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Properties

Label 340.2.bi.a.97.6
Level $340$
Weight $2$
Character 340.97
Analytic conductor $2.715$
Analytic rank $0$
Dimension $72$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [340,2,Mod(37,340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("340.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 4, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.bi (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.71491366872\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label 97.6
Character \(\chi\) \(=\) 340.97
Dual form 340.2.bi.a.333.6

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.192727 - 0.968904i) q^{3} +(-1.91554 + 1.15356i) q^{5} +(2.02209 + 1.35112i) q^{7} +(1.87001 + 0.774583i) q^{9} +(2.10402 - 3.14888i) q^{11} +1.49264 q^{13} +(0.748515 + 2.07830i) q^{15} +(3.30437 + 2.46600i) q^{17} +(0.981257 - 0.406450i) q^{19} +(1.69881 - 1.69881i) q^{21} +(1.64708 - 0.327624i) q^{23} +(2.33859 - 4.41939i) q^{25} +(2.75742 - 4.12676i) q^{27} +(-9.79655 - 1.94866i) q^{29} +(1.58585 + 2.37339i) q^{31} +(-2.64546 - 2.64546i) q^{33} +(-5.43199 - 0.255511i) q^{35} +(1.56116 + 0.310534i) q^{37} +(0.287672 - 1.44623i) q^{39} +(-3.64697 + 0.725426i) q^{41} +(4.34256 + 10.4839i) q^{43} +(-4.47561 + 0.673428i) q^{45} +7.73194i q^{47} +(-0.415456 - 1.00300i) q^{49} +(3.02615 - 2.72635i) q^{51} +(-9.47582 - 3.92501i) q^{53} +(-0.397892 + 8.45893i) q^{55} +(-0.204696 - 1.02908i) q^{57} +(4.01891 - 9.70252i) q^{59} +(0.260515 + 1.30970i) q^{61} +(2.73477 + 4.09287i) q^{63} +(-2.85922 + 1.72186i) q^{65} +(-0.639115 + 0.639115i) q^{67} -1.65900i q^{69} +(-2.88826 + 1.92987i) q^{71} +(-2.40689 + 1.60823i) q^{73} +(-3.83126 - 3.11760i) q^{75} +(8.50902 - 3.52455i) q^{77} +(-6.82340 - 4.55925i) q^{79} +(0.826718 + 0.826718i) q^{81} +(-3.60482 + 8.70280i) q^{83} +(-9.17433 - 0.911919i) q^{85} +(-3.77612 + 9.11635i) q^{87} +(-9.61612 - 9.61612i) q^{89} +(3.01826 + 2.01673i) q^{91} +(2.60522 - 1.07912i) q^{93} +(-1.41077 + 1.91051i) q^{95} +(-3.47250 + 2.32025i) q^{97} +(6.37360 - 4.25870i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 24 q^{15} + 8 q^{25} + 48 q^{27} - 32 q^{31} - 16 q^{33} - 32 q^{37} + 32 q^{39} - 40 q^{41} + 32 q^{53} - 16 q^{55} + 72 q^{57} - 112 q^{59} - 48 q^{63} + 32 q^{67} - 16 q^{71} - 96 q^{73} + 24 q^{75}+ \cdots - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/340\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(e\left(\frac{13}{16}\right)\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.192727 − 0.968904i 0.111271 − 0.559397i −0.884422 − 0.466687i \(-0.845447\pi\)
0.995693 − 0.0927092i \(-0.0295527\pi\)
\(4\) 0 0
\(5\) −1.91554 + 1.15356i −0.856655 + 0.515889i
\(6\) 0 0
\(7\) 2.02209 + 1.35112i 0.764278 + 0.510674i 0.875557 − 0.483115i \(-0.160495\pi\)
−0.111279 + 0.993789i \(0.535495\pi\)
\(8\) 0 0
\(9\) 1.87001 + 0.774583i 0.623336 + 0.258194i
\(10\) 0 0
\(11\) 2.10402 − 3.14888i 0.634385 − 0.949424i −0.365442 − 0.930834i \(-0.619082\pi\)
0.999827 − 0.0185903i \(-0.00591783\pi\)
\(12\) 0 0
\(13\) 1.49264 0.413985 0.206992 − 0.978343i \(-0.433633\pi\)
0.206992 + 0.978343i \(0.433633\pi\)
\(14\) 0 0
\(15\) 0.748515 + 2.07830i 0.193266 + 0.536614i
\(16\) 0 0
\(17\) 3.30437 + 2.46600i 0.801427 + 0.598092i
\(18\) 0 0
\(19\) 0.981257 − 0.406450i 0.225116 − 0.0932460i −0.267275 − 0.963620i \(-0.586123\pi\)
0.492391 + 0.870374i \(0.336123\pi\)
\(20\) 0 0
\(21\) 1.69881 − 1.69881i 0.370711 − 0.370711i
\(22\) 0 0
\(23\) 1.64708 − 0.327624i 0.343440 − 0.0683144i −0.0203570 − 0.999793i \(-0.506480\pi\)
0.363797 + 0.931478i \(0.381480\pi\)
\(24\) 0 0
\(25\) 2.33859 − 4.41939i 0.467717 − 0.883878i
\(26\) 0 0
\(27\) 2.75742 − 4.12676i 0.530665 − 0.794196i
\(28\) 0 0
\(29\) −9.79655 − 1.94866i −1.81917 − 0.361856i −0.836613 − 0.547795i \(-0.815467\pi\)
−0.982561 + 0.185939i \(0.940467\pi\)
\(30\) 0 0
\(31\) 1.58585 + 2.37339i 0.284826 + 0.426273i 0.946101 − 0.323871i \(-0.104984\pi\)
−0.661275 + 0.750144i \(0.729984\pi\)
\(32\) 0 0
\(33\) −2.64546 − 2.64546i −0.460516 − 0.460516i
\(34\) 0 0
\(35\) −5.43199 − 0.255511i −0.918174 − 0.0431892i
\(36\) 0 0
\(37\) 1.56116 + 0.310534i 0.256653 + 0.0510514i 0.321740 − 0.946828i \(-0.395732\pi\)
−0.0650869 + 0.997880i \(0.520732\pi\)
\(38\) 0 0
\(39\) 0.287672 − 1.44623i 0.0460644 − 0.231582i
\(40\) 0 0
\(41\) −3.64697 + 0.725426i −0.569560 + 0.113293i −0.471466 − 0.881885i \(-0.656275\pi\)
−0.0980947 + 0.995177i \(0.531275\pi\)
\(42\) 0 0
\(43\) 4.34256 + 10.4839i 0.662234 + 1.59878i 0.794294 + 0.607534i \(0.207841\pi\)
−0.132059 + 0.991242i \(0.542159\pi\)
\(44\) 0 0
\(45\) −4.47561 + 0.673428i −0.667184 + 0.100389i
\(46\) 0 0
\(47\) 7.73194i 1.12782i 0.825836 + 0.563910i \(0.190704\pi\)
−0.825836 + 0.563910i \(0.809296\pi\)
\(48\) 0 0
\(49\) −0.415456 − 1.00300i −0.0593509 − 0.143286i
\(50\) 0 0
\(51\) 3.02615 − 2.72635i 0.423746 − 0.381765i
\(52\) 0 0
\(53\) −9.47582 − 3.92501i −1.30160 − 0.539142i −0.379180 − 0.925323i \(-0.623794\pi\)
−0.922423 + 0.386181i \(0.873794\pi\)
\(54\) 0 0
\(55\) −0.397892 + 8.45893i −0.0536518 + 1.14060i
\(56\) 0 0
\(57\) −0.204696 − 1.02908i −0.0271127 − 0.136305i
\(58\) 0 0
\(59\) 4.01891 − 9.70252i 0.523218 − 1.26316i −0.412676 − 0.910878i \(-0.635406\pi\)
0.935894 − 0.352282i \(-0.114594\pi\)
\(60\) 0 0
\(61\) 0.260515 + 1.30970i 0.0333555 + 0.167689i 0.993873 − 0.110531i \(-0.0352553\pi\)
−0.960517 + 0.278221i \(0.910255\pi\)
\(62\) 0 0
\(63\) 2.73477 + 4.09287i 0.344549 + 0.515654i
\(64\) 0 0
\(65\) −2.85922 + 1.72186i −0.354642 + 0.213570i
\(66\) 0 0
\(67\) −0.639115 + 0.639115i −0.0780803 + 0.0780803i −0.745068 − 0.666988i \(-0.767583\pi\)
0.666988 + 0.745068i \(0.267583\pi\)
\(68\) 0 0
\(69\) − 1.65900i − 0.199720i
\(70\) 0 0
\(71\) −2.88826 + 1.92987i −0.342773 + 0.229034i −0.715026 − 0.699098i \(-0.753585\pi\)
0.372253 + 0.928131i \(0.378585\pi\)
\(72\) 0 0
\(73\) −2.40689 + 1.60823i −0.281705 + 0.188229i −0.688393 − 0.725338i \(-0.741684\pi\)
0.406688 + 0.913567i \(0.366684\pi\)
\(74\) 0 0
\(75\) −3.83126 − 3.11760i −0.442395 − 0.359989i
\(76\) 0 0
\(77\) 8.50902 − 3.52455i 0.969693 − 0.401660i
\(78\) 0 0
\(79\) −6.82340 − 4.55925i −0.767692 − 0.512956i 0.108980 − 0.994044i \(-0.465242\pi\)
−0.876672 + 0.481088i \(0.840242\pi\)
\(80\) 0 0
\(81\) 0.826718 + 0.826718i 0.0918576 + 0.0918576i
\(82\) 0 0
\(83\) −3.60482 + 8.70280i −0.395680 + 0.955256i 0.592998 + 0.805204i \(0.297944\pi\)
−0.988678 + 0.150052i \(0.952056\pi\)
\(84\) 0 0
\(85\) −9.17433 − 0.911919i −0.995096 − 0.0989115i
\(86\) 0 0
\(87\) −3.77612 + 9.11635i −0.404842 + 0.977376i
\(88\) 0 0
\(89\) −9.61612 − 9.61612i −1.01931 − 1.01931i −0.999810 − 0.0194968i \(-0.993794\pi\)
−0.0194968 − 0.999810i \(-0.506206\pi\)
\(90\) 0 0
\(91\) 3.01826 + 2.01673i 0.316399 + 0.211411i
\(92\) 0 0
\(93\) 2.60522 − 1.07912i 0.270148 − 0.111899i
\(94\) 0 0
\(95\) −1.41077 + 1.91051i −0.144742 + 0.196014i
\(96\) 0 0
\(97\) −3.47250 + 2.32025i −0.352579 + 0.235586i −0.719229 − 0.694773i \(-0.755505\pi\)
0.366650 + 0.930359i \(0.380505\pi\)
\(98\) 0 0
\(99\) 6.37360 − 4.25870i 0.640571 − 0.428016i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 340.2.bi.a.97.6 yes 72
5.3 odd 4 340.2.bd.a.233.6 yes 72
17.10 odd 16 340.2.bd.a.197.6 ✓ 72
85.78 even 16 inner 340.2.bi.a.333.6 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bd.a.197.6 ✓ 72 17.10 odd 16
340.2.bd.a.233.6 yes 72 5.3 odd 4
340.2.bi.a.97.6 yes 72 1.1 even 1 trivial
340.2.bi.a.333.6 yes 72 85.78 even 16 inner