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Properties

Label 252.3.g.b
Level 252252
Weight 33
Character orbit 252.g
Analytic conductor 6.8676.867
Analytic rank 00
Dimension 1212
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,3,Mod(127,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 252=22⋅32⋅7 252 = 2^{2} \cdot 3^{2} \cdot 7
Weight: k k == 3 3
Character orbit: [χ][\chi] == 252.g (of order 22, degree 11, minimal)

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: no
Analytic conductor: 6.866502661886.86650266188
Analytic rank: 00
Dimension: 1212
Coefficient field: 12.0.489494783471841.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x12−3x11+7x10−11x9+18x8−22x7+33x6−44x5+72x4+⋯+64 x^{12} - 3 x^{11} + 7 x^{10} - 11 x^{9} + 18 x^{8} - 22 x^{7} + 33 x^{6} - 44 x^{5} + 72 x^{4} + \cdots + 64 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 219 2^{19}
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{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,…,β111,\beta_1,\ldots,\beta_{11} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q−β3q2−β5q4+(β8−β3)q5−β1q7+(−β7+β2−β1+1)q8+(−β11−β10+β8+⋯+2)q10+(−β11−β10+⋯+3β1)q11+⋯+7β3q98+O(q100) q - \beta_{3} q^{2} - \beta_{5} q^{4} + (\beta_{8} - \beta_{3}) q^{5} - \beta_1 q^{7} + ( - \beta_{7} + \beta_{2} - \beta_1 + 1) q^{8} + ( - \beta_{11} - \beta_{10} + \beta_{8} + \cdots + 2) q^{10} + ( - \beta_{11} - \beta_{10} + \cdots + 3 \beta_1) q^{11}+ \cdots + 7 \beta_{3} q^{98}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 12q−2q2+2q4−8q5+10q8+28q10−24q13+14q14−14q16+40q17+20q20−88q22+180q25−100q26+14q28−72q29−142q32−100q34+⋯+14q98+O(q100) 12 q - 2 q^{2} + 2 q^{4} - 8 q^{5} + 10 q^{8} + 28 q^{10} - 24 q^{13} + 14 q^{14} - 14 q^{16} + 40 q^{17} + 20 q^{20} - 88 q^{22} + 180 q^{25} - 100 q^{26} + 14 q^{28} - 72 q^{29} - 142 q^{32} - 100 q^{34}+ \cdots + 14 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12−3x11+7x10−11x9+18x8−22x7+33x6−44x5+72x4+⋯+64 x^{12} - 3 x^{11} + 7 x^{10} - 11 x^{9} + 18 x^{8} - 22 x^{7} + 33 x^{6} - 44 x^{5} + 72 x^{4} + \cdots + 64 : Copy content Toggle raw display

β1\beta_{1}== (31ν11+23ν10+89ν9+43ν8−66ν7+386ν6+551ν5+⋯+2608)/1264 ( 31 \nu^{11} + 23 \nu^{10} + 89 \nu^{9} + 43 \nu^{8} - 66 \nu^{7} + 386 \nu^{6} + 551 \nu^{5} + \cdots + 2608 ) / 1264 Copy content Toggle raw display
β2\beta_{2}== (33ν11−475ν10+635ν9−2207ν8+1734ν7−4482ν6+2401ν5+⋯−20832)/1264 ( 33 \nu^{11} - 475 \nu^{10} + 635 \nu^{9} - 2207 \nu^{8} + 1734 \nu^{7} - 4482 \nu^{6} + 2401 \nu^{5} + \cdots - 20832 ) / 1264 Copy content Toggle raw display
β3\beta_{3}== (−265ν11+731ν10−1403ν9+2191ν8−2606ν7+4162ν6+⋯+14688)/2528 ( - 265 \nu^{11} + 731 \nu^{10} - 1403 \nu^{9} + 2191 \nu^{8} - 2606 \nu^{7} + 4162 \nu^{6} + \cdots + 14688 ) / 2528 Copy content Toggle raw display
β4\beta_{4}== (−191ν11−157ν10−319ν9−637ν8−628ν7−1206ν6−2243ν5+⋯−8240)/1264 ( - 191 \nu^{11} - 157 \nu^{10} - 319 \nu^{9} - 637 \nu^{8} - 628 \nu^{7} - 1206 \nu^{6} - 2243 \nu^{5} + \cdots - 8240 ) / 1264 Copy content Toggle raw display
β5\beta_{5}== (−423ν11+1521ν10−2193ν9+4877ν8−5134ν7+9534ν6+⋯+39968)/2528 ( - 423 \nu^{11} + 1521 \nu^{10} - 2193 \nu^{9} + 4877 \nu^{8} - 5134 \nu^{7} + 9534 \nu^{6} + \cdots + 39968 ) / 2528 Copy content Toggle raw display
β6\beta_{6}== (443ν11+135ν10+1017ν9+747ν8+1330ν7+2458ν6+3083ν5+⋯+13824)/2528 ( 443 \nu^{11} + 135 \nu^{10} + 1017 \nu^{9} + 747 \nu^{8} + 1330 \nu^{7} + 2458 \nu^{6} + 3083 \nu^{5} + \cdots + 13824 ) / 2528 Copy content Toggle raw display
β7\beta_{7}== (799ν11−1609ν10+2457ν9−4437ν8+6046ν7−8318ν6+⋯−27744)/2528 ( 799 \nu^{11} - 1609 \nu^{10} + 2457 \nu^{9} - 4437 \nu^{8} + 6046 \nu^{7} - 8318 \nu^{6} + \cdots - 27744 ) / 2528 Copy content Toggle raw display
β8\beta_{8}== (−897ν11+2311ν10−4879ν9+6931ν8−10506ν7+12378ν6+⋯+39968)/2528 ( - 897 \nu^{11} + 2311 \nu^{10} - 4879 \nu^{9} + 6931 \nu^{8} - 10506 \nu^{7} + 12378 \nu^{6} + \cdots + 39968 ) / 2528 Copy content Toggle raw display
β9\beta_{9}== (973ν11−1959ν10+3823ν9−6163ν8+9070ν7−10290ν6+⋯−33248)/2528 ( 973 \nu^{11} - 1959 \nu^{10} + 3823 \nu^{9} - 6163 \nu^{8} + 9070 \nu^{7} - 10290 \nu^{6} + \cdots - 33248 ) / 2528 Copy content Toggle raw display
β10\beta_{10}== (32ν11−68ν10+125ν9−213ν8+281ν7−389ν6+528ν5+⋯−1449)/79 ( 32 \nu^{11} - 68 \nu^{10} + 125 \nu^{9} - 213 \nu^{8} + 281 \nu^{7} - 389 \nu^{6} + 528 \nu^{5} + \cdots - 1449 ) / 79 Copy content Toggle raw display
β11\beta_{11}== (−443ν11+734ν10−1728ν9+2176ν8−3937ν7+3704ν6+⋯+12088)/632 ( - 443 \nu^{11} + 734 \nu^{10} - 1728 \nu^{9} + 2176 \nu^{8} - 3937 \nu^{7} + 3704 \nu^{6} + \cdots + 12088 ) / 632 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (−β11−2β10+β8−2β5−β3+β1+2)/8 ( -\beta_{11} - 2\beta_{10} + \beta_{8} - 2\beta_{5} - \beta_{3} + \beta _1 + 2 ) / 8 Copy content Toggle raw display
ν2\nu^{2}== (−β11+β8−2β6+2β5−3β3+β1−2)/8 ( -\beta_{11} + \beta_{8} - 2\beta_{6} + 2\beta_{5} - 3\beta_{3} + \beta _1 - 2 ) / 8 Copy content Toggle raw display
ν3\nu^{3}== (−β11+β10−3β9−β8+2β7−β6+3β5+β4+⋯−1)/8 ( - \beta_{11} + \beta_{10} - 3 \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{4} + \cdots - 1 ) / 8 Copy content Toggle raw display
ν4\nu^{4}== (3β11−4β10+2β9−3β8+4β7+2β6−2β4+⋯−12)/8 ( 3 \beta_{11} - 4 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + \cdots - 12 ) / 8 Copy content Toggle raw display
ν5\nu^{5}== (4β11+3β10+β9−4β8−3β6−β5−5β4+⋯−15)/8 ( 4 \beta_{11} + 3 \beta_{10} + \beta_{9} - 4 \beta_{8} - 3 \beta_{6} - \beta_{5} - 5 \beta_{4} + \cdots - 15 ) / 8 Copy content Toggle raw display
ν6\nu^{6}== (−β11+2β10+3β9+β8−4β7+β6−β5+5β4+⋯−12)/4 ( - \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + \beta_{8} - 4 \beta_{7} + \beta_{6} - \beta_{5} + 5 \beta_{4} + \cdots - 12 ) / 4 Copy content Toggle raw display
ν7\nu^{7}== (−2β11+β10+5β9−2β8−4β7+β6+3β5+⋯+11)/8 ( - 2 \beta_{11} + \beta_{10} + 5 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} + \beta_{6} + 3 \beta_{5} + \cdots + 11 ) / 8 Copy content Toggle raw display
ν8\nu^{8}== (−3β11+6β10−22β9−5β8−4β6−12β5−10β4+⋯+16)/8 ( - 3 \beta_{11} + 6 \beta_{10} - 22 \beta_{9} - 5 \beta_{8} - 4 \beta_{6} - 12 \beta_{5} - 10 \beta_{4} + \cdots + 16 ) / 8 Copy content Toggle raw display
ν9\nu^{9}== (−β11+β9+4β8−11β7+3β6−33β3−2β2−5β1+1)/4 ( -\beta_{11} + \beta_{9} + 4\beta_{8} - 11\beta_{7} + 3\beta_{6} - 33\beta_{3} - 2\beta_{2} - 5\beta _1 + 1 ) / 4 Copy content Toggle raw display
ν10\nu^{10}== (−10β11−5β10−5β9+16β8−10β7+21β6+7β5+⋯+85)/8 ( - 10 \beta_{11} - 5 \beta_{10} - 5 \beta_{9} + 16 \beta_{8} - 10 \beta_{7} + 21 \beta_{6} + 7 \beta_{5} + \cdots + 85 ) / 8 Copy content Toggle raw display
ν11\nu^{11}== (7β11−34β10+10β9+33β8+52β7+54β6−24β5+⋯+2)/8 ( 7 \beta_{11} - 34 \beta_{10} + 10 \beta_{9} + 33 \beta_{8} + 52 \beta_{7} + 54 \beta_{6} - 24 \beta_{5} + \cdots + 2 ) / 8 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

Character values

We give the values of χ\chi on generators for (Z/252Z)×\left(\mathbb{Z}/252\mathbb{Z}\right)^\times.

nn 2929 7373 127127
χ(n)\chi(n) 11 11 −1-1

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}
127.1
0.754714 − 1.19600i
0.754714 + 1.19600i
−1.15503 + 0.816025i
−1.15503 − 0.816025i
−0.434363 + 1.34586i
−0.434363 − 1.34586i
1.10978 − 0.876576i
1.10978 + 0.876576i
0.0311486 − 1.41387i
0.0311486 + 1.41387i
1.19375 + 0.758257i
1.19375 − 0.758257i
−1.98491 − 0.245189i 0 3.87976 + 0.973359i −0.424396 0 − 2.64575i −7.46234 − 2.88331i 0 0.842389 + 0.104057i
127.2 −1.98491 + 0.245189i 0 3.87976 − 0.973359i −0.424396 0 2.64575i −7.46234 + 2.88331i 0 0.842389 − 0.104057i
127.3 −1.42562 − 1.40272i 0 0.0647610 + 3.99948i −1.94731 0 − 2.64575i 5.51781 − 5.79256i 0 2.77611 + 2.73152i
127.4 −1.42562 + 1.40272i 0 0.0647610 − 3.99948i −1.94731 0 2.64575i 5.51781 + 5.79256i 0 2.77611 − 2.73152i
127.5 −0.913644 − 1.77912i 0 −2.33051 + 3.25096i −8.22808 0 2.64575i 7.91309 + 1.17604i 0 7.51753 + 14.6387i
127.6 −0.913644 + 1.77912i 0 −2.33051 − 3.25096i −8.22808 0 − 2.64575i 7.91309 − 1.17604i 0 7.51753 − 14.6387i
127.7 −0.0345996 − 1.99970i 0 −3.99761 + 0.138378i 6.29204 0 2.64575i 0.415030 + 7.98923i 0 −0.217702 − 12.5822i
127.8 −0.0345996 + 1.99970i 0 −3.99761 − 0.138378i 6.29204 0 − 2.64575i 0.415030 − 7.98923i 0 −0.217702 + 12.5822i
127.9 1.51951 − 1.30042i 0 0.617841 − 3.95200i −7.86764 0 2.64575i −4.20042 − 6.80856i 0 −11.9550 + 10.2312i
127.10 1.51951 + 1.30042i 0 0.617841 + 3.95200i −7.86764 0 − 2.64575i −4.20042 + 6.80856i 0 −11.9550 − 10.2312i
127.11 1.83926 − 0.785573i 0 2.76575 − 2.88975i 8.17539 0 − 2.64575i 2.81683 − 7.48769i 0 15.0367 − 6.42236i
127.12 1.83926 + 0.785573i 0 2.76575 + 2.88975i 8.17539 0 2.64575i 2.81683 + 7.48769i 0 15.0367 + 6.42236i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.3.g.b 12
3.b odd 2 1 84.3.g.a ✓ 12
4.b odd 2 1 inner 252.3.g.b 12
12.b even 2 1 84.3.g.a ✓ 12
21.c even 2 1 588.3.g.d 12
24.f even 2 1 1344.3.m.e 12
24.h odd 2 1 1344.3.m.e 12
84.h odd 2 1 588.3.g.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.g.a ✓ 12 3.b odd 2 1
84.3.g.a ✓ 12 12.b even 2 1
252.3.g.b 12 1.a even 1 1 trivial
252.3.g.b 12 4.b odd 2 1 inner
588.3.g.d 12 21.c even 2 1
588.3.g.d 12 84.h odd 2 1
1344.3.m.e 12 24.f even 2 1
1344.3.m.e 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T56+4T55−112T54−384T53+2976T52+7808T5+2752 T_{5}^{6} + 4T_{5}^{5} - 112T_{5}^{4} - 384T_{5}^{3} + 2976T_{5}^{2} + 7808T_{5} + 2752 acting on S3new(252,[χ])S_{3}^{\mathrm{new}}(252, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12+2T11+⋯+4096 T^{12} + 2 T^{11} + \cdots + 4096 Copy content Toggle raw display
33 T12 T^{12} Copy content Toggle raw display
55 (T6+4T5+⋯+2752)2 (T^{6} + 4 T^{5} + \cdots + 2752)^{2} Copy content Toggle raw display
77 (T2+7)6 (T^{2} + 7)^{6} Copy content Toggle raw display
1111 T12+⋯+8305770496 T^{12} + \cdots + 8305770496 Copy content Toggle raw display
1313 (T6+12T5+⋯−1102016)2 (T^{6} + 12 T^{5} + \cdots - 1102016)^{2} Copy content Toggle raw display
1717 (T6−20T5+⋯+39448000)2 (T^{6} - 20 T^{5} + \cdots + 39448000)^{2} Copy content Toggle raw display
1919 T12+⋯+4294967296 T^{12} + \cdots + 4294967296 Copy content Toggle raw display
2323 T12+⋯+12 ⁣⋯ ⁣24 T^{12} + \cdots + 12\!\cdots\!24 Copy content Toggle raw display
2929 (T6+36T5+⋯−12566720)2 (T^{6} + 36 T^{5} + \cdots - 12566720)^{2} Copy content Toggle raw display
3131 T12+⋯+128544076201984 T^{12} + \cdots + 128544076201984 Copy content Toggle raw display
3737 (T6+44T5+⋯−155225024)2 (T^{6} + 44 T^{5} + \cdots - 155225024)^{2} Copy content Toggle raw display
4141 (T6−100T5+⋯+8374720)2 (T^{6} - 100 T^{5} + \cdots + 8374720)^{2} Copy content Toggle raw display
4343 T12+⋯+49 ⁣⋯ ⁣64 T^{12} + \cdots + 49\!\cdots\!64 Copy content Toggle raw display
4747 T12+⋯+47 ⁣⋯ ⁣00 T^{12} + \cdots + 47\!\cdots\!00 Copy content Toggle raw display
5353 (T6+52T5+⋯−11239563200)2 (T^{6} + 52 T^{5} + \cdots - 11239563200)^{2} Copy content Toggle raw display
5959 T12+⋯+90 ⁣⋯ ⁣56 T^{12} + \cdots + 90\!\cdots\!56 Copy content Toggle raw display
6161 (T6−52T5+⋯+44445760)2 (T^{6} - 52 T^{5} + \cdots + 44445760)^{2} Copy content Toggle raw display
6767 T12+⋯+13 ⁣⋯ ⁣24 T^{12} + \cdots + 13\!\cdots\!24 Copy content Toggle raw display
7171 T12+⋯+19 ⁣⋯ ⁣00 T^{12} + \cdots + 19\!\cdots\!00 Copy content Toggle raw display
7373 (T6−156T5+⋯+11256899392)2 (T^{6} - 156 T^{5} + \cdots + 11256899392)^{2} Copy content Toggle raw display
7979 T12+⋯+84 ⁣⋯ ⁣44 T^{12} + \cdots + 84\!\cdots\!44 Copy content Toggle raw display
8383 T12+⋯+25 ⁣⋯ ⁣00 T^{12} + \cdots + 25\!\cdots\!00 Copy content Toggle raw display
8989 (T6−276T5+⋯+733352728000)2 (T^{6} - 276 T^{5} + \cdots + 733352728000)^{2} Copy content Toggle raw display
9797 (T6+132T5+⋯−290193606848)2 (T^{6} + 132 T^{5} + \cdots - 290193606848)^{2} Copy content Toggle raw display
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