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Properties

Label 300.3.f.c
Level $300$
Weight $3$
Character orbit 300.f
Analytic conductor $8.174$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,3,Mod(199,300)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.199");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.f (of order \(2\), degree \(1\), not 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: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 8 x^{14} - 14 x^{13} + 23 x^{12} - 26 x^{11} + 18 x^{10} - 10 x^{9} + 9 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{4} q^{3} + (\beta_{5} + 1) q^{4} + ( - \beta_{9} - 1) q^{6} + (\beta_{13} - \beta_{8} + \cdots + \beta_{3}) q^{7}+ \cdots + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{4} q^{3} + (\beta_{5} + 1) q^{4} + ( - \beta_{9} - 1) q^{6} + (\beta_{13} - \beta_{8} + \cdots + \beta_{3}) q^{7}+ \cdots + (3 \beta_{11} + 3 \beta_{10} + \cdots + 3 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} - 12 q^{6} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} - 12 q^{6} + 48 q^{9} - 44 q^{14} + 80 q^{16} + 48 q^{21} + 72 q^{24} - 132 q^{26} + 64 q^{29} - 248 q^{34} + 48 q^{36} - 32 q^{41} - 80 q^{44} - 152 q^{46} - 32 q^{49} - 36 q^{54} - 344 q^{56} + 272 q^{61} - 32 q^{64} - 216 q^{66} + 192 q^{69} + 216 q^{74} + 240 q^{76} + 144 q^{81} + 288 q^{84} + 428 q^{86} - 256 q^{89} - 24 q^{94} + 192 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 8 x^{14} - 14 x^{13} + 23 x^{12} - 26 x^{11} + 18 x^{10} - 10 x^{9} + 9 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{15} + 14 \nu^{14} - 2 \nu^{13} - 22 \nu^{12} - 11 \nu^{11} + 12 \nu^{10} + 20 \nu^{9} + \cdots + 1408 ) / 320 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11 \nu^{15} - 21 \nu^{14} + 6 \nu^{13} + 2 \nu^{12} - 29 \nu^{11} + 71 \nu^{10} - 82 \nu^{9} + \cdots + 1152 ) / 320 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6 \nu^{15} - 15 \nu^{14} + 26 \nu^{13} - 40 \nu^{12} + 52 \nu^{11} - 17 \nu^{10} - 4 \nu^{9} + \cdots - 512 ) / 320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{15} + 12 \nu^{14} - 17 \nu^{13} + 32 \nu^{12} - 47 \nu^{11} + 32 \nu^{10} - 13 \nu^{9} + \cdots + 640 ) / 160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} - 22 \nu^{14} + 52 \nu^{13} - 86 \nu^{12} + 147 \nu^{11} - 144 \nu^{10} + 82 \nu^{9} + \cdots - 2368 ) / 320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14 \nu^{15} - 17 \nu^{14} + 14 \nu^{13} - 52 \nu^{12} + 24 \nu^{11} + 9 \nu^{10} - 4 \nu^{9} + \cdots + 928 ) / 160 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3 \nu^{15} - 62 \nu^{14} + 68 \nu^{13} - 46 \nu^{12} + 199 \nu^{11} - 144 \nu^{10} - 54 \nu^{9} + \cdots - 5184 ) / 320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33 \nu^{15} - 33 \nu^{14} + 48 \nu^{13} - 150 \nu^{12} + 45 \nu^{11} + 67 \nu^{10} - 48 \nu^{9} + \cdots + 4480 ) / 320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 20 \nu^{15} + 57 \nu^{14} - 102 \nu^{13} + 172 \nu^{12} - 226 \nu^{11} + 179 \nu^{10} - 88 \nu^{9} + \cdots + 2624 ) / 320 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21 \nu^{15} - 55 \nu^{14} + 106 \nu^{13} - 210 \nu^{12} + 357 \nu^{11} - 315 \nu^{10} + 234 \nu^{9} + \cdots - 6016 ) / 320 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 52 \nu^{15} - 57 \nu^{14} + 102 \nu^{13} - 236 \nu^{12} + 194 \nu^{11} - 115 \nu^{10} + 152 \nu^{9} + \cdots + 64 ) / 320 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 52 \nu^{15} - 139 \nu^{14} + 244 \nu^{13} - 432 \nu^{12} + 582 \nu^{11} - 469 \nu^{10} + 126 \nu^{9} + \cdots - 6400 ) / 320 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 32 \nu^{15} - 99 \nu^{14} + 143 \nu^{13} - 302 \nu^{12} + 446 \nu^{11} - 343 \nu^{10} + 223 \nu^{9} + \cdots - 6144 ) / 160 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17 \nu^{15} - 42 \nu^{14} + 66 \nu^{13} - 138 \nu^{12} + 195 \nu^{11} - 144 \nu^{10} + 84 \nu^{9} + \cdots - 2432 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 66 \nu^{15} + 177 \nu^{14} - 270 \nu^{13} + 524 \nu^{12} - 740 \nu^{11} + 559 \nu^{10} + \cdots + 10144 ) / 160 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( - 2 \beta_{15} - 3 \beta_{14} + 2 \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{7} - 3 \beta_{5} + \cdots + 6 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{8} - 2\beta_{4} - \beta_{3} + 2\beta_{2} + 3\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} + 5 \beta_{14} + 5 \beta_{13} + 5 \beta_{12} + 5 \beta_{11} - 11 \beta_{10} + 2 \beta_{9} + \cdots + 24 ) / 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{11} + 3\beta_{10} - 10\beta_{9} - 3\beta_{7} - 12\beta_{5} + 2 ) / 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7 \beta_{15} + 13 \beta_{14} - 5 \beta_{13} - 5 \beta_{12} + 5 \beta_{11} + 5 \beta_{10} - 22 \beta_{9} + \cdots - 68 ) / 40 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12\beta_{14} - 5\beta_{13} - 5\beta_{8} + 70\beta_{4} + 15\beta_{2} ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6 \beta_{15} + 14 \beta_{14} - 20 \beta_{13} - 15 \beta_{12} - 15 \beta_{11} + \beta_{10} + \cdots + 28 ) / 20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9\beta_{15} - 5\beta_{11} + 18\beta_{10} - 19\beta_{9} + 6\beta_{7} + 25\beta_{6} + 8\beta_{5} - 64 ) / 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 61 \beta_{15} + 25 \beta_{14} + 45 \beta_{13} - 55 \beta_{12} + 55 \beta_{11} + 3 \beta_{10} - 126 \beta_{9} + \cdots + 8 ) / 40 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 16\beta_{14} + 5\beta_{13} - 40\beta_{12} + 7\beta_{8} + 134\beta_{4} + 282\beta_{3} - 19\beta_{2} - 26\beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 91 \beta_{15} + 141 \beta_{14} - 95 \beta_{13} - 105 \beta_{12} - 105 \beta_{11} + 175 \beta_{10} + \cdots + 496 ) / 40 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 8\beta_{15} + 27\beta_{11} + 13\beta_{10} + 16\beta_{9} + 27\beta_{7} - 8\beta_{6} + 4\beta_{5} + 114 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 117 \beta_{15} - 118 \beta_{14} - 75 \beta_{13} - 35 \beta_{12} + 35 \beta_{11} + 77 \beta_{10} + \cdots + 686 ) / 20 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 65\beta_{14} - 90\beta_{13} - 15\beta_{12} - 16\beta_{8} - 212\beta_{4} + 84\beta_{3} - 93\beta_{2} + 58\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 453 \beta_{15} + 335 \beta_{14} + 75 \beta_{13} + 435 \beta_{12} + 435 \beta_{11} + 99 \beta_{10} + \cdots + 64 ) / 40 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−0.485936 − 1.32811i
−0.485936 + 1.32811i
1.21868 − 0.717516i
1.21868 + 0.717516i
1.26238 − 0.637499i
1.26238 + 0.637499i
−0.152947 − 1.40592i
−0.152947 + 1.40592i
1.40592 + 0.152947i
1.40592 − 0.152947i
−0.637499 + 1.26238i
−0.637499 − 1.26238i
0.717516 − 1.21868i
0.717516 + 1.21868i
−1.32811 − 0.485936i
−1.32811 + 0.485936i
−1.99209 − 0.177680i −1.73205 3.93686 + 0.707911i 0 3.45040 + 0.307751i −1.19501 −7.71680 − 2.10973i 3.00000 0
199.2 −1.99209 + 0.177680i −1.73205 3.93686 − 0.707911i 0 3.45040 − 0.307751i −1.19501 −7.71680 + 2.10973i 3.00000 0
199.3 −1.92737 − 0.534079i 1.73205 3.42952 + 2.05874i 0 −3.33830 − 0.925051i 11.9716 −5.51043 − 5.79958i 3.00000 0
199.4 −1.92737 + 0.534079i 1.73205 3.42952 − 2.05874i 0 −3.33830 + 0.925051i 11.9716 −5.51043 + 5.79958i 3.00000 0
199.5 −1.49110 − 1.33290i 1.73205 0.446749 + 3.97497i 0 −2.58266 − 2.30865i −6.56834 4.63210 − 6.52255i 3.00000 0
199.6 −1.49110 + 1.33290i 1.73205 0.446749 − 3.97497i 0 −2.58266 + 2.30865i −6.56834 4.63210 + 6.52255i 3.00000 0
199.7 −0.305673 − 1.97650i 1.73205 −3.81313 + 1.20833i 0 −0.529441 − 3.42340i 0.329898 3.55383 + 7.16731i 3.00000 0
199.8 −0.305673 + 1.97650i 1.73205 −3.81313 − 1.20833i 0 −0.529441 + 3.42340i 0.329898 3.55383 − 7.16731i 3.00000 0
199.9 0.305673 − 1.97650i −1.73205 −3.81313 − 1.20833i 0 −0.529441 + 3.42340i −0.329898 −3.55383 + 7.16731i 3.00000 0
199.10 0.305673 + 1.97650i −1.73205 −3.81313 + 1.20833i 0 −0.529441 − 3.42340i −0.329898 −3.55383 − 7.16731i 3.00000 0
199.11 1.49110 − 1.33290i −1.73205 0.446749 − 3.97497i 0 −2.58266 + 2.30865i 6.56834 −4.63210 − 6.52255i 3.00000 0
199.12 1.49110 + 1.33290i −1.73205 0.446749 + 3.97497i 0 −2.58266 − 2.30865i 6.56834 −4.63210 + 6.52255i 3.00000 0
199.13 1.92737 − 0.534079i −1.73205 3.42952 − 2.05874i 0 −3.33830 + 0.925051i −11.9716 5.51043 − 5.79958i 3.00000 0
199.14 1.92737 + 0.534079i −1.73205 3.42952 + 2.05874i 0 −3.33830 − 0.925051i −11.9716 5.51043 + 5.79958i 3.00000 0
199.15 1.99209 − 0.177680i 1.73205 3.93686 − 0.707911i 0 3.45040 − 0.307751i 1.19501 7.71680 − 2.10973i 3.00000 0
199.16 1.99209 + 0.177680i 1.73205 3.93686 + 0.707911i 0 3.45040 + 0.307751i 1.19501 7.71680 + 2.10973i 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.f.c 16
3.b odd 2 1 900.3.f.h 16
4.b odd 2 1 inner 300.3.f.c 16
5.b even 2 1 inner 300.3.f.c 16
5.c odd 4 1 300.3.c.e ✓ 8
5.c odd 4 1 300.3.c.g yes 8
12.b even 2 1 900.3.f.h 16
15.d odd 2 1 900.3.f.h 16
15.e even 4 1 900.3.c.n 8
15.e even 4 1 900.3.c.t 8
20.d odd 2 1 inner 300.3.f.c 16
20.e even 4 1 300.3.c.e ✓ 8
20.e even 4 1 300.3.c.g yes 8
60.h even 2 1 900.3.f.h 16
60.l odd 4 1 900.3.c.n 8
60.l odd 4 1 900.3.c.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.c.e ✓ 8 5.c odd 4 1
300.3.c.e ✓ 8 20.e even 4 1
300.3.c.g yes 8 5.c odd 4 1
300.3.c.g yes 8 20.e even 4 1
300.3.f.c 16 1.a even 1 1 trivial
300.3.f.c 16 4.b odd 2 1 inner
300.3.f.c 16 5.b even 2 1 inner
300.3.f.c 16 20.d odd 2 1 inner
900.3.c.n 8 15.e even 4 1
900.3.c.n 8 60.l odd 4 1
900.3.c.t 8 15.e even 4 1
900.3.c.t 8 60.l odd 4 1
900.3.f.h 16 3.b odd 2 1
900.3.f.h 16 12.b even 2 1
900.3.f.h 16 15.d odd 2 1
900.3.f.h 16 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 188T_{7}^{6} + 6470T_{7}^{4} - 9532T_{7}^{2} + 961 \) acting on \(S_{3}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 8 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 188 T^{6} + \cdots + 961)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 704 T^{6} + \cdots + 30824704)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 852 T^{6} + \cdots + 2430481)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1664 T^{6} + \cdots + 17529760000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1132 T^{6} + \cdots + 1099651921)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 1984 T^{6} + \cdots + 1611540736)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 16 T^{3} + \cdots + 93616)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 5660 T^{6} + \cdots + 819736484449)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 20688524759296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 8 T^{3} + \cdots + 3504448)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 6892 T^{6} + \cdots + 974581609681)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 13610196640000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 24469088997376)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 6192 T^{6} + \cdots + 195562066176)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 68 T^{3} + \cdots + 9745129)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 23066205847441)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 11235904000000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 19\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 2278988775424)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 120362665464064)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 64 T^{3} + \cdots - 1507328)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 243922954789089)^{2} \) Copy content Toggle raw display
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