Properties

Label 575.4.a.r
Level 575575
Weight 44
Character orbit 575.a
Self dual yes
Analytic conductor 33.92633.926
Analytic rank 00
Dimension 1717
CM no
Inner twists 11

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,4,Mod(1,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 575=5223 575 = 5^{2} \cdot 23
Weight: k k == 4 4
Character orbit: [χ][\chi] == 575.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: 33.926098253333.9260982533
Analytic rank: 00
Dimension: 1717
Coefficient field: Q[x]/(x17)\mathbb{Q}[x]/(x^{17} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x174x1696x15+368x14+3705x1313440x1273933x11+248806x10+2150912 x^{17} - 4 x^{16} - 96 x^{15} + 368 x^{14} + 3705 x^{13} - 13440 x^{12} - 73933 x^{11} + 248806 x^{10} + \cdots - 2150912 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 2352 2^{3}\cdot 5^{2}
Twist minimal: no (minimal twist has level 115)
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,,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+β1q2+(β4+1)q3+(β2+4)q4+(β8+2β1)q6+(β3β1+4)q7+(β8+β6β4++1)q8+(β9+β4+β1+9)q9++(9β16+7β14++189)q99+O(q100) q + \beta_1 q^{2} + (\beta_{4} + 1) q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{8} + 2 \beta_1) q^{6} + ( - \beta_{3} - \beta_1 + 4) q^{7} + (\beta_{8} + \beta_{6} - \beta_{4} + \cdots + 1) q^{8} + (\beta_{9} + \beta_{4} + \beta_1 + 9) q^{9}+ \cdots + ( - 9 \beta_{16} + 7 \beta_{14} + \cdots + 189) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 17q+4q2+12q3+72q4+12q6+72q7+48q8+155q94q11+342q12+208q13118q14+220q16+268q17+180q1872q1916q21+318q22++1908q99+O(q100) 17 q + 4 q^{2} + 12 q^{3} + 72 q^{4} + 12 q^{6} + 72 q^{7} + 48 q^{8} + 155 q^{9} - 4 q^{11} + 342 q^{12} + 208 q^{13} - 118 q^{14} + 220 q^{16} + 268 q^{17} + 180 q^{18} - 72 q^{19} - 16 q^{21} + 318 q^{22}+ \cdots + 1908 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x174x1696x15+368x14+3705x1313440x1273933x11+248806x10+2150912 x^{17} - 4 x^{16} - 96 x^{15} + 368 x^{14} + 3705 x^{13} - 13440 x^{12} - 73933 x^{11} + 248806 x^{10} + \cdots - 2150912 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν212 \nu^{2} - 12 Copy content Toggle raw display
β3\beta_{3}== (11 ⁣ ⁣12ν16++20 ⁣ ⁣24)/64 ⁣ ⁣80 ( 11\!\cdots\!12 \nu^{16} + \cdots + 20\!\cdots\!24 ) / 64\!\cdots\!80 Copy content Toggle raw display
β4\beta_{4}== (68 ⁣ ⁣35ν16+23 ⁣ ⁣36)/64 ⁣ ⁣80 ( - 68\!\cdots\!35 \nu^{16} + \cdots - 23\!\cdots\!36 ) / 64\!\cdots\!80 Copy content Toggle raw display
β5\beta_{5}== (16 ⁣ ⁣33ν16++11 ⁣ ⁣88)/36 ⁣ ⁣60 ( 16\!\cdots\!33 \nu^{16} + \cdots + 11\!\cdots\!88 ) / 36\!\cdots\!60 Copy content Toggle raw display
β6\beta_{6}== (38 ⁣ ⁣21ν16++61 ⁣ ⁣64)/64 ⁣ ⁣80 ( 38\!\cdots\!21 \nu^{16} + \cdots + 61\!\cdots\!64 ) / 64\!\cdots\!80 Copy content Toggle raw display
β7\beta_{7}== (16 ⁣ ⁣47ν16+86 ⁣ ⁣68)/25 ⁣ ⁣20 ( 16\!\cdots\!47 \nu^{16} + \cdots - 86\!\cdots\!68 ) / 25\!\cdots\!20 Copy content Toggle raw display
β8\beta_{8}== (44 ⁣ ⁣56ν16+14 ⁣ ⁣20)/64 ⁣ ⁣80 ( - 44\!\cdots\!56 \nu^{16} + \cdots - 14\!\cdots\!20 ) / 64\!\cdots\!80 Copy content Toggle raw display
β9\beta_{9}== (22 ⁣ ⁣41ν16+53 ⁣ ⁣16)/25 ⁣ ⁣20 ( 22\!\cdots\!41 \nu^{16} + \cdots - 53\!\cdots\!16 ) / 25\!\cdots\!20 Copy content Toggle raw display
β10\beta_{10}== (49 ⁣ ⁣41ν16+30 ⁣ ⁣16)/51 ⁣ ⁣40 ( - 49\!\cdots\!41 \nu^{16} + \cdots - 30\!\cdots\!16 ) / 51\!\cdots\!40 Copy content Toggle raw display
β11\beta_{11}== (25 ⁣ ⁣53ν16+10 ⁣ ⁣28)/25 ⁣ ⁣20 ( - 25\!\cdots\!53 \nu^{16} + \cdots - 10\!\cdots\!28 ) / 25\!\cdots\!20 Copy content Toggle raw display
β12\beta_{12}== (56 ⁣ ⁣57ν16++24 ⁣ ⁣84)/51 ⁣ ⁣40 ( 56\!\cdots\!57 \nu^{16} + \cdots + 24\!\cdots\!84 ) / 51\!\cdots\!40 Copy content Toggle raw display
β13\beta_{13}== (22 ⁣ ⁣49ν16++11 ⁣ ⁣00)/17 ⁣ ⁣80 ( 22\!\cdots\!49 \nu^{16} + \cdots + 11\!\cdots\!00 ) / 17\!\cdots\!80 Copy content Toggle raw display
β14\beta_{14}== (74 ⁣ ⁣43ν16+92 ⁣ ⁣20)/51 ⁣ ⁣40 ( 74\!\cdots\!43 \nu^{16} + \cdots - 92\!\cdots\!20 ) / 51\!\cdots\!40 Copy content Toggle raw display
β15\beta_{15}== (26 ⁣ ⁣65ν16++73 ⁣ ⁣84)/17 ⁣ ⁣80 ( 26\!\cdots\!65 \nu^{16} + \cdots + 73\!\cdots\!84 ) / 17\!\cdots\!80 Copy content Toggle raw display
β16\beta_{16}== (12 ⁣ ⁣62ν16++58 ⁣ ⁣72)/64 ⁣ ⁣80 ( 12\!\cdots\!62 \nu^{16} + \cdots + 58\!\cdots\!72 ) / 64\!\cdots\!80 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+12 \beta_{2} + 12 Copy content Toggle raw display
ν3\nu^{3}== β8+β6β4+β2+19β1+1 \beta_{8} + \beta_{6} - \beta_{4} + \beta_{2} + 19\beta _1 + 1 Copy content Toggle raw display
ν4\nu^{4}== β15+β12β11β82β7+β6+5β4β3++237 - \beta_{15} + \beta_{12} - \beta_{11} - \beta_{8} - 2 \beta_{7} + \beta_{6} + 5 \beta_{4} - \beta_{3} + \cdots + 237 Copy content Toggle raw display
ν5\nu^{5}== β15+2β142β13β12β9+32β8β7+34β6++45 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - \beta_{12} - \beta_{9} + 32 \beta_{8} - \beta_{7} + 34 \beta_{6} + \cdots + 45 Copy content Toggle raw display
ν6\nu^{6}== 38β15+β14+3β13+26β1244β11+6β10++5371 - 38 \beta_{15} + \beta_{14} + 3 \beta_{13} + 26 \beta_{12} - 44 \beta_{11} + 6 \beta_{10} + \cdots + 5371 Copy content Toggle raw display
ν7\nu^{7}== 9β16+26β15+97β1494β1335β122β11++1795 9 \beta_{16} + 26 \beta_{15} + 97 \beta_{14} - 94 \beta_{13} - 35 \beta_{12} - 2 \beta_{11} + \cdots + 1795 Copy content Toggle raw display
ν8\nu^{8}== 22β161146β15+72β14+182β13+512β121450β11++129162 - 22 \beta_{16} - 1146 \beta_{15} + 72 \beta_{14} + 182 \beta_{13} + 512 \beta_{12} - 1450 \beta_{11} + \cdots + 129162 Copy content Toggle raw display
ν9\nu^{9}== 488β16+364β15+3446β143150β13968β12126β11++63299 488 \beta_{16} + 364 \beta_{15} + 3446 \beta_{14} - 3150 \beta_{13} - 968 \beta_{12} - 126 \beta_{11} + \cdots + 63299 Copy content Toggle raw display
ν10\nu^{10}== 1138β1632455β15+3532β14+7430β13+8713β12++3193631 - 1138 \beta_{16} - 32455 \beta_{15} + 3532 \beta_{14} + 7430 \beta_{13} + 8713 \beta_{12} + \cdots + 3193631 Copy content Toggle raw display
ν11\nu^{11}== 17614β161447β15+109178β1493096β1326527β12++2049655 17614 \beta_{16} - 1447 \beta_{15} + 109178 \beta_{14} - 93096 \beta_{13} - 26527 \beta_{12} + \cdots + 2049655 Copy content Toggle raw display
ν12\nu^{12}== 37206β16900080β15+144639β14+255299β13+122818β12++80100557 - 37206 \beta_{16} - 900080 \beta_{15} + 144639 \beta_{14} + 255299 \beta_{13} + 122818 \beta_{12} + \cdots + 80100557 Copy content Toggle raw display
ν13\nu^{13}== 536693β16337774β15+3272957β142592662β13765247β12++62885833 536693 \beta_{16} - 337774 \beta_{15} + 3272957 \beta_{14} - 2592662 \beta_{13} - 765247 \beta_{12} + \cdots + 62885833 Copy content Toggle raw display
ν14\nu^{14}== 947998β1624760562β15+5318234β14+7996180β13+955856β12++2025594478 - 947998 \beta_{16} - 24760562 \beta_{15} + 5318234 \beta_{14} + 7996180 \beta_{13} + 955856 \beta_{12} + \cdots + 2025594478 Copy content Toggle raw display
ν15\nu^{15}== 15000142β1616523832β15+95121210β1469937272β13++1863902263 15000142 \beta_{16} - 16523832 \beta_{15} + 95121210 \beta_{14} - 69937272 \beta_{13} + \cdots + 1863902263 Copy content Toggle raw display
ν16\nu^{16}== 19147324β16678427997β15+182102576β14+237025708β13++51495820397 - 19147324 \beta_{16} - 678427997 \beta_{15} + 182102576 \beta_{14} + 237025708 \beta_{13} + \cdots + 51495820397 Copy content Toggle raw display

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
−5.04242
−4.93784
−4.24824
−3.17951
−3.00148
−1.98882
−1.45307
−0.407036
0.231361
0.752118
2.51792
2.81381
3.13266
3.60232
4.94561
5.07225
5.19037
−5.04242 0.0620551 17.4260 0 −0.312908 30.6525 −47.5296 −26.9961 0
1.2 −4.93784 9.92307 16.3822 0 −48.9985 14.6304 −41.3902 71.4673 0
1.3 −4.24824 2.36809 10.0475 0 −10.0602 −12.8082 −8.69841 −21.3922 0
1.4 −3.17951 −4.30546 2.10929 0 13.6892 0.115569 18.7296 −8.46305 0
1.5 −3.00148 −1.85196 1.00888 0 5.55862 4.63746 20.9837 −23.5702 0
1.6 −1.98882 −9.68598 −4.04459 0 19.2637 25.5696 23.9545 66.8182 0
1.7 −1.45307 7.24162 −5.88858 0 −10.5226 −0.356556 20.1811 25.4411 0
1.8 −0.407036 −3.62432 −7.83432 0 1.47523 −11.1245 6.44514 −13.8643 0
1.9 0.231361 2.18914 −7.94647 0 0.506482 −29.8467 −3.68940 −22.2077 0
1.10 0.752118 6.31519 −7.43432 0 4.74977 25.4579 −11.6084 12.8816 0
1.11 2.51792 −6.03797 −1.66008 0 −15.2031 24.0921 −24.3233 9.45712 0
1.12 2.81381 −0.460867 −0.0824876 0 −1.29679 −16.1192 −22.7426 −26.7876 0
1.13 3.13266 −8.19196 1.81356 0 −25.6626 −13.4923 −19.3800 40.1082 0
1.14 3.60232 8.84330 4.97671 0 31.8564 13.0981 −10.8909 51.2040 0
1.15 4.94561 5.42743 16.4591 0 26.8419 28.7748 41.8353 2.45696 0
1.16 5.07225 −3.81839 17.7277 0 −19.3678 14.6908 49.3412 −12.4199 0
1.17 5.19037 7.60701 18.9399 0 39.4832 −25.9715 56.7822 30.8666 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
2323 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 575.4.a.r 17
5.b even 2 1 575.4.a.q 17
5.c odd 4 2 115.4.b.a 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.b.a 34 5.c odd 4 2
575.4.a.q 17 5.b even 2 1
575.4.a.r 17 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 S4new(Γ0(575))S_{4}^{\mathrm{new}}(\Gamma_0(575)):

T2174T21696T215+368T214+3705T21313440T212+2150912 T_{2}^{17} - 4 T_{2}^{16} - 96 T_{2}^{15} + 368 T_{2}^{14} + 3705 T_{2}^{13} - 13440 T_{2}^{12} + \cdots - 2150912 Copy content Toggle raw display
T31712T316235T315+3044T314+20159T313292528T312+1298656000 T_{3}^{17} - 12 T_{3}^{16} - 235 T_{3}^{15} + 3044 T_{3}^{14} + 20159 T_{3}^{13} - 292528 T_{3}^{12} + \cdots - 1298656000 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T174T16+2150912 T^{17} - 4 T^{16} + \cdots - 2150912 Copy content Toggle raw display
33 T17+1298656000 T^{17} + \cdots - 1298656000 Copy content Toggle raw display
55 T17 T^{17} Copy content Toggle raw display
77 T17++17 ⁣ ⁣88 T^{17} + \cdots + 17\!\cdots\!88 Copy content Toggle raw display
1111 T17++66 ⁣ ⁣40 T^{17} + \cdots + 66\!\cdots\!40 Copy content Toggle raw display
1313 T17++92 ⁣ ⁣24 T^{17} + \cdots + 92\!\cdots\!24 Copy content Toggle raw display
1717 T17+57 ⁣ ⁣00 T^{17} + \cdots - 57\!\cdots\!00 Copy content Toggle raw display
1919 T17+15 ⁣ ⁣00 T^{17} + \cdots - 15\!\cdots\!00 Copy content Toggle raw display
2323 (T23)17 (T - 23)^{17} Copy content Toggle raw display
2929 T17+12 ⁣ ⁣00 T^{17} + \cdots - 12\!\cdots\!00 Copy content Toggle raw display
3131 T17+75 ⁣ ⁣00 T^{17} + \cdots - 75\!\cdots\!00 Copy content Toggle raw display
3737 T17+15 ⁣ ⁣44 T^{17} + \cdots - 15\!\cdots\!44 Copy content Toggle raw display
4141 T17++10 ⁣ ⁣60 T^{17} + \cdots + 10\!\cdots\!60 Copy content Toggle raw display
4343 T17++14 ⁣ ⁣00 T^{17} + \cdots + 14\!\cdots\!00 Copy content Toggle raw display
4747 T17+18 ⁣ ⁣00 T^{17} + \cdots - 18\!\cdots\!00 Copy content Toggle raw display
5353 T17++10 ⁣ ⁣04 T^{17} + \cdots + 10\!\cdots\!04 Copy content Toggle raw display
5959 T17+14 ⁣ ⁣00 T^{17} + \cdots - 14\!\cdots\!00 Copy content Toggle raw display
6161 T17+15 ⁣ ⁣92 T^{17} + \cdots - 15\!\cdots\!92 Copy content Toggle raw display
6767 T17++28 ⁣ ⁣00 T^{17} + \cdots + 28\!\cdots\!00 Copy content Toggle raw display
7171 T17+51 ⁣ ⁣00 T^{17} + \cdots - 51\!\cdots\!00 Copy content Toggle raw display
7373 T17++12 ⁣ ⁣60 T^{17} + \cdots + 12\!\cdots\!60 Copy content Toggle raw display
7979 T17+40 ⁣ ⁣60 T^{17} + \cdots - 40\!\cdots\!60 Copy content Toggle raw display
8383 T17++59 ⁣ ⁣48 T^{17} + \cdots + 59\!\cdots\!48 Copy content Toggle raw display
8989 T17+33 ⁣ ⁣00 T^{17} + \cdots - 33\!\cdots\!00 Copy content Toggle raw display
9797 T17++38 ⁣ ⁣00 T^{17} + \cdots + 38\!\cdots\!00 Copy content Toggle raw display
show more
show less