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Properties

Label 105.10.b.a
Level $105$
Weight $10$
Character orbit 105.b
Analytic conductor $54.079$
Analytic rank $0$
Dimension $48$
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] = [105,10,Mod(41,105)] 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("105.41"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 105.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,-106] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(54.0787627972\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 106 q^{3} - 12288 q^{4} - 30000 q^{5} + 3808 q^{6} - 228 q^{7} - 16474 q^{9} - 193002 q^{12} - 471150 q^{14} + 66250 q^{15} + 2919288 q^{16} - 408024 q^{17} - 1530326 q^{18} + 7680000 q^{20} - 91294 q^{21}+ \cdots - 2961709910 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 − 43.8827i −100.039 + 98.3630i −1413.69 −625.000 4316.43 + 4389.96i −4705.38 − 4267.67i 39568.6i 332.435 − 19680.2i 27426.7i
41.2 − 41.7310i 136.280 + 33.3295i −1229.48 −625.000 1390.87 − 5687.09i 5590.24 − 3017.09i 29941.1i 17461.3 + 9084.26i 26081.9i
41.3 − 41.2150i 116.470 − 78.2161i −1186.67 −625.000 −3223.67 − 4800.31i −4635.13 − 4343.86i 27806.7i 7447.50 − 18219.6i 25759.4i
41.4 − 40.9231i −9.88109 − 139.948i −1162.70 −625.000 −5727.09 + 404.365i 4816.82 + 4141.48i 26628.6i −19487.7 + 2765.67i 25576.9i
41.5 − 38.4101i −69.9034 + 121.641i −963.337 −625.000 4672.24 + 2685.00i 5478.84 + 3214.95i 17335.9i −9910.03 − 17006.2i 24006.3i
41.6 − 36.8025i −126.503 − 60.6619i −842.423 −625.000 −2232.51 + 4655.64i −4625.32 + 4354.31i 12160.4i 12323.3 + 15347.9i 23001.6i
41.7 − 34.6222i 54.0957 + 129.447i −686.699 −625.000 4481.76 − 1872.92i −6314.10 + 696.929i 6048.45i −13830.3 + 14005.1i 21638.9i
41.8 − 33.2186i 133.470 − 43.2288i −591.477 −625.000 −1436.00 − 4433.69i −818.322 + 6299.52i 2640.13i 15945.5 − 11539.5i 20761.6i
41.9 − 32.8589i −139.804 − 11.7362i −567.706 −625.000 −385.639 + 4593.81i 2905.91 − 5648.83i 1830.43i 19407.5 + 3281.55i 20536.8i
41.10 − 31.5403i 32.6981 − 136.433i −482.792 −625.000 −4303.13 − 1031.31i −1482.61 − 6177.01i − 921.226i −17544.7 − 8922.17i 19712.7i
41.11 − 30.4947i 22.0477 + 138.553i −417.929 −625.000 4225.13 − 672.340i 3362.45 − 5389.58i − 2868.66i −18710.8 + 6109.55i 19059.2i
41.12 − 23.8452i 94.3975 + 103.789i −56.5947 −625.000 2474.87 − 2250.93i 4976.51 + 3948.16i − 10859.2i −1861.22 + 19594.8i 14903.3i
41.13 − 23.6795i −83.7911 − 112.526i −48.7196 −625.000 −2664.56 + 1984.13i 5716.18 − 2771.09i − 10970.3i −5641.10 + 18857.3i 14799.7i
41.14 − 23.5262i 140.052 + 8.28067i −41.4820 −625.000 194.813 − 3294.88i −6334.06 − 482.954i − 11069.5i 19545.9 + 2319.44i 14703.9i
41.15 − 20.5160i −105.802 + 92.1359i 91.0933 −625.000 1890.26 + 2170.63i −3670.48 + 5184.71i − 12373.1i 2704.96 − 19496.2i 12822.5i
41.16 − 16.5021i −37.2714 − 135.255i 239.680 −625.000 −2231.99 + 615.058i −5846.47 − 2484.44i − 12404.3i −16904.7 + 10082.3i 10313.8i
41.17 − 15.8928i 98.0906 − 100.306i 259.418 −625.000 −1594.14 − 1558.94i 6132.99 + 1655.31i − 12260.0i −439.458 − 19678.1i 9933.02i
41.18 − 14.4961i −135.411 + 36.6978i 301.862 −625.000 531.976 + 1962.94i −5045.95 − 3859.01i − 11797.9i 16989.5 − 9938.60i 9060.09i
41.19 − 13.0875i −86.4807 − 110.472i 340.718 −625.000 −1445.80 + 1131.81i 501.130 + 6332.65i − 11159.9i −4725.18 + 19107.4i 8179.67i
41.20 − 11.9930i −38.9350 + 134.785i 368.167 −625.000 1616.48 + 466.949i 1637.91 − 6137.66i − 10555.9i −16651.1 − 10495.7i 7495.64i
See all 48 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 41.48
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.10.b.a ✓ 48
3.b odd 2 1 105.10.b.b yes 48
7.b odd 2 1 105.10.b.b yes 48
21.c even 2 1 inner 105.10.b.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.10.b.a ✓ 48 1.a even 1 1 trivial
105.10.b.a ✓ 48 21.c even 2 1 inner
105.10.b.b yes 48 3.b odd 2 1
105.10.b.b yes 48 7.b odd 2 1