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Properties

Label 48.7.i.a
Level $48$
Weight $7$
Character orbit 48.i
Analytic conductor $11.043$
Analytic rank $0$
Dimension $92$
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] = [48,7,Mod(5,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.5");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 48.i (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: \(11.0425960138\)
Analytic rank: \(0\)
Dimension: \(92\)
Relative dimension: \(46\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q - 2 q^{3} - 4 q^{4} + 508 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q - 2 q^{3} - 4 q^{4} + 508 q^{6} + 1496 q^{10} + 4288 q^{12} - 4 q^{13} - 4 q^{15} + 10352 q^{16} + 10020 q^{18} - 3940 q^{19} + 1456 q^{21} - 42088 q^{22} + 66376 q^{24} - 34322 q^{27} + 27000 q^{28} - 127836 q^{30} - 8 q^{31} - 4 q^{33} + 128424 q^{34} + 172420 q^{36} - 4 q^{37} - 164392 q^{40} + 221560 q^{42} - 195268 q^{43} - 31252 q^{45} + 150712 q^{46} - 188048 q^{48} - 1142884 q^{49} + 385056 q^{51} - 122368 q^{52} + 151912 q^{54} - 181960 q^{58} + 242304 q^{60} - 326500 q^{61} + 470592 q^{63} - 251752 q^{64} + 489172 q^{66} + 1207676 q^{67} - 1460 q^{69} + 346392 q^{70} - 903480 q^{72} + 1413778 q^{75} - 359992 q^{76} + 1478252 q^{78} + 860920 q^{79} - 4 q^{81} - 2412800 q^{82} - 730640 q^{84} + 434496 q^{85} + 1259360 q^{88} - 291792 q^{90} + 2320992 q^{91} - 1176260 q^{93} - 1412496 q^{94} + 1956200 q^{96} - 8 q^{97} - 4048228 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −7.91573 + 1.15814i −26.8410 − 2.92544i 61.3174 − 18.3351i −131.564 + 131.564i 215.854 − 7.92879i 278.097i −464.137 + 216.150i 711.884 + 157.044i 889.056 − 1193.80i
5.2 −7.90213 + 1.24752i −6.94595 + 26.0913i 60.8874 − 19.7162i −0.540042 + 0.540042i 22.3384 − 214.842i − 321.851i −456.544 + 231.758i −632.508 − 362.457i 3.59377 − 4.94120i
5.3 −7.81038 − 1.73145i −5.41236 − 26.4520i 58.0042 + 27.0465i −54.1549 + 54.1549i −3.52753 + 215.971i − 639.122i −406.205 − 311.675i −670.413 + 286.335i 516.737 − 329.204i
5.4 −7.78639 + 1.83634i −19.2360 − 18.9466i 57.2557 − 28.5969i 148.490 − 148.490i 184.572 + 112.202i 263.428i −393.302 + 327.807i 11.0511 + 728.916i −883.522 + 1428.88i
5.5 −7.66468 − 2.29187i 23.2271 + 13.7659i 53.4947 + 35.1328i 133.976 − 133.976i −146.479 − 158.745i − 25.6967i −329.500 − 391.885i 350.001 + 639.485i −1333.94 + 719.831i
5.6 −7.60798 − 2.47360i 14.3156 − 22.8925i 51.7626 + 37.6382i 4.45378 − 4.45378i −165.539 + 138.754i 601.125i −300.707 − 414.390i −319.129 − 655.437i −44.9011 + 22.8674i
5.7 −7.59869 + 2.50199i 26.4071 + 5.62718i 51.4801 − 38.0237i −81.2421 + 81.2421i −214.738 + 23.3111i 51.0073i −296.046 + 417.733i 665.670 + 297.195i 414.066 − 820.600i
5.8 −6.72424 − 4.33412i −25.2548 + 9.54954i 26.4308 + 58.2873i 70.9576 − 70.9576i 211.208 + 45.2441i − 111.312i 74.8977 − 506.492i 546.612 − 482.344i −784.675 + 169.597i
5.9 −6.67919 − 4.40322i 0.0252819 + 27.0000i 25.2233 + 58.8200i −118.285 + 118.285i 118.718 − 180.449i 448.417i 90.5262 − 503.934i −728.999 + 1.36522i 1310.88 − 269.213i
5.10 −6.62545 + 4.48369i 17.3541 − 20.6842i 23.7931 − 59.4129i 77.8915 − 77.8915i −22.2368 + 214.852i − 310.530i 108.749 + 500.318i −126.674 − 717.910i −166.825 + 865.307i
5.11 −5.62988 − 5.68370i 26.3911 − 5.70179i −0.608985 + 63.9971i −89.3741 + 89.3741i −180.986 − 117.899i − 331.991i 367.169 − 356.834i 663.979 − 300.953i 1011.14 + 4.81082i
5.12 −5.52373 + 5.78692i −24.5429 + 11.2536i −2.97683 − 63.9307i 35.5126 − 35.5126i 70.4446 − 204.190i − 8.76023i 386.405 + 335.909i 475.711 − 552.395i 9.34650 + 401.670i
5.13 −5.38773 + 5.91374i 9.03029 + 25.4451i −5.94475 − 63.7233i 65.5814 − 65.5814i −199.129 − 83.6886i 515.866i 408.872 + 308.168i −565.908 + 459.554i 34.4968 + 741.166i
5.14 −5.29983 + 5.99264i 2.62588 − 26.8720i −7.82358 − 63.5200i −141.081 + 141.081i 147.118 + 158.153i 308.364i 422.117 + 289.761i −715.209 − 141.126i −97.7428 − 1593.15i
5.15 −4.22435 − 6.79374i −18.2268 − 19.9194i −28.3097 + 57.3983i −14.1816 + 14.1816i −58.3307 + 207.975i 191.419i 509.539 − 50.1415i −64.5655 + 726.135i 156.254 + 36.4379i
5.16 −3.66778 + 7.10967i −22.4096 − 15.0602i −37.0947 − 52.1535i −13.2524 + 13.2524i 189.267 − 104.087i − 437.375i 506.849 − 72.4434i 275.382 + 674.986i −45.6131 − 142.827i
5.17 −3.23345 − 7.31743i 9.91420 + 25.1139i −43.0896 + 47.3211i 41.0809 − 41.0809i 151.712 − 153.751i − 200.806i 485.597 + 162.295i −532.417 + 497.969i −433.439 − 167.773i
5.18 −2.77352 − 7.50384i 17.6543 − 20.4285i −48.6151 + 41.6241i 132.844 − 132.844i −202.257 − 75.8159i − 91.8613i 447.176 + 249.355i −105.651 − 721.304i −1365.28 − 628.393i
5.19 −2.70919 + 7.52730i 12.2138 + 24.0795i −49.3206 − 40.7857i −131.879 + 131.879i −214.343 + 26.7009i − 543.561i 440.625 − 260.755i −430.647 + 588.204i −635.408 − 1349.98i
5.20 −1.74289 + 7.80784i 26.7985 − 3.29233i −57.9246 − 27.2165i 59.2583 − 59.2583i −21.0010 + 214.977i − 2.09133i 313.458 − 404.831i 707.321 − 176.459i 359.398 + 565.960i
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.7.i.a ✓ 92
3.b odd 2 1 inner 48.7.i.a ✓ 92
4.b odd 2 1 192.7.i.a 92
12.b even 2 1 192.7.i.a 92
16.e even 4 1 inner 48.7.i.a ✓ 92
16.f odd 4 1 192.7.i.a 92
48.i odd 4 1 inner 48.7.i.a ✓ 92
48.k even 4 1 192.7.i.a 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.7.i.a ✓ 92 1.a even 1 1 trivial
48.7.i.a ✓ 92 3.b odd 2 1 inner
48.7.i.a ✓ 92 16.e even 4 1 inner
48.7.i.a ✓ 92 48.i odd 4 1 inner
192.7.i.a 92 4.b odd 2 1
192.7.i.a 92 12.b even 2 1
192.7.i.a 92 16.f odd 4 1
192.7.i.a 92 48.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(48, [\chi])\).