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Properties

Label 912.3.be.j
Level $912$
Weight $3$
Character orbit 912.be
Analytic conductor $24.850$
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] = [912,3,Mod(145,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.be (of order \(6\), degree \(2\), not 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: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 154 x^{18} - 24 x^{17} + 16374 x^{16} - 4328 x^{15} + 911836 x^{14} - 590088 x^{13} + \cdots + 338560000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{3} + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{15} - 1) q^{7} + 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{3} + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{15} - 1) q^{7} + 3 \beta_{3} q^{9} + (\beta_{19} - \beta_{2}) q^{11} + (\beta_{19} - \beta_{14} + \beta_{11} + \cdots + 1) q^{13}+ \cdots + (3 \beta_{19} + 3 \beta_{11} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 30 q^{3} - 20 q^{7} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 30 q^{3} - 20 q^{7} + 30 q^{9} + 8 q^{11} + 18 q^{13} + 8 q^{17} - 28 q^{19} - 30 q^{21} + 8 q^{23} - 58 q^{25} + 108 q^{29} + 12 q^{33} - 20 q^{35} + 36 q^{39} - 36 q^{41} + 2 q^{43} + 296 q^{49} + 24 q^{51} - 72 q^{53} - 216 q^{55} - 30 q^{57} - 72 q^{59} - 26 q^{61} - 30 q^{63} - 138 q^{67} + 204 q^{71} + 218 q^{73} - 8 q^{77} + 78 q^{79} - 90 q^{81} + 112 q^{83} + 224 q^{85} + 216 q^{87} - 432 q^{89} + 330 q^{91} - 126 q^{93} - 220 q^{95} + 132 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 154 x^{18} - 24 x^{17} + 16374 x^{16} - 4328 x^{15} + 911836 x^{14} - 590088 x^{13} + \cdots + 338560000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 37\!\cdots\!37 \nu^{19} + \cdots + 76\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26\!\cdots\!36 \nu^{19} + \cdots - 17\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!02 \nu^{19} + \cdots - 14\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 36\!\cdots\!53 \nu^{19} + \cdots - 79\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 39\!\cdots\!79 \nu^{19} + \cdots + 19\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\!\cdots\!57 \nu^{19} + \cdots + 39\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 60\!\cdots\!97 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!13 \nu^{19} + \cdots + 37\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17\!\cdots\!67 \nu^{19} + \cdots - 23\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 34\!\cdots\!61 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 25\!\cdots\!79 \nu^{19} + \cdots + 33\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16\!\cdots\!31 \nu^{19} + \cdots + 36\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 36\!\cdots\!69 \nu^{19} + \cdots + 43\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 89\!\cdots\!76 \nu^{19} + \cdots - 21\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 17\!\cdots\!03 \nu^{19} + \cdots - 12\!\cdots\!00 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 21\!\cdots\!28 \nu^{19} + \cdots + 49\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 26\!\cdots\!48 \nu^{19} + \cdots + 29\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 39\!\cdots\!69 \nu^{19} + \cdots - 20\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{19} + 2\beta_{17} - \beta_{15} + 2\beta_{14} - \beta_{12} - 3\beta_{11} + \beta_{9} + 33\beta_{3} - 31 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - 2 \beta_{18} + \beta_{17} - 2 \beta_{16} - 3 \beta_{15} - \beta_{14} + \beta_{12} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 90 \beta_{19} + 4 \beta_{18} - 69 \beta_{17} - \beta_{16} - 16 \beta_{15} - 68 \beta_{14} - 5 \beta_{13} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 218 \beta_{19} + 412 \beta_{18} - 369 \beta_{17} + 333 \beta_{15} + 184 \beta_{14} + 160 \beta_{13} + \cdots + 132 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3712 \beta_{19} + 518 \beta_{18} - 4948 \beta_{17} - 466 \beta_{16} + 6566 \beta_{15} - 4436 \beta_{14} + \cdots + 119506 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 8584 \beta_{19} - 17634 \beta_{18} + 17810 \beta_{17} + 6026 \beta_{16} + 7262 \beta_{15} + \cdots - 7774 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 211170 \beta_{19} - 107592 \beta_{18} + 666214 \beta_{17} - 423524 \beta_{15} + 580176 \beta_{14} + \cdots - 8238722 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2197164 \beta_{19} - 1426144 \beta_{18} + 1014486 \beta_{17} - 311042 \beta_{16} - 3365518 \beta_{15} + \cdots - 6525074 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 42272892 \beta_{19} + 5047356 \beta_{18} - 20171304 \beta_{17} + 7706816 \beta_{16} - 2063872 \beta_{15} + \cdots + 10182492 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 46562976 \beta_{19} + 224957288 \beta_{18} - 185811960 \beta_{17} + 197706352 \beta_{15} + \cdots + 733558100 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2428380356 \beta_{19} + 446682304 \beta_{18} - 2108442808 \beta_{17} - 699301436 \beta_{16} + \cdots + 40292934224 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 14289376640 \beta_{19} - 8755013240 \beta_{18} + 7743792680 \beta_{17} + 661134180 \beta_{16} + \cdots - 932600020 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 46679624616 \beta_{19} - 76266045520 \beta_{18} + 260360219272 \beta_{17} - 197787765176 \beta_{15} + \cdots - 2976798117816 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1398271753536 \beta_{19} - 676404705752 \beta_{18} + 584438324736 \beta_{17} - 17138919832 \beta_{16} + \cdots - 6431362955408 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 18281490387840 \beta_{19} + 3180664346544 \beta_{18} - 7333216127304 \beta_{17} + 4765617065464 \beta_{16} + \cdots + 6020887149240 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 7188886024688 \beta_{19} + 104069237299072 \beta_{18} - 94908213329384 \beta_{17} + \cdots + 572318980969232 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 12\!\cdots\!92 \beta_{19} + 261187118231248 \beta_{18} - 917608495282848 \beta_{17} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 87\!\cdots\!04 \beta_{19} + \cdots - 251547579641824 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
4.40690 − 7.63297i
3.62416 − 6.27723i
1.68077 − 2.91117i
1.60382 − 2.77790i
−0.113972 + 0.197406i
−0.268097 + 0.464358i
−0.328356 + 0.568730i
−2.60694 + 4.51535i
−3.80876 + 6.59697i
−4.18952 + 7.25646i
4.40690 + 7.63297i
3.62416 + 6.27723i
1.68077 + 2.91117i
1.60382 + 2.77790i
−0.113972 − 0.197406i
−0.268097 − 0.464358i
−0.328356 − 0.568730i
−2.60694 − 4.51535i
−3.80876 − 6.59697i
−4.18952 − 7.25646i
0 1.50000 − 0.866025i 0 −4.40690 − 7.63297i 0 −3.21766 0 1.50000 − 2.59808i 0
145.2 0 1.50000 − 0.866025i 0 −3.62416 − 6.27723i 0 0.758627 0 1.50000 − 2.59808i 0
145.3 0 1.50000 − 0.866025i 0 −1.68077 − 2.91117i 0 −10.5431 0 1.50000 − 2.59808i 0
145.4 0 1.50000 − 0.866025i 0 −1.60382 − 2.77790i 0 13.4491 0 1.50000 − 2.59808i 0
145.5 0 1.50000 − 0.866025i 0 0.113972 + 0.197406i 0 2.53976 0 1.50000 − 2.59808i 0
145.6 0 1.50000 − 0.866025i 0 0.268097 + 0.464358i 0 −12.8244 0 1.50000 − 2.59808i 0
145.7 0 1.50000 − 0.866025i 0 0.328356 + 0.568730i 0 2.26614 0 1.50000 − 2.59808i 0
145.8 0 1.50000 − 0.866025i 0 2.60694 + 4.51535i 0 5.77889 0 1.50000 − 2.59808i 0
145.9 0 1.50000 − 0.866025i 0 3.80876 + 6.59697i 0 −10.8827 0 1.50000 − 2.59808i 0
145.10 0 1.50000 − 0.866025i 0 4.18952 + 7.25646i 0 2.67523 0 1.50000 − 2.59808i 0
673.1 0 1.50000 + 0.866025i 0 −4.40690 + 7.63297i 0 −3.21766 0 1.50000 + 2.59808i 0
673.2 0 1.50000 + 0.866025i 0 −3.62416 + 6.27723i 0 0.758627 0 1.50000 + 2.59808i 0
673.3 0 1.50000 + 0.866025i 0 −1.68077 + 2.91117i 0 −10.5431 0 1.50000 + 2.59808i 0
673.4 0 1.50000 + 0.866025i 0 −1.60382 + 2.77790i 0 13.4491 0 1.50000 + 2.59808i 0
673.5 0 1.50000 + 0.866025i 0 0.113972 − 0.197406i 0 2.53976 0 1.50000 + 2.59808i 0
673.6 0 1.50000 + 0.866025i 0 0.268097 − 0.464358i 0 −12.8244 0 1.50000 + 2.59808i 0
673.7 0 1.50000 + 0.866025i 0 0.328356 − 0.568730i 0 2.26614 0 1.50000 + 2.59808i 0
673.8 0 1.50000 + 0.866025i 0 2.60694 − 4.51535i 0 5.77889 0 1.50000 + 2.59808i 0
673.9 0 1.50000 + 0.866025i 0 3.80876 − 6.59697i 0 −10.8827 0 1.50000 + 2.59808i 0
673.10 0 1.50000 + 0.866025i 0 4.18952 − 7.25646i 0 2.67523 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.be.j 20
4.b odd 2 1 456.3.w.a ✓ 20
12.b even 2 1 1368.3.bv.c 20
19.d odd 6 1 inner 912.3.be.j 20
76.f even 6 1 456.3.w.a ✓ 20
228.n odd 6 1 1368.3.bv.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.3.w.a ✓ 20 4.b odd 2 1
456.3.w.a ✓ 20 76.f even 6 1
912.3.be.j 20 1.a even 1 1 trivial
912.3.be.j 20 19.d odd 6 1 inner
1368.3.bv.c 20 12.b even 2 1
1368.3.bv.c 20 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{20} + 154 T_{5}^{18} + 24 T_{5}^{17} + 16374 T_{5}^{16} + 4328 T_{5}^{15} + 911836 T_{5}^{14} + \cdots + 338560000 \) Copy content Toggle raw display
\( T_{7}^{10} + 10 T_{7}^{9} - 269 T_{7}^{8} - 2380 T_{7}^{7} + 19908 T_{7}^{6} + 101364 T_{7}^{5} + \cdots + 4298185 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 338560000 \) Copy content Toggle raw display
$7$ \( (T^{10} + 10 T^{9} + \cdots + 4298185)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} - 4 T^{9} + \cdots - 98906400)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 37\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 28\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 45\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 33\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 19\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 70\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 50\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
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