Properties

Label 9248.2.a.bo
Level 92489248
Weight 22
Character orbit 9248.a
Self dual yes
Analytic conductor 73.84673.846
Analytic rank 11
Dimension 99
CM no
Inner twists 11

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9248,2,Mod(1,9248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9248.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 9248=25172 9248 = 2^{5} \cdot 17^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 9248.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,-9,0,-6,0,-3,0,12,0,-6,0,-12,0,21,0,0,0,0,0,18,0,3,0,9,0, -24,0,-21,0,-24,0,27,0,12,0,-12,0,-3,0,15,0,3,0,-33,0,12,0,-6,0,0,0,9, 0,-21,0,-9,0,-45,0,-3,0,-27,0,51,0,-6,0,18,0,9,0,30,0,-39,0,9,0,18,0,9, 0,9,0,0,0,48,0,0,0,-3,0,33,0,-33,0,33,0,-90,0,-15,0,51,0,-9,0,3,0,-6,0, 21,0,27,0,9,0,24,0,0,0,27,0,-48,0,-27,0,-12,0,6,0,-9,0,27,0,45,0,3,0,0, 0,-51,0,-78,0,57] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(145)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 73.845651789373.8456517893
Analytic rank: 11
Dimension: 99
Coefficient field: Q[x]/(x9)\mathbb{Q}[x]/(x^{9} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x915x77x6+69x5+48x4110x387x2+45x+37 x^{9} - 15x^{7} - 7x^{6} + 69x^{5} + 48x^{4} - 110x^{3} - 87x^{2} + 45x + 37 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β81,\beta_1,\ldots,\beta_{8} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β11)q3+(β6+β5+β11)q5+β7q7+(β2β1+1)q9+(β5β3β2)q11+(2β5β32)q13++(4β8+2β7+11)q99+O(q100) q + (\beta_1 - 1) q^{3} + ( - \beta_{6} + \beta_{5} + \beta_1 - 1) q^{5} + \beta_{7} q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{5} - \beta_{3} - \beta_{2}) q^{11} + (2 \beta_{5} - \beta_{3} - 2) q^{13}+ \cdots + ( - 4 \beta_{8} + 2 \beta_{7} + \cdots - 11) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 9q9q36q53q7+12q96q1112q13+21q15+18q21+3q23+9q2524q2721q2924q31+27q33+12q3512q373q39+15q41+90q99+O(q100) 9 q - 9 q^{3} - 6 q^{5} - 3 q^{7} + 12 q^{9} - 6 q^{11} - 12 q^{13} + 21 q^{15} + 18 q^{21} + 3 q^{23} + 9 q^{25} - 24 q^{27} - 21 q^{29} - 24 q^{31} + 27 q^{33} + 12 q^{35} - 12 q^{37} - 3 q^{39} + 15 q^{41}+ \cdots - 90 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x915x77x6+69x5+48x4110x387x2+45x+37 x^{9} - 15x^{7} - 7x^{6} + 69x^{5} + 48x^{4} - 110x^{3} - 87x^{2} + 45x + 37 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2ν3 \nu^{2} - \nu - 3 Copy content Toggle raw display
β3\beta_{3}== (ν73ν68ν5+22ν4+19ν343ν213ν+19)/2 ( \nu^{7} - 3\nu^{6} - 8\nu^{5} + 22\nu^{4} + 19\nu^{3} - 43\nu^{2} - 13\nu + 19 ) / 2 Copy content Toggle raw display
β4\beta_{4}== (ν83ν78ν6+22ν5+19ν443ν313ν2+19ν)/2 ( \nu^{8} - 3\nu^{7} - 8\nu^{6} + 22\nu^{5} + 19\nu^{4} - 43\nu^{3} - 13\nu^{2} + 19\nu ) / 2 Copy content Toggle raw display
β5\beta_{5}== (3ν88ν727ν6+60ν5+75ν4122ν368ν2+54ν+17)/2 ( 3\nu^{8} - 8\nu^{7} - 27\nu^{6} + 60\nu^{5} + 75\nu^{4} - 122\nu^{3} - 68\nu^{2} + 54\nu + 17 ) / 2 Copy content Toggle raw display
β6\beta_{6}== (3ν86ν733ν6+44ν5+121ν488ν3166ν2+42ν+65)/2 ( 3\nu^{8} - 6\nu^{7} - 33\nu^{6} + 44\nu^{5} + 121\nu^{4} - 88\nu^{3} - 166\nu^{2} + 42\nu + 65 ) / 2 Copy content Toggle raw display
β7\beta_{7}== (4ν87ν746ν6+48ν5+176ν483ν3250ν2+27ν+98)/2 ( 4\nu^{8} - 7\nu^{7} - 46\nu^{6} + 48\nu^{5} + 176\nu^{4} - 83\nu^{3} - 250\nu^{2} + 27\nu + 98 ) / 2 Copy content Toggle raw display
β8\beta_{8}== (5ν89ν757ν6+64ν5+213ν4121ν3290ν2+51ν+111)/2 ( 5\nu^{8} - 9\nu^{7} - 57\nu^{6} + 64\nu^{5} + 213\nu^{4} - 121\nu^{3} - 290\nu^{2} + 51\nu + 111 ) / 2 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+β1+3 \beta_{2} + \beta _1 + 3 Copy content Toggle raw display
ν3\nu^{3}== β8+β7+β52β4+β2+5β1+1 -\beta_{8} + \beta_{7} + \beta_{5} - 2\beta_{4} + \beta_{2} + 5\beta _1 + 1 Copy content Toggle raw display
ν4\nu^{4}== 2β8+2β7+β6+β54β42β3+8β2+9β1+15 -2\beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{5} - 4\beta_{4} - 2\beta_{3} + 8\beta_{2} + 9\beta _1 + 15 Copy content Toggle raw display
ν5\nu^{5}== 10β8+10β7+2β6+9β523β45β3+15β2+36β1+16 -10\beta_{8} + 10\beta_{7} + 2\beta_{6} + 9\beta_{5} - 23\beta_{4} - 5\beta_{3} + 15\beta_{2} + 36\beta _1 + 16 Copy content Toggle raw display
ν6\nu^{6}== 26β8+28β7+10β6+16β560β430β3+69β2+87β1+102 -26\beta_{8} + 28\beta_{7} + 10\beta_{6} + 16\beta_{5} - 60\beta_{4} - 30\beta_{3} + 69\beta_{2} + 87\beta _1 + 102 Copy content Toggle raw display
ν7\nu^{7}== 95β8+101β7+24β6+79β5238β484β3++195 - 95 \beta_{8} + 101 \beta_{7} + 24 \beta_{6} + 79 \beta_{5} - 238 \beta_{4} - 84 \beta_{3} + \cdots + 195 Copy content Toggle raw display
ν8\nu^{8}== 278β8+312β7+89β6+191β5696β4344β3++846 - 278 \beta_{8} + 312 \beta_{7} + 89 \beta_{6} + 191 \beta_{5} - 696 \beta_{4} - 344 \beta_{3} + \cdots + 846 Copy content Toggle raw display

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\iota_m(a_n) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
1.1
−2.10579
−2.09095
−1.62107
−0.894208
−0.775009
0.785165
1.45307
2.04878
3.20001
0 −3.10579 0 −2.03395 0 0.883659 0 6.64591 0
1.2 0 −3.09095 0 −3.92598 0 −3.26034 0 6.55400 0
1.3 0 −2.62107 0 0.309248 0 −4.59618 0 3.87001 0
1.4 0 −1.89421 0 2.78809 0 −0.147450 0 0.588025 0
1.5 0 −1.77501 0 −2.79665 0 3.67713 0 0.150658 0
1.6 0 −0.214835 0 −3.35988 0 −2.68602 0 −2.95385 0
1.7 0 0.453073 0 0.322684 0 0.528404 0 −2.79472 0
1.8 0 1.04878 0 2.71408 0 1.45113 0 −1.90005 0
1.9 0 2.20001 0 −0.0176497 0 1.14966 0 1.84003 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
1717 1 -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 9248.2.a.bo 9
4.b odd 2 1 9248.2.a.bq yes 9
17.b even 2 1 9248.2.a.br yes 9
68.d odd 2 1 9248.2.a.bp yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9248.2.a.bo 9 1.a even 1 1 trivial
9248.2.a.bp yes 9 68.d odd 2 1
9248.2.a.bq yes 9 4.b odd 2 1
9248.2.a.br yes 9 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(9248))S_{2}^{\mathrm{new}}(\Gamma_0(9248)):

T39+9T38+21T3728T36162T35111T34+191T33+177T3260T319 T_{3}^{9} + 9T_{3}^{8} + 21T_{3}^{7} - 28T_{3}^{6} - 162T_{3}^{5} - 111T_{3}^{4} + 191T_{3}^{3} + 177T_{3}^{2} - 60T_{3} - 19 Copy content Toggle raw display
T59+6T589T5797T5633T55+432T54+324T53315T52+51T5+1 T_{5}^{9} + 6T_{5}^{8} - 9T_{5}^{7} - 97T_{5}^{6} - 33T_{5}^{5} + 432T_{5}^{4} + 324T_{5}^{3} - 315T_{5}^{2} + 51T_{5} + 1 Copy content Toggle raw display
T79+3T7824T7749T76+177T75+99T74521T73+378T7248T717 T_{7}^{9} + 3T_{7}^{8} - 24T_{7}^{7} - 49T_{7}^{6} + 177T_{7}^{5} + 99T_{7}^{4} - 521T_{7}^{3} + 378T_{7}^{2} - 48T_{7} - 17 Copy content Toggle raw display
T19957T19741T196+873T195+870T1943862T1931731T192+6663T192447 T_{19}^{9} - 57T_{19}^{7} - 41T_{19}^{6} + 873T_{19}^{5} + 870T_{19}^{4} - 3862T_{19}^{3} - 1731T_{19}^{2} + 6663T_{19} - 2447 Copy content Toggle raw display
T4393T438339T437+1222T436+38238T435144075T434++8890633 T_{43}^{9} - 3 T_{43}^{8} - 339 T_{43}^{7} + 1222 T_{43}^{6} + 38238 T_{43}^{5} - 144075 T_{43}^{4} + \cdots + 8890633 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T9 T^{9} Copy content Toggle raw display
33 T9+9T8+19 T^{9} + 9 T^{8} + \cdots - 19 Copy content Toggle raw display
55 T9+6T8++1 T^{9} + 6 T^{8} + \cdots + 1 Copy content Toggle raw display
77 T9+3T8+17 T^{9} + 3 T^{8} + \cdots - 17 Copy content Toggle raw display
1111 T9+6T8+4672 T^{9} + 6 T^{8} + \cdots - 4672 Copy content Toggle raw display
1313 T9+12T8+29179 T^{9} + 12 T^{8} + \cdots - 29179 Copy content Toggle raw display
1717 T9 T^{9} Copy content Toggle raw display
1919 T957T7+2447 T^{9} - 57 T^{7} + \cdots - 2447 Copy content Toggle raw display
2323 T93T8+71 T^{9} - 3 T^{8} + \cdots - 71 Copy content Toggle raw display
2929 T9+21T8+198917 T^{9} + 21 T^{8} + \cdots - 198917 Copy content Toggle raw display
3131 T9+24T8+239581 T^{9} + 24 T^{8} + \cdots - 239581 Copy content Toggle raw display
3737 T9+12T8+8390231 T^{9} + 12 T^{8} + \cdots - 8390231 Copy content Toggle raw display
4141 T915T8+1549 T^{9} - 15 T^{8} + \cdots - 1549 Copy content Toggle raw display
4343 T93T8++8890633 T^{9} - 3 T^{8} + \cdots + 8890633 Copy content Toggle raw display
4747 T912T8++384067 T^{9} - 12 T^{8} + \cdots + 384067 Copy content Toggle raw display
5353 T99T8+503488 T^{9} - 9 T^{8} + \cdots - 503488 Copy content Toggle raw display
5959 T9+45T8+1677943 T^{9} + 45 T^{8} + \cdots - 1677943 Copy content Toggle raw display
6161 T9+3T8++358085807 T^{9} + 3 T^{8} + \cdots + 358085807 Copy content Toggle raw display
6767 T9+6T8++4191337 T^{9} + 6 T^{8} + \cdots + 4191337 Copy content Toggle raw display
7171 T99T8++41761781 T^{9} - 9 T^{8} + \cdots + 41761781 Copy content Toggle raw display
7373 T930T8+9721 T^{9} - 30 T^{8} + \cdots - 9721 Copy content Toggle raw display
7979 T918T8+6839288 T^{9} - 18 T^{8} + \cdots - 6839288 Copy content Toggle raw display
8383 T99T8++9609011 T^{9} - 9 T^{8} + \cdots + 9609011 Copy content Toggle raw display
8989 T9504T7++10434311 T^{9} - 504 T^{7} + \cdots + 10434311 Copy content Toggle raw display
9797 T933T8+19741184 T^{9} - 33 T^{8} + \cdots - 19741184 Copy content Toggle raw display
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