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Properties

Label 69.2.e.c
Level $69$
Weight $2$
Character orbit 69.e
Analytic conductor $0.551$
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] = [69,2,Mod(4,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.e (of order \(11\), degree \(10\), 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: \(0.550967773947\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 7 x^{19} + 24 x^{18} - 70 x^{17} + 209 x^{16} - 527 x^{15} + 1115 x^{14} - 2187 x^{13} + 4165 x^{12} - 7040 x^{11} + 10649 x^{10} - 13519 x^{9} + 15111 x^{8} - 12101 x^{7} + \cdots + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{19} - \beta_{18} + 2 \beta_{17} + \beta_{15} + \beta_{14} - \beta_{12} + 2 \beta_{11} + \beta_{9} + \cdots + \beta_1) q^{2}+ \cdots + (\beta_{19} - \beta_{17} + \beta_{16} - \beta_{15} + \beta_{13} - \beta_{11} - \beta_{7} - \beta_{6} - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{19} - \beta_{18} + 2 \beta_{17} + \beta_{15} + \beta_{14} - \beta_{12} + 2 \beta_{11} + \beta_{9} + \cdots + \beta_1) q^{2}+ \cdots + ( - \beta_{19} - \beta_{18} + 2 \beta_{17} + \beta_{15} - \beta_{12} + 2 \beta_{11} - \beta_{10} + \beta_{9} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{2} - 2 q^{3} - 6 q^{4} - 6 q^{5} + 7 q^{6} - 6 q^{7} + 10 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{2} - 2 q^{3} - 6 q^{4} - 6 q^{5} + 7 q^{6} - 6 q^{7} + 10 q^{8} - 2 q^{9} - 18 q^{10} - 16 q^{11} - 6 q^{12} + 14 q^{13} - 22 q^{14} + 5 q^{15} - 8 q^{16} + 11 q^{17} - 4 q^{18} - 11 q^{19} + 57 q^{20} - 6 q^{21} + 26 q^{22} - 12 q^{24} - 4 q^{25} - 14 q^{26} - 2 q^{27} - 14 q^{28} + 12 q^{29} - 18 q^{30} + 41 q^{31} - 46 q^{32} - 5 q^{33} - 3 q^{34} - 26 q^{35} + 5 q^{36} - 18 q^{37} + 70 q^{38} + 14 q^{39} - 13 q^{40} + 44 q^{42} - 10 q^{43} - 3 q^{44} + 16 q^{45} - 24 q^{46} + 18 q^{47} + 25 q^{48} - 10 q^{49} + 33 q^{50} + 61 q^{52} - 20 q^{53} + 7 q^{54} - 17 q^{55} + 6 q^{56} + 11 q^{57} - 37 q^{58} + 40 q^{59} - 31 q^{60} - 12 q^{61} - 89 q^{62} - 6 q^{63} - 2 q^{64} - 51 q^{65} + 15 q^{66} - 47 q^{67} - 12 q^{68} - 22 q^{69} + 32 q^{70} - 47 q^{71} + 10 q^{72} + 39 q^{73} - 50 q^{74} - 26 q^{75} - 39 q^{76} + 22 q^{77} - 14 q^{78} - 2 q^{79} + 12 q^{80} - 2 q^{81} + 26 q^{82} - 52 q^{83} - 14 q^{84} + 35 q^{85} + 34 q^{86} - 32 q^{87} + 30 q^{88} + 36 q^{89} - 18 q^{90} + 8 q^{91} - 19 q^{92} - 14 q^{93} + 21 q^{94} + 89 q^{95} - 2 q^{96} + 85 q^{98} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 7 x^{19} + 24 x^{18} - 70 x^{17} + 209 x^{16} - 527 x^{15} + 1115 x^{14} - 2187 x^{13} + 4165 x^{12} - 7040 x^{11} + 10649 x^{10} - 13519 x^{9} + 15111 x^{8} - 12101 x^{7} + \cdots + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 44\!\cdots\!58 \nu^{19} + \cdots + 45\!\cdots\!20 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 75\!\cdots\!64 \nu^{19} + \cdots + 26\!\cdots\!36 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!75 \nu^{19} + \cdots + 23\!\cdots\!82 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!14 \nu^{19} + \cdots + 30\!\cdots\!65 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!92 \nu^{19} + \cdots - 41\!\cdots\!10 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!74 \nu^{19} + \cdots - 80\!\cdots\!94 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25\!\cdots\!75 \nu^{19} + \cdots - 32\!\cdots\!90 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 28\!\cdots\!38 \nu^{19} + \cdots - 25\!\cdots\!46 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 32\!\cdots\!70 \nu^{19} + \cdots + 21\!\cdots\!72 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 35\!\cdots\!92 \nu^{19} + \cdots + 38\!\cdots\!17 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 40\!\cdots\!23 \nu^{19} + \cdots - 23\!\cdots\!35 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 41\!\cdots\!68 \nu^{19} + \cdots + 46\!\cdots\!53 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 43\!\cdots\!69 \nu^{19} + \cdots + 61\!\cdots\!51 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 44\!\cdots\!15 \nu^{19} + \cdots + 35\!\cdots\!20 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 47\!\cdots\!74 \nu^{19} + \cdots + 24\!\cdots\!70 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 50\!\cdots\!84 \nu^{19} + \cdots + 48\!\cdots\!71 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 57\!\cdots\!30 \nu^{19} + \cdots + 12\!\cdots\!30 ) / 24\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 60\!\cdots\!10 \nu^{19} + \cdots + 50\!\cdots\!81 ) / 24\!\cdots\!19 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{19} + \beta_{17} + \beta_{15} + \beta_{11} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{17} + \beta_{15} - \beta_{13} - \beta_{12} + 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - 5 \beta_{4} + 2 \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{19} - \beta_{18} + 2 \beta_{17} - \beta_{16} + \beta_{14} + 5 \beta_{13} - \beta_{12} + 2 \beta_{11} - 7 \beta_{10} + 8 \beta_{9} + 2 \beta_{8} + \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 8 \beta_{4} + \beta_{3} - 5 \beta_{2} + 8 \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{19} + 6 \beta_{17} + 8 \beta_{16} + 7 \beta_{15} + 8 \beta_{14} + 8 \beta_{13} + 7 \beta_{11} - 9 \beta_{10} + 29 \beta_{9} + 10 \beta_{8} + 2 \beta_{6} + 8 \beta_{5} - 17 \beta_{4} - \beta_{2} + 10 \beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15 \beta_{19} - 3 \beta_{18} + 13 \beta_{17} + 29 \beta_{16} + 10 \beta_{15} + 14 \beta_{14} + 7 \beta_{13} + 3 \beta_{11} - 36 \beta_{10} + 50 \beta_{9} + 48 \beta_{8} + 2 \beta_{7} - 9 \beta_{6} + 3 \beta_{5} - 48 \beta_{4} - 9 \beta_{3} - 26 \beta_{2} + 14 \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 21 \beta_{19} - 56 \beta_{18} + 56 \beta_{17} + 61 \beta_{16} - 7 \beta_{15} + 81 \beta_{14} + 102 \beta_{13} + 4 \beta_{12} + 14 \beta_{11} - 100 \beta_{10} + 125 \beta_{9} + 125 \beta_{8} - 23 \beta_{7} + 76 \beta_{6} - 81 \beta_{4} - 4 \beta_{3} + \cdots - 42 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 116 \beta_{19} - 92 \beta_{18} + 19 \beta_{17} + 225 \beta_{16} + 291 \beta_{14} + 262 \beta_{13} + 106 \beta_{12} + 2 \beta_{11} - 28 \beta_{10} + 291 \beta_{9} + 319 \beta_{8} - 133 \beta_{7} + 121 \beta_{6} - 43 \beta_{5} - 92 \beta_{4} + \cdots - 73 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 145 \beta_{19} - 227 \beta_{18} + 13 \beta_{17} + 455 \beta_{16} + 34 \beta_{15} + 509 \beta_{14} + 419 \beta_{13} + 442 \beta_{12} - 228 \beta_{11} + 227 \beta_{9} + 795 \beta_{8} - 192 \beta_{7} + 82 \beta_{6} - 442 \beta_{5} - 380 \beta_{3} + \cdots + 34 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 297 \beta_{19} - 1211 \beta_{18} + 537 \beta_{17} + 674 \beta_{16} - 483 \beta_{15} + 1485 \beta_{14} + 1457 \beta_{13} + 1131 \beta_{12} - 246 \beta_{11} + 274 \beta_{10} + 1485 \beta_{8} - 571 \beta_{7} + 1259 \beta_{6} + \cdots - 1131 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 122 \beta_{19} - 1870 \beta_{18} - 1240 \beta_{17} + 1500 \beta_{16} - 1500 \beta_{15} + 3426 \beta_{14} + 3110 \beta_{13} + 4398 \beta_{12} - 1434 \beta_{11} + 3426 \beta_{10} - 1833 \beta_{9} + 1870 \beta_{8} + \cdots - 118 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4262 \beta_{19} - 2330 \beta_{18} - 4262 \beta_{17} - 1786 \beta_{15} + 2330 \beta_{14} + 1786 \beta_{13} + 12936 \beta_{12} - 6790 \beta_{11} + 9960 \beta_{10} - 12936 \beta_{9} - 5511 \beta_{7} - 4460 \beta_{6} + \cdots + 3102 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 10883 \beta_{19} - 8909 \beta_{18} - 5526 \beta_{17} - 9390 \beta_{16} - 10883 \beta_{15} + 26744 \beta_{12} - 13802 \beta_{11} + 26744 \beta_{10} - 44123 \beta_{9} - 12738 \beta_{8} - 13802 \beta_{7} + \cdots + 9390 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 18468 \beta_{19} - 49417 \beta_{17} - 35000 \beta_{16} - 30770 \beta_{15} - 18468 \beta_{14} - 15115 \beta_{13} + 67948 \beta_{12} - 45605 \beta_{11} + 88470 \beta_{10} - 134683 \beta_{9} - 67948 \beta_{8} + \cdots + 15115 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 67441 \beta_{19} + 53292 \beta_{18} - 140549 \beta_{17} - 138979 \beta_{16} - 35723 \beta_{15} - 141583 \beta_{14} - 134867 \beta_{13} + 141583 \beta_{12} - 140549 \beta_{11} + 222452 \beta_{10} + \cdots + 74142 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 145408 \beta_{19} + 140670 \beta_{18} - 272160 \beta_{17} - 452326 \beta_{16} - 134161 \beta_{15} - 468982 \beta_{14} - 452326 \beta_{13} + 140670 \beta_{12} - 274831 \beta_{11} + 492028 \beta_{10} + \cdots + 203059 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 160116 \beta_{19} + 618272 \beta_{18} - 948695 \beta_{17} - 1216107 \beta_{16} - 330423 \beta_{15} - 1376223 \beta_{14} - 1270704 \beta_{13} - 618272 \beta_{11} + 1177528 \beta_{10} + \cdots + 304386 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 168627 \beta_{19} + 2217072 \beta_{18} - 2217072 \beta_{17} - 3248010 \beta_{16} - 95985 \beta_{15} - 4262288 \beta_{14} - 4093661 \beta_{13} - 1045800 \beta_{12} - 1263141 \beta_{11} + \cdots + 953931 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 241892 \beta_{19} + 5471549 \beta_{18} - 3300861 \beta_{17} - 8086529 \beta_{16} - 11100538 \beta_{14} - 10816524 \beta_{13} - 6475244 \beta_{12} - 894624 \beta_{11} + \cdots + 2170688 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-\beta_{16}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.545935 + 0.160301i
2.55199 + 0.749331i
−0.591994 + 0.380451i
1.76796 − 1.13620i
−0.591994 − 0.380451i
1.76796 + 1.13620i
−0.201819 + 1.40368i
0.124087 − 0.863041i
−0.932269 + 2.04138i
0.480233 − 1.05156i
−0.932269 − 2.04138i
0.480233 + 1.05156i
0.545935 − 0.160301i
2.55199 − 0.749331i
1.05639 − 1.21914i
−1.30051 + 1.50087i
−0.201819 − 1.40368i
0.124087 + 0.863041i
1.05639 + 1.21914i
−1.30051 − 1.50087i
−1.10489 − 2.41937i −0.959493 + 0.281733i −3.32285 + 3.83478i −2.44470 − 1.57111i 1.74175 + 2.01009i −0.326826 − 2.27313i 7.84516 + 2.30355i 0.841254 − 0.540641i −1.09998 + 7.65055i
4.2 −0.236364 − 0.517565i −0.959493 + 0.281733i 1.09772 − 1.26683i 2.27341 + 1.46103i 0.372604 + 0.430008i −0.481879 − 3.35154i −2.00700 − 0.589308i 0.841254 − 0.540641i 0.218827 − 1.52197i
13.1 −1.37624 − 1.58827i 0.841254 + 0.540641i −0.343924 + 2.39205i 1.43308 − 3.13801i −0.299086 − 2.08019i −1.51768 − 0.445631i 0.736612 − 0.473392i 0.415415 + 0.909632i −6.95628 + 2.04255i
13.2 0.460828 + 0.531824i 0.841254 + 0.540641i 0.214155 − 1.48948i −0.700489 + 1.53386i 0.100148 + 0.696541i −3.11747 − 0.915371i 2.07482 − 1.33341i 0.415415 + 0.909632i −1.13855 + 0.334308i
16.1 −1.37624 + 1.58827i 0.841254 − 0.540641i −0.343924 − 2.39205i 1.43308 + 3.13801i −0.299086 + 2.08019i −1.51768 + 0.445631i 0.736612 + 0.473392i 0.415415 − 0.909632i −6.95628 − 2.04255i
16.2 0.460828 − 0.531824i 0.841254 − 0.540641i 0.214155 + 1.48948i −0.700489 − 1.53386i 0.100148 − 0.696541i −3.11747 + 0.915371i 2.07482 + 1.33341i 0.415415 − 0.909632i −1.13855 − 0.334308i
25.1 −0.733503 − 0.471394i −0.142315 − 0.989821i −0.515016 − 1.12773i −1.20424 − 0.353596i −0.362207 + 0.793123i 1.95301 − 2.25389i −0.402011 + 2.79605i −0.959493 + 0.281733i 0.716629 + 0.827034i
25.2 1.19300 + 0.766692i −0.142315 − 0.989821i 0.00459227 + 0.0100557i 0.815022 + 0.239312i 0.589107 − 1.28996i −3.31578 + 3.82661i 0.401407 − 2.79185i −0.959493 + 0.281733i 0.788839 + 0.910368i
31.1 −0.164520 + 1.14426i 0.415415 + 0.909632i 0.636711 + 0.186955i −1.13298 − 1.30753i −1.10920 + 0.325692i −0.589836 − 0.379064i −1.27914 + 2.80093i −0.654861 + 0.755750i 1.68256 − 1.08131i
31.2 0.319381 − 2.22134i 0.415415 + 0.909632i −2.91338 − 0.855446i −1.82263 − 2.10342i 2.15328 − 0.632261i 2.85304 + 1.83354i −0.966182 + 2.11564i −0.654861 + 0.755750i −5.25454 + 3.37689i
49.1 −0.164520 − 1.14426i 0.415415 − 0.909632i 0.636711 − 0.186955i −1.13298 + 1.30753i −1.10920 − 0.325692i −0.589836 + 0.379064i −1.27914 − 2.80093i −0.654861 − 0.755750i 1.68256 + 1.08131i
49.2 0.319381 + 2.22134i 0.415415 − 0.909632i −2.91338 + 0.855446i −1.82263 + 2.10342i 2.15328 + 0.632261i 2.85304 − 1.83354i −0.966182 − 2.11564i −0.654861 − 0.755750i −5.25454 − 3.37689i
52.1 −1.10489 + 2.41937i −0.959493 − 0.281733i −3.32285 − 3.83478i −2.44470 + 1.57111i 1.74175 − 2.01009i −0.326826 + 2.27313i 7.84516 − 2.30355i 0.841254 + 0.540641i −1.09998 − 7.65055i
52.2 −0.236364 + 0.517565i −0.959493 − 0.281733i 1.09772 + 1.26683i 2.27341 − 1.46103i 0.372604 − 0.430008i −0.481879 + 3.35154i −2.00700 + 0.589308i 0.841254 + 0.540641i 0.218827 + 1.52197i
55.1 −1.90549 + 0.559503i −0.654861 − 0.755750i 1.63535 − 1.05098i −0.291446 − 2.02705i 1.67068 + 1.07368i 1.34249 − 2.93963i 0.0729013 − 0.0841325i −0.142315 + 0.989821i 1.68949 + 3.69946i
55.2 1.54781 − 0.454477i −0.654861 − 0.755750i 0.506650 − 0.325604i 0.0749695 + 0.521424i −1.35707 − 0.872135i 0.200934 − 0.439985i −1.47656 + 1.70404i −0.142315 + 0.989821i 0.353014 + 0.772992i
58.1 −0.733503 + 0.471394i −0.142315 + 0.989821i −0.515016 + 1.12773i −1.20424 + 0.353596i −0.362207 − 0.793123i 1.95301 + 2.25389i −0.402011 − 2.79605i −0.959493 − 0.281733i 0.716629 − 0.827034i
58.2 1.19300 − 0.766692i −0.142315 + 0.989821i 0.00459227 − 0.0100557i 0.815022 − 0.239312i 0.589107 + 1.28996i −3.31578 − 3.82661i 0.401407 + 2.79185i −0.959493 − 0.281733i 0.788839 − 0.910368i
64.1 −1.90549 − 0.559503i −0.654861 + 0.755750i 1.63535 + 1.05098i −0.291446 + 2.02705i 1.67068 − 1.07368i 1.34249 + 2.93963i 0.0729013 + 0.0841325i −0.142315 − 0.989821i 1.68949 − 3.69946i
64.2 1.54781 + 0.454477i −0.654861 + 0.755750i 0.506650 + 0.325604i 0.0749695 − 0.521424i −1.35707 + 0.872135i 0.200934 + 0.439985i −1.47656 − 1.70404i −0.142315 − 0.989821i 0.353014 − 0.772992i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.2.e.c ✓ 20
3.b odd 2 1 207.2.i.d 20
23.c even 11 1 inner 69.2.e.c ✓ 20
23.c even 11 1 1587.2.a.u 10
23.d odd 22 1 1587.2.a.t 10
69.g even 22 1 4761.2.a.bu 10
69.h odd 22 1 207.2.i.d 20
69.h odd 22 1 4761.2.a.bt 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.e.c ✓ 20 1.a even 1 1 trivial
69.2.e.c ✓ 20 23.c even 11 1 inner
207.2.i.d 20 3.b odd 2 1
207.2.i.d 20 69.h odd 22 1
1587.2.a.t 10 23.d odd 22 1
1587.2.a.u 10 23.c even 11 1
4761.2.a.bt 10 69.h odd 22 1
4761.2.a.bu 10 69.g even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 4 T_{2}^{19} + 13 T_{2}^{18} + 18 T_{2}^{17} + 11 T_{2}^{16} - 32 T_{2}^{15} - 117 T_{2}^{14} - 130 T_{2}^{13} + 238 T_{2}^{12} + 891 T_{2}^{11} + 584 T_{2}^{10} - 1463 T_{2}^{9} - 3 T_{2}^{8} - 518 T_{2}^{7} + 3360 T_{2}^{6} + \cdots + 529 \) acting on \(S_{2}^{\mathrm{new}}(69, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 4 T^{19} + 13 T^{18} + 18 T^{17} + \cdots + 529 \) Copy content Toggle raw display
$3$ \( (T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + 6 T^{19} + 25 T^{18} + \cdots + 64009 \) Copy content Toggle raw display
$7$ \( T^{20} + 6 T^{19} + 30 T^{18} + \cdots + 5031049 \) Copy content Toggle raw display
$11$ \( T^{20} + 16 T^{19} + 164 T^{18} + \cdots + 212521 \) Copy content Toggle raw display
$13$ \( T^{20} - 14 T^{19} + 137 T^{18} + \cdots + 27867841 \) Copy content Toggle raw display
$17$ \( T^{20} - 11 T^{19} + 93 T^{18} + \cdots + 21743569 \) Copy content Toggle raw display
$19$ \( T^{20} + 11 T^{19} + \cdots + 417211354561 \) Copy content Toggle raw display
$23$ \( T^{20} + 67 T^{18} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} - 12 T^{19} + \cdots + 3983644752649 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 811602438117769 \) Copy content Toggle raw display
$37$ \( T^{20} + 18 T^{19} + \cdots + 185232969769 \) Copy content Toggle raw display
$41$ \( T^{20} + 92 T^{18} + \cdots + 664350515929 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 131462437009849 \) Copy content Toggle raw display
$47$ \( (T^{10} - 9 T^{9} - 247 T^{8} + \cdots + 87731039)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + 20 T^{19} + \cdots + 828516832441 \) Copy content Toggle raw display
$59$ \( T^{20} - 40 T^{19} + \cdots + 57652331881 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 559921052347801 \) Copy content Toggle raw display
$67$ \( T^{20} + 47 T^{19} + 964 T^{18} + \cdots + 2647129 \) Copy content Toggle raw display
$71$ \( T^{20} + 47 T^{19} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$73$ \( T^{20} - 39 T^{19} + 621 T^{18} + \cdots + 11485321 \) Copy content Toggle raw display
$79$ \( T^{20} + 2 T^{19} + \cdots + 39\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{20} + 52 T^{19} + \cdots + 79\!\cdots\!61 \) Copy content Toggle raw display
$89$ \( T^{20} - 36 T^{19} + \cdots + 586753729 \) Copy content Toggle raw display
$97$ \( T^{20} + 12 T^{18} + \cdots + 48681166341721 \) Copy content Toggle raw display
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