Properties

Label 1755.2.i.f
Level $1755$
Weight $2$
Character orbit 1755.i
Analytic conductor $14.014$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1755.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0137455547\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 5 x^{15} + 20 x^{14} - 44 x^{13} + 96 x^{12} - 107 x^{11} + 178 x^{10} - 19 x^{9} + 231 x^{8} + 326 x^{7} + 551 x^{6} + 859 x^{5} + 1118 x^{4} + 1215 x^{3} + 1103 x^{2} + \cdots + 268 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 585)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_1) q^{2} + ( - \beta_{11} + \beta_{2} - 1) q^{4} + (\beta_{2} - 1) q^{5} + ( - \beta_{15} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{12} + \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_1) q^{2} + ( - \beta_{11} + \beta_{2} - 1) q^{4} + (\beta_{2} - 1) q^{5} + ( - \beta_{15} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{12} + \beta_1 + 1) q^{8} + \beta_1 q^{10} + ( - \beta_{15} - \beta_{14} - \beta_{13} + \beta_{5}) q^{11} + ( - \beta_{2} + 1) q^{13} + (\beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{2} + 2) q^{14} + (\beta_{14} + \beta_{13} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{16} + ( - \beta_{15} - \beta_{12} + 2 \beta_{10} - \beta_{9} - \beta_{3} - \beta_1 + 1) q^{17} + ( - \beta_{15} + \beta_{14} + \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{3} - \beta_1 - 2) q^{19} + (\beta_{11} + \beta_{8} - \beta_{2}) q^{20} + (\beta_{13} + \beta_{11} - \beta_{10} + 2 \beta_{9} - 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{2}) q^{22} + ( - \beta_{13} + 2 \beta_{10} - 2 \beta_{5} - \beta_{4} + 2) q^{23} - \beta_{2} q^{25} - \beta_1 q^{26} + (\beta_{12} - 3 \beta_{10} + 2 \beta_{8} - \beta_1 - 6) q^{28} + (\beta_{14} + \beta_{13} - 2 \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{29} + ( - \beta_{13} + \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \cdots + 5) q^{31}+ \cdots + (4 \beta_{15} - \beta_{14} + 3 \beta_{12} - 3 \beta_{10} + 4 \beta_{9} - 3 \beta_{8} - \beta_{3} + \cdots + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3 q^{2} - 9 q^{4} - 8 q^{5} + 11 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 3 q^{2} - 9 q^{4} - 8 q^{5} + 11 q^{7} + 12 q^{8} - 6 q^{10} + 6 q^{11} + 8 q^{13} + 10 q^{14} - 11 q^{16} + 4 q^{17} - 20 q^{19} - 9 q^{20} - 3 q^{22} + 6 q^{23} - 8 q^{25} + 6 q^{26} - 68 q^{28} + 14 q^{29} + 31 q^{31} + q^{32} + 7 q^{34} - 22 q^{35} + 2 q^{37} + 9 q^{38} - 6 q^{40} - 12 q^{41} - 15 q^{43} - 32 q^{44} - 64 q^{46} - 18 q^{47} - 17 q^{49} + 3 q^{50} + 9 q^{52} - 4 q^{53} - 12 q^{55} + 16 q^{56} + 42 q^{58} + 24 q^{59} + 9 q^{61} + 40 q^{62} - 60 q^{64} + 8 q^{65} + 18 q^{67} - 14 q^{68} + 10 q^{70} - 20 q^{71} + 12 q^{73} - 37 q^{74} + 53 q^{76} - 34 q^{77} + 3 q^{79} + 22 q^{80} - 68 q^{82} - 10 q^{83} - 2 q^{85} + 60 q^{86} + 14 q^{88} + 26 q^{89} + 22 q^{91} + 5 q^{92} - 17 q^{94} + 10 q^{95} + 34 q^{97} + 60 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} + 20 x^{14} - 44 x^{13} + 96 x^{12} - 107 x^{11} + 178 x^{10} - 19 x^{9} + 231 x^{8} + 326 x^{7} + 551 x^{6} + 859 x^{5} + 1118 x^{4} + 1215 x^{3} + 1103 x^{2} + \cdots + 268 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{15} + 4 \nu^{14} - 16 \nu^{13} + 28 \nu^{12} - 68 \nu^{11} + 39 \nu^{10} - 139 \nu^{9} - 120 \nu^{8} - 351 \nu^{7} - 677 \nu^{6} - 1228 \nu^{5} - 2087 \nu^{4} - 3205 \nu^{3} - 4420 \nu^{2} + \cdots - 6293 ) / 2187 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 53 \nu^{15} + 1221 \nu^{14} - 4880 \nu^{13} + 14280 \nu^{12} - 18634 \nu^{11} + 24185 \nu^{10} + 23394 \nu^{9} - 18231 \nu^{8} + 152463 \nu^{7} + 42206 \nu^{6} + \cdots + 194118 ) / 68526 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{15} - 7 \nu^{14} + 50 \nu^{13} - 217 \nu^{12} + 469 \nu^{11} - 874 \nu^{10} + 757 \nu^{9} - 840 \nu^{8} - 447 \nu^{7} - 917 \nu^{6} - 1580 \nu^{5} - 2093 \nu^{4} - 2345 \nu^{3} - 607 \nu^{2} + \cdots + 2021 ) / 729 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 547 \nu^{15} - 2825 \nu^{14} + 11696 \nu^{13} - 42716 \nu^{12} + 9478 \nu^{11} + 20721 \nu^{10} - 417232 \nu^{9} + 363147 \nu^{8} - 1222965 \nu^{7} - 271472 \nu^{6} + \cdots - 568472 ) / 205578 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 280 \nu^{15} - 2977 \nu^{14} + 16531 \nu^{13} - 56011 \nu^{12} + 137264 \nu^{11} - 241101 \nu^{10} + 338866 \nu^{9} - 309462 \nu^{8} + 265752 \nu^{7} + 470 \nu^{6} + \cdots + 492218 ) / 102789 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 197 \nu^{15} - 633 \nu^{14} + 2996 \nu^{13} - 7602 \nu^{12} + 28186 \nu^{11} - 53597 \nu^{10} + 131148 \nu^{9} - 144885 \nu^{8} + 281373 \nu^{7} - 81968 \nu^{6} + 367611 \nu^{5} + \cdots + 53568 ) / 68526 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 263 \nu^{15} + 4147 \nu^{14} - 19076 \nu^{13} + 62410 \nu^{12} - 127504 \nu^{11} + 236395 \nu^{10} - 263068 \nu^{9} + 378201 \nu^{8} - 138783 \nu^{7} + 484382 \nu^{6} + \cdots + 299584 ) / 68526 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{15} + 13 \nu^{14} - 51 \nu^{13} + 97 \nu^{12} - 219 \nu^{11} + 170 \nu^{10} - 403 \nu^{9} - 207 \nu^{8} - 780 \nu^{7} - 1407 \nu^{6} - 2383 \nu^{5} - 3732 \nu^{4} - 4999 \nu^{3} + \cdots - 3302 ) / 729 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 197 \nu^{15} + 1103 \nu^{14} - 3278 \nu^{13} + 4688 \nu^{12} - 2101 \nu^{11} - 8913 \nu^{10} + 25879 \nu^{9} - 39966 \nu^{8} + 54630 \nu^{7} - 39292 \nu^{6} + 92848 \nu^{5} + \cdots + 93260 ) / 34263 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14 \nu^{15} + 44 \nu^{14} - 53 \nu^{13} - 331 \nu^{12} + 1253 \nu^{11} - 3357 \nu^{10} + 4663 \nu^{9} - 6243 \nu^{8} + 2448 \nu^{7} - 3007 \nu^{6} - 4508 \nu^{5} + 1643 \nu^{4} + \cdots + 3587 ) / 2187 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1076 \nu^{15} + 10427 \nu^{14} - 44171 \nu^{13} + 127925 \nu^{12} - 248173 \nu^{11} + 410358 \nu^{10} - 433283 \nu^{9} + 561027 \nu^{8} - 211347 \nu^{7} + \cdots + 610868 ) / 102789 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10 \nu^{15} + 48 \nu^{14} - 187 \nu^{13} + 390 \nu^{12} - 833 \nu^{11} + 826 \nu^{10} - 1425 \nu^{9} - 165 \nu^{8} - 1974 \nu^{7} - 3245 \nu^{6} - 5211 \nu^{5} - 7880 \nu^{4} - 9222 \nu^{3} + \cdots - 3849 ) / 729 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1481 \nu^{15} - 5918 \nu^{14} + 18674 \nu^{13} - 21635 \nu^{12} + 30967 \nu^{11} + 47409 \nu^{10} - 22324 \nu^{9} + 252699 \nu^{8} + 105723 \nu^{7} + 543790 \nu^{6} + \cdots + 214288 ) / 102789 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 70 \nu^{15} + 307 \nu^{14} - 1030 \nu^{13} + 1591 \nu^{12} - 2744 \nu^{11} + 426 \nu^{10} - 2350 \nu^{9} - 6591 \nu^{8} - 8262 \nu^{7} - 18500 \nu^{6} - 28558 \nu^{5} - 35156 \nu^{4} + \cdots - 2714 ) / 2187 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1260 \nu^{15} + 5900 \nu^{14} - 21726 \nu^{13} + 42641 \nu^{12} - 89361 \nu^{11} + 88954 \nu^{10} - 167819 \nu^{9} + 22059 \nu^{8} - 276987 \nu^{7} - 278193 \nu^{6} + \cdots - 181150 ) / 34263 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{14} + 2\beta_{13} + \beta_{6} - \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} + 2\beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} - 2\beta_{7} - 2\beta_{5} - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} - \beta_{12} + 4 \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} - 5 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 6 \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{14} - 3 \beta_{13} - \beta_{12} + 4 \beta_{11} + \beta_{10} + 11 \beta_{8} - 2 \beta_{7} + 5 \beta_{6} + \beta_{5} + 3 \beta_{4} - 2 \beta_{2} - 19 \beta _1 - 19 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5 \beta_{15} - 6 \beta_{14} - 9 \beta_{13} - 9 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 15 \beta_{8} + 18 \beta_{7} + 10 \beta_{6} + 12 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} - 21 \beta_{2} - 32 \beta _1 - 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4 \beta_{15} - 2 \beta_{14} - 31 \beta_{13} + 8 \beta_{12} - 47 \beta_{11} - 26 \beta_{10} - 7 \beta_{9} - 10 \beta_{8} + 64 \beta_{7} - 10 \beta_{6} + 31 \beta_{5} - 26 \beta_{4} + \beta_{3} - 24 \beta_{2} - 7 \beta _1 - 26 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 11 \beta_{15} + 9 \beta_{14} - 69 \beta_{13} + 27 \beta_{12} - 105 \beta_{11} - 57 \beta_{10} - 16 \beta_{9} - 87 \beta_{8} + 108 \beta_{7} - 106 \beta_{6} + 48 \beta_{5} - 32 \beta_{4} + 4 \beta_{3} + 108 \beta_{2} + 161 \beta _1 + 45 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 22 \beta_{15} + 10 \beta_{14} - 37 \beta_{13} + 58 \beta_{12} - 121 \beta_{11} - 31 \beta_{10} - 17 \beta_{9} - 230 \beta_{8} + 35 \beta_{7} - 309 \beta_{6} + 14 \beta_{5} + 8 \beta_{4} + 17 \beta_{3} + 484 \beta_{2} + 582 \beta _1 + 354 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 14 \beta_{15} - 40 \beta_{14} + 277 \beta_{13} + 124 \beta_{12} + 98 \beta_{11} + 248 \beta_{10} + 58 \beta_{9} - 413 \beta_{8} - 379 \beta_{7} - 407 \beta_{6} - 211 \beta_{5} + 137 \beta_{4} + 23 \beta_{3} + 831 \beta_{2} + 1048 \beta _1 + 1046 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 98 \beta_{15} - 190 \beta_{14} + 1027 \beta_{13} + 243 \beta_{12} + 1002 \beta_{11} + 981 \beta_{10} + 313 \beta_{9} - 309 \beta_{8} - 1371 \beta_{7} + 483 \beta_{6} - 795 \beta_{5} + 398 \beta_{4} + 32 \beta_{3} - 200 \beta_{2} + \cdots + 1750 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 121 \beta_{15} - 338 \beta_{14} + 1772 \beta_{13} + 75 \beta_{12} + 3174 \beta_{11} + 1770 \beta_{10} + 617 \beta_{9} + 1314 \beta_{8} - 2733 \beta_{7} + 4206 \beta_{6} - 1491 \beta_{5} + 718 \beta_{4} + 73 \beta_{3} + \cdots + 800 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 103 \beta_{15} + 307 \beta_{14} + 749 \beta_{13} - 1995 \beta_{12} + 5949 \beta_{11} + 429 \beta_{10} + 163 \beta_{9} + 6813 \beta_{8} - 2553 \beta_{7} + 12074 \beta_{6} - 798 \beta_{5} + 566 \beta_{4} - 232 \beta_{3} + \cdots - 5494 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 890 \beta_{15} + 4039 \beta_{14} - 5440 \beta_{13} - 9317 \beta_{12} + 3422 \beta_{11} - 7912 \beta_{10} - 2575 \beta_{9} + 17143 \beta_{8} + 4739 \beta_{7} + 17608 \beta_{6} + 4415 \beta_{5} - 1616 \beta_{4} + \cdots - 23142 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 2922 \beta_{15} + 13568 \beta_{14} - 20192 \beta_{13} - 22506 \beta_{12} - 21396 \beta_{11} - 28131 \beta_{10} - 8667 \beta_{9} + 22623 \beta_{8} + 28509 \beta_{7} - 9829 \beta_{6} + 17028 \beta_{5} + \cdots - 58190 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 5826 \beta_{15} + 22246 \beta_{14} - 40255 \beta_{13} - 21812 \beta_{12} - 95482 \beta_{11} - 51508 \beta_{10} - 15069 \beta_{9} - 15488 \beta_{8} + 73283 \beta_{7} - 138565 \beta_{6} + \cdots - 100389 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1755\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(1081\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
586.1
1.79305 + 1.53983i
1.48460 1.66288i
0.466399 + 1.64781i
0.252952 + 1.56266i
0.172467 1.52157i
−0.317019 1.12493i
−0.628312 + 0.590424i
−0.724143 0.165319i
1.79305 1.53983i
1.48460 + 1.66288i
0.466399 1.64781i
0.252952 1.56266i
0.172467 + 1.52157i
−0.317019 + 1.12493i
−0.628312 0.590424i
−0.724143 + 0.165319i
−1.29305 + 2.23963i 0 −2.34397 4.05987i −0.500000 0.866025i 0 2.13745 3.70217i 6.95128 0 2.58610
586.2 −0.984603 + 1.70538i 0 −0.938888 1.62620i −0.500000 0.866025i 0 1.51414 2.62256i −0.240686 0 1.96921
586.3 0.0336011 0.0581988i 0 0.997742 + 1.72814i −0.500000 0.866025i 0 −1.23179 + 2.13352i 0.268505 0 −0.0672022
586.4 0.247048 0.427900i 0 0.877935 + 1.52063i −0.500000 0.866025i 0 2.14269 3.71125i 1.85576 0 −0.494096
586.5 0.327533 0.567303i 0 0.785445 + 1.36043i −0.500000 0.866025i 0 0.388592 0.673061i 2.33917 0 −0.655066
586.6 0.817019 1.41512i 0 −0.335039 0.580304i −0.500000 0.866025i 0 −1.06506 + 1.84473i 2.17314 0 −1.63404
586.7 1.12831 1.95429i 0 −1.54617 2.67805i −0.500000 0.866025i 0 −0.353285 + 0.611908i −2.46502 0 −2.25662
586.8 1.22414 2.12028i 0 −1.99705 3.45900i −0.500000 0.866025i 0 1.96726 3.40740i −4.88214 0 −2.44829
1171.1 −1.29305 2.23963i 0 −2.34397 + 4.05987i −0.500000 + 0.866025i 0 2.13745 + 3.70217i 6.95128 0 2.58610
1171.2 −0.984603 1.70538i 0 −0.938888 + 1.62620i −0.500000 + 0.866025i 0 1.51414 + 2.62256i −0.240686 0 1.96921
1171.3 0.0336011 + 0.0581988i 0 0.997742 1.72814i −0.500000 + 0.866025i 0 −1.23179 2.13352i 0.268505 0 −0.0672022
1171.4 0.247048 + 0.427900i 0 0.877935 1.52063i −0.500000 + 0.866025i 0 2.14269 + 3.71125i 1.85576 0 −0.494096
1171.5 0.327533 + 0.567303i 0 0.785445 1.36043i −0.500000 + 0.866025i 0 0.388592 + 0.673061i 2.33917 0 −0.655066
1171.6 0.817019 + 1.41512i 0 −0.335039 + 0.580304i −0.500000 + 0.866025i 0 −1.06506 1.84473i 2.17314 0 −1.63404
1171.7 1.12831 + 1.95429i 0 −1.54617 + 2.67805i −0.500000 + 0.866025i 0 −0.353285 0.611908i −2.46502 0 −2.25662
1171.8 1.22414 + 2.12028i 0 −1.99705 + 3.45900i −0.500000 + 0.866025i 0 1.96726 + 3.40740i −4.88214 0 −2.44829
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1171.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1755.2.i.f 16
3.b odd 2 1 585.2.i.e 16
9.c even 3 1 inner 1755.2.i.f 16
9.c even 3 1 5265.2.a.ba 8
9.d odd 6 1 585.2.i.e 16
9.d odd 6 1 5265.2.a.bf 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.e 16 3.b odd 2 1
585.2.i.e 16 9.d odd 6 1
1755.2.i.f 16 1.a even 1 1 trivial
1755.2.i.f 16 9.c even 3 1 inner
5265.2.a.ba 8 9.c even 3 1
5265.2.a.bf 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1755, [\chi])\):

\( T_{2}^{16} - 3 T_{2}^{15} + 17 T_{2}^{14} - 38 T_{2}^{13} + 158 T_{2}^{12} - 324 T_{2}^{11} + 875 T_{2}^{10} - 1368 T_{2}^{9} + 2812 T_{2}^{8} - 3873 T_{2}^{7} + 5539 T_{2}^{6} - 4627 T_{2}^{5} + 3027 T_{2}^{4} - 1114 T_{2}^{3} + \cdots + 1 \) Copy content Toggle raw display
\( T_{7}^{16} - 11 T_{7}^{15} + 97 T_{7}^{14} - 476 T_{7}^{13} + 2127 T_{7}^{12} - 6158 T_{7}^{11} + 20644 T_{7}^{10} - 44604 T_{7}^{9} + 140146 T_{7}^{8} - 191361 T_{7}^{7} + 544052 T_{7}^{6} - 359094 T_{7}^{5} + \cdots + 395641 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 3 T^{15} + 17 T^{14} - 38 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} - 11 T^{15} + 97 T^{14} + \cdots + 395641 \) Copy content Toggle raw display
$11$ \( T^{16} - 6 T^{15} + 74 T^{14} + \cdots + 401956 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$17$ \( (T^{8} - 2 T^{7} - 78 T^{6} + 73 T^{5} + \cdots + 892)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 10 T^{7} - 37 T^{6} - 541 T^{5} + \cdots - 1584)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 6 T^{15} + 115 T^{14} + \cdots + 144360225 \) Copy content Toggle raw display
$29$ \( T^{16} - 14 T^{15} + 191 T^{14} + \cdots + 1243225 \) Copy content Toggle raw display
$31$ \( T^{16} - 31 T^{15} + \cdots + 2547422784 \) Copy content Toggle raw display
$37$ \( (T^{8} - T^{7} - 120 T^{6} - 211 T^{5} + \cdots - 33660)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 12 T^{15} + \cdots + 371525625 \) Copy content Toggle raw display
$43$ \( T^{16} + 15 T^{15} + \cdots + 62415940941376 \) Copy content Toggle raw display
$47$ \( T^{16} + 18 T^{15} + \cdots + 424482219529 \) Copy content Toggle raw display
$53$ \( (T^{8} + 2 T^{7} - 282 T^{6} + \cdots + 111438)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} - 24 T^{15} + \cdots + 3524698346724 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 106354483234561 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 974523752673769 \) Copy content Toggle raw display
$71$ \( (T^{8} + 10 T^{7} - 211 T^{6} + \cdots - 127950)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 6 T^{7} - 300 T^{6} + \cdots + 1196532)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} - 3 T^{15} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{16} + 10 T^{15} + 215 T^{14} + \cdots + 469225 \) Copy content Toggle raw display
$89$ \( (T^{8} - 13 T^{7} - 228 T^{6} + \cdots + 1032219)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 122483314259524 \) Copy content Toggle raw display
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