Properties

Label 930.2.n.h
Level $930$
Weight $2$
Character orbit 930.n
Analytic conductor $7.426$
Analytic rank $0$
Dimension $16$
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] = [930,2,Mod(481,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.481");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.n (of order \(5\), degree \(4\), 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: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 5 x^{14} - 60 x^{13} + 480 x^{12} - 1202 x^{11} - 147 x^{10} - 2185 x^{9} + 62400 x^{8} - 289745 x^{7} + 699483 x^{6} - 1042897 x^{5} + 1014415 x^{4} + \cdots + 12851 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + \beta_{3} q^{3} + (\beta_{7} + \beta_{6} - \beta_{3} - 1) q^{4} - q^{5} - q^{6} + ( - \beta_{11} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{4}) q^{7} + \beta_{6} q^{8} - \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + \beta_{3} q^{3} + (\beta_{7} + \beta_{6} - \beta_{3} - 1) q^{4} - q^{5} - q^{6} + ( - \beta_{11} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{4}) q^{7} + \beta_{6} q^{8} - \beta_{6} q^{9} - \beta_{7} q^{10} + ( - \beta_{14} + \beta_{11} - \beta_{9} - \beta_{4}) q^{11} - \beta_{7} q^{12} + ( - \beta_{8} + \beta_{5} - \beta_{2} + \beta_1) q^{13} + ( - \beta_{11} - \beta_{7} - \beta_{6} + 1) q^{14} - \beta_{3} q^{15} + \beta_{3} q^{16} + (\beta_{10} + 2 \beta_{8} - \beta_{7} - \beta_{5} - \beta_{2} - \beta_1) q^{17} - \beta_{3} q^{18} + (\beta_{12} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{2} + \beta_1 - 1) q^{19} + ( - \beta_{7} - \beta_{6} + \beta_{3} + 1) q^{20} + (\beta_{12} - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{3}) q^{21} + (\beta_{11} + \beta_{10}) q^{22} + ( - \beta_{11} + \beta_{10} - \beta_{9}) q^{23} + ( - \beta_{7} - \beta_{6} + \beta_{3} + 1) q^{24} + q^{25} + ( - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_1) q^{26} + (\beta_{7} + \beta_{6} - \beta_{3} - 1) q^{27} + ( - \beta_{6} - \beta_{4} + 1) q^{28} + (\beta_{15} + 3 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{29} + q^{30} + (\beta_{14} - \beta_{9} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{31} - q^{32} + (\beta_{15} - \beta_{12} + \beta_{9}) q^{33} + ( - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{10} - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{3} - \beta_1 + 2) q^{34} + (\beta_{11} - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{4}) q^{35} + q^{36} + ( - \beta_{14} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{6} - \beta_{4} + 2 \beta_{2} - 1) q^{37} + (\beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{38} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}) q^{39} - \beta_{6} q^{40} + ( - \beta_{12} - \beta_{11} + \beta_{9} - \beta_{8} - 4 \beta_{7} + 2 \beta_{5} + \beta_{4} + \beta_1) q^{41} + (\beta_{11} - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{4}) q^{42} + ( - \beta_{14} + \beta_{13} + 2 \beta_{12} + \beta_{11} - 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{4} + \cdots - \beta_1) q^{43}+ \cdots + (\beta_{13} + \beta_{12} + \beta_{7} - \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 4 q^{3} - 4 q^{4} - 16 q^{5} - 16 q^{6} - 7 q^{7} + 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 4 q^{3} - 4 q^{4} - 16 q^{5} - 16 q^{6} - 7 q^{7} + 4 q^{8} - 4 q^{9} - 4 q^{10} + 4 q^{11} - 4 q^{12} + 2 q^{13} + 7 q^{14} + 4 q^{15} - 4 q^{16} + 5 q^{17} + 4 q^{18} - 2 q^{19} + 4 q^{20} + 8 q^{21} - 4 q^{22} - 4 q^{23} + 4 q^{24} + 16 q^{25} + 8 q^{26} - 4 q^{27} + 8 q^{28} + 5 q^{29} + 16 q^{30} - q^{31} - 16 q^{32} - q^{33} - 5 q^{34} + 7 q^{35} + 16 q^{36} - 20 q^{37} - 3 q^{38} + 2 q^{39} - 4 q^{40} - 17 q^{41} + 7 q^{42} - 4 q^{43} - q^{44} + 4 q^{45} - 6 q^{46} + 2 q^{47} - 4 q^{48} - 21 q^{49} + 4 q^{50} + 5 q^{51} + 2 q^{52} + 9 q^{53} + 4 q^{54} - 4 q^{55} + 2 q^{56} - 2 q^{57} - 5 q^{58} - 3 q^{59} + 4 q^{60} + 70 q^{61} - 4 q^{62} - 2 q^{63} - 4 q^{64} - 2 q^{65} - 4 q^{66} - 66 q^{67} - 20 q^{68} - 4 q^{69} - 7 q^{70} - 25 q^{71} + 4 q^{72} - 10 q^{73} - 15 q^{74} - 4 q^{75} + 3 q^{76} + 26 q^{77} - 2 q^{78} + 14 q^{79} + 4 q^{80} - 4 q^{81} + 17 q^{82} + 20 q^{83} - 7 q^{84} - 5 q^{85} - 6 q^{86} - 20 q^{87} + 6 q^{88} - 13 q^{89} - 4 q^{90} - 4 q^{91} - 4 q^{92} + 4 q^{93} - 12 q^{94} + 2 q^{95} + 4 q^{96} + 55 q^{97} - 14 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 5 x^{14} - 60 x^{13} + 480 x^{12} - 1202 x^{11} - 147 x^{10} - 2185 x^{9} + 62400 x^{8} - 289745 x^{7} + 699483 x^{6} - 1042897 x^{5} + 1014415 x^{4} + \cdots + 12851 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!16 \nu^{15} + \cdots - 32\!\cdots\!45 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{15} + \cdots + 24\!\cdots\!56 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18\!\cdots\!38 \nu^{15} + \cdots + 97\!\cdots\!89 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 26\!\cdots\!89 \nu^{15} + \cdots + 57\!\cdots\!82 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!56 \nu^{15} + \cdots + 47\!\cdots\!18 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37\!\cdots\!02 \nu^{15} + \cdots + 91\!\cdots\!39 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 50\!\cdots\!45 \nu^{15} + \cdots + 20\!\cdots\!75 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 51\!\cdots\!37 \nu^{15} + \cdots - 20\!\cdots\!95 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 78\!\cdots\!55 \nu^{15} + \cdots - 31\!\cdots\!01 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 86\!\cdots\!01 \nu^{15} + \cdots - 52\!\cdots\!07 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 38\!\cdots\!97 \nu^{15} + \cdots - 57\!\cdots\!83 ) / 72\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 21\!\cdots\!37 \nu^{15} + \cdots + 31\!\cdots\!34 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!34 \nu^{15} + \cdots + 10\!\cdots\!23 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 19\!\cdots\!21 \nu^{15} + \cdots - 23\!\cdots\!04 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{9} - 2\beta_{8} + 6\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - 2 \beta_{14} + \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + 15 \beta_{8} - 17 \beta_{7} - 21 \beta_{6} - 3 \beta_{5} + 9 \beta_{3} - 12 \beta_{2} - 6 \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 17 \beta_{15} + 21 \beta_{14} - 4 \beta_{13} - 6 \beta_{12} + 9 \beta_{11} + 15 \beta_{10} - 9 \beta_{9} - 14 \beta_{8} + 42 \beta_{7} + 93 \beta_{6} - 24 \beta_{5} - 13 \beta_{4} - 78 \beta_{3} + 7 \beta_{2} + 29 \beta _1 - 114 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 77 \beta_{15} - 47 \beta_{14} + 66 \beta_{13} + 17 \beta_{12} - 14 \beta_{11} - 10 \beta_{10} - 20 \beta_{9} - 123 \beta_{8} + 313 \beta_{7} - 253 \beta_{6} + 194 \beta_{5} - 21 \beta_{4} - 47 \beta_{3} + 105 \beta_{2} - 55 \beta _1 + 66 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 125 \beta_{15} - 106 \beta_{14} - 260 \beta_{13} - 55 \beta_{12} + 60 \beta_{11} - 189 \beta_{10} + 159 \beta_{9} + 744 \beta_{8} - 2198 \beta_{7} - 313 \beta_{6} - 203 \beta_{5} + 47 \beta_{4} + 1503 \beta_{3} - 1154 \beta_{2} + \cdots + 1431 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1461 \beta_{15} + 1747 \beta_{14} - 76 \beta_{13} - 35 \beta_{12} + 497 \beta_{11} + 1004 \beta_{10} - 329 \beta_{9} - 1269 \beta_{8} + 4756 \beta_{7} + 7693 \beta_{6} - 2723 \beta_{5} - 784 \beta_{4} - 10114 \beta_{3} + \cdots - 7413 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9546 \beta_{15} - 6080 \beta_{14} + 5055 \beta_{13} + 1045 \beta_{12} - 3131 \beta_{11} - 1193 \beta_{10} + 448 \beta_{9} - 13782 \beta_{8} + 20068 \beta_{7} - 28538 \beta_{6} + 24166 \beta_{5} + 3225 \beta_{4} + \cdots + 7283 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 15929 \beta_{15} - 13124 \beta_{14} - 22955 \beta_{13} + 5109 \beta_{12} + 5760 \beta_{11} - 18837 \beta_{10} + 5751 \beta_{9} + 85400 \beta_{8} - 194668 \beta_{7} - 6803 \beta_{6} - 71117 \beta_{5} + \cdots + 155369 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 116284 \beta_{15} + 193114 \beta_{14} + 7843 \beta_{13} - 34519 \beta_{12} + 9183 \beta_{11} + 108355 \beta_{10} - 23245 \beta_{9} - 83283 \beta_{8} + 486849 \beta_{7} + 560240 \beta_{6} + \cdots - 900992 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 955552 \beta_{15} - 705841 \beta_{14} + 476916 \beta_{13} + 104737 \beta_{12} - 181288 \beta_{11} - 91126 \beta_{10} - 83936 \beta_{9} - 1588169 \beta_{8} + 2073500 \beta_{7} - 2432456 \beta_{6} + \cdots + 882110 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2111653 \beta_{15} - 781722 \beta_{14} - 2379691 \beta_{13} + 321855 \beta_{12} + 488402 \beta_{11} - 2065085 \beta_{10} + 779820 \beta_{9} + 10703011 \beta_{8} - 21073771 \beta_{7} + \cdots + 16014786 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10935492 \beta_{15} + 20327298 \beta_{14} + 854213 \beta_{13} - 5074260 \beta_{12} + 1981295 \beta_{11} + 13082702 \beta_{10} - 2193955 \beta_{9} - 19308202 \beta_{8} + \cdots - 105329623 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 101049082 \beta_{15} - 85424041 \beta_{14} + 50277050 \beta_{13} + 19745720 \beta_{12} - 18439137 \beta_{11} - 20162415 \beta_{10} - 8696951 \beta_{9} - 142814260 \beta_{8} + \cdots + 180623205 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 264747964 \beta_{15} - 18099316 \beta_{14} - 278040932 \beta_{13} + 2672106 \beta_{12} + 45283488 \beta_{11} - 193091310 \beta_{10} + 107693910 \beta_{9} + \cdots + 1439366547 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
2.41373 + 0.0663824i
−3.22434 + 1.89830i
0.602406 + 0.654917i
1.20820 + 0.458083i
1.36070 + 1.34711i
0.109569 0.374923i
−2.52433 4.00017i
2.05406 + 2.30144i
1.36070 1.34711i
0.109569 + 0.374923i
−2.52433 + 4.00017i
2.05406 2.30144i
2.41373 0.0663824i
−3.22434 1.89830i
0.602406 0.654917i
1.20820 0.458083i
0.809017 0.587785i −0.809017 0.587785i 0.309017 0.951057i −1.00000 −1.00000 −1.42081 + 4.37281i −0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
481.2 0.809017 0.587785i −0.809017 0.587785i 0.309017 0.951057i −1.00000 −1.00000 −0.690690 + 2.12572i −0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
481.3 0.809017 0.587785i −0.809017 0.587785i 0.309017 0.951057i −1.00000 −1.00000 −0.186222 + 0.573132i −0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
481.4 0.809017 0.587785i −0.809017 0.587785i 0.309017 0.951057i −1.00000 −1.00000 1.10674 3.40620i −0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
721.1 −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i −1.00000 −1.00000 −3.70406 2.69116i 0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
721.2 −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i −1.00000 −1.00000 −0.0930287 0.0675893i 0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
721.3 −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i −1.00000 −1.00000 0.170108 + 0.123590i 0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
721.4 −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i −1.00000 −1.00000 1.31796 + 0.957557i 0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
841.1 −0.309017 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i −1.00000 −1.00000 −3.70406 + 2.69116i 0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
841.2 −0.309017 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i −1.00000 −1.00000 −0.0930287 + 0.0675893i 0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
841.3 −0.309017 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i −1.00000 −1.00000 0.170108 0.123590i 0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
841.4 −0.309017 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i −1.00000 −1.00000 1.31796 0.957557i 0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
901.1 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 −1.00000 −1.42081 4.37281i −0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i
901.2 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 −1.00000 −0.690690 2.12572i −0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i
901.3 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 −1.00000 −0.186222 0.573132i −0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i
901.4 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000 −1.00000 1.10674 + 3.40620i −0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 481.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.n.h 16
31.d even 5 1 inner 930.2.n.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.n.h 16 1.a even 1 1 trivial
930.2.n.h 16 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 7 T_{7}^{15} + 49 T_{7}^{14} + 176 T_{7}^{13} + 714 T_{7}^{12} + 1445 T_{7}^{11} + 5737 T_{7}^{10} - 3063 T_{7}^{9} + 16548 T_{7}^{8} - 28408 T_{7}^{7} + 73817 T_{7}^{6} + 6515 T_{7}^{5} + 24239 T_{7}^{4} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 7 T^{15} + 49 T^{14} + 176 T^{13} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{16} - 4 T^{15} + 41 T^{14} + \cdots + 2624400 \) Copy content Toggle raw display
$13$ \( T^{16} - 2 T^{15} + 20 T^{14} + \cdots + 1256641 \) Copy content Toggle raw display
$17$ \( T^{16} - 5 T^{15} + \cdots + 97846342416 \) Copy content Toggle raw display
$19$ \( T^{16} + 2 T^{15} + 19 T^{14} + \cdots + 331776 \) Copy content Toggle raw display
$23$ \( T^{16} + 4 T^{15} + \cdots + 1881824400 \) Copy content Toggle raw display
$29$ \( T^{16} - 5 T^{15} + 26 T^{14} + \cdots + 104976 \) Copy content Toggle raw display
$31$ \( T^{16} + T^{15} + \cdots + 852891037441 \) Copy content Toggle raw display
$37$ \( (T^{8} + 10 T^{7} - 118 T^{6} + \cdots - 223421)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 17 T^{15} + \cdots + 3088876550400 \) Copy content Toggle raw display
$43$ \( T^{16} + 4 T^{15} + \cdots + 210996910336 \) Copy content Toggle raw display
$47$ \( T^{16} - 2 T^{15} + \cdots + 153836528400 \) Copy content Toggle raw display
$53$ \( T^{16} - 9 T^{15} + \cdots + 17398665216 \) Copy content Toggle raw display
$59$ \( T^{16} + 3 T^{15} + \cdots + 42855614816400 \) Copy content Toggle raw display
$61$ \( (T^{8} - 35 T^{7} + 330 T^{6} + \cdots + 356400)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 33 T^{7} + 195 T^{6} + \cdots + 29553984)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 25 T^{15} + \cdots + 23444964000000 \) Copy content Toggle raw display
$73$ \( T^{16} + 10 T^{15} + \cdots + 233420394496 \) Copy content Toggle raw display
$79$ \( T^{16} - 14 T^{15} + \cdots + 8008281121 \) Copy content Toggle raw display
$83$ \( T^{16} - 20 T^{15} + \cdots + 129413376 \) Copy content Toggle raw display
$89$ \( T^{16} + 13 T^{15} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{16} - 55 T^{15} + \cdots + 15953601686416 \) Copy content Toggle raw display
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