LMFDB - Embedded newform 387.3.b.c.343.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,3,Mod(343,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("387.343");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	387.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$10.5449862307$
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			343.4
Root			$1.77533i$ of defining polynomial
Character	$\chi$	$=$	387.343
Dual form			387.3.b.c.343.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.77533i q^{2} +0.848217 q^{4} -2.98959i q^{5} -6.83184i q^{7} +8.60717i q^{8} +O(q^{10})$	\(q+1.77533i q^{2} +0.848217 q^{4} -2.98959i q^{5} -6.83184i q^{7} +8.60717i q^{8} +5.30751 q^{10} -6.88766 q^{11} +12.4014 q^{13} +12.1287 q^{14} -11.8877 q^{16} +15.8212 q^{17} -19.2112i q^{19} -2.53583i q^{20} -12.2278i q^{22} +33.2226 q^{23} +16.0623 q^{25} +22.0165i q^{26} -5.79488i q^{28} -45.8411i q^{29} +14.7278 q^{31} +13.3242i q^{32} +28.0879i q^{34} -20.4244 q^{35} +13.1726i q^{37} +34.1062 q^{38} +25.7319 q^{40} +2.49085 q^{41} +(-40.8836 + 13.3242i) q^{43} -5.84223 q^{44} +58.9810i q^{46} +10.2595 q^{47} +2.32596 q^{49} +28.5159i q^{50} +10.5191 q^{52} +31.2780 q^{53} +20.5913i q^{55} +58.8028 q^{56} +81.3829 q^{58} -64.4112 q^{59} +78.1922i q^{61} +26.1467i q^{62} -71.2054 q^{64} -37.0751i q^{65} -89.3657 q^{67} +13.4198 q^{68} -36.2600i q^{70} +35.5065i q^{71} -35.9603i q^{73} -23.3857 q^{74} -16.2953i q^{76} +47.0554i q^{77} +50.1103 q^{79} +35.5393i q^{80} +4.42207i q^{82} +10.3645 q^{83} -47.2991i q^{85} +(-23.6548 - 72.5817i) q^{86} -59.2832i q^{88} -13.4900i q^{89} -84.7243i q^{91} +28.1800 q^{92} +18.2140i q^{94} -57.4338 q^{95} +66.6322 q^{97} +4.12933i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{4}+O(q^{10}) $ 6 * q - 16 * q^4	$ 6 q - 16 q^{4} - 2 q^{10} - 38 q^{11} + 30 q^{13} - 36 q^{14} - 68 q^{16} + 20 q^{17} + 80 q^{23} - 84 q^{25} - 112 q^{31} - 208 q^{35} - 170 q^{38} + 206 q^{40} + 172 q^{41} + 10 q^{43} + 36 q^{44} - 30 q^{47} - 6 q^{49} - 120 q^{52} + 110 q^{53} + 264 q^{56} + 430 q^{58} + 12 q^{59} + 100 q^{64} - 70 q^{67} + 50 q^{68} + 50 q^{74} + 178 q^{79} - 10 q^{83} + 372 q^{86} - 150 q^{92} + 130 q^{95} - 380 q^{97}+O(q^{100}) $ 6 * q - 16 * q^4 - 2 * q^10 - 38 * q^11 + 30 * q^13 - 36 * q^14 - 68 * q^16 + 20 * q^17 + 80 * q^23 - 84 * q^25 - 112 * q^31 - 208 * q^35 - 170 * q^38 + 206 * q^40 + 172 * q^41 + 10 * q^43 + 36 * q^44 - 30 * q^47 - 6 * q^49 - 120 * q^52 + 110 * q^53 + 264 * q^56 + 430 * q^58 + 12 * q^59 + 100 * q^64 - 70 * q^67 + 50 * q^68 + 50 * q^74 + 178 * q^79 - 10 * q^83 + 372 * q^86 - 150 * q^92 + 130 * q^95 - 380 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/387\mathbb{Z}\right)^\times$.

$n$	$46$	$173$
$\chi(n)$	$-1$	$1$

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 $

$2$			1.77533i			0.887663i	0.896110	+	0.443832i	$0.146381\pi$
$2$			1.77533i			0.887663i	−0.896110	+	0.443832i	$0.853619\pi$
$3$		0			0
$3$		0			0
$4$	0.848217			0.212054
$5$		−	2.98959i		−	0.597919i	−0.954266	−	0.298959i	$-0.903360\pi$
$5$		−	2.98959i		−	0.597919i	0.954266	−	0.298959i	$-0.0966395\pi$
$6$		0			0
$7$		−	6.83184i		−	0.975977i	−0.872850	−	0.487989i	$-0.837731\pi$
$7$		−	6.83184i		−	0.975977i	0.872850	−	0.487989i	$-0.162269\pi$
$8$			8.60717i			1.07590i
$9$		0			0
$10$	5.30751			0.530751
$11$	−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	$0.341733\pi$
$14$	12.1287			0.866339
$15$		0			0
$16$	−11.8877			−0.742979
$17$	15.8212			0.930661			0.465331	−	0.885137i	$-0.345935\pi$
$17$	15.8212			0.930661			0.465331	+	0.885137i	$0.345935\pi$
$18$		0			0
$19$		−	19.2112i		−	1.01112i	−0.862792	−	0.505559i	$-0.831286\pi$
$19$		−	19.2112i		−	1.01112i	0.862792	−	0.505559i	$-0.168714\pi$
$20$		−	2.53583i		−	0.126791i
$21$		0			0
$22$		−	12.2278i		−	0.555811i
$23$	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
$26$			22.0165i			0.846789i
$27$		0			0
$28$		−	5.79488i		−	0.206960i
$29$		−	45.8411i		−	1.58073i	−0.612637	−	0.790364i	$-0.709891\pi$
$29$		−	45.8411i		−	1.58073i	0.612637	−	0.790364i	$-0.290109\pi$
$30$		0			0
$31$	14.7278			0.475092			0.237546	−	0.971376i	$-0.423657\pi$
$31$	14.7278			0.475092			0.237546	+	0.971376i	$0.423657\pi$
$32$			13.3242i			0.416381i
$33$		0			0
$34$			28.0879i			0.826114i
$35$	−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	$0.943034\pi$
$38$	34.1062			0.897532
$39$		0			0
$40$	25.7319			0.643298
$41$	2.49085			0.0607525			0.0303762	−	0.999539i	$-0.490329\pi$
$41$	2.49085			0.0607525			0.0303762	+	0.999539i	$0.490329\pi$
$42$		0			0
$43$	−40.8836	+	13.3242i	−0.950781	+	0.309865i
$43$	−40.8836	+	13.3242i	−0.950781	+	0.309865i
$44$	−5.84223			−0.132778
$45$		0			0
$46$			58.9810i			1.28220i
$47$	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
$50$			28.5159i			0.570317i
$51$		0			0
$52$	10.5191			0.202290
$53$	31.2780			0.590151			0.295075	−	0.955474i	$-0.404655\pi$
$53$	31.2780			0.590151			0.295075	+	0.955474i	$0.404655\pi$
$54$		0			0
$55$			20.5913i			0.374387i
$56$	58.8028			1.05005
$57$		0			0
$58$	81.3829			1.40315
$59$	−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	$0.778553\pi$
$62$			26.1467i			0.421721i
$63$		0			0
$64$	−71.2054			−1.11258
$65$		−	37.0751i		−	0.570387i
$66$		0			0
$67$	−89.3657			−1.33382			−0.666908	−	0.745140i	$-0.732383\pi$
$67$	−89.3657			−1.33382			−0.666908	+	0.745140i	$0.732383\pi$
$68$	13.4198			0.197351
$69$		0			0
$70$		−	36.2600i		−	0.518000i
$71$			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	$-0.0792160\pi$
$74$	−23.3857			−0.316024
$75$		0			0
$76$		−	16.2953i		−	0.214412i
$77$			47.0554i			0.611109i
$78$		0			0
$79$	50.1103			0.634308			0.317154	−	0.948374i	$-0.397273\pi$
$79$	50.1103			0.634308			0.317154	+	0.948374i	$0.397273\pi$
$80$			35.5393i			0.444241i
$81$		0			0
$82$			4.42207i			0.0539277i
$83$	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
$86$	−23.6548	−	72.5817i	−0.275056	−	0.843973i
$87$		0			0
$88$		−	59.2832i		−	0.673673i
$89$		−	13.4900i		−	0.151573i	−0.997124	−	0.0757865i	$-0.975853\pi$
$89$		−	13.4900i		−	0.151573i	0.997124	−	0.0757865i	$-0.0241468\pi$
$90$		0			0
$91$		−	84.7243i		−	0.931037i
$92$	28.1800			0.306304
$93$		0			0
$94$			18.2140i			0.193766i
$95$	−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	$0.388399\pi$
$98$			4.12933i			0.0421360i
$99$		0			0
$100$	13.6243			0.136243
$101$	−73.3449			−0.726187			−0.363094	−	0.931753i	$-0.618279\pi$
$101$	−73.3449			−0.726187			−0.363094	+	0.931753i	$0.618279\pi$
$102$		0			0
$103$	−34.7405			−0.337286			−0.168643	−	0.985677i	$-0.553939\pi$
$103$	−34.7405			−0.337286			−0.168643	+	0.985677i	$0.553939\pi$
$104$			106.741i			1.02635i
$105$		0			0
$106$			55.5286i			0.523855i
$107$	−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	$0.622094\pi$
$110$	−36.5563			−0.332330
$111$		0			0
$112$			81.2146i			0.725130i
$113$			58.7443i			0.519861i	0.965627	+	0.259931i	$0.0836997\pi$
$113$			58.7443i			0.519861i	−0.965627	+	0.259931i	$0.916300\pi$
$114$		0			0
$115$		−	99.3222i		−	0.863671i
$116$		−	38.8832i		−	0.335200i
$117$		0			0
$118$		−	114.351i		−	0.969075i
$119$		−	108.088i		−	0.908304i
$120$		0			0
$121$	−73.5601			−0.607935
$122$	−138.817			−1.13784
$123$		0			0
$124$	12.4924			0.100745
$125$		−	122.760i		−	0.982078i
$126$		0			0
$127$	−17.9828			−0.141597			−0.0707984	−	0.997491i	$-0.522555\pi$
$127$	−17.9828			−0.141597			−0.0707984	+	0.997491i	$0.522555\pi$
$128$		−	73.1161i		−	0.571219i
$129$		0			0
$130$	65.8205			0.506311
$131$			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
$134$		−	158.653i		−	1.18398i
$135$		0			0
$136$			136.176i			1.00129i
$137$		−	140.446i		−	1.02516i	−0.858641	−	0.512578i	$-0.828691\pi$
$137$		−	140.446i		−	1.02516i	0.858641	−	0.512578i	$-0.171309\pi$
$138$		0			0
$139$	248.580			1.78834			0.894172	−	0.447725i	$-0.147765\pi$
$139$	248.580			1.78834			0.894172	+	0.447725i	$0.147765\pi$
$140$	−17.3244			−0.123745
$141$		0			0
$142$	−63.0357			−0.443913
$143$	−85.4166			−0.597319
$144$		0			0
$145$	−137.046			−0.945147
$146$	63.8412			0.437269
$147$		0			0
$148$			11.1733i			0.0754950i
$149$			95.0589i			0.637979i	0.947758	+	0.318990i	$0.103344\pi$
$149$			95.0589i			0.637979i	−0.947758	+	0.318990i	$0.896656\pi$
$150$		0			0
$151$		−	122.926i		−	0.814076i	−0.913411	−	0.407038i	$-0.866562\pi$
$151$		−	122.926i		−	0.814076i	0.913411	−	0.407038i	$-0.133438\pi$
$152$	165.354			1.08786
$153$		0			0
$154$	−83.5387			−0.542459
$155$		−	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	$0.793984\pi$
$158$			88.9621i			0.563051i
$159$		0			0
$160$	39.8339			0.248962
$161$		−	226.972i		−	1.40976i
$162$		0			0
$163$		−	264.440i		−	1.62233i	−0.584817	−	0.811165i	$-0.698834\pi$
$163$		−	264.440i		−	1.62233i	0.584817	−	0.811165i	$-0.301166\pi$
$164$	2.11278			0.0128828
$165$		0			0
$166$			18.4004i			0.110845i
$167$	−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
$170$	83.9713			0.493949
$171$		0			0
$172$	−34.6781	+	11.3018i	−0.201617	+	0.0657082i
$173$	−181.497			−1.04912			−0.524559	−	0.851374i	$-0.675770\pi$
$173$	−181.497			−1.04912			−0.524559	+	0.851374i	$0.675770\pi$
$174$		0			0
$175$		−	109.735i		−	0.627059i
$176$	81.8782			0.465217
$177$		0			0
$178$	23.9492			0.134546
$179$			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	$0.485295\pi$
$182$	150.413			0.826447
$183$		0			0
$184$			285.953i			1.55409i
$185$	39.3809			0.212870
$186$		0			0
$187$	−108.971			−0.582734
$188$	8.70231			0.0462889
$189$		0			0
$190$		−	101.964i		−	0.536651i
$191$			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	$0.386094\pi$
$194$			118.294i			0.609763i
$195$		0			0
$196$	1.97292			0.0100659
$197$	−145.802			−0.740109			−0.370055	−	0.929010i	$-0.620661\pi$
$197$	−145.802			−0.740109			−0.370055	+	0.929010i	$0.620661\pi$
$198$		0			0
$199$			280.657i			1.41034i	0.709039	+	0.705169i	$0.249129\pi$
$199$			280.657i			1.41034i	−0.709039	+	0.705169i	$0.750871\pi$
$200$			138.251i			0.691256i
$201$		0			0
$202$		−	130.211i		−	0.644609i
$203$	−313.179			−1.54275
$204$		0			0
$205$		−	7.44664i		−	0.0363251i
$206$		−	61.6757i		−	0.299396i
$207$		0			0
$208$	−147.424			−0.708767
$209$			132.320i			0.633112i
$210$		0			0
$211$			181.131i			0.858439i	0.903200	+	0.429219i	$0.141211\pi$
$211$			181.131i			0.858439i	−0.903200	+	0.429219i	$0.858789\pi$
$212$	26.5305			0.125144
$213$		0			0
$214$		−	213.961i		−	0.999816i
$215$	39.8339	+	122.225i	0.185274	+	0.568490i
$216$		0			0
$217$		−	100.618i		−	0.463679i
$218$		−	144.836i		−	0.664384i
$219$		0			0
$220$			17.4659i			0.0793905i
$221$	196.205			0.887807
$222$		0			0
$223$			107.757i			0.483213i	0.970374	+	0.241607i	$0.0776744\pi$
$223$			107.757i			0.483213i	−0.970374	+	0.241607i	$0.922326\pi$
$224$	91.0288			0.406378
$225$		0			0
$226$	−104.290			−0.461461
$227$			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	$0.264610\pi$
$230$	176.329			0.766649
$231$		0			0
$232$	394.562			1.70070
$233$			301.133i			1.29242i	0.763161	+	0.646209i	$0.223646\pi$
$233$			301.133i			1.29242i	−0.763161	+	0.646209i	$0.776354\pi$
$234$		0			0
$235$		−	30.6719i		−	0.130519i
$236$	−54.6346			−0.231503
$237$		0			0
$238$	191.892			0.806268
$239$	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	$0.553957\pi$
$242$		−	130.593i		−	0.539642i
$243$		0			0
$244$			66.3240i			0.271820i
$245$		−	6.95367i		−	0.0283823i
$246$		0			0
$247$		−	238.246i		−	0.964559i
$248$			126.765i			0.511149i
$249$		0			0
$250$	217.939			0.871754
$251$	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
$254$		−	31.9253i		−	0.125690i
$255$		0			0
$256$	−155.017			−0.605534
$257$			62.7857i			0.244302i	0.992512	+	0.122151i	$0.0389793\pi$
$257$			62.7857i			0.244302i	−0.992512	+	0.122151i	$0.961021\pi$
$258$		0			0
$259$	89.9934			0.347465
$260$		−	31.4478i		−	0.120953i
$261$		0			0
$262$	−183.505			−0.700401
$263$			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
$266$		−	233.008i		−	0.875970i
$267$		0			0
$268$	−75.8015			−0.282842
$269$	407.934			1.51648			0.758242	−	0.651974i	$-0.226059\pi$
$269$	407.934			1.51648			0.758242	+	0.651974i	$0.226059\pi$
$270$		0			0
$271$	−316.274			−1.16706			−0.583532	−	0.812090i	$-0.698330\pi$
$271$	−316.274			−1.16706			−0.583532	+	0.812090i	$0.698330\pi$
$272$	−188.078			−0.691461
$273$		0			0
$274$	249.338			0.909993
$275$	−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	$-0.123385\pi$
$278$			441.310i			1.58745i
$279$		0			0
$280$		−	175.796i		−	0.627845i
$281$	−36.4182			−0.129602			−0.0648011	−	0.997898i	$-0.520641\pi$
$281$	−36.4182			−0.129602			−0.0648011	+	0.997898i	$0.520641\pi$
$282$		0			0
$283$	−15.3124			−0.0541074			−0.0270537	−	0.999634i	$-0.508613\pi$
$283$	−15.3124			−0.0541074			−0.0270537	+	0.999634i	$0.508613\pi$
$284$			30.1172i			0.106047i
$285$		0			0
$286$		−	151.642i		−	0.530218i
$287$		−	17.0171i		−	0.0592930i
$288$		0			0
$289$	−38.6884			−0.133870
$290$		−	243.302i		−	0.838972i
$291$		0			0
$292$		−	30.5021i		−	0.104459i
$293$	421.851			1.43976			0.719882	−	0.694097i	$-0.244196\pi$
$293$	421.851			1.43976			0.719882	+	0.694097i	$0.244196\pi$
$294$		0			0
$295$			192.563i			0.652757i
$296$	−113.379			−0.383038
$297$		0			0
$298$	−168.761			−0.566311
$299$	412.007			1.37795
$300$		0			0
$301$	91.0288	+	279.310i	0.302421	+	0.927940i
$302$	218.233			0.722625
$303$		0			0
$304$			228.377i			0.751239i
$305$	233.763			0.766436
$306$		0			0
$307$	−239.260			−0.779349			−0.389674	−	0.920953i	$-0.627412\pi$
$307$	−239.260			−0.779349			−0.389674	+	0.920953i	$0.627412\pi$
$308$			39.9132i			0.129588i
$309$		0			0
$310$	78.1681			0.252155
$311$	−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	$0.817148\pi$
$314$	−336.126			−1.07047
$315$		0			0
$316$	42.5044			0.134508
$317$	470.259			1.48347			0.741733	−	0.670695i	$-0.234004\pi$
$317$	470.259			1.48347			0.741733	+	0.670695i	$0.234004\pi$
$318$		0			0
$319$			315.738i			0.989774i
$320$			212.875i			0.665235i
$321$		0			0
$322$	402.949			1.25139
$323$		−	303.945i		−	0.941008i
$324$		0			0
$325$	199.195			0.612908
$326$	469.467			1.44008
$327$		0			0
$328$			21.4392i			0.0653633i
$329$		−	70.0915i		−	0.213044i
$330$		0			0
$331$		−	297.506i		−	0.898809i	−0.893328	−	0.449405i	$-0.851636\pi$
$331$		−	297.506i		−	0.898809i	0.893328	−	0.449405i	$-0.148364\pi$
$332$	8.79134			0.0264799
$333$		0			0
$334$		−	71.0662i		−	0.212773i
$335$			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	$0.242181\pi$
$338$		−	26.9946i		−	0.0798657i
$339$		0			0
$340$		−	40.1199i		−	0.118000i
$341$	−101.440			−0.297479
$342$		0			0
$343$		−	350.651i		−	1.02231i
$344$	−114.684	−	351.892i	−0.333382	−	1.02294i
$345$		0			0
$346$		−	322.217i		−	0.931263i
$347$		−	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	$0.955572\pi$
$350$	194.816			0.556617
$351$		0			0
$352$		−	91.7725i		−	0.260717i
$353$	−355.666			−1.00755			−0.503776	−	0.863834i	$-0.668056\pi$
$353$	−355.666			−1.00755			−0.503776	+	0.863834i	$0.668056\pi$
$354$		0			0
$355$	106.150			0.299014
$356$		−	11.4425i		−	0.0321417i
$357$		0			0
$358$	−111.130			−0.310420
$359$	−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
$362$			29.6796i			0.0819879i
$363$		0			0
$364$		−	71.8646i		−	0.197430i
$365$	−107.507			−0.294539
$366$		0			0
$367$	−379.616			−1.03438			−0.517188	−	0.855872i	$-0.673021\pi$
$367$	−379.616			−1.03438			−0.517188	+	0.855872i	$0.673021\pi$
$368$	−394.939			−1.07320
$369$		0			0
$370$			69.9139i			0.188956i
$371$		−	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	$-0.0821094\pi$
$374$		−	193.460i		−	0.517272i
$375$		0			0
$376$			88.3055i			0.234855i
$377$		−	568.494i		−	1.50794i
$378$		0			0
$379$	−542.592			−1.43164			−0.715820	−	0.698285i	$-0.753947\pi$
$379$	−542.592			−1.43164			−0.715820	+	0.698285i	$0.753947\pi$
$380$	−48.7163			−0.128201
$381$		0			0
$382$	−79.4161			−0.207896
$383$		−	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
$386$			240.024i			0.621823i
$387$		0			0
$388$	56.5186			0.145667
$389$			689.167i			1.77164i	0.464031	+	0.885819i	$0.346403\pi$
$389$			689.167i			1.77164i	−0.464031	+	0.885819i	$0.653597\pi$
$390$		0			0
$391$	525.623			1.34430
$392$			20.0199i			0.0510712i
$393$		0			0
$394$		−	258.845i		−	0.656968i
$395$		−	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	$0.678402\pi$
$398$	−498.258			−1.25191
$399$		0			0
$400$	−190.943			−0.477359
$401$	393.635			0.981632			0.490816	−	0.871263i	$-0.336699\pi$
$401$	393.635			0.981632			0.490816	+	0.871263i	$0.336699\pi$
$402$		0			0
$403$	182.646			0.453215
$404$	−62.2124			−0.153991
$405$		0			0
$406$		−	555.995i		−	1.36945i
$407$		−	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	$-0.127869\pi$
$410$	13.2202			0.0322444
$411$		0			0
$412$	−29.4675			−0.0715229
$413$			440.047i			1.06549i
$414$		0			0
$415$		−	30.9856i		−	0.0746642i
$416$			165.239i			0.397208i
$417$		0			0
$418$	−234.912			−0.561990
$419$			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	$0.870616\pi$
$422$	−321.566			−0.762005
$423$		0			0
$424$			269.215i			0.634941i
$425$	254.126			0.597943
$426$		0			0
$427$	534.197			1.25105
$428$	−102.226			−0.238847
$429$		0			0
$430$	−216.990	+	70.7182i	−0.504627	+	0.164461i
$431$	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	$-0.0704973\pi$
$434$	178.630			0.411590
$435$		0			0
$436$	−69.1997			−0.158715
$437$		−	638.248i		−	1.46052i
$438$		0			0
$439$	−343.705			−0.782927			−0.391463	−	0.920194i	$-0.628031\pi$
$439$	−343.705			−0.782927			−0.391463	+	0.920194i	$0.628031\pi$
$440$	−177.233			−0.402802
$441$		0			0
$442$			348.329i			0.788074i
$443$	−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
$446$	−191.303			−0.428931
$447$		0			0
$448$			486.464i			1.08586i
$449$			598.611i			1.33321i	0.745411	+	0.666605i	$0.232253\pi$
$449$			598.611i			1.33321i	−0.745411	+	0.666605i	$0.767747\pi$
$450$		0			0
$451$	−17.1561			−0.0380402
$452$			49.8279i			0.110239i
$453$		0			0
$454$	−248.087			−0.546448
$455$	−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	$-0.167656\pi$
$458$			547.963i			1.19643i
$459$		0			0
$460$		−	84.2468i		−	0.183145i
$461$	−334.655			−0.725933			−0.362967	−	0.931802i	$-0.618236\pi$
$461$	−334.655			−0.725933			−0.362967	+	0.931802i	$0.618236\pi$
$462$		0			0
$463$			696.646i			1.50464i	0.658801	+	0.752318i	$0.271064\pi$
$463$			696.646i			1.50464i	−0.658801	+	0.752318i	$0.728936\pi$
$464$			544.944i			1.17445i
$465$		0			0
$466$	−534.610			−1.14723
$467$			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
$470$	54.4525			0.115856
$471$		0			0
$472$		−	554.398i		−	1.17457i
$473$	281.592	−	91.7725i	0.595332	−	0.194022i
$474$		0			0
$475$		−	308.577i		−	0.649636i
$476$		−	91.6822i		−	0.192610i
$477$		0			0
$478$			216.660i			0.453263i
$479$	−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
$482$	−843.443			−1.74988
$483$		0			0
$484$	−62.3950			−0.128915
$485$		−	199.203i		−	0.410729i
$486$		0			0
$487$	151.294			0.310666			0.155333	−	0.987862i	$-0.450355\pi$
$487$	151.294			0.310666			0.155333	+	0.987862i	$0.450355\pi$
$488$	−673.014			−1.37913
$489$		0			0
$490$	12.3450			0.0251939
$491$			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
$494$	422.964			0.856203
$495$		0			0
$496$	−175.080			−0.352983
$497$	242.575			0.488078
$498$		0			0
$499$			337.720i			0.676793i	0.941004	+	0.338396i	$0.109884\pi$
$499$			337.720i			0.676793i	−0.941004	+	0.338396i	$0.890116\pi$
$500$		−	104.127i		−	0.208254i
$501$		0			0
$502$			63.8994i			0.127290i
$503$		−	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
$506$		−	406.241i		−	0.802848i
$507$		0			0
$508$	−15.2533			−0.0300262
$509$	−229.831			−0.451535			−0.225767	−	0.974181i	$-0.572489\pi$
$509$	−229.831			−0.451535			−0.225767	+	0.974181i	$0.572489\pi$
$510$		0			0
$511$	−245.675			−0.480773
$512$		−	567.670i		−	1.10873i
$513$		0			0
$514$	−111.465			−0.216858
$515$			103.860i			0.201670i
$516$		0			0
$517$	−70.6642			−0.136681
$518$			159.768i			0.308432i
$519$		0			0
$520$	319.112			0.613677
$521$		−	335.813i		−	0.644555i	−0.946645	−	0.322278i	$-0.895552\pi$
$521$		−	335.813i		−	0.644555i	0.946645	−	0.322278i	$-0.104448\pi$
$522$		0			0
$523$		−	630.020i		−	1.20463i	−0.798259	−	0.602314i	$-0.794246\pi$
$523$		−	630.020i		−	1.20463i	0.798259	−	0.602314i	$-0.205754\pi$
$524$			87.6752i			0.167319i
$525$		0			0
$526$	−30.5073			−0.0579987
$527$	233.013			0.442149
$528$		0			0
$529$	574.743			1.08647
$530$	166.008			0.313223
$531$		0			0
$532$	−111.327			−0.209261
$533$	30.8900			0.0579550
$534$		0			0
$535$			360.303i			0.673464i
$536$		−	769.186i		−	1.43505i
$537$		0			0
$538$			724.216i			1.34613i
$539$	−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	$0.319938\pi$
$542$		−	561.490i		−	1.03596i
$543$		0			0
$544$			210.805i			0.387510i
$545$			243.899i			0.447521i
$546$		0			0
$547$	276.106			0.504764			0.252382	−	0.967628i	$-0.418786\pi$
$547$	276.106			0.504764			0.252382	+	0.967628i	$0.418786\pi$
$548$		−	119.129i		−	0.217389i
$549$		0			0
$550$		−	196.408i		−	0.357105i
$551$	−880.664			−1.59830
$552$		0			0
$553$		−	342.346i		−	0.619070i
$554$	371.767			0.671059
$555$		0			0
$556$	210.850			0.379226
$557$	1.44850			0.00260054			0.00130027	−	0.999999i	$-0.499586\pi$
$557$	1.44850			0.00260054			0.00130027	+	0.999999i	$0.499586\pi$
$558$		0			0
$559$	−507.013	+	165.239i	−0.907000	+	0.295597i
$560$	242.799			0.433569
$561$		0			0
$562$		−	64.6542i		−	0.115043i
$563$	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
$566$		−	27.1845i		−	0.0480292i
$567$		0			0
$568$	−305.611			−0.538047
$569$	657.323			1.15522			0.577612	−	0.816311i	$-0.303985\pi$
$569$	657.323			1.15522			0.577612	+	0.816311i	$0.303985\pi$
$570$		0			0
$571$			321.145i			0.562426i	0.959645	+	0.281213i	$0.0907368\pi$
$571$			321.145i			0.562426i	−0.959645	+	0.281213i	$0.909263\pi$
$572$	−72.4518			−0.126664
$573$		0			0
$574$	30.2109			0.0526322
$575$	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	$-0.124675\pi$
$578$		−	68.6845i		−	0.118831i
$579$		0			0
$580$	−116.245			−0.200423
$581$		−	70.8085i		−	0.121874i
$582$		0			0
$583$	−215.432			−0.369523
$584$	309.516			0.529994
$585$		0			0
$586$			748.923i			1.27803i
$587$			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
$590$	−341.863			−0.579428
$591$		0			0
$592$		−	156.592i		−	0.264513i
$593$		−	14.4319i		−	0.0243370i	−0.999926	−	0.0121685i	$-0.996127\pi$
$593$		−	14.4319i		−	0.0243370i	0.999926	−	0.0121685i	$-0.00387345\pi$
$594$		0			0
$595$	−323.140			−0.543092
$596$			80.6306i			0.135286i
$597$		0			0
$598$			731.447i			1.22316i
$599$	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	$-0.378324\pi$
$602$	−495.866	+	161.606i	−0.823698	+	0.268448i
$603$		0			0
$604$		−	104.268i		−	0.172628i
$605$			219.915i			0.363496i
$606$		0			0
$607$			297.682i			0.490416i	0.969471	+	0.245208i	$0.0788562\pi$
$607$			297.682i			0.490416i	−0.969471	+	0.245208i	$0.921144\pi$
$608$	255.974			0.421010
$609$		0			0
$610$			415.006i			0.680337i
$611$	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	$0.398883\pi$
$614$		−	424.765i		−	0.691799i
$615$		0			0
$616$	−405.014			−0.657490
$617$	702.287			1.13823			0.569114	−	0.822258i	$-0.307286\pi$
$617$	702.287			1.13823			0.569114	+	0.822258i	$0.307286\pi$
$618$		0			0
$619$	−680.743			−1.09975			−0.549873	−	0.835248i	$-0.685324\pi$
$619$	−680.743			−1.09975			−0.549873	+	0.835248i	$0.685324\pi$
$620$		−	37.3472i		−	0.0602375i
$621$		0			0
$622$		−	1003.77i		−	1.61377i
$623$	−92.1616			−0.147932
$624$		0			0
$625$	34.5564			0.0552903
$626$	−603.876			−0.964659
$627$		0			0
$628$			160.595i			0.255724i
$629$			208.408i			0.331332i
$630$		0			0
$631$			1019.35i			1.61546i	0.589555	+	0.807728i	$0.299303\pi$
$631$			1019.35i			1.61546i	−0.589555	+	0.807728i	$0.700697\pi$
$632$			431.308i			0.682449i
$633$		0			0
$634$			834.863i			1.31682i
$635$			53.7613i			0.0846634i
$636$		0			0
$637$	28.8451			0.0452827
$638$	−560.538			−0.878586
$639$		0			0
$640$	−218.587			−0.341543
$641$			815.451i			1.27215i	0.771625	+	0.636077i	$0.219444\pi$
$641$			815.451i			1.27215i	−0.771625	+	0.636077i	$0.780556\pi$
$642$		0			0
$643$	661.567			1.02888			0.514438	−	0.857528i	$-0.328001\pi$
$643$	661.567			1.02888			0.514438	+	0.857528i	$0.328001\pi$
$644$		−	192.521i		−	0.298946i
$645$		0			0
$646$	539.602			0.835298
$647$		−	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
$650$			353.636i			0.544056i
$651$		0			0
$652$		−	224.302i		−	0.344022i
$653$		−	812.108i		−	1.24366i	−0.783153	−	0.621829i	$-0.786390\pi$
$653$		−	812.108i		−	1.24366i	0.783153	−	0.621829i	$-0.213610\pi$
$654$		0			0
$655$	309.017			0.471782
$656$	−29.6104			−0.0451378
$657$		0			0
$658$	124.435			0.189111
$659$	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	$0.370445\pi$
$662$	528.170			0.797840
$663$		0			0
$664$			89.2089i			0.134351i
$665$			392.378i			0.590043i
$666$		0			0
$667$		−	1522.96i		−	2.28330i
$668$	−33.9541			−0.0508295
$669$		0			0
$670$	−474.309			−0.707924
$671$		−	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	$-0.0764803\pi$
$674$			866.629i			1.28580i
$675$		0			0
$676$	−12.8975			−0.0190791
$677$			854.967i			1.26288i	0.775426	+	0.631438i	$0.217535\pi$
$677$			854.967i			1.26288i	−0.775426	+	0.631438i	$0.782465\pi$
$678$		0			0
$679$		−	455.221i		−	0.670428i
$680$	407.111			0.598693
$681$		0			0
$682$		−	180.090i		−	0.264061i
$683$	−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
$686$	622.519			0.907463
$687$		0			0
$688$	486.010	−	158.393i	0.706410	−	0.230223i
$689$	387.891			0.562976
$690$		0			0
$691$		−	568.600i		−	0.822866i	−0.911440	−	0.411433i	$-0.865028\pi$
$691$		−	568.600i		−	0.822866i	0.911440	−	0.411433i	$-0.134972\pi$
$692$	−153.949			−0.222470
$693$		0			0
$694$	1101.12			1.58663
$695$		−	743.152i		−	1.06928i
$696$		0			0
$697$	39.4084			0.0565400
$698$	−172.396			−0.246985
$699$		0			0
$700$		−	93.0793i		−	0.132970i
$701$	−369.079			−0.526504			−0.263252	−	0.964727i	$-0.584795\pi$
$701$	−369.079			−0.526504			−0.263252	+	0.964727i	$0.584795\pi$
$702$		0			0
$703$	253.063			0.359975
$704$	490.439			0.696646
$705$		0			0
$706$		−	631.423i		−	0.894367i
$707$			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	$0.653217\pi$
$710$			188.451i			0.265424i
$711$		0			0
$712$	116.111			0.163077
$713$	489.298			0.686252
$714$		0			0
$715$			255.361i			0.357148i
$716$			53.0960i			0.0741564i
$717$		0			0
$718$		−	328.649i		−	0.457728i
$719$	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
$722$		−	14.3293i		−	0.0198466i
$723$		0			0
$724$	14.1804			0.0195861
$725$		−	736.315i		−	1.01561i
$726$		0			0
$727$			1020.14i			1.40322i	0.712559	+	0.701612i	$0.247536\pi$
$727$			1020.14i			1.40322i	−0.712559	+	0.701612i	$0.752464\pi$
$728$	729.237			1.00170
$729$		0			0
$730$		−	190.859i		−	0.261451i
$731$	−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	$-0.0679987\pi$
$734$		−	673.942i		−	0.918177i
$735$		0			0
$736$			442.665i			0.601447i
$737$	615.521			0.835171
$738$		0			0
$739$		−	1188.34i		−	1.60804i	−0.594603	−	0.804020i	$-0.702691\pi$
$739$		−	1188.34i		−	1.60804i	0.594603	−	0.804020i	$-0.297309\pi$
$740$	33.4035			0.0451399
$741$		0			0
$742$	379.363			0.511271
$743$		−	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
$746$	337.857			0.452891
$747$		0			0
$748$	−92.4313			−0.123571
$749$			823.367i			1.09929i
$750$		0			0
$751$		−	816.888i		−	1.08773i	−0.839171	−	0.543867i	$-0.816960\pi$
$751$		−	816.888i		−	1.08773i	0.839171	−	0.543867i	$-0.183040\pi$
$752$	−121.962			−0.162183
$753$		0			0
$754$	1009.26			1.33854
$755$	−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	$0.820868\pi$
$758$		−	963.277i		−	1.27081i
$759$		0			0
$760$		−	494.342i		−	0.650450i
$761$			589.286i			0.774358i	0.922005	+	0.387179i	$0.126550\pi$
$761$			589.286i			0.774358i	−0.922005	+	0.387179i	$0.873450\pi$
$762$		0			0
$763$			557.359i			0.730483i
$764$			37.9435i			0.0496643i
$765$		0			0
$766$	225.294			0.294118
$767$	−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	$0.189699\pi$
$770$			249.747i			0.324346i
$771$		0			0
$772$	114.679			0.148547
$773$			176.192i			0.227933i	0.993485	+	0.113967i	$0.0363557\pi$
$773$			176.192i			0.227933i	−0.993485	+	0.113967i	$0.963644\pi$
$774$		0			0
$775$	236.563			0.305243
$776$			573.515i			0.739065i
$777$		0			0
$778$	−1223.50			−1.57262
$779$		−	47.8523i		−	0.0614279i
$780$		0			0
$781$		−	244.557i		−	0.313133i
$782$			933.153i			1.19329i
$783$		0			0
$784$	−27.6502			−0.0352681
$785$	566.026			0.721053
$786$		0			0
$787$	−1120.38			−1.42361			−0.711804	−	0.702379i	$-0.752121\pi$
$787$	−1120.38			−1.42361			−0.711804	+	0.702379i	$0.752121\pi$
$788$	−123.671			−0.156943
$789$		0			0
$790$	265.961			0.336659
$791$	401.332			0.507373
$792$		0			0
$793$			969.693i			1.22282i
$794$		−	749.323i		−	0.943732i
$795$		0			0
$796$			238.058i			0.299068i
$797$	872.371			1.09457			0.547284	−	0.836947i	$-0.315662\pi$
$797$	872.371			1.09457			0.547284	+	0.836947i	$0.315662\pi$
$798$		0			0
$799$	162.319			0.203152
$800$			214.018i			0.267522i
$801$		0			0
$802$			698.830i			0.871359i
$803$			247.682i			0.308446i
$804$		0			0
$805$	−678.553			−0.842924
$806$			324.256i			0.402303i
$807$		0			0
$808$		−	631.292i		−	0.781302i
$809$	−273.069			−0.337539			−0.168770	−	0.985656i	$-0.553979\pi$
$809$	−273.069			−0.337539			−0.168770	+	0.985656i	$0.553979\pi$
$810$		0			0
$811$			1420.64i			1.75171i	0.482576	+	0.875854i	$0.339701\pi$
$811$			1420.64i			1.75171i	−0.482576	+	0.875854i	$0.660299\pi$
$812$	−265.644			−0.327148
$813$		0			0
$814$	161.073			0.197878
$815$	−790.568			−0.970022
$816$		0			0
$817$	255.974	+	785.424i	0.313310	+	0.961351i
$818$	567.808			0.694142
$819$		0			0
$820$		−	6.31636i		−	0.00770288i
$821$	−1163.98			−1.41775			−0.708877	−	0.705332i	$-0.750798\pi$
$821$	−1163.98			−1.41775			−0.708877	+	0.705332i	$0.750798\pi$
$822$		0			0
$823$	47.3698			0.0575574			0.0287787	−	0.999586i	$-0.490838\pi$
$823$	47.3698			0.0575574			0.0287787	+	0.999586i	$0.490838\pi$
$824$		−	299.017i		−	0.362885i
$825$		0			0
$826$	−781.227			−0.945795
$827$	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	$0.805090\pi$
$830$	55.0096			0.0662766
$831$		0			0
$832$	−883.047			−1.06135
$833$	36.7995			0.0441771
$834$		0			0
$835$			119.673i			0.143321i
$836$			112.236i			0.134254i
$837$		0			0
$838$	−593.692			−0.708463
$839$			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
$842$	−591.009			−0.701911
$843$		0			0
$844$			153.638i			0.182036i
$845$			45.4581i			0.0537965i
$846$		0			0
$847$			502.551i			0.593331i
$848$	−371.822			−0.438469
$849$		0			0
$850$			451.156i			0.530772i
$851$			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	$0.821701\pi$
$854$			948.374i			1.11051i
$855$		0			0
$856$		−	1037.33i		−	1.21183i
$857$	92.3800			0.107795			0.0538973	−	0.998546i	$-0.482836\pi$
$857$	92.3800			0.107795			0.0538973	+	0.998546i	$0.482836\pi$
$858$		0			0
$859$			786.473i			0.915568i	0.889064	+	0.457784i	$0.151357\pi$
$859$			786.473i			0.915568i	−0.889064	+	0.457784i	$0.848643\pi$
$860$	33.7878	+	103.674i	0.0392882	+	0.120551i
$861$		0			0
$862$			290.099i			0.336542i
$863$		−	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
$866$	337.724			0.389982
$867$		0			0
$868$		−	85.3461i		−	0.0983250i
$869$	−345.143			−0.397172
$870$		0			0
$871$	−1108.26			−1.27240
$872$		−	702.194i		−	0.805269i
$873$		0			0
$874$	1133.10			1.29645
$875$	−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	$0.288592\pi$
$878$		−	610.188i		−	0.694975i
$879$		0			0
$880$		−	244.782i		−	0.278162i
$881$	575.591			0.653338			0.326669	−	0.945139i	$-0.394074\pi$
$881$	575.591			0.653338			0.326669	+	0.945139i	$0.394074\pi$
$882$		0			0
$883$	1401.90			1.58766			0.793828	−	0.608143i	$-0.208085\pi$
$883$	1401.90			1.58766			0.793828	+	0.608143i	$0.208085\pi$
$884$	166.425			0.188263
$885$		0			0
$886$		−	915.234i		−	1.03300i
$887$		−	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
$890$		−	71.5983i		−	0.0804475i
$891$		0			0
$892$			91.4010i			0.102467i
$893$		−	197.098i		−	0.220715i
$894$		0			0
$895$	187.140			0.209095
$896$	−499.517			−0.557497
$897$		0			0
$898$	−1062.73			−1.18344
$899$		−	675.141i		−	0.750991i
$900$		0			0
$901$	494.857			0.549230
$902$		−	30.4577i		−	0.0337669i
$903$		0			0
$904$	−505.622			−0.559316
$905$		−	49.9796i		−	0.0552260i
$906$		0			0
$907$	−252.284			−0.278152			−0.139076	−	0.990282i	$-0.544413\pi$
$907$	−252.284			−0.278152			−0.139076	+	0.990282i	$0.544413\pi$
$908$			118.531i			0.130541i
$909$		0			0
$910$		−	449.675i		−	0.494148i
$911$		−	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
$914$	815.689			0.892439
$915$		0			0
$916$	261.806			0.285815
$917$	706.167			0.770084
$918$		0			0
$919$	−270.734			−0.294597			−0.147298	−	0.989092i	$-0.547058\pi$
$919$	−270.734			−0.294597			−0.147298	+	0.989092i	$0.547058\pi$
$920$	854.883			0.929220
$921$		0			0
$922$		−	594.122i		−	0.644384i
$923$			440.330i			0.477064i
$924$		0			0
$925$			211.583i			0.228739i
$926$	−1236.77			−1.33561
$927$		0			0
$928$	610.796			0.658185
$929$			791.662i			0.852165i	0.904684	+	0.426083i	$0.140107\pi$
$929$			791.662i			0.852165i	−0.904684	+	0.426083i	$0.859893\pi$
$930$		0			0
$931$		−	44.6845i		−	0.0479962i
$932$			255.426i			0.274063i
$933$		0			0
$934$	−218.884			−0.234351
$935$			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	$0.815710\pi$
$938$	−1083.89			−1.15554
$939$		0			0
$940$		−	26.0164i		−	0.0276770i
$941$	−1515.52			−1.61054			−0.805271	−	0.592907i	$-0.797980\pi$
$941$	−1515.52			−1.61054			−0.805271	+	0.592907i	$0.797980\pi$
$942$		0			0
$943$	82.7527			0.0877547
$944$	765.698			0.811121
$945$		0			0
$946$	162.926	+	499.918i	0.172226	+	0.528454i
$947$	−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
$950$	547.825			0.576658
$951$		0			0
$952$	930.333			0.977241
$953$		−	1307.06i		−	1.37152i	−0.727828	−	0.685760i	$-0.759470\pi$
$953$		−	1307.06i		−	1.37152i	0.727828	−	0.685760i	$-0.240530\pi$
$954$		0			0
$955$	133.734			0.140036
$956$	103.516			0.108280
$957$		0			0
$958$		−	358.506i		−	0.374224i
$959$	−959.507			−1.00053
$960$		0			0
$961$	−744.091			−0.774288
$962$	−290.016			−0.301472
$963$		0			0
$964$			402.981i			0.418030i
$965$		−	404.192i		−	0.418852i
$966$		0			0
$967$	−1273.66			−1.31713			−0.658563	−	0.752525i	$-0.728836\pi$
$967$	−1273.66			−1.31713			−0.658563	+	0.752525i	$0.728836\pi$
$968$		−	633.144i		−	0.654075i
$969$		0			0
$970$	353.651			0.364589
$971$	−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
$974$			268.597i			0.275767i
$975$		0			0
$976$		−	929.523i		−	0.952380i
$977$	−1275.34			−1.30537			−0.652684	−	0.757630i	$-0.726357\pi$
$977$	−1275.34			−1.30537			−0.652684	+	0.757630i	$0.726357\pi$
$978$		0			0
$979$			92.9146i			0.0949076i
$980$		−	5.89822i		−	0.00601859i
$981$		0			0
$982$	−100.195			−0.102032
$983$		−	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
$986$	1287.58			1.30586
$987$		0			0
$988$		−	202.084i		−	0.204539i
$989$	−1358.26	+	442.665i	−1.37337	+	0.447588i
$990$		0			0
$991$		−	778.474i		−	0.785544i	−0.919636	−	0.392772i	$-0.871516\pi$
$991$		−	778.474i		−	0.785544i	0.919636	−	0.392772i	$-0.128484\pi$
$992$			196.237i			0.197819i
$993$		0			0
$994$			430.650i			0.433249i
$995$	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	$-0.0736359\pi$
$998$	−599.562			−0.600764
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.b.b.42.3	✓	6	3.2	odd	2
43.3.b.b.42.4	yes	6	129.128	even	2
387.3.b.c.343.3		6	43.42	odd	2	inner
387.3.b.c.343.4		6	1.1	even	1	trivial
688.3.b.e.257.1		6	12.11	even	2
688.3.b.e.257.6		6	516.515	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	387.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.77533i q^{2} +0.848217 q^{4} -2.98959i q^{5} -6.83184i q^{7} +8.60717i q^{8} +O(q^{10})\)	\(q+1.77533i q^{2} +0.848217 q^{4} -2.98959i q^{5} -6.83184i q^{7} +8.60717i q^{8} +5.30751 q^{10} -6.88766 q^{11} +12.4014 q^{13} +12.1287 q^{14} -11.8877 q^{16} +15.8212 q^{17} -19.2112i q^{19} -2.53583i q^{20} -12.2278i q^{22} +33.2226 q^{23} +16.0623 q^{25} +22.0165i q^{26} -5.79488i q^{28} -45.8411i q^{29} +14.7278 q^{31} +13.3242i q^{32} +28.0879i q^{34} -20.4244 q^{35} +13.1726i q^{37} +34.1062 q^{38} +25.7319 q^{40} +2.49085 q^{41} +(-40.8836 + 13.3242i) q^{43} -5.84223 q^{44} +58.9810i q^{46} +10.2595 q^{47} +2.32596 q^{49} +28.5159i q^{50} +10.5191 q^{52} +31.2780 q^{53} +20.5913i q^{55} +58.8028 q^{56} +81.3829 q^{58} -64.4112 q^{59} +78.1922i q^{61} +26.1467i q^{62} -71.2054 q^{64} -37.0751i q^{65} -89.3657 q^{67} +13.4198 q^{68} -36.2600i q^{70} +35.5065i q^{71} -35.9603i q^{73} -23.3857 q^{74} -16.2953i q^{76} +47.0554i q^{77} +50.1103 q^{79} +35.5393i q^{80} +4.42207i q^{82} +10.3645 q^{83} -47.2991i q^{85} +(-23.6548 - 72.5817i) q^{86} -59.2832i q^{88} -13.4900i q^{89} -84.7243i q^{91} +28.1800 q^{92} +18.2140i q^{94} -57.4338 q^{95} +66.6322 q^{97} +4.12933i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{4}+O(q^{10}) \) 6 * q - 16 * q^4	\( 6 q - 16 q^{4} - 2 q^{10} - 38 q^{11} + 30 q^{13} - 36 q^{14} - 68 q^{16} + 20 q^{17} + 80 q^{23} - 84 q^{25} - 112 q^{31} - 208 q^{35} - 170 q^{38} + 206 q^{40} + 172 q^{41} + 10 q^{43} + 36 q^{44} - 30 q^{47} - 6 q^{49} - 120 q^{52} + 110 q^{53} + 264 q^{56} + 430 q^{58} + 12 q^{59} + 100 q^{64} - 70 q^{67} + 50 q^{68} + 50 q^{74} + 178 q^{79} - 10 q^{83} + 372 q^{86} - 150 q^{92} + 130 q^{95} - 380 q^{97}+O(q^{100}) \) 6 * q - 16 * q^4 - 2 * q^10 - 38 * q^11 + 30 * q^13 - 36 * q^14 - 68 * q^16 + 20 * q^17 + 80 * q^23 - 84 * q^25 - 112 * q^31 - 208 * q^35 - 170 * q^38 + 206 * q^40 + 172 * q^41 + 10 * q^43 + 36 * q^44 - 30 * q^47 - 6 * q^49 - 120 * q^52 + 110 * q^53 + 264 * q^56 + 430 * q^58 + 12 * q^59 + 100 * q^64 - 70 * q^67 + 50 * q^68 + 50 * q^74 + 178 * q^79 - 10 * q^83 + 372 * q^86 - 150 * q^92 + 130 * q^95 - 380 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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