LMFDB - Embedded newform 1020.2.g.d.409.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1020,2,Mod(409,1020)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1020.409"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1020, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1020.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0,0,0,0,-10,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$8.14474100617$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} + 2x^{8} + 14x^{7} + 42x^{6} + 2x^{5} + 10x^{4} + 54x^{3} + 121x^{2} + 44x + 8 $ x^10 - 2x^9 + 2x^8 + 14x^7 + 42x^6 + 2x^5 + 10x^4 + 54x^3 + 121x^2 + 44*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			409.7
Root			$-1.29737 + 1.29737i$ of defining polynomial
Character	$\chi$	$=$	1020.409
Dual form			1020.2.g.d.409.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-1.55795 + 1.60399i) q^{5} +3.74956i q^{7} -1.00000 q^{9} -4.55583 q^{11} -5.60969i q^{13} +(-1.60399 - 1.55795i) q^{15} +1.00000i q^{17} -3.76805 q^{19} -3.74956 q^{21} +2.71420i q^{23} +(-0.145568 - 4.99788i) q^{25} -1.00000i q^{27} +1.43993 q^{29} +7.18949 q^{31} -4.55583i q^{33} +(-6.01425 - 5.84163i) q^{35} -4.66640i q^{37} +5.60969 q^{39} +2.32402 q^{41} -8.72845i q^{43} +(1.55795 - 1.60399i) q^{45} -1.13030i q^{47} -7.05919 q^{49} -1.00000 q^{51} +2.25044i q^{53} +(7.09777 - 7.30751i) q^{55} -3.76805i q^{57} -12.1063 q^{59} -5.11877 q^{61} -3.74956i q^{63} +(8.99788 + 8.73962i) q^{65} +12.1063i q^{67} -2.71420 q^{69} -11.2194 q^{71} +9.93371i q^{73} +(4.99788 - 0.145568i) q^{75} -17.0824i q^{77} -16.3790 q^{79} +1.00000 q^{81} +10.3025i q^{83} +(-1.60399 - 1.55795i) q^{85} +1.43993i q^{87} -12.6179 q^{89} +21.0338 q^{91} +7.18949i q^{93} +(5.87045 - 6.04392i) q^{95} -9.03522i q^{97} +4.55583 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{9} + 12 q^{11} + 4 q^{15} - 24 q^{19} - 4 q^{21} + 12 q^{25} - 12 q^{29} + 12 q^{31} + 4 q^{35} + 28 q^{41} - 30 q^{49} - 10 q^{51} + 26 q^{55} - 48 q^{59} - 28 q^{61} + 48 q^{65} - 12 q^{69}+ \cdots - 12 q^{99}+O(q^{100}) $ 10 * q - 10 * q^9 + 12 * q^11 + 4 * q^15 - 24 * q^19 - 4 * q^21 + 12 * q^25 - 12 * q^29 + 12 * q^31 + 4 * q^35 + 28 * q^41 - 30 * q^49 - 10 * q^51 + 26 * q^55 - 48 * q^59 - 28 * q^61 + 48 * q^65 - 12 * q^69 + 8 * q^75 - 44 * q^79 + 10 * q^81 + 4 * q^85 - 36 * q^89 + 40 * q^91 + 44 * q^95 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1020\mathbb{Z}\right)^\times$.

$n$	$241$	$341$	$511$	$817$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	−1.55795	+	1.60399i	−0.696738	+	0.717326i
$5$	−1.55795	+	1.60399i	−0.696738	+	0.717326i
$6$		0			0
$7$			3.74956i			1.41720i	0.705610	+	0.708600i	$0.250673\pi$
$7$			3.74956i			1.41720i	−0.705610	+	0.708600i	$0.749327\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−4.55583			−1.37364			−0.686818	−	0.726830i	$-0.740993\pi$
$11$	−4.55583			−1.37364			−0.686818	+	0.726830i	$0.740993\pi$
$12$		0			0
$13$		−	5.60969i		−	1.55585i	−0.628359	−	0.777923i	$-0.716273\pi$
$13$		−	5.60969i		−	1.55585i	0.628359	−	0.777923i	$-0.283727\pi$
$14$		0			0
$15$	−1.60399	−	1.55795i	−0.414148	−	0.402262i
$16$		0			0
$17$			1.00000i			0.242536i
$17$			1.00000i			0.242536i
$18$		0			0
$19$	−3.76805			−0.864450			−0.432225	−	0.901766i	$-0.642271\pi$
$19$	−3.76805			−0.864450			−0.432225	+	0.901766i	$0.642271\pi$
$20$		0			0
$21$	−3.74956			−0.818221
$22$		0			0
$23$			2.71420i			0.565950i	0.959127	+	0.282975i	$0.0913213\pi$
$23$			2.71420i			0.565950i	−0.959127	+	0.282975i	$0.908679\pi$
$24$		0			0
$25$	−0.145568	−	4.99788i	−0.0291137	−	0.999576i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	1.43993			0.267388			0.133694	−	0.991023i	$-0.457316\pi$
$29$	1.43993			0.267388			0.133694	+	0.991023i	$0.457316\pi$
$30$		0			0
$31$	7.18949			1.29127			0.645635	−	0.763646i	$-0.276593\pi$
$31$	7.18949			1.29127			0.645635	+	0.763646i	$0.276593\pi$
$32$		0			0
$33$		−	4.55583i		−	0.793069i
$34$		0			0
$35$	−6.01425	−	5.84163i	−1.01659	−	0.987416i
$36$		0			0
$37$		−	4.66640i		−	0.767152i	−0.923509	−	0.383576i	$-0.874692\pi$
$37$		−	4.66640i		−	0.767152i	0.923509	−	0.383576i	$-0.125308\pi$
$38$		0			0
$39$	5.60969			0.898269
$40$		0			0
$41$	2.32402			0.362951			0.181476	−	0.983395i	$-0.441913\pi$
$41$	2.32402			0.362951			0.181476	+	0.983395i	$0.441913\pi$
$42$		0			0
$43$		−	8.72845i		−	1.33108i	−0.746364	−	0.665538i	$-0.768202\pi$
$43$		−	8.72845i		−	1.33108i	0.746364	−	0.665538i	$-0.231798\pi$
$44$		0			0
$45$	1.55795	−	1.60399i	0.232246	−	0.239109i
$46$		0			0
$47$		−	1.13030i		−	0.164871i	−0.996596	−	0.0824354i	$-0.973730\pi$
$47$		−	1.13030i		−	0.164871i	0.996596	−	0.0824354i	$-0.0262698\pi$
$48$		0			0
$49$	−7.05919			−1.00846
$50$		0			0
$51$	−1.00000			−0.140028
$52$		0			0
$53$			2.25044i			0.309122i	0.987983	+	0.154561i	$0.0493963\pi$
$53$			2.25044i			0.309122i	−0.987983	+	0.154561i	$0.950604\pi$
$54$		0			0
$55$	7.09777	−	7.30751i	0.957063	−	0.985345i
$56$		0			0
$57$		−	3.76805i		−	0.499091i
$58$		0			0
$59$	−12.1063			−1.57611			−0.788055	−	0.615605i	$-0.788912\pi$
$59$	−12.1063			−1.57611			−0.788055	+	0.615605i	$0.788912\pi$
$60$		0			0
$61$	−5.11877			−0.655391			−0.327696	−	0.944783i	$-0.606272\pi$
$61$	−5.11877			−0.655391			−0.327696	+	0.944783i	$0.606272\pi$
$62$		0			0
$63$		−	3.74956i		−	0.472400i
$64$		0			0
$65$	8.99788	+	8.73962i	1.11605	+	1.08402i
$66$		0			0
$67$			12.1063i			1.47902i	0.673144	+	0.739512i	$0.264944\pi$
$67$			12.1063i			1.47902i	−0.673144	+	0.739512i	$0.735056\pi$
$68$		0			0
$69$	−2.71420			−0.326751
$70$		0			0
$71$	−11.2194			−1.33149			−0.665747	−	0.746177i	$-0.731887\pi$
$71$	−11.2194			−1.33149			−0.665747	+	0.746177i	$0.731887\pi$
$72$		0			0
$73$			9.93371i			1.16265i	0.813670	+	0.581326i	$0.197466\pi$
$73$			9.93371i			1.16265i	−0.813670	+	0.581326i	$0.802534\pi$
$74$		0			0
$75$	4.99788	−	0.145568i	0.577106	−	0.0168088i
$76$		0			0
$77$		−	17.0824i		−	1.94672i
$78$		0			0
$79$	−16.3790			−1.84278			−0.921389	−	0.388641i	$-0.872945\pi$
$79$	−16.3790			−1.84278			−0.921389	+	0.388641i	$0.872945\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			10.3025i			1.13085i	0.824800	+	0.565425i	$0.191288\pi$
$83$			10.3025i			1.13085i	−0.824800	+	0.565425i	$0.808712\pi$
$84$		0			0
$85$	−1.60399	−	1.55795i	−0.173977	−	0.168984i
$86$		0			0
$87$			1.43993i			0.154376i
$88$		0			0
$89$	−12.6179			−1.33749			−0.668747	−	0.743490i	$-0.733169\pi$
$89$	−12.6179			−1.33749			−0.668747	+	0.743490i	$0.733169\pi$
$90$		0			0
$91$	21.0338			2.20495
$92$		0			0
$93$			7.18949i			0.745515i
$94$		0			0
$95$	5.87045	−	6.04392i	0.602295	−	0.620093i
$96$		0			0
$97$		−	9.03522i		−	0.917388i	−0.888594	−	0.458694i	$-0.848317\pi$
$97$		−	9.03522i		−	0.917388i	0.888594	−	0.458694i	$-0.151683\pi$
$98$		0			0
$99$	4.55583			0.457878
$100$		0			0
$101$	−15.9151			−1.58361			−0.791805	−	0.610774i	$-0.790858\pi$
$101$	−15.9151			−1.58361			−0.791805	+	0.610774i	$0.790858\pi$
$102$		0			0
$103$			8.72845i			0.860040i	0.902819	+	0.430020i	$0.141494\pi$
$103$			8.72845i			0.860040i	−0.902819	+	0.430020i	$0.858506\pi$
$104$		0			0
$105$	5.84163	−	6.01425i	0.570085	−	0.586931i
$106$		0			0
$107$		−	6.29114i		−	0.608187i	−0.952642	−	0.304094i	$-0.901646\pi$
$107$		−	6.29114i		−	0.608187i	0.952642	−	0.304094i	$-0.0983536\pi$
$108$		0			0
$109$	−5.53610			−0.530263			−0.265131	−	0.964212i	$-0.585415\pi$
$109$	−5.53610			−0.530263			−0.265131	+	0.964212i	$0.585415\pi$
$110$		0			0
$111$	4.66640			0.442915
$112$		0			0
$113$		−	2.16566i		−	0.203728i	−0.994798	−	0.101864i	$-0.967519\pi$
$113$		−	2.16566i		−	0.203728i	0.994798	−	0.101864i	$-0.0324806\pi$
$114$		0			0
$115$	−4.35355	−	4.22859i	−0.405970	−	0.394318i
$116$		0			0
$117$			5.60969i			0.518616i
$118$		0			0
$119$	−3.74956			−0.343721
$120$		0			0
$121$	9.75561			0.886874
$122$		0			0
$123$			2.32402i			0.209550i
$124$		0			0
$125$	8.24334	+	7.55297i	0.737307	+	0.675558i
$126$		0			0
$127$		−	4.76929i		−	0.423206i	−0.977356	−	0.211603i	$-0.932132\pi$
$127$		−	4.76929i		−	0.423206i	0.977356	−	0.211603i	$-0.0678684\pi$
$128$		0			0
$129$	8.72845			0.768498
$130$		0			0
$131$	15.4554			1.35035			0.675173	−	0.737659i	$-0.264069\pi$
$131$	15.4554			1.35035			0.675173	+	0.737659i	$0.264069\pi$
$132$		0			0
$133$		−	14.1285i		−	1.22510i
$134$		0			0
$135$	1.60399	+	1.55795i	0.138049	+	0.134087i
$136$		0			0
$137$			10.8982i			0.931097i	0.885022	+	0.465549i	$0.154143\pi$
$137$			10.8982i			0.931097i	−0.885022	+	0.465549i	$0.845857\pi$
$138$		0			0
$139$	18.3310			1.55482			0.777409	−	0.628995i	$-0.216534\pi$
$139$	18.3310			1.55482			0.777409	+	0.628995i	$0.216534\pi$
$140$		0			0
$141$	1.13030			0.0951882
$142$		0			0
$143$			25.5568i			2.13717i
$144$		0			0
$145$	−2.24334	+	2.30963i	−0.186299	+	0.191804i
$146$		0			0
$147$		−	7.05919i		−	0.582232i
$148$		0			0
$149$	−13.6424			−1.11763			−0.558816	−	0.829292i	$-0.688744\pi$
$149$	−13.6424			−1.11763			−0.558816	+	0.829292i	$0.688744\pi$
$150$		0			0
$151$	17.9035			1.45697			0.728485	−	0.685062i	$-0.240225\pi$
$151$	17.9035			1.45697			0.728485	+	0.685062i	$0.240225\pi$
$152$		0			0
$153$		−	1.00000i		−	0.0808452i
$154$		0			0
$155$	−11.2009	+	11.5319i	−0.899676	+	0.926262i
$156$		0			0
$157$		−	6.61502i		−	0.527936i	−0.964531	−	0.263968i	$-0.914969\pi$
$157$		−	6.61502i		−	0.527936i	0.964531	−	0.263968i	$-0.0850314\pi$
$158$		0			0
$159$	−2.25044			−0.178472
$160$		0			0
$161$	−10.1770			−0.802064
$162$		0			0
$163$			19.7411i			1.54624i	0.634259	+	0.773120i	$0.281305\pi$
$163$			19.7411i			1.54624i	−0.634259	+	0.773120i	$0.718695\pi$
$164$		0			0
$165$	7.30751	+	7.09777i	0.568889	+	0.552561i
$166$		0			0
$167$			20.5515i			1.59032i	0.606400	+	0.795160i	$0.292613\pi$
$167$			20.5515i			1.59032i	−0.606400	+	0.795160i	$0.707387\pi$
$168$		0			0
$169$	−18.4686			−1.42066
$170$		0			0
$171$	3.76805			0.288150
$172$		0			0
$173$		−	21.5461i		−	1.63812i	−0.573707	−	0.819061i	$-0.694495\pi$
$173$		−	21.5461i		−	1.63812i	0.573707	−	0.819061i	$-0.305505\pi$
$174$		0			0
$175$	18.7398	−	0.545817i	1.41660	−	0.0412599i
$176$		0			0
$177$		−	12.1063i		−	0.909967i
$178$		0			0
$179$	−6.57160			−0.491185			−0.245592	−	0.969373i	$-0.578982\pi$
$179$	−6.57160			−0.491185			−0.245592	+	0.969373i	$0.578982\pi$
$180$		0			0
$181$	−13.8266			−1.02772			−0.513861	−	0.857873i	$-0.671785\pi$
$181$	−13.8266			−1.02772			−0.513861	+	0.857873i	$0.671785\pi$
$182$		0			0
$183$		−	5.11877i		−	0.378390i
$184$		0			0
$185$	7.48486	+	7.27003i	0.550298	+	0.534503i
$186$		0			0
$187$		−	4.55583i		−	0.333156i
$188$		0			0
$189$	3.74956			0.272740
$190$		0			0
$191$	13.1525			0.951681			0.475841	−	0.879531i	$-0.342144\pi$
$191$	13.1525			0.951681			0.475841	+	0.879531i	$0.342144\pi$
$192$		0			0
$193$		−	5.01244i		−	0.360803i	−0.983593	−	0.180402i	$-0.942260\pi$
$193$		−	5.01244i		−	0.360803i	0.983593	−	0.180402i	$-0.0577397\pi$
$194$		0			0
$195$	−8.73962	+	8.99788i	−0.625857	+	0.644352i
$196$		0			0
$197$		−	4.75503i		−	0.338782i	−0.985549	−	0.169391i	$-0.945820\pi$
$197$		−	4.75503i		−	0.338782i	0.985549	−	0.169391i	$-0.0541801\pi$
$198$		0			0
$199$	−17.9630			−1.27336			−0.636682	−	0.771126i	$-0.719694\pi$
$199$	−17.9630			−1.27336			−0.636682	+	0.771126i	$0.719694\pi$
$200$		0			0
$201$	−12.1063			−0.853915
$202$		0			0
$203$			5.39909i			0.378942i
$204$		0			0
$205$	−3.62072	+	3.72771i	−0.252882	+	0.260355i
$206$		0			0
$207$		−	2.71420i		−	0.188650i
$208$		0			0
$209$	17.1666			1.18744
$210$		0			0
$211$	−6.02851			−0.415020			−0.207510	−	0.978233i	$-0.566536\pi$
$211$	−6.02851			−0.415020			−0.207510	+	0.978233i	$0.566536\pi$
$212$		0			0
$213$		−	11.2194i		−	0.768739i
$214$		0			0
$215$	14.0004	+	13.5985i	0.954816	+	0.927411i
$216$		0			0
$217$			26.9574i			1.82999i
$218$		0			0
$219$	−9.93371			−0.671258
$220$		0			0
$221$	5.60969			0.377348
$222$		0			0
$223$		−	24.7200i		−	1.65537i	−0.561191	−	0.827686i	$-0.689657\pi$
$223$		−	24.7200i		−	1.65537i	0.561191	−	0.827686i	$-0.310343\pi$
$224$		0			0
$225$	0.145568	+	4.99788i	0.00970456	+	0.333192i
$226$		0			0
$227$			10.5816i			0.702327i	0.936314	+	0.351163i	$0.114214\pi$
$227$			10.5816i			0.702327i	−0.936314	+	0.351163i	$0.885786\pi$
$228$		0			0
$229$	28.3902			1.87608			0.938039	−	0.346528i	$-0.112640\pi$
$229$	28.3902			1.87608			0.938039	+	0.346528i	$0.112640\pi$
$230$		0			0
$231$	17.0824			1.12394
$232$		0			0
$233$			15.4985i			1.01534i	0.861552	+	0.507669i	$0.169493\pi$
$233$			15.4985i			1.01534i	−0.861552	+	0.507669i	$0.830507\pi$
$234$		0			0
$235$	1.81299	+	1.76095i	0.118266	+	0.114872i
$236$		0			0
$237$		−	16.3790i		−	1.06393i
$238$		0			0
$239$	−11.8266			−0.764998			−0.382499	−	0.923956i	$-0.624936\pi$
$239$	−11.8266			−0.764998			−0.382499	+	0.923956i	$0.624936\pi$
$240$		0			0
$241$	−10.3803			−0.668657			−0.334329	−	0.942457i	$-0.608510\pi$
$241$	−10.3803			−0.668657			−0.334329	+	0.942457i	$0.608510\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	10.9979	−	11.3229i	0.702629	−	0.723392i
$246$		0			0
$247$			21.1376i			1.34495i
$248$		0			0
$249$	−10.3025			−0.652896
$250$		0			0
$251$	6.81051			0.429876			0.214938	−	0.976628i	$-0.431045\pi$
$251$	6.81051			0.429876			0.214938	+	0.976628i	$0.431045\pi$
$252$		0			0
$253$		−	12.3654i		−	0.777408i
$254$		0			0
$255$	1.55795	−	1.60399i	0.0975628	−	0.100446i
$256$		0			0
$257$			1.99152i			0.124228i	0.998069	+	0.0621139i	$0.0197842\pi$
$257$			1.99152i			0.124228i	−0.998069	+	0.0621139i	$0.980216\pi$
$258$		0			0
$259$	17.4969			1.08721
$260$		0			0
$261$	−1.43993			−0.0891293
$262$		0			0
$263$			18.8897i			1.16479i	0.812906	+	0.582395i	$0.197884\pi$
$263$			18.8897i			1.16479i	−0.812906	+	0.582395i	$0.802116\pi$
$264$		0			0
$265$	−3.60969	−	3.50608i	−0.221741	−	0.215377i
$266$		0			0
$267$		−	12.6179i		−	0.772202i
$268$		0			0
$269$	−19.8338			−1.20929			−0.604644	−	0.796496i	$-0.706685\pi$
$269$	−19.8338			−1.20929			−0.604644	+	0.796496i	$0.706685\pi$
$270$		0			0
$271$	−6.50380			−0.395078			−0.197539	−	0.980295i	$-0.563295\pi$
$271$	−6.50380			−0.395078			−0.197539	+	0.980295i	$0.563295\pi$
$272$		0			0
$273$			21.0338i			1.27303i
$274$		0			0
$275$	0.663185	+	22.7695i	0.0399916	+	1.37305i
$276$		0			0
$277$			16.3790i			0.984117i	0.870562	+	0.492059i	$0.163755\pi$
$277$			16.3790i			0.984117i	−0.870562	+	0.492059i	$0.836245\pi$
$278$		0			0
$279$	−7.18949			−0.430423
$280$		0			0
$281$	23.5983			1.40776			0.703880	−	0.710319i	$-0.251449\pi$
$281$	23.5983			1.40776			0.703880	+	0.710319i	$0.251449\pi$
$282$		0			0
$283$			7.65396i			0.454981i	0.973780	+	0.227490i	$0.0730521\pi$
$283$			7.65396i			0.454981i	−0.973780	+	0.227490i	$0.926948\pi$
$284$		0			0
$285$	6.04392	+	5.87045i	0.358011	+	0.347735i
$286$		0			0
$287$			8.71406i			0.514375i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	9.03522			0.529654
$292$		0			0
$293$		−	12.0099i		−	0.701625i	−0.936446	−	0.350812i	$-0.885906\pi$
$293$		−	12.0099i		−	0.701625i	0.936446	−	0.350812i	$-0.114094\pi$
$294$		0			0
$295$	18.8611	−	19.4184i	1.09813	−	1.13058i
$296$		0			0
$297$			4.55583i			0.264356i
$298$		0			0
$299$	15.2258			0.880531
$300$		0			0
$301$	32.7278			1.88640
$302$		0			0
$303$		−	15.9151i		−	0.914297i
$304$		0			0
$305$	7.97480	−	8.21045i	0.456636	−	0.470129i
$306$		0			0
$307$			1.39989i			0.0798959i	0.999202	+	0.0399479i	$0.0127192\pi$
$307$			1.39989i			0.0798959i	−0.999202	+	0.0399479i	$0.987281\pi$
$308$		0			0
$309$	−8.72845			−0.496544
$310$		0			0
$311$	1.29276			0.0733060			0.0366530	−	0.999328i	$-0.488330\pi$
$311$	1.29276			0.0733060			0.0366530	+	0.999328i	$0.488330\pi$
$312$		0			0
$313$			18.3603i			1.03779i	0.854839	+	0.518894i	$0.173656\pi$
$313$			18.3603i			1.03779i	−0.854839	+	0.518894i	$0.826344\pi$
$314$		0			0
$315$	6.01425	+	5.84163i	0.338865	+	0.329139i
$316$		0			0
$317$			7.12371i			0.400108i	0.979785	+	0.200054i	$0.0641117\pi$
$317$			7.12371i			0.400108i	−0.979785	+	0.200054i	$0.935888\pi$
$318$		0			0
$319$	−6.56007			−0.367294
$320$		0			0
$321$	6.29114			0.351137
$322$		0			0
$323$		−	3.76805i		−	0.209660i
$324$		0			0
$325$	−28.0365	+	0.816593i	−1.55519	+	0.0452964i
$326$		0			0
$327$		−	5.53610i		−	0.306147i
$328$		0			0
$329$	4.23812			0.233655
$330$		0			0
$331$	12.5772			0.691305			0.345652	−	0.938363i	$-0.387658\pi$
$331$	12.5772			0.691305			0.345652	+	0.938363i	$0.387658\pi$
$332$		0			0
$333$			4.66640i			0.255717i
$334$		0			0
$335$	−19.4184	−	18.8611i	−1.06094	−	1.03049i
$336$		0			0
$337$			3.85726i			0.210119i	0.994466	+	0.105059i	$0.0335032\pi$
$337$			3.85726i			0.210119i	−0.994466	+	0.105059i	$0.966497\pi$
$338$		0			0
$339$	2.16566			0.117622
$340$		0			0
$341$	−32.7541			−1.77373
$342$		0			0
$343$		−	0.221932i		−	0.0119832i
$344$		0			0
$345$	4.22859	−	4.35355i	0.227660	−	0.234387i
$346$		0			0
$347$		−	21.4920i		−	1.15375i	−0.816832	−	0.576876i	$-0.804272\pi$
$347$		−	21.4920i		−	1.15375i	0.816832	−	0.576876i	$-0.195728\pi$
$348$		0			0
$349$	17.5985			0.942025			0.471013	−	0.882126i	$-0.343889\pi$
$349$	17.5985			0.942025			0.471013	+	0.882126i	$0.343889\pi$
$350$		0			0
$351$	−5.60969			−0.299423
$352$		0			0
$353$		−	17.4328i		−	0.927856i	−0.885873	−	0.463928i	$-0.846440\pi$
$353$		−	17.4328i		−	0.927856i	0.885873	−	0.463928i	$-0.153560\pi$
$354$		0			0
$355$	17.4792	−	17.9958i	0.927702	−	0.955116i
$356$		0			0
$357$		−	3.74956i		−	0.198448i
$358$		0			0
$359$	−21.4021			−1.12956			−0.564781	−	0.825241i	$-0.691039\pi$
$359$	−21.4021			−1.12956			−0.564781	+	0.825241i	$0.691039\pi$
$360$		0			0
$361$	−4.80178			−0.252725
$362$		0			0
$363$			9.75561i			0.512037i
$364$		0			0
$365$	−15.9336	−	15.4762i	−0.834001	−	0.810064i
$366$		0			0
$367$			19.3380i			1.00944i	0.863284	+	0.504718i	$0.168404\pi$
$367$			19.3380i			1.00944i	−0.863284	+	0.504718i	$0.831596\pi$
$368$		0			0
$369$	−2.32402			−0.120984
$370$		0			0
$371$	−8.43816			−0.438088
$372$		0			0
$373$		−	4.50088i		−	0.233047i	−0.993188	−	0.116523i	$-0.962825\pi$
$373$		−	4.50088i		−	0.233047i	0.993188	−	0.116523i	$-0.0371750\pi$
$374$		0			0
$375$	−7.55297	+	8.24334i	−0.390034	+	0.425684i
$376$		0			0
$377$		−	8.07754i		−	0.416015i
$378$		0			0
$379$	21.3377			1.09605			0.548023	−	0.836463i	$-0.315381\pi$
$379$	21.3377			1.09605			0.548023	+	0.836463i	$0.315381\pi$
$380$		0			0
$381$	4.76929			0.244338
$382$		0			0
$383$		−	7.92160i		−	0.404775i	−0.979306	−	0.202387i	$-0.935130\pi$
$383$		−	7.92160i		−	0.404775i	0.979306	−	0.202387i	$-0.0648700\pi$
$384$		0			0
$385$	27.3999	+	26.6135i	1.39643	+	1.35635i
$386$		0			0
$387$			8.72845i			0.443692i
$388$		0			0
$389$	−12.0992			−0.613455			−0.306728	−	0.951797i	$-0.599234\pi$
$389$	−12.0992			−0.613455			−0.306728	+	0.951797i	$0.599234\pi$
$390$		0			0
$391$	−2.71420			−0.137263
$392$		0			0
$393$			15.4554i			0.779623i
$394$		0			0
$395$	25.5177	−	26.2717i	1.28393	−	1.32187i
$396$		0			0
$397$			1.75804i			0.0882333i	0.999026	+	0.0441167i	$0.0140473\pi$
$397$			1.75804i			0.0882333i	−0.999026	+	0.0441167i	$0.985953\pi$
$398$		0			0
$399$	14.1285			0.707311
$400$		0			0
$401$	20.8869			1.04304			0.521520	−	0.853239i	$-0.325365\pi$
$401$	20.8869			1.04304			0.521520	+	0.853239i	$0.325365\pi$
$402$		0			0
$403$		−	40.3308i		−	2.00902i
$404$		0			0
$405$	−1.55795	+	1.60399i	−0.0774153	+	0.0797029i
$406$		0			0
$407$			21.2593i			1.05379i
$408$		0			0
$409$	−11.1009			−0.548902			−0.274451	−	0.961601i	$-0.588496\pi$
$409$	−11.1009			−0.548902			−0.274451	+	0.961601i	$0.588496\pi$
$410$		0			0
$411$	−10.8982			−0.537569
$412$		0			0
$413$		−	45.3934i		−	2.23366i
$414$		0			0
$415$	−16.5252	−	16.0508i	−0.811188	−	0.787905i
$416$		0			0
$417$			18.3310i			0.897675i
$418$		0			0
$419$	−35.6345			−1.74086			−0.870429	−	0.492294i	$-0.836158\pi$
$419$	−35.6345			−1.74086			−0.870429	+	0.492294i	$0.836158\pi$
$420$		0			0
$421$	−12.6476			−0.616408			−0.308204	−	0.951320i	$-0.599728\pi$
$421$	−12.6476			−0.616408			−0.308204	+	0.951320i	$0.599728\pi$
$422$		0			0
$423$			1.13030i			0.0549570i
$424$		0			0
$425$	4.99788	−	0.145568i	0.242433	−	0.00706110i
$426$		0			0
$427$		−	19.1931i		−	0.928820i
$428$		0			0
$429$	−25.5568			−1.23389
$430$		0			0
$431$	11.3265			0.545578			0.272789	−	0.962074i	$-0.412054\pi$
$431$	11.3265			0.545578			0.272789	+	0.962074i	$0.412054\pi$
$432$		0			0
$433$			5.19207i			0.249515i	0.992187	+	0.124758i	$0.0398153\pi$
$433$			5.19207i			0.249515i	−0.992187	+	0.124758i	$0.960185\pi$
$434$		0			0
$435$	−2.30963	−	2.24334i	−0.110738	−	0.107560i
$436$		0			0
$437$		−	10.2272i		−	0.489235i
$438$		0			0
$439$	−27.7417			−1.32404			−0.662019	−	0.749487i	$-0.730300\pi$
$439$	−27.7417			−1.32404			−0.662019	+	0.749487i	$0.730300\pi$
$440$		0			0
$441$	7.05919			0.336152
$442$		0			0
$443$			12.3872i			0.588532i	0.955723	+	0.294266i	$0.0950752\pi$
$443$			12.3872i			0.588532i	−0.955723	+	0.294266i	$0.904925\pi$
$444$		0			0
$445$	19.6581	−	20.2390i	0.931882	−	0.959419i
$446$		0			0
$447$		−	13.6424i		−	0.645265i
$448$		0			0
$449$	−38.3417			−1.80946			−0.904729	−	0.425989i	$-0.859926\pi$
$449$	−38.3417			−1.80946			−0.904729	+	0.425989i	$0.859926\pi$
$450$		0			0
$451$	−10.5879			−0.498563
$452$		0			0
$453$			17.9035i			0.841182i
$454$		0			0
$455$	−32.7697	+	33.7381i	−1.53627	+	1.58167i
$456$		0			0
$457$			18.8539i			0.881950i	0.897519	+	0.440975i	$0.145367\pi$
$457$			18.8539i			0.881950i	−0.897519	+	0.440975i	$0.854633\pi$
$458$		0			0
$459$	1.00000			0.0466760
$460$		0			0
$461$	12.5223			0.583221			0.291611	−	0.956537i	$-0.405809\pi$
$461$	12.5223			0.583221			0.291611	+	0.956537i	$0.405809\pi$
$462$		0			0
$463$		−	19.6830i		−	0.914746i	−0.889275	−	0.457373i	$-0.848790\pi$
$463$		−	19.6830i		−	0.914746i	0.889275	−	0.457373i	$-0.151210\pi$
$464$		0			0
$465$	−11.5319	−	11.2009i	−0.534777	−	0.519428i
$466$		0			0
$467$		−	10.2790i		−	0.475653i	−0.971308	−	0.237827i	$-0.923565\pi$
$467$		−	10.2790i		−	0.475653i	0.971308	−	0.237827i	$-0.0764350\pi$
$468$		0			0
$469$	−45.3934			−2.09607
$470$		0			0
$471$	6.61502			0.304804
$472$		0			0
$473$			39.7654i			1.82841i
$474$		0			0
$475$	0.548509	+	18.8323i	0.0251673	+	0.864084i
$476$		0			0
$477$		−	2.25044i		−	0.103041i
$478$		0			0
$479$	26.1372			1.19424			0.597120	−	0.802152i	$-0.296312\pi$
$479$	26.1372			1.19424			0.597120	+	0.802152i	$0.296312\pi$
$480$		0			0
$481$	−26.1770			−1.19357
$482$		0			0
$483$		−	10.1770i		−	0.463072i
$484$		0			0
$485$	14.4924	+	14.0764i	0.658066	+	0.639178i
$486$		0			0
$487$			33.3013i			1.50902i	0.656286	+	0.754512i	$0.272126\pi$
$487$			33.3013i			1.50902i	−0.656286	+	0.754512i	$0.727874\pi$
$488$		0			0
$489$	−19.7411			−0.892723
$490$		0			0
$491$	13.0690			0.589794			0.294897	−	0.955529i	$-0.404715\pi$
$491$	13.0690			0.589794			0.294897	+	0.955529i	$0.404715\pi$
$492$		0			0
$493$			1.43993i			0.0648511i
$494$		0			0
$495$	−7.09777	+	7.30751i	−0.319021	+	0.328448i
$496$		0			0
$497$		−	42.0677i		−	1.88699i
$498$		0			0
$499$	−28.6993			−1.28476			−0.642378	−	0.766388i	$-0.722052\pi$
$499$	−28.6993			−1.28476			−0.642378	+	0.766388i	$0.722052\pi$
$500$		0			0
$501$	−20.5515			−0.918171
$502$		0			0
$503$		−	38.9674i		−	1.73747i	−0.495276	−	0.868736i	$-0.664933\pi$
$503$		−	38.9674i		−	1.73747i	0.495276	−	0.868736i	$-0.335067\pi$
$504$		0			0
$505$	24.7949	−	25.5276i	1.10336	−	1.13596i
$506$		0			0
$507$		−	18.4686i		−	0.820218i
$508$		0			0
$509$	29.0147			1.28605			0.643027	−	0.765844i	$-0.277678\pi$
$509$	29.0147			1.28605			0.643027	+	0.765844i	$0.277678\pi$
$510$		0			0
$511$	−37.2470			−1.64771
$512$		0			0
$513$			3.76805i			0.166364i
$514$		0			0
$515$	−14.0004	−	13.5985i	−0.616929	−	0.599222i
$516$		0			0
$517$			5.14945i			0.226472i
$518$		0			0
$519$	21.5461			0.945770
$520$		0			0
$521$	19.8338			0.868935			0.434468	−	0.900687i	$-0.356937\pi$
$521$	19.8338			0.868935			0.434468	+	0.900687i	$0.356937\pi$
$522$		0			0
$523$			11.8156i			0.516659i	0.966057	+	0.258329i	$0.0831720\pi$
$523$			11.8156i			0.516659i	−0.966057	+	0.258329i	$0.916828\pi$
$524$		0			0
$525$	0.545817	+	18.7398i	0.0238214	+	0.817874i
$526$		0			0
$527$			7.18949i			0.313179i
$528$		0			0
$529$	15.6331			0.679701
$530$		0			0
$531$	12.1063			0.525370
$532$		0			0
$533$		−	13.0370i		−	0.564697i
$534$		0			0
$535$	10.0909	+	9.80129i	0.436269	+	0.423747i
$536$		0			0
$537$		−	6.57160i		−	0.283586i
$538$		0			0
$539$	32.1605			1.38525
$540$		0			0
$541$	−20.2140			−0.869069			−0.434535	−	0.900655i	$-0.643087\pi$
$541$	−20.2140			−0.869069			−0.434535	+	0.900655i	$0.643087\pi$
$542$		0			0
$543$		−	13.8266i		−	0.593356i
$544$		0			0
$545$	8.62499	−	8.87986i	0.369454	−	0.380371i
$546$		0			0
$547$			24.0362i			1.02771i	0.857876	+	0.513857i	$0.171784\pi$
$547$			24.0362i			1.02771i	−0.857876	+	0.513857i	$0.828216\pi$
$548$		0			0
$549$	5.11877			0.218464
$550$		0			0
$551$	−5.42572			−0.231144
$552$		0			0
$553$		−	61.4139i		−	2.61159i
$554$		0			0
$555$	−7.27003	+	7.48486i	−0.308596	+	0.317715i
$556$		0			0
$557$			5.22510i			0.221395i	0.993854	+	0.110697i	$0.0353084\pi$
$557$			5.22510i			0.221395i	−0.993854	+	0.110697i	$0.964692\pi$
$558$		0			0
$559$	−48.9639			−2.07095
$560$		0			0
$561$	4.55583			0.192347
$562$		0			0
$563$		−	17.6767i		−	0.744986i	−0.928035	−	0.372493i	$-0.878503\pi$
$563$		−	17.6767i		−	0.744986i	0.928035	−	0.372493i	$-0.121497\pi$
$564$		0			0
$565$	3.47369	+	3.37399i	0.146139	+	0.141945i
$566$		0			0
$567$			3.74956i			0.157467i
$568$		0			0
$569$	34.4124			1.44264			0.721322	−	0.692600i	$-0.243535\pi$
$569$	34.4124			1.44264			0.721322	+	0.692600i	$0.243535\pi$
$570$		0			0
$571$	2.76830			0.115850			0.0579250	−	0.998321i	$-0.481552\pi$
$571$	2.76830			0.115850			0.0579250	+	0.998321i	$0.481552\pi$
$572$		0			0
$573$			13.1525i			0.549454i
$574$		0			0
$575$	13.5652	−	0.395102i	0.565710	−	0.0164769i
$576$		0			0
$577$			4.45719i			0.185555i	0.995687	+	0.0927776i	$0.0295745\pi$
$577$			4.45719i			0.185555i	−0.995687	+	0.0927776i	$0.970425\pi$
$578$		0			0
$579$	5.01244			0.208310
$580$		0			0
$581$	−38.6299			−1.60264
$582$		0			0
$583$		−	10.2526i		−	0.424621i
$584$		0			0
$585$	−8.99788	−	8.73962i	−0.372017	−	0.361339i
$586$		0			0
$587$		−	37.4770i		−	1.54684i	−0.633894	−	0.773420i	$-0.718545\pi$
$587$		−	37.4770i		−	1.54684i	0.633894	−	0.773420i	$-0.281455\pi$
$588$		0			0
$589$	−27.0904			−1.11624
$590$		0			0
$591$	4.75503			0.195596
$592$		0			0
$593$			26.0220i			1.06859i	0.845297	+	0.534297i	$0.179424\pi$
$593$			26.0220i			1.06859i	−0.845297	+	0.534297i	$0.820576\pi$
$594$		0			0
$595$	5.84163	−	6.01425i	0.239484	−	0.246560i
$596$		0			0
$597$		−	17.9630i		−	0.735177i
$598$		0			0
$599$	−11.2780			−0.460806			−0.230403	−	0.973095i	$-0.574004\pi$
$599$	−11.2780			−0.460806			−0.230403	+	0.973095i	$0.574004\pi$
$600$		0			0
$601$	−5.58266			−0.227722			−0.113861	−	0.993497i	$-0.536322\pi$
$601$	−5.58266			−0.227722			−0.113861	+	0.993497i	$0.536322\pi$
$602$		0			0
$603$		−	12.1063i		−	0.493008i
$604$		0			0
$605$	−15.1988	+	15.6479i	−0.617918	+	0.636178i
$606$		0			0
$607$		−	29.3534i		−	1.19142i	−0.803201	−	0.595708i	$-0.796872\pi$
$607$		−	29.3534i		−	1.19142i	0.803201	−	0.595708i	$-0.203128\pi$
$608$		0			0
$609$	−5.39909			−0.218782
$610$		0			0
$611$	−6.34062			−0.256514
$612$		0			0
$613$			10.3651i			0.418642i	0.977847	+	0.209321i	$0.0671254\pi$
$613$			10.3651i			0.418642i	−0.977847	+	0.209321i	$0.932875\pi$
$614$		0			0
$615$	−3.72771	−	3.62072i	−0.150316	−	0.146001i
$616$		0			0
$617$		−	19.9698i		−	0.803955i	−0.915649	−	0.401978i	$-0.868323\pi$
$617$		−	19.9698i		−	0.803955i	0.915649	−	0.401978i	$-0.131677\pi$
$618$		0			0
$619$	−6.98756			−0.280854			−0.140427	−	0.990091i	$-0.544848\pi$
$619$	−6.98756			−0.280854			−0.140427	+	0.990091i	$0.544848\pi$
$620$		0			0
$621$	2.71420			0.108917
$622$		0			0
$623$		−	47.3115i		−	1.89550i
$624$		0			0
$625$	−24.9576	+	1.45507i	−0.998305	+	0.0582027i
$626$		0			0
$627$			17.1666i			0.685569i
$628$		0			0
$629$	4.66640			0.186062
$630$		0			0
$631$	−28.5257			−1.13559			−0.567796	−	0.823170i	$-0.692204\pi$
$631$	−28.5257			−1.13559			−0.567796	+	0.823170i	$0.692204\pi$
$632$		0			0
$633$		−	6.02851i		−	0.239612i
$634$		0			0
$635$	7.64989	+	7.43032i	0.303577	+	0.294863i
$636$		0			0
$637$			39.5998i			1.56900i
$638$		0			0
$639$	11.2194			0.443831
$640$		0			0
$641$	46.4035			1.83283			0.916415	−	0.400229i	$-0.131070\pi$
$641$	46.4035			1.83283			0.916415	+	0.400229i	$0.131070\pi$
$642$		0			0
$643$			32.4559i			1.27994i	0.768401	+	0.639969i	$0.221053\pi$
$643$			32.4559i			1.27994i	−0.768401	+	0.639969i	$0.778947\pi$
$644$		0			0
$645$	−13.5985	+	14.0004i	−0.535441	+	0.551263i
$646$		0			0
$647$		−	10.0943i		−	0.396847i	−0.980116	−	0.198424i	$-0.936418\pi$
$647$		−	10.0943i		−	0.396847i	0.980116	−	0.198424i	$-0.0635822\pi$
$648$		0			0
$649$	55.1544			2.16500
$650$		0			0
$651$	−26.9574			−1.05654
$652$		0			0
$653$			49.4313i			1.93440i	0.254022	+	0.967198i	$0.418246\pi$
$653$			49.4313i			1.93440i	−0.254022	+	0.967198i	$0.581754\pi$
$654$		0			0
$655$	−24.0788	+	24.7903i	−0.940837	+	0.968639i
$656$		0			0
$657$		−	9.93371i		−	0.387551i
$658$		0			0
$659$	33.5983			1.30881			0.654403	−	0.756146i	$-0.272920\pi$
$659$	33.5983			1.30881			0.654403	+	0.756146i	$0.272920\pi$
$660$		0			0
$661$	−39.9313			−1.55315			−0.776575	−	0.630025i	$-0.783044\pi$
$661$	−39.9313			−1.55315			−0.776575	+	0.630025i	$0.783044\pi$
$662$		0			0
$663$			5.60969i			0.217862i
$664$		0			0
$665$	22.6620	+	22.0116i	0.878796	+	0.853572i
$666$		0			0
$667$			3.90825i			0.151328i
$668$		0			0
$669$	24.7200			0.955729
$670$		0			0
$671$	23.3203			0.900268
$672$		0			0
$673$			11.4654i			0.441958i	0.975279	+	0.220979i	$0.0709252\pi$
$673$			11.4654i			0.441958i	−0.975279	+	0.220979i	$0.929075\pi$
$674$		0			0
$675$	−4.99788	+	0.145568i	−0.192369	+	0.00560293i
$676$		0			0
$677$		−	46.3844i		−	1.78270i	−0.453320	−	0.891348i	$-0.649760\pi$
$677$		−	46.3844i		−	1.78270i	0.453320	−	0.891348i	$-0.350240\pi$
$678$		0			0
$679$	33.8781			1.30012
$680$		0			0
$681$	−10.5816			−0.405488
$682$		0			0
$683$			26.1558i			1.00082i	0.865787	+	0.500412i	$0.166818\pi$
$683$			26.1558i			1.00082i	−0.865787	+	0.500412i	$0.833182\pi$
$684$		0			0
$685$	−17.4806	−	16.9789i	−0.667900	−	0.648730i
$686$		0			0
$687$			28.3902i			1.08315i
$688$		0			0
$689$	12.6243			0.480946
$690$		0			0
$691$	23.2958			0.886215			0.443107	−	0.896469i	$-0.353876\pi$
$691$	23.2958			0.886215			0.443107	+	0.896469i	$0.353876\pi$
$692$		0			0
$693$			17.0824i			0.648905i
$694$		0			0
$695$	−28.5589	+	29.4028i	−1.08330	+	1.11531i
$696$		0			0
$697$			2.32402i			0.0880287i
$698$		0			0
$699$	−15.4985			−0.586206
$700$		0			0
$701$	−37.9222			−1.43230			−0.716150	−	0.697946i	$-0.754098\pi$
$701$	−37.9222			−1.43230			−0.716150	+	0.697946i	$0.754098\pi$
$702$		0			0
$703$			17.5832i			0.663165i
$704$		0			0
$705$	−1.76095	+	1.81299i	−0.0663212	+	0.0682810i
$706$		0			0
$707$		−	59.6745i		−	2.24429i
$708$		0			0
$709$	29.7775			1.11832			0.559160	−	0.829060i	$-0.311124\pi$
$709$	29.7775			1.11832			0.559160	+	0.829060i	$0.311124\pi$
$710$		0			0
$711$	16.3790			0.614259
$712$		0			0
$713$			19.5137i			0.730794i
$714$		0			0
$715$	−40.9928	−	39.8163i	−1.53305	−	1.48904i
$716$		0			0
$717$		−	11.8266i		−	0.441672i
$718$		0			0
$719$	−1.18930			−0.0443533			−0.0221767	−	0.999754i	$-0.507060\pi$
$719$	−1.18930			−0.0443533			−0.0221767	+	0.999754i	$0.507060\pi$
$720$		0			0
$721$	−32.7278			−1.21885
$722$		0			0
$723$		−	10.3803i		−	0.386049i
$724$		0			0
$725$	−0.209608	−	7.19659i	−0.00778465	−	0.267275i
$726$		0			0
$727$		−	24.5522i		−	0.910590i	−0.890341	−	0.455295i	$-0.849534\pi$
$727$		−	24.5522i		−	0.910590i	0.890341	−	0.455295i	$-0.150466\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	8.72845			0.322834
$732$		0			0
$733$			17.0284i			0.628958i	0.949264	+	0.314479i	$0.101830\pi$
$733$			17.0284i			0.628958i	−0.949264	+	0.314479i	$0.898170\pi$
$734$		0			0
$735$	11.3229	+	10.9979i	0.417650	+	0.405663i
$736$		0			0
$737$		−	55.1544i		−	2.03164i
$738$		0			0
$739$	−37.7887			−1.39008			−0.695039	−	0.718972i	$-0.744613\pi$
$739$	−37.7887			−1.39008			−0.695039	+	0.718972i	$0.744613\pi$
$740$		0			0
$741$	−21.1376			−0.776509
$742$		0			0
$743$			19.0213i			0.697824i	0.937156	+	0.348912i	$0.113449\pi$
$743$			19.0213i			0.697824i	−0.937156	+	0.348912i	$0.886551\pi$
$744$		0			0
$745$	21.2543	−	21.8823i	0.778696	−	0.801707i
$746$		0			0
$747$		−	10.3025i		−	0.376950i
$748$		0			0
$749$	23.5890			0.861923
$750$		0			0
$751$	32.6034			1.18972			0.594858	−	0.803831i	$-0.297208\pi$
$751$	32.6034			1.18972			0.594858	+	0.803831i	$0.297208\pi$
$752$		0			0
$753$			6.81051i			0.248189i
$754$		0			0
$755$	−27.8929	+	28.7171i	−1.01513	+	1.04512i
$756$		0			0
$757$			47.0786i			1.71110i	0.517718	+	0.855551i	$0.326782\pi$
$757$			47.0786i			1.71110i	−0.517718	+	0.855551i	$0.673218\pi$
$758$		0			0
$759$	12.3654			0.448837
$760$		0			0
$761$	−0.0274098			−0.000993605			−0.000496802	−	1.00000i	$-0.500158\pi$
$761$	−0.0274098			−0.000993605			−0.000496802		1.00000i	$0.500158\pi$
$762$		0			0
$763$		−	20.7579i		−	0.751488i
$764$		0			0
$765$	1.60399	+	1.55795i	0.0579924	+	0.0563279i
$766$		0			0
$767$			67.9127i			2.45219i
$768$		0			0
$769$	46.2514			1.66787			0.833935	−	0.551863i	$-0.186083\pi$
$769$	46.2514			1.66787			0.833935	+	0.551863i	$0.186083\pi$
$770$		0			0
$771$	−1.99152			−0.0717229
$772$		0			0
$773$		−	23.6334i		−	0.850033i	−0.905186	−	0.425017i	$-0.860268\pi$
$773$		−	23.6334i		−	0.850033i	0.905186	−	0.425017i	$-0.139732\pi$
$774$		0			0
$775$	−1.04656	−	35.9322i	−0.0375936	−	1.29072i
$776$		0			0
$777$			17.4969i			0.627699i
$778$		0			0
$779$	−8.75704			−0.313754
$780$		0			0
$781$	51.1136			1.82899
$782$		0			0
$783$		−	1.43993i		−	0.0514588i
$784$		0			0
$785$	10.6104	+	10.3059i	0.378702	+	0.367833i
$786$		0			0
$787$		−	13.9213i		−	0.496242i	−0.968729	−	0.248121i	$-0.920187\pi$
$787$		−	13.9213i		−	0.496242i	0.968729	−	0.248121i	$-0.0798130\pi$
$788$		0			0
$789$	−18.8897			−0.672492
$790$		0			0
$791$	8.12026			0.288723
$792$		0			0
$793$			28.7147i			1.01969i
$794$		0			0
$795$	3.50608	−	3.60969i	0.124348	−	0.128022i
$796$		0			0
$797$		−	41.9778i		−	1.48693i	−0.668774	−	0.743466i	$-0.733181\pi$
$797$		−	41.9778i		−	1.48693i	0.668774	−	0.743466i	$-0.266819\pi$
$798$		0			0
$799$	1.13030			0.0399871
$800$		0			0
$801$	12.6179			0.445831
$802$		0			0
$803$		−	45.2563i		−	1.59706i
$804$		0			0
$805$	15.8554	−	16.3239i	0.558828	−	0.575341i
$806$		0			0
$807$		−	19.8338i		−	0.698183i
$808$		0			0
$809$	35.7922			1.25839			0.629193	−	0.777249i	$-0.283385\pi$
$809$	35.7922			1.25839			0.629193	+	0.777249i	$0.283385\pi$
$810$		0			0
$811$	3.77204			0.132454			0.0662271	−	0.997805i	$-0.478904\pi$
$811$	3.77204			0.132454			0.0662271	+	0.997805i	$0.478904\pi$
$812$		0			0
$813$		−	6.50380i		−	0.228098i
$814$		0			0
$815$	−31.6645	−	30.7557i	−1.10916	−	1.07732i
$816$		0			0
$817$			32.8893i			1.15065i
$818$		0			0
$819$	−21.0338			−0.734982
$820$		0			0
$821$	23.3883			0.816259			0.408129	−	0.912924i	$-0.366181\pi$
$821$	23.3883			0.816259			0.408129	+	0.912924i	$0.366181\pi$
$822$		0			0
$823$			20.0362i			0.698418i	0.937045	+	0.349209i	$0.113550\pi$
$823$			20.0362i			0.698418i	−0.937045	+	0.349209i	$0.886450\pi$
$824$		0			0
$825$	−22.7695	+	0.663185i	−0.792733	+	0.0230891i
$826$		0			0
$827$			25.7352i			0.894901i	0.894309	+	0.447451i	$0.147668\pi$
$827$			25.7352i			0.894901i	−0.894309	+	0.447451i	$0.852332\pi$
$828$		0			0
$829$	−6.37955			−0.221571			−0.110786	−	0.993844i	$-0.535337\pi$
$829$	−6.37955			−0.221571			−0.110786	+	0.993844i	$0.535337\pi$
$830$		0			0
$831$	−16.3790			−0.568180
$832$		0			0
$833$		−	7.05919i		−	0.244586i
$834$		0			0
$835$	−32.9643	−	32.0182i	−1.14078	−	1.10804i
$836$		0			0
$837$		−	7.18949i		−	0.248505i
$838$		0			0
$839$	−18.1684			−0.627242			−0.313621	−	0.949548i	$-0.601542\pi$
$839$	−18.1684			−0.627242			−0.313621	+	0.949548i	$0.601542\pi$
$840$		0			0
$841$	−26.9266			−0.928504
$842$		0			0
$843$			23.5983i			0.812770i
$844$		0			0
$845$	28.7732	−	29.6234i	0.989827	−	1.01908i
$846$		0			0
$847$			36.5792i			1.25688i
$848$		0			0
$849$	−7.65396			−0.262683
$850$		0			0
$851$	12.6655			0.434169
$852$		0			0
$853$		−	36.5987i		−	1.25311i	−0.779375	−	0.626557i	$-0.784463\pi$
$853$		−	36.5987i		−	1.25311i	0.779375	−	0.626557i	$-0.215537\pi$
$854$		0			0
$855$	−5.87045	+	6.04392i	−0.200765	+	0.206698i
$856$		0			0
$857$		−	32.3050i		−	1.10352i	−0.834004	−	0.551759i	$-0.813957\pi$
$857$		−	32.3050i		−	1.10352i	0.834004	−	0.551759i	$-0.186043\pi$
$858$		0			0
$859$	−13.3073			−0.454041			−0.227020	−	0.973890i	$-0.572898\pi$
$859$	−13.3073			−0.454041			−0.227020	+	0.973890i	$0.572898\pi$
$860$		0			0
$861$	−8.71406			−0.296974
$862$		0			0
$863$		−	10.1548i		−	0.345675i	−0.984950	−	0.172838i	$-0.944706\pi$
$863$		−	10.1548i		−	0.345675i	0.984950	−	0.172838i	$-0.0552935\pi$
$864$		0			0
$865$	34.5598	+	33.5678i	1.17507	+	1.14134i
$866$		0			0
$867$		−	1.00000i		−	0.0339618i
$868$		0			0
$869$	74.6199			2.53131
$870$		0			0
$871$	67.9127			2.30113
$872$		0			0
$873$			9.03522i			0.305796i
$874$		0			0
$875$	−28.3203	+	30.9089i	−0.957401	+	1.04491i
$876$		0			0
$877$		−	34.6999i		−	1.17173i	−0.810408	−	0.585865i	$-0.800755\pi$
$877$		−	34.6999i		−	1.17173i	0.810408	−	0.585865i	$-0.199245\pi$
$878$		0			0
$879$	12.0099			0.405083
$880$		0			0
$881$	18.3902			0.619582			0.309791	−	0.950805i	$-0.399741\pi$
$881$	18.3902			0.619582			0.309791	+	0.950805i	$0.399741\pi$
$882$		0			0
$883$			34.5369i			1.16226i	0.813811	+	0.581130i	$0.197389\pi$
$883$			34.5369i			1.16226i	−0.813811	+	0.581130i	$0.802611\pi$
$884$		0			0
$885$	19.4184	+	18.8611i	0.652743	+	0.634008i
$886$		0			0
$887$		−	40.3819i		−	1.35589i	−0.735112	−	0.677946i	$-0.762871\pi$
$887$		−	40.3819i		−	1.35589i	0.735112	−	0.677946i	$-0.237129\pi$
$888$		0			0
$889$	17.8827			0.599767
$890$		0			0
$891$	−4.55583			−0.152626
$892$		0			0
$893$			4.25902i			0.142523i
$894$		0			0
$895$	10.2382	−	10.5408i	0.342227	−	0.352340i
$896$		0			0
$897$			15.2258i			0.508375i
$898$		0			0
$899$	10.3523			0.345270
$900$		0			0
$901$	−2.25044			−0.0749731
$902$		0			0
$903$			32.7278i			1.08911i
$904$		0			0
$905$	21.5412	−	22.1777i	0.716052	−	0.737212i
$906$		0			0
$907$			3.86546i			0.128351i	0.997939	+	0.0641753i	$0.0204417\pi$
$907$			3.86546i			0.128351i	−0.997939	+	0.0641753i	$0.979558\pi$
$908$		0			0
$909$	15.9151			0.527870
$910$		0			0
$911$	−48.6071			−1.61043			−0.805213	−	0.592986i	$-0.797949\pi$
$911$	−48.6071			−1.61043			−0.805213	+	0.592986i	$0.797949\pi$
$912$		0			0
$913$		−	46.9366i		−	1.55337i
$914$		0			0
$915$	8.21045	+	7.97480i	0.271429	+	0.263639i
$916$		0			0
$917$			57.9510i			1.91371i
$918$		0			0
$919$	16.1950			0.534222			0.267111	−	0.963666i	$-0.413931\pi$
$919$	16.1950			0.534222			0.267111	+	0.963666i	$0.413931\pi$
$920$		0			0
$921$	−1.39989			−0.0461279
$922$		0			0
$923$			62.9372i			2.07160i
$924$		0			0
$925$	−23.3221	+	0.679281i	−0.766827	+	0.0223346i
$926$		0			0
$927$		−	8.72845i		−	0.286680i
$928$		0			0
$929$	−24.1378			−0.791935			−0.395967	−	0.918265i	$-0.629591\pi$
$929$	−24.1378			−0.791935			−0.395967	+	0.918265i	$0.629591\pi$
$930$		0			0
$931$	26.5994			0.871760
$932$		0			0
$933$			1.29276i			0.0423232i
$934$		0			0
$935$	7.30751	+	7.09777i	0.238981	+	0.232122i
$936$		0			0
$937$			0.0408335i			0.00133397i	1.00000		0.000666986i	$0.000212308\pi$
$937$			0.0408335i			0.00133397i	−1.00000		0.000666986i	$0.999788\pi$
$938$		0			0
$939$	−18.3603			−0.599167
$940$		0			0
$941$	12.8664			0.419434			0.209717	−	0.977762i	$-0.432746\pi$
$941$	12.8664			0.419434			0.209717	+	0.977762i	$0.432746\pi$
$942$		0			0
$943$			6.30786i			0.205412i
$944$		0			0
$945$	−5.84163	+	6.01425i	−0.190028	+	0.195644i
$946$		0			0
$947$			2.66444i			0.0865828i	0.999062	+	0.0432914i	$0.0137844\pi$
$947$			2.66444i			0.0865828i	−0.999062	+	0.0432914i	$0.986216\pi$
$948$		0			0
$949$	55.7250			1.80891
$950$		0			0
$951$	−7.12371			−0.231002
$952$		0			0
$953$			47.0508i			1.52413i	0.647503	+	0.762063i	$0.275813\pi$
$953$			47.0508i			1.52413i	−0.647503	+	0.762063i	$0.724187\pi$
$954$		0			0
$955$	−20.4910	+	21.0965i	−0.663072	+	0.682666i
$956$		0			0
$957$		−	6.56007i		−	0.212057i
$958$		0			0
$959$	−40.8635			−1.31955
$960$		0			0
$961$	20.6887			0.667378
$962$		0			0
$963$			6.29114i			0.202729i
$964$		0			0
$965$	8.03990	+	7.80914i	0.258814	+	0.251385i
$966$		0			0
$967$			30.9954i			0.996744i	0.866963	+	0.498372i	$0.166069\pi$
$967$			30.9954i			0.996744i	−0.866963	+	0.498372i	$0.833931\pi$
$968$		0			0
$969$	3.76805			0.121047
$970$		0			0
$971$	−8.29048			−0.266054			−0.133027	−	0.991112i	$-0.542470\pi$
$971$	−8.29048			−0.266054			−0.133027	+	0.991112i	$0.542470\pi$
$972$		0			0
$973$			68.7333i			2.20349i
$974$		0			0
$975$	−0.816593	−	28.0365i	−0.0261519	−	0.897888i
$976$		0			0
$977$			28.4601i			0.910519i	0.890359	+	0.455259i	$0.150453\pi$
$977$			28.4601i			0.910519i	−0.890359	+	0.455259i	$0.849547\pi$
$978$		0			0
$979$	57.4850			1.83723
$980$		0			0
$981$	5.53610			0.176754
$982$		0			0
$983$			39.5458i			1.26132i	0.776061	+	0.630658i	$0.217215\pi$
$983$			39.5458i			1.26132i	−0.776061	+	0.630658i	$0.782785\pi$
$984$		0			0
$985$	7.62703	+	7.40811i	0.243017	+	0.236042i
$986$		0			0
$987$			4.23812i			0.134901i
$988$		0			0
$989$	23.6908			0.753322
$990$		0			0
$991$	49.3227			1.56679			0.783393	−	0.621526i	$-0.213487\pi$
$991$	49.3227			1.56679			0.783393	+	0.621526i	$0.213487\pi$
$992$		0			0
$993$			12.5772i			0.399125i
$994$		0			0
$995$	27.9855	−	28.8125i	0.887201	−	0.913417i
$996$		0			0
$997$			34.9295i			1.10623i	0.833106	+	0.553114i	$0.186561\pi$
$997$			34.9295i			1.10623i	−0.833106	+	0.553114i	$0.813439\pi$
$998$		0			0
$999$	−4.66640			−0.147638

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1020.2.g.d.409.7	yes	10
3.2	odd	2		3060.2.g.g.2449.7		10
4.3	odd	2		4080.2.m.r.2449.2		10
5.2	odd	4		5100.2.a.bd.1.2		5
5.3	odd	4		5100.2.a.bc.1.4		5
5.4	even	2	inner	1020.2.g.d.409.2	✓	10
15.14	odd	2		3060.2.g.g.2449.8		10
20.19	odd	2		4080.2.m.r.2449.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1020.2.g.d.409.2	✓	10	5.4	even	2	inner
1020.2.g.d.409.7	yes	10	1.1	even	1	trivial
3060.2.g.g.2449.7		10	3.2	odd	2
3060.2.g.g.2449.8		10	15.14	odd	2
4080.2.m.r.2449.2		10	4.3	odd	2
4080.2.m.r.2449.7		10	20.19	odd	2
5100.2.a.bc.1.4		5	5.3	odd	4
5100.2.a.bd.1.2		5	5.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1020.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-1.55795 + 1.60399i) q^{5} +3.74956i q^{7} -1.00000 q^{9} -4.55583 q^{11} -5.60969i q^{13} +(-1.60399 - 1.55795i) q^{15} +1.00000i q^{17} -3.76805 q^{19} -3.74956 q^{21} +2.71420i q^{23} +(-0.145568 - 4.99788i) q^{25} -1.00000i q^{27} +1.43993 q^{29} +7.18949 q^{31} -4.55583i q^{33} +(-6.01425 - 5.84163i) q^{35} -4.66640i q^{37} +5.60969 q^{39} +2.32402 q^{41} -8.72845i q^{43} +(1.55795 - 1.60399i) q^{45} -1.13030i q^{47} -7.05919 q^{49} -1.00000 q^{51} +2.25044i q^{53} +(7.09777 - 7.30751i) q^{55} -3.76805i q^{57} -12.1063 q^{59} -5.11877 q^{61} -3.74956i q^{63} +(8.99788 + 8.73962i) q^{65} +12.1063i q^{67} -2.71420 q^{69} -11.2194 q^{71} +9.93371i q^{73} +(4.99788 - 0.145568i) q^{75} -17.0824i q^{77} -16.3790 q^{79} +1.00000 q^{81} +10.3025i q^{83} +(-1.60399 - 1.55795i) q^{85} +1.43993i q^{87} -12.6179 q^{89} +21.0338 q^{91} +7.18949i q^{93} +(5.87045 - 6.04392i) q^{95} -9.03522i q^{97} +4.55583 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{9} + 12 q^{11} + 4 q^{15} - 24 q^{19} - 4 q^{21} + 12 q^{25} - 12 q^{29} + 12 q^{31} + 4 q^{35} + 28 q^{41} - 30 q^{49} - 10 q^{51} + 26 q^{55} - 48 q^{59} - 28 q^{61} + 48 q^{65} - 12 q^{69}+ \cdots - 12 q^{99}+O(q^{100}) \) 10 * q - 10 * q^9 + 12 * q^11 + 4 * q^15 - 24 * q^19 - 4 * q^21 + 12 * q^25 - 12 * q^29 + 12 * q^31 + 4 * q^35 + 28 * q^41 - 30 * q^49 - 10 * q^51 + 26 * q^55 - 48 * q^59 - 28 * q^61 + 48 * q^65 - 12 * q^69 + 8 * q^75 - 44 * q^79 + 10 * q^81 + 4 * q^85 - 36 * q^89 + 40 * q^91 + 44 * q^95 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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