LMFDB - Embedded newform 4368.2.h.r.337.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))

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

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

chi := DirichletCharacter("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 20x^{8} + 138x^{6} + 364x^{4} + 249x^{2} + 36 $ x^10 + 20x^8 + 138x^6 + 364x^4 + 249x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.9
Root			$2.56826i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.r.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.71697i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.71697i q^{5} -1.00000i q^{7} +1.00000 q^{9} -4.97789i q^{11} +(2.56826 + 2.53062i) q^{13} -2.71697i q^{15} -4.31294 q^{17} +3.75461i q^{19} +1.00000i q^{21} +3.45720 q^{23} -2.38191 q^{25} -1.00000 q^{27} -7.08450 q^{29} -9.39744i q^{31} +4.97789i q^{33} +2.71697 q^{35} +1.12099i q^{37} +(-2.56826 - 2.53062i) q^{39} -9.66353i q^{41} -10.9466 q^{43} +2.71697i q^{45} +12.4905i q^{47} -1.00000 q^{49} +4.31294 q^{51} -2.87561 q^{53} +13.5248 q^{55} -3.75461i q^{57} +3.41034i q^{59} -1.94024 q^{61} -1.00000i q^{63} +(-6.87561 + 6.97789i) q^{65} +11.0811i q^{67} -3.45720 q^{69} -10.0989i q^{71} +6.05202i q^{73} +2.38191 q^{75} -4.97789 q^{77} +8.12298 q^{79} +1.00000 q^{81} -17.6004i q^{83} -11.7181i q^{85} +7.08450 q^{87} +2.19079i q^{89} +(2.53062 - 2.56826i) q^{91} +9.39744i q^{93} -10.2012 q^{95} +11.8454i q^{97} -4.97789i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 10 * q^9	$ 10 q - 10 q^{3} + 10 q^{9} - 4 q^{13} + 10 q^{17} - 8 q^{23} - 2 q^{25} - 10 q^{27} - 44 q^{29} + 4 q^{39} - 20 q^{43} - 10 q^{49} - 10 q^{51} + 10 q^{53} + 2 q^{55} + 18 q^{61} - 30 q^{65} + 8 q^{69} + 2 q^{75} - 2 q^{77} - 2 q^{79} + 10 q^{81} + 44 q^{87} + 6 q^{91} - 44 q^{95}+O(q^{100}) $ 10 * q - 10 * q^3 + 10 * q^9 - 4 * q^13 + 10 * q^17 - 8 * q^23 - 2 * q^25 - 10 * q^27 - 44 * q^29 + 4 * q^39 - 20 * q^43 - 10 * q^49 - 10 * q^51 + 10 * q^53 + 2 * q^55 + 18 * q^61 - 30 * q^65 + 8 * q^69 + 2 * q^75 - 2 * q^77 - 2 * q^79 + 10 * q^81 + 44 * q^87 + 6 * q^91 - 44 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			2.71697i			1.21506i	0.794295	+	0.607532i	$0.207841\pi$
$5$			2.71697i			1.21506i	−0.794295	+	0.607532i	$0.792159\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	4.97789i		−	1.50089i	−0.660933	−	0.750445i	$-0.729839\pi$
$11$		−	4.97789i		−	1.50089i	0.660933	−	0.750445i	$-0.270161\pi$
$12$		0			0
$13$	2.56826	+	2.53062i	0.712308	+	0.701867i
$13$	2.56826	+	2.53062i	0.712308	+	0.701867i
$14$		0			0
$15$		−	2.71697i		−	0.701518i
$16$		0			0
$17$	−4.31294			−1.04604			−0.523021	−	0.852320i	$-0.675195\pi$
$17$	−4.31294			−1.04604			−0.523021	+	0.852320i	$0.675195\pi$
$18$		0			0
$19$			3.75461i			0.861367i	0.902503	+	0.430684i	$0.141728\pi$
$19$			3.75461i			0.861367i	−0.902503	+	0.430684i	$0.858272\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$	3.45720			0.720877			0.360438	−	0.932783i	$-0.382627\pi$
$23$	3.45720			0.720877			0.360438	+	0.932783i	$0.382627\pi$
$24$		0			0
$25$	−2.38191			−0.476383
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−7.08450			−1.31556			−0.657780	−	0.753210i	$-0.728504\pi$
$29$	−7.08450			−1.31556			−0.657780	+	0.753210i	$0.728504\pi$
$30$		0			0
$31$		−	9.39744i		−	1.68783i	−0.536476	−	0.843915i	$-0.680245\pi$
$31$		−	9.39744i		−	1.68783i	0.536476	−	0.843915i	$-0.319755\pi$
$32$		0			0
$33$			4.97789i			0.866539i
$34$		0			0
$35$	2.71697			0.459251
$36$		0			0
$37$			1.12099i			0.184290i	0.995746	+	0.0921451i	$0.0293724\pi$
$37$			1.12099i			0.184290i	−0.995746	+	0.0921451i	$0.970628\pi$
$38$		0			0
$39$	−2.56826	−	2.53062i	−0.411251	−	0.405223i
$40$		0			0
$41$		−	9.66353i		−	1.50919i	−0.656191	−	0.754595i	$-0.727833\pi$
$41$		−	9.66353i		−	1.50919i	0.656191	−	0.754595i	$-0.272167\pi$
$42$		0			0
$43$	−10.9466			−1.66933			−0.834667	−	0.550754i	$-0.814340\pi$
$43$	−10.9466			−1.66933			−0.834667	+	0.550754i	$0.814340\pi$
$44$		0			0
$45$			2.71697i			0.405022i
$46$		0			0
$47$			12.4905i			1.82193i	0.412485	+	0.910964i	$0.364661\pi$
$47$			12.4905i			1.82193i	−0.412485	+	0.910964i	$0.635339\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	4.31294			0.603933
$52$		0			0
$53$	−2.87561			−0.394995			−0.197497	−	0.980303i	$-0.563281\pi$
$53$	−2.87561			−0.394995			−0.197497	+	0.980303i	$0.563281\pi$
$54$		0			0
$55$	13.5248			1.82368
$56$		0			0
$57$		−	3.75461i		−	0.497311i
$58$		0			0
$59$			3.41034i			0.443989i	0.975048	+	0.221995i	$0.0712567\pi$
$59$			3.41034i			0.443989i	−0.975048	+	0.221995i	$0.928743\pi$
$60$		0			0
$61$	−1.94024			−0.248423			−0.124211	−	0.992256i	$-0.539640\pi$
$61$	−1.94024			−0.248423			−0.124211	+	0.992256i	$0.539640\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	−6.87561	+	6.97789i	−0.852814	+	0.865500i
$66$		0			0
$67$			11.0811i			1.35377i	0.736088	+	0.676886i	$0.236671\pi$
$67$			11.0811i			1.35377i	−0.736088	+	0.676886i	$0.763329\pi$
$68$		0			0
$69$	−3.45720			−0.416198
$70$		0			0
$71$		−	10.0989i		−	1.19852i	−0.800556	−	0.599258i	$-0.795462\pi$
$71$		−	10.0989i		−	1.19852i	0.800556	−	0.599258i	$-0.204538\pi$
$72$		0			0
$73$			6.05202i			0.708336i	0.935182	+	0.354168i	$0.115236\pi$
$73$			6.05202i			0.708336i	−0.935182	+	0.354168i	$0.884764\pi$
$74$		0			0
$75$	2.38191			0.275040
$76$		0			0
$77$	−4.97789			−0.567283
$78$		0			0
$79$	8.12298			0.913906			0.456953	−	0.889491i	$-0.348941\pi$
$79$	8.12298			0.913906			0.456953	+	0.889491i	$0.348941\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	17.6004i		−	1.93189i	−0.258745	−	0.965946i	$-0.583309\pi$
$83$		−	17.6004i		−	1.93189i	0.258745	−	0.965946i	$-0.416691\pi$
$84$		0			0
$85$		−	11.7181i		−	1.27101i
$86$		0			0
$87$	7.08450			0.759538
$88$		0			0
$89$			2.19079i			0.232224i	0.993236	+	0.116112i	$0.0370431\pi$
$89$			2.19079i			0.232224i	−0.993236	+	0.116112i	$0.962957\pi$
$90$		0			0
$91$	2.53062	−	2.56826i	0.265281	−	0.269227i
$92$		0			0
$93$			9.39744i			0.974470i
$94$		0			0
$95$	−10.2012			−1.04662
$96$		0			0
$97$			11.8454i			1.20272i	0.798978	+	0.601361i	$0.205375\pi$
$97$			11.8454i			1.20272i	−0.798978	+	0.601361i	$0.794625\pi$
$98$		0			0
$99$		−	4.97789i		−	0.500296i
$100$		0			0
$101$	5.75801			0.572944			0.286472	−	0.958089i	$-0.407518\pi$
$101$	5.75801			0.572944			0.286472	+	0.958089i	$0.407518\pi$
$102$		0			0
$103$	−10.5374			−1.03828			−0.519139	−	0.854690i	$-0.673747\pi$
$103$	−10.5374			−1.03828			−0.519139	+	0.854690i	$0.673747\pi$
$104$		0			0
$105$	−2.71697			−0.265149
$106$		0			0
$107$	−15.5204			−1.50042			−0.750208	−	0.661202i	$-0.770047\pi$
$107$	−15.5204			−1.50042			−0.750208	+	0.661202i	$0.770047\pi$
$108$		0			0
$109$			5.64283i			0.540485i	0.962792	+	0.270243i	$0.0871039\pi$
$109$			5.64283i			0.540485i	−0.962792	+	0.270243i	$0.912896\pi$
$110$		0			0
$111$		−	1.12099i		−	0.106400i
$112$		0			0
$113$	−18.8601			−1.77421			−0.887106	−	0.461566i	$-0.847288\pi$
$113$	−18.8601			−1.77421			−0.887106	+	0.461566i	$0.847288\pi$
$114$		0			0
$115$			9.39311i			0.875912i
$116$		0			0
$117$	2.56826	+	2.53062i	0.237436	+	0.233956i
$118$		0			0
$119$			4.31294i			0.395367i
$120$		0			0
$121$	−13.7794			−1.25267
$122$		0			0
$123$			9.66353i			0.871331i
$124$		0			0
$125$			7.11326i			0.636229i
$126$		0			0
$127$	−6.61469			−0.586959			−0.293479	−	0.955965i	$-0.594813\pi$
$127$	−6.61469			−0.586959			−0.293479	+	0.955965i	$0.594813\pi$
$128$		0			0
$129$	10.9466			0.963791
$130$		0			0
$131$	21.9912			1.92138			0.960689	−	0.277627i	$-0.0895480\pi$
$131$	21.9912			1.92138			0.960689	+	0.277627i	$0.0895480\pi$
$132$		0			0
$133$	3.75461			0.325566
$134$		0			0
$135$		−	2.71697i		−	0.233839i
$136$		0			0
$137$		−	5.12616i		−	0.437957i	−0.975730	−	0.218979i	$-0.929728\pi$
$137$		−	5.12616i		−	0.437957i	0.975730	−	0.218979i	$-0.0702725\pi$
$138$		0			0
$139$	−20.4509			−1.73463			−0.867313	−	0.497763i	$-0.834155\pi$
$139$	−20.4509			−1.73463			−0.867313	+	0.497763i	$0.834155\pi$
$140$		0			0
$141$		−	12.4905i		−	1.05189i
$142$		0			0
$143$	12.5971	−	12.7845i	1.05342	−	1.06910i
$144$		0			0
$145$		−	19.2484i		−	1.59849i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$		−	17.6834i		−	1.44868i	−0.689443	−	0.724340i	$-0.742145\pi$
$149$		−	17.6834i		−	1.44868i	0.689443	−	0.724340i	$-0.257855\pi$
$150$		0			0
$151$		−	11.2273i		−	0.913669i	−0.889552	−	0.456834i	$-0.848983\pi$
$151$		−	11.2273i		−	0.913669i	0.889552	−	0.456834i	$-0.151017\pi$
$152$		0			0
$153$	−4.31294			−0.348681
$154$		0			0
$155$	25.5326			2.05082
$156$		0			0
$157$	−21.8222			−1.74160			−0.870799	−	0.491639i	$-0.836398\pi$
$157$	−21.8222			−1.74160			−0.870799	+	0.491639i	$0.836398\pi$
$158$		0			0
$159$	2.87561			0.228050
$160$		0			0
$161$		−	3.45720i		−	0.272466i
$162$		0			0
$163$		−	6.59713i		−	0.516727i	−0.966048	−	0.258364i	$-0.916817\pi$
$163$		−	6.59713i		−	0.516727i	0.966048	−	0.258364i	$-0.0831832\pi$
$164$		0			0
$165$	−13.5248			−1.05290
$166$		0			0
$167$		−	20.3685i		−	1.57616i	−0.615570	−	0.788082i	$-0.711074\pi$
$167$		−	20.3685i		−	1.57616i	0.615570	−	0.788082i	$-0.288926\pi$
$168$		0			0
$169$	0.191948	+	12.9986i	0.0147652	+	0.999891i
$170$		0			0
$171$			3.75461i			0.287122i
$172$		0			0
$173$	20.7765			1.57961			0.789803	−	0.613361i	$-0.210183\pi$
$173$	20.7765			1.57961			0.789803	+	0.613361i	$0.210183\pi$
$174$		0			0
$175$			2.38191i			0.180056i
$176$		0			0
$177$		−	3.41034i		−	0.256337i
$178$		0			0
$179$	−17.2667			−1.29058			−0.645288	−	0.763939i	$-0.723263\pi$
$179$	−17.2667			−1.29058			−0.645288	+	0.763939i	$0.723263\pi$
$180$		0			0
$181$	17.8737			1.32854			0.664270	−	0.747493i	$-0.268742\pi$
$181$	17.8737			1.32854			0.664270	+	0.747493i	$0.268742\pi$
$182$		0			0
$183$	1.94024			0.143427
$184$		0			0
$185$	−3.04570			−0.223925
$186$		0			0
$187$			21.4693i			1.56999i
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	−18.4553			−1.33538			−0.667689	−	0.744441i	$-0.732716\pi$
$191$	−18.4553			−1.33538			−0.667689	+	0.744441i	$0.732716\pi$
$192$		0			0
$193$		−	16.8702i		−	1.21434i	−0.794571	−	0.607171i	$-0.792304\pi$
$193$		−	16.8702i		−	1.21434i	0.794571	−	0.607171i	$-0.207696\pi$
$194$		0			0
$195$	6.87561	−	6.97789i	0.492372	−	0.499697i
$196$		0			0
$197$		−	3.45753i		−	0.246338i	−0.992386	−	0.123169i	$-0.960694\pi$
$197$		−	3.45753i		−	0.246338i	0.992386	−	0.123169i	$-0.0393058\pi$
$198$		0			0
$199$	−4.79692			−0.340044			−0.170022	−	0.985440i	$-0.554384\pi$
$199$	−4.79692			−0.340044			−0.170022	+	0.985440i	$0.554384\pi$
$200$		0			0
$201$		−	11.0811i		−	0.781601i
$202$		0			0
$203$			7.08450i			0.497235i
$204$		0			0
$205$	26.2555			1.83376
$206$		0			0
$207$	3.45720			0.240292
$208$		0			0
$209$	18.6900			1.29282
$210$		0			0
$211$	0.300810			0.0207086			0.0103543	−	0.999946i	$-0.496704\pi$
$211$	0.300810			0.0207086			0.0103543	+	0.999946i	$0.496704\pi$
$212$		0			0
$213$			10.0989i			0.691964i
$214$		0			0
$215$		−	29.7415i		−	2.02835i
$216$		0			0
$217$	−9.39744			−0.637940
$218$		0			0
$219$		−	6.05202i		−	0.408958i
$220$		0			0
$221$	−11.0768	−	10.9144i	−0.745104	−	0.734182i
$222$		0			0
$223$		−	16.4785i		−	1.10349i	−0.834014	−	0.551743i	$-0.813963\pi$
$223$		−	16.4785i		−	1.10349i	0.834014	−	0.551743i	$-0.186037\pi$
$224$		0			0
$225$	−2.38191			−0.158794
$226$		0			0
$227$		−	22.1012i		−	1.46691i	−0.679739	−	0.733454i	$-0.737907\pi$
$227$		−	22.1012i		−	1.46691i	0.679739	−	0.733454i	$-0.262093\pi$
$228$		0			0
$229$		−	5.22793i		−	0.345472i	−0.984968	−	0.172736i	$-0.944739\pi$
$229$		−	5.22793i		−	0.345472i	0.984968	−	0.172736i	$-0.0552607\pi$
$230$		0			0
$231$	4.97789			0.327521
$232$		0			0
$233$	5.34795			0.350356			0.175178	−	0.984537i	$-0.443950\pi$
$233$	5.34795			0.350356			0.175178	+	0.984537i	$0.443950\pi$
$234$		0			0
$235$	−33.9363			−2.21376
$236$		0			0
$237$	−8.12298			−0.527644
$238$		0			0
$239$		−	27.1358i		−	1.75527i	−0.479333	−	0.877633i	$-0.659121\pi$
$239$		−	27.1358i		−	1.75527i	0.479333	−	0.877633i	$-0.340879\pi$
$240$		0			0
$241$		−	9.02475i		−	0.581335i	−0.956824	−	0.290667i	$-0.906123\pi$
$241$		−	9.02475i		−	0.581335i	0.956824	−	0.290667i	$-0.0938773\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	2.71697i		−	0.173581i
$246$		0			0
$247$	−9.50149	+	9.64283i	−0.604565	+	0.613559i
$248$		0			0
$249$			17.6004i			1.11538i
$250$		0			0
$251$	14.2061			0.896679			0.448339	−	0.893863i	$-0.352016\pi$
$251$	14.2061			0.896679			0.448339	+	0.893863i	$0.352016\pi$
$252$		0			0
$253$		−	17.2096i		−	1.08196i
$254$		0			0
$255$			11.7181i			0.733817i
$256$		0			0
$257$	−14.1535			−0.882873			−0.441437	−	0.897292i	$-0.645531\pi$
$257$	−14.1535			−0.882873			−0.441437	+	0.897292i	$0.645531\pi$
$258$		0			0
$259$	1.12099			0.0696552
$260$		0			0
$261$	−7.08450			−0.438520
$262$		0			0
$263$	−7.50872			−0.463007			−0.231504	−	0.972834i	$-0.574365\pi$
$263$	−7.50872			−0.463007			−0.231504	+	0.972834i	$0.574365\pi$
$264$		0			0
$265$		−	7.81293i		−	0.479944i
$266$		0			0
$267$		−	2.19079i		−	0.134074i
$268$		0			0
$269$	−10.6443			−0.648995			−0.324498	−	0.945887i	$-0.605195\pi$
$269$	−10.6443			−0.648995			−0.324498	+	0.945887i	$0.605195\pi$
$270$		0			0
$271$		−	17.0540i		−	1.03596i	−0.855394	−	0.517979i	$-0.826685\pi$
$271$		−	17.0540i		−	1.03596i	0.855394	−	0.517979i	$-0.173315\pi$
$272$		0			0
$273$	−2.53062	+	2.56826i	−0.153160	+	0.155438i
$274$		0			0
$275$			11.8569i			0.714998i
$276$		0			0
$277$	3.88822			0.233620			0.116810	−	0.993154i	$-0.462733\pi$
$277$	3.88822			0.233620			0.116810	+	0.993154i	$0.462733\pi$
$278$		0			0
$279$		−	9.39744i		−	0.562610i
$280$		0			0
$281$		−	4.12994i		−	0.246372i	−0.992384	−	0.123186i	$-0.960689\pi$
$281$		−	4.12994i		−	0.246372i	0.992384	−	0.123186i	$-0.0393111\pi$
$282$		0			0
$283$	10.8989			0.647874			0.323937	−	0.946079i	$-0.394993\pi$
$283$	10.8989			0.647874			0.323937	+	0.946079i	$0.394993\pi$
$284$		0			0
$285$	10.2012			0.604265
$286$		0			0
$287$	−9.66353			−0.570420
$288$		0			0
$289$	1.60146			0.0942038
$290$		0			0
$291$		−	11.8454i		−	0.694392i
$292$		0			0
$293$			24.0736i			1.40639i	0.710995	+	0.703197i	$0.248245\pi$
$293$			24.0736i			1.40639i	−0.710995	+	0.703197i	$0.751755\pi$
$294$		0			0
$295$	−9.26580			−0.539476
$296$		0			0
$297$			4.97789i			0.288846i
$298$		0			0
$299$	8.87901	+	8.74886i	0.513486	+	0.505960i
$300$		0			0
$301$			10.9466i			0.630949i
$302$		0			0
$303$	−5.75801			−0.330789
$304$		0			0
$305$		−	5.27157i		−	0.301849i
$306$		0			0
$307$			19.2595i			1.09920i	0.835429	+	0.549599i	$0.185219\pi$
$307$			19.2595i			1.09920i	−0.835429	+	0.549599i	$0.814781\pi$
$308$		0			0
$309$	10.5374			0.599450
$310$		0			0
$311$	−18.6443			−1.05722			−0.528611	−	0.848864i	$-0.677287\pi$
$311$	−18.6443			−1.05722			−0.528611	+	0.848864i	$0.677287\pi$
$312$		0			0
$313$	1.28711			0.0727515			0.0363758	−	0.999338i	$-0.488419\pi$
$313$	1.28711			0.0727515			0.0363758	+	0.999338i	$0.488419\pi$
$314$		0			0
$315$	2.71697			0.153084
$316$		0			0
$317$		−	6.90924i		−	0.388062i	−0.980995	−	0.194031i	$-0.937844\pi$
$317$		−	6.90924i		−	0.388062i	0.980995	−	0.194031i	$-0.0621562\pi$
$318$		0			0
$319$			35.2659i			1.97451i
$320$		0			0
$321$	15.5204			0.866265
$322$		0			0
$323$		−	16.1934i		−	0.901026i
$324$		0			0
$325$	−6.11738	−	6.02771i	−0.339331	−	0.334357i
$326$		0			0
$327$		−	5.64283i		−	0.312049i
$328$		0			0
$329$	12.4905			0.688624
$330$		0			0
$331$			16.3955i			0.901179i	0.892731	+	0.450590i	$0.148786\pi$
$331$			16.3955i			0.901179i	−0.892731	+	0.450590i	$0.851214\pi$
$332$		0			0
$333$			1.12099i			0.0614301i
$334$		0			0
$335$	−30.1070			−1.64492
$336$		0			0
$337$	−11.0934			−0.604295			−0.302148	−	0.953261i	$-0.597704\pi$
$337$	−11.0934			−0.604295			−0.302148	+	0.953261i	$0.597704\pi$
$338$		0			0
$339$	18.8601			1.02434
$340$		0			0
$341$	−46.7794			−2.53325
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	9.39311i		−	0.505708i
$346$		0			0
$347$	−14.3572			−0.770736			−0.385368	−	0.922763i	$-0.625926\pi$
$347$	−14.3572			−0.770736			−0.385368	+	0.922763i	$0.625926\pi$
$348$		0			0
$349$			27.0684i			1.44894i	0.689306	+	0.724470i	$0.257916\pi$
$349$			27.0684i			1.44894i	−0.689306	+	0.724470i	$0.742084\pi$
$350$		0			0
$351$	−2.56826	−	2.53062i	−0.137084	−	0.135074i
$352$		0			0
$353$		−	24.2125i		−	1.28870i	−0.764731	−	0.644350i	$-0.777128\pi$
$353$		−	24.2125i		−	1.28870i	0.764731	−	0.644350i	$-0.222872\pi$
$354$		0			0
$355$	27.4383			1.45628
$356$		0			0
$357$		−	4.31294i		−	0.228265i
$358$		0			0
$359$			3.30399i			0.174378i	0.996192	+	0.0871890i	$0.0277884\pi$
$359$			3.30399i			0.174378i	−0.996192	+	0.0871890i	$0.972212\pi$
$360$		0			0
$361$	4.90288			0.258046
$362$		0			0
$363$	13.7794			0.723229
$364$		0			0
$365$	−16.4431			−0.860674
$366$		0			0
$367$	−28.5285			−1.48918			−0.744589	−	0.667523i	$-0.767354\pi$
$367$	−28.5285			−1.48918			−0.744589	+	0.667523i	$0.767354\pi$
$368$		0			0
$369$		−	9.66353i		−	0.503063i
$370$		0			0
$371$			2.87561i			0.149294i
$372$		0			0
$373$	2.05542			0.106426			0.0532129	−	0.998583i	$-0.483054\pi$
$373$	2.05542			0.106426			0.0532129	+	0.998583i	$0.483054\pi$
$374$		0			0
$375$		−	7.11326i		−	0.367327i
$376$		0			0
$377$	−18.1949	−	17.9282i	−0.937083	−	0.923348i
$378$		0			0
$379$			21.5277i			1.10580i	0.833247	+	0.552901i	$0.186479\pi$
$379$			21.5277i			1.10580i	−0.833247	+	0.552901i	$0.813521\pi$
$380$		0			0
$381$	6.61469			0.338881
$382$		0			0
$383$			4.94913i			0.252889i	0.991974	+	0.126444i	$0.0403565\pi$
$383$			4.94913i			0.252889i	−0.991974	+	0.126444i	$0.959643\pi$
$384$		0			0
$385$		−	13.5248i		−	0.689285i
$386$		0			0
$387$	−10.9466			−0.556445
$388$		0			0
$389$	−0.361504			−0.0183290			−0.00916448	−	0.999958i	$-0.502917\pi$
$389$	−0.361504			−0.0183290			−0.00916448	+	0.999958i	$0.502917\pi$
$390$		0			0
$391$	−14.9107			−0.754067
$392$		0			0
$393$	−21.9912			−1.10931
$394$		0			0
$395$			22.0699i			1.11046i
$396$		0			0
$397$		−	1.19144i		−	0.0597968i	−0.999553	−	0.0298984i	$-0.990482\pi$
$397$		−	1.19144i		−	0.0597968i	0.999553	−	0.0298984i	$-0.00951837\pi$
$398$		0			0
$399$	−3.75461			−0.187966
$400$		0			0
$401$			15.3845i			0.768268i	0.923277	+	0.384134i	$0.125500\pi$
$401$			15.3845i			0.768268i	−0.923277	+	0.384134i	$0.874500\pi$
$402$		0			0
$403$	23.7813	−	24.1351i	1.18463	−	1.20226i
$404$		0			0
$405$			2.71697i			0.135007i
$406$		0			0
$407$	5.58018			0.276599
$408$		0			0
$409$		−	33.2596i		−	1.64458i	−0.569069	−	0.822290i	$-0.692696\pi$
$409$		−	33.2596i		−	1.64458i	0.569069	−	0.822290i	$-0.307304\pi$
$410$		0			0
$411$			5.12616i			0.252855i
$412$		0			0
$413$	3.41034			0.167812
$414$		0			0
$415$	47.8196			2.34737
$416$		0			0
$417$	20.4509			1.00149
$418$		0			0
$419$	−25.8251			−1.26164			−0.630820	−	0.775929i	$-0.717281\pi$
$419$	−25.8251			−1.26164			−0.630820	+	0.775929i	$0.717281\pi$
$420$		0			0
$421$		−	17.6317i		−	0.859316i	−0.902992	−	0.429658i	$-0.858634\pi$
$421$		−	17.6317i		−	0.859316i	0.902992	−	0.429658i	$-0.141366\pi$
$422$		0			0
$423$			12.4905i			0.607309i
$424$		0			0
$425$	10.2731			0.498316
$426$		0			0
$427$			1.94024i			0.0938949i
$428$		0			0
$429$	−12.5971	+	12.7845i	−0.608195	+	0.617243i
$430$		0			0
$431$			11.2885i			0.543749i	0.962333	+	0.271874i	$0.0876435\pi$
$431$			11.2885i			0.543749i	−0.962333	+	0.271874i	$0.912356\pi$
$432$		0			0
$433$	29.4280			1.41422			0.707110	−	0.707104i	$-0.249999\pi$
$433$	29.4280			1.41422			0.707110	+	0.707104i	$0.249999\pi$
$434$		0			0
$435$			19.2484i			0.922889i
$436$		0			0
$437$			12.9805i			0.620940i
$438$		0			0
$439$	−20.8989			−0.997449			−0.498724	−	0.866761i	$-0.666198\pi$
$439$	−20.8989			−0.997449			−0.498724	+	0.866761i	$0.666198\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	7.20177			0.342166			0.171083	−	0.985257i	$-0.445273\pi$
$443$	7.20177			0.342166			0.171083	+	0.985257i	$0.445273\pi$
$444$		0			0
$445$	−5.95231			−0.282167
$446$		0			0
$447$			17.6834i			0.836396i
$448$		0			0
$449$		−	27.3734i		−	1.29183i	−0.763409	−	0.645916i	$-0.776476\pi$
$449$		−	27.3734i		−	1.29183i	0.763409	−	0.645916i	$-0.223524\pi$
$450$		0			0
$451$	−48.1040			−2.26513
$452$		0			0
$453$			11.2273i			0.527507i
$454$		0			0
$455$	6.97789	+	6.87561i	0.327128	+	0.322333i
$456$		0			0
$457$			0.918897i			0.0429842i	0.999769	+	0.0214921i	$0.00684167\pi$
$457$			0.918897i			0.0429842i	−0.999769	+	0.0214921i	$0.993158\pi$
$458$		0			0
$459$	4.31294			0.201311
$460$		0			0
$461$			3.10180i			0.144465i	0.997388	+	0.0722326i	$0.0230124\pi$
$461$			3.10180i			0.144465i	−0.997388	+	0.0722326i	$0.976988\pi$
$462$		0			0
$463$			13.2561i			0.616063i	0.951376	+	0.308032i	$0.0996703\pi$
$463$			13.2561i			0.616063i	−0.951376	+	0.308032i	$0.900330\pi$
$464$		0			0
$465$	−25.5326			−1.18404
$466$		0			0
$467$	−18.7366			−0.867025			−0.433513	−	0.901148i	$-0.642726\pi$
$467$	−18.7366			−0.867025			−0.433513	+	0.901148i	$0.642726\pi$
$468$		0			0
$469$	11.0811			0.511678
$470$		0			0
$471$	21.8222			1.00551
$472$		0			0
$473$			54.4907i			2.50549i
$474$		0			0
$475$		−	8.94316i		−	0.410340i
$476$		0			0
$477$	−2.87561			−0.131665
$478$		0			0
$479$			1.96037i			0.0895717i	0.998997	+	0.0447858i	$0.0142605\pi$
$479$			1.96037i			0.0895717i	−0.998997	+	0.0447858i	$0.985739\pi$
$480$		0			0
$481$	−2.83681	+	2.87901i	−0.129347	+	0.131271i
$482$		0			0
$483$			3.45720i			0.157308i
$484$		0			0
$485$	−32.1837			−1.46138
$486$		0			0
$487$			2.01163i			0.0911555i	0.998961	+	0.0455777i	$0.0145129\pi$
$487$			2.01163i			0.0911555i	−0.998961	+	0.0455777i	$0.985487\pi$
$488$		0			0
$489$			6.59713i			0.298332i
$490$		0			0
$491$	−16.3853			−0.739460			−0.369730	−	0.929139i	$-0.620550\pi$
$491$	−16.3853			−0.739460			−0.369730	+	0.929139i	$0.620550\pi$
$492$		0			0
$493$	30.5550			1.37613
$494$		0			0
$495$	13.5248			0.607893
$496$		0			0
$497$	−10.0989			−0.452997
$498$		0			0
$499$			28.7978i			1.28917i	0.764533	+	0.644584i	$0.222970\pi$
$499$			28.7978i			1.28917i	−0.764533	+	0.644584i	$0.777030\pi$
$500$		0			0
$501$			20.3685i			0.909999i
$502$		0			0
$503$	−7.13508			−0.318138			−0.159069	−	0.987267i	$-0.550849\pi$
$503$	−7.13508			−0.318138			−0.159069	+	0.987267i	$0.550849\pi$
$504$		0			0
$505$			15.6443i			0.696164i
$506$		0			0
$507$	−0.191948	−	12.9986i	−0.00852470	−	0.577287i
$508$		0			0
$509$		−	9.82046i		−	0.435284i	−0.976029	−	0.217642i	$-0.930163\pi$
$509$		−	9.82046i		−	0.435284i	0.976029	−	0.217642i	$-0.0698366\pi$
$510$		0			0
$511$	6.05202			0.267726
$512$		0			0
$513$		−	3.75461i		−	0.165770i
$514$		0			0
$515$		−	28.6297i		−	1.26158i
$516$		0			0
$517$	62.1764			2.73451
$518$		0			0
$519$	−20.7765			−0.911986
$520$		0			0
$521$	16.7153			0.732309			0.366154	−	0.930554i	$-0.380674\pi$
$521$	16.7153			0.732309			0.366154	+	0.930554i	$0.380674\pi$
$522$		0			0
$523$	37.3838			1.63468			0.817339	−	0.576157i	$-0.195448\pi$
$523$	37.3838			1.63468			0.817339	+	0.576157i	$0.195448\pi$
$524$		0			0
$525$		−	2.38191i		−	0.103955i
$526$		0			0
$527$			40.5306i			1.76554i
$528$		0			0
$529$	−11.0477			−0.480337
$530$		0			0
$531$			3.41034i			0.147996i
$532$		0			0
$533$	24.4547	−	24.8185i	1.05925	−	1.07501i
$534$		0			0
$535$		−	42.1685i		−	1.82310i
$536$		0			0
$537$	17.2667			0.745115
$538$		0			0
$539$			4.97789i			0.214413i
$540$		0			0
$541$			10.2945i			0.442596i	0.975206	+	0.221298i	$0.0710293\pi$
$541$			10.2945i			0.442596i	−0.975206	+	0.221298i	$0.928971\pi$
$542$		0			0
$543$	−17.8737			−0.767033
$544$		0			0
$545$	−15.3314			−0.656725
$546$		0			0
$547$	33.3820			1.42731			0.713655	−	0.700497i	$-0.247038\pi$
$547$	33.3820			1.42731			0.713655	+	0.700497i	$0.247038\pi$
$548$		0			0
$549$	−1.94024			−0.0828075
$550$		0			0
$551$		−	26.5996i		−	1.13318i
$552$		0			0
$553$		−	8.12298i		−	0.345424i
$554$		0			0
$555$	3.04570			0.129283
$556$		0			0
$557$			3.29950i			0.139804i	0.997554	+	0.0699022i	$0.0222687\pi$
$557$			3.29950i			0.139804i	−0.997554	+	0.0699022i	$0.977731\pi$
$558$		0			0
$559$	−28.1136	−	27.7016i	−1.18908	−	1.17165i
$560$		0			0
$561$		−	21.4693i		−	0.906436i
$562$		0			0
$563$	−37.0807			−1.56277			−0.781383	−	0.624051i	$-0.785486\pi$
$563$	−37.0807			−1.56277			−0.781383	+	0.624051i	$0.785486\pi$
$564$		0			0
$565$		−	51.2424i		−	2.15578i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−31.5171			−1.32126			−0.660632	−	0.750710i	$-0.729712\pi$
$569$	−31.5171			−1.32126			−0.660632	+	0.750710i	$0.729712\pi$
$570$		0			0
$571$	18.9132			0.791495			0.395747	−	0.918359i	$-0.370486\pi$
$571$	18.9132			0.791495			0.395747	+	0.918359i	$0.370486\pi$
$572$		0			0
$573$	18.4553			0.770980
$574$		0			0
$575$	−8.23476			−0.343413
$576$		0			0
$577$			13.2429i			0.551308i	0.961257	+	0.275654i	$0.0888945\pi$
$577$			13.2429i			0.551308i	−0.961257	+	0.275654i	$0.911106\pi$
$578$		0			0
$579$			16.8702i			0.701101i
$580$		0			0
$581$	−17.6004			−0.730186
$582$		0			0
$583$			14.3144i			0.592844i
$584$		0			0
$585$	−6.87561	+	6.97789i	−0.284271	+	0.288500i
$586$		0			0
$587$		−	9.77322i		−	0.403384i	−0.979449	−	0.201692i	$-0.935356\pi$
$587$		−	9.77322i		−	0.403384i	0.979449	−	0.201692i	$-0.0646440\pi$
$588$		0			0
$589$	35.2838			1.45384
$590$		0			0
$591$			3.45753i			0.142224i
$592$		0			0
$593$		−	5.19892i		−	0.213494i	−0.994286	−	0.106747i	$-0.965957\pi$
$593$		−	5.19892i		−	0.213494i	0.994286	−	0.106747i	$-0.0340435\pi$
$594$		0			0
$595$	−11.7181			−0.480396
$596$		0			0
$597$	4.79692			0.196325
$598$		0			0
$599$	6.71281			0.274278			0.137139	−	0.990552i	$-0.456209\pi$
$599$	6.71281			0.274278			0.137139	+	0.990552i	$0.456209\pi$
$600$		0			0
$601$	14.2309			0.580491			0.290245	−	0.956952i	$-0.406263\pi$
$601$	14.2309			0.580491			0.290245	+	0.956952i	$0.406263\pi$
$602$		0			0
$603$			11.0811i			0.451257i
$604$		0			0
$605$		−	37.4381i		−	1.52207i
$606$		0			0
$607$	−2.35861			−0.0957330			−0.0478665	−	0.998854i	$-0.515242\pi$
$607$	−2.35861			−0.0957330			−0.0478665	+	0.998854i	$0.515242\pi$
$608$		0			0
$609$		−	7.08450i		−	0.287079i
$610$		0			0
$611$	−31.6087	+	32.0789i	−1.27875	+	1.29777i
$612$		0			0
$613$		−	7.35366i		−	0.297012i	−0.988912	−	0.148506i	$-0.952554\pi$
$613$		−	7.35366i		−	0.297012i	0.988912	−	0.148506i	$-0.0474464\pi$
$614$		0			0
$615$	−26.2555			−1.05872
$616$		0			0
$617$			22.4090i			0.902152i	0.892486	+	0.451076i	$0.148960\pi$
$617$			22.4090i			0.902152i	−0.892486	+	0.451076i	$0.851040\pi$
$618$		0			0
$619$		−	16.2416i		−	0.652806i	−0.945231	−	0.326403i	$-0.894163\pi$
$619$		−	16.2416i		−	0.652806i	0.945231	−	0.326403i	$-0.105837\pi$
$620$		0			0
$621$	−3.45720			−0.138733
$622$		0			0
$623$	2.19079			0.0877723
$624$		0			0
$625$	−31.2361			−1.24944
$626$		0			0
$627$	−18.6900			−0.746408
$628$		0			0
$629$		−	4.83478i		−	0.192775i
$630$		0			0
$631$		−	5.11232i		−	0.203518i	−0.994809	−	0.101759i	$-0.967553\pi$
$631$		−	5.11232i		−	0.203518i	0.994809	−	0.101759i	$-0.0324471\pi$
$632$		0			0
$633$	−0.300810			−0.0119561
$634$		0			0
$635$		−	17.9719i		−	0.713193i
$636$		0			0
$637$	−2.56826	−	2.53062i	−0.101758	−	0.100267i
$638$		0			0
$639$		−	10.0989i		−	0.399506i
$640$		0			0
$641$	43.4860			1.71759			0.858797	−	0.512316i	$-0.171212\pi$
$641$	43.4860			1.71759			0.858797	+	0.512316i	$0.171212\pi$
$642$		0			0
$643$			32.4481i			1.27963i	0.768531	+	0.639813i	$0.220988\pi$
$643$			32.4481i			1.27963i	−0.768531	+	0.639813i	$0.779012\pi$
$644$		0			0
$645$			29.7415i			1.17107i
$646$		0			0
$647$	−17.0369			−0.669789			−0.334894	−	0.942256i	$-0.608701\pi$
$647$	−17.0369			−0.669789			−0.334894	+	0.942256i	$0.608701\pi$
$648$		0			0
$649$	16.9763			0.666378
$650$		0			0
$651$	9.39744			0.368315
$652$		0			0
$653$	−23.5268			−0.920677			−0.460338	−	0.887744i	$-0.652272\pi$
$653$	−23.5268			−0.920677			−0.460338	+	0.887744i	$0.652272\pi$
$654$		0			0
$655$			59.7493i			2.33460i
$656$		0			0
$657$			6.05202i			0.236112i
$658$		0			0
$659$	24.0616			0.937308			0.468654	−	0.883382i	$-0.344739\pi$
$659$	24.0616			0.937308			0.468654	+	0.883382i	$0.344739\pi$
$660$		0			0
$661$			3.81964i			0.148567i	0.997237	+	0.0742833i	$0.0236669\pi$
$661$			3.81964i			0.148567i	−0.997237	+	0.0742833i	$0.976333\pi$
$662$		0			0
$663$	11.0768	+	10.9144i	0.430186	+	0.423880i
$664$		0			0
$665$			10.2012i			0.395584i
$666$		0			0
$667$	−24.4926			−0.948356
$668$		0			0
$669$			16.4785i			0.637097i
$670$		0			0
$671$			9.65830i			0.372855i
$672$		0			0
$673$	16.8690			0.650251			0.325125	−	0.945671i	$-0.394594\pi$
$673$	16.8690			0.650251			0.325125	+	0.945671i	$0.394594\pi$
$674$		0			0
$675$	2.38191			0.0916799
$676$		0			0
$677$	18.6840			0.718086			0.359043	−	0.933321i	$-0.383103\pi$
$677$	18.6840			0.718086			0.359043	+	0.933321i	$0.383103\pi$
$678$		0			0
$679$	11.8454			0.454586
$680$		0			0
$681$			22.1012i			0.846919i
$682$		0			0
$683$			3.49742i			0.133825i	0.997759	+	0.0669124i	$0.0213148\pi$
$683$			3.49742i			0.133825i	−0.997759	+	0.0669124i	$0.978685\pi$
$684$		0			0
$685$	13.9276			0.532146
$686$		0			0
$687$			5.22793i			0.199458i
$688$		0			0
$689$	−7.38531	−	7.27706i	−0.281358	−	0.277234i
$690$		0			0
$691$			5.85663i			0.222797i	0.993776	+	0.111398i	$0.0355330\pi$
$691$			5.85663i			0.222797i	−0.993776	+	0.111398i	$0.964467\pi$
$692$		0			0
$693$	−4.97789			−0.189094
$694$		0			0
$695$		−	55.5646i		−	2.10768i
$696$		0			0
$697$			41.6782i			1.57868i
$698$		0			0
$699$	−5.34795			−0.202278
$700$		0			0
$701$	0.830735			0.0313764			0.0156882	−	0.999877i	$-0.495006\pi$
$701$	0.830735			0.0313764			0.0156882	+	0.999877i	$0.495006\pi$
$702$		0			0
$703$	−4.20890			−0.158742
$704$		0			0
$705$	33.9363			1.27812
$706$		0			0
$707$		−	5.75801i		−	0.216552i
$708$		0			0
$709$		−	14.7012i		−	0.552114i	−0.961141	−	0.276057i	$-0.910972\pi$
$709$		−	14.7012i		−	0.552114i	0.961141	−	0.276057i	$-0.0890279\pi$
$710$		0			0
$711$	8.12298			0.304635
$712$		0			0
$713$		−	32.4889i		−	1.21672i
$714$		0			0
$715$	34.7351	+	34.2260i	1.29902	+	1.27998i
$716$		0			0
$717$			27.1358i			1.01340i
$718$		0			0
$719$	−3.88728			−0.144971			−0.0724856	−	0.997369i	$-0.523093\pi$
$719$	−3.88728			−0.144971			−0.0724856	+	0.997369i	$0.523093\pi$
$720$		0			0
$721$			10.5374i			0.392432i
$722$		0			0
$723$			9.02475i			0.335634i
$724$		0			0
$725$	16.8747			0.626710
$726$		0			0
$727$	42.3101			1.56920			0.784598	−	0.620005i	$-0.212870\pi$
$727$	42.3101			1.56920			0.784598	+	0.620005i	$0.212870\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	47.2119			1.74619
$732$		0			0
$733$		−	8.53397i		−	0.315209i	−0.987502	−	0.157605i	$-0.949623\pi$
$733$		−	8.53397i		−	0.315209i	0.987502	−	0.157605i	$-0.0503772\pi$
$734$		0			0
$735$			2.71697i			0.100217i
$736$		0			0
$737$	55.1605			2.03186
$738$		0			0
$739$		−	8.93437i		−	0.328656i	−0.986406	−	0.164328i	$-0.947454\pi$
$739$		−	8.93437i		−	0.328656i	0.986406	−	0.164328i	$-0.0525456\pi$
$740$		0			0
$741$	9.50149	−	9.64283i	0.349046	−	0.354238i
$742$		0			0
$743$			24.0954i			0.883975i	0.897021	+	0.441987i	$0.145726\pi$
$743$			24.0954i			0.883975i	−0.897021	+	0.441987i	$0.854274\pi$
$744$		0			0
$745$	48.0452			1.76024
$746$		0			0
$747$		−	17.6004i		−	0.643964i
$748$		0			0
$749$			15.5204i			0.567104i
$750$		0			0
$751$	−35.7813			−1.30568			−0.652840	−	0.757496i	$-0.726423\pi$
$751$	−35.7813			−1.30568			−0.652840	+	0.757496i	$0.726423\pi$
$752$		0			0
$753$	−14.2061			−0.517698
$754$		0			0
$755$	30.5043			1.11017
$756$		0			0
$757$	48.2439			1.75345			0.876727	−	0.480988i	$-0.159722\pi$
$757$	48.2439			1.75345			0.876727	+	0.480988i	$0.159722\pi$
$758$		0			0
$759$			17.2096i			0.624668i
$760$		0			0
$761$			25.4535i			0.922690i	0.887221	+	0.461345i	$0.152633\pi$
$761$			25.4535i			0.922690i	−0.887221	+	0.461345i	$0.847367\pi$
$762$		0			0
$763$	5.64283			0.204284
$764$		0			0
$765$		−	11.7181i		−	0.423670i
$766$		0			0
$767$	−8.63028	+	8.75866i	−0.311621	+	0.316257i
$768$		0			0
$769$			50.6603i			1.82686i	0.407001	+	0.913428i	$0.366575\pi$
$769$			50.6603i			1.82686i	−0.407001	+	0.913428i	$0.633425\pi$
$770$		0			0
$771$	14.1535			0.509727
$772$		0			0
$773$			34.2783i			1.23290i	0.787393	+	0.616452i	$0.211431\pi$
$773$			34.2783i			1.23290i	−0.787393	+	0.616452i	$0.788569\pi$
$774$		0			0
$775$			22.3839i			0.804053i
$776$		0			0
$777$	−1.12099			−0.0402154
$778$		0			0
$779$	36.2828			1.29997
$780$		0			0
$781$	−50.2711			−1.79884
$782$		0			0
$783$	7.08450			0.253179
$784$		0			0
$785$		−	59.2901i		−	2.11616i
$786$		0			0
$787$		−	33.1570i		−	1.18192i	−0.806702	−	0.590959i	$-0.798750\pi$
$787$		−	33.1570i		−	1.18192i	0.806702	−	0.590959i	$-0.201250\pi$
$788$		0			0
$789$	7.50872			0.267318
$790$		0			0
$791$			18.8601i			0.670589i
$792$		0			0
$793$	−4.98305	−	4.91001i	−0.176953	−	0.174360i
$794$		0			0
$795$			7.81293i			0.277096i
$796$		0			0
$797$	−14.6179			−0.517791			−0.258896	−	0.965905i	$-0.583359\pi$
$797$	−14.6179			−0.517791			−0.258896	+	0.965905i	$0.583359\pi$
$798$		0			0
$799$		−	53.8708i		−	1.90581i
$800$		0			0
$801$			2.19079i			0.0774079i
$802$		0			0
$803$	30.1263			1.06313
$804$		0			0
$805$	9.39311			0.331064
$806$		0			0
$807$	10.6443			0.374697
$808$		0			0
$809$	−11.9290			−0.419400			−0.209700	−	0.977766i	$-0.567249\pi$
$809$	−11.9290			−0.419400			−0.209700	+	0.977766i	$0.567249\pi$
$810$		0			0
$811$		−	14.9573i		−	0.525220i	−0.964902	−	0.262610i	$-0.915417\pi$
$811$		−	14.9573i		−	0.525220i	0.964902	−	0.262610i	$-0.0845833\pi$
$812$		0			0
$813$			17.0540i			0.598110i
$814$		0			0
$815$	17.9242			0.627857
$816$		0			0
$817$		−	41.1001i		−	1.43791i
$818$		0			0
$819$	2.53062	−	2.56826i	0.0884269	−	0.0897424i
$820$		0			0
$821$		−	17.5260i		−	0.611662i	−0.952086	−	0.305831i	$-0.901066\pi$
$821$		−	17.5260i		−	0.611662i	0.952086	−	0.305831i	$-0.0989343\pi$
$822$		0			0
$823$	27.9115			0.972935			0.486468	−	0.873699i	$-0.338285\pi$
$823$	27.9115			0.972935			0.486468	+	0.873699i	$0.338285\pi$
$824$		0			0
$825$		−	11.8569i		−	0.412804i
$826$		0			0
$827$			36.3515i			1.26406i	0.774942	+	0.632032i	$0.217779\pi$
$827$			36.3515i			1.26406i	−0.774942	+	0.632032i	$0.782221\pi$
$828$		0			0
$829$	−22.2570			−0.773017			−0.386509	−	0.922286i	$-0.626319\pi$
$829$	−22.2570			−0.773017			−0.386509	+	0.922286i	$0.626319\pi$
$830$		0			0
$831$	−3.88822			−0.134881
$832$		0			0
$833$	4.31294			0.149435
$834$		0			0
$835$	55.3406			1.91514
$836$		0			0
$837$			9.39744i			0.324823i
$838$		0			0
$839$			1.01778i			0.0351376i	0.999846	+	0.0175688i	$0.00559261\pi$
$839$			1.01778i			0.0351376i	−0.999846	+	0.0175688i	$0.994407\pi$
$840$		0			0
$841$	21.1902			0.730696
$842$		0			0
$843$			4.12994i			0.142243i
$844$		0			0
$845$	−35.3167	+	0.521516i	−1.21493	+	0.0179407i
$846$		0			0
$847$			13.7794i			0.473464i
$848$		0			0
$849$	−10.8989			−0.374050
$850$		0			0
$851$			3.87550i			0.132851i
$852$		0			0
$853$			32.3973i			1.10926i	0.832096	+	0.554632i	$0.187141\pi$
$853$			32.3973i			1.10926i	−0.832096	+	0.554632i	$0.812859\pi$
$854$		0			0
$855$	−10.2012			−0.348872
$856$		0			0
$857$	15.1215			0.516540			0.258270	−	0.966073i	$-0.416848\pi$
$857$	15.1215			0.516540			0.258270	+	0.966073i	$0.416848\pi$
$858$		0			0
$859$	0.863905			0.0294760			0.0147380	−	0.999891i	$-0.495309\pi$
$859$	0.863905			0.0294760			0.0147380	+	0.999891i	$0.495309\pi$
$860$		0			0
$861$	9.66353			0.329332
$862$		0			0
$863$		−	20.7603i		−	0.706689i	−0.935493	−	0.353345i	$-0.885044\pi$
$863$		−	20.7603i		−	0.706689i	0.935493	−	0.353345i	$-0.114956\pi$
$864$		0			0
$865$			56.4490i			1.91932i
$866$		0			0
$867$	−1.60146			−0.0543886
$868$		0			0
$869$		−	40.4353i		−	1.37167i
$870$		0			0
$871$	−28.0420	+	28.4592i	−0.950168	+	0.964303i
$872$		0			0
$873$			11.8454i			0.400907i
$874$		0			0
$875$	7.11326			0.240472
$876$		0			0
$877$		−	33.2983i		−	1.12440i	−0.827000	−	0.562202i	$-0.809954\pi$
$877$		−	33.2983i		−	1.12440i	0.827000	−	0.562202i	$-0.190046\pi$
$878$		0			0
$879$		−	24.0736i		−	0.811982i
$880$		0			0
$881$	43.0659			1.45093			0.725464	−	0.688260i	$-0.241625\pi$
$881$	43.0659			1.45093			0.725464	+	0.688260i	$0.241625\pi$
$882$		0			0
$883$	52.3687			1.76235			0.881174	−	0.472793i	$-0.156754\pi$
$883$	52.3687			1.76235			0.881174	+	0.472793i	$0.156754\pi$
$884$		0			0
$885$	9.26580			0.311466
$886$		0			0
$887$	28.7610			0.965700			0.482850	−	0.875703i	$-0.339602\pi$
$887$	28.7610			0.965700			0.482850	+	0.875703i	$0.339602\pi$
$888$		0			0
$889$			6.61469i			0.221849i
$890$		0			0
$891$		−	4.97789i		−	0.166765i
$892$		0			0
$893$	−46.8970			−1.56935
$894$		0			0
$895$		−	46.9132i		−	1.56813i
$896$		0			0
$897$	−8.87901	−	8.74886i	−0.296461	−	0.292116i
$898$		0			0
$899$			66.5762i			2.22044i
$900$		0			0
$901$	12.4023			0.413181
$902$		0			0
$903$		−	10.9466i		−	0.364279i
$904$		0			0
$905$			48.5622i			1.61426i
$906$		0			0
$907$	−6.53782			−0.217085			−0.108542	−	0.994092i	$-0.534618\pi$
$907$	−6.53782			−0.217085			−0.108542	+	0.994092i	$0.534618\pi$
$908$		0			0
$909$	5.75801			0.190981
$910$		0			0
$911$	−27.1394			−0.899168			−0.449584	−	0.893238i	$-0.648428\pi$
$911$	−27.1394			−0.899168			−0.449584	+	0.893238i	$0.648428\pi$
$912$		0			0
$913$	−87.6127			−2.89956
$914$		0			0
$915$			5.27157i			0.174273i
$916$		0			0
$917$		−	21.9912i		−	0.726213i
$918$		0			0
$919$	−33.0830			−1.09131			−0.545653	−	0.838011i	$-0.683718\pi$
$919$	−33.0830			−1.09131			−0.545653	+	0.838011i	$0.683718\pi$
$920$		0			0
$921$		−	19.2595i		−	0.634622i
$922$		0			0
$923$	25.5564	−	25.9366i	0.841199	−	0.853713i
$924$		0			0
$925$		−	2.67011i		−	0.0877927i
$926$		0			0
$927$	−10.5374			−0.346093
$928$		0			0
$929$			4.92137i			0.161465i	0.996736	+	0.0807325i	$0.0257260\pi$
$929$			4.92137i			0.161465i	−0.996736	+	0.0807325i	$0.974274\pi$
$930$		0			0
$931$		−	3.75461i		−	0.123052i
$932$		0			0
$933$	18.6443			0.610387
$934$		0			0
$935$	−58.3315			−1.90764
$936$		0			0
$937$	4.01018			0.131007			0.0655035	−	0.997852i	$-0.479135\pi$
$937$	4.01018			0.131007			0.0655035	+	0.997852i	$0.479135\pi$
$938$		0			0
$939$	−1.28711			−0.0420031
$940$		0			0
$941$			12.4173i			0.404793i	0.979304	+	0.202396i	$0.0648729\pi$
$941$			12.4173i			0.404793i	−0.979304	+	0.202396i	$0.935127\pi$
$942$		0			0
$943$		−	33.4088i		−	1.08794i
$944$		0			0
$945$	−2.71697			−0.0883830
$946$		0			0
$947$			45.1276i			1.46645i	0.679987	+	0.733225i	$0.261986\pi$
$947$			45.1276i			1.46645i	−0.679987	+	0.733225i	$0.738014\pi$
$948$		0			0
$949$	−15.3154	+	15.5432i	−0.497157	+	0.504553i
$950$		0			0
$951$			6.90924i			0.224048i
$952$		0			0
$953$	−1.68597			−0.0546139			−0.0273069	−	0.999627i	$-0.508693\pi$
$953$	−1.68597			−0.0546139			−0.0273069	+	0.999627i	$0.508693\pi$
$954$		0			0
$955$		−	50.1424i		−	1.62257i
$956$		0			0
$957$		−	35.2659i		−	1.13998i
$958$		0			0
$959$	−5.12616			−0.165532
$960$		0			0
$961$	−57.3120			−1.84877
$962$		0			0
$963$	−15.5204			−0.500139
$964$		0			0
$965$	45.8357			1.47550
$966$		0			0
$967$			26.2531i			0.844244i	0.906539	+	0.422122i	$0.138715\pi$
$967$			26.2531i			0.844244i	−0.906539	+	0.422122i	$0.861285\pi$
$968$		0			0
$969$			16.1934i			0.520208i
$970$		0			0
$971$	25.5898			0.821215			0.410608	−	0.911812i	$-0.365317\pi$
$971$	25.5898			0.821215			0.410608	+	0.911812i	$0.365317\pi$
$972$		0			0
$973$			20.4509i			0.655627i
$974$		0			0
$975$	6.11738	+	6.02771i	0.195913	+	0.193041i
$976$		0			0
$977$			48.2850i			1.54477i	0.635153	+	0.772387i	$0.280937\pi$
$977$			48.2850i			1.54477i	−0.635153	+	0.772387i	$0.719063\pi$
$978$		0			0
$979$	10.9055			0.348542
$980$		0			0
$981$			5.64283i			0.180162i
$982$		0			0
$983$			21.2229i			0.676906i	0.940983	+	0.338453i	$0.109904\pi$
$983$			21.2229i			0.676906i	−0.940983	+	0.338453i	$0.890096\pi$
$984$		0			0
$985$	9.39398			0.299317
$986$		0			0
$987$	−12.4905			−0.397577
$988$		0			0
$989$	−37.8445			−1.20338
$990$		0			0
$991$	1.46552			0.0465539			0.0232769	−	0.999729i	$-0.492590\pi$
$991$	1.46552			0.0465539			0.0232769	+	0.999729i	$0.492590\pi$
$992$		0			0
$993$		−	16.3955i		−	0.520296i
$994$		0			0
$995$		−	13.0331i		−	0.413176i
$996$		0			0
$997$	60.3151			1.91020			0.955099	−	0.296287i	$-0.0957484\pi$
$997$	60.3151			1.91020			0.955099	+	0.296287i	$0.0957484\pi$
$998$		0			0
$999$		−	1.12099i		−	0.0354667i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.r.337.9		10
4.3	odd	2		2184.2.h.f.337.9	yes	10
13.12	even	2	inner	4368.2.h.r.337.2		10
52.51	odd	2		2184.2.h.f.337.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.h.f.337.2	✓	10	52.51	odd	2
2184.2.h.f.337.9	yes	10	4.3	odd	2
4368.2.h.r.337.2		10	13.12	even	2	inner
4368.2.h.r.337.9		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.71697i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.71697i q^{5} -1.00000i q^{7} +1.00000 q^{9} -4.97789i q^{11} +(2.56826 + 2.53062i) q^{13} -2.71697i q^{15} -4.31294 q^{17} +3.75461i q^{19} +1.00000i q^{21} +3.45720 q^{23} -2.38191 q^{25} -1.00000 q^{27} -7.08450 q^{29} -9.39744i q^{31} +4.97789i q^{33} +2.71697 q^{35} +1.12099i q^{37} +(-2.56826 - 2.53062i) q^{39} -9.66353i q^{41} -10.9466 q^{43} +2.71697i q^{45} +12.4905i q^{47} -1.00000 q^{49} +4.31294 q^{51} -2.87561 q^{53} +13.5248 q^{55} -3.75461i q^{57} +3.41034i q^{59} -1.94024 q^{61} -1.00000i q^{63} +(-6.87561 + 6.97789i) q^{65} +11.0811i q^{67} -3.45720 q^{69} -10.0989i q^{71} +6.05202i q^{73} +2.38191 q^{75} -4.97789 q^{77} +8.12298 q^{79} +1.00000 q^{81} -17.6004i q^{83} -11.7181i q^{85} +7.08450 q^{87} +2.19079i q^{89} +(2.53062 - 2.56826i) q^{91} +9.39744i q^{93} -10.2012 q^{95} +11.8454i q^{97} -4.97789i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 10 * q^9	\( 10 q - 10 q^{3} + 10 q^{9} - 4 q^{13} + 10 q^{17} - 8 q^{23} - 2 q^{25} - 10 q^{27} - 44 q^{29} + 4 q^{39} - 20 q^{43} - 10 q^{49} - 10 q^{51} + 10 q^{53} + 2 q^{55} + 18 q^{61} - 30 q^{65} + 8 q^{69} + 2 q^{75} - 2 q^{77} - 2 q^{79} + 10 q^{81} + 44 q^{87} + 6 q^{91} - 44 q^{95}+O(q^{100}) \) 10 * q - 10 * q^3 + 10 * q^9 - 4 * q^13 + 10 * q^17 - 8 * q^23 - 2 * q^25 - 10 * q^27 - 44 * q^29 + 4 * q^39 - 20 * q^43 - 10 * q^49 - 10 * q^51 + 10 * q^53 + 2 * q^55 + 18 * q^61 - 30 * q^65 + 8 * q^69 + 2 * q^75 - 2 * q^77 - 2 * q^79 + 10 * q^81 + 44 * q^87 + 6 * q^91 - 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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