LMFDB - Embedded newform 1456.2.k.c.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(337,1456)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			337.2
Root			$0.403032 + 0.403032i$ of defining polynomial
Character	$\chi$	$=$	1456.337
Dual form			1456.2.k.c.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67513 q^{3} +0.675131i q^{5} +1.00000i q^{7} -0.193937 q^{9} +O(q^{10})$	$q-1.67513 q^{3} +0.675131i q^{5} +1.00000i q^{7} -0.193937 q^{9} -4.48119i q^{11} +(3.28726 + 1.48119i) q^{13} -1.13093i q^{15} -3.28726 q^{17} +5.21933i q^{19} -1.67513i q^{21} -4.76845 q^{23} +4.54420 q^{25} +5.35026 q^{27} -9.31265 q^{29} -1.63752i q^{31} +7.50659i q^{33} -0.675131 q^{35} -1.44358i q^{37} +(-5.50659 - 2.48119i) q^{39} -7.92478i q^{41} +4.61213 q^{43} -0.130933i q^{45} -7.86907i q^{47} -1.00000 q^{49} +5.50659 q^{51} -3.15633 q^{53} +3.02539 q^{55} -8.74306i q^{57} +2.54420i q^{59} -2.31265 q^{61} -0.193937i q^{63} +(-1.00000 + 2.21933i) q^{65} -7.35026i q^{67} +7.98778 q^{69} -7.75623i q^{71} -15.1441i q^{73} -7.61213 q^{75} +4.48119 q^{77} -14.6629 q^{79} -8.38058 q^{81} +1.45088i q^{83} -2.21933i q^{85} +15.5999 q^{87} -7.79384i q^{89} +(-1.48119 + 3.28726i) q^{91} +2.74306i q^{93} -3.52373 q^{95} -17.9805i q^{97} +0.869067i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{9}+O(q^{10}) $ 6 * q - 2 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

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
$3$	−1.67513			−0.967137			−0.483569	−	0.875306i	$-0.660660\pi$
$3$	−1.67513			−0.967137			−0.483569	+	0.875306i	$0.660660\pi$
$4$		0			0
$5$			0.675131i			0.301928i	0.988539	+	0.150964i	$0.0482377\pi$
$5$			0.675131i			0.301928i	−0.988539	+	0.150964i	$0.951762\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−0.193937			−0.0646455
$10$		0			0
$11$		−	4.48119i		−	1.35113i	−0.737300	−	0.675565i	$-0.763900\pi$
$11$		−	4.48119i		−	1.35113i	0.737300	−	0.675565i	$-0.236100\pi$
$12$		0			0
$13$	3.28726	+	1.48119i	0.911721	+	0.410809i
$13$	3.28726	+	1.48119i	0.911721	+	0.410809i
$14$		0			0
$15$		−	1.13093i		−	0.292006i
$16$		0			0
$17$	−3.28726			−0.797277			−0.398639	−	0.917108i	$-0.630517\pi$
$17$	−3.28726			−0.797277			−0.398639	+	0.917108i	$0.630517\pi$
$18$		0			0
$19$			5.21933i			1.19740i	0.800975	+	0.598698i	$0.204315\pi$
$19$			5.21933i			1.19740i	−0.800975	+	0.598698i	$0.795685\pi$
$20$		0			0
$21$		−	1.67513i		−	0.365544i
$22$		0			0
$23$	−4.76845			−0.994291			−0.497145	−	0.867667i	$-0.665618\pi$
$23$	−4.76845			−0.994291			−0.497145	+	0.867667i	$0.665618\pi$
$24$		0			0
$25$	4.54420			0.908840
$26$		0			0
$27$	5.35026			1.02966
$28$		0			0
$29$	−9.31265			−1.72932			−0.864658	−	0.502361i	$-0.832465\pi$
$29$	−9.31265			−1.72932			−0.864658	+	0.502361i	$0.832465\pi$
$30$		0			0
$31$		−	1.63752i		−	0.294107i	−0.989129	−	0.147054i	$-0.953021\pi$
$31$		−	1.63752i		−	0.294107i	0.989129	−	0.147054i	$-0.0469790\pi$
$32$		0			0
$33$			7.50659i			1.30673i
$34$		0			0
$35$	−0.675131			−0.114118
$36$		0			0
$37$		−	1.44358i		−	0.237324i	−0.992935	−	0.118662i	$-0.962140\pi$
$37$		−	1.44358i		−	0.237324i	0.992935	−	0.118662i	$-0.0378604\pi$
$38$		0			0
$39$	−5.50659	−	2.48119i	−0.881760	−	0.397309i
$40$		0			0
$41$		−	7.92478i		−	1.23764i	−0.785532	−	0.618821i	$-0.787611\pi$
$41$		−	7.92478i		−	1.23764i	0.785532	−	0.618821i	$-0.212389\pi$
$42$		0			0
$43$	4.61213			0.703343			0.351671	−	0.936124i	$-0.385613\pi$
$43$	4.61213			0.703343			0.351671	+	0.936124i	$0.385613\pi$
$44$		0			0
$45$		−	0.130933i		−	0.0195183i
$46$		0			0
$47$		−	7.86907i		−	1.14782i	−0.818918	−	0.573911i	$-0.805426\pi$
$47$		−	7.86907i		−	1.14782i	0.818918	−	0.573911i	$-0.194574\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	5.50659			0.771076
$52$		0			0
$53$	−3.15633			−0.433555			−0.216777	−	0.976221i	$-0.569555\pi$
$53$	−3.15633			−0.433555			−0.216777	+	0.976221i	$0.569555\pi$
$54$		0			0
$55$	3.02539			0.407944
$56$		0			0
$57$		−	8.74306i		−	1.15805i
$58$		0			0
$59$			2.54420i			0.331226i	0.986191	+	0.165613i	$0.0529603\pi$
$59$			2.54420i			0.331226i	−0.986191	+	0.165613i	$0.947040\pi$
$60$		0			0
$61$	−2.31265			−0.296105			−0.148052	−	0.988980i	$-0.547300\pi$
$61$	−2.31265			−0.296105			−0.148052	+	0.988980i	$0.547300\pi$
$62$		0			0
$63$		−	0.193937i		−	0.0244337i
$64$		0			0
$65$	−1.00000	+	2.21933i	−0.124035	+	0.275274i
$66$		0			0
$67$		−	7.35026i		−	0.897977i	−0.893537	−	0.448989i	$-0.851784\pi$
$67$		−	7.35026i		−	0.897977i	0.893537	−	0.448989i	$-0.148216\pi$
$68$		0			0
$69$	7.98778			0.961616
$70$		0			0
$71$		−	7.75623i		−	0.920496i	−0.887791	−	0.460248i	$-0.847761\pi$
$71$		−	7.75623i		−	0.920496i	0.887791	−	0.460248i	$-0.152239\pi$
$72$		0			0
$73$		−	15.1441i		−	1.77248i	−0.463223	−	0.886242i	$-0.653307\pi$
$73$		−	15.1441i		−	1.77248i	0.463223	−	0.886242i	$-0.346693\pi$
$74$		0			0
$75$	−7.61213			−0.878973
$76$		0			0
$77$	4.48119			0.510679
$78$		0			0
$79$	−14.6629			−1.64971			−0.824853	−	0.565347i	$-0.808742\pi$
$79$	−14.6629			−1.64971			−0.824853	+	0.565347i	$0.808742\pi$
$80$		0			0
$81$	−8.38058			−0.931175
$82$		0			0
$83$			1.45088i			0.159254i	0.996825	+	0.0796272i	$0.0253730\pi$
$83$			1.45088i			0.159254i	−0.996825	+	0.0796272i	$0.974627\pi$
$84$		0			0
$85$		−	2.21933i		−	0.240720i
$86$		0			0
$87$	15.5999			1.67249
$88$		0			0
$89$		−	7.79384i		−	0.826146i	−0.910698	−	0.413073i	$-0.864455\pi$
$89$		−	7.79384i		−	0.826146i	0.910698	−	0.413073i	$-0.135545\pi$
$90$		0			0
$91$	−1.48119	+	3.28726i	−0.155271	+	0.344598i
$92$		0			0
$93$			2.74306i			0.284442i
$94$		0			0
$95$	−3.52373			−0.361527
$96$		0			0
$97$		−	17.9805i		−	1.82564i	−0.408360	−	0.912821i	$-0.633899\pi$
$97$		−	17.9805i		−	1.82564i	0.408360	−	0.912821i	$-0.366101\pi$
$98$		0			0
$99$			0.869067i			0.0873446i
$100$		0			0
$101$	9.33804			0.929170			0.464585	−	0.885529i	$-0.346204\pi$
$101$	9.33804			0.929170			0.464585	+	0.885529i	$0.346204\pi$
$102$		0			0
$103$	−6.23743			−0.614592			−0.307296	−	0.951614i	$-0.599424\pi$
$103$	−6.23743			−0.614592			−0.307296	+	0.951614i	$0.599424\pi$
$104$		0			0
$105$	1.13093			0.110368
$106$		0			0
$107$	−7.24472			−0.700374			−0.350187	−	0.936680i	$-0.613882\pi$
$107$	−7.24472			−0.700374			−0.350187	+	0.936680i	$0.613882\pi$
$108$		0			0
$109$			4.79384i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$109$			4.79384i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$110$		0			0
$111$			2.41819i			0.229524i
$112$		0			0
$113$	−9.34297			−0.878912			−0.439456	−	0.898264i	$-0.644829\pi$
$113$	−9.34297			−0.878912			−0.439456	+	0.898264i	$0.644829\pi$
$114$		0			0
$115$		−	3.21933i		−	0.300204i
$116$		0			0
$117$	−0.637519	−	0.287258i	−0.0589387	−	0.0265570i
$118$		0			0
$119$		−	3.28726i		−	0.301342i
$120$		0			0
$121$	−9.08110			−0.825555
$122$		0			0
$123$			13.2750i			1.19697i
$124$		0			0
$125$			6.44358i			0.576332i
$126$		0			0
$127$	1.38058			0.122507			0.0612533	−	0.998122i	$-0.480490\pi$
$127$	1.38058			0.122507			0.0612533	+	0.998122i	$0.480490\pi$
$128$		0			0
$129$	−7.72592			−0.680229
$130$		0			0
$131$	−12.4993			−1.09207			−0.546034	−	0.837763i	$-0.683863\pi$
$131$	−12.4993			−1.09207			−0.546034	+	0.837763i	$0.683863\pi$
$132$		0			0
$133$	−5.21933			−0.452573
$134$		0			0
$135$			3.61213i			0.310882i
$136$		0			0
$137$			3.20616i			0.273920i	0.990577	+	0.136960i	$0.0437332\pi$
$137$			3.20616i			0.273920i	−0.990577	+	0.136960i	$0.956267\pi$
$138$		0			0
$139$	0.249646			0.0211747			0.0105874	−	0.999944i	$-0.496630\pi$
$139$	0.249646			0.0211747			0.0105874	+	0.999944i	$0.496630\pi$
$140$		0			0
$141$			13.1817i			1.11010i
$142$		0			0
$143$	6.63752	−	14.7308i	0.555057	−	1.23185i
$144$		0			0
$145$		−	6.28726i		−	0.522128i
$146$		0			0
$147$	1.67513			0.138162
$148$		0			0
$149$		−	4.03269i		−	0.330371i	−0.986263	−	0.165185i	$-0.947178\pi$
$149$		−	4.03269i		−	0.330371i	0.986263	−	0.165185i	$-0.0528222\pi$
$150$		0			0
$151$		−	1.56959i		−	0.127731i	−0.997958	−	0.0638657i	$-0.979657\pi$
$151$		−	1.56959i		−	0.127731i	0.997958	−	0.0638657i	$-0.0203429\pi$
$152$		0			0
$153$	0.637519			0.0515404
$154$		0			0
$155$	1.10554			0.0887991
$156$		0			0
$157$	2.89938			0.231396			0.115698	−	0.993284i	$-0.463090\pi$
$157$	2.89938			0.231396			0.115698	+	0.993284i	$0.463090\pi$
$158$		0			0
$159$	5.28726			0.419307
$160$		0			0
$161$		−	4.76845i		−	0.375807i
$162$		0			0
$163$		−	17.2750i		−	1.35309i	−0.736403	−	0.676543i	$-0.763477\pi$
$163$		−	17.2750i		−	1.35309i	0.736403	−	0.676543i	$-0.236523\pi$
$164$		0			0
$165$	−5.06793			−0.394538
$166$		0			0
$167$		−	16.2931i		−	1.26080i	−0.776270	−	0.630400i	$-0.782891\pi$
$167$		−	16.2931i		−	1.26080i	0.776270	−	0.630400i	$-0.217109\pi$
$168$		0			0
$169$	8.61213	+	9.73813i	0.662471	+	0.749087i
$170$		0			0
$171$		−	1.01222i		−	0.0774063i
$172$		0			0
$173$	2.17442			0.165318			0.0826592	−	0.996578i	$-0.473659\pi$
$173$	2.17442			0.165318			0.0826592	+	0.996578i	$0.473659\pi$
$174$		0			0
$175$			4.54420i			0.343509i
$176$		0			0
$177$		−	4.26187i		−	0.320341i
$178$		0			0
$179$	0.551493			0.0412205			0.0206102	−	0.999788i	$-0.493439\pi$
$179$	0.551493			0.0412205			0.0206102	+	0.999788i	$0.493439\pi$
$180$		0			0
$181$	0.511511			0.0380203			0.0190102	−	0.999819i	$-0.493949\pi$
$181$	0.511511			0.0380203			0.0190102	+	0.999819i	$0.493949\pi$
$182$		0			0
$183$	3.87399			0.286374
$184$		0			0
$185$	0.974607			0.0716546
$186$		0			0
$187$			14.7308i			1.07723i
$188$		0			0
$189$			5.35026i			0.389174i
$190$		0			0
$191$	16.5442			1.19710			0.598548	−	0.801087i	$-0.295745\pi$
$191$	16.5442			1.19710			0.598548	+	0.801087i	$0.295745\pi$
$192$		0			0
$193$			7.41090i			0.533448i	0.963773	+	0.266724i	$0.0859412\pi$
$193$			7.41090i			0.533448i	−0.963773	+	0.266724i	$0.914059\pi$
$194$		0			0
$195$	1.67513	−	3.71767i	0.119959	−	0.266228i
$196$		0			0
$197$			7.14903i			0.509347i	0.967027	+	0.254674i	$0.0819681\pi$
$197$			7.14903i			0.509347i	−0.967027	+	0.254674i	$0.918032\pi$
$198$		0			0
$199$	−5.10062			−0.361573			−0.180787	−	0.983522i	$-0.557864\pi$
$199$	−5.10062			−0.361573			−0.180787	+	0.983522i	$0.557864\pi$
$200$		0			0
$201$			12.3127i			0.868467i
$202$		0			0
$203$		−	9.31265i		−	0.653620i
$204$		0			0
$205$	5.35026			0.373678
$206$		0			0
$207$	0.924777			0.0642765
$208$		0			0
$209$	23.3888			1.61784
$210$		0			0
$211$	−0.193937			−0.0133511			−0.00667557	−	0.999978i	$-0.502125\pi$
$211$	−0.193937			−0.0133511			−0.00667557	+	0.999978i	$0.502125\pi$
$212$		0			0
$213$			12.9927i			0.890246i
$214$		0			0
$215$			3.11379i			0.212359i
$216$		0			0
$217$	1.63752			0.111162
$218$		0			0
$219$			25.3684i			1.71423i
$220$		0			0
$221$	−10.8061	−	4.86907i	−0.726894	−	0.327529i
$222$		0			0
$223$		−	5.83875i		−	0.390992i	−0.980705	−	0.195496i	$-0.937368\pi$
$223$		−	5.83875i		−	0.390992i	0.980705	−	0.195496i	$-0.0626316\pi$
$224$		0			0
$225$	−0.881286			−0.0587524
$226$		0			0
$227$			3.68735i			0.244738i	0.992485	+	0.122369i	$0.0390491\pi$
$227$			3.68735i			0.244738i	−0.992485	+	0.122369i	$0.960951\pi$
$228$		0			0
$229$		−	5.65703i		−	0.373827i	−0.982376	−	0.186914i	$-0.940152\pi$
$229$		−	5.65703i		−	0.373827i	0.982376	−	0.186914i	$-0.0598484\pi$
$230$		0			0
$231$	−7.50659			−0.493897
$232$		0			0
$233$	−10.3258			−0.676467			−0.338234	−	0.941062i	$-0.609829\pi$
$233$	−10.3258			−0.676467			−0.338234	+	0.941062i	$0.609829\pi$
$234$		0			0
$235$	5.31265			0.346559
$236$		0			0
$237$	24.5623			1.59549
$238$		0			0
$239$			22.2882i			1.44170i	0.693089	+	0.720852i	$0.256249\pi$
$239$			22.2882i			1.44170i	−0.693089	+	0.720852i	$0.743751\pi$
$240$		0			0
$241$			29.4264i			1.89552i	0.318979	+	0.947762i	$0.396660\pi$
$241$			29.4264i			1.89552i	−0.318979	+	0.947762i	$0.603340\pi$
$242$		0			0
$243$	−2.01222			−0.129084
$244$		0			0
$245$		−	0.675131i		−	0.0431325i
$246$		0			0
$247$	−7.73084	+	17.1573i	−0.491902	+	1.09169i
$248$		0			0
$249$		−	2.43041i		−	0.154021i
$250$		0			0
$251$	21.5247			1.35863			0.679313	−	0.733849i	$-0.262278\pi$
$251$	21.5247			1.35863			0.679313	+	0.733849i	$0.262278\pi$
$252$		0			0
$253$			21.3684i			1.34342i
$254$		0			0
$255$			3.71767i			0.232809i
$256$		0			0
$257$	0.661957			0.0412917			0.0206459	−	0.999787i	$-0.493428\pi$
$257$	0.661957			0.0412917			0.0206459	+	0.999787i	$0.493428\pi$
$258$		0			0
$259$	1.44358			0.0896999
$260$		0			0
$261$	1.80606			0.111793
$262$		0			0
$263$	−5.18664			−0.319822			−0.159911	−	0.987131i	$-0.551121\pi$
$263$	−5.18664			−0.319822			−0.159911	+	0.987131i	$0.551121\pi$
$264$		0			0
$265$		−	2.13093i		−	0.130902i
$266$		0			0
$267$			13.0557i			0.798996i
$268$		0			0
$269$	27.2506			1.66150			0.830749	−	0.556647i	$-0.187912\pi$
$269$	27.2506			1.66150			0.830749	+	0.556647i	$0.187912\pi$
$270$		0			0
$271$			18.7612i			1.13966i	0.821763	+	0.569830i	$0.192991\pi$
$271$			18.7612i			1.13966i	−0.821763	+	0.569830i	$0.807009\pi$
$272$		0			0
$273$	2.48119	−	5.50659i	0.150169	−	0.333274i
$274$		0			0
$275$		−	20.3634i		−	1.22796i
$276$		0			0
$277$	−15.4241			−0.926743			−0.463371	−	0.886164i	$-0.653360\pi$
$277$	−15.4241			−0.926743			−0.463371	+	0.886164i	$0.653360\pi$
$278$		0			0
$279$			0.317575i			0.0190127i
$280$		0			0
$281$		−	24.8446i		−	1.48211i	−0.671446	−	0.741053i	$-0.734327\pi$
$281$		−	24.8446i		−	1.48211i	0.671446	−	0.741053i	$-0.265673\pi$
$282$		0			0
$283$	22.8872			1.36050			0.680250	−	0.732980i	$-0.261871\pi$
$283$	22.8872			1.36050			0.680250	+	0.732980i	$0.261871\pi$
$284$		0			0
$285$	5.90271			0.349646
$286$		0			0
$287$	7.92478			0.467785
$288$		0			0
$289$	−6.19394			−0.364349
$290$		0			0
$291$			30.1197i			1.76565i
$292$		0			0
$293$			25.2193i			1.47333i	0.676258	+	0.736664i	$0.263600\pi$
$293$			25.2193i			1.47333i	−0.676258	+	0.736664i	$0.736400\pi$
$294$		0			0
$295$	−1.71767			−0.100006
$296$		0			0
$297$		−	23.9756i		−	1.39120i
$298$		0			0
$299$	−15.6751	−	7.06300i	−0.906516	−	0.408464i
$300$		0			0
$301$			4.61213i			0.265839i
$302$		0			0
$303$	−15.6424			−0.898635
$304$		0			0
$305$		−	1.56134i		−	0.0894022i
$306$		0			0
$307$		−	7.24965i		−	0.413759i	−0.978366	−	0.206880i	$-0.933669\pi$
$307$		−	7.24965i		−	0.413759i	0.978366	−	0.206880i	$-0.0663308\pi$
$308$		0			0
$309$	10.4485			0.594395
$310$		0			0
$311$	−20.2398			−1.14769			−0.573847	−	0.818963i	$-0.694550\pi$
$311$	−20.2398			−1.14769			−0.573847	+	0.818963i	$0.694550\pi$
$312$		0			0
$313$	−33.1368			−1.87300			−0.936502	−	0.350663i	$-0.885956\pi$
$313$	−33.1368			−1.87300			−0.936502	+	0.350663i	$0.885956\pi$
$314$		0			0
$315$	0.130933			0.00737721
$316$		0			0
$317$			17.0132i			0.955555i	0.878481	+	0.477778i	$0.158557\pi$
$317$			17.0132i			0.955555i	−0.878481	+	0.477778i	$0.841443\pi$
$318$		0			0
$319$			41.7318i			2.33653i
$320$		0			0
$321$	12.1359			0.677357
$322$		0			0
$323$		−	17.1573i		−	0.954657i
$324$		0			0
$325$	14.9380	+	6.73084i	0.828608	+	0.373360i
$326$		0			0
$327$		−	8.03032i		−	0.444078i
$328$		0			0
$329$	7.86907			0.433836
$330$		0			0
$331$			10.8364i			0.595621i	0.954625	+	0.297811i	$0.0962564\pi$
$331$			10.8364i			0.595621i	−0.954625	+	0.297811i	$0.903744\pi$
$332$		0			0
$333$			0.279964i			0.0153419i
$334$		0			0
$335$	4.96239			0.271124
$336$		0			0
$337$	−2.96968			−0.161769			−0.0808845	−	0.996723i	$-0.525774\pi$
$337$	−2.96968			−0.161769			−0.0808845	+	0.996723i	$0.525774\pi$
$338$		0			0
$339$	15.6507			0.850029
$340$		0			0
$341$	−7.33804			−0.397377
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			5.39280i			0.290338i
$346$		0			0
$347$	−30.7367			−1.65003			−0.825017	−	0.565108i	$-0.808834\pi$
$347$	−30.7367			−1.65003			−0.825017	+	0.565108i	$0.808834\pi$
$348$		0			0
$349$			2.00492i			0.107321i	0.998559	+	0.0536606i	$0.0170889\pi$
$349$			2.00492i			0.107321i	−0.998559	+	0.0536606i	$0.982911\pi$
$350$		0			0
$351$	17.5877	+	7.92478i	0.938761	+	0.422993i
$352$		0			0
$353$		−	17.2546i		−	0.918368i	−0.888341	−	0.459184i	$-0.848142\pi$
$353$		−	17.2546i		−	0.918368i	0.888341	−	0.459184i	$-0.151858\pi$
$354$		0			0
$355$	5.23647			0.277923
$356$		0			0
$357$			5.50659i			0.291439i
$358$		0			0
$359$			7.10650i			0.375066i	0.982258	+	0.187533i	$0.0600492\pi$
$359$			7.10650i			0.375066i	−0.982258	+	0.187533i	$0.939951\pi$
$360$		0			0
$361$	−8.24140			−0.433758
$362$		0			0
$363$	15.2120			0.798425
$364$		0			0
$365$	10.2243			0.535162
$366$		0			0
$367$	27.2628			1.42311			0.711554	−	0.702632i	$-0.247992\pi$
$367$	27.2628			1.42311			0.711554	+	0.702632i	$0.247992\pi$
$368$		0			0
$369$			1.53690i			0.0800080i
$370$		0			0
$371$		−	3.15633i		−	0.163868i
$372$		0			0
$373$	−5.91890			−0.306469			−0.153234	−	0.988190i	$-0.548969\pi$
$373$	−5.91890			−0.306469			−0.153234	+	0.988190i	$0.548969\pi$
$374$		0			0
$375$		−	10.7938i		−	0.557392i
$376$		0			0
$377$	−30.6131	−	13.7938i	−1.57665	−	0.710419i
$378$		0			0
$379$		−	28.9706i		−	1.48812i	−0.668112	−	0.744061i	$-0.732897\pi$
$379$		−	28.9706i		−	1.48812i	0.668112	−	0.744061i	$-0.267103\pi$
$380$		0			0
$381$	−2.31265			−0.118481
$382$		0			0
$383$			23.2243i			1.18670i	0.804943	+	0.593352i	$0.202196\pi$
$383$			23.2243i			1.18670i	−0.804943	+	0.593352i	$0.797804\pi$
$384$		0			0
$385$			3.02539i			0.154188i
$386$		0			0
$387$	−0.894460			−0.0454680
$388$		0			0
$389$	−16.1319			−0.817919			−0.408960	−	0.912552i	$-0.634108\pi$
$389$	−16.1319			−0.817919			−0.408960	+	0.912552i	$0.634108\pi$
$390$		0			0
$391$	15.6751			0.792725
$392$		0			0
$393$	20.9380			1.05618
$394$		0			0
$395$		−	9.89938i		−	0.498092i
$396$		0			0
$397$			24.0557i			1.20732i	0.797241	+	0.603661i	$0.206292\pi$
$397$			24.0557i			1.20732i	−0.797241	+	0.603661i	$0.793708\pi$
$398$		0			0
$399$	8.74306			0.437700
$400$		0			0
$401$			2.77575i			0.138614i	0.997595	+	0.0693071i	$0.0220788\pi$
$401$			2.77575i			0.138614i	−0.997595	+	0.0693071i	$0.977921\pi$
$402$		0			0
$403$	2.42548	−	5.38295i	0.120822	−	0.268144i
$404$		0			0
$405$		−	5.65799i		−	0.281148i
$406$		0			0
$407$	−6.46898			−0.320655
$408$		0			0
$409$			20.1309i			0.995411i	0.867346	+	0.497705i	$0.165824\pi$
$409$			20.1309i			0.995411i	−0.867346	+	0.497705i	$0.834176\pi$
$410$		0			0
$411$		−	5.37073i		−	0.264919i
$412$		0			0
$413$	−2.54420			−0.125192
$414$		0			0
$415$	−0.979532			−0.0480833
$416$		0			0
$417$	−0.418190			−0.0204789
$418$		0			0
$419$	0.385503			0.0188331			0.00941654	−	0.999956i	$-0.497003\pi$
$419$	0.385503			0.0188331			0.00941654	+	0.999956i	$0.497003\pi$
$420$		0			0
$421$		−	15.6810i		−	0.764246i	−0.924112	−	0.382123i	$-0.875193\pi$
$421$		−	15.6810i		−	0.764246i	0.924112	−	0.382123i	$-0.124807\pi$
$422$		0			0
$423$			1.52610i			0.0742015i
$424$		0			0
$425$	−14.9380			−0.724597
$426$		0			0
$427$		−	2.31265i		−	0.111917i
$428$		0			0
$429$	−11.1187	+	24.6761i	−0.536817	+	1.19137i
$430$		0			0
$431$			1.53690i			0.0740301i	0.999315	+	0.0370150i	$0.0117849\pi$
$431$			1.53690i			0.0740301i	−0.999315	+	0.0370150i	$0.988215\pi$
$432$		0			0
$433$	26.0362			1.25122			0.625610	−	0.780136i	$-0.284850\pi$
$433$	26.0362			1.25122			0.625610	+	0.780136i	$0.284850\pi$
$434$		0			0
$435$			10.5320i			0.504970i
$436$		0			0
$437$		−	24.8881i		−	1.19056i
$438$		0			0
$439$	19.1246			0.912767			0.456384	−	0.889783i	$-0.349145\pi$
$439$	19.1246			0.912767			0.456384	+	0.889783i	$0.349145\pi$
$440$		0			0
$441$	0.193937			0.00923507
$442$		0			0
$443$	12.6180			0.599500			0.299750	−	0.954018i	$-0.403097\pi$
$443$	12.6180			0.599500			0.299750	+	0.954018i	$0.403097\pi$
$444$		0			0
$445$	5.26187			0.249436
$446$		0			0
$447$			6.75528i			0.319514i
$448$		0			0
$449$		−	1.02302i		−	0.0482794i	−0.999709	−	0.0241397i	$-0.992315\pi$
$449$		−	1.02302i		−	0.0482794i	0.999709	−	0.0241397i	$-0.00768466\pi$
$450$		0			0
$451$	−35.5125			−1.67222
$452$		0			0
$453$			2.62927i			0.123534i
$454$		0			0
$455$	−2.21933	−	1.00000i	−0.104044	−	0.0468807i
$456$		0			0
$457$		−	28.5320i		−	1.33467i	−0.744758	−	0.667335i	$-0.767435\pi$
$457$		−	28.5320i		−	1.33467i	0.744758	−	0.667335i	$-0.232565\pi$
$458$		0			0
$459$	−17.5877			−0.820923
$460$		0			0
$461$		−	25.3503i		−	1.18068i	−0.807155	−	0.590340i	$-0.798994\pi$
$461$		−	25.3503i		−	1.18068i	0.807155	−	0.590340i	$-0.201006\pi$
$462$		0			0
$463$		−	39.6810i		−	1.84413i	−0.387032	−	0.922066i	$-0.626500\pi$
$463$		−	39.6810i		−	1.84413i	0.387032	−	0.922066i	$-0.373500\pi$
$464$		0			0
$465$	−1.85192			−0.0858809
$466$		0			0
$467$	1.95158			0.0903086			0.0451543	−	0.998980i	$-0.485622\pi$
$467$	1.95158			0.0903086			0.0451543	+	0.998980i	$0.485622\pi$
$468$		0			0
$469$	7.35026			0.339404
$470$		0			0
$471$	−4.85685			−0.223792
$472$		0			0
$473$		−	20.6678i		−	0.950308i
$474$		0			0
$475$			23.7177i			1.08824i
$476$		0			0
$477$	0.612127			0.0280274
$478$		0			0
$479$			26.1368i			1.19422i	0.802159	+	0.597111i	$0.203685\pi$
$479$			26.1368i			1.19422i	−0.802159	+	0.597111i	$0.796315\pi$
$480$		0			0
$481$	2.13823	−	4.74543i	0.0974947	−	0.216373i
$482$		0			0
$483$			7.98778i			0.363457i
$484$		0			0
$485$	12.1392			0.551212
$486$		0			0
$487$		−	20.9624i		−	0.949896i	−0.880014	−	0.474948i	$-0.842467\pi$
$487$		−	20.9624i		−	0.949896i	0.880014	−	0.474948i	$-0.157533\pi$
$488$		0			0
$489$			28.9380i			1.30862i
$490$		0			0
$491$	−2.95651			−0.133425			−0.0667127	−	0.997772i	$-0.521251\pi$
$491$	−2.95651			−0.133425			−0.0667127	+	0.997772i	$0.521251\pi$
$492$		0			0
$493$	30.6131			1.37874
$494$		0			0
$495$	−0.586734			−0.0263717
$496$		0			0
$497$	7.75623			0.347915
$498$		0			0
$499$		−	2.85448i		−	0.127784i	−0.997957	−	0.0638920i	$-0.979649\pi$
$499$		−	2.85448i		−	0.127784i	0.997957	−	0.0638920i	$-0.0203513\pi$
$500$		0			0
$501$			27.2931i			1.21937i
$502$		0			0
$503$	−23.8641			−1.06405			−0.532025	−	0.846729i	$-0.678569\pi$
$503$	−23.8641			−1.06405			−0.532025	+	0.846729i	$0.678569\pi$
$504$		0			0
$505$			6.30440i			0.280542i
$506$		0			0
$507$	−14.4264	−	16.3127i	−0.640701	−	0.724470i
$508$		0			0
$509$			11.1949i			0.496205i	0.968734	+	0.248102i	$0.0798070\pi$
$509$			11.1949i			0.496205i	−0.968734	+	0.248102i	$0.920193\pi$
$510$		0			0
$511$	15.1441			0.669936
$512$		0			0
$513$			27.9248i			1.23291i
$514$		0			0
$515$		−	4.21108i		−	0.185562i
$516$		0			0
$517$	−35.2628			−1.55086
$518$		0			0
$519$	−3.64244			−0.159886
$520$		0			0
$521$	−26.4894			−1.16052			−0.580262	−	0.814430i	$-0.697050\pi$
$521$	−26.4894			−1.16052			−0.580262	+	0.814430i	$0.697050\pi$
$522$		0			0
$523$	12.9525			0.566375			0.283188	−	0.959065i	$-0.408608\pi$
$523$	12.9525			0.566375			0.283188	+	0.959065i	$0.408608\pi$
$524$		0			0
$525$		−	7.61213i		−	0.332220i
$526$		0			0
$527$			5.38295i			0.234485i
$528$		0			0
$529$	−0.261865			−0.0113854
$530$		0			0
$531$		−	0.493413i		−	0.0214123i
$532$		0			0
$533$	11.7381	−	26.0508i	0.508435	−	1.12838i
$534$		0			0
$535$		−	4.89114i		−	0.211462i
$536$		0			0
$537$	−0.923822			−0.0398659
$538$		0			0
$539$			4.48119i			0.193019i
$540$		0			0
$541$		−	28.0933i		−	1.20783i	−0.797050	−	0.603913i	$-0.793607\pi$
$541$		−	28.0933i		−	1.20783i	0.797050	−	0.603913i	$-0.206393\pi$
$542$		0			0
$543$	−0.856849			−0.0367709
$544$		0			0
$545$	−3.23647			−0.138635
$546$		0			0
$547$	−14.8192			−0.633625			−0.316812	−	0.948488i	$-0.602613\pi$
$547$	−14.8192			−0.633625			−0.316812	+	0.948488i	$0.602613\pi$
$548$		0			0
$549$	0.448507			0.0191418
$550$		0			0
$551$		−	48.6058i		−	2.07068i
$552$		0			0
$553$		−	14.6629i		−	0.623530i
$554$		0			0
$555$	−1.63259			−0.0692998
$556$		0			0
$557$			18.3879i			0.779119i	0.921001	+	0.389560i	$0.127373\pi$
$557$			18.3879i			0.779119i	−0.921001	+	0.389560i	$0.872627\pi$
$558$		0			0
$559$	15.1612	+	6.83146i	0.641253	+	0.288940i
$560$		0			0
$561$		−	24.6761i		−	1.04183i
$562$		0			0
$563$	−15.3357			−0.646322			−0.323161	−	0.946344i	$-0.604745\pi$
$563$	−15.3357			−0.646322			−0.323161	+	0.946344i	$0.604745\pi$
$564$		0			0
$565$		−	6.30773i		−	0.265368i
$566$		0			0
$567$		−	8.38058i		−	0.351951i
$568$		0			0
$569$	−1.32250			−0.0554421			−0.0277210	−	0.999616i	$-0.508825\pi$
$569$	−1.32250			−0.0554421			−0.0277210	+	0.999616i	$0.508825\pi$
$570$		0			0
$571$	−37.9175			−1.58680			−0.793399	−	0.608702i	$-0.791690\pi$
$571$	−37.9175			−1.58680			−0.793399	+	0.608702i	$0.791690\pi$
$572$		0			0
$573$	−27.7137			−1.15776
$574$		0			0
$575$	−21.6688			−0.903651
$576$		0			0
$577$			12.8627i			0.535482i	0.963491	+	0.267741i	$0.0862772\pi$
$577$			12.8627i			0.535482i	−0.963491	+	0.267741i	$0.913723\pi$
$578$		0			0
$579$		−	12.4142i		−	0.515917i
$580$		0			0
$581$	−1.45088			−0.0601925
$582$		0			0
$583$			14.1441i			0.585789i
$584$		0			0
$585$	0.193937	−	0.430409i	0.00801829	−	0.0177952i
$586$		0			0
$587$		−	31.0240i		−	1.28050i	−0.768168	−	0.640248i	$-0.778831\pi$
$587$		−	31.0240i		−	1.28050i	0.768168	−	0.640248i	$-0.221169\pi$
$588$		0			0
$589$	8.54675			0.352163
$590$		0			0
$591$		−	11.9756i		−	0.492609i
$592$		0			0
$593$		−	11.4119i		−	0.468629i	−0.972161	−	0.234314i	$-0.924716\pi$
$593$		−	11.4119i		−	0.468629i	0.972161	−	0.234314i	$-0.0752845\pi$
$594$		0			0
$595$	2.21933			0.0909836
$596$		0			0
$597$	8.54420			0.349691
$598$		0			0
$599$	−7.37328			−0.301264			−0.150632	−	0.988590i	$-0.548131\pi$
$599$	−7.37328			−0.301264			−0.150632	+	0.988590i	$0.548131\pi$
$600$		0			0
$601$	17.2144			0.702190			0.351095	−	0.936340i	$-0.385809\pi$
$601$	17.2144			0.702190			0.351095	+	0.936340i	$0.385809\pi$
$602$		0			0
$603$			1.42548i			0.0580502i
$604$		0			0
$605$		−	6.13093i		−	0.249258i
$606$		0			0
$607$	−32.4264			−1.31615			−0.658074	−	0.752953i	$-0.728629\pi$
$607$	−32.4264			−1.31615			−0.658074	+	0.752953i	$0.728629\pi$
$608$		0			0
$609$			15.5999i			0.632140i
$610$		0			0
$611$	11.6556	−	25.8677i	0.471536	−	1.04649i
$612$		0			0
$613$			35.6991i			1.44187i	0.693001	+	0.720937i	$0.256288\pi$
$613$			35.6991i			1.44187i	−0.693001	+	0.720937i	$0.743712\pi$
$614$		0			0
$615$	−8.96239			−0.361398
$616$		0			0
$617$		−	20.3733i		−	0.820198i	−0.912041	−	0.410099i	$-0.865494\pi$
$617$		−	20.3733i		−	0.820198i	0.912041	−	0.410099i	$-0.134506\pi$
$618$		0			0
$619$			40.0118i			1.60821i	0.594488	+	0.804104i	$0.297355\pi$
$619$			40.0118i			1.60821i	−0.594488	+	0.804104i	$0.702645\pi$
$620$		0			0
$621$	−25.5125			−1.02378
$622$		0			0
$623$	7.79384			0.312254
$624$		0			0
$625$	18.3707			0.734829
$626$		0			0
$627$	−39.1793			−1.56467
$628$		0			0
$629$			4.74543i			0.189213i
$630$		0			0
$631$			16.3879i			0.652391i	0.945302	+	0.326195i	$0.105767\pi$
$631$			16.3879i			0.652391i	−0.945302	+	0.326195i	$0.894233\pi$
$632$		0			0
$633$	0.324869			0.0129124
$634$		0			0
$635$			0.932071i			0.0369881i
$636$		0			0
$637$	−3.28726	−	1.48119i	−0.130246	−	0.0586871i
$638$		0			0
$639$			1.50422i			0.0595059i
$640$		0			0
$641$	8.23884			0.325415			0.162707	−	0.986674i	$-0.447977\pi$
$641$	8.23884			0.325415			0.162707	+	0.986674i	$0.447977\pi$
$642$		0			0
$643$		−	30.4847i		−	1.20220i	−0.799174	−	0.601100i	$-0.794729\pi$
$643$		−	30.4847i		−	1.20220i	0.799174	−	0.601100i	$-0.205271\pi$
$644$		0			0
$645$		−	5.21600i		−	0.205380i
$646$		0			0
$647$	28.8388			1.13377			0.566884	−	0.823798i	$-0.308149\pi$
$647$	28.8388			1.13377			0.566884	+	0.823798i	$0.308149\pi$
$648$		0			0
$649$	11.4010			0.447530
$650$		0			0
$651$	−2.74306			−0.107509
$652$		0			0
$653$	21.4518			0.839475			0.419738	−	0.907646i	$-0.362122\pi$
$653$	21.4518			0.839475			0.419738	+	0.907646i	$0.362122\pi$
$654$		0			0
$655$		−	8.43866i		−	0.329726i
$656$		0			0
$657$			2.93700i			0.114583i
$658$		0			0
$659$	50.6589			1.97339			0.986696	−	0.162575i	$-0.0519801\pi$
$659$	50.6589			1.97339			0.986696	+	0.162575i	$0.0519801\pi$
$660$		0			0
$661$		−	15.5477i		−	0.604736i	−0.953191	−	0.302368i	$-0.902223\pi$
$661$		−	15.5477i		−	0.604736i	0.953191	−	0.302368i	$-0.0977771\pi$
$662$		0			0
$663$	18.1016	+	8.15633i	0.703007	+	0.316765i
$664$		0			0
$665$		−	3.52373i		−	0.136644i
$666$		0			0
$667$	44.4069			1.71944
$668$		0			0
$669$			9.78067i			0.378143i
$670$		0			0
$671$			10.3634i			0.400076i
$672$		0			0
$673$	26.8700			1.03576			0.517882	−	0.855452i	$-0.326721\pi$
$673$	26.8700			1.03576			0.517882	+	0.855452i	$0.326721\pi$
$674$		0			0
$675$	24.3127			0.935794
$676$		0			0
$677$	16.3757			0.629368			0.314684	−	0.949197i	$-0.398102\pi$
$677$	16.3757			0.629368			0.314684	+	0.949197i	$0.398102\pi$
$678$		0			0
$679$	17.9805			0.690028
$680$		0			0
$681$		−	6.17679i		−	0.236695i
$682$		0			0
$683$			15.7988i			0.604523i	0.953225	+	0.302262i	$0.0977416\pi$
$683$			15.7988i			0.604523i	−0.953225	+	0.302262i	$0.902258\pi$
$684$		0			0
$685$	−2.16457			−0.0827041
$686$		0			0
$687$			9.47627i			0.361542i
$688$		0			0
$689$	−10.3757	−	4.67513i	−0.395281	−	0.178108i
$690$		0			0
$691$			7.56818i			0.287907i	0.989584	+	0.143953i	$0.0459816\pi$
$691$			7.56818i			0.287907i	−0.989584	+	0.143953i	$0.954018\pi$
$692$		0			0
$693$	−0.869067			−0.0330131
$694$		0			0
$695$			0.168544i			0.00639324i
$696$		0			0
$697$			26.0508i			0.986744i
$698$		0			0
$699$	17.2971			0.654237
$700$		0			0
$701$	−32.6629			−1.23366			−0.616831	−	0.787096i	$-0.711584\pi$
$701$	−32.6629			−1.23366			−0.616831	+	0.787096i	$0.711584\pi$
$702$		0			0
$703$	7.53453			0.284170
$704$		0			0
$705$	−8.89938			−0.335170
$706$		0			0
$707$			9.33804i			0.351193i
$708$		0			0
$709$			25.0214i			0.939699i	0.882746	+	0.469850i	$0.155692\pi$
$709$			25.0214i			0.939699i	−0.882746	+	0.469850i	$0.844308\pi$
$710$		0			0
$711$	2.84367			0.106646
$712$		0			0
$713$			7.80843i			0.292428i
$714$		0			0
$715$	9.94525	+	4.48119i	0.371931	+	0.167587i
$716$		0			0
$717$		−	37.3357i		−	1.39433i
$718$		0			0
$719$	−1.85192			−0.0690651			−0.0345326	−	0.999404i	$-0.510994\pi$
$719$	−1.85192			−0.0690651			−0.0345326	+	0.999404i	$0.510994\pi$
$720$		0			0
$721$		−	6.23743i		−	0.232294i
$722$		0			0
$723$		−	49.2931i		−	1.83323i
$724$		0			0
$725$	−42.3185			−1.57167
$726$		0			0
$727$	10.8265			0.401534			0.200767	−	0.979639i	$-0.435657\pi$
$727$	10.8265			0.401534			0.200767	+	0.979639i	$0.435657\pi$
$728$		0			0
$729$	28.5125			1.05602
$730$		0			0
$731$	−15.1612			−0.560759
$732$		0			0
$733$			23.4264i			0.865275i	0.901568	+	0.432638i	$0.142417\pi$
$733$			23.4264i			0.865275i	−0.901568	+	0.432638i	$0.857583\pi$
$734$		0			0
$735$			1.13093i			0.0417151i
$736$		0			0
$737$	−32.9380			−1.21329
$738$		0			0
$739$		−	0.481194i		−	0.0177010i	−0.999961	−	0.00885051i	$-0.997183\pi$
$739$		−	0.481194i		−	0.0177010i	0.999961	−	0.00885051i	$-0.00281724\pi$
$740$		0			0
$741$	12.9502	−	28.7407i	0.475736	−	1.05582i
$742$		0			0
$743$			13.7889i			0.505866i	0.967484	+	0.252933i	$0.0813953\pi$
$743$			13.7889i			0.505866i	−0.967484	+	0.252933i	$0.918605\pi$
$744$		0			0
$745$	2.72259			0.0997480
$746$		0			0
$747$		−	0.281378i		−	0.0102951i
$748$		0			0
$749$		−	7.24472i		−	0.264716i
$750$		0			0
$751$	−26.4617			−0.965600			−0.482800	−	0.875731i	$-0.660380\pi$
$751$	−26.4617			−0.965600			−0.482800	+	0.875731i	$0.660380\pi$
$752$		0			0
$753$	−36.0567			−1.31398
$754$		0			0
$755$	1.05968			0.0385657
$756$		0			0
$757$	−21.1114			−0.767308			−0.383654	−	0.923477i	$-0.625334\pi$
$757$	−21.1114			−0.767308			−0.383654	+	0.923477i	$0.625334\pi$
$758$		0			0
$759$		−	35.7948i		−	1.29927i
$760$		0			0
$761$		−	28.5247i		−	1.03402i	−0.855980	−	0.517010i	$-0.827045\pi$
$761$		−	28.5247i		−	1.03402i	0.855980	−	0.517010i	$-0.172955\pi$
$762$		0			0
$763$	−4.79384			−0.173549
$764$		0			0
$765$			0.430409i			0.0155615i
$766$		0			0
$767$	−3.76845	+	8.36344i	−0.136071	+	0.301986i
$768$		0			0
$769$		−	25.8388i		−	0.931769i	−0.884845	−	0.465885i	$-0.845736\pi$
$769$		−	25.8388i		−	0.931769i	0.884845	−	0.465885i	$-0.154264\pi$
$770$		0			0
$771$	−1.10886			−0.0399348
$772$		0			0
$773$		−	27.4010i		−	0.985547i	−0.870158	−	0.492774i	$-0.835983\pi$
$773$		−	27.4010i		−	0.985547i	0.870158	−	0.492774i	$-0.164017\pi$
$774$		0			0
$775$		−	7.44121i		−	0.267296i
$776$		0			0
$777$	−2.41819			−0.0867521
$778$		0			0
$779$	41.3620			1.48195
$780$		0			0
$781$	−34.7572			−1.24371
$782$		0			0
$783$	−49.8251			−1.78060
$784$		0			0
$785$			1.95746i			0.0698649i
$786$		0			0
$787$		−	31.0240i		−	1.10589i	−0.833219	−	0.552943i	$-0.813505\pi$
$787$		−	31.0240i		−	1.10589i	0.833219	−	0.552943i	$-0.186495\pi$
$788$		0			0
$789$	8.68830			0.309312
$790$		0			0
$791$		−	9.34297i		−	0.332198i
$792$		0			0
$793$	−7.60228	−	3.42548i	−0.269965	−	0.121643i
$794$		0			0
$795$			3.56959i			0.126600i
$796$		0			0
$797$	−16.1378			−0.571629			−0.285815	−	0.958285i	$-0.592264\pi$
$797$	−16.1378			−0.571629			−0.285815	+	0.958285i	$0.592264\pi$
$798$		0			0
$799$			25.8677i			0.915132i
$800$		0			0
$801$			1.51151i			0.0534066i
$802$		0			0
$803$	−67.8637			−2.39486
$804$		0			0
$805$	3.21933			0.113466
$806$		0			0
$807$	−45.6483			−1.60690
$808$		0			0
$809$	−10.3503			−0.363896			−0.181948	−	0.983308i	$-0.558240\pi$
$809$	−10.3503			−0.363896			−0.181948	+	0.983308i	$0.558240\pi$
$810$		0			0
$811$			46.3752i			1.62845i	0.580547	+	0.814227i	$0.302839\pi$
$811$			46.3752i			1.62845i	−0.580547	+	0.814227i	$0.697161\pi$
$812$		0			0
$813$		−	31.4274i		−	1.10221i
$814$		0			0
$815$	11.6629			0.408534
$816$		0			0
$817$			24.0722i			0.842180i
$818$		0			0
$819$	0.287258	−	0.637519i	0.0100376	−	0.0222767i
$820$		0			0
$821$		−	55.5388i		−	1.93832i	−0.246436	−	0.969159i	$-0.579260\pi$
$821$		−	55.5388i		−	1.93832i	0.246436	−	0.969159i	$-0.420740\pi$
$822$		0			0
$823$	−20.8324			−0.726172			−0.363086	−	0.931756i	$-0.618277\pi$
$823$	−20.8324			−0.726172			−0.363086	+	0.931756i	$0.618277\pi$
$824$		0			0
$825$			34.1114i			1.18761i
$826$		0			0
$827$		−	2.47295i		−	0.0859927i	−0.999075	−	0.0429964i	$-0.986310\pi$
$827$		−	2.47295i		−	0.0859927i	0.999075	−	0.0429964i	$-0.0136904\pi$
$828$		0			0
$829$	34.5623			1.20040			0.600199	−	0.799851i	$-0.295088\pi$
$829$	34.5623			1.20040			0.600199	+	0.799851i	$0.295088\pi$
$830$		0			0
$831$	25.8373			0.896287
$832$		0			0
$833$	3.28726			0.113897
$834$		0			0
$835$	11.0000			0.380671
$836$		0			0
$837$		−	8.76116i		−	0.302830i
$838$		0			0
$839$			39.9149i			1.37802i	0.724754	+	0.689008i	$0.241954\pi$
$839$			39.9149i			1.37802i	−0.724754	+	0.689008i	$0.758046\pi$
$840$		0			0
$841$	57.7255			1.99053
$842$		0			0
$843$			41.6180i			1.43340i
$844$		0			0
$845$	−6.57452	+	5.81431i	−0.226170	+	0.200018i
$846$		0			0
$847$		−	9.08110i		−	0.312030i
$848$		0			0
$849$	−38.3390			−1.31579
$850$		0			0
$851$			6.88366i			0.235969i
$852$		0			0
$853$		−	32.6507i		−	1.11794i	−0.829188	−	0.558969i	$-0.811197\pi$
$853$		−	32.6507i		−	1.11794i	0.829188	−	0.558969i	$-0.188803\pi$
$854$		0			0
$855$	0.683380			0.0233711
$856$		0			0
$857$	7.63656			0.260860			0.130430	−	0.991458i	$-0.458364\pi$
$857$	7.63656			0.260860			0.130430	+	0.991458i	$0.458364\pi$
$858$		0			0
$859$	8.96239			0.305793			0.152896	−	0.988242i	$-0.451140\pi$
$859$	8.96239			0.305793			0.152896	+	0.988242i	$0.451140\pi$
$860$		0			0
$861$	−13.2750			−0.452412
$862$		0			0
$863$		−	46.8021i		−	1.59316i	−0.604532	−	0.796581i	$-0.706640\pi$
$863$		−	46.8021i		−	1.59316i	0.604532	−	0.796581i	$-0.293360\pi$
$864$		0			0
$865$			1.46802i			0.0499142i
$866$		0			0
$867$	10.3757			0.352376
$868$		0			0
$869$			65.7074i			2.22897i
$870$		0			0
$871$	10.8872	−	24.1622i	0.368898	−	0.818705i
$872$		0			0
$873$			3.48707i			0.118020i
$874$		0			0
$875$	−6.44358			−0.217833
$876$		0			0
$877$		−	10.3406i		−	0.349177i	−0.984641	−	0.174589i	$-0.944140\pi$
$877$		−	10.3406i		−	0.349177i	0.984641	−	0.174589i	$-0.0558595\pi$
$878$		0			0
$879$		−	42.2457i		−	1.42491i
$880$		0			0
$881$	−15.1538			−0.510543			−0.255272	−	0.966869i	$-0.582165\pi$
$881$	−15.1538			−0.510543			−0.255272	+	0.966869i	$0.582165\pi$
$882$		0			0
$883$	−51.5983			−1.73642			−0.868211	−	0.496196i	$-0.834730\pi$
$883$	−51.5983			−1.73642			−0.868211	+	0.496196i	$0.834730\pi$
$884$		0			0
$885$	2.87732			0.0967199
$886$		0			0
$887$	−4.18664			−0.140574			−0.0702868	−	0.997527i	$-0.522391\pi$
$887$	−4.18664			−0.140574			−0.0702868	+	0.997527i	$0.522391\pi$
$888$		0			0
$889$			1.38058i			0.0463031i
$890$		0			0
$891$			37.5550i			1.25814i
$892$		0			0
$893$	41.0713			1.37440
$894$		0			0
$895$			0.372330i			0.0124456i
$896$		0			0
$897$	26.2579	+	11.8315i	0.876726	+	0.395041i
$898$		0			0
$899$			15.2496i			0.508604i
$900$		0			0
$901$	10.3757			0.345663
$902$		0			0
$903$		−	7.72592i		−	0.257102i
$904$		0			0
$905$			0.345337i			0.0114794i
$906$		0			0
$907$	−9.60483			−0.318923			−0.159462	−	0.987204i	$-0.550976\pi$
$907$	−9.60483			−0.318923			−0.159462	+	0.987204i	$0.550976\pi$
$908$		0			0
$909$	−1.81099			−0.0600667
$910$		0			0
$911$	1.82653			0.0605157			0.0302578	−	0.999542i	$-0.490367\pi$
$911$	1.82653			0.0605157			0.0302578	+	0.999542i	$0.490367\pi$
$912$		0			0
$913$	6.50166			0.215174
$914$		0			0
$915$			2.61545i			0.0864642i
$916$		0			0
$917$		−	12.4993i		−	0.412763i
$918$		0			0
$919$	−15.7842			−0.520672			−0.260336	−	0.965518i	$-0.583833\pi$
$919$	−15.7842			−0.520672			−0.260336	+	0.965518i	$0.583833\pi$
$920$		0			0
$921$			12.1441i			0.400162i
$922$		0			0
$923$	11.4885	−	25.4967i	0.378148	−	0.839235i
$924$		0			0
$925$		−	6.55993i		−	0.215689i
$926$		0			0
$927$	1.20967			0.0397306
$928$		0			0
$929$			16.8011i			0.551227i	0.961268	+	0.275614i	$0.0888811\pi$
$929$			16.8011i			0.551227i	−0.961268	+	0.275614i	$0.911119\pi$
$930$		0			0
$931$		−	5.21933i		−	0.171057i
$932$		0			0
$933$	33.9043			1.10998
$934$		0			0
$935$	−9.94525			−0.325244
$936$		0			0
$937$	−38.3004			−1.25122			−0.625610	−	0.780136i	$-0.715150\pi$
$937$	−38.3004			−1.25122			−0.625610	+	0.780136i	$0.715150\pi$
$938$		0			0
$939$	55.5085			1.81145
$940$		0			0
$941$			45.2496i			1.47510i	0.675295	+	0.737548i	$0.264017\pi$
$941$			45.2496i			1.47510i	−0.675295	+	0.737548i	$0.735983\pi$
$942$		0			0
$943$			37.7889i			1.23058i
$944$		0			0
$945$	−3.61213			−0.117502
$946$		0			0
$947$		−	4.73672i		−	0.153923i	−0.997034	−	0.0769614i	$-0.975478\pi$
$947$		−	4.73672i		−	0.153923i	0.997034	−	0.0769614i	$-0.0245218\pi$
$948$		0			0
$949$	22.4314	−	49.7826i	0.728153	−	1.61601i
$950$		0			0
$951$		−	28.4993i		−	0.924153i
$952$		0			0
$953$	−44.8007			−1.45124			−0.725618	−	0.688098i	$-0.758446\pi$
$953$	−44.8007			−1.45124			−0.725618	+	0.688098i	$0.758446\pi$
$954$		0			0
$955$			11.1695i			0.361436i
$956$		0			0
$957$		−	69.9062i		−	2.25975i
$958$		0			0
$959$	−3.20616			−0.103532
$960$		0			0
$961$	28.3185			0.913501
$962$		0			0
$963$	1.40502			0.0452760
$964$		0			0
$965$	−5.00332			−0.161063
$966$		0			0
$967$		−	40.6843i		−	1.30832i	−0.756356	−	0.654160i	$-0.773022\pi$
$967$		−	40.6843i		−	1.30832i	0.756356	−	0.654160i	$-0.226978\pi$
$968$		0			0
$969$			28.7407i			0.923284i
$970$		0			0
$971$	41.4109			1.32894			0.664469	−	0.747315i	$-0.268658\pi$
$971$	41.4109			1.32894			0.664469	+	0.747315i	$0.268658\pi$
$972$		0			0
$973$			0.249646i			0.00800329i
$974$		0			0
$975$	−25.0230	−	11.2750i	−0.801378	−	0.361090i
$976$		0			0
$977$		−	50.6072i		−	1.61907i	−0.587073	−	0.809534i	$-0.699720\pi$
$977$		−	50.6072i		−	1.61907i	0.587073	−	0.809534i	$-0.300280\pi$
$978$		0			0
$979$	−34.9257			−1.11623
$980$		0			0
$981$		−	0.929702i		−	0.0296831i
$982$		0			0
$983$		−	24.5452i		−	0.782869i	−0.920206	−	0.391434i	$-0.871979\pi$
$983$		−	24.5452i		−	0.782869i	0.920206	−	0.391434i	$-0.128021\pi$
$984$		0			0
$985$	−4.82653			−0.153786
$986$		0			0
$987$	−13.1817			−0.419579
$988$		0			0
$989$	−21.9927			−0.699327
$990$		0			0
$991$	31.6542			1.00553			0.502764	−	0.864423i	$-0.332316\pi$
$991$	31.6542			1.00553			0.502764	+	0.864423i	$0.332316\pi$
$992$		0			0
$993$		−	18.1524i		−	0.576048i
$994$		0			0
$995$		−	3.44358i		−	0.109169i
$996$		0			0
$997$	−29.1852			−0.924305			−0.462153	−	0.886800i	$-0.652923\pi$
$997$	−29.1852			−0.924305			−0.462153	+	0.886800i	$0.652923\pi$
$998$		0			0
$999$		−	7.72355i		−	0.244362i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.k.c.337.2		6
4.3	odd	2		91.2.c.a.64.1	✓	6
12.11	even	2		819.2.c.b.64.6		6
13.12	even	2	inner	1456.2.k.c.337.1		6
28.3	even	6		637.2.r.d.324.6		12
28.11	odd	6		637.2.r.e.324.6		12
28.19	even	6		637.2.r.d.116.1		12
28.23	odd	6		637.2.r.e.116.1		12
28.27	even	2		637.2.c.d.246.1		6
52.31	even	4		1183.2.a.h.1.1		3
52.47	even	4		1183.2.a.j.1.3		3
52.51	odd	2		91.2.c.a.64.6	yes	6
156.155	even	2		819.2.c.b.64.1		6
364.51	odd	6		637.2.r.e.116.6		12
364.83	odd	4		8281.2.a.be.1.1		3
364.103	even	6		637.2.r.d.116.6		12
364.207	odd	6		637.2.r.e.324.1		12
364.307	odd	4		8281.2.a.bi.1.3		3
364.311	even	6		637.2.r.d.324.1		12
364.363	even	2		637.2.c.d.246.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.c.a.64.1	✓	6	4.3	odd	2
91.2.c.a.64.6	yes	6	52.51	odd	2
637.2.c.d.246.1		6	28.27	even	2
637.2.c.d.246.6		6	364.363	even	2
637.2.r.d.116.1		12	28.19	even	6
637.2.r.d.116.6		12	364.103	even	6
637.2.r.d.324.1		12	364.311	even	6
637.2.r.d.324.6		12	28.3	even	6
637.2.r.e.116.1		12	28.23	odd	6
637.2.r.e.116.6		12	364.51	odd	6
637.2.r.e.324.1		12	364.207	odd	6
637.2.r.e.324.6		12	28.11	odd	6
819.2.c.b.64.1		6	156.155	even	2
819.2.c.b.64.6		6	12.11	even	2
1183.2.a.h.1.1		3	52.31	even	4
1183.2.a.j.1.3		3	52.47	even	4
1456.2.k.c.337.1		6	13.12	even	2	inner
1456.2.k.c.337.2		6	1.1	even	1	trivial
8281.2.a.be.1.1		3	364.83	odd	4
8281.2.a.bi.1.3		3	364.307	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.67513 q^{3} +0.675131i q^{5} +1.00000i q^{7} -0.193937 q^{9} +O(q^{10})\)	\(q-1.67513 q^{3} +0.675131i q^{5} +1.00000i q^{7} -0.193937 q^{9} -4.48119i q^{11} +(3.28726 + 1.48119i) q^{13} -1.13093i q^{15} -3.28726 q^{17} +5.21933i q^{19} -1.67513i q^{21} -4.76845 q^{23} +4.54420 q^{25} +5.35026 q^{27} -9.31265 q^{29} -1.63752i q^{31} +7.50659i q^{33} -0.675131 q^{35} -1.44358i q^{37} +(-5.50659 - 2.48119i) q^{39} -7.92478i q^{41} +4.61213 q^{43} -0.130933i q^{45} -7.86907i q^{47} -1.00000 q^{49} +5.50659 q^{51} -3.15633 q^{53} +3.02539 q^{55} -8.74306i q^{57} +2.54420i q^{59} -2.31265 q^{61} -0.193937i q^{63} +(-1.00000 + 2.21933i) q^{65} -7.35026i q^{67} +7.98778 q^{69} -7.75623i q^{71} -15.1441i q^{73} -7.61213 q^{75} +4.48119 q^{77} -14.6629 q^{79} -8.38058 q^{81} +1.45088i q^{83} -2.21933i q^{85} +15.5999 q^{87} -7.79384i q^{89} +(-1.48119 + 3.28726i) q^{91} +2.74306i q^{93} -3.52373 q^{95} -17.9805i q^{97} +0.869067i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{9}+O(q^{10}) \) 6 * q - 2 * q^9	\( 6 q - 2 q^{9} + 8 q^{13} - 8 q^{17} - 6 q^{23} + 8 q^{25} + 12 q^{27} - 14 q^{29} + 6 q^{35} + 8 q^{39} + 26 q^{43} - 6 q^{49} - 8 q^{51} + 2 q^{53} - 12 q^{55} + 28 q^{61} - 6 q^{65} - 4 q^{69} - 44 q^{75} + 16 q^{77} - 26 q^{79} - 26 q^{81} + 40 q^{87} + 2 q^{91} - 58 q^{95}+O(q^{100}) \) 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

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Database

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