Properties

Label 1470.2.n.k
Level $1470$
Weight $2$
Character orbit 1470.n
Analytic conductor $11.738$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 18 x^{14} + 227 x^{12} - 1394 x^{10} + 6177 x^{8} - 14768 x^{6} + 24768 x^{4} - 11264 x^{2} + 4096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 18 x^{14} + 227 x^{12} - 1394 x^{10} + 6177 x^{8} - 14768 x^{6} + 24768 x^{4} - 11264 x^{2} + 4096\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 19 \nu^{14} - 326 \nu^{12} + 3705 \nu^{10} - 18886 \nu^{8} + 51859 \nu^{6} - 85120 \nu^{4} + 38912 \nu^{2} - 716288 \)\()/496640\)
\(\beta_{2}\)\(=\)\((\)\( 3301 \nu^{15} - 53930 \nu^{13} + 643695 \nu^{11} - 3281194 \nu^{9} + 12013925 \nu^{7} - 14788480 \nu^{5} + 6760448 \nu^{3} + 46691840 \nu \)\()/33771520\)
\(\beta_{3}\)\(=\)\((\)\( 269 \nu^{14} - 4686 \nu^{12} + 58575 \nu^{10} - 344566 \nu^{8} + 1506549 \nu^{6} - 3327060 \nu^{4} + 5963712 \nu^{2} - 2710528 \)\()/2110720\)
\(\beta_{4}\)\(=\)\((\)\( -2031 \nu^{15} + 32894 \nu^{13} - 396045 \nu^{11} + 2018814 \nu^{9} - 7668591 \nu^{7} + 9098880 \nu^{5} - 4159488 \nu^{3} - 33238528 \nu \)\()/16885760\)
\(\beta_{5}\)\(=\)\((\)\( -4893 \nu^{15} + 93154 \nu^{13} - 1194855 \nu^{11} + 7811442 \nu^{9} - 35273581 \nu^{7} + 89641160 \nu^{5} - 143948224 \nu^{3} + 65518592 \nu \)\()/33771520\)
\(\beta_{6}\)\(=\)\((\)\( -733 \nu^{14} + 12142 \nu^{12} - 151775 \nu^{10} + 854742 \nu^{8} - 3903653 \nu^{6} + 8620820 \nu^{4} - 16501984 \nu^{2} + 1554176 \)\()/4221440\)
\(\beta_{7}\)\(=\)\((\)\(1385 \nu^{15} + 5482 \nu^{14} - 21810 \nu^{13} - 111332 \nu^{12} + 270075 \nu^{11} + 1452510 \nu^{10} - 1376690 \nu^{9} - 10109828 \nu^{8} + 5905385 \nu^{7} + 46442378 \nu^{6} - 6204800 \nu^{5} - 126049360 \nu^{4} + 2836480 \nu^{3} + 192477056 \nu^{2} + 57592320 \nu - 87668736\)\()/33771520\)
\(\beta_{8}\)\(=\)\((\)\(1385 \nu^{15} - 5482 \nu^{14} - 21810 \nu^{13} + 111332 \nu^{12} + 270075 \nu^{11} - 1452510 \nu^{10} - 1376690 \nu^{9} + 10109828 \nu^{8} + 5905385 \nu^{7} - 46442378 \nu^{6} - 6204800 \nu^{5} + 126049360 \nu^{4} + 2836480 \nu^{3} - 192477056 \nu^{2} + 57592320 \nu + 87668736\)\()/33771520\)
\(\beta_{9}\)\(=\)\((\)\( 3715 \nu^{15} - 69038 \nu^{13} + 878105 \nu^{11} - 5601470 \nu^{9} + 25102547 \nu^{7} - 63272840 \nu^{5} + 101725760 \nu^{3} - 46285824 \nu \)\()/16885760\)
\(\beta_{10}\)\(=\)\((\)\(4339 \nu^{15} + 3301 \nu^{14} - 75250 \nu^{13} - 53930 \nu^{12} + 940625 \nu^{11} + 643695 \nu^{10} - 5519626 \nu^{9} - 3281194 \nu^{8} + 24192875 \nu^{7} + 12013925 \nu^{6} - 53427500 \nu^{5} - 14788480 \nu^{4} + 85946592 \nu^{3} + 6760448 \nu^{2} - 9632000 \nu + 29806080\)\()/16885760\)
\(\beta_{11}\)\(=\)\((\)\(4339 \nu^{15} - 3301 \nu^{14} - 75250 \nu^{13} + 53930 \nu^{12} + 940625 \nu^{11} - 643695 \nu^{10} - 5519626 \nu^{9} + 3281194 \nu^{8} + 24192875 \nu^{7} - 12013925 \nu^{6} - 53427500 \nu^{5} + 14788480 \nu^{4} + 85946592 \nu^{3} - 6760448 \nu^{2} - 9632000 \nu - 29806080\)\()/16885760\)
\(\beta_{12}\)\(=\)\((\)\(6491 \nu^{15} - 761 \nu^{14} - 112738 \nu^{13} + 11858 \nu^{12} + 1409225 \nu^{11} - 148395 \nu^{10} - 8276154 \nu^{9} + 756434 \nu^{8} + 36245267 \nu^{7} - 3323257 \nu^{6} - 80043980 \nu^{5} + 3409280 \nu^{4} + 133656288 \nu^{3} - 1558528 \nu^{2} - 14430464 \nu - 19785216\)\()/16885760\)
\(\beta_{13}\)\(=\)\((\)\(-11597 \nu^{15} + 6596 \nu^{14} + 203666 \nu^{13} - 113560 \nu^{12} - 2548375 \nu^{11} + 1419500 \nu^{10} + 15175618 \nu^{9} - 8295864 \nu^{8} - 66585149 \nu^{7} + 36509540 \nu^{6} + 153883160 \nu^{5} - 80627600 \nu^{4} - 264476096 \nu^{3} + 122123648 \nu^{2} + 120224768 \nu - 14535680\)\()/33771520\)
\(\beta_{14}\)\(=\)\((\)\(-485 \nu^{15} - 194 \nu^{14} + 8730 \nu^{13} + 3340 \nu^{12} - 110095 \nu^{11} - 41750 \nu^{10} + 676090 \nu^{9} + 243996 \nu^{8} - 2995845 \nu^{7} - 1073810 \nu^{6} + 7162480 \nu^{5} + 2371400 \nu^{4} - 12012480 \nu^{3} - 3591872 \nu^{2} + 5463040 \nu + 427520\)\()/993280\)
\(\beta_{15}\)\(=\)\((\)\(10588 \nu^{15} + 761 \nu^{14} - 186280 \nu^{13} - 11858 \nu^{12} + 2328500 \nu^{11} + 148395 \nu^{10} - 13822472 \nu^{9} - 756434 \nu^{8} + 59889020 \nu^{7} + 3323257 \nu^{6} - 132258800 \nu^{5} - 3409280 \nu^{4} + 209010624 \nu^{3} + 1558528 \nu^{2} - 23843840 \nu + 19785216\)\()/16885760\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{8} - \beta_{7} - \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} + 4 \beta_{6} + \beta_{5} + 8 \beta_{3} + 8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{15} + 5 \beta_{12} - 9 \beta_{11} - 9 \beta_{10} + 8 \beta_{9} + 13 \beta_{5} - 8 \beta_{4} - 13 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-17 \beta_{15} - 17 \beta_{14} + 17 \beta_{13} + 17 \beta_{12} + 9 \beta_{8} - 9 \beta_{7} + 44 \beta_{6} + 56 \beta_{3} + 17 \beta_{2} - 44 \beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-37 \beta_{14} - 37 \beta_{13} - 81 \beta_{11} - 81 \beta_{10} + 112 \beta_{9} + 81 \beta_{8} + 81 \beta_{7} + 141 \beta_{5}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-201 \beta_{15} + 201 \beta_{12} - 89 \beta_{11} + 89 \beta_{10} - 201 \beta_{5} + 201 \beta_{2} - 436 \beta_{1} - 472\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-325 \beta_{15} - 325 \beta_{14} - 325 \beta_{13} - 325 \beta_{12} + 761 \beta_{8} + 761 \beta_{7} + 1240 \beta_{4} + 1485 \beta_{2}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(2145 \beta_{14} - 2145 \beta_{13} - 905 \beta_{11} + 905 \beta_{10} - 905 \beta_{8} + 905 \beta_{7} - 4284 \beta_{6} - 2145 \beta_{5} - 4344 \beta_{3} - 4344\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-3077 \beta_{15} - 3077 \beta_{12} + 7361 \beta_{11} + 7361 \beta_{10} - 12864 \beta_{9} - 15277 \beta_{5} + 12864 \beta_{4} + 15277 \beta_{2}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(22041 \beta_{15} + 22041 \beta_{14} - 22041 \beta_{13} - 22041 \beta_{12} - 9177 \beta_{8} + 9177 \beta_{7} - 42308 \beta_{6} - 41752 \beta_{3} - 22041 \beta_{2} + 42308 \beta_{1}\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(30053 \beta_{14} + 30053 \beta_{13} + 72361 \beta_{11} + 72361 \beta_{10} - 130472 \beta_{9} - 72361 \beta_{8} - 72361 \beta_{7} - 154925 \beta_{5}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(222961 \beta_{15} - 222961 \beta_{12} + 92489 \beta_{11} - 92489 \beta_{10} + 222961 \beta_{5} - 222961 \beta_{2} + 419916 \beta_{1} + 409656\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(297317 \beta_{15} + 297317 \beta_{14} + 297317 \beta_{13} + 297317 \beta_{12} - 717233 \beta_{8} - 717233 \beta_{7} - 1311760 \beta_{4} - 1559117 \beta_{2}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-2239977 \beta_{14} + 2239977 \beta_{13} + 928217 \beta_{11} - 928217 \beta_{10} + 928217 \beta_{8} - 928217 \beta_{7} + 4180692 \beta_{6} + 2239977 \beta_{5} + 4058200 \beta_{3} + 4058200\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(2957317 \beta_{15} + 2957317 \beta_{12} - 7138009 \beta_{11} - 7138009 \beta_{10} + 13140600 \beta_{9} + 15630093 \beta_{5} - 13140600 \beta_{4} - 15630093 \beta_{2}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−1.87108 1.08027i
1.46557 + 0.846149i
2.73710 + 1.58027i
−0.599547 0.346149i
−2.73710 1.58027i
−1.46557 0.846149i
0.599547 + 0.346149i
1.87108 + 1.08027i
−1.87108 + 1.08027i
1.46557 0.846149i
2.73710 1.58027i
−0.599547 + 0.346149i
−2.73710 + 1.58027i
−1.46557 + 0.846149i
0.599547 0.346149i
1.87108 1.08027i
−0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.97257 1.05307i 1.00000 0 1.00000i 0.500000 0.866025i 1.18176 + 1.89827i
79.2 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.51486 + 1.64475i 1.00000 0 1.00000i 0.500000 0.866025i 2.13428 0.666969i
79.3 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.506647 + 2.17791i 1.00000 0 1.00000i 0.500000 0.866025i 0.650187 2.13945i
79.4 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 1.98078 1.03755i 1.00000 0 1.00000i 0.500000 0.866025i −2.23418 0.0918501i
79.5 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −2.13945 + 0.650187i 1.00000 0 1.00000i 0.500000 0.866025i −2.17791 0.506647i
79.6 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.666969 + 2.13428i 1.00000 0 1.00000i 0.500000 0.866025i −1.64475 + 1.51486i
79.7 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.0918501 2.23418i 1.00000 0 1.00000i 0.500000 0.866025i 1.03755 1.98078i
79.8 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 1.89827 + 1.18176i 1.00000 0 1.00000i 0.500000 0.866025i 1.05307 + 1.97257i
949.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i −1.97257 + 1.05307i 1.00000 0 1.00000i 0.500000 + 0.866025i 1.18176 1.89827i
949.2 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i −1.51486 1.64475i 1.00000 0 1.00000i 0.500000 + 0.866025i 2.13428 + 0.666969i
949.3 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0.506647 2.17791i 1.00000 0 1.00000i 0.500000 + 0.866025i 0.650187 + 2.13945i
949.4 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i 1.98078 + 1.03755i 1.00000 0 1.00000i 0.500000 + 0.866025i −2.23418 + 0.0918501i
949.5 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −2.13945 0.650187i 1.00000 0 1.00000i 0.500000 + 0.866025i −2.17791 + 0.506647i
949.6 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −0.666969 2.13428i 1.00000 0 1.00000i 0.500000 + 0.866025i −1.64475 1.51486i
949.7 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −0.0918501 + 2.23418i 1.00000 0 1.00000i 0.500000 + 0.866025i 1.03755 + 1.98078i
949.8 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 1.89827 1.18176i 1.00000 0 1.00000i 0.500000 + 0.866025i 1.05307 1.97257i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 949.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.n.k 16
5.b even 2 1 inner 1470.2.n.k 16
7.b odd 2 1 1470.2.n.l 16
7.c even 3 1 1470.2.g.k yes 8
7.c even 3 1 inner 1470.2.n.k 16
7.d odd 6 1 1470.2.g.j 8
7.d odd 6 1 1470.2.n.l 16
35.c odd 2 1 1470.2.n.l 16
35.i odd 6 1 1470.2.g.j 8
35.i odd 6 1 1470.2.n.l 16
35.j even 6 1 1470.2.g.k yes 8
35.j even 6 1 inner 1470.2.n.k 16
35.k even 12 1 7350.2.a.ds 4
35.k even 12 1 7350.2.a.dt 4
35.l odd 12 1 7350.2.a.dr 4
35.l odd 12 1 7350.2.a.du 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.g.j 8 7.d odd 6 1
1470.2.g.j 8 35.i odd 6 1
1470.2.g.k yes 8 7.c even 3 1
1470.2.g.k yes 8 35.j even 6 1
1470.2.n.k 16 1.a even 1 1 trivial
1470.2.n.k 16 5.b even 2 1 inner
1470.2.n.k 16 7.c even 3 1 inner
1470.2.n.k 16 35.j even 6 1 inner
1470.2.n.l 16 7.b odd 2 1
1470.2.n.l 16 7.d odd 6 1
1470.2.n.l 16 35.c odd 2 1
1470.2.n.l 16 35.i odd 6 1
7350.2.a.dr 4 35.l odd 12 1
7350.2.a.ds 4 35.k even 12 1
7350.2.a.dt 4 35.k even 12 1
7350.2.a.du 4 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11}^{8} + 18 T_{11}^{6} - 32 T_{11}^{5} + 292 T_{11}^{4} - 288 T_{11}^{3} + 832 T_{11}^{2} + 512 T_{11} + 1024 \)
\(T_{17}^{16} - \cdots\)
\(T_{19}^{8} - \cdots\)
\(T_{31}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$3$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$5$ \( 390625 + 312500 T + 156250 T^{2} + 50000 T^{3} + 11250 T^{4} + 3500 T^{5} + 5600 T^{6} + 4340 T^{7} + 2399 T^{8} + 868 T^{9} + 224 T^{10} + 28 T^{11} + 18 T^{12} + 16 T^{13} + 10 T^{14} + 4 T^{15} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( ( 1024 + 512 T + 832 T^{2} - 288 T^{3} + 292 T^{4} - 32 T^{5} + 18 T^{6} + T^{8} )^{2} \)
$13$ \( ( 256 + 1472 T^{2} + 644 T^{4} + 52 T^{6} + T^{8} )^{2} \)
$17$ \( 236421376 - 123008000 T^{2} + 43457664 T^{4} - 8227840 T^{6} + 1129520 T^{8} - 90880 T^{10} + 5064 T^{12} - 80 T^{14} + T^{16} \)
$19$ \( ( 200704 - 100352 T + 55552 T^{2} - 8064 T^{3} + 3280 T^{4} - 592 T^{5} + 132 T^{6} - 12 T^{7} + T^{8} )^{2} \)
$23$ \( 268435456 - 150994944 T^{2} + 59506688 T^{4} - 11419648 T^{6} + 1581312 T^{8} - 118144 T^{10} + 6192 T^{12} - 88 T^{14} + T^{16} \)
$29$ \( ( 1568 + 336 T - 66 T^{2} - 8 T^{3} + T^{4} )^{4} \)
$31$ \( ( 4096 + 8192 T + 11392 T^{2} + 7936 T^{3} + 3972 T^{4} + 992 T^{5} + 178 T^{6} + 16 T^{7} + T^{8} )^{2} \)
$37$ \( 1600000000 - 1248000000 T^{2} + 810080000 T^{4} - 115580800 T^{6} + 12021456 T^{8} - 542032 T^{10} + 17820 T^{12} - 148 T^{14} + T^{16} \)
$41$ \( ( -376 + 120 T + 26 T^{2} - 12 T^{3} + T^{4} )^{4} \)
$43$ \( ( 541696 + 297344 T^{2} + 16516 T^{4} + 236 T^{6} + T^{8} )^{2} \)
$47$ \( 1099511627776 - 210453397504 T^{2} + 28600958976 T^{4} - 1808023552 T^{6} + 82107408 T^{8} - 1871152 T^{10} + 30476 T^{12} - 204 T^{14} + T^{16} \)
$53$ \( 16777216 - 48234496 T^{2} + 116588544 T^{4} - 61988864 T^{6} + 26902784 T^{8} - 968576 T^{10} + 28464 T^{12} - 184 T^{14} + T^{16} \)
$59$ \( ( 9048064 - 2213888 T + 698112 T^{2} - 82048 T^{3} + 20432 T^{4} - 2512 T^{5} + 348 T^{6} - 20 T^{7} + T^{8} )^{2} \)
$61$ \( ( 419904 + 979776 T + 2204496 T^{2} + 206064 T^{3} + 34668 T^{4} + 1512 T^{5} + 270 T^{6} + 12 T^{7} + T^{8} )^{2} \)
$67$ \( 1048576 - 1179648 T^{2} + 929792 T^{4} - 356864 T^{6} + 98832 T^{8} - 14768 T^{10} + 1548 T^{12} - 44 T^{14} + T^{16} \)
$71$ \( ( 2944 - 1472 T - 36 T^{2} + 20 T^{3} + T^{4} )^{4} \)
$73$ \( 3782742016 - 15046338560 T^{2} + 58633164544 T^{4} - 4796194432 T^{6} + 313247952 T^{8} - 5756144 T^{10} + 80092 T^{12} - 316 T^{14} + T^{16} \)
$79$ \( ( 12845056 + 6422528 T + 2580480 T^{2} + 372736 T^{3} + 48896 T^{4} + 2176 T^{5} + 240 T^{6} + 8 T^{7} + T^{8} )^{2} \)
$83$ \( ( 16384 + 26624 T^{2} + 6976 T^{4} + 240 T^{6} + T^{8} )^{2} \)
$89$ \( ( 25482304 - 18617024 T + 10279760 T^{2} - 1982480 T^{3} + 265644 T^{4} - 21576 T^{5} + 1278 T^{6} - 44 T^{7} + T^{8} )^{2} \)
$97$ \( ( 107661376 + 4479328 T^{2} + 66804 T^{4} + 428 T^{6} + T^{8} )^{2} \)
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