000	−	6.00000i	0.687118	−	0.343559i
$306$		0			0
$307$			6.00000i			0.342438i	0.985233	+	0.171219i	$0.0547706\pi$
$307$			6.00000i			0.342438i	−0.985233	+	0.171219i	$0.945229\pi$
$308$		0			0
$309$	18.0000			1.02398
$310$		0			0
$311$	−18.0000			−1.02069			−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000			−1.02069			−0.510343	+	0.859971i	$0.670482\pi$
$312$		0			0
$313$		−	10.0000i		−	0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$		−	10.0000i		−	0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$		0			0
$315$	−4.00000	−	8.00000i	−0.225374	−	0.450749i
$316$		0			0
$317$		−	30.0000i		−	1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$		−	30.0000i		−	1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$			4.00000i			0.222566i
$324$		0			0
$325$	−16.0000	−	12.0000i	−0.887520	−	0.665640i
$326$		0			0
$327$			2.00000i			0.110600i
$328$		0			0
$329$	32.0000			1.76422
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$		0			0
$333$		−	2.00000i		−	0.109599i
$334$		0			0
$335$	−2.00000	−	4.00000i	−0.109272	−	0.218543i
$336$		0			0
$337$		−	26.0000i		−	1.41631i	−0.706057	−	0.708155i	$-0.749528\pi$
$337$		−	26.0000i		−	1.41631i	0.706057	−	0.708155i	$-0.250472\pi$
$338$		0			0
$339$	14.0000			0.760376
$340$		0			0
$341$	8.00000			0.433224
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$	−16.0000	+	8.00000i	−0.861411	+	0.430706i
$346$		0			0
$347$			36.0000i			1.93258i	0.257454	+	0.966291i	$0.417117\pi$
$347$			36.0000i			1.93258i	−0.257454	+	0.966291i	$0.582883\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	4.00000			0.213504
$352$		0			0
$353$			26.0000i			1.38384i	0.721974	+	0.691920i	$0.243235\pi$
$353$			26.0000i			1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$354$		0			0
$355$	12.0000	−	6.00000i	0.636894	−	0.318447i
$356$		0			0
$357$			4.00000i			0.211702i
$358$		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
$363$			7.00000i			0.367405i
$364$		0			0
$365$	−10.0000	−	20.0000i	−0.523424	−	1.04685i
$366$		0			0
$367$		−	32.0000i		−	1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$		−	32.0000i		−	1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$		0			0
$369$	−12.0000			−0.624695
$370$		0			0
$371$	−40.0000			−2.07670
$372$		0			0
$373$		−	24.0000i		−	1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$		−	24.0000i		−	1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$		0			0
$375$	11.0000	+	2.00000i	0.568038	+	0.103280i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$	−10.0000			−0.512316
$382$		0			0
$383$			12.0000i			0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$			12.0000i			0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$		0			0
$385$	−8.00000	−	16.0000i	−0.407718	−	0.815436i
$386$		0			0
$387$		−	2.00000i		−	0.101666i
$388$		0			0
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$		−	6.00000i		−	0.302660i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			26.0000i			1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$			26.0000i			1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$		0			0
$399$	16.0000			0.801002
$400$		0			0
$401$	−12.0000			−0.599251			−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000			−0.599251			−0.299626	+	0.954057i	$0.596862\pi$
$402$		0			0
$403$			16.0000i			0.797017i
$404$		0			0
$405$	−2.00000	+	1.00000i	−0.0993808	+	0.0496904i
$406$		0			0
$407$		−	4.00000i		−	0.198273i
$408$		0			0
$409$	6.00000			0.296681			0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000			0.296681			0.148340	+	0.988936i	$0.452607\pi$
$410$		0			0
$411$	2.00000			0.0986527
$412$		0			0
$413$			32.0000i			1.57462i
$414$		0			0
$415$	−12.0000	−	24.0000i	−0.589057	−	1.17811i
$416$		0			0
$417$		−	4.00000i		−	0.195881i
$418$		0			0
$419$	−30.0000			−1.46560			−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000			−1.46560			−0.732798	+	0.680446i	$0.761786\pi$
$420$		0			0
$421$	−30.0000			−1.46211			−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000			−1.46211			−0.731055	+	0.682318i	$0.760972\pi$
$422$		0			0
$423$		−	8.00000i		−	0.388973i
$424$		0			0
$425$	−4.00000	−	3.00000i	−0.194029	−	0.145521i
$426$		0			0
$427$			24.0000i			1.16144i
$428$		0			0
$429$	8.00000			0.386244
$430$		0			0
$431$	34.0000			1.63772			0.818861	−	0.573992i	$-0.194606\pi$
$431$	34.0000			1.63772			0.818861	+	0.573992i	$0.194606\pi$
$432$		0			0
$433$			16.0000i			0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$			16.0000i			0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	32.0000i		−	1.53077i
$438$		0			0
$439$	28.0000			1.33637			0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000			1.33637			0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	4.00000	−	2.00000i	0.189618	−	0.0948091i
$446$		0			0
$447$		−	2.00000i		−	0.0945968i
$448$		0			0
$449$	36.0000			1.69895			0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000			1.69895			0.849473	+	0.527633i	$0.176920\pi$
$450$		0			0
$451$	−24.0000			−1.13012
$452$		0			0
$453$			16.0000i			0.751746i
$454$		0			0
$455$	32.0000	−	16.0000i	1.50018	−	0.750092i
$456$		0			0
$457$		−	28.0000i		−	1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$		−	28.0000i		−	1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$		0			0
$459$	1.00000			0.0466760
$460$		0			0
$461$	−22.0000			−1.02464			−0.512321	−	0.858794i	$-0.671214\pi$
$461$	−22.0000			−1.02464			−0.512321	+	0.858794i	$0.671214\pi$
$462$		0			0
$463$		−	34.0000i		−	1.58011i	−0.613033	−	0.790057i	$-0.710051\pi$
$463$		−	34.0000i		−	1.58011i	0.613033	−	0.790057i	$-0.289949\pi$
$464$		0			0
$465$	−4.00000	−	8.00000i	−0.185496	−	0.370991i
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$	12.0000			0.552931
$472$		0			0
$473$		−	4.00000i		−	0.183920i
$474$		0			0
$475$	−12.0000	+	16.0000i	−0.550598	+	0.734130i
$476$		0			0
$477$			10.0000i			0.457869i
$478$		0			0
$479$	−34.0000			−1.55350			−0.776750	−	0.629809i	$-0.783133\pi$
$479$	−34.0000			−1.55350			−0.776750	+	0.629809i	$0.783133\pi$
$480$		0			0
$481$	8.00000			0.364769
$482$		0			0
$483$		−	32.0000i		−	1.45605i
$484$		0			0
$485$	2.00000	+	4.00000i	0.0908153	+	0.181631i
$486$		0			0
$487$		−	32.0000i		−	1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$		−	32.0000i		−	1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−4.00000	+	2.00000i	−0.179787	+	0.0898933i
$496$		0			0
$497$			24.0000i			1.07655i
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$	−8.00000			−0.357414
$502$		0			0
$503$		−	24.0000i		−	1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$		−	24.0000i		−	1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$		0			0
$505$	28.0000	−	14.0000i	1.24598	−	0.622992i
$506$		0			0
$507$			3.00000i			0.133235i
$508$		0			0
$509$	−10.0000			−0.443242			−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000			−0.443242			−0.221621	+	0.975133i	$0.571135\pi$
$510$		0			0
$511$	40.0000			1.76950
$512$		0			0
$513$		−	4.00000i		−	0.176604i
$514$		0			0
$515$	−18.0000	−	36.0000i	−0.793175	−	1.58635i
$516$		0			0
$517$		−	16.0000i		−	0.703679i
$518$		0			0
$519$	−10.0000			−0.438951
$520$		0			0
$521$	−4.00000			−0.175243			−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000			−0.175243			−0.0876216	+	0.996154i	$0.527927\pi$
$522$		0			0
$523$			2.00000i			0.0874539i	0.999044	+	0.0437269i	$0.0139232\pi$
$523$			2.00000i			0.0874539i	−0.999044	+	0.0437269i	$0.986077\pi$
$524$		0			0
$525$	−12.0000	+	16.0000i	−0.523723	+	0.698297i
$526$		0			0
$527$			4.00000i			0.174243i
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$	8.00000			0.347170
$532$		0			0
$533$		−	48.0000i		−	2.07911i
$534$		0			0
$535$	12.0000	+	24.0000i	0.518805	+	1.03761i
$536$		0			0
$537$			12.0000i			0.517838i
$538$		0			0
$539$	18.0000			0.775315
$540$		0			0
$541$	−14.0000			−0.601907			−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000			−0.601907			−0.300954	+	0.953639i	$0.597305\pi$
$542$		0			0
$543$		−	18.0000i		−	0.772454i
$544$		0			0
$545$	4.00000	−	2.00000i	0.171341	−	0.0856706i
$546$		0			0
$547$		−	20.0000i		−	0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$		−	20.0000i		−	0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$		0			0
$549$	6.00000			0.256074
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−4.00000	+	2.00000i	−0.169791	+	0.0848953i
$556$		0			0
$557$			18.0000i			0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$			18.0000i			0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$	2.00000			0.0844401
$562$		0			0
$563$			16.0000i			0.674320i	0.941447	+	0.337160i	$0.109466\pi$
$563$			16.0000i			0.674320i	−0.941447	+	0.337160i	$0.890534\pi$
$564$		0			0
$565$	−14.0000	−	28.0000i	−0.588984	−	1.17797i
$566$		0			0
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$	40.0000			1.67395			0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000			1.67395			0.836974	+	0.547243i	$0.184323\pi$
$572$		0			0
$573$			16.0000i			0.668410i
$574$		0			0
$575$	32.0000	+	24.0000i	1.33449	+	1.00087i
$576$		0			0
$577$		−	4.00000i		−	0.166522i	−0.996528	−	0.0832611i	$-0.973466\pi$
$577$		−	4.00000i		−	0.166522i	0.996528	−	0.0832611i	$-0.0265335\pi$
$578$		0			0
$579$	14.0000			0.581820
$580$		0			0
$581$	48.0000			1.99138
$582$		0			0
$583$			20.0000i			0.828315i
$584$		0			0
$585$	−4.00000	−	8.00000i	−0.165380	−	0.330759i
$586$		0			0
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$		0			0
$591$	−26.0000			−1.06950
$592$		0			0
$593$			14.0000i			0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$			14.0000i			0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$		0			0
$595$	8.00000	−	4.00000i	0.327968	−	0.163984i
$596$		0			0
$597$		−	24.0000i		−	0.982255i
$598$		0			0
$599$	20.0000			0.817178			0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000			0.817178			0.408589	+	0.912719i	$0.366021\pi$
$600$		0			0
$601$	14.0000			0.571072			0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000			0.571072			0.285536	+	0.958368i	$0.407828\pi$
$602$		0			0
$603$		−	2.00000i		−	0.0814463i
$604$		0			0
$605$	14.0000	−	7.00000i	0.569181	−	0.284590i
$606$		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
$609$		0			0
$610$		0			0
$611$	32.0000			1.29458
$612$		0			0
$613$			20.0000i			0.807792i	0.914805	+	0.403896i	$0.132344\pi$
$613$			20.0000i			0.807792i	−0.914805	+	0.403896i	$0.867656\pi$
$614$		0			0
$615$	12.0000	+	24.0000i	0.483887	+	0.967773i
$616$		0			0
$617$		−	22.0000i		−	0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$		−	22.0000i		−	0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$		0			0
$619$	16.0000			0.643094			0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000			0.643094			0.321547	+	0.946894i	$0.395797\pi$
$620$		0			0
$621$	−8.00000			−0.321029
$622$		0			0
$623$			8.00000i			0.320513i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		−	8.00000i		−	0.319489i
$628$		0			0
$629$	2.00000			0.0797452
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$			8.00000i			0.317971i
$634$		0			0
$635$	10.0000	+	20.0000i	0.396838	+	0.793676i
$636$		0			0
$637$			36.0000i			1.42637i
$638$		0			0
$639$	6.00000			0.237356
$640$		0			0
$641$	48.0000			1.89589			0.947943	−	0.318440i	$-0.103159\pi$
$641$	48.0000			1.89589			0.947943	+	0.318440i	$0.103159\pi$
$642$		0			0
$643$			28.0000i			1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$			28.0000i			1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$		0			0
$645$	−4.00000	+	2.00000i	−0.157500	+	0.0787499i
$646$		0			0
$647$			12.0000i			0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$			12.0000i			0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$		0			0
$649$	16.0000			0.628055
$650$		0			0
$651$	16.0000			0.627089
$652$		0			0
$653$			42.0000i			1.64359i	0.569785	+	0.821794i	$0.307026\pi$
$653$			42.0000i			1.64359i	−0.569785	+	0.821794i	$0.692974\pi$
$654$		0			0
$655$	−12.0000	+	6.00000i	−0.468879	+	0.234439i
$656$		0			0
$657$		−	10.0000i		−	0.390137i
$658$		0			0
$659$	−16.0000			−0.623272			−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000			−0.623272			−0.311636	+	0.950202i	$0.600877\pi$
$660$		0			0
$661$	38.0000			1.47803			0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000			1.47803			0.739014	+	0.673690i	$0.235292\pi$
$662$		0			0
$663$			4.00000i			0.155347i
$664$		0			0
$665$	−16.0000	−	32.0000i	−0.620453	−	1.24091i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−14.0000			−0.541271
$670$		0			0
$671$	12.0000			0.463255
$672$		0			0
$673$			38.0000i			1.46479i	0.680879	+	0.732396i	$0.261598\pi$
$673$			38.0000i			1.46479i	−0.680879	+	0.732396i	$0.738402\pi$
$674$		0			0
$675$	4.00000	+	3.00000i	0.153960	+	0.115470i
$676$		0			0
$677$		−	6.00000i		−	0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$		−	6.00000i		−	0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$		0			0
$679$	−8.00000			−0.307012
$680$		0			0
$681$	20.0000			0.766402
$682$		0			0
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$		0			0
$685$	−2.00000	−	4.00000i	−0.0764161	−	0.152832i
$686$		0			0
$687$			10.0000i			0.381524i
$688$		0			0
$689$	−40.0000			−1.52388
$690$		0			0
$691$	4.00000			0.152167			0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000			0.152167			0.0760836	+	0.997101i	$0.475758\pi$
$692$		0			0
$693$		−	8.00000i		−	0.303895i
$694$		0			0
$695$	−8.00000	+	4.00000i	−0.303457	+	0.151729i
$696$		0			0
$697$		−	12.0000i		−	0.454532i
$698$		0			0
$699$	6.00000			0.226941
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$		−	8.00000i		−	0.301726i
$704$		0			0
$705$	−16.0000	+	8.00000i	−0.602595	+	0.301297i
$706$		0			0
$707$			56.0000i			2.10610i
$708$		0			0
$709$	−22.0000			−0.826227			−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000			−0.826227			−0.413114	+	0.910679i	$0.635559\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	32.0000i		−	1.19841i
$714$		0			0
$715$	−8.00000	−	16.0000i	−0.299183	−	0.598366i
$716$		0			0
$717$			12.0000i			0.448148i
$718$		0			0
$719$	−6.00000			−0.223762			−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000			−0.223762			−0.111881	+	0.993722i	$0.535688\pi$
$720$		0			0
$721$	72.0000			2.68142
$722$		0			0
$723$		−	10.0000i		−	0.371904i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	38.0000i		−	1.40934i	−0.709534	−	0.704671i	$-0.751095\pi$
$727$		−	38.0000i		−	1.40934i	0.709534	−	0.704671i	$-0.248905\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	2.00000			0.0739727
$732$		0			0
$733$		−	36.0000i		−	1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$		−	36.0000i		−	1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$		0			0
$735$	−9.00000	−	18.0000i	−0.331970	−	0.663940i
$736$		0			0
$737$		−	4.00000i		−	0.147342i
$738$		0			0
$739$	12.0000			0.441427			0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000			0.441427			0.220714	+	0.975339i	$0.429161\pi$
$740$		0			0
$741$	16.0000			0.587775
$742$		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
$745$	−4.00000	+	2.00000i	−0.146549	+	0.0732743i
$746$		0			0
$747$		−	12.0000i		−	0.439057i
$748$		0			0
$749$	−48.0000			−1.75388
$750$		0			0
$751$	−48.0000			−1.75154			−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000			−1.75154			−0.875772	+	0.482724i	$0.839647\pi$
$752$		0			0
$753$		−	12.0000i		−	0.437304i
$754$		0			0
$755$	32.0000	−	16.0000i	1.16460	−	0.582300i
$756$		0			0
$757$		−	48.0000i		−	1.74459i	−0.488980	−	0.872295i	$-0.662631\pi$
$757$		−	48.0000i		−	1.74459i	0.488980	−	0.872295i	$-0.337369\pi$
$758$		0			0
$759$	−16.0000			−0.580763
$760$		0			0
$761$	46.0000			1.66750			0.833749	−	0.552143i	$-0.186190\pi$
$761$	46.0000			1.66750			0.833749	+	0.552143i	$0.186190\pi$
$762$		0			0
$763$			8.00000i			0.289619i
$764$		0			0
$765$	−1.00000	−	2.00000i	−0.0361551	−	0.0723102i
$766$		0			0
$767$			32.0000i			1.15545i
$768$		0			0
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$	−14.0000			−0.504198
$772$		0			0
$773$			46.0000i			1.65451i	0.561830	+	0.827253i	$0.310097\pi$
$773$			46.0000i			1.65451i	−0.561830	+	0.827253i	$0.689903\pi$
$774$		0			0
$775$	−12.0000	+	16.0000i	−0.431053	+	0.574737i
$776$		0			0
$777$		−	8.00000i		−	0.286998i
$778$		0			0
$779$	−48.0000			−1.71978
$780$		0			0
$781$	12.0000			0.429394
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−12.0000	−	24.0000i	−0.428298	−	0.856597i
$786$		0			0
$787$			4.00000i			0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$			4.00000i			0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$		0			0
$789$	16.0000			0.569615
$790$		0			0
$791$	56.0000			1.99113
$792$		0			0
$793$			24.0000i			0.852265i
$794$		0			0
$795$	20.0000	−	10.0000i	0.709327	−	0.354663i
$796$		0			0
$797$		−	26.0000i		−	0.920967i	−0.887668	−	0.460484i	$-0.847676\pi$
$797$		−	26.0000i		−	0.920967i	0.887668	−	0.460484i	$-0.152324\pi$
$798$		0			0
$799$	8.00000			0.283020
$800$		0			0
$801$	2.00000			0.0706665
$802$		0			0
$803$		−	20.0000i		−	0.705785i
$804$		0			0
$805$	−64.0000	+	32.0000i	−2.25570	+	1.12785i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−44.0000			−1.54696			−0.773479	−	0.633822i	$-0.781485\pi$
$809$	−44.0000			−1.54696			−0.773479	+	0.633822i	$0.781485\pi$
$810$		0			0
$811$	−52.0000			−1.82597			−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000			−1.82597			−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$			24.0000i			0.841717i
$814$		0			0
$815$	4.00000	+	8.00000i	0.140114	+	0.280228i
$816$		0			0
$817$		−	8.00000i		−	0.279885i
$818$		0			0
$819$	16.0000			0.559085
$820$		0			0
$821$	−52.0000			−1.81481			−0.907406	−	0.420255i	$-0.861941\pi$
$821$	−52.0000			−1.81481			−0.907406	+	0.420255i	$0.861941\pi$
$822$		0			0
$823$			44.0000i			1.53374i	0.641800	+	0.766872i	$0.278188\pi$
$823$			44.0000i			1.53374i	−0.641800	+	0.766872i	$0.721812\pi$
$824$		0			0
$825$	8.00000	+	6.00000i	0.278524	+	0.208893i
$826$		0			0
$827$			12.0000i			0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$			12.0000i			0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$		0			0
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$		0			0
$831$	−26.0000			−0.901930
$832$		0			0
$833$			9.00000i			0.311832i
$834$		0			0
$835$	8.00000	+	16.0000i	0.276851	+	0.553703i
$836$		0			0
$837$		−	4.00000i		−	0.138260i
$838$		0			0
$839$	−50.0000			−1.72619			−0.863096	−	0.505040i	$-0.831478\pi$
$839$	−50.0000			−1.72619			−0.863096	+	0.505040i	$0.831478\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		−	18.0000i		−	0.619953i
$844$		0			0
$845$	6.00000	−	3.00000i	0.206406	−	0.103203i
$846$		0			0
$847$			28.0000i			0.962091i
$848$		0			0
$849$	−16.0000			−0.549119
$850$		0			0
$851$	−16.0000			−0.548473
$852$		0			0
$853$			26.0000i			0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$			26.0000i			0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$		0			0
$855$	−8.00000	+	4.00000i	−0.273594	+	0.136797i
$856$		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
$859$	−28.0000			−0.955348			−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000			−0.955348			−0.477674	+	0.878537i	$0.658520\pi$
$860$		0			0
$861$	−48.0000			−1.63584
$862$		0			0
$863$		−	4.00000i		−	0.136162i	−0.997680	−	0.0680808i	$-0.978312\pi$
$863$		−	4.00000i		−	0.136162i	0.997680	−	0.0680808i	$-0.0216876\pi$
$864$		0			0
$865$	10.0000	+	20.0000i	0.340010	+	0.680020i
$866$		0			0
$867$			1.00000i			0.0339618i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	8.00000			0.271070
$872$		0			0
$873$			2.00000i			0.0676897i
$874$		0			0
$875$	44.0000	+	8.00000i	1.48747	+	0.270449i
$876$		0			0
$877$			50.0000i			1.68838i	0.536044	+	0.844190i	$0.319918\pi$
$877$			50.0000i			1.68838i	−0.536044	+	0.844190i	$0.680082\pi$
$878$		0			0
$879$	10.0000			0.337292
$880$		0			0
$881$	−44.0000			−1.48240			−0.741199	−	0.671286i	$-0.765742\pi$
$881$	−44.0000			−1.48240			−0.741199	+	0.671286i	$0.765742\pi$
$882$		0			0
$883$		−	6.00000i		−	0.201916i	−0.994891	−	0.100958i	$-0.967809\pi$
$883$		−	6.00000i		−	0.201916i	0.994891	−	0.100958i	$-0.0321908\pi$
$884$		0			0
$885$	−8.00000	−	16.0000i	−0.268917	−	0.537834i
$886$		0			0
$887$		−	8.00000i		−	0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$		−	8.00000i		−	0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$		0			0
$889$	−40.0000			−1.34156
$890$		0			0
$891$	−2.00000			−0.0670025
$892$		0			0
$893$		−	32.0000i		−	1.07084i
$894$		0			0
$895$	24.0000	−	12.0000i	0.802232	−	0.401116i
$896$		0			0
$897$		−	32.0000i		−	1.06845i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−10.0000			−0.333148
$902$		0			0
$903$		−	8.00000i		−	0.266223i
$904$		0			0
$905$	−36.0000	+	18.0000i	−1.19668	+	0.598340i
$906$		0			0
$907$		−	32.0000i		−	1.06254i	−0.847202	−	0.531271i	$-0.821714\pi$
$907$		−	32.0000i		−	1.06254i	0.847202	−	0.531271i	$-0.178286\pi$
$908$		0			0
$909$	14.0000			0.464351
$910$		0			0
$911$	14.0000			0.463841			0.231920	−	0.972735i	$-0.425499\pi$
$911$	14.0000			0.463841			0.231920	+	0.972735i	$0.425499\pi$
$912$		0			0
$913$		−	24.0000i		−	0.794284i
$914$		0			0
$915$	−6.00000	−	12.0000i	−0.198354	−	0.396708i
$916$		0			0
$917$		−	24.0000i		−	0.792550i
$918$		0			0
$919$	56.0000			1.84727			0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000			1.84727			0.923635	+	0.383274i	$0.125203\pi$
$920$		0			0
$921$	6.00000			0.197707
$922$		0			0
$923$			24.0000i			0.789970i
$924$		0			0
$925$	8.00000	+	6.00000i	0.263038	+	0.197279i
$926$		0			0
$927$		−	18.0000i		−	0.591198i
$928$		0			0
$929$	−48.0000			−1.57483			−0.787414	−	0.616424i	$-0.788581\pi$
$929$	−48.0000			−1.57483			−0.787414	+	0.616424i	$0.788581\pi$
$930$		0			0
$931$	36.0000			1.17985
$932$		0			0
$933$			18.0000i			0.589294i
$934$		0			0
$935$	−2.00000	−	4.00000i	−0.0654070	−	0.130814i
$936$		0			0
$937$			36.0000i			1.17607i	0.808836	+	0.588034i	$0.200098\pi$
$937$			36.0000i			1.17607i	−0.808836	+	0.588034i	$0.799902\pi$
$938$		0			0
$939$	−10.0000			−0.326338
$940$		0			0
$941$	48.0000			1.56476			0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000			1.56476			0.782378	+	0.622804i	$0.214007\pi$
$942$		0			0
$943$			96.0000i			3.12619i
$944$		0			0
$945$	−8.00000	+	4.00000i	−0.260240	+	0.130120i
$946$		0			0
$947$			36.0000i			1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$			36.0000i			1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$		0			0
$949$	40.0000			1.29845
$950$		0			0
$951$	−30.0000			−0.972817
$952$		0			0
$953$		−	42.0000i		−	1.36051i	−0.732974	−	0.680257i	$-0.761868\pi$
$953$		−	42.0000i		−	1.36051i	0.732974	−	0.680257i	$-0.238132\pi$
$954$		0			0
$955$	32.0000	−	16.0000i	1.03550	−	0.517748i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	8.00000			0.258333
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$			12.0000i			0.386695i
$964$		0			0
$965$	−14.0000	−	28.0000i	−0.450676	−	0.901352i
$966$		0			0
$967$			34.0000i			1.09337i	0.837340	+	0.546683i	$0.184110\pi$
$967$			34.0000i			1.09337i	−0.837340	+	0.546683i	$0.815890\pi$
$968$		0			0
$969$	4.00000			0.128499
$970$		0			0
$971$	−48.0000			−1.54039			−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000			−1.54039			−0.770197	+	0.637806i	$0.779842\pi$
$972$		0			0
$973$		−	16.0000i		−	0.512936i
$974$		0			0
$975$	−12.0000	+	16.0000i	−0.384308	+	0.512410i
$976$		0			0
$977$		−	2.00000i		−	0.0639857i	−0.999488	−	0.0319928i	$-0.989815\pi$
$977$		−	2.00000i		−	0.0639857i	0.999488	−	0.0319928i	$-0.0101854\pi$
$978$		0			0
$979$	4.00000			0.127841
$980$		0			0
$981$	2.00000			0.0638551
$982$		0			0
$983$		−	8.00000i		−	0.255160i	−0.991828	−	0.127580i	$-0.959279\pi$
$983$		−	8.00000i		−	0.255160i	0.991828	−	0.127580i	$-0.0407210\pi$
$984$		0			0
$985$	26.0000	+	52.0000i	0.828429	+	1.65686i
$986$		0			0
$987$		−	32.0000i		−	1.01857i
$988$		0			0
$989$	−16.0000			−0.508770
$990$		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
$993$			20.0000i			0.634681i
$994$		0			0
$995$	−48.0000	+	24.0000i	−1.52170	+	0.760851i
$996$		0			0
$997$			22.0000i			0.696747i	0.937356	+	0.348373i	$0.113266\pi$
$997$			22.0000i			0.696747i	−0.937356	+	0.348373i	$0.886734\pi$
$998$		0			0
$999$	−2.00000			−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2040.2.m.b.409.1	✓	2
4.3	odd	2		4080.2.m.c.2449.2		2
5.4	even	2	inner	2040.2.m.b.409.2	yes	2
20.19	odd	2		4080.2.m.c.2449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.2.m.b.409.1	✓	2	1.1	even	1	trivial
2040.2.m.b.409.2	yes	2	5.4	even	2	inner
4080.2.m.c.2449.1		2	20.19	odd	2
4080.2.m.c.2449.2		2	4.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2040.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-2.00000 + 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} -4.00000i q^{13} +(1.00000 + 2.00000i) q^{15} -1.00000i q^{17} -4.00000 q^{19} -4.00000 q^{21} +8.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} -4.00000 q^{31} +2.00000i q^{33} +(4.00000 + 8.00000i) q^{35} +2.00000i q^{37} -4.00000 q^{39} +12.0000 q^{41} +2.00000i q^{43} +(2.00000 - 1.00000i) q^{45} +8.00000i q^{47} -9.00000 q^{49} -1.00000 q^{51} -10.0000i q^{53} +(4.00000 - 2.00000i) q^{55} +4.00000i q^{57} -8.00000 q^{59} -6.00000 q^{61} +4.00000i q^{63} +(4.00000 + 8.00000i) q^{65} +2.00000i q^{67} +8.00000 q^{69} -6.00000 q^{71} +10.0000i q^{73} +(-4.00000 - 3.00000i) q^{75} +8.00000i q^{77} +1.00000 q^{81} +12.0000i q^{83} +(1.00000 + 2.00000i) q^{85} -2.00000 q^{89} -16.0000 q^{91} +4.00000i q^{93} +(8.00000 - 4.00000i) q^{95} -2.00000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 2 q^{9} - 4 q^{11} + 2 q^{15} - 8 q^{19} - 8 q^{21} + 6 q^{25} - 8 q^{31} + 8 q^{35} - 8 q^{39} + 24 q^{41} + 4 q^{45} - 18 q^{49} - 2 q^{51} + 8 q^{55} - 16 q^{59} - 12 q^{61} + 8 q^{65}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q - 4 * q^5 - 2 * q^9 - 4 * q^11 + 2 * q^15 - 8 * q^19 - 8 * q^21 + 6 * q^25 - 8 * q^31 + 8 * q^35 - 8 * q^39 + 24 * q^41 + 4 * q^45 - 18 * q^49 - 2 * q^51 + 8 * q^55 - 16 * q^59 - 12 * q^61 + 8 * q^65 + 16 * q^69 - 12 * q^71 - 8 * q^75 + 2 * q^81 + 2 * q^85 - 4 * q^89 - 32 * q^91 + 16 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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