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Properties

Label 675.2.r.a.46.16
Level $675$
Weight $2$
Character 675.46
Analytic conductor $5.390$
Analytic rank $0$
Dimension $224$
Inner twists $4$

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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] = [675,2,Mod(46,675)] 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("675.46"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.r (of order \(15\), degree \(8\), not 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(28\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 46.16
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.16

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.220390 + 0.244767i) q^{2} +(0.197717 - 1.88116i) q^{4} +(-2.09765 + 0.774498i) q^{5} +(-1.22023 + 2.11349i) q^{7} +(1.03695 - 0.753387i) q^{8} +(-0.651873 - 0.342746i) q^{10} +(-1.13179 - 1.25698i) q^{11} +(-1.44915 + 1.60944i) q^{13} +(-0.786239 + 0.167120i) q^{14} +(-3.28743 - 0.698765i) q^{16} +(3.09482 - 2.24852i) q^{17} +(-4.15077 + 3.01571i) q^{19} +(1.04221 + 4.09915i) q^{20} +(0.0582329 - 0.554049i) q^{22} +(-8.53438 + 1.81404i) q^{23} +(3.80031 - 3.24926i) q^{25} -0.713316 q^{26} +(3.73455 + 2.71331i) q^{28} +(-5.29950 + 2.35949i) q^{29} +(-6.49762 - 2.89293i) q^{31} +(-1.83522 - 3.17869i) q^{32} +(1.23243 + 0.261961i) q^{34} +(0.922715 - 5.37844i) q^{35} +(2.56039 + 7.88008i) q^{37} +(-1.65293 - 0.351342i) q^{38} +(-1.59166 + 2.38346i) q^{40} +(-2.89941 + 3.22012i) q^{41} +(-3.28981 + 5.69811i) q^{43} +(-2.58834 + 1.88054i) q^{44} +(-2.32491 - 1.68914i) q^{46} +(7.27748 - 3.24014i) q^{47} +(0.522100 + 0.904304i) q^{49} +(1.63286 + 0.214089i) q^{50} +(2.74109 + 3.04429i) q^{52} +(2.04603 + 1.48652i) q^{53} +(3.34762 + 1.76013i) q^{55} +(0.326967 + 3.11088i) q^{56} +(-1.74548 - 0.777139i) q^{58} +(0.645013 - 0.716360i) q^{59} +(-5.83429 - 6.47963i) q^{61} +(-0.723913 - 2.22798i) q^{62} +(-1.70356 + 5.24301i) q^{64} +(1.79330 - 4.49841i) q^{65} +(3.08866 + 1.37516i) q^{67} +(-3.61791 - 6.26641i) q^{68} +(1.51982 - 0.959501i) q^{70} +(0.529305 + 0.384563i) q^{71} +(3.37004 - 10.3719i) q^{73} +(-1.36450 + 2.36339i) q^{74} +(4.85234 + 8.40450i) q^{76} +(4.03764 - 0.858227i) q^{77} +(5.75281 - 2.56132i) q^{79} +(7.43708 - 1.08034i) q^{80} -1.42718 q^{82} +(-1.49547 - 14.2285i) q^{83} +(-4.75039 + 7.11355i) q^{85} +(-2.11975 + 0.450567i) q^{86} +(-2.12059 - 0.450746i) q^{88} +(2.96245 - 9.11749i) q^{89} +(-1.63326 - 5.02664i) q^{91} +(1.72509 + 16.4132i) q^{92} +(2.39696 + 1.06720i) q^{94} +(6.37122 - 9.54068i) q^{95} +(-1.40779 + 0.626788i) q^{97} +(-0.106279 + 0.327092i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.220390 + 0.244767i 0.155839 + 0.173077i 0.816008 − 0.578040i \(-0.196182\pi\)
−0.660169 + 0.751117i \(0.729516\pi\)
\(3\) 0 0
\(4\) 0.197717 − 1.88116i 0.0988587 − 0.940578i
\(5\) −2.09765 + 0.774498i −0.938099 + 0.346366i
\(6\) 0 0
\(7\) −1.22023 + 2.11349i −0.461202 + 0.798825i −0.999021 − 0.0442351i \(-0.985915\pi\)
0.537819 + 0.843060i \(0.319248\pi\)
\(8\) 1.03695 − 0.753387i 0.366616 − 0.266362i
\(9\) 0 0
\(10\) −0.651873 − 0.342746i −0.206140 − 0.108386i
\(11\) −1.13179 − 1.25698i −0.341246 − 0.378992i 0.547956 − 0.836507i \(-0.315406\pi\)
−0.889202 + 0.457515i \(0.848740\pi\)
\(12\) 0 0
\(13\) −1.44915 + 1.60944i −0.401921 + 0.446379i −0.909798 − 0.415051i \(-0.863764\pi\)
0.507877 + 0.861430i \(0.330430\pi\)
\(14\) −0.786239 + 0.167120i −0.210131 + 0.0446648i
\(15\) 0 0
\(16\) −3.28743 − 0.698765i −0.821857 − 0.174691i
\(17\) 3.09482 − 2.24852i 0.750604 − 0.545346i −0.145410 − 0.989372i \(-0.546450\pi\)
0.896014 + 0.444025i \(0.146450\pi\)
\(18\) 0 0
\(19\) −4.15077 + 3.01571i −0.952252 + 0.691851i −0.951338 − 0.308148i \(-0.900291\pi\)
−0.000913382 1.00000i \(0.500291\pi\)
\(20\) 1.04221 + 4.09915i 0.233045 + 0.916597i
\(21\) 0 0
\(22\) 0.0582329 − 0.554049i 0.0124153 − 0.118124i
\(23\) −8.53438 + 1.81404i −1.77954 + 0.378253i −0.976133 − 0.217175i \(-0.930316\pi\)
−0.803409 + 0.595428i \(0.796982\pi\)
\(24\) 0 0
\(25\) 3.80031 − 3.24926i 0.760061 − 0.649852i
\(26\) −0.713316 −0.139893
\(27\) 0 0
\(28\) 3.73455 + 2.71331i 0.705763 + 0.512767i
\(29\) −5.29950 + 2.35949i −0.984093 + 0.438147i −0.834744 − 0.550638i \(-0.814384\pi\)
−0.149349 + 0.988785i \(0.547718\pi\)
\(30\) 0 0
\(31\) −6.49762 − 2.89293i −1.16701 − 0.519585i −0.270545 − 0.962707i \(-0.587204\pi\)
−0.896461 + 0.443123i \(0.853871\pi\)
\(32\) −1.83522 − 3.17869i −0.324424 − 0.561918i
\(33\) 0 0
\(34\) 1.23243 + 0.261961i 0.211360 + 0.0449260i
\(35\) 0.922715 − 5.37844i 0.155967 − 0.909122i
\(36\) 0 0
\(37\) 2.56039 + 7.88008i 0.420926 + 1.29548i 0.906842 + 0.421472i \(0.138486\pi\)
−0.485916 + 0.874006i \(0.661514\pi\)
\(38\) −1.65293 − 0.351342i −0.268141 − 0.0569952i
\(39\) 0 0
\(40\) −1.59166 + 2.38346i −0.251664 + 0.376858i
\(41\) −2.89941 + 3.22012i −0.452812 + 0.502898i −0.925718 − 0.378214i \(-0.876538\pi\)
0.472906 + 0.881113i \(0.343205\pi\)
\(42\) 0 0
\(43\) −3.28981 + 5.69811i −0.501691 + 0.868954i 0.498307 + 0.867001i \(0.333955\pi\)
−0.999998 + 0.00195353i \(0.999378\pi\)
\(44\) −2.58834 + 1.88054i −0.390207 + 0.283502i
\(45\) 0 0
\(46\) −2.32491 − 1.68914i −0.342789 − 0.249051i
\(47\) 7.27748 − 3.24014i 1.06153 − 0.472623i 0.199719 − 0.979853i \(-0.435997\pi\)
0.861811 + 0.507230i \(0.169330\pi\)
\(48\) 0 0
\(49\) 0.522100 + 0.904304i 0.0745857 + 0.129186i
\(50\) 1.63286 + 0.214089i 0.230921 + 0.0302767i
\(51\) 0 0
\(52\) 2.74109 + 3.04429i 0.380120 + 0.422166i
\(53\) 2.04603 + 1.48652i 0.281043 + 0.204190i 0.719372 − 0.694625i \(-0.244430\pi\)
−0.438329 + 0.898815i \(0.644430\pi\)
\(54\) 0 0
\(55\) 3.34762 + 1.76013i 0.451393 + 0.237336i
\(56\) 0.326967 + 3.11088i 0.0436928 + 0.415709i
\(57\) 0 0
\(58\) −1.74548 − 0.777139i −0.229193 − 0.102043i
\(59\) 0.645013 − 0.716360i 0.0839736 − 0.0932621i −0.699694 − 0.714443i \(-0.746680\pi\)
0.783667 + 0.621181i \(0.213347\pi\)
\(60\) 0 0
\(61\) −5.83429 − 6.47963i −0.747004 − 0.829632i 0.243095 − 0.970002i \(-0.421837\pi\)
−0.990099 + 0.140371i \(0.955171\pi\)
\(62\) −0.723913 − 2.22798i −0.0919371 − 0.282953i
\(63\) 0 0
\(64\) −1.70356 + 5.24301i −0.212945 + 0.655376i
\(65\) 1.79330 − 4.49841i 0.222432 − 0.557959i
\(66\) 0 0
\(67\) 3.08866 + 1.37516i 0.377339 + 0.168002i 0.586638 − 0.809850i \(-0.300451\pi\)
−0.209298 + 0.977852i \(0.567118\pi\)
\(68\) −3.61791 − 6.26641i −0.438737 − 0.759914i
\(69\) 0 0
\(70\) 1.51982 − 0.959501i 0.181654 − 0.114682i
\(71\) 0.529305 + 0.384563i 0.0628170 + 0.0456392i 0.618751 − 0.785587i \(-0.287639\pi\)
−0.555934 + 0.831227i \(0.687639\pi\)
\(72\) 0 0
\(73\) 3.37004 − 10.3719i 0.394434 − 1.21394i −0.534968 − 0.844872i \(-0.679676\pi\)
0.929402 − 0.369070i \(-0.120324\pi\)
\(74\) −1.36450 + 2.36339i −0.158620 + 0.274738i
\(75\) 0 0
\(76\) 4.85234 + 8.40450i 0.556602 + 0.964062i
\(77\) 4.03764 − 0.858227i 0.460132 − 0.0978041i
\(78\) 0 0
\(79\) 5.75281 − 2.56132i 0.647242 − 0.288171i −0.0567394 − 0.998389i \(-0.518070\pi\)
0.703982 + 0.710218i \(0.251404\pi\)
\(80\) 7.43708 − 1.08034i 0.831491 − 0.120786i
\(81\) 0 0
\(82\) −1.42718 −0.157606
\(83\) −1.49547 − 14.2285i −0.164150 − 1.56178i −0.697936 − 0.716160i \(-0.745898\pi\)
0.533786 − 0.845620i \(-0.320769\pi\)
\(84\) 0 0
\(85\) −4.75039 + 7.11355i −0.515252 + 0.771573i
\(86\) −2.11975 + 0.450567i −0.228579 + 0.0485859i
\(87\) 0 0
\(88\) −2.12059 − 0.450746i −0.226056 − 0.0480496i
\(89\) 2.96245 − 9.11749i 0.314019 − 0.966452i −0.662137 − 0.749383i \(-0.730350\pi\)
0.976156 − 0.217069i \(-0.0696496\pi\)
\(90\) 0 0
\(91\) −1.63326 − 5.02664i −0.171212 − 0.526935i
\(92\) 1.72509 + 16.4132i 0.179853 + 1.71119i
\(93\) 0 0
\(94\) 2.39696 + 1.06720i 0.247228 + 0.110073i
\(95\) 6.37122 − 9.54068i 0.653673 − 0.978853i
\(96\) 0 0
\(97\) −1.40779 + 0.626788i −0.142939 + 0.0636407i −0.476960 − 0.878925i \(-0.658261\pi\)
0.334020 + 0.942566i \(0.391595\pi\)
\(98\) −0.106279 + 0.327092i −0.0107358 + 0.0330413i
\(99\) 0 0
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 675.2.r.a.46.16 224
3.2 odd 2 225.2.q.a.196.13 yes 224
9.4 even 3 inner 675.2.r.a.496.13 224
9.5 odd 6 225.2.q.a.121.16 yes 224
25.6 even 5 inner 675.2.r.a.181.13 224
75.56 odd 10 225.2.q.a.106.16 yes 224
225.31 even 15 inner 675.2.r.a.631.16 224
225.131 odd 30 225.2.q.a.31.13 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.13 ✓ 224 225.131 odd 30
225.2.q.a.106.16 yes 224 75.56 odd 10
225.2.q.a.121.16 yes 224 9.5 odd 6
225.2.q.a.196.13 yes 224 3.2 odd 2
675.2.r.a.46.16 224 1.1 even 1 trivial
675.2.r.a.181.13 224 25.6 even 5 inner
675.2.r.a.496.13 224 9.4 even 3 inner
675.2.r.a.631.16 224 225.31 even 15 inner