Properties

Label 2100.2.bo.g
Level $2100$
Weight $2$
Character orbit 2100.bo
Analytic conductor $16.769$
Analytic rank $0$
Dimension $20$
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] = [2100,2,Mod(1349,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bo (of order \(6\), degree \(2\), 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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{18} - 16 x^{16} + 87 x^{14} + 91 x^{12} - 1104 x^{10} + 819 x^{8} + 7047 x^{6} - 11664 x^{4} - 19683 x^{2} + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{13} + \beta_{4}) q^{3} + (\beta_{18} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{14} + \beta_{12} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{13} + \beta_{4}) q^{3} + (\beta_{18} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{14} + \beta_{12} - 1) q^{9} + (\beta_{12} + \beta_{8}) q^{11} + (\beta_{18} - \beta_{17} - \beta_{13} + 2 \beta_{10} - 2 \beta_{6} + \beta_{4} + 2 \beta_{2} - \beta_1) q^{13} + ( - 2 \beta_{17} + \beta_{13} + \beta_{10} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1) q^{17} + ( - \beta_{16} - \beta_{14} + \beta_{9} - \beta_{3}) q^{19} + ( - 2 \beta_{16} + \beta_{15} - \beta_{12} + \beta_{9} - \beta_{8} + 3 \beta_{7} - \beta_{3} + 1) q^{21} + ( - \beta_{19} - \beta_{18} + \beta_{10} + \beta_{6} + 2 \beta_{5} - \beta_{2}) q^{23} + (\beta_{19} - 2 \beta_{18} - \beta_{17} + \beta_{13} + 2 \beta_{6} - \beta_{4} - 2 \beta_{2} + \beta_1) q^{27} + ( - \beta_{16} + \beta_{15} - 2 \beta_{12} - \beta_{11} + \beta_{9} - 2 \beta_{7} + \beta_{3} + 2) q^{29} + (\beta_{16} - \beta_{14} - \beta_{12} - \beta_{9} - \beta_{8} + \beta_{7} + 1) q^{31} + (\beta_{19} - 2 \beta_{18} + 2 \beta_{17} - 2 \beta_{10} + \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{2} + \beta_1) q^{33} + ( - \beta_{19} + 2 \beta_{18} + \beta_{17} + \beta_{13} + 2 \beta_{10} - \beta_{6} + 3 \beta_{2} + \beta_1) q^{37} + (\beta_{16} - 2 \beta_{15} + 3 \beta_{12} + 2 \beta_{11} - 2 \beta_{9} + \beta_{8} - \beta_{3}) q^{39} + ( - 3 \beta_{16} - \beta_{14} + 2 \beta_{9} - 3 \beta_{3} + 2) q^{41} + ( - 3 \beta_{19} + 3 \beta_{18} + \beta_{10} + \beta_{6} + 3 \beta_{5}) q^{43} + (3 \beta_{19} - 4 \beta_{18} - 4 \beta_{10} + 2 \beta_{6} + 2 \beta_{4} - 6 \beta_{2} + 2 \beta_1) q^{47} + (\beta_{16} - \beta_{15} + \beta_{14} - 2 \beta_{12} + \beta_{8} + \beta_{7} + 2 \beta_{3} + 1) q^{49} + (\beta_{16} - \beta_{14} + 2 \beta_{11} - 2 \beta_{8} - 3 \beta_{7} - 2 \beta_{3} + 6) q^{51} + (2 \beta_{19} - 2 \beta_{18} - 2 \beta_{13} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_1) q^{53} + (\beta_{10} - \beta_{6} + \beta_{5} + 2 \beta_{2}) q^{57} + (2 \beta_{16} + \beta_{12} + 4 \beta_{11} - \beta_{8} - 2 \beta_{3}) q^{59} + ( - \beta_{16} - \beta_{14} + \beta_{9} - 3 \beta_{7} - \beta_{3} + 6) q^{61} + ( - 2 \beta_{19} + \beta_{18} - \beta_{17} - \beta_{13} - \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + \beta_{2} + \beta_1) q^{63} + (\beta_{17} - 4 \beta_{13} + \beta_{10} - \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{2} - 3 \beta_1) q^{67} + (3 \beta_{16} + \beta_{15} + 3 \beta_{12} - \beta_{9} + \beta_{8} - 3) q^{69} + ( - 2 \beta_{15} - 2 \beta_{14} + 3 \beta_{12} + \beta_{11} + 2 \beta_{9} - \beta_{7} - 1) q^{71} + ( - 2 \beta_{19} + 3 \beta_{18} + 4 \beta_{17} + \beta_{13} + \beta_{10} - 2 \beta_{6} + \cdots + 3 \beta_1) q^{73}+ \cdots + (\beta_{16} + 2 \beta_{15} - \beta_{12} - 4 \beta_{11} - 2 \beta_{9} + 3 \beta_{7} + \beta_{3} - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 6 q^{9} + 12 q^{11} - 6 q^{19} + 24 q^{21} + 30 q^{31} + 42 q^{39} + 16 q^{41} + 26 q^{49} + 80 q^{51} + 84 q^{61} - 28 q^{69} - 2 q^{79} - 26 q^{81} + 56 q^{89} - 22 q^{91} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{18} - 16 x^{16} + 87 x^{14} + 91 x^{12} - 1104 x^{10} + 819 x^{8} + 7047 x^{6} - 11664 x^{4} - 19683 x^{2} + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 29 \nu^{19} - 966 \nu^{17} + 3218 \nu^{15} + 6063 \nu^{13} - 4664 \nu^{11} + 13224 \nu^{9} - 164367 \nu^{7} - 18954 \nu^{5} + 1981422 \nu^{3} - 2119203 \nu ) / 5038848 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 47 \nu^{18} + 654 \nu^{16} - 1354 \nu^{14} + 1797 \nu^{12} + 18808 \nu^{10} - 57192 \nu^{8} - 63765 \nu^{6} + 300834 \nu^{4} - 1217430 \nu^{2} + 1673055 ) / 1679616 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 83 \nu^{19} + 930 \nu^{17} - 86 \nu^{15} - 29625 \nu^{13} + 64280 \nu^{11} + 292920 \nu^{9} - 840303 \nu^{7} - 966978 \nu^{5} + 6617862 \nu^{3} - 1541835 \nu ) / 5038848 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19 \nu^{19} + 222 \nu^{17} - 466 \nu^{15} - 1677 \nu^{13} + 13744 \nu^{11} - 528 \nu^{9} - 97623 \nu^{7} + 99306 \nu^{5} + 477738 \nu^{3} - 448335 \nu ) / 839808 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31 \nu^{19} + 48 \nu^{17} - 908 \nu^{15} - 3921 \nu^{13} + 26636 \nu^{11} + 1212 \nu^{9} - 275409 \nu^{7} + 262764 \nu^{5} + 997272 \nu^{3} - 2512863 \nu ) / 1259712 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 131 \nu^{18} - 138 \nu^{16} + 4094 \nu^{14} - 6303 \nu^{12} - 42728 \nu^{10} + 100920 \nu^{8} + 172431 \nu^{6} - 842886 \nu^{4} - 10206 \nu^{2} + 2224179 ) / 1679616 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 53 \nu^{18} - 366 \nu^{16} + 3098 \nu^{14} + 63 \nu^{12} - 42344 \nu^{10} + 74808 \nu^{8} + 293625 \nu^{6} - 905202 \nu^{4} - 166698 \nu^{2} + 3759453 ) / 559872 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22 \nu^{18} + 111 \nu^{16} + 865 \nu^{14} - 2391 \nu^{12} - 10642 \nu^{10} + 43206 \nu^{8} + 65628 \nu^{6} - 339309 \nu^{4} + 6561 \nu^{2} + 1581201 ) / 209952 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 49 \nu^{19} + 672 \nu^{17} - 2188 \nu^{15} - 6897 \nu^{13} + 32620 \nu^{11} + 30396 \nu^{9} - 228033 \nu^{7} + 48924 \nu^{5} + 437400 \nu^{3} + 426465 \nu ) / 1259712 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 37 \nu^{18} + 138 \nu^{16} + 322 \nu^{14} - 2841 \nu^{12} + 3464 \nu^{10} + 18600 \nu^{8} - 43047 \nu^{6} - 66906 \nu^{4} + 251262 \nu^{2} - 37179 ) / 279936 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 221 \nu^{18} - 1194 \nu^{16} - 4130 \nu^{14} + 32097 \nu^{12} - 39208 \nu^{10} - 303240 \nu^{8} + 714735 \nu^{6} + 720090 \nu^{4} - 5259006 \nu^{2} + 4323699 ) / 1679616 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - \nu^{19} + 3 \nu^{17} + 16 \nu^{15} - 87 \nu^{13} - 91 \nu^{11} + 1104 \nu^{9} - 819 \nu^{7} - 7047 \nu^{5} + 11664 \nu^{3} + 19683 \nu ) / 19683 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 38 \nu^{18} - 33 \nu^{16} - 527 \nu^{14} + 309 \nu^{12} + 3134 \nu^{10} - 1290 \nu^{8} - 15696 \nu^{6} - 30861 \nu^{4} + 169857 \nu^{2} + 137781 ) / 209952 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 59 \nu^{18} - 246 \nu^{16} - 62 \nu^{14} + 4455 \nu^{12} - 9976 \nu^{10} - 20952 \nu^{8} + 150201 \nu^{6} - 142938 \nu^{4} - 557442 \nu^{2} + 1502469 ) / 279936 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 383 \nu^{18} - 150 \nu^{16} - 3374 \nu^{14} + 5187 \nu^{12} + 42872 \nu^{10} - 57576 \nu^{8} - 263691 \nu^{6} + 696438 \nu^{4} + 2576286 \nu^{2} - 4139991 ) / 1679616 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 131 \nu^{19} + 138 \nu^{17} - 4094 \nu^{15} + 6303 \nu^{13} + 42728 \nu^{11} - 100920 \nu^{9} - 172431 \nu^{7} + 842886 \nu^{5} + 10206 \nu^{3} - 2224179 \nu ) / 1679616 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 127 \nu^{19} - 48 \nu^{17} - 4084 \nu^{15} + 3777 \nu^{13} + 48628 \nu^{11} - 119868 \nu^{9} - 339759 \nu^{7} + 1085076 \nu^{5} + 868968 \nu^{3} - 5570289 \nu ) / 1259712 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 65 \nu^{19} - 118 \nu^{17} - 902 \nu^{15} + 3073 \nu^{13} + 6224 \nu^{11} - 33136 \nu^{9} - 4269 \nu^{7} + 144846 \nu^{5} - 37746 \nu^{3} - 277749 \nu ) / 279936 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{16} - \beta_{15} + \beta_{12} + \beta_{11} + \beta_{8} - \beta_{7} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{13} - 2\beta_{10} + \beta_{5} + \beta_{4} - 2\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{14} - \beta_{11} + \beta_{9} - \beta_{8} - 2\beta_{7} - 3\beta_{3} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{19} + 2\beta_{18} + 2\beta_{17} - 5\beta_{13} - 4\beta_{6} - 3\beta_{5} + 4\beta_{4} + 2\beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{16} + \beta_{15} - 5\beta_{14} + 4\beta_{12} + 9\beta_{11} + 3\beta_{9} - 12\beta_{7} - 10\beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{19} - 12 \beta_{18} + 12 \beta_{17} - 10 \beta_{13} - 20 \beta_{10} + 12 \beta_{6} + 9 \beta_{5} + 14 \beta_{4} - 30 \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 19 \beta_{16} + 8 \beta_{15} - 9 \beta_{14} - 21 \beta_{12} - 30 \beta_{11} + 7 \beta_{9} - 5 \beta_{8} - 37 \beta_{7} - 16 \beta_{3} + 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 19 \beta_{19} - 2 \beta_{18} + 37 \beta_{17} - 25 \beta_{13} + 16 \beta_{10} - 26 \beta_{6} - 46 \beta_{5} + 29 \beta_{4} + 42 \beta_{2} - 9 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 32 \beta_{16} + 65 \beta_{15} - 29 \beta_{14} - 48 \beta_{12} + 26 \beta_{11} - 6 \beta_{9} - 19 \beta_{8} - 68 \beta_{7} - 31 \beta_{3} + 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 34 \beta_{19} - 144 \beta_{18} + 68 \beta_{17} - 101 \beta_{13} - 84 \beta_{10} + 168 \beta_{6} + 98 \beta_{5} + 40 \beta_{4} - 124 \beta_{2} - 36 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 28 \beta_{16} - 58 \beta_{15} - 10 \beta_{14} - 120 \beta_{12} - 242 \beta_{11} - 42 \beta_{9} + 176 \beta_{8} - 296 \beta_{7} + 236 \beta_{3} - 183 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 70 \beta_{19} + 20 \beta_{18} + 296 \beta_{17} + 268 \beta_{13} + 184 \beta_{10} - 268 \beta_{6} - 118 \beta_{5} - 164 \beta_{4} + 628 \beta_{2} - 303 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 137 \beta_{16} + 493 \beta_{15} - 330 \beta_{14} - 403 \beta_{12} - 37 \beta_{11} - 162 \beta_{9} - 211 \beta_{8} + 703 \beta_{7} + 336 \beta_{3} + 312 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 42 \beta_{19} - 228 \beta_{18} - 703 \beta_{17} - 907 \beta_{13} + 302 \beta_{10} + 96 \beta_{6} + 995 \beta_{5} - 295 \beta_{4} + 878 \beta_{2} - 91 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1824 \beta_{16} - 1326 \beta_{15} - 232 \beta_{14} + 1056 \beta_{12} + 193 \beta_{11} + 491 \beta_{9} + 979 \beta_{8} - 94 \beta_{7} + 2541 \beta_{3} - 3606 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 849 \beta_{19} + 1834 \beta_{18} + 94 \beta_{17} + 3011 \beta_{13} + 1716 \beta_{10} - 4100 \beta_{6} + 2103 \beta_{5} - 2308 \beta_{4} + 3358 \beta_{2} - 2550 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2062 \beta_{16} + 5951 \beta_{15} - 775 \beta_{14} - 3268 \beta_{12} - 3105 \beta_{11} + 1305 \beta_{9} - 7392 \beta_{8} + 14316 \beta_{7} + 394 \beta_{3} + 2109 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 467 \beta_{19} + 6744 \beta_{18} - 14316 \beta_{17} - 11798 \beta_{13} + 12332 \beta_{10} - 13080 \beta_{6} - 1773 \beta_{5} + 3010 \beta_{4} + 14286 \beta_{2} - 1159 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1349.1
0.268793 1.71107i
1.56233 0.747749i
−0.368412 1.69242i
1.70278 + 0.317079i
1.64368 + 0.546177i
−1.64368 0.546177i
−1.70278 0.317079i
0.368412 + 1.69242i
−1.56233 + 0.747749i
−0.268793 + 1.71107i
0.268793 + 1.71107i
1.56233 + 0.747749i
−0.368412 + 1.69242i
1.70278 0.317079i
1.64368 0.546177i
−1.64368 + 0.546177i
−1.70278 + 0.317079i
0.368412 1.69242i
−1.56233 0.747749i
−0.268793 1.71107i
0 −1.61622 0.622752i 0 0 0 −0.615143 + 2.57325i 0 2.22436 + 2.01301i 0
1349.2 0 −1.42873 + 0.979142i 0 0 0 2.60608 + 0.456468i 0 1.08256 2.79787i 0
1349.3 0 −1.28147 1.16526i 0 0 0 −1.93884 + 1.80025i 0 0.284326 + 2.98650i 0
1349.4 0 −0.576792 + 1.63319i 0 0 0 −1.99797 1.73439i 0 −2.33462 1.88402i 0
1349.5 0 −0.348838 + 1.69656i 0 0 0 −2.41433 + 1.08214i 0 −2.75662 1.18365i 0
1349.6 0 0.348838 1.69656i 0 0 0 2.41433 1.08214i 0 −2.75662 1.18365i 0
1349.7 0 0.576792 1.63319i 0 0 0 1.99797 + 1.73439i 0 −2.33462 1.88402i 0
1349.8 0 1.28147 + 1.16526i 0 0 0 1.93884 1.80025i 0 0.284326 + 2.98650i 0
1349.9 0 1.42873 0.979142i 0 0 0 −2.60608 0.456468i 0 1.08256 2.79787i 0
1349.10 0 1.61622 + 0.622752i 0 0 0 0.615143 2.57325i 0 2.22436 + 2.01301i 0
1949.1 0 −1.61622 + 0.622752i 0 0 0 −0.615143 2.57325i 0 2.22436 2.01301i 0
1949.2 0 −1.42873 0.979142i 0 0 0 2.60608 0.456468i 0 1.08256 + 2.79787i 0
1949.3 0 −1.28147 + 1.16526i 0 0 0 −1.93884 1.80025i 0 0.284326 2.98650i 0
1949.4 0 −0.576792 1.63319i 0 0 0 −1.99797 + 1.73439i 0 −2.33462 + 1.88402i 0
1949.5 0 −0.348838 1.69656i 0 0 0 −2.41433 1.08214i 0 −2.75662 + 1.18365i 0
1949.6 0 0.348838 + 1.69656i 0 0 0 2.41433 + 1.08214i 0 −2.75662 + 1.18365i 0
1949.7 0 0.576792 + 1.63319i 0 0 0 1.99797 1.73439i 0 −2.33462 + 1.88402i 0
1949.8 0 1.28147 1.16526i 0 0 0 1.93884 + 1.80025i 0 0.284326 2.98650i 0
1949.9 0 1.42873 + 0.979142i 0 0 0 −2.60608 + 0.456468i 0 1.08256 + 2.79787i 0
1949.10 0 1.61622 0.622752i 0 0 0 0.615143 + 2.57325i 0 2.22436 2.01301i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1349.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
21.g even 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bo.g 20
3.b odd 2 1 2100.2.bo.h 20
5.b even 2 1 inner 2100.2.bo.g 20
5.c odd 4 1 420.2.bh.b yes 10
5.c odd 4 1 2100.2.bi.j 10
7.d odd 6 1 2100.2.bo.h 20
15.d odd 2 1 2100.2.bo.h 20
15.e even 4 1 420.2.bh.a 10
15.e even 4 1 2100.2.bi.k 10
21.g even 6 1 inner 2100.2.bo.g 20
35.i odd 6 1 2100.2.bo.h 20
35.k even 12 1 420.2.bh.a 10
35.k even 12 1 2100.2.bi.k 10
35.k even 12 1 2940.2.d.b 10
35.l odd 12 1 2940.2.d.a 10
105.p even 6 1 inner 2100.2.bo.g 20
105.w odd 12 1 420.2.bh.b yes 10
105.w odd 12 1 2100.2.bi.j 10
105.w odd 12 1 2940.2.d.a 10
105.x even 12 1 2940.2.d.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bh.a 10 15.e even 4 1
420.2.bh.a 10 35.k even 12 1
420.2.bh.b yes 10 5.c odd 4 1
420.2.bh.b yes 10 105.w odd 12 1
2100.2.bi.j 10 5.c odd 4 1
2100.2.bi.j 10 105.w odd 12 1
2100.2.bi.k 10 15.e even 4 1
2100.2.bi.k 10 35.k even 12 1
2100.2.bo.g 20 1.a even 1 1 trivial
2100.2.bo.g 20 5.b even 2 1 inner
2100.2.bo.g 20 21.g even 6 1 inner
2100.2.bo.g 20 105.p even 6 1 inner
2100.2.bo.h 20 3.b odd 2 1
2100.2.bo.h 20 7.d odd 6 1
2100.2.bo.h 20 15.d odd 2 1
2100.2.bo.h 20 35.i odd 6 1
2940.2.d.a 10 35.l odd 12 1
2940.2.d.a 10 105.w odd 12 1
2940.2.d.b 10 35.k even 12 1
2940.2.d.b 10 105.x even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11}^{10} - 6 T_{11}^{9} - 4 T_{11}^{8} + 96 T_{11}^{7} + 84 T_{11}^{6} - 2280 T_{11}^{5} + 8156 T_{11}^{4} - 14352 T_{11}^{3} + 14128 T_{11}^{2} - 7488 T_{11} + 1728 \) Copy content Toggle raw display
\( T_{13}^{10} - 81T_{13}^{8} + 2183T_{13}^{6} - 22251T_{13}^{4} + 65704T_{13}^{2} - 3888 \) Copy content Toggle raw display
\( T_{19}^{10} + 3 T_{19}^{9} - 12 T_{19}^{8} - 45 T_{19}^{7} + 152 T_{19}^{6} + 519 T_{19}^{5} - 213 T_{19}^{4} - 1320 T_{19}^{3} + 1864 T_{19}^{2} - 960 T_{19} + 192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 3 T^{18} + 11 T^{16} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 13 T^{18} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( (T^{10} - 6 T^{9} - 4 T^{8} + 96 T^{7} + \cdots + 1728)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} - 81 T^{8} + 2183 T^{6} + \cdots - 3888)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} - 104 T^{18} + \cdots + 4857532416 \) Copy content Toggle raw display
$19$ \( (T^{10} + 3 T^{9} - 12 T^{8} - 45 T^{7} + \cdots + 192)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + 130 T^{18} + \cdots + 6063945650064 \) Copy content Toggle raw display
$29$ \( (T^{10} + 142 T^{8} + 6537 T^{6} + \cdots + 8748)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 15 T^{9} + 12 T^{8} + \cdots + 1978032)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} - 135 T^{18} + \cdots + 157351936 \) Copy content Toggle raw display
$41$ \( (T^{5} - 4 T^{4} - 115 T^{3} + 64 T^{2} + \cdots - 1338)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + 253 T^{8} + 19478 T^{6} + \cdots + 2430481)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} - 372 T^{18} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{20} + 432 T^{18} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{10} + 160 T^{8} - 756 T^{7} + \cdots + 125081856)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 42 T^{9} + 807 T^{8} + \cdots + 1338672)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} - 433 T^{18} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{10} + 288 T^{8} + 28832 T^{6} + \cdots + 88259328)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + 509 T^{18} + \cdots + 2472844310784 \) Copy content Toggle raw display
$79$ \( (T^{10} + T^{9} + 194 T^{8} + \cdots + 138485824)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 774 T^{8} + 208853 T^{6} + \cdots + 829094436)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} - 28 T^{9} + 563 T^{8} + \cdots + 272484)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} - 212 T^{8} + 11040 T^{6} + \cdots - 1051392)^{2} \) Copy content Toggle raw display
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