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Properties

Label 6034.2.a.l
Level $6034$
Weight $2$
Character orbit 6034.a
Self dual yes
Analytic conductor $48.182$
Analytic rank $1$
Dimension $20$
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] = [6034,2,Mod(1,6034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6034, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6034.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: \(48.1817325796\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} - 36 x^{18} + 97 x^{17} + 573 x^{16} - 1292 x^{15} - 5329 x^{14} + 9121 x^{13} + \cdots - 21776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} + (\beta_{7} - 1) q^{5} - \beta_1 q^{6} - q^{7} + q^{8} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} + (\beta_{7} - 1) q^{5} - \beta_1 q^{6} - q^{7} + q^{8} + (\beta_{2} + \beta_1 + 1) q^{9} + (\beta_{7} - 1) q^{10} + (\beta_{6} - 1) q^{11} - \beta_1 q^{12} + ( - \beta_{8} - 1) q^{13} - q^{14} + (\beta_{14} - \beta_{10} + \cdots + \beta_1) q^{15}+ \cdots + (\beta_{19} - \beta_{18} - \beta_{17} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{2} - 3 q^{3} + 20 q^{4} - 10 q^{5} - 3 q^{6} - 20 q^{7} + 20 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{2} - 3 q^{3} + 20 q^{4} - 10 q^{5} - 3 q^{6} - 20 q^{7} + 20 q^{8} + 21 q^{9} - 10 q^{10} - 17 q^{11} - 3 q^{12} - 23 q^{13} - 20 q^{14} - 3 q^{15} + 20 q^{16} - 21 q^{17} + 21 q^{18} - 22 q^{19} - 10 q^{20} + 3 q^{21} - 17 q^{22} + 15 q^{23} - 3 q^{24} - 23 q^{26} - 42 q^{27} - 20 q^{28} - 3 q^{29} - 3 q^{30} - 3 q^{31} + 20 q^{32} - 12 q^{33} - 21 q^{34} + 10 q^{35} + 21 q^{36} - 14 q^{37} - 22 q^{38} + q^{39} - 10 q^{40} - 37 q^{41} + 3 q^{42} - 5 q^{43} - 17 q^{44} - 55 q^{45} + 15 q^{46} - 29 q^{47} - 3 q^{48} + 20 q^{49} - 7 q^{51} - 23 q^{52} - 28 q^{53} - 42 q^{54} + 4 q^{55} - 20 q^{56} - 23 q^{57} - 3 q^{58} - 47 q^{59} - 3 q^{60} - 13 q^{61} - 3 q^{62} - 21 q^{63} + 20 q^{64} - 26 q^{65} - 12 q^{66} - 24 q^{67} - 21 q^{68} - 76 q^{69} + 10 q^{70} - 22 q^{71} + 21 q^{72} - 37 q^{73} - 14 q^{74} - 39 q^{75} - 22 q^{76} + 17 q^{77} + q^{78} + 25 q^{79} - 10 q^{80} - 36 q^{81} - 37 q^{82} - 33 q^{83} + 3 q^{84} - 2 q^{85} - 5 q^{86} - 26 q^{87} - 17 q^{88} - 71 q^{89} - 55 q^{90} + 23 q^{91} + 15 q^{92} - 49 q^{93} - 29 q^{94} - 14 q^{95} - 3 q^{96} - 51 q^{97} + 20 q^{98} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} - 36 x^{18} + 97 x^{17} + 573 x^{16} - 1292 x^{15} - 5329 x^{14} + 9121 x^{13} + \cdots - 21776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3960343 \nu^{19} + 614360943 \nu^{18} - 1462616462 \nu^{17} - 19663940787 \nu^{16} + \cdots + 1809016113384 ) / 3040618192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16644951 \nu^{19} - 81044325 \nu^{18} - 577306280 \nu^{17} + 2574973259 \nu^{16} + \cdots + 335027839864 ) / 1520309096 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 34635180 \nu^{19} - 101591649 \nu^{18} - 1492020713 \nu^{17} + 4034068656 \nu^{16} + \cdots - 2198305565808 ) / 760154548 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24749436 \nu^{19} - 91273626 \nu^{18} - 806665861 \nu^{17} + 2893666957 \nu^{16} + \cdots - 434932067360 ) / 380077274 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 100169175 \nu^{19} - 425556827 \nu^{18} - 3269423778 \nu^{17} + 14088317343 \nu^{16} + \cdots - 3456518219736 ) / 1520309096 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 264438933 \nu^{19} - 677141369 \nu^{18} - 9871031970 \nu^{17} + 22739279353 \nu^{16} + \cdots - 6827044858744 ) / 3040618192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 84627259 \nu^{19} - 299043217 \nu^{18} - 2760722388 \nu^{17} + 9521282651 \nu^{16} + \cdots - 1733210336848 ) / 760154548 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 112313458 \nu^{19} - 531994539 \nu^{18} - 3327989547 \nu^{17} + 17119353404 \nu^{16} + \cdots - 2401691382064 ) / 760154548 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 248833715 \nu^{19} - 1291516455 \nu^{18} - 7059152210 \nu^{17} + 41300189627 \nu^{16} + \cdots - 5022246308728 ) / 1520309096 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 315317385 \nu^{19} - 1293022755 \nu^{18} - 9626405628 \nu^{17} + 40646537601 \nu^{16} + \cdots - 3367570379784 ) / 1520309096 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 189297804 \nu^{19} - 776958427 \nu^{18} - 6117797523 \nu^{17} + 25168764824 \nu^{16} + \cdots - 4582432715876 ) / 760154548 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 835590203 \nu^{19} - 3826397139 \nu^{18} - 25262436066 \nu^{17} + 123370536655 \nu^{16} + \cdots - 20980523930824 ) / 3040618192 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1027938729 \nu^{19} + 4718179533 \nu^{18} + 30511731298 \nu^{17} - 149948611701 \nu^{16} + \cdots + 19966247957256 ) / 3040618192 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 519967205 \nu^{19} - 2072767045 \nu^{18} - 15916321158 \nu^{17} + 64771735209 \nu^{16} + \cdots - 6299303348056 ) / 1520309096 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 274087579 \nu^{19} + 1083920107 \nu^{18} + 8622426264 \nu^{17} - 34420744897 \nu^{16} + \cdots + 4753242395944 ) / 760154548 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 290115955 \nu^{19} + 1153750064 \nu^{18} + 9150086029 \nu^{17} - 36865983801 \nu^{16} + \cdots + 5764506969456 ) / 760154548 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 317683195 \nu^{19} + 1370868643 \nu^{18} + 9610163980 \nu^{17} - 43406255205 \nu^{16} + \cdots + 5699083771564 ) / 760154548 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{18} + \beta_{17} - \beta_{16} + \beta_{14} + \beta_{12} - \beta_{10} + \beta_{8} - \beta_{7} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{19} + \beta_{18} - \beta_{16} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} - \beta_{9} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{19} - 12 \beta_{18} + 12 \beta_{17} - 9 \beta_{16} - \beta_{15} + 10 \beta_{14} - \beta_{13} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 16 \beta_{19} + 14 \beta_{18} + 2 \beta_{17} - 12 \beta_{16} + 2 \beta_{15} + 13 \beta_{14} + \cdots + 201 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 30 \beta_{19} - 117 \beta_{18} + 121 \beta_{17} - 72 \beta_{16} - 17 \beta_{15} + 88 \beta_{14} + \cdots + 174 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 184 \beta_{19} + 151 \beta_{18} + 35 \beta_{17} - 113 \beta_{16} + 36 \beta_{15} + 132 \beta_{14} + \cdots + 1650 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 325 \beta_{19} - 1069 \beta_{18} + 1158 \beta_{17} - 573 \beta_{16} - 199 \beta_{15} + 766 \beta_{14} + \cdots + 1449 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1876 \beta_{19} + 1481 \beta_{18} + 454 \beta_{17} - 993 \beta_{16} + 437 \beta_{15} + 1236 \beta_{14} + \cdots + 13856 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3124 \beta_{19} - 9506 \beta_{18} + 10811 \beta_{17} - 4614 \beta_{16} - 2023 \beta_{15} + 6699 \beta_{14} + \cdots + 11824 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 18080 \beta_{19} + 13930 \beta_{18} + 5196 \beta_{17} - 8507 \beta_{16} + 4540 \beta_{15} + \cdots + 117536 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 28391 \beta_{19} - 83477 \beta_{18} + 99476 \beta_{17} - 37639 \beta_{16} - 19267 \beta_{15} + \cdots + 95468 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 169112 \beta_{19} + 128427 \beta_{18} + 55330 \beta_{17} - 72118 \beta_{16} + 43714 \beta_{15} + \cdots + 1002422 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 250476 \beta_{19} - 728577 \beta_{18} + 906580 \beta_{17} - 310590 \beta_{16} - 177563 \beta_{15} + \cdots + 766341 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1554790 \beta_{19} + 1171964 \beta_{18} + 562008 \beta_{17} - 608905 \beta_{16} + 403928 \beta_{15} + \cdots + 8579038 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 2173001 \beta_{19} - 6339442 \beta_{18} + 8205746 \beta_{17} - 2588120 \beta_{16} - 1608507 \beta_{15} + \cdots + 6131100 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 14145184 \beta_{19} + 10634440 \beta_{18} + 5523546 \beta_{17} - 5137667 \beta_{16} + 3644424 \beta_{15} + \cdots + 73611932 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 18666972 \beta_{19} - 55075743 \beta_{18} + 73890253 \beta_{17} - 21746507 \beta_{16} - 14440173 \beta_{15} + \cdots + 48954545 \) 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
2.98166
2.88014
2.84514
2.80072
2.78441
1.59091
1.42043
1.35617
1.32183
−0.395900
−0.860159
−0.903145
−1.01642
−1.35221
−1.37338
−1.76208
−1.78235
−2.22609
−2.34324
−2.96642
1.00000 −2.98166 1.00000 −1.98359 −2.98166 −1.00000 1.00000 5.89028 −1.98359
1.2 1.00000 −2.88014 1.00000 −1.11289 −2.88014 −1.00000 1.00000 5.29519 −1.11289
1.3 1.00000 −2.84514 1.00000 −4.15918 −2.84514 −1.00000 1.00000 5.09480 −4.15918
1.4 1.00000 −2.80072 1.00000 0.235711 −2.80072 −1.00000 1.00000 4.84406 0.235711
1.5 1.00000 −2.78441 1.00000 1.22741 −2.78441 −1.00000 1.00000 4.75292 1.22741
1.6 1.00000 −1.59091 1.00000 3.74979 −1.59091 −1.00000 1.00000 −0.468990 3.74979
1.7 1.00000 −1.42043 1.00000 −4.18942 −1.42043 −1.00000 1.00000 −0.982389 −4.18942
1.8 1.00000 −1.35617 1.00000 1.14574 −1.35617 −1.00000 1.00000 −1.16079 1.14574
1.9 1.00000 −1.32183 1.00000 1.76551 −1.32183 −1.00000 1.00000 −1.25276 1.76551
1.10 1.00000 0.395900 1.00000 0.151677 0.395900 −1.00000 1.00000 −2.84326 0.151677
1.11 1.00000 0.860159 1.00000 −0.340662 0.860159 −1.00000 1.00000 −2.26013 −0.340662
1.12 1.00000 0.903145 1.00000 −3.69995 0.903145 −1.00000 1.00000 −2.18433 −3.69995
1.13 1.00000 1.01642 1.00000 1.24888 1.01642 −1.00000 1.00000 −1.96689 1.24888
1.14 1.00000 1.35221 1.00000 2.46727 1.35221 −1.00000 1.00000 −1.17152 2.46727
1.15 1.00000 1.37338 1.00000 1.18048 1.37338 −1.00000 1.00000 −1.11382 1.18048
1.16 1.00000 1.76208 1.00000 −1.26930 1.76208 −1.00000 1.00000 0.104927 −1.26930
1.17 1.00000 1.78235 1.00000 −0.422890 1.78235 −1.00000 1.00000 0.176787 −0.422890
1.18 1.00000 2.22609 1.00000 −2.43328 2.22609 −1.00000 1.00000 1.95547 −2.43328
1.19 1.00000 2.34324 1.00000 −0.555415 2.34324 −1.00000 1.00000 2.49080 −0.555415
1.20 1.00000 2.96642 1.00000 −3.00590 2.96642 −1.00000 1.00000 5.79966 −3.00590
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(431\) \(-1\)

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 6034.2.a.l ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6034.2.a.l ✓ 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):

\( T_{3}^{20} + 3 T_{3}^{19} - 36 T_{3}^{18} - 97 T_{3}^{17} + 573 T_{3}^{16} + 1292 T_{3}^{15} + \cdots - 21776 \) Copy content Toggle raw display
\( T_{5}^{20} + 10 T_{5}^{19} - 277 T_{5}^{17} - 566 T_{5}^{16} + 2587 T_{5}^{15} + 7633 T_{5}^{14} + \cdots - 128 \) Copy content Toggle raw display
\( T_{11}^{20} + 17 T_{11}^{19} + 42 T_{11}^{18} - 774 T_{11}^{17} - 4654 T_{11}^{16} + 6516 T_{11}^{15} + \cdots - 345720 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 3 T^{19} + \cdots - 21776 \) Copy content Toggle raw display
$5$ \( T^{20} + 10 T^{19} + \cdots - 128 \) Copy content Toggle raw display
$7$ \( (T + 1)^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + 17 T^{19} + \cdots - 345720 \) Copy content Toggle raw display
$13$ \( T^{20} + 23 T^{19} + \cdots - 4949592 \) Copy content Toggle raw display
$17$ \( T^{20} + 21 T^{19} + \cdots + 1401308 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 1695859280 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 597971184 \) Copy content Toggle raw display
$29$ \( T^{20} + 3 T^{19} + \cdots + 36255296 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 413038058464 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 33233417986 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 38665816371776 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 9609523776 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 282678353948960 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 451766512 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 7999222586736 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 9961870290248 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 743708690140232 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 114538969971040 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 32\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 58222444652928 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 11\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 339011361266068 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 19\!\cdots\!48 \) Copy content Toggle raw display
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