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Properties

Label 975.2.bf.a
Level $975$
Weight $2$
Character orbit 975.bf
Analytic conductor $7.785$
Analytic rank $0$
Dimension $288$
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] = [975,2,Mod(64,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bf (of order \(10\), degree \(4\), 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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(72\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 76 q^{4} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 76 q^{4} + 72 q^{9} + 8 q^{10} - 96 q^{16} - 100 q^{22} - 20 q^{23} - 36 q^{25} - 8 q^{26} + 44 q^{29} + 4 q^{30} + 16 q^{35} + 76 q^{36} - 124 q^{40} + 352 q^{49} - 64 q^{51} + 140 q^{52} - 140 q^{53} + 24 q^{55} + 72 q^{56} + 8 q^{61} - 20 q^{62} - 76 q^{64} + 28 q^{65} - 8 q^{66} - 8 q^{69} - 56 q^{74} + 8 q^{75} - 80 q^{77} + 32 q^{79} - 72 q^{81} - 80 q^{87} + 80 q^{88} - 8 q^{90} + 22 q^{91} + 180 q^{92} + 76 q^{94} - 212 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −2.24871 + 1.63378i −0.951057 − 0.309017i 1.76942 − 5.44570i 0.272653 + 2.21938i 2.64352 − 0.858931i −2.65633 3.20033 + 9.84961i 0.809017 + 0.587785i −4.23911 − 4.54529i
64.2 −2.23532 + 1.62406i −0.951057 − 0.309017i 1.74107 − 5.35846i −1.24135 − 1.85985i 2.62778 − 0.853817i 4.93330 3.10296 + 9.54992i 0.809017 + 0.587785i 5.79532 + 2.14135i
64.3 −2.16040 + 1.56962i 0.951057 + 0.309017i 1.58557 − 4.87989i −2.23357 + 0.105631i −2.53970 + 0.825198i 0.0697024 2.58371 + 7.95184i 0.809017 + 0.587785i 4.65960 − 3.73406i
64.4 −2.11825 + 1.53900i 0.951057 + 0.309017i 1.50043 − 4.61783i 1.06846 − 1.96428i −2.49015 + 0.809098i −3.56551 2.31036 + 7.11057i 0.809017 + 0.587785i 0.759755 + 5.80518i
64.5 −2.07416 + 1.50696i 0.951057 + 0.309017i 1.41315 − 4.34924i 1.85083 + 1.25476i −2.43832 + 0.792257i 3.55547 2.03852 + 6.27393i 0.809017 + 0.587785i −5.72979 + 0.186574i
64.6 −2.01179 + 1.46165i 0.951057 + 0.309017i 1.29283 − 3.97893i −0.199331 + 2.22717i −2.36500 + 0.768434i 0.997622 1.67803 + 5.16444i 0.809017 + 0.587785i −2.85432 − 4.77193i
64.7 −2.00431 + 1.45622i −0.951057 − 0.309017i 1.27865 − 3.93529i −0.207022 − 2.22646i 2.35621 − 0.765578i −2.39505 1.63666 + 5.03713i 0.809017 + 0.587785i 3.65715 + 4.16105i
64.8 −1.94832 + 1.41554i −0.951057 − 0.309017i 1.17417 − 3.61373i −1.85965 + 1.24165i 2.29039 − 0.744192i −0.419414 1.33932 + 4.12201i 0.809017 + 0.587785i 1.86560 − 5.05154i
64.9 −1.88180 + 1.36721i 0.951057 + 0.309017i 1.05388 − 3.24352i −1.39643 − 1.74642i −2.21219 + 0.718784i 1.36182 1.01380 + 3.12016i 0.809017 + 0.587785i 5.01553 + 1.37721i
64.10 −1.85355 + 1.34669i −0.951057 − 0.309017i 1.00406 − 3.09019i 2.07812 − 0.825472i 2.17898 − 0.707994i 2.57860 0.884441 + 2.72203i 0.809017 + 0.587785i −2.74026 + 4.32863i
64.11 −1.76661 + 1.28352i −0.951057 − 0.309017i 0.855457 − 2.63283i 1.01923 + 1.99027i 2.07677 − 0.674784i 2.79858 0.518448 + 1.59562i 0.809017 + 0.587785i −4.35512 − 2.20783i
64.12 −1.62207 + 1.17850i −0.951057 − 0.309017i 0.624202 − 1.92110i 2.20260 + 0.385441i 1.90685 − 0.619574i −4.93118 0.0123666 + 0.0380605i 0.809017 + 0.587785i −4.02700 + 1.97055i
64.13 −1.60638 + 1.16711i −0.951057 − 0.309017i 0.600299 − 1.84753i −1.45913 − 1.69438i 1.88842 − 0.613584i −2.30108 −0.0352152 − 0.108381i 0.809017 + 0.587785i 4.32145 + 1.01886i
64.14 −1.53068 + 1.11210i 0.951057 + 0.309017i 0.488170 − 1.50243i 0.263628 − 2.22047i −1.79942 + 0.584667i 4.43915 −0.245705 − 0.756203i 0.809017 + 0.587785i 2.06587 + 3.69201i
64.15 −1.46859 + 1.06699i 0.951057 + 0.309017i 0.400249 − 1.23184i −0.225663 + 2.22465i −1.72643 + 0.560952i −1.69833 −0.395340 − 1.21673i 0.809017 + 0.587785i −2.04228 − 3.50788i
64.16 −1.46493 + 1.06433i 0.951057 + 0.309017i 0.395176 − 1.21623i −1.43057 + 1.71857i −1.72213 + 0.559553i −1.48818 −0.403540 − 1.24197i 0.809017 + 0.587785i 0.266546 − 4.04018i
64.17 −1.45469 + 1.05689i 0.951057 + 0.309017i 0.381062 − 1.17279i 1.85500 − 1.24859i −1.71009 + 0.555642i −2.47899 −0.426098 − 1.31140i 0.809017 + 0.587785i −1.37882 + 3.77684i
64.18 −1.40207 + 1.01867i −0.951057 − 0.309017i 0.310096 − 0.954379i −2.21092 + 0.334393i 1.64824 − 0.535545i 0.709830 −0.533674 − 1.64248i 0.809017 + 0.587785i 2.75924 − 2.72104i
64.19 −1.32837 + 0.965117i 0.951057 + 0.309017i 0.215080 − 0.661950i 2.17539 + 0.517367i −1.56159 + 0.507392i 0.391331 −0.661631 − 2.03629i 0.809017 + 0.587785i −3.38904 + 1.41225i
64.20 −1.26323 + 0.917789i −0.951057 − 0.309017i 0.135375 − 0.416641i −0.751639 + 2.10595i 1.48501 − 0.482510i −3.65676 −0.753641 − 2.31947i 0.809017 + 0.587785i −0.983328 − 3.35015i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
25.e even 10 1 inner
325.p even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bf.a ✓ 288
13.b even 2 1 inner 975.2.bf.a ✓ 288
25.e even 10 1 inner 975.2.bf.a ✓ 288
325.p even 10 1 inner 975.2.bf.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bf.a ✓ 288 1.a even 1 1 trivial
975.2.bf.a ✓ 288 13.b even 2 1 inner
975.2.bf.a ✓ 288 25.e even 10 1 inner
975.2.bf.a ✓ 288 325.p even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(975, [\chi])\).