↑

Properties

Label 49.10.e.a
Level $49$
Weight $10$
Character orbit 49.e
Analytic conductor $25.237$
Analytic rank $0$
Dimension $246$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,10,Mod(8,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([12]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.8");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 49.e (of order \(7\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.2367559720\)
Analytic rank: \(0\)
Dimension: \(246\)
Relative dimension: \(41\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 246 q - 5 q^{2} - 5 q^{3} - 10245 q^{4} + 677 q^{5} + 14971 q^{6} - 2408 q^{7} + 305 q^{8} - 340374 q^{9} + 54469 q^{10} - 75759 q^{11} - 139412 q^{12} + 46305 q^{13} + 1170141 q^{14} + 256627 q^{15} - 3026557 q^{16}+ \cdots - 9994213472 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −27.5021 − 34.4865i −199.512 + 96.0798i −319.024 + 1397.74i −1501.71 + 723.187i 8800.44 + 4238.07i −4765.10 − 4200.88i 36629.2 − 17639.7i 18301.5 − 22949.3i 66240.4 + 31899.7i
8.2 −26.6219 − 33.3828i 49.3817 − 23.7810i −291.754 + 1278.26i −854.899 + 411.698i −2108.51 − 1015.40i 5966.56 + 2180.32i 30742.4 − 14804.8i −10399.1 + 13040.1i 36502.6 + 17578.7i
8.3 −26.0529 − 32.6693i 118.418 − 57.0270i −274.599 + 1203.10i 2200.70 − 1059.80i −4948.16 − 2382.91i −3198.87 + 5488.24i 27182.9 − 13090.6i −1501.47 + 1882.79i −91957.6 − 44284.5i
8.4 −23.5173 − 29.4897i 195.572 − 94.1824i −202.651 + 887.871i −698.746 + 336.498i −7376.72 − 3552.44i −3080.46 − 5555.57i 13549.3 − 6525.01i 17105.8 − 21450.0i 26355.8 + 12692.3i
8.5 −23.2638 − 29.1719i −97.6307 + 47.0165i −195.865 + 858.138i 1758.58 − 846.886i 3642.83 + 1754.29i −183.693 − 6349.79i 12378.1 − 5960.98i −4950.94 + 6208.28i −65616.5 − 31599.3i
8.6 −21.6527 − 27.1516i −188.121 + 90.5945i −154.440 + 676.645i 1069.65 − 515.115i 6533.11 + 3146.18i 3522.82 + 5286.15i 5696.04 − 2743.07i 14910.1 − 18696.7i −37146.9 − 17889.0i
8.7 −21.1744 − 26.5518i −51.7391 + 24.9162i −142.714 + 625.271i 49.5766 − 23.8748i 1757.11 + 846.181i −5980.06 + 2143.01i 3957.86 − 1906.01i −10216.0 + 12810.5i −1683.67 − 810.813i
8.8 −18.8984 − 23.6978i 27.1152 − 13.0580i −90.5066 + 396.535i −2117.49 + 1019.73i −821.877 − 395.795i −4256.71 + 4715.29i −2874.73 + 1384.40i −11707.4 + 14680.7i 64182.3 + 30908.6i
8.9 −18.0662 − 22.6543i −9.41753 + 4.53524i −72.8994 + 319.393i −246.139 + 118.535i 272.882 + 131.413i 3654.09 − 5196.27i −4813.85 + 2318.23i −12204.0 + 15303.4i 7132.13 + 3434.65i
8.10 −17.1277 − 21.4775i 186.442 − 89.7859i −53.9926 + 236.557i 1440.03 − 693.481i −5121.70 − 2466.48i 6242.74 − 1175.50i −6666.72 + 3210.52i 14427.1 − 18091.0i −39558.6 − 19050.4i
8.11 −15.8506 − 19.8760i −168.327 + 81.0619i −29.8833 + 130.927i −2073.17 + 998.385i 4279.26 + 2060.78i 6349.51 + 193.114i −8651.24 + 4166.22i 9490.72 − 11901.0i 52704.7 + 25381.3i
8.12 −15.3789 − 19.2846i 196.531 − 94.6442i −21.4526 + 93.9900i −780.749 + 375.989i −4847.61 − 2334.49i −371.887 + 6341.55i −9235.81 + 4447.73i 17394.6 − 21812.2i 19257.9 + 9274.11i
8.13 −10.6593 − 13.3664i 80.5360 − 38.7841i 48.8920 − 214.210i 1662.55 − 800.640i −1376.86 − 663.062i −5873.41 − 2420.06i −11270.8 + 5427.73i −7290.30 + 9141.75i −28423.3 − 13687.9i
8.14 −10.0609 − 12.6160i −197.170 + 94.9520i 55.9892 − 245.305i 330.844 − 159.326i 3181.63 + 1532.19i −5507.02 + 3166.44i −11101.8 + 5346.34i 17587.9 − 22054.5i −5338.66 − 2570.96i
8.15 −9.39492 − 11.7809i 12.2276 − 5.88849i 63.4066 − 277.803i 653.991 − 314.946i −184.249 − 88.7295i 4089.19 + 4861.29i −10819.4 + 5210.35i −12157.3 + 15244.8i −9854.52 − 4745.69i
8.16 −8.43131 − 10.5725i −178.962 + 86.1834i 73.2393 − 320.883i −655.391 + 315.620i 2420.06 + 1165.44i −2632.21 − 5781.44i −10248.0 + 4935.20i 12327.6 − 15458.3i 8862.71 + 4268.06i
8.17 −6.59322 − 8.26763i 87.9977 − 42.3775i 89.0475 − 390.142i −1655.17 + 797.089i −930.549 − 448.129i −2112.47 − 5990.91i −8690.73 + 4185.24i −6324.41 + 7930.55i 17503.0 + 8428.98i
8.18 −4.37680 − 5.48834i −165.717 + 79.8053i 102.965 − 451.120i 2378.93 − 1145.63i 1163.31 + 560.221i 4498.11 − 4485.61i −6164.79 + 2968.80i 8821.23 − 11061.5i −16699.7 − 8042.17i
8.19 −3.77201 − 4.72995i −18.2261 + 8.77721i 105.786 − 463.480i 840.765 − 404.891i 110.265 + 53.1006i 1565.88 + 6156.43i −5382.03 + 2591.85i −12017.0 + 15068.8i −5086.48 − 2449.52i
8.20 −2.51927 − 3.15906i 165.553 − 79.7261i 110.298 − 483.246i −1650.12 + 794.655i −668.932 − 322.141i 6343.91 − 329.246i −3668.39 + 1766.60i 8779.36 − 11009.0i 6667.46 + 3210.88i
See next 80 embeddings (of 246 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.41
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.10.e.a ✓ 246
49.e even 7 1 inner 49.10.e.a ✓ 246
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.10.e.a ✓ 246 1.a even 1 1 trivial
49.10.e.a ✓ 246 49.e even 7 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(49, [\chi])\).