LMFDB - Embedded newform 1764.4.k.g.1549.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, degree $2$, not 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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1549.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1764.1549
Dual form			1764.4.k.g.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 3.46410i) q^{5} +O(q^{10})$	$q+(-2.00000 + 3.46410i) q^{5} +(-10.0000 - 17.3205i) q^{11} +4.00000 q^{13} +(-12.0000 - 20.7846i) q^{17} +(22.0000 - 38.1051i) q^{19} +(36.0000 - 62.3538i) q^{23} +(54.5000 + 94.3968i) q^{25} +38.0000 q^{29} +(92.0000 + 159.349i) q^{31} +(15.0000 - 25.9808i) q^{37} -216.000 q^{41} -164.000 q^{43} +(-260.000 + 450.333i) q^{47} +(-73.0000 - 126.440i) q^{53} +80.0000 q^{55} +(-230.000 - 398.372i) q^{59} +(314.000 - 543.864i) q^{61} +(-8.00000 + 13.8564i) q^{65} +(-278.000 - 481.510i) q^{67} -592.000 q^{71} +(512.000 + 886.810i) q^{73} +(52.0000 - 90.0666i) q^{79} -324.000 q^{83} +96.0000 q^{85} +(-448.000 + 775.959i) q^{89} +(88.0000 + 152.420i) q^{95} +920.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} - 20 q^{11} + 8 q^{13} - 24 q^{17} + 44 q^{19} + 72 q^{23} + 109 q^{25} + 76 q^{29} + 184 q^{31} + 30 q^{37} - 432 q^{41} - 328 q^{43} - 520 q^{47} - 146 q^{53} + 160 q^{55} - 460 q^{59} + 628 q^{61} - 16 q^{65} - 556 q^{67} - 1184 q^{71} + 1024 q^{73} + 104 q^{79} - 648 q^{83} + 192 q^{85} - 896 q^{89} + 176 q^{95} + 1840 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 20 * q^11 + 8 * q^13 - 24 * q^17 + 44 * q^19 + 72 * q^23 + 109 * q^25 + 76 * q^29 + 184 * q^31 + 30 * q^37 - 432 * q^41 - 328 * q^43 - 520 * q^47 - 146 * q^53 + 160 * q^55 - 460 * q^59 + 628 * q^61 - 16 * q^65 - 556 * q^67 - 1184 * q^71 + 1024 * q^73 + 104 * q^79 - 648 * q^83 + 192 * q^85 - 896 * q^89 + 176 * q^95 + 1840 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−2.00000	+	3.46410i	−0.178885	+	0.309839i	−0.941499	−	0.337016i	$-0.890582\pi$
$5$	−2.00000	+	3.46410i	−0.178885	+	0.309839i	0.762614	+	0.646854i	$0.223916\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−10.0000	−	17.3205i	−0.274101	−	0.474757i	0.695807	−	0.718229i	$-0.255047\pi$
$11$	−10.0000	−	17.3205i	−0.274101	−	0.474757i	−0.969908	+	0.243472i	$0.921714\pi$
$12$		0			0
$13$	4.00000			0.0853385			0.0426692	−	0.999089i	$-0.486414\pi$
$13$	4.00000			0.0853385			0.0426692	+	0.999089i	$0.486414\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12.0000	−	20.7846i	−0.171202	−	0.296530i	0.767639	−	0.640883i	$-0.221432\pi$
$17$	−12.0000	−	20.7846i	−0.171202	−	0.296530i	−0.938840	+	0.344353i	$0.888098\pi$
$18$		0			0
$19$	22.0000	−	38.1051i	0.265639	−	0.460101i	−0.702092	−	0.712087i	$-0.747750\pi$
$19$	22.0000	−	38.1051i	0.265639	−	0.460101i	0.967731	+	0.251986i	$0.0810837\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	36.0000	−	62.3538i	0.326370	−	0.565290i	−0.655418	−	0.755266i	$-0.727508\pi$
$23$	36.0000	−	62.3538i	0.326370	−	0.565290i	0.981789	+	0.189976i	$0.0608410\pi$
$24$		0			0
$25$	54.5000	+	94.3968i	0.436000	+	0.755174i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	38.0000			0.243325			0.121662	−	0.992572i	$-0.461177\pi$
$29$	38.0000			0.243325			0.121662	+	0.992572i	$0.461177\pi$
$30$		0			0
$31$	92.0000	+	159.349i	0.533022	+	0.923222i	0.999256	+	0.0385601i	$0.0122771\pi$
$31$	92.0000	+	159.349i	0.533022	+	0.923222i	−0.466234	+	0.884661i	$0.654390\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	15.0000	−	25.9808i	0.0666482	−	0.115438i	−0.830776	−	0.556607i	$-0.812103\pi$
$37$	15.0000	−	25.9808i	0.0666482	−	0.115438i	0.897424	+	0.441169i	$0.145436\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−216.000			−0.822769			−0.411385	−	0.911462i	$-0.634955\pi$
$41$	−216.000			−0.822769			−0.411385	+	0.911462i	$0.634955\pi$
$42$		0			0
$43$	−164.000			−0.581622			−0.290811	−	0.956780i	$-0.593925\pi$
$43$	−164.000			−0.581622			−0.290811	+	0.956780i	$0.593925\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−260.000	+	450.333i	−0.806913	+	1.39761i	0.108079	+	0.994142i	$0.465530\pi$
$47$	−260.000	+	450.333i	−0.806913	+	1.39761i	−0.914992	+	0.403472i	$0.867803\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−73.0000	−	126.440i	−0.189195	−	0.327695i	0.755787	−	0.654817i	$-0.227254\pi$
$53$	−73.0000	−	126.440i	−0.189195	−	0.327695i	−0.944982	+	0.327122i	$0.893921\pi$
$54$		0			0
$55$	80.0000			0.196131
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−230.000	−	398.372i	−0.507516	−	0.879044i	−0.999962	−	0.00870069i	$-0.997230\pi$
$59$	−230.000	−	398.372i	−0.507516	−	0.879044i	0.492446	−	0.870343i	$-0.336103\pi$
$60$		0			0
$61$	314.000	−	543.864i	0.659075	−	1.14155i	−0.321780	−	0.946814i	$-0.604281\pi$
$61$	314.000	−	543.864i	0.659075	−	1.14155i	0.980855	−	0.194737i	$-0.0623854\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.00000	+	13.8564i	−0.0152658	+	0.0264412i
$66$		0			0
$67$	−278.000	−	481.510i	−0.506912	−	0.877997i	−0.999968	−	0.00799979i	$-0.997454\pi$
$67$	−278.000	−	481.510i	−0.506912	−	0.877997i	0.493056	−	0.869998i	$-0.335880\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−592.000			−0.989542			−0.494771	−	0.869023i	$-0.664748\pi$
$71$	−592.000			−0.989542			−0.494771	+	0.869023i	$0.664748\pi$
$72$		0			0
$73$	512.000	+	886.810i	0.820891	+	1.42183i	0.905019	+	0.425371i	$0.139856\pi$
$73$	512.000	+	886.810i	0.820891	+	1.42183i	−0.0841280	+	0.996455i	$0.526810\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	52.0000	−	90.0666i	0.0740564	−	0.128269i	−0.826619	−	0.562762i	$-0.809739\pi$
$79$	52.0000	−	90.0666i	0.0740564	−	0.128269i	0.900676	+	0.434492i	$0.143072\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−324.000			−0.428477			−0.214239	−	0.976781i	$-0.568727\pi$
$83$	−324.000			−0.428477			−0.214239	+	0.976781i	$0.568727\pi$
$84$		0			0
$85$	96.0000			0.122502
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−448.000	+	775.959i	−0.533572	+	0.924174i	0.465659	+	0.884964i	$0.345817\pi$
$89$	−448.000	+	775.959i	−0.533572	+	0.924174i	−0.999231	+	0.0392095i	$0.987516\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	88.0000	+	152.420i	0.0950380	+	0.164611i
$96$		0			0
$97$	920.000			0.963009			0.481504	−	0.876444i	$-0.340091\pi$
$97$	920.000			0.963009			0.481504	+	0.876444i	$0.340091\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−554.000	−	959.556i	−0.545793	−	0.945341i	−0.998557	−	0.0537102i	$-0.982895\pi$
$101$	−554.000	−	959.556i	−0.545793	−	0.945341i	0.452764	−	0.891630i	$-0.350438\pi$
$102$		0			0
$103$	724.000	−	1254.00i	0.692600	−	1.19962i	−0.278383	−	0.960470i	$-0.589798\pi$
$103$	724.000	−	1254.00i	0.692600	−	1.19962i	0.970983	−	0.239149i	$-0.0768684\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	658.000	−	1139.69i	0.594498	−	1.02970i	−0.399120	−	0.916899i	$-0.630684\pi$
$107$	658.000	−	1139.69i	0.594498	−	1.02970i	0.993618	−	0.112802i	$-0.0359824\pi$
$108$		0			0
$109$	43.0000	+	74.4782i	0.0377858	+	0.0654469i	0.884300	−	0.466919i	$-0.154636\pi$
$109$	43.0000	+	74.4782i	0.0377858	+	0.0654469i	−0.846514	+	0.532366i	$0.821303\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1778.00			−1.48018			−0.740089	−	0.672509i	$-0.765217\pi$
$113$	−1778.00			−1.48018			−0.740089	+	0.672509i	$0.765217\pi$
$114$		0			0
$115$	144.000	+	249.415i	0.116766	+	0.202244i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	465.500	−	806.270i	0.349737	−	0.605762i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−936.000			−0.669747
$126$		0			0
$127$	−928.000			−0.648399			−0.324200	−	0.945989i	$-0.605095\pi$
$127$	−928.000			−0.648399			−0.324200	+	0.945989i	$0.605095\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−702.000	+	1215.90i	−0.468199	+	0.810944i	−0.999339	−	0.0363397i	$-0.988430\pi$
$131$	−702.000	+	1215.90i	−0.468199	+	0.810944i	0.531141	+	0.847284i	$0.321764\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−685.000	−	1186.45i	−0.427179	−	0.739895i	0.569442	−	0.822031i	$-0.307159\pi$
$137$	−685.000	−	1186.45i	−0.427179	−	0.739895i	−0.996621	+	0.0821359i	$0.973826\pi$
$138$		0			0
$139$	−516.000			−0.314867			−0.157434	−	0.987530i	$-0.550322\pi$
$139$	−516.000			−0.314867			−0.157434	+	0.987530i	$0.550322\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−40.0000	−	69.2820i	−0.0233914	−	0.0405151i
$144$		0			0
$145$	−76.0000	+	131.636i	−0.0435273	+	0.0753915i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	695.000	−	1203.78i	0.382125	−	0.661860i	−0.609241	−	0.792985i	$-0.708526\pi$
$149$	695.000	−	1203.78i	0.382125	−	0.661860i	0.991366	+	0.131125i	$0.0418591\pi$
$150$		0			0
$151$	−68.0000	−	117.779i	−0.0366474	−	0.0634752i	0.847120	−	0.531402i	$-0.178335\pi$
$151$	−68.0000	−	117.779i	−0.0366474	−	0.0634752i	−0.883767	+	0.467927i	$0.845001\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−736.000			−0.381400
$156$		0			0
$157$	−74.0000	−	128.172i	−0.0376168	−	0.0651543i	0.846604	−	0.532223i	$-0.178643\pi$
$157$	−74.0000	−	128.172i	−0.0376168	−	0.0651543i	−0.884221	+	0.467069i	$0.845310\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	606.000	−	1049.62i	0.291200	−	0.504373i	−0.682894	−	0.730518i	$-0.739279\pi$
$163$	606.000	−	1049.62i	0.291200	−	0.504373i	0.974094	+	0.226145i	$0.0726122\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1976.00			0.915614			0.457807	−	0.889052i	$-0.348635\pi$
$167$	1976.00			0.915614			0.457807	+	0.889052i	$0.348635\pi$
$168$		0			0
$169$	−2181.00			−0.992717
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1346.00	−	2331.34i	0.591529	−	1.02456i	−0.402498	−	0.915421i	$-0.631858\pi$
$173$	1346.00	−	2331.34i	0.591529	−	1.02456i	0.994027	−	0.109137i	$-0.0348087\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1290.00	−	2234.35i	−0.538654	−	0.932977i	−0.998977	−	0.0452249i	$-0.985600\pi$
$179$	−1290.00	−	2234.35i	−0.538654	−	0.932977i	0.460322	−	0.887752i	$-0.347734\pi$
$180$		0			0
$181$	2036.00			0.836103			0.418052	−	0.908423i	$-0.362713\pi$
$181$	2036.00			0.836103			0.418052	+	0.908423i	$0.362713\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	60.0000	+	103.923i	0.0238448	+	0.0413004i
$186$		0			0
$187$	−240.000	+	415.692i	−0.0938531	+	0.162558i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1980.00	−	3429.46i	0.750093	−	1.29920i	−0.197684	−	0.980266i	$-0.563342\pi$
$191$	1980.00	−	3429.46i	0.750093	−	1.29920i	0.947777	−	0.318933i	$-0.103325\pi$
$192$		0			0
$193$	−1.00000	−	1.73205i	−0.000372962	−	0.000645988i	0.865839	−	0.500323i	$-0.166785\pi$
$193$	−1.00000	−	1.73205i	−0.000372962	−	0.000645988i	−0.866212	+	0.499677i	$0.833452\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3774.00			−1.36491			−0.682453	−	0.730930i	$-0.739087\pi$
$197$	−3774.00			−1.36491			−0.682453	+	0.730930i	$0.739087\pi$
$198$		0			0
$199$	−1780.00	−	3083.05i	−0.634075	−	1.09825i	−0.986710	−	0.162489i	$-0.948048\pi$
$199$	−1780.00	−	3083.05i	−0.634075	−	1.09825i	0.352636	−	0.935761i	$-0.385285\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	432.000	−	748.246i	0.147181	−	0.254926i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−880.000			−0.291248
$210$		0			0
$211$	−2692.00			−0.878317			−0.439159	−	0.898410i	$-0.644723\pi$
$211$	−2692.00			−0.878317			−0.439159	+	0.898410i	$0.644723\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	328.000	−	568.113i	0.104044	−	0.180209i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−48.0000	−	83.1384i	−0.0146101	−	0.0253054i
$222$		0			0
$223$	−4528.00			−1.35972			−0.679859	−	0.733342i	$-0.737959\pi$
$223$	−4528.00			−1.35972			−0.679859	+	0.733342i	$0.737959\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1826.00	+	3162.72i	0.533903	+	0.924746i	0.999216	+	0.0396002i	$0.0126084\pi$
$227$	1826.00	+	3162.72i	0.533903	+	0.924746i	−0.465313	+	0.885146i	$0.654058\pi$
$228$		0			0
$229$	−2402.00	+	4160.39i	−0.693138	+	1.20055i	0.277666	+	0.960678i	$0.410439\pi$
$229$	−2402.00	+	4160.39i	−0.693138	+	1.20055i	−0.970804	+	0.239873i	$0.922894\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1379.00	−	2388.50i	0.387731	−	0.671570i	−0.604413	−	0.796671i	$-0.706592\pi$
$233$	1379.00	−	2388.50i	0.387731	−	0.671570i	0.992144	+	0.125102i	$0.0399257\pi$
$234$		0			0
$235$	−1040.00	−	1801.33i	−0.288690	−	0.500026i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6528.00			−1.76678			−0.883392	−	0.468635i	$-0.844746\pi$
$239$	−6528.00			−1.76678			−0.883392	+	0.468635i	$0.844746\pi$
$240$		0			0
$241$	−28.0000	−	48.4974i	−0.00748398	−	0.0129626i	0.862259	−	0.506467i	$-0.169049\pi$
$241$	−28.0000	−	48.4974i	−0.00748398	−	0.0129626i	−0.869743	+	0.493505i	$0.835716\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	88.0000	−	152.420i	0.0226693	−	0.0392643i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4900.00			1.23221			0.616106	−	0.787663i	$-0.288709\pi$
$251$	4900.00			1.23221			0.616106	+	0.787663i	$0.288709\pi$
$252$		0			0
$253$	−1440.00			−0.357834
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3392.00	−	5875.12i	0.823296	−	1.42599i	−0.0799181	−	0.996801i	$-0.525466\pi$
$257$	3392.00	−	5875.12i	0.823296	−	1.42599i	0.903214	−	0.429190i	$-0.141201\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2272.00	−	3935.22i	−0.532690	−	0.922646i	−0.999271	−	0.0381681i	$-0.987848\pi$
$263$	−2272.00	−	3935.22i	−0.532690	−	0.922646i	0.466581	−	0.884478i	$-0.345486\pi$
$264$		0			0
$265$	584.000			0.135377
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2026.00	+	3509.13i	0.459210	+	0.795374i	0.998919	−	0.0464767i	$-0.0147993\pi$
$269$	2026.00	+	3509.13i	0.459210	+	0.795374i	−0.539710	+	0.841851i	$0.681466\pi$
$270$		0			0
$271$	−1376.00	+	2383.30i	−0.308436	+	0.534226i	−0.978020	−	0.208510i	$-0.933139\pi$
$271$	−1376.00	+	2383.30i	−0.308436	+	0.534226i	0.669585	+	0.742736i	$0.266472\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1090.00	−	1887.94i	0.239016	−	0.413988i
$276$		0			0
$277$	−2183.00	−	3781.07i	−0.473515	−	0.820153i	0.526025	−	0.850469i	$-0.323682\pi$
$277$	−2183.00	−	3781.07i	−0.473515	−	0.820153i	−0.999540	+	0.0303164i	$0.990348\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7734.00			−1.64189			−0.820946	−	0.571006i	$-0.806553\pi$
$281$	−7734.00			−1.64189			−0.820946	+	0.571006i	$0.806553\pi$
$282$		0			0
$283$	−2026.00	−	3509.13i	−0.425559	−	0.737090i	0.570913	−	0.821010i	$-0.306589\pi$
$283$	−2026.00	−	3509.13i	−0.425559	−	0.737090i	−0.996472	+	0.0839204i	$0.973256\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2168.50	−	3755.95i	0.441380	−	0.764493i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3420.00			−0.681906			−0.340953	−	0.940080i	$-0.610750\pi$
$293$	−3420.00			−0.681906			−0.340953	+	0.940080i	$0.610750\pi$
$294$		0			0
$295$	1840.00			0.363149
$296$		0			0
$297$		0			0
$298$		0			0
$299$	144.000	−	249.415i	0.0278520	−	0.0482410i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1256.00	+	2175.46i	0.235798	+	0.408414i
$306$		0			0
$307$	−7324.00			−1.36157			−0.680786	−	0.732482i	$-0.738362\pi$
$307$	−7324.00			−1.36157			−0.680786	+	0.732482i	$0.738362\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2096.00	+	3630.38i	0.382165	+	0.661929i	0.991371	−	0.131083i	$-0.0418453\pi$
$311$	2096.00	+	3630.38i	0.382165	+	0.661929i	−0.609207	+	0.793012i	$0.708512\pi$
$312$		0			0
$313$	−3420.00	+	5923.61i	−0.617603	+	1.06972i	0.372318	+	0.928105i	$0.378563\pi$
$313$	−3420.00	+	5923.61i	−0.617603	+	1.06972i	−0.989922	+	0.141615i	$0.954770\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3315.00	−	5741.75i	0.587347	−	1.01731i	−0.407232	−	0.913325i	$-0.633506\pi$
$317$	3315.00	−	5741.75i	0.587347	−	1.01731i	0.994578	−	0.103990i	$-0.0331609\pi$
$318$		0			0
$319$	−380.000	−	658.179i	−0.0666957	−	0.115520i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1056.00			−0.181911
$324$		0			0
$325$	218.000	+	377.587i	0.0372076	+	0.0644454i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	3434.00	−	5947.86i	0.570241	−	0.987686i	−0.426300	−	0.904582i	$-0.640183\pi$
$331$	3434.00	−	5947.86i	0.570241	−	0.987686i	0.996541	−	0.0831042i	$-0.0264834\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2224.00			0.362717
$336$		0			0
$337$	−7378.00			−1.19260			−0.596299	−	0.802763i	$-0.703363\pi$
$337$	−7378.00			−1.19260			−0.596299	+	0.802763i	$0.703363\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1840.00	−	3186.97i	0.292204	−	0.506112i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1338.00	−	2317.48i	−0.206996	−	0.358528i	0.743771	−	0.668435i	$-0.233035\pi$
$347$	−1338.00	−	2317.48i	−0.206996	−	0.358528i	−0.950767	+	0.309907i	$0.899702\pi$
$348$		0			0
$349$	5124.00			0.785907			0.392953	−	0.919558i	$-0.371453\pi$
$349$	5124.00			0.785907			0.392953	+	0.919558i	$0.371453\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2280.00	−	3949.08i	−0.343774	−	0.595434i	0.641356	−	0.767243i	$-0.278372\pi$
$353$	−2280.00	−	3949.08i	−0.343774	−	0.595434i	−0.985130	+	0.171809i	$0.945039\pi$
$354$		0			0
$355$	1184.00	−	2050.75i	0.177015	−	0.306598i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1828.00	−	3166.19i	0.268741	−	0.465474i	−0.699796	−	0.714343i	$-0.746726\pi$
$359$	1828.00	−	3166.19i	0.268741	−	0.465474i	0.968537	+	0.248869i	$0.0800590\pi$
$360$		0			0
$361$	2461.50	+	4263.44i	0.358872	+	0.621584i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4096.00			−0.587382
$366$		0			0
$367$	−808.000	−	1399.50i	−0.114924	−	0.199055i	0.802825	−	0.596215i	$-0.203329\pi$
$367$	−808.000	−	1399.50i	−0.114924	−	0.199055i	−0.917750	+	0.397160i	$0.869996\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1367.00	+	2367.71i	−0.189760	+	0.328674i	−0.945170	−	0.326578i	$-0.894104\pi$
$373$	−1367.00	+	2367.71i	−0.189760	+	0.328674i	0.755410	+	0.655252i	$0.227438\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	152.000			0.0207650
$378$		0			0
$379$	−1380.00			−0.187034			−0.0935169	−	0.995618i	$-0.529811\pi$
$379$	−1380.00			−0.187034			−0.0935169	+	0.995618i	$0.529811\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3444.00	+	5965.18i	−0.459478	+	0.795840i	−0.998933	−	0.0461746i	$-0.985297\pi$
$383$	−3444.00	+	5965.18i	−0.459478	+	0.795840i	0.539455	+	0.842014i	$0.318630\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1023.00	−	1771.89i	−0.133337	−	0.230947i	0.791624	−	0.611009i	$-0.209236\pi$
$389$	−1023.00	−	1771.89i	−0.133337	−	0.230947i	−0.924961	+	0.380062i	$0.875903\pi$
$390$		0			0
$391$	−1728.00			−0.223501
$392$		0			0
$393$		0			0
$394$		0			0
$395$	208.000	+	360.267i	0.0264952	+	0.0458911i
$396$		0			0
$397$	1558.00	−	2698.54i	0.196962	−	0.341148i	−0.750580	−	0.660779i	$-0.770226\pi$
$397$	1558.00	−	2698.54i	0.196962	−	0.341148i	0.947542	+	0.319632i	$0.103559\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1479.00	−	2561.70i	0.184184	−	0.319016i	−0.759117	−	0.650954i	$-0.774369\pi$
$401$	1479.00	−	2561.70i	0.184184	−	0.319016i	0.943301	+	0.331938i	$0.107702\pi$
$402$		0			0
$403$	368.000	+	637.395i	0.0454873	+	0.0787863i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−600.000			−0.0730735
$408$		0			0
$409$	3972.00	+	6879.71i	0.480202	+	0.831735i	0.999742	−	0.0227114i	$-0.00722990\pi$
$409$	3972.00	+	6879.71i	0.480202	+	0.831735i	−0.519540	+	0.854446i	$0.673897\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	648.000	−	1122.37i	0.0766484	−	0.132759i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4084.00			0.476173			0.238086	−	0.971244i	$-0.423480\pi$
$419$	4084.00			0.476173			0.238086	+	0.971244i	$0.423480\pi$
$420$		0			0
$421$	−6306.00			−0.730013			−0.365007	−	0.931005i	$-0.618933\pi$
$421$	−6306.00			−0.730013			−0.365007	+	0.931005i	$0.618933\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1308.00	−	2265.52i	0.149288	−	0.258574i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5912.00	−	10239.9i	−0.660722	−	1.14440i	−0.980426	−	0.196886i	$-0.936917\pi$
$431$	−5912.00	−	10239.9i	−0.660722	−	1.14440i	0.319705	−	0.947517i	$-0.396416\pi$
$432$		0			0
$433$	4504.00			0.499881			0.249940	−	0.968261i	$-0.419589\pi$
$433$	4504.00			0.499881			0.249940	+	0.968261i	$0.419589\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1584.00	−	2743.57i	−0.173394	−	0.300326i
$438$		0			0
$439$	6528.00	−	11306.8i	0.709714	−	1.22926i	−0.255249	−	0.966875i	$-0.582158\pi$
$439$	6528.00	−	11306.8i	0.709714	−	1.22926i	0.964963	−	0.262385i	$-0.0845092\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	66.0000	−	114.315i	0.00707845	−	0.0122602i	−0.862464	−	0.506118i	$-0.831080\pi$
$443$	66.0000	−	114.315i	0.00707845	−	0.0122602i	0.869543	+	0.493857i	$0.164414\pi$
$444$		0			0
$445$	−1792.00	−	3103.84i	−0.190897	−	0.330642i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4866.00			−0.511449			−0.255725	−	0.966750i	$-0.582314\pi$
$449$	−4866.00			−0.511449			−0.255725	+	0.966750i	$0.582314\pi$
$450$		0			0
$451$	2160.00	+	3741.23i	0.225522	+	0.390616i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5053.00	+	8752.05i	−0.517220	+	0.895851i	0.482580	+	0.875852i	$0.339700\pi$
$457$	−5053.00	+	8752.05i	−0.517220	+	0.895851i	−0.999800	+	0.0199990i	$0.993634\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−18036.0			−1.82217			−0.911085	−	0.412219i	$-0.864754\pi$
$461$	−18036.0			−1.82217			−0.911085	+	0.412219i	$0.864754\pi$
$462$		0			0
$463$	5288.00			0.530787			0.265393	−	0.964140i	$-0.414498\pi$
$463$	5288.00			0.530787			0.265393	+	0.964140i	$0.414498\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7582.00	+	13132.4i	−0.751291	+	1.30128i	0.195906	+	0.980623i	$0.437235\pi$
$467$	−7582.00	+	13132.4i	−0.751291	+	1.30128i	−0.947197	+	0.320652i	$0.896098\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1640.00	+	2840.56i	0.159423	+	0.276129i
$474$		0			0
$475$	4796.00			0.463275
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3948.00	−	6838.14i	−0.376594	−	0.652281i	0.613970	−	0.789329i	$-0.289572\pi$
$479$	−3948.00	−	6838.14i	−0.376594	−	0.652281i	−0.990564	+	0.137049i	$0.956238\pi$
$480$		0			0
$481$	60.0000	−	103.923i	0.00568766	−	0.00985132i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1840.00	+	3186.97i	−0.172268	+	0.298377i
$486$		0			0
$487$	−1460.00	−	2528.79i	−0.135850	−	0.235299i	0.790072	−	0.613014i	$-0.210043\pi$
$487$	−1460.00	−	2528.79i	−0.135850	−	0.235299i	−0.925922	+	0.377715i	$0.876710\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7932.00			0.729055			0.364528	−	0.931193i	$-0.381230\pi$
$491$	7932.00			0.729055			0.364528	+	0.931193i	$0.381230\pi$
$492$		0			0
$493$	−456.000	−	789.815i	−0.0416576	−	0.0721531i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1002.00	−	1735.51i	0.0898911	−	0.155696i	−0.817574	−	0.575824i	$-0.804681\pi$
$499$	1002.00	−	1735.51i	0.0898911	−	0.155696i	0.907465	+	0.420128i	$0.138015\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4496.00			0.398542			0.199271	−	0.979944i	$-0.436143\pi$
$503$	4496.00			0.398542			0.199271	+	0.979944i	$0.436143\pi$
$504$		0			0
$505$	4432.00			0.390537
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6310.00	+	10929.2i	−0.549481	+	0.951729i	0.448829	+	0.893618i	$0.351841\pi$
$509$	−6310.00	+	10929.2i	−0.549481	+	0.951729i	−0.998310	+	0.0581114i	$0.981492\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2896.00	+	5016.02i	0.247792	+	0.429189i
$516$		0			0
$517$	10400.0			0.884703
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9004.00	−	15595.4i	−0.757145	−	1.31141i	−0.944301	−	0.329083i	$-0.893260\pi$
$521$	−9004.00	−	15595.4i	−0.757145	−	1.31141i	0.187156	−	0.982330i	$-0.440073\pi$
$522$		0			0
$523$	6646.00	−	11511.2i	0.555658	−	0.962428i	−0.442194	−	0.896920i	$-0.645800\pi$
$523$	6646.00	−	11511.2i	0.555658	−	0.962428i	0.997852	−	0.0655088i	$-0.0208670\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2208.00	−	3824.37i	0.182509	−	0.316114i
$528$		0			0
$529$	3491.50	+	6047.46i	0.286965	+	0.497038i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−864.000			−0.0702139
$534$		0			0
$535$	2632.00	+	4558.76i	0.212694	+	0.368397i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	4285.00	−	7421.84i	0.340530	−	0.589815i	−0.644002	−	0.765024i	$-0.722727\pi$
$541$	4285.00	−	7421.84i	0.340530	−	0.589815i	0.984531	+	0.175210i	$0.0560603\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−344.000			−0.0270373
$546$		0			0
$547$	−1916.00			−0.149766			−0.0748832	−	0.997192i	$-0.523858\pi$
$547$	−1916.00			−0.149766			−0.0748832	+	0.997192i	$0.523858\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	836.000	−	1447.99i	0.0646367	−	0.111954i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9963.00	+	17256.4i	0.757892	+	1.31271i	0.943924	+	0.330164i	$0.107104\pi$
$557$	9963.00	+	17256.4i	0.757892	+	1.31271i	−0.186032	+	0.982544i	$0.559563\pi$
$558$		0			0
$559$	−656.000			−0.0496348
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2122.00	−	3675.41i	−0.158848	−	0.275133i	0.775605	−	0.631218i	$-0.217445\pi$
$563$	−2122.00	−	3675.41i	−0.158848	−	0.275133i	−0.934454	+	0.356085i	$0.884111\pi$
$564$		0			0
$565$	3556.00	−	6159.17i	0.264782	−	0.458617i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11397.0	+	19740.2i	−0.839696	+	1.45440i	0.0504527	+	0.998726i	$0.483934\pi$
$569$	−11397.0	+	19740.2i	−0.839696	+	1.45440i	−0.890149	+	0.455670i	$0.849400\pi$
$570$		0			0
$571$	−7014.00	−	12148.6i	−0.514057	−	0.890374i	−0.999867	−	0.0163089i	$-0.994809\pi$
$571$	−7014.00	−	12148.6i	−0.514057	−	0.890374i	0.485810	−	0.874065i	$-0.338525\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7848.00			0.569190
$576$		0			0
$577$	4184.00	+	7246.90i	0.301876	+	0.522864i	0.976561	−	0.215242i	$-0.0690540\pi$
$577$	4184.00	+	7246.90i	0.301876	+	0.522864i	−0.674685	+	0.738106i	$0.735721\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1460.00	+	2528.79i	−0.103717	+	0.179643i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	52.0000			0.00365634			0.00182817	−	0.999998i	$-0.499418\pi$
$587$	52.0000			0.00365634			0.00182817	+	0.999998i	$0.499418\pi$
$588$		0			0
$589$	8096.00			0.566366
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2904.00	+	5029.88i	−0.201101	+	0.348317i	−0.948883	−	0.315627i	$-0.897785\pi$
$593$	−2904.00	+	5029.88i	−0.201101	+	0.348317i	0.747782	+	0.663944i	$0.231119\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−5232.00	−	9062.09i	−0.356884	−	0.618142i	0.630554	−	0.776145i	$-0.282828\pi$
$599$	−5232.00	−	9062.09i	−0.356884	−	0.618142i	−0.987439	+	0.158003i	$0.949494\pi$
$600$		0			0
$601$	−1184.00			−0.0803600			−0.0401800	−	0.999192i	$-0.512793\pi$
$601$	−1184.00			−0.0803600			−0.0401800	+	0.999192i	$0.512793\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1862.00	+	3225.08i	0.125126	+	0.216724i
$606$		0			0
$607$	6576.00	−	11390.0i	0.439723	−	0.761622i	−0.557945	−	0.829878i	$-0.688410\pi$
$607$	6576.00	−	11390.0i	0.439723	−	0.761622i	0.997668	+	0.0682559i	$0.0217434\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1040.00	+	1801.33i	−0.0688607	+	0.119270i
$612$		0			0
$613$	9167.00	+	15877.7i	0.603999	+	1.04616i	0.992209	+	0.124586i	$0.0397604\pi$
$613$	9167.00	+	15877.7i	0.603999	+	1.04616i	−0.388209	+	0.921571i	$0.626906\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8122.00			0.529950			0.264975	−	0.964255i	$-0.414636\pi$
$617$	8122.00			0.529950			0.264975	+	0.964255i	$0.414636\pi$
$618$		0			0
$619$	2990.00	+	5178.83i	0.194149	+	0.336276i	0.946621	−	0.322348i	$-0.104472\pi$
$619$	2990.00	+	5178.83i	0.194149	+	0.336276i	−0.752472	+	0.658624i	$0.771139\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−4940.50	+	8557.20i	−0.316192	+	0.547661i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−720.000			−0.0456411
$630$		0			0
$631$	12528.0			0.790383			0.395192	−	0.918599i	$-0.370678\pi$
$631$	12528.0			0.790383			0.395192	+	0.918599i	$0.370678\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1856.00	−	3214.69i	0.115989	−	0.200899i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10399.0	+	18011.6i	0.640773	+	1.10985i	0.985260	+	0.171061i	$0.0547196\pi$
$641$	10399.0	+	18011.6i	0.640773	+	1.10985i	−0.344487	+	0.938791i	$0.611947\pi$
$642$		0			0
$643$	1932.00			0.118492			0.0592462	−	0.998243i	$-0.481130\pi$
$643$	1932.00			0.118492			0.0592462	+	0.998243i	$0.481130\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4212.00	+	7295.40i	0.255936	+	0.443295i	0.965149	−	0.261699i	$-0.0842829\pi$
$647$	4212.00	+	7295.40i	0.255936	+	0.443295i	−0.709213	+	0.704994i	$0.750950\pi$
$648$		0			0
$649$	−4600.00	+	7967.43i	−0.278222	+	0.481894i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8875.00	+	15372.0i	−0.531862	+	0.921211i	0.467447	+	0.884021i	$0.345174\pi$
$653$	−8875.00	+	15372.0i	−0.531862	+	0.921211i	−0.999308	+	0.0371899i	$0.988159\pi$
$654$		0			0
$655$	−2808.00	−	4863.60i	−0.167508	−	0.290132i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	27580.0			1.63029			0.815147	−	0.579254i	$-0.196656\pi$
$659$	27580.0			1.63029			0.815147	+	0.579254i	$0.196656\pi$
$660$		0			0
$661$	−4646.00	−	8047.11i	−0.273386	−	0.473519i	0.696340	−	0.717712i	$-0.254810\pi$
$661$	−4646.00	−	8047.11i	−0.273386	−	0.473519i	−0.969727	+	0.244193i	$0.921477\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1368.00	−	2369.45i	0.0794141	−	0.137549i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12560.0			−0.722613
$672$		0			0
$673$	11486.0			0.657879			0.328940	−	0.944351i	$-0.393309\pi$
$673$	11486.0			0.657879			0.328940	+	0.944351i	$0.393309\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3558.00	+	6162.64i	−0.201987	+	0.349851i	−0.949168	−	0.314768i	$-0.898073\pi$
$677$	−3558.00	+	6162.64i	−0.201987	+	0.349851i	0.747182	+	0.664620i	$0.231406\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3806.00	−	6592.19i	−0.213225	−	0.369316i	0.739497	−	0.673160i	$-0.235063\pi$
$683$	−3806.00	−	6592.19i	−0.213225	−	0.369316i	−0.952722	+	0.303843i	$0.901730\pi$
$684$		0			0
$685$	5480.00			0.305664
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−292.000	−	505.759i	−0.0161456	−	0.0279650i
$690$		0			0
$691$	−10786.0	+	18681.9i	−0.593804	+	1.02850i	0.399910	+	0.916554i	$0.369041\pi$
$691$	−10786.0	+	18681.9i	−0.593804	+	1.02850i	−0.993714	+	0.111945i	$0.964292\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1032.00	−	1787.48i	0.0563252	−	0.0975581i
$696$		0			0
$697$	2592.00	+	4489.48i	0.140859	+	0.243976i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1702.00			0.0917028			0.0458514	−	0.998948i	$-0.485400\pi$
$701$	1702.00			0.0917028			0.0458514	+	0.998948i	$0.485400\pi$
$702$		0			0
$703$	−660.000	−	1143.15i	−0.0354088	−	0.0613298i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3185.00	+	5516.58i	−0.168710	+	0.292214i	−0.937966	−	0.346726i	$-0.887293\pi$
$709$	−3185.00	+	5516.58i	−0.168710	+	0.292214i	0.769257	+	0.638940i	$0.220627\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13248.0			0.695851
$714$		0			0
$715$	320.000			0.0167375
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4404.00	−	7627.95i	0.228430	−	0.395653i	−0.728913	−	0.684607i	$-0.759974\pi$
$719$	4404.00	−	7627.95i	0.228430	−	0.395653i	0.957343	+	0.288954i	$0.0933073\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2071.00	+	3587.08i	0.106090	+	0.183753i
$726$		0			0
$727$	−17768.0			−0.906436			−0.453218	−	0.891400i	$-0.649724\pi$
$727$	−17768.0			−0.906436			−0.453218	+	0.891400i	$0.649724\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1968.00	+	3408.68i	0.0995747	+	0.172468i
$732$		0			0
$733$	−2782.00	+	4818.57i	−0.140185	+	0.242807i	−0.927566	−	0.373659i	$-0.878103\pi$
$733$	−2782.00	+	4818.57i	−0.140185	+	0.242807i	0.787381	+	0.616466i	$0.211436\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5560.00	+	9630.20i	−0.277890	+	0.481320i
$738$		0			0
$739$	8782.00	+	15210.9i	0.437146	+	0.757160i	0.997468	−	0.0711154i	$-0.0226559\pi$
$739$	8782.00	+	15210.9i	0.437146	+	0.757160i	−0.560322	+	0.828275i	$0.689323\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	38280.0			1.89012			0.945059	−	0.326901i	$-0.106004\pi$
$743$	38280.0			1.89012			0.945059	+	0.326901i	$0.106004\pi$
$744$		0			0
$745$	2780.00	+	4815.10i	0.136713	+	0.236794i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−18096.0	+	31343.2i	−0.879271	+	1.52294i	−0.0271284	+	0.999632i	$0.508636\pi$
$751$	−18096.0	+	31343.2i	−0.879271	+	1.52294i	−0.852142	+	0.523310i	$0.824697\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	544.000			0.0262228
$756$		0			0
$757$	−14.0000			−0.000672178			−0.000336089	−	1.00000i	$-0.500107\pi$
$757$	−14.0000			−0.000672178			−0.000336089		1.00000i	$0.500107\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	13252.0	−	22953.1i	0.631254	−	1.09336i	−0.356041	−	0.934470i	$-0.615874\pi$
$761$	13252.0	−	22953.1i	0.631254	−	1.09336i	0.987296	−	0.158894i	$-0.0507930\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−920.000	−	1593.49i	−0.0433107	−	0.0750163i
$768$		0			0
$769$	−40184.0			−1.88436			−0.942180	−	0.335109i	$-0.891227\pi$
$769$	−40184.0			−1.88436			−0.942180	+	0.335109i	$0.891227\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	17670.0	+	30605.3i	0.822181	+	1.42406i	0.904055	+	0.427416i	$0.140576\pi$
$773$	17670.0	+	30605.3i	0.822181	+	1.42406i	−0.0818742	+	0.996643i	$0.526091\pi$
$774$		0			0
$775$	−10028.0	+	17369.0i	−0.464795	+	0.805049i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4752.00	+	8230.71i	−0.218560	+	0.378557i
$780$		0			0
$781$	5920.00	+	10253.7i	0.271235	+	0.469792i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	592.000			0.0269164
$786$		0			0
$787$	−7426.00	−	12862.2i	−0.336351	−	0.582577i	0.647392	−	0.762157i	$-0.275860\pi$
$787$	−7426.00	−	12862.2i	−0.336351	−	0.582577i	−0.983743	+	0.179580i	$0.942526\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1256.00	−	2175.46i	0.0562445	−	0.0974183i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19788.0			0.879457			0.439728	−	0.898131i	$-0.355075\pi$
$797$	19788.0			0.879457			0.439728	+	0.898131i	$0.355075\pi$
$798$		0			0
$799$	12480.0			0.552579
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10240.0	−	17736.2i	0.450015	−	0.779448i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	8493.00	+	14710.3i	0.369095	+	0.639292i	0.989424	−	0.145050i	$-0.0463343\pi$
$809$	8493.00	+	14710.3i	0.369095	+	0.639292i	−0.620329	+	0.784342i	$0.713001\pi$
$810$		0			0
$811$	26596.0			1.15156			0.575778	−	0.817606i	$-0.304699\pi$
$811$	26596.0			1.15156			0.575778	+	0.817606i	$0.304699\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2424.00	+	4198.49i	0.104183	+	0.180450i
$816$		0			0
$817$	−3608.00	+	6249.24i	−0.154502	+	0.267605i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−17449.0	+	30222.6i	−0.741747	+	1.28474i	0.209952	+	0.977712i	$0.432669\pi$
$821$	−17449.0	+	30222.6i	−0.741747	+	1.28474i	−0.951699	+	0.307032i	$0.900664\pi$
$822$		0			0
$823$	−6464.00	−	11196.0i	−0.273780	−	0.474201i	0.696047	−	0.717997i	$-0.254941\pi$
$823$	−6464.00	−	11196.0i	−0.273780	−	0.474201i	−0.969827	+	0.243796i	$0.921607\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	43164.0			1.81494			0.907472	−	0.420112i	$-0.138009\pi$
$827$	43164.0			1.81494			0.907472	+	0.420112i	$0.138009\pi$
$828$		0			0
$829$	−20614.0	−	35704.5i	−0.863635	−	1.49586i	−0.868396	−	0.495872i	$-0.834849\pi$
$829$	−20614.0	−	35704.5i	−0.863635	−	1.49586i	0.00476022	−	0.999989i	$-0.498485\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−3952.00	+	6845.06i	−0.163790	+	0.283692i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1368.00			−0.0562915			−0.0281458	−	0.999604i	$-0.508960\pi$
$839$	−1368.00			−0.0562915			−0.0281458	+	0.999604i	$0.508960\pi$
$840$		0			0
$841$	−22945.0			−0.940793
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4362.00	−	7555.21i	0.177583	−	0.307582i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1080.00	−	1870.61i	−0.0435040	−	0.0753512i
$852$		0			0
$853$	−5276.00			−0.211778			−0.105889	−	0.994378i	$-0.533769\pi$
$853$	−5276.00			−0.211778			−0.105889	+	0.994378i	$0.533769\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−420.000	−	727.461i	−0.0167409	−	0.0289960i	0.857534	−	0.514428i	$-0.171996\pi$
$857$	−420.000	−	727.461i	−0.0167409	−	0.0289960i	−0.874274	+	0.485432i	$0.838662\pi$
$858$		0			0
$859$	−13014.0	+	22540.9i	−0.516917	+	0.895327i	0.482890	+	0.875681i	$0.339587\pi$
$859$	−13014.0	+	22540.9i	−0.516917	+	0.895327i	−0.999807	+	0.0196458i	$0.993746\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14724.0	+	25502.7i	−0.580777	+	1.00594i	0.414610	+	0.909999i	$0.363918\pi$
$863$	−14724.0	+	25502.7i	−0.580777	+	1.00594i	−0.995387	+	0.0959366i	$0.969415\pi$
$864$		0			0
$865$	5384.00	+	9325.36i	0.211632	+	0.366557i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2080.00			−0.0811958
$870$		0			0
$871$	−1112.00	−	1926.04i	−0.0432591	−	0.0749270i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−12933.0	+	22400.6i	−0.497966	+	0.862503i	−0.999997	−	0.00234681i	$-0.999253\pi$
$877$	−12933.0	+	22400.6i	−0.497966	+	0.862503i	0.502031	+	0.864850i	$0.332586\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9472.00			0.362225			0.181112	−	0.983462i	$-0.442030\pi$
$881$	9472.00			0.362225			0.181112	+	0.983462i	$0.442030\pi$
$882$		0			0
$883$	49372.0			1.88165			0.940827	−	0.338888i	$-0.110051\pi$
$883$	49372.0			1.88165			0.940827	+	0.338888i	$0.110051\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5580.00	−	9664.84i	0.211227	−	0.365855i	−0.740872	−	0.671646i	$-0.765588\pi$
$887$	5580.00	−	9664.84i	0.211227	−	0.365855i	0.952099	+	0.305791i	$0.0989208\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	11440.0	+	19814.7i	0.428695	+	0.742522i
$894$		0			0
$895$	10320.0			0.385430
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3496.00	+	6055.25i	0.129698	+	0.224643i
$900$		0			0
$901$	−1752.00	+	3034.55i	−0.0647809	+	0.112204i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4072.00	+	7052.91i	−0.149567	+	0.259057i
$906$		0			0
$907$	−11354.0	−	19665.7i	−0.415660	−	0.719944i	0.579838	−	0.814732i	$-0.303116\pi$
$907$	−11354.0	−	19665.7i	−0.415660	−	0.719944i	−0.995497	+	0.0947882i	$0.969783\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	16192.0			0.588875			0.294437	−	0.955671i	$-0.404868\pi$
$911$	16192.0			0.588875			0.294437	+	0.955671i	$0.404868\pi$
$912$		0			0
$913$	3240.00	+	5611.84i	0.117446	+	0.203423i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	23160.0	−	40114.3i	0.831314	−	1.43988i	−0.0656819	−	0.997841i	$-0.520922\pi$
$919$	23160.0	−	40114.3i	0.831314	−	1.43988i	0.896996	−	0.442038i	$-0.145744\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2368.00			−0.0844460
$924$		0			0
$925$	3270.00			0.116235
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1140.00	−	1974.54i	0.0402607	−	0.0697336i	−0.845193	−	0.534461i	$-0.820514\pi$
$929$	1140.00	−	1974.54i	0.0402607	−	0.0697336i	0.885454	+	0.464728i	$0.153848\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−960.000	−	1662.77i	−0.0335779	−	0.0581587i
$936$		0			0
$937$	−49056.0			−1.71034			−0.855171	−	0.518347i	$-0.826548\pi$
$937$	−49056.0			−1.71034			−0.855171	+	0.518347i	$0.826548\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12438.0	+	21543.2i	0.430890	+	0.746323i	0.996950	−	0.0780409i	$-0.0248665\pi$
$941$	12438.0	+	21543.2i	0.430890	+	0.746323i	−0.566060	+	0.824364i	$0.691533\pi$
$942$		0			0
$943$	−7776.00	+	13468.4i	−0.268527	+	0.465103i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11714.0	−	20289.2i	0.401958	−	0.696211i	−0.592005	−	0.805935i	$-0.701663\pi$
$947$	11714.0	−	20289.2i	0.401958	−	0.696211i	0.993962	+	0.109724i	$0.0349966\pi$
$948$		0			0
$949$	2048.00	+	3547.24i	0.0700536	+	0.121336i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6678.00			0.226990			0.113495	−	0.993539i	$-0.463795\pi$
$953$	6678.00			0.226990			0.113495	+	0.993539i	$0.463795\pi$
$954$		0			0
$955$	7920.00	+	13717.8i	0.268361	+	0.464816i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2032.50	+	3520.39i	−0.0682253	+	0.118170i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.00000			0.000266870
$966$		0			0
$967$	15544.0			0.516920			0.258460	−	0.966022i	$-0.416785\pi$
$967$	15544.0			0.516920			0.258460	+	0.966022i	$0.416785\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	15562.0	−	26954.2i	0.514324	−	0.890835i	−0.485538	−	0.874215i	$-0.661376\pi$
$971$	15562.0	−	26954.2i	0.514324	−	0.890835i	0.999862	−	0.0166194i	$-0.00529035\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25031.0	+	43355.0i	0.819665	+	1.41970i	0.905929	+	0.423429i	$0.139174\pi$
$977$	25031.0	+	43355.0i	0.819665	+	1.41970i	−0.0862643	+	0.996272i	$0.527493\pi$
$978$		0			0
$979$	17920.0			0.585011
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−164.000	−	284.056i	−0.00532125	−	0.00921667i	0.863353	−	0.504601i	$-0.168360\pi$
$983$	−164.000	−	284.056i	−0.00532125	−	0.00921667i	−0.868674	+	0.495385i	$0.835027\pi$
$984$		0			0
$985$	7548.00	−	13073.5i	0.244162	−	0.422900i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−5904.00	+	10226.0i	−0.189824	+	0.328785i
$990$		0			0
$991$	10436.0	+	18075.7i	0.334521	+	0.579408i	0.983393	−	0.181490i	$-0.0580921\pi$
$991$	10436.0	+	18075.7i	0.334521	+	0.579408i	−0.648872	+	0.760898i	$0.724759\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	14240.0			0.453707
$996$		0			0
$997$	−23462.0	−	40637.4i	−0.745285	−	1.29087i	−0.950062	−	0.312063i	$-0.898980\pi$
$997$	−23462.0	−	40637.4i	−0.745285	−	1.29087i	0.204777	−	0.978809i	$-0.434353\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.g.1549.1		2
3.2	odd	2		588.4.i.b.373.1		2
7.2	even	3		1764.4.a.i.1.1		1
7.3	odd	6		1764.4.k.j.361.1		2
7.4	even	3	inner	1764.4.k.g.361.1		2
7.5	odd	6		1764.4.a.d.1.1		1
7.6	odd	2		1764.4.k.j.1549.1		2
21.2	odd	6		588.4.a.e.1.1	yes	1
21.5	even	6		588.4.a.b.1.1	✓	1
21.11	odd	6		588.4.i.b.361.1		2
21.17	even	6		588.4.i.g.361.1		2
21.20	even	2		588.4.i.g.373.1		2
84.23	even	6		2352.4.a.g.1.1		1
84.47	odd	6		2352.4.a.be.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.4.a.b.1.1	✓	1	21.5	even	6
588.4.a.e.1.1	yes	1	21.2	odd	6
588.4.i.b.361.1		2	21.11	odd	6
588.4.i.b.373.1		2	3.2	odd	2
588.4.i.g.361.1		2	21.17	even	6
588.4.i.g.373.1		2	21.20	even	2
1764.4.a.d.1.1		1	7.5	odd	6
1764.4.a.i.1.1		1	7.2	even	3
1764.4.k.g.361.1		2	7.4	even	3	inner
1764.4.k.g.1549.1		2	1.1	even	1	trivial
1764.4.k.j.361.1		2	7.3	odd	6
1764.4.k.j.1549.1		2	7.6	odd	2
2352.4.a.g.1.1		1	84.23	even	6
2352.4.a.be.1.1		1	84.47	odd	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 + 3.46410i) q^{5} +O(q^{10})\)	\(q+(-2.00000 + 3.46410i) q^{5} +(-10.0000 - 17.3205i) q^{11} +4.00000 q^{13} +(-12.0000 - 20.7846i) q^{17} +(22.0000 - 38.1051i) q^{19} +(36.0000 - 62.3538i) q^{23} +(54.5000 + 94.3968i) q^{25} +38.0000 q^{29} +(92.0000 + 159.349i) q^{31} +(15.0000 - 25.9808i) q^{37} -216.000 q^{41} -164.000 q^{43} +(-260.000 + 450.333i) q^{47} +(-73.0000 - 126.440i) q^{53} +80.0000 q^{55} +(-230.000 - 398.372i) q^{59} +(314.000 - 543.864i) q^{61} +(-8.00000 + 13.8564i) q^{65} +(-278.000 - 481.510i) q^{67} -592.000 q^{71} +(512.000 + 886.810i) q^{73} +(52.0000 - 90.0666i) q^{79} -324.000 q^{83} +96.0000 q^{85} +(-448.000 + 775.959i) q^{89} +(88.0000 + 152.420i) q^{95} +920.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} - 20 q^{11} + 8 q^{13} - 24 q^{17} + 44 q^{19} + 72 q^{23} + 109 q^{25} + 76 q^{29} + 184 q^{31} + 30 q^{37} - 432 q^{41} - 328 q^{43} - 520 q^{47} - 146 q^{53} + 160 q^{55} - 460 q^{59} + 628 q^{61} - 16 q^{65} - 556 q^{67} - 1184 q^{71} + 1024 q^{73} + 104 q^{79} - 648 q^{83} + 192 q^{85} - 896 q^{89} + 176 q^{95} + 1840 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 20 * q^11 + 8 * q^13 - 24 * q^17 + 44 * q^19 + 72 * q^23 + 109 * q^25 + 76 * q^29 + 184 * q^31 + 30 * q^37 - 432 * q^41 - 328 * q^43 - 520 * q^47 - 146 * q^53 + 160 * q^55 - 460 * q^59 + 628 * q^61 - 16 * q^65 - 556 * q^67 - 1184 * q^71 + 1024 * q^73 + 104 * q^79 - 648 * q^83 + 192 * q^85 - 896 * q^89 + 176 * q^95 + 1840 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more