Properties

Label 735.2.q.h
Level $735$
Weight $2$
Character orbit 735.q
Analytic conductor $5.869$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,2,Mod(79,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.q (of order \(6\), degree \(2\), not 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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 2 x^{14} - 16 x^{13} + 2 x^{12} + 62 x^{11} + 106 x^{9} + 219 x^{8} - 106 x^{7} - 62 x^{5} + 2 x^{4} + 16 x^{3} + 2 x^{2} + 2 x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{3}) q^{2} - \beta_{4} q^{3} + ( - \beta_{14} + \beta_{10} - \beta_{6} + 1) q^{4} + ( - \beta_{15} + \beta_{4} - \beta_{2}) q^{5} + \beta_1 q^{6} + (\beta_{11} - \beta_{9} + \beta_{8} + 1) q^{8} - \beta_{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{3}) q^{2} - \beta_{4} q^{3} + ( - \beta_{14} + \beta_{10} - \beta_{6} + 1) q^{4} + ( - \beta_{15} + \beta_{4} - \beta_{2}) q^{5} + \beta_1 q^{6} + (\beta_{11} - \beta_{9} + \beta_{8} + 1) q^{8} - \beta_{10} q^{9} + ( - \beta_{15} - 2 \beta_{13} - \beta_{12}) q^{10} + ( - 2 \beta_{14} - 2 \beta_{10} - 2 \beta_{6} - 2) q^{11} + (\beta_{15} + \beta_{13} + \beta_{7} - \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{12} + (2 \beta_{7} - \beta_{5} + \beta_{2}) q^{13} + \beta_{11} q^{15} + ( - 2 \beta_{14} + 2 \beta_{11} - \beta_{10} + 2 \beta_{9} - 2 \beta_{6} - 2) q^{16} + (\beta_{15} + \beta_{13} + 3 \beta_{4}) q^{17} - \beta_{3} q^{18} + (\beta_{15} - \beta_{13} - 2 \beta_{12} + \beta_{5} - \beta_{4} + \beta_{2} + 2 \beta_1) q^{19} + ( - 2 \beta_{7} + \beta_{5} - 2 \beta_{2} - 2 \beta_1) q^{20} + (2 \beta_{11} - 2 \beta_{9} + 2 \beta_{8} + 2) q^{22} + ( - \beta_{14} - \beta_{11} + \beta_{9} + \beta_{6} - 1) q^{23} + ( - \beta_{15} + \beta_{13} + \beta_{12} + \beta_{4}) q^{24} + ( - \beta_{14} - 3 \beta_{10} - \beta_{6} - 2 \beta_{3} - 3) q^{25} + ( - \beta_{15} + \beta_{13} + 4 \beta_{12} - \beta_{5} + \beta_{4} - \beta_{2} - 4 \beta_1) q^{26} - \beta_{7} q^{27} - 2 q^{29} + ( - 2 \beta_{14} + \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} - \beta_{6} + \beta_{3} - 2) q^{30} + (\beta_{15} - \beta_{13} + 2 \beta_{12} - \beta_{4}) q^{31} + \beta_{3} q^{32} + (2 \beta_{15} + 2 \beta_{13} - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{2}) q^{33} + (\beta_{5} + \beta_{2} - 2 \beta_1) q^{34} + ( - \beta_{11} - \beta_{9} + 2) q^{36} + ( - 2 \beta_{14} - 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{6} - 2) q^{37} + (5 \beta_{15} + 5 \beta_{13} - 9 \beta_{4}) q^{38} + ( - \beta_{14} + 2 \beta_{10} - \beta_{6} + 2) q^{39} + ( - \beta_{13} - 3 \beta_{12} - 7 \beta_{7} + \beta_{5} + 7 \beta_{4} + 3 \beta_1) q^{40} + ( - \beta_{5} - \beta_{2} + 4 \beta_1) q^{41} - 4 \beta_{8} q^{43} + ( - 4 \beta_{14} + 4 \beta_{11} + 6 \beta_{10} + 4 \beta_{9} - 4 \beta_{6} - 4) q^{44} + ( - \beta_{15} + \beta_{4}) q^{45} + (3 \beta_{14} + 2 \beta_{10} + 3 \beta_{6} + 2) q^{46} + ( - 4 \beta_{15} - 4 \beta_{13} + 8 \beta_{7} + 4 \beta_{5} - 4 \beta_{4} - 4 \beta_{2}) q^{47} + ( - \beta_{7} - 2 \beta_{5} + 2 \beta_{2}) q^{48} + ( - \beta_{11} - 3 \beta_{9} - \beta_{8} + 9) q^{50} + (\beta_{14} - \beta_{11} + 4 \beta_{10} - \beta_{9} + \beta_{6} + 1) q^{51} + ( - 5 \beta_{15} - 5 \beta_{13} + 11 \beta_{4}) q^{52} + ( - \beta_{14} + \beta_{6} - 2 \beta_{3}) q^{53} + ( - \beta_{12} + \beta_1) q^{54} + ( - 4 \beta_{7} + 2 \beta_{5} - 4 \beta_1) q^{55} + ( - \beta_{11} + \beta_{9} - 2 \beta_{8} - 1) q^{57} + ( - 2 \beta_{8} + 2 \beta_{3}) q^{58} - 4 \beta_{12} q^{59} + (\beta_{14} - 2 \beta_{10} + 2 \beta_{6} + 2 \beta_{3} - 2) q^{60} + (2 \beta_{15} - 2 \beta_{13} + 4 \beta_{12} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - 4 \beta_1) q^{61} + (8 \beta_{7} + \beta_{5} - \beta_{2}) q^{62} + ( - 3 \beta_{11} - 3 \beta_{9} - 2) q^{64} + (\beta_{14} - 3 \beta_{11} - 2 \beta_{10} - \beta_{9} - 2 \beta_{8} + 3 \beta_{6} + 2 \beta_{3} + \cdots + 1) q^{65}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{9}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} + 8 q^{9} - 8 q^{15} - 8 q^{16} - 16 q^{25} - 32 q^{29} - 4 q^{30} + 32 q^{36} + 24 q^{39} - 80 q^{44} - 8 q^{46} + 128 q^{50} - 24 q^{51} - 28 q^{60} - 32 q^{64} + 32 q^{65} - 64 q^{71} - 16 q^{74} + 32 q^{79} - 8 q^{81} - 16 q^{85} + 128 q^{86} + 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} - 16 x^{13} + 2 x^{12} + 62 x^{11} + 106 x^{9} + 219 x^{8} - 106 x^{7} - 62 x^{5} + 2 x^{4} + 16 x^{3} + 2 x^{2} + 2 x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 410563 \nu^{15} + 654388 \nu^{14} - 1334297 \nu^{13} + 7845267 \nu^{12} + 464352 \nu^{11} - 12730071 \nu^{10} - 10533949 \nu^{9} - 97699056 \nu^{8} + \cdots + 47131609 ) / 32621050 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 999808 \nu^{15} - 4061258 \nu^{14} + 5315452 \nu^{13} - 19142897 \nu^{12} + 35866718 \nu^{11} + 67341161 \nu^{10} - 117484566 \nu^{9} + 36181546 \nu^{8} + \cdots - 51150769 ) / 32621050 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 510795 \nu^{15} - 880204 \nu^{14} + 568088 \nu^{13} - 7527742 \nu^{12} - 1689823 \nu^{11} + 35002396 \nu^{10} + 7721953 \nu^{9} + 44843460 \nu^{8} + \cdots - 106392 ) / 6524210 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3400622 \nu^{15} - 6010755 \nu^{14} + 4222770 \nu^{13} - 50697021 \nu^{12} - 8252922 \nu^{11} + 229191358 \nu^{10} + 44084748 \nu^{9} + 301507449 \nu^{8} + \cdots - 698698 ) / 32621050 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 696248 \nu^{15} + 1589638 \nu^{14} - 2684372 \nu^{13} + 13248397 \nu^{12} - 6148818 \nu^{11} - 28733391 \nu^{10} + 11972446 \nu^{9} - 130257126 \nu^{8} + \cdots - 12997151 ) / 6524210 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 483100 \nu^{15} + 595461 \nu^{14} + 324788 \nu^{13} + 5896418 \nu^{12} + 6062697 \nu^{11} - 39478244 \nu^{10} - 21970812 \nu^{9} - 17361695 \nu^{8} + \cdots + 2176428 ) / 3262105 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 63746 \nu^{15} + 209646 \nu^{14} - 303624 \nu^{13} + 1210239 \nu^{12} - 1469466 \nu^{11} - 3591207 \nu^{10} + 5024592 \nu^{9} - 7488702 \nu^{8} + \cdots - 128297 ) / 423650 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6071 \nu^{15} + 20598 \nu^{14} - 29517 \nu^{13} + 115172 \nu^{12} - 148486 \nu^{11} - 351604 \nu^{10} + 519643 \nu^{9} - 675314 \nu^{8} - 429938 \nu^{7} + \cdots - 11088 ) / 28490 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 741 \nu^{15} + 2352 \nu^{14} - 3438 \nu^{13} + 14094 \nu^{12} - 16072 \nu^{11} - 40414 \nu^{10} + 52249 \nu^{9} - 90104 \nu^{8} - 66032 \nu^{7} + 239866 \nu^{6} - 133462 \nu^{5} + \cdots + 2002 ) / 2849 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 183490 \nu^{15} - 392742 \nu^{14} + 434244 \nu^{13} - 3042546 \nu^{12} + 857430 \nu^{11} + 11026292 \nu^{10} - 1188762 \nu^{9} + 20388669 \nu^{8} + 35995518 \nu^{7} + \cdots + 143927 ) / 652421 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4354 \nu^{15} - 12772 \nu^{14} + 19313 \nu^{13} - 82768 \nu^{12} + 78864 \nu^{11} + 221991 \nu^{10} - 246582 \nu^{9} + 613056 \nu^{8} + 519932 \nu^{7} - 1081311 \nu^{6} + \cdots + 15477 ) / 14245 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10615083 \nu^{15} - 22925695 \nu^{14} + 25030430 \nu^{13} - 176305519 \nu^{12} + 53481317 \nu^{11} + 644047062 \nu^{10} - 68875503 \nu^{9} + \cdots + 55825378 ) / 32621050 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3762972 \nu^{15} + 8332125 \nu^{14} - 10212730 \nu^{13} + 64083161 \nu^{12} - 22828178 \nu^{11} - 214268238 \nu^{10} + 45962982 \nu^{9} + \cdots - 15500222 ) / 6524210 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 443169 \nu^{15} - 844836 \nu^{14} + 763892 \nu^{13} - 6905740 \nu^{12} + 97324 \nu^{11} + 28235048 \nu^{10} + 2127491 \nu^{9} + 44579165 \nu^{8} + 102894697 \nu^{7} + \cdots + 325004 ) / 652421 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 678183 \nu^{15} - 1372106 \nu^{14} + 1411548 \nu^{13} - 10929186 \nu^{12} + 1654930 \nu^{11} + 41638864 \nu^{10} - 938729 \nu^{9} + 73345418 \nu^{8} + \cdots + 1488332 ) / 652421 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} - \beta_{10} + \beta_{9} - \beta_{7} + \beta_{2} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{6} + 5\beta_{4} - 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{11} + \beta_{9} - 2\beta_{8} + 9\beta_{7} + 6\beta_{5} - \beta_{2} - 2\beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{15} - 5 \beta_{14} - 4 \beta_{13} + 6 \beta_{12} + 5 \beta_{11} - 19 \beta_{10} + 5 \beta_{9} - 5 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} + 4 \beta_{2} - 6 \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 39 \beta_{15} - 39 \beta_{14} + 6 \beta_{13} + 18 \beta_{12} - 67 \beta_{10} - 6 \beta_{6} + 28 \beta_{4} - 18 \beta_{3} - 67 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 59\beta_{11} - 59\beta_{9} - 78\beta_{8} + 262\beta_{7} + 77\beta_{5} - 77\beta_{2} + 59 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 39 \beta_{15} - 39 \beta_{14} - 268 \beta_{13} + 136 \beta_{12} + 268 \beta_{11} - 483 \beta_{10} + 39 \beta_{9} - 136 \beta_{8} + 483 \beta_{7} - 268 \beta_{6} + 268 \beta_{5} - 444 \beta_{4} + 136 \beta_{3} - 39 \beta_{2} + \cdots - 39 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 212\beta_{15} - 280\beta_{14} - 212\beta_{13} + 268\beta_{12} - 921\beta_{10} - 280\beta_{6} - 212\beta_{4} - 921 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 268 \beta_{11} - 1881 \beta_{9} - 984 \beta_{8} + 3445 \beta_{7} + 268 \beta_{5} - 1881 \beta_{2} + 984 \beta _1 - 1564 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 4005 \beta_{15} + 3021 \beta_{14} - 4005 \beta_{13} + 3021 \beta_{11} - 3021 \beta_{9} - 3762 \beta_{8} + 13030 \beta_{7} - 3021 \beta_{6} + 4005 \beta_{5} - 9025 \beta_{4} + 3762 \beta_{3} - 4005 \beta_{2} + \cdots + 3021 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1881 \beta_{15} - 1881 \beta_{14} - 13298 \beta_{13} + 7026 \beta_{12} - 24485 \beta_{10} - 13298 \beta_{6} - 22604 \beta_{4} + 7026 \beta_{3} - 24485 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -14245\beta_{11} - 14245\beta_{9} - 10732\beta_{5} - 10732\beta_{2} + 13298\beta _1 - 31934 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 94239 \beta_{15} + 94239 \beta_{14} - 13298 \beta_{13} - 49954 \beta_{12} - 13298 \beta_{11} + 173823 \beta_{10} - 94239 \beta_{9} - 49954 \beta_{8} + 173823 \beta_{7} + 13298 \beta_{6} + \cdots + 94239 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -202313\beta_{15} + 152359\beta_{14} - 202313\beta_{13} - 152359\beta_{6} - 452765\beta_{4} + 188478\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 668376 \beta_{11} - 94239 \beta_{9} + 354672 \beta_{8} - 1233527 \beta_{7} - 668376 \beta_{5} + 94239 \beta_{2} + 354672 \beta _1 - 1139288 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(\beta_{10}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.632011 0.169347i
0.395563 + 1.47626i
−0.362630 + 0.0971664i
−0.689408 2.57291i
2.57291 0.689408i
0.0971664 + 0.362630i
−1.47626 + 0.395563i
−0.169347 0.632011i
0.632011 + 0.169347i
0.395563 1.47626i
−0.362630 0.0971664i
−0.689408 + 2.57291i
2.57291 + 0.689408i
0.0971664 0.362630i
−1.47626 0.395563i
−0.169347 + 0.632011i
−2.16265 1.24861i −0.866025 + 0.500000i 2.11803 + 3.66854i 0.629578 + 2.14561i 2.49721 0 5.58394i 0.500000 0.866025i 1.31746 5.42629i
79.2 −2.16265 1.24861i 0.866025 0.500000i 2.11803 + 3.66854i −0.629578 2.14561i −2.49721 0 5.58394i 0.500000 0.866025i −1.31746 + 5.42629i
79.3 −1.15020 0.664066i −0.866025 + 0.500000i −0.118034 0.204441i 0.539247 2.17007i 1.32813 0 2.96979i 0.500000 0.866025i −2.06131 + 2.13791i
79.4 −1.15020 0.664066i 0.866025 0.500000i −0.118034 0.204441i −0.539247 + 2.17007i −1.32813 0 2.96979i 0.500000 0.866025i 2.06131 2.13791i
79.5 1.15020 + 0.664066i −0.866025 + 0.500000i −0.118034 0.204441i −1.60971 + 1.55204i −1.32813 0 2.96979i 0.500000 0.866025i −2.88214 + 0.716191i
79.6 1.15020 + 0.664066i 0.866025 0.500000i −0.118034 0.204441i 1.60971 1.55204i 1.32813 0 2.96979i 0.500000 0.866025i 2.88214 0.716191i
79.7 2.16265 + 1.24861i −0.866025 + 0.500000i 2.11803 + 3.66854i 2.17294 0.527574i −2.49721 0 5.58394i 0.500000 0.866025i 5.35804 + 1.57219i
79.8 2.16265 + 1.24861i 0.866025 0.500000i 2.11803 + 3.66854i −2.17294 + 0.527574i 2.49721 0 5.58394i 0.500000 0.866025i −5.35804 1.57219i
214.1 −2.16265 + 1.24861i −0.866025 0.500000i 2.11803 3.66854i 0.629578 2.14561i 2.49721 0 5.58394i 0.500000 + 0.866025i 1.31746 + 5.42629i
214.2 −2.16265 + 1.24861i 0.866025 + 0.500000i 2.11803 3.66854i −0.629578 + 2.14561i −2.49721 0 5.58394i 0.500000 + 0.866025i −1.31746 5.42629i
214.3 −1.15020 + 0.664066i −0.866025 0.500000i −0.118034 + 0.204441i 0.539247 + 2.17007i 1.32813 0 2.96979i 0.500000 + 0.866025i −2.06131 2.13791i
214.4 −1.15020 + 0.664066i 0.866025 + 0.500000i −0.118034 + 0.204441i −0.539247 2.17007i −1.32813 0 2.96979i 0.500000 + 0.866025i 2.06131 + 2.13791i
214.5 1.15020 0.664066i −0.866025 0.500000i −0.118034 + 0.204441i −1.60971 1.55204i −1.32813 0 2.96979i 0.500000 + 0.866025i −2.88214 0.716191i
214.6 1.15020 0.664066i 0.866025 + 0.500000i −0.118034 + 0.204441i 1.60971 + 1.55204i 1.32813 0 2.96979i 0.500000 + 0.866025i 2.88214 + 0.716191i
214.7 2.16265 1.24861i −0.866025 0.500000i 2.11803 3.66854i 2.17294 + 0.527574i −2.49721 0 5.58394i 0.500000 + 0.866025i 5.35804 1.57219i
214.8 2.16265 1.24861i 0.866025 + 0.500000i 2.11803 3.66854i −2.17294 0.527574i 2.49721 0 5.58394i 0.500000 + 0.866025i −5.35804 + 1.57219i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.q.h 16
5.b even 2 1 inner 735.2.q.h 16
7.b odd 2 1 inner 735.2.q.h 16
7.c even 3 1 735.2.d.c 8
7.c even 3 1 inner 735.2.q.h 16
7.d odd 6 1 735.2.d.c 8
7.d odd 6 1 inner 735.2.q.h 16
21.g even 6 1 2205.2.d.m 8
21.h odd 6 1 2205.2.d.m 8
35.c odd 2 1 inner 735.2.q.h 16
35.i odd 6 1 735.2.d.c 8
35.i odd 6 1 inner 735.2.q.h 16
35.j even 6 1 735.2.d.c 8
35.j even 6 1 inner 735.2.q.h 16
35.k even 12 1 3675.2.a.bt 4
35.k even 12 1 3675.2.a.bv 4
35.l odd 12 1 3675.2.a.bt 4
35.l odd 12 1 3675.2.a.bv 4
105.o odd 6 1 2205.2.d.m 8
105.p even 6 1 2205.2.d.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.d.c 8 7.c even 3 1
735.2.d.c 8 7.d odd 6 1
735.2.d.c 8 35.i odd 6 1
735.2.d.c 8 35.j even 6 1
735.2.q.h 16 1.a even 1 1 trivial
735.2.q.h 16 5.b even 2 1 inner
735.2.q.h 16 7.b odd 2 1 inner
735.2.q.h 16 7.c even 3 1 inner
735.2.q.h 16 7.d odd 6 1 inner
735.2.q.h 16 35.c odd 2 1 inner
735.2.q.h 16 35.i odd 6 1 inner
735.2.q.h 16 35.j even 6 1 inner
2205.2.d.m 8 21.g even 6 1
2205.2.d.m 8 21.h odd 6 1
2205.2.d.m 8 105.o odd 6 1
2205.2.d.m 8 105.p even 6 1
3675.2.a.bt 4 35.k even 12 1
3675.2.a.bt 4 35.l odd 12 1
3675.2.a.bv 4 35.k even 12 1
3675.2.a.bv 4 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{8} - 8T_{2}^{6} + 53T_{2}^{4} - 88T_{2}^{2} + 121 \) Copy content Toggle raw display
\( T_{19}^{8} + 68T_{19}^{6} + 4448T_{19}^{4} + 11968T_{19}^{2} + 30976 \) Copy content Toggle raw display
\( T_{73}^{8} - 60T_{73}^{6} + 3200T_{73}^{4} - 24000T_{73}^{2} + 160000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 8 T^{6} + 53 T^{4} - 88 T^{2} + \cdots + 121)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} + 8 T^{14} + 18 T^{12} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{4} + 20 T^{2} + 400)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 28 T^{2} + 16)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} - 28 T^{6} + 768 T^{4} - 448 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 68 T^{6} + 4448 T^{4} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 28 T^{6} + 608 T^{4} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{16} \) Copy content Toggle raw display
$31$ \( (T^{8} + 52 T^{6} + 2528 T^{4} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 112 T^{6} + 9728 T^{4} + \cdots + 7929856)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 140 T^{2} + 4400)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 128 T^{2} + 2816)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 192 T^{6} + 32768 T^{4} + \cdots + 16777216)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 52 T^{6} + 2528 T^{4} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 128 T^{6} + 13568 T^{4} + \cdots + 7929856)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 208 T^{6} + 40448 T^{4} + \cdots + 7929856)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 112 T^{6} + 9728 T^{4} + \cdots + 7929856)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8 T - 4)^{8} \) Copy content Toggle raw display
$73$ \( (T^{8} - 60 T^{6} + 3200 T^{4} + \cdots + 160000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + 128 T^{2} + 512 T + 4096)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 48 T^{2} + 256)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 252 T^{6} + 49248 T^{4} + \cdots + 203233536)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 252 T^{2} + 15376)^{4} \) Copy content Toggle raw display
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