LMFDB - Embedded newform 1008.3.cd.o.991.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(415,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 2])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.415"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.cd (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.364488705441.8
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 104x^{4} + 1815x^{2} + 14641 $ x^8 + 15x^6 + 104x^4 + 1815*x^2 + 14641
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			991.1
Root			$-0.375047 - 3.29535i$ of defining polynomial
Character	$\chi$	$=$	1008.991
Dual form			1008.3.cd.o.415.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.04138 - 5.26783i) q^{5} +(-2.29129 - 6.61438i) q^{7} +(13.9374 + 8.04674i) q^{11} +14.0000 q^{13} +(-6.08276 + 10.5357i) q^{17} +(4.58258 - 2.64575i) q^{19} +(27.8747 - 16.0935i) q^{23} +(-6.00000 + 10.3923i) q^{25} +42.5793 q^{29} +(-20.6216 - 11.9059i) q^{31} +(-27.8747 + 32.1870i) q^{35} +(-19.0000 - 32.9090i) q^{37} -24.3311 q^{41} -74.0810i q^{43} +(-38.5000 + 30.3109i) q^{49} +(-21.2897 + 36.8748i) q^{53} -97.8928i q^{55} +(-97.5615 - 56.3272i) q^{59} +(-42.5793 - 73.7496i) q^{65} +(-96.2341 - 55.5608i) q^{67} -64.3739i q^{71} +(7.00000 - 12.1244i) q^{73} +(21.2897 - 110.624i) q^{77} +(48.1170 - 27.7804i) q^{79} +112.654i q^{83} +74.0000 q^{85} +(66.9104 + 115.892i) q^{89} +(-32.0780 - 92.6013i) q^{91} +(-27.8747 - 16.0935i) q^{95} +35.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 112 q^{13} - 48 q^{25} - 152 q^{37} - 308 q^{49} + 56 q^{73} + 592 q^{85} + 280 q^{97}+O(q^{100}) $ 8 * q + 112 * q^13 - 48 * q^25 - 152 * q^37 - 308 * q^49 + 56 * q^73 + 592 * q^85 + 280 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$5$	−3.04138	−	5.26783i	−0.608276	−	1.05357i	−0.991524	−	0.129920i	$-0.958528\pi$
$5$	−3.04138	−	5.26783i	−0.608276	−	1.05357i	0.383248	−	0.923645i	$-0.374805\pi$
$6$		0			0
$7$	−2.29129	−	6.61438i	−0.327327	−	0.944911i
$7$	−2.29129	−	6.61438i	−0.327327	−	0.944911i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	13.9374	+	8.04674i	1.26703	+	0.731522i	0.974425	−	0.224711i	$-0.0721438\pi$
$11$	13.9374	+	8.04674i	1.26703	+	0.731522i	0.292607	+	0.956233i	$0.405477\pi$
$12$		0			0
$13$	14.0000			1.07692			0.538462	−	0.842650i	$-0.319006\pi$
$13$	14.0000			1.07692			0.538462	+	0.842650i	$0.319006\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.08276	+	10.5357i	−0.357810	+	0.619744i	−0.987595	−	0.157025i	$-0.949810\pi$
$17$	−6.08276	+	10.5357i	−0.357810	+	0.619744i	0.629785	+	0.776769i	$0.283143\pi$
$18$		0			0
$19$	4.58258	−	2.64575i	0.241188	−	0.139250i	−0.374535	−	0.927213i	$-0.622197\pi$
$19$	4.58258	−	2.64575i	0.241188	−	0.139250i	0.615723	+	0.787963i	$0.288864\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	27.8747	−	16.0935i	1.21194	−	0.699716i	0.248762	−	0.968565i	$-0.419976\pi$
$23$	27.8747	−	16.0935i	1.21194	−	0.699716i	0.963183	+	0.268848i	$0.0866430\pi$
$24$		0			0
$25$	−6.00000	+	10.3923i	−0.240000	+	0.415692i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	42.5793			1.46825			0.734127	−	0.679013i	$-0.237592\pi$
$29$	42.5793			1.46825			0.734127	+	0.679013i	$0.237592\pi$
$30$		0			0
$31$	−20.6216	−	11.9059i	−0.665213	−	0.384061i	0.129048	−	0.991638i	$-0.458808\pi$
$31$	−20.6216	−	11.9059i	−0.665213	−	0.384061i	−0.794260	+	0.607578i	$0.792141\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−27.8747	+	32.1870i	−0.796421	+	0.919627i
$36$		0			0
$37$	−19.0000	−	32.9090i	−0.513514	−	0.889431i	−0.999877	−	0.0156750i	$-0.995010\pi$
$37$	−19.0000	−	32.9090i	−0.513514	−	0.889431i	0.486364	−	0.873757i	$-0.338323\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−24.3311			−0.593440			−0.296720	−	0.954964i	$-0.595893\pi$
$41$	−24.3311			−0.593440			−0.296720	+	0.954964i	$0.595893\pi$
$42$		0			0
$43$		−	74.0810i		−	1.72281i	−0.507915	−	0.861407i	$-0.669583\pi$
$43$		−	74.0810i		−	1.72281i	0.507915	−	0.861407i	$-0.330417\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	−38.5000	+	30.3109i	−0.785714	+	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−21.2897	+	36.8748i	−0.401692	+	0.695751i	−0.993930	−	0.110012i	$-0.964911\pi$
$53$	−21.2897	+	36.8748i	−0.401692	+	0.695751i	0.592238	+	0.805763i	$0.298244\pi$
$54$		0			0
$55$		−	97.8928i		−	1.77987i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−97.5615	−	56.3272i	−1.65359	−	0.954698i	−0.975581	−	0.219640i	$-0.929512\pi$
$59$	−97.5615	−	56.3272i	−1.65359	−	0.954698i	−0.678004	−	0.735058i	$-0.737155\pi$
$60$		0			0
$61$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−42.5793	−	73.7496i	−0.655067	−	1.13461i
$66$		0			0
$67$	−96.2341	−	55.5608i	−1.43633	−	0.829265i	−0.438737	−	0.898615i	$-0.644574\pi$
$67$	−96.2341	−	55.5608i	−1.43633	−	0.829265i	−0.997592	+	0.0693500i	$0.977907\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	64.3739i		−	0.906675i	−0.891339	−	0.453337i	$-0.850233\pi$
$71$		−	64.3739i		−	0.906675i	0.891339	−	0.453337i	$-0.149767\pi$
$72$		0			0
$73$	7.00000	−	12.1244i	0.0958904	−	0.166087i	−0.814089	−	0.580739i	$-0.802763\pi$
$73$	7.00000	−	12.1244i	0.0958904	−	0.166087i	0.909980	+	0.414652i	$0.136097\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	21.2897	−	110.624i	0.276489	−	1.43668i
$78$		0			0
$79$	48.1170	−	27.7804i	0.609077	−	0.351650i	−0.163527	−	0.986539i	$-0.552287\pi$
$79$	48.1170	−	27.7804i	0.609077	−	0.351650i	0.772604	+	0.634888i	$0.218954\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			112.654i			1.35728i	0.734471	+	0.678641i	$0.237431\pi$
$83$			112.654i			1.35728i	−0.734471	+	0.678641i	$0.762569\pi$
$84$		0			0
$85$	74.0000			0.870588
$86$		0			0
$87$		0			0
$88$		0			0
$89$	66.9104	+	115.892i	0.751802	+	1.30216i	0.946949	+	0.321385i	$0.104148\pi$
$89$	66.9104	+	115.892i	0.751802	+	1.30216i	−0.195146	+	0.980774i	$0.562518\pi$
$90$		0			0
$91$	−32.0780	−	92.6013i	−0.352506	−	1.01760i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−27.8747	−	16.0935i	−0.293418	−	0.169405i
$96$		0			0
$97$	35.0000			0.360825			0.180412	−	0.983591i	$-0.442257\pi$
$97$	35.0000			0.360825			0.180412	+	0.983591i	$0.442257\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−79.0759	+	136.963i	−0.782930	+	1.35607i	0.147298	+	0.989092i	$0.452942\pi$
$101$	−79.0759	+	136.963i	−0.782930	+	1.35607i	−0.930228	+	0.366982i	$0.880391\pi$
$102$		0			0
$103$	73.3212	−	42.3320i	0.711856	−	0.410990i	−0.0998916	−	0.994998i	$-0.531850\pi$
$103$	73.3212	−	42.3320i	0.711856	−	0.410990i	0.811748	+	0.584008i	$0.198516\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.9374	−	8.04674i	0.130256	−	0.0752032i	−0.433456	−	0.901175i	$-0.642706\pi$
$107$	13.9374	−	8.04674i	0.130256	−	0.0752032i	0.563712	+	0.825971i	$0.309373\pi$
$108$		0			0
$109$	17.0000	−	29.4449i	0.155963	−	0.270136i	−0.777446	−	0.628950i	$-0.783485\pi$
$109$	17.0000	−	29.4449i	0.155963	−	0.270136i	0.933409	+	0.358813i	$0.116819\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	170.317			1.50723			0.753617	−	0.657314i	$-0.228308\pi$
$113$	170.317			1.50723			0.753617	+	0.657314i	$0.228308\pi$
$114$		0			0
$115$	−169.555	−	97.8928i	−1.47439	−	0.851242i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	83.6242	+	16.0935i	0.702724	+	0.135239i
$120$		0			0
$121$	69.0000	+	119.512i	0.570248	+	0.987698i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−79.0759			−0.632607
$126$		0			0
$127$		−	129.642i		−	1.02080i	−0.859937	−	0.510401i	$-0.829497\pi$
$127$		−	129.642i		−	1.02080i	0.859937	−	0.510401i	$-0.170503\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	97.5615	−	56.3272i	0.744744	−	0.429978i	−0.0790474	−	0.996871i	$-0.525188\pi$
$131$	97.5615	−	56.3272i	0.744744	−	0.429978i	0.823792	+	0.566892i	$0.191855\pi$
$132$		0			0
$133$	−28.0000	−	24.2487i	−0.210526	−	0.182321i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	42.5793	−	73.7496i	0.310798	−	0.538318i	−0.667737	−	0.744397i	$-0.732737\pi$
$137$	42.5793	−	73.7496i	0.310798	−	0.538318i	0.978535	+	0.206079i	$0.0660704\pi$
$138$		0			0
$139$		−	153.454i		−	1.10398i	−0.833850	−	0.551991i	$-0.813868\pi$
$139$		−	153.454i		−	1.10398i	0.833850	−	0.551991i	$-0.186132\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	195.123	+	112.654i	1.36450	+	0.787793i
$144$		0			0
$145$	−129.500	−	224.301i	−0.893103	−	1.54690i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−42.5793	−	73.7496i	−0.285767	−	0.494964i	0.687028	−	0.726631i	$-0.258915\pi$
$149$	−42.5793	−	73.7496i	−0.285767	−	0.494964i	−0.972795	+	0.231668i	$0.925582\pi$
$150$		0			0
$151$	16.0390	+	9.26013i	0.106219	+	0.0613254i	0.552168	−	0.833733i	$-0.313801\pi$
$151$	16.0390	+	9.26013i	0.106219	+	0.0613254i	−0.445950	+	0.895058i	$0.647134\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			144.841i			0.934460i
$156$		0			0
$157$	−126.000	+	218.238i	−0.802548	+	1.39005i	0.115386	+	0.993321i	$0.463189\pi$
$157$	−126.000	+	218.238i	−0.802548	+	1.39005i	−0.917934	+	0.396733i	$0.870144\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−170.317	−	147.499i	−1.05787	−	0.916144i
$162$		0			0
$163$	−32.0780	+	18.5203i	−0.196798	+	0.113621i	−0.595161	−	0.803607i	$-0.702912\pi$
$163$	−32.0780	+	18.5203i	−0.196798	+	0.113621i	0.398363	+	0.917228i	$0.369578\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	225.309i		−	1.34915i	−0.738205	−	0.674577i	$-0.764326\pi$
$167$		−	225.309i		−	1.34915i	0.738205	−	0.674577i	$-0.235674\pi$
$168$		0			0
$169$	27.0000			0.159763
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−18.2483	−	31.6070i	−0.105481	−	0.182699i	0.808453	−	0.588560i	$-0.200305\pi$
$173$	−18.2483	−	31.6070i	−0.105481	−	0.182699i	−0.913935	+	0.405861i	$0.866972\pi$
$174$		0			0
$175$	82.4864	+	15.8745i	0.471351	+	0.0907115i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	55.7494	+	32.1870i	0.311449	+	0.179815i	0.647575	−	0.762002i	$-0.275783\pi$
$179$	55.7494	+	32.1870i	0.311449	+	0.179815i	−0.336125	+	0.941817i	$0.609117\pi$
$180$		0			0
$181$	−168.000			−0.928177			−0.464088	−	0.885789i	$-0.653618\pi$
$181$	−168.000			−0.928177			−0.464088	+	0.885789i	$0.653618\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−115.572	+	200.177i	−0.624716	+	1.08204i
$186$		0			0
$187$	−169.555	+	97.8928i	−0.906713	+	0.523491i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−167.248	+	96.5609i	−0.875646	+	0.505554i	−0.869220	−	0.494425i	$-0.835379\pi$
$191$	−167.248	+	96.5609i	−0.875646	+	0.505554i	−0.00642541	+	0.999979i	$0.502045\pi$
$192$		0			0
$193$	139.500	−	241.621i	0.722798	−	1.25192i	−0.237076	−	0.971491i	$-0.576189\pi$
$193$	139.500	−	241.621i	0.722798	−	1.25192i	0.959874	−	0.280432i	$-0.0904776\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	255.476			1.29683			0.648416	−	0.761286i	$-0.275432\pi$
$197$	255.476			1.29683			0.648416	+	0.761286i	$0.275432\pi$
$198$		0			0
$199$	238.294	+	137.579i	1.19746	+	0.691352i	0.959987	−	0.280043i	$-0.0903487\pi$
$199$	238.294	+	137.579i	1.19746	+	0.691352i	0.237469	+	0.971395i	$0.423682\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−97.5615	−	281.636i	−0.480599	−	1.38737i
$204$		0			0
$205$	74.0000	+	128.172i	0.360976	+	0.625228i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	85.1587			0.407458
$210$		0			0
$211$			74.0810i			0.351095i	0.984471	+	0.175547i	$0.0561696\pi$
$211$			74.0810i			0.351095i	−0.984471	+	0.175547i	$0.943830\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−390.246	+	225.309i	−1.81510	+	1.04795i
$216$		0			0
$217$	−31.5000	+	163.679i	−0.145161	+	0.754280i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−85.1587	+	147.499i	−0.385333	+	0.667417i
$222$		0			0
$223$			336.010i			1.50677i	0.657578	+	0.753387i	$0.271581\pi$
$223$			336.010i			1.50677i	−0.657578	+	0.753387i	$0.728419\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	97.5615	+	56.3272i	0.429786	+	0.248137i	0.699256	−	0.714872i	$-0.253515\pi$
$227$	97.5615	+	56.3272i	0.429786	+	0.248137i	−0.269469	+	0.963009i	$0.586848\pi$
$228$		0			0
$229$	−112.000	−	193.990i	−0.489083	−	0.847117i	0.510838	−	0.859677i	$-0.329335\pi$
$229$	−112.000	−	193.990i	−0.489083	−	0.847117i	−0.999921	+	0.0125604i	$0.996002\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−42.5793	−	73.7496i	−0.182744	−	0.316522i	0.760070	−	0.649841i	$-0.225165\pi$
$233$	−42.5793	−	73.7496i	−0.182744	−	0.316522i	−0.942814	+	0.333319i	$0.891831\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			32.1870i			0.134673i	0.997730	+	0.0673367i	$0.0214502\pi$
$239$			32.1870i			0.134673i	−0.997730	+	0.0673367i	$0.978550\pi$
$240$		0			0
$241$	−87.5000	+	151.554i	−0.363071	+	0.628857i	−0.988465	−	0.151453i	$-0.951605\pi$
$241$	−87.5000	+	151.554i	−0.363071	+	0.628857i	0.625394	+	0.780309i	$0.284938\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	276.766	+	110.624i	1.12966	+	0.451528i
$246$		0			0
$247$	64.1561	−	37.0405i	0.259741	−	0.149962i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	337.963i		−	1.34647i	−0.739430	−	0.673233i	$-0.764905\pi$
$251$		−	337.963i		−	1.34647i	0.739430	−	0.673233i	$-0.235095\pi$
$252$		0			0
$253$	518.000			2.04743
$254$		0			0
$255$		0			0
$256$		0			0
$257$	66.9104	+	115.892i	0.260352	+	0.450942i	0.966335	−	0.257286i	$-0.0828282\pi$
$257$	66.9104	+	115.892i	0.260352	+	0.450942i	−0.705984	+	0.708228i	$0.749495\pi$
$258$		0			0
$259$	−174.138	+	201.077i	−0.672347	+	0.776359i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−222.998	−	128.748i	−0.847900	−	0.489535i	0.0120416	−	0.999927i	$-0.496167\pi$
$263$	−222.998	−	128.748i	−0.847900	−	0.489535i	−0.859942	+	0.510392i	$0.829500\pi$
$264$		0			0
$265$	259.000			0.977358
$266$		0			0
$267$		0			0
$268$		0			0
$269$	185.524	−	321.337i	0.689681	−	1.19456i	−0.282260	−	0.959338i	$-0.591084\pi$
$269$	185.524	−	321.337i	0.689681	−	1.19456i	0.971941	−	0.235225i	$-0.0755827\pi$
$270$		0			0
$271$	249.750	−	144.193i	0.921588	−	0.532079i	0.0374467	−	0.999299i	$-0.488078\pi$
$271$	249.750	−	144.193i	0.921588	−	0.532079i	0.884141	+	0.467219i	$0.154744\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−167.248	+	96.5609i	−0.608176	+	0.351130i
$276$		0			0
$277$	74.0000	−	128.172i	0.267148	−	0.462714i	−0.700976	−	0.713185i	$-0.747252\pi$
$277$	74.0000	−	128.172i	0.267148	−	0.462714i	0.968124	+	0.250471i	$0.0805854\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−425.793			−1.51528			−0.757639	−	0.652673i	$-0.773647\pi$
$281$	−425.793			−1.51528			−0.757639	+	0.652673i	$0.773647\pi$
$282$		0			0
$283$	27.4955	+	15.8745i	0.0971571	+	0.0560937i	0.547791	−	0.836615i	$-0.315469\pi$
$283$	27.4955	+	15.8745i	0.0971571	+	0.0560937i	−0.450634	+	0.892709i	$0.648802\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	55.7494	+	160.935i	0.194249	+	0.560748i
$288$		0			0
$289$	70.5000	+	122.110i	0.243945	+	0.422525i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−322.386			−1.10029			−0.550147	−	0.835068i	$-0.685429\pi$
$293$	−322.386			−1.10029			−0.550147	+	0.835068i	$0.685429\pi$
$294$		0			0
$295$			685.250i			2.32288i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	390.246	−	225.309i	1.30517	−	0.753541i
$300$		0			0
$301$	−490.000	+	169.741i	−1.62791	+	0.563924i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			269.867i			0.879044i	0.898232	+	0.439522i	$0.144852\pi$
$307$			269.867i			0.879044i	−0.898232	+	0.439522i	$0.855148\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	−262.500	−	454.663i	−0.838658	−	1.45260i	−0.891017	−	0.453970i	$-0.850007\pi$
$313$	−262.500	−	454.663i	−0.838658	−	1.45260i	0.0523587	−	0.998628i	$-0.483326\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	63.8690	+	110.624i	0.201480	+	0.348973i	0.949005	−	0.315260i	$-0.102092\pi$
$317$	63.8690	+	110.624i	0.201480	+	0.348973i	−0.747526	+	0.664233i	$0.768758\pi$
$318$		0			0
$319$	593.444	+	342.625i	1.86032	+	1.07406i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			64.3739i			0.199300i
$324$		0			0
$325$	−84.0000	+	145.492i	−0.258462	+	0.447669i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−513.248	+	296.324i	−1.55060	+	0.895239i	−0.552507	+	0.833508i	$0.686329\pi$
$331$	−513.248	+	296.324i	−1.55060	+	0.895239i	−0.998093	+	0.0617307i	$0.980338\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			675.926i			2.01769i
$336$		0			0
$337$	205.000			0.608309			0.304154	−	0.952623i	$-0.401626\pi$
$337$	205.000			0.608309			0.304154	+	0.952623i	$0.401626\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−191.607	−	331.873i	−0.561897	−	0.973235i
$342$		0			0
$343$	288.702	+	185.203i	0.841698	+	0.539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	167.248	+	96.5609i	0.481984	+	0.278273i	0.721243	−	0.692682i	$-0.243571\pi$
$347$	167.248	+	96.5609i	0.481984	+	0.278273i	−0.239259	+	0.970956i	$0.576905\pi$
$348$		0			0
$349$	238.000			0.681948			0.340974	−	0.940073i	$-0.389243\pi$
$349$	238.000			0.681948			0.340974	+	0.940073i	$0.389243\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	121.655	−	210.713i	0.344632	−	0.596921i	−0.640654	−	0.767829i	$-0.721337\pi$
$353$	121.655	−	210.713i	0.344632	−	0.596921i	0.985287	+	0.170908i	$0.0546702\pi$
$354$		0			0
$355$	−339.111	+	195.786i	−0.955241	+	0.551509i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	473.870	−	273.589i	1.31997	−	0.762087i	0.336249	−	0.941773i	$-0.390842\pi$
$359$	473.870	−	273.589i	1.31997	−	0.762087i	0.983724	+	0.179687i	$0.0575083\pi$
$360$		0			0
$361$	−166.500	+	288.386i	−0.461219	+	0.798854i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−85.1587			−0.233311
$366$		0			0
$367$	524.705	+	302.939i	1.42971	+	0.825446i	0.997098	−	0.0761327i	$-0.0242573\pi$
$367$	524.705	+	302.939i	1.42971	+	0.825446i	0.432616	+	0.901578i	$0.357591\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	292.685	+	56.3272i	0.788907	+	0.151825i
$372$		0			0
$373$	−100.000	−	173.205i	−0.268097	−	0.464357i	0.700274	−	0.713874i	$-0.253061\pi$
$373$	−100.000	−	173.205i	−0.268097	−	0.464357i	−0.968370	+	0.249518i	$0.919728\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	596.111			1.58120
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	195.123	−	112.654i	0.509460	−	0.294137i	−0.223152	−	0.974784i	$-0.571635\pi$
$383$	195.123	−	112.654i	0.509460	−	0.294137i	0.732612	+	0.680647i	$0.238301\pi$
$384$		0			0
$385$	−647.500	+	224.301i	−1.68182	+	0.582599i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−42.5793	+	73.7496i	−0.109458	+	0.189588i	−0.915551	−	0.402202i	$-0.868245\pi$
$389$	−42.5793	+	73.7496i	−0.109458	+	0.189588i	0.806093	+	0.591789i	$0.201578\pi$
$390$		0			0
$391$			391.571i			1.00146i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−292.685	−	168.982i	−0.740974	−	0.427801i
$396$		0			0
$397$	−308.000	−	533.472i	−0.775819	−	1.34376i	−0.934333	−	0.356401i	$-0.884004\pi$
$397$	−308.000	−	533.472i	−0.775819	−	1.34376i	0.158515	−	0.987357i	$-0.449330\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	340.635	+	589.997i	0.849463	+	1.47131i	0.881688	+	0.471833i	$0.156407\pi$
$401$	340.635	+	589.997i	0.849463	+	1.47131i	−0.0322251	+	0.999481i	$0.510259\pi$
$402$		0			0
$403$	−288.702	−	166.682i	−0.716383	−	0.413604i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	611.552i		−	1.50259i
$408$		0			0
$409$	−199.500	+	345.544i	−0.487775	+	0.844851i	−0.999901	−	0.0140591i	$-0.995525\pi$
$409$	−199.500	+	345.544i	−0.487775	+	0.844851i	0.512126	+	0.858910i	$0.328858\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−149.028	+	774.371i	−0.360842	+	1.87499i
$414$		0			0
$415$	593.444	−	342.625i	1.42998	−	0.825602i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			450.617i			1.07546i	0.843117	+	0.537730i	$0.180718\pi$
$419$			450.617i			1.07546i	−0.843117	+	0.537730i	$0.819282\pi$
$420$		0			0
$421$	740.000			1.75772			0.878860	−	0.477080i	$-0.158305\pi$
$421$	740.000			1.75772			0.878860	+	0.477080i	$0.158305\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−72.9932	−	126.428i	−0.171749	−	0.297477i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	278.747	+	160.935i	0.646745	+	0.373399i	0.787208	−	0.616687i	$-0.211526\pi$
$431$	278.747	+	160.935i	0.646745	+	0.373399i	−0.140463	+	0.990086i	$0.544859\pi$
$432$		0			0
$433$	322.000			0.743649			0.371824	−	0.928303i	$-0.378732\pi$
$433$	322.000			0.743649			0.371824	+	0.928303i	$0.378732\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	85.1587	−	147.499i	0.194871	−	0.337527i
$438$		0			0
$439$	−364.315	+	210.337i	−0.829874	+	0.479128i	−0.853810	−	0.520585i	$-0.825714\pi$
$439$	−364.315	+	210.337i	−0.829874	+	0.479128i	0.0239354	+	0.999714i	$0.492380\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	41.8121	−	24.1402i	0.0943839	−	0.0544926i	−0.452065	−	0.891985i	$-0.649313\pi$
$443$	41.8121	−	24.1402i	0.0943839	−	0.0544926i	0.546449	+	0.837492i	$0.315979\pi$
$444$		0			0
$445$	407.000	−	704.945i	0.914607	−	1.58415i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−766.428			−1.70697			−0.853483	−	0.521120i	$-0.825514\pi$
$449$	−766.428			−1.70697			−0.853483	+	0.521120i	$0.825514\pi$
$450$		0			0
$451$	−339.111	−	195.786i	−0.751908	−	0.434114i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−390.246	+	450.617i	−0.857684	+	0.990368i
$456$		0			0
$457$	124.500	+	215.640i	0.272429	+	0.471861i	0.969483	−	0.245158i	$-0.0788397\pi$
$457$	124.500	+	215.640i	0.272429	+	0.471861i	−0.697054	+	0.717018i	$0.745506\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	231.145			0.501399			0.250700	−	0.968065i	$-0.419339\pi$
$461$	231.145			0.501399			0.250700	+	0.968065i	$0.419339\pi$
$462$		0			0
$463$		−	74.0810i		−	0.160002i	−0.996795	−	0.0800011i	$-0.974508\pi$
$463$		−	74.0810i		−	0.160002i	0.996795	−	0.0800011i	$-0.0254924\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−390.246	+	225.309i	−0.835645	+	0.482460i	−0.855781	−	0.517338i	$-0.826923\pi$
$467$	−390.246	+	225.309i	−0.835645	+	0.482460i	0.0201367	+	0.999797i	$0.493590\pi$
$468$		0			0
$469$	−147.000	+	763.834i	−0.313433	+	1.62864i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	596.111	−	1032.49i	1.26028	−	2.18286i
$474$		0			0
$475$			63.4980i			0.133680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−780.492	−	450.617i	−1.62942	−	0.940746i	−0.984266	−	0.176693i	$-0.943460\pi$
$479$	−780.492	−	450.617i	−1.62942	−	0.940746i	−0.645154	−	0.764053i	$-0.723207\pi$
$480$		0			0
$481$	−266.000	−	460.726i	−0.553015	−	0.957849i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−106.448	−	184.374i	−0.219481	−	0.380152i
$486$		0			0
$487$	368.897	+	212.983i	0.757489	+	0.437337i	0.828394	−	0.560146i	$-0.189255\pi$
$487$	368.897	+	212.983i	0.757489	+	0.437337i	−0.0709042	+	0.997483i	$0.522588\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			177.028i			0.360546i	0.983617	+	0.180273i	$0.0576982\pi$
$491$			177.028i			0.360546i	−0.983617	+	0.180273i	$0.942302\pi$
$492$		0			0
$493$	−259.000	+	448.601i	−0.525355	+	0.909941i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−425.793	+	147.499i	−0.856727	+	0.296779i
$498$		0			0
$499$	−160.390	+	92.6013i	−0.321423	+	0.185574i	−0.652027	−	0.758196i	$-0.726081\pi$
$499$	−160.390	+	92.6013i	−0.321423	+	0.185574i	0.330604	+	0.943770i	$0.392748\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	675.926i		−	1.34379i	−0.740647	−	0.671895i	$-0.765481\pi$
$503$		−	675.926i		−	1.34379i	0.740647	−	0.671895i	$-0.234519\pi$
$504$		0			0
$505$	962.000			1.90495
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−215.938	−	374.016i	−0.424240	−	0.734805i	0.572109	−	0.820177i	$-0.306125\pi$
$509$	−215.938	−	374.016i	−0.424240	−	0.734805i	−0.996349	+	0.0853725i	$0.972792\pi$
$510$		0			0
$511$	−96.2341	−	18.5203i	−0.188325	−	0.0362432i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−445.996	−	257.496i	−0.866011	−	0.499992i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	121.655	−	210.713i	0.233503	−	0.404440i	−0.725333	−	0.688398i	$-0.758314\pi$
$521$	121.655	−	210.713i	0.233503	−	0.404440i	0.958837	+	0.283958i	$0.0916477\pi$
$522$		0			0
$523$	581.987	−	336.010i	1.11279	−	0.642467i	0.173236	−	0.984880i	$-0.444578\pi$
$523$	581.987	−	336.010i	1.11279	−	0.642467i	0.939550	+	0.342413i	$0.111244\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	250.872	−	144.841i	0.476039	−	0.274841i
$528$		0			0
$529$	253.500	−	439.075i	0.479206	−	0.830009i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−340.635			−0.639089
$534$		0			0
$535$	−84.7777	−	48.9464i	−0.158463	−	0.0914886i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−780.492	+	112.654i	−1.44804	+	0.209006i
$540$		0			0
$541$	268.000	+	464.190i	0.495379	+	0.858021i	0.999986	−	0.00532776i	$-0.00169589\pi$
$541$	268.000	+	464.190i	0.495379	+	0.858021i	−0.504607	+	0.863349i	$0.668363\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−206.814			−0.379475
$546$		0			0
$547$			185.203i			0.338579i	0.985566	+	0.169289i	$0.0541473\pi$
$547$			185.203i			0.338579i	−0.985566	+	0.169289i	$0.945853\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	195.123	−	112.654i	0.354125	−	0.204454i
$552$		0			0
$553$	−294.000	−	254.611i	−0.531646	−	0.460419i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−489.662	+	848.120i	−0.879107	+	1.52266i	−0.0267838	+	0.999641i	$0.508527\pi$
$557$	−489.662	+	848.120i	−0.879107	+	1.52266i	−0.852323	+	0.523016i	$0.824807\pi$
$558$		0			0
$559$		−	1037.13i		−	1.85534i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	292.685	+	168.982i	0.519866	+	0.300145i	0.736880	−	0.676024i	$-0.236298\pi$
$563$	292.685	+	168.982i	0.519866	+	0.300145i	−0.217014	+	0.976169i	$0.569632\pi$
$564$		0			0
$565$	−518.000	−	897.202i	−0.916814	−	1.58797i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	383.214	+	663.746i	0.673487	+	1.16651i	0.976909	+	0.213657i	$0.0685375\pi$
$569$	383.214	+	663.746i	0.673487	+	1.16651i	−0.303422	+	0.952856i	$0.598129\pi$
$570$		0			0
$571$	−705.717	−	407.446i	−1.23593	−	0.713565i	−0.267671	−	0.963510i	$-0.586254\pi$
$571$	−705.717	−	407.446i	−1.23593	−	0.713565i	−0.968260	+	0.249945i	$0.919587\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			386.243i			0.671728i
$576$		0			0
$577$	−479.500	+	830.518i	−0.831023	+	1.43937i	0.0662056	+	0.997806i	$0.478911\pi$
$577$	−479.500	+	830.518i	−0.831023	+	1.43937i	−0.897228	+	0.441567i	$0.854423\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	745.138	−	258.124i	1.28251	−	0.444275i
$582$		0			0
$583$	−593.444	+	342.625i	−1.01791	+	0.587693i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	112.654i		−	0.191915i	−0.995385	−	0.0959577i	$-0.969409\pi$
$587$		−	112.654i		−	0.191915i	0.995385	−	0.0959577i	$-0.0305914\pi$
$588$		0			0
$589$	−126.000			−0.213922
$590$		0			0
$591$		0			0
$592$		0			0
$593$	444.042	+	769.103i	0.748806	+	1.29697i	0.948396	+	0.317090i	$0.102706\pi$
$593$	444.042	+	769.103i	0.748806	+	1.29697i	−0.199590	+	0.979879i	$0.563961\pi$
$594$		0			0
$595$	−169.555	−	489.464i	−0.284967	−	0.822629i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	585.369	+	337.963i	0.977244	+	0.564212i	0.901437	−	0.432910i	$-0.142513\pi$
$599$	585.369	+	337.963i	0.977244	+	0.564212i	0.0758070	+	0.997123i	$0.475847\pi$
$600$		0			0
$601$	−133.000			−0.221298			−0.110649	−	0.993860i	$-0.535293\pi$
$601$	−133.000			−0.221298			−0.110649	+	0.993860i	$0.535293\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	419.711	−	726.960i	0.693737	−	1.20159i
$606$		0			0
$607$	795.077	−	459.038i	1.30985	−	0.756240i	0.327776	−	0.944755i	$-0.393701\pi$
$607$	795.077	−	459.038i	1.30985	−	0.756240i	0.982070	+	0.188515i	$0.0603674\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	193.000	−	334.286i	0.314845	−	0.545328i	−0.664560	−	0.747235i	$-0.731381\pi$
$613$	193.000	−	334.286i	0.314845	−	0.545328i	0.979405	+	0.201908i	$0.0647141\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−681.269			−1.10416			−0.552082	−	0.833790i	$-0.686167\pi$
$617$	−681.269			−1.10416			−0.552082	+	0.833790i	$0.686167\pi$
$618$		0			0
$619$	−568.239	−	328.073i	−0.917996	−	0.530005i	−0.0350007	−	0.999387i	$-0.511143\pi$
$619$	−568.239	−	328.073i	−0.917996	−	0.530005i	−0.882995	+	0.469382i	$0.844477\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	613.244	−	708.113i	0.984340	−	1.13662i
$624$		0			0
$625$	390.500	+	676.366i	0.624800	+	1.08219i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	462.290			0.734960
$630$		0			0
$631$		−	1166.78i		−	1.84909i	−0.381072	−	0.924545i	$-0.624445\pi$
$631$		−	1166.78i		−	1.84909i	0.381072	−	0.924545i	$-0.375555\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−682.931	+	394.290i	−1.07548	+	0.620929i
$636$		0			0
$637$	−539.000	+	424.352i	−0.846154	+	0.666173i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	170.317	−	294.998i	0.265706	−	0.460216i	−0.702043	−	0.712135i	$-0.747728\pi$
$641$	170.317	−	294.998i	0.265706	−	0.460216i	0.967748	+	0.251919i	$0.0810617\pi$
$642$		0			0
$643$		−	703.770i		−	1.09451i	−0.836966	−	0.547255i	$-0.815673\pi$
$643$		−	703.770i		−	1.09451i	0.836966	−	0.547255i	$-0.184327\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	780.492	+	450.617i	1.20632	+	0.696472i	0.961954	−	0.273210i	$-0.0880854\pi$
$647$	780.492	+	450.617i	1.20632	+	0.696472i	0.244370	+	0.969682i	$0.421419\pi$
$648$		0			0
$649$	−906.500	−	1570.10i	−1.39676	−	2.41927i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	21.2897	+	36.8748i	0.0326029	+	0.0564698i	0.881866	−	0.471499i	$-0.156287\pi$
$653$	21.2897	+	36.8748i	0.0326029	+	0.0564698i	−0.849264	+	0.527969i	$0.822954\pi$
$654$		0			0
$655$	−593.444	−	342.625i	−0.906021	−	0.523091i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	450.617i		−	0.683790i	−0.939738	−	0.341895i	$-0.888931\pi$
$659$		−	450.617i		−	0.683790i	0.939738	−	0.341895i	$-0.111069\pi$
$660$		0			0
$661$	−336.000	+	581.969i	−0.508321	+	0.880437i	0.491633	+	0.870803i	$0.336400\pi$
$661$	−336.000	+	581.969i	−0.508321	+	0.880437i	−0.999954	+	0.00963474i	$0.996933\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−42.5793	+	221.249i	−0.0640291	+	0.332705i
$666$		0			0
$667$	1186.89	−	685.250i	1.77944	−	1.02736i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	985.000			1.46360			0.731798	−	0.681522i	$-0.238681\pi$
$673$	985.000			1.46360			0.731798	+	0.681522i	$0.238681\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−209.855	−	363.480i	−0.309978	−	0.536898i	0.668379	−	0.743821i	$-0.266988\pi$
$677$	−209.855	−	363.480i	−0.309978	−	0.536898i	−0.978357	+	0.206923i	$0.933655\pi$
$678$		0			0
$679$	−80.1951	−	231.503i	−0.118108	−	0.340947i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	320.559	+	185.075i	0.469340	+	0.270974i	0.715963	−	0.698138i	$-0.245988\pi$
$683$	320.559	+	185.075i	0.469340	+	0.270974i	−0.246623	+	0.969111i	$0.579321\pi$
$684$		0			0
$685$	−518.000			−0.756204
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−298.055	+	516.247i	−0.432591	+	0.749270i
$690$		0			0
$691$	−1182.30	+	682.604i	−1.71101	+	0.987849i	−0.777795	+	0.628518i	$0.783662\pi$
$691$	−1182.30	+	682.604i	−1.71101	+	0.987849i	−0.933210	+	0.359331i	$0.883005\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−808.367	+	466.711i	−1.16312	+	0.671526i
$696$		0			0
$697$	148.000	−	256.344i	0.212339	−	0.367781i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	723.849			1.03259			0.516297	−	0.856409i	$-0.327310\pi$
$701$	723.849			1.03259			0.516297	+	0.856409i	$0.327310\pi$
$702$		0			0
$703$	−174.138	−	100.539i	−0.247707	−	0.143014i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1087.11	+	209.215i	1.53764	+	0.295920i
$708$		0			0
$709$	495.000	+	857.365i	0.698166	+	1.20926i	0.969102	+	0.246662i	$0.0793338\pi$
$709$	495.000	+	857.365i	0.698166	+	1.20926i	−0.270935	+	0.962598i	$0.587333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−766.428			−1.07493
$714$		0			0
$715$		−	1370.50i		−	1.91678i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	975.615	−	563.272i	1.35691	−	0.783410i	0.367700	−	0.929944i	$-0.380145\pi$
$719$	975.615	−	563.272i	1.35691	−	0.783410i	0.989206	+	0.146535i	$0.0468120\pi$
$720$		0			0
$721$	−448.000	−	387.979i	−0.621359	−	0.538113i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−255.476	+	442.497i	−0.352381	+	0.610341i
$726$		0			0
$727$			394.217i			0.542252i	0.962544	+	0.271126i	$0.0873959\pi$
$727$			394.217i			0.542252i	−0.962544	+	0.271126i	$0.912604\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	780.492	+	450.617i	1.06770	+	0.616440i
$732$		0			0
$733$	−98.0000	−	169.741i	−0.133697	−	0.231570i	0.791402	−	0.611296i	$-0.209352\pi$
$733$	−98.0000	−	169.741i	−0.133697	−	0.231570i	−0.925099	+	0.379726i	$0.876018\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−894.166	−	1548.74i	−1.21325	−	2.10141i
$738$		0			0
$739$	−32.0780	−	18.5203i	−0.0434073	−	0.0250612i	0.478139	−	0.878284i	$-0.341311\pi$
$739$	−32.0780	−	18.5203i	−0.0434073	−	0.0250612i	−0.521547	+	0.853223i	$0.674645\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			997.796i			1.34293i	0.741037	+	0.671464i	$0.234334\pi$
$743$			997.796i			1.34293i	−0.741037	+	0.671464i	$0.765666\pi$
$744$		0			0
$745$	−259.000	+	448.601i	−0.347651	+	0.602149i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−85.1587	−	73.7496i	−0.113696	−	0.0984641i
$750$		0			0
$751$	272.663	−	157.422i	0.363067	−	0.209617i	−0.307358	−	0.951594i	$-0.599445\pi$
$751$	272.663	−	157.422i	0.363067	−	0.209617i	0.670425	+	0.741977i	$0.266112\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	112.654i		−	0.149211i
$756$		0			0
$757$	912.000			1.20476			0.602378	−	0.798211i	$-0.294220\pi$
$757$	912.000			1.20476			0.602378	+	0.798211i	$0.294220\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−188.566	−	326.605i	−0.247787	−	0.429179i	0.715125	−	0.698997i	$-0.246370\pi$
$761$	−188.566	−	326.605i	−0.247787	−	0.429179i	−0.962911	+	0.269818i	$0.913037\pi$
$762$		0			0
$763$	−233.711	−	44.9778i	−0.306306	−	0.0589486i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1365.86	−	788.580i	−1.78078	−	1.02814i
$768$		0			0
$769$	1085.00			1.41092			0.705462	−	0.708748i	$-0.250740\pi$
$769$	1085.00			1.41092			0.705462	+	0.708748i	$0.250740\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−249.393	+	431.962i	−0.322630	+	0.558812i	−0.981030	−	0.193856i	$-0.937900\pi$
$773$	−249.393	+	431.962i	−0.322630	+	0.558812i	0.658400	+	0.752669i	$0.271234\pi$
$774$		0			0
$775$	247.459	−	142.871i	0.319302	−	0.184349i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−111.499	+	64.3739i	−0.143131	+	0.0826366i
$780$		0			0
$781$	518.000	−	897.202i	0.663252	−	1.14879i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1532.86			1.95268
$786$		0			0
$787$	137.477	+	79.3725i	0.174685	+	0.100855i	0.584793	−	0.811182i	$-0.301176\pi$
$787$	137.477	+	79.3725i	0.174685	+	0.100855i	−0.410108	+	0.912037i	$0.634509\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−390.246	−	1126.54i	−0.493358	−	1.42420i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−407.545			−0.511349			−0.255674	−	0.966763i	$-0.582298\pi$
$797$	−407.545			−0.511349			−0.255674	+	0.966763i	$0.582298\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	195.123	−	112.654i	0.242993	−	0.140292i
$804$		0			0
$805$	−259.000	+	1345.80i	−0.321739	+	1.67181i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−85.1587	+	147.499i	−0.105264	+	0.182323i	−0.913846	−	0.406061i	$-0.866902\pi$
$809$	−85.1587	+	147.499i	−0.105264	+	0.182323i	0.808582	+	0.588384i	$0.200235\pi$
$810$		0			0
$811$		−	1285.84i		−	1.58549i	−0.609551	−	0.792747i	$-0.708650\pi$
$811$		−	1285.84i		−	1.58549i	0.609551	−	0.792747i	$-0.291350\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	195.123	+	112.654i	0.239415	+	0.138226i
$816$		0			0
$817$	−196.000	−	339.482i	−0.239902	−	0.415523i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	234.186	+	405.623i	0.285245	+	0.494059i	0.972669	−	0.232198i	$-0.0745916\pi$
$821$	234.186	+	405.623i	0.285245	+	0.494059i	−0.687423	+	0.726257i	$0.741258\pi$
$822$		0			0
$823$	−384.936	−	222.243i	−0.467723	−	0.270040i	0.247563	−	0.968872i	$-0.420370\pi$
$823$	−384.936	−	222.243i	−0.467723	−	0.270040i	−0.715286	+	0.698832i	$0.753704\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	337.963i		−	0.408661i	−0.978902	−	0.204331i	$-0.934498\pi$
$827$		−	337.963i		−	0.408661i	0.978902	−	0.204331i	$-0.0655018\pi$
$828$		0			0
$829$	287.000	−	497.099i	0.346200	−	0.599636i	−0.639371	−	0.768899i	$-0.720805\pi$
$829$	287.000	−	497.099i	0.346200	−	0.599636i	0.985571	+	0.169262i	$0.0541384\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−85.1587	−	589.997i	−0.102231	−	0.708279i
$834$		0			0
$835$	−1186.89	+	685.250i	−1.42142	+	0.820658i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			225.309i			0.268544i	0.990944	+	0.134272i	$0.0428696\pi$
$839$			225.309i			0.268544i	−0.990944	+	0.134272i	$0.957130\pi$
$840$		0			0
$841$	972.000			1.15577
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−82.1173	−	142.231i	−0.0971802	−	0.168321i
$846$		0			0
$847$	632.395	−	730.227i	0.746630	−	0.862134i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1059.24	−	611.552i	−1.24470	−	0.718628i
$852$		0			0
$853$	−518.000			−0.607268			−0.303634	−	0.952789i	$-0.598200\pi$
$853$	−518.000			−0.607268			−0.303634	+	0.952789i	$0.598200\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−462.290	+	800.710i	−0.539428	+	0.934317i	0.459507	+	0.888174i	$0.348026\pi$
$857$	−462.290	+	800.710i	−0.539428	+	0.934317i	−0.998935	+	0.0461427i	$0.985307\pi$
$858$		0			0
$859$	1008.17	−	582.065i	1.17365	−	0.677608i	0.219114	−	0.975699i	$-0.429683\pi$
$859$	1008.17	−	582.065i	1.17365	−	0.677608i	0.954537	+	0.298091i	$0.0963500\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1310.11	+	756.393i	−1.51809	+	0.876470i	−0.518317	+	0.855189i	$0.673441\pi$
$863$	−1310.11	+	756.393i	−1.51809	+	0.876470i	−0.999774	+	0.0212810i	$0.993226\pi$
$864$		0			0
$865$	−111.000	+	192.258i	−0.128324	+	0.222263i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	894.166			1.02896
$870$		0			0
$871$	−1347.28	−	777.851i	−1.54682	−	0.893055i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	181.186	+	523.038i	0.207069	+	0.597758i
$876$		0			0
$877$	−404.000	−	699.749i	−0.460661	−	0.797889i	0.538333	−	0.842732i	$-0.319054\pi$
$877$	−404.000	−	699.749i	−0.460661	−	0.797889i	−0.998994	+	0.0448436i	$0.985721\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1131.39			1.28422			0.642108	−	0.766614i	$-0.278060\pi$
$881$	1131.39			1.28422			0.642108	+	0.766614i	$0.278060\pi$
$882$		0			0
$883$			444.486i			0.503382i	0.967808	+	0.251691i	$0.0809866\pi$
$883$			444.486i			0.503382i	−0.967808	+	0.251691i	$0.919013\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−975.615	+	563.272i	−1.09990	+	0.635030i	−0.936196	−	0.351479i	$-0.885679\pi$
$887$	−975.615	+	563.272i	−1.09990	+	0.635030i	−0.163708	+	0.986509i	$0.552346\pi$
$888$		0			0
$889$	−857.500	+	297.047i	−0.964567	+	0.334136i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	391.571i		−	0.437510i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−878.054	−	506.945i	−0.976700	−	0.563898i
$900$		0			0
$901$	−259.000	−	448.601i	−0.287458	−	0.497893i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	510.952	+	884.995i	0.564588	+	0.977895i
$906$		0			0
$907$	994.419	+	574.128i	1.09638	+	0.632997i	0.935268	−	0.353939i	$-0.115158\pi$
$907$	994.419	+	574.128i	1.09638	+	0.632997i	0.161114	+	0.986936i	$0.448491\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			321.870i			0.353315i	0.984272	+	0.176657i	$0.0565284\pi$
$911$			321.870i			0.353315i	−0.984272	+	0.176657i	$0.943472\pi$
$912$		0			0
$913$	−906.500	+	1570.10i	−0.992881	+	1.71972i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−596.111	−	516.247i	−0.650066	−	0.562974i
$918$		0			0
$919$	64.1561	−	37.0405i	0.0698107	−	0.0403052i	−0.464688	−	0.885474i	$-0.653834\pi$
$919$	64.1561	−	37.0405i	0.0698107	−	0.0403052i	0.534499	+	0.845169i	$0.320500\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	901.235i		−	0.976419i
$924$		0			0
$925$	456.000			0.492973
$926$		0			0
$927$		0			0
$928$		0			0
$929$	535.283	+	927.138i	0.576193	+	0.997995i	0.995911	+	0.0903407i	$0.0287956\pi$
$929$	535.283	+	927.138i	0.576193	+	0.997995i	−0.419718	+	0.907655i	$0.637871\pi$
$930$		0			0
$931$	−96.2341	+	240.763i	−0.103366	+	0.258607i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1031.36	+	595.459i	1.10306	+	0.636854i
$936$		0			0
$937$	1127.00			1.20277			0.601387	−	0.798958i	$-0.294615\pi$
$937$	1127.00			1.20277			0.601387	+	0.798958i	$0.294615\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−538.324	+	932.405i	−0.572077	+	0.990866i	0.424275	+	0.905533i	$0.360529\pi$
$941$	−538.324	+	932.405i	−0.572077	+	0.990866i	−0.996352	+	0.0853333i	$0.972804\pi$
$942$		0			0
$943$	−678.221	+	391.571i	−0.719217	+	0.415240i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	836.242	−	482.804i	0.883043	−	0.509825i	0.0113824	−	0.999935i	$-0.496377\pi$
$947$	836.242	−	482.804i	0.883043	−	0.509825i	0.871660	+	0.490110i	$0.163043\pi$
$948$		0			0
$949$	98.0000	−	169.741i	0.103267	−	0.178863i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	510.952			0.536151			0.268076	−	0.963398i	$-0.413612\pi$
$953$	510.952			0.536151			0.268076	+	0.963398i	$0.413612\pi$
$954$		0			0
$955$	1017.33	+	587.357i	1.06527	+	0.615033i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−585.369	−	112.654i	−0.610395	−	0.117471i
$960$		0			0
$961$	−197.000	−	341.214i	−0.204995	−	0.355061i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1697.09			−1.75864
$966$		0			0
$967$			1314.94i			1.35981i	0.733299	+	0.679906i	$0.237979\pi$
$967$			1314.94i			1.35981i	−0.733299	+	0.679906i	$0.762021\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−487.808	+	281.636i	−0.502377	+	0.290047i	−0.729694	−	0.683773i	$-0.760338\pi$
$971$	−487.808	+	281.636i	−0.502377	+	0.290047i	0.227318	+	0.973821i	$0.427004\pi$
$972$		0			0
$973$	−1015.00	+	351.606i	−1.04317	+	0.361363i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	85.1587	−	147.499i	0.0871634	−	0.150971i	−0.819148	−	0.573583i	$-0.805553\pi$
$977$	85.1587	−	147.499i	0.0871634	−	0.150971i	0.906311	+	0.422611i	$0.138886\pi$
$978$		0			0
$979$			2153.64i			2.19984i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−390.246	−	225.309i	−0.396995	−	0.229205i	0.288192	−	0.957573i	$-0.406946\pi$
$983$	−390.246	−	225.309i	−0.396995	−	0.229205i	−0.685187	+	0.728368i	$0.740279\pi$
$984$		0			0
$985$	−777.000	−	1345.80i	−0.788832	−	1.36630i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1192.22	−	2064.99i	−1.20548	−	2.08796i
$990$		0			0
$991$	−689.678	−	398.186i	−0.695941	−	0.401802i	0.109893	−	0.993943i	$-0.464949\pi$
$991$	−689.678	−	398.186i	−0.695941	−	0.401802i	−0.805834	+	0.592142i	$0.798283\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	1673.72i		−	1.68213i
$996$		0			0
$997$	259.000	−	448.601i	0.259779	−	0.449951i	−0.706403	−	0.707810i	$-0.749684\pi$
$997$	259.000	−	448.601i	0.259779	−	0.449951i	0.966183	+	0.257858i	$0.0830168\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.cd.o.991.1	yes	8
3.2	odd	2	inner	1008.3.cd.o.991.3	yes	8
4.3	odd	2	inner	1008.3.cd.o.991.2	yes	8
7.2	even	3	inner	1008.3.cd.o.415.2	yes	8
12.11	even	2	inner	1008.3.cd.o.991.4	yes	8
21.2	odd	6	inner	1008.3.cd.o.415.4	yes	8
28.23	odd	6	inner	1008.3.cd.o.415.1	✓	8
84.23	even	6	inner	1008.3.cd.o.415.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.3.cd.o.415.1	✓	8	28.23	odd	6	inner
1008.3.cd.o.415.2	yes	8	7.2	even	3	inner
1008.3.cd.o.415.3	yes	8	84.23	even	6	inner
1008.3.cd.o.415.4	yes	8	21.2	odd	6	inner
1008.3.cd.o.991.1	yes	8	1.1	even	1	trivial
1008.3.cd.o.991.2	yes	8	4.3	odd	2	inner
1008.3.cd.o.991.3	yes	8	3.2	odd	2	inner
1008.3.cd.o.991.4	yes	8	12.11	even	2	inner

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cd (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.04138 - 5.26783i) q^{5} +(-2.29129 - 6.61438i) q^{7} +(13.9374 + 8.04674i) q^{11} +14.0000 q^{13} +(-6.08276 + 10.5357i) q^{17} +(4.58258 - 2.64575i) q^{19} +(27.8747 - 16.0935i) q^{23} +(-6.00000 + 10.3923i) q^{25} +42.5793 q^{29} +(-20.6216 - 11.9059i) q^{31} +(-27.8747 + 32.1870i) q^{35} +(-19.0000 - 32.9090i) q^{37} -24.3311 q^{41} -74.0810i q^{43} +(-38.5000 + 30.3109i) q^{49} +(-21.2897 + 36.8748i) q^{53} -97.8928i q^{55} +(-97.5615 - 56.3272i) q^{59} +(-42.5793 - 73.7496i) q^{65} +(-96.2341 - 55.5608i) q^{67} -64.3739i q^{71} +(7.00000 - 12.1244i) q^{73} +(21.2897 - 110.624i) q^{77} +(48.1170 - 27.7804i) q^{79} +112.654i q^{83} +74.0000 q^{85} +(66.9104 + 115.892i) q^{89} +(-32.0780 - 92.6013i) q^{91} +(-27.8747 - 16.0935i) q^{95} +35.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 112 q^{13} - 48 q^{25} - 152 q^{37} - 308 q^{49} + 56 q^{73} + 592 q^{85} + 280 q^{97}+O(q^{100}) \) 8 * q + 112 * q^13 - 48 * q^25 - 152 * q^37 - 308 * q^49 + 56 * q^73 + 592 * q^85 + 280 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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