LMFDB - Embedded newform 4004.2.e.a.3849.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(3849,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4004.3849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.e (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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3849.16
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.a.3849.33

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.04079i q^{3} -4.22530i q^{5} +(-2.02914 + 1.69782i) q^{7} +1.91676 q^{9} +O(q^{10})$	$q-1.04079i q^{3} -4.22530i q^{5} +(-2.02914 + 1.69782i) q^{7} +1.91676 q^{9} +(3.24641 - 0.678817i) q^{11} +1.00000 q^{13} -4.39764 q^{15} -3.87976 q^{17} +7.23838 q^{19} +(1.76708 + 2.11190i) q^{21} +5.94488 q^{23} -12.8531 q^{25} -5.11731i q^{27} +1.09512i q^{29} -5.22134i q^{31} +(-0.706505 - 3.37883i) q^{33} +(7.17381 + 8.57370i) q^{35} +3.09541 q^{37} -1.04079i q^{39} +2.53399 q^{41} -6.88894i q^{43} -8.09887i q^{45} -5.75983i q^{47} +(1.23479 - 6.89023i) q^{49} +4.03802i q^{51} +8.66407 q^{53} +(-2.86820 - 13.7171i) q^{55} -7.53362i q^{57} +6.74791i q^{59} +14.1344 q^{61} +(-3.88936 + 3.25432i) q^{63} -4.22530i q^{65} +2.12772 q^{67} -6.18737i q^{69} -3.96408 q^{71} -14.7968 q^{73} +13.3774i q^{75} +(-5.43491 + 6.88925i) q^{77} +15.2582i q^{79} +0.424235 q^{81} -3.04591 q^{83} +16.3932i q^{85} +1.13979 q^{87} +7.88438i q^{89} +(-2.02914 + 1.69782i) q^{91} -5.43431 q^{93} -30.5843i q^{95} +7.43823i q^{97} +(6.22259 - 1.30113i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 4 q^{7} - 48 q^{9}+O(q^{10}) $ 48 * q - 4 * q^7 - 48 * q^9	$ 48 q - 4 q^{7} - 48 q^{9} + 2 q^{11} + 48 q^{13} + 8 q^{15} + 4 q^{17} + 10 q^{21} + 4 q^{23} - 44 q^{25} + 10 q^{33} - 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} - 12 q^{55} + 16 q^{61} + 16 q^{63} + 4 q^{67} - 16 q^{73} + 2 q^{77} + 64 q^{81} - 4 q^{83} - 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) $ 48 * q - 4 * q^7 - 48 * q^9 + 2 * q^11 + 48 * q^13 + 8 * q^15 + 4 * q^17 + 10 * q^21 + 4 * q^23 - 44 * q^25 + 10 * q^33 - 14 * q^35 - 16 * q^37 + 10 * q^49 - 8 * q^53 - 12 * q^55 + 16 * q^61 + 16 * q^63 + 4 * q^67 - 16 * q^73 + 2 * q^77 + 64 * q^81 - 4 * q^83 - 12 * q^87 - 4 * q^91 + 36 * q^93 - 40 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times$.

$n$	$365$	$925$	$2003$	$3433$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$		−	1.04079i		−	0.600900i	−0.953798	−	0.300450i	$-0.902863\pi$
$3$		−	1.04079i		−	0.600900i	0.953798	−	0.300450i	$-0.0971368\pi$
$4$		0			0
$5$		−	4.22530i		−	1.88961i	−0.327633	−	0.944805i	$-0.606251\pi$
$5$		−	4.22530i		−	1.88961i	0.327633	−	0.944805i	$-0.393749\pi$
$6$		0			0
$7$	−2.02914	+	1.69782i	−0.766941	+	0.641717i
$7$	−2.02914	+	1.69782i	−0.766941	+	0.641717i
$8$		0			0
$9$	1.91676			0.638919
$10$		0			0
$11$	3.24641	−	0.678817i	0.978831	−	0.204671i
$11$	3.24641	−	0.678817i	0.978831	−	0.204671i
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$		0			0
$15$	−4.39764			−1.13547
$16$		0			0
$17$	−3.87976			−0.940981			−0.470490	−	0.882405i	$-0.655923\pi$
$17$	−3.87976			−0.940981			−0.470490	+	0.882405i	$0.655923\pi$
$18$		0			0
$19$	7.23838			1.66060			0.830299	−	0.557319i	$-0.188170\pi$
$19$	7.23838			1.66060			0.830299	+	0.557319i	$0.188170\pi$
$20$		0			0
$21$	1.76708	+	2.11190i	0.385608	+	0.460855i
$22$		0			0
$23$	5.94488			1.23959			0.619797	−	0.784762i	$-0.287215\pi$
$23$	5.94488			1.23959			0.619797	+	0.784762i	$0.287215\pi$
$24$		0			0
$25$	−12.8531			−2.57063
$26$		0			0
$27$		−	5.11731i		−	0.984826i
$28$		0			0
$29$			1.09512i			0.203359i	0.994817	+	0.101679i	$0.0324215\pi$
$29$			1.09512i			0.203359i	−0.994817	+	0.101679i	$0.967578\pi$
$30$		0			0
$31$		−	5.22134i		−	0.937780i	−0.883257	−	0.468890i	$-0.844654\pi$
$31$		−	5.22134i		−	0.937780i	0.883257	−	0.468890i	$-0.155346\pi$
$32$		0			0
$33$	−0.706505	−	3.37883i	−0.122987	−	0.588179i
$34$		0			0
$35$	7.17381	+	8.57370i	1.21259	+	1.44922i
$36$		0			0
$37$	3.09541			0.508882			0.254441	−	0.967088i	$-0.418109\pi$
$37$	3.09541			0.508882			0.254441	+	0.967088i	$0.418109\pi$
$38$		0			0
$39$		−	1.04079i		−	0.166660i
$40$		0			0
$41$	2.53399			0.395743			0.197871	−	0.980228i	$-0.436597\pi$
$41$	2.53399			0.395743			0.197871	+	0.980228i	$0.436597\pi$
$42$		0			0
$43$		−	6.88894i		−	1.05055i	−0.850931	−	0.525277i	$-0.823962\pi$
$43$		−	6.88894i		−	1.05055i	0.850931	−	0.525277i	$-0.176038\pi$
$44$		0			0
$45$		−	8.09887i		−	1.20731i
$46$		0			0
$47$		−	5.75983i		−	0.840157i	−0.907488	−	0.420079i	$-0.862002\pi$
$47$		−	5.75983i		−	0.840157i	0.907488	−	0.420079i	$-0.137998\pi$
$48$		0			0
$49$	1.23479	−	6.89023i	0.176399	−	0.984319i
$50$		0			0
$51$			4.03802i			0.565435i
$52$		0			0
$53$	8.66407			1.19010			0.595051	−	0.803688i	$-0.297132\pi$
$53$	8.66407			1.19010			0.595051	+	0.803688i	$0.297132\pi$
$54$		0			0
$55$	−2.86820	−	13.7171i	−0.386748	−	1.84961i
$56$		0			0
$57$		−	7.53362i		−	0.997853i
$58$		0			0
$59$			6.74791i			0.878503i	0.898364	+	0.439251i	$0.144756\pi$
$59$			6.74791i			0.878503i	−0.898364	+	0.439251i	$0.855244\pi$
$60$		0			0
$61$	14.1344			1.80972			0.904860	−	0.425710i	$-0.139976\pi$
$61$	14.1344			1.80972			0.904860	+	0.425710i	$0.139976\pi$
$62$		0			0
$63$	−3.88936	+	3.25432i	−0.490014	+	0.410005i
$64$		0			0
$65$		−	4.22530i		−	0.524084i
$66$		0			0
$67$	2.12772			0.259942			0.129971	−	0.991518i	$-0.458512\pi$
$67$	2.12772			0.259942			0.129971	+	0.991518i	$0.458512\pi$
$68$		0			0
$69$		−	6.18737i		−	0.744872i
$70$		0			0
$71$	−3.96408			−0.470450			−0.235225	−	0.971941i	$-0.575583\pi$
$71$	−3.96408			−0.470450			−0.235225	+	0.971941i	$0.575583\pi$
$72$		0			0
$73$	−14.7968			−1.73183			−0.865915	−	0.500191i	$-0.833263\pi$
$73$	−14.7968			−1.73183			−0.865915	+	0.500191i	$0.833263\pi$
$74$		0			0
$75$			13.3774i			1.54469i
$76$		0			0
$77$	−5.43491	+	6.88925i	−0.619365	+	0.785103i
$78$		0			0
$79$			15.2582i			1.71668i	0.513082	+	0.858340i	$0.328504\pi$
$79$			15.2582i			1.71668i	−0.513082	+	0.858340i	$0.671496\pi$
$80$		0			0
$81$	0.424235			0.0471372
$82$		0			0
$83$	−3.04591			−0.334332			−0.167166	−	0.985929i	$-0.553462\pi$
$83$	−3.04591			−0.334332			−0.167166	+	0.985929i	$0.553462\pi$
$84$		0			0
$85$			16.3932i			1.77809i
$86$		0			0
$87$	1.13979			0.122198
$88$		0			0
$89$			7.88438i			0.835742i	0.908506	+	0.417871i	$0.137224\pi$
$89$			7.88438i			0.835742i	−0.908506	+	0.417871i	$0.862776\pi$
$90$		0			0
$91$	−2.02914	+	1.69782i	−0.212711	+	0.177980i
$92$		0			0
$93$	−5.43431			−0.563512
$94$		0			0
$95$		−	30.5843i		−	3.13788i
$96$		0			0
$97$			7.43823i			0.755238i	0.925961	+	0.377619i	$0.123257\pi$
$97$			7.43823i			0.755238i	−0.925961	+	0.377619i	$0.876743\pi$
$98$		0			0
$99$	6.22259	−	1.30113i	0.625394	−	0.130768i
$100$		0			0
$101$	5.25530			0.522922			0.261461	−	0.965214i	$-0.415796\pi$
$101$	5.25530			0.522922			0.261461	+	0.965214i	$0.415796\pi$
$102$		0			0
$103$		−	17.6898i		−	1.74303i	−0.490369	−	0.871515i	$-0.663138\pi$
$103$		−	17.6898i		−	1.74303i	0.490369	−	0.871515i	$-0.336862\pi$
$104$		0			0
$105$	8.92342	−	7.46642i	0.870836	−	0.728648i
$106$		0			0
$107$		−	10.1987i		−	0.985946i	−0.870045	−	0.492973i	$-0.835910\pi$
$107$		−	10.1987i		−	0.985946i	0.870045	−	0.492973i	$-0.164090\pi$
$108$		0			0
$109$		−	10.9836i		−	1.05204i	−0.850473	−	0.526019i	$-0.823684\pi$
$109$		−	10.9836i		−	1.05204i	0.850473	−	0.526019i	$-0.176316\pi$
$110$		0			0
$111$		−	3.22167i		−	0.305787i
$112$		0			0
$113$	−9.94091			−0.935162			−0.467581	−	0.883950i	$-0.654874\pi$
$113$	−9.94091			−0.935162			−0.467581	+	0.883950i	$0.654874\pi$
$114$		0			0
$115$		−	25.1189i		−	2.34235i
$116$		0			0
$117$	1.91676			0.177204
$118$		0			0
$119$	7.87257	−	6.58716i	0.721677	−	0.603844i
$120$		0			0
$121$	10.0784	−	4.40744i	0.916220	−	0.400677i
$122$		0			0
$123$		−	2.63735i		−	0.237802i
$124$		0			0
$125$			33.1818i			2.96787i
$126$		0			0
$127$			1.19887i			0.106382i	0.998584	+	0.0531911i	$0.0169393\pi$
$127$			1.19887i			0.106382i	−0.998584	+	0.0531911i	$0.983061\pi$
$128$		0			0
$129$	−7.16994			−0.631278
$130$		0			0
$131$	−16.2740			−1.42186			−0.710932	−	0.703261i	$-0.751726\pi$
$131$	−16.2740			−1.42186			−0.710932	+	0.703261i	$0.751726\pi$
$132$		0			0
$133$	−14.6877	+	12.2895i	−1.27358	+	1.06563i
$134$		0			0
$135$	−21.6221			−1.86094
$136$		0			0
$137$	−3.65984			−0.312682			−0.156341	−	0.987703i	$-0.549970\pi$
$137$	−3.65984			−0.312682			−0.156341	+	0.987703i	$0.549970\pi$
$138$		0			0
$139$	12.1567			1.03112			0.515558	−	0.856855i	$-0.327585\pi$
$139$	12.1567			1.03112			0.515558	+	0.856855i	$0.327585\pi$
$140$		0			0
$141$	−5.99477			−0.504850
$142$		0			0
$143$	3.24641	−	0.678817i	0.271479	−	0.0567655i
$144$		0			0
$145$	4.62720			0.384268
$146$		0			0
$147$	−7.17128	−	1.28516i	−0.591477	−	0.105998i
$148$		0			0
$149$			13.6133i			1.11524i	0.830096	+	0.557621i	$0.188286\pi$
$149$			13.6133i			1.11524i	−0.830096	+	0.557621i	$0.811714\pi$
$150$		0			0
$151$		−	6.83535i		−	0.556253i	−0.960545	−	0.278126i	$-0.910287\pi$
$151$		−	6.83535i		−	0.556253i	0.960545	−	0.278126i	$-0.0897134\pi$
$152$		0			0
$153$	−7.43657			−0.601211
$154$		0			0
$155$	−22.0617			−1.77204
$156$		0			0
$157$			17.6471i			1.40839i	0.710007	+	0.704195i	$0.248692\pi$
$157$			17.6471i			1.40839i	−0.710007	+	0.704195i	$0.751308\pi$
$158$		0			0
$159$		−	9.01747i		−	0.715132i
$160$		0			0
$161$	−12.0630	+	10.0934i	−0.950696	+	0.795468i
$162$		0			0
$163$	−1.69579			−0.132825			−0.0664124	−	0.997792i	$-0.521155\pi$
$163$	−1.69579			−0.132825			−0.0664124	+	0.997792i	$0.521155\pi$
$164$		0			0
$165$	−14.2766	+	2.98519i	−1.11143	+	0.232397i
$166$		0			0
$167$	−12.8023			−0.990675			−0.495338	−	0.868701i	$-0.664956\pi$
$167$	−12.8023			−0.990675			−0.495338	+	0.868701i	$0.664956\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	13.8742			1.06099
$172$		0			0
$173$	−4.92686			−0.374582			−0.187291	−	0.982304i	$-0.559971\pi$
$173$	−4.92686			−0.374582			−0.187291	+	0.982304i	$0.559971\pi$
$174$		0			0
$175$	26.0808	−	21.8223i	1.97152	−	1.64961i
$176$		0			0
$177$	7.02315			0.527892
$178$		0			0
$179$	−10.8659			−0.812156			−0.406078	−	0.913838i	$-0.633104\pi$
$179$	−10.8659			−0.812156			−0.406078	+	0.913838i	$0.633104\pi$
$180$		0			0
$181$		−	7.00030i		−	0.520328i	−0.965564	−	0.260164i	$-0.916223\pi$
$181$		−	7.00030i		−	0.520328i	0.965564	−	0.260164i	$-0.0837767\pi$
$182$		0			0
$183$		−	14.7109i		−	1.08746i
$184$		0			0
$185$		−	13.0790i		−	0.961589i
$186$		0			0
$187$	−12.5953	+	2.63365i	−0.921061	+	0.192592i
$188$		0			0
$189$	8.68829	+	10.3837i	0.631980	+	0.755304i
$190$		0			0
$191$	−25.9684			−1.87901			−0.939503	−	0.342541i	$-0.888713\pi$
$191$	−25.9684			−1.87901			−0.939503	+	0.342541i	$0.888713\pi$
$192$		0			0
$193$		−	8.79949i		−	0.633401i	−0.948526	−	0.316700i	$-0.897425\pi$
$193$		−	8.79949i		−	0.633401i	0.948526	−	0.316700i	$-0.102575\pi$
$194$		0			0
$195$	−4.39764			−0.314922
$196$		0			0
$197$			2.31434i			0.164890i	0.996596	+	0.0824449i	$0.0262728\pi$
$197$			2.31434i			0.164890i	−0.996596	+	0.0824449i	$0.973727\pi$
$198$		0			0
$199$			7.64625i			0.542028i	0.962575	+	0.271014i	$0.0873590\pi$
$199$			7.64625i			0.542028i	−0.962575	+	0.271014i	$0.912641\pi$
$200$		0			0
$201$		−	2.21451i		−	0.156199i
$202$		0			0
$203$	−1.85932	−	2.22215i	−0.130499	−	0.155964i
$204$		0			0
$205$		−	10.7069i		−	0.747799i
$206$		0			0
$207$	11.3949			0.792000
$208$		0			0
$209$	23.4988	−	4.91353i	1.62544	−	0.339876i
$210$		0			0
$211$		−	24.8638i		−	1.71169i	−0.517228	−	0.855847i	$-0.673036\pi$
$211$		−	24.8638i		−	1.71169i	0.517228	−	0.855847i	$-0.326964\pi$
$212$		0			0
$213$			4.12578i			0.282693i
$214$		0			0
$215$	−29.1078			−1.98514
$216$		0			0
$217$	8.86491	+	10.5948i	0.601789	+	0.719223i
$218$		0			0
$219$			15.4003i			1.04066i
$220$		0			0
$221$	−3.87976			−0.260981
$222$		0			0
$223$			10.2035i			0.683275i	0.939832	+	0.341638i	$0.110982\pi$
$223$			10.2035i			0.683275i	−0.939832	+	0.341638i	$0.889018\pi$
$224$		0			0
$225$	−24.6363			−1.64242
$226$		0			0
$227$	−4.27270			−0.283589			−0.141794	−	0.989896i	$-0.545287\pi$
$227$	−4.27270			−0.283589			−0.141794	+	0.989896i	$0.545287\pi$
$228$		0			0
$229$		−	2.23682i		−	0.147813i	−0.997265	−	0.0739066i	$-0.976453\pi$
$229$		−	2.23682i		−	0.147813i	0.997265	−	0.0739066i	$-0.0235467\pi$
$230$		0			0
$231$	7.17026	+	5.65659i	0.471768	+	0.372176i
$232$		0			0
$233$			7.75004i			0.507722i	0.967241	+	0.253861i	$0.0817006\pi$
$233$			7.75004i			0.507722i	−0.967241	+	0.253861i	$0.918299\pi$
$234$		0			0
$235$	−24.3370			−1.58757
$236$		0			0
$237$	15.8806			1.03155
$238$		0			0
$239$			22.7181i			1.46951i	0.678332	+	0.734756i	$0.262703\pi$
$239$			22.7181i			1.46951i	−0.678332	+	0.734756i	$0.737297\pi$
$240$		0			0
$241$	−11.9004			−0.766573			−0.383287	−	0.923629i	$-0.625208\pi$
$241$	−11.9004			−0.766573			−0.383287	+	0.923629i	$0.625208\pi$
$242$		0			0
$243$		−	15.7935i		−	1.01315i
$244$		0			0
$245$	−29.1133	−	5.21735i	−1.85998	−	0.333324i
$246$		0			0
$247$	7.23838			0.460567
$248$		0			0
$249$			3.17015i			0.200900i
$250$		0			0
$251$		−	1.90988i		−	0.120551i	−0.998182	−	0.0602754i	$-0.980802\pi$
$251$		−	1.90988i		−	0.120551i	0.998182	−	0.0602754i	$-0.0191979\pi$
$252$		0			0
$253$	19.2996	−	4.03549i	1.21335	−	0.253709i
$254$		0			0
$255$	17.0618			1.06845
$256$		0			0
$257$			7.52769i			0.469565i	0.972048	+	0.234782i	$0.0754377\pi$
$257$			7.52769i			0.469565i	−0.972048	+	0.234782i	$0.924562\pi$
$258$		0			0
$259$	−6.28101	+	5.25546i	−0.390283	+	0.326558i
$260$		0			0
$261$			2.09908i			0.129930i
$262$		0			0
$263$			10.6408i			0.656139i	0.944654	+	0.328069i	$0.106398\pi$
$263$			10.6408i			0.656139i	−0.944654	+	0.328069i	$0.893602\pi$
$264$		0			0
$265$		−	36.6083i		−	2.24883i
$266$		0			0
$267$	8.20597			0.502197
$268$		0			0
$269$			25.6510i			1.56397i	0.623299	+	0.781984i	$0.285792\pi$
$269$			25.6510i			1.56397i	−0.623299	+	0.781984i	$0.714208\pi$
$270$		0			0
$271$	0.890590			0.0540995			0.0270497	−	0.999634i	$-0.491389\pi$
$271$	0.890590			0.0540995			0.0270497	+	0.999634i	$0.491389\pi$
$272$		0			0
$273$	1.76708	+	2.11190i	0.106948	+	0.127818i
$274$		0			0
$275$	−41.7266	+	8.72492i	−2.51621	+	0.526133i
$276$		0			0
$277$		−	6.75748i		−	0.406018i	−0.979177	−	0.203009i	$-0.934928\pi$
$277$		−	6.75748i		−	0.406018i	0.979177	−	0.203009i	$-0.0650720\pi$
$278$		0			0
$279$		−	10.0080i		−	0.599166i
$280$		0			0
$281$		−	20.6724i		−	1.23321i	−0.787271	−	0.616607i	$-0.788507\pi$
$281$		−	20.6724i		−	1.23321i	0.787271	−	0.616607i	$-0.211493\pi$
$282$		0			0
$283$	30.0715			1.78757			0.893783	−	0.448499i	$-0.148041\pi$
$283$	30.0715			1.78757			0.893783	+	0.448499i	$0.148041\pi$
$284$		0			0
$285$	−31.8318			−1.88555
$286$		0			0
$287$	−5.14181	+	4.30227i	−0.303511	+	0.253955i
$288$		0			0
$289$	−1.94743			−0.114555
$290$		0			0
$291$	7.74163			0.453822
$292$		0			0
$293$	5.88089			0.343565			0.171783	−	0.985135i	$-0.445047\pi$
$293$	5.88089			0.343565			0.171783	+	0.985135i	$0.445047\pi$
$294$		0			0
$295$	28.5119			1.66003
$296$		0			0
$297$	−3.47372	−	16.6129i	−0.201565	−	0.963978i
$298$		0			0
$299$	5.94488			0.343801
$300$		0			0
$301$	11.6962	+	13.9786i	0.674158	+	0.805713i
$302$		0			0
$303$		−	5.46966i		−	0.314224i
$304$		0			0
$305$		−	59.7219i		−	3.41966i
$306$		0			0
$307$	−29.0267			−1.65664			−0.828322	−	0.560253i	$-0.810704\pi$
$307$	−29.0267			−1.65664			−0.828322	+	0.560253i	$0.810704\pi$
$308$		0			0
$309$	−18.4114			−1.04739
$310$		0			0
$311$			28.7132i			1.62817i	0.580743	+	0.814087i	$0.302762\pi$
$311$			28.7132i			1.62817i	−0.580743	+	0.814087i	$0.697238\pi$
$312$		0			0
$313$			12.6871i			0.717116i	0.933507	+	0.358558i	$0.116731\pi$
$313$			12.6871i			0.717116i	−0.933507	+	0.358558i	$0.883269\pi$
$314$		0			0
$315$	13.7505	+	16.4337i	0.774750	+	0.925935i
$316$		0			0
$317$	−4.52722			−0.254274			−0.127137	−	0.991885i	$-0.540579\pi$
$317$	−4.52722			−0.254274			−0.127137	+	0.991885i	$0.540579\pi$
$318$		0			0
$319$	0.743385	+	3.55521i	0.0416216	+	0.199054i
$320$		0			0
$321$	−10.6147			−0.592455
$322$		0			0
$323$	−28.0832			−1.56259
$324$		0			0
$325$	−12.8531			−0.712963
$326$		0			0
$327$	−11.4316			−0.632169
$328$		0			0
$329$	9.77917	+	11.6875i	0.539143	+	0.644352i
$330$		0			0
$331$	−1.93745			−0.106492			−0.0532461	−	0.998581i	$-0.516957\pi$
$331$	−1.93745			−0.106492			−0.0532461	+	0.998581i	$0.516957\pi$
$332$		0			0
$333$	5.93315			0.325135
$334$		0			0
$335$		−	8.99024i		−	0.491189i
$336$		0			0
$337$			22.5809i			1.23006i	0.788503	+	0.615031i	$0.210856\pi$
$337$			22.5809i			1.23006i	−0.788503	+	0.615031i	$0.789144\pi$
$338$		0			0
$339$			10.3464i			0.561939i
$340$		0			0
$341$	−3.54433	−	16.9506i	−0.191936	−	0.917928i
$342$		0			0
$343$	9.19284	+	16.0777i	0.496367	+	0.868113i
$344$		0			0
$345$	−26.1435			−1.40752
$346$		0			0
$347$		−	26.6709i		−	1.43177i	−0.698219	−	0.715884i	$-0.746024\pi$
$347$		−	26.6709i		−	1.43177i	0.698219	−	0.715884i	$-0.253976\pi$
$348$		0			0
$349$	−14.3350			−0.767333			−0.383667	−	0.923472i	$-0.625339\pi$
$349$	−14.3350			−0.767333			−0.383667	+	0.923472i	$0.625339\pi$
$350$		0			0
$351$		−	5.11731i		−	0.273142i
$352$		0			0
$353$		−	31.0974i		−	1.65515i	−0.561358	−	0.827573i	$-0.689721\pi$
$353$		−	31.0974i		−	1.65515i	0.561358	−	0.827573i	$-0.310279\pi$
$354$		0			0
$355$			16.7494i			0.888967i
$356$		0			0
$357$	−6.85584	−	8.19369i	−0.362850	−	0.433656i
$358$		0			0
$359$			12.0723i			0.637152i	0.947897	+	0.318576i	$0.103205\pi$
$359$			12.0723i			0.637152i	−0.947897	+	0.318576i	$0.896795\pi$
$360$		0			0
$361$	33.3941			1.75758
$362$		0			0
$363$	−4.58722	−	10.4895i	−0.240767	−	0.550556i
$364$		0			0
$365$			62.5207i			3.27248i
$366$		0			0
$367$		−	16.2206i		−	0.846709i	−0.905964	−	0.423354i	$-0.860853\pi$
$367$		−	16.2206i		−	0.846709i	0.905964	−	0.423354i	$-0.139147\pi$
$368$		0			0
$369$	4.85704			0.252848
$370$		0			0
$371$	−17.5806	+	14.7101i	−0.912738	+	0.763709i
$372$		0			0
$373$		−	13.3586i		−	0.691681i	−0.938293	−	0.345840i	$-0.887594\pi$
$373$		−	13.3586i		−	0.691681i	0.938293	−	0.345840i	$-0.112406\pi$
$374$		0			0
$375$	34.5353			1.78339
$376$		0			0
$377$			1.09512i			0.0564015i
$378$		0			0
$379$	19.6481			1.00925			0.504627	−	0.863338i	$-0.331630\pi$
$379$	19.6481			1.00925			0.504627	+	0.863338i	$0.331630\pi$
$380$		0			0
$381$	1.24777			0.0639251
$382$		0			0
$383$		−	24.2656i		−	1.23991i	−0.784636	−	0.619957i	$-0.787150\pi$
$383$		−	24.2656i		−	1.23991i	0.784636	−	0.619957i	$-0.212850\pi$
$384$		0			0
$385$	29.1091	+	22.9641i	1.48354	+	1.17036i
$386$		0			0
$387$		−	13.2044i		−	0.671219i
$388$		0			0
$389$	22.5421			1.14293			0.571464	−	0.820627i	$-0.306375\pi$
$389$	22.5421			1.14293			0.571464	+	0.820627i	$0.306375\pi$
$390$		0			0
$391$	−23.0647			−1.16643
$392$		0			0
$393$			16.9378i			0.854398i
$394$		0			0
$395$	64.4703			3.24385
$396$		0			0
$397$		−	4.98437i		−	0.250158i	−0.992147	−	0.125079i	$-0.960082\pi$
$397$		−	4.98437i		−	0.250158i	0.992147	−	0.125079i	$-0.0399185\pi$
$398$		0			0
$399$	12.7908	+	15.2867i	0.640339	+	0.765295i
$400$		0			0
$401$	32.5754			1.62674			0.813369	−	0.581748i	$-0.197631\pi$
$401$	32.5754			1.62674			0.813369	+	0.581748i	$0.197631\pi$
$402$		0			0
$403$		−	5.22134i		−	0.260093i
$404$		0			0
$405$		−	1.79252i		−	0.0890709i
$406$		0			0
$407$	10.0490	−	2.10122i	0.498110	−	0.104153i
$408$		0			0
$409$	−26.9833			−1.33424			−0.667119	−	0.744951i	$-0.732473\pi$
$409$	−26.9833			−1.33424			−0.667119	+	0.744951i	$0.732473\pi$
$410$		0			0
$411$			3.80913i			0.187890i
$412$		0			0
$413$	−11.4568	−	13.6924i	−0.563750	−	0.673760i
$414$		0			0
$415$			12.8699i			0.631757i
$416$		0			0
$417$		−	12.6525i		−	0.619597i
$418$		0			0
$419$			11.7276i			0.572932i	0.958090	+	0.286466i	$0.0924806\pi$
$419$			11.7276i			0.572932i	−0.958090	+	0.286466i	$0.907519\pi$
$420$		0			0
$421$	37.3160			1.81867			0.909335	−	0.416065i	$-0.136591\pi$
$421$	37.3160			1.81867			0.909335	+	0.416065i	$0.136591\pi$
$422$		0			0
$423$		−	11.0402i		−	0.536793i
$424$		0			0
$425$	49.8671			2.41891
$426$		0			0
$427$	−28.6805	+	23.9976i	−1.38795	+	1.16133i
$428$		0			0
$429$	−0.706505	−	3.37883i	−0.0341104	−	0.163132i
$430$		0			0
$431$		−	6.43229i		−	0.309832i	−0.987928	−	0.154916i	$-0.950489\pi$
$431$		−	6.43229i		−	0.309832i	0.987928	−	0.154916i	$-0.0495108\pi$
$432$		0			0
$433$			10.5443i			0.506727i	0.967371	+	0.253364i	$0.0815369\pi$
$433$			10.5443i			0.506727i	−0.967371	+	0.253364i	$0.918463\pi$
$434$		0			0
$435$		−	4.81594i		−	0.230907i
$436$		0			0
$437$	43.0313			2.05847
$438$		0			0
$439$	23.9053			1.14094			0.570469	−	0.821319i	$-0.306761\pi$
$439$	23.9053			1.14094			0.570469	+	0.821319i	$0.306761\pi$
$440$		0			0
$441$	2.36679	−	13.2069i	0.112704	−	0.628900i
$442$		0			0
$443$	−21.6603			−1.02911			−0.514557	−	0.857456i	$-0.672043\pi$
$443$	−21.6603			−1.02911			−0.514557	+	0.857456i	$0.672043\pi$
$444$		0			0
$445$	33.3138			1.57923
$446$		0			0
$447$	14.1685			0.670149
$448$		0			0
$449$	10.4051			0.491047			0.245524	−	0.969391i	$-0.421040\pi$
$449$	10.4051			0.491047			0.245524	+	0.969391i	$0.421040\pi$
$450$		0			0
$451$	8.22638	−	1.72011i	0.387365	−	0.0809970i
$452$		0			0
$453$	−7.11416			−0.334252
$454$		0			0
$455$	7.17381	+	8.57370i	0.336313	+	0.401941i
$456$		0			0
$457$		−	26.1467i		−	1.22309i	−0.791209	−	0.611546i	$-0.790548\pi$
$457$		−	26.1467i		−	1.22309i	0.791209	−	0.611546i	$-0.209452\pi$
$458$		0			0
$459$			19.8539i			0.926703i
$460$		0			0
$461$	−24.4445			−1.13849			−0.569247	−	0.822166i	$-0.692765\pi$
$461$	−24.4445			−1.13849			−0.569247	+	0.822166i	$0.692765\pi$
$462$		0			0
$463$	13.0519			0.606574			0.303287	−	0.952899i	$-0.401916\pi$
$463$	13.0519			0.606574			0.303287	+	0.952899i	$0.401916\pi$
$464$		0			0
$465$			22.9616i			1.06482i
$466$		0			0
$467$		−	2.52615i		−	0.116896i	−0.998290	−	0.0584480i	$-0.981385\pi$
$467$		−	2.52615i		−	0.116896i	0.998290	−	0.0584480i	$-0.0186152\pi$
$468$		0			0
$469$	−4.31743	+	3.61249i	−0.199360	+	0.166809i
$470$		0			0
$471$	18.3669			0.846301
$472$		0			0
$473$	−4.67633	−	22.3644i	−0.215018	−	1.02831i
$474$		0			0
$475$	−93.0358			−4.26877
$476$		0			0
$477$	16.6069			0.760379
$478$		0			0
$479$	20.6964			0.945641			0.472821	−	0.881159i	$-0.343236\pi$
$479$	20.6964			0.945641			0.472821	+	0.881159i	$0.343236\pi$
$480$		0			0
$481$	3.09541			0.141139
$482$		0			0
$483$	10.5051	+	12.5550i	0.477997	+	0.571273i
$484$		0			0
$485$	31.4287			1.42710
$486$		0			0
$487$	1.38430			0.0627285			0.0313642	−	0.999508i	$-0.490015\pi$
$487$	1.38430			0.0627285			0.0313642	+	0.999508i	$0.490015\pi$
$488$		0			0
$489$			1.76496i			0.0798144i
$490$		0			0
$491$		−	17.9851i		−	0.811656i	−0.913949	−	0.405828i	$-0.866983\pi$
$491$		−	17.9851i		−	0.811656i	0.913949	−	0.405828i	$-0.133017\pi$
$492$		0			0
$493$		−	4.24880i		−	0.191357i
$494$		0			0
$495$	−5.49765	−	26.2923i	−0.247101	−	1.18175i
$496$		0			0
$497$	8.04367	−	6.73032i	0.360808	−	0.301896i
$498$		0			0
$499$	−30.7285			−1.37559			−0.687797	−	0.725903i	$-0.741422\pi$
$499$	−30.7285			−1.37559			−0.687797	+	0.725903i	$0.741422\pi$
$500$		0			0
$501$			13.3245i			0.595297i
$502$		0			0
$503$	−11.5177			−0.513547			−0.256773	−	0.966472i	$-0.582659\pi$
$503$	−11.5177			−0.513547			−0.256773	+	0.966472i	$0.582659\pi$
$504$		0			0
$505$		−	22.2052i		−	0.988119i
$506$		0			0
$507$		−	1.04079i		−	0.0462231i
$508$		0			0
$509$			14.5374i			0.644358i	0.946679	+	0.322179i	$0.104415\pi$
$509$			14.5374i			0.644358i	−0.946679	+	0.322179i	$0.895585\pi$
$510$		0			0
$511$	30.0247	−	25.1223i	1.32821	−	1.11135i
$512$		0			0
$513$		−	37.0410i		−	1.63540i
$514$		0			0
$515$	−74.7447			−3.29365
$516$		0			0
$517$	−3.90987	−	18.6988i	−0.171956	−	0.822372i
$518$		0			0
$519$			5.12782i			0.225086i
$520$		0			0
$521$			7.65888i			0.335542i	0.985826	+	0.167771i	$0.0536569\pi$
$521$			7.65888i			0.335542i	−0.985826	+	0.167771i	$0.946343\pi$
$522$		0			0
$523$	34.5129			1.50914			0.754571	−	0.656219i	$-0.227845\pi$
$523$	34.5129			1.50914			0.754571	+	0.656219i	$0.227845\pi$
$524$		0			0
$525$	−22.7125	−	27.1446i	−0.991253	−	1.18469i
$526$		0			0
$527$			20.2576i			0.882433i
$528$		0			0
$529$	12.3416			0.536593
$530$		0			0
$531$			12.9341i			0.561292i
$532$		0			0
$533$	2.53399			0.109759
$534$		0			0
$535$	−43.0925			−1.86305
$536$		0			0
$537$			11.3091i			0.488024i
$538$		0			0
$539$	−0.668567	−	23.2067i	−0.0287972	−	0.999585i
$540$		0			0
$541$		−	10.8668i		−	0.467202i	−0.972333	−	0.233601i	$-0.924949\pi$
$541$		−	10.8668i		−	0.467202i	0.972333	−	0.233601i	$-0.0750510\pi$
$542$		0			0
$543$	−7.28584			−0.312665
$544$		0			0
$545$	−46.4089			−1.98794
$546$		0			0
$547$		−	19.3846i		−	0.828825i	−0.910089	−	0.414413i	$-0.863987\pi$
$547$		−	19.3846i		−	0.828825i	0.910089	−	0.414413i	$-0.136013\pi$
$548$		0			0
$549$	27.0921			1.15626
$550$		0			0
$551$			7.92688i			0.337697i
$552$		0			0
$553$	−25.9057	−	30.9609i	−1.10162	−	1.31659i
$554$		0			0
$555$	−13.6125			−0.577819
$556$		0			0
$557$			39.3119i			1.66570i	0.553501	+	0.832848i	$0.313291\pi$
$557$			39.3119i			1.66570i	−0.553501	+	0.832848i	$0.686709\pi$
$558$		0			0
$559$		−	6.88894i		−	0.291371i
$560$		0			0
$561$	2.74107	+	13.1091i	0.115728	+	0.553466i
$562$		0			0
$563$	40.3326			1.69981			0.849907	−	0.526932i	$-0.176658\pi$
$563$	40.3326			1.69981			0.849907	+	0.526932i	$0.176658\pi$
$564$		0			0
$565$			42.0033i			1.76709i
$566$		0			0
$567$	−0.860830	+	0.720276i	−0.0361515	+	0.0302487i
$568$		0			0
$569$			18.9039i			0.792494i	0.918144	+	0.396247i	$0.129688\pi$
$569$			18.9039i			0.792494i	−0.918144	+	0.396247i	$0.870312\pi$
$570$		0			0
$571$			21.9985i			0.920608i	0.887761	+	0.460304i	$0.152260\pi$
$571$			21.9985i			0.920608i	−0.887761	+	0.460304i	$0.847740\pi$
$572$		0			0
$573$			27.0276i			1.12909i
$574$		0			0
$575$	−76.4104			−3.18653
$576$		0			0
$577$			13.8565i			0.576852i	0.957502	+	0.288426i	$0.0931320\pi$
$577$			13.8565i			0.576852i	−0.957502	+	0.288426i	$0.906868\pi$
$578$		0			0
$579$	−9.15841			−0.380611
$580$		0			0
$581$	6.18056	−	5.17142i	0.256413	−	0.214546i
$582$		0			0
$583$	28.1272	−	5.88132i	1.16491	−	0.243579i
$584$		0			0
$585$		−	8.09887i		−	0.334847i
$586$		0			0
$587$		−	34.9079i		−	1.44080i	−0.693557	−	0.720401i	$-0.743958\pi$
$587$		−	34.9079i		−	1.44080i	0.693557	−	0.720401i	$-0.256042\pi$
$588$		0			0
$589$		−	37.7940i		−	1.55728i
$590$		0			0
$591$	2.40874			0.0990822
$592$		0			0
$593$	15.4485			0.634395			0.317198	−	0.948359i	$-0.397258\pi$
$593$	15.4485			0.634395			0.317198	+	0.948359i	$0.397258\pi$
$594$		0			0
$595$	−27.8327	−	33.2639i	−1.14103	−	1.36369i
$596$		0			0
$597$	7.95813			0.325705
$598$		0			0
$599$	4.57856			0.187075			0.0935375	−	0.995616i	$-0.470182\pi$
$599$	4.57856			0.187075			0.0935375	+	0.995616i	$0.470182\pi$
$600$		0			0
$601$	−6.03610			−0.246218			−0.123109	−	0.992393i	$-0.539286\pi$
$601$	−6.03610			−0.246218			−0.123109	+	0.992393i	$0.539286\pi$
$602$		0			0
$603$	4.07832			0.166082
$604$		0			0
$605$	−18.6228	−	42.5843i	−0.757122	−	1.73130i
$606$		0			0
$607$	22.8513			0.927505			0.463753	−	0.885965i	$-0.346503\pi$
$607$	22.8513			0.927505			0.463753	+	0.885965i	$0.346503\pi$
$608$		0			0
$609$	−2.31279	+	1.93516i	−0.0937188	+	0.0784166i
$610$		0			0
$611$		−	5.75983i		−	0.233018i
$612$		0			0
$613$		−	39.1407i		−	1.58088i	−0.612542	−	0.790438i	$-0.709853\pi$
$613$		−	39.1407i		−	1.58088i	0.612542	−	0.790438i	$-0.290147\pi$
$614$		0			0
$615$	−11.1436			−0.449353
$616$		0			0
$617$	−17.9071			−0.720913			−0.360457	−	0.932776i	$-0.617379\pi$
$617$	−17.9071			−0.720913			−0.360457	+	0.932776i	$0.617379\pi$
$618$		0			0
$619$		−	25.0332i		−	1.00617i	−0.864237	−	0.503084i	$-0.832199\pi$
$619$		−	25.0332i		−	1.00617i	0.864237	−	0.503084i	$-0.167801\pi$
$620$		0			0
$621$		−	30.4218i		−	1.22078i
$622$		0			0
$623$	−13.3863	−	15.9985i	−0.536310	−	0.640965i
$624$		0			0
$625$	75.9373			3.03749
$626$		0			0
$627$	−5.11395	−	24.4573i	−0.204232	−	0.976729i
$628$		0			0
$629$	−12.0095			−0.478848
$630$		0			0
$631$	17.8331			0.709926			0.354963	−	0.934880i	$-0.384493\pi$
$631$	17.8331			0.709926			0.354963	+	0.934880i	$0.384493\pi$
$632$		0			0
$633$	−25.8780			−1.02856
$634$		0			0
$635$	5.06557			0.201021
$636$		0			0
$637$	1.23479	−	6.89023i	0.0489241	−	0.273001i
$638$		0			0
$639$	−7.59819			−0.300580
$640$		0			0
$641$	−12.7048			−0.501808			−0.250904	−	0.968012i	$-0.580728\pi$
$641$	−12.7048			−0.501808			−0.250904	+	0.968012i	$0.580728\pi$
$642$		0			0
$643$		−	24.9403i		−	0.983548i	−0.870723	−	0.491774i	$-0.836349\pi$
$643$		−	24.9403i		−	0.983548i	0.870723	−	0.491774i	$-0.163651\pi$
$644$		0			0
$645$			30.2951i			1.19287i
$646$		0			0
$647$			12.8931i			0.506879i	0.967351	+	0.253440i	$0.0815619\pi$
$647$			12.8931i			0.506879i	−0.967351	+	0.253440i	$0.918438\pi$
$648$		0			0
$649$	4.58059	+	21.9065i	0.179804	+	0.859906i
$650$		0			0
$651$	11.0270	−	9.22651i	0.432181	−	0.361615i
$652$		0			0
$653$	23.7356			0.928845			0.464423	−	0.885614i	$-0.346262\pi$
$653$	23.7356			0.928845			0.464423	+	0.885614i	$0.346262\pi$
$654$		0			0
$655$			68.7623i			2.68677i
$656$		0			0
$657$	−28.3618			−1.10650
$658$		0			0
$659$		−	11.4488i		−	0.445983i	−0.974820	−	0.222992i	$-0.928418\pi$
$659$		−	11.4488i		−	0.445983i	0.974820	−	0.222992i	$-0.0715822\pi$
$660$		0			0
$661$			12.3579i			0.480668i	0.970690	+	0.240334i	$0.0772570\pi$
$661$			12.3579i			0.480668i	−0.970690	+	0.240334i	$0.922743\pi$
$662$		0			0
$663$			4.03802i			0.156824i
$664$		0			0
$665$	51.9267	+	62.0597i	2.01363	+	2.40657i
$666$		0			0
$667$			6.51035i			0.252082i
$668$		0			0
$669$	10.6197			0.410580
$670$		0			0
$671$	45.8860	−	9.59464i	1.77141	−	0.370397i
$672$		0			0
$673$			16.4465i			0.633967i	0.948431	+	0.316984i	$0.102670\pi$
$673$			16.4465i			0.633967i	−0.948431	+	0.316984i	$0.897330\pi$
$674$		0			0
$675$			65.7734i			2.53162i
$676$		0			0
$677$	−28.4667			−1.09406			−0.547032	−	0.837111i	$-0.684243\pi$
$677$	−28.4667			−1.09406			−0.547032	+	0.837111i	$0.684243\pi$
$678$		0			0
$679$	−12.6288	−	15.0932i	−0.484649	−	0.579223i
$680$		0			0
$681$			4.44697i			0.170408i
$682$		0			0
$683$	−10.2263			−0.391299			−0.195649	−	0.980674i	$-0.562681\pi$
$683$	−10.2263			−0.391299			−0.195649	+	0.980674i	$0.562681\pi$
$684$		0			0
$685$			15.4639i			0.590846i
$686$		0			0
$687$	−2.32806			−0.0888210
$688$		0			0
$689$	8.66407			0.330075
$690$		0			0
$691$		−	2.29224i		−	0.0872008i	−0.999049	−	0.0436004i	$-0.986117\pi$
$691$		−	2.29224i		−	0.0872008i	0.999049	−	0.0436004i	$-0.0138828\pi$
$692$		0			0
$693$	−10.4174	+	13.2050i	−0.395724	+	0.501618i
$694$		0			0
$695$		−	51.3655i		−	1.94841i
$696$		0			0
$697$	−9.83128			−0.372386
$698$		0			0
$699$	8.06616			0.305090
$700$		0			0
$701$			25.2242i			0.952706i	0.879254	+	0.476353i	$0.158042\pi$
$701$			25.2242i			0.952706i	−0.879254	+	0.476353i	$0.841958\pi$
$702$		0			0
$703$	22.4057			0.845048
$704$		0			0
$705$			25.3297i			0.953971i
$706$		0			0
$707$	−10.6637	+	8.92258i	−0.401051	+	0.335568i
$708$		0			0
$709$	−18.0689			−0.678593			−0.339297	−	0.940679i	$-0.610189\pi$
$709$	−18.0689			−0.678593			−0.339297	+	0.940679i	$0.610189\pi$
$710$		0			0
$711$			29.2462i			1.09682i
$712$		0			0
$713$		−	31.0402i		−	1.16247i
$714$		0			0
$715$	−2.86820	−	13.7171i	−0.107265	−	0.512989i
$716$		0			0
$717$	23.6447			0.883029
$718$		0			0
$719$			35.6934i			1.33114i	0.746336	+	0.665570i	$0.231811\pi$
$719$			35.6934i			1.33114i	−0.746336	+	0.665570i	$0.768189\pi$
$720$		0			0
$721$	30.0342	+	35.8951i	1.11853	+	1.33680i
$722$		0			0
$723$			12.3858i			0.460634i
$724$		0			0
$725$		−	14.0757i		−	0.522759i
$726$		0			0
$727$			29.3618i			1.08897i	0.838770	+	0.544485i	$0.183275\pi$
$727$			29.3618i			1.08897i	−0.838770	+	0.544485i	$0.816725\pi$
$728$		0			0
$729$	−15.1650			−0.561665
$730$		0			0
$731$			26.7275i			0.988551i
$732$		0			0
$733$	−3.53283			−0.130488			−0.0652441	−	0.997869i	$-0.520783\pi$
$733$	−3.53283			−0.130488			−0.0652441	+	0.997869i	$0.520783\pi$
$734$		0			0
$735$	−5.43016	+	30.3008i	−0.200295	+	1.11766i
$736$		0			0
$737$	6.90745	−	1.44433i	0.254439	−	0.0532026i
$738$		0			0
$739$		−	16.6407i		−	0.612138i	−0.952009	−	0.306069i	$-0.900986\pi$
$739$		−	16.6407i		−	0.612138i	0.952009	−	0.306069i	$-0.0990138\pi$
$740$		0			0
$741$		−	7.53362i		−	0.276755i
$742$		0			0
$743$		−	37.3010i		−	1.36844i	−0.729276	−	0.684220i	$-0.760143\pi$
$743$		−	37.3010i		−	1.36844i	0.729276	−	0.684220i	$-0.239857\pi$
$744$		0			0
$745$	57.5201			2.10737
$746$		0			0
$747$	−5.83827			−0.213611
$748$		0			0
$749$	17.3156	+	20.6946i	0.632698	+	0.756163i
$750$		0			0
$751$	13.9852			0.510326			0.255163	−	0.966898i	$-0.417871\pi$
$751$	13.9852			0.510326			0.255163	+	0.966898i	$0.417871\pi$
$752$		0			0
$753$	−1.98779			−0.0724390
$754$		0			0
$755$	−28.8814			−1.05110
$756$		0			0
$757$	10.2837			0.373769			0.186884	−	0.982382i	$-0.440161\pi$
$757$	10.2837			0.373769			0.186884	+	0.982382i	$0.440161\pi$
$758$		0			0
$759$	−4.20009	−	20.0868i	−0.152454	−	0.729103i
$760$		0			0
$761$	16.6121			0.602189			0.301094	−	0.953594i	$-0.402648\pi$
$761$	16.6121			0.602189			0.301094	+	0.953594i	$0.402648\pi$
$762$		0			0
$763$	18.6482	+	22.2872i	0.675111	+	0.806851i
$764$		0			0
$765$			31.4217i			1.13605i
$766$		0			0
$767$			6.74791i			0.243653i
$768$		0			0
$769$	−33.2510			−1.19906			−0.599530	−	0.800352i	$-0.704646\pi$
$769$	−33.2510			−1.19906			−0.599530	+	0.800352i	$0.704646\pi$
$770$		0			0
$771$	7.83474			0.282161
$772$		0			0
$773$			9.34885i			0.336255i	0.985765	+	0.168127i	$0.0537720\pi$
$773$			9.34885i			0.336255i	−0.985765	+	0.168127i	$0.946228\pi$
$774$		0			0
$775$			67.1105i			2.41068i
$776$		0			0
$777$	5.46983	+	6.53721i	0.196229	+	0.234521i
$778$		0			0
$779$	18.3420			0.657169
$780$		0			0
$781$	−12.8691	+	2.69089i	−0.460491	+	0.0962875i
$782$		0			0
$783$	5.60406			0.200273
$784$		0			0
$785$	74.5641			2.66131
$786$		0			0
$787$	−5.23895			−0.186748			−0.0933741	−	0.995631i	$-0.529765\pi$
$787$	−5.23895			−0.186748			−0.0933741	+	0.995631i	$0.529765\pi$
$788$		0			0
$789$	11.0748			0.394274
$790$		0			0
$791$	20.1715	−	16.8779i	0.717214	−	0.600109i
$792$		0			0
$793$	14.1344			0.501926
$794$		0			0
$795$	−38.1015			−1.35132
$796$		0			0
$797$			37.0880i			1.31373i	0.754010	+	0.656863i	$0.228117\pi$
$797$			37.0880i			1.31373i	−0.754010	+	0.656863i	$0.771883\pi$
$798$		0			0
$799$			22.3468i			0.790572i
$800$		0			0
$801$			15.1124i			0.533972i
$802$		0			0
$803$	−48.0364	+	10.0443i	−1.69517	+	0.354456i
$804$		0			0
$805$	42.6474	+	50.9697i	1.50313	+	1.79644i
$806$		0			0
$807$	26.6973			0.939788
$808$		0			0
$809$			47.2739i			1.66206i	0.556225	+	0.831032i	$0.312249\pi$
$809$			47.2739i			1.66206i	−0.556225	+	0.831032i	$0.687751\pi$
$810$		0			0
$811$	−23.6775			−0.831430			−0.415715	−	0.909495i	$-0.636469\pi$
$811$	−23.6775			−0.831430			−0.415715	+	0.909495i	$0.636469\pi$
$812$		0			0
$813$		−	0.926916i		−	0.0325084i
$814$		0			0
$815$			7.16523i			0.250987i
$816$		0			0
$817$		−	49.8648i		−	1.74455i
$818$		0			0
$819$	−3.88936	+	3.25432i	−0.135905	+	0.113715i
$820$		0			0
$821$		−	0.603496i		−	0.0210622i	−0.999945	−	0.0105311i	$-0.996648\pi$
$821$		−	0.603496i		−	0.0210622i	0.999945	−	0.0105311i	$-0.00335221\pi$
$822$		0			0
$823$	8.53514			0.297516			0.148758	−	0.988874i	$-0.452472\pi$
$823$	8.53514			0.297516			0.148758	+	0.988874i	$0.452472\pi$
$824$		0			0
$825$	9.08080	+	43.4286i	0.316153	+	1.51199i
$826$		0			0
$827$			51.4796i			1.79012i	0.445945	+	0.895060i	$0.352868\pi$
$827$			51.4796i			1.79012i	−0.445945	+	0.895060i	$0.647132\pi$
$828$		0			0
$829$			46.0002i			1.59765i	0.601561	+	0.798827i	$0.294546\pi$
$829$			46.0002i			1.59765i	−0.601561	+	0.798827i	$0.705454\pi$
$830$		0			0
$831$	−7.03312			−0.243976
$832$		0			0
$833$	−4.79069	+	26.7325i	−0.165988	+	0.926225i
$834$		0			0
$835$			54.0937i			1.87199i
$836$		0			0
$837$	−26.7192			−0.923551
$838$		0			0
$839$		−	11.0223i		−	0.380533i	−0.981732	−	0.190267i	$-0.939065\pi$
$839$		−	11.0223i		−	0.380533i	0.981732	−	0.190267i	$-0.0609352\pi$
$840$		0			0
$841$	27.8007			0.958645
$842$		0			0
$843$	−21.5156			−0.741038
$844$		0			0
$845$		−	4.22530i		−	0.145355i
$846$		0			0
$847$	−12.9674	+	26.0547i	−0.445566	+	0.895249i
$848$		0			0
$849$		−	31.2981i		−	1.07415i
$850$		0			0
$851$	18.4018			0.630807
$852$		0			0
$853$	9.54541			0.326829			0.163414	−	0.986558i	$-0.447749\pi$
$853$	9.54541			0.326829			0.163414	+	0.986558i	$0.447749\pi$
$854$		0			0
$855$		−	58.6227i		−	2.00485i
$856$		0			0
$857$	−33.0441			−1.12876			−0.564382	−	0.825514i	$-0.690886\pi$
$857$	−33.0441			−1.12876			−0.564382	+	0.825514i	$0.690886\pi$
$858$		0			0
$859$		−	23.0986i		−	0.788116i	−0.919086	−	0.394058i	$-0.871071\pi$
$859$		−	23.0986i		−	0.788116i	0.919086	−	0.394058i	$-0.128929\pi$
$860$		0			0
$861$	4.47775	+	5.35154i	0.152601	+	0.182380i
$862$		0			0
$863$	−4.88912			−0.166427			−0.0832137	−	0.996532i	$-0.526518\pi$
$863$	−4.88912			−0.166427			−0.0832137	+	0.996532i	$0.526518\pi$
$864$		0			0
$865$			20.8174i			0.707814i
$866$		0			0
$867$			2.02687i			0.0688360i
$868$		0			0
$869$	10.3575	+	49.5344i	0.351355	+	1.68034i
$870$		0			0
$871$	2.12772			0.0720950
$872$		0			0
$873$			14.2573i			0.482536i
$874$		0			0
$875$	−56.3368	−	67.3304i	−1.90453	−	2.27618i
$876$		0			0
$877$		−	10.3368i		−	0.349049i	−0.984653	−	0.174524i	$-0.944161\pi$
$877$		−	10.3368i		−	0.349049i	0.984653	−	0.174524i	$-0.0558388\pi$
$878$		0			0
$879$		−	6.12076i		−	0.206448i
$880$		0			0
$881$		−	11.1044i		−	0.374117i	−0.982349	−	0.187058i	$-0.940105\pi$
$881$		−	11.1044i		−	0.374117i	0.982349	−	0.187058i	$-0.0598954\pi$
$882$		0			0
$883$	−52.6551			−1.77199			−0.885993	−	0.463700i	$-0.846522\pi$
$883$	−52.6551			−1.77199			−0.885993	+	0.463700i	$0.846522\pi$
$884$		0			0
$885$		−	29.6749i		−	0.997510i
$886$		0			0
$887$	14.8310			0.497978			0.248989	−	0.968506i	$-0.419902\pi$
$887$	14.8310			0.497978			0.248989	+	0.968506i	$0.419902\pi$
$888$		0			0
$889$	−2.03547	−	2.43267i	−0.0682673	−	0.0815890i
$890$		0			0
$891$	1.37724	−	0.287978i	0.0461393	−	0.00964761i
$892$		0			0
$893$		−	41.6918i		−	1.39516i
$894$		0			0
$895$			45.9117i			1.53466i
$896$		0			0
$897$		−	6.18737i		−	0.206590i
$898$		0			0
$899$	5.71799			0.190706
$900$		0			0
$901$	−33.6146			−1.11986
$902$		0			0
$903$	14.5488	−	12.1733i	0.484153	−	0.405102i
$904$		0			0
$905$	−29.5784			−0.983218
$906$		0			0
$907$	−0.0368692			−0.00122422			−0.000612111	−	1.00000i	$-0.500195\pi$
$907$	−0.0368692			−0.00122422			−0.000612111		1.00000i	$0.500195\pi$
$908$		0			0
$909$	10.0731			0.334105
$910$		0			0
$911$	−20.1397			−0.667259			−0.333629	−	0.942704i	$-0.608273\pi$
$911$	−20.1397			−0.667259			−0.333629	+	0.942704i	$0.608273\pi$
$912$		0			0
$913$	−9.88828	+	2.06761i	−0.327254	+	0.0684280i
$914$		0			0
$915$	−62.1579			−2.05488
$916$		0			0
$917$	33.0221	−	27.6303i	1.09049	−	0.912434i
$918$		0			0
$919$			35.9160i			1.18476i	0.805659	+	0.592379i	$0.201811\pi$
$919$			35.9160i			1.18476i	−0.805659	+	0.592379i	$0.798189\pi$
$920$		0			0
$921$			30.2107i			0.995477i
$922$		0			0
$923$	−3.96408			−0.130479
$924$		0			0
$925$	−39.7857			−1.30815
$926$		0			0
$927$		−	33.9071i		−	1.11366i
$928$		0			0
$929$		−	50.3872i		−	1.65315i	−0.562825	−	0.826576i	$-0.690286\pi$
$929$		−	50.3872i		−	1.65315i	0.562825	−	0.826576i	$-0.309714\pi$
$930$		0			0
$931$	8.93787	−	49.8741i	0.292927	−	1.63456i
$932$		0			0
$933$	29.8844			0.978370
$934$		0			0
$935$	11.1279	+	53.2190i	0.363923	+	1.74045i
$936$		0			0
$937$	10.5273			0.343910			0.171955	−	0.985105i	$-0.444992\pi$
$937$	10.5273			0.343910			0.171955	+	0.985105i	$0.444992\pi$
$938$		0			0
$939$	13.2046			0.430915
$940$		0			0
$941$	14.5366			0.473880			0.236940	−	0.971524i	$-0.423856\pi$
$941$	14.5366			0.473880			0.236940	+	0.971524i	$0.423856\pi$
$942$		0			0
$943$	15.0643			0.490560
$944$		0			0
$945$	43.8743	−	36.7106i	1.42723	−	1.19420i
$946$		0			0
$947$	33.0726			1.07472			0.537358	−	0.843354i	$-0.319423\pi$
$947$	33.0726			1.07472			0.537358	+	0.843354i	$0.319423\pi$
$948$		0			0
$949$	−14.7968			−0.480323
$950$		0			0
$951$			4.71189i			0.152793i
$952$		0			0
$953$		−	13.0937i		−	0.424146i	−0.977254	−	0.212073i	$-0.931979\pi$
$953$		−	13.0937i		−	0.424146i	0.977254	−	0.212073i	$-0.0680214\pi$
$954$		0			0
$955$			109.724i			3.55059i
$956$		0			0
$957$	3.70022	−	0.773707i	0.119611	−	0.0250104i
$958$		0			0
$959$	7.42632	−	6.21377i	0.239808	−	0.200653i
$960$		0			0
$961$	3.73762			0.120568
$962$		0			0
$963$		−	19.5484i		−	0.629940i
$964$		0			0
$965$	−37.1804			−1.19688
$966$		0			0
$967$			34.2317i			1.10082i	0.834896	+	0.550408i	$0.185528\pi$
$967$			34.2317i			1.10082i	−0.834896	+	0.550408i	$0.814472\pi$
$968$		0			0
$969$			29.2287i			0.938960i
$970$		0			0
$971$		−	6.96437i		−	0.223497i	−0.993737	−	0.111749i	$-0.964355\pi$
$971$		−	6.96437i		−	0.223497i	0.993737	−	0.111749i	$-0.0356451\pi$
$972$		0			0
$973$	−24.6675	+	20.6399i	−0.790805	+	0.661684i
$974$		0			0
$975$			13.3774i			0.428420i
$976$		0			0
$977$	57.1731			1.82913			0.914565	−	0.404438i	$-0.132533\pi$
$977$	57.1731			1.82913			0.914565	+	0.404438i	$0.132533\pi$
$978$		0			0
$979$	5.35205	+	25.5960i	0.171052	+	0.818050i
$980$		0			0
$981$		−	21.0529i		−	0.672167i
$982$		0			0
$983$		−	28.7679i		−	0.917555i	−0.888551	−	0.458777i	$-0.848288\pi$
$983$		−	28.7679i		−	0.917555i	0.888551	−	0.458777i	$-0.151712\pi$
$984$		0			0
$985$	9.77876			0.311577
$986$		0			0
$987$	12.1642	−	10.1781i	0.387191	−	0.323971i
$988$		0			0
$989$		−	40.9540i		−	1.30226i
$990$		0			0
$991$	58.3878			1.85475			0.927374	−	0.374135i	$-0.122060\pi$
$991$	58.3878			1.85475			0.927374	+	0.374135i	$0.122060\pi$
$992$		0			0
$993$			2.01648i			0.0639911i
$994$		0			0
$995$	32.3077			1.02422
$996$		0			0
$997$	41.1774			1.30410			0.652051	−	0.758175i	$-0.273909\pi$
$997$	41.1774			1.30410			0.652051	+	0.758175i	$0.273909\pi$
$998$		0			0
$999$		−	15.8402i		−	0.501161i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.a.3849.16	✓	48
7.6	odd	2		4004.2.e.b.3849.33	yes	48
11.10	odd	2		4004.2.e.b.3849.16	yes	48
77.76	even	2	inner	4004.2.e.a.3849.33	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.16	✓	48	1.1	even	1	trivial
4004.2.e.a.3849.33	yes	48	77.76	even	2	inner
4004.2.e.b.3849.16	yes	48	11.10	odd	2
4004.2.e.b.3849.33	yes	48	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.04079i q^{3} -4.22530i q^{5} +(-2.02914 + 1.69782i) q^{7} +1.91676 q^{9} +O(q^{10})\)	\(q-1.04079i q^{3} -4.22530i q^{5} +(-2.02914 + 1.69782i) q^{7} +1.91676 q^{9} +(3.24641 - 0.678817i) q^{11} +1.00000 q^{13} -4.39764 q^{15} -3.87976 q^{17} +7.23838 q^{19} +(1.76708 + 2.11190i) q^{21} +5.94488 q^{23} -12.8531 q^{25} -5.11731i q^{27} +1.09512i q^{29} -5.22134i q^{31} +(-0.706505 - 3.37883i) q^{33} +(7.17381 + 8.57370i) q^{35} +3.09541 q^{37} -1.04079i q^{39} +2.53399 q^{41} -6.88894i q^{43} -8.09887i q^{45} -5.75983i q^{47} +(1.23479 - 6.89023i) q^{49} +4.03802i q^{51} +8.66407 q^{53} +(-2.86820 - 13.7171i) q^{55} -7.53362i q^{57} +6.74791i q^{59} +14.1344 q^{61} +(-3.88936 + 3.25432i) q^{63} -4.22530i q^{65} +2.12772 q^{67} -6.18737i q^{69} -3.96408 q^{71} -14.7968 q^{73} +13.3774i q^{75} +(-5.43491 + 6.88925i) q^{77} +15.2582i q^{79} +0.424235 q^{81} -3.04591 q^{83} +16.3932i q^{85} +1.13979 q^{87} +7.88438i q^{89} +(-2.02914 + 1.69782i) q^{91} -5.43431 q^{93} -30.5843i q^{95} +7.43823i q^{97} +(6.22259 - 1.30113i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{7} - 48 q^{9}+O(q^{10}) \) 48 * q - 4 * q^7 - 48 * q^9	\( 48 q - 4 q^{7} - 48 q^{9} + 2 q^{11} + 48 q^{13} + 8 q^{15} + 4 q^{17} + 10 q^{21} + 4 q^{23} - 44 q^{25} + 10 q^{33} - 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} - 12 q^{55} + 16 q^{61} + 16 q^{63} + 4 q^{67} - 16 q^{73} + 2 q^{77} + 64 q^{81} - 4 q^{83} - 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) \) 48 * q - 4 * q^7 - 48 * q^9 + 2 * q^11 + 48 * q^13 + 8 * q^15 + 4 * q^17 + 10 * q^21 + 4 * q^23 - 44 * q^25 + 10 * q^33 - 14 * q^35 - 16 * q^37 + 10 * q^49 - 8 * q^53 - 12 * q^55 + 16 * q^61 + 16 * q^63 + 4 * q^67 - 16 * q^73 + 2 * q^77 + 64 * q^81 - 4 * q^83 - 12 * q^87 - 4 * q^91 + 36 * q^93 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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