Properties

Label 1334.2.a.i
Level $1334$
Weight $2$
Character orbit 1334.a
Self dual yes
Analytic conductor $10.652$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1334 = 2 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1334.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6520436296\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 22 x^{6} + 151 x^{4} - 332 x^{2} + 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + q^{2} -\beta_{4} q^{3} + q^{4} -\beta_{1} q^{5} -\beta_{4} q^{6} + ( 1 - \beta_{6} ) q^{7} + q^{8} + ( 1 - \beta_{2} ) q^{9} +O(q^{10})\) \( q + q^{2} -\beta_{4} q^{3} + q^{4} -\beta_{1} q^{5} -\beta_{4} q^{6} + ( 1 - \beta_{6} ) q^{7} + q^{8} + ( 1 - \beta_{2} ) q^{9} -\beta_{1} q^{10} + ( 1 + \beta_{5} + \beta_{7} ) q^{11} -\beta_{4} q^{12} + ( 1 + \beta_{2} - \beta_{5} ) q^{13} + ( 1 - \beta_{6} ) q^{14} + ( \beta_{3} - \beta_{7} ) q^{15} + q^{16} + ( 1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{17} + ( 1 - \beta_{2} ) q^{18} + ( 2 - \beta_{5} + \beta_{6} ) q^{19} -\beta_{1} q^{20} + ( -1 + \beta_{2} - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{21} + ( 1 + \beta_{5} + \beta_{7} ) q^{22} - q^{23} -\beta_{4} q^{24} + ( 1 + \beta_{4} + \beta_{5} + \beta_{6} ) q^{25} + ( 1 + \beta_{2} - \beta_{5} ) q^{26} + ( -\beta_{2} + \beta_{5} + \beta_{6} ) q^{27} + ( 1 - \beta_{6} ) q^{28} - q^{29} + ( \beta_{3} - \beta_{7} ) q^{30} + ( 1 - 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{31} + q^{32} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{33} + ( 1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{34} + ( 1 + \beta_{2} + \beta_{6} + \beta_{7} ) q^{35} + ( 1 - \beta_{2} ) q^{36} + ( 1 - 2 \beta_{5} + \beta_{6} ) q^{37} + ( 2 - \beta_{5} + \beta_{6} ) q^{38} + ( -1 + 2 \beta_{2} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{39} -\beta_{1} q^{40} + ( 1 + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{41} + ( -1 + \beta_{2} - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{42} + ( 5 + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{43} + ( 1 + \beta_{5} + \beta_{7} ) q^{44} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{45} - q^{46} + ( 3 + 2 \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{47} -\beta_{4} q^{48} + ( 1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{49} + ( 1 + \beta_{4} + \beta_{5} + \beta_{6} ) q^{50} + ( 1 - \beta_{2} - 3 \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{51} + ( 1 + \beta_{2} - \beta_{5} ) q^{52} + ( \beta_{1} - 2 \beta_{7} ) q^{53} + ( -\beta_{2} + \beta_{5} + \beta_{6} ) q^{54} + ( 1 - 2 \beta_{1} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{55} + ( 1 - \beta_{6} ) q^{56} + ( -2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{57} - q^{58} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{59} + ( \beta_{3} - \beta_{7} ) q^{60} + ( 1 + 2 \beta_{1} - \beta_{6} + 2 \beta_{7} ) q^{61} + ( 1 - 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{62} + ( 4 + 2 \beta_{1} + 2 \beta_{4} ) q^{63} + q^{64} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{65} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{66} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{67} + ( 1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{68} + \beta_{4} q^{69} + ( 1 + \beta_{2} + \beta_{6} + \beta_{7} ) q^{70} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{71} + ( 1 - \beta_{2} ) q^{72} + ( -1 - 3 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{73} + ( 1 - 2 \beta_{5} + \beta_{6} ) q^{74} + ( -2 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{75} + ( 2 - \beta_{5} + \beta_{6} ) q^{76} + ( -3 - 2 \beta_{1} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{77} + ( -1 + 2 \beta_{2} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{78} + ( -\beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{79} -\beta_{1} q^{80} + ( -1 + 2 \beta_{5} + 2 \beta_{6} ) q^{81} + ( 1 + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{82} + ( -\beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{83} + ( -1 + \beta_{2} - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{84} + ( -5 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{85} + ( 5 + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{86} + \beta_{4} q^{87} + ( 1 + \beta_{5} + \beta_{7} ) q^{88} + ( 3 + \beta_{1} - \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{89} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{90} + ( -1 - 2 \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{91} - q^{92} + ( -1 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{93} + ( 3 + 2 \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{94} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{6} ) q^{95} -\beta_{4} q^{96} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{97} + ( 1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{98} + ( -2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 4q^{3} + 8q^{4} + 4q^{6} + 8q^{7} + 8q^{8} + 12q^{9} + O(q^{10}) \) \( 8q + 8q^{2} + 4q^{3} + 8q^{4} + 4q^{6} + 8q^{7} + 8q^{8} + 12q^{9} + 8q^{11} + 4q^{12} + 4q^{13} + 8q^{14} + 8q^{16} + 8q^{17} + 12q^{18} + 16q^{19} - 4q^{21} + 8q^{22} - 8q^{23} + 4q^{24} + 4q^{25} + 4q^{26} + 4q^{27} + 8q^{28} - 8q^{29} + 8q^{31} + 8q^{32} + 12q^{33} + 8q^{34} + 4q^{35} + 12q^{36} + 8q^{37} + 16q^{38} - 16q^{39} + 8q^{41} - 4q^{42} + 32q^{43} + 8q^{44} - 8q^{46} + 16q^{47} + 4q^{48} + 4q^{49} + 4q^{50} + 12q^{51} + 4q^{52} + 4q^{54} + 8q^{56} + 8q^{57} - 8q^{58} - 4q^{59} + 8q^{61} + 8q^{62} + 24q^{63} + 8q^{64} - 4q^{65} + 12q^{66} + 20q^{67} + 8q^{68} - 4q^{69} + 4q^{70} - 24q^{71} + 12q^{72} - 4q^{73} + 8q^{74} - 16q^{75} + 16q^{76} - 16q^{77} - 16q^{78} + 4q^{79} - 8q^{81} + 8q^{82} - 4q^{83} - 4q^{84} - 44q^{85} + 32q^{86} - 4q^{87} + 8q^{88} + 20q^{89} - 8q^{92} - 4q^{93} + 16q^{94} - 8q^{95} + 4q^{96} + 4q^{97} + 4q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 22 x^{6} + 151 x^{4} - 332 x^{2} + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 15 \nu^{4} + 46 \nu^{2} + 6 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 15 \nu^{5} + 38 \nu^{3} + 70 \nu \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{6} - 83 \nu^{4} + 310 \nu^{2} - 34 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - 5 \nu^{6} + 23 \nu^{5} + 83 \nu^{4} - 174 \nu^{3} - 294 \nu^{2} + 442 \nu - 62 \)\()/32\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 5 \nu^{6} - 23 \nu^{5} + 83 \nu^{4} + 174 \nu^{3} - 294 \nu^{2} - 442 \nu - 62 \)\()/32\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{7} + 53 \nu^{5} - 234 \nu^{3} + 174 \nu \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{4} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 8 \beta_{1}\)
\(\nu^{4}\)\(=\)\(10 \beta_{6} + 10 \beta_{5} + 8 \beta_{4} + 5 \beta_{2} + 52\)
\(\nu^{5}\)\(=\)\(17 \beta_{7} + 15 \beta_{6} - 15 \beta_{5} + 18 \beta_{3} + 72 \beta_{1}\)
\(\nu^{6}\)\(=\)\(104 \beta_{6} + 104 \beta_{5} + 74 \beta_{4} + 83 \beta_{2} + 498\)
\(\nu^{7}\)\(=\)\(217 \beta_{7} + 187 \beta_{6} - 187 \beta_{5} + 240 \beta_{3} + 706 \beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.27484
−2.27484
3.32381
−3.32381
0.134992
−0.134992
2.39983
−2.39983
1.00000 −2.52651 1.00000 −2.27484 −2.52651 4.02057 1.00000 3.38328 −2.27484
1.2 1.00000 −2.52651 1.00000 2.27484 −2.52651 1.33106 1.00000 3.38328 2.27484
1.3 1.00000 −0.153561 1.00000 −3.32381 −0.153561 −3.68051 1.00000 −2.97642 −3.32381
1.4 1.00000 −0.153561 1.00000 3.32381 −0.153561 0.786348 1.00000 −2.97642 3.32381
1.5 1.00000 1.77365 1.00000 −0.134992 1.77365 4.95530 1.00000 0.145840 −0.134992
1.6 1.00000 1.77365 1.00000 0.134992 1.77365 1.25283 1.00000 0.145840 0.134992
1.7 1.00000 2.90642 1.00000 −2.39983 2.90642 0.548309 1.00000 5.44730 −2.39983
1.8 1.00000 2.90642 1.00000 2.39983 2.90642 −1.21390 1.00000 5.44730 2.39983
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(23\) \(1\)
\(29\) \(1\)

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 1334.2.a.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1334.2.a.i 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1334))\):

\( T_{3}^{4} - 2 T_{3}^{3} - 7 T_{3}^{2} + 12 T_{3} + 2 \)
\( T_{5}^{8} - 22 T_{5}^{6} + 151 T_{5}^{4} - 332 T_{5}^{2} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{8} \)
$3$ \( ( 2 + 12 T - 7 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$5$ \( 6 - 332 T^{2} + 151 T^{4} - 22 T^{6} + T^{8} \)
$7$ \( 64 - 256 T + 288 T^{2} + 32 T^{3} - 230 T^{4} + 108 T^{5} + 2 T^{6} - 8 T^{7} + T^{8} \)
$11$ \( -1392 - 4512 T + 6536 T^{2} - 2040 T^{3} - 399 T^{4} + 272 T^{5} - 16 T^{6} - 8 T^{7} + T^{8} \)
$13$ \( -656 + 2272 T + 360 T^{2} - 2008 T^{3} + 497 T^{4} + 188 T^{5} - 48 T^{6} - 4 T^{7} + T^{8} \)
$17$ \( -93348 + 51000 T + 11324 T^{2} - 10064 T^{3} + 432 T^{4} + 516 T^{5} - 54 T^{6} - 8 T^{7} + T^{8} \)
$19$ \( -448 + 1152 T + 3104 T^{2} + 96 T^{3} - 876 T^{4} + 80 T^{5} + 68 T^{6} - 16 T^{7} + T^{8} \)
$23$ \( ( 1 + T )^{8} \)
$29$ \( ( 1 + T )^{8} \)
$31$ \( -306112 + 253568 T + 34832 T^{2} - 42896 T^{3} + 3617 T^{4} + 1120 T^{5} - 128 T^{6} - 8 T^{7} + T^{8} \)
$37$ \( 56896 + 40704 T - 16608 T^{2} - 10976 T^{3} + 1514 T^{4} + 684 T^{5} - 86 T^{6} - 8 T^{7} + T^{8} \)
$41$ \( -1493664 + 875904 T - 23632 T^{2} - 58512 T^{3} + 5994 T^{4} + 1228 T^{5} - 150 T^{6} - 8 T^{7} + T^{8} \)
$43$ \( 2677264 - 1527584 T + 79832 T^{2} + 104728 T^{3} - 22823 T^{4} + 520 T^{5} + 296 T^{6} - 32 T^{7} + T^{8} \)
$47$ \( -180384 + 107904 T + 149120 T^{2} - 51424 T^{3} - 2455 T^{4} + 1880 T^{5} - 72 T^{6} - 16 T^{7} + T^{8} \)
$53$ \( 3174 - 4844 T^{2} + 1655 T^{4} - 150 T^{6} + T^{8} \)
$59$ \( 8418816 - 230400 T - 950912 T^{2} + 32128 T^{3} + 31144 T^{4} - 880 T^{5} - 324 T^{6} + 4 T^{7} + T^{8} \)
$61$ \( 636544 + 828416 T + 40224 T^{2} - 107488 T^{3} + 9146 T^{4} + 1820 T^{5} - 198 T^{6} - 8 T^{7} + T^{8} \)
$67$ \( -10372 + 48936 T + 198052 T^{2} - 37848 T^{3} - 8160 T^{4} + 1932 T^{5} + 6 T^{6} - 20 T^{7} + T^{8} \)
$71$ \( -10564032 - 1418112 T + 1080640 T^{2} + 178464 T^{3} - 17004 T^{4} - 3968 T^{5} - 20 T^{6} + 24 T^{7} + T^{8} \)
$73$ \( 12154528 + 2066560 T - 1929904 T^{2} - 64400 T^{3} + 54586 T^{4} - 428 T^{5} - 418 T^{6} + 4 T^{7} + T^{8} \)
$79$ \( 654854 + 131480 T - 163796 T^{2} - 26856 T^{3} + 9959 T^{4} + 724 T^{5} - 190 T^{6} - 4 T^{7} + T^{8} \)
$83$ \( 3431664 - 887232 T - 715048 T^{2} + 64776 T^{3} + 31198 T^{4} - 944 T^{5} - 340 T^{6} + 4 T^{7} + T^{8} \)
$89$ \( 1308 - 891960 T + 502148 T^{2} - 49624 T^{3} - 16160 T^{4} + 3292 T^{5} - 58 T^{6} - 20 T^{7} + T^{8} \)
$97$ \( 2118256 + 1971296 T - 1770424 T^{2} - 61416 T^{3} + 46590 T^{4} + 928 T^{5} - 380 T^{6} - 4 T^{7} + T^{8} \)
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