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Properties

Label 1064.2.r.i
Level $1064$
Weight $2$
Character orbit 1064.r
Analytic conductor $8.496$
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] = [1064,2,Mod(505,1064)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1064, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1064.505");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1064 = 2^{3} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1064.r (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: \(8.49608277506\)
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: \( x^{14} - 2 x^{13} + 17 x^{12} - 16 x^{11} + 164 x^{10} - 139 x^{9} + 896 x^{8} - 357 x^{7} + 2983 x^{6} + \cdots + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{3} + (\beta_{9} - \beta_{7}) q^{5} - q^{7} + (\beta_{13} - \beta_{11} + \cdots - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1) q^{3} + (\beta_{9} - \beta_{7}) q^{5} - q^{7} + (\beta_{13} - \beta_{11} + \cdots - 2 \beta_1) q^{9}+ \cdots + ( - 3 \beta_{10} + 2 \beta_{8} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{3} - 3 q^{5} - 14 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 2 q^{3} - 3 q^{5} - 14 q^{7} - 9 q^{9} - 14 q^{11} - 4 q^{13} + 9 q^{15} - 2 q^{17} + 14 q^{19} - 2 q^{21} + 4 q^{23} - 22 q^{25} - 22 q^{27} - 21 q^{29} - 14 q^{31} + 12 q^{33} + 3 q^{35} + 26 q^{37} + 16 q^{39} - 14 q^{41} + 7 q^{43} + 18 q^{45} - 21 q^{47} + 14 q^{49} + 30 q^{51} - 20 q^{53} - 11 q^{55} + 15 q^{57} + 18 q^{59} - 7 q^{61} + 9 q^{63} - 32 q^{65} + 2 q^{67} - 24 q^{69} + 16 q^{71} - 6 q^{73} + 78 q^{75} + 14 q^{77} - 4 q^{79} + q^{81} + 26 q^{83} - 9 q^{85} - 54 q^{87} + 15 q^{89} + 4 q^{91} + 12 q^{93} - 4 q^{95} - 7 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 17 x^{12} - 16 x^{11} + 164 x^{10} - 139 x^{9} + 896 x^{8} - 357 x^{7} + 2983 x^{6} + \cdots + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 40101729993 \nu^{13} + 293328643529 \nu^{12} - 1896519332280 \nu^{11} + \cdots - 240214361920212 ) / 202865169372396 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 326020993940 \nu^{13} - 4377869679031 \nu^{12} + 13224933835018 \nu^{11} + \cdots - 332833756699368 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 274094438465 \nu^{13} + 730481782632 \nu^{12} - 5044661807397 \nu^{11} + \cdots + 123194423216976 ) / 405730338744792 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1817824459432 \nu^{13} + 2375983547069 \nu^{12} + 21471867285342 \nu^{11} + \cdots + 36\!\cdots\!84 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2566550483687 \nu^{13} + 4310817651979 \nu^{12} - 41439912874783 \nu^{11} + \cdots - 757652552515152 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2832428056168 \nu^{13} + 3368712490649 \nu^{12} - 40073432221610 \nu^{11} + \cdots - 885785470674120 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4018322671565 \nu^{13} - 7056383447050 \nu^{12} + 63959756287813 \nu^{11} + \cdots - 251273317330080 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 691035132551 \nu^{13} + 1705588527621 \nu^{12} - 11755940375346 \nu^{11} + \cdots + 20936923023432 ) / 202865169372396 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5635636393961 \nu^{13} - 8308397939831 \nu^{12} + 91717681699221 \nu^{11} + \cdots + 722804889851424 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5958297914931 \nu^{13} - 16171165298677 \nu^{12} + 109103186636741 \nu^{11} + \cdots - 58\!\cdots\!28 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2001668063315 \nu^{13} + 5131591171854 \nu^{12} - 37032048847185 \nu^{11} + \cdots + 827110610011488 ) / 405730338744792 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12519172082465 \nu^{13} - 24512684676437 \nu^{12} + 212675317791855 \nu^{11} + \cdots - 35\!\cdots\!52 ) / 12\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} - \beta_{12} - \beta_{10} - \beta_{9} + \beta_{7} - 3\beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} + 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} - \beta_{11} - \beta_{10} + \beta_{6} + \beta_{5} - 8\beta_{4} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{13} - 8\beta_{11} - 2\beta_{8} - 7\beta_{7} + 26\beta_{6} + 9\beta_{3} + 2\beta_{2} - 19\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{13} + 15 \beta_{12} + \beta_{11} + 12 \beta_{10} + 3 \beta_{9} - 11 \beta_{8} - 3 \beta_{7} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{13} + 79 \beta_{12} + 64 \beta_{11} + 60 \beta_{10} + 50 \beta_{9} - 64 \beta_{6} - 64 \beta_{5} + \cdots + 189 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 102 \beta_{13} + 102 \beta_{11} - 21 \beta_{10} + 108 \beta_{8} + 46 \beta_{7} - 215 \beta_{6} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 528 \beta_{13} - 709 \beta_{12} - 73 \beta_{11} - 528 \beta_{10} - 374 \beta_{9} + 296 \beta_{8} + \cdots - 1385 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 288 \beta_{13} - 1775 \beta_{12} - 1191 \beta_{11} - 903 \beta_{10} - 521 \beta_{9} + 1191 \beta_{6} + \cdots - 2502 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3533 \beta_{13} - 3533 \beta_{11} + 950 \beta_{10} - 3029 \beta_{8} - 2918 \beta_{7} + 10807 \beta_{6} + \cdots - 950 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 11228 \beta_{13} + 17621 \beta_{12} + 3364 \beta_{11} + 11228 \beta_{10} + 5281 \beta_{9} - 10036 \beta_{8} + \cdots + 21316 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10817 \beta_{13} + 59795 \beta_{12} + 38942 \beta_{11} + 28125 \beta_{10} + 23625 \beta_{9} + \cdots + 98185 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 68146 \beta_{13} + 68146 \beta_{11} - 36278 \beta_{10} + 95965 \beta_{8} + 50838 \beta_{7} + \cdots + 36278 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1064\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(799\) \(913\) \(1009\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1
−1.19594 − 2.07143i
−1.18152 − 2.04645i
−0.303142 − 0.525058i
0.166004 + 0.287527i
0.640751 + 1.10981i
1.35257 + 2.34271i
1.52128 + 2.63493i
−1.19594 + 2.07143i
−1.18152 + 2.04645i
−0.303142 + 0.525058i
0.166004 − 0.287527i
0.640751 − 1.10981i
1.35257 − 2.34271i
1.52128 − 2.63493i
0 −1.19594 + 2.07143i 0 2.11282 − 3.65951i 0 −1.00000 0 −1.36056 − 2.35656i 0
505.2 0 −1.18152 + 2.04645i 0 −1.30793 + 2.26540i 0 −1.00000 0 −1.29196 − 2.23774i 0
505.3 0 −0.303142 + 0.525058i 0 −1.86373 + 3.22807i 0 −1.00000 0 1.31621 + 2.27974i 0
505.4 0 0.166004 − 0.287527i 0 0.118258 − 0.204828i 0 −1.00000 0 1.44489 + 2.50262i 0
505.5 0 0.640751 − 1.10981i 0 1.43031 − 2.47737i 0 −1.00000 0 0.678876 + 1.17585i 0
505.6 0 1.35257 − 2.34271i 0 −1.52522 + 2.64175i 0 −1.00000 0 −2.15887 − 3.73928i 0
505.7 0 1.52128 − 2.63493i 0 −0.464512 + 0.804558i 0 −1.00000 0 −3.12858 − 5.41886i 0
729.1 0 −1.19594 − 2.07143i 0 2.11282 + 3.65951i 0 −1.00000 0 −1.36056 + 2.35656i 0
729.2 0 −1.18152 − 2.04645i 0 −1.30793 − 2.26540i 0 −1.00000 0 −1.29196 + 2.23774i 0
729.3 0 −0.303142 − 0.525058i 0 −1.86373 − 3.22807i 0 −1.00000 0 1.31621 − 2.27974i 0
729.4 0 0.166004 + 0.287527i 0 0.118258 + 0.204828i 0 −1.00000 0 1.44489 − 2.50262i 0
729.5 0 0.640751 + 1.10981i 0 1.43031 + 2.47737i 0 −1.00000 0 0.678876 − 1.17585i 0
729.6 0 1.35257 + 2.34271i 0 −1.52522 − 2.64175i 0 −1.00000 0 −2.15887 + 3.73928i 0
729.7 0 1.52128 + 2.63493i 0 −0.464512 − 0.804558i 0 −1.00000 0 −3.12858 + 5.41886i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 505.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1064.2.r.i ✓ 14
19.c even 3 1 inner 1064.2.r.i ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.r.i ✓ 14 1.a even 1 1 trivial
1064.2.r.i ✓ 14 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1064, [\chi])\):

\( T_{3}^{14} - 2 T_{3}^{13} + 17 T_{3}^{12} - 16 T_{3}^{11} + 164 T_{3}^{10} - 139 T_{3}^{9} + 896 T_{3}^{8} + \cdots + 144 \) Copy content Toggle raw display
\( T_{11}^{7} + 7T_{11}^{6} - 28T_{11}^{5} - 265T_{11}^{4} - 122T_{11}^{3} + 1880T_{11}^{2} + 2764T_{11} - 96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 2 T^{13} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( T^{14} + 3 T^{13} + \cdots + 6241 \) Copy content Toggle raw display
$7$ \( (T + 1)^{14} \) Copy content Toggle raw display
$11$ \( (T^{7} + 7 T^{6} - 28 T^{5} + \cdots - 96)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 4 T^{13} + \cdots + 95004009 \) Copy content Toggle raw display
$17$ \( T^{14} + 2 T^{13} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 893871739 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 607967649 \) Copy content Toggle raw display
$29$ \( T^{14} + 21 T^{13} + \cdots + 46949904 \) Copy content Toggle raw display
$31$ \( (T^{7} + 7 T^{6} - 28 T^{5} + \cdots - 96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} - 13 T^{6} + \cdots - 27488)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 693082890256 \) Copy content Toggle raw display
$43$ \( T^{14} - 7 T^{13} + \cdots + 2521744 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 727704576 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 307371024 \) Copy content Toggle raw display
$59$ \( T^{14} - 18 T^{13} + \cdots + 13549761 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 669636165969 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 251233517824 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 565953785401 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 24941253184 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 477684736 \) Copy content Toggle raw display
$83$ \( (T^{7} - 13 T^{6} + \cdots - 556852)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 231876919296 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 44306882064 \) Copy content Toggle raw display
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