Properties

Label 7605.2.a.cs
Level $7605$
Weight $2$
Character orbit 7605.a
Self dual yes
Analytic conductor $60.726$
Analytic rank $0$
Dimension $9$
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] = [7605,2,Mod(1,7605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7605, 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("7605.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.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: \(60.7262307372\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 17x^{7} - 9x^{6} + 59x^{5} + 32x^{4} - 44x^{3} - 23x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 845)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 17x^{7} - 9x^{6} + 59x^{5} + 32x^{4} - 44x^{3} - 23x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -27\nu^{8} - 10\nu^{7} + 554\nu^{6} + 310\nu^{5} - 2939\nu^{4} - 926\nu^{3} + 4655\nu^{2} + 904\nu + 446 ) / 1066 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 297 \nu^{8} - 110 \nu^{7} + 5028 \nu^{6} + 4476 \nu^{5} - 16339 \nu^{4} - 14450 \nu^{3} + 10697 \nu^{2} + 6746 \nu - 424 ) / 1066 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 446 \nu^{8} - 27 \nu^{7} + 7572 \nu^{6} + 4568 \nu^{5} - 26004 \nu^{4} - 17211 \nu^{3} + 18698 \nu^{2} + 14913 \nu + 458 ) / 1066 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 480 \nu^{8} + 296 \nu^{7} + 8013 \nu^{6} - 648 \nu^{5} - 28560 \nu^{4} + 2252 \nu^{3} + 22645 \nu^{2} - 1814 \nu - 2435 ) / 1066 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 620 \nu^{8} - 27 \nu^{7} - 10550 \nu^{6} - 5026 \nu^{5} + 36890 \nu^{4} + 16901 \nu^{3} - 28206 \nu^{2} - 9605 \nu + 1524 ) / 1066 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1369 \nu^{8} - 480 \nu^{7} - 22977 \nu^{6} - 4308 \nu^{5} + 80123 \nu^{4} + 15248 \nu^{3} - 57984 \nu^{2} - 8842 \nu + 621 ) / 1066 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1793 \nu^{8} - 777 \nu^{7} - 30295 \nu^{6} - 3096 \nu^{5} + 109615 \nu^{4} + 12477 \nu^{3} - 91090 \nu^{2} - 7897 \nu + 6725 ) / 1066 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{3} - 2\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} - \beta_{6} - 3\beta_{5} - 2\beta_{4} - 3\beta_{2} + 9\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{7} - 27\beta_{6} + 5\beta_{5} - 3\beta_{4} - 16\beta_{3} - 27\beta_{2} + 6\beta _1 + 51 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -16\beta_{8} + 6\beta_{7} - 29\beta_{6} - 44\beta_{5} - 27\beta_{4} - 12\beta_{3} - 64\beta_{2} + 103\beta _1 + 72 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{8} + 103 \beta_{7} - 354 \beta_{6} + 5 \beta_{5} - 64 \beta_{4} - 215 \beta_{3} - 370 \beta_{2} + 154 \beta _1 + 630 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 215 \beta_{8} + 154 \beta_{7} - 613 \beta_{6} - 555 \beta_{5} - 370 \beta_{4} - 316 \beta_{3} - 1091 \beta_{2} + 1318 \beta _1 + 1365 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 316 \beta_{8} + 1318 \beta_{7} - 4741 \beta_{6} - 466 \beta_{5} - 1091 \beta_{4} - 2863 \beta_{3} - 5265 \beta_{2} + 2926 \beta _1 + 8472 \) 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
−3.03152
−1.03381
0.199774
−2.07331
−0.421015
−0.271062
1.76052
3.88295
0.987478
−2.58648 0 4.68989 1.00000 0 −0.858383 −6.95735 0 −2.58648
1.2 −2.28079 0 3.20199 1.00000 0 −2.30369 −2.74149 0 −2.28079
1.3 −1.04721 0 −0.903360 1.00000 0 −3.42366 3.04042 0 −1.04721
1.4 −0.271374 0 −1.92636 1.00000 0 3.38151 1.06551 0 −0.271374
1.5 0.0240266 0 −1.99942 1.00000 0 1.66541 −0.0960927 0 0.0240266
1.6 1.53088 0 0.343581 1.00000 0 −3.86493 −2.53577 0 1.53088
1.7 2.20556 0 2.86449 1.00000 0 −4.60897 1.90669 0 2.20556
1.8 2.63597 0 4.94835 1.00000 0 3.28231 7.77176 0 2.63597
1.9 2.78942 0 5.78084 1.00000 0 −0.269601 10.5463 0 2.78942
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(-1\)

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 7605.2.a.cs 9
3.b odd 2 1 845.2.a.n 9
13.b even 2 1 7605.2.a.cp 9
15.d odd 2 1 4225.2.a.bt 9
39.d odd 2 1 845.2.a.o yes 9
39.f even 4 2 845.2.c.h 18
39.h odd 6 2 845.2.e.o 18
39.i odd 6 2 845.2.e.p 18
39.k even 12 4 845.2.m.j 36
195.e odd 2 1 4225.2.a.bs 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.a.n 9 3.b odd 2 1
845.2.a.o yes 9 39.d odd 2 1
845.2.c.h 18 39.f even 4 2
845.2.e.o 18 39.h odd 6 2
845.2.e.p 18 39.i odd 6 2
845.2.m.j 36 39.k even 12 4
4225.2.a.bs 9 195.e odd 2 1
4225.2.a.bt 9 15.d odd 2 1
7605.2.a.cp 9 13.b even 2 1
7605.2.a.cs 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2}^{9} - 3T_{2}^{8} - 13T_{2}^{7} + 40T_{2}^{6} + 49T_{2}^{5} - 156T_{2}^{4} - 51T_{2}^{3} + 152T_{2}^{2} + 38T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{9} + 7T_{7}^{8} - 15T_{7}^{7} - 176T_{7}^{6} - 58T_{7}^{5} + 1305T_{7}^{4} + 1385T_{7}^{3} - 2321T_{7}^{2} - 2930T_{7} - 601 \) Copy content Toggle raw display
\( T_{11}^{9} + 9 T_{11}^{8} - 12 T_{11}^{7} - 241 T_{11}^{6} - 76 T_{11}^{5} + 2176 T_{11}^{4} + 1216 T_{11}^{3} - 7600 T_{11}^{2} - 2816 T_{11} + 8896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 3 T^{8} - 13 T^{7} + 40 T^{6} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( (T - 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 7 T^{8} - 15 T^{7} - 176 T^{6} + \cdots - 601 \) Copy content Toggle raw display
$11$ \( T^{9} + 9 T^{8} - 12 T^{7} + \cdots + 8896 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - T^{8} - 80 T^{7} + 51 T^{6} + \cdots - 6656 \) Copy content Toggle raw display
$19$ \( T^{9} - 4 T^{8} - 81 T^{7} + \cdots - 10816 \) Copy content Toggle raw display
$23$ \( T^{9} + 14 T^{8} - 20 T^{7} + \cdots + 43693 \) Copy content Toggle raw display
$29$ \( T^{9} + 12 T^{8} - 22 T^{7} - 695 T^{6} + \cdots - 1 \) Copy content Toggle raw display
$31$ \( T^{9} - 7 T^{8} - 110 T^{7} + \cdots + 10816 \) Copy content Toggle raw display
$37$ \( T^{9} - 5 T^{8} - 213 T^{7} + \cdots - 10018112 \) Copy content Toggle raw display
$41$ \( T^{9} + 10 T^{8} - 152 T^{7} + \cdots + 993287 \) Copy content Toggle raw display
$43$ \( T^{9} - 39 T^{8} + 605 T^{7} + \cdots - 71513 \) Copy content Toggle raw display
$47$ \( T^{9} - 36 T^{8} + 460 T^{7} + \cdots + 11677 \) Copy content Toggle raw display
$53$ \( T^{9} - 8 T^{8} - 263 T^{7} + \cdots + 4469312 \) Copy content Toggle raw display
$59$ \( T^{9} + 21 T^{8} - 25 T^{7} + \cdots - 272896 \) Copy content Toggle raw display
$61$ \( T^{9} + 3 T^{8} - 159 T^{7} + \cdots - 183247 \) Copy content Toggle raw display
$67$ \( T^{9} + T^{8} - 210 T^{7} + \cdots - 115123 \) Copy content Toggle raw display
$71$ \( T^{9} + T^{8} - 226 T^{7} + \cdots - 5586496 \) Copy content Toggle raw display
$73$ \( T^{9} - 256 T^{7} + 344 T^{6} + \cdots + 866816 \) Copy content Toggle raw display
$79$ \( T^{9} - 39 T^{8} + 548 T^{7} + \cdots - 10816 \) Copy content Toggle raw display
$83$ \( T^{9} - 7 T^{8} - 275 T^{7} + \cdots - 49784561 \) Copy content Toggle raw display
$89$ \( T^{9} + 19 T^{8} - 74 T^{7} + \cdots - 1049 \) Copy content Toggle raw display
$97$ \( T^{9} - 34 T^{8} + \cdots + 106251776 \) Copy content Toggle raw display
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