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Properties

Label 300.9.g.e
Level 300300
Weight 99
Character orbit 300.g
Analytic conductor 122.214122.214
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] = [300,9,Mod(101,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.101");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: N N == 300=22⋅3⋅52 300 = 2^{2} \cdot 3 \cdot 5^{2}
Weight: k k == 9 9
Character orbit: [χ][\chi] == 300.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: 122.213583018122.213583018
Analytic rank: 00
Dimension: 1010
Coefficient field: Q[x]/(x10−⋯ )\mathbb{Q}[x]/(x^{10} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x10−x9+17581x8−268094x7+129938570x6−2805075950x5+497042336337x4+⋯+51 ⁣⋯ ⁣56 x^{10} - x^{9} + 17581 x^{8} - 268094 x^{7} + 129938570 x^{6} - 2805075950 x^{5} + 497042336337 x^{4} + \cdots + 51\!\cdots\!56 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a17]\Z[a_1, \ldots, a_{17}]
Coefficient ring index: 215⋅317⋅57⋅72 2^{15}\cdot 3^{17}\cdot 5^{7}\cdot 7^{2}
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+(−β1−14)q3+(−β3+2β1−104)q7+(β4−β2+17β1−1240)q9+(−β7−8β1−2)q11+(2β6−β5+2β4+⋯+3430)q13+⋯+(−102β9+528β8+⋯+5247444)q99+O(q100) q + ( - \beta_1 - 14) q^{3} + ( - \beta_{3} + 2 \beta_1 - 104) q^{7} + (\beta_{4} - \beta_{2} + 17 \beta_1 - 1240) q^{9} + ( - \beta_{7} - 8 \beta_1 - 2) q^{11} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \cdots + 3430) q^{13}+ \cdots + ( - 102 \beta_{9} + 528 \beta_{8} + \cdots + 5247444) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 10q−137q3−1048q7−12455q9+34472q13−376030q19−142540q21−1458782q27−410860q31−523275q33−110344q37+3527870q39−6252148q43+13891530q49+⋯+52292505q99+O(q100) 10 q - 137 q^{3} - 1048 q^{7} - 12455 q^{9} + 34472 q^{13} - 376030 q^{19} - 142540 q^{21} - 1458782 q^{27} - 410860 q^{31} - 523275 q^{33} - 110344 q^{37} + 3527870 q^{39} - 6252148 q^{43} + 13891530 q^{49}+ \cdots + 52292505 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x10−x9+17581x8−268094x7+129938570x6−2805075950x5+497042336337x4+⋯+51 ⁣⋯ ⁣56 x^{10} - x^{9} + 17581 x^{8} - 268094 x^{7} + 129938570 x^{6} - 2805075950 x^{5} + 497042336337 x^{4} + \cdots + 51\!\cdots\!56 : Copy content Toggle raw display

β1\beta_{1}== (31 ⁣⋯ ⁣99ν9+⋯+83 ⁣⋯ ⁣94)/20 ⁣⋯ ⁣50 ( 31\!\cdots\!99 \nu^{9} + \cdots + 83\!\cdots\!94 ) / 20\!\cdots\!50 Copy content Toggle raw display
β2\beta_{2}== (−20 ⁣⋯ ⁣89ν9+⋯−99 ⁣⋯ ⁣34)/69 ⁣⋯ ⁣50 ( - 20\!\cdots\!89 \nu^{9} + \cdots - 99\!\cdots\!34 ) / 69\!\cdots\!50 Copy content Toggle raw display
β3\beta_{3}== (43 ⁣⋯ ⁣82ν9+⋯−10 ⁣⋯ ⁣33)/10 ⁣⋯ ⁣25 ( 43\!\cdots\!82 \nu^{9} + \cdots - 10\!\cdots\!33 ) / 10\!\cdots\!25 Copy content Toggle raw display
β4\beta_{4}== (−15 ⁣⋯ ⁣52ν9+⋯+17 ⁣⋯ ⁣06)/20 ⁣⋯ ⁣50 ( - 15\!\cdots\!52 \nu^{9} + \cdots + 17\!\cdots\!06 ) / 20\!\cdots\!50 Copy content Toggle raw display
β5\beta_{5}== (−28 ⁣⋯ ⁣76ν9+⋯−69 ⁣⋯ ⁣70)/20 ⁣⋯ ⁣50 ( - 28\!\cdots\!76 \nu^{9} + \cdots - 69\!\cdots\!70 ) / 20\!\cdots\!50 Copy content Toggle raw display
β6\beta_{6}== (58 ⁣⋯ ⁣66ν9+⋯−67 ⁣⋯ ⁣54)/31 ⁣⋯ ⁣75 ( 58\!\cdots\!66 \nu^{9} + \cdots - 67\!\cdots\!54 ) / 31\!\cdots\!75 Copy content Toggle raw display
β7\beta_{7}== (53 ⁣⋯ ⁣66ν9+⋯+36 ⁣⋯ ⁣24)/20 ⁣⋯ ⁣50 ( 53\!\cdots\!66 \nu^{9} + \cdots + 36\!\cdots\!24 ) / 20\!\cdots\!50 Copy content Toggle raw display
β8\beta_{8}== (−32 ⁣⋯ ⁣36ν9+⋯+29 ⁣⋯ ⁣88)/69 ⁣⋯ ⁣50 ( - 32\!\cdots\!36 \nu^{9} + \cdots + 29\!\cdots\!88 ) / 69\!\cdots\!50 Copy content Toggle raw display
β9\beta_{9}== (43 ⁣⋯ ⁣51ν9+⋯−62 ⁣⋯ ⁣76)/31 ⁣⋯ ⁣75 ( 43\!\cdots\!51 \nu^{9} + \cdots - 62\!\cdots\!76 ) / 31\!\cdots\!75 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (3β6−5β5+10β4−24β3+105β2+3521β1+1645)/5670 ( 3\beta_{6} - 5\beta_{5} + 10\beta_{4} - 24\beta_{3} + 105\beta_{2} + 3521\beta _1 + 1645 ) / 5670 Copy content Toggle raw display
ν2\nu^{2}== (−18β9−126β8−252β7−24β6−593β5+574β4+⋯−19933400)/5670 ( - 18 \beta_{9} - 126 \beta_{8} - 252 \beta_{7} - 24 \beta_{6} - 593 \beta_{5} + 574 \beta_{4} + \cdots - 19933400 ) / 5670 Copy content Toggle raw display
ν3\nu^{3}== (477β9−1197β8−1260β7−24816β6+17531β5+⋯+140784414)/1890 ( 477 \beta_{9} - 1197 \beta_{8} - 1260 \beta_{7} - 24816 \beta_{6} + 17531 \beta_{5} + \cdots + 140784414 ) / 1890 Copy content Toggle raw display
ν4\nu^{4}== (89478β9+979020β8+3251934β7−1069734β6+4077899β5+⋯+56555124980)/5670 ( 89478 \beta_{9} + 979020 \beta_{8} + 3251934 \beta_{7} - 1069734 \beta_{6} + 4077899 \beta_{5} + \cdots + 56555124980 ) / 5670 Copy content Toggle raw display
ν5\nu^{5}== (−12096225β9+20022975β8+60883200β7+540939723β6+⋯−4892621889835)/5670 ( - 12096225 \beta_{9} + 20022975 \beta_{8} + 60883200 \beta_{7} + 540939723 \beta_{6} + \cdots - 4892621889835 ) / 5670 Copy content Toggle raw display
ν6\nu^{6}== (151267842β9−1705873806β8−6682521132β7+656273328β6+⋯+5722741427884)/1890 ( 151267842 \beta_{9} - 1705873806 \beta_{8} - 6682521132 \beta_{7} + 656273328 \beta_{6} + \cdots + 5722741427884 ) / 1890 Copy content Toggle raw display
ν7\nu^{7}== (73494705564β9−25578553329β8−80276140665β7−2593989467772β6+⋯+38 ⁣⋯ ⁣50)/5670 ( 73494705564 \beta_{9} - 25578553329 \beta_{8} - 80276140665 \beta_{7} - 2593989467772 \beta_{6} + \cdots + 38\!\cdots\!50 ) / 5670 Copy content Toggle raw display
ν8\nu^{8}== (−10937784134394β9+17339543872560β8+91365578955378β7+⋯−17 ⁣⋯ ⁣60)/5670 ( - 10937784134394 \beta_{9} + 17339543872560 \beta_{8} + 91365578955378 \beta_{7} + \cdots - 17\!\cdots\!60 ) / 5670 Copy content Toggle raw display
ν9\nu^{9}== (−77721907903605β9−229931190642195β8+⋯−76 ⁣⋯ ⁣00)/1890 ( - 77721907903605 \beta_{9} - 229931190642195 \beta_{8} + \cdots - 76\!\cdots\!00 ) / 1890 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/300Z)×\left(\mathbb{Z}/300\mathbb{Z}\right)^\times.

nn 101101 151151 277277
χ(n)\chi(n) −1-1 11 11

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}
101.1
18.8523 + 21.8350i
18.8523 − 21.8350i
−5.98515 + 56.3224i
−5.98515 − 56.3224i
−16.1674 + 79.4621i
−16.1674 − 79.4621i
−23.3576 + 73.0318i
−23.3576 − 73.0318i
27.1579 + 64.9569i
27.1579 − 64.9569i
0 −78.0015 − 21.8350i 0 0 0 1300.30 0 5607.47 + 3406.32i 0
101.2 0 −78.0015 + 21.8350i 0 0 0 1300.30 0 5607.47 − 3406.32i 0
101.3 0 −58.2133 − 56.3224i 0 0 0 −2295.51 0 216.575 + 6557.42i 0
101.4 0 −58.2133 + 56.3224i 0 0 0 −2295.51 0 216.575 − 6557.42i 0
101.5 0 −15.7090 − 79.4621i 0 0 0 2840.81 0 −6067.46 + 2496.53i 0
101.6 0 −15.7090 + 79.4621i 0 0 0 2840.81 0 −6067.46 − 2496.53i 0
101.7 0 35.0336 − 73.0318i 0 0 0 −4179.12 0 −4106.29 − 5117.14i 0
101.8 0 35.0336 + 73.0318i 0 0 0 −4179.12 0 −4106.29 + 5117.14i 0
101.9 0 48.3901 − 64.9569i 0 0 0 1809.51 0 −1877.79 − 6286.54i 0
101.10 0 48.3901 + 64.9569i 0 0 0 1809.51 0 −1877.79 + 6286.54i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.9.g.e ✓ 10
3.b odd 2 1 inner 300.9.g.e ✓ 10
5.b even 2 1 300.9.g.g yes 10
5.c odd 4 2 300.9.b.e 20
15.d odd 2 1 300.9.g.g yes 10
15.e even 4 2 300.9.b.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.9.b.e 20 5.c odd 4 2
300.9.b.e 20 15.e even 4 2
300.9.g.e ✓ 10 1.a even 1 1 trivial
300.9.g.e ✓ 10 3.b odd 2 1 inner
300.9.g.g yes 10 5.b even 2 1
300.9.g.g yes 10 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T75+524T74−17747597T73+8663960872T72+64044622169696T7−64122838801930496 T_{7}^{5} + 524T_{7}^{4} - 17747597T_{7}^{3} + 8663960872T_{7}^{2} + 64044622169696T_{7} - 64122838801930496 acting on S9new(300,[χ])S_{9}^{\mathrm{new}}(300, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T10 T^{10} Copy content Toggle raw display
33 T10+⋯+12 ⁣⋯ ⁣01 T^{10} + \cdots + 12\!\cdots\!01 Copy content Toggle raw display
55 T10 T^{10} Copy content Toggle raw display
77 (T5+⋯−64 ⁣⋯ ⁣96)2 (T^{5} + \cdots - 64\!\cdots\!96)^{2} Copy content Toggle raw display
1111 T10+⋯+20 ⁣⋯ ⁣00 T^{10} + \cdots + 20\!\cdots\!00 Copy content Toggle raw display
1313 (T5+⋯+17 ⁣⋯ ⁣84)2 (T^{5} + \cdots + 17\!\cdots\!84)^{2} Copy content Toggle raw display
1717 T10+⋯+38 ⁣⋯ ⁣00 T^{10} + \cdots + 38\!\cdots\!00 Copy content Toggle raw display
1919 (T5+⋯+50 ⁣⋯ ⁣11)2 (T^{5} + \cdots + 50\!\cdots\!11)^{2} Copy content Toggle raw display
2323 T10+⋯+86 ⁣⋯ ⁣00 T^{10} + \cdots + 86\!\cdots\!00 Copy content Toggle raw display
2929 T10+⋯+19 ⁣⋯ ⁣00 T^{10} + \cdots + 19\!\cdots\!00 Copy content Toggle raw display
3131 (T5+⋯−82 ⁣⋯ ⁣24)2 (T^{5} + \cdots - 82\!\cdots\!24)^{2} Copy content Toggle raw display
3737 (T5+⋯+97 ⁣⋯ ⁣12)2 (T^{5} + \cdots + 97\!\cdots\!12)^{2} Copy content Toggle raw display
4141 T10+⋯+19 ⁣⋯ ⁣00 T^{10} + \cdots + 19\!\cdots\!00 Copy content Toggle raw display
4343 (T5+⋯+25 ⁣⋯ ⁣04)2 (T^{5} + \cdots + 25\!\cdots\!04)^{2} Copy content Toggle raw display
4747 T10+⋯+17 ⁣⋯ ⁣00 T^{10} + \cdots + 17\!\cdots\!00 Copy content Toggle raw display
5353 T10+⋯+48 ⁣⋯ ⁣00 T^{10} + \cdots + 48\!\cdots\!00 Copy content Toggle raw display
5959 T10+⋯+50 ⁣⋯ ⁣00 T^{10} + \cdots + 50\!\cdots\!00 Copy content Toggle raw display
6161 (T5+⋯−66 ⁣⋯ ⁣84)2 (T^{5} + \cdots - 66\!\cdots\!84)^{2} Copy content Toggle raw display
6767 (T5+⋯+45 ⁣⋯ ⁣29)2 (T^{5} + \cdots + 45\!\cdots\!29)^{2} Copy content Toggle raw display
7171 T10+⋯+11 ⁣⋯ ⁣00 T^{10} + \cdots + 11\!\cdots\!00 Copy content Toggle raw display
7373 (T5+⋯+97 ⁣⋯ ⁣92)2 (T^{5} + \cdots + 97\!\cdots\!92)^{2} Copy content Toggle raw display
7979 (T5+⋯+22 ⁣⋯ ⁣28)2 (T^{5} + \cdots + 22\!\cdots\!28)^{2} Copy content Toggle raw display
8383 T10+⋯+60 ⁣⋯ ⁣00 T^{10} + \cdots + 60\!\cdots\!00 Copy content Toggle raw display
8989 T10+⋯+16 ⁣⋯ ⁣00 T^{10} + \cdots + 16\!\cdots\!00 Copy content Toggle raw display
9797 (T5+⋯+29 ⁣⋯ ⁣64)2 (T^{5} + \cdots + 29\!\cdots\!64)^{2} Copy content Toggle raw display
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