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Properties

Label 300.7.k.d
Level $300$
Weight $7$
Character orbit 300.k
Analytic conductor $69.016$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,7,Mod(157,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.157");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 300.k (of order \(4\), degree \(2\), 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: \(69.0162250860\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 66x^{10} + 1601x^{8} + 17520x^{6} + 84208x^{4} + 136704x^{2} + 14400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{10}\cdot 5^{8} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + (\beta_{10} - \beta_{7} - 5 \beta_{4} + \cdots - 10) q^{7}+ \cdots + 243 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + (\beta_{10} - \beta_{7} - 5 \beta_{4} + \cdots - 10) q^{7}+ \cdots + (729 \beta_{11} + \cdots - 65610 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 120 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 120 q^{7} - 3248 q^{11} + 2100 q^{13} + 5540 q^{17} - 15552 q^{21} + 23840 q^{23} - 127152 q^{31} + 35640 q^{33} - 282900 q^{37} - 320720 q^{41} + 62880 q^{43} - 381600 q^{47} - 145152 q^{51} + 400300 q^{53} + 38880 q^{57} + 807024 q^{61} - 29160 q^{63} - 752160 q^{67} + 202400 q^{71} + 322020 q^{73} + 2448400 q^{77} - 708588 q^{81} - 1894560 q^{83} + 1007640 q^{87} + 2294400 q^{91} - 835920 q^{93} + 3161700 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 66x^{10} + 1601x^{8} + 17520x^{6} + 84208x^{4} + 136704x^{2} + 14400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -1087\nu^{10} - 70434\nu^{8} - 1615175\nu^{6} - 15123164\nu^{4} - 43859608\nu^{2} + 19568256 ) / 95264 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -19\nu^{11} - 1185\nu^{9} - 26441\nu^{7} - 259203\nu^{5} - 1142320\nu^{3} - 2087592\nu ) / 659520 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3789\nu^{10} + 210678\nu^{8} + 3888117\nu^{6} + 26291892\nu^{4} + 42596424\nu^{2} - 26880800 ) / 95264 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 687 \nu^{11} - 1380 \nu^{10} - 53586 \nu^{9} - 79560 \nu^{8} - 1528575 \nu^{7} + \cdots - 49000320 ) / 3810560 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 687 \nu^{11} + 1380 \nu^{10} - 53586 \nu^{9} + 79560 \nu^{8} - 1528575 \nu^{7} + \cdots + 49000320 ) / 3810560 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12977 \nu^{11} + 39900 \nu^{10} + 742716 \nu^{9} + 2145000 \nu^{8} + 14326045 \nu^{7} + \cdots + 236551680 ) / 5715840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12977 \nu^{11} - 39900 \nu^{10} + 742716 \nu^{9} - 2145000 \nu^{8} + 14326045 \nu^{7} + \cdots - 236551680 ) / 5715840 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13177 \nu^{11} - 876174 \nu^{9} - 22728761 \nu^{7} - 293515356 \nu^{5} - 1836484312 \nu^{3} - 3562487520 \nu ) / 3429504 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 34217 \nu^{11} - 1880268 \nu^{9} - 32535349 \nu^{7} - 152524938 \nu^{5} + 651418456 \nu^{3} + 4418541456 \nu ) / 5715840 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 394373 \nu^{11} + 71820 \nu^{10} + 25238694 \nu^{9} + 6004440 \nu^{8} + 588692725 \nu^{7} + \cdots + 13077233280 ) / 34295040 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 394373 \nu^{11} + 71820 \nu^{10} - 25238694 \nu^{9} + 6004440 \nu^{8} - 588692725 \nu^{7} + \cdots + 13077233280 ) / 34295040 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{9} - \beta_{8} - 6\beta_{7} - 6\beta_{6} + 4\beta_{5} + 4\beta_{4} - 239\beta_{2} ) / 720 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{11} + 3\beta_{10} - 25\beta_{5} + 25\beta_{4} + \beta_{3} + 3\beta _1 - 1979 ) / 180 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{11} + 9 \beta_{10} + 48 \beta_{9} + 28 \beta_{8} + 60 \beta_{7} + 60 \beta_{6} + \cdots + 1544 \beta_{2} ) / 360 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 12 \beta_{11} - 12 \beta_{10} - 3 \beta_{7} + 3 \beta_{6} + 152 \beta_{5} - 152 \beta_{4} + \cdots + 5766 ) / 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 84 \beta_{11} - 84 \beta_{10} - 291 \beta_{9} - 199 \beta_{8} - 315 \beta_{7} - 315 \beta_{6} + \cdots - 17333 \beta_{2} ) / 90 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 846 \beta_{11} + 846 \beta_{10} + 393 \beta_{7} - 393 \beta_{6} - 11742 \beta_{5} + 11742 \beta_{4} + \cdots - 345578 ) / 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2460 \beta_{11} + 2460 \beta_{10} + 6864 \beta_{9} + 4934 \beta_{8} + 6837 \beta_{7} + \cdots + 603382 \beta_{2} ) / 90 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6582 \beta_{11} - 6582 \beta_{10} - 4275 \beta_{7} + 4275 \beta_{6} + 93614 \beta_{5} + \cdots + 2466354 ) / 30 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 66186 \beta_{11} - 66186 \beta_{10} - 161520 \beta_{9} - 117470 \beta_{8} - 152157 \beta_{7} + \cdots - 17971750 \beta_{2} ) / 90 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 462738 \beta_{11} + 462738 \beta_{10} + 372273 \beta_{7} - 372273 \beta_{6} - 6590026 \beta_{5} + \cdots - 165056902 ) / 90 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1715178 \beta_{11} + 1715178 \beta_{10} + 3811224 \beta_{9} + 2767798 \beta_{8} + \cdots + 494596862 \beta_{2} ) / 90 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
− 4.27595i
4.93976i
0.336188i
− 3.51909i
1.70867i
2.81042i
4.27595i
− 4.93976i
− 0.336188i
3.51909i
− 1.70867i
− 2.81042i
0 −11.0227 + 11.0227i 0 0 0 −337.242 − 337.242i 0 − 243.000i 0
157.2 0 −11.0227 + 11.0227i 0 0 0 49.0142 + 49.0142i 0 − 243.000i 0
157.3 0 −11.0227 + 11.0227i 0 0 0 434.591 + 434.591i 0 − 243.000i 0
157.4 0 11.0227 − 11.0227i 0 0 0 −312.733 − 312.733i 0 − 243.000i 0
157.5 0 11.0227 − 11.0227i 0 0 0 26.9845 + 26.9845i 0 − 243.000i 0
157.6 0 11.0227 − 11.0227i 0 0 0 79.3849 + 79.3849i 0 − 243.000i 0
193.1 0 −11.0227 − 11.0227i 0 0 0 −337.242 + 337.242i 0 243.000i 0
193.2 0 −11.0227 − 11.0227i 0 0 0 49.0142 − 49.0142i 0 243.000i 0
193.3 0 −11.0227 − 11.0227i 0 0 0 434.591 − 434.591i 0 243.000i 0
193.4 0 11.0227 + 11.0227i 0 0 0 −312.733 + 312.733i 0 243.000i 0
193.5 0 11.0227 + 11.0227i 0 0 0 26.9845 − 26.9845i 0 243.000i 0
193.6 0 11.0227 + 11.0227i 0 0 0 79.3849 − 79.3849i 0 243.000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.7.k.d 12
5.b even 2 1 60.7.k.a ✓ 12
5.c odd 4 1 60.7.k.a ✓ 12
5.c odd 4 1 inner 300.7.k.d 12
15.d odd 2 1 180.7.l.b 12
15.e even 4 1 180.7.l.b 12
20.d odd 2 1 240.7.bg.d 12
20.e even 4 1 240.7.bg.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.k.a ✓ 12 5.b even 2 1
60.7.k.a ✓ 12 5.c odd 4 1
180.7.l.b 12 15.d odd 2 1
180.7.l.b 12 15.e even 4 1
240.7.bg.d 12 20.d odd 2 1
240.7.bg.d 12 20.e even 4 1
300.7.k.d 12 1.a even 1 1 trivial
300.7.k.d 12 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 120 T_{7}^{11} + 7200 T_{7}^{10} + 18658160 T_{7}^{9} + 118648882632 T_{7}^{8} + \cdots + 14\!\cdots\!84 \) acting on \(S_{7}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} + 59049)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 15\!\cdots\!04)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 61\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 20\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 22\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 56\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 91\!\cdots\!64 \) Copy content Toggle raw display
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