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Properties

Label 340.2.bi.a.113.1
Level $340$
Weight $2$
Character 340.113
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 113.1
Character \(\chi\) \(=\) 340.113
Dual form 340.2.bi.a.337.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.76939 + 2.64808i) q^{3} +(-1.59649 + 1.56564i) q^{5} +(-0.489690 + 2.46184i) q^{7} +(-2.73352 - 6.59930i) q^{9} +(3.24485 + 0.645440i) q^{11} -1.18086 q^{13} +(-1.32114 - 6.99784i) q^{15} +(-4.07583 - 0.622605i) q^{17} +(1.37521 - 3.32006i) q^{19} +(-5.65268 - 5.65268i) q^{21} +(-1.22377 + 0.817698i) q^{23} +(0.0975333 - 4.99905i) q^{25} +(12.9413 + 2.57418i) q^{27} +(6.06287 + 4.05108i) q^{29} +(-5.97389 + 1.18828i) q^{31} +(-7.45057 + 7.45057i) q^{33} +(-3.07257 - 4.69697i) q^{35} +(-6.71953 - 4.48985i) q^{37} +(2.08939 - 3.12699i) q^{39} +(-10.2729 + 6.86415i) q^{41} +(8.35071 + 3.45898i) q^{43} +(14.6962 + 6.25598i) q^{45} +2.86764i q^{47} +(0.646305 + 0.267708i) q^{49} +(8.86042 - 9.69147i) q^{51} +(1.07796 + 2.60243i) q^{53} +(-6.19088 + 4.04983i) q^{55} +(6.35849 + 9.51615i) q^{57} +(-12.4276 + 5.14768i) q^{59} +(-1.97526 - 2.95619i) q^{61} +(17.5850 - 3.49787i) q^{63} +(1.88522 - 1.84880i) q^{65} +(5.66190 + 5.66190i) q^{67} -4.68746i q^{69} +(1.73066 + 8.70063i) q^{71} +(-0.761283 - 3.82723i) q^{73} +(13.0653 + 9.10353i) q^{75} +(-3.17794 + 7.67222i) q^{77} +(0.116402 - 0.585193i) q^{79} +(-14.5621 + 14.5621i) q^{81} +(-5.97595 + 2.47532i) q^{83} +(7.48178 - 5.38730i) q^{85} +(-21.4551 + 8.88700i) q^{87} +(5.79498 - 5.79498i) q^{89} +(0.578253 - 2.90708i) q^{91} +(7.42347 - 17.9218i) q^{93} +(3.00252 + 7.45353i) q^{95} +(-3.20159 - 16.0955i) q^{97} +(-4.61040 - 23.1781i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 24 q^{15} + 8 q^{25} + 48 q^{27} - 32 q^{31} - 16 q^{33} - 32 q^{37} + 32 q^{39} - 40 q^{41} + 32 q^{53} - 16 q^{55} + 72 q^{57} - 112 q^{59} - 48 q^{63} + 32 q^{67} - 16 q^{71} - 96 q^{73} + 24 q^{75}+ \cdots - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(e\left(\frac{7}{16}\right)\)

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
\(3\) −1.76939 + 2.64808i −1.02156 + 1.52887i −0.183696 + 0.982983i \(0.558806\pi\)
−0.837861 + 0.545884i \(0.816194\pi\)
\(4\) 0 0
\(5\) −1.59649 + 1.56564i −0.713970 + 0.700176i
\(6\) 0 0
\(7\) −0.489690 + 2.46184i −0.185085 + 0.930487i 0.770874 + 0.636988i \(0.219820\pi\)
−0.955959 + 0.293499i \(0.905180\pi\)
\(8\) 0 0
\(9\) −2.73352 − 6.59930i −0.911174 − 2.19977i
\(10\) 0 0
\(11\) 3.24485 + 0.645440i 0.978358 + 0.194608i 0.658271 − 0.752781i \(-0.271288\pi\)
0.320087 + 0.947388i \(0.396288\pi\)
\(12\) 0 0
\(13\) −1.18086 −0.327510 −0.163755 − 0.986501i \(-0.552361\pi\)
−0.163755 + 0.986501i \(0.552361\pi\)
\(14\) 0 0
\(15\) −1.32114 − 6.99784i −0.341116 − 1.80683i
\(16\) 0 0
\(17\) −4.07583 − 0.622605i −0.988533 − 0.151004i
\(18\) 0 0
\(19\) 1.37521 − 3.32006i 0.315496 − 0.761675i −0.683986 − 0.729495i \(-0.739755\pi\)
0.999482 − 0.0321796i \(-0.0102448\pi\)
\(20\) 0 0
\(21\) −5.65268 − 5.65268i −1.23352 − 1.23352i
\(22\) 0 0
\(23\) −1.22377 + 0.817698i −0.255174 + 0.170502i −0.676577 − 0.736371i \(-0.736538\pi\)
0.421404 + 0.906873i \(0.361538\pi\)
\(24\) 0 0
\(25\) 0.0975333 − 4.99905i 0.0195067 − 0.999810i
\(26\) 0 0
\(27\) 12.9413 + 2.57418i 2.49055 + 0.495401i
\(28\) 0 0
\(29\) 6.06287 + 4.05108i 1.12585 + 0.752266i 0.971802 − 0.235798i \(-0.0757702\pi\)
0.154044 + 0.988064i \(0.450770\pi\)
\(30\) 0 0
\(31\) −5.97389 + 1.18828i −1.07294 + 0.213422i −0.699790 − 0.714348i \(-0.746723\pi\)
−0.373152 + 0.927770i \(0.621723\pi\)
\(32\) 0 0
\(33\) −7.45057 + 7.45057i −1.29698 + 1.29698i
\(34\) 0 0
\(35\) −3.07257 − 4.69697i −0.519360 − 0.793933i
\(36\) 0 0
\(37\) −6.71953 − 4.48985i −1.10468 − 0.738127i −0.137071 − 0.990561i \(-0.543769\pi\)
−0.967614 + 0.252435i \(0.918769\pi\)
\(38\) 0 0
\(39\) 2.08939 − 3.12699i 0.334570 − 0.500720i
\(40\) 0 0
\(41\) −10.2729 + 6.86415i −1.60436 + 1.07200i −0.656023 + 0.754741i \(0.727763\pi\)
−0.948339 + 0.317259i \(0.897237\pi\)
\(42\) 0 0
\(43\) 8.35071 + 3.45898i 1.27347 + 0.527489i 0.914018 − 0.405674i \(-0.132963\pi\)
0.359453 + 0.933163i \(0.382963\pi\)
\(44\) 0 0
\(45\) 14.6962 + 6.25598i 2.19078 + 0.932587i
\(46\) 0 0
\(47\) 2.86764i 0.418289i 0.977885 + 0.209144i \(0.0670679\pi\)
−0.977885 + 0.209144i \(0.932932\pi\)
\(48\) 0 0
\(49\) 0.646305 + 0.267708i 0.0923293 + 0.0382440i
\(50\) 0 0
\(51\) 8.86042 − 9.69147i 1.24071 − 1.35708i
\(52\) 0 0
\(53\) 1.07796 + 2.60243i 0.148070 + 0.357472i 0.980460 − 0.196718i \(-0.0630283\pi\)
−0.832391 + 0.554189i \(0.813028\pi\)
\(54\) 0 0
\(55\) −6.19088 + 4.04983i −0.834778 + 0.546079i
\(56\) 0 0
\(57\) 6.35849 + 9.51615i 0.842202 + 1.26044i
\(58\) 0 0
\(59\) −12.4276 + 5.14768i −1.61793 + 0.670170i −0.993804 − 0.111150i \(-0.964547\pi\)
−0.624130 + 0.781320i \(0.714547\pi\)
\(60\) 0 0
\(61\) −1.97526 − 2.95619i −0.252906 − 0.378501i 0.683192 − 0.730238i \(-0.260591\pi\)
−0.936099 + 0.351737i \(0.885591\pi\)
\(62\) 0 0
\(63\) 17.5850 − 3.49787i 2.21550 − 0.440691i
\(64\) 0 0
\(65\) 1.88522 − 1.84880i 0.233833 − 0.229315i
\(66\) 0 0
\(67\) 5.66190 + 5.66190i 0.691712 + 0.691712i 0.962608 − 0.270897i \(-0.0873202\pi\)
−0.270897 + 0.962608i \(0.587320\pi\)
\(68\) 0 0
\(69\) − 4.68746i − 0.564304i
\(70\) 0 0
\(71\) 1.73066 + 8.70063i 0.205392 + 1.03258i 0.936595 + 0.350415i \(0.113959\pi\)
−0.731203 + 0.682160i \(0.761041\pi\)
\(72\) 0 0
\(73\) −0.761283 − 3.82723i −0.0891014 − 0.447943i −0.999421 − 0.0340362i \(-0.989164\pi\)
0.910319 − 0.413907i \(-0.135836\pi\)
\(74\) 0 0
\(75\) 13.0653 + 9.10353i 1.50865 + 1.05119i
\(76\) 0 0
\(77\) −3.17794 + 7.67222i −0.362160 + 0.874331i
\(78\) 0 0
\(79\) 0.116402 − 0.585193i 0.0130963 − 0.0658394i −0.973685 − 0.227900i \(-0.926814\pi\)
0.986781 + 0.162061i \(0.0518140\pi\)
\(80\) 0 0
\(81\) −14.5621 + 14.5621i −1.61801 + 1.61801i
\(82\) 0 0
\(83\) −5.97595 + 2.47532i −0.655946 + 0.271702i −0.685731 − 0.727855i \(-0.740518\pi\)
0.0297857 + 0.999556i \(0.490518\pi\)
\(84\) 0 0
\(85\) 7.48178 − 5.38730i 0.811513 − 0.584335i
\(86\) 0 0
\(87\) −21.4551 + 8.88700i −2.30023 + 0.952787i
\(88\) 0 0
\(89\) 5.79498 − 5.79498i 0.614267 − 0.614267i −0.329788 − 0.944055i \(-0.606977\pi\)
0.944055 + 0.329788i \(0.106977\pi\)
\(90\) 0 0
\(91\) 0.578253 − 2.90708i 0.0606174 − 0.304744i
\(92\) 0 0
\(93\) 7.42347 − 17.9218i 0.769778 − 1.85841i
\(94\) 0 0
\(95\) 3.00252 + 7.45353i 0.308052 + 0.764716i
\(96\) 0 0
\(97\) −3.20159 − 16.0955i −0.325072 − 1.63425i −0.704981 − 0.709226i \(-0.749045\pi\)
0.379909 − 0.925024i \(-0.375955\pi\)
\(98\) 0 0
\(99\) −4.61040 − 23.1781i −0.463363 − 2.32948i
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.113.1 yes 72
5.2 odd 4 340.2.bd.a.317.9 yes 72
17.14 odd 16 340.2.bd.a.133.9 ✓ 72
85.82 even 16 inner 340.2.bi.a.337.1 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bd.a.133.9 ✓ 72 17.14 odd 16
340.2.bd.a.317.9 yes 72 5.2 odd 4
340.2.bi.a.113.1 yes 72 1.1 even 1 trivial
340.2.bi.a.337.1 yes 72 85.82 even 16 inner