LMFDB - Embedded newform 4004.2.e.b.3849.2

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			3849.2
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.b.3849.47

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.15005i q^{3} -2.09918i q^{5} +(-1.59876 + 2.10807i) q^{7} -6.92283 q^{9} +O(q^{10})$	$q-3.15005i q^{3} -2.09918i q^{5} +(-1.59876 + 2.10807i) q^{7} -6.92283 q^{9} +(1.48288 + 2.96666i) q^{11} -1.00000 q^{13} -6.61253 q^{15} -5.21366 q^{17} +3.50650 q^{19} +(6.64054 + 5.03617i) q^{21} -3.97298 q^{23} +0.593447 q^{25} +12.3571i q^{27} +0.171004i q^{29} +6.22992i q^{31} +(9.34514 - 4.67114i) q^{33} +(4.42522 + 3.35608i) q^{35} -9.42308 q^{37} +3.15005i q^{39} +5.25145 q^{41} +2.92487i q^{43} +14.5323i q^{45} -9.40782i q^{47} +(-1.88794 - 6.74060i) q^{49} +16.4233i q^{51} +5.09027 q^{53} +(6.22755 - 3.11282i) q^{55} -11.0456i q^{57} +9.32514i q^{59} +0.809574 q^{61} +(11.0679 - 14.5938i) q^{63} +2.09918i q^{65} +4.09175 q^{67} +12.5151i q^{69} +10.2221 q^{71} +15.7302 q^{73} -1.86939i q^{75} +(-8.62470 - 1.61696i) q^{77} -2.13543i q^{79} +18.1571 q^{81} +4.96801 q^{83} +10.9444i q^{85} +0.538671 q^{87} +15.0316i q^{89} +(1.59876 - 2.10807i) q^{91} +19.6246 q^{93} -7.36076i q^{95} -1.01911i q^{97} +(-10.2657 - 20.5377i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	3.15005i		−	1.81868i	−0.416049	−	0.909342i	$-0.636586\pi$
$3$		−	3.15005i		−	1.81868i	0.416049	−	0.909342i	$-0.363414\pi$
$4$		0			0
$5$		−	2.09918i		−	0.938781i	−0.882991	−	0.469391i	$-0.844474\pi$
$5$		−	2.09918i		−	0.938781i	0.882991	−	0.469391i	$-0.155526\pi$
$6$		0			0
$7$	−1.59876	+	2.10807i	−0.604274	+	0.796777i
$7$	−1.59876	+	2.10807i	−0.604274	+	0.796777i
$8$		0			0
$9$	−6.92283			−2.30761
$10$		0			0
$11$	1.48288	+	2.96666i	0.447104	+	0.894482i
$11$	1.48288	+	2.96666i	0.447104	+	0.894482i
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	−6.61253			−1.70735
$16$		0			0
$17$	−5.21366			−1.26450			−0.632249	−	0.774765i	$-0.717868\pi$
$17$	−5.21366			−1.26450			−0.632249	+	0.774765i	$0.717868\pi$
$18$		0			0
$19$	3.50650			0.804445			0.402223	−	0.915542i	$-0.368238\pi$
$19$	3.50650			0.804445			0.402223	+	0.915542i	$0.368238\pi$
$20$		0			0
$21$	6.64054	+	5.03617i	1.44909	+	1.09898i
$22$		0			0
$23$	−3.97298			−0.828423			−0.414211	−	0.910181i	$-0.635943\pi$
$23$	−3.97298			−0.828423			−0.414211	+	0.910181i	$0.635943\pi$
$24$		0			0
$25$	0.593447			0.118689
$26$		0			0
$27$			12.3571i			2.37813i
$28$		0			0
$29$			0.171004i			0.0317546i	0.999874	+	0.0158773i	$0.00505412\pi$
$29$			0.171004i			0.0317546i	−0.999874	+	0.0158773i	$0.994946\pi$
$30$		0			0
$31$			6.22992i			1.11893i	0.828855	+	0.559463i	$0.188993\pi$
$31$			6.22992i			1.11893i	−0.828855	+	0.559463i	$0.811007\pi$
$32$		0			0
$33$	9.34514	−	4.67114i	1.62678	−	0.813141i
$34$		0			0
$35$	4.42522	+	3.35608i	0.747999	+	0.567281i
$36$		0			0
$37$	−9.42308			−1.54914			−0.774572	−	0.632485i	$-0.782035\pi$
$37$	−9.42308			−1.54914			−0.774572	+	0.632485i	$0.782035\pi$
$38$		0			0
$39$			3.15005i			0.504412i
$40$		0			0
$41$	5.25145			0.820139			0.410069	−	0.912054i	$-0.365504\pi$
$41$	5.25145			0.820139			0.410069	+	0.912054i	$0.365504\pi$
$42$		0			0
$43$			2.92487i			0.446039i	0.974814	+	0.223019i	$0.0715913\pi$
$43$			2.92487i			0.446039i	−0.974814	+	0.223019i	$0.928409\pi$
$44$		0			0
$45$			14.5323i			2.16634i
$46$		0			0
$47$		−	9.40782i		−	1.37227i	−0.727474	−	0.686135i	$-0.759306\pi$
$47$		−	9.40782i		−	1.37227i	0.727474	−	0.686135i	$-0.240694\pi$
$48$		0			0
$49$	−1.88794	−	6.74060i	−0.269706	−	0.962943i
$50$		0			0
$51$			16.4233i			2.29972i
$52$		0			0
$53$	5.09027			0.699202			0.349601	−	0.936899i	$-0.386317\pi$
$53$	5.09027			0.699202			0.349601	+	0.936899i	$0.386317\pi$
$54$		0			0
$55$	6.22755	−	3.11282i	0.839723	−	0.419733i
$56$		0			0
$57$		−	11.0456i		−	1.46303i
$58$		0			0
$59$			9.32514i			1.21403i	0.794690	+	0.607015i	$0.207633\pi$
$59$			9.32514i			1.21403i	−0.794690	+	0.607015i	$0.792367\pi$
$60$		0			0
$61$	0.809574			0.103655			0.0518277	−	0.998656i	$-0.483495\pi$
$61$	0.809574			0.103655			0.0518277	+	0.998656i	$0.483495\pi$
$62$		0			0
$63$	11.0679	−	14.5938i	1.39443	−	1.83865i
$64$		0			0
$65$			2.09918i			0.260371i
$66$		0			0
$67$	4.09175			0.499887			0.249943	−	0.968260i	$-0.419588\pi$
$67$	4.09175			0.499887			0.249943	+	0.968260i	$0.419588\pi$
$68$		0			0
$69$			12.5151i			1.50664i
$70$		0			0
$71$	10.2221			1.21314			0.606572	−	0.795028i	$-0.292544\pi$
$71$	10.2221			1.21314			0.606572	+	0.795028i	$0.292544\pi$
$72$		0			0
$73$	15.7302			1.84108			0.920541	−	0.390646i	$-0.127748\pi$
$73$	15.7302			1.84108			0.920541	+	0.390646i	$0.127748\pi$
$74$		0			0
$75$		−	1.86939i		−	0.215859i
$76$		0			0
$77$	−8.62470	−	1.61696i	−0.982876	−	0.184270i
$78$		0			0
$79$		−	2.13543i		−	0.240255i	−0.992758	−	0.120127i	$-0.961670\pi$
$79$		−	2.13543i		−	0.240255i	0.992758	−	0.120127i	$-0.0383303\pi$
$80$		0			0
$81$	18.1571			2.01746
$82$		0			0
$83$	4.96801			0.545310			0.272655	−	0.962112i	$-0.412098\pi$
$83$	4.96801			0.545310			0.272655	+	0.962112i	$0.412098\pi$
$84$		0			0
$85$			10.9444i			1.18709i
$86$		0			0
$87$	0.538671			0.0577516
$88$		0			0
$89$			15.0316i			1.59335i	0.604409	+	0.796674i	$0.293409\pi$
$89$			15.0316i			1.59335i	−0.604409	+	0.796674i	$0.706591\pi$
$90$		0			0
$91$	1.59876	−	2.10807i	0.167595	−	0.220986i
$92$		0			0
$93$	19.6246			2.03497
$94$		0			0
$95$		−	7.36076i		−	0.755198i
$96$		0			0
$97$		−	1.01911i		−	0.103475i	−0.998661	−	0.0517375i	$-0.983524\pi$
$97$		−	1.01911i		−	0.103475i	0.998661	−	0.0517375i	$-0.0164759\pi$
$98$		0			0
$99$	−10.2657	−	20.5377i	−1.03174	−	2.06412i
$100$		0			0
$101$	2.73927			0.272568			0.136284	−	0.990670i	$-0.456484\pi$
$101$	2.73927			0.272568			0.136284	+	0.990670i	$0.456484\pi$
$102$		0			0
$103$		−	3.74842i		−	0.369343i	−0.982800	−	0.184671i	$-0.940878\pi$
$103$		−	3.74842i		−	0.369343i	0.982800	−	0.184671i	$-0.0591220\pi$
$104$		0			0
$105$	10.5718	−	13.9397i	1.03170	−	1.36037i
$106$		0			0
$107$			9.64985i			0.932886i	0.884551	+	0.466443i	$0.154465\pi$
$107$			9.64985i			0.932886i	−0.884551	+	0.466443i	$0.845535\pi$
$108$		0			0
$109$			18.7531i			1.79622i	0.439773	+	0.898109i	$0.355059\pi$
$109$			18.7531i			1.79622i	−0.439773	+	0.898109i	$0.644941\pi$
$110$		0			0
$111$			29.6832i			2.81740i
$112$		0			0
$113$	−7.32274			−0.688866			−0.344433	−	0.938811i	$-0.611929\pi$
$113$	−7.32274			−0.688866			−0.344433	+	0.938811i	$0.611929\pi$
$114$		0			0
$115$			8.33999i			0.777708i
$116$		0			0
$117$	6.92283			0.640016
$118$		0			0
$119$	8.33538	−	10.9908i	0.764103	−	1.00752i
$120$		0			0
$121$	−6.60216	+	8.79838i	−0.600196	+	0.799853i
$122$		0			0
$123$		−	16.5423i		−	1.49157i
$124$		0			0
$125$		−	11.7416i		−	1.05020i
$126$		0			0
$127$			0.980063i			0.0869665i	0.999054	+	0.0434833i	$0.0138455\pi$
$127$			0.980063i			0.0869665i	−0.999054	+	0.0434833i	$0.986154\pi$
$128$		0			0
$129$	9.21350			0.811203
$130$		0			0
$131$	−0.170313			−0.0148803			−0.00744015	−	0.999972i	$-0.502368\pi$
$131$	−0.170313			−0.0148803			−0.00744015	+	0.999972i	$0.502368\pi$
$132$		0			0
$133$	−5.60604	+	7.39195i	−0.486105	+	0.640963i
$134$		0			0
$135$	25.9398			2.23255
$136$		0			0
$137$	−1.41851			−0.121191			−0.0605957	−	0.998162i	$-0.519300\pi$
$137$	−1.41851			−0.121191			−0.0605957	+	0.998162i	$0.519300\pi$
$138$		0			0
$139$	−14.8153			−1.25662			−0.628308	−	0.777965i	$-0.716252\pi$
$139$	−14.8153			−1.25662			−0.628308	+	0.777965i	$0.716252\pi$
$140$		0			0
$141$	−29.6351			−2.49573
$142$		0			0
$143$	−1.48288	−	2.96666i	−0.124004	−	0.248085i
$144$		0			0
$145$	0.358968			0.0298106
$146$		0			0
$147$	−21.2332	+	5.94713i	−1.75129	+	0.490511i
$148$		0			0
$149$		−	2.96887i		−	0.243220i	−0.992578	−	0.121610i	$-0.961194\pi$
$149$		−	2.96887i		−	0.243220i	0.992578	−	0.121610i	$-0.0388057\pi$
$150$		0			0
$151$			3.33304i			0.271239i	0.990761	+	0.135619i	$0.0433024\pi$
$151$			3.33304i			0.271239i	−0.990761	+	0.135619i	$0.956698\pi$
$152$		0			0
$153$	36.0933			2.91797
$154$		0			0
$155$	13.0777			1.05043
$156$		0			0
$157$			0.968226i			0.0772728i	0.999253	+	0.0386364i	$0.0123014\pi$
$157$			0.968226i			0.0772728i	−0.999253	+	0.0386364i	$0.987699\pi$
$158$		0			0
$159$		−	16.0346i		−	1.27163i
$160$		0			0
$161$	6.35183	−	8.37532i	0.500594	−	0.660068i
$162$		0			0
$163$	22.0798			1.72943			0.864713	−	0.502267i	$-0.167501\pi$
$163$	22.0798			1.72943			0.864713	+	0.502267i	$0.167501\pi$
$164$		0			0
$165$	−9.80555	−	19.6171i	−0.763361	−	1.52719i
$166$		0			0
$167$	−20.4339			−1.58122			−0.790611	−	0.612319i	$-0.790237\pi$
$167$	−20.4339			−1.58122			−0.790611	+	0.612319i	$0.790237\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−24.2749			−1.85635
$172$		0			0
$173$	−1.29731			−0.0986326			−0.0493163	−	0.998783i	$-0.515704\pi$
$173$	−1.29731			−0.0986326			−0.0493163	+	0.998783i	$0.515704\pi$
$174$		0			0
$175$	−0.948779	+	1.25103i	−0.0717209	+	0.0945690i
$176$		0			0
$177$	29.3747			2.20794
$178$		0			0
$179$	2.47010			0.184624			0.0923118	−	0.995730i	$-0.470574\pi$
$179$	2.47010			0.184624			0.0923118	+	0.995730i	$0.470574\pi$
$180$		0			0
$181$		−	6.20994i		−	0.461581i	−0.973003	−	0.230791i	$-0.925869\pi$
$181$		−	6.20994i		−	0.461581i	0.973003	−	0.230791i	$-0.0741312\pi$
$182$		0			0
$183$		−	2.55020i		−	0.188516i
$184$		0			0
$185$			19.7807i			1.45431i
$186$		0			0
$187$	−7.73121	−	15.4672i	−0.565362	−	1.13107i
$188$		0			0
$189$	−26.0498	−	19.7561i	−1.89484	−	1.43704i
$190$		0			0
$191$	3.53840			0.256029			0.128015	−	0.991772i	$-0.459140\pi$
$191$	3.53840			0.256029			0.128015	+	0.991772i	$0.459140\pi$
$192$		0			0
$193$			15.5634i			1.12028i	0.828398	+	0.560140i	$0.189253\pi$
$193$			15.5634i			1.12028i	−0.828398	+	0.560140i	$0.810747\pi$
$194$		0			0
$195$	6.61253			0.473533
$196$		0			0
$197$		−	6.40788i		−	0.456542i	−0.973598	−	0.228271i	$-0.926693\pi$
$197$		−	6.40788i		−	0.456542i	0.973598	−	0.228271i	$-0.0733073\pi$
$198$		0			0
$199$			25.0981i			1.77916i	0.456779	+	0.889580i	$0.349003\pi$
$199$			25.0981i			1.77916i	−0.456779	+	0.889580i	$0.650997\pi$
$200$		0			0
$201$		−	12.8892i		−	0.909136i
$202$		0			0
$203$	−0.360489	−	0.273394i	−0.0253013	−	0.0191885i
$204$		0			0
$205$		−	11.0237i		−	0.769931i
$206$		0			0
$207$	27.5043			1.91168
$208$		0			0
$209$	5.19970	+	10.4026i	0.359671	+	0.719562i
$210$		0			0
$211$			7.10761i			0.489308i	0.969610	+	0.244654i	$0.0786743\pi$
$211$			7.10761i			0.489308i	−0.969610	+	0.244654i	$0.921326\pi$
$212$		0			0
$213$		−	32.2003i		−	2.20633i
$214$		0			0
$215$	6.13983			0.418733
$216$		0			0
$217$	−13.1331	−	9.96013i	−0.891534	−	0.676138i
$218$		0			0
$219$		−	49.5510i		−	3.34835i
$220$		0			0
$221$	5.21366			0.350709
$222$		0			0
$223$		−	15.3680i		−	1.02911i	−0.857456	−	0.514557i	$-0.827956\pi$
$223$		−	15.3680i		−	1.02911i	0.857456	−	0.514557i	$-0.172044\pi$
$224$		0			0
$225$	−4.10834			−0.273889
$226$		0			0
$227$	−0.822579			−0.0545965			−0.0272982	−	0.999627i	$-0.508690\pi$
$227$	−0.822579			−0.0545965			−0.0272982	+	0.999627i	$0.508690\pi$
$228$		0			0
$229$			18.8070i			1.24280i	0.783492	+	0.621402i	$0.213436\pi$
$229$			18.8070i			1.24280i	−0.783492	+	0.621402i	$0.786564\pi$
$230$		0			0
$231$	−5.09352	+	27.1683i	−0.335129	+	1.78754i
$232$		0			0
$233$			8.47862i			0.555453i	0.960660	+	0.277727i	$0.0895809\pi$
$233$			8.47862i			0.555453i	−0.960660	+	0.277727i	$0.910419\pi$
$234$		0			0
$235$	−19.7487			−1.28826
$236$		0			0
$237$	−6.72672			−0.436948
$238$		0			0
$239$		−	28.3037i		−	1.83081i	−0.402533	−	0.915406i	$-0.631870\pi$
$239$		−	28.3037i		−	1.83081i	0.402533	−	0.915406i	$-0.368130\pi$
$240$		0			0
$241$	−22.6198			−1.45707			−0.728534	−	0.685010i	$-0.759798\pi$
$241$	−22.6198			−1.45707			−0.728534	+	0.685010i	$0.759798\pi$
$242$		0			0
$243$		−	20.1245i		−	1.29099i
$244$		0			0
$245$	−14.1497	+	3.96313i	−0.903993	+	0.253195i
$246$		0			0
$247$	−3.50650			−0.223113
$248$		0			0
$249$		−	15.6495i		−	0.991746i
$250$		0			0
$251$		−	13.9923i		−	0.883187i	−0.897215	−	0.441593i	$-0.854413\pi$
$251$		−	13.9923i		−	0.883187i	0.897215	−	0.441593i	$-0.145587\pi$
$252$		0			0
$253$	−5.89143	−	11.7865i	−0.370391	−	0.741009i
$254$		0			0
$255$	34.4755			2.15894
$256$		0			0
$257$		−	4.25485i		−	0.265410i	−0.991156	−	0.132705i	$-0.957634\pi$
$257$		−	4.25485i		−	0.265410i	0.991156	−	0.132705i	$-0.0423663\pi$
$258$		0			0
$259$	15.0652	−	19.8645i	0.936108	−	1.23432i
$260$		0			0
$261$		−	1.18383i		−	0.0732773i
$262$		0			0
$263$			13.3345i			0.822239i	0.911581	+	0.411120i	$0.134862\pi$
$263$			13.3345i			0.822239i	−0.911581	+	0.411120i	$0.865138\pi$
$264$		0			0
$265$		−	10.6854i		−	0.656398i
$266$		0			0
$267$	47.3504			2.89780
$268$		0			0
$269$		−	13.7799i		−	0.840175i	−0.907484	−	0.420087i	$-0.861999\pi$
$269$		−	13.7799i		−	0.840175i	0.907484	−	0.420087i	$-0.138001\pi$
$270$		0			0
$271$	20.7217			1.25876			0.629378	−	0.777099i	$-0.283310\pi$
$271$	20.7217			1.25876			0.629378	+	0.777099i	$0.283310\pi$
$272$		0			0
$273$	−6.64054	−	5.03617i	−0.401904	−	0.304803i
$274$		0			0
$275$	0.880009	+	1.76056i	0.0530665	+	0.106166i
$276$		0			0
$277$			5.06853i			0.304538i	0.988339	+	0.152269i	$0.0486580\pi$
$277$			5.06853i			0.304538i	−0.988339	+	0.152269i	$0.951342\pi$
$278$		0			0
$279$		−	43.1287i		−	2.58205i
$280$		0			0
$281$			17.9168i			1.06883i	0.845223	+	0.534415i	$0.179468\pi$
$281$			17.9168i			1.06883i	−0.845223	+	0.534415i	$0.820532\pi$
$282$		0			0
$283$	−32.1327			−1.91009			−0.955044	−	0.296463i	$-0.904193\pi$
$283$	−32.1327			−1.91009			−0.955044	+	0.296463i	$0.904193\pi$
$284$		0			0
$285$	−23.1868			−1.37347
$286$		0			0
$287$	−8.39580	+	11.0704i	−0.495588	+	0.653467i
$288$		0			0
$289$	10.1823			0.598957
$290$		0			0
$291$	−3.21025			−0.188188
$292$		0			0
$293$	−2.94308			−0.171937			−0.0859684	−	0.996298i	$-0.527398\pi$
$293$	−2.94308			−0.171937			−0.0859684	+	0.996298i	$0.527398\pi$
$294$		0			0
$295$	19.5751			1.13971
$296$		0			0
$297$	−36.6594	+	18.3241i	−2.12720	+	1.06327i
$298$		0			0
$299$	3.97298			0.229763
$300$		0			0
$301$	−6.16584	−	4.67616i	−0.355393	−	0.269529i
$302$		0			0
$303$		−	8.62885i		−	0.495715i
$304$		0			0
$305$		−	1.69944i		−	0.0973098i
$306$		0			0
$307$	−2.84490			−0.162367			−0.0811836	−	0.996699i	$-0.525870\pi$
$307$	−2.84490			−0.162367			−0.0811836	+	0.996699i	$0.525870\pi$
$308$		0			0
$309$	−11.8077			−0.671717
$310$		0			0
$311$			14.7804i			0.838117i	0.907959	+	0.419059i	$0.137640\pi$
$311$			14.7804i			0.838117i	−0.907959	+	0.419059i	$0.862360\pi$
$312$		0			0
$313$			27.0006i			1.52616i	0.646302	+	0.763082i	$0.276315\pi$
$313$			27.0006i			1.52616i	−0.646302	+	0.763082i	$0.723685\pi$
$314$		0			0
$315$	−30.6351	−	23.2336i	−1.72609	−	1.30906i
$316$		0			0
$317$	−11.4706			−0.644255			−0.322128	−	0.946696i	$-0.604398\pi$
$317$	−11.4706			−0.644255			−0.322128	+	0.946696i	$0.604398\pi$
$318$		0			0
$319$	−0.507311	+	0.253577i	−0.0284039	+	0.0141976i
$320$		0			0
$321$	30.3975			1.69663
$322$		0			0
$323$	−18.2817			−1.01722
$324$		0			0
$325$	−0.593447			−0.0329185
$326$		0			0
$327$	59.0732			3.26675
$328$		0			0
$329$	19.8324	+	15.0408i	1.09339	+	0.829227i
$330$		0			0
$331$	5.33578			0.293281			0.146641	−	0.989190i	$-0.453154\pi$
$331$	5.33578			0.293281			0.146641	+	0.989190i	$0.453154\pi$
$332$		0			0
$333$	65.2344			3.57482
$334$		0			0
$335$		−	8.58932i		−	0.469284i
$336$		0			0
$337$			34.7485i			1.89287i	0.322895	+	0.946435i	$0.395344\pi$
$337$			34.7485i			1.89287i	−0.322895	+	0.946435i	$0.604656\pi$
$338$		0			0
$339$			23.0670i			1.25283i
$340$		0			0
$341$	−18.4821	+	9.23819i	−1.00086	+	0.500276i
$342$		0			0
$343$	17.2280	+	6.79666i	0.930227	+	0.366985i
$344$		0			0
$345$	26.2714			1.41440
$346$		0			0
$347$		−	1.01536i		−	0.0545073i	−0.999629	−	0.0272536i	$-0.991324\pi$
$347$		−	1.01536i		−	0.0545073i	0.999629	−	0.0272536i	$-0.00867618\pi$
$348$		0			0
$349$	−23.6009			−1.26333			−0.631664	−	0.775242i	$-0.717628\pi$
$349$	−23.6009			−1.26333			−0.631664	+	0.775242i	$0.717628\pi$
$350$		0			0
$351$		−	12.3571i		−	0.659575i
$352$		0			0
$353$			10.0321i			0.533954i	0.963703	+	0.266977i	$0.0860248\pi$
$353$			10.0321i			0.533954i	−0.963703	+	0.266977i	$0.913975\pi$
$354$		0			0
$355$		−	21.4581i		−	1.13888i
$356$		0			0
$357$	−34.6215	−	26.2569i	−1.83237	−	1.38966i
$358$		0			0
$359$		−	3.34466i		−	0.176525i	−0.996097	−	0.0882623i	$-0.971869\pi$
$359$		−	3.34466i		−	0.176525i	0.996097	−	0.0882623i	$-0.0281314\pi$
$360$		0			0
$361$	−6.70449			−0.352868
$362$		0			0
$363$	27.7154	+	20.7972i	1.45468	+	1.09157i
$364$		0			0
$365$		−	33.0205i		−	1.72837i
$366$		0			0
$367$		−	27.3321i		−	1.42673i	−0.700795	−	0.713363i	$-0.747171\pi$
$367$		−	27.3321i		−	1.42673i	0.700795	−	0.713363i	$-0.252829\pi$
$368$		0			0
$369$	−36.3549			−1.89256
$370$		0			0
$371$	−8.13810	+	10.7307i	−0.422509	+	0.557108i
$372$		0			0
$373$			8.16687i			0.422865i	0.977393	+	0.211432i	$0.0678128\pi$
$373$			8.16687i			0.422865i	−0.977393	+	0.211432i	$0.932187\pi$
$374$		0			0
$375$	−36.9868			−1.90999
$376$		0			0
$377$		−	0.171004i		−	0.00880715i
$378$		0			0
$379$	−30.7077			−1.57735			−0.788674	−	0.614812i	$-0.789232\pi$
$379$	−30.7077			−1.57735			−0.788674	+	0.614812i	$0.789232\pi$
$380$		0			0
$381$	3.08725			0.158165
$382$		0			0
$383$		−	8.09161i		−	0.413462i	−0.978398	−	0.206731i	$-0.933718\pi$
$383$		−	8.09161i		−	0.413462i	0.978398	−	0.206731i	$-0.0662825\pi$
$384$		0			0
$385$	−3.39430	+	18.1048i	−0.172989	+	0.922705i
$386$		0			0
$387$		−	20.2484i		−	1.02928i
$388$		0			0
$389$	31.9260			1.61871			0.809355	−	0.587320i	$-0.199817\pi$
$389$	31.9260			1.61871			0.809355	+	0.587320i	$0.199817\pi$
$390$		0			0
$391$	20.7138			1.04754
$392$		0			0
$393$			0.536495i			0.0270626i
$394$		0			0
$395$	−4.48265			−0.225547
$396$		0			0
$397$			24.6682i			1.23806i	0.785367	+	0.619031i	$0.212474\pi$
$397$			24.6682i			1.23806i	−0.785367	+	0.619031i	$0.787526\pi$
$398$		0			0
$399$	23.2850	+	17.6593i	1.16571	+	0.884072i
$400$		0			0
$401$	−17.0823			−0.853051			−0.426525	−	0.904476i	$-0.640263\pi$
$401$	−17.0823			−0.853051			−0.426525	+	0.904476i	$0.640263\pi$
$402$		0			0
$403$		−	6.22992i		−	0.310334i
$404$		0			0
$405$		−	38.1151i		−	1.89395i
$406$		0			0
$407$	−13.9733	−	27.9551i	−0.692629	−	1.38568i
$408$		0			0
$409$	22.8408			1.12941			0.564704	−	0.825294i	$-0.308991\pi$
$409$	22.8408			1.12941			0.564704	+	0.825294i	$0.308991\pi$
$410$		0			0
$411$			4.46838i			0.220409i
$412$		0			0
$413$	−19.6581	−	14.9087i	−0.967311	−	0.733607i
$414$		0			0
$415$		−	10.4287i		−	0.511927i
$416$		0			0
$417$			46.6689i			2.28539i
$418$		0			0
$419$			24.8074i			1.21192i	0.795494	+	0.605961i	$0.207211\pi$
$419$			24.8074i			1.21192i	−0.795494	+	0.605961i	$0.792789\pi$
$420$		0			0
$421$	15.3438			0.747809			0.373905	−	0.927467i	$-0.378019\pi$
$421$	15.3438			0.747809			0.373905	+	0.927467i	$0.378019\pi$
$422$		0			0
$423$			65.1288i			3.16667i
$424$		0			0
$425$	−3.09403			−0.150083
$426$		0			0
$427$	−1.29431	+	1.70664i	−0.0626362	+	0.0825902i
$428$		0			0
$429$	−9.34514	+	4.67114i	−0.451188	+	0.225525i
$430$		0			0
$431$		−	25.2364i		−	1.21560i	−0.794092	−	0.607798i	$-0.792053\pi$
$431$		−	25.2364i		−	1.21560i	0.794092	−	0.607798i	$-0.207947\pi$
$432$		0			0
$433$		−	6.20917i		−	0.298394i	−0.988808	−	0.149197i	$-0.952331\pi$
$433$		−	6.20917i		−	0.298394i	0.988808	−	0.149197i	$-0.0476688\pi$
$434$		0			0
$435$		−	1.13077i		−	0.0542161i
$436$		0			0
$437$	−13.9312			−0.666421
$438$		0			0
$439$	15.3128			0.730840			0.365420	−	0.930843i	$-0.380925\pi$
$439$	15.3128			0.730840			0.365420	+	0.930843i	$0.380925\pi$
$440$		0			0
$441$	13.0699	+	46.6640i	0.622378	+	2.22210i
$442$		0			0
$443$	25.8346			1.22744			0.613720	−	0.789523i	$-0.289672\pi$
$443$	25.8346			1.22744			0.613720	+	0.789523i	$0.289672\pi$
$444$		0			0
$445$	31.5540			1.49581
$446$		0			0
$447$	−9.35211			−0.442340
$448$		0			0
$449$	−33.2423			−1.56880			−0.784400	−	0.620255i	$-0.787029\pi$
$449$	−33.2423			−1.56880			−0.784400	+	0.620255i	$0.787029\pi$
$450$		0			0
$451$	7.78725	+	15.5793i	0.366687	+	0.733599i
$452$		0			0
$453$	10.4992			0.493297
$454$		0			0
$455$	−4.42522	−	3.35608i	−0.207458	−	0.157335i
$456$		0			0
$457$			26.5152i			1.24033i	0.784472	+	0.620164i	$0.212934\pi$
$457$			26.5152i			1.24033i	−0.784472	+	0.620164i	$0.787066\pi$
$458$		0			0
$459$		−	64.4259i		−	3.00715i
$460$		0			0
$461$	−29.6582			−1.38132			−0.690659	−	0.723180i	$-0.742680\pi$
$461$	−29.6582			−1.38132			−0.690659	+	0.723180i	$0.742680\pi$
$462$		0			0
$463$	21.9657			1.02083			0.510417	−	0.859927i	$-0.329491\pi$
$463$	21.9657			1.02083			0.510417	+	0.859927i	$0.329491\pi$
$464$		0			0
$465$		−	41.1955i		−	1.91040i
$466$		0			0
$467$		−	16.2309i		−	0.751076i	−0.926807	−	0.375538i	$-0.877458\pi$
$467$		−	16.2309i		−	0.751076i	0.926807	−	0.375538i	$-0.122542\pi$
$468$		0			0
$469$	−6.54172	+	8.62571i	−0.302069	+	0.398298i
$470$		0			0
$471$	3.04996			0.140535
$472$		0			0
$473$	−8.67710	+	4.33722i	−0.398974	+	0.199426i
$474$		0			0
$475$	2.08092			0.0954792
$476$		0			0
$477$	−35.2391			−1.61349
$478$		0			0
$479$	37.0931			1.69483			0.847414	−	0.530933i	$-0.178158\pi$
$479$	37.0931			1.69483			0.847414	+	0.530933i	$0.178158\pi$
$480$		0			0
$481$	9.42308			0.429655
$482$		0			0
$483$	−26.3827	−	20.0086i	−1.20046	−	0.910423i
$484$		0			0
$485$	−2.13930			−0.0971404
$486$		0			0
$487$	40.9088			1.85376			0.926878	−	0.375363i	$-0.122482\pi$
$487$	40.9088			1.85376			0.926878	+	0.375363i	$0.122482\pi$
$488$		0			0
$489$		−	69.5526i		−	3.14528i
$490$		0			0
$491$		−	20.0987i		−	0.907040i	−0.891246	−	0.453520i	$-0.850168\pi$
$491$		−	20.0987i		−	0.907040i	0.891246	−	0.453520i	$-0.149832\pi$
$492$		0			0
$493$		−	0.891556i		−	0.0401537i
$494$		0			0
$495$	−43.1123	+	21.5496i	−1.93775	+	0.968580i
$496$		0			0
$497$	−16.3427	+	21.5490i	−0.733071	+	0.966605i
$498$		0			0
$499$	14.8768			0.665977			0.332988	−	0.942931i	$-0.391943\pi$
$499$	14.8768			0.665977			0.332988	+	0.942931i	$0.391943\pi$
$500$		0			0
$501$			64.3678i			2.87574i
$502$		0			0
$503$	18.6090			0.829733			0.414867	−	0.909882i	$-0.363828\pi$
$503$	18.6090			0.829733			0.414867	+	0.909882i	$0.363828\pi$
$504$		0			0
$505$		−	5.75022i		−	0.255882i
$506$		0			0
$507$		−	3.15005i		−	0.139899i
$508$		0			0
$509$			21.0342i			0.932324i	0.884699	+	0.466162i	$0.154364\pi$
$509$			21.0342i			0.932324i	−0.884699	+	0.466162i	$0.845636\pi$
$510$		0			0
$511$	−25.1488	+	33.1604i	−1.11252	+	1.46693i
$512$		0			0
$513$			43.3302i			1.91308i
$514$		0			0
$515$	−7.86860			−0.346732
$516$		0			0
$517$	27.9098	−	13.9506i	1.22747	−	0.613548i
$518$		0			0
$519$			4.08659i			0.179381i
$520$		0			0
$521$			36.5632i			1.60186i	0.598757	+	0.800931i	$0.295662\pi$
$521$			36.5632i			1.60186i	−0.598757	+	0.800931i	$0.704338\pi$
$522$		0			0
$523$	3.35590			0.146743			0.0733716	−	0.997305i	$-0.476624\pi$
$523$	3.35590			0.146743			0.0733716	+	0.997305i	$0.476624\pi$
$524$		0			0
$525$	3.94081	+	2.98870i	0.171991	+	0.130438i
$526$		0			0
$527$		−	32.4807i		−	1.41488i
$528$		0			0
$529$	−7.21546			−0.313716
$530$		0			0
$531$		−	64.5564i		−	2.80151i
$532$		0			0
$533$	−5.25145			−0.227466
$534$		0			0
$535$	20.2568			0.875776
$536$		0			0
$537$		−	7.78093i		−	0.335772i
$538$		0			0
$539$	17.1975	−	15.5964i	0.740748	−	0.671783i
$540$		0			0
$541$			36.4575i			1.56743i	0.621119	+	0.783716i	$0.286678\pi$
$541$			36.4575i			1.56743i	−0.621119	+	0.783716i	$0.713322\pi$
$542$		0			0
$543$	−19.5616			−0.839470
$544$		0			0
$545$	39.3660			1.68626
$546$		0			0
$547$			23.6889i			1.01286i	0.862280	+	0.506432i	$0.169036\pi$
$547$			23.6889i			1.01286i	−0.862280	+	0.506432i	$0.830964\pi$
$548$		0			0
$549$	−5.60455			−0.239196
$550$		0			0
$551$			0.599624i			0.0255448i
$552$		0			0
$553$	4.50164	+	3.41404i	0.191429	+	0.145180i
$554$		0			0
$555$	62.3104			2.64493
$556$		0			0
$557$			30.9965i			1.31336i	0.754167	+	0.656682i	$0.228041\pi$
$557$			30.9965i			1.31336i	−0.754167	+	0.656682i	$0.771959\pi$
$558$		0			0
$559$		−	2.92487i		−	0.123709i
$560$		0			0
$561$	−48.7224	+	24.3537i	−2.05706	+	1.02822i
$562$		0			0
$563$	39.3122			1.65681			0.828407	−	0.560127i	$-0.189248\pi$
$563$	39.3122			1.65681			0.828407	+	0.560127i	$0.189248\pi$
$564$		0			0
$565$			15.3717i			0.646694i
$566$		0			0
$567$	−29.0289	+	38.2766i	−1.21910	+	1.60747i
$568$		0			0
$569$			31.5812i			1.32395i	0.749525	+	0.661975i	$0.230282\pi$
$569$			31.5812i			1.32395i	−0.749525	+	0.661975i	$0.769718\pi$
$570$		0			0
$571$			44.9066i			1.87928i	0.342158	+	0.939642i	$0.388842\pi$
$571$			44.9066i			1.87928i	−0.342158	+	0.939642i	$0.611158\pi$
$572$		0			0
$573$		−	11.1461i		−	0.465636i
$574$		0			0
$575$	−2.35775			−0.0983250
$576$		0			0
$577$		−	42.3288i		−	1.76217i	−0.472956	−	0.881086i	$-0.656813\pi$
$577$		−	42.3288i		−	1.76217i	0.472956	−	0.881086i	$-0.343187\pi$
$578$		0			0
$579$	49.0256			2.03743
$580$		0			0
$581$	−7.94264	+	10.4729i	−0.329516	+	0.434490i
$582$		0			0
$583$	7.54823	+	15.1011i	0.312616	+	0.625423i
$584$		0			0
$585$		−	14.5323i		−	0.600835i
$586$		0			0
$587$			0.266548i			0.0110016i	0.999985	+	0.00550080i	$0.00175097\pi$
$587$			0.266548i			0.0110016i	−0.999985	+	0.00550080i	$0.998249\pi$
$588$		0			0
$589$			21.8452i			0.900115i
$590$		0			0
$591$	−20.1851			−0.830306
$592$		0			0
$593$	43.7939			1.79840			0.899200	−	0.437537i	$-0.144149\pi$
$593$	43.7939			1.79840			0.899200	+	0.437537i	$0.144149\pi$
$594$		0			0
$595$	−23.0716	−	17.4975i	−0.945844	−	0.717326i
$596$		0			0
$597$	79.0605			3.23573
$598$		0			0
$599$	12.7211			0.519771			0.259885	−	0.965640i	$-0.416315\pi$
$599$	12.7211			0.519771			0.259885	+	0.965640i	$0.416315\pi$
$600$		0			0
$601$	8.82401			0.359939			0.179969	−	0.983672i	$-0.442400\pi$
$601$	8.82401			0.359939			0.179969	+	0.983672i	$0.442400\pi$
$602$		0			0
$603$	−28.3265			−1.15354
$604$		0			0
$605$	18.4694	+	13.8591i	0.750887	+	0.563453i
$606$		0			0
$607$	−25.8452			−1.04902			−0.524512	−	0.851403i	$-0.675752\pi$
$607$	−25.8452			−1.04902			−0.524512	+	0.851403i	$0.675752\pi$
$608$		0			0
$609$	−0.861205	+	1.13556i	−0.0348978	+	0.0460151i
$610$		0			0
$611$			9.40782i			0.380599i
$612$		0			0
$613$		−	17.2470i		−	0.696598i	−0.937383	−	0.348299i	$-0.886759\pi$
$613$		−	17.2470i		−	0.696598i	0.937383	−	0.348299i	$-0.113241\pi$
$614$		0			0
$615$	−34.7253			−1.40026
$616$		0			0
$617$	−43.9135			−1.76789			−0.883944	−	0.467592i	$-0.845122\pi$
$617$	−43.9135			−1.76789			−0.883944	+	0.467592i	$0.845122\pi$
$618$		0			0
$619$			6.60316i			0.265403i	0.991156	+	0.132702i	$0.0423652\pi$
$619$			6.60316i			0.265403i	−0.991156	+	0.132702i	$0.957635\pi$
$620$		0			0
$621$		−	49.0946i		−	1.97010i
$622$		0			0
$623$	−31.6877	−	24.0319i	−1.26954	−	0.962818i
$624$		0			0
$625$	−21.6806			−0.867223
$626$		0			0
$627$	32.7687	−	16.3793i	1.30866	−	0.654127i
$628$		0			0
$629$	49.1288			1.95889
$630$		0			0
$631$	−33.9819			−1.35280			−0.676399	−	0.736535i	$-0.736460\pi$
$631$	−33.9819			−1.35280			−0.676399	+	0.736535i	$0.736460\pi$
$632$		0			0
$633$	22.3893			0.889896
$634$		0			0
$635$	2.05733			0.0816425
$636$		0			0
$637$	1.88794	+	6.74060i	0.0748031	+	0.267072i
$638$		0			0
$639$	−70.7662			−2.79947
$640$		0			0
$641$	40.8625			1.61397			0.806985	−	0.590571i	$-0.201098\pi$
$641$	40.8625			1.61397			0.806985	+	0.590571i	$0.201098\pi$
$642$		0			0
$643$			7.31720i			0.288562i	0.989537	+	0.144281i	$0.0460870\pi$
$643$			7.31720i			0.288562i	−0.989537	+	0.144281i	$0.953913\pi$
$644$		0			0
$645$		−	19.3408i		−	0.761543i
$646$		0			0
$647$		−	15.9627i		−	0.627558i	−0.949496	−	0.313779i	$-0.898405\pi$
$647$		−	15.9627i		−	0.627558i	0.949496	−	0.313779i	$-0.101595\pi$
$648$		0			0
$649$	−27.6645	+	13.8280i	−1.08593	+	0.542798i
$650$		0			0
$651$	−31.3749	+	41.3700i	−1.22968	+	1.62142i
$652$		0			0
$653$	7.94892			0.311065			0.155533	−	0.987831i	$-0.450291\pi$
$653$	7.94892			0.311065			0.155533	+	0.987831i	$0.450291\pi$
$654$		0			0
$655$			0.357517i			0.0139694i
$656$		0			0
$657$	−108.898			−4.24850
$658$		0			0
$659$		−	14.0044i		−	0.545533i	−0.962080	−	0.272767i	$-0.912061\pi$
$659$		−	14.0044i		−	0.545533i	0.962080	−	0.272767i	$-0.0879387\pi$
$660$		0			0
$661$		−	22.4701i		−	0.873985i	−0.899465	−	0.436992i	$-0.856044\pi$
$661$		−	22.4701i		−	0.873985i	0.899465	−	0.436992i	$-0.143956\pi$
$662$		0			0
$663$		−	16.4233i		−	0.637829i
$664$		0			0
$665$	15.5170	+	11.7681i	0.601724	+	0.456346i
$666$		0			0
$667$		−	0.679394i		−	0.0263062i
$668$		0			0
$669$	−48.4099			−1.87163
$670$		0			0
$671$	1.20050	+	2.40173i	0.0463447	+	0.0927179i
$672$		0			0
$673$		−	10.4111i		−	0.401317i	−0.979661	−	0.200659i	$-0.935692\pi$
$673$		−	10.4111i		−	0.401317i	0.979661	−	0.200659i	$-0.0643082\pi$
$674$		0			0
$675$			7.33331i			0.282259i
$676$		0			0
$677$	−24.5256			−0.942596			−0.471298	−	0.881974i	$-0.656214\pi$
$677$	−24.5256			−0.942596			−0.471298	+	0.881974i	$0.656214\pi$
$678$		0			0
$679$	2.14836	+	1.62931i	0.0824465	+	0.0625272i
$680$		0			0
$681$			2.59117i			0.0992938i
$682$		0			0
$683$	−4.82239			−0.184524			−0.0922618	−	0.995735i	$-0.529410\pi$
$683$	−4.82239			−0.184524			−0.0922618	+	0.995735i	$0.529410\pi$
$684$		0			0
$685$			2.97770i			0.113772i
$686$		0			0
$687$	59.2431			2.26027
$688$		0			0
$689$	−5.09027			−0.193924
$690$		0			0
$691$			20.0222i			0.761681i	0.924641	+	0.380841i	$0.124365\pi$
$691$			20.0222i			0.761681i	−0.924641	+	0.380841i	$0.875635\pi$
$692$		0			0
$693$	59.7074	+	11.1940i	2.26810	+	0.425224i
$694$		0			0
$695$			31.0999i			1.17969i
$696$		0			0
$697$	−27.3793			−1.03706
$698$		0			0
$699$	26.7081			1.01019
$700$		0			0
$701$			12.3123i			0.465031i	0.972593	+	0.232515i	$0.0746956\pi$
$701$			12.3123i			0.465031i	−0.972593	+	0.232515i	$0.925304\pi$
$702$		0			0
$703$	−33.0420			−1.24620
$704$		0			0
$705$			62.2094i			2.34294i
$706$		0			0
$707$	−4.37943	+	5.77459i	−0.164706	+	0.217176i
$708$		0			0
$709$	4.34658			0.163239			0.0816196	−	0.996664i	$-0.473991\pi$
$709$	4.34658			0.163239			0.0816196	+	0.996664i	$0.473991\pi$
$710$		0			0
$711$			14.7832i			0.554415i
$712$		0			0
$713$		−	24.7513i		−	0.926944i
$714$		0			0
$715$	−6.22755	+	3.11282i	−0.232897	+	0.116413i
$716$		0			0
$717$	−89.1580			−3.32967
$718$		0			0
$719$			2.17369i			0.0810649i	0.999178	+	0.0405324i	$0.0129054\pi$
$719$			2.17369i			0.0810649i	−0.999178	+	0.0405324i	$0.987095\pi$
$720$		0			0
$721$	7.90194	+	5.99281i	0.294284	+	0.223184i
$722$		0			0
$723$			71.2535i			2.64995i
$724$		0			0
$725$			0.101482i			0.00376894i
$726$		0			0
$727$		−	16.8320i		−	0.624264i	−0.950039	−	0.312132i	$-0.898957\pi$
$727$		−	16.8320i		−	0.624264i	0.950039	−	0.312132i	$-0.101043\pi$
$728$		0			0
$729$	−8.92194			−0.330442
$730$		0			0
$731$		−	15.2493i		−	0.564015i
$732$		0			0
$733$	−20.4963			−0.757050			−0.378525	−	0.925591i	$-0.623569\pi$
$733$	−20.4963			−0.757050			−0.378525	+	0.925591i	$0.623569\pi$
$734$		0			0
$735$	12.4841	+	44.5724i	0.460482	+	1.64408i
$736$		0			0
$737$	6.06756	+	12.1388i	0.223501	+	0.447140i
$738$		0			0
$739$			38.9407i			1.43246i	0.697865	+	0.716229i	$0.254133\pi$
$739$			38.9407i			1.43246i	−0.697865	+	0.716229i	$0.745867\pi$
$740$		0			0
$741$			11.0456i			0.405772i
$742$		0			0
$743$			23.9933i			0.880229i	0.897942	+	0.440114i	$0.145062\pi$
$743$			23.9933i			0.880229i	−0.897942	+	0.440114i	$0.854938\pi$
$744$		0			0
$745$	−6.23220			−0.228330
$746$		0			0
$747$	−34.3927			−1.25836
$748$		0			0
$749$	−20.3426	−	15.4278i	−0.743302	−	0.563719i
$750$		0			0
$751$	−52.2183			−1.90547			−0.952736	−	0.303800i	$-0.901745\pi$
$751$	−52.2183			−1.90547			−0.952736	+	0.303800i	$0.901745\pi$
$752$		0			0
$753$	−44.0765			−1.60624
$754$		0			0
$755$	6.99664			0.254634
$756$		0			0
$757$	−19.0949			−0.694015			−0.347007	−	0.937862i	$-0.612802\pi$
$757$	−19.0949			−0.694015			−0.347007	+	0.937862i	$0.612802\pi$
$758$		0			0
$759$	−37.1280	+	18.5583i	−1.34766	+	0.673624i
$760$		0			0
$761$	−42.6602			−1.54643			−0.773216	−	0.634143i	$-0.781353\pi$
$761$	−42.6602			−1.54643			−0.773216	+	0.634143i	$0.781353\pi$
$762$		0			0
$763$	−39.5328	−	29.9816i	−1.43118	−	1.08541i
$764$		0			0
$765$		−	75.7663i		−	2.73934i
$766$		0			0
$767$		−	9.32514i		−	0.336711i
$768$		0			0
$769$	37.2796			1.34434			0.672168	−	0.740399i	$-0.265363\pi$
$769$	37.2796			1.34434			0.672168	+	0.740399i	$0.265363\pi$
$770$		0			0
$771$	−13.4030			−0.482698
$772$		0			0
$773$			21.9272i			0.788666i	0.918968	+	0.394333i	$0.129024\pi$
$773$			21.9272i			0.788666i	−0.918968	+	0.394333i	$0.870976\pi$
$774$		0			0
$775$			3.69713i			0.132805i
$776$		0			0
$777$	−62.5744	−	47.4563i	−2.24484	−	1.70248i
$778$		0			0
$779$	18.4142			0.659757
$780$		0			0
$781$	15.1582	+	30.3256i	0.542402	+	1.08514i
$782$		0			0
$783$	−2.11312			−0.0755167
$784$		0			0
$785$	2.03248			0.0725423
$786$		0			0
$787$	−53.4270			−1.90447			−0.952234	−	0.305370i	$-0.901220\pi$
$787$	−53.4270			−1.90447			−0.952234	+	0.305370i	$0.901220\pi$
$788$		0			0
$789$	42.0043			1.49539
$790$		0			0
$791$	11.7073	−	15.4369i	0.416263	−	0.548872i
$792$		0			0
$793$	−0.809574			−0.0287488
$794$		0			0
$795$	−33.6595			−1.19378
$796$		0			0
$797$		−	39.6786i		−	1.40549i	−0.711443	−	0.702744i	$-0.751958\pi$
$797$		−	39.6786i		−	1.40549i	0.711443	−	0.702744i	$-0.248042\pi$
$798$		0			0
$799$			49.0492i			1.73523i
$800$		0			0
$801$		−	104.061i		−	3.67683i
$802$		0			0
$803$	23.3260	+	46.6662i	0.823155	+	1.64682i
$804$		0			0
$805$	−17.5813	−	13.3336i	−0.619660	−	0.469948i
$806$		0			0
$807$	−43.4074			−1.52801
$808$		0			0
$809$			15.9594i			0.561102i	0.959839	+	0.280551i	$0.0905172\pi$
$809$			15.9594i			0.561102i	−0.959839	+	0.280551i	$0.909483\pi$
$810$		0			0
$811$	−9.05004			−0.317790			−0.158895	−	0.987295i	$-0.550793\pi$
$811$	−9.05004			−0.317790			−0.158895	+	0.987295i	$0.550793\pi$
$812$		0			0
$813$		−	65.2746i		−	2.28928i
$814$		0			0
$815$		−	46.3495i		−	1.62355i
$816$		0			0
$817$			10.2560i			0.358814i
$818$		0			0
$819$	−11.0679	+	14.5938i	−0.386745	+	0.509950i
$820$		0			0
$821$		−	12.1865i		−	0.425312i	−0.977127	−	0.212656i	$-0.931789\pi$
$821$		−	12.1865i		−	0.425312i	0.977127	−	0.212656i	$-0.0682114\pi$
$822$		0			0
$823$	10.8973			0.379856			0.189928	−	0.981798i	$-0.439174\pi$
$823$	10.8973			0.379856			0.189928	+	0.981798i	$0.439174\pi$
$824$		0			0
$825$	5.54585	−	2.77207i	0.193082	−	0.0965112i
$826$		0			0
$827$		−	12.1714i		−	0.423240i	−0.977352	−	0.211620i	$-0.932126\pi$
$827$		−	12.1714i		−	0.423240i	0.977352	−	0.211620i	$-0.0678740\pi$
$828$		0			0
$829$		−	14.7778i		−	0.513254i	−0.966510	−	0.256627i	$-0.917389\pi$
$829$		−	14.7778i		−	0.513254i	0.966510	−	0.256627i	$-0.0826113\pi$
$830$		0			0
$831$	15.9661			0.553859
$832$		0			0
$833$	9.84311	+	35.1432i	0.341043	+	1.21764i
$834$		0			0
$835$			42.8944i			1.48442i
$836$		0			0
$837$	−76.9840			−2.66095
$838$		0			0
$839$			23.4709i			0.810305i	0.914249	+	0.405152i	$0.132782\pi$
$839$			23.4709i			0.810305i	−0.914249	+	0.405152i	$0.867218\pi$
$840$		0			0
$841$	28.9708			0.998992
$842$		0			0
$843$	56.4390			1.94386
$844$		0			0
$845$		−	2.09918i		−	0.0722140i
$846$		0			0
$847$	−7.99237	−	27.9843i	−0.274621	−	0.961552i
$848$		0			0
$849$			101.220i			3.47385i
$850$		0			0
$851$	37.4377			1.28335
$852$		0			0
$853$	−21.2140			−0.726353			−0.363176	−	0.931720i	$-0.618308\pi$
$853$	−21.2140			−0.726353			−0.363176	+	0.931720i	$0.618308\pi$
$854$		0			0
$855$			50.9573i			1.74270i
$856$		0			0
$857$	10.7861			0.368448			0.184224	−	0.982884i	$-0.441023\pi$
$857$	10.7861			0.368448			0.184224	+	0.982884i	$0.441023\pi$
$858$		0			0
$859$			21.2000i			0.723335i	0.932307	+	0.361668i	$0.117792\pi$
$859$			21.2000i			0.723335i	−0.932307	+	0.361668i	$0.882208\pi$
$860$		0			0
$861$	34.8725	+	26.4472i	1.18845	+	0.901319i
$862$		0			0
$863$	−5.88285			−0.200255			−0.100127	−	0.994975i	$-0.531925\pi$
$863$	−5.88285			−0.200255			−0.100127	+	0.994975i	$0.531925\pi$
$864$		0			0
$865$			2.72328i			0.0925944i
$866$		0			0
$867$		−	32.0747i		−	1.08931i
$868$		0			0
$869$	6.33510	−	3.16658i	0.214904	−	0.107419i
$870$		0			0
$871$	−4.09175			−0.138644
$872$		0			0
$873$			7.05513i			0.238780i
$874$		0			0
$875$	24.7523	+	18.7721i	0.836779	+	0.634611i
$876$		0			0
$877$		−	6.29664i		−	0.212622i	−0.994333	−	0.106311i	$-0.966096\pi$
$877$		−	6.29664i		−	0.212622i	0.994333	−	0.106311i	$-0.0339040\pi$
$878$		0			0
$879$			9.27087i			0.312699i
$880$		0			0
$881$		−	0.791613i		−	0.0266701i	−0.999911	−	0.0133351i	$-0.995755\pi$
$881$		−	0.791613i		−	0.0266701i	0.999911	−	0.0133351i	$-0.00424481\pi$
$882$		0			0
$883$	−41.3874			−1.39280			−0.696399	−	0.717655i	$-0.745216\pi$
$883$	−41.3874			−1.39280			−0.696399	+	0.717655i	$0.745216\pi$
$884$		0			0
$885$		−	61.6628i		−	2.07277i
$886$		0			0
$887$	21.6428			0.726693			0.363346	−	0.931654i	$-0.381634\pi$
$887$	21.6428			0.726693			0.363346	+	0.931654i	$0.381634\pi$
$888$		0			0
$889$	−2.06604	−	1.56688i	−0.0692929	−	0.0525516i
$890$		0			0
$891$	26.9248	+	53.8661i	0.902014	+	1.80458i
$892$		0			0
$893$		−	32.9885i		−	1.10392i
$894$		0			0
$895$		−	5.18517i		−	0.173321i
$896$		0			0
$897$		−	12.5151i		−	0.417867i
$898$		0			0
$899$	−1.06534			−0.0355311
$900$		0			0
$901$	−26.5389			−0.884140
$902$		0			0
$903$	−14.7302	+	19.4227i	−0.490189	+	0.646348i
$904$		0			0
$905$	−13.0358			−0.433324
$906$		0			0
$907$	−35.5319			−1.17982			−0.589908	−	0.807470i	$-0.700836\pi$
$907$	−35.5319			−1.17982			−0.589908	+	0.807470i	$0.700836\pi$
$908$		0			0
$909$	−18.9635			−0.628981
$910$		0			0
$911$	55.2433			1.83029			0.915145	−	0.403124i	$-0.132076\pi$
$911$	55.2433			1.83029			0.915145	+	0.403124i	$0.132076\pi$
$912$		0			0
$913$	7.36694	+	14.7384i	0.243810	+	0.487770i
$914$		0			0
$915$	−5.35333			−0.176976
$916$		0			0
$917$	0.272289	−	0.359032i	0.00899178	−	0.0118563i
$918$		0			0
$919$		−	30.7368i		−	1.01391i	−0.861972	−	0.506956i	$-0.830771\pi$
$919$		−	30.7368i		−	1.01391i	0.861972	−	0.506956i	$-0.169229\pi$
$920$		0			0
$921$			8.96159i			0.295295i
$922$		0			0
$923$	−10.2221			−0.336466
$924$		0			0
$925$	−5.59210			−0.183867
$926$		0			0
$927$			25.9497i			0.852299i
$928$		0			0
$929$			3.25320i			0.106734i	0.998575	+	0.0533670i	$0.0169953\pi$
$929$			3.25320i			0.106734i	−0.998575	+	0.0533670i	$0.983005\pi$
$930$		0			0
$931$	−6.62007	−	23.6359i	−0.216964	−	0.774635i
$932$		0			0
$933$	46.5589			1.52427
$934$		0			0
$935$	−32.4684	+	16.2292i	−1.06183	+	0.530752i
$936$		0			0
$937$	38.2251			1.24876			0.624380	−	0.781121i	$-0.285352\pi$
$937$	38.2251			1.24876			0.624380	+	0.781121i	$0.285352\pi$
$938$		0			0
$939$	85.0533			2.77561
$940$		0			0
$941$	48.4166			1.57833			0.789167	−	0.614178i	$-0.210512\pi$
$941$	48.4166			1.57833			0.789167	+	0.614178i	$0.210512\pi$
$942$		0			0
$943$	−20.8639			−0.679422
$944$		0			0
$945$	−41.4715	+	54.6831i	−1.34907	+	1.77884i
$946$		0			0
$947$	37.1470			1.20711			0.603557	−	0.797320i	$-0.293750\pi$
$947$	37.1470			1.20711			0.603557	+	0.797320i	$0.293750\pi$
$948$		0			0
$949$	−15.7302			−0.510624
$950$		0			0
$951$			36.1331i			1.17170i
$952$		0			0
$953$			29.0590i			0.941312i	0.882317	+	0.470656i	$0.155983\pi$
$953$			29.0590i			0.941312i	−0.882317	+	0.470656i	$0.844017\pi$
$954$		0			0
$955$		−	7.42773i		−	0.240356i
$956$		0			0
$957$	0.798782	+	1.59806i	0.0258210	+	0.0516578i
$958$		0			0
$959$	2.26785	−	2.99032i	0.0732327	−	0.0965624i
$960$		0			0
$961$	−7.81187			−0.251996
$962$		0			0
$963$		−	66.8043i		−	2.15274i
$964$		0			0
$965$	32.6704			1.05170
$966$		0			0
$967$			19.5929i			0.630064i	0.949081	+	0.315032i	$0.102015\pi$
$967$			19.5929i			0.630064i	−0.949081	+	0.315032i	$0.897985\pi$
$968$		0			0
$969$			57.5883i			1.85000i
$970$		0			0
$971$		−	36.0517i		−	1.15696i	−0.815698	−	0.578478i	$-0.803647\pi$
$971$		−	36.0517i		−	1.15696i	0.815698	−	0.578478i	$-0.196353\pi$
$972$		0			0
$973$	23.6860	−	31.2317i	0.759340	−	1.00124i
$974$		0			0
$975$			1.86939i			0.0598684i
$976$		0			0
$977$	−15.7501			−0.503889			−0.251945	−	0.967742i	$-0.581070\pi$
$977$	−15.7501			−0.503889			−0.251945	+	0.967742i	$0.581070\pi$
$978$		0			0
$979$	−44.5937	+	22.2900i	−1.42522	+	0.712392i
$980$		0			0
$981$		−	129.824i		−	4.14497i
$982$		0			0
$983$			15.6199i			0.498196i	0.968478	+	0.249098i	$0.0801342\pi$
$983$			15.6199i			0.498196i	−0.968478	+	0.249098i	$0.919866\pi$
$984$		0			0
$985$	−13.4513			−0.428593
$986$		0			0
$987$	47.3794	−	62.4730i	1.50810	−	1.98854i
$988$		0			0
$989$		−	11.6204i		−	0.369508i
$990$		0			0
$991$	−4.64602			−0.147586			−0.0737928	−	0.997274i	$-0.523510\pi$
$991$	−4.64602			−0.147586			−0.0737928	+	0.997274i	$0.523510\pi$
$992$		0			0
$993$		−	16.8080i		−	0.533386i
$994$		0			0
$995$	52.6855			1.67024
$996$		0			0
$997$	21.8642			0.692446			0.346223	−	0.938152i	$-0.387464\pi$
$997$	21.8642			0.692446			0.346223	+	0.938152i	$0.387464\pi$
$998$		0			0
$999$		−	116.442i		−	3.68407i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.b.3849.2	yes	48
7.6	odd	2		4004.2.e.a.3849.47	yes	48
11.10	odd	2		4004.2.e.a.3849.2	✓	48
77.76	even	2	inner	4004.2.e.b.3849.47	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.2	✓	48	11.10	odd	2
4004.2.e.a.3849.47	yes	48	7.6	odd	2
4004.2.e.b.3849.2	yes	48	1.1	even	1	trivial
4004.2.e.b.3849.47	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)

\(f(q)\)	\(=\)	\(q-3.15005i q^{3} -2.09918i q^{5} +(-1.59876 + 2.10807i) q^{7} -6.92283 q^{9} +O(q^{10})\)	\(q-3.15005i q^{3} -2.09918i q^{5} +(-1.59876 + 2.10807i) q^{7} -6.92283 q^{9} +(1.48288 + 2.96666i) q^{11} -1.00000 q^{13} -6.61253 q^{15} -5.21366 q^{17} +3.50650 q^{19} +(6.64054 + 5.03617i) q^{21} -3.97298 q^{23} +0.593447 q^{25} +12.3571i q^{27} +0.171004i q^{29} +6.22992i q^{31} +(9.34514 - 4.67114i) q^{33} +(4.42522 + 3.35608i) q^{35} -9.42308 q^{37} +3.15005i q^{39} +5.25145 q^{41} +2.92487i q^{43} +14.5323i q^{45} -9.40782i q^{47} +(-1.88794 - 6.74060i) q^{49} +16.4233i q^{51} +5.09027 q^{53} +(6.22755 - 3.11282i) q^{55} -11.0456i q^{57} +9.32514i q^{59} +0.809574 q^{61} +(11.0679 - 14.5938i) q^{63} +2.09918i q^{65} +4.09175 q^{67} +12.5151i q^{69} +10.2221 q^{71} +15.7302 q^{73} -1.86939i q^{75} +(-8.62470 - 1.61696i) q^{77} -2.13543i q^{79} +18.1571 q^{81} +4.96801 q^{83} +10.9444i q^{85} +0.538671 q^{87} +15.0316i q^{89} +(1.59876 - 2.10807i) q^{91} +19.6246 q^{93} -7.36076i q^{95} -1.01911i q^{97} +(-10.2657 - 20.5377i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{7} - 48 q^{9}+O(q^{10}) \) 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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