Properties

Label 8034.2.a.bb
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
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] = [8034,2,Mod(1,8034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, 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("8034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.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: \(64.1518129839\)
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: \( x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} + \cdots - 2048 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - \beta_1 q^{5} - q^{6} + \beta_{6} q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - \beta_1 q^{5} - q^{6} + \beta_{6} q^{7} - q^{8} + q^{9} + \beta_1 q^{10} + (\beta_{9} - 1) q^{11} + q^{12} - q^{13} - \beta_{6} q^{14} - \beta_1 q^{15} + q^{16} + ( - \beta_{10} - \beta_{6} - 1) q^{17} - q^{18} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_1) q^{19} - \beta_1 q^{20} + \beta_{6} q^{21} + ( - \beta_{9} + 1) q^{22} + ( - \beta_{6} - \beta_{5} - 1) q^{23} - q^{24} + (\beta_{13} + \beta_{11} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{25} + q^{26} + q^{27} + \beta_{6} q^{28} + ( - \beta_{12} + \beta_{11} - \beta_{7}) q^{29} + \beta_1 q^{30} + (\beta_{11} + \beta_{6} + \beta_{5} + \beta_{2}) q^{31} - q^{32} + (\beta_{9} - 1) q^{33} + (\beta_{10} + \beta_{6} + 1) q^{34} + ( - \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{9} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \cdots - 2) q^{35}+ \cdots + (\beta_{9} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} + 14 q^{3} + 14 q^{4} - 6 q^{5} - 14 q^{6} - 4 q^{7} - 14 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{2} + 14 q^{3} + 14 q^{4} - 6 q^{5} - 14 q^{6} - 4 q^{7} - 14 q^{8} + 14 q^{9} + 6 q^{10} - 8 q^{11} + 14 q^{12} - 14 q^{13} + 4 q^{14} - 6 q^{15} + 14 q^{16} - 4 q^{17} - 14 q^{18} - q^{19} - 6 q^{20} - 4 q^{21} + 8 q^{22} - 9 q^{23} - 14 q^{24} + 24 q^{25} + 14 q^{26} + 14 q^{27} - 4 q^{28} - 10 q^{29} + 6 q^{30} - 5 q^{31} - 14 q^{32} - 8 q^{33} + 4 q^{34} - 16 q^{35} + 14 q^{36} - 4 q^{37} + q^{38} - 14 q^{39} + 6 q^{40} - 24 q^{41} + 4 q^{42} - 8 q^{44} - 6 q^{45} + 9 q^{46} - 32 q^{47} + 14 q^{48} + 24 q^{49} - 24 q^{50} - 4 q^{51} - 14 q^{52} - 5 q^{53} - 14 q^{54} - 8 q^{55} + 4 q^{56} - q^{57} + 10 q^{58} - 13 q^{59} - 6 q^{60} + 2 q^{61} + 5 q^{62} - 4 q^{63} + 14 q^{64} + 6 q^{65} + 8 q^{66} - 16 q^{67} - 4 q^{68} - 9 q^{69} + 16 q^{70} - 29 q^{71} - 14 q^{72} + 4 q^{74} + 24 q^{75} - q^{76} - 9 q^{77} + 14 q^{78} - 21 q^{79} - 6 q^{80} + 14 q^{81} + 24 q^{82} - 40 q^{83} - 4 q^{84} - 7 q^{85} - 10 q^{87} + 8 q^{88} - 48 q^{89} + 6 q^{90} + 4 q^{91} - 9 q^{92} - 5 q^{93} + 32 q^{94} - 26 q^{95} - 14 q^{96} + 18 q^{97} - 24 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} + \cdots - 2048 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17211124339 \nu^{13} + 48838930211 \nu^{12} - 1083388884247 \nu^{11} - 1278700159484 \nu^{10} + 21896289725388 \nu^{9} + \cdots + 118321024199296 ) / 165587393151104 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 88374380939 \nu^{13} + 341475795883 \nu^{12} + 3738713297463 \nu^{11} - 13680291423778 \nu^{10} + \cdots + 823088103896320 ) / 165587393151104 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 406907271083 \nu^{13} + 2304692170158 \nu^{12} + 11004973389543 \nu^{11} - 75016193619169 \nu^{10} + \cdots + 39\!\cdots\!20 ) / 662349572604416 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 113401884803 \nu^{13} - 883549115785 \nu^{12} - 2323893522087 \nu^{11} + 29836393456848 \nu^{10} - 852019922224 \nu^{9} + \cdots - 10\!\cdots\!04 ) / 165587393151104 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 736531555801 \nu^{13} - 3946026327910 \nu^{12} - 23999796454509 \nu^{11} + 138185242548231 \nu^{10} + \cdots - 32\!\cdots\!20 ) / 662349572604416 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1012895827421 \nu^{13} - 5401357076630 \nu^{12} - 30269645318433 \nu^{11} + 182412406997691 \nu^{10} + \cdots - 804113198053376 ) / 662349572604416 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 161946406606 \nu^{13} + 993100465781 \nu^{12} + 4568699998010 \nu^{11} - 33836085473159 \nu^{10} + \cdots + 821129125068864 ) / 82793696575552 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 339042867519 \nu^{13} + 2193728871065 \nu^{12} + 8513285647275 \nu^{11} - 73385354065404 \nu^{10} + \cdots + 21\!\cdots\!32 ) / 165587393151104 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2206595272859 \nu^{13} + 14412109193718 \nu^{12} + 54849889967511 \nu^{11} - 474790047743417 \nu^{10} + \cdots + 67\!\cdots\!36 ) / 662349572604416 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2312493819663 \nu^{13} + 15946662074178 \nu^{12} + 56053557441147 \nu^{11} - 536833570831273 \nu^{10} + \cdots + 10\!\cdots\!76 ) / 662349572604416 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 155979666211 \nu^{13} + 1092105641867 \nu^{12} + 3542117432817 \nu^{11} - 36139437565114 \nu^{10} + \cdots + 564375014073344 ) / 41396848287776 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2734835840775 \nu^{13} - 17117209536866 \nu^{12} - 75341966167987 \nu^{11} + 586439935888449 \nu^{10} + \cdots - 17\!\cdots\!84 ) / 662349572604416 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{11} + \beta_{3} - \beta_{2} + \beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - 2\beta_{12} + 3\beta_{11} + \beta_{8} - \beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} + 10\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13 \beta_{13} - 4 \beta_{12} + 19 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + 5 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} + 14 \beta_{3} - 14 \beta_{2} + 17 \beta _1 + 88 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 26 \beta_{13} - 37 \beta_{12} + 65 \beta_{11} - 3 \beta_{10} - 5 \beta_{9} + 18 \beta_{8} - 20 \beta_{7} + 33 \beta_{6} + 16 \beta_{5} + 31 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + 123 \beta _1 + 148 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 167 \beta_{13} - 103 \beta_{12} + 335 \beta_{11} + 18 \beta_{10} - 41 \beta_{9} - 37 \beta_{8} - 17 \beta_{7} + 111 \beta_{6} + 113 \beta_{5} + 108 \beta_{4} + 177 \beta_{3} - 177 \beta_{2} + 276 \beta _1 + 1263 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 479 \beta_{13} - 611 \beta_{12} + 1200 \beta_{11} - 93 \beta_{10} - 162 \beta_{9} + 242 \beta_{8} - 350 \beta_{7} + 483 \beta_{6} + 315 \beta_{5} + 679 \beta_{4} + 107 \beta_{3} - 18 \beta_{2} + 1682 \beta _1 + 2702 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2222 \beta_{13} - 2060 \beta_{12} + 5775 \beta_{11} + 203 \beta_{10} - 1006 \beta_{9} - 885 \beta_{8} - 542 \beta_{7} + 1884 \beta_{6} + 2415 \beta_{5} + 2350 \beta_{4} + 2137 \beta_{3} - 2159 \beta_{2} + \cdots + 19087 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 7842 \beta_{13} - 10023 \beta_{12} + 21287 \beta_{11} - 2124 \beta_{10} - 3977 \beta_{9} + 2676 \beta_{8} - 5991 \beta_{7} + 6985 \beta_{6} + 6668 \beta_{5} + 13401 \beta_{4} + 1459 \beta_{3} + 285 \beta_{2} + \cdots + 47413 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 30616 \beta_{13} - 37773 \beta_{12} + 98893 \beta_{11} + 986 \beta_{10} - 21437 \beta_{9} - 18052 \beta_{8} - 12588 \beta_{7} + 29338 \beta_{6} + 46972 \beta_{5} + 47547 \beta_{4} + 24544 \beta_{3} + \cdots + 296729 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 122139 \beta_{13} - 166504 \beta_{12} + 373041 \beta_{11} - 43026 \beta_{10} - 86704 \beta_{9} + 20835 \beta_{8} - 102258 \beta_{7} + 101217 \beta_{6} + 138595 \beta_{5} + 252965 \beta_{4} + \cdots + 819867 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 434104 \beta_{13} - 667453 \beta_{12} + 1693145 \beta_{11} - 27689 \beta_{10} - 430192 \beta_{9} - 343448 \beta_{8} - 259336 \beta_{7} + 440956 \beta_{6} + 876463 \beta_{5} + 925564 \beta_{4} + \cdots + 4704078 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1860057 \beta_{13} - 2802582 \beta_{12} + 6513435 \beta_{11} - 823424 \beta_{10} - 1769865 \beta_{9} - 35291 \beta_{8} - 1747648 \beta_{7} + 1471030 \beta_{6} + 2780674 \beta_{5} + \cdots + 14098951 \) 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
4.19123
3.81560
3.75223
2.57230
2.22430
1.73527
0.453790
−0.290663
−0.480706
−1.34813
−1.50270
−2.36221
−3.07028
−3.69005
−1.00000 1.00000 1.00000 −4.19123 −1.00000 −2.48485 −1.00000 1.00000 4.19123
1.2 −1.00000 1.00000 1.00000 −3.81560 −1.00000 4.94203 −1.00000 1.00000 3.81560
1.3 −1.00000 1.00000 1.00000 −3.75223 −1.00000 1.91373 −1.00000 1.00000 3.75223
1.4 −1.00000 1.00000 1.00000 −2.57230 −1.00000 −4.81484 −1.00000 1.00000 2.57230
1.5 −1.00000 1.00000 1.00000 −2.22430 −1.00000 1.37131 −1.00000 1.00000 2.22430
1.6 −1.00000 1.00000 1.00000 −1.73527 −1.00000 −3.07502 −1.00000 1.00000 1.73527
1.7 −1.00000 1.00000 1.00000 −0.453790 −1.00000 3.87720 −1.00000 1.00000 0.453790
1.8 −1.00000 1.00000 1.00000 0.290663 −1.00000 −3.15490 −1.00000 1.00000 −0.290663
1.9 −1.00000 1.00000 1.00000 0.480706 −1.00000 0.765483 −1.00000 1.00000 −0.480706
1.10 −1.00000 1.00000 1.00000 1.34813 −1.00000 2.71387 −1.00000 1.00000 −1.34813
1.11 −1.00000 1.00000 1.00000 1.50270 −1.00000 0.210711 −1.00000 1.00000 −1.50270
1.12 −1.00000 1.00000 1.00000 2.36221 −1.00000 −4.26027 −1.00000 1.00000 −2.36221
1.13 −1.00000 1.00000 1.00000 3.07028 −1.00000 −1.20257 −1.00000 1.00000 −3.07028
1.14 −1.00000 1.00000 1.00000 3.69005 −1.00000 −0.801879 −1.00000 1.00000 −3.69005
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(13\) \(1\)
\(103\) \(-1\)

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 8034.2.a.bb 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.bb 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{14} + 6 T_{5}^{13} - 29 T_{5}^{12} - 207 T_{5}^{11} + 269 T_{5}^{10} + 2601 T_{5}^{9} - 847 T_{5}^{8} - 14851 T_{5}^{7} + 678 T_{5}^{6} + 39390 T_{5}^{5} - 3280 T_{5}^{4} - 42456 T_{5}^{3} + 10816 T_{5}^{2} + \cdots - 2048 \) Copy content Toggle raw display
\( T_{7}^{14} + 4 T_{7}^{13} - 53 T_{7}^{12} - 214 T_{7}^{11} + 963 T_{7}^{10} + 3920 T_{7}^{9} - 7497 T_{7}^{8} - 30365 T_{7}^{7} + 27271 T_{7}^{6} + 101391 T_{7}^{5} - 49359 T_{7}^{4} - 129678 T_{7}^{3} + 37432 T_{7}^{2} + \cdots - 10496 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{14} \) Copy content Toggle raw display
$3$ \( (T - 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 6 T^{13} - 29 T^{12} + \cdots - 2048 \) Copy content Toggle raw display
$7$ \( T^{14} + 4 T^{13} - 53 T^{12} + \cdots - 10496 \) Copy content Toggle raw display
$11$ \( T^{14} + 8 T^{13} - 39 T^{12} + \cdots - 234648 \) Copy content Toggle raw display
$13$ \( (T + 1)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + 4 T^{13} - 126 T^{12} + \cdots - 309504 \) Copy content Toggle raw display
$19$ \( T^{14} + T^{13} - 160 T^{12} + \cdots + 1835136 \) Copy content Toggle raw display
$23$ \( T^{14} + 9 T^{13} - 104 T^{12} + \cdots - 3987232 \) Copy content Toggle raw display
$29$ \( T^{14} + 10 T^{13} + \cdots - 151832576 \) Copy content Toggle raw display
$31$ \( T^{14} + 5 T^{13} - 188 T^{12} + \cdots - 4772992 \) Copy content Toggle raw display
$37$ \( T^{14} + 4 T^{13} + \cdots + 1682701888 \) Copy content Toggle raw display
$41$ \( T^{14} + 24 T^{13} + \cdots + 104915276544 \) Copy content Toggle raw display
$43$ \( T^{14} - 231 T^{12} + \cdots + 37069568 \) Copy content Toggle raw display
$47$ \( T^{14} + 32 T^{13} + 279 T^{12} + \cdots + 23612928 \) Copy content Toggle raw display
$53$ \( T^{14} + 5 T^{13} + \cdots + 74233155072 \) Copy content Toggle raw display
$59$ \( T^{14} + 13 T^{13} + \cdots + 10577117773824 \) Copy content Toggle raw display
$61$ \( T^{14} - 2 T^{13} - 323 T^{12} + \cdots + 67974144 \) Copy content Toggle raw display
$67$ \( T^{14} + 16 T^{13} + \cdots - 6677658624 \) Copy content Toggle raw display
$71$ \( T^{14} + 29 T^{13} + \cdots + 3192115560448 \) Copy content Toggle raw display
$73$ \( T^{14} - 428 T^{12} + \cdots + 13746176 \) Copy content Toggle raw display
$79$ \( T^{14} + 21 T^{13} + \cdots + 562577408 \) Copy content Toggle raw display
$83$ \( T^{14} + 40 T^{13} + \cdots - 663287808 \) Copy content Toggle raw display
$89$ \( T^{14} + 48 T^{13} + \cdots - 416381859328 \) Copy content Toggle raw display
$97$ \( T^{14} - 18 T^{13} + \cdots + 6667509669504 \) Copy content Toggle raw display
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