LMFDB - Embedded newform 927.2.c.a.926.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [927,2,Mod(926,927)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(927, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("927.926");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 927 = 3^{2} \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	927.c (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:	$7.40213226737$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 34x^{14} + 564x^{12} - 5204x^{10} + 29740x^{8} - 99088x^{6} + 214644x^{4} - 222312x^{2} + 84100 $ x^16 - 34x^14 + 564x^12 - 5204x^10 + 29740x^8 - 99088x^6 + 214644x^4 - 222312*x^2 + 84100
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			926.2
Root			$1.54287 + 0.912316i$ of defining polynomial
Character	$\chi$	$=$	927.926
Dual form			927.2.c.a.926.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.10850i q^{2} -2.44579 q^{4} +3.56582 q^{5} +0.463913 q^{7} +0.939944i q^{8} +O(q^{10})$	\(q-2.10850i q^{2} -2.44579 q^{4} +3.56582 q^{5} +0.463913 q^{7} +0.939944i q^{8} -7.51854i q^{10} +3.08574 q^{11} -0.329281 q^{13} -0.978162i q^{14} -2.90970 q^{16} +3.45887i q^{17} +5.56229 q^{19} -8.72123 q^{20} -6.50630i q^{22} +2.57418i q^{23} +7.71505 q^{25} +0.694290i q^{26} -1.13463 q^{28} -0.157325i q^{29} -3.35167i q^{31} +8.01500i q^{32} +7.29303 q^{34} +1.65423 q^{35} -5.82738i q^{37} -11.7281i q^{38} +3.35167i q^{40} -2.20543i q^{41} -4.81514i q^{43} -7.54707 q^{44} +5.42766 q^{46} -3.08574 q^{47} -6.78478 q^{49} -16.2672i q^{50} +0.805351 q^{52} -7.54707 q^{53} +11.0032 q^{55} +0.436052i q^{56} -0.331721 q^{58} +4.85205i q^{59} -0.391738 q^{61} -7.06700 q^{62} +11.0803 q^{64} -1.17416 q^{65} +8.98201i q^{67} -8.45965i q^{68} -3.48795i q^{70} -2.39166 q^{71} -4.84580i q^{73} -12.2870 q^{74} -13.6042 q^{76} +1.43152 q^{77} -13.7207 q^{79} -10.3755 q^{80} -4.65015 q^{82} +9.77694i q^{83} +12.3337i q^{85} -10.1527 q^{86} +2.90043i q^{88} +15.7882 q^{89} -0.152758 q^{91} -6.29589i q^{92} +6.50630i q^{94} +19.8341 q^{95} -16.2084 q^{97} +14.3057i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{7}+O(q^{10}) $ 16 * q + 16 * q^7	$ 16 q + 16 q^{7} - 8 q^{13} - 16 q^{16} + 8 q^{19} + 96 q^{25} - 24 q^{28} + 24 q^{34} - 32 q^{49} + 16 q^{52} - 8 q^{55} - 40 q^{58} - 32 q^{61} + 40 q^{64} - 80 q^{76} - 40 q^{79} + 8 q^{82} - 56 q^{91} + 16 q^{97}+O(q^{100}) $ 16 * q + 16 * q^7 - 8 * q^13 - 16 * q^16 + 8 * q^19 + 96 * q^25 - 24 * q^28 + 24 * q^34 - 32 * q^49 + 16 * q^52 - 8 * q^55 - 40 * q^58 - 32 * q^61 + 40 * q^64 - 80 * q^76 - 40 * q^79 + 8 * q^82 - 56 * q^91 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/927\mathbb{Z}\right)^\times$.

$n$	$722$	$829$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	2.10850i		−	1.49094i	−0.666541	−	0.745469i	$-0.732226\pi$
$2$		−	2.10850i		−	1.49094i	0.666541	−	0.745469i	$-0.267774\pi$
$3$		0			0
$3$		0			0
$4$	−2.44579			−1.22289
$5$	3.56582			1.59468			0.797341	−	0.603529i	$-0.206239\pi$
$5$	3.56582			1.59468			0.797341	+	0.603529i	$0.206239\pi$
$6$		0			0
$7$	0.463913			0.175343			0.0876713	−	0.996149i	$-0.472057\pi$
$7$	0.463913			0.175343			0.0876713	+	0.996149i	$0.472057\pi$
$8$			0.939944i			0.332320i
$9$		0			0
$10$		−	7.51854i		−	2.37757i
$11$	3.08574			0.930387			0.465193	−	0.885209i	$-0.345985\pi$
$11$	3.08574			0.930387			0.465193	+	0.885209i	$0.345985\pi$
$12$		0			0
$13$	−0.329281			−0.0913261			−0.0456630	−	0.998957i	$-0.514540\pi$
$13$	−0.329281			−0.0913261			−0.0456630	+	0.998957i	$0.514540\pi$
$14$		−	0.978162i		−	0.261425i
$15$		0			0
$16$	−2.90970			−0.727425
$17$			3.45887i			0.838898i	0.907779	+	0.419449i	$0.137777\pi$
$17$			3.45887i			0.838898i	−0.907779	+	0.419449i	$0.862223\pi$
$18$		0			0
$19$	5.56229			1.27608			0.638039	−	0.770004i	$-0.279746\pi$
$19$	5.56229			1.27608			0.638039	+	0.770004i	$0.279746\pi$
$20$	−8.72123			−1.95013
$21$		0			0
$22$		−	6.50630i		−	1.38715i
$23$			2.57418i			0.536753i	0.963314	+	0.268377i	$0.0864871\pi$
$23$			2.57418i			0.536753i	−0.963314	+	0.268377i	$0.913513\pi$
$24$		0			0
$25$	7.71505			1.54301
$26$			0.694290i			0.136161i
$27$		0			0
$28$	−1.13463			−0.214425
$29$		−	0.157325i		−	0.0292146i	−0.999893	−	0.0146073i	$-0.995350\pi$
$29$		−	0.157325i		−	0.0292146i	0.999893	−	0.0146073i	$-0.00464981\pi$
$30$		0			0
$31$		−	3.35167i		−	0.601977i	−0.953628	−	0.300989i	$-0.902683\pi$
$31$		−	3.35167i		−	0.601977i	0.953628	−	0.300989i	$-0.0973166\pi$
$32$			8.01500i			1.41687i
$33$		0			0
$34$	7.29303			1.25074
$35$	1.65423			0.279616
$36$		0			0
$37$		−	5.82738i		−	0.958015i	−0.877811	−	0.479007i	$-0.840997\pi$
$37$		−	5.82738i		−	0.958015i	0.877811	−	0.479007i	$-0.159003\pi$
$38$		−	11.7281i		−	1.90255i
$39$		0			0
$40$			3.35167i			0.529945i
$41$		−	2.20543i		−	0.344430i	−0.985059	−	0.172215i	$-0.944908\pi$
$41$		−	2.20543i		−	0.344430i	0.985059	−	0.172215i	$-0.0550924\pi$
$42$		0			0
$43$		−	4.81514i		−	0.734302i	−0.930161	−	0.367151i	$-0.880333\pi$
$43$		−	4.81514i		−	0.734302i	0.930161	−	0.367151i	$-0.119667\pi$
$44$	−7.54707			−1.13776
$45$		0			0
$46$	5.42766			0.800265
$47$	−3.08574			−0.450102			−0.225051	−	0.974347i	$-0.572255\pi$
$47$	−3.08574			−0.450102			−0.225051	+	0.974347i	$0.572255\pi$
$48$		0			0
$49$	−6.78478			−0.969255
$50$		−	16.2672i		−	2.30053i
$51$		0			0
$52$	0.805351			0.111682
$53$	−7.54707			−1.03667			−0.518335	−	0.855178i	$-0.673448\pi$
$53$	−7.54707			−1.03667			−0.518335	+	0.855178i	$0.673448\pi$
$54$		0			0
$55$	11.0032			1.48367
$56$			0.436052i			0.0582699i
$57$		0			0
$58$	−0.331721			−0.0435571
$59$			4.85205i			0.631684i	0.948812	+	0.315842i	$0.102287\pi$
$59$			4.85205i			0.631684i	−0.948812	+	0.315842i	$0.897713\pi$
$60$		0			0
$61$	−0.391738			−0.0501570			−0.0250785	−	0.999685i	$-0.507984\pi$
$61$	−0.391738			−0.0501570			−0.0250785	+	0.999685i	$0.507984\pi$
$62$	−7.06700			−0.897510
$63$		0			0
$64$	11.0803			1.38503
$65$	−1.17416			−0.145636
$66$		0			0
$67$			8.98201i			1.09733i	0.836043	+	0.548664i	$0.184863\pi$
$67$			8.98201i			1.09733i	−0.836043	+	0.548664i	$0.815137\pi$
$68$		−	8.45965i		−	1.02588i
$69$		0			0
$70$		−	3.48795i		−	0.416889i
$71$	−2.39166			−0.283838			−0.141919	−	0.989878i	$-0.545327\pi$
$71$	−2.39166			−0.283838			−0.141919	+	0.989878i	$0.545327\pi$
$72$		0			0
$73$		−	4.84580i		−	0.567158i	−0.958949	−	0.283579i	$-0.908478\pi$
$73$		−	4.84580i		−	0.567158i	0.958949	−	0.283579i	$-0.0915217\pi$
$74$	−12.2870			−1.42834
$75$		0			0
$76$	−13.6042			−1.56051
$77$	1.43152			0.163136
$78$		0			0
$79$	−13.7207			−1.54370			−0.771849	−	0.635806i	$-0.780668\pi$
$79$	−13.7207			−1.54370			−0.771849	+	0.635806i	$0.780668\pi$
$80$	−10.3755			−1.16001
$81$		0			0
$82$	−4.65015			−0.513523
$83$			9.77694i			1.07316i	0.843850	+	0.536579i	$0.180284\pi$
$83$			9.77694i			1.07316i	−0.843850	+	0.536579i	$0.819716\pi$
$84$		0			0
$85$			12.3337i			1.33778i
$86$	−10.1527			−1.09480
$87$		0			0
$88$			2.90043i			0.309186i
$89$	15.7882			1.67355			0.836775	−	0.547548i	$-0.184439\pi$
$89$	15.7882			1.67355			0.836775	+	0.547548i	$0.184439\pi$
$90$		0			0
$91$	−0.152758			−0.0160133
$92$		−	6.29589i		−	0.656392i
$93$		0			0
$94$			6.50630i			0.671074i
$95$	19.8341			2.03494
$96$		0			0
$97$	−16.2084			−1.64571			−0.822855	−	0.568251i	$-0.807620\pi$
$97$	−16.2084			−1.64571			−0.822855	+	0.568251i	$0.807620\pi$
$98$			14.3057i			1.44510i
$99$		0			0
$100$	−18.8694			−1.88694
$101$	−13.4612			−1.33944			−0.669720	−	0.742614i	$-0.733586\pi$
$101$	−13.4612			−1.33944			−0.669720	+	0.742614i	$0.733586\pi$
$102$		0			0
$103$	−4.72477	−	8.98201i	−0.465545	−	0.885024i
$103$	−4.72477	−	8.98201i	−0.465545	−	0.885024i
$104$		−	0.309505i		−	0.0303495i
$105$		0			0
$106$			15.9130i			1.54561i
$107$			3.47989i			0.336414i	0.985752	+	0.168207i	$0.0537977\pi$
$107$			3.47989i			0.336414i	−0.985752	+	0.168207i	$0.946202\pi$
$108$		0			0
$109$		−	15.9130i		−	1.52419i	−0.647464	−	0.762096i	$-0.724170\pi$
$109$		−	15.9130i		−	1.52419i	0.647464	−	0.762096i	$-0.275830\pi$
$110$		−	23.2003i		−	2.21206i
$111$		0			0
$112$	−1.34985			−0.127549
$113$	17.2844			1.62598			0.812989	−	0.582279i	$-0.197839\pi$
$113$	17.2844			1.62598			0.812989	+	0.582279i	$0.197839\pi$
$114$		0			0
$115$			9.17904i			0.855950i
$116$			0.384784i			0.0357263i
$117$		0			0
$118$	10.2306			0.941800
$119$			1.60461i			0.147095i
$120$		0			0
$121$	−1.47818			−0.134380
$122$			0.825982i			0.0747809i
$123$		0			0
$124$			8.19746i			0.736154i
$125$	9.68137			0.865928
$126$		0			0
$127$			16.9253i			1.50187i	0.660374	+	0.750937i	$0.270398\pi$
$127$			16.9253i			1.50187i	−0.660374	+	0.750937i	$0.729602\pi$
$128$		−	7.33276i		−	0.648130i
$129$		0			0
$130$			2.47571i			0.217134i
$131$		−	0.973553i		−	0.0850597i	−0.999095	−	0.0425299i	$-0.986458\pi$
$131$		−	0.973553i		−	0.0850597i	0.999095	−	0.0425299i	$-0.0135418\pi$
$132$		0			0
$133$	2.58042			0.223751
$134$	18.9386			1.63605
$135$		0			0
$136$	−3.25114			−0.278783
$137$		−	4.79664i		−	0.409805i	−0.978782	−	0.204902i	$-0.934312\pi$
$137$		−	4.79664i		−	0.409805i	0.978782	−	0.204902i	$-0.0656877\pi$
$138$		0			0
$139$	17.4619			1.48110			0.740552	−	0.672000i	$-0.234564\pi$
$139$	17.4619			1.48110			0.740552	+	0.672000i	$0.234564\pi$
$140$	−4.04589			−0.341940
$141$		0			0
$142$			5.04283i			0.423185i
$143$	−1.01608			−0.0849686
$144$		0			0
$145$		−	0.560994i		−	0.0465880i
$146$	−10.2174			−0.845596
$147$		0			0
$148$			14.2525i			1.17155i
$149$			2.13529i			0.174930i	0.996168	+	0.0874651i	$0.0278766\pi$
$149$			2.13529i			0.174930i	−0.996168	+	0.0874651i	$0.972123\pi$
$150$		0			0
$151$		−	11.7462i		−	0.955889i	−0.878390	−	0.477944i	$-0.841382\pi$
$151$		−	11.7462i		−	0.955889i	0.878390	−	0.477944i	$-0.158618\pi$
$152$			5.22824i			0.424066i
$153$		0			0
$154$		−	3.01836i		−	0.243226i
$155$		−	11.9514i		−	0.959962i
$156$		0			0
$157$			18.8400i			1.50359i	0.659394	+	0.751797i	$0.270813\pi$
$157$			18.8400i			1.50359i	−0.659394	+	0.751797i	$0.729187\pi$
$158$			28.9301i			2.30156i
$159$		0			0
$160$			28.5800i			2.25945i
$161$			1.19419i			0.0941156i
$162$		0			0
$163$	−8.40146			−0.658053			−0.329026	−	0.944321i	$-0.606721\pi$
$163$	−8.40146			−0.658053			−0.329026	+	0.944321i	$0.606721\pi$
$164$			5.39401i			0.421201i
$165$		0			0
$166$	20.6147			1.60001
$167$			5.23223i			0.404882i	0.979294	+	0.202441i	$0.0648874\pi$
$167$			5.23223i			0.404882i	−0.979294	+	0.202441i	$0.935113\pi$
$168$		0			0
$169$	−12.8916			−0.991660
$170$	26.0056			1.99454
$171$		0			0
$172$			11.7768i			0.897974i
$173$	−0.737433			−0.0560660			−0.0280330	−	0.999607i	$-0.508924\pi$
$173$	−0.737433			−0.0560660			−0.0280330	+	0.999607i	$0.508924\pi$
$174$		0			0
$175$	3.57911			0.270555
$176$	−8.97859			−0.676787
$177$		0			0
$178$		−	33.2895i		−	2.49516i
$179$			26.3976i			1.97305i	0.163609	+	0.986525i	$0.447686\pi$
$179$			26.3976i			1.97305i	−0.163609	+	0.986525i	$0.552314\pi$
$180$		0			0
$181$		−	17.4821i		−	1.29944i	−0.760175	−	0.649718i	$-0.774887\pi$
$181$		−	17.4821i		−	1.29944i	0.760175	−	0.649718i	$-0.225113\pi$
$182$			0.322090i			0.0238749i
$183$		0			0
$184$	−2.41958			−0.178374
$185$		−	20.7794i		−	1.52773i
$186$		0			0
$187$			10.6732i			0.780500i
$188$	7.54707			0.550427
$189$		0			0
$190$		−	41.8203i		−	3.03396i
$191$	8.65660			0.626369			0.313185	−	0.949692i	$-0.398604\pi$
$191$	8.65660			0.626369			0.313185	+	0.949692i	$0.398604\pi$
$192$		0			0
$193$		−	1.43695i		−	0.103434i	−0.998662	−	0.0517169i	$-0.983531\pi$
$193$		−	1.43695i		−	0.103434i	0.998662	−	0.0517169i	$-0.0164694\pi$
$194$			34.1754i			2.45365i
$195$		0			0
$196$	16.5941			1.18530
$197$	−9.68137			−0.689769			−0.344885	−	0.938645i	$-0.612082\pi$
$197$	−9.68137			−0.689769			−0.344885	+	0.938645i	$0.612082\pi$
$198$		0			0
$199$			1.91472i			0.135731i	0.997694	+	0.0678654i	$0.0216188\pi$
$199$			1.91472i			0.135731i	−0.997694	+	0.0678654i	$0.978381\pi$
$200$			7.25171i			0.512773i
$201$		0			0
$202$			28.3830i			1.99702i
$203$		−	0.0729853i		−	0.00512256i
$204$		0			0
$205$		−	7.86415i		−	0.549256i
$206$	−18.9386	+	9.96219i	−1.31952	+	0.694099i
$207$		0			0
$208$	0.958109			0.0664329
$209$	17.1638			1.18725
$210$		0			0
$211$			18.6430i			1.28343i	0.766942	+	0.641717i	$0.221778\pi$
$211$			18.6430i			1.28343i	−0.766942	+	0.641717i	$0.778222\pi$
$212$	18.4585			1.26774
$213$		0			0
$214$	7.33736			0.501572
$215$		−	17.1699i		−	1.17098i
$216$		0			0
$217$		−	1.55488i		−	0.105552i
$218$	−33.5527			−2.27247
$219$		0			0
$220$	−26.9115			−1.81437
$221$		−	1.13894i		−	0.0766133i
$222$		0			0
$223$	14.3046			0.957909			0.478955	−	0.877840i	$-0.341016\pi$
$223$	14.3046			0.957909			0.478955	+	0.877840i	$0.341016\pi$
$224$			3.71826i			0.248437i
$225$		0			0
$226$		−	36.4442i		−	2.42423i
$227$	16.2037			1.07548			0.537738	−	0.843112i	$-0.319279\pi$
$227$	16.2037			1.07548			0.537738	+	0.843112i	$0.319279\pi$
$228$		0			0
$229$	24.6601			1.62959			0.814793	−	0.579751i	$-0.196850\pi$
$229$	24.6601			1.62959			0.814793	+	0.579751i	$0.196850\pi$
$230$	19.3540			1.27617
$231$		0			0
$232$	0.147877			0.00970860
$233$	5.47741			0.358837			0.179418	−	0.983773i	$-0.442578\pi$
$233$	5.47741			0.358837			0.179418	+	0.983773i	$0.442578\pi$
$234$		0			0
$235$	−11.0032			−0.717770
$236$		−	11.8671i		−	0.772482i
$237$		0			0
$238$	3.38333			0.219309
$239$			18.7320i			1.21167i	0.795589	+	0.605837i	$0.207162\pi$
$239$			18.7320i			1.21167i	−0.795589	+	0.605837i	$0.792838\pi$
$240$		0			0
$241$		−	29.2896i		−	1.88671i	−0.331787	−	0.943354i	$-0.607651\pi$
$241$		−	29.2896i		−	1.88671i	0.331787	−	0.943354i	$-0.392349\pi$
$242$			3.11675i			0.200352i
$243$		0			0
$244$	0.958109			0.0613366
$245$	−24.1933			−1.54565
$246$		0			0
$247$	−1.83156			−0.116539
$248$	3.15038			0.200049
$249$		0			0
$250$		−	20.4132i		−	1.29104i
$251$	11.2710			0.711417			0.355709	−	0.934597i	$-0.384240\pi$
$251$	11.2710			0.711417			0.355709	+	0.934597i	$0.384240\pi$
$252$		0			0
$253$			7.94325i			0.499388i
$254$	35.6870			2.23920
$255$		0			0
$256$	6.69937			0.418710
$257$	10.4188			0.649907			0.324954	−	0.945730i	$-0.394651\pi$
$257$	10.4188			0.649907			0.324954	+	0.945730i	$0.394651\pi$
$258$		0			0
$259$		−	2.70339i		−	0.167981i
$260$	2.87173			0.178097
$261$		0			0
$262$	−2.05274			−0.126819
$263$	9.93874			0.612849			0.306424	−	0.951895i	$-0.400867\pi$
$263$	9.93874			0.612849			0.306424	+	0.951895i	$0.400867\pi$
$264$		0			0
$265$	−26.9115			−1.65316
$266$		−	5.44082i		−	0.333598i
$267$		0			0
$268$		−	21.9681i		−	1.34191i
$269$		−	17.5532i		−	1.07024i	−0.844777	−	0.535118i	$-0.820267\pi$
$269$		−	17.5532i		−	1.07024i	0.844777	−	0.535118i	$-0.179733\pi$
$270$		0			0
$271$			6.16069i			0.374235i	0.982338	+	0.187118i	$0.0599145\pi$
$271$			6.16069i			0.374235i	−0.982338	+	0.187118i	$0.940085\pi$
$272$		−	10.0643i		−	0.610235i
$273$		0			0
$274$	−10.1137			−0.610993
$275$	23.8067			1.43560
$276$		0			0
$277$		−	13.7058i		−	0.823499i	−0.911297	−	0.411749i	$-0.864918\pi$
$277$		−	13.7058i		−	0.823499i	0.911297	−	0.411749i	$-0.135082\pi$
$278$		−	36.8186i		−	2.20823i
$279$		0			0
$280$			1.55488i			0.0929219i
$281$	−15.0295			−0.896586			−0.448293	−	0.893887i	$-0.647968\pi$
$281$	−15.0295			−0.896586			−0.448293	+	0.893887i	$0.647968\pi$
$282$		0			0
$283$			16.1672i			0.961043i	0.876983	+	0.480521i	$0.159553\pi$
$283$			16.1672i			0.961043i	−0.876983	+	0.480521i	$0.840447\pi$
$284$	5.84949			0.347104
$285$		0			0
$286$			2.14240i			0.126683i
$287$		−	1.02313i		−	0.0603932i
$288$		0			0
$289$	5.03625			0.296250
$290$	−1.18286			−0.0694598
$291$		0			0
$292$			11.8518i			0.693573i
$293$	−1.31095			−0.0765866			−0.0382933	−	0.999267i	$-0.512192\pi$
$293$	−1.31095			−0.0765866			−0.0382933	+	0.999267i	$0.512192\pi$
$294$		0			0
$295$			17.3015i			1.00733i
$296$	5.47741			0.318368
$297$		0			0
$298$	4.50228			0.260810
$299$		−	0.847627i		−	0.0490196i
$300$		0			0
$301$		−	2.23381i		−	0.128754i
$302$	−24.7668			−1.42517
$303$		0			0
$304$	−16.1846			−0.928251
$305$	−1.39687			−0.0799844
$306$		0			0
$307$			24.4131i			1.39333i	0.717396	+	0.696666i	$0.245334\pi$
$307$			24.4131i			1.39333i	−0.717396	+	0.696666i	$0.754666\pi$
$308$	−3.50118			−0.199498
$309$		0			0
$310$	−25.1996			−1.43124
$311$		−	11.8206i		−	0.670284i	−0.942168	−	0.335142i	$-0.891216\pi$
$311$		−	11.8206i		−	0.670284i	0.942168	−	0.335142i	$-0.108784\pi$
$312$		0			0
$313$	−4.64477			−0.262538			−0.131269	−	0.991347i	$-0.541905\pi$
$313$	−4.64477			−0.262538			−0.131269	+	0.991347i	$0.541905\pi$
$314$	39.7242			2.24176
$315$		0			0
$316$	33.5579			1.88778
$317$			21.2873i			1.19562i	0.801640	+	0.597808i	$0.203961\pi$
$317$			21.2873i			1.19562i	−0.801640	+	0.597808i	$0.796039\pi$
$318$		0			0
$319$		−	0.485466i		−	0.0271809i
$320$	39.5102			2.20869
$321$		0			0
$322$	2.51796			0.140321
$323$			19.2392i			1.07050i
$324$		0			0
$325$	−2.54042			−0.140917
$326$			17.7145i			0.981115i
$327$		0			0
$328$	2.07298			0.114461
$329$	−1.43152			−0.0789220
$330$		0			0
$331$		−	11.7768i		−	0.647312i	−0.946175	−	0.323656i	$-0.895088\pi$
$331$		−	11.7768i		−	0.647312i	0.946175	−	0.323656i	$-0.104912\pi$
$332$		−	23.9123i		−	1.31236i
$333$		0			0
$334$	11.0322			0.603654
$335$			32.0282i			1.74989i
$336$		0			0
$337$	2.10890			0.114879			0.0574396	−	0.998349i	$-0.481706\pi$
$337$	2.10890			0.114879			0.0574396	+	0.998349i	$0.481706\pi$
$338$			27.1819i			1.47850i
$339$		0			0
$340$		−	30.1656i		−	1.63596i
$341$		−	10.3424i		−	0.560072i
$342$		0			0
$343$	−6.39494			−0.345294
$344$	4.52596			0.244024
$345$		0			0
$346$			1.55488i			0.0835909i
$347$			22.1641i			1.18983i	0.803788	+	0.594916i	$0.202815\pi$
$347$			22.1641i			1.18983i	−0.803788	+	0.594916i	$0.797185\pi$
$348$		0			0
$349$			26.5597i			1.42171i	0.703340	+	0.710853i	$0.251691\pi$
$349$			26.5597i			1.42171i	−0.703340	+	0.710853i	$0.748309\pi$
$350$		−	7.54657i		−	0.403381i
$351$		0			0
$352$			24.7322i			1.31823i
$353$	−32.6012			−1.73519			−0.867594	−	0.497272i	$-0.834335\pi$
$353$	−32.6012			−1.73519			−0.867594	+	0.497272i	$0.834335\pi$
$354$		0			0
$355$	−8.52823			−0.452631
$356$	−38.6146			−2.04657
$357$		0			0
$358$	55.6595			2.94169
$359$			24.3300i			1.28409i	0.766667	+	0.642045i	$0.221914\pi$
$359$			24.3300i			1.28409i	−0.766667	+	0.642045i	$0.778086\pi$
$360$		0			0
$361$	11.9391			0.628374
$362$	−36.8611			−1.93738
$363$		0			0
$364$	0.373613			0.0195826
$365$		−	17.2792i		−	0.904436i
$366$		0			0
$367$	−23.8705			−1.24603			−0.623014	−	0.782211i	$-0.714092\pi$
$367$	−23.8705			−1.24603			−0.623014	+	0.782211i	$0.714092\pi$
$368$		−	7.49008i		−	0.390448i
$369$		0			0
$370$	−43.8134			−2.27775
$371$	−3.50118			−0.181772
$372$		0			0
$373$	−25.4114			−1.31575			−0.657877	−	0.753125i	$-0.728545\pi$
$373$	−25.4114			−1.31575			−0.657877	+	0.753125i	$0.728545\pi$
$374$	22.5044			1.16368
$375$		0			0
$376$		−	2.90043i		−	0.149578i
$377$			0.0518043i			0.00266806i
$378$		0			0
$379$		−	9.96359i		−	0.511795i	−0.966704	−	0.255898i	$-0.917629\pi$
$379$		−	9.96359i		−	0.511795i	0.966704	−	0.255898i	$-0.0823710\pi$
$380$	−48.5100			−2.48851
$381$		0			0
$382$		−	18.2525i		−	0.933877i
$383$	−29.8288			−1.52418			−0.762090	−	0.647471i	$-0.775827\pi$
$383$	−29.8288			−1.52418			−0.762090	+	0.647471i	$0.775827\pi$
$384$		0			0
$385$	5.10452			0.260151
$386$	−3.02981			−0.154213
$387$		0			0
$388$	39.6422			2.01253
$389$	−6.15891			−0.312269			−0.156135	−	0.987736i	$-0.549903\pi$
$389$	−6.15891			−0.312269			−0.156135	+	0.987736i	$0.549903\pi$
$390$		0			0
$391$	−8.90373			−0.450281
$392$		−	6.37731i		−	0.322103i
$393$		0			0
$394$			20.4132i			1.02840i
$395$	−48.9255			−2.46171
$396$		0			0
$397$			27.9175i			1.40114i	0.713583	+	0.700570i	$0.247071\pi$
$397$			27.9175i			1.40114i	−0.713583	+	0.700570i	$0.752929\pi$
$398$	4.03719			0.202366
$399$		0			0
$400$	−22.4485			−1.12242
$401$		−	3.23784i		−	0.161690i	−0.996727	−	0.0808451i	$-0.974238\pi$
$401$		−	3.23784i		−	0.161690i	0.996727	−	0.0808451i	$-0.0257619\pi$
$402$		0			0
$403$			1.10364i			0.0549762i
$404$	32.9232			1.63799
$405$		0			0
$406$	−0.153890			−0.00763742
$407$		−	17.9818i		−	0.891325i
$408$		0			0
$409$	−15.5906			−0.770906			−0.385453	−	0.922727i	$-0.625955\pi$
$409$	−15.5906			−0.770906			−0.385453	+	0.922727i	$0.625955\pi$
$410$	−16.5816			−0.818907
$411$		0			0
$412$	11.5558	+	21.9681i	0.569312	+	1.08229i
$413$			2.25093i			0.110761i
$414$		0			0
$415$			34.8628i			1.71135i
$416$		−	2.63919i		−	0.129397i
$417$		0			0
$418$		−	36.1900i		−	1.77011i
$419$		−	38.5792i		−	1.88472i	−0.334604	−	0.942359i	$-0.608603\pi$
$419$		−	38.5792i		−	1.88472i	0.334604	−	0.942359i	$-0.391397\pi$
$420$		0			0
$421$	19.4628			0.948556			0.474278	−	0.880375i	$-0.342709\pi$
$421$	19.4628			0.948556			0.474278	+	0.880375i	$0.342709\pi$
$422$	39.3087			1.91352
$423$		0			0
$424$		−	7.09382i		−	0.344507i
$425$			26.6853i			1.29443i
$426$		0			0
$427$	−0.181732			−0.00879465
$428$		−	8.51107i		−	0.411398i
$429$		0			0
$430$	−36.2028			−1.74586
$431$		−	29.0104i		−	1.39738i	−0.715424	−	0.698691i	$-0.753766\pi$
$431$		−	29.0104i		−	1.39738i	0.715424	−	0.698691i	$-0.246234\pi$
$432$		0			0
$433$			18.7078i			0.899041i	0.893270	+	0.449520i	$0.148405\pi$
$433$			18.7078i			0.899041i	−0.893270	+	0.449520i	$0.851595\pi$
$434$	−3.27847			−0.157372
$435$		0			0
$436$			38.9199i			1.86392i
$437$			14.3183i			0.684938i
$438$		0			0
$439$		−	4.06125i		−	0.193833i	−0.995293	−	0.0969163i	$-0.969102\pi$
$439$		−	4.06125i		−	0.193833i	0.995293	−	0.0969163i	$-0.0308979\pi$
$440$			10.3424i			0.493054i
$441$		0			0
$442$	−2.40146			−0.114226
$443$	35.5289			1.68803			0.844015	−	0.536320i	$-0.180186\pi$
$443$	35.5289			1.68803			0.844015	+	0.536320i	$0.180186\pi$
$444$		0			0
$445$	56.2979			2.66878
$446$		−	30.1614i		−	1.42818i
$447$		0			0
$448$	5.14027			0.242855
$449$	−22.0677			−1.04144			−0.520720	−	0.853728i	$-0.674336\pi$
$449$	−22.0677			−1.04144			−0.520720	+	0.853728i	$0.674336\pi$
$450$		0			0
$451$		−	6.80539i		−	0.320453i
$452$	−42.2739			−1.98840
$453$		0			0
$454$		−	34.1655i		−	1.60347i
$455$	−0.544706			−0.0255362
$456$		0			0
$457$			6.11581i			0.286086i	0.989716	+	0.143043i	$0.0456887\pi$
$457$			6.11581i			0.286086i	−0.989716	+	0.143043i	$0.954311\pi$
$458$		−	51.9960i		−	2.42961i
$459$		0			0
$460$		−	22.4500i		−	1.04674i
$461$			9.57694i			0.446043i	0.974814	+	0.223021i	$0.0715919\pi$
$461$			9.57694i			0.446043i	−0.974814	+	0.223021i	$0.928408\pi$
$462$		0			0
$463$			18.6914i			0.868663i	0.900753	+	0.434331i	$0.143015\pi$
$463$			18.6914i			0.868663i	−0.900753	+	0.434331i	$0.856985\pi$
$464$			0.457770i			0.0212514i
$465$		0			0
$466$		−	11.5491i		−	0.535003i
$467$		−	30.8613i		−	1.42809i	−0.700100	−	0.714045i	$-0.746861\pi$
$467$		−	30.8613i		−	1.42809i	0.700100	−	0.714045i	$-0.253139\pi$
$468$		0			0
$469$			4.16687i			0.192408i
$470$			23.2003i			1.07015i
$471$		0			0
$472$	−4.56066			−0.209921
$473$		−	14.8583i		−	0.683185i
$474$		0			0
$475$	42.9134			1.96900
$476$		−	3.92454i		−	0.179881i
$477$		0			0
$478$	39.4966			1.80653
$479$	11.6922			0.534232			0.267116	−	0.963664i	$-0.413929\pi$
$479$	11.6922			0.534232			0.267116	+	0.963664i	$0.413929\pi$
$480$		0			0
$481$			1.91884i			0.0874918i
$482$	−61.7572			−2.81296
$483$		0			0
$484$	3.61532			0.164333
$485$	−57.7961			−2.62438
$486$		0			0
$487$		−	19.8563i		−	0.899777i	−0.893085	−	0.449888i	$-0.851464\pi$
$487$		−	19.8563i		−	0.899777i	0.893085	−	0.449888i	$-0.148536\pi$
$488$		−	0.368212i		−	0.0166682i
$489$		0			0
$490$			51.0117i			2.30447i
$491$			32.8239i			1.48132i	0.671879	+	0.740661i	$0.265487\pi$
$491$			32.8239i			1.48132i	−0.671879	+	0.740661i	$0.734513\pi$
$492$		0			0
$493$	0.544167			0.0245081
$494$			3.86184i			0.173753i
$495$		0			0
$496$			9.75234i			0.437893i
$497$	−1.10952			−0.0497689
$498$		0			0
$499$		−	28.4479i		−	1.27350i	−0.771070	−	0.636751i	$-0.780278\pi$
$499$		−	28.4479i		−	1.27350i	0.771070	−	0.636751i	$-0.219722\pi$
$500$	−23.6786			−1.05894
$501$		0			0
$502$		−	23.7649i		−	1.06068i
$503$		−	17.3702i		−	0.774501i	−0.921975	−	0.387251i	$-0.873425\pi$
$503$		−	17.3702i		−	0.774501i	0.921975	−	0.387251i	$-0.126575\pi$
$504$		0			0
$505$	−48.0002			−2.13598
$506$	16.7484			0.744556
$507$		0			0
$508$		−	41.3956i		−	1.83663i
$509$		−	9.34372i		−	0.414153i	−0.978325	−	0.207076i	$-0.933605\pi$
$509$		−	9.34372i		−	0.414153i	0.978325	−	0.207076i	$-0.0663949\pi$
$510$		0			0
$511$		−	2.24803i		−	0.0994469i
$512$		−	28.7911i		−	1.27240i
$513$		0			0
$514$		−	21.9681i		−	0.968971i
$515$	−16.8477	−	32.0282i	−0.742396	−	1.41133i
$516$		0			0
$517$	−9.52182			−0.418769
$518$	−5.70012			−0.250449
$519$		0			0
$520$		−	1.10364i		−	0.0483978i
$521$	−21.5172			−0.942684			−0.471342	−	0.881951i	$-0.656230\pi$
$521$	−21.5172			−0.942684			−0.471342	+	0.881951i	$0.656230\pi$
$522$		0			0
$523$	−2.12003			−0.0927026			−0.0463513	−	0.998925i	$-0.514759\pi$
$523$	−2.12003			−0.0927026			−0.0463513	+	0.998925i	$0.514759\pi$
$524$			2.38110i			0.104019i
$525$		0			0
$526$		−	20.9559i		−	0.913719i
$527$	11.5930			0.504997
$528$		0			0
$529$	16.3736			0.711896
$530$			56.7430i			2.46476i
$531$		0			0
$532$	−6.31116			−0.273623
$533$			0.726205i			0.0314554i
$534$		0			0
$535$			12.4087i			0.536473i
$536$	−8.44259			−0.364664
$537$		0			0
$538$	−37.0110			−1.59566
$539$	−20.9361			−0.901782
$540$		0			0
$541$	16.2008			0.696528			0.348264	−	0.937396i	$-0.386771\pi$
$541$	16.2008			0.696528			0.348264	+	0.937396i	$0.386771\pi$
$542$	12.9898			0.557961
$543$		0			0
$544$	−27.7228			−1.18861
$545$		−	56.7430i		−	2.43060i
$546$		0			0
$547$	−11.1635			−0.477316			−0.238658	−	0.971104i	$-0.576708\pi$
$547$	−11.1635			−0.477316			−0.238658	+	0.971104i	$0.576708\pi$
$548$			11.7316i			0.501148i
$549$		0			0
$550$		−	50.1965i		−	2.14038i
$551$		−	0.875090i		−	0.0372801i
$552$		0			0
$553$	−6.36520			−0.270676
$554$	−28.8986			−1.22778
$555$		0			0
$556$	−42.7082			−1.81123
$557$	−15.5742			−0.659901			−0.329950	−	0.943998i	$-0.607032\pi$
$557$	−15.5742			−0.659901			−0.329950	+	0.943998i	$0.607032\pi$
$558$		0			0
$559$			1.58553i			0.0670610i
$560$	−4.81331			−0.203399
$561$		0			0
$562$			31.6898i			1.33675i
$563$	−2.16311			−0.0911644			−0.0455822	−	0.998961i	$-0.514514\pi$
$563$	−2.16311			−0.0911644			−0.0455822	+	0.998961i	$0.514514\pi$
$564$		0			0
$565$	61.6329			2.59292
$566$	34.0887			1.43285
$567$		0			0
$568$		−	2.24803i		−	0.0943251i
$569$	29.3054			1.22854			0.614272	−	0.789094i	$-0.289450\pi$
$569$	29.3054			1.22854			0.614272	+	0.789094i	$0.289450\pi$
$570$		0			0
$571$	−21.5600			−0.902258			−0.451129	−	0.892459i	$-0.648979\pi$
$571$	−21.5600			−0.902258			−0.451129	+	0.892459i	$0.648979\pi$
$572$	2.48511			0.103908
$573$		0			0
$574$	−2.15727			−0.0900425
$575$			19.8599i			0.828215i
$576$		0			0
$577$		−	6.08572i		−	0.253352i	−0.991944	−	0.126676i	$-0.959569\pi$
$577$		−	6.08572i		−	0.253352i	0.991944	−	0.126676i	$-0.0404308\pi$
$578$		−	10.6190i		−	0.441690i
$579$		0			0
$580$			1.37207i			0.0569721i
$581$			4.53565i			0.188170i
$582$		0			0
$583$	−23.2883			−0.964504
$584$	4.55477			0.188478
$585$		0			0
$586$			2.76415i			0.114186i
$587$		−	19.7929i		−	0.816941i	−0.912772	−	0.408471i	$-0.866062\pi$
$587$		−	19.7929i		−	0.816941i	0.912772	−	0.408471i	$-0.133938\pi$
$588$		0			0
$589$		−	18.6430i		−	0.768169i
$590$	36.4803			1.50187
$591$		0			0
$592$			16.9559i			0.696884i
$593$	−40.2129			−1.65135			−0.825674	−	0.564148i	$-0.809205\pi$
$593$	−40.2129			−1.65135			−0.825674	+	0.564148i	$0.809205\pi$
$594$		0			0
$595$			5.72175i			0.234569i
$596$		−	5.22248i		−	0.213921i
$597$		0			0
$598$	−1.78723			−0.0730851
$599$	42.3385			1.72991			0.864953	−	0.501853i	$-0.167348\pi$
$599$	42.3385			1.72991			0.864953	+	0.501853i	$0.167348\pi$
$600$		0			0
$601$		−	41.7289i		−	1.70216i	−0.525038	−	0.851079i	$-0.675949\pi$
$601$		−	41.7289i		−	1.70216i	0.525038	−	0.851079i	$-0.324051\pi$
$602$	−4.70999			−0.191965
$603$		0			0
$604$			28.7286i			1.16895i
$605$	−5.27093			−0.214294
$606$		0			0
$607$	−23.0083			−0.933878			−0.466939	−	0.884289i	$-0.654643\pi$
$607$	−23.0083			−0.933878			−0.466939	+	0.884289i	$0.654643\pi$
$608$			44.5818i			1.80803i
$609$		0			0
$610$			2.94530i			0.119252i
$611$	1.01608			0.0411061
$612$		0			0
$613$	10.8502			0.438237			0.219118	−	0.975698i	$-0.429682\pi$
$613$	10.8502			0.438237			0.219118	+	0.975698i	$0.429682\pi$
$614$	51.4752			2.07737
$615$		0			0
$616$			1.34554i			0.0542135i
$617$	9.30929			0.374778			0.187389	−	0.982286i	$-0.439998\pi$
$617$	9.30929			0.374778			0.187389	+	0.982286i	$0.439998\pi$
$618$		0			0
$619$	−6.29194			−0.252894			−0.126447	−	0.991973i	$-0.540357\pi$
$619$	−6.29194			−0.252894			−0.126447	+	0.991973i	$0.540357\pi$
$620$			29.2307i			1.17393i
$621$		0			0
$622$	−24.9237			−0.999351
$623$	7.32436			0.293444
$624$		0			0
$625$	−4.05325			−0.162130
$626$			9.79351i			0.391427i
$627$		0			0
$628$		−	46.0786i		−	1.83874i
$629$	20.1561			0.803677
$630$		0			0
$631$	−14.5182			−0.577961			−0.288981	−	0.957335i	$-0.593316\pi$
$631$	−14.5182			−0.577961			−0.288981	+	0.957335i	$0.593316\pi$
$632$		−	12.8967i		−	0.513002i
$633$		0			0
$634$	44.8844			1.78259
$635$			60.3524i			2.39501i
$636$		0			0
$637$	2.23410			0.0885183
$638$	−1.02361			−0.0405250
$639$		0			0
$640$		−	26.1473i		−	1.03356i
$641$		−	45.4231i		−	1.79411i	−0.441924	−	0.897053i	$-0.645704\pi$
$641$		−	45.4231i		−	1.79411i	0.441924	−	0.897053i	$-0.354296\pi$
$642$		0			0
$643$	−2.58883			−0.102093			−0.0510467	−	0.998696i	$-0.516256\pi$
$643$	−2.58883			−0.102093			−0.0510467	+	0.998696i	$0.516256\pi$
$644$		−	2.92074i		−	0.115093i
$645$		0			0
$646$	40.5660			1.59605
$647$		−	21.4687i		−	0.844020i	−0.906591	−	0.422010i	$-0.861325\pi$
$647$		−	21.4687i		−	0.844020i	0.906591	−	0.422010i	$-0.138675\pi$
$648$		0			0
$649$			14.9722i			0.587710i
$650$			5.35648i			0.210099i
$651$		0			0
$652$	20.5482			0.804728
$653$	−28.4261			−1.11240			−0.556199	−	0.831049i	$-0.687741\pi$
$653$	−28.4261			−1.11240			−0.556199	+	0.831049i	$0.687741\pi$
$654$		0			0
$655$		−	3.47151i		−	0.135643i
$656$			6.41713i			0.250547i
$657$		0			0
$658$			3.01836i			0.117668i
$659$			5.66246i			0.220578i	0.993900	+	0.110289i	$0.0351777\pi$
$659$			5.66246i			0.220578i	−0.993900	+	0.110289i	$0.964822\pi$
$660$		0			0
$661$			9.30110i			0.361771i	0.983504	+	0.180885i	$0.0578963\pi$
$661$			9.30110i			0.361771i	−0.983504	+	0.180885i	$0.942104\pi$
$662$	−24.8315			−0.965102
$663$		0			0
$664$	−9.18977			−0.356632
$665$	9.20130			0.356811
$666$		0			0
$667$	0.404983			0.0156810
$668$		−	12.7969i		−	0.495128i
$669$		0			0
$670$	67.5316			2.60897
$671$	−1.20880			−0.0466654
$672$		0			0
$673$	43.1135			1.66190			0.830951	−	0.556346i	$-0.187797\pi$
$673$	43.1135			1.66190			0.830951	+	0.556346i	$0.187797\pi$
$674$		−	4.44662i		−	0.171278i
$675$		0			0
$676$	31.5300			1.21269
$677$			21.4645i			0.824948i	0.910969	+	0.412474i	$0.135335\pi$
$677$			21.4645i			0.824948i	−0.910969	+	0.412474i	$0.864665\pi$
$678$		0			0
$679$	−7.51927			−0.288563
$680$	−11.5930			−0.444570
$681$		0			0
$682$	−21.8070			−0.835032
$683$	24.5182			0.938161			0.469081	−	0.883155i	$-0.344585\pi$
$683$	24.5182			0.938161			0.469081	+	0.883155i	$0.344585\pi$
$684$		0			0
$685$		−	17.1039i		−	0.653508i
$686$			13.4837i			0.514812i
$687$		0			0
$688$			14.0106i			0.534150i
$689$	2.48511			0.0946750
$690$		0			0
$691$		−	42.1966i		−	1.60523i	−0.596495	−	0.802617i	$-0.703440\pi$
$691$		−	42.1966i		−	1.60523i	0.596495	−	0.802617i	$-0.296560\pi$
$692$	1.80361			0.0685628
$693$		0			0
$694$	46.7331			1.77396
$695$	62.2661			2.36189
$696$		0			0
$697$	7.62828			0.288942
$698$	56.0012			2.11968
$699$		0			0
$700$	−8.75374			−0.330860
$701$			33.5523i			1.26725i	0.773639	+	0.633627i	$0.218435\pi$
$701$			33.5523i			1.26725i	−0.773639	+	0.633627i	$0.781565\pi$
$702$		0			0
$703$		−	32.4136i		−	1.22250i
$704$	34.1908			1.28862
$705$		0			0
$706$			68.7398i			2.58706i
$707$	−6.24482			−0.234861
$708$		0			0
$709$	−9.79748			−0.367952			−0.183976	−	0.982931i	$-0.558897\pi$
$709$	−9.79748			−0.367952			−0.183976	+	0.982931i	$0.558897\pi$
$710$			17.9818i			0.674845i
$711$		0			0
$712$			14.8400i			0.556154i
$713$	8.62778			0.323113
$714$		0			0
$715$	−3.62314			−0.135498
$716$		−	64.5629i		−	2.41283i
$717$		0			0
$718$	51.2999			1.91450
$719$	47.9220			1.78719			0.893594	−	0.448876i	$-0.148176\pi$
$719$	47.9220			1.78719			0.893594	+	0.448876i	$0.148176\pi$
$720$		0			0
$721$	−2.19188	−	4.16687i	−0.0816299	−	0.155182i
$722$		−	25.1736i		−	0.936866i
$723$		0			0
$724$			42.7576i			1.58907i
$725$		−	1.21377i		−	0.0450784i
$726$		0			0
$727$			9.99012i			0.370513i	0.982690	+	0.185257i	$0.0593116\pi$
$727$			9.99012i			0.370513i	−0.982690	+	0.185257i	$0.940688\pi$
$728$		−	0.143584i		−	0.00532156i
$729$		0			0
$730$	−36.4333			−1.34846
$731$	16.6549			0.616005
$732$		0			0
$733$		−	35.1471i		−	1.29819i	−0.760709	−	0.649093i	$-0.775148\pi$
$733$		−	35.1471i		−	1.29819i	0.760709	−	0.649093i	$-0.224852\pi$
$734$			50.3309i			1.85775i
$735$		0			0
$736$	−20.6320			−0.760507
$737$			27.7162i			1.02094i
$738$		0			0
$739$	−46.4078			−1.70714			−0.853569	−	0.520980i	$-0.825567\pi$
$739$	−46.4078			−1.70714			−0.853569	+	0.520980i	$0.825567\pi$
$740$			50.8219i			1.86825i
$741$		0			0
$742$			7.38226i			0.271011i
$743$	−2.38583			−0.0875274			−0.0437637	−	0.999042i	$-0.513935\pi$
$743$	−2.38583			−0.0875274			−0.0437637	+	0.999042i	$0.513935\pi$
$744$		0			0
$745$			7.61407i			0.278958i
$746$			53.5801i			1.96171i
$747$		0			0
$748$		−	26.1043i		−	0.954468i
$749$			1.61437i			0.0589876i
$750$		0			0
$751$	−1.39814			−0.0510188			−0.0255094	−	0.999675i	$-0.508121\pi$
$751$	−1.39814			−0.0510188			−0.0255094	+	0.999675i	$0.508121\pi$
$752$	8.97859			0.327415
$753$		0			0
$754$	0.109229			0.00397790
$755$		−	41.8847i		−	1.52434i
$756$		0			0
$757$	48.7659			1.77243			0.886214	−	0.463276i	$-0.153326\pi$
$757$	48.7659			1.77243			0.886214	+	0.463276i	$0.153326\pi$
$758$	−21.0083			−0.763055
$759$		0			0
$760$			18.6430i			0.676251i
$761$	−17.4136			−0.631244			−0.315622	−	0.948885i	$-0.602213\pi$
$761$	−17.4136			−0.631244			−0.315622	+	0.948885i	$0.602213\pi$
$762$		0			0
$763$		−	7.38226i		−	0.267256i
$764$	−21.1722			−0.765983
$765$		0			0
$766$			62.8941i			2.27246i
$767$		−	1.59769i		−	0.0576892i
$768$		0			0
$769$		−	19.7507i		−	0.712229i	−0.934442	−	0.356115i	$-0.884101\pi$
$769$		−	19.7507i		−	0.712229i	0.934442	−	0.356115i	$-0.115899\pi$
$770$		−	10.7629i		−	0.387868i
$771$		0			0
$772$			3.51447i			0.126489i
$773$		−	33.3915i		−	1.20101i	−0.799622	−	0.600504i	$-0.794967\pi$
$773$		−	33.3915i		−	1.20101i	0.799622	−	0.600504i	$-0.205033\pi$
$774$		0			0
$775$		−	25.8583i		−	0.928857i
$776$		−	15.2350i		−	0.546903i
$777$		0			0
$778$			12.9861i			0.465573i
$779$		−	12.2672i		−	0.439519i
$780$		0			0
$781$	−7.38006			−0.264079
$782$			18.7735i			0.671341i
$783$		0			0
$784$	19.7417			0.705060
$785$			67.1799i			2.39775i
$786$		0			0
$787$	−6.38788			−0.227703			−0.113852	−	0.993498i	$-0.536319\pi$
$787$	−6.38788			−0.227703			−0.113852	+	0.993498i	$0.536319\pi$
$788$	23.6786			0.843514
$789$		0			0
$790$			103.160i			3.67025i
$791$	8.01844			0.285103
$792$		0			0
$793$	0.128992			0.00458064
$794$	58.8642			2.08901
$795$		0			0
$796$		−	4.68299i		−	0.165984i
$797$			33.3121i			1.17998i	0.807412	+	0.589988i	$0.200868\pi$
$797$			33.3121i			1.17998i	−0.807412	+	0.589988i	$0.799132\pi$
$798$		0			0
$799$		−	10.6732i		−	0.377590i
$800$			61.8361i			2.18624i
$801$		0			0
$802$	−6.82700			−0.241070
$803$		−	14.9529i		−	0.527676i
$804$		0			0
$805$			4.25828i			0.150085i
$806$	2.32703			0.0819661
$807$		0			0
$808$		−	12.6528i		−	0.445123i
$809$	−10.3379			−0.363463			−0.181731	−	0.983348i	$-0.558170\pi$
$809$	−10.3379			−0.363463			−0.181731	+	0.983348i	$0.558170\pi$
$810$		0			0
$811$		−	38.5866i		−	1.35496i	−0.735542	−	0.677479i	$-0.763072\pi$
$811$		−	38.5866i		−	1.35496i	0.735542	−	0.677479i	$-0.236928\pi$
$812$			0.178506i			0.00626435i
$813$		0			0
$814$	−37.9147			−1.32891
$815$	−29.9581			−1.04938
$816$		0			0
$817$		−	26.7832i		−	0.937027i
$818$			32.8729i			1.14937i
$819$		0			0
$820$			19.2340i			0.671682i
$821$			18.4581i			0.644193i	0.946707	+	0.322096i	$0.104388\pi$
$821$			18.4581i			0.644193i	−0.946707	+	0.322096i	$0.895612\pi$
$822$		0			0
$823$			22.1002i			0.770366i	0.922840	+	0.385183i	$0.125862\pi$
$823$			22.1002i			0.770366i	−0.922840	+	0.385183i	$0.874138\pi$
$824$	8.44259	−	4.44101i	0.294111	−	0.154710i
$825$		0			0
$826$	4.74609			0.165138
$827$	−38.8374			−1.35051			−0.675254	−	0.737586i	$-0.735966\pi$
$827$	−38.8374			−1.35051			−0.675254	+	0.737586i	$0.735966\pi$
$828$		0			0
$829$		−	29.7102i		−	1.03188i	−0.856625	−	0.515939i	$-0.827443\pi$
$829$		−	29.7102i		−	1.03188i	0.856625	−	0.515939i	$-0.172557\pi$
$830$	73.5083			2.55151
$831$		0			0
$832$	−3.64852			−0.126490
$833$		−	23.4677i		−	0.813106i
$834$		0			0
$835$			18.6572i			0.645658i
$836$	−41.9790			−1.45188
$837$		0			0
$838$	−81.3444			−2.81000
$839$			15.2721i			0.527251i	0.964625	+	0.263626i	$0.0849184\pi$
$839$			15.2721i			0.527251i	−0.964625	+	0.263626i	$0.915082\pi$
$840$		0			0
$841$	28.9752			0.999147
$842$		−	41.0373i		−	1.41424i
$843$		0			0
$844$		−	45.5967i		−	1.56950i
$845$	−45.9690			−1.58138
$846$		0			0
$847$	−0.685747			−0.0235626
$848$	21.9597			0.754100
$849$		0			0
$850$	56.2661			1.92991
$851$	15.0007			0.514217
$852$		0			0
$853$	−18.2525			−0.624953			−0.312477	−	0.949925i	$-0.601159\pi$
$853$	−18.2525			−0.624953			−0.312477	+	0.949925i	$0.601159\pi$
$854$			0.383183i			0.0131123i
$855$		0			0
$856$	−3.27090			−0.111797
$857$		−	30.7422i		−	1.05013i	−0.851061	−	0.525067i	$-0.824040\pi$
$857$		−	30.7422i		−	1.05013i	0.851061	−	0.525067i	$-0.175960\pi$
$858$		0			0
$859$			12.9171i			0.440725i	0.975418	+	0.220362i	$0.0707240\pi$
$859$			12.9171i			0.440725i	−0.975418	+	0.220362i	$0.929276\pi$
$860$			41.9940i			1.43198i
$861$		0			0
$862$	−61.1685			−2.08341
$863$	−46.1058			−1.56946			−0.784730	−	0.619838i	$-0.787198\pi$
$863$	−46.1058			−1.56946			−0.784730	+	0.619838i	$0.787198\pi$
$864$		0			0
$865$	−2.62955			−0.0894075
$866$	39.4455			1.34041
$867$		0			0
$868$			3.80291i			0.129079i
$869$	−42.3385			−1.43624
$870$		0			0
$871$		−	2.95761i		−	0.100215i
$872$	14.9574			0.506520
$873$		0			0
$874$	30.1902			1.02120
$875$	4.49131			0.151834
$876$		0			0
$877$		−	56.2841i		−	1.90058i	−0.311368	−	0.950289i	$-0.600787\pi$
$877$		−	56.2841i		−	1.90058i	0.311368	−	0.950289i	$-0.399213\pi$
$878$	−8.56315			−0.288992
$879$		0			0
$880$	−32.0160			−1.07926
$881$	13.7832			0.464368			0.232184	−	0.972672i	$-0.425413\pi$
$881$	13.7832			0.464368			0.232184	+	0.972672i	$0.425413\pi$
$882$		0			0
$883$	−36.2843			−1.22106			−0.610531	−	0.791992i	$-0.709044\pi$
$883$	−36.2843			−1.22106			−0.610531	+	0.791992i	$0.709044\pi$
$884$			2.78560i			0.0936899i
$885$		0			0
$886$		−	74.9128i		−	2.51675i
$887$			9.07781i			0.304803i	0.988319	+	0.152402i	$0.0487007\pi$
$887$			9.07781i			0.304803i	−0.988319	+	0.152402i	$0.951299\pi$
$888$		0			0
$889$			7.85185i			0.263343i
$890$		−	118.704i		−	3.97898i
$891$		0			0
$892$	−34.9861			−1.17142
$893$	−17.1638			−0.574365
$894$		0			0
$895$			94.1291i			3.14639i
$896$		−	3.40176i		−	0.113645i
$897$		0			0
$898$			46.5298i			1.55272i
$899$	−0.527302			−0.0175865
$900$		0			0
$901$		−	26.1043i		−	0.869661i
$902$	−14.3492			−0.477776
$903$		0			0
$904$			16.2463i			0.540345i
$905$		−	62.3381i		−	2.07219i
$906$		0			0
$907$	28.5157			0.946847			0.473424	−	0.880835i	$-0.343018\pi$
$907$	28.5157			0.946847			0.473424	+	0.880835i	$0.343018\pi$
$908$	−39.6307			−1.31519
$909$		0			0
$910$			1.14851i			0.0380729i
$911$	11.3327			0.375470			0.187735	−	0.982220i	$-0.439885\pi$
$911$	11.3327			0.375470			0.187735	+	0.982220i	$0.439885\pi$
$912$		0			0
$913$			30.1691i			0.998452i
$914$	12.8952			0.426536
$915$		0			0
$916$	−60.3134			−1.99281
$917$		−	0.451644i		−	0.0149146i
$918$		0			0
$919$		−	9.63854i		−	0.317946i	−0.987283	−	0.158973i	$-0.949182\pi$
$919$		−	9.63854i		−	0.317946i	0.987283	−	0.158973i	$-0.0508183\pi$
$920$	−8.62778			−0.284450
$921$		0			0
$922$	20.1930			0.665021
$923$	0.787528			0.0259218
$924$		0			0
$925$		−	44.9585i		−	1.47823i
$926$	39.4109			1.29512
$927$		0			0
$928$	1.26096			0.0413931
$929$			36.6056i			1.20099i	0.799628	+	0.600496i	$0.205030\pi$
$929$			36.6056i			1.20099i	−0.799628	+	0.600496i	$0.794970\pi$
$930$		0			0
$931$	−37.7390			−1.23684
$932$	−13.3966			−0.438819
$933$		0			0
$934$	−65.0711			−2.12919
$935$			38.0586i			1.24465i
$936$		0			0
$937$			39.4461i			1.28865i	0.764753	+	0.644324i	$0.222861\pi$
$937$			39.4461i			1.28865i	−0.764753	+	0.644324i	$0.777139\pi$
$938$	8.78586			0.286869
$939$		0			0
$940$	26.9115			0.877756
$941$		−	34.7821i		−	1.13387i	−0.823764	−	0.566933i	$-0.808130\pi$
$941$		−	34.7821i		−	1.13387i	0.823764	−	0.566933i	$-0.191870\pi$
$942$		0			0
$943$	5.67716			0.184874
$944$		−	14.1180i		−	0.459502i
$945$		0			0
$946$	−31.3288			−1.01859
$947$	53.6529			1.74348			0.871742	−	0.489965i	$-0.162991\pi$
$947$	53.6529			1.74348			0.871742	+	0.489965i	$0.162991\pi$
$948$		0			0
$949$			1.59563i			0.0517963i
$950$		−	90.4830i		−	2.93566i
$951$		0			0
$952$	−1.50824			−0.0488825
$953$			21.1602i			0.685445i	0.939437	+	0.342722i	$0.111349\pi$
$953$			21.1602i			0.685445i	−0.939437	+	0.342722i	$0.888651\pi$
$954$		0			0
$955$	30.8678			0.998860
$956$		−	45.8146i		−	1.48175i
$957$		0			0
$958$		−	24.6531i		−	0.796507i
$959$		−	2.22522i		−	0.0718562i
$960$		0			0
$961$	19.7663			0.637624
$962$	4.04589			0.130445
$963$		0			0
$964$			71.6361i			2.30724i
$965$		−	5.12390i		−	0.164944i
$966$		0			0
$967$		−	6.10811i		−	0.196424i	−0.995166	−	0.0982119i	$-0.968688\pi$
$967$		−	6.10811i		−	0.196424i	0.995166	−	0.0982119i	$-0.0313123\pi$
$968$		−	1.38941i		−	0.0446572i
$969$		0			0
$970$			121.863i			3.91279i
$971$	52.7428			1.69260			0.846299	−	0.532709i	$-0.178826\pi$
$971$	52.7428			1.69260			0.846299	+	0.532709i	$0.178826\pi$
$972$		0			0
$973$	8.10082			0.259700
$974$	−41.8672			−1.34151
$975$		0			0
$976$	1.13984			0.0364854
$977$		−	41.3547i		−	1.32305i	−0.749921	−	0.661527i	$-0.769909\pi$
$977$		−	41.3547i		−	1.32305i	0.749921	−	0.661527i	$-0.230091\pi$
$978$		0			0
$979$	48.7184			1.55705
$980$	59.1717			1.89017
$981$		0			0
$982$	69.2093			2.20856
$983$		−	35.9511i		−	1.14666i	−0.819324	−	0.573331i	$-0.805651\pi$
$983$		−	35.9511i		−	1.14666i	0.819324	−	0.573331i	$-0.194349\pi$
$984$		0			0
$985$	−34.5220			−1.09996
$986$		−	1.14738i		−	0.0365400i
$987$		0			0
$988$	4.47960			0.142515
$989$	12.3950			0.394139
$990$		0			0
$991$	28.9615			0.919993			0.459997	−	0.887921i	$-0.347851\pi$
$991$	28.9615			0.919993			0.459997	+	0.887921i	$0.347851\pi$
$992$	26.8636			0.852920
$993$		0			0
$994$			2.33943i			0.0742023i
$995$			6.82753i			0.216447i
$996$		0			0
$997$			7.36995i			0.233409i	0.993167	+	0.116704i	$0.0372330\pi$
$997$			7.36995i			0.233409i	−0.993167	+	0.116704i	$0.962767\pi$
$998$	−59.9824			−1.89871
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.c.a.926.2	yes	16
3.2	odd	2	inner	927.2.c.a.926.15	yes	16
103.102	odd	2	inner	927.2.c.a.926.1	✓	16
309.308	even	2	inner	927.2.c.a.926.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.c.a.926.1	✓	16	103.102	odd	2	inner
927.2.c.a.926.2	yes	16	1.1	even	1	trivial
927.2.c.a.926.15	yes	16	3.2	odd	2	inner
927.2.c.a.926.16	yes	16	309.308	even	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:	\( N \)	\(=\)	\( 927 = 3^{2} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	927.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.10850i q^{2} -2.44579 q^{4} +3.56582 q^{5} +0.463913 q^{7} +0.939944i q^{8} +O(q^{10})\)	\(q-2.10850i q^{2} -2.44579 q^{4} +3.56582 q^{5} +0.463913 q^{7} +0.939944i q^{8} -7.51854i q^{10} +3.08574 q^{11} -0.329281 q^{13} -0.978162i q^{14} -2.90970 q^{16} +3.45887i q^{17} +5.56229 q^{19} -8.72123 q^{20} -6.50630i q^{22} +2.57418i q^{23} +7.71505 q^{25} +0.694290i q^{26} -1.13463 q^{28} -0.157325i q^{29} -3.35167i q^{31} +8.01500i q^{32} +7.29303 q^{34} +1.65423 q^{35} -5.82738i q^{37} -11.7281i q^{38} +3.35167i q^{40} -2.20543i q^{41} -4.81514i q^{43} -7.54707 q^{44} +5.42766 q^{46} -3.08574 q^{47} -6.78478 q^{49} -16.2672i q^{50} +0.805351 q^{52} -7.54707 q^{53} +11.0032 q^{55} +0.436052i q^{56} -0.331721 q^{58} +4.85205i q^{59} -0.391738 q^{61} -7.06700 q^{62} +11.0803 q^{64} -1.17416 q^{65} +8.98201i q^{67} -8.45965i q^{68} -3.48795i q^{70} -2.39166 q^{71} -4.84580i q^{73} -12.2870 q^{74} -13.6042 q^{76} +1.43152 q^{77} -13.7207 q^{79} -10.3755 q^{80} -4.65015 q^{82} +9.77694i q^{83} +12.3337i q^{85} -10.1527 q^{86} +2.90043i q^{88} +15.7882 q^{89} -0.152758 q^{91} -6.29589i q^{92} +6.50630i q^{94} +19.8341 q^{95} -16.2084 q^{97} +14.3057i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{7}+O(q^{10}) \) 16 * q + 16 * q^7	\( 16 q + 16 q^{7} - 8 q^{13} - 16 q^{16} + 8 q^{19} + 96 q^{25} - 24 q^{28} + 24 q^{34} - 32 q^{49} + 16 q^{52} - 8 q^{55} - 40 q^{58} - 32 q^{61} + 40 q^{64} - 80 q^{76} - 40 q^{79} + 8 q^{82} - 56 q^{91} + 16 q^{97}+O(q^{100}) \) 16 * q + 16 * q^7 - 8 * q^13 - 16 * q^16 + 8 * q^19 + 96 * q^25 - 24 * q^28 + 24 * q^34 - 32 * q^49 + 16 * q^52 - 8 * q^55 - 40 * q^58 - 32 * q^61 + 40 * q^64 - 80 * q^76 - 40 * q^79 + 8 * q^82 - 56 * q^91 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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