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Properties

Label 105.10.b.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 106 q^{3} - 12288 q^{4} + 30000 q^{5} - 3808 q^{6} - 228 q^{7} - 16474 q^{9} + 193002 q^{12} + 471150 q^{14} + 66250 q^{15} + 2919288 q^{16} + 408024 q^{17} - 1530326 q^{18} - 7680000 q^{20} + 964582 q^{21}+ \cdots - 2961709910 q^{99}+O(q^{100}) \) 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.b yes 48
3.b odd 2 1 105.10.b.a ✓ 48
7.b odd 2 1 105.10.b.a ✓ 48
21.c even 2 1 inner 105.10.b.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.10.b.a ✓ 48 3.b odd 2 1
105.10.b.a ✓ 48 7.b odd 2 1
105.10.b.b yes 48 1.a even 1 1 trivial
105.10.b.b yes 48 21.c even 2 1 inner