↑

Properties

Label 51.7.j.a
Level $51$
Weight $7$
Character orbit 51.j
Analytic conductor $11.733$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [51,7,Mod(7,51)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(51, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("51.7");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 51.j (of order \(16\), degree \(8\), 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: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 7488 q^{10} + 8960 q^{11} - 5760 q^{13} - 24320 q^{14} + 12800 q^{17} + 23040 q^{19} + 39424 q^{20} - 80640 q^{22} - 58240 q^{23} - 73872 q^{24} + 75840 q^{25} + 429056 q^{26} - 42480 q^{28} - 147680 q^{29} - 326592 q^{30} - 153792 q^{31} - 235600 q^{32} + 53328 q^{34} + 400768 q^{35} + 128304 q^{36} + 255360 q^{37} + 462800 q^{38} + 46656 q^{39} + 97728 q^{40} - 434240 q^{41} - 758160 q^{42} + 53760 q^{43} - 596288 q^{44} + 85536 q^{45} + 324624 q^{46} - 224256 q^{49} + 933888 q^{52} - 1777776 q^{53} + 2107392 q^{55} + 2688400 q^{56} + 1987152 q^{58} + 289280 q^{59} - 2771520 q^{61} - 3335920 q^{62} - 2662176 q^{64} - 2796240 q^{65} - 1360208 q^{68} - 673920 q^{69} + 7264080 q^{70} + 4567552 q^{71} + 1492992 q^{72} + 5703024 q^{73} + 5190240 q^{74} + 248832 q^{75} - 1748496 q^{76} - 5175488 q^{77} - 3193344 q^{78} - 6474240 q^{79} - 18375504 q^{80} - 3905616 q^{82} + 263936 q^{83} + 8060688 q^{85} + 2665984 q^{86} + 2830464 q^{87} + 13258944 q^{88} + 3717120 q^{89} + 606528 q^{90} + 996480 q^{91} + 7598080 q^{92} + 684288 q^{93} - 7744800 q^{94} - 3001856 q^{95} - 5308416 q^{96} - 6505344 q^{97} - 9680832 q^{98} - 2177280 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −5.72406 + 13.8191i 8.66048 − 12.9613i −112.948 − 112.948i 45.9841 + 231.178i 129.541 + 193.872i −103.823 + 521.951i 1322.94 − 547.979i −92.9921 − 224.503i −3457.89 − 687.816i
7.2 −5.46901 + 13.2034i −8.66048 + 12.9613i −99.1640 − 99.1640i −10.3407 − 51.9862i −123.769 − 185.233i −2.55084 + 12.8239i 1006.61 − 416.952i −92.9921 − 224.503i 742.947 + 147.781i
7.3 −3.95009 + 9.53637i 8.66048 − 12.9613i −30.0843 − 30.0843i 2.66004 + 13.3729i 89.3943 + 133.788i 54.5944 − 274.465i −204.597 + 84.7467i −92.9921 − 224.503i −138.036 − 27.4572i
7.4 −3.76220 + 9.08275i −8.66048 + 12.9613i −23.0874 − 23.0874i 15.0546 + 75.6844i −85.1421 − 127.424i −66.9598 + 336.629i −284.739 + 117.943i −92.9921 − 224.503i −744.061 − 148.003i
7.5 −3.50589 + 8.46397i 8.66048 − 12.9613i −14.0927 − 14.0927i −9.76725 − 49.1033i 79.3416 + 118.743i −1.87827 + 9.44269i −373.007 + 154.504i −92.9921 − 224.503i 449.851 + 89.4810i
7.6 −1.94127 + 4.68665i −8.66048 + 12.9613i 27.0587 + 27.0587i −15.9376 − 80.1239i −43.9328 − 65.7501i 119.728 − 601.916i −479.288 + 198.528i −92.9921 − 224.503i 406.452 + 80.8482i
7.7 −0.851789 + 2.05640i 8.66048 − 12.9613i 41.7516 + 41.7516i 43.7276 + 219.833i 19.2768 + 28.8497i 53.8363 − 270.654i −253.031 + 104.809i −92.9921 − 224.503i −489.312 − 97.3302i
7.8 −0.655545 + 1.58263i −8.66048 + 12.9613i 43.1799 + 43.1799i 39.3037 + 197.593i −14.8356 − 22.2030i 4.36618 − 21.9503i −197.932 + 81.9861i −92.9921 − 224.503i −338.481 − 67.3281i
7.9 −0.152447 + 0.368040i 8.66048 − 12.9613i 45.1426 + 45.1426i −29.5046 − 148.330i 3.45002 + 5.16332i 34.6983 − 174.440i −47.0507 + 19.4890i −92.9921 − 224.503i 59.0892 + 11.7536i
7.10 0.0466865 − 0.112711i 8.66048 − 12.9613i 45.2443 + 45.2443i −6.17076 − 31.0225i −1.05656 − 1.58125i −125.510 + 630.980i 14.4254 − 5.97518i −92.9921 − 224.503i −3.78467 − 0.752818i
7.11 0.729773 − 1.76183i −8.66048 + 12.9613i 42.6834 + 42.6834i 2.18942 + 11.0070i 16.5155 + 24.7171i −22.2209 + 111.712i 219.107 − 90.7571i −92.9921 − 224.503i 20.9902 + 4.17520i
7.12 2.29147 − 5.53209i −8.66048 + 12.9613i 19.9016 + 19.9016i −28.7528 − 144.550i 51.8580 + 77.6110i −58.2045 + 292.614i 509.755 − 211.148i −92.9921 − 224.503i −865.549 − 172.168i
7.13 2.79425 − 6.74592i 8.66048 − 12.9613i 7.55525 + 7.55525i 14.6073 + 73.4356i −63.2365 − 94.6401i 56.2741 − 282.909i 503.817 − 208.688i −92.9921 − 224.503i 536.207 + 106.658i
7.14 4.21230 − 10.1694i 8.66048 − 12.9613i −40.4183 − 40.4183i −34.2708 − 172.291i −95.3283 − 142.669i −22.0776 + 110.992i 69.5570 − 28.8114i −92.9921 − 224.503i −1896.45 − 377.228i
7.15 4.30206 − 10.3861i −8.66048 + 12.9613i −44.1082 − 44.1082i 32.7127 + 164.458i 97.3596 + 145.709i 105.363 − 529.694i 16.8418 − 6.97610i −92.9921 − 224.503i 1848.81 + 367.751i
7.16 4.48081 − 10.8176i −8.66048 + 12.9613i −51.6887 − 51.6887i −22.9697 − 115.477i 101.405 + 151.763i 59.6740 − 300.001i −98.4287 + 40.7705i −92.9921 − 224.503i −1352.11 − 268.951i
7.17 5.49543 − 13.2671i −8.66048 + 12.9613i −100.563 − 100.563i 16.3517 + 82.2055i 124.367 + 186.128i −123.588 + 621.317i −1037.71 + 429.836i −92.9921 − 224.503i 1180.49 + 234.815i
7.18 5.98910 − 14.4590i 8.66048 − 12.9613i −127.938 − 127.938i 16.5653 + 83.2792i −135.539 − 202.848i 69.4925 − 349.362i −1690.71 + 700.314i −92.9921 − 224.503i 1303.34 + 259.251i
10.1 −5.83790 + 14.0939i 12.9613 + 8.66048i −119.303 − 119.303i 223.087 − 44.3748i −197.727 + 132.117i 10.4271 + 2.07408i 1475.92 − 611.344i 92.9921 + 224.503i −676.944 + 3403.23i
10.2 −5.36244 + 12.9461i −12.9613 − 8.66048i −93.5901 − 93.5901i −122.478 + 24.3623i 181.623 − 121.357i −504.743 − 100.400i 884.946 − 366.557i 92.9921 + 224.503i 341.383 − 1716.25i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.7.j.a ✓ 144
17.e odd 16 1 inner 51.7.j.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.7.j.a ✓ 144 1.a even 1 1 trivial
51.7.j.a ✓ 144 17.e odd 16 1 inner

Hecke kernels

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