↑

Properties

Label 309.4.a.d
Level $309$
Weight $4$
Character orbit 309.a
Self dual yes
Analytic conductor $18.232$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [309,4,Mod(1,309)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 309.a (trivial)

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: yes
Analytic conductor: \(18.2315901918\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 81 x^{12} + 144 x^{11} + 2483 x^{10} - 4087 x^{9} - 35705 x^{8} + 57997 x^{7} + \cdots + 412320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{5} + \beta_1 - 2) q^{5} + 3 \beta_1 q^{6} + (\beta_{7} + \beta_{6} + \cdots + \beta_{3}) q^{7}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{5} + \beta_1 - 2) q^{5} + 3 \beta_1 q^{6} + (\beta_{7} + \beta_{6} + \cdots + \beta_{3}) q^{7}+ \cdots + (9 \beta_{13} - 9 \beta_{11} + \cdots - 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} - 42 q^{3} + 54 q^{4} - 20 q^{5} + 6 q^{6} + 5 q^{7} - 30 q^{8} + 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} - 42 q^{3} + 54 q^{4} - 20 q^{5} + 6 q^{6} + 5 q^{7} - 30 q^{8} + 126 q^{9} - 103 q^{10} - 145 q^{11} - 162 q^{12} + q^{13} - 134 q^{14} + 60 q^{15} + 262 q^{16} - 93 q^{17} - 18 q^{18} - 254 q^{19} - 145 q^{20} - 15 q^{21} - 65 q^{22} - 154 q^{23} + 90 q^{24} + 470 q^{25} - 542 q^{26} - 378 q^{27} + 59 q^{28} - 760 q^{29} + 309 q^{30} - 559 q^{31} - 1687 q^{32} + 435 q^{33} - 970 q^{34} - 1058 q^{35} + 486 q^{36} + 313 q^{37} - 1166 q^{38} - 3 q^{39} - 2315 q^{40} - 1211 q^{41} + 402 q^{42} - 648 q^{43} - 3137 q^{44} - 180 q^{45} - 185 q^{46} - 1030 q^{47} - 786 q^{48} + 443 q^{49} - 1705 q^{50} + 279 q^{51} - 1749 q^{52} - 694 q^{53} + 54 q^{54} - 714 q^{55} - 2937 q^{56} + 762 q^{57} + 547 q^{58} - 1867 q^{59} + 435 q^{60} - 559 q^{61} - 1408 q^{62} + 45 q^{63} + 1798 q^{64} - 1964 q^{65} + 195 q^{66} + 160 q^{67} - 2577 q^{68} + 462 q^{69} - 2472 q^{70} - 2044 q^{71} - 270 q^{72} + 768 q^{73} - 3547 q^{74} - 1410 q^{75} - 3483 q^{76} - 3046 q^{77} + 1626 q^{78} - 991 q^{79} + 695 q^{80} + 1134 q^{81} + 3995 q^{82} - 2017 q^{83} - 177 q^{84} + 2754 q^{85} + 3829 q^{86} + 2280 q^{87} + 3937 q^{88} - 1341 q^{89} - 927 q^{90} - 466 q^{91} + 5884 q^{92} + 1677 q^{93} + 644 q^{94} + 184 q^{95} + 5061 q^{96} + 1812 q^{97} + 1893 q^{98} - 1305 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 81 x^{12} + 144 x^{11} + 2483 x^{10} - 4087 x^{9} - 35705 x^{8} + 57997 x^{7} + \cdots + 412320 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!73 \nu^{13} + \cdots - 15\!\cdots\!00 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 45\!\cdots\!51 \nu^{13} + 54030878197623 \nu^{12} + \cdots - 33\!\cdots\!44 ) / 32\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 48\!\cdots\!39 \nu^{13} + \cdots - 20\!\cdots\!04 ) / 32\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 63\!\cdots\!23 \nu^{13} + \cdots - 29\!\cdots\!00 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 32\!\cdots\!71 \nu^{13} + \cdots + 18\!\cdots\!80 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 71\!\cdots\!13 \nu^{13} + \cdots + 26\!\cdots\!00 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 76\!\cdots\!57 \nu^{13} + \cdots + 44\!\cdots\!40 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 89\!\cdots\!73 \nu^{13} + \cdots - 33\!\cdots\!80 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!91 \nu^{13} + \cdots + 11\!\cdots\!00 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12\!\cdots\!89 \nu^{13} + \cdots + 53\!\cdots\!40 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!57 \nu^{13} + \cdots + 44\!\cdots\!20 ) / 32\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2} + 20\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{13} + 4 \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} + 3 \beta_{7} + 3 \beta_{6} + \cdots + 240 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 28 \beta_{13} + 10 \beta_{12} - 5 \beta_{11} - 6 \beta_{10} - 12 \beta_{9} - 40 \beta_{8} + 47 \beta_{7} + \cdots + 146 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 42 \beta_{13} + 178 \beta_{12} - 42 \beta_{11} - 7 \beta_{10} - 56 \beta_{9} - 77 \beta_{8} + \cdots + 5581 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 645 \beta_{13} + 606 \beta_{12} - 227 \beta_{11} - 310 \beta_{10} - 626 \beta_{9} - 1332 \beta_{8} + \cdots + 7003 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1313 \beta_{13} + 6278 \beta_{12} - 1512 \beta_{11} - 683 \beta_{10} - 2318 \beta_{9} - 3580 \beta_{8} + \cdots + 139618 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 13562 \beta_{13} + 25740 \beta_{12} - 7820 \beta_{11} - 12192 \beta_{10} - 23941 \beta_{9} + \cdots + 268293 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 38988 \beta_{13} + 207400 \beta_{12} - 50667 \beta_{11} - 37233 \beta_{10} - 87212 \beta_{9} + \cdots + 3669021 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 258477 \beta_{13} + 954444 \beta_{12} - 248793 \beta_{11} - 436485 \beta_{10} - 820078 \beta_{9} + \cdots + 9407170 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1171843 \beta_{13} + 6698576 \beta_{12} - 1628461 \beta_{11} - 1616849 \beta_{10} - 3127526 \beta_{9} + \cdots + 100203068 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4106068 \beta_{13} + 33154010 \beta_{12} - 7746521 \beta_{11} - 14953151 \beta_{10} - 26774690 \beta_{9} + \cdots + 315759449 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.62572
4.91372
3.99880
2.48961
2.26086
1.61990
0.895197
0.551438
−1.16634
−2.45027
−3.11138
−4.20195
−4.54276
−4.88254
−5.62572 −3.00000 23.6487 13.0903 16.8772 26.4341 −88.0352 9.00000 −73.6421
1.2 −4.91372 −3.00000 16.1446 −16.2219 14.7412 1.09686 −40.0203 9.00000 79.7096
1.3 −3.99880 −3.00000 7.99043 18.6423 11.9964 −16.9381 0.0382794 9.00000 −74.5467
1.4 −2.48961 −3.00000 −1.80185 −17.7300 7.46882 −13.8438 24.4028 9.00000 44.1407
1.5 −2.26086 −3.00000 −2.88851 6.96640 6.78258 −21.2259 24.6174 9.00000 −15.7501
1.6 −1.61990 −3.00000 −5.37594 −16.2736 4.85969 29.2303 21.6676 9.00000 26.3615
1.7 −0.895197 −3.00000 −7.19862 −4.43685 2.68559 14.2417 13.6058 9.00000 3.97186
1.8 −0.551438 −3.00000 −7.69592 7.20906 1.65431 −10.0029 8.65532 9.00000 −3.97535
1.9 1.16634 −3.00000 −6.63964 1.75004 −3.49903 19.3344 −17.0748 9.00000 2.04114
1.10 2.45027 −3.00000 −1.99619 14.2962 −7.35080 −7.42545 −24.4933 9.00000 35.0295
1.11 3.11138 −3.00000 1.68069 −4.20802 −9.33414 15.6033 −19.6618 9.00000 −13.0927
1.12 4.20195 −3.00000 9.65641 2.34049 −12.6059 −22.7944 6.96014 9.00000 9.83461
1.13 4.54276 −3.00000 12.6367 −3.10062 −13.6283 −29.1412 21.0632 9.00000 −14.0854
1.14 4.88254 −3.00000 15.8392 −22.3238 −14.6476 20.4311 38.2750 9.00000 −108.997
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(103\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 309.4.a.d ✓ 14
3.b odd 2 1 927.4.a.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.4.a.d ✓ 14 1.a even 1 1 trivial
927.4.a.e 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 2 T_{2}^{13} - 81 T_{2}^{12} - 144 T_{2}^{11} + 2483 T_{2}^{10} + 4087 T_{2}^{9} + \cdots + 412320 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(309))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + \cdots + 412320 \) Copy content Toggle raw display
$3$ \( (T + 3)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 4340775106560 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 18\!\cdots\!60 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots - 15\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 71\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 21\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 10\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 17\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 59\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 29\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 32\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 24\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 16\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 19\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 76\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 10\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 55\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 70\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 15\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 14\!\cdots\!50 \) Copy content Toggle raw display
show more
show less