LMFDB - Embedded newform 688.3.b.e.257.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,3,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, 3, names="a")

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

chi := DirichletCharacter("688.257");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 3 $
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:	$18.7466421880$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 20x^{4} + 121x^{2} + 214 $ x^6 + 20x^4 + 121x^2 + 214
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.4
Root			$-3.18991i$ of defining polynomial
Character	$\chi$	$=$	688.257
Dual form			688.3.b.e.257.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.724539i q^{3} +7.66434i q^{5} +10.1297i q^{7} +8.47504 q^{9} +O(q^{10})$	$q+0.724539i q^{3} +7.66434i q^{5} +10.1297i q^{7} +8.47504 q^{9} +2.43517 q^{11} +18.1491 q^{13} -5.55311 q^{15} -1.13567 q^{17} +12.3269i q^{19} -7.33937 q^{21} +24.2847 q^{23} -33.7420 q^{25} +12.6614i q^{27} -35.5218i q^{29} +12.9823 q^{31} +1.76438i q^{33} -77.6375 q^{35} +41.5116i q^{37} +13.1497i q^{39} -65.0667 q^{41} +(-23.6064 - 35.9408i) q^{43} +64.9556i q^{45} -51.0402 q^{47} -53.6110 q^{49} -0.822839i q^{51} +56.2026 q^{53} +18.6640i q^{55} -8.93130 q^{57} +88.8355 q^{59} -65.1393i q^{61} +85.8497i q^{63} +139.101i q^{65} -45.3618 q^{67} +17.5952i q^{69} -63.7983i q^{71} +8.80247i q^{73} -24.4474i q^{75} +24.6676i q^{77} -31.8496 q^{79} +67.1017 q^{81} +68.4740 q^{83} -8.70417i q^{85} +25.7370 q^{87} -5.90432i q^{89} +183.845i q^{91} +9.40616i q^{93} -94.4773 q^{95} -100.657 q^{97} +20.6382 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 36 q^{9}+O(q^{10}) $ 6 * q - 36 * q^9	$ 6 q - 36 q^{9} - 38 q^{11} + 30 q^{13} - 28 q^{15} - 20 q^{17} + 56 q^{21} + 80 q^{23} - 84 q^{25} + 112 q^{31} - 208 q^{35} - 172 q^{41} - 10 q^{43} - 30 q^{47} - 6 q^{49} - 110 q^{53} + 420 q^{57} + 12 q^{59} + 70 q^{67} - 178 q^{79} + 382 q^{81} - 10 q^{83} + 510 q^{87} + 130 q^{95} - 380 q^{97} + 466 q^{99}+O(q^{100}) $ 6 * q - 36 * q^9 - 38 * q^11 + 30 * q^13 - 28 * q^15 - 20 * q^17 + 56 * q^21 + 80 * q^23 - 84 * q^25 + 112 * q^31 - 208 * q^35 - 172 * q^41 - 10 * q^43 - 30 * q^47 - 6 * q^49 - 110 * q^53 + 420 * q^57 + 12 * q^59 + 70 * q^67 - 178 * q^79 + 382 * q^81 - 10 * q^83 + 510 * q^87 + 130 * q^95 - 380 * q^97 + 466 * 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
$3$			0.724539i			0.241513i	0.992682	+	0.120757i	$0.0385320\pi$
$3$			0.724539i			0.241513i	−0.992682	+	0.120757i	$0.961468\pi$
$4$		0			0
$5$			7.66434i			1.53287i	0.642324	+	0.766434i	$0.277971\pi$
$5$			7.66434i			1.53287i	−0.642324	+	0.766434i	$0.722029\pi$
$6$		0			0
$7$			10.1297i			1.44710i	0.690271	+	0.723551i	$0.257491\pi$
$7$			10.1297i			1.44710i	−0.690271	+	0.723551i	$0.742509\pi$
$8$		0			0
$9$	8.47504			0.941671
$10$		0			0
$11$	2.43517			0.221379			0.110690	−	0.993855i	$-0.464694\pi$
$11$	2.43517			0.221379			0.110690	+	0.993855i	$0.464694\pi$
$12$		0			0
$13$	18.1491			1.39608			0.698041	−	0.716058i	$-0.254055\pi$
$13$	18.1491			1.39608			0.698041	+	0.716058i	$0.254055\pi$
$14$		0			0
$15$	−5.55311			−0.370207
$16$		0			0
$17$	−1.13567			−0.0668042			−0.0334021	−	0.999442i	$-0.510634\pi$
$17$	−1.13567			−0.0668042			−0.0334021	+	0.999442i	$0.510634\pi$
$18$		0			0
$19$			12.3269i			0.648783i	0.945923	+	0.324391i	$0.105159\pi$
$19$			12.3269i			0.648783i	−0.945923	+	0.324391i	$0.894841\pi$
$20$		0			0
$21$	−7.33937			−0.349494
$22$		0			0
$23$	24.2847			1.05586			0.527929	−	0.849288i	$-0.322969\pi$
$23$	24.2847			1.05586			0.527929	+	0.849288i	$0.322969\pi$
$24$		0			0
$25$	−33.7420			−1.34968
$26$		0			0
$27$			12.6614i			0.468939i
$28$		0			0
$29$		−	35.5218i		−	1.22489i	−0.790513	−	0.612445i	$-0.790186\pi$
$29$		−	35.5218i		−	1.22489i	0.790513	−	0.612445i	$-0.209814\pi$
$30$		0			0
$31$	12.9823			0.418783			0.209391	−	0.977832i	$-0.432852\pi$
$31$	12.9823			0.418783			0.209391	+	0.977832i	$0.432852\pi$
$32$		0			0
$33$			1.76438i			0.0534660i
$34$		0			0
$35$	−77.6375			−2.21821
$36$		0			0
$37$			41.5116i			1.12193i	0.827838	+	0.560967i	$0.189571\pi$
$37$			41.5116i			1.12193i	−0.827838	+	0.560967i	$0.810429\pi$
$38$		0			0
$39$			13.1497i			0.337172i
$40$		0			0
$41$	−65.0667			−1.58699			−0.793496	−	0.608576i	$-0.791741\pi$
$41$	−65.0667			−1.58699			−0.793496	+	0.608576i	$0.791741\pi$
$42$		0			0
$43$	−23.6064	−	35.9408i	−0.548986	−	0.835832i
$43$	−23.6064	−	35.9408i	−0.548986	−	0.835832i
$44$		0			0
$45$			64.9556i			1.44346i
$46$		0			0
$47$	−51.0402			−1.08596			−0.542981	−	0.839745i	$-0.682704\pi$
$47$	−51.0402			−1.08596			−0.542981	+	0.839745i	$0.682704\pi$
$48$		0			0
$49$	−53.6110			−1.09410
$50$		0			0
$51$		−	0.822839i		−	0.0161341i
$52$		0			0
$53$	56.2026			1.06043			0.530214	−	0.847864i	$-0.322112\pi$
$53$	56.2026			1.06043			0.530214	+	0.847864i	$0.322112\pi$
$54$		0			0
$55$			18.6640i			0.339345i
$56$		0			0
$57$	−8.93130			−0.156689
$58$		0			0
$59$	88.8355			1.50569			0.752843	−	0.658200i	$-0.228682\pi$
$59$	88.8355			1.50569			0.752843	+	0.658200i	$0.228682\pi$
$60$		0			0
$61$		−	65.1393i		−	1.06786i	−0.845529	−	0.533929i	$-0.820715\pi$
$61$		−	65.1393i		−	1.06786i	0.845529	−	0.533929i	$-0.179285\pi$
$62$		0			0
$63$			85.8497i			1.36269i
$64$		0			0
$65$			139.101i			2.14001i
$66$		0			0
$67$	−45.3618			−0.677042			−0.338521	−	0.940959i	$-0.609927\pi$
$67$	−45.3618			−0.677042			−0.338521	+	0.940959i	$0.609927\pi$
$68$		0			0
$69$			17.5952i			0.255004i
$70$		0			0
$71$		−	63.7983i		−	0.898567i	−0.893389	−	0.449284i	$-0.851679\pi$
$71$		−	63.7983i		−	0.898567i	0.893389	−	0.449284i	$-0.148321\pi$
$72$		0			0
$73$			8.80247i			0.120582i	0.998181	+	0.0602909i	$0.0192028\pi$
$73$			8.80247i			0.120582i	−0.998181	+	0.0602909i	$0.980797\pi$
$74$		0			0
$75$		−	24.4474i		−	0.325966i
$76$		0			0
$77$			24.6676i			0.320358i
$78$		0			0
$79$	−31.8496			−0.403159			−0.201580	−	0.979472i	$-0.564607\pi$
$79$	−31.8496			−0.403159			−0.201580	+	0.979472i	$0.564607\pi$
$80$		0			0
$81$	67.1017			0.828417
$82$		0			0
$83$	68.4740			0.824988			0.412494	−	0.910960i	$-0.364658\pi$
$83$	68.4740			0.824988			0.412494	+	0.910960i	$0.364658\pi$
$84$		0			0
$85$		−	8.70417i		−	0.102402i
$86$		0			0
$87$	25.7370			0.295827
$88$		0			0
$89$		−	5.90432i		−	0.0663406i	−0.999450	−	0.0331703i	$-0.989440\pi$
$89$		−	5.90432i		−	0.0663406i	0.999450	−	0.0331703i	$-0.0105604\pi$
$90$		0			0
$91$			183.845i			2.02027i
$92$		0			0
$93$			9.40616i			0.101142i
$94$		0			0
$95$	−94.4773			−0.994498
$96$		0			0
$97$	−100.657			−1.03770			−0.518849	−	0.854866i	$-0.673639\pi$
$97$	−100.657			−1.03770			−0.518849	+	0.854866i	$0.673639\pi$
$98$		0			0
$99$	20.6382			0.208466
$100$		0			0
$101$	13.7062			0.135705			0.0678525	−	0.997695i	$-0.478385\pi$
$101$	13.7062			0.135705			0.0678525	+	0.997695i	$0.478385\pi$
$102$		0			0
$103$	96.0402			0.932429			0.466214	−	0.884672i	$-0.345618\pi$
$103$	96.0402			0.932429			0.466214	+	0.884672i	$0.345618\pi$
$104$		0			0
$105$		−	56.2514i		−	0.535728i
$106$		0			0
$107$	2.08036			0.0194427			0.00972133	−	0.999953i	$-0.496906\pi$
$107$	2.08036			0.0194427			0.00972133	+	0.999953i	$0.496906\pi$
$108$		0			0
$109$	−53.1078			−0.487228			−0.243614	−	0.969872i	$-0.578333\pi$
$109$	−53.1078			−0.487228			−0.243614	+	0.969872i	$0.578333\pi$
$110$		0			0
$111$	−30.0768			−0.270962
$112$		0			0
$113$		−	28.8791i		−	0.255568i	−0.991802	−	0.127784i	$-0.959214\pi$
$113$		−	28.8791i		−	0.255568i	0.991802	−	0.127784i	$-0.0407864\pi$
$114$		0			0
$115$			186.126i			1.61849i
$116$		0			0
$117$	153.814			1.31465
$118$		0			0
$119$		−	11.5040i		−	0.0966725i
$120$		0			0
$121$	−115.070			−0.950991
$122$		0			0
$123$		−	47.1433i		−	0.383279i
$124$		0			0
$125$		−	67.0020i		−	0.536016i
$126$		0			0
$127$	−148.673			−1.17066			−0.585328	−	0.810797i	$-0.699034\pi$
$127$	−148.673			−1.17066			−0.585328	+	0.810797i	$0.699034\pi$
$128$		0			0
$129$	26.0405	−	17.1037i	0.201864	−	0.132587i
$130$		0			0
$131$		−	157.214i		−	1.20011i	−0.799960	−	0.600054i	$-0.795146\pi$
$131$		−	157.214i		−	1.20011i	0.799960	−	0.600054i	$-0.204854\pi$
$132$		0			0
$133$	−124.868			−0.938854
$134$		0			0
$135$	−97.0409			−0.718821
$136$		0			0
$137$			121.604i			0.887618i	0.896122	+	0.443809i	$0.146373\pi$
$137$			121.604i			0.887618i	−0.896122	+	0.443809i	$0.853627\pi$
$138$		0			0
$139$	27.8992			0.200714			0.100357	−	0.994952i	$-0.468002\pi$
$139$	27.8992			0.200714			0.100357	+	0.994952i	$0.468002\pi$
$140$		0			0
$141$		−	36.9806i		−	0.262274i
$142$		0			0
$143$	44.1961			0.309063
$144$		0			0
$145$	272.251			1.87759
$146$		0			0
$147$		−	38.8433i		−	0.264240i
$148$		0			0
$149$		−	212.121i		−	1.42363i	−0.702366	−	0.711816i	$-0.747873\pi$
$149$		−	212.121i		−	1.42363i	0.702366	−	0.711816i	$-0.252127\pi$
$150$		0			0
$151$		−	108.847i		−	0.720842i	−0.932790	−	0.360421i	$-0.882633\pi$
$151$		−	108.847i		−	0.720842i	0.932790	−	0.360421i	$-0.117367\pi$
$152$		0			0
$153$	−9.62487			−0.0629077
$154$		0			0
$155$			99.5005i			0.641938i
$156$		0			0
$157$		−	106.772i		−	0.680075i	−0.940412	−	0.340038i	$-0.889560\pi$
$157$		−	106.772i		−	0.680075i	0.940412	−	0.340038i	$-0.110440\pi$
$158$		0			0
$159$			40.7210i			0.256107i
$160$		0			0
$161$			245.997i			1.52793i
$162$		0			0
$163$		−	202.231i		−	1.24068i	−0.784334	−	0.620339i	$-0.786995\pi$
$163$		−	202.231i		−	1.24068i	0.784334	−	0.620339i	$-0.213005\pi$
$164$		0			0
$165$	−13.5228			−0.0819562
$166$		0			0
$167$	41.0703			0.245930			0.122965	−	0.992411i	$-0.460760\pi$
$167$	41.0703			0.245930			0.122965	+	0.992411i	$0.460760\pi$
$168$		0			0
$169$	160.389			0.949045
$170$		0			0
$171$			104.471i			0.610940i
$172$		0			0
$173$	−24.1006			−0.139310			−0.0696549	−	0.997571i	$-0.522190\pi$
$173$	−24.1006			−0.139310			−0.0696549	+	0.997571i	$0.522190\pi$
$174$		0			0
$175$		−	341.797i		−	1.95313i
$176$		0			0
$177$			64.3648i			0.363643i
$178$		0			0
$179$			83.0659i			0.464055i	0.972709	+	0.232028i	$0.0745360\pi$
$179$			83.0659i			0.464055i	−0.972709	+	0.232028i	$0.925464\pi$
$180$		0			0
$181$	−112.273			−0.620291			−0.310145	−	0.950689i	$-0.600378\pi$
$181$	−112.273			−0.620291			−0.310145	+	0.950689i	$0.600378\pi$
$182$		0			0
$183$	47.1960			0.257902
$184$		0			0
$185$	−318.159			−1.71978
$186$		0			0
$187$	−2.76556			−0.0147891
$188$		0			0
$189$	−128.256			−0.678602
$190$		0			0
$191$		−	43.7078i		−	0.228836i	−0.993433	−	0.114418i	$-0.963500\pi$
$191$		−	43.7078i		−	0.228836i	0.993433	−	0.114418i	$-0.0365004\pi$
$192$		0			0
$193$	236.100			1.22332			0.611659	−	0.791122i	$-0.290503\pi$
$193$	236.100			1.22332			0.611659	+	0.791122i	$0.290503\pi$
$194$		0			0
$195$	−100.784			−0.516840
$196$		0			0
$197$	350.134			1.77733			0.888665	−	0.458557i	$-0.151634\pi$
$197$	350.134			1.77733			0.888665	+	0.458557i	$0.151634\pi$
$198$		0			0
$199$			353.573i			1.77675i	0.459122	+	0.888373i	$0.348164\pi$
$199$			353.573i			1.77675i	−0.459122	+	0.888373i	$0.651836\pi$
$200$		0			0
$201$		−	32.8664i		−	0.163514i
$202$		0			0
$203$	359.826			1.77254
$204$		0			0
$205$		−	498.693i		−	2.43265i
$206$		0			0
$207$	205.814			0.994272
$208$		0			0
$209$			30.0180i			0.143627i
$210$		0			0
$211$		−	281.519i		−	1.33421i	−0.744962	−	0.667107i	$-0.767532\pi$
$211$		−	281.519i		−	1.33421i	0.744962	−	0.667107i	$-0.232468\pi$
$212$		0			0
$213$	46.2243			0.217016
$214$		0			0
$215$	275.462	−	180.927i	1.28122	−	0.841522i
$216$		0			0
$217$			131.507i			0.606021i
$218$		0			0
$219$	−6.37774			−0.0291221
$220$		0			0
$221$	−20.6114			−0.0932642
$222$		0			0
$223$		−	95.1942i		−	0.426880i	−0.976956	−	0.213440i	$-0.931533\pi$
$223$		−	95.1942i		−	0.426880i	0.976956	−	0.213440i	$-0.0684668\pi$
$224$		0			0
$225$	−285.965			−1.27096
$226$		0			0
$227$		−	106.481i		−	0.469079i	−0.972107	−	0.234540i	$-0.924642\pi$
$227$		−	106.481i		−	0.469079i	0.972107	−	0.234540i	$-0.0753583\pi$
$228$		0			0
$229$	170.352			0.743896			0.371948	−	0.928254i	$-0.378690\pi$
$229$	170.352			0.743896			0.371948	+	0.928254i	$0.378690\pi$
$230$		0			0
$231$	−17.8726			−0.0773707
$232$		0			0
$233$			55.4970i			0.238184i	0.992883	+	0.119092i	$0.0379984\pi$
$233$			55.4970i			0.238184i	−0.992883	+	0.119092i	$0.962002\pi$
$234$		0			0
$235$		−	391.189i		−	1.66463i
$236$		0			0
$237$		−	23.0763i		−	0.0973682i
$238$		0			0
$239$	20.7695			0.0869017			0.0434509	−	0.999056i	$-0.486165\pi$
$239$	20.7695			0.0869017			0.0434509	+	0.999056i	$0.486165\pi$
$240$		0			0
$241$			303.282i			1.25843i	0.777231	+	0.629216i	$0.216624\pi$
$241$			303.282i			1.25843i	−0.777231	+	0.629216i	$0.783376\pi$
$242$		0			0
$243$			162.570i			0.669012i
$244$		0			0
$245$		−	410.893i		−	1.67711i
$246$		0			0
$247$			223.721i			0.905754i
$248$		0			0
$249$			49.6121i			0.199245i
$250$		0			0
$251$	7.25467			0.0289031			0.0144515	−	0.999896i	$-0.495400\pi$
$251$	7.25467			0.0289031			0.0144515	+	0.999896i	$0.495400\pi$
$252$		0			0
$253$	59.1375			0.233745
$254$		0			0
$255$	6.30651			0.0247314
$256$		0			0
$257$		−	316.956i		−	1.23329i	−0.787241	−	0.616645i	$-0.788491\pi$
$257$		−	316.956i		−	1.23329i	0.787241	−	0.616645i	$-0.211509\pi$
$258$		0			0
$259$	−420.500			−1.62355
$260$		0			0
$261$		−	301.049i		−	1.15344i
$262$		0			0
$263$			229.845i			0.873934i	0.899478	+	0.436967i	$0.143947\pi$
$263$			229.845i			0.873934i	−0.899478	+	0.436967i	$0.856053\pi$
$264$		0			0
$265$			430.756i			1.62549i
$266$		0			0
$267$	4.27791			0.0160221
$268$		0			0
$269$	285.119			1.05992			0.529961	−	0.848022i	$-0.322206\pi$
$269$	285.119			1.05992			0.529961	+	0.848022i	$0.322206\pi$
$270$		0			0
$271$	−78.3270			−0.289030			−0.144515	−	0.989503i	$-0.546162\pi$
$271$	−78.3270			−0.289030			−0.144515	+	0.989503i	$0.546162\pi$
$272$		0			0
$273$	−133.203			−0.487922
$274$		0			0
$275$	−82.1677			−0.298791
$276$		0			0
$277$			119.631i			0.431880i	0.976407	+	0.215940i	$0.0692815\pi$
$277$			119.631i			0.431880i	−0.976407	+	0.215940i	$0.930718\pi$
$278$		0			0
$279$	110.025			0.394356
$280$		0			0
$281$	−150.423			−0.535314			−0.267657	−	0.963514i	$-0.586249\pi$
$281$	−150.423			−0.535314			−0.267657	+	0.963514i	$0.586249\pi$
$282$		0			0
$283$	261.144			0.922771			0.461385	−	0.887200i	$-0.347353\pi$
$283$	261.144			0.922771			0.461385	+	0.887200i	$0.347353\pi$
$284$		0			0
$285$		−	68.4525i		−	0.240184i
$286$		0			0
$287$		−	659.106i		−	2.29654i
$288$		0			0
$289$	−287.710			−0.995537
$290$		0			0
$291$		−	72.9297i		−	0.250618i
$292$		0			0
$293$	358.409			1.22324			0.611619	−	0.791152i	$-0.290518\pi$
$293$	358.409			1.22324			0.611619	+	0.791152i	$0.290518\pi$
$294$		0			0
$295$			680.865i			2.30802i
$296$		0			0
$297$			30.8326i			0.103813i
$298$		0			0
$299$	440.745			1.47406
$300$		0			0
$301$	364.070	−	239.126i	1.20953	−	0.794438i
$302$		0			0
$303$			9.93068i			0.0327745i
$304$		0			0
$305$	499.250			1.63688
$306$		0			0
$307$	−509.730			−1.66036			−0.830180	−	0.557496i	$-0.811762\pi$
$307$	−509.730			−1.66036			−0.830180	+	0.557496i	$0.811762\pi$
$308$		0			0
$309$			69.5849i			0.225194i
$310$		0			0
$311$	27.7820			0.0893312			0.0446656	−	0.999002i	$-0.485778\pi$
$311$	27.7820			0.0893312			0.0446656	+	0.999002i	$0.485778\pi$
$312$		0			0
$313$			38.5686i			0.123222i	0.998100	+	0.0616111i	$0.0196239\pi$
$313$			38.5686i			0.123222i	−0.998100	+	0.0616111i	$0.980376\pi$
$314$		0			0
$315$	−657.981			−2.08883
$316$		0			0
$317$	85.8945			0.270961			0.135480	−	0.990780i	$-0.456742\pi$
$317$	85.8945			0.270961			0.135480	+	0.990780i	$0.456742\pi$
$318$		0			0
$319$		−	86.5017i		−	0.271165i
$320$		0			0
$321$			1.50731i			0.00469566i
$322$		0			0
$323$		−	13.9993i		−	0.0433414i
$324$		0			0
$325$	−612.387			−1.88427
$326$		0			0
$327$		−	38.4787i		−	0.117672i
$328$		0			0
$329$		−	517.022i		−	1.57150i
$330$		0			0
$331$			363.666i			1.09869i	0.835596	+	0.549345i	$0.185123\pi$
$331$			363.666i			1.09869i	−0.835596	+	0.549345i	$0.814877\pi$
$332$		0			0
$333$			351.813i			1.05649i
$334$		0			0
$335$		−	347.668i		−	1.03782i
$336$		0			0
$337$	8.64293			0.0256467			0.0128233	−	0.999918i	$-0.495918\pi$
$337$	8.64293			0.0256467			0.0128233	+	0.999918i	$0.495918\pi$
$338$		0			0
$339$	20.9241			0.0617229
$340$		0			0
$341$	31.6140			0.0927098
$342$		0			0
$343$		−	46.7084i		−	0.136176i
$344$		0			0
$345$	−134.856			−0.390887
$346$		0			0
$347$		−	278.736i		−	0.803273i	−0.915799	−	0.401636i	$-0.868442\pi$
$347$		−	278.736i		−	0.803273i	0.915799	−	0.401636i	$-0.131558\pi$
$348$		0			0
$349$			169.755i			0.486403i	0.969976	+	0.243201i	$0.0781976\pi$
$349$			169.755i			0.486403i	−0.969976	+	0.243201i	$0.921802\pi$
$350$		0			0
$351$			229.792i			0.654677i
$352$		0			0
$353$	26.8357			0.0760219			0.0380109	−	0.999277i	$-0.487898\pi$
$353$	26.8357			0.0760219			0.0380109	+	0.999277i	$0.487898\pi$
$354$		0			0
$355$	488.971			1.37738
$356$		0			0
$357$	8.33512			0.0233477
$358$		0			0
$359$	−169.169			−0.471223			−0.235611	−	0.971847i	$-0.575709\pi$
$359$	−169.169			−0.471223			−0.235611	+	0.971847i	$0.575709\pi$
$360$		0			0
$361$	209.048			0.579081
$362$		0			0
$363$		−	83.3727i		−	0.229677i
$364$		0			0
$365$	−67.4651			−0.184836
$366$		0			0
$367$	540.524			1.47282			0.736409	−	0.676536i	$-0.236520\pi$
$367$	540.524			1.47282			0.736409	+	0.676536i	$0.236520\pi$
$368$		0			0
$369$	−551.443			−1.49442
$370$		0			0
$371$			569.316i			1.53455i
$372$		0			0
$373$		−	26.7152i		−	0.0716226i	−0.999359	−	0.0358113i	$-0.988598\pi$
$373$		−	26.7152i		−	0.0716226i	0.999359	−	0.0358113i	$-0.0114015\pi$
$374$		0			0
$375$	48.5456			0.129455
$376$		0			0
$377$		−	644.688i		−	1.71005i
$378$		0			0
$379$	−315.193			−0.831644			−0.415822	−	0.909446i	$-0.636506\pi$
$379$	−315.193			−0.831644			−0.415822	+	0.909446i	$0.636506\pi$
$380$		0			0
$381$		−	107.720i		−	0.282729i
$382$		0			0
$383$			131.099i			0.342295i	0.985245	+	0.171148i	$0.0547475\pi$
$383$			131.099i			0.342295i	−0.985245	+	0.171148i	$0.945252\pi$
$384$		0			0
$385$	−189.061			−0.491066
$386$		0			0
$387$	−200.065	−	304.600i	−0.516964	−	0.787079i
$388$		0			0
$389$		−	110.050i		−	0.282904i	−0.989945	−	0.141452i	$-0.954823\pi$
$389$		−	110.050i		−	0.282904i	0.989945	−	0.141452i	$-0.0451771\pi$
$390$		0			0
$391$	−27.5795			−0.0705358
$392$		0			0
$393$	113.908			0.289841
$394$		0			0
$395$		−	244.106i		−	0.617989i
$396$		0			0
$397$	68.3215			0.172094			0.0860472	−	0.996291i	$-0.472576\pi$
$397$	68.3215			0.172094			0.0860472	+	0.996291i	$0.472576\pi$
$398$		0			0
$399$		−	90.4715i		−	0.226746i
$400$		0			0
$401$	−675.508			−1.68456			−0.842279	−	0.539042i	$-0.818787\pi$
$401$	−675.508			−1.68456			−0.842279	+	0.539042i	$0.818787\pi$
$402$		0			0
$403$	235.616			0.584655
$404$		0			0
$405$			514.290i			1.26985i
$406$		0			0
$407$			101.088i			0.248373i
$408$		0			0
$409$		−	1.76834i		−	0.00432356i	−0.999998	−	0.00216178i	$-0.999312\pi$
$409$		−	1.76834i		−	0.00432356i	0.999998	−	0.00216178i	$-0.000688116\pi$
$410$		0			0
$411$	−88.1066			−0.214371
$412$		0			0
$413$			899.878i			2.17888i
$414$		0			0
$415$			524.808i			1.26460i
$416$		0			0
$417$			20.2141i			0.0484749i
$418$		0			0
$419$			444.028i			1.05973i	0.848081	+	0.529867i	$0.177758\pi$
$419$			444.028i			1.05973i	−0.848081	+	0.529867i	$0.822242\pi$
$420$		0			0
$421$			141.195i			0.335381i	0.985840	+	0.167690i	$0.0536309\pi$
$421$			141.195i			0.335381i	−0.985840	+	0.167690i	$0.946369\pi$
$422$		0			0
$423$	−432.568			−1.02262
$424$		0			0
$425$	38.3199			0.0901645
$426$		0			0
$427$	659.843			1.54530
$428$		0			0
$429$			32.0218i			0.0746429i
$430$		0			0
$431$	−302.066			−0.700849			−0.350424	−	0.936591i	$-0.613963\pi$
$431$	−302.066			−0.700849			−0.350424	+	0.936591i	$0.613963\pi$
$432$		0			0
$433$			380.497i			0.878747i	0.898304	+	0.439374i	$0.144800\pi$
$433$			380.497i			0.878747i	−0.898304	+	0.439374i	$0.855200\pi$
$434$		0			0
$435$			197.257i			0.453464i
$436$		0			0
$437$			299.355i			0.685022i
$438$		0			0
$439$	−558.769			−1.27282			−0.636412	−	0.771350i	$-0.719582\pi$
$439$	−558.769			−1.27282			−0.636412	+	0.771350i	$0.719582\pi$
$440$		0			0
$441$	−454.356			−1.03029
$442$		0			0
$443$	346.462			0.782082			0.391041	−	0.920373i	$-0.372115\pi$
$443$	346.462			0.782082			0.391041	+	0.920373i	$0.372115\pi$
$444$		0			0
$445$	45.2527			0.101691
$446$		0			0
$447$	153.690			0.343826
$448$		0			0
$449$			250.699i			0.558350i	0.960240	+	0.279175i	$0.0900609\pi$
$449$			250.699i			0.558350i	−0.960240	+	0.279175i	$0.909939\pi$
$450$		0			0
$451$	−158.448			−0.351327
$452$		0			0
$453$	78.8640			0.174093
$454$		0			0
$455$	−1409.05			−3.09681
$456$		0			0
$457$			745.356i			1.63098i	0.578773	+	0.815488i	$0.303532\pi$
$457$			745.356i			1.63098i	−0.578773	+	0.815488i	$0.696468\pi$
$458$		0			0
$459$		−	14.3791i		−	0.0313271i
$460$		0			0
$461$	6.73204			0.0146031			0.00730156	−	0.999973i	$-0.497676\pi$
$461$	6.73204			0.0146031			0.00730156	+	0.999973i	$0.497676\pi$
$462$		0			0
$463$		−	448.198i		−	0.968031i	−0.875059	−	0.484016i	$-0.839178\pi$
$463$		−	448.198i		−	0.968031i	0.875059	−	0.484016i	$-0.160822\pi$
$464$		0			0
$465$	−72.0920			−0.155037
$466$		0			0
$467$			506.048i			1.08361i	0.840503	+	0.541807i	$0.182260\pi$
$467$			506.048i			1.08361i	−0.840503	+	0.541807i	$0.817740\pi$
$468$		0			0
$469$		−	459.502i		−	0.979749i
$470$		0			0
$471$	77.3603			0.164247
$472$		0			0
$473$	−57.4856	−	87.5219i	−0.121534	−	0.185036i
$474$		0			0
$475$		−	415.934i		−	0.875650i
$476$		0			0
$477$	476.320			0.998574
$478$		0			0
$479$	772.714			1.61318			0.806591	−	0.591110i	$-0.201310\pi$
$479$	772.714			1.61318			0.806591	+	0.591110i	$0.201310\pi$
$480$		0			0
$481$			753.397i			1.56631i
$482$		0			0
$483$	−178.235			−0.369016
$484$		0			0
$485$		−	771.467i		−	1.59065i
$486$		0			0
$487$	762.454			1.56561			0.782807	−	0.622265i	$-0.213787\pi$
$487$	762.454			1.56561			0.782807	+	0.622265i	$0.213787\pi$
$488$		0			0
$489$	146.524			0.299640
$490$		0			0
$491$		−	928.398i		−	1.89083i	−0.325868	−	0.945415i	$-0.605657\pi$
$491$		−	928.398i		−	1.89083i	0.325868	−	0.945415i	$-0.394343\pi$
$492$		0			0
$493$			40.3411i			0.0818279i
$494$		0			0
$495$			158.178i			0.319551i
$496$		0			0
$497$	646.258			1.30032
$498$		0			0
$499$			669.978i			1.34264i	0.741167	+	0.671320i	$0.234272\pi$
$499$			669.978i			1.34264i	−0.741167	+	0.671320i	$0.765728\pi$
$500$		0			0
$501$			29.7570i			0.0593952i
$502$		0			0
$503$		−	274.168i		−	0.545065i	−0.962147	−	0.272532i	$-0.912139\pi$
$503$		−	274.168i		−	0.545065i	0.962147	−	0.272532i	$-0.0878612\pi$
$504$		0			0
$505$			105.049i			0.208018i
$506$		0			0
$507$			116.208i			0.229207i
$508$		0			0
$509$	703.988			1.38308			0.691540	−	0.722338i	$-0.256933\pi$
$509$	703.988			1.38308			0.691540	+	0.722338i	$0.256933\pi$
$510$		0			0
$511$	−89.1665			−0.174494
$512$		0			0
$513$	−156.075			−0.304239
$514$		0			0
$515$			736.084i			1.42929i
$516$		0			0
$517$	−124.292			−0.240409
$518$		0			0
$519$		−	17.4618i		−	0.0336451i
$520$		0			0
$521$			588.189i			1.12896i	0.825446	+	0.564480i	$0.190923\pi$
$521$			588.189i			1.12896i	−0.825446	+	0.564480i	$0.809077\pi$
$522$		0			0
$523$			743.907i			1.42238i	0.702998	+	0.711192i	$0.251844\pi$
$523$			743.907i			1.42238i	−0.702998	+	0.711192i	$0.748156\pi$
$524$		0			0
$525$	247.645			0.471706
$526$		0			0
$527$	−14.7436			−0.0279765
$528$		0			0
$529$	60.7485			0.114837
$530$		0			0
$531$	752.885			1.41786
$532$		0			0
$533$	−1180.90			−2.21557
$534$		0			0
$535$			15.9446i			0.0298030i
$536$		0			0
$537$	−60.1845			−0.112075
$538$		0			0
$539$	−130.552			−0.242212
$540$		0			0
$541$	−2.77685			−0.00513282			−0.00256641	−	0.999997i	$-0.500817\pi$
$541$	−2.77685			−0.00513282			−0.00256641	+	0.999997i	$0.500817\pi$
$542$		0			0
$543$		−	81.3459i		−	0.149808i
$544$		0			0
$545$		−	407.036i		−	0.746856i
$546$		0			0
$547$	−700.748			−1.28108			−0.640538	−	0.767927i	$-0.721289\pi$
$547$	−700.748			−1.28108			−0.640538	+	0.767927i	$0.721289\pi$
$548$		0			0
$549$		−	552.059i		−	1.00557i
$550$		0			0
$551$	437.873			0.794688
$552$		0			0
$553$		−	322.627i		−	0.583412i
$554$		0			0
$555$		−	230.518i		−	0.415349i
$556$		0			0
$557$	−590.822			−1.06072			−0.530361	−	0.847772i	$-0.677944\pi$
$557$	−590.822			−1.06072			−0.530361	+	0.847772i	$0.677944\pi$
$558$		0			0
$559$	−428.434	−	652.291i	−0.766429	−	1.16689i
$560$		0			0
$561$		−	2.00375i		−	0.00357175i
$562$		0			0
$563$	306.648			0.544668			0.272334	−	0.962203i	$-0.412205\pi$
$563$	306.648			0.544668			0.272334	+	0.962203i	$0.412205\pi$
$564$		0			0
$565$	221.339			0.391751
$566$		0			0
$567$			679.721i			1.19880i
$568$		0			0
$569$	−5.67367			−0.00997130			−0.00498565	−	0.999988i	$-0.501587\pi$
$569$	−5.67367			−0.00997130			−0.00498565	+	0.999988i	$0.501587\pi$
$570$		0			0
$571$			811.264i			1.42078i	0.703810	+	0.710389i	$0.251481\pi$
$571$			811.264i			1.42078i	−0.703810	+	0.710389i	$0.748519\pi$
$572$		0			0
$573$	31.6680			0.0552670
$574$		0			0
$575$	−819.417			−1.42507
$576$		0			0
$577$		−	976.820i		−	1.69293i	−0.532445	−	0.846464i	$-0.678727\pi$
$577$		−	976.820i		−	1.69293i	0.532445	−	0.846464i	$-0.321273\pi$
$578$		0			0
$579$			171.064i			0.295447i
$580$		0			0
$581$			693.622i			1.19384i
$582$		0			0
$583$	136.863			0.234757
$584$		0			0
$585$			1178.88i			2.01518i
$586$		0			0
$587$		−	681.557i		−	1.16109i	−0.814230	−	0.580543i	$-0.802840\pi$
$587$		−	681.557i		−	1.16109i	0.814230	−	0.580543i	$-0.197160\pi$
$588$		0			0
$589$			160.031i			0.271699i
$590$		0			0
$591$			253.686i			0.429248i
$592$		0			0
$593$		−	1030.13i		−	1.73715i	−0.495561	−	0.868573i	$-0.665038\pi$
$593$		−	1030.13i		−	1.73715i	0.495561	−	0.868573i	$-0.334962\pi$
$594$		0			0
$595$	88.1708			0.148186
$596$		0			0
$597$	−256.177			−0.429107
$598$		0			0
$599$	−1028.30			−1.71669			−0.858345	−	0.513073i	$-0.828507\pi$
$599$	−1028.30			−1.71669			−0.858345	+	0.513073i	$0.828507\pi$
$600$		0			0
$601$			655.529i			1.09073i	0.838198	+	0.545365i	$0.183609\pi$
$601$			655.529i			1.09073i	−0.838198	+	0.545365i	$0.816391\pi$
$602$		0			0
$603$	−384.443			−0.637551
$604$		0			0
$605$		−	881.935i		−	1.45774i
$606$		0			0
$607$			1016.42i			1.67450i	0.546822	+	0.837249i	$0.315837\pi$
$607$			1016.42i			1.67450i	−0.546822	+	0.837249i	$0.684163\pi$
$608$		0			0
$609$			260.708i			0.428092i
$610$		0			0
$611$	−926.332			−1.51609
$612$		0			0
$613$	−1029.51			−1.67946			−0.839728	−	0.543007i	$-0.817286\pi$
$613$	−1029.51			−1.67946			−0.839728	+	0.543007i	$0.817286\pi$
$614$		0			0
$615$	361.322			0.587516
$616$		0			0
$617$	426.603			0.691415			0.345708	−	0.938342i	$-0.387639\pi$
$617$	426.603			0.691415			0.345708	+	0.938342i	$0.387639\pi$
$618$		0			0
$619$	503.515			0.813432			0.406716	−	0.913555i	$-0.366674\pi$
$619$	503.515			0.813432			0.406716	+	0.913555i	$0.366674\pi$
$620$		0			0
$621$			307.478i			0.495133i
$622$		0			0
$623$	59.8090			0.0960016
$624$		0			0
$625$	−330.025			−0.528041
$626$		0			0
$627$	−21.7492			−0.0346878
$628$		0			0
$629$		−	47.1436i		−	0.0749500i
$630$		0			0
$631$			299.884i			0.475252i	0.971357	+	0.237626i	$0.0763692\pi$
$631$			299.884i			0.475252i	−0.971357	+	0.237626i	$0.923631\pi$
$632$		0			0
$633$	203.972			0.322230
$634$		0			0
$635$		−	1139.48i		−	1.79446i
$636$		0			0
$637$	−972.990			−1.52746
$638$		0			0
$639$		−	540.693i		−	0.846155i
$640$		0			0
$641$		−	724.346i		−	1.13002i	−0.825082	−	0.565012i	$-0.808871\pi$
$641$		−	724.346i		−	1.13002i	0.825082	−	0.565012i	$-0.191129\pi$
$642$		0			0
$643$	−743.353			−1.15607			−0.578035	−	0.816012i	$-0.696180\pi$
$643$	−743.353			−1.15607			−0.578035	+	0.816012i	$0.696180\pi$
$644$		0			0
$645$	131.089	+	199.583i	0.203239	+	0.309431i
$646$		0			0
$647$			61.7177i			0.0953905i	0.998862	+	0.0476953i	$0.0151876\pi$
$647$			61.7177i			0.0953905i	−0.998862	+	0.0476953i	$0.984812\pi$
$648$		0			0
$649$	216.330			0.333328
$650$		0			0
$651$	−95.2817			−0.146362
$652$		0			0
$653$		−	422.488i		−	0.646996i	−0.946229	−	0.323498i	$-0.895141\pi$
$653$		−	422.488i		−	0.646996i	0.946229	−	0.323498i	$-0.104859\pi$
$654$		0			0
$655$	1204.94			1.83960
$656$		0			0
$657$			74.6013i			0.113548i
$658$		0			0
$659$	620.921			0.942217			0.471108	−	0.882075i	$-0.343854\pi$
$659$	620.921			0.942217			0.471108	+	0.882075i	$0.343854\pi$
$660$		0			0
$661$	496.134			0.750581			0.375291	−	0.926907i	$-0.377543\pi$
$661$	496.134			0.750581			0.375291	+	0.926907i	$0.377543\pi$
$662$		0			0
$663$		−	14.9338i		−	0.0225245i
$664$		0			0
$665$		−	957.027i		−	1.43914i
$666$		0			0
$667$		−	862.638i		−	1.29331i
$668$		0			0
$669$	68.9719			0.103097
$670$		0			0
$671$		−	158.625i		−	0.236402i
$672$		0			0
$673$		−	453.069i		−	0.673209i	−0.941646	−	0.336604i	$-0.890722\pi$
$673$		−	453.069i		−	0.673209i	0.941646	−	0.336604i	$-0.109278\pi$
$674$		0			0
$675$		−	427.220i		−	0.632918i
$676$		0			0
$677$		−	233.751i		−	0.345275i	−0.984985	−	0.172638i	$-0.944771\pi$
$677$		−	233.751i		−	0.345275i	0.984985	−	0.172638i	$-0.0552289\pi$
$678$		0			0
$679$		−	1019.62i		−	1.50165i
$680$		0			0
$681$	77.1497			0.113289
$682$		0			0
$683$	−578.370			−0.846808			−0.423404	−	0.905941i	$-0.639165\pi$
$683$	−578.370			−0.846808			−0.423404	+	0.905941i	$0.639165\pi$
$684$		0			0
$685$	−932.011			−1.36060
$686$		0			0
$687$			123.427i			0.179660i
$688$		0			0
$689$	1020.03			1.48044
$690$		0			0
$691$		−	127.474i		−	0.184477i	−0.995737	−	0.0922387i	$-0.970598\pi$
$691$		−	127.474i		−	0.184477i	0.995737	−	0.0922387i	$-0.0294023\pi$
$692$		0			0
$693$			209.059i			0.301672i
$694$		0			0
$695$			213.829i			0.307667i
$696$		0			0
$697$	73.8944			0.106018
$698$		0			0
$699$	−40.2097			−0.0575247
$700$		0			0
$701$	126.686			0.180721			0.0903606	−	0.995909i	$-0.471198\pi$
$701$	126.686			0.180721			0.0903606	+	0.995909i	$0.471198\pi$
$702$		0			0
$703$	−511.708			−0.727892
$704$		0			0
$705$	283.432			0.402031
$706$		0			0
$707$			138.840i			0.196379i
$708$		0			0
$709$	733.535			1.03461			0.517303	−	0.855803i	$-0.326936\pi$
$709$	733.535			1.03461			0.517303	+	0.855803i	$0.326936\pi$
$710$		0			0
$711$	−269.926			−0.379643
$712$		0			0
$713$	315.271			0.442175
$714$		0			0
$715$			338.734i			0.473753i
$716$		0			0
$717$			15.0483i			0.0209879i
$718$		0			0
$719$	885.641			1.23177			0.615884	−	0.787837i	$-0.288799\pi$
$719$	885.641			1.23177			0.615884	+	0.787837i	$0.288799\pi$
$720$		0			0
$721$			972.859i			1.34932i
$722$		0			0
$723$	−219.740			−0.303928
$724$		0			0
$725$			1198.58i			1.65321i
$726$		0			0
$727$			244.282i			0.336014i	0.985786	+	0.168007i	$0.0537331\pi$
$727$			244.282i			0.336014i	−0.985786	+	0.168007i	$0.946267\pi$
$728$		0			0
$729$	486.127			0.666841
$730$		0			0
$731$	26.8091	+	40.8169i	0.0366746	+	0.0558371i
$732$		0			0
$733$			1317.81i			1.79783i	0.438120	+	0.898917i	$0.355645\pi$
$733$			1317.81i			1.79783i	−0.438120	+	0.898917i	$0.644355\pi$
$734$		0			0
$735$	297.708			0.405045
$736$		0			0
$737$	−110.464			−0.149883
$738$		0			0
$739$		−	126.907i		−	0.171728i	−0.996307	−	0.0858641i	$-0.972635\pi$
$739$		−	126.907i		−	0.171728i	0.996307	−	0.0858641i	$-0.0273651\pi$
$740$		0			0
$741$	−162.095			−0.218751
$742$		0			0
$743$		−	483.885i		−	0.651259i	−0.945497	−	0.325629i	$-0.894424\pi$
$743$		−	483.885i		−	0.651259i	0.945497	−	0.325629i	$-0.105576\pi$
$744$		0			0
$745$	1625.77			2.18224
$746$		0			0
$747$	580.320			0.776867
$748$		0			0
$749$			21.0735i			0.0281355i
$750$		0			0
$751$		−	11.4506i		−	0.0152471i	−0.999971	−	0.00762354i	$-0.997573\pi$
$751$		−	11.4506i		−	0.0152471i	0.999971	−	0.00762354i	$-0.00242667\pi$
$752$		0			0
$753$			5.25629i			0.00698047i
$754$		0			0
$755$	834.241			1.10495
$756$		0			0
$757$		−	995.145i		−	1.31459i	−0.753633	−	0.657295i	$-0.771701\pi$
$757$		−	995.145i		−	1.31459i	0.753633	−	0.657295i	$-0.228299\pi$
$758$		0			0
$759$			42.8474i			0.0564525i
$760$		0			0
$761$		−	1067.64i		−	1.40295i	−0.712695	−	0.701474i	$-0.752526\pi$
$761$		−	1067.64i		−	1.40295i	0.712695	−	0.701474i	$-0.247474\pi$
$762$		0			0
$763$		−	537.967i		−	0.705068i
$764$		0			0
$765$		−	73.7682i		−	0.0964291i
$766$		0			0
$767$	1612.28			2.10206
$768$		0			0
$769$	−361.976			−0.470710			−0.235355	−	0.971909i	$-0.575625\pi$
$769$	−361.976			−0.470710			−0.235355	+	0.971909i	$0.575625\pi$
$770$		0			0
$771$	229.647			0.297856
$772$		0			0
$773$			579.373i			0.749513i	0.927123	+	0.374756i	$0.122274\pi$
$773$			579.373i			0.749513i	−0.927123	+	0.374756i	$0.877726\pi$
$774$		0			0
$775$	−438.048			−0.565224
$776$		0			0
$777$		−	304.669i		−	0.392109i
$778$		0			0
$779$		−	802.068i		−	1.02961i
$780$		0			0
$781$		−	155.360i		−	0.198924i
$782$		0			0
$783$	449.754			0.574399
$784$		0			0
$785$	818.335			1.04246
$786$		0			0
$787$	1046.34			1.32954			0.664768	−	0.747050i	$-0.268530\pi$
$787$	1046.34			1.32954			0.664768	+	0.747050i	$0.268530\pi$
$788$		0			0
$789$	−166.531			−0.211066
$790$		0			0
$791$	292.537			0.369832
$792$		0			0
$793$		−	1182.22i		−	1.49082i
$794$		0			0
$795$	−312.100			−0.392578
$796$		0			0
$797$	−959.219			−1.20354			−0.601769	−	0.798670i	$-0.705537\pi$
$797$	−959.219			−1.20354			−0.601769	+	0.798670i	$0.705537\pi$
$798$		0			0
$799$	57.9649			0.0725468
$800$		0			0
$801$		−	50.0393i		−	0.0624711i
$802$		0			0
$803$			21.4355i			0.0266943i
$804$		0			0
$805$	−1885.41			−2.34212
$806$		0			0
$807$			206.580i			0.255985i
$808$		0			0
$809$	639.565			0.790562			0.395281	−	0.918560i	$-0.370647\pi$
$809$	639.565			0.790562			0.395281	+	0.918560i	$0.370647\pi$
$810$		0			0
$811$			515.867i			0.636087i	0.948076	+	0.318044i	$0.103026\pi$
$811$			515.867i			0.636087i	−0.948076	+	0.318044i	$0.896974\pi$
$812$		0			0
$813$		−	56.7510i		−	0.0698044i
$814$		0			0
$815$	1549.96			1.90180
$816$		0			0
$817$	443.037	−	290.993i	0.542273	−	0.356172i
$818$		0			0
$819$			1558.09i			1.90243i
$820$		0			0
$821$	1288.57			1.56951			0.784756	−	0.619805i	$-0.212788\pi$
$821$	1288.57			1.56951			0.784756	+	0.619805i	$0.212788\pi$
$822$		0			0
$823$	−516.675			−0.627795			−0.313897	−	0.949457i	$-0.601635\pi$
$823$	−516.675			−0.627795			−0.313897	+	0.949457i	$0.601635\pi$
$824$		0			0
$825$		−	59.5337i		−	0.0721620i
$826$		0			0
$827$	−35.6545			−0.0431131			−0.0215565	−	0.999768i	$-0.506862\pi$
$827$	−35.6545			−0.0431131			−0.0215565	+	0.999768i	$0.506862\pi$
$828$		0			0
$829$			617.780i			0.745212i	0.927990	+	0.372606i	$0.121536\pi$
$829$			617.780i			0.745212i	−0.927990	+	0.372606i	$0.878464\pi$
$830$		0			0
$831$	−86.6771			−0.104305
$832$		0			0
$833$	60.8846			0.0730907
$834$		0			0
$835$			314.776i			0.376978i
$836$		0			0
$837$			164.373i			0.196384i
$838$		0			0
$839$			98.1185i			0.116947i	0.998289	+	0.0584735i	$0.0186233\pi$
$839$			98.1185i			0.116947i	−0.998289	+	0.0584735i	$0.981377\pi$
$840$		0			0
$841$	−420.800			−0.500357
$842$		0			0
$843$		−	108.988i		−	0.129285i
$844$		0			0
$845$			1229.27i			1.45476i
$846$		0			0
$847$		−	1165.63i		−	1.37618i
$848$		0			0
$849$			189.209i			0.222861i
$850$		0			0
$851$			1008.10i			1.18460i
$852$		0			0
$853$	564.129			0.661347			0.330674	−	0.943745i	$-0.392724\pi$
$853$	564.129			0.661347			0.330674	+	0.943745i	$0.392724\pi$
$854$		0			0
$855$	−800.699			−0.936490
$856$		0			0
$857$	−699.603			−0.816340			−0.408170	−	0.912906i	$-0.633833\pi$
$857$	−699.603			−0.816340			−0.408170	+	0.912906i	$0.633833\pi$
$858$		0			0
$859$		−	1395.67i		−	1.62476i	−0.583128	−	0.812380i	$-0.698171\pi$
$859$		−	1395.67i		−	1.62476i	0.583128	−	0.812380i	$-0.301829\pi$
$860$		0			0
$861$	477.548			0.554644
$862$		0			0
$863$		−	768.514i		−	0.890515i	−0.895403	−	0.445257i	$-0.853112\pi$
$863$		−	768.514i		−	0.890515i	0.895403	−	0.445257i	$-0.146888\pi$
$864$		0			0
$865$		−	184.715i		−	0.213543i
$866$		0			0
$867$		−	208.457i		−	0.240435i
$868$		0			0
$869$	−77.5591			−0.0892510
$870$		0			0
$871$	−823.275			−0.945206
$872$		0			0
$873$	−853.070			−0.977170
$874$		0			0
$875$	678.711			0.775670
$876$		0			0
$877$	1318.08			1.50294			0.751470	−	0.659767i	$-0.229345\pi$
$877$	1318.08			1.50294			0.751470	+	0.659767i	$0.229345\pi$
$878$		0			0
$879$			259.681i			0.295428i
$880$		0			0
$881$	−691.467			−0.784867			−0.392433	−	0.919780i	$-0.628367\pi$
$881$	−691.467			−0.784867			−0.392433	+	0.919780i	$0.628367\pi$
$882$		0			0
$883$	173.332			0.196299			0.0981493	−	0.995172i	$-0.468708\pi$
$883$	173.332			0.196299			0.0981493	+	0.995172i	$0.468708\pi$
$884$		0			0
$885$	−493.313			−0.557416
$886$		0			0
$887$		−	142.316i		−	0.160446i	−0.996777	−	0.0802232i	$-0.974437\pi$
$887$		−	142.316i		−	0.160446i	0.996777	−	0.0802232i	$-0.0255633\pi$
$888$		0			0
$889$		−	1506.02i		−	1.69406i
$890$		0			0
$891$	163.404			0.183394
$892$		0			0
$893$		−	629.166i		−	0.704553i
$894$		0			0
$895$	−636.645			−0.711335
$896$		0			0
$897$			319.337i			0.356006i
$898$		0			0
$899$		−	461.154i		−	0.512963i
$900$		0			0
$901$	−63.8278			−0.0708410
$902$		0			0
$903$	173.256	+	263.783i	0.191867	+	0.292118i
$904$		0			0
$905$		−	860.495i		−	0.950823i
$906$		0			0
$907$	−1795.52			−1.97963			−0.989814	−	0.142370i	$-0.954528\pi$
$907$	−1795.52			−1.97963			−0.989814	+	0.142370i	$0.954528\pi$
$908$		0			0
$909$	116.161			0.127790
$910$		0			0
$911$			365.632i			0.401353i	0.979658	+	0.200676i	$0.0643140\pi$
$911$			365.632i			0.401353i	−0.979658	+	0.200676i	$0.935686\pi$
$912$		0			0
$913$	166.746			0.182635
$914$		0			0
$915$			361.726i			0.395329i
$916$		0			0
$917$	1592.53			1.73668
$918$		0			0
$919$	−1083.62			−1.17913			−0.589566	−	0.807720i	$-0.700701\pi$
$919$	−1083.62			−1.17913			−0.589566	+	0.807720i	$0.700701\pi$
$920$		0			0
$921$		−	369.320i		−	0.400998i
$922$		0			0
$923$		−	1157.88i		−	1.25447i
$924$		0			0
$925$		−	1400.69i		−	1.51426i
$926$		0			0
$927$	813.945			0.878042
$928$		0			0
$929$		−	1400.56i		−	1.50760i	−0.657105	−	0.753799i	$-0.728219\pi$
$929$		−	1400.56i		−	1.50760i	0.657105	−	0.753799i	$-0.271781\pi$
$930$		0			0
$931$		−	660.856i		−	0.709835i
$932$		0			0
$933$			20.1292i			0.0215747i
$934$		0			0
$935$		−	21.1962i		−	0.0226697i
$936$		0			0
$937$		−	846.473i		−	0.903387i	−0.892173	−	0.451693i	$-0.850820\pi$
$937$		−	846.473i		−	0.903387i	0.892173	−	0.451693i	$-0.149180\pi$
$938$		0			0
$939$	−27.9444			−0.0297598
$940$		0			0
$941$	−1354.63			−1.43956			−0.719780	−	0.694203i	$-0.755757\pi$
$941$	−1354.63			−1.43956			−0.719780	+	0.694203i	$0.755757\pi$
$942$		0			0
$943$	−1580.13			−1.67564
$944$		0			0
$945$		−	982.996i		−	1.04021i
$946$		0			0
$947$	209.288			0.221001			0.110501	−	0.993876i	$-0.464755\pi$
$947$	209.288			0.221001			0.110501	+	0.993876i	$0.464755\pi$
$948$		0			0
$949$			159.757i			0.168342i
$950$		0			0
$951$			62.2339i			0.0654405i
$952$		0			0
$953$		−	293.406i		−	0.307876i	−0.988080	−	0.153938i	$-0.950804\pi$
$953$		−	293.406i		−	0.307876i	0.988080	−	0.153938i	$-0.0491957\pi$
$954$		0			0
$955$	334.991			0.350776
$956$		0			0
$957$	62.6739			0.0654899
$958$		0			0
$959$	−1231.81			−1.28447
$960$		0			0
$961$	−792.461			−0.824621
$962$		0			0
$963$	17.6312			0.0183086
$964$		0			0
$965$			1809.55i			1.87518i
$966$		0			0
$967$	1028.49			1.06359			0.531794	−	0.846874i	$-0.321518\pi$
$967$	1028.49			1.06359			0.531794	+	0.846874i	$0.321518\pi$
$968$		0			0
$969$	10.1430			0.0104675
$970$		0			0
$971$	438.558			0.451656			0.225828	−	0.974167i	$-0.427491\pi$
$971$	438.558			0.451656			0.225828	+	0.974167i	$0.427491\pi$
$972$		0			0
$973$			282.611i			0.290453i
$974$		0			0
$975$		−	443.698i		−	0.455075i
$976$		0			0
$977$	−626.212			−0.640954			−0.320477	−	0.947256i	$-0.603843\pi$
$977$	−626.212			−0.640954			−0.320477	+	0.947256i	$0.603843\pi$
$978$		0			0
$979$		−	14.3780i		−	0.0146864i
$980$		0			0
$981$	−450.091			−0.458808
$982$		0			0
$983$		−	1720.98i		−	1.75074i	−0.483454	−	0.875370i	$-0.660618\pi$
$983$		−	1720.98i		−	1.75074i	0.483454	−	0.875370i	$-0.339382\pi$
$984$		0			0
$985$			2683.54i			2.72441i
$986$		0			0
$987$	374.603			0.379537
$988$		0			0
$989$	−573.275	−	872.812i	−0.579651	−	0.882520i
$990$		0			0
$991$		−	753.187i		−	0.760027i	−0.924981	−	0.380014i	$-0.875919\pi$
$991$		−	753.187i		−	0.760027i	0.924981	−	0.380014i	$-0.124081\pi$
$992$		0			0
$993$	−263.490			−0.265348
$994$		0			0
$995$	−2709.90			−2.72352
$996$		0			0
$997$		−	533.031i		−	0.534635i	−0.963609	−	0.267317i	$-0.913863\pi$
$997$		−	533.031i		−	0.534635i	0.963609	−	0.267317i	$-0.0861372\pi$
$998$		0			0
$999$	−525.593			−0.526119

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.3.b.e.257.4		6
4.3	odd	2		43.3.b.b.42.6	yes	6
12.11	even	2		387.3.b.c.343.1		6
43.42	odd	2	inner	688.3.b.e.257.3		6
172.171	even	2		43.3.b.b.42.1	✓	6
516.515	odd	2		387.3.b.c.343.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.b.b.42.1	✓	6	172.171	even	2
43.3.b.b.42.6	yes	6	4.3	odd	2
387.3.b.c.343.1		6	12.11	even	2
387.3.b.c.343.6		6	516.515	odd	2
688.3.b.e.257.3		6	43.42	odd	2	inner
688.3.b.e.257.4		6	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	688.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.724539i q^{3} +7.66434i q^{5} +10.1297i q^{7} +8.47504 q^{9} +O(q^{10})\)	\(q+0.724539i q^{3} +7.66434i q^{5} +10.1297i q^{7} +8.47504 q^{9} +2.43517 q^{11} +18.1491 q^{13} -5.55311 q^{15} -1.13567 q^{17} +12.3269i q^{19} -7.33937 q^{21} +24.2847 q^{23} -33.7420 q^{25} +12.6614i q^{27} -35.5218i q^{29} +12.9823 q^{31} +1.76438i q^{33} -77.6375 q^{35} +41.5116i q^{37} +13.1497i q^{39} -65.0667 q^{41} +(-23.6064 - 35.9408i) q^{43} +64.9556i q^{45} -51.0402 q^{47} -53.6110 q^{49} -0.822839i q^{51} +56.2026 q^{53} +18.6640i q^{55} -8.93130 q^{57} +88.8355 q^{59} -65.1393i q^{61} +85.8497i q^{63} +139.101i q^{65} -45.3618 q^{67} +17.5952i q^{69} -63.7983i q^{71} +8.80247i q^{73} -24.4474i q^{75} +24.6676i q^{77} -31.8496 q^{79} +67.1017 q^{81} +68.4740 q^{83} -8.70417i q^{85} +25.7370 q^{87} -5.90432i q^{89} +183.845i q^{91} +9.40616i q^{93} -94.4773 q^{95} -100.657 q^{97} +20.6382 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 36 q^{9}+O(q^{10}) \) 6 * q - 36 * q^9	\( 6 q - 36 q^{9} - 38 q^{11} + 30 q^{13} - 28 q^{15} - 20 q^{17} + 56 q^{21} + 80 q^{23} - 84 q^{25} + 112 q^{31} - 208 q^{35} - 172 q^{41} - 10 q^{43} - 30 q^{47} - 6 q^{49} - 110 q^{53} + 420 q^{57} + 12 q^{59} + 70 q^{67} - 178 q^{79} + 382 q^{81} - 10 q^{83} + 510 q^{87} + 130 q^{95} - 380 q^{97} + 466 q^{99}+O(q^{100}) \) 6 * q - 36 * q^9 - 38 * q^11 + 30 * q^13 - 28 * q^15 - 20 * q^17 + 56 * q^21 + 80 * q^23 - 84 * q^25 + 112 * q^31 - 208 * q^35 - 172 * q^41 - 10 * q^43 - 30 * q^47 - 6 * q^49 - 110 * q^53 + 420 * q^57 + 12 * q^59 + 70 * q^67 - 178 * q^79 + 382 * q^81 - 10 * q^83 + 510 * q^87 + 130 * q^95 - 380 * q^97 + 466 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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