LMFDB - Embedded newform 4004.2.e.b.3849.4

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			3849.4
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.b.3849.45

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.99006i q^{3} +1.25381i q^{5} +(2.36377 + 1.18853i) q^{7} -5.94047 q^{9} +O(q^{10})$	$q-2.99006i q^{3} +1.25381i q^{5} +(2.36377 + 1.18853i) q^{7} -5.94047 q^{9} +(-3.16026 - 1.00636i) q^{11} -1.00000 q^{13} +3.74896 q^{15} +5.49148 q^{17} +3.45158 q^{19} +(3.55378 - 7.06781i) q^{21} +0.0769856 q^{23} +3.42796 q^{25} +8.79219i q^{27} +1.19887i q^{29} +10.2714i q^{31} +(-3.00909 + 9.44937i) q^{33} +(-1.49019 + 2.96371i) q^{35} +2.73571 q^{37} +2.99006i q^{39} -4.91709 q^{41} +9.53368i q^{43} -7.44821i q^{45} +0.917396i q^{47} +(4.17479 + 5.61882i) q^{49} -16.4199i q^{51} +7.45254 q^{53} +(1.26179 - 3.96236i) q^{55} -10.3204i q^{57} -10.7603i q^{59} -11.0348 q^{61} +(-14.0419 - 7.06043i) q^{63} -1.25381i q^{65} -9.82223 q^{67} -0.230192i q^{69} +16.6445 q^{71} +14.4452 q^{73} -10.2498i q^{75} +(-6.27402 - 6.13487i) q^{77} +7.54913i q^{79} +8.46779 q^{81} +7.11686 q^{83} +6.88526i q^{85} +3.58469 q^{87} +3.72868i q^{89} +(-2.36377 - 1.18853i) q^{91} +30.7120 q^{93} +4.32762i q^{95} +7.07399i q^{97} +(18.7734 + 5.97827i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.99006i		−	1.72631i	−0.504936	−	0.863157i	$-0.668484\pi$
$3$		−	2.99006i		−	1.72631i	0.504936	−	0.863157i	$-0.331516\pi$
$4$		0			0
$5$			1.25381i			0.560720i	0.959895	+	0.280360i	$0.0904539\pi$
$5$			1.25381i			0.560720i	−0.959895	+	0.280360i	$0.909546\pi$
$6$		0			0
$7$	2.36377	+	1.18853i	0.893420	+	0.449222i
$7$	2.36377	+	1.18853i	0.893420	+	0.449222i
$8$		0			0
$9$	−5.94047			−1.98016
$10$		0			0
$11$	−3.16026	−	1.00636i	−0.952854	−	0.303430i
$11$	−3.16026	−	1.00636i	−0.952854	−	0.303430i
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	3.74896			0.967979
$16$		0			0
$17$	5.49148			1.33188			0.665939	−	0.746006i	$-0.268031\pi$
$17$	5.49148			1.33188			0.665939	+	0.746006i	$0.268031\pi$
$18$		0			0
$19$	3.45158			0.791847			0.395924	−	0.918283i	$-0.370425\pi$
$19$	3.45158			0.791847			0.395924	+	0.918283i	$0.370425\pi$
$20$		0			0
$21$	3.55378	−	7.06781i	0.775498	−	1.54232i
$22$		0			0
$23$	0.0769856			0.0160526			0.00802630	−	0.999968i	$-0.497445\pi$
$23$	0.0769856			0.0160526			0.00802630	+	0.999968i	$0.497445\pi$
$24$		0			0
$25$	3.42796			0.685593
$26$		0			0
$27$			8.79219i			1.69206i
$28$		0			0
$29$			1.19887i			0.222624i	0.993786	+	0.111312i	$0.0355053\pi$
$29$			1.19887i			0.222624i	−0.993786	+	0.111312i	$0.964495\pi$
$30$		0			0
$31$			10.2714i			1.84479i	0.386248	+	0.922395i	$0.373771\pi$
$31$			10.2714i			1.84479i	−0.386248	+	0.922395i	$0.626229\pi$
$32$		0			0
$33$	−3.00909	+	9.44937i	−0.523815	+	1.64492i
$34$		0			0
$35$	−1.49019	+	2.96371i	−0.251888	+	0.500959i
$36$		0			0
$37$	2.73571			0.449748			0.224874	−	0.974388i	$-0.427803\pi$
$37$	2.73571			0.449748			0.224874	+	0.974388i	$0.427803\pi$
$38$		0			0
$39$			2.99006i			0.478793i
$40$		0			0
$41$	−4.91709			−0.767920			−0.383960	−	0.923350i	$-0.625440\pi$
$41$	−4.91709			−0.767920			−0.383960	+	0.923350i	$0.625440\pi$
$42$		0			0
$43$			9.53368i			1.45387i	0.686705	+	0.726936i	$0.259056\pi$
$43$			9.53368i			1.45387i	−0.686705	+	0.726936i	$0.740944\pi$
$44$		0			0
$45$		−	7.44821i		−	1.11031i
$46$		0			0
$47$			0.917396i			0.133816i	0.997759	+	0.0669080i	$0.0213134\pi$
$47$			0.917396i			0.133816i	−0.997759	+	0.0669080i	$0.978687\pi$
$48$		0			0
$49$	4.17479	+	5.61882i	0.596399	+	0.802688i
$50$		0			0
$51$		−	16.4199i		−	2.29924i
$52$		0			0
$53$	7.45254			1.02369			0.511843	−	0.859079i	$-0.328963\pi$
$53$	7.45254			1.02369			0.511843	+	0.859079i	$0.328963\pi$
$54$		0			0
$55$	1.26179	−	3.96236i	0.170139	−	0.534284i
$56$		0			0
$57$		−	10.3204i		−	1.36698i
$58$		0			0
$59$		−	10.7603i		−	1.40087i	−0.713715	−	0.700436i	$-0.752989\pi$
$59$		−	10.7603i		−	1.40087i	0.713715	−	0.700436i	$-0.247011\pi$
$60$		0			0
$61$	−11.0348			−1.41286			−0.706432	−	0.707781i	$-0.749696\pi$
$61$	−11.0348			−1.41286			−0.706432	+	0.707781i	$0.749696\pi$
$62$		0			0
$63$	−14.0419	−	7.06043i	−1.76911	−	0.889530i
$64$		0			0
$65$		−	1.25381i		−	0.155516i
$66$		0			0
$67$	−9.82223			−1.19998			−0.599988	−	0.800009i	$-0.704828\pi$
$67$	−9.82223			−1.19998			−0.599988	+	0.800009i	$0.704828\pi$
$68$		0			0
$69$		−	0.230192i		−	0.0277118i
$70$		0			0
$71$	16.6445			1.97534			0.987671	−	0.156546i	$-0.0500358\pi$
$71$	16.6445			1.97534			0.987671	+	0.156546i	$0.0500358\pi$
$72$		0			0
$73$	14.4452			1.69069			0.845343	−	0.534224i	$-0.179396\pi$
$73$	14.4452			1.69069			0.845343	+	0.534224i	$0.179396\pi$
$74$		0			0
$75$		−	10.2498i		−	1.18355i
$76$		0			0
$77$	−6.27402	−	6.13487i	−0.714991	−	0.699133i
$78$		0			0
$79$			7.54913i			0.849344i	0.905347	+	0.424672i	$0.139610\pi$
$79$			7.54913i			0.849344i	−0.905347	+	0.424672i	$0.860390\pi$
$80$		0			0
$81$	8.46779			0.940865
$82$		0			0
$83$	7.11686			0.781176			0.390588	−	0.920566i	$-0.372272\pi$
$83$	7.11686			0.781176			0.390588	+	0.920566i	$0.372272\pi$
$84$		0			0
$85$			6.88526i			0.746811i
$86$		0			0
$87$	3.58469			0.384319
$88$		0			0
$89$			3.72868i			0.395239i	0.980279	+	0.197620i	$0.0633211\pi$
$89$			3.72868i			0.395239i	−0.980279	+	0.197620i	$0.936679\pi$
$90$		0			0
$91$	−2.36377	−	1.18853i	−0.247790	−	0.124592i
$92$		0			0
$93$	30.7120			3.18468
$94$		0			0
$95$			4.32762i			0.444005i
$96$		0			0
$97$			7.07399i			0.718254i	0.933289	+	0.359127i	$0.116926\pi$
$97$			7.07399i			0.718254i	−0.933289	+	0.359127i	$0.883074\pi$
$98$		0			0
$99$	18.7734	+	5.97827i	1.88680	+	0.600839i
$100$		0			0
$101$	8.51025			0.846802			0.423401	−	0.905942i	$-0.360836\pi$
$101$	8.51025			0.846802			0.423401	+	0.905942i	$0.360836\pi$
$102$		0			0
$103$		−	4.35642i		−	0.429251i	−0.976696	−	0.214626i	$-0.931147\pi$
$103$		−	4.35642i		−	0.429251i	0.976696	−	0.214626i	$-0.0688531\pi$
$104$		0			0
$105$	8.86168	+	4.45576i	0.864812	+	0.434837i
$106$		0			0
$107$		−	8.18371i		−	0.791149i	−0.918434	−	0.395574i	$-0.870546\pi$
$107$		−	8.18371i		−	0.791149i	0.918434	−	0.395574i	$-0.129454\pi$
$108$		0			0
$109$		−	12.3652i		−	1.18437i	−0.805802	−	0.592184i	$-0.798266\pi$
$109$		−	12.3652i		−	1.18437i	0.805802	−	0.592184i	$-0.201734\pi$
$110$		0			0
$111$		−	8.17995i		−	0.776406i
$112$		0			0
$113$	9.37456			0.881885			0.440942	−	0.897535i	$-0.354644\pi$
$113$	9.37456			0.881885			0.440942	+	0.897535i	$0.354644\pi$
$114$		0			0
$115$			0.0965252i			0.00900102i
$116$		0			0
$117$	5.94047			0.549197
$118$		0			0
$119$	12.9806	+	6.52678i	1.18993	+	0.598309i
$120$		0			0
$121$	8.97446	+	6.36074i	0.815860	+	0.578249i
$122$		0			0
$123$			14.7024i			1.32567i
$124$		0			0
$125$			10.5671i			0.945146i
$126$		0			0
$127$		−	15.2897i		−	1.35674i	−0.734720	−	0.678371i	$-0.762686\pi$
$127$		−	15.2897i		−	1.35674i	0.734720	−	0.678371i	$-0.237314\pi$
$128$		0			0
$129$	28.5063			2.50984
$130$		0			0
$131$	−16.7169			−1.46056			−0.730282	−	0.683146i	$-0.760611\pi$
$131$	−16.7169			−1.46056			−0.730282	+	0.683146i	$0.760611\pi$
$132$		0			0
$133$	8.15874	+	4.10231i	0.707452	+	0.355715i
$134$		0			0
$135$	−11.0237			−0.948771
$136$		0			0
$137$	17.9066			1.52986			0.764930	−	0.644113i	$-0.222773\pi$
$137$	17.9066			1.52986			0.764930	+	0.644113i	$0.222773\pi$
$138$		0			0
$139$	−8.51822			−0.722506			−0.361253	−	0.932468i	$-0.617651\pi$
$139$	−8.51822			−0.722506			−0.361253	+	0.932468i	$0.617651\pi$
$140$		0			0
$141$	2.74307			0.231008
$142$		0			0
$143$	3.16026	+	1.00636i	0.264274	+	0.0841564i
$144$		0			0
$145$	−1.50315			−0.124830
$146$		0			0
$147$	16.8006	−	12.4829i	1.38569	−	1.02957i
$148$		0			0
$149$		−	22.9290i		−	1.87842i	−0.343348	−	0.939208i	$-0.611561\pi$
$149$		−	22.9290i		−	1.87842i	0.343348	−	0.939208i	$-0.388439\pi$
$150$		0			0
$151$		−	20.3612i		−	1.65697i	−0.560013	−	0.828484i	$-0.689204\pi$
$151$		−	20.3612i		−	1.65697i	0.560013	−	0.828484i	$-0.310796\pi$
$152$		0			0
$153$	−32.6220			−2.63733
$154$		0			0
$155$	−12.8783			−1.03441
$156$		0			0
$157$			23.2708i			1.85722i	0.371063	+	0.928608i	$0.378993\pi$
$157$			23.2708i			1.85722i	−0.371063	+	0.928608i	$0.621007\pi$
$158$		0			0
$159$		−	22.2836i		−	1.76720i
$160$		0			0
$161$	0.181976	+	0.0914997i	0.0143417	+	0.00721118i
$162$		0			0
$163$	−19.9615			−1.56350			−0.781752	−	0.623589i	$-0.785674\pi$
$163$	−19.9615			−1.56350			−0.781752	+	0.623589i	$0.785674\pi$
$164$		0			0
$165$	−11.8477	−	3.77282i	−0.922342	−	0.293714i
$166$		0			0
$167$	−10.7222			−0.829712			−0.414856	−	0.909887i	$-0.636168\pi$
$167$	−10.7222			−0.829712			−0.414856	+	0.909887i	$0.636168\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−20.5040			−1.56798
$172$		0			0
$173$	19.2326			1.46223			0.731113	−	0.682257i	$-0.239001\pi$
$173$	19.2326			1.46223			0.731113	+	0.682257i	$0.239001\pi$
$174$		0			0
$175$	8.10291	+	4.07424i	0.612523	+	0.307983i
$176$		0			0
$177$	−32.1740			−2.41835
$178$		0			0
$179$	5.91295			0.441955			0.220977	−	0.975279i	$-0.429075\pi$
$179$	5.91295			0.441955			0.220977	+	0.975279i	$0.429075\pi$
$180$		0			0
$181$			1.33656i			0.0993458i	0.998766	+	0.0496729i	$0.0158179\pi$
$181$			1.33656i			0.0993458i	−0.998766	+	0.0496729i	$0.984182\pi$
$182$		0			0
$183$			32.9948i			2.43905i
$184$		0			0
$185$			3.43006i			0.252183i
$186$		0			0
$187$	−17.3545	−	5.52642i	−1.26909	−	0.404132i
$188$		0			0
$189$	−10.4498	+	20.7827i	−0.760110	+	1.51172i
$190$		0			0
$191$	17.9978			1.30227			0.651137	−	0.758960i	$-0.274292\pi$
$191$	17.9978			1.30227			0.651137	+	0.758960i	$0.274292\pi$
$192$		0			0
$193$			12.1798i			0.876719i	0.898800	+	0.438360i	$0.144440\pi$
$193$			12.1798i			0.876719i	−0.898800	+	0.438360i	$0.855560\pi$
$194$		0			0
$195$	−3.74896			−0.268469
$196$		0			0
$197$			4.33356i			0.308753i	0.988012	+	0.154377i	$0.0493369\pi$
$197$			4.33356i			0.308753i	−0.988012	+	0.154377i	$0.950663\pi$
$198$		0			0
$199$		−	21.2266i		−	1.50471i	−0.658758	−	0.752355i	$-0.728918\pi$
$199$		−	21.2266i		−	1.50471i	0.658758	−	0.752355i	$-0.271082\pi$
$200$		0			0
$201$			29.3691i			2.07154i
$202$		0			0
$203$	−1.42489	+	2.83384i	−0.100008	+	0.198897i
$204$		0			0
$205$		−	6.16509i		−	0.430588i
$206$		0			0
$207$	−0.457331			−0.0317867
$208$		0			0
$209$	−10.9079	−	3.47355i	−0.754515	−	0.240270i
$210$		0			0
$211$			11.5267i			0.793529i	0.917920	+	0.396764i	$0.129867\pi$
$211$			11.5267i			0.793529i	−0.917920	+	0.396764i	$0.870133\pi$
$212$		0			0
$213$		−	49.7682i		−	3.41006i
$214$		0			0
$215$	−11.9534			−0.815216
$216$		0			0
$217$	−12.2078	+	24.2791i	−0.828720	+	1.64817i
$218$		0			0
$219$		−	43.1921i		−	2.91865i
$220$		0			0
$221$	−5.49148			−0.369397
$222$		0			0
$223$		−	9.05742i		−	0.606530i	−0.952906	−	0.303265i	$-0.901923\pi$
$223$		−	9.05742i		−	0.606530i	0.952906	−	0.303265i	$-0.0980767\pi$
$224$		0			0
$225$	−20.3637			−1.35758
$226$		0			0
$227$	21.8559			1.45063			0.725314	−	0.688418i	$-0.241694\pi$
$227$	21.8559			1.45063			0.725314	+	0.688418i	$0.241694\pi$
$228$		0			0
$229$		−	22.7219i		−	1.50150i	−0.660584	−	0.750752i	$-0.729691\pi$
$229$		−	22.7219i		−	1.50150i	0.660584	−	0.750752i	$-0.270309\pi$
$230$		0			0
$231$	−18.3436	+	18.7597i	−1.20692	+	1.23430i
$232$		0			0
$233$		−	10.8005i		−	0.707563i	−0.935328	−	0.353781i	$-0.884896\pi$
$233$		−	10.8005i		−	0.707563i	0.935328	−	0.353781i	$-0.115104\pi$
$234$		0			0
$235$	−1.15024			−0.0750333
$236$		0			0
$237$	22.5724			1.46623
$238$		0			0
$239$		−	4.15966i		−	0.269066i	−0.990909	−	0.134533i	$-0.957047\pi$
$239$		−	4.15966i		−	0.269066i	0.990909	−	0.134533i	$-0.0429534\pi$
$240$		0			0
$241$	16.9864			1.09419			0.547096	−	0.837070i	$-0.315733\pi$
$241$	16.9864			1.09419			0.547096	+	0.837070i	$0.315733\pi$
$242$		0			0
$243$			1.05737i			0.0678300i
$244$		0			0
$245$	−7.04492	+	5.23439i	−0.450083	+	0.334413i
$246$		0			0
$247$	−3.45158			−0.219619
$248$		0			0
$249$		−	21.2798i		−	1.34856i
$250$		0			0
$251$			17.9028i			1.13001i	0.825086	+	0.565007i	$0.191127\pi$
$251$			17.9028i			1.13001i	−0.825086	+	0.565007i	$0.808873\pi$
$252$		0			0
$253$	−0.243294	−	0.0774755i	−0.0152958	−	0.00487084i
$254$		0			0
$255$	20.5874			1.28923
$256$		0			0
$257$			1.47955i			0.0922916i	0.998935	+	0.0461458i	$0.0146939\pi$
$257$			1.47955i			0.0922916i	−0.998935	+	0.0461458i	$0.985306\pi$
$258$		0			0
$259$	6.46659	+	3.25148i	0.401814	+	0.202037i
$260$		0			0
$261$		−	7.12184i		−	0.440831i
$262$		0			0
$263$			20.9381i			1.29110i	0.763718	+	0.645550i	$0.223372\pi$
$263$			20.9381i			1.29110i	−0.763718	+	0.645550i	$0.776628\pi$
$264$		0			0
$265$			9.34406i			0.574001i
$266$		0			0
$267$	11.1490			0.682307
$268$		0			0
$269$			25.0727i			1.52871i	0.644798	+	0.764353i	$0.276942\pi$
$269$			25.0727i			1.52871i	−0.644798	+	0.764353i	$0.723058\pi$
$270$		0			0
$271$	−4.95297			−0.300872			−0.150436	−	0.988620i	$-0.548068\pi$
$271$	−4.95297			−0.300872			−0.150436	+	0.988620i	$0.548068\pi$
$272$		0			0
$273$	−3.55378	+	7.06781i	−0.215084	+	0.427763i
$274$		0			0
$275$	−10.8333	−	3.44978i	−0.653270	−	0.208029i
$276$		0			0
$277$			2.39854i			0.144114i	0.997400	+	0.0720572i	$0.0229564\pi$
$277$			2.39854i			0.144114i	−0.997400	+	0.0720572i	$0.977044\pi$
$278$		0			0
$279$		−	61.0167i		−	3.65297i
$280$		0			0
$281$		−	12.3876i		−	0.738981i	−0.929234	−	0.369491i	$-0.879532\pi$
$281$		−	12.3876i		−	0.738981i	0.929234	−	0.369491i	$-0.120468\pi$
$282$		0			0
$283$	−16.1678			−0.961077			−0.480538	−	0.876974i	$-0.659559\pi$
$283$	−16.1678			−0.961077			−0.480538	+	0.876974i	$0.659559\pi$
$284$		0			0
$285$	12.9399			0.766491
$286$		0			0
$287$	−11.6229	−	5.84411i	−0.686075	−	0.344967i
$288$		0			0
$289$	13.1563			0.773901
$290$		0			0
$291$	21.1517			1.23993
$292$		0			0
$293$	−2.99490			−0.174964			−0.0874819	−	0.996166i	$-0.527882\pi$
$293$	−2.99490			−0.174964			−0.0874819	+	0.996166i	$0.527882\pi$
$294$		0			0
$295$	13.4914			0.785498
$296$		0			0
$297$	8.84814	−	27.7856i	0.513421	−	1.61228i
$298$		0			0
$299$	−0.0769856			−0.00445219
$300$		0			0
$301$	−11.3311	+	22.5354i	−0.653112	+	1.29892i
$302$		0			0
$303$		−	25.4462i		−	1.46185i
$304$		0			0
$305$		−	13.8356i		−	0.792221i
$306$		0			0
$307$	31.2065			1.78105			0.890523	−	0.454938i	$-0.150339\pi$
$307$	31.2065			1.78105			0.890523	+	0.454938i	$0.150339\pi$
$308$		0			0
$309$	−13.0260			−0.741022
$310$		0			0
$311$			3.74497i			0.212358i	0.994347	+	0.106179i	$0.0338616\pi$
$311$			3.74497i			0.212358i	−0.994347	+	0.106179i	$0.966138\pi$
$312$		0			0
$313$			9.22433i			0.521390i	0.965421	+	0.260695i	$0.0839517\pi$
$313$			9.22433i			0.521390i	−0.965421	+	0.260695i	$0.916048\pi$
$314$		0			0
$315$	8.85242	−	17.6058i	0.498778	−	0.991977i
$316$		0			0
$317$	−16.4357			−0.923120			−0.461560	−	0.887109i	$-0.652710\pi$
$317$	−16.4357			−0.923120			−0.461560	+	0.887109i	$0.652710\pi$
$318$		0			0
$319$	1.20650	−	3.78873i	0.0675508	−	0.212128i
$320$		0			0
$321$	−24.4698			−1.36577
$322$		0			0
$323$	18.9543			1.05464
$324$		0			0
$325$	−3.42796			−0.190149
$326$		0			0
$327$	−36.9726			−2.04459
$328$		0			0
$329$	−1.09035	+	2.16851i	−0.0601131	+	0.119554i
$330$		0			0
$331$	5.41091			0.297410			0.148705	−	0.988882i	$-0.452489\pi$
$331$	5.41091			0.297410			0.148705	+	0.988882i	$0.452489\pi$
$332$		0			0
$333$	−16.2514			−0.890572
$334$		0			0
$335$		−	12.3152i		−	0.672851i
$336$		0			0
$337$			12.9666i			0.706337i	0.935560	+	0.353169i	$0.114896\pi$
$337$			12.9666i			0.706337i	−0.935560	+	0.353169i	$0.885104\pi$
$338$		0			0
$339$		−	28.0305i		−	1.52241i
$340$		0			0
$341$	10.3367	−	32.4601i	0.559765	−	1.75781i
$342$		0			0
$343$	3.19011	+	18.2434i	0.172250	+	0.985053i
$344$		0			0
$345$	0.288616			0.0155386
$346$		0			0
$347$			12.7630i			0.685155i	0.939490	+	0.342577i	$0.111300\pi$
$347$			12.7630i			0.685155i	−0.939490	+	0.342577i	$0.888700\pi$
$348$		0			0
$349$	−15.4714			−0.828167			−0.414084	−	0.910239i	$-0.635898\pi$
$349$	−15.4714			−0.828167			−0.414084	+	0.910239i	$0.635898\pi$
$350$		0			0
$351$		−	8.79219i		−	0.469293i
$352$		0			0
$353$		−	1.39325i		−	0.0741555i	−0.999312	−	0.0370777i	$-0.988195\pi$
$353$		−	1.39325i		−	0.0741555i	0.999312	−	0.0370777i	$-0.0118049\pi$
$354$		0			0
$355$			20.8690i			1.10761i
$356$		0			0
$357$	19.5155	−	38.8127i	1.03287	−	2.05419i
$358$		0			0
$359$		−	20.1323i		−	1.06254i	−0.847202	−	0.531271i	$-0.821715\pi$
$359$		−	20.1323i		−	1.06254i	0.847202	−	0.531271i	$-0.178285\pi$
$360$		0			0
$361$	−7.08658			−0.372978
$362$		0			0
$363$	19.0190	−	26.8342i	0.998239	−	1.40843i
$364$		0			0
$365$			18.1115i			0.948002i
$366$		0			0
$367$		−	28.8871i		−	1.50790i	−0.656934	−	0.753948i	$-0.728147\pi$
$367$		−	28.8871i		−	1.50790i	0.656934	−	0.753948i	$-0.271853\pi$
$368$		0			0
$369$	29.2098			1.52060
$370$		0			0
$371$	17.6161	+	8.85757i	0.914581	+	0.459862i
$372$		0			0
$373$		−	15.2010i		−	0.787080i	−0.919307	−	0.393540i	$-0.871250\pi$
$373$		−	15.2010i		−	0.787080i	0.919307	−	0.393540i	$-0.128750\pi$
$374$		0			0
$375$	31.5961			1.63162
$376$		0			0
$377$		−	1.19887i		−	0.0617448i
$378$		0			0
$379$	−9.43768			−0.484781			−0.242391	−	0.970179i	$-0.577932\pi$
$379$	−9.43768			−0.484781			−0.242391	+	0.970179i	$0.577932\pi$
$380$		0			0
$381$	−45.7171			−2.34216
$382$		0			0
$383$			11.6925i			0.597460i	0.954338	+	0.298730i	$0.0965630\pi$
$383$			11.6925i			0.597460i	−0.954338	+	0.298730i	$0.903437\pi$
$384$		0			0
$385$	7.69195	−	7.86642i	0.392018	−	0.400910i
$386$		0			0
$387$		−	56.6346i		−	2.87890i
$388$		0			0
$389$	30.2622			1.53436			0.767178	−	0.641435i	$-0.221660\pi$
$389$	30.2622			1.53436			0.767178	+	0.641435i	$0.221660\pi$
$390$		0			0
$391$	0.422764			0.0213801
$392$		0			0
$393$			49.9846i			2.52139i
$394$		0			0
$395$	−9.46516			−0.476244
$396$		0			0
$397$		−	1.00414i		−	0.0503964i	−0.999682	−	0.0251982i	$-0.991978\pi$
$397$		−	1.00414i		−	0.0503964i	0.999682	−	0.0251982i	$-0.00802168\pi$
$398$		0			0
$399$	12.2662	−	24.3951i	0.614076	−	1.22128i
$400$		0			0
$401$	−11.6705			−0.582795			−0.291398	−	0.956602i	$-0.594120\pi$
$401$	−11.6705			−0.582795			−0.291398	+	0.956602i	$0.594120\pi$
$402$		0			0
$403$		−	10.2714i		−	0.511653i
$404$		0			0
$405$			10.6170i			0.527562i
$406$		0			0
$407$	−8.64556	−	2.75312i	−0.428544	−	0.136467i
$408$		0			0
$409$	−34.5956			−1.71064			−0.855322	−	0.518097i	$-0.826640\pi$
$409$	−34.5956			−1.71064			−0.855322	+	0.518097i	$0.826640\pi$
$410$		0			0
$411$		−	53.5417i		−	2.64102i
$412$		0			0
$413$	12.7889	−	25.4349i	0.629303	−	1.25157i
$414$		0			0
$415$			8.92317i			0.438021i
$416$		0			0
$417$			25.4700i			1.24727i
$418$		0			0
$419$		−	8.22048i		−	0.401597i	−0.979633	−	0.200798i	$-0.935646\pi$
$419$		−	8.22048i		−	0.401597i	0.979633	−	0.200798i	$-0.0643536\pi$
$420$		0			0
$421$	9.32914			0.454674			0.227337	−	0.973816i	$-0.426998\pi$
$421$	9.32914			0.454674			0.227337	+	0.973816i	$0.426998\pi$
$422$		0			0
$423$		−	5.44976i		−	0.264977i
$424$		0			0
$425$	18.8246			0.913127
$426$		0			0
$427$	−26.0838	−	13.1152i	−1.26228	−	0.634690i
$428$		0			0
$429$	3.00909	−	9.44937i	0.145280	−	0.456220i
$430$		0			0
$431$		−	33.8711i		−	1.63151i	−0.578395	−	0.815757i	$-0.696321\pi$
$431$		−	33.8711i		−	1.63151i	0.578395	−	0.815757i	$-0.303679\pi$
$432$		0			0
$433$			33.4609i			1.60803i	0.594610	+	0.804015i	$0.297307\pi$
$433$			33.4609i			1.60803i	−0.594610	+	0.804015i	$0.702693\pi$
$434$		0			0
$435$			4.49451i			0.215495i
$436$		0			0
$437$	0.265722			0.0127112
$438$		0			0
$439$	19.0081			0.907205			0.453602	−	0.891204i	$-0.350139\pi$
$439$	19.0081			0.907205			0.453602	+	0.891204i	$0.350139\pi$
$440$		0			0
$441$	−24.8002	−	33.3784i	−1.18096	−	1.58945i
$442$		0			0
$443$	7.07681			0.336229			0.168115	−	0.985767i	$-0.446232\pi$
$443$	7.07681			0.336229			0.168115	+	0.985767i	$0.446232\pi$
$444$		0			0
$445$	−4.67505			−0.221619
$446$		0			0
$447$	−68.5591			−3.24273
$448$		0			0
$449$	38.6487			1.82395			0.911973	−	0.410251i	$-0.134559\pi$
$449$	38.6487			1.82395			0.911973	+	0.410251i	$0.134559\pi$
$450$		0			0
$451$	15.5393	+	4.94838i	0.731716	+	0.233010i
$452$		0			0
$453$	−60.8811			−2.86044
$454$		0			0
$455$	1.49019	−	2.96371i	0.0698611	−	0.138941i
$456$		0			0
$457$			1.24408i			0.0581957i	0.999577	+	0.0290979i	$0.00926344\pi$
$457$			1.24408i			0.0581957i	−0.999577	+	0.0290979i	$0.990737\pi$
$458$		0			0
$459$			48.2821i			2.25362i
$460$		0			0
$461$	−14.8782			−0.692949			−0.346475	−	0.938059i	$-0.612621\pi$
$461$	−14.8782			−0.692949			−0.346475	+	0.938059i	$0.612621\pi$
$462$		0			0
$463$	−10.3684			−0.481858			−0.240929	−	0.970543i	$-0.577452\pi$
$463$	−10.3684			−0.481858			−0.240929	+	0.970543i	$0.577452\pi$
$464$		0			0
$465$			38.5069i			1.78572i
$466$		0			0
$467$			33.0310i			1.52849i	0.644924	+	0.764247i	$0.276889\pi$
$467$			33.0310i			1.52849i	−0.644924	+	0.764247i	$0.723111\pi$
$468$		0			0
$469$	−23.2175	−	11.6740i	−1.07208	−	0.539056i
$470$		0			0
$471$	69.5813			3.20613
$472$		0			0
$473$	9.59435	−	30.1289i	0.441149	−	1.38533i
$474$		0			0
$475$	11.8319			0.542885
$476$		0			0
$477$	−44.2716			−2.02706
$478$		0			0
$479$	−32.1483			−1.46889			−0.734446	−	0.678667i	$-0.762558\pi$
$479$	−32.1483			−1.46889			−0.734446	+	0.678667i	$0.762558\pi$
$480$		0			0
$481$	−2.73571			−0.124738
$482$		0			0
$483$	0.273590	−	0.544120i	0.0124488	−	0.0247583i
$484$		0			0
$485$	−8.86942			−0.402740
$486$		0			0
$487$	−26.4886			−1.20031			−0.600155	−	0.799883i	$-0.704895\pi$
$487$	−26.4886			−1.20031			−0.600155	+	0.799883i	$0.704895\pi$
$488$		0			0
$489$			59.6861i			2.69910i
$490$		0			0
$491$			27.2841i			1.23132i	0.788014	+	0.615658i	$0.211110\pi$
$491$			27.2841i			1.23132i	−0.788014	+	0.615658i	$0.788890\pi$
$492$		0			0
$493$			6.58355i			0.296508i
$494$		0			0
$495$	−7.49561	+	23.5383i	−0.336903	+	1.05797i
$496$		0			0
$497$	39.3438	+	19.7825i	1.76481	+	0.887367i
$498$		0			0
$499$	0.597069			0.0267285			0.0133642	−	0.999911i	$-0.495746\pi$
$499$	0.597069			0.0267285			0.0133642	+	0.999911i	$0.495746\pi$
$500$		0			0
$501$			32.0602i			1.43234i
$502$		0			0
$503$	21.2614			0.948001			0.474000	−	0.880525i	$-0.342810\pi$
$503$	21.2614			0.948001			0.474000	+	0.880525i	$0.342810\pi$
$504$		0			0
$505$			10.6702i			0.474819i
$506$		0			0
$507$		−	2.99006i		−	0.132793i
$508$		0			0
$509$		−	5.94442i		−	0.263482i	−0.991284	−	0.131741i	$-0.957943\pi$
$509$		−	5.94442i		−	0.263482i	0.991284	−	0.131741i	$-0.0420567\pi$
$510$		0			0
$511$	34.1452	+	17.1686i	1.51049	+	0.759494i
$512$		0			0
$513$			30.3470i			1.33985i
$514$		0			0
$515$	5.46212			0.240690
$516$		0			0
$517$	0.923234	−	2.89921i	0.0406038	−	0.127507i
$518$		0			0
$519$		−	57.5066i		−	2.52426i
$520$		0			0
$521$			44.1076i			1.93239i	0.257817	+	0.966194i	$0.416997\pi$
$521$			44.1076i			1.93239i	−0.257817	+	0.966194i	$0.583003\pi$
$522$		0			0
$523$	−15.9389			−0.696961			−0.348480	−	0.937316i	$-0.613302\pi$
$523$	−15.9389			−0.696961			−0.348480	+	0.937316i	$0.613302\pi$
$524$		0			0
$525$	12.1822	−	24.2282i	0.531676	−	1.05741i
$526$		0			0
$527$			56.4049i			2.45704i
$528$		0			0
$529$	−22.9941			−0.999742
$530$		0			0
$531$			63.9213i			2.77395i
$532$		0			0
$533$	4.91709			0.212983
$534$		0			0
$535$	10.2608			0.443613
$536$		0			0
$537$		−	17.6801i		−	0.762952i
$538$		0			0
$539$	−7.53885	−	21.9583i	−0.324721	−	0.945810i
$540$		0			0
$541$			35.3885i			1.52147i	0.649061	+	0.760736i	$0.275162\pi$
$541$			35.3885i			1.52147i	−0.649061	+	0.760736i	$0.724838\pi$
$542$		0			0
$543$	3.99640			0.171502
$544$		0			0
$545$	15.5036			0.664100
$546$		0			0
$547$			8.54239i			0.365246i	0.983183	+	0.182623i	$0.0584588\pi$
$547$			8.54239i			0.365246i	−0.983183	+	0.182623i	$0.941541\pi$
$548$		0			0
$549$	65.5520			2.79769
$550$		0			0
$551$			4.13799i			0.176284i
$552$		0			0
$553$	−8.97237	+	17.8444i	−0.381544	+	0.758821i
$554$		0			0
$555$	10.2561			0.435347
$556$		0			0
$557$		−	32.1700i		−	1.36309i	−0.731777	−	0.681544i	$-0.761309\pi$
$557$		−	32.1700i		−	1.36309i	0.731777	−	0.681544i	$-0.238691\pi$
$558$		0			0
$559$		−	9.53368i		−	0.403232i
$560$		0			0
$561$	−16.5243	+	51.8910i	−0.697658	+	2.19084i
$562$		0			0
$563$	−42.4427			−1.78875			−0.894373	−	0.447322i	$-0.852378\pi$
$563$	−42.4427			−1.78875			−0.894373	+	0.447322i	$0.852378\pi$
$564$		0			0
$565$			11.7539i			0.494490i
$566$		0			0
$567$	20.0159	+	10.0642i	0.840588	+	0.422657i
$568$		0			0
$569$		−	16.8084i		−	0.704644i	−0.935879	−	0.352322i	$-0.885392\pi$
$569$		−	16.8084i		−	0.704644i	0.935879	−	0.352322i	$-0.114608\pi$
$570$		0			0
$571$		−	13.6447i		−	0.571014i	−0.958377	−	0.285507i	$-0.907838\pi$
$571$		−	13.6447i		−	0.571014i	0.958377	−	0.285507i	$-0.0921620\pi$
$572$		0			0
$573$		−	53.8145i		−	2.24813i
$574$		0			0
$575$	0.263904			0.0110056
$576$		0			0
$577$			35.3012i			1.46961i	0.678279	+	0.734805i	$0.262726\pi$
$577$			35.3012i			1.46961i	−0.678279	+	0.734805i	$0.737274\pi$
$578$		0			0
$579$	36.4183			1.51349
$580$		0			0
$581$	16.8226	+	8.45860i	0.697919	+	0.350922i
$582$		0			0
$583$	−23.5520	−	7.49997i	−0.975423	−	0.310617i
$584$		0			0
$585$			7.44821i			0.307946i
$586$		0			0
$587$		−	30.0791i		−	1.24150i	−0.784010	−	0.620748i	$-0.786829\pi$
$587$		−	30.0791i		−	1.24150i	0.784010	−	0.620748i	$-0.213171\pi$
$588$		0			0
$589$			35.4524i			1.46079i
$590$		0			0
$591$	12.9576			0.533005
$592$		0			0
$593$	−8.09168			−0.332286			−0.166143	−	0.986102i	$-0.553131\pi$
$593$	−8.09168			−0.332286			−0.166143	+	0.986102i	$0.553131\pi$
$594$		0			0
$595$	−8.18334	+	16.2752i	−0.335484	+	0.667216i
$596$		0			0
$597$	−63.4687			−2.59760
$598$		0			0
$599$	−17.6806			−0.722409			−0.361204	−	0.932487i	$-0.617634\pi$
$599$	−17.6806			−0.722409			−0.361204	+	0.932487i	$0.617634\pi$
$600$		0			0
$601$	4.16310			0.169816			0.0849082	−	0.996389i	$-0.472940\pi$
$601$	4.16310			0.169816			0.0849082	+	0.996389i	$0.472940\pi$
$602$		0			0
$603$	58.3487			2.37614
$604$		0			0
$605$	−7.97515	+	11.2523i	−0.324236	+	0.457469i
$606$		0			0
$607$	14.3746			0.583445			0.291723	−	0.956503i	$-0.405772\pi$
$607$	14.3746			0.583445			0.291723	+	0.956503i	$0.405772\pi$
$608$		0			0
$609$	8.47337	+	4.26051i	0.343358	+	0.172645i
$610$		0			0
$611$		−	0.917396i		−	0.0371139i
$612$		0			0
$613$		−	18.2231i		−	0.736023i	−0.929821	−	0.368012i	$-0.880039\pi$
$613$		−	18.2231i		−	0.736023i	0.929821	−	0.368012i	$-0.119961\pi$
$614$		0			0
$615$	−18.4340			−0.743330
$616$		0			0
$617$	−9.33728			−0.375905			−0.187952	−	0.982178i	$-0.560185\pi$
$617$	−9.33728			−0.375905			−0.187952	+	0.982178i	$0.560185\pi$
$618$		0			0
$619$			9.91369i			0.398465i	0.979952	+	0.199232i	$0.0638449\pi$
$619$			9.91369i			0.398465i	−0.979952	+	0.199232i	$0.936155\pi$
$620$		0			0
$621$			0.676872i			0.0271619i
$622$		0			0
$623$	−4.43165	+	8.81374i	−0.177550	+	0.353115i
$624$		0			0
$625$	3.89076			0.155631
$626$		0			0
$627$	−10.3861	+	32.6153i	−0.414782	+	1.30253i
$628$		0			0
$629$	15.0231			0.599010
$630$		0			0
$631$	5.05308			0.201160			0.100580	−	0.994929i	$-0.467930\pi$
$631$	5.05308			0.201160			0.100580	+	0.994929i	$0.467930\pi$
$632$		0			0
$633$	34.4655			1.36988
$634$		0			0
$635$	19.1703			0.760752
$636$		0			0
$637$	−4.17479	−	5.61882i	−0.165411	−	0.222626i
$638$		0			0
$639$	−98.8763			−3.91149
$640$		0			0
$641$	−4.14028			−0.163531			−0.0817656	−	0.996652i	$-0.526056\pi$
$641$	−4.14028			−0.163531			−0.0817656	+	0.996652i	$0.526056\pi$
$642$		0			0
$643$			37.6929i			1.48646i	0.669034	+	0.743232i	$0.266708\pi$
$643$			37.6929i			1.48646i	−0.669034	+	0.743232i	$0.733292\pi$
$644$		0			0
$645$			35.7414i			1.40732i
$646$		0			0
$647$			22.8145i			0.896932i	0.893800	+	0.448466i	$0.148030\pi$
$647$			22.8145i			0.896932i	−0.893800	+	0.448466i	$0.851970\pi$
$648$		0			0
$649$	−10.8288	+	34.0054i	−0.425067	+	1.33483i
$650$		0			0
$651$	72.5960	+	36.5021i	2.84526	+	1.43063i
$652$		0			0
$653$	−33.0589			−1.29369			−0.646847	−	0.762620i	$-0.723913\pi$
$653$	−33.0589			−1.29369			−0.646847	+	0.762620i	$0.723913\pi$
$654$		0			0
$655$		−	20.9598i		−	0.818967i
$656$		0			0
$657$	−85.8115			−3.34782
$658$		0			0
$659$		−	19.9332i		−	0.776488i	−0.921557	−	0.388244i	$-0.873082\pi$
$659$		−	19.9332i		−	0.776488i	0.921557	−	0.388244i	$-0.126918\pi$
$660$		0			0
$661$		−	13.6296i		−	0.530130i	−0.964231	−	0.265065i	$-0.914607\pi$
$661$		−	13.6296i		−	0.530130i	0.964231	−	0.265065i	$-0.0853934\pi$
$662$		0			0
$663$			16.4199i			0.637694i
$664$		0			0
$665$	−5.14351	+	10.2295i	−0.199457	+	0.396683i
$666$		0			0
$667$			0.0922955i			0.00357370i
$668$		0			0
$669$	−27.0822			−1.04706
$670$		0			0
$671$	34.8729	+	11.1050i	1.34625	+	0.428705i
$672$		0			0
$673$		−	23.1362i		−	0.891836i	−0.895074	−	0.445918i	$-0.852877\pi$
$673$		−	23.1362i		−	0.891836i	0.895074	−	0.445918i	$-0.147123\pi$
$674$		0			0
$675$			30.1393i			1.16006i
$676$		0			0
$677$	−16.5897			−0.637595			−0.318798	−	0.947823i	$-0.603279\pi$
$677$	−16.5897			−0.637595			−0.318798	+	0.947823i	$0.603279\pi$
$678$		0			0
$679$	−8.40764	+	16.7213i	−0.322656	+	0.641703i
$680$		0			0
$681$		−	65.3506i		−	2.50424i
$682$		0			0
$683$	21.1016			0.807431			0.403716	−	0.914885i	$-0.367719\pi$
$683$	21.1016			0.807431			0.403716	+	0.914885i	$0.367719\pi$
$684$		0			0
$685$			22.4514i			0.857824i
$686$		0			0
$687$	−67.9399			−2.59207
$688$		0			0
$689$	−7.45254			−0.283919
$690$		0			0
$691$			25.6549i			0.975958i	0.872855	+	0.487979i	$0.162266\pi$
$691$			25.6549i			0.975958i	−0.872855	+	0.487979i	$0.837734\pi$
$692$		0			0
$693$	37.2707	+	36.4440i	1.41580	+	1.38439i
$694$		0			0
$695$		−	10.6802i		−	0.405124i
$696$		0			0
$697$	−27.0021			−1.02278
$698$		0			0
$699$	−32.2941			−1.22147
$700$		0			0
$701$			29.6214i			1.11879i	0.828903	+	0.559393i	$0.188966\pi$
$701$			29.6214i			1.11879i	−0.828903	+	0.559393i	$0.811034\pi$
$702$		0			0
$703$	9.44253			0.356132
$704$		0			0
$705$			3.43929i			0.129531i
$706$		0			0
$707$	20.1163	+	10.1147i	0.756550	+	0.380402i
$708$		0			0
$709$	22.7845			0.855689			0.427844	−	0.903852i	$-0.359273\pi$
$709$	22.7845			0.855689			0.427844	+	0.903852i	$0.359273\pi$
$710$		0			0
$711$		−	44.8454i		−	1.68183i
$712$		0			0
$713$			0.790746i			0.0296137i
$714$		0			0
$715$	−1.26179	+	3.96236i	−0.0471882	+	0.148184i
$716$		0			0
$717$	−12.4376			−0.464492
$718$		0			0
$719$			12.2982i			0.458646i	0.973350	+	0.229323i	$0.0736513\pi$
$719$			12.2982i			0.458646i	−0.973350	+	0.229323i	$0.926349\pi$
$720$		0			0
$721$	5.17774	−	10.2976i	0.192829	−	0.383501i
$722$		0			0
$723$		−	50.7905i		−	1.88892i
$724$		0			0
$725$			4.10967i			0.152629i
$726$		0			0
$727$		−	7.19162i		−	0.266722i	−0.991067	−	0.133361i	$-0.957423\pi$
$727$		−	7.19162i		−	0.266722i	0.991067	−	0.133361i	$-0.0425770\pi$
$728$		0			0
$729$	28.5650			1.05796
$730$		0			0
$731$			52.3540i			1.93638i
$732$		0			0
$733$	22.5531			0.833018			0.416509	−	0.909132i	$-0.363253\pi$
$733$	22.5531			0.833018			0.416509	+	0.909132i	$0.363253\pi$
$734$		0			0
$735$	15.6512	+	21.0647i	0.577301	+	0.776985i
$736$		0			0
$737$	31.0408	+	9.88474i	1.14340	+	0.364109i
$738$		0			0
$739$		−	14.5491i		−	0.535197i	−0.963531	−	0.267598i	$-0.913770\pi$
$739$		−	14.5491i		−	0.535197i	0.963531	−	0.267598i	$-0.0862300\pi$
$740$		0			0
$741$			10.3204i			0.379131i
$742$		0			0
$743$		−	26.9876i		−	0.990079i	−0.868870	−	0.495040i	$-0.835154\pi$
$743$		−	26.9876i		−	0.990079i	0.868870	−	0.495040i	$-0.164846\pi$
$744$		0			0
$745$	28.7486			1.05327
$746$		0			0
$747$	−42.2775			−1.54685
$748$		0			0
$749$	9.72658	−	19.3444i	0.355402	−	0.706828i
$750$		0			0
$751$	12.4922			0.455848			0.227924	−	0.973679i	$-0.426806\pi$
$751$	12.4922			0.455848			0.227924	+	0.973679i	$0.426806\pi$
$752$		0			0
$753$	53.5305			1.95076
$754$		0			0
$755$	25.5290			0.929095
$756$		0			0
$757$	−5.92448			−0.215329			−0.107664	−	0.994187i	$-0.534337\pi$
$757$	−5.92448			−0.215329			−0.107664	+	0.994187i	$0.534337\pi$
$758$		0			0
$759$	−0.231656	+	0.727465i	−0.00840860	+	0.0264053i
$760$		0			0
$761$	24.0785			0.872846			0.436423	−	0.899742i	$-0.356245\pi$
$761$	24.0785			0.872846			0.436423	+	0.899742i	$0.356245\pi$
$762$		0			0
$763$	14.6964	−	29.2284i	0.532045	−	1.05814i
$764$		0			0
$765$		−	40.9017i		−	1.47880i
$766$		0			0
$767$			10.7603i			0.388532i
$768$		0			0
$769$	−25.3869			−0.915475			−0.457738	−	0.889087i	$-0.651340\pi$
$769$	−25.3869			−0.915475			−0.457738	+	0.889087i	$0.651340\pi$
$770$		0			0
$771$	4.42394			0.159324
$772$		0			0
$773$		−	18.7110i		−	0.672987i	−0.941686	−	0.336494i	$-0.890759\pi$
$773$		−	18.7110i		−	0.672987i	0.941686	−	0.336494i	$-0.109241\pi$
$774$		0			0
$775$			35.2098i			1.26477i
$776$		0			0
$777$	9.72211	−	19.3355i	0.348779	−	0.693657i
$778$		0			0
$779$	−16.9717			−0.608075
$780$		0			0
$781$	−52.6010	−	16.7504i	−1.88221	−	0.599378i
$782$		0			0
$783$	−10.5407			−0.376693
$784$		0			0
$785$	−29.1772			−1.04138
$786$		0			0
$787$	22.4572			0.800511			0.400256	−	0.916404i	$-0.368921\pi$
$787$	22.4572			0.800511			0.400256	+	0.916404i	$0.368921\pi$
$788$		0			0
$789$	62.6063			2.22884
$790$		0			0
$791$	22.1593	+	11.1419i	0.787894	+	0.396162i
$792$		0			0
$793$	11.0348			0.391858
$794$		0			0
$795$	27.9393			0.990906
$796$		0			0
$797$		−	0.427862i		−	0.0151557i	−0.999971	−	0.00757783i	$-0.997588\pi$
$797$		−	0.427862i		−	0.0151557i	0.999971	−	0.00757783i	$-0.00241212\pi$
$798$		0			0
$799$			5.03786i			0.178227i
$800$		0			0
$801$		−	22.1501i		−	0.782636i
$802$		0			0
$803$	−45.6507	−	14.5372i	−1.61098	−	0.513005i
$804$		0			0
$805$	−0.114723	+	0.228163i	−0.00404346	+	0.00804169i
$806$		0			0
$807$	74.9688			2.63903
$808$		0			0
$809$			7.81966i			0.274925i	0.990507	+	0.137462i	$0.0438946\pi$
$809$			7.81966i			0.274925i	−0.990507	+	0.137462i	$0.956105\pi$
$810$		0			0
$811$	−27.3324			−0.959771			−0.479885	−	0.877331i	$-0.659322\pi$
$811$	−27.3324			−0.959771			−0.479885	+	0.877331i	$0.659322\pi$
$812$		0			0
$813$			14.8097i			0.519399i
$814$		0			0
$815$		−	25.0279i		−	0.876689i
$816$		0			0
$817$			32.9063i			1.15124i
$818$		0			0
$819$	14.0419	+	7.06043i	0.490663	+	0.246711i
$820$		0			0
$821$			8.40960i			0.293497i	0.989174	+	0.146748i	$0.0468808\pi$
$821$			8.40960i			0.293497i	−0.989174	+	0.146748i	$0.953119\pi$
$822$		0			0
$823$	10.6270			0.370432			0.185216	−	0.982698i	$-0.440701\pi$
$823$	10.6270			0.370432			0.185216	+	0.982698i	$0.440701\pi$
$824$		0			0
$825$	−10.3151	+	32.3921i	−0.359124	+	1.12775i
$826$		0			0
$827$		−	19.4102i		−	0.674959i	−0.941333	−	0.337480i	$-0.890426\pi$
$827$		−	19.4102i		−	0.674959i	0.941333	−	0.337480i	$-0.109574\pi$
$828$		0			0
$829$		−	10.2946i		−	0.357548i	−0.983890	−	0.178774i	$-0.942787\pi$
$829$		−	10.2946i		−	0.357548i	0.983890	−	0.178774i	$-0.0572130\pi$
$830$		0			0
$831$	7.17179			0.248787
$832$		0			0
$833$	22.9258	+	30.8556i	0.794331	+	1.06908i
$834$		0			0
$835$		−	13.4436i		−	0.465236i
$836$		0			0
$837$	−90.3077			−3.12149
$838$		0			0
$839$			50.0711i			1.72865i	0.502935	+	0.864324i	$0.332253\pi$
$839$			50.0711i			1.72865i	−0.502935	+	0.864324i	$0.667747\pi$
$840$		0			0
$841$	27.5627			0.950439
$842$		0			0
$843$	−37.0397			−1.27571
$844$		0			0
$845$			1.25381i			0.0431323i
$846$		0			0
$847$	13.6536	+	25.7017i	0.469144	+	0.883122i
$848$		0			0
$849$			48.3428i			1.65912i
$850$		0			0
$851$	0.210610			0.00721963
$852$		0			0
$853$	−13.5132			−0.462685			−0.231342	−	0.972872i	$-0.574312\pi$
$853$	−13.5132			−0.462685			−0.231342	+	0.972872i	$0.574312\pi$
$854$		0			0
$855$		−	25.7081i		−	0.879199i
$856$		0			0
$857$	−1.09494			−0.0374023			−0.0187011	−	0.999825i	$-0.505953\pi$
$857$	−1.09494			−0.0374023			−0.0187011	+	0.999825i	$0.505953\pi$
$858$		0			0
$859$		−	9.53122i		−	0.325201i	−0.986692	−	0.162601i	$-0.948012\pi$
$859$		−	9.53122i		−	0.325201i	0.986692	−	0.162601i	$-0.0519882\pi$
$860$		0			0
$861$	−17.4742	+	34.7531i	−0.595521	+	1.18438i
$862$		0			0
$863$	−24.9340			−0.848764			−0.424382	−	0.905483i	$-0.639509\pi$
$863$	−24.9340			−0.848764			−0.424382	+	0.905483i	$0.639509\pi$
$864$		0			0
$865$			24.1140i			0.819899i
$866$		0			0
$867$		−	39.3382i		−	1.33599i
$868$		0			0
$869$	7.59717	−	23.8572i	0.257716	−	0.809300i
$870$		0			0
$871$	9.82223			0.332814
$872$		0			0
$873$		−	42.0228i		−	1.42226i
$874$		0			0
$875$	−12.5593	+	24.9781i	−0.424580	+	0.844412i
$876$		0			0
$877$			37.4767i			1.26550i	0.774356	+	0.632750i	$0.218074\pi$
$877$			37.4767i			1.26550i	−0.774356	+	0.632750i	$0.781926\pi$
$878$		0			0
$879$			8.95493i			0.302042i
$880$		0			0
$881$			34.5046i			1.16249i	0.813729	+	0.581245i	$0.197434\pi$
$881$			34.5046i			1.16249i	−0.813729	+	0.581245i	$0.802566\pi$
$882$		0			0
$883$	−6.27161			−0.211057			−0.105528	−	0.994416i	$-0.533653\pi$
$883$	−6.27161			−0.211057			−0.105528	+	0.994416i	$0.533653\pi$
$884$		0			0
$885$		−	40.3400i		−	1.35601i
$886$		0			0
$887$	22.5832			0.758270			0.379135	−	0.925341i	$-0.376222\pi$
$887$	22.5832			0.758270			0.379135	+	0.925341i	$0.376222\pi$
$888$		0			0
$889$	18.1723	−	36.1413i	0.609478	−	1.21214i
$890$		0			0
$891$	−26.7604	−	8.52167i	−0.896507	−	0.285487i
$892$		0			0
$893$			3.16647i			0.105962i
$894$		0			0
$895$			7.41370i			0.247813i
$896$		0			0
$897$			0.230192i			0.00768588i
$898$		0			0
$899$	−12.3140			−0.410695
$900$		0			0
$901$	40.9255			1.36342
$902$		0			0
$903$	67.3823	+	33.8806i	2.24234	+	1.12748i
$904$		0			0
$905$	−1.67579			−0.0557052
$906$		0			0
$907$	−34.3309			−1.13994			−0.569970	−	0.821666i	$-0.693045\pi$
$907$	−34.3309			−1.13994			−0.569970	+	0.821666i	$0.693045\pi$
$908$		0			0
$909$	−50.5549			−1.67680
$910$		0			0
$911$	24.8755			0.824162			0.412081	−	0.911147i	$-0.364802\pi$
$911$	24.8755			0.824162			0.412081	+	0.911147i	$0.364802\pi$
$912$		0			0
$913$	−22.4911	−	7.16214i	−0.744347	−	0.237032i
$914$		0			0
$915$	−41.3692			−1.36762
$916$		0			0
$917$	−39.5149	−	19.8685i	−1.30490	−	0.656117i
$918$		0			0
$919$		−	30.1947i		−	0.996030i	−0.867168	−	0.498015i	$-0.834062\pi$
$919$		−	30.1947i		−	0.996030i	0.867168	−	0.498015i	$-0.165938\pi$
$920$		0			0
$921$		−	93.3092i		−	3.07464i
$922$		0			0
$923$	−16.6445			−0.547861
$924$		0			0
$925$	9.37792			0.308344
$926$		0			0
$927$			25.8792i			0.849984i
$928$		0			0
$929$			17.0899i			0.560701i	0.959898	+	0.280351i	$0.0904508\pi$
$929$			17.0899i			0.560701i	−0.959898	+	0.280351i	$0.909549\pi$
$930$		0			0
$931$	14.4096	+	19.3938i	0.472257	+	0.635606i
$932$		0			0
$933$	11.1977			0.366596
$934$		0			0
$935$	6.92907	−	21.7592i	0.226605	−	0.711602i
$936$		0			0
$937$	34.1429			1.11540			0.557700	−	0.830043i	$-0.311684\pi$
$937$	34.1429			1.11540			0.557700	+	0.830043i	$0.311684\pi$
$938$		0			0
$939$	27.5813			0.900083
$940$		0			0
$941$	−32.6546			−1.06451			−0.532255	−	0.846584i	$-0.678655\pi$
$941$	−32.6546			−1.06451			−0.532255	+	0.846584i	$0.678655\pi$
$942$		0			0
$943$	−0.378545			−0.0123271
$944$		0			0
$945$	−26.0575	−	13.1020i	−0.847651	−	0.426209i
$946$		0			0
$947$	43.4276			1.41121			0.705604	−	0.708606i	$-0.250676\pi$
$947$	43.4276			1.41121			0.705604	+	0.708606i	$0.250676\pi$
$948$		0			0
$949$	−14.4452			−0.468912
$950$		0			0
$951$			49.1437i			1.59359i
$952$		0			0
$953$			58.8326i			1.90578i	0.303322	+	0.952888i	$0.401904\pi$
$953$			58.8326i			1.90578i	−0.303322	+	0.952888i	$0.598096\pi$
$954$		0			0
$955$			22.5658i			0.730212i
$956$		0			0
$957$	−11.3285	−	3.60750i	−0.366200	−	0.116614i
$958$		0			0
$959$	42.3270	+	21.2825i	1.36681	+	0.687247i
$960$		0			0
$961$	−74.5007			−2.40325
$962$		0			0
$963$			48.6151i			1.56660i
$964$		0			0
$965$	−15.2711			−0.491594
$966$		0			0
$967$		−	53.1571i		−	1.70942i	−0.519108	−	0.854708i	$-0.673736\pi$
$967$		−	53.1571i		−	1.70942i	0.519108	−	0.854708i	$-0.326264\pi$
$968$		0			0
$969$		−	56.6745i		−	1.82065i
$970$		0			0
$971$			16.5488i			0.531077i	0.964100	+	0.265538i	$0.0855497\pi$
$971$			16.5488i			0.531077i	−0.964100	+	0.265538i	$0.914450\pi$
$972$		0			0
$973$	−20.1351	−	10.1242i	−0.645502	−	0.324566i
$974$		0			0
$975$			10.2498i			0.328257i
$976$		0			0
$977$	33.9265			1.08540			0.542702	−	0.839925i	$-0.317401\pi$
$977$	33.9265			1.08540			0.542702	+	0.839925i	$0.317401\pi$
$978$		0			0
$979$	3.75241	−	11.7836i	0.119928	−	0.376605i
$980$		0			0
$981$			73.4550i			2.34524i
$982$		0			0
$983$		−	6.22213i		−	0.198455i	−0.995065	−	0.0992275i	$-0.968363\pi$
$983$		−	6.22213i		−	0.198455i	0.995065	−	0.0992275i	$-0.0316372\pi$
$984$		0			0
$985$	−5.43346			−0.173124
$986$		0			0
$987$	6.48398	+	3.26022i	0.206387	+	0.103774i
$988$		0			0
$989$			0.733956i			0.0233384i
$990$		0			0
$991$	−16.4757			−0.523369			−0.261684	−	0.965154i	$-0.584278\pi$
$991$	−16.4757			−0.523369			−0.261684	+	0.965154i	$0.584278\pi$
$992$		0			0
$993$		−	16.1790i		−	0.513424i
$994$		0			0
$995$	26.6140			0.843722
$996$		0			0
$997$	48.4183			1.53342			0.766711	−	0.641992i	$-0.221892\pi$
$997$	48.4183			1.53342			0.766711	+	0.641992i	$0.221892\pi$
$998$		0			0
$999$			24.0529i			0.761000i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.b.3849.4	yes	48
7.6	odd	2		4004.2.e.a.3849.45	yes	48
11.10	odd	2		4004.2.e.a.3849.4	✓	48
77.76	even	2	inner	4004.2.e.b.3849.45	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.4	✓	48	11.10	odd	2
4004.2.e.a.3849.45	yes	48	7.6	odd	2
4004.2.e.b.3849.4	yes	48	1.1	even	1	trivial
4004.2.e.b.3849.45	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)

\(f(q)\)	\(=\)	\(q-2.99006i q^{3} +1.25381i q^{5} +(2.36377 + 1.18853i) q^{7} -5.94047 q^{9} +O(q^{10})\)	\(q-2.99006i q^{3} +1.25381i q^{5} +(2.36377 + 1.18853i) q^{7} -5.94047 q^{9} +(-3.16026 - 1.00636i) q^{11} -1.00000 q^{13} +3.74896 q^{15} +5.49148 q^{17} +3.45158 q^{19} +(3.55378 - 7.06781i) q^{21} +0.0769856 q^{23} +3.42796 q^{25} +8.79219i q^{27} +1.19887i q^{29} +10.2714i q^{31} +(-3.00909 + 9.44937i) q^{33} +(-1.49019 + 2.96371i) q^{35} +2.73571 q^{37} +2.99006i q^{39} -4.91709 q^{41} +9.53368i q^{43} -7.44821i q^{45} +0.917396i q^{47} +(4.17479 + 5.61882i) q^{49} -16.4199i q^{51} +7.45254 q^{53} +(1.26179 - 3.96236i) q^{55} -10.3204i q^{57} -10.7603i q^{59} -11.0348 q^{61} +(-14.0419 - 7.06043i) q^{63} -1.25381i q^{65} -9.82223 q^{67} -0.230192i q^{69} +16.6445 q^{71} +14.4452 q^{73} -10.2498i q^{75} +(-6.27402 - 6.13487i) q^{77} +7.54913i q^{79} +8.46779 q^{81} +7.11686 q^{83} +6.88526i q^{85} +3.58469 q^{87} +3.72868i q^{89} +(-2.36377 - 1.18853i) q^{91} +30.7120 q^{93} +4.32762i q^{95} +7.07399i q^{97} +(18.7734 + 5.97827i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{7} - 48 q^{9}+O(q^{10}) \) 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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