Properties

Label 912.3.m.b
Level $912$
Weight $3$
Character orbit 912.m
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(799,912)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.799");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.m (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: \(24.8502001097\)
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: \( x^{12} - 2 x^{11} - 2 x^{10} - 28 x^{9} - 400 x^{8} - 520 x^{7} + 17067 x^{6} - 3250 x^{5} - 195494 x^{4} + 302996 x^{3} + 602332 x^{2} - 2263536 x + 2052928 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + ( - \beta_{2} - 1) q^{5} + \beta_{10} q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + ( - \beta_{2} - 1) q^{5} + \beta_{10} q^{7} - 3 q^{9} + ( - \beta_{9} - \beta_{8} - 2 \beta_{5}) q^{11} + (\beta_{4} - \beta_{2} + 3) q^{13} + ( - \beta_{8} + \beta_{7} + \beta_{5}) q^{15} + ( - \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{17} + \beta_{8} q^{19} + ( - \beta_{6} - \beta_{4} + 1) q^{21} + (2 \beta_{11} + \beta_{10} + 2 \beta_{9} - 4 \beta_{8} + \beta_{7}) q^{23} + ( - \beta_{4} + 2 \beta_{2} + 1) q^{25} + 3 \beta_{5} q^{27} + (3 \beta_{6} - 4 \beta_{4} + 4 \beta_{2} + \beta_1 - 8) q^{29} + ( - \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 5 \beta_{8} - 4 \beta_{7} - 2 \beta_{5}) q^{31} + ( - 2 \beta_{6} + \beta_{4} - \beta_1 - 7) q^{33} + ( - \beta_{11} - 4 \beta_{8} - 3 \beta_{7}) q^{35} + (3 \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1) q^{37} + (2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - 4 \beta_{5}) q^{39} + ( - 4 \beta_{6} + 2 \beta_{4} + \beta_{3} - 7 \beta_{2} - 2 \beta_1 - 1) q^{41} + (\beta_{10} - 2 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} + 2 \beta_{5}) q^{43} + (3 \beta_{2} + 3) q^{45} + (\beta_{11} - \beta_{10} + 2 \beta_{9} - 10 \beta_{8} + 6 \beta_{5}) q^{47} + (2 \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1 - 9) q^{49} + ( - 3 \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{5}) q^{51} + ( - 2 \beta_{6} + 6 \beta_{4} - \beta_{3} + \beta_{2} + 19) q^{53} + (2 \beta_{11} + 3 \beta_{10} + 4 \beta_{9} - 8 \beta_{8} + 4 \beta_{7} - 12 \beta_{5}) q^{55} + \beta_1 q^{57} + ( - 3 \beta_{11} - 5 \beta_{10} - 4 \beta_{8} + \beta_{7} + 12 \beta_{5}) q^{59} + (2 \beta_{6} + \beta_{4} + 2 \beta_{3} - 6 \beta_{2} - 8 \beta_1 + 30) q^{61} - 3 \beta_{10} q^{63} + (3 \beta_{6} - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 13) q^{65} + (5 \beta_{11} + \beta_{10} + 4 \beta_{9} - 4 \beta_{8} + 7 \beta_{7} + 12 \beta_{5}) q^{67} + (3 \beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{69} + (\beta_{11} - 5 \beta_{10} - 4 \beta_{9} - 12 \beta_{8} - 3 \beta_{7} + 4 \beta_{5}) q^{71} + ( - 3 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 14) q^{73} + ( - 2 \beta_{10} - \beta_{9} + 2 \beta_{8} - 2 \beta_{7}) q^{75} + ( - \beta_{6} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - 5 \beta_1 - 25) q^{77} + ( - 6 \beta_{11} - 2 \beta_{10} - 3 \beta_{9} - 5 \beta_{8} - 3 \beta_{7} + 6 \beta_{5}) q^{79} + 9 q^{81} + ( - \beta_{11} + 6 \beta_{10} + 3 \beta_{9} - 5 \beta_{8} - \beta_{7} - 2 \beta_{5}) q^{83} + (4 \beta_{6} + 5 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 10 \beta_1 + 10) q^{85} + ( - 5 \beta_{10} - 7 \beta_{9} + \beta_{8} - 4 \beta_{7} + 12 \beta_{5}) q^{87} + ( - 4 \beta_{6} + 2 \beta_{4} - 3 \beta_{3} - 9 \beta_{2} + 2 \beta_1 - 35) q^{89} + ( - 3 \beta_{11} + 7 \beta_{10} + 2 \beta_{9} - 6 \beta_{8} + \beta_{7} - 32 \beta_{5}) q^{91} + ( - 3 \beta_{6} + 5 \beta_{4} + \beta_{3} - 10 \beta_{2} - 9 \beta_1 - 12) q^{93} + ( - \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} + 4 \beta_{5}) q^{95} + ( - \beta_{6} - 3 \beta_{3} + 13 \beta_{2} + 3 \beta_1 + 21) q^{97} + (3 \beta_{9} + 3 \beta_{8} + 6 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{5} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{5} - 36 q^{9} + 36 q^{13} + 14 q^{17} + 18 q^{21} + 10 q^{25} - 108 q^{29} - 78 q^{33} - 16 q^{37} + 16 q^{41} + 30 q^{45} - 118 q^{49} + 220 q^{53} + 366 q^{61} + 140 q^{65} + 24 q^{69} - 158 q^{73} - 286 q^{77} + 108 q^{81} + 98 q^{85} - 396 q^{89} - 120 q^{93} + 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 2 x^{10} - 28 x^{9} - 400 x^{8} - 520 x^{7} + 17067 x^{6} - 3250 x^{5} - 195494 x^{4} + 302996 x^{3} + 602332 x^{2} - 2263536 x + 2052928 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 651569787185 \nu^{11} + 1209190969407 \nu^{10} + 4989114003428 \nu^{9} - 6889549986684 \nu^{8} - 277551169868310 \nu^{7} + \cdots + 13\!\cdots\!02 ) / 21\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 51\!\cdots\!21 \nu^{11} + \cdots - 73\!\cdots\!44 ) / 64\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 64\!\cdots\!87 \nu^{11} + \cdots - 12\!\cdots\!48 ) / 64\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 99\!\cdots\!53 \nu^{11} + \cdots + 13\!\cdots\!44 ) / 64\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 157524610518805 \nu^{11} + 6067399667026 \nu^{10} - 201453765859666 \nu^{9} + \cdots - 15\!\cdots\!52 ) / 80\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 78\!\cdots\!35 \nu^{11} + \cdots + 98\!\cdots\!14 ) / 32\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16\!\cdots\!15 \nu^{11} + \cdots + 69\!\cdots\!96 ) / 64\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 237830692375 \nu^{11} + 6844355458 \nu^{10} - 326852665758 \nu^{9} - 7584570552796 \nu^{8} - 109922313318736 \nu^{7} + \cdots - 23\!\cdots\!12 ) / 48\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 88\!\cdots\!41 \nu^{11} + \cdots - 61\!\cdots\!92 ) / 12\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 88\!\cdots\!89 \nu^{11} + \cdots + 76\!\cdots\!96 ) / 64\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 74\!\cdots\!88 \nu^{11} + \cdots - 68\!\cdots\!28 ) / 32\!\cdots\!54 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} - \beta_{7} + \beta_{4} + \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} - 3 \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{7} - 5 \beta_{6} - 4 \beta_{5} + 5 \beta_{4} + \beta_{3} - 10 \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{11} + 7 \beta_{10} + 10 \beta_{9} - 23 \beta_{8} + 7 \beta_{7} - 4 \beta_{6} + 61 \beta_{5} + 6 \beta_{4} - 6 \beta_{2} - 6 \beta _1 + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 19 \beta_{11} - 55 \beta_{10} - \beta_{9} - 49 \beta_{8} + 42 \beta_{7} + 165 \beta_{6} + 38 \beta_{5} - 75 \beta_{4} + 13 \beta_{3} + 306 \beta_{2} - 13 \beta _1 + 718 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 173 \beta_{11} - 461 \beta_{10} - 943 \beta_{9} + 979 \beta_{8} - 898 \beta_{7} - 83 \beta_{6} - 1512 \beta_{5} - 43 \beta_{4} - 55 \beta_{3} - 246 \beta_{2} - 363 \beta _1 + 2464 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 20 \beta_{11} - 244 \beta_{10} - 852 \beta_{9} - 80 \beta_{8} - 840 \beta_{7} - 1907 \beta_{6} + 628 \beta_{5} + 1917 \beta_{4} + 316 \beta_{3} - 2841 \beta_{2} + 774 \beta _1 - 13560 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 989 \beta_{11} + 1759 \beta_{10} + 24805 \beta_{9} - 33945 \beta_{8} + 28894 \beta_{7} - 7801 \beta_{6} + 62032 \beta_{5} + 5227 \beta_{4} + 127 \beta_{3} - 13858 \beta_{2} - 1065 \beta _1 - 20864 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10239 \beta_{11} + 23999 \beta_{10} + 27437 \beta_{9} - 92599 \beta_{8} + 19282 \beta_{7} + 133925 \beta_{6} + 211174 \beta_{5} - 108035 \beta_{4} - 10767 \beta_{3} + 221390 \beta_{2} + \cdots + 798374 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 73222 \beta_{11} - 213437 \beta_{10} - 430818 \beta_{9} + 552189 \beta_{8} - 392265 \beta_{7} + 162346 \beta_{6} - 1043371 \beta_{5} - 92634 \beta_{4} + 2770 \beta_{3} + 303036 \beta_{2} + \cdots + 437110 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 263331 \beta_{11} - 835819 \beta_{10} - 1880729 \beta_{9} + 1984355 \beta_{8} - 1756674 \beta_{7} - 4759831 \beta_{6} - 3536502 \beta_{5} + 3235969 \beta_{4} + 132573 \beta_{3} + \cdots - 21212802 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3045003 \beta_{11} + 10000603 \beta_{10} + 25128717 \beta_{9} - 36690833 \beta_{8} + 24330242 \beta_{7} - 8278379 \beta_{6} + 71479104 \beta_{5} + 7683217 \beta_{4} + \cdots - 66587876 ) / 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
0.198491 5.66831i
1.30395 + 1.41343i
4.06962 + 0.767506i
2.21305 + 0.177784i
−3.01700 + 0.588235i
−3.76811 + 2.72136i
0.198491 + 5.66831i
1.30395 1.41343i
4.06962 0.767506i
2.21305 0.177784i
−3.01700 0.588235i
−3.76811 2.72136i
0 1.73205i 0 −6.54540 0 0.687253i 0 −3.00000 0
799.2 0 1.73205i 0 −6.08630 0 8.85163i 0 −3.00000 0
799.3 0 1.73205i 0 −4.65777 0 4.49506i 0 −3.00000 0
799.4 0 1.73205i 0 2.67107 0 11.9371i 0 −3.00000 0
799.5 0 1.73205i 0 4.69015 0 3.50408i 0 −3.00000 0
799.6 0 1.73205i 0 4.92825 0 9.95984i 0 −3.00000 0
799.7 0 1.73205i 0 −6.54540 0 0.687253i 0 −3.00000 0
799.8 0 1.73205i 0 −6.08630 0 8.85163i 0 −3.00000 0
799.9 0 1.73205i 0 −4.65777 0 4.49506i 0 −3.00000 0
799.10 0 1.73205i 0 2.67107 0 11.9371i 0 −3.00000 0
799.11 0 1.73205i 0 4.69015 0 3.50408i 0 −3.00000 0
799.12 0 1.73205i 0 4.92825 0 9.95984i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 799.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.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.m.b 12
3.b odd 2 1 2736.3.m.e 12
4.b odd 2 1 inner 912.3.m.b 12
12.b even 2 1 2736.3.m.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.3.m.b 12 1.a even 1 1 trivial
912.3.m.b 12 4.b odd 2 1 inner
2736.3.m.e 12 3.b odd 2 1
2736.3.m.e 12 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 5T_{5}^{5} - 65T_{5}^{4} - 245T_{5}^{3} + 1468T_{5}^{2} + 2964T_{5} - 11456 \) 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
$3$ \( (T^{2} + 3)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + 5 T^{5} - 65 T^{4} - 245 T^{3} + \cdots - 11456)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 353 T^{10} + \cdots + 129777664 \) Copy content Toggle raw display
$11$ \( T^{12} + 717 T^{10} + \cdots + 67566724096 \) Copy content Toggle raw display
$13$ \( (T^{6} - 18 T^{5} - 104 T^{4} + 1568 T^{3} + \cdots - 7808)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 7 T^{5} - 1443 T^{4} + \cdots - 65017064)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + 6520 T^{10} + \cdots + 75\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{6} + 54 T^{5} - 2384 T^{4} + \cdots + 351804928)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 10576 T^{10} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{6} + 8 T^{5} - 3532 T^{4} + \cdots + 343318528)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 8 T^{5} - 5820 T^{4} + \cdots - 622678016)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 7105 T^{10} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{12} + 17145 T^{10} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{6} - 110 T^{5} - 4184 T^{4} + \cdots - 5005174784)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 16688 T^{10} + \cdots + 34\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{6} - 183 T^{5} + \cdots + 21645030952)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 33328 T^{10} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{12} + 30160 T^{10} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{6} + 79 T^{5} - 10287 T^{4} + \cdots + 30337444648)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 35772 T^{10} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{12} + 21676 T^{10} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{6} + 198 T^{5} + \cdots - 12103330176)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 112 T^{5} + \cdots - 12366236864)^{2} \) Copy content Toggle raw display
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