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Properties

Label 300.3.f.b
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: \( x^{16} + 5x^{14} + 12x^{12} + 25x^{10} + 53x^{8} + 100x^{6} + 192x^{4} + 320x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} - \beta_{9} q^{3} + (\beta_{4} - 1) q^{4} + ( - \beta_{2} - 1) q^{6} + (\beta_{15} - \beta_{14} + \cdots + \beta_{9}) q^{7}+ \cdots + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{2} - \beta_{9} q^{3} + (\beta_{4} - 1) q^{4} + ( - \beta_{2} - 1) q^{6} + (\beta_{15} - \beta_{14} + \cdots + \beta_{9}) q^{7}+ \cdots + (6 \beta_{10} + 3 \beta_{7} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 20 q^{4} - 12 q^{6} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 20 q^{4} - 12 q^{6} + 48 q^{9} + 40 q^{14} + 68 q^{16} - 96 q^{21} - 36 q^{24} - 72 q^{26} - 128 q^{29} + 184 q^{34} - 60 q^{36} - 32 q^{41} - 344 q^{44} + 304 q^{46} + 112 q^{49} - 36 q^{54} + 232 q^{56} - 352 q^{61} + 220 q^{64} + 216 q^{66} + 192 q^{69} - 264 q^{74} - 48 q^{76} + 144 q^{81} + 72 q^{84} - 400 q^{86} - 160 q^{89} + 192 q^{94} - 348 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 5x^{14} + 12x^{12} + 25x^{10} + 53x^{8} + 100x^{6} + 192x^{4} + 320x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{14} - \nu^{12} - 8\nu^{10} - 25\nu^{8} - 17\nu^{6} - 96\nu^{4} - 160\nu^{2} - 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{14} - 3\nu^{12} + 16\nu^{10} + 5\nu^{8} - 67\nu^{6} + 24\nu^{4} + 248\nu^{2} - 336 ) / 304 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -17\nu^{14} - 81\nu^{12} - 176\nu^{10} - 321\nu^{8} - 593\nu^{6} - 1176\nu^{4} - 2120\nu^{2} - 3296 ) / 304 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{14} - 16\nu^{12} - 35\nu^{10} - 81\nu^{8} - 180\nu^{6} - 271\nu^{4} - 552\nu^{2} - 804 ) / 76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 165\nu^{14} + 357\nu^{12} + 680\nu^{10} + 1533\nu^{8} + 2805\nu^{6} + 4896\nu^{4} + 8640\nu^{2} + 9280 ) / 2432 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41\nu^{15} + 97\nu^{13} + 192\nu^{11} + 193\nu^{9} + 241\nu^{7} - 1384\nu^{5} + 1760\nu^{3} + 832\nu ) / 2432 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 267\nu^{14} + 843\nu^{12} + 1432\nu^{10} + 3763\nu^{8} + 6971\nu^{6} + 11040\nu^{4} + 25920\nu^{2} + 32704 ) / 2432 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 71\nu^{15} + 231\nu^{13} + 440\nu^{11} + 1135\nu^{9} + 1815\nu^{7} + 3168\nu^{5} + 7200\nu^{3} + 9152\nu ) / 2432 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\nu^{15} + 47\nu^{13} + 88\nu^{11} + 215\nu^{9} + 479\nu^{7} + 704\nu^{5} + 1536\nu^{3} + 1984\nu ) / 512 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 407 \nu^{14} - 1367 \nu^{12} - 2488 \nu^{10} - 5727 \nu^{8} - 11783 \nu^{6} - 20832 \nu^{4} + \cdots - 56128 ) / 2432 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 353 \nu^{15} - 1217 \nu^{13} - 2984 \nu^{11} - 4761 \nu^{9} - 10257 \nu^{7} - 19904 \nu^{5} + \cdots - 53312 \nu ) / 9728 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -103\nu^{15} - 303\nu^{13} - 512\nu^{11} - 1167\nu^{9} - 2207\nu^{7} - 4264\nu^{5} - 8544\nu^{3} - 9920\nu ) / 2432 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 449 \nu^{15} - 1889 \nu^{13} - 3048 \nu^{11} - 7289 \nu^{9} - 15537 \nu^{7} - 27904 \nu^{5} + \cdots - 99392 \nu ) / 9728 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 527 \nu^{15} + 1295 \nu^{13} + 2872 \nu^{11} + 5847 \nu^{9} + 10175 \nu^{7} + 21408 \nu^{5} + \cdots + 48064 \nu ) / 9728 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 427 \nu^{15} + 1051 \nu^{13} + 2248 \nu^{11} + 5139 \nu^{9} + 9995 \nu^{7} + 19408 \nu^{5} + \cdots + 56000 \nu ) / 4864 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{15} - 2\beta_{14} - \beta_{13} + \beta_{12} - 3\beta_{9} + \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} + \beta_{7} - 4\beta_{5} + \beta_{3} + 4\beta_{2} + \beta _1 - 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} + 3\beta_{14} - \beta_{13} + 2\beta_{12} + \beta_{11} + \beta_{9} - \beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{10} - 7\beta_{7} + 4\beta_{5} + 4\beta_{4} + \beta_{3} - 6\beta_{2} - 5\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{13} - 7\beta_{12} - 2\beta_{11} - 5\beta_{9} - 2\beta_{8} - 7\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{7} - 6\beta_{4} + 2\beta_{3} - \beta_{2} + 3\beta _1 - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2\beta_{15} - 2\beta_{14} + 3\beta_{13} + 5\beta_{12} + 41\beta_{9} - 20\beta_{8} + \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 21\beta_{10} + 27\beta_{7} + 4\beta_{5} - 5\beta_{3} - 20\beta_{2} - 21\beta _1 - 12 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( \beta_{15} - 3\beta_{14} + 5\beta_{13} + 10\beta_{12} + 7\beta_{11} - 5\beta_{9} + 41\beta_{8} - 4\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 21\beta_{10} - 9\beta_{7} - 4\beta_{5} - 36\beta_{4} - 33\beta_{3} + 54\beta_{2} + 21\beta _1 - 16 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 23\beta_{13} + 47\beta_{12} - 78\beta_{11} - 19\beta_{9} + 18\beta_{8} + 15\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -12\beta_{10} + 14\beta_{7} + 54\beta_{4} - 14\beta_{3} - 31\beta_{2} + 17\beta _1 - 46 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -94\beta_{15} - 30\beta_{14} - 67\beta_{13} - 213\beta_{12} - 73\beta_{9} - 12\beta_{8} + 15\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -29\beta_{10} - 51\beta_{7} + 188\beta_{5} + 93\beta_{3} + 100\beta_{2} + 29\beta _1 + 476 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 135 \beta_{15} - 53 \beta_{14} + 63 \beta_{13} - 134 \beta_{12} + 73 \beta_{11} - 63 \beta_{9} + \cdots + 72 \beta_{6} ) / 4 \) 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
1.28061 − 0.600040i
1.28061 + 0.600040i
−0.120653 + 1.40906i
−0.120653 − 1.40906i
0.422403 − 1.34966i
0.422403 + 1.34966i
−0.957636 − 1.04064i
−0.957636 + 1.04064i
0.957636 − 1.04064i
0.957636 + 1.04064i
−0.422403 − 1.34966i
−0.422403 + 1.34966i
0.120653 + 1.40906i
0.120653 − 1.40906i
−1.28061 − 0.600040i
−1.28061 + 0.600040i
−1.95141 − 0.438172i 1.73205 3.61601 + 1.71011i 0 −3.37994 − 0.758935i −6.33166 −6.30701 − 4.92155i 3.00000 0
199.2 −1.95141 + 0.438172i 1.73205 3.61601 − 1.71011i 0 −3.37994 + 0.758935i −6.33166 −6.30701 + 4.92155i 3.00000 0
199.3 −1.08539 − 1.67986i −1.73205 −1.64388 + 3.64660i 0 1.87994 + 2.90961i 0.596540 7.91002 − 1.19648i 3.00000 0
199.4 −1.08539 + 1.67986i −1.73205 −1.64388 − 3.64660i 0 1.87994 − 2.90961i 0.596540 7.91002 + 1.19648i 3.00000 0
199.5 −0.696577 − 1.87477i 1.73205 −3.02956 + 2.61185i 0 −1.20651 − 3.24721i 5.46770 7.00695 + 3.86039i 3.00000 0
199.6 −0.696577 + 1.87477i 1.73205 −3.02956 − 2.61185i 0 −1.20651 + 3.24721i 5.46770 7.00695 − 3.86039i 3.00000 0
199.7 −0.169449 − 1.99281i 1.73205 −3.94257 + 0.675358i 0 −0.293494 − 3.45165i −12.3959 2.01392 + 7.74236i 3.00000 0
199.8 −0.169449 + 1.99281i 1.73205 −3.94257 − 0.675358i 0 −0.293494 + 3.45165i −12.3959 2.01392 − 7.74236i 3.00000 0
199.9 0.169449 − 1.99281i −1.73205 −3.94257 − 0.675358i 0 −0.293494 + 3.45165i 12.3959 −2.01392 + 7.74236i 3.00000 0
199.10 0.169449 + 1.99281i −1.73205 −3.94257 + 0.675358i 0 −0.293494 − 3.45165i 12.3959 −2.01392 − 7.74236i 3.00000 0
199.11 0.696577 − 1.87477i −1.73205 −3.02956 − 2.61185i 0 −1.20651 + 3.24721i −5.46770 −7.00695 + 3.86039i 3.00000 0
199.12 0.696577 + 1.87477i −1.73205 −3.02956 + 2.61185i 0 −1.20651 − 3.24721i −5.46770 −7.00695 − 3.86039i 3.00000 0
199.13 1.08539 − 1.67986i 1.73205 −1.64388 − 3.64660i 0 1.87994 − 2.90961i −0.596540 −7.91002 − 1.19648i 3.00000 0
199.14 1.08539 + 1.67986i 1.73205 −1.64388 + 3.64660i 0 1.87994 + 2.90961i −0.596540 −7.91002 + 1.19648i 3.00000 0
199.15 1.95141 − 0.438172i −1.73205 3.61601 − 1.71011i 0 −3.37994 + 0.758935i 6.33166 6.30701 − 4.92155i 3.00000 0
199.16 1.95141 + 0.438172i −1.73205 3.61601 + 1.71011i 0 −3.37994 − 0.758935i 6.33166 6.30701 + 4.92155i 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.b 16
3.b odd 2 1 900.3.f.f 16
4.b odd 2 1 inner 300.3.f.b 16
5.b even 2 1 inner 300.3.f.b 16
5.c odd 4 1 60.3.c.a ✓ 8
5.c odd 4 1 300.3.c.d 8
12.b even 2 1 900.3.f.f 16
15.d odd 2 1 900.3.f.f 16
15.e even 4 1 180.3.c.b 8
15.e even 4 1 900.3.c.u 8
20.d odd 2 1 inner 300.3.f.b 16
20.e even 4 1 60.3.c.a ✓ 8
20.e even 4 1 300.3.c.d 8
40.i odd 4 1 960.3.e.c 8
40.k even 4 1 960.3.e.c 8
60.h even 2 1 900.3.f.f 16
60.l odd 4 1 180.3.c.b 8
60.l odd 4 1 900.3.c.u 8
120.q odd 4 1 2880.3.e.j 8
120.w even 4 1 2880.3.e.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.c.a ✓ 8 5.c odd 4 1
60.3.c.a ✓ 8 20.e even 4 1
180.3.c.b 8 15.e even 4 1
180.3.c.b 8 60.l odd 4 1
300.3.c.d 8 5.c odd 4 1
300.3.c.d 8 20.e even 4 1
300.3.f.b 16 1.a even 1 1 trivial
300.3.f.b 16 4.b odd 2 1 inner
300.3.f.b 16 5.b even 2 1 inner
300.3.f.b 16 20.d odd 2 1 inner
900.3.c.u 8 15.e even 4 1
900.3.c.u 8 60.l odd 4 1
900.3.f.f 16 3.b odd 2 1
900.3.f.f 16 12.b even 2 1
900.3.f.f 16 15.d odd 2 1
900.3.f.f 16 60.h even 2 1
960.3.e.c 8 40.i odd 4 1
960.3.e.c 8 40.k even 4 1
2880.3.e.j 8 120.q odd 4 1
2880.3.e.j 8 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 224T_{7}^{6} + 12032T_{7}^{4} - 188416T_{7}^{2} + 65536 \) acting on \(S_{3}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 10 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 224 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 208 T^{2} + 10496)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 1008 T^{6} + \cdots + 155351296)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 848 T^{6} + \cdots + 77721856)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1696 T^{6} + \cdots + 6544162816)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 3616 T^{6} + \cdots + 101419319296)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 32 T^{3} + \cdots + 1334416)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 5408 T^{6} + \cdots + 59895709696)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 6192 T^{6} + \cdots + 59919206656)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 8 T^{3} + \cdots + 87184)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 33624411406336)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 1056981385216)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 11472 T^{6} + \cdots + 228545188096)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 4896 T^{6} + \cdots + 173909016576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 88 T^{3} + \cdots - 2142704)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 16064 T^{6} + \cdots + 281086590976)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 16079971680256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 24622079140096)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 31\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 42\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 40 T^{3} + \cdots + 70652944)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 35\!\cdots\!76)^{2} \) Copy content Toggle raw display
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