LMFDB - Embedded newform 4140.3.d.b.2161.15

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

Embedding invariants

Embedding label			2161.15
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.b.2161.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +9.55915i q^{7} +O(q^{10})$	$q-2.23607i q^{5} +9.55915i q^{7} +18.8089i q^{11} +20.1043 q^{13} -28.4427i q^{17} -28.9585i q^{19} +(-18.3876 + 13.8165i) q^{23} -5.00000 q^{25} +1.59078 q^{29} -26.9023 q^{31} +21.3749 q^{35} -51.7035i q^{37} -54.4792 q^{41} -81.2956i q^{43} +27.5628 q^{47} -42.3773 q^{49} -37.2235i q^{53} +42.0579 q^{55} +53.7093 q^{59} +39.8548i q^{61} -44.9547i q^{65} -42.9728i q^{67} +84.4430 q^{71} -77.4277 q^{73} -179.797 q^{77} -80.2412i q^{79} +119.214i q^{83} -63.5997 q^{85} +22.1863i q^{89} +192.180i q^{91} -64.7531 q^{95} +99.9018i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 24 q^{13} - 160 q^{25} - 28 q^{31} - 260 q^{49} + 120 q^{55} - 296 q^{73} - 60 q^{85}+O(q^{100}) $ 32 * q - 24 * q^13 - 160 * q^25 - 28 * q^31 - 260 * q^49 + 120 * q^55 - 296 * q^73 - 60 * q^85

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
$7$			9.55915i			1.36559i	0.730609	+	0.682796i	$0.239236\pi$
$7$			9.55915i			1.36559i	−0.730609	+	0.682796i	$0.760764\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			18.8089i			1.70990i	0.518714	+	0.854948i	$0.326411\pi$
$11$			18.8089i			1.70990i	−0.518714	+	0.854948i	$0.673589\pi$
$12$		0			0
$13$	20.1043			1.54649			0.773244	−	0.634109i	$-0.218633\pi$
$13$	20.1043			1.54649			0.773244	+	0.634109i	$0.218633\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	28.4427i		−	1.67310i	−0.547892	−	0.836549i	$-0.684570\pi$
$17$		−	28.4427i		−	1.67310i	0.547892	−	0.836549i	$-0.315430\pi$
$18$		0			0
$19$		−	28.9585i		−	1.52413i	−0.647501	−	0.762065i	$-0.724186\pi$
$19$		−	28.9585i		−	1.52413i	0.647501	−	0.762065i	$-0.275814\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−18.3876	+	13.8165i	−0.799461	+	0.600719i
$23$	−18.3876	+	13.8165i	−0.799461	+	0.600719i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.59078			0.0548544			0.0274272	−	0.999624i	$-0.491269\pi$
$29$	1.59078			0.0548544			0.0274272	+	0.999624i	$0.491269\pi$
$30$		0			0
$31$	−26.9023			−0.867815			−0.433907	−	0.900957i	$-0.642866\pi$
$31$	−26.9023			−0.867815			−0.433907	+	0.900957i	$0.642866\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	21.3749			0.610712
$36$		0			0
$37$		−	51.7035i		−	1.39739i	−0.715419	−	0.698695i	$-0.753764\pi$
$37$		−	51.7035i		−	1.39739i	0.715419	−	0.698695i	$-0.246236\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−54.4792			−1.32876			−0.664381	−	0.747394i	$-0.731305\pi$
$41$	−54.4792			−1.32876			−0.664381	+	0.747394i	$0.731305\pi$
$42$		0			0
$43$		−	81.2956i		−	1.89060i	−0.326209	−	0.945298i	$-0.605771\pi$
$43$		−	81.2956i		−	1.89060i	0.326209	−	0.945298i	$-0.394229\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	27.5628			0.586443			0.293222	−	0.956045i	$-0.405273\pi$
$47$	27.5628			0.586443			0.293222	+	0.956045i	$0.405273\pi$
$48$		0			0
$49$	−42.3773			−0.864843
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	37.2235i		−	0.702330i	−0.936314	−	0.351165i	$-0.885786\pi$
$53$		−	37.2235i		−	0.702330i	0.936314	−	0.351165i	$-0.114214\pi$
$54$		0			0
$55$	42.0579			0.764689
$56$		0			0
$57$		0			0
$58$		0			0
$59$	53.7093			0.910327			0.455164	−	0.890408i	$-0.349581\pi$
$59$	53.7093			0.910327			0.455164	+	0.890408i	$0.349581\pi$
$60$		0			0
$61$			39.8548i			0.653357i	0.945136	+	0.326679i	$0.105929\pi$
$61$			39.8548i			0.653357i	−0.945136	+	0.326679i	$0.894071\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	44.9547i		−	0.691611i
$66$		0			0
$67$		−	42.9728i		−	0.641385i	−0.947183	−	0.320693i	$-0.896084\pi$
$67$		−	42.9728i		−	0.641385i	0.947183	−	0.320693i	$-0.103916\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	84.4430			1.18934			0.594669	−	0.803970i	$-0.297283\pi$
$71$	84.4430			1.18934			0.594669	+	0.803970i	$0.297283\pi$
$72$		0			0
$73$	−77.4277			−1.06065			−0.530327	−	0.847793i	$-0.677931\pi$
$73$	−77.4277			−1.06065			−0.530327	+	0.847793i	$0.677931\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−179.797			−2.33502
$78$		0			0
$79$		−	80.2412i		−	1.01571i	−0.861442	−	0.507856i	$-0.830438\pi$
$79$		−	80.2412i		−	1.01571i	0.861442	−	0.507856i	$-0.169562\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			119.214i			1.43632i	0.695880	+	0.718158i	$0.255015\pi$
$83$			119.214i			1.43632i	−0.695880	+	0.718158i	$0.744985\pi$
$84$		0			0
$85$	−63.5997			−0.748232
$86$		0			0
$87$		0			0
$88$		0			0
$89$			22.1863i			0.249285i	0.992202	+	0.124642i	$0.0397783\pi$
$89$			22.1863i			0.249285i	−0.992202	+	0.124642i	$0.960222\pi$
$90$		0			0
$91$			192.180i			2.11187i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−64.7531			−0.681612
$96$		0			0
$97$			99.9018i			1.02992i	0.857216	+	0.514958i	$0.172192\pi$
$97$			99.9018i			1.02992i	−0.857216	+	0.514958i	$0.827808\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	65.3352			0.646884			0.323442	−	0.946248i	$-0.395160\pi$
$101$	65.3352			0.646884			0.323442	+	0.946248i	$0.395160\pi$
$102$		0			0
$103$		−	202.325i		−	1.96432i	−0.188042	−	0.982161i	$-0.560214\pi$
$103$		−	202.325i		−	1.96432i	0.188042	−	0.982161i	$-0.439786\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	168.608i		−	1.57577i	−0.615820	−	0.787887i	$-0.711175\pi$
$107$		−	168.608i		−	1.57577i	0.615820	−	0.787887i	$-0.288825\pi$
$108$		0			0
$109$		−	59.8713i		−	0.549278i	−0.961547	−	0.274639i	$-0.911442\pi$
$109$		−	59.8713i		−	0.549278i	0.961547	−	0.274639i	$-0.0885583\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			105.020i			0.929376i	0.885474	+	0.464688i	$0.153834\pi$
$113$			105.020i			0.929376i	−0.885474	+	0.464688i	$0.846166\pi$
$114$		0			0
$115$	30.8947	+	41.1159i	0.268650	+	0.357530i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	271.888			2.28477
$120$		0			0
$121$	−232.773			−1.92374
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	72.4497			0.570470			0.285235	−	0.958458i	$-0.407928\pi$
$127$	72.4497			0.570470			0.285235	+	0.958458i	$0.407928\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	149.778			1.14335			0.571673	−	0.820482i	$-0.306295\pi$
$131$	149.778			1.14335			0.571673	+	0.820482i	$0.306295\pi$
$132$		0			0
$133$	276.818			2.08134
$134$		0			0
$135$		0			0
$136$		0			0
$137$			112.515i			0.821276i	0.911798	+	0.410638i	$0.134694\pi$
$137$			112.515i			0.821276i	−0.911798	+	0.410638i	$0.865306\pi$
$138$		0			0
$139$	156.250			1.12410			0.562052	−	0.827102i	$-0.310012\pi$
$139$	156.250			1.12410			0.562052	+	0.827102i	$0.310012\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			378.140i			2.64433i
$144$		0			0
$145$		−	3.55709i		−	0.0245317i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	113.365i		−	0.760837i	−0.924814	−	0.380418i	$-0.875780\pi$
$149$		−	113.365i		−	0.760837i	0.924814	−	0.380418i	$-0.124220\pi$
$150$		0			0
$151$	−90.5230			−0.599490			−0.299745	−	0.954019i	$-0.596902\pi$
$151$	−90.5230			−0.599490			−0.299745	+	0.954019i	$0.596902\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			60.1553i			0.388099i
$156$		0			0
$157$		−	79.0800i		−	0.503694i	−0.967767	−	0.251847i	$-0.918962\pi$
$157$		−	79.0800i		−	0.503694i	0.967767	−	0.251847i	$-0.0810380\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−132.074	−	175.770i	−0.820337	−	1.09174i
$162$		0			0
$163$	210.029			1.28852			0.644260	−	0.764807i	$-0.277166\pi$
$163$	210.029			1.28852			0.644260	+	0.764807i	$0.277166\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	162.144			0.970921			0.485461	−	0.874259i	$-0.338652\pi$
$167$	162.144			0.970921			0.485461	+	0.874259i	$0.338652\pi$
$168$		0			0
$169$	235.185			1.39163
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−186.737			−1.07940			−0.539701	−	0.841857i	$-0.681463\pi$
$173$	−186.737			−1.07940			−0.539701	+	0.841857i	$0.681463\pi$
$174$		0			0
$175$		−	47.7957i		−	0.273119i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.54664			0.0254002			0.0127001	−	0.999919i	$-0.495957\pi$
$179$	4.54664			0.0254002			0.0127001	+	0.999919i	$0.495957\pi$
$180$		0			0
$181$			69.4440i			0.383669i	0.981427	+	0.191834i	$0.0614436\pi$
$181$			69.4440i			0.383669i	−0.981427	+	0.191834i	$0.938556\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−115.612			−0.624932
$186$		0			0
$187$	534.974			2.86082
$188$		0			0
$189$		0			0
$190$		0			0
$191$			216.144i			1.13164i	0.824527	+	0.565822i	$0.191441\pi$
$191$			216.144i			1.13164i	−0.824527	+	0.565822i	$0.808559\pi$
$192$		0			0
$193$	238.065			1.23350			0.616748	−	0.787161i	$-0.288450\pi$
$193$	238.065			1.23350			0.616748	+	0.787161i	$0.288450\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	88.2296			0.447866			0.223933	−	0.974605i	$-0.428110\pi$
$197$	88.2296			0.447866			0.223933	+	0.974605i	$0.428110\pi$
$198$		0			0
$199$			74.1586i			0.372656i	0.982488	+	0.186328i	$0.0596587\pi$
$199$			74.1586i			0.372656i	−0.982488	+	0.186328i	$0.940341\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			15.2065i			0.0749088i
$204$		0			0
$205$			121.819i			0.594240i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	544.676			2.60610
$210$		0			0
$211$	150.488			0.713216			0.356608	−	0.934254i	$-0.383933\pi$
$211$	150.488			0.713216			0.356608	+	0.934254i	$0.383933\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−181.782			−0.845500
$216$		0			0
$217$		−	257.163i		−	1.18508i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	571.821i		−	2.58743i
$222$		0			0
$223$	−98.5288			−0.441833			−0.220917	−	0.975293i	$-0.570905\pi$
$223$	−98.5288			−0.441833			−0.220917	+	0.975293i	$0.570905\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	52.6756i		−	0.232051i	−0.993246	−	0.116026i	$-0.962985\pi$
$227$		−	52.6756i		−	0.232051i	0.993246	−	0.116026i	$-0.0370155\pi$
$228$		0			0
$229$		−	372.613i		−	1.62713i	−0.581473	−	0.813566i	$-0.697523\pi$
$229$		−	372.613i		−	1.62713i	0.581473	−	0.813566i	$-0.302477\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−44.5159			−0.191055			−0.0955277	−	0.995427i	$-0.530454\pi$
$233$	−44.5159			−0.191055			−0.0955277	+	0.995427i	$0.530454\pi$
$234$		0			0
$235$		−	61.6324i		−	0.262265i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−433.189			−1.81251			−0.906253	−	0.422736i	$-0.861070\pi$
$239$	−433.189			−1.81251			−0.906253	+	0.422736i	$0.861070\pi$
$240$		0			0
$241$		−	212.470i		−	0.881618i	−0.897601	−	0.440809i	$-0.854692\pi$
$241$		−	212.470i		−	0.881618i	0.897601	−	0.440809i	$-0.145308\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			94.7586i			0.386770i
$246$		0			0
$247$		−	582.191i		−	2.35705i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	468.549i		−	1.86673i	−0.358932	−	0.933364i	$-0.616859\pi$
$251$		−	468.549i		−	1.86673i	0.358932	−	0.933364i	$-0.383141\pi$
$252$		0			0
$253$	−259.873	−	345.849i	−1.02717	−	1.36699i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−449.301			−1.74825			−0.874127	−	0.485697i	$-0.838566\pi$
$257$	−449.301			−1.74825			−0.874127	+	0.485697i	$0.838566\pi$
$258$		0			0
$259$	494.241			1.90827
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	181.830i		−	0.691369i	−0.938351	−	0.345685i	$-0.887647\pi$
$263$		−	181.830i		−	0.691369i	0.938351	−	0.345685i	$-0.112353\pi$
$264$		0			0
$265$	−83.2342			−0.314091
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−214.110			−0.795949			−0.397975	−	0.917396i	$-0.630287\pi$
$269$	−214.110			−0.795949			−0.397975	+	0.917396i	$0.630287\pi$
$270$		0			0
$271$	−337.430			−1.24513			−0.622564	−	0.782569i	$-0.713909\pi$
$271$	−337.430			−1.24513			−0.622564	+	0.782569i	$0.713909\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	94.0443i		−	0.341979i
$276$		0			0
$277$	−153.931			−0.555708			−0.277854	−	0.960623i	$-0.589623\pi$
$277$	−153.931			−0.555708			−0.277854	+	0.960623i	$0.589623\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			204.395i			0.727386i	0.931519	+	0.363693i	$0.118484\pi$
$281$			204.395i			0.727386i	−0.931519	+	0.363693i	$0.881516\pi$
$282$		0			0
$283$			372.530i			1.31636i	0.752861	+	0.658180i	$0.228673\pi$
$283$			372.530i			1.31636i	−0.752861	+	0.658180i	$0.771327\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	520.775i		−	1.81455i
$288$		0			0
$289$	−519.985			−1.79926
$290$		0			0
$291$		0			0
$292$		0			0
$293$			98.4000i			0.335836i	0.985801	+	0.167918i	$0.0537044\pi$
$293$			98.4000i			0.335836i	−0.985801	+	0.167918i	$0.946296\pi$
$294$		0			0
$295$		−	120.098i		−	0.407111i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−369.671	+	277.772i	−1.23636	+	0.929004i
$300$		0			0
$301$	777.117			2.58178
$302$		0			0
$303$		0			0
$304$		0			0
$305$	89.1180			0.292190
$306$		0			0
$307$	290.060			0.944822			0.472411	−	0.881378i	$-0.343384\pi$
$307$	290.060			0.944822			0.472411	+	0.881378i	$0.343384\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−319.010			−1.02576			−0.512878	−	0.858462i	$-0.671421\pi$
$311$	−319.010			−1.02576			−0.512878	+	0.858462i	$0.671421\pi$
$312$		0			0
$313$			52.6956i			0.168356i	0.996451	+	0.0841782i	$0.0268265\pi$
$313$			52.6956i			0.168356i	−0.996451	+	0.0841782i	$0.973174\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	61.8574			0.195134			0.0975668	−	0.995229i	$-0.468894\pi$
$317$	61.8574			0.195134			0.0975668	+	0.995229i	$0.468894\pi$
$318$		0			0
$319$			29.9207i			0.0937954i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−823.656			−2.55002
$324$		0			0
$325$	−100.522			−0.309298
$326$		0			0
$327$		0			0
$328$		0			0
$329$			263.477i			0.800842i
$330$		0			0
$331$	616.092			1.86131			0.930653	−	0.365902i	$-0.119240\pi$
$331$	616.092			1.86131			0.930653	+	0.365902i	$0.119240\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−96.0901			−0.286836
$336$		0			0
$337$		−	330.744i		−	0.981436i	−0.871318	−	0.490718i	$-0.836735\pi$
$337$		−	330.744i		−	0.981436i	0.871318	−	0.490718i	$-0.163265\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	506.001i		−	1.48387i
$342$		0			0
$343$			63.3072i			0.184569i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	551.866			1.59039			0.795197	−	0.606352i	$-0.207368\pi$
$347$	551.866			1.59039			0.795197	+	0.606352i	$0.207368\pi$
$348$		0			0
$349$	−26.0766			−0.0747181			−0.0373590	−	0.999302i	$-0.511895\pi$
$349$	−26.0766			−0.0747181			−0.0373590	+	0.999302i	$0.511895\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	367.348			1.04064			0.520322	−	0.853970i	$-0.325812\pi$
$353$	367.348			1.04064			0.520322	+	0.853970i	$0.325812\pi$
$354$		0			0
$355$		−	188.820i		−	0.531888i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			666.079i			1.85537i	0.373359	+	0.927687i	$0.378206\pi$
$359$			666.079i			1.85537i	−0.373359	+	0.927687i	$0.621794\pi$
$360$		0			0
$361$	−477.593			−1.32297
$362$		0			0
$363$		0			0
$364$		0			0
$365$			173.134i			0.474339i
$366$		0			0
$367$		−	387.158i		−	1.05493i	−0.849578	−	0.527463i	$-0.823143\pi$
$367$		−	387.158i		−	1.05493i	0.849578	−	0.527463i	$-0.176857\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	355.825			0.959096
$372$		0			0
$373$		−	631.246i		−	1.69235i	−0.532906	−	0.846174i	$-0.678900\pi$
$373$		−	631.246i		−	1.69235i	0.532906	−	0.846174i	$-0.321100\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	31.9816			0.0848318
$378$		0			0
$379$		−	199.324i		−	0.525920i	−0.964807	−	0.262960i	$-0.915301\pi$
$379$		−	199.324i		−	0.525920i	0.964807	−	0.262960i	$-0.0846987\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	255.191i		−	0.666294i	−0.942875	−	0.333147i	$-0.891889\pi$
$383$		−	255.191i		−	0.666294i	0.942875	−	0.333147i	$-0.108111\pi$
$384$		0			0
$385$			402.037i			1.04425i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	203.813i		−	0.523940i	−0.965076	−	0.261970i	$-0.915628\pi$
$389$		−	203.813i		−	0.523940i	0.965076	−	0.261970i	$-0.0843721\pi$
$390$		0			0
$391$	392.979	+	522.992i	1.00506	+	1.33758i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−179.425			−0.454240
$396$		0			0
$397$	−33.1882			−0.0835975			−0.0417987	−	0.999126i	$-0.513309\pi$
$397$	−33.1882			−0.0835975			−0.0417987	+	0.999126i	$0.513309\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	55.4248i		−	0.138216i	−0.997609	−	0.0691082i	$-0.977985\pi$
$401$		−	55.4248i		−	0.138216i	0.997609	−	0.0691082i	$-0.0220154\pi$
$402$		0			0
$403$	−540.852			−1.34207
$404$		0			0
$405$		0			0
$406$		0			0
$407$	972.482			2.38939
$408$		0			0
$409$	28.5349			0.0697675			0.0348838	−	0.999391i	$-0.488894\pi$
$409$	28.5349			0.0697675			0.0348838	+	0.999391i	$0.488894\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			513.415i			1.24314i
$414$		0			0
$415$	266.571			0.642340
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	271.192i		−	0.647236i	−0.946188	−	0.323618i	$-0.895101\pi$
$419$		−	271.192i		−	0.647236i	0.946188	−	0.323618i	$-0.104899\pi$
$420$		0			0
$421$		−	792.990i		−	1.88359i	−0.336192	−	0.941793i	$-0.609139\pi$
$421$		−	792.990i		−	1.88359i	0.336192	−	0.941793i	$-0.390861\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			142.213i			0.334620i
$426$		0			0
$427$	−380.978			−0.892220
$428$		0			0
$429$		0			0
$430$		0			0
$431$			583.988i			1.35496i	0.735540	+	0.677481i	$0.236928\pi$
$431$			583.988i			1.35496i	−0.735540	+	0.677481i	$0.763072\pi$
$432$		0			0
$433$			543.770i			1.25582i	0.778286	+	0.627910i	$0.216089\pi$
$433$			543.770i			1.25582i	−0.778286	+	0.627910i	$0.783911\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	400.105	+	532.477i	0.915573	+	1.21848i
$438$		0			0
$439$	554.291			1.26262			0.631311	−	0.775530i	$-0.282517\pi$
$439$	554.291			1.26262			0.631311	+	0.775530i	$0.282517\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	154.056			0.347756			0.173878	−	0.984767i	$-0.444370\pi$
$443$	154.056			0.347756			0.173878	+	0.984767i	$0.444370\pi$
$444$		0			0
$445$	49.6102			0.111483
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−216.232			−0.481586			−0.240793	−	0.970576i	$-0.577408\pi$
$449$	−216.232			−0.481586			−0.240793	+	0.970576i	$0.577408\pi$
$450$		0			0
$451$		−	1024.69i		−	2.27204i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	429.728			0.944458
$456$		0			0
$457$			368.540i			0.806434i	0.915104	+	0.403217i	$0.132108\pi$
$457$			368.540i			0.806434i	−0.915104	+	0.403217i	$0.867892\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	343.678			0.745506			0.372753	−	0.927931i	$-0.378414\pi$
$461$	343.678			0.745506			0.372753	+	0.927931i	$0.378414\pi$
$462$		0			0
$463$	−24.2165			−0.0523033			−0.0261517	−	0.999658i	$-0.508325\pi$
$463$	−24.2165			−0.0523033			−0.0261517	+	0.999658i	$0.508325\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			21.3382i			0.0456921i	0.999739	+	0.0228460i	$0.00727275\pi$
$467$			21.3382i			0.0456921i	−0.999739	+	0.0228460i	$0.992727\pi$
$468$		0			0
$469$	410.784			0.875871
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1529.08			3.23272
$474$		0			0
$475$			144.792i			0.304826i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	353.583i		−	0.738168i	−0.929396	−	0.369084i	$-0.879671\pi$
$479$		−	353.583i		−	0.738168i	0.929396	−	0.369084i	$-0.120329\pi$
$480$		0			0
$481$		−	1039.46i		−	2.16105i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	223.387			0.460592
$486$		0			0
$487$	−832.579			−1.70961			−0.854804	−	0.518951i	$-0.826323\pi$
$487$	−832.579			−1.70961			−0.854804	+	0.518951i	$0.826323\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−200.501			−0.408352			−0.204176	−	0.978934i	$-0.565451\pi$
$491$	−200.501			−0.408352			−0.204176	+	0.978934i	$0.565451\pi$
$492$		0			0
$493$		−	45.2460i		−	0.0917768i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			807.203i			1.62415i
$498$		0			0
$499$	−790.773			−1.58471			−0.792357	−	0.610057i	$-0.791146\pi$
$499$	−790.773			−1.58471			−0.792357	+	0.610057i	$0.791146\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	207.441i		−	0.412408i	−0.978509	−	0.206204i	$-0.933889\pi$
$503$		−	207.441i		−	0.412408i	0.978509	−	0.206204i	$-0.0661110\pi$
$504$		0			0
$505$		−	146.094i		−	0.289295i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	240.857			0.473197			0.236598	−	0.971608i	$-0.423967\pi$
$509$	240.857			0.473197			0.236598	+	0.971608i	$0.423967\pi$
$510$		0			0
$511$		−	740.143i		−	1.44842i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−452.413			−0.878472
$516$		0			0
$517$			518.425i			1.00276i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	549.887i		−	1.05545i	−0.849417	−	0.527723i	$-0.823046\pi$
$521$		−	549.887i		−	1.05545i	0.849417	−	0.527723i	$-0.176954\pi$
$522$		0			0
$523$			173.718i			0.332156i	0.986113	+	0.166078i	$0.0531104\pi$
$523$			173.718i			0.332156i	−0.986113	+	0.166078i	$0.946890\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			765.172i			1.45194i
$528$		0			0
$529$	147.207	−	508.105i	0.278274	−	0.960502i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1095.27			−2.05491
$534$		0			0
$535$	−377.018			−0.704707
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	797.069i		−	1.47879i
$540$		0			0
$541$	346.955			0.641322			0.320661	−	0.947194i	$-0.396095\pi$
$541$	346.955			0.641322			0.320661	+	0.947194i	$0.396095\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−133.876			−0.245644
$546$		0			0
$547$	707.916			1.29418			0.647090	−	0.762414i	$-0.275986\pi$
$547$	707.916			1.29418			0.647090	+	0.762414i	$0.275986\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	46.0665i		−	0.0836053i
$552$		0			0
$553$	767.037			1.38705
$554$		0			0
$555$		0			0
$556$		0			0
$557$			688.947i			1.23689i	0.785829	+	0.618444i	$0.212237\pi$
$557$			688.947i			1.23689i	−0.785829	+	0.618444i	$0.787763\pi$
$558$		0			0
$559$		−	1634.39i		−	2.92378i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	373.375i		−	0.663187i	−0.943422	−	0.331594i	$-0.892414\pi$
$563$		−	373.375i		−	0.663187i	0.943422	−	0.331594i	$-0.107586\pi$
$564$		0			0
$565$	234.831			0.415630
$566$		0			0
$567$		0			0
$568$		0			0
$569$			133.254i			0.234190i	0.993121	+	0.117095i	$0.0373582\pi$
$569$			133.254i			0.234190i	−0.993121	+	0.117095i	$0.962642\pi$
$570$		0			0
$571$		−	83.6044i		−	0.146418i	−0.997317	−	0.0732088i	$-0.976676\pi$
$571$		−	83.6044i		−	0.146418i	0.997317	−	0.0732088i	$-0.0233239\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	91.9380	−	69.0826i	0.159892	−	0.120144i
$576$		0			0
$577$	926.137			1.60509			0.802545	−	0.596591i	$-0.203479\pi$
$577$	926.137			1.60509			0.802545	+	0.596591i	$0.203479\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1139.59			−1.96142
$582$		0			0
$583$	700.131			1.20091
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−588.228			−1.00209			−0.501046	−	0.865421i	$-0.667051\pi$
$587$	−588.228			−1.00209			−0.501046	+	0.865421i	$0.667051\pi$
$588$		0			0
$589$			779.048i			1.32266i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−832.912			−1.40457			−0.702287	−	0.711894i	$-0.747838\pi$
$593$	−832.912			−1.40457			−0.702287	+	0.711894i	$0.747838\pi$
$594$		0			0
$595$		−	607.959i		−	1.02178i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	594.374			0.992277			0.496139	−	0.868243i	$-0.334751\pi$
$599$	594.374			0.992277			0.496139	+	0.868243i	$0.334751\pi$
$600$		0			0
$601$	180.418			0.300196			0.150098	−	0.988671i	$-0.452041\pi$
$601$	180.418			0.300196			0.150098	+	0.988671i	$0.452041\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			520.496i			0.860324i
$606$		0			0
$607$	854.913			1.40842			0.704212	−	0.709990i	$-0.251300\pi$
$607$	854.913			1.40842			0.704212	+	0.709990i	$0.251300\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	554.133			0.906927
$612$		0			0
$613$		−	637.880i		−	1.04059i	−0.853987	−	0.520294i	$-0.825822\pi$
$613$		−	637.880i		−	1.04059i	0.853987	−	0.520294i	$-0.174178\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	625.473i		−	1.01373i	−0.862025	−	0.506866i	$-0.830804\pi$
$617$		−	625.473i		−	1.01373i	0.862025	−	0.506866i	$-0.169196\pi$
$618$		0			0
$619$			330.557i			0.534017i	0.963694	+	0.267009i	$0.0860352\pi$
$619$			330.557i			0.534017i	−0.963694	+	0.267009i	$0.913965\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−212.082			−0.340421
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1470.58			−2.33797
$630$		0			0
$631$		−	38.1193i		−	0.0604109i	−0.999544	−	0.0302055i	$-0.990384\pi$
$631$		−	38.1193i		−	0.0604109i	0.999544	−	0.0302055i	$-0.00961616\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	162.003i		−	0.255122i
$636$		0			0
$637$	−851.968			−1.33747
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	133.794i		−	0.208727i	−0.994539	−	0.104363i	$-0.966720\pi$
$641$		−	133.794i		−	0.208727i	0.994539	−	0.104363i	$-0.0332805\pi$
$642$		0			0
$643$		−	195.216i		−	0.303601i	−0.988411	−	0.151801i	$-0.951493\pi$
$643$		−	195.216i		−	0.303601i	0.988411	−	0.151801i	$-0.0485072\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	973.923			1.50529			0.752645	−	0.658426i	$-0.228777\pi$
$647$	973.923			1.50529			0.752645	+	0.658426i	$0.228777\pi$
$648$		0			0
$649$			1010.21i			1.55656i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	587.480			0.899663			0.449831	−	0.893114i	$-0.351484\pi$
$653$	587.480			0.899663			0.449831	+	0.893114i	$0.351484\pi$
$654$		0			0
$655$		−	334.914i		−	0.511320i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	322.810i		−	0.489849i	−0.969542	−	0.244924i	$-0.921237\pi$
$659$		−	322.810i		−	0.489849i	0.969542	−	0.244924i	$-0.0787631\pi$
$660$		0			0
$661$			463.530i			0.701256i	0.936515	+	0.350628i	$0.114032\pi$
$661$			463.530i			0.701256i	−0.936515	+	0.350628i	$0.885968\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	618.985i		−	0.930804i
$666$		0			0
$667$	−29.2506	+	21.9790i	−0.0438540	+	0.0329521i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−749.623			−1.11717
$672$		0			0
$673$	312.153			0.463823			0.231912	−	0.972737i	$-0.425502\pi$
$673$	312.153			0.463823			0.231912	+	0.972737i	$0.425502\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	386.425i		−	0.570790i	−0.958410	−	0.285395i	$-0.907875\pi$
$677$		−	386.425i		−	0.570790i	0.958410	−	0.285395i	$-0.0921249\pi$
$678$		0			0
$679$	−954.976			−1.40645
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−889.807			−1.30279			−0.651396	−	0.758738i	$-0.725816\pi$
$683$	−889.807			−1.30279			−0.651396	+	0.758738i	$0.725816\pi$
$684$		0			0
$685$	251.591			0.367286
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	748.354i		−	1.08614i
$690$		0			0
$691$	211.397			0.305928			0.152964	−	0.988232i	$-0.451118\pi$
$691$	211.397			0.305928			0.152964	+	0.988232i	$0.451118\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	349.387i		−	0.502714i
$696$		0			0
$697$			1549.53i			2.22315i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1169.73i		−	1.66865i	−0.551269	−	0.834327i	$-0.685856\pi$
$701$		−	1169.73i		−	1.66865i	0.551269	−	0.834327i	$-0.314144\pi$
$702$		0			0
$703$	−1497.25			−2.12980
$704$		0			0
$705$		0			0
$706$		0			0
$707$			624.549i			0.883379i
$708$		0			0
$709$			16.6676i			0.0235086i	0.999931	+	0.0117543i	$0.00374160\pi$
$709$			16.6676i			0.0235086i	−0.999931	+	0.0117543i	$0.996258\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	494.668	−	371.696i	0.693784	−	0.521312i
$714$		0			0
$715$	845.546			1.18258
$716$		0			0
$717$		0			0
$718$		0			0
$719$	673.301			0.936442			0.468221	−	0.883612i	$-0.344895\pi$
$719$	673.301			0.936442			0.468221	+	0.883612i	$0.344895\pi$
$720$		0			0
$721$	1934.06			2.68246
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.95389			−0.0109709
$726$		0			0
$727$			456.897i			0.628469i	0.949345	+	0.314235i	$0.101748\pi$
$727$			456.897i			0.628469i	−0.949345	+	0.314235i	$0.898252\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2312.26			−3.16315
$732$		0			0
$733$		−	945.078i		−	1.28933i	−0.764466	−	0.644664i	$-0.776997\pi$
$733$		−	945.078i		−	1.28933i	0.764466	−	0.644664i	$-0.223003\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	808.269			1.09670
$738$		0			0
$739$	1245.76			1.68574			0.842872	−	0.538114i	$-0.180863\pi$
$739$	1245.76			1.68574			0.842872	+	0.538114i	$0.180863\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1452.42i			1.95480i	0.211401	+	0.977399i	$0.432197\pi$
$743$			1452.42i			1.95480i	−0.211401	+	0.977399i	$0.567803\pi$
$744$		0			0
$745$	−253.491			−0.340257
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1611.75			2.15187
$750$		0			0
$751$			269.832i			0.359297i	0.983731	+	0.179649i	$0.0574960\pi$
$751$			269.832i			0.359297i	−0.983731	+	0.179649i	$0.942504\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			202.416i			0.268100i
$756$		0			0
$757$			624.541i			0.825021i	0.910953	+	0.412510i	$0.135348\pi$
$757$			624.541i			0.825021i	−0.910953	+	0.412510i	$0.864652\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−236.248			−0.310445			−0.155222	−	0.987880i	$-0.549609\pi$
$761$	−236.248			−0.310445			−0.155222	+	0.987880i	$0.549609\pi$
$762$		0			0
$763$	572.318			0.750090
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1079.79			1.40781
$768$		0			0
$769$			1361.59i			1.77060i	0.465023	+	0.885298i	$0.346046\pi$
$769$			1361.59i			1.77060i	−0.465023	+	0.885298i	$0.653954\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			289.140i			0.374049i	0.982355	+	0.187025i	$0.0598844\pi$
$773$			289.140i			0.374049i	−0.982355	+	0.187025i	$0.940116\pi$
$774$		0			0
$775$	134.511			0.173563
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1577.63i			2.02521i
$780$		0			0
$781$			1588.28i			2.03364i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−176.828			−0.225259
$786$		0			0
$787$			958.162i			1.21749i	0.793367	+	0.608744i	$0.208326\pi$
$787$			958.162i			1.21749i	−0.793367	+	0.608744i	$0.791674\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1003.90			−1.26915
$792$		0			0
$793$			801.254i			1.01041i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			147.380i			0.184919i	0.995716	+	0.0924594i	$0.0294728\pi$
$797$			147.380i			0.184919i	−0.995716	+	0.0924594i	$0.970527\pi$
$798$		0			0
$799$		−	783.960i		−	0.981177i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	1456.33i		−	1.81361i
$804$		0			0
$805$	−393.033	+	295.327i	−0.488240	+	0.366866i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	339.448			0.419589			0.209795	−	0.977745i	$-0.432720\pi$
$809$	339.448			0.419589			0.209795	+	0.977745i	$0.432720\pi$
$810$		0			0
$811$	−763.230			−0.941097			−0.470548	−	0.882374i	$-0.655944\pi$
$811$	−763.230			−0.941097			−0.470548	+	0.882374i	$0.655944\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	469.638i		−	0.576243i
$816$		0			0
$817$	−2354.20			−2.88151
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−26.5246			−0.0323076			−0.0161538	−	0.999870i	$-0.505142\pi$
$821$	−26.5246			−0.0323076			−0.0161538	+	0.999870i	$0.505142\pi$
$822$		0			0
$823$	419.120			0.509259			0.254629	−	0.967039i	$-0.418046\pi$
$823$	419.120			0.509259			0.254629	+	0.967039i	$0.418046\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			131.070i			0.158489i	0.996855	+	0.0792443i	$0.0252507\pi$
$827$			131.070i			0.158489i	−0.996855	+	0.0792443i	$0.974749\pi$
$828$		0			0
$829$	−570.854			−0.688606			−0.344303	−	0.938859i	$-0.611885\pi$
$829$	−570.854			−0.688606			−0.344303	+	0.938859i	$0.611885\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1205.32i			1.44697i
$834$		0			0
$835$		−	362.565i		−	0.434209i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			419.908i			0.500486i	0.968183	+	0.250243i	$0.0805106\pi$
$839$			419.908i			0.500486i	−0.968183	+	0.250243i	$0.919489\pi$
$840$		0			0
$841$	−838.469			−0.996991
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	525.889i		−	0.622354i
$846$		0			0
$847$		−	2225.11i		−	2.62705i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	714.362	+	950.702i	0.839438	+	1.11716i
$852$		0			0
$853$	1378.65			1.61624			0.808119	−	0.589020i	$-0.200486\pi$
$853$	1378.65			1.61624			0.808119	+	0.589020i	$0.200486\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1310.60			−1.52929			−0.764646	−	0.644451i	$-0.777086\pi$
$857$	−1310.60			−1.52929			−0.764646	+	0.644451i	$0.777086\pi$
$858$		0			0
$859$	532.378			0.619765			0.309882	−	0.950775i	$-0.399710\pi$
$859$	532.378			0.619765			0.309882	+	0.950775i	$0.399710\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	904.616			1.04822			0.524111	−	0.851650i	$-0.324398\pi$
$863$	904.616			1.04822			0.524111	+	0.851650i	$0.324398\pi$
$864$		0			0
$865$			417.556i			0.482724i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1509.24			1.73676
$870$		0			0
$871$		−	863.940i		−	0.991895i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−106.875			−0.122142
$876$		0			0
$877$	750.929			0.856248			0.428124	−	0.903720i	$-0.359175\pi$
$877$	750.929			0.856248			0.428124	+	0.903720i	$0.359175\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			888.261i			1.00824i	0.863633	+	0.504121i	$0.168183\pi$
$881$			888.261i			1.00824i	−0.863633	+	0.504121i	$0.831817\pi$
$882$		0			0
$883$	−96.1065			−0.108841			−0.0544205	−	0.998518i	$-0.517331\pi$
$883$	−96.1065			−0.108841			−0.0544205	+	0.998518i	$0.517331\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	714.733			0.805787			0.402894	−	0.915247i	$-0.368004\pi$
$887$	714.733			0.805787			0.402894	+	0.915247i	$0.368004\pi$
$888$		0			0
$889$			692.558i			0.779030i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	798.177i		−	0.893816i
$894$		0			0
$895$		−	10.1666i		−	0.0113593i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−42.7955			−0.0476035
$900$		0			0
$901$	−1058.73			−1.17507
$902$		0			0
$903$		0			0
$904$		0			0
$905$	155.282			0.171582
$906$		0			0
$907$			876.164i			0.966003i	0.875620	+	0.483001i	$0.160453\pi$
$907$			876.164i			0.966003i	−0.875620	+	0.483001i	$0.839547\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			286.256i			0.314222i	0.987581	+	0.157111i	$0.0502180\pi$
$911$			286.256i			0.314222i	−0.987581	+	0.157111i	$0.949782\pi$
$912$		0			0
$913$	−2242.28			−2.45595
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1431.75i			1.56134i
$918$		0			0
$919$		−	701.426i		−	0.763249i	−0.924317	−	0.381625i	$-0.875365\pi$
$919$		−	701.426i		−	0.763249i	0.924317	−	0.381625i	$-0.124635\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1697.67			1.83930
$924$		0			0
$925$			258.517i			0.279478i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−651.581			−0.701379			−0.350690	−	0.936492i	$-0.614053\pi$
$929$	−651.581			−0.701379			−0.350690	+	0.936492i	$0.614053\pi$
$930$		0			0
$931$			1227.18i			1.31813i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	1196.24i		−	1.27940i
$936$		0			0
$937$		−	81.1240i		−	0.0865784i	−0.999063	−	0.0432892i	$-0.986216\pi$
$937$		−	81.1240i		−	0.0865784i	0.999063	−	0.0432892i	$-0.0137837\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			158.773i			0.168728i	0.996435	+	0.0843641i	$0.0268859\pi$
$941$			158.773i			0.168728i	−0.996435	+	0.0843641i	$0.973114\pi$
$942$		0			0
$943$	1001.74	−	752.714i	1.06229	−	0.798212i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	557.127			0.588307			0.294154	−	0.955758i	$-0.404962\pi$
$947$	557.127			0.588307			0.294154	+	0.955758i	$0.404962\pi$
$948$		0			0
$949$	−1556.63			−1.64029
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	982.429i		−	1.03088i	−0.856926	−	0.515440i	$-0.827629\pi$
$953$		−	982.429i		−	1.03088i	0.856926	−	0.515440i	$-0.172371\pi$
$954$		0			0
$955$	483.313			0.506086
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1075.55			−1.12153
$960$		0			0
$961$	−237.268			−0.246897
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	532.329i		−	0.551636i
$966$		0			0
$967$	−993.341			−1.02724			−0.513620	−	0.858018i	$-0.671696\pi$
$967$	−993.341			−1.02724			−0.513620	+	0.858018i	$0.671696\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	903.976i		−	0.930975i	−0.885055	−	0.465487i	$-0.845879\pi$
$971$		−	903.976i		−	0.930975i	0.885055	−	0.465487i	$-0.154121\pi$
$972$		0			0
$973$			1493.62i			1.53507i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			681.912i			0.697965i	0.937129	+	0.348982i	$0.113473\pi$
$977$			681.912i			0.697965i	−0.937129	+	0.348982i	$0.886527\pi$
$978$		0			0
$979$	−417.299			−0.426251
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	849.167i		−	0.863852i	−0.901909	−	0.431926i	$-0.857834\pi$
$983$		−	849.167i		−	0.863852i	0.901909	−	0.431926i	$-0.142166\pi$
$984$		0			0
$985$		−	197.287i		−	0.200292i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1123.22	+	1494.83i	1.13572	+	1.51146i
$990$		0			0
$991$	−321.586			−0.324507			−0.162253	−	0.986749i	$-0.551876\pi$
$991$	−321.586			−0.324507			−0.162253	+	0.986749i	$0.551876\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	165.824			0.166657
$996$		0			0
$997$	−1057.30			−1.06049			−0.530243	−	0.847846i	$-0.677899\pi$
$997$	−1057.30			−1.06049			−0.530243	+	0.847846i	$0.677899\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.d.b.2161.15	yes	32
3.2	odd	2	inner	4140.3.d.b.2161.31	yes	32
23.22	odd	2	inner	4140.3.d.b.2161.18	yes	32
69.68	even	2	inner	4140.3.d.b.2161.2	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.3.d.b.2161.2	✓	32	69.68	even	2	inner
4140.3.d.b.2161.15	yes	32	1.1	even	1	trivial
4140.3.d.b.2161.18	yes	32	23.22	odd	2	inner
4140.3.d.b.2161.31	yes	32	3.2	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)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +9.55915i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} +9.55915i q^{7} +18.8089i q^{11} +20.1043 q^{13} -28.4427i q^{17} -28.9585i q^{19} +(-18.3876 + 13.8165i) q^{23} -5.00000 q^{25} +1.59078 q^{29} -26.9023 q^{31} +21.3749 q^{35} -51.7035i q^{37} -54.4792 q^{41} -81.2956i q^{43} +27.5628 q^{47} -42.3773 q^{49} -37.2235i q^{53} +42.0579 q^{55} +53.7093 q^{59} +39.8548i q^{61} -44.9547i q^{65} -42.9728i q^{67} +84.4430 q^{71} -77.4277 q^{73} -179.797 q^{77} -80.2412i q^{79} +119.214i q^{83} -63.5997 q^{85} +22.1863i q^{89} +192.180i q^{91} -64.7531 q^{95} +99.9018i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 24 q^{13} - 160 q^{25} - 28 q^{31} - 260 q^{49} + 120 q^{55} - 296 q^{73} - 60 q^{85}+O(q^{100}) \) 32 * q - 24 * q^13 - 160 * q^25 - 28 * q^31 - 260 * q^49 + 120 * q^55 - 296 * q^73 - 60 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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