Properties

Label 2300.4.c.c
Level $2300$
Weight $4$
Character orbit 2300.c
Analytic conductor $135.704$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2300,4,Mod(1749,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2300.1749");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2300.c (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: \(135.704393013\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 114x^{8} + 3951x^{6} + 38578x^{4} + 50049x^{2} + 5184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 460)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{4}) q^{3} + (\beta_{8} + 2 \beta_{7} + 5 \beta_{4} - \beta_{3}) q^{7} + ( - 2 \beta_{5} + \beta_{2} - 3 \beta_1 - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{4}) q^{3} + (\beta_{8} + 2 \beta_{7} + 5 \beta_{4} - \beta_{3}) q^{7} + ( - 2 \beta_{5} + \beta_{2} - 3 \beta_1 - 6) q^{9} + (2 \beta_{6} - 3 \beta_{5} - \beta_{2} - 12) q^{11} + (2 \beta_{9} - 8 \beta_{8} + \beta_{7} - 22 \beta_{4}) q^{13} + (2 \beta_{9} + 6 \beta_{8} - 7 \beta_{7} + 10 \beta_{4} + 3 \beta_{3}) q^{17} + ( - 3 \beta_{6} - 5 \beta_{5} + 2 \beta_{2} + 7 \beta_1 + 44) q^{19} + (17 \beta_{6} - 6 \beta_{5} + 4 \beta_{2} - \beta_1 - 19) q^{21} + 23 \beta_{4} q^{23} + ( - 3 \beta_{9} - 3 \beta_{7} + 4 \beta_{4} - \beta_{3}) q^{27} + ( - 3 \beta_{6} - 6 \beta_{5} - 11 \beta_{2} - 13 \beta_1 - 43) q^{29} + (4 \beta_{6} - 3 \beta_{5} + 5 \beta_{2} - 10 \beta_1 - 107) q^{31} + ( - 9 \beta_{9} - 21 \beta_{8} - 15 \beta_{7} - 36 \beta_{4} - 6 \beta_{3}) q^{33} + (21 \beta_{9} - 24 \beta_{8} - \beta_{7} + 16 \beta_{4} + 26 \beta_{3}) q^{37} + ( - 28 \beta_{6} + 3 \beta_{5} + 31 \beta_1 + 211) q^{39} + ( - 23 \beta_{6} - 9 \beta_{5} - \beta_{2} - \beta_1 - 22) q^{41} + (16 \beta_{9} - 12 \beta_{8} + 6 \beta_{7} - 63 \beta_{4} + 31 \beta_{3}) q^{43} + ( - 9 \beta_{9} - 25 \beta_{8} - 52 \beta_{7} + 48 \beta_{4}) q^{47} + (14 \beta_{6} + 2 \beta_{5} - 21 \beta_{2} - 7 \beta_1 + 29) q^{49} + ( - 27 \beta_{6} - 7 \beta_{5} + \beta_{2} - 19 \beta_1 - 223) q^{51} + ( - 5 \beta_{9} + 46 \beta_{8} + \beta_{7} + 69 \beta_{4} - 45 \beta_{3}) q^{53} + ( - 56 \beta_{9} + 81 \beta_{8} - 13 \beta_{7} + 151 \beta_{4} - 34 \beta_{3}) q^{57} + (38 \beta_{6} - 3 \beta_{5} + 21 \beta_{2} + 15 \beta_1 + 295) q^{59} + ( - 42 \beta_{6} + 15 \beta_{5} + 7 \beta_{2} - 30 \beta_1 - 144) q^{61} + ( - \beta_{9} - 23 \beta_{8} - 14 \beta_{7} - 277 \beta_{4} - 30 \beta_{3}) q^{63} + (63 \beta_{9} - 92 \beta_{8} - 31 \beta_{7} + 84 \beta_{4} + 30 \beta_{3}) q^{67} + (23 \beta_{6} + 23) q^{69} + ( - 95 \beta_{6} + 35 \beta_{5} - 26 \beta_{2} + 45 \beta_1 - 309) q^{71} + (\beta_{9} - 79 \beta_{8} + 26 \beta_{7} - 163 \beta_{4} + 57 \beta_{3}) q^{73} + (30 \beta_{9} - 95 \beta_{8} - 51 \beta_{7} + 132 \beta_{4} + 41 \beta_{3}) q^{77} + ( - 6 \beta_{6} + 6 \beta_{5} - 47 \beta_{2} - 6 \beta_1 + 599) q^{79} + ( - 9 \beta_{6} - 28 \beta_{5} + 11 \beta_{2} - 93 \beta_1 - 105) q^{81} + ( - 9 \beta_{9} - 84 \beta_{8} - \beta_{7} - 87 \beta_{4} + 95 \beta_{3}) q^{83} + (25 \beta_{9} - 141 \beta_{8} + 4 \beta_{7} + 6 \beta_{4} + 16 \beta_{3}) q^{87} + (112 \beta_{6} + 32 \beta_{5} + 89 \beta_{2} + 34 \beta_1 + 243) q^{89} + ( - 196 \beta_{6} + 57 \beta_{5} - 75 \beta_{2} + 64 \beta_1 + 76) q^{91} + (13 \beta_{9} - 172 \beta_{8} - 11 \beta_{7} + 49 \beta_{4} + 10 \beta_{3}) q^{93} + ( - 25 \beta_{9} + 16 \beta_{8} + 148 \beta_{7} + 308 \beta_{4} - 35 \beta_{3}) q^{97} + ( - 81 \beta_{6} + 63 \beta_{5} - 111 \beta_{2} + 21 \beta_1 + 531) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 60 q^{9} - 126 q^{11} + 398 q^{19} - 136 q^{21} - 462 q^{29} - 1036 q^{31} + 1948 q^{39} - 348 q^{41} + 326 q^{49} - 2326 q^{51} + 3102 q^{59} - 1474 q^{61} + 322 q^{69} - 3472 q^{71} + 5908 q^{79} - 990 q^{81} + 3116 q^{89} - 74 q^{91} + 4974 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 114x^{8} + 3951x^{6} + 38578x^{4} + 50049x^{2} + 5184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} + 128\nu^{6} + 5188\nu^{4} + 71805\nu^{2} + 222264 ) / 19980 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 17\nu^{6} - 2027\nu^{4} - 20214\nu^{2} + 19467 ) / 12987 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{9} + 566\nu^{7} + 19687\nu^{5} + 200998\nu^{3} + 538893\nu ) / 103896 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{9} + 566\nu^{7} + 19687\nu^{5} + 200998\nu^{3} + 331101\nu ) / 103896 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -29\nu^{8} - 4822\nu^{6} - 222602\nu^{4} - 2742795\nu^{2} - 2629476 ) / 259740 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 106\nu^{8} + 11903\nu^{6} + 398413\nu^{4} + 3460485\nu^{2} + 1881684 ) / 259740 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 157\nu^{9} + 18061\nu^{7} + 631736\nu^{5} + 6245825\nu^{3} + 8969733\nu ) / 259740 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -313\nu^{9} - 35624\nu^{7} - 1227019\nu^{5} - 11629710\nu^{3} - 10179747\nu ) / 519480 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 387\nu^{9} + 43616\nu^{7} + 1485871\nu^{5} + 13955900\nu^{3} + 15351903\nu ) / 519480 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} - \beta_{2} + 6\beta _1 - 45 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{9} + 6\beta_{8} + 3\beta_{7} + 14\beta_{4} - 23\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -12\beta_{6} - 128\beta_{5} + 73\beta_{2} - 300\beta _1 + 2019 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -378\beta_{9} - 684\beta_{8} - 354\beta_{7} - 499\beta_{4} + 2233\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 390\beta_{6} + 3331\beta_{5} - 2075\beta_{2} + 7443\beta _1 - 48792 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19128\beta_{9} + 34872\beta_{8} + 19800\beta_{7} + 2257\beta_{4} - 110449\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -37584\beta_{6} - 332282\beta_{5} + 224281\beta_{2} - 739878\beta _1 + 4802877 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -459075\beta_{9} - 868362\beta_{8} - 544359\beta_{7} + 345730\beta_{4} + 2746787\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1749.1
3.65968i
1.17944i
0.336706i
7.11066i
6.96712i
6.96712i
7.11066i
0.336706i
1.17944i
3.65968i
0 7.99061i 0 0 0 4.09549i 0 −36.8498 0
1749.2 0 7.44221i 0 0 0 23.3042i 0 −28.3866 0
1749.3 0 6.75371i 0 0 0 19.0133i 0 −18.6126 0
1749.4 0 0.380607i 0 0 0 24.3526i 0 26.8551 0
1749.5 0 0.0785019i 0 0 0 6.13060i 0 26.9938 0
1749.6 0 0.0785019i 0 0 0 6.13060i 0 26.9938 0
1749.7 0 0.380607i 0 0 0 24.3526i 0 26.8551 0
1749.8 0 6.75371i 0 0 0 19.0133i 0 −18.6126 0
1749.9 0 7.44221i 0 0 0 23.3042i 0 −28.3866 0
1749.10 0 7.99061i 0 0 0 4.09549i 0 −36.8498 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1749.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.4.c.c 10
5.b even 2 1 inner 2300.4.c.c 10
5.c odd 4 1 460.4.a.a 5
5.c odd 4 1 2300.4.a.d 5
20.e even 4 1 1840.4.a.q 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.4.a.a 5 5.c odd 4 1
1840.4.a.q 5 20.e even 4 1
2300.4.a.d 5 5.c odd 4 1
2300.4.c.c 10 1.a even 1 1 trivial
2300.4.c.c 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 165T_{3}^{8} + 9000T_{3}^{6} + 162661T_{3}^{4} + 24369T_{3}^{2} + 144 \) acting on \(S_{4}^{\mathrm{new}}(2300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 165 T^{8} + 9000 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 1552 T^{8} + \cdots + 73399813776 \) Copy content Toggle raw display
$11$ \( (T^{5} + 63 T^{4} - 1803 T^{3} + \cdots + 72719208)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 207155282552100 \) Copy content Toggle raw display
$17$ \( T^{10} + 24648 T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} - 199 T^{4} - 4473 T^{3} + \cdots + 800499864)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 529)^{5} \) Copy content Toggle raw display
$29$ \( (T^{5} + 231 T^{4} + \cdots - 43256062404)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 518 T^{4} + 91393 T^{3} + \cdots - 9011869285)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 412901 T^{8} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{5} + 174 T^{4} + \cdots + 12760146261)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 502076 T^{8} + \cdots + 64\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{10} + 1051206 T^{8} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{10} + 897433 T^{8} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{5} - 1551 T^{4} + \cdots + 897857553600)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 737 T^{4} + \cdots + 310836562128)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 2673629 T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + 1736 T^{4} + \cdots - 108175489939041)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 2104066 T^{8} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{5} - 2954 T^{4} + \cdots + 8954177264640)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 4330785 T^{8} + \cdots + 84\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{5} - 1558 T^{4} + \cdots - 180704830832640)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 6167551 T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
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