Properties

Label 196.2.f.d
Level $196$
Weight $2$
Character orbit 196.f
Analytic conductor $1.565$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 196.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.56506787962\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 4 x^{14} + 6 x^{12} + 8 x^{10} + 20 x^{8} + 32 x^{6} + 96 x^{4} + 256 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{6} - \beta_{11} ) q^{2} + ( 2 \beta_{1} + \beta_{3} ) q^{3} + ( \beta_{4} - \beta_{5} - \beta_{8} - \beta_{11} + \beta_{14} ) q^{4} + ( \beta_{3} - \beta_{13} + \beta_{15} ) q^{5} + ( -\beta_{2} - \beta_{9} - 2 \beta_{10} - 2 \beta_{15} ) q^{6} + ( 2 \beta_{4} - \beta_{6} ) q^{8} + ( -2 - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{6} - \beta_{11} ) q^{2} + ( 2 \beta_{1} + \beta_{3} ) q^{3} + ( \beta_{4} - \beta_{5} - \beta_{8} - \beta_{11} + \beta_{14} ) q^{4} + ( \beta_{3} - \beta_{13} + \beta_{15} ) q^{5} + ( -\beta_{2} - \beta_{9} - 2 \beta_{10} - 2 \beta_{15} ) q^{6} + ( 2 \beta_{4} - \beta_{6} ) q^{8} + ( -2 - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} ) q^{9} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{10} - \beta_{12} - \beta_{15} ) q^{10} + ( -2 \beta_{4} + 2 \beta_{8} - \beta_{11} - \beta_{14} ) q^{11} + ( \beta_{1} + 3 \beta_{3} + \beta_{9} + 2 \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{12} + ( \beta_{10} + \beta_{13} + \beta_{15} ) q^{13} + ( -2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{14} ) q^{15} + ( -2 - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} - \beta_{11} ) q^{16} + ( -\beta_{3} - 2 \beta_{10} ) q^{17} + ( -\beta_{4} - \beta_{5} + \beta_{8} + 3 \beta_{11} - \beta_{14} ) q^{18} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{9} - 2 \beta_{12} - \beta_{15} ) q^{19} + ( \beta_{2} + \beta_{10} - 2 \beta_{13} + \beta_{15} ) q^{20} + ( 4 - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 3 \beta_{14} ) q^{22} + ( -2 - 2 \beta_{5} - 4 \beta_{8} ) q^{23} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{10} - \beta_{12} - \beta_{15} ) q^{24} + ( \beta_{5} - 2 \beta_{11} + 2 \beta_{14} ) q^{25} + ( \beta_{12} + 2 \beta_{15} ) q^{26} + ( 2 \beta_{2} + \beta_{10} + \beta_{13} + \beta_{15} ) q^{27} + ( -6 - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{14} ) q^{29} + ( 8 + 8 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} ) q^{30} + ( 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{10} + 4 \beta_{12} + 4 \beta_{15} ) q^{31} + ( 6 \beta_{5} + \beta_{11} + 2 \beta_{14} ) q^{32} + ( -5 \beta_{3} + 5 \beta_{13} + \beta_{15} ) q^{33} + ( \beta_{2} + \beta_{9} - 2 \beta_{10} - 2 \beta_{15} ) q^{34} + ( -4 - 4 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{14} ) q^{36} + ( 4 + 4 \beta_{5} ) q^{37} + ( \beta_{1} - 3 \beta_{3} + \beta_{10} ) q^{38} + ( 4 \beta_{4} - 2 \beta_{5} - 4 \beta_{8} ) q^{39} + ( -2 \beta_{1} - 4 \beta_{3} - 2 \beta_{9} - 3 \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{40} + ( 3 \beta_{10} + 3 \beta_{15} ) q^{41} + ( -1 - 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{14} ) q^{43} + ( -2 - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - 2 \beta_{11} ) q^{44} + ( -\beta_{3} + 3 \beta_{10} ) q^{45} + ( 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{8} + 2 \beta_{11} - 4 \beta_{14} ) q^{46} + ( 4 \beta_{1} + 2 \beta_{3} + 4 \beta_{9} + 2 \beta_{13} ) q^{47} + ( -4 \beta_{2} + \beta_{9} + 3 \beta_{10} - 2 \beta_{13} + 3 \beta_{15} ) q^{48} + ( 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{14} ) q^{50} + ( 3 + 3 \beta_{5} + \beta_{6} + \beta_{7} + 6 \beta_{8} + \beta_{11} ) q^{51} + ( -2 \beta_{1} + \beta_{2} - \beta_{10} + \beta_{12} + \beta_{15} ) q^{52} + ( 2 \beta_{5} + 6 \beta_{11} - 6 \beta_{14} ) q^{53} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{9} - \beta_{12} + 4 \beta_{13} ) q^{54} + ( -4 \beta_{2} + 4 \beta_{9} - 2 \beta_{10} - 2 \beta_{15} ) q^{55} + ( 9 + 5 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} + 5 \beta_{14} ) q^{57} + ( 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{11} ) q^{58} + ( -2 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - 3 \beta_{10} - 6 \beta_{12} - 6 \beta_{15} ) q^{59} + ( -4 \beta_{4} + 4 \beta_{8} - 6 \beta_{11} ) q^{60} + ( 5 \beta_{3} - 5 \beta_{13} - 5 \beta_{15} ) q^{61} + ( 2 \beta_{2} + 4 \beta_{9} + 2 \beta_{10} + 8 \beta_{13} + 2 \beta_{15} ) q^{62} + ( -6 - \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{14} ) q^{64} + ( 2 + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{11} ) q^{65} + ( 4 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} - \beta_{10} + 5 \beta_{12} + 5 \beta_{15} ) q^{66} + ( -4 \beta_{4} + 4 \beta_{5} + 4 \beta_{8} + 2 \beta_{11} + 2 \beta_{14} ) q^{67} + ( 3 \beta_{1} + \beta_{3} + 3 \beta_{9} - 2 \beta_{12} + 2 \beta_{13} - 3 \beta_{15} ) q^{68} + ( -8 \beta_{10} - 6 \beta_{13} - 8 \beta_{15} ) q^{69} + ( -2 - 2 \beta_{5} + \beta_{6} - \beta_{7} - 7 \beta_{8} + \beta_{11} ) q^{72} + ( 8 \beta_{3} + \beta_{10} ) q^{73} -4 \beta_{11} q^{74} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{9} + 4 \beta_{12} - 3 \beta_{13} + 2 \beta_{15} ) q^{75} + ( 3 \beta_{2} - 5 \beta_{9} ) q^{76} + ( 4 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 4 \beta_{14} ) q^{78} + ( 2 + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 2 \beta_{11} ) q^{79} + ( 4 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{10} + \beta_{12} + \beta_{15} ) q^{80} + ( -7 \beta_{5} - 3 \beta_{11} + 3 \beta_{14} ) q^{81} + ( -3 \beta_{1} - 3 \beta_{3} - 3 \beta_{9} + 3 \beta_{15} ) q^{82} + ( 2 \beta_{2} - 6 \beta_{9} + \beta_{10} - 2 \beta_{13} + \beta_{15} ) q^{83} + ( -2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{14} ) q^{85} + ( -12 - 12 \beta_{5} - 2 \beta_{6} + \beta_{7} - 5 \beta_{8} - 2 \beta_{11} ) q^{86} + ( -8 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} - 2 \beta_{10} - 4 \beta_{12} - 4 \beta_{15} ) q^{87} + ( \beta_{4} - 7 \beta_{5} - \beta_{8} + 3 \beta_{14} ) q^{88} + ( 2 \beta_{3} - 2 \beta_{13} + 3 \beta_{15} ) q^{89} + ( \beta_{2} - 4 \beta_{9} + 3 \beta_{10} + 3 \beta_{15} ) q^{90} + ( 4 + 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} - 6 \beta_{14} ) q^{92} + ( -8 - 8 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} - 12 \beta_{11} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{10} - 2 \beta_{12} - 2 \beta_{15} ) q^{94} + ( 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{8} + 4 \beta_{11} + 4 \beta_{14} ) q^{95} + ( -9 \beta_{1} - \beta_{3} - 9 \beta_{9} + \beta_{12} - 8 \beta_{13} + 5 \beta_{15} ) q^{96} + 5 \beta_{13} q^{97} + ( 3 + 6 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 4q^{2} + 4q^{4} - 8q^{8} - 24q^{9} + O(q^{10}) \) \( 16q + 4q^{2} + 4q^{4} - 8q^{8} - 24q^{9} - 4q^{16} + 20q^{18} + 32q^{22} - 8q^{25} - 64q^{29} + 40q^{30} - 36q^{32} + 8q^{36} + 32q^{37} - 24q^{44} + 8q^{46} - 40q^{50} - 16q^{53} + 64q^{57} - 8q^{60} - 104q^{64} + 4q^{72} - 16q^{74} + 16q^{78} + 56q^{81} + 32q^{85} - 64q^{86} + 64q^{88} + 112q^{92} + 32q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 4 x^{14} + 6 x^{12} + 8 x^{10} + 20 x^{8} + 32 x^{6} + 96 x^{4} + 256 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} + 6 \nu^{13} + 4 \nu^{11} - 12 \nu^{9} + 24 \nu^{7} + 24 \nu^{5} + 88 \nu^{3} + 320 \nu \)\()/224\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{15} - 44 \nu^{13} - 6 \nu^{11} - 24 \nu^{9} - 148 \nu^{7} - 64 \nu^{5} - 384 \nu^{3} - 2944 \nu \)\()/896\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{14} - 12 \nu^{12} + 6 \nu^{10} + 24 \nu^{8} - 76 \nu^{6} + 64 \nu^{4} + 384 \nu^{2} - 1088 \)\()/448\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{14} - 3 \nu^{12} - 2 \nu^{10} + 6 \nu^{8} - 12 \nu^{6} - 12 \nu^{4} + 96 \nu^{2} - 384 \)\()/224\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{14} + 16 \nu^{12} + 6 \nu^{10} + 24 \nu^{8} + 36 \nu^{6} + 64 \nu^{4} + 384 \nu^{2} + 704 \)\()/224\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{14} + 9 \nu^{12} + 6 \nu^{10} + 38 \nu^{8} + 36 \nu^{6} + 36 \nu^{4} + 272 \nu^{2} + 480 \)\()/224\)
\(\beta_{8}\)\(=\)\((\)\( 5 \nu^{14} + 9 \nu^{12} + 6 \nu^{10} + 38 \nu^{8} + 36 \nu^{6} + 36 \nu^{4} + 496 \nu^{2} + 480 \)\()/224\)
\(\beta_{9}\)\(=\)\((\)\( 3 \nu^{15} + 18 \nu^{13} + 12 \nu^{11} + 20 \nu^{9} + 72 \nu^{7} + 72 \nu^{5} + 264 \nu^{3} + 960 \nu \)\()/224\)
\(\beta_{10}\)\(=\)\((\)\( 15 \nu^{15} + 20 \nu^{13} + 18 \nu^{11} + 72 \nu^{9} - 4 \nu^{7} + 192 \nu^{5} + 1152 \nu^{3} + 768 \nu \)\()/896\)
\(\beta_{11}\)\(=\)\((\)\( 9 \nu^{14} + 26 \nu^{12} + 22 \nu^{10} + 60 \nu^{8} + 132 \nu^{6} + 104 \nu^{4} + 736 \nu^{2} + 1536 \)\()/224\)
\(\beta_{12}\)\(=\)\((\)\( 11 \nu^{15} + 24 \nu^{13} + 2 \nu^{11} + 64 \nu^{9} + 124 \nu^{7} + 208 \nu^{5} + 800 \nu^{3} + 1280 \nu \)\()/448\)
\(\beta_{13}\)\(=\)\((\)\( -6 \nu^{15} - 15 \nu^{13} - 10 \nu^{11} - 26 \nu^{9} - 60 \nu^{7} - 60 \nu^{5} - 360 \nu^{3} - 800 \nu \)\()/224\)
\(\beta_{14}\)\(=\)\((\)\( 29 \nu^{14} + 76 \nu^{12} + 46 \nu^{10} + 184 \nu^{8} + 388 \nu^{6} + 416 \nu^{4} + 2272 \nu^{2} + 4352 \)\()/448\)
\(\beta_{15}\)\(=\)\((\)\( 29 \nu^{15} + 76 \nu^{13} + 46 \nu^{11} + 184 \nu^{9} + 388 \nu^{7} + 192 \nu^{5} + 2272 \nu^{3} + 4352 \nu \)\()/896\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} - \beta_{7}\)
\(\nu^{3}\)\(=\)\(\beta_{15} + 2 \beta_{13} + \beta_{10} + \beta_{2}\)
\(\nu^{4}\)\(=\)\(2 \beta_{14} - 2 \beta_{11} - 2 \beta_{8} - 2 \beta_{5} + 2 \beta_{4}\)
\(\nu^{5}\)\(=\)\(-2 \beta_{15} + 2 \beta_{12} + 2 \beta_{9} + 2 \beta_{3} + 2 \beta_{1}\)
\(\nu^{6}\)\(=\)\(2 \beta_{14} - 2 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 4\)
\(\nu^{7}\)\(=\)\(2 \beta_{15} + 2 \beta_{12} - 6 \beta_{10} + 4 \beta_{3} + 2 \beta_{2}\)
\(\nu^{8}\)\(=\)\(-4 \beta_{11} + 4 \beta_{8} + 12 \beta_{7} - 4 \beta_{6} - 8 \beta_{5} - 8\)
\(\nu^{9}\)\(=\)\(4 \beta_{9} - 12 \beta_{2}\)
\(\nu^{10}\)\(=\)\(-12 \beta_{14} + 24 \beta_{11} - 12 \beta_{8} - 4 \beta_{5} + 12 \beta_{4}\)
\(\nu^{11}\)\(=\)\(-8 \beta_{13} - 12 \beta_{12} + 16 \beta_{9} + 24 \beta_{3} + 16 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-8 \beta_{14} + 8 \beta_{7} + 24 \beta_{6} - 8 \beta_{5} - 8 \beta_{4} - 48\)
\(\nu^{13}\)\(=\)\(-8 \beta_{15} - 8 \beta_{12} + 16 \beta_{10} - 40 \beta_{3} - 8 \beta_{2} - 72 \beta_{1}\)
\(\nu^{14}\)\(=\)\(16 \beta_{11} - 56 \beta_{8} + 8 \beta_{7} + 16 \beta_{6} + 80 \beta_{5} + 80\)
\(\nu^{15}\)\(=\)\(-40 \beta_{15} - 144 \beta_{13} - 40 \beta_{10} - 64 \beta_{9} - 8 \beta_{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\) \(99\) \(101\)
\(\chi(n)\) \(-1\) \(1 + \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.264742 1.38921i
−0.264742 + 1.38921i
−1.01214 0.987711i
1.01214 + 0.987711i
−1.33546 0.465333i
1.33546 + 0.465333i
−0.349313 1.37039i
0.349313 + 1.37039i
0.264742 + 1.38921i
−0.264742 1.38921i
−1.01214 + 0.987711i
1.01214 0.987711i
−1.33546 + 0.465333i
1.33546 0.465333i
−0.349313 + 1.37039i
0.349313 1.37039i
−1.31509 0.520123i −1.07072 1.85455i 1.45894 + 1.36802i −2.26303 1.30656i 0.443508 + 2.99581i 0 −1.20711 2.55791i −0.792893 + 1.37333i 2.29653 + 2.89531i
19.2 −1.31509 0.520123i 1.07072 + 1.85455i 1.45894 + 1.36802i 2.26303 + 1.30656i −0.443508 2.99581i 0 −1.20711 2.55791i −0.792893 + 1.37333i −2.29653 2.89531i
19.3 −0.0345453 1.41379i −1.36145 2.35811i −1.99761 + 0.0976797i −0.937379 0.541196i −3.28684 + 2.00627i 0 0.207107 + 2.82083i −2.20711 + 3.82282i −0.732756 + 1.34395i
19.4 −0.0345453 1.41379i 1.36145 + 2.35811i −1.99761 + 0.0976797i 0.937379 + 0.541196i 3.28684 2.00627i 0 0.207107 + 2.82083i −2.20711 + 3.82282i 0.732756 1.34395i
19.5 1.10799 + 0.878843i −1.07072 1.85455i 0.455270 + 1.94749i 2.26303 + 1.30656i 0.443508 2.99581i 0 −1.20711 + 2.55791i −0.792893 + 1.37333i 1.35915 + 3.43651i
19.6 1.10799 + 0.878843i 1.07072 + 1.85455i 0.455270 + 1.94749i −2.26303 1.30656i −0.443508 + 2.99581i 0 −1.20711 + 2.55791i −0.792893 + 1.37333i −1.35915 3.43651i
19.7 1.24165 0.676979i −1.36145 2.35811i 1.08340 1.68114i 0.937379 + 0.541196i −3.28684 2.00627i 0 0.207107 2.82083i −2.20711 + 3.82282i 1.53028 + 0.0373915i
19.8 1.24165 0.676979i 1.36145 + 2.35811i 1.08340 1.68114i −0.937379 0.541196i 3.28684 + 2.00627i 0 0.207107 2.82083i −2.20711 + 3.82282i −1.53028 0.0373915i
31.1 −1.31509 + 0.520123i −1.07072 + 1.85455i 1.45894 1.36802i −2.26303 + 1.30656i 0.443508 2.99581i 0 −1.20711 + 2.55791i −0.792893 1.37333i 2.29653 2.89531i
31.2 −1.31509 + 0.520123i 1.07072 1.85455i 1.45894 1.36802i 2.26303 1.30656i −0.443508 + 2.99581i 0 −1.20711 + 2.55791i −0.792893 1.37333i −2.29653 + 2.89531i
31.3 −0.0345453 + 1.41379i −1.36145 + 2.35811i −1.99761 0.0976797i −0.937379 + 0.541196i −3.28684 2.00627i 0 0.207107 2.82083i −2.20711 3.82282i −0.732756 1.34395i
31.4 −0.0345453 + 1.41379i 1.36145 2.35811i −1.99761 0.0976797i 0.937379 0.541196i 3.28684 + 2.00627i 0 0.207107 2.82083i −2.20711 3.82282i 0.732756 + 1.34395i
31.5 1.10799 0.878843i −1.07072 + 1.85455i 0.455270 1.94749i 2.26303 1.30656i 0.443508 + 2.99581i 0 −1.20711 2.55791i −0.792893 1.37333i 1.35915 3.43651i
31.6 1.10799 0.878843i 1.07072 1.85455i 0.455270 1.94749i −2.26303 + 1.30656i −0.443508 2.99581i 0 −1.20711 2.55791i −0.792893 1.37333i −1.35915 + 3.43651i
31.7 1.24165 + 0.676979i −1.36145 + 2.35811i 1.08340 + 1.68114i 0.937379 0.541196i −3.28684 + 2.00627i 0 0.207107 + 2.82083i −2.20711 3.82282i 1.53028 0.0373915i
31.8 1.24165 + 0.676979i 1.36145 2.35811i 1.08340 + 1.68114i −0.937379 + 0.541196i 3.28684 2.00627i 0 0.207107 + 2.82083i −2.20711 3.82282i −1.53028 + 0.0373915i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.2.f.d 16
4.b odd 2 1 inner 196.2.f.d 16
7.b odd 2 1 inner 196.2.f.d 16
7.c even 3 1 196.2.d.c 8
7.c even 3 1 inner 196.2.f.d 16
7.d odd 6 1 196.2.d.c 8
7.d odd 6 1 inner 196.2.f.d 16
21.g even 6 1 1764.2.b.k 8
21.h odd 6 1 1764.2.b.k 8
28.d even 2 1 inner 196.2.f.d 16
28.f even 6 1 196.2.d.c 8
28.f even 6 1 inner 196.2.f.d 16
28.g odd 6 1 196.2.d.c 8
28.g odd 6 1 inner 196.2.f.d 16
56.j odd 6 1 3136.2.f.i 8
56.k odd 6 1 3136.2.f.i 8
56.m even 6 1 3136.2.f.i 8
56.p even 6 1 3136.2.f.i 8
84.j odd 6 1 1764.2.b.k 8
84.n even 6 1 1764.2.b.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.d.c 8 7.c even 3 1
196.2.d.c 8 7.d odd 6 1
196.2.d.c 8 28.f even 6 1
196.2.d.c 8 28.g odd 6 1
196.2.f.d 16 1.a even 1 1 trivial
196.2.f.d 16 4.b odd 2 1 inner
196.2.f.d 16 7.b odd 2 1 inner
196.2.f.d 16 7.c even 3 1 inner
196.2.f.d 16 7.d odd 6 1 inner
196.2.f.d 16 28.d even 2 1 inner
196.2.f.d 16 28.f even 6 1 inner
196.2.f.d 16 28.g odd 6 1 inner
1764.2.b.k 8 21.g even 6 1
1764.2.b.k 8 21.h odd 6 1
1764.2.b.k 8 84.j odd 6 1
1764.2.b.k 8 84.n even 6 1
3136.2.f.i 8 56.j odd 6 1
3136.2.f.i 8 56.k odd 6 1
3136.2.f.i 8 56.m even 6 1
3136.2.f.i 8 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(196, [\chi])\):

\( T_{3}^{8} + 12 T_{3}^{6} + 110 T_{3}^{4} + 408 T_{3}^{2} + 1156 \)
\( T_{5}^{8} - 8 T_{5}^{6} + 56 T_{5}^{4} - 64 T_{5}^{2} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 - 16 T + 4 T^{2} + 4 T^{3} - 3 T^{4} + 2 T^{5} + T^{6} - 2 T^{7} + T^{8} )^{2} \)
$3$ \( ( 1156 + 408 T^{2} + 110 T^{4} + 12 T^{6} + T^{8} )^{2} \)
$5$ \( ( 64 - 64 T^{2} + 56 T^{4} - 8 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 4624 - 1360 T^{2} + 332 T^{4} - 20 T^{6} + T^{8} )^{2} \)
$13$ \( ( 8 + 8 T^{2} + T^{4} )^{4} \)
$17$ \( ( 4 - 40 T^{2} + 398 T^{4} - 20 T^{6} + T^{8} )^{2} \)
$19$ \( ( 1156 + 952 T^{2} + 750 T^{4} + 28 T^{6} + T^{8} )^{2} \)
$23$ \( ( 73984 - 15232 T^{2} + 2864 T^{4} - 56 T^{6} + T^{8} )^{2} \)
$29$ \( ( 8 + 8 T + T^{2} )^{8} \)
$31$ \( ( 4734976 + 208896 T^{2} + 7040 T^{4} + 96 T^{6} + T^{8} )^{2} \)
$37$ \( ( 16 - 4 T + T^{2} )^{8} \)
$41$ \( ( 162 + 36 T^{2} + T^{4} )^{4} \)
$43$ \( ( 3332 + 116 T^{2} + T^{4} )^{4} \)
$47$ \( ( 295936 + 26112 T^{2} + 1760 T^{4} + 48 T^{6} + T^{8} )^{2} \)
$53$ \( ( 4624 - 272 T + 84 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$59$ \( ( 96550276 + 2004504 T^{2} + 31790 T^{4} + 204 T^{6} + T^{8} )^{2} \)
$61$ \( ( 25000000 - 1000000 T^{2} + 35000 T^{4} - 200 T^{6} + T^{8} )^{2} \)
$67$ \( ( 1183744 - 121856 T^{2} + 11456 T^{4} - 112 T^{6} + T^{8} )^{2} \)
$71$ \( T^{16} \)
$73$ \( ( 155800324 - 3245320 T^{2} + 55118 T^{4} - 260 T^{6} + T^{8} )^{2} \)
$79$ \( ( 1183744 - 87040 T^{2} + 5312 T^{4} - 80 T^{6} + T^{8} )^{2} \)
$83$ \( ( 1666 - 108 T^{2} + T^{4} )^{4} \)
$89$ \( ( 334084 - 30056 T^{2} + 2126 T^{4} - 52 T^{6} + T^{8} )^{2} \)
$97$ \( ( 1250 + 100 T^{2} + T^{4} )^{4} \)
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