LMFDB - Embedded newform 161.4.c.a.160.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [161,4,Mod(160,161)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(161, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("161.160");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 161 = 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	161.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.49930751092$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 2 $ x^2 - x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			160.2
Root			$0.500000 + 1.32288i$ of defining polynomial
Character	$\chi$	$=$	161.160
Dual form			161.4.c.a.160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{2} +17.0000 q^{4} +18.5203i q^{7} +45.0000 q^{8} +27.0000 q^{9} -26.4575i q^{11} +92.6013i q^{14} +89.0000 q^{16} +135.000 q^{18} -132.288i q^{22} +(-20.0000 - 108.476i) q^{23} -125.000 q^{25} +314.844i q^{28} -166.000 q^{29} +85.0000 q^{32} +459.000 q^{36} -10.5830i q^{37} +534.442i q^{43} -449.778i q^{44} +(-100.000 - 542.379i) q^{46} -343.000 q^{49} -625.000 q^{50} -497.401i q^{53} +833.412i q^{56} -830.000 q^{58} +500.047i q^{63} -287.000 q^{64} -809.600i q^{67} +688.000 q^{71} +1215.00 q^{72} -52.9150i q^{74} +490.000 q^{77} +238.118i q^{79} +729.000 q^{81} +2672.21i q^{86} -1190.59i q^{88} +(-340.000 - 1844.09i) q^{92} -1715.00 q^{98} -714.353i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{2} + 34 q^{4} + 90 q^{8} + 54 q^{9} + 178 q^{16} + 270 q^{18} - 40 q^{23} - 250 q^{25} - 332 q^{29} + 170 q^{32} + 918 q^{36} - 200 q^{46} - 686 q^{49} - 1250 q^{50} - 1660 q^{58} - 574 q^{64}+ \cdots - 3430 q^{98}+O(q^{100}) $ 2 * q + 10 * q^2 + 34 * q^4 + 90 * q^8 + 54 * q^9 + 178 * q^16 + 270 * q^18 - 40 * q^23 - 250 * q^25 - 332 * q^29 + 170 * q^32 + 918 * q^36 - 200 * q^46 - 686 * q^49 - 1250 * q^50 - 1660 * q^58 - 574 * q^64 + 1376 * q^71 + 2430 * q^72 + 980 * q^77 + 1458 * q^81 - 680 * q^92 - 3430 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/161\mathbb{Z}\right)^\times$.

$n$	$24$	$120$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	5.00000			1.76777			0.883883	−	0.467707i	$-0.154920\pi$
$2$	5.00000			1.76777			0.883883	+	0.467707i	$0.154920\pi$
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	17.0000			2.12500
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$			18.5203i			1.00000i
$7$			18.5203i			1.00000i
$8$	45.0000			1.98874
$9$	27.0000			1.00000
$10$		0			0
$11$		−	26.4575i		−	0.725204i	−0.931944	−	0.362602i	$-0.881889\pi$
$11$		−	26.4575i		−	0.725204i	0.931944	−	0.362602i	$-0.118111\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$			92.6013i			1.76777i
$15$		0			0
$16$	89.0000			1.39062
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	135.000			1.76777
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		−	132.288i		−	1.28199i
$23$	−20.0000	−	108.476i	−0.181317	−	0.983425i
$23$	−20.0000	−	108.476i	−0.181317	−	0.983425i
$24$		0			0
$25$	−125.000			−1.00000
$26$		0			0
$27$		0			0
$28$			314.844i			2.12500i
$29$	−166.000			−1.06295			−0.531473	−	0.847075i	$-0.678361\pi$
$29$	−166.000			−1.06295			−0.531473	+	0.847075i	$0.678361\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	85.0000			0.469563
$33$		0			0
$34$		0			0
$35$		0			0
$36$	459.000			2.12500
$37$		−	10.5830i		−	0.0470226i	−0.999724	−	0.0235113i	$-0.992515\pi$
$37$		−	10.5830i		−	0.0470226i	0.999724	−	0.0235113i	$-0.00748457\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$			534.442i			1.89539i	0.319183	+	0.947693i	$0.396592\pi$
$43$			534.442i			1.89539i	−0.319183	+	0.947693i	$0.603408\pi$
$44$		−	449.778i		−	1.54106i
$45$		0			0
$46$	−100.000	−	542.379i	−0.320526	−	1.73847i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−343.000			−1.00000
$50$	−625.000			−1.76777
$51$		0			0
$52$		0			0
$53$		−	497.401i		−	1.28912i	−0.764554	−	0.644560i	$-0.777041\pi$
$53$		−	497.401i		−	1.28912i	0.764554	−	0.644560i	$-0.222959\pi$
$54$		0			0
$55$		0			0
$56$			833.412i			1.98874i
$57$		0			0
$58$	−830.000			−1.87904
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$			500.047i			1.00000i
$64$	−287.000			−0.560547
$65$		0			0
$66$		0			0
$67$		−	809.600i		−	1.47624i	−0.674667	−	0.738122i	$-0.735713\pi$
$67$		−	809.600i		−	1.47624i	0.674667	−	0.738122i	$-0.264287\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	688.000			1.15001			0.575004	−	0.818151i	$-0.305000\pi$
$71$	688.000			1.15001			0.575004	+	0.818151i	$0.305000\pi$
$72$	1215.00			1.98874
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		−	52.9150i		−	0.0831250i
$75$		0			0
$76$		0			0
$77$	490.000			0.725204
$78$		0			0
$79$			238.118i			0.339118i	0.985520	+	0.169559i	$0.0542343\pi$
$79$			238.118i			0.339118i	−0.985520	+	0.169559i	$0.945766\pi$
$80$		0			0
$81$	729.000			1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$			2672.21i			3.35060i
$87$		0			0
$88$		−	1190.59i		−	1.44224i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	−340.000	−	1844.09i	−0.385298	−	2.08978i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−1715.00			−1.76777
$99$		−	714.353i		−	0.725204i
$100$	−2125.00			−2.12500
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		−	2487.01i		−	2.27886i
$107$			1550.41i			1.40078i	0.713759	+	0.700392i	$0.246991\pi$
$107$			1550.41i			1.40078i	−0.713759	+	0.700392i	$0.753009\pi$
$108$		0			0
$109$			2275.35i			1.99944i	0.0237260	+	0.999718i	$0.492447\pi$
$109$			2275.35i			1.99944i	−0.0237260	+	0.999718i	$0.507553\pi$
$110$		0			0
$111$		0			0
$112$			1648.30i			1.39062i
$113$		−	2307.10i		−	1.92065i	−0.278886	−	0.960324i	$-0.589965\pi$
$113$		−	2307.10i		−	1.92065i	0.278886	−	0.960324i	$-0.410035\pi$
$114$		0			0
$115$		0			0
$116$	−2822.00			−2.25876
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	631.000			0.474080
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$			2500.23i			1.76777i
$127$	2000.00			1.39741			0.698706	−	0.715409i	$-0.253760\pi$
$127$	2000.00			1.39741			0.698706	+	0.715409i	$0.253760\pi$
$128$	−2115.00			−1.46048
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		−	4048.00i		−	2.60966i
$135$		0			0
$136$		0			0
$137$		−	783.142i		−	0.488382i	−0.969727	−	0.244191i	$-0.921478\pi$
$137$		−	783.142i		−	0.488382i	0.969727	−	0.244191i	$-0.0785224\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	3440.00			2.03295
$143$		0			0
$144$	2403.00			1.39062
$145$		0			0
$146$		0			0
$147$		0			0
$148$		−	179.911i		−	0.0999230i
$149$			3545.31i			1.94928i	0.223777	+	0.974640i	$0.428161\pi$
$149$			3545.31i			1.94928i	−0.223777	+	0.974640i	$0.571839\pi$
$150$		0			0
$151$	2952.00			1.59093			0.795465	−	0.606000i	$-0.207227\pi$
$151$	2952.00			1.59093			0.795465	+	0.606000i	$0.207227\pi$
$152$		0			0
$153$		0			0
$154$	2450.00			1.28199
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$			1190.59i			0.599482i
$159$		0			0
$160$		0			0
$161$	2009.00	−	370.405i	0.983425	−	0.181317i
$162$	3645.00			1.76777
$163$	1780.00			0.855340			0.427670	−	0.903935i	$-0.359335\pi$
$163$	1780.00			0.855340			0.427670	+	0.903935i	$0.359335\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	2197.00			1.00000
$170$		0			0
$171$		0			0
$172$			9085.51i			4.02770i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		−	2315.03i		−	1.00000i
$176$		−	2354.72i		−	1.00849i
$177$		0			0
$178$		0			0
$179$	2084.00			0.870198			0.435099	−	0.900383i	$-0.356713\pi$
$179$	2084.00			0.870198			0.435099	+	0.900383i	$0.356713\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	−900.000	−	4881.41i	−0.360592	−	1.95577i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			3360.10i			1.27292i	0.771308	+	0.636462i	$0.219603\pi$
$191$			3360.10i			1.27292i	−0.771308	+	0.636462i	$0.780397\pi$
$192$		0			0
$193$	−4590.00			−1.71189			−0.855947	−	0.517064i	$-0.827025\pi$
$193$	−4590.00			−1.71189			−0.855947	+	0.517064i	$0.827025\pi$
$194$		0			0
$195$		0			0
$196$	−5831.00			−2.12500
$197$	−2210.00			−0.799269			−0.399634	−	0.916675i	$-0.630863\pi$
$197$	−2210.00			−0.799269			−0.399634	+	0.916675i	$0.630863\pi$
$198$		−	3571.76i		−	1.28199i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−5625.00			−1.98874
$201$		0			0
$202$		0			0
$203$		−	3074.36i		−	1.06295i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−540.000	−	2928.85i	−0.181317	−	0.983425i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−5868.00			−1.91455			−0.957274	−	0.289181i	$-0.906617\pi$
$211$	−5868.00			−1.91455			−0.957274	+	0.289181i	$0.906617\pi$
$212$		−	8455.82i		−	2.73938i
$213$		0			0
$214$			7752.05i			2.47626i
$215$		0			0
$216$		0			0
$217$		0			0
$218$			11376.7i			3.53454i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$			1574.22i			0.469563i
$225$	−3375.00			−1.00000
$226$		−	11535.5i		−	3.39526i
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$	−7470.00			−2.11392
$233$	−4730.00			−1.32993			−0.664963	−	0.746877i	$-0.731553\pi$
$233$	−4730.00			−1.32993			−0.664963	+	0.746877i	$0.731553\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7376.00			1.99629			0.998146	−	0.0608655i	$-0.0193861\pi$
$239$	7376.00			1.99629			0.998146	+	0.0608655i	$0.0193861\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	3155.00			0.838062
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$			8500.80i			2.12500i
$253$	−2870.00	+	529.150i	−0.713183	+	0.131492i
$254$	10000.0			2.47030
$255$		0			0
$256$	−8279.00			−2.02124
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	196.000			0.0470226
$260$		0			0
$261$	−4482.00			−1.06295
$262$		0			0
$263$			4026.83i			0.944126i	0.881565	+	0.472063i	$0.156491\pi$
$263$			4026.83i			0.944126i	−0.881565	+	0.472063i	$0.843509\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		−	13763.2i		−	3.13702i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		−	3915.71i		−	0.863346i
$275$			3307.19i			0.725204i
$276$		0			0
$277$	−7310.00			−1.58561			−0.792807	−	0.609472i	$-0.791381\pi$
$277$	−7310.00			−1.58561			−0.792807	+	0.609472i	$0.791381\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	8360.57i		−	1.77491i	−0.460893	−	0.887456i	$-0.652471\pi$
$281$		−	8360.57i		−	1.77491i	0.460893	−	0.887456i	$-0.347529\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	11696.0			2.44377
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2295.00			0.469563
$289$	−4913.00			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		−	476.235i		−	0.0935156i
$297$		0			0
$298$			17726.5i			3.44587i
$299$		0			0
$300$		0			0
$301$	−9898.00			−1.89539
$302$	14760.0			2.81239
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	8330.00			1.54106
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$			4048.00i			0.720626i
$317$	6970.00			1.23493			0.617467	−	0.786597i	$-0.288159\pi$
$317$	6970.00			1.23493			0.617467	+	0.786597i	$0.288159\pi$
$318$		0			0
$319$			4391.95i			0.770852i
$320$		0			0
$321$		0			0
$322$	10045.0	−	1852.03i	1.73847	−	0.320526i
$323$		0			0
$324$	12393.0			2.12500
$325$		0			0
$326$	8900.00			1.51204
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−10908.0			−1.81135			−0.905677	−	0.423969i	$-0.860636\pi$
$331$	−10908.0			−1.81135			−0.905677	+	0.423969i	$0.860636\pi$
$332$		0			0
$333$		−	285.741i		−	0.0470226i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	11916.5i		−	1.92621i	−0.269135	−	0.963103i	$-0.586738\pi$
$337$		−	11916.5i		−	1.92621i	0.269135	−	0.963103i	$-0.413262\pi$
$338$	10985.0			1.76777
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	6352.45i		−	1.00000i
$344$			24049.9i			3.76943i
$345$		0			0
$346$		0			0
$347$	4100.00			0.634293			0.317146	−	0.948377i	$-0.397275\pi$
$347$	4100.00			0.634293			0.317146	+	0.948377i	$0.397275\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		−	11575.2i		−	1.76777i
$351$		0			0
$352$		−	2248.89i		−	0.340529i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	10420.0			1.53831
$359$			10927.0i			1.60641i	0.595700	+	0.803207i	$0.296875\pi$
$359$			10927.0i			1.60641i	−0.595700	+	0.803207i	$0.703125\pi$
$360$		0			0
$361$	−6859.00			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−1780.00	−	9654.35i	−0.252144	−	1.36758i
$369$		0			0
$370$		0			0
$371$	9212.00			1.28912
$372$		0			0
$373$		−	3524.14i		−	0.489204i	−0.969624	−	0.244602i	$-0.921343\pi$
$373$		−	3524.14i		−	0.489204i	0.969624	−	0.244602i	$-0.0786573\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	8704.52i		−	1.17974i	−0.807498	−	0.589870i	$-0.799179\pi$
$379$		−	8704.52i		−	1.17974i	0.807498	−	0.589870i	$-0.200821\pi$
$380$		0			0
$381$		0			0
$382$			16800.5i			2.25023i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	−22950.0			−3.02623
$387$			14429.9i			1.89539i
$388$		0			0
$389$		−	11165.1i		−	1.45525i	−0.685976	−	0.727624i	$-0.740625\pi$
$389$		−	11165.1i		−	1.45525i	0.685976	−	0.727624i	$-0.259375\pi$
$390$		0			0
$391$		0			0
$392$	−15435.0			−1.98874
$393$		0			0
$394$	−11050.0			−1.41292
$395$		0			0
$396$		−	12144.0i		−	1.54106i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−11125.0			−1.39062
$401$			15980.3i			1.99007i	0.0995016	+	0.995037i	$0.468275\pi$
$401$			15980.3i			1.99007i	−0.0995016	+	0.995037i	$0.531725\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		−	15371.8i		−	1.87904i
$407$	−280.000			−0.0341009
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−2700.00	−	14644.2i	−0.320526	−	1.73847i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$			8096.00i			0.937232i	0.883402	+	0.468616i	$0.155247\pi$
$421$			8096.00i			0.937232i	−0.883402	+	0.468616i	$0.844753\pi$
$422$	−29340.0			−3.38448
$423$		0			0
$424$		−	22383.1i		−	2.56372i
$425$		0			0
$426$		0			0
$427$		0			0
$428$			26357.0i			2.97666i
$429$		0			0
$430$		0			0
$431$		−	15689.3i		−	1.75343i	−0.481012	−	0.876714i	$-0.659731\pi$
$431$		−	15689.3i		−	1.75343i	0.481012	−	0.876714i	$-0.340269\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$			38680.9i			4.24880i
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−9261.00			−1.00000
$442$		0			0
$443$	18580.0			1.99269			0.996346	−	0.0854102i	$-0.0272201\pi$
$443$	18580.0			1.99269			0.996346	+	0.0854102i	$0.0272201\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		−	5315.31i		−	0.560547i
$449$	2686.00			0.282317			0.141158	−	0.989987i	$-0.454917\pi$
$449$	2686.00			0.282317			0.141158	+	0.989987i	$0.454917\pi$
$450$	−16875.0			−1.76777
$451$		0			0
$452$		−	39220.6i		−	4.08138i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			17821.8i			1.82422i	0.409947	+	0.912109i	$0.365547\pi$
$457$			17821.8i			1.82422i	−0.409947	+	0.912109i	$0.634453\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	8440.00			0.847171			0.423585	−	0.905856i	$-0.360771\pi$
$463$	8440.00			0.847171			0.423585	+	0.905856i	$0.360771\pi$
$464$	−14774.0			−1.47816
$465$		0			0
$466$	−23650.0			−2.35100
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	14994.0			1.47624
$470$		0			0
$471$		0			0
$472$		0			0
$473$	14140.0			1.37454
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	13429.8i		−	1.28912i
$478$	36880.0			3.52898
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	10727.0			1.00742
$485$		0			0
$486$		0			0
$487$	21240.0			1.97634			0.988169	−	0.153371i	$-0.0490130\pi$
$487$	21240.0			1.97634			0.988169	+	0.153371i	$0.0490130\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	20372.0			1.87246			0.936228	−	0.351394i	$-0.114292\pi$
$491$	20372.0			1.87246			0.936228	+	0.351394i	$0.114292\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			12741.9i			1.15001i
$498$		0			0
$499$	7236.00			0.649154			0.324577	−	0.945859i	$-0.394778\pi$
$499$	7236.00			0.649154			0.324577	+	0.945859i	$0.394778\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$			22502.1i			1.98874i
$505$		0			0
$506$	−14350.0	+	2645.75i	−1.26074	+	0.232447i
$507$		0			0
$508$	34000.0			2.96950
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	−24475.0			−2.11260
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	980.000			0.0831250
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	−22410.0			−1.87904
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$			20134.2i			1.66900i
$527$		0			0
$528$		0			0
$529$	−11367.0	+	4339.03i	−0.934248	+	0.356623i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		−	36432.0i		−	2.93586i
$537$		0			0
$538$		0			0
$539$			9074.93i			0.725204i
$540$		0			0
$541$	−15878.0			−1.26183			−0.630914	−	0.775853i	$-0.717320\pi$
$541$	−15878.0			−1.26183			−0.630914	+	0.775853i	$0.717320\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−12980.0			−1.01460			−0.507299	−	0.861770i	$-0.669356\pi$
$547$	−12980.0			−1.01460			−0.507299	+	0.861770i	$0.669356\pi$
$548$		−	13313.4i		−	1.03781i
$549$		0			0
$550$			16535.9i			1.28199i
$551$		0			0
$552$		0			0
$553$	−4410.00			−0.339118
$554$	−36550.0			−2.80300
$555$		0			0
$556$		0			0
$557$		−	16498.9i		−	1.25508i	−0.778583	−	0.627541i	$-0.784061\pi$
$557$		−	16498.9i		−	1.25508i	0.778583	−	0.627541i	$-0.215939\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		−	41802.9i		−	3.13763i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			13501.3i			1.00000i
$568$	30960.0			2.28706
$569$		−	3598.22i		−	0.265106i	−0.991176	−	0.132553i	$-0.957683\pi$
$569$		−	3598.22i		−	0.265106i	0.991176	−	0.132553i	$-0.0423175\pi$
$570$		0			0
$571$		−	26431.1i		−	1.93714i	−0.248747	−	0.968569i	$-0.580019\pi$
$571$		−	26431.1i		−	1.93714i	0.248747	−	0.968569i	$-0.419981\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2500.00	+	13559.5i	0.181317	+	0.983425i
$576$	−7749.00			−0.560547
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−24565.0			−1.76777
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−13160.0			−0.934874
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		−	941.887i		−	0.0653908i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$			60270.2i			4.14222i
$597$		0			0
$598$		0			0
$599$	24736.0			1.68729			0.843644	−	0.536903i	$-0.180406\pi$
$599$	24736.0			1.68729			0.843644	+	0.536903i	$0.180406\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−49490.0			−3.35060
$603$		−	21859.2i		−	1.47624i
$604$	50184.0			3.38073
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			26383.4i			1.73836i	0.494493	+	0.869182i	$0.335354\pi$
$613$			26383.4i			1.73836i	−0.494493	+	0.869182i	$0.664646\pi$
$614$		0			0
$615$		0			0
$616$	22050.0			1.44224
$617$			2497.59i			0.162965i	0.996675	+	0.0814823i	$0.0259654\pi$
$617$			2497.59i			0.162965i	−0.996675	+	0.0814823i	$0.974035\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15625.0			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	17858.8i		−	1.12670i	−0.826218	−	0.563351i	$-0.809512\pi$
$631$		−	17858.8i		−	1.12670i	0.826218	−	0.563351i	$-0.190488\pi$
$632$			10715.3i			0.674417i
$633$		0			0
$634$	34850.0			2.18308
$635$		0			0
$636$		0			0
$637$		0			0
$638$			21959.7i			1.36269i
$639$	18576.0			1.15001
$640$		0			0
$641$		−	31219.9i		−	1.92373i	−0.273526	−	0.961865i	$-0.588190\pi$
$641$		−	31219.9i		−	1.92373i	0.273526	−	0.961865i	$-0.411810\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	34153.0	−	6296.89i	2.08978	−	0.385298i
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	32805.0			1.98874
$649$		0			0
$650$		0			0
$651$		0			0
$652$	30260.0			1.81760
$653$	−27050.0			−1.62105			−0.810527	−	0.585701i	$-0.800819\pi$
$653$	−27050.0			−1.62105			−0.810527	+	0.585701i	$0.800819\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			33786.2i			1.99716i	0.0533186	+	0.998578i	$0.483020\pi$
$659$			33786.2i			1.99716i	−0.0533186	+	0.998578i	$0.516980\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−54540.0			−3.20205
$663$		0			0
$664$		0			0
$665$		0			0
$666$		−	1428.71i		−	0.0831250i
$667$	3320.00	+	18007.0i	0.192730	+	1.04533i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−33570.0			−1.92278			−0.961388	−	0.275196i	$-0.911257\pi$
$673$	−33570.0			−1.92278			−0.961388	+	0.275196i	$0.911257\pi$
$674$		−	59582.3i		−	3.40508i
$675$		0			0
$676$	37349.0			2.12500
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−34060.0			−1.90815			−0.954077	−	0.299560i	$-0.903160\pi$
$683$	−34060.0			−1.90815			−0.954077	+	0.299560i	$0.903160\pi$
$684$		0			0
$685$		0			0
$686$		−	31762.2i		−	1.76777i
$687$		0			0
$688$			47565.3i			2.63577i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$	13230.0			0.725204
$694$	20500.0			1.12128
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		−	39355.6i		−	2.12500i
$701$			36881.8i			1.98717i	0.113093	+	0.993584i	$0.463924\pi$
$701$			36881.8i			1.98717i	−0.113093	+	0.993584i	$0.536076\pi$
$702$		0			0
$703$		0			0
$704$			7593.31i			0.406511i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	35611.8i		−	1.88636i	−0.332281	−	0.943180i	$-0.607818\pi$
$709$		−	35611.8i		−	1.88636i	0.332281	−	0.943180i	$-0.392182\pi$
$710$		0			0
$711$			6429.18i			0.339118i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	35428.0			1.84917
$717$		0			0
$718$			54634.8i			2.83977i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−34295.0			−1.76777
$723$		0			0
$724$		0			0
$725$	20750.0			1.06295
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	19683.0			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$	−1700.00	−	9220.44i	−0.0851397	−	0.461780i
$737$	−21420.0			−1.07058
$738$		0			0
$739$	−25324.0			−1.26057			−0.630283	−	0.776365i	$-0.717061\pi$
$739$	−25324.0			−1.26057			−0.630283	+	0.776365i	$0.717061\pi$
$740$		0			0
$741$		0			0
$742$	46060.0			2.27886
$743$		−	31743.7i		−	1.56738i	−0.621151	−	0.783691i	$-0.713335\pi$
$743$		−	31743.7i		−	1.56738i	0.621151	−	0.783691i	$-0.286665\pi$
$744$		0			0
$745$		0			0
$746$		−	17620.7i		−	0.864799i
$747$		0			0
$748$		0			0
$749$	−28714.0			−1.40078
$750$		0			0
$751$			41088.5i			1.99646i	0.0594732	+	0.998230i	$0.481058\pi$
$751$			41088.5i			1.99646i	−0.0594732	+	0.998230i	$0.518942\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	22848.7i		−	1.09703i	−0.836141	−	0.548514i	$-0.815194\pi$
$757$		−	22848.7i		−	1.09703i	0.836141	−	0.548514i	$-0.184806\pi$
$758$		−	43522.6i		−	2.08550i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−42140.0			−1.99944
$764$			57121.8i			2.70496i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	−78030.0			−3.63777
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$			72149.6i			3.35060i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		−	55825.4i		−	2.57254i
$779$		0			0
$780$		0			0
$781$		−	18202.8i		−	0.833990i
$782$		0			0
$783$		0			0
$784$	−30527.0			−1.39062
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	−37570.0			−1.69845
$789$		0			0
$790$		0			0
$791$	42728.0			1.92065
$792$		−	32145.9i		−	1.44224i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−10625.0			−0.469563
$801$		0			0
$802$			79901.7i			3.51799i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	37354.0			1.62336			0.811679	−	0.584104i	$-0.198554\pi$
$809$	37354.0			1.62336			0.811679	+	0.584104i	$0.198554\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		−	52264.2i		−	2.25876i
$813$		0			0
$814$	−1400.00			−0.0602825
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	43538.0			1.85078			0.925388	−	0.379022i	$-0.123739\pi$
$821$	43538.0			1.85078			0.925388	+	0.379022i	$0.123739\pi$
$822$		0			0
$823$	−46240.0			−1.95848			−0.979238	−	0.202716i	$-0.935023\pi$
$823$	−46240.0			−1.95848			−0.979238	+	0.202716i	$0.935023\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	41077.9i		−	1.72723i	−0.504151	−	0.863615i	$-0.668195\pi$
$827$		−	41077.9i		−	1.72723i	0.504151	−	0.863615i	$-0.331805\pi$
$828$	−9180.00	−	49790.4i	−0.385298	−	2.08978i
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	3167.00			0.129854
$842$			40480.0i			1.65681i
$843$		0			0
$844$	−99756.0			−4.06842
$845$		0			0
$846$		0			0
$847$			11686.3i			0.474080i
$848$		−	44268.7i		−	1.79268i
$849$		0			0
$850$		0			0
$851$	−1148.00	+	211.660i	−0.0462432	+	0.00852599i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$			69768.5i			2.78579i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		−	78446.5i		−	3.09965i
$863$	−20200.0			−0.796774			−0.398387	−	0.917217i	$-0.630430\pi$
$863$	−20200.0			−0.796774			−0.398387	+	0.917217i	$0.630430\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6300.00			0.245930
$870$		0			0
$871$		0			0
$872$			102391.i			3.97636i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−6550.00			−0.252198			−0.126099	−	0.992018i	$-0.540246\pi$
$877$	−6550.00			−0.252198			−0.126099	+	0.992018i	$0.540246\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−46305.0			−1.76777
$883$	−30060.0			−1.14564			−0.572820	−	0.819681i	$-0.694150\pi$
$883$	−30060.0			−1.14564			−0.572820	+	0.819681i	$0.694150\pi$
$884$		0			0
$885$		0			0
$886$	92900.0			3.52261
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$			37040.5i			1.39741i
$890$		0			0
$891$		−	19287.5i		−	0.725204i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		−	39170.3i		−	1.46048i
$897$		0			0
$898$	13430.0			0.499070
$899$		0			0
$900$	−57375.0			−2.12500
$901$		0			0
$902$		0			0
$903$		0			0
$904$		−	103819.i		−	3.81967i
$905$		0			0
$906$		0			0
$907$			14250.0i			0.521680i	0.965382	+	0.260840i	$0.0839996\pi$
$907$			14250.0i			0.521680i	−0.965382	+	0.260840i	$0.916000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			38125.3i			1.38655i	0.720673	+	0.693275i	$0.243833\pi$
$911$			38125.3i			1.38655i	−0.720673	+	0.693275i	$0.756167\pi$
$912$		0			0
$913$		0			0
$914$			89108.9i			3.22479i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	51301.1i		−	1.84142i	−0.390244	−	0.920711i	$-0.627609\pi$
$919$		−	51301.1i		−	1.84142i	0.390244	−	0.920711i	$-0.372391\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$			1322.88i			0.0470226i
$926$	42200.0			1.49760
$927$		0			0
$928$	−14110.0			−0.499120
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	−80410.0			−2.82609
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$	74970.0			2.60966
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	70700.0			2.42987
$947$	−48820.0			−1.67522			−0.837612	−	0.546266i	$-0.816049\pi$
$947$	−48820.0			−1.67522			−0.837612	+	0.546266i	$0.816049\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			51031.3i			1.73459i	0.497794	+	0.867295i	$0.334143\pi$
$953$			51031.3i			1.73459i	−0.497794	+	0.867295i	$0.665857\pi$
$954$		−	67149.2i		−	2.27886i
$955$		0			0
$956$	125392.			4.24212
$957$		0			0
$958$		0			0
$959$	14504.0			0.488382
$960$		0			0
$961$	29791.0			1.00000
$962$		0			0
$963$			41861.1i			1.40078i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	52040.0			1.73060			0.865302	−	0.501251i	$-0.167127\pi$
$967$	52040.0			1.73060			0.865302	+	0.501251i	$0.167127\pi$
$968$	28395.0			0.942820
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$	106200.			3.49370
$975$		0			0
$976$		0			0
$977$		−	48216.2i		−	1.57889i	−0.613824	−	0.789443i	$-0.710369\pi$
$977$		−	48216.2i		−	1.57889i	0.613824	−	0.789443i	$-0.289631\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			61434.3i			1.99944i
$982$	101860.			3.31006
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	57974.0	−	10688.8i	1.86397	−	0.343666i
$990$		0			0
$991$	57528.0			1.84403			0.922017	−	0.387150i	$-0.126540\pi$
$991$	57528.0			1.84403			0.922017	+	0.387150i	$0.126540\pi$
$992$		0			0
$993$		0			0
$994$			63709.7i			2.03295i
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	36180.0			1.14755
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.4.c.a.160.2	yes	2
7.6	odd	2	CM	161.4.c.a.160.2	yes	2
23.22	odd	2	inner	161.4.c.a.160.1	✓	2
161.160	even	2	inner	161.4.c.a.160.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.4.c.a.160.1	✓	2	23.22	odd	2	inner
161.4.c.a.160.1	✓	2	161.160	even	2	inner
161.4.c.a.160.2	yes	2	1.1	even	1	trivial
161.4.c.a.160.2	yes	2	7.6	odd	2	CM

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 161 = 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	161.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.00000 q^{2} +17.0000 q^{4} +18.5203i q^{7} +45.0000 q^{8} +27.0000 q^{9} -26.4575i q^{11} +92.6013i q^{14} +89.0000 q^{16} +135.000 q^{18} -132.288i q^{22} +(-20.0000 - 108.476i) q^{23} -125.000 q^{25} +314.844i q^{28} -166.000 q^{29} +85.0000 q^{32} +459.000 q^{36} -10.5830i q^{37} +534.442i q^{43} -449.778i q^{44} +(-100.000 - 542.379i) q^{46} -343.000 q^{49} -625.000 q^{50} -497.401i q^{53} +833.412i q^{56} -830.000 q^{58} +500.047i q^{63} -287.000 q^{64} -809.600i q^{67} +688.000 q^{71} +1215.00 q^{72} -52.9150i q^{74} +490.000 q^{77} +238.118i q^{79} +729.000 q^{81} +2672.21i q^{86} -1190.59i q^{88} +(-340.000 - 1844.09i) q^{92} -1715.00 q^{98} -714.353i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{2} + 34 q^{4} + 90 q^{8} + 54 q^{9} + 178 q^{16} + 270 q^{18} - 40 q^{23} - 250 q^{25} - 332 q^{29} + 170 q^{32} + 918 q^{36} - 200 q^{46} - 686 q^{49} - 1250 q^{50} - 1660 q^{58} - 574 q^{64}+ \cdots - 3430 q^{98}+O(q^{100}) \) 2 * q + 10 * q^2 + 34 * q^4 + 90 * q^8 + 54 * q^9 + 178 * q^16 + 270 * q^18 - 40 * q^23 - 250 * q^25 - 332 * q^29 + 170 * q^32 + 918 * q^36 - 200 * q^46 - 686 * q^49 - 1250 * q^50 - 1660 * q^58 - 574 * q^64 + 1376 * q^71 + 2430 * q^72 + 980 * q^77 + 1458 * q^81 - 680 * q^92 - 3430 * q^98

Introduction

L-functions

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