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Properties

Label 375.2.i.c
Level $375$
Weight $2$
Character orbit 375.i
Analytic conductor $2.994$
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] = [375,2,Mod(49,375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(375, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("375.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.i (of order \(10\), degree \(4\), 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: \(2.99439007580\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9 \nu^{15} - 11 \nu^{14} + 158 \nu^{13} - 194 \nu^{12} + 1016 \nu^{11} - 1257 \nu^{10} + 2975 \nu^{9} - 3731 \nu^{8} + 4097 \nu^{7} - 5277 \nu^{6} + 2332 \nu^{5} - 3223 \nu^{4} + 270 \nu^{3} - 565 \nu^{2} + \cdots - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15 \nu^{15} - 6 \nu^{14} + 263 \nu^{13} - 106 \nu^{12} + 1688 \nu^{11} - 689 \nu^{10} + 4929 \nu^{9} - 2058 \nu^{8} + 6764 \nu^{7} - 2948 \nu^{6} + 3837 \nu^{5} - 1847 \nu^{4} + 444 \nu^{3} - 339 \nu^{2} + \cdots + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9 \nu^{15} - 11 \nu^{14} - 158 \nu^{13} - 194 \nu^{12} - 1016 \nu^{11} - 1257 \nu^{10} - 2975 \nu^{9} - 3731 \nu^{8} - 4097 \nu^{7} - 5277 \nu^{6} - 2332 \nu^{5} - 3223 \nu^{4} - 270 \nu^{3} + \cdots - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15 \nu^{15} - 6 \nu^{14} - 263 \nu^{13} - 106 \nu^{12} - 1688 \nu^{11} - 689 \nu^{10} - 4929 \nu^{9} - 2058 \nu^{8} - 6764 \nu^{7} - 2948 \nu^{6} - 3837 \nu^{5} - 1847 \nu^{4} - 444 \nu^{3} + \cdots + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -24\nu^{14} - 423\nu^{12} - 2739\nu^{10} - 8128\nu^{8} - 11513\nu^{6} - 7066\nu^{4} - 1247\nu^{2} - 2\nu - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -24\nu^{14} - 423\nu^{12} - 2739\nu^{10} - 8128\nu^{8} - 11513\nu^{6} - 7066\nu^{4} - 1247\nu^{2} + 2\nu - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7 \nu^{15} - 32 \nu^{14} + 124 \nu^{13} - 564 \nu^{12} + 810 \nu^{11} - 3652 \nu^{10} + 2444 \nu^{9} - 10838 \nu^{8} + 3583 \nu^{7} - 15360 \nu^{6} + 2400 \nu^{5} - 9462 \nu^{4} + 595 \nu^{3} + \cdots - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7 \nu^{15} - 32 \nu^{14} - 124 \nu^{13} - 564 \nu^{12} - 810 \nu^{11} - 3652 \nu^{10} - 2444 \nu^{9} - 10838 \nu^{8} - 3583 \nu^{7} - 15360 \nu^{6} - 2400 \nu^{5} - 9462 \nu^{4} - 595 \nu^{3} + \cdots - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12 \nu^{15} - 45 \nu^{14} + 208 \nu^{13} - 794 \nu^{12} + 1308 \nu^{11} - 5150 \nu^{10} + 3668 \nu^{9} - 15324 \nu^{8} + 4594 \nu^{7} - 21803 \nu^{6} + 1920 \nu^{5} - 13514 \nu^{4} - 330 \nu^{3} + \cdots - 17 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10 \nu^{15} + 45 \nu^{14} - 177 \nu^{13} + 794 \nu^{12} - 1154 \nu^{11} + 5150 \nu^{10} - 3465 \nu^{9} + 15324 \nu^{8} - 5013 \nu^{7} + 21803 \nu^{6} - 3225 \nu^{5} + 13514 \nu^{4} + \cdots + 23 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6 \nu^{15} - 45 \nu^{14} + 108 \nu^{13} - 794 \nu^{12} + 724 \nu^{11} - 5150 \nu^{10} + 2282 \nu^{9} - 15324 \nu^{8} + 3600 \nu^{7} - 21803 \nu^{6} + 2744 \nu^{5} - 13514 \nu^{4} + 870 \nu^{3} + \cdots - 17 ) / 4 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 22 \nu^{15} - 29 \nu^{14} - 387 \nu^{13} - 512 \nu^{12} - 2498 \nu^{11} - 3324 \nu^{10} - 7373 \nu^{9} - 9905 \nu^{8} - 10347 \nu^{7} - 14123 \nu^{6} - 6237 \nu^{5} - 8782 \nu^{4} - 1041 \nu^{3} + \cdots - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10 \nu^{15} + 45 \nu^{14} + 177 \nu^{13} + 794 \nu^{12} + 1154 \nu^{11} + 5150 \nu^{10} + 3465 \nu^{9} + 15324 \nu^{8} + 5013 \nu^{7} + 21803 \nu^{6} + 3225 \nu^{5} + 13514 \nu^{4} + 675 \nu^{3} + \cdots + 23 ) / 4 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 29 \nu^{15} + 32 \nu^{14} + 512 \nu^{13} + 564 \nu^{12} + 3325 \nu^{11} + 3652 \nu^{10} + 9921 \nu^{9} + 10838 \nu^{8} + 14211 \nu^{7} + 15360 \nu^{6} + 8977 \nu^{5} + 9462 \nu^{4} + 1785 \nu^{3} + \cdots + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 45 \nu^{15} - 17 \nu^{14} - 794 \nu^{13} - 299 \nu^{12} - 5150 \nu^{11} - 1930 \nu^{10} - 15324 \nu^{9} - 5701 \nu^{8} - 21803 \nu^{7} - 8032 \nu^{6} - 13514 \nu^{5} - 4923 \nu^{4} - 2487 \nu^{3} + \cdots - 8 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_{6} - \beta_{5} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{3} - \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{13} - \beta_{12} + \beta_{10} + 2\beta_{8} - 5\beta_{6} + 4\beta_{5} + \beta_{4} + \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} - 9 \beta_{13} + 2 \beta_{12} - 9 \beta_{11} - 5 \beta_{10} - 9 \beta_{9} + 3 \beta_{8} + 3 \beta_{7} - 2 \beta_{5} - 7 \beta_{3} + 2 \beta_{2} + 7 \beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{15} + 7 \beta_{13} + 8 \beta_{12} - 7 \beta_{10} - 17 \beta_{8} + \beta_{7} + 28 \beta_{6} - 20 \beta_{5} - 7 \beta_{4} - 10 \beta_{3} - \beta_{2} + 10 \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19 \beta_{15} + 63 \beta_{13} - 19 \beta_{12} + 62 \beta_{11} + 25 \beta_{10} + 62 \beta_{9} - 28 \beta_{8} - 28 \beta_{7} + 19 \beta_{5} + 3 \beta_{4} + 44 \beta_{3} - 16 \beta_{2} - 42 \beta _1 - 130 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 57 \beta_{15} - 8 \beta_{14} - 43 \beta_{13} - 57 \beta_{12} + \beta_{11} + 43 \beta_{10} - \beta_{9} + 116 \beta_{8} - 10 \beta_{7} - 167 \beta_{6} + 110 \beta_{5} + 43 \beta_{4} + 78 \beta_{3} + 14 \beta_{2} - 78 \beta _1 + 53 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 144 \beta_{15} - 413 \beta_{13} + 144 \beta_{12} - 399 \beta_{11} - 125 \beta_{10} - 399 \beta_{9} + 206 \beta_{8} + 206 \beta_{7} + 4 \beta_{6} - 140 \beta_{5} - 42 \beta_{4} - 269 \beta_{3} + 102 \beta_{2} + 241 \beta _1 + 826 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 395 \beta_{15} + 120 \beta_{14} + 259 \beta_{13} + 395 \beta_{12} - 18 \beta_{11} - 259 \beta_{10} + 18 \beta_{9} - 752 \beta_{8} + 82 \beta_{7} + 1024 \beta_{6} - 629 \beta_{5} - 255 \beta_{4} - 563 \beta_{3} - 140 \beta_{2} + \cdots - 335 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1018 \beta_{15} + 2654 \beta_{13} - 1018 \beta_{12} + 2518 \beta_{11} + 618 \beta_{10} + 2518 \beta_{9} - 1415 \beta_{8} - 1415 \beta_{7} - 60 \beta_{6} + 958 \beta_{5} + 406 \beta_{4} + 1640 \beta_{3} - 612 \beta_{2} + \cdots - 5209 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2699 \beta_{15} - 1212 \beta_{14} - 1564 \beta_{13} - 2699 \beta_{12} + 200 \beta_{11} + 1564 \beta_{10} - 200 \beta_{9} + 4811 \beta_{8} - 625 \beta_{7} - 6359 \beta_{6} + 3660 \beta_{5} + 1496 \beta_{4} + \cdots + 2093 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6987 \beta_{15} - 16954 \beta_{13} + 6987 \beta_{12} - 15819 \beta_{11} - 2980 \beta_{10} - 15819 \beta_{9} + 9477 \beta_{8} + 9477 \beta_{7} + 606 \beta_{6} - 6381 \beta_{5} - 3379 \beta_{4} + \cdots + 32816 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 18251 \beta_{15} + 10376 \beta_{14} + 9506 \beta_{13} + 18251 \beta_{12} - 1809 \beta_{11} - 9506 \beta_{10} + 1809 \beta_{9} - 30691 \beta_{8} + 4565 \beta_{7} + 39759 \beta_{6} - 21508 \beta_{5} + \cdots - 13063 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 47209 \beta_{15} + 108172 \beta_{13} - 47209 \beta_{12} + 99427 \beta_{11} + 13754 \beta_{10} + 99427 \beta_{9} - 62762 \beta_{8} - 62762 \beta_{7} - 5188 \beta_{6} + 42021 \beta_{5} + 26005 \beta_{4} + \cdots - 207014 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 122384 \beta_{15} - 81360 \beta_{14} - 58148 \beta_{13} - 122384 \beta_{12} + 14675 \beta_{11} + 58148 \beta_{10} - 14675 \beta_{9} + 195806 \beta_{8} - 32398 \beta_{7} - 249699 \beta_{6} + \cdots + 81704 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/375\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(-\beta_{7}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
2.53767i
0.0898194i
− 1.08982i
− 1.53767i
− 2.53767i
− 0.0898194i
1.08982i
1.53767i
1.53655i
1.35083i
− 0.536547i
− 2.35083i
− 1.53655i
− 1.35083i
0.536547i
2.35083i
−1.49161 + 2.05302i 0.951057 − 0.309017i −1.37197 − 4.22249i 0 −0.784184 + 2.41347i − 1.04054i 5.88835 + 1.91324i 0.809017 − 0.587785i 0
49.2 −0.0527945 + 0.0726655i −0.951057 + 0.309017i 0.615541 + 1.89444i 0 0.0277557 − 0.0854234i 4.36070i −0.341004 − 0.110799i 0.809017 − 0.587785i 0
49.3 0.640580 − 0.881682i −0.951057 + 0.309017i 0.251013 + 0.772537i 0 −0.336773 + 1.03648i − 3.08724i 2.91489 + 0.947104i 0.809017 − 0.587785i 0
49.4 0.903822 − 1.24400i 0.951057 − 0.309017i −0.112618 − 0.346603i 0 0.475167 − 1.46241i − 1.68601i 2.39187 + 0.777165i 0.809017 − 0.587785i 0
199.1 −1.49161 − 2.05302i 0.951057 + 0.309017i −1.37197 + 4.22249i 0 −0.784184 − 2.41347i 1.04054i 5.88835 − 1.91324i 0.809017 + 0.587785i 0
199.2 −0.0527945 − 0.0726655i −0.951057 − 0.309017i 0.615541 − 1.89444i 0 0.0277557 + 0.0854234i − 4.36070i −0.341004 + 0.110799i 0.809017 + 0.587785i 0
199.3 0.640580 + 0.881682i −0.951057 − 0.309017i 0.251013 − 0.772537i 0 −0.336773 − 1.03648i 3.08724i 2.91489 − 0.947104i 0.809017 + 0.587785i 0
199.4 0.903822 + 1.24400i 0.951057 + 0.309017i −0.112618 + 0.346603i 0 0.475167 + 1.46241i 1.68601i 2.39187 − 0.777165i 0.809017 + 0.587785i 0
274.1 −1.46134 − 0.474819i −0.587785 + 0.809017i 0.292036 + 0.212177i 0 1.24309 − 0.903160i − 1.49550i 1.48030 + 2.03746i −0.309017 − 0.951057i 0
274.2 −1.28472 − 0.417429i 0.587785 − 0.809017i −0.141788 − 0.103015i 0 −1.09284 + 0.793998i 1.59580i 1.72715 + 2.37722i −0.309017 − 0.951057i 0
274.3 0.510286 + 0.165802i −0.587785 + 0.809017i −1.38513 − 1.00636i 0 −0.434076 + 0.315374i 2.57318i −1.17071 − 1.61134i −0.309017 − 0.951057i 0
274.4 2.23577 + 0.726446i 0.587785 − 0.809017i 2.85292 + 2.07277i 0 1.90186 − 1.38178i 3.48189i 2.10915 + 2.90300i −0.309017 − 0.951057i 0
349.1 −1.46134 + 0.474819i −0.587785 − 0.809017i 0.292036 − 0.212177i 0 1.24309 + 0.903160i 1.49550i 1.48030 − 2.03746i −0.309017 + 0.951057i 0
349.2 −1.28472 + 0.417429i 0.587785 + 0.809017i −0.141788 + 0.103015i 0 −1.09284 − 0.793998i − 1.59580i 1.72715 − 2.37722i −0.309017 + 0.951057i 0
349.3 0.510286 − 0.165802i −0.587785 − 0.809017i −1.38513 + 1.00636i 0 −0.434076 − 0.315374i − 2.57318i −1.17071 + 1.61134i −0.309017 + 0.951057i 0
349.4 2.23577 − 0.726446i 0.587785 + 0.809017i 2.85292 − 2.07277i 0 1.90186 + 1.38178i − 3.48189i 2.10915 − 2.90300i −0.309017 + 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.i.c 16
5.b even 2 1 75.2.i.a ✓ 16
5.c odd 4 1 375.2.g.d 16
5.c odd 4 1 375.2.g.e 16
15.d odd 2 1 225.2.m.b 16
25.d even 5 1 75.2.i.a ✓ 16
25.d even 5 1 1875.2.b.h 16
25.e even 10 1 inner 375.2.i.c 16
25.e even 10 1 1875.2.b.h 16
25.f odd 20 1 375.2.g.d 16
25.f odd 20 1 375.2.g.e 16
25.f odd 20 1 1875.2.a.m 8
25.f odd 20 1 1875.2.a.p 8
75.j odd 10 1 225.2.m.b 16
75.l even 20 1 5625.2.a.t 8
75.l even 20 1 5625.2.a.bd 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.i.a ✓ 16 5.b even 2 1
75.2.i.a ✓ 16 25.d even 5 1
225.2.m.b 16 15.d odd 2 1
225.2.m.b 16 75.j odd 10 1
375.2.g.d 16 5.c odd 4 1
375.2.g.d 16 25.f odd 20 1
375.2.g.e 16 5.c odd 4 1
375.2.g.e 16 25.f odd 20 1
375.2.i.c 16 1.a even 1 1 trivial
375.2.i.c 16 25.e even 10 1 inner
1875.2.a.m 8 25.f odd 20 1
1875.2.a.p 8 25.f odd 20 1
1875.2.b.h 16 25.d even 5 1
1875.2.b.h 16 25.e even 10 1
5625.2.a.t 8 75.l even 20 1
5625.2.a.bd 8 75.l even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 5 T_{2}^{14} - 10 T_{2}^{13} + 26 T_{2}^{12} + 50 T_{2}^{11} - 95 T_{2}^{10} - 20 T_{2}^{9} + 381 T_{2}^{8} - 30 T_{2}^{7} - 365 T_{2}^{6} + 410 T_{2}^{5} + 226 T_{2}^{4} - 360 T_{2}^{3} + 85 T_{2}^{2} + 10 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 5 T^{14} - 10 T^{13} + 26 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 56 T^{14} + 1236 T^{12} + \cdots + 255025 \) Copy content Toggle raw display
$11$ \( T^{16} + 6 T^{15} + 48 T^{14} + \cdots + 27888961 \) Copy content Toggle raw display
$13$ \( T^{16} - 30 T^{14} - 210 T^{13} + \cdots + 78961 \) Copy content Toggle raw display
$17$ \( T^{16} + 10 T^{15} - 13 T^{14} + \cdots + 53860921 \) Copy content Toggle raw display
$19$ \( T^{16} + 2 T^{15} + 23 T^{14} + \cdots + 6375625 \) Copy content Toggle raw display
$23$ \( T^{16} - 20 T^{15} + 89 T^{14} + \cdots + 4389025 \) Copy content Toggle raw display
$29$ \( T^{16} - 16 T^{15} + 217 T^{14} + \cdots + 156025 \) Copy content Toggle raw display
$31$ \( T^{16} - 6 T^{15} + 97 T^{14} + \cdots + 15625 \) Copy content Toggle raw display
$37$ \( T^{16} - 10 T^{15} + \cdots + 8653650625 \) Copy content Toggle raw display
$41$ \( T^{16} + 14 T^{15} + 82 T^{14} + \cdots + 22137025 \) Copy content Toggle raw display
$43$ \( T^{16} + 388 T^{14} + \cdots + 527207521 \) Copy content Toggle raw display
$47$ \( T^{16} + 40 T^{15} + \cdots + 36687479166361 \) Copy content Toggle raw display
$53$ \( T^{16} + 10 T^{15} + \cdots + 40398990025 \) Copy content Toggle raw display
$59$ \( T^{16} - 12 T^{15} + 188 T^{14} + \cdots + 12924025 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 275701483356121 \) Copy content Toggle raw display
$67$ \( T^{16} - 40 T^{15} + 762 T^{14} + \cdots + 13980121 \) Copy content Toggle raw display
$71$ \( T^{16} + 8 T^{15} + \cdots + 25529328841 \) Copy content Toggle raw display
$73$ \( T^{16} - 20 T^{15} + \cdots + 757413387025 \) Copy content Toggle raw display
$79$ \( T^{16} + 20 T^{15} + \cdots + 3940125750625 \) Copy content Toggle raw display
$83$ \( T^{16} + 10 T^{15} + \cdots + 2356228681 \) Copy content Toggle raw display
$89$ \( T^{16} - 18 T^{15} + 68 T^{14} + 1034 T^{13} + \cdots + 25 \) Copy content Toggle raw display
$97$ \( T^{16} + 40 T^{15} + \cdots + 216648961 \) Copy content Toggle raw display
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