# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 252.i (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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{3} -\beta_{9} q^{5} + \beta_{10} q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{12} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{3} -\beta_{9} q^{5} + \beta_{10} q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{12} ) q^{9} + ( \beta_{8} - \beta_{11} ) 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_{2} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{13} ) q^{17} + ( 1 - \beta_{2} + \beta_{5} + \beta_{8} - \beta_{11} ) q^{19} + ( -1 - \beta_{1} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} ) q^{21} + ( 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{23} + ( \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{12} ) 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} + ( -\beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} ) q^{33} + ( -3 + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{35} + ( 3 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{37} + ( 1 - \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} ) 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_{3} + \beta_{4} - 3 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{45} + ( 1 + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{47} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{49} + ( -\beta_{3} - \beta_{5} - \beta_{8} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{51} + ( 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + \beta_{13} ) 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} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{59} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{13} ) q^{61} + ( 3 - \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{63} + ( -1 + 6 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 4 \beta_{12} - \beta_{13} ) q^{65} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} - \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} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} ) q^{73} + ( -3 + 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{75} + ( -4 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{9} - \beta_{11} + \beta_{13} ) q^{77} + ( 3 - \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{79} + ( -2 - \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) 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} + ( -1 + \beta_{1} + 4 \beta_{2} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{87} + ( -2 - 2 \beta_{1} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} + 2 \beta_{11} + 2 \beta_{12} ) q^{89} + ( -3 + \beta_{2} + 5 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{12} ) q^{91} + ( 5 - \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{7} + \beta_{8} - 4 \beta_{9} - 4 \beta_{10} - 5 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{93} + ( 4 - 5 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{7} - 2 \beta_{9} + 2 \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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q + 3 q^{3} - 2 q^{5} + 6 q^{7} - 5 q^{9} + O(q^{10})$$ $$14 q + 3 q^{3} - 2 q^{5} + 6 q^{7} - 5 q^{9} + 2 q^{11} + 2 q^{13} + 7 q^{15} + 2 q^{17} + 7 q^{19} - 11 q^{21} + 11 q^{23} - 9 q^{25} + 9 q^{27} + q^{29} + 2 q^{31} - 4 q^{33} - 19 q^{35} + 10 q^{37} - 2 q^{39} - 33 q^{41} + 7 q^{43} - 10 q^{45} + 6 q^{47} - 4 q^{49} - 13 q^{51} - 15 q^{53} - 28 q^{55} - 18 q^{57} + 28 q^{59} + 20 q^{61} + 33 q^{63} - 30 q^{65} - 12 q^{67} - 43 q^{69} + 2 q^{71} + 21 q^{73} - 44 q^{75} - 47 q^{77} + 20 q^{79} - 29 q^{81} - 25 q^{83} + 8 q^{85} + 28 q^{87} - 6 q^{89} + 2 q^{91} + 22 q^{93} + 56 q^{95} - 18 q^{97} + 7 q^{99} + O(q^{100})$$

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 $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 1.64515 − 0.541745i −0.473632 − 1.66604i 1.68442 + 0.403398i 1.13119 + 1.31165i −1.73040 − 0.0755709i −1.58203 + 0.705117i −0.674693 + 1.59524i 1.64515 + 0.541745i −0.473632 + 1.66604i 1.68442 − 0.403398i 1.13119 − 1.31165i −1.73040 + 0.0755709i −1.58203 − 0.705117i −0.674693 − 1.59524i
0 −1.29174 1.15387i 0 0.381918 0.661502i 0 2.62892 0.297968i 0 0.337180 + 2.98099i 0
25.2 0 −1.20601 + 1.24319i 0 0.951504 1.64805i 0 2.11495 + 1.58965i 0 −0.0910656 2.99862i 0
25.3 0 −0.492857 1.66045i 0 −1.80173 + 3.12069i 0 −1.02133 + 2.44067i 0 −2.51418 + 1.63673i 0
25.4 0 0.570327 1.63546i 0 0.764702 1.32450i 0 −1.91978 1.82056i 0 −2.34945 1.86549i 0
25.5 0 0.799754 + 1.53636i 0 −0.483929 + 0.838189i 0 −1.52054 + 2.16517i 0 −1.72079 + 2.45742i 0
25.6 0 1.40166 + 1.01752i 0 1.26013 2.18261i 0 0.527655 2.59260i 0 0.929318 + 2.85243i 0
25.7 0 1.71886 0.213318i 0 −2.07260 + 3.58985i 0 2.19013 1.48437i 0 2.90899 0.733330i 0
121.1 0 −1.29174 + 1.15387i 0 0.381918 + 0.661502i 0 2.62892 + 0.297968i 0 0.337180 2.98099i 0
121.2 0 −1.20601 1.24319i 0 0.951504 + 1.64805i 0 2.11495 1.58965i 0 −0.0910656 + 2.99862i 0
121.3 0 −0.492857 + 1.66045i 0 −1.80173 3.12069i 0 −1.02133 2.44067i 0 −2.51418 1.63673i 0
121.4 0 0.570327 + 1.63546i 0 0.764702 + 1.32450i 0 −1.91978 + 1.82056i 0 −2.34945 + 1.86549i 0
121.5 0 0.799754 1.53636i 0 −0.483929 0.838189i 0 −1.52054 2.16517i 0 −1.72079 2.45742i 0
121.6 0 1.40166 1.01752i 0 1.26013 + 2.18261i 0 0.527655 + 2.59260i 0 0.929318 2.85243i 0
121.7 0 1.71886 + 0.213318i 0 −2.07260 3.58985i 0 2.19013 + 1.48437i 0 2.90899 + 0.733330i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{14} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14}$$
$3$ $$2187 - 2187 T + 1701 T^{2} - 1215 T^{3} + 918 T^{4} - 702 T^{5} + 432 T^{6} - 252 T^{7} + 144 T^{8} - 78 T^{9} + 34 T^{10} - 15 T^{11} + 7 T^{12} - 3 T^{13} + T^{14}$$
$5$ $$6561 - 8748 T + 18225 T^{2} - 12150 T^{3} + 21465 T^{4} - 14661 T^{5} + 13833 T^{6} - 4923 T^{7} + 2670 T^{8} - 357 T^{9} + 295 T^{10} - 16 T^{11} + 24 T^{12} + 2 T^{13} + T^{14}$$
$7$ $$823543 - 705894 T + 336140 T^{2} - 184877 T^{3} + 105987 T^{4} - 47089 T^{5} + 18942 T^{6} - 7572 T^{7} + 2706 T^{8} - 961 T^{9} + 309 T^{10} - 77 T^{11} + 20 T^{12} - 6 T^{13} + T^{14}$$
$11$ $$6561 + 17496 T + 40824 T^{2} + 43740 T^{3} + 48843 T^{4} + 23193 T^{5} + 27144 T^{6} + 7497 T^{7} + 12183 T^{8} - 2451 T^{9} + 1657 T^{10} - 68 T^{11} + 45 T^{12} - 2 T^{13} + T^{14}$$
$13$ $$150626529 - 59904513 T + 57525819 T^{2} - 13891926 T^{3} + 11286787 T^{4} - 2477084 T^{5} + 1261032 T^{6} - 166125 T^{7} + 72459 T^{8} - 6792 T^{9} + 3006 T^{10} - 150 T^{11} + 66 T^{12} - 2 T^{13} + T^{14}$$
$17$ $$6561 - 34992 T + 171315 T^{2} - 163782 T^{3} + 252315 T^{4} + 117693 T^{5} + 229455 T^{6} + 35055 T^{7} + 25764 T^{8} + 717 T^{9} + 2053 T^{10} + 40 T^{11} + 54 T^{12} - 2 T^{13} + T^{14}$$
$19$ $$4084441 + 2718245 T + 6386590 T^{2} - 3636557 T^{3} + 4434668 T^{4} - 1055750 T^{5} + 592827 T^{6} - 107232 T^{7} + 52224 T^{8} - 7631 T^{9} + 2483 T^{10} - 284 T^{11} + 79 T^{12} - 7 T^{13} + T^{14}$$
$23$ $$105822369 - 184425336 T + 244569294 T^{2} - 150895710 T^{3} + 77195754 T^{4} - 16527618 T^{5} + 6276375 T^{6} - 1215225 T^{7} + 339450 T^{8} - 46164 T^{9} + 8911 T^{10} - 932 T^{11} + 153 T^{12} - 11 T^{13} + T^{14}$$
$29$ $$145660761 - 14663835 T + 120307599 T^{2} + 85897584 T^{3} + 91122165 T^{4} + 31944915 T^{5} + 11256606 T^{6} + 1678950 T^{7} + 388671 T^{8} + 35634 T^{9} + 9532 T^{10} + 461 T^{11} + 114 T^{12} - T^{13} + T^{14}$$
$31$ $$( -117504 - 57600 T + 5737 T^{2} + 5003 T^{3} - 8 T^{4} - 131 T^{5} - T^{6} + T^{7} )^{2}$$
$37$ $$1566893056 - 582676480 T + 616635136 T^{2} - 22905344 T^{3} + 108195008 T^{4} - 5679392 T^{5} + 9392352 T^{6} + 84720 T^{7} + 457224 T^{8} - 15644 T^{9} + 9977 T^{10} - 554 T^{11} + 175 T^{12} - 10 T^{13} + T^{14}$$
$41$ $$1108290681 - 5174719749 T + 27318068505 T^{2} + 15810186042 T^{3} + 6482676969 T^{4} + 1620277101 T^{5} + 339156396 T^{6} + 54852066 T^{7} + 8343351 T^{8} + 1058184 T^{9} + 122094 T^{10} + 10647 T^{11} + 750 T^{12} + 33 T^{13} + T^{14}$$
$43$ $$4084441 + 2718245 T + 6386590 T^{2} - 3636557 T^{3} + 4434668 T^{4} - 1055750 T^{5} + 592827 T^{6} - 107232 T^{7} + 52224 T^{8} - 7631 T^{9} + 2483 T^{10} - 284 T^{11} + 79 T^{12} - 7 T^{13} + T^{14}$$
$47$ $$( -11664 - 13608 T + 1215 T^{2} + 3132 T^{3} + 135 T^{4} - 105 T^{5} - 3 T^{6} + T^{7} )^{2}$$
$53$ $$952401321 + 2277264051 T + 4569369084 T^{2} + 2291724495 T^{3} + 1102507524 T^{4} + 196816392 T^{5} + 74649195 T^{6} + 13241718 T^{7} + 3279960 T^{8} + 325539 T^{9} + 43173 T^{10} + 2412 T^{11} + 327 T^{12} + 15 T^{13} + T^{14}$$
$59$ $$( 26244 - 192456 T - 68175 T^{2} + 7506 T^{3} + 2331 T^{4} - 176 T^{5} - 14 T^{6} + T^{7} )^{2}$$
$61$ $$( -12192 + 22152 T - 9743 T^{2} - 358 T^{3} + 679 T^{4} - 50 T^{5} - 10 T^{6} + T^{7} )^{2}$$
$67$ $$( 10816 + 10464 T - 647 T^{2} - 3650 T^{3} - 1298 T^{4} - 125 T^{5} + 6 T^{6} + T^{7} )^{2}$$
$71$ $$( -972 + 1620 T + 4509 T^{2} + 2169 T^{3} + 9 T^{4} - 116 T^{5} - T^{6} + T^{7} )^{2}$$
$73$ $$2748590329 - 132587883 T + 1197851843 T^{2} + 280079096 T^{3} + 449343003 T^{4} + 56145421 T^{5} + 29169996 T^{6} - 713634 T^{7} + 983931 T^{8} - 68306 T^{9} + 28230 T^{10} - 3133 T^{11} + 404 T^{12} - 21 T^{13} + T^{14}$$
$79$ $$( -233232 - 123096 T + 24703 T^{2} + 17570 T^{3} + 970 T^{4} - 227 T^{5} - 10 T^{6} + T^{7} )^{2}$$
$83$ $$901054679121 - 614158582239 T + 552490013322 T^{2} + 38563914105 T^{3} + 31939205499 T^{4} + 4248171090 T^{5} + 1673845947 T^{6} + 219863025 T^{7} + 33382563 T^{8} + 2232114 T^{9} + 196147 T^{10} + 9625 T^{11} + 738 T^{12} + 25 T^{13} + T^{14}$$
$89$ $$16524331209 + 12932085294 T + 13851839079 T^{2} - 765782424 T^{3} + 1553505561 T^{4} + 61446195 T^{5} + 79129143 T^{6} + 1816587 T^{7} + 2286090 T^{8} + 125469 T^{9} + 34641 T^{10} + 900 T^{11} + 228 T^{12} + 6 T^{13} + T^{14}$$
$97$ $$767677849 - 1012081296 T + 992279576 T^{2} - 522939832 T^{3} + 225655953 T^{4} - 52605197 T^{5} + 13339572 T^{6} - 1272273 T^{7} + 601413 T^{8} - 13043 T^{9} + 15987 T^{10} + 1448 T^{11} + 347 T^{12} + 18 T^{13} + T^{14}$$