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Properties

Label 912.2.ci.e
Level $912$
Weight $2$
Character orbit 912.ci
Analytic conductor $7.282$
Analytic rank $0$
Dimension $18$
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] = [912,2,Mod(79,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 0, 0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.ci (of order \(18\), degree \(6\), 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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} - 3 x^{16} + 100 x^{15} - 171 x^{14} - 471 x^{13} + 1537 x^{12} + 321 x^{11} + \cdots + 1367631 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + (\beta_{16} + \beta_{12} + \cdots - \beta_1) q^{5}+ \cdots + \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + (\beta_{16} + \beta_{12} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{11} - \beta_{10} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 3 q^{13} + 12 q^{17} - 3 q^{19} + 15 q^{21} - 6 q^{23} + 24 q^{25} + 9 q^{27} + 12 q^{29} - 12 q^{31} + 6 q^{33} + 36 q^{35} + 12 q^{41} - 21 q^{43} - 6 q^{45} + 24 q^{47} - 3 q^{49} + 6 q^{51} + 6 q^{53} - 12 q^{55} - 54 q^{59} - 24 q^{61} - 12 q^{63} + 36 q^{65} + 21 q^{67} + 15 q^{73} - 18 q^{75} + 60 q^{79} + 54 q^{85} - 18 q^{87} + 36 q^{89} - 24 q^{91} + 24 q^{93} + 6 q^{95} - 6 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} - 3 x^{16} + 100 x^{15} - 171 x^{14} - 471 x^{13} + 1537 x^{12} + 321 x^{11} + \cdots + 1367631 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 97\!\cdots\!17 \nu^{17} + \cdots - 23\!\cdots\!81 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!28 \nu^{17} + \cdots - 24\!\cdots\!68 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\!\cdots\!57 \nu^{17} + \cdots + 21\!\cdots\!99 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22\!\cdots\!40 \nu^{17} + \cdots + 18\!\cdots\!98 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36\!\cdots\!58 \nu^{17} + \cdots + 16\!\cdots\!11 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 41\!\cdots\!50 \nu^{17} + \cdots + 38\!\cdots\!37 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21\!\cdots\!87 \nu^{17} + \cdots - 68\!\cdots\!41 ) / 32\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!92 \nu^{17} + \cdots + 68\!\cdots\!91 ) / 17\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 57\!\cdots\!91 \nu^{17} + \cdots + 19\!\cdots\!65 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 66\!\cdots\!55 \nu^{17} + \cdots + 58\!\cdots\!23 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 68\!\cdots\!21 \nu^{17} + \cdots - 50\!\cdots\!22 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\!\cdots\!65 \nu^{17} + \cdots - 13\!\cdots\!54 ) / 17\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 78\!\cdots\!46 \nu^{17} + \cdots - 83\!\cdots\!67 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 23\!\cdots\!44 \nu^{17} + \cdots - 17\!\cdots\!25 ) / 17\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 13\!\cdots\!64 \nu^{17} + \cdots - 78\!\cdots\!81 ) / 87\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 31\!\cdots\!36 \nu^{17} + \cdots - 21\!\cdots\!33 ) / 17\!\cdots\!37 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - 3 \beta_{6} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{17} - \beta_{16} - \beta_{15} - 3 \beta_{13} - 2 \beta_{12} - \beta_{11} + 2 \beta_{10} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{17} - 4 \beta_{16} + \beta_{15} + 3 \beta_{14} - 6 \beta_{13} - 10 \beta_{12} - 4 \beta_{11} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15 \beta_{17} - 8 \beta_{16} - 7 \beta_{15} + 8 \beta_{14} - 14 \beta_{13} - 14 \beta_{12} - 6 \beta_{11} + \cdots - 44 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{17} - 20 \beta_{16} - 7 \beta_{15} + 27 \beta_{14} - 3 \beta_{13} - 50 \beta_{12} - 13 \beta_{11} + \cdots - 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{17} + 59 \beta_{16} - 20 \beta_{15} + 16 \beta_{14} + \beta_{13} - 17 \beta_{12} - 32 \beta_{11} + \cdots - 292 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 121 \beta_{17} + 135 \beta_{16} - 144 \beta_{15} + 78 \beta_{14} + 249 \beta_{13} + 44 \beta_{12} + \cdots - 256 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 208 \beta_{17} + 952 \beta_{16} - 496 \beta_{15} - 327 \beta_{14} + 693 \beta_{13} + 688 \beta_{12} + \cdots - 1434 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1427 \beta_{17} + 1967 \beta_{16} - 1439 \beta_{15} - 416 \beta_{14} + 2290 \beta_{13} + 1723 \beta_{12} + \cdots - 298 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1510 \beta_{17} + 7056 \beta_{16} - 4053 \beta_{15} - 3543 \beta_{14} + 4563 \beta_{13} + 6119 \beta_{12} + \cdots - 1030 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6211 \beta_{17} + 7531 \beta_{16} - 6772 \beta_{15} - 2826 \beta_{14} + 9171 \beta_{13} + 10096 \beta_{12} + \cdots + 9582 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 755 \beta_{17} + 22457 \beta_{16} - 12245 \beta_{15} - 14276 \beta_{14} + 5827 \beta_{13} + \cdots + 37664 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11569 \beta_{17} - 47250 \beta_{16} + 33870 \beta_{15} + 4347 \beta_{14} - 13107 \beta_{13} + \cdots + 97034 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 25748 \beta_{17} - 116042 \beta_{16} + 136559 \beta_{15} - 31065 \beta_{14} - 93582 \beta_{13} + \cdots + 373716 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 47696 \beta_{17} - 956557 \beta_{16} + 854071 \beta_{15} + 63757 \beta_{14} - 322373 \beta_{13} + \cdots + 372131 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 150682 \beta_{17} - 1835490 \beta_{16} + 2049249 \beta_{15} - 317313 \beta_{14} - 557694 \beta_{13} + \cdots + 2261204 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
1.12178 + 1.48644i
−0.489413 − 1.79677i
−1.33811 + 1.29514i
1.12178 − 1.48644i
−0.489413 + 1.79677i
−1.33811 − 1.29514i
2.46004 + 0.909722i
−2.61262 − 0.231530i
2.26592 − 1.32098i
2.46004 − 0.909722i
−2.61262 + 0.231530i
2.26592 + 1.32098i
1.98709 − 0.839158i
1.76137 + 1.24510i
−2.15607 − 0.0639189i
1.98709 + 0.839158i
1.76137 − 1.24510i
−2.15607 + 0.0639189i
0 0.939693 − 0.342020i 0 −0.189533 + 1.07490i 0 −1.46125 + 0.843655i 0 0.766044 − 0.642788i 0
79.2 0 0.939693 − 0.342020i 0 −0.0537040 + 0.304570i 0 4.22544 − 2.43956i 0 0.766044 − 0.642788i 0
79.3 0 0.939693 − 0.342020i 0 0.590533 − 3.34908i 0 −1.12990 + 0.652348i 0 0.766044 − 0.642788i 0
127.1 0 0.939693 + 0.342020i 0 −0.189533 − 1.07490i 0 −1.46125 − 0.843655i 0 0.766044 + 0.642788i 0
127.2 0 0.939693 + 0.342020i 0 −0.0537040 − 0.304570i 0 4.22544 + 2.43956i 0 0.766044 + 0.642788i 0
127.3 0 0.939693 + 0.342020i 0 0.590533 + 3.34908i 0 −1.12990 − 0.652348i 0 0.766044 + 0.642788i 0
223.1 0 −0.173648 + 0.984808i 0 −0.718121 − 0.602575i 0 −0.983288 + 0.567702i 0 −0.939693 − 0.342020i 0
223.2 0 −0.173648 + 0.984808i 0 −0.345737 − 0.290108i 0 0.993418 − 0.573550i 0 −0.939693 − 0.342020i 0
223.3 0 −0.173648 + 0.984808i 0 2.59595 + 2.17826i 0 2.88040 − 1.66300i 0 −0.939693 − 0.342020i 0
319.1 0 −0.173648 − 0.984808i 0 −0.718121 + 0.602575i 0 −0.983288 − 0.567702i 0 −0.939693 + 0.342020i 0
319.2 0 −0.173648 − 0.984808i 0 −0.345737 + 0.290108i 0 0.993418 + 0.573550i 0 −0.939693 + 0.342020i 0
319.3 0 −0.173648 − 0.984808i 0 2.59595 − 2.17826i 0 2.88040 + 1.66300i 0 −0.939693 + 0.342020i 0
751.1 0 −0.766044 + 0.642788i 0 −3.87454 − 1.41022i 0 −3.15920 − 1.82397i 0 0.173648 − 0.984808i 0
751.2 0 −0.766044 + 0.642788i 0 −1.03170 − 0.375509i 0 0.450834 + 0.260289i 0 0.173648 − 0.984808i 0
751.3 0 −0.766044 + 0.642788i 0 3.02685 + 1.10168i 0 −1.81645 − 1.04873i 0 0.173648 − 0.984808i 0
895.1 0 −0.766044 − 0.642788i 0 −3.87454 + 1.41022i 0 −3.15920 + 1.82397i 0 0.173648 + 0.984808i 0
895.2 0 −0.766044 − 0.642788i 0 −1.03170 + 0.375509i 0 0.450834 − 0.260289i 0 0.173648 + 0.984808i 0
895.3 0 −0.766044 − 0.642788i 0 3.02685 − 1.10168i 0 −1.81645 + 1.04873i 0 0.173648 + 0.984808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.k even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.ci.e ✓ 18
4.b odd 2 1 912.2.ci.f yes 18
19.f odd 18 1 912.2.ci.f yes 18
76.k even 18 1 inner 912.2.ci.e ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ci.e ✓ 18 1.a even 1 1 trivial
912.2.ci.e ✓ 18 76.k even 18 1 inner
912.2.ci.f yes 18 4.b odd 2 1
912.2.ci.f yes 18 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{18} - 12 T_{5}^{16} + 38 T_{5}^{15} + 48 T_{5}^{14} - 486 T_{5}^{13} + 2238 T_{5}^{12} + \cdots + 576 \) Copy content Toggle raw display
\( T_{7}^{18} - 30 T_{7}^{16} + 675 T_{7}^{14} + 855 T_{7}^{13} - 5477 T_{7}^{12} - 10431 T_{7}^{11} + \cdots + 34347 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{3} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} - 12 T^{16} + \cdots + 576 \) Copy content Toggle raw display
$7$ \( T^{18} - 30 T^{16} + \cdots + 34347 \) Copy content Toggle raw display
$11$ \( T^{18} - 33 T^{16} + \cdots + 1728 \) Copy content Toggle raw display
$13$ \( T^{18} - 3 T^{17} + \cdots + 162867 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 268697664 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 9032809152 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 122712760512 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 544949574849 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 13733220843 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 108865728 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 11432149083 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 7316338368 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 3614768832 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 7831445534784 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 573075721 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 7453904896 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 4653477324864 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 855648672669729 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 26\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 123685470912 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 56816030267643 \) Copy content Toggle raw display
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