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Properties

Label 245.10.a.m
Level 245245
Weight 1010
Character orbit 245.a
Self dual yes
Analytic conductor 126.184126.184
Analytic rank 00
Dimension 1313
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] = [245,10,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: N N == 245=5⋅72 245 = 5 \cdot 7^{2}
Weight: k k == 10 10
Character orbit: [χ][\chi] == 245.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: 126.183779860126.183779860
Analytic rank: 00
Dimension: 1313
Coefficient field: Q[x]/(x13−⋯ )\mathbb{Q}[x]/(x^{13} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x13−x12−5109x11+3203x10+9635922x9+242128x8−8405086048x7+⋯−96 ⁣⋯ ⁣52 x^{13} - x^{12} - 5109 x^{11} + 3203 x^{10} + 9635922 x^{9} + 242128 x^{8} - 8405086048 x^{7} + \cdots - 96\!\cdots\!52 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 214⋅33⋅53⋅77 2^{14}\cdot 3^{3}\cdot 5^{3}\cdot 7^{7}
Twist minimal: no (minimal twist has level 35)
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,…,β121,\beta_1,\ldots,\beta_{12} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q−β1q2+(β3+21)q3+(β3+β2+β1+274)q4+625q5+(−β4+β3+β2+⋯+238)q6+(−β5−β4+8β3+⋯−334)q8+⋯+(−1334β12+17371β11+⋯+405275727)q99+O(q100) q - \beta_1 q^{2} + (\beta_{3} + 21) q^{3} + (\beta_{3} + \beta_{2} + \beta_1 + 274) q^{4} + 625 q^{5} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \cdots + 238) q^{6} + ( - \beta_{5} - \beta_{4} + 8 \beta_{3} + \cdots - 334) q^{8}+ \cdots + ( - 1334 \beta_{12} + 17371 \beta_{11} + \cdots + 405275727) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 13q−q2+268q3+3563q4+8125q5+3040q6−4695q8+82107q9−625q10+129087q11+356068q12+35889q13+167500q15+1379187q16+251650q17+391089q18+⋯+5266142099q99+O(q100) 13 q - q^{2} + 268 q^{3} + 3563 q^{4} + 8125 q^{5} + 3040 q^{6} - 4695 q^{8} + 82107 q^{9} - 625 q^{10} + 129087 q^{11} + 356068 q^{12} + 35889 q^{13} + 167500 q^{15} + 1379187 q^{16} + 251650 q^{17} + 391089 q^{18}+ \cdots + 5266142099 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x13−x12−5109x11+3203x10+9635922x9+242128x8−8405086048x7+⋯−96 ⁣⋯ ⁣52 x^{13} - x^{12} - 5109 x^{11} + 3203 x^{10} + 9635922 x^{9} + 242128 x^{8} - 8405086048 x^{7} + \cdots - 96\!\cdots\!52 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (−15 ⁣⋯ ⁣47ν12+⋯−42 ⁣⋯ ⁣80)/46 ⁣⋯ ⁣12 ( - 15\!\cdots\!47 \nu^{12} + \cdots - 42\!\cdots\!80 ) / 46\!\cdots\!12 Copy content Toggle raw display
β3\beta_{3}== (15 ⁣⋯ ⁣47ν12+⋯+60 ⁣⋯ ⁣48)/46 ⁣⋯ ⁣12 ( 15\!\cdots\!47 \nu^{12} + \cdots + 60\!\cdots\!48 ) / 46\!\cdots\!12 Copy content Toggle raw display
β4\beta_{4}== (−11 ⁣⋯ ⁣89ν12+⋯+30 ⁣⋯ ⁣92)/11 ⁣⋯ ⁣28 ( - 11\!\cdots\!89 \nu^{12} + \cdots + 30\!\cdots\!92 ) / 11\!\cdots\!28 Copy content Toggle raw display
β5\beta_{5}== (68 ⁣⋯ ⁣19ν12+⋯−87 ⁣⋯ ⁣76)/23 ⁣⋯ ⁣56 ( 68\!\cdots\!19 \nu^{12} + \cdots - 87\!\cdots\!76 ) / 23\!\cdots\!56 Copy content Toggle raw display
β6\beta_{6}== (19 ⁣⋯ ⁣43ν12+⋯−14 ⁣⋯ ⁣72)/15 ⁣⋯ ⁣04 ( 19\!\cdots\!43 \nu^{12} + \cdots - 14\!\cdots\!72 ) / 15\!\cdots\!04 Copy content Toggle raw display
β7\beta_{7}== (31 ⁣⋯ ⁣25ν12+⋯−36 ⁣⋯ ⁣08)/11 ⁣⋯ ⁣28 ( 31\!\cdots\!25 \nu^{12} + \cdots - 36\!\cdots\!08 ) / 11\!\cdots\!28 Copy content Toggle raw display
β8\beta_{8}== (55 ⁣⋯ ⁣27ν12+⋯−31 ⁣⋯ ⁣00)/15 ⁣⋯ ⁣04 ( 55\!\cdots\!27 \nu^{12} + \cdots - 31\!\cdots\!00 ) / 15\!\cdots\!04 Copy content Toggle raw display
β9\beta_{9}== (−58 ⁣⋯ ⁣75ν12+⋯+32 ⁣⋯ ⁣88)/15 ⁣⋯ ⁣04 ( - 58\!\cdots\!75 \nu^{12} + \cdots + 32\!\cdots\!88 ) / 15\!\cdots\!04 Copy content Toggle raw display
β10\beta_{10}== (−60 ⁣⋯ ⁣87ν12+⋯+46 ⁣⋯ ⁣76)/58 ⁣⋯ ⁣64 ( - 60\!\cdots\!87 \nu^{12} + \cdots + 46\!\cdots\!76 ) / 58\!\cdots\!64 Copy content Toggle raw display
β11\beta_{11}== (−62 ⁣⋯ ⁣43ν12+⋯+26 ⁣⋯ ⁣40)/46 ⁣⋯ ⁣12 ( - 62\!\cdots\!43 \nu^{12} + \cdots + 26\!\cdots\!40 ) / 46\!\cdots\!12 Copy content Toggle raw display
β12\beta_{12}== (−10 ⁣⋯ ⁣19ν12+⋯+38 ⁣⋯ ⁣80)/46 ⁣⋯ ⁣12 ( - 10\!\cdots\!19 \nu^{12} + \cdots + 38\!\cdots\!80 ) / 46\!\cdots\!12 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β3+β2+β1+786 \beta_{3} + \beta_{2} + \beta _1 + 786 Copy content Toggle raw display
ν3\nu^{3}== β5+β4−8β3−2β2+1335β1+334 \beta_{5} + \beta_{4} - 8\beta_{3} - 2\beta_{2} + 1335\beta _1 + 334 Copy content Toggle raw display
ν4\nu^{4}== −2β12+5β11+3β9+β8+6β7+2β6−7β5+⋯+1051468 - 2 \beta_{12} + 5 \beta_{11} + 3 \beta_{9} + \beta_{8} + 6 \beta_{7} + 2 \beta_{6} - 7 \beta_{5} + \cdots + 1051468 Copy content Toggle raw display
ν5\nu^{5}== 14β12−90β11+102β10+2β9+68β8−140β7+⋯−219792 14 \beta_{12} - 90 \beta_{11} + 102 \beta_{10} + 2 \beta_{9} + 68 \beta_{8} - 140 \beta_{7} + \cdots - 219792 Copy content Toggle raw display
ν6\nu^{6}== −4960β12+16113β11−1250β10+7535β9+4191β8+⋯+1675490862 - 4960 \beta_{12} + 16113 \beta_{11} - 1250 \beta_{10} + 7535 \beta_{9} + 4191 \beta_{8} + \cdots + 1675490862 Copy content Toggle raw display
ν7\nu^{7}== 38956β12−298852β11+327268β10−17980β9+204948β8+⋯−3460687922 38956 \beta_{12} - 298852 \beta_{11} + 327268 \beta_{10} - 17980 \beta_{9} + 204948 \beta_{8} + \cdots - 3460687922 Copy content Toggle raw display
ν8\nu^{8}== −10078794β12+39066893β11−4060376β10+15408491β9+⋯+2892994683596 - 10078794 \beta_{12} + 39066893 \beta_{11} - 4060376 \beta_{10} + 15408491 \beta_{9} + \cdots + 2892994683596 Copy content Toggle raw display
ν9\nu^{9}== 80096150β12−751957078β11+799945494β10−92840322β9+⋯−12799081766744 80096150 \beta_{12} - 751957078 \beta_{11} + 799945494 \beta_{10} - 92840322 \beta_{9} + \cdots - 12799081766744 Copy content Toggle raw display
ν10\nu^{10}== −19599857440β12+85663414677β11−9925013066β10+29962185483β9+⋯+52 ⁣⋯ ⁣54 - 19599857440 \beta_{12} + 85663414677 \beta_{11} - 9925013066 \beta_{10} + 29962185483 \beta_{9} + \cdots + 52\!\cdots\!54 Copy content Toggle raw display
ν11\nu^{11}== 156594515668β12−1729478338264β11+1777006887972β10−294213318376β9+⋯−35 ⁣⋯ ⁣34 156594515668 \beta_{12} - 1729478338264 \beta_{11} + 1777006887972 \beta_{10} - 294213318376 \beta_{9} + \cdots - 35\!\cdots\!34 Copy content Toggle raw display
ν12\nu^{12}== −37853573041914β12+179558986649977β11−22373125097184β10+⋯+97 ⁣⋯ ⁣96 - 37853573041914 \beta_{12} + 179558986649977 \beta_{11} - 22373125097184 \beta_{10} + \cdots + 97\!\cdots\!96 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

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
43.3172
36.4811
34.1284
21.3241
15.1795
9.52028
2.56137
−13.9930
−15.4429
−24.0875
−26.9956
−35.9424
−45.0506
−43.3172 196.236 1364.38 625.000 −8500.37 0 −36922.6 18825.4 −27073.2
1.2 −36.4811 −102.189 818.871 625.000 3727.97 0 −11195.0 −9240.37 −22800.7
1.3 −34.1284 −128.345 652.750 625.000 4380.23 0 −4803.59 −3210.47 −21330.3
1.4 −21.3241 190.064 −57.2810 625.000 −4052.95 0 12139.4 16441.3 −13327.6
1.5 −15.1795 45.9146 −281.582 625.000 −696.962 0 12046.2 −17574.9 −9487.21
1.6 −9.52028 −175.497 −421.364 625.000 1670.78 0 8885.89 11116.2 −5950.18
1.7 −2.56137 177.556 −505.439 625.000 −454.786 0 2606.03 11843.1 −1600.85
1.8 13.9930 −255.495 −316.195 625.000 −3575.14 0 −11589.0 45594.6 8745.64
1.9 15.4429 −46.3250 −273.516 625.000 −715.393 0 −12130.7 −17537.0 9651.82
1.10 24.0875 256.792 68.2094 625.000 6185.48 0 −10689.8 46259.1 15054.7
1.11 26.9956 40.9716 216.763 625.000 1106.05 0 −7970.10 −18004.3 16872.3
1.12 35.9424 −97.4212 779.854 625.000 −3501.55 0 9627.32 −10192.1 22464.0
1.13 45.0506 165.739 1517.55 625.000 7466.64 0 45300.8 7786.42 28156.6
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
55 −1 -1
77 +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 245.10.a.m 13
7.b odd 2 1 245.10.a.l 13
7.c even 3 2 35.10.e.b ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.e.b ✓ 26 7.c even 3 2
245.10.a.l 13 7.b odd 2 1
245.10.a.m 13 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 S10new(Γ0(245))S_{10}^{\mathrm{new}}(\Gamma_0(245)):

T213+T212−5109T211−3203T210+9635922T29−242128T28+⋯+96 ⁣⋯ ⁣52 T_{2}^{13} + T_{2}^{12} - 5109 T_{2}^{11} - 3203 T_{2}^{10} + 9635922 T_{2}^{9} - 242128 T_{2}^{8} + \cdots + 96\!\cdots\!52 Copy content Toggle raw display
T313−268T312−133081T311+36954174T310+6155444118T39+⋯−14 ⁣⋯ ⁣28 T_{3}^{13} - 268 T_{3}^{12} - 133081 T_{3}^{11} + 36954174 T_{3}^{10} + 6155444118 T_{3}^{9} + \cdots - 14\!\cdots\!28 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T13+⋯+96 ⁣⋯ ⁣52 T^{13} + \cdots + 96\!\cdots\!52 Copy content Toggle raw display
33 T13+⋯−14 ⁣⋯ ⁣28 T^{13} + \cdots - 14\!\cdots\!28 Copy content Toggle raw display
55 (T−625)13 (T - 625)^{13} Copy content Toggle raw display
77 T13 T^{13} Copy content Toggle raw display
1111 T13+⋯+29 ⁣⋯ ⁣00 T^{13} + \cdots + 29\!\cdots\!00 Copy content Toggle raw display
1313 T13+⋯+17 ⁣⋯ ⁣28 T^{13} + \cdots + 17\!\cdots\!28 Copy content Toggle raw display
1717 T13+⋯−12 ⁣⋯ ⁣04 T^{13} + \cdots - 12\!\cdots\!04 Copy content Toggle raw display
1919 T13+⋯+19 ⁣⋯ ⁣72 T^{13} + \cdots + 19\!\cdots\!72 Copy content Toggle raw display
2323 T13+⋯−12 ⁣⋯ ⁣81 T^{13} + \cdots - 12\!\cdots\!81 Copy content Toggle raw display
2929 T13+⋯−32 ⁣⋯ ⁣00 T^{13} + \cdots - 32\!\cdots\!00 Copy content Toggle raw display
3131 T13+⋯−33 ⁣⋯ ⁣48 T^{13} + \cdots - 33\!\cdots\!48 Copy content Toggle raw display
3737 T13+⋯−18 ⁣⋯ ⁣00 T^{13} + \cdots - 18\!\cdots\!00 Copy content Toggle raw display
4141 T13+⋯+38 ⁣⋯ ⁣25 T^{13} + \cdots + 38\!\cdots\!25 Copy content Toggle raw display
4343 T13+⋯−56 ⁣⋯ ⁣48 T^{13} + \cdots - 56\!\cdots\!48 Copy content Toggle raw display
4747 T13+⋯+75 ⁣⋯ ⁣00 T^{13} + \cdots + 75\!\cdots\!00 Copy content Toggle raw display
5353 T13+⋯−18 ⁣⋯ ⁣68 T^{13} + \cdots - 18\!\cdots\!68 Copy content Toggle raw display
5959 T13+⋯−13 ⁣⋯ ⁣00 T^{13} + \cdots - 13\!\cdots\!00 Copy content Toggle raw display
6161 T13+⋯+26 ⁣⋯ ⁣48 T^{13} + \cdots + 26\!\cdots\!48 Copy content Toggle raw display
6767 T13+⋯+23 ⁣⋯ ⁣00 T^{13} + \cdots + 23\!\cdots\!00 Copy content Toggle raw display
7171 T13+⋯−17 ⁣⋯ ⁣68 T^{13} + \cdots - 17\!\cdots\!68 Copy content Toggle raw display
7373 T13+⋯+18 ⁣⋯ ⁣72 T^{13} + \cdots + 18\!\cdots\!72 Copy content Toggle raw display
7979 T13+⋯+24 ⁣⋯ ⁣36 T^{13} + \cdots + 24\!\cdots\!36 Copy content Toggle raw display
8383 T13+⋯+71 ⁣⋯ ⁣04 T^{13} + \cdots + 71\!\cdots\!04 Copy content Toggle raw display
8989 T13+⋯−15 ⁣⋯ ⁣52 T^{13} + \cdots - 15\!\cdots\!52 Copy content Toggle raw display
9797 T13+⋯+26 ⁣⋯ ⁣16 T^{13} + \cdots + 26\!\cdots\!16 Copy content Toggle raw display
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