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Properties

Label 833.2.g.g
Level $833$
Weight $2$
Character orbit 833.g
Analytic conductor $6.652$
Analytic rank $0$
Dimension $16$
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] = [833,2,Mod(344,833)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(833, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("833.344");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.g (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: \(6.65153848837\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 230x^{12} + 1126x^{10} + 2987x^{8} + 4170x^{6} + 2679x^{4} + 502x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 119)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{11} q^{3} + ( - \beta_{11} - \beta_{8} + \beta_{4} - 1) q^{4} + (\beta_{15} - \beta_{13} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{13} - \beta_{12} + \beta_{7}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{11} q^{3} + ( - \beta_{11} - \beta_{8} + \beta_{4} - 1) q^{4} + (\beta_{15} - \beta_{13} + \cdots + \beta_1) q^{5}+ \cdots + (2 \beta_{15} - 3 \beta_{9} - \beta_{7} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 8 q^{5} - 6 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 8 q^{5} - 6 q^{6} + 6 q^{10} + 18 q^{11} - 26 q^{12} + 8 q^{16} - 10 q^{17} + 40 q^{18} - 12 q^{20} + 26 q^{22} + 12 q^{23} - 14 q^{24} + 30 q^{27} - 16 q^{29} + 44 q^{30} + 6 q^{31} + 8 q^{33} - 20 q^{34} + 2 q^{37} + 32 q^{38} - 14 q^{39} - 72 q^{40} + 2 q^{41} - 24 q^{44} - 4 q^{46} + 36 q^{47} + 36 q^{48} - 40 q^{50} + 20 q^{51} - 12 q^{52} + 18 q^{54} + 36 q^{55} - 24 q^{57} - 8 q^{58} + 52 q^{61} + 18 q^{62} + 12 q^{64} + 2 q^{65} + 28 q^{67} - 24 q^{68} - 4 q^{69} + 40 q^{71} - 36 q^{72} + 16 q^{73} - 60 q^{74} + 2 q^{75} + 32 q^{78} + 6 q^{79} - 30 q^{80} + 40 q^{82} + 34 q^{85} + 4 q^{86} - 30 q^{88} - 8 q^{89} - 10 q^{90} - 20 q^{92} + 12 q^{95} + 30 q^{96} - 38 q^{97} - 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 230x^{12} + 1126x^{10} + 2987x^{8} + 4170x^{6} + 2679x^{4} + 502x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{14} + 101\nu^{12} + 764\nu^{10} + 2685\nu^{8} + 4435\nu^{6} + 3029\nu^{4} + 490\nu^{2} - 1 ) / 44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 29\nu^{13} + 331\nu^{11} + 1890\nu^{9} + 5672\nu^{7} + 8605\nu^{5} + 5708\nu^{3} + 992\nu ) / 44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{14} - 123\nu^{12} - 1171\nu^{10} - 5358\nu^{8} - 11750\nu^{6} - 10289\nu^{4} - 1590\nu^{2} + 166 ) / 44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - 40 \nu^{13} + 47 \nu^{12} - 551 \nu^{11} + 431 \nu^{10} - 3540 \nu^{9} + \cdots + 136 ) / 88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{14} - 195\nu^{12} - 1626\nu^{10} - 6549\nu^{8} - 13043\nu^{6} - 11467\nu^{4} - 3038\nu^{2} - 161 ) / 44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{13} + 23\nu^{11} + 205\nu^{9} + 884\nu^{7} + 1860\nu^{5} + 1641\nu^{3} + 346\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8 \nu^{15} - 5 \nu^{14} - 177 \nu^{13} - 123 \nu^{12} - 1504 \nu^{11} - 1171 \nu^{10} - 6100 \nu^{9} + \cdots + 34 ) / 88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{15} + 4 \nu^{14} + 76 \nu^{13} + 83 \nu^{12} + 773 \nu^{11} + 653 \nu^{10} + 4031 \nu^{9} + \cdots - 91 ) / 44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{15} + 4 \nu^{14} - 76 \nu^{13} + 83 \nu^{12} - 773 \nu^{11} + 653 \nu^{10} - 4031 \nu^{9} + \cdots - 91 ) / 44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8 \nu^{15} - 5 \nu^{14} + 177 \nu^{13} - 123 \nu^{12} + 1504 \nu^{11} - 1171 \nu^{10} + 6100 \nu^{9} + \cdots + 34 ) / 88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10 \nu^{15} + 13 \nu^{14} + 224 \nu^{13} + 278 \nu^{12} + 1957 \nu^{11} + 2279 \nu^{10} + 8439 \nu^{9} + \cdots + 59 ) / 88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10 \nu^{15} - 13 \nu^{14} + 224 \nu^{13} - 278 \nu^{12} + 1957 \nu^{11} - 2279 \nu^{10} + 8439 \nu^{9} + \cdots - 59 ) / 88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11 \nu^{15} - 15 \nu^{14} - 264 \nu^{13} - 325 \nu^{12} - 2508 \nu^{11} - 2710 \nu^{10} - 11979 \nu^{9} + \cdots - 195 ) / 88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14\nu^{15} + 318\nu^{13} + 2841\nu^{11} + 12721\nu^{9} + 30161\nu^{7} + 36650\nu^{5} + 20138\nu^{3} + 3493\nu ) / 44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{8} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} + 2\beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + \beta_{13} + 7 \beta_{11} - \beta_{10} - \beta_{9} + 7 \beta_{8} - 2 \beta_{6} - \beta_{5} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{15} - \beta_{14} + 10 \beta_{13} + 9 \beta_{12} - 7 \beta_{10} + 7 \beta_{9} + \cdots + 12 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{14} - 11 \beta_{13} - 46 \beta_{11} + 8 \beta_{10} + 8 \beta_{9} - 46 \beta_{8} + 21 \beta_{6} + \cdots - 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 68 \beta_{15} + 11 \beta_{14} - 78 \beta_{13} - 67 \beta_{12} - 3 \beta_{11} + 43 \beta_{10} + \cdots - 56 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 89 \beta_{14} + 93 \beta_{13} - 4 \beta_{12} + 298 \beta_{11} - 51 \beta_{10} - 51 \beta_{9} + \cdots + 462 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 484 \beta_{15} - 93 \beta_{14} + 557 \beta_{13} + 464 \beta_{12} + 48 \beta_{11} - 260 \beta_{10} + \cdots + 292 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 646 \beta_{14} - 714 \beta_{13} + 68 \beta_{12} - 1922 \beta_{11} + 305 \beta_{10} + 305 \beta_{9} + \cdots - 2904 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3354 \beta_{15} + 718 \beta_{14} - 3823 \beta_{13} - 3105 \beta_{12} - 523 \beta_{11} + \cdots - 1653 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4465 \beta_{14} + 5237 \beta_{13} - 772 \beta_{12} + 12385 \beta_{11} - 1776 \beta_{10} - 1776 \beta_{9} + \cdots + 18731 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 22935 \beta_{15} - 5313 \beta_{14} + 25737 \beta_{13} + 20424 \beta_{12} + 4841 \beta_{11} + \cdots + 9920 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 30126 \beta_{14} - 37478 \beta_{13} + 7352 \beta_{12} - 79862 \beta_{11} + 10168 \beta_{10} + \cdots - 122617 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 155742 \beta_{15} + 38402 \beta_{14} - 171616 \beta_{13} - 133214 \beta_{12} - 40980 \beta_{11} + \cdots - 61913 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/833\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(785\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
344.1
− 2.59345i
− 1.92639i
− 0.545814i
0.0448732i
1.28565i
1.32517i
1.94979i
2.46018i
− 2.46018i
− 1.94979i
− 1.32517i
− 1.28565i
− 0.0448732i
0.545814i
1.92639i
2.59345i
− 2.59345i 0.755051 − 0.755051i −4.72596 −1.71665 + 1.71665i −1.95819 − 1.95819i 0 7.06964i 1.85979i 4.45205 + 4.45205i
344.2 − 1.92639i 1.09360 − 1.09360i −1.71099 2.70734 − 2.70734i −2.10671 − 2.10671i 0 − 0.556738i 0.608066i −5.21541 − 5.21541i
344.3 − 0.545814i −1.37781 + 1.37781i 1.70209 2.54797 − 2.54797i 0.752026 + 0.752026i 0 − 2.02065i − 0.796704i −1.39071 − 1.39071i
344.4 0.0448732i 0.423280 − 0.423280i 1.99799 −0.872695 + 0.872695i 0.0189939 + 0.0189939i 0 0.179402i 2.64167i −0.0391606 − 0.0391606i
344.5 1.28565i −2.27435 + 2.27435i 0.347116 −1.87424 + 1.87424i −2.92401 − 2.92401i 0 3.01756i − 7.34535i −2.40961 − 2.40961i
344.6 1.32517i 0.651644 − 0.651644i 0.243930 −0.421668 + 0.421668i 0.863538 + 0.863538i 0 2.97358i 2.15072i −0.558781 − 0.558781i
344.7 1.94979i −1.10092 + 1.10092i −1.80166 1.50610 − 1.50610i −2.14656 − 2.14656i 0 0.386712i 0.575954i 2.93657 + 2.93657i
344.8 2.46018i 1.82950 − 1.82950i −4.05250 2.12385 − 2.12385i 4.50090 + 4.50090i 0 − 5.04951i − 3.69415i 5.22506 + 5.22506i
540.1 − 2.46018i 1.82950 + 1.82950i −4.05250 2.12385 + 2.12385i 4.50090 − 4.50090i 0 5.04951i 3.69415i 5.22506 − 5.22506i
540.2 − 1.94979i −1.10092 − 1.10092i −1.80166 1.50610 + 1.50610i −2.14656 + 2.14656i 0 − 0.386712i − 0.575954i 2.93657 − 2.93657i
540.3 − 1.32517i 0.651644 + 0.651644i 0.243930 −0.421668 − 0.421668i 0.863538 − 0.863538i 0 − 2.97358i − 2.15072i −0.558781 + 0.558781i
540.4 − 1.28565i −2.27435 − 2.27435i 0.347116 −1.87424 − 1.87424i −2.92401 + 2.92401i 0 − 3.01756i 7.34535i −2.40961 + 2.40961i
540.5 − 0.0448732i 0.423280 + 0.423280i 1.99799 −0.872695 − 0.872695i 0.0189939 − 0.0189939i 0 − 0.179402i − 2.64167i −0.0391606 + 0.0391606i
540.6 0.545814i −1.37781 − 1.37781i 1.70209 2.54797 + 2.54797i 0.752026 − 0.752026i 0 2.02065i 0.796704i −1.39071 + 1.39071i
540.7 1.92639i 1.09360 + 1.09360i −1.71099 2.70734 + 2.70734i −2.10671 + 2.10671i 0 0.556738i − 0.608066i −5.21541 + 5.21541i
540.8 2.59345i 0.755051 + 0.755051i −4.72596 −1.71665 − 1.71665i −1.95819 + 1.95819i 0 − 7.06964i − 1.85979i 4.45205 − 4.45205i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 344.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 833.2.g.g 16
7.b odd 2 1 833.2.g.f 16
7.c even 3 2 119.2.n.b ✓ 32
7.d odd 6 2 833.2.o.d 32
17.c even 4 1 inner 833.2.g.g 16
119.f odd 4 1 833.2.g.f 16
119.m odd 12 2 833.2.o.d 32
119.n even 12 2 119.2.n.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.2.n.b ✓ 32 7.c even 3 2
119.2.n.b ✓ 32 119.n even 12 2
833.2.g.f 16 7.b odd 2 1
833.2.g.f 16 119.f odd 4 1
833.2.g.g 16 1.a even 1 1 trivial
833.2.g.g 16 17.c even 4 1 inner
833.2.o.d 32 7.d odd 6 2
833.2.o.d 32 119.m odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(833, [\chi])\):

\( T_{2}^{16} + 24T_{2}^{14} + 230T_{2}^{12} + 1126T_{2}^{10} + 2987T_{2}^{8} + 4170T_{2}^{6} + 2679T_{2}^{4} + 502T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{16} - 10 T_{3}^{13} + 90 T_{3}^{12} - 72 T_{3}^{11} + 50 T_{3}^{10} - 316 T_{3}^{9} + 1335 T_{3}^{8} + \cdots + 529 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 24 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} - 10 T^{13} + \cdots + 529 \) Copy content Toggle raw display
$5$ \( T^{16} - 8 T^{15} + \cdots + 174724 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} - 18 T^{15} + \cdots + 25 \) Copy content Toggle raw display
$13$ \( (T^{8} - 49 T^{6} + \cdots - 361)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{16} + 74 T^{14} + \cdots + 1444 \) Copy content Toggle raw display
$23$ \( T^{16} - 12 T^{15} + \cdots + 16384 \) Copy content Toggle raw display
$29$ \( T^{16} + 16 T^{15} + \cdots + 17322244 \) Copy content Toggle raw display
$31$ \( T^{16} - 6 T^{15} + \cdots + 169744 \) Copy content Toggle raw display
$37$ \( T^{16} - 2 T^{15} + \cdots + 70023424 \) Copy content Toggle raw display
$41$ \( T^{16} - 2 T^{15} + \cdots + 1444 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 4388267536 \) Copy content Toggle raw display
$47$ \( (T^{8} - 18 T^{7} + \cdots + 289882)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 193627225 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2073818885776 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1733223424 \) Copy content Toggle raw display
$67$ \( (T^{8} - 14 T^{7} + \cdots - 7821278)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 7742064121 \) Copy content Toggle raw display
$73$ \( T^{16} - 16 T^{15} + \cdots + 85414564 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 31488857401 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 177989484544 \) Copy content Toggle raw display
$89$ \( (T^{8} + 4 T^{7} + \cdots - 46664)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 22705034400400 \) Copy content Toggle raw display
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