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Properties

Label 1226.2.a.e
Level $1226$
Weight $2$
Character orbit 1226.a
Self dual yes
Analytic conductor $9.790$
Analytic rank $0$
Dimension $17$
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] = [1226,2,Mod(1,1226)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1226, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1226.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1226 = 2 \cdot 613 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1226.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: \(9.78965928781\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 4 x^{16} - 28 x^{15} + 120 x^{14} + 291 x^{13} - 1382 x^{12} - 1398 x^{11} + 7700 x^{10} + \cdots - 320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{9} q^{5} - \beta_1 q^{6} - \beta_{12} q^{7} - q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{9} q^{5} - \beta_1 q^{6} - \beta_{12} q^{7} - q^{8} + (\beta_{2} + 1) q^{9} - \beta_{9} q^{10} + (\beta_{10} - \beta_{7} - \beta_{3} - \beta_1) q^{11} + \beta_1 q^{12} + ( - \beta_{16} - \beta_{9} + \beta_{6} + \cdots + 1) q^{13}+ \cdots + (2 \beta_{16} - \beta_{15} - 3 \beta_{14} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 17 q^{2} + 4 q^{3} + 17 q^{4} - 5 q^{5} - 4 q^{6} + 7 q^{7} - 17 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - 17 q^{2} + 4 q^{3} + 17 q^{4} - 5 q^{5} - 4 q^{6} + 7 q^{7} - 17 q^{8} + 21 q^{9} + 5 q^{10} + 8 q^{11} + 4 q^{12} + 9 q^{13} - 7 q^{14} - 4 q^{15} + 17 q^{16} - q^{17} - 21 q^{18} + 32 q^{19} - 5 q^{20} + 6 q^{21} - 8 q^{22} - 5 q^{23} - 4 q^{24} + 30 q^{25} - 9 q^{26} + 16 q^{27} + 7 q^{28} + 3 q^{29} + 4 q^{30} + 27 q^{31} - 17 q^{32} - 14 q^{33} + q^{34} + 25 q^{35} + 21 q^{36} + 7 q^{37} - 32 q^{38} + 27 q^{39} + 5 q^{40} - 2 q^{41} - 6 q^{42} + 36 q^{43} + 8 q^{44} - q^{45} + 5 q^{46} - 3 q^{47} + 4 q^{48} + 52 q^{49} - 30 q^{50} + 40 q^{51} + 9 q^{52} - 20 q^{53} - 16 q^{54} + 48 q^{55} - 7 q^{56} + 12 q^{57} - 3 q^{58} + 34 q^{59} - 4 q^{60} + 49 q^{61} - 27 q^{62} + 27 q^{63} + 17 q^{64} - 6 q^{65} + 14 q^{66} + 36 q^{67} - q^{68} + 18 q^{69} - 25 q^{70} - q^{71} - 21 q^{72} + 24 q^{73} - 7 q^{74} + 35 q^{75} + 32 q^{76} - 6 q^{77} - 27 q^{78} + 43 q^{79} - 5 q^{80} + 37 q^{81} + 2 q^{82} + 10 q^{83} + 6 q^{84} + 16 q^{85} - 36 q^{86} + 28 q^{87} - 8 q^{88} - 12 q^{89} + q^{90} + 42 q^{91} - 5 q^{92} + 3 q^{93} + 3 q^{94} - 10 q^{95} - 4 q^{96} + 26 q^{97} - 52 q^{98} + 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 4 x^{16} - 28 x^{15} + 120 x^{14} + 291 x^{13} - 1382 x^{12} - 1398 x^{11} + 7700 x^{10} + \cdots - 320 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6410869 \nu^{16} + 31454554 \nu^{15} + 150495136 \nu^{14} - 903533720 \nu^{13} + \cdots + 2002536928 ) / 63799808 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7113859 \nu^{16} - 36075798 \nu^{15} - 160236128 \nu^{14} + 1024099112 \nu^{13} + \cdots - 1970724128 ) / 63799808 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8027319 \nu^{16} - 38499966 \nu^{15} - 194168544 \nu^{14} + 1118728456 \nu^{13} + \cdots - 2331848864 ) / 63799808 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9204915 \nu^{16} - 43151990 \nu^{15} - 229179104 \nu^{14} + 1268757032 \nu^{13} + \cdots - 3912131360 ) / 63799808 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12323149 \nu^{16} - 62600714 \nu^{15} - 277269280 \nu^{14} + 1777499864 \nu^{13} + \cdots - 3830748384 ) / 63799808 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13296709 \nu^{16} + 68342010 \nu^{15} + 295056032 \nu^{14} - 1934785304 \nu^{13} + \cdots + 4114727392 ) / 63799808 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3630363 \nu^{16} + 18327238 \nu^{15} + 82390368 \nu^{14} - 521739752 \nu^{13} + \cdots + 1041602464 ) / 15949952 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16728525 \nu^{16} + 85438474 \nu^{15} + 373232672 \nu^{14} - 2418634712 \nu^{13} + \cdots + 4745183968 ) / 63799808 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 23493833 \nu^{16} - 120098370 \nu^{15} - 524556960 \nu^{14} + 3402019576 \nu^{13} + \cdots - 6903340896 ) / 63799808 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13370579 \nu^{16} - 65848534 \nu^{15} - 313444000 \nu^{14} + 1894414632 \nu^{13} + \cdots - 4429469472 ) / 31899904 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 33298539 \nu^{16} - 165695718 \nu^{15} - 770001760 \nu^{14} + 4744006760 \nu^{13} + \cdots - 10830330912 ) / 63799808 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 17777837 \nu^{16} + 89678666 \nu^{15} + 404087840 \nu^{14} - 2554662040 \nu^{13} + \cdots + 5400476384 ) / 31899904 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 9549461 \nu^{16} - 47642218 \nu^{15} - 220344896 \nu^{14} + 1363831512 \nu^{13} + \cdots - 3185682272 ) / 15949952 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 23466867 \nu^{16} - 116501014 \nu^{15} - 544379552 \nu^{14} + 3339801384 \nu^{13} + \cdots - 7932876320 ) / 31899904 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{14} + \beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} + \cdots + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{15} + 15 \beta_{14} - 4 \beta_{12} + 11 \beta_{11} + \beta_{10} - 15 \beta_{9} - 14 \beta_{8} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{16} + 2 \beta_{15} + 36 \beta_{14} - 16 \beta_{13} - 21 \beta_{12} + 18 \beta_{11} - 26 \beta_{9} + \cdots + 206 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 73 \beta_{15} + 181 \beta_{14} + 4 \beta_{13} - 80 \beta_{12} + 109 \beta_{11} + 15 \beta_{10} + \cdots - 92 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 34 \beta_{16} + 32 \beta_{15} + 474 \beta_{14} - 188 \beta_{13} - 315 \beta_{12} + 236 \beta_{11} + \cdots + 1634 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6 \beta_{16} - 594 \beta_{15} + 2042 \beta_{14} + 104 \beta_{13} - 1168 \beta_{12} + 1072 \beta_{11} + \cdots - 1062 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 428 \beta_{16} + 344 \beta_{15} + 5622 \beta_{14} - 1951 \beta_{13} - 4164 \beta_{12} + 2779 \beta_{11} + \cdots + 13181 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 128 \beta_{16} - 4945 \beta_{15} + 22445 \beta_{14} + 1762 \beta_{13} - 15126 \beta_{12} + \cdots - 11212 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4866 \beta_{16} + 3026 \beta_{15} + 63836 \beta_{14} - 18911 \beta_{13} - 51635 \beta_{12} + \cdots + 107103 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1670 \beta_{16} - 42284 \beta_{15} + 244164 \beta_{14} + 24746 \beta_{13} - 184442 \beta_{12} + \cdots - 114602 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 53068 \beta_{16} + 22320 \beta_{15} + 710574 \beta_{14} - 175051 \beta_{13} - 616650 \beta_{12} + \cdots + 870945 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 16016 \beta_{16} - 371756 \beta_{15} + 2646196 \beta_{14} + 313634 \beta_{13} - 2172178 \beta_{12} + \cdots - 1159492 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 568954 \beta_{16} + 123246 \beta_{15} + 7836254 \beta_{14} - 1560508 \beta_{13} - 7188194 \beta_{12} + \cdots + 7045926 \) 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
−3.01090
−2.85621
−2.70602
−1.14543
−1.11874
−0.961867
−0.926840
−0.573426
0.106437
0.722845
1.14779
1.60312
2.22792
2.44833
2.79660
2.93820
3.30820
−1.00000 −3.01090 1.00000 1.77003 3.01090 0.608967 −1.00000 6.06551 −1.77003
1.2 −1.00000 −2.85621 1.00000 −1.33964 2.85621 −1.59636 −1.00000 5.15796 1.33964
1.3 −1.00000 −2.70602 1.00000 −1.85368 2.70602 4.20807 −1.00000 4.32254 1.85368
1.4 −1.00000 −1.14543 1.00000 3.29370 1.14543 −1.02782 −1.00000 −1.68799 −3.29370
1.5 −1.00000 −1.11874 1.00000 −0.440297 1.11874 0.464735 −1.00000 −1.74842 0.440297
1.6 −1.00000 −0.961867 1.00000 2.82340 0.961867 4.64973 −1.00000 −2.07481 −2.82340
1.7 −1.00000 −0.926840 1.00000 0.613322 0.926840 −4.30069 −1.00000 −2.14097 −0.613322
1.8 −1.00000 −0.573426 1.00000 −4.28838 0.573426 −1.62799 −1.00000 −2.67118 4.28838
1.9 −1.00000 0.106437 1.00000 −3.26505 −0.106437 −4.74384 −1.00000 −2.98867 3.26505
1.10 −1.00000 0.722845 1.00000 −2.20965 −0.722845 2.70732 −1.00000 −2.47750 2.20965
1.11 −1.00000 1.14779 1.00000 −3.86057 −1.14779 3.93842 −1.00000 −1.68259 3.86057
1.12 −1.00000 1.60312 1.00000 3.81766 −1.60312 2.54246 −1.00000 −0.430013 −3.81766
1.13 −1.00000 2.22792 1.00000 2.67030 −2.22792 1.51453 −1.00000 1.96364 −2.67030
1.14 −1.00000 2.44833 1.00000 0.914976 −2.44833 2.07967 −1.00000 2.99430 −0.914976
1.15 −1.00000 2.79660 1.00000 1.18952 −2.79660 −2.91390 −1.00000 4.82099 −1.18952
1.16 −1.00000 2.93820 1.00000 −3.59255 −2.93820 −4.18406 −1.00000 5.63304 3.59255
1.17 −1.00000 3.30820 1.00000 −1.24310 −3.30820 4.68076 −1.00000 7.94416 1.24310
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(613\) \(-1\)

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 1226.2.a.e ✓ 17
4.b odd 2 1 9808.2.a.f 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1226.2.a.e ✓ 17 1.a even 1 1 trivial
9808.2.a.f 17 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1226))\):

\( T_{3}^{17} - 4 T_{3}^{16} - 28 T_{3}^{15} + 120 T_{3}^{14} + 291 T_{3}^{13} - 1382 T_{3}^{12} + \cdots - 320 \) Copy content Toggle raw display
\( T_{5}^{17} + 5 T_{5}^{16} - 45 T_{5}^{15} - 234 T_{5}^{14} + 789 T_{5}^{13} + 4315 T_{5}^{12} + \cdots + 65328 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{17} \) Copy content Toggle raw display
$3$ \( T^{17} - 4 T^{16} + \cdots - 320 \) Copy content Toggle raw display
$5$ \( T^{17} + 5 T^{16} + \cdots + 65328 \) Copy content Toggle raw display
$7$ \( T^{17} - 7 T^{16} + \cdots + 1470464 \) Copy content Toggle raw display
$11$ \( T^{17} - 8 T^{16} + \cdots - 488448 \) Copy content Toggle raw display
$13$ \( T^{17} - 9 T^{16} + \cdots - 418084 \) Copy content Toggle raw display
$17$ \( T^{17} + T^{16} + \cdots + 905664 \) Copy content Toggle raw display
$19$ \( T^{17} - 32 T^{16} + \cdots + 2364160 \) Copy content Toggle raw display
$23$ \( T^{17} + 5 T^{16} + \cdots - 3072000 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots + 2585075712 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots + 64681746460 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots + 953335808 \) Copy content Toggle raw display
$41$ \( T^{17} + 2 T^{16} + \cdots - 41960355 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 4355948336 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 3740209152 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 201706006272 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 4951223159808 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 7701165838336 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 15773024744656 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 461374586880 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 11070767104 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots + 7473765257216 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 47380113408 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 10\!\cdots\!67 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 21464492749201 \) Copy content Toggle raw display
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