Properties

Label 1336.2.a.b
Level $1336$
Weight $2$
Character orbit 1336.a
Self dual yes
Analytic conductor $10.668$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6680137100\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - x^{6} - 9 x^{5} + 5 x^{4} + 25 x^{3} - 4 x^{2} - 17 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -\beta_{4} + \beta_{5} ) q^{5} + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -\beta_{4} + \beta_{5} ) q^{5} + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} + ( -\beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( -2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{13} + ( -1 - \beta_{1} - \beta_{2} + 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{15} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{17} + ( -1 + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{19} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{21} + ( 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{23} + ( 1 + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{25} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{27} + ( -1 - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{29} + ( -4 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{31} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{33} + ( -2 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{35} + ( 3 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{37} + ( -5 + \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{39} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{41} + ( -5 - \beta_{1} + \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{43} + ( 1 + \beta_{1} + \beta_{2} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{45} + ( 1 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} ) q^{47} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{49} + ( -3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{51} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{6} ) q^{53} + ( -1 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} ) q^{55} + ( 1 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{57} + ( -2 - 2 \beta_{1} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{59} + ( 5 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{61} + ( -4 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{63} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{65} + ( -5 - 3 \beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{67} + ( -3 - \beta_{1} + \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 5 \beta_{6} ) q^{69} + ( -6 + 2 \beta_{1} + 2 \beta_{3} ) q^{71} + ( -5 - 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{73} + ( -4 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{75} + ( -2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{77} + ( -4 - 4 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{79} + ( -3 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{81} + ( 1 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} ) q^{83} + ( -3 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{85} + ( 4 + \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} + 4 \beta_{6} ) q^{87} + ( 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} ) q^{89} + ( 4 \beta_{1} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{91} + ( 9 - 3 \beta_{1} - \beta_{2} + 5 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} + 7 \beta_{6} ) q^{93} + ( -8 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} ) q^{95} + ( 3 + \beta_{1} + 2 \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{97} + ( -2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 6q^{3} - q^{7} + 3q^{9} + O(q^{10}) \) \( 7q - 6q^{3} - q^{7} + 3q^{9} - 6q^{11} - 2q^{13} - 9q^{15} - 9q^{17} - 3q^{19} + 7q^{21} - 10q^{23} + 3q^{25} - 12q^{27} - 5q^{29} - 21q^{31} + 8q^{33} - 12q^{35} + 19q^{37} - 27q^{39} - 22q^{41} - 19q^{43} + 13q^{45} - 13q^{47} - 14q^{49} + 4q^{51} + 5q^{53} - 17q^{55} + 5q^{57} - 18q^{59} + 26q^{61} - 20q^{63} - 20q^{65} - 27q^{67} - 3q^{69} - 46q^{71} - 25q^{73} - 19q^{75} - 19q^{77} - 22q^{79} - 9q^{81} + q^{83} - 11q^{85} + 9q^{87} - 3q^{89} + 33q^{93} - 40q^{95} + 11q^{97} - 21q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 9 x^{5} + 5 x^{4} + 25 x^{3} - 4 x^{2} - 17 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 4 \nu^{2} + 2 \nu + 1 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - \nu^{4} - 5 \nu^{3} + 3 \nu^{2} + 4 \nu - 2 \)
\(\beta_{6}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 6 \nu^{4} + 9 \nu^{3} + 11 \nu^{2} - 7 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 5 \beta_{2} + 7 \beta_{1} + 13\)
\(\nu^{5}\)\(=\)\(\beta_{5} + \beta_{4} + 6 \beta_{3} + 7 \beta_{2} + 25 \beta_{1} + 16\)
\(\nu^{6}\)\(=\)\(\beta_{6} + 2 \beta_{5} + 8 \beta_{4} + 9 \beta_{3} + 24 \beta_{2} + 43 \beta_{1} + 63\)

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
−1.83013
−1.71785
−1.07427
0.0583180
0.960446
2.19297
2.41051
0 −2.83013 0 −1.66287 0 −2.94376 0 5.00961 0
1.2 0 −2.71785 0 2.12216 0 1.38168 0 4.38672 0
1.3 0 −2.07427 0 3.79451 0 −2.17413 0 1.30261 0
1.4 0 −0.941682 0 −2.86037 0 1.50273 0 −2.11324 0
1.5 0 −0.0395540 0 0.949625 0 1.23977 0 −2.99844 0
1.6 0 1.19297 0 −2.67228 0 2.81671 0 −1.57681 0
1.7 0 1.41051 0 0.329231 0 −2.82300 0 −1.01045 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(167\) \(1\)

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 1336.2.a.b 7
4.b odd 2 1 2672.2.a.l 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1336.2.a.b 7 1.a even 1 1 trivial
2672.2.a.l 7 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} + 6 T_{3}^{6} + 6 T_{3}^{5} - 20 T_{3}^{4} - 30 T_{3}^{3} + 17 T_{3}^{2} + 26 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1336))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 6 T + 27 T^{2} + 88 T^{3} + 249 T^{4} + 587 T^{5} + 1241 T^{6} + 2263 T^{7} + 3723 T^{8} + 5283 T^{9} + 6723 T^{10} + 7128 T^{11} + 6561 T^{12} + 4374 T^{13} + 2187 T^{14} \)
$5$ \( 1 + 16 T^{2} - 6 T^{3} + 142 T^{4} - 104 T^{5} + 893 T^{6} - 708 T^{7} + 4465 T^{8} - 2600 T^{9} + 17750 T^{10} - 3750 T^{11} + 50000 T^{12} + 78125 T^{14} \)
$7$ \( 1 + T + 32 T^{2} + 34 T^{3} + 529 T^{4} + 495 T^{5} + 5487 T^{6} + 4415 T^{7} + 38409 T^{8} + 24255 T^{9} + 181447 T^{10} + 81634 T^{11} + 537824 T^{12} + 117649 T^{13} + 823543 T^{14} \)
$11$ \( 1 + 6 T + 57 T^{2} + 280 T^{3} + 1477 T^{4} + 6073 T^{5} + 23343 T^{6} + 81859 T^{7} + 256773 T^{8} + 734833 T^{9} + 1965887 T^{10} + 4099480 T^{11} + 9179907 T^{12} + 10629366 T^{13} + 19487171 T^{14} \)
$13$ \( 1 + 2 T + 54 T^{2} + 78 T^{3} + 1412 T^{4} + 1590 T^{5} + 24481 T^{6} + 22964 T^{7} + 318253 T^{8} + 268710 T^{9} + 3102164 T^{10} + 2227758 T^{11} + 20049822 T^{12} + 9653618 T^{13} + 62748517 T^{14} \)
$17$ \( 1 + 9 T + 122 T^{2} + 798 T^{3} + 6216 T^{4} + 31223 T^{5} + 174989 T^{6} + 688740 T^{7} + 2974813 T^{8} + 9023447 T^{9} + 30539208 T^{10} + 66649758 T^{11} + 173222554 T^{12} + 217238121 T^{13} + 410338673 T^{14} \)
$19$ \( 1 + 3 T + 52 T^{2} + 68 T^{3} + 1203 T^{4} - 321 T^{5} + 16531 T^{6} - 37799 T^{7} + 314089 T^{8} - 115881 T^{9} + 8251377 T^{10} + 8861828 T^{11} + 128757148 T^{12} + 141137643 T^{13} + 893871739 T^{14} \)
$23$ \( 1 + 10 T + 156 T^{2} + 1070 T^{3} + 9810 T^{4} + 52254 T^{5} + 355503 T^{6} + 1519732 T^{7} + 8176569 T^{8} + 27642366 T^{9} + 119358270 T^{10} + 299429870 T^{11} + 1004069508 T^{12} + 1480358890 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 + 5 T + 126 T^{2} + 582 T^{3} + 7819 T^{4} + 31203 T^{5} + 320931 T^{6} + 1074715 T^{7} + 9306999 T^{8} + 26241723 T^{9} + 190697591 T^{10} + 411637542 T^{11} + 2584404774 T^{12} + 2974116605 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 + 21 T + 316 T^{2} + 3540 T^{3} + 33161 T^{4} + 259869 T^{5} + 1781471 T^{6} + 10560667 T^{7} + 55225601 T^{8} + 249734109 T^{9} + 987899351 T^{10} + 3269264340 T^{11} + 9046811716 T^{12} + 18637577301 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 - 19 T + 360 T^{2} - 4286 T^{3} + 46858 T^{4} - 399525 T^{5} + 3092377 T^{6} - 19755620 T^{7} + 114417949 T^{8} - 546949725 T^{9} + 2373498274 T^{10} - 8032654046 T^{11} + 24963824520 T^{12} - 48748801771 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 + 22 T + 270 T^{2} + 2078 T^{3} + 13932 T^{4} + 101594 T^{5} + 873197 T^{6} + 6232516 T^{7} + 35801077 T^{8} + 170779514 T^{9} + 960207372 T^{10} + 5871931358 T^{11} + 31281174270 T^{12} + 104502293302 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 + 19 T + 330 T^{2} + 3778 T^{3} + 40164 T^{4} + 339037 T^{5} + 2712855 T^{6} + 18187404 T^{7} + 116652765 T^{8} + 626879413 T^{9} + 3193319148 T^{10} + 12916230178 T^{11} + 48512786190 T^{12} + 120105897931 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 + 13 T + 179 T^{2} + 2011 T^{3} + 20590 T^{4} + 165352 T^{5} + 1351646 T^{6} + 9741796 T^{7} + 63527362 T^{8} + 365262568 T^{9} + 2137715570 T^{10} + 9813038491 T^{11} + 41052756253 T^{12} + 140129799277 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 - 5 T + 292 T^{2} - 1358 T^{3} + 39830 T^{4} - 162507 T^{5} + 3267729 T^{6} - 11087924 T^{7} + 173189637 T^{8} - 456482163 T^{9} + 5929770910 T^{10} - 10715273198 T^{11} + 122113083956 T^{12} - 110821805645 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 + 18 T + 418 T^{2} + 5368 T^{3} + 72948 T^{4} + 714158 T^{5} + 7100527 T^{6} + 54300560 T^{7} + 418931093 T^{8} + 2485983998 T^{9} + 14981987292 T^{10} + 65045993848 T^{11} + 298838356982 T^{12} + 759249605538 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 - 26 T + 537 T^{2} - 6694 T^{3} + 72271 T^{4} - 555381 T^{5} + 4325959 T^{6} - 29289133 T^{7} + 263883499 T^{8} - 2066572701 T^{9} + 16404143851 T^{10} - 92684059654 T^{11} + 453548213637 T^{12} - 1339529733386 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 + 27 T + 586 T^{2} + 8768 T^{3} + 117100 T^{4} + 1283653 T^{5} + 12863415 T^{6} + 109990832 T^{7} + 861848805 T^{8} + 5762318317 T^{9} + 35219347300 T^{10} + 176685028928 T^{11} + 791173312702 T^{12} + 2442376318563 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 + 46 T + 1357 T^{2} + 28004 T^{3} + 457241 T^{4} + 6006866 T^{5} + 65915325 T^{6} + 603704184 T^{7} + 4679988075 T^{8} + 30280611506 T^{9} + 163651583551 T^{10} + 711628714724 T^{11} + 2448339229307 T^{12} + 5892613060366 T^{13} + 9095120158391 T^{14} \)
$73$ \( 1 + 25 T + 468 T^{2} + 5668 T^{3} + 55638 T^{4} + 413863 T^{5} + 2707213 T^{6} + 19340664 T^{7} + 197626549 T^{8} + 2205475927 T^{9} + 21644127846 T^{10} + 160961229988 T^{11} + 970197505524 T^{12} + 3783355657225 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 + 22 T + 460 T^{2} + 7210 T^{3} + 96802 T^{4} + 1132698 T^{5} + 11996871 T^{6} + 110259820 T^{7} + 947752809 T^{8} + 7069168218 T^{9} + 47727161278 T^{10} + 280830084010 T^{11} + 1415445943540 T^{12} + 5347924021462 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 - T + 344 T^{2} + 140 T^{3} + 59570 T^{4} + 62633 T^{5} + 6942631 T^{6} + 7393352 T^{7} + 576238373 T^{8} + 431478737 T^{9} + 34061351590 T^{10} + 6644164940 T^{11} + 1355029981192 T^{12} - 326940373369 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 + 3 T + 21 T^{2} + 135 T^{3} + 2712 T^{4} - 6152 T^{5} + 710964 T^{6} + 8512188 T^{7} + 63275796 T^{8} - 48729992 T^{9} + 1911875928 T^{10} + 8470202535 T^{11} + 117265248429 T^{12} + 1490943872883 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 - 11 T + 517 T^{2} - 5591 T^{3} + 125912 T^{4} - 1249988 T^{5} + 18637548 T^{6} - 157305852 T^{7} + 1807842156 T^{8} - 11761137092 T^{9} + 114916482776 T^{10} - 494967210071 T^{11} + 4439654912869 T^{12} - 9162692054219 T^{13} + 80798284478113 T^{14} \)
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