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Properties

Label 152.5.f.a
Level $152$
Weight $5$
Character orbit 152.f
Analytic conductor $15.712$
Analytic rank $0$
Dimension $72$
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] = [152,5,Mod(115,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.115");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 152.f (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: \(15.7122343887\)
Analytic rank: \(0\)
Dimension: \(72\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 6 q^{2} - 6 q^{4} + 66 q^{6} - 216 q^{8} + 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 6 q^{2} - 6 q^{4} + 66 q^{6} - 216 q^{8} + 1944 q^{9} - 152 q^{10} + 192 q^{11} + 620 q^{12} + 1080 q^{14} - 582 q^{16} + 240 q^{17} - 806 q^{18} - 600 q^{20} - 820 q^{22} - 1850 q^{24} - 8664 q^{25} + 702 q^{26} + 3648 q^{27} - 1470 q^{28} + 716 q^{30} - 516 q^{32} + 5972 q^{34} - 10032 q^{36} + 5500 q^{40} + 720 q^{41} + 4146 q^{42} + 14528 q^{43} + 924 q^{44} - 6404 q^{46} - 10324 q^{48} - 24696 q^{49} - 7050 q^{50} - 13760 q^{51} + 3800 q^{52} + 4338 q^{54} - 576 q^{56} - 1090 q^{58} + 20544 q^{59} - 15852 q^{60} + 10728 q^{62} - 4506 q^{64} - 1344 q^{65} + 13904 q^{66} - 18880 q^{67} + 35766 q^{68} + 38616 q^{70} + 6564 q^{72} - 13200 q^{73} + 1440 q^{74} - 2304 q^{75} - 4720 q^{78} - 36972 q^{80} + 57544 q^{81} - 1208 q^{82} + 10560 q^{83} - 8136 q^{84} + 23472 q^{86} + 4904 q^{88} + 29232 q^{89} - 4124 q^{90} + 17280 q^{91} + 4446 q^{92} - 64816 q^{94} + 42354 q^{96} - 15440 q^{97} - 31086 q^{98} + 2624 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1 −3.99347 − 0.228504i −1.75665 15.8956 + 1.82505i − 39.1101i 7.01512 + 0.401402i − 92.9431i −63.0614 − 10.9205i −77.9142 −8.93682 + 156.185i
115.2 −3.99347 + 0.228504i −1.75665 15.8956 − 1.82505i 39.1101i 7.01512 − 0.401402i 92.9431i −63.0614 + 10.9205i −77.9142 −8.93682 − 156.185i
115.3 −3.95998 − 0.564427i −9.24411 15.3628 + 4.47024i 21.9570i 36.6065 + 5.21763i − 54.4201i −58.3134 − 26.3733i 4.45361 12.3932 − 86.9494i
115.4 −3.95998 + 0.564427i −9.24411 15.3628 − 4.47024i − 21.9570i 36.6065 − 5.21763i 54.4201i −58.3134 + 26.3733i 4.45361 12.3932 + 86.9494i
115.5 −3.91672 − 0.811991i 13.6499 14.6813 + 6.36068i − 12.5496i −53.4628 − 11.0836i 28.4088i −52.3378 − 36.8341i 105.320 −10.1901 + 49.1531i
115.6 −3.91672 + 0.811991i 13.6499 14.6813 − 6.36068i 12.5496i −53.4628 + 11.0836i − 28.4088i −52.3378 + 36.8341i 105.320 −10.1901 − 49.1531i
115.7 −3.87300 − 0.999949i 7.42172 14.0002 + 7.74560i − 24.4567i −28.7443 − 7.42134i 41.8080i −46.4775 − 43.9982i −25.9181 −24.4554 + 94.7205i
115.8 −3.87300 + 0.999949i 7.42172 14.0002 − 7.74560i 24.4567i −28.7443 + 7.42134i − 41.8080i −46.4775 + 43.9982i −25.9181 −24.4554 − 94.7205i
115.9 −3.71829 − 1.47456i −15.1933 11.6513 + 10.9657i 13.7401i 56.4930 + 22.4035i 21.2157i −27.1533 − 57.9543i 149.836 20.2607 − 51.0898i
115.10 −3.71829 + 1.47456i −15.1933 11.6513 − 10.9657i − 13.7401i 56.4930 − 22.4035i − 21.2157i −27.1533 + 57.9543i 149.836 20.2607 + 51.0898i
115.11 −3.64196 − 1.65412i 1.35813 10.5278 + 12.0485i 18.1667i −4.94627 − 2.24651i − 25.9695i −18.4121 − 61.2943i −79.1555 30.0499 − 66.1624i
115.12 −3.64196 + 1.65412i 1.35813 10.5278 − 12.0485i − 18.1667i −4.94627 + 2.24651i 25.9695i −18.4121 + 61.2943i −79.1555 30.0499 + 66.1624i
115.13 −3.37629 − 2.14491i −8.68255 6.79870 + 14.4837i − 37.1834i 29.3148 + 18.6233i 48.9321i 8.11185 − 63.4838i −5.61327 −79.7551 + 125.542i
115.14 −3.37629 + 2.14491i −8.68255 6.79870 − 14.4837i 37.1834i 29.3148 − 18.6233i − 48.9321i 8.11185 + 63.4838i −5.61327 −79.7551 − 125.542i
115.15 −3.26305 − 2.31355i 16.5635 5.29500 + 15.0984i 49.3873i −54.0474 − 38.3203i 24.3419i 17.6531 − 61.5172i 193.348 114.260 − 161.153i
115.16 −3.26305 + 2.31355i 16.5635 5.29500 − 15.0984i − 49.3873i −54.0474 + 38.3203i − 24.3419i 17.6531 + 61.5172i 193.348 114.260 + 161.153i
115.17 −3.17537 − 2.43250i 3.35681 4.16589 + 15.4482i 19.6849i −10.6591 − 8.16543i − 23.0835i 24.3494 − 59.1870i −69.7319 47.8835 − 62.5068i
115.18 −3.17537 + 2.43250i 3.35681 4.16589 − 15.4482i − 19.6849i −10.6591 + 8.16543i 23.0835i 24.3494 + 59.1870i −69.7319 47.8835 + 62.5068i
115.19 −2.63125 − 3.01273i 12.3405 −2.15306 + 15.8545i − 23.4970i −32.4709 − 37.1785i − 60.3290i 53.4304 − 35.2305i 71.2873 −70.7901 + 61.8265i
115.20 −2.63125 + 3.01273i 12.3405 −2.15306 − 15.8545i 23.4970i −32.4709 + 37.1785i 60.3290i 53.4304 + 35.2305i 71.2873 −70.7901 − 61.8265i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.5.f.a ✓ 72
4.b odd 2 1 608.5.f.a 72
8.b even 2 1 608.5.f.a 72
8.d odd 2 1 inner 152.5.f.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.5.f.a ✓ 72 1.a even 1 1 trivial
152.5.f.a ✓ 72 8.d odd 2 1 inner
608.5.f.a 72 4.b odd 2 1
608.5.f.a 72 8.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(152, [\chi])\).