Properties

Label 51.7.f.a
Level $51$
Weight $7$
Character orbit 51.f
Analytic conductor $11.733$
Analytic rank $0$
Dimension $68$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [51,7,Mod(38,51)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(51, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("51.38");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 51.f (of order \(4\), degree \(2\), 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: \(11.7327582646\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 18 q^{3} + 2040 q^{4} - 290 q^{6} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 18 q^{3} + 2040 q^{4} - 290 q^{6} - 4 q^{7} - 4256 q^{10} + 4182 q^{12} - 1832 q^{13} + 54976 q^{16} + 10940 q^{18} - 1604 q^{21} + 19576 q^{22} + 15110 q^{24} + 37314 q^{27} - 63120 q^{28} + 51360 q^{30} - 88660 q^{31} - 128604 q^{33} + 80200 q^{34} + 335180 q^{37} + 31460 q^{39} - 286828 q^{40} + 45484 q^{45} - 85244 q^{46} + 111922 q^{48} - 590566 q^{51} - 904320 q^{52} - 1215638 q^{54} - 731624 q^{55} + 753144 q^{57} - 328292 q^{58} + 751532 q^{61} + 88664 q^{63} + 870368 q^{64} + 174232 q^{67} + 1420428 q^{69} + 1158660 q^{72} + 2397332 q^{73} + 1415042 q^{75} - 1868044 q^{78} - 1438708 q^{79} - 2973932 q^{81} - 617036 q^{82} + 4901976 q^{84} + 2375468 q^{85} + 521156 q^{88} - 3185088 q^{90} + 1741672 q^{91} + 4082570 q^{96} + 2753300 q^{97} + 203004 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1 −15.2598 −17.0979 20.8965i 168.862 23.5746 + 23.5746i 260.911 + 318.876i −238.814 238.814i −1600.18 −144.325 + 714.571i −359.744 359.744i
38.2 −14.7512 3.59419 + 26.7597i 153.599 −68.2416 68.2416i −53.0188 394.739i 124.784 + 124.784i −1321.70 −703.164 + 192.359i 1006.65 + 1006.65i
38.3 −14.0069 25.1159 9.90914i 132.195 154.796 + 154.796i −351.797 + 138.797i 321.666 + 321.666i −955.198 532.618 497.754i −2168.22 2168.22i
38.4 −13.3013 20.6191 17.4313i 112.924 −135.530 135.530i −274.261 + 231.859i −104.927 104.927i −650.746 121.297 718.838i 1802.72 + 1802.72i
38.5 −12.6359 −17.0115 + 20.9669i 95.6654 152.781 + 152.781i 214.955 264.935i −234.270 234.270i −400.120 −150.221 713.355i −1930.52 1930.52i
38.6 −11.5147 −26.9464 + 1.69997i 68.5886 −61.6662 61.6662i 310.280 19.5747i 157.734 + 157.734i −52.8363 723.220 91.6164i 710.068 + 710.068i
38.7 −10.6271 24.8978 + 10.4451i 48.9356 5.72766 + 5.72766i −264.592 111.002i −324.888 324.888i 160.091 510.798 + 520.121i −60.8685 60.8685i
38.8 −9.96367 −2.68583 26.8661i 35.2747 −16.8097 16.8097i 26.7607 + 267.685i 296.612 + 296.612i 286.209 −714.573 + 144.315i 167.486 + 167.486i
38.9 −8.95432 15.4594 + 22.1361i 16.1799 34.6227 + 34.6227i −138.428 198.214i 218.156 + 218.156i 428.197 −251.014 + 684.422i −310.022 310.022i
38.10 −7.21375 −16.6299 + 21.2708i −11.9618 −104.798 104.798i 119.964 153.442i −266.693 266.693i 547.970 −175.895 707.462i 755.989 + 755.989i
38.11 −7.18422 8.83916 25.5121i −12.3869 72.9187 + 72.9187i −63.5025 + 183.285i −433.048 433.048i 548.781 −572.739 451.012i −523.864 523.864i
38.12 −6.18050 −23.8985 12.5644i −25.8014 111.665 + 111.665i 147.704 + 77.6542i 7.14097 + 7.14097i 555.018 413.272 + 600.539i −690.145 690.145i
38.13 −4.10952 26.3021 + 6.09898i −47.1119 −147.997 147.997i −108.089 25.0638i 245.141 + 245.141i 456.616 654.605 + 320.832i 608.198 + 608.198i
38.14 −3.73857 −13.3008 + 23.4965i −50.0231 94.5565 + 94.5565i 49.7261 87.8435i 434.343 + 434.343i 426.283 −375.175 625.048i −353.506 353.506i
38.15 −3.17896 24.3955 11.5698i −53.8942 27.6632 + 27.6632i −77.5524 + 36.7800i 99.5174 + 99.5174i 374.781 461.280 564.501i −87.9404 87.9404i
38.16 −2.39460 −13.6109 23.3183i −58.2659 −151.447 151.447i 32.5926 + 55.8380i −127.576 127.576i 292.778 −358.488 + 634.765i 362.655 + 362.655i
38.17 −1.10526 2.63593 + 26.8710i −62.7784 −6.38746 6.38746i −2.91338 29.6994i −175.878 175.878i 140.123 −715.104 + 141.660i 7.05978 + 7.05978i
38.18 1.10526 −26.8710 2.63593i −62.7784 6.38746 + 6.38746i −29.6994 2.91338i −175.878 175.878i −140.123 715.104 + 141.660i 7.05978 + 7.05978i
38.19 2.39460 23.3183 + 13.6109i −58.2659 151.447 + 151.447i 55.8380 + 32.5926i −127.576 127.576i −292.778 358.488 + 634.765i 362.655 + 362.655i
38.20 3.17896 11.5698 24.3955i −53.8942 −27.6632 27.6632i 36.7800 77.5524i 99.5174 + 99.5174i −374.781 −461.280 564.501i −87.9404 87.9404i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 38.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.c even 4 1 inner
51.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.7.f.a 68
3.b odd 2 1 inner 51.7.f.a 68
17.c even 4 1 inner 51.7.f.a 68
51.f odd 4 1 inner 51.7.f.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.7.f.a 68 1.a even 1 1 trivial
51.7.f.a 68 3.b odd 2 1 inner
51.7.f.a 68 17.c even 4 1 inner
51.7.f.a 68 51.f odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(51, [\chi])\).