Properties

Label 700.2.g.l
Level $700$
Weight $2$
Character orbit 700.g
Analytic conductor $5.590$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 17 x^{12} - 104 x^{10} + 713 x^{8} + 238 x^{6} + 1004 x^{4} - 152 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{10} q^{3} + ( 1 - \beta_{4} ) q^{4} + \beta_{14} q^{6} + ( -\beta_{7} + \beta_{8} ) q^{7} + ( \beta_{2} - \beta_{6} ) q^{8} + ( -\beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + \beta_{10} q^{3} + ( 1 - \beta_{4} ) q^{4} + \beta_{14} q^{6} + ( -\beta_{7} + \beta_{8} ) q^{7} + ( \beta_{2} - \beta_{6} ) q^{8} + ( -\beta_{3} - \beta_{4} ) q^{9} + ( \beta_{3} - \beta_{9} ) q^{11} + ( -\beta_{8} + \beta_{10} + \beta_{12} ) q^{12} + ( \beta_{12} - \beta_{15} ) q^{13} + ( \beta_{9} - \beta_{11} ) q^{14} + ( -2 \beta_{3} - 2 \beta_{4} + \beta_{9} ) q^{16} + ( \beta_{8} + \beta_{10} - \beta_{12} - \beta_{15} ) q^{17} + ( \beta_{5} - \beta_{6} ) q^{18} + ( -\beta_{1} - \beta_{11} - \beta_{13} + 2 \beta_{14} ) q^{19} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{11} + \beta_{13} ) q^{21} + ( -\beta_{6} + 2 \beta_{7} ) q^{22} + ( \beta_{2} - \beta_{5} - 2 \beta_{7} ) q^{23} + ( \beta_{11} + \beta_{13} + \beta_{14} ) q^{24} + ( 2 \beta_{1} + \beta_{13} - \beta_{14} ) q^{26} + ( -\beta_{8} - 2 \beta_{10} ) q^{27} + ( -\beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{28} + ( 1 + \beta_{3} + \beta_{4} ) q^{29} + ( -2 \beta_{1} - 2 \beta_{11} - \beta_{13} ) q^{31} + ( \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{32} + ( \beta_{8} + \beta_{10} - \beta_{12} - \beta_{15} ) q^{33} + ( 2 \beta_{1} - \beta_{11} - \beta_{13} ) q^{34} + ( 2 - 2 \beta_{3} + \beta_{9} ) q^{36} + ( -3 \beta_{2} - \beta_{5} ) q^{37} + ( \beta_{8} + 2 \beta_{10} + 2 \beta_{12} - \beta_{15} ) q^{38} + ( \beta_{3} - \beta_{9} ) q^{39} -3 \beta_{13} q^{41} + ( -\beta_{2} - \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{10} + 2 \beta_{12} + \beta_{15} ) q^{42} + ( -\beta_{2} + \beta_{5} ) q^{43} + ( -1 - 2 \beta_{3} - \beta_{4} - \beta_{9} ) q^{44} + ( -2 - 2 \beta_{4} + 2 \beta_{9} ) q^{46} + ( -\beta_{8} + 4 \beta_{10} ) q^{47} + ( -3 \beta_{8} + 2 \beta_{12} + \beta_{15} ) q^{48} + ( -1 + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{13} ) q^{49} + ( \beta_{3} + 2 \beta_{4} - 3 \beta_{9} ) q^{51} + ( -3 \beta_{8} - \beta_{10} - \beta_{12} ) q^{52} + ( 2 \beta_{2} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{53} + ( \beta_{11} - 2 \beta_{14} ) q^{54} + ( -2 + 2 \beta_{1} + 2 \beta_{3} + \beta_{9} + \beta_{13} - 2 \beta_{14} ) q^{56} + ( 3 \beta_{2} + \beta_{5} ) q^{57} + ( \beta_{2} - \beta_{5} + \beta_{6} ) q^{58} + ( \beta_{1} + \beta_{11} + 2 \beta_{13} - 6 \beta_{14} ) q^{59} + ( -\beta_{1} + \beta_{11} - \beta_{13} ) q^{61} + ( 4 \beta_{8} - 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{15} ) q^{62} + ( \beta_{2} - \beta_{5} - \beta_{7} - 2 \beta_{8} - 2 \beta_{10} ) q^{63} + ( 2 - 2 \beta_{3} + 3 \beta_{9} ) q^{64} + ( 2 \beta_{1} - \beta_{11} - \beta_{13} ) q^{66} + 2 \beta_{7} q^{67} + ( 3 \beta_{10} - 3 \beta_{12} - \beta_{15} ) q^{68} + ( -2 \beta_{1} + 2 \beta_{11} + 3 \beta_{13} ) q^{69} + ( -2 \beta_{3} + 4 \beta_{4} - 2 \beta_{9} ) q^{71} + ( 2 \beta_{2} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{72} + ( -2 \beta_{8} - 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{15} ) q^{73} + ( -6 + 2 \beta_{4} ) q^{74} + ( 2 \beta_{1} - \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{76} + ( 3 \beta_{2} + \beta_{5} - \beta_{8} - \beta_{10} + 3 \beta_{12} - \beta_{15} ) q^{77} + ( -\beta_{6} + 2 \beta_{7} ) q^{78} + ( \beta_{3} + 2 \beta_{4} - 3 \beta_{9} ) q^{79} + ( -5 + 4 \beta_{3} + 4 \beta_{4} ) q^{81} + ( 6 \beta_{8} + 6 \beta_{10} - 6 \beta_{12} ) q^{82} + ( 5 \beta_{8} + \beta_{10} ) q^{83} + ( -3 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{9} + \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{84} + ( 2 + 2 \beta_{4} ) q^{86} + \beta_{8} q^{87} + ( -\beta_{2} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{88} -\beta_{13} q^{89} + ( -\beta_{1} + 3 \beta_{3} - 2 \beta_{4} - \beta_{9} - \beta_{11} - \beta_{13} + 2 \beta_{14} ) q^{91} + ( -2 \beta_{2} - 2 \beta_{5} - 4 \beta_{7} ) q^{92} + ( -5 \beta_{2} - 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{93} + ( \beta_{11} + 4 \beta_{14} ) q^{94} + ( -2 \beta_{1} + 3 \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{96} + ( -3 \beta_{8} - 3 \beta_{10} + 3 \beta_{12} + 3 \beta_{15} ) q^{97} + ( -\beta_{2} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{8} + 4 \beta_{10} - 4 \beta_{12} ) q^{98} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 12q^{4} - 8q^{9} + O(q^{10}) \) \( 16q + 12q^{4} - 8q^{9} + 4q^{14} - 12q^{16} - 8q^{21} + 24q^{29} + 28q^{36} - 32q^{44} - 32q^{46} - 20q^{56} + 36q^{64} - 88q^{74} - 48q^{81} - 40q^{84} + 40q^{86} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 17 x^{12} - 104 x^{10} + 713 x^{8} + 238 x^{6} + 1004 x^{4} - 152 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -1739863 \nu^{15} - 15207626 \nu^{13} + 18409195 \nu^{11} + 443618586 \nu^{9} + 534122349 \nu^{7} - 10160239700 \nu^{5} - 13245387916 \nu^{3} - 15043137568 \nu \)\()/ 3284689024 \)
\(\beta_{2}\)\(=\)\((\)\(59557735 \nu^{14} - 533150 \nu^{12} - 1053470715 \nu^{10} - 6261947986 \nu^{8} + 43163441379 \nu^{6} + 18860238060 \nu^{4} + 35114940892 \nu^{2} - 37634726944\)\()/ 26277512192 \)
\(\beta_{3}\)\(=\)\((\)\(-107882115 \nu^{14} - 77610858 \nu^{12} + 1877392743 \nu^{10} + 12516347930 \nu^{8} - 70055295983 \nu^{6} - 85825593884 \nu^{4} - 89210141260 \nu^{2} - 17971784544\)\()/ 26277512192 \)
\(\beta_{4}\)\(=\)\((\)\(157920899 \nu^{14} - 22764694 \nu^{12} - 2755245415 \nu^{10} - 15956221978 \nu^{8} + 116401708015 \nu^{6} + 28168725020 \nu^{4} + 100553570892 \nu^{2} - 26166657184\)\()/ 26277512192 \)
\(\beta_{5}\)\(=\)\((\)\(-158998017 \nu^{14} - 50557646 \nu^{12} + 2795857485 \nu^{10} + 17374927454 \nu^{8} - 109290094885 \nu^{6} - 81527049396 \nu^{4} - 100430658244 \nu^{2} - 32954999072\)\()/ 26277512192 \)
\(\beta_{6}\)\(=\)\((\)\(-111135863 \nu^{14} + 35207038 \nu^{12} + 1894572235 \nu^{10} + 10887840882 \nu^{8} - 82988272691 \nu^{6} - 677905516 \nu^{4} - 86864792220 \nu^{2} + 16400486944\)\()/ 13138756096 \)
\(\beta_{7}\)\(=\)\((\)\(-58429897 \nu^{14} - 4454734 \nu^{12} + 986022581 \nu^{10} + 6149655454 \nu^{8} - 41069887437 \nu^{6} - 16730498212 \nu^{4} - 64300182404 \nu^{2} + 4129925088\)\()/ 6569378048 \)
\(\beta_{8}\)\(=\)\((\)\(54968795 \nu^{15} - 11713990 \nu^{13} - 949732575 \nu^{11} - 5499231466 \nu^{9} + 40692711847 \nu^{7} + 6048096764 \nu^{5} + 38674762028 \nu^{3} - 15359983776 \nu\)\()/ 6569378048 \)
\(\beta_{9}\)\(=\)\((\)\(-305660455 \nu^{14} - 81621346 \nu^{12} + 5174279611 \nu^{10} + 33239289426 \nu^{8} - 209185441443 \nu^{6} - 127577110892 \nu^{4} - 347003644892 \nu^{2} + 150348832\)\()/ 26277512192 \)
\(\beta_{10}\)\(=\)\((\)\(-146835231 \nu^{15} + 25824526 \nu^{13} + 2503166931 \nu^{11} + 14845603426 \nu^{9} - 107472197627 \nu^{7} - 17672134092 \nu^{5} - 137231331644 \nu^{3} + 55358945312 \nu\)\()/ 13138756096 \)
\(\beta_{11}\)\(=\)\((\)\(171891871 \nu^{15} - 47519246 \nu^{13} - 2943150035 \nu^{11} - 17039166818 \nu^{9} + 127754563067 \nu^{7} + 8119855564 \nu^{5} + 144633757372 \nu^{3} - 39715733280 \nu\)\()/ 13138756096 \)
\(\beta_{12}\)\(=\)\((\)\(-354912961 \nu^{15} - 119115470 \nu^{13} + 6034586637 \nu^{11} + 39017889374 \nu^{9} - 240529045221 \nu^{7} - 170796167476 \nu^{5} - 394053088964 \nu^{3} - 16283111712 \nu\)\()/ 26277512192 \)
\(\beta_{13}\)\(=\)\((\)\( 74983 \nu^{15} + 31874 \nu^{13} - 1281915 \nu^{11} - 8349682 \nu^{9} + 50321507 \nu^{7} + 41578380 \nu^{5} + 78299292 \nu^{3} + 9807584 \nu \)\()/4257536\)
\(\beta_{14}\)\(=\)\((\)\(525461139 \nu^{15} - 22220726 \nu^{13} - 8971096375 \nu^{11} - 54288962554 \nu^{9} + 377737969983 \nu^{7} + 113217186140 \nu^{5} + 498241271180 \nu^{3} - 137015149728 \nu\)\()/ 26277512192 \)
\(\beta_{15}\)\(=\)\((\)\(-529467953 \nu^{15} - 68532718 \nu^{13} + 8933979069 \nu^{11} + 56185200254 \nu^{9} - 369146557717 \nu^{7} - 168419959540 \nu^{5} - 598285870468 \nu^{3} - 33389699360 \nu\)\()/ 26277512192 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} - \beta_{8}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{9} + 2 \beta_{6} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 1\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{15} + \beta_{14} + 2 \beta_{10} - 2 \beta_{8} + 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(11 \beta_{9} - 16 \beta_{7} + 4 \beta_{6} - \beta_{4} - 4 \beta_{3} + 4 \beta_{2} + 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{15} - 28 \beta_{14} + 9 \beta_{13} + 5 \beta_{12} - 10 \beta_{11} - 33 \beta_{10} + 2 \beta_{8} + 14 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(-13 \beta_{9} + 24 \beta_{7} - 6 \beta_{6} + 30 \beta_{5} + 14 \beta_{4} - 21 \beta_{3} + 10 \beta_{2} + 44\)
\(\nu^{7}\)\(=\)\((\)\(3 \beta_{15} + 60 \beta_{14} + 113 \beta_{13} + 199 \beta_{12} + 40 \beta_{11} - 15 \beta_{10} - 136 \beta_{8} - 10 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(109 \beta_{9} - 208 \beta_{7} + 176 \beta_{6} - 120 \beta_{5} + 303 \beta_{4} + 86 \beta_{3} - 568 \beta_{2} - 693\)\()/2\)
\(\nu^{9}\)\(=\)\(-267 \beta_{15} + 130 \beta_{14} - 247 \beta_{13} - 239 \beta_{12} - 343 \beta_{11} + 413 \beta_{10} + 287 \beta_{8} + 294 \beta_{1}\)
\(\nu^{10}\)\(=\)\((\)\(1241 \beta_{9} - 948 \beta_{7} - 1260 \beta_{6} + 1182 \beta_{5} - 1955 \beta_{4} - 2440 \beta_{3} + 1866 \beta_{2} + 2701\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(225 \beta_{15} - 2808 \beta_{14} + 2487 \beta_{13} + 3957 \beta_{12} - 622 \beta_{11} - 6345 \beta_{10} + 778 \beta_{8} - 1974 \beta_{1}\)\()/2\)
\(\nu^{12}\)\(=\)\(-4228 \beta_{9} + 6376 \beta_{7} - 700 \beta_{6} + 1976 \beta_{5} + 3935 \beta_{4} - 397 \beta_{3} - 5168 \beta_{2} - 4387\)
\(\nu^{13}\)\(=\)\((\)\(-475 \beta_{15} + 30988 \beta_{14} - 3947 \beta_{13} + 7849 \beta_{12} - 1324 \beta_{11} + 36311 \beta_{10} - 3780 \beta_{8} - 1986 \beta_{1}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(49285 \beta_{9} - 68244 \beta_{7} + 2456 \beta_{6} - 35158 \beta_{5} - 24397 \beta_{4} - 3598 \beta_{3} - 40506 \beta_{2} - 89305\)\()/2\)
\(\nu^{15}\)\(=\)\(-24620 \beta_{15} - 29462 \beta_{14} - 45828 \beta_{13} - 62684 \beta_{12} - 53945 \beta_{11} - 3806 \beta_{10} + 86647 \beta_{8} + 13672 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−0.409646 + 0.286988i
0.409646 0.286988i
−0.409646 0.286988i
0.409646 + 0.286988i
−0.645096 0.854135i
0.645096 + 0.854135i
−0.645096 + 0.854135i
0.645096 0.854135i
2.15532 + 0.457057i
−2.15532 0.457057i
2.15532 0.457057i
−2.15532 + 0.457057i
0.877859 + 2.23141i
−0.877859 2.23141i
0.877859 2.23141i
−0.877859 + 2.23141i
−1.37491 0.331077i −2.13578 1.78078 + 0.910404i 0 2.93651 + 0.707107i 1.19935 2.35829i −2.14700 1.84130i 1.56155 0
251.2 −1.37491 0.331077i 2.13578 1.78078 + 0.910404i 0 −2.93651 0.707107i −1.19935 2.35829i −2.14700 1.84130i 1.56155 0
251.3 −1.37491 + 0.331077i −2.13578 1.78078 0.910404i 0 2.93651 0.707107i 1.19935 + 2.35829i −2.14700 + 1.84130i 1.56155 0
251.4 −1.37491 + 0.331077i 2.13578 1.78078 0.910404i 0 −2.93651 + 0.707107i −1.19935 + 2.35829i −2.14700 + 1.84130i 1.56155 0
251.5 −0.927153 1.06789i −0.662153 −0.280776 + 1.98019i 0 0.613917 + 0.707107i −2.35829 + 1.19935i 2.37495 1.53610i −2.56155 0
251.6 −0.927153 1.06789i 0.662153 −0.280776 + 1.98019i 0 −0.613917 0.707107i 2.35829 + 1.19935i 2.37495 1.53610i −2.56155 0
251.7 −0.927153 + 1.06789i −0.662153 −0.280776 1.98019i 0 0.613917 0.707107i −2.35829 1.19935i 2.37495 + 1.53610i −2.56155 0
251.8 −0.927153 + 1.06789i 0.662153 −0.280776 1.98019i 0 −0.613917 + 0.707107i 2.35829 1.19935i 2.37495 + 1.53610i −2.56155 0
251.9 0.927153 1.06789i −0.662153 −0.280776 1.98019i 0 −0.613917 + 0.707107i −2.35829 + 1.19935i −2.37495 1.53610i −2.56155 0
251.10 0.927153 1.06789i 0.662153 −0.280776 1.98019i 0 0.613917 0.707107i 2.35829 + 1.19935i −2.37495 1.53610i −2.56155 0
251.11 0.927153 + 1.06789i −0.662153 −0.280776 + 1.98019i 0 −0.613917 0.707107i −2.35829 1.19935i −2.37495 + 1.53610i −2.56155 0
251.12 0.927153 + 1.06789i 0.662153 −0.280776 + 1.98019i 0 0.613917 + 0.707107i 2.35829 1.19935i −2.37495 + 1.53610i −2.56155 0
251.13 1.37491 0.331077i −2.13578 1.78078 0.910404i 0 −2.93651 + 0.707107i 1.19935 2.35829i 2.14700 1.84130i 1.56155 0
251.14 1.37491 0.331077i 2.13578 1.78078 0.910404i 0 2.93651 0.707107i −1.19935 2.35829i 2.14700 1.84130i 1.56155 0
251.15 1.37491 + 0.331077i −2.13578 1.78078 + 0.910404i 0 −2.93651 0.707107i 1.19935 + 2.35829i 2.14700 + 1.84130i 1.56155 0
251.16 1.37491 + 0.331077i 2.13578 1.78078 + 0.910404i 0 2.93651 + 0.707107i −1.19935 + 2.35829i 2.14700 + 1.84130i 1.56155 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.g.l 16
4.b odd 2 1 inner 700.2.g.l 16
5.b even 2 1 inner 700.2.g.l 16
5.c odd 4 2 140.2.c.b 16
7.b odd 2 1 inner 700.2.g.l 16
20.d odd 2 1 inner 700.2.g.l 16
20.e even 4 2 140.2.c.b 16
28.d even 2 1 inner 700.2.g.l 16
35.c odd 2 1 inner 700.2.g.l 16
35.f even 4 2 140.2.c.b 16
35.k even 12 4 980.2.s.f 32
35.l odd 12 4 980.2.s.f 32
40.i odd 4 2 2240.2.e.f 16
40.k even 4 2 2240.2.e.f 16
140.c even 2 1 inner 700.2.g.l 16
140.j odd 4 2 140.2.c.b 16
140.w even 12 4 980.2.s.f 32
140.x odd 12 4 980.2.s.f 32
280.s even 4 2 2240.2.e.f 16
280.y odd 4 2 2240.2.e.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.c.b 16 5.c odd 4 2
140.2.c.b 16 20.e even 4 2
140.2.c.b 16 35.f even 4 2
140.2.c.b 16 140.j odd 4 2
700.2.g.l 16 1.a even 1 1 trivial
700.2.g.l 16 4.b odd 2 1 inner
700.2.g.l 16 5.b even 2 1 inner
700.2.g.l 16 7.b odd 2 1 inner
700.2.g.l 16 20.d odd 2 1 inner
700.2.g.l 16 28.d even 2 1 inner
700.2.g.l 16 35.c odd 2 1 inner
700.2.g.l 16 140.c even 2 1 inner
980.2.s.f 32 35.k even 12 4
980.2.s.f 32 35.l odd 12 4
980.2.s.f 32 140.w even 12 4
980.2.s.f 32 140.x odd 12 4
2240.2.e.f 16 40.i odd 4 2
2240.2.e.f 16 40.k even 4 2
2240.2.e.f 16 280.s even 4 2
2240.2.e.f 16 280.y odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{4} - 5 T_{3}^{2} + 2 \)
\( T_{11}^{4} + 15 T_{11}^{2} + 52 \)
\( T_{19}^{4} - 38 T_{19}^{2} + 208 \)
\( T_{37}^{4} - 44 T_{37}^{2} + 416 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 - 12 T^{2} + 6 T^{4} - 3 T^{6} + T^{8} )^{2} \)
$3$ \( ( 2 - 5 T^{2} + T^{4} )^{4} \)
$5$ \( T^{16} \)
$7$ \( ( 2401 + 30 T^{4} + T^{8} )^{2} \)
$11$ \( ( 52 + 15 T^{2} + T^{4} )^{4} \)
$13$ \( ( 26 + 23 T^{2} + T^{4} )^{4} \)
$17$ \( ( 104 + 29 T^{2} + T^{4} )^{4} \)
$19$ \( ( 208 - 38 T^{2} + T^{4} )^{4} \)
$23$ \( ( 128 + 40 T^{2} + T^{4} )^{4} \)
$29$ \( ( -2 - 3 T + T^{2} )^{8} \)
$31$ \( ( 3328 - 120 T^{2} + T^{4} )^{4} \)
$37$ \( ( 416 - 44 T^{2} + T^{4} )^{4} \)
$41$ \( ( 72 + T^{2} )^{8} \)
$43$ \( ( 32 + 20 T^{2} + T^{4} )^{4} \)
$47$ \( ( 8 - 95 T^{2} + T^{4} )^{4} \)
$53$ \( ( 6656 - 164 T^{2} + T^{4} )^{4} \)
$59$ \( ( 13312 - 270 T^{2} + T^{4} )^{4} \)
$61$ \( ( 16 + 42 T^{2} + T^{4} )^{4} \)
$67$ \( ( 128 + 28 T^{2} + T^{4} )^{4} \)
$71$ \( ( 13312 + 236 T^{2} + T^{4} )^{4} \)
$73$ \( ( 1664 + 116 T^{2} + T^{4} )^{4} \)
$79$ \( ( 208 + 115 T^{2} + T^{4} )^{4} \)
$83$ \( ( 2312 - 170 T^{2} + T^{4} )^{4} \)
$89$ \( ( 8 + T^{2} )^{8} \)
$97$ \( ( 8424 + 261 T^{2} + T^{4} )^{4} \)
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