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Properties

Label 325.10.a.h
Level 325325
Weight 1010
Character orbit 325.a
Self dual yes
Analytic conductor 167.387167.387
Analytic rank 11
Dimension 1717
CM no
Inner twists 11

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [325,10,Mod(1,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("325.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: N N == 325=52⋅13 325 = 5^{2} \cdot 13
Weight: k k == 10 10
Character orbit: [χ][\chi] == 325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [17,33] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 167.386646753167.386646753
Analytic rank: 11
Dimension: 1717
Coefficient field: Q[x]/(x17−⋯ )\mathbb{Q}[x]/(x^{17} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x17−x16−6453x15−11965x14+16673200x13+68278926x12−22023799708x11+⋯+22 ⁣⋯ ⁣80 x^{17} - x^{16} - 6453 x^{15} - 11965 x^{14} + 16673200 x^{13} + 68278926 x^{12} - 22023799708 x^{11} + \cdots + 22\!\cdots\!80 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a19]\Z[a_1, \ldots, a_{19}]
Coefficient ring index: 216⋅34⋅58 2^{16}\cdot 3^{4}\cdot 5^{8}
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,…,β161,\beta_1,\ldots,\beta_{16} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(−β1+2)q2+(−β3−4)q3+(β2+251)q4+(−β5−7β3+⋯−63)q6+(β6+2β2+24β1+626)q7+(−β4−4β3+⋯−659)q8+⋯+(1934β16−3910β15+⋯−178130303)q99+O(q100) q + ( - \beta_1 + 2) q^{2} + ( - \beta_{3} - 4) q^{3} + (\beta_{2} + 251) q^{4} + ( - \beta_{5} - 7 \beta_{3} + \cdots - 63) q^{6} + (\beta_{6} + 2 \beta_{2} + 24 \beta_1 + 626) q^{7} + ( - \beta_{4} - 4 \beta_{3} + \cdots - 659) q^{8}+ \cdots + (1934 \beta_{16} - 3910 \beta_{15} + \cdots - 178130303) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 17q+33q2−73q3+4267q4−1103q6+10670q7−11481q8+47590q9−130917q11−32239q12−485537q13−292206q14+1064251q16+193953q17+2026286q18+⋯−3023832936q99+O(q100) 17 q + 33 q^{2} - 73 q^{3} + 4267 q^{4} - 1103 q^{6} + 10670 q^{7} - 11481 q^{8} + 47590 q^{9} - 130917 q^{11} - 32239 q^{12} - 485537 q^{13} - 292206 q^{14} + 1064251 q^{16} + 193953 q^{17} + 2026286 q^{18}+ \cdots - 3023832936 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x17−x16−6453x15−11965x14+16673200x13+68278926x12−22023799708x11+⋯+22 ⁣⋯ ⁣80 x^{17} - x^{16} - 6453 x^{15} - 11965 x^{14} + 16673200 x^{13} + 68278926 x^{12} - 22023799708 x^{11} + \cdots + 22\!\cdots\!80 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2−4ν−759 \nu^{2} - 4\nu - 759 Copy content Toggle raw display
β3\beta_{3}== (23 ⁣⋯ ⁣71ν16+⋯+98 ⁣⋯ ⁣20)/39 ⁣⋯ ⁣00 ( 23\!\cdots\!71 \nu^{16} + \cdots + 98\!\cdots\!20 ) / 39\!\cdots\!00 Copy content Toggle raw display
β4\beta_{4}== (−23 ⁣⋯ ⁣71ν16+⋯−62 ⁣⋯ ⁣20)/99 ⁣⋯ ⁣00 ( - 23\!\cdots\!71 \nu^{16} + \cdots - 62\!\cdots\!20 ) / 99\!\cdots\!00 Copy content Toggle raw display
β5\beta_{5}== (38 ⁣⋯ ⁣31ν16+⋯+35 ⁣⋯ ⁣80)/39 ⁣⋯ ⁣00 ( 38\!\cdots\!31 \nu^{16} + \cdots + 35\!\cdots\!80 ) / 39\!\cdots\!00 Copy content Toggle raw display
β6\beta_{6}== (−23 ⁣⋯ ⁣99ν16+⋯−63 ⁣⋯ ⁣00)/44 ⁣⋯ ⁣00 ( - 23\!\cdots\!99 \nu^{16} + \cdots - 63\!\cdots\!00 ) / 44\!\cdots\!00 Copy content Toggle raw display
β7\beta_{7}== (−27 ⁣⋯ ⁣35ν16+⋯−18 ⁣⋯ ⁣20)/33 ⁣⋯ ⁣00 ( - 27\!\cdots\!35 \nu^{16} + \cdots - 18\!\cdots\!20 ) / 33\!\cdots\!00 Copy content Toggle raw display
β8\beta_{8}== (62 ⁣⋯ ⁣51ν16+⋯+14 ⁣⋯ ⁣60)/66 ⁣⋯ ⁣00 ( 62\!\cdots\!51 \nu^{16} + \cdots + 14\!\cdots\!60 ) / 66\!\cdots\!00 Copy content Toggle raw display
β9\beta_{9}== (−23 ⁣⋯ ⁣19ν16+⋯−17 ⁣⋯ ⁣20)/19 ⁣⋯ ⁣00 ( - 23\!\cdots\!19 \nu^{16} + \cdots - 17\!\cdots\!20 ) / 19\!\cdots\!00 Copy content Toggle raw display
β10\beta_{10}== (−17 ⁣⋯ ⁣09ν16+⋯+44 ⁣⋯ ⁣40)/14 ⁣⋯ ⁣00 ( - 17\!\cdots\!09 \nu^{16} + \cdots + 44\!\cdots\!40 ) / 14\!\cdots\!00 Copy content Toggle raw display
β11\beta_{11}== (69 ⁣⋯ ⁣51ν16+⋯+14 ⁣⋯ ⁣80)/49 ⁣⋯ ⁣00 ( 69\!\cdots\!51 \nu^{16} + \cdots + 14\!\cdots\!80 ) / 49\!\cdots\!00 Copy content Toggle raw display
β12\beta_{12}== (−90 ⁣⋯ ⁣67ν16+⋯−42 ⁣⋯ ⁣40)/39 ⁣⋯ ⁣00 ( - 90\!\cdots\!67 \nu^{16} + \cdots - 42\!\cdots\!40 ) / 39\!\cdots\!00 Copy content Toggle raw display
β13\beta_{13}== (−10 ⁣⋯ ⁣27ν16+⋯−57 ⁣⋯ ⁣60)/39 ⁣⋯ ⁣00 ( - 10\!\cdots\!27 \nu^{16} + \cdots - 57\!\cdots\!60 ) / 39\!\cdots\!00 Copy content Toggle raw display
β14\beta_{14}== (−14 ⁣⋯ ⁣55ν16+⋯+93 ⁣⋯ ⁣20)/39 ⁣⋯ ⁣00 ( - 14\!\cdots\!55 \nu^{16} + \cdots + 93\!\cdots\!20 ) / 39\!\cdots\!00 Copy content Toggle raw display
β15\beta_{15}== (16 ⁣⋯ ⁣83ν16+⋯+33 ⁣⋯ ⁣60)/19 ⁣⋯ ⁣00 ( 16\!\cdots\!83 \nu^{16} + \cdots + 33\!\cdots\!60 ) / 19\!\cdots\!00 Copy content Toggle raw display
β16\beta_{16}== (40 ⁣⋯ ⁣77ν16+⋯+11 ⁣⋯ ⁣20)/39 ⁣⋯ ⁣00 ( 40\!\cdots\!77 \nu^{16} + \cdots + 11\!\cdots\!20 ) / 39\!\cdots\!00 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}== β2+4β1+759 \beta_{2} + 4\beta _1 + 759 Copy content Toggle raw display
ν3\nu^{3}== β4+4β3+9β2+1289β1+3173 \beta_{4} + 4\beta_{3} + 9\beta_{2} + 1289\beta _1 + 3173 Copy content Toggle raw display
ν4\nu^{4}== β16−β15−2β12−3β11−2β8−11β5+15β4+⋯+979399 \beta_{16} - \beta_{15} - 2 \beta_{12} - 3 \beta_{11} - 2 \beta_{8} - 11 \beta_{5} + 15 \beta_{4} + \cdots + 979399 Copy content Toggle raw display
ν5\nu^{5}== 42β16−16β15+21β14−73β13+22β12+11β11+⋯+9855909 42 \beta_{16} - 16 \beta_{15} + 21 \beta_{14} - 73 \beta_{13} + 22 \beta_{12} + 11 \beta_{11} + \cdots + 9855909 Copy content Toggle raw display
ν6\nu^{6}== 3583β16−3023β15+104β14−212β13−5926β12+⋯+1482284811 3583 \beta_{16} - 3023 \beta_{15} + 104 \beta_{14} - 212 \beta_{13} - 5926 \beta_{12} + \cdots + 1482284811 Copy content Toggle raw display
ν7\nu^{7}== 156122β16−67060β15+48215β14−199779β13+39810β12+⋯+24145260029 156122 \beta_{16} - 67060 \beta_{15} + 48215 \beta_{14} - 199779 \beta_{13} + 39810 \beta_{12} + \cdots + 24145260029 Copy content Toggle raw display
ν8\nu^{8}== 9358991β16−6937831β15+391748β14−1127576β13−13607806β12+⋯+2446213626667 9358991 \beta_{16} - 6937831 \beta_{15} + 391748 \beta_{14} - 1127576 \beta_{13} - 13607806 \beta_{12} + \cdots + 2446213626667 Copy content Toggle raw display
ν9\nu^{9}== 419684982β16−195352416β15+83794423β14−418523443β13+⋯+54033229558469 419684982 \beta_{16} - 195352416 \beta_{15} + 83794423 \beta_{14} - 418523443 \beta_{13} + \cdots + 54033229558469 Copy content Toggle raw display
ν10\nu^{10}== 21855852387β16−14786754339β15+907756048β14−3761389716β13+⋯+42 ⁣⋯ ⁣23 21855852387 \beta_{16} - 14786754339 \beta_{15} + 907756048 \beta_{14} - 3761389716 \beta_{13} + \cdots + 42\!\cdots\!23 Copy content Toggle raw display
ν11\nu^{11}== 998996362906β16−490312250156β15+128805747539β14−808236954127β13+⋯+11 ⁣⋯ ⁣69 998996362906 \beta_{16} - 490312250156 \beta_{15} + 128805747539 \beta_{14} - 808236954127 \beta_{13} + \cdots + 11\!\cdots\!69 Copy content Toggle raw display
ν12\nu^{12}== 48620319227551β16−30867905424551β15+1606771497484β14−10234724463120β13+⋯+78 ⁣⋯ ⁣19 48620319227551 \beta_{16} - 30867905424551 \beta_{15} + 1606771497484 \beta_{14} - 10234724463120 \beta_{13} + \cdots + 78\!\cdots\!19 Copy content Toggle raw display
ν13\nu^{13}== 22 ⁣⋯ ⁣46β16+⋯+24 ⁣⋯ ⁣61 22\!\cdots\!46 \beta_{16} + \cdots + 24\!\cdots\!61 Copy content Toggle raw display
ν14\nu^{14}== 10 ⁣⋯ ⁣39β16+⋯+14 ⁣⋯ ⁣67 10\!\cdots\!39 \beta_{16} + \cdots + 14\!\cdots\!67 Copy content Toggle raw display
ν15\nu^{15}== 49 ⁣⋯ ⁣18β16+⋯+50 ⁣⋯ ⁣57 49\!\cdots\!18 \beta_{16} + \cdots + 50\!\cdots\!57 Copy content Toggle raw display
ν16\nu^{16}== 22 ⁣⋯ ⁣91β16+⋯+28 ⁣⋯ ⁣91 22\!\cdots\!91 \beta_{16} + \cdots + 28\!\cdots\!91 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.

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
45.5365
41.9783
32.7069
32.3076
22.0262
13.6975
12.0171
8.33068
−6.66848
−8.60741
−11.1845
−15.0241
−27.7222
−30.4531
−33.1624
−35.1331
−39.6455
−43.5365 −112.372 1383.43 0 4892.27 6926.24 −37939.1 −7055.63 0
1.2 −39.9783 152.057 1086.26 0 −6078.99 3359.20 −22958.1 3438.43 0
1.3 −30.7069 −116.164 430.914 0 3567.03 −1957.66 2489.89 −6188.99 0
1.4 −30.3076 7.32501 406.551 0 −222.003 −9118.61 3195.89 −19629.3 0
1.5 −20.0262 143.807 −110.950 0 −2879.91 11821.9 12475.3 997.339 0
1.6 −11.6975 −207.589 −375.168 0 2428.27 5519.45 10377.7 23410.1 0
1.7 −10.0171 174.740 −411.658 0 −1750.39 −417.700 9252.35 10851.2 0
1.8 −6.33068 −33.0585 −471.922 0 209.283 180.411 6228.90 −18590.1 0
1.9 8.66848 −65.6463 −436.857 0 −569.054 −11239.1 −8225.15 −15373.6 0
1.10 10.6074 39.6173 −399.483 0 420.237 7059.84 −9668.47 −18113.5 0
1.11 13.1845 −236.239 −338.168 0 −3114.70 −5988.15 −11209.1 36125.9 0
1.12 17.0241 228.743 −222.181 0 3894.14 −5864.90 −12498.8 32640.5 0
1.13 29.7222 192.679 371.411 0 5726.85 7108.54 −4178.62 17442.2 0
1.14 32.4531 −140.165 541.204 0 −4548.78 −1977.44 947.752 −36.8521 0
1.15 35.1624 −237.080 724.396 0 −8336.29 11662.6 7468.37 36523.7 0
1.16 37.1331 92.8669 866.866 0 3448.44 −3610.83 13177.3 −11058.7 0
1.17 41.6455 43.4764 1222.35 0 1810.60 −2793.78 29582.9 −17792.8 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
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 325.10.a.h yes 17
5.b even 2 1 325.10.a.g ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.10.a.g ✓ 17 5.b even 2 1
325.10.a.h yes 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T217−33T216−5941T215+200595T214+13661520T213+⋯−26 ⁣⋯ ⁣00 T_{2}^{17} - 33 T_{2}^{16} - 5941 T_{2}^{15} + 200595 T_{2}^{14} + 13661520 T_{2}^{13} + \cdots - 26\!\cdots\!00 acting on S10new(Γ0(325))S_{10}^{\mathrm{new}}(\Gamma_0(325)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T17+⋯−26 ⁣⋯ ⁣00 T^{17} + \cdots - 26\!\cdots\!00 Copy content Toggle raw display
33 T17+⋯−91 ⁣⋯ ⁣92 T^{17} + \cdots - 91\!\cdots\!92 Copy content Toggle raw display
55 T17 T^{17} Copy content Toggle raw display
77 T17+⋯+94 ⁣⋯ ⁣48 T^{17} + \cdots + 94\!\cdots\!48 Copy content Toggle raw display
1111 T17+⋯−16 ⁣⋯ ⁣64 T^{17} + \cdots - 16\!\cdots\!64 Copy content Toggle raw display
1313 (T+28561)17 (T + 28561)^{17} Copy content Toggle raw display
1717 T17+⋯+10 ⁣⋯ ⁣72 T^{17} + \cdots + 10\!\cdots\!72 Copy content Toggle raw display
1919 T17+⋯−34 ⁣⋯ ⁣60 T^{17} + \cdots - 34\!\cdots\!60 Copy content Toggle raw display
2323 T17+⋯−13 ⁣⋯ ⁣52 T^{17} + \cdots - 13\!\cdots\!52 Copy content Toggle raw display
2929 T17+⋯+11 ⁣⋯ ⁣60 T^{17} + \cdots + 11\!\cdots\!60 Copy content Toggle raw display
3131 T17+⋯−13 ⁣⋯ ⁣20 T^{17} + \cdots - 13\!\cdots\!20 Copy content Toggle raw display
3737 T17+⋯−49 ⁣⋯ ⁣36 T^{17} + \cdots - 49\!\cdots\!36 Copy content Toggle raw display
4141 T17+⋯−23 ⁣⋯ ⁣12 T^{17} + \cdots - 23\!\cdots\!12 Copy content Toggle raw display
4343 T17+⋯−14 ⁣⋯ ⁣40 T^{17} + \cdots - 14\!\cdots\!40 Copy content Toggle raw display
4747 T17+⋯+13 ⁣⋯ ⁣00 T^{17} + \cdots + 13\!\cdots\!00 Copy content Toggle raw display
5353 T17+⋯−10 ⁣⋯ ⁣72 T^{17} + \cdots - 10\!\cdots\!72 Copy content Toggle raw display
5959 T17+⋯+74 ⁣⋯ ⁣80 T^{17} + \cdots + 74\!\cdots\!80 Copy content Toggle raw display
6161 T17+⋯−73 ⁣⋯ ⁣44 T^{17} + \cdots - 73\!\cdots\!44 Copy content Toggle raw display
6767 T17+⋯−24 ⁣⋯ ⁣72 T^{17} + \cdots - 24\!\cdots\!72 Copy content Toggle raw display
7171 T17+⋯+66 ⁣⋯ ⁣32 T^{17} + \cdots + 66\!\cdots\!32 Copy content Toggle raw display
7373 T17+⋯+11 ⁣⋯ ⁣92 T^{17} + \cdots + 11\!\cdots\!92 Copy content Toggle raw display
7979 T17+⋯−29 ⁣⋯ ⁣80 T^{17} + \cdots - 29\!\cdots\!80 Copy content Toggle raw display
8383 T17+⋯+33 ⁣⋯ ⁣88 T^{17} + \cdots + 33\!\cdots\!88 Copy content Toggle raw display
8989 T17+⋯−16 ⁣⋯ ⁣00 T^{17} + \cdots - 16\!\cdots\!00 Copy content Toggle raw display
9797 T17+⋯+64 ⁣⋯ ⁣00 T^{17} + \cdots + 64\!\cdots\!00 Copy content Toggle raw display
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