LMFDB - Embedded newform 1764.4.k.c.361.1

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 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.4.k.c.1549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.00000 - 12.1244i) q^{5} +O(q^{10})$	$q+(-7.00000 - 12.1244i) q^{5} +(2.00000 - 3.46410i) q^{11} -54.0000 q^{13} +(7.00000 - 12.1244i) q^{17} +(46.0000 + 79.6743i) q^{19} +(-76.0000 - 131.636i) q^{23} +(-35.5000 + 61.4878i) q^{25} +106.000 q^{29} +(-72.0000 + 124.708i) q^{31} +(-79.0000 - 136.832i) q^{37} -390.000 q^{41} -508.000 q^{43} +(264.000 + 457.261i) q^{47} +(303.000 - 524.811i) q^{53} -56.0000 q^{55} +(182.000 - 315.233i) q^{59} +(339.000 + 587.165i) q^{61} +(378.000 + 654.715i) q^{65} +(-422.000 + 730.925i) q^{67} +8.00000 q^{71} +(-211.000 + 365.463i) q^{73} +(-192.000 - 332.554i) q^{79} -548.000 q^{83} -196.000 q^{85} +(-597.000 - 1034.03i) q^{89} +(644.000 - 1115.44i) q^{95} +1502.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{5}+O(q^{10}) $ 2 * q - 14 * q^5	$ 2 q - 14 q^{5} + 4 q^{11} - 108 q^{13} + 14 q^{17} + 92 q^{19} - 152 q^{23} - 71 q^{25} + 212 q^{29} - 144 q^{31} - 158 q^{37} - 780 q^{41} - 1016 q^{43} + 528 q^{47} + 606 q^{53} - 112 q^{55} + 364 q^{59} + 678 q^{61} + 756 q^{65} - 844 q^{67} + 16 q^{71} - 422 q^{73} - 384 q^{79} - 1096 q^{83} - 392 q^{85} - 1194 q^{89} + 1288 q^{95} + 3004 q^{97}+O(q^{100}) $ 2 * q - 14 * q^5 + 4 * q^11 - 108 * q^13 + 14 * q^17 + 92 * q^19 - 152 * q^23 - 71 * q^25 + 212 * q^29 - 144 * q^31 - 158 * q^37 - 780 * q^41 - 1016 * q^43 + 528 * q^47 + 606 * q^53 - 112 * q^55 + 364 * q^59 + 678 * q^61 + 756 * q^65 - 844 * q^67 + 16 * q^71 - 422 * q^73 - 384 * q^79 - 1096 * q^83 - 392 * q^85 - 1194 * q^89 + 1288 * q^95 + 3004 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−7.00000	−	12.1244i	−0.626099	−	1.08444i	−0.988327	−	0.152346i	$-0.951317\pi$
$5$	−7.00000	−	12.1244i	−0.626099	−	1.08444i	0.362228	−	0.932089i	$-0.382016\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000	−	3.46410i	0.0548202	−	0.0949514i	−0.837313	−	0.546724i	$-0.815875\pi$
$11$	2.00000	−	3.46410i	0.0548202	−	0.0949514i	0.892133	+	0.451772i	$0.149208\pi$
$12$		0			0
$13$	−54.0000			−1.15207			−0.576035	−	0.817425i	$-0.695401\pi$
$13$	−54.0000			−1.15207			−0.576035	+	0.817425i	$0.695401\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.00000	−	12.1244i	0.0998676	−	0.172976i	−0.811762	−	0.583988i	$-0.801491\pi$
$17$	7.00000	−	12.1244i	0.0998676	−	0.172976i	0.911630	+	0.411012i	$0.134825\pi$
$18$		0			0
$19$	46.0000	+	79.6743i	0.555428	+	0.962029i	0.997870	+	0.0652319i	$0.0207787\pi$
$19$	46.0000	+	79.6743i	0.555428	+	0.962029i	−0.442443	+	0.896797i	$0.645888\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−76.0000	−	131.636i	−0.689004	−	1.19339i	−0.972160	−	0.234316i	$-0.924715\pi$
$23$	−76.0000	−	131.636i	−0.689004	−	1.19339i	0.283156	−	0.959074i	$-0.408619\pi$
$24$		0			0
$25$	−35.5000	+	61.4878i	−0.284000	+	0.491902i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	106.000			0.678748			0.339374	−	0.940651i	$-0.389785\pi$
$29$	106.000			0.678748			0.339374	+	0.940651i	$0.389785\pi$
$30$		0			0
$31$	−72.0000	+	124.708i	−0.417148	+	0.722521i	−0.995651	−	0.0931587i	$-0.970304\pi$
$31$	−72.0000	+	124.708i	−0.417148	+	0.722521i	0.578503	+	0.815680i	$0.303637\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−79.0000	−	136.832i	−0.351014	−	0.607974i	0.635413	−	0.772172i	$-0.280830\pi$
$37$	−79.0000	−	136.832i	−0.351014	−	0.607974i	−0.986427	+	0.164198i	$0.947496\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−390.000			−1.48556			−0.742778	−	0.669538i	$-0.766492\pi$
$41$	−390.000			−1.48556			−0.742778	+	0.669538i	$0.766492\pi$
$42$		0			0
$43$	−508.000			−1.80161			−0.900806	−	0.434223i	$-0.857023\pi$
$43$	−508.000			−1.80161			−0.900806	+	0.434223i	$0.857023\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	264.000	+	457.261i	0.819327	+	1.41912i	0.906179	+	0.422894i	$0.138986\pi$
$47$	264.000	+	457.261i	0.819327	+	1.41912i	−0.0868522	+	0.996221i	$0.527681\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	303.000	−	524.811i	0.785288	−	1.36016i	−0.143539	−	0.989645i	$-0.545848\pi$
$53$	303.000	−	524.811i	0.785288	−	1.36016i	0.928827	−	0.370514i	$-0.120818\pi$
$54$		0			0
$55$	−56.0000			−0.137292
$56$		0			0
$57$		0			0
$58$		0			0
$59$	182.000	−	315.233i	0.401600	−	0.695591i	−0.592319	−	0.805703i	$-0.701788\pi$
$59$	182.000	−	315.233i	0.401600	−	0.695591i	0.993919	+	0.110112i	$0.0351210\pi$
$60$		0			0
$61$	339.000	+	587.165i	0.711549	+	1.23244i	0.964275	+	0.264902i	$0.0853395\pi$
$61$	339.000	+	587.165i	0.711549	+	1.23244i	−0.252726	+	0.967538i	$0.581327\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	378.000	+	654.715i	0.721310	+	1.24935i
$66$		0			0
$67$	−422.000	+	730.925i	−0.769485	+	1.33279i	0.168357	+	0.985726i	$0.446154\pi$
$67$	−422.000	+	730.925i	−0.769485	+	1.33279i	−0.937842	+	0.347061i	$0.887180\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	8.00000			0.0133722			0.00668609	−	0.999978i	$-0.497872\pi$
$71$	8.00000			0.0133722			0.00668609	+	0.999978i	$0.497872\pi$
$72$		0			0
$73$	−211.000	+	365.463i	−0.338297	+	0.585948i	−0.984113	−	0.177546i	$-0.943184\pi$
$73$	−211.000	+	365.463i	−0.338297	+	0.585948i	0.645816	+	0.763494i	$0.276518\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−192.000	−	332.554i	−0.273439	−	0.473610i	0.696301	−	0.717750i	$-0.254828\pi$
$79$	−192.000	−	332.554i	−0.273439	−	0.473610i	−0.969740	+	0.244139i	$0.921495\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−548.000			−0.724709			−0.362354	−	0.932040i	$-0.618027\pi$
$83$	−548.000			−0.724709			−0.362354	+	0.932040i	$0.618027\pi$
$84$		0			0
$85$	−196.000			−0.250108
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−597.000	−	1034.03i	−0.711032	−	1.23154i	−0.964470	−	0.264192i	$-0.914895\pi$
$89$	−597.000	−	1034.03i	−0.711032	−	1.23154i	0.253438	−	0.967352i	$-0.418439\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	644.000	−	1115.44i	0.695505	−	1.20465i
$96$		0			0
$97$	1502.00			1.57222			0.786108	−	0.618089i	$-0.212093\pi$
$97$	1502.00			1.57222			0.786108	+	0.618089i	$0.212093\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−199.000	+	344.678i	−0.196052	+	0.339572i	−0.947245	−	0.320511i	$-0.896145\pi$
$101$	−199.000	+	344.678i	−0.196052	+	0.339572i	0.751193	+	0.660083i	$0.229479\pi$
$102$		0			0
$103$	580.000	+	1004.59i	0.554846	+	0.961021i	0.997916	+	0.0645337i	$0.0205560\pi$
$103$	580.000	+	1004.59i	0.554846	+	0.961021i	−0.443070	+	0.896487i	$0.646111\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	162.000	+	280.592i	0.146366	+	0.253513i	0.929882	−	0.367859i	$-0.119909\pi$
$107$	162.000	+	280.592i	0.146366	+	0.253513i	−0.783516	+	0.621372i	$0.786576\pi$
$108$		0			0
$109$	469.000	−	812.332i	0.412129	−	0.713828i	−0.582993	−	0.812477i	$-0.698119\pi$
$109$	469.000	−	812.332i	0.412129	−	0.713828i	0.995122	+	0.0986487i	$0.0314520\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	622.000			0.517813			0.258906	−	0.965902i	$-0.416638\pi$
$113$	622.000			0.517813			0.258906	+	0.965902i	$0.416638\pi$
$114$		0			0
$115$	−1064.00	+	1842.90i	−0.862770	+	1.49436i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	657.500	+	1138.82i	0.493989	+	0.855615i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−756.000			−0.540950
$126$		0			0
$127$	1200.00			0.838447			0.419224	−	0.907883i	$-0.362302\pi$
$127$	1200.00			0.838447			0.419224	+	0.907883i	$0.362302\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	698.000	+	1208.97i	0.465531	+	0.806323i	0.999225	−	0.0393543i	$-0.0125301\pi$
$131$	698.000	+	1208.97i	0.465531	+	0.806323i	−0.533694	+	0.845677i	$0.679197\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1405.00	−	2433.53i	0.876184	−	1.51760i	0.0206885	−	0.999786i	$-0.493414\pi$
$137$	1405.00	−	2433.53i	0.876184	−	1.51760i	0.855496	−	0.517810i	$-0.173253\pi$
$138$		0			0
$139$	−4.00000			−0.00244083			−0.00122042	−	0.999999i	$-0.500388\pi$
$139$	−4.00000			−0.00244083			−0.00122042	+	0.999999i	$0.500388\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−108.000	+	187.061i	−0.0631567	+	0.109391i
$144$		0			0
$145$	−742.000	−	1285.18i	−0.424964	−	0.736059i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	687.000	+	1189.92i	0.377726	+	0.654241i	0.990731	−	0.135838i	$-0.0433727\pi$
$149$	687.000	+	1189.92i	0.377726	+	0.654241i	−0.613005	+	0.790079i	$0.710039\pi$
$150$		0			0
$151$	−1052.00	+	1822.12i	−0.566957	+	0.981999i	0.429907	+	0.902873i	$0.358546\pi$
$151$	−1052.00	+	1822.12i	−0.566957	+	0.981999i	−0.996865	+	0.0791258i	$0.974787\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2016.00			1.04470
$156$		0			0
$157$	1603.00	−	2776.48i	0.814862	−	1.41138i	−0.0945650	−	0.995519i	$-0.530146\pi$
$157$	1603.00	−	2776.48i	0.814862	−	1.41138i	0.909427	−	0.415864i	$-0.136521\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−166.000	−	287.520i	−0.0797676	−	0.138162i	0.823382	−	0.567488i	$-0.192084\pi$
$163$	−166.000	−	287.520i	−0.0797676	−	0.138162i	−0.903150	+	0.429326i	$0.858751\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1496.00			0.693197			0.346599	−	0.938014i	$-0.387337\pi$
$167$	1496.00			0.693197			0.346599	+	0.938014i	$0.387337\pi$
$168$		0			0
$169$	719.000			0.327264
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1661.00	+	2876.94i	0.729962	+	1.26433i	0.956899	+	0.290422i	$0.0937958\pi$
$173$	1661.00	+	2876.94i	0.729962	+	1.26433i	−0.226936	+	0.973910i	$0.572871\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−450.000	+	779.423i	−0.187903	+	0.325457i	−0.944551	−	0.328365i	$-0.893502\pi$
$179$	−450.000	+	779.423i	−0.187903	+	0.325457i	0.756648	+	0.653822i	$0.226836\pi$
$180$		0			0
$181$	−1902.00			−0.781075			−0.390537	−	0.920587i	$-0.627711\pi$
$181$	−1902.00			−0.781075			−0.390537	+	0.920587i	$0.627711\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1106.00	+	1915.65i	−0.439539	+	0.761304i
$186$		0			0
$187$	−28.0000	−	48.4974i	−0.0109495	−	0.0189651i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2064.00	+	3574.95i	0.781915	+	1.35432i	0.930825	+	0.365466i	$0.119090\pi$
$191$	2064.00	+	3574.95i	0.781915	+	1.35432i	−0.148910	+	0.988851i	$0.547576\pi$
$192$		0			0
$193$	671.000	−	1162.21i	0.250257	−	0.433458i	−0.713339	−	0.700819i	$-0.752818\pi$
$193$	671.000	−	1162.21i	0.250257	−	0.433458i	0.963597	+	0.267361i	$0.0861515\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3506.00			1.26798			0.633990	−	0.773341i	$-0.281416\pi$
$197$	3506.00			1.26798			0.633990	+	0.773341i	$0.281416\pi$
$198$		0			0
$199$	340.000	−	588.897i	0.121115	−	0.209778i	−0.799092	−	0.601208i	$-0.794686\pi$
$199$	340.000	−	588.897i	0.121115	−	0.209778i	0.920208	+	0.391430i	$0.128020\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2730.00	+	4728.50i	0.930105	+	1.61099i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	368.000			0.121795
$210$		0			0
$211$	5372.00			1.75272			0.876360	−	0.481657i	$-0.159965\pi$
$211$	5372.00			1.75272			0.876360	+	0.481657i	$0.159965\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3556.00	+	6159.17i	1.12799	+	1.95373i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−378.000	+	654.715i	−0.115054	+	0.199280i
$222$		0			0
$223$	1072.00			0.321912			0.160956	−	0.986962i	$-0.448542\pi$
$223$	1072.00			0.321912			0.160956	+	0.986962i	$0.448542\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1434.00	−	2483.76i	0.419286	−	0.726225i	−0.576582	−	0.817039i	$-0.695614\pi$
$227$	1434.00	−	2483.76i	0.419286	−	0.726225i	0.995868	+	0.0908148i	$0.0289471\pi$
$228$		0			0
$229$	2399.00	+	4155.19i	0.692272	+	1.19905i	0.971092	+	0.238707i	$0.0767237\pi$
$229$	2399.00	+	4155.19i	0.692272	+	1.19905i	−0.278819	+	0.960344i	$0.589943\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2563.00	−	4439.25i	−0.720634	−	1.24817i	−0.960746	−	0.277429i	$-0.910518\pi$
$233$	−2563.00	−	4439.25i	−0.720634	−	1.24817i	0.240112	−	0.970745i	$-0.422816\pi$
$234$		0			0
$235$	3696.00	−	6401.66i	1.02596	−	1.77701i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	528.000			0.142902			0.0714508	−	0.997444i	$-0.477237\pi$
$239$	528.000			0.142902			0.0714508	+	0.997444i	$0.477237\pi$
$240$		0			0
$241$	−407.000	+	704.945i	−0.108785	+	0.188421i	−0.915278	−	0.402822i	$-0.868029\pi$
$241$	−407.000	+	704.945i	−0.108785	+	0.188421i	0.806493	+	0.591243i	$0.201363\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2484.00	−	4302.41i	−0.639891	−	1.10832i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1932.00			−0.485844			−0.242922	−	0.970046i	$-0.578106\pi$
$251$	−1932.00			−0.485844			−0.242922	+	0.970046i	$0.578106\pi$
$252$		0			0
$253$	−608.000			−0.151086
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1647.00	+	2852.69i	0.399755	+	0.692396i	0.993695	−	0.112113i	$-0.0357619\pi$
$257$	1647.00	+	2852.69i	0.399755	+	0.692396i	−0.593940	+	0.804509i	$0.702429\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3540.00	+	6131.46i	−0.829984	+	1.43757i	0.0680662	+	0.997681i	$0.478317\pi$
$263$	−3540.00	+	6131.46i	−0.829984	+	1.43757i	−0.898050	+	0.439893i	$0.855016\pi$
$264$		0			0
$265$	−8484.00			−1.96667
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3907.00	+	6767.12i	−0.885554	+	1.53382i	−0.0404764	+	0.999180i	$0.512888\pi$
$269$	−3907.00	+	6767.12i	−0.885554	+	1.53382i	−0.845078	+	0.534644i	$0.820446\pi$
$270$		0			0
$271$	1584.00	+	2743.57i	0.355060	+	0.614981i	0.987128	−	0.159931i	$-0.0511272\pi$
$271$	1584.00	+	2743.57i	0.355060	+	0.614981i	−0.632069	+	0.774912i	$0.717794\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	142.000	+	245.951i	0.0311379	+	0.0539324i
$276$		0			0
$277$	3929.00	−	6805.23i	0.852241	−	1.47612i	−0.0269403	−	0.999637i	$-0.508576\pi$
$277$	3929.00	−	6805.23i	0.852241	−	1.47612i	0.879181	−	0.476488i	$-0.158090\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6730.00			−1.42875			−0.714374	−	0.699764i	$-0.753288\pi$
$281$	−6730.00			−1.42875			−0.714374	+	0.699764i	$0.753288\pi$
$282$		0			0
$283$	−1510.00	+	2615.40i	−0.317174	+	0.549361i	−0.979897	−	0.199503i	$-0.936067\pi$
$283$	−1510.00	+	2615.40i	−0.317174	+	0.549361i	0.662723	+	0.748864i	$0.269401\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2358.50	+	4085.04i	0.480053	+	0.831476i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6834.00			−1.36262			−0.681308	−	0.731997i	$-0.738589\pi$
$293$	−6834.00			−1.36262			−0.681308	+	0.731997i	$0.738589\pi$
$294$		0			0
$295$	−5096.00			−1.00576
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4104.00	+	7108.34i	0.793781	+	1.37487i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4746.00	−	8220.31i	0.891001	−	1.54326i
$306$		0			0
$307$	−2332.00			−0.433532			−0.216766	−	0.976224i	$-0.569551\pi$
$307$	−2332.00			−0.433532			−0.216766	+	0.976224i	$0.569551\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4420.00	+	7655.66i	−0.805901	+	1.39586i	0.109780	+	0.993956i	$0.464985\pi$
$311$	−4420.00	+	7655.66i	−0.805901	+	1.39586i	−0.915681	+	0.401906i	$0.868348\pi$
$312$		0			0
$313$	−523.000	−	905.863i	−0.0944464	−	0.163586i	0.814931	−	0.579558i	$-0.196775\pi$
$313$	−523.000	−	905.863i	−0.0944464	−	0.163586i	−0.909377	+	0.415972i	$0.863441\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3771.00	+	6531.56i	0.668140	+	1.15725i	0.978424	+	0.206609i	$0.0662427\pi$
$317$	3771.00	+	6531.56i	0.668140	+	1.15725i	−0.310283	+	0.950644i	$0.600424\pi$
$318$		0			0
$319$	212.000	−	367.195i	0.0372092	−	0.0644482i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1288.00			0.221877
$324$		0			0
$325$	1917.00	−	3320.34i	0.327188	−	0.566706i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1378.00	−	2386.77i	−0.228827	−	0.396340i	0.728634	−	0.684904i	$-0.240156\pi$
$331$	−1378.00	−	2386.77i	−0.228827	−	0.396340i	−0.957461	+	0.288564i	$0.906822\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11816.0			1.92710
$336$		0			0
$337$	3954.00			0.639134			0.319567	−	0.947564i	$-0.396463\pi$
$337$	3954.00			0.639134			0.319567	+	0.947564i	$0.396463\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	288.000	+	498.831i	0.0457363	+	0.0792176i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3450.00	−	5975.58i	0.533734	−	0.924454i	−0.465489	−	0.885053i	$-0.654122\pi$
$347$	3450.00	−	5975.58i	0.533734	−	0.924454i	0.999223	−	0.0394010i	$-0.0125450\pi$
$348$		0			0
$349$	2426.00			0.372094			0.186047	−	0.982541i	$-0.440432\pi$
$349$	2426.00			0.372094			0.186047	+	0.982541i	$0.440432\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	735.000	−	1273.06i	0.110822	−	0.191949i	−0.805280	−	0.592895i	$-0.797985\pi$
$353$	735.000	−	1273.06i	0.110822	−	0.191949i	0.916102	+	0.400946i	$0.131318\pi$
$354$		0			0
$355$	−56.0000	−	96.9948i	−0.00837231	−	0.0145013i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3436.00	+	5951.33i	0.505140	+	0.874928i	0.999982	+	0.00594499i	$0.00189236\pi$
$359$	3436.00	+	5951.33i	0.505140	+	0.874928i	−0.494843	+	0.868983i	$0.664774\pi$
$360$		0			0
$361$	−802.500	+	1389.97i	−0.117000	+	0.202649i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5908.00			0.847230
$366$		0			0
$367$	−3536.00	+	6124.53i	−0.502937	+	0.871112i	0.497058	+	0.867717i	$0.334414\pi$
$367$	−3536.00	+	6124.53i	−0.502937	+	0.871112i	−0.999994	+	0.00339411i	$0.998920\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	409.000	+	708.409i	0.0567754	+	0.0983378i	0.893016	−	0.450025i	$-0.148585\pi$
$373$	409.000	+	708.409i	0.0567754	+	0.0983378i	−0.836241	+	0.548362i	$0.815251\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5724.00			−0.781966
$378$		0			0
$379$	−5132.00			−0.695549			−0.347775	−	0.937578i	$-0.613063\pi$
$379$	−5132.00			−0.695549			−0.347775	+	0.937578i	$0.613063\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4288.00	+	7427.03i	0.572080	+	0.990871i	0.996352	+	0.0853367i	$0.0271966\pi$
$383$	4288.00	+	7427.03i	0.572080	+	0.990871i	−0.424272	+	0.905535i	$0.639470\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1865.00	+	3230.27i	−0.243083	+	0.421032i	−0.961591	−	0.274487i	$-0.911492\pi$
$389$	−1865.00	+	3230.27i	−0.243083	+	0.421032i	0.718508	+	0.695519i	$0.244825\pi$
$390$		0			0
$391$	−2128.00			−0.275237
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2688.00	+	4655.75i	−0.342400	+	0.593054i
$396$		0			0
$397$	3339.00	+	5783.32i	0.422115	+	0.731124i	0.996146	−	0.0877090i	$-0.0279546\pi$
$397$	3339.00	+	5783.32i	0.422115	+	0.731124i	−0.574031	+	0.818833i	$0.694621\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1527.00	−	2644.84i	−0.190161	−	0.329369i	0.755142	−	0.655561i	$-0.227568\pi$
$401$	−1527.00	−	2644.84i	−0.190161	−	0.329369i	−0.945304	+	0.326192i	$0.894234\pi$
$402$		0			0
$403$	3888.00	−	6734.21i	0.480583	−	0.832395i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−632.000			−0.0769707
$408$		0			0
$409$	133.000	−	230.363i	0.0160793	−	0.0278501i	−0.857874	−	0.513860i	$-0.828215\pi$
$409$	133.000	−	230.363i	0.0160793	−	0.0278501i	0.873953	+	0.486010i	$0.161548\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	3836.00	+	6644.15i	0.453739	+	0.785900i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8844.00			1.03116			0.515582	−	0.856840i	$-0.327576\pi$
$419$	8844.00			1.03116			0.515582	+	0.856840i	$0.327576\pi$
$420$		0			0
$421$	−4482.00			−0.518858			−0.259429	−	0.965762i	$-0.583534\pi$
$421$	−4482.00			−0.518858			−0.259429	+	0.965762i	$0.583534\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	497.000	+	860.829i	0.0567248	+	0.0982502i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4968.00	−	8604.83i	0.555221	−	0.961671i	−0.442666	−	0.896687i	$-0.645967\pi$
$431$	4968.00	−	8604.83i	0.555221	−	0.961671i	0.997886	−	0.0649838i	$-0.0206996\pi$
$432$		0			0
$433$	11758.0			1.30497			0.652487	−	0.757800i	$-0.273726\pi$
$433$	11758.0			1.30497			0.652487	+	0.757800i	$0.273726\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6992.00	−	12110.5i	0.765384	−	1.32568i
$438$		0			0
$439$	−2052.00	−	3554.17i	−0.223090	−	0.386404i	0.732655	−	0.680601i	$-0.238281\pi$
$439$	−2052.00	−	3554.17i	−0.223090	−	0.386404i	−0.955745	+	0.294197i	$0.904948\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4874.00	+	8442.02i	0.522733	+	0.905400i	0.999650	+	0.0264519i	$0.00842087\pi$
$443$	4874.00	+	8442.02i	0.522733	+	0.905400i	−0.476917	+	0.878948i	$0.658246\pi$
$444$		0			0
$445$	−8358.00	+	14476.5i	−0.890353	+	1.54214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	478.000			0.0502410			0.0251205	−	0.999684i	$-0.492003\pi$
$449$	478.000			0.0502410			0.0251205	+	0.999684i	$0.492003\pi$
$450$		0			0
$451$	−780.000	+	1351.00i	−0.0814385	+	0.141056i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5587.00	+	9676.97i	0.571879	+	0.990524i	0.996373	+	0.0850931i	$0.0271188\pi$
$457$	5587.00	+	9676.97i	0.571879	+	0.990524i	−0.424494	+	0.905431i	$0.639548\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11674.0			−1.17942			−0.589710	−	0.807615i	$-0.700758\pi$
$461$	−11674.0			−1.17942			−0.589710	+	0.807615i	$0.700758\pi$
$462$		0			0
$463$	10528.0			1.05676			0.528378	−	0.849009i	$-0.322801\pi$
$463$	10528.0			1.05676			0.528378	+	0.849009i	$0.322801\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8302.00	−	14379.5i	−0.822635	−	1.42485i	−0.903713	−	0.428139i	$-0.859170\pi$
$467$	−8302.00	−	14379.5i	−0.822635	−	1.42485i	0.0810777	−	0.996708i	$-0.474164\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1016.00	+	1759.76i	−0.0987648	+	0.171066i
$474$		0			0
$475$	−6532.00			−0.630966
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4288.00	−	7427.03i	0.409027	−	0.708455i	−0.585754	−	0.810489i	$-0.699202\pi$
$479$	4288.00	−	7427.03i	0.409027	−	0.708455i	0.994781	+	0.102034i	$0.0325350\pi$
$480$		0			0
$481$	4266.00	+	7388.93i	0.404393	+	0.700429i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10514.0	−	18210.8i	−0.984363	−	1.70497i
$486$		0			0
$487$	−4852.00	+	8403.91i	−0.451468	+	0.781966i	−0.998478	−	0.0551605i	$-0.982433\pi$
$487$	−4852.00	+	8403.91i	−0.451468	+	0.781966i	0.547009	+	0.837127i	$0.315766\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4092.00			0.376109			0.188054	−	0.982159i	$-0.439782\pi$
$491$	4092.00			0.376109			0.188054	+	0.982159i	$0.439782\pi$
$492$		0			0
$493$	742.000	−	1285.18i	0.0677850	−	0.117407i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−8942.00	−	15488.0i	−0.802202	−	1.38945i	−0.918164	−	0.396201i	$-0.870328\pi$
$499$	−8942.00	−	15488.0i	−0.802202	−	1.38945i	0.115961	−	0.993254i	$-0.463005\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−7704.00			−0.682911			−0.341456	−	0.939898i	$-0.610920\pi$
$503$	−7704.00			−0.682911			−0.341456	+	0.939898i	$0.610920\pi$
$504$		0			0
$505$	5572.00			0.490992
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7179.00	−	12434.4i	−0.625154	−	1.08280i	−0.988511	−	0.151149i	$-0.951703\pi$
$509$	−7179.00	−	12434.4i	−0.625154	−	1.08280i	0.363357	−	0.931650i	$-0.381631\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8120.00	−	14064.3i	0.694777	−	1.20339i
$516$		0			0
$517$	2112.00			0.179663
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2541.00	+	4401.14i	−0.213672	+	0.370091i	−0.952861	−	0.303407i	$-0.901876\pi$
$521$	−2541.00	+	4401.14i	−0.213672	+	0.370091i	0.739189	+	0.673498i	$0.235209\pi$
$522$		0			0
$523$	−878.000	−	1520.74i	−0.0734078	−	0.127146i	0.826985	−	0.562224i	$-0.190054\pi$
$523$	−878.000	−	1520.74i	−0.0734078	−	0.127146i	−0.900393	+	0.435078i	$0.856721\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1008.00	+	1745.91i	0.0833191	+	0.144313i
$528$		0			0
$529$	−5468.50	+	9471.72i	−0.449453	+	0.778476i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	21060.0			1.71146
$534$		0			0
$535$	2268.00	−	3928.29i	0.183279	−	0.317448i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8115.00	−	14055.6i	−0.644900	−	1.11700i	−0.984325	−	0.176367i	$-0.943566\pi$
$541$	−8115.00	−	14055.6i	−0.644900	−	1.11700i	0.339424	−	0.940633i	$-0.389768\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−13132.0			−1.03213
$546$		0			0
$547$	17676.0			1.38167			0.690833	−	0.723014i	$-0.257244\pi$
$547$	17676.0			1.38167			0.690833	+	0.723014i	$0.257244\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4876.00	+	8445.48i	0.376996	+	0.652976i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6125.00	+	10608.8i	−0.465933	+	0.807019i	−0.999243	−	0.0389004i	$-0.987614\pi$
$557$	−6125.00	+	10608.8i	−0.465933	+	0.807019i	0.533310	+	0.845920i	$0.320948\pi$
$558$		0			0
$559$	27432.0			2.07558
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5026.00	−	8705.29i	0.376236	−	0.651659i	−0.614276	−	0.789092i	$-0.710552\pi$
$563$	5026.00	−	8705.29i	0.376236	−	0.651659i	0.990511	+	0.137432i	$0.0438850\pi$
$564$		0			0
$565$	−4354.00	−	7541.35i	−0.324202	−	0.561534i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12837.0	+	22234.3i	0.945791	+	1.63816i	0.754160	+	0.656691i	$0.228044\pi$
$569$	12837.0	+	22234.3i	0.945791	+	1.63816i	0.191631	+	0.981467i	$0.438622\pi$
$570$		0			0
$571$	−1866.00	+	3232.01i	−0.136759	+	0.236874i	−0.926268	−	0.376865i	$-0.877002\pi$
$571$	−1866.00	+	3232.01i	−0.136759	+	0.236874i	0.789509	+	0.613739i	$0.210335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	10792.0			0.782709
$576$		0			0
$577$	−607.000	+	1051.35i	−0.0437950	+	0.0758552i	−0.887092	−	0.461593i	$-0.847278\pi$
$577$	−607.000	+	1051.35i	−0.0437950	+	0.0758552i	0.843297	+	0.537448i	$0.180612\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1212.00	−	2099.25i	−0.0860993	−	0.149128i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7108.00			0.499793			0.249897	−	0.968273i	$-0.419603\pi$
$587$	7108.00			0.499793			0.249897	+	0.968273i	$0.419603\pi$
$588$		0			0
$589$	−13248.0			−0.926782
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3081.00	−	5336.45i	−0.213358	−	0.369548i	0.739405	−	0.673261i	$-0.235107\pi$
$593$	−3081.00	−	5336.45i	−0.213358	−	0.369548i	−0.952763	+	0.303713i	$0.901774\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1236.00	−	2140.81i	0.0843098	−	0.146029i	−0.820787	−	0.571234i	$-0.806465\pi$
$599$	1236.00	−	2140.81i	0.0843098	−	0.146029i	0.905097	+	0.425205i	$0.139798\pi$
$600$		0			0
$601$	13750.0			0.933235			0.466617	−	0.884459i	$-0.345472\pi$
$601$	13750.0			0.933235			0.466617	+	0.884459i	$0.345472\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9205.00	−	15943.5i	0.618573	−	1.07140i
$606$		0			0
$607$	−5688.00	−	9851.90i	−0.380344	−	0.658775i	0.610767	−	0.791810i	$-0.290861\pi$
$607$	−5688.00	−	9851.90i	−0.380344	−	0.658775i	−0.991111	+	0.133035i	$0.957528\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−14256.0	−	24692.1i	−0.943921	−	1.63492i
$612$		0			0
$613$	−10191.0	+	17651.3i	−0.671469	+	1.16302i	0.306018	+	0.952026i	$0.401003\pi$
$613$	−10191.0	+	17651.3i	−0.671469	+	1.16302i	−0.977488	+	0.210993i	$0.932330\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−21178.0			−1.38184			−0.690919	−	0.722932i	$-0.742794\pi$
$617$	−21178.0			−1.38184			−0.690919	+	0.722932i	$0.742794\pi$
$618$		0			0
$619$	−2350.00	+	4070.32i	−0.152592	+	0.264297i	−0.932180	−	0.361996i	$-0.882095\pi$
$619$	−2350.00	+	4070.32i	−0.152592	+	0.264297i	0.779588	+	0.626293i	$0.215429\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9729.50	+	16852.0i	0.622688	+	1.07853i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2212.00			−0.140220
$630$		0			0
$631$	−21736.0			−1.37131			−0.685655	−	0.727927i	$-0.740484\pi$
$631$	−21736.0			−1.37131			−0.685655	+	0.727927i	$0.740484\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8400.00	−	14549.2i	−0.524951	−	0.909242i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6511.00	+	11277.4i	−0.401200	+	0.694898i	−0.993871	−	0.110546i	$-0.964740\pi$
$641$	−6511.00	+	11277.4i	−0.401200	+	0.694898i	0.592671	+	0.805444i	$0.298073\pi$
$642$		0			0
$643$	−3308.00			−0.202885			−0.101442	−	0.994841i	$-0.532346\pi$
$643$	−3308.00			−0.202885			−0.101442	+	0.994841i	$0.532346\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6900.00	−	11951.2i	0.419269	−	0.726195i	−0.576597	−	0.817029i	$-0.695620\pi$
$647$	6900.00	−	11951.2i	0.419269	−	0.726195i	0.995866	+	0.0908335i	$0.0289531\pi$
$648$		0			0
$649$	−728.000	−	1260.93i	−0.0440316	−	0.0762649i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1341.00	−	2322.68i	−0.0803635	−	0.139194i	0.823043	−	0.567980i	$-0.192275\pi$
$653$	−1341.00	−	2322.68i	−0.0803635	−	0.139194i	−0.903406	+	0.428786i	$0.858941\pi$
$654$		0			0
$655$	9772.00	−	16925.6i	0.582937	−	1.00968i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−23836.0			−1.40898			−0.704491	−	0.709713i	$-0.748825\pi$
$659$	−23836.0			−1.40898			−0.704491	+	0.709713i	$0.748825\pi$
$660$		0			0
$661$	−5641.00	+	9770.50i	−0.331936	+	0.574929i	−0.982891	−	0.184186i	$-0.941035\pi$
$661$	−5641.00	+	9770.50i	−0.331936	+	0.574929i	0.650956	+	0.759116i	$0.274368\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8056.00	−	13953.4i	−0.467661	−	0.810012i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2712.00			0.156029
$672$		0			0
$673$	−13726.0			−0.786179			−0.393089	−	0.919500i	$-0.628594\pi$
$673$	−13726.0			−0.786179			−0.393089	+	0.919500i	$0.628594\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2487.00	−	4307.61i	−0.141186	−	0.244542i	0.786757	−	0.617263i	$-0.211758\pi$
$677$	−2487.00	−	4307.61i	−0.141186	−	0.244542i	−0.927944	+	0.372721i	$0.878425\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4494.00	+	7783.84i	−0.251769	+	0.436076i	−0.964013	−	0.265855i	$-0.914346\pi$
$683$	−4494.00	+	7783.84i	−0.251769	+	0.436076i	0.712244	+	0.701932i	$0.247679\pi$
$684$		0			0
$685$	−39340.0			−2.19431
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−16362.0	+	28339.8i	−0.904706	+	1.56700i
$690$		0			0
$691$	5086.00	+	8809.21i	0.280001	+	0.484976i	0.971385	−	0.237512i	$-0.0763319\pi$
$691$	5086.00	+	8809.21i	0.280001	+	0.484976i	−0.691384	+	0.722488i	$0.742999\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	28.0000	+	48.4974i	0.00152820	+	0.00264692i
$696$		0			0
$697$	−2730.00	+	4728.50i	−0.148359	+	0.256965i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−27446.0			−1.47877			−0.739387	−	0.673280i	$-0.764885\pi$
$701$	−27446.0			−1.47877			−0.739387	+	0.673280i	$0.764885\pi$
$702$		0			0
$703$	7268.00	−	12588.5i	0.389926	−	0.675371i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1999.00	−	3462.37i	−0.105887	−	0.183402i	0.808213	−	0.588890i	$-0.200435\pi$
$709$	−1999.00	−	3462.37i	−0.105887	−	0.183402i	−0.914100	+	0.405488i	$0.867102\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	21888.0			1.14967
$714$		0			0
$715$	3024.00			0.158169
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12936.0	−	22405.8i	−0.670976	−	1.16216i	−0.977628	−	0.210342i	$-0.932542\pi$
$719$	−12936.0	−	22405.8i	−0.670976	−	1.16216i	0.306652	−	0.951822i	$-0.400791\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3763.00	+	6517.71i	−0.192765	+	0.333878i
$726$		0			0
$727$	−12088.0			−0.616670			−0.308335	−	0.951278i	$-0.599772\pi$
$727$	−12088.0			−0.616670			−0.308335	+	0.951278i	$0.599772\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3556.00	+	6159.17i	−0.179923	+	0.311635i
$732$		0			0
$733$	3987.00	+	6905.69i	0.200905	+	0.347977i	0.948820	−	0.315817i	$-0.102278\pi$
$733$	3987.00	+	6905.69i	0.200905	+	0.347977i	−0.747915	+	0.663794i	$0.768945\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1688.00	+	2923.70i	0.0843667	+	0.146127i
$738$		0			0
$739$	15882.0	−	27508.4i	0.790567	−	1.36930i	−0.135050	−	0.990839i	$-0.543119\pi$
$739$	15882.0	−	27508.4i	0.790567	−	1.36930i	0.925616	−	0.378463i	$-0.123547\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−888.000			−0.0438460			−0.0219230	−	0.999760i	$-0.506979\pi$
$743$	−888.000			−0.0438460			−0.0219230	+	0.999760i	$0.506979\pi$
$744$		0			0
$745$	9618.00	−	16658.9i	0.472988	−	0.819240i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	17328.0	+	30013.0i	0.841954	+	1.45831i	0.888240	+	0.459379i	$0.151928\pi$
$751$	17328.0	+	30013.0i	0.841954	+	1.45831i	−0.0462858	+	0.998928i	$0.514738\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	29456.0			1.41989
$756$		0			0
$757$	−22866.0			−1.09786			−0.548929	−	0.835869i	$-0.684964\pi$
$757$	−22866.0			−1.09786			−0.548929	+	0.835869i	$0.684964\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11285.0	−	19546.2i	−0.537557	−	0.931076i	−0.999035	−	0.0439244i	$-0.986014\pi$
$761$	−11285.0	−	19546.2i	−0.537557	−	0.931076i	0.461478	−	0.887152i	$-0.347319\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9828.00	+	17022.6i	−0.462671	+	0.801369i
$768$		0			0
$769$	1790.00			0.0839389			0.0419695	−	0.999119i	$-0.486637\pi$
$769$	1790.00			0.0839389			0.0419695	+	0.999119i	$0.486637\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1495.00	+	2589.42i	−0.0695620	+	0.120485i	−0.898709	−	0.438546i	$-0.855493\pi$
$773$	−1495.00	+	2589.42i	−0.0695620	+	0.120485i	0.829147	+	0.559031i	$0.188827\pi$
$774$		0			0
$775$	−5112.00	−	8854.24i	−0.236940	−	0.410392i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−17940.0	−	31073.0i	−0.825118	−	1.42915i
$780$		0			0
$781$	16.0000	−	27.7128i	0.000733067	−	0.00126971i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−44884.0			−2.04074
$786$		0			0
$787$	−15378.0	+	26635.5i	−0.696527	+	1.20642i	0.273137	+	0.961975i	$0.411939\pi$
$787$	−15378.0	+	26635.5i	−0.696527	+	1.20642i	−0.969663	+	0.244444i	$0.921394\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−18306.0	−	31706.9i	−0.819754	−	1.41986i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15126.0			0.672259			0.336129	−	0.941816i	$-0.390882\pi$
$797$	15126.0			0.672259			0.336129	+	0.941816i	$0.390882\pi$
$798$		0			0
$799$	7392.00			0.327297
$800$		0			0
$801$		0			0
$802$		0			0
$803$	844.000	+	1461.85i	0.0370911	+	0.0642436i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3251.00	+	5630.90i	−0.141284	+	0.244712i	−0.927981	−	0.372629i	$-0.878457\pi$
$809$	−3251.00	+	5630.90i	−0.141284	+	0.244712i	0.786696	+	0.617340i	$0.211790\pi$
$810$		0			0
$811$	8252.00			0.357296			0.178648	−	0.983913i	$-0.442828\pi$
$811$	8252.00			0.357296			0.178648	+	0.983913i	$0.442828\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2324.00	+	4025.29i	−0.0998849	+	0.173006i
$816$		0			0
$817$	−23368.0	−	40474.6i	−1.00066	−	1.73320i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10993.0	−	19040.4i	−0.467306	−	0.809398i	0.531996	−	0.846747i	$-0.321442\pi$
$821$	−10993.0	−	19040.4i	−0.467306	−	0.809398i	−0.999302	+	0.0373488i	$0.988109\pi$
$822$		0			0
$823$	−1868.00	+	3235.47i	−0.0791183	+	0.137037i	−0.902870	−	0.429914i	$-0.858544\pi$
$823$	−1868.00	+	3235.47i	−0.0791183	+	0.137037i	0.823751	+	0.566951i	$0.191877\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	23820.0			1.00158			0.500788	−	0.865570i	$-0.333044\pi$
$827$	23820.0			1.00158			0.500788	+	0.865570i	$0.333044\pi$
$828$		0			0
$829$	3971.00	−	6877.97i	0.166367	−	0.288157i	−0.770773	−	0.637110i	$-0.780130\pi$
$829$	3971.00	−	6877.97i	0.166367	−	0.288157i	0.937140	+	0.348954i	$0.113463\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−10472.0	−	18138.0i	−0.434010	−	0.751728i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21016.0			0.864783			0.432391	−	0.901686i	$-0.357670\pi$
$839$	21016.0			0.864783			0.432391	+	0.901686i	$0.357670\pi$
$840$		0			0
$841$	−13153.0			−0.539301
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5033.00	−	8717.41i	−0.204900	−	0.354897i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12008.0	+	20798.5i	−0.483700	+	0.837793i
$852$		0			0
$853$	−24878.0			−0.998601			−0.499300	−	0.866429i	$-0.666410\pi$
$853$	−24878.0			−0.998601			−0.499300	+	0.866429i	$0.666410\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3195.00	−	5533.90i	0.127350	−	0.220577i	−0.795299	−	0.606217i	$-0.792686\pi$
$857$	3195.00	−	5533.90i	0.127350	−	0.220577i	0.922649	+	0.385640i	$0.126019\pi$
$858$		0			0
$859$	−23222.0	−	40221.7i	−0.922380	−	1.59761i	−0.795722	−	0.605663i	$-0.792908\pi$
$859$	−23222.0	−	40221.7i	−0.922380	−	1.59761i	−0.126658	−	0.991946i	$-0.540425\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12704.0	−	22004.0i	−0.501100	−	0.867930i	−0.999999	−	0.00127049i	$-0.999596\pi$
$863$	−12704.0	−	22004.0i	−0.501100	−	0.867930i	0.498899	−	0.866660i	$-0.333738\pi$
$864$		0			0
$865$	23254.0	−	40277.1i	0.914057	−	1.58319i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1536.00			−0.0599600
$870$		0			0
$871$	22788.0	−	39470.0i	0.886500	−	1.53546i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−539.000	−	933.575i	−0.0207534	−	0.0359460i	0.855462	−	0.517865i	$-0.173273\pi$
$877$	−539.000	−	933.575i	−0.0207534	−	0.0359460i	−0.876216	+	0.481919i	$0.839940\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−45006.0			−1.72110			−0.860551	−	0.509364i	$-0.829881\pi$
$881$	−45006.0			−1.72110			−0.860551	+	0.509364i	$0.829881\pi$
$882$		0			0
$883$	4028.00			0.153514			0.0767571	−	0.997050i	$-0.475543\pi$
$883$	4028.00			0.153514			0.0767571	+	0.997050i	$0.475543\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	14652.0	+	25378.0i	0.554640	+	0.960665i	0.997931	+	0.0642876i	$0.0204775\pi$
$887$	14652.0	+	25378.0i	0.554640	+	0.960665i	−0.443291	+	0.896378i	$0.646189\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−24288.0	+	42068.1i	−0.910153	+	1.57643i
$894$		0			0
$895$	12600.0			0.470583
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−7632.00	+	13219.0i	−0.283138	+	0.490410i
$900$		0			0
$901$	−4242.00	−	7347.36i	−0.156850	−	0.271672i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	13314.0	+	23060.5i	0.489030	+	0.847025i
$906$		0			0
$907$	−25458.0	+	44094.5i	−0.931995	+	1.61426i	−0.152087	+	0.988367i	$0.548599\pi$
$907$	−25458.0	+	44094.5i	−0.931995	+	1.61426i	−0.779908	+	0.625895i	$0.784734\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	24432.0			0.888549			0.444275	−	0.895891i	$-0.353461\pi$
$911$	24432.0			0.888549			0.444275	+	0.895891i	$0.353461\pi$
$912$		0			0
$913$	−1096.00	+	1898.33i	−0.0397287	+	0.0688121i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	10180.0	+	17632.3i	0.365405	+	0.632900i	0.988841	−	0.148974i	$-0.0475972\pi$
$919$	10180.0	+	17632.3i	0.365405	+	0.632900i	−0.623436	+	0.781874i	$0.714264\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−432.000			−0.0154057
$924$		0			0
$925$	11218.0			0.398752
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−11601.0	−	20093.5i	−0.409706	−	0.709631i	0.585151	−	0.810924i	$-0.301035\pi$
$929$	−11601.0	−	20093.5i	−0.409706	−	0.709631i	−0.994857	+	0.101293i	$0.967702\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−392.000	+	678.964i	−0.0137110	+	0.0237481i
$936$		0			0
$937$	1990.00			0.0693815			0.0346908	−	0.999398i	$-0.488955\pi$
$937$	1990.00			0.0693815			0.0346908	+	0.999398i	$0.488955\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	25565.0	−	44279.9i	0.885648	−	1.53399i	0.0406798	−	0.999172i	$-0.487048\pi$
$941$	25565.0	−	44279.9i	0.885648	−	1.53399i	0.844969	−	0.534816i	$-0.179619\pi$
$942$		0			0
$943$	29640.0	+	51338.0i	1.02355	+	1.77285i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−23522.0	−	40741.3i	−0.807141	−	1.39801i	−0.914836	−	0.403825i	$-0.867681\pi$
$947$	−23522.0	−	40741.3i	−0.807141	−	1.39801i	0.107696	−	0.994184i	$-0.465653\pi$
$948$		0			0
$949$	11394.0	−	19735.0i	0.389742	−	0.675052i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−46858.0			−1.59274			−0.796369	−	0.604811i	$-0.793249\pi$
$953$	−46858.0			−1.59274			−0.796369	+	0.604811i	$0.793249\pi$
$954$		0			0
$955$	28896.0	−	50049.3i	0.979113	−	1.69587i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	4527.50	+	7841.86i	0.151975	+	0.263229i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18788.0			−0.626743
$966$		0			0
$967$	30632.0			1.01867			0.509337	−	0.860567i	$-0.329890\pi$
$967$	30632.0			1.01867			0.509337	+	0.860567i	$0.329890\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1902.00	+	3294.36i	0.0628611	+	0.108879i	0.895743	−	0.444572i	$-0.146644\pi$
$971$	1902.00	+	3294.36i	0.0628611	+	0.108879i	−0.832882	+	0.553450i	$0.813311\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−24663.0	+	42717.6i	−0.807614	+	1.39883i	0.106897	+	0.994270i	$0.465908\pi$
$977$	−24663.0	+	42717.6i	−0.807614	+	1.39883i	−0.914512	+	0.404559i	$0.867425\pi$
$978$		0			0
$979$	−4776.00			−0.155916
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−5556.00	+	9623.27i	−0.180274	+	0.312243i	−0.941974	−	0.335687i	$-0.891032\pi$
$983$	−5556.00	+	9623.27i	−0.180274	+	0.312243i	0.761700	+	0.647930i	$0.224365\pi$
$984$		0			0
$985$	−24542.0	−	42508.0i	−0.793881	−	1.37504i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	38608.0	+	66871.0i	1.24132	+	2.15002i
$990$		0			0
$991$	−6808.00	+	11791.8i	−0.218227	+	0.377981i	−0.954266	−	0.298959i	$-0.903361\pi$
$991$	−6808.00	+	11791.8i	−0.218227	+	0.377981i	0.736039	+	0.676939i	$0.236694\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9520.00			−0.303321
$996$		0			0
$997$	−28337.0	+	49081.1i	−0.900142	+	1.55909i	−0.0728338	+	0.997344i	$0.523204\pi$
$997$	−28337.0	+	49081.1i	−0.900142	+	1.55909i	−0.827308	+	0.561748i	$0.810129\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.c.361.1		2
3.2	odd	2		588.4.i.h.361.1		2
7.2	even	3	inner	1764.4.k.c.1549.1		2
7.3	odd	6		252.4.a.a.1.1		1
7.4	even	3		1764.4.a.l.1.1		1
7.5	odd	6		1764.4.k.n.1549.1		2
7.6	odd	2		1764.4.k.n.361.1		2
21.2	odd	6		588.4.i.h.373.1		2
21.5	even	6		588.4.i.a.373.1		2
21.11	odd	6		588.4.a.a.1.1		1
21.17	even	6		84.4.a.b.1.1	✓	1
21.20	even	2		588.4.i.a.361.1		2
28.3	even	6		1008.4.a.d.1.1		1
84.11	even	6		2352.4.a.v.1.1		1
84.59	odd	6		336.4.a.e.1.1		1
105.17	odd	12		2100.4.k.g.1849.1		2
105.38	odd	12		2100.4.k.g.1849.2		2
105.59	even	6		2100.4.a.g.1.1		1
168.59	odd	6		1344.4.a.p.1.1		1
168.101	even	6		1344.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.a.b.1.1	✓	1	21.17	even	6
252.4.a.a.1.1		1	7.3	odd	6
336.4.a.e.1.1		1	84.59	odd	6
588.4.a.a.1.1		1	21.11	odd	6
588.4.i.a.361.1		2	21.20	even	2
588.4.i.a.373.1		2	21.5	even	6
588.4.i.h.361.1		2	3.2	odd	2
588.4.i.h.373.1		2	21.2	odd	6
1008.4.a.d.1.1		1	28.3	even	6
1344.4.a.b.1.1		1	168.101	even	6
1344.4.a.p.1.1		1	168.59	odd	6
1764.4.a.l.1.1		1	7.4	even	3
1764.4.k.c.361.1		2	1.1	even	1	trivial
1764.4.k.c.1549.1		2	7.2	even	3	inner
1764.4.k.n.361.1		2	7.6	odd	2
1764.4.k.n.1549.1		2	7.5	odd	6
2100.4.a.g.1.1		1	105.59	even	6
2100.4.k.g.1849.1		2	105.17	odd	12
2100.4.k.g.1849.2		2	105.38	odd	12
2352.4.a.v.1.1		1	84.11	even	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)

\(f(q)\)	\(=\)	\(q+(-7.00000 - 12.1244i) q^{5} +O(q^{10})\)	\(q+(-7.00000 - 12.1244i) q^{5} +(2.00000 - 3.46410i) q^{11} -54.0000 q^{13} +(7.00000 - 12.1244i) q^{17} +(46.0000 + 79.6743i) q^{19} +(-76.0000 - 131.636i) q^{23} +(-35.5000 + 61.4878i) q^{25} +106.000 q^{29} +(-72.0000 + 124.708i) q^{31} +(-79.0000 - 136.832i) q^{37} -390.000 q^{41} -508.000 q^{43} +(264.000 + 457.261i) q^{47} +(303.000 - 524.811i) q^{53} -56.0000 q^{55} +(182.000 - 315.233i) q^{59} +(339.000 + 587.165i) q^{61} +(378.000 + 654.715i) q^{65} +(-422.000 + 730.925i) q^{67} +8.00000 q^{71} +(-211.000 + 365.463i) q^{73} +(-192.000 - 332.554i) q^{79} -548.000 q^{83} -196.000 q^{85} +(-597.000 - 1034.03i) q^{89} +(644.000 - 1115.44i) q^{95} +1502.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{5}+O(q^{10}) \) 2 * q - 14 * q^5	\( 2 q - 14 q^{5} + 4 q^{11} - 108 q^{13} + 14 q^{17} + 92 q^{19} - 152 q^{23} - 71 q^{25} + 212 q^{29} - 144 q^{31} - 158 q^{37} - 780 q^{41} - 1016 q^{43} + 528 q^{47} + 606 q^{53} - 112 q^{55} + 364 q^{59} + 678 q^{61} + 756 q^{65} - 844 q^{67} + 16 q^{71} - 422 q^{73} - 384 q^{79} - 1096 q^{83} - 392 q^{85} - 1194 q^{89} + 1288 q^{95} + 3004 q^{97}+O(q^{100}) \) 2 * q - 14 * q^5 + 4 * q^11 - 108 * q^13 + 14 * q^17 + 92 * q^19 - 152 * q^23 - 71 * q^25 + 212 * q^29 - 144 * q^31 - 158 * q^37 - 780 * q^41 - 1016 * q^43 + 528 * q^47 + 606 * q^53 - 112 * q^55 + 364 * q^59 + 678 * q^61 + 756 * q^65 - 844 * q^67 + 16 * q^71 - 422 * q^73 - 384 * q^79 - 1096 * q^83 - 392 * q^85 - 1194 * q^89 + 1288 * q^95 + 3004 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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