LMFDB - Embedded newform 4004.2.e.b.3849.12

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			3849.12
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.b.3849.37

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77770i q^{3} +3.16364i q^{5} +(-0.435138 - 2.60972i) q^{7} -0.160220 q^{9} +O(q^{10})$	$q-1.77770i q^{3} +3.16364i q^{5} +(-0.435138 - 2.60972i) q^{7} -0.160220 q^{9} +(-1.86786 + 2.74064i) q^{11} -1.00000 q^{13} +5.62400 q^{15} -2.82452 q^{17} -2.08776 q^{19} +(-4.63931 + 0.773546i) q^{21} +7.03278 q^{23} -5.00860 q^{25} -5.04828i q^{27} -8.40307i q^{29} +7.59748i q^{31} +(4.87203 + 3.32050i) q^{33} +(8.25622 - 1.37662i) q^{35} -6.03705 q^{37} +1.77770i q^{39} -1.64149 q^{41} +8.16795i q^{43} -0.506878i q^{45} -8.73531i q^{47} +(-6.62131 + 2.27118i) q^{49} +5.02116i q^{51} +0.279662 q^{53} +(-8.67038 - 5.90924i) q^{55} +3.71142i q^{57} +4.51485i q^{59} -13.5553 q^{61} +(0.0697178 + 0.418130i) q^{63} -3.16364i q^{65} +5.22397 q^{67} -12.5022i q^{69} -15.8883 q^{71} +5.00046 q^{73} +8.90379i q^{75} +(7.96508 + 3.68205i) q^{77} -5.00022i q^{79} -9.45499 q^{81} -2.92898 q^{83} -8.93577i q^{85} -14.9381 q^{87} +10.7229i q^{89} +(0.435138 + 2.60972i) q^{91} +13.5060 q^{93} -6.60492i q^{95} -12.9209i q^{97} +(0.299269 - 0.439104i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 22 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 + 22 * 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
$3$		−	1.77770i		−	1.02636i	−0.858282	−	0.513178i	$-0.828468\pi$
$3$		−	1.77770i		−	1.02636i	0.858282	−	0.513178i	$-0.171532\pi$
$4$		0			0
$5$			3.16364i			1.41482i	0.706803	+	0.707411i	$0.250137\pi$
$5$			3.16364i			1.41482i	−0.706803	+	0.707411i	$0.749863\pi$
$6$		0			0
$7$	−0.435138	−	2.60972i	−0.164467	−	0.986383i
$7$	−0.435138	−	2.60972i	−0.164467	−	0.986383i
$8$		0			0
$9$	−0.160220			−0.0534066
$10$		0			0
$11$	−1.86786	+	2.74064i	−0.563182	+	0.826333i
$11$	−1.86786	+	2.74064i	−0.563182	+	0.826333i
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	5.62400			1.45211
$16$		0			0
$17$	−2.82452			−0.685048			−0.342524	−	0.939509i	$-0.611282\pi$
$17$	−2.82452			−0.685048			−0.342524	+	0.939509i	$0.611282\pi$
$18$		0			0
$19$	−2.08776			−0.478965			−0.239483	−	0.970901i	$-0.576978\pi$
$19$	−2.08776			−0.478965			−0.239483	+	0.970901i	$0.576978\pi$
$20$		0			0
$21$	−4.63931	+	0.773546i	−1.01238	+	0.168802i
$22$		0			0
$23$	7.03278			1.46644			0.733218	−	0.679994i	$-0.238018\pi$
$23$	7.03278			1.46644			0.733218	+	0.679994i	$0.238018\pi$
$24$		0			0
$25$	−5.00860			−1.00172
$26$		0			0
$27$		−	5.04828i		−	0.971542i
$28$		0			0
$29$		−	8.40307i		−	1.56041i	−0.625523	−	0.780206i	$-0.715114\pi$
$29$		−	8.40307i		−	1.56041i	0.625523	−	0.780206i	$-0.284886\pi$
$30$		0			0
$31$			7.59748i			1.36455i	0.731097	+	0.682274i	$0.239009\pi$
$31$			7.59748i			1.36455i	−0.731097	+	0.682274i	$0.760991\pi$
$32$		0			0
$33$	4.87203	+	3.32050i	0.848112	+	0.578025i
$34$		0			0
$35$	8.25622	−	1.37662i	1.39556	−	0.232691i
$36$		0			0
$37$	−6.03705			−0.992484			−0.496242	−	0.868184i	$-0.665287\pi$
$37$	−6.03705			−0.992484			−0.496242	+	0.868184i	$0.665287\pi$
$38$		0			0
$39$			1.77770i			0.284660i
$40$		0			0
$41$	−1.64149			−0.256358			−0.128179	−	0.991751i	$-0.540913\pi$
$41$	−1.64149			−0.256358			−0.128179	+	0.991751i	$0.540913\pi$
$42$		0			0
$43$			8.16795i			1.24560i	0.782381	+	0.622800i	$0.214005\pi$
$43$			8.16795i			1.24560i	−0.782381	+	0.622800i	$0.785995\pi$
$44$		0			0
$45$		−	0.506878i		−	0.0755608i
$46$		0			0
$47$		−	8.73531i		−	1.27418i	−0.770791	−	0.637088i	$-0.780139\pi$
$47$		−	8.73531i		−	1.27418i	0.770791	−	0.637088i	$-0.219861\pi$
$48$		0			0
$49$	−6.62131	+	2.27118i	−0.945901	+	0.324454i
$50$		0			0
$51$			5.02116i			0.703103i
$52$		0			0
$53$	0.279662			0.0384146			0.0192073	−	0.999816i	$-0.493886\pi$
$53$	0.279662			0.0384146			0.0192073	+	0.999816i	$0.493886\pi$
$54$		0			0
$55$	−8.67038	−	5.90924i	−1.16911	−	0.796802i
$56$		0			0
$57$			3.71142i			0.491589i
$58$		0			0
$59$			4.51485i			0.587783i	0.955839	+	0.293891i	$0.0949504\pi$
$59$			4.51485i			0.587783i	−0.955839	+	0.293891i	$0.905050\pi$
$60$		0			0
$61$	−13.5553			−1.73558			−0.867788	−	0.496935i	$-0.834459\pi$
$61$	−13.5553			−1.73558			−0.867788	+	0.496935i	$0.834459\pi$
$62$		0			0
$63$	0.0697178	+	0.418130i	0.00878362	+	0.0526794i
$64$		0			0
$65$		−	3.16364i		−	0.392401i
$66$		0			0
$67$	5.22397			0.638209			0.319105	−	0.947719i	$-0.396618\pi$
$67$	5.22397			0.638209			0.319105	+	0.947719i	$0.396618\pi$
$68$		0			0
$69$		−	12.5022i		−	1.50509i
$70$		0			0
$71$	−15.8883			−1.88559			−0.942795	−	0.333372i	$-0.891813\pi$
$71$	−15.8883			−1.88559			−0.942795	+	0.333372i	$0.891813\pi$
$72$		0			0
$73$	5.00046			0.585260			0.292630	−	0.956226i	$-0.405470\pi$
$73$	5.00046			0.585260			0.292630	+	0.956226i	$0.405470\pi$
$74$		0			0
$75$			8.90379i			1.02812i
$76$		0			0
$77$	7.96508	+	3.68205i	0.907705	+	0.419608i
$78$		0			0
$79$		−	5.00022i		−	0.562569i	−0.959624	−	0.281285i	$-0.909240\pi$
$79$		−	5.00022i		−	0.562569i	0.959624	−	0.281285i	$-0.0907605\pi$
$80$		0			0
$81$	−9.45499			−1.05055
$82$		0			0
$83$	−2.92898			−0.321498			−0.160749	−	0.986995i	$-0.551391\pi$
$83$	−2.92898			−0.321498			−0.160749	+	0.986995i	$0.551391\pi$
$84$		0			0
$85$		−	8.93577i		−	0.969220i
$86$		0			0
$87$	−14.9381			−1.60154
$88$		0			0
$89$			10.7229i			1.13663i	0.822812	+	0.568314i	$0.192404\pi$
$89$			10.7229i			1.13663i	−0.822812	+	0.568314i	$0.807596\pi$
$90$		0			0
$91$	0.435138	+	2.60972i	0.0456149	+	0.273573i
$92$		0			0
$93$	13.5060			1.40051
$94$		0			0
$95$		−	6.60492i		−	0.677650i
$96$		0			0
$97$		−	12.9209i		−	1.31192i	−0.754795	−	0.655960i	$-0.772264\pi$
$97$		−	12.9209i		−	1.31192i	0.754795	−	0.655960i	$-0.227736\pi$
$98$		0			0
$99$	0.299269	−	0.439104i	0.0300776	−	0.0441317i
$100$		0			0
$101$	−7.83904			−0.780014			−0.390007	−	0.920812i	$-0.627527\pi$
$101$	−7.83904			−0.780014			−0.390007	+	0.920812i	$0.627527\pi$
$102$		0			0
$103$			15.9810i			1.57466i	0.616535	+	0.787328i	$0.288536\pi$
$103$			15.9810i			1.57466i	−0.616535	+	0.787328i	$0.711464\pi$
$104$		0			0
$105$	−2.44722	−	14.6771i	−0.238824	−	1.43234i
$106$		0			0
$107$		−	9.09349i		−	0.879100i	−0.898218	−	0.439550i	$-0.855138\pi$
$107$		−	9.09349i		−	0.879100i	0.898218	−	0.439550i	$-0.144862\pi$
$108$		0			0
$109$		−	12.6269i		−	1.20944i	−0.796439	−	0.604719i	$-0.793286\pi$
$109$		−	12.6269i		−	1.20944i	0.796439	−	0.604719i	$-0.206714\pi$
$110$		0			0
$111$			10.7321i			1.01864i
$112$		0			0
$113$	1.27300			0.119754			0.0598771	−	0.998206i	$-0.480929\pi$
$113$	1.27300			0.119754			0.0598771	+	0.998206i	$0.480929\pi$
$114$		0			0
$115$			22.2492i			2.07474i
$116$		0			0
$117$	0.160220			0.0148123
$118$		0			0
$119$	1.22906	+	7.37123i	0.112668	+	0.675719i
$120$		0			0
$121$	−4.02218	−	10.2383i	−0.365653	−	0.930751i
$122$		0			0
$123$			2.91808i			0.263114i
$124$		0			0
$125$		−	0.0272027i		−	0.00243308i
$126$		0			0
$127$		−	1.63222i		−	0.144836i	−0.997374	−	0.0724181i	$-0.976928\pi$
$127$		−	1.63222i		−	0.144836i	0.997374	−	0.0724181i	$-0.0230716\pi$
$128$		0			0
$129$	14.5202			1.27843
$130$		0			0
$131$	−17.2454			−1.50674			−0.753369	−	0.657598i	$-0.771573\pi$
$131$	−17.2454			−1.50674			−0.753369	+	0.657598i	$0.771573\pi$
$132$		0			0
$133$	0.908465	+	5.44848i	0.0787739	+	0.472443i
$134$		0			0
$135$	15.9709			1.37456
$136$		0			0
$137$	−16.0856			−1.37429			−0.687144	−	0.726521i	$-0.741136\pi$
$137$	−16.0856			−1.37429			−0.687144	+	0.726521i	$0.741136\pi$
$138$		0			0
$139$	−16.8865			−1.43229			−0.716147	−	0.697949i	$-0.754096\pi$
$139$	−16.8865			−1.43229			−0.716147	+	0.697949i	$0.754096\pi$
$140$		0			0
$141$	−15.5288			−1.30776
$142$		0			0
$143$	1.86786	−	2.74064i	0.156199	−	0.229184i
$144$		0			0
$145$	26.5843			2.20770
$146$		0			0
$147$	4.03748	+	11.7707i	0.333006	+	0.970831i
$148$		0			0
$149$		−	9.36375i		−	0.767108i	−0.923518	−	0.383554i	$-0.874700\pi$
$149$		−	9.36375i		−	0.767108i	0.923518	−	0.383554i	$-0.125300\pi$
$150$		0			0
$151$			20.5486i			1.67222i	0.548560	+	0.836111i	$0.315176\pi$
$151$			20.5486i			1.67222i	−0.548560	+	0.836111i	$0.684824\pi$
$152$		0			0
$153$	0.452545			0.0365861
$154$		0			0
$155$	−24.0357			−1.93059
$156$		0			0
$157$			12.4372i			0.992596i	0.868152	+	0.496298i	$0.165308\pi$
$157$			12.4372i			0.992596i	−0.868152	+	0.496298i	$0.834692\pi$
$158$		0			0
$159$		−	0.497156i		−	0.0394270i
$160$		0			0
$161$	−3.06023	−	18.3536i	−0.241180	−	1.44647i
$162$		0			0
$163$	0.746974			0.0585075			0.0292538	−	0.999572i	$-0.490687\pi$
$163$	0.746974			0.0585075			0.0292538	+	0.999572i	$0.490687\pi$
$164$		0			0
$165$	−10.5049	+	15.4133i	−0.817802	+	1.19993i
$166$		0			0
$167$	1.35353			0.104740			0.0523698	−	0.998628i	$-0.483323\pi$
$167$	1.35353			0.104740			0.0523698	+	0.998628i	$0.483323\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	0.334501			0.0255799
$172$		0			0
$173$	−22.3315			−1.69783			−0.848917	−	0.528526i	$-0.822745\pi$
$173$	−22.3315			−1.69783			−0.848917	+	0.528526i	$0.822745\pi$
$174$		0			0
$175$	2.17943	+	13.0711i	0.164750	+	0.988079i
$176$		0			0
$177$	8.02605			0.603275
$178$		0			0
$179$	−10.5187			−0.786205			−0.393103	−	0.919495i	$-0.628598\pi$
$179$	−10.5187			−0.786205			−0.393103	+	0.919495i	$0.628598\pi$
$180$		0			0
$181$			0.990863i			0.0736502i	0.999322	+	0.0368251i	$0.0117245\pi$
$181$			0.990863i			0.0736502i	−0.999322	+	0.0368251i	$0.988276\pi$
$182$		0			0
$183$			24.0972i			1.78132i
$184$		0			0
$185$		−	19.0990i		−	1.40419i
$186$		0			0
$187$	5.27582	−	7.74100i	0.385806	−	0.566078i
$188$		0			0
$189$	−13.1746	+	2.19670i	−0.958312	+	0.159786i
$190$		0			0
$191$	22.5588			1.63230			0.816148	−	0.577843i	$-0.196106\pi$
$191$	22.5588			1.63230			0.816148	+	0.577843i	$0.196106\pi$
$192$		0			0
$193$			1.52662i			0.109889i	0.998489	+	0.0549444i	$0.0174981\pi$
$193$			1.52662i			0.109889i	−0.998489	+	0.0549444i	$0.982502\pi$
$194$		0			0
$195$	−5.62400			−0.402743
$196$		0			0
$197$		−	6.93875i		−	0.494365i	−0.968969	−	0.247183i	$-0.920495\pi$
$197$		−	6.93875i		−	0.494365i	0.968969	−	0.247183i	$-0.0795048\pi$
$198$		0			0
$199$		−	10.9781i		−	0.778216i	−0.921192	−	0.389108i	$-0.872783\pi$
$199$		−	10.9781i		−	0.778216i	0.921192	−	0.389108i	$-0.127217\pi$
$200$		0			0
$201$		−	9.28665i		−	0.655030i
$202$		0			0
$203$	−21.9297	+	3.65650i	−1.53916	+	0.256636i
$204$		0			0
$205$		−	5.19308i		−	0.362701i
$206$		0			0
$207$	−1.12679			−0.0783174
$208$		0			0
$209$	3.89965	−	5.72180i	0.269745	−	0.395785i
$210$		0			0
$211$		−	12.4233i		−	0.855255i	−0.903955	−	0.427627i	$-0.859350\pi$
$211$		−	12.4233i		−	0.855255i	0.903955	−	0.427627i	$-0.140650\pi$
$212$		0			0
$213$			28.2446i			1.93529i
$214$		0			0
$215$	−25.8404			−1.76230
$216$		0			0
$217$	19.8273	−	3.30595i	1.34597	−	0.224423i
$218$		0			0
$219$		−	8.88932i		−	0.600685i
$220$		0			0
$221$	2.82452			0.189998
$222$		0			0
$223$			17.3722i			1.16333i	0.813428	+	0.581665i	$0.197599\pi$
$223$			17.3722i			1.16333i	−0.813428	+	0.581665i	$0.802401\pi$
$224$		0			0
$225$	0.802477			0.0534985
$226$		0			0
$227$	−16.7339			−1.11067			−0.555333	−	0.831628i	$-0.687409\pi$
$227$	−16.7339			−1.11067			−0.555333	+	0.831628i	$0.687409\pi$
$228$		0			0
$229$			8.37897i			0.553698i	0.960913	+	0.276849i	$0.0892902\pi$
$229$			8.37897i			0.553698i	−0.960913	+	0.276849i	$0.910710\pi$
$230$		0			0
$231$	6.54558	−	14.1595i	0.430667	−	0.931629i
$232$		0			0
$233$		−	25.8688i		−	1.69472i	−0.531019	−	0.847360i	$-0.678191\pi$
$233$		−	25.8688i		−	1.69472i	0.531019	−	0.847360i	$-0.321809\pi$
$234$		0			0
$235$	27.6354			1.80273
$236$		0			0
$237$	−8.88890			−0.577396
$238$		0			0
$239$			18.6350i			1.20539i	0.797970	+	0.602697i	$0.205907\pi$
$239$			18.6350i			1.20539i	−0.797970	+	0.602697i	$0.794093\pi$
$240$		0			0
$241$	24.5237			1.57971			0.789855	−	0.613294i	$-0.210156\pi$
$241$	24.5237			1.57971			0.789855	+	0.613294i	$0.210156\pi$
$242$		0			0
$243$			1.66330i			0.106701i
$244$		0			0
$245$	−7.18519	−	20.9474i	−0.459045	−	1.33828i
$246$		0			0
$247$	2.08776			0.132841
$248$		0			0
$249$			5.20686i			0.329971i
$250$		0			0
$251$			11.9735i			0.755761i	0.925854	+	0.377881i	$0.123347\pi$
$251$			11.9735i			0.755761i	−0.925854	+	0.377881i	$0.876653\pi$
$252$		0			0
$253$	−13.1363	+	19.2743i	−0.825870	+	1.21176i
$254$		0			0
$255$	−15.8851			−0.994765
$256$		0			0
$257$		−	4.55259i		−	0.283983i	−0.989868	−	0.141991i	$-0.954649\pi$
$257$		−	4.55259i		−	0.283983i	0.989868	−	0.141991i	$-0.0453505\pi$
$258$		0			0
$259$	2.62695	+	15.7550i	0.163231	+	0.978969i
$260$		0			0
$261$			1.34634i			0.0833363i
$262$		0			0
$263$		−	25.4594i		−	1.56989i	−0.619564	−	0.784946i	$-0.712691\pi$
$263$		−	25.4594i		−	1.56989i	0.619564	−	0.784946i	$-0.287309\pi$
$264$		0			0
$265$			0.884750i			0.0543497i
$266$		0			0
$267$	19.0621			1.16658
$268$		0			0
$269$		−	10.8883i		−	0.663872i	−0.943302	−	0.331936i	$-0.892298\pi$
$269$		−	10.8883i		−	0.663872i	0.943302	−	0.331936i	$-0.107702\pi$
$270$		0			0
$271$	7.20612			0.437740			0.218870	−	0.975754i	$-0.429763\pi$
$271$	7.20612			0.437740			0.218870	+	0.975754i	$0.429763\pi$
$272$		0			0
$273$	4.63931	−	0.773546i	0.280784	−	0.0468171i
$274$		0			0
$275$	9.35537	−	13.7267i	0.564150	−	0.827754i
$276$		0			0
$277$			17.8986i			1.07542i	0.843130	+	0.537710i	$0.180710\pi$
$277$			17.8986i			1.07542i	−0.843130	+	0.537710i	$0.819290\pi$
$278$		0			0
$279$		−	1.21727i		−	0.0728759i
$280$		0			0
$281$			7.81897i			0.466441i	0.972424	+	0.233220i	$0.0749264\pi$
$281$			7.81897i			0.466441i	−0.972424	+	0.233220i	$0.925074\pi$
$282$		0			0
$283$	−4.82471			−0.286800			−0.143400	−	0.989665i	$-0.545803\pi$
$283$	−4.82471			−0.286800			−0.143400	+	0.989665i	$0.545803\pi$
$284$		0			0
$285$	−11.7416			−0.695511
$286$		0			0
$287$	0.714276	+	4.28384i	0.0421624	+	0.252867i
$288$		0			0
$289$	−9.02206			−0.530709
$290$		0			0
$291$	−22.9695			−1.34650
$292$		0			0
$293$	−9.78277			−0.571516			−0.285758	−	0.958302i	$-0.592245\pi$
$293$	−9.78277			−0.571516			−0.285758	+	0.958302i	$0.592245\pi$
$294$		0			0
$295$	−14.2833			−0.831608
$296$		0			0
$297$	13.8355	+	9.42949i	0.802817	+	0.547155i
$298$		0			0
$299$	−7.03278			−0.406716
$300$		0			0
$301$	21.3161	−	3.55419i	1.22864	−	0.204860i
$302$		0			0
$303$			13.9355i			0.800572i
$304$		0			0
$305$		−	42.8840i		−	2.45553i
$306$		0			0
$307$	−4.91927			−0.280758			−0.140379	−	0.990098i	$-0.544832\pi$
$307$	−4.91927			−0.280758			−0.140379	+	0.990098i	$0.544832\pi$
$308$		0			0
$309$	28.4094			1.61616
$310$		0			0
$311$			25.5920i			1.45119i	0.688123	+	0.725594i	$0.258435\pi$
$311$			25.5920i			1.45119i	−0.688123	+	0.725594i	$0.741565\pi$
$312$		0			0
$313$			4.07887i			0.230551i	0.993334	+	0.115276i	$0.0367751\pi$
$313$			4.07887i			0.230551i	−0.993334	+	0.115276i	$0.963225\pi$
$314$		0			0
$315$	−1.32281	+	0.220562i	−0.0745319	+	0.0124273i
$316$		0			0
$317$	−14.6993			−0.825594			−0.412797	−	0.910823i	$-0.635448\pi$
$317$	−14.6993			−0.825594			−0.412797	+	0.910823i	$0.635448\pi$
$318$		0			0
$319$	23.0298	+	15.6958i	1.28942	+	0.878795i
$320$		0			0
$321$	−16.1655			−0.902270
$322$		0			0
$323$	5.89693			0.328114
$324$		0			0
$325$	5.00860			0.277827
$326$		0			0
$327$	−22.4468			−1.24131
$328$		0			0
$329$	−22.7967	+	3.80107i	−1.25683	+	0.209560i
$330$		0			0
$331$	27.1713			1.49347			0.746735	−	0.665122i	$-0.231620\pi$
$331$	27.1713			1.49347			0.746735	+	0.665122i	$0.231620\pi$
$332$		0			0
$333$	0.967255			0.0530052
$334$		0			0
$335$			16.5267i			0.902952i
$336$		0			0
$337$			15.3836i			0.837997i	0.907987	+	0.418999i	$0.137619\pi$
$337$			15.3836i			0.837997i	−0.907987	+	0.418999i	$0.862381\pi$
$338$		0			0
$339$		−	2.26302i		−	0.122910i
$340$		0			0
$341$	−20.8219	−	14.1910i	−1.12757	−	0.768488i
$342$		0			0
$343$	8.80834	+	16.2915i	0.475606	+	0.879659i
$344$		0			0
$345$	39.5523			2.12943
$346$		0			0
$347$			23.3795i			1.25508i	0.778585	+	0.627539i	$0.215938\pi$
$347$			23.3795i			1.25508i	−0.778585	+	0.627539i	$0.784062\pi$
$348$		0			0
$349$	10.8422			0.580370			0.290185	−	0.956971i	$-0.406283\pi$
$349$	10.8422			0.580370			0.290185	+	0.956971i	$0.406283\pi$
$350$		0			0
$351$			5.04828i			0.269457i
$352$		0			0
$353$			19.5308i			1.03952i	0.854312	+	0.519761i	$0.173979\pi$
$353$			19.5308i			1.03952i	−0.854312	+	0.519761i	$0.826021\pi$
$354$		0			0
$355$		−	50.2647i		−	2.66777i
$356$		0			0
$357$	13.1038	−	2.18490i	0.693529	−	0.115637i
$358$		0			0
$359$			24.3761i			1.28652i	0.765648	+	0.643260i	$0.222419\pi$
$359$			24.3761i			1.28652i	−0.765648	+	0.643260i	$0.777581\pi$
$360$		0			0
$361$	−14.6413			−0.770592
$362$		0			0
$363$	−18.2006	+	7.15023i	−0.955282	+	0.375290i
$364$		0			0
$365$			15.8196i			0.828038i
$366$		0			0
$367$		−	24.2926i		−	1.26806i	−0.773306	−	0.634032i	$-0.781399\pi$
$367$		−	24.2926i		−	1.26806i	0.773306	−	0.634032i	$-0.218601\pi$
$368$		0			0
$369$	0.262999			0.0136912
$370$		0			0
$371$	−0.121692	−	0.729841i	−0.00631792	−	0.0378915i
$372$		0			0
$373$			4.86824i			0.252068i	0.992026	+	0.126034i	$0.0402249\pi$
$373$			4.86824i			0.252068i	−0.992026	+	0.126034i	$0.959775\pi$
$374$		0			0
$375$	−0.0483583			−0.00249721
$376$		0			0
$377$			8.40307i			0.432780i
$378$		0			0
$379$	−31.9883			−1.64313			−0.821563	−	0.570117i	$-0.806898\pi$
$379$	−31.9883			−1.64313			−0.821563	+	0.570117i	$0.806898\pi$
$380$		0			0
$381$	−2.90160			−0.148654
$382$		0			0
$383$			12.4345i			0.635374i	0.948196	+	0.317687i	$0.102906\pi$
$383$			12.4345i			0.635374i	−0.948196	+	0.317687i	$0.897094\pi$
$384$		0			0
$385$	−11.6487	+	25.1986i	−0.593671	+	1.28424i
$386$		0			0
$387$		−	1.30867i		−	0.0665233i
$388$		0			0
$389$	21.9605			1.11344			0.556720	−	0.830700i	$-0.312059\pi$
$389$	21.9605			1.11344			0.556720	+	0.830700i	$0.312059\pi$
$390$		0			0
$391$	−19.8643			−1.00458
$392$		0			0
$393$			30.6572i			1.54645i
$394$		0			0
$395$	15.8189			0.795935
$396$		0			0
$397$		−	27.5853i		−	1.38447i	−0.721674	−	0.692234i	$-0.756627\pi$
$397$		−	27.5853i		−	1.38447i	0.721674	−	0.692234i	$-0.243373\pi$
$398$		0			0
$399$	9.68577	−	1.61498i	0.484895	−	0.0808501i
$400$		0			0
$401$	−20.5578			−1.02661			−0.513305	−	0.858207i	$-0.671579\pi$
$401$	−20.5578			−1.02661			−0.513305	+	0.858207i	$0.671579\pi$
$402$		0			0
$403$		−	7.59748i		−	0.378457i
$404$		0			0
$405$		−	29.9122i		−	1.48635i
$406$		0			0
$407$	11.2764	−	16.5453i	0.558949	−	0.820122i
$408$		0			0
$409$	−18.7556			−0.927406			−0.463703	−	0.885991i	$-0.653479\pi$
$409$	−18.7556			−0.927406			−0.463703	+	0.885991i	$0.653479\pi$
$410$		0			0
$411$			28.5954i			1.41051i
$412$		0			0
$413$	11.7825	−	1.96458i	0.579779	−	0.0966708i
$414$		0			0
$415$		−	9.26624i		−	0.454862i
$416$		0			0
$417$			30.0191i			1.47004i
$418$		0			0
$419$		−	25.1697i		−	1.22962i	−0.788676	−	0.614810i	$-0.789233\pi$
$419$		−	25.1697i		−	1.22962i	0.788676	−	0.614810i	$-0.210767\pi$
$420$		0			0
$421$	−20.8432			−1.01584			−0.507918	−	0.861406i	$-0.669585\pi$
$421$	−20.8432			−1.01584			−0.507918	+	0.861406i	$0.669585\pi$
$422$		0			0
$423$			1.39957i			0.0680495i
$424$		0			0
$425$	14.1469			0.686226
$426$		0			0
$427$	5.89842	+	35.3755i	0.285445	+	1.71194i
$428$		0			0
$429$	−4.87203	−	3.32050i	−0.235224	−	0.160315i
$430$		0			0
$431$			5.87599i			0.283036i	0.989936	+	0.141518i	$0.0451984\pi$
$431$			5.87599i			0.283036i	−0.989936	+	0.141518i	$0.954802\pi$
$432$		0			0
$433$		−	13.0075i		−	0.625098i	−0.949902	−	0.312549i	$-0.898817\pi$
$433$		−	13.0075i		−	0.625098i	0.949902	−	0.312549i	$-0.101183\pi$
$434$		0			0
$435$		−	47.2589i		−	2.26589i
$436$		0			0
$437$	−14.6828			−0.702372
$438$		0			0
$439$	15.9417			0.760856			0.380428	−	0.924810i	$-0.375777\pi$
$439$	15.9417			0.760856			0.380428	+	0.924810i	$0.375777\pi$
$440$		0			0
$441$	1.06087	−	0.363888i	0.0505174	−	0.0173280i
$442$		0			0
$443$	19.7187			0.936865			0.468432	−	0.883499i	$-0.344819\pi$
$443$	19.7187			0.936865			0.468432	+	0.883499i	$0.344819\pi$
$444$		0			0
$445$	−33.9234			−1.60812
$446$		0			0
$447$	−16.6459			−0.787326
$448$		0			0
$449$	−10.4391			−0.492653			−0.246327	−	0.969187i	$-0.579224\pi$
$449$	−10.4391			−0.492653			−0.246327	+	0.969187i	$0.579224\pi$
$450$		0			0
$451$	3.06608	−	4.49873i	0.144376	−	0.211837i
$452$		0			0
$453$	36.5293			1.71630
$454$		0			0
$455$	−8.25622	+	1.37662i	−0.387057	+	0.0645369i
$456$		0			0
$457$			8.96189i			0.419219i	0.977785	+	0.209610i	$0.0672193\pi$
$457$			8.96189i			0.419219i	−0.977785	+	0.209610i	$0.932781\pi$
$458$		0			0
$459$			14.2590i			0.665553i
$460$		0			0
$461$	5.84772			0.272356			0.136178	−	0.990684i	$-0.456518\pi$
$461$	5.84772			0.272356			0.136178	+	0.990684i	$0.456518\pi$
$462$		0			0
$463$	3.41997			0.158940			0.0794698	−	0.996837i	$-0.474677\pi$
$463$	3.41997			0.158940			0.0794698	+	0.996837i	$0.474677\pi$
$464$		0			0
$465$			42.7282i			1.98147i
$466$		0			0
$467$			4.28804i			0.198427i	0.995066	+	0.0992134i	$0.0316327\pi$
$467$			4.28804i			0.198427i	−0.995066	+	0.0992134i	$0.968367\pi$
$468$		0			0
$469$	−2.27315	−	13.6331i	−0.104964	−	0.629519i
$470$		0			0
$471$	22.1096			1.01876
$472$		0			0
$473$	−22.3854	−	15.2566i	−1.02928	−	0.701500i
$474$		0			0
$475$	10.4568			0.479789
$476$		0			0
$477$	−0.0448074			−0.00205159
$478$		0			0
$479$	2.54744			0.116396			0.0581978	−	0.998305i	$-0.481465\pi$
$479$	2.54744			0.116396			0.0581978	+	0.998305i	$0.481465\pi$
$480$		0			0
$481$	6.03705			0.275266
$482$		0			0
$483$	−32.6272	+	5.44018i	−1.48459	+	0.247537i
$484$		0			0
$485$	40.8771			1.85613
$486$		0			0
$487$	−0.887480			−0.0402156			−0.0201078	−	0.999798i	$-0.506401\pi$
$487$	−0.887480			−0.0402156			−0.0201078	+	0.999798i	$0.506401\pi$
$488$		0			0
$489$		−	1.32790i		−	0.0600495i
$490$		0			0
$491$		−	5.52910i		−	0.249525i	−0.992187	−	0.124762i	$-0.960183\pi$
$491$		−	5.52910i		−	0.249525i	0.992187	−	0.124762i	$-0.0398169\pi$
$492$		0			0
$493$			23.7347i			1.06896i
$494$		0			0
$495$	1.38917	+	0.946778i	0.0624384	+	0.0425545i
$496$		0			0
$497$	6.91360	+	41.4640i	0.310117	+	1.85991i
$498$		0			0
$499$	37.3836			1.67352			0.836759	−	0.547571i	$-0.184447\pi$
$499$	37.3836			1.67352			0.836759	+	0.547571i	$0.184447\pi$
$500$		0			0
$501$		−	2.40618i		−	0.107500i
$502$		0			0
$503$	30.3677			1.35403			0.677015	−	0.735969i	$-0.263273\pi$
$503$	30.3677			1.35403			0.677015	+	0.735969i	$0.263273\pi$
$504$		0			0
$505$		−	24.7999i		−	1.10358i
$506$		0			0
$507$		−	1.77770i		−	0.0789505i
$508$		0			0
$509$			10.1115i			0.448183i	0.974568	+	0.224092i	$0.0719415\pi$
$509$			10.1115i			0.448183i	−0.974568	+	0.224092i	$0.928059\pi$
$510$		0			0
$511$	−2.17589	−	13.0498i	−0.0962558	−	0.577290i
$512$		0			0
$513$			10.5396i			0.465335i
$514$		0			0
$515$	−50.5581			−2.22786
$516$		0			0
$517$	23.9403	+	16.3164i	1.05289	+	0.717593i
$518$		0			0
$519$			39.6988i			1.74258i
$520$		0			0
$521$		−	38.3565i		−	1.68043i	−0.542253	−	0.840215i	$-0.682429\pi$
$521$		−	38.3565i		−	1.68043i	0.542253	−	0.840215i	$-0.317571\pi$
$522$		0			0
$523$	9.77819			0.427570			0.213785	−	0.976881i	$-0.431421\pi$
$523$	9.77819			0.427570			0.213785	+	0.976881i	$0.431421\pi$
$524$		0			0
$525$	23.2364	−	3.87438i	1.01412	−	0.169092i
$526$		0			0
$527$		−	21.4593i		−	0.934780i
$528$		0			0
$529$	26.4600			1.15043
$530$		0			0
$531$		−	0.723368i		−	0.0313915i
$532$		0			0
$533$	1.64149			0.0711009
$534$		0			0
$535$	28.7685			1.24377
$536$		0			0
$537$			18.6991i			0.806927i
$538$		0			0
$539$	6.14321	−	22.3889i	0.264607	−	0.964356i
$540$		0			0
$541$		−	25.0967i		−	1.07899i	−0.841988	−	0.539496i	$-0.818615\pi$
$541$		−	25.0967i		−	1.07899i	0.841988	−	0.539496i	$-0.181385\pi$
$542$		0			0
$543$	1.76146			0.0755914
$544$		0			0
$545$	39.9469			1.71114
$546$		0			0
$547$			11.2380i			0.480500i	0.970711	+	0.240250i	$0.0772294\pi$
$547$			11.2380i			0.480500i	−0.970711	+	0.240250i	$0.922771\pi$
$548$		0			0
$549$	2.17183			0.0926912
$550$		0			0
$551$			17.5436i			0.747383i
$552$		0			0
$553$	−13.0492	+	2.17579i	−0.554908	+	0.0925240i
$554$		0			0
$555$	−33.9523			−1.44120
$556$		0			0
$557$			5.83798i			0.247363i	0.992322	+	0.123682i	$0.0394701\pi$
$557$			5.83798i			0.247363i	−0.992322	+	0.123682i	$0.960530\pi$
$558$		0			0
$559$		−	8.16795i		−	0.345467i
$560$		0			0
$561$	−13.7612	−	9.37884i	−0.580997	−	0.395975i
$562$		0			0
$563$	−7.13370			−0.300650			−0.150325	−	0.988637i	$-0.548032\pi$
$563$	−7.13370			−0.300650			−0.150325	+	0.988637i	$0.548032\pi$
$564$		0			0
$565$			4.02732i			0.169431i
$566$		0			0
$567$	4.11423	+	24.6749i	0.172781	+	1.03625i
$568$		0			0
$569$		−	5.16832i		−	0.216667i	−0.994115	−	0.108334i	$-0.965449\pi$
$569$		−	5.16832i		−	0.216667i	0.994115	−	0.108334i	$-0.0345515\pi$
$570$		0			0
$571$			9.55317i			0.399788i	0.979818	+	0.199894i	$0.0640597\pi$
$571$			9.55317i			0.399788i	−0.979818	+	0.199894i	$0.935940\pi$
$572$		0			0
$573$		−	40.1027i		−	1.67532i
$574$		0			0
$575$	−35.2244			−1.46896
$576$		0			0
$577$		−	17.2574i		−	0.718435i	−0.933254	−	0.359218i	$-0.883044\pi$
$577$		−	17.2574i		−	0.718435i	0.933254	−	0.359218i	$-0.116956\pi$
$578$		0			0
$579$	2.71388			0.112785
$580$		0			0
$581$	1.27451	+	7.64384i	0.0528757	+	0.317120i
$582$		0			0
$583$	−0.522371	+	0.766453i	−0.0216344	+	0.0317432i
$584$		0			0
$585$			0.506878i			0.0209568i
$586$		0			0
$587$			35.5925i			1.46906i	0.678577	+	0.734529i	$0.262597\pi$
$587$			35.5925i			1.46906i	−0.678577	+	0.734529i	$0.737403\pi$
$588$		0			0
$589$		−	15.8617i		−	0.653571i
$590$		0			0
$591$	−12.3350			−0.507395
$592$		0			0
$593$	40.3021			1.65501			0.827503	−	0.561461i	$-0.189760\pi$
$593$	40.3021			1.65501			0.827503	+	0.561461i	$0.189760\pi$
$594$		0			0
$595$	−23.3199	+	3.88830i	−0.956022	+	0.159405i
$596$		0			0
$597$	−19.5158			−0.798727
$598$		0			0
$599$	−15.8797			−0.648827			−0.324413	−	0.945915i	$-0.605167\pi$
$599$	−15.8797			−0.648827			−0.324413	+	0.945915i	$0.605167\pi$
$600$		0			0
$601$	−0.574763			−0.0234451			−0.0117225	−	0.999931i	$-0.503731\pi$
$601$	−0.574763			−0.0234451			−0.0117225	+	0.999931i	$0.503731\pi$
$602$		0			0
$603$	−0.836984			−0.0340846
$604$		0			0
$605$	32.3902	−	12.7247i	1.31685	−	0.517333i
$606$		0			0
$607$	15.8059			0.641540			0.320770	−	0.947157i	$-0.396058\pi$
$607$	15.8059			0.641540			0.320770	+	0.947157i	$0.396058\pi$
$608$		0			0
$609$	6.50016	+	38.9844i	0.263400	+	1.57973i
$610$		0			0
$611$			8.73531i			0.353393i
$612$		0			0
$613$			23.5245i			0.950146i	0.879946	+	0.475073i	$0.157578\pi$
$613$			23.5245i			0.950146i	−0.879946	+	0.475073i	$0.842422\pi$
$614$		0			0
$615$	−9.23174			−0.372260
$616$		0			0
$617$	35.5782			1.43232			0.716162	−	0.697934i	$-0.245897\pi$
$617$	35.5782			1.43232			0.716162	+	0.697934i	$0.245897\pi$
$618$		0			0
$619$		−	30.2625i		−	1.21635i	−0.793801	−	0.608177i	$-0.791901\pi$
$619$		−	30.2625i		−	1.21635i	0.793801	−	0.608177i	$-0.208099\pi$
$620$		0			0
$621$		−	35.5034i		−	1.42470i
$622$		0			0
$623$	27.9839	−	4.66595i	1.12115	−	0.186938i
$624$		0			0
$625$	−24.9569			−0.998277
$626$		0			0
$627$	−10.1716	−	6.93241i	−0.406216	−	0.276854i
$628$		0			0
$629$	17.0518			0.679899
$630$		0			0
$631$	33.6080			1.33791			0.668957	−	0.743301i	$-0.266741\pi$
$631$	33.6080			1.33791			0.668957	+	0.743301i	$0.266741\pi$
$632$		0			0
$633$	−22.0849			−0.877796
$634$		0			0
$635$	5.16376			0.204917
$636$		0			0
$637$	6.62131	−	2.27118i	0.262346	−	0.0899875i
$638$		0			0
$639$	2.54562			0.100703
$640$		0			0
$641$	47.8962			1.89178			0.945892	−	0.324481i	$-0.105190\pi$
$641$	47.8962			1.89178			0.945892	+	0.324481i	$0.105190\pi$
$642$		0			0
$643$		−	48.8186i		−	1.92522i	−0.270898	−	0.962608i	$-0.587321\pi$
$643$		−	48.8186i		−	1.92522i	0.270898	−	0.962608i	$-0.412679\pi$
$644$		0			0
$645$			45.9365i			1.80875i
$646$		0			0
$647$			3.90795i			0.153637i	0.997045	+	0.0768186i	$0.0244762\pi$
$647$			3.90795i			0.153637i	−0.997045	+	0.0768186i	$0.975524\pi$
$648$		0			0
$649$	−12.3736	−	8.43311i	−0.485704	−	0.331029i
$650$		0			0
$651$	−5.87700	−	35.2470i	−0.230338	−	1.38144i
$652$		0			0
$653$	−27.2824			−1.06764			−0.533821	−	0.845597i	$-0.679244\pi$
$653$	−27.2824			−1.06764			−0.533821	+	0.845597i	$0.679244\pi$
$654$		0			0
$655$		−	54.5582i		−	2.13177i
$656$		0			0
$657$	−0.801173			−0.0312567
$658$		0			0
$659$		−	41.4076i		−	1.61301i	−0.591227	−	0.806505i	$-0.701356\pi$
$659$		−	41.4076i		−	1.61301i	0.591227	−	0.806505i	$-0.298644\pi$
$660$		0			0
$661$			12.2355i			0.475904i	0.971277	+	0.237952i	$0.0764761\pi$
$661$			12.2355i			0.475904i	−0.971277	+	0.237952i	$0.923524\pi$
$662$		0			0
$663$		−	5.02116i		−	0.195006i
$664$		0			0
$665$	−17.2370	+	2.87405i	−0.668423	+	0.111451i
$666$		0			0
$667$		−	59.0970i		−	2.28824i
$668$		0			0
$669$	30.8826			1.19399
$670$		0			0
$671$	25.3194	−	37.1501i	0.977445	−	1.43416i
$672$		0			0
$673$			11.0632i			0.426454i	0.977003	+	0.213227i	$0.0683974\pi$
$673$			11.0632i			0.426454i	−0.977003	+	0.213227i	$0.931603\pi$
$674$		0			0
$675$			25.2848i			0.973213i
$676$		0			0
$677$	−6.17730			−0.237413			−0.118706	−	0.992929i	$-0.537875\pi$
$677$	−6.17730			−0.237413			−0.118706	+	0.992929i	$0.537875\pi$
$678$		0			0
$679$	−33.7200	+	5.62239i	−1.29406	+	0.215767i
$680$		0			0
$681$			29.7478i			1.13994i
$682$		0			0
$683$	−28.2629			−1.08145			−0.540724	−	0.841200i	$-0.681850\pi$
$683$	−28.2629			−1.08145			−0.540724	+	0.841200i	$0.681850\pi$
$684$		0			0
$685$		−	50.8891i		−	1.94437i
$686$		0			0
$687$	14.8953			0.568291
$688$		0			0
$689$	−0.279662			−0.0106543
$690$		0			0
$691$			10.3534i			0.393863i	0.980417	+	0.196932i	$0.0630977\pi$
$691$			10.3534i			0.393863i	−0.980417	+	0.196932i	$0.936902\pi$
$692$		0			0
$693$	−1.27616	−	0.589937i	−0.0484775	−	0.0224099i
$694$		0			0
$695$		−	53.4228i		−	2.02644i
$696$		0			0
$697$	4.63643			0.175617
$698$		0			0
$699$	−45.9870			−1.73939
$700$		0			0
$701$		−	24.6806i		−	0.932173i	−0.884739	−	0.466087i	$-0.845664\pi$
$701$		−	24.6806i		−	0.932173i	0.884739	−	0.466087i	$-0.154336\pi$
$702$		0			0
$703$	12.6039			0.475365
$704$		0			0
$705$		−	49.1274i		−	1.85024i
$706$		0			0
$707$	3.41107	+	20.4577i	0.128286	+	0.769392i
$708$		0			0
$709$	17.6947			0.664538			0.332269	−	0.943185i	$-0.392186\pi$
$709$	17.6947			0.664538			0.332269	+	0.943185i	$0.392186\pi$
$710$		0			0
$711$			0.801135i			0.0300449i
$712$		0			0
$713$			53.4314i			2.00102i
$714$		0			0
$715$	8.67038	+	5.90924i	0.324254	+	0.220993i
$716$		0			0
$717$	33.1274			1.23716
$718$		0			0
$719$		−	6.71283i		−	0.250346i	−0.992135	−	0.125173i	$-0.960051\pi$
$719$		−	6.71283i		−	0.250346i	0.992135	−	0.125173i	$-0.0399486\pi$
$720$		0			0
$721$	41.7060	−	6.95395i	1.55321	−	0.258979i
$722$		0			0
$723$		−	43.5958i		−	1.62134i
$724$		0			0
$725$			42.0876i			1.56309i
$726$		0			0
$727$		−	18.6146i		−	0.690377i	−0.938533	−	0.345188i	$-0.887815\pi$
$727$		−	18.6146i		−	0.690377i	0.938533	−	0.345188i	$-0.112185\pi$
$728$		0			0
$729$	−25.4081			−0.941041
$730$		0			0
$731$		−	23.0706i		−	0.853296i
$732$		0			0
$733$	46.8732			1.73130			0.865651	−	0.500648i	$-0.166905\pi$
$733$	46.8732			1.73130			0.865651	+	0.500648i	$0.166905\pi$
$734$		0			0
$735$	−37.2382	+	12.7731i	−1.37355	+	0.471144i
$736$		0			0
$737$	−9.75765	+	14.3170i	−0.359428	+	0.527373i
$738$		0			0
$739$		−	13.8328i		−	0.508849i	−0.967093	−	0.254425i	$-0.918114\pi$
$739$		−	13.8328i		−	0.508849i	0.967093	−	0.254425i	$-0.0818861\pi$
$740$		0			0
$741$		−	3.71142i		−	0.136342i
$742$		0			0
$743$		−	51.1913i		−	1.87803i	−0.343879	−	0.939014i	$-0.611741\pi$
$743$		−	51.1913i		−	1.87803i	0.343879	−	0.939014i	$-0.388259\pi$
$744$		0			0
$745$	29.6235			1.08532
$746$		0			0
$747$	0.469281			0.0171701
$748$		0			0
$749$	−23.7315	+	3.95692i	−0.867129	+	0.144583i
$750$		0			0
$751$	−36.0843			−1.31673			−0.658367	−	0.752697i	$-0.728753\pi$
$751$	−36.0843			−1.31673			−0.658367	+	0.752697i	$0.728753\pi$
$752$		0			0
$753$	21.2853			0.775680
$754$		0			0
$755$	−65.0083			−2.36590
$756$		0			0
$757$	−13.7503			−0.499764			−0.249882	−	0.968276i	$-0.580392\pi$
$757$	−13.7503			−0.499764			−0.249882	+	0.968276i	$0.580392\pi$
$758$		0			0
$759$	34.2639	+	23.3523i	1.24370	+	0.847637i
$760$		0			0
$761$	21.5501			0.781192			0.390596	−	0.920562i	$-0.372269\pi$
$761$	21.5501			0.781192			0.390596	+	0.920562i	$0.372269\pi$
$762$		0			0
$763$	−32.9527	+	5.49445i	−1.19297	+	0.198912i
$764$		0			0
$765$			1.43169i			0.0517628i
$766$		0			0
$767$		−	4.51485i		−	0.163022i
$768$		0			0
$769$	−48.3906			−1.74501			−0.872506	−	0.488604i	$-0.837506\pi$
$769$	−48.3906			−1.74501			−0.872506	+	0.488604i	$0.837506\pi$
$770$		0			0
$771$	−8.09314			−0.291467
$772$		0			0
$773$		−	29.0710i		−	1.04561i	−0.852452	−	0.522805i	$-0.824885\pi$
$773$		−	29.0710i		−	1.04561i	0.852452	−	0.522805i	$-0.175115\pi$
$774$		0			0
$775$		−	38.0527i		−	1.36689i
$776$		0			0
$777$	28.0077	−	4.66993i	1.00477	−	0.167533i
$778$		0			0
$779$	3.42704			0.122786
$780$		0			0
$781$	29.6771	−	43.5440i	1.06193	−	1.55813i
$782$		0			0
$783$	−42.4211			−1.51600
$784$		0			0
$785$	−39.3468			−1.40435
$786$		0			0
$787$	−21.8465			−0.778743			−0.389371	−	0.921081i	$-0.627308\pi$
$787$	−21.8465			−0.778743			−0.389371	+	0.921081i	$0.627308\pi$
$788$		0			0
$789$	−45.2591			−1.61127
$790$		0			0
$791$	−0.553933	−	3.32219i	−0.0196956	−	0.118123i
$792$		0			0
$793$	13.5553			0.481362
$794$		0			0
$795$	1.57282			0.0557822
$796$		0			0
$797$		−	43.0993i		−	1.52666i	−0.646011	−	0.763328i	$-0.723564\pi$
$797$		−	43.0993i		−	1.52666i	0.646011	−	0.763328i	$-0.276436\pi$
$798$		0			0
$799$			24.6731i			0.872872i
$800$		0			0
$801$		−	1.71803i		−	0.0607034i
$802$		0			0
$803$	−9.34017	+	13.7044i	−0.329608	+	0.483619i
$804$		0			0
$805$	58.0641	−	9.68146i	2.04649	−	0.341227i
$806$		0			0
$807$	−19.3562			−0.681369
$808$		0			0
$809$		−	22.0087i		−	0.773785i	−0.922125	−	0.386892i	$-0.873548\pi$
$809$		−	22.0087i		−	0.773785i	0.922125	−	0.386892i	$-0.126452\pi$
$810$		0			0
$811$	39.8072			1.39782			0.698911	−	0.715209i	$-0.253669\pi$
$811$	39.8072			1.39782			0.698911	+	0.715209i	$0.253669\pi$
$812$		0			0
$813$		−	12.8103i		−	0.449277i
$814$		0			0
$815$			2.36315i			0.0827777i
$816$		0			0
$817$		−	17.0527i		−	0.596599i
$818$		0			0
$819$	−0.0697178	−	0.418130i	−0.00243614	−	0.0146106i
$820$		0			0
$821$			17.0555i			0.595242i	0.954684	+	0.297621i	$0.0961931\pi$
$821$			17.0555i			0.595242i	−0.954684	+	0.297621i	$0.903807\pi$
$822$		0			0
$823$	40.7345			1.41991			0.709957	−	0.704245i	$-0.248714\pi$
$823$	40.7345			1.41991			0.709957	+	0.704245i	$0.248714\pi$
$824$		0			0
$825$	−24.4021	−	16.6311i	−0.849570	−	0.579019i
$826$		0			0
$827$		−	6.99140i		−	0.243115i	−0.992584	−	0.121557i	$-0.961211\pi$
$827$		−	6.99140i		−	0.243115i	0.992584	−	0.121557i	$-0.0387889\pi$
$828$		0			0
$829$		−	11.1334i		−	0.386678i	−0.981132	−	0.193339i	$-0.938068\pi$
$829$		−	11.1334i		−	0.386678i	0.981132	−	0.193339i	$-0.0619318\pi$
$830$		0			0
$831$	31.8183			1.10376
$832$		0			0
$833$	18.7021	−	6.41501i	0.647988	−	0.222267i
$834$		0			0
$835$			4.28209i			0.148188i
$836$		0			0
$837$	38.3542			1.32571
$838$		0			0
$839$		−	42.2782i		−	1.45961i	−0.683657	−	0.729804i	$-0.739612\pi$
$839$		−	42.2782i		−	1.45961i	0.683657	−	0.729804i	$-0.260388\pi$
$840$		0			0
$841$	−41.6116			−1.43488
$842$		0			0
$843$	13.8998			0.478734
$844$		0			0
$845$			3.16364i			0.108832i
$846$		0			0
$847$	−24.9688	+	14.9518i	−0.857939	+	0.513751i
$848$		0			0
$849$			8.57690i			0.294358i
$850$		0			0
$851$	−42.4572			−1.45541
$852$		0			0
$853$	−37.7618			−1.29294			−0.646470	−	0.762940i	$-0.723755\pi$
$853$	−37.7618			−1.29294			−0.646470	+	0.762940i	$0.723755\pi$
$854$		0			0
$855$			1.05824i			0.0361910i
$856$		0			0
$857$	−35.3446			−1.20735			−0.603674	−	0.797231i	$-0.706297\pi$
$857$	−35.3446			−1.20735			−0.603674	+	0.797231i	$0.706297\pi$
$858$		0			0
$859$		−	6.87083i		−	0.234430i	−0.993107	−	0.117215i	$-0.962603\pi$
$859$		−	6.87083i		−	0.234430i	0.993107	−	0.117215i	$-0.0373966\pi$
$860$		0			0
$861$	7.61538	−	1.26977i	0.259531	−	0.0432736i
$862$		0			0
$863$	−7.06134			−0.240371			−0.120185	−	0.992751i	$-0.538349\pi$
$863$	−7.06134			−0.240371			−0.120185	+	0.992751i	$0.538349\pi$
$864$		0			0
$865$		−	70.6488i		−	2.40213i
$866$		0			0
$867$			16.0385i			0.544697i
$868$		0			0
$869$	13.7038	+	9.33973i	0.464870	+	0.316829i
$870$		0			0
$871$	−5.22397			−0.177007
$872$		0			0
$873$			2.07019i			0.0700652i
$874$		0			0
$875$	−0.0709915	+	0.0118369i	−0.00239995	+	0.000400162i
$876$		0			0
$877$			52.8222i			1.78368i	0.452351	+	0.891840i	$0.350585\pi$
$877$			52.8222i			1.78368i	−0.452351	+	0.891840i	$0.649415\pi$
$878$		0			0
$879$			17.3908i			0.586579i
$880$		0			0
$881$		−	4.39443i		−	0.148052i	−0.997256	−	0.0740261i	$-0.976415\pi$
$881$		−	4.39443i		−	0.148052i	0.997256	−	0.0740261i	$-0.0235848\pi$
$882$		0			0
$883$	−34.7396			−1.16908			−0.584541	−	0.811364i	$-0.698725\pi$
$883$	−34.7396			−1.16908			−0.584541	+	0.811364i	$0.698725\pi$
$884$		0			0
$885$			25.3915i			0.853526i
$886$		0			0
$887$	−28.4387			−0.954879			−0.477439	−	0.878665i	$-0.658435\pi$
$887$	−28.4387			−0.954879			−0.477439	+	0.878665i	$0.658435\pi$
$888$		0			0
$889$	−4.25965	+	0.710242i	−0.142864	+	0.0238208i
$890$		0			0
$891$	17.6606	−	25.9127i	0.591653	−	0.868108i
$892$		0			0
$893$			18.2373i			0.610286i
$894$		0			0
$895$		−	33.2774i		−	1.11234i
$896$		0			0
$897$			12.5022i			0.417436i
$898$		0			0
$899$	63.8421			2.12925
$900$		0			0
$901$	−0.789913			−0.0263158
$902$		0			0
$903$	−6.31828	−	37.8936i	−0.210259	−	1.26102i
$904$		0			0
$905$	−3.13473			−0.104202
$906$		0			0
$907$	−21.5716			−0.716274			−0.358137	−	0.933669i	$-0.616588\pi$
$907$	−21.5716			−0.716274			−0.358137	+	0.933669i	$0.616588\pi$
$908$		0			0
$909$	1.25597			0.0416579
$910$		0			0
$911$	14.4030			0.477193			0.238596	−	0.971119i	$-0.423313\pi$
$911$	14.4030			0.477193			0.238596	+	0.971119i	$0.423313\pi$
$912$		0			0
$913$	5.47094	−	8.02728i	0.181062	−	0.265664i
$914$		0			0
$915$	−76.2349			−2.52025
$916$		0			0
$917$	7.50414	+	45.0057i	0.247808	+	1.48622i
$918$		0			0
$919$			53.9729i			1.78040i	0.455568	+	0.890201i	$0.349436\pi$
$919$			53.9729i			1.78040i	−0.455568	+	0.890201i	$0.650564\pi$
$920$		0			0
$921$			8.74499i			0.288157i
$922$		0			0
$923$	15.8883			0.522969
$924$		0			0
$925$	30.2371			0.994191
$926$		0			0
$927$		−	2.56048i		−	0.0840970i
$928$		0			0
$929$			8.86182i			0.290747i	0.989377	+	0.145373i	$0.0464384\pi$
$929$			8.86182i			0.290747i	−0.989377	+	0.145373i	$0.953562\pi$
$930$		0			0
$931$	13.8237	−	4.74168i	0.453054	−	0.155402i
$932$		0			0
$933$	45.4949			1.48944
$934$		0			0
$935$	24.4897	+	16.6908i	0.800899	+	0.545847i
$936$		0			0
$937$	18.7145			0.611377			0.305689	−	0.952132i	$-0.401113\pi$
$937$	18.7145			0.611377			0.305689	+	0.952132i	$0.401113\pi$
$938$		0			0
$939$	7.25101			0.236628
$940$		0			0
$941$	31.4105			1.02395			0.511977	−	0.858999i	$-0.328913\pi$
$941$	31.4105			1.02395			0.511977	+	0.858999i	$0.328913\pi$
$942$		0			0
$943$	−11.5442			−0.375932
$944$		0			0
$945$	−6.94956	−	41.6797i	−0.226069	−	1.35584i
$946$		0			0
$947$	30.3290			0.985559			0.492779	−	0.870154i	$-0.335981\pi$
$947$	30.3290			0.985559			0.492779	+	0.870154i	$0.335981\pi$
$948$		0			0
$949$	−5.00046			−0.162322
$950$		0			0
$951$			26.1309i			0.847353i
$952$		0			0
$953$		−	45.9597i		−	1.48878i	−0.667744	−	0.744391i	$-0.732740\pi$
$953$		−	45.9597i		−	1.48878i	0.667744	−	0.744391i	$-0.267260\pi$
$954$		0			0
$955$			71.3678i			2.30941i
$956$		0			0
$957$	27.9024	−	40.9400i	0.901957	−	1.32340i
$958$		0			0
$959$	6.99948	+	41.9790i	0.226025	+	1.35557i
$960$		0			0
$961$	−26.7216			−0.861989
$962$		0			0
$963$			1.45696i			0.0469498i
$964$		0			0
$965$	−4.82968			−0.155473
$966$		0			0
$967$			24.4685i			0.786855i	0.919356	+	0.393428i	$0.128711\pi$
$967$			24.4685i			0.786855i	−0.919356	+	0.393428i	$0.871289\pi$
$968$		0			0
$969$		−	10.4830i		−	0.336762i
$970$		0			0
$971$		−	46.1110i		−	1.47977i	−0.672732	−	0.739886i	$-0.734879\pi$
$971$		−	46.1110i		−	1.47977i	0.672732	−	0.739886i	$-0.265121\pi$
$972$		0			0
$973$	7.34796	+	44.0691i	0.235565	+	1.41279i
$974$		0			0
$975$		−	8.90379i		−	0.285149i
$976$		0			0
$977$	19.9243			0.637436			0.318718	−	0.947850i	$-0.396748\pi$
$977$	19.9243			0.637436			0.318718	+	0.947850i	$0.396748\pi$
$978$		0			0
$979$	−29.3876	−	20.0289i	−0.939233	−	0.640128i
$980$		0			0
$981$			2.02308i			0.0645920i
$982$		0			0
$983$			50.1749i			1.60033i	0.599779	+	0.800165i	$0.295255\pi$
$983$			50.1749i			1.60033i	−0.599779	+	0.800165i	$0.704745\pi$
$984$		0			0
$985$	21.9517			0.699439
$986$		0			0
$987$	6.75716	+	40.5258i	0.215083	+	1.28995i
$988$		0			0
$989$			57.4434i			1.82659i
$990$		0			0
$991$	40.5572			1.28834			0.644171	−	0.764881i	$-0.277202\pi$
$991$	40.5572			1.28834			0.644171	+	0.764881i	$0.277202\pi$
$992$		0			0
$993$		−	48.3024i		−	1.53283i
$994$		0			0
$995$	34.7307			1.10104
$996$		0			0
$997$	0.413347			0.0130908			0.00654542	−	0.999979i	$-0.497917\pi$
$997$	0.413347			0.0130908			0.00654542	+	0.999979i	$0.497917\pi$
$998$		0			0
$999$			30.4767i			0.964240i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.b.3849.12	yes	48
7.6	odd	2		4004.2.e.a.3849.37	yes	48
11.10	odd	2		4004.2.e.a.3849.12	✓	48
77.76	even	2	inner	4004.2.e.b.3849.37	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.12	✓	48	11.10	odd	2
4004.2.e.a.3849.37	yes	48	7.6	odd	2
4004.2.e.b.3849.12	yes	48	1.1	even	1	trivial
4004.2.e.b.3849.37	yes	48	77.76	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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.77770i q^{3} +3.16364i q^{5} +(-0.435138 - 2.60972i) q^{7} -0.160220 q^{9} +O(q^{10})\)	\(q-1.77770i q^{3} +3.16364i q^{5} +(-0.435138 - 2.60972i) q^{7} -0.160220 q^{9} +(-1.86786 + 2.74064i) q^{11} -1.00000 q^{13} +5.62400 q^{15} -2.82452 q^{17} -2.08776 q^{19} +(-4.63931 + 0.773546i) q^{21} +7.03278 q^{23} -5.00860 q^{25} -5.04828i q^{27} -8.40307i q^{29} +7.59748i q^{31} +(4.87203 + 3.32050i) q^{33} +(8.25622 - 1.37662i) q^{35} -6.03705 q^{37} +1.77770i q^{39} -1.64149 q^{41} +8.16795i q^{43} -0.506878i q^{45} -8.73531i q^{47} +(-6.62131 + 2.27118i) q^{49} +5.02116i q^{51} +0.279662 q^{53} +(-8.67038 - 5.90924i) q^{55} +3.71142i q^{57} +4.51485i q^{59} -13.5553 q^{61} +(0.0697178 + 0.418130i) q^{63} -3.16364i q^{65} +5.22397 q^{67} -12.5022i q^{69} -15.8883 q^{71} +5.00046 q^{73} +8.90379i q^{75} +(7.96508 + 3.68205i) q^{77} -5.00022i q^{79} -9.45499 q^{81} -2.92898 q^{83} -8.93577i q^{85} -14.9381 q^{87} +10.7229i q^{89} +(0.435138 + 2.60972i) q^{91} +13.5060 q^{93} -6.60492i q^{95} -12.9209i q^{97} +(0.299269 - 0.439104i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{7} - 48 q^{9}+O(q^{10}) \) 48 * q + 4 * q^7 - 48 * q^9	\( 48 q + 4 q^{7} - 48 q^{9} + 2 q^{11} - 48 q^{13} + 8 q^{15} - 4 q^{17} - 10 q^{21} + 4 q^{23} - 44 q^{25} - 10 q^{33} + 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} + 12 q^{55} - 16 q^{61} - 16 q^{63} + 4 q^{67} + 16 q^{73} + 22 q^{77} + 64 q^{81} + 4 q^{83} + 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) \) 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 + 22 * 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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