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Properties

Label 49.7.h.a
Level $49$
Weight $7$
Character orbit 49.h
Analytic conductor $11.273$
Analytic rank $0$
Dimension $324$
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] = [49,7,Mod(3,49)] 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("49.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.h (of order \(42\), degree \(12\), 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: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(324\)
Relative dimension: \(27\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 324 q - 13 q^{2} - 11 q^{3} + 819 q^{4} - 179 q^{5} + 770 q^{6} + 392 q^{7} + 828 q^{8} - 1160 q^{9} - 2594 q^{10} - 5305 q^{11} + 7497 q^{12} - 14 q^{13} - 11403 q^{14} - 6196 q^{15} + 27903 q^{16} - 5107 q^{17}+ \cdots - 4449616 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −5.64737 + 14.3893i 39.0389 + 2.92556i −128.243 − 118.992i 88.5491 − 129.878i −262.564 + 545.219i −318.627 − 126.988i 1545.11 − 744.086i 794.620 + 119.770i 1368.77 + 2007.62i
3.2 −5.28359 + 13.4624i 11.4646 + 0.859156i −106.404 − 98.7284i −126.721 + 185.865i −72.1408 + 149.802i 71.8347 + 335.393i 1057.40 − 509.218i −590.158 − 88.9520i −1832.64 − 2688.00i
3.3 −5.28100 + 13.4558i −41.7729 − 3.13045i −106.253 − 98.5887i 0.579414 − 0.849844i 262.725 − 545.555i −337.162 − 63.0124i 1054.21 − 507.679i 1014.32 + 152.884i 8.37542 + 12.2845i
3.4 −4.73204 + 12.0571i −20.1535 − 1.51030i −76.0649 − 70.5779i 43.5678 − 63.9023i 113.577 − 235.845i 342.839 + 10.5003i 464.042 − 223.471i −316.975 − 47.7763i 564.308 + 827.688i
3.5 −4.21887 + 10.7495i 22.3732 + 1.67664i −50.8375 − 47.1703i −46.9023 + 68.7930i −112.413 + 233.427i 65.6154 − 336.665i 55.6673 − 26.8080i −223.109 − 33.6282i −541.616 − 794.405i
3.6 −3.59619 + 9.16295i 6.31557 + 0.473286i −24.1117 − 22.3724i 88.0567 − 129.155i −27.0487 + 56.1672i −147.850 + 309.499i −275.882 + 132.858i −681.195 − 102.674i 866.776 + 1271.33i
3.7 −3.19117 + 8.13097i −23.4226 − 1.75528i −9.01384 − 8.36362i −44.5842 + 65.3931i 89.0179 − 184.848i −186.492 − 287.871i −406.896 + 195.951i −175.318 − 26.4250i −389.433 − 571.194i
3.8 −3.12007 + 7.94981i 45.1855 + 3.38619i −6.54926 − 6.07683i 24.2678 − 35.5943i −167.901 + 348.651i 341.606 + 30.8894i −423.699 + 204.043i 1309.41 + 197.361i 207.251 + 303.981i
3.9 −2.24612 + 5.72303i −33.4096 − 2.50371i 19.2073 + 17.8218i −77.8645 + 114.206i 89.3709 − 185.581i −16.1750 + 342.618i −499.644 + 240.616i 389.077 + 58.6439i −478.712 − 702.142i
3.10 −1.80046 + 4.58749i 34.7561 + 2.60461i 29.1119 + 27.0119i −64.1993 + 94.1630i −74.5254 + 154.754i −320.804 + 121.382i −460.499 + 221.764i 480.345 + 72.4004i −316.384 − 464.049i
3.11 −1.66144 + 4.23327i −49.8303 − 3.73426i 31.7551 + 29.4644i 72.7081 − 106.643i 98.5979 − 204.741i 331.343 − 88.6605i −439.716 + 211.756i 1748.25 + 263.507i 330.650 + 484.974i
3.12 −1.34783 + 3.43420i −1.28182 − 0.0960589i 36.9382 + 34.2736i 103.584 − 151.930i 2.05755 − 4.27255i −113.036 − 323.839i −380.217 + 183.103i −719.224 − 108.406i 382.145 + 560.504i
3.13 −0.399133 + 1.01698i −2.92834 − 0.219449i 46.0404 + 42.7192i −88.1806 + 129.337i 1.39197 − 2.89046i 305.864 − 155.229i −124.816 + 60.1083i −712.331 − 107.367i −96.3370 − 141.300i
3.14 0.314138 − 0.800410i 19.5533 + 1.46532i 46.3733 + 43.0282i 43.3923 − 63.6448i 7.31531 − 15.1904i 268.107 + 213.933i 98.5884 − 47.4777i −340.671 − 51.3480i −37.3108 − 54.7249i
3.15 0.472580 − 1.20411i −31.3476 − 2.34917i 45.6888 + 42.3930i 70.8216 − 103.876i −17.6429 + 36.6359i −336.701 + 65.4302i 147.225 − 70.8999i 256.293 + 38.6300i −91.6099 − 134.367i
3.16 1.12891 − 2.87641i 48.9686 + 3.66969i 39.9160 + 37.0366i 69.9503 − 102.598i 65.8366 − 136.711i −159.163 − 303.836i 329.771 − 158.809i 1663.60 + 250.747i −216.147 − 317.030i
3.17 1.55186 − 3.95409i 1.47718 + 0.110699i 33.6888 + 31.2586i −51.2166 + 75.1210i 2.73010 − 5.66911i −331.151 + 89.3766i 420.812 − 202.652i −718.688 − 108.325i 217.554 + 319.093i
3.18 2.25296 − 5.74046i −30.3956 − 2.27784i 19.0383 + 17.6650i 25.7577 − 37.7795i −81.5560 + 169.353i 83.9581 + 332.566i 499.884 − 240.732i 197.847 + 29.8207i −158.841 − 232.977i
3.19 2.29501 − 5.84759i −46.2118 − 3.46309i 17.9881 + 16.6905i −95.8768 + 140.625i −126.307 + 262.280i −87.9175 − 331.541i 501.105 − 241.319i 1402.68 + 211.419i 602.282 + 883.385i
3.20 2.90184 − 7.39377i 22.3918 + 1.67803i 0.668154 + 0.619956i −33.9718 + 49.8275i 77.3843 − 160.690i −28.4067 − 341.822i 464.522 − 223.702i −222.282 − 33.5037i 269.833 + 395.772i
See next 80 embeddings (of 324 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.27
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.h odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.7.h.a ✓ 324
49.h odd 42 1 inner 49.7.h.a ✓ 324
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.7.h.a ✓ 324 1.a even 1 1 trivial
49.7.h.a ✓ 324 49.h odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(49, [\chi])\).