LMFDB - Embedded newform 2340.2.bi.c.161.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(161,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 0, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.161"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.bi (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 10 x^{10} - 16 x^{9} + 50 x^{8} - 32 x^{7} - 110 x^{6} + 40 x^{5} + 417 x^{4} + 712 x^{3} + \cdots + 32 $ x^12 + 10x^10 - 16x^9 + 50x^8 - 32x^7 - 110x^6 + 40x^5 + 417x^4 + 712x^3 + 600x^2 + 224x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			161.6
Root			$0.981031 + 2.36842i$ of defining polynomial
Character	$\chi$	$=$	2340.161
Dual form			2340.2.bi.c.1061.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +(2.50560 + 2.50560i) q^{7} +(1.31223 - 1.31223i) q^{11} +(1.64983 + 3.20594i) q^{13} +3.11967 q^{17} +3.74743 q^{23} +1.00000i q^{25} -4.03867i q^{29} +(-3.29966 + 3.29966i) q^{31} +3.54346i q^{35} +(-2.94950 - 2.94950i) q^{37} +(-0.525773 - 0.525773i) q^{41} +0.288455i q^{43} +(1.90943 - 1.90943i) q^{47} +5.55611i q^{49} +11.4621i q^{53} +1.85577 q^{55} +(5.24892 - 5.24892i) q^{59} -4.04322 q^{61} +(-1.10033 + 3.43355i) q^{65} +(-3.01121 + 3.01121i) q^{67} +(-1.10826 - 1.10826i) q^{71} +(5.26765 + 5.26765i) q^{73} +6.57586 q^{77} -5.75476 q^{79} +(2.55304 + 2.55304i) q^{83} +(2.20594 + 2.20594i) q^{85} +(1.09241 - 1.09241i) q^{89} +(-3.89899 + 12.1666i) q^{91} +(4.66104 - 4.66104i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{7} + 12 q^{13} - 24 q^{31} - 12 q^{37} - 8 q^{55} - 32 q^{61} + 40 q^{67} - 12 q^{73} + 8 q^{79} + 4 q^{85} - 28 q^{97}+O(q^{100}) $ 12 * q - 8 * q^7 + 12 * q^13 - 24 * q^31 - 12 * q^37 - 8 * q^55 - 32 * q^61 + 40 * q^67 - 12 * q^73 + 8 * q^79 + 4 * q^85 - 28 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$	2.50560	+	2.50560i	0.947030	+	0.947030i	0.998666	−	0.0516364i	$-0.0164437\pi$
$7$	2.50560	+	2.50560i	0.947030	+	0.947030i	−0.0516364	+	0.998666i	$0.516444\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.31223	−	1.31223i	0.395652	−	0.395652i	−0.481044	−	0.876696i	$-0.659742\pi$
$11$	1.31223	−	1.31223i	0.395652	−	0.395652i	0.876696	+	0.481044i	$0.159742\pi$
$12$		0			0
$13$	1.64983	+	3.20594i	0.457581	+	0.889168i
$13$	1.64983	+	3.20594i	0.457581	+	0.889168i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.11967			0.756631			0.378316	−	0.925677i	$-0.376503\pi$
$17$	3.11967			0.756631			0.378316	+	0.925677i	$0.376503\pi$
$18$		0			0
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.74743			0.781393			0.390696	−	0.920520i	$-0.372234\pi$
$23$	3.74743			0.781393			0.390696	+	0.920520i	$0.372234\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.03867i		−	0.749963i	−0.927032	−	0.374981i	$-0.877649\pi$
$29$		−	4.03867i		−	0.749963i	0.927032	−	0.374981i	$-0.122351\pi$
$30$		0			0
$31$	−3.29966	+	3.29966i	−0.592637	+	0.592637i	−0.938343	−	0.345706i	$-0.887640\pi$
$31$	−3.29966	+	3.29966i	−0.592637	+	0.592637i	0.345706	+	0.938343i	$0.387640\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			3.54346i			0.598954i
$36$		0			0
$37$	−2.94950	−	2.94950i	−0.484894	−	0.484894i	0.421796	−	0.906691i	$-0.361400\pi$
$37$	−2.94950	−	2.94950i	−0.484894	−	0.484894i	−0.906691	+	0.421796i	$0.861400\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.525773	−	0.525773i	−0.0821120	−	0.0821120i	0.664858	−	0.746970i	$-0.268492\pi$
$41$	−0.525773	−	0.525773i	−0.0821120	−	0.0821120i	−0.746970	+	0.664858i	$0.768492\pi$
$42$		0			0
$43$			0.288455i			0.0439890i	0.999758	+	0.0219945i	$0.00700163\pi$
$43$			0.288455i			0.0439890i	−0.999758	+	0.0219945i	$0.992998\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.90943	−	1.90943i	0.278518	−	0.278518i	−0.553999	−	0.832517i	$-0.686899\pi$
$47$	1.90943	−	1.90943i	0.278518	−	0.278518i	0.832517	+	0.553999i	$0.186899\pi$
$48$		0			0
$49$			5.55611i			0.793730i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			11.4621i			1.57444i	0.616671	+	0.787221i	$0.288481\pi$
$53$			11.4621i			1.57444i	−0.616671	+	0.787221i	$0.711519\pi$
$54$		0			0
$55$	1.85577			0.250232
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.24892	−	5.24892i	0.683351	−	0.683351i	−0.277403	−	0.960754i	$-0.589474\pi$
$59$	5.24892	−	5.24892i	0.683351	−	0.683351i	0.960754	+	0.277403i	$0.0894737\pi$
$60$		0			0
$61$	−4.04322			−0.517681			−0.258841	−	0.965920i	$-0.583340\pi$
$61$	−4.04322			−0.517681			−0.258841	+	0.965920i	$0.583340\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.10033	+	3.43355i	−0.136480	+	0.425879i
$66$		0			0
$67$	−3.01121	+	3.01121i	−0.367878	+	0.367878i	−0.866703	−	0.498825i	$-0.833765\pi$
$67$	−3.01121	+	3.01121i	−0.367878	+	0.367878i	0.498825	+	0.866703i	$0.333765\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.10826	−	1.10826i	−0.131526	−	0.131526i	0.638279	−	0.769805i	$-0.279647\pi$
$71$	−1.10826	−	1.10826i	−0.131526	−	0.131526i	−0.769805	+	0.638279i	$0.779647\pi$
$72$		0			0
$73$	5.26765	+	5.26765i	0.616532	+	0.616532i	0.944640	−	0.328108i	$-0.106411\pi$
$73$	5.26765	+	5.26765i	0.616532	+	0.616532i	−0.328108	+	0.944640i	$0.606411\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.57586			0.749388
$78$		0			0
$79$	−5.75476			−0.647462			−0.323731	−	0.946149i	$-0.604937\pi$
$79$	−5.75476			−0.647462			−0.323731	+	0.946149i	$0.604937\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.55304	+	2.55304i	0.280232	+	0.280232i	0.833202	−	0.552969i	$-0.186505\pi$
$83$	2.55304	+	2.55304i	0.280232	+	0.280232i	−0.552969	+	0.833202i	$0.686505\pi$
$84$		0			0
$85$	2.20594	+	2.20594i	0.239268	+	0.239268i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.09241	−	1.09241i	0.115795	−	0.115795i	−0.646835	−	0.762630i	$-0.723908\pi$
$89$	1.09241	−	1.09241i	0.115795	−	0.115795i	0.762630	+	0.646835i	$0.223908\pi$
$90$		0			0
$91$	−3.89899	+	12.1666i	−0.408725	+	1.27541i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.66104	−	4.66104i	0.473257	−	0.473257i	−0.429710	−	0.902967i	$-0.641384\pi$
$97$	4.66104	−	4.66104i	0.473257	−	0.473257i	0.902967	+	0.429710i	$0.141384\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.91865			−0.787935			−0.393967	−	0.919124i	$-0.628898\pi$
$101$	−7.91865			−0.787935			−0.393967	+	0.919124i	$0.628898\pi$
$102$		0			0
$103$		−	0.187447i		−	0.0184697i	−0.999957	−	0.00923487i	$-0.997060\pi$
$103$		−	0.187447i		−	0.0184697i	0.999957	−	0.00923487i	$-0.00293959\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	13.3977i		−	1.29520i	−0.761979	−	0.647602i	$-0.775772\pi$
$107$		−	13.3977i		−	1.29520i	0.761979	−	0.647602i	$-0.224228\pi$
$108$		0			0
$109$	−6.56732	+	6.56732i	−0.629035	+	0.629035i	−0.947825	−	0.318790i	$-0.896723\pi$
$109$	−6.56732	+	6.56732i	−0.629035	+	0.629035i	0.318790	+	0.947825i	$0.396723\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	3.04825i		−	0.286755i	−0.989668	−	0.143378i	$-0.954204\pi$
$113$		−	3.04825i		−	0.286755i	0.989668	−	0.143378i	$-0.0457963\pi$
$114$		0			0
$115$	2.64983	+	2.64983i	0.247098	+	0.247098i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.81666	+	7.81666i	0.716552	+	0.716552i
$120$		0			0
$121$			7.55611i			0.686919i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$			13.2997i			1.18015i	0.807347	+	0.590077i	$0.200903\pi$
$127$			13.2997i			1.18015i	−0.807347	+	0.590077i	$0.799097\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	1.25552i		−	0.109695i	−0.998495	−	0.0548474i	$-0.982533\pi$
$131$		−	1.25552i		−	0.109695i	0.998495	−	0.0548474i	$-0.0174672\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.79238	−	8.79238i	0.751183	−	0.751183i	−0.223517	−	0.974700i	$-0.571754\pi$
$137$	8.79238	−	8.79238i	0.751183	−	0.751183i	0.974700	+	0.223517i	$0.0717537\pi$
$138$		0			0
$139$	14.2901			1.21207			0.606034	−	0.795439i	$-0.292760\pi$
$139$	14.2901			1.21207			0.606034	+	0.795439i	$0.292760\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.37189	+	2.04197i	0.532844	+	0.170758i
$144$		0			0
$145$	2.85577	−	2.85577i	0.237159	−	0.237159i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.525773	−	0.525773i	−0.0430730	−	0.0430730i	0.685242	−	0.728315i	$-0.259696\pi$
$149$	−0.525773	−	0.525773i	−0.0430730	−	0.0430730i	−0.728315	+	0.685242i	$0.759696\pi$
$150$		0			0
$151$	11.8558	+	11.8558i	0.964809	+	0.964809i	0.999402	−	0.0345924i	$-0.0110133\pi$
$151$	11.8558	+	11.8558i	0.964809	+	0.964809i	−0.0345924	+	0.999402i	$0.511013\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.66643			−0.374817
$156$		0			0
$157$	−22.0224			−1.75758			−0.878790	−	0.477208i	$-0.841649\pi$
$157$	−22.0224			−1.75758			−0.878790	+	0.477208i	$0.841649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.38957	+	9.38957i	0.740002	+	0.740002i
$162$		0			0
$163$	5.51681	+	5.51681i	0.432110	+	0.432110i	0.889346	−	0.457235i	$-0.151160\pi$
$163$	5.51681	+	5.51681i	0.432110	+	0.432110i	−0.457235	+	0.889346i	$0.651160\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.73785	−	4.73785i	0.366626	−	0.366626i	−0.499619	−	0.866245i	$-0.666527\pi$
$167$	4.73785	−	4.73785i	0.366626	−	0.366626i	0.866245	+	0.499619i	$0.166527\pi$
$168$		0			0
$169$	−7.55611	+	10.5785i	−0.581239	+	0.813733i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.1352			−0.770563			−0.385281	−	0.922799i	$-0.625896\pi$
$173$	−10.1352			−0.770563			−0.385281	+	0.922799i	$0.625896\pi$
$174$		0			0
$175$	−2.50560	+	2.50560i	−0.189406	+	0.189406i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	16.7666			1.25319			0.626597	−	0.779343i	$-0.284447\pi$
$179$	16.7666			1.25319			0.626597	+	0.779343i	$0.284447\pi$
$180$		0			0
$181$		−	2.86698i		−	0.213101i	−0.994307	−	0.106551i	$-0.966019\pi$
$181$		−	2.86698i		−	0.213101i	0.994307	−	0.106551i	$-0.0339806\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	4.17122i		−	0.306674i
$186$		0			0
$187$	4.09372	−	4.09372i	0.299363	−	0.299363i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			9.85422i			0.713027i	0.934290	+	0.356513i	$0.116035\pi$
$191$			9.85422i			0.713027i	−0.934290	+	0.356513i	$0.883965\pi$
$192$		0			0
$193$	1.64983	+	1.64983i	0.118758	+	0.118758i	0.763988	−	0.645230i	$-0.223239\pi$
$193$	1.64983	+	1.64983i	0.118758	+	0.118758i	−0.645230	+	0.763988i	$0.723239\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.17122	+	4.17122i	0.297187	+	0.297187i	0.839911	−	0.542724i	$-0.182607\pi$
$197$	4.17122	+	4.17122i	0.297187	+	0.297187i	−0.542724	+	0.839911i	$0.682607\pi$
$198$		0			0
$199$		−	16.3333i		−	1.15784i	−0.815386	−	0.578918i	$-0.803475\pi$
$199$		−	16.3333i		−	1.15784i	0.815386	−	0.578918i	$-0.196525\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.1193	−	10.1193i	0.710237	−	0.710237i
$204$		0			0
$205$		−	0.743556i		−	0.0519322i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	23.1346			1.59265			0.796327	−	0.604866i	$-0.206773\pi$
$211$	23.1346			1.59265			0.796327	+	0.604866i	$0.206773\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.203968	+	0.203968i	−0.0139105	+	0.0139105i
$216$		0			0
$217$	−16.5353			−1.12249
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.14693	+	10.0015i	0.346220	+	0.672772i
$222$		0			0
$223$	−3.61054	+	3.61054i	−0.241779	+	0.241779i	−0.817586	−	0.575807i	$-0.804688\pi$
$223$	−3.61054	+	3.61054i	−0.241779	+	0.241779i	0.575807	+	0.817586i	$0.304688\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.23934	−	6.23934i	−0.414120	−	0.414120i	0.469051	−	0.883171i	$-0.344596\pi$
$227$	−6.23934	−	6.23934i	−0.414120	−	0.414120i	−0.883171	+	0.469051i	$0.844596\pi$
$228$		0			0
$229$	6.29966	+	6.29966i	0.416294	+	0.416294i	0.883924	−	0.467631i	$-0.154892\pi$
$229$	6.29966	+	6.29966i	0.416294	+	0.416294i	−0.467631	+	0.883924i	$0.654892\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	17.4364			1.14229			0.571147	−	0.820848i	$-0.306499\pi$
$233$	17.4364			1.14229			0.571147	+	0.820848i	$0.306499\pi$
$234$		0			0
$235$	2.70034			0.176150
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−14.0560	−	14.0560i	−0.909207	−	0.909207i	0.0870009	−	0.996208i	$-0.472272\pi$
$239$	−14.0560	−	14.0560i	−0.909207	−	0.909207i	−0.996208	+	0.0870009i	$0.972272\pi$
$240$		0			0
$241$	6.87819	+	6.87819i	0.443063	+	0.443063i	0.893040	−	0.449977i	$-0.148568\pi$
$241$	6.87819	+	6.87819i	0.443063	+	0.443063i	−0.449977	+	0.893040i	$0.648568\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.92876	+	3.92876i	−0.250999	+	0.250999i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.8727			−1.38059			−0.690295	−	0.723528i	$-0.742519\pi$
$251$	−21.8727			−1.38059			−0.690295	+	0.723528i	$0.742519\pi$
$252$		0			0
$253$	4.91749	−	4.91749i	0.309160	−	0.309160i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	16.2000			1.01053			0.505263	−	0.862966i	$-0.331396\pi$
$257$	16.2000			1.01053			0.505263	+	0.862966i	$0.331396\pi$
$258$		0			0
$259$		−	14.7805i		−	0.918418i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	17.8181i		−	1.09871i	−0.835588	−	0.549357i	$-0.814873\pi$
$263$		−	17.8181i		−	1.09871i	0.835588	−	0.549357i	$-0.185127\pi$
$264$		0			0
$265$	−8.10493	+	8.10493i	−0.497882	+	0.497882i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			7.77580i			0.474099i	0.971498	+	0.237049i	$0.0761803\pi$
$269$			7.77580i			0.474099i	−0.971498	+	0.237049i	$0.923820\pi$
$270$		0			0
$271$	4.96799	+	4.96799i	0.301784	+	0.301784i	0.841711	−	0.539928i	$-0.181548\pi$
$271$	4.96799	+	4.96799i	0.301784	+	0.301784i	−0.539928	+	0.841711i	$0.681548\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.31223	+	1.31223i	0.0791304	+	0.0791304i
$276$		0			0
$277$		−	5.89899i		−	0.354436i	−0.984172	−	0.177218i	$-0.943290\pi$
$277$		−	5.89899i		−	0.354436i	0.984172	−	0.177218i	$-0.0567098\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.3587	+	16.3587i	−0.975876	+	0.975876i	−0.999716	−	0.0238401i	$-0.992411\pi$
$281$	−16.3587	+	16.3587i	−0.975876	+	0.975876i	0.0238401	+	0.999716i	$0.492411\pi$
$282$		0			0
$283$		−	16.1458i		−	0.959771i	−0.877331	−	0.479885i	$-0.840678\pi$
$283$		−	16.1458i		−	0.959771i	0.877331	−	0.479885i	$-0.159322\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	2.63476i		−	0.155525i
$288$		0			0
$289$	−7.26765			−0.427509
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.7762	+	15.7762i	−0.921654	+	0.921654i	−0.997146	−	0.0754927i	$-0.975947\pi$
$293$	−15.7762	+	15.7762i	−0.921654	+	0.921654i	0.0754927	+	0.997146i	$0.475947\pi$
$294$		0			0
$295$	7.42309			0.432189
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.18263	+	12.0140i	0.357551	+	0.694789i
$300$		0			0
$301$	−0.722754	+	0.722754i	−0.0416589	+	0.0416589i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.85899	−	2.85899i	−0.163705	−	0.163705i
$306$		0			0
$307$	−10.9175	−	10.9175i	−0.623094	−	0.623094i	0.323227	−	0.946321i	$-0.395232\pi$
$307$	−10.9175	−	10.9175i	−0.623094	−	0.623094i	−0.946321	+	0.323227i	$0.895232\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.47571			−0.537318			−0.268659	−	0.963235i	$-0.586580\pi$
$311$	−9.47571			−0.537318			−0.268659	+	0.963235i	$0.586580\pi$
$312$		0			0
$313$	−26.2693			−1.48483			−0.742413	−	0.669942i	$-0.766319\pi$
$313$	−26.2693			−1.48483			−0.742413	+	0.669942i	$0.766319\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.67271	+	5.67271i	0.318611	+	0.318611i	0.848233	−	0.529623i	$-0.177666\pi$
$317$	5.67271	+	5.67271i	0.318611	+	0.318611i	−0.529623	+	0.848233i	$0.677666\pi$
$318$		0			0
$319$	−5.29966	−	5.29966i	−0.296724	−	0.296724i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−3.20594	+	1.64983i	−0.177834	+	0.0915162i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.56853			0.527530
$330$		0			0
$331$	23.1346	−	23.1346i	1.27159	−	1.27159i	0.326343	−	0.945251i	$-0.394183\pi$
$331$	23.1346	−	23.1346i	1.27159	−	1.27159i	0.945251	−	0.326343i	$-0.105817\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.25849			−0.232666
$336$		0			0
$337$		−	18.3109i		−	0.997457i	−0.866758	−	0.498728i	$-0.833801\pi$
$337$		−	18.3109i		−	0.997457i	0.866758	−	0.498728i	$-0.166199\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			8.65983i			0.468956i
$342$		0			0
$343$	3.61782	−	3.61782i	0.195344	−	0.195344i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	7.74083i		−	0.415550i	−0.978177	−	0.207775i	$-0.933378\pi$
$347$		−	7.74083i		−	0.415550i	0.978177	−	0.207775i	$-0.0666221\pi$
$348$		0			0
$349$	−17.4343	−	17.4343i	−0.933237	−	0.933237i	0.0646701	−	0.997907i	$-0.479401\pi$
$349$	−17.4343	−	17.4343i	−0.933237	−	0.933237i	−0.997907	+	0.0646701i	$0.979401\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−16.9308	−	16.9308i	−0.901138	−	0.901138i	0.0943968	−	0.995535i	$-0.469908\pi$
$353$	−16.9308	−	16.9308i	−0.901138	−	0.901138i	−0.995535	+	0.0943968i	$0.969908\pi$
$354$		0			0
$355$		−	1.56732i		−	0.0831846i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.23934	−	6.23934i	0.329300	−	0.329300i	−0.523020	−	0.852320i	$-0.675195\pi$
$359$	6.23934	−	6.23934i	0.329300	−	0.329300i	0.852320	+	0.523020i	$0.175195\pi$
$360$		0			0
$361$			19.0000i			1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			7.44959i			0.389929i
$366$		0			0
$367$	17.9214			0.935490			0.467745	−	0.883863i	$-0.345067\pi$
$367$	17.9214			0.935490			0.467745	+	0.883863i	$0.345067\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−28.7195	+	28.7195i	−1.49104	+	1.49104i
$372$		0			0
$373$	−37.2581			−1.92915			−0.964575	−	0.263810i	$-0.915021\pi$
$373$	−37.2581			−1.92915			−0.964575	+	0.263810i	$0.915021\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.9477	−	6.66313i	0.666843	−	0.343169i
$378$		0			0
$379$	19.8990	−	19.8990i	1.02214	−	1.02214i	0.0223933	−	0.999749i	$-0.492871\pi$
$379$	19.8990	−	19.8990i	1.02214	−	1.02214i	0.999749	−	0.0223933i	$-0.00712862\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−18.2681	−	18.2681i	−0.933455	−	0.933455i	0.0644646	−	0.997920i	$-0.479466\pi$
$383$	−18.2681	−	18.2681i	−0.933455	−	0.933455i	−0.997920	+	0.0644646i	$0.979466\pi$
$384$		0			0
$385$	4.64983	+	4.64983i	0.236977	+	0.236977i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−36.7877			−1.86521			−0.932605	−	0.360899i	$-0.882470\pi$
$389$	−36.7877			−1.86521			−0.932605	+	0.360899i	$0.882470\pi$
$390$		0			0
$391$	11.6907			0.591226
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.06923	−	4.06923i	−0.204745	−	0.204745i
$396$		0			0
$397$	−14.7620	−	14.7620i	−0.740886	−	0.740886i	0.231863	−	0.972749i	$-0.425518\pi$
$397$	−14.7620	−	14.7620i	−0.740886	−	0.740886i	−0.972749	+	0.231863i	$0.925518\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−25.4423	+	25.4423i	−1.27053	+	1.27053i	−0.324715	+	0.945812i	$0.605268\pi$
$401$	−25.4423	+	25.4423i	−1.27053	+	1.27053i	−0.945812	+	0.324715i	$0.894732\pi$
$402$		0			0
$403$	−16.0224	−	5.13464i	−0.798134	−	0.255774i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7.74083			−0.383699
$408$		0			0
$409$	−5.78054	+	5.78054i	−0.285829	+	0.285829i	−0.835429	−	0.549599i	$-0.814780\pi$
$409$	−5.78054	+	5.78054i	−0.285829	+	0.285829i	0.549599	+	0.835429i	$0.314780\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	26.3034			1.29431
$414$		0			0
$415$			3.61054i			0.177234i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	20.8211i		−	1.01718i	−0.861009	−	0.508589i	$-0.830167\pi$
$419$		−	20.8211i		−	1.01718i	0.861009	−	0.508589i	$-0.169833\pi$
$420$		0			0
$421$	7.41188	−	7.41188i	0.361233	−	0.361233i	−0.503034	−	0.864267i	$-0.667783\pi$
$421$	7.41188	−	7.41188i	0.361233	−	0.361233i	0.864267	+	0.503034i	$0.167783\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			3.11967i			0.151326i
$426$		0			0
$427$	−10.1307	−	10.1307i	−0.490259	−	0.490259i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.05452	+	4.05452i	0.195300	+	0.195300i	0.797982	−	0.602682i	$-0.205901\pi$
$431$	4.05452	+	4.05452i	0.195300	+	0.195300i	−0.602682	+	0.797982i	$0.705901\pi$
$432$		0			0
$433$		−	0.677918i		−	0.0325786i	−0.999867	−	0.0162893i	$-0.994815\pi$
$433$		−	0.677918i		−	0.0325786i	0.999867	−	0.0162893i	$-0.00518528\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			21.4663i			1.02453i	0.858827	+	0.512266i	$0.171194\pi$
$439$			21.4663i			1.02453i	−0.858827	+	0.512266i	$0.828806\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			2.63476i			0.125181i	0.998039	+	0.0625906i	$0.0199362\pi$
$443$			2.63476i			0.125181i	−0.998039	+	0.0625906i	$0.980064\pi$
$444$		0			0
$445$	1.54490			0.0732352
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.5343	−	18.5343i	0.874688	−	0.874688i	−0.118291	−	0.992979i	$-0.537741\pi$
$449$	18.5343	−	18.5343i	0.874688	−	0.874688i	0.992979	+	0.118291i	$0.0377414\pi$
$450$		0			0
$451$	−1.37987			−0.0649756
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−11.3601	+	5.84611i	−0.532571	+	0.274070i
$456$		0			0
$457$	26.4590	−	26.4590i	1.23770	−	1.23770i	0.276763	−	0.960938i	$-0.410738\pi$
$457$	26.4590	−	26.4590i	1.23770	−	1.23770i	0.960938	−	0.276763i	$-0.0892616\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.19550	−	3.19550i	−0.148829	−	0.148829i	0.628766	−	0.777595i	$-0.283561\pi$
$461$	−3.19550	−	3.19550i	−0.148829	−	0.148829i	−0.777595	+	0.628766i	$0.783561\pi$
$462$		0			0
$463$	6.50560	+	6.50560i	0.302341	+	0.302341i	0.841929	−	0.539588i	$-0.181420\pi$
$463$	6.50560	+	6.50560i	0.302341	+	0.302341i	−0.539588	+	0.841929i	$0.681420\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.00592			0.370470			0.185235	−	0.982694i	$-0.440695\pi$
$467$	8.00592			0.370470			0.185235	+	0.982694i	$0.440695\pi$
$468$		0			0
$469$	−15.0898			−0.696782
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.378519	+	0.378519i	0.0174043	+	0.0174043i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	19.1621	−	19.1621i	0.875538	−	0.875538i	−0.117532	−	0.993069i	$-0.537498\pi$
$479$	19.1621	−	19.1621i	0.875538	−	0.875538i	0.993069	+	0.117532i	$0.0374981\pi$
$480$		0			0
$481$	4.58974	−	14.3221i	0.209274	−	0.653031i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.59171			0.299314
$486$		0			0
$487$	9.49440	−	9.49440i	0.430232	−	0.430232i	−0.458475	−	0.888707i	$-0.651604\pi$
$487$	9.49440	−	9.49440i	0.430232	−	0.430232i	0.888707	+	0.458475i	$0.151604\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−22.0155			−0.993546			−0.496773	−	0.867880i	$-0.665482\pi$
$491$	−22.0155			−0.993546			−0.496773	+	0.867880i	$0.665482\pi$
$492$		0			0
$493$		−	12.5993i		−	0.567445i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	5.55373i		−	0.249119i
$498$		0			0
$499$	12.5561	−	12.5561i	0.562089	−	0.562089i	−0.367812	−	0.929900i	$-0.619893\pi$
$499$	12.5561	−	12.5561i	0.562089	−	0.562089i	0.929900	+	0.367812i	$0.119893\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	42.0819i		−	1.87634i	−0.346177	−	0.938169i	$-0.612520\pi$
$503$		−	42.0819i		−	1.87634i	0.346177	−	0.938169i	$-0.387480\pi$
$504$		0			0
$505$	−5.59933	−	5.59933i	−0.249167	−	0.249167i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.36117	−	7.36117i	−0.326278	−	0.326278i	0.524891	−	0.851169i	$-0.324106\pi$
$509$	−7.36117	−	7.36117i	−0.326278	−	0.326278i	−0.851169	+	0.524891i	$0.824106\pi$
$510$		0			0
$511$			26.3973i			1.16775i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.132545	−	0.132545i	0.00584064	−	0.00584064i
$516$		0			0
$517$		−	5.01121i		−	0.220393i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			24.6241i			1.07880i	0.842049	+	0.539401i	$0.181349\pi$
$521$			24.6241i			1.07880i	−0.842049	+	0.539401i	$0.818651\pi$
$522$		0			0
$523$	18.4983			0.808875			0.404438	−	0.914566i	$-0.367467\pi$
$523$	18.4983			0.808875			0.404438	+	0.914566i	$0.367467\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.2939	+	10.2939i	−0.448408	+	0.448408i
$528$		0			0
$529$	−8.95678			−0.389425
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.818160	−	2.55304i	0.0354385	−	0.110584i
$534$		0			0
$535$	9.47359	−	9.47359i	0.409579	−	0.409579i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7.29089	+	7.29089i	0.314041	+	0.314041i
$540$		0			0
$541$	20.1554	+	20.1554i	0.866550	+	0.866550i	0.992089	−	0.125539i	$-0.0400660\pi$
$541$	20.1554	+	20.1554i	0.866550	+	0.866550i	−0.125539	+	0.992089i	$0.540066\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.28759			−0.397837
$546$		0			0
$547$	10.1089			0.432224			0.216112	−	0.976369i	$-0.430662\pi$
$547$	10.1089			0.432224			0.216112	+	0.976369i	$0.430662\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−14.4192	−	14.4192i	−0.613165	−	0.613165i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	23.0059	−	23.0059i	0.974793	−	0.974793i	−0.0248972	−	0.999690i	$-0.507926\pi$
$557$	23.0059	−	23.0059i	0.974793	−	0.974793i	0.999690	+	0.0248972i	$0.00792585\pi$
$558$		0			0
$559$	−0.924770	+	0.475902i	−0.0391136	+	0.0201285i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−25.7335			−1.08454			−0.542269	−	0.840205i	$-0.682435\pi$
$563$	−25.7335			−1.08454			−0.542269	+	0.840205i	$0.682435\pi$
$564$		0			0
$565$	2.15544	−	2.15544i	0.0906799	−	0.0906799i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−32.7943			−1.37481			−0.687404	−	0.726275i	$-0.741250\pi$
$569$	−32.7943			−1.37481			−0.687404	+	0.726275i	$0.741250\pi$
$570$		0			0
$571$		−	5.95678i		−	0.249283i	−0.992202	−	0.124642i	$-0.960222\pi$
$571$		−	5.95678i		−	0.249283i	0.992202	−	0.124642i	$-0.0397782\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			3.74743i			0.156279i
$576$		0			0
$577$	0.826070	−	0.826070i	0.0343898	−	0.0343898i	−0.689703	−	0.724093i	$-0.742259\pi$
$577$	0.826070	−	0.826070i	0.0343898	−	0.0343898i	0.724093	+	0.689703i	$0.242259\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			12.7938i			0.530776i
$582$		0			0
$583$	15.0409	+	15.0409i	0.622931	+	0.622931i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−25.7629	−	25.7629i	−1.06335	−	1.06335i	−0.997853	−	0.0654976i	$-0.979137\pi$
$587$	−25.7629	−	25.7629i	−1.06335	−	1.06335i	−0.0654976	−	0.997853i	$-0.520863\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.63440	−	9.63440i	0.395638	−	0.395638i	−0.481054	−	0.876691i	$-0.659746\pi$
$593$	9.63440	−	9.63440i	0.395638	−	0.395638i	0.876691	+	0.481054i	$0.159746\pi$
$594$		0			0
$595$			11.0544i			0.453187i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	0.929302i		−	0.0379703i	−0.999820	−	0.0189851i	$-0.993956\pi$
$599$		−	0.929302i		−	0.0379703i	0.999820	−	0.0189851i	$-0.00604352\pi$
$600$		0			0
$601$	5.13302			0.209380			0.104690	−	0.994505i	$-0.466615\pi$
$601$	5.13302			0.209380			0.104690	+	0.994505i	$0.466615\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.34298	+	5.34298i	−0.217223	+	0.217223i
$606$		0			0
$607$	−25.1987			−1.02278			−0.511391	−	0.859348i	$-0.670870\pi$
$607$	−25.1987			−1.02278			−0.511391	+	0.859348i	$0.670870\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.27174	+	2.97127i	0.375094	+	0.120205i
$612$		0			0
$613$	29.6723	−	29.6723i	1.19845	−	1.19845i	0.223821	−	0.974630i	$-0.428147\pi$
$613$	29.6723	−	29.6723i	1.19845	−	1.19845i	0.974630	−	0.223821i	$-0.0718529\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.70840	+	6.70840i	0.270070	+	0.270070i	0.829128	−	0.559058i	$-0.188837\pi$
$617$	6.70840	+	6.70840i	0.270070	+	0.270070i	−0.559058	+	0.829128i	$0.688837\pi$
$618$		0			0
$619$	−28.4135	−	28.4135i	−1.14203	−	1.14203i	−0.988077	−	0.153957i	$-0.950798\pi$
$619$	−28.4135	−	28.4135i	−1.14203	−	1.14203i	−0.153957	−	0.988077i	$-0.549202\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.47429			0.219323
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9.20146	−	9.20146i	−0.366886	−	0.366886i
$630$		0			0
$631$	−19.9214	−	19.9214i	−0.793059	−	0.793059i	0.188931	−	0.981990i	$-0.439498\pi$
$631$	−19.9214	−	19.9214i	−0.793059	−	0.793059i	−0.981990	+	0.188931i	$0.939498\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.40428	+	9.40428i	−0.373198	+	0.373198i
$636$		0			0
$637$	−17.8126	+	9.16665i	−0.705759	+	0.363196i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.6860			−0.422070			−0.211035	−	0.977479i	$-0.567683\pi$
$641$	−10.6860			−0.422070			−0.211035	+	0.977479i	$0.567683\pi$
$642$		0			0
$643$	0.0713054	−	0.0713054i	0.00281201	−	0.00281201i	−0.705699	−	0.708511i	$-0.749367\pi$
$643$	0.0713054	−	0.0713054i	0.00281201	−	0.00281201i	0.708511	+	0.705699i	$0.249367\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	32.7887			1.28906			0.644529	−	0.764580i	$-0.277053\pi$
$647$	32.7887			1.28906			0.644529	+	0.764580i	$0.277053\pi$
$648$		0			0
$649$		−	13.7756i		−	0.540738i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			38.8908i			1.52191i	0.648802	+	0.760957i	$0.275270\pi$
$653$			38.8908i			1.52191i	−0.648802	+	0.760957i	$0.724730\pi$
$654$		0			0
$655$	0.887783	−	0.887783i	0.0346886	−	0.0346886i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			43.5119i			1.69498i	0.530808	+	0.847492i	$0.321889\pi$
$659$			43.5119i			1.69498i	−0.530808	+	0.847492i	$0.678111\pi$
$660$		0			0
$661$	−16.4231	−	16.4231i	−0.638784	−	0.638784i	0.311471	−	0.950256i	$-0.399178\pi$
$661$	−16.4231	−	16.4231i	−0.638784	−	0.638784i	−0.950256	+	0.311471i	$0.899178\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	15.1346i		−	0.586015i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5.30563	+	5.30563i	−0.204822	+	0.204822i
$672$		0			0
$673$		−	3.54652i		−	0.136708i	−0.997661	−	0.0683541i	$-0.978225\pi$
$673$		−	3.54652i		−	0.136708i	0.997661	−	0.0683541i	$-0.0217748\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	19.5712i		−	0.752181i	−0.926583	−	0.376090i	$-0.877268\pi$
$677$		−	19.5712i		−	0.752181i	0.926583	−	0.376090i	$-0.122732\pi$
$678$		0			0
$679$	23.3575			0.896377
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−28.6231	+	28.6231i	−1.09523	+	1.09523i	−0.100271	+	0.994960i	$0.531971\pi$
$683$	−28.6231	+	28.6231i	−1.09523	+	1.09523i	−0.994960	+	0.100271i	$0.968029\pi$
$684$		0			0
$685$	12.4343			0.475090
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−36.7468	+	18.9105i	−1.39994	+	0.720435i
$690$		0			0
$691$	−9.65376	+	9.65376i	−0.367246	+	0.367246i	−0.866472	−	0.499226i	$-0.833618\pi$
$691$	−9.65376	+	9.65376i	−0.367246	+	0.367246i	0.499226	+	0.866472i	$0.333618\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10.1046	+	10.1046i	0.383290	+	0.383290i
$696$		0			0
$697$	−1.64024	−	1.64024i	−0.0621285	−	0.0621285i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−21.3060			−0.804718			−0.402359	−	0.915482i	$-0.631810\pi$
$701$	−21.3060			−0.804718			−0.402359	+	0.915482i	$0.631810\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−19.8410	−	19.8410i	−0.746197	−	0.746197i
$708$		0			0
$709$	−32.7772	−	32.7772i	−1.23097	−	1.23097i	−0.963590	−	0.267382i	$-0.913841\pi$
$709$	−32.7772	−	32.7772i	−1.23097	−	1.23097i	−0.267382	−	0.963590i	$-0.586159\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12.3653	+	12.3653i	−0.463082	+	0.463082i
$714$		0			0
$715$	3.06171	+	5.94950i	0.114502	+	0.222499i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	49.6585			1.85195			0.925974	−	0.377587i	$-0.123246\pi$
$719$	49.6585			1.85195			0.925974	+	0.377587i	$0.123246\pi$
$720$		0			0
$721$	0.469669	−	0.469669i	0.0174914	−	0.0174914i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.03867			0.149993
$726$		0			0
$727$		−	6.88778i		−	0.255454i	−0.991809	−	0.127727i	$-0.959232\pi$
$727$		−	6.88778i		−	0.255454i	0.991809	−	0.127727i	$-0.0407681\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			0.899885i			0.0332834i
$732$		0			0
$733$	30.4814	−	30.4814i	1.12586	−	1.12586i	0.135014	−	0.990844i	$-0.456892\pi$
$733$	30.4814	−	30.4814i	1.12586	−	1.12586i	0.990844	−	0.135014i	$-0.0431080\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			7.90279i			0.291103i
$738$		0			0
$739$	5.09765	+	5.09765i	0.187520	+	0.187520i	0.794623	−	0.607103i	$-0.207668\pi$
$739$	5.09765	+	5.09765i	0.187520	+	0.187520i	−0.607103	+	0.794623i	$0.707668\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.25552	+	1.25552i	0.0460604	+	0.0460604i	0.729762	−	0.683701i	$-0.239631\pi$
$743$	1.25552	+	1.25552i	0.0460604	+	0.0460604i	−0.683701	+	0.729762i	$0.739631\pi$
$744$		0			0
$745$		−	0.743556i		−	0.0272418i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	33.5693	−	33.5693i	1.22660	−	1.22660i
$750$		0			0
$751$			51.7772i			1.88938i	0.327970	+	0.944688i	$0.393635\pi$
$751$			51.7772i			1.88938i	−0.327970	+	0.944688i	$0.606365\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			16.7666i			0.610199i
$756$		0			0
$757$	−28.4791			−1.03509			−0.517546	−	0.855655i	$-0.673154\pi$
$757$	−28.4791			−1.03509			−0.517546	+	0.855655i	$0.673154\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−36.4544	+	36.4544i	−1.32147	+	1.32147i	−0.408888	+	0.912585i	$0.634083\pi$
$761$	−36.4544	+	36.4544i	−1.32147	+	1.32147i	−0.912585	+	0.408888i	$0.865917\pi$
$762$		0			0
$763$	−32.9102			−1.19143
$764$		0			0
$765$		0			0
$766$		0			0
$767$	25.4875	+	8.16788i	0.920302	+	0.294925i
$768$		0			0
$769$	25.2693	−	25.2693i	0.911233	−	0.911233i	−0.0851361	−	0.996369i	$-0.527133\pi$
$769$	25.2693	−	25.2693i	0.911233	−	0.911233i	0.996369	+	0.0851361i	$0.0271325\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	17.7173	+	17.7173i	0.637247	+	0.637247i	0.949875	−	0.312628i	$-0.101209\pi$
$773$	17.7173	+	17.7173i	0.637247	+	0.637247i	−0.312628	+	0.949875i	$0.601209\pi$
$774$		0			0
$775$	−3.29966	−	3.29966i	−0.118527	−	0.118527i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−2.90858			−0.104077
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−15.5722	−	15.5722i	−0.555796	−	0.555796i
$786$		0			0
$787$	−36.8462	−	36.8462i	−1.31342	−	1.31342i	−0.918874	−	0.394550i	$-0.870900\pi$
$787$	−36.8462	−	36.8462i	−1.31342	−	1.31342i	−0.394550	−	0.918874i	$-0.629100\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7.63770	−	7.63770i	0.271565	−	0.271565i
$792$		0			0
$793$	−6.67063	−	12.9623i	−0.236881	−	0.460306i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.12265			0.216875			0.108438	−	0.994103i	$-0.465415\pi$
$797$	6.12265			0.216875			0.108438	+	0.994103i	$0.465415\pi$
$798$		0			0
$799$	5.95678	−	5.95678i	0.210736	−	0.210736i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13.8247			0.487864
$804$		0			0
$805$			13.2789i			0.468018i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	41.3907i		−	1.45522i	−0.685991	−	0.727610i	$-0.740631\pi$
$809$		−	41.3907i		−	1.45522i	0.685991	−	0.727610i	$-0.259369\pi$
$810$		0			0
$811$	20.1458	−	20.1458i	0.707416	−	0.707416i	−0.258575	−	0.965991i	$-0.583253\pi$
$811$	20.1458	−	20.1458i	0.707416	−	0.707416i	0.965991	+	0.258575i	$0.0832529\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			7.80195i			0.273290i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−38.2357	−	38.2357i	−1.33444	−	1.33444i	−0.901355	−	0.433081i	$-0.857426\pi$
$821$	−38.2357	−	38.2357i	−1.33444	−	1.33444i	−0.433081	−	0.901355i	$-0.642574\pi$
$822$		0			0
$823$			10.9472i			0.381595i	0.981629	+	0.190797i	$0.0611074\pi$
$823$			10.9472i			0.381595i	−0.981629	+	0.190797i	$0.938893\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	31.7983	−	31.7983i	1.10574	−	1.10574i	0.112031	−	0.993705i	$-0.464264\pi$
$827$	31.7983	−	31.7983i	1.10574	−	1.10574i	0.993705	−	0.112031i	$-0.0357356\pi$
$828$		0			0
$829$			35.3591i			1.22807i	0.789278	+	0.614036i	$0.210455\pi$
$829$			35.3591i			1.22807i	−0.789278	+	0.614036i	$0.789545\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			17.3332i			0.600561i
$834$		0			0
$835$	6.70034			0.231875
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5.80411	+	5.80411i	−0.200380	+	0.200380i	−0.800163	−	0.599783i	$-0.795254\pi$
$839$	−5.80411	+	5.80411i	−0.200380	+	0.200380i	0.599783	+	0.800163i	$0.295254\pi$
$840$		0			0
$841$	12.6891			0.437556
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.8231	+	2.13717i	−0.441129	+	0.0735210i
$846$		0			0
$847$	−18.9326	+	18.9326i	−0.650533	+	0.650533i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−11.0530	−	11.0530i	−0.378893	−	0.378893i
$852$		0			0
$853$	30.3950	+	30.3950i	1.04071	+	1.04071i	0.999136	+	0.0415695i	$0.0132358\pi$
$853$	30.3950	+	30.3950i	1.04071	+	1.04071i	0.0415695	+	0.999136i	$0.486764\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.0998			−0.652437			−0.326219	−	0.945294i	$-0.605775\pi$
$857$	−19.0998			−0.652437			−0.326219	+	0.945294i	$0.605775\pi$
$858$		0			0
$859$	36.9758			1.26160			0.630800	−	0.775946i	$-0.282727\pi$
$859$	36.9758			1.26160			0.630800	+	0.775946i	$0.282727\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.48161	+	2.48161i	0.0844751	+	0.0844751i	0.748082	−	0.663607i	$-0.230975\pi$
$863$	2.48161	+	2.48161i	0.0844751	+	0.0844751i	−0.663607	+	0.748082i	$0.730975\pi$
$864$		0			0
$865$	−7.16665	−	7.16665i	−0.243673	−	0.243673i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.55157	+	7.55157i	−0.256170	+	0.256170i
$870$		0			0
$871$	−14.6217	−	4.68577i	−0.495439	−	0.158771i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.54346			−0.119791
$876$		0			0
$877$	−13.5785	+	13.5785i	−0.458514	+	0.458514i	−0.898168	−	0.439653i	$-0.855101\pi$
$877$	−13.5785	+	13.5785i	−0.458514	+	0.458514i	0.439653	+	0.898168i	$0.355101\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	58.6604			1.97632			0.988159	−	0.153435i	$-0.0490334\pi$
$881$	58.6604			1.97632			0.988159	+	0.153435i	$0.0490334\pi$
$882$		0			0
$883$		−	1.79013i		−	0.0602428i	−0.999546	−	0.0301214i	$-0.990411\pi$
$883$		−	1.79013i		−	0.0602428i	0.999546	−	0.0301214i	$-0.00958939\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	19.9021i		−	0.668248i	−0.942529	−	0.334124i	$-0.891560\pi$
$887$		−	19.9021i		−	0.668248i	0.942529	−	0.334124i	$-0.108440\pi$
$888$		0			0
$889$	−33.3237	+	33.3237i	−1.11764	+	1.11764i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	11.8558	+	11.8558i	0.396295	+	0.396295i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	13.3263	+	13.3263i	0.444456	+	0.444456i
$900$		0			0
$901$			35.7580i			1.19127i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.02726	−	2.02726i	0.0673885	−	0.0673885i
$906$		0			0
$907$			18.6139i			0.618064i	0.951052	+	0.309032i	$0.100005\pi$
$907$			18.6139i			0.618064i	−0.951052	+	0.309032i	$0.899995\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	45.6968i		−	1.51400i	−0.653414	−	0.757001i	$-0.726664\pi$
$911$		−	45.6968i		−	1.51400i	0.653414	−	0.757001i	$-0.273336\pi$
$912$		0			0
$913$	6.70034			0.221749
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.14582	−	3.14582i	0.103884	−	0.103884i
$918$		0			0
$919$	14.3349			0.472865			0.236432	−	0.971648i	$-0.424022\pi$
$919$	14.3349			0.472865			0.236432	+	0.971648i	$0.424022\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.72457	−	5.38146i	0.0567650	−	0.177133i
$924$		0			0
$925$	2.94950	−	2.94950i	0.0969788	−	0.0969788i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5.63184	+	5.63184i	0.184775	+	0.184775i	0.793433	−	0.608658i	$-0.208292\pi$
$929$	5.63184	+	5.63184i	0.184775	+	0.184775i	−0.608658	+	0.793433i	$0.708292\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.78940			0.189334
$936$		0			0
$937$	−4.74194			−0.154912			−0.0774562	−	0.996996i	$-0.524680\pi$
$937$	−4.74194			−0.154912			−0.0774562	+	0.996996i	$0.524680\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4.20178	−	4.20178i	−0.136974	−	0.136974i	0.635295	−	0.772269i	$-0.280878\pi$
$941$	−4.20178	−	4.20178i	−0.136974	−	0.136974i	−0.772269	+	0.635295i	$0.780878\pi$
$942$		0			0
$943$	−1.97030	−	1.97030i	−0.0641617	−	0.0641617i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.1864	−	18.1864i	0.590977	−	0.590977i	−0.346918	−	0.937895i	$-0.612772\pi$
$947$	18.1864	−	18.1864i	0.590977	−	0.590977i	0.937895	+	0.346918i	$0.112772\pi$
$948$		0			0
$949$	−8.19704	+	25.5785i	−0.266087	+	0.830314i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	45.7206			1.48104			0.740518	−	0.672036i	$-0.234580\pi$
$953$	45.7206			1.48104			0.740518	+	0.672036i	$0.234580\pi$
$954$		0			0
$955$	−6.96799	+	6.96799i	−0.225479	+	0.225479i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	44.0604			1.42279
$960$		0			0
$961$			9.22443i			0.297562i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			2.33321i			0.0751088i
$966$		0			0
$967$	5.83497	−	5.83497i	0.187640	−	0.187640i	−0.607035	−	0.794675i	$-0.707641\pi$
$967$	5.83497	−	5.83497i	0.187640	−	0.187640i	0.794675	+	0.607035i	$0.207641\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			18.1333i			0.581924i	0.956735	+	0.290962i	$0.0939753\pi$
$971$			18.1333i			0.581924i	−0.956735	+	0.290962i	$0.906025\pi$
$972$		0			0
$973$	35.8053	+	35.8053i	1.14786	+	1.14786i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−15.8317	−	15.8317i	−0.506502	−	0.506502i	0.406949	−	0.913451i	$-0.366593\pi$
$977$	−15.8317	−	15.8317i	−0.506502	−	0.506502i	−0.913451	+	0.406949i	$0.866593\pi$
$978$		0			0
$979$		−	2.86698i		−	0.0916291i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16.8594	+	16.8594i	−0.537732	+	0.537732i	−0.922862	−	0.385130i	$-0.874157\pi$
$983$	−16.8594	+	16.8594i	−0.537732	+	0.537732i	0.385130	+	0.922862i	$0.374157\pi$
$984$		0			0
$985$			5.89899i			0.187958i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1.08096i			0.0343727i
$990$		0			0
$991$	35.4023			1.12459			0.562295	−	0.826936i	$-0.309918\pi$
$991$	35.4023			1.12459			0.562295	+	0.826936i	$0.309918\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	11.5494	−	11.5494i	0.366140	−	0.366140i
$996$		0			0
$997$	15.0976			0.478147			0.239074	−	0.971001i	$-0.423156\pi$
$997$	15.0976			0.478147			0.239074	+	0.971001i	$0.423156\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.bi.c.161.6	yes	12
3.2	odd	2	inner	2340.2.bi.c.161.3	✓	12
13.8	odd	4	inner	2340.2.bi.c.1061.3	yes	12
39.8	even	4	inner	2340.2.bi.c.1061.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.bi.c.161.3	✓	12	3.2	odd	2	inner
2340.2.bi.c.161.6	yes	12	1.1	even	1	trivial
2340.2.bi.c.1061.3	yes	12	13.8	odd	4	inner
2340.2.bi.c.1061.6	yes	12	39.8	even	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.bi (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +(2.50560 + 2.50560i) q^{7} +(1.31223 - 1.31223i) q^{11} +(1.64983 + 3.20594i) q^{13} +3.11967 q^{17} +3.74743 q^{23} +1.00000i q^{25} -4.03867i q^{29} +(-3.29966 + 3.29966i) q^{31} +3.54346i q^{35} +(-2.94950 - 2.94950i) q^{37} +(-0.525773 - 0.525773i) q^{41} +0.288455i q^{43} +(1.90943 - 1.90943i) q^{47} +5.55611i q^{49} +11.4621i q^{53} +1.85577 q^{55} +(5.24892 - 5.24892i) q^{59} -4.04322 q^{61} +(-1.10033 + 3.43355i) q^{65} +(-3.01121 + 3.01121i) q^{67} +(-1.10826 - 1.10826i) q^{71} +(5.26765 + 5.26765i) q^{73} +6.57586 q^{77} -5.75476 q^{79} +(2.55304 + 2.55304i) q^{83} +(2.20594 + 2.20594i) q^{85} +(1.09241 - 1.09241i) q^{89} +(-3.89899 + 12.1666i) q^{91} +(4.66104 - 4.66104i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{7} + 12 q^{13} - 24 q^{31} - 12 q^{37} - 8 q^{55} - 32 q^{61} + 40 q^{67} - 12 q^{73} + 8 q^{79} + 4 q^{85} - 28 q^{97}+O(q^{100}) \) 12 * q - 8 * q^7 + 12 * q^13 - 24 * q^31 - 12 * q^37 - 8 * q^55 - 32 * q^61 + 40 * q^67 - 12 * q^73 + 8 * q^79 + 4 * q^85 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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