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Properties

Label 340.2.bd.a.57.6
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 57.6
Character \(\chi\) \(=\) 340.57
Dual form 340.2.bd.a.173.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.349423 - 0.522949i) q^{3} +(1.09010 - 1.95235i) q^{5} +(0.0348214 - 0.175059i) q^{7} +(0.996671 + 2.40618i) q^{9} +(0.707055 - 3.55461i) q^{11} -2.05870i q^{13} +(-0.640072 - 1.25227i) q^{15} +(-1.65503 + 3.77636i) q^{17} +(2.61374 - 6.31013i) q^{19} +(-0.0793796 - 0.0793796i) q^{21} +(-0.577982 + 0.386195i) q^{23} +(-2.62335 - 4.25653i) q^{25} +(3.45715 + 0.687670i) q^{27} +(-0.304090 + 0.455102i) q^{29} +(0.666895 + 3.35271i) q^{31} +(-1.61182 - 1.61182i) q^{33} +(-0.303818 - 0.258816i) q^{35} +(4.14826 + 2.77178i) q^{37} +(-1.07660 - 0.719358i) q^{39} +(4.71458 + 7.05587i) q^{41} +(0.920283 - 2.22176i) q^{43} +(5.78418 + 0.677131i) q^{45} -10.9567 q^{47} +(6.43772 + 2.66659i) q^{49} +(1.39654 + 2.18505i) q^{51} +(-6.78119 + 2.80886i) q^{53} +(-6.16908 - 5.25531i) q^{55} +(-2.38658 - 3.57176i) q^{57} +(3.07891 - 1.27533i) q^{59} +(-8.11858 + 5.42466i) q^{61} +(0.455929 - 0.0906898i) q^{63} +(-4.01930 - 2.24420i) q^{65} +(-7.46930 + 7.46930i) q^{67} +0.437201i q^{69} +(-4.57239 + 0.909505i) q^{71} +(1.82819 + 9.19094i) q^{73} +(-3.14261 - 0.115455i) q^{75} +(-0.597645 - 0.247553i) q^{77} +(12.8682 + 2.55964i) q^{79} +(-3.95720 + 3.95720i) q^{81} +(0.00760730 + 0.0183656i) q^{83} +(5.56861 + 7.34782i) q^{85} +(0.131739 + 0.318047i) q^{87} +(2.21165 - 2.21165i) q^{89} +(-0.360394 - 0.0716868i) q^{91} +(1.98633 + 0.822763i) q^{93} +(-9.47034 - 11.9816i) q^{95} +(2.30306 + 11.5782i) q^{97} +(9.25772 - 1.84147i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 24 q^{15} + 8 q^{25} - 48 q^{27} - 32 q^{31} + 16 q^{33} + 32 q^{37} - 32 q^{39} - 40 q^{41} + 80 q^{47} - 40 q^{53} + 16 q^{55} + 8 q^{57} + 112 q^{59} - 48 q^{63} - 32 q^{67} - 16 q^{71} + 8 q^{73}+ \cdots + 96 q^{99}+O(q^{100}) \) 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
\(3\) 0.349423 − 0.522949i 0.201740 − 0.301925i −0.716780 − 0.697299i \(-0.754385\pi\)
0.918520 + 0.395374i \(0.129385\pi\)
\(4\) 0 0
\(5\) 1.09010 − 1.95235i 0.487509 − 0.873118i
\(6\) 0 0
\(7\) 0.0348214 − 0.175059i 0.0131613 − 0.0661661i −0.973647 − 0.228062i \(-0.926761\pi\)
0.986808 + 0.161896i \(0.0517610\pi\)
\(8\) 0 0
\(9\) 0.996671 + 2.40618i 0.332224 + 0.802059i
\(10\) 0 0
\(11\) 0.707055 − 3.55461i 0.213185 − 1.07175i −0.714855 − 0.699273i \(-0.753507\pi\)
0.928040 − 0.372481i \(-0.121493\pi\)
\(12\) 0 0
\(13\) − 2.05870i − 0.570981i −0.958382 − 0.285490i \(-0.907844\pi\)
0.958382 − 0.285490i \(-0.0921564\pi\)
\(14\) 0 0
\(15\) −0.640072 − 1.25227i −0.165266 − 0.323334i
\(16\) 0 0
\(17\) −1.65503 + 3.77636i −0.401404 + 0.915901i
\(18\) 0 0
\(19\) 2.61374 − 6.31013i 0.599634 − 1.44764i −0.274322 − 0.961638i \(-0.588453\pi\)
0.873956 − 0.486006i \(-0.161547\pi\)
\(20\) 0 0
\(21\) −0.0793796 − 0.0793796i −0.0173220 − 0.0173220i
\(22\) 0 0
\(23\) −0.577982 + 0.386195i −0.120518 + 0.0805272i −0.614368 − 0.789020i \(-0.710589\pi\)
0.493851 + 0.869547i \(0.335589\pi\)
\(24\) 0 0
\(25\) −2.62335 − 4.25653i −0.524669 − 0.851306i
\(26\) 0 0
\(27\) 3.45715 + 0.687670i 0.665329 + 0.132342i
\(28\) 0 0
\(29\) −0.304090 + 0.455102i −0.0564680 + 0.0845104i −0.858629 − 0.512597i \(-0.828683\pi\)
0.802161 + 0.597108i \(0.203683\pi\)
\(30\) 0 0
\(31\) 0.666895 + 3.35271i 0.119778 + 0.602164i 0.993318 + 0.115412i \(0.0368187\pi\)
−0.873540 + 0.486753i \(0.838181\pi\)
\(32\) 0 0
\(33\) −1.61182 − 1.61182i −0.280581 − 0.280581i
\(34\) 0 0
\(35\) −0.303818 − 0.258816i −0.0513546 − 0.0437479i
\(36\) 0 0
\(37\) 4.14826 + 2.77178i 0.681971 + 0.455678i 0.847688 − 0.530495i \(-0.177994\pi\)
−0.165717 + 0.986173i \(0.552994\pi\)
\(38\) 0 0
\(39\) −1.07660 − 0.719358i −0.172393 − 0.115189i
\(40\) 0 0
\(41\) 4.71458 + 7.05587i 0.736294 + 1.10194i 0.990861 + 0.134887i \(0.0430672\pi\)
−0.254567 + 0.967055i \(0.581933\pi\)
\(42\) 0 0
\(43\) 0.920283 − 2.22176i 0.140342 − 0.338815i −0.838044 − 0.545602i \(-0.816301\pi\)
0.978386 + 0.206787i \(0.0663009\pi\)
\(44\) 0 0
\(45\) 5.78418 + 0.677131i 0.862254 + 0.100941i
\(46\) 0 0
\(47\) −10.9567 −1.59820 −0.799100 − 0.601198i \(-0.794690\pi\)
−0.799100 + 0.601198i \(0.794690\pi\)
\(48\) 0 0
\(49\) 6.43772 + 2.66659i 0.919675 + 0.380942i
\(50\) 0 0
\(51\) 1.39654 + 2.18505i 0.195554 + 0.305968i
\(52\) 0 0
\(53\) −6.78119 + 2.80886i −0.931468 + 0.385827i −0.796235 − 0.604987i \(-0.793178\pi\)
−0.135233 + 0.990814i \(0.543178\pi\)
\(54\) 0 0
\(55\) −6.16908 − 5.25531i −0.831838 − 0.708626i
\(56\) 0 0
\(57\) −2.38658 − 3.57176i −0.316110 − 0.473092i
\(58\) 0 0
\(59\) 3.07891 − 1.27533i 0.400840 − 0.166034i −0.173150 − 0.984895i \(-0.555395\pi\)
0.573991 + 0.818862i \(0.305395\pi\)
\(60\) 0 0
\(61\) −8.11858 + 5.42466i −1.03948 + 0.694556i −0.953393 − 0.301732i \(-0.902435\pi\)
−0.0860841 + 0.996288i \(0.527435\pi\)
\(62\) 0 0
\(63\) 0.455929 − 0.0906898i 0.0574416 − 0.0114258i
\(64\) 0 0
\(65\) −4.01930 − 2.24420i −0.498533 − 0.278358i
\(66\) 0 0
\(67\) −7.46930 + 7.46930i −0.912520 + 0.912520i −0.996470 − 0.0839496i \(-0.973247\pi\)
0.0839496 + 0.996470i \(0.473247\pi\)
\(68\) 0 0
\(69\) 0.437201i 0.0526328i
\(70\) 0 0
\(71\) −4.57239 + 0.909505i −0.542643 + 0.107938i −0.458798 − 0.888540i \(-0.651720\pi\)
−0.0838446 + 0.996479i \(0.526720\pi\)
\(72\) 0 0
\(73\) 1.82819 + 9.19094i 0.213974 + 1.07572i 0.927138 + 0.374721i \(0.122261\pi\)
−0.713164 + 0.700997i \(0.752739\pi\)
\(74\) 0 0
\(75\) −3.14261 − 0.115455i −0.362877 − 0.0133315i
\(76\) 0 0
\(77\) −0.597645 − 0.247553i −0.0681080 − 0.0282113i
\(78\) 0 0
\(79\) 12.8682 + 2.55964i 1.44778 + 0.287982i 0.855522 − 0.517766i \(-0.173236\pi\)
0.592259 + 0.805748i \(0.298236\pi\)
\(80\) 0 0
\(81\) −3.95720 + 3.95720i −0.439689 + 0.439689i
\(82\) 0 0
\(83\) 0.00760730 + 0.0183656i 0.000835010 + 0.00201589i 0.924296 − 0.381675i \(-0.124653\pi\)
−0.923461 + 0.383691i \(0.874653\pi\)
\(84\) 0 0
\(85\) 5.56861 + 7.34782i 0.604001 + 0.796983i
\(86\) 0 0
\(87\) 0.131739 + 0.318047i 0.0141239 + 0.0340982i
\(88\) 0 0
\(89\) 2.21165 − 2.21165i 0.234435 − 0.234435i −0.580106 − 0.814541i \(-0.696989\pi\)
0.814541 + 0.580106i \(0.196989\pi\)
\(90\) 0 0
\(91\) −0.360394 − 0.0716868i −0.0377796 − 0.00751482i
\(92\) 0 0
\(93\) 1.98633 + 0.822763i 0.205972 + 0.0853165i
\(94\) 0 0
\(95\) −9.47034 − 11.9816i −0.971637 − 1.22929i
\(96\) 0 0
\(97\) 2.30306 + 11.5782i 0.233840 + 1.17559i 0.902054 + 0.431623i \(0.142059\pi\)
−0.668214 + 0.743969i \(0.732941\pi\)
\(98\) 0 0
\(99\) 9.25772 − 1.84147i 0.930435 − 0.185075i
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.57.6 ✓ 72
5.3 odd 4 340.2.bi.a.193.6 yes 72
17.3 odd 16 340.2.bi.a.37.6 yes 72
85.3 even 16 inner 340.2.bd.a.173.6 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bd.a.57.6 ✓ 72 1.1 even 1 trivial
340.2.bd.a.173.6 yes 72 85.3 even 16 inner
340.2.bi.a.37.6 yes 72 17.3 odd 16
340.2.bi.a.193.6 yes 72 5.3 odd 4