LMFDB - Embedded newform 4004.2.m.c.2157.33

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(2157,4004)]

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, 0, 0, 1]))

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

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

chi := DirichletCharacter("4004.2157");

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.m (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:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2157.33
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.c.2157.34

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.97211 q^{3} -4.37654i q^{5} +1.00000i q^{7} +5.83342 q^{9} +O(q^{10})$	$q+2.97211 q^{3} -4.37654i q^{5} +1.00000i q^{7} +5.83342 q^{9} -1.00000i q^{11} +(1.31026 - 3.35905i) q^{13} -13.0075i q^{15} -5.05132 q^{17} -0.365210i q^{19} +2.97211i q^{21} -3.66878 q^{23} -14.1541 q^{25} +8.42121 q^{27} +8.10081 q^{29} +7.09407i q^{31} -2.97211i q^{33} +4.37654 q^{35} -6.97230i q^{37} +(3.89424 - 9.98345i) q^{39} -3.30611i q^{41} +4.30504 q^{43} -25.5301i q^{45} -10.8852i q^{47} -1.00000 q^{49} -15.0131 q^{51} +8.81346 q^{53} -4.37654 q^{55} -1.08544i q^{57} +8.55100i q^{59} -1.88126 q^{61} +5.83342i q^{63} +(-14.7010 - 5.73441i) q^{65} -12.4780i q^{67} -10.9040 q^{69} -5.58236i q^{71} +1.87964i q^{73} -42.0674 q^{75} +1.00000 q^{77} +0.669936 q^{79} +7.52850 q^{81} +12.6010i q^{83} +22.1073i q^{85} +24.0765 q^{87} -8.69604i q^{89} +(3.35905 + 1.31026i) q^{91} +21.0843i q^{93} -1.59836 q^{95} +12.2029i q^{97} -5.83342i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) $ 36 * q - 4 * q^3 + 40 * q^9	$ 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) $ 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

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
$3$	2.97211			1.71595			0.857973	−	0.513694i	$-0.171724\pi$
$3$	2.97211			1.71595			0.857973	+	0.513694i	$0.171724\pi$
$4$		0			0
$5$		−	4.37654i		−	1.95725i	−0.205663	−	0.978623i	$-0.565935\pi$
$5$		−	4.37654i		−	1.95725i	0.205663	−	0.978623i	$-0.434065\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	5.83342			1.94447
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	1.31026	−	3.35905i	0.363401	−	0.931633i
$13$	1.31026	−	3.35905i	0.363401	−	0.931633i
$14$		0			0
$15$		−	13.0075i		−	3.35853i
$16$		0			0
$17$	−5.05132			−1.22513			−0.612563	−	0.790422i	$-0.709861\pi$
$17$	−5.05132			−1.22513			−0.612563	+	0.790422i	$0.709861\pi$
$18$		0			0
$19$		−	0.365210i		−	0.0837850i	−0.999122	−	0.0418925i	$-0.986661\pi$
$19$		−	0.365210i		−	0.0837850i	0.999122	−	0.0418925i	$-0.0133387\pi$
$20$		0			0
$21$			2.97211i			0.648567i
$22$		0			0
$23$	−3.66878			−0.764993			−0.382496	−	0.923957i	$-0.624936\pi$
$23$	−3.66878			−0.764993			−0.382496	+	0.923957i	$0.624936\pi$
$24$		0			0
$25$	−14.1541			−2.83081
$26$		0			0
$27$	8.42121			1.62066
$28$		0			0
$29$	8.10081			1.50428			0.752142	−	0.659001i	$-0.229021\pi$
$29$	8.10081			1.50428			0.752142	+	0.659001i	$0.229021\pi$
$30$		0			0
$31$			7.09407i			1.27413i	0.770809	+	0.637067i	$0.219852\pi$
$31$			7.09407i			1.27413i	−0.770809	+	0.637067i	$0.780148\pi$
$32$		0			0
$33$		−	2.97211i		−	0.517377i
$34$		0			0
$35$	4.37654			0.739769
$36$		0			0
$37$		−	6.97230i		−	1.14624i	−0.819472	−	0.573120i	$-0.805733\pi$
$37$		−	6.97230i		−	1.14624i	0.819472	−	0.573120i	$-0.194267\pi$
$38$		0			0
$39$	3.89424	−	9.98345i	0.623577	−	1.59863i
$40$		0			0
$41$		−	3.30611i		−	0.516327i	−0.966101	−	0.258164i	$-0.916883\pi$
$41$		−	3.30611i		−	0.516327i	0.966101	−	0.258164i	$-0.0831174\pi$
$42$		0			0
$43$	4.30504			0.656512			0.328256	−	0.944589i	$-0.393539\pi$
$43$	4.30504			0.656512			0.328256	+	0.944589i	$0.393539\pi$
$44$		0			0
$45$		−	25.5301i		−	3.80581i
$46$		0			0
$47$		−	10.8852i		−	1.58777i	−0.608069	−	0.793884i	$-0.708056\pi$
$47$		−	10.8852i		−	1.58777i	0.608069	−	0.793884i	$-0.291944\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−15.0131			−2.10225
$52$		0			0
$53$	8.81346			1.21062			0.605311	−	0.795989i	$-0.293049\pi$
$53$	8.81346			1.21062			0.605311	+	0.795989i	$0.293049\pi$
$54$		0			0
$55$	−4.37654			−0.590132
$56$		0			0
$57$		−	1.08544i		−	0.143770i
$58$		0			0
$59$			8.55100i			1.11325i	0.830766	+	0.556623i	$0.187903\pi$
$59$			8.55100i			1.11325i	−0.830766	+	0.556623i	$0.812097\pi$
$60$		0			0
$61$	−1.88126			−0.240870			−0.120435	−	0.992721i	$-0.538429\pi$
$61$	−1.88126			−0.240870			−0.120435	+	0.992721i	$0.538429\pi$
$62$		0			0
$63$			5.83342i			0.734941i
$64$		0			0
$65$	−14.7010	−	5.73441i	−1.82343	−	0.711265i
$66$		0			0
$67$		−	12.4780i		−	1.52443i	−0.647321	−	0.762217i	$-0.724111\pi$
$67$		−	12.4780i		−	1.52443i	0.647321	−	0.762217i	$-0.275889\pi$
$68$		0			0
$69$	−10.9040			−1.31269
$70$		0			0
$71$		−	5.58236i		−	0.662504i	−0.943542	−	0.331252i	$-0.892529\pi$
$71$		−	5.58236i		−	0.662504i	0.943542	−	0.331252i	$-0.107471\pi$
$72$		0			0
$73$			1.87964i			0.219995i	0.993932	+	0.109997i	$0.0350843\pi$
$73$			1.87964i			0.219995i	−0.993932	+	0.109997i	$0.964916\pi$
$74$		0			0
$75$	−42.0674			−4.85752
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	0.669936			0.0753737			0.0376869	−	0.999290i	$-0.488001\pi$
$79$	0.669936			0.0753737			0.0376869	+	0.999290i	$0.488001\pi$
$80$		0			0
$81$	7.52850			0.836499
$82$		0			0
$83$			12.6010i			1.38314i	0.722308	+	0.691571i	$0.243081\pi$
$83$			12.6010i			1.38314i	−0.722308	+	0.691571i	$0.756919\pi$
$84$		0			0
$85$			22.1073i			2.39787i
$86$		0			0
$87$	24.0765			2.58127
$88$		0			0
$89$		−	8.69604i		−	0.921779i	−0.887458	−	0.460889i	$-0.847530\pi$
$89$		−	8.69604i		−	0.921779i	0.887458	−	0.460889i	$-0.152470\pi$
$90$		0			0
$91$	3.35905	+	1.31026i	0.352124	+	0.137353i
$92$		0			0
$93$			21.0843i			2.18634i
$94$		0			0
$95$	−1.59836			−0.163988
$96$		0			0
$97$			12.2029i			1.23902i	0.784990	+	0.619508i	$0.212668\pi$
$97$			12.2029i			1.23902i	−0.784990	+	0.619508i	$0.787332\pi$
$98$		0			0
$99$		−	5.83342i		−	0.586280i
$100$		0			0
$101$	3.40873			0.339181			0.169591	−	0.985515i	$-0.445755\pi$
$101$	3.40873			0.339181			0.169591	+	0.985515i	$0.445755\pi$
$102$		0			0
$103$	−9.89078			−0.974567			−0.487284	−	0.873244i	$-0.662012\pi$
$103$	−9.89078			−0.974567			−0.487284	+	0.873244i	$0.662012\pi$
$104$		0			0
$105$	13.0075			1.26940
$106$		0			0
$107$	−9.80621			−0.948002			−0.474001	−	0.880524i	$-0.657191\pi$
$107$	−9.80621			−0.948002			−0.474001	+	0.880524i	$0.657191\pi$
$108$		0			0
$109$			0.426645i			0.0408652i	0.999791	+	0.0204326i	$0.00650435\pi$
$109$			0.426645i			0.0408652i	−0.999791	+	0.0204326i	$0.993496\pi$
$110$		0			0
$111$		−	20.7224i		−	1.96688i
$112$		0			0
$113$	−13.3622			−1.25701			−0.628504	−	0.777806i	$-0.716332\pi$
$113$	−13.3622			−1.25701			−0.628504	+	0.777806i	$0.716332\pi$
$114$		0			0
$115$			16.0565i			1.49728i
$116$		0			0
$117$	7.64330	−	19.5947i	0.706623	−	1.81153i
$118$		0			0
$119$		−	5.05132i		−	0.463054i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	9.82610i		−	0.885990i
$124$		0			0
$125$			40.0631i			3.58335i
$126$		0			0
$127$	11.6941			1.03769			0.518843	−	0.854869i	$-0.326363\pi$
$127$	11.6941			1.03769			0.518843	+	0.854869i	$0.326363\pi$
$128$		0			0
$129$	12.7950			1.12654
$130$		0			0
$131$	22.1155			1.93224			0.966118	−	0.258100i	$-0.0830964\pi$
$131$	22.1155			1.93224			0.966118	+	0.258100i	$0.0830964\pi$
$132$		0			0
$133$	0.365210			0.0316677
$134$		0			0
$135$		−	36.8557i		−	3.17204i
$136$		0			0
$137$			0.304431i			0.0260093i	0.999915	+	0.0130046i	$0.00413962\pi$
$137$			0.304431i			0.0260093i	−0.999915	+	0.0130046i	$0.995860\pi$
$138$		0			0
$139$	8.45124			0.716825			0.358413	−	0.933563i	$-0.383318\pi$
$139$	8.45124			0.716825			0.358413	+	0.933563i	$0.383318\pi$
$140$		0			0
$141$		−	32.3519i		−	2.72452i
$142$		0			0
$143$	−3.35905	−	1.31026i	−0.280898	−	0.109570i
$144$		0			0
$145$		−	35.4535i		−	2.94425i
$146$		0			0
$147$	−2.97211			−0.245135
$148$		0			0
$149$			15.9615i			1.30762i	0.756658	+	0.653811i	$0.226831\pi$
$149$			15.9615i			1.30762i	−0.756658	+	0.653811i	$0.773169\pi$
$150$		0			0
$151$			7.95070i			0.647018i	0.946225	+	0.323509i	$0.104863\pi$
$151$			7.95070i			0.647018i	−0.946225	+	0.323509i	$0.895137\pi$
$152$		0			0
$153$	−29.4665			−2.38222
$154$		0			0
$155$	31.0475			2.49379
$156$		0			0
$157$	−13.3991			−1.06937			−0.534683	−	0.845053i	$-0.679569\pi$
$157$	−13.3991			−1.06937			−0.534683	+	0.845053i	$0.679569\pi$
$158$		0			0
$159$	26.1945			2.07736
$160$		0			0
$161$		−	3.66878i		−	0.289140i
$162$		0			0
$163$		−	0.522685i		−	0.0409398i	−0.999790	−	0.0204699i	$-0.993484\pi$
$163$		−	0.522685i		−	0.0409398i	0.999790	−	0.0204699i	$-0.00651623\pi$
$164$		0			0
$165$	−13.0075			−1.01263
$166$		0			0
$167$		−	7.17381i		−	0.555126i	−0.960707	−	0.277563i	$-0.910473\pi$
$167$		−	7.17381i		−	0.555126i	0.960707	−	0.277563i	$-0.0895267\pi$
$168$		0			0
$169$	−9.56643	−	8.80247i	−0.735879	−	0.677113i
$170$		0			0
$171$		−	2.13042i		−	0.162918i
$172$		0			0
$173$	20.8269			1.58344			0.791719	−	0.610885i	$-0.209186\pi$
$173$	20.8269			1.58344			0.791719	+	0.610885i	$0.209186\pi$
$174$		0			0
$175$		−	14.1541i		−	1.06995i
$176$		0			0
$177$			25.4145i			1.91027i
$178$		0			0
$179$	6.94915			0.519404			0.259702	−	0.965689i	$-0.416376\pi$
$179$	6.94915			0.519404			0.259702	+	0.965689i	$0.416376\pi$
$180$		0			0
$181$	−12.1185			−0.900758			−0.450379	−	0.892838i	$-0.648711\pi$
$181$	−12.1185			−0.900758			−0.450379	+	0.892838i	$0.648711\pi$
$182$		0			0
$183$	−5.59130			−0.413321
$184$		0			0
$185$	−30.5145			−2.24347
$186$		0			0
$187$			5.05132i			0.369389i
$188$		0			0
$189$			8.42121i			0.612553i
$190$		0			0
$191$	14.8420			1.07393			0.536964	−	0.843605i	$-0.319571\pi$
$191$	14.8420			1.07393			0.536964	+	0.843605i	$0.319571\pi$
$192$		0			0
$193$			5.88239i			0.423424i	0.977332	+	0.211712i	$0.0679038\pi$
$193$			5.88239i			0.423424i	−0.977332	+	0.211712i	$0.932096\pi$
$194$		0			0
$195$	−43.6929	−	17.0433i	−3.12892	−	1.22049i
$196$		0			0
$197$			21.8083i			1.55378i	0.629638	+	0.776889i	$0.283203\pi$
$197$			21.8083i			1.55378i	−0.629638	+	0.776889i	$0.716797\pi$
$198$		0			0
$199$	3.18379			0.225693			0.112847	−	0.993612i	$-0.464003\pi$
$199$	3.18379			0.225693			0.112847	+	0.993612i	$0.464003\pi$
$200$		0			0
$201$		−	37.0860i		−	2.61585i
$202$		0			0
$203$			8.10081i			0.568566i
$204$		0			0
$205$	−14.4693			−1.01058
$206$		0			0
$207$	−21.4015			−1.48751
$208$		0			0
$209$	−0.365210			−0.0252621
$210$		0			0
$211$	3.13346			0.215716			0.107858	−	0.994166i	$-0.465601\pi$
$211$	3.13346			0.215716			0.107858	+	0.994166i	$0.465601\pi$
$212$		0			0
$213$		−	16.5914i		−	1.13682i
$214$		0			0
$215$		−	18.8411i		−	1.28496i
$216$		0			0
$217$	−7.09407			−0.481577
$218$		0			0
$219$			5.58648i			0.377500i
$220$		0			0
$221$	−6.61855	+	16.9676i	−0.445212	+	1.14137i
$222$		0			0
$223$			3.00513i			0.201239i	0.994925	+	0.100619i	$0.0320824\pi$
$223$			3.00513i			0.201239i	−0.994925	+	0.100619i	$0.967918\pi$
$224$		0			0
$225$	−82.5665			−5.50443
$226$		0			0
$227$		−	24.4148i		−	1.62046i	−0.586109	−	0.810232i	$-0.699341\pi$
$227$		−	24.4148i		−	1.62046i	0.586109	−	0.810232i	$-0.300659\pi$
$228$		0			0
$229$			19.4735i			1.28685i	0.765510	+	0.643424i	$0.222487\pi$
$229$			19.4735i			1.28685i	−0.765510	+	0.643424i	$0.777513\pi$
$230$		0			0
$231$	2.97211			0.195550
$232$		0			0
$233$	4.77487			0.312812			0.156406	−	0.987693i	$-0.450009\pi$
$233$	4.77487			0.312812			0.156406	+	0.987693i	$0.450009\pi$
$234$		0			0
$235$	−47.6394			−3.10765
$236$		0			0
$237$	1.99112			0.129337
$238$		0			0
$239$			14.0234i			0.907097i	0.891232	+	0.453548i	$0.149842\pi$
$239$			14.0234i			0.907097i	−0.891232	+	0.453548i	$0.850158\pi$
$240$		0			0
$241$		−	28.3006i		−	1.82300i	−0.411300	−	0.911500i	$-0.634925\pi$
$241$		−	28.3006i		−	1.82300i	0.411300	−	0.911500i	$-0.365075\pi$
$242$		0			0
$243$	−2.88815			−0.185275
$244$		0			0
$245$			4.37654i			0.279607i
$246$		0			0
$247$	−1.22676	−	0.478521i	−0.0780568	−	0.0304475i
$248$		0			0
$249$			37.4516i			2.37340i
$250$		0			0
$251$	19.8130			1.25059			0.625294	−	0.780390i	$-0.284979\pi$
$251$	19.8130			1.25059			0.625294	+	0.780390i	$0.284979\pi$
$252$		0			0
$253$			3.66878i			0.230654i
$254$		0			0
$255$			65.7052i			4.11462i
$256$		0			0
$257$	4.38190			0.273335			0.136668	−	0.990617i	$-0.456361\pi$
$257$	4.38190			0.273335			0.136668	+	0.990617i	$0.456361\pi$
$258$		0			0
$259$	6.97230			0.433238
$260$		0			0
$261$	47.2554			2.92504
$262$		0			0
$263$	16.2974			1.00494			0.502472	−	0.864594i	$-0.332424\pi$
$263$	16.2974			1.00494			0.502472	+	0.864594i	$0.332424\pi$
$264$		0			0
$265$		−	38.5724i		−	2.36948i
$266$		0			0
$267$		−	25.8456i		−	1.58172i
$268$		0			0
$269$	17.5407			1.06948			0.534739	−	0.845017i	$-0.320410\pi$
$269$	17.5407			1.06948			0.534739	+	0.845017i	$0.320410\pi$
$270$		0			0
$271$			9.03371i			0.548759i	0.961621	+	0.274379i	$0.0884724\pi$
$271$			9.03371i			0.548759i	−0.961621	+	0.274379i	$0.911528\pi$
$272$		0			0
$273$	9.98345	+	3.89424i	0.604226	+	0.235690i
$274$		0			0
$275$			14.1541i			0.853522i
$276$		0			0
$277$	−25.7696			−1.54834			−0.774172	−	0.632975i	$-0.781833\pi$
$277$	−25.7696			−1.54834			−0.774172	+	0.632975i	$0.781833\pi$
$278$		0			0
$279$			41.3827i			2.47752i
$280$		0			0
$281$		−	3.52275i		−	0.210150i	−0.994464	−	0.105075i	$-0.966492\pi$
$281$		−	3.52275i		−	0.210150i	0.994464	−	0.105075i	$-0.0335082\pi$
$282$		0			0
$283$	22.5438			1.34009			0.670044	−	0.742321i	$-0.266275\pi$
$283$	22.5438			1.34009			0.670044	+	0.742321i	$0.266275\pi$
$284$		0			0
$285$	−4.75048			−0.281394
$286$		0			0
$287$	3.30611			0.195153
$288$		0			0
$289$	8.51585			0.500932
$290$		0			0
$291$			36.2683i			2.12609i
$292$		0			0
$293$		−	11.0431i		−	0.645143i	−0.946545	−	0.322572i	$-0.895453\pi$
$293$		−	11.0431i		−	0.645143i	0.946545	−	0.322572i	$-0.104547\pi$
$294$		0			0
$295$	37.4237			2.17889
$296$		0			0
$297$		−	8.42121i		−	0.488648i
$298$		0			0
$299$	−4.80706	+	12.3236i	−0.277999	+	0.712692i
$300$		0			0
$301$			4.30504i			0.248138i
$302$		0			0
$303$	10.1311			0.582017
$304$		0			0
$305$			8.23339i			0.471443i
$306$		0			0
$307$		−	19.4400i		−	1.10950i	−0.832018	−	0.554749i	$-0.812814\pi$
$307$		−	19.4400i		−	1.10950i	0.832018	−	0.554749i	$-0.187186\pi$
$308$		0			0
$309$	−29.3964			−1.67230
$310$		0			0
$311$	25.4133			1.44105			0.720527	−	0.693427i	$-0.243900\pi$
$311$	25.4133			1.44105			0.720527	+	0.693427i	$0.243900\pi$
$312$		0			0
$313$	−11.4944			−0.649704			−0.324852	−	0.945765i	$-0.605315\pi$
$313$	−11.4944			−0.649704			−0.324852	+	0.945765i	$0.605315\pi$
$314$		0			0
$315$	25.5301			1.43846
$316$		0			0
$317$		−	29.9397i		−	1.68158i	−0.541359	−	0.840792i	$-0.682090\pi$
$317$		−	29.9397i		−	1.68158i	0.541359	−	0.840792i	$-0.317910\pi$
$318$		0			0
$319$		−	8.10081i		−	0.453559i
$320$		0			0
$321$	−29.1451			−1.62672
$322$		0			0
$323$			1.84479i			0.102647i
$324$		0			0
$325$	−18.5455	+	47.5442i	−1.02872	+	2.63728i
$326$		0			0
$327$			1.26804i			0.0701225i
$328$		0			0
$329$	10.8852			0.600120
$330$		0			0
$331$		−	20.6712i		−	1.13619i	−0.822962	−	0.568096i	$-0.807680\pi$
$331$		−	20.6712i		−	1.13619i	0.822962	−	0.568096i	$-0.192320\pi$
$332$		0			0
$333$		−	40.6723i		−	2.22883i
$334$		0			0
$335$	−54.6106			−2.98369
$336$		0			0
$337$	1.80964			0.0985772			0.0492886	−	0.998785i	$-0.484305\pi$
$337$	1.80964			0.0985772			0.0492886	+	0.998785i	$0.484305\pi$
$338$		0			0
$339$	−39.7138			−2.15696
$340$		0			0
$341$	7.09407			0.384166
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			47.7217i			2.56925i
$346$		0			0
$347$	17.5981			0.944718			0.472359	−	0.881406i	$-0.343403\pi$
$347$	17.5981			0.944718			0.472359	+	0.881406i	$0.343403\pi$
$348$		0			0
$349$		−	12.0035i		−	0.642535i	−0.946988	−	0.321268i	$-0.895891\pi$
$349$		−	12.0035i		−	0.642535i	0.946988	−	0.321268i	$-0.104109\pi$
$350$		0			0
$351$	11.0340	−	28.2873i	0.588951	−	1.50986i
$352$		0			0
$353$		−	15.2431i		−	0.811307i	−0.914027	−	0.405654i	$-0.867044\pi$
$353$		−	15.2431i		−	0.811307i	0.914027	−	0.405654i	$-0.132956\pi$
$354$		0			0
$355$	−24.4314			−1.29668
$356$		0			0
$357$		−	15.0131i		−	0.794576i
$358$		0			0
$359$			12.7953i			0.675309i	0.941270	+	0.337654i	$0.109633\pi$
$359$			12.7953i			0.675309i	−0.941270	+	0.337654i	$0.890367\pi$
$360$		0			0
$361$	18.8666			0.992980
$362$		0			0
$363$	−2.97211			−0.155995
$364$		0			0
$365$	8.22630			0.430584
$366$		0			0
$367$	−19.5221			−1.01905			−0.509523	−	0.860457i	$-0.670178\pi$
$367$	−19.5221			−1.01905			−0.509523	+	0.860457i	$0.670178\pi$
$368$		0			0
$369$		−	19.2859i		−	1.00398i
$370$		0			0
$371$			8.81346i			0.457572i
$372$		0			0
$373$	32.8112			1.69890			0.849449	−	0.527671i	$-0.176935\pi$
$373$	32.8112			1.69890			0.849449	+	0.527671i	$0.176935\pi$
$374$		0			0
$375$			119.072i			6.14883i
$376$		0			0
$377$	10.6142	−	27.2110i	0.546658	−	1.40144i
$378$		0			0
$379$			2.24744i			0.115443i	0.998333	+	0.0577216i	$0.0183836\pi$
$379$			2.24744i			0.115443i	−0.998333	+	0.0577216i	$0.981616\pi$
$380$		0			0
$381$	34.7562			1.78061
$382$		0			0
$383$			6.41373i			0.327726i	0.986483	+	0.163863i	$0.0523955\pi$
$383$			6.41373i			0.327726i	−0.986483	+	0.163863i	$0.947604\pi$
$384$		0			0
$385$		−	4.37654i		−	0.223049i
$386$		0			0
$387$	25.1131			1.27657
$388$		0			0
$389$	−2.79047			−0.141483			−0.0707413	−	0.997495i	$-0.522536\pi$
$389$	−2.79047			−0.141483			−0.0707413	+	0.997495i	$0.522536\pi$
$390$		0			0
$391$	18.5322			0.937212
$392$		0			0
$393$	65.7295			3.31561
$394$		0			0
$395$		−	2.93200i		−	0.147525i
$396$		0			0
$397$			8.53813i			0.428516i	0.976777	+	0.214258i	$0.0687334\pi$
$397$			8.53813i			0.428516i	−0.976777	+	0.214258i	$0.931267\pi$
$398$		0			0
$399$	1.08544			0.0543401
$400$		0			0
$401$			27.1699i			1.35680i	0.734693	+	0.678400i	$0.237326\pi$
$401$			27.1699i			1.35680i	−0.734693	+	0.678400i	$0.762674\pi$
$402$		0			0
$403$	23.8293	+	9.29509i	1.18702	+	0.463021i
$404$		0			0
$405$		−	32.9487i		−	1.63724i
$406$		0			0
$407$	−6.97230			−0.345604
$408$		0			0
$409$			13.2682i			0.656070i	0.944666	+	0.328035i	$0.106386\pi$
$409$			13.2682i			0.656070i	−0.944666	+	0.328035i	$0.893614\pi$
$410$		0			0
$411$			0.904801i			0.0446305i
$412$		0			0
$413$	−8.55100			−0.420767
$414$		0			0
$415$	55.1488			2.70715
$416$		0			0
$417$	25.1180			1.23003
$418$		0			0
$419$	8.00787			0.391210			0.195605	−	0.980683i	$-0.437333\pi$
$419$	8.00787			0.391210			0.195605	+	0.980683i	$0.437333\pi$
$420$		0			0
$421$		−	3.92460i		−	0.191273i	−0.995416	−	0.0956367i	$-0.969511\pi$
$421$		−	3.92460i		−	0.191273i	0.995416	−	0.0956367i	$-0.0304887\pi$
$422$		0			0
$423$		−	63.4978i		−	3.08737i
$424$		0			0
$425$	71.4967			3.46810
$426$		0			0
$427$		−	1.88126i		−	0.0910405i
$428$		0			0
$429$	−9.98345	−	3.89424i	−0.482006	−	0.188016i
$430$		0			0
$431$			18.4567i			0.889029i	0.895772	+	0.444515i	$0.146624\pi$
$431$			18.4567i			0.889029i	−0.895772	+	0.444515i	$0.853376\pi$
$432$		0			0
$433$	−23.7708			−1.14235			−0.571176	−	0.820827i	$-0.693513\pi$
$433$	−23.7708			−1.14235			−0.571176	+	0.820827i	$0.693513\pi$
$434$		0			0
$435$		−	105.372i		−	5.05218i
$436$		0			0
$437$			1.33987i			0.0640949i
$438$		0			0
$439$	35.6058			1.69937			0.849687	−	0.527288i	$-0.176791\pi$
$439$	35.6058			1.69937			0.849687	+	0.527288i	$0.176791\pi$
$440$		0			0
$441$	−5.83342			−0.277782
$442$		0			0
$443$	1.83490			0.0871785			0.0435892	−	0.999050i	$-0.486121\pi$
$443$	1.83490			0.0871785			0.0435892	+	0.999050i	$0.486121\pi$
$444$		0			0
$445$	−38.0585			−1.80415
$446$		0			0
$447$			47.4394i			2.24381i
$448$		0			0
$449$			25.4778i			1.20237i	0.799109	+	0.601186i	$0.205305\pi$
$449$			25.4778i			1.20237i	−0.799109	+	0.601186i	$0.794695\pi$
$450$		0			0
$451$	−3.30611			−0.155678
$452$		0			0
$453$			23.6303i			1.11025i
$454$		0			0
$455$	5.73441	−	14.7010i	0.268833	−	0.689193i
$456$		0			0
$457$			21.7753i			1.01861i	0.860587	+	0.509303i	$0.170097\pi$
$457$			21.7753i			1.01861i	−0.860587	+	0.509303i	$0.829903\pi$
$458$		0			0
$459$	−42.5383			−1.98552
$460$		0			0
$461$		−	12.3915i		−	0.577131i	−0.957460	−	0.288566i	$-0.906822\pi$
$461$		−	12.3915i		−	0.577131i	0.957460	−	0.288566i	$-0.0931784\pi$
$462$		0			0
$463$			0.686488i			0.0319038i	0.999873	+	0.0159519i	$0.00507787\pi$
$463$			0.686488i			0.0319038i	−0.999873	+	0.0159519i	$0.994922\pi$
$464$		0			0
$465$	92.2764			4.27921
$466$		0			0
$467$	−28.3829			−1.31340			−0.656702	−	0.754151i	$-0.728049\pi$
$467$	−28.3829			−1.31340			−0.656702	+	0.754151i	$0.728049\pi$
$468$		0			0
$469$	12.4780			0.576182
$470$		0			0
$471$	−39.8236			−1.83497
$472$		0			0
$473$		−	4.30504i		−	0.197946i
$474$		0			0
$475$			5.16921i			0.237179i
$476$		0			0
$477$	51.4126			2.35402
$478$		0			0
$479$		−	0.390732i		−	0.0178530i	−0.999960	−	0.00892649i	$-0.997159\pi$
$479$		−	0.390732i		−	0.0178530i	0.999960	−	0.00892649i	$-0.00284143\pi$
$480$		0			0
$481$	−23.4203	−	9.13554i	−1.06787	−	0.416545i
$482$		0			0
$483$		−	10.9040i		−	0.496149i
$484$		0			0
$485$	53.4064			2.42506
$486$		0			0
$487$			26.7504i			1.21217i	0.795398	+	0.606087i	$0.207262\pi$
$487$			26.7504i			1.21217i	−0.795398	+	0.606087i	$0.792738\pi$
$488$		0			0
$489$		−	1.55348i		−	0.0702506i
$490$		0			0
$491$	40.0180			1.80599			0.902994	−	0.429653i	$-0.141364\pi$
$491$	40.0180			1.80599			0.902994	+	0.429653i	$0.141364\pi$
$492$		0			0
$493$	−40.9198			−1.84294
$494$		0			0
$495$	−25.5301			−1.14749
$496$		0			0
$497$	5.58236			0.250403
$498$		0			0
$499$			3.83434i			0.171649i	0.996310	+	0.0858243i	$0.0273524\pi$
$499$			3.83434i			0.171649i	−0.996310	+	0.0858243i	$0.972648\pi$
$500$		0			0
$501$		−	21.3213i		−	0.952566i
$502$		0			0
$503$	−1.01192			−0.0451194			−0.0225597	−	0.999745i	$-0.507182\pi$
$503$	−1.01192			−0.0451194			−0.0225597	+	0.999745i	$0.507182\pi$
$504$		0			0
$505$		−	14.9184i		−	0.663861i
$506$		0			0
$507$	−28.4324	−	26.1619i	−1.26273	−	1.16189i
$508$		0			0
$509$		−	27.8812i		−	1.23581i	−0.786253	−	0.617905i	$-0.787982\pi$
$509$		−	27.8812i		−	1.23581i	0.786253	−	0.617905i	$-0.212018\pi$
$510$		0			0
$511$	−1.87964			−0.0831503
$512$		0			0
$513$		−	3.07551i		−	0.135787i
$514$		0			0
$515$			43.2873i			1.90747i
$516$		0			0
$517$	−10.8852			−0.478730
$518$		0			0
$519$	61.8997			2.71709
$520$		0			0
$521$	2.43277			0.106582			0.0532908	−	0.998579i	$-0.483029\pi$
$521$	2.43277			0.106582			0.0532908	+	0.998579i	$0.483029\pi$
$522$		0			0
$523$	−26.6725			−1.16631			−0.583153	−	0.812362i	$-0.698181\pi$
$523$	−26.6725			−1.16631			−0.583153	+	0.812362i	$0.698181\pi$
$524$		0			0
$525$		−	42.0674i		−	1.83597i
$526$		0			0
$527$		−	35.8344i		−	1.56097i
$528$		0			0
$529$	−9.54008			−0.414786
$530$		0			0
$531$			49.8815i			2.16467i
$532$		0			0
$533$	−11.1054	−	4.33186i	−0.481027	−	0.187634i
$534$		0			0
$535$			42.9172i			1.85547i
$536$		0			0
$537$	20.6536			0.891270
$538$		0			0
$539$			1.00000i			0.0430730i
$540$		0			0
$541$			3.65306i			0.157057i	0.996912	+	0.0785287i	$0.0250222\pi$
$541$			3.65306i			0.157057i	−0.996912	+	0.0785287i	$0.974978\pi$
$542$		0			0
$543$	−36.0173			−1.54565
$544$		0			0
$545$	1.86723			0.0799833
$546$		0			0
$547$	−29.8746			−1.27735			−0.638674	−	0.769478i	$-0.720517\pi$
$547$	−29.8746			−1.27735			−0.638674	+	0.769478i	$0.720517\pi$
$548$		0			0
$549$	−10.9742			−0.468366
$550$		0			0
$551$		−	2.95850i		−	0.126036i
$552$		0			0
$553$			0.669936i			0.0284886i
$554$		0			0
$555$	−90.6924			−3.84968
$556$		0			0
$557$			37.8282i			1.60283i	0.598108	+	0.801415i	$0.295919\pi$
$557$			37.8282i			1.60283i	−0.598108	+	0.801415i	$0.704081\pi$
$558$		0			0
$559$	5.64072	−	14.4608i	0.238577	−	0.611628i
$560$		0			0
$561$			15.0131i			0.633852i
$562$		0			0
$563$	2.39597			0.100978			0.0504891	−	0.998725i	$-0.483922\pi$
$563$	2.39597			0.100978			0.0504891	+	0.998725i	$0.483922\pi$
$564$		0			0
$565$			58.4800i			2.46027i
$566$		0			0
$567$			7.52850i			0.316167i
$568$		0			0
$569$	−31.8603			−1.33565			−0.667827	−	0.744316i	$-0.732775\pi$
$569$	−31.8603			−1.33565			−0.667827	+	0.744316i	$0.732775\pi$
$570$		0			0
$571$	5.15719			0.215822			0.107911	−	0.994161i	$-0.465584\pi$
$571$	5.15719			0.215822			0.107911	+	0.994161i	$0.465584\pi$
$572$		0			0
$573$	44.1119			1.84280
$574$		0			0
$575$	51.9281			2.16555
$576$		0			0
$577$			13.7534i			0.572563i	0.958146	+	0.286282i	$0.0924193\pi$
$577$			13.7534i			0.572563i	−0.958146	+	0.286282i	$0.907581\pi$
$578$		0			0
$579$			17.4831i			0.726572i
$580$		0			0
$581$	−12.6010			−0.522779
$582$		0			0
$583$		−	8.81346i		−	0.365016i
$584$		0			0
$585$	−85.7570	−	33.4512i	−3.54562	−	1.38304i
$586$		0			0
$587$		−	41.7726i		−	1.72414i	−0.506788	−	0.862070i	$-0.669167\pi$
$587$		−	41.7726i		−	1.72414i	0.506788	−	0.862070i	$-0.330833\pi$
$588$		0			0
$589$	2.59083			0.106753
$590$		0			0
$591$			64.8166i			2.66620i
$592$		0			0
$593$		−	16.2387i		−	0.666844i	−0.942778	−	0.333422i	$-0.891797\pi$
$593$		−	16.2387i		−	0.666844i	0.942778	−	0.333422i	$-0.108203\pi$
$594$		0			0
$595$	−22.1073			−0.906310
$596$		0			0
$597$	9.46258			0.387277
$598$		0			0
$599$	16.4777			0.673261			0.336631	−	0.941637i	$-0.390713\pi$
$599$	16.4777			0.673261			0.336631	+	0.941637i	$0.390713\pi$
$600$		0			0
$601$	−19.5466			−0.797322			−0.398661	−	0.917098i	$-0.630525\pi$
$601$	−19.5466			−0.797322			−0.398661	+	0.917098i	$0.630525\pi$
$602$		0			0
$603$		−	72.7896i		−	2.96422i
$604$		0			0
$605$			4.37654i			0.177931i
$606$		0			0
$607$	−45.8970			−1.86290			−0.931452	−	0.363865i	$-0.881457\pi$
$607$	−45.8970			−1.86290			−0.931452	+	0.363865i	$0.881457\pi$
$608$		0			0
$609$			24.0765i			0.975628i
$610$		0			0
$611$	−36.5639	−	14.2624i	−1.47922	−	0.576997i
$612$		0			0
$613$			30.0636i			1.21426i	0.794604	+	0.607128i	$0.207678\pi$
$613$			30.0636i			1.21426i	−0.794604	+	0.607128i	$0.792322\pi$
$614$		0			0
$615$	−43.0043			−1.73410
$616$		0			0
$617$			13.2390i			0.532983i	0.963837	+	0.266491i	$0.0858644\pi$
$617$			13.2390i			0.532983i	−0.963837	+	0.266491i	$0.914136\pi$
$618$		0			0
$619$		−	3.38558i		−	0.136078i	−0.997683	−	0.0680390i	$-0.978326\pi$
$619$		−	3.38558i		−	0.136078i	0.997683	−	0.0680390i	$-0.0216742\pi$
$620$		0			0
$621$	−30.8956			−1.23980
$622$		0			0
$623$	8.69604			0.348400
$624$		0			0
$625$	104.567			4.18268
$626$		0			0
$627$	−1.08544			−0.0433484
$628$		0			0
$629$			35.2193i			1.40429i
$630$		0			0
$631$		−	45.0471i		−	1.79330i	−0.442743	−	0.896648i	$-0.645995\pi$
$631$		−	45.0471i		−	1.79330i	0.442743	−	0.896648i	$-0.354005\pi$
$632$		0			0
$633$	9.31297			0.370157
$634$		0			0
$635$		−	51.1798i		−	2.03101i
$636$		0			0
$637$	−1.31026	+	3.35905i	−0.0519144	+	0.133090i
$638$		0			0
$639$		−	32.5642i		−	1.28822i
$640$		0			0
$641$	−25.9217			−1.02384			−0.511922	−	0.859032i	$-0.671066\pi$
$641$	−25.9217			−1.02384			−0.511922	+	0.859032i	$0.671066\pi$
$642$		0			0
$643$			6.63578i			0.261690i	0.991403	+	0.130845i	$0.0417690\pi$
$643$			6.63578i			0.261690i	−0.991403	+	0.130845i	$0.958231\pi$
$644$		0			0
$645$		−	55.9979i		−	2.20491i
$646$		0			0
$647$	40.6450			1.59792			0.798959	−	0.601385i	$-0.205384\pi$
$647$	40.6450			1.59792			0.798959	+	0.601385i	$0.205384\pi$
$648$		0			0
$649$	8.55100			0.335656
$650$		0			0
$651$	−21.0843			−0.826361
$652$		0			0
$653$	36.7087			1.43652			0.718261	−	0.695774i	$-0.244938\pi$
$653$	36.7087			1.43652			0.718261	+	0.695774i	$0.244938\pi$
$654$		0			0
$655$		−	96.7891i		−	3.78186i
$656$		0			0
$657$			10.9647i			0.427774i
$658$		0			0
$659$	−42.4542			−1.65378			−0.826890	−	0.562364i	$-0.809892\pi$
$659$	−42.4542			−1.65378			−0.826890	+	0.562364i	$0.809892\pi$
$660$		0			0
$661$		−	13.7370i		−	0.534306i	−0.963654	−	0.267153i	$-0.913917\pi$
$661$		−	13.7370i		−	0.534306i	0.963654	−	0.267153i	$-0.0860828\pi$
$662$		0			0
$663$	−19.6710	+	50.4296i	−0.763960	+	1.95852i
$664$		0			0
$665$		−	1.59836i		−	0.0619816i
$666$		0			0
$667$	−29.7201			−1.15077
$668$		0			0
$669$			8.93157i			0.345315i
$670$		0			0
$671$			1.88126i			0.0726252i
$672$		0			0
$673$	31.6543			1.22018			0.610092	−	0.792331i	$-0.291133\pi$
$673$	31.6543			1.22018			0.610092	+	0.792331i	$0.291133\pi$
$674$		0			0
$675$	−119.194			−4.58779
$676$		0			0
$677$	35.3747			1.35956			0.679779	−	0.733417i	$-0.262075\pi$
$677$	35.3747			1.35956			0.679779	+	0.733417i	$0.262075\pi$
$678$		0			0
$679$	−12.2029			−0.468304
$680$		0			0
$681$		−	72.5633i		−	2.78063i
$682$		0			0
$683$		−	1.57174i		−	0.0601411i	−0.999548	−	0.0300706i	$-0.990427\pi$
$683$		−	1.57174i		−	0.0601411i	0.999548	−	0.0300706i	$-0.00957320\pi$
$684$		0			0
$685$	1.33235			0.0509066
$686$		0			0
$687$			57.8775i			2.20816i
$688$		0			0
$689$	11.5479	−	29.6048i	0.439941	−	1.12785i
$690$		0			0
$691$			21.6781i			0.824672i	0.911032	+	0.412336i	$0.135287\pi$
$691$			21.6781i			0.824672i	−0.911032	+	0.412336i	$0.864713\pi$
$692$		0			0
$693$	5.83342			0.221593
$694$		0			0
$695$		−	36.9872i		−	1.40300i
$696$		0			0
$697$			16.7002i			0.632565i
$698$		0			0
$699$	14.1914			0.536769
$700$		0			0
$701$	15.5541			0.587472			0.293736	−	0.955887i	$-0.405101\pi$
$701$	15.5541			0.587472			0.293736	+	0.955887i	$0.405101\pi$
$702$		0			0
$703$	−2.54635			−0.0960376
$704$		0			0
$705$	−141.589			−5.33256
$706$		0			0
$707$			3.40873i			0.128198i
$708$		0			0
$709$			19.1915i			0.720753i	0.932807	+	0.360376i	$0.117352\pi$
$709$			19.1915i			0.720753i	−0.932807	+	0.360376i	$0.882648\pi$
$710$		0			0
$711$	3.90802			0.146562
$712$		0			0
$713$		−	26.0266i		−	0.974703i
$714$		0			0
$715$	−5.73441	+	14.7010i	−0.214455	+	0.549786i
$716$		0			0
$717$			41.6790i			1.55653i
$718$		0			0
$719$	−0.196763			−0.00733803			−0.00366902	−	0.999993i	$-0.501168\pi$
$719$	−0.196763			−0.00733803			−0.00366902	+	0.999993i	$0.501168\pi$
$720$		0			0
$721$		−	9.89078i		−	0.368352i
$722$		0			0
$723$		−	84.1123i		−	3.12817i
$724$		0			0
$725$	−114.659			−4.25834
$726$		0			0
$727$	18.9703			0.703569			0.351784	−	0.936081i	$-0.385575\pi$
$727$	18.9703			0.703569			0.351784	+	0.936081i	$0.385575\pi$
$728$		0			0
$729$	−31.1694			−1.15442
$730$		0			0
$731$	−21.7461			−0.804309
$732$		0			0
$733$			5.98245i			0.220967i	0.993878	+	0.110483i	$0.0352399\pi$
$733$			5.98245i			0.220967i	−0.993878	+	0.110483i	$0.964760\pi$
$734$		0			0
$735$			13.0075i			0.479790i
$736$		0			0
$737$	−12.4780			−0.459634
$738$		0			0
$739$			21.3883i			0.786780i	0.919372	+	0.393390i	$0.128698\pi$
$739$			21.3883i			0.786780i	−0.919372	+	0.393390i	$0.871302\pi$
$740$		0			0
$741$	−3.64606	−	1.42221i	−0.133941	−	0.0522464i
$742$		0			0
$743$			32.4297i			1.18973i	0.803826	+	0.594865i	$0.202794\pi$
$743$			32.4297i			1.18973i	−0.803826	+	0.594865i	$0.797206\pi$
$744$		0			0
$745$	69.8563			2.55934
$746$		0			0
$747$			73.5070i			2.68948i
$748$		0			0
$749$		−	9.80621i		−	0.358311i
$750$		0			0
$751$	34.1195			1.24504			0.622519	−	0.782605i	$-0.286109\pi$
$751$	34.1195			1.24504			0.622519	+	0.782605i	$0.286109\pi$
$752$		0			0
$753$	58.8864			2.14594
$754$		0			0
$755$	34.7965			1.26637
$756$		0			0
$757$	22.7448			0.826675			0.413337	−	0.910578i	$-0.364363\pi$
$757$	22.7448			0.826675			0.413337	+	0.910578i	$0.364363\pi$
$758$		0			0
$759$			10.9040i			0.395790i
$760$		0			0
$761$		−	35.0219i		−	1.26954i	−0.772700	−	0.634771i	$-0.781094\pi$
$761$		−	35.0219i		−	1.26954i	0.772700	−	0.634771i	$-0.218906\pi$
$762$		0			0
$763$	−0.426645			−0.0154456
$764$		0			0
$765$			128.961i			4.66259i
$766$		0			0
$767$	28.7232	+	11.2040i	1.03714	+	0.404555i
$768$		0			0
$769$			16.6468i			0.600299i	0.953892	+	0.300149i	$0.0970366\pi$
$769$			16.6468i			0.600299i	−0.953892	+	0.300149i	$0.902963\pi$
$770$		0			0
$771$	13.0235			0.469029
$772$		0			0
$773$			10.5398i			0.379089i	0.981872	+	0.189544i	$0.0607011\pi$
$773$			10.5398i			0.379089i	−0.981872	+	0.189544i	$0.939299\pi$
$774$		0			0
$775$		−	100.410i		−	3.60683i
$776$		0			0
$777$	20.7224			0.743413
$778$		0			0
$779$	−1.20742			−0.0432604
$780$		0			0
$781$	−5.58236			−0.199752
$782$		0			0
$783$	68.2187			2.43794
$784$		0			0
$785$			58.6417i			2.09301i
$786$		0			0
$787$		−	42.9678i		−	1.53164i	−0.643057	−	0.765819i	$-0.722334\pi$
$787$		−	42.9678i		−	1.53164i	0.643057	−	0.765819i	$-0.277666\pi$
$788$		0			0
$789$	48.4377			1.72443
$790$		0			0
$791$		−	13.3622i		−	0.475104i
$792$		0			0
$793$	−2.46494	+	6.31924i	−0.0875326	+	0.224403i
$794$		0			0
$795$		−	114.641i		−	4.06591i
$796$		0			0
$797$	16.0948			0.570107			0.285054	−	0.958512i	$-0.407989\pi$
$797$	16.0948			0.570107			0.285054	+	0.958512i	$0.407989\pi$
$798$		0			0
$799$			54.9846i			1.94521i
$800$		0			0
$801$		−	50.7276i		−	1.79237i
$802$		0			0
$803$	1.87964			0.0663310
$804$		0			0
$805$	−16.0565			−0.565918
$806$		0			0
$807$	52.1329			1.83517
$808$		0			0
$809$	−32.0110			−1.12545			−0.562723	−	0.826645i	$-0.690246\pi$
$809$	−32.0110			−1.12545			−0.562723	+	0.826645i	$0.690246\pi$
$810$		0			0
$811$			32.7581i			1.15029i	0.818050	+	0.575146i	$0.195055\pi$
$811$			32.7581i			1.15029i	−0.818050	+	0.575146i	$0.804945\pi$
$812$		0			0
$813$			26.8492i			0.941641i
$814$		0			0
$815$	−2.28755			−0.0801293
$816$		0			0
$817$		−	1.57224i		−	0.0550058i
$818$		0			0
$819$	19.5947	+	7.64330i	0.684695	+	0.267079i
$820$		0			0
$821$		−	2.91947i		−	0.101890i	−0.998701	−	0.0509451i	$-0.983777\pi$
$821$		−	2.91947i		−	0.101890i	0.998701	−	0.0509451i	$-0.0162233\pi$
$822$		0			0
$823$	−38.7566			−1.35097			−0.675484	−	0.737374i	$-0.736065\pi$
$823$	−38.7566			−1.35097			−0.675484	+	0.737374i	$0.736065\pi$
$824$		0			0
$825$			42.0674i			1.46460i
$826$		0			0
$827$			30.7485i			1.06923i	0.845095	+	0.534616i	$0.179544\pi$
$827$			30.7485i			1.06923i	−0.845095	+	0.534616i	$0.820456\pi$
$828$		0			0
$829$	−13.3787			−0.464662			−0.232331	−	0.972637i	$-0.574635\pi$
$829$	−13.3787			−0.464662			−0.232331	+	0.972637i	$0.574635\pi$
$830$		0			0
$831$	−76.5899			−2.65687
$832$		0			0
$833$	5.05132			0.175018
$834$		0			0
$835$	−31.3964			−1.08652
$836$		0			0
$837$			59.7407i			2.06494i
$838$		0			0
$839$		−	11.0368i		−	0.381033i	−0.981684	−	0.190516i	$-0.938984\pi$
$839$		−	11.0368i		−	0.381033i	0.981684	−	0.190516i	$-0.0610162\pi$
$840$		0			0
$841$	36.6232			1.26287
$842$		0			0
$843$		−	10.4700i		−	0.360606i
$844$		0			0
$845$	−38.5243	+	41.8678i	−1.32528	+	1.44030i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$	67.0025			2.29952
$850$		0			0
$851$			25.5798i			0.876865i
$852$		0			0
$853$			12.9506i			0.443420i	0.975113	+	0.221710i	$0.0711639\pi$
$853$			12.9506i			0.443420i	−0.975113	+	0.221710i	$0.928836\pi$
$854$		0			0
$855$	−9.32387			−0.318870
$856$		0			0
$857$	−47.0677			−1.60780			−0.803902	−	0.594762i	$-0.797246\pi$
$857$	−47.0677			−1.60780			−0.803902	+	0.594762i	$0.797246\pi$
$858$		0			0
$859$	40.7423			1.39011			0.695055	−	0.718956i	$-0.255380\pi$
$859$	40.7423			1.39011			0.695055	+	0.718956i	$0.255380\pi$
$860$		0			0
$861$	9.82610			0.334873
$862$		0			0
$863$		−	5.23560i		−	0.178222i	−0.996022	−	0.0891110i	$-0.971597\pi$
$863$		−	5.23560i		−	0.178222i	0.996022	−	0.0891110i	$-0.0284026\pi$
$864$		0			0
$865$		−	91.1496i		−	3.09918i
$866$		0			0
$867$	25.3100			0.859573
$868$		0			0
$869$		−	0.669936i		−	0.0227260i
$870$		0			0
$871$	−41.9143	−	16.3495i	−1.42021	−	0.553981i
$872$		0			0
$873$			71.1846i			2.40923i
$874$		0			0
$875$	−40.0631			−1.35438
$876$		0			0
$877$		−	27.2979i		−	0.921785i	−0.887456	−	0.460892i	$-0.847529\pi$
$877$		−	27.2979i		−	0.921785i	0.887456	−	0.460892i	$-0.152471\pi$
$878$		0			0
$879$		−	32.8212i		−	1.10703i
$880$		0			0
$881$	−17.3070			−0.583089			−0.291545	−	0.956557i	$-0.594169\pi$
$881$	−17.3070			−0.583089			−0.291545	+	0.956557i	$0.594169\pi$
$882$		0			0
$883$	−25.5191			−0.858787			−0.429394	−	0.903117i	$-0.641273\pi$
$883$	−25.5191			−0.858787			−0.429394	+	0.903117i	$0.641273\pi$
$884$		0			0
$885$	111.227			3.73887
$886$		0			0
$887$	33.7338			1.13267			0.566335	−	0.824175i	$-0.308361\pi$
$887$	33.7338			1.13267			0.566335	+	0.824175i	$0.308361\pi$
$888$		0			0
$889$			11.6941i			0.392209i
$890$		0			0
$891$		−	7.52850i		−	0.252214i
$892$		0			0
$893$	−3.97538			−0.133031
$894$		0			0
$895$		−	30.4132i		−	1.01660i
$896$		0			0
$897$	−14.2871	+	36.6271i	−0.477032	+	1.22294i
$898$		0			0
$899$			57.4678i			1.91666i
$900$		0			0
$901$	−44.5196			−1.48316
$902$		0			0
$903$			12.7950i			0.425792i
$904$		0			0
$905$			53.0368i			1.76300i
$906$		0			0
$907$	6.75199			0.224196			0.112098	−	0.993697i	$-0.464243\pi$
$907$	6.75199			0.224196			0.112098	+	0.993697i	$0.464243\pi$
$908$		0			0
$909$	19.8845			0.659528
$910$		0			0
$911$	55.2106			1.82921			0.914604	−	0.404350i	$-0.132502\pi$
$911$	55.2106			1.82921			0.914604	+	0.404350i	$0.132502\pi$
$912$		0			0
$913$	12.6010			0.417033
$914$		0			0
$915$			24.4705i			0.808971i
$916$		0			0
$917$			22.1155i			0.730317i
$918$		0			0
$919$	25.1731			0.830382			0.415191	−	0.909734i	$-0.363715\pi$
$919$	25.1731			0.830382			0.415191	+	0.909734i	$0.363715\pi$
$920$		0			0
$921$		−	57.7776i		−	1.90384i
$922$		0			0
$923$	−18.7514	−	7.31435i	−0.617210	−	0.240755i
$924$		0			0
$925$			98.6864i			3.24479i
$926$		0			0
$927$	−57.6970			−1.89502
$928$		0			0
$929$			45.5508i			1.49447i	0.664557	+	0.747237i	$0.268620\pi$
$929$			45.5508i			1.49447i	−0.664557	+	0.747237i	$0.731380\pi$
$930$		0			0
$931$			0.365210i			0.0119693i
$932$		0			0
$933$	75.5309			2.47277
$934$		0			0
$935$	22.1073			0.722985
$936$		0			0
$937$	−46.1218			−1.50673			−0.753367	−	0.657600i	$-0.771572\pi$
$937$	−46.1218			−1.50673			−0.753367	+	0.657600i	$0.771572\pi$
$938$		0			0
$939$	−34.1627			−1.11486
$940$		0			0
$941$			45.2412i			1.47482i	0.675445	+	0.737410i	$0.263951\pi$
$941$			45.2412i			1.47482i	−0.675445	+	0.737410i	$0.736049\pi$
$942$		0			0
$943$			12.1294i			0.394987i
$944$		0			0
$945$	36.8557			1.19892
$946$		0			0
$947$		−	11.3895i		−	0.370110i	−0.982728	−	0.185055i	$-0.940754\pi$
$947$		−	11.3895i		−	0.370110i	0.982728	−	0.185055i	$-0.0592463\pi$
$948$		0			0
$949$	6.31380	+	2.46282i	0.204955	+	0.0799464i
$950$		0			0
$951$		−	88.9841i		−	2.88551i
$952$		0			0
$953$	−35.7292			−1.15738			−0.578691	−	0.815547i	$-0.696436\pi$
$953$	−35.7292			−1.15738			−0.578691	+	0.815547i	$0.696436\pi$
$954$		0			0
$955$		−	64.9564i		−	2.10194i
$956$		0			0
$957$		−	24.0765i		−	0.778282i
$958$		0			0
$959$	−0.304431			−0.00983059
$960$		0			0
$961$	−19.3259			−0.623416
$962$		0			0
$963$	−57.2037			−1.84336
$964$		0			0
$965$	25.7445			0.828744
$966$		0			0
$967$		−	5.16544i		−	0.166109i	−0.996545	−	0.0830546i	$-0.973532\pi$
$967$		−	5.16544i		−	0.166109i	0.996545	−	0.0830546i	$-0.0264676\pi$
$968$		0			0
$969$			5.48292i			0.176137i
$970$		0			0
$971$	31.3918			1.00741			0.503706	−	0.863875i	$-0.331970\pi$
$971$	31.3918			1.00741			0.503706	+	0.863875i	$0.331970\pi$
$972$		0			0
$973$			8.45124i			0.270934i
$974$		0			0
$975$	−55.1193	+	141.306i	−1.76523	+	4.52543i
$976$		0			0
$977$			20.1707i			0.645318i	0.946515	+	0.322659i	$0.104577\pi$
$977$			20.1707i			0.645318i	−0.946515	+	0.322659i	$0.895423\pi$
$978$		0			0
$979$	−8.69604			−0.277927
$980$		0			0
$981$			2.48880i			0.0794613i
$982$		0			0
$983$			18.5169i			0.590598i	0.955405	+	0.295299i	$0.0954193\pi$
$983$			18.5169i			0.590598i	−0.955405	+	0.295299i	$0.904581\pi$
$984$		0			0
$985$	95.4448			3.04113
$986$		0			0
$987$	32.3519			1.02977
$988$		0			0
$989$	−15.7942			−0.502227
$990$		0			0
$991$	−2.84381			−0.0903367			−0.0451684	−	0.998979i	$-0.514382\pi$
$991$	−2.84381			−0.0903367			−0.0451684	+	0.998979i	$0.514382\pi$
$992$		0			0
$993$		−	61.4371i		−	1.94965i
$994$		0			0
$995$		−	13.9340i		−	0.441737i
$996$		0			0
$997$	51.6184			1.63477			0.817385	−	0.576092i	$-0.195423\pi$
$997$	51.6184			1.63477			0.817385	+	0.576092i	$0.195423\pi$
$998$		0			0
$999$		−	58.7152i		−	1.85767i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.c.2157.33	✓	36
13.12	even	2	inner	4004.2.m.c.2157.34	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.c.2157.33	✓	36	1.1	even	1	trivial
4004.2.m.c.2157.34	yes	36	13.12	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.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.97211 q^{3} -4.37654i q^{5} +1.00000i q^{7} +5.83342 q^{9} +O(q^{10})\)	\(q+2.97211 q^{3} -4.37654i q^{5} +1.00000i q^{7} +5.83342 q^{9} -1.00000i q^{11} +(1.31026 - 3.35905i) q^{13} -13.0075i q^{15} -5.05132 q^{17} -0.365210i q^{19} +2.97211i q^{21} -3.66878 q^{23} -14.1541 q^{25} +8.42121 q^{27} +8.10081 q^{29} +7.09407i q^{31} -2.97211i q^{33} +4.37654 q^{35} -6.97230i q^{37} +(3.89424 - 9.98345i) q^{39} -3.30611i q^{41} +4.30504 q^{43} -25.5301i q^{45} -10.8852i q^{47} -1.00000 q^{49} -15.0131 q^{51} +8.81346 q^{53} -4.37654 q^{55} -1.08544i q^{57} +8.55100i q^{59} -1.88126 q^{61} +5.83342i q^{63} +(-14.7010 - 5.73441i) q^{65} -12.4780i q^{67} -10.9040 q^{69} -5.58236i q^{71} +1.87964i q^{73} -42.0674 q^{75} +1.00000 q^{77} +0.669936 q^{79} +7.52850 q^{81} +12.6010i q^{83} +22.1073i q^{85} +24.0765 q^{87} -8.69604i q^{89} +(3.35905 + 1.31026i) q^{91} +21.0843i q^{93} -1.59836 q^{95} +12.2029i q^{97} -5.83342i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) 36 * q - 4 * q^3 + 40 * q^9	\( 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) \) 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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