Properties

Label 300.7.k.b
Level $300$
Weight $7$
Character orbit 300.k
Analytic conductor $69.016$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.469950251728896.64
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 98x^{6} - 12x^{5} + 3571x^{4} + 576x^{3} + 53874x^{2} + 20916x + 343098 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 \beta_1 q^{3} + ( - \beta_{6} - 7 \beta_{3}) q^{7} + 243 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 \beta_1 q^{3} + ( - \beta_{6} - 7 \beta_{3}) q^{7} + 243 \beta_{2} q^{9} + (2 \beta_{7} + 606) q^{11} + (7 \beta_{4} - 375 \beta_1) q^{13} + (18 \beta_{6} - 1242 \beta_{3}) q^{17} + ( - 8 \beta_{5} - 3695 \beta_{2}) q^{19} + (9 \beta_{7} + 189) q^{21} + ( - 24 \beta_{4} + 5226 \beta_1) q^{23} + 2187 \beta_{3} q^{27} + (40 \beta_{5} - 3318 \beta_{2}) q^{29} + ( - 51 \beta_{7} - 2051) q^{31} + (54 \beta_{4} + 5454 \beta_1) q^{33} + ( - 118 \beta_{6} - 26396 \beta_{3}) q^{37} + ( - 63 \beta_{5} - 10125 \beta_{2}) q^{39} + (82 \beta_{7} + 32460) q^{41} + ( - 72 \beta_{4} + 29591 \beta_1) q^{43} + ( - 78 \beta_{6} - 32898 \beta_{3}) q^{47} + (14 \beta_{5} + 1102 \beta_{2}) q^{49} + ( - 162 \beta_{7} + 33534) q^{51} + ( - 210 \beta_{4} + 20892 \beta_1) q^{53} + (216 \beta_{6} - 33255 \beta_{3}) q^{57} + ( - 4 \beta_{5} + 52242 \beta_{2}) q^{59} + (67 \beta_{7} + 105647) q^{61} + (243 \beta_{4} + 1701 \beta_1) q^{63} + (938 \beta_{6} - 49413 \beta_{3}) q^{67} + (216 \beta_{5} + 141102 \beta_{2}) q^{69} + (490 \beta_{7} + 74220) q^{71} + (1144 \beta_{4} + 4148 \beta_1) q^{73} + ( - 648 \beta_{6} - 237042 \beta_{3}) q^{77} + ( - 272 \beta_{5} + 757814 \beta_{2}) q^{79} - 59049 q^{81} + ( - 804 \beta_{4} - 38274 \beta_1) q^{83} + ( - 1080 \beta_{6} - 29862 \beta_{3}) q^{87} + ( - 76 \beta_{5} + 1066740 \beta_{2}) q^{89} + ( - 326 \beta_{7} + 806925) q^{91} + ( - 1377 \beta_{4} - 18459 \beta_1) q^{93} + ( - 1538 \beta_{6} - 477029 \beta_{3}) q^{97} + ( - 486 \beta_{5} + 147258 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4848 q^{11} + 1512 q^{21} - 16408 q^{31} + 259680 q^{41} + 268272 q^{51} + 845176 q^{61} + 593760 q^{71} - 472392 q^{81} + 6455400 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 98x^{6} - 12x^{5} + 3571x^{4} + 576x^{3} + 53874x^{2} + 20916x + 343098 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1992230 \nu^{7} + 6919103 \nu^{6} + 273872758 \nu^{5} + 823212151 \nu^{4} + 13841503414 \nu^{3} + 28892850095 \nu^{2} + \cdots + 341852125167 ) / 195543175135 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1490 \nu^{7} - 4586 \nu^{6} + 109894 \nu^{5} - 575137 \nu^{4} + 2716082 \nu^{3} - 17767115 \nu^{2} + 15960150 \nu - 174345129 ) / 61407885 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 40406470 \nu^{7} + 4736544 \nu^{6} - 2908689491 \nu^{5} + 1002126303 \nu^{4} - 66384131413 \nu^{3} - 19303151040 \nu^{2} + \cdots - 97724147409 ) / 586629525405 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 488 \nu^{7} + 290524 \nu^{6} + 29712 \nu^{5} + 21402716 \nu^{4} - 4687192 \nu^{3} + 513139420 \nu^{2} + 124879372 \nu + 3235482684 ) / 4093859 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14964200 \nu^{7} - 1311680 \nu^{6} + 1538887960 \nu^{5} - 317149060 \nu^{4} + 62283286760 \nu^{3} + 2073057700 \nu^{2} + \cdots + 220339276380 ) / 5586947861 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 36224 \nu^{7} - 89472 \nu^{6} - 3547756 \nu^{5} - 3517188 \nu^{4} - 107094404 \nu^{3} + 201588480 \nu^{2} - 1013561676 \nu + 5942434428 ) / 12281577 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1152 \nu^{7} - 4656 \nu^{6} - 112608 \nu^{5} - 326664 \nu^{4} - 3341952 \nu^{3} - 8668080 \nu^{2} - 30894048 \nu - 72152556 ) / 66871 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - 60\beta_{2} - 120\beta_1 ) / 120 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 6\beta_{6} + 120\beta_{3} + 360\beta_{2} - 2940 ) / 120 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{7} - 25\beta_{5} + 9\beta_{4} - 360\beta_{3} + 4380\beta_{2} + 8820\beta _1 + 540 ) / 120 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 49\beta_{7} - 300\beta_{6} + 18\beta_{5} - 36\beta_{4} - 17520\beta_{3} - 52920\beta_{2} - 720\beta _1 + 73860 ) / 120 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 750 \beta_{7} + 90 \beta_{6} + 604 \beta_{5} - 735 \beta_{4} + 88200 \beta_{3} - 177360 \beta_{2} - 373620 \beta _1 - 131400 ) / 120 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 338 \beta_{7} + 2304 \beta_{6} - 441 \beta_{5} + 900 \beta_{4} + 215424 \beta_{3} + 673812 \beta_{2} + 52560 \beta _1 - 316344 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 40698 \beta_{7} - 15435 \beta_{6} - 9526 \beta_{5} + 37758 \beta_{4} - 7865460 \beta_{3} + 4920360 \beta_{2} + 13294680 \beta _1 + 11332440 ) / 120 \) Copy content Toggle raw display

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
1.22474 + 4.19968i
1.22474 5.64917i
−1.22474 3.19968i
−1.22474 + 6.64917i
1.22474 4.19968i
1.22474 + 5.64917i
−1.22474 + 3.19968i
−1.22474 6.64917i
0 −11.0227 + 11.0227i 0 0 0 −249.820 249.820i 0 243.000i 0
157.2 0 −11.0227 + 11.0227i 0 0 0 232.674 + 232.674i 0 243.000i 0
157.3 0 11.0227 11.0227i 0 0 0 −232.674 232.674i 0 243.000i 0
157.4 0 11.0227 11.0227i 0 0 0 249.820 + 249.820i 0 243.000i 0
193.1 0 −11.0227 11.0227i 0 0 0 −249.820 + 249.820i 0 243.000i 0
193.2 0 −11.0227 11.0227i 0 0 0 232.674 232.674i 0 243.000i 0
193.3 0 11.0227 + 11.0227i 0 0 0 −232.674 + 232.674i 0 243.000i 0
193.4 0 11.0227 + 11.0227i 0 0 0 249.820 249.820i 0 243.000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.7.k.b 8
5.b even 2 1 inner 300.7.k.b 8
5.c odd 4 2 inner 300.7.k.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.7.k.b 8 1.a even 1 1 trivial
300.7.k.b 8 5.b even 2 1 inner
300.7.k.b 8 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 27303292818T_{7}^{4} + 182648738100865680081 \) acting on \(S_{7}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 59049)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 27303292818 T^{4} + \cdots + 18\!\cdots\!81 \) Copy content Toggle raw display
$11$ \( (T^{2} - 1212 T - 1029564)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 94292537951250 T^{4} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{4} + 72003650 T^{2} + \cdots + 75616502850625)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 49\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + 1139458248 T^{2} + \cdots + 29\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4102 T - 904062599)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{2} - 64920 T - 1294369200)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{4} + 5469627528 T^{2} + \cdots + 74\!\cdots\!96)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 211294 T + 9593729809)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 81\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{2} - 148440 T - 78334311600)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 53\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{4} + 1200234542792 T^{2} + \cdots + 30\!\cdots\!16)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{4} + 2279902413600 T^{2} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 27\!\cdots\!41 \) Copy content Toggle raw display
show more
show less