Properties

Label 1450.2.b.k
Level $1450$
Weight $2$
Character orbit 1450.b
Analytic conductor $11.578$
Analytic rank $0$
Dimension $6$
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] = [1450,2,Mod(349,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1450.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.b (of order \(2\), degree \(1\), 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: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_{5} q^{3} - q^{4} + \beta_{3} q^{6} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{7} + \beta_{2} q^{8} - \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - \beta_{5} q^{3} - q^{4} + \beta_{3} q^{6} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{7} + \beta_{2} q^{8} - \beta_1 q^{9} + ( - 2 \beta_{3} - \beta_1 + 1) q^{11} + \beta_{5} q^{12} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{13} + ( - \beta_{3} + \beta_1 - 1) q^{14} + q^{16} + (2 \beta_{5} + \beta_{4} - 3 \beta_{2}) q^{17} - \beta_{4} q^{18} - \beta_{3} q^{19} + (2 \beta_1 + 4) q^{21} + ( - 2 \beta_{5} - \beta_{4} - \beta_{2}) q^{22} + (\beta_{5} + 3 \beta_{4} - 3 \beta_{2}) q^{23} - \beta_{3} q^{24} + (\beta_{3} + \beta_1 + 1) q^{26} + ( - 2 \beta_{5} - \beta_{4} + \beta_{2}) q^{27} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{28} - q^{29} + ( - 4 \beta_{3} - 3 \beta_1) q^{31} - \beta_{2} q^{32} + (\beta_{4} - 5 \beta_{2}) q^{33} + ( - 2 \beta_{3} - \beta_1 - 3) q^{34} + \beta_1 q^{36} + (\beta_{5} + 2 \beta_{4} - 5 \beta_{2}) q^{37} - \beta_{5} q^{38} + ( - 2 \beta_{3} - 2) q^{39} + (\beta_{3} + 2 \beta_1 + 6) q^{41} + (2 \beta_{4} - 4 \beta_{2}) q^{42} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{43} + (2 \beta_{3} + \beta_1 - 1) q^{44} + ( - \beta_{3} - 3 \beta_1 - 3) q^{46} + (3 \beta_{4} - 4 \beta_{2}) q^{47} - \beta_{5} q^{48} + (2 \beta_{3} - 3) q^{49} + (4 \beta_{3} + \beta_1 + 5) q^{51} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{52} + ( - 3 \beta_{5} - 3 \beta_{4} - \beta_{2}) q^{53} + (2 \beta_{3} + \beta_1 + 1) q^{54} + (\beta_{3} - \beta_1 + 1) q^{56} + (\beta_{4} - 3 \beta_{2}) q^{57} + \beta_{2} q^{58} + (\beta_{3} + 2 \beta_1 - 5) q^{59} + ( - 5 \beta_{3} - 1) q^{61} + ( - 4 \beta_{5} - 3 \beta_{4}) q^{62} + ( - 3 \beta_{5} - \beta_{4} - 5 \beta_{2}) q^{63} - q^{64} + ( - \beta_1 - 5) q^{66} + ( - \beta_{5} - 3 \beta_{4}) q^{67} + ( - 2 \beta_{5} - \beta_{4} + 3 \beta_{2}) q^{68} + (6 \beta_{3} - 2 \beta_1) q^{69} + (\beta_{3} + 3 \beta_1 + 3) q^{71} + \beta_{4} q^{72} + (5 \beta_{5} - 6 \beta_{2}) q^{73} + ( - \beta_{3} - 2 \beta_1 - 5) q^{74} + \beta_{3} q^{76} + ( - 2 \beta_{5} - 6 \beta_{4} + 2 \beta_{2}) q^{77} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{78} + 6 \beta_1 q^{79} + ( - 2 \beta_{3} - 4 \beta_1 - 5) q^{81} + (\beta_{5} + 2 \beta_{4} - 6 \beta_{2}) q^{82} + (4 \beta_{5} + 3 \beta_{4} - 5 \beta_{2}) q^{83} + ( - 2 \beta_1 - 4) q^{84} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{86} + \beta_{5} q^{87} + (2 \beta_{5} + \beta_{4} + \beta_{2}) q^{88} + ( - \beta_{3} + 2 \beta_1 - 4) q^{89} + (4 \beta_{3} + 2 \beta_1) q^{91} + ( - \beta_{5} - 3 \beta_{4} + 3 \beta_{2}) q^{92} + (3 \beta_{5} + \beta_{4} - 9 \beta_{2}) q^{93} + ( - 3 \beta_1 - 4) q^{94} + \beta_{3} q^{96} + ( - 3 \beta_{5} - \beta_{4} - 7 \beta_{2}) q^{97} + (2 \beta_{5} + 3 \beta_{2}) q^{98} + ( - 4 \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 2 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} - 2 q^{6} + 10 q^{11} - 4 q^{14} + 6 q^{16} + 2 q^{19} + 24 q^{21} + 2 q^{24} + 4 q^{26} - 6 q^{29} + 8 q^{31} - 14 q^{34} - 8 q^{39} + 34 q^{41} - 10 q^{44} - 16 q^{46} - 22 q^{49} + 22 q^{51} + 2 q^{54} + 4 q^{56} - 32 q^{59} + 4 q^{61} - 6 q^{64} - 30 q^{66} - 12 q^{69} + 16 q^{71} - 28 q^{74} - 2 q^{76} - 26 q^{81} - 24 q^{84} - 8 q^{86} - 22 q^{89} - 8 q^{91} - 24 q^{94} - 2 q^{96} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 2\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 2\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{3} - 5\beta_{2} - \beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
−0.671462 1.24464i
0.264658 + 1.38923i
1.40680 0.144584i
1.40680 + 0.144584i
0.264658 1.38923i
−0.671462 + 1.24464i
1.00000i 2.34292i −1.00000 0 −2.34292 3.83221i 1.00000i −2.48929 0
349.2 1.00000i 0.470683i −1.00000 0 −0.470683 3.30777i 1.00000i 2.77846 0
349.3 1.00000i 1.81361i −1.00000 0 1.81361 2.52444i 1.00000i −0.289169 0
349.4 1.00000i 1.81361i −1.00000 0 1.81361 2.52444i 1.00000i −0.289169 0
349.5 1.00000i 0.470683i −1.00000 0 −0.470683 3.30777i 1.00000i 2.77846 0
349.6 1.00000i 2.34292i −1.00000 0 −2.34292 3.83221i 1.00000i −2.48929 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1450.2.b.k 6
5.b even 2 1 inner 1450.2.b.k 6
5.c odd 4 1 1450.2.a.q 3
5.c odd 4 1 1450.2.a.s yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1450.2.a.q 3 5.c odd 4 1
1450.2.a.s yes 3 5.c odd 4 1
1450.2.b.k 6 1.a even 1 1 trivial
1450.2.b.k 6 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1450, [\chi])\):

\( T_{3}^{6} + 9T_{3}^{4} + 20T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 32T_{7}^{4} + 324T_{7}^{2} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 9 T^{4} + 20 T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 32 T^{4} + 324 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( (T^{3} - 5 T^{2} - 8 T + 44)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 16 T^{4} + 68 T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 49 T^{4} + 56 T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{3} - T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 132 T^{4} + 5444 T^{2} + \cdots + 71824 \) Copy content Toggle raw display
$29$ \( (T + 1)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 79 T + 158)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 114 T^{4} + 3305 T^{2} + \cdots + 3364 \) Copy content Toggle raw display
$41$ \( (T^{3} - 17 T^{2} + 72 T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 128 T^{4} + 5184 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$47$ \( T^{6} + 174 T^{4} + 6033 T^{2} + \cdots + 58564 \) Copy content Toggle raw display
$53$ \( T^{6} + 144 T^{4} + 3300 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$59$ \( (T^{3} + 16 T^{2} + 61 T + 68)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 2 T^{2} - 107 T + 146)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 111 T^{4} + 2783 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$71$ \( (T^{3} - 8 T^{2} - 34 T + 268)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 273 T^{4} + 10868 T^{2} + \cdots + 98596 \) Copy content Toggle raw display
$79$ \( (T^{3} - 252 T + 432)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 209 T^{4} + 6600 T^{2} + \cdots + 44944 \) Copy content Toggle raw display
$89$ \( (T^{3} + 11 T^{2} - 158)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 260 T^{4} + 17092 T^{2} + \cdots + 26896 \) Copy content Toggle raw display
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