Properties

Label 261.3.d.a
Level $261$
Weight $3$
Character orbit 261.d
Analytic conductor $7.112$
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] = [261,3,Mod(260,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.260");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 261.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: \(7.11173489980\)
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: \( x^{20} - 40 x^{18} + 842 x^{16} - 9842 x^{14} + 77885 x^{12} - 385712 x^{10} + 1978501 x^{8} - 3421162 x^{6} + 20772594 x^{4} + 5678964 x^{2} + 140920641 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + (\beta_1 + 2) q^{4} - \beta_{13} q^{5} + (\beta_{2} - 1) q^{7} + (\beta_{12} + 2 \beta_{10}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + (\beta_1 + 2) q^{4} - \beta_{13} q^{5} + (\beta_{2} - 1) q^{7} + (\beta_{12} + 2 \beta_{10}) q^{8} + (\beta_{6} + \beta_{3}) q^{10} + (\beta_{18} + \beta_{10}) q^{11} + ( - \beta_{5} + \beta_1) q^{13} + ( - \beta_{18} - \beta_{14} - 2 \beta_{10}) q^{14} + ( - \beta_{5} - \beta_{4} + 2 \beta_1 + 4) q^{16} + ( - \beta_{18} + \beta_{16}) q^{17} + ( - \beta_{7} - \beta_{3}) q^{19} + ( - \beta_{19} + \beta_{17} + \beta_{15} - 4 \beta_{13} + \beta_{11}) q^{20} + ( - 2 \beta_{5} - \beta_{4} - 2 \beta_{2} + 3 \beta_1 + 7) q^{22} + ( - \beta_{17} - \beta_{15} + 2 \beta_{13}) q^{23} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 - 6) q^{25} + (\beta_{18} + \beta_{12} + 2 \beta_{10}) q^{26} + (2 \beta_{5} + 4 \beta_{4} + 3 \beta_{2} - \beta_1 - 8) q^{28} + ( - \beta_{16} - \beta_{15} - \beta_{13} - \beta_{12} - \beta_{11} - 3 \beta_{10}) q^{29} + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{31} + (\beta_{18} + \beta_{16} + \beta_{14} - \beta_{12} + 3 \beta_{10}) q^{32} + (\beta_{5} - 2 \beta_{4} + 3 \beta_{2} - 2 \beta_1 + 2) q^{34} + (\beta_{19} - 2 \beta_{15} + 4 \beta_{13} - \beta_{11}) q^{35} + (\beta_{8} - \beta_{7} - \beta_{6} - \beta_{3}) q^{37} + (2 \beta_{19} - \beta_{17} - \beta_{15} + 3 \beta_{13} - 4 \beta_{11}) q^{38} + (\beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} + 4 \beta_{3}) q^{40} + (2 \beta_{16} + \beta_{14} - \beta_{12} - 6 \beta_{10}) q^{41} + ( - \beta_{9} + 2 \beta_{6} + 3 \beta_{3}) q^{43} + (\beta_{16} + 3 \beta_{14} + 4 \beta_{12} + 14 \beta_{10}) q^{44} + ( - \beta_{9} - \beta_{8} - 4 \beta_{6} - 3 \beta_{3}) q^{46} + ( - \beta_{18} + 2 \beta_{16} + 3 \beta_{14} + 3 \beta_{12} + 5 \beta_{10}) q^{47} + (3 \beta_{4} - 3 \beta_{2} + 4 \beta_1 + 2) q^{49} + ( - 2 \beta_{16} - 3 \beta_{14} - 3 \beta_{12} - 15 \beta_{10}) q^{50} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 4 \beta_1 + 13) q^{52} + (\beta_{19} + \beta_{17} - \beta_{15} + 5 \beta_{13} - 5 \beta_{11}) q^{53} + (\beta_{9} + 2 \beta_{8} + \beta_{6} + 2 \beta_{3}) q^{55} + ( - \beta_{18} - 4 \beta_{16} - 3 \beta_{14} - 5 \beta_{12} - 7 \beta_{10}) q^{56} + ( - \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - \beta_{3} - \beta_{2} - 7 \beta_1 - 21) q^{58} + ( - 3 \beta_{19} + \beta_{17} + \beta_{15} + 2 \beta_{13} + 6 \beta_{11}) q^{59} + (\beta_{9} - \beta_{8} + 2 \beta_{3}) q^{61} + (\beta_{15} - 6 \beta_{13} + 3 \beta_{11}) q^{62} + (2 \beta_{5} - 2 \beta_{4} - 6 \beta_{2} - 10 \beta_1 + 5) q^{64} + ( - 2 \beta_{19} + 2 \beta_{17} + 3 \beta_{15} + 2 \beta_{13}) q^{65} + ( - \beta_{5} - \beta_{4} + 4 \beta_{2} - 3 \beta_1 - 24) q^{67} + ( - 2 \beta_{16} - \beta_{14} - 5 \beta_{10}) q^{68} + ( - 2 \beta_{8} + \beta_{7} - 4 \beta_{6} - 9 \beta_{3}) q^{70} + (2 \beta_{19} - 2 \beta_{17} - \beta_{15} - 6 \beta_{13} - 6 \beta_{11}) q^{71} + (\beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{3}) q^{73} + ( - \beta_{19} - 2 \beta_{17} + 4 \beta_{15} + 8 \beta_{13} + 4 \beta_{11}) q^{74} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} - 5 \beta_{6} - 8 \beta_{3}) q^{76} + (\beta_{18} - 2 \beta_{14} - 4 \beta_{12} - 21 \beta_{10}) q^{77} - 5 \beta_{3} q^{79} + ( - 4 \beta_{19} + 3 \beta_{17} + 4 \beta_{15} - 7 \beta_{13} + 10 \beta_{11}) q^{80} + ( - \beta_{5} - 8 \beta_{4} - 3 \beta_{2} - 13 \beta_1 - 31) q^{82} + (5 \beta_{19} - 2 \beta_{15} - 7 \beta_{13} - 2 \beta_{11}) q^{83} + ( - \beta_{9} - 3 \beta_{8} + \beta_{7} + 4 \beta_{3}) q^{85} + ( - 3 \beta_{19} - 2 \beta_{17} + 4 \beta_{15} - 16 \beta_{13} + 9 \beta_{11}) q^{86} + (3 \beta_{5} - 12 \beta_{4} - 6 \beta_{2} + 9 \beta_1 + 56) q^{88} + (\beta_{18} - 3 \beta_{16} - 3 \beta_{14} - 3 \beta_{12} + 4 \beta_{10}) q^{89} + ( - \beta_{5} + 4 \beta_{4} - 5 \beta_1) q^{91} + (6 \beta_{19} - 4 \beta_{17} - 3 \beta_{15} + 26 \beta_{13}) q^{92} + ( - 3 \beta_{5} - 17 \beta_{4} - 11 \beta_{2} + 6 \beta_1 + 32) q^{94} + (3 \beta_{16} + 2 \beta_{14} + 8 \beta_{12} + 17 \beta_{10}) q^{95} + (\beta_{8} + 2 \beta_{7} + 5 \beta_{6} + 3 \beta_{3}) q^{97} + (3 \beta_{18} - 3 \beta_{16} + \beta_{12} + 18 \beta_{10}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 40 q^{4} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 40 q^{4} - 16 q^{7} - 8 q^{13} + 72 q^{16} + 116 q^{22} - 124 q^{25} - 132 q^{28} + 60 q^{34} + 28 q^{49} + 260 q^{52} - 408 q^{58} + 92 q^{64} - 472 q^{67} - 640 q^{82} + 1120 q^{88} - 8 q^{91} + 572 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 40 x^{18} + 842 x^{16} - 9842 x^{14} + 77885 x^{12} - 385712 x^{10} + 1978501 x^{8} - 3421162 x^{6} + 20772594 x^{4} + 5678964 x^{2} + 140920641 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11\!\cdots\!35 \nu^{18} + \cdots - 23\!\cdots\!83 ) / 46\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!07 \nu^{18} + \cdots + 62\!\cdots\!39 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!35 \nu^{18} + \cdots + 51\!\cdots\!59 ) / 30\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 74\!\cdots\!17 \nu^{18} + \cdots - 77\!\cdots\!51 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21\!\cdots\!57 \nu^{18} + \cdots + 26\!\cdots\!91 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 53\!\cdots\!23 \nu^{18} + \cdots + 37\!\cdots\!25 ) / 77\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 39\!\cdots\!21 \nu^{18} + \cdots + 95\!\cdots\!35 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!81 \nu^{18} + \cdots - 12\!\cdots\!65 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14\!\cdots\!79 \nu^{18} + \cdots - 15\!\cdots\!75 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37\!\cdots\!05 \nu^{19} + \cdots + 11\!\cdots\!47 \nu ) / 36\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 37\!\cdots\!05 \nu^{19} + \cdots + 24\!\cdots\!37 \nu ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!25 \nu^{19} + \cdots - 43\!\cdots\!95 \nu ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 57\!\cdots\!76 \nu^{19} + \cdots - 22\!\cdots\!91 \nu ) / 49\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14\!\cdots\!03 \nu^{19} + \cdots - 87\!\cdots\!67 \nu ) / 91\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 97\!\cdots\!39 \nu^{19} + \cdots - 40\!\cdots\!81 \nu ) / 61\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 53\!\cdots\!07 \nu^{19} + \cdots + 40\!\cdots\!97 \nu ) / 18\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 94\!\cdots\!71 \nu^{19} + \cdots - 39\!\cdots\!51 \nu ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 11\!\cdots\!23 \nu^{19} + \cdots + 22\!\cdots\!87 \nu ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 74\!\cdots\!97 \nu^{19} + \cdots + 99\!\cdots\!82 \nu ) / 15\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} + 3\beta_{10} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + 3\beta _1 + 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{19} + 3\beta_{15} - 3\beta_{13} + 3\beta_{12} + 16\beta_{11} + 12\beta_{10} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{8} + 4\beta_{6} - 3\beta_{5} - 3\beta_{4} + 32\beta_{3} + 6\beta _1 - 24 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 45 \beta_{19} + 3 \beta_{18} + 5 \beta_{17} + 3 \beta_{16} + 55 \beta_{15} + 3 \beta_{14} - 75 \beta_{13} - 15 \beta_{12} + 174 \beta_{11} - 195 \beta_{10} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 6 \beta_{9} + 50 \beta_{8} - 6 \beta_{7} + 86 \beta_{6} + 36 \beta_{5} + 24 \beta_{4} + 332 \beta_{3} - 18 \beta_{2} - 558 \beta _1 - 2265 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 308 \beta_{19} - 54 \beta_{18} + 98 \beta_{17} - 60 \beta_{16} + 378 \beta_{15} - 42 \beta_{14} - 812 \beta_{13} - 882 \beta_{12} + 979 \beta_{11} - 6315 \beta_{10} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 80 \beta_{9} + 80 \beta_{8} + 64 \beta_{7} + 608 \beta_{6} + 1386 \beta_{5} + 1746 \beta_{4} + 632 \beta_{3} + 510 \beta_{2} - 11757 \beta _1 - 44964 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2691 \beta_{19} - 2376 \beta_{18} + 360 \beta_{17} - 2130 \beta_{16} - 3429 \beta_{15} - 2928 \beta_{14} + 1863 \beta_{13} - 14175 \beta_{12} - 13040 \beta_{11} - 93540 \beta_{10} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 432 \beta_{9} - 7970 \beta_{8} + 1092 \beta_{7} - 10742 \beta_{6} + 20301 \beta_{5} + 32553 \beta_{4} - 57556 \beta_{3} + 20178 \beta_{2} - 126390 \beta _1 - 490494 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 122463 \beta_{19} - 37887 \beta_{18} - 24871 \beta_{17} - 29121 \beta_{16} - 145871 \beta_{15} - 50619 \beta_{14} + 254133 \beta_{13} - 111603 \beta_{12} - 459084 \beta_{11} - 670503 \beta_{10} ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 37500 \beta_{9} - 178828 \beta_{8} - 10332 \beta_{7} - 416572 \beta_{6} + 119940 \beta_{5} + 257040 \beta_{4} - 1370968 \beta_{3} + 247080 \beta_{2} - 262164 \beta _1 - 1079805 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 2047552 \beta_{19} - 142020 \beta_{18} - 699244 \beta_{17} - 67128 \beta_{16} - 2354976 \beta_{15} - 218544 \beta_{14} + 5346640 \beta_{13} + 614340 \beta_{12} - 6809843 \beta_{11} + 5072043 \beta_{10} ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 746584 \beta_{9} - 2124688 \beta_{8} - 495896 \beta_{7} - 6800512 \beta_{6} - 2276508 \beta_{5} - 2914392 \beta_{4} - 17083954 \beta_{3} - 872496 \beta_{2} + 20017515 \beta _1 + 77764632 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 17749413 \beta_{19} + 7628508 \beta_{18} - 9233076 \beta_{17} + 6347736 \beta_{16} - 19529469 \beta_{15} + 9705192 \beta_{14} + 58642005 \beta_{13} + 37118835 \beta_{12} + \cdots + 244302336 \beta_{10} ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 6690240 \beta_{9} - 7848752 \beta_{8} - 6638400 \beta_{7} - 49374512 \beta_{6} - 80749851 \beta_{5} - 133798719 \beta_{4} - 72808384 \beta_{3} - 90807432 \beta_{2} + \cdots + 1859515140 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 58382709 \beta_{19} + 211698723 \beta_{18} - 25588111 \beta_{17} + 161995839 \beta_{16} + 79577935 \beta_{15} + 279460551 \beta_{14} + 25121325 \beta_{13} + \cdots + 4117844637 \beta_{10} ) / 3 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 44978130 \beta_{9} + 280215278 \beta_{8} + 115134 \beta_{7} + 565796186 \beta_{6} - 1263852912 \beta_{5} - 2289245784 \beta_{4} + 2121224828 \beta_{3} + \cdots + 23269807935 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 5106951404 \beta_{19} + 2797431390 \beta_{18} + 1768257230 \beta_{17} + 2048758140 \beta_{16} + 5863708218 \beta_{15} + 3762018870 \beta_{14} + \cdots + 36590492085 \beta_{10} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).

\(n\) \(118\) \(146\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
260.1
−3.59888 1.41421i
−3.59888 + 1.41421i
−3.25982 + 1.41421i
−3.25982 1.41421i
−1.79139 + 1.41421i
−1.79139 1.41421i
−1.60400 1.41421i
−1.60400 + 1.41421i
−0.799859 1.41421i
−0.799859 + 1.41421i
0.799859 1.41421i
0.799859 + 1.41421i
1.60400 1.41421i
1.60400 + 1.41421i
1.79139 + 1.41421i
1.79139 1.41421i
3.25982 + 1.41421i
3.25982 1.41421i
3.59888 1.41421i
3.59888 + 1.41421i
−3.59888 0 8.95192 7.92043i 0 −8.60311 −17.8214 0 28.5047i
260.2 −3.59888 0 8.95192 7.92043i 0 −8.60311 −17.8214 0 28.5047i
260.3 −3.25982 0 6.62643 3.95390i 0 8.06717 −8.56167 0 12.8890i
260.4 −3.25982 0 6.62643 3.95390i 0 8.06717 −8.56167 0 12.8890i
260.5 −1.79139 0 −0.790922 4.65286i 0 −9.55150 8.58241 0 8.33508i
260.6 −1.79139 0 −0.790922 4.65286i 0 −9.55150 8.58241 0 8.33508i
260.7 −1.60400 0 −1.42720 7.47702i 0 1.78921 8.70520 0 11.9931i
260.8 −1.60400 0 −1.42720 7.47702i 0 1.78921 8.70520 0 11.9931i
260.9 −0.799859 0 −3.36023 0.280379i 0 4.29824 5.88714 0 0.224264i
260.10 −0.799859 0 −3.36023 0.280379i 0 4.29824 5.88714 0 0.224264i
260.11 0.799859 0 −3.36023 0.280379i 0 4.29824 −5.88714 0 0.224264i
260.12 0.799859 0 −3.36023 0.280379i 0 4.29824 −5.88714 0 0.224264i
260.13 1.60400 0 −1.42720 7.47702i 0 1.78921 −8.70520 0 11.9931i
260.14 1.60400 0 −1.42720 7.47702i 0 1.78921 −8.70520 0 11.9931i
260.15 1.79139 0 −0.790922 4.65286i 0 −9.55150 −8.58241 0 8.33508i
260.16 1.79139 0 −0.790922 4.65286i 0 −9.55150 −8.58241 0 8.33508i
260.17 3.25982 0 6.62643 3.95390i 0 8.06717 8.56167 0 12.8890i
260.18 3.25982 0 6.62643 3.95390i 0 8.06717 8.56167 0 12.8890i
260.19 3.59888 0 8.95192 7.92043i 0 −8.60311 17.8214 0 28.5047i
260.20 3.59888 0 8.95192 7.92043i 0 −8.60311 17.8214 0 28.5047i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 260.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.b even 2 1 inner
87.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.3.d.a 20
3.b odd 2 1 inner 261.3.d.a 20
29.b even 2 1 inner 261.3.d.a 20
87.d odd 2 1 inner 261.3.d.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.3.d.a 20 1.a even 1 1 trivial
261.3.d.a 20 3.b odd 2 1 inner
261.3.d.a 20 29.b even 2 1 inner
261.3.d.a 20 87.d odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(261, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - 30 T^{8} + 301 T^{6} - 1171 T^{4} + \cdots - 727)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + 156 T^{8} + 8281 T^{6} + \cdots + 93312)^{2} \) Copy content Toggle raw display
$7$ \( (T^{5} + 4 T^{4} - 118 T^{3} - 194 T^{2} + \cdots - 5098)^{4} \) Copy content Toggle raw display
$11$ \( (T^{10} - 818 T^{8} + 226597 T^{6} + \cdots - 898967488)^{2} \) Copy content Toggle raw display
$13$ \( (T^{5} + 2 T^{4} - 263 T^{3} + 550 T^{2} + \cdots - 2972)^{4} \) Copy content Toggle raw display
$17$ \( (T^{10} - 1392 T^{8} + \cdots - 6181721712)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + 2228 T^{8} + \cdots + 1373715936)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + 4578 T^{8} + \cdots + 67020410880000)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + 694 T^{18} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( (T^{10} + 2736 T^{8} + \cdots + 776675246976)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 9390 T^{8} + \cdots + 64389156781176)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} - 6230 T^{8} + \cdots - 107423962720000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 18582 T^{8} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 16104 T^{8} + \cdots - 71\!\cdots\!28)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 13766 T^{8} + \cdots + 23\!\cdots\!72)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 16580 T^{8} + \cdots + 58\!\cdots\!12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 27024 T^{8} + \cdots + 58\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + 118 T^{4} + 2497 T^{3} + \cdots - 12068384)^{4} \) Copy content Toggle raw display
$71$ \( (T^{10} + 21018 T^{8} + \cdots + 4723920000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 16526 T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 13500 T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 46134 T^{8} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} - 18422 T^{8} + \cdots - 15\!\cdots\!88)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 51210 T^{8} + \cdots + 71\!\cdots\!24)^{2} \) Copy content Toggle raw display
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