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Properties

Label 300.7.b.e
Level $300$
Weight $7$
Character orbit 300.b
Analytic conductor $69.016$
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] = [300,7,Mod(149,300)]
 
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, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.149");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 300.b (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: \(69.0162250860\)
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: \( x^{16} + 406 x^{14} + 67561 x^{12} + 5921226 x^{10} + 291565644 x^{8} + 7924637994 x^{6} + \cdots + 276002078881 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{18}\cdot 5^{26} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{5} + \beta_{4}) q^{7} + (\beta_{8} - 187) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{5} + \beta_{4}) q^{7} + (\beta_{8} - 187) q^{9} + (\beta_{14} - \beta_{11} + \beta_{8}) q^{11} + (\beta_{12} + 2 \beta_{7} + \cdots - 4 \beta_{3}) q^{13}+ \cdots + (78 \beta_{15} - 255 \beta_{14} + \cdots - 331212) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2984 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2984 q^{9} + 30544 q^{19} - 1736 q^{21} + 70064 q^{31} - 79216 q^{39} - 405120 q^{49} + 858240 q^{51} - 271216 q^{61} + 509880 q^{69} - 1415408 q^{79} - 2396224 q^{81} - 4008224 q^{91} - 5300160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 406 x^{14} + 67561 x^{12} + 5921226 x^{10} + 291565644 x^{8} + 7924637994 x^{6} + \cdots + 276002078881 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12\!\cdots\!05 \nu^{14} + \cdots + 20\!\cdots\!65 ) / 13\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31\!\cdots\!24 \nu^{14} + \cdots + 54\!\cdots\!07 ) / 17\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30\!\cdots\!07 \nu^{15} + \cdots + 93\!\cdots\!90 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 486211813880 \nu^{15} - 167875593922015 \nu^{13} + \cdots - 58\!\cdots\!95 \nu ) / 74\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 84\!\cdots\!07 \nu^{15} + \cdots - 23\!\cdots\!63 \nu ) / 76\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 21\!\cdots\!49 \nu^{15} + \cdots + 11\!\cdots\!10 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!26 \nu^{15} + \cdots - 46\!\cdots\!50 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!96 \nu^{15} + \cdots - 74\!\cdots\!28 ) / 35\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 43\!\cdots\!97 \nu^{15} + \cdots - 62\!\cdots\!00 ) / 11\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 42\!\cdots\!31 \nu^{15} + \cdots - 15\!\cdots\!90 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 28\!\cdots\!25 \nu^{15} + \cdots - 40\!\cdots\!80 ) / 71\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 38\!\cdots\!99 \nu^{15} + \cdots + 25\!\cdots\!30 ) / 80\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!71 \nu^{15} + \cdots + 38\!\cdots\!20 ) / 35\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!99 \nu^{15} + \cdots + 10\!\cdots\!76 ) / 71\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 88\!\cdots\!54 \nu^{15} + \cdots + 16\!\cdots\!08 ) / 35\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 339 \beta_{15} - 561 \beta_{14} - 855 \beta_{13} - 950 \beta_{12} - 1167 \beta_{11} - 330 \beta_{9} + \cdots - 411 ) / 270000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 465 \beta_{15} + 40 \beta_{14} - 850 \beta_{13} + 1244 \beta_{12} + 890 \beta_{11} + \cdots - 6851210 ) / 135000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 29217 \beta_{15} + 33483 \beta_{14} + 57765 \beta_{13} + 39950 \beta_{12} + 86901 \beta_{11} + \cdots + 41733 ) / 270000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 21855 \beta_{15} + 2620 \beta_{14} + 48950 \beta_{13} - 60453 \beta_{12} - 46330 \beta_{11} + \cdots + 250714495 ) / 67500 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2699667 \beta_{15} - 3510433 \beta_{14} - 4850615 \beta_{13} - 2092750 \beta_{12} + \cdots - 4169083 ) / 270000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4261155 \beta_{15} - 822320 \beta_{14} - 10166950 \beta_{13} + 9589078 \beta_{12} + \cdots - 40463629820 ) / 135000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 50272101 \beta_{15} + 75351319 \beta_{14} + 89164625 \beta_{13} + 23604290 \beta_{12} + \cdots + 81694189 ) / 54000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 69224880 \beta_{15} + 21207220 \beta_{14} + 180864200 \beta_{13} - 116217423 \beta_{12} + \cdots + 573075962845 ) / 22500 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 23543580147 \beta_{15} - 39230971953 \beta_{14} - 42709222815 \beta_{13} - 6979773050 \beta_{12} + \cdots - 40020821703 ) / 270000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 7979228265 \beta_{15} - 3719985160 \beta_{14} - 23398426850 \beta_{13} + 9382226752 \beta_{12} + \cdots - 60473560985410 ) / 27000 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2221526278353 \beta_{15} + 3994892201547 \beta_{14} + 4175505191085 \beta_{13} + \cdots + 3942081164697 ) / 270000 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1891483563315 \beta_{15} + 1245313063360 \beta_{14} + 6273593253350 \beta_{13} + \cdots + 13\!\cdots\!60 ) / 67500 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 211290899286339 \beta_{15} - 401297883975361 \beta_{14} - 412273074365255 \beta_{13} + \cdots - 390467511207211 ) / 270000 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 355620497542395 \beta_{15} - 310094154421880 \beta_{14} + \cdots - 25\!\cdots\!30 ) / 135000 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 20\!\cdots\!09 \beta_{15} + \cdots + 38\!\cdots\!41 ) / 270000 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
− 4.99240i
4.99240i
8.61329i
− 8.61329i
9.98918i
− 9.98918i
0.745166i
− 0.745166i
6.94307i
− 6.94307i
− 7.89868i
7.89868i
3.23723i
− 3.23723i
9.24517i
− 9.24517i
0 −24.5453 − 11.2485i 0 0 0 − 414.697i 0 475.944 + 552.194i 0
149.2 0 −24.5453 + 11.2485i 0 0 0 414.697i 0 475.944 − 552.194i 0
149.3 0 −15.3993 − 22.1779i 0 0 0 − 437.053i 0 −254.720 + 683.051i 0
149.4 0 −15.3993 + 22.1779i 0 0 0 437.053i 0 −254.720 − 683.051i 0
149.5 0 −12.0655 − 24.1542i 0 0 0 436.815i 0 −437.848 + 582.864i 0
149.6 0 −12.0655 + 24.1542i 0 0 0 − 436.815i 0 −437.848 − 582.864i 0
149.7 0 −9.99060 − 25.0836i 0 0 0 134.460i 0 −529.376 + 501.201i 0
149.8 0 −9.99060 + 25.0836i 0 0 0 − 134.460i 0 −529.376 − 501.201i 0
149.9 0 9.99060 − 25.0836i 0 0 0 134.460i 0 −529.376 − 501.201i 0
149.10 0 9.99060 + 25.0836i 0 0 0 − 134.460i 0 −529.376 + 501.201i 0
149.11 0 12.0655 − 24.1542i 0 0 0 436.815i 0 −437.848 − 582.864i 0
149.12 0 12.0655 + 24.1542i 0 0 0 − 436.815i 0 −437.848 + 582.864i 0
149.13 0 15.3993 − 22.1779i 0 0 0 − 437.053i 0 −254.720 − 683.051i 0
149.14 0 15.3993 + 22.1779i 0 0 0 437.053i 0 −254.720 + 683.051i 0
149.15 0 24.5453 − 11.2485i 0 0 0 − 414.697i 0 475.944 − 552.194i 0
149.16 0 24.5453 + 11.2485i 0 0 0 414.697i 0 475.944 + 552.194i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.7.b.e 16
3.b odd 2 1 inner 300.7.b.e 16
5.b even 2 1 inner 300.7.b.e 16
5.c odd 4 1 60.7.g.a ✓ 8
5.c odd 4 1 300.7.g.h 8
15.d odd 2 1 inner 300.7.b.e 16
15.e even 4 1 60.7.g.a ✓ 8
15.e even 4 1 300.7.g.h 8
20.e even 4 1 240.7.l.c 8
60.l odd 4 1 240.7.l.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.g.a ✓ 8 5.c odd 4 1
60.7.g.a ✓ 8 15.e even 4 1
240.7.l.c 8 20.e even 4 1
240.7.l.c 8 60.l odd 4 1
300.7.b.e 16 1.a even 1 1 trivial
300.7.b.e 16 3.b odd 2 1 inner
300.7.b.e 16 5.b even 2 1 inner
300.7.b.e 16 15.d odd 2 1 inner
300.7.g.h 8 5.c odd 4 1
300.7.g.h 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 571876T_{7}^{6} + 112122973716T_{7}^{4} + 8114052473491936T_{7}^{2} + 113320643614501228096 \) acting on \(S_{7}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 79\!\cdots\!61 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 68\!\cdots\!36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 11\!\cdots\!36)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 43\!\cdots\!56)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 25\!\cdots\!36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 67\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 48\!\cdots\!76)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 87\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 89\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 53\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 63\!\cdots\!36)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 37\!\cdots\!76)^{2} \) Copy content Toggle raw display
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