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Properties

Label 138.7.c.a
Level $138$
Weight $7$
Character orbit 138.c
Analytic conductor $31.747$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [138,7,Mod(47,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("138.47");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 138.c (of order \(2\), degree \(1\), 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: \(31.7474635395\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 20 q^{3} - 1408 q^{4} + 160 q^{6} - 8 q^{7} - 1204 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 20 q^{3} - 1408 q^{4} + 160 q^{6} - 8 q^{7} - 1204 q^{9} - 3840 q^{10} + 640 q^{12} + 8152 q^{13} - 15392 q^{15} + 45056 q^{16} + 7360 q^{18} - 28712 q^{19} + 12832 q^{21} + 17088 q^{22} - 5120 q^{24} - 121972 q^{25} + 73900 q^{27} + 256 q^{28} + 3936 q^{30} - 4184 q^{31} - 120768 q^{33} + 98880 q^{34} + 38528 q^{36} - 156296 q^{37} + 83328 q^{39} + 122880 q^{40} + 17408 q^{42} - 123320 q^{43} - 560576 q^{45} - 20480 q^{48} + 1138980 q^{49} + 744704 q^{51} - 260864 q^{52} + 31552 q^{54} - 562896 q^{55} - 39208 q^{57} + 403200 q^{58} + 492544 q^{60} - 238856 q^{61} - 908688 q^{63} - 1441792 q^{64} - 371904 q^{66} - 151928 q^{67} + 133824 q^{70} - 235520 q^{72} + 1106584 q^{73} + 68948 q^{75} + 918784 q^{76} + 95968 q^{78} - 2156984 q^{79} + 872468 q^{81} - 388224 q^{82} - 410624 q^{84} + 124656 q^{85} + 3271232 q^{87} - 546816 q^{88} - 1310848 q^{90} + 4516448 q^{91} + 1293216 q^{93} - 906240 q^{94} + 163840 q^{96} - 1521416 q^{97} - 2086728 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 − 5.65685i −26.9854 + 0.888393i −32.0000 − 89.0819i 5.02551 + 152.652i 49.7238 181.019i 727.422 − 47.9472i −503.924
47.2 − 5.65685i −25.1240 − 9.88856i −32.0000 − 80.0092i −55.9381 + 142.123i −445.140 181.019i 533.433 + 496.881i −452.600
47.3 − 5.65685i −24.4008 − 11.5586i −32.0000 189.287i −65.3855 + 138.032i 188.135 181.019i 461.796 + 564.079i 1070.77
47.4 − 5.65685i −23.4759 + 13.3373i −32.0000 68.7891i 75.4473 + 132.800i −294.785 181.019i 373.232 − 626.210i 389.130
47.5 − 5.65685i −21.8045 − 15.9237i −32.0000 − 197.773i −90.0783 + 123.345i 237.581 181.019i 221.868 + 694.417i −1118.77
47.6 − 5.65685i −18.6531 + 19.5208i −32.0000 84.1700i 110.426 + 105.518i 192.450 181.019i −33.1213 − 728.247i 476.137
47.7 − 5.65685i −15.5150 + 22.0972i −32.0000 − 208.480i 125.001 + 87.7659i −555.651 181.019i −247.572 − 685.674i −1179.34
47.8 − 5.65685i −11.0297 − 24.6444i −32.0000 86.0362i −139.410 + 62.3935i −420.942 181.019i −485.691 + 543.641i 486.694
47.9 − 5.65685i −9.54334 + 25.2572i −32.0000 − 176.597i 142.876 + 53.9853i 562.561 181.019i −546.849 − 482.075i −998.986
47.10 − 5.65685i −5.77576 − 26.3750i −32.0000 − 87.2265i −149.200 + 32.6726i 545.022 181.019i −662.281 + 304.671i −493.427
47.11 − 5.65685i −5.43275 − 26.4478i −32.0000 67.0725i −149.611 + 30.7323i 284.104 181.019i −669.970 + 287.368i 379.419
47.12 − 5.65685i −3.42372 + 26.7820i −32.0000 133.253i 151.502 + 19.3675i −412.521 181.019i −705.556 − 183.389i 753.792
47.13 − 5.65685i 4.92561 + 26.5469i −32.0000 218.477i 150.172 − 27.8634i 597.880 181.019i −680.477 + 261.519i 1235.89
47.14 − 5.65685i 9.91752 − 25.1126i −32.0000 − 182.947i −142.058 − 56.1019i −194.119 181.019i −532.286 − 498.109i −1034.90
47.15 − 5.65685i 12.4419 − 23.9625i −32.0000 11.4170i −135.552 − 70.3820i −566.403 181.019i −419.399 − 596.277i 64.5844
47.16 − 5.65685i 12.6992 + 23.8271i −32.0000 − 48.6667i 134.786 − 71.8373i −33.2033 181.019i −406.462 + 605.169i −275.300
47.17 − 5.65685i 15.9438 + 21.7898i −32.0000 − 66.5783i 123.262 − 90.1916i 119.852 181.019i −220.592 + 694.824i −376.624
47.18 − 5.65685i 20.5410 − 17.5233i −32.0000 − 38.1931i −99.1269 − 116.198i 320.362 181.019i 114.867 − 719.894i −216.053
47.19 − 5.65685i 24.1537 + 12.0665i −32.0000 204.160i 68.2582 − 136.634i −361.059 181.019i 437.801 + 582.899i 1154.90
47.20 − 5.65685i 26.5602 + 4.85367i −32.0000 − 35.5126i 27.4565 − 150.247i −378.171 181.019i 681.884 + 257.828i −200.890
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.44
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.7.c.a ✓ 44
3.b odd 2 1 inner 138.7.c.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.7.c.a ✓ 44 1.a even 1 1 trivial
138.7.c.a ✓ 44 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(138, [\chi])\).