LMFDB - Embedded newform 2940.2.f.b.1469.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2940,2,Mod(1469,2940)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2940, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2940.1469");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2940.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$23.4760181943$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1469.8
Character	$\chi$	$=$	2940.1469
Dual form			2940.2.f.b.1469.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.57897 + 0.711939i) q^{3} +(2.16481 + 0.559982i) q^{5} +(1.98629 - 2.24826i) q^{9} +O(q^{10})$	$q+(-1.57897 + 0.711939i) q^{3} +(2.16481 + 0.559982i) q^{5} +(1.98629 - 2.24826i) q^{9} -2.19829i q^{11} -5.07095 q^{13} +(-3.81685 + 0.657022i) q^{15} -2.90787i q^{17} -0.155105i q^{19} -3.63484 q^{23} +(4.37284 + 2.42451i) q^{25} +(-1.53566 + 4.96405i) q^{27} +3.88854i q^{29} +9.79837i q^{31} +(1.56505 + 3.47103i) q^{33} -4.01287i q^{37} +(8.00688 - 3.61021i) q^{39} -7.97945 q^{41} +2.59370i q^{43} +(5.55892 - 3.75478i) q^{45} +12.2525i q^{47} +(2.07023 + 4.59144i) q^{51} +9.42504 q^{53} +(1.23100 - 4.75889i) q^{55} +(0.110425 + 0.244905i) q^{57} +6.80649 q^{59} +6.97180i q^{61} +(-10.9777 - 2.83964i) q^{65} +6.51121i q^{67} +(5.73930 - 2.58778i) q^{69} +2.22868i q^{71} +13.0608 q^{73} +(-8.63069 - 0.715035i) q^{75} -3.68597 q^{79} +(-1.10934 - 8.93137i) q^{81} +13.6861i q^{83} +(1.62836 - 6.29500i) q^{85} +(-2.76841 - 6.13989i) q^{87} +11.3301 q^{89} +(-6.97585 - 15.4713i) q^{93} +(0.0868557 - 0.335772i) q^{95} +1.73455 q^{97} +(-4.94233 - 4.36643i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 8 q^{9}+O(q^{10}) $ 48 * q - 8 * q^9	$ 48 q - 8 q^{9} - 16 q^{15} + 16 q^{25} + 56 q^{39} + 8 q^{51} + 48 q^{79} - 24 q^{81} + 32 q^{85} + 88 q^{99}+O(q^{100}) $ 48 * q - 8 * q^9 - 16 * q^15 + 16 * q^25 + 56 * q^39 + 8 * q^51 + 48 * q^79 - 24 * q^81 + 32 * q^85 + 88 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times$.

$n$	$1081$	$1177$	$1471$	$1961$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	−1.57897	+	0.711939i	−0.911618	+	0.411038i
$3$	−1.57897	+	0.711939i	−0.911618	+	0.411038i
$4$		0			0
$5$	2.16481	+	0.559982i	0.968134	+	0.250432i
$5$	2.16481	+	0.559982i	0.968134	+	0.250432i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.98629	−	2.24826i	0.662095	−	0.749420i
$10$		0			0
$11$		−	2.19829i		−	0.662810i	−0.943489	−	0.331405i	$-0.892477\pi$
$11$		−	2.19829i		−	0.662810i	0.943489	−	0.331405i	$-0.107523\pi$
$12$		0			0
$13$	−5.07095			−1.40643			−0.703215	−	0.710978i	$-0.748253\pi$
$13$	−5.07095			−1.40643			−0.703215	+	0.710978i	$0.748253\pi$
$14$		0			0
$15$	−3.81685	+	0.657022i	−0.985506	+	0.169642i
$16$		0			0
$17$		−	2.90787i		−	0.705262i	−0.935762	−	0.352631i	$-0.885287\pi$
$17$		−	2.90787i		−	0.705262i	0.935762	−	0.352631i	$-0.114713\pi$
$18$		0			0
$19$		−	0.155105i		−	0.0355834i	−0.999842	−	0.0177917i	$-0.994336\pi$
$19$		−	0.155105i		−	0.0355834i	0.999842	−	0.0177917i	$-0.00566358\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.63484			−0.757916			−0.378958	−	0.925414i	$-0.623718\pi$
$23$	−3.63484			−0.757916			−0.378958	+	0.925414i	$0.623718\pi$
$24$		0			0
$25$	4.37284	+	2.42451i	0.874568	+	0.484903i
$26$		0			0
$27$	−1.53566	+	4.96405i	−0.295538	+	0.955331i
$28$		0			0
$29$			3.88854i			0.722084i	0.932549	+	0.361042i	$0.117579\pi$
$29$			3.88854i			0.722084i	−0.932549	+	0.361042i	$0.882421\pi$
$30$		0			0
$31$			9.79837i			1.75984i	0.475122	+	0.879920i	$0.342404\pi$
$31$			9.79837i			1.75984i	−0.475122	+	0.879920i	$0.657596\pi$
$32$		0			0
$33$	1.56505	+	3.47103i	0.272440	+	0.604229i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	4.01287i		−	0.659712i	−0.944031	−	0.329856i	$-0.893000\pi$
$37$		−	4.01287i		−	0.659712i	0.944031	−	0.329856i	$-0.107000\pi$
$38$		0			0
$39$	8.00688	−	3.61021i	1.28213	−	0.578096i
$40$		0			0
$41$	−7.97945			−1.24618			−0.623090	−	0.782150i	$-0.714123\pi$
$41$	−7.97945			−1.24618			−0.623090	+	0.782150i	$0.714123\pi$
$42$		0			0
$43$			2.59370i			0.395536i	0.980249	+	0.197768i	$0.0633692\pi$
$43$			2.59370i			0.395536i	−0.980249	+	0.197768i	$0.936631\pi$
$44$		0			0
$45$	5.55892	−	3.75478i	0.828675	−	0.559730i
$46$		0			0
$47$			12.2525i			1.78722i	0.448848	+	0.893608i	$0.351834\pi$
$47$			12.2525i			1.78722i	−0.448848	+	0.893608i	$0.648166\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	2.07023	+	4.59144i	0.289890	+	0.642930i
$52$		0			0
$53$	9.42504			1.29463			0.647314	−	0.762223i	$-0.275892\pi$
$53$	9.42504			1.29463			0.647314	+	0.762223i	$0.275892\pi$
$54$		0			0
$55$	1.23100	−	4.75889i	0.165988	−	0.641689i
$56$		0			0
$57$	0.110425	+	0.244905i	0.0146261	+	0.0324385i
$58$		0			0
$59$	6.80649			0.886129			0.443064	−	0.896490i	$-0.353891\pi$
$59$	6.80649			0.886129			0.443064	+	0.896490i	$0.353891\pi$
$60$		0			0
$61$			6.97180i			0.892647i	0.894872	+	0.446324i	$0.147267\pi$
$61$			6.97180i			0.892647i	−0.894872	+	0.446324i	$0.852733\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−10.9777	−	2.83964i	−1.36161	−	0.352214i
$66$		0			0
$67$			6.51121i			0.795471i	0.917500	+	0.397736i	$0.130204\pi$
$67$			6.51121i			0.795471i	−0.917500	+	0.397736i	$0.869796\pi$
$68$		0			0
$69$	5.73930	−	2.58778i	0.690930	−	0.311533i
$70$		0			0
$71$			2.22868i			0.264495i	0.991217	+	0.132248i	$0.0422194\pi$
$71$			2.22868i			0.264495i	−0.991217	+	0.132248i	$0.957781\pi$
$72$		0			0
$73$	13.0608			1.52865			0.764325	−	0.644831i	$-0.223072\pi$
$73$	13.0608			1.52865			0.764325	+	0.644831i	$0.223072\pi$
$74$		0			0
$75$	−8.63069	−	0.715035i	−0.996586	−	0.0825652i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−3.68597			−0.414705			−0.207352	−	0.978266i	$-0.566485\pi$
$79$	−3.68597			−0.414705			−0.207352	+	0.978266i	$0.566485\pi$
$80$		0			0
$81$	−1.10934	−	8.93137i	−0.123260	−	0.992374i
$82$		0			0
$83$			13.6861i			1.50225i	0.660162	+	0.751123i	$0.270488\pi$
$83$			13.6861i			1.50225i	−0.660162	+	0.751123i	$0.729512\pi$
$84$		0			0
$85$	1.62836	−	6.29500i	0.176620	−	0.682789i
$86$		0			0
$87$	−2.76841	−	6.13989i	−0.296804	−	0.658265i
$88$		0			0
$89$	11.3301			1.20098			0.600492	−	0.799631i	$-0.294971\pi$
$89$	11.3301			1.20098			0.600492	+	0.799631i	$0.294971\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−6.97585	−	15.4713i	−0.723362	−	1.60430i
$94$		0			0
$95$	0.0868557	−	0.335772i	0.00891121	−	0.0344495i
$96$		0			0
$97$	1.73455			0.176117			0.0880586	−	0.996115i	$-0.471934\pi$
$97$	1.73455			0.176117			0.0880586	+	0.996115i	$0.471934\pi$
$98$		0			0
$99$	−4.94233	−	4.36643i	−0.496723	−	0.438843i
$100$		0			0
$101$	−13.6985			−1.36305			−0.681524	−	0.731796i	$-0.738683\pi$
$101$	−13.6985			−1.36305			−0.681524	+	0.731796i	$0.738683\pi$
$102$		0			0
$103$	−9.45213			−0.931346			−0.465673	−	0.884957i	$-0.654188\pi$
$103$	−9.45213			−0.931346			−0.465673	+	0.884957i	$0.654188\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.87241			−0.181013			−0.0905065	−	0.995896i	$-0.528849\pi$
$107$	−1.87241			−0.181013			−0.0905065	+	0.995896i	$0.528849\pi$
$108$		0			0
$109$	−7.67338			−0.734976			−0.367488	−	0.930028i	$-0.619782\pi$
$109$	−7.67338			−0.734976			−0.367488	+	0.930028i	$0.619782\pi$
$110$		0			0
$111$	2.85692	+	6.33619i	0.271167	+	0.601405i
$112$		0			0
$113$	−3.38523			−0.318455			−0.159228	−	0.987242i	$-0.550900\pi$
$113$	−3.38523			−0.318455			−0.159228	+	0.987242i	$0.550900\pi$
$114$		0			0
$115$	−7.86875	−	2.03544i	−0.733765	−	0.189806i
$116$		0			0
$117$	−10.0724	+	11.4008i	−0.931190	+	1.05401i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	6.16751			0.560683
$122$		0			0
$123$	12.5993	−	5.68088i	1.13604	−	0.512228i
$124$		0			0
$125$	8.10870	+	7.69733i	0.725264	+	0.688470i
$126$		0			0
$127$		−	11.6585i		−	1.03452i	−0.855827	−	0.517262i	$-0.826951\pi$
$127$		−	11.6585i		−	1.03452i	0.855827	−	0.517262i	$-0.173049\pi$
$128$		0			0
$129$	−1.84656	−	4.09537i	−0.162580	−	0.360577i
$130$		0			0
$131$	10.0471			0.877821			0.438911	−	0.898531i	$-0.355364\pi$
$131$	10.0471			0.877821			0.438911	+	0.898531i	$0.355364\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−6.10419	+	9.88630i	−0.525365	+	0.850877i
$136$		0			0
$137$	19.4230			1.65942			0.829708	−	0.558198i	$-0.188507\pi$
$137$	19.4230			1.65942			0.829708	+	0.558198i	$0.188507\pi$
$138$		0			0
$139$		−	4.71667i		−	0.400063i	−0.979789	−	0.200031i	$-0.935896\pi$
$139$		−	4.71667i		−	0.400063i	0.979789	−	0.200031i	$-0.0641044\pi$
$140$		0			0
$141$	−8.72306	−	19.3464i	−0.734614	−	1.62926i
$142$		0			0
$143$			11.1474i			0.932195i
$144$		0			0
$145$	−2.17751	+	8.41797i	−0.180833	+	0.699075i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			2.56445i			0.210088i	0.994468	+	0.105044i	$0.0334983\pi$
$149$			2.56445i			0.210088i	−0.994468	+	0.105044i	$0.966502\pi$
$150$		0			0
$151$	−5.38227			−0.438003			−0.219001	−	0.975725i	$-0.570280\pi$
$151$	−5.38227			−0.438003			−0.219001	+	0.975725i	$0.570280\pi$
$152$		0			0
$153$	−6.53765	−	5.77586i	−0.528538	−	0.466951i
$154$		0			0
$155$	−5.48691	+	21.2117i	−0.440719	+	1.70376i
$156$		0			0
$157$	−5.74803			−0.458742			−0.229371	−	0.973339i	$-0.573667\pi$
$157$	−5.74803			−0.458742			−0.229371	+	0.973339i	$0.573667\pi$
$158$		0			0
$159$	−14.8818	+	6.71005i	−1.18021	+	0.532142i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	2.31538i		−	0.181355i	−0.995880	−	0.0906774i	$-0.971097\pi$
$163$		−	2.31538i		−	0.181355i	0.995880	−	0.0906774i	$-0.0289032\pi$
$164$		0			0
$165$	1.44433	+	8.39054i	0.112441	+	0.653203i
$166$		0			0
$167$			20.3572i			1.57529i	0.616132	+	0.787643i	$0.288699\pi$
$167$			20.3572i			1.57529i	−0.616132	+	0.787643i	$0.711301\pi$
$168$		0			0
$169$	12.7146			0.978043
$170$		0			0
$171$	−0.348715	−	0.308082i	−0.0266669	−	0.0235596i
$172$		0			0
$173$			21.7254i			1.65175i	0.563850	+	0.825877i	$0.309319\pi$
$173$			21.7254i			1.65175i	−0.563850	+	0.825877i	$0.690681\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−10.7472	+	4.84580i	−0.807811	+	0.364233i
$178$		0			0
$179$			16.9273i			1.26521i	0.774475	+	0.632604i	$0.218014\pi$
$179$			16.9273i			1.26521i	−0.774475	+	0.632604i	$0.781986\pi$
$180$		0			0
$181$			24.5548i			1.82515i	0.408912	+	0.912574i	$0.365908\pi$
$181$			24.5548i			1.82515i	−0.408912	+	0.912574i	$0.634092\pi$
$182$		0			0
$183$	−4.96350	−	11.0083i	−0.366912	−	0.813753i
$184$		0			0
$185$	2.24713	−	8.68712i	0.165213	−	0.638689i
$186$		0			0
$187$	−6.39235			−0.467455
$188$		0			0
$189$		0			0
$190$		0			0
$191$			18.6786i			1.35154i	0.737113	+	0.675770i	$0.236189\pi$
$191$			18.6786i			1.35154i	−0.737113	+	0.675770i	$0.763811\pi$
$192$		0			0
$193$		−	18.2568i		−	1.31415i	−0.753824	−	0.657076i	$-0.771793\pi$
$193$		−	18.2568i		−	1.31415i	0.753824	−	0.657076i	$-0.228207\pi$
$194$		0			0
$195$	19.3551	−	3.33173i	1.38604	−	0.238590i
$196$		0			0
$197$	5.28939			0.376853			0.188427	−	0.982087i	$-0.439661\pi$
$197$	5.28939			0.376853			0.188427	+	0.982087i	$0.439661\pi$
$198$		0			0
$199$		−	5.33067i		−	0.377881i	−0.981988	−	0.188941i	$-0.939495\pi$
$199$		−	5.33067i		−	0.377881i	0.981988	−	0.188941i	$-0.0605054\pi$
$200$		0			0
$201$	−4.63559	−	10.2810i	−0.326969	−	0.725166i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−17.2740	−	4.46835i	−1.20647	−	0.312083i
$206$		0			0
$207$	−7.21983	+	8.17206i	−0.501813	+	0.567998i
$208$		0			0
$209$	−0.340965			−0.0235850
$210$		0			0
$211$	−20.2924			−1.39698			−0.698492	−	0.715618i	$-0.746145\pi$
$211$	−20.2924			−1.39698			−0.698492	+	0.715618i	$0.746145\pi$
$212$		0			0
$213$	−1.58668	−	3.51901i	−0.108718	−	0.241119i
$214$		0			0
$215$	−1.45243	+	5.61488i	−0.0990546	+	0.382932i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−20.6226	+	9.29849i	−1.39355	+	0.628334i
$220$		0			0
$221$			14.7457i			0.991901i
$222$		0			0
$223$	−27.4665			−1.83929			−0.919645	−	0.392750i	$-0.871524\pi$
$223$	−27.4665			−1.83929			−0.919645	+	0.392750i	$0.871524\pi$
$224$		0			0
$225$	14.1366	−	5.01550i	0.942443	−	0.334367i
$226$		0			0
$227$			5.40469i			0.358722i	0.983783	+	0.179361i	$0.0574029\pi$
$227$			5.40469i			0.358722i	−0.983783	+	0.179361i	$0.942597\pi$
$228$		0			0
$229$		−	4.74357i		−	0.313464i	−0.987641	−	0.156732i	$-0.949904\pi$
$229$		−	4.74357i		−	0.313464i	0.987641	−	0.156732i	$-0.0500958\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.81595			−0.118967			−0.0594834	−	0.998229i	$-0.518945\pi$
$233$	−1.81595			−0.118967			−0.0594834	+	0.998229i	$0.518945\pi$
$234$		0			0
$235$	−6.86120	+	26.5245i	−0.447575	+	1.73027i
$236$		0			0
$237$	5.82004	−	2.62419i	0.378052	−	0.170459i
$238$		0			0
$239$			16.5077i			1.06779i	0.845549	+	0.533897i	$0.179273\pi$
$239$			16.5077i			1.06779i	−0.845549	+	0.533897i	$0.820727\pi$
$240$		0			0
$241$		−	15.6532i		−	1.00831i	−0.863612	−	0.504156i	$-0.831804\pi$
$241$		−	15.6532i		−	1.00831i	0.863612	−	0.504156i	$-0.168196\pi$
$242$		0			0
$243$	8.11021	+	13.3126i	0.520270	+	0.854002i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			0.786528i			0.0500456i
$248$		0			0
$249$	−9.74368	−	21.6100i	−0.617481	−	1.36948i
$250$		0			0
$251$	−22.9733			−1.45006			−0.725031	−	0.688716i	$-0.758175\pi$
$251$	−22.9733			−1.45006			−0.725031	+	0.688716i	$0.758175\pi$
$252$		0			0
$253$			7.99044i			0.502354i
$254$		0			0
$255$	1.91053	+	11.0989i	0.119642	+	0.695040i
$256$		0			0
$257$		−	9.08162i		−	0.566496i	−0.959047	−	0.283248i	$-0.908588\pi$
$257$		−	9.08162i		−	0.566496i	0.959047	−	0.283248i	$-0.0914120\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	8.74246	+	7.72376i	0.541144	+	0.478089i
$262$		0			0
$263$	18.1174			1.11717			0.558584	−	0.829448i	$-0.311345\pi$
$263$	18.1174			1.11717			0.558584	+	0.829448i	$0.311345\pi$
$264$		0			0
$265$	20.4035	+	5.27785i	1.25337	+	0.324216i
$266$		0			0
$267$	−17.8898	+	8.06632i	−1.09484	+	0.493650i
$268$		0			0
$269$	−18.7746			−1.14471			−0.572353	−	0.820007i	$-0.693969\pi$
$269$	−18.7746			−1.14471			−0.572353	+	0.820007i	$0.693969\pi$
$270$		0			0
$271$		−	17.5534i		−	1.06629i	−0.846022	−	0.533147i	$-0.821009\pi$
$271$		−	17.5534i		−	1.06629i	0.846022	−	0.533147i	$-0.178991\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.32979	−	9.61278i	0.321398	−	0.579672i
$276$		0			0
$277$			19.2815i			1.15851i	0.815145	+	0.579257i	$0.196657\pi$
$277$			19.2815i			1.15851i	−0.815145	+	0.579257i	$0.803343\pi$
$278$		0			0
$279$	22.0293	+	19.4624i	1.31886	+	1.16518i
$280$		0			0
$281$			6.16244i			0.367620i	0.982962	+	0.183810i	$0.0588432\pi$
$281$			6.16244i			0.367620i	−0.982962	+	0.183810i	$0.941157\pi$
$282$		0			0
$283$	23.2606			1.38270			0.691350	−	0.722520i	$-0.257016\pi$
$283$	23.2606			1.38270			0.691350	+	0.722520i	$0.257016\pi$
$284$		0			0
$285$	0.101907	+	0.592010i	0.00603645	+	0.0350677i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.54429			0.502605
$290$		0			0
$291$	−2.73880	+	1.23490i	−0.160552	+	0.0723909i
$292$		0			0
$293$		−	30.1092i		−	1.75900i	−0.475900	−	0.879499i	$-0.657878\pi$
$293$		−	30.1092i		−	1.75900i	0.475900	−	0.879499i	$-0.342122\pi$
$294$		0			0
$295$	14.7348	+	3.81151i	0.857892	+	0.221915i
$296$		0			0
$297$	10.9124	+	3.37583i	0.633203	+	0.195885i
$298$		0			0
$299$	18.4321			1.06596
$300$		0			0
$301$		0			0
$302$		0			0
$303$	21.6294	−	9.75247i	1.24258	−	0.560265i
$304$		0			0
$305$	−3.90408	+	15.0926i	−0.223547	+	0.864203i
$306$		0			0
$307$	0.575026			0.0328185			0.0164092	−	0.999865i	$-0.494777\pi$
$307$	0.575026			0.0328185			0.0164092	+	0.999865i	$0.494777\pi$
$308$		0			0
$309$	14.9246	−	6.72934i	0.849032	−	0.382819i
$310$		0			0
$311$	−7.04265			−0.399352			−0.199676	−	0.979862i	$-0.563989\pi$
$311$	−7.04265			−0.399352			−0.199676	+	0.979862i	$0.563989\pi$
$312$		0			0
$313$	8.21607			0.464400			0.232200	−	0.972668i	$-0.425408\pi$
$313$	8.21607			0.464400			0.232200	+	0.972668i	$0.425408\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−29.0153			−1.62966			−0.814832	−	0.579698i	$-0.803171\pi$
$317$	−29.0153			−1.62966			−0.814832	+	0.579698i	$0.803171\pi$
$318$		0			0
$319$	8.54815			0.478605
$320$		0			0
$321$	2.95648	−	1.33304i	0.165015	−	0.0744033i
$322$		0			0
$323$	−0.451024			−0.0250956
$324$		0			0
$325$	−22.1745	−	12.2946i	−1.23002	−	0.681981i
$326$		0			0
$327$	12.1160	−	5.46298i	0.670018	−	0.302103i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.92177			0.105630			0.0528150	−	0.998604i	$-0.483181\pi$
$331$	1.92177			0.105630			0.0528150	+	0.998604i	$0.483181\pi$
$332$		0			0
$333$	−9.02197	−	7.97070i	−0.494401	−	0.436792i
$334$		0			0
$335$	−3.64616	+	14.0956i	−0.199211	+	0.770123i
$336$		0			0
$337$			23.9897i			1.30680i	0.757011	+	0.653402i	$0.226659\pi$
$337$			23.9897i			1.30680i	−0.757011	+	0.653402i	$0.773341\pi$
$338$		0			0
$339$	5.34517	−	2.41007i	0.290310	−	0.130897i
$340$		0			0
$341$	21.5397			1.16644
$342$		0			0
$343$		0			0
$344$		0			0
$345$	13.8736	−	2.38817i	0.746931	−	0.128575i
$346$		0			0
$347$	−18.5388			−0.995213			−0.497606	−	0.867403i	$-0.665788\pi$
$347$	−18.5388			−0.995213			−0.497606	+	0.867403i	$0.665788\pi$
$348$		0			0
$349$		−	25.7711i		−	1.37949i	−0.724051	−	0.689747i	$-0.757722\pi$
$349$		−	25.7711i		−	1.37949i	0.724051	−	0.689747i	$-0.242278\pi$
$350$		0			0
$351$	7.78725	−	25.1724i	0.415653	−	1.34361i
$352$		0			0
$353$		−	12.1537i		−	0.646874i	−0.946250	−	0.323437i	$-0.895162\pi$
$353$		−	12.1537i		−	0.646874i	0.946250	−	0.323437i	$-0.104838\pi$
$354$		0			0
$355$	−1.24802	+	4.82467i	−0.0662379	+	0.256067i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	21.0223i		−	1.10952i	−0.832011	−	0.554759i	$-0.812811\pi$
$359$		−	21.0223i		−	1.10952i	0.832011	−	0.554759i	$-0.187189\pi$
$360$		0			0
$361$	18.9759			0.998734
$362$		0			0
$363$	−9.73831	+	4.39089i	−0.511129	+	0.230462i
$364$		0			0
$365$	28.2742	+	7.31381i	1.47994	+	0.382822i
$366$		0			0
$367$	−23.5306			−1.22829			−0.614145	−	0.789194i	$-0.710499\pi$
$367$	−23.5306			−1.22829			−0.614145	+	0.789194i	$0.710499\pi$
$368$		0			0
$369$	−15.8495	+	17.9399i	−0.825090	+	0.933913i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	35.8517i		−	1.85633i	−0.372166	−	0.928166i	$-0.621385\pi$
$373$		−	35.8517i		−	1.85633i	0.372166	−	0.928166i	$-0.378615\pi$
$374$		0			0
$375$	−18.2834	−	6.38095i	−0.944152	−	0.329511i
$376$		0			0
$377$		−	19.7186i		−	1.01556i
$378$		0			0
$379$	−12.3860			−0.636227			−0.318114	−	0.948053i	$-0.603049\pi$
$379$	−12.3860			−0.636227			−0.318114	+	0.948053i	$0.603049\pi$
$380$		0			0
$381$	8.30013	+	18.4084i	0.425229	+	0.943090i
$382$		0			0
$383$		−	15.7586i		−	0.805226i	−0.915370	−	0.402613i	$-0.868102\pi$
$383$		−	15.7586i		−	0.805226i	0.915370	−	0.402613i	$-0.131898\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	5.83131	+	5.15183i	0.296422	+	0.261882i
$388$		0			0
$389$			13.9576i			0.707681i	0.935306	+	0.353840i	$0.115124\pi$
$389$			13.9576i			0.707681i	−0.935306	+	0.353840i	$0.884876\pi$
$390$		0			0
$391$			10.5696i			0.534530i
$392$		0			0
$393$	−15.8641	+	7.15294i	−0.800238	+	0.360818i
$394$		0			0
$395$	−7.97945	−	2.06408i	−0.401490	−	0.103855i
$396$		0			0
$397$	−3.75124			−0.188269			−0.0941347	−	0.995559i	$-0.530008\pi$
$397$	−3.75124			−0.188269			−0.0941347	+	0.995559i	$0.530008\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			7.09180i			0.354147i	0.984198	+	0.177074i	$0.0566631\pi$
$401$			7.09180i			0.354147i	−0.984198	+	0.177074i	$0.943337\pi$
$402$		0			0
$403$		−	49.6871i		−	2.47509i
$404$		0			0
$405$	2.59989	−	19.9560i	0.129189	−	0.991620i
$406$		0			0
$407$	−8.82146			−0.437263
$408$		0			0
$409$			13.3712i			0.661163i	0.943777	+	0.330582i	$0.107245\pi$
$409$			13.3712i			0.661163i	−0.943777	+	0.330582i	$0.892755\pi$
$410$		0			0
$411$	−30.6683	+	13.8280i	−1.51275	+	0.682084i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−7.66398	+	29.6279i	−0.376210	+	1.45438i
$416$		0			0
$417$	3.35798	+	7.44748i	0.164441	+	0.364705i
$418$		0			0
$419$	−20.5013			−1.00155			−0.500777	−	0.865576i	$-0.666952\pi$
$419$	−20.5013			−1.00155			−0.500777	+	0.865576i	$0.666952\pi$
$420$		0			0
$421$	−8.45042			−0.411848			−0.205924	−	0.978568i	$-0.566020\pi$
$421$	−8.45042			−0.411848			−0.205924	+	0.978568i	$0.566020\pi$
$422$		0			0
$423$	27.5469	+	24.3370i	1.33938	+	1.18331i
$424$		0			0
$425$	7.05017	−	12.7157i	0.341984	−	0.616800i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−7.93629	−	17.6014i	−0.383168	−	0.849806i
$430$		0			0
$431$		−	14.9646i		−	0.720820i	−0.932794	−	0.360410i	$-0.882637\pi$
$431$		−	14.9646i		−	0.720820i	0.932794	−	0.360410i	$-0.117363\pi$
$432$		0			0
$433$	−1.78058			−0.0855692			−0.0427846	−	0.999084i	$-0.513623\pi$
$433$	−1.78058			−0.0855692			−0.0427846	+	0.999084i	$0.513623\pi$
$434$		0			0
$435$	−2.55486	−	14.8420i	−0.122496	−	0.711618i
$436$		0			0
$437$			0.563780i			0.0269693i
$438$		0			0
$439$		−	33.2628i		−	1.58755i	−0.608213	−	0.793774i	$-0.708113\pi$
$439$		−	33.2628i		−	1.58755i	0.608213	−	0.793774i	$-0.291887\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	35.4666			1.68507			0.842535	−	0.538641i	$-0.181062\pi$
$443$	35.4666			1.68507			0.842535	+	0.538641i	$0.181062\pi$
$444$		0			0
$445$	24.5275	+	6.34463i	1.16271	+	0.300764i
$446$		0			0
$447$	−1.82573	−	4.04918i	−0.0863541	−	0.191520i
$448$		0			0
$449$			22.2177i			1.04852i	0.851559	+	0.524258i	$0.175657\pi$
$449$			22.2177i			1.04852i	−0.851559	+	0.524258i	$0.824343\pi$
$450$		0			0
$451$			17.5412i			0.825981i
$452$		0			0
$453$	8.49843	−	3.83185i	0.399291	−	0.180036i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			38.6337i			1.80721i	0.428368	+	0.903604i	$0.359089\pi$
$457$			38.6337i			1.80721i	−0.428368	+	0.903604i	$0.640911\pi$
$458$		0			0
$459$	14.4348	+	4.46550i	0.673759	+	0.208432i
$460$		0			0
$461$	−28.3198			−1.31898			−0.659491	−	0.751712i	$-0.729228\pi$
$461$	−28.3198			−1.31898			−0.659491	+	0.751712i	$0.729228\pi$
$462$		0			0
$463$			0.569792i			0.0264805i	0.999912	+	0.0132402i	$0.00421462\pi$
$463$			0.569792i			0.0264805i	−0.999912	+	0.0132402i	$0.995785\pi$
$464$		0			0
$465$	−6.43775	−	37.3989i	−0.298543	−	1.73433i
$466$		0			0
$467$		−	32.2741i		−	1.49347i	−0.665123	−	0.746734i	$-0.731621\pi$
$467$		−	32.2741i		−	1.49347i	0.665123	−	0.746734i	$-0.268379\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	9.07595	−	4.09224i	0.418198	−	0.188561i
$472$		0			0
$473$	5.70171			0.262165
$474$		0			0
$475$	0.376053	−	0.678247i	0.0172545	−	0.0311201i
$476$		0			0
$477$	18.7208	−	21.1899i	0.857167	−	0.970220i
$478$		0			0
$479$	42.9641			1.96308			0.981541	−	0.191253i	$-0.0612552\pi$
$479$	42.9641			1.96308			0.981541	+	0.191253i	$0.0612552\pi$
$480$		0			0
$481$			20.3491i			0.927838i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.75498	+	0.971318i	0.170505	+	0.0441053i
$486$		0			0
$487$		−	20.3089i		−	0.920284i	−0.887845	−	0.460142i	$-0.847799\pi$
$487$		−	20.3089i		−	0.920284i	0.887845	−	0.460142i	$-0.152201\pi$
$488$		0			0
$489$	1.64841	+	3.65592i	0.0745438	+	0.165326i
$490$		0			0
$491$			27.4781i			1.24007i	0.784574	+	0.620036i	$0.212882\pi$
$491$			27.4781i			1.24007i	−0.784574	+	0.620036i	$0.787118\pi$
$492$		0			0
$493$	11.3074			0.509259
$494$		0			0
$495$	−8.25410	−	12.2201i	−0.370994	−	0.549254i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	36.6767			1.64188			0.820938	−	0.571017i	$-0.193451\pi$
$499$	36.6767			1.64188			0.820938	+	0.571017i	$0.193451\pi$
$500$		0			0
$501$	−14.4931	−	32.1433i	−0.647503	−	1.43606i
$502$		0			0
$503$		−	3.23782i		−	0.144367i	−0.997391	−	0.0721837i	$-0.977003\pi$
$503$		−	3.23782i		−	0.144367i	0.997391	−	0.0721837i	$-0.0229968\pi$
$504$		0			0
$505$	−29.6546	−	7.67089i	−1.31961	−	0.341350i
$506$		0			0
$507$	−20.0759	+	9.05199i	−0.891602	+	0.402013i
$508$		0			0
$509$	−13.0171			−0.576975			−0.288487	−	0.957484i	$-0.593152\pi$
$509$	−13.0171			−0.576975			−0.288487	+	0.957484i	$0.593152\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	0.769946	+	0.238188i	0.0339939	+	0.0105162i
$514$		0			0
$515$	−20.4621	−	5.29302i	−0.901668	−	0.233239i
$516$		0			0
$517$	26.9346			1.18458
$518$		0			0
$519$	−15.4672	−	34.3038i	−0.678934	−	1.50577i
$520$		0			0
$521$	−8.53166			−0.373779			−0.186889	−	0.982381i	$-0.559841\pi$
$521$	−8.53166			−0.373779			−0.186889	+	0.982381i	$0.559841\pi$
$522$		0			0
$523$	27.3558			1.19618			0.598092	−	0.801427i	$-0.295926\pi$
$523$	27.3558			1.19618			0.598092	+	0.801427i	$0.295926\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	28.4924			1.24115
$528$		0			0
$529$	−9.78794			−0.425563
$530$		0			0
$531$	13.5196	−	15.3027i	0.586702	−	0.664083i
$532$		0			0
$533$	40.4634			1.75267
$534$		0			0
$535$	−4.05343	−	1.04852i	−0.175245	−	0.0453314i
$536$		0			0
$537$	−12.0512	−	26.7277i	−0.520049	−	1.15339i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5.53907			−0.238143			−0.119072	−	0.992886i	$-0.537992\pi$
$541$	−5.53907			−0.238143			−0.119072	+	0.992886i	$0.537992\pi$
$542$		0			0
$543$	−17.4816	−	38.7713i	−0.750205	−	1.66384i
$544$		0			0
$545$	−16.6114	−	4.29695i	−0.711556	−	0.184061i
$546$		0			0
$547$		−	27.1333i		−	1.16014i	−0.814568	−	0.580068i	$-0.803026\pi$
$547$		−	27.1333i		−	1.16014i	0.814568	−	0.580068i	$-0.196974\pi$
$548$		0			0
$549$	15.6744	+	13.8480i	0.668968	+	0.591017i
$550$		0			0
$551$	0.603131			0.0256942
$552$		0			0
$553$		0			0
$554$		0			0
$555$	2.63654	+	15.3165i	0.111915	+	0.650150i
$556$		0			0
$557$	−36.8859			−1.56291			−0.781453	−	0.623964i	$-0.785521\pi$
$557$	−36.8859			−1.56291			−0.781453	+	0.623964i	$0.785521\pi$
$558$		0			0
$559$		−	13.1525i		−	0.556293i
$560$		0			0
$561$	10.0933	−	4.55096i	0.426140	−	0.192142i
$562$		0			0
$563$			32.8085i			1.38271i	0.722513	+	0.691357i	$0.242987\pi$
$563$			32.8085i			1.38271i	−0.722513	+	0.691357i	$0.757013\pi$
$564$		0			0
$565$	−7.32838	−	1.89567i	−0.308307	−	0.0797512i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			14.2653i			0.598032i	0.954248	+	0.299016i	$0.0966584\pi$
$569$			14.2653i			0.598032i	−0.954248	+	0.299016i	$0.903342\pi$
$570$		0			0
$571$	−10.8302			−0.453229			−0.226614	−	0.973985i	$-0.572766\pi$
$571$	−10.8302			−0.453229			−0.226614	+	0.973985i	$0.572766\pi$
$572$		0			0
$573$	−13.2981	−	29.4930i	−0.555534	−	1.23209i
$574$		0			0
$575$	−15.8946	−	8.81272i	−0.662849	−	0.367516i
$576$		0			0
$577$	26.4905			1.10281			0.551407	−	0.834236i	$-0.314091\pi$
$577$	26.4905			1.10281			0.551407	+	0.834236i	$0.314091\pi$
$578$		0			0
$579$	12.9977	+	28.8269i	0.540167	+	1.19800i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	20.7190i		−	0.858092i
$584$		0			0
$585$	−28.1890	+	19.0403i	−1.16547	+	0.787220i
$586$		0			0
$587$			18.9901i			0.783804i	0.920007	+	0.391902i	$0.128183\pi$
$587$			18.9901i			0.783804i	−0.920007	+	0.391902i	$0.871817\pi$
$588$		0			0
$589$	1.51977			0.0626211
$590$		0			0
$591$	−8.35178	+	3.76572i	−0.343546	+	0.154901i
$592$		0			0
$593$		−	31.5973i		−	1.29754i	−0.760983	−	0.648772i	$-0.775283\pi$
$593$		−	31.5973i		−	1.29754i	0.760983	−	0.648772i	$-0.224717\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	3.79512	+	8.41697i	0.155324	+	0.344484i
$598$		0			0
$599$		−	1.72344i		−	0.0704178i	−0.999380	−	0.0352089i	$-0.988790\pi$
$599$		−	1.72344i		−	0.0704178i	0.999380	−	0.0352089i	$-0.0112097\pi$
$600$		0			0
$601$		−	41.9659i		−	1.71183i	−0.517120	−	0.855913i	$-0.672996\pi$
$601$		−	41.9659i		−	1.71183i	0.517120	−	0.855913i	$-0.327004\pi$
$602$		0			0
$603$	14.6389	+	12.9331i	0.596142	+	0.526678i
$604$		0			0
$605$	13.3515	+	3.45370i	0.542817	+	0.140413i
$606$		0			0
$607$	7.99600			0.324548			0.162274	−	0.986746i	$-0.448117\pi$
$607$	7.99600			0.324548			0.162274	+	0.986746i	$0.448117\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	62.1320i		−	2.51359i
$612$		0			0
$613$		−	35.9232i		−	1.45092i	−0.688262	−	0.725462i	$-0.741626\pi$
$613$		−	35.9232i		−	1.45092i	0.688262	−	0.725462i	$-0.258374\pi$
$614$		0			0
$615$	30.4563	−	5.24267i	1.22812	−	0.211405i
$616$		0			0
$617$	14.5615			0.586225			0.293112	−	0.956078i	$-0.405309\pi$
$617$	14.5615			0.586225			0.293112	+	0.956078i	$0.405309\pi$
$618$		0			0
$619$		−	26.7421i		−	1.07486i	−0.843309	−	0.537429i	$-0.819396\pi$
$619$		−	26.7421i		−	1.07486i	0.843309	−	0.537429i	$-0.180604\pi$
$620$		0			0
$621$	5.58187	−	18.0435i	0.223993	−	0.724061i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	13.2435	+	21.2040i	0.529739	+	0.848161i
$626$		0			0
$627$	0.538373	−	0.242746i	0.0215005	−	0.00969435i
$628$		0			0
$629$	−11.6689			−0.465270
$630$		0			0
$631$	30.0412			1.19592			0.597961	−	0.801525i	$-0.295978\pi$
$631$	30.0412			1.19592			0.597961	+	0.801525i	$0.295978\pi$
$632$		0			0
$633$	32.0410	−	14.4469i	1.27352	−	0.574214i
$634$		0			0
$635$	6.52854	−	25.2384i	0.259077	−	1.00156i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	5.01064	+	4.42679i	0.198218	+	0.175121i
$640$		0			0
$641$			38.9572i			1.53872i	0.638818	+	0.769358i	$0.279424\pi$
$641$			38.9572i			1.53872i	−0.638818	+	0.769358i	$0.720576\pi$
$642$		0			0
$643$	−33.9814			−1.34010			−0.670049	−	0.742317i	$-0.733727\pi$
$643$	−33.9814			−1.34010			−0.670049	+	0.742317i	$0.733727\pi$
$644$		0			0
$645$	−1.70412	−	9.89976i	−0.0670996	−	0.389803i
$646$		0			0
$647$		−	11.1558i		−	0.438579i	−0.975660	−	0.219289i	$-0.929626\pi$
$647$		−	11.1558i		−	0.438579i	0.975660	−	0.219289i	$-0.0703738\pi$
$648$		0			0
$649$		−	14.9626i		−	0.587335i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	23.4880			0.919157			0.459579	−	0.888137i	$-0.348000\pi$
$653$	23.4880			0.919157			0.459579	+	0.888137i	$0.348000\pi$
$654$		0			0
$655$	21.7502	+	5.62621i	0.849849	+	0.219834i
$656$		0			0
$657$	25.9425	−	29.3641i	1.01211	−	1.14560i
$658$		0			0
$659$			8.07402i			0.314519i	0.987557	+	0.157259i	$0.0502659\pi$
$659$			8.07402i			0.314519i	−0.987557	+	0.157259i	$0.949734\pi$
$660$		0			0
$661$			15.4885i			0.602433i	0.953556	+	0.301217i	$0.0973927\pi$
$661$			15.4885i			0.602433i	−0.953556	+	0.301217i	$0.902607\pi$
$662$		0			0
$663$	−10.4980	−	23.2830i	−0.407709	−	0.904235i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	14.1342i		−	0.547280i
$668$		0			0
$669$	43.3687	−	19.5545i	1.67673	−	0.756019i
$670$		0			0
$671$	15.3260			0.591655
$672$		0			0
$673$		−	15.4028i		−	0.593735i	−0.954919	−	0.296867i	$-0.904058\pi$
$673$		−	15.4028i		−	0.593735i	0.954919	−	0.296867i	$-0.0959419\pi$
$674$		0			0
$675$	−18.7506	+	17.9838i	−0.721710	+	0.692195i
$676$		0			0
$677$		−	20.1263i		−	0.773518i	−0.922181	−	0.386759i	$-0.873595\pi$
$677$		−	20.1263i		−	0.773518i	0.922181	−	0.386759i	$-0.126405\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−3.84781	−	8.53383i	−0.147448	−	0.327017i
$682$		0			0
$683$	8.80121			0.336769			0.168384	−	0.985721i	$-0.446145\pi$
$683$	8.80121			0.336769			0.168384	+	0.985721i	$0.446145\pi$
$684$		0			0
$685$	42.0471	+	10.8765i	1.60654	+	0.415570i
$686$		0			0
$687$	3.37713	+	7.48994i	0.128846	+	0.285759i
$688$		0			0
$689$	−47.7939			−1.82080
$690$		0			0
$691$			36.3368i			1.38232i	0.722703	+	0.691159i	$0.242900\pi$
$691$			36.3368i			1.38232i	−0.722703	+	0.691159i	$0.757100\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.64125	−	10.2107i	0.100188	−	0.387315i
$696$		0			0
$697$			23.2032i			0.878884i
$698$		0			0
$699$	2.86733	−	1.29285i	0.108452	−	0.0488999i
$700$		0			0
$701$		−	7.52551i		−	0.284235i	−0.989850	−	0.142117i	$-0.954609\pi$
$701$		−	7.52551i		−	0.284235i	0.989850	−	0.142117i	$-0.0453910\pi$
$702$		0			0
$703$	−0.622414			−0.0234748
$704$		0			0
$705$	−8.05018	−	46.7661i	−0.303187	−	1.76131i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−10.9451			−0.411051			−0.205525	−	0.978652i	$-0.565890\pi$
$709$	−10.9451			−0.411051			−0.205525	+	0.978652i	$0.565890\pi$
$710$		0			0
$711$	−7.32140	+	8.28703i	−0.274574	+	0.310788i
$712$		0			0
$713$		−	35.6155i		−	1.33381i
$714$		0			0
$715$	−6.24236	+	24.1321i	−0.233451	+	0.902490i
$716$		0			0
$717$	−11.7525	−	26.0652i	−0.438905	−	0.973421i
$718$		0			0
$719$	16.4203			0.612373			0.306186	−	0.951972i	$-0.400947\pi$
$719$	16.4203			0.612373			0.306186	+	0.951972i	$0.400947\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	11.1441	+	24.7160i	0.414455	+	0.919196i
$724$		0			0
$725$	−9.42783	+	17.0040i	−0.350141	+	0.631512i
$726$		0			0
$727$	30.1583			1.11851			0.559254	−	0.828996i	$-0.311088\pi$
$727$	30.1583			1.11851			0.559254	+	0.828996i	$0.311088\pi$
$728$		0			0
$729$	−22.2835	−	15.2462i	−0.825315	−	0.564673i
$730$		0			0
$731$	7.54215			0.278956
$732$		0			0
$733$	41.2975			1.52536			0.762680	−	0.646776i	$-0.223883\pi$
$733$	41.2975			1.52536			0.762680	+	0.646776i	$0.223883\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14.3135			0.527246
$738$		0			0
$739$	−48.0567			−1.76779			−0.883897	−	0.467682i	$-0.845089\pi$
$739$	−48.0567			−1.76779			−0.883897	+	0.467682i	$0.845089\pi$
$740$		0			0
$741$	−0.559960	−	1.24190i	−0.0205706	−	0.0456224i
$742$		0			0
$743$	−25.1299			−0.921927			−0.460964	−	0.887419i	$-0.652496\pi$
$743$	−25.1299			−0.921927			−0.460964	+	0.887419i	$0.652496\pi$
$744$		0			0
$745$	−1.43604	+	5.55155i	−0.0526126	+	0.203393i
$746$		0			0
$747$	30.7699	+	27.1845i	1.12581	+	0.994630i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−2.37050			−0.0865008			−0.0432504	−	0.999064i	$-0.513771\pi$
$751$	−2.37050			−0.0865008			−0.0432504	+	0.999064i	$0.513771\pi$
$752$		0			0
$753$	36.2741	−	16.3556i	1.32190	−	0.596031i
$754$		0			0
$755$	−11.6516	−	3.01397i	−0.424045	−	0.109690i
$756$		0			0
$757$			43.4331i			1.57860i	0.614006	+	0.789301i	$0.289557\pi$
$757$			43.4331i			1.57860i	−0.614006	+	0.789301i	$0.710443\pi$
$758$		0			0
$759$	−5.68870	−	12.6166i	−0.206487	−	0.457955i
$760$		0			0
$761$	−30.7460			−1.11454			−0.557271	−	0.830331i	$-0.688152\pi$
$761$	−30.7460			−1.11454			−0.557271	+	0.830331i	$0.688152\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−10.9184	−	16.1646i	−0.394756	−	0.584433i
$766$		0			0
$767$	−34.5154			−1.24628
$768$		0			0
$769$			22.3354i			0.805435i	0.915324	+	0.402717i	$0.131934\pi$
$769$			22.3354i			0.805435i	−0.915324	+	0.402717i	$0.868066\pi$
$770$		0			0
$771$	6.46556	+	14.3396i	0.232852	+	0.516428i
$772$		0			0
$773$			29.0010i			1.04309i	0.853222	+	0.521547i	$0.174645\pi$
$773$			29.0010i			1.04309i	−0.853222	+	0.521547i	$0.825355\pi$
$774$		0			0
$775$	−23.7563	+	42.8467i	−0.853351	+	1.53910i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1.23765i			0.0443434i
$780$		0			0
$781$	4.89928			0.175310
$782$		0			0
$783$	−19.3029	−	5.97148i	−0.689830	−	0.213403i
$784$		0			0
$785$	−12.4434	−	3.21879i	−0.444124	−	0.114884i
$786$		0			0
$787$	−26.8511			−0.957140			−0.478570	−	0.878050i	$-0.658845\pi$
$787$	−26.8511			−0.957140			−0.478570	+	0.878050i	$0.658845\pi$
$788$		0			0
$789$	−28.6068	+	12.8985i	−1.01843	+	0.459198i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	35.3537i		−	1.25545i
$794$		0			0
$795$	−35.9739	+	6.19246i	−1.27586	+	0.219624i
$796$		0			0
$797$		−	12.6894i		−	0.449480i	−0.974419	−	0.224740i	$-0.927847\pi$
$797$		−	12.6894i		−	0.449480i	0.974419	−	0.224740i	$-0.0721533\pi$
$798$		0			0
$799$	35.6288			1.26046
$800$		0			0
$801$	22.5047	−	25.4729i	0.795166	−	0.900041i
$802$		0			0
$803$		−	28.7114i		−	1.01320i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	29.6445	−	13.3664i	1.04353	−	0.470518i
$808$		0			0
$809$		−	2.38958i		−	0.0840133i	−0.999117	−	0.0420067i	$-0.986625\pi$
$809$		−	2.38958i		−	0.0840133i	0.999117	−	0.0420067i	$-0.0133751\pi$
$810$		0			0
$811$			26.9826i			0.947486i	0.880663	+	0.473743i	$0.157097\pi$
$811$			26.9826i			0.947486i	−0.880663	+	0.473743i	$0.842903\pi$
$812$		0			0
$813$	12.4970	+	27.7163i	0.438288	+	0.972054i
$814$		0			0
$815$	1.29657	−	5.01237i	0.0454170	−	0.175576i
$816$		0			0
$817$	0.402295			0.0140745
$818$		0			0
$819$		0			0
$820$		0			0
$821$			31.1133i			1.08586i	0.839778	+	0.542930i	$0.182685\pi$
$821$			31.1133i			1.08586i	−0.839778	+	0.542930i	$0.817315\pi$
$822$		0			0
$823$		−	17.4887i		−	0.609616i	−0.952414	−	0.304808i	$-0.901408\pi$
$823$		−	17.4887i		−	0.609616i	0.952414	−	0.304808i	$-0.0985923\pi$
$824$		0			0
$825$	−1.57186	+	18.9728i	−0.0547250	+	0.660547i
$826$		0			0
$827$	24.1404			0.839445			0.419722	−	0.907653i	$-0.362127\pi$
$827$	24.1404			0.839445			0.419722	+	0.907653i	$0.362127\pi$
$828$		0			0
$829$			23.2654i			0.808042i	0.914750	+	0.404021i	$0.132388\pi$
$829$			23.2654i			0.808042i	−0.914750	+	0.404021i	$0.867612\pi$
$830$		0			0
$831$	−13.7273	−	30.4449i	−0.476193	−	1.05612i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−11.3996	+	44.0695i	−0.394501	+	1.52509i
$836$		0			0
$837$	−48.6396	−	15.0470i	−1.68123	−	0.520099i
$838$		0			0
$839$	−20.2640			−0.699593			−0.349796	−	0.936826i	$-0.613749\pi$
$839$	−20.2640			−0.699593			−0.349796	+	0.936826i	$0.613749\pi$
$840$		0			0
$841$	13.8792			0.478594
$842$		0			0
$843$	−4.38728	−	9.73030i	−0.151106	−	0.335129i
$844$		0			0
$845$	27.5247	+	7.11993i	0.946877	+	0.244933i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−36.7278	+	16.5601i	−1.26049	+	0.568343i
$850$		0			0
$851$			14.5861i			0.500006i
$852$		0			0
$853$	−16.3638			−0.560288			−0.280144	−	0.959958i	$-0.590382\pi$
$853$	−16.3638			−0.560288			−0.280144	+	0.959958i	$0.590382\pi$
$854$		0			0
$855$	−0.582383	−	0.862214i	−0.0199171	−	0.0294871i
$856$		0			0
$857$		−	30.8994i		−	1.05550i	−0.849398	−	0.527752i	$-0.823035\pi$
$857$		−	30.8994i		−	1.05550i	0.849398	−	0.527752i	$-0.176965\pi$
$858$		0			0
$859$		−	49.7980i		−	1.69909i	−0.527520	−	0.849543i	$-0.676878\pi$
$859$		−	49.7980i		−	1.69909i	0.527520	−	0.849543i	$-0.323122\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.00483			−0.0682451			−0.0341225	−	0.999418i	$-0.510864\pi$
$863$	−2.00483			−0.0682451			−0.0341225	+	0.999418i	$0.510864\pi$
$864$		0			0
$865$	−12.1658	+	47.0315i	−0.413651	+	1.59912i
$866$		0			0
$867$	−13.4912	+	6.08301i	−0.458184	+	0.206590i
$868$		0			0
$869$			8.10285i			0.274870i
$870$		0			0
$871$		−	33.0180i		−	1.11877i
$872$		0			0
$873$	3.44532	−	3.89972i	0.116606	−	0.131986i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			37.4955i			1.26613i	0.774097	+	0.633067i	$0.218204\pi$
$877$			37.4955i			1.26613i	−0.774097	+	0.633067i	$0.781796\pi$
$878$		0			0
$879$	21.4359	+	47.5415i	0.723016	+	1.60354i
$880$		0			0
$881$	44.7858			1.50887			0.754436	−	0.656374i	$-0.227911\pi$
$881$	44.7858			1.50887			0.754436	+	0.656374i	$0.227911\pi$
$882$		0			0
$883$			5.59317i			0.188225i	0.995562	+	0.0941127i	$0.0300014\pi$
$883$			5.59317i			0.188225i	−0.995562	+	0.0941127i	$0.969999\pi$
$884$		0			0
$885$	−25.9793	+	4.47201i	−0.873285	+	0.150325i
$886$		0			0
$887$			12.4118i			0.416748i	0.978049	+	0.208374i	$0.0668171\pi$
$887$			12.4118i			0.416748i	−0.978049	+	0.208374i	$0.933183\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−19.6338	+	2.43866i	−0.657756	+	0.0816981i
$892$		0			0
$893$	1.90042			0.0635952
$894$		0			0
$895$	−9.47900	+	36.6445i	−0.316848	+	1.22489i
$896$		0			0
$897$	−29.1037	+	13.1225i	−0.971745	+	0.438149i
$898$		0			0
$899$	−38.1014			−1.27075
$900$		0			0
$901$		−	27.4068i		−	0.913052i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−13.7503	+	53.1567i	−0.457074	+	1.76699i
$906$		0			0
$907$			39.7981i			1.32147i	0.750617	+	0.660737i	$0.229756\pi$
$907$			39.7981i			1.32147i	−0.750617	+	0.660737i	$0.770244\pi$
$908$		0			0
$909$	−27.2091	+	30.7977i	−0.902467	+	1.02150i
$910$		0			0
$911$			32.2818i			1.06954i	0.844996	+	0.534772i	$0.179602\pi$
$911$			32.2818i			1.06954i	−0.844996	+	0.534772i	$0.820398\pi$
$912$		0			0
$913$	30.0861			0.995704
$914$		0			0
$915$	−4.58062	−	26.6103i	−0.151431	−	0.879709i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	21.6592			0.714471			0.357235	−	0.934014i	$-0.383719\pi$
$919$	21.6592			0.714471			0.357235	+	0.934014i	$0.383719\pi$
$920$		0			0
$921$	−0.907948	+	0.409384i	−0.0299179	+	0.0134896i
$922$		0			0
$923$		−	11.3015i		−	0.371994i
$924$		0			0
$925$	9.72926	−	17.5476i	0.319896	−	0.576963i
$926$		0			0
$927$	−18.7746	+	21.2509i	−0.616640	+	0.697970i
$928$		0			0
$929$	−36.8987			−1.21061			−0.605303	−	0.795995i	$-0.706948\pi$
$929$	−36.8987			−1.21061			−0.605303	+	0.795995i	$0.706948\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	11.1201	−	5.01394i	0.364057	−	0.164149i
$934$		0			0
$935$	−13.8382	−	3.57960i	−0.452559	−	0.117065i
$936$		0			0
$937$	−39.9185			−1.30408			−0.652041	−	0.758184i	$-0.726087\pi$
$937$	−39.9185			−1.30408			−0.652041	+	0.758184i	$0.726087\pi$
$938$		0			0
$939$	−12.9729	+	5.84934i	−0.423355	+	0.190886i
$940$		0			0
$941$	50.6169			1.65006			0.825032	−	0.565086i	$-0.191157\pi$
$941$	50.6169			1.65006			0.825032	+	0.565086i	$0.191157\pi$
$942$		0			0
$943$	29.0040			0.944501
$944$		0			0
$945$		0			0
$946$		0			0
$947$	31.8954			1.03646			0.518230	−	0.855241i	$-0.326591\pi$
$947$	31.8954			1.03646			0.518230	+	0.855241i	$0.326591\pi$
$948$		0			0
$949$	−66.2307			−2.14994
$950$		0			0
$951$	45.8143	−	20.6572i	1.48563	−	0.669854i
$952$		0			0
$953$	3.68959			0.119518			0.0597588	−	0.998213i	$-0.480967\pi$
$953$	3.68959			0.119518			0.0597588	+	0.998213i	$0.480967\pi$
$954$		0			0
$955$	−10.4597	+	40.4358i	−0.338468	+	1.30847i
$956$		0			0
$957$	−13.4973	+	6.08577i	−0.436305	+	0.196725i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−65.0081			−2.09704
$962$		0			0
$963$	−3.71915	+	4.20967i	−0.119848	+	0.135655i
$964$		0			0
$965$	10.2235	−	39.5226i	0.329105	−	1.27228i
$966$		0			0
$967$		−	43.8775i		−	1.41100i	−0.708708	−	0.705502i	$-0.750721\pi$
$967$		−	43.8775i		−	1.41100i	0.708708	−	0.705502i	$-0.249279\pi$
$968$		0			0
$969$	0.712153	−	0.321102i	0.0228776	−	0.0103153i
$970$		0			0
$971$	27.6348			0.886842			0.443421	−	0.896313i	$-0.353765\pi$
$971$	27.6348			0.886842			0.443421	+	0.896313i	$0.353765\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	43.7658	+	3.62591i	1.40163	+	0.116122i
$976$		0			0
$977$	48.8293			1.56219			0.781094	−	0.624413i	$-0.214662\pi$
$977$	48.8293			1.56219			0.781094	+	0.624413i	$0.214662\pi$
$978$		0			0
$979$		−	24.9068i		−	0.796024i
$980$		0			0
$981$	−15.2415	+	17.2517i	−0.486624	+	0.550806i
$982$		0			0
$983$			20.9766i			0.669050i	0.942387	+	0.334525i	$0.108576\pi$
$983$			20.9766i			0.669050i	−0.942387	+	0.334525i	$0.891424\pi$
$984$		0			0
$985$	11.4505	+	2.96196i	0.364845	+	0.0943760i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	9.42769i		−	0.299783i
$990$		0			0
$991$	28.7338			0.912759			0.456379	−	0.889785i	$-0.349146\pi$
$991$	28.7338			0.912759			0.456379	+	0.889785i	$0.349146\pi$
$992$		0			0
$993$	−3.03442	+	1.36818i	−0.0962943	+	0.0434180i
$994$		0			0
$995$	2.98508	−	11.5399i	0.0946334	−	0.365840i
$996$		0			0
$997$	−15.2318			−0.482395			−0.241198	−	0.970476i	$-0.577540\pi$
$997$	−15.2318			−0.482395			−0.241198	+	0.970476i	$0.577540\pi$
$998$		0			0
$999$	19.9201	+	6.16240i	0.630243	+	0.194970i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2940.2.f.b.1469.8	yes	48
3.2	odd	2	inner	2940.2.f.b.1469.5	✓	48
5.4	even	2	inner	2940.2.f.b.1469.42	yes	48
7.6	odd	2	inner	2940.2.f.b.1469.41	yes	48
15.14	odd	2	inner	2940.2.f.b.1469.43	yes	48
21.20	even	2	inner	2940.2.f.b.1469.44	yes	48
35.34	odd	2	inner	2940.2.f.b.1469.7	yes	48
105.104	even	2	inner	2940.2.f.b.1469.6	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2940.2.f.b.1469.5	✓	48	3.2	odd	2	inner
2940.2.f.b.1469.6	yes	48	105.104	even	2	inner
2940.2.f.b.1469.7	yes	48	35.34	odd	2	inner
2940.2.f.b.1469.8	yes	48	1.1	even	1	trivial
2940.2.f.b.1469.41	yes	48	7.6	odd	2	inner
2940.2.f.b.1469.42	yes	48	5.4	even	2	inner
2940.2.f.b.1469.43	yes	48	15.14	odd	2	inner
2940.2.f.b.1469.44	yes	48	21.20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2940.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.57897 + 0.711939i) q^{3} +(2.16481 + 0.559982i) q^{5} +(1.98629 - 2.24826i) q^{9} +O(q^{10})\)	\(q+(-1.57897 + 0.711939i) q^{3} +(2.16481 + 0.559982i) q^{5} +(1.98629 - 2.24826i) q^{9} -2.19829i q^{11} -5.07095 q^{13} +(-3.81685 + 0.657022i) q^{15} -2.90787i q^{17} -0.155105i q^{19} -3.63484 q^{23} +(4.37284 + 2.42451i) q^{25} +(-1.53566 + 4.96405i) q^{27} +3.88854i q^{29} +9.79837i q^{31} +(1.56505 + 3.47103i) q^{33} -4.01287i q^{37} +(8.00688 - 3.61021i) q^{39} -7.97945 q^{41} +2.59370i q^{43} +(5.55892 - 3.75478i) q^{45} +12.2525i q^{47} +(2.07023 + 4.59144i) q^{51} +9.42504 q^{53} +(1.23100 - 4.75889i) q^{55} +(0.110425 + 0.244905i) q^{57} +6.80649 q^{59} +6.97180i q^{61} +(-10.9777 - 2.83964i) q^{65} +6.51121i q^{67} +(5.73930 - 2.58778i) q^{69} +2.22868i q^{71} +13.0608 q^{73} +(-8.63069 - 0.715035i) q^{75} -3.68597 q^{79} +(-1.10934 - 8.93137i) q^{81} +13.6861i q^{83} +(1.62836 - 6.29500i) q^{85} +(-2.76841 - 6.13989i) q^{87} +11.3301 q^{89} +(-6.97585 - 15.4713i) q^{93} +(0.0868557 - 0.335772i) q^{95} +1.73455 q^{97} +(-4.94233 - 4.36643i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 8 q^{9}+O(q^{10}) \) 48 * q - 8 * q^9	\( 48 q - 8 q^{9} - 16 q^{15} + 16 q^{25} + 56 q^{39} + 8 q^{51} + 48 q^{79} - 24 q^{81} + 32 q^{85} + 88 q^{99}+O(q^{100}) \) 48 * q - 8 * q^9 - 16 * q^15 + 16 * q^25 + 56 * q^39 + 8 * q^51 + 48 * q^79 - 24 * q^81 + 32 * q^85 + 88 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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