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Properties

Label 8025.2.a.bk
Level $8025$
Weight $2$
Character orbit 8025.a
Self dual yes
Analytic conductor $64.080$
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] = [8025,2,Mod(1,8025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8025, 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("8025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8025.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: \(64.0799476221\)
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: \( x^{17} - 4 x^{16} - 17 x^{15} + 81 x^{14} + 94 x^{13} - 635 x^{12} - 112 x^{11} + 2429 x^{10} - 643 x^{9} + \cdots - 8 \) 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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 4 x^{16} - 17 x^{15} + 81 x^{14} + 94 x^{13} - 635 x^{12} - 112 x^{11} + 2429 x^{10} - 643 x^{9} + \cdots - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1748469 \nu^{16} - 25367918 \nu^{15} + 20418323 \nu^{14} + 508467127 \nu^{13} - 839155584 \nu^{12} + \cdots - 583221244 ) / 128123212 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 91417 \nu^{16} - 601608 \nu^{15} - 1244239 \nu^{14} + 12687713 \nu^{13} + 2687626 \nu^{12} + \cdots - 10639235 ) / 4575829 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2231864 \nu^{16} + 1036215 \nu^{15} + 59480346 \nu^{14} - 24244003 \nu^{13} - 639963479 \nu^{12} + \cdots - 363479666 ) / 64061606 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1075027 \nu^{16} - 2827746 \nu^{15} - 20633779 \nu^{14} + 54142245 \nu^{13} + 145391508 \nu^{12} + \cdots - 43480832 ) / 18303316 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8266425 \nu^{16} + 38998826 \nu^{15} + 117833185 \nu^{14} - 775070667 \nu^{13} + \cdots + 987245460 ) / 128123212 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8604963 \nu^{16} - 25353646 \nu^{15} - 166047935 \nu^{14} + 501649925 \nu^{13} + \cdots - 523215888 ) / 128123212 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4736786 \nu^{16} - 20853033 \nu^{15} - 81543814 \nu^{14} + 436817227 \nu^{13} + 462493921 \nu^{12} + \cdots - 88998402 ) / 64061606 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7990051 \nu^{16} + 22312303 \nu^{15} + 161800507 \nu^{14} - 450212084 \nu^{13} + \cdots - 30309134 ) / 64061606 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 18022307 \nu^{16} - 49788790 \nu^{15} - 363087443 \nu^{14} + 1008481397 \nu^{13} + \cdots + 87485268 ) / 128123212 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 20155081 \nu^{16} - 85234102 \nu^{15} - 340040685 \nu^{14} + 1738405627 \nu^{13} + \cdots + 351365640 ) / 128123212 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10602351 \nu^{16} - 36139681 \nu^{15} - 195925887 \nu^{14} + 732084206 \nu^{13} + 1307118167 \nu^{12} + \cdots - 62105646 ) / 64061606 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 24367461 \nu^{16} - 88392212 \nu^{15} - 444722769 \nu^{14} + 1800674141 \nu^{13} + \cdots + 405494252 ) / 128123212 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 14885671 \nu^{16} - 57517008 \nu^{15} - 259308417 \nu^{14} + 1163584443 \nu^{13} + \cdots - 103617912 ) / 64061606 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} + 6\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + \beta_{13} + \beta_{12} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 9\beta_{3} + \beta_{2} + 28\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9 \beta_{15} - 11 \beta_{14} + \beta_{13} + 12 \beta_{12} + 11 \beta_{11} - \beta_{9} + 10 \beta_{8} + \cdots + 73 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{16} - 12 \beta_{15} - 3 \beta_{14} + 11 \beta_{13} + 14 \beta_{12} + \beta_{11} + 2 \beta_{10} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3 \beta_{16} + 61 \beta_{15} - 94 \beta_{14} + 15 \beta_{13} + 108 \beta_{12} + 88 \beta_{11} + \cdots + 395 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 46 \beta_{16} - 105 \beta_{15} - 52 \beta_{14} + 94 \beta_{13} + 144 \beta_{12} + 17 \beta_{11} + \cdots - 109 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 54 \beta_{16} + 369 \beta_{15} - 737 \beta_{14} + 157 \beta_{13} + 880 \beta_{12} + 626 \beta_{11} + \cdots + 2159 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 482 \beta_{16} - 822 \beta_{15} - 610 \beta_{14} + 749 \beta_{13} + 1320 \beta_{12} + 188 \beta_{11} + \cdots - 910 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 652 \beta_{16} + 2069 \beta_{15} - 5574 \beta_{14} + 1431 \beta_{13} + 6867 \beta_{12} + 4212 \beta_{11} + \cdots + 11737 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4346 \beta_{16} - 6142 \beta_{15} - 6070 \beta_{14} + 5825 \beta_{13} + 11417 \beta_{12} + 1726 \beta_{11} + \cdots - 7384 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 6650 \beta_{16} + 10702 \beta_{15} - 41526 \beta_{14} + 12190 \beta_{13} + 52521 \beta_{12} + \cdots + 62550 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 36440 \beta_{16} - 45041 \beta_{15} - 55317 \beta_{14} + 44848 \beta_{13} + 95412 \beta_{12} + \cdots - 59378 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 61951 \beta_{16} + 48739 \beta_{15} - 307818 \beta_{14} + 100033 \beta_{13} + 398100 \beta_{12} + \cdots + 320328 \) 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
2.77157
2.51457
2.22568
2.11632
1.63863
1.55192
0.983746
0.562840
0.327080
−0.114567
−0.294160
−0.786610
−1.07783
−1.58348
−2.00883
−2.36160
−2.46528
−2.77157 −1.00000 5.68163 0 2.77157 0.100745 −10.2039 1.00000 0
1.2 −2.51457 −1.00000 4.32307 0 2.51457 −3.29454 −5.84152 1.00000 0
1.3 −2.22568 −1.00000 2.95367 0 2.22568 2.90739 −2.12258 1.00000 0
1.4 −2.11632 −1.00000 2.47880 0 2.11632 −2.52450 −1.01329 1.00000 0
1.5 −1.63863 −1.00000 0.685124 0 1.63863 −4.99661 2.15460 1.00000 0
1.6 −1.55192 −1.00000 0.408445 0 1.55192 1.99208 2.46996 1.00000 0
1.7 −0.983746 −1.00000 −1.03224 0 0.983746 −1.25907 2.98296 1.00000 0
1.8 −0.562840 −1.00000 −1.68321 0 0.562840 −0.979667 2.07306 1.00000 0
1.9 −0.327080 −1.00000 −1.89302 0 0.327080 2.73512 1.27333 1.00000 0
1.10 0.114567 −1.00000 −1.98687 0 −0.114567 0.982937 −0.456763 1.00000 0
1.11 0.294160 −1.00000 −1.91347 0 −0.294160 −4.05246 −1.15119 1.00000 0
1.12 0.786610 −1.00000 −1.38125 0 −0.786610 −1.88827 −2.65972 1.00000 0
1.13 1.07783 −1.00000 −0.838278 0 −1.07783 2.37765 −3.05919 1.00000 0
1.14 1.58348 −1.00000 0.507395 0 −1.58348 −1.83980 −2.36350 1.00000 0
1.15 2.00883 −1.00000 2.03542 0 −2.00883 2.62425 0.0711429 1.00000 0
1.16 2.36160 −1.00000 3.57718 0 −2.36160 −4.08057 3.72467 1.00000 0
1.17 2.46528 −1.00000 4.07762 0 −2.46528 2.19531 5.12192 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(107\) \(-1\)

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 8025.2.a.bk ✓ 17
5.b even 2 1 8025.2.a.bl yes 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8025.2.a.bk ✓ 17 1.a even 1 1 trivial
8025.2.a.bl yes 17 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):

\( T_{2}^{17} + 4 T_{2}^{16} - 17 T_{2}^{15} - 81 T_{2}^{14} + 94 T_{2}^{13} + 635 T_{2}^{12} - 112 T_{2}^{11} + \cdots + 8 \) Copy content Toggle raw display
\( T_{7}^{17} + 9 T_{7}^{16} - 21 T_{7}^{15} - 355 T_{7}^{14} - 34 T_{7}^{13} + 5863 T_{7}^{12} + \cdots + 63276 \) Copy content Toggle raw display
\( T_{11}^{17} - 7 T_{11}^{16} - 86 T_{11}^{15} + 675 T_{11}^{14} + 2617 T_{11}^{13} - 25565 T_{11}^{12} + \cdots + 18711366 \) Copy content Toggle raw display
\( T_{13}^{17} + 8 T_{13}^{16} - 81 T_{13}^{15} - 748 T_{13}^{14} + 1830 T_{13}^{13} + 24347 T_{13}^{12} + \cdots + 17920 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + 4 T^{16} + \cdots + 8 \) Copy content Toggle raw display
$3$ \( (T + 1)^{17} \) Copy content Toggle raw display
$5$ \( T^{17} \) Copy content Toggle raw display
$7$ \( T^{17} + 9 T^{16} + \cdots + 63276 \) Copy content Toggle raw display
$11$ \( T^{17} - 7 T^{16} + \cdots + 18711366 \) Copy content Toggle raw display
$13$ \( T^{17} + 8 T^{16} + \cdots + 17920 \) Copy content Toggle raw display
$17$ \( T^{17} + 7 T^{16} + \cdots - 57176 \) Copy content Toggle raw display
$19$ \( T^{17} - 4 T^{16} + \cdots + 39321959 \) Copy content Toggle raw display
$23$ \( T^{17} + 25 T^{16} + \cdots + 10364544 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 438456210 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots + 2915005644 \) Copy content Toggle raw display
$37$ \( T^{17} + 15 T^{16} + \cdots + 87440206 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots + 21572953384 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 229923252964 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 274054377072 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 36141643372 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 9442357383528 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 1546818780029 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 3098393263296 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 34058607087776 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 6622245000 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 2426161152 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 53778954221154 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 3199270520304 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 50846123432508 \) Copy content Toggle raw display
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