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Properties

Label 644.2.i.a
Level $644$
Weight $2$
Character orbit 644.i
Analytic conductor $5.142$
Analytic rank $0$
Dimension $14$
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] = [644,2,Mod(93,644)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(644, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("644.93");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 644.i (of order \(3\), 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: \(5.14236589017\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 13 x^{12} - 8 x^{11} + 130 x^{10} - 78 x^{9} + 505 x^{8} - 519 x^{7} + 1508 x^{6} - 955 x^{5} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{3} + ( - \beta_{11} - \beta_{2}) q^{5} + ( - \beta_{8} - \beta_{3}) q^{7} + ( - \beta_{11} - \beta_{10} - \beta_{7} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{3} + ( - \beta_{11} - \beta_{2}) q^{5} + ( - \beta_{8} - \beta_{3}) q^{7} + ( - \beta_{11} - \beta_{10} - \beta_{7} + \cdots - 1) q^{9}+ \cdots + (\beta_{8} + 3 \beta_{6} - 2 \beta_{5} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{3} - 2 q^{5} + 6 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 3 q^{3} - 2 q^{5} + 6 q^{7} - 12 q^{9} + 12 q^{13} + 2 q^{15} - 4 q^{17} - 11 q^{19} + 7 q^{23} - 3 q^{25} + 48 q^{27} - 2 q^{29} - 24 q^{31} + 13 q^{33} + 5 q^{35} - 11 q^{37} + 16 q^{39} - 18 q^{41} + 10 q^{43} - 38 q^{45} - 8 q^{47} + 20 q^{49} - 23 q^{51} + 20 q^{53} + 50 q^{55} - 8 q^{57} - 13 q^{59} + 2 q^{61} + 26 q^{63} - 21 q^{65} + 4 q^{67} - 6 q^{69} - 16 q^{71} - 11 q^{73} + 10 q^{75} + 70 q^{77} - 28 q^{79} - 3 q^{81} + 42 q^{83} - 46 q^{85} - 59 q^{87} + 9 q^{89} + 14 q^{91} - 31 q^{93} - 12 q^{95} + 2 q^{97} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 13 x^{12} - 8 x^{11} + 130 x^{10} - 78 x^{9} + 505 x^{8} - 519 x^{7} + 1508 x^{6} - 955 x^{5} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 21989300411 \nu^{13} - 40431457751 \nu^{12} - 121386533509 \nu^{11} + \cdots - 28361882211929 ) / 17051458105375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 114811762871 \nu^{13} + 43872435659 \nu^{12} - 1506262589884 \nu^{11} + \cdots + 1576314697776 ) / 17051458105375 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 855671348766 \nu^{13} + 291802197544 \nu^{12} - 11065898536954 \nu^{11} + \cdots + 43016111908326 ) / 17051458105375 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 892221909748 \nu^{13} - 335748057007 \nu^{12} + 11678064398612 \nu^{11} + \cdots + 19148230877122 ) / 17051458105375 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1024633915058 \nu^{13} - 378098736962 \nu^{12} + 13368725563572 \nu^{11} + \cdots - 35811814125328 ) / 17051458105375 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1576314697776 \nu^{13} - 114811762871 \nu^{12} - 20448218635429 \nu^{11} + \cdots - 4515342840944 ) / 17051458105375 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1698943231713 \nu^{13} + 556956033702 \nu^{12} - 21915521774727 \nu^{11} + \cdots + 44969246674753 ) / 17051458105375 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 114993859747 \nu^{13} + 29037401042 \nu^{12} + 1485408021048 \nu^{11} - 545673690198 \nu^{10} + \cdots - 648064057627 ) / 1003026947375 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2129756329766 \nu^{13} - 258086197471 \nu^{12} - 27654049937709 \nu^{11} + \cdots - 6155746211264 ) / 17051458105375 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2551800926191 \nu^{13} + 72592960761 \nu^{12} + 32926082107139 \nu^{11} + \cdots + 7216930687304 ) / 17051458105375 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 612962220086 \nu^{13} + 76068353439 \nu^{12} + 7974346459040 \nu^{11} - 3917686570126 \nu^{10} + \cdots + 1775864671607 ) / 3410291621075 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 4049907538174 \nu^{13} - 48950026801 \nu^{12} + 52719392217111 \nu^{11} + \cdots - 38006779141524 ) / 17051458105375 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} + \beta_{12} - \beta_{11} - \beta_{9} - 3\beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} - 6\beta_{3} - \beta_{2} - 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9 \beta_{13} - 12 \beta_{12} + 8 \beta_{11} - 2 \beta_{10} + 9 \beta_{9} + 9 \beta_{8} + 22 \beta_{7} + \cdots + 10 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{13} - 6 \beta_{12} + 10 \beta_{11} + 15 \beta_{10} - 12 \beta_{9} - 24 \beta_{7} - 15 \beta_{6} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -76\beta_{8} + 31\beta_{6} - 82\beta_{5} + 113\beta_{4} - 90\beta_{3} + 62\beta_{2} - 90\beta _1 + 184 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 68 \beta_{13} + 17 \beta_{12} - 85 \beta_{11} - 164 \beta_{10} + 127 \beta_{9} - 68 \beta_{8} + \cdots + 387 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 636 \beta_{13} + 1012 \beta_{12} - 489 \beta_{11} + 359 \beta_{10} - 746 \beta_{9} - 1605 \beta_{7} + \cdots - 1605 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 471\beta_{8} + 1628\beta_{6} - 1287\beta_{5} + 145\beta_{4} - 3392\beta_{3} - 683\beta_{2} - 3392\beta _1 + 2434 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5346 \beta_{13} - 8944 \beta_{12} + 3930 \beta_{11} - 3743 \beta_{10} + 6778 \beta_{9} + \cdots + 7572 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2975 \beta_{13} + 3616 \beta_{12} + 5302 \beta_{11} + 15535 \beta_{10} - 12747 \beta_{9} - 23961 \beta_{7} + \cdots - 23961 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 45253 \beta_{8} + 37200 \beta_{6} - 61616 \beta_{5} + 78837 \beta_{4} - 70896 \beta_{3} + \cdots + 127636 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 15994 \beta_{13} - 50777 \beta_{12} - 39986 \beta_{11} - 145608 \beta_{10} + 124459 \beta_{9} + \cdots + 274972 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/644\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\) \(323\)
\(\chi(n)\) \(-1 - \beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
−1.11314 − 1.92802i
1.41340 + 2.44808i
0.0453148 + 0.0784875i
0.615856 + 1.06669i
−0.160672 − 0.278292i
0.725556 + 1.25670i
−1.52632 − 2.64366i
−1.11314 + 1.92802i
1.41340 − 2.44808i
0.0453148 − 0.0784875i
0.615856 − 1.06669i
−0.160672 + 0.278292i
0.725556 − 1.25670i
−1.52632 + 2.64366i
0 −1.73002 − 2.99648i 0 −1.14277 + 1.97934i 0 1.81184 − 1.92802i 0 −4.48593 + 7.76986i 0
93.2 0 −1.17316 − 2.03197i 0 −0.992835 + 1.71964i 0 1.00344 + 2.44808i 0 −1.25259 + 2.16955i 0
93.3 0 −1.03959 − 1.80062i 0 0.832793 − 1.44244i 0 −2.64459 + 0.0784875i 0 −0.661478 + 1.14571i 0
93.4 0 −0.331159 − 0.573585i 0 1.49597 − 2.59110i 0 2.42119 + 1.06669i 0 1.28067 − 2.21818i 0
93.5 0 0.637005 + 1.10333i 0 0.712116 − 1.23342i 0 2.63107 − 0.278292i 0 0.688449 − 1.19243i 0
93.6 0 1.04242 + 1.80553i 0 −1.94126 + 3.36236i 0 −2.32824 + 1.25670i 0 −0.673281 + 1.16616i 0
93.7 0 1.09449 + 1.89572i 0 0.0359834 − 0.0623251i 0 0.105282 − 2.64366i 0 −0.895837 + 1.55164i 0
277.1 0 −1.73002 + 2.99648i 0 −1.14277 − 1.97934i 0 1.81184 + 1.92802i 0 −4.48593 − 7.76986i 0
277.2 0 −1.17316 + 2.03197i 0 −0.992835 − 1.71964i 0 1.00344 − 2.44808i 0 −1.25259 − 2.16955i 0
277.3 0 −1.03959 + 1.80062i 0 0.832793 + 1.44244i 0 −2.64459 − 0.0784875i 0 −0.661478 − 1.14571i 0
277.4 0 −0.331159 + 0.573585i 0 1.49597 + 2.59110i 0 2.42119 − 1.06669i 0 1.28067 + 2.21818i 0
277.5 0 0.637005 − 1.10333i 0 0.712116 + 1.23342i 0 2.63107 + 0.278292i 0 0.688449 + 1.19243i 0
277.6 0 1.04242 − 1.80553i 0 −1.94126 − 3.36236i 0 −2.32824 − 1.25670i 0 −0.673281 − 1.16616i 0
277.7 0 1.09449 − 1.89572i 0 0.0359834 + 0.0623251i 0 0.105282 + 2.64366i 0 −0.895837 − 1.55164i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 644.2.i.a ✓ 14
7.c even 3 1 inner 644.2.i.a ✓ 14
7.c even 3 1 4508.2.a.m 7
7.d odd 6 1 4508.2.a.l 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.i.a ✓ 14 1.a even 1 1 trivial
644.2.i.a ✓ 14 7.c even 3 1 inner
4508.2.a.l 7 7.d odd 6 1
4508.2.a.m 7 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 3 T_{3}^{13} + 21 T_{3}^{12} + 26 T_{3}^{11} + 189 T_{3}^{10} + 177 T_{3}^{9} + 1130 T_{3}^{8} + \cdots + 4225 \) acting on \(S_{2}^{\mathrm{new}}(644, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 3 T^{13} + \cdots + 4225 \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{14} - 6 T^{13} + \cdots + 823543 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 104714289 \) Copy content Toggle raw display
$13$ \( (T^{7} - 6 T^{6} + \cdots + 933)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + 4 T^{13} + \cdots + 13689 \) Copy content Toggle raw display
$19$ \( T^{14} + 11 T^{13} + \cdots + 58660281 \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{7} \) Copy content Toggle raw display
$29$ \( (T^{7} + T^{6} - 68 T^{5} + \cdots - 975)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 2798092609 \) Copy content Toggle raw display
$37$ \( T^{14} + 11 T^{13} + \cdots + 531441 \) Copy content Toggle raw display
$41$ \( (T^{7} + 9 T^{6} + \cdots - 1893)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} - 5 T^{6} + \cdots + 87797)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 492917730561 \) Copy content Toggle raw display
$53$ \( T^{14} - 20 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 108735722001 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 5611957569 \) Copy content Toggle raw display
$67$ \( T^{14} - 4 T^{13} + \cdots + 221841 \) Copy content Toggle raw display
$71$ \( (T^{7} + 8 T^{6} + \cdots + 21195)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 520524225 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 13964385241 \) Copy content Toggle raw display
$83$ \( (T^{7} - 21 T^{6} + \cdots - 452721)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 1813993281 \) Copy content Toggle raw display
$97$ \( (T^{7} - T^{6} + \cdots + 2066909)^{2} \) Copy content Toggle raw display
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