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Properties

Label 1014.3.f.h
Level $1014$
Weight $3$
Character orbit 1014.f
Analytic conductor $27.629$
Analytic rank $0$
Dimension $8$
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] = [1014,3,Mod(577,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1014.577");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1014.f (of order \(4\), degree \(2\), 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: \(27.6294988061\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 82x^{5} + 5053x^{4} - 6736x^{3} + 6728x^{2} + 275384x + 5635876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + (\beta_{6} - \beta_{3}) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{5} + (2 \beta_{3} + \beta_{2} - 1) q^{6} + (2 \beta_{6} + \beta_{5} + \beta_{2} - 1) q^{7} + ( - 2 \beta_{2} + 2) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + (\beta_{6} - \beta_{3}) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{5} + (2 \beta_{3} + \beta_{2} - 1) q^{6} + (2 \beta_{6} + \beta_{5} + \beta_{2} - 1) q^{7} + ( - 2 \beta_{2} + 2) q^{8} + 3 q^{9} + (\beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{10} + ( - 4 \beta_{6} - 2 \beta_{5}) q^{11} + ( - 2 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{12} + ( - 2 \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_1) q^{14} + (\beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{15} - 4 q^{16} + ( - 6 \beta_{6} + \beta_{5} + \beta_{4} - 7 \beta_{3} + \beta_{2} + \beta_1 + 6) q^{17} + ( - 3 \beta_{2} - 3) q^{18} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 6) q^{19} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 2) q^{20} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} + 3) q^{21} + (4 \beta_{6} + 2 \beta_{5} - 4 \beta_{3} - 2 \beta_1 + 4) q^{22} + (2 \beta_{6} + 2 \beta_{4} + 20 \beta_{2} - 2) q^{23} + (4 \beta_{6} + 2 \beta_{2} - 2) q^{24} + (5 \beta_{6} + 2 \beta_{5} + \beta_{4} + 4 \beta_{3} + 30 \beta_{2} + 2 \beta_1 - 5) q^{25} + (3 \beta_{6} - 3 \beta_{3}) q^{27} + ( - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{28} + (3 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 5 \beta_{3} + 2 \beta_1 - 10) q^{29} + ( - 3 \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - 4 \beta_{2} - \beta_1 + 3) q^{30} + (\beta_{7} + \beta_{6} + \beta_{4} + 17 \beta_{3} - 3 \beta_{2} + \beta_1 - 21) q^{31} + (4 \beta_{2} + 4) q^{32} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + \cdots - 4) q^{33}+ \cdots + ( - 12 \beta_{6} - 6 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 6 q^{5} - 2 q^{7} + 16 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 6 q^{5} - 2 q^{7} + 16 q^{8} + 24 q^{9} - 12 q^{11} + 4 q^{14} + 6 q^{15} - 32 q^{16} - 24 q^{18} - 44 q^{19} + 12 q^{20} + 18 q^{21} + 24 q^{22} - 4 q^{28} - 72 q^{29} - 94 q^{31} + 32 q^{32} - 36 q^{33} + 60 q^{34} + 408 q^{35} - 46 q^{37} - 24 q^{40} - 30 q^{41} - 36 q^{42} - 24 q^{44} - 18 q^{45} + 144 q^{46} + 300 q^{47} + 208 q^{50} + 84 q^{53} - 792 q^{55} + 24 q^{57} + 72 q^{58} - 12 q^{59} - 12 q^{60} + 180 q^{61} - 6 q^{63} + 72 q^{66} - 74 q^{67} - 120 q^{68} - 408 q^{70} + 156 q^{71} + 48 q^{72} + 16 q^{73} + 92 q^{74} + 88 q^{76} - 96 q^{79} + 24 q^{80} + 72 q^{81} + 36 q^{84} + 234 q^{85} + 168 q^{86} - 60 q^{87} + 228 q^{89} - 288 q^{92} - 198 q^{93} - 600 q^{94} + 2 q^{97} - 32 q^{98} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 82x^{5} + 5053x^{4} - 6736x^{3} + 6728x^{2} + 275384x + 5635876 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4081829 \nu^{7} + 125878448 \nu^{6} - 175318185 \nu^{5} - 167602931 \nu^{4} - 13767350850 \nu^{3} + 914252484693 \nu^{2} + \cdots - 561922481148 ) / 28390156409030 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 207487050 \nu^{7} + 9646046383 \nu^{6} + 263049796262 \nu^{5} + 452388400182 \nu^{4} - 699596394446 \nu^{3} + \cdots + 20\!\cdots\!94 ) / 965265317907020 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 186368592 \nu^{7} + 13656504841 \nu^{6} + 24877044110 \nu^{5} + 30851132578 \nu^{4} + 372305475560 \nu^{3} + \cdots + 132247184234574 ) / 50803437784580 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 99170 \nu^{7} - 140821 \nu^{6} + 140781 \nu^{5} + 5777701 \nu^{4} + 747057527 \nu^{3} - 473888448 \nu^{2} + 473587124 \nu + 19380524092 ) / 23917570690 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5690069035 \nu^{7} + 3991178383 \nu^{6} - 3261499808 \nu^{5} - 237335849778 \nu^{4} + 16756634760889 \nu^{3} + \cdots - 378059897144086 ) / 965265317907020 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15163829403 \nu^{7} + 4995591495 \nu^{6} + 440871714386 \nu^{5} + 11542895672784 \nu^{4} - 43834774990983 \nu^{3} + \cdots + 30\!\cdots\!86 ) / 965265317907020 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{6} - \beta_{4} + 3\beta_{3} + 52\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 44\beta_{6} + 53\beta_{5} + 2\beta_{4} - 2\beta_{3} - 3\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 95\beta_{7} + 251\beta_{6} - 5\beta_{5} - 156\beta_{3} + 5\beta _1 - 2612 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -161\beta_{7} - 161\beta_{6} - 161\beta_{4} + 4239\beta_{3} + 545\beta_{2} - 2863\beta _1 - 3533 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9054\beta_{6} + 706\beta_{5} + 6941\beta_{4} - 15995\beta_{3} - 149842\beta_{2} + 706\beta _1 + 9054 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9760\beta_{7} + 313762\beta_{6} - 156783\beta_{5} - 9760\beta_{4} + 9760\beta_{3} + 57535\beta_{2} - 57535 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
5.02578 − 5.02578i
−5.39181 + 5.39181i
5.41254 − 5.41254i
−4.04651 + 4.04651i
5.02578 + 5.02578i
−5.39181 − 5.39181i
5.41254 + 5.41254i
−4.04651 − 4.04651i
−1.00000 + 1.00000i −1.73205 − 2.00000i −6.39181 + 6.39181i 1.73205 − 1.73205i −6.75784 − 6.75784i 2.00000 + 2.00000i 3.00000 − 12.7836i
577.2 −1.00000 + 1.00000i −1.73205 − 2.00000i 4.02578 − 4.02578i 1.73205 − 1.73205i 3.65976 + 3.65976i 2.00000 + 2.00000i 3.00000 8.05157i
577.3 −1.00000 + 1.00000i 1.73205 − 2.00000i −5.04651 + 5.04651i −1.73205 + 1.73205i −3.68049 − 3.68049i 2.00000 + 2.00000i 3.00000 − 10.0930i
577.4 −1.00000 + 1.00000i 1.73205 − 2.00000i 4.41254 − 4.41254i −1.73205 + 1.73205i 5.77857 + 5.77857i 2.00000 + 2.00000i 3.00000 8.82508i
775.1 −1.00000 − 1.00000i −1.73205 2.00000i −6.39181 − 6.39181i 1.73205 + 1.73205i −6.75784 + 6.75784i 2.00000 − 2.00000i 3.00000 12.7836i
775.2 −1.00000 − 1.00000i −1.73205 2.00000i 4.02578 + 4.02578i 1.73205 + 1.73205i 3.65976 − 3.65976i 2.00000 − 2.00000i 3.00000 − 8.05157i
775.3 −1.00000 − 1.00000i 1.73205 2.00000i −5.04651 − 5.04651i −1.73205 − 1.73205i −3.68049 + 3.68049i 2.00000 − 2.00000i 3.00000 10.0930i
775.4 −1.00000 − 1.00000i 1.73205 2.00000i 4.41254 + 4.41254i −1.73205 − 1.73205i 5.77857 − 5.77857i 2.00000 − 2.00000i 3.00000 − 8.82508i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 775.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.3.f.h 8
13.b even 2 1 1014.3.f.j 8
13.c even 3 1 78.3.l.c ✓ 8
13.d odd 4 1 inner 1014.3.f.h 8
13.d odd 4 1 1014.3.f.j 8
13.f odd 12 1 78.3.l.c ✓ 8
39.i odd 6 1 234.3.bb.d 8
39.k even 12 1 234.3.bb.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.3.l.c ✓ 8 13.c even 3 1
78.3.l.c ✓ 8 13.f odd 12 1
234.3.bb.d 8 39.i odd 6 1
234.3.bb.d 8 39.k even 12 1
1014.3.f.h 8 1.a even 1 1 trivial
1014.3.f.h 8 13.d odd 4 1 inner
1014.3.f.j 8 13.b even 2 1
1014.3.f.j 8 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1014, [\chi])\):

\( T_{5}^{8} + 6T_{5}^{7} + 18T_{5}^{6} - 282T_{5}^{5} + 4065T_{5}^{4} + 11916T_{5}^{3} + 38088T_{5}^{2} - 632592T_{5} + 5253264 \) Copy content Toggle raw display
\( T_{7}^{8} + 2T_{7}^{7} + 2T_{7}^{6} - 154T_{7}^{5} + 6817T_{7}^{4} + 1672T_{7}^{3} + 1568T_{7}^{2} - 117824T_{7} + 4426816 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 5253264 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 4426816 \) Copy content Toggle raw display
$11$ \( T^{8} + 12 T^{7} + \cdots + 973440000 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 1242 T^{6} + \cdots + 4567597056 \) Copy content Toggle raw display
$19$ \( T^{8} + 44 T^{7} + 968 T^{6} + \cdots + 1763584 \) Copy content Toggle raw display
$23$ \( T^{8} + 2760 T^{6} + \cdots + 17831863296 \) Copy content Toggle raw display
$29$ \( (T^{4} + 36 T^{3} - 987 T^{2} + \cdots + 220452)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 94 T^{7} + \cdots + 33544655104 \) Copy content Toggle raw display
$37$ \( T^{8} + 46 T^{7} + \cdots + 10744151716 \) Copy content Toggle raw display
$41$ \( T^{8} + 30 T^{7} + \cdots + 370617958656 \) Copy content Toggle raw display
$43$ \( T^{8} + 7158 T^{6} + \cdots + 1819196698176 \) Copy content Toggle raw display
$47$ \( T^{8} - 300 T^{7} + \cdots + 20494380893184 \) Copy content Toggle raw display
$53$ \( (T^{4} - 42 T^{3} - 2211 T^{2} + \cdots - 109128)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{7} + \cdots + 15611728564224 \) Copy content Toggle raw display
$61$ \( (T^{4} - 90 T^{3} - 5682 T^{2} + \cdots - 17180643)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 74 T^{7} + \cdots + 42600893548096 \) Copy content Toggle raw display
$71$ \( T^{8} - 156 T^{7} + \cdots + 1623606027264 \) Copy content Toggle raw display
$73$ \( T^{8} - 16 T^{7} + \cdots + 261324417601 \) Copy content Toggle raw display
$79$ \( (T^{4} + 48 T^{3} - 11667 T^{2} + \cdots - 8209344)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 682176 T^{5} + \cdots + 13865554427904 \) Copy content Toggle raw display
$89$ \( T^{8} - 228 T^{7} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{8} - 2 T^{7} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
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