Properties

Label 2808.2.c.f
Level $2808$
Weight $2$
Character orbit 2808.c
Analytic conductor $22.422$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2808,2,Mod(649,2808)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2808, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2808.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2808.c (of order \(2\), degree \(1\), 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: \(22.4219928876\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 44x^{12} + 708x^{10} + 5026x^{8} + 15252x^{6} + 19688x^{4} + 8857x^{2} + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + \beta_{4} q^{7} + \beta_{7} q^{11} - \beta_{10} q^{13} + ( - \beta_{8} + 1) q^{17} + ( - \beta_{3} + \beta_1) q^{19} + ( - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - 1) q^{23} + ( - \beta_{13} + \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - 1) q^{25} + ( - \beta_{9} + 1) q^{29} + ( - \beta_{7} - \beta_{2}) q^{31} + (\beta_{13} + \beta_{10} - \beta_{8} + \beta_{6} + 1) q^{35} + (\beta_{7} - \beta_{5} + \beta_{3} - \beta_{2}) q^{37} + ( - \beta_{13} + \beta_{10} + \beta_{7} + \beta_{3} - \beta_{2}) q^{41} + (\beta_{12} + \beta_{11} + \beta_{10} + \beta_{6} - 1) q^{43} + (\beta_{13} - \beta_{10} - \beta_{7} - \beta_{5} - 2 \beta_{4} - \beta_{3}) q^{47} + ( - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - 1) q^{49} + ( - \beta_{11} - \beta_{10} - \beta_{6}) q^{53} + ( - \beta_{13} - \beta_{12} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - 1) q^{55} + ( - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2}) q^{59} + ( - \beta_{13} - \beta_{12} - \beta_{10} + \beta_{9} + 2 \beta_{8} - 1) q^{61} + ( - \beta_{13} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 1) q^{65}+ \cdots + ( - \beta_{13} + \beta_{10} - \beta_{7} + 2 \beta_{4} - \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 16 q^{17} - 8 q^{23} - 18 q^{25} + 14 q^{29} + 16 q^{35} - 12 q^{43} - 10 q^{49} - 4 q^{53} - 4 q^{55} - 16 q^{61} + 8 q^{65} - 34 q^{79} - 32 q^{91} - 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 44x^{12} + 708x^{10} + 5026x^{8} + 15252x^{6} + 19688x^{4} + 8857x^{2} + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 447 \nu^{13} + 20043 \nu^{11} + 332228 \nu^{9} + 2474986 \nu^{7} + 8071279 \nu^{5} + 10529392 \nu^{3} + 5374061 \nu ) / 365370 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1891 \nu^{13} - 82420 \nu^{11} - 1305636 \nu^{9} - 9076878 \nu^{7} - 27680780 \nu^{5} - 44611592 \nu^{3} - 28927107 \nu ) / 1461480 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 447 \nu^{13} - 20043 \nu^{11} - 332228 \nu^{9} - 2474986 \nu^{7} - 8071279 \nu^{5} - 10346707 \nu^{3} - 3181841 \nu ) / 182685 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 571 \nu^{13} - 24050 \nu^{11} - 359544 \nu^{9} - 2213668 \nu^{7} - 4828134 \nu^{5} - 3826544 \nu^{3} - 1317985 \nu ) / 146148 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1789 \nu^{12} + 77356 \nu^{10} + 1211271 \nu^{8} + 8183937 \nu^{6} + 22200188 \nu^{4} + 21860909 \nu^{2} + 3105852 ) / 365370 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3263 \nu^{13} - 140506 \nu^{11} - 2175180 \nu^{9} - 14274734 \nu^{7} - 35832678 \nu^{5} - 31215532 \nu^{3} - 4283015 \nu ) / 730740 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2408 \nu^{12} - 100207 \nu^{10} - 1465492 \nu^{8} - 8593354 \nu^{6} - 16069391 \nu^{4} - 8603328 \nu^{2} - 1581234 ) / 365370 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4718 \nu^{12} + 196265 \nu^{10} + 2877573 \nu^{8} + 17041614 \nu^{6} + 32878795 \nu^{4} + 16171381 \nu^{2} + 103356 ) / 365370 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10763 \nu^{13} - 9894 \nu^{12} + 455546 \nu^{11} - 418134 \nu^{10} + 6864250 \nu^{9} - 6266982 \nu^{8} + 42879344 \nu^{7} - 38569122 \nu^{6} + 96229278 \nu^{5} + \cdots - 9075024 ) / 1461480 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10763 \nu^{13} + 11002 \nu^{12} - 455546 \nu^{11} + 464546 \nu^{10} - 6864250 \nu^{9} + 6966794 \nu^{8} - 42879344 \nu^{7} + 43003286 \nu^{6} + \cdots + 7500528 ) / 1461480 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5468 \nu^{12} - 227769 \nu^{10} - 3334301 \nu^{8} - 19548884 \nu^{6} - 35971137 \nu^{4} - 14251535 \nu^{2} + 2238438 ) / 365370 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10763 \nu^{13} - 9894 \nu^{12} - 455546 \nu^{11} - 418134 \nu^{10} - 6864250 \nu^{9} - 6266982 \nu^{8} - 42879344 \nu^{7} - 38569122 \nu^{6} + \cdots - 9075024 ) / 1461480 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} + \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 2\beta_{2} - 12\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 15\beta_{13} - 14\beta_{12} + 2\beta_{11} + 17\beta_{10} - 16\beta_{9} - 19\beta_{8} + 15\beta_{6} + 69 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{13} - 4\beta_{10} - 3\beta_{7} - 12\beta_{5} - 17\beta_{4} + 2\beta_{3} - 33\beta_{2} + 159\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -212\beta_{13} + 195\beta_{12} - 63\beta_{11} - 275\beta_{10} + 239\beta_{9} + 297\beta_{8} - 211\beta_{6} - 884 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 108 \beta_{13} + 108 \beta_{10} + 82 \beta_{7} + 326 \beta_{5} + 264 \beta_{4} - 72 \beta_{3} + 492 \beta_{2} - 2153 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3007 \beta_{13} - 2727 \beta_{12} + 1342 \beta_{11} + 4349 \beta_{10} - 3517 \beta_{9} - 4507 \beta_{8} + 2981 \beta_{6} + 11646 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2158 \beta_{13} - 2158 \beta_{10} - 1556 \beta_{7} - 6610 \beta_{5} - 4093 \beta_{4} + 1664 \beta_{3} - 7234 \beta_{2} + 29490 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 43083 \beta_{13} + 38280 \beta_{12} - 24600 \beta_{11} - 67683 \beta_{10} + 51386 \beta_{9} + 67875 \beta_{8} - 42481 \beta_{6} - 155747 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 38308 \beta_{13} + 38308 \beta_{10} + 25511 \beta_{7} + 119476 \beta_{5} + 63581 \beta_{4} - 32594 \beta_{3} + 106155 \beta_{2} - 407729 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 622856 \beta_{13} - 539395 \beta_{12} + 418457 \beta_{11} + 1041313 \beta_{10} - 747677 \beta_{9} - 1018461 \beta_{8} + 610059 \beta_{6} + 2108636 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 639536 \beta_{13} - 639536 \beta_{10} - 387260 \beta_{7} - 2032708 \beta_{5} - 987156 \beta_{4} + 587272 \beta_{3} - 1557856 \beta_{2} + 5684561 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(1081\) \(1405\) \(2081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
3.85219i
3.58975i
3.33574i
1.50472i
1.43413i
0.865229i
0.278658i
0.278658i
0.865229i
1.43413i
1.50472i
3.33574i
3.58975i
3.85219i
0 0 0 3.85219i 0 2.37854i 0 0 0
649.2 0 0 0 3.58975i 0 1.30066i 0 0 0
649.3 0 0 0 3.33574i 0 2.76871i 0 0 0
649.4 0 0 0 1.50472i 0 0.0279770i 0 0 0
649.5 0 0 0 1.43413i 0 1.94535i 0 0 0
649.6 0 0 0 0.865229i 0 4.63756i 0 0 0
649.7 0 0 0 0.278658i 0 3.70040i 0 0 0
649.8 0 0 0 0.278658i 0 3.70040i 0 0 0
649.9 0 0 0 0.865229i 0 4.63756i 0 0 0
649.10 0 0 0 1.43413i 0 1.94535i 0 0 0
649.11 0 0 0 1.50472i 0 0.0279770i 0 0 0
649.12 0 0 0 3.33574i 0 2.76871i 0 0 0
649.13 0 0 0 3.58975i 0 1.30066i 0 0 0
649.14 0 0 0 3.85219i 0 2.37854i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2808.2.c.f yes 14
3.b odd 2 1 2808.2.c.e 14
13.b even 2 1 inner 2808.2.c.f yes 14
39.d odd 2 1 2808.2.c.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2808.2.c.e 14 3.b odd 2 1
2808.2.c.e 14 39.d odd 2 1
2808.2.c.f yes 14 1.a even 1 1 trivial
2808.2.c.f yes 14 13.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 \(S_{2}^{\mathrm{new}}(2808, [\chi])\):

\( T_{5}^{14} + 44T_{5}^{12} + 708T_{5}^{10} + 5026T_{5}^{8} + 15252T_{5}^{6} + 19688T_{5}^{4} + 8857T_{5}^{2} + 576 \) Copy content Toggle raw display
\( T_{17}^{7} - 8T_{17}^{6} - 7T_{17}^{5} + 224T_{17}^{4} - 701T_{17}^{3} + 742T_{17}^{2} - 198T_{17} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 44 T^{12} + 708 T^{10} + \cdots + 576 \) Copy content Toggle raw display
$7$ \( T^{14} + 54 T^{12} + 1079 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{14} + 96 T^{12} + 3236 T^{10} + \cdots + 145924 \) Copy content Toggle raw display
$13$ \( T^{14} - 3 T^{12} - 96 T^{11} + \cdots + 62748517 \) Copy content Toggle raw display
$17$ \( (T^{7} - 8 T^{6} - 7 T^{5} + 224 T^{4} + \cdots - 12)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + 171 T^{12} + \cdots + 193432464 \) Copy content Toggle raw display
$23$ \( (T^{7} + 4 T^{6} - 95 T^{5} - 96 T^{4} + \cdots + 5172)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} - 7 T^{6} - 66 T^{5} + 306 T^{4} + \cdots + 2988)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + 258 T^{12} + \cdots + 12181095424 \) Copy content Toggle raw display
$37$ \( T^{14} + 281 T^{12} + 28062 T^{10} + \cdots + 1507984 \) Copy content Toggle raw display
$41$ \( T^{14} + 341 T^{12} + \cdots + 43369728516 \) Copy content Toggle raw display
$43$ \( (T^{7} + 6 T^{6} - 211 T^{5} - 1070 T^{4} + \cdots - 49904)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + 423 T^{12} + \cdots + 5319076624 \) Copy content Toggle raw display
$53$ \( (T^{7} + 2 T^{6} - 106 T^{5} - 578 T^{4} + \cdots + 34582)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + 312 T^{12} + \cdots + 961496064 \) Copy content Toggle raw display
$61$ \( (T^{7} + 8 T^{6} - 244 T^{5} - 2592 T^{4} + \cdots - 2656)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 522 T^{12} + \cdots + 25423664704 \) Copy content Toggle raw display
$71$ \( T^{14} + 541 T^{12} + \cdots + 339738624 \) Copy content Toggle raw display
$73$ \( T^{14} + 324 T^{12} + \cdots + 1352327076 \) Copy content Toggle raw display
$79$ \( (T^{7} + 17 T^{6} - 243 T^{5} + \cdots - 9654192)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 918 T^{12} + \cdots + 80035317492516 \) Copy content Toggle raw display
$89$ \( T^{14} + 525 T^{12} + \cdots + 21233664 \) Copy content Toggle raw display
$97$ \( T^{14} + 418 T^{12} + \cdots + 870354653184 \) Copy content Toggle raw display
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