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Properties

Label 828.2.u.a.343.8
Level $828$
Weight $2$
Character 828.343
Analytic conductor $6.612$
Analytic rank $0$
Dimension $100$
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] = [828,2,Mod(19,828)] 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("828.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(828, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 0, 15])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 828.u (of order \(22\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [100,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.61161328736\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 92)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 343.8
Character \(\chi\) \(=\) 828.343
Dual form 828.2.u.a.379.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.04738 - 0.950264i) q^{2} +(0.193997 - 1.99057i) q^{4} +(-0.934364 + 3.18215i) q^{5} +(-0.148303 + 0.171151i) q^{7} +(-1.68838 - 2.26922i) q^{8} +(2.04525 + 4.22080i) q^{10} +(3.48603 - 2.24034i) q^{11} +(3.81497 + 4.40271i) q^{13} +(0.00730929 + 0.320186i) q^{14} +(-3.92473 - 0.772328i) q^{16} +(0.449525 + 0.205291i) q^{17} +(0.608905 + 1.33332i) q^{19} +(6.15303 + 2.47724i) q^{20} +(1.52228 - 5.65913i) q^{22} +(4.22490 + 2.26941i) q^{23} +(-5.04678 - 3.24337i) q^{25} +(8.17944 + 0.986067i) q^{26} +(0.311917 + 0.328410i) q^{28} +(0.299010 - 0.654741i) q^{29} +(9.23576 + 1.32790i) q^{31} +(-4.84459 + 2.92061i) q^{32} +(0.665903 - 0.212150i) q^{34} +(-0.406059 - 0.631840i) q^{35} +(-1.33496 - 4.54647i) q^{37} +(1.90476 + 0.817864i) q^{38} +(8.79858 - 3.25239i) q^{40} +(2.79241 + 0.819926i) q^{41} +(-0.508752 - 3.53845i) q^{43} +(-3.78327 - 7.37380i) q^{44} +(6.58160 - 1.63784i) q^{46} +6.02195i q^{47} +(0.988905 + 6.87799i) q^{49} +(-8.36794 + 1.39875i) q^{50} +(9.50398 - 6.73984i) q^{52} +(-4.56115 - 3.95226i) q^{53} +(3.87187 + 13.1864i) q^{55} +(0.638771 + 0.0475654i) q^{56} +(-0.309000 - 0.969899i) q^{58} +(4.34794 - 3.76751i) q^{59} +(-9.72972 - 1.39892i) q^{61} +(10.9352 - 7.38560i) q^{62} +(-2.29876 + 7.66262i) q^{64} +(-17.5746 + 8.02607i) q^{65} +(-10.2383 - 6.57976i) q^{67} +(0.495852 - 0.854984i) q^{68} +(-1.02571 - 0.275911i) q^{70} +(-2.54626 + 3.96206i) q^{71} +(-2.85291 - 6.24699i) q^{73} +(-5.71855 - 3.49330i) q^{74} +(2.77218 - 0.953409i) q^{76} +(-0.133554 + 0.928886i) q^{77} +(-1.34769 - 1.55532i) q^{79} +(6.12479 - 11.7675i) q^{80} +(3.70385 - 1.79476i) q^{82} +(-6.97591 + 2.04831i) q^{83} +(-1.07329 + 1.23864i) q^{85} +(-3.89531 - 3.22264i) q^{86} +(-10.9696 - 4.12805i) q^{88} +(-14.1914 + 2.04041i) q^{89} -1.31930 q^{91} +(5.33704 - 7.96969i) q^{92} +(5.72245 + 6.30725i) q^{94} +(-4.81175 + 0.691826i) q^{95} +(-0.265360 + 0.903735i) q^{97} +(7.57166 + 6.26412i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 100 q + 7 q^{2} - 11 q^{4} + 22 q^{5} + 10 q^{8} - 11 q^{10} - 18 q^{13} + 11 q^{14} + 5 q^{16} + 22 q^{17} + 11 q^{20} - 16 q^{25} - 12 q^{26} - 11 q^{28} + 42 q^{29} + 27 q^{32} + 11 q^{34} - 22 q^{37}+ \cdots + 71 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/828\mathbb{Z}\right)^\times\).

\(n\) \(415\) \(461\) \(649\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{13}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.04738 − 0.950264i 0.740607 − 0.671938i
\(3\) 0 0
\(4\) 0.193997 − 1.99057i 0.0969984 − 0.995285i
\(5\) −0.934364 + 3.18215i −0.417860 + 1.42310i 0.434749 + 0.900552i \(0.356837\pi\)
−0.852609 + 0.522550i \(0.824981\pi\)
\(6\) 0 0
\(7\) −0.148303 + 0.171151i −0.0560533 + 0.0646889i −0.783084 − 0.621916i \(-0.786354\pi\)
0.727030 + 0.686605i \(0.240900\pi\)
\(8\) −1.68838 − 2.26922i −0.596932 − 0.802292i
\(9\) 0 0
\(10\) 2.04525 + 4.22080i 0.646766 + 1.33474i
\(11\) 3.48603 − 2.24034i 1.05108 − 0.675487i 0.103376 − 0.994642i \(-0.467036\pi\)
0.947702 + 0.319156i \(0.103399\pi\)
\(12\) 0 0
\(13\) 3.81497 + 4.40271i 1.05808 + 1.22109i 0.974452 + 0.224594i \(0.0721055\pi\)
0.0836287 + 0.996497i \(0.473349\pi\)
\(14\) 0.00730929 + 0.320186i 0.00195349 + 0.0855734i
\(15\) 0 0
\(16\) −3.92473 − 0.772328i −0.981183 − 0.193082i
\(17\) 0.449525 + 0.205291i 0.109026 + 0.0497904i 0.469181 − 0.883102i \(-0.344549\pi\)
−0.360155 + 0.932892i \(0.617276\pi\)
\(18\) 0 0
\(19\) 0.608905 + 1.33332i 0.139692 + 0.305884i 0.966529 − 0.256559i \(-0.0825889\pi\)
−0.826836 + 0.562443i \(0.809862\pi\)
\(20\) 6.15303 + 2.47724i 1.37586 + 0.553928i
\(21\) 0 0
\(22\) 1.52228 − 5.65913i 0.324551 − 1.20653i
\(23\) 4.22490 + 2.26941i 0.880952 + 0.473206i
\(24\) 0 0
\(25\) −5.04678 − 3.24337i −1.00936 − 0.648674i
\(26\) 8.17944 + 0.986067i 1.60412 + 0.193384i
\(27\) 0 0
\(28\) 0.311917 + 0.328410i 0.0589468 + 0.0620637i
\(29\) 0.299010 − 0.654741i 0.0555248 − 0.121582i −0.879836 − 0.475277i \(-0.842348\pi\)
0.935361 + 0.353695i \(0.115075\pi\)
\(30\) 0 0
\(31\) 9.23576 + 1.32790i 1.65879 + 0.238498i 0.907069 − 0.420981i \(-0.138314\pi\)
0.751723 + 0.659479i \(0.229223\pi\)
\(32\) −4.84459 + 2.92061i −0.856410 + 0.516296i
\(33\) 0 0
\(34\) 0.665903 − 0.212150i 0.114201 − 0.0363834i
\(35\) −0.406059 − 0.631840i −0.0686364 − 0.106800i
\(36\) 0 0
\(37\) −1.33496 − 4.54647i −0.219467 − 0.747435i −0.993454 − 0.114233i \(-0.963559\pi\)
0.773987 − 0.633201i \(-0.218259\pi\)
\(38\) 1.90476 + 0.817864i 0.308992 + 0.132675i
\(39\) 0 0
\(40\) 8.79858 − 3.25239i 1.39118 − 0.514249i
\(41\) 2.79241 + 0.819926i 0.436101 + 0.128051i 0.492412 − 0.870362i \(-0.336115\pi\)
−0.0563103 + 0.998413i \(0.517934\pi\)
\(42\) 0 0
\(43\) −0.508752 − 3.53845i −0.0775839 − 0.539608i −0.991132 − 0.132878i \(-0.957578\pi\)
0.913548 − 0.406730i \(-0.133331\pi\)
\(44\) −3.78327 − 7.37380i −0.570349 − 1.11164i
\(45\) 0 0
\(46\) 6.58160 − 1.63784i 0.970404 − 0.241486i
\(47\) 6.02195i 0.878392i 0.898391 + 0.439196i \(0.144737\pi\)
−0.898391 + 0.439196i \(0.855263\pi\)
\(48\) 0 0
\(49\) 0.988905 + 6.87799i 0.141272 + 0.982569i
\(50\) −8.36794 + 1.39875i −1.18341 + 0.197812i
\(51\) 0 0
\(52\) 9.50398 − 6.73984i 1.31796 − 0.934648i
\(53\) −4.56115 − 3.95226i −0.626522 − 0.542885i 0.282693 − 0.959210i \(-0.408772\pi\)
−0.909216 + 0.416326i \(0.863318\pi\)
\(54\) 0 0
\(55\) 3.87187 + 13.1864i 0.522083 + 1.77805i
\(56\) 0.638771 + 0.0475654i 0.0853594 + 0.00635620i
\(57\) 0 0
\(58\) −0.309000 − 0.969899i −0.0405737 − 0.127354i
\(59\) 4.34794 − 3.76751i 0.566053 − 0.490488i −0.324178 − 0.945996i \(-0.605088\pi\)
0.890232 + 0.455508i \(0.150542\pi\)
\(60\) 0 0
\(61\) −9.72972 − 1.39892i −1.24576 − 0.179114i −0.512278 − 0.858820i \(-0.671198\pi\)
−0.733485 + 0.679706i \(0.762107\pi\)
\(62\) 10.9352 − 7.38560i 1.38877 − 0.937972i
\(63\) 0 0
\(64\) −2.29876 + 7.66262i −0.287345 + 0.957827i
\(65\) −17.5746 + 8.02607i −2.17987 + 0.995511i
\(66\) 0 0
\(67\) −10.2383 − 6.57976i −1.25081 − 0.803846i −0.263810 − 0.964575i \(-0.584979\pi\)
−0.986999 + 0.160729i \(0.948615\pi\)
\(68\) 0.495852 − 0.854984i 0.0601309 − 0.103682i
\(69\) 0 0
\(70\) −1.02571 − 0.275911i −0.122596 − 0.0329777i
\(71\) −2.54626 + 3.96206i −0.302186 + 0.470210i −0.958826 − 0.283993i \(-0.908341\pi\)
0.656640 + 0.754204i \(0.271977\pi\)
\(72\) 0 0
\(73\) −2.85291 − 6.24699i −0.333907 − 0.731155i 0.665983 − 0.745967i \(-0.268012\pi\)
−0.999890 + 0.0148119i \(0.995285\pi\)
\(74\) −5.71855 − 3.49330i −0.664768 − 0.406088i
\(75\) 0 0
\(76\) 2.77218 − 0.953409i 0.317991 − 0.109363i
\(77\) −0.133554 + 0.928886i −0.0152198 + 0.105856i
\(78\) 0 0
\(79\) −1.34769 − 1.55532i −0.151627 − 0.174987i 0.674854 − 0.737951i \(-0.264206\pi\)
−0.826482 + 0.562964i \(0.809661\pi\)
\(80\) 6.12479 − 11.7675i 0.684772 − 1.31564i
\(81\) 0 0
\(82\) 3.70385 − 1.79476i 0.409022 − 0.198198i
\(83\) −6.97591 + 2.04831i −0.765706 + 0.224832i −0.641186 − 0.767386i \(-0.721557\pi\)
−0.124520 + 0.992217i \(0.539739\pi\)
\(84\) 0 0
\(85\) −1.07329 + 1.23864i −0.116414 + 0.134349i
\(86\) −3.89531 − 3.22264i −0.420042 − 0.347506i
\(87\) 0 0
\(88\) −10.9696 − 4.12805i −1.16936 − 0.440052i
\(89\) −14.1914 + 2.04041i −1.50428 + 0.216283i −0.844655 − 0.535311i \(-0.820194\pi\)
−0.659626 + 0.751594i \(0.729285\pi\)
\(90\) 0 0
\(91\) −1.31930 −0.138300
\(92\) 5.33704 − 7.96969i 0.556425 − 0.830898i
\(93\) 0 0
\(94\) 5.72245 + 6.30725i 0.590225 + 0.650544i
\(95\) −4.81175 + 0.691826i −0.493675 + 0.0709798i
\(96\) 0 0
\(97\) −0.265360 + 0.903735i −0.0269433 + 0.0917604i −0.971867 − 0.235530i \(-0.924317\pi\)
0.944924 + 0.327290i \(0.106136\pi\)
\(98\) 7.57166 + 6.26412i 0.764853 + 0.632772i
\(99\) 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 828.2.u.a.343.8 100
3.2 odd 2 92.2.h.a.67.3 yes 100
4.3 odd 2 inner 828.2.u.a.343.4 100
12.11 even 2 92.2.h.a.67.7 yes 100
23.11 odd 22 inner 828.2.u.a.379.4 100
69.11 even 22 92.2.h.a.11.7 yes 100
92.11 even 22 inner 828.2.u.a.379.8 100
276.11 odd 22 92.2.h.a.11.3 ✓ 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
92.2.h.a.11.3 ✓ 100 276.11 odd 22
92.2.h.a.11.7 yes 100 69.11 even 22
92.2.h.a.67.3 yes 100 3.2 odd 2
92.2.h.a.67.7 yes 100 12.11 even 2
828.2.u.a.343.4 100 4.3 odd 2 inner
828.2.u.a.343.8 100 1.1 even 1 trivial
828.2.u.a.379.4 100 23.11 odd 22 inner
828.2.u.a.379.8 100 92.11 even 22 inner