Properties

Label 896.2.q.c
Level $896$
Weight $2$
Character orbit 896.q
Analytic conductor $7.155$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} + 4 x^{14} - 24 x^{13} + 104 x^{12} - 196 x^{11} + 312 x^{10} - 236 x^{9} + 31 x^{8} + 236 x^{7} + 312 x^{6} + 196 x^{5} + 104 x^{4} + 24 x^{3} + 4 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + \beta_{9} q^{5} -\beta_{14} q^{7} + ( 2 \beta_{6} + 2 \beta_{7} + \beta_{13} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{3} + \beta_{9} q^{5} -\beta_{14} q^{7} + ( 2 \beta_{6} + 2 \beta_{7} + \beta_{13} ) q^{9} + ( -\beta_{3} + \beta_{12} ) q^{11} + ( -\beta_{1} - 2 \beta_{2} + \beta_{8} ) q^{13} + ( \beta_{4} + \beta_{10} + \beta_{11} + \beta_{14} ) q^{15} + ( 1 + \beta_{6} + \beta_{7} ) q^{17} + ( -2 \beta_{3} + \beta_{12} ) q^{19} + ( 2 \beta_{1} + 3 \beta_{2} - \beta_{8} - \beta_{9} ) q^{21} + ( \beta_{4} - \beta_{10} + \beta_{14} ) q^{23} + ( 2 - 2 \beta_{6} ) q^{25} + ( -\beta_{3} + 2 \beta_{5} + 2 \beta_{12} - 2 \beta_{15} ) q^{27} + ( -\beta_{1} + 2 \beta_{2} - \beta_{8} - 4 \beta_{9} ) q^{29} + ( -3 \beta_{4} + \beta_{10} - 2 \beta_{11} - 2 \beta_{14} ) q^{31} + ( -2 + \beta_{6} + 2 \beta_{13} ) q^{33} + ( -\beta_{3} + \beta_{5} + \beta_{12} + \beta_{15} ) q^{35} + ( -2 \beta_{2} - 3 \beta_{8} + \beta_{9} ) q^{37} + ( -3 \beta_{4} - 4 \beta_{10} - \beta_{11} + \beta_{14} ) q^{39} + ( -3 \beta_{7} - 3 \beta_{13} ) q^{41} + ( -2 \beta_{3} + 2 \beta_{5} + 2 \beta_{15} ) q^{43} + ( 3 \beta_{1} - 2 \beta_{2} + 6 \beta_{8} + 2 \beta_{9} ) q^{45} + ( 2 \beta_{4} + 2 \beta_{10} + \beta_{11} + \beta_{14} ) q^{47} + ( -1 - 2 \beta_{6} - \beta_{7} - 3 \beta_{13} ) q^{49} + ( 4 \beta_{5} + \beta_{12} - 2 \beta_{15} ) q^{51} + ( -\beta_{1} + \beta_{2} + \beta_{9} ) q^{53} + ( \beta_{4} - 2 \beta_{10} - \beta_{11} + 2 \beta_{14} ) q^{55} + ( -3 - 2 \beta_{7} + 2 \beta_{13} ) q^{57} + \beta_{5} q^{59} + ( -6 \beta_{1} - 3 \beta_{8} + 3 \beta_{9} ) q^{61} + ( \beta_{4} + 5 \beta_{10} + \beta_{11} - 2 \beta_{14} ) q^{63} + ( -6 \beta_{6} - 2 \beta_{7} - \beta_{13} ) q^{65} + ( -2 \beta_{3} + \beta_{5} + 2 \beta_{12} - 2 \beta_{15} ) q^{67} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{8} ) q^{69} + ( 4 \beta_{4} + 4 \beta_{11} ) q^{71} + ( -5 - 5 \beta_{6} - 2 \beta_{7} ) q^{73} + 2 \beta_{15} q^{75} + ( 5 \beta_{1} - 3 \beta_{2} + \beta_{8} + \beta_{9} ) q^{77} + ( -\beta_{4} + \beta_{10} + 2 \beta_{11} + \beta_{14} ) q^{79} + ( -7 + 7 \beta_{6} + \beta_{7} + 2 \beta_{13} ) q^{81} + ( -4 \beta_{5} + 4 \beta_{15} ) q^{83} + ( 3 \beta_{1} - \beta_{2} + 3 \beta_{8} + 2 \beta_{9} ) q^{85} + ( -3 \beta_{4} - 2 \beta_{10} - 5 \beta_{11} - 5 \beta_{14} ) q^{87} + ( 2 - \beta_{6} - 4 \beta_{13} ) q^{89} + ( -\beta_{3} + \beta_{5} - 5 \beta_{15} ) q^{91} + ( 6 \beta_{2} - 11 \beta_{8} - 3 \beta_{9} ) q^{93} + ( -3 \beta_{10} - 3 \beta_{11} + 3 \beta_{14} ) q^{95} + ( 2 - 4 \beta_{6} - \beta_{7} - \beta_{13} ) q^{97} + ( \beta_{3} - 3 \beta_{5} - 3 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{9} + O(q^{10}) \) \( 16q + 16q^{9} + 24q^{17} + 16q^{25} - 24q^{33} - 32q^{49} - 48q^{57} - 48q^{65} - 120q^{73} - 56q^{81} + 24q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} + 4 x^{14} - 24 x^{13} + 104 x^{12} - 196 x^{11} + 312 x^{10} - 236 x^{9} + 31 x^{8} + 236 x^{7} + 312 x^{6} + 196 x^{5} + 104 x^{4} + 24 x^{3} + 4 x^{2} + 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-24578404 \nu^{15} + 112409521 \nu^{14} - 161766434 \nu^{13} + 676914588 \nu^{12} - 2932745926 \nu^{11} + 6464372519 \nu^{10} - 11224587756 \nu^{9} + 11842175116 \nu^{8} - 6808155680 \nu^{7} - 2875070153 \nu^{6} - 5236873676 \nu^{5} - 1970820116 \nu^{4} - 1399477502 \nu^{3} + 116148784 \nu^{2} - 266195456 \nu - 80530744\)\()/55881261\)
\(\beta_{2}\)\(=\)\((\)\(478 \nu^{15} - 1916 \nu^{14} + 1922 \nu^{13} - 11453 \nu^{12} + 49716 \nu^{11} - 93806 \nu^{10} + 148878 \nu^{9} - 111096 \nu^{8} + 8892 \nu^{7} + 123734 \nu^{6} + 134994 \nu^{5} + 101866 \nu^{4} + 45050 \nu^{3} + 8050 \nu^{2} - 572 \nu + 1412\)\()/969\)
\(\beta_{3}\)\(=\)\((\)\(-713760 \nu^{15} + 2873130 \nu^{14} - 2934707 \nu^{13} + 17216192 \nu^{12} - 74582450 \nu^{11} + 141664463 \nu^{10} - 226302900 \nu^{9} + 172573690 \nu^{8} - 19779888 \nu^{7} - 183883199 \nu^{6} - 193251327 \nu^{5} - 157627568 \nu^{4} - 67531547 \nu^{3} - 10955821 \nu^{2} + 433452 \nu - 2172105\)\()/1095711\)
\(\beta_{4}\)\(=\)\((\)\(41571440 \nu^{15} - 185423177 \nu^{14} + 254219263 \nu^{13} - 1125645507 \nu^{12} + 4853270456 \nu^{11} - 10441949317 \nu^{10} + 18054484920 \nu^{9} - 18663563720 \nu^{8} + 10686848023 \nu^{7} + 4323827851 \nu^{6} + 10838029165 \nu^{5} + 4051741093 \nu^{4} + 3030984061 \nu^{3} - 282979166 \nu^{2} + 21421645 \nu + 5669660\)\()/55881261\)
\(\beta_{5}\)\(=\)\((\)\(16331260 \nu^{15} - 74991976 \nu^{14} + 108507436 \nu^{13} - 450127912 \nu^{12} + 1954868524 \nu^{11} - 4322761180 \nu^{10} + 7497148468 \nu^{9} - 7919869405 \nu^{8} + 4541015683 \nu^{7} + 1915043380 \nu^{6} + 3495942116 \nu^{5} + 1025333980 \nu^{4} + 934847368 \nu^{3} - 77706604 \nu^{2} + 177822799 \nu + 53802093\)\()/18627087\)
\(\beta_{6}\)\(=\)\((\)\(-16361188 \nu^{15} + 72139004 \nu^{14} - 94401360 \nu^{13} + 428892200 \nu^{12} - 1874038268 \nu^{11} + 3958996896 \nu^{10} - 6661892204 \nu^{9} + 6457440527 \nu^{8} - 2929890732 \nu^{7} - 2870483616 \nu^{6} - 3833618044 \nu^{5} - 1548767784 \nu^{4} - 916399440 \nu^{3} + 54074084 \nu^{2} - 25648164 \nu - 32692448\)\()/18627087\)
\(\beta_{7}\)\(=\)\((\)\(-16819000 \nu^{15} + 72297973 \nu^{14} - 89870960 \nu^{13} + 435551452 \nu^{12} - 1887492644 \nu^{11} + 3888935115 \nu^{10} - 6539006616 \nu^{9} + 6229582792 \nu^{8} - 2911659902 \nu^{7} - 2510098475 \nu^{6} - 4825415772 \nu^{5} - 2001171996 \nu^{4} - 1263419608 \nu^{3} + 101493346 \nu^{2} - 27011014 \nu - 19377740\)\()/18627087\)
\(\beta_{8}\)\(=\)\((\)\(-57962620 \nu^{15} + 244034719 \nu^{14} - 284439440 \nu^{13} + 1456967568 \nu^{12} - 6343717582 \nu^{11} + 12731116736 \nu^{10} - 20919252180 \nu^{9} + 18436863304 \nu^{8} - 6312521300 \nu^{7} - 11645397440 \nu^{6} - 16068699614 \nu^{5} - 8122374800 \nu^{4} - 4493625488 \nu^{3} - 627232847 \nu^{2} - 154199804 \nu - 168001120\)\()/55881261\)
\(\beta_{9}\)\(=\)\((\)\(-23022602 \nu^{15} + 100944114 \nu^{14} - 130859040 \nu^{13} + 602140523 \nu^{12} - 2623735034 \nu^{11} + 5518742544 \nu^{10} - 9288708034 \nu^{9} + 8944701020 \nu^{8} - 4051325654 \nu^{7} - 4009850208 \nu^{6} - 5588137678 \nu^{5} - 2281853173 \nu^{4} - 1667581296 \nu^{3} - 87881958 \nu^{2} - 141488302 \nu - 73290672\)\()/18627087\)
\(\beta_{10}\)\(=\)\((\)\(81436312 \nu^{15} - 349229857 \nu^{14} + 425779916 \nu^{13} - 2076724695 \nu^{12} + 9074758729 \nu^{11} - 18573123677 \nu^{10} + 30748771548 \nu^{9} - 28193204368 \nu^{8} + 10926609317 \nu^{7} + 15431269295 \nu^{6} + 21654724265 \nu^{5} + 9289838258 \nu^{4} + 5192070428 \nu^{3} - 313967458 \nu^{2} + 174107807 \nu + 120487072\)\()/55881261\)
\(\beta_{11}\)\(=\)\((\)\(81657397 \nu^{15} - 361433545 \nu^{14} + 473165969 \nu^{13} - 2128224750 \nu^{12} + 9356454580 \nu^{11} - 19798644830 \nu^{10} + 33056593947 \nu^{9} - 31549953604 \nu^{8} + 12982980755 \nu^{7} + 16563936710 \nu^{6} + 17306567999 \nu^{5} + 7099394417 \nu^{4} + 3743469443 \nu^{3} - 126506317 \nu^{2} + 151097009 \nu + 304152748\)\()/55881261\)
\(\beta_{12}\)\(=\)\((\)\(28053456 \nu^{15} - 123097333 \nu^{14} + 159579226 \nu^{13} - 732793144 \nu^{12} + 3197206163 \nu^{11} - 6727519330 \nu^{10} + 11304474312 \nu^{9} - 10850615079 \nu^{8} + 4828872624 \nu^{7} + 5010528130 \nu^{6} + 6755327387 \nu^{5} + 2636843224 \nu^{4} + 2078302474 \nu^{3} + 108463237 \nu^{2} + 169601736 \nu + 90494094\)\()/18627087\)
\(\beta_{13}\)\(=\)\((\)\(-29904546 \nu^{15} + 129757565 \nu^{14} - 161347800 \nu^{13} + 762815256 \nu^{12} - 3357607152 \nu^{11} + 6943520122 \nu^{10} - 11435934986 \nu^{9} + 10437065588 \nu^{8} - 3662556090 \nu^{7} - 6508066522 \nu^{6} - 6938376848 \nu^{5} - 2996702536 \nu^{4} - 1496150424 \nu^{3} + 50381843 \nu^{2} - 69368690 \nu - 107903336\)\()/18627087\)
\(\beta_{14}\)\(=\)\((\)\(95630831 \nu^{15} - 411561677 \nu^{14} + 503317981 \nu^{13} - 2431707663 \nu^{12} + 10669553489 \nu^{11} - 21886755922 \nu^{10} + 36060331821 \nu^{9} - 32758079528 \nu^{8} + 11742438997 \nu^{7} + 19770274810 \nu^{6} + 23948074300 \nu^{5} + 10370368018 \nu^{4} + 5444666191 \nu^{3} - 258493073 \nu^{2} + 220080667 \nu + 371054036\)\()/55881261\)
\(\beta_{15}\)\(=\)\((\)\(-41682527 \nu^{15} + 175603960 \nu^{14} - 205133396 \nu^{13} + 1048417004 \nu^{12} - 4563459364 \nu^{11} + 9167107496 \nu^{10} - 15069743507 \nu^{9} + 13276126963 \nu^{8} - 4498087181 \nu^{7} - 8537548760 \nu^{6} - 11303294252 \nu^{5} - 5984840948 \nu^{4} - 3300359084 \nu^{3} - 432772592 \nu^{2} - 114281729 \nu - 123561984\)\()/18627087\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - 3 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \beta_{2} + 3 \beta_{1} + 3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(6 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} + 2 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} - 9 \beta_{8} + 6 \beta_{7} + 6 \beta_{6} + 4 \beta_{4} + 4 \beta_{2}\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(-6 \beta_{15} - 11 \beta_{14} - 9 \beta_{13} + 6 \beta_{12} - 5 \beta_{11} + 11 \beta_{10} + 14 \beta_{9} + 6 \beta_{8} + 9 \beta_{7} + 6 \beta_{5} + 5 \beta_{4} - 3 \beta_{3} - 7 \beta_{2} + 6 \beta_{1} + 33\)\()/6\)
\(\nu^{4}\)\(=\)\(2 \beta_{12} + 7 \beta_{9} + 10 \beta_{5} + 2 \beta_{3} + 7 \beta_{2} + 12 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(66 \beta_{15} + 37 \beta_{14} + 75 \beta_{13} - 15 \beta_{12} + 103 \beta_{11} + 140 \beta_{10} - 49 \beta_{9} - 78 \beta_{8} + 150 \beta_{7} + 303 \beta_{6} + 140 \beta_{4} + 30 \beta_{3} + 98 \beta_{2}\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(-402 \beta_{15} - 230 \beta_{14} - 171 \beta_{13} + 96 \beta_{12} - 86 \beta_{11} + 230 \beta_{10} + 452 \beta_{9} + 513 \beta_{8} + 171 \beta_{7} + 402 \beta_{5} + 86 \beta_{4} - 48 \beta_{3} - 226 \beta_{2} + 513 \beta_{1} + 678\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(509 \beta_{14} + 504 \beta_{13} + 204 \beta_{12} + 509 \beta_{11} + 103 \beta_{10} + 895 \beta_{9} + 252 \beta_{7} + 1173 \beta_{6} + 1527 \beta_{5} + 406 \beta_{4} + 204 \beta_{3} + 895 \beta_{2} + 1929 \beta_{1} - 1173\)\()/6\)
\(\nu^{8}\)\(=\)\(152 \beta_{14} + 336 \beta_{13} + 500 \beta_{11} + 652 \beta_{10} + 672 \beta_{7} + 1457 \beta_{6} + 652 \beta_{4}\)
\(\nu^{9}\)\(=\)\((\)\(-13263 \beta_{15} - 3406 \beta_{14} - 2268 \beta_{13} + 3624 \beta_{12} - 1015 \beta_{11} + 3406 \beta_{10} + 15682 \beta_{9} + 16719 \beta_{8} + 2268 \beta_{7} + 13263 \beta_{5} + 1015 \beta_{4} - 1812 \beta_{3} - 7841 \beta_{2} + 16719 \beta_{1} + 9915\)\()/6\)
\(\nu^{10}\)\(=\)\((\)\(24068 \beta_{14} + 25206 \beta_{13} + 4080 \beta_{12} + 24068 \beta_{11} + 5914 \beta_{10} + 17698 \beta_{9} + 12603 \beta_{7} + 53094 \beta_{6} + 29982 \beta_{5} + 18154 \beta_{4} + 4080 \beta_{3} + 17698 \beta_{2} + 37809 \beta_{1} - 53094\)\()/6\)
\(\nu^{11}\)\(=\)\((\)\(-47658 \beta_{15} + 22597 \beta_{14} + 48477 \beta_{13} + 6711 \beta_{12} + 70255 \beta_{11} + 92852 \beta_{10} + 28511 \beta_{9} + 59898 \beta_{8} + 96954 \beta_{7} + 205329 \beta_{6} + 92852 \beta_{4} - 13422 \beta_{3} - 57022 \beta_{2}\)\()/6\)
\(\nu^{12}\)\(=\)\(-61406 \beta_{15} + 17036 \beta_{12} + 73038 \beta_{9} + 77292 \beta_{8} + 61406 \beta_{5} - 8518 \beta_{3} - 36519 \beta_{2} + 77292 \beta_{1}\)
\(\nu^{13}\)\(=\)\((\)\(809128 \beta_{14} + 844350 \beta_{13} + 57819 \beta_{12} + 809128 \beta_{11} + 196517 \beta_{10} + 247625 \beta_{9} + 422175 \beta_{7} + 1790175 \beta_{6} + 416094 \beta_{5} + 612611 \beta_{4} + 57819 \beta_{3} + 247625 \beta_{2} + 523650 \beta_{1} - 1790175\)\()/6\)
\(\nu^{14}\)\(=\)\((\)\(-2272170 \beta_{15} + 444142 \beta_{14} + 953907 \beta_{13} + 313248 \beta_{12} + 1383886 \beta_{11} + 1828028 \beta_{10} + 1348034 \beta_{9} + 2861721 \beta_{8} + 1907814 \beta_{7} + 4044102 \beta_{6} + 1828028 \beta_{4} - 626496 \beta_{3} - 2696068 \beta_{2}\)\()/6\)
\(\nu^{15}\)\(=\)\((\)\(-8761899 \beta_{15} + 2213258 \beta_{14} + 1522584 \beta_{13} + 2418672 \beta_{12} + 707375 \beta_{11} - 2213258 \beta_{10} + 10401214 \beta_{9} + 11034069 \beta_{8} - 1522584 \beta_{7} + 8761899 \beta_{5} - 707375 \beta_{4} - 1209336 \beta_{3} - 5200607 \beta_{2} + 11034069 \beta_{1} - 6466317\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(-1\) \(\beta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
−1.79700 2.34189i
−0.335868 0.0442178i
−0.472726 + 0.362736i
0.219056 1.66389i
1.33145 1.02165i
−0.0777751 + 0.590761i
0.206228 + 0.268761i
2.92664 + 0.385299i
−1.79700 + 2.34189i
−0.335868 + 0.0442178i
−0.472726 0.362736i
0.219056 + 1.66389i
1.33145 + 1.02165i
−0.0777751 0.590761i
0.206228 0.268761i
2.92664 0.385299i
0 −2.63287 1.52009i 0 −0.866025 1.50000i 0 2.14973 1.54230i 0 3.12132 + 5.40629i 0
703.2 0 −2.63287 1.52009i 0 0.866025 + 1.50000i 0 −2.14973 + 1.54230i 0 3.12132 + 5.40629i 0
703.3 0 −0.753671 0.435132i 0 −0.866025 1.50000i 0 −0.615370 2.57319i 0 −1.12132 1.94218i 0
703.4 0 −0.753671 0.435132i 0 0.866025 + 1.50000i 0 0.615370 + 2.57319i 0 −1.12132 1.94218i 0
703.5 0 0.753671 + 0.435132i 0 −0.866025 1.50000i 0 0.615370 + 2.57319i 0 −1.12132 1.94218i 0
703.6 0 0.753671 + 0.435132i 0 0.866025 + 1.50000i 0 −0.615370 2.57319i 0 −1.12132 1.94218i 0
703.7 0 2.63287 + 1.52009i 0 −0.866025 1.50000i 0 −2.14973 + 1.54230i 0 3.12132 + 5.40629i 0
703.8 0 2.63287 + 1.52009i 0 0.866025 + 1.50000i 0 2.14973 1.54230i 0 3.12132 + 5.40629i 0
831.1 0 −2.63287 + 1.52009i 0 −0.866025 + 1.50000i 0 2.14973 + 1.54230i 0 3.12132 5.40629i 0
831.2 0 −2.63287 + 1.52009i 0 0.866025 1.50000i 0 −2.14973 1.54230i 0 3.12132 5.40629i 0
831.3 0 −0.753671 + 0.435132i 0 −0.866025 + 1.50000i 0 −0.615370 + 2.57319i 0 −1.12132 + 1.94218i 0
831.4 0 −0.753671 + 0.435132i 0 0.866025 1.50000i 0 0.615370 2.57319i 0 −1.12132 + 1.94218i 0
831.5 0 0.753671 0.435132i 0 −0.866025 + 1.50000i 0 0.615370 2.57319i 0 −1.12132 + 1.94218i 0
831.6 0 0.753671 0.435132i 0 0.866025 1.50000i 0 −0.615370 + 2.57319i 0 −1.12132 + 1.94218i 0
831.7 0 2.63287 1.52009i 0 −0.866025 + 1.50000i 0 −2.14973 1.54230i 0 3.12132 5.40629i 0
831.8 0 2.63287 1.52009i 0 0.866025 1.50000i 0 2.14973 + 1.54230i 0 3.12132 5.40629i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 831.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.f even 6 1 inner
56.j odd 6 1 inner
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.q.c 16
4.b odd 2 1 inner 896.2.q.c 16
7.d odd 6 1 inner 896.2.q.c 16
8.b even 2 1 inner 896.2.q.c 16
8.d odd 2 1 inner 896.2.q.c 16
28.f even 6 1 inner 896.2.q.c 16
56.j odd 6 1 inner 896.2.q.c 16
56.m even 6 1 inner 896.2.q.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.q.c 16 1.a even 1 1 trivial
896.2.q.c 16 4.b odd 2 1 inner
896.2.q.c 16 7.d odd 6 1 inner
896.2.q.c 16 8.b even 2 1 inner
896.2.q.c 16 8.d odd 2 1 inner
896.2.q.c 16 28.f even 6 1 inner
896.2.q.c 16 56.j odd 6 1 inner
896.2.q.c 16 56.m even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 10 T_{3}^{6} + 93 T_{3}^{4} - 70 T_{3}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 49 - 70 T^{2} + 93 T^{4} - 10 T^{6} + T^{8} )^{2} \)
$5$ \( ( 9 + 3 T^{2} + T^{4} )^{4} \)
$7$ \( ( 2401 + 392 T^{2} + 42 T^{4} + 8 T^{6} + T^{8} )^{2} \)
$11$ \( ( 3969 + 1134 T^{2} + 261 T^{4} + 18 T^{6} + T^{8} )^{2} \)
$13$ \( ( 36 - 36 T^{2} + T^{4} )^{4} \)
$17$ \( ( 9 + 18 T + 9 T^{2} - 6 T^{3} + T^{4} )^{4} \)
$19$ \( ( 321489 - 30618 T^{2} + 2349 T^{4} - 54 T^{6} + T^{8} )^{2} \)
$23$ \( ( 3969 - 1134 T^{2} + 261 T^{4} - 18 T^{6} + T^{8} )^{2} \)
$29$ \( ( 1156 + 76 T^{2} + T^{4} )^{4} \)
$31$ \( ( 13712209 + 451766 T^{2} + 11181 T^{4} + 122 T^{6} + T^{8} )^{2} \)
$37$ \( ( 6561 - 4374 T^{2} + 2835 T^{4} - 54 T^{6} + T^{8} )^{2} \)
$41$ \( ( 54 + T^{2} )^{8} \)
$43$ \( ( 4032 - 144 T^{2} + T^{4} )^{4} \)
$47$ \( ( 321489 + 51030 T^{2} + 7533 T^{4} + 90 T^{6} + T^{8} )^{2} \)
$53$ \( ( 2401 - 1078 T^{2} + 435 T^{4} - 22 T^{6} + T^{8} )^{2} \)
$59$ \( ( 49 - 70 T^{2} + 93 T^{4} - 10 T^{6} + T^{8} )^{2} \)
$61$ \( ( 531441 + 118098 T^{2} + 25515 T^{4} + 162 T^{6} + T^{8} )^{2} \)
$67$ \( ( 9529569 + 388962 T^{2} + 12789 T^{4} + 126 T^{6} + T^{8} )^{2} \)
$71$ \( ( 16128 + 288 T^{2} + T^{4} )^{4} \)
$73$ \( ( 2601 + 1530 T + 351 T^{2} + 30 T^{3} + T^{4} )^{4} \)
$79$ \( ( 9529569 - 351918 T^{2} + 9909 T^{4} - 114 T^{6} + T^{8} )^{2} \)
$83$ \( ( 1792 + 160 T^{2} + T^{4} )^{4} \)
$89$ \( ( 8649 + 558 T - 81 T^{2} - 6 T^{3} + T^{4} )^{4} \)
$97$ \( ( 36 + 36 T^{2} + T^{4} )^{4} \)
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