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Properties

Label 115.7.d.a
Level $115$
Weight $7$
Character orbit 115.d
Analytic conductor $26.456$
Analytic rank $0$
Dimension $48$
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] = [115,7,Mod(91,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.91");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 115.d (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: \(26.4562196163\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 20 q^{2} + 1584 q^{4} + 410 q^{6} - 1210 q^{8} + 12896 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 20 q^{2} + 1584 q^{4} + 410 q^{6} - 1210 q^{8} + 12896 q^{9} + 4290 q^{12} - 1440 q^{13} + 65400 q^{16} + 4610 q^{18} + 26600 q^{23} + 14940 q^{24} - 150000 q^{25} + 47594 q^{26} + 16080 q^{27} + 131800 q^{29} - 1392 q^{31} - 225040 q^{32} + 5000 q^{35} + 658786 q^{36} - 236320 q^{39} - 351496 q^{41} + 382692 q^{46} + 395680 q^{47} + 1042550 q^{48} - 637848 q^{49} + 62500 q^{50} + 523890 q^{52} - 241250 q^{54} - 402000 q^{55} - 479130 q^{58} - 466312 q^{59} - 1124330 q^{62} + 837582 q^{64} + 1021060 q^{69} - 396000 q^{70} - 114336 q^{71} - 1960750 q^{72} - 498720 q^{73} + 3610400 q^{77} - 1104610 q^{78} + 972888 q^{81} + 124950 q^{82} - 246000 q^{85} - 2090960 q^{87} + 4913480 q^{92} + 3234320 q^{93} - 5550378 q^{94} - 1664000 q^{95} - 776990 q^{96} + 9993220 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −15.5617 −20.6736 178.167 55.9017i 321.716 270.279i −1776.64 −301.604 − 869.927i
91.2 −15.5617 −20.6736 178.167 − 55.9017i 321.716 − 270.279i −1776.64 −301.604 869.927i
91.3 −15.2752 41.9187 169.333 − 55.9017i −640.318 492.500i −1608.98 1028.18 853.912i
91.4 −15.2752 41.9187 169.333 55.9017i −640.318 − 492.500i −1608.98 1028.18 − 853.912i
91.5 −12.7662 22.2634 98.9763 55.9017i −284.219 111.835i −446.515 −233.341 − 713.653i
91.6 −12.7662 22.2634 98.9763 − 55.9017i −284.219 − 111.835i −446.515 −233.341 713.653i
91.7 −12.5973 −47.4272 94.6924 55.9017i 597.456 − 458.638i −386.642 1520.34 − 704.211i
91.8 −12.5973 −47.4272 94.6924 − 55.9017i 597.456 458.638i −386.642 1520.34 704.211i
91.9 −11.9049 −28.4011 77.7265 55.9017i 338.112 365.722i −163.412 77.6213 − 665.504i
91.10 −11.9049 −28.4011 77.7265 − 55.9017i 338.112 − 365.722i −163.412 77.6213 665.504i
91.11 −10.2854 36.0105 41.7887 − 55.9017i −370.381 − 481.571i 228.451 567.758 574.969i
91.12 −10.2854 36.0105 41.7887 55.9017i −370.381 481.571i 228.451 567.758 − 574.969i
91.13 −8.10890 −4.25404 1.75429 − 55.9017i 34.4956 124.794i 504.744 −710.903 453.301i
91.14 −8.10890 −4.25404 1.75429 55.9017i 34.4956 − 124.794i 504.744 −710.903 − 453.301i
91.15 −6.76022 −30.6083 −18.2994 − 55.9017i 206.919 − 356.010i 556.362 207.867 377.908i
91.16 −6.76022 −30.6083 −18.2994 55.9017i 206.919 356.010i 556.362 207.867 − 377.908i
91.17 −6.73099 3.68399 −18.6938 55.9017i −24.7969 − 614.798i 556.611 −715.428 − 376.274i
91.18 −6.73099 3.68399 −18.6938 − 55.9017i −24.7969 614.798i 556.611 −715.428 376.274i
91.19 −4.89108 32.2424 −40.0773 55.9017i −157.700 6.78629i 509.051 310.574 − 273.420i
91.20 −4.89108 32.2424 −40.0773 − 55.9017i −157.700 − 6.78629i 509.051 310.574 273.420i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.48
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.7.d.a ✓ 48
23.b odd 2 1 inner 115.7.d.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.7.d.a ✓ 48 1.a even 1 1 trivial
115.7.d.a ✓ 48 23.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(115, [\chi])\).