Properties

Label 2400.2.o.k
Level $2400$
Weight $2$
Character orbit 2400.o
Analytic conductor $19.164$
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] = [2400,2,Mod(2399,2400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 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("2400.2399");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.o (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: \(19.1640964851\)
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: \( x^{16} - 4 x^{15} + 8 x^{14} - 16 x^{13} + 41 x^{12} - 64 x^{11} + 56 x^{10} - 100 x^{9} + 256 x^{8} - 300 x^{7} + 504 x^{6} - 1728 x^{5} + 3321 x^{4} - 3888 x^{3} + 5832 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} - \beta_{13} q^{7} + ( - \beta_{13} + \beta_{9}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{3} - \beta_{13} q^{7} + ( - \beta_{13} + \beta_{9}) q^{9} - \beta_{15} q^{11} + (\beta_{12} - \beta_{7} + \beta_{2}) q^{13} + (\beta_{15} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{2} - \beta_1) q^{17} + ( - \beta_{11} + 2 \beta_{7} + \beta_{3} - \beta_1) q^{19} + (\beta_{14} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} + 1) q^{21} + (\beta_{14} + 2 \beta_{9} + \beta_{8} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4}) q^{23} + (\beta_{14} + \beta_{13} - 2 \beta_{6}) q^{27} + ( - \beta_{14} + \beta_{8} - \beta_{5} - \beta_{4}) q^{29} + ( - 2 \beta_{12} - \beta_{3} - 2 \beta_{2}) q^{31} + (\beta_{15} + \beta_{12} + \beta_{11} - \beta_{10} - \beta_{7} - \beta_{3} + \beta_1) q^{33} + ( - 2 \beta_{11} + 2 \beta_{7} - 2 \beta_1) q^{37} + (\beta_{15} - 2 \beta_{12} + \beta_{11} + 2 \beta_{10} + 2 \beta_{7} - \beta_{3} - \beta_1) q^{39} + (\beta_{14} + \beta_{9} + 3 \beta_{8} - \beta_{6} - \beta_{5} - 3 \beta_{4}) q^{41} + ( - \beta_{13} + 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} + 2 \beta_{4} + 6) q^{43} + ( - 2 \beta_{9} + 2 \beta_{6}) q^{47} + (2 \beta_{13} + \beta_{9} + 2 \beta_{8} + \beta_{6} + 2 \beta_{4} + 2) q^{49} + ( - 2 \beta_{12} + \beta_{3} - \beta_1) q^{51} + (\beta_{15} + 2 \beta_{12} - 3 \beta_{11} - \beta_{10} - 2 \beta_{2} + 3 \beta_1) q^{53} + ( - \beta_{15} + \beta_{11} - \beta_{10} + 2 \beta_{7} - \beta_{3} - \beta_{2} - \beta_1) q^{57} + ( - 2 \beta_{12} + 2 \beta_{11} + 2 \beta_{2} - 2 \beta_1) q^{59} + (\beta_{9} + 4 \beta_{8} + \beta_{6} + 4 \beta_{4} + 3) q^{61} + (\beta_{13} + 2 \beta_{9} + 2 \beta_{8} + 6) q^{63} + (3 \beta_{13} - 2 \beta_{9} - \beta_{8} - 2 \beta_{6} - \beta_{4} - 2) q^{67} + (\beta_{14} + 2 \beta_{13} + 2 \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} + 2) q^{69} + ( - \beta_{15} + \beta_{11} + 2 \beta_{10} - \beta_1) q^{71} + (3 \beta_{12} + 4 \beta_{11} - 4 \beta_{7} + 2 \beta_{3} + 3 \beta_{2} + 4 \beta_1) q^{73} + (\beta_{15} - 2 \beta_{12} - \beta_{11} + 3 \beta_{10} + 2 \beta_{2} + \beta_1) q^{77} + ( - 2 \beta_{12} - 2 \beta_{11} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{79} + (\beta_{13} + 2 \beta_{9} + 2 \beta_{8} - \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + 2) q^{81} + (2 \beta_{9} + 3 \beta_{8} - 2 \beta_{6} - 4 \beta_{5} - 3 \beta_{4}) q^{83} + (\beta_{14} - 2 \beta_{13} - \beta_{8} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2) q^{87} + (\beta_{9} - \beta_{6} - 4 \beta_{5}) q^{89} + ( - 2 \beta_{11} + 6 \beta_{7} + 3 \beta_{3} - 2 \beta_1) q^{91} + ( - \beta_{15} + 3 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} - 3 \beta_{7} + 2 \beta_{3} + \cdots + 3 \beta_1) q^{93}+ \cdots + ( - 2 \beta_{12} - \beta_{11} + 4 \beta_{10} - 2 \beta_{7} + \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} - 4 q^{9} + 8 q^{21} + 8 q^{27} + 64 q^{43} + 8 q^{49} + 8 q^{61} + 80 q^{63} - 8 q^{67} + 24 q^{69} + 36 q^{81} - 40 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 8 x^{14} - 16 x^{13} + 41 x^{12} - 64 x^{11} + 56 x^{10} - 100 x^{9} + 256 x^{8} - 300 x^{7} + 504 x^{6} - 1728 x^{5} + 3321 x^{4} - 3888 x^{3} + 5832 x^{2} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 145 \nu^{15} + 53066 \nu^{14} - 194947 \nu^{13} + 208328 \nu^{12} - 238312 \nu^{11} + 1114004 \nu^{10} - 685408 \nu^{9} - 975904 \nu^{8} - 793136 \nu^{7} + \cdots + 22263660 ) / 45822024 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2569 \nu^{15} + 70928 \nu^{14} - 166771 \nu^{13} + 274232 \nu^{12} - 837544 \nu^{11} + 1478060 \nu^{10} - 828592 \nu^{9} + 1200608 \nu^{8} - 6256304 \nu^{7} + \cdots + 258564636 ) / 45822024 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2569 \nu^{15} - 70928 \nu^{14} + 166771 \nu^{13} - 274232 \nu^{12} + 837544 \nu^{11} - 1478060 \nu^{10} + 828592 \nu^{9} - 1200608 \nu^{8} + \cdots - 258564636 ) / 45822024 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2053 \nu^{15} + 24280 \nu^{14} - 68231 \nu^{13} + 119578 \nu^{12} - 304424 \nu^{11} + 440620 \nu^{10} - 503816 \nu^{9} + 662344 \nu^{8} - 952648 \nu^{7} + \cdots + 39107934 ) / 15274008 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 67 \nu^{15} - 1196 \nu^{14} + 1045 \nu^{13} + 1195 \nu^{12} + 5188 \nu^{11} - 12284 \nu^{10} - 10892 \nu^{9} + 15904 \nu^{8} + 70964 \nu^{7} - 53220 \nu^{6} - 33516 \nu^{5} + \cdots - 828873 ) / 424278 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7399 \nu^{15} + 4840 \nu^{14} + 36259 \nu^{13} - 321224 \nu^{12} + 725896 \nu^{11} - 657740 \nu^{10} + 940576 \nu^{9} - 3252872 \nu^{8} + 437312 \nu^{7} + \cdots - 203819652 ) / 45822024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4634 \nu^{15} + 2117 \nu^{14} - 14398 \nu^{13} + 66416 \nu^{12} - 91540 \nu^{11} - 16432 \nu^{10} - 177052 \nu^{9} + 484040 \nu^{8} + 103696 \nu^{7} + \cdots - 2064528 ) / 11455506 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20321 \nu^{15} - 145784 \nu^{14} + 164923 \nu^{13} - 366494 \nu^{12} + 1271008 \nu^{11} - 1583372 \nu^{10} - 158504 \nu^{9} - 2835200 \nu^{8} + \cdots - 227636082 ) / 45822024 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41737 \nu^{15} - 9496 \nu^{14} - 77581 \nu^{13} + 37208 \nu^{12} + 261176 \nu^{11} + 590228 \nu^{10} - 512224 \nu^{9} - 1465984 \nu^{8} + 1578736 \nu^{7} + \cdots + 15440220 ) / 45822024 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 203 \nu^{15} - 341 \nu^{14} + 433 \nu^{13} - 1172 \nu^{12} + 2416 \nu^{11} + 1036 \nu^{10} - 272 \nu^{9} - 7280 \nu^{8} + 6956 \nu^{7} - 11028 \nu^{6} + 63792 \nu^{5} - 97956 \nu^{4} + \cdots + 227448 ) / 118098 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 28405 \nu^{15} + 52738 \nu^{14} - 66017 \nu^{13} + 262408 \nu^{12} - 602720 \nu^{11} + 312100 \nu^{10} - 338144 \nu^{9} + 2248432 \nu^{8} - 2272408 \nu^{7} + \cdots + 47020500 ) / 15274008 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 86377 \nu^{15} - 201448 \nu^{14} + 252269 \nu^{13} - 563848 \nu^{12} + 1592888 \nu^{11} - 1308556 \nu^{10} - 158128 \nu^{9} - 3660256 \nu^{8} + \cdots - 108011556 ) / 45822024 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1697 \nu^{15} - 5048 \nu^{14} + 7147 \nu^{13} - 18164 \nu^{12} + 47416 \nu^{11} - 45548 \nu^{10} + 24208 \nu^{9} - 141344 \nu^{8} + 263168 \nu^{7} - 120108 \nu^{6} + \cdots - 6543504 ) / 472392 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3070 \nu^{15} + 7672 \nu^{14} - 7190 \nu^{13} + 21013 \nu^{12} - 71384 \nu^{11} + 52840 \nu^{10} + 1960 \nu^{9} + 262180 \nu^{8} - 449608 \nu^{7} + 37848 \nu^{6} + \cdots + 9865557 ) / 848556 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1070 \nu^{15} - 3089 \nu^{14} + 4030 \nu^{13} - 12632 \nu^{12} + 30628 \nu^{11} - 27956 \nu^{10} + 20884 \nu^{9} - 97832 \nu^{8} + 141248 \nu^{7} - 63912 \nu^{6} + \cdots - 4076568 ) / 236196 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{12} + \beta_{10} + 2\beta_{9} + 2\beta_{8} + 2\beta_{7} - \beta_{5} - 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{15} - 2 \beta_{14} - 4 \beta_{13} + 2 \beta_{11} + \beta_{10} - 2 \beta_{8} - 4 \beta_{7} + \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} + \beta_{9} + 4\beta_{8} - 2\beta_{6} - 2\beta_{5} - 2\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{11} - 9\beta_{10} - 30\beta_{7} - 6\beta_{6} + 9\beta_{5} + 2\beta_{4} + 6\beta_{2} - 30 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{15} + 6\beta_{12} + 12\beta_{11} - 2\beta_{10} - 13\beta_{7} + 2\beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 16 \beta_{15} + 16 \beta_{14} + 32 \beta_{13} + 2 \beta_{12} - 21 \beta_{10} + 2 \beta_{9} + 42 \beta_{8} - 42 \beta_{7} - 48 \beta_{6} - 21 \beta_{5} - 32 \beta_{3} - 48 \beta_{2} + 42 \beta _1 + 42 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16\beta_{14} + 9\beta_{13} + 22\beta_{9} + 14\beta_{8} - \beta_{6} + 2\beta_{5} + 4\beta_{4} - 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 42 \beta_{15} + 42 \beta_{14} + 60 \beta_{13} + 16 \beta_{12} + 90 \beta_{11} - 11 \beta_{10} - 16 \beta_{9} - 166 \beta_{8} + 68 \beta_{7} - 48 \beta_{6} + 11 \beta_{5} - 90 \beta_{4} + 60 \beta_{3} + 48 \beta_{2} + 166 \beta _1 + 68 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -39\beta_{12} + 12\beta_{11} + 60\beta_{10} + 12\beta_{7} + \beta_{3} - 44\beta_{2} + 24\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 112 \beta_{15} + 112 \beta_{14} - 160 \beta_{13} + 144 \beta_{12} - 502 \beta_{11} + 61 \beta_{10} + 144 \beta_{9} + 256 \beta_{8} + 278 \beta_{7} + 42 \beta_{6} + 61 \beta_{5} - 502 \beta_{4} + 160 \beta_{3} + 42 \beta_{2} + \cdots - 278 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -108\beta_{14} - 64\beta_{13} - 20\beta_{9} - 296\beta_{8} + 52\beta_{6} + 148\beta_{5} - 152\beta_{4} - 185 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 624 \beta_{15} - 624 \beta_{14} + 96 \beta_{13} + 738 \beta_{12} + 1024 \beta_{11} + 471 \beta_{10} - 738 \beta_{9} - 66 \beta_{8} - 1122 \beta_{7} + 144 \beta_{6} - 471 \beta_{5} - 1024 \beta_{4} + 96 \beta_{3} + \cdots - 1122 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 448 \beta_{15} + 468 \beta_{12} - 936 \beta_{11} - 476 \beta_{10} - 628 \beta_{7} - 9 \beta_{3} - 529 \beta_{2} - 20 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 82 \beta_{15} + 82 \beta_{14} + 116 \beta_{13} + 3152 \beta_{12} - 162 \beta_{11} - 1737 \beta_{10} + 3152 \beta_{9} - 2142 \beta_{8} + 3060 \beta_{7} - 144 \beta_{6} - 1737 \beta_{5} - 162 \beta_{4} - 116 \beta_{3} + \cdots - 3060 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2399.1
−0.901513 + 1.47894i
1.47894 + 0.901513i
−0.901513 1.47894i
1.47894 0.901513i
−0.672666 1.59610i
−1.59610 + 0.672666i
−0.672666 + 1.59610i
−1.59610 0.672666i
1.32799 1.11195i
1.11195 + 1.32799i
1.32799 + 1.11195i
1.11195 1.32799i
−0.427163 1.67855i
1.67855 0.427163i
−0.427163 + 1.67855i
1.67855 + 0.427163i
0 −1.68324 0.408305i 0 0 0 1.40591 0 2.66657 + 1.37455i 0
2399.2 0 −1.68324 0.408305i 0 0 0 1.40591 0 2.66657 + 1.37455i 0
2399.3 0 −1.68324 + 0.408305i 0 0 0 1.40591 0 2.66657 1.37455i 0
2399.4 0 −1.68324 + 0.408305i 0 0 0 1.40591 0 2.66657 1.37455i 0
2399.5 0 −0.652963 1.60426i 0 0 0 0.468520 0 −2.14728 + 2.09504i 0
2399.6 0 −0.652963 1.60426i 0 0 0 0.468520 0 −2.14728 + 2.09504i 0
2399.7 0 −0.652963 + 1.60426i 0 0 0 0.468520 0 −2.14728 2.09504i 0
2399.8 0 −0.652963 + 1.60426i 0 0 0 0.468520 0 −2.14728 2.09504i 0
2399.9 0 −0.152764 1.72530i 0 0 0 −4.54603 0 −2.95333 + 0.527129i 0
2399.10 0 −0.152764 1.72530i 0 0 0 −4.54603 0 −2.95333 + 0.527129i 0
2399.11 0 −0.152764 + 1.72530i 0 0 0 −4.54603 0 −2.95333 0.527129i 0
2399.12 0 −0.152764 + 1.72530i 0 0 0 −4.54603 0 −2.95333 0.527129i 0
2399.13 0 1.48896 0.884864i 0 0 0 2.67161 0 1.43403 2.63506i 0
2399.14 0 1.48896 0.884864i 0 0 0 2.67161 0 1.43403 2.63506i 0
2399.15 0 1.48896 + 0.884864i 0 0 0 2.67161 0 1.43403 + 2.63506i 0
2399.16 0 1.48896 + 0.884864i 0 0 0 2.67161 0 1.43403 + 2.63506i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2399.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b 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 2400.2.o.k 16
3.b odd 2 1 inner 2400.2.o.k 16
4.b odd 2 1 2400.2.o.l 16
5.b even 2 1 2400.2.o.l 16
5.c odd 4 1 2400.2.h.f 16
5.c odd 4 1 2400.2.h.g yes 16
12.b even 2 1 2400.2.o.l 16
15.d odd 2 1 2400.2.o.l 16
15.e even 4 1 2400.2.h.f 16
15.e even 4 1 2400.2.h.g yes 16
20.d odd 2 1 inner 2400.2.o.k 16
20.e even 4 1 2400.2.h.f 16
20.e even 4 1 2400.2.h.g yes 16
60.h even 2 1 inner 2400.2.o.k 16
60.l odd 4 1 2400.2.h.f 16
60.l odd 4 1 2400.2.h.g yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.2.h.f 16 5.c odd 4 1
2400.2.h.f 16 15.e even 4 1
2400.2.h.f 16 20.e even 4 1
2400.2.h.f 16 60.l odd 4 1
2400.2.h.g yes 16 5.c odd 4 1
2400.2.h.g yes 16 15.e even 4 1
2400.2.h.g yes 16 20.e even 4 1
2400.2.h.g yes 16 60.l odd 4 1
2400.2.o.k 16 1.a even 1 1 trivial
2400.2.o.k 16 3.b odd 2 1 inner
2400.2.o.k 16 20.d odd 2 1 inner
2400.2.o.k 16 60.h even 2 1 inner
2400.2.o.l 16 4.b odd 2 1
2400.2.o.l 16 5.b even 2 1
2400.2.o.l 16 12.b even 2 1
2400.2.o.l 16 15.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 \(S_{2}^{\mathrm{new}}(2400, [\chi])\):

\( T_{7}^{4} - 15T_{7}^{2} + 24T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{8} - 58T_{11}^{6} + 833T_{11}^{4} - 2656T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{8} - 54T_{17}^{6} + 1001T_{17}^{4} - 6912T_{17}^{2} + 10000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 2 T^{7} + 3 T^{6} + 2 T^{5} - 4 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} - 15 T^{2} + 24 T - 8)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 58 T^{6} + 833 T^{4} - 2656 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 50 T^{6} + 857 T^{4} + 5528 T^{2} + \cdots + 8464)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 54 T^{6} + 1001 T^{4} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 52 T^{6} + 550 T^{4} + 1620 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 136 T^{6} + 5792 T^{4} + \cdots + 350464)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 104 T^{6} + 2848 T^{4} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 150 T^{6} + 5377 T^{4} + \cdots + 71824)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 88 T^{6} + 1616 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 182 T^{6} + 8969 T^{4} + \cdots + 364816)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 16 T^{3} + 17 T^{2} + 700 T - 2636)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 216 T^{6} + 14224 T^{4} + \cdots + 1048576)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 408 T^{6} + 54176 T^{4} + \cdots + 8386816)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 240 T^{6} + 8768 T^{4} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 2 T^{3} - 147 T^{2} - 80 T + 3656)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 2 T^{3} - 112 T^{2} - 142 T + 2959)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} - 184 T^{6} + 8480 T^{4} + \cdots + 246016)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 458 T^{6} + 53921 T^{4} + \cdots + 425104)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 200 T^{6} + 8272 T^{4} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 394 T^{6} + 43649 T^{4} + \cdots + 6310144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 438 T^{6} + 62713 T^{4} + \cdots + 56070144)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 458 T^{6} + 62305 T^{4} + \cdots + 810000)^{2} \) Copy content Toggle raw display
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