LMFDB - Embedded newform 2340.2.u.f.577.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(73,2340)] 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, 0, 3, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.73"); 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.u (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,6,0,-8,0,0,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			577.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	2340.577
Dual form			2340.2.u.f.73.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +(2.61803 + 2.61803i) q^{11} +(-2.00000 + 3.00000i) q^{13} +(2.23607 - 2.23607i) q^{17} +(5.85410 + 5.85410i) q^{19} +(-3.38197 - 3.38197i) q^{23} -5.00000 q^{25} +5.23607i q^{29} +(5.85410 - 5.85410i) q^{31} +9.70820i q^{37} +(-3.76393 + 3.76393i) q^{41} +(-3.85410 - 3.85410i) q^{43} +8.94427i q^{47} +7.00000 q^{49} +(-1.47214 + 1.47214i) q^{53} +(5.85410 - 5.85410i) q^{55} +(3.38197 - 3.38197i) q^{59} +5.70820 q^{61} +(6.70820 + 4.47214i) q^{65} -0.291796 q^{67} +(-2.61803 + 2.61803i) q^{71} +11.7082 q^{73} -3.70820i q^{79} -10.4721i q^{83} +(-5.00000 - 5.00000i) q^{85} +(2.23607 - 2.23607i) q^{89} +(13.0902 - 13.0902i) q^{95} +15.4164 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{11} - 8 q^{13} + 10 q^{19} - 18 q^{23} - 20 q^{25} + 10 q^{31} - 24 q^{41} - 2 q^{43} + 28 q^{49} + 12 q^{53} + 10 q^{55} + 18 q^{59} - 4 q^{61} - 28 q^{67} - 6 q^{71} + 20 q^{73} - 20 q^{85}+ \cdots + 8 q^{97}+O(q^{100}) $ 4 * q + 6 * q^11 - 8 * q^13 + 10 * q^19 - 18 * q^23 - 20 * q^25 + 10 * q^31 - 24 * q^41 - 2 * q^43 + 28 * q^49 + 12 * q^53 + 10 * q^55 + 18 * q^59 - 4 * q^61 - 28 * q^67 - 6 * q^71 + 20 * q^73 - 20 * q^85 + 30 * q^95 + 8 * q^97

Character values

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

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

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
$5$		−	2.23607i		−	1.00000i
$5$		−	2.23607i		−	1.00000i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.61803	+	2.61803i	0.789367	+	0.789367i	0.981390	−	0.192023i	$-0.0615050\pi$
$11$	2.61803	+	2.61803i	0.789367	+	0.789367i	−0.192023	+	0.981390i	$0.561505\pi$
$12$		0			0
$13$	−2.00000	+	3.00000i	−0.554700	+	0.832050i
$13$	−2.00000	+	3.00000i	−0.554700	+	0.832050i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.23607	−	2.23607i	0.542326	−	0.542326i	−0.381884	−	0.924210i	$-0.624725\pi$
$17$	2.23607	−	2.23607i	0.542326	−	0.542326i	0.924210	+	0.381884i	$0.124725\pi$
$18$		0			0
$19$	5.85410	+	5.85410i	1.34302	+	1.34302i	0.893034	+	0.449989i	$0.148572\pi$
$19$	5.85410	+	5.85410i	1.34302	+	1.34302i	0.449989	+	0.893034i	$0.351428\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.38197	−	3.38197i	−0.705189	−	0.705189i	0.260331	−	0.965519i	$-0.416168\pi$
$23$	−3.38197	−	3.38197i	−0.705189	−	0.705189i	−0.965519	+	0.260331i	$0.916168\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			5.23607i			0.972313i	0.873872	+	0.486157i	$0.161602\pi$
$29$			5.23607i			0.972313i	−0.873872	+	0.486157i	$0.838398\pi$
$30$		0			0
$31$	5.85410	−	5.85410i	1.05143	−	1.05143i	0.0528239	−	0.998604i	$-0.483178\pi$
$31$	5.85410	−	5.85410i	1.05143	−	1.05143i	0.998604	−	0.0528239i	$-0.0168222\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			9.70820i			1.59602i	0.602645	+	0.798009i	$0.294114\pi$
$37$			9.70820i			1.59602i	−0.602645	+	0.798009i	$0.705886\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.76393	+	3.76393i	−0.587827	+	0.587827i	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−3.76393	+	3.76393i	−0.587827	+	0.587827i	0.349215	+	0.937043i	$0.386448\pi$
$42$		0			0
$43$	−3.85410	−	3.85410i	−0.587745	−	0.587745i	0.349275	−	0.937020i	$-0.386428\pi$
$43$	−3.85410	−	3.85410i	−0.587745	−	0.587745i	−0.937020	+	0.349275i	$0.886428\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.94427i			1.30466i	0.757937	+	0.652328i	$0.226208\pi$
$47$			8.94427i			1.30466i	−0.757937	+	0.652328i	$0.773792\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.47214	+	1.47214i	−0.202213	+	0.202213i	−0.800948	−	0.598734i	$-0.795671\pi$
$53$	−1.47214	+	1.47214i	−0.202213	+	0.202213i	0.598734	+	0.800948i	$0.295671\pi$
$54$		0			0
$55$	5.85410	−	5.85410i	0.789367	−	0.789367i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.38197	−	3.38197i	0.440294	−	0.440294i	−0.451816	−	0.892111i	$-0.649224\pi$
$59$	3.38197	−	3.38197i	0.440294	−	0.440294i	0.892111	+	0.451816i	$0.149224\pi$
$60$		0			0
$61$	5.70820			0.730861			0.365430	−	0.930839i	$-0.380922\pi$
$61$	5.70820			0.730861			0.365430	+	0.930839i	$0.380922\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.70820	+	4.47214i	0.832050	+	0.554700i
$66$		0			0
$67$	−0.291796			−0.0356486			−0.0178243	−	0.999841i	$-0.505674\pi$
$67$	−0.291796			−0.0356486			−0.0178243	+	0.999841i	$0.505674\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.61803	+	2.61803i	−0.310703	+	0.310703i	−0.845182	−	0.534479i	$-0.820508\pi$
$71$	−2.61803	+	2.61803i	−0.310703	+	0.310703i	0.534479	+	0.845182i	$0.320508\pi$
$72$		0			0
$73$	11.7082			1.37034			0.685171	−	0.728382i	$-0.259728\pi$
$73$	11.7082			1.37034			0.685171	+	0.728382i	$0.259728\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	3.70820i		−	0.417206i	−0.978000	−	0.208603i	$-0.933108\pi$
$79$		−	3.70820i		−	0.417206i	0.978000	−	0.208603i	$-0.0668916\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	10.4721i		−	1.14947i	−0.818341	−	0.574733i	$-0.805106\pi$
$83$		−	10.4721i		−	1.14947i	0.818341	−	0.574733i	$-0.194894\pi$
$84$		0			0
$85$	−5.00000	−	5.00000i	−0.542326	−	0.542326i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.23607	−	2.23607i	0.237023	−	0.237023i	−0.578593	−	0.815616i	$-0.696398\pi$
$89$	2.23607	−	2.23607i	0.237023	−	0.237023i	0.815616	+	0.578593i	$0.196398\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13.0902	−	13.0902i	1.34302	−	1.34302i
$96$		0			0
$97$	15.4164			1.56530			0.782650	−	0.622463i	$-0.213868\pi$
$97$	15.4164			1.56530			0.782650	+	0.622463i	$0.213868\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			1.52786i			0.152028i	0.997107	+	0.0760141i	$0.0242194\pi$
$101$			1.52786i			0.152028i	−0.997107	+	0.0760141i	$0.975781\pi$
$102$		0			0
$103$	−3.85410	−	3.85410i	−0.379756	−	0.379756i	0.491258	−	0.871014i	$-0.336537\pi$
$103$	−3.85410	−	3.85410i	−0.379756	−	0.379756i	−0.871014	+	0.491258i	$0.836537\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.61803	+	2.61803i	0.253095	+	0.253095i	0.822238	−	0.569143i	$-0.192725\pi$
$107$	2.61803	+	2.61803i	0.253095	+	0.253095i	−0.569143	+	0.822238i	$0.692725\pi$
$108$		0			0
$109$	4.70820	+	4.70820i	0.450964	+	0.450964i	0.895674	−	0.444710i	$-0.146693\pi$
$109$	4.70820	+	4.70820i	0.450964	+	0.450964i	−0.444710	+	0.895674i	$0.646693\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	11.9443	−	11.9443i	1.12362	−	1.12362i	0.132430	−	0.991192i	$-0.457722\pi$
$113$	11.9443	−	11.9443i	1.12362	−	1.12362i	0.991192	−	0.132430i	$-0.0422781\pi$
$114$		0			0
$115$	−7.56231	+	7.56231i	−0.705189	+	0.705189i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$			2.70820i			0.246200i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			1.00000i
$126$		0			0
$127$	−0.145898	+	0.145898i	−0.0129464	+	0.0129464i	−0.713550	−	0.700604i	$-0.752914\pi$
$127$	−0.145898	+	0.145898i	−0.0129464	+	0.0129464i	0.700604	+	0.713550i	$0.252914\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.41641			0.647975			0.323987	−	0.946061i	$-0.394976\pi$
$131$	7.41641			0.647975			0.323987	+	0.946061i	$0.394976\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	16.4721i		−	1.40731i	−0.710542	−	0.703655i	$-0.751550\pi$
$137$		−	16.4721i		−	1.40731i	0.710542	−	0.703655i	$-0.248450\pi$
$138$		0			0
$139$			23.1246i			1.96140i	0.195509	+	0.980702i	$0.437364\pi$
$139$			23.1246i			1.96140i	−0.195509	+	0.980702i	$0.562636\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−13.0902	+	2.61803i	−1.09466	+	0.218931i
$144$		0			0
$145$	11.7082			0.972313
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.94427	−	5.94427i	−0.486974	−	0.486974i	0.420376	−	0.907350i	$-0.361898\pi$
$149$	−5.94427	−	5.94427i	−0.486974	−	0.486974i	−0.907350	+	0.420376i	$0.861898\pi$
$150$		0			0
$151$	−0.145898	−	0.145898i	−0.0118730	−	0.0118730i	0.701145	−	0.713018i	$-0.252672\pi$
$151$	−0.145898	−	0.145898i	−0.0118730	−	0.0118730i	−0.713018	+	0.701145i	$0.752672\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−13.0902	−	13.0902i	−1.05143	−	1.05143i
$156$		0			0
$157$	−5.00000	−	5.00000i	−0.399043	−	0.399043i	0.478852	−	0.877896i	$-0.341053\pi$
$157$	−5.00000	−	5.00000i	−0.399043	−	0.399043i	−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.29180			0.336159			0.168080	−	0.985773i	$-0.446243\pi$
$163$	4.29180			0.336159			0.168080	+	0.985773i	$0.446243\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			13.5279i			1.04682i	0.852082	+	0.523409i	$0.175340\pi$
$167$			13.5279i			1.04682i	−0.852082	+	0.523409i	$0.824660\pi$
$168$		0			0
$169$	−5.00000	−	12.0000i	−0.384615	−	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	17.1803	+	17.1803i	1.30620	+	1.30620i	0.924138	+	0.382059i	$0.124785\pi$
$173$	17.1803	+	17.1803i	1.30620	+	1.30620i	0.382059	+	0.924138i	$0.375215\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$			8.29180i			0.616324i	0.951334	+	0.308162i	$0.0997139\pi$
$181$			8.29180i			0.616324i	−0.951334	+	0.308162i	$0.900286\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	21.7082			1.59602
$186$		0			0
$187$	11.7082			0.856189
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.41641			−0.536632			−0.268316	−	0.963331i	$-0.586467\pi$
$191$	−7.41641			−0.536632			−0.268316	+	0.963331i	$0.586467\pi$
$192$		0			0
$193$	3.41641			0.245918			0.122959	−	0.992412i	$-0.460762\pi$
$193$	3.41641			0.245918			0.122959	+	0.992412i	$0.460762\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	−11.4164			−0.809288			−0.404644	−	0.914474i	$-0.632605\pi$
$199$	−11.4164			−0.809288			−0.404644	+	0.914474i	$0.632605\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	8.41641	+	8.41641i	0.587827	+	0.587827i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			30.6525i			2.12028i
$210$		0			0
$211$	3.41641			0.235195			0.117598	−	0.993061i	$-0.462481\pi$
$211$	3.41641			0.235195			0.117598	+	0.993061i	$0.462481\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.61803	+	8.61803i	−0.587745	+	0.587745i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.23607	+	11.1803i	0.150414	+	0.752071i
$222$		0			0
$223$		−	7.41641i		−	0.496639i	−0.968678	−	0.248320i	$-0.920122\pi$
$223$		−	7.41641i		−	0.496639i	0.968678	−	0.248320i	$-0.0798784\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.29180			−0.550346			−0.275173	−	0.961395i	$-0.588735\pi$
$227$	−8.29180			−0.550346			−0.275173	+	0.961395i	$0.588735\pi$
$228$		0			0
$229$	8.41641	−	8.41641i	0.556172	−	0.556172i	−0.372043	−	0.928215i	$-0.621343\pi$
$229$	8.41641	−	8.41641i	0.556172	−	0.556172i	0.928215	+	0.372043i	$0.121343\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.47214	+	7.47214i	0.489516	+	0.489516i	0.908153	−	0.418638i	$-0.137492\pi$
$233$	7.47214	+	7.47214i	0.489516	+	0.489516i	−0.418638	+	0.908153i	$0.637492\pi$
$234$		0			0
$235$	20.0000			1.30466
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.61803	+	8.61803i	0.557454	+	0.557454i	0.928582	−	0.371128i	$-0.121029\pi$
$239$	8.61803	+	8.61803i	0.557454	+	0.557454i	−0.371128	+	0.928582i	$0.621029\pi$
$240$		0			0
$241$	−2.70820	−	2.70820i	−0.174451	−	0.174451i	0.614481	−	0.788932i	$-0.289365\pi$
$241$	−2.70820	−	2.70820i	−0.174451	−	0.174451i	−0.788932	+	0.614481i	$0.789365\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	15.6525i		−	1.00000i
$246$		0			0
$247$	−29.2705	+	5.85410i	−1.86244	+	0.372488i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	26.1803i		−	1.65249i	−0.563312	−	0.826244i	$-0.690473\pi$
$251$		−	26.1803i		−	1.65249i	0.563312	−	0.826244i	$-0.309527\pi$
$252$		0			0
$253$		−	17.7082i		−	1.11331i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−17.1803	+	17.1803i	−1.07168	+	1.07168i	−0.0744558	+	0.997224i	$0.523722\pi$
$257$	−17.1803	+	17.1803i	−1.07168	+	1.07168i	−0.997224	+	0.0744558i	$0.976278\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−22.0344	+	22.0344i	−1.35870	+	1.35870i	−0.483182	+	0.875520i	$0.660519\pi$
$263$	−22.0344	+	22.0344i	−1.35870	+	1.35870i	−0.875520	+	0.483182i	$0.839481\pi$
$264$		0			0
$265$	3.29180	+	3.29180i	0.202213	+	0.202213i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			13.5279i			0.824808i	0.911001	+	0.412404i	$0.135311\pi$
$269$			13.5279i			0.824808i	−0.911001	+	0.412404i	$0.864689\pi$
$270$		0			0
$271$	−7.56231	−	7.56231i	−0.459377	−	0.459377i	0.439074	−	0.898451i	$-0.355307\pi$
$271$	−7.56231	−	7.56231i	−0.459377	−	0.459377i	−0.898451	+	0.439074i	$0.855307\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−13.0902	−	13.0902i	−0.789367	−	0.789367i
$276$		0			0
$277$	−2.70820	+	2.70820i	−0.162720	+	0.162720i	−0.783771	−	0.621050i	$-0.786706\pi$
$277$	−2.70820	+	2.70820i	−0.162720	+	0.162720i	0.621050	+	0.783771i	$0.286706\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.23607	−	2.23607i	−0.133393	−	0.133393i	0.637258	−	0.770651i	$-0.280069\pi$
$281$	−2.23607	−	2.23607i	−0.133393	−	0.133393i	−0.770651	+	0.637258i	$0.780069\pi$
$282$		0			0
$283$	−15.8541	−	15.8541i	−0.942429	−	0.942429i	0.0560021	−	0.998431i	$-0.482165\pi$
$283$	−15.8541	−	15.8541i	−0.942429	−	0.942429i	−0.998431	+	0.0560021i	$0.982165\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			7.00000i			0.411765i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−27.7082			−1.61873			−0.809365	−	0.587306i	$-0.800189\pi$
$293$	−27.7082			−1.61873			−0.809365	+	0.587306i	$0.800189\pi$
$294$		0			0
$295$	−7.56231	−	7.56231i	−0.440294	−	0.440294i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	16.9098	−	3.38197i	0.977921	−	0.195584i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	12.7639i		−	0.730861i
$306$		0			0
$307$		−	12.0000i		−	0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$		−	12.0000i		−	0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	2.18034i		−	0.123636i	−0.998087	−	0.0618179i	$-0.980310\pi$
$311$		−	2.18034i		−	0.123636i	0.998087	−	0.0618179i	$-0.0196898\pi$
$312$		0			0
$313$	−14.7082	+	14.7082i	−0.831357	+	0.831357i	−0.987702	−	0.156346i	$-0.950029\pi$
$313$	−14.7082	+	14.7082i	−0.831357	+	0.831357i	0.156346	+	0.987702i	$0.450029\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.70820			0.208273			0.104137	−	0.994563i	$-0.466792\pi$
$317$	3.70820			0.208273			0.104137	+	0.994563i	$0.466792\pi$
$318$		0			0
$319$	−13.7082	+	13.7082i	−0.767512	+	0.767512i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	26.1803			1.45671
$324$		0			0
$325$	10.0000	−	15.0000i	0.554700	−	0.832050i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−15.8541	+	15.8541i	−0.871420	+	0.871420i	−0.992627	−	0.121207i	$-0.961324\pi$
$331$	−15.8541	+	15.8541i	−0.871420	+	0.871420i	0.121207	+	0.992627i	$0.461324\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			0.652476i			0.0356486i
$336$		0			0
$337$	14.4164	−	14.4164i	0.785312	−	0.785312i	−0.195410	−	0.980722i	$-0.562604\pi$
$337$	14.4164	−	14.4164i	0.785312	−	0.785312i	0.980722	+	0.195410i	$0.0626037\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	30.6525			1.65992
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.09017	−	7.09017i	−0.380620	−	0.380620i	0.490705	−	0.871326i	$-0.336739\pi$
$347$	−7.09017	−	7.09017i	−0.380620	−	0.380620i	−0.871326	+	0.490705i	$0.836739\pi$
$348$		0			0
$349$	16.7082	−	16.7082i	0.894370	−	0.894370i	−0.100561	−	0.994931i	$-0.532064\pi$
$349$	16.7082	−	16.7082i	0.894370	−	0.894370i	0.994931	+	0.100561i	$0.0320638\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	12.7639i		−	0.679356i	−0.940542	−	0.339678i	$-0.889682\pi$
$353$		−	12.7639i		−	0.679356i	0.940542	−	0.339678i	$-0.110318\pi$
$354$		0			0
$355$	5.85410	+	5.85410i	0.310703	+	0.310703i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	13.0902	−	13.0902i	0.690873	−	0.690873i	−0.271551	−	0.962424i	$-0.587537\pi$
$359$	13.0902	−	13.0902i	0.690873	−	0.690873i	0.962424	+	0.271551i	$0.0875366\pi$
$360$		0			0
$361$			49.5410i			2.60742i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	26.1803i		−	1.37034i
$366$		0			0
$367$	11.8541	+	11.8541i	0.618779	+	0.618779i	0.945218	−	0.326439i	$-0.105849\pi$
$367$	11.8541	+	11.8541i	0.618779	+	0.618779i	−0.326439	+	0.945218i	$0.605849\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.416408	+	0.416408i	−0.0215608	+	0.0215608i	−0.717805	−	0.696244i	$-0.754853\pi$
$373$	−0.416408	+	0.416408i	−0.0215608	+	0.0215608i	0.696244	+	0.717805i	$0.254853\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−15.7082	−	10.4721i	−0.809014	−	0.539342i
$378$		0			0
$379$	−3.85410	−	3.85410i	−0.197972	−	0.197972i	0.601158	−	0.799130i	$-0.294706\pi$
$379$	−3.85410	−	3.85410i	−0.197972	−	0.197972i	−0.799130	+	0.601158i	$0.794706\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	17.8885i		−	0.914062i	−0.889451	−	0.457031i	$-0.848913\pi$
$383$		−	17.8885i		−	0.914062i	0.889451	−	0.457031i	$-0.151087\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	25.4164			1.28866			0.644332	−	0.764746i	$-0.277136\pi$
$389$	25.4164			1.28866			0.644332	+	0.764746i	$0.277136\pi$
$390$		0			0
$391$	−15.1246			−0.764884
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.29180			−0.417206
$396$		0			0
$397$		−	5.12461i		−	0.257197i	−0.991697	−	0.128598i	$-0.958952\pi$
$397$		−	5.12461i		−	0.257197i	0.991697	−	0.128598i	$-0.0410478\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.23607	−	2.23607i	−0.111664	−	0.111664i	0.649067	−	0.760731i	$-0.275159\pi$
$401$	−2.23607	−	2.23607i	−0.111664	−	0.111664i	−0.760731	+	0.649067i	$0.775159\pi$
$402$		0			0
$403$	5.85410	+	29.2705i	0.291614	+	1.45807i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−25.4164	+	25.4164i	−1.25984	+	1.25984i
$408$		0			0
$409$	3.29180	+	3.29180i	0.162769	+	0.162769i	0.783792	−	0.621023i	$-0.213283\pi$
$409$	3.29180	+	3.29180i	0.162769	+	0.162769i	−0.621023	+	0.783792i	$0.713283\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−23.4164			−1.14947
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	21.5967i		−	1.05507i	−0.849533	−	0.527535i	$-0.823116\pi$
$419$		−	21.5967i		−	1.05507i	0.849533	−	0.527535i	$-0.176884\pi$
$420$		0			0
$421$	3.29180	−	3.29180i	0.160432	−	0.160432i	−0.622326	−	0.782758i	$-0.713812\pi$
$421$	3.29180	−	3.29180i	0.160432	−	0.160432i	0.782758	+	0.622326i	$0.213812\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−11.1803	+	11.1803i	−0.542326	+	0.542326i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.9098	+	16.9098i	−0.814518	+	0.814518i	−0.985308	−	0.170790i	$-0.945368\pi$
$431$	−16.9098	+	16.9098i	−0.814518	+	0.814518i	0.170790	+	0.985308i	$0.445368\pi$
$432$		0			0
$433$	−17.0000	−	17.0000i	−0.816968	−	0.816968i	0.168700	−	0.985668i	$-0.446043\pi$
$433$	−17.0000	−	17.0000i	−0.816968	−	0.816968i	−0.985668	+	0.168700i	$0.946043\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	39.5967i		−	1.89417i
$438$		0			0
$439$	20.0000			0.954548			0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000			0.954548			0.477274	+	0.878755i	$0.341625\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.326238	+	0.326238i	−0.0155000	+	0.0155000i	−0.714814	−	0.699314i	$-0.753489\pi$
$443$	−0.326238	+	0.326238i	−0.0155000	+	0.0155000i	0.699314	+	0.714814i	$0.253489\pi$
$444$		0			0
$445$	−5.00000	−	5.00000i	−0.237023	−	0.237023i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.5279	−	10.5279i	0.496841	−	0.496841i	−0.413612	−	0.910453i	$-0.635733\pi$
$449$	10.5279	−	10.5279i	0.496841	−	0.496841i	0.910453	+	0.413612i	$0.135733\pi$
$450$		0			0
$451$	−19.7082			−0.928023
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−23.4164			−1.09537			−0.547687	−	0.836684i	$-0.684491\pi$
$457$	−23.4164			−1.09537			−0.547687	+	0.836684i	$0.684491\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.1803	+	17.1803i	−0.800168	+	0.800168i	−0.983122	−	0.182953i	$-0.941434\pi$
$461$	−17.1803	+	17.1803i	−0.800168	+	0.800168i	0.182953	+	0.983122i	$0.441434\pi$
$462$		0			0
$463$	4.29180			0.199457			0.0997283	−	0.995015i	$-0.468203\pi$
$463$	4.29180			0.199457			0.0997283	+	0.995015i	$0.468203\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.0344	+	16.0344i	−0.741985	+	0.741985i	−0.972960	−	0.230974i	$-0.925809\pi$
$467$	−16.0344	+	16.0344i	−0.741985	+	0.741985i	0.230974	+	0.972960i	$0.425809\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	20.1803i		−	0.927893i
$474$		0			0
$475$	−29.2705	−	29.2705i	−1.34302	−	1.34302i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−25.7426	+	25.7426i	−1.17621	+	1.17621i	−0.195510	+	0.980702i	$0.562636\pi$
$479$	−25.7426	+	25.7426i	−1.17621	+	1.17621i	−0.980702	+	0.195510i	$0.937364\pi$
$480$		0			0
$481$	−29.1246	−	19.4164i	−1.32797	−	0.885312i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	34.4721i		−	1.56530i
$486$		0			0
$487$	−15.1246			−0.685362			−0.342681	−	0.939452i	$-0.611335\pi$
$487$	−15.1246			−0.685362			−0.342681	+	0.939452i	$0.611335\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			24.6525i			1.11255i	0.830998	+	0.556275i	$0.187770\pi$
$491$			24.6525i			1.11255i	−0.830998	+	0.556275i	$0.812230\pi$
$492$		0			0
$493$	11.7082	+	11.7082i	0.527311	+	0.527311i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−20.9787	−	20.9787i	−0.939136	−	0.939136i	0.0591150	−	0.998251i	$-0.481172\pi$
$499$	−20.9787	−	20.9787i	−0.939136	−	0.939136i	−0.998251	+	0.0591150i	$0.981172\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	11.6738	−	11.6738i	0.520507	−	0.520507i	−0.397217	−	0.917725i	$-0.630024\pi$
$503$	11.6738	−	11.6738i	0.520507	−	0.520507i	0.917725	+	0.397217i	$0.130024\pi$
$504$		0			0
$505$	3.41641			0.152028
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.76393	+	9.76393i	0.432779	+	0.432779i	0.889573	−	0.456794i	$-0.151002\pi$
$509$	9.76393	+	9.76393i	0.432779	+	0.432779i	−0.456794	+	0.889573i	$0.651002\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.61803	+	8.61803i	−0.379756	+	0.379756i
$516$		0			0
$517$	−23.4164	+	23.4164i	−1.02985	+	1.02985i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−41.1246			−1.80170			−0.900851	−	0.434128i	$-0.857056\pi$
$521$	−41.1246			−1.80170			−0.900851	+	0.434128i	$0.857056\pi$
$522$		0			0
$523$	25.2705	−	25.2705i	1.10500	−	1.10500i	0.111205	−	0.993798i	$-0.464529\pi$
$523$	25.2705	−	25.2705i	1.10500	−	1.10500i	0.993798	−	0.111205i	$-0.0354709\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	26.1803i		−	1.14043i
$528$		0			0
$529$		−	0.124612i		−	0.00541790i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.76393	−	18.8197i	−0.163034	−	0.815170i
$534$		0			0
$535$	5.85410	−	5.85410i	0.253095	−	0.253095i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18.3262	+	18.3262i	0.789367	+	0.789367i
$540$		0			0
$541$	14.4164	+	14.4164i	0.619810	+	0.619810i	0.945483	−	0.325673i	$-0.105591\pi$
$541$	14.4164	+	14.4164i	0.619810	+	0.619810i	−0.325673	+	0.945483i	$0.605591\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.5279	−	10.5279i	0.450964	−	0.450964i
$546$		0			0
$547$	−7.56231	−	7.56231i	−0.323341	−	0.323341i	0.526706	−	0.850047i	$-0.323427\pi$
$547$	−7.56231	−	7.56231i	−0.323341	−	0.323341i	−0.850047	+	0.526706i	$0.823427\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−30.6525	+	30.6525i	−1.30584	+	1.30584i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$			35.2361i			1.49300i	0.665385	+	0.746500i	$0.268267\pi$
$557$			35.2361i			1.49300i	−0.665385	+	0.746500i	$0.731733\pi$
$558$		0			0
$559$	19.2705	−	3.85410i	0.815056	−	0.163011i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.09017	−	1.09017i	−0.0459452	−	0.0459452i	0.683761	−	0.729706i	$-0.260343\pi$
$563$	−1.09017	−	1.09017i	−0.0459452	−	0.0459452i	−0.729706	+	0.683761i	$0.760343\pi$
$564$		0			0
$565$	−26.7082	−	26.7082i	−1.12362	−	1.12362i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.1246			0.717901			0.358951	−	0.933357i	$-0.383135\pi$
$569$	17.1246			0.717901			0.358951	+	0.933357i	$0.383135\pi$
$570$		0			0
$571$			8.29180i			0.347001i	0.984834	+	0.173500i	$0.0555078\pi$
$571$			8.29180i			0.347001i	−0.984834	+	0.173500i	$0.944492\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	16.9098	+	16.9098i	0.705189	+	0.705189i
$576$		0			0
$577$	−27.1246			−1.12921			−0.564606	−	0.825360i	$-0.690972\pi$
$577$	−27.1246			−1.12921			−0.564606	+	0.825360i	$0.690972\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−7.70820			−0.319241
$584$		0			0
$585$		0			0
$586$		0			0
$587$	23.1246			0.954455			0.477227	−	0.878780i	$-0.341642\pi$
$587$	23.1246			0.954455			0.477227	+	0.878780i	$0.341642\pi$
$588$		0			0
$589$	68.5410			2.82418
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16.5836			−0.681007			−0.340503	−	0.940243i	$-0.610597\pi$
$593$	−16.5836			−0.681007			−0.340503	+	0.940243i	$0.610597\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	41.0132i		−	1.67575i	−0.545861	−	0.837876i	$-0.683797\pi$
$599$		−	41.0132i		−	1.67575i	0.545861	−	0.837876i	$-0.316203\pi$
$600$		0			0
$601$	14.0000			0.571072			0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000			0.571072			0.285536	+	0.958368i	$0.407828\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.05573			0.246200
$606$		0			0
$607$	−29.2705	−	29.2705i	−1.18805	−	1.18805i	−0.977605	−	0.210448i	$-0.932508\pi$
$607$	−29.2705	−	29.2705i	−1.18805	−	1.18805i	−0.210448	−	0.977605i	$-0.567492\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−26.8328	−	17.8885i	−1.08554	−	0.723693i
$612$		0			0
$613$		−	13.4164i		−	0.541884i	−0.962596	−	0.270942i	$-0.912665\pi$
$613$		−	13.4164i		−	0.541884i	0.962596	−	0.270942i	$-0.0873351\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.58359			−0.184528			−0.0922642	−	0.995735i	$-0.529410\pi$
$617$	−4.58359			−0.184528			−0.0922642	+	0.995735i	$0.529410\pi$
$618$		0			0
$619$	−17.2705	+	17.2705i	−0.694160	+	0.694160i	−0.963145	−	0.268984i	$-0.913312\pi$
$619$	−17.2705	+	17.2705i	−0.694160	+	0.694160i	0.268984	+	0.963145i	$0.413312\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	21.7082	+	21.7082i	0.865563	+	0.865563i
$630$		0			0
$631$	−29.2705	−	29.2705i	−1.16524	−	1.16524i	−0.983312	−	0.181929i	$-0.941766\pi$
$631$	−29.2705	−	29.2705i	−1.16524	−	1.16524i	−0.181929	−	0.983312i	$-0.558234\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.326238	+	0.326238i	0.0129464	+	0.0129464i
$636$		0			0
$637$	−14.0000	+	21.0000i	−0.554700	+	0.832050i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			32.9443i			1.30122i	0.759412	+	0.650610i	$0.225487\pi$
$641$			32.9443i			1.30122i	−0.759412	+	0.650610i	$0.774513\pi$
$642$		0			0
$643$		−	43.4164i		−	1.71218i	−0.516830	−	0.856088i	$-0.672888\pi$
$643$		−	43.4164i		−	1.71218i	0.516830	−	0.856088i	$-0.327112\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.09017	−	1.09017i	0.0428590	−	0.0428590i	−0.685352	−	0.728211i	$-0.740352\pi$
$647$	1.09017	−	1.09017i	0.0428590	−	0.0428590i	0.728211	+	0.685352i	$0.240352\pi$
$648$		0			0
$649$	17.7082			0.695108
$650$		0			0
$651$		0			0
$652$		0			0
$653$	29.0689	−	29.0689i	1.13755	−	1.13755i	0.148666	−	0.988887i	$-0.452502\pi$
$653$	29.0689	−	29.0689i	1.13755	−	1.13755i	0.988887	−	0.148666i	$-0.0474979\pi$
$654$		0			0
$655$		−	16.5836i		−	0.647975i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			32.0689i			1.24923i	0.780934	+	0.624613i	$0.214743\pi$
$659$			32.0689i			1.24923i	−0.780934	+	0.624613i	$0.785257\pi$
$660$		0			0
$661$	10.7082	+	10.7082i	0.416501	+	0.416501i	0.883996	−	0.467495i	$-0.154843\pi$
$661$	10.7082	+	10.7082i	0.416501	+	0.416501i	−0.467495	+	0.883996i	$0.654843\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	17.7082	−	17.7082i	0.685664	−	0.685664i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.9443	+	14.9443i	0.576917	+	0.576917i
$672$		0			0
$673$	30.1246	+	30.1246i	1.16122	+	1.16122i	0.984209	+	0.177009i	$0.0566423\pi$
$673$	30.1246	+	30.1246i	1.16122	+	1.16122i	0.177009	+	0.984209i	$0.443358\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	0.0557281	+	0.0557281i	0.00214180	+	0.00214180i	0.708177	−	0.706035i	$-0.249518\pi$
$677$	0.0557281	+	0.0557281i	0.00214180	+	0.00214180i	−0.706035	+	0.708177i	$0.749518\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−35.1246			−1.34401			−0.672003	−	0.740548i	$-0.734566\pi$
$683$	−35.1246			−1.34401			−0.672003	+	0.740548i	$0.734566\pi$
$684$		0			0
$685$	−36.8328			−1.40731
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.47214	−	7.36068i	−0.0560839	−	0.280420i
$690$		0			0
$691$	34.9787	−	34.9787i	1.33065	−	1.33065i	0.425867	−	0.904786i	$-0.359969\pi$
$691$	34.9787	−	34.9787i	1.33065	−	1.33065i	0.904786	−	0.425867i	$-0.140031\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	51.7082			1.96140
$696$		0			0
$697$			16.8328i			0.637588i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			28.3607i			1.07117i	0.844482	+	0.535584i	$0.179908\pi$
$701$			28.3607i			1.07117i	−0.844482	+	0.535584i	$0.820092\pi$
$702$		0			0
$703$	−56.8328	+	56.8328i	−2.14349	+	2.14349i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−18.4164	+	18.4164i	−0.691643	+	0.691643i	−0.962593	−	0.270951i	$-0.912662\pi$
$709$	−18.4164	+	18.4164i	−0.691643	+	0.691643i	0.270951	+	0.962593i	$0.412662\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−39.5967			−1.48291
$714$		0			0
$715$	5.85410	+	29.2705i	0.218931	+	1.09466i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−31.4164			−1.17163			−0.585817	−	0.810443i	$-0.699226\pi$
$719$	−31.4164			−1.17163			−0.585817	+	0.810443i	$0.699226\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	26.1803i		−	0.972313i
$726$		0			0
$727$	−21.8541	+	21.8541i	−0.810524	+	0.810524i	−0.984712	−	0.174189i	$-0.944270\pi$
$727$	−21.8541	+	21.8541i	−0.810524	+	0.810524i	0.174189	+	0.984712i	$0.444270\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−17.2361			−0.637499
$732$		0			0
$733$			30.0000i			1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$			30.0000i			1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.763932	−	0.763932i	−0.0281398	−	0.0281398i
$738$		0			0
$739$	−7.56231	+	7.56231i	−0.278184	+	0.278184i	−0.832384	−	0.554200i	$-0.813024\pi$
$739$	−7.56231	+	7.56231i	−0.278184	+	0.278184i	0.554200	+	0.832384i	$0.313024\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	37.3050i		−	1.36859i	−0.729207	−	0.684293i	$-0.760111\pi$
$743$		−	37.3050i		−	1.36859i	0.729207	−	0.684293i	$-0.239889\pi$
$744$		0			0
$745$	−13.2918	+	13.2918i	−0.486974	+	0.486974i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$			11.1246i			0.405943i	0.979185	+	0.202971i	$0.0650599\pi$
$751$			11.1246i			0.405943i	−0.979185	+	0.202971i	$0.934940\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.326238	+	0.326238i	−0.0118730	+	0.0118730i
$756$		0			0
$757$	−13.2918	−	13.2918i	−0.483099	−	0.483099i	0.423021	−	0.906120i	$-0.360970\pi$
$757$	−13.2918	−	13.2918i	−0.483099	−	0.483099i	−0.906120	+	0.423021i	$0.860970\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	25.4721	+	25.4721i	0.923364	+	0.923364i	0.997266	−	0.0739013i	$-0.0235450\pi$
$761$	25.4721	+	25.4721i	0.923364	+	0.923364i	−0.0739013	+	0.997266i	$0.523545\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.38197	+	16.9098i	0.122116	+	0.610579i
$768$		0			0
$769$	−5.00000	−	5.00000i	−0.180305	−	0.180305i	0.611184	−	0.791489i	$-0.290694\pi$
$769$	−5.00000	−	5.00000i	−0.180305	−	0.180305i	−0.791489	+	0.611184i	$0.790694\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			30.6525i			1.10249i	0.834342	+	0.551246i	$0.185848\pi$
$773$			30.6525i			1.10249i	−0.834342	+	0.551246i	$0.814152\pi$
$774$		0			0
$775$	−29.2705	+	29.2705i	−1.05143	+	1.05143i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−44.0689			−1.57893
$780$		0			0
$781$	−13.7082			−0.490518
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11.1803	+	11.1803i	−0.399043	+	0.399043i
$786$		0			0
$787$			16.5836i			0.591141i	0.955321	+	0.295571i	$0.0955098\pi$
$787$			16.5836i			0.591141i	−0.955321	+	0.295571i	$0.904490\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−11.4164	+	17.1246i	−0.405409	+	0.608113i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−26.8885	+	26.8885i	−0.952441	+	0.952441i	−0.998919	−	0.0464782i	$-0.985200\pi$
$797$	−26.8885	+	26.8885i	−0.952441	+	0.952441i	0.0464782	+	0.998919i	$0.485200\pi$
$798$		0			0
$799$	20.0000	+	20.0000i	0.707549	+	0.707549i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	30.6525	+	30.6525i	1.08170	+	1.08170i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	26.1803i		−	0.920452i	−0.887802	−	0.460226i	$-0.847768\pi$
$809$		−	26.1803i		−	0.920452i	0.887802	−	0.460226i	$-0.152232\pi$
$810$		0			0
$811$	3.56231	−	3.56231i	0.125089	−	0.125089i	−0.641791	−	0.766880i	$-0.721808\pi$
$811$	3.56231	−	3.56231i	0.125089	−	0.125089i	0.766880	+	0.641791i	$0.221808\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	9.59675i		−	0.336159i
$816$		0			0
$817$		−	45.1246i		−	1.57871i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	37.3607	−	37.3607i	1.30390	−	1.30390i	0.378154	−	0.925743i	$-0.376559\pi$
$821$	37.3607	−	37.3607i	1.30390	−	1.30390i	0.925743	−	0.378154i	$-0.123441\pi$
$822$		0			0
$823$	22.9787	+	22.9787i	0.800988	+	0.800988i	0.983250	−	0.182262i	$-0.0583420\pi$
$823$	22.9787	+	22.9787i	0.800988	+	0.800988i	−0.182262	+	0.983250i	$0.558342\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	22.4721i		−	0.781433i	−0.920511	−	0.390716i	$-0.872227\pi$
$827$		−	22.4721i		−	0.781433i	0.920511	−	0.390716i	$-0.127773\pi$
$828$		0			0
$829$	−13.7082			−0.476106			−0.238053	−	0.971252i	$-0.576509\pi$
$829$	−13.7082			−0.476106			−0.238053	+	0.971252i	$0.576509\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	15.6525	−	15.6525i	0.542326	−	0.542326i
$834$		0			0
$835$	30.2492			1.04682
$836$		0			0
$837$		0			0
$838$		0			0
$839$	22.7984	−	22.7984i	0.787087	−	0.787087i	−0.193928	−	0.981016i	$-0.562123\pi$
$839$	22.7984	−	22.7984i	0.787087	−	0.787087i	0.981016	+	0.193928i	$0.0621230\pi$
$840$		0			0
$841$	1.58359			0.0546066
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−26.8328	+	11.1803i	−0.923077	+	0.384615i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	32.8328	−	32.8328i	1.12549	−	1.12549i
$852$		0			0
$853$	−4.87539			−0.166930			−0.0834651	−	0.996511i	$-0.526599\pi$
$853$	−4.87539			−0.166930			−0.0834651	+	0.996511i	$0.526599\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.65248	−	9.65248i	0.329722	−	0.329722i	−0.522759	−	0.852481i	$-0.675097\pi$
$857$	9.65248	−	9.65248i	0.329722	−	0.329722i	0.852481	+	0.522759i	$0.175097\pi$
$858$		0			0
$859$		−	51.7082i		−	1.76426i	−0.471005	−	0.882131i	$-0.656109\pi$
$859$		−	51.7082i		−	1.76426i	0.471005	−	0.882131i	$-0.343891\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	34.4721i		−	1.17345i	−0.809788	−	0.586723i	$-0.800418\pi$
$863$		−	34.4721i		−	1.17345i	0.809788	−	0.586723i	$-0.199582\pi$
$864$		0			0
$865$	38.4164	−	38.4164i	1.30620	−	1.30620i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.70820	−	9.70820i	0.329328	−	0.329328i
$870$		0			0
$871$	0.583592	−	0.875388i	0.0197743	−	0.0296614i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	46.8328			1.58143			0.790716	−	0.612183i	$-0.209709\pi$
$877$	46.8328			1.58143			0.790716	+	0.612183i	$0.209709\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			20.0689i			0.676138i	0.941121	+	0.338069i	$0.109774\pi$
$881$			20.0689i			0.676138i	−0.941121	+	0.338069i	$0.890226\pi$
$882$		0			0
$883$	−27.8541	−	27.8541i	−0.937365	−	0.937365i	0.0607857	−	0.998151i	$-0.480639\pi$
$883$	−27.8541	−	27.8541i	−0.937365	−	0.937365i	−0.998151	+	0.0607857i	$0.980639\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21.3820	−	21.3820i	−0.717936	−	0.717936i	0.250246	−	0.968182i	$-0.419488\pi$
$887$	−21.3820	−	21.3820i	−0.717936	−	0.717936i	−0.968182	+	0.250246i	$0.919488\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−52.3607	+	52.3607i	−1.75218	+	1.75218i
$894$		0			0
$895$		−	26.8328i		−	0.896922i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	30.6525	+	30.6525i	1.02232	+	1.02232i
$900$		0			0
$901$			6.58359i			0.219331i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	18.5410			0.616324
$906$		0			0
$907$	−5.27051	+	5.27051i	−0.175004	+	0.175004i	−0.789174	−	0.614170i	$-0.789491\pi$
$907$	−5.27051	+	5.27051i	−0.175004	+	0.175004i	0.614170	+	0.789174i	$0.289491\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−50.8328			−1.68417			−0.842083	−	0.539348i	$-0.818671\pi$
$911$	−50.8328			−1.68417			−0.842083	+	0.539348i	$0.818671\pi$
$912$		0			0
$913$	27.4164	−	27.4164i	0.907351	−	0.907351i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	35.1246i		−	1.15865i	−0.815095	−	0.579327i	$-0.803315\pi$
$919$		−	35.1246i		−	1.15865i	0.815095	−	0.579327i	$-0.196685\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.61803	−	13.0902i	−0.0861736	−	0.430868i
$924$		0			0
$925$		−	48.5410i		−	1.59602i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.88854	+	2.88854i	0.0947700	+	0.0947700i	0.752902	−	0.658132i	$-0.228653\pi$
$929$	2.88854	+	2.88854i	0.0947700	+	0.0947700i	−0.658132	+	0.752902i	$0.728653\pi$
$930$		0			0
$931$	40.9787	+	40.9787i	1.34302	+	1.34302i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	26.1803i		−	0.856189i
$936$		0			0
$937$	32.4164	+	32.4164i	1.05900	+	1.05900i	0.998147	+	0.0608510i	$0.0193815\pi$
$937$	32.4164	+	32.4164i	1.05900	+	1.05900i	0.0608510	+	0.998147i	$0.480619\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13.3607	−	13.3607i	0.435546	−	0.435546i	−0.454964	−	0.890510i	$-0.650348\pi$
$941$	13.3607	−	13.3607i	0.435546	−	0.435546i	0.890510	+	0.454964i	$0.150348\pi$
$942$		0			0
$943$	25.4590			0.829058
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	3.05573i		−	0.0992978i	−0.998767	−	0.0496489i	$-0.984190\pi$
$947$		−	3.05573i		−	0.0992978i	0.998767	−	0.0496489i	$-0.0158102\pi$
$948$		0			0
$949$	−23.4164	+	35.1246i	−0.760129	+	1.14019i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	27.7639	+	27.7639i	0.899362	+	0.899362i	0.995380	−	0.0960177i	$-0.0306105\pi$
$953$	27.7639	+	27.7639i	0.899362	+	0.899362i	−0.0960177	+	0.995380i	$0.530611\pi$
$954$		0			0
$955$			16.5836i			0.536632i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	37.5410i		−	1.21100i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	7.63932i		−	0.245918i
$966$		0			0
$967$	7.12461			0.229112			0.114556	−	0.993417i	$-0.463455\pi$
$967$	7.12461			0.229112			0.114556	+	0.993417i	$0.463455\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.70820			−0.118636			−0.0593180	−	0.998239i	$-0.518893\pi$
$977$	−3.70820			−0.118636			−0.0593180	+	0.998239i	$0.518893\pi$
$978$		0			0
$979$	11.7082			0.374196
$980$		0			0
$981$		0			0
$982$		0			0
$983$	32.2918			1.02995			0.514974	−	0.857206i	$-0.327802\pi$
$983$	32.2918			1.02995			0.514974	+	0.857206i	$0.327802\pi$
$984$		0			0
$985$			26.8328i			0.854965i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			26.0689i			0.828942i
$990$		0			0
$991$	−20.5836			−0.653859			−0.326930	−	0.945049i	$-0.606014\pi$
$991$	−20.5836			−0.653859			−0.326930	+	0.945049i	$0.606014\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			25.5279i			0.809288i
$996$		0			0
$997$	−29.0000	−	29.0000i	−0.918439	−	0.918439i	0.0784767	−	0.996916i	$-0.474994\pi$
$997$	−29.0000	−	29.0000i	−0.918439	−	0.918439i	−0.996916	+	0.0784767i	$0.974994\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.u.f.577.1		4
3.2	odd	2		260.2.m.b.57.1	✓	4
5.3	odd	4		2340.2.bp.e.1513.2		4
12.11	even	2		1040.2.bg.j.577.2		4
13.8	odd	4		2340.2.bp.e.1477.2		4
15.2	even	4		1300.2.r.b.993.2		4
15.8	even	4		260.2.r.b.213.1	yes	4
15.14	odd	2		1300.2.m.b.57.2		4
39.8	even	4		260.2.r.b.177.1	yes	4
60.23	odd	4		1040.2.cd.j.993.2		4
65.8	even	4	inner	2340.2.u.f.73.2		4
156.47	odd	4		1040.2.cd.j.177.2		4
195.8	odd	4		260.2.m.b.73.1	yes	4
195.47	odd	4		1300.2.m.b.593.2		4
195.164	even	4		1300.2.r.b.957.2		4
780.203	even	4		1040.2.bg.j.593.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.m.b.57.1	✓	4	3.2	odd	2
260.2.m.b.73.1	yes	4	195.8	odd	4
260.2.r.b.177.1	yes	4	39.8	even	4
260.2.r.b.213.1	yes	4	15.8	even	4
1040.2.bg.j.577.2		4	12.11	even	2
1040.2.bg.j.593.2		4	780.203	even	4
1040.2.cd.j.177.2		4	156.47	odd	4
1040.2.cd.j.993.2		4	60.23	odd	4
1300.2.m.b.57.2		4	15.14	odd	2
1300.2.m.b.593.2		4	195.47	odd	4
1300.2.r.b.957.2		4	195.164	even	4
1300.2.r.b.993.2		4	15.2	even	4
2340.2.u.f.73.2		4	65.8	even	4	inner
2340.2.u.f.577.1		4	1.1	even	1	trivial
2340.2.bp.e.1477.2		4	13.8	odd	4
2340.2.bp.e.1513.2		4	5.3	odd	4

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.u (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +(2.61803 + 2.61803i) q^{11} +(-2.00000 + 3.00000i) q^{13} +(2.23607 - 2.23607i) q^{17} +(5.85410 + 5.85410i) q^{19} +(-3.38197 - 3.38197i) q^{23} -5.00000 q^{25} +5.23607i q^{29} +(5.85410 - 5.85410i) q^{31} +9.70820i q^{37} +(-3.76393 + 3.76393i) q^{41} +(-3.85410 - 3.85410i) q^{43} +8.94427i q^{47} +7.00000 q^{49} +(-1.47214 + 1.47214i) q^{53} +(5.85410 - 5.85410i) q^{55} +(3.38197 - 3.38197i) q^{59} +5.70820 q^{61} +(6.70820 + 4.47214i) q^{65} -0.291796 q^{67} +(-2.61803 + 2.61803i) q^{71} +11.7082 q^{73} -3.70820i q^{79} -10.4721i q^{83} +(-5.00000 - 5.00000i) q^{85} +(2.23607 - 2.23607i) q^{89} +(13.0902 - 13.0902i) q^{95} +15.4164 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{11} - 8 q^{13} + 10 q^{19} - 18 q^{23} - 20 q^{25} + 10 q^{31} - 24 q^{41} - 2 q^{43} + 28 q^{49} + 12 q^{53} + 10 q^{55} + 18 q^{59} - 4 q^{61} - 28 q^{67} - 6 q^{71} + 20 q^{73} - 20 q^{85}+ \cdots + 8 q^{97}+O(q^{100}) \) 4 * q + 6 * q^11 - 8 * q^13 + 10 * q^19 - 18 * q^23 - 20 * q^25 + 10 * q^31 - 24 * q^41 - 2 * q^43 + 28 * q^49 + 12 * q^53 + 10 * q^55 + 18 * q^59 - 4 * q^61 - 28 * q^67 - 6 * q^71 + 20 * q^73 - 20 * q^85 + 30 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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