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Properties

Label 309.6.a.a
Level $309$
Weight $6$
Character orbit 309.a
Self dual yes
Analytic conductor $49.559$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [309,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(49.5586003222\)
Analytic rank: \(1\)
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: \( x^{17} - 2 x^{16} - 361 x^{15} + 760 x^{14} + 51755 x^{13} - 106962 x^{12} - 3783559 x^{11} + \cdots + 35152033792 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 9 q^{3} + (\beta_{2} + 11) q^{4} + (\beta_{4} - 2 \beta_1 - 6) q^{5} + 9 \beta_1 q^{6} + ( - \beta_{15} - \beta_{10} + \cdots - 22) q^{7}+ \cdots + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 9 q^{3} + (\beta_{2} + 11) q^{4} + (\beta_{4} - 2 \beta_1 - 6) q^{5} + 9 \beta_1 q^{6} + ( - \beta_{15} - \beta_{10} + \cdots - 22) q^{7}+ \cdots + (81 \beta_{15} - 243 \beta_{14} + \cdots - 4293) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 2 q^{2} + 153 q^{3} + 182 q^{4} - 103 q^{5} + 18 q^{6} - 382 q^{7} - 234 q^{8} + 1377 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 2 q^{2} + 153 q^{3} + 182 q^{4} - 103 q^{5} + 18 q^{6} - 382 q^{7} - 234 q^{8} + 1377 q^{9} - 1307 q^{10} - 950 q^{11} + 1638 q^{12} - 2217 q^{13} - 2882 q^{14} - 927 q^{15} + 1046 q^{16} - 3166 q^{17} + 162 q^{18} - 8472 q^{19} - 5273 q^{20} - 3438 q^{21} - 9059 q^{22} - 781 q^{23} - 2106 q^{24} + 334 q^{25} - 9921 q^{26} + 12393 q^{27} - 23190 q^{28} - 22866 q^{29} - 11763 q^{30} - 23583 q^{31} - 32346 q^{32} - 8550 q^{33} - 16894 q^{34} - 28526 q^{35} + 14742 q^{36} - 27728 q^{37} - 5241 q^{38} - 19953 q^{39} - 49863 q^{40} - 16232 q^{41} - 25938 q^{42} - 53897 q^{43} - 29023 q^{44} - 8343 q^{45} - 42474 q^{46} - 1176 q^{47} + 9414 q^{48} + 4435 q^{49} + 12907 q^{50} - 28494 q^{51} - 44605 q^{52} - 50104 q^{53} + 1458 q^{54} - 39774 q^{55} + 522 q^{56} - 76248 q^{57} - 1947 q^{58} - 68415 q^{59} - 47457 q^{60} - 97857 q^{61} + 26078 q^{62} - 30942 q^{63} - 58786 q^{64} - 22881 q^{65} - 81531 q^{66} - 31613 q^{67} + 140258 q^{68} - 7029 q^{69} + 66706 q^{70} + 29386 q^{71} - 18954 q^{72} - 170816 q^{73} + 236925 q^{74} + 3006 q^{75} - 235751 q^{76} + 4564 q^{77} - 89289 q^{78} - 206374 q^{79} - 3321 q^{80} + 111537 q^{81} - 363798 q^{82} - 215543 q^{83} - 208710 q^{84} - 266940 q^{85} - 216285 q^{86} - 205794 q^{87} - 410265 q^{88} + 33930 q^{89} - 105867 q^{90} - 381078 q^{91} - 386458 q^{92} - 212247 q^{93} - 264080 q^{94} - 83334 q^{95} - 291114 q^{96} - 299241 q^{97} - 5672 q^{98} - 76950 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 2 x^{16} - 361 x^{15} + 760 x^{14} + 51755 x^{13} - 106962 x^{12} - 3783559 x^{11} + \cdots + 35152033792 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 43 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 48\!\cdots\!15 \nu^{16} + \cdots - 15\!\cdots\!80 ) / 69\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\!\cdots\!21 \nu^{16} + \cdots - 10\!\cdots\!56 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 68\!\cdots\!59 \nu^{16} + \cdots - 33\!\cdots\!56 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!76 \nu^{16} + \cdots - 10\!\cdots\!24 ) / 43\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27\!\cdots\!73 \nu^{16} + \cdots - 14\!\cdots\!32 ) / 87\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!39 \nu^{16} + \cdots + 89\!\cdots\!76 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31\!\cdots\!97 \nu^{16} + \cdots + 21\!\cdots\!28 ) / 87\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13\!\cdots\!71 \nu^{16} + \cdots - 40\!\cdots\!84 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 27\!\cdots\!41 \nu^{16} + \cdots - 21\!\cdots\!32 ) / 69\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 68\!\cdots\!79 \nu^{16} + \cdots + 11\!\cdots\!76 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 30\!\cdots\!27 \nu^{16} + \cdots + 17\!\cdots\!24 ) / 69\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 34\!\cdots\!37 \nu^{16} + \cdots - 29\!\cdots\!16 ) / 69\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 18\!\cdots\!87 \nu^{16} + \cdots - 31\!\cdots\!68 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 49\!\cdots\!03 \nu^{16} + \cdots + 10\!\cdots\!92 ) / 87\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 43 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} - 2\beta_{3} - \beta_{2} + 74\beta _1 - 17 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + 4 \beta_{15} + \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - \beta_{11} + 6 \beta_{10} + \cdots + 3176 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 29 \beta_{16} - 12 \beta_{15} + 116 \beta_{14} - 113 \beta_{13} - 6 \beta_{12} + \beta_{11} + \cdots - 4112 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 152 \beta_{16} + 604 \beta_{15} + 120 \beta_{14} - 162 \beta_{13} - 192 \beta_{12} - 124 \beta_{11} + \cdots + 273565 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 5482 \beta_{16} - 1892 \beta_{15} + 11607 \beta_{14} - 11585 \beta_{13} - 1460 \beta_{12} + \cdots - 618055 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 21607 \beta_{16} + 66748 \beta_{15} + 8559 \beta_{14} - 9274 \beta_{13} - 14334 \beta_{12} + \cdots + 25390666 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 755591 \beta_{16} - 207100 \beta_{15} + 1140330 \beta_{14} - 1164009 \beta_{13} - 219522 \beta_{12} + \cdots - 81205390 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2959166 \beta_{16} + 6475148 \beta_{15} + 250254 \beta_{14} - 278202 \beta_{13} - 909676 \beta_{12} + \cdots + 2458256543 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 92919700 \beta_{16} - 18730948 \beta_{15} + 112868813 \beta_{14} - 116444505 \beta_{13} + \cdots - 10033334485 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 391725997 \beta_{16} + 578856620 \beta_{15} - 48409019 \beta_{14} + 31008414 \beta_{13} + \cdots + 244542379852 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10844585089 \beta_{16} - 1401667260 \beta_{15} + 11312056016 \beta_{14} - 11658797009 \beta_{13} + \cdots - 1196915211372 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 50405486020 \beta_{16} + 48065579996 \beta_{15} - 12943684252 \beta_{14} + 8666649038 \beta_{13} + \cdots + 24799878774145 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1232439000830 \beta_{16} - 71914803108 \beta_{15} + 1147786400131 \beta_{14} - 1170743454097 \beta_{13} + \cdots - 139635402783587 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 6336233489235 \beta_{16} + 3629068235228 \beta_{15} - 2164019316869 \beta_{14} + 1418929134294 \beta_{13} + \cdots + 25\!\cdots\!54 \) 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
−10.5670
−9.87464
−7.74891
−6.60069
−4.91408
−3.82308
−1.71791
−1.43716
−0.533743
0.470241
3.67061
5.41627
5.71438
6.60979
8.54654
9.06987
9.71954
−10.5670 9.00000 79.6619 12.4127 −95.1032 11.2678 −503.644 81.0000 −131.165
1.2 −9.87464 9.00000 65.5085 15.4723 −88.8718 −200.470 −330.885 81.0000 −152.784
1.3 −7.74891 9.00000 28.0457 33.7039 −69.7402 124.131 30.6418 81.0000 −261.169
1.4 −6.60069 9.00000 11.5691 −52.1901 −59.4062 202.006 134.858 81.0000 344.491
1.5 −4.91408 9.00000 −7.85181 −77.7833 −44.2267 −96.5746 195.835 81.0000 382.234
1.6 −3.82308 9.00000 −17.3840 95.6922 −34.4078 −152.095 188.799 81.0000 −365.839
1.7 −1.71791 9.00000 −29.0488 78.3594 −15.4612 92.5818 104.876 81.0000 −134.614
1.8 −1.43716 9.00000 −29.9346 −15.1781 −12.9345 19.1878 89.0100 81.0000 21.8134
1.9 −0.533743 9.00000 −31.7151 −104.402 −4.80369 95.5777 34.0075 81.0000 55.7237
1.10 0.470241 9.00000 −31.7789 10.6308 4.23217 −128.909 −29.9915 81.0000 4.99904
1.11 3.67061 9.00000 −18.5266 −21.9954 33.0355 72.6173 −185.464 81.0000 −80.7367
1.12 5.41627 9.00000 −2.66401 62.3562 48.7464 −201.663 −187.750 81.0000 337.738
1.13 5.71438 9.00000 0.654085 −2.42685 51.4294 106.404 −179.122 81.0000 −13.8679
1.14 6.60979 9.00000 11.6894 28.6036 59.4881 −29.3506 −134.249 81.0000 189.064
1.15 8.54654 9.00000 41.0433 −45.1056 76.9188 −100.068 77.2890 81.0000 −385.497
1.16 9.06987 9.00000 50.2625 −92.5588 81.6288 41.3200 165.638 81.0000 −839.496
1.17 9.71954 9.00000 62.4695 −28.5913 87.4759 −237.963 296.150 81.0000 −277.894
\(n\): 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 309.6.a.a ✓ 17
3.b odd 2 1 927.6.a.a 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.6.a.a ✓ 17 1.a even 1 1 trivial
927.6.a.a 17 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 2 T_{2}^{16} - 361 T_{2}^{15} + 760 T_{2}^{14} + 51755 T_{2}^{13} - 106962 T_{2}^{12} + \cdots + 35152033792 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(309))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + \cdots + 35152033792 \) Copy content Toggle raw display
$3$ \( (T - 9)^{17} \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( T^{17} + \cdots - 81\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots - 23\!\cdots\!80 \) Copy content Toggle raw display
$13$ \( T^{17} + \cdots - 11\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots + 19\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 38\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots - 55\!\cdots\!75 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 10\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots + 57\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 56\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 56\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 83\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots - 23\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 51\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 34\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 22\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 13\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 58\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 47\!\cdots\!21 \) Copy content Toggle raw display
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