LMFDB - Embedded newform 1456.2.k.c.337.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1456,2,Mod(337,1456)] 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("1456.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1456, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$11.6262185343$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.6
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	1456.337
Dual form			1456.2.k.c.337.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21432 q^{3} +3.21432i q^{5} -1.00000i q^{7} +1.90321 q^{9} +2.68889i q^{11} +(3.59210 + 0.311108i) q^{13} +7.11753i q^{15} -3.59210 q^{17} +8.54617i q^{19} -2.21432i q^{21} -3.28100 q^{23} -5.33185 q^{25} -2.42864 q^{27} +2.05086 q^{29} -5.83654i q^{31} +5.95407i q^{33} +3.21432 q^{35} -3.93332i q^{37} +(7.95407 + 0.688892i) q^{39} +0.755569i q^{41} +8.80642 q^{43} +6.11753i q^{45} +1.88247i q^{47} -1.00000 q^{49} -7.95407 q^{51} +2.52543 q^{53} -8.64296 q^{55} +18.9240i q^{57} +7.33185i q^{59} +9.05086 q^{61} -1.90321i q^{63} +(-1.00000 + 11.5462i) q^{65} -0.428639i q^{67} -7.26517 q^{69} -8.98418i q^{71} -5.79060i q^{73} -11.8064 q^{75} +2.68889 q^{77} +4.47949 q^{79} -11.0874 q^{81} +10.8272i q^{83} -11.5462i q^{85} +4.54125 q^{87} -5.36196i q^{89} +(0.311108 - 3.59210i) q^{91} -12.9240i q^{93} -27.4701 q^{95} +9.62867i q^{97} +5.11753i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{9} + 8 q^{13} - 8 q^{17} - 6 q^{23} + 8 q^{25} + 12 q^{27} - 14 q^{29} + 6 q^{35} + 8 q^{39} + 26 q^{43} - 6 q^{49} - 8 q^{51} + 2 q^{53} - 12 q^{55} + 28 q^{61} - 6 q^{65} - 4 q^{69} - 44 q^{75}+ \cdots - 58 q^{95}+O(q^{100}) $ 6 * q - 2 * q^9 + 8 * q^13 - 8 * q^17 - 6 * q^23 + 8 * q^25 + 12 * q^27 - 14 * q^29 + 6 * q^35 + 8 * q^39 + 26 * q^43 - 6 * q^49 - 8 * q^51 + 2 * q^53 - 12 * q^55 + 28 * q^61 - 6 * q^65 - 4 * q^69 - 44 * q^75 + 16 * q^77 - 26 * q^79 - 26 * q^81 + 40 * q^87 + 2 * q^91 - 58 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$.

$n$	$561$	$911$	$1093$	$1249$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	2.21432			1.27844			0.639219	−	0.769025i	$-0.279258\pi$
$3$	2.21432			1.27844			0.639219	+	0.769025i	$0.279258\pi$
$4$		0			0
$5$			3.21432i			1.43749i	0.695275	+	0.718744i	$0.255283\pi$
$5$			3.21432i			1.43749i	−0.695275	+	0.718744i	$0.744717\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.90321			0.634404
$10$		0			0
$11$			2.68889i			0.810731i	0.914155	+	0.405366i	$0.132856\pi$
$11$			2.68889i			0.810731i	−0.914155	+	0.405366i	$0.867144\pi$
$12$		0			0
$13$	3.59210	+	0.311108i	0.996270	+	0.0862858i
$13$	3.59210	+	0.311108i	0.996270	+	0.0862858i
$14$		0			0
$15$			7.11753i			1.83774i
$16$		0			0
$17$	−3.59210			−0.871213			−0.435607	−	0.900137i	$-0.643466\pi$
$17$	−3.59210			−0.871213			−0.435607	+	0.900137i	$0.643466\pi$
$18$		0			0
$19$			8.54617i			1.96063i	0.197449	+	0.980313i	$0.436734\pi$
$19$			8.54617i			1.96063i	−0.197449	+	0.980313i	$0.563266\pi$
$20$		0			0
$21$		−	2.21432i		−	0.483204i
$22$		0			0
$23$	−3.28100			−0.684135			−0.342068	−	0.939675i	$-0.611127\pi$
$23$	−3.28100			−0.684135			−0.342068	+	0.939675i	$0.611127\pi$
$24$		0			0
$25$	−5.33185			−1.06637
$26$		0			0
$27$	−2.42864			−0.467392
$28$		0			0
$29$	2.05086			0.380834			0.190417	−	0.981703i	$-0.439016\pi$
$29$	2.05086			0.380834			0.190417	+	0.981703i	$0.439016\pi$
$30$		0			0
$31$		−	5.83654i		−	1.04827i	−0.851634	−	0.524136i	$-0.824388\pi$
$31$		−	5.83654i		−	1.04827i	0.851634	−	0.524136i	$-0.175612\pi$
$32$		0			0
$33$			5.95407i			1.03647i
$34$		0			0
$35$	3.21432			0.543319
$36$		0			0
$37$		−	3.93332i		−	0.646634i	−0.946291	−	0.323317i	$-0.895202\pi$
$37$		−	3.93332i		−	0.646634i	0.946291	−	0.323317i	$-0.104798\pi$
$38$		0			0
$39$	7.95407	+	0.688892i	1.27367	+	0.110311i
$40$		0			0
$41$			0.755569i			0.118000i	0.998258	+	0.0590000i	$0.0187912\pi$
$41$			0.755569i			0.118000i	−0.998258	+	0.0590000i	$0.981209\pi$
$42$		0			0
$43$	8.80642			1.34297			0.671484	−	0.741019i	$-0.265657\pi$
$43$	8.80642			1.34297			0.671484	+	0.741019i	$0.265657\pi$
$44$		0			0
$45$			6.11753i			0.911948i
$46$		0			0
$47$			1.88247i			0.274586i	0.990530	+	0.137293i	$0.0438402\pi$
$47$			1.88247i			0.274586i	−0.990530	+	0.137293i	$0.956160\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−7.95407			−1.11379
$52$		0			0
$53$	2.52543			0.346894			0.173447	−	0.984843i	$-0.444509\pi$
$53$	2.52543			0.346894			0.173447	+	0.984843i	$0.444509\pi$
$54$		0			0
$55$	−8.64296			−1.16542
$56$		0			0
$57$			18.9240i			2.50654i
$58$		0			0
$59$			7.33185i			0.954526i	0.878761	+	0.477263i	$0.158371\pi$
$59$			7.33185i			0.954526i	−0.878761	+	0.477263i	$0.841629\pi$
$60$		0			0
$61$	9.05086			1.15884			0.579422	−	0.815028i	$-0.303278\pi$
$61$	9.05086			1.15884			0.579422	+	0.815028i	$0.303278\pi$
$62$		0			0
$63$		−	1.90321i		−	0.239782i
$64$		0			0
$65$	−1.00000	+	11.5462i	−0.124035	+	1.43213i
$66$		0			0
$67$		−	0.428639i		−	0.0523666i	−0.999657	−	0.0261833i	$-0.991665\pi$
$67$		−	0.428639i		−	0.0523666i	0.999657	−	0.0261833i	$-0.00833536\pi$
$68$		0			0
$69$	−7.26517			−0.874624
$70$		0			0
$71$		−	8.98418i		−	1.06623i	−0.846044	−	0.533113i	$-0.821022\pi$
$71$		−	8.98418i		−	1.06623i	0.846044	−	0.533113i	$-0.178978\pi$
$72$		0			0
$73$		−	5.79060i		−	0.677739i	−0.940833	−	0.338869i	$-0.889956\pi$
$73$		−	5.79060i		−	0.677739i	0.940833	−	0.338869i	$-0.110044\pi$
$74$		0			0
$75$	−11.8064			−1.36329
$76$		0			0
$77$	2.68889			0.306428
$78$		0			0
$79$	4.47949			0.503983			0.251991	−	0.967730i	$-0.418915\pi$
$79$	4.47949			0.503983			0.251991	+	0.967730i	$0.418915\pi$
$80$		0			0
$81$	−11.0874			−1.23194
$82$		0			0
$83$			10.8272i			1.18844i	0.804304	+	0.594218i	$0.202538\pi$
$83$			10.8272i			1.18844i	−0.804304	+	0.594218i	$0.797462\pi$
$84$		0			0
$85$		−	11.5462i		−	1.25236i
$86$		0			0
$87$	4.54125			0.486873
$88$		0			0
$89$		−	5.36196i		−	0.568367i	−0.958770	−	0.284183i	$-0.908278\pi$
$89$		−	5.36196i		−	0.568367i	0.958770	−	0.284183i	$-0.0917225\pi$
$90$		0			0
$91$	0.311108	−	3.59210i	0.0326130	−	0.376555i
$92$		0			0
$93$		−	12.9240i		−	1.34015i
$94$		0			0
$95$	−27.4701			−2.81838
$96$		0			0
$97$			9.62867i			0.977643i	0.872384	+	0.488822i	$0.162573\pi$
$97$			9.62867i			0.977643i	−0.872384	+	0.488822i	$0.837427\pi$
$98$		0			0
$99$			5.11753i			0.514331i
$100$		0			0
$101$	−13.6938			−1.36259			−0.681293	−	0.732011i	$-0.738582\pi$
$101$	−13.6938			−1.36259			−0.681293	+	0.732011i	$0.738582\pi$
$102$		0			0
$103$	12.2953			1.21149			0.605745	−	0.795659i	$-0.292875\pi$
$103$	12.2953			1.21149			0.605745	+	0.795659i	$0.292875\pi$
$104$		0			0
$105$	7.11753			0.694600
$106$		0			0
$107$	18.1891			1.75841			0.879205	−	0.476444i	$-0.158075\pi$
$107$	18.1891			1.75841			0.879205	+	0.476444i	$0.158075\pi$
$108$		0			0
$109$			8.36196i			0.800931i	0.916312	+	0.400465i	$0.131152\pi$
$109$			8.36196i			0.800931i	−0.916312	+	0.400465i	$0.868848\pi$
$110$		0			0
$111$		−	8.70964i		−	0.826682i
$112$		0			0
$113$	−8.46520			−0.796339			−0.398170	−	0.917312i	$-0.630354\pi$
$113$	−8.46520			−0.796339			−0.398170	+	0.917312i	$0.630354\pi$
$114$		0			0
$115$		−	10.5462i		−	0.983436i
$116$		0			0
$117$	6.83654	+	0.592104i	0.632038	+	0.0547400i
$118$		0			0
$119$			3.59210i			0.329288i
$120$		0			0
$121$	3.76986			0.342714
$122$		0			0
$123$			1.67307i			0.150856i
$124$		0			0
$125$		−	1.06668i		−	0.0954065i
$126$		0			0
$127$	4.08742			0.362700			0.181350	−	0.983419i	$-0.441953\pi$
$127$	4.08742			0.362700			0.181350	+	0.983419i	$0.441953\pi$
$128$		0			0
$129$	19.5002			1.71690
$130$		0			0
$131$	−5.93978			−0.518961			−0.259480	−	0.965748i	$-0.583551\pi$
$131$	−5.93978			−0.518961			−0.259480	+	0.965748i	$0.583551\pi$
$132$		0			0
$133$	8.54617			0.741047
$134$		0			0
$135$		−	7.80642i		−	0.671870i
$136$		0			0
$137$		−	16.3620i		−	1.39790i	−0.715172	−	0.698948i	$-0.753652\pi$
$137$		−	16.3620i		−	1.39790i	0.715172	−	0.698948i	$-0.246348\pi$
$138$		0			0
$139$	−3.03011			−0.257011			−0.128505	−	0.991709i	$-0.541018\pi$
$139$	−3.03011			−0.257011			−0.128505	+	0.991709i	$0.541018\pi$
$140$		0			0
$141$			4.16839i			0.351041i
$142$		0			0
$143$	−0.836535	+	9.65878i	−0.0699546	+	0.807708i
$144$		0			0
$145$			6.59210i			0.547444i
$146$		0			0
$147$	−2.21432			−0.182634
$148$		0			0
$149$		−	14.5368i		−	1.19090i	−0.803392	−	0.595451i	$-0.796974\pi$
$149$		−	14.5368i		−	1.19090i	0.803392	−	0.595451i	$-0.203026\pi$
$150$		0			0
$151$		−	19.9748i		−	1.62553i	−0.582594	−	0.812764i	$-0.697962\pi$
$151$		−	19.9748i		−	1.62553i	0.582594	−	0.812764i	$-0.302038\pi$
$152$		0			0
$153$	−6.83654			−0.552701
$154$		0			0
$155$	18.7605			1.50688
$156$		0			0
$157$	7.39853			0.590467			0.295233	−	0.955425i	$-0.404603\pi$
$157$	7.39853			0.590467			0.295233	+	0.955425i	$0.404603\pi$
$158$		0			0
$159$	5.59210			0.443483
$160$		0			0
$161$			3.28100i			0.258579i
$162$		0			0
$163$			2.32693i			0.182259i	0.995839	+	0.0911296i	$0.0290477\pi$
$163$			2.32693i			0.182259i	−0.995839	+	0.0911296i	$0.970952\pi$
$164$		0			0
$165$	−19.1383			−1.48991
$166$		0			0
$167$		−	3.42219i		−	0.264817i	−0.991195	−	0.132408i	$-0.957729\pi$
$167$		−	3.42219i		−	0.264817i	0.991195	−	0.132408i	$-0.0422710\pi$
$168$		0			0
$169$	12.8064	+	2.23506i	0.985110	+	0.171928i
$170$		0			0
$171$			16.2652i			1.24383i
$172$		0			0
$173$	−8.27454			−0.629102			−0.314551	−	0.949241i	$-0.601854\pi$
$173$	−8.27454			−0.629102			−0.314551	+	0.949241i	$0.601854\pi$
$174$		0			0
$175$			5.33185i			0.403050i
$176$		0			0
$177$			16.2351i			1.22030i
$178$		0			0
$179$	−16.2257			−1.21277			−0.606383	−	0.795173i	$-0.707380\pi$
$179$	−16.2257			−1.21277			−0.606383	+	0.795173i	$0.707380\pi$
$180$		0			0
$181$	9.20495			0.684199			0.342099	−	0.939664i	$-0.388862\pi$
$181$	9.20495			0.684199			0.342099	+	0.939664i	$0.388862\pi$
$182$		0			0
$183$	20.0415			1.48151
$184$		0			0
$185$	12.6430			0.929529
$186$		0			0
$187$		−	9.65878i		−	0.706320i
$188$		0			0
$189$			2.42864i			0.176658i
$190$		0			0
$191$	6.66815			0.482490			0.241245	−	0.970464i	$-0.422444\pi$
$191$	6.66815			0.482490			0.241245	+	0.970464i	$0.422444\pi$
$192$		0			0
$193$		−	20.6035i		−	1.48307i	−0.670914	−	0.741535i	$-0.734098\pi$
$193$		−	20.6035i		−	1.48307i	0.670914	−	0.741535i	$-0.265902\pi$
$194$		0			0
$195$	−2.21432	+	25.5669i	−0.158571	+	1.83088i
$196$		0			0
$197$		−	8.36842i		−	0.596225i	−0.954531	−	0.298112i	$-0.903643\pi$
$197$		−	8.36842i		−	0.596225i	0.954531	−	0.298112i	$-0.0963571\pi$
$198$		0			0
$199$	−0.601472			−0.0426372			−0.0213186	−	0.999773i	$-0.506786\pi$
$199$	−0.601472			−0.0426372			−0.0213186	+	0.999773i	$0.506786\pi$
$200$		0			0
$201$		−	0.949145i		−	0.0669475i
$202$		0			0
$203$		−	2.05086i		−	0.143942i
$204$		0			0
$205$	−2.42864			−0.169624
$206$		0			0
$207$	−6.24443			−0.434018
$208$		0			0
$209$	−22.9797			−1.58954
$210$		0			0
$211$	1.90321			0.131023			0.0655113	−	0.997852i	$-0.479132\pi$
$211$	1.90321			0.131023			0.0655113	+	0.997852i	$0.479132\pi$
$212$		0			0
$213$		−	19.8938i		−	1.36310i
$214$		0			0
$215$			28.3067i			1.93050i
$216$		0			0
$217$	−5.83654			−0.396210
$218$		0			0
$219$		−	12.8222i		−	0.866447i
$220$		0			0
$221$	−12.9032	−	1.11753i	−0.867964	−	0.0751733i
$222$		0			0
$223$		−	10.6336i		−	0.712078i	−0.934471	−	0.356039i	$-0.884127\pi$
$223$		−	10.6336i		−	0.712078i	0.934471	−	0.356039i	$-0.115873\pi$
$224$		0			0
$225$	−10.1476			−0.676510
$226$		0			0
$227$		−	15.0509i		−	0.998960i	−0.866325	−	0.499480i	$-0.833524\pi$
$227$		−	15.0509i		−	0.998960i	0.866325	−	0.499480i	$-0.166476\pi$
$228$		0			0
$229$			6.53480i			0.431831i	0.976412	+	0.215916i	$0.0692736\pi$
$229$			6.53480i			0.431831i	−0.976412	+	0.215916i	$0.930726\pi$
$230$		0			0
$231$	5.95407			0.391749
$232$		0			0
$233$	27.9590			1.83165			0.915827	−	0.401573i	$-0.131536\pi$
$233$	27.9590			1.83165			0.915827	+	0.401573i	$0.131536\pi$
$234$		0			0
$235$	−6.05086			−0.394714
$236$		0			0
$237$	9.91903			0.644310
$238$		0			0
$239$			19.5812i			1.26660i	0.773905	+	0.633301i	$0.218301\pi$
$239$			19.5812i			1.26660i	−0.773905	+	0.633301i	$0.781699\pi$
$240$		0			0
$241$			13.3575i			0.860433i	0.902726	+	0.430217i	$0.141563\pi$
$241$			13.3575i			0.860433i	−0.902726	+	0.430217i	$0.858437\pi$
$242$		0			0
$243$	−17.2652			−1.10756
$244$		0			0
$245$		−	3.21432i		−	0.205355i
$246$		0			0
$247$	−2.65878	+	30.6987i	−0.169174	+	1.95331i
$248$		0			0
$249$			23.9748i			1.51934i
$250$		0			0
$251$	3.29682			0.208093			0.104047	−	0.994572i	$-0.466821\pi$
$251$	3.29682			0.208093			0.104047	+	0.994572i	$0.466821\pi$
$252$		0			0
$253$		−	8.82225i		−	0.554650i
$254$		0			0
$255$		−	25.5669i		−	1.60106i
$256$		0			0
$257$	23.6938			1.47798			0.738990	−	0.673717i	$-0.235303\pi$
$257$	23.6938			1.47798			0.738990	+	0.673717i	$0.235303\pi$
$258$		0			0
$259$	−3.93332			−0.244405
$260$		0			0
$261$	3.90321			0.241603
$262$		0			0
$263$	−9.99063			−0.616049			−0.308024	−	0.951378i	$-0.599668\pi$
$263$	−9.99063			−0.616049			−0.308024	+	0.951378i	$0.599668\pi$
$264$		0			0
$265$			8.11753i			0.498656i
$266$		0			0
$267$		−	11.8731i		−	0.726622i
$268$		0			0
$269$	−18.2034			−1.10988			−0.554941	−	0.831890i	$-0.687259\pi$
$269$	−18.2034			−1.10988			−0.554941	+	0.831890i	$0.687259\pi$
$270$		0			0
$271$		−	24.1748i		−	1.46852i	−0.678870	−	0.734258i	$-0.737530\pi$
$271$		−	24.1748i		−	1.46852i	0.678870	−	0.734258i	$-0.262470\pi$
$272$		0			0
$273$	0.688892	−	7.95407i	0.0416937	−	0.481402i
$274$		0			0
$275$		−	14.3368i		−	0.864540i
$276$		0			0
$277$	−1.69535			−0.101863			−0.0509317	−	0.998702i	$-0.516219\pi$
$277$	−1.69535			−0.101863			−0.0509317	+	0.998702i	$0.516219\pi$
$278$		0			0
$279$		−	11.1082i		−	0.665028i
$280$		0			0
$281$		−	11.6479i		−	0.694854i	−0.937707	−	0.347427i	$-0.887055\pi$
$281$		−	11.6479i		−	0.694854i	0.937707	−	0.347427i	$-0.112945\pi$
$282$		0			0
$283$	12.1334			0.721253			0.360626	−	0.932710i	$-0.382563\pi$
$283$	12.1334			0.721253			0.360626	+	0.932710i	$0.382563\pi$
$284$		0			0
$285$	−60.8276			−3.60312
$286$		0			0
$287$	0.755569			0.0445998
$288$		0			0
$289$	−4.09679			−0.240988
$290$		0			0
$291$			21.3210i			1.24986i
$292$		0			0
$293$		−	11.4538i		−	0.669140i	−0.942371	−	0.334570i	$-0.891409\pi$
$293$		−	11.4538i		−	0.669140i	0.942371	−	0.334570i	$-0.108591\pi$
$294$		0			0
$295$	−23.5669			−1.37212
$296$		0			0
$297$		−	6.53035i		−	0.378929i
$298$		0			0
$299$	−11.7857	−	1.02074i	−0.681583	−	0.0590311i
$300$		0			0
$301$		−	8.80642i		−	0.507594i
$302$		0			0
$303$	−30.3225			−1.74198
$304$		0			0
$305$			29.0923i			1.66582i
$306$		0			0
$307$			3.96989i			0.226574i	0.993562	+	0.113287i	$0.0361379\pi$
$307$			3.96989i			0.226574i	−0.993562	+	0.113287i	$0.963862\pi$
$308$		0			0
$309$	27.2257			1.54882
$310$		0			0
$311$	27.3481			1.55077			0.775386	−	0.631488i	$-0.217556\pi$
$311$	27.3481			1.55077			0.775386	+	0.631488i	$0.217556\pi$
$312$		0			0
$313$	−19.1032			−1.07978			−0.539890	−	0.841736i	$-0.681534\pi$
$313$	−19.1032			−1.07978			−0.539890	+	0.841736i	$0.681534\pi$
$314$		0			0
$315$	6.11753			0.344684
$316$		0			0
$317$			9.90813i			0.556496i	0.960509	+	0.278248i	$0.0897538\pi$
$317$			9.90813i			0.556496i	−0.960509	+	0.278248i	$0.910246\pi$
$318$		0			0
$319$			5.51453i			0.308754i
$320$		0			0
$321$	40.2766			2.24802
$322$		0			0
$323$		−	30.6987i		−	1.70812i
$324$		0			0
$325$	−19.1526	−	1.65878i	−1.06239	−	0.0920126i
$326$		0			0
$327$			18.5161i			1.02394i
$328$		0			0
$329$	1.88247			0.103784
$330$		0			0
$331$		−	23.4193i		−	1.28724i	−0.765345	−	0.643620i	$-0.777432\pi$
$331$		−	23.4193i		−	1.28724i	0.765345	−	0.643620i	$-0.222568\pi$
$332$		0			0
$333$		−	7.48595i		−	0.410227i
$334$		0			0
$335$	1.37778			0.0752764
$336$		0			0
$337$	7.51606			0.409426			0.204713	−	0.978822i	$-0.434374\pi$
$337$	7.51606			0.409426			0.204713	+	0.978822i	$0.434374\pi$
$338$		0			0
$339$	−18.7447			−1.01807
$340$		0			0
$341$	15.6938			0.849868
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	23.3526i		−	1.25726i
$346$		0			0
$347$	−5.64449			−0.303012			−0.151506	−	0.988456i	$-0.548412\pi$
$347$	−5.64449			−0.303012			−0.151506	+	0.988456i	$0.548412\pi$
$348$		0			0
$349$		−	24.1590i		−	1.29320i	−0.762828	−	0.646601i	$-0.776190\pi$
$349$		−	24.1590i		−	1.29320i	0.762828	−	0.646601i	$-0.223810\pi$
$350$		0			0
$351$	−8.72393	−	0.755569i	−0.465649	−	0.0403293i
$352$		0			0
$353$			36.1289i			1.92295i	0.274896	+	0.961474i	$0.411356\pi$
$353$			36.1289i			1.92295i	−0.274896	+	0.961474i	$0.588644\pi$
$354$		0			0
$355$	28.8780			1.53269
$356$		0			0
$357$			7.95407i			0.420974i
$358$		0			0
$359$			17.4128i			0.919013i	0.888174	+	0.459507i	$0.151974\pi$
$359$			17.4128i			0.919013i	−0.888174	+	0.459507i	$0.848026\pi$
$360$		0			0
$361$	−54.0370			−2.84405
$362$		0			0
$363$	8.34767			0.438139
$364$		0			0
$365$	18.6128			0.974241
$366$		0			0
$367$	−2.93825			−0.153375			−0.0766876	−	0.997055i	$-0.524434\pi$
$367$	−2.93825			−0.153375			−0.0766876	+	0.997055i	$0.524434\pi$
$368$		0			0
$369$			1.43801i			0.0748597i
$370$		0			0
$371$		−	2.52543i		−	0.131114i
$372$		0			0
$373$	−18.7699			−0.971866			−0.485933	−	0.873996i	$-0.661520\pi$
$373$	−18.7699			−0.971866			−0.485933	+	0.873996i	$0.661520\pi$
$374$		0			0
$375$		−	2.36196i		−	0.121971i
$376$		0			0
$377$	7.36689	+	0.638037i	0.379414	+	0.0328606i
$378$		0			0
$379$		−	23.6894i		−	1.21684i	−0.793615	−	0.608421i	$-0.791803\pi$
$379$		−	23.6894i		−	1.21684i	0.793615	−	0.608421i	$-0.208197\pi$
$380$		0			0
$381$	9.05086			0.463689
$382$		0			0
$383$		−	31.6128i		−	1.61534i	−0.589634	−	0.807671i	$-0.700728\pi$
$383$		−	31.6128i		−	1.61534i	0.589634	−	0.807671i	$-0.299272\pi$
$384$		0			0
$385$			8.64296i			0.440486i
$386$		0			0
$387$	16.7605			0.851984
$388$		0			0
$389$	20.0558			1.01687			0.508434	−	0.861101i	$-0.330225\pi$
$389$	20.0558			1.01687			0.508434	+	0.861101i	$0.330225\pi$
$390$		0			0
$391$	11.7857			0.596027
$392$		0			0
$393$	−13.1526			−0.663459
$394$		0			0
$395$			14.3985i			0.724469i
$396$		0			0
$397$		−	22.8731i		−	1.14797i	−0.818867	−	0.573984i	$-0.805397\pi$
$397$		−	22.8731i		−	1.14797i	0.818867	−	0.573984i	$-0.194603\pi$
$398$		0			0
$399$	18.9240			0.947383
$400$		0			0
$401$			5.61285i			0.280292i	0.990131	+	0.140146i	$0.0447572\pi$
$401$			5.61285i			0.280292i	−0.990131	+	0.140146i	$0.955243\pi$
$402$		0			0
$403$	1.81579	−	20.9654i	0.0904510	−	1.04436i
$404$		0			0
$405$		−	35.6385i		−	1.77089i
$406$		0			0
$407$	10.5763			0.524247
$408$		0			0
$409$		−	26.1175i		−	1.29143i	−0.763579	−	0.645714i	$-0.776560\pi$
$409$		−	26.1175i		−	1.29143i	0.763579	−	0.645714i	$-0.223440\pi$
$410$		0			0
$411$		−	36.2306i		−	1.78712i
$412$		0			0
$413$	7.33185			0.360777
$414$		0			0
$415$	−34.8020			−1.70836
$416$		0			0
$417$	−6.70964			−0.328572
$418$		0			0
$419$	25.2464			1.23337			0.616685	−	0.787210i	$-0.288475\pi$
$419$	25.2464			1.23337			0.616685	+	0.787210i	$0.288475\pi$
$420$		0			0
$421$		−	8.22861i		−	0.401038i	−0.979690	−	0.200519i	$-0.935737\pi$
$421$		−	8.22861i		−	0.401038i	0.979690	−	0.200519i	$-0.0642628\pi$
$422$		0			0
$423$			3.58274i			0.174199i
$424$		0			0
$425$	19.1526			0.929036
$426$		0			0
$427$		−	9.05086i		−	0.438002i
$428$		0			0
$429$	−1.85236	+	21.3876i	−0.0894326	+	1.03260i
$430$		0			0
$431$			1.43801i			0.0692664i	0.999400	+	0.0346332i	$0.0110263\pi$
$431$			1.43801i			0.0692664i	−0.999400	+	0.0346332i	$0.988974\pi$
$432$		0			0
$433$	16.5018			0.793024			0.396512	−	0.918029i	$-0.370220\pi$
$433$	16.5018			0.793024			0.396512	+	0.918029i	$0.370220\pi$
$434$		0			0
$435$			14.5970i			0.699874i
$436$		0			0
$437$		−	28.0400i		−	1.34133i
$438$		0			0
$439$	−10.1619			−0.485003			−0.242501	−	0.970151i	$-0.577968\pi$
$439$	−10.1619			−0.485003			−0.242501	+	0.970151i	$0.577968\pi$
$440$		0			0
$441$	−1.90321			−0.0906291
$442$		0			0
$443$	−3.20787			−0.152410			−0.0762052	−	0.997092i	$-0.524280\pi$
$443$	−3.20787			−0.152410			−0.0762052	+	0.997092i	$0.524280\pi$
$444$		0			0
$445$	17.2351			0.817020
$446$		0			0
$447$		−	32.1891i		−	1.52249i
$448$		0			0
$449$			18.4099i			0.868817i	0.900716	+	0.434409i	$0.143043\pi$
$449$			18.4099i			0.868817i	−0.900716	+	0.434409i	$0.856957\pi$
$450$		0			0
$451$	−2.03164			−0.0956663
$452$		0			0
$453$		−	44.2306i		−	2.07814i
$454$		0			0
$455$	11.5462	+	1.00000i	0.541293	+	0.0468807i
$456$		0			0
$457$			3.40297i			0.159184i	0.996828	+	0.0795922i	$0.0253618\pi$
$457$			3.40297i			0.159184i	−0.996828	+	0.0795922i	$0.974638\pi$
$458$		0			0
$459$	8.72393			0.407198
$460$		0			0
$461$			17.5714i			0.818380i	0.912449	+	0.409190i	$0.134189\pi$
$461$			17.5714i			0.818380i	−0.912449	+	0.409190i	$0.865811\pi$
$462$		0			0
$463$			15.7714i			0.732959i	0.930426	+	0.366479i	$0.119437\pi$
$463$			15.7714i			0.732959i	−0.930426	+	0.366479i	$0.880563\pi$
$464$		0			0
$465$	41.5417			1.92645
$466$		0			0
$467$	−3.76694			−0.174313			−0.0871567	−	0.996195i	$-0.527778\pi$
$467$	−3.76694			−0.174313			−0.0871567	+	0.996195i	$0.527778\pi$
$468$		0			0
$469$	−0.428639			−0.0197927
$470$		0			0
$471$	16.3827			0.754875
$472$		0			0
$473$			23.6795i			1.08879i
$474$		0			0
$475$		−	45.5669i		−	2.09075i
$476$		0			0
$477$	4.80642			0.220071
$478$		0			0
$479$		−	12.1032i		−	0.553011i	−0.961012	−	0.276506i	$-0.910824\pi$
$479$		−	12.1032i		−	0.553011i	0.961012	−	0.276506i	$-0.0891764\pi$
$480$		0			0
$481$	1.22369	−	14.1289i	0.0557954	−	0.644223i
$482$		0			0
$483$			7.26517i			0.330577i
$484$		0			0
$485$	−30.9496			−1.40535
$486$		0			0
$487$			17.3778i			0.787463i	0.919226	+	0.393731i	$0.128816\pi$
$487$			17.3778i			0.787463i	−0.919226	+	0.393731i	$0.871184\pi$
$488$		0			0
$489$			5.15257i			0.233007i
$490$		0			0
$491$	−19.3921			−0.875152			−0.437576	−	0.899181i	$-0.644163\pi$
$491$	−19.3921			−0.875152			−0.437576	+	0.899181i	$0.644163\pi$
$492$		0			0
$493$	−7.36689			−0.331788
$494$		0			0
$495$	−16.4494			−0.739345
$496$		0			0
$497$	−8.98418			−0.402995
$498$		0			0
$499$			10.6702i			0.477662i	0.971061	+	0.238831i	$0.0767642\pi$
$499$			10.6702i			0.477662i	−0.971061	+	0.238831i	$0.923236\pi$
$500$		0			0
$501$		−	7.57781i		−	0.338552i
$502$		0			0
$503$	4.27655			0.190682			0.0953410	−	0.995445i	$-0.469606\pi$
$503$	4.27655			0.190682			0.0953410	+	0.995445i	$0.469606\pi$
$504$		0			0
$505$		−	44.0163i		−	1.95870i
$506$		0			0
$507$	28.3575	+	4.94914i	1.25940	+	0.219799i
$508$		0			0
$509$			33.0765i			1.46609i	0.680180	+	0.733046i	$0.261902\pi$
$509$			33.0765i			1.46609i	−0.680180	+	0.733046i	$0.738098\pi$
$510$		0			0
$511$	−5.79060			−0.256161
$512$		0			0
$513$		−	20.7556i		−	0.916381i
$514$		0			0
$515$			39.5210i			1.74150i
$516$		0			0
$517$	−5.06175			−0.222616
$518$		0			0
$519$	−18.3225			−0.804268
$520$		0			0
$521$	24.3783			1.06803			0.534015	−	0.845475i	$-0.320682\pi$
$521$	24.3783			1.06803			0.534015	+	0.845475i	$0.320682\pi$
$522$		0			0
$523$	−34.9403			−1.52783			−0.763915	−	0.645317i	$-0.776725\pi$
$523$	−34.9403			−1.52783			−0.763915	+	0.645317i	$0.776725\pi$
$524$		0			0
$525$			11.8064i			0.515275i
$526$		0			0
$527$			20.9654i			0.913269i
$528$		0			0
$529$	−12.2351			−0.531959
$530$		0			0
$531$			13.9541i			0.605555i
$532$		0			0
$533$	−0.235063	+	2.71408i	−0.0101817	+	0.117560i
$534$		0			0
$535$			58.4657i			2.52769i
$536$		0			0
$537$	−35.9289			−1.55045
$538$		0			0
$539$		−	2.68889i		−	0.115819i
$540$		0			0
$541$			30.4953i			1.31110i	0.755153	+	0.655548i	$0.227562\pi$
$541$			30.4953i			1.31110i	−0.755153	+	0.655548i	$0.772438\pi$
$542$		0			0
$543$	20.3827			0.874706
$544$		0			0
$545$	−26.8780			−1.15133
$546$		0			0
$547$	10.0049			0.427780			0.213890	−	0.976858i	$-0.431387\pi$
$547$	10.0049			0.427780			0.213890	+	0.976858i	$0.431387\pi$
$548$		0			0
$549$	17.2257			0.735175
$550$		0			0
$551$			17.5270i			0.746674i
$552$		0			0
$553$		−	4.47949i		−	0.190487i
$554$		0			0
$555$	27.9956			1.18835
$556$		0			0
$557$		−	14.1936i		−	0.601401i	−0.953719	−	0.300701i	$-0.902780\pi$
$557$		−	14.1936i		−	0.601401i	0.953719	−	0.300701i	$-0.0972205\pi$
$558$		0			0
$559$	31.6336	+	2.73975i	1.33796	+	0.115879i
$560$		0			0
$561$		−	21.3876i		−	0.902986i
$562$		0			0
$563$	−21.3590			−0.900177			−0.450088	−	0.892984i	$-0.648607\pi$
$563$	−21.3590			−0.900177			−0.450088	+	0.892984i	$0.648607\pi$
$564$		0			0
$565$		−	27.2099i		−	1.14473i
$566$		0			0
$567$			11.0874i			0.465628i
$568$		0			0
$569$	−34.2672			−1.43656			−0.718278	−	0.695757i	$-0.755069\pi$
$569$	−34.2672			−1.43656			−0.718278	+	0.695757i	$0.755069\pi$
$570$		0			0
$571$	−37.6494			−1.57558			−0.787789	−	0.615945i	$-0.788774\pi$
$571$	−37.6494			−1.57558			−0.787789	+	0.615945i	$0.788774\pi$
$572$		0			0
$573$	14.7654			0.616834
$574$		0			0
$575$	17.4938			0.729541
$576$		0			0
$577$			28.3970i			1.18218i	0.806605	+	0.591091i	$0.201303\pi$
$577$			28.3970i			1.18218i	−0.806605	+	0.591091i	$0.798697\pi$
$578$		0			0
$579$		−	45.6227i		−	1.89601i
$580$		0			0
$581$	10.8272			0.449187
$582$		0			0
$583$			6.79060i			0.281238i
$584$		0			0
$585$	−1.90321	+	21.9748i	−0.0786881	+	0.908547i
$586$		0			0
$587$			6.23659i			0.257412i	0.991683	+	0.128706i	$0.0410823\pi$
$587$			6.23659i			0.257412i	−0.991683	+	0.128706i	$0.958918\pi$
$588$		0			0
$589$	49.8800			2.05527
$590$		0			0
$591$		−	18.5303i		−	0.762237i
$592$		0			0
$593$		−	17.5698i		−	0.721506i	−0.932661	−	0.360753i	$-0.882520\pi$
$593$		−	17.5698i		−	0.721506i	0.932661	−	0.360753i	$-0.117480\pi$
$594$		0			0
$595$	−11.5462			−0.473347
$596$		0			0
$597$	−1.33185			−0.0545090
$598$		0			0
$599$	−16.9813			−0.693836			−0.346918	−	0.937896i	$-0.612772\pi$
$599$	−16.9813			−0.693836			−0.346918	+	0.937896i	$0.612772\pi$
$600$		0			0
$601$	−18.7052			−0.763001			−0.381500	−	0.924369i	$-0.624592\pi$
$601$	−18.7052			−0.763001			−0.381500	+	0.924369i	$0.624592\pi$
$602$		0			0
$603$		−	0.815792i		−	0.0332216i
$604$		0			0
$605$			12.1175i			0.492648i
$606$		0			0
$607$	10.3575			0.420399			0.210199	−	0.977659i	$-0.432589\pi$
$607$	10.3575			0.420399			0.210199	+	0.977659i	$0.432589\pi$
$608$		0			0
$609$		−	4.54125i		−	0.184021i
$610$		0			0
$611$	−0.585651	+	6.76202i	−0.0236929	+	0.273562i
$612$		0			0
$613$		−	7.02227i		−	0.283627i	−0.989893	−	0.141814i	$-0.954707\pi$
$613$		−	7.02227i		−	0.283627i	0.989893	−	0.141814i	$-0.0452933\pi$
$614$		0			0
$615$	−5.37778			−0.216853
$616$		0			0
$617$			29.9813i			1.20700i	0.797363	+	0.603500i	$0.206228\pi$
$617$			29.9813i			1.20700i	−0.797363	+	0.603500i	$0.793772\pi$
$618$		0			0
$619$			0.0285802i			0.00114874i	1.00000		0.000574368i	$0.000182827\pi$
$619$			0.0285802i			0.00114874i	−1.00000		0.000574368i	$0.999817\pi$
$620$		0			0
$621$	7.96836			0.319759
$622$		0			0
$623$	−5.36196			−0.214823
$624$		0			0
$625$	−23.2306			−0.929225
$626$		0			0
$627$	−50.8845			−2.03213
$628$		0			0
$629$			14.1289i			0.563356i
$630$		0			0
$631$		−	12.1936i		−	0.485419i	−0.970099	−	0.242709i	$-0.921964\pi$
$631$		−	12.1936i		−	0.485419i	0.970099	−	0.242709i	$-0.0780361\pi$
$632$		0			0
$633$	4.21432			0.167504
$634$		0			0
$635$			13.1383i			0.521377i
$636$		0			0
$637$	−3.59210	−	0.311108i	−0.142324	−	0.0123265i
$638$		0			0
$639$		−	17.0988i		−	0.676418i
$640$		0			0
$641$	2.82516			0.111587			0.0557935	−	0.998442i	$-0.482231\pi$
$641$	2.82516			0.111587			0.0557935	+	0.998442i	$0.482231\pi$
$642$		0			0
$643$			37.7275i			1.48783i	0.668276	+	0.743913i	$0.267032\pi$
$643$			37.7275i			1.48783i	−0.668276	+	0.743913i	$0.732968\pi$
$644$		0			0
$645$			62.6800i			2.46802i
$646$		0			0
$647$	12.3664			0.486174			0.243087	−	0.970005i	$-0.421840\pi$
$647$	12.3664			0.486174			0.243087	+	0.970005i	$0.421840\pi$
$648$		0			0
$649$	−19.7146			−0.773864
$650$		0			0
$651$	−12.9240			−0.506530
$652$		0			0
$653$	−33.0005			−1.29141			−0.645704	−	0.763588i	$-0.723436\pi$
$653$	−33.0005			−1.29141			−0.645704	+	0.763588i	$0.723436\pi$
$654$		0			0
$655$		−	19.0923i		−	0.746000i
$656$		0			0
$657$		−	11.0207i		−	0.429960i
$658$		0			0
$659$	−32.8118			−1.27817			−0.639084	−	0.769137i	$-0.720686\pi$
$659$	−32.8118			−1.27817			−0.639084	+	0.769137i	$0.720686\pi$
$660$		0			0
$661$			14.7067i			0.572025i	0.958226	+	0.286013i	$0.0923299\pi$
$661$			14.7067i			0.572025i	−0.958226	+	0.286013i	$0.907670\pi$
$662$		0			0
$663$	−28.5718	−	2.47457i	−1.10964	−	0.0961044i
$664$		0			0
$665$			27.4701i			1.06525i
$666$		0			0
$667$	−6.72885			−0.260542
$668$		0			0
$669$		−	23.5462i		−	0.910348i
$670$		0			0
$671$			24.3368i			0.939511i
$672$		0			0
$673$	−21.2908			−0.820702			−0.410351	−	0.911928i	$-0.634594\pi$
$673$	−21.2908			−0.820702			−0.410351	+	0.911928i	$0.634594\pi$
$674$		0			0
$675$	12.9491			0.498413
$676$		0			0
$677$	−3.07160			−0.118051			−0.0590256	−	0.998256i	$-0.518799\pi$
$677$	−3.07160			−0.118051			−0.0590256	+	0.998256i	$0.518799\pi$
$678$		0			0
$679$	9.62867			0.369514
$680$		0			0
$681$		−	33.3274i		−	1.27711i
$682$		0			0
$683$		−	24.7971i		−	0.948833i	−0.880301	−	0.474416i	$-0.842659\pi$
$683$		−	24.7971i		−	0.948833i	0.880301	−	0.474416i	$-0.157341\pi$
$684$		0			0
$685$	52.5926			2.00946
$686$		0			0
$687$			14.4701i			0.552070i
$688$		0			0
$689$	9.07160	+	0.785680i	0.345600	+	0.0299320i
$690$		0			0
$691$			27.0953i			1.03075i	0.856964	+	0.515376i	$0.172348\pi$
$691$			27.0953i			1.03075i	−0.856964	+	0.515376i	$0.827652\pi$
$692$		0			0
$693$	5.11753			0.194399
$694$		0			0
$695$		−	9.73975i		−	0.369450i
$696$		0			0
$697$		−	2.71408i		−	0.102803i
$698$		0			0
$699$	61.9101			2.34166
$700$		0			0
$701$	−13.5205			−0.510662			−0.255331	−	0.966854i	$-0.582185\pi$
$701$	−13.5205			−0.510662			−0.255331	+	0.966854i	$0.582185\pi$
$702$		0			0
$703$	33.6149			1.26781
$704$		0			0
$705$	−13.3985			−0.504618
$706$		0			0
$707$			13.6938i			0.515009i
$708$		0			0
$709$			50.9753i			1.91442i	0.289399	+	0.957209i	$0.406545\pi$
$709$			50.9753i			1.91442i	−0.289399	+	0.957209i	$0.593455\pi$
$710$		0			0
$711$	8.52543			0.319729
$712$		0			0
$713$			19.1497i			0.717160i
$714$		0			0
$715$	−31.0464	−	2.68889i	−1.16107	−	0.100559i
$716$		0			0
$717$			43.3590i			1.61927i
$718$		0			0
$719$	41.5417			1.54924			0.774622	−	0.632424i	$-0.217940\pi$
$719$	41.5417			1.54924			0.774622	+	0.632424i	$0.217940\pi$
$720$		0			0
$721$		−	12.2953i		−	0.457900i
$722$		0			0
$723$			29.5778i			1.10001i
$724$		0			0
$725$	−10.9349			−0.406110
$726$		0			0
$727$	−20.8988			−0.775092			−0.387546	−	0.921850i	$-0.626677\pi$
$727$	−20.8988			−0.775092			−0.387546	+	0.921850i	$0.626677\pi$
$728$		0			0
$729$	−4.96836			−0.184013
$730$		0			0
$731$	−31.6336			−1.17001
$732$		0			0
$733$			19.3575i			0.714986i	0.933916	+	0.357493i	$0.116368\pi$
$733$			19.3575i			0.714986i	−0.933916	+	0.357493i	$0.883632\pi$
$734$		0			0
$735$		−	7.11753i		−	0.262534i
$736$		0			0
$737$	1.15257			0.0424553
$738$		0			0
$739$		−	1.31111i		−	0.0482299i	−0.999709	−	0.0241149i	$-0.992323\pi$
$739$		−	1.31111i		−	0.0482299i	0.999709	−	0.0241149i	$-0.00767677\pi$
$740$		0			0
$741$	−5.88739	+	67.9768i	−0.216279	+	2.49719i
$742$		0			0
$743$			21.5210i			0.789528i	0.918783	+	0.394764i	$0.129174\pi$
$743$			21.5210i			0.789528i	−0.918783	+	0.394764i	$0.870826\pi$
$744$		0			0
$745$	46.7259			1.71191
$746$		0			0
$747$			20.6064i			0.753949i
$748$		0			0
$749$		−	18.1891i		−	0.664616i
$750$		0			0
$751$	−16.3176			−0.595436			−0.297718	−	0.954654i	$-0.596226\pi$
$751$	−16.3176			−0.595436			−0.297718	+	0.954654i	$0.596226\pi$
$752$		0			0
$753$	7.30021			0.266034
$754$		0			0
$755$	64.2054			2.33667
$756$		0			0
$757$	−18.7462			−0.681342			−0.340671	−	0.940183i	$-0.610654\pi$
$757$	−18.7462			−0.681342			−0.340671	+	0.940183i	$0.610654\pi$
$758$		0			0
$759$		−	19.5353i		−	0.709085i
$760$		0			0
$761$			10.2968i			0.373259i	0.982430	+	0.186630i	$0.0597564\pi$
$761$			10.2968i			0.373259i	−0.982430	+	0.186630i	$0.940244\pi$
$762$		0			0
$763$	8.36196			0.302723
$764$		0			0
$765$		−	21.9748i		−	0.794501i
$766$		0			0
$767$	−2.28100	+	26.3368i	−0.0823620	+	0.950966i
$768$		0			0
$769$			9.36641i			0.337761i	0.985637	+	0.168881i	$0.0540153\pi$
$769$			9.36641i			0.337761i	−0.985637	+	0.168881i	$0.945985\pi$
$770$		0			0
$771$	52.4657			1.88951
$772$		0			0
$773$		−	3.71456i		−	0.133603i	−0.997766	−	0.0668017i	$-0.978721\pi$
$773$		−	3.71456i		−	0.133603i	0.997766	−	0.0668017i	$-0.0212795\pi$
$774$		0			0
$775$			31.1195i			1.11785i
$776$		0			0
$777$	−8.70964			−0.312456
$778$		0			0
$779$	−6.45722			−0.231354
$780$		0			0
$781$	24.1575			0.864423
$782$		0			0
$783$	−4.98079			−0.177999
$784$		0			0
$785$			23.7812i			0.848789i
$786$		0			0
$787$			6.23659i			0.222311i	0.993803	+	0.111155i	$0.0354551\pi$
$787$			6.23659i			0.222311i	−0.993803	+	0.111155i	$0.964545\pi$
$788$		0			0
$789$	−22.1225			−0.787580
$790$		0			0
$791$			8.46520i			0.300988i
$792$		0			0
$793$	32.5116	+	2.81579i	1.15452	+	0.0999917i
$794$		0			0
$795$			17.9748i			0.637501i
$796$		0			0
$797$	40.0701			1.41935			0.709677	−	0.704527i	$-0.248841\pi$
$797$	40.0701			1.41935			0.709677	+	0.704527i	$0.248841\pi$
$798$		0			0
$799$		−	6.76202i		−	0.239223i
$800$		0			0
$801$		−	10.2050i		−	0.360574i
$802$		0			0
$803$	15.5703			0.549464
$804$		0			0
$805$	−10.5462			−0.371704
$806$		0			0
$807$	−40.3082			−1.41892
$808$		0			0
$809$	−2.57136			−0.0904042			−0.0452021	−	0.998978i	$-0.514393\pi$
$809$	−2.57136			−0.0904042			−0.0452021	+	0.998978i	$0.514393\pi$
$810$		0			0
$811$			28.3654i			0.996042i	0.867165	+	0.498021i	$0.165940\pi$
$811$			28.3654i			0.996042i	−0.867165	+	0.498021i	$0.834060\pi$
$812$		0			0
$813$		−	53.5308i		−	1.87741i
$814$		0			0
$815$	−7.47949			−0.261995
$816$		0			0
$817$			75.2612i			2.63306i
$818$		0			0
$819$	0.592104	−	6.83654i	0.0206898	−	0.238888i
$820$		0			0
$821$		−	31.7846i		−	1.10929i	−0.832087	−	0.554646i	$-0.812854\pi$
$821$		−	31.7846i		−	1.10929i	0.832087	−	0.554646i	$-0.187146\pi$
$822$		0			0
$823$	30.9131			1.07756			0.538781	−	0.842446i	$-0.318885\pi$
$823$	30.9131			1.07756			0.538781	+	0.842446i	$0.318885\pi$
$824$		0			0
$825$		−	31.7462i		−	1.10526i
$826$		0			0
$827$			49.7560i			1.73019i	0.501610	+	0.865094i	$0.332741\pi$
$827$			49.7560i			1.73019i	−0.501610	+	0.865094i	$0.667259\pi$
$828$		0			0
$829$	19.9190			0.691817			0.345908	−	0.938268i	$-0.387571\pi$
$829$	19.9190			0.691817			0.345908	+	0.938268i	$0.387571\pi$
$830$		0			0
$831$	−3.75404			−0.130226
$832$		0			0
$833$	3.59210			0.124459
$834$		0			0
$835$	11.0000			0.380671
$836$		0			0
$837$			14.1748i			0.489954i
$838$		0			0
$839$			11.5625i			0.399181i	0.979879	+	0.199590i	$0.0639611\pi$
$839$			11.5625i			0.399181i	−0.979879	+	0.199590i	$0.936039\pi$
$840$		0			0
$841$	−24.7940			−0.854965
$842$		0			0
$843$		−	25.7921i		−	0.888328i
$844$		0			0
$845$	−7.18421	+	41.1639i	−0.247144	+	1.41608i
$846$		0			0
$847$		−	3.76986i		−	0.129534i
$848$		0			0
$849$	26.8671			0.922077
$850$		0			0
$851$			12.9052i			0.442385i
$852$		0			0
$853$		−	1.74467i		−	0.0597363i	−0.999554	−	0.0298682i	$-0.990491\pi$
$853$		−	1.74467i		−	0.0597363i	0.999554	−	0.0298682i	$-0.00950875\pi$
$854$		0			0
$855$	−52.2815			−1.78799
$856$		0			0
$857$	42.3368			1.44620			0.723098	−	0.690745i	$-0.242717\pi$
$857$	42.3368			1.44620			0.723098	+	0.690745i	$0.242717\pi$
$858$		0			0
$859$	5.37778			0.183488			0.0917438	−	0.995783i	$-0.470756\pi$
$859$	5.37778			0.183488			0.0917438	+	0.995783i	$0.470756\pi$
$860$		0			0
$861$	1.67307			0.0570181
$862$		0			0
$863$		−	15.4291i		−	0.525213i	−0.964903	−	0.262607i	$-0.915418\pi$
$863$		−	15.4291i		−	0.525213i	0.964903	−	0.262607i	$-0.0845821\pi$
$864$		0			0
$865$		−	26.5970i		−	0.904326i
$866$		0			0
$867$	−9.07160			−0.308088
$868$		0			0
$869$			12.0449i			0.408595i
$870$		0			0
$871$	0.133353	−	1.53972i	0.00451850	−	0.0521713i
$872$		0			0
$873$			18.3254i			0.620221i
$874$		0			0
$875$	−1.06668			−0.0360602
$876$		0			0
$877$			38.5181i			1.30066i	0.759651	+	0.650331i	$0.225370\pi$
$877$			38.5181i			1.30066i	−0.759651	+	0.650331i	$0.774630\pi$
$878$		0			0
$879$		−	25.3624i		−	0.855454i
$880$		0			0
$881$	41.7373			1.40617			0.703083	−	0.711108i	$-0.251806\pi$
$881$	41.7373			1.40617			0.703083	+	0.711108i	$0.251806\pi$
$882$		0			0
$883$	52.8439			1.77834			0.889170	−	0.457577i	$-0.151283\pi$
$883$	52.8439			1.77834			0.889170	+	0.457577i	$0.151283\pi$
$884$		0			0
$885$	−52.1847			−1.75417
$886$		0			0
$887$	−8.99063			−0.301876			−0.150938	−	0.988543i	$-0.548229\pi$
$887$	−8.99063			−0.301876			−0.150938	+	0.988543i	$0.548229\pi$
$888$		0			0
$889$		−	4.08742i		−	0.137088i
$890$		0			0
$891$		−	29.8129i		−	0.998769i
$892$		0			0
$893$	−16.0879			−0.538361
$894$		0			0
$895$		−	52.1546i		−	1.74334i
$896$		0			0
$897$	−26.0973	−	2.26025i	−0.871362	−	0.0754676i
$898$		0			0
$899$		−	11.9699i		−	0.399218i
$900$		0			0
$901$	−9.07160			−0.302219
$902$		0			0
$903$		−	19.5002i		−	0.648927i
$904$		0			0
$905$			29.5877i			0.983527i
$906$		0			0
$907$	−20.7003			−0.687341			−0.343671	−	0.939090i	$-0.611670\pi$
$907$	−20.7003			−0.687341			−0.343671	+	0.939090i	$0.611670\pi$
$908$		0			0
$909$	−26.0622			−0.864430
$910$		0			0
$911$	−29.8988			−0.990590			−0.495295	−	0.868725i	$-0.664940\pi$
$911$	−29.8988			−0.990590			−0.495295	+	0.868725i	$0.664940\pi$
$912$		0			0
$913$	−29.1131			−0.963503
$914$		0			0
$915$			64.4197i			2.12965i
$916$		0			0
$917$			5.93978i			0.196149i
$918$		0			0
$919$	−38.5847			−1.27279			−0.636397	−	0.771362i	$-0.719576\pi$
$919$	−38.5847			−1.27279			−0.636397	+	0.771362i	$0.719576\pi$
$920$		0			0
$921$			8.79060i			0.289660i
$922$		0			0
$923$	2.79505	−	32.2721i	0.0920001	−	1.06225i
$924$		0			0
$925$			20.9719i			0.689552i
$926$		0			0
$927$	23.4005			0.768574
$928$		0			0
$929$			3.25581i			0.106820i	0.998573	+	0.0534098i	$0.0170090\pi$
$929$			3.25581i			0.106820i	−0.998573	+	0.0534098i	$0.982991\pi$
$930$		0			0
$931$		−	8.54617i		−	0.280089i
$932$		0			0
$933$	60.5575			1.98257
$934$		0			0
$935$	31.0464			1.01533
$936$		0			0
$937$	−11.6840			−0.381699			−0.190849	−	0.981619i	$-0.561124\pi$
$937$	−11.6840			−0.381699			−0.190849	+	0.981619i	$0.561124\pi$
$938$		0			0
$939$	−42.3007			−1.38043
$940$		0			0
$941$		−	41.9699i		−	1.36818i	−0.729398	−	0.684090i	$-0.760200\pi$
$941$		−	41.9699i		−	1.36818i	0.729398	−	0.684090i	$-0.239800\pi$
$942$		0			0
$943$		−	2.47902i		−	0.0807279i
$944$		0			0
$945$	−7.80642			−0.253943
$946$		0			0
$947$		−	20.3555i		−	0.661465i	−0.943725	−	0.330733i	$-0.892704\pi$
$947$		−	20.3555i		−	0.661465i	0.943725	−	0.330733i	$-0.107296\pi$
$948$		0			0
$949$	1.80150	−	20.8004i	0.0584792	−	0.675211i
$950$		0			0
$951$			21.9398i			0.711446i
$952$		0			0
$953$	30.5496			0.989597			0.494799	−	0.869008i	$-0.335242\pi$
$953$	30.5496			0.989597			0.494799	+	0.869008i	$0.335242\pi$
$954$		0			0
$955$			21.4336i			0.693574i
$956$		0			0
$957$			12.2109i			0.394723i
$958$		0			0
$959$	−16.3620			−0.528355
$960$		0			0
$961$	−3.06515			−0.0988757
$962$		0			0
$963$	34.6178			1.11554
$964$		0			0
$965$	66.2262			2.13190
$966$		0			0
$967$		−	54.4548i		−	1.75115i	−0.483084	−	0.875574i	$-0.660484\pi$
$967$		−	54.4548i		−	1.75115i	0.483084	−	0.875574i	$-0.339516\pi$
$968$		0			0
$969$		−	67.9768i		−	2.18373i
$970$		0			0
$971$	54.6035			1.75231			0.876155	−	0.482030i	$-0.160101\pi$
$971$	54.6035			1.75231			0.876155	+	0.482030i	$0.160101\pi$
$972$		0			0
$973$			3.03011i			0.0971409i
$974$		0			0
$975$	−42.4099	−	3.67307i	−1.35820	−	0.117632i
$976$		0			0
$977$			32.6474i			1.04448i	0.852798	+	0.522242i	$0.174904\pi$
$977$			32.6474i			1.04448i	−0.852798	+	0.522242i	$0.825096\pi$
$978$		0			0
$979$	14.4177			0.460793
$980$		0			0
$981$			15.9146i			0.508114i
$982$		0			0
$983$		−	27.5052i		−	0.877278i	−0.898663	−	0.438639i	$-0.855461\pi$
$983$		−	27.5052i		−	0.877278i	0.898663	−	0.438639i	$-0.144539\pi$
$984$		0			0
$985$	26.8988			0.857066
$986$		0			0
$987$	4.16839			0.132681
$988$		0			0
$989$	−28.8938			−0.918771
$990$		0			0
$991$	6.29390			0.199932			0.0999662	−	0.994991i	$-0.468127\pi$
$991$	6.29390			0.199932			0.0999662	+	0.994991i	$0.468127\pi$
$992$		0			0
$993$		−	51.8578i		−	1.64566i
$994$		0			0
$995$		−	1.93332i		−	0.0612905i
$996$		0			0
$997$	−20.8702			−0.660965			−0.330483	−	0.943812i	$-0.607212\pi$
$997$	−20.8702			−0.660965			−0.330483	+	0.943812i	$0.607212\pi$
$998$		0			0
$999$			9.55262i			0.302232i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.k.c.337.6		6
4.3	odd	2		91.2.c.a.64.4	yes	6
12.11	even	2		819.2.c.b.64.3		6
13.12	even	2	inner	1456.2.k.c.337.5		6
28.3	even	6		637.2.r.d.324.3		12
28.11	odd	6		637.2.r.e.324.3		12
28.19	even	6		637.2.r.d.116.4		12
28.23	odd	6		637.2.r.e.116.4		12
28.27	even	2		637.2.c.d.246.4		6
52.31	even	4		1183.2.a.j.1.2		3
52.47	even	4		1183.2.a.h.1.2		3
52.51	odd	2		91.2.c.a.64.3	✓	6
156.155	even	2		819.2.c.b.64.4		6
364.51	odd	6		637.2.r.e.116.3		12
364.83	odd	4		8281.2.a.bi.1.2		3
364.103	even	6		637.2.r.d.116.3		12
364.207	odd	6		637.2.r.e.324.4		12
364.307	odd	4		8281.2.a.be.1.2		3
364.311	even	6		637.2.r.d.324.4		12
364.363	even	2		637.2.c.d.246.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.c.a.64.3	✓	6	52.51	odd	2
91.2.c.a.64.4	yes	6	4.3	odd	2
637.2.c.d.246.3		6	364.363	even	2
637.2.c.d.246.4		6	28.27	even	2
637.2.r.d.116.3		12	364.103	even	6
637.2.r.d.116.4		12	28.19	even	6
637.2.r.d.324.3		12	28.3	even	6
637.2.r.d.324.4		12	364.311	even	6
637.2.r.e.116.3		12	364.51	odd	6
637.2.r.e.116.4		12	28.23	odd	6
637.2.r.e.324.3		12	28.11	odd	6
637.2.r.e.324.4		12	364.207	odd	6
819.2.c.b.64.3		6	12.11	even	2
819.2.c.b.64.4		6	156.155	even	2
1183.2.a.h.1.2		3	52.47	even	4
1183.2.a.j.1.2		3	52.31	even	4
1456.2.k.c.337.5		6	13.12	even	2	inner
1456.2.k.c.337.6		6	1.1	even	1	trivial
8281.2.a.be.1.2		3	364.307	odd	4
8281.2.a.bi.1.2		3	364.83	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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.21432 q^{3} +3.21432i q^{5} -1.00000i q^{7} +1.90321 q^{9} +2.68889i q^{11} +(3.59210 + 0.311108i) q^{13} +7.11753i q^{15} -3.59210 q^{17} +8.54617i q^{19} -2.21432i q^{21} -3.28100 q^{23} -5.33185 q^{25} -2.42864 q^{27} +2.05086 q^{29} -5.83654i q^{31} +5.95407i q^{33} +3.21432 q^{35} -3.93332i q^{37} +(7.95407 + 0.688892i) q^{39} +0.755569i q^{41} +8.80642 q^{43} +6.11753i q^{45} +1.88247i q^{47} -1.00000 q^{49} -7.95407 q^{51} +2.52543 q^{53} -8.64296 q^{55} +18.9240i q^{57} +7.33185i q^{59} +9.05086 q^{61} -1.90321i q^{63} +(-1.00000 + 11.5462i) q^{65} -0.428639i q^{67} -7.26517 q^{69} -8.98418i q^{71} -5.79060i q^{73} -11.8064 q^{75} +2.68889 q^{77} +4.47949 q^{79} -11.0874 q^{81} +10.8272i q^{83} -11.5462i q^{85} +4.54125 q^{87} -5.36196i q^{89} +(0.311108 - 3.59210i) q^{91} -12.9240i q^{93} -27.4701 q^{95} +9.62867i q^{97} +5.11753i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{9} + 8 q^{13} - 8 q^{17} - 6 q^{23} + 8 q^{25} + 12 q^{27} - 14 q^{29} + 6 q^{35} + 8 q^{39} + 26 q^{43} - 6 q^{49} - 8 q^{51} + 2 q^{53} - 12 q^{55} + 28 q^{61} - 6 q^{65} - 4 q^{69} - 44 q^{75}+ \cdots - 58 q^{95}+O(q^{100}) \) 6 * q - 2 * q^9 + 8 * q^13 - 8 * q^17 - 6 * q^23 + 8 * q^25 + 12 * q^27 - 14 * q^29 + 6 * q^35 + 8 * q^39 + 26 * q^43 - 6 * q^49 - 8 * q^51 + 2 * q^53 - 12 * q^55 + 28 * q^61 - 6 * q^65 - 4 * q^69 - 44 * q^75 + 16 * q^77 - 26 * q^79 - 26 * q^81 + 40 * q^87 + 2 * q^91 - 58 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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