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Properties

Label 650.2.j.h
Level $650$
Weight $2$
Character orbit 650.j
Analytic conductor $5.190$
Analytic rank $0$
Dimension $12$
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] = [650,2,Mod(307,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([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("650.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.j (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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 110x^{8} - 24x^{7} + 336x^{5} + 1513x^{4} + 1032x^{3} + 288x^{2} - 288x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_1 q^{3} - q^{4} + \beta_{5} q^{6} - \beta_{10} q^{7} - \beta_{4} q^{8} + ( - \beta_{11} + \beta_{9} - \beta_{8} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_1 q^{3} - q^{4} + \beta_{5} q^{6} - \beta_{10} q^{7} - \beta_{4} q^{8} + ( - \beta_{11} + \beta_{9} - \beta_{8} + \cdots + 1) q^{9}+ \cdots + ( - 4 \beta_{11} - 2 \beta_{10} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} - 4 q^{7} + 16 q^{11} - 12 q^{13} + 12 q^{16} + 8 q^{17} + 20 q^{18} + 8 q^{19} - 16 q^{21} + 16 q^{22} - 20 q^{23} - 4 q^{26} + 4 q^{28} + 4 q^{31} + 8 q^{34} + 8 q^{37} + 8 q^{38} + 16 q^{41} - 16 q^{42} - 12 q^{43} - 16 q^{44} + 20 q^{46} - 44 q^{47} + 56 q^{49} + 12 q^{52} - 4 q^{53} - 12 q^{58} + 24 q^{59} - 4 q^{61} - 4 q^{62} - 12 q^{64} - 48 q^{66} - 8 q^{68} - 48 q^{69} + 8 q^{71} - 20 q^{72} - 8 q^{76} + 44 q^{77} - 32 q^{78} - 44 q^{81} - 16 q^{82} - 4 q^{83} + 16 q^{84} + 12 q^{86} + 48 q^{87} - 16 q^{88} + 8 q^{89} + 16 q^{91} + 20 q^{92} + 120 q^{93} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 110x^{8} - 24x^{7} + 336x^{5} + 1513x^{4} + 1032x^{3} + 288x^{2} - 288x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4751849 \nu^{11} + 39750666 \nu^{10} - 129079206 \nu^{9} + 114321696 \nu^{8} + \cdots + 2811941916 ) / 18918544250 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27883627 \nu^{11} - 29152818 \nu^{10} - 68136762 \nu^{9} - 25511933 \nu^{8} + \cdots - 14401006068 ) / 37837088500 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 168108751 \nu^{11} - 177623809 \nu^{10} + 50875644 \nu^{9} - 14652504 \nu^{8} + \cdots + 26669280816 ) / 113511265500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 177623809 \nu^{11} - 50875644 \nu^{10} + 14652504 \nu^{9} - 8155764 \nu^{8} + \cdots - 24207660144 ) / 113511265500 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 737002777 \nu^{11} - 868172632 \nu^{10} - 425027838 \nu^{9} + 890093658 \nu^{8} + \cdots - 664304690232 ) / 227022531000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1265634313 \nu^{11} - 130381608 \nu^{10} + 42675228 \nu^{9} - 272856198 \nu^{8} + \cdots - 272625018408 ) / 227022531000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1611718637 \nu^{11} - 925655892 \nu^{10} + 245100672 \nu^{9} + 908399448 \nu^{8} + \cdots - 112271553792 ) / 227022531000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1715997043 \nu^{11} - 1456236838 \nu^{10} + 1539228258 \nu^{9} - 427628628 \nu^{8} + \cdots - 175261964688 ) / 227022531000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1799122433 \nu^{11} + 491323128 \nu^{10} + 1030993602 \nu^{9} - 2705264232 \nu^{8} + \cdots + 123299440728 ) / 227022531000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2708700571 \nu^{11} + 95402514 \nu^{10} + 900220176 \nu^{9} - 191954316 \nu^{8} + \cdots - 675636100536 ) / 227022531000 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} - 4\beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} + 2\beta_{8} - 2\beta_{7} + 7\beta_{5} - \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{11} - 9 \beta_{10} - \beta_{8} + 12 \beta_{7} - 9 \beta_{6} - 2 \beta_{5} - 9 \beta_{4} + \cdots - 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{10} + 14 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + 6 \beta_{4} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 83 \beta_{11} - 83 \beta_{9} + 97 \beta_{8} - 42 \beta_{7} - 83 \beta_{6} - 43 \beta_{5} + 260 \beta_{4} + \cdots - 83 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 80 \beta_{11} - 165 \beta_{10} + 165 \beta_{9} - 318 \beta_{8} + 234 \beta_{7} - 497 \beta_{5} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 787 \beta_{11} + 811 \beta_{10} + 177 \beta_{8} - 1270 \beta_{7} + 787 \beta_{6} + 558 \beta_{5} + \cdots + 2374 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1828 \beta_{10} - 1828 \beta_{9} + 582 \beta_{8} + 582 \beta_{7} - 1164 \beta_{6} - 582 \beta_{5} + \cdots - 1592 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7605 \beta_{11} + 8229 \beta_{9} - 9805 \beta_{8} + 5232 \beta_{7} + 7605 \beta_{6} + 781 \beta_{5} + \cdots + 7605 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 14896 \beta_{11} + 19661 \beta_{10} - 19661 \beta_{9} + 35650 \beta_{8} - 27682 \beta_{7} + 42711 \beta_{5} + \cdots + 8259 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(\beta_{4}\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
2.03014 − 2.03014i
1.72749 − 1.72749i
0.263073 − 0.263073i
−0.671174 + 0.671174i
−1.05632 + 1.05632i
−2.29321 + 2.29321i
2.03014 + 2.03014i
1.72749 + 1.72749i
0.263073 + 0.263073i
−0.671174 − 0.671174i
−1.05632 − 1.05632i
−2.29321 − 2.29321i
1.00000i −2.03014 + 2.03014i −1.00000 0 −2.03014 − 2.03014i 4.50439 − 1.00000i − 5.24293i 0
307.2 1.00000i −1.72749 + 1.72749i −1.00000 0 −1.72749 − 1.72749i −3.06667 − 1.00000i − 2.96846i 0
307.3 1.00000i −0.263073 + 0.263073i −1.00000 0 −0.263073 − 0.263073i −1.87836 − 1.00000i 2.86159i 0
307.4 1.00000i 0.671174 − 0.671174i −1.00000 0 0.671174 + 0.671174i 4.01098 − 1.00000i 2.09905i 0
307.5 1.00000i 1.05632 − 1.05632i −1.00000 0 1.05632 + 1.05632i −4.39380 − 1.00000i 0.768383i 0
307.6 1.00000i 2.29321 − 2.29321i −1.00000 0 2.29321 + 2.29321i −1.17655 − 1.00000i − 7.51764i 0
343.1 − 1.00000i −2.03014 − 2.03014i −1.00000 0 −2.03014 + 2.03014i 4.50439 1.00000i 5.24293i 0
343.2 − 1.00000i −1.72749 − 1.72749i −1.00000 0 −1.72749 + 1.72749i −3.06667 1.00000i 2.96846i 0
343.3 − 1.00000i −0.263073 − 0.263073i −1.00000 0 −0.263073 + 0.263073i −1.87836 1.00000i − 2.86159i 0
343.4 − 1.00000i 0.671174 + 0.671174i −1.00000 0 0.671174 − 0.671174i 4.01098 1.00000i − 2.09905i 0
343.5 − 1.00000i 1.05632 + 1.05632i −1.00000 0 1.05632 − 1.05632i −4.39380 1.00000i − 0.768383i 0
343.6 − 1.00000i 2.29321 + 2.29321i −1.00000 0 2.29321 − 2.29321i −1.17655 1.00000i 7.51764i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.6
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 650.2.j.h yes 12
5.b even 2 1 650.2.j.i yes 12
5.c odd 4 1 650.2.g.h ✓ 12
5.c odd 4 1 650.2.g.i yes 12
13.d odd 4 1 650.2.g.i yes 12
65.f even 4 1 inner 650.2.j.h yes 12
65.g odd 4 1 650.2.g.h ✓ 12
65.k even 4 1 650.2.j.i yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.g.h ✓ 12 5.c odd 4 1
650.2.g.h ✓ 12 65.g odd 4 1
650.2.g.i yes 12 5.c odd 4 1
650.2.g.i yes 12 13.d odd 4 1
650.2.j.h yes 12 1.a even 1 1 trivial
650.2.j.h yes 12 65.f even 4 1 inner
650.2.j.i yes 12 5.b even 2 1
650.2.j.i yes 12 65.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\):

\( T_{3}^{12} + 110T_{3}^{8} + 24T_{3}^{7} - 336T_{3}^{5} + 1513T_{3}^{4} - 1032T_{3}^{3} + 288T_{3}^{2} + 288T_{3} + 144 \) Copy content Toggle raw display
\( T_{7}^{6} + 2T_{7}^{5} - 33T_{7}^{4} - 80T_{7}^{3} + 234T_{7}^{2} + 788T_{7} + 538 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 110 T^{8} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 2 T^{5} + \cdots + 538)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 16 T^{11} + \cdots + 194481 \) Copy content Toggle raw display
$13$ \( T^{12} + 12 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 8 T^{11} + \cdots + 4473225 \) Copy content Toggle raw display
$19$ \( T^{12} - 8 T^{11} + \cdots + 11451456 \) Copy content Toggle raw display
$23$ \( T^{12} + 20 T^{11} + \cdots + 32400 \) Copy content Toggle raw display
$29$ \( T^{12} + 126 T^{10} + \cdots + 419904 \) Copy content Toggle raw display
$31$ \( T^{12} - 4 T^{11} + \cdots + 360000 \) Copy content Toggle raw display
$37$ \( (T^{6} - 4 T^{5} + \cdots - 5108)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 16 T^{11} + \cdots + 1327104 \) Copy content Toggle raw display
$43$ \( T^{12} + 12 T^{11} + \cdots + 3686400 \) Copy content Toggle raw display
$47$ \( (T^{6} + 22 T^{5} + \cdots + 1440)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 4032758016 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 940771584 \) Copy content Toggle raw display
$61$ \( (T^{6} + 2 T^{5} + \cdots + 4734)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 292717881 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 4199040000 \) Copy content Toggle raw display
$73$ \( T^{12} + 488 T^{10} + \cdots + 5062500 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 6766707600 \) Copy content Toggle raw display
$83$ \( (T^{6} + 2 T^{5} + \cdots - 231741)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 8 T^{11} + \cdots + 71571600 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 4287630400 \) Copy content Toggle raw display
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