Properties

Label 241.2.a.b
Level $241$
Weight $2$
Character orbit 241.a
Self dual yes
Analytic conductor $1.924$
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] = [241,2,Mod(1,241)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(241, 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("241.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 241.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: \(1.92439468871\)
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: \( x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} - 328 x^{3} - 45 x^{2} + 18 x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} - 328 x^{3} - 45 x^{2} + 18 x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + \nu^{10} - 23 \nu^{9} - 13 \nu^{8} + 184 \nu^{7} + 54 \nu^{6} - 611 \nu^{5} - 94 \nu^{4} + 768 \nu^{3} + 94 \nu^{2} - 213 \nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4 \nu^{11} + 9 \nu^{10} + 57 \nu^{9} - 119 \nu^{8} - 273 \nu^{7} + 488 \nu^{6} + 538 \nu^{5} - 663 \nu^{4} - 422 \nu^{3} + 168 \nu^{2} + 34 \nu + 7 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7 \nu^{11} - 10 \nu^{10} - 130 \nu^{9} + 172 \nu^{8} + 869 \nu^{7} - 1046 \nu^{6} - 2491 \nu^{5} + 2625 \nu^{4} + 2774 \nu^{3} - 2230 \nu^{2} - 745 \nu + 85 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13 \nu^{11} + 48 \nu^{10} + 160 \nu^{9} - 690 \nu^{8} - 569 \nu^{7} + 3330 \nu^{6} + 597 \nu^{5} - 6481 \nu^{4} - 470 \nu^{3} + 4634 \nu^{2} + 951 \nu - 169 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13 \nu^{11} - 44 \nu^{10} - 180 \nu^{9} + 662 \nu^{8} + 829 \nu^{7} - 3426 \nu^{6} - 1629 \nu^{5} + 7333 \nu^{4} + 1798 \nu^{3} - 5698 \nu^{2} - 1335 \nu + 141 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11 \nu^{11} - 30 \nu^{10} - 158 \nu^{9} + 432 \nu^{8} + 773 \nu^{7} - 2086 \nu^{6} - 1631 \nu^{5} + 4025 \nu^{4} + 1654 \nu^{3} - 2750 \nu^{2} - 741 \nu + 93 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11 \nu^{11} + 34 \nu^{10} + 150 \nu^{9} - 492 \nu^{8} - 669 \nu^{7} + 2398 \nu^{6} + 1223 \nu^{5} - 4717 \nu^{4} - 1178 \nu^{3} + 3354 \nu^{2} + 737 \nu - 109 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25 \nu^{11} - 76 \nu^{10} - 340 \nu^{9} + 1086 \nu^{8} + 1513 \nu^{7} - 5178 \nu^{6} - 2777 \nu^{5} + 9809 \nu^{4} + 2718 \nu^{3} - 6602 \nu^{2} - 1643 \nu + 201 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 31 \nu^{11} - 90 \nu^{10} - 450 \nu^{9} + 1340 \nu^{8} + 2237 \nu^{7} - 6822 \nu^{6} - 4819 \nu^{5} + 14249 \nu^{4} + 5014 \nu^{3} - 10758 \nu^{2} - 2497 \nu + 381 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + \beta_{9} - \beta_{8} - 2\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11 \beta_{11} - 9 \beta_{10} + 10 \beta_{9} - \beta_{7} - 8 \beta_{6} - 11 \beta_{5} - 9 \beta_{4} + 9 \beta_{3} + \beta_{2} + 29 \beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20 \beta_{11} + 2 \beta_{10} + 10 \beta_{9} - 14 \beta_{8} - 19 \beta_{7} - 10 \beta_{6} - 9 \beta_{5} - 2 \beta_{4} + 11 \beta_{3} + 56 \beta_{2} + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 93 \beta_{11} - 67 \beta_{10} + 78 \beta_{9} - 3 \beta_{8} - 12 \beta_{7} - 54 \beta_{6} - 94 \beta_{5} - 68 \beta_{4} + 71 \beta_{3} + 11 \beta_{2} + 181 \beta _1 - 77 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 163 \beta_{11} + 24 \beta_{10} + 76 \beta_{9} - 138 \beta_{8} - 146 \beta_{7} - 78 \beta_{6} - 71 \beta_{5} - 30 \beta_{4} + 98 \beta_{3} + 379 \beta_{2} + 4 \beta _1 + 558 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 720 \beta_{11} - 478 \beta_{10} + 557 \beta_{9} - 59 \beta_{8} - 106 \beta_{7} - 355 \beta_{6} - 733 \beta_{5} - 499 \beta_{4} + 542 \beta_{3} + 88 \beta_{2} + 1179 \beta _1 - 544 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1256 \beta_{11} + 191 \beta_{10} + 522 \beta_{9} - 1189 \beta_{8} - 1056 \beta_{7} - 566 \beta_{6} - 561 \beta_{5} - 323 \beta_{4} + 822 \beta_{3} + 2542 \beta_{2} + 76 \beta _1 + 3665 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5372 \beta_{11} - 3384 \beta_{10} + 3821 \beta_{9} - 748 \beta_{8} - 845 \beta_{7} - 2351 \beta_{6} - 5486 \beta_{5} - 3655 \beta_{4} + 4093 \beta_{3} + 630 \beta_{2} + 7881 \beta _1 - 3713 \) 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
−2.59703
−2.02418
−1.32986
−1.28632
−0.342147
0.0822506
0.115670
1.54879
1.63125
2.01020
2.49073
2.70063
−2.59703 −1.20534 4.74454 3.49051 3.13029 −0.744578 −7.12764 −1.54716 −9.06493
1.2 −2.02418 2.93498 2.09729 1.44091 −5.94092 0.381245 −0.196936 5.61411 −2.91665
1.3 −1.32986 −2.18147 −0.231473 −3.40432 2.90104 −3.83334 2.96755 1.75880 4.52727
1.4 −1.28632 −0.126224 −0.345373 0.612768 0.162365 1.03110 3.01691 −2.98407 −0.788217
1.5 −0.342147 2.18519 −1.88294 −0.548903 −0.747658 1.82459 1.32853 1.77508 0.187805
1.6 0.0822506 1.81824 −1.99323 4.31963 0.149552 0.690569 −0.328446 0.306010 0.355292
1.7 0.115670 −3.28295 −1.98662 −1.31091 −0.379739 3.19647 −0.461133 7.77775 −0.151633
1.8 1.54879 2.81087 0.398765 0.334961 4.35346 −4.24623 −2.47998 4.90098 0.518786
1.9 1.63125 −1.16790 0.660992 1.75438 −1.90514 5.06139 −2.18426 −1.63601 2.86183
1.10 2.01020 0.500591 2.04092 1.92585 1.00629 −0.852319 0.0822476 −2.74941 3.87135
1.11 2.49073 1.22208 4.20371 −3.14843 3.04385 0.136122 5.48885 −1.50653 −7.84189
1.12 2.70063 −2.50808 5.29342 0.533570 −6.77340 0.354992 8.89432 3.29045 1.44098
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(241\) \(-1\)

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 241.2.a.b 12
3.b odd 2 1 2169.2.a.h 12
4.b odd 2 1 3856.2.a.n 12
5.b even 2 1 6025.2.a.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.a.b 12 1.a even 1 1 trivial
2169.2.a.h 12 3.b odd 2 1
3856.2.a.n 12 4.b odd 2 1
6025.2.a.h 12 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 3 T_{2}^{11} - 14 T_{2}^{10} + 44 T_{2}^{9} + 65 T_{2}^{8} - 219 T_{2}^{7} - 123 T_{2}^{6} + 444 T_{2}^{5} + 105 T_{2}^{4} - 328 T_{2}^{3} - 45 T_{2}^{2} + 18 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(241))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 3 T^{11} - 14 T^{10} + 44 T^{9} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{12} - T^{11} - 25 T^{10} + 25 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} - 14 T^{10} + 134 T^{9} + \cdots + 62 \) Copy content Toggle raw display
$7$ \( T^{12} - 3 T^{11} - 33 T^{10} + 96 T^{9} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{12} - 22 T^{11} + 177 T^{10} + \cdots + 128 \) Copy content Toggle raw display
$13$ \( T^{12} + 5 T^{11} - 62 T^{10} + \cdots - 52672 \) Copy content Toggle raw display
$17$ \( T^{12} + 4 T^{11} - 97 T^{10} + \cdots + 154144 \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} - 86 T^{10} + \cdots - 3556280 \) Copy content Toggle raw display
$23$ \( T^{12} - 32 T^{11} + \cdots - 116949436 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{11} - 213 T^{10} + \cdots + 58109390 \) Copy content Toggle raw display
$31$ \( T^{12} - 8 T^{11} - 262 T^{10} + \cdots - 318193616 \) Copy content Toggle raw display
$37$ \( T^{12} + 8 T^{11} - 159 T^{10} + \cdots + 50796928 \) Copy content Toggle raw display
$41$ \( T^{12} + T^{11} - 262 T^{10} + \cdots - 63338 \) Copy content Toggle raw display
$43$ \( T^{12} + 2 T^{11} - 237 T^{10} + \cdots + 12503272 \) Copy content Toggle raw display
$47$ \( T^{12} - 34 T^{11} + 332 T^{10} + \cdots + 53297792 \) Copy content Toggle raw display
$53$ \( T^{12} - 5 T^{11} - 195 T^{10} + \cdots - 3014 \) Copy content Toggle raw display
$59$ \( T^{12} - 26 T^{11} + 22 T^{10} + \cdots - 25476160 \) Copy content Toggle raw display
$61$ \( T^{12} + 26 T^{11} + 20 T^{10} + \cdots + 10893274 \) Copy content Toggle raw display
$67$ \( T^{12} - 6 T^{11} + \cdots + 4538509504 \) Copy content Toggle raw display
$71$ \( T^{12} - 94 T^{11} + \cdots - 12017198348 \) Copy content Toggle raw display
$73$ \( T^{12} + 22 T^{11} - 208 T^{10} + \cdots + 2219968 \) Copy content Toggle raw display
$79$ \( T^{12} - 9 T^{11} + \cdots - 1277319040 \) Copy content Toggle raw display
$83$ \( T^{12} + 8 T^{11} + \cdots + 98860915136 \) Copy content Toggle raw display
$89$ \( T^{12} + 3 T^{11} + \cdots - 1500609440 \) Copy content Toggle raw display
$97$ \( T^{12} + 29 T^{11} + \cdots + 107861318 \) Copy content Toggle raw display
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