# Properties

 Label 336.4.bc.d Level $336$ Weight $4$ Character orbit 336.bc Analytic conductor $19.825$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 336.bc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.8246417619$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - x^{11} - 29 x^{9} + 6 x^{8} - 49 x^{7} + 1564 x^{6} - 441 x^{5} + 486 x^{4} - 21141 x^{3} - 59049 x + 531441$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{4} q^{5} + ( 4 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{10} + 3 \beta_{11} ) q^{7} + ( 2 \beta_{1} + 3 \beta_{2} + \beta_{4} + \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{4} q^{5} + ( 4 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{10} + 3 \beta_{11} ) q^{7} + ( 2 \beta_{1} + 3 \beta_{2} + \beta_{4} + \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + \beta_{11} ) q^{9} + ( \beta_{1} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{9} + 3 \beta_{11} ) q^{11} + ( 9 - 3 \beta_{1} - 3 \beta_{2} + 18 \beta_{3} + 5 \beta_{6} - 3 \beta_{10} + 2 \beta_{11} ) q^{13} + ( -3 - \beta_{1} - \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} - 3 \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{15} + ( -8 \beta_{1} - \beta_{5} - 9 \beta_{6} - 9 \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{17} + ( -34 + \beta_{1} + \beta_{2} - 17 \beta_{3} + \beta_{6} - 2 \beta_{10} + 2 \beta_{11} ) q^{19} + ( 44 + 4 \beta_{1} + 5 \beta_{2} + 31 \beta_{3} - 3 \beta_{4} + \beta_{5} - 12 \beta_{6} + \beta_{7} + 4 \beta_{8} - \beta_{9} - 4 \beta_{10} + 3 \beta_{11} ) q^{21} + ( -5 \beta_{1} - 10 \beta_{6} - 5 \beta_{7} - \beta_{8} + 5 \beta_{11} ) q^{23} + ( -13 + 11 \beta_{1} - 13 \beta_{3} - 4 \beta_{6} - 11 \beta_{7} + 15 \beta_{10} + 7 \beta_{11} ) q^{25} + ( 21 - 3 \beta_{1} + 3 \beta_{2} + 42 \beta_{3} - 3 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} + 6 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} ) q^{27} + ( \beta_{1} + 5 \beta_{4} + 13 \beta_{5} + 7 \beta_{6} + 14 \beta_{7} - 13 \beta_{8} - 5 \beta_{9} - 6 \beta_{11} ) q^{29} + ( 55 - 12 \beta_{1} - 4 \beta_{2} - 55 \beta_{3} + 10 \beta_{6} + 8 \beta_{7} + 2 \beta_{10} - 2 \beta_{11} ) q^{31} + ( -98 - 5 \beta_{1} + 10 \beta_{2} - 49 \beta_{3} - 5 \beta_{4} + 8 \beta_{5} + 10 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} - 20 \beta_{10} + 5 \beta_{11} ) q^{33} + ( -3 \beta_{1} + 3 \beta_{4} + 6 \beta_{5} + 9 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - 13 \beta_{9} - 12 \beta_{11} ) q^{35} + ( -15 \beta_{1} - 11 \beta_{2} - 135 \beta_{3} + 19 \beta_{6} + 4 \beta_{7} + 4 \beta_{11} ) q^{37} + ( 63 + 10 \beta_{1} + 63 \beta_{3} - 4 \beta_{4} - 9 \beta_{5} + 20 \beta_{6} - 5 \beta_{7} - 2 \beta_{9} - 15 \beta_{10} + 26 \beta_{11} ) q^{39} + ( -8 \beta_{1} - 16 \beta_{4} + 8 \beta_{5} - 6 \beta_{6} + 8 \beta_{8} - 16 \beta_{9} - 2 \beta_{11} ) q^{41} + ( 87 + 5 \beta_{1} + 31 \beta_{2} + 13 \beta_{6} + 26 \beta_{7} - 31 \beta_{10} + 18 \beta_{11} ) q^{43} + ( 135 + 9 \beta_{1} + 30 \beta_{2} - 135 \beta_{3} + 15 \beta_{6} + 30 \beta_{7} - 12 \beta_{9} - 15 \beta_{10} + 15 \beta_{11} ) q^{45} + ( 17 \beta_{1} - 14 \beta_{4} + 10 \beta_{5} + 27 \beta_{7} - 5 \beta_{8} + 17 \beta_{11} ) q^{47} + ( -133 + 28 \beta_{1} + 7 \beta_{2} - 175 \beta_{3} + 14 \beta_{6} - 21 \beta_{7} - 35 \beta_{10} + 42 \beta_{11} ) q^{49} + ( -26 \beta_{1} - 39 \beta_{2} - 219 \beta_{3} - 5 \beta_{4} - 5 \beta_{6} - 22 \beta_{7} + 3 \beta_{8} - 10 \beta_{9} - 13 \beta_{11} ) q^{51} + ( 38 \beta_{1} - 6 \beta_{4} - 10 \beta_{5} - 28 \beta_{6} + 28 \beta_{7} - 3 \beta_{9} + 66 \beta_{11} ) q^{53} + ( -150 + 19 \beta_{1} + 19 \beta_{2} - 300 \beta_{3} + 9 \beta_{6} + 19 \beta_{10} + 28 \beta_{11} ) q^{55} + ( 30 - 31 \beta_{1} + 6 \beta_{2} - \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + \beta_{9} - 6 \beta_{10} - 11 \beta_{11} ) q^{57} + ( -50 \beta_{1} - 4 \beta_{5} - 54 \beta_{6} - 54 \beta_{7} + 8 \beta_{8} - 19 \beta_{9} + 4 \beta_{11} ) q^{59} + ( 228 + 51 \beta_{1} - 38 \beta_{2} + 114 \beta_{3} - 38 \beta_{6} - 89 \beta_{7} + 76 \beta_{10} + 13 \beta_{11} ) q^{61} + ( -237 + 35 \beta_{1} + 6 \beta_{2} - 282 \beta_{3} + 25 \beta_{4} - 6 \beta_{5} + 7 \beta_{6} + 20 \beta_{7} + 18 \beta_{8} + 20 \beta_{9} - 30 \beta_{10} + 46 \beta_{11} ) q^{63} + ( 6 \beta_{1} + 14 \beta_{4} + 12 \beta_{6} + 6 \beta_{7} + 21 \beta_{8} + 28 \beta_{9} - 6 \beta_{11} ) q^{65} + ( -181 + 62 \beta_{1} - 181 \beta_{3} + 23 \beta_{6} - 62 \beta_{7} + 39 \beta_{10} + 85 \beta_{11} ) q^{67} + ( -143 - 10 \beta_{1} - 11 \beta_{2} - 286 \beta_{3} - 19 \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{8} - 19 \beta_{9} - 11 \beta_{10} + \beta_{11} ) q^{69} + ( 19 \beta_{1} - 4 \beta_{4} + 17 \beta_{5} + 18 \beta_{6} + 36 \beta_{7} - 17 \beta_{8} + 4 \beta_{9} + \beta_{11} ) q^{71} + ( -145 - 38 \beta_{1} + 22 \beta_{2} + 145 \beta_{3} + 49 \beta_{6} + 60 \beta_{7} - 11 \beta_{10} + 11 \beta_{11} ) q^{73} + ( 294 - 28 \beta_{1} + 3 \beta_{2} + 147 \beta_{3} - 18 \beta_{4} + 30 \beta_{5} + 3 \beta_{6} + 81 \beta_{7} - 15 \beta_{8} - 6 \beta_{10} - 25 \beta_{11} ) q^{75} + ( -30 \beta_{1} + 23 \beta_{4} - 24 \beta_{5} - 15 \beta_{6} - 54 \beta_{7} + 37 \beta_{8} - 18 \beta_{9} - 15 \beta_{11} ) q^{77} + ( 38 \beta_{1} - 12 \beta_{2} + 325 \beta_{3} - 88 \beta_{6} - 50 \beta_{7} - 50 \beta_{11} ) q^{79} + ( 144 - 24 \beta_{1} + 144 \beta_{3} + 42 \beta_{4} + 18 \beta_{5} + 24 \beta_{6} + 57 \beta_{7} + 21 \beta_{9} - 81 \beta_{10} + 51 \beta_{11} ) q^{81} + ( -14 \beta_{1} - 13 \beta_{4} + 14 \beta_{5} + 27 \beta_{6} + 14 \beta_{8} - 13 \beta_{9} - 41 \beta_{11} ) q^{83} + ( 54 - 46 \beta_{1} + 12 \beta_{2} + 29 \beta_{6} + 58 \beta_{7} - 12 \beta_{10} - 17 \beta_{11} ) q^{85} + ( -100 + 60 \beta_{1} + 146 \beta_{2} + 100 \beta_{3} + 28 \beta_{5} - 39 \beta_{6} + 34 \beta_{7} - 56 \beta_{8} - 16 \beta_{9} - 73 \beta_{10} + 45 \beta_{11} ) q^{87} + ( -103 \beta_{1} - 48 \beta_{4} + 52 \beta_{5} - 51 \beta_{7} - 26 \beta_{8} - 103 \beta_{11} ) q^{89} + ( 356 - 46 \beta_{1} - 55 \beta_{2} + 499 \beta_{3} + 75 \beta_{6} - 9 \beta_{7} + 2 \beta_{10} + 29 \beta_{11} ) q^{91} + ( 33 \beta_{1} - 36 \beta_{2} + 276 \beta_{3} - 10 \beta_{4} + 8 \beta_{6} - 14 \beta_{7} - 6 \beta_{8} - 20 \beta_{9} - 69 \beta_{11} ) q^{93} + ( -12 \beta_{1} - 32 \beta_{4} + 9 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 16 \beta_{9} - 15 \beta_{11} ) q^{95} + ( 156 + 81 \beta_{1} + 81 \beta_{2} + 312 \beta_{3} - 44 \beta_{6} + 81 \beta_{10} + 37 \beta_{11} ) q^{97} + ( 303 - 66 \beta_{1} + 87 \beta_{2} + 18 \beta_{4} - 33 \beta_{5} + 12 \beta_{6} + 24 \beta_{7} + 33 \beta_{8} - 18 \beta_{9} - 87 \beta_{10} + 27 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 3 q^{3} + 56 q^{7} - 3 q^{9} + O(q^{10})$$ $$12 q + 3 q^{3} + 56 q^{7} - 3 q^{9} - 6 q^{15} - 300 q^{19} + 357 q^{21} - 42 q^{25} + 930 q^{31} - 855 q^{33} + 764 q^{37} + 426 q^{39} + 1012 q^{43} + 2367 q^{45} - 336 q^{49} + 1341 q^{51} + 270 q^{57} + 2358 q^{61} - 1071 q^{63} - 792 q^{67} - 2904 q^{73} + 2418 q^{75} - 1674 q^{79} + 837 q^{81} + 348 q^{85} - 1638 q^{87} + 1218 q^{91} - 1479 q^{93} + 3354 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} - 29 x^{9} + 6 x^{8} - 49 x^{7} + 1564 x^{6} - 441 x^{5} + 486 x^{4} - 21141 x^{3} - 59049 x + 531441$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$538 \nu^{11} + 22601 \nu^{10} - 146502 \nu^{9} + 1327 \nu^{8} - 161148 \nu^{7} + 632573 \nu^{6} - 6980468 \nu^{5} + 17528769 \nu^{4} - 61874280 \nu^{3} + 81309015 \nu^{2} + 145536102 \nu - 142839531$$$$)/ 489398112$$ $$\beta_{2}$$ $$=$$ $$($$$$-109 \nu^{11} + 3322 \nu^{10} + 26109 \nu^{9} - 69172 \nu^{8} - 83625 \nu^{7} + 33772 \nu^{6} - 26593 \nu^{5} + 1141056 \nu^{4} + 12229137 \nu^{3} - 61499898 \nu^{2} + 15136227 \nu + 271271106$$$$)/69914016$$ $$\beta_{3}$$ $$=$$ $$($$$$857 \nu^{11} - 5426 \nu^{10} + 627 \nu^{9} - 6088 \nu^{8} + 24405 \nu^{7} - 289700 \nu^{6} + 658001 \nu^{5} - 2301324 \nu^{4} + 3432699 \nu^{3} + 5390226 \nu^{2} - 4113747 \nu - 336854862$$$$)/ 163132704$$ $$\beta_{4}$$ $$=$$ $$($$$$4301 \nu^{11} - 145853 \nu^{10} + 231543 \nu^{9} + 1385435 \nu^{8} + 5417499 \nu^{7} - 15945431 \nu^{6} - 12125077 \nu^{5} - 198180063 \nu^{4} + 245384073 \nu^{3} + 938815677 \nu^{2} + 3646715337 \nu - 8693961417$$$$)/ 489398112$$ $$\beta_{5}$$ $$=$$ $$($$$$4280 \nu^{11} + 50035 \nu^{10} + 126396 \nu^{9} + 330857 \nu^{8} - 3202098 \nu^{7} - 6882797 \nu^{6} - 4401802 \nu^{5} + 45805959 \nu^{4} + 124247358 \nu^{3} + 425453877 \nu^{2} - 1006011252 \nu - 3237833817$$$$)/ 244699056$$ $$\beta_{6}$$ $$=$$ $$($$$$-1523 \nu^{11} + 209 \nu^{10} + 6255 \nu^{9} + 6421 \nu^{8} - 82569 \nu^{7} - 227449 \nu^{6} - 641129 \nu^{5} + 1005399 \nu^{4} + 7836021 \nu^{3} - 1371249 \nu^{2} + 13338513 \nu - 151814979$$$$)/54377568$$ $$\beta_{7}$$ $$=$$ $$($$$$-1523 \nu^{11} + 209 \nu^{10} + 6255 \nu^{9} + 6421 \nu^{8} - 82569 \nu^{7} - 227449 \nu^{6} - 641129 \nu^{5} + 1005399 \nu^{4} + 7836021 \nu^{3} - 1371249 \nu^{2} - 149794191 \nu - 151814979$$$$)/54377568$$ $$\beta_{8}$$ $$=$$ $$($$$$-1228 \nu^{11} + 12349 \nu^{10} + 50088 \nu^{9} + 39527 \nu^{8} - 725238 \nu^{7} - 1298243 \nu^{6} - 1880782 \nu^{5} + 12758757 \nu^{4} + 57560058 \nu^{3} + 63733311 \nu^{2} - 328627368 \nu - 585313371$$$$)/40783176$$ $$\beta_{9}$$ $$=$$ $$($$$$22376 \nu^{11} + 116845 \nu^{10} - 61380 \nu^{9} - 2138089 \nu^{8} - 3679350 \nu^{7} + 8057845 \nu^{6} + 34199906 \nu^{5} + 126399249 \nu^{4} - 75013614 \nu^{3} - 784745901 \nu^{2} - 2180220300 \nu + 5512873689$$$$)/ 489398112$$ $$\beta_{10}$$ $$=$$ $$($$$$-8011 \nu^{11} - 4808 \nu^{10} - 1005 \nu^{9} - 51046 \nu^{8} - 321723 \nu^{7} + 1816582 \nu^{6} - 1810579 \nu^{5} - 5875446 \nu^{4} - 11721861 \nu^{3} - 137629368 \nu^{2} - 328734531 \nu + 864004968$$$$)/ 163132704$$ $$\beta_{11}$$ $$=$$ $$($$$$25402 \nu^{11} - 2263 \nu^{10} - 146502 \nu^{9} - 719729 \nu^{8} - 11964 \nu^{7} - 585763 \nu^{6} + 31906828 \nu^{5} + 6563745 \nu^{4} - 49790376 \nu^{3} - 444340809 \nu^{2} + 145536102 \nu - 1611033867$$$$)/ 489398112$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{11} + 2 \beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} - 3 \beta_{2} - \beta_{1}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{11} + 3 \beta_{10} - \beta_{9} + 6 \beta_{8} - 6 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} + \beta_{4} - 3 \beta_{2} + 5 \beta_{1} + 21$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{11} - 27 \beta_{10} - 7 \beta_{9} + 2 \beta_{7} + 25 \beta_{6} - 6 \beta_{5} - 14 \beta_{4} + 48 \beta_{3} - 13 \beta_{1} + 48$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$33 \beta_{11} + 12 \beta_{9} - 10 \beta_{7} + 52 \beta_{6} + 6 \beta_{4} - 216 \beta_{3} - 72 \beta_{2} - 105 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-44 \beta_{11} + 114 \beta_{10} + 8 \beta_{9} + 90 \beta_{8} - 280 \beta_{7} - 140 \beta_{6} - 90 \beta_{5} - 8 \beta_{4} - 114 \beta_{2} - 64 \beta_{1} - 1629$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-32 \beta_{11} - 552 \beta_{10} - 10 \beta_{9} + 681 \beta_{7} - 129 \beta_{6} - 228 \beta_{5} - 20 \beta_{4} - 984 \beta_{3} - 178 \beta_{1} - 984$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$1282 \beta_{11} - 1582 \beta_{9} + 294 \beta_{8} - 493 \beta_{7} - 1013 \beta_{6} - 791 \beta_{4} - 3648 \beta_{3} + 27 \beta_{2} - 1255 \beta_{1}$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$-3930 \beta_{11} - 2709 \beta_{10} + 681 \beta_{9} - 2106 \beta_{8} + 974 \beta_{7} + 487 \beta_{6} + 2106 \beta_{5} - 681 \beta_{4} + 2709 \beta_{2} - 8463 \beta_{1} - 29619$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$7579 \beta_{11} + 4467 \beta_{10} + 3905 \beta_{9} + 8606 \beta_{7} - 13073 \beta_{6} + 2772 \beta_{5} + 7810 \beta_{4} - 114192 \beta_{3} + 4637 \beta_{1} - 114192$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$3799 \beta_{11} - 20608 \beta_{9} + 10020 \beta_{8} - 25200 \beta_{7} - 96240 \beta_{6} - 10304 \beta_{4} + 144912 \beta_{3} + 45840 \beta_{2} + 42041 \beta_{1}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 −2.59957 + 1.49740i 0.00299931 + 3.00000i −2.23014 − 2.00661i −0.232749 − 2.99096i 2.85284 + 0.928053i 2.70662 − 1.29391i −2.59957 − 1.49740i 0.00299931 − 3.00000i −2.23014 + 2.00661i −0.232749 + 2.99096i 2.85284 − 0.928053i 2.70662 + 1.29391i
0 −5.19615 + 0.00519496i 0 8.05907 + 13.9587i 0 5.67909 + 17.6280i 0 26.9999 0.0539876i 0
17.2 0 −2.59358 4.50260i 0 −8.05907 13.9587i 0 5.67909 + 17.6280i 0 −13.5467 + 23.3556i 0
17.3 0 −1.60743 + 4.94127i 0 −0.623706 1.08029i 0 −10.0808 15.5363i 0 −21.8323 15.8855i 0
17.4 0 2.24112 + 4.68800i 0 −5.80193 10.0492i 0 18.4018 2.09174i 0 −16.9548 + 21.0128i 0
17.5 0 3.47555 3.86271i 0 0.623706 + 1.08029i 0 −10.0808 15.5363i 0 −2.84113 26.8501i 0
17.6 0 5.18049 0.403134i 0 5.80193 + 10.0492i 0 18.4018 2.09174i 0 26.6750 4.17686i 0
257.1 0 −5.19615 0.00519496i 0 8.05907 13.9587i 0 5.67909 17.6280i 0 26.9999 + 0.0539876i 0
257.2 0 −2.59358 + 4.50260i 0 −8.05907 + 13.9587i 0 5.67909 17.6280i 0 −13.5467 23.3556i 0
257.3 0 −1.60743 4.94127i 0 −0.623706 + 1.08029i 0 −10.0808 + 15.5363i 0 −21.8323 + 15.8855i 0
257.4 0 2.24112 4.68800i 0 −5.80193 + 10.0492i 0 18.4018 + 2.09174i 0 −16.9548 21.0128i 0
257.5 0 3.47555 + 3.86271i 0 0.623706 1.08029i 0 −10.0808 + 15.5363i 0 −2.84113 + 26.8501i 0
257.6 0 5.18049 + 0.403134i 0 5.80193 10.0492i 0 18.4018 + 2.09174i 0 26.6750 + 4.17686i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.bc.d 12
3.b odd 2 1 inner 336.4.bc.d 12
4.b odd 2 1 21.4.g.a 12
7.d odd 6 1 inner 336.4.bc.d 12
12.b even 2 1 21.4.g.a 12
21.g even 6 1 inner 336.4.bc.d 12
28.d even 2 1 147.4.g.d 12
28.f even 6 1 21.4.g.a 12
28.f even 6 1 147.4.c.a 12
28.g odd 6 1 147.4.c.a 12
28.g odd 6 1 147.4.g.d 12
84.h odd 2 1 147.4.g.d 12
84.j odd 6 1 21.4.g.a 12
84.j odd 6 1 147.4.c.a 12
84.n even 6 1 147.4.c.a 12
84.n even 6 1 147.4.g.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.g.a 12 4.b odd 2 1
21.4.g.a 12 12.b even 2 1
21.4.g.a 12 28.f even 6 1
21.4.g.a 12 84.j odd 6 1
147.4.c.a 12 28.f even 6 1
147.4.c.a 12 28.g odd 6 1
147.4.c.a 12 84.j odd 6 1
147.4.c.a 12 84.n even 6 1
147.4.g.d 12 28.d even 2 1
147.4.g.d 12 28.g odd 6 1
147.4.g.d 12 84.h odd 2 1
147.4.g.d 12 84.n even 6 1
336.4.bc.d 12 1.a even 1 1 trivial
336.4.bc.d 12 3.b odd 2 1 inner
336.4.bc.d 12 7.d odd 6 1 inner
336.4.bc.d 12 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}^{12} + 396 T_{5}^{10} + 121221 T_{5}^{8} + 13986756 T_{5}^{6} + 1245448953 T_{5}^{4} + 1937507040 T_{5}^{2} + 2962842624$$ $$T_{13}^{6} + 4335 T_{13}^{4} + 1731204 T_{13}^{2} + 82121472$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$387420489 - 43046721 T + 3188646 T^{2} - 177147 T^{3} - 144342 T^{4} + 69255 T^{5} - 36018 T^{6} + 2565 T^{7} - 198 T^{8} - 9 T^{9} + 6 T^{10} - 3 T^{11} + T^{12}$$
$5$ $$2962842624 + 1937507040 T^{2} + 1245448953 T^{4} + 13986756 T^{6} + 121221 T^{8} + 396 T^{10} + T^{12}$$
$7$ $$( 40353607 - 3294172 T + 163268 T^{2} - 10780 T^{3} + 476 T^{4} - 28 T^{5} + T^{6} )^{2}$$
$11$ $$453620224546062336 - 2045138759862912 T^{2} + 7035594641593 T^{4} - 8503453924 T^{6} + 7487013 T^{8} - 3244 T^{10} + T^{12}$$
$13$ $$( 82121472 + 1731204 T^{2} + 4335 T^{4} + T^{6} )^{2}$$
$17$ $$17696515773733945344 + 61355067231370752 T^{2} + 161143714802448 T^{4} + 170413288668 T^{6} + 135747117 T^{8} + 12261 T^{10} + T^{12}$$
$19$ $$( 243972972 + 60952662 T + 6428709 T^{2} + 337950 T^{3} + 9753 T^{4} + 150 T^{5} + T^{6} )^{2}$$
$23$ $$89\!\cdots\!76$$$$- 6200136676834232832 T^{2} + 2919015626694160 T^{4} - 746172164500 T^{6} + 139414053 T^{8} - 14311 T^{10} + T^{12}$$
$29$ $$( 14683734245376 + 3697274560 T^{2} + 120001 T^{4} + T^{6} )^{2}$$
$31$ $$( 33414175107 + 4755813831 T + 176555736 T^{2} - 6984765 T^{3} + 87096 T^{4} - 465 T^{5} + T^{6} )^{2}$$
$37$ $$( 3418867564324 + 49455684446 T + 1421726885 T^{2} - 13915390 T^{3} + 119177 T^{4} - 382 T^{5} + T^{6} )^{2}$$
$41$ $$( -4591113633792 + 4941510336 T^{2} - 172788 T^{4} + T^{6} )^{2}$$
$43$ $$( 6662944 - 23284 T - 253 T^{2} + T^{3} )^{4}$$
$47$ $$11\!\cdots\!04$$$$+$$$$85\!\cdots\!56$$$$T^{2} + 47280242457173456640 T^{4} + 1306125205474320 T^{6} + 26259713937 T^{8} + 185553 T^{10} + T^{12}$$
$53$ $$13\!\cdots\!76$$$$-$$$$30\!\cdots\!92$$$$T^{2} +$$$$50\!\cdots\!89$$$$T^{4} - 37040421831506548 T^{6} + 198439484133 T^{8} - 531100 T^{10} + T^{12}$$
$59$ $$38\!\cdots\!24$$$$+$$$$44\!\cdots\!36$$$$T^{2} +$$$$47\!\cdots\!89$$$$T^{4} + 39639172481816796 T^{6} + 253702703277 T^{8} + 570420 T^{10} + T^{12}$$
$61$ $$( 6477700166618112 - 28202272530048 T - 13856716944 T^{2} + 238521132 T^{3} + 261039 T^{4} - 1179 T^{5} + T^{6} )^{2}$$
$67$ $$( 9665143560577444 - 52759337639610 T + 249067250073 T^{2} - 409138304 T^{3} + 693471 T^{4} + 396 T^{5} + T^{6} )^{2}$$
$71$ $$( 6720226523136 + 3717765184 T^{2} + 225148 T^{4} + T^{6} )^{2}$$
$73$ $$( 4047431204396592 - 4635670445244 T - 51563164023 T^{2} + 61084188 T^{3} + 744837 T^{4} + 1452 T^{5} + T^{6} )^{2}$$
$79$ $$( 363201760969609 + 4951859118549 T + 83464610850 T^{2} - 179364515 T^{3} + 960402 T^{4} + 837 T^{5} + T^{6} )^{2}$$
$83$ $$( -388952511994368 + 75760581456 T^{2} - 567987 T^{4} + T^{6} )^{2}$$
$89$ $$88\!\cdots\!24$$$$+$$$$70\!\cdots\!32$$$$T^{2} +$$$$53\!\cdots\!72$$$$T^{4} + 1923321552966019836 T^{6} + 5981516897085 T^{8} + 2594253 T^{10} + T^{12}$$
$97$ $$( 9887068459035648 + 312291915984 T^{2} + 2159691 T^{4} + T^{6} )^{2}$$