↑

Properties

Label 340.2.l.b.103.18
Level $340$
Weight $2$
Character 340.103
Analytic conductor $2.715$
Analytic rank $0$
Dimension $88$
Inner twists $4$

Related objects

Downloads

Learn more

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(103,340)] 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("340.103"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [88] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.71491366872\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(44\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 103.18
Character \(\chi\) \(=\) 340.103
Dual form 340.2.l.b.307.18

$q$-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+(-0.479077 + 1.33060i) q^{2} +(-0.597940 + 0.597940i) q^{3} +(-1.54097 - 1.27491i) q^{4} +(1.85193 - 1.25314i) q^{5} +(-0.509157 - 1.08208i) q^{6} +(-1.46544 - 1.46544i) q^{7} +(2.43464 - 1.43963i) q^{8} +2.28494i q^{9} +(0.780205 + 3.06452i) q^{10} +3.69287i q^{11} +(1.68373 - 0.159086i) q^{12} +(3.68263 + 3.68263i) q^{13} +(2.65196 - 1.24785i) q^{14} +(-0.358041 + 1.85664i) q^{15} +(0.749185 + 3.92921i) q^{16} +(0.707107 - 0.707107i) q^{17} +(-3.04033 - 1.09466i) q^{18} +7.57626 q^{19} +(-4.45142 - 0.430002i) q^{20} +1.75248 q^{21} +(-4.91372 - 1.76917i) q^{22} +(-4.86168 + 4.86168i) q^{23} +(-0.594957 + 2.31658i) q^{24} +(1.85929 - 4.64145i) q^{25} +(-6.66436 + 3.13583i) q^{26} +(-3.16007 - 3.16007i) q^{27} +(0.389889 + 4.12650i) q^{28} +1.05234i q^{29} +(-2.29891 - 1.36588i) q^{30} +0.268307i q^{31} +(-5.58711 - 0.885532i) q^{32} +(-2.20812 - 2.20812i) q^{33} +(0.602115 + 1.27963i) q^{34} +(-4.55028 - 0.877490i) q^{35} +(2.91310 - 3.52102i) q^{36} +(-3.62712 + 3.62712i) q^{37} +(-3.62961 + 10.0809i) q^{38} -4.40399 q^{39} +(2.70473 - 5.71703i) q^{40} +11.1898 q^{41} +(-0.839575 + 2.33185i) q^{42} +(1.70602 - 1.70602i) q^{43} +(4.70810 - 5.69061i) q^{44} +(2.86334 + 4.23154i) q^{45} +(-4.13982 - 8.79805i) q^{46} +(0.582701 + 0.582701i) q^{47} +(-2.79740 - 1.90147i) q^{48} -2.70499i q^{49} +(5.28515 + 4.69757i) q^{50} +0.845615i q^{51} +(-0.979789 - 10.3699i) q^{52} +(-4.22583 - 4.22583i) q^{53} +(5.71870 - 2.69086i) q^{54} +(4.62768 + 6.83894i) q^{55} +(-5.67749 - 1.45813i) q^{56} +(-4.53015 + 4.53015i) q^{57} +(-1.40023 - 0.504149i) q^{58} +11.1501 q^{59} +(2.91879 - 2.40456i) q^{60} -8.34263 q^{61} +(-0.357009 - 0.128540i) q^{62} +(3.34843 - 3.34843i) q^{63} +(3.85494 - 7.00995i) q^{64} +(11.4348 + 2.20513i) q^{65} +(3.99597 - 1.88025i) q^{66} +(-8.48559 - 8.48559i) q^{67} +(-1.99113 + 0.188130i) q^{68} -5.81398i q^{69} +(3.34752 - 5.63420i) q^{70} +11.1822i q^{71} +(3.28946 + 5.56300i) q^{72} +(-1.53007 - 1.53007i) q^{73} +(-3.08857 - 6.56390i) q^{74} +(1.66357 + 3.88705i) q^{75} +(-11.6748 - 9.65909i) q^{76} +(5.41167 - 5.41167i) q^{77} +(2.10985 - 5.85993i) q^{78} +0.527241 q^{79} +(6.31129 + 6.33780i) q^{80} -3.07574 q^{81} +(-5.36077 + 14.8891i) q^{82} +(5.87580 - 5.87580i) q^{83} +(-2.70053 - 2.23427i) q^{84} +(0.423409 - 2.19561i) q^{85} +(1.45271 + 3.08735i) q^{86} +(-0.629233 - 0.629233i) q^{87} +(5.31637 + 8.99082i) q^{88} +5.86790i q^{89} +(-7.00223 + 1.78272i) q^{90} -10.7933i q^{91} +(13.6899 - 1.29348i) q^{92} +(-0.160432 - 0.160432i) q^{93} +(-1.05450 + 0.496182i) q^{94} +(14.0307 - 9.49411i) q^{95} +(3.87025 - 2.81126i) q^{96} +(-2.93054 + 2.93054i) q^{97} +(3.59925 + 1.29590i) q^{98} -8.43798 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q - 4 q^{2} + 16 q^{5} - 20 q^{6} - 16 q^{8} - 4 q^{10} - 28 q^{12} - 16 q^{13} + 4 q^{16} + 28 q^{18} + 16 q^{20} - 48 q^{21} - 16 q^{22} - 24 q^{25} - 52 q^{26} + 16 q^{28} + 36 q^{30} - 4 q^{32} + 56 q^{33}+ \cdots - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.479077 + 1.33060i −0.338758 + 0.940873i
\(3\) −0.597940 + 0.597940i −0.345221 + 0.345221i −0.858326 − 0.513105i \(-0.828495\pi\)
0.513105 + 0.858326i \(0.328495\pi\)
\(4\) −1.54097 − 1.27491i −0.770486 − 0.637457i
\(5\) 1.85193 − 1.25314i 0.828208 − 0.560421i
\(6\) −0.509157 − 1.08208i −0.207863 − 0.441755i
\(7\) −1.46544 − 1.46544i −0.553883 − 0.553883i 0.373676 − 0.927559i \(-0.378097\pi\)
−0.927559 + 0.373676i \(0.878097\pi\)
\(8\) 2.43464 − 1.43963i 0.860775 − 0.508985i
\(9\) 2.28494i 0.761645i
\(10\) 0.780205 + 3.06452i 0.246723 + 0.969086i
\(11\) 3.69287i 1.11344i 0.830699 + 0.556722i \(0.187941\pi\)
−0.830699 + 0.556722i \(0.812059\pi\)
\(12\) 1.68373 − 0.159086i 0.486051 − 0.0459241i
\(13\) 3.68263 + 3.68263i 1.02138 + 1.02138i 0.999766 + 0.0216127i \(0.00688009\pi\)
0.0216127 + 0.999766i \(0.493120\pi\)
\(14\) 2.65196 − 1.24785i 0.708766 − 0.333501i
\(15\) −0.358041 + 1.85664i −0.0924458 + 0.479383i
\(16\) 0.749185 + 3.92921i 0.187296 + 0.982303i
\(17\) 0.707107 − 0.707107i 0.171499 − 0.171499i
\(18\) −3.04033 − 1.09466i −0.716612 − 0.258014i
\(19\) 7.57626 1.73811 0.869057 − 0.494712i \(-0.164726\pi\)
0.869057 + 0.494712i \(0.164726\pi\)
\(20\) −4.45142 − 0.430002i −0.995367 − 0.0961513i
\(21\) 1.75248 0.382424
\(22\) −4.91372 − 1.76917i −1.04761 − 0.377188i
\(23\) −4.86168 + 4.86168i −1.01373 + 1.01373i −0.0138262 + 0.999904i \(0.504401\pi\)
−0.999904 + 0.0138262i \(0.995599\pi\)
\(24\) −0.594957 + 2.31658i −0.121445 + 0.472870i
\(25\) 1.85929 − 4.64145i 0.371857 − 0.928290i
\(26\) −6.66436 + 3.13583i −1.30699 + 0.614988i
\(27\) −3.16007 − 3.16007i −0.608156 − 0.608156i
\(28\) 0.389889 + 4.12650i 0.0736820 + 0.779835i
\(29\) 1.05234i 0.195414i 0.995215 + 0.0977069i \(0.0311508\pi\)
−0.995215 + 0.0977069i \(0.968849\pi\)
\(30\) −2.29891 − 1.36588i −0.419722 − 0.249375i
\(31\) 0.268307i 0.0481894i 0.999710 + 0.0240947i \(0.00767032\pi\)
−0.999710 + 0.0240947i \(0.992330\pi\)
\(32\) −5.58711 − 0.885532i −0.987671 − 0.156541i
\(33\) −2.20812 − 2.20812i −0.384384 − 0.384384i
\(34\) 0.602115 + 1.27963i 0.103262 + 0.219455i
\(35\) −4.55028 − 0.877490i −0.769138 − 0.148323i
\(36\) 2.91310 − 3.52102i 0.485516 − 0.586837i
\(37\) −3.62712 + 3.62712i −0.596295 + 0.596295i −0.939325 − 0.343029i \(-0.888547\pi\)
0.343029 + 0.939325i \(0.388547\pi\)
\(38\) −3.62961 + 10.0809i −0.588801 + 1.63535i
\(39\) −4.40399 −0.705202
\(40\) 2.70473 − 5.71703i 0.427655 − 0.903942i
\(41\) 11.1898 1.74755 0.873776 − 0.486328i \(-0.161664\pi\)
0.873776 + 0.486328i \(0.161664\pi\)
\(42\) −0.839575 + 2.33185i −0.129549 + 0.359812i
\(43\) 1.70602 − 1.70602i 0.260166 − 0.260166i −0.564955 − 0.825122i \(-0.691107\pi\)
0.825122 + 0.564955i \(0.191107\pi\)
\(44\) 4.70810 − 5.69061i 0.709773 − 0.857892i
\(45\) 2.86334 + 4.23154i 0.426842 + 0.630801i
\(46\) −4.13982 − 8.79805i −0.610383 − 1.29720i
\(47\) 0.582701 + 0.582701i 0.0849957 + 0.0849957i 0.748326 − 0.663331i \(-0.230858\pi\)
−0.663331 + 0.748326i \(0.730858\pi\)
\(48\) −2.79740 − 1.90147i −0.403770 − 0.274453i
\(49\) − 2.70499i − 0.386428i
\(50\) 5.28515 + 4.69757i 0.747434 + 0.664337i
\(51\) 0.845615i 0.118410i
\(52\) −0.979789 − 10.3699i −0.135872 − 1.43804i
\(53\) −4.22583 − 4.22583i −0.580462 − 0.580462i 0.354568 − 0.935030i \(-0.384628\pi\)
−0.935030 + 0.354568i \(0.884628\pi\)
\(54\) 5.71870 − 2.69086i 0.778216 − 0.366180i
\(55\) 4.62768 + 6.83894i 0.623997 + 0.922163i
\(56\) −5.67749 − 1.45813i −0.758687 − 0.194850i
\(57\) −4.53015 + 4.53015i −0.600033 + 0.600033i
\(58\) −1.40023 − 0.504149i −0.183860 − 0.0661981i
\(59\) 11.1501 1.45162 0.725811 − 0.687894i \(-0.241465\pi\)
0.725811 + 0.687894i \(0.241465\pi\)
\(60\) 2.91879 − 2.40456i 0.376815 − 0.310428i
\(61\) −8.34263 −1.06816 −0.534082 − 0.845432i \(-0.679343\pi\)
−0.534082 + 0.845432i \(0.679343\pi\)
\(62\) −0.357009 − 0.128540i −0.0453401 − 0.0163246i
\(63\) 3.34843 − 3.34843i 0.421862 − 0.421862i
\(64\) 3.85494 − 7.00995i 0.481868 − 0.876244i
\(65\) 11.4348 + 2.20513i 1.41832 + 0.273512i
\(66\) 3.99597 − 1.88025i 0.491870 − 0.231443i
\(67\) −8.48559 − 8.48559i −1.03668 − 1.03668i −0.999301 − 0.0373783i \(-0.988099\pi\)
−0.0373783 − 0.999301i \(-0.511901\pi\)
\(68\) −1.99113 + 0.188130i −0.241460 + 0.0228142i
\(69\) − 5.81398i − 0.699922i
\(70\) 3.34752 − 5.63420i 0.400105 − 0.673415i
\(71\) 11.1822i 1.32709i 0.748138 + 0.663543i \(0.230948\pi\)
−0.748138 + 0.663543i \(0.769052\pi\)
\(72\) 3.28946 + 5.56300i 0.387666 + 0.655605i
\(73\) −1.53007 − 1.53007i −0.179081 − 0.179081i 0.611874 − 0.790955i \(-0.290416\pi\)
−0.790955 + 0.611874i \(0.790416\pi\)
\(74\) −3.08857 − 6.56390i −0.359038 − 0.763038i
\(75\) 1.66357 + 3.88705i 0.192092 + 0.448838i
\(76\) −11.6748 − 9.65909i −1.33919 − 1.10797i
\(77\) 5.41167 − 5.41167i 0.616717 − 0.616717i
\(78\) 2.10985 − 5.85993i 0.238893 − 0.663506i
\(79\) 0.527241 0.0593192 0.0296596 − 0.999560i \(-0.490558\pi\)
0.0296596 + 0.999560i \(0.490558\pi\)
\(80\) 6.31129 + 6.33780i 0.705623 + 0.708587i
\(81\) −3.07574 −0.341749
\(82\) −5.36077 + 14.8891i −0.591998 + 1.64423i
\(83\) 5.87580 − 5.87580i 0.644953 − 0.644953i −0.306816 − 0.951769i \(-0.599264\pi\)
0.951769 + 0.306816i \(0.0992637\pi\)
\(84\) −2.70053 − 2.23427i −0.294652 − 0.243779i
\(85\) 0.423409 − 2.19561i 0.0459252 − 0.238148i
\(86\) 1.45271 + 3.08735i 0.156650 + 0.332917i
\(87\) −0.629233 − 0.629233i −0.0674609 − 0.0674609i
\(88\) 5.31637 + 8.99082i 0.566726 + 0.958424i
\(89\) 5.86790i 0.621996i 0.950411 + 0.310998i \(0.100663\pi\)
−0.950411 + 0.310998i \(0.899337\pi\)
\(90\) −7.00223 + 1.78272i −0.738100 + 0.187915i
\(91\) − 10.7933i − 1.13145i
\(92\) 13.6899 − 1.29348i 1.42727 − 0.134855i
\(93\) −0.160432 − 0.160432i −0.0166360 − 0.0166360i
\(94\) −1.05450 + 0.496182i −0.108763 + 0.0511772i
\(95\) 14.0307 − 9.49411i 1.43952 − 0.974075i
\(96\) 3.87025 − 2.81126i 0.395006 − 0.286923i
\(97\) −2.93054 + 2.93054i −0.297551 + 0.297551i −0.840054 − 0.542503i \(-0.817477\pi\)
0.542503 + 0.840054i \(0.317477\pi\)
\(98\) 3.59925 + 1.29590i 0.363580 + 0.130906i
\(99\) −8.43798 −0.848049
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.l.b.103.18 yes 88
4.3 odd 2 inner 340.2.l.b.103.4 ✓ 88
5.2 odd 4 inner 340.2.l.b.307.4 yes 88
20.7 even 4 inner 340.2.l.b.307.18 yes 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.l.b.103.4 ✓ 88 4.3 odd 2 inner
340.2.l.b.103.18 yes 88 1.1 even 1 trivial
340.2.l.b.307.4 yes 88 5.2 odd 4 inner
340.2.l.b.307.18 yes 88 20.7 even 4 inner