Properties

Label 550.2.k.c
Level $550$
Weight $2$
Character orbit 550.k
Analytic conductor $4.392$
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] = [550,2,Mod(111,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.k (of order \(5\), 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: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} - 7 x^{18} + 37 x^{17} + 4 x^{16} - 208 x^{15} + 152 x^{14} + 1068 x^{13} - 2777 x^{12} + 498 x^{11} + 11367 x^{10} - 32054 x^{9} + 50370 x^{8} - 50274 x^{7} + \cdots + 3455 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{7} + \beta_{6} + 1) q^{2} + ( - \beta_{19} - \beta_{16} + \beta_{14} - \beta_{13} + \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \cdots + 2) q^{3}+ \cdots + ( - \beta_{19} - \beta_{18} - \beta_{15} - \beta_{13} + \beta_{11} + \beta_{10} + \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} - \beta_{7} + \beta_{6} + 1) q^{2} + ( - \beta_{19} - \beta_{16} + \beta_{14} - \beta_{13} + \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \cdots + 2) q^{3}+ \cdots + ( - \beta_{19} + \beta_{18} - 2 \beta_{16} + \beta_{15} + 2 \beta_{14} - \beta_{10} + 2 \beta_{9} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 5 q^{2} + 6 q^{3} - 5 q^{4} - 14 q^{5} - 6 q^{6} + 8 q^{7} + 5 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 5 q^{2} + 6 q^{3} - 5 q^{4} - 14 q^{5} - 6 q^{6} + 8 q^{7} + 5 q^{8} - q^{9} - 6 q^{10} - 5 q^{11} - 9 q^{12} + 6 q^{13} + 2 q^{14} + 11 q^{15} - 5 q^{16} - 21 q^{17} - 14 q^{18} + 11 q^{19} + 6 q^{20} + 33 q^{21} + 5 q^{22} - 13 q^{23} - 6 q^{24} + 4 q^{25} - 16 q^{26} - 18 q^{27} - 2 q^{28} - 14 q^{29} - q^{30} + 3 q^{31} - 20 q^{32} + 6 q^{33} - 14 q^{34} + 20 q^{35} - q^{36} - 7 q^{37} - 11 q^{38} + 13 q^{39} - 6 q^{40} - 33 q^{42} - 12 q^{43} - 5 q^{44} + 38 q^{45} + 8 q^{46} - 20 q^{47} - 9 q^{48} - 4 q^{49} - 4 q^{50} - 2 q^{51} + 6 q^{52} + 15 q^{53} - 27 q^{54} + 6 q^{55} + 2 q^{56} + 70 q^{57} + 14 q^{58} - 14 q^{59} - 9 q^{60} - 20 q^{61} - 8 q^{62} + 24 q^{63} - 5 q^{64} + 27 q^{65} + 9 q^{66} - 40 q^{67} + 14 q^{68} - 26 q^{69} + 30 q^{70} - 13 q^{71} + 6 q^{72} - 13 q^{73} - 8 q^{74} - 61 q^{75} - 44 q^{76} - 2 q^{77} + 2 q^{78} + 21 q^{79} - 4 q^{80} + 4 q^{81} + 30 q^{82} - 7 q^{83} - 22 q^{84} + 58 q^{85} - 3 q^{86} - 24 q^{87} + 5 q^{88} + 26 q^{89} + 12 q^{90} + 43 q^{91} - 8 q^{92} - 30 q^{93} + 20 q^{94} + 38 q^{95} - 6 q^{96} - 13 q^{97} - q^{98} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} - 7 x^{18} + 37 x^{17} + 4 x^{16} - 208 x^{15} + 152 x^{14} + 1068 x^{13} - 2777 x^{12} + 498 x^{11} + 11367 x^{10} - 32054 x^{9} + 50370 x^{8} - 50274 x^{7} + \cdots + 3455 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 41\!\cdots\!58 \nu^{19} + \cdots - 85\!\cdots\!85 ) / 56\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 71\!\cdots\!32 \nu^{19} + \cdots - 92\!\cdots\!40 ) / 81\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 75\!\cdots\!42 \nu^{19} + \cdots + 35\!\cdots\!85 ) / 81\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!92 \nu^{19} + \cdots + 25\!\cdots\!30 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 62\!\cdots\!18 \nu^{19} + \cdots - 15\!\cdots\!15 ) / 56\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\!\cdots\!82 \nu^{19} + \cdots - 12\!\cdots\!65 ) / 48\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 77\!\cdots\!41 \nu^{19} + \cdots - 23\!\cdots\!70 ) / 48\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23\!\cdots\!75 \nu^{19} + \cdots - 25\!\cdots\!70 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!93 \nu^{19} + \cdots + 41\!\cdots\!15 ) / 48\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\!\cdots\!84 \nu^{19} + \cdots - 15\!\cdots\!30 ) / 56\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18\!\cdots\!61 \nu^{19} + \cdots - 12\!\cdots\!95 ) / 56\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!96 \nu^{19} + \cdots + 13\!\cdots\!20 ) / 56\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 23\!\cdots\!29 \nu^{19} + \cdots + 71\!\cdots\!30 ) / 56\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 34\!\cdots\!64 \nu^{19} + \cdots - 40\!\cdots\!45 ) / 56\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 44\!\cdots\!44 \nu^{19} + \cdots - 29\!\cdots\!80 ) / 56\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 59\!\cdots\!76 \nu^{19} + \cdots - 22\!\cdots\!20 ) / 56\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 13\!\cdots\!51 \nu^{19} + \cdots + 16\!\cdots\!70 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 66\!\cdots\!06 \nu^{19} + \cdots + 14\!\cdots\!55 ) / 56\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 80\!\cdots\!73 \nu^{19} + \cdots + 10\!\cdots\!35 ) / 56\!\cdots\!75 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3 \beta_{19} - \beta_{18} + \beta_{17} + 5 \beta_{16} - 2 \beta_{15} - 5 \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + 3 \beta_{10} - 5 \beta_{9} - 3 \beta_{8} + 4 \beta_{7} - 9 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 4 \beta _1 - 9 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{19} + \beta_{18} + 2 \beta_{17} + \beta_{14} - \beta_{13} + \beta_{11} - \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13 \beta_{19} + 14 \beta_{18} + 11 \beta_{17} + 10 \beta_{16} + 13 \beta_{15} - 5 \beta_{14} + 9 \beta_{13} + 21 \beta_{12} - 9 \beta_{11} - 7 \beta_{10} - 18 \beta_{8} + 24 \beta_{7} - 9 \beta_{6} + 22 \beta_{5} + 4 \beta_{4} + 12 \beta_{3} + 8 \beta_{2} + 14 \beta _1 - 14 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{19} - \beta_{18} + 4 \beta_{17} - 2 \beta_{16} + \beta_{15} - \beta_{14} - 3 \beta_{13} - \beta_{12} + 6 \beta_{11} - 6 \beta_{10} - 13 \beta_{9} - 5 \beta_{8} - 16 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 6 \beta_{4} - 8 \beta_{3} + \beta_{2} - 5 \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 68 \beta_{19} + 99 \beta_{18} + 56 \beta_{17} + 95 \beta_{16} - 12 \beta_{15} + 10 \beta_{14} + 119 \beta_{13} + 81 \beta_{12} - 69 \beta_{11} + 13 \beta_{10} - 15 \beta_{9} - 53 \beta_{8} + 114 \beta_{7} + 76 \beta_{6} + 27 \beta_{5} + 69 \beta_{4} + \cdots - 14 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 40 \beta_{19} - 6 \beta_{18} + 11 \beta_{17} - 72 \beta_{16} + 65 \beta_{15} + 22 \beta_{14} - 45 \beta_{13} + 17 \beta_{12} + 14 \beta_{11} - 35 \beta_{10} + 24 \beta_{9} - 25 \beta_{8} - 53 \beta_{7} - 24 \beta_{6} + 35 \beta_{5} - 78 \beta_{4} + \cdots + 43 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 598 \beta_{19} + 134 \beta_{18} - 124 \beta_{17} + 1330 \beta_{16} - 1162 \beta_{15} - 835 \beta_{14} + 809 \beta_{13} + 16 \beta_{12} + 171 \beta_{11} + 228 \beta_{10} - 1675 \beta_{9} - 198 \beta_{8} + 789 \beta_{7} - 1064 \beta_{6} + \cdots - 2089 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 491 \beta_{19} + 60 \beta_{18} + 269 \beta_{17} - 848 \beta_{16} + 557 \beta_{15} + 726 \beta_{14} - 529 \beta_{13} + 147 \beta_{12} + 7 \beta_{11} + 127 \beta_{10} + 1192 \beta_{9} + 363 \beta_{8} - 377 \beta_{7} + 968 \beta_{6} + \cdots + 1434 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8083 \beta_{19} + 719 \beta_{18} - 4549 \beta_{17} + 10505 \beta_{16} - 5387 \beta_{15} - 10490 \beta_{14} + 7259 \beta_{13} + 581 \beta_{12} - 684 \beta_{11} - 2672 \beta_{10} - 16910 \beta_{9} - 5148 \beta_{8} + \cdots - 22879 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5000 \beta_{19} - 1805 \beta_{18} + 2159 \beta_{17} - 4504 \beta_{16} - 173 \beta_{15} + 4483 \beta_{14} - 4630 \beta_{13} - 859 \beta_{12} + 2334 \beta_{11} + 3574 \beta_{10} + 7844 \beta_{9} + 4789 \beta_{8} + \cdots + 9397 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 65388 \beta_{19} + 45984 \beta_{18} - 21334 \beta_{17} + 41380 \beta_{16} + 24373 \beta_{15} - 27880 \beta_{14} + 62154 \beta_{13} + 21831 \beta_{12} - 51049 \beta_{11} - 44237 \beta_{10} - 61430 \beta_{9} + \cdots - 60679 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 29231 \beta_{19} - 36791 \beta_{18} + 1386 \beta_{17} - 12099 \beta_{16} - 22699 \beta_{15} - 4726 \beta_{14} - 33725 \beta_{13} - 12648 \beta_{12} + 33040 \beta_{11} + 25033 \beta_{10} + 19281 \beta_{9} + \cdots - 3155 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 277483 \beta_{19} + 527684 \beta_{18} + 55596 \beta_{17} + 137365 \beta_{16} + 221583 \beta_{15} + 234060 \beta_{14} + 436204 \beta_{13} + 85071 \beta_{12} - 432924 \beta_{11} - 260357 \beta_{10} + \cdots + 432676 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 92799 \beta_{19} - 266367 \beta_{18} - 55121 \beta_{17} - 89206 \beta_{16} - 31628 \beta_{15} - 135082 \beta_{14} - 210653 \beta_{13} + 6161 \beta_{12} + 162683 \beta_{11} + 98520 \beta_{10} + \cdots - 224216 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 747803 \beta_{19} + 2674004 \beta_{18} + 445366 \beta_{17} + 1791510 \beta_{16} - 852997 \beta_{15} + 986315 \beta_{14} + 2654919 \beta_{13} - 1020494 \beta_{12} - 1164224 \beta_{11} + \cdots + 1605656 ) / 5 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 399876 \beta_{19} - 965138 \beta_{18} + 63738 \beta_{17} - 1214461 \beta_{16} + 834626 \beta_{15} + 212890 \beta_{14} - 1244985 \beta_{13} + 762573 \beta_{12} + 159113 \beta_{11} + \cdots + 604830 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 6560713 \beta_{19} + 9513579 \beta_{18} - 5026984 \beta_{17} + 14544265 \beta_{16} - 7132252 \beta_{15} - 9022220 \beta_{14} + 15388324 \beta_{13} - 10216014 \beta_{12} + \cdots - 19852444 ) / 5 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 4385408 \beta_{19} - 5614160 \beta_{18} + 3520248 \beta_{17} - 4710700 \beta_{16} - 1679215 \beta_{15} + 4652639 \beta_{14} - 7124648 \beta_{13} + 3515599 \beta_{12} + \cdots + 9203461 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 49906503 \beta_{19} + 112035479 \beta_{18} - 31499699 \beta_{17} - 17398080 \beta_{16} + 122501228 \beta_{15} + 6708370 \beta_{14} + 76630834 \beta_{13} - 13767139 \beta_{12} + \cdots + 12925721 ) / 5 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/550\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
0.984234 + 1.23415i
1.96518 1.10400i
0.976097 1.18440i
−0.573401 + 0.235766i
−2.04310 + 1.76953i
0.0925249 + 1.32171i
0.815489 + 0.869137i
1.95488 0.468188i
0.0602581 0.730161i
−2.73217 0.404712i
0.0925249 1.32171i
0.815489 0.869137i
1.95488 + 0.468188i
0.0602581 + 0.730161i
−2.73217 + 0.404712i
0.984234 1.23415i
1.96518 + 1.10400i
0.976097 + 1.18440i
−0.573401 0.235766i
−2.04310 1.76953i
0.809017 0.587785i −0.440447 + 1.35556i 0.309017 0.951057i −2.23539 + 0.0548903i 0.440447 + 1.35556i −2.36313 −0.309017 0.951057i 0.783508 + 0.569252i −1.77621 + 1.35834i
111.2 0.809017 0.587785i −0.0815421 + 0.250961i 0.309017 0.951057i 1.20277 1.88503i 0.0815421 + 0.250961i 0.510766 −0.309017 0.951057i 2.37072 + 1.72243i −0.134938 2.23199i
111.3 0.809017 0.587785i 0.442208 1.36098i 0.309017 0.951057i −1.07360 + 1.96147i −0.442208 1.36098i 4.92733 −0.309017 0.951057i 0.770342 + 0.559686i 0.284363 + 2.21791i
111.4 0.809017 0.587785i 0.701053 2.15762i 0.309017 0.951057i 1.95017 1.09399i −0.701053 2.15762i 0.746726 −0.309017 0.951057i −1.73680 1.26186i 0.934690 2.03134i
111.5 0.809017 0.587785i 0.878728 2.70445i 0.309017 0.951057i −2.22591 0.212906i −0.878728 2.70445i −1.82169 −0.309017 0.951057i −4.11482 2.98959i −1.92594 + 1.13611i
221.1 −0.309017 + 0.951057i −1.58016 1.14806i −0.809017 0.587785i −1.08136 + 1.95721i 1.58016 1.14806i −4.33989 0.809017 0.587785i 0.251833 + 0.775064i −1.52725 1.63325i
221.2 −0.309017 + 0.951057i −1.24521 0.904697i −0.809017 0.587785i −1.86845 1.22837i 1.24521 0.904697i 1.36061 0.809017 0.587785i −0.194983 0.600096i 1.74563 1.39742i
221.3 −0.309017 + 0.951057i 0.243026 + 0.176569i −0.809017 0.587785i −1.85272 + 1.25196i −0.243026 + 0.176569i 0.0996140 0.809017 0.587785i −0.899166 2.76735i −0.618164 2.14892i
221.4 −0.309017 + 0.951057i 1.59347 + 1.15773i −0.809017 0.587785i 0.968560 + 2.01541i −1.59347 + 1.15773i 3.27607 0.809017 0.587785i 0.271775 + 0.836439i −2.21607 + 0.298358i
221.5 −0.309017 + 0.951057i 2.48887 + 1.80827i −0.809017 0.587785i −0.784052 2.09410i −2.48887 + 1.80827i 1.60360 0.809017 0.587785i 1.99759 + 6.14795i 2.23389 0.0985649i
331.1 −0.309017 0.951057i −1.58016 + 1.14806i −0.809017 + 0.587785i −1.08136 1.95721i 1.58016 + 1.14806i −4.33989 0.809017 + 0.587785i 0.251833 0.775064i −1.52725 + 1.63325i
331.2 −0.309017 0.951057i −1.24521 + 0.904697i −0.809017 + 0.587785i −1.86845 + 1.22837i 1.24521 + 0.904697i 1.36061 0.809017 + 0.587785i −0.194983 + 0.600096i 1.74563 + 1.39742i
331.3 −0.309017 0.951057i 0.243026 0.176569i −0.809017 + 0.587785i −1.85272 1.25196i −0.243026 0.176569i 0.0996140 0.809017 + 0.587785i −0.899166 + 2.76735i −0.618164 + 2.14892i
331.4 −0.309017 0.951057i 1.59347 1.15773i −0.809017 + 0.587785i 0.968560 2.01541i −1.59347 1.15773i 3.27607 0.809017 + 0.587785i 0.271775 0.836439i −2.21607 0.298358i
331.5 −0.309017 0.951057i 2.48887 1.80827i −0.809017 + 0.587785i −0.784052 + 2.09410i −2.48887 1.80827i 1.60360 0.809017 + 0.587785i 1.99759 6.14795i 2.23389 + 0.0985649i
441.1 0.809017 + 0.587785i −0.440447 1.35556i 0.309017 + 0.951057i −2.23539 0.0548903i 0.440447 1.35556i −2.36313 −0.309017 + 0.951057i 0.783508 0.569252i −1.77621 1.35834i
441.2 0.809017 + 0.587785i −0.0815421 0.250961i 0.309017 + 0.951057i 1.20277 + 1.88503i 0.0815421 0.250961i 0.510766 −0.309017 + 0.951057i 2.37072 1.72243i −0.134938 + 2.23199i
441.3 0.809017 + 0.587785i 0.442208 + 1.36098i 0.309017 + 0.951057i −1.07360 1.96147i −0.442208 + 1.36098i 4.92733 −0.309017 + 0.951057i 0.770342 0.559686i 0.284363 2.21791i
441.4 0.809017 + 0.587785i 0.701053 + 2.15762i 0.309017 + 0.951057i 1.95017 + 1.09399i −0.701053 + 2.15762i 0.746726 −0.309017 + 0.951057i −1.73680 + 1.26186i 0.934690 + 2.03134i
441.5 0.809017 + 0.587785i 0.878728 + 2.70445i 0.309017 + 0.951057i −2.22591 + 0.212906i −0.878728 + 2.70445i −1.82169 −0.309017 + 0.951057i −4.11482 + 2.98959i −1.92594 1.13611i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.2.k.c 20
25.d even 5 1 inner 550.2.k.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.k.c 20 1.a even 1 1 trivial
550.2.k.c 20 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 6 T_{3}^{19} + 26 T_{3}^{18} - 60 T_{3}^{17} + 121 T_{3}^{16} - 122 T_{3}^{15} + 480 T_{3}^{14} - 825 T_{3}^{13} + 2980 T_{3}^{12} - 2805 T_{3}^{11} + 6752 T_{3}^{10} - 3637 T_{3}^{9} + 19082 T_{3}^{8} - 3240 T_{3}^{7} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} - 6 T^{19} + 26 T^{18} - 60 T^{17} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( T^{20} + 14 T^{19} + 96 T^{18} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T^{10} - 4 T^{9} - 26 T^{8} + 105 T^{7} + \cdots - 25)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{20} - 6 T^{19} + \cdots + 20263807201 \) Copy content Toggle raw display
$17$ \( T^{20} + 21 T^{19} + \cdots + 535043161 \) Copy content Toggle raw display
$19$ \( T^{20} - 11 T^{19} + \cdots + 166851825625 \) Copy content Toggle raw display
$23$ \( T^{20} + 13 T^{19} + \cdots + 4819275241 \) Copy content Toggle raw display
$29$ \( T^{20} + 14 T^{19} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{20} - 3 T^{19} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{20} + 7 T^{19} + \cdots + 106936194121 \) Copy content Toggle raw display
$41$ \( T^{20} + 142 T^{18} + \cdots + 16\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( (T^{10} + 6 T^{9} - 141 T^{8} + \cdots + 3815555)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + 20 T^{19} + \cdots + 27163286609281 \) Copy content Toggle raw display
$53$ \( T^{20} - 15 T^{19} + \cdots + 85241733696025 \) Copy content Toggle raw display
$59$ \( T^{20} + 14 T^{19} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{20} + 20 T^{19} + \cdots + 8672265625 \) Copy content Toggle raw display
$67$ \( T^{20} + 40 T^{19} + \cdots + 605752446601 \) Copy content Toggle raw display
$71$ \( T^{20} + 13 T^{19} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 540580588651321 \) Copy content Toggle raw display
$79$ \( T^{20} - 21 T^{19} + \cdots + 51\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{20} + 7 T^{19} + \cdots + 28\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( T^{20} - 26 T^{19} + \cdots + 61\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 310462320843481 \) Copy content Toggle raw display
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