Properties

Label 850.2.l.i
Level $850$
Weight $2$
Character orbit 850.l
Analytic conductor $6.787$
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] = [850,2,Mod(151,850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(850, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("850.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 850.l (of order \(8\), degree \(4\), 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: \(6.78728417181\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 16 x^{15} + 52 x^{14} + 992 x^{13} + 6181 x^{12} + 8952 x^{11} + 6244 x^{10} - 11448 x^{9} - 14520 x^{8} + 27936 x^{7} + 27880 x^{6} - 121104 x^{5} + 187460 x^{4} + \cdots + 2048 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 170)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + \beta_{2} q^{3} + \beta_{14} q^{4} + \beta_1 q^{6} + (\beta_{19} - \beta_{11} - \beta_{9} + \beta_{4} - \beta_{3}) q^{7} + \beta_{8} q^{8} + (\beta_{19} - \beta_{13} - \beta_{11} - 2 \beta_{8} + \beta_{6} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{2} + \beta_{2} q^{3} + \beta_{14} q^{4} + \beta_1 q^{6} + (\beta_{19} - \beta_{11} - \beta_{9} + \beta_{4} - \beta_{3}) q^{7} + \beta_{8} q^{8} + (\beta_{19} - \beta_{13} - \beta_{11} - 2 \beta_{8} + \beta_{6} + \beta_{4}) q^{9} - \beta_{15} q^{11} - \beta_{7} q^{12} + ( - \beta_{14} + \beta_{11} + \beta_{10} + \beta_{8}) q^{13} + (\beta_{12} + \beta_{11} + \beta_{10} + \beta_{9}) q^{14} - q^{16} + ( - \beta_{19} - \beta_{15} + \beta_{12} + 2 \beta_{11} + \beta_{9} - \beta_{6} - \beta_{4} + \beta_{2} + 1) q^{17} + ( - \beta_{3} + 1) q^{18} + ( - \beta_{19} - \beta_{15} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{8} + \beta_{5} + \beta_{2}) q^{19} + (\beta_{15} + \beta_{12} - \beta_{11} - \beta_{8} + \beta_{6} + \beta_{4} + \beta_{2} - \beta_1) q^{21} + ( - \beta_{17} - \beta_{11} - \beta_{8} + \beta_{4} - 1) q^{22} + (\beta_{15} + \beta_{12} - \beta_{8} - \beta_{7} + \beta_{4} + \beta_{2}) q^{23} + \beta_{5} q^{24} + ( - \beta_{13} + \beta_{12} - 2 \beta_{8} + \beta_{4} - 1) q^{26} + (2 \beta_{19} - \beta_{17} + \beta_{16} - \beta_{13} - 2 \beta_{11} + \beta_{10} - 2 \beta_{8} + \beta_{6} + \cdots + 2 \beta_{4}) q^{27}+ \cdots + ( - 3 \beta_{19} + \beta_{17} + \beta_{15} - \beta_{14} + 2 \beta_{13} + \beta_{12} + 3 \beta_{11} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 8 q^{11} - 20 q^{16} + 20 q^{17} + 28 q^{18} - 8 q^{22} + 8 q^{23} - 12 q^{26} + 12 q^{29} + 8 q^{31} - 32 q^{33} - 8 q^{34} - 28 q^{37} + 4 q^{41} + 8 q^{42} - 16 q^{43} + 8 q^{44} + 16 q^{46} + 56 q^{49} - 8 q^{51} + 24 q^{52} + 44 q^{53} + 24 q^{54} - 24 q^{57} - 12 q^{58} - 16 q^{59} + 8 q^{61} + 8 q^{62} - 80 q^{63} - 8 q^{66} - 48 q^{67} + 16 q^{69} + 8 q^{71} + 16 q^{73} - 28 q^{74} - 56 q^{79} - 16 q^{82} - 16 q^{84} + 48 q^{86} + 72 q^{87} + 8 q^{88} - 24 q^{91} - 16 q^{92} - 72 q^{93} - 32 q^{94} - 12 q^{97} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 16 x^{15} + 52 x^{14} + 992 x^{13} + 6181 x^{12} + 8952 x^{11} + 6244 x^{10} - 11448 x^{9} - 14520 x^{8} + 27936 x^{7} + 27880 x^{6} - 121104 x^{5} + 187460 x^{4} + \cdots + 2048 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24\!\cdots\!67 \nu^{19} + \cdots + 24\!\cdots\!36 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16\!\cdots\!93 \nu^{19} + \cdots + 51\!\cdots\!36 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!48 \nu^{19} + \cdots + 71\!\cdots\!24 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!03 \nu^{19} + \cdots - 17\!\cdots\!44 ) / 15\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38\!\cdots\!01 \nu^{19} + \cdots - 13\!\cdots\!32 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 38\!\cdots\!43 \nu^{19} + \cdots + 19\!\cdots\!44 ) / 25\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!32 \nu^{19} + \cdots - 17\!\cdots\!56 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 51\!\cdots\!17 \nu^{19} + \cdots - 38\!\cdots\!64 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 56\!\cdots\!45 \nu^{19} + \cdots - 61\!\cdots\!08 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 74\!\cdots\!99 \nu^{19} + \cdots - 33\!\cdots\!32 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 25\!\cdots\!33 \nu^{19} + \cdots - 49\!\cdots\!04 ) / 64\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 42\!\cdots\!69 \nu^{19} + \cdots + 16\!\cdots\!48 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\!\cdots\!31 \nu^{19} + \cdots - 12\!\cdots\!56 ) / 30\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 24\!\cdots\!87 \nu^{19} + \cdots - 34\!\cdots\!36 ) / 42\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 34\!\cdots\!27 \nu^{19} + \cdots + 60\!\cdots\!96 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 27\!\cdots\!23 \nu^{19} + \cdots - 52\!\cdots\!72 ) / 21\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 13\!\cdots\!13 \nu^{19} + \cdots + 95\!\cdots\!04 ) / 68\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 43\!\cdots\!67 \nu^{19} + \cdots - 36\!\cdots\!28 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{18} + 5\beta_{11} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{18} + \beta_{17} - \beta_{16} + 2 \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{8} - 7 \beta_{7} - 2 \beta_{6} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{19} + 9 \beta_{18} + 10 \beta_{17} - 10 \beta_{16} + 19 \beta_{14} + 11 \beta_{11} - 9 \beta_{10} + 11 \beta_{8} - 3 \beta_{7} - 10 \beta_{6} + 3 \beta_{5} - 10 \beta_{4} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 26 \beta_{19} + 2 \beta_{18} + 13 \beta_{17} - 13 \beta_{16} + 10 \beta_{14} + 13 \beta_{13} + 22 \beta_{11} - 13 \beta_{10} + 38 \beta_{8} - \beta_{7} - 15 \beta_{6} + 57 \beta_{5} - 26 \beta_{4} - \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 84 \beta_{19} + \beta_{18} + \beta_{17} - 14 \beta_{16} + 5 \beta_{14} + 83 \beta_{13} + 9 \beta_{12} + 95 \beta_{11} - 3 \beta_{10} + 327 \beta_{8} + 15 \beta_{7} - 95 \beta_{6} + 51 \beta_{5} - 86 \beta_{4} + 3 \beta_{3} - 51 \beta_{2} + 15 \beta _1 - 17 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 42 \beta_{19} - 104 \beta_{17} - \beta_{15} + 44 \beta_{14} + 146 \beta_{13} - 97 \beta_{12} + 106 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 454 \beta_{8} - 147 \beta_{6} + 22 \beta_{5} - 146 \beta_{4} + 146 \beta_{3} - 499 \beta_{2} + \cdots - 308 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 16 \beta_{19} - 16 \beta_{18} - 49 \beta_{17} - 49 \beta_{16} - 54 \beta_{15} + 68 \beta_{13} + 54 \beta_{12} - 184 \beta_{11} - 68 \beta_{9} + 184 \beta_{8} - 167 \beta_{7} - 5 \beta_{6} - 167 \beta_{5} + 5 \beta_{4} + 793 \beta_{3} + \cdots - 2094 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 618 \beta_{18} - 20 \beta_{17} - 949 \beta_{16} + 803 \beta_{15} - 281 \beta_{14} - 50 \beta_{13} + 18 \beta_{12} - 5124 \beta_{11} - 50 \beta_{10} - 1567 \beta_{9} - 1354 \beta_{8} + 343 \beta_{7} + 1567 \beta_{6} + \cdots - 3557 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 235 \beta_{19} - 8160 \beta_{18} - 1898 \beta_{17} + 345 \beta_{16} - 187 \beta_{15} - 1791 \beta_{14} + 110 \beta_{12} - 28767 \beta_{11} + 1057 \beta_{10} - 7757 \beta_{9} - 9111 \beta_{8} + 7403 \beta_{7} + 8924 \beta_{6} + \cdots - 2925 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7922 \beta_{19} - 32118 \beta_{18} - 17990 \beta_{17} + 16713 \beta_{16} - 199 \beta_{15} - 22087 \beta_{14} + 513 \beta_{13} - 64712 \beta_{11} + 16514 \beta_{10} - 16514 \beta_{9} - 27602 \beta_{8} + \cdots + 5311 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 82813 \beta_{19} - 82813 \beta_{18} - 86733 \beta_{17} + 86733 \beta_{16} - 1591 \beta_{15} - 114125 \beta_{14} - 14070 \beta_{13} - 1591 \beta_{12} - 140959 \beta_{11} + 77063 \beta_{10} - 14070 \beta_{9} + \cdots + 15097 \beta_1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 331942 \beta_{19} - 94782 \beta_{18} - 174291 \beta_{17} + 191543 \beta_{16} - 238314 \beta_{14} - 172631 \beta_{13} - 1660 \beta_{12} - 301906 \beta_{11} + 172631 \beta_{10} + 3540 \beta_{9} + \cdots - 43136 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 848546 \beta_{19} - 49943 \beta_{18} - 69075 \beta_{17} + 241486 \beta_{16} + 19132 \beta_{15} - 285339 \beta_{14} - 773659 \beta_{13} + 29605 \beta_{12} - 936175 \beta_{11} + 172989 \beta_{10} + \cdots + 409591 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1090742 \beta_{19} + 707214 \beta_{17} + 180224 \beta_{16} + 9895 \beta_{15} - 185866 \beta_{14} - 1797956 \beta_{13} + 479981 \beta_{12} - 1628070 \beta_{11} - 6085 \beta_{10} - 6085 \beta_{9} + \cdots + 4334562 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 683616 \beta_{19} + 683616 \beta_{18} + 1317975 \beta_{17} + 1317975 \beta_{16} - 440432 \beta_{15} - 2032620 \beta_{13} + 440432 \beta_{12} + 5342456 \beta_{11} + 2032620 \beta_{9} + \cdots + 19468618 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 12259266 \beta_{18} + 2539356 \beta_{17} + 6436269 \beta_{16} - 4307055 \beta_{15} + 2218577 \beta_{14} - 330322 \beta_{13} - 12030 \beta_{12} + 63959300 \beta_{11} - 330322 \beta_{10} + \cdots + 45263765 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 8931491 \beta_{19} + 90481344 \beta_{18} + 30086562 \beta_{17} - 11036645 \beta_{16} - 5424885 \beta_{15} + 37389111 \beta_{14} - 2105154 \beta_{12} + 292497267 \beta_{11} + \cdots + 53087193 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 135715274 \beta_{19} + 362298902 \beta_{18} + 220871206 \beta_{17} - 193432777 \beta_{16} - 858473 \beta_{15} + 277867367 \beta_{14} + 7913799 \beta_{13} + 827886624 \beta_{11} + \cdots - 3698703 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/850\mathbb{Z}\right)^\times\).

\(n\) \(477\) \(751\)
\(\chi(n)\) \(1\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
−0.969032 + 2.33945i
−0.826884 + 1.99627i
0.254075 0.613391i
0.355063 0.857197i
1.18678 2.86514i
−2.32088 + 0.961341i
−1.86202 + 0.771273i
0.236338 0.0978946i
0.953222 0.394838i
2.99334 1.23988i
−0.969032 2.33945i
−0.826884 1.99627i
0.254075 + 0.613391i
0.355063 + 0.857197i
1.18678 + 2.86514i
−2.32088 0.961341i
−1.86202 0.771273i
0.236338 + 0.0978946i
0.953222 + 0.394838i
2.99334 + 1.23988i
−0.707107 0.707107i −0.969032 2.33945i 1.00000i 0 −0.969032 + 2.33945i 3.06021 + 1.26758i 0.707107 0.707107i −2.41268 + 2.41268i 0
151.2 −0.707107 0.707107i −0.826884 1.99627i 1.00000i 0 −0.826884 + 1.99627i −3.20595 1.32795i 0.707107 0.707107i −1.18005 + 1.18005i 0
151.3 −0.707107 0.707107i 0.254075 + 0.613391i 1.00000i 0 0.254075 0.613391i −4.76369 1.97319i 0.707107 0.707107i 1.80963 1.80963i 0
151.4 −0.707107 0.707107i 0.355063 + 0.857197i 1.00000i 0 0.355063 0.857197i 2.26926 + 0.939960i 0.707107 0.707107i 1.51260 1.51260i 0
151.5 −0.707107 0.707107i 1.18678 + 2.86514i 1.00000i 0 1.18678 2.86514i 2.64018 + 1.09360i 0.707107 0.707107i −4.67924 + 4.67924i 0
451.1 0.707107 0.707107i −2.32088 0.961341i 1.00000i 0 −2.32088 + 0.961341i −0.0515871 0.124542i −0.707107 0.707107i 2.34100 + 2.34100i 0
451.2 0.707107 0.707107i −1.86202 0.771273i 1.00000i 0 −1.86202 + 0.771273i 1.02468 + 2.47378i −0.707107 0.707107i 0.750924 + 0.750924i 0
451.3 0.707107 0.707107i 0.236338 + 0.0978946i 1.00000i 0 0.236338 0.0978946i −1.44332 3.48449i −0.707107 0.707107i −2.07505 2.07505i 0
451.4 0.707107 0.707107i 0.953222 + 0.394838i 1.00000i 0 0.953222 0.394838i −0.150759 0.363965i −0.707107 0.707107i −1.36858 1.36858i 0
451.5 0.707107 0.707107i 2.99334 + 1.23988i 1.00000i 0 2.99334 1.23988i 0.620992 + 1.49921i −0.707107 0.707107i 5.30145 + 5.30145i 0
501.1 −0.707107 + 0.707107i −0.969032 + 2.33945i 1.00000i 0 −0.969032 2.33945i 3.06021 1.26758i 0.707107 + 0.707107i −2.41268 2.41268i 0
501.2 −0.707107 + 0.707107i −0.826884 + 1.99627i 1.00000i 0 −0.826884 1.99627i −3.20595 + 1.32795i 0.707107 + 0.707107i −1.18005 1.18005i 0
501.3 −0.707107 + 0.707107i 0.254075 0.613391i 1.00000i 0 0.254075 + 0.613391i −4.76369 + 1.97319i 0.707107 + 0.707107i 1.80963 + 1.80963i 0
501.4 −0.707107 + 0.707107i 0.355063 0.857197i 1.00000i 0 0.355063 + 0.857197i 2.26926 0.939960i 0.707107 + 0.707107i 1.51260 + 1.51260i 0
501.5 −0.707107 + 0.707107i 1.18678 2.86514i 1.00000i 0 1.18678 + 2.86514i 2.64018 1.09360i 0.707107 + 0.707107i −4.67924 4.67924i 0
801.1 0.707107 + 0.707107i −2.32088 + 0.961341i 1.00000i 0 −2.32088 0.961341i −0.0515871 + 0.124542i −0.707107 + 0.707107i 2.34100 2.34100i 0
801.2 0.707107 + 0.707107i −1.86202 + 0.771273i 1.00000i 0 −1.86202 0.771273i 1.02468 2.47378i −0.707107 + 0.707107i 0.750924 0.750924i 0
801.3 0.707107 + 0.707107i 0.236338 0.0978946i 1.00000i 0 0.236338 + 0.0978946i −1.44332 + 3.48449i −0.707107 + 0.707107i −2.07505 + 2.07505i 0
801.4 0.707107 + 0.707107i 0.953222 0.394838i 1.00000i 0 0.953222 + 0.394838i −0.150759 + 0.363965i −0.707107 + 0.707107i −1.36858 + 1.36858i 0
801.5 0.707107 + 0.707107i 2.99334 1.23988i 1.00000i 0 2.99334 + 1.23988i 0.620992 1.49921i −0.707107 + 0.707107i 5.30145 5.30145i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 850.2.l.i 20
5.b even 2 1 850.2.l.h 20
5.c odd 4 1 170.2.n.a 20
5.c odd 4 1 170.2.n.b yes 20
17.d even 8 1 inner 850.2.l.i 20
85.k odd 8 1 170.2.n.a 20
85.m even 8 1 850.2.l.h 20
85.n odd 8 1 170.2.n.b yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.n.a 20 5.c odd 4 1
170.2.n.a 20 85.k odd 8 1
170.2.n.b yes 20 5.c odd 4 1
170.2.n.b yes 20 85.n odd 8 1
850.2.l.h 20 5.b even 2 1
850.2.l.h 20 85.m even 8 1
850.2.l.i 20 1.a even 1 1 trivial
850.2.l.i 20 17.d even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 16 T_{3}^{15} + 52 T_{3}^{14} + 992 T_{3}^{13} + 6181 T_{3}^{12} + 8952 T_{3}^{11} + 6244 T_{3}^{10} - 11448 T_{3}^{9} - 14520 T_{3}^{8} + 27936 T_{3}^{7} + 27880 T_{3}^{6} - 121104 T_{3}^{5} + 187460 T_{3}^{4} + \cdots + 2048 \) acting on \(S_{2}^{\mathrm{new}}(850, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} - 16 T^{15} + 52 T^{14} + \cdots + 2048 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 28 T^{18} + 8 T^{17} + \cdots + 131072 \) Copy content Toggle raw display
$11$ \( T^{20} + 8 T^{19} + 32 T^{18} + \cdots + 102760448 \) Copy content Toggle raw display
$13$ \( T^{20} + 152 T^{18} + 9418 T^{16} + \cdots + 430336 \) Copy content Toggle raw display
$17$ \( T^{20} - 20 T^{19} + \cdots + 2015993900449 \) Copy content Toggle raw display
$19$ \( T^{20} + 104 T^{17} + \cdots + 554696704 \) Copy content Toggle raw display
$23$ \( T^{20} - 8 T^{19} + \cdots + 4933025792 \) Copy content Toggle raw display
$29$ \( T^{20} - 12 T^{19} + \cdots + 1394342432 \) Copy content Toggle raw display
$31$ \( T^{20} - 8 T^{19} + 4 T^{18} + \cdots + 104603648 \) Copy content Toggle raw display
$37$ \( T^{20} + 28 T^{19} + \cdots + 22002985088 \) Copy content Toggle raw display
$41$ \( T^{20} - 4 T^{19} + \cdots + 496557944352 \) Copy content Toggle raw display
$43$ \( T^{20} + 16 T^{19} + \cdots + 56865079296 \) Copy content Toggle raw display
$47$ \( T^{20} + 564 T^{18} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{20} - 44 T^{19} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{20} + 16 T^{19} + \cdots + 24599695396864 \) Copy content Toggle raw display
$61$ \( T^{20} - 8 T^{19} + \cdots + 44\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( (T^{10} + 24 T^{9} - 116 T^{8} + \cdots + 4189696)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 215737327296512 \) Copy content Toggle raw display
$73$ \( T^{20} - 16 T^{19} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + 56 T^{19} + \cdots + 694577266688 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 740139447894016 \) Copy content Toggle raw display
$89$ \( T^{20} + 832 T^{18} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{20} + 12 T^{19} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
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