Properties

Label 195.6.a.i
Level 195195
Weight 66
Character orbit 195.a
Self dual yes
Analytic conductor 31.27531.275
Analytic rank 00
Dimension 77
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [195,6,Mod(1,195)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(195, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("195.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 195=3513 195 = 3 \cdot 5 \cdot 13
Weight: k k == 6 6
Character orbit: [χ][\chi] == 195.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: 31.274844863531.2748448635
Analytic rank: 00
Dimension: 77
Coefficient field: Q[x]/(x7)\mathbb{Q}[x]/(x^{7} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x72x6201x5+480x4+11274x329276x2162072x+455280 x^{7} - 2x^{6} - 201x^{5} + 480x^{4} + 11274x^{3} - 29276x^{2} - 162072x + 455280 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 24 2^{4}
Twist minimal: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

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

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β61,\beta_1,\ldots,\beta_{6} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q29q3+(β2β1+26)q425q59β1q6+(β5+β3+β2++4)q7+(β52β4+2β3+56)q8++(81β5+243β4+10530)q99+O(q100) q + \beta_1 q^{2} - 9 q^{3} + (\beta_{2} - \beta_1 + 26) q^{4} - 25 q^{5} - 9 \beta_1 q^{6} + (\beta_{5} + \beta_{3} + \beta_{2} + \cdots + 4) q^{7} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \cdots - 56) q^{8}+ \cdots + ( - 81 \beta_{5} + 243 \beta_{4} + \cdots - 10530) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 7q+2q263q3+182q4175q518q6+32q7354q8+567q950q10910q111638q12+1183q13+1278q14+1575q15+3290q16+142q17+73710q99+O(q100) 7 q + 2 q^{2} - 63 q^{3} + 182 q^{4} - 175 q^{5} - 18 q^{6} + 32 q^{7} - 354 q^{8} + 567 q^{9} - 50 q^{10} - 910 q^{11} - 1638 q^{12} + 1183 q^{13} + 1278 q^{14} + 1575 q^{15} + 3290 q^{16} + 142 q^{17}+ \cdots - 73710 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x72x6201x5+480x4+11274x329276x2162072x+455280 x^{7} - 2x^{6} - 201x^{5} + 480x^{4} + 11274x^{3} - 29276x^{2} - 162072x + 455280 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2+ν58 \nu^{2} + \nu - 58 Copy content Toggle raw display
β3\beta_{3}== (33ν696ν5+4465ν4+12866ν3113758ν2440704ν+721272)/37328 ( -33\nu^{6} - 96\nu^{5} + 4465\nu^{4} + 12866\nu^{3} - 113758\nu^{2} - 440704\nu + 721272 ) / 37328 Copy content Toggle raw display
β4\beta_{4}== (11ν6+32ν52266ν45844ν3+132017ν2+257330ν1882856)/4666 ( 11\nu^{6} + 32\nu^{5} - 2266\nu^{4} - 5844\nu^{3} + 132017\nu^{2} + 257330\nu - 1882856 ) / 4666 Copy content Toggle raw display
β5\beta_{5}== (121ν6+352ν522593ν440954ν3+1151230ν2+856912ν13656424)/18664 ( 121\nu^{6} + 352\nu^{5} - 22593\nu^{4} - 40954\nu^{3} + 1151230\nu^{2} + 856912\nu - 13656424 ) / 18664 Copy content Toggle raw display
β6\beta_{6}== (443ν6+408ν5+84259ν472178ν34181362ν2+1537040ν+43030008)/37328 ( -443\nu^{6} + 408\nu^{5} + 84259\nu^{4} - 72178\nu^{3} - 4181362\nu^{2} + 1537040\nu + 43030008 ) / 37328 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2β1+58 \beta_{2} - \beta _1 + 58 Copy content Toggle raw display
ν3\nu^{3}== β52β4+2β3+β2+87β156 \beta_{5} - 2\beta_{4} + 2\beta_{3} + \beta_{2} + 87\beta _1 - 56 Copy content Toggle raw display
ν4\nu^{4}== 2β52β420β3+119β2153β1+5018 -2\beta_{5} - 2\beta_{4} - 20\beta_{3} + 119\beta_{2} - 153\beta _1 + 5018 Copy content Toggle raw display
ν5\nu^{5}== 22β6+173β5260β4+280β3+3β2+8793β18930 22\beta_{6} + 173\beta_{5} - 260\beta_{4} + 280\beta_{3} + 3\beta_{2} + 8793\beta _1 - 8930 Copy content Toggle raw display
ν6\nu^{6}== 64β6384β5294β43872β3+13035β222269β1+505014 -64\beta_{6} - 384\beta_{5} - 294\beta_{4} - 3872\beta_{3} + 13035\beta_{2} - 22269\beta _1 + 505014 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.

comment: embeddings in the coefficient field
 
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
−11.0349
−7.90544
−4.78396
3.01237
4.36956
8.06197
10.2804
−11.0349 −9.00000 89.7686 −25.0000 99.3139 −171.189 −637.470 81.0000 275.872
1.2 −7.90544 −9.00000 30.4961 −25.0000 71.1490 176.088 11.8893 81.0000 197.636
1.3 −4.78396 −9.00000 −9.11374 −25.0000 43.0556 73.5517 196.686 81.0000 119.599
1.4 3.01237 −9.00000 −22.9256 −25.0000 −27.1114 −242.606 −165.456 81.0000 −75.3093
1.5 4.36956 −9.00000 −12.9069 −25.0000 −39.3261 51.6597 −196.224 81.0000 −109.239
1.6 8.06197 −9.00000 32.9954 −25.0000 −72.5577 −68.7454 8.02471 81.0000 −201.549
1.7 10.2804 −9.00000 73.6862 −25.0000 −92.5234 213.240 428.550 81.0000 −257.009
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 +1 +1
55 +1 +1
1313 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 195.6.a.i 7
3.b odd 2 1 585.6.a.n 7
5.b even 2 1 975.6.a.m 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.6.a.i 7 1.a even 1 1 trivial
585.6.a.n 7 3.b odd 2 1
975.6.a.m 7 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T272T26201T25+480T24+11274T2329276T22162072T2+455280 T_{2}^{7} - 2T_{2}^{6} - 201T_{2}^{5} + 480T_{2}^{4} + 11274T_{2}^{3} - 29276T_{2}^{2} - 162072T_{2} + 455280 acting on S6new(Γ0(195))S_{6}^{\mathrm{new}}(\Gamma_0(195)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T72T6++455280 T^{7} - 2 T^{6} + \cdots + 455280 Copy content Toggle raw display
33 (T+9)7 (T + 9)^{7} Copy content Toggle raw display
55 (T+25)7 (T + 25)^{7} Copy content Toggle raw display
77 T7++407345549010432 T^{7} + \cdots + 407345549010432 Copy content Toggle raw display
1111 T7++32 ⁣ ⁣56 T^{7} + \cdots + 32\!\cdots\!56 Copy content Toggle raw display
1313 (T169)7 (T - 169)^{7} Copy content Toggle raw display
1717 T7++62 ⁣ ⁣44 T^{7} + \cdots + 62\!\cdots\!44 Copy content Toggle raw display
1919 T7+28 ⁣ ⁣40 T^{7} + \cdots - 28\!\cdots\!40 Copy content Toggle raw display
2323 T7++29 ⁣ ⁣20 T^{7} + \cdots + 29\!\cdots\!20 Copy content Toggle raw display
2929 T7++35 ⁣ ⁣20 T^{7} + \cdots + 35\!\cdots\!20 Copy content Toggle raw display
3131 T7+17 ⁣ ⁣16 T^{7} + \cdots - 17\!\cdots\!16 Copy content Toggle raw display
3737 T7+48 ⁣ ⁣32 T^{7} + \cdots - 48\!\cdots\!32 Copy content Toggle raw display
4141 T7+52 ⁣ ⁣04 T^{7} + \cdots - 52\!\cdots\!04 Copy content Toggle raw display
4343 T7++84 ⁣ ⁣08 T^{7} + \cdots + 84\!\cdots\!08 Copy content Toggle raw display
4747 T7++24 ⁣ ⁣00 T^{7} + \cdots + 24\!\cdots\!00 Copy content Toggle raw display
5353 T7++72 ⁣ ⁣64 T^{7} + \cdots + 72\!\cdots\!64 Copy content Toggle raw display
5959 T7+11 ⁣ ⁣28 T^{7} + \cdots - 11\!\cdots\!28 Copy content Toggle raw display
6161 T7++44 ⁣ ⁣08 T^{7} + \cdots + 44\!\cdots\!08 Copy content Toggle raw display
6767 T7+81 ⁣ ⁣12 T^{7} + \cdots - 81\!\cdots\!12 Copy content Toggle raw display
7171 T7++57 ⁣ ⁣20 T^{7} + \cdots + 57\!\cdots\!20 Copy content Toggle raw display
7373 T7++34 ⁣ ⁣52 T^{7} + \cdots + 34\!\cdots\!52 Copy content Toggle raw display
7979 T7++26 ⁣ ⁣96 T^{7} + \cdots + 26\!\cdots\!96 Copy content Toggle raw display
8383 T7+33 ⁣ ⁣36 T^{7} + \cdots - 33\!\cdots\!36 Copy content Toggle raw display
8989 T7++20 ⁣ ⁣40 T^{7} + \cdots + 20\!\cdots\!40 Copy content Toggle raw display
9797 T7+28 ⁣ ⁣00 T^{7} + \cdots - 28\!\cdots\!00 Copy content Toggle raw display
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