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Properties

Label 289.4.b.d
Level $289$
Weight $4$
Character orbit 289.b
Analytic conductor $17.052$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,4,Mod(288,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.288");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(17.0515519917\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 43x^{6} + 505x^{4} + 1528x^{2} + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{6} - \beta_{5} + \beta_1) q^{3} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 3) q^{4} + (2 \beta_{7} + \beta_{6} + 4 \beta_{5} - \beta_1) q^{5} + ( - \beta_{7} + 3 \beta_{6} - 11 \beta_{5} + 2 \beta_1) q^{6} + (5 \beta_{6} - 10 \beta_{5} - \beta_1) q^{7} + (3 \beta_{4} - \beta_{3} + 2 \beta_{2} - 15) q^{8} + (2 \beta_{4} + 3 \beta_{3} - 9 \beta_{2} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{6} - \beta_{5} + \beta_1) q^{3} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 3) q^{4} + (2 \beta_{7} + \beta_{6} + 4 \beta_{5} - \beta_1) q^{5} + ( - \beta_{7} + 3 \beta_{6} - 11 \beta_{5} + 2 \beta_1) q^{6} + (5 \beta_{6} - 10 \beta_{5} - \beta_1) q^{7} + (3 \beta_{4} - \beta_{3} + 2 \beta_{2} - 15) q^{8} + (2 \beta_{4} + 3 \beta_{3} - 9 \beta_{2} - 5) q^{9} + ( - 3 \beta_{7} + \beta_{6} + 3 \beta_{5} - 9 \beta_1) q^{10} + (4 \beta_{7} - 2 \beta_{6} - \beta_{5}) q^{11} + ( - 10 \beta_{5} + 11 \beta_1) q^{12} + (2 \beta_{4} + 5 \beta_{3} + 9 \beta_{2} - 5) q^{13} + (\beta_{7} + 9 \beta_{6} + 11 \beta_{5} + 21 \beta_1) q^{14} + ( - 10 \beta_{3} + 8 \beta_{2} - 9) q^{15} + (4 \beta_{3} - 19 \beta_{2} + 10) q^{16} + (5 \beta_{4} + 15 \beta_{3} - 10 \beta_{2} - 91) q^{18} + (4 \beta_{4} - 11 \beta_{3} - 13 \beta_{2} - 7) q^{19} + ( - \beta_{7} - 15 \beta_{6} + 79 \beta_{5} + 28 \beta_1) q^{20} + (4 \beta_{4} + 2 \beta_{3} - 38 \beta_{2} - 99) q^{21} + ( - 8 \beta_{7} - 4 \beta_{6} - 16 \beta_{5} - 19 \beta_1) q^{22} + (4 \beta_{7} - \beta_{6} - 100 \beta_{5} + 25 \beta_1) q^{23} + ( - 3 \beta_{7} - 13 \beta_{6} - 33 \beta_{5} - 17 \beta_1) q^{24} + ( - 18 \beta_{4} - 7 \beta_{3} + 25 \beta_{2} - 82) q^{25} + ( - 13 \beta_{4} + \beta_{3} - 32 \beta_{2} + 107) q^{26} + (12 \beta_{7} - 4 \beta_{6} + 133 \beta_{5} - 22 \beta_1) q^{27} + ( - 23 \beta_{7} - \beta_{6} - 155 \beta_{5} - 10 \beta_1) q^{28} + (6 \beta_{7} - \beta_{6} - 15 \beta_{5} - 11 \beta_1) q^{29} + ( - 8 \beta_{4} - 28 \beta_{3} + 3 \beta_{2} + 88) q^{30} + ( - 8 \beta_{7} - 16 \beta_{6} + 93 \beta_{5} - 2 \beta_1) q^{31} + ( - 5 \beta_{4} + 35 \beta_{3} + 5 \beta_{2} - 89) q^{32} + ( - 2 \beta_{4} - \beta_{3} + 27 \beta_{2} + 31) q^{33} + ( - 28 \beta_{4} - 51 \beta_{3} - 9 \beta_{2} - 43) q^{35} + ( - 16 \beta_{4} + 16 \beta_{3} - 59 \beta_{2} - 50) q^{36} + (4 \beta_{7} + 46 \beta_{6} + 108 \beta_{5} + 38 \beta_1) q^{37} + (5 \beta_{4} - 9 \beta_{3} + 12 \beta_{2} - 127) q^{38} + ( - 4 \beta_{7} + 25 \beta_{6} + 2 \beta_{5} - 25 \beta_1) q^{39} + ( - 2 \beta_{7} - 10 \beta_{6} - 328 \beta_{5} - 61 \beta_1) q^{40} + (12 \beta_{7} + 4 \beta_{6} - 210 \beta_{5} + 8 \beta_1) q^{41} + (30 \beta_{4} + 42 \beta_{3} - 81 \beta_{2} - 402) q^{42} + (24 \beta_{4} - 32 \beta_{3} + 50 \beta_{2} + 43) q^{43} + (3 \beta_{7} - 11 \beta_{6} + 249 \beta_{5} + 59 \beta_1) q^{44} + (36 \beta_{7} + 32 \beta_{6} - 171 \beta_{5} + 40 \beta_1) q^{45} + ( - 33 \beta_{7} + 23 \beta_{6} - 291 \beta_{5} + 57 \beta_1) q^{46} + (4 \beta_{4} + 42 \beta_{3} - 32 \beta_{2} - 17) q^{47} + (23 \beta_{7} - 43 \beta_{6} + 279 \beta_{5} - 52 \beta_1) q^{48} + (26 \beta_{4} - 19 \beta_{3} - 59 \beta_{2} - 268) q^{49} + (11 \beta_{4} - 39 \beta_{3} - 21 \beta_{2} + 203) q^{50} + (42 \beta_{4} - 6 \beta_{3} + 117 \beta_{2} - 364) q^{52} + ( - 2 \beta_{4} + 65 \beta_{3} + 93 \beta_{2} - 5) q^{53} + ( - 2 \beta_{7} - 30 \beta_{6} + 194 \beta_{5} - 167 \beta_1) q^{54} + ( - 44 \beta_{4} + 45 \beta_{3} + 85 \beta_{2} - 308) q^{55} + (48 \beta_{7} - 84 \beta_{6} + 114 \beta_{5} + 87 \beta_1) q^{56} + (2 \beta_{7} - 61 \beta_{6} - 78 \beta_{5} + 17 \beta_1) q^{57} + ( - \beta_{7} - 13 \beta_{6} + 97 \beta_{5}) q^{58} + (12 \beta_{4} - 64 \beta_{3} + 56 \beta_{2} - 222) q^{59} + (13 \beta_{4} + 21 \beta_{3} + 109 \beta_{2} + 73) q^{60} + ( - 60 \beta_{7} + 46 \beta_{6} - 139 \beta_{5} + 2 \beta_1) q^{61} + (18 \beta_{7} - 34 \beta_{6} + 54 \beta_{5} - 91 \beta_1) q^{62} + (40 \beta_{7} - 80 \beta_{6} + 279 \beta_{5} - 230 \beta_1) q^{63} + (5 \beta_{4} + 33 \beta_{3} + 8 \beta_{2} - 45) q^{64} + ( - 20 \beta_{7} + 74 \beta_{6} - 85 \beta_{5} - 126 \beta_1) q^{65} + ( - 23 \beta_{4} - 29 \beta_{3} + 14 \beta_{2} + 289) q^{66} + ( - 16 \beta_{4} + 10 \beta_{3} + 158 \beta_{2} - 176) q^{67} + (24 \beta_{4} + 172 \beta_{3} - 128 \beta_{2} - 363) q^{69} + (65 \beta_{4} - 93 \beta_{3} + 180 \beta_{2} - 211) q^{70} + ( - 4 \beta_{7} - 44 \beta_{6} + 298 \beta_{5} - 36 \beta_1) q^{71} + (51 \beta_{4} - 29 \beta_{3} + 121 \beta_{2} + 15) q^{72} + ( - 16 \beta_{7} + 62 \beta_{6} + 83 \beta_{5} + 130 \beta_1) q^{73} + ( - 46 \beta_{7} + 130 \beta_{6} - 434 \beta_{5} - 70 \beta_1) q^{74} + ( - 32 \beta_{7} + 4 \beta_{6} - 297 \beta_{5} + 82 \beta_1) q^{75} + ( - 54 \beta_{4} + 58 \beta_{3} - 37 \beta_{2} + 208) q^{76} + ( - 78 \beta_{4} + 9 \beta_{3} + 21 \beta_{2} + 246) q^{77} + (33 \beta_{7} + 25 \beta_{6} + 291 \beta_{5} + 89 \beta_1) q^{78} + ( - 76 \beta_{7} - 52 \beta_{6} - 294 \beta_{5} + 212 \beta_1) q^{79} + (73 \beta_{7} + 39 \beta_{6} + 47 \beta_{5} + 153 \beta_1) q^{80} + (28 \beta_{4} - 126 \beta_{3} + 6 \beta_{2} + 296) q^{81} + ( - 32 \beta_{7} + 16 \beta_{6} - 136 \beta_{5} + 162 \beta_1) q^{82} + ( - 84 \beta_{4} + 3 \beta_{3} + 37 \beta_{2} - 183) q^{83} + ( - 11 \beta_{4} + 149 \beta_{3} - 221 \beta_{2} + 21) q^{84} + ( - 98 \beta_{4} - 114 \beta_{3} - 39 \beta_{2} + 646) q^{86} + ( - 12 \beta_{4} - 23 \beta_{3} + 37 \beta_{2} + 114) q^{87} + ( - \beta_{7} + 69 \beta_{6} - 533 \beta_{5} - 190 \beta_1) q^{88} + (44 \beta_{4} - 112 \beta_{3} + 20 \beta_{2} + 218) q^{89} + ( - 112 \beta_{7} + 104 \beta_{6} - 584 \beta_{5} + 51 \beta_1) q^{90} + (68 \beta_{7} + 106 \beta_{6} + 677 \beta_{5} + 112 \beta_1) q^{91} + ( - 23 \beta_{7} + 111 \beta_{6} + 305 \beta_{5} + 212 \beta_1) q^{92} + ( - 18 \beta_{4} - 75 \beta_{3} + 161 \beta_{2} + 451) q^{93} + (24 \beta_{4} + 116 \beta_{3} - 85 \beta_{2} - 336) q^{94} + (80 \beta_{7} - 132 \beta_{6} - 587 \beta_{5} + 66 \beta_1) q^{95} + (30 \beta_{7} - 34 \beta_{6} + 744 \beta_{5} - 269 \beta_1) q^{96} + ( - 162 \beta_{7} + 149 \beta_{6} - 224 \beta_{5} + 67 \beta_1) q^{97} + (7 \beta_{4} + 21 \beta_{3} - 275 \beta_{2} - 545) q^{98} + (80 \beta_{7} + 59 \beta_{6} - 371 \beta_{5} + 99 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9} - 44 q^{13} - 108 q^{15} + 126 q^{16} - 668 q^{18} - 44 q^{19} - 704 q^{21} - 756 q^{25} + 896 q^{26} + 626 q^{30} - 662 q^{32} + 188 q^{33} - 484 q^{35} - 282 q^{36} - 1048 q^{38} - 2910 q^{42} + 228 q^{43} + 20 q^{47} - 2012 q^{49} + 1610 q^{50} - 3074 q^{52} - 100 q^{53} - 2632 q^{55} - 1992 q^{59} + 434 q^{60} - 300 q^{64} + 2180 q^{66} - 1736 q^{67} - 2256 q^{69} - 2104 q^{70} - 78 q^{72} + 1746 q^{76} + 1788 q^{77} + 2160 q^{81} - 1700 q^{83} + 886 q^{84} + 4822 q^{86} + 768 q^{87} + 1568 q^{89} + 3100 q^{93} - 2238 q^{94} - 3754 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 43x^{6} + 505x^{4} + 1528x^{2} + 1296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 47\nu^{4} - 557\nu^{2} - 900 ) / 136 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -19\nu^{6} - 757\nu^{4} - 7319\nu^{2} - 9756 ) / 544 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\nu^{6} + 945\nu^{4} + 10091\nu^{2} + 19340 ) / 544 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25\nu^{7} + 1039\nu^{5} + 10933\nu^{3} + 18148\nu ) / 4896 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\nu^{7} + 757\nu^{5} + 7319\nu^{3} + 8668\nu ) / 1088 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -307\nu^{7} - 12661\nu^{5} - 134551\nu^{3} - 266236\nu ) / 9792 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} - \beta_{6} - 15\beta_{5} - 18\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -24\beta_{4} - 20\beta_{3} - 43\beta_{2} + 210 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 91\beta_{7} - 3\beta_{6} + 569\beta_{5} + 389\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 571\beta_{4} + 383\beta_{3} + 1328\beta_{2} - 4643 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2470\beta_{7} + 562\beta_{6} - 16892\beta_{5} - 9021\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
288.1
− 5.04171i
5.04171i
1.22501i
− 1.22501i
1.58184i
− 1.58184i
− 3.68488i
3.68488i
−5.04171 − 2.60975i 17.4188 − 13.4166i 13.1576i 22.2015i −47.4869 20.1892 67.6428i
288.2 −5.04171 2.60975i 17.4188 13.4166i − 13.1576i − 22.2015i −47.4869 20.1892 − 67.6428i
288.3 −1.22501 − 0.534684i −6.49935 − 20.9528i 0.654993i − 15.0235i 17.7618 26.7141 25.6674i
288.4 −1.22501 0.534684i −6.49935 20.9528i − 0.654993i 15.0235i 17.7618 26.7141 − 25.6674i
288.5 1.58184 − 4.98387i −5.49778 − 14.4471i − 7.88368i − 29.4104i −21.3513 2.16104 − 22.8530i
288.6 1.58184 4.98387i −5.49778 14.4471i 7.88368i 29.4104i −21.3513 2.16104 22.8530i
288.7 3.68488 − 9.05894i 5.57832 7.08909i − 33.3811i − 28.1854i −8.92359 −55.0643 26.1224i
288.8 3.68488 9.05894i 5.57832 − 7.08909i 33.3811i 28.1854i −8.92359 −55.0643 − 26.1224i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 288.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.4.b.d 8
17.b even 2 1 inner 289.4.b.d 8
17.c even 4 1 289.4.a.c ✓ 4
17.c even 4 1 289.4.a.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.4.a.c ✓ 4 17.c even 4 1
289.4.a.d yes 4 17.c even 4 1
289.4.b.d 8 1.a even 1 1 trivial
289.4.b.d 8 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} - 21T_{2}^{2} + 4T_{2} + 36 \) acting on \(S_{4}^{\mathrm{new}}(289, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} - 21 T^{2} + 4 T + 36)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 114 T^{6} + 2799 T^{4} + \cdots + 3969 \) Copy content Toggle raw display
$5$ \( T^{8} + 878 T^{6} + \cdots + 828921681 \) Copy content Toggle raw display
$7$ \( T^{8} + 2378 T^{6} + \cdots + 76446167121 \) Copy content Toggle raw display
$11$ \( T^{8} + 3612 T^{6} + \cdots + 100151362089 \) Copy content Toggle raw display
$13$ \( (T^{4} + 22 T^{3} - 3509 T^{2} + \cdots - 26579)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 22 T^{3} - 9191 T^{2} + \cdots + 4371507)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 69986 T^{6} + \cdots + 31\!\cdots\!89 \) Copy content Toggle raw display
$29$ \( T^{8} + 11494 T^{6} + \cdots + 11623211721 \) Copy content Toggle raw display
$31$ \( T^{8} + 64652 T^{6} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{8} + 285656 T^{6} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{8} + 211696 T^{6} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{4} - 114 T^{3} - 156808 T^{2} + \cdots + 4351627359)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 10 T^{3} - 85816 T^{2} + \cdots - 153320769)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 50 T^{3} - 372241 T^{2} + \cdots + 3826783197)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 996 T^{3} + 136768 T^{2} + \cdots - 7997430672)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 959060 T^{6} + \cdots + 71\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( (T^{4} + 868 T^{3} + \cdots + 30297331184)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 512176 T^{6} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{8} + 1015948 T^{6} + \cdots + 21\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( T^{8} + 2963344 T^{6} + \cdots + 52\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{4} + 850 T^{3} - 402171 T^{2} + \cdots + 1637812071)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 784 T^{3} + \cdots + 24704450064)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 7019870 T^{6} + \cdots + 48\!\cdots\!01 \) Copy content Toggle raw display
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