LMFDB - Embedded newform 342.3.d.a.37.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,3,Mod(37,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("342.37");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	342.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.31882504112$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.2
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	342.37
Dual form			342.3.d.a.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} +1.00000 q^{5} +5.00000 q^{7} -2.82843i q^{8} +O(q^{10})$	\(q+1.41421i q^{2} -2.00000 q^{4} +1.00000 q^{5} +5.00000 q^{7} -2.82843i q^{8} +1.41421i q^{10} -5.00000 q^{11} +16.9706i q^{13} +7.07107i q^{14} +4.00000 q^{16} +25.0000 q^{17} +19.0000 q^{19} -2.00000 q^{20} -7.07107i q^{22} +10.0000 q^{23} -24.0000 q^{25} -24.0000 q^{26} -10.0000 q^{28} +42.4264i q^{29} +42.4264i q^{31} +5.65685i q^{32} +35.3553i q^{34} +5.00000 q^{35} +25.4558i q^{37} +26.8701i q^{38} -2.82843i q^{40} -42.4264i q^{41} +5.00000 q^{43} +10.0000 q^{44} +14.1421i q^{46} -5.00000 q^{47} -24.0000 q^{49} -33.9411i q^{50} -33.9411i q^{52} -25.4558i q^{53} -5.00000 q^{55} -14.1421i q^{56} -60.0000 q^{58} +84.8528i q^{59} +95.0000 q^{61} -60.0000 q^{62} -8.00000 q^{64} +16.9706i q^{65} -110.309i q^{67} -50.0000 q^{68} +7.07107i q^{70} -25.0000 q^{73} -36.0000 q^{74} -38.0000 q^{76} -25.0000 q^{77} -42.4264i q^{79} +4.00000 q^{80} +60.0000 q^{82} +130.000 q^{83} +25.0000 q^{85} +7.07107i q^{86} +14.1421i q^{88} -127.279i q^{89} +84.8528i q^{91} -20.0000 q^{92} -7.07107i q^{94} +19.0000 q^{95} -16.9706i q^{97} -33.9411i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 2 q^{5} + 10 q^{7}+O(q^{10}) $ 2 * q - 4 * q^4 + 2 * q^5 + 10 * q^7	$ 2 q - 4 q^{4} + 2 q^{5} + 10 q^{7} - 10 q^{11} + 8 q^{16} + 50 q^{17} + 38 q^{19} - 4 q^{20} + 20 q^{23} - 48 q^{25} - 48 q^{26} - 20 q^{28} + 10 q^{35} + 10 q^{43} + 20 q^{44} - 10 q^{47} - 48 q^{49} - 10 q^{55} - 120 q^{58} + 190 q^{61} - 120 q^{62} - 16 q^{64} - 100 q^{68} - 50 q^{73} - 72 q^{74} - 76 q^{76} - 50 q^{77} + 8 q^{80} + 120 q^{82} + 260 q^{83} + 50 q^{85} - 40 q^{92} + 38 q^{95}+O(q^{100}) $ 2 * q - 4 * q^4 + 2 * q^5 + 10 * q^7 - 10 * q^11 + 8 * q^16 + 50 * q^17 + 38 * q^19 - 4 * q^20 + 20 * q^23 - 48 * q^25 - 48 * q^26 - 20 * q^28 + 10 * q^35 + 10 * q^43 + 20 * q^44 - 10 * q^47 - 48 * q^49 - 10 * q^55 - 120 * q^58 + 190 * q^61 - 120 * q^62 - 16 * q^64 - 100 * q^68 - 50 * q^73 - 72 * q^74 - 76 * q^76 - 50 * q^77 + 8 * q^80 + 120 * q^82 + 260 * q^83 + 50 * q^85 - 40 * q^92 + 38 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	1.00000	0.200000	0.100000	−	0.994987i	$-0.468116\pi$
$5$	1.00000	0.200000	0.100000	+	0.994987i	$0.468116\pi$
$6$	0	0
$7$	5.00000	0.714286	0.357143	−	0.934050i	$-0.383751\pi$
$7$	5.00000	0.714286	0.357143	+	0.934050i	$0.383751\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	1.41421i	0.141421i
$11$	−5.00000	−0.454545	−0.227273	−	0.973831i	$-0.572981\pi$
$11$	−5.00000	−0.454545	−0.227273	+	0.973831i	$0.572981\pi$
$12$	0	0
$13$	16.9706i	1.30543i	0.757604	+	0.652714i	$0.226370\pi$
$13$	16.9706i	1.30543i	−0.757604	+	0.652714i	$0.773630\pi$
$14$	7.07107i	0.505076i
$15$	0	0
$16$	4.00000	0.250000
$17$	25.0000	1.47059	0.735294	−	0.677748i	$-0.237044\pi$
$17$	25.0000	1.47059	0.735294	+	0.677748i	$0.237044\pi$
$18$	0	0
$19$	19.0000	1.00000
$19$	19.0000	1.00000
$20$	−2.00000	−0.100000
$21$	0	0
$22$	− 7.07107i	− 0.321412i
$23$	10.0000	0.434783	0.217391	−	0.976085i	$-0.430245\pi$
$23$	10.0000	0.434783	0.217391	+	0.976085i	$0.430245\pi$
$24$	0	0
$25$	−24.0000	−0.960000
$26$	−24.0000	−0.923077
$27$	0	0
$28$	−10.0000	−0.357143
$29$	42.4264i	1.46298i	0.681852	+	0.731490i	$0.261175\pi$
$29$	42.4264i	1.46298i	−0.681852	+	0.731490i	$0.738825\pi$
$30$	0	0
$31$	42.4264i	1.36859i	0.729204	+	0.684297i	$0.239891\pi$
$31$	42.4264i	1.36859i	−0.729204	+	0.684297i	$0.760109\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	35.3553i	1.03986i
$35$	5.00000	0.142857
$36$	0	0
$37$	25.4558i	0.687996i	0.938970	+	0.343998i	$0.111781\pi$
$37$	25.4558i	0.687996i	−0.938970	+	0.343998i	$0.888219\pi$
$38$	26.8701i	0.707107i
$39$	0	0
$40$	− 2.82843i	− 0.0707107i
$41$	− 42.4264i	− 1.03479i	−0.855747	−	0.517395i	$-0.826902\pi$
$41$	− 42.4264i	− 1.03479i	0.855747	−	0.517395i	$-0.173098\pi$
$42$	0	0
$43$	5.00000	0.116279	0.0581395	−	0.998308i	$-0.481483\pi$
$43$	5.00000	0.116279	0.0581395	+	0.998308i	$0.481483\pi$
$44$	10.0000	0.227273
$45$	0	0
$46$	14.1421i	0.307438i
$47$	−5.00000	−0.106383	−0.0531915	−	0.998584i	$-0.516939\pi$
$47$	−5.00000	−0.106383	−0.0531915	+	0.998584i	$0.516939\pi$
$48$	0	0
$49$	−24.0000	−0.489796
$50$	− 33.9411i	− 0.678823i
$51$	0	0
$52$	− 33.9411i	− 0.652714i
$53$	− 25.4558i	− 0.480299i	−0.970736	−	0.240149i	$-0.922804\pi$
$53$	− 25.4558i	− 0.480299i	0.970736	−	0.240149i	$-0.0771965\pi$
$54$	0	0
$55$	−5.00000	−0.0909091
$56$	− 14.1421i	− 0.252538i
$57$	0	0
$58$	−60.0000	−1.03448
$59$	84.8528i	1.43818i	0.694915	+	0.719092i	$0.255442\pi$
$59$	84.8528i	1.43818i	−0.694915	+	0.719092i	$0.744558\pi$
$60$	0	0
$61$	95.0000	1.55738	0.778689	−	0.627411i	$-0.215885\pi$
$61$	95.0000	1.55738	0.778689	+	0.627411i	$0.215885\pi$
$62$	−60.0000	−0.967742
$63$	0	0
$64$	−8.00000	−0.125000
$65$	16.9706i	0.261086i
$66$	0	0
$67$	− 110.309i	− 1.64640i	−0.567753	−	0.823199i	$-0.692187\pi$
$67$	− 110.309i	− 1.64640i	0.567753	−	0.823199i	$-0.307813\pi$
$68$	−50.0000	−0.735294
$69$	0	0
$70$	7.07107i	0.101015i
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−25.0000	−0.342466	−0.171233	−	0.985231i	$-0.554775\pi$
$73$	−25.0000	−0.342466	−0.171233	+	0.985231i	$0.554775\pi$
$74$	−36.0000	−0.486486
$75$	0	0
$76$	−38.0000	−0.500000
$77$	−25.0000	−0.324675
$78$	0	0
$79$	− 42.4264i	− 0.537043i	−0.963274	−	0.268522i	$-0.913465\pi$
$79$	− 42.4264i	− 0.537043i	0.963274	−	0.268522i	$-0.0865351\pi$
$80$	4.00000	0.0500000
$81$	0	0
$82$	60.0000	0.731707
$83$	130.000	1.56627	0.783133	−	0.621855i	$-0.213621\pi$
$83$	130.000	1.56627	0.783133	+	0.621855i	$0.213621\pi$
$84$	0	0
$85$	25.0000	0.294118
$86$	7.07107i	0.0822217i
$87$	0	0
$88$	14.1421i	0.160706i
$89$	− 127.279i	− 1.43010i	−0.699071	−	0.715052i	$-0.746403\pi$
$89$	− 127.279i	− 1.43010i	0.699071	−	0.715052i	$-0.253597\pi$
$90$	0	0
$91$	84.8528i	0.932449i
$92$	−20.0000	−0.217391
$93$	0	0
$94$	− 7.07107i	− 0.0752241i
$95$	19.0000	0.200000
$96$	0	0
$97$	− 16.9706i	− 0.174954i	−0.996167	−	0.0874771i	$-0.972120\pi$
$97$	− 16.9706i	− 0.174954i	0.996167	−	0.0874771i	$-0.0278805\pi$
$98$	− 33.9411i	− 0.346338i
$99$	0	0
$100$	48.0000	0.480000
$101$	−50.0000	−0.495050	−0.247525	−	0.968882i	$-0.579617\pi$
$101$	−50.0000	−0.495050	−0.247525	+	0.968882i	$0.579617\pi$
$102$	0	0
$103$	16.9706i	0.164763i	0.996601	+	0.0823814i	$0.0262526\pi$
$103$	16.9706i	0.164763i	−0.996601	+	0.0823814i	$0.973747\pi$
$104$	48.0000	0.461538
$105$	0	0
$106$	36.0000	0.339623
$107$	− 101.823i	− 0.951620i	−0.879548	−	0.475810i	$-0.842155\pi$
$107$	− 101.823i	− 0.951620i	0.879548	−	0.475810i	$-0.157845\pi$
$108$	0	0
$109$	− 127.279i	− 1.16770i	−0.811862	−	0.583850i	$-0.801546\pi$
$109$	− 127.279i	− 1.16770i	0.811862	−	0.583850i	$-0.198454\pi$
$110$	− 7.07107i	− 0.0642824i
$111$	0	0
$112$	20.0000	0.178571
$113$	110.309i	0.976183i	0.872793	+	0.488091i	$0.162307\pi$
$113$	110.309i	0.976183i	−0.872793	+	0.488091i	$0.837693\pi$
$114$	0	0
$115$	10.0000	0.0869565
$116$	− 84.8528i	− 0.731490i
$117$	0	0
$118$	−120.000	−1.01695
$119$	125.000	1.05042
$120$	0	0
$121$	−96.0000	−0.793388
$122$	134.350i	1.10123i
$123$	0	0
$124$	− 84.8528i	− 0.684297i
$125$	−49.0000	−0.392000
$126$	0	0
$127$	− 229.103i	− 1.80396i	−0.431780	−	0.901979i	$-0.642114\pi$
$127$	− 229.103i	− 1.80396i	0.431780	−	0.901979i	$-0.357886\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	−24.0000	−0.184615
$131$	163.000	1.24427	0.622137	−	0.782908i	$-0.286265\pi$
$131$	163.000	1.24427	0.622137	+	0.782908i	$0.286265\pi$
$132$	0	0
$133$	95.0000	0.714286
$134$	156.000	1.16418
$135$	0	0
$136$	− 70.7107i	− 0.519931i
$137$	−95.0000	−0.693431	−0.346715	−	0.937970i	$-0.612703\pi$
$137$	−95.0000	−0.693431	−0.346715	+	0.937970i	$0.612703\pi$
$138$	0	0
$139$	125.000	0.899281	0.449640	−	0.893210i	$-0.351552\pi$
$139$	125.000	0.899281	0.449640	+	0.893210i	$0.351552\pi$
$140$	−10.0000	−0.0714286
$141$	0	0
$142$	0	0
$143$	− 84.8528i	− 0.593376i
$144$	0	0
$145$	42.4264i	0.292596i
$146$	− 35.3553i	− 0.242160i
$147$	0	0
$148$	− 50.9117i	− 0.343998i
$149$	−215.000	−1.44295	−0.721477	−	0.692439i	$-0.756536\pi$
$149$	−215.000	−1.44295	−0.721477	+	0.692439i	$0.756536\pi$
$150$	0	0
$151$	84.8528i	0.561939i	0.959717	+	0.280970i	$0.0906560\pi$
$151$	84.8528i	0.561939i	−0.959717	+	0.280970i	$0.909344\pi$
$152$	− 53.7401i	− 0.353553i
$153$	0	0
$154$	− 35.3553i	− 0.229580i
$155$	42.4264i	0.273719i
$156$	0	0
$157$	−190.000	−1.21019	−0.605096	−	0.796153i	$-0.706865\pi$
$157$	−190.000	−1.21019	−0.605096	+	0.796153i	$0.706865\pi$
$158$	60.0000	0.379747
$159$	0	0
$160$	5.65685i	0.0353553i
$161$	50.0000	0.310559
$162$	0	0
$163$	110.000	0.674847	0.337423	−	0.941353i	$-0.390445\pi$
$163$	110.000	0.674847	0.337423	+	0.941353i	$0.390445\pi$
$164$	84.8528i	0.517395i
$165$	0	0
$166$	183.848i	1.10752i
$167$	59.3970i	0.355670i	0.984060	+	0.177835i	$0.0569094\pi$
$167$	59.3970i	0.355670i	−0.984060	+	0.177835i	$0.943091\pi$
$168$	0	0
$169$	−119.000	−0.704142
$170$	35.3553i	0.207973i
$171$	0	0
$172$	−10.0000	−0.0581395
$173$	− 186.676i	− 1.07905i	−0.841969	−	0.539527i	$-0.818603\pi$
$173$	− 186.676i	− 1.07905i	0.841969	−	0.539527i	$-0.181397\pi$
$174$	0	0
$175$	−120.000	−0.685714
$176$	−20.0000	−0.113636
$177$	0	0
$178$	180.000	1.01124
$179$	− 127.279i	− 0.711057i	−0.934665	−	0.355529i	$-0.884301\pi$
$179$	− 127.279i	− 0.711057i	0.934665	−	0.355529i	$-0.115699\pi$
$180$	0	0
$181$	254.558i	1.40640i	0.710992	+	0.703200i	$0.248246\pi$
$181$	254.558i	1.40640i	−0.710992	+	0.703200i	$0.751754\pi$
$182$	−120.000	−0.659341
$183$	0	0
$184$	− 28.2843i	− 0.153719i
$185$	25.4558i	0.137599i
$186$	0	0
$187$	−125.000	−0.668449
$188$	10.0000	0.0531915
$189$	0	0
$190$	26.8701i	0.141421i
$191$	−293.000	−1.53403	−0.767016	−	0.641628i	$-0.778259\pi$
$191$	−293.000	−1.53403	−0.767016	+	0.641628i	$0.778259\pi$
$192$	0	0
$193$	− 59.3970i	− 0.307756i	−0.988090	−	0.153878i	$-0.950824\pi$
$193$	− 59.3970i	− 0.307756i	0.988090	−	0.153878i	$-0.0491763\pi$
$194$	24.0000	0.123711
$195$	0	0
$196$	48.0000	0.244898
$197$	70.0000	0.355330	0.177665	−	0.984091i	$-0.443146\pi$
$197$	70.0000	0.355330	0.177665	+	0.984091i	$0.443146\pi$
$198$	0	0
$199$	173.000	0.869347	0.434673	−	0.900588i	$-0.356864\pi$
$199$	173.000	0.869347	0.434673	+	0.900588i	$0.356864\pi$
$200$	67.8823i	0.339411i
$201$	0	0
$202$	− 70.7107i	− 0.350053i
$203$	212.132i	1.04499i
$204$	0	0
$205$	− 42.4264i	− 0.206958i
$206$	−24.0000	−0.116505
$207$	0	0
$208$	67.8823i	0.326357i
$209$	−95.0000	−0.454545
$210$	0	0
$211$	− 84.8528i	− 0.402146i	−0.979576	−	0.201073i	$-0.935557\pi$
$211$	− 84.8528i	− 0.402146i	0.979576	−	0.201073i	$-0.0644429\pi$
$212$	50.9117i	0.240149i
$213$	0	0
$214$	144.000	0.672897
$215$	5.00000	0.0232558
$216$	0	0
$217$	212.132i	0.977567i
$218$	180.000	0.825688
$219$	0	0
$220$	10.0000	0.0454545
$221$	424.264i	1.91975i
$222$	0	0
$223$	364.867i	1.63618i	0.575094	+	0.818088i	$0.304966\pi$
$223$	364.867i	1.63618i	−0.575094	+	0.818088i	$0.695034\pi$
$224$	28.2843i	0.126269i
$225$	0	0
$226$	−156.000	−0.690265
$227$	67.8823i	0.299041i	0.988759	+	0.149520i	$0.0477730\pi$
$227$	67.8823i	0.299041i	−0.988759	+	0.149520i	$0.952227\pi$
$228$	0	0
$229$	−145.000	−0.633188	−0.316594	−	0.948561i	$-0.602539\pi$
$229$	−145.000	−0.633188	−0.316594	+	0.948561i	$0.602539\pi$
$230$	14.1421i	0.0614875i
$231$	0	0
$232$	120.000	0.517241
$233$	−335.000	−1.43777	−0.718884	−	0.695130i	$-0.755347\pi$
$233$	−335.000	−1.43777	−0.718884	+	0.695130i	$0.755347\pi$
$234$	0	0
$235$	−5.00000	−0.0212766
$236$	− 169.706i	− 0.719092i
$237$	0	0
$238$	176.777i	0.742759i
$239$	−197.000	−0.824268	−0.412134	−	0.911123i	$-0.635216\pi$
$239$	−197.000	−0.824268	−0.412134	+	0.911123i	$0.635216\pi$
$240$	0	0
$241$	− 296.985i	− 1.23230i	−0.787628	−	0.616151i	$-0.788691\pi$
$241$	− 296.985i	− 1.23230i	0.787628	−	0.616151i	$-0.211309\pi$
$242$	− 135.765i	− 0.561010i
$243$	0	0
$244$	−190.000	−0.778689
$245$	−24.0000	−0.0979592
$246$	0	0
$247$	322.441i	1.30543i
$248$	120.000	0.483871
$249$	0	0
$250$	− 69.2965i	− 0.277186i
$251$	−173.000	−0.689243	−0.344622	−	0.938742i	$-0.611993\pi$
$251$	−173.000	−0.689243	−0.344622	+	0.938742i	$0.611993\pi$
$252$	0	0
$253$	−50.0000	−0.197628
$254$	324.000	1.27559
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 67.8823i	− 0.264133i	−0.991241	−	0.132067i	$-0.957839\pi$
$257$	− 67.8823i	− 0.264133i	0.991241	−	0.132067i	$-0.0421613\pi$
$258$	0	0
$259$	127.279i	0.491426i
$260$	− 33.9411i	− 0.130543i
$261$	0	0
$262$	230.517i	0.879835i
$263$	355.000	1.34981	0.674905	−	0.737905i	$-0.264185\pi$
$263$	355.000	1.34981	0.674905	+	0.737905i	$0.264185\pi$
$264$	0	0
$265$	− 25.4558i	− 0.0960598i
$266$	134.350i	0.505076i
$267$	0	0
$268$	220.617i	0.823199i
$269$	− 381.838i	− 1.41947i	−0.704468	−	0.709735i	$-0.748814\pi$
$269$	− 381.838i	− 1.41947i	0.704468	−	0.709735i	$-0.251186\pi$
$270$	0	0
$271$	110.000	0.405904	0.202952	−	0.979189i	$-0.434946\pi$
$271$	110.000	0.405904	0.202952	+	0.979189i	$0.434946\pi$
$272$	100.000	0.367647
$273$	0	0
$274$	− 134.350i	− 0.490330i
$275$	120.000	0.436364
$276$	0	0
$277$	−265.000	−0.956679	−0.478339	−	0.878175i	$-0.658761\pi$
$277$	−265.000	−0.956679	−0.478339	+	0.878175i	$0.658761\pi$
$278$	176.777i	0.635887i
$279$	0	0
$280$	− 14.1421i	− 0.0505076i
$281$	424.264i	1.50984i	0.655819	+	0.754918i	$0.272323\pi$
$281$	424.264i	1.50984i	−0.655819	+	0.754918i	$0.727677\pi$
$282$	0	0
$283$	125.000	0.441696	0.220848	−	0.975308i	$-0.429118\pi$
$283$	125.000	0.441696	0.220848	+	0.975308i	$0.429118\pi$
$284$	0	0
$285$	0	0
$286$	120.000	0.419580
$287$	− 212.132i	− 0.739136i
$288$	0	0
$289$	336.000	1.16263
$290$	−60.0000	−0.206897
$291$	0	0
$292$	50.0000	0.171233
$293$	186.676i	0.637120i	0.947903	+	0.318560i	$0.103199\pi$
$293$	186.676i	0.637120i	−0.947903	+	0.318560i	$0.896801\pi$
$294$	0	0
$295$	84.8528i	0.287637i
$296$	72.0000	0.243243
$297$	0	0
$298$	− 304.056i	− 1.02032i
$299$	169.706i	0.567577i
$300$	0	0
$301$	25.0000	0.0830565
$302$	−120.000	−0.397351
$303$	0	0
$304$	76.0000	0.250000
$305$	95.0000	0.311475
$306$	0	0
$307$	− 280.014i	− 0.912099i	−0.889955	−	0.456049i	$-0.849264\pi$
$307$	− 280.014i	− 0.912099i	0.889955	−	0.456049i	$-0.150736\pi$
$308$	50.0000	0.162338
$309$	0	0
$310$	−60.0000	−0.193548
$311$	235.000	0.755627	0.377814	−	0.925882i	$-0.376676\pi$
$311$	235.000	0.755627	0.377814	+	0.925882i	$0.376676\pi$
$312$	0	0
$313$	−310.000	−0.990415	−0.495208	−	0.868775i	$-0.664908\pi$
$313$	−310.000	−0.990415	−0.495208	+	0.868775i	$0.664908\pi$
$314$	− 268.701i	− 0.855734i
$315$	0	0
$316$	84.8528i	0.268522i
$317$	− 186.676i	− 0.588884i	−0.955669	−	0.294442i	$-0.904866\pi$
$317$	− 186.676i	− 0.588884i	0.955669	−	0.294442i	$-0.0951338\pi$
$318$	0	0
$319$	− 212.132i	− 0.664991i
$320$	−8.00000	−0.0250000
$321$	0	0
$322$	70.7107i	0.219598i
$323$	475.000	1.47059
$324$	0	0
$325$	− 407.294i	− 1.25321i
$326$	155.563i	0.477189i
$327$	0	0
$328$	−120.000	−0.365854
$329$	−25.0000	−0.0759878
$330$	0	0
$331$	− 296.985i	− 0.897235i	−0.893724	−	0.448618i	$-0.851917\pi$
$331$	− 296.985i	− 0.897235i	0.893724	−	0.448618i	$-0.148083\pi$
$332$	−260.000	−0.783133
$333$	0	0
$334$	−84.0000	−0.251497
$335$	− 110.309i	− 0.329280i
$336$	0	0
$337$	526.087i	1.56109i	0.625099	+	0.780545i	$0.285058\pi$
$337$	526.087i	1.56109i	−0.625099	+	0.780545i	$0.714942\pi$
$338$	− 168.291i	− 0.497904i
$339$	0	0
$340$	−50.0000	−0.147059
$341$	− 212.132i	− 0.622088i
$342$	0	0
$343$	−365.000	−1.06414
$344$	− 14.1421i	− 0.0411109i
$345$	0	0
$346$	264.000	0.763006
$347$	−125.000	−0.360231	−0.180115	−	0.983646i	$-0.557647\pi$
$347$	−125.000	−0.360231	−0.180115	+	0.983646i	$0.557647\pi$
$348$	0	0
$349$	23.0000	0.0659026	0.0329513	−	0.999457i	$-0.489509\pi$
$349$	23.0000	0.0659026	0.0329513	+	0.999457i	$0.489509\pi$
$350$	− 169.706i	− 0.484873i
$351$	0	0
$352$	− 28.2843i	− 0.0803530i
$353$	−410.000	−1.16147	−0.580737	−	0.814092i	$-0.697235\pi$
$353$	−410.000	−1.16147	−0.580737	+	0.814092i	$0.697235\pi$
$354$	0	0
$355$	0	0
$356$	254.558i	0.715052i
$357$	0	0
$358$	180.000	0.502793
$359$	475.000	1.32312	0.661560	−	0.749892i	$-0.269895\pi$
$359$	475.000	1.32312	0.661560	+	0.749892i	$0.269895\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	−360.000	−0.994475
$363$	0	0
$364$	− 169.706i	− 0.466224i
$365$	−25.0000	−0.0684932
$366$	0	0
$367$	230.000	0.626703	0.313351	−	0.949637i	$-0.398548\pi$
$367$	230.000	0.626703	0.313351	+	0.949637i	$0.398548\pi$
$368$	40.0000	0.108696
$369$	0	0
$370$	−36.0000	−0.0972973
$371$	− 127.279i	− 0.343071i
$372$	0	0
$373$	67.8823i	0.181990i	0.995851	+	0.0909950i	$0.0290047\pi$
$373$	67.8823i	0.181990i	−0.995851	+	0.0909950i	$0.970995\pi$
$374$	− 176.777i	− 0.472665i
$375$	0	0
$376$	14.1421i	0.0376121i
$377$	−720.000	−1.90981
$378$	0	0
$379$	254.558i	0.671658i	0.941923	+	0.335829i	$0.109016\pi$
$379$	254.558i	0.671658i	−0.941923	+	0.335829i	$0.890984\pi$
$380$	−38.0000	−0.100000
$381$	0	0
$382$	− 414.365i	− 1.08472i
$383$	− 144.250i	− 0.376631i	−0.982109	−	0.188316i	$-0.939697\pi$
$383$	− 144.250i	− 0.376631i	0.982109	−	0.188316i	$-0.0603028\pi$
$384$	0	0
$385$	−25.0000	−0.0649351
$386$	84.0000	0.217617
$387$	0	0
$388$	33.9411i	0.0874771i
$389$	553.000	1.42159	0.710797	−	0.703397i	$-0.248334\pi$
$389$	553.000	1.42159	0.710797	+	0.703397i	$0.248334\pi$
$390$	0	0
$391$	250.000	0.639386
$392$	67.8823i	0.173169i
$393$	0	0
$394$	98.9949i	0.251256i
$395$	− 42.4264i	− 0.107409i
$396$	0	0
$397$	335.000	0.843829	0.421914	−	0.906636i	$-0.361358\pi$
$397$	335.000	0.843829	0.421914	+	0.906636i	$0.361358\pi$
$398$	244.659i	0.614721i
$399$	0	0
$400$	−96.0000	−0.240000
$401$	212.132i	0.529008i	0.964385	+	0.264504i	$0.0852082\pi$
$401$	212.132i	0.529008i	−0.964385	+	0.264504i	$0.914792\pi$
$402$	0	0
$403$	−720.000	−1.78660
$404$	100.000	0.247525
$405$	0	0
$406$	−300.000	−0.738916
$407$	− 127.279i	− 0.312725i
$408$	0	0
$409$	− 721.249i	− 1.76344i	−0.471769	−	0.881722i	$-0.656384\pi$
$409$	− 721.249i	− 1.76344i	0.471769	−	0.881722i	$-0.343616\pi$
$410$	60.0000	0.146341
$411$	0	0
$412$	− 33.9411i	− 0.0823814i
$413$	424.264i	1.02727i
$414$	0	0
$415$	130.000	0.313253
$416$	−96.0000	−0.230769
$417$	0	0
$418$	− 134.350i	− 0.321412i
$419$	−62.0000	−0.147971	−0.0739857	−	0.997259i	$-0.523572\pi$
$419$	−62.0000	−0.147971	−0.0739857	+	0.997259i	$0.523572\pi$
$420$	0	0
$421$	− 296.985i	− 0.705427i	−0.935731	−	0.352714i	$-0.885259\pi$
$421$	− 296.985i	− 0.705427i	0.935731	−	0.352714i	$-0.114741\pi$
$422$	120.000	0.284360
$423$	0	0
$424$	−72.0000	−0.169811
$425$	−600.000	−1.41176
$426$	0	0
$427$	475.000	1.11241
$428$	203.647i	0.475810i
$429$	0	0
$430$	7.07107i	0.0164443i
$431$	− 509.117i	− 1.18125i	−0.806948	−	0.590623i	$-0.798882\pi$
$431$	− 509.117i	− 1.18125i	0.806948	−	0.590623i	$-0.201118\pi$
$432$	0	0
$433$	229.103i	0.529105i	0.964371	+	0.264553i	$0.0852243\pi$
$433$	229.103i	0.529105i	−0.964371	+	0.264553i	$0.914776\pi$
$434$	−300.000	−0.691244
$435$	0	0
$436$	254.558i	0.583850i
$437$	190.000	0.434783
$438$	0	0
$439$	− 806.102i	− 1.83622i	−0.396323	−	0.918111i	$-0.629714\pi$
$439$	− 806.102i	− 1.83622i	0.396323	−	0.918111i	$-0.370286\pi$
$440$	14.1421i	0.0321412i
$441$	0	0
$442$	−600.000	−1.35747
$443$	−365.000	−0.823928	−0.411964	−	0.911200i	$-0.635157\pi$
$443$	−365.000	−0.823928	−0.411964	+	0.911200i	$0.635157\pi$
$444$	0	0
$445$	− 127.279i	− 0.286021i
$446$	−516.000	−1.15695
$447$	0	0
$448$	−40.0000	−0.0892857
$449$	− 763.675i	− 1.70084i	−0.526108	−	0.850418i	$-0.676349\pi$
$449$	− 763.675i	− 1.70084i	0.526108	−	0.850418i	$-0.323651\pi$
$450$	0	0
$451$	212.132i	0.470359i
$452$	− 220.617i	− 0.488091i
$453$	0	0
$454$	−96.0000	−0.211454
$455$	84.8528i	0.186490i
$456$	0	0
$457$	−265.000	−0.579869	−0.289934	−	0.957047i	$-0.593633\pi$
$457$	−265.000	−0.579869	−0.289934	+	0.957047i	$0.593633\pi$
$458$	− 205.061i	− 0.447731i
$459$	0	0
$460$	−20.0000	−0.0434783
$461$	553.000	1.19957	0.599783	−	0.800163i	$-0.295254\pi$
$461$	553.000	1.19957	0.599783	+	0.800163i	$0.295254\pi$
$462$	0	0
$463$	485.000	1.04752	0.523758	−	0.851867i	$-0.324530\pi$
$463$	485.000	1.04752	0.523758	+	0.851867i	$0.324530\pi$
$464$	169.706i	0.365745i
$465$	0	0
$466$	− 473.762i	− 1.01666i
$467$	115.000	0.246253	0.123126	−	0.992391i	$-0.460708\pi$
$467$	115.000	0.246253	0.123126	+	0.992391i	$0.460708\pi$
$468$	0	0
$469$	− 551.543i	− 1.17600i
$470$	− 7.07107i	− 0.0150448i
$471$	0	0
$472$	240.000	0.508475
$473$	−25.0000	−0.0528541
$474$	0	0
$475$	−456.000	−0.960000
$476$	−250.000	−0.525210
$477$	0	0
$478$	− 278.600i	− 0.582845i
$479$	490.000	1.02296	0.511482	−	0.859294i	$-0.329097\pi$
$479$	490.000	1.02296	0.511482	+	0.859294i	$0.329097\pi$
$480$	0	0
$481$	−432.000	−0.898129
$482$	420.000	0.871369
$483$	0	0
$484$	192.000	0.396694
$485$	− 16.9706i	− 0.0349909i
$486$	0	0
$487$	610.940i	1.25450i	0.778819	+	0.627249i	$0.215819\pi$
$487$	610.940i	1.25450i	−0.778819	+	0.627249i	$0.784181\pi$
$488$	− 268.701i	− 0.550616i
$489$	0	0
$490$	− 33.9411i	− 0.0692676i
$491$	82.0000	0.167006	0.0835031	−	0.996508i	$-0.473389\pi$
$491$	82.0000	0.167006	0.0835031	+	0.996508i	$0.473389\pi$
$492$	0	0
$493$	1060.66i	2.15144i
$494$	−456.000	−0.923077
$495$	0	0
$496$	169.706i	0.342148i
$497$	0	0
$498$	0	0
$499$	485.000	0.971944	0.485972	−	0.873974i	$-0.338466\pi$
$499$	485.000	0.971944	0.485972	+	0.873974i	$0.338466\pi$
$500$	98.0000	0.196000
$501$	0	0
$502$	− 244.659i	− 0.487368i
$503$	250.000	0.497018	0.248509	−	0.968630i	$-0.420059\pi$
$503$	250.000	0.497018	0.248509	+	0.968630i	$0.420059\pi$
$504$	0	0
$505$	−50.0000	−0.0990099
$506$	− 70.7107i	− 0.139744i
$507$	0	0
$508$	458.205i	0.901979i
$509$	− 169.706i	− 0.333410i	−0.986007	−	0.166705i	$-0.946687\pi$
$509$	− 169.706i	− 0.333410i	0.986007	−	0.166705i	$-0.0533127\pi$
$510$	0	0
$511$	−125.000	−0.244618
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	96.0000	0.186770
$515$	16.9706i	0.0329525i
$516$	0	0
$517$	25.0000	0.0483559
$518$	−180.000	−0.347490
$519$	0	0
$520$	48.0000	0.0923077
$521$	− 127.279i	− 0.244298i	−0.992512	−	0.122149i	$-0.961021\pi$
$521$	− 127.279i	− 0.244298i	0.992512	−	0.122149i	$-0.0389786\pi$
$522$	0	0
$523$	− 356.382i	− 0.681418i	−0.940169	−	0.340709i	$-0.889333\pi$
$523$	− 356.382i	− 0.681418i	0.940169	−	0.340709i	$-0.110667\pi$
$524$	−326.000	−0.622137
$525$	0	0
$526$	502.046i	0.954460i
$527$	1060.66i	2.01264i
$528$	0	0
$529$	−429.000	−0.810964
$530$	36.0000	0.0679245
$531$	0	0
$532$	−190.000	−0.357143
$533$	720.000	1.35084
$534$	0	0
$535$	− 101.823i	− 0.190324i
$536$	−312.000	−0.582090
$537$	0	0
$538$	540.000	1.00372
$539$	120.000	0.222635
$540$	0	0
$541$	−25.0000	−0.0462107	−0.0231054	−	0.999733i	$-0.507355\pi$
$541$	−25.0000	−0.0462107	−0.0231054	+	0.999733i	$0.507355\pi$
$542$	155.563i	0.287018i
$543$	0	0
$544$	141.421i	0.259966i
$545$	− 127.279i	− 0.233540i
$546$	0	0
$547$	− 16.9706i	− 0.0310248i	−0.999880	−	0.0155124i	$-0.995062\pi$
$547$	− 16.9706i	− 0.0310248i	0.999880	−	0.0155124i	$-0.00493795\pi$
$548$	190.000	0.346715
$549$	0	0
$550$	169.706i	0.308556i
$551$	806.102i	1.46298i
$552$	0	0
$553$	− 212.132i	− 0.383602i
$554$	− 374.767i	− 0.676474i
$555$	0	0
$556$	−250.000	−0.449640
$557$	745.000	1.33752	0.668761	−	0.743477i	$-0.266825\pi$
$557$	745.000	1.33752	0.668761	+	0.743477i	$0.266825\pi$
$558$	0	0
$559$	84.8528i	0.151794i
$560$	20.0000	0.0357143
$561$	0	0
$562$	−600.000	−1.06762
$563$	313.955i	0.557647i	0.960342	+	0.278824i	$0.0899445\pi$
$563$	313.955i	0.557647i	−0.960342	+	0.278824i	$0.910056\pi$
$564$	0	0
$565$	110.309i	0.195237i
$566$	176.777i	0.312326i
$567$	0	0
$568$	0	0
$569$	424.264i	0.745631i	0.927905	+	0.372816i	$0.121608\pi$
$569$	424.264i	0.745631i	−0.927905	+	0.372816i	$0.878392\pi$
$570$	0	0
$571$	1070.00	1.87391	0.936953	−	0.349456i	$-0.113634\pi$
$571$	1070.00	1.87391	0.936953	+	0.349456i	$0.113634\pi$
$572$	169.706i	0.296688i
$573$	0	0
$574$	300.000	0.522648
$575$	−240.000	−0.417391
$576$	0	0
$577$	−25.0000	−0.0433276	−0.0216638	−	0.999765i	$-0.506896\pi$
$577$	−25.0000	−0.0433276	−0.0216638	+	0.999765i	$0.506896\pi$
$578$	475.176i	0.822103i
$579$	0	0
$580$	− 84.8528i	− 0.146298i
$581$	650.000	1.11876
$582$	0	0
$583$	127.279i	0.218318i
$584$	70.7107i	0.121080i
$585$	0	0
$586$	−264.000	−0.450512
$587$	−725.000	−1.23509	−0.617547	−	0.786534i	$-0.711873\pi$
$587$	−725.000	−1.23509	−0.617547	+	0.786534i	$0.711873\pi$
$588$	0	0
$589$	806.102i	1.36859i
$590$	−120.000	−0.203390
$591$	0	0
$592$	101.823i	0.171999i
$593$	−650.000	−1.09612	−0.548061	−	0.836439i	$-0.684634\pi$
$593$	−650.000	−1.09612	−0.548061	+	0.836439i	$0.684634\pi$
$594$	0	0
$595$	125.000	0.210084
$596$	430.000	0.721477
$597$	0	0
$598$	−240.000	−0.401338
$599$	− 296.985i	− 0.495801i	−0.968785	−	0.247901i	$-0.920259\pi$
$599$	− 296.985i	− 0.495801i	0.968785	−	0.247901i	$-0.0797406\pi$
$600$	0	0
$601$	848.528i	1.41186i	0.708281	+	0.705930i	$0.249471\pi$
$601$	848.528i	1.41186i	−0.708281	+	0.705930i	$0.750529\pi$
$602$	35.3553i	0.0587298i
$603$	0	0
$604$	− 169.706i	− 0.280970i
$605$	−96.0000	−0.158678
$606$	0	0
$607$	271.529i	0.447329i	0.974666	+	0.223665i	$0.0718021\pi$
$607$	271.529i	0.447329i	−0.974666	+	0.223665i	$0.928198\pi$
$608$	107.480i	0.176777i
$609$	0	0
$610$	134.350i	0.220246i
$611$	− 84.8528i	− 0.138875i
$612$	0	0
$613$	1055.00	1.72104	0.860522	−	0.509413i	$-0.170138\pi$
$613$	1055.00	1.72104	0.860522	+	0.509413i	$0.170138\pi$
$614$	396.000	0.644951
$615$	0	0
$616$	70.7107i	0.114790i
$617$	505.000	0.818476	0.409238	−	0.912428i	$-0.365794\pi$
$617$	505.000	0.818476	0.409238	+	0.912428i	$0.365794\pi$
$618$	0	0
$619$	−130.000	−0.210016	−0.105008	−	0.994471i	$-0.533487\pi$
$619$	−130.000	−0.210016	−0.105008	+	0.994471i	$0.533487\pi$
$620$	− 84.8528i	− 0.136859i
$621$	0	0
$622$	332.340i	0.534309i
$623$	− 636.396i	− 1.02150i
$624$	0	0
$625$	551.000	0.881600
$626$	− 438.406i	− 0.700329i
$627$	0	0
$628$	380.000	0.605096
$629$	636.396i	1.01176i
$630$	0	0
$631$	−475.000	−0.752773	−0.376387	−	0.926463i	$-0.622834\pi$
$631$	−475.000	−0.752773	−0.376387	+	0.926463i	$0.622834\pi$
$632$	−120.000	−0.189873
$633$	0	0
$634$	264.000	0.416404
$635$	− 229.103i	− 0.360791i
$636$	0	0
$637$	− 407.294i	− 0.639393i
$638$	300.000	0.470219
$639$	0	0
$640$	− 11.3137i	− 0.0176777i
$641$	− 848.528i	− 1.32376i	−0.749611	−	0.661878i	$-0.769760\pi$
$641$	− 848.528i	− 1.32376i	0.749611	−	0.661878i	$-0.230240\pi$
$642$	0	0
$643$	−955.000	−1.48523	−0.742613	−	0.669721i	$-0.766414\pi$
$643$	−955.000	−1.48523	−0.742613	+	0.669721i	$0.766414\pi$
$644$	−100.000	−0.155280
$645$	0	0
$646$	671.751i	1.03986i
$647$	−965.000	−1.49150	−0.745750	−	0.666226i	$-0.767908\pi$
$647$	−965.000	−1.49150	−0.745750	+	0.666226i	$0.767908\pi$
$648$	0	0
$649$	− 424.264i	− 0.653720i
$650$	576.000	0.886154
$651$	0	0
$652$	−220.000	−0.337423
$653$	−935.000	−1.43185	−0.715926	−	0.698176i	$-0.753995\pi$
$653$	−935.000	−1.43185	−0.715926	+	0.698176i	$0.753995\pi$
$654$	0	0
$655$	163.000	0.248855
$656$	− 169.706i	− 0.258698i
$657$	0	0
$658$	− 35.3553i	− 0.0537315i
$659$	− 84.8528i	− 0.128760i	−0.997925	−	0.0643800i	$-0.979493\pi$
$659$	− 84.8528i	− 0.128760i	0.997925	−	0.0643800i	$-0.0205070\pi$
$660$	0	0
$661$	678.823i	1.02696i	0.858101	+	0.513481i	$0.171644\pi$
$661$	678.823i	1.02696i	−0.858101	+	0.513481i	$0.828356\pi$
$662$	420.000	0.634441
$663$	0	0
$664$	− 367.696i	− 0.553758i
$665$	95.0000	0.142857
$666$	0	0
$667$	424.264i	0.636078i
$668$	− 118.794i	− 0.177835i
$669$	0	0
$670$	156.000	0.232836
$671$	−475.000	−0.707899
$672$	0	0
$673$	186.676i	0.277379i	0.990336	+	0.138690i	$0.0442890\pi$
$673$	186.676i	0.277379i	−0.990336	+	0.138690i	$0.955711\pi$
$674$	−744.000	−1.10386
$675$	0	0
$676$	238.000	0.352071
$677$	907.925i	1.34110i	0.741864	+	0.670550i	$0.233942\pi$
$677$	907.925i	1.34110i	−0.741864	+	0.670550i	$0.766058\pi$
$678$	0	0
$679$	− 84.8528i	− 0.124967i
$680$	− 70.7107i	− 0.103986i
$681$	0	0
$682$	300.000	0.439883
$683$	1120.06i	1.63991i	0.572429	+	0.819954i	$0.306001\pi$
$683$	1120.06i	1.63991i	−0.572429	+	0.819954i	$0.693999\pi$
$684$	0	0
$685$	−95.0000	−0.138686
$686$	− 516.188i	− 0.752461i
$687$	0	0
$688$	20.0000	0.0290698
$689$	432.000	0.626996
$690$	0	0
$691$	−715.000	−1.03473	−0.517366	−	0.855764i	$-0.673087\pi$
$691$	−715.000	−1.03473	−0.517366	+	0.855764i	$0.673087\pi$
$692$	373.352i	0.539527i
$693$	0	0
$694$	− 176.777i	− 0.254721i
$695$	125.000	0.179856
$696$	0	0
$697$	− 1060.66i	− 1.52175i
$698$	32.5269i	0.0466002i
$699$	0	0
$700$	240.000	0.342857
$701$	430.000	0.613409	0.306705	−	0.951805i	$-0.400774\pi$
$701$	430.000	0.613409	0.306705	+	0.951805i	$0.400774\pi$
$702$	0	0
$703$	483.661i	0.687996i
$704$	40.0000	0.0568182
$705$	0	0
$706$	− 579.828i	− 0.821285i
$707$	−250.000	−0.353607
$708$	0	0
$709$	−382.000	−0.538787	−0.269394	−	0.963030i	$-0.586823\pi$
$709$	−382.000	−0.538787	−0.269394	+	0.963030i	$0.586823\pi$
$710$	0	0
$711$	0	0
$712$	−360.000	−0.505618
$713$	424.264i	0.595041i
$714$	0	0
$715$	− 84.8528i	− 0.118675i
$716$	254.558i	0.355529i
$717$	0	0
$718$	671.751i	0.935587i
$719$	115.000	0.159944	0.0799722	−	0.996797i	$-0.474517\pi$
$719$	115.000	0.159944	0.0799722	+	0.996797i	$0.474517\pi$
$720$	0	0
$721$	84.8528i	0.117688i
$722$	510.531i	0.707107i
$723$	0	0
$724$	− 509.117i	− 0.703200i
$725$	− 1018.23i	− 1.40446i
$726$	0	0
$727$	−1075.00	−1.47868	−0.739340	−	0.673333i	$-0.764862\pi$
$727$	−1075.00	−1.47868	−0.739340	+	0.673333i	$0.764862\pi$
$728$	240.000	0.329670
$729$	0	0
$730$	− 35.3553i	− 0.0484320i
$731$	125.000	0.170999
$732$	0	0
$733$	530.000	0.723056	0.361528	−	0.932361i	$-0.382255\pi$
$733$	530.000	0.723056	0.361528	+	0.932361i	$0.382255\pi$
$734$	325.269i	0.443146i
$735$	0	0
$736$	56.5685i	0.0768594i
$737$	551.543i	0.748363i
$738$	0	0
$739$	−547.000	−0.740189	−0.370095	−	0.928994i	$-0.620675\pi$
$739$	−547.000	−0.740189	−0.370095	+	0.928994i	$0.620675\pi$
$740$	− 50.9117i	− 0.0687996i
$741$	0	0
$742$	180.000	0.242588
$743$	− 958.837i	− 1.29049i	−0.763974	−	0.645247i	$-0.776755\pi$
$743$	− 958.837i	− 1.29049i	0.763974	−	0.645247i	$-0.223245\pi$
$744$	0	0
$745$	−215.000	−0.288591
$746$	−96.0000	−0.128686
$747$	0	0
$748$	250.000	0.334225
$749$	− 509.117i	− 0.679729i
$750$	0	0
$751$	169.706i	0.225973i	0.993597	+	0.112986i	$0.0360417\pi$
$751$	169.706i	0.225973i	−0.993597	+	0.112986i	$0.963958\pi$
$752$	−20.0000	−0.0265957
$753$	0	0
$754$	− 1018.23i	− 1.35044i
$755$	84.8528i	0.112388i
$756$	0	0
$757$	1055.00	1.39366	0.696830	−	0.717237i	$-0.254593\pi$
$757$	1055.00	1.39366	0.696830	+	0.717237i	$0.254593\pi$
$758$	−360.000	−0.474934
$759$	0	0
$760$	− 53.7401i	− 0.0707107i
$761$	−215.000	−0.282523	−0.141261	−	0.989972i	$-0.545116\pi$
$761$	−215.000	−0.282523	−0.141261	+	0.989972i	$0.545116\pi$
$762$	0	0
$763$	− 636.396i	− 0.834071i
$764$	586.000	0.767016
$765$	0	0
$766$	204.000	0.266319
$767$	−1440.00	−1.87744
$768$	0	0
$769$	−145.000	−0.188557	−0.0942783	−	0.995546i	$-0.530054\pi$
$769$	−145.000	−0.188557	−0.0942783	+	0.995546i	$0.530054\pi$
$770$	− 35.3553i	− 0.0459160i
$771$	0	0
$772$	118.794i	0.153878i
$773$	407.294i	0.526900i	0.964673	+	0.263450i	$0.0848604\pi$
$773$	407.294i	0.526900i	−0.964673	+	0.263450i	$0.915140\pi$
$774$	0	0
$775$	− 1018.23i	− 1.31385i
$776$	−48.0000	−0.0618557
$777$	0	0
$778$	782.060i	1.00522i
$779$	− 806.102i	− 1.03479i
$780$	0	0
$781$	0	0
$782$	353.553i	0.452114i
$783$	0	0
$784$	−96.0000	−0.122449
$785$	−190.000	−0.242038
$786$	0	0
$787$	− 186.676i	− 0.237200i	−0.992942	−	0.118600i	$-0.962159\pi$
$787$	− 186.676i	− 0.237200i	0.992942	−	0.118600i	$-0.0378406\pi$
$788$	−140.000	−0.177665
$789$	0	0
$790$	60.0000	0.0759494
$791$	551.543i	0.697273i
$792$	0	0
$793$	1612.20i	2.03304i
$794$	473.762i	0.596677i
$795$	0	0
$796$	−346.000	−0.434673
$797$	− 704.278i	− 0.883662i	−0.897098	−	0.441831i	$-0.854329\pi$
$797$	− 704.278i	− 0.883662i	0.897098	−	0.441831i	$-0.145671\pi$
$798$	0	0
$799$	−125.000	−0.156446
$800$	− 135.765i	− 0.169706i
$801$	0	0
$802$	−300.000	−0.374065
$803$	125.000	0.155666
$804$	0	0
$805$	50.0000	0.0621118
$806$	− 1018.23i	− 1.26332i
$807$	0	0
$808$	141.421i	0.175026i
$809$	457.000	0.564895	0.282447	−	0.959283i	$-0.408854\pi$
$809$	457.000	0.564895	0.282447	+	0.959283i	$0.408854\pi$
$810$	0	0
$811$	− 509.117i	− 0.627764i	−0.949462	−	0.313882i	$-0.898370\pi$
$811$	− 509.117i	− 0.627764i	0.949462	−	0.313882i	$-0.101630\pi$
$812$	− 424.264i	− 0.522493i
$813$	0	0
$814$	180.000	0.221130
$815$	110.000	0.134969
$816$	0	0
$817$	95.0000	0.116279
$818$	1020.00	1.24694
$819$	0	0
$820$	84.8528i	0.103479i
$821$	−167.000	−0.203410	−0.101705	−	0.994815i	$-0.532430\pi$
$821$	−167.000	−0.203410	−0.101705	+	0.994815i	$0.532430\pi$
$822$	0	0
$823$	−1315.00	−1.59781	−0.798906	−	0.601455i	$-0.794588\pi$
$823$	−1315.00	−1.59781	−0.798906	+	0.601455i	$0.794588\pi$
$824$	48.0000	0.0582524
$825$	0	0
$826$	−600.000	−0.726392
$827$	− 534.573i	− 0.646400i	−0.946331	−	0.323200i	$-0.895241\pi$
$827$	− 534.573i	− 0.646400i	0.946331	−	0.323200i	$-0.104759\pi$
$828$	0	0
$829$	− 763.675i	− 0.921201i	−0.887608	−	0.460600i	$-0.847634\pi$
$829$	− 763.675i	− 0.921201i	0.887608	−	0.460600i	$-0.152366\pi$
$830$	183.848i	0.221503i
$831$	0	0
$832$	− 135.765i	− 0.163178i
$833$	−600.000	−0.720288
$834$	0	0
$835$	59.3970i	0.0711341i
$836$	190.000	0.227273
$837$	0	0
$838$	− 87.6812i	− 0.104632i
$839$	− 339.411i	− 0.404543i	−0.979330	−	0.202271i	$-0.935168\pi$
$839$	− 339.411i	− 0.404543i	0.979330	−	0.202271i	$-0.0648323\pi$
$840$	0	0
$841$	−959.000	−1.14031
$842$	420.000	0.498812
$843$	0	0
$844$	169.706i	0.201073i
$845$	−119.000	−0.140828
$846$	0	0
$847$	−480.000	−0.566706
$848$	− 101.823i	− 0.120075i
$849$	0	0
$850$	− 848.528i	− 0.998268i
$851$	254.558i	0.299129i
$852$	0	0
$853$	770.000	0.902696	0.451348	−	0.892348i	$-0.350943\pi$
$853$	770.000	0.902696	0.451348	+	0.892348i	$0.350943\pi$
$854$	671.751i	0.786594i
$855$	0	0
$856$	−288.000	−0.336449
$857$	− 1255.82i	− 1.46537i	−0.680568	−	0.732685i	$-0.738267\pi$
$857$	− 1255.82i	− 1.46537i	0.680568	−	0.732685i	$-0.261733\pi$
$858$	0	0
$859$	557.000	0.648428	0.324214	−	0.945984i	$-0.394900\pi$
$859$	557.000	0.648428	0.324214	+	0.945984i	$0.394900\pi$
$860$	−10.0000	−0.0116279
$861$	0	0
$862$	720.000	0.835267
$863$	992.778i	1.15038i	0.818020	+	0.575190i	$0.195072\pi$
$863$	992.778i	1.15038i	−0.818020	+	0.575190i	$0.804928\pi$
$864$	0	0
$865$	− 186.676i	− 0.215811i
$866$	−324.000	−0.374134
$867$	0	0
$868$	− 424.264i	− 0.488783i
$869$	212.132i	0.244111i
$870$	0	0
$871$	1872.00	2.14925
$872$	−360.000	−0.412844
$873$	0	0
$874$	268.701i	0.307438i
$875$	−245.000	−0.280000
$876$	0	0
$877$	− 186.676i	− 0.212858i	−0.994320	−	0.106429i	$-0.966058\pi$
$877$	− 186.676i	− 0.212858i	0.994320	−	0.106429i	$-0.0339416\pi$
$878$	1140.00	1.29841
$879$	0	0
$880$	−20.0000	−0.0227273
$881$	25.0000	0.0283768	0.0141884	−	0.999899i	$-0.495484\pi$
$881$	25.0000	0.0283768	0.0141884	+	0.999899i	$0.495484\pi$
$882$	0	0
$883$	965.000	1.09287	0.546433	−	0.837503i	$-0.315985\pi$
$883$	965.000	1.09287	0.546433	+	0.837503i	$0.315985\pi$
$884$	− 848.528i	− 0.959873i
$885$	0	0
$886$	− 516.188i	− 0.582605i
$887$	− 780.646i	− 0.880097i	−0.897974	−	0.440048i	$-0.854961\pi$
$887$	− 780.646i	− 0.880097i	0.897974	−	0.440048i	$-0.145039\pi$
$888$	0	0
$889$	− 1145.51i	− 1.28854i
$890$	180.000	0.202247
$891$	0	0
$892$	− 729.734i	− 0.818088i
$893$	−95.0000	−0.106383
$894$	0	0
$895$	− 127.279i	− 0.142211i
$896$	− 56.5685i	− 0.0631345i
$897$	0	0
$898$	1080.00	1.20267
$899$	−1800.00	−2.00222
$900$	0	0
$901$	− 636.396i	− 0.706322i
$902$	−300.000	−0.332594
$903$	0	0
$904$	312.000	0.345133
$905$	254.558i	0.281280i
$906$	0	0
$907$	313.955i	0.346147i	0.984909	+	0.173074i	$0.0553698\pi$
$907$	313.955i	0.346147i	−0.984909	+	0.173074i	$0.944630\pi$
$908$	− 135.765i	− 0.149520i
$909$	0	0
$910$	−120.000	−0.131868
$911$	933.381i	1.02457i	0.858816	+	0.512284i	$0.171200\pi$
$911$	933.381i	1.02457i	−0.858816	+	0.512284i	$0.828800\pi$
$912$	0	0
$913$	−650.000	−0.711939
$914$	− 374.767i	− 0.410029i
$915$	0	0
$916$	290.000	0.316594
$917$	815.000	0.888768
$918$	0	0
$919$	−538.000	−0.585419	−0.292709	−	0.956201i	$-0.594557\pi$
$919$	−538.000	−0.585419	−0.292709	+	0.956201i	$0.594557\pi$
$920$	− 28.2843i	− 0.0307438i
$921$	0	0
$922$	782.060i	0.848221i
$923$	0	0
$924$	0	0
$925$	− 610.940i	− 0.660476i
$926$	685.894i	0.740706i
$927$	0	0
$928$	−240.000	−0.258621
$929$	742.000	0.798708	0.399354	−	0.916797i	$-0.369234\pi$
$929$	742.000	0.798708	0.399354	+	0.916797i	$0.369234\pi$
$930$	0	0
$931$	−456.000	−0.489796
$932$	670.000	0.718884
$933$	0	0
$934$	162.635i	0.174127i
$935$	−125.000	−0.133690
$936$	0	0
$937$	335.000	0.357524	0.178762	−	0.983892i	$-0.442791\pi$
$937$	335.000	0.357524	0.178762	+	0.983892i	$0.442791\pi$
$938$	780.000	0.831557
$939$	0	0
$940$	10.0000	0.0106383
$941$	− 424.264i	− 0.450865i	−0.974259	−	0.225433i	$-0.927620\pi$
$941$	− 424.264i	− 0.450865i	0.974259	−	0.225433i	$-0.0723795\pi$
$942$	0	0
$943$	− 424.264i	− 0.449909i
$944$	339.411i	0.359546i
$945$	0	0
$946$	− 35.3553i	− 0.0373735i
$947$	1210.00	1.27772	0.638860	−	0.769323i	$-0.279406\pi$
$947$	1210.00	1.27772	0.638860	+	0.769323i	$0.279406\pi$
$948$	0	0
$949$	− 424.264i	− 0.447064i
$950$	− 644.881i	− 0.678823i
$951$	0	0
$952$	− 353.553i	− 0.371380i
$953$	992.778i	1.04174i	0.853636	+	0.520870i	$0.174392\pi$
$953$	992.778i	1.04174i	−0.853636	+	0.520870i	$0.825608\pi$
$954$	0	0
$955$	−293.000	−0.306806
$956$	394.000	0.412134
$957$	0	0
$958$	692.965i	0.723345i
$959$	−475.000	−0.495308
$960$	0	0
$961$	−839.000	−0.873049
$962$	− 610.940i	− 0.635073i
$963$	0	0
$964$	593.970i	0.616151i
$965$	− 59.3970i	− 0.0615513i
$966$	0	0
$967$	350.000	0.361944	0.180972	−	0.983488i	$-0.442076\pi$
$967$	350.000	0.361944	0.180972	+	0.983488i	$0.442076\pi$
$968$	271.529i	0.280505i
$969$	0	0
$970$	24.0000	0.0247423
$971$	254.558i	0.262161i	0.991372	+	0.131081i	$0.0418447\pi$
$971$	254.558i	0.262161i	−0.991372	+	0.131081i	$0.958155\pi$
$972$	0	0
$973$	625.000	0.642343
$974$	−864.000	−0.887064
$975$	0	0
$976$	380.000	0.389344
$977$	− 398.808i	− 0.408197i	−0.978950	−	0.204098i	$-0.934574\pi$
$977$	− 398.808i	− 0.408197i	0.978950	−	0.204098i	$-0.0654262\pi$
$978$	0	0
$979$	636.396i	0.650047i
$980$	48.0000	0.0489796
$981$	0	0
$982$	115.966i	0.118091i
$983$	− 695.793i	− 0.707826i	−0.935278	−	0.353913i	$-0.884851\pi$
$983$	− 695.793i	− 0.707826i	0.935278	−	0.353913i	$-0.115149\pi$
$984$	0	0
$985$	70.0000	0.0710660
$986$	−1500.00	−1.52130
$987$	0	0
$988$	− 644.881i	− 0.652714i
$989$	50.0000	0.0505561
$990$	0	0
$991$	− 381.838i	− 0.385305i	−0.981267	−	0.192653i	$-0.938291\pi$
$991$	− 381.838i	− 0.385305i	0.981267	−	0.192653i	$-0.0617091\pi$
$992$	−240.000	−0.241935
$993$	0	0
$994$	0	0
$995$	173.000	0.173869
$996$	0	0
$997$	−265.000	−0.265797	−0.132899	−	0.991130i	$-0.542428\pi$
$997$	−265.000	−0.265797	−0.132899	+	0.991130i	$0.542428\pi$
$998$	685.894i	0.687268i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.3.d.a.37.2		2
3.2	odd	2		38.3.b.a.37.1	✓	2
4.3	odd	2		2736.3.o.h.721.2		2
12.11	even	2		304.3.e.c.113.2		2
15.2	even	4		950.3.d.a.949.4		4
15.8	even	4		950.3.d.a.949.1		4
15.14	odd	2		950.3.c.a.151.2		2
19.18	odd	2	inner	342.3.d.a.37.1		2
24.5	odd	2		1216.3.e.j.1025.2		2
24.11	even	2		1216.3.e.i.1025.1		2
57.56	even	2		38.3.b.a.37.2	yes	2
76.75	even	2		2736.3.o.h.721.1		2
228.227	odd	2		304.3.e.c.113.1		2
285.113	odd	4		950.3.d.a.949.3		4
285.227	odd	4		950.3.d.a.949.2		4
285.284	even	2		950.3.c.a.151.1		2
456.227	odd	2		1216.3.e.i.1025.2		2
456.341	even	2		1216.3.e.j.1025.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.b.a.37.1	✓	2	3.2	odd	2
38.3.b.a.37.2	yes	2	57.56	even	2
304.3.e.c.113.1		2	228.227	odd	2
304.3.e.c.113.2		2	12.11	even	2
342.3.d.a.37.1		2	19.18	odd	2	inner
342.3.d.a.37.2		2	1.1	even	1	trivial
950.3.c.a.151.1		2	285.284	even	2
950.3.c.a.151.2		2	15.14	odd	2
950.3.d.a.949.1		4	15.8	even	4
950.3.d.a.949.2		4	285.227	odd	4
950.3.d.a.949.3		4	285.113	odd	4
950.3.d.a.949.4		4	15.2	even	4
1216.3.e.i.1025.1		2	24.11	even	2
1216.3.e.i.1025.2		2	456.227	odd	2
1216.3.e.j.1025.1		2	456.341	even	2
1216.3.e.j.1025.2		2	24.5	odd	2
2736.3.o.h.721.1		2	76.75	even	2
2736.3.o.h.721.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	342.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +1.00000 q^{5} +5.00000 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} +1.00000 q^{5} +5.00000 q^{7} -2.82843i q^{8} +1.41421i q^{10} -5.00000 q^{11} +16.9706i q^{13} +7.07107i q^{14} +4.00000 q^{16} +25.0000 q^{17} +19.0000 q^{19} -2.00000 q^{20} -7.07107i q^{22} +10.0000 q^{23} -24.0000 q^{25} -24.0000 q^{26} -10.0000 q^{28} +42.4264i q^{29} +42.4264i q^{31} +5.65685i q^{32} +35.3553i q^{34} +5.00000 q^{35} +25.4558i q^{37} +26.8701i q^{38} -2.82843i q^{40} -42.4264i q^{41} +5.00000 q^{43} +10.0000 q^{44} +14.1421i q^{46} -5.00000 q^{47} -24.0000 q^{49} -33.9411i q^{50} -33.9411i q^{52} -25.4558i q^{53} -5.00000 q^{55} -14.1421i q^{56} -60.0000 q^{58} +84.8528i q^{59} +95.0000 q^{61} -60.0000 q^{62} -8.00000 q^{64} +16.9706i q^{65} -110.309i q^{67} -50.0000 q^{68} +7.07107i q^{70} -25.0000 q^{73} -36.0000 q^{74} -38.0000 q^{76} -25.0000 q^{77} -42.4264i q^{79} +4.00000 q^{80} +60.0000 q^{82} +130.000 q^{83} +25.0000 q^{85} +7.07107i q^{86} +14.1421i q^{88} -127.279i q^{89} +84.8528i q^{91} -20.0000 q^{92} -7.07107i q^{94} +19.0000 q^{95} -16.9706i q^{97} -33.9411i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 2 q^{5} + 10 q^{7}+O(q^{10}) \) 2 * q - 4 * q^4 + 2 * q^5 + 10 * q^7	\( 2 q - 4 q^{4} + 2 q^{5} + 10 q^{7} - 10 q^{11} + 8 q^{16} + 50 q^{17} + 38 q^{19} - 4 q^{20} + 20 q^{23} - 48 q^{25} - 48 q^{26} - 20 q^{28} + 10 q^{35} + 10 q^{43} + 20 q^{44} - 10 q^{47} - 48 q^{49} - 10 q^{55} - 120 q^{58} + 190 q^{61} - 120 q^{62} - 16 q^{64} - 100 q^{68} - 50 q^{73} - 72 q^{74} - 76 q^{76} - 50 q^{77} + 8 q^{80} + 120 q^{82} + 260 q^{83} + 50 q^{85} - 40 q^{92} + 38 q^{95}+O(q^{100}) \) 2 * q - 4 * q^4 + 2 * q^5 + 10 * q^7 - 10 * q^11 + 8 * q^16 + 50 * q^17 + 38 * q^19 - 4 * q^20 + 20 * q^23 - 48 * q^25 - 48 * q^26 - 20 * q^28 + 10 * q^35 + 10 * q^43 + 20 * q^44 - 10 * q^47 - 48 * q^49 - 10 * q^55 - 120 * q^58 + 190 * q^61 - 120 * q^62 - 16 * q^64 - 100 * q^68 - 50 * q^73 - 72 * q^74 - 76 * q^76 - 50 * q^77 + 8 * q^80 + 120 * q^82 + 260 * q^83 + 50 * q^85 - 40 * q^92 + 38 * q^95

Introduction

L-functions

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