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Properties

Label 3330.2.e.h
Level $3330$
Weight $2$
Character orbit 3330.e
Analytic conductor $26.590$
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] = [3330,2,Mod(739,3330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3330, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3330.739");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.e (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: \(26.5901838731\)
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: \( x^{20} - 18 x^{18} + 293 x^{16} - 2544 x^{14} + 11440 x^{12} - 168560 x^{10} + 1259572 x^{8} + \cdots + 71098624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{11} \)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - \beta_{10} q^{5} - \beta_{11} q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - \beta_{10} q^{5} - \beta_{11} q^{7} + q^{8} - \beta_{10} q^{10} - \beta_{15} q^{11} + \beta_{12} q^{13} - \beta_{11} q^{14} + q^{16} + (\beta_{3} - 1) q^{17} + \beta_{5} q^{19} - \beta_{10} q^{20} - \beta_{15} q^{22} + ( - \beta_{4} + 1) q^{23} + ( - \beta_{13} + \beta_{4}) q^{25} + \beta_{12} q^{26} - \beta_{11} q^{28} + (\beta_{14} - \beta_{11}) q^{29} + (\beta_{6} + \beta_1) q^{31} + q^{32} + (\beta_{3} - 1) q^{34} + (\beta_{14} + \beta_{8} - \beta_{3}) q^{35} + ( - \beta_{19} - \beta_{13} + \cdots + \beta_{9}) q^{37}+ \cdots + (\beta_{8} - \beta_{7} - \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{2} + 20 q^{4} + 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{2} + 20 q^{4} + 20 q^{8} + 20 q^{16} - 12 q^{17} + 16 q^{23} + 4 q^{25} + 20 q^{32} - 12 q^{34} - 12 q^{35} + 16 q^{46} - 16 q^{49} + 4 q^{50} + 20 q^{64} - 12 q^{68} - 12 q^{70} - 8 q^{85} + 16 q^{92} - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 18 x^{18} + 293 x^{16} - 2544 x^{14} + 11440 x^{12} - 168560 x^{10} + 1259572 x^{8} + \cdots + 71098624 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 48\!\cdots\!95 \nu^{18} + \cdots + 87\!\cdots\!36 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 65\!\cdots\!51 \nu^{18} + \cdots - 29\!\cdots\!04 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 74\!\cdots\!55 \nu^{18} + \cdots - 52\!\cdots\!44 ) / 78\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40\!\cdots\!69 \nu^{18} + \cdots - 19\!\cdots\!12 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 88\!\cdots\!05 \nu^{18} + \cdots - 75\!\cdots\!64 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!59 \nu^{18} + \cdots - 91\!\cdots\!96 ) / 46\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 54\!\cdots\!39 \nu^{19} + \cdots + 50\!\cdots\!72 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 54\!\cdots\!39 \nu^{19} + \cdots - 50\!\cdots\!72 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!63 \nu^{19} + \cdots + 14\!\cdots\!20 ) / 49\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12\!\cdots\!63 \nu^{19} + \cdots + 14\!\cdots\!20 ) / 49\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 46\!\cdots\!11 \nu^{19} + \cdots + 13\!\cdots\!04 \nu ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23\!\cdots\!57 \nu^{19} + \cdots + 12\!\cdots\!16 \nu ) / 82\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 23\!\cdots\!57 \nu^{19} + \cdots + 37\!\cdots\!24 \nu ) / 82\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 21\!\cdots\!65 \nu^{19} + \cdots + 11\!\cdots\!48 \nu ) / 49\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 19\!\cdots\!39 \nu^{19} + \cdots + 77\!\cdots\!28 \nu ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 22\!\cdots\!37 \nu^{19} + \cdots - 20\!\cdots\!92 \nu ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 41\!\cdots\!05 \nu^{19} + \cdots - 51\!\cdots\!52 ) / 58\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 41\!\cdots\!05 \nu^{19} + \cdots - 51\!\cdots\!52 ) / 58\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 32\!\cdots\!41 \nu^{19} + \cdots + 11\!\cdots\!64 \nu ) / 24\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} + \beta_{12} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{18} + \beta_{17} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8 \beta_{19} - 3 \beta_{18} + 3 \beta_{17} + 2 \beta_{16} - 8 \beta_{15} - 5 \beta_{14} + \cdots - \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9 \beta_{18} + 9 \beta_{17} - 18 \beta_{10} - 18 \beta_{9} + 10 \beta_{8} - 10 \beta_{7} + 11 \beta_{6} + \cdots - 41 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 100 \beta_{19} - 67 \beta_{18} + 67 \beta_{17} + 76 \beta_{16} + 38 \beta_{15} - 57 \beta_{14} + \cdots - 27 \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 209 \beta_{18} - 209 \beta_{17} - 262 \beta_{10} - 262 \beta_{9} + 220 \beta_{8} - 220 \beta_{7} + \cdots - 285 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 476 \beta_{19} - 325 \beta_{18} + 325 \beta_{17} + 396 \beta_{16} + 162 \beta_{15} + 425 \beta_{14} + \cdots - 17 \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4335 \beta_{18} - 4335 \beta_{17} + 240 \beta_{10} + 240 \beta_{9} - 1600 \beta_{8} + 1600 \beta_{7} + \cdots + 5495 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 11520 \beta_{19} + 8165 \beta_{18} - 8165 \beta_{17} - 8692 \beta_{16} - 10754 \beta_{15} + \cdots + 2829 \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2765 \beta_{18} - 2765 \beta_{17} + 26880 \beta_{10} + 26880 \beta_{9} - 58126 \beta_{8} + \cdots + 202207 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 196932 \beta_{19} + 123727 \beta_{18} - 123727 \beta_{17} - 113736 \beta_{16} + \cdots - 26001 \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 784141 \beta_{18} + 784141 \beta_{17} + 78144 \beta_{10} + 78144 \beta_{9} - 203508 \beta_{8} + \cdots + 885551 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1183904 \beta_{19} - 781247 \beta_{18} + 781247 \beta_{17} + 775404 \beta_{16} + \cdots - 1154191 \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 6371087 \beta_{18} + 6371087 \beta_{17} - 4503156 \beta_{10} - 4503156 \beta_{9} + 9911082 \beta_{8} + \cdots - 36468785 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 51432356 \beta_{19} - 32695765 \beta_{18} + 32695765 \beta_{17} + 30668888 \beta_{16} + \cdots - 8025437 \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 118806215 \beta_{18} - 118806215 \beta_{17} - 59222672 \beta_{10} - 59222672 \beta_{9} + \cdots - 489092393 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 164956952 \beta_{19} - 109096035 \beta_{18} + 109096035 \beta_{17} + 104540036 \beta_{16} + \cdots + 124954269 \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 2465443397 \beta_{18} - 2465443397 \beta_{17} + 534857036 \beta_{10} + 534857036 \beta_{9} + \cdots + 3073784007 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 9525370804 \beta_{19} + 6051619719 \beta_{18} - 6051619719 \beta_{17} - 5824927808 \beta_{16} + \cdots + 2476936959 \beta_{7} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
739.1
1.03229 − 1.59315i
−1.03229 − 1.59315i
1.03229 + 1.59315i
−1.03229 + 1.59315i
−3.06888 − 2.24659i
3.06888 − 2.24659i
−3.06888 + 2.24659i
3.06888 + 2.24659i
1.16403 − 0.828995i
−1.16403 − 0.828995i
1.16403 + 0.828995i
−1.16403 + 0.828995i
1.41163 + 2.46145i
−1.41163 + 2.46145i
1.41163 − 2.46145i
−1.41163 − 2.46145i
−3.10945 + 0.410768i
3.10945 + 0.410768i
−3.10945 − 0.410768i
3.10945 − 0.410768i
1.00000 0 1.00000 −2.10395 − 0.757219i 0 1.70662i 1.00000 0 −2.10395 − 0.757219i
739.2 1.00000 0 1.00000 −2.10395 − 0.757219i 0 1.70662i 1.00000 0 −2.10395 − 0.757219i
739.3 1.00000 0 1.00000 −2.10395 + 0.757219i 0 − 1.70662i 1.00000 0 −2.10395 + 0.757219i
739.4 1.00000 0 1.00000 −2.10395 + 0.757219i 0 − 1.70662i 1.00000 0 −2.10395 + 0.757219i
739.5 1.00000 0 1.00000 −1.18460 − 1.89650i 0 − 4.56092i 1.00000 0 −1.18460 − 1.89650i
739.6 1.00000 0 1.00000 −1.18460 − 1.89650i 0 − 4.56092i 1.00000 0 −1.18460 − 1.89650i
739.7 1.00000 0 1.00000 −1.18460 + 1.89650i 0 4.56092i 1.00000 0 −1.18460 + 1.89650i
739.8 1.00000 0 1.00000 −1.18460 + 1.89650i 0 4.56092i 1.00000 0 −1.18460 + 1.89650i
739.9 1.00000 0 1.00000 −0.376096 − 2.20421i 0 2.07862i 1.00000 0 −0.376096 − 2.20421i
739.10 1.00000 0 1.00000 −0.376096 − 2.20421i 0 2.07862i 1.00000 0 −0.376096 − 2.20421i
739.11 1.00000 0 1.00000 −0.376096 + 2.20421i 0 − 2.07862i 1.00000 0 −0.376096 + 2.20421i
739.12 1.00000 0 1.00000 −0.376096 + 2.20421i 0 − 2.07862i 1.00000 0 −0.376096 + 2.20421i
739.13 1.00000 0 1.00000 1.43619 − 1.71387i 0 0.224501i 1.00000 0 1.43619 − 1.71387i
739.14 1.00000 0 1.00000 1.43619 − 1.71387i 0 0.224501i 1.00000 0 1.43619 − 1.71387i
739.15 1.00000 0 1.00000 1.43619 + 1.71387i 0 − 0.224501i 1.00000 0 1.43619 + 1.71387i
739.16 1.00000 0 1.00000 1.43619 + 1.71387i 0 − 0.224501i 1.00000 0 1.43619 + 1.71387i
739.17 1.00000 0 1.00000 2.22846 − 0.184329i 0 − 3.30370i 1.00000 0 2.22846 − 0.184329i
739.18 1.00000 0 1.00000 2.22846 − 0.184329i 0 − 3.30370i 1.00000 0 2.22846 − 0.184329i
739.19 1.00000 0 1.00000 2.22846 + 0.184329i 0 3.30370i 1.00000 0 2.22846 + 0.184329i
739.20 1.00000 0 1.00000 2.22846 + 0.184329i 0 3.30370i 1.00000 0 2.22846 + 0.184329i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 739.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
111.d odd 2 1 inner
185.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3330.2.e.h yes 20
3.b odd 2 1 3330.2.e.g ✓ 20
5.b even 2 1 3330.2.e.g ✓ 20
15.d odd 2 1 inner 3330.2.e.h yes 20
37.b even 2 1 3330.2.e.g ✓ 20
111.d odd 2 1 inner 3330.2.e.h yes 20
185.d even 2 1 inner 3330.2.e.h yes 20
555.b odd 2 1 3330.2.e.g ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3330.2.e.g ✓ 20 3.b odd 2 1
3330.2.e.g ✓ 20 5.b even 2 1
3330.2.e.g ✓ 20 37.b even 2 1
3330.2.e.g ✓ 20 555.b odd 2 1
3330.2.e.h yes 20 1.a even 1 1 trivial
3330.2.e.h yes 20 15.d odd 2 1 inner
3330.2.e.h yes 20 111.d odd 2 1 inner
3330.2.e.h yes 20 185.d 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 \(S_{2}^{\mathrm{new}}(3330, [\chi])\):

\( T_{7}^{10} + 39T_{7}^{8} + 471T_{7}^{6} + 2065T_{7}^{4} + 2960T_{7}^{2} + 144 \) Copy content Toggle raw display
\( T_{13}^{10} - 94T_{13}^{8} + 2905T_{13}^{6} - 33560T_{13}^{4} + 160160T_{13}^{2} - 268288 \) Copy content Toggle raw display
\( T_{17}^{5} + 3T_{17}^{4} - 47T_{17}^{3} - 45T_{17}^{2} + 392T_{17} - 180 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} - T^{8} + \cdots + 3125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 39 T^{8} + \cdots + 144)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} - 65 T^{8} + \cdots - 84888)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} - 94 T^{8} + \cdots - 268288)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} + 3 T^{4} + \cdots - 180)^{4} \) Copy content Toggle raw display
$19$ \( (T^{10} + 94 T^{8} + \cdots + 150912)^{2} \) Copy content Toggle raw display
$23$ \( (T^{5} - 4 T^{4} + \cdots - 384)^{4} \) Copy content Toggle raw display
$29$ \( (T^{10} + 125 T^{8} + \cdots + 565504)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 239 T^{8} + \cdots + 9658368)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 48\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( (T^{10} - 323 T^{8} + \cdots - 73041408)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} - 255 T^{8} + \cdots - 41920000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + 220 T^{8} + \cdots + 8870272)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 305 T^{8} + \cdots + 602143072)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 184 T^{8} + \cdots + 1763584)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 603 T^{8} + \cdots + 8334115200)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 412 T^{8} + \cdots + 35426304)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 452 T^{8} + \cdots - 782327808)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 330 T^{8} + \cdots + 2359296)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 528 T^{8} + \cdots + 440059392)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 442 T^{8} + \cdots + 67072)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 130 T^{8} + \cdots + 64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} - 495 T^{8} + \cdots - 4898607712)^{2} \) Copy content Toggle raw display
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