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Properties

Label 1238.2.a.f
Level $1238$
Weight $2$
Character orbit 1238.a
Self dual yes
Analytic conductor $9.885$
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] = [1238,2,Mod(1,1238)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1238, 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("1238.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1238 = 2 \cdot 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1238.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: \(9.88547977023\)
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: \( x^{17} - 5 x^{16} - 23 x^{15} + 140 x^{14} + 166 x^{13} - 1505 x^{12} - 239 x^{11} + 8043 x^{10} + \cdots - 1712 \) 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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 5 x^{16} - 23 x^{15} + 140 x^{14} + 166 x^{13} - 1505 x^{12} - 239 x^{11} + 8043 x^{10} + \cdots - 1712 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 195620161 \nu^{16} - 1521108399 \nu^{15} - 2096605807 \nu^{14} + 40441722168 \nu^{13} + \cdots - 881971841008 ) / 8115061000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3726980 \nu^{16} + 5283857 \nu^{15} + 136254027 \nu^{14} - 166418377 \nu^{13} + \cdots - 11789394388 ) / 97380732 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 159912751 \nu^{16} + 831475309 \nu^{15} + 3350737587 \nu^{14} - 22437025163 \nu^{13} + \cdots + 237800623228 ) / 2434518300 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 393283969 \nu^{16} + 3074704771 \nu^{15} + 3872705628 \nu^{14} - 80661305772 \nu^{13} + \cdots + 1828327386132 ) / 4057530500 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 490389679 \nu^{16} + 1940182111 \nu^{15} + 12669167073 \nu^{14} - 52812815552 \nu^{13} + \cdots - 105340220288 ) / 4869036600 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1376693228 \nu^{16} - 4374082652 \nu^{15} - 40123737861 \nu^{14} + 121596759064 \nu^{13} + \cdots + 1613974548616 ) / 12172591500 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 192555197 \nu^{16} - 1198883523 \nu^{15} - 3163889789 \nu^{14} + 31708897486 \nu^{13} + \cdots - 495817940016 ) / 1623012200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3147812453 \nu^{16} + 16312103927 \nu^{15} + 65262452811 \nu^{14} - 435672966364 \nu^{13} + \cdots + 4201277941184 ) / 24345183000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3472702249 \nu^{16} - 21868459291 \nu^{15} - 56050576863 \nu^{14} + 577856848412 \nu^{13} + \cdots - 9045725574472 ) / 24345183000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1821602678 \nu^{16} + 10184030702 \nu^{15} + 34954498011 \nu^{14} - 271423127914 \nu^{13} + \cdots + 2980521757484 ) / 12172591500 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1914986753 \nu^{16} + 13498077002 \nu^{15} + 25096805661 \nu^{14} - 355744557514 \nu^{13} + \cdots + 6801510768584 ) / 12172591500 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 706382432 \nu^{16} + 3882253413 \nu^{15} + 13826352309 \nu^{14} - 103804275016 \nu^{13} + \cdots + 1201065634596 ) / 4057530500 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 901207907 \nu^{16} + 4916857763 \nu^{15} + 17751106209 \nu^{14} - 131251046116 \nu^{13} + \cdots + 1442396753096 ) / 4869036600 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 2884047371 \nu^{16} + 14785829289 \nu^{15} + 60900833327 \nu^{14} - 396561464148 \nu^{13} + \cdots + 3484160427888 ) / 8115061000 \) 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
\(\nu^{3}\)\(=\) \( \beta_{15} - 2\beta_{13} + 2\beta_{10} - \beta_{7} - 2\beta_{4} - 2\beta_{3} + \beta_{2} + 8\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{15} - \beta_{14} - \beta_{13} - 2 \beta_{12} - 2 \beta_{10} - 2 \beta_{9} - \beta_{8} + \cdots + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{16} + 13 \beta_{15} - 2 \beta_{14} - 22 \beta_{13} - 2 \beta_{12} + \beta_{11} + 22 \beta_{10} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5 \beta_{16} + 60 \beta_{15} - 10 \beta_{14} - 20 \beta_{13} - 28 \beta_{12} - 3 \beta_{11} + \cdots + 225 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 21 \beta_{16} + 148 \beta_{15} - 33 \beta_{14} - 223 \beta_{13} - 34 \beta_{12} + 19 \beta_{11} + \cdots - 70 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 104 \beta_{16} + 721 \beta_{15} - 83 \beta_{14} - 289 \beta_{13} - 324 \beta_{12} - 72 \beta_{11} + \cdots + 2072 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 329 \beta_{16} + 1655 \beta_{15} - 431 \beta_{14} - 2277 \beta_{13} - 442 \beta_{12} + 241 \beta_{11} + \cdots - 327 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1558 \beta_{16} + 8179 \beta_{15} - 706 \beta_{14} - 3716 \beta_{13} - 3562 \beta_{12} - 1150 \beta_{11} + \cdots + 20057 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4546 \beta_{16} + 18538 \beta_{15} - 5229 \beta_{14} - 23631 \beta_{13} - 5280 \beta_{12} + \cdots + 533 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 20591 \beta_{16} + 91069 \beta_{15} - 6686 \beta_{14} - 45070 \beta_{13} - 38470 \beta_{12} + \cdots + 199670 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 58665 \beta_{16} + 208560 \beta_{15} - 61424 \beta_{14} - 248528 \beta_{13} - 61034 \beta_{12} + \cdots + 42885 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 255683 \beta_{16} + 1007224 \beta_{15} - 71765 \beta_{14} - 528399 \beta_{13} - 412786 \beta_{12} + \cdots + 2023988 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 726534 \beta_{16} + 2354431 \beta_{15} - 709593 \beta_{14} - 2639043 \beta_{13} - 695280 \beta_{12} + \cdots + 804036 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3064551 \beta_{16} + 11112769 \beta_{15} - 845859 \beta_{14} - 6066187 \beta_{13} - 4420160 \beta_{12} + \cdots + 20783881 \) 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
−3.16740
−2.71917
−1.83849
−1.79922
−1.32266
−0.832595
−0.824108
−0.483466
0.465842
0.921017
1.64089
1.67922
1.88714
2.04914
2.79833
3.20072
3.34482
−1.00000 −3.16740 1.00000 2.75542 3.16740 0.158293 −1.00000 7.03243 −2.75542
1.2 −1.00000 −2.71917 1.00000 −1.60149 2.71917 3.13184 −1.00000 4.39390 1.60149
1.3 −1.00000 −1.83849 1.00000 −1.07018 1.83849 −2.91409 −1.00000 0.380062 1.07018
1.4 −1.00000 −1.79922 1.00000 3.54390 1.79922 −3.63586 −1.00000 0.237188 −3.54390
1.5 −1.00000 −1.32266 1.00000 1.13495 1.32266 4.39966 −1.00000 −1.25058 −1.13495
1.6 −1.00000 −0.832595 1.00000 −3.36307 0.832595 −2.08481 −1.00000 −2.30679 3.36307
1.7 −1.00000 −0.824108 1.00000 3.98915 0.824108 −0.894864 −1.00000 −2.32085 −3.98915
1.8 −1.00000 −0.483466 1.00000 −3.00899 0.483466 2.21267 −1.00000 −2.76626 3.00899
1.9 −1.00000 0.465842 1.00000 1.23920 −0.465842 2.06882 −1.00000 −2.78299 −1.23920
1.10 −1.00000 0.921017 1.00000 −0.665858 −0.921017 −4.34055 −1.00000 −2.15173 0.665858
1.11 −1.00000 1.64089 1.00000 −4.29784 −1.64089 −1.48364 −1.00000 −0.307465 4.29784
1.12 −1.00000 1.67922 1.00000 3.95654 −1.67922 4.60036 −1.00000 −0.180223 −3.95654
1.13 −1.00000 1.88714 1.00000 3.87284 −1.88714 2.40197 −1.00000 0.561288 −3.87284
1.14 −1.00000 2.04914 1.00000 −2.80507 −2.04914 4.42468 −1.00000 1.19896 2.80507
1.15 −1.00000 2.79833 1.00000 0.772526 −2.79833 −0.0293689 −1.00000 4.83066 −0.772526
1.16 −1.00000 3.20072 1.00000 3.36658 −3.20072 −3.23160 −1.00000 7.24459 −3.36658
1.17 −1.00000 3.34482 1.00000 −0.818610 −3.34482 4.21648 −1.00000 8.18781 0.818610
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(619\) \(-1\)

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 1238.2.a.f ✓ 17
4.b odd 2 1 9904.2.a.h 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1238.2.a.f ✓ 17 1.a even 1 1 trivial
9904.2.a.h 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1238))\):

\( T_{3}^{17} - 5 T_{3}^{16} - 23 T_{3}^{15} + 140 T_{3}^{14} + 166 T_{3}^{13} - 1505 T_{3}^{12} + \cdots - 1712 \) Copy content Toggle raw display
\( T_{5}^{17} - 7 T_{5}^{16} - 42 T_{5}^{15} + 369 T_{5}^{14} + 534 T_{5}^{13} - 7537 T_{5}^{12} + \cdots - 248832 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{17} \) Copy content Toggle raw display
$3$ \( T^{17} - 5 T^{16} + \cdots - 1712 \) Copy content Toggle raw display
$5$ \( T^{17} - 7 T^{16} + \cdots - 248832 \) Copy content Toggle raw display
$7$ \( T^{17} - 9 T^{16} + \cdots - 24867 \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots - 302812740 \) Copy content Toggle raw display
$13$ \( T^{17} - 14 T^{16} + \cdots + 16001476 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots + 409522176 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots + 253056404 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots - 115063995312 \) Copy content Toggle raw display
$29$ \( T^{17} - 18 T^{16} + \cdots - 4310496 \) Copy content Toggle raw display
$31$ \( T^{17} - 15 T^{16} + \cdots + 75269175 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots + 28364928864256 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots + 12065950755 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 151158784 \) Copy content Toggle raw display
$47$ \( T^{17} + 2 T^{16} + \cdots + 37343232 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 398793798647808 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots - 1124694010548 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 38163931592704 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 15\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 26\!\cdots\!97 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 379402432704 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots + 8772647630183 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 11876811791616 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 337971143346831 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 68\!\cdots\!04 \) Copy content Toggle raw display
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