LMFDB - Embedded newform 2340.2.bi.c.1061.6

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			1061.6
Root			$0.981031 - 2.36842i$ of defining polynomial
Character	$\chi$	$=$	2340.1061
Dual form			2340.2.bi.c.161.6

$q$-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 $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

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

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	−0.0516364	−	0.998666i	$-0.516444\pi$
$7$	2.50560	−	2.50560i	0.947030	−	0.947030i	0.998666	+	0.0516364i	$0.0164437\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.31223	+	1.31223i	0.395652	+	0.395652i	0.876696	−	0.481044i	$-0.159742\pi$
$11$	1.31223	+	1.31223i	0.395652	+	0.395652i	−0.481044	+	0.876696i	$0.659742\pi$
$12$		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	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$		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	$0.122351\pi$
$29$			4.03867i			0.749963i	−0.927032	+	0.374981i	$0.877649\pi$
$30$		0			0
$31$	−3.29966	−	3.29966i	−0.592637	−	0.592637i	0.345706	−	0.938343i	$-0.387640\pi$
$31$	−3.29966	−	3.29966i	−0.592637	−	0.592637i	−0.938343	+	0.345706i	$0.887640\pi$
$32$		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	−0.906691	−	0.421796i	$-0.861400\pi$
$37$	−2.94950	+	2.94950i	−0.484894	+	0.484894i	0.421796	+	0.906691i	$0.361400\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.525773	+	0.525773i	−0.0821120	+	0.0821120i	−0.746970	−	0.664858i	$-0.768492\pi$
$41$	−0.525773	+	0.525773i	−0.0821120	+	0.0821120i	0.664858	+	0.746970i	$0.268492\pi$
$42$		0			0
$43$		−	0.288455i		−	0.0439890i	−0.999758	−	0.0219945i	$-0.992998\pi$
$43$		−	0.288455i		−	0.0439890i	0.999758	−	0.0219945i	$-0.00700163\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.90943	+	1.90943i	0.278518	+	0.278518i	0.832517	−	0.553999i	$-0.186899\pi$
$47$	1.90943	+	1.90943i	0.278518	+	0.278518i	−0.553999	+	0.832517i	$0.686899\pi$
$48$		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	$-0.711519\pi$
$53$		−	11.4621i		−	1.57444i	0.616671	−	0.787221i	$-0.288481\pi$
$54$		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	0.960754	−	0.277403i	$-0.0894737\pi$
$59$	5.24892	+	5.24892i	0.683351	+	0.683351i	−0.277403	+	0.960754i	$0.589474\pi$
$60$		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	0.498825	−	0.866703i	$-0.333765\pi$
$67$	−3.01121	−	3.01121i	−0.367878	−	0.367878i	−0.866703	+	0.498825i	$0.833765\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.10826	+	1.10826i	−0.131526	+	0.131526i	−0.769805	−	0.638279i	$-0.779647\pi$
$71$	−1.10826	+	1.10826i	−0.131526	+	0.131526i	0.638279	+	0.769805i	$0.279647\pi$
$72$		0			0
$73$	5.26765	−	5.26765i	0.616532	−	0.616532i	−0.328108	−	0.944640i	$-0.606411\pi$
$73$	5.26765	−	5.26765i	0.616532	−	0.616532i	0.944640	+	0.328108i	$0.106411\pi$
$74$		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	−0.552969	−	0.833202i	$-0.686505\pi$
$83$	2.55304	−	2.55304i	0.280232	−	0.280232i	0.833202	+	0.552969i	$0.186505\pi$
$84$		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	0.762630	−	0.646835i	$-0.223908\pi$
$89$	1.09241	+	1.09241i	0.115795	+	0.115795i	−0.646835	+	0.762630i	$0.723908\pi$
$90$		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	0.902967	−	0.429710i	$-0.141384\pi$
$97$	4.66104	+	4.66104i	0.473257	+	0.473257i	−0.429710	+	0.902967i	$0.641384\pi$
$98$		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	$0.00293959\pi$
$103$			0.187447i			0.0184697i	−0.999957	+	0.00923487i	$0.997060\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			13.3977i			1.29520i	0.761979	+	0.647602i	$0.224228\pi$
$107$			13.3977i			1.29520i	−0.761979	+	0.647602i	$0.775772\pi$
$108$		0			0
$109$	−6.56732	−	6.56732i	−0.629035	−	0.629035i	0.318790	−	0.947825i	$-0.396723\pi$
$109$	−6.56732	−	6.56732i	−0.629035	−	0.629035i	−0.947825	+	0.318790i	$0.896723\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			3.04825i			0.286755i	0.989668	+	0.143378i	$0.0457963\pi$
$113$			3.04825i			0.286755i	−0.989668	+	0.143378i	$0.954204\pi$
$114$		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	$-0.799097\pi$
$127$		−	13.2997i		−	1.18015i	0.807347	−	0.590077i	$-0.200903\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			1.25552i			0.109695i	0.998495	+	0.0548474i	$0.0174672\pi$
$131$			1.25552i			0.109695i	−0.998495	+	0.0548474i	$0.982533\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.79238	+	8.79238i	0.751183	+	0.751183i	0.974700	−	0.223517i	$-0.0717537\pi$
$137$	8.79238	+	8.79238i	0.751183	+	0.751183i	−0.223517	+	0.974700i	$0.571754\pi$
$138$		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	−0.728315	−	0.685242i	$-0.759696\pi$
$149$	−0.525773	+	0.525773i	−0.0430730	+	0.0430730i	0.685242	+	0.728315i	$0.259696\pi$
$150$		0			0
$151$	11.8558	−	11.8558i	0.964809	−	0.964809i	−0.0345924	−	0.999402i	$-0.511013\pi$
$151$	11.8558	−	11.8558i	0.964809	−	0.964809i	0.999402	+	0.0345924i	$0.0110133\pi$
$152$		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	−0.457235	−	0.889346i	$-0.651160\pi$
$163$	5.51681	−	5.51681i	0.432110	−	0.432110i	0.889346	+	0.457235i	$0.151160\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.73785	+	4.73785i	0.366626	+	0.366626i	0.866245	−	0.499619i	$-0.166527\pi$
$167$	4.73785	+	4.73785i	0.366626	+	0.366626i	−0.499619	+	0.866245i	$0.666527\pi$
$168$		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	$0.0339806\pi$
$181$			2.86698i			0.213101i	−0.994307	+	0.106551i	$0.966019\pi$
$182$		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	$-0.883965\pi$
$191$		−	9.85422i		−	0.713027i	0.934290	−	0.356513i	$-0.116035\pi$
$192$		0			0
$193$	1.64983	−	1.64983i	0.118758	−	0.118758i	−0.645230	−	0.763988i	$-0.723239\pi$
$193$	1.64983	−	1.64983i	0.118758	−	0.118758i	0.763988	+	0.645230i	$0.223239\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.17122	−	4.17122i	0.297187	−	0.297187i	−0.542724	−	0.839911i	$-0.682607\pi$
$197$	4.17122	−	4.17122i	0.297187	−	0.297187i	0.839911	+	0.542724i	$0.182607\pi$
$198$		0			0
$199$			16.3333i			1.15784i	0.815386	+	0.578918i	$0.196525\pi$
$199$			16.3333i			1.15784i	−0.815386	+	0.578918i	$0.803475\pi$
$200$		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	0.575807	−	0.817586i	$-0.304688\pi$
$223$	−3.61054	−	3.61054i	−0.241779	−	0.241779i	−0.817586	+	0.575807i	$0.804688\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.23934	+	6.23934i	−0.414120	+	0.414120i	−0.883171	−	0.469051i	$-0.844596\pi$
$227$	−6.23934	+	6.23934i	−0.414120	+	0.414120i	0.469051	+	0.883171i	$0.344596\pi$
$228$		0			0
$229$	6.29966	−	6.29966i	0.416294	−	0.416294i	−0.467631	−	0.883924i	$-0.654892\pi$
$229$	6.29966	−	6.29966i	0.416294	−	0.416294i	0.883924	+	0.467631i	$0.154892\pi$
$230$		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	−0.996208	−	0.0870009i	$-0.972272\pi$
$239$	−14.0560	+	14.0560i	−0.909207	+	0.909207i	0.0870009	+	0.996208i	$0.472272\pi$
$240$		0			0
$241$	6.87819	−	6.87819i	0.443063	−	0.443063i	−0.449977	−	0.893040i	$-0.648568\pi$
$241$	6.87819	−	6.87819i	0.443063	−	0.443063i	0.893040	+	0.449977i	$0.148568\pi$
$242$		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	$0.185127\pi$
$263$			17.8181i			1.09871i	−0.835588	+	0.549357i	$0.814873\pi$
$264$		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	$-0.923820\pi$
$269$		−	7.77580i		−	0.474099i	0.971498	−	0.237049i	$-0.0761803\pi$
$270$		0			0
$271$	4.96799	−	4.96799i	0.301784	−	0.301784i	−0.539928	−	0.841711i	$-0.681548\pi$
$271$	4.96799	−	4.96799i	0.301784	−	0.301784i	0.841711	+	0.539928i	$0.181548\pi$
$272$		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	$0.0567098\pi$
$277$			5.89899i			0.354436i	−0.984172	+	0.177218i	$0.943290\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.3587	−	16.3587i	−0.975876	−	0.975876i	0.0238401	−	0.999716i	$-0.492411\pi$
$281$	−16.3587	−	16.3587i	−0.975876	−	0.975876i	−0.999716	+	0.0238401i	$0.992411\pi$
$282$		0			0
$283$			16.1458i			0.959771i	0.877331	+	0.479885i	$0.159322\pi$
$283$			16.1458i			0.959771i	−0.877331	+	0.479885i	$0.840678\pi$
$284$		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	0.0754927	−	0.997146i	$-0.475947\pi$
$293$	−15.7762	−	15.7762i	−0.921654	−	0.921654i	−0.997146	+	0.0754927i	$0.975947\pi$
$294$		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	−0.946321	−	0.323227i	$-0.895232\pi$
$307$	−10.9175	+	10.9175i	−0.623094	+	0.623094i	0.323227	+	0.946321i	$0.395232\pi$
$308$		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	−0.529623	−	0.848233i	$-0.677666\pi$
$317$	5.67271	−	5.67271i	0.318611	−	0.318611i	0.848233	+	0.529623i	$0.177666\pi$
$318$		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	0.945251	+	0.326343i	$0.105817\pi$
$331$	23.1346	+	23.1346i	1.27159	+	1.27159i	0.326343	+	0.945251i	$0.394183\pi$
$332$		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	$0.166199\pi$
$337$			18.3109i			0.997457i	−0.866758	+	0.498728i	$0.833801\pi$
$338$		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	$0.0666221\pi$
$347$			7.74083i			0.415550i	−0.978177	+	0.207775i	$0.933378\pi$
$348$		0			0
$349$	−17.4343	+	17.4343i	−0.933237	+	0.933237i	−0.997907	−	0.0646701i	$-0.979401\pi$
$349$	−17.4343	+	17.4343i	−0.933237	+	0.933237i	0.0646701	+	0.997907i	$0.479401\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−16.9308	+	16.9308i	−0.901138	+	0.901138i	−0.995535	−	0.0943968i	$-0.969908\pi$
$353$	−16.9308	+	16.9308i	−0.901138	+	0.901138i	0.0943968	+	0.995535i	$0.469908\pi$
$354$		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	0.852320	−	0.523020i	$-0.175195\pi$
$359$	6.23934	+	6.23934i	0.329300	+	0.329300i	−0.523020	+	0.852320i	$0.675195\pi$
$360$		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	0.999749	+	0.0223933i	$0.00712862\pi$
$379$	19.8990	+	19.8990i	1.02214	+	1.02214i	0.0223933	+	0.999749i	$0.492871\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−18.2681	+	18.2681i	−0.933455	+	0.933455i	−0.997920	−	0.0644646i	$-0.979466\pi$
$383$	−18.2681	+	18.2681i	−0.933455	+	0.933455i	0.0644646	+	0.997920i	$0.479466\pi$
$384$		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	−0.972749	−	0.231863i	$-0.925518\pi$
$397$	−14.7620	+	14.7620i	−0.740886	+	0.740886i	0.231863	+	0.972749i	$0.425518\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−25.4423	−	25.4423i	−1.27053	−	1.27053i	−0.945812	−	0.324715i	$-0.894732\pi$
$401$	−25.4423	−	25.4423i	−1.27053	−	1.27053i	−0.324715	−	0.945812i	$-0.605268\pi$
$402$		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	0.549599	−	0.835429i	$-0.314780\pi$
$409$	−5.78054	−	5.78054i	−0.285829	−	0.285829i	−0.835429	+	0.549599i	$0.814780\pi$
$410$		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	$0.169833\pi$
$419$			20.8211i			1.01718i	−0.861009	+	0.508589i	$0.830167\pi$
$420$		0			0
$421$	7.41188	+	7.41188i	0.361233	+	0.361233i	0.864267	−	0.503034i	$-0.167783\pi$
$421$	7.41188	+	7.41188i	0.361233	+	0.361233i	−0.503034	+	0.864267i	$0.667783\pi$
$422$		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	−0.602682	−	0.797982i	$-0.705901\pi$
$431$	4.05452	−	4.05452i	0.195300	−	0.195300i	0.797982	+	0.602682i	$0.205901\pi$
$432$		0			0
$433$			0.677918i			0.0325786i	0.999867	+	0.0162893i	$0.00518528\pi$
$433$			0.677918i			0.0325786i	−0.999867	+	0.0162893i	$0.994815\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	21.4663i		−	1.02453i	−0.858827	−	0.512266i	$-0.828806\pi$
$439$		−	21.4663i		−	1.02453i	0.858827	−	0.512266i	$-0.171194\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	2.63476i		−	0.125181i	−0.998039	−	0.0625906i	$-0.980064\pi$
$443$		−	2.63476i		−	0.125181i	0.998039	−	0.0625906i	$-0.0199362\pi$
$444$		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	0.992979	−	0.118291i	$-0.0377414\pi$
$449$	18.5343	+	18.5343i	0.874688	+	0.874688i	−0.118291	+	0.992979i	$0.537741\pi$
$450$		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	0.960938	+	0.276763i	$0.0892616\pi$
$457$	26.4590	+	26.4590i	1.23770	+	1.23770i	0.276763	+	0.960938i	$0.410738\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.19550	+	3.19550i	−0.148829	+	0.148829i	−0.777595	−	0.628766i	$-0.783561\pi$
$461$	−3.19550	+	3.19550i	−0.148829	+	0.148829i	0.628766	+	0.777595i	$0.283561\pi$
$462$		0			0
$463$	6.50560	−	6.50560i	0.302341	−	0.302341i	−0.539588	−	0.841929i	$-0.681420\pi$
$463$	6.50560	−	6.50560i	0.302341	−	0.302341i	0.841929	+	0.539588i	$0.181420\pi$
$464$		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	0.993069	−	0.117532i	$-0.0374981\pi$
$479$	19.1621	+	19.1621i	0.875538	+	0.875538i	−0.117532	+	0.993069i	$0.537498\pi$
$480$		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	0.888707	−	0.458475i	$-0.151604\pi$
$487$	9.49440	+	9.49440i	0.430232	+	0.430232i	−0.458475	+	0.888707i	$0.651604\pi$
$488$		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	0.929900	−	0.367812i	$-0.119893\pi$
$499$	12.5561	+	12.5561i	0.562089	+	0.562089i	−0.367812	+	0.929900i	$0.619893\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			42.0819i			1.87634i	0.346177	+	0.938169i	$0.387480\pi$
$503$			42.0819i			1.87634i	−0.346177	+	0.938169i	$0.612520\pi$
$504$		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	−0.851169	−	0.524891i	$-0.824106\pi$
$509$	−7.36117	+	7.36117i	−0.326278	+	0.326278i	0.524891	+	0.851169i	$0.324106\pi$
$510$		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	$-0.818651\pi$
$521$		−	24.6241i		−	1.07880i	0.842049	−	0.539401i	$-0.181349\pi$
$522$		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	−0.125539	−	0.992089i	$-0.540066\pi$
$541$	20.1554	−	20.1554i	0.866550	−	0.866550i	0.992089	+	0.125539i	$0.0400660\pi$
$542$		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	0.999690	−	0.0248972i	$-0.00792585\pi$
$557$	23.0059	+	23.0059i	0.974793	+	0.974793i	−0.0248972	+	0.999690i	$0.507926\pi$
$558$		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	$0.0397782\pi$
$571$			5.95678i			0.249283i	−0.992202	+	0.124642i	$0.960222\pi$
$572$		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	0.724093	−	0.689703i	$-0.242259\pi$
$577$	0.826070	+	0.826070i	0.0343898	+	0.0343898i	−0.689703	+	0.724093i	$0.742259\pi$
$578$		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	−0.0654976	+	0.997853i	$0.520863\pi$
$587$	−25.7629	+	25.7629i	−1.06335	+	1.06335i	−0.997853	+	0.0654976i	$0.979137\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.63440	+	9.63440i	0.395638	+	0.395638i	0.876691	−	0.481054i	$-0.159746\pi$
$593$	9.63440	+	9.63440i	0.395638	+	0.395638i	−0.481054	+	0.876691i	$0.659746\pi$
$594$		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	$0.00604352\pi$
$599$			0.929302i			0.0379703i	−0.999820	+	0.0189851i	$0.993956\pi$
$600$		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	0.974630	+	0.223821i	$0.0718529\pi$
$613$	29.6723	+	29.6723i	1.19845	+	1.19845i	0.223821	+	0.974630i	$0.428147\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.70840	−	6.70840i	0.270070	−	0.270070i	−0.559058	−	0.829128i	$-0.688837\pi$
$617$	6.70840	−	6.70840i	0.270070	−	0.270070i	0.829128	+	0.559058i	$0.188837\pi$
$618$		0			0
$619$	−28.4135	+	28.4135i	−1.14203	+	1.14203i	−0.153957	+	0.988077i	$0.549202\pi$
$619$	−28.4135	+	28.4135i	−1.14203	+	1.14203i	−0.988077	+	0.153957i	$0.950798\pi$
$620$		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	−0.981990	−	0.188931i	$-0.939498\pi$
$631$	−19.9214	+	19.9214i	−0.793059	+	0.793059i	0.188931	+	0.981990i	$0.439498\pi$
$632$		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	0.708511	−	0.705699i	$-0.249367\pi$
$643$	0.0713054	+	0.0713054i	0.00281201	+	0.00281201i	−0.705699	+	0.708511i	$0.749367\pi$
$644$		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	$-0.724730\pi$
$653$		−	38.8908i		−	1.52191i	0.648802	−	0.760957i	$-0.275270\pi$
$654$		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	$-0.678111\pi$
$659$		−	43.5119i		−	1.69498i	0.530808	−	0.847492i	$-0.321889\pi$
$660$		0			0
$661$	−16.4231	+	16.4231i	−0.638784	+	0.638784i	−0.950256	−	0.311471i	$-0.899178\pi$
$661$	−16.4231	+	16.4231i	−0.638784	+	0.638784i	0.311471	+	0.950256i	$0.399178\pi$
$662$		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	$0.0217748\pi$
$673$			3.54652i			0.136708i	−0.997661	+	0.0683541i	$0.978225\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			19.5712i			0.752181i	0.926583	+	0.376090i	$0.122732\pi$
$677$			19.5712i			0.752181i	−0.926583	+	0.376090i	$0.877268\pi$
$678$		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	−0.994960	−	0.100271i	$-0.968029\pi$
$683$	−28.6231	−	28.6231i	−1.09523	−	1.09523i	−0.100271	−	0.994960i	$-0.531971\pi$
$684$		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	0.499226	−	0.866472i	$-0.333618\pi$
$691$	−9.65376	−	9.65376i	−0.367246	−	0.367246i	−0.866472	+	0.499226i	$0.833618\pi$
$692$		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	−0.267382	+	0.963590i	$0.586159\pi$
$709$	−32.7772	+	32.7772i	−1.23097	+	1.23097i	−0.963590	+	0.267382i	$0.913841\pi$
$710$		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	$0.0407681\pi$
$727$			6.88778i			0.255454i	−0.991809	+	0.127727i	$0.959232\pi$
$728$		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	0.990844	+	0.135014i	$0.0431080\pi$
$733$	30.4814	+	30.4814i	1.12586	+	1.12586i	0.135014	+	0.990844i	$0.456892\pi$
$734$		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	−0.607103	−	0.794623i	$-0.707668\pi$
$739$	5.09765	−	5.09765i	0.187520	−	0.187520i	0.794623	+	0.607103i	$0.207668\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.25552	−	1.25552i	0.0460604	−	0.0460604i	−0.683701	−	0.729762i	$-0.739631\pi$
$743$	1.25552	−	1.25552i	0.0460604	−	0.0460604i	0.729762	+	0.683701i	$0.239631\pi$
$744$		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	$-0.606365\pi$
$751$		−	51.7772i		−	1.88938i	0.327970	−	0.944688i	$-0.393635\pi$
$752$		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	−0.912585	−	0.408888i	$-0.865917\pi$
$761$	−36.4544	−	36.4544i	−1.32147	−	1.32147i	−0.408888	−	0.912585i	$-0.634083\pi$
$762$		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	0.996369	−	0.0851361i	$-0.0271325\pi$
$769$	25.2693	+	25.2693i	0.911233	+	0.911233i	−0.0851361	+	0.996369i	$0.527133\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	17.7173	−	17.7173i	0.637247	−	0.637247i	−0.312628	−	0.949875i	$-0.601209\pi$
$773$	17.7173	−	17.7173i	0.637247	−	0.637247i	0.949875	+	0.312628i	$0.101209\pi$
$774$		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	−0.394550	+	0.918874i	$0.629100\pi$
$787$	−36.8462	+	36.8462i	−1.31342	+	1.31342i	−0.918874	+	0.394550i	$0.870900\pi$
$788$		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	$0.259369\pi$
$809$			41.3907i			1.45522i	−0.685991	+	0.727610i	$0.740631\pi$
$810$		0			0
$811$	20.1458	+	20.1458i	0.707416	+	0.707416i	0.965991	−	0.258575i	$-0.0832529\pi$
$811$	20.1458	+	20.1458i	0.707416	+	0.707416i	−0.258575	+	0.965991i	$0.583253\pi$
$812$		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	−0.433081	+	0.901355i	$0.642574\pi$
$821$	−38.2357	+	38.2357i	−1.33444	+	1.33444i	−0.901355	+	0.433081i	$0.857426\pi$
$822$		0			0
$823$		−	10.9472i		−	0.381595i	−0.981629	−	0.190797i	$-0.938893\pi$
$823$		−	10.9472i		−	0.381595i	0.981629	−	0.190797i	$-0.0611074\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	31.7983	+	31.7983i	1.10574	+	1.10574i	0.993705	+	0.112031i	$0.0357356\pi$
$827$	31.7983	+	31.7983i	1.10574	+	1.10574i	0.112031	+	0.993705i	$0.464264\pi$
$828$		0			0
$829$		−	35.3591i		−	1.22807i	−0.789278	−	0.614036i	$-0.789545\pi$
$829$		−	35.3591i		−	1.22807i	0.789278	−	0.614036i	$-0.210455\pi$
$830$		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	0.599783	−	0.800163i	$-0.295254\pi$
$839$	−5.80411	−	5.80411i	−0.200380	−	0.200380i	−0.800163	+	0.599783i	$0.795254\pi$
$840$		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	0.0415695	−	0.999136i	$-0.486764\pi$
$853$	30.3950	−	30.3950i	1.04071	−	1.04071i	0.999136	−	0.0415695i	$-0.0132358\pi$
$854$		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	−0.663607	−	0.748082i	$-0.730975\pi$
$863$	2.48161	−	2.48161i	0.0844751	−	0.0844751i	0.748082	+	0.663607i	$0.230975\pi$
$864$		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	0.439653	−	0.898168i	$-0.355101\pi$
$877$	−13.5785	−	13.5785i	−0.458514	−	0.458514i	−0.898168	+	0.439653i	$0.855101\pi$
$878$		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	$0.00958939\pi$
$883$			1.79013i			0.0602428i	−0.999546	+	0.0301214i	$0.990411\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			19.9021i			0.668248i	0.942529	+	0.334124i	$0.108440\pi$
$887$			19.9021i			0.668248i	−0.942529	+	0.334124i	$0.891560\pi$
$888$		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	$-0.899995\pi$
$907$		−	18.6139i		−	0.618064i	0.951052	−	0.309032i	$-0.100005\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			45.6968i			1.51400i	0.653414	+	0.757001i	$0.273336\pi$
$911$			45.6968i			1.51400i	−0.653414	+	0.757001i	$0.726664\pi$
$912$		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	−0.608658	−	0.793433i	$-0.708292\pi$
$929$	5.63184	−	5.63184i	0.184775	−	0.184775i	0.793433	+	0.608658i	$0.208292\pi$
$930$		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	−0.772269	−	0.635295i	$-0.780878\pi$
$941$	−4.20178	+	4.20178i	−0.136974	+	0.136974i	0.635295	+	0.772269i	$0.280878\pi$
$942$		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	0.937895	−	0.346918i	$-0.112772\pi$
$947$	18.1864	+	18.1864i	0.590977	+	0.590977i	−0.346918	+	0.937895i	$0.612772\pi$
$948$		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	0.794675	−	0.607035i	$-0.207641\pi$
$967$	5.83497	+	5.83497i	0.187640	+	0.187640i	−0.607035	+	0.794675i	$0.707641\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	18.1333i		−	0.581924i	−0.956735	−	0.290962i	$-0.906025\pi$
$971$		−	18.1333i		−	0.581924i	0.956735	−	0.290962i	$-0.0939753\pi$
$972$		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	−0.913451	−	0.406949i	$-0.866593\pi$
$977$	−15.8317	+	15.8317i	−0.506502	+	0.506502i	0.406949	+	0.913451i	$0.366593\pi$
$978$		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	0.385130	−	0.922862i	$-0.374157\pi$
$983$	−16.8594	−	16.8594i	−0.537732	−	0.537732i	−0.922862	+	0.385130i	$0.874157\pi$
$984$		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.1061.6	yes	12
3.2	odd	2	inner	2340.2.bi.c.1061.3	yes	12
13.5	odd	4	inner	2340.2.bi.c.161.3	✓	12
39.5	even	4	inner	2340.2.bi.c.161.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.bi.c.161.3	✓	12	13.5	odd	4	inner
2340.2.bi.c.161.6	yes	12	39.5	even	4	inner
2340.2.bi.c.1061.3	yes	12	3.2	odd	2	inner
2340.2.bi.c.1061.6	yes	12	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 $\F_{q}$
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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