Properties

Label 1650.2.f.g
Level $1650$
Weight $2$
Character orbit 1650.f
Analytic conductor $13.175$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1650,2,Mod(1649,1650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1650.1649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1650.f (of order \(2\), degree \(1\), not 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: \(13.1753163335\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{14} + 10x^{12} + 39x^{10} + 178x^{8} + 351x^{6} + 810x^{4} + 3645x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 5x^{14} + 10x^{12} + 39x^{10} + 178x^{8} + 351x^{6} + 810x^{4} + 3645x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} - 8\nu^{13} + 8\nu^{11} - 19\nu^{9} - 185\nu^{7} + 8\nu^{5} - 72\nu^{3} - 2997\nu ) / 3888 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -17\nu^{14} - 40\nu^{12} - 26\nu^{10} - 861\nu^{8} - 2081\nu^{6} - 1602\nu^{4} - 13608\nu^{2} - 50301 ) / 15552 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{15} - 5\nu^{13} - 10\nu^{11} - 39\nu^{9} - 178\nu^{7} - 351\nu^{5} - 810\nu^{3} - 3645\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23\nu^{15} + 64\nu^{13} + 110\nu^{11} + 819\nu^{9} + 1511\nu^{7} + 1830\nu^{5} + 13824\nu^{3} + 43011\nu ) / 46656 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{14} - 5\nu^{12} - 10\nu^{10} - 39\nu^{8} - 178\nu^{6} - 351\nu^{4} - 810\nu^{2} - 2916 ) / 729 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\nu^{14} + 64\nu^{12} + 110\nu^{10} + 819\nu^{8} + 1511\nu^{6} + 1830\nu^{4} + 13824\nu^{2} + 43011 ) / 15552 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -7\nu^{15} - 14\nu^{13} + 8\nu^{11} - 441\nu^{9} - 1183\nu^{7} - 744\nu^{5} - 6750\nu^{3} - 26973\nu ) / 11664 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3\nu^{15} - 2\nu^{13} + 8\nu^{11} - 41\nu^{9} - 135\nu^{7} - 440\nu^{5} - 1458\nu^{3} - 3969\nu ) / 3888 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 107\nu^{14} + 112\nu^{12} + 494\nu^{10} + 2535\nu^{8} + 6275\nu^{6} + 6246\nu^{4} + 58320\nu^{2} + 133407 ) / 46656 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -109\nu^{14} - 320\nu^{12} - 370\nu^{10} - 3297\nu^{8} - 14677\nu^{6} - 6714\nu^{4} - 77760\nu^{2} - 292329 ) / 46656 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -35\nu^{15} - 76\nu^{13} + 64\nu^{11} - 51\nu^{9} - 1721\nu^{7} + 720\nu^{5} + 20412\nu^{3} - 18225\nu ) / 34992 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -4\nu^{14} - 11\nu^{12} + 5\nu^{10} - 66\nu^{8} - 361\nu^{6} - 531\nu^{4} - 2268\nu^{2} - 8748 ) / 1458 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} + 2\nu^{13} + 4\nu^{11} + 27\nu^{9} + 97\nu^{7} + 60\nu^{5} + 630\nu^{3} + 1755\nu ) / 648 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{13} - \beta_{10} + \beta_{9} + \beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} + 3\beta_{12} + \beta_{11} - 2\beta_{8} - 2\beta_{7} - 2\beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{15} - 3\beta_{10} - \beta_{9} - 6\beta_{5} + 2\beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} - 5\beta_{12} + \beta_{11} - 10\beta_{8} - \beta_{7} - 2\beta_{4} - 2\beta_{2} - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{15} - 2\beta_{13} + 5\beta_{10} - 5\beta_{9} - 32\beta_{6} - 5\beta_{5} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5\beta_{14} + 9\beta_{12} - 5\beta_{11} + 10\beta_{8} + \beta_{7} - 30\beta_{4} - 5\beta_{2} - 15 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -33\beta_{15} - 5\beta_{13} + 20\beta_{10} - 40\beta_{9} + 64\beta_{6} - 2\beta_{5} + 19\beta_{3} - 33\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 40\beta_{14} - 22\beta_{12} + 30\beta_{11} + 20\beta_{8} - 25\beta_{7} + 68\beta_{4} + 2\beta_{2} + 79 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 36\beta_{15} + 2\beta_{13} + 118\beta_{10} + 30\beta_{9} - 73\beta_{5} + 86\beta_{3} + 38\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -30\beta_{14} - 76\beta_{12} - 236\beta_{11} + 384\beta_{8} + 24\beta_{7} + 232\beta_{4} + 10\beta_{2} - 207 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -44\beta_{15} + 10\beta_{13} - 100\beta_{10} + 272\beta_{9} + 608\beta_{6} + 82\beta_{5} - 382\beta_{3} - 67\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -272\beta_{14} + 86\beta_{12} + 546\beta_{11} - 28\beta_{8} + 242\beta_{7} + 388\beta_{4} - 329\beta_{2} - 25 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 317 \beta_{15} - 329 \beta_{13} - 487 \beta_{10} + 331 \beta_{9} + 160 \beta_{6} + 397 \beta_{5} + 417 \beta_{3} + 74 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(727\) \(1201\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1649.1
−1.60300 + 0.656040i
−1.35457 1.07942i
−0.549154 + 1.64269i
−0.209658 1.71931i
0.209658 1.71931i
0.549154 + 1.64269i
1.35457 1.07942i
1.60300 + 0.656040i
−1.60300 0.656040i
−1.35457 + 1.07942i
−0.549154 1.64269i
−0.209658 + 1.71931i
0.209658 + 1.71931i
0.549154 1.64269i
1.35457 + 1.07942i
1.60300 0.656040i
1.00000i −1.60300 + 0.656040i −1.00000 0 0.656040 + 1.60300i 0.623830 1.00000i 2.13922 2.10327i 0
1649.2 1.00000i −1.35457 1.07942i −1.00000 0 −1.07942 + 1.35457i 0.738241 1.00000i 0.669725 + 2.92429i 0
1649.3 1.00000i −0.549154 + 1.64269i −1.00000 0 1.64269 + 0.549154i 1.82098 1.00000i −2.39686 1.80418i 0
1649.4 1.00000i −0.209658 1.71931i −1.00000 0 −1.71931 + 0.209658i 4.76968 1.00000i −2.91209 + 0.720935i 0
1649.5 1.00000i 0.209658 1.71931i −1.00000 0 −1.71931 0.209658i −4.76968 1.00000i −2.91209 0.720935i 0
1649.6 1.00000i 0.549154 + 1.64269i −1.00000 0 1.64269 0.549154i −1.82098 1.00000i −2.39686 + 1.80418i 0
1649.7 1.00000i 1.35457 1.07942i −1.00000 0 −1.07942 1.35457i −0.738241 1.00000i 0.669725 2.92429i 0
1649.8 1.00000i 1.60300 + 0.656040i −1.00000 0 0.656040 1.60300i −0.623830 1.00000i 2.13922 + 2.10327i 0
1649.9 1.00000i −1.60300 0.656040i −1.00000 0 0.656040 1.60300i 0.623830 1.00000i 2.13922 + 2.10327i 0
1649.10 1.00000i −1.35457 + 1.07942i −1.00000 0 −1.07942 1.35457i 0.738241 1.00000i 0.669725 2.92429i 0
1649.11 1.00000i −0.549154 1.64269i −1.00000 0 1.64269 0.549154i 1.82098 1.00000i −2.39686 + 1.80418i 0
1649.12 1.00000i −0.209658 + 1.71931i −1.00000 0 −1.71931 0.209658i 4.76968 1.00000i −2.91209 0.720935i 0
1649.13 1.00000i 0.209658 + 1.71931i −1.00000 0 −1.71931 + 0.209658i −4.76968 1.00000i −2.91209 + 0.720935i 0
1649.14 1.00000i 0.549154 1.64269i −1.00000 0 1.64269 + 0.549154i −1.82098 1.00000i −2.39686 1.80418i 0
1649.15 1.00000i 1.35457 + 1.07942i −1.00000 0 −1.07942 + 1.35457i −0.738241 1.00000i 0.669725 + 2.92429i 0
1649.16 1.00000i 1.60300 0.656040i −1.00000 0 0.656040 + 1.60300i −0.623830 1.00000i 2.13922 2.10327i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1649.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
33.d even 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1650.2.f.g 16
3.b odd 2 1 1650.2.f.h 16
5.b even 2 1 inner 1650.2.f.g 16
5.c odd 4 1 1650.2.d.e yes 8
5.c odd 4 1 1650.2.d.g yes 8
11.b odd 2 1 1650.2.f.h 16
15.d odd 2 1 1650.2.f.h 16
15.e even 4 1 1650.2.d.d 8
15.e even 4 1 1650.2.d.h yes 8
33.d even 2 1 inner 1650.2.f.g 16
55.d odd 2 1 1650.2.f.h 16
55.e even 4 1 1650.2.d.d 8
55.e even 4 1 1650.2.d.h yes 8
165.d even 2 1 inner 1650.2.f.g 16
165.l odd 4 1 1650.2.d.e yes 8
165.l odd 4 1 1650.2.d.g yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1650.2.d.d 8 15.e even 4 1
1650.2.d.d 8 55.e even 4 1
1650.2.d.e yes 8 5.c odd 4 1
1650.2.d.e yes 8 165.l odd 4 1
1650.2.d.g yes 8 5.c odd 4 1
1650.2.d.g yes 8 165.l odd 4 1
1650.2.d.h yes 8 15.e even 4 1
1650.2.d.h yes 8 55.e even 4 1
1650.2.f.g 16 1.a even 1 1 trivial
1650.2.f.g 16 5.b even 2 1 inner
1650.2.f.g 16 33.d even 2 1 inner
1650.2.f.g 16 165.d even 2 1 inner
1650.2.f.h 16 3.b odd 2 1
1650.2.f.h 16 11.b odd 2 1
1650.2.f.h 16 15.d odd 2 1
1650.2.f.h 16 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1650, [\chi])\):

\( T_{7}^{8} - 27T_{7}^{6} + 100T_{7}^{4} - 76T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{23}^{8} - 125T_{23}^{6} + 4351T_{23}^{4} - 35423T_{23}^{2} + 75076 \) Copy content Toggle raw display
\( T_{29}^{4} + 4T_{29}^{3} - 49T_{29}^{2} - 82T_{29} + 440 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 5 T^{14} + 10 T^{12} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 27 T^{6} + 100 T^{4} - 76 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 3 T^{7} + 12 T^{6} + 63 T^{5} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 38 T^{6} + 241 T^{4} - 480 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 90 T^{6} + 2465 T^{4} + \cdots + 19600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 56 T^{6} + 678 T^{4} + 2248 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 125 T^{6} + 4351 T^{4} + \cdots + 75076)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} - 49 T^{2} - 82 T + 440)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} - 49 T^{2} + 130 T + 392)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 121 T^{6} + 4164 T^{4} + \cdots + 204304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 2 T^{3} - 69 T^{2} - 236 T - 86)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 178 T^{6} + 7465 T^{4} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 163 T^{6} + 6256 T^{4} + \cdots + 200704)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 400 T^{6} + 51156 T^{4} + \cdots + 20070400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 299 T^{6} + 30796 T^{4} + \cdots + 14961424)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 196 T^{6} + 3632 T^{4} + \cdots + 25600)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 73 T^{6} + 1688 T^{4} + \cdots + 50176)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 401 T^{6} + 45427 T^{4} + \cdots + 1000000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 455 T^{6} + 67016 T^{4} + \cdots + 58369600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 319 T^{6} + 31596 T^{4} + \cdots + 7795264)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 299 T^{6} + 27155 T^{4} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 157 T^{6} + 7591 T^{4} + \cdots + 71824)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 295 T^{6} + 7471 T^{4} + \cdots + 20164)^{2} \) Copy content Toggle raw display
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