Properties

Label 700.2.c.k
Level $700$
Weight $2$
Character orbit 700.c
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.29960650073923649536.7
Defining polynomial: \(x^{16} - 7 x^{12} + 40 x^{8} - 112 x^{4} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta_{7} q^{2} + ( -\beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{3} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{4} + ( \beta_{3} + \beta_{4} - \beta_{10} + \beta_{11} - \beta_{13} - 2 \beta_{15} ) q^{6} + ( -\beta_{1} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{14} ) q^{7} + ( -\beta_{7} + 2 \beta_{14} ) q^{8} + ( -2 + 2 \beta_{2} + 2 \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{7} q^{2} + ( -\beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{3} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{4} + ( \beta_{3} + \beta_{4} - \beta_{10} + \beta_{11} - \beta_{13} - 2 \beta_{15} ) q^{6} + ( -\beta_{1} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{14} ) q^{7} + ( -\beta_{7} + 2 \beta_{14} ) q^{8} + ( -2 + 2 \beta_{2} + 2 \beta_{4} ) q^{9} + ( 1 + \beta_{2} - \beta_{4} ) q^{11} + ( \beta_{1} + \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{12} - \beta_{14} ) q^{12} + ( -2 \beta_{1} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{14} ) q^{13} + ( -1 - \beta_{3} + \beta_{4} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{14} + ( \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{16} + ( \beta_{5} - \beta_{8} + \beta_{9} - 2 \beta_{12} ) q^{17} + ( 2 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{14} ) q^{18} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{10} + \beta_{13} - 3 \beta_{15} ) q^{19} + ( -3 + \beta_{2} - \beta_{3} - 2 \beta_{11} + 2 \beta_{13} + \beta_{15} ) q^{21} + ( -3 \beta_{1} - \beta_{6} + \beta_{7} + \beta_{14} ) q^{22} + ( \beta_{6} + \beta_{7} - 2 \beta_{14} ) q^{23} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{10} - \beta_{11} + 3 \beta_{13} ) q^{24} + ( -4 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{10} + 2 \beta_{11} - 4 \beta_{13} + 2 \beta_{15} ) q^{26} + ( 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{27} + ( 3 \beta_{1} + \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{14} ) q^{28} + ( -4 + 3 \beta_{2} + 3 \beta_{4} ) q^{29} + ( -\beta_{2} - \beta_{4} + 2 \beta_{15} ) q^{31} + ( 6 \beta_{1} - 2 \beta_{6} - \beta_{7} ) q^{32} + ( -\beta_{5} + \beta_{8} - \beta_{9} + 2 \beta_{12} ) q^{33} + ( -2 \beta_{3} - 2 \beta_{4} + 3 \beta_{10} + \beta_{11} + \beta_{13} ) q^{34} + ( 4 - 2 \beta_{2} - 4 \beta_{3} ) q^{36} + ( 2 \beta_{1} + 3 \beta_{6} - 3 \beta_{7} ) q^{37} + ( 2 \beta_{1} + 3 \beta_{5} - \beta_{6} + 3 \beta_{7} + 4 \beta_{9} - \beta_{12} - 2 \beta_{14} ) q^{38} + ( -6 - \beta_{2} + 10 \beta_{3} + \beta_{4} ) q^{39} + ( \beta_{2} - \beta_{3} + \beta_{10} - 2 \beta_{11} + \beta_{13} + \beta_{15} ) q^{41} + ( -2 \beta_{1} + 3 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + \beta_{9} - 2 \beta_{12} + \beta_{14} ) q^{42} + ( 2 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} ) q^{43} + ( 2 + 3 \beta_{2} - 2 \beta_{3} ) q^{44} + ( -2 - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{46} + ( -6 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{47} + ( -\beta_{1} + 5 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - 3 \beta_{12} + \beta_{14} ) q^{48} + ( -3 \beta_{2} - \beta_{3} - 4 \beta_{4} + \beta_{10} - 2 \beta_{11} + \beta_{13} + \beta_{15} ) q^{49} + ( 1 + 3 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} ) q^{51} + ( -\beta_{1} - 6 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} + 4 \beta_{8} - 6 \beta_{9} + 4 \beta_{12} + \beta_{14} ) q^{52} + ( -8 \beta_{1} - 3 \beta_{6} + 3 \beta_{7} ) q^{53} + ( -\beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} + 3 \beta_{13} + 2 \beta_{15} ) q^{54} + ( 3 - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{15} ) q^{56} + ( 3 \beta_{1} - 8 \beta_{6} + 8 \beta_{7} ) q^{57} + ( 3 \beta_{1} - 3 \beta_{6} + 4 \beta_{7} + 3 \beta_{14} ) q^{58} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{10} + 2 \beta_{13} + 2 \beta_{15} ) q^{59} + ( -2 \beta_{2} - 2 \beta_{4} + 4 \beta_{10} - 4 \beta_{13} ) q^{61} + ( -\beta_{1} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + \beta_{14} ) q^{62} + ( 4 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - 6 \beta_{14} ) q^{63} + ( 4 - 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{64} + ( 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{10} - \beta_{11} - \beta_{13} ) q^{66} + ( -3 \beta_{1} - \beta_{6} - \beta_{7} + 8 \beta_{14} ) q^{67} + ( 2 \beta_{1} - 3 \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{12} - 2 \beta_{14} ) q^{68} + ( -\beta_{2} - \beta_{4} + 2 \beta_{10} - 2 \beta_{13} ) q^{69} + ( -2 - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{71} + ( 6 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} - 6 \beta_{14} ) q^{72} + ( -3 \beta_{5} + 3 \beta_{8} - 3 \beta_{9} + 6 \beta_{12} ) q^{73} + ( -6 - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{74} + ( 4 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + 5 \beta_{10} - 3 \beta_{11} + 7 \beta_{13} + 2 \beta_{15} ) q^{76} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} - 2 \beta_{9} + 2 \beta_{12} + \beta_{14} ) q^{77} + ( -7 \beta_{1} + 11 \beta_{6} - 6 \beta_{7} + 9 \beta_{14} ) q^{78} + ( 4 + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} ) q^{79} + ( -3 - 2 \beta_{2} - 2 \beta_{4} ) q^{81} + ( -\beta_{1} + \beta_{5} - 2 \beta_{6} + \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - \beta_{12} + \beta_{14} ) q^{82} + ( -\beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{83} + ( -1 - 6 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} + 3 \beta_{10} + \beta_{11} + 5 \beta_{15} ) q^{84} + ( 4 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{86} + ( 10 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} ) q^{87} + ( -\beta_{1} - 5 \beta_{6} + 3 \beta_{7} + \beta_{14} ) q^{88} + ( 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{10} - 6 \beta_{11} + 3 \beta_{13} + 3 \beta_{15} ) q^{89} + ( -7 - \beta_{2} + 7 \beta_{3} + 8 \beta_{4} - \beta_{10} - \beta_{13} - 7 \beta_{15} ) q^{91} + ( -5 \beta_{1} + 3 \beta_{6} + 2 \beta_{7} - \beta_{14} ) q^{92} + ( -4 \beta_{1} + 5 \beta_{6} - 5 \beta_{7} ) q^{93} + ( 3 \beta_{3} + 3 \beta_{4} - 6 \beta_{13} - 6 \beta_{15} ) q^{94} + ( -2 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 3 \beta_{10} + 3 \beta_{11} + 3 \beta_{13} + 4 \beta_{15} ) q^{96} + ( 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - 4 \beta_{12} ) q^{97} + ( -5 \beta_{1} + \beta_{5} + 2 \beta_{6} + \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - \beta_{12} - 3 \beta_{14} ) q^{98} + ( 2 - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 32 q^{9} + O(q^{10}) \) \( 16 q + 8 q^{4} - 32 q^{9} - 24 q^{14} + 8 q^{16} - 48 q^{21} - 64 q^{29} + 48 q^{36} - 8 q^{44} - 32 q^{46} + 40 q^{56} + 104 q^{64} - 32 q^{74} - 48 q^{81} - 40 q^{84} + 80 q^{86} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 7 x^{12} + 40 x^{8} - 112 x^{4} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{10} - 3 \nu^{6} + 12 \nu^{2} \)\()/16\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} - 4 \nu^{13} + 2 \nu^{12} - 3 \nu^{11} + 28 \nu^{9} + 10 \nu^{8} + 28 \nu^{7} - 96 \nu^{5} - 56 \nu^{4} + 256 \nu + 128 \)\()/256\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{15} + 2 \nu^{13} - 8 \nu^{12} + 11 \nu^{11} + 10 \nu^{9} + 24 \nu^{8} - 20 \nu^{7} - 56 \nu^{5} - 224 \nu^{4} + 384 \nu + 512 \)\()/512\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} - \nu^{13} - \nu^{12} + 3 \nu^{11} + 3 \nu^{9} - 5 \nu^{8} - 12 \nu^{7} - 28 \nu^{5} + 28 \nu^{4} + 16 \nu^{3} - 64 \)\()/128\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{15} + 2 \nu^{13} - 12 \nu^{12} - 17 \nu^{11} - 22 \nu^{9} + 4 \nu^{8} + 76 \nu^{7} + 168 \nu^{5} - 112 \nu^{4} - 384 \nu - 256 \)\()/512\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{15} + \nu^{14} + 2 \nu^{13} + 6 \nu^{11} + 5 \nu^{10} - 6 \nu^{9} - 24 \nu^{7} - 28 \nu^{6} + 56 \nu^{5} + 32 \nu^{3} + 192 \nu^{2} \)\()/256\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{15} + \nu^{14} - 4 \nu^{13} + 3 \nu^{11} + 5 \nu^{10} + 28 \nu^{9} - 28 \nu^{7} - 28 \nu^{6} - 96 \nu^{5} + 192 \nu^{2} + 256 \nu \)\()/256\)
\(\beta_{8}\)\(=\)\((\)\( 4 \nu^{15} + 3 \nu^{14} + 2 \nu^{13} - 2 \nu^{12} - 44 \nu^{11} - 33 \nu^{10} + 10 \nu^{9} + 22 \nu^{8} + 144 \nu^{7} + 124 \nu^{6} - 56 \nu^{5} - 168 \nu^{4} - 448 \nu^{3} - 448 \nu^{2} - 128 \nu + 384 \)\()/512\)
\(\beta_{9}\)\(=\)\((\)\( 5 \nu^{15} + 3 \nu^{14} + 8 \nu^{13} + 2 \nu^{12} - 39 \nu^{11} - 33 \nu^{10} + 8 \nu^{9} - 22 \nu^{8} + 180 \nu^{7} + 124 \nu^{6} + 168 \nu^{4} - 448 \nu^{3} - 448 \nu^{2} + 256 \nu - 384 \)\()/512\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{15} - 5 \nu^{14} - 4 \nu^{13} - 6 \nu^{12} - 21 \nu^{11} + 23 \nu^{10} + 12 \nu^{9} + 2 \nu^{8} + 76 \nu^{7} - 68 \nu^{6} - 112 \nu^{5} - 56 \nu^{4} - 384 \nu^{3} + 64 \nu^{2} + 512 \nu + 128 \)\()/512\)
\(\beta_{11}\)\(=\)\((\)\( -2 \nu^{15} + 3 \nu^{14} - 4 \nu^{13} + 6 \nu^{11} - 17 \nu^{10} + 28 \nu^{9} - 24 \nu^{7} + 140 \nu^{6} - 96 \nu^{5} + 32 \nu^{3} - 192 \nu^{2} + 256 \nu \)\()/256\)
\(\beta_{12}\)\(=\)\((\)\( \nu^{15} + 2 \nu^{13} - 18 \nu^{12} - 3 \nu^{11} - 6 \nu^{9} + 102 \nu^{8} + 28 \nu^{7} + 56 \nu^{5} - 328 \nu^{4} + 640 \)\()/256\)
\(\beta_{13}\)\(=\)\((\)\( -5 \nu^{15} - 5 \nu^{14} + 8 \nu^{13} - 6 \nu^{12} + 39 \nu^{11} + 23 \nu^{10} + 8 \nu^{9} + 2 \nu^{8} - 180 \nu^{7} - 68 \nu^{6} - 56 \nu^{4} + 448 \nu^{3} + 64 \nu^{2} + 256 \nu + 128 \)\()/512\)
\(\beta_{14}\)\(=\)\((\)\( \nu^{15} + 8 \nu^{14} + 2 \nu^{13} - 11 \nu^{11} - 24 \nu^{10} + 10 \nu^{9} + 20 \nu^{7} + 96 \nu^{6} - 56 \nu^{5} - 128 \nu^{2} + 384 \nu \)\()/512\)
\(\beta_{15}\)\(=\)\((\)\( -5 \nu^{15} + 6 \nu^{14} - 10 \nu^{13} + 23 \nu^{11} - 34 \nu^{10} + 46 \nu^{9} - 68 \nu^{7} + 152 \nu^{6} - 136 \nu^{5} + 64 \nu^{3} - 512 \nu^{2} + 128 \nu \)\()/512\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{14} + \beta_{10} - \beta_{8} - \beta_{4} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{15} + 2 \beta_{14} + \beta_{13} - \beta_{9} - 2 \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{12} + 2 \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{2} + 4\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(3 \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} - 3 \beta_{3} - \beta_{2} - 2 \beta_{1} + 3\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-4 \beta_{15} - 3 \beta_{14} - \beta_{13} + 7 \beta_{11} - \beta_{10} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_{1}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-\beta_{15} - 2 \beta_{14} - 3 \beta_{13} - 2 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} - 10 \beta_{7} + 7 \beta_{6} + \beta_{5} + 10 \beta_{4} + 4 \beta_{3} + 11 \beta_{2} + 3 \beta_{1} + 1\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(7 \beta_{12} + 2 \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} - 11 \beta_{5} - \beta_{4} - 13 \beta_{3} + \beta_{2} - 8\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(9 \beta_{15} - 15 \beta_{14} + 11 \beta_{13} - 6 \beta_{10} + 11 \beta_{9} + 6 \beta_{8} + 14 \beta_{7} + 6 \beta_{6} + 9 \beta_{5} - 9 \beta_{4} - 9 \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 9\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(12 \beta_{15} + 3 \beta_{14} + 9 \beta_{13} + 9 \beta_{11} + 9 \beta_{10} - 15 \beta_{7} - 15 \beta_{6} - 15 \beta_{4} - 9 \beta_{3} - 6 \beta_{2} + 53 \beta_{1}\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(9 \beta_{15} - 30 \beta_{14} - 21 \beta_{13} - 22 \beta_{10} + 21 \beta_{9} - 22 \beta_{8} - 6 \beta_{7} - 7 \beta_{6} - 9 \beta_{5} + 6 \beta_{4} + 52 \beta_{3} - 3 \beta_{2} + 21 \beta_{1} - 9\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(-7 \beta_{12} - 18 \beta_{9} + 18 \beta_{8} - 31 \beta_{7} + 31 \beta_{6} - 37 \beta_{5} - 31 \beta_{4} - 19 \beta_{3} + 31 \beta_{2} - 40\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-25 \beta_{15} - 17 \beta_{14} + 37 \beta_{13} - 42 \beta_{10} + 37 \beta_{9} + 42 \beta_{8} - 14 \beta_{7} + 26 \beta_{6} - 25 \beta_{5} + 41 \beta_{4} + 25 \beta_{3} + 11 \beta_{2} + 42 \beta_{1} - 25\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(84 \beta_{15} + 93 \beta_{14} - 9 \beta_{13} - 41 \beta_{11} - 9 \beta_{10} - 17 \beta_{7} - 17 \beta_{6} - 17 \beta_{4} + 9 \beta_{3} - 26 \beta_{2} + 139 \beta_{1}\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(-9 \beta_{15} - 34 \beta_{14} - 43 \beta_{13} - 74 \beta_{10} + 43 \beta_{9} - 74 \beta_{8} + 134 \beta_{7} - 217 \beta_{6} + 9 \beta_{5} - 134 \beta_{4} + 108 \beta_{3} - 125 \beta_{2} + 43 \beta_{1} + 9\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
699.1
−1.32968 0.481610i
1.38588 0.281691i
1.38588 + 0.281691i
−1.32968 + 0.481610i
0.281691 + 1.38588i
1.32968 0.481610i
1.32968 + 0.481610i
0.281691 1.38588i
0.481610 1.32968i
−1.38588 0.281691i
−1.38588 + 0.281691i
0.481610 + 1.32968i
−0.281691 + 1.38588i
−0.481610 1.32968i
−0.481610 + 1.32968i
−0.281691 1.38588i
−1.39897 0.207107i 1.47363i 1.91421 + 0.579471i 0 −0.305198 + 2.06155i 0.819496 2.51564i −2.55791 1.20711i 0.828427 0
699.2 −1.39897 0.207107i 1.47363i 1.91421 + 0.579471i 0 0.305198 2.06155i 0.819496 + 2.51564i −2.55791 1.20711i 0.828427 0
699.3 −1.39897 + 0.207107i 1.47363i 1.91421 0.579471i 0 0.305198 + 2.06155i 0.819496 2.51564i −2.55791 + 1.20711i 0.828427 0
699.4 −1.39897 + 0.207107i 1.47363i 1.91421 0.579471i 0 −0.305198 2.06155i 0.819496 + 2.51564i −2.55791 + 1.20711i 0.828427 0
699.5 −0.736813 1.20711i 2.79793i −0.914214 + 1.77882i 0 −3.37740 + 2.06155i 2.51564 0.819496i 2.82083 0.207107i −4.82843 0
699.6 −0.736813 1.20711i 2.79793i −0.914214 + 1.77882i 0 3.37740 2.06155i 2.51564 + 0.819496i 2.82083 0.207107i −4.82843 0
699.7 −0.736813 + 1.20711i 2.79793i −0.914214 1.77882i 0 3.37740 + 2.06155i 2.51564 0.819496i 2.82083 + 0.207107i −4.82843 0
699.8 −0.736813 + 1.20711i 2.79793i −0.914214 1.77882i 0 −3.37740 2.06155i 2.51564 + 0.819496i 2.82083 + 0.207107i −4.82843 0
699.9 0.736813 1.20711i 2.79793i −0.914214 1.77882i 0 −3.37740 2.06155i −2.51564 0.819496i −2.82083 0.207107i −4.82843 0
699.10 0.736813 1.20711i 2.79793i −0.914214 1.77882i 0 3.37740 + 2.06155i −2.51564 + 0.819496i −2.82083 0.207107i −4.82843 0
699.11 0.736813 + 1.20711i 2.79793i −0.914214 + 1.77882i 0 3.37740 2.06155i −2.51564 0.819496i −2.82083 + 0.207107i −4.82843 0
699.12 0.736813 + 1.20711i 2.79793i −0.914214 + 1.77882i 0 −3.37740 + 2.06155i −2.51564 + 0.819496i −2.82083 + 0.207107i −4.82843 0
699.13 1.39897 0.207107i 1.47363i 1.91421 0.579471i 0 −0.305198 2.06155i −0.819496 2.51564i 2.55791 1.20711i 0.828427 0
699.14 1.39897 0.207107i 1.47363i 1.91421 0.579471i 0 0.305198 + 2.06155i −0.819496 + 2.51564i 2.55791 1.20711i 0.828427 0
699.15 1.39897 + 0.207107i 1.47363i 1.91421 + 0.579471i 0 0.305198 2.06155i −0.819496 2.51564i 2.55791 + 1.20711i 0.828427 0
699.16 1.39897 + 0.207107i 1.47363i 1.91421 + 0.579471i 0 −0.305198 + 2.06155i −0.819496 + 2.51564i 2.55791 + 1.20711i 0.828427 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 699.16
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.c.k 16
4.b odd 2 1 inner 700.2.c.k 16
5.b even 2 1 inner 700.2.c.k 16
5.c odd 4 1 700.2.g.i 8
5.c odd 4 1 700.2.g.k yes 8
7.b odd 2 1 inner 700.2.c.k 16
20.d odd 2 1 inner 700.2.c.k 16
20.e even 4 1 700.2.g.i 8
20.e even 4 1 700.2.g.k yes 8
28.d even 2 1 inner 700.2.c.k 16
35.c odd 2 1 inner 700.2.c.k 16
35.f even 4 1 700.2.g.i 8
35.f even 4 1 700.2.g.k yes 8
140.c even 2 1 inner 700.2.c.k 16
140.j odd 4 1 700.2.g.i 8
140.j odd 4 1 700.2.g.k yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.c.k 16 1.a even 1 1 trivial
700.2.c.k 16 4.b odd 2 1 inner
700.2.c.k 16 5.b even 2 1 inner
700.2.c.k 16 7.b odd 2 1 inner
700.2.c.k 16 20.d odd 2 1 inner
700.2.c.k 16 28.d even 2 1 inner
700.2.c.k 16 35.c odd 2 1 inner
700.2.c.k 16 140.c even 2 1 inner
700.2.g.i 8 5.c odd 4 1
700.2.g.i 8 20.e even 4 1
700.2.g.i 8 35.f even 4 1
700.2.g.i 8 140.j odd 4 1
700.2.g.k yes 8 5.c odd 4 1
700.2.g.k yes 8 20.e even 4 1
700.2.g.k yes 8 35.f even 4 1
700.2.g.k yes 8 140.j odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 10 T_{3}^{2} + 17 \)
\( T_{11}^{4} + 10 T_{11}^{2} + 17 \)
\( T_{13}^{2} - 34 \)
\( T_{19}^{4} - 58 T_{19}^{2} + 833 \)
\( T_{23}^{4} - 20 T_{23}^{2} + 68 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 - 8 T^{2} + T^{4} - 2 T^{6} + T^{8} )^{2} \)
$3$ \( ( 17 + 10 T^{2} + T^{4} )^{4} \)
$5$ \( T^{16} \)
$7$ \( ( 2401 - 30 T^{4} + T^{8} )^{2} \)
$11$ \( ( 17 + 10 T^{2} + T^{4} )^{4} \)
$13$ \( ( -34 + T^{2} )^{8} \)
$17$ \( ( -17 + T^{2} )^{8} \)
$19$ \( ( 833 - 58 T^{2} + T^{4} )^{4} \)
$23$ \( ( 68 - 20 T^{2} + T^{4} )^{4} \)
$29$ \( ( -2 + 8 T + T^{2} )^{8} \)
$31$ \( ( 68 - 20 T^{2} + T^{4} )^{4} \)
$37$ \( ( 196 + 44 T^{2} + T^{4} )^{4} \)
$41$ \( ( 17 + T^{2} )^{8} \)
$43$ \( ( 272 - 40 T^{2} + T^{4} )^{4} \)
$47$ \( ( 5508 + 180 T^{2} + T^{4} )^{4} \)
$53$ \( ( 2116 + 164 T^{2} + T^{4} )^{4} \)
$59$ \( ( 272 - 40 T^{2} + T^{4} )^{4} \)
$61$ \( ( 136 + T^{2} )^{8} \)
$67$ \( ( 8993 - 218 T^{2} + T^{4} )^{4} \)
$71$ \( ( 3332 + 116 T^{2} + T^{4} )^{4} \)
$73$ \( ( -153 + T^{2} )^{8} \)
$79$ \( ( 1088 + 80 T^{2} + T^{4} )^{4} \)
$83$ \( ( 17 + 10 T^{2} + T^{4} )^{4} \)
$89$ \( ( 153 + T^{2} )^{8} \)
$97$ \( ( -68 + T^{2} )^{8} \)
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