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Properties

Label 40.12.k.a
Level $40$
Weight $12$
Character orbit 40.k
Analytic conductor $30.734$
Analytic rank $0$
Dimension $128$
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] = [40,12,Mod(3,40)] 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("40.3"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 40.k (of order \(4\), degree \(2\), 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: \(30.7337272224\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(64\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 2 q^{2} - 4 q^{3} + 22352 q^{6} + 74836 q^{8} - 141690 q^{10} - 8 q^{11} - 114212 q^{12} - 1529272 q^{16} - 5279472 q^{17} - 7492118 q^{18} - 19078400 q^{20} + 110378148 q^{22} - 34182200 q^{25}+ \cdots - 277418419174 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −45.2548 + 0.0166695i −512.817 − 512.817i 2048.00 − 1.50875i 2579.37 + 6494.23i 23216.0 + 23198.9i −48901.7 − 48901.7i −92681.8 + 102.417i 348815.i −116837. − 293852.i
3.2 −45.1450 − 3.15148i 407.682 + 407.682i 2028.14 + 284.547i −6950.12 + 723.894i −17120.0 − 19689.6i −28654.9 − 28654.9i −90663.4 − 19237.5i 155262.i 316044. − 10777.0i
3.3 −44.8665 + 5.91579i −242.674 − 242.674i 1978.01 − 530.842i −5127.25 + 4747.57i 12323.5 + 9452.33i 59475.8 + 59475.8i −85605.9 + 35518.5i − 59365.5i 201956. − 243339.i
3.4 −44.7090 − 7.00721i 536.038 + 536.038i 1949.80 + 626.571i 4766.70 + 5109.47i −20209.6 − 27721.9i 17355.0 + 17355.0i −82783.1 − 41676.0i 397526.i −177312. − 261841.i
3.5 −44.5442 − 7.98827i 7.55508 + 7.55508i 1920.38 + 711.662i 6527.10 − 2495.01i −276.183 − 396.887i −8488.40 − 8488.40i −79856.7 − 47040.9i − 177033.i −310675. + 58998.1i
3.6 −43.9230 + 10.8982i 267.607 + 267.607i 1810.46 − 957.367i 220.376 − 6984.24i −14670.5 − 8837.65i 48495.9 + 48495.9i −69087.0 + 61781.2i − 33920.1i 66436.4 + 309170.i
3.7 −43.4260 − 12.7352i −347.850 − 347.850i 1723.63 + 1106.08i −4198.50 − 5585.76i 10675.8 + 19535.7i −6139.06 − 6139.06i −60764.3 − 69983.1i 64852.8i 111188. + 296036.i
3.8 −41.4860 + 18.0807i 33.8917 + 33.8917i 1394.18 − 1500.19i −5788.81 − 3913.80i −2018.82 − 793.244i −45957.4 − 45957.4i −30714.2 + 87444.7i − 174850.i 310919. + 57702.0i
3.9 −41.2436 + 18.6269i −478.390 − 478.390i 1354.08 − 1536.48i 1373.13 − 6851.47i 28641.5 + 10819.6i 4736.00 + 4736.00i −27227.0 + 88592.5i 280567.i 70988.8 + 308157.i
3.10 −41.2233 + 18.6718i 106.766 + 106.766i 1350.73 − 1539.43i 2141.26 + 6651.55i −6394.79 − 2407.75i −11151.9 − 11151.9i −26937.6 + 88680.9i − 154349.i −212466. − 234218.i
3.11 −39.1257 − 22.7416i −106.549 − 106.549i 1013.64 + 1779.56i −5069.71 + 4808.97i 1745.71 + 6591.88i −23188.0 − 23188.0i 810.535 − 92678.4i − 154442.i 307719. − 72861.1i
3.12 −35.7238 − 27.7815i 299.701 + 299.701i 504.382 + 1984.92i −5247.28 − 4614.56i −2380.33 − 19032.6i 37379.9 + 37379.9i 37125.5 − 84921.3i 2494.06i 59253.5 + 310627.i
3.13 −35.6957 − 27.8176i −465.247 − 465.247i 500.363 + 1985.94i 6743.24 + 1832.15i 3665.25 + 29549.4i 56964.7 + 56964.7i 37383.2 − 84808.2i 255762.i −189739. − 252981.i
3.14 −35.0127 − 28.6725i 161.775 + 161.775i 403.781 + 2007.80i 1170.45 + 6888.99i −1025.70 − 10302.7i 18427.4 + 18427.4i 43431.1 − 81876.0i − 124804.i 156544. − 274762.i
3.15 −34.4421 + 29.3555i −280.418 − 280.418i 324.511 − 2022.13i 6970.18 − 494.633i 17890.0 + 1426.37i 9905.30 + 9905.30i 48183.7 + 79172.4i − 19878.4i −225547. + 221649.i
3.16 −33.9596 + 29.9123i 462.074 + 462.074i 258.514 − 2031.62i 5907.12 − 3732.84i −29513.6 − 1870.19i −32730.0 − 32730.0i 51991.2 + 76725.8i 249879.i −88945.8 + 303461.i
3.17 −32.4144 − 31.5801i 420.354 + 420.354i 53.3909 + 2047.30i 2247.72 − 6616.33i −350.704 − 26900.4i −46629.5 − 46629.5i 62923.5 − 68048.3i 176249.i −281803. + 143481.i
3.18 −29.9123 + 33.9596i 462.074 + 462.074i −258.514 − 2031.62i −5907.12 + 3732.84i −29513.6 + 1870.19i 32730.0 + 32730.0i 76725.8 + 51991.2i 249879.i 49929.2 − 312261.i
3.19 −29.3555 + 34.4421i −280.418 − 280.418i −324.511 − 2022.13i −6970.18 + 494.633i 17890.0 − 1426.37i −9905.30 − 9905.30i 79172.4 + 48183.7i − 19878.4i 187577. − 254588.i
3.20 −26.3884 − 36.7648i −240.149 − 240.149i −655.299 + 1940.33i 1087.38 − 6902.59i −2491.86 + 15166.2i −4728.67 − 4728.67i 88628.2 − 27110.4i − 61804.2i −282466. + 142171.i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.12.k.a ✓ 128
5.c odd 4 1 inner 40.12.k.a ✓ 128
8.d odd 2 1 inner 40.12.k.a ✓ 128
40.k even 4 1 inner 40.12.k.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.12.k.a ✓ 128 1.a even 1 1 trivial
40.12.k.a ✓ 128 5.c odd 4 1 inner
40.12.k.a ✓ 128 8.d odd 2 1 inner
40.12.k.a ✓ 128 40.k even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(40, [\chi])\).