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Properties

Label 912.3.h.b
Level $912$
Weight $3$
Character orbit 912.h
Analytic conductor $24.850$
Analytic rank $0$
Dimension $12$
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(305,912)]
 
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([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.305");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.h (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 13 x^{10} - 32 x^{9} + 145 x^{8} - 542 x^{7} + 1722 x^{6} - 4878 x^{5} + \cdots + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 228)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{10} q^{5} + ( - \beta_{8} + \beta_{7} + 1) q^{7} + (\beta_{4} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{10} q^{5} + ( - \beta_{8} + \beta_{7} + 1) q^{7} + (\beta_{4} - 2) q^{9} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \cdots - 1) q^{11}+ \cdots + (2 \beta_{11} - 8 \beta_{10} + \cdots + 28) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 8 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} + 8 q^{7} - 22 q^{9} + 32 q^{13} + 22 q^{15} + 42 q^{21} - 144 q^{25} - 26 q^{27} - 80 q^{31} + 94 q^{33} - 80 q^{37} + 114 q^{39} + 52 q^{43} + 64 q^{45} + 252 q^{49} + 70 q^{51} - 84 q^{55} - 108 q^{61} + 46 q^{63} + 112 q^{67} - 92 q^{69} + 88 q^{73} - 288 q^{75} - 244 q^{79} - 178 q^{81} - 36 q^{85} + 134 q^{87} + 164 q^{91} - 136 q^{93} - 84 q^{97} + 196 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 13 x^{10} - 32 x^{9} + 145 x^{8} - 542 x^{7} + 1722 x^{6} - 4878 x^{5} + \cdots + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} - 13 \nu^{9} + 32 \nu^{8} - 145 \nu^{7} + 542 \nu^{6} - 1722 \nu^{5} + \cdots + 59049 ) / 59049 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 2 \nu^{10} + 13 \nu^{9} - 32 \nu^{8} + 145 \nu^{7} - 542 \nu^{6} + 1722 \nu^{5} + \cdots - 118098 ) / 59049 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2 \nu^{11} - 5 \nu^{10} - 8 \nu^{9} - 53 \nu^{8} - 2 \nu^{7} - 221 \nu^{6} + 1434 \nu^{5} + \cdots - 413343 ) / 59049 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17 \nu^{11} - 160 \nu^{10} + 878 \nu^{9} - 2992 \nu^{8} + 9575 \nu^{7} - 22948 \nu^{6} + \cdots - 236196 ) / 708588 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13 \nu^{11} + \nu^{10} - 128 \nu^{9} + 664 \nu^{8} - 1895 \nu^{7} + 8533 \nu^{6} - 23595 \nu^{5} + \cdots + 2421009 ) / 236196 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 59 \nu^{11} + 46 \nu^{10} - 56 \nu^{9} - 1640 \nu^{8} + 10597 \nu^{7} - 28682 \nu^{6} + \cdots - 16179426 ) / 708588 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 59 \nu^{11} - 82 \nu^{10} + 128 \nu^{9} + 2630 \nu^{8} - 12361 \nu^{7} + 42416 \nu^{6} + \cdots + 19604268 ) / 708588 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 92 \nu^{11} + 301 \nu^{10} - 620 \nu^{9} - 2258 \nu^{8} + 12400 \nu^{7} - 36365 \nu^{6} + \cdots - 23442453 ) / 708588 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12 \nu^{11} + 79 \nu^{10} - 374 \nu^{9} + 1234 \nu^{8} - 3770 \nu^{7} + 11293 \nu^{6} + \cdots - 203391 ) / 78732 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 179 \nu^{11} + 1456 \nu^{10} - 7844 \nu^{9} + 31018 \nu^{8} - 97703 \nu^{7} + 296890 \nu^{6} + \cdots + 18305190 ) / 708588 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} - \beta_{2} + 3\beta _1 - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{10} - 3\beta_{9} - 6\beta_{8} - 3\beta_{7} + 9\beta_{5} - 2\beta_{3} - 8\beta_{2} + 6\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9 \beta_{11} - 21 \beta_{10} + 3 \beta_{9} - 21 \beta_{8} - 15 \beta_{7} + 9 \beta_{5} - 9 \beta_{4} + \cdots - 43 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 36 \beta_{11} - 102 \beta_{10} + 75 \beta_{9} + 51 \beta_{8} - 6 \beta_{7} - 9 \beta_{6} - 36 \beta_{5} + \cdots + 277 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9 \beta_{11} - 33 \beta_{10} + 141 \beta_{9} + 255 \beta_{8} + 87 \beta_{7} + 54 \beta_{6} + 45 \beta_{5} + \cdots - 118 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 36 \beta_{10} - 81 \beta_{9} - 63 \beta_{8} + 342 \beta_{7} + 333 \beta_{6} + 108 \beta_{5} + \cdots - 1424 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 459 \beta_{11} - 423 \beta_{10} - 927 \beta_{9} - 1287 \beta_{8} + 639 \beta_{7} - 324 \beta_{6} + \cdots + 6716 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 864 \beta_{11} - 1005 \beta_{10} - 372 \beta_{9} + 147 \beta_{8} - 291 \beta_{7} - 4131 \beta_{6} + \cdots - 26054 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1530 \beta_{11} + 2634 \beta_{10} + 9120 \beta_{9} + 12570 \beta_{8} - 2616 \beta_{7} - 6102 \beta_{6} + \cdots - 49786 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 12816 \beta_{11} + 30102 \beta_{10} + 2298 \beta_{9} + 9600 \beta_{8} - 11040 \beta_{7} + 27612 \beta_{6} + \cdots + 3778 ) / 3 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
2.86047 − 0.904275i
2.86047 + 0.904275i
1.82499 − 2.38105i
1.82499 + 2.38105i
0.814348 − 2.88736i
0.814348 + 2.88736i
−0.189969 − 2.99398i
−0.189969 + 2.99398i
−2.13934 − 2.10314i
−2.13934 + 2.10314i
−2.17050 − 2.07097i
−2.17050 + 2.07097i
0 −2.86047 − 0.904275i 0 − 4.85658i 0 −3.82052 0 7.36457 + 5.17330i 0
305.2 0 −2.86047 + 0.904275i 0 4.85658i 0 −3.82052 0 7.36457 − 5.17330i 0
305.3 0 −1.82499 − 2.38105i 0 1.25646i 0 9.60340 0 −2.33880 + 8.69080i 0
305.4 0 −1.82499 + 2.38105i 0 − 1.25646i 0 9.60340 0 −2.33880 − 8.69080i 0
305.5 0 −0.814348 − 2.88736i 0 6.89560i 0 −12.4554 0 −7.67367 + 4.70263i 0
305.6 0 −0.814348 + 2.88736i 0 − 6.89560i 0 −12.4554 0 −7.67367 − 4.70263i 0
305.7 0 0.189969 − 2.99398i 0 − 6.94968i 0 2.94391 0 −8.92782 − 1.13752i 0
305.8 0 0.189969 + 2.99398i 0 6.94968i 0 2.94391 0 −8.92782 + 1.13752i 0
305.9 0 2.13934 − 2.10314i 0 − 3.09276i 0 −3.86580 0 0.153570 − 8.99869i 0
305.10 0 2.13934 + 2.10314i 0 3.09276i 0 −3.86580 0 0.153570 + 8.99869i 0
305.11 0 2.17050 − 2.07097i 0 9.56151i 0 11.5944 0 0.422149 − 8.99009i 0
305.12 0 2.17050 + 2.07097i 0 − 9.56151i 0 11.5944 0 0.422149 + 8.99009i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.h.b 12
3.b odd 2 1 inner 912.3.h.b 12
4.b odd 2 1 228.3.e.a ✓ 12
12.b even 2 1 228.3.e.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.e.a ✓ 12 4.b odd 2 1
228.3.e.a ✓ 12 12.b even 2 1
912.3.h.b 12 1.a even 1 1 trivial
912.3.h.b 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 222T_{5}^{10} + 17841T_{5}^{8} + 646448T_{5}^{6} + 10432308T_{5}^{4} + 62294688T_{5}^{2} + 74779200 \) acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{12} + 222 T^{10} + \cdots + 74779200 \) Copy content Toggle raw display
$7$ \( (T^{6} - 4 T^{5} + \cdots - 60300)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 369215154432 \) Copy content Toggle raw display
$13$ \( (T^{6} - 16 T^{5} + \cdots - 94524)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 248099438592 \) Copy content Toggle raw display
$19$ \( (T^{2} - 19)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 364480287040512 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 2295350535168 \) Copy content Toggle raw display
$31$ \( (T^{6} + 40 T^{5} + \cdots - 222246432)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 40 T^{5} + \cdots + 957692224)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 18\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( (T^{6} - 26 T^{5} + \cdots - 682910400)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 16\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 37\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( (T^{6} + 54 T^{5} + \cdots - 7311543728)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 56 T^{5} + \cdots - 4631040)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 12\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( (T^{6} - 44 T^{5} + \cdots - 1042920)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 122 T^{5} + \cdots - 126628655872)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 52\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( (T^{6} + 42 T^{5} + \cdots - 4314805696)^{2} \) Copy content Toggle raw display
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