LMFDB - Embedded newform 4140.3.d.c.2161.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,3,Mod(2161,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4140.2161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4140.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:	$112.806829445$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2161.12
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.c.2161.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +8.75590i q^{7} +O(q^{10})$	$q-2.23607i q^{5} +8.75590i q^{7} -16.6170i q^{11} -4.06836 q^{13} +6.82034i q^{17} +24.2082i q^{19} +(-0.407838 - 22.9964i) q^{23} -5.00000 q^{25} -3.59320 q^{29} +18.0876 q^{31} +19.5788 q^{35} +2.42726i q^{37} +23.8095 q^{41} -15.8533i q^{43} -24.3529 q^{47} -27.6658 q^{49} +37.7048i q^{53} -37.1567 q^{55} -66.7659 q^{59} -17.2797i q^{61} +9.09713i q^{65} -34.6259i q^{67} +89.2001 q^{71} +19.8949 q^{73} +145.497 q^{77} +20.3152i q^{79} +93.5247i q^{83} +15.2507 q^{85} -94.8577i q^{89} -35.6222i q^{91} +54.1311 q^{95} +95.4642i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 24 q^{13} - 64 q^{23} - 160 q^{25} + 60 q^{29} - 4 q^{31} + 60 q^{35} + 108 q^{41} - 136 q^{47} - 428 q^{49} + 120 q^{55} + 84 q^{59} - 188 q^{71} + 472 q^{73} + 120 q^{77} + 60 q^{85} - 80 q^{95}+O(q^{100}) $ 32 * q + 24 * q^13 - 64 * q^23 - 160 * q^25 + 60 * q^29 - 4 * q^31 + 60 * q^35 + 108 * q^41 - 136 * q^47 - 428 * q^49 + 120 * q^55 + 84 * q^59 - 188 * q^71 + 472 * q^73 + 120 * q^77 + 60 * q^85 - 80 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$			8.75590i			1.25084i	0.780287	+	0.625422i	$0.215073\pi$
$7$			8.75590i			1.25084i	−0.780287	+	0.625422i	$0.784927\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	16.6170i		−	1.51063i	−0.655359	−	0.755317i	$-0.727483\pi$
$11$		−	16.6170i		−	1.51063i	0.655359	−	0.755317i	$-0.272517\pi$
$12$		0			0
$13$	−4.06836			−0.312951			−0.156475	−	0.987682i	$-0.550013\pi$
$13$	−4.06836			−0.312951			−0.156475	+	0.987682i	$0.550013\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.82034i			0.401196i	0.979674	+	0.200598i	$0.0642886\pi$
$17$			6.82034i			0.401196i	−0.979674	+	0.200598i	$0.935711\pi$
$18$		0			0
$19$			24.2082i			1.27411i	0.770817	+	0.637057i	$0.219849\pi$
$19$			24.2082i			1.27411i	−0.770817	+	0.637057i	$0.780151\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.407838	−	22.9964i	−0.0177321	−	0.999843i
$23$	−0.407838	−	22.9964i	−0.0177321	−	0.999843i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.59320			−0.123904			−0.0619518	−	0.998079i	$-0.519733\pi$
$29$	−3.59320			−0.123904			−0.0619518	+	0.998079i	$0.519733\pi$
$30$		0			0
$31$	18.0876			0.583472			0.291736	−	0.956499i	$-0.405767\pi$
$31$	18.0876			0.583472			0.291736	+	0.956499i	$0.405767\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	19.5788			0.559394
$36$		0			0
$37$			2.42726i			0.0656016i	0.999462	+	0.0328008i	$0.0104427\pi$
$37$			2.42726i			0.0656016i	−0.999462	+	0.0328008i	$0.989557\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	23.8095			0.580720			0.290360	−	0.956917i	$-0.406225\pi$
$41$	23.8095			0.580720			0.290360	+	0.956917i	$0.406225\pi$
$42$		0			0
$43$		−	15.8533i		−	0.368682i	−0.982862	−	0.184341i	$-0.940985\pi$
$43$		−	15.8533i		−	0.368682i	0.982862	−	0.184341i	$-0.0590150\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−24.3529			−0.518147			−0.259074	−	0.965858i	$-0.583417\pi$
$47$	−24.3529			−0.518147			−0.259074	+	0.965858i	$0.583417\pi$
$48$		0			0
$49$	−27.6658			−0.564609
$50$		0			0
$51$		0			0
$52$		0			0
$53$			37.7048i			0.711411i	0.934598	+	0.355705i	$0.115759\pi$
$53$			37.7048i			0.711411i	−0.934598	+	0.355705i	$0.884241\pi$
$54$		0			0
$55$	−37.1567			−0.675576
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−66.7659			−1.13163			−0.565813	−	0.824534i	$-0.691438\pi$
$59$	−66.7659			−1.13163			−0.565813	+	0.824534i	$0.691438\pi$
$60$		0			0
$61$		−	17.2797i		−	0.283274i	−0.989919	−	0.141637i	$-0.954763\pi$
$61$		−	17.2797i		−	0.283274i	0.989919	−	0.141637i	$-0.0452365\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			9.09713i			0.139956i
$66$		0			0
$67$		−	34.6259i		−	0.516805i	−0.966037	−	0.258402i	$-0.916804\pi$
$67$		−	34.6259i		−	0.516805i	0.966037	−	0.258402i	$-0.0831960\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	89.2001			1.25634			0.628170	−	0.778076i	$-0.283804\pi$
$71$	89.2001			1.25634			0.628170	+	0.778076i	$0.283804\pi$
$72$		0			0
$73$	19.8949			0.272533			0.136266	−	0.990672i	$-0.456490\pi$
$73$	19.8949			0.272533			0.136266	+	0.990672i	$0.456490\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	145.497			1.88957
$78$		0			0
$79$			20.3152i			0.257154i	0.991700	+	0.128577i	$0.0410410\pi$
$79$			20.3152i			0.257154i	−0.991700	+	0.128577i	$0.958959\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			93.5247i			1.12680i	0.826183	+	0.563402i	$0.190508\pi$
$83$			93.5247i			1.12680i	−0.826183	+	0.563402i	$0.809492\pi$
$84$		0			0
$85$	15.2507			0.179420
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	94.8577i		−	1.06582i	−0.846173	−	0.532908i	$-0.821099\pi$
$89$		−	94.8577i		−	1.06582i	0.846173	−	0.532908i	$-0.178901\pi$
$90$		0			0
$91$		−	35.6222i		−	0.391452i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	54.1311			0.569801
$96$		0			0
$97$			95.4642i			0.984167i	0.870548	+	0.492083i	$0.163765\pi$
$97$			95.4642i			0.984167i	−0.870548	+	0.492083i	$0.836235\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	188.528			1.86661			0.933305	−	0.359083i	$-0.116911\pi$
$101$	188.528			1.86661			0.933305	+	0.359083i	$0.116911\pi$
$102$		0			0
$103$		−	25.6654i		−	0.249178i	−0.992208	−	0.124589i	$-0.960239\pi$
$103$		−	25.6654i		−	0.249178i	0.992208	−	0.124589i	$-0.0397613\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	9.87458i		−	0.0922858i	−0.998935	−	0.0461429i	$-0.985307\pi$
$107$		−	9.87458i		−	0.0922858i	0.998935	−	0.0461429i	$-0.0146930\pi$
$108$		0			0
$109$		−	0.323967i		−	0.00297218i	−0.999999	−	0.00148609i	$-0.999527\pi$
$109$		−	0.323967i		−	0.00297218i	0.999999	−	0.00148609i	$-0.000473037\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	143.741i		−	1.27204i	−0.771671	−	0.636022i	$-0.780579\pi$
$113$		−	143.741i		−	1.27204i	0.771671	−	0.636022i	$-0.219421\pi$
$114$		0			0
$115$	−51.4215	+	0.911954i	−0.447143	+	0.00793004i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−59.7182			−0.501834
$120$		0			0
$121$	−155.124			−1.28202
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	0.539714			0.00424971			0.00212486	−	0.999998i	$-0.499324\pi$
$127$	0.539714			0.00424971			0.00212486	+	0.999998i	$0.499324\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	230.200			1.75725			0.878627	−	0.477510i	$-0.158460\pi$
$131$	230.200			1.75725			0.878627	+	0.477510i	$0.158460\pi$
$132$		0			0
$133$	−211.964			−1.59372
$134$		0			0
$135$		0			0
$136$		0			0
$137$			136.988i			0.999912i	0.866051	+	0.499956i	$0.166651\pi$
$137$			136.988i			0.999912i	−0.866051	+	0.499956i	$0.833349\pi$
$138$		0			0
$139$	−76.8842			−0.553124			−0.276562	−	0.960996i	$-0.589195\pi$
$139$	−76.8842			−0.553124			−0.276562	+	0.960996i	$0.589195\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			67.6039i			0.472754i
$144$		0			0
$145$			8.03465i			0.0554114i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	135.525i		−	0.909562i	−0.890603	−	0.454781i	$-0.849718\pi$
$149$		−	135.525i		−	0.909562i	0.890603	−	0.454781i	$-0.150282\pi$
$150$		0			0
$151$	261.020			1.72861			0.864305	−	0.502969i	$-0.167759\pi$
$151$	261.020			1.72861			0.864305	+	0.502969i	$0.167759\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	40.4452i		−	0.260937i
$156$		0			0
$157$		−	247.968i		−	1.57941i	−0.613484	−	0.789707i	$-0.710233\pi$
$157$		−	247.968i		−	1.57941i	0.613484	−	0.789707i	$-0.289767\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	201.354	−	3.57099i	1.25065	−	0.0221801i
$162$		0			0
$163$	−101.295			−0.621444			−0.310722	−	0.950501i	$-0.600571\pi$
$163$	−101.295			−0.621444			−0.310722	+	0.950501i	$0.600571\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−22.0914			−0.132284			−0.0661419	−	0.997810i	$-0.521069\pi$
$167$	−22.0914			−0.132284			−0.0661419	+	0.997810i	$0.521069\pi$
$168$		0			0
$169$	−152.448			−0.902062
$170$		0			0
$171$		0			0
$172$		0			0
$173$	63.5786			0.367506			0.183753	−	0.982972i	$-0.441175\pi$
$173$	63.5786			0.367506			0.183753	+	0.982972i	$0.441175\pi$
$174$		0			0
$175$		−	43.7795i		−	0.250169i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	232.356			1.29808			0.649040	−	0.760754i	$-0.275171\pi$
$179$	232.356			1.29808			0.649040	+	0.760754i	$0.275171\pi$
$180$		0			0
$181$			162.593i			0.898303i	0.893456	+	0.449152i	$0.148274\pi$
$181$			162.593i			0.898303i	−0.893456	+	0.449152i	$0.851726\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.42752			0.0293379
$186$		0			0
$187$	113.333			0.606061
$188$		0			0
$189$		0			0
$190$		0			0
$191$			187.030i			0.979213i	0.871944	+	0.489606i	$0.162859\pi$
$191$			187.030i			0.979213i	−0.871944	+	0.489606i	$0.837141\pi$
$192$		0			0
$193$	236.066			1.22314			0.611569	−	0.791191i	$-0.290539\pi$
$193$	236.066			1.22314			0.611569	+	0.791191i	$0.290539\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−168.603			−0.855854			−0.427927	−	0.903813i	$-0.640756\pi$
$197$	−168.603			−0.855854			−0.427927	+	0.903813i	$0.640756\pi$
$198$		0			0
$199$			81.6522i			0.410313i	0.978729	+	0.205156i	$0.0657703\pi$
$199$			81.6522i			0.410313i	−0.978729	+	0.205156i	$0.934230\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	31.4617i		−	0.154984i
$204$		0			0
$205$		−	53.2397i		−	0.259706i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	402.267			1.92472
$210$		0			0
$211$	208.725			0.989217			0.494609	−	0.869116i	$-0.335311\pi$
$211$	208.725			0.989217			0.494609	+	0.869116i	$0.335311\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−35.4491			−0.164880
$216$		0			0
$217$			158.373i			0.729832i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	27.7476i		−	0.125555i
$222$		0			0
$223$	−130.446			−0.584958			−0.292479	−	0.956272i	$-0.594480\pi$
$223$	−130.446			−0.584958			−0.292479	+	0.956272i	$0.594480\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			315.921i			1.39172i	0.718176	+	0.695861i	$0.244977\pi$
$227$			315.921i			1.39172i	−0.718176	+	0.695861i	$0.755023\pi$
$228$		0			0
$229$			397.591i			1.73621i	0.496385	+	0.868103i	$0.334661\pi$
$229$			397.591i			1.73621i	−0.496385	+	0.868103i	$0.665339\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	287.974			1.23594			0.617971	−	0.786201i	$-0.287955\pi$
$233$	287.974			1.23594			0.617971	+	0.786201i	$0.287955\pi$
$234$		0			0
$235$			54.4548i			0.231722i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	87.1342			0.364578			0.182289	−	0.983245i	$-0.441649\pi$
$239$	87.1342			0.364578			0.182289	+	0.983245i	$0.441649\pi$
$240$		0			0
$241$		−	308.442i		−	1.27984i	−0.768441	−	0.639921i	$-0.778967\pi$
$241$		−	308.442i		−	1.27984i	0.768441	−	0.639921i	$-0.221033\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			61.8627i			0.252501i
$246$		0			0
$247$		−	98.4876i		−	0.398735i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			28.1041i			0.111969i	0.998432	+	0.0559843i	$0.0178297\pi$
$251$			28.1041i			0.111969i	−0.998432	+	0.0559843i	$0.982170\pi$
$252$		0			0
$253$	−382.130	+	6.77704i	−1.51040	+	0.0267867i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−232.393			−0.904255			−0.452127	−	0.891953i	$-0.649335\pi$
$257$	−232.393			−0.904255			−0.452127	+	0.891953i	$0.649335\pi$
$258$		0			0
$259$	−21.2529			−0.0820574
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	120.124i		−	0.456747i	−0.973574	−	0.228373i	$-0.926659\pi$
$263$		−	120.124i		−	0.456747i	0.973574	−	0.228373i	$-0.0733407\pi$
$264$		0			0
$265$	84.3104			0.318152
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−74.0358			−0.275226			−0.137613	−	0.990486i	$-0.543943\pi$
$269$	−74.0358			−0.275226			−0.137613	+	0.990486i	$0.543943\pi$
$270$		0			0
$271$	381.929			1.40933			0.704666	−	0.709539i	$-0.251097\pi$
$271$	381.929			1.40933			0.704666	+	0.709539i	$0.251097\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			83.0849i			0.302127i
$276$		0			0
$277$	−208.429			−0.752452			−0.376226	−	0.926528i	$-0.622778\pi$
$277$	−208.429			−0.752452			−0.376226	+	0.926528i	$0.622778\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3.52824i			0.0125560i	0.999980	+	0.00627800i	$0.00199836\pi$
$281$			3.52824i			0.0125560i	−0.999980	+	0.00627800i	$0.998002\pi$
$282$		0			0
$283$			170.463i			0.602344i	0.953570	+	0.301172i	$0.0973779\pi$
$283$			170.463i			0.602344i	−0.953570	+	0.301172i	$0.902622\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			208.474i			0.726390i
$288$		0			0
$289$	242.483			0.839042
$290$		0			0
$291$		0			0
$292$		0			0
$293$			319.088i			1.08904i	0.838749	+	0.544519i	$0.183288\pi$
$293$			319.088i			1.08904i	−0.838749	+	0.544519i	$0.816712\pi$
$294$		0			0
$295$			149.293i			0.506079i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.65923	+	93.5576i	0.00554928	+	0.312902i
$300$		0			0
$301$	138.810			0.461163
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−38.6386			−0.126684
$306$		0			0
$307$	351.884			1.14620			0.573101	−	0.819485i	$-0.305740\pi$
$307$	351.884			1.14620			0.573101	+	0.819485i	$0.305740\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	68.1579			0.219157			0.109579	−	0.993978i	$-0.465050\pi$
$311$	68.1579			0.219157			0.109579	+	0.993978i	$0.465050\pi$
$312$		0			0
$313$		−	168.431i		−	0.538117i	−0.963124	−	0.269058i	$-0.913288\pi$
$313$		−	168.431i		−	0.538117i	0.963124	−	0.269058i	$-0.0867124\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	519.008			1.63725			0.818625	−	0.574328i	$-0.194737\pi$
$317$	519.008			1.63725			0.818625	+	0.574328i	$0.194737\pi$
$318$		0			0
$319$			59.7082i			0.187173i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−165.108			−0.511170
$324$		0			0
$325$	20.3418			0.0625902
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	213.232i		−	0.648121i
$330$		0			0
$331$	544.399			1.64471			0.822355	−	0.568975i	$-0.192660\pi$
$331$	544.399			1.64471			0.822355	+	0.568975i	$0.192660\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−77.4259			−0.231122
$336$		0			0
$337$		−	331.555i		−	0.983843i	−0.870640	−	0.491922i	$-0.836295\pi$
$337$		−	331.555i		−	0.983843i	0.870640	−	0.491922i	$-0.163705\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	300.562i		−	0.881413i
$342$		0			0
$343$			186.800i			0.544606i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	240.842			0.694070			0.347035	−	0.937852i	$-0.387189\pi$
$347$	240.842			0.694070			0.347035	+	0.937852i	$0.387189\pi$
$348$		0			0
$349$	583.434			1.67173			0.835865	−	0.548935i	$-0.184967\pi$
$349$	583.434			1.67173			0.835865	+	0.548935i	$0.184967\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−144.124			−0.408283			−0.204142	−	0.978941i	$-0.565440\pi$
$353$	−144.124			−0.408283			−0.204142	+	0.978941i	$0.565440\pi$
$354$		0			0
$355$		−	199.457i		−	0.561852i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	395.272i		−	1.10104i	−0.834823	−	0.550518i	$-0.814430\pi$
$359$		−	395.272i		−	1.10104i	0.834823	−	0.550518i	$-0.185570\pi$
$360$		0			0
$361$	−225.036			−0.623368
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	44.4863i		−	0.121880i
$366$		0			0
$367$		−	213.726i		−	0.582360i	−0.956668	−	0.291180i	$-0.905952\pi$
$367$		−	213.726i		−	0.582360i	0.956668	−	0.291180i	$-0.0940479\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−330.139			−0.889863
$372$		0			0
$373$		−	2.66084i		−	0.00713362i	−0.999994	−	0.00356681i	$-0.998865\pi$
$373$		−	2.66084i		−	0.00713362i	0.999994	−	0.00356681i	$-0.00113535\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	14.6185			0.0387757
$378$		0			0
$379$			597.273i			1.57592i	0.615728	+	0.787959i	$0.288862\pi$
$379$			597.273i			1.57592i	−0.615728	+	0.787959i	$0.711138\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	24.4361i		−	0.0638019i	−0.999491	−	0.0319009i	$-0.989844\pi$
$383$		−	24.4361i		−	0.0638019i	0.999491	−	0.0319009i	$-0.0101561\pi$
$384$		0			0
$385$		−	325.340i		−	0.845040i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	186.463i		−	0.479340i	−0.970854	−	0.239670i	$-0.922961\pi$
$389$		−	186.463i		−	0.479340i	0.970854	−	0.239670i	$-0.0770392\pi$
$390$		0			0
$391$	156.843	−	2.78160i	0.401133	−	0.00711405i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	45.4262			0.115003
$396$		0			0
$397$	−162.586			−0.409537			−0.204769	−	0.978810i	$-0.565644\pi$
$397$	−162.586			−0.409537			−0.204769	+	0.978810i	$0.565644\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	223.488i		−	0.557328i	−0.960389	−	0.278664i	$-0.910109\pi$
$401$		−	223.488i		−	0.557328i	0.960389	−	0.278664i	$-0.0898915\pi$
$402$		0			0
$403$	−73.5870			−0.182598
$404$		0			0
$405$		0			0
$406$		0			0
$407$	40.3337			0.0991001
$408$		0			0
$409$	669.739			1.63750			0.818752	−	0.574147i	$-0.194666\pi$
$409$	669.739			1.63750			0.818752	+	0.574147i	$0.194666\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	584.596i		−	1.41549i
$414$		0			0
$415$	209.128			0.503922
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	800.096i		−	1.90954i	−0.297352	−	0.954768i	$-0.596103\pi$
$419$		−	800.096i		−	1.90954i	0.297352	−	0.954768i	$-0.403897\pi$
$420$		0			0
$421$		−	464.919i		−	1.10432i	−0.833738	−	0.552160i	$-0.813804\pi$
$421$		−	464.919i		−	1.10432i	0.833738	−	0.552160i	$-0.186196\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	34.1017i		−	0.0802393i
$426$		0			0
$427$	151.299			0.354331
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	618.727i		−	1.43556i	−0.696269	−	0.717781i	$-0.745158\pi$
$431$		−	618.727i		−	1.43556i	0.696269	−	0.717781i	$-0.254842\pi$
$432$		0			0
$433$		−	140.163i		−	0.323703i	−0.986815	−	0.161852i	$-0.948253\pi$
$433$		−	140.163i		−	0.323703i	0.986815	−	0.161852i	$-0.0517466\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	556.700	−	9.87302i	1.27391	−	0.0225927i
$438$		0			0
$439$	268.392			0.611371			0.305685	−	0.952133i	$-0.401114\pi$
$439$	268.392			0.611371			0.305685	+	0.952133i	$0.401114\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	59.8523			0.135107			0.0675534	−	0.997716i	$-0.478481\pi$
$443$	59.8523			0.135107			0.0675534	+	0.997716i	$0.478481\pi$
$444$		0			0
$445$	−212.108			−0.476648
$446$		0			0
$447$		0			0
$448$		0			0
$449$	158.627			0.353289			0.176644	−	0.984275i	$-0.443476\pi$
$449$	158.627			0.353289			0.176644	+	0.984275i	$0.443476\pi$
$450$		0			0
$451$		−	395.642i		−	0.877256i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−79.6536			−0.175063
$456$		0			0
$457$		−	532.128i		−	1.16439i	−0.813048	−	0.582197i	$-0.802193\pi$
$457$		−	532.128i		−	1.16439i	0.813048	−	0.582197i	$-0.197807\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−252.930			−0.548655			−0.274327	−	0.961636i	$-0.588455\pi$
$461$	−252.930			−0.548655			−0.274327	+	0.961636i	$0.588455\pi$
$462$		0			0
$463$	864.114			1.86634			0.933169	−	0.359439i	$-0.117032\pi$
$463$	864.114			1.86634			0.933169	+	0.359439i	$0.117032\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			496.118i			1.06235i	0.847262	+	0.531176i	$0.178250\pi$
$467$			496.118i			1.06235i	−0.847262	+	0.531176i	$0.821750\pi$
$468$		0			0
$469$	303.181			0.646441
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−263.434			−0.556944
$474$		0			0
$475$		−	121.041i		−	0.254823i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			387.267i			0.808490i	0.914651	+	0.404245i	$0.132466\pi$
$479$			387.267i			0.808490i	−0.914651	+	0.404245i	$0.867534\pi$
$480$		0			0
$481$		−	9.87497i		−	0.0205301i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	213.464			0.440133
$486$		0			0
$487$	−227.685			−0.467527			−0.233763	−	0.972294i	$-0.575104\pi$
$487$	−227.685			−0.467527			−0.233763	+	0.972294i	$0.575104\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	411.634			0.838358			0.419179	−	0.907904i	$-0.362318\pi$
$491$	411.634			0.838358			0.419179	+	0.907904i	$0.362318\pi$
$492$		0			0
$493$		−	24.5069i		−	0.0497097i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			781.027i			1.57148i
$498$		0			0
$499$	−308.271			−0.617778			−0.308889	−	0.951098i	$-0.599957\pi$
$499$	−308.271			−0.617778			−0.308889	+	0.951098i	$0.599957\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	617.324i		−	1.22728i	−0.789584	−	0.613642i	$-0.789704\pi$
$503$		−	617.324i		−	1.22728i	0.789584	−	0.613642i	$-0.210296\pi$
$504$		0			0
$505$		−	421.561i		−	0.834774i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	166.710			0.327525			0.163762	−	0.986500i	$-0.447637\pi$
$509$	166.710			0.327525			0.163762	+	0.986500i	$0.447637\pi$
$510$		0			0
$511$			174.198i			0.340895i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−57.3895			−0.111436
$516$		0			0
$517$			404.672i			0.782731i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			797.142i			1.53002i	0.644016	+	0.765012i	$0.277267\pi$
$521$			797.142i			1.53002i	−0.644016	+	0.765012i	$0.722733\pi$
$522$		0			0
$523$			382.596i			0.731540i	0.930705	+	0.365770i	$0.119194\pi$
$523$			382.596i			0.731540i	−0.930705	+	0.365770i	$0.880806\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			123.364i			0.234087i
$528$		0			0
$529$	−528.667	+	18.7576i	−0.999371	+	0.0354586i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−96.8657			−0.181737
$534$		0			0
$535$	−22.0802			−0.0412715
$536$		0			0
$537$		0			0
$538$		0			0
$539$			459.723i			0.852918i
$540$		0			0
$541$	−495.570			−0.916025			−0.458012	−	0.888946i	$-0.651439\pi$
$541$	−495.570			−0.916025			−0.458012	+	0.888946i	$0.651439\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.724413			−0.00132920
$546$		0			0
$547$	465.624			0.851233			0.425616	−	0.904904i	$-0.360057\pi$
$547$	465.624			0.851233			0.425616	+	0.904904i	$0.360057\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	86.9849i		−	0.157867i
$552$		0			0
$553$	−177.878			−0.321660
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	31.8462i		−	0.0571745i	−0.999591	−	0.0285873i	$-0.990899\pi$
$557$		−	31.8462i		−	0.0571745i	0.999591	−	0.0285873i	$-0.00910085\pi$
$558$		0			0
$559$			64.4970i			0.115379i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			618.955i			1.09939i	0.835367	+	0.549693i	$0.185255\pi$
$563$			618.955i			1.09939i	−0.835367	+	0.549693i	$0.814745\pi$
$564$		0			0
$565$	−321.415			−0.568875
$566$		0			0
$567$		0			0
$568$		0			0
$569$			80.8333i			0.142062i	0.997474	+	0.0710310i	$0.0226289\pi$
$569$			80.8333i			0.142062i	−0.997474	+	0.0710310i	$0.977371\pi$
$570$		0			0
$571$			239.307i			0.419101i	0.977798	+	0.209550i	$0.0672000\pi$
$571$			239.307i			0.419101i	−0.977798	+	0.209550i	$0.932800\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.03919	+	114.982i	0.00354642	+	0.199969i
$576$		0			0
$577$	433.630			0.751525			0.375762	−	0.926716i	$-0.377381\pi$
$577$	433.630			0.751525			0.375762	+	0.926716i	$0.377381\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−818.893			−1.40946
$582$		0			0
$583$	626.539			1.07468
$584$		0			0
$585$		0			0
$586$		0			0
$587$	487.008			0.829656			0.414828	−	0.909900i	$-0.363842\pi$
$587$	487.008			0.829656			0.414828	+	0.909900i	$0.363842\pi$
$588$		0			0
$589$			437.868i			0.743410i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−66.3106			−0.111822			−0.0559111	−	0.998436i	$-0.517806\pi$
$593$	−66.3106			−0.111822			−0.0559111	+	0.998436i	$0.517806\pi$
$594$		0			0
$595$			133.534i			0.224427i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−275.622			−0.460137			−0.230068	−	0.973174i	$-0.573895\pi$
$599$	−275.622			−0.460137			−0.230068	+	0.973174i	$0.573895\pi$
$600$		0			0
$601$	−122.738			−0.204223			−0.102112	−	0.994773i	$-0.532560\pi$
$601$	−122.738			−0.204223			−0.102112	+	0.994773i	$0.532560\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			346.868i			0.573336i
$606$		0			0
$607$	1117.38			1.84082			0.920411	−	0.390951i	$-0.127854\pi$
$607$	1117.38			1.84082			0.920411	+	0.390951i	$0.127854\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	99.0764			0.162155
$612$		0			0
$613$			303.835i			0.495652i	0.968805	+	0.247826i	$0.0797161\pi$
$613$			303.835i			0.495652i	−0.968805	+	0.247826i	$0.920284\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			56.8354i			0.0921158i	0.998939	+	0.0460579i	$0.0146659\pi$
$617$			56.8354i			0.0921158i	−0.998939	+	0.0460579i	$0.985334\pi$
$618$		0			0
$619$			164.762i			0.266174i	0.991104	+	0.133087i	$0.0424890\pi$
$619$			164.762i			0.266174i	−0.991104	+	0.133087i	$0.957511\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	830.564			1.33317
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−16.5547			−0.0263191
$630$		0			0
$631$			121.020i			0.191790i	0.995391	+	0.0958952i	$0.0305714\pi$
$631$			121.020i			0.191790i	−0.995391	+	0.0958952i	$0.969429\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	1.20684i		−	0.00190053i
$636$		0			0
$637$	112.555			0.176695
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	850.798i		−	1.32730i	−0.748044	−	0.663649i	$-0.769007\pi$
$641$		−	850.798i		−	1.32730i	0.748044	−	0.663649i	$-0.230993\pi$
$642$		0			0
$643$			1226.41i			1.90733i	0.300866	+	0.953666i	$0.402724\pi$
$643$			1226.41i			1.90733i	−0.300866	+	0.953666i	$0.597276\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−234.413			−0.362307			−0.181154	−	0.983455i	$-0.557983\pi$
$647$	−234.413			−0.362307			−0.181154	+	0.983455i	$0.557983\pi$
$648$		0			0
$649$			1109.45i			1.70947i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	478.868			0.733335			0.366667	−	0.930352i	$-0.380499\pi$
$653$	478.868			0.733335			0.366667	+	0.930352i	$0.380499\pi$
$654$		0			0
$655$		−	514.743i		−	0.785867i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			91.3047i			0.138550i	0.997598	+	0.0692752i	$0.0220686\pi$
$659$			91.3047i			0.138550i	−0.997598	+	0.0692752i	$0.977931\pi$
$660$		0			0
$661$			742.457i			1.12323i	0.827398	+	0.561616i	$0.189820\pi$
$661$			742.457i			1.12323i	−0.827398	+	0.561616i	$0.810180\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			473.967i			0.712732i
$666$		0			0
$667$	1.46545	+	82.6307i	0.00219707	+	0.123884i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−287.136			−0.427923
$672$		0			0
$673$	−387.832			−0.576273			−0.288136	−	0.957589i	$-0.593036\pi$
$673$	−387.832			−0.576273			−0.288136	+	0.957589i	$0.593036\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			10.3505i			0.0152887i	0.999971	+	0.00764437i	$0.00243330\pi$
$677$			10.3505i			0.0152887i	−0.999971	+	0.00764437i	$0.997567\pi$
$678$		0			0
$679$	−835.875			−1.23104
$680$		0			0
$681$		0			0
$682$		0			0
$683$	413.572			0.605523			0.302762	−	0.953066i	$-0.402091\pi$
$683$	413.572			0.605523			0.302762	+	0.953066i	$0.402091\pi$
$684$		0			0
$685$	306.314			0.447174
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	153.397i		−	0.222637i
$690$		0			0
$691$	−936.023			−1.35459			−0.677296	−	0.735711i	$-0.736848\pi$
$691$	−936.023			−1.35459			−0.677296	+	0.735711i	$0.736848\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			171.918i			0.247364i
$696$		0			0
$697$			162.389i			0.232983i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	297.821i		−	0.424851i	−0.977177	−	0.212426i	$-0.931864\pi$
$701$		−	297.821i		−	0.424851i	0.977177	−	0.212426i	$-0.0681363\pi$
$702$		0			0
$703$	−58.7595			−0.0835840
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1650.73i			2.33484i
$708$		0			0
$709$			611.365i			0.862292i	0.902282	+	0.431146i	$0.141891\pi$
$709$			611.365i			0.862292i	−0.902282	+	0.431146i	$0.858109\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.37683	−	415.950i	−0.0103462	−	0.583380i
$714$		0			0
$715$	151.167			0.211422
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−326.454			−0.454039			−0.227020	−	0.973890i	$-0.572898\pi$
$719$	−326.454			−0.454039			−0.227020	+	0.973890i	$0.572898\pi$
$720$		0			0
$721$	224.723			0.311683
$722$		0			0
$723$		0			0
$724$		0			0
$725$	17.9660			0.0247807
$726$		0			0
$727$		−	915.246i		−	1.25894i	−0.777027	−	0.629468i	$-0.783273\pi$
$727$		−	915.246i		−	1.25894i	0.777027	−	0.629468i	$-0.216727\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	108.125			0.147914
$732$		0			0
$733$			411.079i			0.560817i	0.959881	+	0.280409i	$0.0904699\pi$
$733$			411.079i			0.560817i	−0.959881	+	0.280409i	$0.909530\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−575.378			−0.780703
$738$		0			0
$739$	531.025			0.718573			0.359286	−	0.933227i	$-0.383020\pi$
$739$	531.025			0.718573			0.359286	+	0.933227i	$0.383020\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	915.120i		−	1.23165i	−0.787881	−	0.615827i	$-0.788822\pi$
$743$		−	915.120i		−	1.23165i	0.787881	−	0.615827i	$-0.211178\pi$
$744$		0			0
$745$	−303.042			−0.406768
$746$		0			0
$747$		0			0
$748$		0			0
$749$	86.4609			0.115435
$750$		0			0
$751$		−	930.746i		−	1.23934i	−0.784861	−	0.619671i	$-0.787266\pi$
$751$		−	930.746i		−	1.23934i	0.784861	−	0.619671i	$-0.212734\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	583.658i		−	0.773058i
$756$		0			0
$757$			555.566i			0.733905i	0.930240	+	0.366953i	$0.119599\pi$
$757$			555.566i			0.733905i	−0.930240	+	0.366953i	$0.880401\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	716.027			0.940902			0.470451	−	0.882426i	$-0.344091\pi$
$761$	716.027			0.940902			0.470451	+	0.882426i	$0.344091\pi$
$762$		0			0
$763$	2.83663			0.00371773
$764$		0			0
$765$		0			0
$766$		0			0
$767$	271.628			0.354143
$768$		0			0
$769$		−	586.636i		−	0.762856i	−0.924398	−	0.381428i	$-0.875432\pi$
$769$		−	586.636i		−	0.762856i	0.924398	−	0.381428i	$-0.124568\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			1249.30i			1.61618i	0.589062	+	0.808088i	$0.299497\pi$
$773$			1249.30i			1.61618i	−0.589062	+	0.808088i	$0.700503\pi$
$774$		0			0
$775$	−90.4381			−0.116694
$776$		0			0
$777$		0			0
$778$		0			0
$779$			576.385i			0.739904i
$780$		0			0
$781$		−	1482.24i		−	1.89787i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−554.473			−0.706335
$786$		0			0
$787$			565.736i			0.718851i	0.933174	+	0.359426i	$0.117027\pi$
$787$			565.736i			0.718851i	−0.933174	+	0.359426i	$0.882973\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1258.58			1.59113
$792$		0			0
$793$			70.3000i			0.0886507i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			153.101i			0.192097i	0.995377	+	0.0960484i	$0.0306203\pi$
$797$			153.101i			0.192097i	−0.995377	+	0.0960484i	$0.969380\pi$
$798$		0			0
$799$		−	166.095i		−	0.207879i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	330.593i		−	0.411697i
$804$		0			0
$805$	−7.98498	−	450.241i	−0.00991924	−	0.559306i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	970.408			1.19952			0.599758	−	0.800182i	$-0.295264\pi$
$809$	970.408			1.19952			0.599758	+	0.800182i	$0.295264\pi$
$810$		0			0
$811$	−443.122			−0.546390			−0.273195	−	0.961959i	$-0.588080\pi$
$811$	−443.122			−0.546390			−0.273195	+	0.961959i	$0.588080\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			226.503i			0.277918i
$816$		0			0
$817$	383.780			0.469743
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1460.92			−1.77944			−0.889722	−	0.456502i	$-0.849102\pi$
$821$	−1460.92			−1.77944			−0.889722	+	0.456502i	$0.849102\pi$
$822$		0			0
$823$	1138.38			1.38320			0.691601	−	0.722279i	$-0.256906\pi$
$823$	1138.38			1.38320			0.691601	+	0.722279i	$0.256906\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			8.50521i			0.0102844i	0.999987	+	0.00514221i	$0.00163682\pi$
$827$			8.50521i			0.0102844i	−0.999987	+	0.00514221i	$0.998363\pi$
$828$		0			0
$829$	−474.404			−0.572261			−0.286130	−	0.958191i	$-0.592369\pi$
$829$	−474.404			−0.572261			−0.286130	+	0.958191i	$0.592369\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	188.690i		−	0.226519i
$834$		0			0
$835$			49.3978i			0.0591591i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1394.78i		−	1.66243i	−0.555953	−	0.831214i	$-0.687646\pi$
$839$		−	1394.78i		−	1.66243i	0.555953	−	0.831214i	$-0.312354\pi$
$840$		0			0
$841$	−828.089			−0.984648
$842$		0			0
$843$		0			0
$844$		0			0
$845$			340.885i			0.403414i
$846$		0			0
$847$		−	1358.25i		−	1.60360i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	55.8182	−	0.989930i	0.0655913	−	0.00116326i
$852$		0			0
$853$	−677.350			−0.794080			−0.397040	−	0.917801i	$-0.629963\pi$
$853$	−677.350			−0.794080			−0.397040	+	0.917801i	$0.629963\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−516.881			−0.603129			−0.301564	−	0.953446i	$-0.597509\pi$
$857$	−516.881			−0.603129			−0.301564	+	0.953446i	$0.597509\pi$
$858$		0			0
$859$	−598.549			−0.696797			−0.348399	−	0.937346i	$-0.613274\pi$
$859$	−598.549			−0.696797			−0.348399	+	0.937346i	$0.613274\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	104.216			0.120760			0.0603799	−	0.998175i	$-0.480769\pi$
$863$	104.216			0.120760			0.0603799	+	0.998175i	$0.480769\pi$
$864$		0			0
$865$		−	142.166i		−	0.164354i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	337.577			0.388466
$870$		0			0
$871$			140.871i			0.161734i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−97.8940			−0.111879
$876$		0			0
$877$	−1443.37			−1.64580			−0.822900	−	0.568186i	$-0.807646\pi$
$877$	−1443.37			−1.64580			−0.822900	+	0.568186i	$0.807646\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			621.206i			0.705114i	0.935790	+	0.352557i	$0.114688\pi$
$881$			621.206i			0.705114i	−0.935790	+	0.352557i	$0.885312\pi$
$882$		0			0
$883$	67.3745			0.0763019			0.0381509	−	0.999272i	$-0.487853\pi$
$883$	67.3745			0.0763019			0.0381509	+	0.999272i	$0.487853\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−278.838			−0.314361			−0.157181	−	0.987570i	$-0.550240\pi$
$887$	−278.838			−0.314361			−0.157181	+	0.987570i	$0.550240\pi$
$888$		0			0
$889$			4.72568i			0.00531572i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	589.539i		−	0.660179i
$894$		0			0
$895$		−	519.565i		−	0.580519i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−64.9925			−0.0722943
$900$		0			0
$901$	−257.159			−0.285415
$902$		0			0
$903$		0			0
$904$		0			0
$905$	363.569			0.401733
$906$		0			0
$907$		−	37.9757i		−	0.0418695i	−0.999781	−	0.0209348i	$-0.993336\pi$
$907$		−	37.9757i		−	0.0418695i	0.999781	−	0.0209348i	$-0.00666423\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	507.088i		−	0.556628i	−0.960490	−	0.278314i	$-0.910224\pi$
$911$		−	507.088i		−	0.556628i	0.960490	−	0.278314i	$-0.0897756\pi$
$912$		0			0
$913$	1554.10			1.70219
$914$		0			0
$915$		0			0
$916$		0			0
$917$			2015.61i			2.19805i
$918$		0			0
$919$		−	276.526i		−	0.300899i	−0.988618	−	0.150449i	$-0.951928\pi$
$919$		−	276.526i		−	0.300899i	0.988618	−	0.150449i	$-0.0480720\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−362.898			−0.393172
$924$		0			0
$925$		−	12.1363i		−	0.0131203i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	299.273			0.322145			0.161073	−	0.986943i	$-0.448505\pi$
$929$	299.273			0.322145			0.161073	+	0.986943i	$0.448505\pi$
$930$		0			0
$931$		−	669.739i		−	0.719376i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	253.421i		−	0.271039i
$936$		0			0
$937$			245.487i			0.261993i	0.991383	+	0.130996i	$0.0418176\pi$
$937$			245.487i			0.261993i	−0.991383	+	0.130996i	$0.958182\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	120.178i		−	0.127713i	−0.997959	−	0.0638566i	$-0.979660\pi$
$941$		−	120.178i		−	0.127713i	0.997959	−	0.0638566i	$-0.0203400\pi$
$942$		0			0
$943$	−9.71044	−	547.533i	−0.0102974	−	0.580629i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1449.83			1.53098			0.765488	−	0.643451i	$-0.222498\pi$
$947$	1449.83			1.53098			0.765488	+	0.643451i	$0.222498\pi$
$948$		0			0
$949$	−80.9395			−0.0852893
$950$		0			0
$951$		0			0
$952$		0			0
$953$			437.649i			0.459232i	0.973281	+	0.229616i	$0.0737471\pi$
$953$			437.649i			0.459232i	−0.973281	+	0.229616i	$0.926253\pi$
$954$		0			0
$955$	418.211			0.437917
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1199.45			−1.25073
$960$		0			0
$961$	−633.838			−0.659561
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	527.859i		−	0.547004i
$966$		0			0
$967$	76.8365			0.0794586			0.0397293	−	0.999210i	$-0.487350\pi$
$967$	76.8365			0.0794586			0.0397293	+	0.999210i	$0.487350\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	559.459i		−	0.576168i	−0.957605	−	0.288084i	$-0.906982\pi$
$971$		−	559.459i		−	0.576168i	0.957605	−	0.288084i	$-0.0930182\pi$
$972$		0			0
$973$		−	673.191i		−	0.691871i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1144.07i		−	1.17100i	−0.810673	−	0.585499i	$-0.800899\pi$
$977$		−	1144.07i		−	1.17100i	0.810673	−	0.585499i	$-0.199101\pi$
$978$		0			0
$979$	−1576.25			−1.61006
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1026.66i		−	1.04442i	−0.852818	−	0.522208i	$-0.825108\pi$
$983$		−	1026.66i		−	1.04442i	0.852818	−	0.522208i	$-0.174892\pi$
$984$		0			0
$985$			377.008i			0.382749i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−364.569	+	6.46559i	−0.368624	+	0.00653750i
$990$		0			0
$991$	−895.905			−0.904042			−0.452021	−	0.892007i	$-0.649297\pi$
$991$	−895.905			−0.904042			−0.452021	+	0.892007i	$0.649297\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	182.580			0.183497
$996$		0			0
$997$	−1886.02			−1.89169			−0.945847	−	0.324614i	$-0.894766\pi$
$997$	−1886.02			−1.89169			−0.945847	+	0.324614i	$0.894766\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.d.c.2161.12		32
3.2	odd	2		1380.3.d.a.781.15	yes	32
23.22	odd	2	inner	4140.3.d.c.2161.21		32
69.68	even	2		1380.3.d.a.781.2	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.3.d.a.781.2	✓	32	69.68	even	2
1380.3.d.a.781.15	yes	32	3.2	odd	2
4140.3.d.c.2161.12		32	1.1	even	1	trivial
4140.3.d.c.2161.21		32	23.22	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4140.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +8.75590i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} +8.75590i q^{7} -16.6170i q^{11} -4.06836 q^{13} +6.82034i q^{17} +24.2082i q^{19} +(-0.407838 - 22.9964i) q^{23} -5.00000 q^{25} -3.59320 q^{29} +18.0876 q^{31} +19.5788 q^{35} +2.42726i q^{37} +23.8095 q^{41} -15.8533i q^{43} -24.3529 q^{47} -27.6658 q^{49} +37.7048i q^{53} -37.1567 q^{55} -66.7659 q^{59} -17.2797i q^{61} +9.09713i q^{65} -34.6259i q^{67} +89.2001 q^{71} +19.8949 q^{73} +145.497 q^{77} +20.3152i q^{79} +93.5247i q^{83} +15.2507 q^{85} -94.8577i q^{89} -35.6222i q^{91} +54.1311 q^{95} +95.4642i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 24 q^{13} - 64 q^{23} - 160 q^{25} + 60 q^{29} - 4 q^{31} + 60 q^{35} + 108 q^{41} - 136 q^{47} - 428 q^{49} + 120 q^{55} + 84 q^{59} - 188 q^{71} + 472 q^{73} + 120 q^{77} + 60 q^{85} - 80 q^{95}+O(q^{100}) \) 32 * q + 24 * q^13 - 64 * q^23 - 160 * q^25 + 60 * q^29 - 4 * q^31 + 60 * q^35 + 108 * q^41 - 136 * q^47 - 428 * q^49 + 120 * q^55 + 84 * q^59 - 188 * q^71 + 472 * q^73 + 120 * q^77 + 60 * q^85 - 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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