Properties

Label 300.2.h.c
Level $300$
Weight $2$
Character orbit 300.h
Analytic conductor $2.396$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,2,Mod(299,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, 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("300.299");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.h (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: \(2.39551206064\)
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: \( x^{16} - 2 x^{15} + 2 x^{14} + 10 x^{13} - 42 x^{11} + 134 x^{10} + 110 x^{9} + 92 x^{8} + 142 x^{7} + 1514 x^{6} + 1102 x^{5} + 249 x^{4} - 1056 x^{3} + 392 x^{2} - 280 x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{10} q^{3} - \beta_{4} q^{4} - \beta_{12} q^{6} + ( - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_1) q^{8} + (\beta_{8} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{10} q^{3} - \beta_{4} q^{4} - \beta_{12} q^{6} + ( - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_1) q^{8} + (\beta_{8} + \beta_{4}) q^{9} + (\beta_{14} + \beta_{13} - \beta_{12} + \beta_{8} - \beta_{5} + \beta_{4}) q^{11} + (\beta_{11} + \beta_{3}) q^{12} + ( - \beta_{15} + \beta_{7} - \beta_{2}) q^{13} + ( - \beta_{14} - \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4}) q^{14} + ( - \beta_{13} - \beta_{12} - 2 \beta_{5}) q^{16} + ( - \beta_{11} - \beta_{7} + \beta_{3} - \beta_1) q^{17} + (\beta_{15} + \beta_{11} - 2 \beta_{9} - \beta_{7} - \beta_1) q^{18} + (\beta_{13} + \beta_{12} + \beta_{4} + 1) q^{19} + (\beta_{14} - \beta_{13} + \beta_{12} - \beta_{8} + \beta_{5} + \beta_{4} + 1) q^{21} + ( - 2 \beta_{10} - 2 \beta_{9} + \beta_{7} + \beta_{3}) q^{22} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{3} + \beta_1) q^{23} + ( - \beta_{13} + \beta_{6} + \beta_{5} - \beta_{4} - 2) q^{24} + ( - \beta_{13} + \beta_{12} - 2 \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4}) q^{26} + ( - \beta_{11} + 2 \beta_{9} + \beta_{7} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{27} + ( - \beta_{15} + 2 \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{3} - 2 \beta_{2}) q^{28} + (\beta_{14} - \beta_{13} + \beta_{12} - \beta_{8}) q^{29} + (\beta_{13} + \beta_{12} - \beta_{5} + 2 \beta_{4} + 1) q^{31} + ( - 2 \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{3} + 2 \beta_1) q^{32} + (2 \beta_{15} - \beta_{10} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{33} + ( - \beta_{13} - \beta_{12} - 4 \beta_{5} + 2 \beta_{4} - 4) q^{34} + ( - \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{8} - \beta_{6} + \beta_{5} - 2) q^{36} + (2 \beta_{15} - 2 \beta_{7} + 2 \beta_{2}) q^{37} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{7} - \beta_{3}) q^{38} + (\beta_{14} + 2 \beta_{13} + \beta_{8} + \beta_{5} + 1) q^{39} + ( - \beta_{14} + \beta_{13} - \beta_{12} + 3 \beta_{8} - \beta_{5} + \beta_{4}) q^{41} + ( - 2 \beta_{15} + 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - \beta_{7} + \beta_{3} - \beta_1) q^{42} + (2 \beta_{10} + 2 \beta_{9} + \beta_{3} + \beta_{2}) q^{43} + (\beta_{13} - \beta_{12} + 2 \beta_{6}) q^{44} + ( - 2 \beta_{4} - 4) q^{46} + (2 \beta_{10} - 2 \beta_{9}) q^{47} + (\beta_{15} + \beta_{11} - 3 \beta_{7} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{48} + ( - \beta_{5} - \beta_{4} + 2) q^{49} + ( - \beta_{14} - \beta_{13} + 3 \beta_{12} - 2 \beta_{8} + 2 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + \cdots + 1) q^{51}+ \cdots + ( - \beta_{14} - 3 \beta_{13} - 3 \beta_{12} - 2 \beta_{6} - 3 \beta_{5} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} + 6 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} + 6 q^{6} - 4 q^{9} + 20 q^{16} + 8 q^{21} - 26 q^{24} - 44 q^{34} - 42 q^{36} - 56 q^{46} + 40 q^{49} + 56 q^{54} - 40 q^{61} + 76 q^{64} + 58 q^{66} + 24 q^{69} + 36 q^{76} - 60 q^{81} - 80 q^{84} - 24 q^{94} - 78 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} + 10 x^{13} - 42 x^{11} + 134 x^{10} + 110 x^{9} + 92 x^{8} + 142 x^{7} + 1514 x^{6} + 1102 x^{5} + 249 x^{4} - 1056 x^{3} + 392 x^{2} - 280 x + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 86636170065458 \nu^{15} - 139763456245741 \nu^{14} + 128073458413399 \nu^{13} + 900030238567851 \nu^{12} + \cdots - 10\!\cdots\!80 ) / 55\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 226346343508237 \nu^{15} + 357846443441218 \nu^{14} - 336891024322778 \nu^{13} + \cdots + 60\!\cdots\!60 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19\!\cdots\!77 \nu^{15} + \cdots - 36\!\cdots\!00 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 587554941526996 \nu^{15} + 914739690969668 \nu^{14} - 779859476137840 \nu^{13} + \cdots + 98\!\cdots\!00 ) / 13\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 604292511642446 \nu^{15} - 975202941656918 \nu^{14} + 895309719750590 \nu^{13} + \cdots - 14\!\cdots\!75 ) / 13\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 34\!\cdots\!93 \nu^{15} + \cdots + 80\!\cdots\!00 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 917800794270695 \nu^{15} + \cdots + 20\!\cdots\!00 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!59 \nu^{15} + \cdots + 38\!\cdots\!20 ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 47453684427081 \nu^{15} - 72993794820788 \nu^{14} + 60782462586860 \nu^{13} + 505465180962810 \nu^{12} + \cdots - 10\!\cdots\!00 ) / 558775600728200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10024607033433 \nu^{15} + 15519853661140 \nu^{14} - 13088749471584 \nu^{13} - 105861785006214 \nu^{12} + \cdots + 21\!\cdots\!00 ) / 111755120145640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!77 \nu^{15} + \cdots - 25\!\cdots\!80 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 63\!\cdots\!53 \nu^{15} + \cdots + 13\!\cdots\!00 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!95 \nu^{15} + \cdots + 30\!\cdots\!80 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 43\!\cdots\!33 \nu^{15} + \cdots - 90\!\cdots\!40 ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 57\!\cdots\!07 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 27\!\cdots\!50 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3 \beta_{15} + 2 \beta_{14} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{7} - \beta_{6} - \beta_{5} - 4 \beta_{4} + 2 \beta_{2} - 5 \beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7 \beta_{15} + 2 \beta_{14} - 2 \beta_{10} - 2 \beta_{9} - 10 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14 \beta_{15} + 25 \beta_{14} + 5 \beta_{13} + 35 \beta_{12} - 5 \beta_{11} - 39 \beta_{10} + 31 \beta_{9} - 5 \beta_{8} + 19 \beta_{7} + 20 \beta_{6} + 10 \beta_{5} + 20 \beta_{4} - 35 \beta_{3} - 24 \beta_{2} + 35 \beta _1 - 20 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5 \beta_{13} + 5 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} - 4 \beta_{7} - 4 \beta_{5} + 9 \beta_{4} + 4 \beta_{3} + 12 \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 24 \beta_{15} - 79 \beta_{14} + 25 \beta_{13} + 195 \beta_{12} + 25 \beta_{11} - 241 \beta_{10} - 31 \beta_{9} - 25 \beta_{8} - 19 \beta_{7} - 78 \beta_{6} + 92 \beta_{5} + 108 \beta_{4} + 195 \beta_{3} + 44 \beta_{2} + 55 \beta _1 + 260 ) / 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 38 \beta_{15} - 50 \beta_{14} - 35 \beta_{13} + 35 \beta_{12} - 48 \beta_{10} - 48 \beta_{9} - 50 \beta_{8} - 12 \beta_{7} - 100 \beta_{6} + 50 \beta_{5} - 50 \beta_{4} + 115 \beta_{3} + 127 \beta_{2} ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 279 \beta_{15} - 189 \beta_{14} - 485 \beta_{13} - 35 \beta_{12} - 465 \beta_{11} - 319 \beta_{10} - 389 \beta_{9} - 465 \beta_{8} + 79 \beta_{7} - 173 \beta_{6} + 567 \beta_{5} - 402 \beta_{4} + 35 \beta_{3} + 406 \beta_{2} + \cdots - 380 ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 119 \beta_{13} - 119 \beta_{12} - 296 \beta_{11} - 88 \beta_{10} + 88 \beta_{9} + 248 \beta_{7} + 170 \beta_{5} - 69 \beta_{4} - 248 \beta_{3} + 344 \beta _1 - 580 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4602 \beta_{15} - 1715 \beta_{14} + 985 \beta_{13} - 4465 \beta_{12} - 4015 \beta_{11} + 1427 \beta_{10} + 3877 \beta_{9} + 4015 \beta_{8} + 2793 \beta_{7} + 1640 \beta_{6} - 170 \beta_{5} + 3100 \beta_{4} + \cdots - 10740 ) / 20 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 9123 \beta_{15} - 3442 \beta_{14} + 3400 \beta_{13} - 3400 \beta_{12} + 2958 \beta_{10} + 2958 \beta_{9} + 8530 \beta_{8} + 2667 \beta_{7} + 696 \beta_{6} - 2544 \beta_{5} + 2544 \beta_{4} - 645 \beta_{3} - 3312 \beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 28892 \beta_{15} - 17087 \beta_{14} + 6585 \beta_{13} - 21405 \beta_{12} + 26495 \beta_{11} + 40697 \beta_{10} - 17273 \beta_{9} + 26495 \beta_{8} - 9237 \beta_{7} - 7194 \beta_{6} - 28584 \beta_{5} + \cdots + 36100 ) / 20 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1979 \beta_{13} - 1979 \beta_{12} + 2412 \beta_{11} + 4652 \beta_{10} - 4652 \beta_{9} - 1324 \beta_{7} - 2227 \beta_{5} - 3895 \beta_{4} + 1324 \beta_{3} - 9256 \beta _1 + 6093 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 56593 \beta_{15} + 69762 \beta_{14} - 41490 \beta_{13} - 50530 \beta_{12} + 22640 \beta_{11} + 103592 \beta_{10} - 47828 \beta_{9} - 22640 \beta_{8} - 7392 \beta_{7} + 29509 \beta_{6} - 53601 \beta_{5} + \cdots + 31440 ) / 10 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 118157 \beta_{15} + 212602 \beta_{14} + 41880 \beta_{13} - 41880 \beta_{12} + 115318 \beta_{10} + 115318 \beta_{9} - 26370 \beta_{8} - 35853 \beta_{7} + 159364 \beta_{6} - 93116 \beta_{5} + \cdots - 174862 \beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 69354 \beta_{15} + 691355 \beta_{14} + 1028615 \beta_{13} + 501225 \beta_{12} + 295825 \beta_{11} - 231109 \beta_{10} + 1303021 \beta_{9} + 295825 \beta_{8} - 216751 \beta_{7} + \cdots + 485380 ) / 20 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1
2.13477 1.10359i
1.10359 + 2.13477i
2.13477 + 1.10359i
1.10359 2.13477i
0.994127 + 1.77516i
−1.77516 + 0.994127i
0.994127 1.77516i
−1.77516 0.994127i
−0.869987 1.05054i
−1.05054 + 0.869987i
−0.869987 + 1.05054i
−1.05054 0.869987i
0.474040 0.0108430i
−0.0108430 0.474040i
0.474040 + 0.0108430i
−0.0108430 + 0.474040i
−1.38758 0.273147i −0.758030 1.55737i 1.85078 + 0.758030i 0 0.626440 + 2.36803i 3.56393 −2.36106 1.55737i −1.85078 + 2.36106i 0
299.2 −1.38758 0.273147i 0.758030 1.55737i 1.85078 + 0.758030i 0 −1.47722 + 1.95392i −3.56393 −2.36106 1.55737i −1.85078 2.36106i 0
299.3 −1.38758 + 0.273147i −0.758030 + 1.55737i 1.85078 0.758030i 0 0.626440 2.36803i 3.56393 −2.36106 + 1.55737i −1.85078 2.36106i 0
299.4 −1.38758 + 0.273147i 0.758030 + 1.55737i 1.85078 0.758030i 0 −1.47722 1.95392i −3.56393 −2.36106 + 1.55737i −1.85078 + 2.36106i 0
299.5 −0.569745 1.29437i −1.47492 + 0.908080i −1.35078 + 1.47492i 0 2.01572 + 1.39172i −2.50967 2.67869 + 0.908080i 1.35078 2.67869i 0
299.6 −0.569745 1.29437i 1.47492 + 0.908080i −1.35078 + 1.47492i 0 0.335062 2.42647i 2.50967 2.67869 + 0.908080i 1.35078 + 2.67869i 0
299.7 −0.569745 + 1.29437i −1.47492 0.908080i −1.35078 1.47492i 0 2.01572 1.39172i −2.50967 2.67869 0.908080i 1.35078 + 2.67869i 0
299.8 −0.569745 + 1.29437i 1.47492 0.908080i −1.35078 1.47492i 0 0.335062 + 2.42647i 2.50967 2.67869 0.908080i 1.35078 2.67869i 0
299.9 0.569745 1.29437i −1.47492 + 0.908080i −1.35078 1.47492i 0 0.335062 + 2.42647i −2.50967 −2.67869 + 0.908080i 1.35078 2.67869i 0
299.10 0.569745 1.29437i 1.47492 + 0.908080i −1.35078 1.47492i 0 2.01572 1.39172i 2.50967 −2.67869 + 0.908080i 1.35078 + 2.67869i 0
299.11 0.569745 + 1.29437i −1.47492 0.908080i −1.35078 + 1.47492i 0 0.335062 2.42647i −2.50967 −2.67869 0.908080i 1.35078 + 2.67869i 0
299.12 0.569745 + 1.29437i 1.47492 0.908080i −1.35078 + 1.47492i 0 2.01572 + 1.39172i 2.50967 −2.67869 0.908080i 1.35078 2.67869i 0
299.13 1.38758 0.273147i −0.758030 1.55737i 1.85078 0.758030i 0 −1.47722 1.95392i 3.56393 2.36106 1.55737i −1.85078 + 2.36106i 0
299.14 1.38758 0.273147i 0.758030 1.55737i 1.85078 0.758030i 0 0.626440 2.36803i −3.56393 2.36106 1.55737i −1.85078 2.36106i 0
299.15 1.38758 + 0.273147i −0.758030 + 1.55737i 1.85078 + 0.758030i 0 −1.47722 + 1.95392i 3.56393 2.36106 + 1.55737i −1.85078 2.36106i 0
299.16 1.38758 + 0.273147i 0.758030 + 1.55737i 1.85078 + 0.758030i 0 0.626440 + 2.36803i −3.56393 2.36106 + 1.55737i −1.85078 + 2.36106i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.h.c 16
3.b odd 2 1 inner 300.2.h.c 16
4.b odd 2 1 inner 300.2.h.c 16
5.b even 2 1 inner 300.2.h.c 16
5.c odd 4 1 300.2.e.d 8
5.c odd 4 1 300.2.e.e yes 8
12.b even 2 1 inner 300.2.h.c 16
15.d odd 2 1 inner 300.2.h.c 16
15.e even 4 1 300.2.e.d 8
15.e even 4 1 300.2.e.e yes 8
20.d odd 2 1 inner 300.2.h.c 16
20.e even 4 1 300.2.e.d 8
20.e even 4 1 300.2.e.e yes 8
60.h even 2 1 inner 300.2.h.c 16
60.l odd 4 1 300.2.e.d 8
60.l odd 4 1 300.2.e.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.e.d 8 5.c odd 4 1
300.2.e.d 8 15.e even 4 1
300.2.e.d 8 20.e even 4 1
300.2.e.d 8 60.l odd 4 1
300.2.e.e yes 8 5.c odd 4 1
300.2.e.e yes 8 15.e even 4 1
300.2.e.e yes 8 20.e even 4 1
300.2.e.e yes 8 60.l odd 4 1
300.2.h.c 16 1.a even 1 1 trivial
300.2.h.c 16 3.b odd 2 1 inner
300.2.h.c 16 4.b odd 2 1 inner
300.2.h.c 16 5.b even 2 1 inner
300.2.h.c 16 12.b even 2 1 inner
300.2.h.c 16 15.d odd 2 1 inner
300.2.h.c 16 20.d odd 2 1 inner
300.2.h.c 16 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{4} - 19T_{7}^{2} + 80 \) Copy content Toggle raw display
\( T_{17}^{4} - 59T_{17}^{2} + 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{6} - 2 T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + T^{6} + 8 T^{4} + 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} - 19 T^{2} + 80)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 29 T^{2} + 200)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 21 T^{2} + 100)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 59 T^{2} + 40)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 26 T^{2} + 5)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 28 T^{2} + 32)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 36 T^{2} + 160)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 55 T^{2} + 500)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 84 T^{2} + 1600)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 115 T^{2} + 1000)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 71 T^{2} + 20)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 52 T^{2} + 512)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 36 T^{2} + 160)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T - 4)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 34 T^{2} + 125)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 284 T^{2} + 20000)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 61 T^{2} + 100)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 76 T^{2} + 1280)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 13 T^{2} + 32)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 51 T^{2} + 640)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 189 T^{2} + 8100)^{4} \) Copy content Toggle raw display
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