LMFDB - Embedded newform 688.5.b.d.257.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,5,Mod(257,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, 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("688.257");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	688.b (of order $2$, degree $1$, not 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:	$71.1185346017$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 142x^{10} + 7173x^{8} + 157368x^{6} + 1510016x^{4} + 5098688x^{2} + 90352 $ x^12 + 142x^10 + 7173x^8 + 157368x^6 + 1510016x^4 + 5098688*x^2 + 90352
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.9
Root			$-6.72223i$ of defining polynomial
Character	$\chi$	$=$	688.257
Dual form			688.5.b.d.257.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.31879i q^{3} +1.48242i q^{5} +13.7963i q^{7} +11.7977 q^{9} +O(q^{10})$	$q+8.31879i q^{3} +1.48242i q^{5} +13.7963i q^{7} +11.7977 q^{9} +10.2025 q^{11} +98.4183 q^{13} -12.3320 q^{15} -286.515 q^{17} +367.004i q^{19} -114.768 q^{21} +242.039 q^{23} +622.802 q^{25} +771.965i q^{27} -1147.40i q^{29} -895.225 q^{31} +84.8728i q^{33} -20.4519 q^{35} +2295.69i q^{37} +818.722i q^{39} +1692.26 q^{41} +(1546.34 + 1013.73i) q^{43} +17.4892i q^{45} -743.419 q^{47} +2210.66 q^{49} -2383.46i q^{51} +99.3399 q^{53} +15.1245i q^{55} -3053.03 q^{57} -3286.35 q^{59} +3223.22i q^{61} +162.764i q^{63} +145.898i q^{65} -5556.36 q^{67} +2013.47i q^{69} +2953.33i q^{71} +3618.34i q^{73} +5180.96i q^{75} +140.757i q^{77} -2380.74 q^{79} -5466.20 q^{81} +6427.91 q^{83} -424.736i q^{85} +9544.98 q^{87} -3293.36i q^{89} +1357.80i q^{91} -7447.19i q^{93} -544.055 q^{95} -9329.95 q^{97} +120.367 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 462 q^{9}+O(q^{10}) $ 12 * q - 462 * q^9	$ 12 q - 462 q^{9} + 180 q^{11} - 216 q^{13} + 92 q^{15} + 678 q^{17} - 2392 q^{21} - 1566 q^{23} - 174 q^{25} - 5710 q^{31} - 936 q^{35} + 4878 q^{41} + 1108 q^{43} + 5526 q^{47} - 8544 q^{49} + 1212 q^{53} - 7692 q^{57} - 14016 q^{59} + 1088 q^{67} - 24302 q^{79} - 23660 q^{81} + 7032 q^{83} - 17850 q^{87} - 606 q^{95} - 5842 q^{97} + 25924 q^{99}+O(q^{100}) $ 12 * q - 462 * q^9 + 180 * q^11 - 216 * q^13 + 92 * q^15 + 678 * q^17 - 2392 * q^21 - 1566 * q^23 - 174 * q^25 - 5710 * q^31 - 936 * q^35 + 4878 * q^41 + 1108 * q^43 + 5526 * q^47 - 8544 * q^49 + 1212 * q^53 - 7692 * q^57 - 14016 * q^59 + 1088 * q^67 - 24302 * q^79 - 23660 * q^81 + 7032 * q^83 - 17850 * q^87 - 606 * q^95 - 5842 * q^97 + 25924 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			8.31879i			0.924310i	0.886799	+	0.462155i	$0.152924\pi$
$3$			8.31879i			0.924310i	−0.886799	+	0.462155i	$0.847076\pi$
$4$		0			0
$5$			1.48242i			0.0592969i	0.999560	+	0.0296484i	$0.00943877\pi$
$5$			1.48242i			0.0592969i	−0.999560	+	0.0296484i	$0.990561\pi$
$6$		0			0
$7$			13.7963i			0.281556i	0.990041	+	0.140778i	$0.0449604\pi$
$7$			13.7963i			0.281556i	−0.990041	+	0.140778i	$0.955040\pi$
$8$		0			0
$9$	11.7977			0.145651
$10$		0			0
$11$	10.2025			0.0843185			0.0421592	−	0.999111i	$-0.486576\pi$
$11$	10.2025			0.0843185			0.0421592	+	0.999111i	$0.486576\pi$
$12$		0			0
$13$	98.4183			0.582357			0.291179	−	0.956669i	$-0.405953\pi$
$13$	98.4183			0.582357			0.291179	+	0.956669i	$0.405953\pi$
$14$		0			0
$15$	−12.3320			−0.0548087
$16$		0			0
$17$	−286.515			−0.991400			−0.495700	−	0.868494i	$-0.665089\pi$
$17$	−286.515			−0.991400			−0.495700	+	0.868494i	$0.665089\pi$
$18$		0			0
$19$			367.004i			1.01663i	0.861171	+	0.508316i	$0.169732\pi$
$19$			367.004i			1.01663i	−0.861171	+	0.508316i	$0.830268\pi$
$20$		0			0
$21$	−114.768			−0.260245
$22$		0			0
$23$	242.039			0.457540			0.228770	−	0.973480i	$-0.426530\pi$
$23$	242.039			0.457540			0.228770	+	0.973480i	$0.426530\pi$
$24$		0			0
$25$	622.802			0.996484
$26$		0			0
$27$			771.965i			1.05894i
$28$		0			0
$29$		−	1147.40i		−	1.36433i	−0.731199	−	0.682164i	$-0.761039\pi$
$29$		−	1147.40i		−	1.36433i	0.731199	−	0.682164i	$-0.238961\pi$
$30$		0			0
$31$	−895.225			−0.931556			−0.465778	−	0.884902i	$-0.654225\pi$
$31$	−895.225			−0.931556			−0.465778	+	0.884902i	$0.654225\pi$
$32$		0			0
$33$			84.8728i			0.0779364i
$34$		0			0
$35$	−20.4519			−0.0166954
$36$		0			0
$37$			2295.69i			1.67691i	0.544969	+	0.838456i	$0.316541\pi$
$37$			2295.69i			1.67691i	−0.544969	+	0.838456i	$0.683459\pi$
$38$		0			0
$39$			818.722i			0.538279i
$40$		0			0
$41$	1692.26			1.00670			0.503348	−	0.864084i	$-0.332101\pi$
$41$	1692.26			1.00670			0.503348	+	0.864084i	$0.332101\pi$
$42$		0			0
$43$	1546.34	+	1013.73i	0.836310	+	0.548257i
$43$	1546.34	+	1013.73i	0.836310	+	0.548257i
$44$		0			0
$45$			17.4892i			0.00863663i
$46$		0			0
$47$	−743.419			−0.336541			−0.168271	−	0.985741i	$-0.553818\pi$
$47$	−743.419			−0.336541			−0.168271	+	0.985741i	$0.553818\pi$
$48$		0			0
$49$	2210.66			0.920726
$50$		0			0
$51$		−	2383.46i		−	0.916361i
$52$		0			0
$53$	99.3399			0.0353649			0.0176824	−	0.999844i	$-0.494371\pi$
$53$	99.3399			0.0353649			0.0176824	+	0.999844i	$0.494371\pi$
$54$		0			0
$55$			15.1245i			0.00499982i
$56$		0			0
$57$	−3053.03			−0.939683
$58$		0			0
$59$	−3286.35			−0.944082			−0.472041	−	0.881577i	$-0.656483\pi$
$59$	−3286.35			−0.944082			−0.472041	+	0.881577i	$0.656483\pi$
$60$		0			0
$61$			3223.22i			0.866225i	0.901340	+	0.433112i	$0.142585\pi$
$61$			3223.22i			0.866225i	−0.901340	+	0.433112i	$0.857415\pi$
$62$		0			0
$63$			162.764i			0.0410089i
$64$		0			0
$65$			145.898i			0.0345320i
$66$		0			0
$67$	−5556.36			−1.23777			−0.618886	−	0.785481i	$-0.712416\pi$
$67$	−5556.36			−1.23777			−0.618886	+	0.785481i	$0.712416\pi$
$68$		0			0
$69$			2013.47i			0.422909i
$70$		0			0
$71$			2953.33i			0.585861i	0.956134	+	0.292930i	$0.0946305\pi$
$71$			2953.33i			0.585861i	−0.956134	+	0.292930i	$0.905370\pi$
$72$		0			0
$73$			3618.34i			0.678991i	0.940608	+	0.339496i	$0.110256\pi$
$73$			3618.34i			0.678991i	−0.940608	+	0.339496i	$0.889744\pi$
$74$		0			0
$75$			5180.96i			0.921060i
$76$		0			0
$77$			140.757i			0.0237404i
$78$		0			0
$79$	−2380.74			−0.381468			−0.190734	−	0.981642i	$-0.561087\pi$
$79$	−2380.74			−0.381468			−0.190734	+	0.981642i	$0.561087\pi$
$80$		0			0
$81$	−5466.20			−0.833135
$82$		0			0
$83$	6427.91			0.933069			0.466535	−	0.884503i	$-0.345502\pi$
$83$	6427.91			0.933069			0.466535	+	0.884503i	$0.345502\pi$
$84$		0			0
$85$		−	424.736i		−	0.0587869i
$86$		0			0
$87$	9544.98			1.26106
$88$		0			0
$89$		−	3293.36i		−	0.415776i	−0.978153	−	0.207888i	$-0.933341\pi$
$89$		−	3293.36i		−	0.415776i	0.978153	−	0.207888i	$-0.0666590\pi$
$90$		0			0
$91$			1357.80i			0.163966i
$92$		0			0
$93$		−	7447.19i		−	0.861046i
$94$		0			0
$95$	−544.055			−0.0602831
$96$		0			0
$97$	−9329.95			−0.991599			−0.495799	−	0.868437i	$-0.665125\pi$
$97$	−9329.95			−0.991599			−0.495799	+	0.868437i	$0.665125\pi$
$98$		0			0
$99$	120.367			0.0122811
$100$		0			0
$101$	−13572.8			−1.33054			−0.665270	−	0.746603i	$-0.731683\pi$
$101$	−13572.8			−1.33054			−0.665270	+	0.746603i	$0.731683\pi$
$102$		0			0
$103$	5527.90			0.521057			0.260529	−	0.965466i	$-0.416103\pi$
$103$	5527.90			0.521057			0.260529	+	0.965466i	$0.416103\pi$
$104$		0			0
$105$		−	170.135i		−	0.0154317i
$106$		0			0
$107$	2435.69			0.212742			0.106371	−	0.994326i	$-0.466077\pi$
$107$	2435.69			0.212742			0.106371	+	0.994326i	$0.466077\pi$
$108$		0			0
$109$	3235.93			0.272362			0.136181	−	0.990684i	$-0.456517\pi$
$109$	3235.93			0.272362			0.136181	+	0.990684i	$0.456517\pi$
$110$		0			0
$111$	−19097.4			−1.54999
$112$		0			0
$113$			14680.9i			1.14973i	0.818247	+	0.574867i	$0.194946\pi$
$113$			14680.9i			1.14973i	−0.818247	+	0.574867i	$0.805054\pi$
$114$		0			0
$115$			358.803i			0.0271307i
$116$		0			0
$117$	1161.11			0.0848207
$118$		0			0
$119$		−	3952.83i		−	0.279135i
$120$		0			0
$121$	−14536.9			−0.992890
$122$		0			0
$123$			14077.5i			0.930499i
$124$		0			0
$125$			1849.77i			0.118385i
$126$		0			0
$127$	−27000.0			−1.67400			−0.837001	−	0.547201i	$-0.815693\pi$
$127$	−27000.0			−1.67400			−0.837001	+	0.547201i	$0.815693\pi$
$128$		0			0
$129$	−8432.98	+	12863.7i	−0.506759	+	0.773010i
$130$		0			0
$131$		−	17222.3i		−	1.00357i	−0.864991	−	0.501787i	$-0.832676\pi$
$131$		−	17222.3i		−	1.00357i	0.864991	−	0.501787i	$-0.167324\pi$
$132$		0			0
$133$	−5063.28			−0.286239
$134$		0			0
$135$	−1144.38			−0.0627916
$136$		0			0
$137$		−	1880.48i		−	0.100191i	−0.998744	−	0.0500955i	$-0.984047\pi$
$137$		−	1880.48i		−	0.100191i	0.998744	−	0.0500955i	$-0.0159526\pi$
$138$		0			0
$139$	−20668.9			−1.06977			−0.534883	−	0.844926i	$-0.679644\pi$
$139$	−20668.9			−1.06977			−0.534883	+	0.844926i	$0.679644\pi$
$140$		0			0
$141$		−	6184.35i		−	0.311068i
$142$		0			0
$143$	1004.12			0.0491035
$144$		0			0
$145$	1700.93			0.0809004
$146$		0			0
$147$			18390.0i			0.851036i
$148$		0			0
$149$		−	13651.5i		−	0.614904i	−0.951564	−	0.307452i	$-0.900524\pi$
$149$		−	13651.5i		−	0.614904i	0.951564	−	0.307452i	$-0.0994763\pi$
$150$		0			0
$151$		−	41870.6i		−	1.83635i	−0.396175	−	0.918175i	$-0.629662\pi$
$151$		−	41870.6i		−	1.83635i	0.396175	−	0.918175i	$-0.370338\pi$
$152$		0			0
$153$	−3380.22			−0.144398
$154$		0			0
$155$		−	1327.10i		−	0.0552383i
$156$		0			0
$157$			41348.7i			1.67750i	0.544517	+	0.838750i	$0.316713\pi$
$157$			41348.7i			1.67750i	−0.544517	+	0.838750i	$0.683287\pi$
$158$		0			0
$159$			826.388i			0.0326881i
$160$		0			0
$161$			3339.23i			0.128823i
$162$		0			0
$163$			6111.31i			0.230017i	0.993365	+	0.115008i	$0.0366895\pi$
$163$			6111.31i			0.230017i	−0.993365	+	0.115008i	$0.963311\pi$
$164$		0			0
$165$	−125.817			−0.00462139
$166$		0			0
$167$	−34991.0			−1.25465			−0.627326	−	0.778757i	$-0.715851\pi$
$167$	−34991.0			−1.25465			−0.627326	+	0.778757i	$0.715851\pi$
$168$		0			0
$169$	−18874.8			−0.660860
$170$		0			0
$171$			4329.81i			0.148073i
$172$		0			0
$173$	−4597.15			−0.153602			−0.0768009	−	0.997046i	$-0.524471\pi$
$173$	−4597.15			−0.153602			−0.0768009	+	0.997046i	$0.524471\pi$
$174$		0			0
$175$			8592.34i			0.280566i
$176$		0			0
$177$		−	27338.5i		−	0.872624i
$178$		0			0
$179$			26315.3i			0.821301i	0.911793	+	0.410650i	$0.134698\pi$
$179$			26315.3i			0.821301i	−0.911793	+	0.410650i	$0.865302\pi$
$180$		0			0
$181$	48133.4			1.46923			0.734614	−	0.678485i	$-0.237363\pi$
$181$	48133.4			1.46923			0.734614	+	0.678485i	$0.237363\pi$
$182$		0			0
$183$	−26813.3			−0.800660
$184$		0			0
$185$	−3403.19			−0.0994357
$186$		0			0
$187$	−2923.18			−0.0835934
$188$		0			0
$189$	−10650.2			−0.298150
$190$		0			0
$191$			46954.1i			1.28708i	0.765411	+	0.643542i	$0.222536\pi$
$191$			46954.1i			1.28708i	−0.765411	+	0.643542i	$0.777464\pi$
$192$		0			0
$193$	18967.8			0.509215			0.254608	−	0.967044i	$-0.418054\pi$
$193$	18967.8			0.509215			0.254608	+	0.967044i	$0.418054\pi$
$194$		0			0
$195$	−1213.69			−0.0319182
$196$		0			0
$197$	−77177.1			−1.98864			−0.994320	−	0.106435i	$-0.966056\pi$
$197$	−77177.1			−1.98864			−0.994320	+	0.106435i	$0.966056\pi$
$198$		0			0
$199$		−	2250.86i		−	0.0568385i	−0.999596	−	0.0284192i	$-0.990953\pi$
$199$		−	2250.86i		−	0.0568385i	0.999596	−	0.0284192i	$-0.00904734\pi$
$200$		0			0
$201$		−	46222.2i		−	1.14409i
$202$		0			0
$203$	15829.8			0.384135
$204$		0			0
$205$			2508.64i			0.0596939i
$206$		0			0
$207$	2855.50			0.0666411
$208$		0			0
$209$			3744.37i			0.0857208i
$210$		0			0
$211$			64198.9i			1.44199i	0.692939	+	0.720996i	$0.256315\pi$
$211$			64198.9i			1.44199i	−0.692939	+	0.720996i	$0.743685\pi$
$212$		0			0
$213$	−24568.1			−0.541517
$214$		0			0
$215$	−1502.77	+	2292.32i	−0.0325099	+	0.0495906i
$216$		0			0
$217$		−	12350.8i		−	0.262285i
$218$		0			0
$219$	−30100.2			−0.627598
$220$		0			0
$221$	−28198.3			−0.577349
$222$		0			0
$223$		−	53407.6i		−	1.07397i	−0.843591	−	0.536987i	$-0.819563\pi$
$223$		−	53407.6i		−	1.07397i	0.843591	−	0.536987i	$-0.180437\pi$
$224$		0			0
$225$	7347.64			0.145139
$226$		0			0
$227$			43494.8i			0.844084i	0.906576	+	0.422042i	$0.138687\pi$
$227$			43494.8i			0.844084i	−0.906576	+	0.422042i	$0.861313\pi$
$228$		0			0
$229$	67145.4			1.28040			0.640199	−	0.768209i	$-0.278852\pi$
$229$	67145.4			1.28040			0.640199	+	0.768209i	$0.278852\pi$
$230$		0			0
$231$	−1170.93			−0.0219435
$232$		0			0
$233$			76734.0i			1.41344i	0.707496	+	0.706718i	$0.249825\pi$
$233$			76734.0i			1.41344i	−0.707496	+	0.706718i	$0.750175\pi$
$234$		0			0
$235$		−	1102.06i		−	0.0199558i
$236$		0			0
$237$		−	19804.9i		−	0.352595i
$238$		0			0
$239$	26767.6			0.468613			0.234307	−	0.972163i	$-0.424718\pi$
$239$	26767.6			0.468613			0.234307	+	0.972163i	$0.424718\pi$
$240$		0			0
$241$			100445.i			1.72939i	0.502297	+	0.864695i	$0.332488\pi$
$241$			100445.i			1.72939i	−0.502297	+	0.864695i	$0.667512\pi$
$242$		0			0
$243$			17057.0i			0.288861i
$244$		0			0
$245$			3277.14i			0.0545962i
$246$		0			0
$247$			36119.9i			0.592043i
$248$		0			0
$249$			53472.5i			0.862445i
$250$		0			0
$251$	71737.6			1.13867			0.569337	−	0.822104i	$-0.307200\pi$
$251$	71737.6			1.13867			0.569337	+	0.822104i	$0.307200\pi$
$252$		0			0
$253$	2469.41			0.0385791
$254$		0			0
$255$	3533.29			0.0543374
$256$		0			0
$257$		−	7267.88i		−	0.110038i	−0.998485	−	0.0550189i	$-0.982478\pi$
$257$		−	7267.88i		−	0.110038i	0.998485	−	0.0550189i	$-0.0175219\pi$
$258$		0			0
$259$	−31672.0			−0.472145
$260$		0			0
$261$		−	13536.7i		−	0.198715i
$262$		0			0
$263$		−	73838.4i		−	1.06751i	−0.845640	−	0.533754i	$-0.820781\pi$
$263$		−	73838.4i		−	1.06751i	0.845640	−	0.533754i	$-0.179219\pi$
$264$		0			0
$265$			147.264i			0.00209703i
$266$		0			0
$267$	27396.8			0.384306
$268$		0			0
$269$	127266.			1.75877			0.879384	−	0.476113i	$-0.157955\pi$
$269$	127266.			1.75877			0.879384	+	0.476113i	$0.157955\pi$
$270$		0			0
$271$	−44182.9			−0.601610			−0.300805	−	0.953686i	$-0.597255\pi$
$271$	−44182.9			−0.601610			−0.300805	+	0.953686i	$0.597255\pi$
$272$		0			0
$273$	−11295.3			−0.151556
$274$		0			0
$275$	6354.17			0.0840220
$276$		0			0
$277$		−	31197.7i		−	0.406596i	−0.979117	−	0.203298i	$-0.934834\pi$
$277$		−	31197.7i		−	0.406596i	0.979117	−	0.203298i	$-0.0651660\pi$
$278$		0			0
$279$	−10561.6			−0.135682
$280$		0			0
$281$	−140212.			−1.77572			−0.887859	−	0.460116i	$-0.847808\pi$
$281$	−140212.			−1.77572			−0.887859	+	0.460116i	$0.847808\pi$
$282$		0			0
$283$	−2024.17			−0.0252740			−0.0126370	−	0.999920i	$-0.504023\pi$
$283$	−2024.17			−0.0252740			−0.0126370	+	0.999920i	$0.504023\pi$
$284$		0			0
$285$		−	4525.88i		−	0.0557203i
$286$		0			0
$287$			23346.8i			0.283442i
$288$		0			0
$289$	−1430.31			−0.0171252
$290$		0			0
$291$		−	77613.9i		−	0.916545i
$292$		0			0
$293$	75732.9			0.882164			0.441082	−	0.897467i	$-0.354595\pi$
$293$	75732.9			0.882164			0.441082	+	0.897467i	$0.354595\pi$
$294$		0			0
$295$		−	4871.76i		−	0.0559811i
$296$		0			0
$297$			7876.00i			0.0892879i
$298$		0			0
$299$	23821.0			0.266452
$300$		0			0
$301$	−13985.6	+	21333.7i	−0.154365	+	0.235468i
$302$		0			0
$303$		−	112910.i		−	1.22983i
$304$		0			0
$305$	−4778.18			−0.0513644
$306$		0			0
$307$	158838.			1.68530			0.842652	−	0.538458i	$-0.180993\pi$
$307$	158838.			1.68530			0.842652	+	0.538458i	$0.180993\pi$
$308$		0			0
$309$			45985.4i			0.481619i
$310$		0			0
$311$	49344.1			0.510169			0.255085	−	0.966919i	$-0.417897\pi$
$311$	49344.1			0.510169			0.255085	+	0.966919i	$0.417897\pi$
$312$		0			0
$313$		−	66350.6i		−	0.677261i	−0.940919	−	0.338631i	$-0.890036\pi$
$313$		−	66350.6i		−	0.677261i	0.940919	−	0.338631i	$-0.109964\pi$
$314$		0			0
$315$	−241.285			−0.00243170
$316$		0			0
$317$	15717.4			0.156409			0.0782046	−	0.996937i	$-0.475081\pi$
$317$	15717.4			0.156409			0.0782046	+	0.996937i	$0.475081\pi$
$318$		0			0
$319$		−	11706.4i		−	0.115038i
$320$		0			0
$321$			20262.0i			0.196640i
$322$		0			0
$323$		−	105152.i		−	1.00789i
$324$		0			0
$325$	61295.2			0.580309
$326$		0			0
$327$			26919.1i			0.251747i
$328$		0			0
$329$		−	10256.4i		−	0.0947553i
$330$		0			0
$331$			85452.6i			0.779954i	0.920825	+	0.389977i	$0.127517\pi$
$331$			85452.6i			0.779954i	−0.920825	+	0.389977i	$0.872483\pi$
$332$		0			0
$333$			27083.9i			0.244244i
$334$		0			0
$335$		−	8236.87i		−	0.0733960i
$336$		0			0
$337$	−146925.			−1.29371			−0.646855	−	0.762613i	$-0.723916\pi$
$337$	−146925.			−1.29371			−0.646855	+	0.762613i	$0.723916\pi$
$338$		0			0
$339$	−122128.			−1.06271
$340$		0			0
$341$	−9133.57			−0.0785474
$342$		0			0
$343$			63623.7i			0.540793i
$344$		0			0
$345$	−2984.81			−0.0250772
$346$		0			0
$347$			78623.4i			0.652969i	0.945203	+	0.326485i	$0.105864\pi$
$347$			78623.4i			0.652969i	−0.945203	+	0.326485i	$0.894136\pi$
$348$		0			0
$349$		−	165981.i		−	1.36273i	−0.731945	−	0.681363i	$-0.761387\pi$
$349$		−	165981.i		−	1.36273i	0.731945	−	0.681363i	$-0.238613\pi$
$350$		0			0
$351$			75975.5i			0.616679i
$352$		0			0
$353$	53106.9			0.426188			0.213094	−	0.977032i	$-0.431646\pi$
$353$	53106.9			0.426188			0.213094	+	0.977032i	$0.431646\pi$
$354$		0			0
$355$	−4378.07			−0.0347397
$356$		0			0
$357$	32882.8			0.258007
$358$		0			0
$359$	120101.			0.931879			0.465939	−	0.884817i	$-0.345716\pi$
$359$	120101.			0.931879			0.465939	+	0.884817i	$0.345716\pi$
$360$		0			0
$361$	−4370.92			−0.0335396
$362$		0			0
$363$		−	120930.i		−	0.917739i
$364$		0			0
$365$	−5363.91			−0.0402620
$366$		0			0
$367$	37702.0			0.279919			0.139959	−	0.990157i	$-0.455303\pi$
$367$	37702.0			0.279919			0.139959	+	0.990157i	$0.455303\pi$
$368$		0			0
$369$	19964.7			0.146626
$370$		0			0
$371$			1370.52i			0.00995720i
$372$		0			0
$373$		−	139388.i		−	1.00186i	−0.865488	−	0.500930i	$-0.832991\pi$
$373$		−	139388.i		−	1.00186i	0.865488	−	0.500930i	$-0.167009\pi$
$374$		0			0
$375$	−15387.8			−0.109425
$376$		0			0
$377$		−	112925.i		−	0.794526i
$378$		0			0
$379$	88126.5			0.613519			0.306759	−	0.951787i	$-0.400755\pi$
$379$	88126.5			0.613519			0.306759	+	0.951787i	$0.400755\pi$
$380$		0			0
$381$		−	224607.i		−	1.54730i
$382$		0			0
$383$		−	250818.i		−	1.70986i	−0.518743	−	0.854930i	$-0.673600\pi$
$383$		−	250818.i		−	1.70986i	0.518743	−	0.854930i	$-0.326400\pi$
$384$		0			0
$385$	−208.661			−0.00140773
$386$		0			0
$387$	18243.2	+	11959.7i	0.121809	+	0.0798540i
$388$		0			0
$389$			34015.7i			0.224792i	0.993664	+	0.112396i	$0.0358524\pi$
$389$			34015.7i			0.224792i	−0.993664	+	0.112396i	$0.964148\pi$
$390$		0			0
$391$	−69347.7			−0.453605
$392$		0			0
$393$	143269.			0.927614
$394$		0			0
$395$		−	3529.26i		−	0.0226199i
$396$		0			0
$397$	206559.			1.31058			0.655291	−	0.755377i	$-0.272546\pi$
$397$	206559.			1.31058			0.655291	+	0.755377i	$0.272546\pi$
$398$		0			0
$399$		−	42120.4i		−	0.264574i
$400$		0			0
$401$	85162.0			0.529611			0.264806	−	0.964302i	$-0.414692\pi$
$401$	85162.0			0.529611			0.264806	+	0.964302i	$0.414692\pi$
$402$		0			0
$403$	−88106.5			−0.542498
$404$		0			0
$405$		−	8103.21i		−	0.0494023i
$406$		0			0
$407$			23421.9i			0.141395i
$408$		0			0
$409$			216421.i			1.29375i	0.762594	+	0.646877i	$0.223925\pi$
$409$			216421.i			1.29375i	−0.762594	+	0.646877i	$0.776075\pi$
$410$		0			0
$411$	15643.3			0.0926075
$412$		0			0
$413$		−	45339.3i		−	0.265812i
$414$		0			0
$415$			9528.88i			0.0553281i
$416$		0			0
$417$		−	171941.i		−	0.988795i
$418$		0			0
$419$		−	195727.i		−	1.11487i	−0.830222	−	0.557433i	$-0.811786\pi$
$419$		−	195727.i		−	1.11487i	0.830222	−	0.557433i	$-0.188214\pi$
$420$		0			0
$421$		−	216454.i		−	1.22124i	−0.791924	−	0.610620i	$-0.790920\pi$
$421$		−	216454.i		−	1.22124i	0.791924	−	0.610620i	$-0.209080\pi$
$422$		0			0
$423$	−8770.65			−0.0490175
$424$		0			0
$425$	−178442.			−0.987915
$426$		0			0
$427$	−44468.4			−0.243891
$428$		0			0
$429$			8353.04i			0.0453868i
$430$		0			0
$431$	219971.			1.18416			0.592082	−	0.805878i	$-0.298306\pi$
$431$	219971.			1.18416			0.592082	+	0.805878i	$0.298306\pi$
$432$		0			0
$433$			141123.i			0.752699i	0.926478	+	0.376349i	$0.122821\pi$
$433$			141123.i			0.752699i	−0.926478	+	0.376349i	$0.877179\pi$
$434$		0			0
$435$			14149.7i			0.0747770i
$436$		0			0
$437$			88829.2i			0.465150i
$438$		0			0
$439$	18660.8			0.0968279			0.0484139	−	0.998827i	$-0.484583\pi$
$439$	18660.8			0.0968279			0.0484139	+	0.998827i	$0.484583\pi$
$440$		0			0
$441$	26080.8			0.134104
$442$		0			0
$443$	−21366.2			−0.108873			−0.0544366	−	0.998517i	$-0.517336\pi$
$443$	−21366.2			−0.108873			−0.0544366	+	0.998517i	$0.517336\pi$
$444$		0			0
$445$	4882.15			0.0246542
$446$		0			0
$447$	113564.			0.568362
$448$		0			0
$449$			101517.i			0.503553i	0.967785	+	0.251777i	$0.0810148\pi$
$449$			101517.i			0.503553i	−0.967785	+	0.251777i	$0.918985\pi$
$450$		0			0
$451$	17265.3			0.0848831
$452$		0			0
$453$	348313.			1.69736
$454$		0			0
$455$	−2012.84			−0.00972269
$456$		0			0
$457$		−	308003.i		−	1.47477i	−0.675475	−	0.737383i	$-0.736061\pi$
$457$		−	308003.i		−	1.47477i	0.675475	−	0.737383i	$-0.263939\pi$
$458$		0			0
$459$		−	221179.i		−	1.04983i
$460$		0			0
$461$	156343.			0.735660			0.367830	−	0.929893i	$-0.380101\pi$
$461$	156343.			0.735660			0.367830	+	0.929893i	$0.380101\pi$
$462$		0			0
$463$			203393.i			0.948800i	0.880309	+	0.474400i	$0.157335\pi$
$463$			203393.i			0.948800i	−0.880309	+	0.474400i	$0.842665\pi$
$464$		0			0
$465$	11039.9			0.0510573
$466$		0			0
$467$			208975.i			0.958209i	0.877758	+	0.479104i	$0.159038\pi$
$467$			208975.i			0.958209i	−0.877758	+	0.479104i	$0.840962\pi$
$468$		0			0
$469$		−	76657.0i		−	0.348503i
$470$		0			0
$471$	−343971.			−1.55053
$472$		0			0
$473$	15776.6	+	10342.6i	0.0705164	+	0.0462282i
$474$		0			0
$475$			228571.i			1.01306i
$476$		0			0
$477$	1171.98			0.00515092
$478$		0			0
$479$	−98547.6			−0.429512			−0.214756	−	0.976668i	$-0.568896\pi$
$479$	−98547.6			−0.429512			−0.214756	+	0.976668i	$0.568896\pi$
$480$		0			0
$481$			225938.i			0.976562i
$482$		0			0
$483$	−27778.3			−0.119073
$484$		0			0
$485$		−	13830.9i		−	0.0587987i
$486$		0			0
$487$	319902.			1.34884			0.674418	−	0.738350i	$-0.264395\pi$
$487$	319902.			1.34884			0.674418	+	0.738350i	$0.264395\pi$
$488$		0			0
$489$	−50838.7			−0.212607
$490$		0			0
$491$		−	454263.i		−	1.88428i	−0.335226	−	0.942138i	$-0.608813\pi$
$491$		−	454263.i		−	1.88428i	0.335226	−	0.942138i	$-0.391187\pi$
$492$		0			0
$493$			328747.i			1.35260i
$494$		0			0
$495$			178.434i			0.000728228i
$496$		0			0
$497$	−40744.8			−0.164953
$498$		0			0
$499$		−	336510.i		−	1.35144i	−0.737158	−	0.675720i	$-0.763833\pi$
$499$		−	336510.i		−	1.35144i	0.737158	−	0.675720i	$-0.236167\pi$
$500$		0			0
$501$		−	291083.i		−	1.15969i
$502$		0			0
$503$			313054.i			1.23732i	0.785658	+	0.618661i	$0.212325\pi$
$503$			313054.i			1.23732i	−0.785658	+	0.618661i	$0.787675\pi$
$504$		0			0
$505$		−	20120.7i		−	0.0788968i
$506$		0			0
$507$		−	157016.i		−	0.610840i
$508$		0			0
$509$	406043.			1.56724			0.783621	−	0.621239i	$-0.213370\pi$
$509$	406043.			1.56724			0.783621	+	0.621239i	$0.213370\pi$
$510$		0			0
$511$	−49919.6			−0.191174
$512$		0			0
$513$	−283314.			−1.07655
$514$		0			0
$515$			8194.68i			0.0308971i
$516$		0			0
$517$	−7584.76			−0.0283766
$518$		0			0
$519$		−	38242.7i		−	0.141976i
$520$		0			0
$521$			221529.i			0.816122i	0.912955	+	0.408061i	$0.133795\pi$
$521$			221529.i			0.816122i	−0.912955	+	0.408061i	$0.866205\pi$
$522$		0			0
$523$		−	146666.i		−	0.536199i	−0.963391	−	0.268099i	$-0.913604\pi$
$523$		−	146666.i		−	0.536199i	0.963391	−	0.268099i	$-0.0863955\pi$
$524$		0			0
$525$	−71477.9			−0.259330
$526$		0			0
$527$	256495.			0.923545
$528$		0			0
$529$	−221258.			−0.790657
$530$		0			0
$531$	−38771.4			−0.137506
$532$		0			0
$533$	166549.			0.586257
$534$		0			0
$535$			3610.71i			0.0126150i
$536$		0			0
$537$	−218911.			−0.759137
$538$		0			0
$539$	22554.4			0.0776342
$540$		0			0
$541$	38164.1			0.130395			0.0651974	−	0.997872i	$-0.479232\pi$
$541$	38164.1			0.130395			0.0651974	+	0.997872i	$0.479232\pi$
$542$		0			0
$543$			400411.i			1.35802i
$544$		0			0
$545$			4797.02i			0.0161502i
$546$		0			0
$547$	−105758.			−0.353459			−0.176730	−	0.984259i	$-0.556552\pi$
$547$	−105758.			−0.353459			−0.176730	+	0.984259i	$0.556552\pi$
$548$		0			0
$549$			38026.6i			0.126166i
$550$		0			0
$551$	421100.			1.38702
$552$		0			0
$553$		−	32845.3i		−	0.107405i
$554$		0			0
$555$		−	28310.4i		−	0.0919094i
$556$		0			0
$557$	280135.			0.902936			0.451468	−	0.892287i	$-0.350900\pi$
$557$	280135.			0.902936			0.451468	+	0.892287i	$0.350900\pi$
$558$		0			0
$559$	152188.	+	99769.3i	0.487031	+	0.319281i
$560$		0			0
$561$		−	24317.3i		−	0.0772662i
$562$		0			0
$563$	403332.			1.27247			0.636233	−	0.771497i	$-0.280492\pi$
$563$	403332.			1.27247			0.636233	+	0.771497i	$0.280492\pi$
$564$		0			0
$565$	−21763.3			−0.0681756
$566$		0			0
$567$		−	75413.1i		−	0.234574i
$568$		0			0
$569$	−72865.2			−0.225059			−0.112529	−	0.993648i	$-0.535895\pi$
$569$	−72865.2			−0.225059			−0.112529	+	0.993648i	$0.535895\pi$
$570$		0			0
$571$			280074.i			0.859015i	0.903063	+	0.429507i	$0.141313\pi$
$571$			280074.i			0.859015i	−0.903063	+	0.429507i	$0.858687\pi$
$572$		0			0
$573$	−390601.			−1.18966
$574$		0			0
$575$	150742.			0.455931
$576$		0			0
$577$			205999.i			0.618748i	0.950940	+	0.309374i	$0.100119\pi$
$577$			205999.i			0.618748i	−0.950940	+	0.309374i	$0.899881\pi$
$578$		0			0
$579$			157789.i			0.470673i
$580$		0			0
$581$			88681.2i			0.262712i
$582$		0			0
$583$	1013.52			0.00298191
$584$		0			0
$585$			1721.26i			0.00502960i
$586$		0			0
$587$			466185.i			1.35295i	0.736464	+	0.676476i	$0.236494\pi$
$587$			466185.i			1.35295i	−0.736464	+	0.676476i	$0.763506\pi$
$588$		0			0
$589$		−	328551.i		−	0.947049i
$590$		0			0
$591$		−	642020.i		−	1.83812i
$592$		0			0
$593$			284311.i			0.808507i	0.914647	+	0.404253i	$0.132469\pi$
$593$			284311.i			0.808507i	−0.914647	+	0.404253i	$0.867531\pi$
$594$		0			0
$595$	5859.76			0.0165518
$596$		0			0
$597$	18724.4			0.0525364
$598$		0			0
$599$	−438880.			−1.22318			−0.611592	−	0.791173i	$-0.709471\pi$
$599$	−438880.			−1.22318			−0.611592	+	0.791173i	$0.709471\pi$
$600$		0			0
$601$			52157.2i			0.144399i	0.997390	+	0.0721997i	$0.0230019\pi$
$601$			52157.2i			0.144399i	−0.997390	+	0.0721997i	$0.976998\pi$
$602$		0			0
$603$	−65552.3			−0.180282
$604$		0			0
$605$		−	21549.8i		−	0.0588753i
$606$		0			0
$607$			328500.i			0.891575i	0.895139	+	0.445788i	$0.147076\pi$
$607$			328500.i			0.891575i	−0.895139	+	0.445788i	$0.852924\pi$
$608$		0			0
$609$			131685.i			0.355060i
$610$		0			0
$611$	−73166.1			−0.195987
$612$		0			0
$613$	−194249.			−0.516937			−0.258468	−	0.966020i	$-0.583218\pi$
$613$	−194249.			−0.516937			−0.258468	+	0.966020i	$0.583218\pi$
$614$		0			0
$615$	−20868.8			−0.0551757
$616$		0			0
$617$	−203384.			−0.534253			−0.267126	−	0.963662i	$-0.586074\pi$
$617$	−203384.			−0.534253			−0.267126	+	0.963662i	$0.586074\pi$
$618$		0			0
$619$	−504248.			−1.31602			−0.658011	−	0.753008i	$-0.728602\pi$
$619$	−504248.			−1.31602			−0.658011	+	0.753008i	$0.728602\pi$
$620$		0			0
$621$			186845.i			0.484506i
$622$		0			0
$623$	45436.1			0.117064
$624$		0			0
$625$	386509.			0.989464
$626$		0			0
$627$	−31148.6			−0.0792326
$628$		0			0
$629$		−	657750.i		−	1.66249i
$630$		0			0
$631$			612948.i			1.53945i	0.638376	+	0.769724i	$0.279606\pi$
$631$			612948.i			1.53945i	−0.638376	+	0.769724i	$0.720394\pi$
$632$		0			0
$633$	−534057.			−1.33285
$634$		0			0
$635$		−	40025.4i		−	0.0992631i
$636$		0			0
$637$	217570.			0.536191
$638$		0			0
$639$			34842.5i			0.0853311i
$640$		0			0
$641$			571747.i			1.39151i	0.718277	+	0.695757i	$0.244931\pi$
$641$			571747.i			1.39151i	−0.718277	+	0.695757i	$0.755069\pi$
$642$		0			0
$643$	530650.			1.28347			0.641736	−	0.766926i	$-0.278215\pi$
$643$	530650.			1.28347			0.641736	+	0.766926i	$0.278215\pi$
$644$		0			0
$645$	−19069.4	−	12501.2i	−0.0458371	−	0.0300492i
$646$		0			0
$647$		−	481379.i		−	1.14995i	−0.818171	−	0.574975i	$-0.805012\pi$
$647$		−	481379.i		−	1.14995i	0.818171	−	0.574975i	$-0.194988\pi$
$648$		0			0
$649$	−33529.1			−0.0796036
$650$		0			0
$651$	102743.			0.242433
$652$		0			0
$653$			293363.i			0.687984i	0.938973	+	0.343992i	$0.111779\pi$
$653$			293363.i			0.687984i	−0.938973	+	0.343992i	$0.888221\pi$
$654$		0			0
$655$	25530.8			0.0595088
$656$		0			0
$657$			42688.2i			0.0988956i
$658$		0			0
$659$	−657906.			−1.51493			−0.757466	−	0.652875i	$-0.773563\pi$
$659$	−657906.			−1.51493			−0.757466	+	0.652875i	$0.773563\pi$
$660$		0			0
$661$	−267844.			−0.613026			−0.306513	−	0.951866i	$-0.599162\pi$
$661$	−267844.			−0.613026			−0.306513	+	0.951866i	$0.599162\pi$
$662$		0			0
$663$		−	234576.i		−	0.533650i
$664$		0			0
$665$		−	7505.92i		−	0.0169731i
$666$		0			0
$667$		−	277715.i		−	0.624235i
$668$		0			0
$669$	444287.			0.992685
$670$		0			0
$671$			32885.0i			0.0730388i
$672$		0			0
$673$		−	48089.2i		−	0.106174i	−0.998590	−	0.0530869i	$-0.983094\pi$
$673$		−	48089.2i		−	0.106174i	0.998590	−	0.0530869i	$-0.0169060\pi$
$674$		0			0
$675$			480782.i			1.05521i
$676$		0			0
$677$			84879.3i			0.185193i	0.995704	+	0.0925964i	$0.0295166\pi$
$677$			84879.3i			0.185193i	−0.995704	+	0.0925964i	$0.970483\pi$
$678$		0			0
$679$		−	128718.i		−	0.279191i
$680$		0			0
$681$	−361824.			−0.780195
$682$		0			0
$683$	707010.			1.51560			0.757799	−	0.652488i	$-0.226275\pi$
$683$	707010.			1.51560			0.757799	+	0.652488i	$0.226275\pi$
$684$		0			0
$685$	2787.67			0.00594101
$686$		0			0
$687$			558568.i			1.18348i
$688$		0			0
$689$	9776.87			0.0205950
$690$		0			0
$691$		−	316021.i		−	0.661851i	−0.943657	−	0.330925i	$-0.892639\pi$
$691$		−	316021.i		−	0.661851i	0.943657	−	0.330925i	$-0.107361\pi$
$692$		0			0
$693$			1660.61i			0.00345781i
$694$		0			0
$695$		−	30640.1i		−	0.0634338i
$696$		0			0
$697$	−484856.			−0.998039
$698$		0			0
$699$	−638334.			−1.30645
$700$		0			0
$701$	790342.			1.60835			0.804173	−	0.594396i	$-0.202609\pi$
$701$	790342.			1.60835			0.804173	+	0.594396i	$0.202609\pi$
$702$		0			0
$703$	−842529.			−1.70480
$704$		0			0
$705$	9167.81			0.0184454
$706$		0			0
$707$		−	187254.i		−	0.374622i
$708$		0			0
$709$	218246.			0.434165			0.217082	−	0.976153i	$-0.430346\pi$
$709$	218246.			0.434165			0.217082	+	0.976153i	$0.430346\pi$
$710$		0			0
$711$	−28087.3			−0.0555611
$712$		0			0
$713$	−216679.			−0.426224
$714$		0			0
$715$			1488.52i			0.00291168i
$716$		0			0
$717$			222674.i			0.433144i
$718$		0			0
$719$	−733807.			−1.41946			−0.709732	−	0.704472i	$-0.751184\pi$
$719$	−733807.			−1.41946			−0.709732	+	0.704472i	$0.751184\pi$
$720$		0			0
$721$			76264.3i			0.146707i
$722$		0			0
$723$	−835578.			−1.59849
$724$		0			0
$725$		−	714603.i		−	1.35953i
$726$		0			0
$727$		−	250259.i		−	0.473501i	−0.971570	−	0.236750i	$-0.923918\pi$
$727$		−	250259.i		−	0.473501i	0.971570	−	0.236750i	$-0.0760824\pi$
$728$		0			0
$729$	−584656.			−1.10013
$730$		0			0
$731$	−443048.	−	290448.i	−0.829118	−	0.543542i
$732$		0			0
$733$			610120.i			1.13555i	0.823183	+	0.567776i	$0.192196\pi$
$733$			610120.i			1.13555i	−0.823183	+	0.567776i	$0.807804\pi$
$734$		0			0
$735$	−27261.8			−0.0504638
$736$		0			0
$737$	−56689.0			−0.104367
$738$		0			0
$739$		−	67964.4i		−	0.124449i	−0.998062	−	0.0622246i	$-0.980180\pi$
$739$		−	67964.4i		−	0.124449i	0.998062	−	0.0622246i	$-0.0198195\pi$
$740$		0			0
$741$	−300474.			−0.547231
$742$		0			0
$743$			707585.i			1.28174i	0.767648	+	0.640871i	$0.221427\pi$
$743$			707585.i			1.28174i	−0.767648	+	0.640871i	$0.778573\pi$
$744$		0			0
$745$	20237.3			0.0364619
$746$		0			0
$747$	75834.7			0.135902
$748$		0			0
$749$			33603.4i			0.0598989i
$750$		0			0
$751$		−	613506.i		−	1.08778i	−0.839158	−	0.543888i	$-0.816952\pi$
$751$		−	613506.i		−	1.08778i	0.839158	−	0.543888i	$-0.183048\pi$
$752$		0			0
$753$			596770.i			1.05249i
$754$		0			0
$755$	62069.9			0.108890
$756$		0			0
$757$			132879.i			0.231880i	0.993256	+	0.115940i	$0.0369881\pi$
$757$			132879.i			0.231880i	−0.993256	+	0.115940i	$0.963012\pi$
$758$		0			0
$759$			20542.5i			0.0356590i
$760$		0			0
$761$		−	790123.i		−	1.36435i	−0.731190	−	0.682174i	$-0.761035\pi$
$761$		−	790123.i		−	1.36435i	0.731190	−	0.682174i	$-0.238965\pi$
$762$		0			0
$763$			44643.8i			0.0766853i
$764$		0			0
$765$		−	5010.91i		−	0.00856236i
$766$		0			0
$767$	−323437.			−0.549793
$768$		0			0
$769$	−204245.			−0.345382			−0.172691	−	0.984976i	$-0.555246\pi$
$769$	−204245.			−0.345382			−0.172691	+	0.984976i	$0.555246\pi$
$770$		0			0
$771$	60460.0			0.101709
$772$		0			0
$773$		−	824077.i		−	1.37914i	−0.724218	−	0.689571i	$-0.757799\pi$
$773$		−	824077.i		−	1.37914i	0.724218	−	0.689571i	$-0.242201\pi$
$774$		0			0
$775$	−557548.			−0.928280
$776$		0			0
$777$		−	263473.i		−	0.436409i
$778$		0			0
$779$			621065.i			1.02344i
$780$		0			0
$781$			30131.4i			0.0493989i
$782$		0			0
$783$	885752.			1.44474
$784$		0			0
$785$	−61296.2			−0.0994705
$786$		0			0
$787$	−347326.			−0.560774			−0.280387	−	0.959887i	$-0.590463\pi$
$787$	−347326.			−0.560774			−0.280387	+	0.959887i	$0.590463\pi$
$788$		0			0
$789$	614246.			0.986708
$790$		0			0
$791$	−202542.			−0.323715
$792$		0			0
$793$			317224.i			0.504452i
$794$		0			0
$795$	−1225.06			−0.00193830
$796$		0			0
$797$	−550267.			−0.866278			−0.433139	−	0.901327i	$-0.642594\pi$
$797$	−550267.			−0.866278			−0.433139	+	0.901327i	$0.642594\pi$
$798$		0			0
$799$	213001.			0.333647
$800$		0			0
$801$		−	38854.1i		−	0.0605581i
$802$		0			0
$803$			36916.3i			0.0572515i
$804$		0			0
$805$	−4950.15			−0.00763882
$806$		0			0
$807$			1.05870e6i			1.62565i
$808$		0			0
$809$	−237489.			−0.362866			−0.181433	−	0.983403i	$-0.558074\pi$
$809$	−237489.			−0.362866			−0.181433	+	0.983403i	$0.558074\pi$
$810$		0			0
$811$			1.00555e6i			1.52884i	0.644717	+	0.764422i	$0.276975\pi$
$811$			1.00555e6i			1.52884i	−0.644717	+	0.764422i	$0.723025\pi$
$812$		0			0
$813$		−	367548.i		−	0.556074i
$814$		0			0
$815$	−9059.54			−0.0136393
$816$		0			0
$817$	−372042.	+	567512.i	−0.557375	+	0.850219i
$818$		0			0
$819$			16019.0i			0.0238818i
$820$		0			0
$821$	−259037.			−0.384305			−0.192152	−	0.981365i	$-0.561547\pi$
$821$	−259037.			−0.384305			−0.192152	+	0.981365i	$0.561547\pi$
$822$		0			0
$823$	70048.9			0.103419			0.0517097	−	0.998662i	$-0.483533\pi$
$823$	70048.9			0.103419			0.0517097	+	0.998662i	$0.483533\pi$
$824$		0			0
$825$			52859.0i			0.0776624i
$826$		0			0
$827$	771588.			1.12817			0.564085	−	0.825717i	$-0.309229\pi$
$827$	771588.			1.12817			0.564085	+	0.825717i	$0.309229\pi$
$828$		0			0
$829$		−	131193.i		−	0.190898i	−0.995434	−	0.0954489i	$-0.969571\pi$
$829$		−	131193.i		−	0.190898i	0.995434	−	0.0954489i	$-0.0304286\pi$
$830$		0			0
$831$	259527.			0.375821
$832$		0			0
$833$	−633388.			−0.912808
$834$		0			0
$835$		−	51871.4i		−	0.0743970i
$836$		0			0
$837$		−	691082.i		−	0.986458i
$838$		0			0
$839$		−	473423.i		−	0.672552i	−0.941764	−	0.336276i	$-0.890833\pi$
$839$		−	473423.i		−	0.672552i	0.941764	−	0.336276i	$-0.109167\pi$
$840$		0			0
$841$	−609245.			−0.861390
$842$		0			0
$843$		−	1.16640e6i		−	1.64131i
$844$		0			0
$845$		−	27980.5i		−	0.0391869i
$846$		0			0
$847$		−	200555.i		−	0.279555i
$848$		0			0
$849$		−	16838.6i		−	0.0233610i
$850$		0			0
$851$			555647.i			0.767255i
$852$		0			0
$853$	−535509.			−0.735985			−0.367992	−	0.929829i	$-0.619955\pi$
$853$	−535509.			−0.735985			−0.367992	+	0.929829i	$0.619955\pi$
$854$		0			0
$855$	−6418.60			−0.00878027
$856$		0			0
$857$	1.28191e6			1.74540			0.872699	−	0.488258i	$-0.162368\pi$
$857$	1.28191e6			1.74540			0.872699	+	0.488258i	$0.162368\pi$
$858$		0			0
$859$		−	683886.i		−	0.926825i	−0.886143	−	0.463412i	$-0.846625\pi$
$859$		−	683886.i		−	0.926825i	0.886143	−	0.463412i	$-0.153375\pi$
$860$		0			0
$861$	−194217.			−0.261988
$862$		0			0
$863$		−	423128.i		−	0.568134i	−0.958804	−	0.284067i	$-0.908316\pi$
$863$		−	423128.i		−	0.568134i	0.958804	−	0.284067i	$-0.0916838\pi$
$864$		0			0
$865$		−	6814.91i		−	0.00910810i
$866$		0			0
$867$		−	11898.5i		−	0.0158290i
$868$		0			0
$869$	−24289.6			−0.0321648
$870$		0			0
$871$	−546848.			−0.720825
$872$		0			0
$873$	−110072.			−0.144427
$874$		0			0
$875$	−25519.9			−0.0333321
$876$		0			0
$877$	543060.			0.706071			0.353036	−	0.935610i	$-0.385149\pi$
$877$	543060.			0.706071			0.353036	+	0.935610i	$0.385149\pi$
$878$		0			0
$879$			630006.i			0.815393i
$880$		0			0
$881$	−747183.			−0.962665			−0.481333	−	0.876538i	$-0.659847\pi$
$881$	−747183.			−0.962665			−0.481333	+	0.876538i	$0.659847\pi$
$882$		0			0
$883$	−908500.			−1.16521			−0.582604	−	0.812756i	$-0.697966\pi$
$883$	−908500.			−1.16521			−0.582604	+	0.812756i	$0.697966\pi$
$884$		0			0
$885$	40527.1			0.0517439
$886$		0			0
$887$			283934.i			0.360886i	0.983585	+	0.180443i	$0.0577531\pi$
$887$			283934.i			0.360886i	−0.983585	+	0.180443i	$0.942247\pi$
$888$		0			0
$889$		−	372499.i		−	0.471326i
$890$		0			0
$891$	−55769.1			−0.0702487
$892$		0			0
$893$		−	272838.i		−	0.342138i
$894$		0			0
$895$	−39010.4			−0.0487006
$896$		0			0
$897$			198162.i			0.246284i
$898$		0			0
$899$			1.02718e6i			1.27095i
$900$		0			0
$901$	−28462.3			−0.0350607
$902$		0			0
$903$	−177470.	−	116344.i	−0.217646	−	0.142681i
$904$		0			0
$905$			71354.0i			0.0871206i
$906$		0			0
$907$	329783.			0.400880			0.200440	−	0.979706i	$-0.435763\pi$
$907$	329783.			0.400880			0.200440	+	0.979706i	$0.435763\pi$
$908$		0			0
$909$	−160128.			−0.193794
$910$		0			0
$911$		−	1.20039e6i		−	1.44639i	−0.690642	−	0.723197i	$-0.742672\pi$
$911$		−	1.20039e6i		−	1.44639i	0.690642	−	0.723197i	$-0.257328\pi$
$912$		0			0
$913$	65581.0			0.0786750
$914$		0			0
$915$		−	39748.6i		−	0.0474767i
$916$		0			0
$917$	237604.			0.282563
$918$		0			0
$919$	775338.			0.918037			0.459018	−	0.888427i	$-0.348201\pi$
$919$	775338.			0.918037			0.459018	+	0.888427i	$0.348201\pi$
$920$		0			0
$921$			1.32134e6i			1.55774i
$922$		0			0
$923$			290661.i			0.341180i
$924$		0			0
$925$			1.42976e6i			1.67102i
$926$		0			0
$927$	65216.5			0.0758924
$928$		0			0
$929$			763445.i			0.884599i	0.896867	+	0.442300i	$0.145837\pi$
$929$			763445.i			0.884599i	−0.896867	+	0.442300i	$0.854163\pi$
$930$		0			0
$931$			811322.i			0.936039i
$932$		0			0
$933$			410483.i			0.471555i
$934$		0			0
$935$		−	4333.38i		−	0.00495683i
$936$		0			0
$937$			1.72023e6i			1.95932i	0.200654	+	0.979662i	$0.435693\pi$
$937$			1.72023e6i			1.95932i	−0.200654	+	0.979662i	$0.564307\pi$
$938$		0			0
$939$	551957.			0.625999
$940$		0			0
$941$	657189.			0.742183			0.371092	−	0.928596i	$-0.378984\pi$
$941$	657189.			0.742183			0.371092	+	0.928596i	$0.378984\pi$
$942$		0			0
$943$	409591.			0.460604
$944$		0			0
$945$		−	15788.1i		−	0.0176794i
$946$		0			0
$947$	829200.			0.924612			0.462306	−	0.886721i	$-0.347022\pi$
$947$	829200.			0.924612			0.462306	+	0.886721i	$0.347022\pi$
$948$		0			0
$949$			356111.i			0.395415i
$950$		0			0
$951$			130750.i			0.144571i
$952$		0			0
$953$		−	1.33377e6i		−	1.46858i	−0.678838	−	0.734288i	$-0.737516\pi$
$953$		−	1.33377e6i		−	1.46858i	0.678838	−	0.734288i	$-0.262484\pi$
$954$		0			0
$955$	−69605.8			−0.0763200
$956$		0			0
$957$	97383.0			0.106331
$958$		0			0
$959$	25943.6			0.0282094
$960$		0			0
$961$	−122093.			−0.132204
$962$		0			0
$963$	28735.5			0.0309861
$964$		0			0
$965$			28118.2i			0.0301949i
$966$		0			0
$967$	−788889.			−0.843652			−0.421826	−	0.906677i	$-0.638611\pi$
$967$	−788889.			−0.843652			−0.421826	+	0.906677i	$0.638611\pi$
$968$		0			0
$969$	874738.			0.931602
$970$		0			0
$971$	1.20251e6			1.27541			0.637704	−	0.770282i	$-0.279885\pi$
$971$	1.20251e6			1.27541			0.637704	+	0.770282i	$0.279885\pi$
$972$		0			0
$973$		−	285154.i		−	0.301199i
$974$		0			0
$975$			509902.i			0.536386i
$976$		0			0
$977$	634645.			0.664878			0.332439	−	0.943125i	$-0.392128\pi$
$977$	634645.			0.664878			0.332439	+	0.943125i	$0.392128\pi$
$978$		0			0
$979$		−	33600.7i		−	0.0350576i
$980$		0			0
$981$	38176.6			0.0396698
$982$		0			0
$983$		−	1.63591e6i		−	1.69298i	−0.532403	−	0.846491i	$-0.678711\pi$
$983$		−	1.63591e6i		−	1.69298i	0.532403	−	0.846491i	$-0.321289\pi$
$984$		0			0
$985$		−	114409.i		−	0.117920i
$986$		0			0
$987$	85320.9			0.0875832
$988$		0			0
$989$	374273.	+	245361.i	0.382645	+	0.250849i
$990$		0			0
$991$		−	1.20041e6i		−	1.22231i	−0.791510	−	0.611156i	$-0.790705\pi$
$991$		−	1.20041e6i		−	1.22231i	0.791510	−	0.611156i	$-0.209295\pi$
$992$		0			0
$993$	−710862.			−0.720920
$994$		0			0
$995$	3336.72			0.00337034
$996$		0			0
$997$			1.25293e6i			1.26048i	0.776401	+	0.630239i	$0.217043\pi$
$997$			1.25293e6i			1.26048i	−0.776401	+	0.630239i	$0.782957\pi$
$998$		0			0
$999$	−1.77219e6			−1.77574

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.5.b.d.257.9		12
4.3	odd	2		43.5.b.b.42.2	✓	12
12.11	even	2		387.5.b.c.343.11		12
43.42	odd	2	inner	688.5.b.d.257.4		12
172.171	even	2		43.5.b.b.42.11	yes	12
516.515	odd	2		387.5.b.c.343.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.b.b.42.2	✓	12	4.3	odd	2
43.5.b.b.42.11	yes	12	172.171	even	2
387.5.b.c.343.2		12	516.515	odd	2
387.5.b.c.343.11		12	12.11	even	2
688.5.b.d.257.4		12	43.42	odd	2	inner
688.5.b.d.257.9		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	688.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.31879i q^{3} +1.48242i q^{5} +13.7963i q^{7} +11.7977 q^{9} +O(q^{10})\)	\(q+8.31879i q^{3} +1.48242i q^{5} +13.7963i q^{7} +11.7977 q^{9} +10.2025 q^{11} +98.4183 q^{13} -12.3320 q^{15} -286.515 q^{17} +367.004i q^{19} -114.768 q^{21} +242.039 q^{23} +622.802 q^{25} +771.965i q^{27} -1147.40i q^{29} -895.225 q^{31} +84.8728i q^{33} -20.4519 q^{35} +2295.69i q^{37} +818.722i q^{39} +1692.26 q^{41} +(1546.34 + 1013.73i) q^{43} +17.4892i q^{45} -743.419 q^{47} +2210.66 q^{49} -2383.46i q^{51} +99.3399 q^{53} +15.1245i q^{55} -3053.03 q^{57} -3286.35 q^{59} +3223.22i q^{61} +162.764i q^{63} +145.898i q^{65} -5556.36 q^{67} +2013.47i q^{69} +2953.33i q^{71} +3618.34i q^{73} +5180.96i q^{75} +140.757i q^{77} -2380.74 q^{79} -5466.20 q^{81} +6427.91 q^{83} -424.736i q^{85} +9544.98 q^{87} -3293.36i q^{89} +1357.80i q^{91} -7447.19i q^{93} -544.055 q^{95} -9329.95 q^{97} +120.367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 462 q^{9}+O(q^{10}) \) 12 * q - 462 * q^9	\( 12 q - 462 q^{9} + 180 q^{11} - 216 q^{13} + 92 q^{15} + 678 q^{17} - 2392 q^{21} - 1566 q^{23} - 174 q^{25} - 5710 q^{31} - 936 q^{35} + 4878 q^{41} + 1108 q^{43} + 5526 q^{47} - 8544 q^{49} + 1212 q^{53} - 7692 q^{57} - 14016 q^{59} + 1088 q^{67} - 24302 q^{79} - 23660 q^{81} + 7032 q^{83} - 17850 q^{87} - 606 q^{95} - 5842 q^{97} + 25924 q^{99}+O(q^{100}) \) 12 * q - 462 * q^9 + 180 * q^11 - 216 * q^13 + 92 * q^15 + 678 * q^17 - 2392 * q^21 - 1566 * q^23 - 174 * q^25 - 5710 * q^31 - 936 * q^35 + 4878 * q^41 + 1108 * q^43 + 5526 * q^47 - 8544 * q^49 + 1212 * q^53 - 7692 * q^57 - 14016 * q^59 + 1088 * q^67 - 24302 * q^79 - 23660 * q^81 + 7032 * q^83 - 17850 * q^87 - 606 * q^95 - 5842 * q^97 + 25924 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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