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Properties

Label 975.2.k.d
Level $975$
Weight $2$
Character orbit 975.k
Analytic conductor $7.785$
Analytic rank $0$
Dimension $28$
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] = [975,2,Mod(307,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.k (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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 195)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 28 q^{4} - 8 q^{11} - 8 q^{12} + 12 q^{13} + 28 q^{16} + 28 q^{17} + 4 q^{18} + 8 q^{21} + 32 q^{22} - 8 q^{23} - 16 q^{31} + 28 q^{34} - 32 q^{37} + 8 q^{39} + 4 q^{41} + 40 q^{44} - 16 q^{46} + 24 q^{47} + 16 q^{48} - 28 q^{49} - 52 q^{52} - 20 q^{53} + 8 q^{58} + 32 q^{59} + 8 q^{61} - 72 q^{62} - 28 q^{64} + 16 q^{66} - 60 q^{68} + 8 q^{69} + 40 q^{71} - 12 q^{72} - 40 q^{76} + 48 q^{77} - 8 q^{78} - 28 q^{81} - 4 q^{82} + 104 q^{83} - 32 q^{84} + 16 q^{86} + 24 q^{87} - 72 q^{88} - 36 q^{89} - 56 q^{91} + 32 q^{92} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 − 2.48675i 0.707107 − 0.707107i −4.18390 0 −1.75839 − 1.75839i 0.242414 5.43081i − 1.00000i 0
307.2 − 2.21780i −0.707107 + 0.707107i −2.91862 0 1.56822 + 1.56822i −4.11325 2.03732i − 1.00000i 0
307.3 − 1.97160i 0.707107 − 0.707107i −1.88719 0 −1.39413 − 1.39413i 0.616758 − 0.222418i − 1.00000i 0
307.4 − 1.58074i −0.707107 + 0.707107i −0.498726 0 1.11775 + 1.11775i −0.974287 − 2.37312i − 1.00000i 0
307.5 − 0.750656i 0.707107 − 0.707107i 1.43651 0 −0.530794 − 0.530794i −3.56892 − 2.57964i − 1.00000i 0
307.6 − 0.709689i −0.707107 + 0.707107i 1.49634 0 0.501826 + 0.501826i 3.37036 − 2.48132i − 1.00000i 0
307.7 0.147953i 0.707107 − 0.707107i 1.97811 0 0.104618 + 0.104618i 1.74764 0.588572i − 1.00000i 0
307.8 0.470635i −0.707107 + 0.707107i 1.77850 0 −0.332789 − 0.332789i 1.17941 1.77829i − 1.00000i 0
307.9 0.792814i −0.707107 + 0.707107i 1.37145 0 −0.560604 − 0.560604i −1.67222 2.67293i − 1.00000i 0
307.10 1.38150i 0.707107 − 0.707107i 0.0914500 0 0.976870 + 0.976870i 3.94352 2.88934i − 1.00000i 0
307.11 1.67997i −0.707107 + 0.707107i −0.822299 0 −1.18792 − 1.18792i −2.35789 1.97850i − 1.00000i 0
307.12 1.94332i 0.707107 − 0.707107i −1.77647 0 1.37413 + 1.37413i −2.33552 0.434380i − 1.00000i 0
307.13 2.56480i −0.707107 + 0.707107i −4.57821 0 −1.81359 − 1.81359i 1.73944 − 6.61261i − 1.00000i 0
307.14 2.73623i 0.707107 − 0.707107i −5.48694 0 1.93480 + 1.93480i 2.18253 − 9.54105i − 1.00000i 0
343.1 − 2.73623i 0.707107 + 0.707107i −5.48694 0 1.93480 − 1.93480i 2.18253 9.54105i 1.00000i 0
343.2 − 2.56480i −0.707107 − 0.707107i −4.57821 0 −1.81359 + 1.81359i 1.73944 6.61261i 1.00000i 0
343.3 − 1.94332i 0.707107 + 0.707107i −1.77647 0 1.37413 − 1.37413i −2.33552 − 0.434380i 1.00000i 0
343.4 − 1.67997i −0.707107 − 0.707107i −0.822299 0 −1.18792 + 1.18792i −2.35789 − 1.97850i 1.00000i 0
343.5 − 1.38150i 0.707107 + 0.707107i 0.0914500 0 0.976870 − 0.976870i 3.94352 − 2.88934i 1.00000i 0
343.6 − 0.792814i −0.707107 − 0.707107i 1.37145 0 −0.560604 + 0.560604i −1.67222 − 2.67293i 1.00000i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.k.d 28
5.b even 2 1 195.2.k.a ✓ 28
5.c odd 4 1 195.2.t.a yes 28
5.c odd 4 1 975.2.t.d 28
13.d odd 4 1 975.2.t.d 28
15.d odd 2 1 585.2.n.g 28
15.e even 4 1 585.2.w.g 28
65.f even 4 1 inner 975.2.k.d 28
65.g odd 4 1 195.2.t.a yes 28
65.k even 4 1 195.2.k.a ✓ 28
195.j odd 4 1 585.2.n.g 28
195.n even 4 1 585.2.w.g 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.k.a ✓ 28 5.b even 2 1
195.2.k.a ✓ 28 65.k even 4 1
195.2.t.a yes 28 5.c odd 4 1
195.2.t.a yes 28 65.g odd 4 1
585.2.n.g 28 15.d odd 2 1
585.2.n.g 28 195.j odd 4 1
585.2.w.g 28 15.e even 4 1
585.2.w.g 28 195.n even 4 1
975.2.k.d 28 1.a even 1 1 trivial
975.2.k.d 28 65.f even 4 1 inner
975.2.t.d 28 5.c odd 4 1
975.2.t.d 28 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{28} + 42 T_{2}^{26} + 777 T_{2}^{24} + 8344 T_{2}^{22} + 57706 T_{2}^{20} + 269580 T_{2}^{18} + \cdots + 256 \) Copy content Toggle raw display
\( T_{7}^{14} - 42 T_{7}^{12} + 8 T_{7}^{11} + 649 T_{7}^{10} - 248 T_{7}^{9} - 4640 T_{7}^{8} + \cdots + 2048 \) Copy content Toggle raw display