LMFDB - Embedded newform 684.5.h.e.37.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,5,Mod(37,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("684.37");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	684.h (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:	$70.7050547493$
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} + 588x^{6} + 3710486x^{4} - 3505922196x^{2} + 813472313329 $ x^8 + 588x^6 + 3710486x^4 - 3505922196*x^2 + 813472313329
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.7
Root			$-20.2540 - 2.77638i$ of defining polynomial
Character	$\chi$	$=$	684.37
Dual form			684.5.h.e.37.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+26.4233 q^{5} -31.3965 q^{7} +O(q^{10})$	$q+26.4233 q^{5} -31.3965 q^{7} +7.91544 q^{11} -252.076i q^{13} -375.164 q^{17} +(346.569 + 101.051i) q^{19} -472.942 q^{23} +73.1895 q^{25} +1166.35i q^{29} +1310.36i q^{31} -829.599 q^{35} -604.102i q^{37} -3161.64i q^{41} -1819.43 q^{43} +726.582 q^{47} -1415.26 q^{49} +5494.33i q^{53} +209.152 q^{55} +1995.30i q^{59} -3506.67 q^{61} -6660.68i q^{65} -4078.79i q^{67} -2670.09i q^{71} -4258.84 q^{73} -248.517 q^{77} +8218.57i q^{79} +11479.7 q^{83} -9913.07 q^{85} +3161.64i q^{89} +7914.31i q^{91} +(9157.48 + 2670.09i) q^{95} +15181.2i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 68 q^{7}+O(q^{10}) $ 8 * q - 68 * q^7	$ 8 q - 68 q^{7} + 1124 q^{19} + 36 q^{25} + 5044 q^{43} - 14436 q^{49} - 11332 q^{55} - 10652 q^{61} - 4580 q^{73} - 20140 q^{85}+O(q^{100}) $ 8 * q - 68 * q^7 + 1124 * q^19 + 36 * q^25 + 5044 * q^43 - 14436 * q^49 - 11332 * q^55 - 10652 * q^61 - 4580 * q^73 - 20140 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	26.4233			1.05693			0.528466	−	0.848955i	$-0.322768\pi$
$5$	26.4233			1.05693			0.528466	+	0.848955i	$0.322768\pi$
$6$		0			0
$7$	−31.3965			−0.640745			−0.320373	−	0.947292i	$-0.603808\pi$
$7$	−31.3965			−0.640745			−0.320373	+	0.947292i	$0.603808\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.91544			0.0654169			0.0327084	−	0.999465i	$-0.489587\pi$
$11$	7.91544			0.0654169			0.0327084	+	0.999465i	$0.489587\pi$
$12$		0			0
$13$		−	252.076i		−	1.49158i	−0.666184	−	0.745788i	$-0.732073\pi$
$13$		−	252.076i		−	1.49158i	0.666184	−	0.745788i	$-0.267927\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−375.164			−1.29815			−0.649073	−	0.760726i	$-0.724843\pi$
$17$	−375.164			−1.29815			−0.649073	+	0.760726i	$0.724843\pi$
$18$		0			0
$19$	346.569	+	101.051i	0.960024	+	0.279919i
$19$	346.569	+	101.051i	0.960024	+	0.279919i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−472.942			−0.894030			−0.447015	−	0.894526i	$-0.647513\pi$
$23$	−472.942			−0.894030			−0.447015	+	0.894526i	$0.647513\pi$
$24$		0			0
$25$	73.1895			0.117103
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1166.35i			1.38686i	0.720526	+	0.693428i	$0.243901\pi$
$29$			1166.35i			1.38686i	−0.720526	+	0.693428i	$0.756099\pi$
$30$		0			0
$31$			1310.36i			1.36353i	0.731569	+	0.681767i	$0.238788\pi$
$31$			1310.36i			1.36353i	−0.731569	+	0.681767i	$0.761212\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−829.599			−0.677223
$36$		0			0
$37$		−	604.102i		−	0.441273i	−0.975356	−	0.220636i	$-0.929187\pi$
$37$		−	604.102i		−	0.441273i	0.975356	−	0.220636i	$-0.0708134\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	3161.64i		−	1.88081i	−0.340057	−	0.940405i	$-0.610446\pi$
$41$		−	3161.64i		−	1.88081i	0.340057	−	0.940405i	$-0.389554\pi$
$42$		0			0
$43$	−1819.43			−0.984006			−0.492003	−	0.870594i	$-0.663735\pi$
$43$	−1819.43			−0.984006			−0.492003	+	0.870594i	$0.663735\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	726.582			0.328919			0.164460	−	0.986384i	$-0.447412\pi$
$47$	726.582			0.328919			0.164460	+	0.986384i	$0.447412\pi$
$48$		0			0
$49$	−1415.26			−0.589446
$50$		0			0
$51$		0			0
$52$		0			0
$53$			5494.33i			1.95597i	0.208664	+	0.977987i	$0.433089\pi$
$53$			5494.33i			1.95597i	−0.208664	+	0.977987i	$0.566911\pi$
$54$		0			0
$55$	209.152			0.0691411
$56$		0			0
$57$		0			0
$58$		0			0
$59$			1995.30i			0.573196i	0.958051	+	0.286598i	$0.0925244\pi$
$59$			1995.30i			0.573196i	−0.958051	+	0.286598i	$0.907476\pi$
$60$		0			0
$61$	−3506.67			−0.942399			−0.471200	−	0.882027i	$-0.656179\pi$
$61$	−3506.67			−0.942399			−0.471200	+	0.882027i	$0.656179\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	6660.68i		−	1.57649i
$66$		0			0
$67$		−	4078.79i		−	0.908619i	−0.890844	−	0.454310i	$-0.849886\pi$
$67$		−	4078.79i		−	0.908619i	0.890844	−	0.454310i	$-0.150114\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2670.09i		−	0.529675i	−0.964293	−	0.264837i	$-0.914682\pi$
$71$		−	2670.09i		−	0.529675i	0.964293	−	0.264837i	$-0.0853182\pi$
$72$		0			0
$73$	−4258.84			−0.799181			−0.399591	−	0.916694i	$-0.630848\pi$
$73$	−4258.84			−0.799181			−0.399591	+	0.916694i	$0.630848\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−248.517			−0.0419155
$78$		0			0
$79$			8218.57i			1.31687i	0.752639	+	0.658433i	$0.228781\pi$
$79$			8218.57i			1.31687i	−0.752639	+	0.658433i	$0.771219\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11479.7			1.66638			0.833191	−	0.552986i	$-0.186512\pi$
$83$	11479.7			1.66638			0.833191	+	0.552986i	$0.186512\pi$
$84$		0			0
$85$	−9913.07			−1.37205
$86$		0			0
$87$		0			0
$88$		0			0
$89$			3161.64i			0.399147i	0.979883	+	0.199573i	$0.0639556\pi$
$89$			3161.64i			0.399147i	−0.979883	+	0.199573i	$0.936044\pi$
$90$		0			0
$91$			7914.31i			0.955719i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9157.48	+	2670.09i	1.01468	+	0.295855i
$96$		0			0
$97$			15181.2i			1.61347i	0.590912	+	0.806736i	$0.298768\pi$
$97$			15181.2i			1.61347i	−0.590912	+	0.806736i	$0.701232\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9373.18			−0.918849			−0.459424	−	0.888217i	$-0.651944\pi$
$101$	−9373.18			−0.918849			−0.459424	+	0.888217i	$0.651944\pi$
$102$		0			0
$103$		−	13603.3i		−	1.28224i	−0.767440	−	0.641121i	$-0.778470\pi$
$103$		−	13603.3i		−	1.28224i	0.767440	−	0.641121i	$-0.221530\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	17986.7i		−	1.57103i	−0.618841	−	0.785516i	$-0.712398\pi$
$107$		−	17986.7i		−	1.57103i	0.618841	−	0.785516i	$-0.287602\pi$
$108$		0			0
$109$		−	2963.93i		−	0.249468i	−0.992190	−	0.124734i	$-0.960192\pi$
$109$		−	2963.93i		−	0.249468i	0.992190	−	0.124734i	$-0.0398078\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			15162.5i			1.18745i	0.804669	+	0.593723i	$0.202343\pi$
$113$			15162.5i			1.18745i	−0.804669	+	0.593723i	$0.797657\pi$
$114$		0			0
$115$	−12496.7			−0.944928
$116$		0			0
$117$		0			0
$118$		0			0
$119$	11778.9			0.831781
$120$		0			0
$121$	−14578.3			−0.995721
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−14580.6			−0.933161
$126$		0			0
$127$		−	19903.0i		−	1.23399i	−0.786967	−	0.616995i	$-0.788350\pi$
$127$		−	19903.0i		−	1.23399i	0.786967	−	0.616995i	$-0.211650\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−21308.4			−1.24167			−0.620837	−	0.783940i	$-0.713207\pi$
$131$	−21308.4			−1.24167			−0.620837	+	0.783940i	$0.713207\pi$
$132$		0			0
$133$	−10881.0	−	3172.64i	−0.615130	−	0.179357i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6091.45			0.324549			0.162274	−	0.986746i	$-0.448117\pi$
$137$	6091.45			0.324549			0.162274	+	0.986746i	$0.448117\pi$
$138$		0			0
$139$	−30957.9			−1.60229			−0.801147	−	0.598468i	$-0.795776\pi$
$139$	−30957.9			−1.60229			−0.801147	+	0.598468i	$0.795776\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	1995.30i		−	0.0975742i
$144$		0			0
$145$			30818.7i			1.46581i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−41071.7			−1.84999			−0.924997	−	0.379974i	$-0.875933\pi$
$149$	−41071.7			−1.84999			−0.924997	+	0.379974i	$0.875933\pi$
$150$		0			0
$151$			31122.9i			1.36498i	0.730894	+	0.682491i	$0.239103\pi$
$151$			31122.9i			1.36498i	−0.730894	+	0.682491i	$0.760897\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			34623.9i			1.44116i
$156$		0			0
$157$	−23889.7			−0.969195			−0.484597	−	0.874737i	$-0.661034\pi$
$157$	−23889.7			−0.969195			−0.484597	+	0.874737i	$0.661034\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	14848.7			0.572845
$162$		0			0
$163$	−33141.8			−1.24739			−0.623694	−	0.781669i	$-0.714369\pi$
$163$	−33141.8			−1.24739			−0.623694	+	0.781669i	$0.714369\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	13967.1i		−	0.500809i	−0.968141	−	0.250405i	$-0.919436\pi$
$167$		−	13967.1i		−	0.500809i	0.968141	−	0.250405i	$-0.0805636\pi$
$168$		0			0
$169$	−34981.4			−1.22480
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	29129.6i		−	0.973289i	−0.873600	−	0.486644i	$-0.838221\pi$
$173$		−	29129.6i		−	0.973289i	0.873600	−	0.486644i	$-0.161779\pi$
$174$		0			0
$175$	−2297.90			−0.0750333
$176$		0			0
$177$		0			0
$178$		0			0
$179$			31953.8i			0.997279i	0.866810	+	0.498639i	$0.166167\pi$
$179$			31953.8i			0.997279i	−0.866810	+	0.498639i	$0.833833\pi$
$180$		0			0
$181$			26124.8i			0.797435i	0.917074	+	0.398718i	$0.130545\pi$
$181$			26124.8i			0.797435i	−0.917074	+	0.398718i	$0.869455\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	15962.4i		−	0.466395i
$186$		0			0
$187$	−2969.59			−0.0849207
$188$		0			0
$189$		0			0
$190$		0			0
$191$	28794.4			0.789299			0.394650	−	0.918832i	$-0.370866\pi$
$191$	28794.4			0.789299			0.394650	+	0.918832i	$0.370866\pi$
$192$		0			0
$193$			26394.5i			0.708595i	0.935133	+	0.354298i	$0.115280\pi$
$193$			26394.5i			0.708595i	−0.935133	+	0.354298i	$0.884720\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	25023.7			0.644791			0.322396	−	0.946605i	$-0.395512\pi$
$197$	25023.7			0.644791			0.322396	+	0.946605i	$0.395512\pi$
$198$		0			0
$199$	−18965.2			−0.478906			−0.239453	−	0.970908i	$-0.576968\pi$
$199$	−18965.2			−0.478906			−0.239453	+	0.970908i	$0.576968\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	36619.2i		−	0.888621i
$204$		0			0
$205$		−	83540.9i		−	1.98789i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2743.24	+	799.861i	0.0628018	+	0.0183114i
$210$		0			0
$211$			77289.7i			1.73603i	0.496541	+	0.868013i	$0.334603\pi$
$211$			77289.7i			1.73603i	−0.496541	+	0.868013i	$0.665397\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−48075.2			−1.04003
$216$		0			0
$217$		−	41140.6i		−	0.873678i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			94570.0i			1.93628i
$222$		0			0
$223$			4319.86i			0.0868680i	0.999056	+	0.0434340i	$0.0138298\pi$
$223$			4319.86i			0.0868680i	−0.999056	+	0.0434340i	$0.986170\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	63320.1i		−	1.22882i	−0.788985	−	0.614412i	$-0.789393\pi$
$227$		−	63320.1i		−	1.22882i	0.788985	−	0.614412i	$-0.210607\pi$
$228$		0			0
$229$	−78778.6			−1.50223			−0.751116	−	0.660170i	$-0.770484\pi$
$229$	−78778.6			−1.50223			−0.751116	+	0.660170i	$0.770484\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	102208.			1.88266			0.941330	−	0.337488i	$-0.109577\pi$
$233$	102208.			1.88266			0.941330	+	0.337488i	$0.109577\pi$
$234$		0			0
$235$	19198.7			0.347645
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−37523.3			−0.656910			−0.328455	−	0.944520i	$-0.606528\pi$
$239$	−37523.3			−0.656910			−0.328455	+	0.944520i	$0.606528\pi$
$240$		0			0
$241$		−	29327.6i		−	0.504943i	−0.967604	−	0.252471i	$-0.918757\pi$
$241$		−	29327.6i		−	0.504943i	0.967604	−	0.252471i	$-0.0812434\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−37395.8			−0.623004
$246$		0			0
$247$	25472.5	−	87361.7i	0.417520	−	1.43195i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	77485.0			1.22990			0.614951	−	0.788566i	$-0.289176\pi$
$251$	77485.0			1.22990			0.614951	+	0.788566i	$0.289176\pi$
$252$		0			0
$253$	−3743.55			−0.0584847
$254$		0			0
$255$		0			0
$256$		0			0
$257$			19124.0i			0.289543i	0.989465	+	0.144771i	$0.0462446\pi$
$257$			19124.0i			0.289543i	−0.989465	+	0.144771i	$0.953755\pi$
$258$		0			0
$259$			18966.7i			0.282743i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	54940.8			0.794298			0.397149	−	0.917754i	$-0.370000\pi$
$263$	54940.8			0.794298			0.397149	+	0.917754i	$0.370000\pi$
$264$		0			0
$265$			145178.i			2.06733i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	122379.i		−	1.69123i	−0.533793	−	0.845615i	$-0.679234\pi$
$269$		−	122379.i		−	1.69123i	0.533793	−	0.845615i	$-0.320766\pi$
$270$		0			0
$271$	26810.6			0.365063			0.182531	−	0.983200i	$-0.441571\pi$
$271$	26810.6			0.365063			0.182531	+	0.983200i	$0.441571\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	579.327			0.00766053
$276$		0			0
$277$	−10201.0			−0.132949			−0.0664745	−	0.997788i	$-0.521175\pi$
$277$	−10201.0			−0.132949			−0.0664745	+	0.997788i	$0.521175\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	53864.3i		−	0.682163i	−0.940034	−	0.341081i	$-0.889207\pi$
$281$		−	53864.3i		−	0.682163i	0.940034	−	0.341081i	$-0.110793\pi$
$282$		0			0
$283$	104035.			1.29899			0.649493	−	0.760367i	$-0.274981\pi$
$283$	104035.			1.29899			0.649493	+	0.760367i	$0.274981\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			99264.5i			1.20512i
$288$		0			0
$289$	57227.3			0.685185
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	42701.1i		−	0.497397i	−0.968581	−	0.248699i	$-0.919997\pi$
$293$		−	42701.1i		−	0.497397i	0.968581	−	0.248699i	$-0.0800029\pi$
$294$		0			0
$295$			52722.2i			0.605829i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			119217.i			1.33351i
$300$		0			0
$301$	57123.6			0.630497
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−92657.7			−0.996051
$306$		0			0
$307$		−	75410.4i		−	0.800119i	−0.916489	−	0.400060i	$-0.868989\pi$
$307$		−	75410.4i		−	0.800119i	0.916489	−	0.400060i	$-0.131011\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−76788.2			−0.793914			−0.396957	−	0.917837i	$-0.629934\pi$
$311$	−76788.2			−0.793914			−0.396957	+	0.917837i	$0.629934\pi$
$312$		0			0
$313$	−113481.			−1.15834			−0.579170	−	0.815206i	$-0.696623\pi$
$313$	−113481.			−1.15834			−0.579170	+	0.815206i	$0.696623\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			150342.i			1.49611i	0.663638	+	0.748054i	$0.269011\pi$
$317$			150342.i			1.49611i	−0.663638	+	0.748054i	$0.730989\pi$
$318$		0			0
$319$			9232.15i			0.0907238i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−130020.	−	37910.6i	−1.24625	−	0.363376i
$324$		0			0
$325$		−	18449.3i		−	0.174668i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−22812.1			−0.210753
$330$		0			0
$331$		−	94854.9i		−	0.865772i	−0.901449	−	0.432886i	$-0.857495\pi$
$331$		−	94854.9i		−	0.865772i	0.901449	−	0.432886i	$-0.142505\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	107775.i		−	0.960348i
$336$		0			0
$337$		−	68777.2i		−	0.605598i	−0.953054	−	0.302799i	$-0.902079\pi$
$337$		−	68777.2i		−	0.605598i	0.953054	−	0.302799i	$-0.0979211\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			10372.0i			0.0891981i
$342$		0			0
$343$	119817.			1.01843
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10932.9			−0.0907979			−0.0453990	−	0.998969i	$-0.514456\pi$
$347$	−10932.9			−0.0907979			−0.0453990	+	0.998969i	$0.514456\pi$
$348$		0			0
$349$	25643.6			0.210537			0.105268	−	0.994444i	$-0.466430\pi$
$349$	25643.6			0.210537			0.105268	+	0.994444i	$0.466430\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	39119.2			0.313935			0.156968	−	0.987604i	$-0.449828\pi$
$353$	39119.2			0.313935			0.156968	+	0.987604i	$0.449828\pi$
$354$		0			0
$355$		−	70552.5i		−	0.559829i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−174197.			−1.35161			−0.675806	−	0.737079i	$-0.736204\pi$
$359$	−174197.			−1.35161			−0.675806	+	0.737079i	$0.736204\pi$
$360$		0			0
$361$	109899.	+	70042.0i	0.843291	+	0.537457i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−112532.			−0.844680
$366$		0			0
$367$	65235.3			0.484340			0.242170	−	0.970234i	$-0.422141\pi$
$367$	65235.3			0.484340			0.242170	+	0.970234i	$0.422141\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	172503.i		−	1.25328i
$372$		0			0
$373$			50922.5i			0.366009i	0.983112	+	0.183005i	$0.0585823\pi$
$373$			50922.5i			0.366009i	−0.983112	+	0.183005i	$0.941418\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	294008.			2.06860
$378$		0			0
$379$		−	144198.i		−	1.00388i	−0.864904	−	0.501938i	$-0.832621\pi$
$379$		−	144198.i		−	1.00388i	0.864904	−	0.501938i	$-0.167379\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			136588.i			0.931137i	0.885012	+	0.465569i	$0.154150\pi$
$383$			136588.i			0.931137i	−0.885012	+	0.465569i	$0.845850\pi$
$384$		0			0
$385$	−6566.64			−0.0443018
$386$		0			0
$387$		0			0
$388$		0			0
$389$	271749.			1.79584			0.897922	−	0.440155i	$-0.145077\pi$
$389$	271749.			1.79584			0.897922	+	0.440155i	$0.145077\pi$
$390$		0			0
$391$	177431.			1.16058
$392$		0			0
$393$		0			0
$394$		0			0
$395$			217161.i			1.39184i
$396$		0			0
$397$	78558.0			0.498436			0.249218	−	0.968447i	$-0.419826\pi$
$397$	78558.0			0.498436			0.249218	+	0.968447i	$0.419826\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	17978.0i		−	0.111803i	−0.998436	−	0.0559015i	$-0.982197\pi$
$401$		−	17978.0i		−	0.111803i	0.998436	−	0.0559015i	$-0.0178033\pi$
$402$		0			0
$403$	330310.			2.03381
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4781.74i		−	0.0288667i
$408$		0			0
$409$		−	244521.i		−	1.46174i	−0.682516	−	0.730870i	$-0.739114\pi$
$409$		−	244521.i		−	1.46174i	0.682516	−	0.730870i	$-0.260886\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	62645.3i		−	0.367272i
$414$		0			0
$415$	303331.			1.76125
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−51988.5			−0.296128			−0.148064	−	0.988978i	$-0.547304\pi$
$419$	−51988.5			−0.296128			−0.148064	+	0.988978i	$0.547304\pi$
$420$		0			0
$421$			243118.i			1.37168i	0.727753	+	0.685839i	$0.240565\pi$
$421$			243118.i			1.37168i	−0.727753	+	0.685839i	$0.759435\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−27458.1			−0.152017
$426$		0			0
$427$	110097.			0.603838
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	233857.i		−	1.25891i	−0.777036	−	0.629456i	$-0.783278\pi$
$431$		−	233857.i		−	1.25891i	0.777036	−	0.629456i	$-0.216722\pi$
$432$		0			0
$433$			22977.2i			0.122552i	0.998121	+	0.0612762i	$0.0195171\pi$
$433$			22977.2i			0.122552i	−0.998121	+	0.0612762i	$0.980483\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−163907.	−	47791.1i	−0.858290	−	0.250256i
$438$		0			0
$439$		−	65319.9i		−	0.338935i	−0.985536	−	0.169468i	$-0.945795\pi$
$439$		−	65319.9i		−	0.338935i	0.985536	−	0.169468i	$-0.0542048\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	150356.			0.766150			0.383075	−	0.923717i	$-0.374865\pi$
$443$	150356.			0.766150			0.383075	+	0.923717i	$0.374865\pi$
$444$		0			0
$445$			83540.9i			0.421871i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	227158.i		−	1.12677i	−0.826194	−	0.563386i	$-0.809498\pi$
$449$		−	227158.i		−	1.12677i	0.826194	−	0.563386i	$-0.190502\pi$
$450$		0			0
$451$		−	25025.8i		−	0.123037i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			209122.i			1.01013i
$456$		0			0
$457$	−206105.			−0.986863			−0.493431	−	0.869785i	$-0.664258\pi$
$457$	−206105.			−0.986863			−0.493431	+	0.869785i	$0.664258\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	111128.			0.522902			0.261451	−	0.965217i	$-0.415799\pi$
$461$	111128.			0.522902			0.261451	+	0.965217i	$0.415799\pi$
$462$		0			0
$463$	378581.			1.76602			0.883012	−	0.469350i	$-0.155512\pi$
$463$	378581.			1.76602			0.883012	+	0.469350i	$0.155512\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−335537.			−1.53853			−0.769266	−	0.638929i	$-0.779378\pi$
$467$	−335537.			−1.53853			−0.769266	+	0.638929i	$0.779378\pi$
$468$		0			0
$469$			128060.i			0.582193i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14401.6			−0.0643706
$474$		0			0
$475$	25365.2	+	7395.85i	0.112422	+	0.0327794i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	275390.			1.20026			0.600132	−	0.799901i	$-0.295115\pi$
$479$	275390.			1.20026			0.600132	+	0.799901i	$0.295115\pi$
$480$		0			0
$481$	−152280.			−0.658191
$482$		0			0
$483$		0			0
$484$		0			0
$485$			401136.i			1.70533i
$486$		0			0
$487$		−	204690.i		−	0.863056i	−0.902100	−	0.431528i	$-0.857975\pi$
$487$		−	204690.i		−	0.863056i	0.902100	−	0.431528i	$-0.142025\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	449100.			1.86286			0.931430	−	0.363919i	$-0.118562\pi$
$491$	449100.			1.86286			0.931430	+	0.363919i	$0.118562\pi$
$492$		0			0
$493$		−	437572.i		−	1.80034i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			83831.5i			0.339386i
$498$		0			0
$499$	225107.			0.904041			0.452021	−	0.892008i	$-0.350703\pi$
$499$	225107.			0.904041			0.452021	+	0.892008i	$0.350703\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	384096.			1.51811			0.759056	−	0.651025i	$-0.225661\pi$
$503$	384096.			1.51811			0.759056	+	0.651025i	$0.225661\pi$
$504$		0			0
$505$	−247670.			−0.971160
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	168771.i		−	0.651422i	−0.945469	−	0.325711i	$-0.894396\pi$
$509$		−	168771.i		−	0.651422i	0.945469	−	0.325711i	$-0.105604\pi$
$510$		0			0
$511$	133713.			0.512071
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	359444.i		−	1.35524i
$516$		0			0
$517$	5751.22			0.0215169
$518$		0			0
$519$		0			0
$520$		0			0
$521$			137579.i			0.506848i	0.967355	+	0.253424i	$0.0815568\pi$
$521$			137579.i			0.506848i	−0.967355	+	0.253424i	$0.918443\pi$
$522$		0			0
$523$		−	20831.0i		−	0.0761564i	−0.999275	−	0.0380782i	$-0.987876\pi$
$523$		−	20831.0i		−	0.0761564i	0.999275	−	0.0380782i	$-0.0121236\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	491599.i		−	1.77007i
$528$		0			0
$529$	−56166.8			−0.200710
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−796975.			−2.80537
$534$		0			0
$535$		−	475269.i		−	1.66047i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−11202.4			−0.0385597
$540$		0			0
$541$	−108590.			−0.371020			−0.185510	−	0.982642i	$-0.559394\pi$
$541$	−108590.			−0.371020			−0.185510	+	0.982642i	$0.559394\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	78316.8i		−	0.263671i
$546$		0			0
$547$			555311.i			1.85593i	0.372668	+	0.927965i	$0.378443\pi$
$547$			555311.i			1.85593i	−0.372668	+	0.927965i	$0.621557\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−117860.	+	404219.i	−0.388207	+	1.33141i
$552$		0			0
$553$		−	258034.i		−	0.843776i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	40941.2			0.131962			0.0659812	−	0.997821i	$-0.478982\pi$
$557$	40941.2			0.131962			0.0659812	+	0.997821i	$0.478982\pi$
$558$		0			0
$559$			458634.i			1.46772i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			611200.i			1.92826i	0.265421	+	0.964132i	$0.414489\pi$
$563$			611200.i			1.92826i	−0.265421	+	0.964132i	$0.585511\pi$
$564$		0			0
$565$			400643.i			1.25505i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	5965.51i		−	0.0184257i	−0.999958	−	0.00921283i	$-0.997067\pi$
$569$		−	5965.51i		−	0.0184257i	0.999958	−	0.00921283i	$-0.00293258\pi$
$570$		0			0
$571$	−107707.			−0.330349			−0.165174	−	0.986264i	$-0.552819\pi$
$571$	−107707.			−0.330349			−0.165174	+	0.986264i	$0.552819\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−34614.4			−0.104694
$576$		0			0
$577$	−116991.			−0.351401			−0.175700	−	0.984444i	$-0.556219\pi$
$577$	−116991.			−0.351401			−0.175700	+	0.984444i	$0.556219\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−360423.			−1.06773
$582$		0			0
$583$			43490.1i			0.127954i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−390111.			−1.13217			−0.566085	−	0.824347i	$-0.691543\pi$
$587$	−390111.			−1.13217			−0.566085	+	0.824347i	$0.691543\pi$
$588$		0			0
$589$	−132412.	+	454128.i	−0.381679	+	1.30902i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−20112.3			−0.0571943			−0.0285972	−	0.999591i	$-0.509104\pi$
$593$	−20112.3			−0.0571943			−0.0285972	+	0.999591i	$0.509104\pi$
$594$		0			0
$595$	311236.			0.879135
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	60237.0i		−	0.167884i	−0.996471	−	0.0839421i	$-0.973249\pi$
$599$		−	60237.0i		−	0.167884i	0.996471	−	0.0839421i	$-0.0267511\pi$
$600$		0			0
$601$		−	355921.i		−	0.985383i	−0.870204	−	0.492691i	$-0.836013\pi$
$601$		−	355921.i		−	0.985383i	0.870204	−	0.492691i	$-0.163987\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−385208.			−1.05241
$606$		0			0
$607$		−	57673.1i		−	0.156529i	−0.996933	−	0.0782647i	$-0.975062\pi$
$607$		−	57673.1i		−	0.156529i	0.996933	−	0.0782647i	$-0.0249379\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	183154.i		−	0.490608i
$612$		0			0
$613$	−387169.			−1.03034			−0.515169	−	0.857088i	$-0.672271\pi$
$613$	−387169.			−1.03034			−0.515169	+	0.857088i	$0.672271\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	377203.			0.990842			0.495421	−	0.868653i	$-0.335014\pi$
$617$	377203.			0.990842			0.495421	+	0.868653i	$0.335014\pi$
$618$		0			0
$619$	−216140.			−0.564096			−0.282048	−	0.959400i	$-0.591014\pi$
$619$	−216140.			−0.564096			−0.282048	+	0.959400i	$0.591014\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	99264.5i		−	0.255751i
$624$		0			0
$625$	−431012.			−1.10339
$626$		0			0
$627$		0			0
$628$		0			0
$629$			226638.i			0.572837i
$630$		0			0
$631$	318330.			0.799500			0.399750	−	0.916624i	$-0.369097\pi$
$631$	318330.			0.799500			0.399750	+	0.916624i	$0.369097\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	525903.i		−	1.30424i
$636$		0			0
$637$			356753.i			0.879203i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			377480.i			0.918709i	0.888253	+	0.459355i	$0.151919\pi$
$641$			377480.i			0.918709i	−0.888253	+	0.459355i	$0.848081\pi$
$642$		0			0
$643$	38260.5			0.0925398			0.0462699	−	0.998929i	$-0.485267\pi$
$643$	38260.5			0.0925398			0.0462699	+	0.998929i	$0.485267\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	402210.			0.960825			0.480412	−	0.877043i	$-0.340487\pi$
$647$	402210.			0.960825			0.480412	+	0.877043i	$0.340487\pi$
$648$		0			0
$649$			15793.6i			0.0374967i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	216444.			0.507596			0.253798	−	0.967257i	$-0.418320\pi$
$653$	216444.			0.507596			0.253798	+	0.967257i	$0.418320\pi$
$654$		0			0
$655$	−563036.			−1.31236
$656$		0			0
$657$		0			0
$658$		0			0
$659$			683147.i			1.57305i	0.617556	+	0.786527i	$0.288123\pi$
$659$			683147.i			1.57305i	−0.617556	+	0.786527i	$0.711877\pi$
$660$		0			0
$661$		−	558513.i		−	1.27829i	−0.769085	−	0.639147i	$-0.779288\pi$
$661$		−	558513.i		−	1.27829i	0.769085	−	0.639147i	$-0.220712\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−287513.	−	83831.5i	−0.650150	−	0.189567i
$666$		0			0
$667$		−	551614.i		−	1.23989i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−27756.8			−0.0616488
$672$		0			0
$673$			61317.1i			0.135379i	0.997706	+	0.0676895i	$0.0215627\pi$
$673$			61317.1i			0.135379i	−0.997706	+	0.0676895i	$0.978437\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	309675.i		−	0.675661i	−0.941207	−	0.337830i	$-0.890307\pi$
$677$		−	309675.i		−	0.675661i	0.941207	−	0.337830i	$-0.109693\pi$
$678$		0			0
$679$		−	476635.i		−	1.03382i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	586960.i		−	1.25825i	−0.777304	−	0.629125i	$-0.783413\pi$
$683$		−	586960.i		−	1.25825i	0.777304	−	0.629125i	$-0.216587\pi$
$684$		0			0
$685$	160956.			0.343026
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.38499e6			2.91748
$690$		0			0
$691$	−403564.			−0.845193			−0.422597	−	0.906318i	$-0.638881\pi$
$691$	−403564.			−0.845193			−0.422597	+	0.906318i	$0.638881\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−818009.			−1.69351
$696$		0			0
$697$			1.18614e6i			2.44157i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−300140.			−0.610785			−0.305392	−	0.952227i	$-0.598788\pi$
$701$	−300140.			−0.610785			−0.305392	+	0.952227i	$0.598788\pi$
$702$		0			0
$703$	61044.9	−	209363.i	0.123520	−	0.423632i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	294285.			0.588748
$708$		0			0
$709$	−673441.			−1.33970			−0.669850	−	0.742497i	$-0.733641\pi$
$709$	−673441.			−1.33970			−0.669850	+	0.742497i	$0.733641\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	619722.i		−	1.21904i
$714$		0			0
$715$		−	52722.2i		−	0.103129i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18595.8			0.0359713			0.0179857	−	0.999838i	$-0.494275\pi$
$719$	18595.8			0.0359713			0.0179857	+	0.999838i	$0.494275\pi$
$720$		0			0
$721$			427096.i			0.821590i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			85364.3i			0.162405i
$726$		0			0
$727$	−421735.			−0.797940			−0.398970	−	0.916964i	$-0.630632\pi$
$727$	−421735.			−0.797940			−0.398970	+	0.916964i	$0.630632\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	682584.			1.27738
$732$		0			0
$733$	−171245.			−0.318720			−0.159360	−	0.987221i	$-0.550943\pi$
$733$	−171245.			−0.318720			−0.159360	+	0.987221i	$0.550943\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	32285.4i		−	0.0594390i
$738$		0			0
$739$	−33928.6			−0.0621266			−0.0310633	−	0.999517i	$-0.509889\pi$
$739$	−33928.6			−0.0621266			−0.0310633	+	0.999517i	$0.509889\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			609356.i			1.10381i	0.833908	+	0.551904i	$0.186099\pi$
$743$			609356.i			1.10381i	−0.833908	+	0.551904i	$0.813901\pi$
$744$		0			0
$745$	−1.08525e6			−1.95532
$746$		0			0
$747$		0			0
$748$		0			0
$749$			564721.i			1.00663i
$750$		0			0
$751$			398449.i			0.706469i	0.935535	+	0.353234i	$0.114918\pi$
$751$			398449.i			0.706469i	−0.935535	+	0.353234i	$0.885082\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			822370.i			1.44269i
$756$		0			0
$757$	−379180.			−0.661689			−0.330844	−	0.943685i	$-0.607334\pi$
$757$	−379180.			−0.661689			−0.330844	+	0.943685i	$0.607334\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−275671.			−0.476017			−0.238008	−	0.971263i	$-0.576495\pi$
$761$	−275671.			−0.476017			−0.238008	+	0.971263i	$0.576495\pi$
$762$		0			0
$763$			93057.1i			0.159845i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	502966.			0.854965
$768$		0			0
$769$	−936342.			−1.58337			−0.791684	−	0.610931i	$-0.790795\pi$
$769$	−936342.			−1.58337			−0.791684	+	0.610931i	$0.790795\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	675931.i		−	1.13121i	−0.824676	−	0.565605i	$-0.808643\pi$
$773$		−	675931.i		−	1.13121i	0.824676	−	0.565605i	$-0.191357\pi$
$774$		0			0
$775$			95904.3i			0.159674i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	319486.	−	1.09573e6i	0.526474	−	1.80562i
$780$		0			0
$781$		−	21134.9i		−	0.0346497i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−631244.			−1.02437
$786$		0			0
$787$			1.20498e6i			1.94549i	0.231880	+	0.972744i	$0.425512\pi$
$787$			1.20498e6i			1.94549i	−0.231880	+	0.972744i	$0.574488\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	476050.i		−	0.760850i
$792$		0			0
$793$			883948.i			1.40566i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			381506.i			0.600599i	0.953845	+	0.300299i	$0.0970866\pi$
$797$			381506.i			0.600599i	−0.953845	+	0.300299i	$0.902913\pi$
$798$		0			0
$799$	−272588.			−0.426985
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−33710.6			−0.0522800
$804$		0			0
$805$	392352.			0.605458
$806$		0			0
$807$		0			0
$808$		0			0
$809$	637181.			0.973567			0.486783	−	0.873523i	$-0.338170\pi$
$809$	637181.			0.973567			0.486783	+	0.873523i	$0.338170\pi$
$810$		0			0
$811$		−	803565.i		−	1.22174i	−0.791730	−	0.610871i	$-0.790819\pi$
$811$		−	803565.i		−	1.22174i	0.791730	−	0.610871i	$-0.209181\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−875716.			−1.31840
$816$		0			0
$817$	−630556.	−	183854.i	−0.944669	−	0.275442i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	339737.			0.504030			0.252015	−	0.967723i	$-0.418907\pi$
$821$	339737.			0.504030			0.252015	+	0.967723i	$0.418907\pi$
$822$		0			0
$823$	591981.			0.873993			0.436996	−	0.899463i	$-0.356042\pi$
$823$	591981.			0.873993			0.436996	+	0.899463i	$0.356042\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			421263.i			0.615946i	0.951395	+	0.307973i	$0.0996506\pi$
$827$			421263.i			0.615946i	−0.951395	+	0.307973i	$0.900349\pi$
$828$		0			0
$829$			157114.i			0.228615i	0.993445	+	0.114308i	$0.0364649\pi$
$829$			157114.i			0.228615i	−0.993445	+	0.114308i	$0.963535\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	530955.			0.765187
$834$		0			0
$835$		−	369056.i		−	0.529321i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	565047.i		−	0.802713i	−0.915922	−	0.401357i	$-0.868539\pi$
$839$		−	565047.i		−	0.802713i	0.915922	−	0.401357i	$-0.131461\pi$
$840$		0			0
$841$	−653082.			−0.923370
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−924324.			−1.29453
$846$		0			0
$847$	457709.			0.638003
$848$		0			0
$849$		0			0
$850$		0			0
$851$			285705.i			0.394511i
$852$		0			0
$853$	−1.07874e6			−1.48259			−0.741294	−	0.671180i	$-0.765788\pi$
$853$	−1.07874e6			−1.48259			−0.741294	+	0.671180i	$0.765788\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	222673.i		−	0.303184i	−0.988443	−	0.151592i	$-0.951560\pi$
$857$		−	222673.i		−	0.303184i	0.988443	−	0.151592i	$-0.0484400\pi$
$858$		0			0
$859$	409435.			0.554880			0.277440	−	0.960743i	$-0.410514\pi$
$859$	409435.			0.554880			0.277440	+	0.960743i	$0.410514\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			770405.i			1.03442i	0.855858	+	0.517211i	$0.173030\pi$
$863$			770405.i			1.03442i	−0.855858	+	0.517211i	$0.826970\pi$
$864$		0			0
$865$		−	769699.i		−	1.02870i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			65053.6i			0.0861453i
$870$		0			0
$871$	−1.02817e6			−1.35527
$872$		0			0
$873$		0			0
$874$		0			0
$875$	457781.			0.597918
$876$		0			0
$877$		−	452418.i		−	0.588221i	−0.955771	−	0.294110i	$-0.904977\pi$
$877$		−	452418.i		−	0.588221i	0.955771	−	0.294110i	$-0.0950233\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−246744.			−0.317903			−0.158952	−	0.987286i	$-0.550811\pi$
$881$	−246744.			−0.317903			−0.158952	+	0.987286i	$0.550811\pi$
$882$		0			0
$883$	853177.			1.09425			0.547127	−	0.837050i	$-0.315722\pi$
$883$	853177.			1.09425			0.547127	+	0.837050i	$0.315722\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			34676.3i			0.0440743i	0.999757	+	0.0220372i	$0.00701521\pi$
$887$			34676.3i			0.0440743i	−0.999757	+	0.0220372i	$0.992985\pi$
$888$		0			0
$889$			624885.i			0.790672i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	251811.	+	73421.6i	0.315770	+	0.0920706i
$894$		0			0
$895$			844324.i			1.05405i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.52833e6			−1.89103
$900$		0			0
$901$		−	2.06128e6i		−	2.53914i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			690302.i			0.842834i
$906$		0			0
$907$		−	663156.i		−	0.806123i	−0.915173	−	0.403061i	$-0.867946\pi$
$907$		−	663156.i		−	0.806123i	0.915173	−	0.403061i	$-0.132054\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	51912.5i		−	0.0625512i	−0.999511	−	0.0312756i	$-0.990043\pi$
$911$		−	51912.5i		−	0.0625512i	0.999511	−	0.0312756i	$-0.00995695\pi$
$912$		0			0
$913$	90866.9			0.109009
$914$		0			0
$915$		0			0
$916$		0			0
$917$	669008.			0.795596
$918$		0			0
$919$	569306.			0.674085			0.337042	−	0.941489i	$-0.390573\pi$
$919$	569306.			0.674085			0.337042	+	0.941489i	$0.390573\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−673066.			−0.790049
$924$		0			0
$925$		−	44214.0i		−	0.0516745i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.11325e6			−1.28991			−0.644957	−	0.764218i	$-0.723125\pi$
$929$	−1.11325e6			−1.28991			−0.644957	+	0.764218i	$0.723125\pi$
$930$		0			0
$931$	−490484.	−	143013.i	−0.565882	−	0.164997i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−78466.4			−0.0897553
$936$		0			0
$937$	999850.			1.13882			0.569411	−	0.822053i	$-0.307171\pi$
$937$	999850.			1.13882			0.569411	+	0.822053i	$0.307171\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	879829.i		−	0.993617i	−0.867860	−	0.496809i	$-0.834505\pi$
$941$		−	879829.i		−	0.993617i	0.867860	−	0.496809i	$-0.165495\pi$
$942$		0			0
$943$			1.49527e6i			1.68150i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−383733.			−0.427888			−0.213944	−	0.976846i	$-0.568631\pi$
$947$	−383733.			−0.427888			−0.213944	+	0.976846i	$0.568631\pi$
$948$		0			0
$949$			1.07355e6i			1.19204i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			696035.i			0.766382i	0.923669	+	0.383191i	$0.125175\pi$
$953$			696035.i			0.766382i	−0.923669	+	0.383191i	$0.874825\pi$
$954$		0			0
$955$	760843.			0.834235
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−191250.			−0.207953
$960$		0			0
$961$	−793512.			−0.859224
$962$		0			0
$963$		0			0
$964$		0			0
$965$			697428.i			0.748937i
$966$		0			0
$967$	−313275.			−0.335021			−0.167511	−	0.985870i	$-0.553573\pi$
$967$	−313275.			−0.335021			−0.167511	+	0.985870i	$0.553573\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	328846.i		−	0.348782i	−0.984677	−	0.174391i	$-0.944204\pi$
$971$		−	328846.i		−	0.348782i	0.984677	−	0.174391i	$-0.0557956\pi$
$972$		0			0
$973$	971970.			1.02666
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.08690e6i		−	1.13867i	−0.822104	−	0.569337i	$-0.807200\pi$
$977$		−	1.08690e6i		−	1.13867i	0.822104	−	0.569337i	$-0.192800\pi$
$978$		0			0
$979$			25025.8i			0.0261109i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1.27016e6i		−	1.31447i	−0.753685	−	0.657236i	$-0.771726\pi$
$983$		−	1.27016e6i		−	1.31447i	0.753685	−	0.657236i	$-0.228274\pi$
$984$		0			0
$985$	661208.			0.681500
$986$		0			0
$987$		0			0
$988$		0			0
$989$	860483.			0.879731
$990$		0			0
$991$			1.01645e6i			1.03500i	0.855683	+	0.517500i	$0.173137\pi$
$991$			1.01645e6i			1.03500i	−0.855683	+	0.517500i	$0.826863\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−501121.			−0.506170
$996$		0			0
$997$	370598.			0.372832			0.186416	−	0.982471i	$-0.440313\pi$
$997$	370598.			0.372832			0.186416	+	0.982471i	$0.440313\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.5.h.e.37.7	yes	8
3.2	odd	2	inner	684.5.h.e.37.1	✓	8
19.18	odd	2	inner	684.5.h.e.37.8	yes	8
57.56	even	2	inner	684.5.h.e.37.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.5.h.e.37.1	✓	8	3.2	odd	2	inner
684.5.h.e.37.2	yes	8	57.56	even	2	inner
684.5.h.e.37.7	yes	8	1.1	even	1	trivial
684.5.h.e.37.8	yes	8	19.18	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+26.4233 q^{5} -31.3965 q^{7} +O(q^{10})\)	\(q+26.4233 q^{5} -31.3965 q^{7} +7.91544 q^{11} -252.076i q^{13} -375.164 q^{17} +(346.569 + 101.051i) q^{19} -472.942 q^{23} +73.1895 q^{25} +1166.35i q^{29} +1310.36i q^{31} -829.599 q^{35} -604.102i q^{37} -3161.64i q^{41} -1819.43 q^{43} +726.582 q^{47} -1415.26 q^{49} +5494.33i q^{53} +209.152 q^{55} +1995.30i q^{59} -3506.67 q^{61} -6660.68i q^{65} -4078.79i q^{67} -2670.09i q^{71} -4258.84 q^{73} -248.517 q^{77} +8218.57i q^{79} +11479.7 q^{83} -9913.07 q^{85} +3161.64i q^{89} +7914.31i q^{91} +(9157.48 + 2670.09i) q^{95} +15181.2i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 68 q^{7}+O(q^{10}) \) 8 * q - 68 * q^7	\( 8 q - 68 q^{7} + 1124 q^{19} + 36 q^{25} + 5044 q^{43} - 14436 q^{49} - 11332 q^{55} - 10652 q^{61} - 4580 q^{73} - 20140 q^{85}+O(q^{100}) \) 8 * q - 68 * q^7 + 1124 * q^19 + 36 * q^25 + 5044 * q^43 - 14436 * q^49 - 11332 * q^55 - 10652 * q^61 - 4580 * q^73 - 20140 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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