LMFDB - Embedded newform 3420.2.f.e.1369.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3420,2,Mod(1369,3420)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3420.1369"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$27.3088374913$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 2x^{8} + 18x^{7} - 7x^{6} - 48x^{5} - 35x^{4} + 450x^{3} - 250x^{2} - 1250x + 3125 $ x^10 - 2x^9 - 2x^8 + 18x^7 - 7x^6 - 48x^5 - 35x^4 + 450x^3 - 250x^2 - 1250*x + 3125
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1369.8
Root			$-1.38485 + 1.75562i$ of defining polynomial
Character	$\chi$	$=$	3420.1369
Dual form			3420.2.f.e.1369.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.75562 + 1.38485i) q^{5} +0.408758i q^{7} -5.98891 q^{11} -5.17846i q^{13} -2.06892i q^{17} +1.00000 q^{19} +4.32169i q^{23} +(1.16437 + 4.86253i) q^{25} +8.83460 q^{29} -2.83863 q^{31} +(-0.566069 + 0.717622i) q^{35} -10.3388i q^{37} +5.50015 q^{41} -8.61371i q^{43} -1.25141i q^{47} +6.83292 q^{49} -1.65614i q^{53} +(-10.5142 - 8.29376i) q^{55} +5.02246 q^{59} -2.14491 q^{61} +(7.17140 - 9.09139i) q^{65} -5.02246i q^{67} +14.6997 q^{71} -10.1042i q^{73} -2.44802i q^{77} +0.955365 q^{79} -6.67861i q^{83} +(2.86515 - 3.63224i) q^{85} -3.81214 q^{89} +2.11674 q^{91} +(1.75562 + 1.38485i) q^{95} +13.8218i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{5} - 12 q^{11} + 10 q^{19} - 8 q^{25} + 16 q^{31} + 22 q^{35} - 24 q^{41} - 6 q^{49} - 10 q^{55} - 12 q^{59} + 32 q^{65} - 4 q^{71} - 24 q^{79} - 10 q^{85} - 12 q^{89} + 32 q^{91} + 2 q^{95}+O(q^{100}) $ 10 * q + 2 * q^5 - 12 * q^11 + 10 * q^19 - 8 * q^25 + 16 * q^31 + 22 * q^35 - 24 * q^41 - 6 * q^49 - 10 * q^55 - 12 * q^59 + 32 * q^65 - 4 * q^71 - 24 * q^79 - 10 * q^85 - 12 * q^89 + 32 * q^91 + 2 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$.

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.75562	+	1.38485i	0.785135	+	0.619324i
$5$	1.75562	+	1.38485i	0.785135	+	0.619324i
$6$		0			0
$7$			0.408758i			0.154496i	0.997012	+	0.0772480i	$0.0246133\pi$
$7$			0.408758i			0.154496i	−0.997012	+	0.0772480i	$0.975387\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.98891			−1.80573			−0.902863	−	0.429929i	$-0.858539\pi$
$11$	−5.98891			−1.80573			−0.902863	+	0.429929i	$0.858539\pi$
$12$		0			0
$13$		−	5.17846i		−	1.43625i	−0.695916	−	0.718123i	$-0.745001\pi$
$13$		−	5.17846i		−	1.43625i	0.695916	−	0.718123i	$-0.254999\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.06892i		−	0.501788i	−0.968015	−	0.250894i	$-0.919275\pi$
$17$		−	2.06892i		−	0.501788i	0.968015	−	0.250894i	$-0.0807245\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.32169i			0.901134i	0.892743	+	0.450567i	$0.148778\pi$
$23$			4.32169i			0.901134i	−0.892743	+	0.450567i	$0.851222\pi$
$24$		0			0
$25$	1.16437	+	4.86253i	0.232875	+	0.972507i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	8.83460			1.64054			0.820272	−	0.571973i	$-0.193822\pi$
$29$	8.83460			1.64054			0.820272	+	0.571973i	$0.193822\pi$
$30$		0			0
$31$	−2.83863			−0.509832			−0.254916	−	0.966963i	$-0.582048\pi$
$31$	−2.83863			−0.509832			−0.254916	+	0.966963i	$0.582048\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.566069	+	0.717622i	−0.0956832	+	0.121300i
$36$		0			0
$37$		−	10.3388i		−	1.69968i	−0.527038	−	0.849842i	$-0.676697\pi$
$37$		−	10.3388i		−	1.69968i	0.527038	−	0.849842i	$-0.323303\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.50015			0.858978			0.429489	−	0.903072i	$-0.358694\pi$
$41$	5.50015			0.858978			0.429489	+	0.903072i	$0.358694\pi$
$42$		0			0
$43$		−	8.61371i		−	1.31358i	−0.754075	−	0.656789i	$-0.771914\pi$
$43$		−	8.61371i		−	1.31358i	0.754075	−	0.656789i	$-0.228086\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	1.25141i		−	0.182537i	−0.995826	−	0.0912683i	$-0.970908\pi$
$47$		−	1.25141i		−	0.182537i	0.995826	−	0.0912683i	$-0.0290921\pi$
$48$		0			0
$49$	6.83292			0.976131
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.65614i		−	0.227489i	−0.993510	−	0.113744i	$-0.963716\pi$
$53$		−	1.65614i		−	0.227489i	0.993510	−	0.113744i	$-0.0362845\pi$
$54$		0			0
$55$	−10.5142	−	8.29376i	−1.41774	−	1.11833i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.02246			0.653869			0.326935	−	0.945047i	$-0.393984\pi$
$59$	5.02246			0.653869			0.326935	+	0.945047i	$0.393984\pi$
$60$		0			0
$61$	−2.14491			−0.274628			−0.137314	−	0.990528i	$-0.543847\pi$
$61$	−2.14491			−0.274628			−0.137314	+	0.990528i	$0.543847\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.17140	−	9.09139i	0.889502	−	1.12765i
$66$		0			0
$67$		−	5.02246i		−	0.613592i	−0.951775	−	0.306796i	$-0.900743\pi$
$67$		−	5.02246i		−	0.613592i	0.951775	−	0.306796i	$-0.0992569\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	14.6997			1.74454			0.872268	−	0.489029i	$-0.162649\pi$
$71$	14.6997			1.74454			0.872268	+	0.489029i	$0.162649\pi$
$72$		0			0
$73$		−	10.1042i		−	1.18260i	−0.806451	−	0.591301i	$-0.798614\pi$
$73$		−	10.1042i		−	1.18260i	0.806451	−	0.591301i	$-0.201386\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.44802i		−	0.278977i
$78$		0			0
$79$	0.955365			0.107487			0.0537435	−	0.998555i	$-0.482885\pi$
$79$	0.955365			0.107487			0.0537435	+	0.998555i	$0.482885\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	6.67861i		−	0.733072i	−0.930404	−	0.366536i	$-0.880544\pi$
$83$		−	6.67861i		−	0.733072i	0.930404	−	0.366536i	$-0.119456\pi$
$84$		0			0
$85$	2.86515	−	3.63224i	0.310769	−	0.393971i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.81214			−0.404086			−0.202043	−	0.979377i	$-0.564758\pi$
$89$	−3.81214			−0.404086			−0.202043	+	0.979377i	$0.564758\pi$
$90$		0			0
$91$	2.11674			0.221894
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.75562	+	1.38485i	0.180122	+	0.142083i
$96$		0			0
$97$			13.8218i			1.40339i	0.712475	+	0.701697i	$0.247574\pi$
$97$			13.8218i			1.40339i	−0.712475	+	0.701697i	$0.752426\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−11.2982			−1.12421			−0.562104	−	0.827066i	$-0.690008\pi$
$101$	−11.2982			−1.12421			−0.562104	+	0.827066i	$0.690008\pi$
$102$		0			0
$103$			0.137848i			0.0135826i	0.999977	+	0.00679130i	$0.00216176\pi$
$103$			0.137848i			0.0135826i	−0.999977	+	0.00679130i	$0.997838\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			14.4948i			1.40126i	0.713524	+	0.700631i	$0.247098\pi$
$107$			14.4948i			1.40126i	−0.713524	+	0.700631i	$0.752902\pi$
$108$		0			0
$109$	13.6658			1.30895			0.654475	−	0.756084i	$-0.272890\pi$
$109$	13.6658			1.30895			0.654475	+	0.756084i	$0.272890\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	8.01306i		−	0.753806i	−0.926253	−	0.376903i	$-0.876989\pi$
$113$		−	8.01306i		−	0.753806i	0.926253	−	0.376903i	$-0.123011\pi$
$114$		0			0
$115$	−5.98489	+	7.58722i	−0.558094	+	0.707512i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.845690			0.0775242
$120$		0			0
$121$	24.8671			2.26064
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.68969	+	10.1492i	−0.419459	+	0.907774i
$126$		0			0
$127$			6.79534i			0.602989i	0.953468	+	0.301495i	$0.0974855\pi$
$127$			6.79534i			0.602989i	−0.953468	+	0.301495i	$0.902514\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.7108			1.11055			0.555274	−	0.831667i	$-0.312613\pi$
$131$	12.7108			1.11055			0.555274	+	0.831667i	$0.312613\pi$
$132$		0			0
$133$			0.408758i			0.0354438i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1.81434i		−	0.155009i	−0.996992	−	0.0775047i	$-0.975305\pi$
$137$		−	1.81434i		−	0.155009i	0.996992	−	0.0775047i	$-0.0246953\pi$
$138$		0			0
$139$	2.70784			0.229676			0.114838	−	0.993384i	$-0.463365\pi$
$139$	2.70784			0.229676			0.114838	+	0.993384i	$0.463365\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			31.0134i			2.59347i
$144$		0			0
$145$	15.5102	+	12.2346i	1.28805	+	1.01603i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.22478			−0.591877			−0.295939	−	0.955207i	$-0.595632\pi$
$149$	−7.22478			−0.591877			−0.295939	+	0.955207i	$0.595632\pi$
$150$		0			0
$151$	7.50417			0.610681			0.305340	−	0.952243i	$-0.401230\pi$
$151$	7.50417			0.610681			0.305340	+	0.952243i	$0.401230\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.98354	−	3.93108i	−0.400287	−	0.315752i
$156$		0			0
$157$		−	17.0566i		−	1.36127i	−0.732624	−	0.680634i	$-0.761704\pi$
$157$		−	17.0566i		−	1.36127i	0.732624	−	0.680634i	$-0.238296\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.76652			−0.139222
$162$		0			0
$163$		−	8.49879i		−	0.665677i	−0.942984	−	0.332838i	$-0.891994\pi$
$163$		−	8.49879i		−	0.665677i	0.942984	−	0.332838i	$-0.108006\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			14.9792i			1.15912i	0.814928	+	0.579562i	$0.196776\pi$
$167$			14.9792i			1.15912i	−0.814928	+	0.579562i	$0.803224\pi$
$168$		0			0
$169$	−13.8165			−1.06280
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	5.34415i		−	0.406308i	−0.979147	−	0.203154i	$-0.934881\pi$
$173$		−	5.34415i		−	0.406308i	0.979147	−	0.203154i	$-0.0651192\pi$
$174$		0			0
$175$	−1.98760	+	0.475947i	−0.150248	+	0.0359782i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.6800			−1.02249			−0.511244	−	0.859436i	$-0.670815\pi$
$179$	−13.6800			−1.02249			−0.511244	+	0.859436i	$0.670815\pi$
$180$		0			0
$181$	−7.91073			−0.588000			−0.294000	−	0.955805i	$-0.594987\pi$
$181$	−7.91073			−0.588000			−0.294000	+	0.955805i	$0.594987\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	14.3177	−	18.1509i	1.05266	−	1.33448i
$186$		0			0
$187$			12.3906i			0.906091i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−15.3683			−1.11201			−0.556006	−	0.831179i	$-0.687667\pi$
$191$	−15.3683			−1.11201			−0.556006	+	0.831179i	$0.687667\pi$
$192$		0			0
$193$		−	6.95939i		−	0.500948i	−0.968123	−	0.250474i	$-0.919414\pi$
$193$		−	6.95939i		−	0.500948i	0.968123	−	0.250474i	$-0.0805864\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	25.5491i		−	1.82030i	−0.414281	−	0.910149i	$-0.635967\pi$
$197$		−	25.5491i		−	1.82030i	0.414281	−	0.910149i	$-0.364033\pi$
$198$		0			0
$199$	−18.9835			−1.34571			−0.672854	−	0.739776i	$-0.734932\pi$
$199$	−18.9835			−1.34571			−0.672854	+	0.739776i	$0.734932\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			3.61122i			0.253458i
$204$		0			0
$205$	9.65614	+	7.61688i	0.674414	+	0.531986i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.98891			−0.414262
$210$		0			0
$211$	24.0342			1.65458			0.827290	−	0.561774i	$-0.189881\pi$
$211$	24.0342			1.65458			0.827290	+	0.561774i	$0.189881\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	11.9287	−	15.1224i	0.813531	−	1.03134i
$216$		0			0
$217$		−	1.16031i		−	0.0787671i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−10.7138			−0.720691
$222$		0			0
$223$		−	7.03659i		−	0.471205i	−0.971850	−	0.235602i	$-0.924294\pi$
$223$		−	7.03659i		−	0.471205i	0.971850	−	0.235602i	$-0.0757063\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	20.6564i		−	1.37102i	−0.728065	−	0.685508i	$-0.759580\pi$
$227$		−	20.6564i		−	1.37102i	0.728065	−	0.685508i	$-0.240420\pi$
$228$		0			0
$229$	23.5806			1.55825			0.779126	−	0.626867i	$-0.215663\pi$
$229$	23.5806			1.55825			0.779126	+	0.626867i	$0.215663\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			0.586927i			0.0384509i	0.999815	+	0.0192254i	$0.00612003\pi$
$233$			0.586927i			0.0384509i	−0.999815	+	0.0192254i	$0.993880\pi$
$234$		0			0
$235$	1.73301	−	2.19699i	0.113049	−	0.143316i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.9889			−0.775498			−0.387749	−	0.921765i	$-0.626747\pi$
$239$	−11.9889			−0.775498			−0.387749	+	0.921765i	$0.626747\pi$
$240$		0			0
$241$	−12.2643			−0.790012			−0.395006	−	0.918679i	$-0.629257\pi$
$241$	−12.2643			−0.790012			−0.395006	+	0.918679i	$0.629257\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.9960	+	9.46257i	0.766395	+	0.604542i
$246$		0			0
$247$		−	5.17846i		−	0.329498i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	17.6239			1.11241			0.556207	−	0.831044i	$-0.312256\pi$
$251$	17.6239			1.11241			0.556207	+	0.831044i	$0.312256\pi$
$252$		0			0
$253$		−	25.8822i		−	1.62720i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	19.1955i		−	1.19739i	−0.800979	−	0.598693i	$-0.795687\pi$
$257$		−	19.1955i		−	1.19739i	0.800979	−	0.598693i	$-0.204313\pi$
$258$		0			0
$259$	4.22606			0.262594
$260$		0			0
$261$		0			0
$262$		0			0
$263$			16.9314i			1.04403i	0.852935	+	0.522016i	$0.174820\pi$
$263$			16.9314i			1.04403i	−0.852935	+	0.522016i	$0.825180\pi$
$264$		0			0
$265$	2.29351	−	2.90755i	0.140889	−	0.178609i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−22.8205			−1.39139			−0.695695	−	0.718338i	$-0.744903\pi$
$269$	−22.8205			−1.39139			−0.695695	+	0.718338i	$0.744903\pi$
$270$		0			0
$271$	17.3150			1.05181			0.525905	−	0.850543i	$-0.323727\pi$
$271$	17.3150			1.05181			0.525905	+	0.850543i	$0.323727\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.97334	−	29.1213i	−0.420508	−	1.75608i
$276$		0			0
$277$			28.1413i			1.69085i	0.534096	+	0.845424i	$0.320652\pi$
$277$			28.1413i			1.69085i	−0.534096	+	0.845424i	$0.679348\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.58942			0.0948166			0.0474083	−	0.998876i	$-0.484904\pi$
$281$	1.58942			0.0948166			0.0474083	+	0.998876i	$0.484904\pi$
$282$		0			0
$283$		−	9.73984i		−	0.578974i	−0.957182	−	0.289487i	$-0.906515\pi$
$283$		−	9.73984i		−	0.578974i	0.957182	−	0.289487i	$-0.0934846\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			2.24823i			0.132709i
$288$		0			0
$289$	12.7196			0.748209
$290$		0			0
$291$		0			0
$292$		0			0
$293$			17.7940i			1.03954i	0.854307	+	0.519768i	$0.173982\pi$
$293$			17.7940i			1.03954i	−0.854307	+	0.519768i	$0.826018\pi$
$294$		0			0
$295$	8.81752	+	6.95536i	0.513376	+	0.404957i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	22.3797			1.29425
$300$		0			0
$301$	3.52092			0.202943
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.76564	−	2.97038i	−0.215620	−	0.170084i
$306$		0			0
$307$			18.7668i			1.07108i	0.844510	+	0.535539i	$0.179892\pi$
$307$			18.7668i			1.07108i	−0.844510	+	0.535539i	$0.820108\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.3344			1.37988			0.689939	−	0.723867i	$-0.257637\pi$
$311$	24.3344			1.37988			0.689939	+	0.723867i	$0.257637\pi$
$312$		0			0
$313$		−	3.93290i		−	0.222301i	−0.993804	−	0.111150i	$-0.964546\pi$
$313$		−	3.93290i		−	0.222301i	0.993804	−	0.111150i	$-0.0354535\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	24.2851i		−	1.36399i	−0.731358	−	0.681994i	$-0.761113\pi$
$317$		−	24.2851i		−	1.36399i	0.731358	−	0.681994i	$-0.238887\pi$
$318$		0			0
$319$	−52.9097			−2.96237
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2.06892i		−	0.115118i
$324$		0			0
$325$	25.1804	−	6.02966i	1.39676	−	0.334466i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.511523			0.0282012
$330$		0			0
$331$	−8.17579			−0.449382			−0.224691	−	0.974430i	$-0.572137\pi$
$331$	−8.17579			−0.449382			−0.224691	+	0.974430i	$0.572137\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.95536	−	8.81752i	0.380012	−	0.481752i
$336$		0			0
$337$		−	22.4686i		−	1.22394i	−0.790880	−	0.611971i	$-0.790377\pi$
$337$		−	22.4686i		−	1.22394i	0.790880	−	0.611971i	$-0.209623\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	17.0003			0.920617
$342$		0			0
$343$			5.65432i			0.305304i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	17.1055i		−	0.918272i	−0.888366	−	0.459136i	$-0.848159\pi$
$347$		−	17.1055i		−	0.918272i	0.888366	−	0.459136i	$-0.151841\pi$
$348$		0			0
$349$	−1.88256			−0.100771			−0.0503855	−	0.998730i	$-0.516045\pi$
$349$	−1.88256			−0.100771			−0.0503855	+	0.998730i	$0.516045\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			29.6578i			1.57852i	0.614056	+	0.789262i	$0.289537\pi$
$353$			29.6578i			1.57852i	−0.614056	+	0.789262i	$0.710463\pi$
$354$		0			0
$355$	25.8071	+	20.3569i	1.36970	+	1.08043i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−26.6323			−1.40560			−0.702799	−	0.711388i	$-0.748067\pi$
$359$	−26.6323			−1.40560			−0.702799	+	0.711388i	$0.748067\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$	13.9928	−	17.7390i	0.732414	−	0.928503i
$366$		0			0
$367$		−	32.4430i		−	1.69351i	−0.531981	−	0.846756i	$-0.678552\pi$
$367$		−	32.4430i		−	1.69351i	0.531981	−	0.846756i	$-0.321448\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.676962			0.0351461
$372$		0			0
$373$			37.3276i			1.93275i	0.257133	+	0.966376i	$0.417222\pi$
$373$			37.3276i			1.93275i	−0.257133	+	0.966376i	$0.582778\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	45.7496i		−	2.35623i
$378$		0			0
$379$	14.1429			0.726470			0.363235	−	0.931698i	$-0.381672\pi$
$379$	14.1429			0.726470			0.363235	+	0.931698i	$0.381672\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	24.8073i		−	1.26760i	−0.773499	−	0.633798i	$-0.781495\pi$
$383$		−	24.8073i		−	1.26760i	0.773499	−	0.633798i	$-0.218505\pi$
$384$		0			0
$385$	3.39014	−	4.29778i	0.172778	−	0.219035i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.5759			−0.941836			−0.470918	−	0.882177i	$-0.656077\pi$
$389$	−18.5759			−0.941836			−0.470918	+	0.882177i	$0.656077\pi$
$390$		0			0
$391$	8.94124			0.452178
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.67725	+	1.32304i	0.0843918	+	0.0665693i
$396$		0			0
$397$			23.4023i			1.17453i	0.809395	+	0.587264i	$0.199795\pi$
$397$			23.4023i			1.17453i	−0.809395	+	0.587264i	$0.800205\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.13489			0.206486			0.103243	−	0.994656i	$-0.467078\pi$
$401$	4.13489			0.206486			0.103243	+	0.994656i	$0.467078\pi$
$402$		0			0
$403$			14.6997i			0.732245i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			61.9180i			3.06916i
$408$		0			0
$409$	−12.1234			−0.599466			−0.299733	−	0.954023i	$-0.596898\pi$
$409$	−12.1234			−0.599466			−0.299733	+	0.954023i	$0.596898\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			2.05297i			0.101020i
$414$		0			0
$415$	9.24888	−	11.7251i	0.454009	−	0.575561i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−22.9439			−1.12088			−0.560442	−	0.828194i	$-0.689369\pi$
$419$	−22.9439			−1.12088			−0.560442	+	0.828194i	$0.689369\pi$
$420$		0			0
$421$	−0.401265			−0.0195565			−0.00977823	−	0.999952i	$-0.503113\pi$
$421$	−0.401265			−0.0195565			−0.00977823	+	0.999952i	$0.503113\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	10.0602	−	2.40900i	0.487992	−	0.116854i
$426$		0			0
$427$		−	0.876750i		−	0.0424289i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.6578			−0.754209			−0.377105	−	0.926171i	$-0.623080\pi$
$431$	−15.6578			−0.754209			−0.377105	+	0.926171i	$0.623080\pi$
$432$		0			0
$433$		−	18.5015i		−	0.889125i	−0.895748	−	0.444563i	$-0.853359\pi$
$433$		−	18.5015i		−	0.889125i	0.895748	−	0.444563i	$-0.146641\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			4.32169i			0.206734i
$438$		0			0
$439$	−25.3713			−1.21091			−0.605454	−	0.795880i	$-0.707008\pi$
$439$	−25.3713			−1.21091			−0.605454	+	0.795880i	$0.707008\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	16.3895i		−	0.778691i	−0.921092	−	0.389345i	$-0.872701\pi$
$443$		−	16.3895i		−	0.778691i	0.921092	−	0.389345i	$-0.127299\pi$
$444$		0			0
$445$	−6.69265	−	5.27925i	−0.317262	−	0.250260i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−39.6155			−1.86957			−0.934787	−	0.355210i	$-0.884409\pi$
$449$	−39.6155			−1.86957			−0.934787	+	0.355210i	$0.884409\pi$
$450$		0			0
$451$	−32.9399			−1.55108
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.71618	+	2.93137i	0.174217	+	0.137425i
$456$		0			0
$457$			31.1494i			1.45711i	0.684989	+	0.728553i	$0.259807\pi$
$457$			31.1494i			1.45711i	−0.684989	+	0.728553i	$0.740193\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	20.8738			0.972189			0.486095	−	0.873906i	$-0.338421\pi$
$461$	20.8738			0.972189			0.486095	+	0.873906i	$0.338421\pi$
$462$		0			0
$463$			8.53519i			0.396664i	0.980135	+	0.198332i	$0.0635524\pi$
$463$			8.53519i			0.396664i	−0.980135	+	0.198332i	$0.936448\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			37.9994i			1.75840i	0.476449	+	0.879202i	$0.341924\pi$
$467$			37.9994i			1.75840i	−0.476449	+	0.879202i	$0.658076\pi$
$468$		0			0
$469$	2.05297			0.0947975
$470$		0			0
$471$		0			0
$472$		0			0
$473$			51.5867i			2.37196i
$474$		0			0
$475$	1.16437	+	4.86253i	0.0534251	+	0.223108i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.52019			0.343606			0.171803	−	0.985131i	$-0.445041\pi$
$479$	7.52019			0.343606			0.171803	+	0.985131i	$0.445041\pi$
$480$		0			0
$481$	−53.5389			−2.44116
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−19.1412	+	24.2658i	−0.869156	+	1.10185i
$486$		0			0
$487$		−	26.8547i		−	1.21690i	−0.793592	−	0.608451i	$-0.791791\pi$
$487$		−	26.8547i		−	1.21690i	0.793592	−	0.608451i	$-0.208209\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.81485			0.172162			0.0860808	−	0.996288i	$-0.472566\pi$
$491$	3.81485			0.172162			0.0860808	+	0.996288i	$0.472566\pi$
$492$		0			0
$493$		−	18.2781i		−	0.823205i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			6.00863i			0.269524i
$498$		0			0
$499$	−20.6186			−0.923014			−0.461507	−	0.887137i	$-0.652691\pi$
$499$	−20.6186			−0.923014			−0.461507	+	0.887137i	$0.652691\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	2.27676i		−	0.101516i	−0.998711	−	0.0507578i	$-0.983836\pi$
$503$		−	2.27676i		−	0.101516i	0.998711	−	0.0507578i	$-0.0161637\pi$
$504$		0			0
$505$	−19.8352	−	15.6463i	−0.882656	−	0.696250i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	38.6802			1.71447			0.857236	−	0.514925i	$-0.172180\pi$
$509$	38.6802			1.71447			0.857236	+	0.514925i	$0.172180\pi$
$510$		0			0
$511$	4.13016			0.182707
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.190900	+	0.242009i	−0.00841204	+	0.0106642i
$516$		0			0
$517$			7.49457i			0.329611i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11.8604			0.519615			0.259808	−	0.965660i	$-0.416341\pi$
$521$	11.8604			0.519615			0.259808	+	0.965660i	$0.416341\pi$
$522$		0			0
$523$		−	0.892661i		−	0.0390333i	−0.999810	−	0.0195167i	$-0.993787\pi$
$523$		−	0.892661i		−	0.0390333i	0.999810	−	0.0195167i	$-0.00621274\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			5.87290i			0.255828i
$528$		0			0
$529$	4.32304			0.187958
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	28.4823i		−	1.23370i
$534$		0			0
$535$	−20.0731	+	25.4472i	−0.867836	+	1.10018i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−40.9218			−1.76262
$540$		0			0
$541$	25.4639			1.09478			0.547389	−	0.836878i	$-0.315622\pi$
$541$	25.4639			1.09478			0.547389	+	0.836878i	$0.315622\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	23.9920	+	18.9251i	1.02770	+	0.810664i
$546$		0			0
$547$		−	41.8264i		−	1.78837i	−0.447698	−	0.894185i	$-0.647756\pi$
$547$		−	41.8264i		−	1.78837i	0.447698	−	0.894185i	$-0.352244\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.83460			0.376367
$552$		0			0
$553$			0.390513i			0.0166063i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			43.8854i			1.85948i	0.368212	+	0.929742i	$0.379970\pi$
$557$			43.8854i			1.85948i	−0.368212	+	0.929742i	$0.620030\pi$
$558$		0			0
$559$	−44.6057			−1.88662
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	0.258981i		−	0.0109148i	−0.999985	−	0.00545738i	$-0.998263\pi$
$563$		−	0.258981i		−	0.0109148i	0.999985	−	0.00545738i	$-0.00173715\pi$
$564$		0			0
$565$	11.0969	−	14.0679i	0.466850	−	0.591839i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.9134			0.667125			0.333563	−	0.942728i	$-0.391749\pi$
$569$	15.9134			0.667125			0.333563	+	0.942728i	$0.391749\pi$
$570$		0			0
$571$	−25.2504			−1.05670			−0.528349	−	0.849027i	$-0.677189\pi$
$571$	−25.2504			−1.05670			−0.528349	+	0.849027i	$0.677189\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−21.0143	+	5.03206i	−0.876359	+	0.209851i
$576$		0			0
$577$			22.7838i			0.948503i	0.880389	+	0.474252i	$0.157281\pi$
$577$			22.7838i			0.948503i	−0.880389	+	0.474252i	$0.842719\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.72993			0.113257
$582$		0			0
$583$			9.91850i			0.410782i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	5.61469i		−	0.231743i	−0.993264	−	0.115871i	$-0.963034\pi$
$587$		−	5.61469i		−	0.231743i	0.993264	−	0.115871i	$-0.0369661\pi$
$588$		0			0
$589$	−2.83863			−0.116964
$590$		0			0
$591$		0			0
$592$		0			0
$593$			9.68867i			0.397866i	0.980013	+	0.198933i	$0.0637476\pi$
$593$			9.68867i			0.397866i	−0.980013	+	0.198933i	$0.936252\pi$
$594$		0			0
$595$	1.48471	+	1.17115i	0.0608670	+	0.0480126i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.13249			−0.127990			−0.0639951	−	0.997950i	$-0.520384\pi$
$599$	−3.13249			−0.127990			−0.0639951	+	0.997950i	$0.520384\pi$
$600$		0			0
$601$	36.3326			1.48204			0.741019	−	0.671484i	$-0.234343\pi$
$601$	36.3326			1.48204			0.741019	+	0.671484i	$0.234343\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	43.6571	+	34.4372i	1.77491	+	1.40007i
$606$		0			0
$607$		−	1.17048i		−	0.0475084i	−0.999718	−	0.0237542i	$-0.992438\pi$
$607$		−	1.17048i		−	0.0475084i	0.999718	−	0.0237542i	$-0.00756190\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.48037			−0.262168
$612$		0			0
$613$		−	3.88144i		−	0.156770i	−0.996923	−	0.0783849i	$-0.975024\pi$
$613$		−	3.88144i		−	0.156770i	0.996923	−	0.0783849i	$-0.0249763\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	24.9233i		−	1.00338i	−0.865049	−	0.501688i	$-0.832713\pi$
$617$		−	24.9233i		−	1.00338i	0.865049	−	0.501688i	$-0.167287\pi$
$618$		0			0
$619$	−10.0530			−0.404063			−0.202032	−	0.979379i	$-0.564754\pi$
$619$	−10.0530			−0.404063			−0.202032	+	0.979379i	$0.564754\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	1.55824i		−	0.0624297i
$624$		0			0
$625$	−22.2885	+	11.3236i	−0.891539	+	0.452945i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−21.3901			−0.852881
$630$		0			0
$631$	7.63495			0.303943			0.151971	−	0.988385i	$-0.451438\pi$
$631$	7.63495			0.303943			0.151971	+	0.988385i	$0.451438\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.41054	+	11.9300i	−0.373446	+	0.473428i
$636$		0			0
$637$		−	35.3840i		−	1.40196i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−24.3770			−0.962833			−0.481417	−	0.876492i	$-0.659878\pi$
$641$	−24.3770			−0.962833			−0.481417	+	0.876492i	$0.659878\pi$
$642$		0			0
$643$		−	5.63645i		−	0.222280i	−0.993805	−	0.111140i	$-0.964550\pi$
$643$		−	5.63645i		−	0.222280i	0.993805	−	0.111140i	$-0.0354502\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	8.72305i		−	0.342938i	−0.985189	−	0.171469i	$-0.945149\pi$
$647$		−	8.72305i		−	0.342938i	0.985189	−	0.171469i	$-0.0548514\pi$
$648$		0			0
$649$	−30.0791			−1.18071
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	18.2736i		−	0.715101i	−0.933894	−	0.357550i	$-0.883612\pi$
$653$		−	18.2736i		−	0.715101i	0.933894	−	0.357550i	$-0.116388\pi$
$654$		0			0
$655$	22.3153	+	17.6026i	0.871930	+	0.687789i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−28.8373			−1.12334			−0.561671	−	0.827361i	$-0.689841\pi$
$659$	−28.8373			−1.12334			−0.561671	+	0.827361i	$0.689841\pi$
$660$		0			0
$661$	9.22176			0.358685			0.179343	−	0.983787i	$-0.442603\pi$
$661$	9.22176			0.358685			0.179343	+	0.983787i	$0.442603\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.566069	+	0.717622i	−0.0219512	+	0.0278282i
$666$		0			0
$667$			38.1804i			1.47835i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12.8457			0.495902
$672$		0			0
$673$			3.68465i			0.142033i	0.997475	+	0.0710164i	$0.0226243\pi$
$673$			3.68465i			0.142033i	−0.997475	+	0.0710164i	$0.977376\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			21.2522i			0.816787i	0.912806	+	0.408394i	$0.133911\pi$
$677$			21.2522i			0.816787i	−0.912806	+	0.408394i	$0.866089\pi$
$678$		0			0
$679$	−5.64979			−0.216819
$680$		0			0
$681$		0			0
$682$		0			0
$683$			23.2086i			0.888053i	0.896014	+	0.444026i	$0.146450\pi$
$683$			23.2086i			0.888053i	−0.896014	+	0.444026i	$0.853550\pi$
$684$		0			0
$685$	2.51259	−	3.18528i	0.0960011	−	0.121703i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−8.57627			−0.326730
$690$		0			0
$691$	38.1744			1.45222			0.726111	−	0.687578i	$-0.241326\pi$
$691$	38.1744			1.45222			0.726111	+	0.687578i	$0.241326\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4.75393	+	3.74996i	0.180327	+	0.142244i
$696$		0			0
$697$		−	11.3794i		−	0.431025i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21.3545			0.806549			0.403274	−	0.915079i	$-0.367872\pi$
$701$	21.3545			0.806549			0.403274	+	0.915079i	$0.367872\pi$
$702$		0			0
$703$		−	10.3388i		−	0.389934i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	4.61822i		−	0.173686i
$708$		0			0
$709$	39.6825			1.49031			0.745155	−	0.666892i	$-0.232376\pi$
$709$	39.6825			1.49031			0.745155	+	0.666892i	$0.232376\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	12.2677i		−	0.459427i
$714$		0			0
$715$	−42.9489	+	54.4475i	−1.60620	+	2.03622i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.6098			−0.805910			−0.402955	−	0.915220i	$-0.632017\pi$
$719$	−21.6098			−0.805910			−0.402955	+	0.915220i	$0.632017\pi$
$720$		0			0
$721$	−0.0563467			−0.00209846
$722$		0			0
$723$		0			0
$724$		0			0
$725$	10.2868	+	42.9586i	0.382042	+	1.59544i
$726$		0			0
$727$			22.0497i			0.817779i	0.912584	+	0.408889i	$0.134084\pi$
$727$			22.0497i			0.817779i	−0.912584	+	0.408889i	$0.865916\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−17.8211			−0.659137
$732$		0			0
$733$		−	53.7342i		−	1.98472i	−0.123387	−	0.992359i	$-0.539376\pi$
$733$		−	53.7342i		−	1.98472i	0.123387	−	0.992359i	$-0.460624\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			30.0791i			1.10798i
$738$		0			0
$739$	−33.7980			−1.24328			−0.621640	−	0.783303i	$-0.713533\pi$
$739$	−33.7980			−1.24328			−0.621640	+	0.783303i	$0.713533\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	25.4016i		−	0.931893i	−0.884813	−	0.465946i	$-0.845714\pi$
$743$		−	25.4016i		−	0.931893i	0.884813	−	0.465946i	$-0.154286\pi$
$744$		0			0
$745$	−12.6839	−	10.0052i	−0.464704	−	0.366564i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.92486			−0.216489
$750$		0			0
$751$	−45.8986			−1.67486			−0.837431	−	0.546543i	$-0.815943\pi$
$751$	−45.8986			−1.67486			−0.837431	+	0.546543i	$0.815943\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	13.1744	+	10.3922i	0.479467	+	0.378209i
$756$		0			0
$757$		−	5.75924i		−	0.209323i	−0.994508	−	0.104662i	$-0.966624\pi$
$757$		−	5.75924i		−	0.209323i	0.994508	−	0.104662i	$-0.0333759\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.0788741			−0.00285919			−0.00142959	−	0.999999i	$-0.500455\pi$
$761$	−0.0788741			−0.00285919			−0.00142959	+	0.999999i	$0.500455\pi$
$762$		0			0
$763$			5.58602i			0.202228i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	26.0086i		−	0.939117i
$768$		0			0
$769$	17.7396			0.639706			0.319853	−	0.947467i	$-0.396366\pi$
$769$	17.7396			0.639706			0.319853	+	0.947467i	$0.396366\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	9.82224i		−	0.353282i	−0.984275	−	0.176641i	$-0.943477\pi$
$773$		−	9.82224i		−	0.353282i	0.984275	−	0.176641i	$-0.0565231\pi$
$774$		0			0
$775$	−3.30522	−	13.8029i	−0.118727	−	0.495815i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.50015			0.197063
$780$		0			0
$781$	−88.0353			−3.15015
$782$		0			0
$783$		0			0
$784$		0			0
$785$	23.6209	−	29.9449i	0.843066	−	1.06878i
$786$		0			0
$787$			55.4900i			1.97800i	0.147905	+	0.989002i	$0.452747\pi$
$787$			55.4900i			1.97800i	−0.147905	+	0.989002i	$0.547253\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.27541			0.116460
$792$		0			0
$793$			11.1073i			0.394433i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			7.60757i			0.269474i	0.990881	+	0.134737i	$0.0430189\pi$
$797$			7.60757i			0.269474i	−0.990881	+	0.134737i	$0.956981\pi$
$798$		0			0
$799$	−2.58907			−0.0915946
$800$		0			0
$801$		0			0
$802$		0			0
$803$			60.5129i			2.13546i
$804$		0			0
$805$	−3.10134	−	2.44637i	−0.109308	−	0.0862233i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	27.5125			0.967287			0.483644	−	0.875265i	$-0.339313\pi$
$809$	27.5125			0.967287			0.483644	+	0.875265i	$0.339313\pi$
$810$		0			0
$811$	−20.2033			−0.709432			−0.354716	−	0.934974i	$-0.615423\pi$
$811$	−20.2033			−0.709432			−0.354716	+	0.934974i	$0.615423\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	11.7696	−	14.9206i	0.412270	−	0.522646i
$816$		0			0
$817$		−	8.61371i		−	0.301355i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−47.2039			−1.64743			−0.823713	−	0.567007i	$-0.808101\pi$
$821$	−47.2039			−1.64743			−0.823713	+	0.567007i	$0.808101\pi$
$822$		0			0
$823$			52.0980i			1.81602i	0.418946	+	0.908011i	$0.362400\pi$
$823$			52.0980i			1.81602i	−0.418946	+	0.908011i	$0.637600\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	3.07481i		−	0.106922i	−0.998570	−	0.0534608i	$-0.982975\pi$
$827$		−	3.07481i		−	0.106922i	0.998570	−	0.0534608i	$-0.0170252\pi$
$828$		0			0
$829$	−1.33446			−0.0463476			−0.0231738	−	0.999731i	$-0.507377\pi$
$829$	−1.33446			−0.0463476			−0.0231738	+	0.999731i	$0.507377\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	14.1368i		−	0.489811i
$834$		0			0
$835$	−20.7439	+	26.2977i	−0.717874	+	0.910069i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21.2397			0.733276			0.366638	−	0.930364i	$-0.380509\pi$
$839$	21.2397			0.733276			0.366638	+	0.930364i	$0.380509\pi$
$840$		0			0
$841$	49.0502			1.69139
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−24.2564	−	19.1337i	−0.834445	−	0.658220i
$846$		0			0
$847$			10.1646i			0.349261i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	44.6809			1.53164
$852$		0			0
$853$			13.8195i			0.473171i	0.971611	+	0.236585i	$0.0760283\pi$
$853$			13.8195i			0.473171i	−0.971611	+	0.236585i	$0.923972\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.12237i		−	0.0383394i	−0.999816	−	0.0191697i	$-0.993898\pi$
$857$		−	1.12237i		−	0.0383394i	0.999816	−	0.0191697i	$-0.00610228\pi$
$858$		0			0
$859$	39.2847			1.34038			0.670188	−	0.742191i	$-0.266213\pi$
$859$	39.2847			1.34038			0.670188	+	0.742191i	$0.266213\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			12.7480i			0.433948i	0.976177	+	0.216974i	$0.0696186\pi$
$863$			12.7480i			0.433948i	−0.976177	+	0.216974i	$0.930381\pi$
$864$		0			0
$865$	7.40085	−	9.38227i	0.251636	−	0.319007i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5.72160			−0.194092
$870$		0			0
$871$	−26.0086			−0.881269
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.14858	−	1.91695i	−0.140248	−	0.0648047i
$876$		0			0
$877$			7.48438i			0.252729i	0.991984	+	0.126365i	$0.0403310\pi$
$877$			7.48438i			0.252729i	−0.991984	+	0.126365i	$0.959669\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−24.7242			−0.832981			−0.416490	−	0.909140i	$-0.636740\pi$
$881$	−24.7242			−0.832981			−0.416490	+	0.909140i	$0.636740\pi$
$882$		0			0
$883$			31.2232i			1.05074i	0.850872	+	0.525372i	$0.176074\pi$
$883$			31.2232i			1.05074i	−0.850872	+	0.525372i	$0.823926\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			31.8215i			1.06846i	0.845339	+	0.534230i	$0.179398\pi$
$887$			31.8215i			1.06846i	−0.845339	+	0.534230i	$0.820602\pi$
$888$		0			0
$889$	−2.77765			−0.0931595
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	1.25141i		−	0.0418768i
$894$		0			0
$895$	−24.0168	−	18.9447i	−0.802792	−	0.633252i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−25.0781			−0.836403
$900$		0			0
$901$	−3.42643			−0.114151
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−13.8882	−	10.9552i	−0.461659	−	0.364163i
$906$		0			0
$907$		−	30.5699i		−	1.01506i	−0.861635	−	0.507529i	$-0.830559\pi$
$907$		−	30.5699i		−	1.01506i	0.861635	−	0.507529i	$-0.169441\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.715809			−0.0237158			−0.0118579	−	0.999930i	$-0.503775\pi$
$911$	−0.715809			−0.0237158			−0.0118579	+	0.999930i	$0.503775\pi$
$912$		0			0
$913$			39.9976i			1.32373i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			5.19564i			0.171575i
$918$		0			0
$919$	53.8718			1.77707			0.888533	−	0.458813i	$-0.151725\pi$
$919$	53.8718			1.77707			0.888533	+	0.458813i	$0.151725\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	76.1219i		−	2.50558i
$924$		0			0
$925$	50.2726	−	12.0382i	1.65295	−	0.395814i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−28.6575			−0.940222			−0.470111	−	0.882607i	$-0.655786\pi$
$929$	−28.6575			−0.940222			−0.470111	+	0.882607i	$0.655786\pi$
$930$		0			0
$931$	6.83292			0.223940
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−17.1592	+	21.7532i	−0.561164	+	0.711404i
$936$		0			0
$937$			37.2057i			1.21546i	0.794144	+	0.607729i	$0.207919\pi$
$937$			37.2057i			1.21546i	−0.794144	+	0.607729i	$0.792081\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13.5814			0.442740			0.221370	−	0.975190i	$-0.428947\pi$
$941$	13.5814			0.442740			0.221370	+	0.975190i	$0.428947\pi$
$942$		0			0
$943$			23.7699i			0.774054i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	14.7512i		−	0.479350i	−0.970853	−	0.239675i	$-0.922959\pi$
$947$		−	14.7512i		−	0.479350i	0.970853	−	0.239675i	$-0.0770408\pi$
$948$		0			0
$949$	−52.3240			−1.69851
$950$		0			0
$951$		0			0
$952$		0			0
$953$			24.3978i			0.790322i	0.918612	+	0.395161i	$0.129311\pi$
$953$			24.3978i			0.790322i	−0.918612	+	0.395161i	$0.870689\pi$
$954$		0			0
$955$	−26.9808	−	21.2828i	−0.873079	−	0.688695i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.741625			0.0239483
$960$		0			0
$961$	−22.9422			−0.740071
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.63772	−	12.2180i	0.310249	−	0.393312i
$966$		0			0
$967$		−	30.4732i		−	0.979954i	−0.871735	−	0.489977i	$-0.837005\pi$
$967$		−	30.4732i		−	0.979954i	0.871735	−	0.489977i	$-0.162995\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−35.8154			−1.14937			−0.574686	−	0.818374i	$-0.694876\pi$
$971$	−35.8154			−1.14937			−0.574686	+	0.818374i	$0.694876\pi$
$972$		0			0
$973$			1.10685i			0.0354841i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			9.99427i			0.319745i	0.987138	+	0.159872i	$0.0511083\pi$
$977$			9.99427i			0.319745i	−0.987138	+	0.159872i	$0.948892\pi$
$978$		0			0
$979$	22.8306			0.729669
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	9.83029i		−	0.313538i	−0.987635	−	0.156769i	$-0.949892\pi$
$983$		−	9.83029i		−	0.313538i	0.987635	−	0.156769i	$-0.0501077\pi$
$984$		0			0
$985$	35.3817	−	44.8544i	1.12735	−	1.42918i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	37.2257			1.18371
$990$		0			0
$991$	−12.2811			−0.390122			−0.195061	−	0.980791i	$-0.562491\pi$
$991$	−12.2811			−0.390122			−0.195061	+	0.980791i	$0.562491\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−33.3278	−	26.2894i	−1.05656	−	0.833429i
$996$		0			0
$997$			56.8743i			1.80123i	0.434622	+	0.900613i	$0.356882\pi$
$997$			56.8743i			1.80123i	−0.434622	+	0.900613i	$0.643118\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.f.e.1369.8		10
3.2	odd	2		1140.2.f.b.229.2	✓	10
5.4	even	2	inner	3420.2.f.e.1369.7		10
15.2	even	4		5700.2.a.bc.1.3		5
15.8	even	4		5700.2.a.bd.1.3		5
15.14	odd	2		1140.2.f.b.229.7	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.f.b.229.2	✓	10	3.2	odd	2
1140.2.f.b.229.7	yes	10	15.14	odd	2
3420.2.f.e.1369.7		10	5.4	even	2	inner
3420.2.f.e.1369.8		10	1.1	even	1	trivial
5700.2.a.bc.1.3		5	15.2	even	4
5700.2.a.bd.1.3		5	15.8	even	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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.75562 + 1.38485i) q^{5} +0.408758i q^{7} -5.98891 q^{11} -5.17846i q^{13} -2.06892i q^{17} +1.00000 q^{19} +4.32169i q^{23} +(1.16437 + 4.86253i) q^{25} +8.83460 q^{29} -2.83863 q^{31} +(-0.566069 + 0.717622i) q^{35} -10.3388i q^{37} +5.50015 q^{41} -8.61371i q^{43} -1.25141i q^{47} +6.83292 q^{49} -1.65614i q^{53} +(-10.5142 - 8.29376i) q^{55} +5.02246 q^{59} -2.14491 q^{61} +(7.17140 - 9.09139i) q^{65} -5.02246i q^{67} +14.6997 q^{71} -10.1042i q^{73} -2.44802i q^{77} +0.955365 q^{79} -6.67861i q^{83} +(2.86515 - 3.63224i) q^{85} -3.81214 q^{89} +2.11674 q^{91} +(1.75562 + 1.38485i) q^{95} +13.8218i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{5} - 12 q^{11} + 10 q^{19} - 8 q^{25} + 16 q^{31} + 22 q^{35} - 24 q^{41} - 6 q^{49} - 10 q^{55} - 12 q^{59} + 32 q^{65} - 4 q^{71} - 24 q^{79} - 10 q^{85} - 12 q^{89} + 32 q^{91} + 2 q^{95}+O(q^{100}) \) 10 * q + 2 * q^5 - 12 * q^11 + 10 * q^19 - 8 * q^25 + 16 * q^31 + 22 * q^35 - 24 * q^41 - 6 * q^49 - 10 * q^55 - 12 * q^59 + 32 * q^65 - 4 * q^71 - 24 * q^79 - 10 * q^85 - 12 * q^89 + 32 * q^91 + 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more