Properties

Label 325.2.e.c
Level 325325
Weight 22
Character orbit 325.e
Analytic conductor 2.5952.595
Analytic rank 00
Dimension 1010
Inner twists 22

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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] = [325,2,Mod(126,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) 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("325.126"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 325=5213 325 = 5^{2} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 325.e (of order 33, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 2.595138065692.59513806569
Analytic rank: 00
Dimension: 1010
Relative dimension: 55 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x10+)\mathbb{Q}[x]/(x^{10} + \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x10+8x82x7+52x65x5+97x4+60x3+141x2+36x+9 x^{10} + 8x^{8} - 2x^{7} + 52x^{6} - 5x^{5} + 97x^{4} + 60x^{3} + 141x^{2} + 36x + 9 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 3 3
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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,,β91,\beta_1,\ldots,\beta_{9} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2+(β8+β6β41)q3+(β9β6)q4+(β7β6++β2)q6+(β7β5+β1)q7+(β4+β3β2)q8++(3β5+5β4β3++9)q99+O(q100) q + \beta_1 q^{2} + (\beta_{8} + \beta_{6} - \beta_{4} - 1) q^{3} + (\beta_{9} - \beta_{6}) q^{4} + ( - \beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{6} + ( - \beta_{7} - \beta_{5} + \beta_1) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2}) q^{8}+ \cdots + (3 \beta_{5} + 5 \beta_{4} - \beta_{3} + \cdots + 9) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 10q3q36q43q6+2q7+6q84q93q11+4q12+10q1316q144q164q174q18+4q1916q218q2215q23+14q24++56q99+O(q100) 10 q - 3 q^{3} - 6 q^{4} - 3 q^{6} + 2 q^{7} + 6 q^{8} - 4 q^{9} - 3 q^{11} + 4 q^{12} + 10 q^{13} - 16 q^{14} - 4 q^{16} - 4 q^{17} - 4 q^{18} + 4 q^{19} - 16 q^{21} - 8 q^{22} - 15 q^{23} + 14 q^{24}+ \cdots + 56 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x10+8x82x7+52x65x5+97x4+60x3+141x2+36x+9 x^{10} + 8x^{8} - 2x^{7} + 52x^{6} - 5x^{5} + 97x^{4} + 60x^{3} + 141x^{2} + 36x + 9 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (1304ν9+1776ν81628ν7+8447ν614800ν5+93980ν4+242496)/983355 ( 1304 \nu^{9} + 1776 \nu^{8} - 1628 \nu^{7} + 8447 \nu^{6} - 14800 \nu^{5} + 93980 \nu^{4} + \cdots - 242496 ) / 983355 Copy content Toggle raw display
β3\beta_{3}== (592ν94020ν8+3685ν727536ν6+33500ν5212725ν4+987267)/327785 ( 592 \nu^{9} - 4020 \nu^{8} + 3685 \nu^{7} - 27536 \nu^{6} + 33500 \nu^{5} - 212725 \nu^{4} + \cdots - 987267 ) / 327785 Copy content Toggle raw display
β4\beta_{4}== (4744ν920940ν8+19195ν7124843ν6+174500ν51108075ν4+1749321)/983355 ( - 4744 \nu^{9} - 20940 \nu^{8} + 19195 \nu^{7} - 124843 \nu^{6} + 174500 \nu^{5} - 1108075 \nu^{4} + \cdots - 1749321 ) / 983355 Copy content Toggle raw display
β5\beta_{5}== (3213ν912516ν8+11473ν7129958ν6+104300ν5662305ν4+553661)/327785 ( 3213 \nu^{9} - 12516 \nu^{8} + 11473 \nu^{7} - 129958 \nu^{6} + 104300 \nu^{5} - 662305 \nu^{4} + \cdots - 553661 ) / 327785 Copy content Toggle raw display
β6\beta_{6}== (26944ν9+1304ν8+217328ν755516ν6+1409535ν5149520ν4++1012608)/983355 ( 26944 \nu^{9} + 1304 \nu^{8} + 217328 \nu^{7} - 55516 \nu^{6} + 1409535 \nu^{5} - 149520 \nu^{4} + \cdots + 1012608 ) / 983355 Copy content Toggle raw display
β7\beta_{7}== (47884ν9+55843ν8351659ν7+476704ν62541330ν5+2272140ν4++13443)/983355 ( - 47884 \nu^{9} + 55843 \nu^{8} - 351659 \nu^{7} + 476704 \nu^{6} - 2541330 \nu^{5} + 2272140 \nu^{4} + \cdots + 13443 ) / 983355 Copy content Toggle raw display
β8\beta_{8}== (75292ν99639ν8564788ν7+116165ν63525310ν5+1809360)/983355 ( - 75292 \nu^{9} - 9639 \nu^{8} - 564788 \nu^{7} + 116165 \nu^{6} - 3525310 \nu^{5} + \cdots - 1809360 ) / 983355 Copy content Toggle raw display
β9\beta_{9}== (26944ν9+1304ν8+217328ν755516ν6+1409535ν5149520ν4++1012608)/327785 ( 26944 \nu^{9} + 1304 \nu^{8} + 217328 \nu^{7} - 55516 \nu^{6} + 1409535 \nu^{5} - 149520 \nu^{4} + \cdots + 1012608 ) / 327785 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β93β6 \beta_{9} - 3\beta_{6} Copy content Toggle raw display
ν3\nu^{3}== β4+β35β2 -\beta_{4} + \beta_{3} - 5\beta_{2} Copy content Toggle raw display
ν4\nu^{4}== 6β9β8β7+14β6+β46β3+3β114 -6\beta_{9} - \beta_{8} - \beta_{7} + 14\beta_{6} + \beta_{4} - 6\beta_{3} + 3\beta _1 - 14 Copy content Toggle raw display
ν5\nu^{5}== 9β9+8β86β6+28β228β1 9\beta_{9} + 8\beta_{8} - 6\beta_{6} + 28\beta_{2} - 28\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 8β59β4+37β324β2+76 -8\beta_{5} - 9\beta_{4} + 37\beta_{3} - 24\beta_{2} + 76 Copy content Toggle raw display
ν7\nu^{7}== 69β953β8β7+71β6+53β469β3+167β171 -69\beta_{9} - 53\beta_{8} - \beta_{7} + 71\beta_{6} + 53\beta_{4} - 69\beta_{3} + 167\beta _1 - 71 Copy content Toggle raw display
ν8\nu^{8}== 236β9+71β8+52β7446β6+52β5+211β2263β1 236\beta_{9} + 71\beta_{8} + 52\beta_{7} - 446\beta_{6} + 52\beta_{5} + 211\beta_{2} - 263\beta_1 Copy content Toggle raw display
ν9\nu^{9}== 19β5340β4+499β31022β2+614 -19\beta_{5} - 340\beta_{4} + 499\beta_{3} - 1022\beta_{2} + 614 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/325Z)×\left(\mathbb{Z}/325\mathbb{Z}\right)^\times.

nn 2727 301301
χ(n)\chi(n) 11 β6-\beta_{6}

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}
126.1
−1.30241 + 2.25583i
−0.547998 + 0.949161i
−0.134432 + 0.232843i
0.904178 1.56608i
1.08066 1.87176i
−1.30241 2.25583i
−0.547998 0.949161i
−0.134432 0.232843i
0.904178 + 1.56608i
1.08066 + 1.87176i
−1.30241 + 2.25583i −0.0676602 + 0.117191i −2.39253 4.14398i 0 −0.176242 0.305261i 1.62616 + 2.81660i 7.25455 1.49084 + 2.58222i 0
126.2 −0.547998 + 0.949161i −1.68234 + 2.91389i 0.399395 + 0.691773i 0 −1.84383 3.19362i −0.795836 1.37843i −3.06747 −4.16051 7.20621i 0
126.3 −0.134432 + 0.232843i 0.301414 0.522064i 0.963856 + 1.66945i 0 0.0810394 + 0.140364i 0.715471 + 1.23923i −1.05602 1.31830 + 2.28336i 0
126.4 0.904178 1.56608i 0.929015 1.60910i −0.635076 1.09998i 0 −1.67999 2.90983i −2.08417 3.60988i 1.31983 −0.226138 0.391682i 0
126.5 1.08066 1.87176i −0.980433 + 1.69816i −1.33565 2.31341i 0 2.11903 + 3.67026i 1.53837 + 2.66453i −1.45089 −0.422497 0.731787i 0
276.1 −1.30241 2.25583i −0.0676602 0.117191i −2.39253 + 4.14398i 0 −0.176242 + 0.305261i 1.62616 2.81660i 7.25455 1.49084 2.58222i 0
276.2 −0.547998 0.949161i −1.68234 2.91389i 0.399395 0.691773i 0 −1.84383 + 3.19362i −0.795836 + 1.37843i −3.06747 −4.16051 + 7.20621i 0
276.3 −0.134432 0.232843i 0.301414 + 0.522064i 0.963856 1.66945i 0 0.0810394 0.140364i 0.715471 1.23923i −1.05602 1.31830 2.28336i 0
276.4 0.904178 + 1.56608i 0.929015 + 1.60910i −0.635076 + 1.09998i 0 −1.67999 + 2.90983i −2.08417 + 3.60988i 1.31983 −0.226138 + 0.391682i 0
276.5 1.08066 + 1.87176i −0.980433 1.69816i −1.33565 + 2.31341i 0 2.11903 3.67026i 1.53837 2.66453i −1.45089 −0.422497 + 0.731787i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 126.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.e.c 10
5.b even 2 1 325.2.e.d yes 10
5.c odd 4 2 325.2.o.c 20
13.c even 3 1 inner 325.2.e.c 10
13.c even 3 1 4225.2.a.bp 5
13.e even 6 1 4225.2.a.bo 5
65.l even 6 1 4225.2.a.bm 5
65.n even 6 1 325.2.e.d yes 10
65.n even 6 1 4225.2.a.bn 5
65.q odd 12 2 325.2.o.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.e.c 10 1.a even 1 1 trivial
325.2.e.c 10 13.c even 3 1 inner
325.2.e.d yes 10 5.b even 2 1
325.2.e.d yes 10 65.n even 6 1
325.2.o.c 20 5.c odd 4 2
325.2.o.c 20 65.q odd 12 2
4225.2.a.bm 5 65.l even 6 1
4225.2.a.bn 5 65.n even 6 1
4225.2.a.bo 5 13.e even 6 1
4225.2.a.bp 5 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T210+8T282T27+52T265T25+97T24+60T23+141T22+36T2+9 T_{2}^{10} + 8T_{2}^{8} - 2T_{2}^{7} + 52T_{2}^{6} - 5T_{2}^{5} + 97T_{2}^{4} + 60T_{2}^{3} + 141T_{2}^{2} + 36T_{2} + 9 acting on S2new(325,[χ])S_{2}^{\mathrm{new}}(325, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T10+8T8++9 T^{10} + 8 T^{8} + \cdots + 9 Copy content Toggle raw display
33 T10+3T9++1 T^{10} + 3 T^{9} + \cdots + 1 Copy content Toggle raw display
55 T10 T^{10} Copy content Toggle raw display
77 T102T9++9025 T^{10} - 2 T^{9} + \cdots + 9025 Copy content Toggle raw display
1111 T10+3T9++9 T^{10} + 3 T^{9} + \cdots + 9 Copy content Toggle raw display
1313 T1010T9++371293 T^{10} - 10 T^{9} + \cdots + 371293 Copy content Toggle raw display
1717 T10+4T9++239121 T^{10} + 4 T^{9} + \cdots + 239121 Copy content Toggle raw display
1919 T104T9++225 T^{10} - 4 T^{9} + \cdots + 225 Copy content Toggle raw display
2323 T10+15T9++281961 T^{10} + 15 T^{9} + \cdots + 281961 Copy content Toggle raw display
2929 T10T9++881721 T^{10} - T^{9} + \cdots + 881721 Copy content Toggle raw display
3131 (T557T3++225)2 (T^{5} - 57 T^{3} + \cdots + 225)^{2} Copy content Toggle raw display
3737 T10+17T9++826281 T^{10} + 17 T^{9} + \cdots + 826281 Copy content Toggle raw display
4141 T10+6T9++50625 T^{10} + 6 T^{9} + \cdots + 50625 Copy content Toggle raw display
4343 T10+12T9++14190289 T^{10} + 12 T^{9} + \cdots + 14190289 Copy content Toggle raw display
4747 (T5+12T4++2025)2 (T^{5} + 12 T^{4} + \cdots + 2025)^{2} Copy content Toggle raw display
5353 (T58T4++6075)2 (T^{5} - 8 T^{4} + \cdots + 6075)^{2} Copy content Toggle raw display
5959 T1012T9++4782969 T^{10} - 12 T^{9} + \cdots + 4782969 Copy content Toggle raw display
6161 T10+5T9++403225 T^{10} + 5 T^{9} + \cdots + 403225 Copy content Toggle raw display
6767 T10++2567550241 T^{10} + \cdots + 2567550241 Copy content Toggle raw display
7171 T10++101787921 T^{10} + \cdots + 101787921 Copy content Toggle raw display
7373 (T5+8T4++10125)2 (T^{5} + 8 T^{4} + \cdots + 10125)^{2} Copy content Toggle raw display
7979 (T5+14T4++16875)2 (T^{5} + 14 T^{4} + \cdots + 16875)^{2} Copy content Toggle raw display
8383 (T5+7T4+53775)2 (T^{5} + 7 T^{4} + \cdots - 53775)^{2} Copy content Toggle raw display
8989 T1010T9++2537649 T^{10} - 10 T^{9} + \cdots + 2537649 Copy content Toggle raw display
9797 T10++1683378841 T^{10} + \cdots + 1683378841 Copy content Toggle raw display
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