LMFDB - Embedded newform 40.6.c.a.9.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [40,6,Mod(9,40)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(40, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("40.9");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 40 = 2^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	40.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:	$6.41535279252$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 41x^{6} + 460x^{4} + 969x^{2} + 9 $ x^8 + 41x^6 + 460x^4 + 969*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{31}\cdot 5^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			9.7
Root			$0.0965878i$ of defining polynomial
Character	$\chi$	$=$	40.9
Dual form			40.6.c.a.9.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+24.1383i q^{3} +(46.7401 - 30.6653i) q^{5} +179.876i q^{7} -339.657 q^{9} +O(q^{10})$	$q+24.1383i q^{3} +(46.7401 - 30.6653i) q^{5} +179.876i q^{7} -339.657 q^{9} -653.681 q^{11} +284.851i q^{13} +(740.209 + 1128.23i) q^{15} +383.672i q^{17} +2563.29 q^{19} -4341.90 q^{21} -948.124i q^{23} +(1244.27 - 2866.60i) q^{25} -2333.13i q^{27} +1524.26 q^{29} +3103.88 q^{31} -15778.7i q^{33} +(5515.97 + 8407.43i) q^{35} +9991.75i q^{37} -6875.83 q^{39} +15120.7 q^{41} +1754.92i q^{43} +(-15875.6 + 10415.7i) q^{45} -14760.7i q^{47} -15548.5 q^{49} -9261.18 q^{51} -8704.12i q^{53} +(-30553.1 + 20045.4i) q^{55} +61873.3i q^{57} +12632.1 q^{59} +43098.3 q^{61} -61096.2i q^{63} +(8735.07 + 13314.0i) q^{65} +26125.5i q^{67} +22886.1 q^{69} -46285.8 q^{71} -51303.2i q^{73} +(69194.9 + 30034.6i) q^{75} -117582. i q^{77} -39314.0 q^{79} -26218.8 q^{81} -69544.8i q^{83} +(11765.4 + 17932.8i) q^{85} +36793.1i q^{87} +13092.2 q^{89} -51238.0 q^{91} +74922.5i q^{93} +(119808. - 78604.0i) q^{95} +26258.1i q^{97} +222027. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5} - 1000 q^{9}+O(q^{10}) $ 8 * q + 8 * q^5 - 1000 * q^9	$ 8 q + 8 q^{5} - 1000 q^{9} - 736 q^{11} - 992 q^{15} + 1376 q^{19} + 1984 q^{21} - 2136 q^{25} + 5872 q^{29} + 4224 q^{31} + 19232 q^{35} - 3008 q^{39} + 23600 q^{41} - 28328 q^{45} - 45000 q^{49} - 124800 q^{51} + 15008 q^{55} + 91680 q^{59} + 123856 q^{61} - 72064 q^{65} - 76736 q^{69} - 125632 q^{71} + 222784 q^{75} + 43264 q^{79} + 409672 q^{81} - 293760 q^{85} - 41904 q^{89} - 487616 q^{91} + 442592 q^{95} + 266848 q^{99}+O(q^{100}) $ 8 * q + 8 * q^5 - 1000 * q^9 - 736 * q^11 - 992 * q^15 + 1376 * q^19 + 1984 * q^21 - 2136 * q^25 + 5872 * q^29 + 4224 * q^31 + 19232 * q^35 - 3008 * q^39 + 23600 * q^41 - 28328 * q^45 - 45000 * q^49 - 124800 * q^51 + 15008 * q^55 + 91680 * q^59 + 123856 * q^61 - 72064 * q^65 - 76736 * q^69 - 125632 * q^71 + 222784 * q^75 + 43264 * q^79 + 409672 * q^81 - 293760 * q^85 - 41904 * q^89 - 487616 * q^91 + 442592 * q^95 + 266848 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/40\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			24.1383i			1.54847i	0.632897	+	0.774236i	$0.281866\pi$
$3$			24.1383i			1.54847i	−0.632897	+	0.774236i	$0.718134\pi$
$4$		0			0
$5$	46.7401	−	30.6653i	0.836112	−	0.548558i
$5$	46.7401	−	30.6653i	0.836112	−	0.548558i
$6$		0			0
$7$			179.876i			1.38749i	0.720223	+	0.693743i	$0.244040\pi$
$7$			179.876i			1.38749i	−0.720223	+	0.693743i	$0.755960\pi$
$8$		0			0
$9$	−339.657			−1.39777
$10$		0			0
$11$	−653.681			−1.62886			−0.814431	−	0.580260i	$-0.802951\pi$
$11$	−653.681			−1.62886			−0.814431	+	0.580260i	$0.802951\pi$
$12$		0			0
$13$			284.851i			0.467477i	0.972300	+	0.233738i	$0.0750959\pi$
$13$			284.851i			0.467477i	−0.972300	+	0.233738i	$0.924904\pi$
$14$		0			0
$15$	740.209	+	1128.23i	0.849427	+	1.29470i
$16$		0			0
$17$			383.672i			0.321986i	0.986956	+	0.160993i	$0.0514697\pi$
$17$			383.672i			0.321986i	−0.986956	+	0.160993i	$0.948530\pi$
$18$		0			0
$19$	2563.29			1.62897			0.814485	−	0.580185i	$-0.197020\pi$
$19$	2563.29			1.62897			0.814485	+	0.580185i	$0.197020\pi$
$20$		0			0
$21$	−4341.90			−2.14848
$22$		0			0
$23$		−	948.124i		−	0.373719i	−0.982387	−	0.186860i	$-0.940169\pi$
$23$		−	948.124i		−	0.373719i	0.982387	−	0.186860i	$-0.0598309\pi$
$24$		0			0
$25$	1244.27	−	2866.60i	0.398168	−	0.917313i
$26$		0			0
$27$		−	2333.13i		−	0.615929i
$28$		0			0
$29$	1524.26			0.336561			0.168281	−	0.985739i	$-0.446178\pi$
$29$	1524.26			0.336561			0.168281	+	0.985739i	$0.446178\pi$
$30$		0			0
$31$	3103.88			0.580098			0.290049	−	0.957012i	$-0.406328\pi$
$31$	3103.88			0.580098			0.290049	+	0.957012i	$0.406328\pi$
$32$		0			0
$33$		−	15778.7i		−	2.52225i
$34$		0			0
$35$	5515.97	+	8407.43i	0.761117	+	1.16009i
$36$		0			0
$37$			9991.75i			1.19988i	0.800046	+	0.599939i	$0.204809\pi$
$37$			9991.75i			1.19988i	−0.800046	+	0.599939i	$0.795191\pi$
$38$		0			0
$39$	−6875.83			−0.723875
$40$		0			0
$41$	15120.7			1.40479			0.702394	−	0.711788i	$-0.252114\pi$
$41$	15120.7			1.40479			0.702394	+	0.711788i	$0.252114\pi$
$42$		0			0
$43$			1754.92i			0.144739i	0.997378	+	0.0723696i	$0.0230561\pi$
$43$			1754.92i			0.144739i	−0.997378	+	0.0723696i	$0.976944\pi$
$44$		0			0
$45$	−15875.6	+	10415.7i	−1.16869	+	0.766756i
$46$		0			0
$47$		−	14760.7i		−	0.974682i	−0.873212	−	0.487341i	$-0.837967\pi$
$47$		−	14760.7i		−	0.974682i	0.873212	−	0.487341i	$-0.162033\pi$
$48$		0			0
$49$	−15548.5			−0.925118
$50$		0			0
$51$	−9261.18			−0.498587
$52$		0			0
$53$		−	8704.12i		−	0.425633i	−0.977092	−	0.212816i	$-0.931736\pi$
$53$		−	8704.12i		−	0.425633i	0.977092	−	0.212816i	$-0.0682636\pi$
$54$		0			0
$55$	−30553.1	+	20045.4i	−1.36191	+	0.893526i
$56$		0			0
$57$			61873.3i			2.52241i
$58$		0			0
$59$	12632.1			0.472438			0.236219	−	0.971700i	$-0.424092\pi$
$59$	12632.1			0.472438			0.236219	+	0.971700i	$0.424092\pi$
$60$		0			0
$61$	43098.3			1.48298			0.741490	−	0.670964i	$-0.234119\pi$
$61$	43098.3			1.48298			0.741490	+	0.670964i	$0.234119\pi$
$62$		0			0
$63$		−	61096.2i		−	1.93938i
$64$		0			0
$65$	8735.07	+	13314.0i	0.256438	+	0.390863i
$66$		0			0
$67$			26125.5i			0.711014i	0.934674	+	0.355507i	$0.115692\pi$
$67$			26125.5i			0.711014i	−0.934674	+	0.355507i	$0.884308\pi$
$68$		0			0
$69$	22886.1			0.578694
$70$		0			0
$71$	−46285.8			−1.08969			−0.544844	−	0.838538i	$-0.683411\pi$
$71$	−46285.8			−1.08969			−0.544844	+	0.838538i	$0.683411\pi$
$72$		0			0
$73$		−	51303.2i		−	1.12678i	−0.826192	−	0.563388i	$-0.809498\pi$
$73$		−	51303.2i		−	1.12678i	0.826192	−	0.563388i	$-0.190502\pi$
$74$		0			0
$75$	69194.9	+	30034.6i	1.42043	+	0.616551i
$76$		0			0
$77$		−	117582.i		−	2.26002i
$78$		0			0
$79$	−39314.0			−0.708729			−0.354364	−	0.935107i	$-0.615303\pi$
$79$	−39314.0			−0.708729			−0.354364	+	0.935107i	$0.615303\pi$
$80$		0			0
$81$	−26218.8			−0.444017
$82$		0			0
$83$		−	69544.8i		−	1.10808i	−0.832492	−	0.554038i	$-0.813086\pi$
$83$		−	69544.8i		−	1.10808i	0.832492	−	0.554038i	$-0.186914\pi$
$84$		0			0
$85$	11765.4	+	17932.8i	0.176628	+	0.269217i
$86$		0			0
$87$			36793.1i			0.521156i
$88$		0			0
$89$	13092.2			0.175201			0.0876007	−	0.996156i	$-0.472080\pi$
$89$	13092.2			0.175201			0.0876007	+	0.996156i	$0.472080\pi$
$90$		0			0
$91$	−51238.0			−0.648618
$92$		0			0
$93$			74922.5i			0.898265i
$94$		0			0
$95$	119808.	−	78604.0i	1.36200	−	0.893585i
$96$		0			0
$97$			26258.1i			0.283357i	0.989913	+	0.141678i	$0.0452499\pi$
$97$			26258.1i			0.283357i	−0.989913	+	0.141678i	$0.954750\pi$
$98$		0			0
$99$	222027.			2.27677
$100$		0			0
$101$	7382.54			0.0720116			0.0360058	−	0.999352i	$-0.488537\pi$
$101$	7382.54			0.0720116			0.0360058	+	0.999352i	$0.488537\pi$
$102$		0			0
$103$			44518.9i			0.413477i	0.978396	+	0.206738i	$0.0662849\pi$
$103$			44518.9i			0.413477i	−0.978396	+	0.206738i	$0.933715\pi$
$104$		0			0
$105$	−202941.	+	133146.i	−1.79637	+	1.17857i
$106$		0			0
$107$			194929.i			1.64595i	0.568079	+	0.822974i	$0.307687\pi$
$107$			194929.i			1.64595i	−0.568079	+	0.822974i	$0.692313\pi$
$108$		0			0
$109$	−213783.			−1.72348			−0.861739	−	0.507351i	$-0.830625\pi$
$109$	−213783.			−1.72348			−0.861739	+	0.507351i	$0.830625\pi$
$110$		0			0
$111$	−241184.			−1.85798
$112$		0			0
$113$		−	148946.i		−	1.09732i	−0.836046	−	0.548660i	$-0.815138\pi$
$113$		−	148946.i		−	1.09732i	0.836046	−	0.548660i	$-0.184862\pi$
$114$		0			0
$115$	−29074.5	−	44315.4i	−0.205007	−	0.312471i
$116$		0			0
$117$		−	96751.8i		−	0.653423i
$118$		0			0
$119$	−69013.4			−0.446751
$120$		0			0
$121$	266248.			1.65319
$122$		0			0
$123$			364987.i			2.17528i
$124$		0			0
$125$	−29747.8	−	172141.i	−0.170287	−	0.985395i
$126$		0			0
$127$		−	116757.i		−	0.642351i	−0.947020	−	0.321175i	$-0.895922\pi$
$127$		−	116757.i		−	0.642351i	0.947020	−	0.321175i	$-0.104078\pi$
$128$		0			0
$129$	−42360.8			−0.224125
$130$		0			0
$131$	349111.			1.77740			0.888700	−	0.458490i	$-0.151610\pi$
$131$	349111.			1.77740			0.888700	+	0.458490i	$0.151610\pi$
$132$		0			0
$133$			461074.i			2.26017i
$134$		0			0
$135$	−71546.4	−	109051.i	−0.337873	−	0.514986i
$136$		0			0
$137$		−	219776.i		−	1.00041i	−0.865907	−	0.500205i	$-0.833258\pi$
$137$		−	219776.i		−	1.00041i	0.865907	−	0.500205i	$-0.166742\pi$
$138$		0			0
$139$	−126594.			−0.555744			−0.277872	−	0.960618i	$-0.589629\pi$
$139$	−126594.			−0.555744			−0.277872	+	0.960618i	$0.589629\pi$
$140$		0			0
$141$	356299.			1.50927
$142$		0			0
$143$		−	186202.i		−	0.761455i
$144$		0			0
$145$	71244.1	−	46742.0i	0.281403	−	0.184624i
$146$		0			0
$147$		−	375313.i		−	1.43252i
$148$		0			0
$149$	25809.9			0.0952403			0.0476202	−	0.998866i	$-0.484836\pi$
$149$	25809.9			0.0952403			0.0476202	+	0.998866i	$0.484836\pi$
$150$		0			0
$151$	−311270.			−1.11095			−0.555475	−	0.831533i	$-0.687464\pi$
$151$	−311270.			−1.11095			−0.555475	+	0.831533i	$0.687464\pi$
$152$		0			0
$153$		−	130317.i		−	0.450061i
$154$		0			0
$155$	145076.	−	95181.7i	0.485027	−	0.318217i
$156$		0			0
$157$			429151.i			1.38951i	0.719247	+	0.694755i	$0.244487\pi$
$157$			429151.i			1.38951i	−0.719247	+	0.694755i	$0.755513\pi$
$158$		0			0
$159$	210102.			0.659080
$160$		0			0
$161$	170545.			0.518530
$162$		0			0
$163$		−	162467.i		−	0.478957i	−0.970902	−	0.239479i	$-0.923023\pi$
$163$		−	162467.i		−	0.478957i	0.970902	−	0.239479i	$-0.0769765\pi$
$164$		0			0
$165$	−483861.	−	737500.i	−1.38360	−	2.10888i
$166$		0			0
$167$		−	280057.i		−	0.777060i	−0.921436	−	0.388530i	$-0.872983\pi$
$167$		−	280057.i		−	0.777060i	0.921436	−	0.388530i	$-0.127017\pi$
$168$		0			0
$169$	290153.			0.781465
$170$		0			0
$171$	−870638.			−2.27692
$172$		0			0
$173$			742577.i			1.88637i	0.332271	+	0.943184i	$0.392185\pi$
$173$			742577.i			1.88637i	−0.332271	+	0.943184i	$0.607815\pi$
$174$		0			0
$175$	515634.	+	223815.i	1.27276	+	0.552452i
$176$		0			0
$177$			304917.i			0.731557i
$178$		0			0
$179$	14556.0			0.0339554			0.0169777	−	0.999856i	$-0.494596\pi$
$179$	14556.0			0.0339554			0.0169777	+	0.999856i	$0.494596\pi$
$180$		0			0
$181$	−243243.			−0.551878			−0.275939	−	0.961175i	$-0.588989\pi$
$181$	−243243.			−0.551878			−0.275939	+	0.961175i	$0.588989\pi$
$182$		0			0
$183$			1.04032e6i			2.29635i
$184$		0			0
$185$	306400.	+	467015.i	0.658203	+	1.00323i
$186$		0			0
$187$		−	250799.i		−	0.524471i
$188$		0			0
$189$	419675.			0.854592
$190$		0			0
$191$	75871.1			0.150485			0.0752425	−	0.997165i	$-0.476027\pi$
$191$	75871.1			0.150485			0.0752425	+	0.997165i	$0.476027\pi$
$192$		0			0
$193$		−	370161.i		−	0.715314i	−0.933853	−	0.357657i	$-0.883576\pi$
$193$		−	370161.i		−	0.715314i	0.933853	−	0.357657i	$-0.116424\pi$
$194$		0			0
$195$	−321377.	+	210850.i	−0.605241	+	0.397087i
$196$		0			0
$197$		−	213848.i		−	0.392590i	−0.980545	−	0.196295i	$-0.937109\pi$
$197$		−	213848.i		−	0.392590i	0.980545	−	0.196295i	$-0.0628910\pi$
$198$		0			0
$199$	−466075.			−0.834302			−0.417151	−	0.908837i	$-0.636971\pi$
$199$	−466075.			−0.834302			−0.417151	+	0.908837i	$0.636971\pi$
$200$		0			0
$201$	−630626.			−1.10099
$202$		0			0
$203$			274178.i			0.466974i
$204$		0			0
$205$	706741.	−	463680.i	1.17456	−	0.770609i
$206$		0			0
$207$			322037.i			0.522372i
$208$		0			0
$209$	−1.67557e6			−2.65337
$210$		0			0
$211$	−228211.			−0.352883			−0.176441	−	0.984311i	$-0.556459\pi$
$211$	−228211.			−0.352883			−0.176441	+	0.984311i	$0.556459\pi$
$212$		0			0
$213$		−	1.11726e6i		−	1.68735i
$214$		0			0
$215$	53815.2	+	82025.1i	0.0793979	+	0.121018i
$216$		0			0
$217$			558315.i			0.804878i
$218$		0			0
$219$	1.23837e6			1.74478
$220$		0			0
$221$	−109289.			−0.150521
$222$		0			0
$223$		−	972750.i		−	1.30990i	−0.755671	−	0.654951i	$-0.772689\pi$
$223$		−	972750.i		−	1.30990i	0.755671	−	0.654951i	$-0.227311\pi$
$224$		0			0
$225$	−422626.	+	973661.i	−0.556545	+	1.28219i
$226$		0			0
$227$			357063.i			0.459917i	0.973200	+	0.229959i	$0.0738591\pi$
$227$			357063.i			0.459917i	−0.973200	+	0.229959i	$0.926141\pi$
$228$		0			0
$229$	295196.			0.371982			0.185991	−	0.982551i	$-0.440451\pi$
$229$	295196.			0.371982			0.185991	+	0.982551i	$0.440451\pi$
$230$		0			0
$231$	2.83822e6			3.49958
$232$		0			0
$233$		−	492373.i		−	0.594162i	−0.954852	−	0.297081i	$-0.903987\pi$
$233$		−	492373.i		−	0.594162i	0.954852	−	0.297081i	$-0.0960131\pi$
$234$		0			0
$235$	−452643.	−	689918.i	−0.534670	−	0.814944i
$236$		0			0
$237$		−	948974.i		−	1.09745i
$238$		0			0
$239$	71796.0			0.0813028			0.0406514	−	0.999173i	$-0.487057\pi$
$239$	71796.0			0.0813028			0.0406514	+	0.999173i	$0.487057\pi$
$240$		0			0
$241$	614304.			0.681304			0.340652	−	0.940190i	$-0.389352\pi$
$241$	614304.			0.681304			0.340652	+	0.940190i	$0.389352\pi$
$242$		0			0
$243$		−	1.19983e6i		−	1.30348i
$244$		0			0
$245$	−726736.	+	476799.i	−0.773502	+	0.507481i
$246$		0			0
$247$			730156.i			0.761505i
$248$		0			0
$249$	1.67869e6			1.71582
$250$		0			0
$251$	−244276.			−0.244735			−0.122368	−	0.992485i	$-0.539049\pi$
$251$	−244276.			−0.244735			−0.122368	+	0.992485i	$0.539049\pi$
$252$		0			0
$253$			619771.i			0.608737i
$254$		0			0
$255$	−432868.	+	283997.i	−0.416875	+	0.273504i
$256$		0			0
$257$		−	951728.i		−	0.898835i	−0.893322	−	0.449418i	$-0.851632\pi$
$257$		−	951728.i		−	0.898835i	0.893322	−	0.449418i	$-0.148368\pi$
$258$		0			0
$259$	−1.79728e6			−1.66481
$260$		0			0
$261$	−517726.			−0.470434
$262$		0			0
$263$		−	258512.i		−	0.230458i	−0.993339	−	0.115229i	$-0.963240\pi$
$263$		−	258512.i		−	0.230458i	0.993339	−	0.115229i	$-0.0367602\pi$
$264$		0			0
$265$	−266915.	−	406831.i	−0.233484	−	0.355877i
$266$		0			0
$267$			316023.i			0.271295i
$268$		0			0
$269$	−894716.			−0.753884			−0.376942	−	0.926237i	$-0.623024\pi$
$269$	−894716.			−0.753884			−0.376942	+	0.926237i	$0.623024\pi$
$270$		0			0
$271$	474149.			0.392186			0.196093	−	0.980585i	$-0.437175\pi$
$271$	474149.			0.392186			0.196093	+	0.980585i	$0.437175\pi$
$272$		0			0
$273$		−	1.23680e6i		−	1.00437i
$274$		0			0
$275$	−813358.	+	1.87384e6i	−0.648560	+	1.49418i
$276$		0			0
$277$		−	142752.i		−	0.111785i	−0.998437	−	0.0558923i	$-0.982200\pi$
$277$		−	142752.i		−	0.111785i	0.998437	−	0.0558923i	$-0.0178003\pi$
$278$		0			0
$279$	−1.05426e6			−0.810841
$280$		0			0
$281$	−427838.			−0.323231			−0.161616	−	0.986854i	$-0.551670\pi$
$281$	−427838.			−0.323231			−0.161616	+	0.986854i	$0.551670\pi$
$282$		0			0
$283$		−	2.30233e6i		−	1.70884i	−0.519585	−	0.854419i	$-0.673913\pi$
$283$		−	2.30233e6i		−	1.70884i	0.519585	−	0.854419i	$-0.326087\pi$
$284$		0			0
$285$	1.89737e6	+	2.89197e6i	1.38369	+	2.10902i
$286$		0			0
$287$			2.71985e6i			1.94913i
$288$		0			0
$289$	1.27265e6			0.896325
$290$		0			0
$291$	−633825.			−0.438770
$292$		0			0
$293$			1.59390e6i			1.08465i	0.840168	+	0.542326i	$0.182456\pi$
$293$			1.59390e6i			1.08465i	−0.840168	+	0.542326i	$0.817544\pi$
$294$		0			0
$295$	590424.	−	387367.i	0.395011	−	0.259160i
$296$		0			0
$297$			1.52513e6i			1.00326i
$298$		0			0
$299$	270074.			0.174705
$300$		0			0
$301$	−315668.			−0.200824
$302$		0			0
$303$			178202.i			0.111508i
$304$		0			0
$305$	2.01442e6	−	1.32162e6i	1.23994	−	0.813501i
$306$		0			0
$307$			322680.i			0.195401i	0.995216	+	0.0977005i	$0.0311487\pi$
$307$			322680.i			0.195401i	−0.995216	+	0.0977005i	$0.968851\pi$
$308$		0			0
$309$	−1.07461e6			−0.640257
$310$		0			0
$311$	−660204.			−0.387059			−0.193529	−	0.981094i	$-0.561994\pi$
$311$	−660204.			−0.387059			−0.193529	+	0.981094i	$0.561994\pi$
$312$		0			0
$313$			1.15378e6i			0.665677i	0.942984	+	0.332838i	$0.108006\pi$
$313$			1.15378e6i			0.665677i	−0.942984	+	0.332838i	$0.891994\pi$
$314$		0			0
$315$	−1.87354e6	−	2.85564e6i	−1.06386	−	1.62154i
$316$		0			0
$317$			1.20296e6i			0.672364i	0.941797	+	0.336182i	$0.109136\pi$
$317$			1.20296e6i			0.672364i	−0.941797	+	0.336182i	$0.890864\pi$
$318$		0			0
$319$	−996381.			−0.548212
$320$		0			0
$321$	−4.70524e6			−2.54870
$322$		0			0
$323$			983460.i			0.524506i
$324$		0			0
$325$	816556.	+	354433.i	0.428822	+	0.186134i
$326$		0			0
$327$		−	5.16034e6i		−	2.66876i
$328$		0			0
$329$	2.65510e6			1.35236
$330$		0			0
$331$	1.82424e6			0.915193			0.457597	−	0.889160i	$-0.348710\pi$
$331$	1.82424e6			0.915193			0.457597	+	0.889160i	$0.348710\pi$
$332$		0			0
$333$		−	3.39377e6i		−	1.67715i
$334$		0			0
$335$	801149.	+	1.22111e6i	0.390033	+	0.594488i
$336$		0			0
$337$		−	485867.i		−	0.233047i	−0.993188	−	0.116523i	$-0.962825\pi$
$337$		−	485867.i		−	0.233047i	0.993188	−	0.116523i	$-0.0371750\pi$
$338$		0			0
$339$	3.59530e6			1.69917
$340$		0			0
$341$	−2.02895e6			−0.944899
$342$		0			0
$343$			226383.i			0.103898i
$344$		0			0
$345$	1.06970e6	−	701809.i	0.483853	−	0.317447i
$346$		0			0
$347$		−	2.35066e6i		−	1.04801i	−0.851715	−	0.524006i	$-0.824437\pi$
$347$		−	2.35066e6i		−	1.04801i	0.851715	−	0.524006i	$-0.175563\pi$
$348$		0			0
$349$	2.62295e6			1.15273			0.576364	−	0.817193i	$-0.304471\pi$
$349$	2.62295e6			1.15273			0.576364	+	0.817193i	$0.304471\pi$
$350$		0			0
$351$	664597.			0.287932
$352$		0			0
$353$			583638.i			0.249291i	0.992201	+	0.124646i	$0.0397794\pi$
$353$			583638.i			0.249291i	−0.992201	+	0.124646i	$0.960221\pi$
$354$		0			0
$355$	−2.16340e6	+	1.41937e6i	−0.911101	+	0.597757i
$356$		0			0
$357$		−	1.66587e6i		−	0.691782i
$358$		0			0
$359$	−1.64503e6			−0.673654			−0.336827	−	0.941567i	$-0.609354\pi$
$359$	−1.64503e6			−0.673654			−0.336827	+	0.941567i	$0.609354\pi$
$360$		0			0
$361$	4.09433e6			1.65354
$362$		0			0
$363$			6.42678e6i			2.55992i
$364$		0			0
$365$	−1.57323e6	−	2.39792e6i	−0.618102	−	0.942111i
$366$		0			0
$367$			2.62251e6i			1.01637i	0.861247	+	0.508186i	$0.169684\pi$
$367$			2.62251e6i			1.01637i	−0.861247	+	0.508186i	$0.830316\pi$
$368$		0			0
$369$	−5.13584e6			−1.96357
$370$		0			0
$371$	1.56566e6			0.590559
$372$		0			0
$373$		−	3.73068e6i		−	1.38840i	−0.719780	−	0.694202i	$-0.755757\pi$
$373$		−	3.73068e6i		−	1.38840i	0.719780	−	0.694202i	$-0.244243\pi$
$374$		0			0
$375$	4.15520e6	−	718062.i	1.52586	−	0.263684i
$376$		0			0
$377$			434188.i			0.157335i
$378$		0			0
$379$	3.00539e6			1.07474			0.537368	−	0.843348i	$-0.319418\pi$
$379$	3.00539e6			1.07474			0.537368	+	0.843348i	$0.319418\pi$
$380$		0			0
$381$	2.81831e6			0.994662
$382$		0			0
$383$		−	343090.i		−	0.119512i	−0.998213	−	0.0597560i	$-0.980968\pi$
$383$		−	343090.i		−	0.119512i	0.998213	−	0.0597560i	$-0.0190323\pi$
$384$		0			0
$385$	−3.60568e6	−	5.49578e6i	−1.23975	−	1.88963i
$386$		0			0
$387$		−	596071.i		−	0.202311i
$388$		0			0
$389$	−190583.			−0.0638572			−0.0319286	−	0.999490i	$-0.510165\pi$
$389$	−190583.			−0.0638572			−0.0319286	+	0.999490i	$0.510165\pi$
$390$		0			0
$391$	363768.			0.120332
$392$		0			0
$393$			8.42694e6i			2.75225i
$394$		0			0
$395$	−1.83754e6	+	1.20558e6i	−0.592577	+	0.388779i
$396$		0			0
$397$			5.22551e6i			1.66400i	0.554778	+	0.831999i	$0.312803\pi$
$397$			5.22551e6i			1.66400i	−0.554778	+	0.831999i	$0.687197\pi$
$398$		0			0
$399$	−1.11295e7			−3.49981
$400$		0			0
$401$	−1.60312e6			−0.497859			−0.248929	−	0.968522i	$-0.580079\pi$
$401$	−1.60312e6			−0.497859			−0.248929	+	0.968522i	$0.580079\pi$
$402$		0			0
$403$			884146.i			0.271182i
$404$		0			0
$405$	−1.22547e6	+	804008.i	−0.371248	+	0.243569i
$406$		0			0
$407$		−	6.53142e6i		−	1.95444i
$408$		0			0
$409$	−4.58732e6			−1.35597			−0.677986	−	0.735075i	$-0.737147\pi$
$409$	−4.58732e6			−1.35597			−0.677986	+	0.735075i	$0.737147\pi$
$410$		0			0
$411$	5.30501e6			1.54911
$412$		0			0
$413$			2.27221e6i			0.655501i
$414$		0			0
$415$	−2.13261e6	−	3.25053e6i	−0.607844	−	0.926476i
$416$		0			0
$417$		−	3.05575e6i		−	0.860554i
$418$		0			0
$419$	6.16854e6			1.71651			0.858256	−	0.513221i	$-0.171548\pi$
$419$	6.16854e6			1.71651			0.858256	+	0.513221i	$0.171548\pi$
$420$		0			0
$421$	−6.15589e6			−1.69272			−0.846361	−	0.532610i	$-0.821211\pi$
$421$	−6.15589e6			−1.69272			−0.846361	+	0.532610i	$0.821211\pi$
$422$		0			0
$423$			5.01358e6i			1.36238i
$424$		0			0
$425$	1.09983e6	+	477393.i	0.295362	+	0.128205i
$426$		0			0
$427$			7.75236e6i			2.05761i
$428$		0			0
$429$	4.49460e6			1.17909
$430$		0			0
$431$	1.15900e6			0.300532			0.150266	−	0.988646i	$-0.451987\pi$
$431$	1.15900e6			0.300532			0.150266	+	0.988646i	$0.451987\pi$
$432$		0			0
$433$			6.73906e6i			1.72735i	0.504052	+	0.863673i	$0.331842\pi$
$433$			6.73906e6i			1.72735i	−0.504052	+	0.863673i	$0.668158\pi$
$434$		0			0
$435$	1.12827e6	+	1.71971e6i	0.285884	+	0.435745i
$436$		0			0
$437$		−	2.43031e6i		−	0.608777i
$438$		0			0
$439$	3.69068e6			0.913998			0.456999	−	0.889467i	$-0.348924\pi$
$439$	3.69068e6			0.913998			0.456999	+	0.889467i	$0.348924\pi$
$440$		0			0
$441$	5.28114e6			1.29310
$442$		0			0
$443$		−	4.55124e6i		−	1.10185i	−0.834556	−	0.550923i	$-0.814276\pi$
$443$		−	4.55124e6i		−	1.10185i	0.834556	−	0.550923i	$-0.185724\pi$
$444$		0			0
$445$	611931.	−	401477.i	0.146488	−	0.0961082i
$446$		0			0
$447$			623007.i			0.147477i
$448$		0			0
$449$	3.31419e6			0.775822			0.387911	−	0.921697i	$-0.373197\pi$
$449$	3.31419e6			0.775822			0.387911	+	0.921697i	$0.373197\pi$
$450$		0			0
$451$	−9.88409e6			−2.28821
$452$		0			0
$453$		−	7.51352e6i		−	1.72028i
$454$		0			0
$455$	−2.39487e6	+	1.57123e6i	−0.542317	+	0.355805i
$456$		0			0
$457$		−	4.10489e6i		−	0.919414i	−0.888071	−	0.459707i	$-0.847954\pi$
$457$		−	4.10489e6i		−	0.919414i	0.888071	−	0.459707i	$-0.152046\pi$
$458$		0			0
$459$	895158.			0.198321
$460$		0			0
$461$	8.16525e6			1.78944			0.894720	−	0.446628i	$-0.147375\pi$
$461$	8.16525e6			1.78944			0.894720	+	0.446628i	$0.147375\pi$
$462$		0			0
$463$		−	5.65614e6i		−	1.22622i	−0.789998	−	0.613109i	$-0.789919\pi$
$463$		−	5.65614e6i		−	1.22622i	0.789998	−	0.613109i	$-0.210081\pi$
$464$		0			0
$465$	2.29752e6	+	3.50188e6i	0.492751	+	0.751051i
$466$		0			0
$467$		−	8.77216e6i		−	1.86129i	−0.365921	−	0.930646i	$-0.619246\pi$
$467$		−	8.77216e6i		−	1.86129i	0.365921	−	0.930646i	$-0.380754\pi$
$468$		0			0
$469$	−4.69936e6			−0.986522
$470$		0			0
$471$	−1.03590e7			−2.15162
$472$		0			0
$473$		−	1.14716e6i		−	0.235760i
$474$		0			0
$475$	3.18943e6	−	7.34792e6i	0.648603	−	1.49427i
$476$		0			0
$477$			2.95641e6i			0.594935i
$478$		0			0
$479$	−417978.			−0.0832367			−0.0416184	−	0.999134i	$-0.513251\pi$
$479$	−417978.			−0.0832367			−0.0416184	+	0.999134i	$0.513251\pi$
$480$		0			0
$481$	−2.84616e6			−0.560915
$482$		0			0
$483$			4.11666e6i			0.802929i
$484$		0			0
$485$	805213.	+	1.22731e6i	0.155438	+	0.236918i
$486$		0			0
$487$			7.07076e6i			1.35096i	0.737376	+	0.675482i	$0.236064\pi$
$487$			7.07076e6i			1.35096i	−0.737376	+	0.675482i	$0.763936\pi$
$488$		0			0
$489$	3.92168e6			0.741652
$490$		0			0
$491$	−1.00127e7			−1.87433			−0.937166	−	0.348883i	$-0.886561\pi$
$491$	−1.00127e7			−1.87433			−0.937166	+	0.348883i	$0.886561\pi$
$492$		0			0
$493$			584816.i			0.108368i
$494$		0			0
$495$	1.03776e7	−	6.80855e6i	1.90363	−	1.24894i
$496$		0			0
$497$		−	8.32571e6i		−	1.51193i
$498$		0			0
$499$	7.74452e6			1.39233			0.696166	−	0.717881i	$-0.254888\pi$
$499$	7.74452e6			1.39233			0.696166	+	0.717881i	$0.254888\pi$
$500$		0			0
$501$	6.76009e6			1.20326
$502$		0			0
$503$		−	2.20376e6i		−	0.388370i	−0.980965	−	0.194185i	$-0.937794\pi$
$503$		−	2.20376e6i		−	0.388370i	0.980965	−	0.194185i	$-0.0622061\pi$
$504$		0			0
$505$	345061.	−	226388.i	0.0602098	−	0.0395025i
$506$		0			0
$507$			7.00379e6i			1.21008i
$508$		0			0
$509$	−1.40053e6			−0.239607			−0.119803	−	0.992798i	$-0.538226\pi$
$509$	−1.40053e6			−0.239607			−0.119803	+	0.992798i	$0.538226\pi$
$510$		0			0
$511$	9.22823e6			1.56339
$512$		0			0
$513$		−	5.98049e6i		−	1.00333i
$514$		0			0
$515$	1.36519e6	+	2.08082e6i	0.226816	+	0.345713i
$516$		0			0
$517$			9.64881e6i			1.58762i
$518$		0			0
$519$	−1.79245e7			−2.92099
$520$		0			0
$521$	1.60571e6			0.259162			0.129581	−	0.991569i	$-0.458637\pi$
$521$	1.60571e6			0.259162			0.129581	+	0.991569i	$0.458637\pi$
$522$		0			0
$523$		−	5.80485e6i		−	0.927977i	−0.885841	−	0.463988i	$-0.846418\pi$
$523$		−	5.80485e6i		−	0.927977i	0.885841	−	0.463988i	$-0.153582\pi$
$524$		0			0
$525$	−5.40252e6	+	1.24465e7i	−0.855456	+	1.97083i
$526$		0			0
$527$			1.19087e6i			0.186784i
$528$		0			0
$529$	5.53740e6			0.860334
$530$		0			0
$531$	−4.29057e6			−0.660357
$532$		0			0
$533$			4.30714e6i			0.656706i
$534$		0			0
$535$	5.97755e6	+	9.11098e6i	0.902898	+	1.37620i
$536$		0			0
$537$			351356.i			0.0525790i
$538$		0			0
$539$	1.01637e7			1.50689
$540$		0			0
$541$	−9.37930e6			−1.37777			−0.688886	−	0.724870i	$-0.741900\pi$
$541$	−9.37930e6			−1.37777			−0.688886	+	0.724870i	$0.741900\pi$
$542$		0			0
$543$		−	5.87146e6i		−	0.854568i
$544$		0			0
$545$	−9.99222e6	+	6.55571e6i	−1.44102	+	0.945429i
$546$		0			0
$547$			2.33685e6i			0.333936i	0.985962	+	0.166968i	$0.0533976\pi$
$547$			2.33685e6i			0.333936i	−0.985962	+	0.166968i	$0.946602\pi$
$548$		0			0
$549$	−1.46386e7			−2.07286
$550$		0			0
$551$	3.90712e6			0.548248
$552$		0			0
$553$		−	7.07166e6i		−	0.983351i
$554$		0			0
$555$	−1.12730e7	+	7.39598e6i	−1.55348	+	1.01921i
$556$		0			0
$557$		−	7.02238e6i		−	0.959061i	−0.877525	−	0.479530i	$-0.840807\pi$
$557$		−	7.02238e6i		−	0.959061i	0.877525	−	0.479530i	$-0.159193\pi$
$558$		0			0
$559$	−499891.			−0.0676622
$560$		0			0
$561$	6.05386e6			0.812129
$562$		0			0
$563$		−	1.81010e6i		−	0.240675i	−0.992733	−	0.120338i	$-0.961602\pi$
$563$		−	1.81010e6i		−	0.240675i	0.992733	−	0.120338i	$-0.0383977\pi$
$564$		0			0
$565$	−4.56748e6	−	6.96175e6i	−0.601943	−	0.917482i
$566$		0			0
$567$		−	4.71613e6i		−	0.616068i
$568$		0			0
$569$	−5.62751e6			−0.728678			−0.364339	−	0.931266i	$-0.618705\pi$
$569$	−5.62751e6			−0.728678			−0.364339	+	0.931266i	$0.618705\pi$
$570$		0			0
$571$	−2.16702e6			−0.278145			−0.139073	−	0.990282i	$-0.544412\pi$
$571$	−2.16702e6			−0.278145			−0.139073	+	0.990282i	$0.544412\pi$
$572$		0			0
$573$			1.83140e6i			0.233022i
$574$		0			0
$575$	−2.71789e6	−	1.17973e6i	−0.342817	−	0.148803i
$576$		0			0
$577$			1.04992e7i			1.31286i	0.754387	+	0.656430i	$0.227934\pi$
$577$			1.04992e7i			1.31286i	−0.754387	+	0.656430i	$0.772066\pi$
$578$		0			0
$579$	8.93504e6			1.10764
$580$		0			0
$581$	1.25095e7			1.53744
$582$		0			0
$583$			5.68972e6i			0.693297i
$584$		0			0
$585$	−2.96693e6	−	4.52219e6i	−0.358441	−	0.546335i
$586$		0			0
$587$			1.47459e7i			1.76634i	0.469050	+	0.883172i	$0.344596\pi$
$587$			1.47459e7i			1.76634i	−0.469050	+	0.883172i	$0.655404\pi$
$588$		0			0
$589$	7.95614e6			0.944962
$590$		0			0
$591$	5.16192e6			0.607915
$592$		0			0
$593$		−	4.85519e6i		−	0.566982i	−0.958975	−	0.283491i	$-0.908507\pi$
$593$		−	4.85519e6i		−	0.566982i	0.958975	−	0.283491i	$-0.0914926\pi$
$594$		0			0
$595$	−3.22569e6	+	2.11632e6i	−0.373534	+	0.245069i
$596$		0			0
$597$		−	1.12503e7i		−	1.29189i
$598$		0			0
$599$	4.48204e6			0.510398			0.255199	−	0.966889i	$-0.417859\pi$
$599$	4.48204e6			0.510398			0.255199	+	0.966889i	$0.417859\pi$
$600$		0			0
$601$	4.29562e6			0.485110			0.242555	−	0.970138i	$-0.422015\pi$
$601$	4.29562e6			0.485110			0.242555	+	0.970138i	$0.422015\pi$
$602$		0			0
$603$		−	8.87372e6i		−	0.993831i
$604$		0			0
$605$	1.24445e7	−	8.16459e6i	1.38225	−	0.906872i
$606$		0			0
$607$			1.05706e7i			1.16447i	0.813020	+	0.582235i	$0.197822\pi$
$607$			1.05706e7i			1.16447i	−0.813020	+	0.582235i	$0.802178\pi$
$608$		0			0
$609$	−6.61820e6			−0.723097
$610$		0			0
$611$	4.20461e6			0.455641
$612$		0			0
$613$		−	1.30960e7i		−	1.40763i	−0.710383	−	0.703816i	$-0.751478\pi$
$613$		−	1.30960e7i		−	1.40763i	0.710383	−	0.703816i	$-0.248522\pi$
$614$		0			0
$615$	1.11924e7	+	1.70595e7i	1.19327	+	1.81878i
$616$		0			0
$617$			1.69519e7i			1.79270i	0.443351	+	0.896348i	$0.353789\pi$
$617$			1.69519e7i			1.79270i	−0.443351	+	0.896348i	$0.646211\pi$
$618$		0			0
$619$	−1.27214e7			−1.33447			−0.667236	−	0.744846i	$-0.732523\pi$
$619$	−1.27214e7			−1.33447			−0.667236	+	0.744846i	$0.732523\pi$
$620$		0			0
$621$	−2.21210e6			−0.230184
$622$		0			0
$623$			2.35498e6i			0.243090i
$624$		0			0
$625$	−6.66919e6	−	7.13368e6i	−0.682925	−	0.730488i
$626$		0			0
$627$		−	4.04454e7i		−	4.10866i
$628$		0			0
$629$	−3.83355e6			−0.386344
$630$		0			0
$631$	8.32823e6			0.832682			0.416341	−	0.909209i	$-0.363312\pi$
$631$	8.32823e6			0.832682			0.416341	+	0.909209i	$0.363312\pi$
$632$		0			0
$633$		−	5.50862e6i		−	0.546429i
$634$		0			0
$635$	−3.58038e6	−	5.45722e6i	−0.352367	−	0.537078i
$636$		0			0
$637$		−	4.42900e6i		−	0.432471i
$638$		0			0
$639$	1.57213e7			1.52313
$640$		0			0
$641$	−4.18862e6			−0.402648			−0.201324	−	0.979525i	$-0.564524\pi$
$641$	−4.18862e6			−0.402648			−0.201324	+	0.979525i	$0.564524\pi$
$642$		0			0
$643$		−	271198.i		−	0.0258677i	−0.999916	−	0.0129339i	$-0.995883\pi$
$643$		−	271198.i		−	0.0258677i	0.999916	−	0.0129339i	$-0.00411709\pi$
$644$		0			0
$645$	−1.97995e6	+	1.29901e6i	−0.187393	+	0.122945i
$646$		0			0
$647$		−	1.01692e7i		−	0.955052i	−0.878618	−	0.477526i	$-0.841534\pi$
$647$		−	1.01692e7i		−	0.955052i	0.878618	−	0.477526i	$-0.158466\pi$
$648$		0			0
$649$	−8.25735e6			−0.769536
$650$		0			0
$651$	−1.34768e7			−1.24633
$652$		0			0
$653$			8.95062e6i			0.821429i	0.911764	+	0.410714i	$0.134721\pi$
$653$			8.95062e6i			0.821429i	−0.911764	+	0.410714i	$0.865279\pi$
$654$		0			0
$655$	1.63175e7	−	1.07056e7i	1.48611	−	0.975007i
$656$		0			0
$657$			1.74255e7i			1.57497i
$658$		0			0
$659$	−1.07182e7			−0.961409			−0.480705	−	0.876883i	$-0.659619\pi$
$659$	−1.07182e7			−0.961409			−0.480705	+	0.876883i	$0.659619\pi$
$660$		0			0
$661$	4.65131e6			0.414068			0.207034	−	0.978334i	$-0.433619\pi$
$661$	4.65131e6			0.414068			0.207034	+	0.978334i	$0.433619\pi$
$662$		0			0
$663$		−	2.63806e6i		−	0.233078i
$664$		0			0
$665$	1.41390e7	+	2.15506e7i	1.23984	+	1.88976i
$666$		0			0
$667$		−	1.44519e6i		−	0.125779i
$668$		0			0
$669$	2.34805e7			2.02835
$670$		0			0
$671$	−2.81725e7			−2.41557
$672$		0			0
$673$		−	9.92139e6i		−	0.844374i	−0.906509	−	0.422187i	$-0.861263\pi$
$673$		−	9.92139e6i		−	0.844374i	0.906509	−	0.422187i	$-0.138737\pi$
$674$		0			0
$675$	−6.68817e6	−	2.90306e6i	−0.564999	−	0.245243i
$676$		0			0
$677$		−	6.45936e6i		−	0.541649i	−0.962629	−	0.270824i	$-0.912704\pi$
$677$		−	6.45936e6i		−	0.541649i	0.962629	−	0.270824i	$-0.0872963\pi$
$678$		0			0
$679$	−4.72321e6			−0.393154
$680$		0			0
$681$	−8.61888e6			−0.712169
$682$		0			0
$683$			1.26684e7i			1.03913i	0.854432	+	0.519564i	$0.173905\pi$
$683$			1.26684e7i			1.03913i	−0.854432	+	0.519564i	$0.826095\pi$
$684$		0			0
$685$	−6.73950e6	−	1.02723e7i	−0.548783	−	0.836455i
$686$		0			0
$687$			7.12552e6i			0.576003i
$688$		0			0
$689$	2.47938e6			0.198973
$690$		0			0
$691$	4.07160e6			0.324392			0.162196	−	0.986759i	$-0.448142\pi$
$691$	4.07160e6			0.324392			0.162196	+	0.986759i	$0.448142\pi$
$692$		0			0
$693$			3.99375e7i			3.15898i
$694$		0			0
$695$	−5.91700e6	+	3.88204e6i	−0.464665	+	0.304858i
$696$		0			0
$697$			5.80137e6i			0.452323i
$698$		0			0
$699$	1.18850e7			0.920043
$700$		0			0
$701$	−2.11804e7			−1.62795			−0.813973	−	0.580903i	$-0.802700\pi$
$701$	−2.11804e7			−1.62795			−0.813973	+	0.580903i	$0.802700\pi$
$702$		0			0
$703$			2.56117e7i			1.95457i
$704$		0			0
$705$	1.66534e7	−	1.09260e7i	1.26192	−	0.827921i
$706$		0			0
$707$			1.32794e6i			0.0999150i
$708$		0			0
$709$	1.87573e7			1.40138			0.700689	−	0.713467i	$-0.252876\pi$
$709$	1.87573e7			1.40138			0.700689	+	0.713467i	$0.252876\pi$
$710$		0			0
$711$	1.33533e7			0.990637
$712$		0			0
$713$		−	2.94287e6i		−	0.216794i
$714$		0			0
$715$	−5.70995e6	−	8.70310e6i	−0.417703	−	0.636662i
$716$		0			0
$717$			1.73303e6i			0.125895i
$718$		0			0
$719$	−1.82800e7			−1.31872			−0.659362	−	0.751826i	$-0.729174\pi$
$719$	−1.82800e7			−1.31872			−0.659362	+	0.751826i	$0.729174\pi$
$720$		0			0
$721$	−8.00788e6			−0.573693
$722$		0			0
$723$			1.48282e7i			1.05498i
$724$		0			0
$725$	1.89660e6	−	4.36945e6i	0.134008	−	0.308732i
$726$		0			0
$727$		−	6.45855e6i		−	0.453210i	−0.973987	−	0.226605i	$-0.927237\pi$
$727$		−	6.45855e6i		−	0.453210i	0.973987	−	0.226605i	$-0.0727626\pi$
$728$		0			0
$729$	2.25906e7			1.57438
$730$		0			0
$731$	−673313.			−0.0466040
$732$		0			0
$733$			1.88365e6i			0.129491i	0.997902	+	0.0647454i	$0.0206235\pi$
$733$			1.88365e6i			0.129491i	−0.997902	+	0.0647454i	$0.979376\pi$
$734$		0			0
$735$	−1.15091e7	−	1.75422e7i	−0.785820	−	1.19775i
$736$		0			0
$737$		−	1.70778e7i		−	1.15814i
$738$		0			0
$739$	−2.24002e7			−1.50883			−0.754417	−	0.656395i	$-0.772080\pi$
$739$	−2.24002e7			−1.50883			−0.754417	+	0.656395i	$0.772080\pi$
$740$		0			0
$741$	−1.76247e7			−1.17917
$742$		0			0
$743$		−	2.22827e6i		−	0.148080i	−0.997255	−	0.0740400i	$-0.976411\pi$
$743$		−	2.22827e6i		−	0.148080i	0.997255	−	0.0740400i	$-0.0235893\pi$
$744$		0			0
$745$	1.20636e6	−	791470.i	0.0796316	−	0.0522449i
$746$		0			0
$747$			2.36214e7i			1.54883i
$748$		0			0
$749$	−3.50630e7			−2.28373
$750$		0			0
$751$	1.92986e7			1.24861			0.624303	−	0.781183i	$-0.285383\pi$
$751$	1.92986e7			1.24861			0.624303	+	0.781183i	$0.285383\pi$
$752$		0			0
$753$		−	5.89641e6i		−	0.378966i
$754$		0			0
$755$	−1.45488e7	+	9.54520e6i	−0.928879	+	0.609421i
$756$		0			0
$757$			7.22211e6i			0.458062i	0.973419	+	0.229031i	$0.0735557\pi$
$757$			7.22211e6i			0.458062i	−0.973419	+	0.229031i	$0.926444\pi$
$758$		0			0
$759$	−1.49602e7			−0.942612
$760$		0			0
$761$	1.08001e7			0.676028			0.338014	−	0.941141i	$-0.390245\pi$
$761$	1.08001e7			0.676028			0.338014	+	0.941141i	$0.390245\pi$
$762$		0			0
$763$		−	3.84544e7i		−	2.39130i
$764$		0			0
$765$	−3.99621e6	−	6.09102e6i	−0.246885	−	0.376302i
$766$		0			0
$767$			3.59827e6i			0.220854i
$768$		0			0
$769$	−5.50354e6			−0.335603			−0.167802	−	0.985821i	$-0.553667\pi$
$769$	−5.50354e6			−0.335603			−0.167802	+	0.985821i	$0.553667\pi$
$770$		0			0
$771$	2.29731e7			1.39182
$772$		0			0
$773$			2.91304e7i			1.75347i	0.480977	+	0.876733i	$0.340282\pi$
$773$			2.91304e7i			1.75347i	−0.480977	+	0.876733i	$0.659718\pi$
$774$		0			0
$775$	3.86208e6	−	8.89760e6i	0.230976	−	0.532131i
$776$		0			0
$777$		−	4.33832e7i		−	2.57792i
$778$		0			0
$779$	3.87586e7			2.28836
$780$		0			0
$781$	3.02562e7			1.77495
$782$		0			0
$783$		−	3.55631e6i		−	0.207298i
$784$		0			0
$785$	1.31601e7	+	2.00586e7i	0.762227	+	1.16179i
$786$		0			0
$787$		−	5.40251e6i		−	0.310927i	−0.987842	−	0.155464i	$-0.950313\pi$
$787$		−	5.40251e6i		−	0.310927i	0.987842	−	0.155464i	$-0.0496871\pi$
$788$		0			0
$789$	6.24005e6			0.356858
$790$		0			0
$791$	2.67919e7			1.52251
$792$		0			0
$793$			1.22766e7i			0.693259i
$794$		0			0
$795$	9.82021e6	−	6.44286e6i	0.551065	−	0.361544i
$796$		0			0
$797$		−	2.19820e7i		−	1.22580i	−0.790159	−	0.612902i	$-0.790002\pi$
$797$		−	2.19820e7i		−	1.22580i	0.790159	−	0.612902i	$-0.209998\pi$
$798$		0			0
$799$	5.66327e6			0.313834
$800$		0			0
$801$	−4.44686e6			−0.244891
$802$		0			0
$803$			3.35360e7i			1.83536i
$804$		0			0
$805$	7.97128e6	−	5.22982e6i	0.433549	−	0.284444i
$806$		0			0
$807$		−	2.15969e7i		−	1.16737i
$808$		0			0
$809$	−3.37377e7			−1.81236			−0.906178	−	0.422896i	$-0.861014\pi$
$809$	−3.37377e7			−1.81236			−0.906178	+	0.422896i	$0.861014\pi$
$810$		0			0
$811$	3.30907e6			0.176666			0.0883331	−	0.996091i	$-0.471846\pi$
$811$	3.30907e6			0.176666			0.0883331	+	0.996091i	$0.471846\pi$
$812$		0			0
$813$			1.14451e7i			0.607289i
$814$		0			0
$815$	−4.98211e6	−	7.59373e6i	−0.262736	−	0.400462i
$816$		0			0
$817$			4.49836e6i			0.235776i
$818$		0			0
$819$	1.74033e7			0.906615
$820$		0			0
$821$	4.00294e6			0.207263			0.103631	−	0.994616i	$-0.466954\pi$
$821$	4.00294e6			0.207263			0.103631	+	0.994616i	$0.466954\pi$
$822$		0			0
$823$		−	1.17703e7i		−	0.605740i	−0.953032	−	0.302870i	$-0.902055\pi$
$823$		−	1.17703e7i		−	0.605740i	0.953032	−	0.302870i	$-0.0979448\pi$
$824$		0			0
$825$	−4.52314e7	−	1.96331e7i	−2.31369	−	1.00428i
$826$		0			0
$827$			1.12347e7i			0.571214i	0.958347	+	0.285607i	$0.0921952\pi$
$827$			1.12347e7i			0.571214i	−0.958347	+	0.285607i	$0.907805\pi$
$828$		0			0
$829$	−2.44928e7			−1.23781			−0.618903	−	0.785468i	$-0.712423\pi$
$829$	−2.44928e7			−1.23781			−0.618903	+	0.785468i	$0.712423\pi$
$830$		0			0
$831$	3.44578e6			0.173095
$832$		0			0
$833$		−	5.96550e6i		−	0.297875i
$834$		0			0
$835$	−8.58803e6	−	1.30899e7i	−0.426263	−	0.649710i
$836$		0			0
$837$		−	7.24178e6i		−	0.357299i
$838$		0			0
$839$	1.20271e6			0.0589868			0.0294934	−	0.999565i	$-0.490611\pi$
$839$	1.20271e6			0.0589868			0.0294934	+	0.999565i	$0.490611\pi$
$840$		0			0
$841$	−1.81878e7			−0.886726
$842$		0			0
$843$		−	1.03273e7i		−	0.500515i
$844$		0			0
$845$	1.35618e7	−	8.89763e6i	0.653393	−	0.428679i
$846$		0			0
$847$			4.78917e7i			2.29378i
$848$		0			0
$849$	5.55742e7			2.64609
$850$		0			0
$851$	9.47341e6			0.448418
$852$		0			0
$853$		−	3.92510e7i		−	1.84705i	−0.383543	−	0.923523i	$-0.625296\pi$
$853$		−	3.92510e7i		−	1.84705i	0.383543	−	0.923523i	$-0.374704\pi$
$854$		0			0
$855$	−4.06937e7	+	2.66984e7i	−1.90376	+	1.24902i
$856$		0			0
$857$		−	1.29324e6i		−	0.0601488i	−0.999548	−	0.0300744i	$-0.990426\pi$
$857$		−	1.29324e6i		−	0.0601488i	0.999548	−	0.0300744i	$-0.00957443\pi$
$858$		0			0
$859$	−2.87256e7			−1.32827			−0.664134	−	0.747613i	$-0.731200\pi$
$859$	−2.87256e7			−1.32827			−0.664134	+	0.747613i	$0.731200\pi$
$860$		0			0
$861$	−6.56525e7			−3.01817
$862$		0			0
$863$		−	8.57964e6i		−	0.392141i	−0.980590	−	0.196070i	$-0.937182\pi$
$863$		−	8.57964e6i		−	0.392141i	0.980590	−	0.196070i	$-0.0628181\pi$
$864$		0			0
$865$	2.27714e7	+	3.47081e7i	1.03478	+	1.57722i
$866$		0			0
$867$			3.07197e7i			1.38793i
$868$		0			0
$869$	2.56989e7			1.15442
$870$		0			0
$871$	−7.44190e6			−0.332383
$872$		0			0
$873$		−	8.91875e6i		−	0.396067i
$874$		0			0
$875$	3.09641e7	−	5.35093e6i	1.36722	−	0.236270i
$876$		0			0
$877$			2.79556e7i			1.22735i	0.789558	+	0.613676i	$0.210310\pi$
$877$			2.79556e7i			1.22735i	−0.789558	+	0.613676i	$0.789690\pi$
$878$		0			0
$879$	−3.84739e7			−1.67955
$880$		0			0
$881$	−5.36409e6			−0.232839			−0.116420	−	0.993200i	$-0.537142\pi$
$881$	−5.36409e6			−0.232839			−0.116420	+	0.993200i	$0.537142\pi$
$882$		0			0
$883$			4.40226e7i			1.90009i	0.312117	+	0.950044i	$0.398962\pi$
$883$			4.40226e7i			1.90009i	−0.312117	+	0.950044i	$0.601038\pi$
$884$		0			0
$885$	9.35037e6	+	1.42518e7i	0.401302	+	0.611664i
$886$		0			0
$887$		−	1.88424e7i		−	0.804133i	−0.915610	−	0.402067i	$-0.868292\pi$
$887$		−	1.88424e7i		−	0.804133i	0.915610	−	0.402067i	$-0.131708\pi$
$888$		0			0
$889$	2.10017e7			0.891253
$890$		0			0
$891$	1.71387e7			0.723243
$892$		0			0
$893$		−	3.78359e7i		−	1.58773i
$894$		0			0
$895$	680348.	−	446364.i	0.0283905	−	0.0186265i
$896$		0			0
$897$			6.51913e6i			0.270526i
$898$		0			0
$899$	4.73113e6			0.195239
$900$		0			0
$901$	3.33952e6			0.137048
$902$		0			0
$903$		−	7.61969e6i		−	0.310970i
$904$		0			0
$905$	−1.13692e7	+	7.45911e6i	−0.461432	+	0.302737i
$906$		0			0
$907$		−	4.40217e7i		−	1.77684i	−0.459029	−	0.888421i	$-0.651803\pi$
$907$		−	4.40217e7i		−	1.77684i	0.459029	−	0.888421i	$-0.348197\pi$
$908$		0			0
$909$	−2.50753e6			−0.100655
$910$		0			0
$911$	−3.02447e7			−1.20741			−0.603703	−	0.797209i	$-0.706309\pi$
$911$	−3.02447e7			−1.20741			−0.603703	+	0.797209i	$0.706309\pi$
$912$		0			0
$913$			4.54601e7i			1.80490i
$914$		0			0
$915$	3.19017e7	+	4.86246e7i	1.25968	+	1.92001i
$916$		0			0
$917$			6.27967e7i			2.46612i
$918$		0			0
$919$	2.35173e7			0.918543			0.459272	−	0.888296i	$-0.348110\pi$
$919$	2.35173e7			0.918543			0.459272	+	0.888296i	$0.348110\pi$
$920$		0			0
$921$	−7.78895e6			−0.302573
$922$		0			0
$923$		−	1.31846e7i		−	0.509404i
$924$		0			0
$925$	2.86424e7	+	1.24325e7i	1.10066	+	0.477753i
$926$		0			0
$927$		−	1.51211e7i		−	0.577943i
$928$		0			0
$929$	−4.07876e7			−1.55056			−0.775281	−	0.631616i	$-0.782392\pi$
$929$	−4.07876e7			−1.55056			−0.775281	+	0.631616i	$0.782392\pi$
$930$		0			0
$931$	−3.98551e7			−1.50699
$932$		0			0
$933$		−	1.59362e7i		−	0.599350i
$934$		0			0
$935$	−7.69083e6	−	1.17224e7i	−0.287703	−	0.438517i
$936$		0			0
$937$			1.42613e7i			0.530654i	0.964159	+	0.265327i	$0.0854799\pi$
$937$			1.42613e7i			0.530654i	−0.964159	+	0.265327i	$0.914520\pi$
$938$		0			0
$939$	−2.78504e7			−1.03078
$940$		0			0
$941$	2.24934e7			0.828095			0.414048	−	0.910255i	$-0.364115\pi$
$941$	2.24934e7			0.828095			0.414048	+	0.910255i	$0.364115\pi$
$942$		0			0
$943$		−	1.43363e7i		−	0.524997i
$944$		0			0
$945$	1.96157e7	−	1.28695e7i	0.714535	−	0.468794i
$946$		0			0
$947$		−	2.45401e7i		−	0.889205i	−0.895728	−	0.444603i	$-0.853345\pi$
$947$		−	2.45401e7i		−	0.889205i	0.895728	−	0.444603i	$-0.146655\pi$
$948$		0			0
$949$	1.46138e7			0.526742
$950$		0			0
$951$	−2.90375e7			−1.04114
$952$		0			0
$953$		−	4.06466e7i		−	1.44975i	−0.688882	−	0.724873i	$-0.741898\pi$
$953$		−	4.06466e7i		−	1.44975i	0.688882	−	0.724873i	$-0.258102\pi$
$954$		0			0
$955$	3.54622e6	−	2.32661e6i	0.125822	−	0.0825497i
$956$		0			0
$957$		−	2.40509e7i		−	0.848891i
$958$		0			0
$959$	3.95324e7			1.38806
$960$		0			0
$961$	−1.89951e7			−0.663486
$962$		0			0
$963$		−	6.62089e7i		−	2.30065i
$964$		0			0
$965$	−1.13511e7	−	1.73013e7i	−0.392392	−	0.598083i
$966$		0			0
$967$		−	4.22263e7i		−	1.45217i	−0.687606	−	0.726084i	$-0.741338\pi$
$967$		−	4.22263e7i		−	1.45217i	0.687606	−	0.726084i	$-0.258662\pi$
$968$		0			0
$969$	−2.37390e7			−0.812183
$970$		0			0
$971$	4.00999e7			1.36488			0.682442	−	0.730940i	$-0.260918\pi$
$971$	4.00999e7			1.36488			0.682442	+	0.730940i	$0.260918\pi$
$972$		0			0
$973$		−	2.27712e7i		−	0.771087i
$974$		0			0
$975$	−8.55541e6	+	1.97103e7i	−0.288223	+	0.664019i
$976$		0			0
$977$			1.75634e7i			0.588671i	0.955702	+	0.294336i	$0.0950983\pi$
$977$			1.75634e7i			0.588671i	−0.955702	+	0.294336i	$0.904902\pi$
$978$		0			0
$979$	−8.55813e6			−0.285379
$980$		0			0
$981$	7.26127e7			2.40902
$982$		0			0
$983$		−	3.46862e6i		−	0.114491i	−0.998360	−	0.0572456i	$-0.981768\pi$
$983$		−	3.46862e6i		−	0.114491i	0.998360	−	0.0572456i	$-0.0182318\pi$
$984$		0			0
$985$	−6.55772e6	−	9.99527e6i	−0.215358	−	0.328249i
$986$		0			0
$987$			6.40896e7i			2.09409i
$988$		0			0
$989$	1.66388e6			0.0540918
$990$		0			0
$991$	−1.88098e7			−0.608414			−0.304207	−	0.952606i	$-0.598391\pi$
$991$	−1.88098e7			−0.608414			−0.304207	+	0.952606i	$0.598391\pi$
$992$		0			0
$993$			4.40341e7i			1.41715i
$994$		0			0
$995$	−2.17844e7	+	1.42924e7i	−0.697570	+	0.457663i
$996$		0			0
$997$		−	1.29454e7i		−	0.412455i	−0.978504	−	0.206227i	$-0.933881\pi$
$997$		−	1.29454e7i		−	0.412455i	0.978504	−	0.206227i	$-0.0661187\pi$
$998$		0			0
$999$	2.33121e7			0.739039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.6.c.a.9.7	yes	8
3.2	odd	2		360.6.f.b.289.2		8
4.3	odd	2		80.6.c.d.49.2		8
5.2	odd	4		200.6.a.j.1.4		4
5.3	odd	4		200.6.a.k.1.1		4
5.4	even	2	inner	40.6.c.a.9.2	✓	8
8.3	odd	2		320.6.c.i.129.7		8
8.5	even	2		320.6.c.j.129.2		8
12.11	even	2		720.6.f.n.289.2		8
15.14	odd	2		360.6.f.b.289.1		8
20.3	even	4		400.6.a.z.1.4		4
20.7	even	4		400.6.a.ba.1.1		4
20.19	odd	2		80.6.c.d.49.7		8
40.19	odd	2		320.6.c.i.129.2		8
40.29	even	2		320.6.c.j.129.7		8
60.59	even	2		720.6.f.n.289.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.c.a.9.2	✓	8	5.4	even	2	inner
40.6.c.a.9.7	yes	8	1.1	even	1	trivial
80.6.c.d.49.2		8	4.3	odd	2
80.6.c.d.49.7		8	20.19	odd	2
200.6.a.j.1.4		4	5.2	odd	4
200.6.a.k.1.1		4	5.3	odd	4
320.6.c.i.129.2		8	40.19	odd	2
320.6.c.i.129.7		8	8.3	odd	2
320.6.c.j.129.2		8	8.5	even	2
320.6.c.j.129.7		8	40.29	even	2
360.6.f.b.289.1		8	15.14	odd	2
360.6.f.b.289.2		8	3.2	odd	2
400.6.a.z.1.4		4	20.3	even	4
400.6.a.ba.1.1		4	20.7	even	4
720.6.f.n.289.1		8	60.59	even	2
720.6.f.n.289.2		8	12.11	even	2

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:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	40.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+24.1383i q^{3} +(46.7401 - 30.6653i) q^{5} +179.876i q^{7} -339.657 q^{9} +O(q^{10})\)	\(q+24.1383i q^{3} +(46.7401 - 30.6653i) q^{5} +179.876i q^{7} -339.657 q^{9} -653.681 q^{11} +284.851i q^{13} +(740.209 + 1128.23i) q^{15} +383.672i q^{17} +2563.29 q^{19} -4341.90 q^{21} -948.124i q^{23} +(1244.27 - 2866.60i) q^{25} -2333.13i q^{27} +1524.26 q^{29} +3103.88 q^{31} -15778.7i q^{33} +(5515.97 + 8407.43i) q^{35} +9991.75i q^{37} -6875.83 q^{39} +15120.7 q^{41} +1754.92i q^{43} +(-15875.6 + 10415.7i) q^{45} -14760.7i q^{47} -15548.5 q^{49} -9261.18 q^{51} -8704.12i q^{53} +(-30553.1 + 20045.4i) q^{55} +61873.3i q^{57} +12632.1 q^{59} +43098.3 q^{61} -61096.2i q^{63} +(8735.07 + 13314.0i) q^{65} +26125.5i q^{67} +22886.1 q^{69} -46285.8 q^{71} -51303.2i q^{73} +(69194.9 + 30034.6i) q^{75} -117582. i q^{77} -39314.0 q^{79} -26218.8 q^{81} -69544.8i q^{83} +(11765.4 + 17932.8i) q^{85} +36793.1i q^{87} +13092.2 q^{89} -51238.0 q^{91} +74922.5i q^{93} +(119808. - 78604.0i) q^{95} +26258.1i q^{97} +222027. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} - 1000 q^{9}+O(q^{10}) \) 8 * q + 8 * q^5 - 1000 * q^9	\( 8 q + 8 q^{5} - 1000 q^{9} - 736 q^{11} - 992 q^{15} + 1376 q^{19} + 1984 q^{21} - 2136 q^{25} + 5872 q^{29} + 4224 q^{31} + 19232 q^{35} - 3008 q^{39} + 23600 q^{41} - 28328 q^{45} - 45000 q^{49} - 124800 q^{51} + 15008 q^{55} + 91680 q^{59} + 123856 q^{61} - 72064 q^{65} - 76736 q^{69} - 125632 q^{71} + 222784 q^{75} + 43264 q^{79} + 409672 q^{81} - 293760 q^{85} - 41904 q^{89} - 487616 q^{91} + 442592 q^{95} + 266848 q^{99}+O(q^{100}) \) 8 * q + 8 * q^5 - 1000 * q^9 - 736 * q^11 - 992 * q^15 + 1376 * q^19 + 1984 * q^21 - 2136 * q^25 + 5872 * q^29 + 4224 * q^31 + 19232 * q^35 - 3008 * q^39 + 23600 * q^41 - 28328 * q^45 - 45000 * q^49 - 124800 * q^51 + 15008 * q^55 + 91680 * q^59 + 123856 * q^61 - 72064 * q^65 - 76736 * q^69 - 125632 * q^71 + 222784 * q^75 + 43264 * q^79 + 409672 * q^81 - 293760 * q^85 - 41904 * q^89 - 487616 * q^91 + 442592 * q^95 + 266848 * q^99

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