LMFDB - Embedded newform 1008.4.b.k.559.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(559,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1008.559");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.b (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:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 158x^{6} + 8461x^{4} + 180672x^{2} + 1232100 $ x^8 + 158x^6 + 8461x^4 + 180672*x^2 + 1232100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.7
Root			$5.92762i$ of defining polynomial
Character	$\chi$	$=$	1008.559
Dual form			1008.4.b.k.559.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+17.7376i q^{5} +(-2.09706 - 18.4011i) q^{7} +O(q^{10})$	$q+17.7376i q^{5} +(-2.09706 - 18.4011i) q^{7} +32.9212i q^{11} +53.7866i q^{13} +48.1047i q^{17} -110.800 q^{19} -157.820i q^{23} -189.621 q^{25} +72.0082 q^{29} -184.189 q^{31} +(326.391 - 37.1968i) q^{35} +422.967 q^{37} +346.448i q^{41} -198.168i q^{43} -29.0729 q^{47} +(-334.205 + 77.1768i) q^{49} -682.553 q^{53} -583.941 q^{55} +305.320 q^{59} -172.135i q^{61} -954.042 q^{65} -109.033i q^{67} +741.453i q^{71} -752.602i q^{73} +(605.787 - 69.0378i) q^{77} +449.132i q^{79} +262.988 q^{83} -853.261 q^{85} -674.294i q^{89} +(989.735 - 112.794i) q^{91} -1965.32i q^{95} -1459.49i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7}+O(q^{10}) $ 8 * q + 4 * q^7	$ 8 q + 4 q^{7} + 56 q^{19} - 656 q^{25} - 240 q^{29} - 320 q^{31} + 600 q^{35} + 392 q^{37} - 816 q^{47} - 16 q^{49} - 288 q^{53} + 456 q^{55} - 1824 q^{59} + 816 q^{65} + 2064 q^{77} + 1680 q^{83} + 2568 q^{85} + 864 q^{91}+O(q^{100}) $ 8 * q + 4 * q^7 + 56 * q^19 - 656 * q^25 - 240 * q^29 - 320 * q^31 + 600 * q^35 + 392 * q^37 - 816 * q^47 - 16 * q^49 - 288 * q^53 + 456 * q^55 - 1824 * q^59 + 816 * q^65 + 2064 * q^77 + 1680 * q^83 + 2568 * q^85 + 864 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$			17.7376i			1.58650i	0.608899	+	0.793248i	$0.291611\pi$
$5$			17.7376i			1.58650i	−0.608899	+	0.793248i	$0.708389\pi$
$6$		0			0
$7$	−2.09706	−	18.4011i	−0.113231	−	0.993569i
$7$	−2.09706	−	18.4011i	−0.113231	−	0.993569i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			32.9212i			0.902373i	0.892430	+	0.451186i	$0.148999\pi$
$11$			32.9212i			0.902373i	−0.892430	+	0.451186i	$0.851001\pi$
$12$		0			0
$13$			53.7866i			1.14752i	0.819025	+	0.573758i	$0.194515\pi$
$13$			53.7866i			1.14752i	−0.819025	+	0.573758i	$0.805485\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			48.1047i			0.686301i	0.939280	+	0.343150i	$0.111494\pi$
$17$			48.1047i			0.686301i	−0.939280	+	0.343150i	$0.888506\pi$
$18$		0			0
$19$	−110.800			−1.33785			−0.668927	−	0.743328i	$-0.733246\pi$
$19$	−110.800			−1.33785			−0.668927	+	0.743328i	$0.733246\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	157.820i		−	1.43077i	−0.698732	−	0.715384i	$-0.746252\pi$
$23$		−	157.820i		−	1.43077i	0.698732	−	0.715384i	$-0.253748\pi$
$24$		0			0
$25$	−189.621			−1.51697
$26$		0			0
$27$		0			0
$28$		0			0
$29$	72.0082			0.461089			0.230545	−	0.973062i	$-0.425949\pi$
$29$	72.0082			0.461089			0.230545	+	0.973062i	$0.425949\pi$
$30$		0			0
$31$	−184.189			−1.06714			−0.533570	−	0.845756i	$-0.679150\pi$
$31$	−184.189			−1.06714			−0.533570	+	0.845756i	$0.679150\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	326.391	−	37.1968i	1.57629	−	0.179640i
$36$		0			0
$37$	422.967			1.87933			0.939667	−	0.342091i	$-0.111135\pi$
$37$	422.967			1.87933			0.939667	+	0.342091i	$0.111135\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			346.448i			1.31966i	0.751415	+	0.659830i	$0.229372\pi$
$41$			346.448i			1.31966i	−0.751415	+	0.659830i	$0.770628\pi$
$42$		0			0
$43$		−	198.168i		−	0.702800i	−0.936225	−	0.351400i	$-0.885706\pi$
$43$		−	198.168i		−	0.702800i	0.936225	−	0.351400i	$-0.114294\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−29.0729			−0.0902281			−0.0451141	−	0.998982i	$-0.514365\pi$
$47$	−29.0729			−0.0902281			−0.0451141	+	0.998982i	$0.514365\pi$
$48$		0			0
$49$	−334.205	+	77.1768i	−0.974358	+	0.225005i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−682.553			−1.76898			−0.884490	−	0.466560i	$-0.845493\pi$
$53$	−682.553			−1.76898			−0.884490	+	0.466560i	$0.845493\pi$
$54$		0			0
$55$	−583.941			−1.43161
$56$		0			0
$57$		0			0
$58$		0			0
$59$	305.320			0.673716			0.336858	−	0.941555i	$-0.390636\pi$
$59$	305.320			0.673716			0.336858	+	0.941555i	$0.390636\pi$
$60$		0			0
$61$		−	172.135i		−	0.361306i	−0.983547	−	0.180653i	$-0.942179\pi$
$61$		−	172.135i		−	0.361306i	0.983547	−	0.180653i	$-0.0578211\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−954.042			−1.82053
$66$		0			0
$67$		−	109.033i		−	0.198814i	−0.995047	−	0.0994071i	$-0.968305\pi$
$67$		−	109.033i		−	0.198814i	0.995047	−	0.0994071i	$-0.0316946\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			741.453i			1.23936i	0.784856	+	0.619678i	$0.212737\pi$
$71$			741.453i			1.23936i	−0.784856	+	0.619678i	$0.787263\pi$
$72$		0			0
$73$		−	752.602i		−	1.20665i	−0.797496	−	0.603325i	$-0.793842\pi$
$73$		−	752.602i		−	1.20665i	0.797496	−	0.603325i	$-0.206158\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	605.787	−	69.0378i	0.896569	−	0.102176i
$78$		0			0
$79$			449.132i			0.639637i	0.947479	+	0.319818i	$0.103622\pi$
$79$			449.132i			0.639637i	−0.947479	+	0.319818i	$0.896378\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	262.988			0.347791			0.173895	−	0.984764i	$-0.444365\pi$
$83$	262.988			0.347791			0.173895	+	0.984764i	$0.444365\pi$
$84$		0			0
$85$	−853.261			−1.08881
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	674.294i		−	0.803090i	−0.915839	−	0.401545i	$-0.868473\pi$
$89$		−	674.294i		−	0.803090i	0.915839	−	0.401545i	$-0.131527\pi$
$90$		0			0
$91$	989.735	−	112.794i	1.14014	−	0.129934i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	1965.32i		−	2.12250i
$96$		0			0
$97$		−	1459.49i		−	1.52771i	−0.645385	−	0.763857i	$-0.723303\pi$
$97$		−	1459.49i		−	1.52771i	0.645385	−	0.763857i	$-0.276697\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	226.672i		−	0.223314i	−0.993747	−	0.111657i	$-0.964384\pi$
$101$		−	226.672i		−	0.223314i	0.993747	−	0.111657i	$-0.0356158\pi$
$102$		0			0
$103$	−712.681			−0.681772			−0.340886	−	0.940105i	$-0.610727\pi$
$103$	−712.681			−0.681772			−0.340886	+	0.940105i	$0.610727\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	759.426i		−	0.686135i	−0.939311	−	0.343068i	$-0.888534\pi$
$107$		−	759.426i		−	0.686135i	0.939311	−	0.343068i	$-0.111466\pi$
$108$		0			0
$109$	−1081.36			−0.950231			−0.475115	−	0.879923i	$-0.657594\pi$
$109$	−1081.36			−0.950231			−0.475115	+	0.879923i	$0.657594\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1934.09			−1.61013			−0.805063	−	0.593189i	$-0.797869\pi$
$113$	−1934.09			−1.61013			−0.805063	+	0.593189i	$0.797869\pi$
$114$		0			0
$115$	2799.33			2.26991
$116$		0			0
$117$		0			0
$118$		0			0
$119$	885.183	−	100.879i	0.681887	−	0.0777104i
$120$		0			0
$121$	247.198			0.185723
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1146.22i		−	0.820166i
$126$		0			0
$127$		−	2518.01i		−	1.75935i	−0.475574	−	0.879676i	$-0.657760\pi$
$127$		−	2518.01i		−	1.75935i	0.475574	−	0.879676i	$-0.342240\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1114.33			−0.743204			−0.371602	−	0.928392i	$-0.621191\pi$
$131$	−1114.33			−0.743204			−0.371602	+	0.928392i	$0.621191\pi$
$132$		0			0
$133$	232.355	+	2038.85i	0.151486	+	1.32925i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−260.914			−0.162711			−0.0813555	−	0.996685i	$-0.525925\pi$
$137$	−260.914			−0.162711			−0.0813555	+	0.996685i	$0.525925\pi$
$138$		0			0
$139$	−505.973			−0.308749			−0.154374	−	0.988012i	$-0.549336\pi$
$139$	−505.973			−0.308749			−0.154374	+	0.988012i	$0.549336\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1770.72			−1.03549
$144$		0			0
$145$			1277.25i			0.731516i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1397.28			−0.768254			−0.384127	−	0.923280i	$-0.625497\pi$
$149$	−1397.28			−0.768254			−0.384127	+	0.923280i	$0.625497\pi$
$150$		0			0
$151$		−	344.187i		−	0.185493i	−0.995690	−	0.0927467i	$-0.970435\pi$
$151$		−	344.187i		−	0.185493i	0.995690	−	0.0927467i	$-0.0295647\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	3267.06i		−	1.69301i
$156$		0			0
$157$			2450.31i			1.24558i	0.782390	+	0.622789i	$0.214000\pi$
$157$			2450.31i			1.24558i	−0.782390	+	0.622789i	$0.786000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2904.06	+	330.958i	−1.42157	+	0.162007i
$162$		0			0
$163$		−	1749.52i		−	0.840693i	−0.907364	−	0.420346i	$-0.861909\pi$
$163$		−	1749.52i		−	0.840693i	0.907364	−	0.420346i	$-0.138091\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4072.62			−1.88712			−0.943560	−	0.331202i	$-0.892546\pi$
$167$	−4072.62			−1.88712			−0.943560	+	0.331202i	$0.892546\pi$
$168$		0			0
$169$	−695.995			−0.316793
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	3988.51i		−	1.75284i	−0.481550	−	0.876419i	$-0.659926\pi$
$173$		−	3988.51i		−	1.75284i	0.481550	−	0.876419i	$-0.340074\pi$
$174$		0			0
$175$	397.647	+	3489.24i	0.171767	+	1.50721i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			32.2713i			0.0134753i	0.999977	+	0.00673763i	$0.00214467\pi$
$179$			32.2713i			0.0134753i	−0.999977	+	0.00673763i	$0.997855\pi$
$180$		0			0
$181$		−	233.112i		−	0.0957299i	−0.998854	−	0.0478650i	$-0.984758\pi$
$181$		−	233.112i		−	0.0957299i	0.998854	−	0.0478650i	$-0.0152417\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			7502.40i			2.98155i
$186$		0			0
$187$	−1583.66			−0.619299
$188$		0			0
$189$		0			0
$190$		0			0
$191$			111.172i			0.0421158i	0.999778	+	0.0210579i	$0.00670344\pi$
$191$			111.172i			0.0421158i	−0.999778	+	0.0210579i	$0.993297\pi$
$192$		0			0
$193$	−1713.45			−0.639052			−0.319526	−	0.947577i	$-0.603524\pi$
$193$	−1713.45			−0.639052			−0.319526	+	0.947577i	$0.603524\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4045.10			−1.46295			−0.731475	−	0.681868i	$-0.761168\pi$
$197$	−4045.10			−1.46295			−0.731475	+	0.681868i	$0.761168\pi$
$198$		0			0
$199$	4480.63			1.59610			0.798048	−	0.602594i	$-0.205866\pi$
$199$	4480.63			1.59610			0.798048	+	0.602594i	$0.205866\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−151.006	−	1325.03i	−0.0522096	−	0.458124i
$204$		0			0
$205$	−6145.14			−2.09363
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3647.66i		−	1.20724i
$210$		0			0
$211$			3937.96i			1.28483i	0.766355	+	0.642417i	$0.222068\pi$
$211$			3937.96i			1.28483i	−0.766355	+	0.642417i	$0.777932\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3515.02			1.11499
$216$		0			0
$217$	386.256	+	3389.29i	0.120833	+	1.06028i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2587.39			−0.787541
$222$		0			0
$223$	−6231.07			−1.87114			−0.935568	−	0.353147i	$-0.885112\pi$
$223$	−6231.07			−1.87114			−0.935568	+	0.353147i	$0.885112\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2627.92			0.768377			0.384188	−	0.923255i	$-0.374481\pi$
$227$	2627.92			0.768377			0.384188	+	0.923255i	$0.374481\pi$
$228$		0			0
$229$			1930.43i			0.557059i	0.960428	+	0.278530i	$0.0898471\pi$
$229$			1930.43i			0.557059i	−0.960428	+	0.278530i	$0.910153\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5138.78			1.44486			0.722431	−	0.691443i	$-0.243025\pi$
$233$	5138.78			1.44486			0.722431	+	0.691443i	$0.243025\pi$
$234$		0			0
$235$		−	515.683i		−	0.143147i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			5724.04i			1.54919i	0.632455	+	0.774597i	$0.282047\pi$
$239$			5724.04i			1.54919i	−0.632455	+	0.774597i	$0.717953\pi$
$240$		0			0
$241$		−	5731.84i		−	1.53204i	−0.642820	−	0.766018i	$-0.722236\pi$
$241$		−	5731.84i		−	1.53204i	0.642820	−	0.766018i	$-0.277764\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1368.93	−	5927.97i	−0.356970	−	1.54581i
$246$		0			0
$247$		−	5959.55i		−	1.53521i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2433.43			−0.611940			−0.305970	−	0.952041i	$-0.598981\pi$
$251$	−2433.43			−0.611940			−0.305970	+	0.952041i	$0.598981\pi$
$252$		0			0
$253$	5195.60			1.29109
$254$		0			0
$255$		0			0
$256$		0			0
$257$			2495.76i			0.605762i	0.953028	+	0.302881i	$0.0979485\pi$
$257$			2495.76i			0.605762i	−0.953028	+	0.302881i	$0.902051\pi$
$258$		0			0
$259$	−886.989	−	7783.08i	−0.212799	−	1.86725i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			1255.35i			0.294328i	0.989112	+	0.147164i	$0.0470146\pi$
$263$			1255.35i			0.294328i	−0.989112	+	0.147164i	$0.952985\pi$
$264$		0			0
$265$		−	12106.8i		−	2.80648i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			2843.99i			0.644614i	0.946635	+	0.322307i	$0.104458\pi$
$269$			2843.99i			0.644614i	−0.946635	+	0.322307i	$0.895542\pi$
$270$		0			0
$271$	−688.100			−0.154240			−0.0771201	−	0.997022i	$-0.524572\pi$
$271$	−688.100			−0.154240			−0.0771201	+	0.997022i	$0.524572\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	6242.54i		−	1.36887i
$276$		0			0
$277$	3641.63			0.789906			0.394953	−	0.918701i	$-0.370761\pi$
$277$	3641.63			0.789906			0.394953	+	0.918701i	$0.370761\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−471.011			−0.0999935			−0.0499967	−	0.998749i	$-0.515921\pi$
$281$	−471.011			−0.0999935			−0.0499967	+	0.998749i	$0.515921\pi$
$282$		0			0
$283$	6928.51			1.45533			0.727663	−	0.685935i	$-0.240607\pi$
$283$	6928.51			1.45533			0.727663	+	0.685935i	$0.240607\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6375.04	−	726.523i	1.31117	−	0.149426i
$288$		0			0
$289$	2598.93			0.528991
$290$		0			0
$291$		0			0
$292$		0			0
$293$			6739.54i			1.34378i	0.740650	+	0.671891i	$0.234518\pi$
$293$			6739.54i			1.34378i	−0.740650	+	0.671891i	$0.765482\pi$
$294$		0			0
$295$			5415.63i			1.06885i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8488.57			1.64183
$300$		0			0
$301$	−3646.53	+	415.572i	−0.698280	+	0.0795787i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3053.26			0.573210
$306$		0			0
$307$	−7974.24			−1.48246			−0.741228	−	0.671253i	$-0.765756\pi$
$307$	−7974.24			−1.48246			−0.741228	+	0.671253i	$0.765756\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−461.001			−0.0840546			−0.0420273	−	0.999116i	$-0.513382\pi$
$311$	−461.001			−0.0840546			−0.0420273	+	0.999116i	$0.513382\pi$
$312$		0			0
$313$			1977.23i			0.357059i	0.983935	+	0.178530i	$0.0571340\pi$
$313$			1977.23i			0.357059i	−0.983935	+	0.178530i	$0.942866\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1570.62			0.278280			0.139140	−	0.990273i	$-0.455566\pi$
$317$	1570.62			0.278280			0.139140	+	0.990273i	$0.455566\pi$
$318$		0			0
$319$			2370.59i			0.416075i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5330.00i		−	0.918171i
$324$		0			0
$325$		−	10199.1i		−	1.74074i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	60.9678	+	534.975i	0.0102166	+	0.0896479i
$330$		0			0
$331$			5222.32i			0.867204i	0.901104	+	0.433602i	$0.142758\pi$
$331$			5222.32i			0.867204i	−0.901104	+	0.433602i	$0.857242\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1933.99			0.315418
$336$		0			0
$337$	−3136.18			−0.506940			−0.253470	−	0.967343i	$-0.581572\pi$
$337$	−3136.18			−0.506940			−0.253470	+	0.967343i	$0.581572\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	6063.72i		−	0.962958i
$342$		0			0
$343$	2120.99	+	5987.90i	0.333886	+	0.942614i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			2160.06i			0.334173i	0.985942	+	0.167087i	$0.0534359\pi$
$347$			2160.06i			0.334173i	−0.985942	+	0.167087i	$0.946564\pi$
$348$		0			0
$349$			4717.67i			0.723584i	0.932259	+	0.361792i	$0.117835\pi$
$349$			4717.67i			0.723584i	−0.932259	+	0.361792i	$0.882165\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			2083.75i			0.314184i	0.987584	+	0.157092i	$0.0502119\pi$
$353$			2083.75i			0.314184i	−0.987584	+	0.157092i	$0.949788\pi$
$354$		0			0
$355$	−13151.6			−1.96623
$356$		0			0
$357$		0			0
$358$		0			0
$359$			2144.44i			0.315263i	0.987498	+	0.157631i	$0.0503858\pi$
$359$			2144.44i			0.315263i	−0.987498	+	0.157631i	$0.949614\pi$
$360$		0			0
$361$	5417.62			0.789855
$362$		0			0
$363$		0			0
$364$		0			0
$365$	13349.3			1.91434
$366$		0			0
$367$	2435.16			0.346361			0.173181	−	0.984890i	$-0.444596\pi$
$367$	2435.16			0.346361			0.173181	+	0.984890i	$0.444596\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1431.36	+	12559.8i	0.200303	+	1.75760i
$372$		0			0
$373$	9190.90			1.27584			0.637918	−	0.770105i	$-0.279796\pi$
$373$	9190.90			1.27584			0.637918	+	0.770105i	$0.279796\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3873.08i			0.529108i
$378$		0			0
$379$		−	3771.98i		−	0.511223i	−0.966780	−	0.255612i	$-0.917723\pi$
$379$		−	3771.98i		−	0.511223i	0.966780	−	0.255612i	$-0.0822769\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3974.64			−0.530273			−0.265136	−	0.964211i	$-0.585417\pi$
$383$	−3974.64			−0.530273			−0.265136	+	0.964211i	$0.585417\pi$
$384$		0			0
$385$	1224.56	+	10745.2i	0.162102	+	1.42240i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8337.10			1.08665			0.543326	−	0.839522i	$-0.317165\pi$
$389$	8337.10			1.08665			0.543326	+	0.839522i	$0.317165\pi$
$390$		0			0
$391$	7591.87			0.981937
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7966.51			−1.01478
$396$		0			0
$397$			8116.13i			1.02604i	0.858378	+	0.513019i	$0.171473\pi$
$397$			8116.13i			1.02604i	−0.858378	+	0.513019i	$0.828527\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8543.46			−1.06394			−0.531970	−	0.846763i	$-0.678548\pi$
$401$	−8543.46			−1.06394			−0.531970	+	0.846763i	$0.678548\pi$
$402$		0			0
$403$		−	9906.90i		−	1.22456i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			13924.6i			1.69586i
$408$		0			0
$409$		−	4970.55i		−	0.600923i	−0.953794	−	0.300462i	$-0.902859\pi$
$409$		−	4970.55i		−	0.600923i	0.953794	−	0.300462i	$-0.0971408\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−640.276	−	5618.24i	−0.0762855	−	0.669383i
$414$		0			0
$415$			4664.76i			0.551768i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11967.9			−1.39540			−0.697699	−	0.716391i	$-0.745793\pi$
$419$	−11967.9			−1.39540			−0.697699	+	0.716391i	$0.745793\pi$
$420$		0			0
$421$	−15762.7			−1.82476			−0.912381	−	0.409341i	$-0.865759\pi$
$421$	−15762.7			−1.82476			−0.912381	+	0.409341i	$0.865759\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	9121.66i		−	1.04110i
$426$		0			0
$427$	−3167.48	+	360.979i	−0.358982	+	0.0409110i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			3980.94i			0.444908i	0.974943	+	0.222454i	$0.0714067\pi$
$431$			3980.94i			0.444908i	−0.974943	+	0.222454i	$0.928593\pi$
$432$		0			0
$433$			7828.52i			0.868856i	0.900707	+	0.434428i	$0.143049\pi$
$433$			7828.52i			0.868856i	−0.900707	+	0.434428i	$0.856951\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			17486.4i			1.91416i
$438$		0			0
$439$	8174.69			0.888739			0.444370	−	0.895844i	$-0.353428\pi$
$439$	8174.69			0.888739			0.444370	+	0.895844i	$0.353428\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	13439.8i		−	1.44141i	−0.693243	−	0.720704i	$-0.743819\pi$
$443$		−	13439.8i		−	1.44141i	0.693243	−	0.720704i	$-0.256181\pi$
$444$		0			0
$445$	11960.3			1.27410
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8966.31			0.942419			0.471210	−	0.882021i	$-0.343818\pi$
$449$	8966.31			0.942419			0.471210	+	0.882021i	$0.343818\pi$
$450$		0			0
$451$	−11405.5			−1.19082
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2000.69	+	17555.5i	0.206140	+	1.80882i
$456$		0			0
$457$	7444.29			0.761990			0.380995	−	0.924577i	$-0.375582\pi$
$457$	7444.29			0.761990			0.380995	+	0.924577i	$0.375582\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			5552.31i			0.560947i	0.959862	+	0.280474i	$0.0904915\pi$
$461$			5552.31i			0.560947i	−0.959862	+	0.280474i	$0.909508\pi$
$462$		0			0
$463$			3923.67i			0.393841i	0.980419	+	0.196921i	$0.0630941\pi$
$463$			3923.67i			0.393841i	−0.980419	+	0.196921i	$0.936906\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8843.37			−0.876279			−0.438140	−	0.898907i	$-0.644362\pi$
$467$	−8843.37			−0.876279			−0.438140	+	0.898907i	$0.644362\pi$
$468$		0			0
$469$	−2006.34	+	228.650i	−0.197535	+	0.0225119i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6523.93			0.634188
$474$		0			0
$475$	21010.0			2.02948
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19687.6			−1.87798			−0.938988	−	0.343951i	$-0.888235\pi$
$479$	−19687.6			−1.87798			−0.938988	+	0.343951i	$0.888235\pi$
$480$		0			0
$481$			22749.9i			2.15657i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	25887.7			2.42371
$486$		0			0
$487$			17969.6i			1.67203i	0.548706	+	0.836015i	$0.315121\pi$
$487$			17969.6i			1.67203i	−0.548706	+	0.836015i	$0.684879\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			2351.82i			0.216163i	0.994142	+	0.108082i	$0.0344708\pi$
$491$			2351.82i			0.216163i	−0.994142	+	0.108082i	$0.965529\pi$
$492$		0			0
$493$			3463.94i			0.316446i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13643.6	−	1554.88i	1.23139	−	0.140333i
$498$		0			0
$499$		−	1548.25i		−	0.138896i	−0.997586	−	0.0694479i	$-0.977876\pi$
$499$		−	1548.25i		−	0.138896i	0.997586	−	0.0694479i	$-0.0221238\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	17661.5			1.56558			0.782791	−	0.622284i	$-0.213795\pi$
$503$	17661.5			1.56558			0.782791	+	0.622284i	$0.213795\pi$
$504$		0			0
$505$	4020.61			0.354287
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	7607.03i		−	0.662427i	−0.943556	−	0.331214i	$-0.892542\pi$
$509$		−	7607.03i		−	0.662427i	0.943556	−	0.331214i	$-0.107458\pi$
$510$		0			0
$511$	−13848.7	+	1578.25i	−1.19889	+	0.136630i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	12641.2i		−	1.08163i
$516$		0			0
$517$		−	957.114i		−	0.0814194i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	1121.16i		−	0.0942781i	−0.998888	−	0.0471391i	$-0.984990\pi$
$521$		−	1121.16i		−	0.0942781i	0.998888	−	0.0471391i	$-0.0150104\pi$
$522$		0			0
$523$	10129.6			0.846918			0.423459	−	0.905915i	$-0.360816\pi$
$523$	10129.6			0.846918			0.423459	+	0.905915i	$0.360816\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	8860.37i		−	0.732379i
$528$		0			0
$529$	−12740.0			−1.04710
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−18634.2			−1.51433
$534$		0			0
$535$	13470.4			1.08855
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2540.75	−	11002.4i	−0.203039	−	0.879234i
$540$		0			0
$541$	14846.0			1.17981			0.589907	−	0.807471i	$-0.299164\pi$
$541$	14846.0			1.17981			0.589907	+	0.807471i	$0.299164\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	19180.6i		−	1.50754i
$546$		0			0
$547$			5857.85i			0.457886i	0.973440	+	0.228943i	$0.0735269\pi$
$547$			5857.85i			0.457886i	−0.973440	+	0.228943i	$0.926473\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7978.51			−0.616871
$552$		0			0
$553$	8264.55	−	941.859i	0.635523	−	0.0724266i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8589.49			−0.653408			−0.326704	−	0.945127i	$-0.605938\pi$
$557$	−8589.49			−0.653408			−0.326704	+	0.945127i	$0.605938\pi$
$558$		0			0
$559$	10658.8			0.806474
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6159.17			−0.461062			−0.230531	−	0.973065i	$-0.574046\pi$
$563$	−6159.17			−0.461062			−0.230531	+	0.973065i	$0.574046\pi$
$564$		0			0
$565$		−	34306.1i		−	2.55446i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	14756.9			1.08724			0.543620	−	0.839331i	$-0.317053\pi$
$569$	14756.9			1.08724			0.543620	+	0.839331i	$0.317053\pi$
$570$		0			0
$571$		−	11216.5i		−	0.822059i	−0.911622	−	0.411030i	$-0.865169\pi$
$571$		−	11216.5i		−	0.822059i	0.911622	−	0.411030i	$-0.134831\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			29925.9i			2.17043i
$576$		0			0
$577$		−	25252.3i		−	1.82195i	−0.412461	−	0.910975i	$-0.635331\pi$
$577$		−	25252.3i		−	1.82195i	0.412461	−	0.910975i	$-0.364669\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−551.502	−	4839.27i	−0.0393807	−	0.345554i
$582$		0			0
$583$		−	22470.4i		−	1.59628i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16488.5			1.15938			0.579688	−	0.814839i	$-0.303174\pi$
$587$	16488.5			1.15938			0.579688	+	0.814839i	$0.303174\pi$
$588$		0			0
$589$	20408.1			1.42768
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	8899.79i		−	0.616308i	−0.951336	−	0.308154i	$-0.900289\pi$
$593$		−	8899.79i		−	0.616308i	0.951336	−	0.308154i	$-0.0997112\pi$
$594$		0			0
$595$	1789.34	+	15701.0i	0.123287	+	1.08181i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			10099.6i			0.688911i	0.938803	+	0.344456i	$0.111936\pi$
$599$			10099.6i			0.688911i	−0.938803	+	0.344456i	$0.888064\pi$
$600$		0			0
$601$			8719.01i			0.591773i	0.955223	+	0.295887i	$0.0956150\pi$
$601$			8719.01i			0.591773i	−0.955223	+	0.295887i	$0.904385\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			4384.69i			0.294649i
$606$		0			0
$607$	−15841.9			−1.05931			−0.529655	−	0.848213i	$-0.677679\pi$
$607$	−15841.9			−1.05931			−0.529655	+	0.848213i	$0.677679\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1563.73i		−	0.103538i
$612$		0			0
$613$	−6841.48			−0.450775			−0.225387	−	0.974269i	$-0.572365\pi$
$613$	−6841.48			−0.450775			−0.225387	+	0.974269i	$0.572365\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7676.90			−0.500908			−0.250454	−	0.968129i	$-0.580580\pi$
$617$	−7676.90			−0.500908			−0.250454	+	0.968129i	$0.580580\pi$
$618$		0			0
$619$	−1213.39			−0.0787886			−0.0393943	−	0.999224i	$-0.512543\pi$
$619$	−1213.39			−0.0787886			−0.0393943	+	0.999224i	$0.512543\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12407.8	+	1414.04i	−0.797925	+	0.0909346i
$624$		0			0
$625$	−3371.53			−0.215778
$626$		0			0
$627$		0			0
$628$		0			0
$629$			20346.7i			1.28979i
$630$		0			0
$631$			26695.3i			1.68419i	0.539328	+	0.842096i	$0.318678\pi$
$631$			26695.3i			1.68419i	−0.539328	+	0.842096i	$0.681322\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	44663.4			2.79120
$636$		0			0
$637$	−4151.08	−	17975.7i	−0.258197	−	1.11809i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	17576.8			1.08306			0.541530	−	0.840681i	$-0.317845\pi$
$641$	17576.8			1.08306			0.541530	+	0.840681i	$0.317845\pi$
$642$		0			0
$643$	26186.4			1.60605			0.803025	−	0.595945i	$-0.203223\pi$
$643$	26186.4			1.60605			0.803025	+	0.595945i	$0.203223\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9990.55			0.607062			0.303531	−	0.952822i	$-0.401834\pi$
$647$	9990.55			0.607062			0.303531	+	0.952822i	$0.401834\pi$
$648$		0			0
$649$			10051.5i			0.607943i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	28277.7			1.69463			0.847315	−	0.531091i	$-0.178218\pi$
$653$	28277.7			1.69463			0.847315	+	0.531091i	$0.178218\pi$
$654$		0			0
$655$		−	19765.6i		−	1.17909i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			3090.10i			0.182661i	0.995821	+	0.0913303i	$0.0291119\pi$
$659$			3090.10i			0.182661i	−0.995821	+	0.0913303i	$0.970888\pi$
$660$		0			0
$661$			11860.9i			0.697937i	0.937134	+	0.348969i	$0.113468\pi$
$661$			11860.9i			0.697937i	−0.937134	+	0.348969i	$0.886532\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−36164.1	+	4121.40i	−2.10885	+	0.240332i
$666$		0			0
$667$		−	11364.3i		−	0.659712i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5666.89			0.326032
$672$		0			0
$673$	−6269.15			−0.359076			−0.179538	−	0.983751i	$-0.557460\pi$
$673$	−6269.15			−0.359076			−0.179538	+	0.983751i	$0.557460\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			30107.8i			1.70921i	0.519275	+	0.854607i	$0.326202\pi$
$677$			30107.8i			1.70921i	−0.519275	+	0.854607i	$0.673798\pi$
$678$		0			0
$679$	−26856.2	+	3060.64i	−1.51789	+	0.172984i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	21336.3i		−	1.19533i	−0.801746	−	0.597665i	$-0.796095\pi$
$683$		−	21336.3i		−	1.19533i	0.801746	−	0.597665i	$-0.203905\pi$
$684$		0			0
$685$		−	4627.98i		−	0.258140i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	36712.2i		−	2.02993i
$690$		0			0
$691$	−8597.07			−0.473297			−0.236648	−	0.971595i	$-0.576049\pi$
$691$	−8597.07			−0.473297			−0.236648	+	0.971595i	$0.576049\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	8974.73i		−	0.489829i
$696$		0			0
$697$	−16665.8			−0.905683
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−14251.7			−0.767872			−0.383936	−	0.923360i	$-0.625432\pi$
$701$	−14251.7			−0.767872			−0.383936	+	0.923360i	$0.625432\pi$
$702$		0			0
$703$	−46864.7			−2.51427
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4171.03	+	475.346i	−0.221878	+	0.0252860i
$708$		0			0
$709$	−21610.0			−1.14468			−0.572342	−	0.820015i	$-0.693965\pi$
$709$	−21610.0			−1.14468			−0.572342	+	0.820015i	$0.693965\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			29068.6i			1.52683i
$714$		0			0
$715$		−	31408.2i		−	1.64280i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−32462.5			−1.68379			−0.841896	−	0.539639i	$-0.818561\pi$
$719$	−32462.5			−1.68379			−0.841896	+	0.539639i	$0.818561\pi$
$720$		0			0
$721$	1494.54	+	13114.1i	0.0771977	+	0.677388i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−13654.3			−0.699458
$726$		0			0
$727$	−27608.0			−1.40842			−0.704211	−	0.709991i	$-0.748699\pi$
$727$	−27608.0			−1.40842			−0.704211	+	0.709991i	$0.748699\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	9532.84			0.482332
$732$		0			0
$733$			6821.15i			0.343718i	0.985122	+	0.171859i	$0.0549773\pi$
$733$			6821.15i			0.343718i	−0.985122	+	0.171859i	$0.945023\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3589.50			0.179404
$738$		0			0
$739$			17123.2i			0.852351i	0.904641	+	0.426175i	$0.140139\pi$
$739$			17123.2i			0.852351i	−0.904641	+	0.426175i	$0.859861\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			2411.20i			0.119055i	0.998227	+	0.0595277i	$0.0189595\pi$
$743$			2411.20i			0.119055i	−0.998227	+	0.0595277i	$0.981041\pi$
$744$		0			0
$745$		−	24784.4i		−	1.21883i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−13974.3	+	1592.57i	−0.681723	+	0.0776917i
$750$		0			0
$751$			9873.14i			0.479728i	0.970806	+	0.239864i	$0.0771029\pi$
$751$			9873.14i			0.479728i	−0.970806	+	0.239864i	$0.922897\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6105.03			0.294284
$756$		0			0
$757$	17357.4			0.833376			0.416688	−	0.909050i	$-0.363191\pi$
$757$	17357.4			0.833376			0.416688	+	0.909050i	$0.363191\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			26432.1i			1.25909i	0.776966	+	0.629543i	$0.216758\pi$
$761$			26432.1i			1.25909i	−0.776966	+	0.629543i	$0.783242\pi$
$762$		0			0
$763$	2267.67	+	19898.2i	0.107595	+	0.944120i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			16422.1i			0.773100i
$768$		0			0
$769$			28143.2i			1.31973i	0.751385	+	0.659864i	$0.229386\pi$
$769$			28143.2i			1.31973i	−0.751385	+	0.659864i	$0.770614\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			15072.7i			0.701328i	0.936501	+	0.350664i	$0.114044\pi$
$773$			15072.7i			0.701328i	−0.936501	+	0.350664i	$0.885956\pi$
$774$		0			0
$775$	34926.1			1.61882
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	38386.4i		−	1.76551i
$780$		0			0
$781$	−24409.5			−1.11836
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−43462.4			−1.97610
$786$		0			0
$787$	−22371.5			−1.01329			−0.506645	−	0.862155i	$-0.669114\pi$
$787$	−22371.5			−1.01329			−0.506645	+	0.862155i	$0.669114\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4055.92	+	35589.6i	0.182316	+	1.59977i
$792$		0			0
$793$	9258.56			0.414604
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	28223.2i		−	1.25435i	−0.778878	−	0.627175i	$-0.784211\pi$
$797$		−	28223.2i		−	1.25435i	0.778878	−	0.627175i	$-0.215789\pi$
$798$		0			0
$799$		−	1398.55i		−	0.0619236i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	24776.5			1.08885
$804$		0			0
$805$	−5870.38	−	51510.9i	−0.257023	−	2.25531i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−750.987			−0.0326370			−0.0163185	−	0.999867i	$-0.505195\pi$
$809$	−750.987			−0.0326370			−0.0163185	+	0.999867i	$0.505195\pi$
$810$		0			0
$811$	5362.03			0.232166			0.116083	−	0.993240i	$-0.462966\pi$
$811$	5362.03			0.232166			0.116083	+	0.993240i	$0.462966\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	31032.2			1.33376
$816$		0			0
$817$			21957.0i			0.940244i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−34039.9			−1.44702			−0.723508	−	0.690316i	$-0.757471\pi$
$821$	−34039.9			−1.44702			−0.723508	+	0.690316i	$0.757471\pi$
$822$		0			0
$823$			3953.11i			0.167432i	0.996490	+	0.0837161i	$0.0266789\pi$
$823$			3953.11i			0.167432i	−0.996490	+	0.0837161i	$0.973321\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			22122.3i			0.930190i	0.885261	+	0.465095i	$0.153980\pi$
$827$			22122.3i			0.930190i	−0.885261	+	0.465095i	$0.846020\pi$
$828$		0			0
$829$			45342.1i			1.89963i	0.312805	+	0.949817i	$0.398731\pi$
$829$			45342.1i			1.89963i	−0.312805	+	0.949817i	$0.601269\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3712.57	−	16076.8i	−0.154421	−	0.668702i
$834$		0			0
$835$		−	72238.4i		−	2.99391i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−29527.4			−1.21502			−0.607508	−	0.794313i	$-0.707831\pi$
$839$	−29527.4			−1.21502			−0.607508	+	0.794313i	$0.707831\pi$
$840$		0			0
$841$	−19203.8			−0.787396
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	12345.2i		−	0.502591i
$846$		0			0
$847$	−518.390	−	4548.72i	−0.0210296	−	0.184529i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	66752.5i		−	2.68889i
$852$		0			0
$853$			18339.4i			0.736140i	0.929798	+	0.368070i	$0.119981\pi$
$853$			18339.4i			0.736140i	−0.929798	+	0.368070i	$0.880019\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	14576.0i		−	0.580988i	−0.956877	−	0.290494i	$-0.906180\pi$
$857$		−	14576.0i		−	0.580988i	0.956877	−	0.290494i	$-0.0938196\pi$
$858$		0			0
$859$	3972.38			0.157783			0.0788917	−	0.996883i	$-0.474862\pi$
$859$	3972.38			0.157783			0.0788917	+	0.996883i	$0.474862\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	40644.4i		−	1.60319i	−0.597869	−	0.801594i	$-0.703986\pi$
$863$		−	40644.4i		−	1.60319i	0.597869	−	0.801594i	$-0.296014\pi$
$864$		0			0
$865$	70746.4			2.78087
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−14785.9			−0.577191
$870$		0			0
$871$	5864.53			0.228142
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−21091.7	+	2403.69i	−0.814891	+	0.0928681i
$876$		0			0
$877$	−9602.49			−0.369730			−0.184865	−	0.982764i	$-0.559185\pi$
$877$	−9602.49			−0.369730			−0.184865	+	0.982764i	$0.559185\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			5204.78i			0.199039i	0.995036	+	0.0995196i	$0.0317306\pi$
$881$			5204.78i			0.199039i	−0.995036	+	0.0995196i	$0.968269\pi$
$882$		0			0
$883$			22108.6i			0.842596i	0.906922	+	0.421298i	$0.138425\pi$
$883$			22108.6i			0.842596i	−0.906922	+	0.421298i	$0.861575\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21305.3			0.806497			0.403248	−	0.915091i	$-0.367881\pi$
$887$	21305.3			0.806497			0.403248	+	0.915091i	$0.367881\pi$
$888$		0			0
$889$	−46334.4	+	5280.44i	−1.74804	+	0.199213i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3221.28			0.120712
$894$		0			0
$895$	−572.414			−0.0213784
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−13263.1			−0.492047
$900$		0			0
$901$		−	32834.1i		−	1.21405i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4134.85			0.151875
$906$		0			0
$907$		−	5073.71i		−	0.185744i	−0.995678	−	0.0928720i	$-0.970395\pi$
$907$		−	5073.71i		−	0.185744i	0.995678	−	0.0928720i	$-0.0296047\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	22711.5i		−	0.825977i	−0.910736	−	0.412989i	$-0.864485\pi$
$911$		−	22711.5i		−	0.825977i	0.910736	−	0.412989i	$-0.135515\pi$
$912$		0			0
$913$			8657.85i			0.313837i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2336.83	+	20505.0i	0.0841537	+	0.738425i
$918$		0			0
$919$			8104.63i			0.290911i	0.989365	+	0.145455i	$0.0464647\pi$
$919$			8104.63i			0.290911i	−0.989365	+	0.145455i	$0.953535\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−39880.2			−1.42218
$924$		0			0
$925$	−80203.4			−2.85089
$926$		0			0
$927$		0			0
$928$		0			0
$929$			28055.3i			0.990814i	0.868661	+	0.495407i	$0.164981\pi$
$929$			28055.3i			0.990814i	−0.868661	+	0.495407i	$0.835019\pi$
$930$		0			0
$931$	37029.8	−	8551.18i	1.30355	−	0.301024i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	28090.3i		−	0.982515i
$936$		0			0
$937$		−	26420.7i		−	0.921160i	−0.887618	−	0.460580i	$-0.847641\pi$
$937$		−	26420.7i		−	0.921160i	0.887618	−	0.460580i	$-0.152359\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	48331.7i		−	1.67435i	−0.546931	−	0.837177i	$-0.684204\pi$
$941$		−	48331.7i		−	1.67435i	0.546931	−	0.837177i	$-0.315796\pi$
$942$		0			0
$943$	54676.2			1.88813
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	56278.5i		−	1.93116i	−0.260115	−	0.965578i	$-0.583760\pi$
$947$		−	56278.5i		−	1.93116i	0.260115	−	0.965578i	$-0.416240\pi$
$948$		0			0
$949$	40479.9			1.38465
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−6765.72			−0.229972			−0.114986	−	0.993367i	$-0.536682\pi$
$953$	−6765.72			−0.229972			−0.114986	+	0.993367i	$0.536682\pi$
$954$		0			0
$955$	−1971.92			−0.0668166
$956$		0			0
$957$		0			0
$958$		0			0
$959$	547.154	+	4801.12i	0.0184239	+	0.161665i
$960$		0			0
$961$	4134.61			0.138787
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	30392.5i		−	1.01385i
$966$		0			0
$967$			30443.1i			1.01239i	0.862418	+	0.506196i	$0.168949\pi$
$967$			30443.1i			1.01239i	−0.862418	+	0.506196i	$0.831051\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−50571.9			−1.67140			−0.835701	−	0.549185i	$-0.814938\pi$
$971$	−50571.9			−1.67140			−0.835701	+	0.549185i	$0.814938\pi$
$972$		0			0
$973$	1061.06	+	9310.49i	0.0349599	+	0.306763i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3249.34			−0.106403			−0.0532014	−	0.998584i	$-0.516943\pi$
$977$	−3249.34			−0.106403			−0.0532014	+	0.998584i	$0.516943\pi$
$978$		0			0
$979$	22198.5			0.724686
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−29605.3			−0.960592			−0.480296	−	0.877106i	$-0.659471\pi$
$983$	−29605.3			−0.960592			−0.480296	+	0.877106i	$0.659471\pi$
$984$		0			0
$985$		−	71750.1i		−	2.32096i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−31274.9			−1.00554
$990$		0			0
$991$		−	40111.2i		−	1.28575i	−0.765972	−	0.642873i	$-0.777742\pi$
$991$		−	40111.2i		−	1.28575i	0.765972	−	0.642873i	$-0.222258\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			79475.4i			2.53220i
$996$		0			0
$997$			50086.1i			1.59102i	0.605943	+	0.795508i	$0.292796\pi$
$997$			50086.1i			1.59102i	−0.605943	+	0.795508i	$0.707204\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.b.k.559.7		8
3.2	odd	2		336.4.b.f.223.2	yes	8
4.3	odd	2		1008.4.b.i.559.7		8
7.6	odd	2		1008.4.b.i.559.2		8
12.11	even	2		336.4.b.e.223.2	✓	8
21.20	even	2		336.4.b.e.223.7	yes	8
24.5	odd	2		1344.4.b.e.895.7		8
24.11	even	2		1344.4.b.f.895.7		8
28.27	even	2	inner	1008.4.b.k.559.2		8
84.83	odd	2		336.4.b.f.223.7	yes	8
168.83	odd	2		1344.4.b.e.895.2		8
168.125	even	2		1344.4.b.f.895.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.b.e.223.2	✓	8	12.11	even	2
336.4.b.e.223.7	yes	8	21.20	even	2
336.4.b.f.223.2	yes	8	3.2	odd	2
336.4.b.f.223.7	yes	8	84.83	odd	2
1008.4.b.i.559.2		8	7.6	odd	2
1008.4.b.i.559.7		8	4.3	odd	2
1008.4.b.k.559.2		8	28.27	even	2	inner
1008.4.b.k.559.7		8	1.1	even	1	trivial
1344.4.b.e.895.2		8	168.83	odd	2
1344.4.b.e.895.7		8	24.5	odd	2
1344.4.b.f.895.2		8	168.125	even	2
1344.4.b.f.895.7		8	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+17.7376i q^{5} +(-2.09706 - 18.4011i) q^{7} +O(q^{10})\)	\(q+17.7376i q^{5} +(-2.09706 - 18.4011i) q^{7} +32.9212i q^{11} +53.7866i q^{13} +48.1047i q^{17} -110.800 q^{19} -157.820i q^{23} -189.621 q^{25} +72.0082 q^{29} -184.189 q^{31} +(326.391 - 37.1968i) q^{35} +422.967 q^{37} +346.448i q^{41} -198.168i q^{43} -29.0729 q^{47} +(-334.205 + 77.1768i) q^{49} -682.553 q^{53} -583.941 q^{55} +305.320 q^{59} -172.135i q^{61} -954.042 q^{65} -109.033i q^{67} +741.453i q^{71} -752.602i q^{73} +(605.787 - 69.0378i) q^{77} +449.132i q^{79} +262.988 q^{83} -853.261 q^{85} -674.294i q^{89} +(989.735 - 112.794i) q^{91} -1965.32i q^{95} -1459.49i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7}+O(q^{10}) \) 8 * q + 4 * q^7	\( 8 q + 4 q^{7} + 56 q^{19} - 656 q^{25} - 240 q^{29} - 320 q^{31} + 600 q^{35} + 392 q^{37} - 816 q^{47} - 16 q^{49} - 288 q^{53} + 456 q^{55} - 1824 q^{59} + 816 q^{65} + 2064 q^{77} + 1680 q^{83} + 2568 q^{85} + 864 q^{91}+O(q^{100}) \) 8 * q + 4 * q^7 + 56 * q^19 - 656 * q^25 - 240 * q^29 - 320 * q^31 + 600 * q^35 + 392 * q^37 - 816 * q^47 - 16 * q^49 - 288 * q^53 + 456 * q^55 - 1824 * q^59 + 816 * q^65 + 2064 * q^77 + 1680 * q^83 + 2568 * q^85 + 864 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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