Properties

Label 1183.2.a.q
Level $1183$
Weight $2$
Character orbit 1183.a
Self dual yes
Analytic conductor $9.446$
Analytic rank $0$
Dimension $12$
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] = [1183,2,Mod(1,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1183.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.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: \(9.44630255912\)
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: \( x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 80 x^{8} - 246 x^{7} - 199 x^{6} + 562 x^{5} + 262 x^{4} - 542 x^{3} - 157 x^{2} + 183 x + 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{6} + 1) q^{3} + (\beta_{5} - \beta_{4} + 2) q^{4} + ( - \beta_{11} - \beta_{7} - \beta_{4} + 1) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{4} + \beta_{2} - \beta_1) q^{6} - q^{7} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{8} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{6} + 1) q^{3} + (\beta_{5} - \beta_{4} + 2) q^{4} + ( - \beta_{11} - \beta_{7} - \beta_{4} + 1) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{4} + \beta_{2} - \beta_1) q^{6} - q^{7} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{8} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_1 + 3) q^{9} + (\beta_{10} - \beta_{9} - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{10} + (\beta_{10} - \beta_{9} + \beta_{6} - \beta_{5} - \beta_{2} - 1) q^{11} + (\beta_{11} - \beta_{10} + 3 \beta_{9} + 2 \beta_{8} + \beta_{5} - \beta_{4} - \beta_{3} + 1) q^{12} + \beta_1 q^{14} + ( - 2 \beta_{10} + 4 \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 - 2) q^{15} + ( - \beta_{9} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{16} + ( - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 3) q^{17} + ( - \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + \cdots + 2) q^{18}+ \cdots + ( - 2 \beta_{11} - 4 \beta_{10} - 6 \beta_{9} - 6 \beta_{8} + \beta_{7} + 2 \beta_{6} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} + 8 q^{3} + 15 q^{4} + 4 q^{5} - 2 q^{6} - 12 q^{7} - 12 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} + 8 q^{3} + 15 q^{4} + 4 q^{5} - 2 q^{6} - 12 q^{7} - 12 q^{8} + 26 q^{9} - 6 q^{10} - 12 q^{11} + 13 q^{12} + 3 q^{14} - 11 q^{15} + 13 q^{16} + 31 q^{17} + 29 q^{18} + 3 q^{19} + 18 q^{20} - 8 q^{21} - 4 q^{22} + 18 q^{23} + 6 q^{24} + 32 q^{25} + 32 q^{27} - 15 q^{28} + 15 q^{29} - 10 q^{30} + 21 q^{31} + 3 q^{32} + 29 q^{33} - 3 q^{34} - 4 q^{35} + 49 q^{36} - 5 q^{37} + 45 q^{38} - 20 q^{40} + 16 q^{41} + 2 q^{42} - 22 q^{43} - 35 q^{44} - 5 q^{45} - 2 q^{46} + 4 q^{47} + 11 q^{48} + 12 q^{49} - 13 q^{50} + 18 q^{51} + 53 q^{53} + 5 q^{54} - 26 q^{55} + 12 q^{56} + 8 q^{57} - 32 q^{58} + 26 q^{59} - 38 q^{60} + 22 q^{61} + 19 q^{62} - 26 q^{63} + 2 q^{64} - 34 q^{66} - 12 q^{67} + 34 q^{68} + 3 q^{69} + 6 q^{70} - 21 q^{71} + 4 q^{72} + 15 q^{73} - 40 q^{74} + 15 q^{75} - 43 q^{76} + 12 q^{77} + 2 q^{79} - 13 q^{80} + 36 q^{81} - 32 q^{82} + 9 q^{83} - 13 q^{84} + 39 q^{85} - 44 q^{86} + 27 q^{87} - 48 q^{88} + 22 q^{89} - 26 q^{90} + 52 q^{92} + 53 q^{93} - 44 q^{94} + 29 q^{95} + 114 q^{96} - 9 q^{97} - 3 q^{98} - 37 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 80 x^{8} - 246 x^{7} - 199 x^{6} + 562 x^{5} + 262 x^{4} - 542 x^{3} - 157 x^{2} + 183 x + 29 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 202 \nu^{11} + 1816 \nu^{10} - 5010 \nu^{9} - 29197 \nu^{8} + 39197 \nu^{7} + 161452 \nu^{6} - 110863 \nu^{5} - 346540 \nu^{4} + 77279 \nu^{3} + 193411 \nu^{2} + \cdots + 25085 ) / 14629 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 202 \nu^{11} + 1816 \nu^{10} - 5010 \nu^{9} - 29197 \nu^{8} + 39197 \nu^{7} + 161452 \nu^{6} - 110863 \nu^{5} - 346540 \nu^{4} + 91908 \nu^{3} + 193411 \nu^{2} + \cdots + 10456 ) / 14629 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 266 \nu^{11} + 1812 \nu^{10} - 9639 \nu^{9} - 26136 \nu^{8} + 98110 \nu^{7} + 125845 \nu^{6} - 384542 \nu^{5} - 239507 \nu^{4} + 573947 \nu^{3} + 165757 \nu^{2} + \cdots - 14765 ) / 14629 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 266 \nu^{11} + 1812 \nu^{10} - 9639 \nu^{9} - 26136 \nu^{8} + 98110 \nu^{7} + 125845 \nu^{6} - 384542 \nu^{5} - 239507 \nu^{4} + 573947 \nu^{3} + 180386 \nu^{2} + \cdots - 73281 ) / 14629 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 721 \nu^{11} - 2788 \nu^{10} - 8033 \nu^{9} + 41570 \nu^{8} + 13387 \nu^{7} - 212101 \nu^{6} + 95671 \nu^{5} + 447600 \nu^{4} - 310269 \nu^{3} - 374170 \nu^{2} + \cdots + 63537 ) / 14629 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1025 \nu^{11} - 2807 \nu^{10} - 19049 \nu^{9} + 45138 \nu^{8} + 135962 \nu^{7} - 245916 \nu^{6} - 469197 \nu^{5} + 506165 \nu^{4} + 761552 \nu^{3} - 264148 \nu^{2} + \cdots - 16033 ) / 14629 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1181 \nu^{11} - 6474 \nu^{10} - 9303 \nu^{9} + 90086 \nu^{8} - 23075 \nu^{7} - 408596 \nu^{6} + 301924 \nu^{5} + 705799 \nu^{4} - 620810 \nu^{3} - 446758 \nu^{2} + \cdots + 60552 ) / 14629 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1429 \nu^{11} + 825 \nu^{10} - 29069 \nu^{9} - 13256 \nu^{8} + 214356 \nu^{7} + 76988 \nu^{6} - 690923 \nu^{5} - 201544 \nu^{4} + 930739 \nu^{3} + 225077 \nu^{2} + \cdots - 68266 ) / 14629 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2354 \nu^{11} - 5633 \nu^{10} - 34485 \nu^{9} + 79215 \nu^{8} + 175064 \nu^{7} - 364728 \nu^{6} - 391458 \nu^{5} + 632025 \nu^{4} + 415204 \nu^{3} - 345129 \nu^{2} + \cdots + 5396 ) / 14629 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4628 \nu^{11} + 11261 \nu^{10} + 66841 \nu^{9} - 158261 \nu^{8} - 324031 \nu^{7} + 725177 \nu^{6} + 627337 \nu^{5} - 1236319 \nu^{4} - 443637 \nu^{3} + \cdots + 5226 ) / 14629 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + \beta_{7} + 7\beta_{5} - 7\beta_{4} + \beta_{3} + \beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{11} - \beta_{10} - 3\beta_{9} - 2\beta_{8} - \beta_{6} + \beta_{4} + 8\beta_{3} - 7\beta_{2} + 28\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{11} - 3 \beta_{10} - 10 \beta_{9} - \beta_{8} + 11 \beta_{7} - 2 \beta_{6} + 45 \beta_{5} - 48 \beta_{4} + 10 \beta_{3} + 11 \beta_{2} - \beta _1 + 131 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 23 \beta_{11} - 9 \beta_{10} - 37 \beta_{9} - 25 \beta_{8} + \beta_{7} - 13 \beta_{6} + 10 \beta_{4} + 57 \beta_{3} - 43 \beta_{2} + 166 \beta _1 + 43 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 27 \beta_{11} - 36 \beta_{10} - 84 \beta_{9} - 19 \beta_{8} + 93 \beta_{7} - 24 \beta_{6} + 289 \beta_{5} - 330 \beta_{4} + 80 \beta_{3} + 94 \beta_{2} - 14 \beta _1 + 812 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 198 \beta_{11} - 61 \beta_{10} - 331 \beta_{9} - 231 \beta_{8} + 14 \beta_{7} - 120 \beta_{6} - \beta_{5} + 67 \beta_{4} + 396 \beta_{3} - 259 \beta_{2} + 1021 \beta _1 + 265 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 257 \beta_{11} - 306 \beta_{10} - 658 \beta_{9} - 224 \beta_{8} + 710 \beta_{7} - 212 \beta_{6} + 1874 \beta_{5} - 2277 \beta_{4} + 593 \beta_{3} + 734 \beta_{2} - 132 \beta _1 + 5167 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1539 \beta_{11} - 370 \beta_{10} - 2625 \beta_{9} - 1909 \beta_{8} + 136 \beta_{7} - 967 \beta_{6} - 16 \beta_{5} + 356 \beta_{4} + 2724 \beta_{3} - 1555 \beta_{2} + 6448 \beta _1 + 1686 \) Copy content Toggle raw display

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
2.62803
2.47725
2.23724
2.06743
0.983820
0.842530
−0.149660
−0.961590
−1.10989
−1.35819
−2.07140
−2.58557
−2.62803 −1.76762 4.90656 2.94291 4.64535 −1.00000 −7.63852 0.124466 −7.73406
1.2 −2.47725 0.982981 4.13677 −1.35413 −2.43509 −1.00000 −5.29330 −2.03375 3.35452
1.3 −2.23724 3.02592 3.00523 3.28547 −6.76971 −1.00000 −2.24893 6.15622 −7.35037
1.4 −2.06743 2.11889 2.27425 −2.43928 −4.38065 −1.00000 −0.566992 1.48970 5.04303
1.5 −0.983820 −1.57171 −1.03210 −0.398447 1.54628 −1.00000 2.98304 −0.529731 0.392000
1.6 −0.842530 0.161973 −1.29014 3.72786 −0.136467 −1.00000 2.77204 −2.97376 −3.14083
1.7 0.149660 2.76031 −1.97760 −4.13443 0.413107 −1.00000 −0.595288 4.61930 −0.618759
1.8 0.961590 −1.98737 −1.07534 −3.39320 −1.91103 −1.00000 −2.95722 0.949635 −3.26287
1.9 1.10989 0.955760 −0.768150 3.55862 1.06079 −1.00000 −3.07233 −2.08652 3.94966
1.10 1.35819 3.39737 −0.155322 0.772491 4.61428 −1.00000 −2.92733 8.54215 1.04919
1.11 2.07140 −3.01646 2.29068 2.69245 −6.24828 −1.00000 0.602118 6.09903 5.57713
1.12 2.58557 2.93994 4.68518 −1.26031 7.60143 −1.00000 6.94272 5.64327 −3.25863
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(13\) \(-1\)

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 1183.2.a.q 12
7.b odd 2 1 8281.2.a.cn 12
13.b even 2 1 1183.2.a.r yes 12
13.d odd 4 2 1183.2.c.j 24
91.b odd 2 1 8281.2.a.cq 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.2.a.q 12 1.a even 1 1 trivial
1183.2.a.r yes 12 13.b even 2 1
1183.2.c.j 24 13.d odd 4 2
8281.2.a.cn 12 7.b odd 2 1
8281.2.a.cq 12 91.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1183))\):

\( T_{2}^{12} + 3 T_{2}^{11} - 15 T_{2}^{10} - 46 T_{2}^{9} + 80 T_{2}^{8} + 246 T_{2}^{7} - 199 T_{2}^{6} - 562 T_{2}^{5} + 262 T_{2}^{4} + 542 T_{2}^{3} - 157 T_{2}^{2} - 183 T_{2} + 29 \) Copy content Toggle raw display
\( T_{11}^{12} + 12 T_{11}^{11} - 13 T_{11}^{10} - 613 T_{11}^{9} - 994 T_{11}^{8} + 12005 T_{11}^{7} + 30684 T_{11}^{6} - 112394 T_{11}^{5} - 339321 T_{11}^{4} + 505685 T_{11}^{3} + 1617506 T_{11}^{2} - 900676 T_{11} - 2701133 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} - 15 T^{10} - 46 T^{9} + \cdots + 29 \) Copy content Toggle raw display
$3$ \( T^{12} - 8 T^{11} + T^{10} + 136 T^{9} + \cdots + 448 \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} - 38 T^{10} + \cdots + 6208 \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 12 T^{11} - 13 T^{10} + \cdots - 2701133 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 31 T^{11} + 364 T^{10} + \cdots + 110272 \) Copy content Toggle raw display
$19$ \( T^{12} - 3 T^{11} - 131 T^{10} + \cdots + 1395008 \) Copy content Toggle raw display
$23$ \( T^{12} - 18 T^{11} + 12 T^{10} + \cdots + 7017331 \) Copy content Toggle raw display
$29$ \( T^{12} - 15 T^{11} - 24 T^{10} + \cdots + 177673 \) Copy content Toggle raw display
$31$ \( T^{12} - 21 T^{11} + 7 T^{10} + \cdots - 15261184 \) Copy content Toggle raw display
$37$ \( T^{12} + 5 T^{11} - 252 T^{10} + \cdots + 545930729 \) Copy content Toggle raw display
$41$ \( T^{12} - 16 T^{11} + \cdots - 101827648 \) Copy content Toggle raw display
$43$ \( T^{12} + 22 T^{11} - 28 T^{10} + \cdots - 9002449 \) Copy content Toggle raw display
$47$ \( T^{12} - 4 T^{11} - 262 T^{10} + \cdots + 17156608 \) Copy content Toggle raw display
$53$ \( T^{12} - 53 T^{11} + 1054 T^{10} + \cdots + 62267981 \) Copy content Toggle raw display
$59$ \( T^{12} - 26 T^{11} + \cdots + 913870784 \) Copy content Toggle raw display
$61$ \( T^{12} - 22 T^{11} - 129 T^{10} + \cdots - 69982144 \) Copy content Toggle raw display
$67$ \( T^{12} + 12 T^{11} - 199 T^{10} + \cdots + 1207037 \) Copy content Toggle raw display
$71$ \( T^{12} + 21 T^{11} - 125 T^{10} + \cdots - 22571863 \) Copy content Toggle raw display
$73$ \( T^{12} - 15 T^{11} + \cdots - 4111861312 \) Copy content Toggle raw display
$79$ \( T^{12} - 2 T^{11} - 364 T^{10} + \cdots - 33746539 \) Copy content Toggle raw display
$83$ \( T^{12} - 9 T^{11} - 248 T^{10} + \cdots - 478100288 \) Copy content Toggle raw display
$89$ \( T^{12} - 22 T^{11} + \cdots - 639815168 \) Copy content Toggle raw display
$97$ \( T^{12} + 9 T^{11} + \cdots - 6873362944 \) Copy content Toggle raw display
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