Properties

Label 435.2.c.e
Level 435435
Weight 22
Character orbit 435.c
Analytic conductor 3.4733.473
Analytic rank 00
Dimension 1010
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [435,2,Mod(349,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("435.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 435=3529 435 = 3 \cdot 5 \cdot 29
Weight: k k == 2 2
Character orbit: [χ][\chi] == 435.c (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: 3.473492487933.47349248793
Analytic rank: 00
Dimension: 1010
Coefficient field: 10.0.3899266318336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x102x9+2x8+6x7+19x612x5+4x4+2x3+9x26x+2 x^{10} - 2x^{9} + 2x^{8} + 6x^{7} + 19x^{6} - 12x^{5} + 4x^{4} + 2x^{3} + 9x^{2} - 6x + 2 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: yes
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,,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+(β6β4)q2β6q3+(β5β21)q4+(β8β7++β1)q5+(β2+1)q6+(β7+2β6)q7++(β9+β8β5+4)q99+O(q100) q + (\beta_{6} - \beta_{4}) q^{2} - \beta_{6} q^{3} + (\beta_{5} - \beta_{2} - 1) q^{4} + ( - \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{5} + (\beta_{2} + 1) q^{6} + (\beta_{7} + 2 \beta_{6}) q^{7}+ \cdots + ( - \beta_{9} + \beta_{8} - \beta_{5} + \cdots - 4) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 10q10q4+6q610q9+4q10+24q1112q14+2q15+2q16+4q19+8q20+16q2118q24+2q2510q2918q30+4q318q34+2q35+24q99+O(q100) 10 q - 10 q^{4} + 6 q^{6} - 10 q^{9} + 4 q^{10} + 24 q^{11} - 12 q^{14} + 2 q^{15} + 2 q^{16} + 4 q^{19} + 8 q^{20} + 16 q^{21} - 18 q^{24} + 2 q^{25} - 10 q^{29} - 18 q^{30} + 4 q^{31} - 8 q^{34} + 2 q^{35}+ \cdots - 24 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x102x9+2x8+6x7+19x612x5+4x4+2x3+9x26x+2 x^{10} - 2x^{9} + 2x^{8} + 6x^{7} + 19x^{6} - 12x^{5} + 4x^{4} + 2x^{3} + 9x^{2} - 6x + 2 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (4381ν9+19149ν824628ν715014ν67864ν5+263961ν4+8160)/57533 ( - 4381 \nu^{9} + 19149 \nu^{8} - 24628 \nu^{7} - 15014 \nu^{6} - 7864 \nu^{5} + 263961 \nu^{4} + \cdots - 8160 ) / 57533 Copy content Toggle raw display
β3\beta_{3}== (4394ν99304ν8+11897ν7+22741ν6+82151ν549465ν4+25960)/57533 ( 4394 \nu^{9} - 9304 \nu^{8} + 11897 \nu^{7} + 22741 \nu^{6} + 82151 \nu^{5} - 49465 \nu^{4} + \cdots - 25960 ) / 57533 Copy content Toggle raw display
β4\beta_{4}== (4908ν9+9502ν813053ν714036ν6116141ν5+39251ν4++20774)/57533 ( - 4908 \nu^{9} + 9502 \nu^{8} - 13053 \nu^{7} - 14036 \nu^{6} - 116141 \nu^{5} + 39251 \nu^{4} + \cdots + 20774 ) / 57533 Copy content Toggle raw display
β5\beta_{5}== (9291ν9+31562ν840148ν739090ν686752ν5+380853ν4++124078)/57533 ( - 9291 \nu^{9} + 31562 \nu^{8} - 40148 \nu^{7} - 39090 \nu^{6} - 86752 \nu^{5} + 380853 \nu^{4} + \cdots + 124078 ) / 57533 Copy content Toggle raw display
β6\beta_{6}== (12980ν921566ν8+16656ν7+89777ν6+269361ν573609ν4+38951)/57533 ( 12980 \nu^{9} - 21566 \nu^{8} + 16656 \nu^{7} + 89777 \nu^{6} + 269361 \nu^{5} - 73609 \nu^{4} + \cdots - 38951 ) / 57533 Copy content Toggle raw display
β7\beta_{7}== (35262ν961938ν8+58262ν7+216331ν6+737014ν5235934ν4+124636)/57533 ( 35262 \nu^{9} - 61938 \nu^{8} + 58262 \nu^{7} + 216331 \nu^{6} + 737014 \nu^{5} - 235934 \nu^{4} + \cdots - 124636 ) / 57533 Copy content Toggle raw display
β8\beta_{8}== (49059ν9+93221ν875860ν7322605ν6948470ν5+584107ν4++303094)/57533 ( - 49059 \nu^{9} + 93221 \nu^{8} - 75860 \nu^{7} - 322605 \nu^{6} - 948470 \nu^{5} + 584107 \nu^{4} + \cdots + 303094 ) / 57533 Copy content Toggle raw display
β9\beta_{9}== (49338ν9+67808ν846042ν7342393ν61131012ν562807ν4+26786)/57533 ( - 49338 \nu^{9} + 67808 \nu^{8} - 46042 \nu^{7} - 342393 \nu^{6} - 1131012 \nu^{5} - 62807 \nu^{4} + \cdots - 26786 ) / 57533 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β7+2β6β4+β3+β1 -\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{3} + \beta_1 Copy content Toggle raw display
ν3\nu^{3}== β82β7+2β5β4+7β3β2 -\beta_{8} - 2\beta_{7} + 2\beta_{5} - \beta_{4} + 7\beta_{3} - \beta_{2} Copy content Toggle raw display
ν4\nu^{4}== β9β8+8β5+10β37β210β18 \beta_{9} - \beta_{8} + 8\beta_{5} + 10\beta_{3} - 7\beta_{2} - 10\beta _1 - 8 Copy content Toggle raw display
ν5\nu^{5}== 7β9+16β75β6+16β5+10β410β248β15 7\beta_{9} + 16\beta_{7} - 5\beta_{6} + 16\beta_{5} + 10\beta_{4} - 10\beta_{2} - 48\beta _1 - 5 Copy content Toggle raw display
ν6\nu^{6}== 10β9+10β8+61β744β6+48β482β382β1 10\beta_{9} + 10\beta_{8} + 61\beta_{7} - 44\beta_{6} + 48\beta_{4} - 82\beta_{3} - 82\beta_1 Copy content Toggle raw display
ν7\nu^{7}== 48β8+120β755β6120β5+82β4345β3+82β2+55 48\beta_{8} + 120\beta_{7} - 55\beta_{6} - 120\beta_{5} + 82\beta_{4} - 345\beta_{3} + 82\beta_{2} + 55 Copy content Toggle raw display
ν8\nu^{8}== 82β9+82β8461β5636β3+345β2+636β1+286 -82\beta_{9} + 82\beta_{8} - 461\beta_{5} - 636\beta_{3} + 345\beta_{2} + 636\beta _1 + 286 Copy content Toggle raw display
ν9\nu^{9}== 345β9899β7+466β6899β5636β4+636β2+2545β1+466 -345\beta_{9} - 899\beta_{7} + 466\beta_{6} - 899\beta_{5} - 636\beta_{4} + 636\beta_{2} + 2545\beta _1 + 466 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/435Z)×\left(\mathbb{Z}/435\mathbb{Z}\right)^\times.

nn 3131 146146 262262
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}
349.1
−1.20964 + 1.20964i
1.93833 1.93833i
−0.604479 0.604479i
0.313948 0.313948i
0.561843 + 0.561843i
0.561843 0.561843i
0.313948 + 0.313948i
−0.604479 + 0.604479i
1.93833 + 1.93833i
−1.20964 1.20964i
2.51908i 1.00000i −4.34577 −1.27413 + 1.83755i 2.51908 0.173311i 5.90919i −1.00000 4.62893 + 3.20964i
349.2 2.15351i 1.00000i −2.63760 −0.0286357 2.23588i 2.15351 1.51591i 1.37308i −1.00000 −4.81500 + 0.0616673i
349.3 1.71457i 1.00000i −0.939748 1.51903 + 1.64090i −1.71457 0.654317i 1.81788i −1.00000 2.81344 2.60448i
349.4 0.754474i 1.00000i 1.43077 −2.23474 + 0.0770824i 0.754474 4.18524i 2.58843i −1.00000 0.0581566 + 1.68605i
349.5 0.712495i 1.00000i 1.49235 2.01848 0.962154i −0.712495 2.77986i 2.48828i −1.00000 −0.685530 1.43816i
349.6 0.712495i 1.00000i 1.49235 2.01848 + 0.962154i −0.712495 2.77986i 2.48828i −1.00000 −0.685530 + 1.43816i
349.7 0.754474i 1.00000i 1.43077 −2.23474 0.0770824i 0.754474 4.18524i 2.58843i −1.00000 0.0581566 1.68605i
349.8 1.71457i 1.00000i −0.939748 1.51903 1.64090i −1.71457 0.654317i 1.81788i −1.00000 2.81344 + 2.60448i
349.9 2.15351i 1.00000i −2.63760 −0.0286357 + 2.23588i 2.15351 1.51591i 1.37308i −1.00000 −4.81500 0.0616673i
349.10 2.51908i 1.00000i −4.34577 −1.27413 1.83755i 2.51908 0.173311i 5.90919i −1.00000 4.62893 3.20964i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 435.2.c.e 10
3.b odd 2 1 1305.2.c.j 10
5.b even 2 1 inner 435.2.c.e 10
5.c odd 4 1 2175.2.a.w 5
5.c odd 4 1 2175.2.a.z 5
15.d odd 2 1 1305.2.c.j 10
15.e even 4 1 6525.2.a.bl 5
15.e even 4 1 6525.2.a.bs 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.c.e 10 1.a even 1 1 trivial
435.2.c.e 10 5.b even 2 1 inner
1305.2.c.j 10 3.b odd 2 1
1305.2.c.j 10 15.d odd 2 1
2175.2.a.w 5 5.c odd 4 1
2175.2.a.z 5 5.c odd 4 1
6525.2.a.bl 5 15.e even 4 1
6525.2.a.bs 5 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(435,[χ])S_{2}^{\mathrm{new}}(435, [\chi]):

T210+15T28+77T26+157T24+111T22+25 T_{2}^{10} + 15T_{2}^{8} + 77T_{2}^{6} + 157T_{2}^{4} + 111T_{2}^{2} + 25 Copy content Toggle raw display
T710+28T78+206T76+400T74+145T72+4 T_{7}^{10} + 28T_{7}^{8} + 206T_{7}^{6} + 400T_{7}^{4} + 145T_{7}^{2} + 4 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T10+15T8++25 T^{10} + 15 T^{8} + \cdots + 25 Copy content Toggle raw display
33 (T2+1)5 (T^{2} + 1)^{5} Copy content Toggle raw display
55 T10T8++3125 T^{10} - T^{8} + \cdots + 3125 Copy content Toggle raw display
77 T10+28T8++4 T^{10} + 28 T^{8} + \cdots + 4 Copy content Toggle raw display
1111 (T512T4+48T3+4)2 (T^{5} - 12 T^{4} + 48 T^{3} + \cdots - 4)^{2} Copy content Toggle raw display
1313 T10+16T8++16 T^{10} + 16 T^{8} + \cdots + 16 Copy content Toggle raw display
1717 T10+56T8++88804 T^{10} + 56 T^{8} + \cdots + 88804 Copy content Toggle raw display
1919 (T52T4++304)2 (T^{5} - 2 T^{4} + \cdots + 304)^{2} Copy content Toggle raw display
2323 T10+76T8++1600 T^{10} + 76 T^{8} + \cdots + 1600 Copy content Toggle raw display
2929 (T+1)10 (T + 1)^{10} Copy content Toggle raw display
3131 (T52T4+6304)2 (T^{5} - 2 T^{4} + \cdots - 6304)^{2} Copy content Toggle raw display
3737 T10+180T8++341056 T^{10} + 180 T^{8} + \cdots + 341056 Copy content Toggle raw display
4141 (T5+14T4++6176)2 (T^{5} + 14 T^{4} + \cdots + 6176)^{2} Copy content Toggle raw display
4343 T10+292T8++46895104 T^{10} + 292 T^{8} + \cdots + 46895104 Copy content Toggle raw display
4747 T10+220T8++7246864 T^{10} + 220 T^{8} + \cdots + 7246864 Copy content Toggle raw display
5353 T10++226683136 T^{10} + \cdots + 226683136 Copy content Toggle raw display
5959 (T5+4T4+2000)2 (T^{5} + 4 T^{4} + \cdots - 2000)^{2} Copy content Toggle raw display
6161 (T5+12T4++6872)2 (T^{5} + 12 T^{4} + \cdots + 6872)^{2} Copy content Toggle raw display
6767 T10+228T8++1716100 T^{10} + 228 T^{8} + \cdots + 1716100 Copy content Toggle raw display
7171 (T530T4++6592)2 (T^{5} - 30 T^{4} + \cdots + 6592)^{2} Copy content Toggle raw display
7373 T10+420T8++11343424 T^{10} + 420 T^{8} + \cdots + 11343424 Copy content Toggle raw display
7979 (T518T4+52048)2 (T^{5} - 18 T^{4} + \cdots - 52048)^{2} Copy content Toggle raw display
8383 T10+216T8++64 T^{10} + 216 T^{8} + \cdots + 64 Copy content Toggle raw display
8989 (T522T4+40682)2 (T^{5} - 22 T^{4} + \cdots - 40682)^{2} Copy content Toggle raw display
9797 T10+112T8++107584 T^{10} + 112 T^{8} + \cdots + 107584 Copy content Toggle raw display
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