LMFDB - Embedded newform 688.3.b.e.257.1

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			257.1
Root			$1.77533i$ of defining polynomial
Character	$\chi$	$=$	688.257
Dual form			688.3.b.e.257.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.61757i q^{3} +2.98959i q^{5} +6.83184i q^{7} -22.5571 q^{9} +O(q^{10})$	$q-5.61757i q^{3} +2.98959i q^{5} +6.83184i q^{7} -22.5571 q^{9} -6.88766 q^{11} +12.4014 q^{13} +16.7943 q^{15} -15.8212 q^{17} +19.2112i q^{19} +38.3784 q^{21} +33.2226 q^{23} +16.0623 q^{25} +76.1581i q^{27} +45.8411i q^{29} -14.7278 q^{31} +38.6919i q^{33} -20.4244 q^{35} +13.1726i q^{37} -69.6657i q^{39} -2.49085 q^{41} +(40.8836 - 13.3242i) q^{43} -67.4366i q^{45} +10.2595 q^{47} +2.32596 q^{49} +88.8770i q^{51} -31.2780 q^{53} -20.5913i q^{55} +107.920 q^{57} -64.4112 q^{59} +78.1922i q^{61} -154.107i q^{63} +37.0751i q^{65} +89.3657 q^{67} -186.631i q^{69} +35.5065i q^{71} -35.9603i q^{73} -90.2313i q^{75} -47.0554i q^{77} -50.1103 q^{79} +224.809 q^{81} +10.3645 q^{83} -47.2991i q^{85} +257.516 q^{87} +13.4900i q^{89} +84.7243i q^{91} +82.7347i q^{93} -57.4338 q^{95} +66.6322 q^{97} +155.366 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	5.61757i		−	1.87252i	−0.351302	−	0.936262i	$-0.614261\pi$
$3$		−	5.61757i		−	1.87252i	0.351302	−	0.936262i	$-0.385739\pi$
$4$		0			0
$5$			2.98959i			0.597919i	0.954266	+	0.298959i	$0.0966395\pi$
$5$			2.98959i			0.597919i	−0.954266	+	0.298959i	$0.903360\pi$
$6$		0			0
$7$			6.83184i			0.975977i	0.872850	+	0.487989i	$0.162269\pi$
$7$			6.83184i			0.975977i	−0.872850	+	0.487989i	$0.837731\pi$
$8$		0			0
$9$	−22.5571			−2.50635
$10$		0			0
$11$	−6.88766			−0.626151			−0.313075	−	0.949728i	$-0.601359\pi$
$11$	−6.88766			−0.626151			−0.313075	+	0.949728i	$0.601359\pi$
$12$		0			0
$13$	12.4014			0.953953			0.476977	−	0.878916i	$-0.341733\pi$
$13$	12.4014			0.953953			0.476977	+	0.878916i	$0.341733\pi$
$14$		0			0
$15$	16.7943			1.11962
$16$		0			0
$17$	−15.8212			−0.930661			−0.465331	−	0.885137i	$-0.654065\pi$
$17$	−15.8212			−0.930661			−0.465331	+	0.885137i	$0.654065\pi$
$18$		0			0
$19$			19.2112i			1.01112i	0.862792	+	0.505559i	$0.168714\pi$
$19$			19.2112i			1.01112i	−0.862792	+	0.505559i	$0.831286\pi$
$20$		0			0
$21$	38.3784			1.82754
$22$		0			0
$23$	33.2226			1.44446			0.722231	−	0.691652i	$-0.243117\pi$
$23$	33.2226			1.44446			0.722231	+	0.691652i	$0.243117\pi$
$24$		0			0
$25$	16.0623			0.642493
$26$		0			0
$27$			76.1581i			2.82067i
$28$		0			0
$29$			45.8411i			1.58073i	0.612637	+	0.790364i	$0.290109\pi$
$29$			45.8411i			1.58073i	−0.612637	+	0.790364i	$0.709891\pi$
$30$		0			0
$31$	−14.7278			−0.475092			−0.237546	−	0.971376i	$-0.576343\pi$
$31$	−14.7278			−0.475092			−0.237546	+	0.971376i	$0.576343\pi$
$32$		0			0
$33$			38.6919i			1.17248i
$34$		0			0
$35$	−20.4244			−0.583555
$36$		0			0
$37$			13.1726i			0.356017i	0.984029	+	0.178009i	$0.0569655\pi$
$37$			13.1726i			0.356017i	−0.984029	+	0.178009i	$0.943034\pi$
$38$		0			0
$39$		−	69.6657i		−	1.78630i
$40$		0			0
$41$	−2.49085			−0.0607525			−0.0303762	−	0.999539i	$-0.509671\pi$
$41$	−2.49085			−0.0607525			−0.0303762	+	0.999539i	$0.509671\pi$
$42$		0			0
$43$	40.8836	−	13.3242i	0.950781	−	0.309865i
$43$	40.8836	−	13.3242i	0.950781	−	0.309865i
$44$		0			0
$45$		−	67.4366i		−	1.49859i
$46$		0			0
$47$	10.2595			0.218288			0.109144	−	0.994026i	$-0.465189\pi$
$47$	10.2595			0.218288			0.109144	+	0.994026i	$0.465189\pi$
$48$		0			0
$49$	2.32596			0.0474685
$50$		0			0
$51$			88.8770i			1.74269i
$52$		0			0
$53$	−31.2780			−0.590151			−0.295075	−	0.955474i	$-0.595345\pi$
$53$	−31.2780			−0.590151			−0.295075	+	0.955474i	$0.595345\pi$
$54$		0			0
$55$		−	20.5913i		−	0.374387i
$56$		0			0
$57$	107.920			1.89334
$58$		0			0
$59$	−64.4112			−1.09171			−0.545857	−	0.837878i	$-0.683796\pi$
$59$	−64.4112			−1.09171			−0.545857	+	0.837878i	$0.683796\pi$
$60$		0			0
$61$			78.1922i			1.28184i	0.767608	+	0.640920i	$0.221447\pi$
$61$			78.1922i			1.28184i	−0.767608	+	0.640920i	$0.778553\pi$
$62$		0			0
$63$		−	154.107i		−	2.44614i
$64$		0			0
$65$			37.0751i			0.570387i
$66$		0			0
$67$	89.3657			1.33382			0.666908	−	0.745140i	$-0.267617\pi$
$67$	89.3657			1.33382			0.666908	+	0.745140i	$0.267617\pi$
$68$		0			0
$69$		−	186.631i		−	2.70479i
$70$		0			0
$71$			35.5065i			0.500092i	0.968234	+	0.250046i	$0.0804457\pi$
$71$			35.5065i			0.500092i	−0.968234	+	0.250046i	$0.919554\pi$
$72$		0			0
$73$		−	35.9603i		−	0.492607i	−0.969193	−	0.246303i	$-0.920784\pi$
$73$		−	35.9603i		−	0.492607i	0.969193	−	0.246303i	$-0.0792160\pi$
$74$		0			0
$75$		−	90.2313i		−	1.20308i
$76$		0			0
$77$		−	47.0554i		−	0.611109i
$78$		0			0
$79$	−50.1103			−0.634308			−0.317154	−	0.948374i	$-0.602727\pi$
$79$	−50.1103			−0.634308			−0.317154	+	0.948374i	$0.602727\pi$
$80$		0			0
$81$	224.809			2.77543
$82$		0			0
$83$	10.3645			0.124873			0.0624367	−	0.998049i	$-0.480113\pi$
$83$	10.3645			0.124873			0.0624367	+	0.998049i	$0.480113\pi$
$84$		0			0
$85$		−	47.2991i		−	0.556460i
$86$		0			0
$87$	257.516			2.95995
$88$		0			0
$89$			13.4900i			0.151573i	0.997124	+	0.0757865i	$0.0241468\pi$
$89$			13.4900i			0.151573i	−0.997124	+	0.0757865i	$0.975853\pi$
$90$		0			0
$91$			84.7243i			0.931037i
$92$		0			0
$93$			82.7347i			0.889621i
$94$		0			0
$95$	−57.4338			−0.604566
$96$		0			0
$97$	66.6322			0.686930			0.343465	−	0.939165i	$-0.388399\pi$
$97$	66.6322			0.686930			0.343465	+	0.939165i	$0.388399\pi$
$98$		0			0
$99$	155.366			1.56935
$100$		0			0
$101$	73.3449			0.726187			0.363094	−	0.931753i	$-0.381721\pi$
$101$	73.3449			0.726187			0.363094	+	0.931753i	$0.381721\pi$
$102$		0			0
$103$	34.7405			0.337286			0.168643	−	0.985677i	$-0.446061\pi$
$103$	34.7405			0.337286			0.168643	+	0.985677i	$0.446061\pi$
$104$		0			0
$105$			114.736i			1.09272i
$106$		0			0
$107$	−120.519			−1.12635			−0.563173	−	0.826339i	$-0.690420\pi$
$107$	−120.519			−1.12635			−0.563173	+	0.826339i	$0.690420\pi$
$108$		0			0
$109$	−81.5825			−0.748464			−0.374232	−	0.927335i	$-0.622094\pi$
$109$	−81.5825			−0.748464			−0.374232	+	0.927335i	$0.622094\pi$
$110$		0			0
$111$	73.9983			0.666651
$112$		0			0
$113$		−	58.7443i		−	0.519861i	−0.965627	−	0.259931i	$-0.916300\pi$
$113$		−	58.7443i		−	0.519861i	0.965627	−	0.259931i	$-0.0836997\pi$
$114$		0			0
$115$			99.3222i			0.863671i
$116$		0			0
$117$	−279.740			−2.39094
$118$		0			0
$119$		−	108.088i		−	0.908304i
$120$		0			0
$121$	−73.5601			−0.607935
$122$		0			0
$123$			13.9925i			0.113760i
$124$		0			0
$125$			122.760i			0.982078i
$126$		0			0
$127$	17.9828			0.141597			0.0707984	−	0.997491i	$-0.477445\pi$
$127$	17.9828			0.141597			0.0707984	+	0.997491i	$0.477445\pi$
$128$		0			0
$129$	−74.8496	−	229.666i	−0.580230	−	1.78036i
$130$		0			0
$131$			103.364i			0.789039i	0.918887	+	0.394520i	$0.129089\pi$
$131$			103.364i			0.789039i	−0.918887	+	0.394520i	$0.870911\pi$
$132$		0			0
$133$	−131.248			−0.986827
$134$		0			0
$135$	−227.682			−1.68653
$136$		0			0
$137$			140.446i			1.02516i	0.858641	+	0.512578i	$0.171309\pi$
$137$			140.446i			1.02516i	−0.858641	+	0.512578i	$0.828691\pi$
$138$		0			0
$139$	−248.580			−1.78834			−0.894172	−	0.447725i	$-0.852235\pi$
$139$	−248.580			−1.78834			−0.894172	+	0.447725i	$0.852235\pi$
$140$		0			0
$141$		−	57.6337i		−	0.408750i
$142$		0			0
$143$	−85.4166			−0.597319
$144$		0			0
$145$	−137.046			−0.945147
$146$		0			0
$147$		−	13.0662i		−	0.0888859i
$148$		0			0
$149$		−	95.0589i		−	0.637979i	−0.947758	−	0.318990i	$-0.896656\pi$
$149$		−	95.0589i		−	0.637979i	0.947758	−	0.318990i	$-0.103344\pi$
$150$		0			0
$151$			122.926i			0.814076i	0.913411	+	0.407038i	$0.133438\pi$
$151$			122.926i			0.814076i	−0.913411	+	0.407038i	$0.866562\pi$
$152$		0			0
$153$	356.882			2.33256
$154$		0			0
$155$		−	44.0303i		−	0.284066i
$156$		0			0
$157$			189.332i			1.20594i	0.797765	+	0.602969i	$0.206016\pi$
$157$			189.332i			1.20594i	−0.797765	+	0.602969i	$0.793984\pi$
$158$		0			0
$159$			175.706i			1.10507i
$160$		0			0
$161$			226.972i			1.40976i
$162$		0			0
$163$			264.440i			1.62233i	0.584817	+	0.811165i	$0.301166\pi$
$163$			264.440i			1.62233i	−0.584817	+	0.811165i	$0.698834\pi$
$164$		0			0
$165$	−115.673			−0.701049
$166$		0			0
$167$	−40.0299			−0.239700			−0.119850	−	0.992792i	$-0.538241\pi$
$167$	−40.0299			−0.239700			−0.119850	+	0.992792i	$0.538241\pi$
$168$		0			0
$169$	−15.2054			−0.0899729
$170$		0			0
$171$		−	433.350i		−	2.53421i
$172$		0			0
$173$	181.497			1.04912			0.524559	−	0.851374i	$-0.324230\pi$
$173$	181.497			1.04912			0.524559	+	0.851374i	$0.324230\pi$
$174$		0			0
$175$			109.735i			0.627059i
$176$		0			0
$177$			361.834i			2.04426i
$178$		0			0
$179$			62.5972i			0.349705i	0.984595	+	0.174852i	$0.0559448\pi$
$179$			62.5972i			0.349705i	−0.984595	+	0.174852i	$0.944055\pi$
$180$		0			0
$181$	16.7178			0.0923638			0.0461819	−	0.998933i	$-0.485295\pi$
$181$	16.7178			0.0923638			0.0461819	+	0.998933i	$0.485295\pi$
$182$		0			0
$183$	439.251			2.40028
$184$		0			0
$185$	−39.3809			−0.212870
$186$		0			0
$187$	108.971			0.582734
$188$		0			0
$189$	−520.300			−2.75291
$190$		0			0
$191$			44.7333i			0.234206i	0.993120	+	0.117103i	$0.0373607\pi$
$191$			44.7333i			0.234206i	−0.993120	+	0.117103i	$0.962639\pi$
$192$		0			0
$193$	135.200			0.700516			0.350258	−	0.936653i	$-0.386094\pi$
$193$	135.200			0.700516			0.350258	+	0.936653i	$0.386094\pi$
$194$		0			0
$195$	208.272			1.06806
$196$		0			0
$197$	145.802			0.740109			0.370055	−	0.929010i	$-0.379339\pi$
$197$	145.802			0.740109			0.370055	+	0.929010i	$0.379339\pi$
$198$		0			0
$199$		−	280.657i		−	1.41034i	−0.709039	−	0.705169i	$-0.750871\pi$
$199$		−	280.657i		−	1.41034i	0.709039	−	0.705169i	$-0.249129\pi$
$200$		0			0
$201$		−	502.018i		−	2.49760i
$202$		0			0
$203$	−313.179			−1.54275
$204$		0			0
$205$		−	7.44664i		−	0.0363251i
$206$		0			0
$207$	−749.407			−3.62032
$208$		0			0
$209$		−	132.320i		−	0.633112i
$210$		0			0
$211$		−	181.131i		−	0.858439i	−0.903200	−	0.429219i	$-0.858789\pi$
$211$		−	181.131i		−	0.858439i	0.903200	−	0.429219i	$-0.141211\pi$
$212$		0			0
$213$	199.460			0.936434
$214$		0			0
$215$	39.8339	+	122.225i	0.185274	+	0.568490i
$216$		0			0
$217$		−	100.618i		−	0.463679i
$218$		0			0
$219$	−202.010			−0.922418
$220$		0			0
$221$	−196.205			−0.887807
$222$		0			0
$223$		−	107.757i		−	0.483213i	−0.970374	−	0.241607i	$-0.922326\pi$
$223$		−	107.757i		−	0.483213i	0.970374	−	0.241607i	$-0.0776744\pi$
$224$		0			0
$225$	−362.320			−1.61031
$226$		0			0
$227$			139.742i			0.615603i	0.951451	+	0.307801i	$0.0995932\pi$
$227$			139.742i			0.615603i	−0.951451	+	0.307801i	$0.900407\pi$
$228$		0			0
$229$	308.655			1.34784			0.673919	−	0.738805i	$-0.264610\pi$
$229$	308.655			1.34784			0.673919	+	0.738805i	$0.264610\pi$
$230$		0			0
$231$	−264.337			−1.14432
$232$		0			0
$233$		−	301.133i		−	1.29242i	−0.763161	−	0.646209i	$-0.776354\pi$
$233$		−	301.133i		−	1.29242i	0.763161	−	0.646209i	$-0.223646\pi$
$234$		0			0
$235$			30.6719i			0.130519i
$236$		0			0
$237$			281.498i			1.18776i
$238$		0			0
$239$	122.039			0.510625			0.255312	−	0.966859i	$-0.417822\pi$
$239$	122.039			0.510625			0.255312	+	0.966859i	$0.417822\pi$
$240$		0			0
$241$			475.092i			1.97133i	0.168701	+	0.985667i	$0.446043\pi$
$241$			475.092i			1.97133i	−0.168701	+	0.985667i	$0.553957\pi$
$242$		0			0
$243$		−	577.461i		−	2.37638i
$244$		0			0
$245$			6.95367i			0.0283823i
$246$		0			0
$247$			238.246i			0.964559i
$248$		0			0
$249$		−	58.2233i		−	0.233828i
$250$		0			0
$251$	35.9930			0.143399			0.0716993	−	0.997426i	$-0.477158\pi$
$251$	35.9930			0.143399			0.0716993	+	0.997426i	$0.477158\pi$
$252$		0			0
$253$	−228.826			−0.904451
$254$		0			0
$255$	−265.706			−1.04198
$256$		0			0
$257$		−	62.7857i		−	0.244302i	−0.992512	−	0.122151i	$-0.961021\pi$
$257$		−	62.7857i		−	0.244302i	0.992512	−	0.122151i	$-0.0389793\pi$
$258$		0			0
$259$	−89.9934			−0.347465
$260$		0			0
$261$		−	1034.04i		−	3.96185i
$262$		0			0
$263$			17.1841i			0.0653386i	0.999466	+	0.0326693i	$0.0104008\pi$
$263$			17.1841i			0.0653386i	−0.999466	+	0.0326693i	$0.989599\pi$
$264$		0			0
$265$		−	93.5085i		−	0.352862i
$266$		0			0
$267$	75.7811			0.283824
$268$		0			0
$269$	−407.934			−1.51648			−0.758242	−	0.651974i	$-0.773941\pi$
$269$	−407.934			−1.51648			−0.758242	+	0.651974i	$0.773941\pi$
$270$		0			0
$271$	316.274			1.16706			0.583532	−	0.812090i	$-0.301670\pi$
$271$	316.274			1.16706			0.583532	+	0.812090i	$0.301670\pi$
$272$		0			0
$273$	475.945			1.74339
$274$		0			0
$275$	−110.632			−0.402298
$276$		0			0
$277$		−	209.408i		−	0.755984i	−0.925809	−	0.377992i	$-0.876615\pi$
$277$		−	209.408i		−	0.755984i	0.925809	−	0.377992i	$-0.123385\pi$
$278$		0			0
$279$	332.218			1.19074
$280$		0			0
$281$	36.4182			0.129602			0.0648011	−	0.997898i	$-0.479359\pi$
$281$	36.4182			0.129602			0.0648011	+	0.997898i	$0.479359\pi$
$282$		0			0
$283$	15.3124			0.0541074			0.0270537	−	0.999634i	$-0.491387\pi$
$283$	15.3124			0.0541074			0.0270537	+	0.999634i	$0.491387\pi$
$284$		0			0
$285$			322.638i			1.13206i
$286$		0			0
$287$		−	17.0171i		−	0.0592930i
$288$		0			0
$289$	−38.6884			−0.133870
$290$		0			0
$291$		−	374.311i		−	1.28629i
$292$		0			0
$293$	−421.851			−1.43976			−0.719882	−	0.694097i	$-0.755804\pi$
$293$	−421.851			−1.43976			−0.719882	+	0.694097i	$0.755804\pi$
$294$		0			0
$295$		−	192.563i		−	0.652757i
$296$		0			0
$297$		−	524.551i		−	1.76616i
$298$		0			0
$299$	412.007			1.37795
$300$		0			0
$301$	91.0288	+	279.310i	0.302421	+	0.927940i
$302$		0			0
$303$		−	412.020i		−	1.35980i
$304$		0			0
$305$	−233.763			−0.766436
$306$		0			0
$307$	239.260			0.779349			0.389674	−	0.920953i	$-0.372588\pi$
$307$	239.260			0.779349			0.389674	+	0.920953i	$0.372588\pi$
$308$		0			0
$309$		−	195.157i		−	0.631576i
$310$		0			0
$311$	−565.399			−1.81800			−0.909001	−	0.416793i	$-0.863154\pi$
$311$	−565.399			−1.81800			−0.909001	+	0.416793i	$0.863154\pi$
$312$		0			0
$313$			340.149i			1.08674i	0.839493	+	0.543370i	$0.182852\pi$
$313$			340.149i			1.08674i	−0.839493	+	0.543370i	$0.817148\pi$
$314$		0			0
$315$	460.716			1.46259
$316$		0			0
$317$	−470.259			−1.48347			−0.741733	−	0.670695i	$-0.765996\pi$
$317$	−470.259			−1.48347			−0.741733	+	0.670695i	$0.765996\pi$
$318$		0			0
$319$		−	315.738i		−	0.989774i
$320$		0			0
$321$			677.025i			2.10911i
$322$		0			0
$323$		−	303.945i		−	0.941008i
$324$		0			0
$325$	199.195			0.612908
$326$		0			0
$327$			458.296i			1.40152i
$328$		0			0
$329$			70.0915i			0.213044i
$330$		0			0
$331$			297.506i			0.898809i	0.893328	+	0.449405i	$0.148364\pi$
$331$			297.506i			0.898809i	−0.893328	+	0.449405i	$0.851636\pi$
$332$		0			0
$333$		−	297.137i		−	0.892303i
$334$		0			0
$335$			267.167i			0.797514i
$336$		0			0
$337$	488.152			1.44852			0.724261	−	0.689526i	$-0.242181\pi$
$337$	488.152			1.44852			0.724261	+	0.689526i	$0.242181\pi$
$338$		0			0
$339$	−330.000			−0.973452
$340$		0			0
$341$	101.440			0.297479
$342$		0			0
$343$			350.651i			1.02231i
$344$		0			0
$345$	557.950			1.61725
$346$		0			0
$347$		−	620.235i		−	1.78742i	−0.448646	−	0.893710i	$-0.648093\pi$
$347$		−	620.235i		−	1.78742i	0.448646	−	0.893710i	$-0.351907\pi$
$348$		0			0
$349$			97.1065i			0.278242i	0.990275	+	0.139121i	$0.0444277\pi$
$349$			97.1065i			0.278242i	−0.990275	+	0.139121i	$0.955572\pi$
$350$		0			0
$351$			944.466i			2.69079i
$352$		0			0
$353$	355.666			1.00755			0.503776	−	0.863834i	$-0.331944\pi$
$353$	355.666			1.00755			0.503776	+	0.863834i	$0.331944\pi$
$354$		0			0
$355$	−106.150			−0.299014
$356$		0			0
$357$	−607.193			−1.70082
$358$		0			0
$359$	−185.120			−0.515655			−0.257827	−	0.966191i	$-0.583007\pi$
$359$	−185.120			−0.515655			−0.257827	+	0.966191i	$0.583007\pi$
$360$		0			0
$361$	−8.07135			−0.0223583
$362$		0			0
$363$			413.229i			1.13837i
$364$		0			0
$365$	107.507			0.294539
$366$		0			0
$367$	379.616			1.03438			0.517188	−	0.855872i	$-0.326979\pi$
$367$	379.616			1.03438			0.517188	+	0.855872i	$0.326979\pi$
$368$		0			0
$369$	56.1864			0.152267
$370$		0			0
$371$		−	213.686i		−	0.575974i
$372$		0			0
$373$		−	190.307i		−	0.510206i	−0.966914	−	0.255103i	$-0.917891\pi$
$373$		−	190.307i		−	0.510206i	0.966914	−	0.255103i	$-0.0821094\pi$
$374$		0			0
$375$	689.611			1.83896
$376$		0			0
$377$			568.494i			1.50794i
$378$		0			0
$379$	542.592			1.43164			0.715820	−	0.698285i	$-0.246053\pi$
$379$	542.592			1.43164			0.715820	+	0.698285i	$0.246053\pi$
$380$		0			0
$381$		−	101.020i		−	0.265143i
$382$		0			0
$383$		−	126.903i		−	0.331340i	−0.986181	−	0.165670i	$-0.947021\pi$
$383$		−	126.903i		−	0.331340i	0.986181	−	0.165670i	$-0.0529786\pi$
$384$		0			0
$385$	140.677			0.365394
$386$		0			0
$387$	−922.215	+	300.555i	−2.38299	+	0.776629i
$388$		0			0
$389$		−	689.167i		−	1.77164i	−0.464031	−	0.885819i	$-0.653597\pi$
$389$		−	689.167i		−	1.77164i	0.464031	−	0.885819i	$-0.346403\pi$
$390$		0			0
$391$	−525.623			−1.34430
$392$		0			0
$393$	580.656			1.47750
$394$		0			0
$395$		−	149.809i		−	0.379264i
$396$		0			0
$397$	−422.076			−1.06316			−0.531582	−	0.847007i	$-0.678402\pi$
$397$	−422.076			−1.06316			−0.531582	+	0.847007i	$0.678402\pi$
$398$		0			0
$399$			737.295i			1.84786i
$400$		0			0
$401$	−393.635			−0.981632			−0.490816	−	0.871263i	$-0.663301\pi$
$401$	−393.635			−0.981632			−0.490816	+	0.871263i	$0.663301\pi$
$402$		0			0
$403$	−182.646			−0.453215
$404$		0			0
$405$			672.089i			1.65948i
$406$		0			0
$407$		−	90.7287i		−	0.222921i
$408$		0			0
$409$		−	319.833i		−	0.781988i	−0.920393	−	0.390994i	$-0.872131\pi$
$409$		−	319.833i		−	0.781988i	0.920393	−	0.390994i	$-0.127869\pi$
$410$		0			0
$411$	788.967			1.91963
$412$		0			0
$413$		−	440.047i		−	1.06549i
$414$		0			0
$415$			30.9856i			0.0746642i
$416$		0			0
$417$			1396.41i			3.34872i
$418$		0			0
$419$			334.413i			0.798121i	0.916925	+	0.399061i	$0.130664\pi$
$419$			334.413i			0.798121i	−0.916925	+	0.399061i	$0.869336\pi$
$420$		0			0
$421$			332.902i			0.790740i	0.918522	+	0.395370i	$0.129384\pi$
$421$			332.902i			0.790740i	−0.918522	+	0.395370i	$0.870616\pi$
$422$		0			0
$423$	−231.426			−0.547105
$424$		0			0
$425$	−254.126			−0.597943
$426$		0			0
$427$	−534.197			−1.25105
$428$		0			0
$429$			479.834i			1.11849i
$430$		0			0
$431$	163.406			0.379132			0.189566	−	0.981868i	$-0.439292\pi$
$431$	163.406			0.379132			0.189566	+	0.981868i	$0.439292\pi$
$432$		0			0
$433$		−	190.232i		−	0.439335i	−0.975575	−	0.219668i	$-0.929503\pi$
$433$		−	190.232i		−	0.439335i	0.975575	−	0.219668i	$-0.0704973\pi$
$434$		0			0
$435$			769.868i			1.76981i
$436$		0			0
$437$			638.248i			1.46052i
$438$		0			0
$439$	343.705			0.782927			0.391463	−	0.920194i	$-0.371969\pi$
$439$	343.705			0.782927			0.391463	+	0.920194i	$0.371969\pi$
$440$		0			0
$441$	−52.4669			−0.118973
$442$		0			0
$443$	−515.530			−1.16373			−0.581863	−	0.813287i	$-0.697676\pi$
$443$	−515.530			−1.16373			−0.581863	+	0.813287i	$0.697676\pi$
$444$		0			0
$445$	−40.3296			−0.0906284
$446$		0			0
$447$	−534.000			−1.19463
$448$		0			0
$449$		−	598.611i		−	1.33321i	−0.745411	−	0.666605i	$-0.767747\pi$
$449$		−	598.611i		−	1.33321i	0.745411	−	0.666605i	$-0.232253\pi$
$450$		0			0
$451$	17.1561			0.0380402
$452$		0			0
$453$	690.543			1.52438
$454$		0			0
$455$	−253.291			−0.556684
$456$		0			0
$457$		−	459.459i		−	1.00538i	−0.864467	−	0.502690i	$-0.832344\pi$
$457$		−	459.459i		−	1.00538i	0.864467	−	0.502690i	$-0.167656\pi$
$458$		0			0
$459$		−	1204.92i		−	2.62509i
$460$		0			0
$461$	334.655			0.725933			0.362967	−	0.931802i	$-0.381764\pi$
$461$	334.655			0.725933			0.362967	+	0.931802i	$0.381764\pi$
$462$		0			0
$463$		−	696.646i		−	1.50464i	−0.658801	−	0.752318i	$-0.728936\pi$
$463$		−	696.646i		−	1.50464i	0.658801	−	0.752318i	$-0.271064\pi$
$464$		0			0
$465$	−247.343			−0.531921
$466$		0			0
$467$			123.292i			0.264009i	0.991249	+	0.132004i	$0.0421413\pi$
$467$			123.292i			0.264009i	−0.991249	+	0.132004i	$0.957859\pi$
$468$		0			0
$469$			610.532i			1.30177i
$470$		0			0
$471$	1063.59			2.25815
$472$		0			0
$473$	−281.592	+	91.7725i	−0.595332	+	0.194022i
$474$		0			0
$475$			308.577i			0.649636i
$476$		0			0
$477$	705.541			1.47912
$478$		0			0
$479$	−201.938			−0.421583			−0.210791	−	0.977531i	$-0.567604\pi$
$479$	−201.938			−0.421583			−0.210791	+	0.977531i	$0.567604\pi$
$480$		0			0
$481$			163.359i			0.339624i
$482$		0			0
$483$	1275.03			2.63981
$484$		0			0
$485$			199.203i			0.410729i
$486$		0			0
$487$	−151.294			−0.310666			−0.155333	−	0.987862i	$-0.549645\pi$
$487$	−151.294			−0.310666			−0.155333	+	0.987862i	$0.549645\pi$
$488$		0			0
$489$	1485.51			3.03785
$490$		0			0
$491$			56.4377i			0.114944i	0.998347	+	0.0574722i	$0.0183041\pi$
$491$			56.4377i			0.114944i	−0.998347	+	0.0574722i	$0.981696\pi$
$492$		0			0
$493$		−	725.263i		−	1.47112i
$494$		0			0
$495$			464.481i			0.938345i
$496$		0			0
$497$	−242.575			−0.488078
$498$		0			0
$499$		−	337.720i		−	0.676793i	−0.941004	−	0.338396i	$-0.890116\pi$
$499$		−	337.720i		−	0.676793i	0.941004	−	0.338396i	$-0.109884\pi$
$500$		0			0
$501$			224.871i			0.448844i
$502$		0			0
$503$		−	124.026i		−	0.246573i	−0.992371	−	0.123287i	$-0.960657\pi$
$503$		−	124.026i		−	0.246573i	0.992371	−	0.123287i	$-0.0393434\pi$
$504$		0			0
$505$			219.272i			0.434201i
$506$		0			0
$507$			85.4176i			0.168476i
$508$		0			0
$509$	229.831			0.451535			0.225767	−	0.974181i	$-0.427511\pi$
$509$	229.831			0.451535			0.225767	+	0.974181i	$0.427511\pi$
$510$		0			0
$511$	245.675			0.480773
$512$		0			0
$513$	−1463.09			−2.85203
$514$		0			0
$515$			103.860i			0.201670i
$516$		0			0
$517$	−70.6642			−0.136681
$518$		0			0
$519$		−	1019.57i		−	1.96450i
$520$		0			0
$521$			335.813i			0.644555i	0.946645	+	0.322278i	$0.104448\pi$
$521$			335.813i			0.644555i	−0.946645	+	0.322278i	$0.895552\pi$
$522$		0			0
$523$			630.020i			1.20463i	0.798259	+	0.602314i	$0.205754\pi$
$523$			630.020i			1.20463i	−0.798259	+	0.602314i	$0.794246\pi$
$524$		0			0
$525$	616.446			1.17418
$526$		0			0
$527$	233.013			0.442149
$528$		0			0
$529$	574.743			1.08647
$530$		0			0
$531$	1452.93			2.73621
$532$		0			0
$533$	−30.8900			−0.0579550
$534$		0			0
$535$		−	360.303i		−	0.673464i
$536$		0			0
$537$	351.644			0.654831
$538$		0			0
$539$	−16.0204			−0.0297224
$540$		0			0
$541$	579.942			1.07198			0.535991	−	0.844224i	$-0.319938\pi$
$541$	579.942			1.07198			0.535991	+	0.844224i	$0.319938\pi$
$542$		0			0
$543$		−	93.9137i		−	0.172953i
$544$		0			0
$545$		−	243.899i		−	0.447521i
$546$		0			0
$547$	−276.106			−0.504764			−0.252382	−	0.967628i	$-0.581214\pi$
$547$	−276.106			−0.504764			−0.252382	+	0.967628i	$0.581214\pi$
$548$		0			0
$549$		−	1763.79i		−	3.21273i
$550$		0			0
$551$	−880.664			−1.59830
$552$		0			0
$553$		−	342.346i		−	0.619070i
$554$		0			0
$555$			221.225i			0.398603i
$556$		0			0
$557$	−1.44850			−0.00260054			−0.00130027	−	0.999999i	$-0.500414\pi$
$557$	−1.44850			−0.00260054			−0.00130027	+	0.999999i	$0.500414\pi$
$558$		0			0
$559$	507.013	−	165.239i	0.907000	−	0.295597i
$560$		0			0
$561$		−	612.154i		−	1.09118i
$562$		0			0
$563$	87.6035			0.155601			0.0778006	−	0.996969i	$-0.475210\pi$
$563$	87.6035			0.155601			0.0778006	+	0.996969i	$0.475210\pi$
$564$		0			0
$565$	175.622			0.310835
$566$		0			0
$567$			1535.86i			2.70875i
$568$		0			0
$569$	−657.323			−1.15522			−0.577612	−	0.816311i	$-0.696015\pi$
$569$	−657.323			−1.15522			−0.577612	+	0.816311i	$0.696015\pi$
$570$		0			0
$571$		−	321.145i		−	0.562426i	−0.959645	−	0.281213i	$-0.909263\pi$
$571$		−	321.145i		−	0.562426i	0.959645	−	0.281213i	$-0.0907368\pi$
$572$		0			0
$573$	251.292			0.438556
$574$		0			0
$575$	533.633			0.928057
$576$		0			0
$577$		−	440.527i		−	0.763479i	−0.924270	−	0.381740i	$-0.875325\pi$
$577$		−	440.527i		−	0.763479i	0.924270	−	0.381740i	$-0.124675\pi$
$578$		0			0
$579$		−	759.494i		−	1.31173i
$580$		0			0
$581$			70.8085i			0.121874i
$582$		0			0
$583$	215.432			0.369523
$584$		0			0
$585$		−	836.308i		−	1.42959i
$586$		0			0
$587$			975.567i			1.66195i	0.556307	+	0.830977i	$0.312218\pi$
$587$			975.567i			1.66195i	−0.556307	+	0.830977i	$0.687782\pi$
$588$		0			0
$589$		−	282.940i		−	0.480373i
$590$		0			0
$591$		−	819.051i		−	1.38587i
$592$		0			0
$593$			14.4319i			0.0243370i	0.999926	+	0.0121685i	$0.00387345\pi$
$593$			14.4319i			0.0243370i	−0.999926	+	0.0121685i	$0.996127\pi$
$594$		0			0
$595$	323.140			0.543092
$596$		0			0
$597$	−1576.61			−2.64089
$598$		0			0
$599$	947.716			1.58216			0.791082	−	0.611710i	$-0.209518\pi$
$599$	947.716			1.58216			0.791082	+	0.611710i	$0.209518\pi$
$600$		0			0
$601$		−	1115.25i		−	1.85565i	−0.373016	−	0.927825i	$-0.621676\pi$
$601$		−	1115.25i		−	1.85565i	0.373016	−	0.927825i	$-0.378324\pi$
$602$		0			0
$603$	−2015.83			−3.34301
$604$		0			0
$605$		−	219.915i		−	0.363496i
$606$		0			0
$607$		−	297.682i		−	0.490416i	−0.969471	−	0.245208i	$-0.921144\pi$
$607$		−	297.682i		−	0.490416i	0.969471	−	0.245208i	$-0.0788562\pi$
$608$		0			0
$609$			1759.31i			2.88885i
$610$		0			0
$611$	127.233			0.208237
$612$		0			0
$613$	382.944			0.624705			0.312352	−	0.949966i	$-0.398883\pi$
$613$	382.944			0.624705			0.312352	+	0.949966i	$0.398883\pi$
$614$		0			0
$615$	−41.8320			−0.0680195
$616$		0			0
$617$	−702.287			−1.13823			−0.569114	−	0.822258i	$-0.692714\pi$
$617$	−702.287			−1.13823			−0.569114	+	0.822258i	$0.692714\pi$
$618$		0			0
$619$	680.743			1.09975			0.549873	−	0.835248i	$-0.314676\pi$
$619$	680.743			1.09975			0.549873	+	0.835248i	$0.314676\pi$
$620$		0			0
$621$			2530.17i			4.07435i
$622$		0			0
$623$	−92.1616			−0.147932
$624$		0			0
$625$	34.5564			0.0552903
$626$		0			0
$627$	−743.319			−1.18552
$628$		0			0
$629$		−	208.408i		−	0.331332i
$630$		0			0
$631$		−	1019.35i		−	1.61546i	−0.589555	−	0.807728i	$-0.700697\pi$
$631$		−	1019.35i		−	1.61546i	0.589555	−	0.807728i	$-0.299303\pi$
$632$		0			0
$633$	−1017.51			−1.60745
$634$		0			0
$635$			53.7613i			0.0846634i
$636$		0			0
$637$	28.8451			0.0452827
$638$		0			0
$639$		−	800.925i		−	1.25340i
$640$		0			0
$641$		−	815.451i		−	1.27215i	−0.771625	−	0.636077i	$-0.780556\pi$
$641$		−	815.451i		−	1.27215i	0.771625	−	0.636077i	$-0.219444\pi$
$642$		0			0
$643$	−661.567			−1.02888			−0.514438	−	0.857528i	$-0.671999\pi$
$643$	−661.567			−1.02888			−0.514438	+	0.857528i	$0.671999\pi$
$644$		0			0
$645$	686.609	−	223.770i	1.06451	−	0.346930i
$646$		0			0
$647$		−	1013.73i		−	1.56681i	−0.621512	−	0.783405i	$-0.713481\pi$
$647$		−	1013.73i		−	1.56681i	0.621512	−	0.783405i	$-0.286519\pi$
$648$		0			0
$649$	443.642			0.683578
$650$		0			0
$651$	−565.230			−0.868249
$652$		0			0
$653$			812.108i			1.24366i	0.783153	+	0.621829i	$0.213610\pi$
$653$			812.108i			1.24366i	−0.783153	+	0.621829i	$0.786390\pi$
$654$		0			0
$655$	−309.017			−0.471782
$656$		0			0
$657$			811.160i			1.23464i
$658$		0			0
$659$	1106.83			1.67955			0.839777	−	0.542932i	$-0.182686\pi$
$659$	1106.83			1.67955			0.839777	+	0.542932i	$0.182686\pi$
$660$		0			0
$661$	523.333			0.791729			0.395865	−	0.918309i	$-0.370445\pi$
$661$	523.333			0.791729			0.395865	+	0.918309i	$0.370445\pi$
$662$		0			0
$663$			1102.20i			1.66244i
$664$		0			0
$665$		−	392.378i		−	0.590043i
$666$		0			0
$667$			1522.96i			2.28330i
$668$		0			0
$669$	−605.330			−0.904828
$670$		0			0
$671$		−	538.562i		−	0.802625i
$672$		0			0
$673$		−	320.300i		−	0.475929i	−0.971274	−	0.237965i	$-0.923520\pi$
$673$		−	320.300i		−	0.475929i	0.971274	−	0.237965i	$-0.0764803\pi$
$674$		0			0
$675$			1223.28i			1.81226i
$676$		0			0
$677$		−	854.967i		−	1.26288i	−0.775426	−	0.631438i	$-0.782465\pi$
$677$		−	854.967i		−	1.26288i	0.775426	−	0.631438i	$-0.217535\pi$
$678$		0			0
$679$			455.221i			0.670428i
$680$		0			0
$681$	785.010			1.15273
$682$		0			0
$683$	−1331.16			−1.94898			−0.974492	−	0.224420i	$-0.927951\pi$
$683$	−1331.16			−1.94898			−0.974492	+	0.224420i	$0.927951\pi$
$684$		0			0
$685$	−419.877			−0.612960
$686$		0			0
$687$		−	1733.89i		−	2.52386i
$688$		0			0
$689$	−387.891			−0.562976
$690$		0			0
$691$			568.600i			0.822866i	0.911440	+	0.411433i	$0.134972\pi$
$691$			568.600i			0.822866i	−0.911440	+	0.411433i	$0.865028\pi$
$692$		0			0
$693$			1061.43i			1.53165i
$694$		0			0
$695$		−	743.152i		−	1.06928i
$696$		0			0
$697$	39.4084			0.0565400
$698$		0			0
$699$	−1691.64			−2.42008
$700$		0			0
$701$	369.079			0.526504			0.263252	−	0.964727i	$-0.415205\pi$
$701$	369.079			0.526504			0.263252	+	0.964727i	$0.415205\pi$
$702$		0			0
$703$	−253.063			−0.359975
$704$		0			0
$705$	172.301			0.244399
$706$		0			0
$707$			501.081i			0.708742i
$708$		0			0
$709$	−656.495			−0.925945			−0.462972	−	0.886373i	$-0.653217\pi$
$709$	−656.495			−0.925945			−0.462972	+	0.886373i	$0.653217\pi$
$710$		0			0
$711$	1130.34			1.58979
$712$		0			0
$713$	−489.298			−0.686252
$714$		0			0
$715$		−	255.361i		−	0.357148i
$716$		0			0
$717$		−	685.564i		−	0.956157i
$718$		0			0
$719$	497.631			0.692115			0.346057	−	0.938213i	$-0.387520\pi$
$719$	497.631			0.692115			0.346057	+	0.938213i	$0.387520\pi$
$720$		0			0
$721$			237.341i			0.329183i
$722$		0			0
$723$	2668.86			3.69137
$724$		0			0
$725$			736.315i			1.01561i
$726$		0			0
$727$		−	1020.14i		−	1.40322i	−0.712559	−	0.701612i	$-0.752464\pi$
$727$		−	1020.14i		−	1.40322i	0.712559	−	0.701612i	$-0.247536\pi$
$728$		0			0
$729$	−1220.64			−1.67441
$730$		0			0
$731$	−646.829	+	210.805i	−0.884855	+	0.288379i
$732$		0			0
$733$		−	310.797i		−	0.424007i	−0.977269	−	0.212003i	$-0.932001\pi$
$733$		−	310.797i		−	0.424007i	0.977269	−	0.212003i	$-0.0679987\pi$
$734$		0			0
$735$	39.0627			0.0531466
$736$		0			0
$737$	−615.521			−0.835171
$738$		0			0
$739$			1188.34i			1.60804i	0.594603	+	0.804020i	$0.297309\pi$
$739$			1188.34i			1.60804i	−0.594603	+	0.804020i	$0.702691\pi$
$740$		0			0
$741$	1338.36			1.80616
$742$		0			0
$743$		−	258.515i		−	0.347934i	−0.984752	−	0.173967i	$-0.944341\pi$
$743$		−	258.515i		−	0.347934i	0.984752	−	0.173967i	$-0.0556585\pi$
$744$		0			0
$745$	284.188			0.381460
$746$		0			0
$747$	−233.793			−0.312976
$748$		0			0
$749$		−	823.367i		−	1.09929i
$750$		0			0
$751$			816.888i			1.08773i	0.839171	+	0.543867i	$0.183040\pi$
$751$			816.888i			1.08773i	−0.839171	+	0.543867i	$0.816960\pi$
$752$		0			0
$753$		−	202.193i		−	0.268517i
$754$		0			0
$755$	−367.497			−0.486752
$756$		0			0
$757$			807.752i			1.06704i	0.845786	+	0.533522i	$0.179132\pi$
$757$			807.752i			1.06704i	−0.845786	+	0.533522i	$0.820868\pi$
$758$		0			0
$759$			1285.45i			1.69361i
$760$		0			0
$761$		−	589.286i		−	0.774358i	−0.922005	−	0.387179i	$-0.873450\pi$
$761$		−	589.286i		−	0.774358i	0.922005	−	0.387179i	$-0.126550\pi$
$762$		0			0
$763$		−	557.359i		−	0.730483i
$764$		0			0
$765$			1066.93i			1.39468i
$766$		0			0
$767$	−798.788			−1.04144
$768$		0			0
$769$	1272.87			1.65522			0.827612	−	0.561300i	$-0.189699\pi$
$769$	1272.87			1.65522			0.827612	+	0.561300i	$0.189699\pi$
$770$		0			0
$771$	−352.703			−0.457462
$772$		0			0
$773$		−	176.192i		−	0.227933i	−0.993485	−	0.113967i	$-0.963644\pi$
$773$		−	176.192i		−	0.227933i	0.993485	−	0.113967i	$-0.0363557\pi$
$774$		0			0
$775$	−236.563			−0.305243
$776$		0			0
$777$			505.544i			0.650636i
$778$		0			0
$779$		−	47.8523i		−	0.0614279i
$780$		0			0
$781$		−	244.557i		−	0.313133i
$782$		0			0
$783$	−3491.17			−4.45871
$784$		0			0
$785$	−566.026			−0.721053
$786$		0			0
$787$	1120.38			1.42361			0.711804	−	0.702379i	$-0.247879\pi$
$787$	1120.38			1.42361			0.711804	+	0.702379i	$0.247879\pi$
$788$		0			0
$789$	96.5327			0.122348
$790$		0			0
$791$	401.332			0.507373
$792$		0			0
$793$			969.693i			1.22282i
$794$		0			0
$795$	−525.291			−0.660743
$796$		0			0
$797$	−872.371			−1.09457			−0.547284	−	0.836947i	$-0.684338\pi$
$797$	−872.371			−1.09457			−0.547284	+	0.836947i	$0.684338\pi$
$798$		0			0
$799$	−162.319			−0.203152
$800$		0			0
$801$		−	304.296i		−	0.379895i
$802$		0			0
$803$			247.682i			0.308446i
$804$		0			0
$805$	−678.553			−0.842924
$806$		0			0
$807$			2291.60i			2.83965i
$808$		0			0
$809$	273.069			0.337539			0.168770	−	0.985656i	$-0.446021\pi$
$809$	273.069			0.337539			0.168770	+	0.985656i	$0.446021\pi$
$810$		0			0
$811$		−	1420.64i		−	1.75171i	−0.482576	−	0.875854i	$-0.660299\pi$
$811$		−	1420.64i		−	1.75171i	0.482576	−	0.875854i	$-0.339701\pi$
$812$		0			0
$813$		−	1776.69i		−	2.18536i
$814$		0			0
$815$	−790.568			−0.970022
$816$		0			0
$817$	255.974	+	785.424i	0.313310	+	0.961351i
$818$		0			0
$819$		−	1911.14i		−	2.33350i
$820$		0			0
$821$	1163.98			1.41775			0.708877	−	0.705332i	$-0.249202\pi$
$821$	1163.98			1.41775			0.708877	+	0.705332i	$0.249202\pi$
$822$		0			0
$823$	−47.3698			−0.0575574			−0.0287787	−	0.999586i	$-0.509162\pi$
$823$	−47.3698			−0.0575574			−0.0287787	+	0.999586i	$0.509162\pi$
$824$		0			0
$825$			621.482i			0.753312i
$826$		0			0
$827$	662.925			0.801602			0.400801	−	0.916165i	$-0.368732\pi$
$827$	662.925			0.801602			0.400801	+	0.916165i	$0.368732\pi$
$828$		0			0
$829$			952.975i			1.14955i	0.818312	+	0.574774i	$0.194910\pi$
$829$			952.975i			1.14955i	−0.818312	+	0.574774i	$0.805090\pi$
$830$		0			0
$831$	−1176.36			−1.41560
$832$		0			0
$833$	−36.7995			−0.0441771
$834$		0			0
$835$		−	119.673i		−	0.143321i
$836$		0			0
$837$		−	1121.64i		−	1.34008i
$838$		0			0
$839$			218.560i			0.260501i	0.991481	+	0.130250i	$0.0415781\pi$
$839$			218.560i			0.260501i	−0.991481	+	0.130250i	$0.958422\pi$
$840$		0			0
$841$	−1260.41			−1.49870
$842$		0			0
$843$		−	204.582i		−	0.242683i
$844$		0			0
$845$		−	45.4581i		−	0.0537965i
$846$		0			0
$847$		−	502.551i		−	0.593331i
$848$		0			0
$849$		−	86.0185i		−	0.101317i
$850$		0			0
$851$			437.630i			0.514254i
$852$		0			0
$853$	−1445.29			−1.69436			−0.847180	−	0.531306i	$-0.821701\pi$
$853$	−1445.29			−1.69436			−0.847180	+	0.531306i	$0.821701\pi$
$854$		0			0
$855$	1295.54			1.51525
$856$		0			0
$857$	−92.3800			−0.107795			−0.0538973	−	0.998546i	$-0.517164\pi$
$857$	−92.3800			−0.107795			−0.0538973	+	0.998546i	$0.517164\pi$
$858$		0			0
$859$		−	786.473i		−	0.915568i	−0.889064	−	0.457784i	$-0.848643\pi$
$859$		−	786.473i		−	0.915568i	0.889064	−	0.457784i	$-0.151357\pi$
$860$		0			0
$861$	−95.5948			−0.111028
$862$		0			0
$863$		−	1277.18i		−	1.47993i	−0.672646	−	0.739965i	$-0.734842\pi$
$863$		−	1277.18i		−	1.47993i	0.672646	−	0.739965i	$-0.265158\pi$
$864$		0			0
$865$			542.603i			0.627287i
$866$		0			0
$867$			217.335i			0.250674i
$868$		0			0
$869$	345.143			0.397172
$870$		0			0
$871$	1108.26			1.27240
$872$		0			0
$873$	−1503.03			−1.72169
$874$		0			0
$875$	−838.675			−0.958485
$876$		0			0
$877$	1081.16			1.23279			0.616396	−	0.787436i	$-0.288592\pi$
$877$	1081.16			1.23279			0.616396	+	0.787436i	$0.288592\pi$
$878$		0			0
$879$			2369.78i			2.69599i
$880$		0			0
$881$	−575.591			−0.653338			−0.326669	−	0.945139i	$-0.605926\pi$
$881$	−575.591			−0.653338			−0.326669	+	0.945139i	$0.605926\pi$
$882$		0			0
$883$	−1401.90			−1.58766			−0.793828	−	0.608143i	$-0.791915\pi$
$883$	−1401.90			−1.58766			−0.793828	+	0.608143i	$0.791915\pi$
$884$		0			0
$885$	−1081.74			−1.22230
$886$		0			0
$887$		−	205.103i		−	0.231232i	−0.993294	−	0.115616i	$-0.963116\pi$
$887$		−	205.103i		−	0.231232i	0.993294	−	0.115616i	$-0.0368842\pi$
$888$		0			0
$889$			122.856i			0.138195i
$890$		0			0
$891$	−1548.41			−1.73783
$892$		0			0
$893$			197.098i			0.220715i
$894$		0			0
$895$	−187.140			−0.209095
$896$		0			0
$897$		−	2314.48i		−	2.58024i
$898$		0			0
$899$		−	675.141i		−	0.750991i
$900$		0			0
$901$	494.857			0.549230
$902$		0			0
$903$	1569.04	−	511.361i	1.73759	−	0.566291i
$904$		0			0
$905$			49.9796i			0.0552260i
$906$		0			0
$907$	252.284			0.278152			0.139076	−	0.990282i	$-0.455587\pi$
$907$	252.284			0.278152			0.139076	+	0.990282i	$0.455587\pi$
$908$		0			0
$909$	−1654.45			−1.82008
$910$		0			0
$911$		−	1367.68i		−	1.50130i	−0.660702	−	0.750649i	$-0.729741\pi$
$911$		−	1367.68i		−	1.50130i	0.660702	−	0.750649i	$-0.270259\pi$
$912$		0			0
$913$	−71.3871			−0.0781896
$914$		0			0
$915$			1313.18i			1.43517i
$916$		0			0
$917$	−706.167			−0.770084
$918$		0			0
$919$	270.734			0.294597			0.147298	−	0.989092i	$-0.452942\pi$
$919$	270.734			0.294597			0.147298	+	0.989092i	$0.452942\pi$
$920$		0			0
$921$		−	1344.06i		−	1.45935i
$922$		0			0
$923$			440.330i			0.477064i
$924$		0			0
$925$			211.583i			0.228739i
$926$		0			0
$927$	−783.645			−0.845356
$928$		0			0
$929$		−	791.662i		−	0.852165i	−0.904684	−	0.426083i	$-0.859893\pi$
$929$		−	791.662i		−	0.852165i	0.904684	−	0.426083i	$-0.140107\pi$
$930$		0			0
$931$			44.6845i			0.0479962i
$932$		0			0
$933$			3176.17i			3.40425i
$934$		0			0
$935$			325.780i			0.348428i
$936$		0			0
$937$			1025.37i			1.09431i	0.837030	+	0.547157i	$0.184290\pi$
$937$			1025.37i			1.09431i	−0.837030	+	0.547157i	$0.815710\pi$
$938$		0			0
$939$	1910.81			2.03495
$940$		0			0
$941$	1515.52			1.61054			0.805271	−	0.592907i	$-0.202020\pi$
$941$	1515.52			1.61054			0.805271	+	0.592907i	$0.202020\pi$
$942$		0			0
$943$	−82.7527			−0.0877547
$944$		0			0
$945$		−	1555.49i		−	1.64602i
$946$		0			0
$947$	−804.042			−0.849042			−0.424521	−	0.905418i	$-0.639557\pi$
$947$	−804.042			−0.849042			−0.424521	+	0.905418i	$0.639557\pi$
$948$		0			0
$949$		−	445.958i		−	0.469924i
$950$		0			0
$951$			2641.71i			2.77783i
$952$		0			0
$953$			1307.06i			1.37152i	0.727828	+	0.685760i	$0.240530\pi$
$953$			1307.06i			1.37152i	−0.727828	+	0.685760i	$0.759470\pi$
$954$		0			0
$955$	−133.734			−0.140036
$956$		0			0
$957$	−1773.68			−1.85338
$958$		0			0
$959$	−959.507			−1.00053
$960$		0			0
$961$	−744.091			−0.774288
$962$		0			0
$963$	2718.56			2.82301
$964$		0			0
$965$			404.192i			0.418852i
$966$		0			0
$967$	1273.66			1.31713			0.658563	−	0.752525i	$-0.271164\pi$
$967$	1273.66			1.31713			0.658563	+	0.752525i	$0.271164\pi$
$968$		0			0
$969$	−1707.44			−1.76206
$970$		0			0
$971$	−1511.12			−1.55625			−0.778123	−	0.628112i	$-0.783828\pi$
$971$	−1511.12			−1.55625			−0.778123	+	0.628112i	$0.783828\pi$
$972$		0			0
$973$		−	1698.26i		−	1.74538i
$974$		0			0
$975$		−	1118.99i		−	1.14769i
$976$		0			0
$977$	1275.34			1.30537			0.652684	−	0.757630i	$-0.273643\pi$
$977$	1275.34			1.30537			0.652684	+	0.757630i	$0.273643\pi$
$978$		0			0
$979$		−	92.9146i		−	0.0949076i
$980$		0			0
$981$	1840.27			1.87591
$982$		0			0
$983$		−	643.947i		−	0.655084i	−0.944837	−	0.327542i	$-0.893780\pi$
$983$		−	643.947i		−	0.655084i	0.944837	−	0.327542i	$-0.106220\pi$
$984$		0			0
$985$			435.887i			0.442525i
$986$		0			0
$987$	393.744			0.398930
$988$		0			0
$989$	1358.26	−	442.665i	1.37337	−	0.447588i
$990$		0			0
$991$			778.474i			0.785544i	0.919636	+	0.392772i	$0.128484\pi$
$991$			778.474i			0.785544i	−0.919636	+	0.392772i	$0.871516\pi$
$992$		0			0
$993$	1671.26			1.68304
$994$		0			0
$995$	839.052			0.843268
$996$		0			0
$997$		−	457.177i		−	0.458552i	−0.973361	−	0.229276i	$-0.926364\pi$
$997$		−	457.177i		−	0.458552i	0.973361	−	0.229276i	$-0.0736359\pi$
$998$		0			0
$999$	−1003.20			−1.00421

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.3.b.e.257.1		6
4.3	odd	2		43.3.b.b.42.3	✓	6
12.11	even	2		387.3.b.c.343.4		6
43.42	odd	2	inner	688.3.b.e.257.6		6
172.171	even	2		43.3.b.b.42.4	yes	6
516.515	odd	2		387.3.b.c.343.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.b.b.42.3	✓	6	4.3	odd	2
43.3.b.b.42.4	yes	6	172.171	even	2
387.3.b.c.343.3		6	516.515	odd	2
387.3.b.c.343.4		6	12.11	even	2
688.3.b.e.257.1		6	1.1	even	1	trivial
688.3.b.e.257.6		6	43.42	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	688.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.61757i q^{3} +2.98959i q^{5} +6.83184i q^{7} -22.5571 q^{9} +O(q^{10})\)	\(q-5.61757i q^{3} +2.98959i q^{5} +6.83184i q^{7} -22.5571 q^{9} -6.88766 q^{11} +12.4014 q^{13} +16.7943 q^{15} -15.8212 q^{17} +19.2112i q^{19} +38.3784 q^{21} +33.2226 q^{23} +16.0623 q^{25} +76.1581i q^{27} +45.8411i q^{29} -14.7278 q^{31} +38.6919i q^{33} -20.4244 q^{35} +13.1726i q^{37} -69.6657i q^{39} -2.49085 q^{41} +(40.8836 - 13.3242i) q^{43} -67.4366i q^{45} +10.2595 q^{47} +2.32596 q^{49} +88.8770i q^{51} -31.2780 q^{53} -20.5913i q^{55} +107.920 q^{57} -64.4112 q^{59} +78.1922i q^{61} -154.107i q^{63} +37.0751i q^{65} +89.3657 q^{67} -186.631i q^{69} +35.5065i q^{71} -35.9603i q^{73} -90.2313i q^{75} -47.0554i q^{77} -50.1103 q^{79} +224.809 q^{81} +10.3645 q^{83} -47.2991i q^{85} +257.516 q^{87} +13.4900i q^{89} +84.7243i q^{91} +82.7347i q^{93} -57.4338 q^{95} +66.6322 q^{97} +155.366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 36 q^{9}+O(q^{10}) \) 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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