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Properties

Label 833.4.a.g
Level $833$
Weight $4$
Character orbit 833.a
Self dual yes
Analytic conductor $49.149$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [833,4,Mod(1,833)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(833, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("833.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 833.a (trivial)

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: yes
Analytic conductor: \(49.1485910348\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 53x^{7} + 90x^{6} + 880x^{5} - 1087x^{4} - 4674x^{3} + 2515x^{2} + 1814x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 119)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} - 1) q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{6} - \beta_1 + 1) q^{5} + (\beta_{3} - \beta_1 - 1) q^{6} + (\beta_{8} + \beta_{7} + \beta_{4} + \cdots + 2) q^{8}+ \cdots + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{5} - 1) q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{6} - \beta_1 + 1) q^{5} + (\beta_{3} - \beta_1 - 1) q^{6} + (\beta_{8} + \beta_{7} + \beta_{4} + \cdots + 2) q^{8}+ \cdots + ( - 11 \beta_{8} + 15 \beta_{7} + \cdots - 302) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 2 q^{2} - 11 q^{3} + 38 q^{4} + 3 q^{5} - 9 q^{6} + 24 q^{8} + 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 2 q^{2} - 11 q^{3} + 38 q^{4} + 3 q^{5} - 9 q^{6} + 24 q^{8} + 74 q^{9} - 134 q^{10} - 8 q^{11} - 56 q^{12} - 164 q^{13} + 34 q^{15} + 178 q^{16} - 153 q^{17} + 98 q^{18} - 244 q^{19} + 41 q^{20} - 80 q^{22} - 14 q^{23} - 298 q^{24} + 684 q^{25} - 326 q^{26} - 218 q^{27} - 234 q^{29} - 335 q^{30} - 555 q^{31} - 181 q^{32} - 458 q^{33} - 34 q^{34} - 1221 q^{36} - 364 q^{37} + 714 q^{38} - 52 q^{39} - 123 q^{40} + 45 q^{41} - 135 q^{43} - 748 q^{44} + 844 q^{45} - 1576 q^{46} + 172 q^{47} + 949 q^{48} - 2901 q^{50} + 187 q^{51} + 1596 q^{52} + 101 q^{53} + 1163 q^{54} - 1260 q^{55} - 602 q^{57} + 1062 q^{58} - 280 q^{59} - 1727 q^{60} - 639 q^{61} + 1708 q^{62} - 2390 q^{64} + 638 q^{65} + 2476 q^{66} + 35 q^{67} - 646 q^{68} - 1288 q^{69} - 1616 q^{71} + 1335 q^{72} - 1049 q^{73} - 370 q^{74} - 1260 q^{75} - 4964 q^{76} - 4714 q^{78} + 2304 q^{79} + 3996 q^{80} - 791 q^{81} + 215 q^{82} - 2508 q^{83} - 51 q^{85} + 623 q^{86} - 166 q^{87} - 416 q^{88} - 2762 q^{89} - 2935 q^{90} - 2392 q^{92} + 2784 q^{93} + 862 q^{94} - 3462 q^{95} - 2928 q^{96} - 3107 q^{97} - 2396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 53x^{7} + 90x^{6} + 880x^{5} - 1087x^{4} - 4674x^{3} + 2515x^{2} + 1814x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 725 \nu^{8} - 40561 \nu^{7} - 35990 \nu^{6} + 1728180 \nu^{5} + 2133388 \nu^{4} + \cdots + 2882012 ) / 3388208 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6667 \nu^{8} - 15479 \nu^{7} + 358366 \nu^{6} + 843788 \nu^{5} - 5557212 \nu^{4} + \cdots + 22936788 ) / 3388208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14061 \nu^{8} + 28847 \nu^{7} + 785794 \nu^{6} - 1229500 \nu^{5} - 14101860 \nu^{4} + \cdots - 31983300 ) / 3388208 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 37659 \nu^{8} + 68585 \nu^{7} + 1904318 \nu^{6} - 2952436 \nu^{5} - 29732012 \nu^{4} + \cdots - 38173324 ) / 3388208 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 41373 \nu^{8} + 96807 \nu^{7} + 2163922 \nu^{6} - 4509364 \nu^{5} - 35178740 \nu^{4} + \cdots - 33613820 ) / 1694104 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 89413 \nu^{8} - 178135 \nu^{7} - 4686210 \nu^{6} + 8174940 \nu^{5} + 75914692 \nu^{4} + \cdots + 78172932 ) / 3388208 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{4} + \beta_{2} + 20\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - \beta_{6} - 5\beta_{5} + 3\beta_{4} - 2\beta_{3} + 25\beta_{2} + 246 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 39\beta_{8} + 35\beta_{7} + 8\beta_{6} + \beta_{5} + 42\beta_{4} - 6\beta_{3} + 36\beta_{2} + 436\beta _1 + 47 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -41\beta_{8} - 60\beta_{6} - 153\beta_{5} + 120\beta_{4} - 76\beta_{3} + 617\beta_{2} - 29\beta _1 + 5482 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1227 \beta_{8} + 1050 \beta_{7} + 382 \beta_{6} - 15 \beta_{5} + 1334 \beta_{4} - 334 \beta_{3} + \cdots + 756 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1246 \beta_{8} + 29 \beta_{7} - 2266 \beta_{6} - 3895 \beta_{5} + 3697 \beta_{4} - 2402 \beta_{3} + \cdots + 128026 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.08433
−4.05606
−2.64062
−0.411462
−0.0204472
0.923542
3.78610
4.42628
5.07700
−5.08433 2.28838 17.8505 18.7142 −11.6349 0 −50.0830 −21.7633 −95.1492
1.2 −4.05606 −2.41760 8.45161 −20.0958 9.80593 0 −1.83177 −21.1552 81.5097
1.3 −2.64062 −8.72657 −1.02714 19.5792 23.0435 0 23.8372 49.1531 −51.7011
1.4 −0.411462 2.90636 −7.83070 0.206478 −1.19585 0 6.51373 −18.5531 −0.0849579
1.5 −0.0204472 8.34282 −7.99958 11.9677 −0.170588 0 0.327147 42.6027 −0.244705
1.6 0.923542 −8.63083 −7.14707 −12.8089 −7.97094 0 −13.9890 47.4913 −11.8296
1.7 3.78610 −5.19121 6.33458 −4.98519 −19.6544 0 −6.30544 −0.0513763 −18.8744
1.8 4.42628 5.22352 11.5919 −16.9037 23.1207 0 15.8988 0.285172 −74.8205
1.9 5.07700 −4.79487 17.7759 7.32611 −24.3435 0 49.6323 −4.00927 37.1947
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(17\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 833.4.a.g 9
7.b odd 2 1 119.4.a.e ✓ 9
21.c even 2 1 1071.4.a.r 9
28.d even 2 1 1904.4.a.s 9
119.d odd 2 1 2023.4.a.h 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.4.a.e ✓ 9 7.b odd 2 1
833.4.a.g 9 1.a even 1 1 trivial
1071.4.a.r 9 21.c even 2 1
1904.4.a.s 9 28.d even 2 1
2023.4.a.h 9 119.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(833))\):

\( T_{2}^{9} - 2T_{2}^{8} - 53T_{2}^{7} + 90T_{2}^{6} + 880T_{2}^{5} - 1087T_{2}^{4} - 4674T_{2}^{3} + 2515T_{2}^{2} + 1814T_{2} + 36 \) Copy content Toggle raw display
\( T_{3}^{9} + 11 T_{3}^{8} - 98 T_{3}^{7} - 1251 T_{3}^{6} + 1978 T_{3}^{5} + 39497 T_{3}^{4} + \cdots + 1313648 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 2 T^{8} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{9} + 11 T^{8} + \cdots + 1313648 \) Copy content Toggle raw display
$5$ \( T^{9} - 3 T^{8} + \cdots - 143880840 \) Copy content Toggle raw display
$7$ \( T^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 9140353044480 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 221595169961984 \) Copy content Toggle raw display
$17$ \( (T + 17)^{9} \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 42\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots - 53\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots - 99\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 92\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 14\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots - 64\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 16\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 53\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 27\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 60\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 17\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 87\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 42\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 22\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
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