Properties

Label 966.2.q.d
Level $966$
Weight $2$
Character orbit 966.q
Analytic conductor $7.714$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(85,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.q (of order \(11\), degree \(10\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 6 x^{19} + 10 x^{18} - 14 x^{17} + 77 x^{16} + 12 x^{15} - 226 x^{14} - 793 x^{13} + 690 x^{12} + 6105 x^{11} + 13883 x^{10} + 24365 x^{9} + 36461 x^{8} + 41912 x^{7} + \cdots + 1849 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} - \beta_{19} q^{3} - \beta_{15} q^{4} + ( - \beta_{19} + \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{10} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{5} + \beta_{2} q^{6} + \beta_{16} q^{7} - \beta_{18} q^{8} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} - \beta_{19} q^{3} - \beta_{15} q^{4} + ( - \beta_{19} + \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{10} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{5} + \beta_{2} q^{6} + \beta_{16} q^{7} - \beta_{18} q^{8} - \beta_{3} q^{9} + ( - \beta_{15} + \beta_{8} - \beta_{7}) q^{10} + ( - \beta_{19} - \beta_{16} - \beta_{14} + \beta_{12} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{3} + \beta_{2} + \cdots - 1) q^{11}+ \cdots + (\beta_{19} + \beta_{16} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} - \beta_{5} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 12 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 12 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} - 2 q^{9} - 10 q^{10} - 7 q^{11} + 2 q^{12} - 8 q^{13} - 2 q^{14} - q^{15} - 2 q^{16} + 13 q^{17} - 2 q^{18} - 8 q^{19} + q^{20} + 2 q^{21} + 4 q^{22} - 20 q^{24} + 16 q^{25} - 8 q^{26} + 2 q^{27} - 2 q^{28} + 3 q^{29} + 10 q^{30} + 9 q^{31} - 2 q^{32} - 4 q^{33} - 9 q^{34} + q^{35} - 2 q^{36} - q^{37} + 3 q^{38} + 8 q^{39} + q^{40} + 10 q^{41} + 2 q^{42} - 15 q^{43} - 7 q^{44} - 10 q^{45} - 28 q^{47} + 2 q^{48} - 2 q^{49} + 5 q^{50} - 2 q^{51} + 3 q^{52} + 38 q^{53} + 2 q^{54} - 8 q^{55} - 2 q^{56} - 14 q^{57} - 19 q^{58} - 10 q^{59} - 12 q^{60} - 6 q^{61} + 20 q^{62} - 2 q^{63} - 2 q^{64} - 36 q^{65} - 4 q^{66} + 36 q^{67} - 20 q^{68} + 11 q^{69} - 10 q^{70} - q^{71} - 2 q^{72} + 65 q^{73} - q^{74} - 5 q^{75} + 14 q^{76} + 4 q^{77} - 14 q^{78} + 10 q^{79} + q^{80} - 2 q^{81} + 10 q^{82} - 14 q^{83} + 2 q^{84} - 5 q^{85} + 51 q^{86} - 3 q^{87} - 7 q^{88} - 18 q^{89} + q^{90} - 30 q^{91} - 11 q^{92} + 2 q^{93} + 38 q^{94} - 73 q^{95} + 2 q^{96} - 20 q^{97} - 2 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6 x^{19} + 10 x^{18} - 14 x^{17} + 77 x^{16} + 12 x^{15} - 226 x^{14} - 793 x^{13} + 690 x^{12} + 6105 x^{11} + 13883 x^{10} + 24365 x^{9} + 36461 x^{8} + 41912 x^{7} + \cdots + 1849 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 68\!\cdots\!47 \nu^{19} + \cdots + 41\!\cdots\!66 ) / 57\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23\!\cdots\!05 \nu^{19} + \cdots + 11\!\cdots\!87 ) / 57\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 60\!\cdots\!14 \nu^{19} + \cdots - 50\!\cdots\!27 ) / 13\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\!\cdots\!75 \nu^{19} + \cdots - 31\!\cdots\!72 ) / 57\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!48 \nu^{19} + \cdots - 32\!\cdots\!11 ) / 57\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34\!\cdots\!90 \nu^{19} + \cdots + 17\!\cdots\!49 ) / 57\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 83\!\cdots\!51 \nu^{19} + \cdots + 33\!\cdots\!22 ) / 13\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 36\!\cdots\!04 \nu^{19} + \cdots + 12\!\cdots\!09 ) / 57\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!03 \nu^{19} + \cdots + 79\!\cdots\!84 ) / 13\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 53\!\cdots\!45 \nu^{19} + \cdots + 15\!\cdots\!03 ) / 58\!\cdots\!77 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!75 \nu^{19} + \cdots + 46\!\cdots\!54 ) / 13\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15\!\cdots\!00 \nu^{19} + \cdots - 29\!\cdots\!21 ) / 13\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 17\!\cdots\!99 \nu^{19} + \cdots - 41\!\cdots\!67 ) / 13\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 76\!\cdots\!54 \nu^{19} + \cdots + 38\!\cdots\!59 ) / 57\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 94\!\cdots\!52 \nu^{19} + \cdots + 14\!\cdots\!27 ) / 57\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 22\!\cdots\!79 \nu^{19} + \cdots + 58\!\cdots\!09 ) / 13\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 10\!\cdots\!78 \nu^{19} + \cdots - 96\!\cdots\!46 ) / 57\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 11\!\cdots\!89 \nu^{19} + \cdots - 19\!\cdots\!17 ) / 57\!\cdots\!53 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2 \beta_{19} - \beta_{17} - \beta_{15} + \beta_{14} - \beta_{12} + \beta_{10} + 3 \beta_{9} + \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 11 \beta_{19} + \beta_{18} - 7 \beta_{17} + \beta_{16} - 8 \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + 4 \beta_{9} + 7 \beta_{8} - 8 \beta_{7} - 8 \beta_{6} - 8 \beta_{5} + \beta_{3} - 12 \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 29 \beta_{19} + \beta_{18} - 14 \beta_{17} - \beta_{16} - 15 \beta_{15} - 6 \beta_{14} - 14 \beta_{13} - \beta_{12} + 6 \beta_{11} + 14 \beta_{9} + 14 \beta_{8} - 15 \beta_{7} - 15 \beta_{6} - 25 \beta_{5} - 10 \beta_{4} - 39 \beta_{2} + 10 \beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 64 \beta_{19} + 7 \beta_{18} - 17 \beta_{17} - 21 \beta_{16} + 10 \beta_{15} - 8 \beta_{14} - 69 \beta_{13} + 8 \beta_{12} - 17 \beta_{11} + 35 \beta_{9} + 36 \beta_{8} - 3 \beta_{7} - 11 \beta_{6} - 46 \beta_{5} - 42 \beta_{4} + 8 \beta_{3} - 64 \beta_{2} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 149 \beta_{19} + 148 \beta_{18} - 133 \beta_{16} + 148 \beta_{15} + 13 \beta_{14} - 183 \beta_{13} + 34 \beta_{12} - 134 \beta_{11} - 17 \beta_{10} + 134 \beta_{8} + 133 \beta_{7} - 70 \beta_{5} - 183 \beta_{4} + 111 \beta_{3} - 70 \beta_{2} + \cdots - 111 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 747 \beta_{18} + 240 \beta_{17} - 587 \beta_{16} + 915 \beta_{15} - 535 \beta_{13} + 17 \beta_{12} - 535 \beta_{11} - 295 \beta_{10} - 404 \beta_{9} + 278 \beta_{8} + 992 \beta_{7} + 245 \beta_{6} - 765 \beta_{4} + 404 \beta_{3} + \cdots - 832 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2565 \beta_{19} + 2871 \beta_{18} + 1956 \beta_{17} - 2343 \beta_{16} + 5697 \beta_{15} - 198 \beta_{14} - 1773 \beta_{13} - 2367 \beta_{11} - 1773 \beta_{10} - 2343 \beta_{9} + 307 \beta_{8} + 5697 \beta_{7} + \cdots - 5436 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 16436 \beta_{19} + 11137 \beta_{18} + 8882 \beta_{17} - 8052 \beta_{16} + 26583 \beta_{15} - 913 \beta_{14} - 4395 \beta_{13} + 78 \beta_{12} - 8882 \beta_{11} - 7579 \beta_{10} - 11137 \beta_{9} + 913 \beta_{8} + \cdots - 26583 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 75250 \beta_{19} + 38542 \beta_{18} + 30704 \beta_{17} - 21783 \beta_{16} + 101258 \beta_{15} - 8988 \beta_{14} - 8988 \beta_{13} + 1806 \beta_{12} - 23522 \beta_{11} - 32510 \beta_{10} - 47791 \beta_{9} + \cdots - 105731 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 313555 \beta_{19} + 102085 \beta_{18} + 103091 \beta_{17} - 40451 \beta_{16} + 360429 \beta_{15} - 67186 \beta_{14} - 23525 \beta_{13} + 23525 \beta_{12} - 41193 \beta_{11} - 131642 \beta_{10} + \cdots - 395560 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1201324 \beta_{19} + 185477 \beta_{18} + 338433 \beta_{17} + 1201324 \beta_{15} - 338433 \beta_{14} - 57090 \beta_{13} + 168682 \beta_{12} - 458450 \beta_{10} - 621768 \beta_{9} + \cdots - 1418651 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4147140 \beta_{19} + 999504 \beta_{17} + 521377 \beta_{16} + 3466498 \beta_{15} - 1408112 \beta_{14} + 878365 \beta_{12} + 529747 \beta_{11} - 1408112 \beta_{10} - 2040815 \beta_{9} + \cdots - 4599753 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 13098153 \beta_{19} - 2467606 \beta_{18} + 2727662 \beta_{17} + 3555288 \beta_{16} + 8021991 \beta_{15} - 5294329 \beta_{14} + 922370 \beta_{13} + 3798522 \beta_{12} + \cdots - 13098153 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 37638988 \beta_{19} - 17315737 \beta_{18} + 7073276 \beta_{17} + 17315737 \beta_{16} + 10881693 \beta_{15} - 17414930 \beta_{14} + 7073276 \beta_{13} + 14334018 \beta_{12} + \cdots - 31073356 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 90164625 \beta_{19} - 86028536 \beta_{18} + 14293796 \beta_{17} + 72372259 \beta_{16} - 30137932 \beta_{15} - 46873408 \beta_{14} + 41640299 \beta_{13} + 46873408 \beta_{12} + \cdots - 42234327 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 125514925 \beta_{19} - 365683870 \beta_{18} + 276298682 \beta_{16} - 365683870 \beta_{15} - 89053662 \beta_{14} + 209154735 \beta_{13} + 126882870 \beta_{12} + \cdots + 135671513 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 337802687 \beta_{19} - 1421774709 \beta_{18} - 214501895 \beta_{17} + 981157816 \beta_{16} - 2180733691 \beta_{15} + 915891844 \beta_{13} + 243384338 \beta_{12} + \cdots + 1563814066 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 4141128378 \beta_{19} - 5132324382 \beta_{18} - 1606872268 \beta_{17} + 3236739490 \beta_{16} - 10461611385 \beta_{15} + 1170889285 \beta_{14} + 3591921664 \beta_{13} + \cdots + 9273452760 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{7}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1
0.474546 1.03911i
−0.511167 + 1.11930i
3.56300 + 1.04619i
−1.42457 0.418291i
0.287778 0.332113i
−1.18676 + 1.36960i
2.68238 1.72386i
−0.665158 + 0.427471i
0.287778 + 0.332113i
−1.18676 1.36960i
0.150891 + 1.04947i
−0.370938 2.57993i
2.68238 + 1.72386i
−0.665158 0.427471i
3.56300 1.04619i
−1.42457 + 0.418291i
0.474546 + 1.03911i
−0.511167 1.11930i
0.150891 1.04947i
−0.370938 + 2.57993i
−0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i −0.129407 0.283361i 0.142315 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i 0.298893 + 0.0877630i
85.2 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i 0.856306 + 1.87505i 0.142315 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −1.97783 0.580743i
127.1 0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i −1.72174 + 0.505549i −0.415415 0.909632i −0.654861 0.755750i −0.142315 0.989821i −0.959493 0.281733i −1.17510 + 1.35614i
127.2 0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i 3.26582 0.958931i −0.415415 0.909632i −0.654861 0.755750i −0.142315 0.989821i −0.959493 0.281733i 2.22895 2.57234i
169.1 −0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i 0.569907 + 0.657708i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i −0.732120 + 0.470505i
169.2 −0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i 2.04445 + 2.35942i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i −2.62636 + 1.68786i
211.1 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i −1.26697 0.814230i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.214333 1.49072i
211.2 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i 2.08057 + 1.33710i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i 0.841254 0.540641i −0.351971 + 2.44801i
463.1 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.569907 0.657708i 0.959493 + 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i −0.732120 0.470505i
463.2 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 2.04445 2.35942i 0.959493 + 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i −2.62636 1.68786i
547.1 −0.959493 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i −0.110384 + 0.767734i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i −0.142315 0.989821i 0.322208 0.705537i
547.2 −0.959493 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i 0.411445 2.86166i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i −0.142315 0.989821i −1.20100 + 2.62983i
673.1 0.415415 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i −1.26697 + 0.814230i 0.654861 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.214333 + 1.49072i
673.2 0.415415 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i 2.08057 1.33710i 0.654861 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.351971 2.44801i
715.1 0.841254 + 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i −1.72174 0.505549i −0.415415 + 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i −1.17510 1.35614i
715.2 0.841254 + 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i 3.26582 + 0.958931i −0.415415 + 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i 2.22895 + 2.57234i
841.1 −0.654861 0.755750i −0.841254 0.540641i −0.142315 + 0.989821i −0.129407 + 0.283361i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i 0.298893 0.0877630i
841.2 −0.654861 0.755750i −0.841254 0.540641i −0.142315 + 0.989821i 0.856306 1.87505i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i −1.97783 + 0.580743i
883.1 −0.959493 + 0.281733i 0.654861 + 0.755750i 0.841254 0.540641i −0.110384 0.767734i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.322208 + 0.705537i
883.2 −0.959493 + 0.281733i 0.654861 + 0.755750i 0.841254 0.540641i 0.411445 + 2.86166i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i −1.20100 2.62983i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.q.d 20
23.c even 11 1 inner 966.2.q.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.d 20 1.a even 1 1 trivial
966.2.q.d 20 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} - 12 T_{5}^{19} + 69 T_{5}^{18} - 236 T_{5}^{17} + 462 T_{5}^{16} - 186 T_{5}^{15} - 1695 T_{5}^{14} + 4623 T_{5}^{13} - 1380 T_{5}^{12} - 14652 T_{5}^{11} + 30119 T_{5}^{10} - 2937 T_{5}^{9} - 46066 T_{5}^{8} + \cdots + 7921 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} - 12 T^{19} + 69 T^{18} + \cdots + 7921 \) Copy content Toggle raw display
$7$ \( (T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + 7 T^{19} + 60 T^{18} + 277 T^{17} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{20} + 8 T^{19} + 7 T^{18} - 246 T^{17} + \cdots + 529 \) Copy content Toggle raw display
$17$ \( T^{20} - 13 T^{19} + 39 T^{18} + \cdots + 19562929 \) Copy content Toggle raw display
$19$ \( T^{20} + 8 T^{19} + 81 T^{18} + \cdots + 591361 \) Copy content Toggle raw display
$23$ \( T^{20} - 32 T^{18} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} - 3 T^{19} + 29 T^{18} + \cdots + 4280761 \) Copy content Toggle raw display
$31$ \( T^{20} - 9 T^{19} + 95 T^{18} + \cdots + 113401201 \) Copy content Toggle raw display
$37$ \( T^{20} + T^{19} + 9 T^{18} + \cdots + 73462139521 \) Copy content Toggle raw display
$41$ \( T^{20} - 10 T^{19} + 98 T^{18} + \cdots + 529 \) Copy content Toggle raw display
$43$ \( T^{20} + 15 T^{19} + 158 T^{18} + \cdots + 982081 \) Copy content Toggle raw display
$47$ \( (T^{10} + 14 T^{9} - 189 T^{8} + \cdots + 2025893)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} - 38 T^{19} + \cdots + 47475035185369 \) Copy content Toggle raw display
$59$ \( T^{20} + 10 T^{19} + \cdots + 195345436441 \) Copy content Toggle raw display
$61$ \( T^{20} + 6 T^{19} + \cdots + 381272905729 \) Copy content Toggle raw display
$67$ \( T^{20} - 36 T^{19} + \cdots + 43\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( T^{20} + T^{19} + \cdots + 603311046361 \) Copy content Toggle raw display
$73$ \( T^{20} - 65 T^{19} + \cdots + 29094577935721 \) Copy content Toggle raw display
$79$ \( T^{20} - 10 T^{19} + \cdots + 2135456521 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 272358943896601 \) Copy content Toggle raw display
$89$ \( T^{20} + 18 T^{19} + \cdots + 21708582607009 \) Copy content Toggle raw display
$97$ \( (T^{10} + 10 T^{9} + 243 T^{8} + \cdots + 3659569)^{2} \) Copy content Toggle raw display
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