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Properties

Label 8.12.b.a
Level $8$
Weight $12$
Character orbit 8.b
Analytic conductor $6.147$
Analytic rank $0$
Dimension $10$
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] = [8,12,Mod(5,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.5");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 8.b (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: \(6.14674544448\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 48 x^{8} + 8836 x^{7} - 92554 x^{6} - 2200342 x^{5} - 88894588 x^{4} + \cdots + 34431558027300 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{2}\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 2) q^{2} + (\beta_{3} + \beta_1) q^{3} + (\beta_{2} + 2 \beta_1 + 43) q^{4} + (\beta_{4} - 2 \beta_{3} - \beta_1) q^{5} + (\beta_{7} + \beta_{2} + 2 \beta_1 - 2431) q^{6} + (\beta_{8} - \beta_{7} - \beta_{4} + \cdots - 3385) q^{7}+ \cdots + (2 \beta_{9} + 6 \beta_{7} + \cdots - 47267) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 2) q^{2} + (\beta_{3} + \beta_1) q^{3} + (\beta_{2} + 2 \beta_1 + 43) q^{4} + (\beta_{4} - 2 \beta_{3} - \beta_1) q^{5} + (\beta_{7} + \beta_{2} + 2 \beta_1 - 2431) q^{6} + (\beta_{8} - \beta_{7} - \beta_{4} + \cdots - 3385) q^{7}+ \cdots + (707040 \beta_{9} - 707040 \beta_{8} + \cdots + 181894176) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 22 q^{2} + 436 q^{4} - 24308 q^{6} - 33616 q^{7} - 208472 q^{8} - 472394 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 22 q^{2} + 436 q^{4} - 24308 q^{6} - 33616 q^{7} - 208472 q^{8} - 472394 q^{9} + 29864 q^{10} - 476216 q^{12} + 2264272 q^{14} + 3391792 q^{15} + 3757072 q^{16} + 2639732 q^{17} + 2329130 q^{18} + 8296432 q^{20} - 24201892 q^{22} - 45357808 q^{23} + 10668944 q^{24} - 41502654 q^{25} - 101931080 q^{26} - 95722144 q^{28} + 261362768 q^{30} + 442035392 q^{31} + 343908512 q^{32} - 54732216 q^{33} - 187103796 q^{34} - 431195700 q^{36} + 375100844 q^{38} - 431459312 q^{39} + 1519138784 q^{40} - 654907964 q^{41} - 3697737056 q^{42} - 4127012952 q^{44} + 6028867440 q^{46} - 560226528 q^{47} + 9357606048 q^{48} + 1143661722 q^{49} - 9626584226 q^{50} - 11595427248 q^{52} + 20050494008 q^{54} + 5632783984 q^{55} + 18698733760 q^{56} - 783862360 q^{57} - 22828679464 q^{58} - 43681325472 q^{60} + 40774631744 q^{62} - 14483624624 q^{63} + 55266728512 q^{64} - 80291360 q^{65} - 83128448712 q^{66} - 83303223768 q^{68} + 100120831168 q^{70} - 11769460176 q^{71} + 112862051672 q^{72} + 32304890372 q^{73} - 102103996184 q^{74} - 97501928568 q^{76} + 144445560944 q^{78} + 57291670496 q^{79} + 128322038976 q^{80} - 23172996094 q^{81} - 123309929604 q^{82} - 199990865984 q^{84} + 130201908124 q^{86} - 73844404272 q^{87} + 169216119632 q^{88} - 108276011804 q^{89} - 228506434984 q^{90} - 141139403232 q^{92} + 145336130016 q^{94} + 70146835376 q^{95} + 109508068928 q^{96} + 60995282900 q^{97} - 6194726778 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 48 x^{8} + 8836 x^{7} - 92554 x^{6} - 2200342 x^{5} - 88894588 x^{4} + \cdots + 34431558027300 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} + 4\nu - 39 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 73 \nu^{9} + 4718 \nu^{8} + 80354 \nu^{7} - 2051134 \nu^{6} - 24450588 \nu^{5} + \cdots + 403377232281156 ) / 360777252864 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 41 \nu^{9} - 13102 \nu^{8} - 155170 \nu^{7} + 2037758 \nu^{6} + 23134236 \nu^{5} + \cdots - 643828474284996 ) / 45097156608 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 823 \nu^{9} - 6510 \nu^{8} - 121570 \nu^{7} + 10009406 \nu^{6} - 66436068 \nu^{5} + \cdots - 10\!\cdots\!40 ) / 120259084288 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 817 \nu^{9} + 36510 \nu^{8} - 446926 \nu^{7} + 8893202 \nu^{6} + 459904388 \nu^{5} + \cdots - 14\!\cdots\!28 ) / 60129542144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19 \nu^{9} + 346 \nu^{8} - 10218 \nu^{7} - 77130 \nu^{6} - 1064180 \nu^{5} + \cdots - 6502213267284 ) / 704643072 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2899 \nu^{9} - 9690 \nu^{8} + 835178 \nu^{7} + 607178 \nu^{6} - 145501964 \nu^{5} + \cdots + 17\!\cdots\!32 ) / 45097156608 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12521 \nu^{9} - 116434 \nu^{8} - 5184222 \nu^{7} + 31910658 \nu^{6} + 698838116 \nu^{5} + \cdots - 11\!\cdots\!64 ) / 180388626432 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 2\beta _1 + 39 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} + \beta_{8} - \beta_{5} - 3\beta_{4} - 5\beta_{3} - 4\beta_{2} + 67\beta _1 - 21104 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 5 \beta_{9} - 21 \beta_{8} - 8 \beta_{7} - 20 \beta_{6} - 33 \beta_{5} - 109 \beta_{4} + \cdots + 273442 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 95 \beta_{9} - 145 \beta_{8} - 408 \beta_{7} + 244 \beta_{6} - 1177 \beta_{5} - 129 \beta_{4} + \cdots + 3711000 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1037 \beta_{9} + 1587 \beta_{8} - 128 \beta_{7} + 1392 \beta_{6} + 31845 \beta_{5} - 19209 \beta_{4} + \cdots + 320843183 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 32187 \beta_{9} - 84261 \beta_{8} - 485424 \beta_{7} + 12840 \beta_{6} + 339441 \beta_{5} + \cdots - 14601281944 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 251631 \beta_{9} + 2142913 \beta_{8} + 4932632 \beta_{7} + 1984220 \beta_{6} + 2358769 \beta_{5} + \cdots - 111706686750 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 28743150 \beta_{9} - 3619858 \beta_{8} - 27438368 \beta_{7} - 3161440 \beta_{6} + 33384786 \beta_{5} + \cdots - 461581578530 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−23.4302 − 2.98090i
−23.4302 + 2.98090i
−13.4162 − 18.9166i
−13.4162 + 18.9166i
7.36621 − 21.0240i
7.36621 + 21.0240i
9.13304 − 20.2317i
9.13304 + 20.2317i
20.8472 − 5.89062i
20.8472 + 5.89062i
−44.8604 − 5.96180i − 614.710i 1976.91 + 534.897i − 4397.69i −3664.78 + 27576.1i −41556.6 −85496.2 − 35781.7i −200721. −26218.2 + 197282.i
5.2 −44.8604 + 5.96180i 614.710i 1976.91 − 534.897i 4397.69i −3664.78 − 27576.1i −41556.6 −85496.2 + 35781.7i −200721. −26218.2 − 197282.i
5.3 −24.8325 − 37.8331i 102.287i −814.694 + 1878.98i 2319.82i 3869.83 − 2540.04i 38165.0 91318.7 − 15837.4i 166684. 87766.1 − 57606.9i
5.4 −24.8325 + 37.8331i − 102.287i −814.694 − 1878.98i − 2319.82i 3869.83 + 2540.04i 38165.0 91318.7 + 15837.4i 166684. 87766.1 + 57606.9i
5.5 16.7324 − 42.0479i 284.352i −1488.05 − 1407.13i − 10467.0i 11956.4 + 4757.90i −70503.2 −84065.4 + 39024.9i 96290.8 −440117. − 175139.i
5.6 16.7324 + 42.0479i − 284.352i −1488.05 + 1407.13i 10467.0i 11956.4 − 4757.90i −70503.2 −84065.4 − 39024.9i 96290.8 −440117. + 175139.i
5.7 20.2661 − 40.4634i − 712.065i −1226.57 − 1640.07i 7122.63i −28812.6 − 14430.8i 46911.8 −91220.5 + 16393.5i −329890. 288206. + 144348.i
5.8 20.2661 + 40.4634i 712.065i −1226.57 + 1640.07i − 7122.63i −28812.6 + 14430.8i 46911.8 −91220.5 − 16393.5i −329890. 288206. − 144348.i
5.9 43.6944 − 11.7812i 381.717i 1770.40 − 1029.55i 8937.55i 4497.10 + 16678.9i 10175.0 65227.5 − 65843.1i 31439.0 105295. + 390521.i
5.10 43.6944 + 11.7812i − 381.717i 1770.40 + 1029.55i − 8937.55i 4497.10 − 16678.9i 10175.0 65227.5 + 65843.1i 31439.0 105295. − 390521.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.12.b.a ✓ 10
3.b odd 2 1 72.12.d.b 10
4.b odd 2 1 32.12.b.a 10
8.b even 2 1 inner 8.12.b.a ✓ 10
8.d odd 2 1 32.12.b.a 10
16.e even 4 2 256.12.a.p 10
16.f odd 4 2 256.12.a.o 10
24.h odd 2 1 72.12.d.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.12.b.a ✓ 10 1.a even 1 1 trivial
8.12.b.a ✓ 10 8.b even 2 1 inner
32.12.b.a 10 4.b odd 2 1
32.12.b.a 10 8.d odd 2 1
72.12.d.b 10 3.b odd 2 1
72.12.d.b 10 24.h odd 2 1
256.12.a.o 10 16.f odd 4 2
256.12.a.p 10 16.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 36\!\cdots\!68 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots - 53\!\cdots\!92)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots - 18\!\cdots\!08)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 57\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots - 62\!\cdots\!32)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 52\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots - 14\!\cdots\!72)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 45\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 95\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 58\!\cdots\!68)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 38\!\cdots\!20)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 61\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 91\!\cdots\!60)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 31\!\cdots\!32)^{2} \) Copy content Toggle raw display
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