↑

Properties

Label 120.2.m.b
Level $120$
Weight $2$
Character orbit 120.m
Analytic conductor $0.958$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 120.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.958204824255\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{13} q^{3} - \beta_{11} q^{4} + \beta_{7} q^{5} + ( - \beta_{9} + \beta_{5}) q^{6} + ( - \beta_{15} + \beta_{14}) q^{7} + (\beta_{13} - \beta_{12} + \beta_{8} - \beta_{7}) q^{8} + (\beta_{11} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{13} q^{3} - \beta_{11} q^{4} + \beta_{7} q^{5} + ( - \beta_{9} + \beta_{5}) q^{6} + ( - \beta_{15} + \beta_{14}) q^{7} + (\beta_{13} - \beta_{12} + \beta_{8} - \beta_{7}) q^{8} + (\beta_{11} + \beta_1 - 1) q^{9} + ( - \beta_{14} + \beta_{11} + \beta_{10} - \beta_{9} + 1) q^{10} + ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{5} + \beta_{4} - \beta_1) q^{11} + ( - \beta_{15} - \beta_{12} + \beta_{11} - \beta_{9} - \beta_{6} - \beta_{2}) q^{12} + (\beta_{15} - \beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6}) q^{13} + (\beta_{10} - \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{14} + (\beta_{11} - \beta_{10} + 2 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4}) q^{15} + 2 \beta_{9} q^{16} + ( - \beta_{13} + \beta_{12} + \beta_{3} - \beta_{2}) q^{17} + (\beta_{15} + \beta_{12} - \beta_{10} - \beta_{6} + \beta_{3} - \beta_{2}) q^{18} + ( - \beta_{11} + \beta_{10} - 2) q^{19} + ( - \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{20} + (\beta_{11} - 2 \beta_{9} - \beta_{5} - \beta_{4}) q^{21} + (\beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6}) q^{22} + ( - \beta_{8} + \beta_{7}) q^{23} + ( - \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{24} + (\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{10} + \beta_{9} + 1) q^{25} + (2 \beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{26} + (\beta_{15} + \beta_{14} + 2 \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{3} + \beta_{2}) q^{27} + (2 \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{10} + \beta_{8} - \beta_{7} - 2 \beta_{6}) q^{28} + ( - \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2}) q^{29} + (\beta_{15} - \beta_{14} - \beta_{13} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_{4} - \beta_1 + 1) q^{30} + (\beta_{11} + \beta_{10} + 2 \beta_{9}) q^{31} + ( - 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{8} - 2 \beta_{7}) q^{32} + ( - \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - 2 \beta_{3} + 2 \beta_{2}) q^{33} + (\beta_{11} - \beta_{10} - \beta_{9} - 2) q^{34} + (\beta_{13} - \beta_{12} - \beta_{9} + \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{35} + ( - \beta_{10} + \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{36} + (\beta_{15} - \beta_{14} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - 2 \beta_{6}) q^{37} + (\beta_{13} - \beta_{12} + \beta_{8} - \beta_{7} + 2 \beta_{3} + 2 \beta_{2}) q^{38} + ( - 2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{3} + \beta_{2}) q^{39} + ( - \beta_{13} - \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - \beta_{8} + \beta_{7} + 2 \beta_{6} + \cdots + 2) q^{40}+ \cdots + (3 \beta_{11} - 3 \beta_{10} - \beta_{9} + \beta_{5} - \beta_{4} - \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} + 16 q^{10} - 32 q^{19} - 32 q^{24} + 16 q^{25} + 16 q^{30} - 32 q^{34} - 32 q^{36} + 32 q^{40} + 16 q^{49} + 32 q^{51} + 32 q^{54} + 64 q^{66} - 64 q^{70} + 32 q^{75} + 64 q^{76} - 48 q^{81} + 32 q^{84} - 16 q^{90} - 64 q^{91} + 64 q^{96} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{13} + 69\nu^{11} + 506\nu^{9} + 1488\nu^{7} + 1638\nu^{5} + 594\nu^{3} + 44\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7 \nu^{14} + 3 \nu^{13} - 164 \nu^{12} + 69 \nu^{11} - 1250 \nu^{10} + 506 \nu^{9} - 3984 \nu^{8} + 1488 \nu^{7} - 5334 \nu^{6} + 1638 \nu^{5} - 3024 \nu^{4} + 594 \nu^{3} - 596 \nu^{2} + \cdots - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7 \nu^{14} + 3 \nu^{13} + 164 \nu^{12} + 69 \nu^{11} + 1250 \nu^{10} + 506 \nu^{9} + 3984 \nu^{8} + 1488 \nu^{7} + 5334 \nu^{6} + 1638 \nu^{5} + 3024 \nu^{4} + 594 \nu^{3} + 596 \nu^{2} + 28 \nu + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{15} - 20 \nu^{14} + 107 \nu^{13} - 474 \nu^{12} + 657 \nu^{11} - 3698 \nu^{10} + 1074 \nu^{9} - 12334 \nu^{8} - 1686 \nu^{7} - 18152 \nu^{6} - 4770 \nu^{5} - 12164 \nu^{4} - 3014 \nu^{3} + \cdots - 300 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5 \nu^{15} - 20 \nu^{14} - 107 \nu^{13} - 474 \nu^{12} - 657 \nu^{11} - 3698 \nu^{10} - 1074 \nu^{9} - 12334 \nu^{8} + 1686 \nu^{7} - 18152 \nu^{6} + 4770 \nu^{5} - 12164 \nu^{4} + \cdots - 300 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8 \nu^{15} - 3 \nu^{14} + 190 \nu^{13} - 70 \nu^{12} + 1487 \nu^{11} - 530 \nu^{10} + 4970 \nu^{9} - 1674 \nu^{8} + 7248 \nu^{7} - 2198 \nu^{6} + 4556 \nu^{5} - 1156 \nu^{4} + 1086 \nu^{3} - 180 \nu^{2} + \cdots - 20 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19 \nu^{15} + 6 \nu^{14} - 448 \nu^{13} + 138 \nu^{12} - 3460 \nu^{11} + 1012 \nu^{10} - 11326 \nu^{9} + 2976 \nu^{8} - 16094 \nu^{7} + 3276 \nu^{6} - 10328 \nu^{5} + 1188 \nu^{4} + \cdots + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19 \nu^{15} + 6 \nu^{14} + 454 \nu^{13} + 138 \nu^{12} + 3598 \nu^{11} + 1012 \nu^{10} + 12338 \nu^{9} + 2976 \nu^{8} + 19070 \nu^{7} + 3276 \nu^{6} + 13604 \nu^{5} + 1188 \nu^{4} + 4092 \nu^{3} + \cdots + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\nu^{15} + 261\nu^{13} + 2042\nu^{11} + 6862\nu^{9} + 10334\nu^{7} + 7410\nu^{5} + 2412\nu^{3} + 252\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11 \nu^{15} + 12 \nu^{14} + 258 \nu^{13} + 282 \nu^{12} + 1971 \nu^{11} + 2164 \nu^{10} + 6311 \nu^{9} + 7004 \nu^{8} + 8530 \nu^{7} + 9752 \nu^{6} + 4916 \nu^{5} + 6092 \nu^{4} + 1046 \nu^{3} + \cdots + 136 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11 \nu^{15} - 12 \nu^{14} + 258 \nu^{13} - 282 \nu^{12} + 1971 \nu^{11} - 2164 \nu^{10} + 6311 \nu^{9} - 7004 \nu^{8} + 8530 \nu^{7} - 9752 \nu^{6} + 4916 \nu^{5} - 6092 \nu^{4} + 1046 \nu^{3} + \cdots - 136 ) / 8 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 38 \nu^{15} + \nu^{14} - 896 \nu^{13} + 20 \nu^{12} - 6921 \nu^{11} + 100 \nu^{10} - 22674 \nu^{9} - 4 \nu^{8} - 32332 \nu^{7} - 918 \nu^{6} - 20960 \nu^{5} - 1440 \nu^{4} - 5842 \nu^{3} + \cdots - 72 ) / 16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 38 \nu^{15} - \nu^{14} - 896 \nu^{13} - 20 \nu^{12} - 6921 \nu^{11} - 100 \nu^{10} - 22674 \nu^{9} + 4 \nu^{8} - 32332 \nu^{7} + 918 \nu^{6} - 20960 \nu^{5} + 1440 \nu^{4} - 5842 \nu^{3} + \cdots + 72 ) / 16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 38 \nu^{15} + 15 \nu^{14} + 891 \nu^{13} + 352 \nu^{12} + 6803 \nu^{11} + 2692 \nu^{10} + 21762 \nu^{9} + 8638 \nu^{8} + 29356 \nu^{7} + 11726 \nu^{6} + 16838 \nu^{5} + 6808 \nu^{4} + 3518 \nu^{3} + \cdots + 76 ) / 16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 38 \nu^{15} - 15 \nu^{14} + 891 \nu^{13} - 352 \nu^{12} + 6803 \nu^{11} - 2692 \nu^{10} + 21762 \nu^{9} - 8638 \nu^{8} + 29356 \nu^{7} - 11726 \nu^{6} + 16838 \nu^{5} - 6808 \nu^{4} + 3518 \nu^{3} + \cdots - 76 ) / 16 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{9} + 2\beta_{8} - 2\beta_{6} - 2\beta_{2} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} - 3 \beta_{14} + 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} + 9 \beta_{3} + 9 \beta_{2} - 8 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 8 \beta_{15} + 8 \beta_{14} - 20 \beta_{13} + 20 \beta_{12} + 5 \beta_{11} - 5 \beta_{10} + 12 \beta_{9} - 22 \beta_{8} + 2 \beta_{7} + 24 \beta_{6} - 8 \beta_{5} - 8 \beta_{4} - 2 \beta_{3} + 22 \beta_{2} + 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 40 \beta_{15} + 40 \beta_{14} - 10 \beta_{13} - 10 \beta_{12} - 31 \beta_{11} - 31 \beta_{10} - 44 \beta_{9} - 31 \beta_{8} + 31 \beta_{7} + 29 \beta_{5} - 29 \beta_{4} - 95 \beta_{3} - 95 \beta_{2} + 83 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37 \beta_{15} - 37 \beta_{14} + 140 \beta_{13} - 140 \beta_{12} - 42 \beta_{11} + 42 \beta_{10} - 72 \beta_{9} + 129 \beta_{8} - 15 \beta_{7} - 144 \beta_{6} + 67 \beta_{5} + 67 \beta_{4} + 10 \beta_{3} - 124 \beta_{2} - 252 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 497 \beta_{15} - 497 \beta_{14} + 175 \beta_{13} + 175 \beta_{12} + 404 \beta_{11} + 404 \beta_{10} + 631 \beta_{9} + 413 \beta_{8} - 413 \beta_{7} - 372 \beta_{5} + 372 \beta_{4} + 1091 \beta_{3} + 1091 \beta_{2} - 956 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 396 \beta_{15} + 396 \beta_{14} - 1792 \beta_{13} + 1792 \beta_{12} + 564 \beta_{11} - 564 \beta_{10} + 876 \beta_{9} - 1556 \beta_{8} + 196 \beta_{7} + 1752 \beta_{6} - 900 \beta_{5} - 900 \beta_{4} - 104 \beta_{3} + \cdots + 2910 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3054 \beta_{15} + 3054 \beta_{14} - 1194 \beta_{13} - 1194 \beta_{12} - 2520 \beta_{11} - 2520 \beta_{10} - 4074 \beta_{9} - 2610 \beta_{8} + 2610 \beta_{7} + 2316 \beta_{5} - 2316 \beta_{4} - 6517 \beta_{3} + \cdots + 5723 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4587 \beta_{15} - 4587 \beta_{14} + 22287 \beta_{13} - 22287 \beta_{12} - 7128 \beta_{11} + 7128 \beta_{10} - 10707 \beta_{9} + 18950 \beta_{8} - 2464 \beta_{7} - 21414 \beta_{6} + 11376 \beta_{5} + \cdots - 34858 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 37433 \beta_{15} - 37433 \beta_{14} + 15180 \beta_{13} + 15180 \beta_{12} + 31030 \beta_{11} + 31030 \beta_{10} + 50850 \beta_{9} + 32346 \beta_{8} - 32346 \beta_{7} - 28542 \beta_{5} + \cdots - 69508 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 54976 \beta_{15} + 54976 \beta_{14} - 274428 \beta_{13} + 274428 \beta_{12} + 88283 \beta_{11} - 88283 \beta_{10} + 131044 \beta_{9} - 231582 \beta_{8} + 30506 \beta_{7} + 262088 \beta_{6} + \cdots + 423384 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 458484 \beta_{15} + 458484 \beta_{14} - 188422 \beta_{13} - 188422 \beta_{12} - 380577 \beta_{11} - 380577 \beta_{10} - 627076 \beta_{9} - 397897 \beta_{8} + 397897 \beta_{7} + 350371 \beta_{5} + \cdots + 848621 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 667774 \beta_{15} - 667774 \beta_{14} + 3367136 \beta_{13} - 3367136 \beta_{12} - 1085560 \beta_{11} + 1085560 \beta_{10} - 1604464 \beta_{9} + 2833662 \beta_{8} - 375266 \beta_{7} + \cdots - 5168936 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 5614479 \beta_{15} - 5614479 \beta_{14} + 2318785 \beta_{13} + 2318785 \beta_{12} + 4662232 \beta_{11} + 4662232 \beta_{10} + 7698833 \beta_{9} + 4880807 \beta_{8} + \cdots - 10380392 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
− 0.724535i
1.05636i
− 1.05636i
0.724535i
0.886177i
− 2.08509i
2.08509i
− 0.886177i
− 3.49930i
− 0.528036i
0.528036i
3.49930i
− 0.357857i
− 2.13875i
2.13875i
0.357857i
−1.30656 − 0.541196i 0.541196 − 1.64533i 1.41421 + 1.41421i 1.25928 + 1.84776i −1.59755 + 1.85683i 3.29066 −1.08239 − 2.61313i −2.41421 − 1.78089i −0.645329 − 3.09573i
59.2 −1.30656 − 0.541196i 0.541196 + 1.64533i 1.41421 + 1.41421i −1.25928 + 1.84776i 0.183339 − 2.44262i −3.29066 −1.08239 − 2.61313i −2.41421 + 1.78089i 2.64533 − 1.73270i
59.3 −1.30656 + 0.541196i 0.541196 − 1.64533i 1.41421 − 1.41421i −1.25928 − 1.84776i 0.183339 + 2.44262i −3.29066 −1.08239 + 2.61313i −2.41421 − 1.78089i 2.64533 + 1.73270i
59.4 −1.30656 + 0.541196i 0.541196 + 1.64533i 1.41421 − 1.41421i 1.25928 − 1.84776i −1.59755 − 1.85683i 3.29066 −1.08239 + 2.61313i −2.41421 + 1.78089i −0.645329 + 3.09573i
59.5 −0.541196 − 1.30656i −1.30656 − 1.13705i −1.41421 + 1.41421i −2.10100 + 0.765367i −0.778527 + 2.32248i −2.27411 2.61313 + 1.08239i 0.414214 + 2.97127i 2.13705 + 2.33088i
59.6 −0.541196 − 1.30656i −1.30656 + 1.13705i −1.41421 + 1.41421i 2.10100 + 0.765367i 2.19274 + 1.09174i 2.27411 2.61313 + 1.08239i 0.414214 − 2.97127i −0.137055 − 3.15931i
59.7 −0.541196 + 1.30656i −1.30656 − 1.13705i −1.41421 − 1.41421i 2.10100 − 0.765367i 2.19274 − 1.09174i 2.27411 2.61313 − 1.08239i 0.414214 + 2.97127i −0.137055 + 3.15931i
59.8 −0.541196 + 1.30656i −1.30656 + 1.13705i −1.41421 − 1.41421i −2.10100 − 0.765367i −0.778527 − 2.32248i −2.27411 2.61313 − 1.08239i 0.414214 − 2.97127i 2.13705 − 2.33088i
59.9 0.541196 − 1.30656i 1.30656 − 1.13705i −1.41421 − 1.41421i −2.10100 + 0.765367i −0.778527 − 2.32248i 2.27411 −2.61313 + 1.08239i 0.414214 − 2.97127i −0.137055 + 3.15931i
59.10 0.541196 − 1.30656i 1.30656 + 1.13705i −1.41421 − 1.41421i 2.10100 + 0.765367i 2.19274 − 1.09174i −2.27411 −2.61313 + 1.08239i 0.414214 + 2.97127i 2.13705 − 2.33088i
59.11 0.541196 + 1.30656i 1.30656 − 1.13705i −1.41421 + 1.41421i 2.10100 − 0.765367i 2.19274 + 1.09174i −2.27411 −2.61313 − 1.08239i 0.414214 − 2.97127i 2.13705 + 2.33088i
59.12 0.541196 + 1.30656i 1.30656 + 1.13705i −1.41421 + 1.41421i −2.10100 − 0.765367i −0.778527 + 2.32248i 2.27411 −2.61313 − 1.08239i 0.414214 + 2.97127i −0.137055 − 3.15931i
59.13 1.30656 − 0.541196i −0.541196 − 1.64533i 1.41421 − 1.41421i 1.25928 + 1.84776i −1.59755 − 1.85683i −3.29066 1.08239 − 2.61313i −2.41421 + 1.78089i 2.64533 + 1.73270i
59.14 1.30656 − 0.541196i −0.541196 + 1.64533i 1.41421 − 1.41421i −1.25928 + 1.84776i 0.183339 + 2.44262i 3.29066 1.08239 − 2.61313i −2.41421 − 1.78089i −0.645329 + 3.09573i
59.15 1.30656 + 0.541196i −0.541196 − 1.64533i 1.41421 + 1.41421i −1.25928 − 1.84776i 0.183339 − 2.44262i 3.29066 1.08239 + 2.61313i −2.41421 + 1.78089i −0.645329 − 3.09573i
59.16 1.30656 + 0.541196i −0.541196 + 1.64533i 1.41421 + 1.41421i 1.25928 − 1.84776i −1.59755 + 1.85683i −3.29066 1.08239 + 2.61313i −2.41421 − 1.78089i 2.64533 − 1.73270i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
8.d odd 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.2.m.b ✓ 16
3.b odd 2 1 inner 120.2.m.b ✓ 16
4.b odd 2 1 480.2.m.b 16
5.b even 2 1 inner 120.2.m.b ✓ 16
5.c odd 4 2 600.2.b.i 16
8.b even 2 1 480.2.m.b 16
8.d odd 2 1 inner 120.2.m.b ✓ 16
12.b even 2 1 480.2.m.b 16
15.d odd 2 1 inner 120.2.m.b ✓ 16
15.e even 4 2 600.2.b.i 16
20.d odd 2 1 480.2.m.b 16
20.e even 4 2 2400.2.b.i 16
24.f even 2 1 inner 120.2.m.b ✓ 16
24.h odd 2 1 480.2.m.b 16
40.e odd 2 1 inner 120.2.m.b ✓ 16
40.f even 2 1 480.2.m.b 16
40.i odd 4 2 2400.2.b.i 16
40.k even 4 2 600.2.b.i 16
60.h even 2 1 480.2.m.b 16
60.l odd 4 2 2400.2.b.i 16
120.i odd 2 1 480.2.m.b 16
120.m even 2 1 inner 120.2.m.b ✓ 16
120.q odd 4 2 600.2.b.i 16
120.w even 4 2 2400.2.b.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.m.b ✓ 16 1.a even 1 1 trivial
120.2.m.b ✓ 16 3.b odd 2 1 inner
120.2.m.b ✓ 16 5.b even 2 1 inner
120.2.m.b ✓ 16 8.d odd 2 1 inner
120.2.m.b ✓ 16 15.d odd 2 1 inner
120.2.m.b ✓ 16 24.f even 2 1 inner
120.2.m.b ✓ 16 40.e odd 2 1 inner
120.2.m.b ✓ 16 120.m even 2 1 inner
480.2.m.b 16 4.b odd 2 1
480.2.m.b 16 8.b even 2 1
480.2.m.b 16 12.b even 2 1
480.2.m.b 16 20.d odd 2 1
480.2.m.b 16 24.h odd 2 1
480.2.m.b 16 40.f even 2 1
480.2.m.b 16 60.h even 2 1
480.2.m.b 16 120.i odd 2 1
600.2.b.i 16 5.c odd 4 2
600.2.b.i 16 15.e even 4 2
600.2.b.i 16 40.k even 4 2
600.2.b.i 16 120.q odd 4 2
2400.2.b.i 16 20.e even 4 2
2400.2.b.i 16 40.i odd 4 2
2400.2.b.i 16 60.l odd 4 2
2400.2.b.i 16 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 16T_{7}^{2} + 56 \) acting on \(S_{2}^{\mathrm{new}}(120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 4 T^{6} + 14 T^{4} + 36 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 4 T^{6} + 22 T^{4} - 100 T^{2} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 16 T^{2} + 56)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 24 T^{2} + 112)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 32 T^{2} + 224)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 16 T^{2} + 32)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 4)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 8 T^{2} + 8)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 40 T^{2} + 112)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 48 T^{2} + 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 64 T^{2} + 224)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 80 T^{2} + 448)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 112 T^{2} + 2744)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{2} + 8)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 144 T^{2} + 2592)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 24 T^{2} + 112)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 72)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 16 T^{2} + 56)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 192 T^{2} + 7168)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 64 T^{2} + 896)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 272 T^{2} + 64)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 136 T^{2} + 4232)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 96 T^{2} + 1792)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 128 T^{2} + 896)^{4} \) Copy content Toggle raw display
show more
show less