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Properties

Label 5635.2.a.be
Level $5635$
Weight $2$
Character orbit 5635.a
Self dual yes
Analytic conductor $44.996$
Analytic rank $1$
Dimension $13$
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] = [5635,2,Mod(1,5635)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5635, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5635.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5635 = 5 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5635.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: \(44.9957015390\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 5 x^{12} - 7 x^{11} + 65 x^{10} - 20 x^{9} - 287 x^{8} + 238 x^{7} + 509 x^{6} - 543 x^{5} + \cdots + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 5 x^{12} - 7 x^{11} + 65 x^{10} - 20 x^{9} - 287 x^{8} + 238 x^{7} + 509 x^{6} - 543 x^{5} + \cdots + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1161 \nu^{12} + 1600 \nu^{11} + 20706 \nu^{10} - 25519 \nu^{9} - 140454 \nu^{8} + 157683 \nu^{7} + \cdots - 110200 ) / 35639 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1782 \nu^{12} - 1627 \nu^{11} - 37583 \nu^{10} + 33367 \nu^{9} + 289345 \nu^{8} - 237881 \nu^{7} + \cdots + 134334 ) / 35639 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2515 \nu^{12} + 10956 \nu^{11} + 25423 \nu^{10} - 155229 \nu^{9} - 48153 \nu^{8} + 796137 \nu^{7} + \cdots + 118561 ) / 35639 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2953 \nu^{12} + 7876 \nu^{11} + 55060 \nu^{10} - 139071 \nu^{9} - 385424 \nu^{8} + 892645 \nu^{7} + \cdots - 132611 ) / 35639 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3307 \nu^{12} - 15639 \nu^{11} - 26287 \nu^{10} + 197778 \nu^{9} - 9364 \nu^{8} - 826624 \nu^{7} + \cdots - 23218 ) / 35639 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5822 \nu^{12} - 26595 \nu^{11} - 51710 \nu^{10} + 353007 \nu^{9} + 38789 \nu^{8} - 1622761 \nu^{7} + \cdots + 107694 ) / 35639 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5822 \nu^{12} - 26595 \nu^{11} - 51710 \nu^{10} + 353007 \nu^{9} + 38789 \nu^{8} - 1622761 \nu^{7} + \cdots + 72055 ) / 35639 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7522 \nu^{12} - 32867 \nu^{11} - 73004 \nu^{10} + 441637 \nu^{9} + 121785 \nu^{8} - 2054130 \nu^{7} + \cdots - 101824 ) / 35639 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12840 \nu^{12} - 53242 \nu^{11} - 130644 \nu^{10} + 709249 \nu^{9} + 294373 \nu^{8} - 3271941 \nu^{7} + \cdots - 83962 ) / 35639 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14265 \nu^{12} - 56403 \nu^{11} - 151638 \nu^{10} + 748951 \nu^{9} + 403775 \nu^{8} - 3416027 \nu^{7} + \cdots - 13259 ) / 35639 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{7} + \beta_{5} + 7\beta_{2} + 8\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 2 \beta_{7} + \beta_{6} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{12} + 3 \beta_{11} + 4 \beta_{10} + 10 \beta_{9} - 2 \beta_{8} - 12 \beta_{7} + \beta_{6} + \cdots + 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 27 \beta_{12} + 25 \beta_{11} + 31 \beta_{10} + 56 \beta_{9} - 56 \beta_{8} - 27 \beta_{7} + \cdots + 109 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 52 \beta_{12} + 48 \beta_{11} + 68 \beta_{10} + 82 \beta_{9} - 29 \beta_{8} - 112 \beta_{7} + \cdots + 589 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 274 \beta_{12} + 241 \beta_{11} + 337 \beta_{10} + 385 \beta_{9} - 382 \beta_{8} - 275 \beta_{7} + \cdots + 902 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 605 \beta_{12} + 533 \beta_{11} + 792 \beta_{10} + 637 \beta_{9} - 300 \beta_{8} - 974 \beta_{7} + \cdots + 4035 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2498 \beta_{12} + 2129 \beta_{11} + 3188 \beta_{10} + 2659 \beta_{9} - 2606 \beta_{8} - 2522 \beta_{7} + \cdots + 7170 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5970 \beta_{12} + 5115 \beta_{11} + 7870 \beta_{10} + 4848 \beta_{9} - 2725 \beta_{8} - 8212 \beta_{7} + \cdots + 28381 \) 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
2.77405
2.53905
2.45535
1.63772
1.43005
0.815121
0.663262
0.0245790
−0.631146
−0.807792
−1.61708
−1.86015
−2.42302
−2.77405 2.92735 5.69533 1.00000 −8.12060 0 −10.2510 5.56938 −2.77405
1.2 −2.53905 −1.59479 4.44677 1.00000 4.04926 0 −6.21247 −0.456633 −2.53905
1.3 −2.45535 −1.73135 4.02876 1.00000 4.25107 0 −4.98133 −0.00243873 −2.45535
1.4 −1.63772 1.98612 0.682139 1.00000 −3.25272 0 2.15829 0.944672 −1.63772
1.5 −1.43005 1.29435 0.0450556 1.00000 −1.85099 0 2.79568 −1.32466 −1.43005
1.6 −0.815121 −2.98702 −1.33558 1.00000 2.43478 0 2.71890 5.92227 −0.815121
1.7 −0.663262 −0.140921 −1.56008 1.00000 0.0934675 0 2.36127 −2.98014 −0.663262
1.8 −0.0245790 −2.41981 −1.99940 1.00000 0.0594766 0 0.0983013 2.85547 −0.0245790
1.9 0.631146 1.49200 −1.60165 1.00000 0.941671 0 −2.27317 −0.773934 0.631146
1.10 0.807792 0.871452 −1.34747 1.00000 0.703952 0 −2.70406 −2.24057 0.807792
1.11 1.61708 −1.90180 0.614949 1.00000 −3.07537 0 −2.23974 0.616861 1.61708
1.12 1.86015 1.29570 1.46016 1.00000 2.41019 0 −1.00418 −1.32117 1.86015
1.13 2.42302 −1.09128 3.87102 1.00000 −2.64419 0 4.53352 −1.80911 2.42302
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

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 5635.2.a.be 13
7.b odd 2 1 5635.2.a.bf 13
7.c even 3 2 805.2.i.d ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.i.d ✓ 26 7.c even 3 2
5635.2.a.be 13 1.a even 1 1 trivial
5635.2.a.bf 13 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5635))\):

\( T_{2}^{13} + 5 T_{2}^{12} - 7 T_{2}^{11} - 65 T_{2}^{10} - 20 T_{2}^{9} + 287 T_{2}^{8} + 238 T_{2}^{7} + \cdots - 2 \) Copy content Toggle raw display
\( T_{3}^{13} + 2 T_{3}^{12} - 20 T_{3}^{11} - 37 T_{3}^{10} + 149 T_{3}^{9} + 242 T_{3}^{8} - 552 T_{3}^{7} + \cdots + 74 \) Copy content Toggle raw display
\( T_{11}^{13} + 17 T_{11}^{12} + 69 T_{11}^{11} - 294 T_{11}^{10} - 2627 T_{11}^{9} - 2570 T_{11}^{8} + \cdots - 2590 \) Copy content Toggle raw display
\( T_{17}^{13} - 2 T_{17}^{12} - 102 T_{17}^{11} + 97 T_{17}^{10} + 3578 T_{17}^{9} - 1220 T_{17}^{8} + \cdots - 8938 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + 5 T^{12} + \cdots - 2 \) Copy content Toggle raw display
$3$ \( T^{13} + 2 T^{12} + \cdots + 74 \) Copy content Toggle raw display
$5$ \( (T - 1)^{13} \) Copy content Toggle raw display
$7$ \( T^{13} \) Copy content Toggle raw display
$11$ \( T^{13} + 17 T^{12} + \cdots - 2590 \) Copy content Toggle raw display
$13$ \( T^{13} - T^{12} + \cdots + 718316 \) Copy content Toggle raw display
$17$ \( T^{13} - 2 T^{12} + \cdots - 8938 \) Copy content Toggle raw display
$19$ \( T^{13} + 13 T^{12} + \cdots - 4007182 \) Copy content Toggle raw display
$23$ \( (T - 1)^{13} \) Copy content Toggle raw display
$29$ \( T^{13} + 26 T^{12} + \cdots - 43740665 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 1555046825 \) Copy content Toggle raw display
$37$ \( T^{13} + 29 T^{12} + \cdots - 361963 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots - 2248187465 \) Copy content Toggle raw display
$43$ \( T^{13} + 10 T^{12} + \cdots - 69091367 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots + 209548726 \) Copy content Toggle raw display
$53$ \( T^{13} + 43 T^{12} + \cdots + 12646534 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots - 3977942272 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 1653945118 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 488147485 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 2148316048 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 183316252 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 15827927812 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 32670323891 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 138612092906 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 6280485043 \) Copy content Toggle raw display
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