Properties

Label 750.2.h.a
Level $750$
Weight $2$
Character orbit 750.h
Analytic conductor $5.989$
Analytic rank $0$
Dimension $8$
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] = [750,2,Mod(49,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("750.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.h (of order \(10\), degree \(4\), not 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: \(5.98878015160\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{20} q^{2} + \zeta_{20}^{7} q^{3} + \zeta_{20}^{2} q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{6} - 2 \zeta_{20}^{5} q^{7} + \zeta_{20}^{3} q^{8} - \zeta_{20}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{20} q^{2} + \zeta_{20}^{7} q^{3} + \zeta_{20}^{2} q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{6} - 2 \zeta_{20}^{5} q^{7} + \zeta_{20}^{3} q^{8} - \zeta_{20}^{4} q^{9} + (4 \zeta_{20}^{6} + 2 \zeta_{20}^{2} - 2) q^{11} + (\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}) q^{12} + ( - 3 \zeta_{20}^{7} + 3 \zeta_{20}) q^{13} - 2 \zeta_{20}^{6} q^{14} + \zeta_{20}^{4} q^{16} + (3 \zeta_{20}^{5} + 3 \zeta_{20}^{3} + 3 \zeta_{20}) q^{17} - \zeta_{20}^{5} q^{18} + (4 \zeta_{20}^{6} - 6 \zeta_{20}^{4} + 6 \zeta_{20}^{2} - 4) q^{19} + 2 \zeta_{20}^{2} q^{21} + (4 \zeta_{20}^{7} + 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{22} + 6 \zeta_{20} q^{23} - q^{24} + ( - 3 \zeta_{20}^{6} + 3 \zeta_{20}^{4} + 3) q^{26} + \zeta_{20} q^{27} - 2 \zeta_{20}^{7} q^{28} + (\zeta_{20}^{4} - 3 \zeta_{20}^{2} + 1) q^{29} + (6 \zeta_{20}^{4} - 6 \zeta_{20}^{2}) q^{31} + \zeta_{20}^{5} q^{32} + ( - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{33} + (3 \zeta_{20}^{6} + 3 \zeta_{20}^{4} + 3 \zeta_{20}^{2}) q^{34} - \zeta_{20}^{6} q^{36} + (4 \zeta_{20}^{7} - 7 \zeta_{20}^{5} + 7 \zeta_{20}^{3} - 4 \zeta_{20}) q^{37} + (4 \zeta_{20}^{7} - 6 \zeta_{20}^{5} + 6 \zeta_{20}^{3} - 4 \zeta_{20}) q^{38} + (3 \zeta_{20}^{6} + 3 \zeta_{20}^{2} - 3) q^{39} + ( - 5 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 5 \zeta_{20}^{2}) q^{41} + 2 \zeta_{20}^{3} q^{42} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{3}) q^{43} + (4 \zeta_{20}^{6} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2} - 4) q^{44} + 6 \zeta_{20}^{2} q^{46} + (6 \zeta_{20}^{7} - 2 \zeta_{20}^{3} + 2 \zeta_{20}) q^{47} - \zeta_{20} q^{48} + 3 q^{49} + (3 \zeta_{20}^{6} - 3 \zeta_{20}^{4} - 6) q^{51} + ( - 3 \zeta_{20}^{7} + 3 \zeta_{20}^{5} + 3 \zeta_{20}) q^{52} + ( - 3 \zeta_{20}^{7} - 9 \zeta_{20}^{3} + 9 \zeta_{20}) q^{53} + \zeta_{20}^{2} q^{54} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2} + 2) q^{56} + (2 \zeta_{20}^{7} - 6 \zeta_{20}^{5} + 2 \zeta_{20}^{3}) q^{57} + (\zeta_{20}^{5} - 3 \zeta_{20}^{3} + \zeta_{20}) q^{58} + ( - 8 \zeta_{20}^{6} + 4 \zeta_{20}^{4} - 8 \zeta_{20}^{2}) q^{59} + (7 \zeta_{20}^{6} + 3 \zeta_{20}^{2} - 3) q^{61} + (6 \zeta_{20}^{5} - 6 \zeta_{20}^{3}) q^{62} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{63} + \zeta_{20}^{6} q^{64} + ( - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2}) q^{66} + ( - 6 \zeta_{20}^{5} - 6 \zeta_{20}) q^{67} + (3 \zeta_{20}^{7} + 3 \zeta_{20}^{5} + 3 \zeta_{20}^{3}) q^{68} + (6 \zeta_{20}^{6} - 6 \zeta_{20}^{4} + 6 \zeta_{20}^{2} - 6) q^{69} + (10 \zeta_{20}^{4} - 2 \zeta_{20}^{2} + 10) q^{71} - \zeta_{20}^{7} q^{72} + ( - 3 \zeta_{20}^{7} + 3 \zeta_{20}^{5} - 5 \zeta_{20}) q^{73} + ( - 3 \zeta_{20}^{6} + 3 \zeta_{20}^{4} - 4) q^{74} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 4) q^{76} + ( - 4 \zeta_{20}^{7} + 4 \zeta_{20}^{5} + 8 \zeta_{20}) q^{77} + (3 \zeta_{20}^{7} + 3 \zeta_{20}^{3} - 3 \zeta_{20}) q^{78} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{81} + ( - 5 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 5 \zeta_{20}^{3}) q^{82} - 6 \zeta_{20}^{3} q^{83} + 2 \zeta_{20}^{4} q^{84} + ( - 2 \zeta_{20}^{2} + 2) q^{86} + ( - 2 \zeta_{20}^{7} + 3 \zeta_{20}^{5} - 3 \zeta_{20}^{3} + 2 \zeta_{20}) q^{87} + (4 \zeta_{20}^{7} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3} - 4 \zeta_{20}) q^{88} + (2 \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{89} + ( - 6 \zeta_{20}^{6} - 6 \zeta_{20}^{2}) q^{91} + 6 \zeta_{20}^{3} q^{92} + ( - 6 \zeta_{20}^{7} + 6 \zeta_{20}^{5} - 6 \zeta_{20}^{3}) q^{93} + (6 \zeta_{20}^{6} - 8 \zeta_{20}^{4} + 8 \zeta_{20}^{2} - 6) q^{94} - \zeta_{20}^{2} q^{96} + ( - 12 \zeta_{20}^{7} - 3 \zeta_{20}^{3} + 3 \zeta_{20}) q^{97} + 3 \zeta_{20} q^{98} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4} + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} - 2 q^{6} + 2 q^{9} - 4 q^{11} - 4 q^{14} - 2 q^{16} + 4 q^{21} - 8 q^{24} + 12 q^{26} - 24 q^{31} + 6 q^{34} - 2 q^{36} - 12 q^{39} - 24 q^{41} - 16 q^{44} + 12 q^{46} + 24 q^{49} - 36 q^{51} + 2 q^{54} + 4 q^{56} - 40 q^{59} - 4 q^{61} + 2 q^{64} - 4 q^{66} - 12 q^{69} + 56 q^{71} - 44 q^{74} - 40 q^{76} - 2 q^{81} - 4 q^{84} + 12 q^{86} + 10 q^{89} - 24 q^{91} - 4 q^{94} - 2 q^{96} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(\zeta_{20}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.587785 + 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i 0 0.309017 0.951057i 2.00000i 0.951057 + 0.309017i 0.809017 0.587785i 0
49.2 0.587785 0.809017i 0.951057 0.309017i −0.309017 0.951057i 0 0.309017 0.951057i 2.00000i −0.951057 0.309017i 0.809017 0.587785i 0
199.1 −0.587785 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i 0 0.309017 + 0.951057i 2.00000i 0.951057 0.309017i 0.809017 + 0.587785i 0
199.2 0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 0 0.309017 + 0.951057i 2.00000i −0.951057 + 0.309017i 0.809017 + 0.587785i 0
349.1 −0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i 0 −0.809017 0.587785i 2.00000i −0.587785 + 0.809017i −0.309017 + 0.951057i 0
349.2 0.951057 0.309017i −0.587785 0.809017i 0.809017 0.587785i 0 −0.809017 0.587785i 2.00000i 0.587785 0.809017i −0.309017 + 0.951057i 0
649.1 −0.951057 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i 0 −0.809017 + 0.587785i 2.00000i −0.587785 0.809017i −0.309017 0.951057i 0
649.2 0.951057 + 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 0 −0.809017 + 0.587785i 2.00000i 0.587785 + 0.809017i −0.309017 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 750.2.h.a 8
5.b even 2 1 inner 750.2.h.a 8
5.c odd 4 1 150.2.g.b 4
5.c odd 4 1 750.2.g.a 4
15.e even 4 1 450.2.h.b 4
25.d even 5 1 inner 750.2.h.a 8
25.d even 5 1 3750.2.c.c 4
25.e even 10 1 inner 750.2.h.a 8
25.e even 10 1 3750.2.c.c 4
25.f odd 20 1 150.2.g.b 4
25.f odd 20 1 750.2.g.a 4
25.f odd 20 1 3750.2.a.b 2
25.f odd 20 1 3750.2.a.g 2
75.l even 20 1 450.2.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.b 4 5.c odd 4 1
150.2.g.b 4 25.f odd 20 1
450.2.h.b 4 15.e even 4 1
450.2.h.b 4 75.l even 20 1
750.2.g.a 4 5.c odd 4 1
750.2.g.a 4 25.f odd 20 1
750.2.h.a 8 1.a even 1 1 trivial
750.2.h.a 8 5.b even 2 1 inner
750.2.h.a 8 25.d even 5 1 inner
750.2.h.a 8 25.e even 10 1 inner
3750.2.a.b 2 25.f odd 20 1
3750.2.a.g 2 25.f odd 20 1
3750.2.c.c 4 25.d even 5 1
3750.2.c.c 4 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} + 24 T^{2} - 32 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 36 T^{6} + 486 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$17$ \( T^{8} + 36 T^{6} + 3726 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$19$ \( (T^{4} + 40 T^{2} + 200 T + 400)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 36 T^{6} + 1296 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$29$ \( (T^{4} + 10 T^{2} + 25 T + 25)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 12 T^{3} + 144 T^{2} + 648 T + 1296)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 41 T^{6} + 5806 T^{4} + \cdots + 130321 \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + 94 T^{2} + 403 T + 961)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 12 T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 124 T^{6} + 5856 T^{4} + \cdots + 3748096 \) Copy content Toggle raw display
$53$ \( T^{8} - 171 T^{6} + \cdots + 96059601 \) Copy content Toggle raw display
$59$ \( (T^{4} + 20 T^{3} + 240 T^{2} + 1600 T + 6400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{3} + 64 T^{2} - 247 T + 361)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 36 T^{6} + 7776 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$71$ \( (T^{4} - 28 T^{3} + 384 T^{2} + \cdots + 13456)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 44 T^{6} + 8566 T^{4} + \cdots + 923521 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} - 36 T^{6} + 1296 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$89$ \( (T^{4} - 5 T^{3} + 10 T^{2} + 25)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 36 T^{6} + 29646 T^{4} + \cdots + 96059601 \) Copy content Toggle raw display
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