↑

Properties

Label 309.4.a.e
Level $309$
Weight $4$
Character orbit 309.a
Self dual yes
Analytic conductor $18.232$
Analytic rank $0$
Dimension $17$
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: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 2 x^{16} - 111 x^{15} + 218 x^{14} + 4961 x^{13} - 9211 x^{12} - 115289 x^{11} + \cdots - 27667968 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{7} + 1) q^{5} + 3 \beta_1 q^{6} + (\beta_{5} + \beta_1 + 3) q^{7} + (\beta_{3} - \beta_{2} + 5 \beta_1 - 1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{7} + 1) q^{5} + 3 \beta_1 q^{6} + (\beta_{5} + \beta_1 + 3) q^{7} + (\beta_{3} - \beta_{2} + 5 \beta_1 - 1) q^{8} + 9 q^{9} + ( - \beta_{16} - \beta_{15} + \beta_{12} + \cdots + 7) q^{10}+ \cdots + (9 \beta_{16} - 9 \beta_{12} + \cdots + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 2 q^{2} + 51 q^{3} + 90 q^{4} + 20 q^{5} + 6 q^{6} + 61 q^{7} - 12 q^{8} + 153 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 2 q^{2} + 51 q^{3} + 90 q^{4} + 20 q^{5} + 6 q^{6} + 61 q^{7} - 12 q^{8} + 153 q^{9} + 113 q^{10} + 97 q^{11} + 270 q^{12} + 161 q^{13} + 208 q^{14} + 60 q^{15} + 510 q^{16} + 77 q^{17} + 18 q^{18} + 494 q^{19} + 319 q^{20} + 183 q^{21} + 411 q^{22} + 134 q^{23} - 36 q^{24} + 941 q^{25} + 144 q^{26} + 459 q^{27} + 555 q^{28} + 96 q^{29} + 339 q^{30} + 495 q^{31} - 1479 q^{32} + 291 q^{33} - 298 q^{34} + 370 q^{35} + 810 q^{36} + 679 q^{37} - 1236 q^{38} + 483 q^{39} + 225 q^{40} + 115 q^{41} + 624 q^{42} + 1230 q^{43} - 685 q^{44} + 180 q^{45} + 899 q^{46} + 42 q^{47} + 1530 q^{48} + 1678 q^{49} - 173 q^{50} + 231 q^{51} + 1451 q^{52} - 154 q^{53} + 54 q^{54} + 1322 q^{55} + 273 q^{56} + 1482 q^{57} - 97 q^{58} + 1623 q^{59} + 957 q^{60} + 2141 q^{61} - 52 q^{62} + 549 q^{63} + 4310 q^{64} + 1092 q^{65} + 1233 q^{66} + 1774 q^{67} - 181 q^{68} + 402 q^{69} + 176 q^{70} + 724 q^{71} - 108 q^{72} + 3134 q^{73} - 771 q^{74} + 2823 q^{75} + 4417 q^{76} + 910 q^{77} + 432 q^{78} + 1161 q^{79} - 1599 q^{80} + 1377 q^{81} - 3437 q^{82} + 1141 q^{83} + 1665 q^{84} - 246 q^{85} - 6339 q^{86} + 288 q^{87} + 1233 q^{88} + 357 q^{89} + 1017 q^{90} + 2106 q^{91} - 6034 q^{92} + 1485 q^{93} + 1956 q^{94} - 288 q^{95} - 4437 q^{96} - 1200 q^{97} - 575 q^{98} + 873 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 2 x^{16} - 111 x^{15} + 218 x^{14} + 4961 x^{13} - 9211 x^{12} - 115289 x^{11} + \cdots - 27667968 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 21\nu - 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22\!\cdots\!39 \nu^{16} + \cdots + 23\!\cdots\!92 ) / 79\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 50\!\cdots\!17 \nu^{16} + \cdots - 31\!\cdots\!44 ) / 15\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 77\!\cdots\!65 \nu^{16} + \cdots + 31\!\cdots\!28 ) / 15\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!91 \nu^{16} + \cdots + 30\!\cdots\!44 ) / 15\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!79 \nu^{16} + \cdots + 67\!\cdots\!72 ) / 15\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22\!\cdots\!35 \nu^{16} + \cdots + 68\!\cdots\!64 ) / 15\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25\!\cdots\!29 \nu^{16} + \cdots - 47\!\cdots\!48 ) / 15\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\!\cdots\!62 \nu^{16} + \cdots + 64\!\cdots\!00 ) / 79\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!13 \nu^{16} + \cdots + 23\!\cdots\!80 ) / 46\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 88\!\cdots\!01 \nu^{16} + \cdots - 19\!\cdots\!36 ) / 39\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 46\!\cdots\!71 \nu^{16} + \cdots - 92\!\cdots\!92 ) / 15\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 28\!\cdots\!61 \nu^{16} + \cdots - 70\!\cdots\!52 ) / 79\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 31\!\cdots\!22 \nu^{16} + \cdots + 77\!\cdots\!84 ) / 79\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 21\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \cdots + 279 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{16} + 2 \beta_{15} - 2 \beta_{14} - 3 \beta_{13} - 2 \beta_{12} - \beta_{11} - 2 \beta_{10} + \cdots - 118 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{16} + 48 \beta_{15} - 36 \beta_{14} - 47 \beta_{13} + 55 \beta_{12} - 38 \beta_{11} + \cdots + 6936 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 62 \beta_{16} + 89 \beta_{15} - 132 \beta_{14} - 144 \beta_{13} - 143 \beta_{12} - 37 \beta_{11} + \cdots - 5972 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 157 \beta_{16} + 1724 \beta_{15} - 960 \beta_{14} - 1657 \beta_{13} + 2289 \beta_{12} - 1120 \beta_{11} + \cdots + 184857 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2586 \beta_{16} + 2916 \beta_{15} - 5729 \beta_{14} - 5053 \beta_{13} - 6954 \beta_{12} + \cdots - 235970 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5764 \beta_{16} + 55310 \beta_{15} - 21688 \beta_{14} - 52366 \beta_{13} + 85699 \beta_{12} + \cdots + 5122708 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 92407 \beta_{16} + 84563 \beta_{15} - 213228 \beta_{14} - 156159 \beta_{13} - 289270 \beta_{12} + \cdots - 8443591 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 183769 \beta_{16} + 1674359 \beta_{15} - 397066 \beta_{14} - 1568097 \beta_{13} + 3040790 \beta_{12} + \cdots + 145397742 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3052043 \beta_{16} + 2295847 \beta_{15} - 7408186 \beta_{14} - 4510281 \beta_{13} - 11101002 \beta_{12} + \cdots - 288177120 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5443089 \beta_{16} + 49000596 \beta_{15} - 3899115 \beta_{14} - 45591990 \beta_{13} + 104567731 \beta_{12} + \cdots + 4192720960 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 96259756 \beta_{16} + 59741133 \beta_{15} - 248882128 \beta_{14} - 124954928 \beta_{13} + \cdots - 9584524257 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 154131596 \beta_{16} + 1404040527 \beta_{15} + 125813850 \beta_{14} - 1303508164 \beta_{13} + \cdots + 122264823953 \) 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
−5.60121
−5.40301
−4.49448
−3.09156
−2.87312
−2.20589
−2.18410
−0.896650
0.818839
1.04099
1.05228
3.45669
3.71592
3.87636
4.40063
5.11513
5.27318
−5.60121 3.00000 23.3736 −9.16774 −16.8036 −18.3927 −86.1105 9.00000 51.3505
1.2 −5.40301 3.00000 21.1925 14.0845 −16.2090 3.88038 −71.2790 9.00000 −76.0984
1.3 −4.49448 3.00000 12.2003 −8.80652 −13.4834 27.2714 −18.8782 9.00000 39.5807
1.4 −3.09156 3.00000 1.55773 14.9596 −9.27467 −3.72803 19.9166 9.00000 −46.2484
1.5 −2.87312 3.00000 0.254836 −20.5579 −8.61937 12.1672 22.2528 9.00000 59.0653
1.6 −2.20589 3.00000 −3.13405 10.4010 −6.61767 −35.8707 24.5605 9.00000 −22.9435
1.7 −2.18410 3.00000 −3.22971 10.4492 −6.55230 32.5641 24.5268 9.00000 −22.8221
1.8 −0.896650 3.00000 −7.19602 −17.7489 −2.68995 −22.1026 13.6255 9.00000 15.9145
1.9 0.818839 3.00000 −7.32950 −15.3510 2.45652 −19.9954 −12.5524 9.00000 −12.5700
1.10 1.04099 3.00000 −6.91635 −1.61584 3.12296 16.1859 −15.5277 9.00000 −1.68207
1.11 1.05228 3.00000 −6.89271 15.0185 3.15684 9.27436 −15.6713 9.00000 15.8036
1.12 3.45669 3.00000 3.94870 10.0296 10.3701 −10.5855 −14.0041 9.00000 34.6694
1.13 3.71592 3.00000 5.80804 20.9909 11.1478 33.3392 −8.14515 9.00000 78.0004
1.14 3.87636 3.00000 7.02617 −17.3755 11.6291 30.0673 −3.77492 9.00000 −67.3537
1.15 4.40063 3.00000 11.3655 11.3478 13.2019 4.26433 14.8103 9.00000 49.9373
1.16 5.11513 3.00000 18.1646 −4.88354 15.3454 17.4113 51.9931 9.00000 −24.9800
1.17 5.27318 3.00000 19.8065 8.22585 15.8195 −14.7506 62.2576 9.00000 43.3764
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
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.e ✓ 17
3.b odd 2 1 927.4.a.g 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.4.a.e ✓ 17 1.a even 1 1 trivial
927.4.a.g 17 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 2 T_{2}^{16} - 111 T_{2}^{15} + 218 T_{2}^{14} + 4961 T_{2}^{13} - 9211 T_{2}^{12} + \cdots - 27667968 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(309))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} - 2 T^{16} + \cdots - 27667968 \) Copy content Toggle raw display
$3$ \( (T - 3)^{17} \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots - 41\!\cdots\!72 \) Copy content Toggle raw display
$7$ \( T^{17} + \cdots + 79\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots - 71\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{17} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 17\!\cdots\!88 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 64\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots + 53\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots + 51\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 41\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 15\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 30\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 51\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 41\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 44\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 83\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 31\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 52\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 11\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
show more
show less