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Properties

Label 3332.2.b.f
Level $3332$
Weight $2$
Character orbit 3332.b
Analytic conductor $26.606$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,2,Mod(2549,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3332.2549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3332.b (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: \(26.6061539535\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 28 x^{18} + 462 x^{16} + 3896 x^{14} + 26475 x^{12} + 79940 x^{10} + 417406 x^{8} + \cdots + 32285124 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{3} - \beta_{14} q^{5} + (\beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{3} - \beta_{14} q^{5} + (\beta_{3} - 2) q^{9} + \beta_{4} q^{11} - \beta_{16} q^{13} + ( - \beta_{8} - \beta_{7} - \beta_{3} + 1) q^{15} + \beta_{18} q^{17} + \beta_{19} q^{19} + (\beta_{6} - \beta_1) q^{23} - \beta_{7} q^{25} + (\beta_{17} + \beta_{15} + \cdots - 2 \beta_{10}) q^{27}+ \cdots + (\beta_{6} - \beta_{5} - 2 \beta_{4} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 36 q^{9} + 8 q^{15} - 4 q^{25} - 40 q^{43} + 24 q^{51} - 8 q^{53} - 80 q^{67} + 108 q^{81} - 12 q^{85} - 64 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 28 x^{18} + 462 x^{16} + 3896 x^{14} + 26475 x^{12} + 79940 x^{10} + 417406 x^{8} + \cdots + 32285124 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 65\!\cdots\!29 \nu^{18} + \cdots - 54\!\cdots\!38 ) / 47\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 41\!\cdots\!17 \nu^{18} + \cdots + 10\!\cdots\!88 ) / 12\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 41\!\cdots\!17 \nu^{18} + \cdots + 29\!\cdots\!42 ) / 62\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 69\!\cdots\!14 \nu^{18} + \cdots - 37\!\cdots\!72 ) / 70\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 34\!\cdots\!49 \nu^{18} + \cdots + 23\!\cdots\!98 ) / 35\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22\!\cdots\!08 \nu^{18} + \cdots + 22\!\cdots\!66 ) / 70\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 76\!\cdots\!42 \nu^{18} + \cdots - 55\!\cdots\!14 ) / 11\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37\!\cdots\!37 \nu^{18} + \cdots + 68\!\cdots\!78 ) / 35\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 79\!\cdots\!65 \nu^{18} + \cdots - 55\!\cdots\!82 ) / 35\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!81 \nu^{19} + \cdots + 87\!\cdots\!74 \nu ) / 35\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!81 \nu^{19} + \cdots + 34\!\cdots\!02 \nu ) / 35\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 21\!\cdots\!93 \nu^{19} + \cdots + 18\!\cdots\!58 \nu ) / 26\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 27\!\cdots\!71 \nu^{19} + \cdots - 10\!\cdots\!06 \nu ) / 26\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\!\cdots\!23 \nu^{19} + \cdots - 17\!\cdots\!06 \nu ) / 67\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 70\!\cdots\!61 \nu^{19} + \cdots - 16\!\cdots\!52 \nu ) / 33\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 36\!\cdots\!81 \nu^{19} + \cdots - 31\!\cdots\!44 \nu ) / 13\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 13\!\cdots\!67 \nu^{19} + \cdots + 12\!\cdots\!86 \nu ) / 33\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 10\!\cdots\!57 \nu^{19} + \cdots + 12\!\cdots\!74 \nu ) / 26\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 10\!\cdots\!72 \nu^{19} + \cdots + 90\!\cdots\!28 \nu ) / 13\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{11} + \beta_{10} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 2\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{15} - 2\beta_{14} + 3\beta_{13} + 3\beta_{12} - 10\beta_{11} - 2\beta_{10} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - 2\beta_{8} - \beta_{7} - 8\beta_{6} + 4\beta_{5} + 8\beta_{4} - \beta_{3} + 20\beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{19} + 7 \beta_{18} + 7 \beta_{17} - 20 \beta_{16} + 4 \beta_{15} - 2 \beta_{14} + \cdots - 63 \beta_{10} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 14 \beta_{9} - 22 \beta_{8} - 7 \beta_{7} + 76 \beta_{6} - 110 \beta_{5} - 112 \beta_{4} - 173 \beta_{3} + \cdots + 560 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 84 \beta_{19} - 46 \beta_{18} - 249 \beta_{17} + 252 \beta_{16} - 328 \beta_{15} + \cdots + 1354 \beta_{10} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 412 \beta_{9} + 1031 \beta_{8} + 644 \beta_{7} - 48 \beta_{6} + 464 \beta_{5} + 128 \beta_{4} + \cdots - 8802 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 696 \beta_{19} - 1316 \beta_{18} + 3553 \beta_{17} + 1086 \beta_{16} + 4844 \beta_{15} + \cdots - 16082 \beta_{10} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4148 \beta_{9} - 12904 \beta_{8} - 10151 \beta_{7} - 13652 \beta_{6} + 17010 \beta_{5} + 18544 \beta_{4} + \cdots + 77619 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 26840 \beta_{19} + 36568 \beta_{18} - 21824 \beta_{17} - 73480 \beta_{16} - 27981 \beta_{15} + \cdots + 83088 \beta_{10} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1141 \beta_{9} + 24011 \beta_{8} + 34101 \beta_{7} + 277728 \beta_{6} - 405524 \beta_{5} - 366008 \beta_{4} + \cdots + 42603 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 368290 \beta_{19} - 386587 \beta_{18} - 259600 \beta_{17} + 1136798 \beta_{16} + \cdots + 1306405 \beta_{10} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 723978 \beta_{9} + 2055750 \beta_{8} + 1499286 \beta_{7} - 2785088 \beta_{6} + 4270144 \beta_{5} + \cdots - 15320151 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2115692 \beta_{19} - 230966 \beta_{18} + 9189621 \beta_{17} - 7213876 \beta_{16} + \cdots - 41601658 \beta_{10} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 14088293 \beta_{9} - 43242538 \beta_{8} - 34270061 \beta_{7} + 1341792 \beta_{6} - 3756528 \beta_{5} + \cdots + 295161931 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 27223018 \beta_{19} + 78483663 \beta_{18} - 129044989 \beta_{17} - 77630228 \beta_{16} + \cdots + 573928193 \beta_{10} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 138582810 \beta_{9} + 436136106 \beta_{8} + 354280797 \beta_{7} + 499658516 \beta_{6} + \cdots - 2908467828 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 925478980 \beta_{19} - 1428561706 \beta_{18} + 770363279 \beta_{17} + 2832075020 \beta_{16} + \cdots - 3389926142 \beta_{10} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2549.1
−1.41421 − 3.42554i
1.41421 − 3.42554i
−1.41421 − 2.53485i
1.41421 − 2.53485i
1.41421 − 2.04724i
−1.41421 − 2.04724i
1.41421 − 1.02037i
−1.41421 − 1.02037i
−1.41421 − 0.779660i
1.41421 − 0.779660i
1.41421 + 0.779660i
−1.41421 + 0.779660i
−1.41421 + 1.02037i
1.41421 + 1.02037i
−1.41421 + 2.04724i
1.41421 + 2.04724i
1.41421 + 2.53485i
−1.41421 + 2.53485i
1.41421 + 3.42554i
−1.41421 + 3.42554i
0 − 3.42554i 0 2.65330i 0 0 0 −8.73429 0
2549.2 0 − 3.42554i 0 2.65330i 0 0 0 −8.73429 0
2549.3 0 − 2.53485i 0 − 3.38990i 0 0 0 −3.42547 0
2549.4 0 − 2.53485i 0 − 3.38990i 0 0 0 −3.42547 0
2549.5 0 − 2.04724i 0 − 0.203168i 0 0 0 −1.19120 0
2549.6 0 − 2.04724i 0 − 0.203168i 0 0 0 −1.19120 0
2549.7 0 − 1.02037i 0 2.57140i 0 0 0 1.95884 0
2549.8 0 − 1.02037i 0 2.57140i 0 0 0 1.95884 0
2549.9 0 − 0.779660i 0 − 0.902896i 0 0 0 2.39213 0
2549.10 0 − 0.779660i 0 − 0.902896i 0 0 0 2.39213 0
2549.11 0 0.779660i 0 0.902896i 0 0 0 2.39213 0
2549.12 0 0.779660i 0 0.902896i 0 0 0 2.39213 0
2549.13 0 1.02037i 0 − 2.57140i 0 0 0 1.95884 0
2549.14 0 1.02037i 0 − 2.57140i 0 0 0 1.95884 0
2549.15 0 2.04724i 0 0.203168i 0 0 0 −1.19120 0
2549.16 0 2.04724i 0 0.203168i 0 0 0 −1.19120 0
2549.17 0 2.53485i 0 3.38990i 0 0 0 −3.42547 0
2549.18 0 2.53485i 0 3.38990i 0 0 0 −3.42547 0
2549.19 0 3.42554i 0 − 2.65330i 0 0 0 −8.73429 0
2549.20 0 3.42554i 0 − 2.65330i 0 0 0 −8.73429 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2549.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
17.b even 2 1 inner
119.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.2.b.f ✓ 20
7.b odd 2 1 inner 3332.2.b.f ✓ 20
17.b even 2 1 inner 3332.2.b.f ✓ 20
119.d odd 2 1 inner 3332.2.b.f ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.2.b.f ✓ 20 1.a even 1 1 trivial
3332.2.b.f ✓ 20 7.b odd 2 1 inner
3332.2.b.f ✓ 20 17.b even 2 1 inner
3332.2.b.f ✓ 20 119.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3332, [\chi])\):

\( T_{3}^{10} + 24T_{3}^{8} + 189T_{3}^{6} + 580T_{3}^{4} + 617T_{3}^{2} + 200 \) Copy content Toggle raw display
\( T_{13}^{10} - 60T_{13}^{8} + 1116T_{13}^{6} - 8336T_{13}^{4} + 22032T_{13}^{2} - 2592 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + 24 T^{8} + \cdots + 200)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + 26 T^{8} + \cdots + 18)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( (T^{10} + 64 T^{8} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} - 60 T^{8} + \cdots - 2592)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 2015993900449 \) Copy content Toggle raw display
$19$ \( (T^{10} - 158 T^{8} + \cdots - 10396800)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + 158 T^{8} + \cdots + 614656)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 96 T^{8} + \cdots + 23104)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 180 T^{8} + \cdots + 749088)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 286 T^{8} + \cdots + 112021056)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 270 T^{8} + \cdots + 78100002)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 10 T^{4} + \cdots + 9996)^{4} \) Copy content Toggle raw display
$47$ \( (T^{10} - 224 T^{8} + \cdots - 53747712)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + 2 T^{4} + \cdots + 23850)^{4} \) Copy content Toggle raw display
$59$ \( (T^{10} - 244 T^{8} + \cdots - 508032)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 382 T^{8} + \cdots + 374777442)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + 20 T^{4} + \cdots - 23256)^{4} \) Copy content Toggle raw display
$71$ \( (T^{10} + 294 T^{8} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 342 T^{8} + \cdots + 94311378)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 448 T^{8} + \cdots + 418120704)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} - 558 T^{8} + \cdots - 3333197952)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} - 418 T^{8} + \cdots - 131220000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 602 T^{8} + \cdots + 2294082)^{2} \) Copy content Toggle raw display
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