LMFDB - Embedded newform 95.5.d.a.94.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [95,5,Mod(94,95)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("95.94"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 95 = 5 \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	95.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$9.82014649297$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-19}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 5 $ x^2 - x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			94.2
Root			$0.500000 + 2.17945i$ of defining polynomial
Character	$\chi$	$=$	95.94
Dual form			95.5.d.a.94.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-16.0000 q^{4} +(15.5000 + 19.6150i) q^{5} -65.3835i q^{7} -81.0000 q^{9} +233.000 q^{11} +256.000 q^{16} -457.684i q^{17} +361.000 q^{19} +(-248.000 - 313.841i) q^{20} -1046.14i q^{23} +(-144.500 + 608.066i) q^{25} +1046.14i q^{28} +(1282.50 - 1013.44i) q^{35} +1296.00 q^{36} -1111.52i q^{43} -3728.00 q^{44} +(-1255.50 - 1588.82i) q^{45} +4249.93i q^{47} -1874.00 q^{49} +(3611.50 + 4570.31i) q^{55} -3167.00 q^{61} +5296.06i q^{63} -4096.00 q^{64} +7322.95i q^{68} -3596.09i q^{73} -5776.00 q^{76} -15234.4i q^{77} +(3968.00 + 5021.45i) q^{80} +6561.00 q^{81} -12553.6i q^{83} +(8977.50 - 7094.11i) q^{85} +16738.2i q^{92} +(5595.50 + 7081.03i) q^{95} -18873.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 31 q^{5} - 162 q^{9} + 466 q^{11} + 512 q^{16} + 722 q^{19} - 496 q^{20} - 289 q^{25} + 2565 q^{35} + 2592 q^{36} - 7456 q^{44} - 2511 q^{45} - 3748 q^{49} + 7223 q^{55} - 6334 q^{61} - 8192 q^{64}+ \cdots - 37746 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 31 * q^5 - 162 * q^9 + 466 * q^11 + 512 * q^16 + 722 * q^19 - 496 * q^20 - 289 * q^25 + 2565 * q^35 + 2592 * q^36 - 7456 * q^44 - 2511 * q^45 - 3748 * q^49 + 7223 * q^55 - 6334 * q^61 - 8192 * q^64 - 11552 * q^76 + 7936 * q^80 + 13122 * q^81 + 17955 * q^85 + 11191 * q^95 - 37746 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$.

$n$	$21$	$77$
$\chi(n)$	$-1$	$-1$

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 $

$2$		0			0			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	−16.0000			−1.00000
$5$	15.5000	+	19.6150i	0.620000	+	0.784602i
$5$	15.5000	+	19.6150i	0.620000	+	0.784602i
$6$		0			0
$7$		−	65.3835i		−	1.33436i	−0.744898	−	0.667178i	$-0.767502\pi$
$7$		−	65.3835i		−	1.33436i	0.744898	−	0.667178i	$-0.232498\pi$
$8$		0			0
$9$	−81.0000			−1.00000
$10$		0			0
$11$	233.000			1.92562			0.962810	−	0.270180i	$-0.0870831\pi$
$11$	233.000			1.92562			0.962810	+	0.270180i	$0.0870831\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	256.000			1.00000
$17$		−	457.684i		−	1.58368i	−0.610727	−	0.791842i	$-0.709123\pi$
$17$		−	457.684i		−	1.58368i	0.610727	−	0.791842i	$-0.290877\pi$
$18$		0			0
$19$	361.000			1.00000
$19$	361.000			1.00000
$20$	−248.000	−	313.841i	−0.620000	−	0.784602i
$21$		0			0
$22$		0			0
$23$		−	1046.14i		−	1.97757i	−0.149338	−	0.988786i	$-0.547714\pi$
$23$		−	1046.14i		−	1.97757i	0.149338	−	0.988786i	$-0.452286\pi$
$24$		0			0
$25$	−144.500	+	608.066i	−0.231200	+	0.972906i
$26$		0			0
$27$		0			0
$28$			1046.14i			1.33436i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1282.50	−	1013.44i	1.04694	−	0.827301i
$36$	1296.00			1.00000
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		−	1111.52i		−	0.601146i	−0.953759	−	0.300573i	$-0.902822\pi$
$43$		−	1111.52i		−	0.601146i	0.953759	−	0.300573i	$-0.0971779\pi$
$44$	−3728.00			−1.92562
$45$	−1255.50	−	1588.82i	−0.620000	−	0.784602i
$46$		0			0
$47$			4249.93i			1.92391i	0.273201	+	0.961957i	$0.411918\pi$
$47$			4249.93i			1.92391i	−0.273201	+	0.961957i	$0.588082\pi$
$48$		0			0
$49$	−1874.00			−0.780508
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	3611.50	+	4570.31i	1.19388	+	1.51084i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	−3167.00			−0.851115			−0.425558	−	0.904931i	$-0.639922\pi$
$61$	−3167.00			−0.851115			−0.425558	+	0.904931i	$0.639922\pi$
$62$		0			0
$63$			5296.06i			1.33436i
$64$	−4096.00			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$			7322.95i			1.58368i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	3596.09i		−	0.674815i	−0.941359	−	0.337408i	$-0.890450\pi$
$73$		−	3596.09i		−	0.674815i	0.941359	−	0.337408i	$-0.109550\pi$
$74$		0			0
$75$		0			0
$76$	−5776.00			−1.00000
$77$		−	15234.4i		−	2.56946i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	3968.00	+	5021.45i	0.620000	+	0.784602i
$81$	6561.00			1.00000
$82$		0			0
$83$		−	12553.6i		−	1.82227i	−0.412106	−	0.911136i	$-0.635207\pi$
$83$		−	12553.6i		−	1.82227i	0.412106	−	0.911136i	$-0.364793\pi$
$84$		0			0
$85$	8977.50	−	7094.11i	1.24256	−	0.981883i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$			16738.2i			1.97757i
$93$		0			0
$94$		0			0
$95$	5595.50	+	7081.03i	0.620000	+	0.784602i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$	−18873.0			−1.92562
$100$	2312.00	−	9729.06i	0.231200	−	0.972906i
$101$	9998.00			0.980100			0.490050	−	0.871694i	$-0.336979\pi$
$101$	9998.00			0.980100			0.490050	+	0.871694i	$0.336979\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		−	16738.2i		−	1.33436i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	20520.0	−	16215.1i	1.55161	−	1.22609i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−29925.0			−2.11320
$120$		0			0
$121$	39648.0			2.70801
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−14167.0	+	6590.66i	−0.906688	+	0.421802i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	11047.0			0.643727			0.321864	−	0.946786i	$-0.395691\pi$
$131$	11047.0			0.643727			0.321864	+	0.946786i	$0.395691\pi$
$132$		0			0
$133$		−	23603.4i		−	1.33436i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			25564.9i			1.36208i	0.732245	+	0.681042i	$0.238473\pi$
$137$			25564.9i			1.36208i	−0.732245	+	0.681042i	$0.761527\pi$
$138$		0			0
$139$	−167.000			−0.00864344			−0.00432172	−	0.999991i	$-0.501376\pi$
$139$	−167.000			−0.00864344			−0.00432172	+	0.999991i	$0.501376\pi$
$140$	−20520.0	+	16215.1i	−1.04694	+	0.827301i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−20736.0			−1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13073.0			0.588847			0.294424	−	0.955675i	$-0.404872\pi$
$149$	13073.0			0.588847			0.294424	+	0.955675i	$0.404872\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$			37072.4i			1.58368i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	3138.41i		−	0.127324i	−0.997972	−	0.0636620i	$-0.979722\pi$
$157$		−	3138.41i		−	0.127324i	0.997972	−	0.0636620i	$-0.0202779\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−68400.0			−2.63879
$162$		0			0
$163$			52306.8i			1.96871i	0.176183	+	0.984357i	$0.443625\pi$
$163$			52306.8i			1.96871i	−0.176183	+	0.984357i	$0.556375\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−28561.0			−1.00000
$170$		0			0
$171$	−29241.0			−1.00000
$172$			17784.3i			0.601146i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	39757.5	+	9447.91i	1.29820	+	0.308503i
$176$	59648.0			1.92562
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	20088.0	+	25421.1i	0.620000	+	0.784602i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	106640.i		−	3.04957i
$188$		−	67998.8i		−	1.92391i
$189$		0			0
$190$		0			0
$191$	−64313.0			−1.76292			−0.881459	−	0.472261i	$-0.843438\pi$
$191$	−64313.0			−1.76292			−0.881459	+	0.472261i	$0.843438\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	29984.0			0.780508
$197$			34522.5i			0.889548i	0.895643	+	0.444774i	$0.146716\pi$
$197$			34522.5i			0.889548i	−0.895643	+	0.444774i	$0.853284\pi$
$198$		0			0
$199$	−27673.0			−0.698795			−0.349398	−	0.936975i	$-0.613614\pi$
$199$	−27673.0			−0.698795			−0.349398	+	0.936975i	$0.613614\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$			84737.0i			1.97757i
$208$		0			0
$209$	84113.0			1.92562
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	21802.5	−	17228.5i	0.471660	−	0.372711i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−57784.0	−	73124.9i	−1.19388	−	1.51084i
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	11704.5	−	49253.4i	0.231200	−	0.972906i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	104593.			1.99449			0.997245	−	0.0741847i	$-0.0236354\pi$
$229$	104593.			1.99449			0.997245	+	0.0741847i	$0.0236354\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			14188.2i			0.261346i	0.991426	+	0.130673i	$0.0417138\pi$
$233$			14188.2i			0.261346i	−0.991426	+	0.130673i	$0.958286\pi$
$234$		0			0
$235$	−83362.5	+	65873.9i	−1.50951	+	1.19283i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	90967.0			1.59253			0.796266	−	0.604947i	$-0.206806\pi$
$239$	90967.0			1.59253			0.796266	+	0.604947i	$0.206806\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$	50672.0			0.851115
$245$	−29047.0	−	36758.6i	−0.483915	−	0.612388i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−125273.			−1.98843			−0.994214	−	0.107414i	$-0.965743\pi$
$251$	−125273.			−1.98843			−0.994214	+	0.107414i	$0.965743\pi$
$252$		−	84737.0i		−	1.33436i
$253$		−	243750.i		−	3.80805i
$254$		0			0
$255$		0			0
$256$	65536.0			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$			135932.i			1.96522i	0.185683	+	0.982610i	$0.440550\pi$
$263$			135932.i			1.96522i	−0.185683	+	0.982610i	$0.559450\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	126718.			1.72544			0.862720	−	0.505682i	$-0.168759\pi$
$271$	126718.			1.72544			0.862720	+	0.505682i	$0.168759\pi$
$272$		−	117167.i		−	1.58368i
$273$		0			0
$274$		0			0
$275$	−33668.5	+	141679.i	−0.445203	+	1.87345i
$276$		0			0
$277$			76956.4i			1.00296i	0.865168	+	0.501482i	$0.167211\pi$
$277$			76956.4i			1.00296i	−0.865168	+	0.501482i	$0.832789\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$			160124.i			1.99933i	0.0259274	+	0.999664i	$0.491746\pi$
$283$			160124.i			1.99933i	−0.0259274	+	0.999664i	$0.508254\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−125954.			−1.50805
$290$		0			0
$291$		0			0
$292$			57537.5i			0.674815i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−72675.0			−0.802143
$302$		0			0
$303$		0			0
$304$	92416.0			1.00000
$305$	−49088.5	−	62120.8i	−0.527691	−	0.667787i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$			243750.i			2.56946i
$309$		0			0
$310$		0			0
$311$	−170167.			−1.75936			−0.879680	−	0.475567i	$-0.842243\pi$
$311$	−170167.			−1.75936			−0.879680	+	0.475567i	$0.842243\pi$
$312$		0			0
$313$		−	123444.i		−	1.26003i	−0.776582	−	0.630016i	$-0.783048\pi$
$313$		−	123444.i		−	1.26003i	0.776582	−	0.630016i	$-0.216952\pi$
$314$		0			0
$315$	−103882.	+	82089.0i	−1.04694	+	0.827301i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$	−63488.0	−	80343.2i	−0.620000	−	0.784602i
$321$		0			0
$322$		0			0
$323$		−	165224.i		−	1.58368i
$324$	−104976.			−1.00000
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	277875.			2.56719
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			200858.i			1.82227i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$		0			0
$340$	−143640.	+	113506.i	−1.24256	+	0.981883i
$341$		0			0
$342$		0			0
$343$		−	34457.1i		−	0.292880i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	108864.i		−	0.904114i	−0.891989	−	0.452057i	$-0.850690\pi$
$347$		−	108864.i		−	0.904114i	0.891989	−	0.452057i	$-0.149310\pi$
$348$		0			0
$349$	34127.0			0.280187			0.140093	−	0.990138i	$-0.455260\pi$
$349$	34127.0			0.280187			0.140093	+	0.990138i	$0.455260\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			248980.i			1.99809i	0.0436646	+	0.999046i	$0.486097\pi$
$353$			248980.i			1.99809i	−0.0436646	+	0.999046i	$0.513903\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	198713.			1.54183			0.770917	−	0.636936i	$-0.219799\pi$
$359$	198713.			1.54183			0.770917	+	0.636936i	$0.219799\pi$
$360$		0			0
$361$	130321.			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	70537.5	−	55739.4i	0.529461	−	0.418386i
$366$		0			0
$367$			36614.8i			0.271847i	0.990719	+	0.135923i	$0.0434001\pi$
$367$			36614.8i			0.271847i	−0.990719	+	0.135923i	$0.956600\pi$
$368$		−	267811.i		−	1.97757i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	−89528.0	−	113297.i	−0.620000	−	0.784602i
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	298822.	−	236132.i	2.01601	−	1.59307i
$386$		0			0
$387$			90033.1i			0.601146i
$388$		0			0
$389$	−279233.			−1.84530			−0.922651	−	0.385636i	$-0.873982\pi$
$389$	−279233.			−1.84530			−0.922651	+	0.385636i	$0.873982\pi$
$390$		0			0
$391$	−478800.			−3.13185
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	301968.			1.92562
$397$		−	204585.i		−	1.29805i	−0.760766	−	0.649027i	$-0.775176\pi$
$397$		−	204585.i		−	1.29805i	0.760766	−	0.649027i	$-0.224824\pi$
$398$		0			0
$399$		0			0
$400$	−36992.0	+	155665.i	−0.231200	+	0.972906i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	−159968.			−0.980100
$405$	101696.	+	128694.i	0.620000	+	0.784602i
$406$		0			0
$407$		0			0
$408$		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
$414$		0			0
$415$	246240.	−	194581.i	1.42976	−	1.12981i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	229522.			1.30736			0.653682	−	0.756770i	$-0.273224\pi$
$419$	229522.			1.30736			0.653682	+	0.756770i	$0.273224\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		−	344244.i		−	1.92391i
$424$		0			0
$425$	278302.	+	66135.4i	1.54078	+	0.366148i
$426$		0			0
$427$			207069.i			1.13569i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	377655.i		−	1.97757i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	151794.			0.780508
$442$		0			0
$443$		−	39818.5i		−	0.202898i	−0.994841	−	0.101449i	$-0.967652\pi$
$443$		−	39818.5i		−	0.202898i	0.994841	−	0.101449i	$-0.0323479\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$			267811.i			1.33436i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	416820.i		−	1.99579i	−0.0648148	−	0.997897i	$-0.520646\pi$
$457$		−	416820.i		−	1.99579i	0.0648148	−	0.997897i	$-0.479354\pi$
$458$		0			0
$459$		0			0
$460$	−328320.	+	259442.i	−1.55161	+	1.22609i
$461$	−225233.			−1.05982			−0.529908	−	0.848055i	$-0.677773\pi$
$461$	−225233.			−1.05982			−0.529908	+	0.848055i	$0.677773\pi$
$462$		0			0
$463$			404789.i			1.88828i	0.329542	+	0.944141i	$0.393106\pi$
$463$			404789.i			1.88828i	−0.329542	+	0.944141i	$0.606894\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			171501.i			0.786380i	0.919457	+	0.393190i	$0.128629\pi$
$467$			171501.i			0.786380i	−0.919457	+	0.393190i	$0.871371\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	258984.i		−	1.15758i
$474$		0			0
$475$	−52164.5	+	219512.i	−0.231200	+	0.972906i
$476$	478800.			2.11320
$477$		0			0
$478$		0			0
$479$	−428482.			−1.86750			−0.933752	−	0.357921i	$-0.883486\pi$
$479$	−428482.			−1.86750			−0.933752	+	0.357921i	$0.883486\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−634368.			−2.70801
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	360562.			1.49561			0.747803	−	0.663921i	$-0.231109\pi$
$491$	360562.			1.49561			0.747803	+	0.663921i	$0.231109\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−292532.	−	370195.i	−1.19388	−	1.51084i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	224473.			0.901494			0.450747	−	0.892652i	$-0.351158\pi$
$499$	224473.			0.901494			0.450747	+	0.892652i	$0.351158\pi$
$500$	226672.	−	105450.i	0.906688	−	0.421802i
$501$		0			0
$502$		0			0
$503$			356732.i			1.40996i	0.709228	+	0.704979i	$0.249044\pi$
$503$			356732.i			1.40996i	−0.709228	+	0.704979i	$0.750956\pi$
$504$		0			0
$505$	154969.	+	196111.i	0.607662	+	0.768988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	−235125.			−0.900445
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			990233.i			3.70473i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	−176752.			−0.643727
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−814559.			−2.91079
$530$		0			0
$531$		0			0
$532$			377655.i			1.33436i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−436642.			−1.50296
$540$		0			0
$541$	376513.			1.28643			0.643214	−	0.765687i	$-0.277601\pi$
$541$	376513.			1.28643			0.643214	+	0.765687i	$0.277601\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		−	409039.i		−	1.36208i
$549$	256527.			0.851115
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	2672.00			0.00864344
$557$		−	224331.i		−	0.723067i	−0.932359	−	0.361533i	$-0.882253\pi$
$557$		−	224331.i		−	0.723067i	0.932359	−	0.361533i	$-0.117747\pi$
$558$		0			0
$559$		0			0
$560$	328320.	−	259442.i	1.04694	−	0.827301i
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	428981.i		−	1.33436i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	442318.			1.35663			0.678317	−	0.734770i	$-0.262710\pi$
$571$	442318.			1.35663			0.678317	+	0.734770i	$0.262710\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	636120.	+	151167.i	1.92399	+	0.457215i
$576$	331776.			1.00000
$577$		−	164701.i		−	0.494703i	−0.968926	−	0.247352i	$-0.920440\pi$
$577$		−	164701.i		−	0.494703i	0.968926	−	0.247352i	$-0.0795602\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−820800.			−2.43156
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			377590.i			1.09583i	0.836533	+	0.547916i	$0.184579\pi$
$587$			377590.i			1.09583i	−0.836533	+	0.547916i	$0.815421\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			35568.6i			0.101148i	0.998720	+	0.0505740i	$0.0161051\pi$
$593$			35568.6i			0.101148i	−0.998720	+	0.0505740i	$0.983895\pi$
$594$		0			0
$595$	−463838.	−	586980.i	−1.31018	−	1.65802i
$596$	−209168.			−0.588847
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	614544.	+	777697.i	1.67897	+	2.12471i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		−	593159.i		−	1.58368i
$613$			351044.i			0.934201i	0.884204	+	0.467101i	$0.154701\pi$
$613$			351044.i			0.934201i	−0.884204	+	0.467101i	$0.845299\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	663839.i		−	1.74378i	−0.489700	−	0.871891i	$-0.662894\pi$
$617$		−	663839.i		−	1.74378i	0.489700	−	0.871891i	$-0.337106\pi$
$618$		0			0
$619$	328078.			0.856241			0.428120	−	0.903722i	$-0.359176\pi$
$619$	328078.			0.856241			0.428120	+	0.903722i	$0.359176\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−348864.	−	175731.i	−0.893093	−	0.449872i
$626$		0			0
$627$		0			0
$628$			50214.5i			0.127324i
$629$		0			0
$630$		0			0
$631$	−279047.			−0.700840			−0.350420	−	0.936593i	$-0.613961\pi$
$631$	−279047.			−0.700840			−0.350420	+	0.936593i	$0.613961\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		−	714445.i		−	1.72801i	−0.503480	−	0.864007i	$-0.667947\pi$
$643$		−	714445.i		−	1.72801i	0.503480	−	0.864007i	$-0.332053\pi$
$644$	1.09440e6			2.63879
$645$		0			0
$646$		0			0
$647$			819190.i			1.95693i	0.206406	+	0.978466i	$0.433823\pi$
$647$			819190.i			1.95693i	−0.206406	+	0.978466i	$0.566177\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		−	836909.i		−	1.96871i
$653$		−	469126.i		−	1.10018i	−0.835105	−	0.550090i	$-0.814593\pi$
$653$		−	469126.i		−	1.10018i	0.835105	−	0.550090i	$-0.185407\pi$
$654$		0			0
$655$	171228.	+	216687.i	0.399111	+	0.505069i
$656$		0			0
$657$			291283.i			0.674815i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	462982.	−	365853.i	1.04694	−	0.827301i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−737911.			−1.63892
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	456976.			1.00000
$677$		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
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	467856.			1.00000
$685$	−501458.	+	396257.i	−1.06869	+	0.844492i
$686$		0			0
$687$		0			0
$688$		−	284549.i		−	0.601146i
$689$		0			0
$690$		0			0
$691$	930313.			1.94838			0.974189	−	0.225736i	$-0.0724787\pi$
$691$	930313.			1.94838			0.974189	+	0.225736i	$0.0724787\pi$
$692$		0			0
$693$			1.23398e6i			2.56946i
$694$		0			0
$695$	−2588.50	−	3275.71i	−0.00535894	−	0.00678166i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−636120.	−	151167.i	−1.29820	−	0.308503i
$701$	−222802.			−0.453402			−0.226701	−	0.973964i	$-0.572794\pi$
$701$	−222802.			−0.453402			−0.226701	+	0.973964i	$0.572794\pi$
$702$		0			0
$703$		0			0
$704$	−954368.			−1.92562
$705$		0			0
$706$		0			0
$707$		−	653704.i		−	1.30780i
$708$		0			0
$709$	731762.			1.45572			0.727859	−	0.685727i	$-0.240515\pi$
$709$	731762.			1.45572			0.727859	+	0.685727i	$0.240515\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	106553.			0.206114			0.103057	−	0.994675i	$-0.467138\pi$
$719$	106553.			0.206114			0.103057	+	0.994675i	$0.467138\pi$
$720$	−321408.	−	406738.i	−0.620000	−	0.784602i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	123379.i		−	0.233438i	−0.993165	−	0.116719i	$-0.962762\pi$
$727$		−	123379.i		−	0.233438i	0.993165	−	0.116719i	$-0.0372377\pi$
$728$		0			0
$729$	−531441.			−1.00000
$730$		0			0
$731$	−508725.			−0.952025
$732$		0			0
$733$		−	930015.i		−	1.73094i	−0.500961	−	0.865470i	$-0.667020\pi$
$733$		−	930015.i		−	1.73094i	0.500961	−	0.865470i	$-0.332980\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−793033.			−1.45212			−0.726060	−	0.687632i	$-0.758650\pi$
$739$	−793033.			−1.45212			−0.726060	+	0.687632i	$0.758650\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	202632.	+	256427.i	0.365085	+	0.462011i
$746$		0			0
$747$			1.01684e6i			1.82227i
$748$			1.70625e6i			3.04957i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			1.08798e6i			1.92391i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			1.01626e6i			1.77342i	0.462328	+	0.886709i	$0.347014\pi$
$757$			1.01626e6i			1.77342i	−0.462328	+	0.886709i	$0.652986\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.10077e6			−1.90075			−0.950377	−	0.311099i	$-0.899303\pi$
$761$	−1.10077e6			−1.90075			−0.950377	+	0.311099i	$0.899303\pi$
$762$		0			0
$763$		0			0
$764$	1.02901e6			1.76292
$765$	−727178.	+	574623.i	−1.24256	+	0.981883i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	52753.0			0.0892061			0.0446030	−	0.999005i	$-0.485798\pi$
$769$	52753.0			0.0892061			0.0446030	+	0.999005i	$0.485798\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−479744.			−0.780508
$785$	61560.0	−	48645.3i	0.0998986	−	0.0789408i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		−	552360.i		−	0.889548i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	442768.			0.698795
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	1.94512e6			3.04687
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	837889.i		−	1.29944i
$804$		0			0
$805$	−1.06020e6	−	1.34167e6i	−1.63605	−	2.07040i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.22869e6			1.87735			0.938673	−	0.344809i	$-0.112056\pi$
$809$	1.22869e6			1.87735			0.938673	+	0.344809i	$0.112056\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.02600e6	+	810755.i	−1.54466	+	1.22060i
$816$		0			0
$817$		−	401258.i		−	0.601146i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13807.0			0.0204839			0.0102420	−	0.999948i	$-0.496740\pi$
$821$	13807.0			0.0204839			0.0102420	+	0.999948i	$0.496740\pi$
$822$		0			0
$823$			798136.i			1.17836i	0.808002	+	0.589179i	$0.200549\pi$
$823$			798136.i			1.17836i	−0.808002	+	0.589179i	$0.799451\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		−	1.35579e6i		−	1.97757i
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			857701.i			1.23608i
$834$		0			0
$835$		0			0
$836$	−1.34581e6			−1.92562
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	707281.			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−442696.	−	560225.i	−0.620000	−	0.784602i
$846$		0			0
$847$		−	2.59232e6i		−	3.61345i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			1.40078e6i			1.92518i	0.270968	+	0.962588i	$0.412656\pi$
$853$			1.40078e6i			1.92518i	−0.270968	+	0.962588i	$0.587344\pi$
$854$		0			0
$855$	−453236.	−	573564.i	−0.620000	−	0.784602i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	753287.			1.02088			0.510439	−	0.859914i	$-0.329483\pi$
$859$	753287.			1.02088			0.510439	+	0.859914i	$0.329483\pi$
$860$	−348840.	+	275657.i	−0.471660	+	0.372711i
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	430920.	+	926288.i	0.562834	+	1.20985i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$	924544.	+	1.17000e6i	1.19388	+	1.51084i
$881$	1.26395e6			1.62847			0.814234	−	0.580537i	$-0.197157\pi$
$881$	1.26395e6			1.62847			0.814234	+	0.580537i	$0.197157\pi$
$882$		0			0
$883$		−	1.29937e6i		−	1.66652i	−0.552883	−	0.833259i	$-0.686472\pi$
$883$		−	1.29937e6i		−	1.66652i	0.552883	−	0.833259i	$-0.313528\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.52871e6			1.92562
$892$		0			0
$893$			1.53422e6i			1.92391i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−187272.	+	788054.i	−0.231200	+	0.972906i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$	−809838.			−0.980100
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		−	2.92500e6i		−	3.50900i
$914$		0			0
$915$		0			0
$916$	−1.67349e6			−1.99449
$917$		−	722291.i		−	0.858962i
$918$		0			0
$919$	1.41552e6			1.67604			0.838022	−	0.545636i	$-0.183712\pi$
$919$	1.41552e6			1.67604			0.838022	+	0.545636i	$0.183712\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.31392e6			−1.52243			−0.761214	−	0.648501i	$-0.775396\pi$
$929$	−1.31392e6			−1.52243			−0.761214	+	0.648501i	$0.775396\pi$
$930$		0			0
$931$	−676514.			−0.780508
$932$		−	227011.i		−	0.261346i
$933$		0			0
$934$		0			0
$935$	2.09176e6	−	1.65293e6i	2.39270	−	1.89073i
$936$		0			0
$937$			617678.i			0.703530i	0.936088	+	0.351765i	$0.114418\pi$
$937$			617678.i			0.703530i	−0.936088	+	0.351765i	$0.885582\pi$
$938$		0			0
$939$		0			0
$940$	1.33380e6	−	1.05398e6i	1.50951	−	1.19283i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			893400.i			0.996199i	0.867120	+	0.498099i	$0.165969\pi$
$947$			893400.i			0.996199i	−0.867120	+	0.498099i	$0.834031\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−996851.	−	1.26150e6i	−1.09301	−	1.38319i
$956$	−1.45547e6			−1.59253
$957$		0			0
$958$		0			0
$959$	1.67152e6			1.81751
$960$		0			0
$961$	923521.			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1.31081e6i		−	1.40180i	−0.713259	−	0.700900i	$-0.752782\pi$
$967$		−	1.31081e6i		−	1.40180i	0.713259	−	0.700900i	$-0.247218\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$			10919.0i			0.0115334i
$974$		0			0
$975$		0			0
$976$	−810752.			−0.851115
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$	464752.	+	588138.i	0.483915	+	0.612388i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	−677160.	+	535098.i	−0.697941	+	0.551520i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.16280e6			−1.18881
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−428932.	−	542807.i	−0.433253	−	0.548276i
$996$		0			0
$997$			542356.i			0.545625i	0.962067	+	0.272812i	$0.0879538\pi$
$997$			542356.i			0.545625i	−0.962067	+	0.272812i	$0.912046\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.5.d.a.94.2	yes	2
5.4	even	2	inner	95.5.d.a.94.1	✓	2
19.18	odd	2	CM	95.5.d.a.94.2	yes	2
95.94	odd	2	inner	95.5.d.a.94.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.5.d.a.94.1	✓	2	5.4	even	2	inner
95.5.d.a.94.1	✓	2	95.94	odd	2	inner
95.5.d.a.94.2	yes	2	1.1	even	1	trivial
95.5.d.a.94.2	yes	2	19.18	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:	\( N \)	\(=\)	\( 95 = 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	95.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-16.0000 q^{4} +(15.5000 + 19.6150i) q^{5} -65.3835i q^{7} -81.0000 q^{9} +233.000 q^{11} +256.000 q^{16} -457.684i q^{17} +361.000 q^{19} +(-248.000 - 313.841i) q^{20} -1046.14i q^{23} +(-144.500 + 608.066i) q^{25} +1046.14i q^{28} +(1282.50 - 1013.44i) q^{35} +1296.00 q^{36} -1111.52i q^{43} -3728.00 q^{44} +(-1255.50 - 1588.82i) q^{45} +4249.93i q^{47} -1874.00 q^{49} +(3611.50 + 4570.31i) q^{55} -3167.00 q^{61} +5296.06i q^{63} -4096.00 q^{64} +7322.95i q^{68} -3596.09i q^{73} -5776.00 q^{76} -15234.4i q^{77} +(3968.00 + 5021.45i) q^{80} +6561.00 q^{81} -12553.6i q^{83} +(8977.50 - 7094.11i) q^{85} +16738.2i q^{92} +(5595.50 + 7081.03i) q^{95} -18873.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 31 q^{5} - 162 q^{9} + 466 q^{11} + 512 q^{16} + 722 q^{19} - 496 q^{20} - 289 q^{25} + 2565 q^{35} + 2592 q^{36} - 7456 q^{44} - 2511 q^{45} - 3748 q^{49} + 7223 q^{55} - 6334 q^{61} - 8192 q^{64}+ \cdots - 37746 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 31 * q^5 - 162 * q^9 + 466 * q^11 + 512 * q^16 + 722 * q^19 - 496 * q^20 - 289 * q^25 + 2565 * q^35 + 2592 * q^36 - 7456 * q^44 - 2511 * q^45 - 3748 * q^49 + 7223 * q^55 - 6334 * q^61 - 8192 * q^64 - 11552 * q^76 + 7936 * q^80 + 13122 * q^81 + 17955 * q^85 + 11191 * q^95 - 37746 * q^99

Introduction

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