LMFDB - Embedded newform 4140.3.d.c.2161.11

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			2161.11
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.c.2161.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +4.95389i q^{7} +O(q^{10})$	$q-2.23607i q^{5} +4.95389i q^{7} +10.4673i q^{11} -12.6514 q^{13} -22.4509i q^{17} -11.3001i q^{19} +(-22.4154 - 5.15265i) q^{23} -5.00000 q^{25} +40.2069 q^{29} -11.9214 q^{31} +11.0772 q^{35} -13.6705i q^{37} +24.9888 q^{41} -8.14323i q^{43} -31.5359 q^{47} +24.4590 q^{49} +53.4925i q^{53} +23.4056 q^{55} +99.1504 q^{59} -88.9543i q^{61} +28.2893i q^{65} +126.262i q^{67} -54.6003 q^{71} +98.8788 q^{73} -51.8539 q^{77} +47.2228i q^{79} +38.9940i q^{83} -50.2017 q^{85} +56.1888i q^{89} -62.6734i q^{91} -25.2677 q^{95} -83.0908i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			4.95389i			0.707698i	0.935303	+	0.353849i	$0.115127\pi$
$7$			4.95389i			0.707698i	−0.935303	+	0.353849i	$0.884873\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			10.4673i			0.951574i	0.879560	+	0.475787i	$0.157837\pi$
$11$			10.4673i			0.951574i	−0.879560	+	0.475787i	$0.842163\pi$
$12$		0			0
$13$	−12.6514			−0.973181			−0.486591	−	0.873630i	$-0.661760\pi$
$13$	−12.6514			−0.973181			−0.486591	+	0.873630i	$0.661760\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	22.4509i		−	1.32064i	−0.750984	−	0.660320i	$-0.770421\pi$
$17$		−	22.4509i		−	1.32064i	0.750984	−	0.660320i	$-0.229579\pi$
$18$		0			0
$19$		−	11.3001i		−	0.594740i	−0.954762	−	0.297370i	$-0.903891\pi$
$19$		−	11.3001i		−	0.594740i	0.954762	−	0.297370i	$-0.0961095\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−22.4154	−	5.15265i	−0.974583	−	0.224028i
$23$	−22.4154	−	5.15265i	−0.974583	−	0.224028i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	40.2069			1.38644			0.693222	−	0.720725i	$-0.256191\pi$
$29$	40.2069			1.38644			0.693222	+	0.720725i	$0.256191\pi$
$30$		0			0
$31$	−11.9214			−0.384561			−0.192281	−	0.981340i	$-0.561588\pi$
$31$	−11.9214			−0.384561			−0.192281	+	0.981340i	$0.561588\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	11.0772			0.316492
$36$		0			0
$37$		−	13.6705i		−	0.369474i	−0.982788	−	0.184737i	$-0.940857\pi$
$37$		−	13.6705i		−	0.369474i	0.982788	−	0.184737i	$-0.0591433\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	24.9888			0.609482			0.304741	−	0.952435i	$-0.401430\pi$
$41$	24.9888			0.609482			0.304741	+	0.952435i	$0.401430\pi$
$42$		0			0
$43$		−	8.14323i		−	0.189377i	−0.995507	−	0.0946887i	$-0.969814\pi$
$43$		−	8.14323i		−	0.189377i	0.995507	−	0.0946887i	$-0.0301856\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−31.5359			−0.670976			−0.335488	−	0.942045i	$-0.608901\pi$
$47$	−31.5359			−0.670976			−0.335488	+	0.942045i	$0.608901\pi$
$48$		0			0
$49$	24.4590			0.499163
$50$		0			0
$51$		0			0
$52$		0			0
$53$			53.4925i			1.00929i	0.863326	+	0.504646i	$0.168377\pi$
$53$			53.4925i			1.00929i	−0.863326	+	0.504646i	$0.831623\pi$
$54$		0			0
$55$	23.4056			0.425557
$56$		0			0
$57$		0			0
$58$		0			0
$59$	99.1504			1.68051			0.840257	−	0.542188i	$-0.182404\pi$
$59$	99.1504			1.68051			0.840257	+	0.542188i	$0.182404\pi$
$60$		0			0
$61$		−	88.9543i		−	1.45827i	−0.684371	−	0.729134i	$-0.739923\pi$
$61$		−	88.9543i		−	1.45827i	0.684371	−	0.729134i	$-0.260077\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			28.2893i			0.435220i
$66$		0			0
$67$			126.262i			1.88451i	0.334897	+	0.942255i	$0.391299\pi$
$67$			126.262i			1.88451i	−0.334897	+	0.942255i	$0.608701\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−54.6003			−0.769018			−0.384509	−	0.923121i	$-0.625629\pi$
$71$	−54.6003			−0.769018			−0.384509	+	0.923121i	$0.625629\pi$
$72$		0			0
$73$	98.8788			1.35450			0.677252	−	0.735751i	$-0.263171\pi$
$73$	98.8788			1.35450			0.677252	+	0.735751i	$0.263171\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−51.8539			−0.673427
$78$		0			0
$79$			47.2228i			0.597756i	0.954291	+	0.298878i	$0.0966124\pi$
$79$			47.2228i			0.597756i	−0.954291	+	0.298878i	$0.903388\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			38.9940i			0.469807i	0.972019	+	0.234903i	$0.0754774\pi$
$83$			38.9940i			0.469807i	−0.972019	+	0.234903i	$0.924523\pi$
$84$		0			0
$85$	−50.2017			−0.590608
$86$		0			0
$87$		0			0
$88$		0			0
$89$			56.1888i			0.631335i	0.948870	+	0.315668i	$0.102228\pi$
$89$			56.1888i			0.631335i	−0.948870	+	0.315668i	$0.897772\pi$
$90$		0			0
$91$		−	62.6734i		−	0.688719i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−25.2677			−0.265976
$96$		0			0
$97$		−	83.0908i		−	0.856606i	−0.903635	−	0.428303i	$-0.859112\pi$
$97$		−	83.0908i		−	0.856606i	0.903635	−	0.428303i	$-0.140888\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	92.9981			0.920773			0.460387	−	0.887718i	$-0.347711\pi$
$101$	92.9981			0.920773			0.460387	+	0.887718i	$0.347711\pi$
$102$		0			0
$103$		−	106.509i		−	1.03406i	−0.855966	−	0.517032i	$-0.827037\pi$
$103$		−	106.509i		−	1.03406i	0.855966	−	0.517032i	$-0.172963\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			174.424i			1.63013i	0.579366	+	0.815067i	$0.303300\pi$
$107$			174.424i			1.63013i	−0.579366	+	0.815067i	$0.696700\pi$
$108$		0			0
$109$		−	31.9315i		−	0.292950i	−0.989214	−	0.146475i	$-0.953207\pi$
$109$		−	31.9315i		−	0.292950i	0.989214	−	0.146475i	$-0.0467927\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			224.239i			1.98442i	0.124583	+	0.992209i	$0.460241\pi$
$113$			224.239i			1.98442i	−0.124583	+	0.992209i	$0.539759\pi$
$114$		0			0
$115$	−11.5217	+	50.1224i	−0.100188	+	0.435847i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	111.219			0.934615
$120$		0			0
$121$	11.4353			0.0945067
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−213.610			−1.68197			−0.840985	−	0.541059i	$-0.818024\pi$
$127$	−213.610			−1.68197			−0.840985	+	0.541059i	$0.818024\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−119.321			−0.910845			−0.455423	−	0.890275i	$-0.650512\pi$
$131$	−119.321			−0.910845			−0.455423	+	0.890275i	$0.650512\pi$
$132$		0			0
$133$	55.9792			0.420896
$134$		0			0
$135$		0			0
$136$		0			0
$137$			231.465i			1.68953i	0.535141	+	0.844763i	$0.320259\pi$
$137$			231.465i			1.68953i	−0.535141	+	0.844763i	$0.679741\pi$
$138$		0			0
$139$	−168.368			−1.21128			−0.605641	−	0.795738i	$-0.707083\pi$
$139$	−168.368			−1.21128			−0.605641	+	0.795738i	$0.707083\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	132.426i		−	0.926054i
$144$		0			0
$145$		−	89.9053i		−	0.620036i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			204.197i			1.37045i	0.728331	+	0.685225i	$0.240296\pi$
$149$			204.197i			1.37045i	−0.728331	+	0.685225i	$0.759704\pi$
$150$		0			0
$151$	−103.969			−0.688535			−0.344267	−	0.938872i	$-0.611873\pi$
$151$	−103.969			−0.688535			−0.344267	+	0.938872i	$0.611873\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			26.6571i			0.171981i
$156$		0			0
$157$		−	35.7851i		−	0.227931i	−0.993485	−	0.113965i	$-0.963645\pi$
$157$		−	35.7851i		−	0.227931i	0.993485	−	0.113965i	$-0.0363553\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	25.5256	−	111.043i	0.158544	−	0.689710i
$162$		0			0
$163$	−302.615			−1.85653			−0.928267	−	0.371915i	$-0.878701\pi$
$163$	−302.615			−1.85653			−0.928267	+	0.371915i	$0.878701\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−109.541			−0.655935			−0.327967	−	0.944689i	$-0.606364\pi$
$167$	−109.541			−0.655935			−0.327967	+	0.944689i	$0.606364\pi$
$168$		0			0
$169$	−8.94316			−0.0529181
$170$		0			0
$171$		0			0
$172$		0			0
$173$	305.451			1.76561			0.882806	−	0.469739i	$-0.155652\pi$
$173$	305.451			1.76561			0.882806	+	0.469739i	$0.155652\pi$
$174$		0			0
$175$		−	24.7694i		−	0.141540i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−171.695			−0.959192			−0.479596	−	0.877489i	$-0.659217\pi$
$179$	−171.695			−0.959192			−0.479596	+	0.877489i	$0.659217\pi$
$180$		0			0
$181$			46.3083i			0.255847i	0.991784	+	0.127923i	$0.0408312\pi$
$181$			46.3083i			0.255847i	−0.991784	+	0.127923i	$0.959169\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−30.5683			−0.165234
$186$		0			0
$187$	235.000			1.25669
$188$		0			0
$189$		0			0
$190$		0			0
$191$			103.122i			0.539907i	0.962873	+	0.269954i	$0.0870084\pi$
$191$			103.122i			0.539907i	−0.962873	+	0.269954i	$0.912992\pi$
$192$		0			0
$193$	−67.8233			−0.351416			−0.175708	−	0.984442i	$-0.556221\pi$
$193$	−67.8233			−0.351416			−0.175708	+	0.984442i	$0.556221\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−77.7356			−0.394597			−0.197298	−	0.980343i	$-0.563217\pi$
$197$	−77.7356			−0.394597			−0.197298	+	0.980343i	$0.563217\pi$
$198$		0			0
$199$			15.3243i			0.0770065i	0.999258	+	0.0385032i	$0.0122590\pi$
$199$			15.3243i			0.0770065i	−0.999258	+	0.0385032i	$0.987741\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			199.180i			0.981183i
$204$		0			0
$205$		−	55.8766i		−	0.272569i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	118.281			0.565939
$210$		0			0
$211$	264.369			1.25294			0.626468	−	0.779447i	$-0.284500\pi$
$211$	264.369			1.25294			0.626468	+	0.779447i	$0.284500\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−18.2088			−0.0846922
$216$		0			0
$217$		−	59.0573i		−	0.272153i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			284.034i			1.28522i
$222$		0			0
$223$	270.415			1.21262			0.606312	−	0.795227i	$-0.292648\pi$
$223$	270.415			1.21262			0.606312	+	0.795227i	$0.292648\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			218.072i			0.960671i	0.877085	+	0.480335i	$0.159485\pi$
$227$			218.072i			0.960671i	−0.877085	+	0.480335i	$0.840515\pi$
$228$		0			0
$229$			163.213i			0.712720i	0.934349	+	0.356360i	$0.115982\pi$
$229$			163.213i			0.712720i	−0.934349	+	0.356360i	$0.884018\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−176.562			−0.757777			−0.378888	−	0.925442i	$-0.623694\pi$
$233$	−176.562			−0.757777			−0.378888	+	0.925442i	$0.623694\pi$
$234$		0			0
$235$			70.5163i			0.300069i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−177.541			−0.742849			−0.371425	−	0.928463i	$-0.621131\pi$
$239$	−177.541			−0.742849			−0.371425	+	0.928463i	$0.621131\pi$
$240$		0			0
$241$			289.787i			1.20244i	0.799085	+	0.601218i	$0.205318\pi$
$241$			289.787i			1.20244i	−0.799085	+	0.601218i	$0.794682\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	54.6920i		−	0.223233i
$246$		0			0
$247$			142.961i			0.578790i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	117.024i		−	0.466232i	−0.972449	−	0.233116i	$-0.925108\pi$
$251$		−	117.024i		−	0.466232i	0.972449	−	0.233116i	$-0.0748923\pi$
$252$		0			0
$253$	53.9344	−	234.629i	0.213179	−	0.927388i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	176.964			0.688575			0.344287	−	0.938864i	$-0.388121\pi$
$257$	176.964			0.688575			0.344287	+	0.938864i	$0.388121\pi$
$258$		0			0
$259$	67.7223			0.261476
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	309.435i		−	1.17656i	−0.808658	−	0.588279i	$-0.799806\pi$
$263$		−	309.435i		−	1.17656i	0.808658	−	0.588279i	$-0.200194\pi$
$264$		0			0
$265$	119.613			0.451369
$266$		0			0
$267$		0			0
$268$		0			0
$269$	484.424			1.80083			0.900416	−	0.435030i	$-0.143262\pi$
$269$	484.424			1.80083			0.900416	+	0.435030i	$0.143262\pi$
$270$		0			0
$271$	165.353			0.610158			0.305079	−	0.952327i	$-0.401317\pi$
$271$	165.353			0.610158			0.305079	+	0.952327i	$0.401317\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	52.3366i		−	0.190315i
$276$		0			0
$277$	413.441			1.49257			0.746284	−	0.665628i	$-0.231836\pi$
$277$	413.441			1.49257			0.746284	+	0.665628i	$0.231836\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			222.395i			0.791443i	0.918371	+	0.395722i	$0.129505\pi$
$281$			222.395i			0.791443i	−0.918371	+	0.395722i	$0.870495\pi$
$282$		0			0
$283$			245.986i			0.869207i	0.900622	+	0.434604i	$0.143112\pi$
$283$			245.986i			0.869207i	−0.900622	+	0.434604i	$0.856888\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			123.792i			0.431329i
$288$		0			0
$289$	−215.042			−0.744090
$290$		0			0
$291$		0			0
$292$		0			0
$293$			102.022i			0.348197i	0.984728	+	0.174099i	$0.0557011\pi$
$293$			102.022i			0.348197i	−0.984728	+	0.174099i	$0.944299\pi$
$294$		0			0
$295$		−	221.707i		−	0.751549i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	283.585	+	65.1880i	0.948446	+	0.218020i
$300$		0			0
$301$	40.3406			0.134022
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−198.908			−0.652157
$306$		0			0
$307$	0.583336			0.00190012			0.000950058	−	1.00000i	$-0.499698\pi$
$307$	0.583336			0.00190012			0.000950058		1.00000i	$0.499698\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	582.237			1.87215			0.936073	−	0.351807i	$-0.114433\pi$
$311$	582.237			1.87215			0.936073	+	0.351807i	$0.114433\pi$
$312$		0			0
$313$		−	329.931i		−	1.05409i	−0.849837	−	0.527046i	$-0.823300\pi$
$313$		−	329.931i		−	1.05409i	0.849837	−	0.527046i	$-0.176700\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	85.3975			0.269393			0.134696	−	0.990887i	$-0.456994\pi$
$317$	85.3975			0.269393			0.134696	+	0.990887i	$0.456994\pi$
$318$		0			0
$319$			420.858i			1.31930i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−253.696			−0.785437
$324$		0			0
$325$	63.2568			0.194636
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	156.225i		−	0.474848i
$330$		0			0
$331$	256.162			0.773903			0.386951	−	0.922100i	$-0.373528\pi$
$331$	256.162			0.773903			0.386951	+	0.922100i	$0.373528\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	282.331			0.842778
$336$		0			0
$337$		−	155.918i		−	0.462664i	−0.972875	−	0.231332i	$-0.925692\pi$
$337$		−	155.918i		−	0.462664i	0.972875	−	0.231332i	$-0.0743084\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	124.785i		−	0.365939i
$342$		0			0
$343$			363.908i			1.06096i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−494.666			−1.42555			−0.712775	−	0.701393i	$-0.752562\pi$
$347$	−494.666			−1.42555			−0.712775	+	0.701393i	$0.752562\pi$
$348$		0			0
$349$	−618.486			−1.77216			−0.886082	−	0.463528i	$-0.846583\pi$
$349$	−618.486			−1.77216			−0.886082	+	0.463528i	$0.846583\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	114.300			0.323797			0.161898	−	0.986807i	$-0.448238\pi$
$353$	114.300			0.323797			0.161898	+	0.986807i	$0.448238\pi$
$354$		0			0
$355$			122.090i			0.343915i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			636.043i			1.77171i	0.463963	+	0.885854i	$0.346427\pi$
$359$			636.043i			1.77171i	−0.463963	+	0.885854i	$0.653573\pi$
$360$		0			0
$361$	233.309			0.646284
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	221.100i		−	0.605752i
$366$		0			0
$367$			688.016i			1.87470i	0.348384	+	0.937352i	$0.386731\pi$
$367$			688.016i			1.87470i	−0.348384	+	0.937352i	$0.613269\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−264.996			−0.714275
$372$		0			0
$373$			441.781i			1.18440i	0.805791	+	0.592200i	$0.201740\pi$
$373$			441.781i			1.18440i	−0.805791	+	0.592200i	$0.798260\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−508.671			−1.34926
$378$		0			0
$379$			7.23361i			0.0190860i	0.999954	+	0.00954302i	$0.00303768\pi$
$379$			7.23361i			0.0190860i	−0.999954	+	0.00954302i	$0.996962\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	311.404i		−	0.813065i	−0.913636	−	0.406532i	$-0.866738\pi$
$383$		−	311.404i		−	0.813065i	0.913636	−	0.406532i	$-0.133262\pi$
$384$		0			0
$385$			115.949i			0.301166i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	313.380i		−	0.805604i	−0.915287	−	0.402802i	$-0.868036\pi$
$389$		−	313.380i		−	0.805604i	0.915287	−	0.402802i	$-0.131964\pi$
$390$		0			0
$391$	−115.682	+	503.246i	−0.295861	+	1.28707i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	105.593			0.267325
$396$		0			0
$397$	124.378			0.313295			0.156648	−	0.987655i	$-0.449931\pi$
$397$	124.378			0.313295			0.156648	+	0.987655i	$0.449931\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	116.000i		−	0.289276i	−0.989485	−	0.144638i	$-0.953798\pi$
$401$		−	116.000i		−	0.289276i	0.989485	−	0.144638i	$-0.0462017\pi$
$402$		0			0
$403$	150.822			0.374248
$404$		0			0
$405$		0			0
$406$		0			0
$407$	143.094			0.351582
$408$		0			0
$409$	48.5546			0.118716			0.0593578	−	0.998237i	$-0.481095\pi$
$409$	48.5546			0.118716			0.0593578	+	0.998237i	$0.481095\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			491.180i			1.18930i
$414$		0			0
$415$	87.1932			0.210104
$416$		0			0
$417$		0			0
$418$		0			0
$419$			101.954i			0.243328i	0.992571	+	0.121664i	$0.0388230\pi$
$419$			101.954i			0.243328i	−0.992571	+	0.121664i	$0.961177\pi$
$420$		0			0
$421$			145.029i			0.344486i	0.985055	+	0.172243i	$0.0551014\pi$
$421$			145.029i			0.344486i	−0.985055	+	0.172243i	$0.944899\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			112.254i			0.264128i
$426$		0			0
$427$	440.670			1.03201
$428$		0			0
$429$		0			0
$430$		0			0
$431$			374.528i			0.868975i	0.900678	+	0.434487i	$0.143070\pi$
$431$			374.528i			0.868975i	−0.900678	+	0.434487i	$0.856930\pi$
$432$		0			0
$433$			96.0734i			0.221879i	0.993827	+	0.110939i	$0.0353859\pi$
$433$			96.0734i			0.221879i	−0.993827	+	0.110939i	$0.964614\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−58.2252	+	253.295i	−0.133238	+	0.579623i
$438$		0			0
$439$	−555.994			−1.26650			−0.633251	−	0.773946i	$-0.718280\pi$
$439$	−555.994			−1.26650			−0.633251	+	0.773946i	$0.718280\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	409.529			0.924444			0.462222	−	0.886764i	$-0.347052\pi$
$443$	409.529			0.924444			0.462222	+	0.886764i	$0.347052\pi$
$444$		0			0
$445$	125.642			0.282342
$446$		0			0
$447$		0			0
$448$		0			0
$449$	247.673			0.551611			0.275805	−	0.961213i	$-0.411055\pi$
$449$	247.673			0.551611			0.275805	+	0.961213i	$0.411055\pi$
$450$		0			0
$451$			261.565i			0.579968i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−140.142			−0.308004
$456$		0			0
$457$			320.343i			0.700969i	0.936569	+	0.350485i	$0.113983\pi$
$457$			320.343i			0.700969i	−0.936569	+	0.350485i	$0.886017\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−112.651			−0.244362			−0.122181	−	0.992508i	$-0.538989\pi$
$461$	−112.651			−0.244362			−0.122181	+	0.992508i	$0.538989\pi$
$462$		0			0
$463$	−883.113			−1.90737			−0.953685	−	0.300806i	$-0.902744\pi$
$463$	−883.113			−1.90737			−0.953685	+	0.300806i	$0.902744\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			589.482i			1.26227i	0.775672	+	0.631137i	$0.217411\pi$
$467$			589.482i			1.26227i	−0.775672	+	0.631137i	$0.782589\pi$
$468$		0			0
$469$	−625.488			−1.33366
$470$		0			0
$471$		0			0
$472$		0			0
$473$	85.2377			0.180207
$474$		0			0
$475$			56.5003i			0.118948i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	724.628i		−	1.51279i	−0.654113	−	0.756396i	$-0.726958\pi$
$479$		−	724.628i		−	1.51279i	0.654113	−	0.756396i	$-0.273042\pi$
$480$		0			0
$481$			172.951i			0.359565i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−185.797			−0.383086
$486$		0			0
$487$	367.381			0.754375			0.377187	−	0.926137i	$-0.376891\pi$
$487$	367.381			0.754375			0.377187	+	0.926137i	$0.376891\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	451.658			0.919873			0.459937	−	0.887952i	$-0.347872\pi$
$491$	451.658			0.919873			0.459937	+	0.887952i	$0.347872\pi$
$492$		0			0
$493$		−	902.679i		−	1.83099i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	270.484i		−	0.544233i
$498$		0			0
$499$	−126.441			−0.253389			−0.126694	−	0.991942i	$-0.540437\pi$
$499$	−126.441			−0.253389			−0.126694	+	0.991942i	$0.540437\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	61.6336i		−	0.122532i	−0.998121	−	0.0612661i	$-0.980486\pi$
$503$		−	61.6336i		−	0.122532i	0.998121	−	0.0612661i	$-0.0195138\pi$
$504$		0			0
$505$		−	207.950i		−	0.411782i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	69.5576			0.136655			0.0683277	−	0.997663i	$-0.478234\pi$
$509$	69.5576			0.136655			0.0683277	+	0.997663i	$0.478234\pi$
$510$		0			0
$511$			489.834i			0.958580i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−238.160			−0.462447
$516$		0			0
$517$		−	330.096i		−	0.638483i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			751.778i			1.44295i	0.692440	+	0.721476i	$0.256536\pi$
$521$			751.778i			1.44295i	−0.692440	+	0.721476i	$0.743464\pi$
$522$		0			0
$523$			860.039i			1.64443i	0.569174	+	0.822217i	$0.307263\pi$
$523$			860.039i			1.64443i	−0.569174	+	0.822217i	$0.692737\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			267.646i			0.507867i
$528$		0			0
$529$	475.900	+	230.997i	0.899623	+	0.436668i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−316.142			−0.593137
$534$		0			0
$535$	390.025			0.729018
$536$		0			0
$537$		0			0
$538$		0			0
$539$			256.020i			0.474991i
$540$		0			0
$541$	22.5472			0.0416769			0.0208384	−	0.999783i	$-0.493366\pi$
$541$	22.5472			0.0416769			0.0208384	+	0.999783i	$0.493366\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−71.4010			−0.131011
$546$		0			0
$547$	597.063			1.09152			0.545762	−	0.837940i	$-0.316240\pi$
$547$	597.063			1.09152			0.545762	+	0.837940i	$0.316240\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	454.340i		−	0.824573i
$552$		0			0
$553$	−233.936			−0.423031
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	585.625i		−	1.05139i	−0.850673	−	0.525696i	$-0.823805\pi$
$557$		−	585.625i		−	1.05139i	0.850673	−	0.525696i	$-0.176195\pi$
$558$		0			0
$559$			103.023i			0.184299i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	791.609i		−	1.40606i	−0.711162	−	0.703028i	$-0.751831\pi$
$563$		−	791.609i		−	1.40606i	0.711162	−	0.703028i	$-0.248169\pi$
$564$		0			0
$565$	501.414			0.887459
$566$		0			0
$567$		0			0
$568$		0			0
$569$			452.157i			0.794652i	0.917677	+	0.397326i	$0.130062\pi$
$569$			452.157i			0.794652i	−0.917677	+	0.397326i	$0.869938\pi$
$570$		0			0
$571$			216.845i			0.379764i	0.981807	+	0.189882i	$0.0608106\pi$
$571$			216.845i			0.379764i	−0.981807	+	0.189882i	$0.939189\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	112.077	+	25.7632i	0.194917	+	0.0448056i
$576$		0			0
$577$	−842.044			−1.45935			−0.729674	−	0.683795i	$-0.760328\pi$
$577$	−842.044			−1.45935			−0.729674	+	0.683795i	$0.760328\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−193.172			−0.332481
$582$		0			0
$583$	−559.923			−0.960417
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−911.800			−1.55332			−0.776661	−	0.629919i	$-0.783088\pi$
$587$	−911.800			−1.55332			−0.776661	+	0.629919i	$0.783088\pi$
$588$		0			0
$589$			134.713i			0.228714i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	169.996			0.286671			0.143335	−	0.989674i	$-0.454217\pi$
$593$	169.996			0.286671			0.143335	+	0.989674i	$0.454217\pi$
$594$		0			0
$595$		−	248.694i		−	0.417972i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1076.66			1.79744			0.898718	−	0.438527i	$-0.144500\pi$
$599$	1076.66			1.79744			0.898718	+	0.438527i	$0.144500\pi$
$600$		0			0
$601$	−508.812			−0.846609			−0.423304	−	0.905988i	$-0.639130\pi$
$601$	−508.812			−0.846609			−0.423304	+	0.905988i	$0.639130\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	25.5701i		−	0.0422647i
$606$		0			0
$607$	1134.44			1.86893			0.934464	−	0.356057i	$-0.115879\pi$
$607$	1134.44			1.86893			0.934464	+	0.356057i	$0.115879\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	398.971			0.652981
$612$		0			0
$613$		−	200.659i		−	0.327340i	−0.986515	−	0.163670i	$-0.947667\pi$
$613$		−	200.659i		−	0.327340i	0.986515	−	0.163670i	$-0.0523332\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	751.585i		−	1.21813i	−0.793121	−	0.609064i	$-0.791545\pi$
$617$		−	751.585i		−	1.21813i	0.793121	−	0.609064i	$-0.208455\pi$
$618$		0			0
$619$		−	564.131i		−	0.911359i	−0.890144	−	0.455679i	$-0.849396\pi$
$619$		−	564.131i		−	0.911359i	0.890144	−	0.455679i	$-0.150604\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−278.353			−0.446795
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−306.916			−0.487942
$630$		0			0
$631$		−	250.333i		−	0.396724i	−0.980129	−	0.198362i	$-0.936438\pi$
$631$		−	250.333i		−	0.396724i	0.980129	−	0.198362i	$-0.0635621\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			477.647i			0.752199i
$636$		0			0
$637$	−309.440			−0.485776
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1198.22i			1.86930i	0.355576	+	0.934648i	$0.384285\pi$
$641$			1198.22i			1.86930i	−0.355576	+	0.934648i	$0.615715\pi$
$642$		0			0
$643$		−	286.680i		−	0.445848i	−0.974836	−	0.222924i	$-0.928440\pi$
$643$		−	286.680i		−	0.445848i	0.974836	−	0.222924i	$-0.0715602\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−625.965			−0.967488			−0.483744	−	0.875209i	$-0.660723\pi$
$647$	−625.965			−0.967488			−0.483744	+	0.875209i	$0.660723\pi$
$648$		0			0
$649$			1037.84i			1.59913i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−53.4205			−0.0818078			−0.0409039	−	0.999163i	$-0.513024\pi$
$653$	−53.4205			−0.0818078			−0.0409039	+	0.999163i	$0.513024\pi$
$654$		0			0
$655$			266.809i			0.407342i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	122.051i		−	0.185207i	−0.995703	−	0.0926035i	$-0.970481\pi$
$659$		−	122.051i		−	0.185207i	0.995703	−	0.0926035i	$-0.0295189\pi$
$660$		0			0
$661$		−	1015.33i		−	1.53604i	−0.640423	−	0.768022i	$-0.721241\pi$
$661$		−	1015.33i		−	1.53604i	0.640423	−	0.768022i	$-0.278759\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	125.173i		−	0.188231i
$666$		0			0
$667$	−901.253	−	207.172i	−1.35120	−	0.310602i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	931.113			1.38765
$672$		0			0
$673$	95.0999			0.141307			0.0706537	−	0.997501i	$-0.477491\pi$
$673$	95.0999			0.141307			0.0706537	+	0.997501i	$0.477491\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	166.036i		−	0.245253i	−0.992453	−	0.122627i	$-0.960868\pi$
$677$		−	166.036i		−	0.245253i	0.992453	−	0.122627i	$-0.0391317\pi$
$678$		0			0
$679$	411.622			0.606219
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1021.81			1.49606			0.748030	−	0.663665i	$-0.231000\pi$
$683$	1021.81			1.49606			0.748030	+	0.663665i	$0.231000\pi$
$684$		0			0
$685$	517.572			0.755579
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	676.753i		−	0.982225i
$690$		0			0
$691$	694.464			1.00501			0.502506	−	0.864574i	$-0.332411\pi$
$691$	694.464			1.00501			0.502506	+	0.864574i	$0.332411\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			376.483i			0.541702i
$696$		0			0
$697$		−	561.020i		−	0.804907i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	468.889i		−	0.668885i	−0.942416	−	0.334443i	$-0.891452\pi$
$701$		−	468.889i		−	0.668885i	0.942416	−	0.334443i	$-0.108548\pi$
$702$		0			0
$703$	−154.478			−0.219741
$704$		0			0
$705$		0			0
$706$		0			0
$707$			460.702i			0.651630i
$708$		0			0
$709$		−	949.107i		−	1.33866i	−0.742967	−	0.669328i	$-0.766582\pi$
$709$		−	949.107i		−	1.33866i	0.742967	−	0.669328i	$-0.233418\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	267.223	+	61.4268i	0.374787	+	0.0861526i
$714$		0			0
$715$	−296.113			−0.414144
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−159.541			−0.221893			−0.110946	−	0.993826i	$-0.535388\pi$
$719$	−159.541			−0.221893			−0.110946	+	0.993826i	$0.535388\pi$
$720$		0			0
$721$	527.631			0.731805
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−201.034			−0.277289
$726$		0			0
$727$			385.956i			0.530889i	0.964126	+	0.265444i	$0.0855186\pi$
$727$			385.956i			0.530889i	−0.964126	+	0.265444i	$0.914481\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−182.823			−0.250099
$732$		0			0
$733$			1307.93i			1.78435i	0.451687	+	0.892177i	$0.350822\pi$
$733$			1307.93i			1.78435i	−0.451687	+	0.892177i	$0.649178\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1321.63			−1.79325
$738$		0			0
$739$	−562.916			−0.761727			−0.380863	−	0.924631i	$-0.624373\pi$
$739$	−562.916			−0.761727			−0.380863	+	0.924631i	$0.624373\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			791.374i			1.06511i	0.846396	+	0.532553i	$0.178767\pi$
$743$			791.374i			1.06511i	−0.846396	+	0.532553i	$0.821233\pi$
$744$		0			0
$745$	456.599			0.612884
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−864.079			−1.15364
$750$		0			0
$751$		−	671.687i		−	0.894391i	−0.894436	−	0.447195i	$-0.852423\pi$
$751$		−	671.687i		−	0.894391i	0.894436	−	0.447195i	$-0.147577\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			232.481i			0.307922i
$756$		0			0
$757$		−	681.158i		−	0.899812i	−0.893076	−	0.449906i	$-0.851457\pi$
$757$		−	681.158i		−	0.899812i	0.893076	−	0.449906i	$-0.148543\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	592.938			0.779156			0.389578	−	0.920993i	$-0.372621\pi$
$761$	592.938			0.779156			0.389578	+	0.920993i	$0.372621\pi$
$762$		0			0
$763$	158.185			0.207320
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1254.39			−1.63545
$768$		0			0
$769$			1225.72i			1.59391i	0.604040	+	0.796954i	$0.293557\pi$
$769$			1225.72i			1.59391i	−0.604040	+	0.796954i	$0.706443\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			95.0322i			0.122939i	0.998109	+	0.0614697i	$0.0195788\pi$
$773$			95.0322i			0.122939i	−0.998109	+	0.0614697i	$0.980421\pi$
$774$		0			0
$775$	59.6070			0.0769123
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	282.375i		−	0.362483i
$780$		0			0
$781$		−	571.519i		−	0.731778i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−80.0179			−0.101934
$786$		0			0
$787$			785.408i			0.997977i	0.866609	+	0.498989i	$0.166295\pi$
$787$			785.408i			0.997977i	−0.866609	+	0.498989i	$0.833705\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1110.86			−1.40437
$792$		0			0
$793$			1125.39i			1.41916i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1397.41i			1.75334i	0.481090	+	0.876671i	$0.340241\pi$
$797$			1397.41i			1.75334i	−0.481090	+	0.876671i	$0.659759\pi$
$798$		0			0
$799$			708.008i			0.886118i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			1035.00i			1.28891i
$804$		0			0
$805$	−248.301	−	57.0771i	−0.308448	−	0.0709032i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	120.833			0.149361			0.0746804	−	0.997208i	$-0.476206\pi$
$809$	120.833			0.149361			0.0746804	+	0.997208i	$0.476206\pi$
$810$		0			0
$811$	−217.352			−0.268005			−0.134003	−	0.990981i	$-0.542783\pi$
$811$	−217.352			−0.268005			−0.134003	+	0.990981i	$0.542783\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			676.668i			0.830267i
$816$		0			0
$817$	−92.0189			−0.112630
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1395.38			1.69961			0.849807	−	0.527095i	$-0.176719\pi$
$821$	1395.38			1.69961			0.849807	+	0.527095i	$0.176719\pi$
$822$		0			0
$823$	−833.124			−1.01230			−0.506151	−	0.862445i	$-0.668932\pi$
$823$	−833.124			−1.01230			−0.506151	+	0.862445i	$0.668932\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	911.805i		−	1.10255i	−0.834325	−	0.551273i	$-0.814142\pi$
$827$		−	911.805i		−	1.10255i	0.834325	−	0.551273i	$-0.185858\pi$
$828$		0			0
$829$	−1441.89			−1.73931			−0.869657	−	0.493656i	$-0.835660\pi$
$829$	−1441.89			−1.73931			−0.869657	+	0.493656i	$0.835660\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	549.126i		−	0.659215i
$834$		0			0
$835$			244.941i			0.293343i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			289.761i			0.345365i	0.984978	+	0.172682i	$0.0552435\pi$
$839$			289.761i			0.345365i	−0.984978	+	0.172682i	$0.944757\pi$
$840$		0			0
$841$	775.591			0.922225
$842$		0			0
$843$		0			0
$844$		0			0
$845$			19.9975i			0.0236657i
$846$		0			0
$847$			56.6492i			0.0668822i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−70.4395	+	306.431i	−0.0827726	+	0.360083i
$852$		0			0
$853$	193.079			0.226353			0.113176	−	0.993575i	$-0.463897\pi$
$853$	193.079			0.226353			0.113176	+	0.993575i	$0.463897\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	592.010			0.690794			0.345397	−	0.938457i	$-0.387744\pi$
$857$	592.010			0.690794			0.345397	+	0.938457i	$0.387744\pi$
$858$		0			0
$859$	−146.555			−0.170611			−0.0853055	−	0.996355i	$-0.527187\pi$
$859$	−146.555			−0.170611			−0.0853055	+	0.996355i	$0.527187\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1202.49			1.39338			0.696691	−	0.717372i	$-0.254655\pi$
$863$	1202.49			1.39338			0.696691	+	0.717372i	$0.254655\pi$
$864$		0			0
$865$		−	683.009i		−	0.789605i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−494.295			−0.568810
$870$		0			0
$871$		−	1597.39i		−	1.83397i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−55.3861			−0.0632984
$876$		0			0
$877$	−46.7267			−0.0532801			−0.0266401	−	0.999645i	$-0.508481\pi$
$877$	−46.7267			−0.0532801			−0.0266401	+	0.999645i	$0.508481\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	112.423i		−	0.127609i	−0.997962	−	0.0638043i	$-0.979677\pi$
$881$		−	112.423i		−	0.127609i	0.997962	−	0.0638043i	$-0.0203234\pi$
$882$		0			0
$883$	364.160			0.412412			0.206206	−	0.978509i	$-0.433888\pi$
$883$	364.160			0.412412			0.206206	+	0.978509i	$0.433888\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	435.974			0.491516			0.245758	−	0.969331i	$-0.420963\pi$
$887$	435.974			0.491516			0.245758	+	0.969331i	$0.420963\pi$
$888$		0			0
$889$		−	1058.20i		−	1.19033i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			356.357i			0.399056i
$894$		0			0
$895$			383.923i			0.428964i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−479.322			−0.533172
$900$		0			0
$901$	1200.95			1.33291
$902$		0			0
$903$		0			0
$904$		0			0
$905$	103.549			0.114418
$906$		0			0
$907$		−	1032.65i		−	1.13853i	−0.822154	−	0.569266i	$-0.807228\pi$
$907$		−	1032.65i		−	1.13853i	0.822154	−	0.569266i	$-0.192772\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	699.303i		−	0.767621i	−0.923412	−	0.383811i	$-0.874612\pi$
$911$		−	699.303i		−	0.767621i	0.923412	−	0.383811i	$-0.125388\pi$
$912$		0			0
$913$	−408.162			−0.447056
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	591.101i		−	0.644604i
$918$		0			0
$919$			259.391i			0.282253i	0.989992	+	0.141127i	$0.0450725\pi$
$919$			259.391i			0.282253i	−0.989992	+	0.141127i	$0.954928\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	690.768			0.748394
$924$		0			0
$925$			68.3527i			0.0738948i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1581.62			1.70250			0.851251	−	0.524760i	$-0.175845\pi$
$929$	1581.62			1.70250			0.851251	+	0.524760i	$0.175845\pi$
$930$		0			0
$931$		−	276.388i		−	0.296872i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	525.477i		−	0.562007i
$936$		0			0
$937$		−	505.998i		−	0.540019i	−0.962858	−	0.270009i	$-0.912973\pi$
$937$		−	505.998i		−	0.540019i	0.962858	−	0.270009i	$-0.0870268\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1822.51i		−	1.93678i	−0.249432	−	0.968392i	$-0.580244\pi$
$941$		−	1822.51i		−	1.93678i	0.249432	−	0.968392i	$-0.419756\pi$
$942$		0			0
$943$	−560.133	−	128.758i	−0.593991	−	0.136541i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−371.810			−0.392619			−0.196309	−	0.980542i	$-0.562896\pi$
$947$	−371.810			−0.392619			−0.196309	+	0.980542i	$0.562896\pi$
$948$		0			0
$949$	−1250.95			−1.31818
$950$		0			0
$951$		0			0
$952$		0			0
$953$			439.838i			0.461530i	0.973010	+	0.230765i	$0.0741229\pi$
$953$			439.838i			0.461530i	−0.973010	+	0.230765i	$0.925877\pi$
$954$		0			0
$955$	230.589			0.241454
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1146.65			−1.19567
$960$		0			0
$961$	−818.880			−0.852113
$962$		0			0
$963$		0			0
$964$		0			0
$965$			151.658i			0.157158i
$966$		0			0
$967$	207.130			0.214198			0.107099	−	0.994248i	$-0.465844\pi$
$967$	207.130			0.214198			0.107099	+	0.994248i	$0.465844\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1019.54i		−	1.04999i	−0.851106	−	0.524994i	$-0.824067\pi$
$971$		−	1019.54i		−	1.04999i	0.851106	−	0.524994i	$-0.175933\pi$
$972$		0			0
$973$		−	834.077i		−	0.857222i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			957.325i			0.979862i	0.871761	+	0.489931i	$0.162978\pi$
$977$			957.325i			0.979862i	−0.871761	+	0.489931i	$0.837022\pi$
$978$		0			0
$979$	−588.146			−0.600762
$980$		0			0
$981$		0			0
$982$		0			0
$983$			179.004i			0.182100i	0.995846	+	0.0910500i	$0.0290223\pi$
$983$			179.004i			0.182100i	−0.995846	+	0.0910500i	$0.970978\pi$
$984$		0			0
$985$			173.822i			0.176469i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−41.9592	+	182.534i	−0.0424259	+	0.184564i
$990$		0			0
$991$	964.970			0.973734			0.486867	−	0.873476i	$-0.338140\pi$
$991$	964.970			0.973734			0.486867	+	0.873476i	$0.338140\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	34.2662			0.0344384
$996$		0			0
$997$	−1092.14			−1.09543			−0.547713	−	0.836666i	$-0.684501\pi$
$997$	−1092.14			−1.09543			−0.547713	+	0.836666i	$0.684501\pi$
$998$		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.2161.11		32
3.2	odd	2		1380.3.d.a.781.14	yes	32
23.22	odd	2	inner	4140.3.d.c.2161.22		32
69.68	even	2		1380.3.d.a.781.3	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.3.d.a.781.3	✓	32	69.68	even	2
1380.3.d.a.781.14	yes	32	3.2	odd	2
4140.3.d.c.2161.11		32	1.1	even	1	trivial
4140.3.d.c.2161.22		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)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +4.95389i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} +4.95389i q^{7} +10.4673i q^{11} -12.6514 q^{13} -22.4509i q^{17} -11.3001i q^{19} +(-22.4154 - 5.15265i) q^{23} -5.00000 q^{25} +40.2069 q^{29} -11.9214 q^{31} +11.0772 q^{35} -13.6705i q^{37} +24.9888 q^{41} -8.14323i q^{43} -31.5359 q^{47} +24.4590 q^{49} +53.4925i q^{53} +23.4056 q^{55} +99.1504 q^{59} -88.9543i q^{61} +28.2893i q^{65} +126.262i q^{67} -54.6003 q^{71} +98.8788 q^{73} -51.8539 q^{77} +47.2228i q^{79} +38.9940i q^{83} -50.2017 q^{85} +56.1888i q^{89} -62.6734i q^{91} -25.2677 q^{95} -83.0908i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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