Properties

Label 2010.2.d.d
Level $2010$
Weight $2$
Character orbit 2010.d
Analytic conductor $16.050$
Analytic rank $0$
Dimension $20$
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] = [2010,2,Mod(401,2010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2010.401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2010 = 2 \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2010.d (of order \(2\), degree \(1\), 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: \(16.0499308063\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 2 x^{18} + 2 x^{17} - 9 x^{16} + 4 x^{15} + 14 x^{14} - 28 x^{13} - 16 x^{12} + 188 x^{11} - 288 x^{10} + 564 x^{9} - 144 x^{8} - 756 x^{7} + 1134 x^{6} + 972 x^{5} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{4} q^{3} + q^{4} + q^{5} + \beta_{4} q^{6} + \beta_{13} q^{7} + q^{8} + \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_{4} q^{3} + q^{4} + q^{5} + \beta_{4} q^{6} + \beta_{13} q^{7} + q^{8} + \beta_{6} q^{9} + q^{10} - \beta_{12} q^{11} + \beta_{4} q^{12} - \beta_{15} q^{13} + \beta_{13} q^{14} + \beta_{4} q^{15} + q^{16} + ( - \beta_{17} - \beta_{4} - \beta_1) q^{17} + \beta_{6} q^{18} + (\beta_{11} + 1) q^{19} + q^{20} + (\beta_{14} + \beta_{9}) q^{21} - \beta_{12} q^{22} + ( - \beta_{15} + \beta_{6} + \beta_{4} - \beta_{2} + \beta_1) q^{23} + \beta_{4} q^{24} + q^{25} - \beta_{15} q^{26} + ( - \beta_{18} - \beta_{12} + \beta_{7} - \beta_{2}) q^{27} + \beta_{13} q^{28} + (\beta_{13} - \beta_{4} - \beta_{3} - \beta_1) q^{29} + \beta_{4} q^{30} + ( - \beta_{17} - \beta_{15}) q^{31} + q^{32} + ( - \beta_{17} + \beta_{15} + \beta_{12} + \beta_{8} - \beta_{4} + \beta_{2} - \beta_1) q^{33} + ( - \beta_{17} - \beta_{4} - \beta_1) q^{34} + \beta_{13} q^{35} + \beta_{6} q^{36} + (\beta_{9} + 1) q^{37} + (\beta_{11} + 1) q^{38} + (\beta_{9} - \beta_{7} + \beta_{2} - \beta_1 + 1) q^{39} + q^{40} + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{41} + (\beta_{14} + \beta_{9}) q^{42} + (\beta_{19} + \beta_{18} - \beta_{16} + 2 \beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} + 2 \beta_{10} - \beta_{9} + \cdots - 2 \beta_1) q^{43}+ \cdots + (\beta_{17} + \beta_{16} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{2} - 2 q^{3} + 20 q^{4} + 20 q^{5} - 2 q^{6} + 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{2} - 2 q^{3} + 20 q^{4} + 20 q^{5} - 2 q^{6} + 20 q^{8} + 20 q^{10} - 2 q^{12} - 2 q^{15} + 20 q^{16} + 16 q^{19} + 20 q^{20} - 2 q^{21} - 2 q^{24} + 20 q^{25} + 10 q^{27} - 2 q^{30} + 20 q^{32} + 2 q^{33} + 20 q^{37} + 16 q^{38} + 16 q^{39} + 20 q^{40} + 8 q^{41} - 2 q^{42} - 2 q^{48} - 48 q^{49} + 20 q^{50} + 32 q^{51} - 36 q^{53} + 10 q^{54} + 16 q^{57} - 2 q^{60} - 4 q^{63} + 20 q^{64} + 2 q^{66} + 16 q^{67} - 28 q^{69} - 4 q^{73} + 20 q^{74} - 2 q^{75} + 16 q^{76} + 16 q^{78} + 20 q^{80} + 12 q^{81} + 8 q^{82} - 2 q^{84} + 32 q^{87} + 12 q^{91} - 12 q^{93} + 16 q^{95} - 2 q^{96} - 48 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 2 x^{18} + 2 x^{17} - 9 x^{16} + 4 x^{15} + 14 x^{14} - 28 x^{13} - 16 x^{12} + 188 x^{11} - 288 x^{10} + 564 x^{9} - 144 x^{8} - 756 x^{7} + 1134 x^{6} + 972 x^{5} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 134 \nu^{19} - 481 \nu^{18} + 732 \nu^{17} - 2507 \nu^{16} + 2780 \nu^{15} + 10510 \nu^{14} + 24540 \nu^{13} - 14504 \nu^{12} + 10168 \nu^{11} - 94592 \nu^{10} + \cdots - 1620567 ) / 7978176 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{19} - 2 \nu^{18} + 2 \nu^{17} + 2 \nu^{16} - 9 \nu^{15} + 4 \nu^{14} + 14 \nu^{13} - 28 \nu^{12} - 16 \nu^{11} + 188 \nu^{10} - 288 \nu^{9} + 564 \nu^{8} - 144 \nu^{7} - 756 \nu^{6} + 1134 \nu^{5} + \cdots - 39366 ) / 19683 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 317 \nu^{19} - 554 \nu^{18} + 1379 \nu^{17} + 4084 \nu^{16} - 4918 \nu^{15} + 3468 \nu^{14} + 3672 \nu^{13} - 2552 \nu^{12} - 23976 \nu^{11} + 26004 \nu^{10} - 70600 \nu^{9} + \cdots - 18257076 ) / 5318784 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2 \nu^{19} + \nu^{18} + 2 \nu^{17} - 10 \nu^{16} + 12 \nu^{15} + 19 \nu^{14} - 40 \nu^{13} + 14 \nu^{12} + 116 \nu^{11} - 328 \nu^{10} + 12 \nu^{9} - 264 \nu^{8} - 1404 \nu^{7} + 1944 \nu^{6} + \cdots + 39366 ) / 19683 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3305 \nu^{19} - 17630 \nu^{18} + 28865 \nu^{17} + 48356 \nu^{16} - 86946 \nu^{15} + 32644 \nu^{14} + 262904 \nu^{13} + 9608 \nu^{12} - 378568 \nu^{11} + \cdots - 338153940 ) / 47869056 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1229 \nu^{19} + 17986 \nu^{18} - 11851 \nu^{17} - 3604 \nu^{16} + 66678 \nu^{15} + 50020 \nu^{14} - 90472 \nu^{13} + 96104 \nu^{12} + 640760 \nu^{11} + \cdots + 333115092 ) / 47869056 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4789 \nu^{19} - 7514 \nu^{18} + 3947 \nu^{17} + 38852 \nu^{16} - 11334 \nu^{15} - 24404 \nu^{14} + 116504 \nu^{13} - 73624 \nu^{12} - 382312 \nu^{11} + 545108 \nu^{10} + \cdots - 161479332 ) / 47869056 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 418 \nu^{19} - 19349 \nu^{18} - 7300 \nu^{17} + 30185 \nu^{16} - 55188 \nu^{15} - 50042 \nu^{14} + 136460 \nu^{13} + 290264 \nu^{12} - 79336 \nu^{11} + 878144 \nu^{10} + \cdots - 53242515 ) / 47869056 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 99 \nu^{19} + 1670 \nu^{18} - 1405 \nu^{17} - 3104 \nu^{16} + 8362 \nu^{15} + 2844 \nu^{14} - 23848 \nu^{13} + 7216 \nu^{12} + 14872 \nu^{11} - 47492 \nu^{10} + \cdots + 26611416 ) / 3989088 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4235 \nu^{19} - 15334 \nu^{18} + 31189 \nu^{17} + 53548 \nu^{16} - 67962 \nu^{15} + 28820 \nu^{14} + 237736 \nu^{13} - 130760 \nu^{12} - 464600 \nu^{11} + \cdots - 378622188 ) / 47869056 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1396 \nu^{19} - 1789 \nu^{18} - 1458 \nu^{17} + 5425 \nu^{16} - 5488 \nu^{15} - 7418 \nu^{14} + 7140 \nu^{13} - 4592 \nu^{12} - 50624 \nu^{11} + 91936 \nu^{10} - 171100 \nu^{9} + \cdots - 17078283 ) / 7978176 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 9995 \nu^{19} - 7202 \nu^{18} + 17843 \nu^{17} - 23020 \nu^{16} - 39462 \nu^{15} + 102364 \nu^{14} - 55864 \nu^{13} + 90872 \nu^{12} + 548264 \nu^{11} + \cdots + 73771884 ) / 47869056 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4384 \nu^{19} - 6851 \nu^{18} - 4282 \nu^{17} + 24743 \nu^{16} - 14904 \nu^{15} - 43286 \nu^{14} + 23900 \nu^{13} + 20960 \nu^{12} - 291616 \nu^{11} + 610496 \nu^{10} + \cdots - 79657101 ) / 23934528 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1705 \nu^{19} - 5786 \nu^{18} + 2393 \nu^{17} + 9818 \nu^{16} - 21726 \nu^{15} - 29612 \nu^{14} + 50060 \nu^{13} - 113476 \nu^{12} - 96004 \nu^{11} + 248144 \nu^{10} + \cdots - 87116958 ) / 11967264 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 3298 \nu^{19} + 6527 \nu^{18} + 4480 \nu^{17} - 12491 \nu^{16} + 1092 \nu^{15} + 36494 \nu^{14} - 64916 \nu^{13} - 33080 \nu^{12} + 110632 \nu^{11} - 452576 \nu^{10} + \cdots + 14335785 ) / 15956352 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 3101 \nu^{19} - 1790 \nu^{18} - 1189 \nu^{17} - 82 \nu^{16} - 12042 \nu^{15} + 17980 \nu^{14} + 23924 \nu^{13} - 3820 \nu^{12} + 66500 \nu^{11} - 59152 \nu^{10} + \cdots - 13817466 ) / 11967264 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 6191 \nu^{19} - 36046 \nu^{18} + 45625 \nu^{17} + 68500 \nu^{16} - 125682 \nu^{15} - 43420 \nu^{14} + 340696 \nu^{13} - 550616 \nu^{12} - 566504 \nu^{11} + \cdots - 631351908 ) / 47869056 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} + \beta_{16} + \beta_{12} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} + \beta_{12} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} + \beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} + 2 \beta_{15} - 3 \beta_{13} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{19} - \beta_{18} + 2 \beta_{17} - \beta_{16} + 6 \beta_{15} - 2 \beta_{13} - 3 \beta_{12} - 4 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5 \beta_{19} + \beta_{17} - \beta_{16} - 3 \beta_{14} - \beta_{13} + 6 \beta_{11} + 5 \beta_{10} - 10 \beta_{9} + \beta_{8} - 5 \beta_{6} + 3 \beta_{5} - 5 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3 \beta_{19} - 2 \beta_{18} + 4 \beta_{17} + 2 \beta_{16} - 4 \beta_{15} - 11 \beta_{14} + 4 \beta_{13} - 7 \beta_{12} + 7 \beta_{11} + 8 \beta_{10} - 9 \beta_{9} + 3 \beta_{8} + 11 \beta_{7} + 8 \beta_{6} + 2 \beta_{5} + 11 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 5 \beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6 \beta_{19} - 7 \beta_{18} - 3 \beta_{17} + 6 \beta_{16} - 4 \beta_{15} + 20 \beta_{14} - \beta_{13} - 13 \beta_{12} + 3 \beta_{11} - 7 \beta_{10} + 25 \beta_{9} - 2 \beta_{8} + 7 \beta_{7} - 20 \beta_{6} + \beta_{5} - 5 \beta_{4} + 3 \beta_{3} - 6 \beta_{2} + 11 \beta _1 - 36 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2 \beta_{19} + 7 \beta_{18} - 4 \beta_{17} + 3 \beta_{16} + 18 \beta_{15} + 2 \beta_{14} - 46 \beta_{13} + 7 \beta_{12} + 16 \beta_{11} + 6 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} - 4 \beta_{7} - 21 \beta_{6} - 7 \beta_{5} + 69 \beta_{4} - 7 \beta_{3} + 24 \beta_{2} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 14 \beta_{19} - 25 \beta_{17} + 16 \beta_{16} - 6 \beta_{15} + 7 \beta_{14} - 21 \beta_{13} + 17 \beta_{12} - 4 \beta_{11} + 15 \beta_{10} - 64 \beta_{9} + 61 \beta_{8} - 16 \beta_{7} + 5 \beta_{5} + 30 \beta_{4} - 17 \beta_{3} - 25 \beta_{2} - 9 \beta _1 - 254 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 31 \beta_{19} - 58 \beta_{18} + 3 \beta_{17} - 102 \beta_{16} + 4 \beta_{15} + 9 \beta_{14} - 66 \beta_{12} - 53 \beta_{11} + 47 \beta_{10} - 6 \beta_{9} - 13 \beta_{8} + 24 \beta_{7} - 40 \beta_{6} + 9 \beta_{5} - 25 \beta_{4} + 2 \beta_{3} - 13 \beta_{2} + \cdots - 219 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 60 \beta_{19} + 3 \beta_{18} - 82 \beta_{17} + 24 \beta_{16} - 52 \beta_{15} - 58 \beta_{14} - 62 \beta_{13} + 17 \beta_{12} - 96 \beta_{11} - 42 \beta_{10} - 68 \beta_{9} + 18 \beta_{8} + 49 \beta_{7} - 56 \beta_{6} + 22 \beta_{5} - 34 \beta_{4} + \cdots + 104 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 166 \beta_{19} + 144 \beta_{18} + 122 \beta_{17} - 332 \beta_{16} + 16 \beta_{15} + 22 \beta_{14} + 280 \beta_{13} - 60 \beta_{12} + 150 \beta_{11} + 2 \beta_{10} + 128 \beta_{9} + 2 \beta_{8} - 16 \beta_{7} - 179 \beta_{6} - 82 \beta_{5} + \cdots + 392 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 252 \beta_{19} + 114 \beta_{18} - 532 \beta_{17} - 88 \beta_{16} - 120 \beta_{15} - 196 \beta_{14} - 428 \beta_{13} - 10 \beta_{12} + 460 \beta_{11} - 12 \beta_{10} + 20 \beta_{9} + 36 \beta_{8} + 2 \beta_{7} + 376 \beta_{6} + 248 \beta_{5} + \cdots - 1324 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 32 \beta_{19} + 256 \beta_{18} - 284 \beta_{17} + 272 \beta_{16} + 64 \beta_{15} - 76 \beta_{14} + 776 \beta_{13} + 324 \beta_{12} + 488 \beta_{11} - 364 \beta_{10} - 500 \beta_{9} + 724 \beta_{8} + 748 \beta_{7} + 386 \beta_{6} + \cdots - 831 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 902 \beta_{19} - 908 \beta_{18} - 1000 \beta_{17} + 730 \beta_{16} - 1968 \beta_{15} + 1120 \beta_{14} - 64 \beta_{13} + 1398 \beta_{12} - 76 \beta_{11} - 592 \beta_{10} + 1436 \beta_{9} - 592 \beta_{8} + 40 \beta_{7} + \cdots + 1052 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 280 \beta_{19} + 248 \beta_{18} + 1314 \beta_{17} - 1172 \beta_{16} - 1432 \beta_{15} - 1920 \beta_{14} - 32 \beta_{13} + 962 \beta_{12} - 2116 \beta_{11} + 34 \beta_{10} + 974 \beta_{9} + 160 \beta_{8} - 846 \beta_{7} + \cdots + 3904 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 2657 \beta_{19} - 2376 \beta_{18} - 2066 \beta_{17} - 1051 \beta_{16} + 532 \beta_{15} + 4552 \beta_{14} + 1922 \beta_{13} - 975 \beta_{12} - 5814 \beta_{11} - 5034 \beta_{10} + 4834 \beta_{9} - 632 \beta_{8} + \cdots - 2936 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2010\mathbb{Z}\right)^\times\).

\(n\) \(671\) \(1141\) \(1207\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.68704 0.392308i
1.68704 + 0.392308i
1.63938 0.558950i
1.63938 + 0.558950i
1.16984 1.27729i
1.16984 + 1.27729i
0.929804 1.46132i
0.929804 + 1.46132i
0.315540 1.70307i
0.315540 + 1.70307i
0.244204 1.71475i
0.244204 + 1.71475i
−0.492519 1.66055i
−0.492519 + 1.66055i
−1.28042 1.16641i
−1.28042 + 1.16641i
−1.48239 0.895838i
−1.48239 + 0.895838i
−1.73048 0.0738163i
−1.73048 + 0.0738163i
1.00000 −1.68704 0.392308i 1.00000 1.00000 −1.68704 0.392308i 2.72263i 1.00000 2.69219 + 1.32367i 1.00000
401.2 1.00000 −1.68704 + 0.392308i 1.00000 1.00000 −1.68704 + 0.392308i 2.72263i 1.00000 2.69219 1.32367i 1.00000
401.3 1.00000 −1.63938 0.558950i 1.00000 1.00000 −1.63938 0.558950i 2.20327i 1.00000 2.37515 + 1.83267i 1.00000
401.4 1.00000 −1.63938 + 0.558950i 1.00000 1.00000 −1.63938 + 0.558950i 2.20327i 1.00000 2.37515 1.83267i 1.00000
401.5 1.00000 −1.16984 1.27729i 1.00000 1.00000 −1.16984 1.27729i 1.10570i 1.00000 −0.262965 + 2.98845i 1.00000
401.6 1.00000 −1.16984 + 1.27729i 1.00000 1.00000 −1.16984 + 1.27729i 1.10570i 1.00000 −0.262965 2.98845i 1.00000
401.7 1.00000 −0.929804 1.46132i 1.00000 1.00000 −0.929804 1.46132i 4.47955i 1.00000 −1.27093 + 2.71749i 1.00000
401.8 1.00000 −0.929804 + 1.46132i 1.00000 1.00000 −0.929804 + 1.46132i 4.47955i 1.00000 −1.27093 2.71749i 1.00000
401.9 1.00000 −0.315540 1.70307i 1.00000 1.00000 −0.315540 1.70307i 0.516210i 1.00000 −2.80087 + 1.07477i 1.00000
401.10 1.00000 −0.315540 + 1.70307i 1.00000 1.00000 −0.315540 + 1.70307i 0.516210i 1.00000 −2.80087 1.07477i 1.00000
401.11 1.00000 −0.244204 1.71475i 1.00000 1.00000 −0.244204 1.71475i 4.63764i 1.00000 −2.88073 + 0.837496i 1.00000
401.12 1.00000 −0.244204 + 1.71475i 1.00000 1.00000 −0.244204 + 1.71475i 4.63764i 1.00000 −2.88073 0.837496i 1.00000
401.13 1.00000 0.492519 1.66055i 1.00000 1.00000 0.492519 1.66055i 1.30548i 1.00000 −2.51485 1.63570i 1.00000
401.14 1.00000 0.492519 + 1.66055i 1.00000 1.00000 0.492519 + 1.66055i 1.30548i 1.00000 −2.51485 + 1.63570i 1.00000
401.15 1.00000 1.28042 1.16641i 1.00000 1.00000 1.28042 1.16641i 4.84956i 1.00000 0.278954 2.98700i 1.00000
401.16 1.00000 1.28042 + 1.16641i 1.00000 1.00000 1.28042 + 1.16641i 4.84956i 1.00000 0.278954 + 2.98700i 1.00000
401.17 1.00000 1.48239 0.895838i 1.00000 1.00000 1.48239 0.895838i 3.66624i 1.00000 1.39495 2.65596i 1.00000
401.18 1.00000 1.48239 + 0.895838i 1.00000 1.00000 1.48239 + 0.895838i 3.66624i 1.00000 1.39495 + 2.65596i 1.00000
401.19 1.00000 1.73048 0.0738163i 1.00000 1.00000 1.73048 0.0738163i 0.0771495i 1.00000 2.98910 0.255475i 1.00000
401.20 1.00000 1.73048 + 0.0738163i 1.00000 1.00000 1.73048 + 0.0738163i 0.0771495i 1.00000 2.98910 + 0.255475i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
201.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2010.2.d.d yes 20
3.b odd 2 1 2010.2.d.c 20
67.b odd 2 1 2010.2.d.c 20
201.d even 2 1 inner 2010.2.d.d yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2010.2.d.c 20 3.b odd 2 1
2010.2.d.c 20 67.b odd 2 1
2010.2.d.d yes 20 1.a even 1 1 trivial
2010.2.d.d yes 20 201.d 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 \(S_{2}^{\mathrm{new}}(2010, [\chi])\):

\( T_{7}^{20} + 94 T_{7}^{18} + 3577 T_{7}^{16} + 70708 T_{7}^{14} + 776801 T_{7}^{12} + 4736402 T_{7}^{10} + 15306235 T_{7}^{8} + 23877820 T_{7}^{6} + 15710118 T_{7}^{4} + 2818448 T_{7}^{2} + \cdots + 16224 \) Copy content Toggle raw display
\( T_{11}^{10} - 65 T_{11}^{8} + 10 T_{11}^{7} + 1289 T_{11}^{6} - 128 T_{11}^{5} - 8471 T_{11}^{4} + 98 T_{11}^{3} + 13846 T_{11}^{2} - 3300 T_{11} - 2808 \) Copy content Toggle raw display
\( T_{53}^{10} + 18 T_{53}^{9} - 56 T_{53}^{8} - 2888 T_{53}^{7} - 14984 T_{53}^{6} + 47828 T_{53}^{5} + 546262 T_{53}^{4} + 665000 T_{53}^{3} - 3830704 T_{53}^{2} - 8294880 T_{53} + 1238112 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 2 T^{19} + 2 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( (T - 1)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 94 T^{18} + 3577 T^{16} + \cdots + 16224 \) Copy content Toggle raw display
$11$ \( (T^{10} - 65 T^{8} + 10 T^{7} + 1289 T^{6} + \cdots - 2808)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + 104 T^{18} + \cdots + 87005184 \) Copy content Toggle raw display
$17$ \( T^{20} + 124 T^{18} + \cdots + 2369378304 \) Copy content Toggle raw display
$19$ \( (T^{10} - 8 T^{9} - 60 T^{8} + 448 T^{7} + \cdots + 50176)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + 180 T^{18} + 11554 T^{16} + \cdots + 301056 \) Copy content Toggle raw display
$29$ \( T^{20} + 264 T^{18} + \cdots + 1738899456 \) Copy content Toggle raw display
$31$ \( T^{20} + 200 T^{18} + \cdots + 181236096 \) Copy content Toggle raw display
$37$ \( (T^{10} - 10 T^{9} - 69 T^{8} + 626 T^{7} + \cdots + 71568)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} - 4 T^{9} - 248 T^{8} + \cdots + 8513232)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 769106539708416 \) Copy content Toggle raw display
$47$ \( T^{20} + 528 T^{18} + \cdots + 23380534656 \) Copy content Toggle raw display
$53$ \( (T^{10} + 18 T^{9} - 56 T^{8} + \cdots + 1238112)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 421674691762176 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 151788741097056 \) Copy content Toggle raw display
$67$ \( T^{20} - 16 T^{19} + \cdots + 18\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( T^{20} + 1126 T^{18} + \cdots + 72\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{10} + 2 T^{9} - 340 T^{8} + \cdots + 29161344)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + 976 T^{18} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{20} + 726 T^{18} + \cdots + 35419602917376 \) Copy content Toggle raw display
$89$ \( T^{20} + 1202 T^{18} + \cdots + 68\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{20} + 1374 T^{18} + \cdots + 97\!\cdots\!84 \) Copy content Toggle raw display
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