Properties

Label 252.2.l.b
Level $252$
Weight $2$
Character orbit 252.l
Analytic conductor $2.012$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 5 x^{12} - 3 x^{11} + 7 x^{10} + 30 x^{9} - 117 x^{7} + 270 x^{5} + 189 x^{4} - 243 x^{3} - 1215 x^{2} + 2187\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -\beta_{5} + \beta_{9} ) q^{5} + \beta_{6} q^{7} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -\beta_{5} + \beta_{9} ) q^{5} + \beta_{6} q^{7} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} ) q^{9} + ( -1 + \beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{11} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{12} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{8} - \beta_{9} - \beta_{13} ) q^{15} + ( 1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{8} - \beta_{12} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{13} ) q^{19} + ( -2 \beta_{2} + \beta_{4} + \beta_{9} + \beta_{13} ) q^{21} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{12} ) q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} ) q^{25} + ( \beta_{1} - \beta_{2} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{27} + ( -\beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{29} + ( 1 - \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{31} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{10} + \beta_{11} + \beta_{13} ) q^{33} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{35} + ( -2 + \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{37} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{39} + ( -3 + 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{6} + \beta_{9} - \beta_{10} ) q^{41} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{13} ) q^{43} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{45} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{47} + ( 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{49} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} ) q^{51} + ( -3 + 3 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - 3 \beta_{12} ) q^{53} + ( -2 - 3 \beta_{1} - \beta_{2} + \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{12} + \beta_{13} ) q^{55} + ( -\beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{57} + ( 1 - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{59} + ( -1 + \beta_{2} + \beta_{5} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{61} + ( -1 - \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{7} - 4 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} ) q^{63} + ( -6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - \beta_{5} - 4 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + 3 \beta_{12} ) q^{65} + ( -2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{67} + ( -5 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{12} ) q^{69} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{12} ) q^{71} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{73} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{75} + ( -1 + 5 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{13} ) q^{77} + ( -3 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} ) q^{79} + ( \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{81} + ( 2 - 3 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{9} + \beta_{10} + 3 \beta_{11} - \beta_{12} - \beta_{13} ) q^{83} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{85} + ( 3 - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 5 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} - 3 \beta_{12} ) q^{87} + ( -2 \beta_{2} + 2 \beta_{6} + \beta_{9} + 2 \beta_{10} ) q^{89} + ( 1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} - \beta_{12} - \beta_{13} ) q^{91} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} + 4 \beta_{11} - \beta_{12} ) q^{93} + ( -2 + \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{12} ) q^{95} + ( 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{97} + ( -3 + 3 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 4 q^{5} - 3 q^{7} + 10 q^{9} + O(q^{10}) \) \( 14 q + 4 q^{5} - 3 q^{7} + 10 q^{9} - 4 q^{11} + 2 q^{13} + 7 q^{15} + 2 q^{17} + 7 q^{19} - 2 q^{21} - 22 q^{23} + 18 q^{25} + 9 q^{27} + q^{29} - q^{31} + 5 q^{33} - 19 q^{35} + 10 q^{37} - 20 q^{39} - 33 q^{41} + 7 q^{43} + 5 q^{45} - 3 q^{47} - 13 q^{49} + 20 q^{51} - 15 q^{53} - 28 q^{55} - 18 q^{57} - 14 q^{59} - 10 q^{61} - 39 q^{63} + 15 q^{65} + 6 q^{67} - 43 q^{69} + 2 q^{71} + 21 q^{73} + q^{75} + 19 q^{77} - 10 q^{79} + 22 q^{81} - 25 q^{83} + 8 q^{85} - 2 q^{87} - 6 q^{89} + 2 q^{91} + 16 q^{93} - 28 q^{95} - 18 q^{97} + 7 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 5 x^{12} - 3 x^{11} + 7 x^{10} + 30 x^{9} - 117 x^{7} + 270 x^{5} + 189 x^{4} - 243 x^{3} - 1215 x^{2} + 2187\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 35 \nu^{13} - 72 \nu^{12} - 157 \nu^{11} - 312 \nu^{10} + 290 \nu^{9} + 1383 \nu^{8} - 1143 \nu^{7} - 3393 \nu^{6} + 2025 \nu^{5} + 8802 \nu^{4} + 9288 \nu^{3} - 23814 \nu^{2} - 63180 \nu + 46656 \)\()/43011\)
\(\beta_{3}\)\(=\)\((\)\( -8 \nu^{13} + 2 \nu^{12} - 23 \nu^{11} + 5 \nu^{10} + 37 \nu^{9} - 127 \nu^{8} + 78 \nu^{7} + 225 \nu^{6} - 72 \nu^{5} + 297 \nu^{4} - 1701 \nu^{3} - 2295 \nu^{2} + 5184 \nu - 8505 \)\()/4779\)
\(\beta_{4}\)\(=\)\((\)\( 28 \nu^{13} - 15 \nu^{12} - 191 \nu^{11} - 135 \nu^{10} + 289 \nu^{9} + 546 \nu^{8} + 498 \nu^{7} - 2151 \nu^{6} - 1152 \nu^{5} + 8370 \nu^{4} + 2916 \nu^{3} + 1701 \nu^{2} - 24381 \nu - 40581 \)\()/14337\)
\(\beta_{5}\)\(=\)\((\)\( 26 \nu^{13} + 7 \nu^{12} - 70 \nu^{11} - 266 \nu^{10} - 301 \nu^{9} + 955 \nu^{8} + 846 \nu^{7} - 2529 \nu^{6} - 2574 \nu^{5} - 864 \nu^{4} + 14985 \nu^{3} + 11124 \nu^{2} - 48438 \nu - 30618 \)\()/14337\)
\(\beta_{6}\)\(=\)\((\)\( -56 \nu^{13} + 441 \nu^{12} - 62 \nu^{11} - 1227 \nu^{10} - 761 \nu^{9} + 165 \nu^{8} + 8622 \nu^{7} + 1638 \nu^{6} - 26190 \nu^{5} + 2538 \nu^{4} + 35424 \nu^{3} + 61722 \nu^{2} + 22113 \nu - 293058 \)\()/43011\)
\(\beta_{7}\)\(=\)\((\)\( -83 \nu^{13} + 423 \nu^{12} + 406 \nu^{11} - 354 \nu^{10} - 2237 \nu^{9} - 2364 \nu^{8} + 7902 \nu^{7} + 7065 \nu^{6} - 17037 \nu^{5} - 20142 \nu^{4} + 7965 \nu^{3} + 116154 \nu^{2} + 87966 \nu - 237654 \)\()/43011\)
\(\beta_{8}\)\(=\)\((\)\( 97 \nu^{13} - 675 \nu^{12} + \nu^{11} + 1491 \nu^{10} + 2002 \nu^{9} + 1209 \nu^{8} - 10584 \nu^{7} - 2736 \nu^{6} + 42714 \nu^{5} - 1269 \nu^{4} - 51003 \nu^{3} - 89424 \nu^{2} - 59292 \nu + 454896 \)\()/43011\)
\(\beta_{9}\)\(=\)\((\)\( -73 \nu^{13} - 360 \nu^{12} - 40 \nu^{11} + 804 \nu^{10} + 1703 \nu^{9} - 417 \nu^{8} - 8694 \nu^{7} + 1413 \nu^{6} + 19845 \nu^{5} - 2295 \nu^{4} - 32022 \nu^{3} - 98901 \nu^{2} - 11178 \nu + 239841 \)\()/43011\)
\(\beta_{10}\)\(=\)\((\)\( 5 \nu^{13} + 203 \nu^{12} + 68 \nu^{11} - 463 \nu^{10} - 760 \nu^{9} - 247 \nu^{8} + 3780 \nu^{7} + 1044 \nu^{6} - 11700 \nu^{5} - 999 \nu^{4} + 21789 \nu^{3} + 41445 \nu^{2} + 11583 \nu - 116397 \)\()/14337\)
\(\beta_{11}\)\(=\)\((\)\( 239 \nu^{13} - 324 \nu^{12} - 385 \nu^{11} + 93 \nu^{10} + 1133 \nu^{9} + 3687 \nu^{8} - 6129 \nu^{7} - 13221 \nu^{6} + 20358 \nu^{5} + 15120 \nu^{4} - 4239 \nu^{3} - 46413 \nu^{2} - 113724 \nu + 267543 \)\()/43011\)
\(\beta_{12}\)\(=\)\((\)\( 179 \nu^{13} + 471 \nu^{12} - 769 \nu^{11} - 2352 \nu^{10} - 763 \nu^{9} + 5184 \nu^{8} + 9288 \nu^{7} - 16785 \nu^{6} - 31266 \nu^{5} + 33021 \nu^{4} + 101142 \nu^{3} + 55485 \nu^{2} - 162081 \nu - 324405 \)\()/43011\)
\(\beta_{13}\)\(=\)\((\)\( 500 \nu^{13} - 81 \nu^{12} - 1096 \nu^{11} - 1392 \nu^{10} - 145 \nu^{9} + 10167 \nu^{8} - 27936 \nu^{6} + 5319 \nu^{5} + 40797 \nu^{4} + 89964 \nu^{3} + 243 \nu^{2} - 335097 \nu + 97686 \)\()/43011\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{2} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{12} - 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + \beta_{1}\)
\(\nu^{5}\)\(=\)\(3 \beta_{12} - \beta_{11} + \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 8\)
\(\nu^{6}\)\(=\)\(5 \beta_{13} - 9 \beta_{12} - 2 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} - \beta_{8} + 6 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} + 6 \beta_{2} + 2 \beta_{1} - 1\)
\(\nu^{7}\)\(=\)\(-3 \beta_{13} + 5 \beta_{12} - 4 \beta_{11} - 2 \beta_{10} - 18 \beta_{9} + 10 \beta_{8} - 10 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 8 \beta_{4} + 10 \beta_{3} - 5 \beta_{2} - 10 \beta_{1} + 11\)
\(\nu^{8}\)\(=\)\(13 \beta_{13} - 13 \beta_{12} - 14 \beta_{11} - 22 \beta_{10} - \beta_{9} - 8 \beta_{8} + 2 \beta_{7} + 23 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - 23 \beta_{3} + 2 \beta_{2} + 27 \beta_{1} - 5\)
\(\nu^{9}\)\(=\)\(-9 \beta_{13} + 10 \beta_{12} + 14 \beta_{11} + 10 \beta_{10} + 22 \beta_{9} + 19 \beta_{8} - 9 \beta_{7} + 32 \beta_{6} + 9 \beta_{4} - 18 \beta_{3} - 58 \beta_{2} - 20 \beta_{1} - 9\)
\(\nu^{10}\)\(=\)\(21 \beta_{13} + 22 \beta_{12} - 65 \beta_{11} - 46 \beta_{10} - 28 \beta_{9} - 6 \beta_{8} - 32 \beta_{7} - 26 \beta_{6} - 14 \beta_{5} - 5 \beta_{4} - 35 \beta_{3} - 137 \beta_{2} - 44 \beta_{1} + 57\)
\(\nu^{11}\)\(=\)\(-54 \beta_{13} + 66 \beta_{12} - 10 \beta_{11} - 53 \beta_{10} - 115 \beta_{9} + 31 \beta_{8} - 41 \beta_{7} - 10 \beta_{6} - 38 \beta_{5} - 55 \beta_{4} - 115 \beta_{3} + 66 \beta_{2} - 54 \beta_{1} - 242\)
\(\nu^{12}\)\(=\)\(77 \beta_{13} + 81 \beta_{12} + 7 \beta_{11} - 58 \beta_{10} + 167 \beta_{9} - 19 \beta_{8} + 135 \beta_{7} + 60 \beta_{6} - 181 \beta_{5} - 78 \beta_{4} - 26 \beta_{3} - 183 \beta_{2} - 304 \beta_{1} - 109\)
\(\nu^{13}\)\(=\)\(51 \beta_{13} - 310 \beta_{12} + 329 \beta_{11} + 367 \beta_{10} - 117 \beta_{9} - 62 \beta_{8} - 325 \beta_{7} + 146 \beta_{6} + 38 \beta_{5} - 89 \beta_{4} + 307 \beta_{3} - 167 \beta_{2} - 325 \beta_{1} - 457\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−1.73040 + 0.0755709i
−1.58203 0.705117i
−0.674693 1.59524i
−0.473632 + 1.66604i
1.13119 1.31165i
1.64515 + 0.541745i
1.68442 0.403398i
−1.73040 0.0755709i
−1.58203 + 0.705117i
−0.674693 + 1.59524i
−0.473632 1.66604i
1.13119 + 1.31165i
1.64515 0.541745i
1.68442 + 0.403398i
0 −1.73040 + 0.0755709i 0 0.967857 0 −1.11482 2.39941i 0 2.98858 0.261536i 0
193.2 0 −1.58203 0.705117i 0 −2.52026 0 1.98143 + 1.75326i 0 2.00562 + 2.23103i 0
193.3 0 −0.674693 1.59524i 0 4.14520 0 0.190437 + 2.63889i 0 −2.08958 + 2.15260i 0
193.4 0 −0.473632 + 1.66604i 0 −1.90301 0 −2.43415 + 1.03677i 0 −2.55135 1.57817i 0
193.5 0 1.13119 1.31165i 0 −1.52940 0 2.53654 0.752299i 0 −0.440838 2.96743i 0
193.6 0 1.64515 + 0.541745i 0 −0.763837 0 −1.05641 + 2.42569i 0 2.41302 + 1.78250i 0
193.7 0 1.68442 0.403398i 0 3.60346 0 −1.60302 2.10483i 0 2.67454 1.35898i 0
205.1 0 −1.73040 0.0755709i 0 0.967857 0 −1.11482 + 2.39941i 0 2.98858 + 0.261536i 0
205.2 0 −1.58203 + 0.705117i 0 −2.52026 0 1.98143 1.75326i 0 2.00562 2.23103i 0
205.3 0 −0.674693 + 1.59524i 0 4.14520 0 0.190437 2.63889i 0 −2.08958 2.15260i 0
205.4 0 −0.473632 1.66604i 0 −1.90301 0 −2.43415 1.03677i 0 −2.55135 + 1.57817i 0
205.5 0 1.13119 + 1.31165i 0 −1.52940 0 2.53654 + 0.752299i 0 −0.440838 + 2.96743i 0
205.6 0 1.64515 0.541745i 0 −0.763837 0 −1.05641 2.42569i 0 2.41302 1.78250i 0
205.7 0 1.68442 + 0.403398i 0 3.60346 0 −1.60302 + 2.10483i 0 2.67454 + 1.35898i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 205.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.l.b yes 14
3.b odd 2 1 756.2.l.b 14
4.b odd 2 1 1008.2.t.j 14
7.b odd 2 1 1764.2.l.i 14
7.c even 3 1 252.2.i.b 14
7.c even 3 1 1764.2.j.g 14
7.d odd 6 1 1764.2.i.i 14
7.d odd 6 1 1764.2.j.h 14
9.c even 3 1 252.2.i.b 14
9.c even 3 1 2268.2.k.e 14
9.d odd 6 1 756.2.i.b 14
9.d odd 6 1 2268.2.k.f 14
12.b even 2 1 3024.2.t.j 14
21.c even 2 1 5292.2.l.i 14
21.g even 6 1 5292.2.i.i 14
21.g even 6 1 5292.2.j.g 14
21.h odd 6 1 756.2.i.b 14
21.h odd 6 1 5292.2.j.h 14
28.g odd 6 1 1008.2.q.j 14
36.f odd 6 1 1008.2.q.j 14
36.h even 6 1 3024.2.q.j 14
63.g even 3 1 inner 252.2.l.b yes 14
63.h even 3 1 1764.2.j.g 14
63.h even 3 1 2268.2.k.e 14
63.i even 6 1 5292.2.j.g 14
63.j odd 6 1 2268.2.k.f 14
63.j odd 6 1 5292.2.j.h 14
63.k odd 6 1 1764.2.l.i 14
63.l odd 6 1 1764.2.i.i 14
63.n odd 6 1 756.2.l.b 14
63.o even 6 1 5292.2.i.i 14
63.s even 6 1 5292.2.l.i 14
63.t odd 6 1 1764.2.j.h 14
84.n even 6 1 3024.2.q.j 14
252.o even 6 1 3024.2.t.j 14
252.bl odd 6 1 1008.2.t.j 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.b 14 7.c even 3 1
252.2.i.b 14 9.c even 3 1
252.2.l.b yes 14 1.a even 1 1 trivial
252.2.l.b yes 14 63.g even 3 1 inner
756.2.i.b 14 9.d odd 6 1
756.2.i.b 14 21.h odd 6 1
756.2.l.b 14 3.b odd 2 1
756.2.l.b 14 63.n odd 6 1
1008.2.q.j 14 28.g odd 6 1
1008.2.q.j 14 36.f odd 6 1
1008.2.t.j 14 4.b odd 2 1
1008.2.t.j 14 252.bl odd 6 1
1764.2.i.i 14 7.d odd 6 1
1764.2.i.i 14 63.l odd 6 1
1764.2.j.g 14 7.c even 3 1
1764.2.j.g 14 63.h even 3 1
1764.2.j.h 14 7.d odd 6 1
1764.2.j.h 14 63.t odd 6 1
1764.2.l.i 14 7.b odd 2 1
1764.2.l.i 14 63.k odd 6 1
2268.2.k.e 14 9.c even 3 1
2268.2.k.e 14 63.h even 3 1
2268.2.k.f 14 9.d odd 6 1
2268.2.k.f 14 63.j odd 6 1
3024.2.q.j 14 36.h even 6 1
3024.2.q.j 14 84.n even 6 1
3024.2.t.j 14 12.b even 2 1
3024.2.t.j 14 252.o even 6 1
5292.2.i.i 14 21.g even 6 1
5292.2.i.i 14 63.o even 6 1
5292.2.j.g 14 21.g even 6 1
5292.2.j.g 14 63.i even 6 1
5292.2.j.h 14 21.h odd 6 1
5292.2.j.h 14 63.j odd 6 1
5292.2.l.i 14 21.c even 2 1
5292.2.l.i 14 63.s even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{7} - 2 T_{5}^{6} - 20 T_{5}^{5} + 12 T_{5}^{4} + 129 T_{5}^{3} + 81 T_{5}^{2} - 108 T_{5} - 81 \) acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \)
$3$ \( 2187 - 1215 T^{2} - 243 T^{3} + 189 T^{4} + 270 T^{5} - 117 T^{7} + 30 T^{9} + 7 T^{10} - 3 T^{11} - 5 T^{12} + T^{14} \)
$5$ \( ( -81 - 108 T + 81 T^{2} + 129 T^{3} + 12 T^{4} - 20 T^{5} - 2 T^{6} + T^{7} )^{2} \)
$7$ \( 823543 + 352947 T + 184877 T^{2} - 12005 T^{3} - 5145 T^{4} - 4312 T^{5} + 2184 T^{6} + 753 T^{7} + 312 T^{8} - 88 T^{9} - 15 T^{10} - 5 T^{11} + 11 T^{12} + 3 T^{13} + T^{14} \)
$11$ \( ( 81 - 216 T + 72 T^{2} + 174 T^{3} - 75 T^{4} - 41 T^{5} + 2 T^{6} + T^{7} )^{2} \)
$13$ \( 150626529 - 59904513 T + 57525819 T^{2} - 13891926 T^{3} + 11286787 T^{4} - 2477084 T^{5} + 1261032 T^{6} - 166125 T^{7} + 72459 T^{8} - 6792 T^{9} + 3006 T^{10} - 150 T^{11} + 66 T^{12} - 2 T^{13} + T^{14} \)
$17$ \( 6561 - 34992 T + 171315 T^{2} - 163782 T^{3} + 252315 T^{4} + 117693 T^{5} + 229455 T^{6} + 35055 T^{7} + 25764 T^{8} + 717 T^{9} + 2053 T^{10} + 40 T^{11} + 54 T^{12} - 2 T^{13} + T^{14} \)
$19$ \( 4084441 + 2718245 T + 6386590 T^{2} - 3636557 T^{3} + 4434668 T^{4} - 1055750 T^{5} + 592827 T^{6} - 107232 T^{7} + 52224 T^{8} - 7631 T^{9} + 2483 T^{10} - 284 T^{11} + 79 T^{12} - 7 T^{13} + T^{14} \)
$23$ \( ( 10287 + 17928 T + 7470 T^{2} - 825 T^{3} - 642 T^{4} - 32 T^{5} + 11 T^{6} + T^{7} )^{2} \)
$29$ \( 145660761 - 14663835 T + 120307599 T^{2} + 85897584 T^{3} + 91122165 T^{4} + 31944915 T^{5} + 11256606 T^{6} + 1678950 T^{7} + 388671 T^{8} + 35634 T^{9} + 9532 T^{10} + 461 T^{11} + 114 T^{12} - T^{13} + T^{14} \)
$31$ \( 13807190016 - 6768230400 T + 3991880448 T^{2} - 845293824 T^{3} + 320145937 T^{4} - 43173635 T^{5} + 17765313 T^{6} - 1403166 T^{7} + 545994 T^{8} - 16791 T^{9} + 12150 T^{10} - 147 T^{11} + 132 T^{12} + T^{13} + T^{14} \)
$37$ \( 1566893056 - 582676480 T + 616635136 T^{2} - 22905344 T^{3} + 108195008 T^{4} - 5679392 T^{5} + 9392352 T^{6} + 84720 T^{7} + 457224 T^{8} - 15644 T^{9} + 9977 T^{10} - 554 T^{11} + 175 T^{12} - 10 T^{13} + T^{14} \)
$41$ \( 1108290681 - 5174719749 T + 27318068505 T^{2} + 15810186042 T^{3} + 6482676969 T^{4} + 1620277101 T^{5} + 339156396 T^{6} + 54852066 T^{7} + 8343351 T^{8} + 1058184 T^{9} + 122094 T^{10} + 10647 T^{11} + 750 T^{12} + 33 T^{13} + T^{14} \)
$43$ \( 4084441 + 2718245 T + 6386590 T^{2} - 3636557 T^{3} + 4434668 T^{4} - 1055750 T^{5} + 592827 T^{6} - 107232 T^{7} + 52224 T^{8} - 7631 T^{9} + 2483 T^{10} - 284 T^{11} + 79 T^{12} - 7 T^{13} + T^{14} \)
$47$ \( 136048896 - 158723712 T + 199349424 T^{2} - 56529576 T^{3} + 45671121 T^{4} - 8704260 T^{5} + 8286543 T^{6} - 707130 T^{7} + 323514 T^{8} - 5832 T^{9} + 8298 T^{10} - 45 T^{11} + 114 T^{12} + 3 T^{13} + T^{14} \)
$53$ \( 952401321 + 2277264051 T + 4569369084 T^{2} + 2291724495 T^{3} + 1102507524 T^{4} + 196816392 T^{5} + 74649195 T^{6} + 13241718 T^{7} + 3279960 T^{8} + 325539 T^{9} + 43173 T^{10} + 2412 T^{11} + 327 T^{12} + 15 T^{13} + T^{14} \)
$59$ \( 688747536 + 5050815264 T + 38828496636 T^{2} - 12726712872 T^{3} + 6031230597 T^{4} - 380889378 T^{5} + 180648873 T^{6} + 3780486 T^{7} + 5415255 T^{8} + 268263 T^{9} + 56104 T^{10} + 2198 T^{11} + 372 T^{12} + 14 T^{13} + T^{14} \)
$61$ \( 148644864 + 270077184 T + 371924448 T^{2} + 224556408 T^{3} + 111134833 T^{4} + 25984822 T^{5} + 8095101 T^{6} + 1451094 T^{7} + 390015 T^{8} + 50853 T^{9} + 9648 T^{10} + 858 T^{11} + 150 T^{12} + 10 T^{13} + T^{14} \)
$67$ \( 116985856 - 113178624 T + 116493248 T^{2} - 72186592 T^{3} + 52651377 T^{4} - 28174094 T^{5} + 13920486 T^{6} - 4649550 T^{7} + 1253364 T^{8} - 205403 T^{9} + 27063 T^{10} - 1846 T^{11} + 161 T^{12} - 6 T^{13} + T^{14} \)
$71$ \( ( -972 + 1620 T + 4509 T^{2} + 2169 T^{3} + 9 T^{4} - 116 T^{5} - T^{6} + T^{7} )^{2} \)
$73$ \( 2748590329 - 132587883 T + 1197851843 T^{2} + 280079096 T^{3} + 449343003 T^{4} + 56145421 T^{5} + 29169996 T^{6} - 713634 T^{7} + 983931 T^{8} - 68306 T^{9} + 28230 T^{10} - 3133 T^{11} + 404 T^{12} - 21 T^{13} + T^{14} \)
$79$ \( 54397165824 - 28709926272 T + 20914155312 T^{2} - 5154931992 T^{3} + 2999269969 T^{4} - 725781614 T^{5} + 261464838 T^{6} - 29255790 T^{7} + 4930128 T^{8} - 155913 T^{9} + 43659 T^{10} - 330 T^{11} + 327 T^{12} + 10 T^{13} + T^{14} \)
$83$ \( 901054679121 - 614158582239 T + 552490013322 T^{2} + 38563914105 T^{3} + 31939205499 T^{4} + 4248171090 T^{5} + 1673845947 T^{6} + 219863025 T^{7} + 33382563 T^{8} + 2232114 T^{9} + 196147 T^{10} + 9625 T^{11} + 738 T^{12} + 25 T^{13} + T^{14} \)
$89$ \( 16524331209 + 12932085294 T + 13851839079 T^{2} - 765782424 T^{3} + 1553505561 T^{4} + 61446195 T^{5} + 79129143 T^{6} + 1816587 T^{7} + 2286090 T^{8} + 125469 T^{9} + 34641 T^{10} + 900 T^{11} + 228 T^{12} + 6 T^{13} + T^{14} \)
$97$ \( 767677849 - 1012081296 T + 992279576 T^{2} - 522939832 T^{3} + 225655953 T^{4} - 52605197 T^{5} + 13339572 T^{6} - 1272273 T^{7} + 601413 T^{8} - 13043 T^{9} + 15987 T^{10} + 1448 T^{11} + 347 T^{12} + 18 T^{13} + T^{14} \)
show more
show less