LMFDB - Embedded newform 4140.3.d.b.2161.19

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			2161.19
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.b.2161.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{5} -8.25712i q^{7} +O(q^{10})$	$q+2.23607i q^{5} -8.25712i q^{7} -1.54491i q^{11} -9.82497 q^{13} +30.3657i q^{17} -35.7016i q^{19} +(-6.71778 + 21.9971i) q^{23} -5.00000 q^{25} +26.2090 q^{29} +20.1852 q^{31} +18.4635 q^{35} -29.5761i q^{37} +4.35033 q^{41} +14.5062i q^{43} -21.5540 q^{47} -19.1801 q^{49} -82.8682i q^{53} +3.45452 q^{55} +13.9428 q^{59} -91.8245i q^{61} -21.9693i q^{65} +88.3187i q^{67} -77.9367 q^{71} -101.521 q^{73} -12.7565 q^{77} +41.8245i q^{79} +83.7907i q^{83} -67.8998 q^{85} -62.4761i q^{89} +81.1260i q^{91} +79.8312 q^{95} -12.9698i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	8.25712i		−	1.17959i	−0.807553	−	0.589794i	$-0.799209\pi$
$7$		−	8.25712i		−	1.17959i	0.807553	−	0.589794i	$-0.200791\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.54491i		−	0.140446i	−0.997531	−	0.0702232i	$-0.977629\pi$
$11$		−	1.54491i		−	0.140446i	0.997531	−	0.0702232i	$-0.0223711\pi$
$12$		0			0
$13$	−9.82497			−0.755767			−0.377883	−	0.925853i	$-0.623348\pi$
$13$	−9.82497			−0.755767			−0.377883	+	0.925853i	$0.623348\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			30.3657i			1.78622i	0.449841	+	0.893109i	$0.351481\pi$
$17$			30.3657i			1.78622i	−0.449841	+	0.893109i	$0.648519\pi$
$18$		0			0
$19$		−	35.7016i		−	1.87903i	−0.342505	−	0.939516i	$-0.611275\pi$
$19$		−	35.7016i		−	1.87903i	0.342505	−	0.939516i	$-0.388725\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.71778	+	21.9971i	−0.292077	+	0.956395i
$23$	−6.71778	+	21.9971i	−0.292077	+	0.956395i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	26.2090			0.903759			0.451880	−	0.892079i	$-0.350754\pi$
$29$	26.2090			0.903759			0.451880	+	0.892079i	$0.350754\pi$
$30$		0			0
$31$	20.1852			0.651135			0.325567	−	0.945519i	$-0.394445\pi$
$31$	20.1852			0.651135			0.325567	+	0.945519i	$0.394445\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	18.4635			0.527528
$36$		0			0
$37$		−	29.5761i		−	0.799354i	−0.916656	−	0.399677i	$-0.869122\pi$
$37$		−	29.5761i		−	0.799354i	0.916656	−	0.399677i	$-0.130878\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.35033			0.106106			0.0530528	−	0.998592i	$-0.483105\pi$
$41$	4.35033			0.106106			0.0530528	+	0.998592i	$0.483105\pi$
$42$		0			0
$43$			14.5062i			0.337355i	0.985671	+	0.168677i	$0.0539496\pi$
$43$			14.5062i			0.337355i	−0.985671	+	0.168677i	$0.946050\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−21.5540			−0.458596			−0.229298	−	0.973356i	$-0.573643\pi$
$47$	−21.5540			−0.458596			−0.229298	+	0.973356i	$0.573643\pi$
$48$		0			0
$49$	−19.1801			−0.391430
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	82.8682i		−	1.56355i	−0.623560	−	0.781775i	$-0.714314\pi$
$53$		−	82.8682i		−	1.56355i	0.623560	−	0.781775i	$-0.285686\pi$
$54$		0			0
$55$	3.45452			0.0628095
$56$		0			0
$57$		0			0
$58$		0			0
$59$	13.9428			0.236318			0.118159	−	0.992995i	$-0.462301\pi$
$59$	13.9428			0.236318			0.118159	+	0.992995i	$0.462301\pi$
$60$		0			0
$61$		−	91.8245i		−	1.50532i	−0.658410	−	0.752659i	$-0.728771\pi$
$61$		−	91.8245i		−	1.50532i	0.658410	−	0.752659i	$-0.271229\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	21.9693i		−	0.337989i
$66$		0			0
$67$			88.3187i			1.31819i	0.752060	+	0.659095i	$0.229061\pi$
$67$			88.3187i			1.31819i	−0.752060	+	0.659095i	$0.770939\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−77.9367			−1.09770			−0.548850	−	0.835921i	$-0.684934\pi$
$71$	−77.9367			−1.09770			−0.548850	+	0.835921i	$0.684934\pi$
$72$		0			0
$73$	−101.521			−1.39070			−0.695348	−	0.718674i	$-0.744750\pi$
$73$	−101.521			−1.39070			−0.695348	+	0.718674i	$0.744750\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.7565			−0.165669
$78$		0			0
$79$			41.8245i			0.529425i	0.964327	+	0.264712i	$0.0852770\pi$
$79$			41.8245i			0.529425i	−0.964327	+	0.264712i	$0.914723\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			83.7907i			1.00953i	0.863258	+	0.504763i	$0.168420\pi$
$83$			83.7907i			1.00953i	−0.863258	+	0.504763i	$0.831580\pi$
$84$		0			0
$85$	−67.8998			−0.798821
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	62.4761i		−	0.701979i	−0.936379	−	0.350989i	$-0.885845\pi$
$89$		−	62.4761i		−	0.701979i	0.936379	−	0.350989i	$-0.114155\pi$
$90$		0			0
$91$			81.1260i			0.891494i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	79.8312			0.840329
$96$		0			0
$97$		−	12.9698i		−	0.133710i	−0.997763	−	0.0668548i	$-0.978704\pi$
$97$		−	12.9698i		−	0.133710i	0.997763	−	0.0668548i	$-0.0212964\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−121.482			−1.20279			−0.601395	−	0.798952i	$-0.705388\pi$
$101$	−121.482			−1.20279			−0.601395	+	0.798952i	$0.705388\pi$
$102$		0			0
$103$			72.1346i			0.700336i	0.936687	+	0.350168i	$0.113875\pi$
$103$			72.1346i			0.700336i	−0.936687	+	0.350168i	$0.886125\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			90.1030i			0.842084i	0.907041	+	0.421042i	$0.138335\pi$
$107$			90.1030i			0.842084i	−0.907041	+	0.421042i	$0.861665\pi$
$108$		0			0
$109$		−	136.256i		−	1.25006i	−0.780602	−	0.625029i	$-0.785087\pi$
$109$		−	136.256i		−	1.25006i	0.780602	−	0.625029i	$-0.214913\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			10.0375i			0.0888273i	0.999013	+	0.0444136i	$0.0141419\pi$
$113$			10.0375i			0.0888273i	−0.999013	+	0.0444136i	$0.985858\pi$
$114$		0			0
$115$	−49.1870	−	15.0214i	−0.427713	−	0.130621i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	250.733			2.10700
$120$		0			0
$121$	118.613			0.980275
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−236.137			−1.85934			−0.929672	−	0.368389i	$-0.879909\pi$
$127$	−236.137			−1.85934			−0.929672	+	0.368389i	$0.879909\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−199.418			−1.52228			−0.761139	−	0.648589i	$-0.775360\pi$
$131$	−199.418			−1.52228			−0.761139	+	0.648589i	$0.775360\pi$
$132$		0			0
$133$	−294.793			−2.21649
$134$		0			0
$135$		0			0
$136$		0			0
$137$			121.308i			0.885463i	0.896654	+	0.442731i	$0.145990\pi$
$137$			121.308i			0.885463i	−0.896654	+	0.442731i	$0.854010\pi$
$138$		0			0
$139$	58.6966			0.422278			0.211139	−	0.977456i	$-0.432283\pi$
$139$	58.6966			0.422278			0.211139	+	0.977456i	$0.432283\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			15.1787i			0.106145i
$144$		0			0
$145$			58.6052i			0.404174i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	78.0370i		−	0.523738i	−0.965103	−	0.261869i	$-0.915661\pi$
$149$		−	78.0370i		−	0.523738i	0.965103	−	0.261869i	$-0.0843389\pi$
$150$		0			0
$151$	46.1110			0.305371			0.152685	−	0.988275i	$-0.451208\pi$
$151$	46.1110			0.305371			0.152685	+	0.988275i	$0.451208\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			45.1354i			0.291196i
$156$		0			0
$157$			64.2552i			0.409269i	0.978838	+	0.204634i	$0.0656006\pi$
$157$			64.2552i			0.409269i	−0.978838	+	0.204634i	$0.934399\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	181.633	+	55.4696i	1.12815	+	0.344531i
$162$		0			0
$163$	142.643			0.875110			0.437555	−	0.899192i	$-0.355845\pi$
$163$	142.643			0.875110			0.437555	+	0.899192i	$0.355845\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	243.385			1.45740			0.728699	−	0.684835i	$-0.240126\pi$
$167$	243.385			1.45740			0.728699	+	0.684835i	$0.240126\pi$
$168$		0			0
$169$	−72.4700			−0.428816
$170$		0			0
$171$		0			0
$172$		0			0
$173$	110.046			0.636106			0.318053	−	0.948073i	$-0.396971\pi$
$173$	110.046			0.636106			0.318053	+	0.948073i	$0.396971\pi$
$174$		0			0
$175$			41.2856i			0.235918i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−217.476			−1.21495			−0.607474	−	0.794340i	$-0.707817\pi$
$179$	−217.476			−1.21495			−0.607474	+	0.794340i	$0.707817\pi$
$180$		0			0
$181$			18.3140i			0.101182i	0.998719	+	0.0505912i	$0.0161106\pi$
$181$			18.3140i			0.101182i	−0.998719	+	0.0505912i	$0.983889\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	66.1341			0.357482
$186$		0			0
$187$	46.9123			0.250868
$188$		0			0
$189$		0			0
$190$		0			0
$191$			332.407i			1.74035i	0.492741	+	0.870176i	$0.335995\pi$
$191$			332.407i			1.74035i	−0.492741	+	0.870176i	$0.664005\pi$
$192$		0			0
$193$	−74.8827			−0.387993			−0.193997	−	0.981002i	$-0.562145\pi$
$193$	−74.8827			−0.387993			−0.193997	+	0.981002i	$0.562145\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−274.529			−1.39355			−0.696774	−	0.717290i	$-0.745382\pi$
$197$	−274.529			−1.39355			−0.696774	+	0.717290i	$0.745382\pi$
$198$		0			0
$199$			325.119i			1.63376i	0.576806	+	0.816881i	$0.304299\pi$
$199$			325.119i			1.63376i	−0.576806	+	0.816881i	$0.695701\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	216.411i		−	1.06606i
$204$		0			0
$205$			9.72764i			0.0474519i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−55.1558			−0.263903
$210$		0			0
$211$	−323.434			−1.53286			−0.766430	−	0.642328i	$-0.777969\pi$
$211$	−323.434			−1.53286			−0.766430	+	0.642328i	$0.777969\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−32.4369			−0.150870
$216$		0			0
$217$		−	166.672i		−	0.768072i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	298.342i		−	1.34996i
$222$		0			0
$223$	−219.020			−0.982151			−0.491076	−	0.871117i	$-0.663396\pi$
$223$	−219.020			−0.982151			−0.491076	+	0.871117i	$0.663396\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			52.6040i			0.231736i	0.993265	+	0.115868i	$0.0369649\pi$
$227$			52.6040i			0.231736i	−0.993265	+	0.115868i	$0.963035\pi$
$228$		0			0
$229$			298.061i			1.30158i	0.759259	+	0.650788i	$0.225562\pi$
$229$			298.061i			1.30158i	−0.759259	+	0.650788i	$0.774438\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	40.1511			0.172322			0.0861612	−	0.996281i	$-0.472540\pi$
$233$	40.1511			0.172322			0.0861612	+	0.996281i	$0.472540\pi$
$234$		0			0
$235$		−	48.1963i		−	0.205091i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−57.3844			−0.240102			−0.120051	−	0.992768i	$-0.538306\pi$
$239$	−57.3844			−0.240102			−0.120051	+	0.992768i	$0.538306\pi$
$240$		0			0
$241$			225.395i			0.935250i	0.883927	+	0.467625i	$0.154890\pi$
$241$			225.395i			0.935250i	−0.883927	+	0.467625i	$0.845110\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	42.8880i		−	0.175053i
$246$		0			0
$247$			350.767i			1.42011i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	456.663i		−	1.81937i	−0.415295	−	0.909687i	$-0.636322\pi$
$251$		−	456.663i		−	1.81937i	0.415295	−	0.909687i	$-0.363678\pi$
$252$		0			0
$253$	33.9835	+	10.3784i	0.134322	+	0.0410212i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−201.760			−0.785057			−0.392528	−	0.919740i	$-0.628400\pi$
$257$	−201.760			−0.785057			−0.392528	+	0.919740i	$0.628400\pi$
$258$		0			0
$259$	−244.213			−0.942909
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	130.499i		−	0.496192i	−0.968735	−	0.248096i	$-0.920195\pi$
$263$		−	130.499i		−	0.496192i	0.968735	−	0.248096i	$-0.0798049\pi$
$264$		0			0
$265$	185.299			0.699241
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−377.216			−1.40229			−0.701145	−	0.713018i	$-0.747328\pi$
$269$	−377.216			−1.40229			−0.701145	+	0.713018i	$0.747328\pi$
$270$		0			0
$271$	−130.338			−0.480951			−0.240475	−	0.970655i	$-0.577303\pi$
$271$	−130.338			−0.480951			−0.240475	+	0.970655i	$0.577303\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			7.72455i			0.0280893i
$276$		0			0
$277$	345.291			1.24654			0.623268	−	0.782008i	$-0.285805\pi$
$277$	345.291			1.24654			0.623268	+	0.782008i	$0.285805\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			77.5390i			0.275939i	0.990436	+	0.137970i	$0.0440577\pi$
$281$			77.5390i			0.275939i	−0.990436	+	0.137970i	$0.955942\pi$
$282$		0			0
$283$			63.7549i			0.225282i	0.993636	+	0.112641i	$0.0359311\pi$
$283$			63.7549i			0.225282i	−0.993636	+	0.112641i	$0.964069\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	35.9212i		−	0.125161i
$288$		0			0
$289$	−633.075			−2.19057
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	117.299i		−	0.400338i	−0.979761	−	0.200169i	$-0.935851\pi$
$293$		−	117.299i		−	0.400338i	0.979761	−	0.200169i	$-0.0641491\pi$
$294$		0			0
$295$			31.1769i			0.105685i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	66.0020	−	216.121i	0.220742	−	0.722811i
$300$		0			0
$301$	119.780			0.397940
$302$		0			0
$303$		0			0
$304$		0			0
$305$	205.326			0.673199
$306$		0			0
$307$	−453.883			−1.47845			−0.739223	−	0.673461i	$-0.764807\pi$
$307$	−453.883			−1.47845			−0.739223	+	0.673461i	$0.764807\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−289.628			−0.931279			−0.465640	−	0.884974i	$-0.654176\pi$
$311$	−289.628			−0.931279			−0.465640	+	0.884974i	$0.654176\pi$
$312$		0			0
$313$			586.270i			1.87307i	0.350579	+	0.936533i	$0.385985\pi$
$313$			586.270i			1.87307i	−0.350579	+	0.936533i	$0.614015\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	183.164			0.577806			0.288903	−	0.957358i	$-0.406710\pi$
$317$	183.164			0.577806			0.288903	+	0.957358i	$0.406710\pi$
$318$		0			0
$319$		−	40.4906i		−	0.126930i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1084.10			3.35636
$324$		0			0
$325$	49.1248			0.151153
$326$		0			0
$327$		0			0
$328$		0			0
$329$			177.974i			0.540955i
$330$		0			0
$331$	−117.331			−0.354473			−0.177237	−	0.984168i	$-0.556716\pi$
$331$	−117.331			−0.354473			−0.177237	+	0.984168i	$0.556716\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−197.487			−0.589513
$336$		0			0
$337$		−	534.552i		−	1.58621i	−0.609086	−	0.793104i	$-0.708464\pi$
$337$		−	534.552i		−	1.58621i	0.609086	−	0.793104i	$-0.291536\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	31.1843i		−	0.0914495i
$342$		0			0
$343$		−	246.227i		−	0.717862i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−242.924			−0.700070			−0.350035	−	0.936737i	$-0.613830\pi$
$347$	−242.924			−0.700070			−0.350035	+	0.936737i	$0.613830\pi$
$348$		0			0
$349$	−219.666			−0.629417			−0.314708	−	0.949188i	$-0.601907\pi$
$349$	−219.666			−0.629417			−0.314708	+	0.949188i	$0.601907\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	426.201			1.20737			0.603684	−	0.797224i	$-0.293699\pi$
$353$	426.201			1.20737			0.603684	+	0.797224i	$0.293699\pi$
$354$		0			0
$355$		−	174.272i		−	0.490906i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			280.197i			0.780494i	0.920710	+	0.390247i	$0.127610\pi$
$359$			280.197i			0.780494i	−0.920710	+	0.390247i	$0.872390\pi$
$360$		0			0
$361$	−913.605			−2.53076
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	227.007i		−	0.621938i
$366$		0			0
$367$			122.724i			0.334398i	0.985923	+	0.167199i	$0.0534722\pi$
$367$			122.724i			0.334398i	−0.985923	+	0.167199i	$0.946528\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−684.253			−1.84435
$372$		0			0
$373$		−	548.909i		−	1.47161i	−0.677196	−	0.735803i	$-0.736805\pi$
$373$		−	548.909i		−	1.47161i	0.677196	−	0.735803i	$-0.263195\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−257.503			−0.683031
$378$		0			0
$379$		−	242.303i		−	0.639322i	−0.947532	−	0.319661i	$-0.896431\pi$
$379$		−	242.303i		−	0.639322i	0.947532	−	0.319661i	$-0.103569\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	78.0097i		−	0.203681i	−0.994801	−	0.101840i	$-0.967527\pi$
$383$		−	78.0097i		−	0.203681i	0.994801	−	0.101840i	$-0.0324731\pi$
$384$		0			0
$385$		−	28.5244i		−	0.0740894i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			403.808i			1.03807i	0.854754	+	0.519033i	$0.173708\pi$
$389$			403.808i			1.03807i	−0.854754	+	0.519033i	$0.826292\pi$
$390$		0			0
$391$	−667.956	−	203.990i	−1.70833	−	0.521714i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−93.5225			−0.236766
$396$		0			0
$397$	−404.295			−1.01837			−0.509187	−	0.860656i	$-0.670054\pi$
$397$	−404.295			−1.01837			−0.509187	+	0.860656i	$0.670054\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			163.662i			0.408135i	0.978957	+	0.204067i	$0.0654162\pi$
$401$			163.662i			0.408135i	−0.978957	+	0.204067i	$0.934584\pi$
$402$		0			0
$403$	−198.319			−0.492106
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−45.6924			−0.112266
$408$		0			0
$409$	−121.485			−0.297029			−0.148514	−	0.988910i	$-0.547449\pi$
$409$	−121.485			−0.297029			−0.148514	+	0.988910i	$0.547449\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	115.127i		−	0.278758i
$414$		0			0
$415$	−187.362			−0.451474
$416$		0			0
$417$		0			0
$418$		0			0
$419$			648.991i			1.54891i	0.632632	+	0.774453i	$0.281975\pi$
$419$			648.991i			1.54891i	−0.632632	+	0.774453i	$0.718025\pi$
$420$		0			0
$421$		−	245.177i		−	0.582369i	−0.956667	−	0.291184i	$-0.905951\pi$
$421$		−	245.177i		−	0.582369i	0.956667	−	0.291184i	$-0.0940493\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	151.828i		−	0.357243i
$426$		0			0
$427$	−758.206			−1.77566
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	32.4641i		−	0.0753228i	−0.999291	−	0.0376614i	$-0.988009\pi$
$431$		−	32.4641i		−	0.0753228i	0.999291	−	0.0376614i	$-0.0119908\pi$
$432$		0			0
$433$		−	656.422i		−	1.51599i	−0.652262	−	0.757993i	$-0.726180\pi$
$433$		−	656.422i		−	1.51599i	0.652262	−	0.757993i	$-0.273820\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	785.331	+	239.836i	1.79710	+	0.548823i
$438$		0			0
$439$	−165.051			−0.375970			−0.187985	−	0.982172i	$-0.560196\pi$
$439$	−165.051			−0.375970			−0.187985	+	0.982172i	$0.560196\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	301.111			0.679710			0.339855	−	0.940478i	$-0.389622\pi$
$443$	301.111			0.679710			0.339855	+	0.940478i	$0.389622\pi$
$444$		0			0
$445$	139.701			0.313934
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−148.645			−0.331059			−0.165529	−	0.986205i	$-0.552933\pi$
$449$	−148.645			−0.331059			−0.165529	+	0.986205i	$0.552933\pi$
$450$		0			0
$451$		−	6.72087i		−	0.0149022i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−181.403			−0.398688
$456$		0			0
$457$			560.260i			1.22595i	0.790102	+	0.612976i	$0.210028\pi$
$457$			560.260i			1.22595i	−0.790102	+	0.612976i	$0.789972\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−118.810			−0.257721			−0.128861	−	0.991663i	$-0.541132\pi$
$461$	−118.810			−0.257721			−0.128861	+	0.991663i	$0.541132\pi$
$462$		0			0
$463$	441.958			0.954553			0.477277	−	0.878753i	$-0.341624\pi$
$463$	441.958			0.954553			0.477277	+	0.878753i	$0.341624\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	352.531i		−	0.754884i	−0.926033	−	0.377442i	$-0.876804\pi$
$467$		−	352.531i		−	0.754884i	0.926033	−	0.377442i	$-0.123196\pi$
$468$		0			0
$469$	729.259			1.55492
$470$		0			0
$471$		0			0
$472$		0			0
$473$	22.4108			0.0473802
$474$		0			0
$475$			178.508i			0.375806i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	474.158i		−	0.989891i	−0.868924	−	0.494946i	$-0.835188\pi$
$479$		−	474.158i		−	0.989891i	0.868924	−	0.494946i	$-0.164812\pi$
$480$		0			0
$481$			290.584i			0.604125i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	29.0014			0.0597967
$486$		0			0
$487$	−847.581			−1.74041			−0.870206	−	0.492687i	$-0.836015\pi$
$487$	−847.581			−1.74041			−0.870206	+	0.492687i	$0.836015\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	213.009			0.433826			0.216913	−	0.976191i	$-0.430401\pi$
$491$	213.009			0.433826			0.216913	+	0.976191i	$0.430401\pi$
$492$		0			0
$493$			795.855i			1.61431i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			643.533i			1.29483i
$498$		0			0
$499$	−136.037			−0.272619			−0.136309	−	0.990666i	$-0.543524\pi$
$499$	−136.037			−0.272619			−0.136309	+	0.990666i	$0.543524\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			723.337i			1.43805i	0.694986	+	0.719023i	$0.255410\pi$
$503$			723.337i			1.43805i	−0.694986	+	0.719023i	$0.744590\pi$
$504$		0			0
$505$		−	271.641i		−	0.537904i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−688.135			−1.35194			−0.675968	−	0.736931i	$-0.736274\pi$
$509$	−688.135			−1.35194			−0.675968	+	0.736931i	$0.736274\pi$
$510$		0			0
$511$			838.269i			1.64045i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−161.298			−0.313200
$516$		0			0
$517$			33.2990i			0.0644082i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	536.837i		−	1.03040i	−0.857071	−	0.515199i	$-0.827718\pi$
$521$		−	536.837i		−	1.03040i	0.857071	−	0.515199i	$-0.172282\pi$
$522$		0			0
$523$			185.934i			0.355515i	0.984074	+	0.177758i	$0.0568843\pi$
$523$			185.934i			0.355515i	−0.984074	+	0.177758i	$0.943116\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			612.937i			1.16307i
$528$		0			0
$529$	−438.743	−	295.543i	−0.829381	−	0.558683i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−42.7419			−0.0801911
$534$		0			0
$535$	−201.476			−0.376591
$536$		0			0
$537$		0			0
$538$		0			0
$539$			29.6315i			0.0549749i
$540$		0			0
$541$	142.351			0.263126			0.131563	−	0.991308i	$-0.458000\pi$
$541$	142.351			0.263126			0.131563	+	0.991308i	$0.458000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	304.678			0.559042
$546$		0			0
$547$	18.9760			0.0346911			0.0173455	−	0.999850i	$-0.494478\pi$
$547$	18.9760			0.0346911			0.0173455	+	0.999850i	$0.494478\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	935.704i		−	1.69819i
$552$		0			0
$553$	345.350			0.624503
$554$		0			0
$555$		0			0
$556$		0			0
$557$			366.555i			0.658088i	0.944315	+	0.329044i	$0.106726\pi$
$557$			366.555i			0.658088i	−0.944315	+	0.329044i	$0.893274\pi$
$558$		0			0
$559$		−	142.523i		−	0.254961i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			225.193i			0.399988i	0.979797	+	0.199994i	$0.0640923\pi$
$563$			225.193i			0.399988i	−0.979797	+	0.199994i	$0.935908\pi$
$564$		0			0
$565$	−22.4445			−0.0397248
$566$		0			0
$567$		0			0
$568$		0			0
$569$			271.777i			0.477639i	0.971064	+	0.238820i	$0.0767604\pi$
$569$			271.777i			0.477639i	−0.971064	+	0.238820i	$0.923240\pi$
$570$		0			0
$571$			250.599i			0.438878i	0.975626	+	0.219439i	$0.0704227\pi$
$571$			250.599i			0.438878i	−0.975626	+	0.219439i	$0.929577\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	33.5889	−	109.985i	0.0584155	−	0.191279i
$576$		0			0
$577$	664.990			1.15250			0.576248	−	0.817275i	$-0.304516\pi$
$577$	664.990			1.15250			0.576248	+	0.817275i	$0.304516\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	691.870			1.19083
$582$		0			0
$583$	−128.024			−0.219595
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−104.744			−0.178440			−0.0892198	−	0.996012i	$-0.528437\pi$
$587$	−104.744			−0.178440			−0.0892198	+	0.996012i	$0.528437\pi$
$588$		0			0
$589$		−	720.644i		−	1.22350i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−846.406			−1.42733			−0.713665	−	0.700488i	$-0.752966\pi$
$593$	−846.406			−1.42733			−0.713665	+	0.700488i	$0.752966\pi$
$594$		0			0
$595$			560.657i			0.942280i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	846.157			1.41262			0.706308	−	0.707905i	$-0.250359\pi$
$599$	846.157			1.41262			0.706308	+	0.707905i	$0.250359\pi$
$600$		0			0
$601$	−122.784			−0.204299			−0.102150	−	0.994769i	$-0.532572\pi$
$601$	−122.784			−0.204299			−0.102150	+	0.994769i	$0.532572\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			265.227i			0.438392i
$606$		0			0
$607$	312.344			0.514570			0.257285	−	0.966336i	$-0.417172\pi$
$607$	312.344			0.514570			0.257285	+	0.966336i	$0.417172\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	211.768			0.346592
$612$		0			0
$613$			483.988i			0.789540i	0.918780	+	0.394770i	$0.129176\pi$
$613$			483.988i			0.789540i	−0.918780	+	0.394770i	$0.870824\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	655.050i		−	1.06167i	−0.847475	−	0.530835i	$-0.821878\pi$
$617$		−	655.050i		−	1.06167i	0.847475	−	0.530835i	$-0.178122\pi$
$618$		0			0
$619$			778.464i			1.25762i	0.777561	+	0.628808i	$0.216457\pi$
$619$			778.464i			1.25762i	−0.777561	+	0.628808i	$0.783543\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−515.873			−0.828046
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	898.098			1.42782
$630$		0			0
$631$			0.479764i			0.000760323i	1.00000		0.000380161i	$0.000121009\pi$
$631$			0.479764i			0.000760323i	−1.00000		0.000380161i	$0.999879\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	528.018i		−	0.831524i
$636$		0			0
$637$	188.444			0.295830
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	431.973i		−	0.673904i	−0.941522	−	0.336952i	$-0.890604\pi$
$641$		−	431.973i		−	0.673904i	0.941522	−	0.336952i	$-0.109396\pi$
$642$		0			0
$643$		−	28.5576i		−	0.0444131i	−0.999753	−	0.0222065i	$-0.992931\pi$
$643$		−	28.5576i		−	0.0444131i	0.999753	−	0.0222065i	$-0.00706914\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−54.7403			−0.0846063			−0.0423032	−	0.999105i	$-0.513470\pi$
$647$	−54.7403			−0.0846063			−0.0423032	+	0.999105i	$0.513470\pi$
$648$		0			0
$649$		−	21.5403i		−	0.0331900i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−992.932			−1.52057			−0.760285	−	0.649590i	$-0.774941\pi$
$653$	−992.932			−1.52057			−0.760285	+	0.649590i	$0.774941\pi$
$654$		0			0
$655$		−	445.913i		−	0.680784i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			91.3775i			0.138661i	0.997594	+	0.0693304i	$0.0220863\pi$
$659$			91.3775i			0.138661i	−0.997594	+	0.0693304i	$0.977914\pi$
$660$		0			0
$661$		−	112.254i		−	0.169825i	−0.996388	−	0.0849126i	$-0.972939\pi$
$661$		−	112.254i		−	0.169825i	0.996388	−	0.0849126i	$-0.0270611\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	659.176i		−	0.991243i
$666$		0			0
$667$	−176.067	+	576.522i	−0.263968	+	0.864351i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−141.861			−0.211417
$672$		0			0
$673$	−183.306			−0.272371			−0.136186	−	0.990683i	$-0.543484\pi$
$673$	−183.306			−0.272371			−0.136186	+	0.990683i	$0.543484\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1119.70i		−	1.65391i	−0.562268	−	0.826955i	$-0.690071\pi$
$677$		−	1119.70i		−	1.65391i	0.562268	−	0.826955i	$-0.309929\pi$
$678$		0			0
$679$	−107.093			−0.157722
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−350.783			−0.513591			−0.256796	−	0.966466i	$-0.582667\pi$
$683$	−350.783			−0.513591			−0.256796	+	0.966466i	$0.582667\pi$
$684$		0			0
$685$	−271.254			−0.395991
$686$		0			0
$687$		0			0
$688$		0			0
$689$			814.177i			1.18168i
$690$		0			0
$691$	401.127			0.580501			0.290251	−	0.956951i	$-0.406261\pi$
$691$	401.127			0.580501			0.290251	+	0.956951i	$0.406261\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			131.250i			0.188848i
$696$		0			0
$697$			132.101i			0.189528i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	131.941i		−	0.188218i	−0.995562	−	0.0941090i	$-0.970000\pi$
$701$		−	131.941i		−	0.188218i	0.995562	−	0.0941090i	$-0.0300002\pi$
$702$		0			0
$703$	−1055.91			−1.50201
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1003.09i			1.41880i
$708$		0			0
$709$			1385.55i			1.95424i	0.212694	+	0.977119i	$0.431776\pi$
$709$			1385.55i			1.95424i	−0.212694	+	0.977119i	$0.568224\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−135.600	+	444.015i	−0.190182	+	0.622742i
$714$		0			0
$715$	−33.9406			−0.0474694
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1325.76			1.84389			0.921945	−	0.387321i	$-0.126600\pi$
$719$	1325.76			1.84389			0.921945	+	0.387321i	$0.126600\pi$
$720$		0			0
$721$	595.624			0.826109
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−131.045			−0.180752
$726$		0			0
$727$		−	121.055i		−	0.166513i	−0.996528	−	0.0832567i	$-0.973468\pi$
$727$		−	121.055i		−	0.166513i	0.996528	−	0.0832567i	$-0.0265322\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−440.492			−0.602588
$732$		0			0
$733$		−	719.733i		−	0.981900i	−0.871188	−	0.490950i	$-0.836650\pi$
$733$		−	719.733i		−	0.981900i	0.871188	−	0.490950i	$-0.163350\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	136.445			0.185135
$738$		0			0
$739$	150.535			0.203700			0.101850	−	0.994800i	$-0.467524\pi$
$739$	150.535			0.203700			0.101850	+	0.994800i	$0.467524\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			883.407i			1.18897i	0.804106	+	0.594486i	$0.202645\pi$
$743$			883.407i			1.18897i	−0.804106	+	0.594486i	$0.797355\pi$
$744$		0			0
$745$	174.496			0.234223
$746$		0			0
$747$		0			0
$748$		0			0
$749$	743.991			0.993313
$750$		0			0
$751$		−	1090.55i		−	1.45213i	−0.687625	−	0.726066i	$-0.741346\pi$
$751$		−	1090.55i		−	1.45213i	0.687625	−	0.726066i	$-0.258654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			103.107i			0.136566i
$756$		0			0
$757$		−	659.664i		−	0.871418i	−0.900088	−	0.435709i	$-0.856498\pi$
$757$		−	659.664i		−	0.871418i	0.900088	−	0.435709i	$-0.143502\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−85.9148			−0.112897			−0.0564486	−	0.998406i	$-0.517978\pi$
$761$	−85.9148			−0.112897			−0.0564486	+	0.998406i	$0.517978\pi$
$762$		0			0
$763$	−1125.08			−1.47455
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−136.987			−0.178601
$768$		0			0
$769$		−	288.068i		−	0.374600i	−0.982303	−	0.187300i	$-0.940026\pi$
$769$		−	288.068i		−	0.374600i	0.982303	−	0.187300i	$-0.0599737\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	616.977i		−	0.798159i	−0.916916	−	0.399079i	$-0.869330\pi$
$773$		−	616.977i		−	0.798159i	0.916916	−	0.399079i	$-0.130670\pi$
$774$		0			0
$775$	−100.926			−0.130227
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	155.314i		−	0.199376i
$780$		0			0
$781$			120.405i			0.154168i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−143.679			−0.183031
$786$		0			0
$787$			1026.07i			1.30377i	0.758318	+	0.651885i	$0.226022\pi$
$787$			1026.07i			1.30377i	−0.758318	+	0.651885i	$0.773978\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	82.8807			0.104780
$792$		0			0
$793$			902.172i			1.13767i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			343.185i			0.430596i	0.976548	+	0.215298i	$0.0690723\pi$
$797$			343.185i			0.430596i	−0.976548	+	0.215298i	$0.930928\pi$
$798$		0			0
$799$		−	654.503i		−	0.819153i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			156.840i			0.195318i
$804$		0			0
$805$	−124.034	+	406.143i	−0.154079	+	0.504525i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1142.71			−1.41249			−0.706246	−	0.707967i	$-0.749613\pi$
$809$	−1142.71			−1.41249			−0.706246	+	0.707967i	$0.749613\pi$
$810$		0			0
$811$	−614.448			−0.757643			−0.378821	−	0.925470i	$-0.623671\pi$
$811$	−614.448			−0.757643			−0.378821	+	0.925470i	$0.623671\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			318.959i			0.391361i
$816$		0			0
$817$	517.896			0.633900
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1243.62			−1.51476			−0.757381	−	0.652974i	$-0.773521\pi$
$821$	−1243.62			−1.51476			−0.757381	+	0.652974i	$0.773521\pi$
$822$		0			0
$823$	−215.102			−0.261364			−0.130682	−	0.991424i	$-0.541717\pi$
$823$	−215.102			−0.261364			−0.130682	+	0.991424i	$0.541717\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			857.854i			1.03731i	0.854984	+	0.518654i	$0.173567\pi$
$827$			857.854i			1.03731i	−0.854984	+	0.518654i	$0.826433\pi$
$828$		0			0
$829$	791.727			0.955038			0.477519	−	0.878621i	$-0.341536\pi$
$829$	791.727			0.955038			0.477519	+	0.878621i	$0.341536\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	582.416i		−	0.699179i
$834$		0			0
$835$			544.226i			0.651768i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	710.695i		−	0.847074i	−0.905879	−	0.423537i	$-0.860788\pi$
$839$		−	710.695i		−	0.847074i	0.905879	−	0.423537i	$-0.139212\pi$
$840$		0			0
$841$	−154.087			−0.183219
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	162.048i		−	0.191773i
$846$		0			0
$847$		−	979.404i		−	1.15632i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	650.587	+	198.686i	0.764498	+	0.233473i
$852$		0			0
$853$	−567.558			−0.665366			−0.332683	−	0.943039i	$-0.607954\pi$
$853$	−567.558			−0.665366			−0.332683	+	0.943039i	$0.607954\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1392.65			1.62503			0.812515	−	0.582940i	$-0.198098\pi$
$857$	1392.65			1.62503			0.812515	+	0.582940i	$0.198098\pi$
$858$		0			0
$859$	120.946			0.140799			0.0703993	−	0.997519i	$-0.477573\pi$
$859$	120.946			0.140799			0.0703993	+	0.997519i	$0.477573\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1523.91			−1.76583			−0.882915	−	0.469534i	$-0.844422\pi$
$863$	−1523.91			−1.76583			−0.882915	+	0.469534i	$0.844422\pi$
$864$		0			0
$865$			246.071i			0.284475i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	64.6152			0.0743558
$870$		0			0
$871$		−	867.729i		−	0.996244i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−92.3174			−0.105506
$876$		0			0
$877$	535.968			0.611138			0.305569	−	0.952170i	$-0.401153\pi$
$877$	535.968			0.611138			0.305569	+	0.952170i	$0.401153\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	275.381i		−	0.312577i	−0.987711	−	0.156289i	$-0.950047\pi$
$881$		−	275.381i		−	0.312577i	0.987711	−	0.156289i	$-0.0499530\pi$
$882$		0			0
$883$	1085.01			1.22877			0.614386	−	0.789006i	$-0.289404\pi$
$883$	1085.01			1.22877			0.614386	+	0.789006i	$0.289404\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	598.564			0.674819			0.337409	−	0.941358i	$-0.390449\pi$
$887$	598.564			0.674819			0.337409	+	0.941358i	$0.390449\pi$
$888$		0			0
$889$			1949.81i			2.19326i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			769.514i			0.861717i
$894$		0			0
$895$		−	486.290i		−	0.543341i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	529.034			0.588469
$900$		0			0
$901$	2516.35			2.79284
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−40.9514			−0.0452501
$906$		0			0
$907$		−	1134.72i		−	1.25106i	−0.780198	−	0.625532i	$-0.784882\pi$
$907$		−	1134.72i		−	1.25106i	0.780198	−	0.625532i	$-0.215118\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1617.84i		−	1.77589i	−0.459946	−	0.887947i	$-0.652131\pi$
$911$		−	1617.84i		−	1.77589i	0.459946	−	0.887947i	$-0.347869\pi$
$912$		0			0
$913$	129.449			0.141784
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1646.62i			1.79566i
$918$		0			0
$919$		−	1138.10i		−	1.23841i	−0.785228	−	0.619206i	$-0.787454\pi$
$919$		−	1138.10i		−	1.23841i	0.785228	−	0.619206i	$-0.212546\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	765.726			0.829605
$924$		0			0
$925$			147.880i			0.159871i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−169.583			−0.182544			−0.0912720	−	0.995826i	$-0.529093\pi$
$929$	−169.583			−0.182544			−0.0912720	+	0.995826i	$0.529093\pi$
$930$		0			0
$931$			684.760i			0.735510i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			104.899i			0.112191i
$936$		0			0
$937$			902.471i			0.963149i	0.876405	+	0.481575i	$0.159935\pi$
$937$			902.471i			0.963149i	−0.876405	+	0.481575i	$0.840065\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1203.32i		−	1.27877i	−0.768887	−	0.639385i	$-0.779189\pi$
$941$		−	1203.32i		−	1.27877i	0.768887	−	0.639385i	$-0.220811\pi$
$942$		0			0
$943$	−29.2246	+	95.6946i	−0.0309911	+	0.101479i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1055.03			1.11408			0.557040	−	0.830486i	$-0.311937\pi$
$947$	1055.03			1.11408			0.557040	+	0.830486i	$0.311937\pi$
$948$		0			0
$949$	997.438			1.05104
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	695.343i		−	0.729636i	−0.931079	−	0.364818i	$-0.881131\pi$
$953$		−	695.343i		−	0.729636i	0.931079	−	0.364818i	$-0.118869\pi$
$954$		0			0
$955$	−743.285			−0.778309
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1001.66			1.04448
$960$		0			0
$961$	−553.558			−0.576023
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	167.443i		−	0.173516i
$966$		0			0
$967$	−516.823			−0.534461			−0.267230	−	0.963633i	$-0.586108\pi$
$967$	−516.823			−0.534461			−0.267230	+	0.963633i	$0.586108\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1272.12i		−	1.31011i	−0.755580	−	0.655057i	$-0.772645\pi$
$971$		−	1272.12i		−	1.31011i	0.755580	−	0.655057i	$-0.227355\pi$
$972$		0			0
$973$		−	484.665i		−	0.498114i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1301.33i			1.33196i	0.745968	+	0.665982i	$0.231987\pi$
$977$			1301.33i			1.33196i	−0.745968	+	0.665982i	$0.768013\pi$
$978$		0			0
$979$	−96.5200			−0.0985904
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1644.23i			1.67266i	0.548226	+	0.836330i	$0.315303\pi$
$983$			1644.23i			1.67266i	−0.548226	+	0.836330i	$0.684697\pi$
$984$		0			0
$985$		−	613.866i		−	0.623214i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−319.095	−	97.4498i	−0.322644	−	0.0985337i
$990$		0			0
$991$	924.866			0.933266			0.466633	−	0.884451i	$-0.345467\pi$
$991$	924.866			0.933266			0.466633	+	0.884451i	$0.345467\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−726.988			−0.730641
$996$		0			0
$997$	748.988			0.751241			0.375621	−	0.926773i	$-0.377430\pi$
$997$	748.988			0.751241			0.375621	+	0.926773i	$0.377430\pi$
$998$		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.2161.19	yes	32
3.2	odd	2	inner	4140.3.d.b.2161.3	✓	32
23.22	odd	2	inner	4140.3.d.b.2161.14	yes	32
69.68	even	2	inner	4140.3.d.b.2161.30	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.3.d.b.2161.3	✓	32	3.2	odd	2	inner
4140.3.d.b.2161.14	yes	32	23.22	odd	2	inner
4140.3.d.b.2161.19	yes	32	1.1	even	1	trivial
4140.3.d.b.2161.30	yes	32	69.68	even	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)

\(f(q)\)	\(=\)	\(q+2.23607i q^{5} -8.25712i q^{7} +O(q^{10})\)	\(q+2.23607i q^{5} -8.25712i q^{7} -1.54491i q^{11} -9.82497 q^{13} +30.3657i q^{17} -35.7016i q^{19} +(-6.71778 + 21.9971i) q^{23} -5.00000 q^{25} +26.2090 q^{29} +20.1852 q^{31} +18.4635 q^{35} -29.5761i q^{37} +4.35033 q^{41} +14.5062i q^{43} -21.5540 q^{47} -19.1801 q^{49} -82.8682i q^{53} +3.45452 q^{55} +13.9428 q^{59} -91.8245i q^{61} -21.9693i q^{65} +88.3187i q^{67} -77.9367 q^{71} -101.521 q^{73} -12.7565 q^{77} +41.8245i q^{79} +83.7907i q^{83} -67.8998 q^{85} -62.4761i q^{89} +81.1260i q^{91} +79.8312 q^{95} -12.9698i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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