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Properties

Label 2695.2.a.z
Level $2695$
Weight $2$
Character orbit 2695.a
Self dual yes
Analytic conductor $21.520$
Analytic rank $0$
Dimension $10$
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] = [2695,2,Mod(1,2695)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2695, 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("2695.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2695 = 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2695.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: \(21.5196833447\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} - 13x^{8} + 41x^{7} + 51x^{6} - 184x^{5} - 45x^{4} + 297x^{3} - 59x^{2} - 109x + 21 \) 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 385)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 13x^{8} + 41x^{7} + 51x^{6} - 184x^{5} - 45x^{4} + 297x^{3} - 59x^{2} - 109x + 21 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{9} - 5\nu^{8} - 37\nu^{7} + 52\nu^{6} + 153\nu^{5} - 140\nu^{4} - 287\nu^{3} + 36\nu^{2} + 235\nu + 21 ) / 39 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{9} - 11\nu^{8} - 58\nu^{7} + 143\nu^{6} + 295\nu^{5} - 594\nu^{4} - 595\nu^{3} + 841\nu^{2} + 335\nu - 219 ) / 39 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{9} + 16\nu^{8} + 95\nu^{7} - 195\nu^{6} - 409\nu^{5} + 734\nu^{4} + 531\nu^{3} - 916\nu^{2} + 54\nu + 198 ) / 39 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8 \nu^{9} + 9 \nu^{8} + 129 \nu^{7} - 104 \nu^{6} - 707 \nu^{5} + 356 \nu^{4} + 1450 \nu^{3} + \cdots + 87 ) / 39 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12 \nu^{9} + 20 \nu^{8} + 187 \nu^{7} - 247 \nu^{6} - 1002 \nu^{5} + 911 \nu^{4} + 2084 \nu^{3} + \cdots + 33 ) / 39 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9 \nu^{9} - 15 \nu^{8} - 137 \nu^{7} + 195 \nu^{6} + 706 \nu^{5} - 810 \nu^{4} - 1355 \nu^{3} + \cdots - 223 ) / 13 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} + \beta_{7} - \beta_{5} + \beta_{3} + 8\beta_{2} + \beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} + 9\beta_{3} + 10\beta_{2} + 29\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} - 10\beta_{8} + 12\beta_{7} + \beta_{6} - 11\beta_{5} + 11\beta_{3} + 59\beta_{2} + 13\beta _1 + 135 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{9} + 3\beta_{7} + 11\beta_{6} + 8\beta_{5} + 14\beta_{4} + 68\beta_{3} + 84\beta_{2} + 183\beta _1 + 124 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{9} - 76 \beta_{8} + 104 \beta_{7} + 16 \beta_{6} - 97 \beta_{5} + 5 \beta_{4} + 97 \beta_{3} + \cdots + 878 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 20 \beta_{9} + 49 \beta_{7} + 94 \beta_{6} + 30 \beta_{5} + 143 \beta_{4} + 493 \beta_{3} + 672 \beta_{2} + \cdots + 1057 \) 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
−2.47520
−2.10414
−1.66564
−0.650789
0.190943
1.06607
1.29002
1.93394
2.63370
2.78109
−2.47520 3.19739 4.12660 1.00000 −7.91417 0 −5.26377 7.22329 −2.47520
1.2 −2.10414 −1.86644 2.42740 1.00000 3.92725 0 −0.899305 0.483603 −2.10414
1.3 −1.66564 −0.562253 0.774372 1.00000 0.936513 0 2.04146 −2.68387 −1.66564
1.4 −0.650789 1.14162 −1.57647 1.00000 −0.742952 0 2.32753 −1.69671 −0.650789
1.5 0.190943 3.31339 −1.96354 1.00000 0.632670 0 −0.756812 7.97855 0.190943
1.6 1.06607 −3.07491 −0.863486 1.00000 −3.27808 0 −3.05269 6.45507 1.06607
1.7 1.29002 −0.350856 −0.335851 1.00000 −0.452611 0 −3.01329 −2.87690 1.29002
1.8 1.93394 2.14512 1.74012 1.00000 4.14853 0 −0.502587 1.60154 1.93394
1.9 2.63370 −2.48035 4.93639 1.00000 −6.53251 0 7.73356 3.15215 2.63370
1.10 2.78109 1.53729 5.73447 1.00000 4.27536 0 10.3859 −0.636727 2.78109
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 2695.2.a.z 10
7.b odd 2 1 2695.2.a.y 10
7.d odd 6 2 385.2.i.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.i.d ✓ 20 7.d odd 6 2
2695.2.a.y 10 7.b odd 2 1
2695.2.a.z 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2695))\):

\( T_{2}^{10} - 3T_{2}^{9} - 13T_{2}^{8} + 41T_{2}^{7} + 51T_{2}^{6} - 184T_{2}^{5} - 45T_{2}^{4} + 297T_{2}^{3} - 59T_{2}^{2} - 109T_{2} + 21 \) Copy content Toggle raw display
\( T_{3}^{10} - 3 T_{3}^{9} - 20 T_{3}^{8} + 58 T_{3}^{7} + 128 T_{3}^{6} - 357 T_{3}^{5} - 281 T_{3}^{4} + \cdots - 112 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 3 T^{9} + \cdots + 21 \) Copy content Toggle raw display
$3$ \( T^{10} - 3 T^{9} + \cdots - 112 \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T - 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} + \cdots - 56056 \) Copy content Toggle raw display
$17$ \( T^{10} - 5 T^{9} + \cdots + 52416 \) Copy content Toggle raw display
$19$ \( T^{10} + T^{9} + \cdots + 57024 \) Copy content Toggle raw display
$23$ \( T^{10} - 18 T^{9} + \cdots + 1087536 \) Copy content Toggle raw display
$29$ \( T^{10} - 14 T^{9} + \cdots - 2074896 \) Copy content Toggle raw display
$31$ \( T^{10} + 10 T^{9} + \cdots - 166824 \) Copy content Toggle raw display
$37$ \( T^{10} - 13 T^{9} + \cdots + 6559424 \) Copy content Toggle raw display
$41$ \( T^{10} - 7 T^{9} + \cdots - 4402944 \) Copy content Toggle raw display
$43$ \( T^{10} - 6 T^{9} + \cdots - 13653189 \) Copy content Toggle raw display
$47$ \( T^{10} + T^{9} + \cdots + 2496 \) Copy content Toggle raw display
$53$ \( T^{10} - 16 T^{9} + \cdots + 2303616 \) Copy content Toggle raw display
$59$ \( T^{10} + 13 T^{9} + \cdots - 60061224 \) Copy content Toggle raw display
$61$ \( T^{10} + 18 T^{9} + \cdots - 7149584 \) Copy content Toggle raw display
$67$ \( T^{10} - 29 T^{9} + \cdots - 2847728 \) Copy content Toggle raw display
$71$ \( T^{10} - 19 T^{9} + \cdots - 16214508 \) Copy content Toggle raw display
$73$ \( T^{10} - 31 T^{9} + \cdots + 4541548 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 120362368 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 423475857 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 1911845607 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 326335744 \) Copy content Toggle raw display
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