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Properties

Label 912.4.k.d
Level $912$
Weight $4$
Character orbit 912.k
Analytic conductor $53.810$
Analytic rank $0$
Dimension $20$
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] = [912,4,Mod(607,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.607");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 912.k (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: \(53.8097419252\)
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: \( x^{20} - 4 x^{19} + 674 x^{18} - 3288 x^{17} + 315871 x^{16} - 1436972 x^{15} + 72752722 x^{14} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{51}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + 3 q^{3} + \beta_{2} q^{5} + \beta_{10} q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + \beta_{2} q^{5} + \beta_{10} q^{7} + 9 q^{9} - \beta_{3} q^{11} + (\beta_{14} - \beta_{5}) q^{13} + 3 \beta_{2} q^{15} + (\beta_{8} + \beta_{2} + 6) q^{17} + ( - \beta_{13} - 2) q^{19} + 3 \beta_{10} q^{21} + (\beta_{17} - \beta_{10} + \cdots + \beta_{3}) q^{23}+ \cdots - 9 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 60 q^{3} + 8 q^{5} + 180 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 60 q^{3} + 8 q^{5} + 180 q^{9} + 24 q^{15} + 128 q^{17} - 36 q^{19} + 164 q^{25} + 540 q^{27} - 136 q^{31} + 72 q^{45} - 1436 q^{49} + 384 q^{51} - 108 q^{57} - 1120 q^{59} - 192 q^{61} + 1368 q^{67} - 640 q^{71} - 216 q^{73} + 492 q^{75} - 16 q^{77} + 1472 q^{79} + 1620 q^{81} + 2576 q^{85} - 736 q^{91} - 408 q^{93} - 248 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 674 x^{18} - 3288 x^{17} + 315871 x^{16} - 1436972 x^{15} + 72752722 x^{14} + \cdots + 33\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 68\!\cdots\!53 \nu^{19} + \cdots - 17\!\cdots\!00 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 45\!\cdots\!19 \nu^{19} + \cdots - 13\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!31 \nu^{19} + \cdots - 14\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 87\!\cdots\!41 \nu^{19} + \cdots + 10\!\cdots\!00 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23\!\cdots\!14 \nu^{19} + \cdots - 25\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 55\!\cdots\!93 \nu^{19} + \cdots - 97\!\cdots\!00 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34\!\cdots\!74 \nu^{19} + \cdots + 53\!\cdots\!00 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!98 \nu^{19} + \cdots - 11\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 36\!\cdots\!48 \nu^{19} + \cdots - 28\!\cdots\!00 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 88\!\cdots\!57 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!91 \nu^{19} + \cdots - 48\!\cdots\!00 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 40\!\cdots\!61 \nu^{19} + \cdots + 25\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 77\!\cdots\!07 \nu^{19} + \cdots - 25\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!19 \nu^{19} + \cdots + 11\!\cdots\!00 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 50\!\cdots\!61 \nu^{19} + \cdots + 29\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 60\!\cdots\!92 \nu^{19} + \cdots + 80\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 28\!\cdots\!96 \nu^{19} + \cdots - 26\!\cdots\!00 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 15\!\cdots\!43 \nu^{19} + \cdots - 83\!\cdots\!00 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 40\!\cdots\!98 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} - \beta_{10} + \beta_{5} + 8\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4 \beta_{18} - 8 \beta_{15} - 4 \beta_{14} - 8 \beta_{13} + \beta_{12} - 8 \beta_{11} + 11 \beta_{10} + \cdots - 1072 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{16} + 3 \beta_{15} - 3 \beta_{13} + 3 \beta_{9} - 9 \beta_{8} + 3 \beta_{7} - \beta_{6} + \cdots + 194 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 24 \beta_{19} - 1204 \beta_{18} + 456 \beta_{17} - 280 \beta_{16} + 3680 \beta_{15} + 3484 \beta_{14} + \cdots - 280120 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 232 \beta_{19} - 22068 \beta_{18} - 19200 \beta_{17} + 10448 \beta_{16} - 59744 \beta_{15} + \cdots - 1130024 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1605 \beta_{19} + 15394 \beta_{16} - 18801 \beta_{15} + 18801 \beta_{13} + 16846 \beta_{9} + \cdots + 11468888 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 41720 \beta_{19} + 7377444 \beta_{18} + 7318128 \beta_{17} + 3831016 \beta_{16} + 18756392 \beta_{15} + \cdots - 602057184 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4940664 \beta_{19} + 151476052 \beta_{18} - 149374440 \beta_{17} - 43175976 \beta_{16} + \cdots - 32897752552 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 255207 \beta_{19} - 168857868 \beta_{16} + 152252868 \beta_{15} - 152252868 \beta_{13} + \cdots + 35850822551 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1809779976 \beta_{19} - 52282122532 \beta_{18} + 63868211160 \beta_{17} - 13979501920 \beta_{16} + \cdots - 12283585275568 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5927999912 \beta_{19} - 785363824260 \beta_{18} - 1147424843856 \beta_{17} + 471476345912 \beta_{16} + \cdots - 129336215881904 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 83248710363 \beta_{19} + 539313406519 \beta_{16} - 1039343544330 \beta_{15} + 1039343544330 \beta_{13} + \cdots + 585622869346115 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 3777672039256 \beta_{19} + 252439227898740 \beta_{18} + 464867713151040 \beta_{17} + \cdots - 56\!\cdots\!72 ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 248450897796840 \beta_{19} + \cdots - 18\!\cdots\!08 ) / 16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 240130122571977 \beta_{19} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 93\!\cdots\!04 \beta_{19} + \cdots - 70\!\cdots\!24 ) / 16 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 89\!\cdots\!48 \beta_{19} + \cdots - 10\!\cdots\!08 ) / 16 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 44\!\cdots\!53 \beta_{19} + \cdots + 34\!\cdots\!46 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 39\!\cdots\!92 \beta_{19} + \cdots - 42\!\cdots\!72 ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1
−10.0465 + 17.4010i
−10.0465 − 17.4010i
−6.15578 + 10.6621i
−6.15578 − 10.6621i
−3.91808 − 6.78632i
−3.91808 + 6.78632i
−3.24614 − 5.62248i
−3.24614 + 5.62248i
−0.484035 − 0.838373i
−0.484035 + 0.838373i
1.26056 + 2.18335i
1.26056 − 2.18335i
4.19044 − 7.25805i
4.19044 + 7.25805i
4.54519 − 7.87250i
4.54519 + 7.87250i
6.79497 − 11.7692i
6.79497 + 11.7692i
9.05938 + 15.6913i
9.05938 − 15.6913i
0 3.00000 0 −20.0930 0 − 6.95813i 0 9.00000 0
607.2 0 3.00000 0 −20.0930 0 6.95813i 0 9.00000 0
607.3 0 3.00000 0 −12.3116 0 − 31.5108i 0 9.00000 0
607.4 0 3.00000 0 −12.3116 0 31.5108i 0 9.00000 0
607.5 0 3.00000 0 −7.83617 0 − 10.4823i 0 9.00000 0
607.6 0 3.00000 0 −7.83617 0 10.4823i 0 9.00000 0
607.7 0 3.00000 0 −6.49228 0 − 5.92108i 0 9.00000 0
607.8 0 3.00000 0 −6.49228 0 5.92108i 0 9.00000 0
607.9 0 3.00000 0 −0.968069 0 − 28.0796i 0 9.00000 0
607.10 0 3.00000 0 −0.968069 0 28.0796i 0 9.00000 0
607.11 0 3.00000 0 2.52111 0 − 13.8757i 0 9.00000 0
607.12 0 3.00000 0 2.52111 0 13.8757i 0 9.00000 0
607.13 0 3.00000 0 8.38088 0 − 16.8956i 0 9.00000 0
607.14 0 3.00000 0 8.38088 0 16.8956i 0 9.00000 0
607.15 0 3.00000 0 9.09038 0 − 31.7709i 0 9.00000 0
607.16 0 3.00000 0 9.09038 0 31.7709i 0 9.00000 0
607.17 0 3.00000 0 13.5899 0 − 11.8822i 0 9.00000 0
607.18 0 3.00000 0 13.5899 0 11.8822i 0 9.00000 0
607.19 0 3.00000 0 18.1188 0 − 23.3385i 0 9.00000 0
607.20 0 3.00000 0 18.1188 0 23.3385i 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.4.k.d yes 20
4.b odd 2 1 912.4.k.c ✓ 20
19.b odd 2 1 912.4.k.c ✓ 20
76.d even 2 1 inner 912.4.k.d yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.4.k.c ✓ 20 4.b odd 2 1
912.4.k.c ✓ 20 19.b odd 2 1
912.4.k.d yes 20 1.a even 1 1 trivial
912.4.k.d yes 20 76.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 \(S_{4}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{10} - 4 T_{5}^{9} - 658 T_{5}^{8} + 2960 T_{5}^{7} + 128933 T_{5}^{6} - 520756 T_{5}^{5} + \cdots - 576201600 \) Copy content Toggle raw display
\( T_{31}^{10} + 68 T_{31}^{9} - 126852 T_{31}^{8} - 9384224 T_{31}^{7} + 5533653008 T_{31}^{6} + \cdots - 17\!\cdots\!60 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T - 3)^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} - 4 T^{9} + \cdots - 576201600)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 62\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 27\!\cdots\!96)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 17\!\cdots\!60)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 12\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots - 48\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 75\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 87\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 96\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 78\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
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