Properties

Label 3009.2.a.d
Level $3009$
Weight $2$
Character orbit 3009.a
Self dual yes
Analytic conductor $24.027$
Analytic rank $1$
Dimension $14$
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] = [3009,2,Mod(1,3009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3009, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3009 = 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3009.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: \(24.0269859682\)
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: \( x^{14} - 4 x^{13} - 11 x^{12} + 57 x^{11} + 33 x^{10} - 311 x^{9} + 33 x^{8} + 805 x^{7} - 330 x^{6} - 981 x^{5} + 542 x^{4} + 454 x^{3} - 283 x^{2} - 7 x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_{10} q^{5} - \beta_1 q^{6} + ( - \beta_{7} - 1) q^{7} + ( - \beta_{3} - \beta_{2} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_{10} q^{5} - \beta_1 q^{6} + ( - \beta_{7} - 1) q^{7} + ( - \beta_{3} - \beta_{2} - 1) q^{8} + q^{9} + ( - \beta_{6} - \beta_{5} - \beta_{3} - \beta_1 - 1) q^{10} + ( - \beta_{13} + \beta_{10} + \beta_1 - 1) q^{11} + (\beta_{2} + 1) q^{12} + (\beta_{12} - \beta_{11} - \beta_{9} - \beta_{6} + \beta_{2} - \beta_1 - 1) q^{13} + (\beta_{10} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{14} - \beta_{10} q^{15} + (\beta_{7} + \beta_{6} + 2 \beta_{3} + 2 \beta_1 - 1) q^{16} + q^{17} - \beta_1 q^{18} + (\beta_{9} - \beta_{8} + \beta_{7} - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{19} + ( - \beta_{9} - \beta_{8} + \beta_{6} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{20} + ( - \beta_{7} - 1) q^{21} + (\beta_{13} - 2 \beta_{12} + \beta_{11} + \beta_{10} + \beta_{8} + 2 \beta_{6} + 2 \beta_{5} + \beta_{3} + \cdots - 2) q^{22}+ \cdots + ( - \beta_{13} + \beta_{10} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{2} + 14 q^{3} + 10 q^{4} - 3 q^{5} - 4 q^{6} - 11 q^{7} - 9 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{2} + 14 q^{3} + 10 q^{4} - 3 q^{5} - 4 q^{6} - 11 q^{7} - 9 q^{8} + 14 q^{9} - 10 q^{10} - 11 q^{11} + 10 q^{12} - 23 q^{13} - 3 q^{14} - 3 q^{15} - 14 q^{16} + 14 q^{17} - 4 q^{18} - 13 q^{19} + 5 q^{20} - 11 q^{21} - 7 q^{22} - 24 q^{23} - 9 q^{24} - q^{25} + 7 q^{26} + 14 q^{27} - 8 q^{29} - 10 q^{30} - 29 q^{31} - 12 q^{32} - 11 q^{33} - 4 q^{34} - 3 q^{35} + 10 q^{36} - 24 q^{37} - 5 q^{38} - 23 q^{39} - 3 q^{40} - 6 q^{41} - 3 q^{42} - 8 q^{43} - 19 q^{44} - 3 q^{45} - 6 q^{46} - 36 q^{47} - 14 q^{48} - q^{49} - 21 q^{50} + 14 q^{51} - 18 q^{52} - 12 q^{53} - 4 q^{54} - 39 q^{55} - 29 q^{56} - 13 q^{57} - 21 q^{58} - 14 q^{59} + 5 q^{60} - 16 q^{61} + 39 q^{62} - 11 q^{63} - 19 q^{64} - 19 q^{65} - 7 q^{66} - 34 q^{67} + 10 q^{68} - 24 q^{69} - 32 q^{70} - 30 q^{71} - 9 q^{72} - 39 q^{73} + 35 q^{74} - q^{75} - 44 q^{76} + 17 q^{77} + 7 q^{78} - 17 q^{79} + 39 q^{80} + 14 q^{81} - 32 q^{82} - 5 q^{83} - 3 q^{85} + 16 q^{86} - 8 q^{87} - 7 q^{88} - 15 q^{89} - 10 q^{90} - 32 q^{91} - 2 q^{92} - 29 q^{93} - 11 q^{94} - 2 q^{95} - 12 q^{96} - 2 q^{97} + 34 q^{98} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} - 11 x^{12} + 57 x^{11} + 33 x^{10} - 311 x^{9} + 33 x^{8} + 805 x^{7} - 330 x^{6} - 981 x^{5} + 542 x^{4} + 454 x^{3} - 283 x^{2} - 7 x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19 \nu^{13} - 47 \nu^{12} - 255 \nu^{11} + 608 \nu^{10} + 1229 \nu^{9} - 2712 \nu^{8} - 2603 \nu^{7} + 4639 \nu^{6} + 2758 \nu^{5} - 1724 \nu^{4} - 2671 \nu^{3} - 1233 \nu^{2} + 1972 \nu - 100 ) / 163 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 31 \nu^{13} + 111 \nu^{12} + 356 \nu^{11} - 1481 \nu^{10} - 1439 \nu^{9} + 7539 \nu^{8} + 2291 \nu^{7} - 18121 \nu^{6} - 708 \nu^{5} + 20271 \nu^{4} - 1450 \nu^{3} - 8283 \nu^{2} + \cdots + 146 ) / 163 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 48 \nu^{13} + 93 \nu^{12} + 730 \nu^{11} - 1210 \nu^{10} - 4426 \nu^{9} + 5942 \nu^{8} + 13259 \nu^{7} - 13667 \nu^{6} - 19510 \nu^{5} + 14693 \nu^{4} + 11389 \nu^{3} - 6116 \nu^{2} + \cdots + 184 ) / 163 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 48 \nu^{13} - 93 \nu^{12} - 730 \nu^{11} + 1210 \nu^{10} + 4426 \nu^{9} - 5942 \nu^{8} - 13259 \nu^{7} + 13667 \nu^{6} + 19510 \nu^{5} - 14530 \nu^{4} - 11715 \nu^{3} + 5464 \nu^{2} + \cdots - 21 ) / 163 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21 \nu^{13} + 112 \nu^{12} + 136 \nu^{11} - 1487 \nu^{10} + 366 \nu^{9} + 7673 \nu^{8} - 4784 \nu^{7} - 19437 \nu^{6} + 12797 \nu^{5} + 24837 \nu^{4} - 12061 \nu^{3} - 13882 \nu^{2} + \cdots + 1466 ) / 163 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 60 \nu^{13} + 157 \nu^{12} + 831 \nu^{11} - 2083 \nu^{10} - 4473 \nu^{9} + 10280 \nu^{8} + 11643 \nu^{7} - 22911 \nu^{6} - 14526 \nu^{5} + 21830 \nu^{4} + 6779 \nu^{3} - 6178 \nu^{2} + \cdots - 259 ) / 163 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 63 \nu^{13} - 173 \nu^{12} - 897 \nu^{11} + 2505 \nu^{10} + 4770 \nu^{9} - 13728 \nu^{8} - 11402 \nu^{7} + 35165 \nu^{6} + 10998 \nu^{5} - 41585 \nu^{4} - 818 \nu^{3} + 18500 \nu^{2} + \cdots - 812 ) / 163 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 54 \nu^{13} + 125 \nu^{12} + 862 \nu^{11} - 1891 \nu^{10} - 5346 \nu^{9} + 10882 \nu^{8} + 16037 \nu^{7} - 29536 \nu^{6} - 23212 \nu^{5} + 37577 \nu^{4} + 13159 \nu^{3} + \cdots + 1185 ) / 163 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 79 \nu^{13} + 204 \nu^{12} + 1086 \nu^{11} - 2691 \nu^{10} - 5865 \nu^{9} + 13481 \nu^{8} + 15550 \nu^{7} - 31788 \nu^{6} - 20381 \nu^{5} + 35127 \nu^{4} + 11080 \nu^{3} + \cdots + 1145 ) / 163 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 111 \nu^{13} - 266 \nu^{12} - 1627 \nu^{11} + 3715 \nu^{10} + 9196 \nu^{9} - 19670 \nu^{8} - 24661 \nu^{7} + 48995 \nu^{6} + 30345 \nu^{5} - 57419 \nu^{4} - 11718 \nu^{3} + \cdots - 1811 ) / 163 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} + 2\beta_{3} + 6\beta_{2} + 2\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{12} + \beta_{7} + 2\beta_{6} + \beta_{5} + 9\beta_{3} + 8\beta_{2} + 20\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{10} + 8\beta_{7} + 10\beta_{6} + \beta_{5} + 20\beta_{3} + 34\beta_{2} + 20\beta _1 + 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{13} - 11 \beta_{12} + \beta_{11} - \beta_{10} + \beta_{8} + 11 \beta_{7} + 24 \beta_{6} + 10 \beta_{5} - \beta_{4} + 66 \beta_{3} + 55 \beta_{2} + 111 \beta _1 + 76 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13 \beta_{13} - 16 \beta_{12} + 2 \beta_{11} - 11 \beta_{10} + \beta_{9} + 3 \beta_{8} + 52 \beta_{7} + 81 \beta_{6} + 12 \beta_{5} - \beta_{4} + 154 \beta_{3} + 194 \beta_{2} + 158 \beta _1 + 340 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 29 \beta_{13} - 92 \beta_{12} + 14 \beta_{11} - 15 \beta_{10} + 4 \beta_{9} + 17 \beta_{8} + 88 \beta_{7} + 208 \beta_{6} + 71 \beta_{5} - 12 \beta_{4} + 451 \beta_{3} + 359 \beta_{2} + 651 \beta _1 + 526 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 121 \beta_{13} - 169 \beta_{12} + 33 \beta_{11} - 87 \beta_{10} + 22 \beta_{9} + 50 \beta_{8} + 324 \beta_{7} + 611 \beta_{6} + 99 \beta_{5} - 16 \beta_{4} + 1084 \beta_{3} + 1124 \beta_{2} + 1147 \beta _1 + 1901 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 290 \beta_{13} - 705 \beta_{12} + 143 \beta_{11} - 146 \beta_{10} + 75 \beta_{9} + 191 \beta_{8} + 631 \beta_{7} + 1605 \beta_{6} + 444 \beta_{5} - 102 \beta_{4} + 2993 \beta_{3} + 2283 \beta_{2} + 3958 \beta _1 + 3481 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 995 \beta_{13} - 1496 \beta_{12} + 365 \beta_{11} - 614 \beta_{10} + 288 \beta_{9} + 551 \beta_{8} + 2018 \beta_{7} + 4446 \beta_{6} + 699 \beta_{5} - 167 \beta_{4} + 7337 \beta_{3} + 6606 \beta_{2} + 7988 \beta _1 + 11010 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2491 \beta_{13} - 5205 \beta_{12} + 1283 \beta_{11} - 1187 \beta_{10} + 899 \beta_{9} + 1792 \beta_{8} + 4322 \beta_{7} + 11741 \beta_{6} + 2621 \beta_{5} - 765 \beta_{4} + 19614 \beta_{3} + 14305 \beta_{2} + \cdots + 22483 \) Copy content Toggle raw display

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
2.55803
2.41923
1.93259
1.80987
1.63095
1.21108
0.516004
0.290090
−0.204332
−1.07223
−1.28664
−1.77724
−1.99436
−2.03304
−2.55803 1.00000 4.54352 2.13160 −2.55803 2.16070 −6.50640 1.00000 −5.45270
1.2 −2.41923 1.00000 3.85265 1.06966 −2.41923 −2.11008 −4.48198 1.00000 −2.58775
1.3 −1.93259 1.00000 1.73492 −3.60381 −1.93259 −3.30664 0.512291 1.00000 6.96471
1.4 −1.80987 1.00000 1.27562 −0.420571 −1.80987 2.61231 1.31103 1.00000 0.761178
1.5 −1.63095 1.00000 0.659996 −0.781986 −1.63095 0.585172 2.18548 1.00000 1.27538
1.6 −1.21108 1.00000 −0.533291 3.92043 −1.21108 −1.14805 3.06801 1.00000 −4.74795
1.7 −0.516004 1.00000 −1.73374 −1.64789 −0.516004 −2.94857 1.92662 1.00000 0.850319
1.8 −0.290090 1.00000 −1.91585 2.92267 −0.290090 −3.99462 1.13595 1.00000 −0.847839
1.9 0.204332 1.00000 −1.95825 −3.44302 0.204332 −0.996968 −0.808797 1.00000 −0.703520
1.10 1.07223 1.00000 −0.850330 0.491774 1.07223 0.441186 −3.05620 1.00000 0.527293
1.11 1.28664 1.00000 −0.344561 −1.08073 1.28664 4.13388 −3.01660 1.00000 −1.39051
1.12 1.77724 1.00000 1.15857 −2.35884 1.77724 0.213215 −1.49542 1.00000 −4.19221
1.13 1.99436 1.00000 1.97748 1.32527 1.99436 −4.63060 −0.0449167 1.00000 2.64306
1.14 2.03304 1.00000 2.13326 −1.52455 2.03304 −2.01094 0.270928 1.00000 −3.09947
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

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 3009.2.a.d 14
3.b odd 2 1 9027.2.a.l 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3009.2.a.d 14 1.a even 1 1 trivial
9027.2.a.l 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3009))\):

\( T_{2}^{14} + 4 T_{2}^{13} - 11 T_{2}^{12} - 57 T_{2}^{11} + 33 T_{2}^{10} + 311 T_{2}^{9} + 33 T_{2}^{8} - 805 T_{2}^{7} - 330 T_{2}^{6} + 981 T_{2}^{5} + 542 T_{2}^{4} - 454 T_{2}^{3} - 283 T_{2}^{2} + 7 T_{2} + 13 \) Copy content Toggle raw display
\( T_{5}^{14} + 3 T_{5}^{13} - 30 T_{5}^{12} - 95 T_{5}^{11} + 283 T_{5}^{10} + 995 T_{5}^{9} - 906 T_{5}^{8} - 4270 T_{5}^{7} + 257 T_{5}^{6} + 7619 T_{5}^{5} + 2332 T_{5}^{4} - 5258 T_{5}^{3} - 2464 T_{5}^{2} + 911 T_{5} + 445 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 4 T^{13} - 11 T^{12} - 57 T^{11} + \cdots + 13 \) Copy content Toggle raw display
$3$ \( (T - 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 3 T^{13} - 30 T^{12} - 95 T^{11} + \cdots + 445 \) Copy content Toggle raw display
$7$ \( T^{14} + 11 T^{13} + 12 T^{12} + \cdots + 1125 \) Copy content Toggle raw display
$11$ \( T^{14} + 11 T^{13} - 7 T^{12} + \cdots - 9865 \) Copy content Toggle raw display
$13$ \( T^{14} + 23 T^{13} + 164 T^{12} + \cdots - 125061 \) Copy content Toggle raw display
$17$ \( (T - 1)^{14} \) Copy content Toggle raw display
$19$ \( T^{14} + 13 T^{13} - 87 T^{12} + \cdots + 35675443 \) Copy content Toggle raw display
$23$ \( T^{14} + 24 T^{13} + 169 T^{12} + \cdots + 6824059 \) Copy content Toggle raw display
$29$ \( T^{14} + 8 T^{13} - 139 T^{12} + \cdots - 398709 \) Copy content Toggle raw display
$31$ \( T^{14} + 29 T^{13} + \cdots - 278873887 \) Copy content Toggle raw display
$37$ \( T^{14} + 24 T^{13} + \cdots + 163991691 \) Copy content Toggle raw display
$41$ \( T^{14} + 6 T^{13} - 259 T^{12} + \cdots - 1117281 \) Copy content Toggle raw display
$43$ \( T^{14} + 8 T^{13} - 245 T^{12} + \cdots + 95528509 \) Copy content Toggle raw display
$47$ \( T^{14} + 36 T^{13} + \cdots + 15280156029 \) Copy content Toggle raw display
$53$ \( T^{14} + 12 T^{13} + \cdots + 1349330169 \) Copy content Toggle raw display
$59$ \( (T + 1)^{14} \) Copy content Toggle raw display
$61$ \( T^{14} + 16 T^{13} + \cdots - 3428396765 \) Copy content Toggle raw display
$67$ \( T^{14} + 34 T^{13} + \cdots - 344652703 \) Copy content Toggle raw display
$71$ \( T^{14} + 30 T^{13} + \cdots - 255539999 \) Copy content Toggle raw display
$73$ \( T^{14} + 39 T^{13} + \cdots + 363535602511 \) Copy content Toggle raw display
$79$ \( T^{14} + 17 T^{13} + \cdots - 10366133921 \) Copy content Toggle raw display
$83$ \( T^{14} + 5 T^{13} + \cdots - 5090048627 \) Copy content Toggle raw display
$89$ \( T^{14} + 15 T^{13} + \cdots - 2158155569979 \) Copy content Toggle raw display
$97$ \( T^{14} + 2 T^{13} + \cdots + 601509799201 \) Copy content Toggle raw display
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