Properties

Label 693.2.bu.d
Level $693$
Weight $2$
Character orbit 693.bu
Analytic conductor $5.534$
Analytic rank $0$
Dimension $16$
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] = [693,2,Mod(118,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.bu (of order \(10\), 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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 20x^{14} + 260x^{12} + 2030x^{10} + 11605x^{8} + 42100x^{6} + 106925x^{4} + 113575x^{2} + 87025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{12} - \beta_{11} - \beta_{9} - \beta_{8}) q^{2} + ( - \beta_{14} + \beta_{12} - 2 \beta_{11} - \beta_{9} + \beta_{4} - 1) q^{4} + (\beta_{15} - \beta_{13} + \beta_{10} + 2 \beta_{7} + 2 \beta_{6} + \beta_{3} + \beta_{2} + 3 \beta_1) q^{5} + (\beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{4} - \beta_{2}) q^{7} + ( - 2 \beta_{14} - \beta_{12} - \beta_{11} - \beta_{9} + \beta_{8} - \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{12} - \beta_{11} - \beta_{9} - \beta_{8}) q^{2} + ( - \beta_{14} + \beta_{12} - 2 \beta_{11} - \beta_{9} + \beta_{4} - 1) q^{4} + (\beta_{15} - \beta_{13} + \beta_{10} + 2 \beta_{7} + 2 \beta_{6} + \beta_{3} + \beta_{2} + 3 \beta_1) q^{5} + (\beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{4} - \beta_{2}) q^{7} + ( - 2 \beta_{14} - \beta_{12} - \beta_{11} - \beta_{9} + \beta_{8} - \beta_{5}) q^{8} + (\beta_{15} - 2 \beta_{13} + 2 \beta_{7} + 2 \beta_{6} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{10} + (\beta_{14} + 2 \beta_{12} + 2 \beta_{11} + \beta_{9}) q^{11} + (\beta_{3} + \beta_{2}) q^{13} + (\beta_{15} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{2}) q^{14} + ( - 2 \beta_{12} + 2 \beta_{11} + \beta_{9} + \beta_{8} - 2 \beta_{5} + 1) q^{16} + (\beta_{15} - \beta_{13} - \beta_{10} + \beta_{7} + \beta_{3} + \beta_{2}) q^{17} + (2 \beta_{15} - 2 \beta_{13} + \beta_{10} + \beta_{7} + \beta_{6} + \beta_{3} + \beta_1) q^{19} + ( - 3 \beta_{15} - \beta_{10} - \beta_{6} + 2 \beta_{3} + \beta_{2} - 4 \beta_1) q^{20} + (3 \beta_{14} + 3 \beta_{12} - \beta_{11} - \beta_{8} - \beta_{5} - \beta_{4} + 2) q^{22} + ( - 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{8} + 3 \beta_{4} + 2) q^{23} + ( - \beta_{14} + 5 \beta_{11} + 4 \beta_{9} + 4 \beta_{8} - \beta_{5} + \beta_{4} + 1) q^{25} + (\beta_{15} + \beta_{10} + \beta_1) q^{26} + ( - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{11} - 2 \beta_{8} + \beta_{7} - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{28} + (2 \beta_{14} + 2 \beta_{12} - \beta_{8} - \beta_{5} - 2 \beta_{4} - 1) q^{29} + ( - \beta_{10} - \beta_{6} + \beta_{3} - \beta_{2} - \beta_1) q^{31} + (\beta_{12} - \beta_{11} + 2 \beta_{9} + 3 \beta_{8} + 2 \beta_{5} + 3 \beta_{4}) q^{32} + ( - \beta_{15} - \beta_{10} - \beta_{7} - 2 \beta_{6} + \beta_{3} + \beta_{2} - 4 \beta_1) q^{34} + ( - \beta_{15} - 2 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{10} + \beta_{9} - 4 \beta_{7} + \cdots - 3) q^{35}+ \cdots + (5 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} + 7 \beta_{5} - 2 \beta_{4} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 10 q^{2} - 10 q^{4} - 10 q^{7} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 10 q^{2} - 10 q^{4} - 10 q^{7} - 10 q^{8} - 2 q^{11} - 8 q^{14} - 14 q^{16} + 42 q^{22} + 8 q^{23} - 30 q^{25} - 10 q^{28} - 10 q^{29} - 40 q^{35} + 4 q^{37} + 10 q^{44} - 10 q^{46} + 8 q^{49} + 60 q^{50} + 4 q^{56} - 2 q^{58} - 38 q^{64} - 4 q^{67} + 56 q^{71} - 90 q^{74} - 2 q^{77} + 50 q^{79} + 80 q^{85} - 6 q^{86} - 86 q^{88} - 30 q^{91} - 20 q^{92} + 60 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 20x^{14} + 260x^{12} + 2030x^{10} + 11605x^{8} + 42100x^{6} + 106925x^{4} + 113575x^{2} + 87025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 473414 \nu^{15} - 121962046 \nu^{13} - 1914724533 \nu^{11} - 19540502651 \nu^{9} - 110976178125 \nu^{7} - 427139590595 \nu^{5} + \cdots - 700136714820 \nu ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1553143 \nu^{15} - 83115610 \nu^{13} - 738702950 \nu^{11} - 5169909042 \nu^{9} - 7773695495 \nu^{7} + 12526757215 \nu^{5} + \cdots + 389963137595 \nu ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2026557 \nu^{14} + 205077656 \nu^{12} + 2653427483 \nu^{10} + 24710411693 \nu^{8} + 118749873620 \nu^{6} + 414612833380 \nu^{4} + \cdots + 310173577225 ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2307412 \nu^{14} - 42954924 \nu^{12} - 546435360 \nu^{10} - 4022687807 \nu^{8} - 22443730260 \nu^{6} - 74163373020 \nu^{4} + \cdots - 70674046800 ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2307412 \nu^{15} + 42954924 \nu^{13} + 546435360 \nu^{11} + 4022687807 \nu^{9} + 22443730260 \nu^{7} + 74163373020 \nu^{5} + \cdots + 70674046800 \nu ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8126617 \nu^{15} + 107242387 \nu^{13} + 1727473773 \nu^{11} + 21511248812 \nu^{9} + 103285230840 \nu^{7} + 401725517900 \nu^{5} + \cdots + 534354646485 \nu ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9624825 \nu^{14} + 326813127 \nu^{12} + 4112826637 \nu^{10} + 33537013348 \nu^{8} + 159160366925 \nu^{6} + 503717613335 \nu^{4} + \cdots + 564950170875 ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 85328 \nu^{14} - 3129227 \nu^{12} - 37034577 \nu^{10} - 295472690 \nu^{8} - 1274305820 \nu^{6} - 3749548520 \nu^{4} - 3135087840 \nu^{2} + \cdots - 2416417865 ) / 1080638985 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 85328 \nu^{15} - 3129227 \nu^{13} - 37034577 \nu^{11} - 295472690 \nu^{9} - 1274305820 \nu^{7} - 3749548520 \nu^{5} - 3135087840 \nu^{3} + \cdots - 2416417865 \nu ) / 1080638985 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19576647 \nu^{14} + 220271693 \nu^{12} + 1820750374 \nu^{10} + 4608222307 \nu^{8} + 1502139270 \nu^{6} - 102149491525 \nu^{4} + \cdots - 312330974765 ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23957031 \nu^{14} - 304074935 \nu^{12} - 2842215611 \nu^{10} - 12038362652 \nu^{8} - 33691071920 \nu^{6} + 17998043615 \nu^{4} + \cdots + 147206723390 ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 198348 \nu^{15} - 3886661 \nu^{13} - 41951474 \nu^{11} - 280524995 \nu^{9} - 1163671340 \nu^{7} - 3027593815 \nu^{5} - 2666658230 \nu^{3} + \cdots - 1244021785 \nu ) / 1080638985 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 26536505 \nu^{14} - 264615202 \nu^{12} - 2085345531 \nu^{10} - 3473599188 \nu^{8} + 6011890130 \nu^{6} + 135449500340 \nu^{4} + \cdots + 29496038150 ) / 141563707035 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 38187887 \nu^{15} - 796505985 \nu^{13} - 8851599651 \nu^{11} - 61721024229 \nu^{9} - 271898350415 \nu^{7} - 782880946375 \nu^{5} + \cdots - 987191416985 \nu ) / 141563707035 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} - 2\beta_{11} - 2\beta_{9} - 2\beta_{8} - 5\beta_{5} - \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - 2\beta_{13} + \beta_{10} + 2\beta_{7} + 7\beta_{6} + \beta_{3} + 2\beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{14} + 8\beta_{12} + \beta_{11} + 7\beta_{9} + 16\beta_{8} + 37\beta_{5} - 2\beta_{4} - 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{15} + 9\beta_{13} - 2\beta_{10} - \beta_{7} - 38\beta_{6} + 9\beta_{3} + \beta_{2} - 47\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 84\beta_{14} - 29\beta_{12} + 98\beta_{11} + 42\beta_{9} - 13\beta_{8} - 42\beta_{5} + 56\beta_{4} + 231 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -14\beta_{15} + 69\beta_{13} - 29\beta_{10} - 98\beta_{7} - 56\beta_{6} - 138\beta_{3} - 111\beta_{2} + 146\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -245\beta_{14} - 260\beta_{12} - 750\beta_{11} - 490\beta_{9} - 630\beta_{8} - 1210\beta_{5} - 125\beta_{4} - 245 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 505 \beta_{15} - 1010 \beta_{13} + 385 \beta_{10} + 750 \beta_{7} + 1960 \beta_{6} + 505 \beta_{3} + 630 \beta_{2} + 1135 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1420 \beta_{14} + 2645 \beta_{12} + 965 \beta_{11} + 1420 \beta_{9} + 5030 \beta_{8} + 9235 \beta_{5} - 2190 \beta_{4} - 7815 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2385 \beta_{15} + 3610 \beta_{13} - 2190 \beta_{10} - 965 \beta_{7} - 10200 \beta_{6} + 3610 \beta_{3} + 965 \beta_{2} - 13810 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 16410 \beta_{14} - 995 \beta_{12} + 26405 \beta_{11} + 8205 \beta_{9} - 7210 \beta_{8} - 8205 \beta_{5} + 18200 \beta_{4} + 59020 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9995 \beta_{15} + 25410 \beta_{13} - 995 \beta_{10} - 26405 \beta_{7} - 18200 \beta_{6} - 50820 \beta_{3} - 33615 \beta_{2} + 39825 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 47235 \beta_{14} - 129635 \beta_{12} - 224105 \beta_{11} - 94470 \beta_{9} - 172640 \beta_{8} - 332295 \beta_{5} + 4230 \beta_{4} - 47235 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 176870 \beta_{15} - 353740 \beta_{13} + 125405 \beta_{10} + 224105 \beta_{7} + 556400 \beta_{6} + 176870 \beta_{3} + 172640 \beta_{2} + 349510 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(442\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
−0.551501 + 0.955228i
0.551501 0.955228i
1.27939 + 2.21596i
−1.27939 2.21596i
1.17141 + 2.02895i
−1.17141 2.02895i
1.29877 2.24954i
−1.29877 + 2.24954i
−0.551501 0.955228i
0.551501 + 0.955228i
1.27939 2.21596i
−1.27939 + 2.21596i
1.17141 2.02895i
−1.17141 + 2.02895i
1.29877 + 2.24954i
−1.29877 2.24954i
0.395472 + 0.544320i 0 0.478148 1.47159i −2.08654 + 2.87188i 0 −1.94632 1.79216i 2.26988 0.737529i 0 −2.38839
118.2 0.395472 + 0.544320i 0 0.478148 1.47159i 2.08654 2.87188i 0 −2.62801 + 0.305873i 2.26988 0.737529i 0 2.38839
118.3 1.41355 + 1.94558i 0 −1.16913 + 3.59821i −1.97962 + 2.72471i 0 −1.43059 + 2.22563i −4.07890 + 1.32531i 0 −8.09942
118.4 1.41355 + 1.94558i 0 −1.16913 + 3.59821i 1.97962 2.72471i 0 0.150818 2.64145i −4.07890 + 1.32531i 0 8.09942
244.1 −0.478148 0.155360i 0 −1.41355 1.02700i −0.572621 + 0.186056i 0 2.61795 + 0.382556i 1.10735 + 1.52414i 0 0.302703
244.2 −0.478148 0.155360i 0 −1.41355 1.02700i 0.572621 0.186056i 0 −1.17282 2.37160i 1.10735 + 1.52414i 0 −0.302703
244.3 1.16913 + 0.379874i 0 −0.395472 0.287327i −2.26926 + 0.737329i 0 1.80595 + 1.93354i −1.79833 2.47520i 0 −2.93316
244.4 1.16913 + 0.379874i 0 −0.395472 0.287327i 2.26926 0.737329i 0 −2.39697 1.12006i −1.79833 2.47520i 0 2.93316
370.1 0.395472 0.544320i 0 0.478148 + 1.47159i −2.08654 2.87188i 0 −1.94632 + 1.79216i 2.26988 + 0.737529i 0 −2.38839
370.2 0.395472 0.544320i 0 0.478148 + 1.47159i 2.08654 + 2.87188i 0 −2.62801 0.305873i 2.26988 + 0.737529i 0 2.38839
370.3 1.41355 1.94558i 0 −1.16913 3.59821i −1.97962 2.72471i 0 −1.43059 2.22563i −4.07890 1.32531i 0 −8.09942
370.4 1.41355 1.94558i 0 −1.16913 3.59821i 1.97962 + 2.72471i 0 0.150818 + 2.64145i −4.07890 1.32531i 0 8.09942
622.1 −0.478148 + 0.155360i 0 −1.41355 + 1.02700i −0.572621 0.186056i 0 2.61795 0.382556i 1.10735 1.52414i 0 0.302703
622.2 −0.478148 + 0.155360i 0 −1.41355 + 1.02700i 0.572621 + 0.186056i 0 −1.17282 + 2.37160i 1.10735 1.52414i 0 −0.302703
622.3 1.16913 0.379874i 0 −0.395472 + 0.287327i −2.26926 0.737329i 0 1.80595 1.93354i −1.79833 + 2.47520i 0 −2.93316
622.4 1.16913 0.379874i 0 −0.395472 + 0.287327i 2.26926 + 0.737329i 0 −2.39697 + 1.12006i −1.79833 + 2.47520i 0 2.93316
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.d odd 10 1 inner
77.l even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.bu.d 16
3.b odd 2 1 77.2.l.b 16
7.b odd 2 1 inner 693.2.bu.d 16
11.d odd 10 1 inner 693.2.bu.d 16
21.c even 2 1 77.2.l.b 16
21.g even 6 1 539.2.s.b 16
21.g even 6 1 539.2.s.c 16
21.h odd 6 1 539.2.s.b 16
21.h odd 6 1 539.2.s.c 16
33.d even 2 1 847.2.l.i 16
33.f even 10 1 77.2.l.b 16
33.f even 10 1 847.2.b.f 16
33.f even 10 1 847.2.l.e 16
33.f even 10 1 847.2.l.j 16
33.h odd 10 1 847.2.b.f 16
33.h odd 10 1 847.2.l.e 16
33.h odd 10 1 847.2.l.i 16
33.h odd 10 1 847.2.l.j 16
77.l even 10 1 inner 693.2.bu.d 16
231.h odd 2 1 847.2.l.i 16
231.r odd 10 1 77.2.l.b 16
231.r odd 10 1 847.2.b.f 16
231.r odd 10 1 847.2.l.e 16
231.r odd 10 1 847.2.l.j 16
231.u even 10 1 847.2.b.f 16
231.u even 10 1 847.2.l.e 16
231.u even 10 1 847.2.l.i 16
231.u even 10 1 847.2.l.j 16
231.be even 30 1 539.2.s.b 16
231.be even 30 1 539.2.s.c 16
231.bf odd 30 1 539.2.s.b 16
231.bf odd 30 1 539.2.s.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.l.b 16 3.b odd 2 1
77.2.l.b 16 21.c even 2 1
77.2.l.b 16 33.f even 10 1
77.2.l.b 16 231.r odd 10 1
539.2.s.b 16 21.g even 6 1
539.2.s.b 16 21.h odd 6 1
539.2.s.b 16 231.be even 30 1
539.2.s.b 16 231.bf odd 30 1
539.2.s.c 16 21.g even 6 1
539.2.s.c 16 21.h odd 6 1
539.2.s.c 16 231.be even 30 1
539.2.s.c 16 231.bf odd 30 1
693.2.bu.d 16 1.a even 1 1 trivial
693.2.bu.d 16 7.b odd 2 1 inner
693.2.bu.d 16 11.d odd 10 1 inner
693.2.bu.d 16 77.l even 10 1 inner
847.2.b.f 16 33.f even 10 1
847.2.b.f 16 33.h odd 10 1
847.2.b.f 16 231.r odd 10 1
847.2.b.f 16 231.u even 10 1
847.2.l.e 16 33.f even 10 1
847.2.l.e 16 33.h odd 10 1
847.2.l.e 16 231.r odd 10 1
847.2.l.e 16 231.u even 10 1
847.2.l.i 16 33.d even 2 1
847.2.l.i 16 33.h odd 10 1
847.2.l.i 16 231.h odd 2 1
847.2.l.i 16 231.u even 10 1
847.2.l.j 16 33.f even 10 1
847.2.l.j 16 33.h odd 10 1
847.2.l.j 16 231.r odd 10 1
847.2.l.j 16 231.u even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 5T_{2}^{7} + 13T_{2}^{6} - 15T_{2}^{5} + 4T_{2}^{4} + 5T_{2}^{3} - 3T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 5 T^{7} + 13 T^{6} - 15 T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 5 T^{14} + 235 T^{12} + \cdots + 87025 \) Copy content Toggle raw display
$7$ \( T^{16} + 10 T^{15} + 46 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} + T^{7} - 20 T^{6} - T^{5} + 309 T^{4} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 15 T^{14} + 125 T^{12} + \cdots + 87025 \) Copy content Toggle raw display
$17$ \( T^{16} + 15 T^{14} + 425 T^{12} + \cdots + 54390625 \) Copy content Toggle raw display
$19$ \( T^{16} - 55 T^{14} + 5745 T^{12} + \cdots + 87025 \) Copy content Toggle raw display
$23$ \( (T^{2} - T - 11)^{8} \) Copy content Toggle raw display
$29$ \( (T^{8} + 5 T^{7} + 8 T^{6} - 25 T^{5} + \cdots + 961)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} - 85 T^{14} + 3120 T^{12} + \cdots + 87025 \) Copy content Toggle raw display
$37$ \( (T^{8} - 2 T^{7} + 43 T^{6} - 319 T^{5} + \cdots + 477481)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 110 T^{14} + \cdots + 61551129025 \) Copy content Toggle raw display
$43$ \( (T^{2} + 3)^{8} \) Copy content Toggle raw display
$47$ \( T^{16} - 15 T^{14} + \cdots + 570971025 \) Copy content Toggle raw display
$53$ \( (T^{8} + 35 T^{6} - 115 T^{5} + 390 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} - 55 T^{14} + \cdots + 1204934313025 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 283945921983025 \) Copy content Toggle raw display
$67$ \( (T^{4} + T^{3} - 109 T^{2} - 439 T + 241)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} - 28 T^{7} + 453 T^{6} + \cdots + 8288641)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + 315 T^{14} + \cdots + 1204934313025 \) Copy content Toggle raw display
$79$ \( (T^{8} - 25 T^{7} + 343 T^{6} + \cdots + 44521)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 235 T^{14} + \cdots + 1204934313025 \) Copy content Toggle raw display
$89$ \( (T^{8} + 165 T^{6} + 3150 T^{4} + \cdots + 23895)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 10 T^{14} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
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