Properties

Label 1183.2.a.m
Level $1183$
Weight $2$
Character orbit 1183.a
Self dual yes
Analytic conductor $9.446$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.44630255912\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7674048.1
Defining polynomial: \(x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 4 \nu^{2} + 2 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 6 \nu^{2} + 4 \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} + 18 \beta_{1} + 8\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.70320
−1.10939
−0.276564
0.879640
1.82356
2.38595
−2.70320 −0.345949 5.30727 −3.25812 0.935168 1.00000 −8.94020 −2.88032 8.80735
1.2 −2.10939 2.26165 2.44952 −3.60178 −4.77070 1.00000 −0.948212 2.11505 7.59755
1.3 −1.27656 −1.16793 −0.370384 1.81487 1.49093 1.00000 3.02595 −1.63595 −2.31680
1.4 −0.120360 −0.582292 −1.98551 1.68817 0.0700846 1.00000 0.479696 −2.66094 −0.203187
1.5 0.823556 2.66029 −1.32176 −3.16209 2.19090 1.00000 −2.73565 4.07715 −2.60416
1.6 1.38595 −2.82577 −0.0791355 0.518957 −3.91639 1.00000 −2.88158 4.98500 0.719250
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.m 6
7.b odd 2 1 8281.2.a.by 6
13.b even 2 1 1183.2.a.p 6
13.d odd 4 2 1183.2.c.i 12
13.f odd 12 2 91.2.q.a 12
39.k even 12 2 819.2.ct.a 12
52.l even 12 2 1456.2.cc.c 12
91.b odd 2 1 8281.2.a.ch 6
91.w even 12 2 637.2.u.i 12
91.x odd 12 2 637.2.k.h 12
91.ba even 12 2 637.2.k.g 12
91.bc even 12 2 637.2.q.h 12
91.bd odd 12 2 637.2.u.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a 12 13.f odd 12 2
637.2.k.g 12 91.ba even 12 2
637.2.k.h 12 91.x odd 12 2
637.2.q.h 12 91.bc even 12 2
637.2.u.h 12 91.bd odd 12 2
637.2.u.i 12 91.w even 12 2
819.2.ct.a 12 39.k even 12 2
1183.2.a.m 6 1.a even 1 1 trivial
1183.2.a.p 6 13.b even 2 1
1183.2.c.i 12 13.d odd 4 2
1456.2.cc.c 12 52.l even 12 2
8281.2.a.by 6 7.b odd 2 1
8281.2.a.ch 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1183))\):

\( T_{2}^{6} + 4 T_{2}^{5} - 12 T_{2}^{3} - 4 T_{2}^{2} + 8 T_{2} + 1 \)
\( T_{11}^{6} + 4 T_{11}^{5} - 17 T_{11}^{4} - 16 T_{11}^{3} + 85 T_{11}^{2} - 72 T_{11} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 8 T - 4 T^{2} - 12 T^{3} + 4 T^{5} + T^{6} \)
$3$ \( 4 + 20 T + 25 T^{2} - 2 T^{3} - 11 T^{4} + T^{6} \)
$5$ \( -59 + 128 T - 2 T^{2} - 50 T^{3} - 2 T^{4} + 6 T^{5} + T^{6} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( 16 - 72 T + 85 T^{2} - 16 T^{3} - 17 T^{4} + 4 T^{5} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( -491 + 224 T + 167 T^{2} - 60 T^{3} - 21 T^{4} + 4 T^{5} + T^{6} \)
$19$ \( -236 - 32 T + 149 T^{2} - 27 T^{4} + 2 T^{5} + T^{6} \)
$23$ \( 6208 + 1472 T - 1616 T^{2} - 608 T^{3} - 20 T^{4} + 12 T^{5} + T^{6} \)
$29$ \( 3169 + 336 T - 1412 T^{2} - 566 T^{3} - 44 T^{4} + 8 T^{5} + T^{6} \)
$31$ \( -956 - 1360 T - 295 T^{2} + 198 T^{3} + 30 T^{4} - 14 T^{5} + T^{6} \)
$37$ \( -41904 + 22464 T + 2916 T^{2} - 1044 T^{3} - 87 T^{4} + 12 T^{5} + T^{6} \)
$41$ \( -29744 - 20224 T - 3228 T^{2} + 636 T^{3} + 257 T^{4} + 28 T^{5} + T^{6} \)
$43$ \( 1552 + 3776 T + 2421 T^{2} + 90 T^{3} - 109 T^{4} - 2 T^{5} + T^{6} \)
$47$ \( 3076 + 6696 T - 443 T^{2} - 758 T^{3} - 38 T^{4} + 14 T^{5} + T^{6} \)
$53$ \( -2339 + 7302 T - 3353 T^{2} - 700 T^{3} + 91 T^{4} + 22 T^{5} + T^{6} \)
$59$ \( -67616 - 7192 T + 6845 T^{2} + 210 T^{3} - 162 T^{4} - 2 T^{5} + T^{6} \)
$61$ \( 2368 + 1600 T - 1888 T^{2} - 1416 T^{3} - 87 T^{4} + 14 T^{5} + T^{6} \)
$67$ \( 24772 - 19256 T - 11855 T^{2} - 1408 T^{3} + 94 T^{4} + 24 T^{5} + T^{6} \)
$71$ \( -6848 + 2496 T + 1072 T^{2} - 256 T^{3} - 68 T^{4} + 4 T^{5} + T^{6} \)
$73$ \( -37232 - 12112 T + 4924 T^{2} + 2788 T^{3} + 481 T^{4} + 36 T^{5} + T^{6} \)
$79$ \( -512 + 1664 T - 1584 T^{2} + 192 T^{3} + 212 T^{4} + 28 T^{5} + T^{6} \)
$83$ \( -11888 - 25544 T - 8335 T^{2} - 330 T^{3} + 186 T^{4} + 26 T^{5} + T^{6} \)
$89$ \( 42832 - 28352 T - 10775 T^{2} + 1640 T^{3} + 553 T^{4} + 42 T^{5} + T^{6} \)
$97$ \( 7312 + 14224 T - 3815 T^{2} - 934 T^{3} + 97 T^{4} + 24 T^{5} + T^{6} \)
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