LMFDB - Embedded newform 88.3.h.a.65.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [88,3,Mod(65,88)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(88, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("88.65"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 88 = 2^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	88.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$2.39782632637$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1750426112.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 21x^{4} + 4x^{3} + 228x^{2} + 368x + 548 $ x^6 - 2x^5 - 21x^4 + 4x^3 + 228x^2 + 368*x + 548
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.2
Root			$4.50331 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	88.65
Dual form			88.3.h.a.65.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.88824 q^{3} +5.11838 q^{5} +12.7373i q^{7} +6.11838 q^{9} +(10.0066 + 4.56811i) q^{11} +14.9148i q^{13} -19.9015 q^{15} +1.42357i q^{17} -29.0756i q^{19} -49.5255i q^{21} -11.8882 q^{23} +1.19781 q^{25} +11.2044 q^{27} +16.3383i q^{29} -16.8088 q^{31} +(-38.9081 - 17.7619i) q^{33} +65.1942i q^{35} +56.2507 q^{37} -57.9921i q^{39} +16.8489i q^{41} -36.1186i q^{43} +31.3162 q^{45} +21.8544 q^{47} -113.238 q^{49} -5.53517i q^{51} -0.185315 q^{53} +(51.2177 + 23.3813i) q^{55} +113.053i q^{57} +17.9264 q^{59} -77.0185i q^{61} +77.9315i q^{63} +76.3394i q^{65} -10.9412 q^{67} +46.2243 q^{69} +139.271 q^{71} -82.2022i q^{73} -4.65735 q^{75} +(-58.1853 + 127.457i) q^{77} +30.8506i q^{79} -98.6308 q^{81} -74.1713i q^{83} +7.28637i q^{85} -63.5273i q^{87} +9.98603 q^{89} -189.973 q^{91} +65.3566 q^{93} -148.820i q^{95} +82.3831 q^{97} +(61.2243 + 27.9494i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 6 q^{9} + 10 q^{11} - 52 q^{23} + 22 q^{25} + 32 q^{27} - 36 q^{31} - 64 q^{33} - 48 q^{37} + 172 q^{45} - 60 q^{47} - 170 q^{49} + 108 q^{53} + 172 q^{55} + 236 q^{59} - 292 q^{67} + 92 q^{69}+ \cdots + 182 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + 6 * q^9 + 10 * q^11 - 52 * q^23 + 22 * q^25 + 32 * q^27 - 36 * q^31 - 64 * q^33 - 48 * q^37 + 172 * q^45 - 60 * q^47 - 170 * q^49 + 108 * q^53 + 172 * q^55 + 236 * q^59 - 292 * q^67 + 92 * q^69 + 300 * q^71 - 244 * q^75 - 240 * q^77 - 362 * q^81 + 384 * q^89 - 48 * q^91 - 148 * q^93 + 400 * q^97 + 182 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/88\mathbb{Z}\right)^\times$.

$n$	$23$	$45$	$57$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−3.88824			−1.29608			−0.648039	−	0.761607i	$-0.724411\pi$
$3$	−3.88824			−1.29608			−0.648039	+	0.761607i	$0.724411\pi$
$4$		0			0
$5$	5.11838			1.02368			0.511838	−	0.859082i	$-0.328965\pi$
$5$	5.11838			1.02368			0.511838	+	0.859082i	$0.328965\pi$
$6$		0			0
$7$			12.7373i			1.81961i	0.415035	+	0.909806i	$0.363769\pi$
$7$			12.7373i			1.81961i	−0.415035	+	0.909806i	$0.636231\pi$
$8$		0			0
$9$	6.11838			0.679820
$10$		0			0
$11$	10.0066	+	4.56811i	0.909692	+	0.415283i
$11$	10.0066	+	4.56811i	0.909692	+	0.415283i
$12$		0			0
$13$			14.9148i			1.14729i	0.819104	+	0.573645i	$0.194471\pi$
$13$			14.9148i			1.14729i	−0.819104	+	0.573645i	$0.805529\pi$
$14$		0			0
$15$	−19.9015			−1.32676
$16$		0			0
$17$			1.42357i			0.0837394i	0.999123	+	0.0418697i	$0.0133314\pi$
$17$			1.42357i			0.0837394i	−0.999123	+	0.0418697i	$0.986669\pi$
$18$		0			0
$19$		−	29.0756i		−	1.53030i	−0.643855	−	0.765148i	$-0.722666\pi$
$19$		−	29.0756i		−	1.53030i	0.643855	−	0.765148i	$-0.277334\pi$
$20$		0			0
$21$		−	49.5255i		−	2.35836i
$22$		0			0
$23$	−11.8882			−0.516880			−0.258440	−	0.966027i	$-0.583208\pi$
$23$	−11.8882			−0.516880			−0.258440	+	0.966027i	$0.583208\pi$
$24$		0			0
$25$	1.19781			0.0479123
$26$		0			0
$27$	11.2044			0.414979
$28$		0			0
$29$			16.3383i			0.563391i	0.959504	+	0.281695i	$0.0908967\pi$
$29$			16.3383i			0.563391i	−0.959504	+	0.281695i	$0.909103\pi$
$30$		0			0
$31$	−16.8088			−0.542220			−0.271110	−	0.962548i	$-0.587391\pi$
$31$	−16.8088			−0.542220			−0.271110	+	0.962548i	$0.587391\pi$
$32$		0			0
$33$	−38.9081	−	17.7619i	−1.17903	−	0.538239i
$34$		0			0
$35$			65.1942i			1.86269i
$36$		0			0
$37$	56.2507			1.52029			0.760145	−	0.649754i	$-0.225128\pi$
$37$	56.2507			1.52029			0.760145	+	0.649754i	$0.225128\pi$
$38$		0			0
$39$		−	57.9921i		−	1.48698i
$40$		0			0
$41$			16.8489i			0.410948i	0.978663	+	0.205474i	$0.0658736\pi$
$41$			16.8489i			0.410948i	−0.978663	+	0.205474i	$0.934126\pi$
$42$		0			0
$43$		−	36.1186i		−	0.839968i	−0.907532	−	0.419984i	$-0.862036\pi$
$43$		−	36.1186i		−	0.839968i	0.907532	−	0.419984i	$-0.137964\pi$
$44$		0			0
$45$	31.3162			0.695915
$46$		0			0
$47$	21.8544			0.464987			0.232493	−	0.972598i	$-0.425312\pi$
$47$	21.8544			0.464987			0.232493	+	0.972598i	$0.425312\pi$
$48$		0			0
$49$	−113.238			−2.31098
$50$		0			0
$51$		−	5.53517i		−	0.108533i
$52$		0			0
$53$	−0.185315			−0.00349651			−0.00174826	−	0.999998i	$-0.500556\pi$
$53$	−0.185315			−0.00349651			−0.00174826	+	0.999998i	$0.500556\pi$
$54$		0			0
$55$	51.2177	+	23.3813i	0.931230	+	0.425115i
$56$		0			0
$57$			113.053i			1.98338i
$58$		0			0
$59$	17.9264			0.303838			0.151919	−	0.988393i	$-0.451455\pi$
$59$	17.9264			0.303838			0.151919	+	0.988393i	$0.451455\pi$
$60$		0			0
$61$		−	77.0185i		−	1.26260i	−0.775540	−	0.631299i	$-0.782522\pi$
$61$		−	77.0185i		−	1.26260i	0.775540	−	0.631299i	$-0.217478\pi$
$62$		0			0
$63$			77.9315i			1.23701i
$64$		0			0
$65$			76.3394i			1.17445i
$66$		0			0
$67$	−10.9412			−0.163302			−0.0816508	−	0.996661i	$-0.526019\pi$
$67$	−10.9412			−0.163302			−0.0816508	+	0.996661i	$0.526019\pi$
$68$		0			0
$69$	46.2243			0.669917
$70$		0			0
$71$	139.271			1.96156			0.980779	−	0.195123i	$-0.0625107\pi$
$71$	139.271			1.96156			0.980779	+	0.195123i	$0.0625107\pi$
$72$		0			0
$73$		−	82.2022i		−	1.12606i	−0.826437	−	0.563029i	$-0.809636\pi$
$73$		−	82.2022i		−	1.12606i	0.826437	−	0.563029i	$-0.190364\pi$
$74$		0			0
$75$	−4.65735			−0.0620980
$76$		0			0
$77$	−58.1853	+	127.457i	−0.755653	+	1.65529i
$78$		0			0
$79$			30.8506i			0.390514i	0.980752	+	0.195257i	$0.0625541\pi$
$79$			30.8506i			0.390514i	−0.980752	+	0.195257i	$0.937446\pi$
$80$		0			0
$81$	−98.6308			−1.21766
$82$		0			0
$83$		−	74.1713i		−	0.893631i	−0.894626	−	0.446815i	$-0.852558\pi$
$83$		−	74.1713i		−	0.893631i	0.894626	−	0.446815i	$-0.147442\pi$
$84$		0			0
$85$			7.28637i			0.0857220i
$86$		0			0
$87$		−	63.5273i		−	0.730199i
$88$		0			0
$89$	9.98603			0.112203			0.0561013	−	0.998425i	$-0.482133\pi$
$89$	9.98603			0.112203			0.0561013	+	0.998425i	$0.482133\pi$
$90$		0			0
$91$	−189.973			−2.08762
$92$		0			0
$93$	65.3566			0.702759
$94$		0			0
$95$		−	148.820i		−	1.56653i
$96$		0			0
$97$	82.3831			0.849311			0.424655	−	0.905355i	$-0.360395\pi$
$97$	82.3831			0.849311			0.424655	+	0.905355i	$0.360395\pi$
$98$		0			0
$99$	61.2243	+	27.9494i	0.618427	+	0.282318i
$100$		0			0
$101$		−	128.202i		−	1.26932i	−0.772791	−	0.634661i	$-0.781140\pi$
$101$		−	128.202i		−	1.26932i	0.772791	−	0.634661i	$-0.218860\pi$
$102$		0			0
$103$	−48.0662			−0.466662			−0.233331	−	0.972397i	$-0.574963\pi$
$103$	−48.0662			−0.466662			−0.233331	+	0.972397i	$0.574963\pi$
$104$		0			0
$105$		−	253.491i		−	2.41420i
$106$		0			0
$107$			201.703i			1.88508i	0.334098	+	0.942538i	$0.391568\pi$
$107$			201.703i			1.88508i	−0.334098	+	0.942538i	$0.608432\pi$
$108$		0			0
$109$		−	164.512i		−	1.50929i	−0.656134	−	0.754644i	$-0.727809\pi$
$109$		−	164.512i		−	1.50929i	0.656134	−	0.754644i	$-0.272191\pi$
$110$		0			0
$111$	−218.716			−1.97042
$112$		0			0
$113$	85.1713			0.753728			0.376864	−	0.926269i	$-0.377002\pi$
$113$	85.1713			0.753728			0.376864	+	0.926269i	$0.377002\pi$
$114$		0			0
$115$	−60.8485			−0.529117
$116$		0			0
$117$			91.2542i			0.779950i
$118$		0			0
$119$	−18.1324			−0.152373
$120$		0			0
$121$	79.2647	+	91.4227i	0.655080	+	0.755559i
$122$		0			0
$123$		−	65.5124i		−	0.532621i
$124$		0			0
$125$	−121.829			−0.974629
$126$		0			0
$127$			62.1786i			0.489595i	0.969574	+	0.244797i	$0.0787215\pi$
$127$			62.1786i			0.489595i	−0.969574	+	0.244797i	$0.921279\pi$
$128$		0			0
$129$			140.438i			1.08866i
$130$		0			0
$131$		−	65.6205i		−	0.500920i	−0.968127	−	0.250460i	$-0.919418\pi$
$131$		−	65.6205i		−	0.500920i	0.968127	−	0.250460i	$-0.0805818\pi$
$132$		0			0
$133$	370.344			2.78454
$134$		0			0
$135$	57.3485			0.424804
$136$		0			0
$137$	−123.279			−0.899845			−0.449922	−	0.893068i	$-0.648548\pi$
$137$	−123.279			−0.899845			−0.449922	+	0.893068i	$0.648548\pi$
$138$		0			0
$139$			76.6909i			0.551733i	0.961196	+	0.275866i	$0.0889647\pi$
$139$			76.6909i			0.551733i	−0.961196	+	0.275866i	$0.911035\pi$
$140$		0			0
$141$	−84.9750			−0.602659
$142$		0			0
$143$	−68.1323	+	149.246i	−0.476450	+	1.04368i
$144$		0			0
$145$			83.6258i			0.576729i
$146$		0			0
$147$	440.297			2.99522
$148$		0			0
$149$			203.286i			1.36433i	0.731196	+	0.682167i	$0.238962\pi$
$149$			203.286i			1.36433i	−0.731196	+	0.682167i	$0.761038\pi$
$150$		0			0
$151$		−	109.100i		−	0.722519i	−0.932465	−	0.361259i	$-0.882347\pi$
$151$		−	109.100i		−	0.722519i	0.932465	−	0.361259i	$-0.117653\pi$
$152$		0			0
$153$			8.70993i			0.0569277i
$154$		0			0
$155$	−86.0339			−0.555057
$156$		0			0
$157$	−72.5435			−0.462060			−0.231030	−	0.972947i	$-0.574210\pi$
$157$	−72.5435			−0.462060			−0.231030	+	0.972947i	$0.574210\pi$
$158$		0			0
$159$	0.720549			0.00453175
$160$		0			0
$161$		−	151.424i		−	0.940520i
$162$		0			0
$163$	−144.357			−0.885628			−0.442814	−	0.896613i	$-0.646020\pi$
$163$	−144.357			−0.885628			−0.442814	+	0.896613i	$0.646020\pi$
$164$		0			0
$165$	−199.146	−	90.9121i	−1.20695	−	0.550983i
$166$		0			0
$167$			120.985i			0.724461i	0.932089	+	0.362231i	$0.117985\pi$
$167$			120.985i			0.724461i	−0.932089	+	0.362231i	$0.882015\pi$
$168$		0			0
$169$	−53.4501			−0.316273
$170$		0			0
$171$		−	177.896i		−	1.04033i
$172$		0			0
$173$		−	18.3329i		−	0.105970i	−0.998595	−	0.0529852i	$-0.983126\pi$
$173$		−	18.3329i		−	0.105970i	0.998595	−	0.0529852i	$-0.0168736\pi$
$174$		0			0
$175$			15.2568i			0.0871817i
$176$		0			0
$177$	−69.7023			−0.393798
$178$		0			0
$179$	257.827			1.44037			0.720186	−	0.693781i	$-0.244057\pi$
$179$	257.827			1.44037			0.720186	+	0.693781i	$0.244057\pi$
$180$		0			0
$181$	83.1244			0.459251			0.229625	−	0.973279i	$-0.426250\pi$
$181$	83.1244			0.459251			0.229625	+	0.973279i	$0.426250\pi$
$182$		0			0
$183$			299.466i			1.63643i
$184$		0			0
$185$	287.913			1.55628
$186$		0			0
$187$	−6.50302	+	14.2451i	−0.0347755	+	0.0761770i
$188$		0			0
$189$			142.714i			0.755100i
$190$		0			0
$191$	158.747			0.831137			0.415568	−	0.909562i	$-0.363583\pi$
$191$	158.747			0.831137			0.415568	+	0.909562i	$0.363583\pi$
$192$		0			0
$193$			212.675i			1.10194i	0.834524	+	0.550971i	$0.185743\pi$
$193$			212.675i			1.10194i	−0.834524	+	0.550971i	$0.814257\pi$
$194$		0			0
$195$		−	296.826i		−	1.52218i
$196$		0			0
$197$			36.6530i			0.186056i	0.995664	+	0.0930279i	$0.0296546\pi$
$197$			36.6530i			0.186056i	−0.995664	+	0.0930279i	$0.970345\pi$
$198$		0			0
$199$	−141.828			−0.712703			−0.356352	−	0.934352i	$-0.615979\pi$
$199$	−141.828			−0.712703			−0.356352	+	0.934352i	$0.615979\pi$
$200$		0			0
$201$	42.5420			0.211652
$202$		0			0
$203$	−208.106			−1.02515
$204$		0			0
$205$			86.2390i			0.420678i
$206$		0			0
$207$	−72.7367			−0.351385
$208$		0			0
$209$	132.821	−	290.948i	0.635505	−	1.39210i
$210$		0			0
$211$			28.0451i			0.132915i	0.997789	+	0.0664576i	$0.0211697\pi$
$211$			28.0451i			0.132915i	−0.997789	+	0.0664576i	$0.978830\pi$
$212$		0			0
$213$	−541.517			−2.54233
$214$		0			0
$215$		−	184.869i		−	0.859855i
$216$		0			0
$217$		−	214.098i		−	0.986629i
$218$		0			0
$219$			319.622i			1.45946i
$220$		0			0
$221$	−21.2322			−0.0960733
$222$		0			0
$223$	−93.6765			−0.420074			−0.210037	−	0.977693i	$-0.567358\pi$
$223$	−93.6765			−0.420074			−0.210037	+	0.977693i	$0.567358\pi$
$224$		0			0
$225$	7.32863			0.0325717
$226$		0			0
$227$		−	325.742i		−	1.43499i	−0.696565	−	0.717494i	$-0.745289\pi$
$227$		−	325.742i		−	1.43499i	0.696565	−	0.717494i	$-0.254711\pi$
$228$		0			0
$229$	−31.5964			−0.137976			−0.0689878	−	0.997618i	$-0.521977\pi$
$229$	−31.5964			−0.137976			−0.0689878	+	0.997618i	$0.521977\pi$
$230$		0			0
$231$	226.238	−	495.583i	0.979386	−	2.14538i
$232$		0			0
$233$		−	85.6680i		−	0.367674i	−0.982957	−	0.183837i	$-0.941148\pi$
$233$		−	85.6680i		−	0.367674i	0.982957	−	0.183837i	$-0.0588518\pi$
$234$		0			0
$235$	111.859			0.475996
$236$		0			0
$237$		−	119.955i		−	0.506137i
$238$		0			0
$239$		−	264.111i		−	1.10507i	−0.833491	−	0.552533i	$-0.813661\pi$
$239$		−	264.111i		−	1.10507i	0.833491	−	0.552533i	$-0.186339\pi$
$240$		0			0
$241$		−	209.659i		−	0.869955i	−0.900441	−	0.434978i	$-0.856756\pi$
$241$		−	209.659i		−	0.869955i	0.900441	−	0.434978i	$-0.143244\pi$
$242$		0			0
$243$	282.660			1.16321
$244$		0			0
$245$	−579.596			−2.36570
$246$		0			0
$247$	433.656			1.75569
$248$		0			0
$249$			288.396i			1.15822i
$250$		0			0
$251$	−406.438			−1.61928			−0.809638	−	0.586930i	$-0.800336\pi$
$251$	−406.438			−1.61928			−0.809638	+	0.586930i	$0.800336\pi$
$252$		0			0
$253$	−118.961	−	54.3068i	−0.470202	−	0.214651i
$254$		0			0
$255$		−	28.3311i		−	0.111102i
$256$		0			0
$257$	−200.291			−0.779343			−0.389672	−	0.920954i	$-0.627411\pi$
$257$	−200.291			−0.779343			−0.389672	+	0.920954i	$0.627411\pi$
$258$		0			0
$259$			716.481i			2.76634i
$260$		0			0
$261$			99.9641i			0.383004i
$262$		0			0
$263$			86.1641i			0.327620i	0.986492	+	0.163810i	$0.0523784\pi$
$263$			86.1641i			0.327620i	−0.986492	+	0.163810i	$0.947622\pi$
$264$		0			0
$265$	−0.948513			−0.00357929
$266$		0			0
$267$	−38.8280			−0.145423
$268$		0			0
$269$	312.768			1.16270			0.581352	−	0.813652i	$-0.302524\pi$
$269$	312.768			1.16270			0.581352	+	0.813652i	$0.302524\pi$
$270$		0			0
$271$			64.5578i			0.238221i	0.992881	+	0.119110i	$0.0380042\pi$
$271$			64.5578i			0.238221i	−0.992881	+	0.119110i	$0.961996\pi$
$272$		0			0
$273$	738.662			2.70572
$274$		0			0
$275$	11.9860	+	5.47171i	0.0435854	+	0.0198971i
$276$		0			0
$277$			145.687i			0.525945i	0.964803	+	0.262973i	$0.0847029\pi$
$277$			145.687i			0.525945i	−0.964803	+	0.262973i	$0.915297\pi$
$278$		0			0
$279$	−102.843			−0.368612
$280$		0			0
$281$			335.693i			1.19464i	0.802005	+	0.597318i	$0.203767\pi$
$281$			335.693i			1.19464i	−0.802005	+	0.597318i	$0.796233\pi$
$282$		0			0
$283$			73.8343i			0.260899i	0.991455	+	0.130449i	$0.0416420\pi$
$283$			73.8343i			0.260899i	−0.991455	+	0.130449i	$0.958358\pi$
$284$		0			0
$285$			578.647i			2.03034i
$286$		0			0
$287$	−214.609			−0.747766
$288$		0			0
$289$	286.973			0.992988
$290$		0			0
$291$	−320.325			−1.10077
$292$		0			0
$293$		−	335.182i		−	1.14397i	−0.820265	−	0.571983i	$-0.806174\pi$
$293$		−	335.182i		−	1.14397i	0.820265	−	0.571983i	$-0.193826\pi$
$294$		0			0
$295$	91.7544			0.311032
$296$		0			0
$297$	112.118	+	51.1831i	0.377503	+	0.172334i
$298$		0			0
$299$		−	177.310i		−	0.593011i
$300$		0			0
$301$	460.053			1.52841
$302$		0			0
$303$			498.478i			1.64514i
$304$		0			0
$305$		−	394.210i		−	1.29249i
$306$		0			0
$307$		−	289.197i		−	0.942011i	−0.882130	−	0.471005i	$-0.843891\pi$
$307$		−	289.197i		−	0.942011i	0.882130	−	0.471005i	$-0.156109\pi$
$308$		0			0
$309$	186.893			0.604830
$310$		0			0
$311$	86.6486			0.278613			0.139306	−	0.990249i	$-0.455513\pi$
$311$	86.6486			0.278613			0.139306	+	0.990249i	$0.455513\pi$
$312$		0			0
$313$	−394.046			−1.25893			−0.629467	−	0.777027i	$-0.716727\pi$
$313$	−394.046			−1.25893			−0.629467	+	0.777027i	$0.716727\pi$
$314$		0			0
$315$			398.883i			1.26630i
$316$		0			0
$317$	−207.808			−0.655546			−0.327773	−	0.944756i	$-0.606298\pi$
$317$	−207.808			−0.655546			−0.327773	+	0.944756i	$0.606298\pi$
$318$		0			0
$319$	−74.6353	+	163.491i	−0.233967	+	0.512512i
$320$		0			0
$321$		−	784.270i		−	2.44321i
$322$		0			0
$323$	41.3911			0.128146
$324$		0			0
$325$			17.8650i			0.0549692i
$326$		0			0
$327$			639.663i			1.95616i
$328$		0			0
$329$			278.365i			0.846095i
$330$		0			0
$331$	−487.682			−1.47336			−0.736680	−	0.676241i	$-0.763608\pi$
$331$	−487.682			−1.47336			−0.736680	+	0.676241i	$0.763608\pi$
$332$		0			0
$333$	344.163			1.03352
$334$		0			0
$335$	−56.0012			−0.167168
$336$		0			0
$337$			388.113i			1.15167i	0.817566	+	0.575835i	$0.195323\pi$
$337$			388.113i			1.15167i	−0.817566	+	0.575835i	$0.804677\pi$
$338$		0			0
$339$	−331.166			−0.976891
$340$		0			0
$341$	−168.199	−	76.7845i	−0.493253	−	0.225175i
$342$		0			0
$343$		−	818.220i		−	2.38548i
$344$		0			0
$345$	236.593			0.685778
$346$		0			0
$347$		−	107.644i		−	0.310212i	−0.987898	−	0.155106i	$-0.950428\pi$
$347$		−	107.644i		−	0.310212i	0.987898	−	0.155106i	$-0.0495719\pi$
$348$		0			0
$349$		−	304.548i		−	0.872629i	−0.899794	−	0.436315i	$-0.856283\pi$
$349$		−	304.548i		−	0.872629i	0.899794	−	0.436315i	$-0.143717\pi$
$350$		0			0
$351$			167.111i			0.476100i
$352$		0			0
$353$	300.542			0.851393			0.425697	−	0.904866i	$-0.360029\pi$
$353$	300.542			0.851393			0.425697	+	0.904866i	$0.360029\pi$
$354$		0			0
$355$	712.840			2.00800
$356$		0			0
$357$	70.5030			0.197487
$358$		0			0
$359$		−	254.389i		−	0.708605i	−0.935131	−	0.354302i	$-0.884718\pi$
$359$		−	254.389i		−	0.708605i	0.935131	−	0.354302i	$-0.115282\pi$
$360$		0			0
$361$	−484.391			−1.34180
$362$		0			0
$363$	−308.200	−	355.473i	−0.849035	−	0.979264i
$364$		0			0
$365$		−	420.742i		−	1.15272i
$366$		0			0
$367$	−421.179			−1.14763			−0.573814	−	0.818986i	$-0.694537\pi$
$367$	−421.179			−1.14763			−0.573814	+	0.818986i	$0.694537\pi$
$368$		0			0
$369$			103.088i			0.279371i
$370$		0			0
$371$		−	2.36041i		−	0.00636229i
$372$		0			0
$373$			147.279i			0.394850i	0.980318	+	0.197425i	$0.0632578\pi$
$373$			147.279i			0.394850i	−0.980318	+	0.197425i	$0.936742\pi$
$374$		0			0
$375$	473.699			1.26320
$376$		0			0
$377$	−243.682			−0.646372
$378$		0			0
$379$	76.7735			0.202569			0.101284	−	0.994858i	$-0.467705\pi$
$379$	76.7735			0.202569			0.101284	+	0.994858i	$0.467705\pi$
$380$		0			0
$381$		−	241.765i		−	0.634554i
$382$		0			0
$383$	434.879			1.13546			0.567728	−	0.823216i	$-0.307823\pi$
$383$	434.879			1.13546			0.567728	+	0.823216i	$0.307823\pi$
$384$		0			0
$385$	−297.815	+	652.373i	−0.773544	+	1.69448i
$386$		0			0
$387$		−	220.987i		−	0.571027i
$388$		0			0
$389$	506.077			1.30097			0.650485	−	0.759519i	$-0.274566\pi$
$389$	506.077			1.30097			0.650485	+	0.759519i	$0.274566\pi$
$390$		0			0
$391$		−	16.9237i		−	0.0432832i
$392$		0			0
$393$			255.148i			0.649232i
$394$		0			0
$395$			157.905i			0.399760i
$396$		0			0
$397$	−275.285			−0.693413			−0.346707	−	0.937974i	$-0.612700\pi$
$397$	−275.285			−0.693413			−0.346707	+	0.937974i	$0.612700\pi$
$398$		0			0
$399$	−1439.99			−3.60899
$400$		0			0
$401$	−469.509			−1.17085			−0.585423	−	0.810728i	$-0.699071\pi$
$401$	−469.509			−1.17085			−0.585423	+	0.810728i	$0.699071\pi$
$402$		0			0
$403$		−	250.699i		−	0.622083i
$404$		0			0
$405$	−504.830			−1.24649
$406$		0			0
$407$	562.879	+	256.960i	1.38300	+	0.631350i
$408$		0			0
$409$		−	682.090i		−	1.66770i	−0.551990	−	0.833851i	$-0.686131\pi$
$409$		−	682.090i		−	1.66770i	0.551990	−	0.833851i	$-0.313869\pi$
$410$		0			0
$411$	479.337			1.16627
$412$		0			0
$413$			228.334i			0.552867i
$414$		0			0
$415$		−	379.637i		−	0.914788i
$416$		0			0
$417$		−	298.192i		−	0.715089i
$418$		0			0
$419$	212.596			0.507388			0.253694	−	0.967284i	$-0.418354\pi$
$419$	212.596			0.507388			0.253694	+	0.967284i	$0.418354\pi$
$420$		0			0
$421$	347.762			0.826037			0.413019	−	0.910723i	$-0.364474\pi$
$421$	347.762			0.826037			0.413019	+	0.910723i	$0.364474\pi$
$422$		0			0
$423$	133.713			0.316107
$424$		0			0
$425$			1.70516i			0.00401214i
$426$		0			0
$427$	981.006			2.29744
$428$		0			0
$429$	264.914	−	580.305i	0.617516	−	1.35269i
$430$		0			0
$431$			392.374i			0.910381i	0.890394	+	0.455191i	$0.150429\pi$
$431$			392.374i			0.910381i	−0.890394	+	0.455191i	$0.849571\pi$
$432$		0			0
$433$	−554.325			−1.28020			−0.640099	−	0.768293i	$-0.721107\pi$
$433$	−554.325			−1.28020			−0.640099	+	0.768293i	$0.721107\pi$
$434$		0			0
$435$		−	325.157i		−	0.747487i
$436$		0			0
$437$			345.658i			0.790979i
$438$		0			0
$439$			130.153i			0.296477i	0.988952	+	0.148239i	$0.0473603\pi$
$439$			130.153i			0.296477i	−0.988952	+	0.148239i	$0.952640\pi$
$440$		0			0
$441$	−692.834			−1.57105
$442$		0			0
$443$	107.762			0.243254			0.121627	−	0.992576i	$-0.461189\pi$
$443$	107.762			0.243254			0.121627	+	0.992576i	$0.461189\pi$
$444$		0			0
$445$	51.1123			0.114859
$446$		0			0
$447$		−	790.424i		−	1.76829i
$448$		0			0
$449$	−86.8082			−0.193337			−0.0966683	−	0.995317i	$-0.530819\pi$
$449$	−86.8082			−0.193337			−0.0966683	+	0.995317i	$0.530819\pi$
$450$		0			0
$451$	−76.9676	+	168.600i	−0.170660	+	0.373837i
$452$		0			0
$453$			424.208i			0.936441i
$454$		0			0
$455$	−972.356			−2.13705
$456$		0			0
$457$			358.902i			0.785344i	0.919679	+	0.392672i	$0.128449\pi$
$457$			358.902i			0.785344i	−0.919679	+	0.392672i	$0.871551\pi$
$458$		0			0
$459$			15.9503i			0.0347500i
$460$		0			0
$461$			463.660i			1.00577i	0.864353	+	0.502885i	$0.167728\pi$
$461$			463.660i			1.00577i	−0.864353	+	0.502885i	$0.832272\pi$
$462$		0			0
$463$	−21.6620			−0.0467862			−0.0233931	−	0.999726i	$-0.507447\pi$
$463$	−21.6620			−0.0467862			−0.0233931	+	0.999726i	$0.507447\pi$
$464$		0			0
$465$	334.520			0.719398
$466$		0			0
$467$	383.773			0.821784			0.410892	−	0.911684i	$-0.365217\pi$
$467$	383.773			0.821784			0.410892	+	0.911684i	$0.365217\pi$
$468$		0			0
$469$		−	139.361i		−	0.297145i
$470$		0			0
$471$	282.066			0.598866
$472$		0			0
$473$	164.994	−	361.425i	0.348824	−	0.764112i
$474$		0			0
$475$		−	34.8269i		−	0.0733199i
$476$		0			0
$477$	−1.13383			−0.00237700
$478$		0			0
$479$			437.216i			0.912768i	0.889783	+	0.456384i	$0.150856\pi$
$479$			437.216i			0.912768i	−0.889783	+	0.456384i	$0.849144\pi$
$480$		0			0
$481$			838.966i			1.74421i
$482$		0			0
$483$			588.771i			1.21899i
$484$		0			0
$485$	421.668			0.869419
$486$		0			0
$487$	−158.744			−0.325963			−0.162981	−	0.986629i	$-0.552111\pi$
$487$	−158.744			−0.325963			−0.162981	+	0.986629i	$0.552111\pi$
$488$		0			0
$489$	561.296			1.14784
$490$		0			0
$491$			899.040i			1.83104i	0.402273	+	0.915520i	$0.368220\pi$
$491$			899.040i			1.83104i	−0.402273	+	0.915520i	$0.631780\pi$
$492$		0			0
$493$	−23.2587			−0.0471780
$494$		0			0
$495$	313.369	+	143.056i	0.633069	+	0.289002i
$496$		0			0
$497$			1773.93i			3.56927i
$498$		0			0
$499$	−501.946			−1.00590			−0.502951	−	0.864315i	$-0.667753\pi$
$499$	−501.946			−1.00590			−0.502951	+	0.864315i	$0.667753\pi$
$500$		0			0
$501$		−	470.418i		−	0.938959i
$502$		0			0
$503$		−	245.501i		−	0.488074i	−0.969766	−	0.244037i	$-0.921528\pi$
$503$		−	245.501i		−	0.488074i	0.969766	−	0.244037i	$-0.0784718\pi$
$504$		0			0
$505$		−	656.184i		−	1.29937i
$506$		0			0
$507$	207.827			0.409914
$508$		0			0
$509$	−9.33172			−0.0183334			−0.00916672	−	0.999958i	$-0.502918\pi$
$509$	−9.33172			−0.0183334			−0.00916672	+	0.999958i	$0.502918\pi$
$510$		0			0
$511$	1047.03			2.04899
$512$		0			0
$513$		−	325.775i		−	0.635040i
$514$		0			0
$515$	−246.021			−0.477710
$516$		0			0
$517$	218.688	+	99.8332i	0.422995	+	0.193101i
$518$		0			0
$519$			71.2826i			0.137346i
$520$		0			0
$521$	862.330			1.65514			0.827572	−	0.561360i	$-0.189722\pi$
$521$	862.330			1.65514			0.827572	+	0.561360i	$0.189722\pi$
$522$		0			0
$523$		−	391.948i		−	0.749423i	−0.927142	−	0.374711i	$-0.877742\pi$
$523$		−	391.948i		−	0.749423i	0.927142	−	0.374711i	$-0.122258\pi$
$524$		0			0
$525$		−	59.3220i		−	0.112994i
$526$		0			0
$527$		−	23.9285i		−	0.0454051i
$528$		0			0
$529$	−387.670			−0.732835
$530$		0			0
$531$	109.681			0.206555
$532$		0			0
$533$	−251.297			−0.471477
$534$		0			0
$535$			1032.39i			1.92971i
$536$		0			0
$537$	−1002.49			−1.86683
$538$		0			0
$539$	−1133.13	−	517.285i	−2.10228	−	0.959712i
$540$		0			0
$541$		−	238.052i		−	0.440023i	−0.975497	−	0.220011i	$-0.929391\pi$
$541$		−	238.052i		−	0.440023i	0.975497	−	0.220011i	$-0.0706095\pi$
$542$		0			0
$543$	−323.207			−0.595225
$544$		0			0
$545$		−	842.037i		−	1.54502i
$546$		0			0
$547$		−	417.066i		−	0.762461i	−0.924480	−	0.381230i	$-0.875500\pi$
$547$		−	417.066i		−	0.762461i	0.924480	−	0.381230i	$-0.124500\pi$
$548$		0			0
$549$		−	471.228i		−	0.858339i
$550$		0			0
$551$	475.047			0.862154
$552$		0			0
$553$	−392.953			−0.710584
$554$		0			0
$555$	−1119.47			−2.01707
$556$		0			0
$557$		−	151.232i		−	0.271513i	−0.990742	−	0.135756i	$-0.956654\pi$
$557$		−	151.232i		−	0.271513i	0.990742	−	0.135756i	$-0.0433464\pi$
$558$		0			0
$559$	538.700			0.963686
$560$		0			0
$561$	25.2853	−	55.3883i	0.0450718	−	0.0987314i
$562$		0			0
$563$			875.506i			1.55507i	0.628838	+	0.777536i	$0.283531\pi$
$563$			875.506i			1.55507i	−0.628838	+	0.777536i	$0.716469\pi$
$564$		0			0
$565$	435.939			0.771573
$566$		0			0
$567$		−	1256.29i		−	2.21568i
$568$		0			0
$569$		−	706.404i		−	1.24148i	−0.784015	−	0.620741i	$-0.786832\pi$
$569$		−	706.404i		−	1.24148i	0.784015	−	0.620741i	$-0.213168\pi$
$570$		0			0
$571$		−	944.530i		−	1.65417i	−0.562078	−	0.827084i	$-0.689998\pi$
$571$		−	944.530i		−	1.65417i	0.562078	−	0.827084i	$-0.310002\pi$
$572$		0			0
$573$	−617.246			−1.07722
$574$		0			0
$575$	−14.2398			−0.0247649
$576$		0			0
$577$	−638.796			−1.10710			−0.553549	−	0.832816i	$-0.686727\pi$
$577$	−638.796			−1.10710			−0.553549	+	0.832816i	$0.686727\pi$
$578$		0			0
$579$		−	826.930i		−	1.42820i
$580$		0			0
$581$	944.741			1.62606
$582$		0			0
$583$	−1.85438	−	0.846540i	−0.00318075	−	0.00145204i
$584$		0			0
$585$			467.073i			0.798416i
$586$		0			0
$587$	499.575			0.851065			0.425532	−	0.904943i	$-0.360087\pi$
$587$	499.575			0.851065			0.425532	+	0.904943i	$0.360087\pi$
$588$		0			0
$589$			488.726i			0.829756i
$590$		0			0
$591$		−	142.515i		−	0.241143i
$592$		0			0
$593$		−	71.9478i		−	0.121329i	−0.998158	−	0.0606643i	$-0.980678\pi$
$593$		−	71.9478i		−	0.121329i	0.998158	−	0.0606643i	$-0.0193219\pi$
$594$		0			0
$595$	−92.8085			−0.155981
$596$		0			0
$597$	551.460			0.923719
$598$		0			0
$599$	−770.834			−1.28687			−0.643434	−	0.765502i	$-0.722491\pi$
$599$	−770.834			−1.28687			−0.643434	+	0.765502i	$0.722491\pi$
$600$		0			0
$601$			545.840i			0.908219i	0.890946	+	0.454110i	$0.150043\pi$
$601$			545.840i			0.908219i	−0.890946	+	0.454110i	$0.849957\pi$
$602$		0			0
$603$	−66.9424			−0.111016
$604$		0			0
$605$	405.707	+	467.936i	0.670590	+	0.773448i
$606$		0			0
$607$			659.902i			1.08715i	0.839360	+	0.543576i	$0.182930\pi$
$607$			659.902i			1.08715i	−0.839360	+	0.543576i	$0.817070\pi$
$608$		0			0
$609$	809.165			1.32868
$610$		0			0
$611$			325.953i			0.533474i
$612$		0			0
$613$		−	752.909i		−	1.22824i	−0.789214	−	0.614118i	$-0.789512\pi$
$613$		−	752.909i		−	1.22824i	0.789214	−	0.614118i	$-0.210488\pi$
$614$		0			0
$615$		−	335.317i		−	0.545232i
$616$		0			0
$617$	−49.7497			−0.0806316			−0.0403158	−	0.999187i	$-0.512836\pi$
$617$	−49.7497			−0.0806316			−0.0403158	+	0.999187i	$0.512836\pi$
$618$		0			0
$619$	501.926			0.810866			0.405433	−	0.914125i	$-0.367121\pi$
$619$	501.926			0.810866			0.405433	+	0.914125i	$0.367121\pi$
$620$		0			0
$621$	−133.201			−0.214494
$622$		0			0
$623$			127.195i			0.204165i
$624$		0			0
$625$	−653.510			−1.04562
$626$		0			0
$627$	−516.438	+	1131.28i	−0.823665	+	1.80427i
$628$		0			0
$629$			80.0768i			0.127308i
$630$		0			0
$631$	−545.597			−0.864655			−0.432327	−	0.901717i	$-0.642307\pi$
$631$	−545.597			−0.864655			−0.432327	+	0.901717i	$0.642307\pi$
$632$		0			0
$633$		−	109.046i		−	0.172268i
$634$		0			0
$635$			318.253i			0.501187i
$636$		0			0
$637$		−	1688.92i		−	2.65137i
$638$		0			0
$639$	852.110			1.33351
$640$		0			0
$641$	−450.576			−0.702927			−0.351463	−	0.936202i	$-0.614316\pi$
$641$	−450.576			−0.702927			−0.351463	+	0.936202i	$0.614316\pi$
$642$		0			0
$643$	−864.073			−1.34382			−0.671908	−	0.740635i	$-0.734525\pi$
$643$	−864.073			−1.34382			−0.671908	+	0.740635i	$0.734525\pi$
$644$		0			0
$645$			718.813i			1.11444i
$646$		0			0
$647$	−1200.90			−1.85610			−0.928052	−	0.372450i	$-0.878518\pi$
$647$	−1200.90			−1.85610			−0.928052	+	0.372450i	$0.878518\pi$
$648$		0			0
$649$	179.383	+	81.8900i	0.276399	+	0.126179i
$650$		0			0
$651$			832.465i			1.27875i
$652$		0			0
$653$	272.815			0.417787			0.208894	−	0.977938i	$-0.433014\pi$
$653$	272.815			0.417787			0.208894	+	0.977938i	$0.433014\pi$
$654$		0			0
$655$		−	335.871i		−	0.512780i
$656$		0			0
$657$		−	502.944i		−	0.765516i
$658$		0			0
$659$		−	328.580i		−	0.498604i	−0.968426	−	0.249302i	$-0.919799\pi$
$659$		−	328.580i		−	0.498604i	0.968426	−	0.249302i	$-0.0802011\pi$
$660$		0			0
$661$	123.900			0.187444			0.0937220	−	0.995598i	$-0.470124\pi$
$661$	123.900			0.187444			0.0937220	+	0.995598i	$0.470124\pi$
$662$		0			0
$663$	82.5558			0.124519
$664$		0			0
$665$	1895.56			2.85047
$666$		0			0
$667$		−	194.234i		−	0.291205i
$668$		0			0
$669$	364.236			0.544449
$670$		0			0
$671$	351.829	−	770.694i	0.524335	−	1.14858i
$672$		0			0
$673$		−	782.630i		−	1.16290i	−0.813583	−	0.581449i	$-0.802486\pi$
$673$		−	782.630i		−	1.16290i	0.813583	−	0.581449i	$-0.197514\pi$
$674$		0			0
$675$	13.4207			0.0198826
$676$		0			0
$677$		−	1209.30i		−	1.78626i	−0.449803	−	0.893128i	$-0.648506\pi$
$677$		−	1209.30i		−	1.78626i	0.449803	−	0.893128i	$-0.351494\pi$
$678$		0			0
$679$			1049.34i			1.54541i
$680$		0			0
$681$			1266.56i			1.85986i
$682$		0			0
$683$	113.737			0.166525			0.0832625	−	0.996528i	$-0.473466\pi$
$683$	113.737			0.166525			0.0832625	+	0.996528i	$0.473466\pi$
$684$		0			0
$685$	−630.987			−0.921149
$686$		0			0
$687$	122.854			0.178827
$688$		0			0
$689$		−	2.76393i		−	0.00401151i
$690$		0			0
$691$	−460.880			−0.666975			−0.333487	−	0.942755i	$-0.608225\pi$
$691$	−460.880			−0.666975			−0.333487	+	0.942755i	$0.608225\pi$
$692$		0			0
$693$	−356.000	+	779.830i	−0.513708	+	1.12530i
$694$		0			0
$695$			392.533i			0.564795i
$696$		0			0
$697$	−23.9855			−0.0344125
$698$		0			0
$699$			333.097i			0.476534i
$700$		0			0
$701$			1209.73i			1.72572i	0.505446	+	0.862858i	$0.331328\pi$
$701$			1209.73i			1.72572i	−0.505446	+	0.862858i	$0.668672\pi$
$702$		0			0
$703$		−	1635.52i		−	2.32649i
$704$		0			0
$705$	−434.934			−0.616928
$706$		0			0
$707$	1632.94			2.30967
$708$		0			0
$709$	346.827			0.489178			0.244589	−	0.969627i	$-0.421347\pi$
$709$	346.827			0.489178			0.244589	+	0.969627i	$0.421347\pi$
$710$		0			0
$711$			188.756i			0.265479i
$712$		0			0
$713$	199.827			0.280262
$714$		0			0
$715$	−348.727	+	763.899i	−0.487730	+	1.06839i
$716$		0			0
$717$			1026.92i			1.43225i
$718$		0			0
$719$	−1053.75			−1.46558			−0.732788	−	0.680457i	$-0.761781\pi$
$719$	−1053.75			−1.46558			−0.732788	+	0.680457i	$0.761781\pi$
$720$		0			0
$721$		−	612.232i		−	0.849143i
$722$		0			0
$723$			815.205i			1.12753i
$724$		0			0
$725$			19.5702i			0.0269933i
$726$		0			0
$727$	649.906			0.893956			0.446978	−	0.894545i	$-0.352500\pi$
$727$	649.906			0.893956			0.446978	+	0.894545i	$0.352500\pi$
$728$		0			0
$729$	−211.372			−0.289948
$730$		0			0
$731$	51.4173			0.0703383
$732$		0			0
$733$			1115.68i			1.52207i	0.648710	+	0.761036i	$0.275309\pi$
$733$			1115.68i			1.52207i	−0.648710	+	0.761036i	$0.724691\pi$
$734$		0			0
$735$	2253.61			3.06613
$736$		0			0
$737$	−109.484	−	49.9806i	−0.148554	−	0.0678163i
$738$		0			0
$739$		−	1274.73i		−	1.72494i	−0.506106	−	0.862471i	$-0.668915\pi$
$739$		−	1274.73i		−	1.72494i	0.506106	−	0.862471i	$-0.331085\pi$
$740$		0			0
$741$	−1686.16			−2.27551
$742$		0			0
$743$			347.381i			0.467538i	0.972292	+	0.233769i	$0.0751059\pi$
$743$			347.381i			0.467538i	−0.972292	+	0.233769i	$0.924894\pi$
$744$		0			0
$745$			1040.49i			1.39664i
$746$		0			0
$747$		−	453.808i		−	0.607508i
$748$		0			0
$749$	−2569.15			−3.43011
$750$		0			0
$751$	−863.715			−1.15009			−0.575043	−	0.818123i	$-0.695015\pi$
$751$	−863.715			−1.15009			−0.575043	+	0.818123i	$0.695015\pi$
$752$		0			0
$753$	1580.33			2.09871
$754$		0			0
$755$		−	558.417i		−	0.739625i
$756$		0			0
$757$	−6.91483			−0.00913452			−0.00456726	−	0.999990i	$-0.501454\pi$
$757$	−6.91483			−0.00913452			−0.00456726	+	0.999990i	$0.501454\pi$
$758$		0			0
$759$	462.548	+	211.158i	0.609418	+	0.278205i
$760$		0			0
$761$			734.052i			0.964589i	0.876009	+	0.482295i	$0.160197\pi$
$761$			734.052i			0.964589i	−0.876009	+	0.482295i	$0.839803\pi$
$762$		0			0
$763$	2095.44			2.74632
$764$		0			0
$765$			44.5807i			0.0582755i
$766$		0			0
$767$			267.369i			0.348590i
$768$		0			0
$769$			1277.23i			1.66090i	0.557096	+	0.830448i	$0.311915\pi$
$769$			1277.23i			1.66090i	−0.557096	+	0.830448i	$0.688085\pi$
$770$		0			0
$771$	778.779			1.01009
$772$		0			0
$773$	−1138.74			−1.47315			−0.736573	−	0.676359i	$-0.763557\pi$
$773$	−1138.74			−1.47315			−0.736573	+	0.676359i	$0.763557\pi$
$774$		0			0
$775$	−20.1337			−0.0259790
$776$		0			0
$777$		−	2785.85i		−	3.58539i
$778$		0			0
$779$	489.891			0.628872
$780$		0			0
$781$	1393.63	+	636.204i	1.78441	+	0.814601i
$782$		0			0
$783$			183.062i			0.233795i
$784$		0			0
$785$	−371.305			−0.473000
$786$		0			0
$787$		−	715.623i		−	0.909305i	−0.890669	−	0.454653i	$-0.849763\pi$
$787$		−	715.623i		−	0.909305i	0.890669	−	0.454653i	$-0.150237\pi$
$788$		0			0
$789$		−	335.026i		−	0.424622i
$790$		0			0
$791$			1084.85i			1.37149i
$792$		0			0
$793$	1148.71			1.44857
$794$		0			0
$795$	3.68804			0.00463905
$796$		0			0
$797$	−1348.15			−1.69153			−0.845767	−	0.533553i	$-0.820857\pi$
$797$	−1348.15			−1.69153			−0.845767	+	0.533553i	$0.820857\pi$
$798$		0			0
$799$			31.1112i			0.0389377i
$800$		0			0
$801$	61.0983			0.0762776
$802$		0			0
$803$	375.509	−	822.566i	0.467632	−	1.02437i
$804$		0			0
$805$		−	775.044i		−	0.962788i
$806$		0			0
$807$	−1216.11			−1.50696
$808$		0			0
$809$		−	779.307i		−	0.963297i	−0.876365	−	0.481648i	$-0.840038\pi$
$809$		−	779.307i		−	0.963297i	0.876365	−	0.481648i	$-0.159962\pi$
$810$		0			0
$811$			407.625i			0.502620i	0.967907	+	0.251310i	$0.0808613\pi$
$811$			407.625i			0.502620i	−0.967907	+	0.251310i	$0.919139\pi$
$812$		0			0
$813$		−	251.016i		−	0.308753i
$814$		0			0
$815$	−738.876			−0.906596
$816$		0			0
$817$	−1050.17			−1.28540
$818$		0			0
$819$	−1162.33			−1.41921
$820$		0			0
$821$		−	144.497i		−	0.176001i	−0.996120	−	0.0880007i	$-0.971952\pi$
$821$		−	144.497i		−	0.176001i	0.996120	−	0.0880007i	$-0.0280478\pi$
$822$		0			0
$823$	−1257.65			−1.52813			−0.764063	−	0.645142i	$-0.776798\pi$
$823$	−1257.65			−1.52813			−0.764063	+	0.645142i	$0.776798\pi$
$824$		0			0
$825$	−46.6043	−	21.2753i	−0.0564901	−	0.0257883i
$826$		0			0
$827$		−	1058.91i		−	1.28043i	−0.768198	−	0.640213i	$-0.778846\pi$
$827$		−	1058.91i		−	1.28043i	0.768198	−	0.640213i	$-0.221154\pi$
$828$		0			0
$829$	237.316			0.286267			0.143134	−	0.989703i	$-0.454282\pi$
$829$	237.316			0.286267			0.143134	+	0.989703i	$0.454282\pi$
$830$		0			0
$831$		−	566.465i		−	0.681666i
$832$		0			0
$833$		−	161.202i		−	0.193520i
$834$		0			0
$835$			619.247i			0.741613i
$836$		0			0
$837$	−188.333			−0.225010
$838$		0			0
$839$	920.950			1.09768			0.548838	−	0.835929i	$-0.315070\pi$
$839$	920.950			1.09768			0.548838	+	0.835929i	$0.315070\pi$
$840$		0			0
$841$	574.059			0.682591
$842$		0			0
$843$		−	1305.25i		−	1.54834i
$844$		0			0
$845$	−273.578			−0.323761
$846$		0			0
$847$	−1164.48	+	1009.62i	−1.37482	+	1.19199i
$848$		0			0
$849$		−	287.085i		−	0.338145i
$850$		0			0
$851$	−668.722			−0.785807
$852$		0			0
$853$			751.983i			0.881574i	0.897612	+	0.440787i	$0.145301\pi$
$853$			751.983i			0.881574i	−0.897612	+	0.440787i	$0.854699\pi$
$854$		0			0
$855$		−	910.537i		−	1.06496i
$856$		0			0
$857$		−	1447.30i		−	1.68880i	−0.535715	−	0.844399i	$-0.679958\pi$
$857$		−	1447.30i		−	1.68880i	0.535715	−	0.844399i	$-0.320042\pi$
$858$		0			0
$859$	1153.12			1.34240			0.671198	−	0.741278i	$-0.265780\pi$
$859$	1153.12			1.34240			0.671198	+	0.741278i	$0.265780\pi$
$860$		0			0
$861$	834.450			0.969164
$862$		0			0
$863$	1231.34			1.42681			0.713405	−	0.700752i	$-0.247152\pi$
$863$	1231.34			1.42681			0.713405	+	0.700752i	$0.247152\pi$
$864$		0			0
$865$		−	93.8347i		−	0.108479i
$866$		0			0
$867$	−1115.82			−1.28699
$868$		0			0
$869$	−140.929	+	308.710i	−0.162174	+	0.355248i
$870$		0			0
$871$		−	163.185i		−	0.187354i
$872$		0			0
$873$	504.051			0.577378
$874$		0			0
$875$		−	1551.77i		−	1.77345i
$876$		0			0
$877$			148.205i			0.168991i	0.996424	+	0.0844953i	$0.0269278\pi$
$877$			148.205i			0.168991i	−0.996424	+	0.0844953i	$0.973072\pi$
$878$		0			0
$879$			1303.27i			1.48267i
$880$		0			0
$881$	1371.29			1.55652			0.778260	−	0.627942i	$-0.216102\pi$
$881$	1371.29			1.55652			0.778260	+	0.627942i	$0.216102\pi$
$882$		0			0
$883$	−183.895			−0.208262			−0.104131	−	0.994564i	$-0.533206\pi$
$883$	−183.895			−0.208262			−0.104131	+	0.994564i	$0.533206\pi$
$884$		0			0
$885$	−356.763			−0.403122
$886$		0			0
$887$		−	1524.66i		−	1.71890i	−0.511222	−	0.859448i	$-0.670807\pi$
$887$		−	1524.66i		−	1.71890i	0.511222	−	0.859448i	$-0.329193\pi$
$888$		0			0
$889$	−791.986			−0.890872
$890$		0			0
$891$	−986.961	−	450.557i	−1.10770	−	0.505675i
$892$		0			0
$893$		−	635.429i		−	0.711567i
$894$		0			0
$895$	1319.65			1.47447
$896$		0			0
$897$			689.424i			0.768589i
$898$		0			0
$899$		−	274.628i		−	0.305482i
$900$		0			0
$901$		−	0.263809i		−	0.000292796i
$902$		0			0
$903$	−1788.79			−1.98095
$904$		0			0
$905$	425.462			0.470124
$906$		0			0
$907$	−1274.87			−1.40559			−0.702797	−	0.711390i	$-0.748066\pi$
$907$	−1274.87			−1.40559			−0.702797	+	0.711390i	$0.748066\pi$
$908$		0			0
$909$		−	784.386i		−	0.862910i
$910$		0			0
$911$	659.601			0.724041			0.362020	−	0.932170i	$-0.382087\pi$
$911$	659.601			0.724041			0.362020	+	0.932170i	$0.382087\pi$
$912$		0			0
$913$	338.823	−	742.204i	0.371110	−	0.812929i
$914$		0			0
$915$			1532.78i			1.67517i
$916$		0			0
$917$	835.827			0.911479
$918$		0			0
$919$			1094.37i			1.19082i	0.803420	+	0.595412i	$0.203011\pi$
$919$			1094.37i			1.19082i	−0.803420	+	0.595412i	$0.796989\pi$
$920$		0			0
$921$			1124.47i			1.22092i
$922$		0			0
$923$			2077.19i			2.25047i
$924$		0			0
$925$	67.3775			0.0728405
$926$		0			0
$927$	−294.087			−0.317246
$928$		0			0
$929$	−693.167			−0.746143			−0.373072	−	0.927803i	$-0.621695\pi$
$929$	−693.167			−0.746143			−0.373072	+	0.927803i	$0.621695\pi$
$930$		0			0
$931$			3292.47i			3.53649i
$932$		0			0
$933$	−336.910			−0.361104
$934$		0			0
$935$	−33.2849	+	72.9119i	−0.0355989	+	0.0779806i
$936$		0			0
$937$			394.132i			0.420632i	0.977633	+	0.210316i	$0.0674493\pi$
$937$			394.132i			0.420632i	−0.977633	+	0.210316i	$0.932551\pi$
$938$		0			0
$939$	1532.15			1.63168
$940$		0			0
$941$		−	387.265i		−	0.411547i	−0.978600	−	0.205773i	$-0.934029\pi$
$941$		−	387.265i		−	0.411547i	0.978600	−	0.205773i	$-0.0659710\pi$
$942$		0			0
$943$		−	200.303i		−	0.212411i
$944$		0			0
$945$			730.463i			0.772977i
$946$		0			0
$947$	41.2644			0.0435738			0.0217869	−	0.999763i	$-0.493064\pi$
$947$	41.2644			0.0435738			0.0217869	+	0.999763i	$0.493064\pi$
$948$		0			0
$949$	1226.03			1.29191
$950$		0			0
$951$	808.007			0.849639
$952$		0			0
$953$			1082.94i			1.13635i	0.822909	+	0.568173i	$0.192350\pi$
$953$			1082.94i			1.13635i	−0.822909	+	0.568173i	$0.807650\pi$
$954$		0			0
$955$	812.528			0.850814
$956$		0			0
$957$	290.200	−	635.693i	0.303239	−	0.664256i
$958$		0			0
$959$		−	1570.24i		−	1.63737i
$960$		0			0
$961$	−678.464			−0.705998
$962$		0			0
$963$			1234.10i			1.28151i
$964$		0			0
$965$			1088.55i			1.12803i
$966$		0			0
$967$		−	1004.94i		−	1.03924i	−0.854399	−	0.519618i	$-0.826074\pi$
$967$		−	1004.94i		−	1.03924i	0.854399	−	0.519618i	$-0.173926\pi$
$968$		0			0
$969$	−160.938			−0.166087
$970$		0			0
$971$	−123.788			−0.127485			−0.0637425	−	0.997966i	$-0.520304\pi$
$971$	−123.788			−0.127485			−0.0637425	+	0.997966i	$0.520304\pi$
$972$		0			0
$973$	−976.833			−1.00394
$974$		0			0
$975$		−	69.4633i		−	0.0712444i
$976$		0			0
$977$	1471.46			1.50610			0.753051	−	0.657962i	$-0.228581\pi$
$977$	1471.46			1.50610			0.753051	+	0.657962i	$0.228581\pi$
$978$		0			0
$979$	99.9264	+	45.6173i	0.102070	+	0.0465958i
$980$		0			0
$981$		−	1006.55i		−	1.02604i
$982$		0			0
$983$	−524.027			−0.533089			−0.266545	−	0.963823i	$-0.585882\pi$
$983$	−524.027			−0.533089			−0.266545	+	0.963823i	$0.585882\pi$
$984$		0			0
$985$			187.604i			0.190461i
$986$		0			0
$987$		−	1082.35i		−	1.09661i
$988$		0			0
$989$			429.387i			0.434162i
$990$		0			0
$991$	−392.423			−0.395986			−0.197993	−	0.980203i	$-0.563442\pi$
$991$	−392.423			−0.395986			−0.197993	+	0.980203i	$0.563442\pi$
$992$		0			0
$993$	1896.22			1.90959
$994$		0			0
$995$	−725.929			−0.729577
$996$		0			0
$997$			1166.17i			1.16967i	0.811151	+	0.584837i	$0.198842\pi$
$997$			1166.17i			1.16967i	−0.811151	+	0.584837i	$0.801158\pi$
$998$		0			0
$999$	630.257			0.630888

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	88.3.h.a.65.2	yes	6
3.2	odd	2		792.3.j.a.505.2		6
4.3	odd	2		176.3.h.d.65.5		6
8.3	odd	2		704.3.h.g.65.1		6
8.5	even	2		704.3.h.h.65.6		6
11.10	odd	2	inner	88.3.h.a.65.1	✓	6
12.11	even	2		1584.3.j.k.1297.1		6
33.32	even	2		792.3.j.a.505.1		6
44.43	even	2		176.3.h.d.65.6		6
88.21	odd	2		704.3.h.h.65.5		6
88.43	even	2		704.3.h.g.65.2		6
132.131	odd	2		1584.3.j.k.1297.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.3.h.a.65.1	✓	6	11.10	odd	2	inner
88.3.h.a.65.2	yes	6	1.1	even	1	trivial
176.3.h.d.65.5		6	4.3	odd	2
176.3.h.d.65.6		6	44.43	even	2
704.3.h.g.65.1		6	8.3	odd	2
704.3.h.g.65.2		6	88.43	even	2
704.3.h.h.65.5		6	88.21	odd	2
704.3.h.h.65.6		6	8.5	even	2
792.3.j.a.505.1		6	33.32	even	2
792.3.j.a.505.2		6	3.2	odd	2
1584.3.j.k.1297.1		6	12.11	even	2
1584.3.j.k.1297.2		6	132.131	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	88.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.88824 q^{3} +5.11838 q^{5} +12.7373i q^{7} +6.11838 q^{9} +(10.0066 + 4.56811i) q^{11} +14.9148i q^{13} -19.9015 q^{15} +1.42357i q^{17} -29.0756i q^{19} -49.5255i q^{21} -11.8882 q^{23} +1.19781 q^{25} +11.2044 q^{27} +16.3383i q^{29} -16.8088 q^{31} +(-38.9081 - 17.7619i) q^{33} +65.1942i q^{35} +56.2507 q^{37} -57.9921i q^{39} +16.8489i q^{41} -36.1186i q^{43} +31.3162 q^{45} +21.8544 q^{47} -113.238 q^{49} -5.53517i q^{51} -0.185315 q^{53} +(51.2177 + 23.3813i) q^{55} +113.053i q^{57} +17.9264 q^{59} -77.0185i q^{61} +77.9315i q^{63} +76.3394i q^{65} -10.9412 q^{67} +46.2243 q^{69} +139.271 q^{71} -82.2022i q^{73} -4.65735 q^{75} +(-58.1853 + 127.457i) q^{77} +30.8506i q^{79} -98.6308 q^{81} -74.1713i q^{83} +7.28637i q^{85} -63.5273i q^{87} +9.98603 q^{89} -189.973 q^{91} +65.3566 q^{93} -148.820i q^{95} +82.3831 q^{97} +(61.2243 + 27.9494i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 6 q^{9} + 10 q^{11} - 52 q^{23} + 22 q^{25} + 32 q^{27} - 36 q^{31} - 64 q^{33} - 48 q^{37} + 172 q^{45} - 60 q^{47} - 170 q^{49} + 108 q^{53} + 172 q^{55} + 236 q^{59} - 292 q^{67} + 92 q^{69}+ \cdots + 182 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + 6 * q^9 + 10 * q^11 - 52 * q^23 + 22 * q^25 + 32 * q^27 - 36 * q^31 - 64 * q^33 - 48 * q^37 + 172 * q^45 - 60 * q^47 - 170 * q^49 + 108 * q^53 + 172 * q^55 + 236 * q^59 - 292 * q^67 + 92 * q^69 + 300 * q^71 - 244 * q^75 - 240 * q^77 - 362 * q^81 + 384 * q^89 - 48 * q^91 - 148 * q^93 + 400 * q^97 + 182 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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