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Properties

Label 340.2.bd.a.57.7
Level $340$
Weight $2$
Character 340.57
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(57,340)] 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("340.57"); 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, 15])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.bd (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 57.7
Character \(\chi\) \(=\) 340.57
Dual form 340.2.bd.a.173.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.841077 - 1.25876i) q^{3} +(-0.704990 + 2.12202i) q^{5} +(-0.993774 + 4.99604i) q^{7} +(0.270982 + 0.654209i) q^{9} +(-0.0182977 + 0.0919885i) q^{11} +1.69499i q^{13} +(2.07817 + 2.67220i) q^{15} +(-4.04310 - 0.808285i) q^{17} +(0.555392 - 1.34083i) q^{19} +(5.45298 + 5.45298i) q^{21} +(7.06718 - 4.72214i) q^{23} +(-4.00598 - 2.99201i) q^{25} +(5.50584 + 1.09518i) q^{27} +(1.26422 - 1.89204i) q^{29} +(-1.33938 - 6.73352i) q^{31} +(0.100402 + 0.100402i) q^{33} +(-9.90112 - 5.63097i) q^{35} +(7.49668 + 5.00912i) q^{37} +(2.13358 + 1.42561i) q^{39} +(0.409242 + 0.612474i) q^{41} +(-3.51206 + 8.47887i) q^{43} +(-1.57929 + 0.113820i) q^{45} +9.58734 q^{47} +(-17.5057 - 7.25109i) q^{49} +(-4.41800 + 4.40947i) q^{51} +(-8.55750 + 3.54463i) q^{53} +(-0.182302 - 0.103679i) q^{55} +(-1.22066 - 1.82685i) q^{57} +(8.64350 - 3.58026i) q^{59} +(-1.11782 + 0.746906i) q^{61} +(-3.53775 + 0.703702i) q^{63} +(-3.59680 - 1.19495i) q^{65} +(-1.45408 + 1.45408i) q^{67} -12.8676i q^{69} +(7.11492 - 1.41525i) q^{71} +(-0.164343 - 0.826206i) q^{73} +(-7.13556 + 2.52606i) q^{75} +(-0.441395 - 0.182832i) q^{77} +(5.67544 + 1.12892i) q^{79} +(4.50727 - 4.50727i) q^{81} +(0.540204 + 1.30417i) q^{83} +(4.56555 - 8.00973i) q^{85} +(-1.31832 - 3.18271i) q^{87} +(2.30681 - 2.30681i) q^{89} +(-8.46822 - 1.68443i) q^{91} +(-9.60242 - 3.97745i) q^{93} +(2.45374 + 2.12383i) q^{95} +(-2.23441 - 11.2332i) q^{97} +(-0.0651380 + 0.0129568i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 24 q^{15} + 8 q^{25} - 48 q^{27} - 32 q^{31} + 16 q^{33} + 32 q^{37} - 32 q^{39} - 40 q^{41} + 80 q^{47} - 40 q^{53} + 16 q^{55} + 8 q^{57} + 112 q^{59} - 48 q^{63} - 32 q^{67} - 16 q^{71} + 8 q^{73}+ \cdots + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(e\left(\frac{15}{16}\right)\)

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
\(3\) 0.841077 − 1.25876i 0.485596 − 0.726746i −0.505067 − 0.863080i \(-0.668532\pi\)
0.990663 + 0.136334i \(0.0435321\pi\)
\(4\) 0 0
\(5\) −0.704990 + 2.12202i −0.315281 + 0.948998i
\(6\) 0 0
\(7\) −0.993774 + 4.99604i −0.375611 + 1.88833i 0.0777048 + 0.996976i \(0.475241\pi\)
−0.453316 + 0.891350i \(0.649759\pi\)
\(8\) 0 0
\(9\) 0.270982 + 0.654209i 0.0903274 + 0.218070i
\(10\) 0 0
\(11\) −0.0182977 + 0.0919885i −0.00551695 + 0.0277356i −0.983446 − 0.181203i \(-0.942001\pi\)
0.977929 + 0.208939i \(0.0670009\pi\)
\(12\) 0 0
\(13\) 1.69499i 0.470105i 0.971983 + 0.235052i \(0.0755261\pi\)
−0.971983 + 0.235052i \(0.924474\pi\)
\(14\) 0 0
\(15\) 2.07817 + 2.67220i 0.536581 + 0.689959i
\(16\) 0 0
\(17\) −4.04310 − 0.808285i −0.980596 − 0.196038i
\(18\) 0 0
\(19\) 0.555392 − 1.34083i 0.127416 − 0.307609i −0.847279 − 0.531148i \(-0.821761\pi\)
0.974695 + 0.223539i \(0.0717610\pi\)
\(20\) 0 0
\(21\) 5.45298 + 5.45298i 1.18994 + 1.18994i
\(22\) 0 0
\(23\) 7.06718 − 4.72214i 1.47361 − 0.984634i 0.479350 − 0.877624i \(-0.340872\pi\)
0.994258 − 0.107010i \(-0.0341277\pi\)
\(24\) 0 0
\(25\) −4.00598 − 2.99201i −0.801196 − 0.598402i
\(26\) 0 0
\(27\) 5.50584 + 1.09518i 1.05960 + 0.210767i
\(28\) 0 0
\(29\) 1.26422 − 1.89204i 0.234760 − 0.351343i −0.695320 − 0.718700i \(-0.744737\pi\)
0.930080 + 0.367357i \(0.119737\pi\)
\(30\) 0 0
\(31\) −1.33938 − 6.73352i −0.240560 − 1.20938i −0.892479 − 0.451088i \(-0.851036\pi\)
0.651919 − 0.758288i \(-0.273964\pi\)
\(32\) 0 0
\(33\) 0.100402 + 0.100402i 0.0174777 + 0.0174777i
\(34\) 0 0
\(35\) −9.90112 − 5.63097i −1.67360 − 0.951808i
\(36\) 0 0
\(37\) 7.49668 + 5.00912i 1.23245 + 0.823495i 0.989215 − 0.146473i \(-0.0467923\pi\)
0.243232 + 0.969968i \(0.421792\pi\)
\(38\) 0 0
\(39\) 2.13358 + 1.42561i 0.341647 + 0.228281i
\(40\) 0 0
\(41\) 0.409242 + 0.612474i 0.0639128 + 0.0956523i 0.862049 − 0.506824i \(-0.169181\pi\)
−0.798137 + 0.602477i \(0.794181\pi\)
\(42\) 0 0
\(43\) −3.51206 + 8.47887i −0.535585 + 1.29302i 0.392193 + 0.919883i \(0.371716\pi\)
−0.927778 + 0.373133i \(0.878284\pi\)
\(44\) 0 0
\(45\) −1.57929 + 0.113820i −0.235426 + 0.0169673i
\(46\) 0 0
\(47\) 9.58734 1.39846 0.699229 − 0.714898i \(-0.253527\pi\)
0.699229 + 0.714898i \(0.253527\pi\)
\(48\) 0 0
\(49\) −17.5057 − 7.25109i −2.50081 − 1.03587i
\(50\) 0 0
\(51\) −4.41800 + 4.40947i −0.618644 + 0.617449i
\(52\) 0 0
\(53\) −8.55750 + 3.54463i −1.17546 + 0.486893i −0.882995 − 0.469382i \(-0.844477\pi\)
−0.292468 + 0.956275i \(0.594477\pi\)
\(54\) 0 0
\(55\) −0.182302 − 0.103679i −0.0245816 − 0.0139801i
\(56\) 0 0
\(57\) −1.22066 − 1.82685i −0.161681 − 0.241972i
\(58\) 0 0
\(59\) 8.64350 − 3.58026i 1.12529 − 0.466110i 0.259111 − 0.965847i \(-0.416570\pi\)
0.866177 + 0.499738i \(0.166570\pi\)
\(60\) 0 0
\(61\) −1.11782 + 0.746906i −0.143123 + 0.0956316i −0.625070 − 0.780569i \(-0.714930\pi\)
0.481947 + 0.876200i \(0.339930\pi\)
\(62\) 0 0
\(63\) −3.53775 + 0.703702i −0.445714 + 0.0886581i
\(64\) 0 0
\(65\) −3.59680 − 1.19495i −0.446128 − 0.148215i
\(66\) 0 0
\(67\) −1.45408 + 1.45408i −0.177644 + 0.177644i −0.790328 − 0.612684i \(-0.790090\pi\)
0.612684 + 0.790328i \(0.290090\pi\)
\(68\) 0 0
\(69\) − 12.8676i − 1.54907i
\(70\) 0 0
\(71\) 7.11492 − 1.41525i 0.844385 − 0.167959i 0.246103 − 0.969244i \(-0.420850\pi\)
0.598283 + 0.801285i \(0.295850\pi\)
\(72\) 0 0
\(73\) −0.164343 − 0.826206i −0.0192348 − 0.0967001i 0.969975 − 0.243203i \(-0.0781982\pi\)
−0.989210 + 0.146503i \(0.953198\pi\)
\(74\) 0 0
\(75\) −7.13556 + 2.52606i −0.823944 + 0.291684i
\(76\) 0 0
\(77\) −0.441395 − 0.182832i −0.0503016 − 0.0208356i
\(78\) 0 0
\(79\) 5.67544 + 1.12892i 0.638537 + 0.127013i 0.503733 − 0.863859i \(-0.331960\pi\)
0.134804 + 0.990872i \(0.456960\pi\)
\(80\) 0 0
\(81\) 4.50727 − 4.50727i 0.500808 − 0.500808i
\(82\) 0 0
\(83\) 0.540204 + 1.30417i 0.0592951 + 0.143151i 0.950750 − 0.309958i \(-0.100315\pi\)
−0.891455 + 0.453109i \(0.850315\pi\)
\(84\) 0 0
\(85\) 4.56555 − 8.00973i 0.495203 − 0.868777i
\(86\) 0 0
\(87\) −1.31832 − 3.18271i −0.141339 − 0.341222i
\(88\) 0 0
\(89\) 2.30681 − 2.30681i 0.244522 − 0.244522i −0.574196 − 0.818718i \(-0.694685\pi\)
0.818718 + 0.574196i \(0.194685\pi\)
\(90\) 0 0
\(91\) −8.46822 − 1.68443i −0.887711 − 0.176577i
\(92\) 0 0
\(93\) −9.60242 − 3.97745i −0.995724 − 0.412443i
\(94\) 0 0
\(95\) 2.45374 + 2.12383i 0.251748 + 0.217900i
\(96\) 0 0
\(97\) −2.23441 − 11.2332i −0.226870 − 1.14055i −0.911385 − 0.411555i \(-0.864986\pi\)
0.684515 − 0.728999i \(-0.260014\pi\)
\(98\) 0 0
\(99\) −0.0651380 + 0.0129568i −0.00654662 + 0.00130220i
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.bd.a.57.7 ✓ 72
5.3 odd 4 340.2.bi.a.193.7 yes 72
17.3 odd 16 340.2.bi.a.37.7 yes 72
85.3 even 16 inner 340.2.bd.a.173.7 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bd.a.57.7 ✓ 72 1.1 even 1 trivial
340.2.bd.a.173.7 yes 72 85.3 even 16 inner
340.2.bi.a.37.7 yes 72 17.3 odd 16
340.2.bi.a.193.7 yes 72 5.3 odd 4