Properties

Label 675.2.q.b
Level $675$
Weight $2$
Character orbit 675.q
Analytic conductor $5.390$
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] = [675,2,Mod(143,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([2, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.143");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.q (of order \(12\), degree \(4\), 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.9349208943630483456.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + ( - \beta_{9} - \beta_{7}) q^{4} + (\beta_{11} - 2 \beta_{6}) q^{7} + ( - \beta_{13} + \beta_{5} + \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{2} + ( - \beta_{9} - \beta_{7}) q^{4} + (\beta_{11} - 2 \beta_{6}) q^{7} + ( - \beta_{13} + \beta_{5} + \beta_{3}) q^{8} + (\beta_{15} - \beta_{4}) q^{11} + ( - \beta_{13} + 2 \beta_{3}) q^{13} + ( - 6 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} - \beta_{7}) q^{14} - \beta_1 q^{16} + ( - \beta_{14} - \beta_{2}) q^{17} + ( - 2 \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7}) q^{19} + ( - 2 \beta_{14} + 2 \beta_{11} + 2 \beta_{6} - \beta_{2}) q^{22} + ( - 2 \beta_{13} - \beta_{12}) q^{23} + ( - 2 \beta_{15} - \beta_{4}) q^{26} + (\beta_{13} - 6 \beta_{12} + 3 \beta_{5} + \beta_{3}) q^{28} + (3 \beta_{10} + 6 \beta_{9}) q^{29} + (\beta_{15} + 2 \beta_1 + 2) q^{31} + ( - \beta_{11} + \beta_{6} - 2 \beta_{2}) q^{32} + \beta_{8} q^{34} + ( - \beta_{14} - 4 \beta_{11} + 2 \beta_{6} + \beta_{2}) q^{37} + ( - 4 \beta_{12} + 4 \beta_{5} + \beta_{3}) q^{38} + (\beta_{15} + 2 \beta_{4} - 3 \beta_1 + 3) q^{41} + ( - 2 \beta_{13} + \beta_{3}) q^{43} + (6 \beta_{10} - 6 \beta_{9} + \beta_{8} - \beta_{7}) q^{44} + ( - \beta_{4} - 3) q^{46} + ( - \beta_{14} - \beta_{11}) q^{47} + (2 \beta_{9} + 3 \beta_{7}) q^{49} + ( - \beta_{14} + 2 \beta_{11} - 4 \beta_{6} - 2 \beta_{2}) q^{52} + ( - 2 \beta_{13} + 2 \beta_{5} + 2 \beta_{3}) q^{53} + ( - 2 \beta_{15} + 2 \beta_{4} - 3 \beta_1 - 6) q^{56} + (3 \beta_{12} - 6 \beta_{5}) q^{58} + (4 \beta_{8} + 2 \beta_{7}) q^{59} + (\beta_{15} + \beta_{4} + 5 \beta_1) q^{61} + ( - \beta_{14} + 4 \beta_{6} - \beta_{2}) q^{62} + (5 \beta_{10} + 5 \beta_{9} + 3 \beta_{8} + 3 \beta_{7}) q^{64} + (2 \beta_{14} + \beta_{11} + \beta_{6} + \beta_{2}) q^{67} + (\beta_{13} + 2 \beta_{12}) q^{68} + (2 \beta_{15} + \beta_{4} + 12 \beta_1 + 6) q^{71} + (6 \beta_{10} + 12 \beta_{9} + \beta_{8} + 2 \beta_{7}) q^{74} + ( - 3 \beta_{15} - 8 \beta_1 - 8) q^{76} + (6 \beta_{11} - 6 \beta_{6} + 3 \beta_{2}) q^{77} + ( - 2 \beta_{10} - 4 \beta_{8}) q^{79} + (\beta_{14} + 2 \beta_{11} - \beta_{6} - \beta_{2}) q^{82} + (3 \beta_{12} - 3 \beta_{5} - 2 \beta_{3}) q^{83} + ( - \beta_{15} - 2 \beta_{4}) q^{86} + (2 \beta_{13} + 4 \beta_{12} + 4 \beta_{5} - \beta_{3}) q^{88} + (3 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7}) q^{89} + 3 \beta_{4} q^{91} + (3 \beta_{14} + \beta_{11}) q^{92} + 3 \beta_{9} q^{94} + (2 \beta_{14} + 2 \beta_{11} - 4 \beta_{6} + 4 \beta_{2}) q^{97} + (3 \beta_{13} - 8 \beta_{5} - 3 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{16} + 16 q^{31} + 72 q^{41} - 48 q^{46} - 72 q^{56} - 40 q^{61} - 64 q^{76}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3456 \nu^{15} + 25920 \nu^{14} - 151876 \nu^{13} + 594074 \nu^{12} - 1879372 \nu^{11} + 4666596 \nu^{10} - 9554736 \nu^{9} + 15945783 \nu^{8} - 21928484 \nu^{7} + \cdots + 125460 ) / 17095 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1757 \nu^{15} + 29034 \nu^{14} - 217536 \nu^{13} + 1408900 \nu^{12} - 5469225 \nu^{11} + 17560899 \nu^{10} - 42890877 \nu^{9} + 87005139 \nu^{8} - 141294613 \nu^{7} + \cdots + 2553985 ) / 17095 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1757 \nu^{15} - 55389 \nu^{14} + 373425 \nu^{13} - 2022461 \nu^{12} + 7436448 \nu^{11} - 22510833 \nu^{10} + 53250351 \nu^{9} - 104737641 \nu^{8} + 166646564 \nu^{7} + \cdots - 2875650 ) / 17095 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 54 \nu^{14} + 378 \nu^{13} - 2193 \nu^{12} + 8244 \nu^{11} - 25569 \nu^{10} + 61284 \nu^{9} - 122021 \nu^{8} + 195620 \nu^{7} - 258293 \nu^{6} + 273134 \nu^{5} - 230109 \nu^{4} + \cdots - 3308 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13659 \nu^{15} - 66806 \nu^{14} + 357125 \nu^{13} - 945718 \nu^{12} + 2245378 \nu^{11} - 2537530 \nu^{10} + 267574 \nu^{9} + 10564902 \nu^{8} - 29075066 \nu^{7} + \cdots + 1260620 ) / 17095 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13659 \nu^{15} + 138079 \nu^{14} - 856036 \nu^{13} + 3832406 \nu^{12} - 13079663 \nu^{11} + 36027161 \nu^{10} - 80292162 \nu^{9} + 148038300 \nu^{8} + \cdots + 2779060 ) / 17095 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22230 \nu^{15} - 144633 \nu^{14} + 820486 \nu^{13} - 2915351 \nu^{12} + 8665028 \nu^{11} - 19406977 \nu^{10} + 35943838 \nu^{9} - 51803400 \nu^{8} + 59433982 \nu^{7} + \cdots + 599315 ) / 17095 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22230 \nu^{15} - 188817 \nu^{14} + 1129774 \nu^{13} - 4704014 \nu^{12} + 15376262 \nu^{11} - 40133218 \nu^{10} + 85426762 \nu^{9} - 149690685 \nu^{8} + \cdots - 1317955 ) / 17095 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 26630 \nu^{15} + 182630 \nu^{14} - 1056740 \nu^{13} + 3923413 \nu^{12} - 12097498 \nu^{11} + 28666706 \nu^{10} - 56590650 \nu^{9} + 89528958 \nu^{8} + \cdots - 218885 ) / 17095 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 26630 \nu^{15} + 216820 \nu^{14} - 1296070 \nu^{13} + 5311527 \nu^{12} - 17314892 \nu^{11} + 44845414 \nu^{10} - 95362110 \nu^{9} + 166733397 \nu^{8} + \cdots + 2071845 ) / 17095 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 63850 \nu^{15} - 464147 \nu^{14} + 2710563 \nu^{13} - 10426206 \nu^{12} + 32745418 \nu^{11} - 80184009 \nu^{10} + 162524611 \nu^{9} - 267170328 \nu^{8} + \cdots - 667700 ) / 17095 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 63850 \nu^{15} + 493603 \nu^{14} - 2916755 \nu^{13} + 11625486 \nu^{12} - 37260602 \nu^{11} + 94237414 \nu^{10} - 196316692 \nu^{9} + 334828656 \nu^{8} + \cdots + 3005770 ) / 17095 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 97569 \nu^{15} + 712700 \nu^{14} - 4168110 \nu^{13} + 16084516 \nu^{12} - 50637049 \nu^{11} + 124422927 \nu^{10} - 253084785 \nu^{9} + 418004961 \nu^{8} + \cdots + 1608045 ) / 17095 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 97569 \nu^{15} - 750835 \nu^{14} + 4435055 \nu^{13} - 17639109 \nu^{12} + 56494322 \nu^{11} - 142673023 \nu^{10} + 297005785 \nu^{9} - 505980039 \nu^{8} + \cdots - 4286700 ) / 17095 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 162552 \nu^{15} + 1254645 \nu^{14} - 7410690 \nu^{13} + 29505615 \nu^{12} - 94503294 \nu^{11} + 238745115 \nu^{10} - 496891020 \nu^{9} + 846295960 \nu^{8} + \cdots + 7071655 ) / 17095 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2 \beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} - \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{15} - 2 \beta_{13} + 2 \beta_{12} + 6 \beta_{11} + 6 \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{5} + \beta_{4} + 4 \beta_{3} - 12 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 10 \beta_{15} - 6 \beta_{14} + 3 \beta_{13} + 15 \beta_{12} - 3 \beta_{11} - 21 \beta_{10} - 12 \beta_{9} - 6 \beta_{8} - 3 \beta_{7} + 12 \beta_{6} - 9 \beta_{5} - 5 \beta_{4} - 3 \beta_{3} - 9 \beta_{2} - 9 \beta _1 - 24 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11 \beta_{15} - 2 \beta_{14} + 8 \beta_{13} - 2 \beta_{12} - 22 \beta_{11} - 9 \beta_{10} - 27 \beta_{9} - 9 \beta_{7} + 8 \beta_{6} - 14 \beta_{5} - 10 \beta_{4} - 10 \beta_{3} - 4 \beta_{2} - 9 \beta _1 + 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 43 \beta_{15} + 28 \beta_{14} + 7 \beta_{13} - 121 \beta_{12} - 19 \beta_{11} + 138 \beta_{10} + 33 \beta_{9} + 55 \beta_{8} + 5 \beta_{7} - 85 \beta_{6} + 50 \beta_{5} - \beta_{4} - 2 \beta_{3} + 38 \beta_{2} + 75 \beta _1 + 183 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 185 \beta_{15} + 24 \beta_{14} - 90 \beta_{13} - 96 \beta_{12} + 312 \beta_{11} + 258 \beta_{10} + 438 \beta_{9} + 70 \beta_{8} + 155 \beta_{7} - 222 \beta_{6} + 294 \beta_{5} + 160 \beta_{4} + 102 \beta_{3} + 78 \beta_{2} + 270 \beta _1 - 30 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 128 \beta_{15} - 180 \beta_{14} - 177 \beta_{13} + 822 \beta_{12} + 438 \beta_{11} - 801 \beta_{10} + 207 \beta_{9} - 364 \beta_{8} + 112 \beta_{7} + 459 \beta_{6} - 81 \beta_{5} + 251 \beta_{4} + 75 \beta_{3} - 156 \beta_{2} + \cdots - 1299 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 714 \beta_{15} - 108 \beta_{14} + 188 \beta_{13} + 784 \beta_{12} - 984 \beta_{11} - 1338 \beta_{10} - 1560 \beta_{9} - 486 \beta_{8} - 564 \beta_{7} + 1116 \beta_{6} - 1220 \beta_{5} - 462 \beta_{4} - 292 \beta_{3} + \cdots - 606 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 379 \beta_{15} + 1278 \beta_{14} + 1737 \beta_{13} - 4707 \beta_{12} - 5265 \beta_{11} + 3675 \beta_{10} - 4830 \beta_{9} + 1806 \beta_{8} - 1968 \beta_{7} - 1404 \beta_{6} - 1953 \beta_{5} - 3119 \beta_{4} + \cdots + 8202 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 10223 \beta_{15} + 2460 \beta_{14} - 586 \beta_{13} - 16730 \beta_{12} + 10062 \beta_{11} + 24006 \beta_{10} + 18930 \beta_{9} + 9581 \beta_{8} + 7024 \beta_{7} - 19002 \beta_{6} + 17128 \beta_{5} + \cdots + 17628 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12443 \beta_{15} - 8283 \beta_{14} - 13376 \beta_{13} + 19613 \beta_{12} + 50589 \beta_{11} - 5778 \beta_{10} + 56559 \beta_{9} - 3993 \beta_{8} + 22341 \beta_{7} - 7959 \beta_{6} + 32984 \beta_{5} + \cdots - 43491 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 32900 \beta_{15} - 13056 \beta_{14} - 5274 \beta_{13} + 74346 \beta_{12} - 14082 \beta_{11} - 95841 \beta_{10} - 44046 \beta_{9} - 39501 \beta_{8} - 16731 \beta_{7} + 70494 \beta_{6} - 49980 \beta_{5} + \cdots - 89109 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 158381 \beta_{15} + 42692 \beta_{14} + 88595 \beta_{13} - 4160 \beta_{12} - 420446 \beta_{11} - 140085 \beta_{10} - 525717 \beta_{9} - 50609 \beta_{8} - 208546 \beta_{7} + 203251 \beta_{6} + \cdots + 151791 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 352742 \beta_{15} + 244178 \beta_{14} + 172508 \beta_{13} - 1157114 \beta_{12} - 198050 \beta_{11} + 1355046 \beta_{10} + 149892 \beta_{9} + 563485 \beta_{8} + 56225 \beta_{7} + \cdots + 1521186 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1567315 \beta_{15} - 107706 \beta_{14} - 494328 \beta_{13} - 1118625 \beta_{12} + 3071523 \beta_{11} + 2418363 \beta_{10} + 4219254 \beta_{9} + 968762 \beta_{8} + 1689964 \beta_{7} + \cdots + 356820 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(1 + \beta_{1}\) \(-\beta_{9} - \beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
0.500000 + 2.74530i
0.500000 1.74530i
0.500000 + 1.33108i
0.500000 0.331082i
0.500000 0.589118i
0.500000 2.00333i
0.500000 0.410882i
0.500000 + 1.00333i
0.500000 + 0.589118i
0.500000 + 2.00333i
0.500000 + 0.410882i
0.500000 1.00333i
0.500000 2.74530i
0.500000 + 1.74530i
0.500000 1.33108i
0.500000 + 0.331082i
−2.25487 + 0.604191i 0 2.98735 1.72474i 0 0 1.04649 + 3.90555i −2.39264 + 2.39264i 0 0
143.2 −0.716682 + 0.192034i 0 −1.25529 + 0.724745i 0 0 0.332613 + 1.24133i 1.80977 1.80977i 0 0
143.3 0.716682 0.192034i 0 −1.25529 + 0.724745i 0 0 −0.332613 1.24133i −1.80977 + 1.80977i 0 0
143.4 2.25487 0.604191i 0 2.98735 1.72474i 0 0 −1.04649 3.90555i 2.39264 2.39264i 0 0
332.1 −0.604191 2.25487i 0 −2.98735 + 1.72474i 0 0 −3.90555 + 1.04649i 2.39264 + 2.39264i 0 0
332.2 −0.192034 0.716682i 0 1.25529 0.724745i 0 0 −1.24133 + 0.332613i −1.80977 1.80977i 0 0
332.3 0.192034 + 0.716682i 0 1.25529 0.724745i 0 0 1.24133 0.332613i 1.80977 + 1.80977i 0 0
332.4 0.604191 + 2.25487i 0 −2.98735 + 1.72474i 0 0 3.90555 1.04649i −2.39264 2.39264i 0 0
368.1 −0.604191 + 2.25487i 0 −2.98735 1.72474i 0 0 −3.90555 1.04649i 2.39264 2.39264i 0 0
368.2 −0.192034 + 0.716682i 0 1.25529 + 0.724745i 0 0 −1.24133 0.332613i −1.80977 + 1.80977i 0 0
368.3 0.192034 0.716682i 0 1.25529 + 0.724745i 0 0 1.24133 + 0.332613i 1.80977 1.80977i 0 0
368.4 0.604191 2.25487i 0 −2.98735 1.72474i 0 0 3.90555 + 1.04649i −2.39264 + 2.39264i 0 0
557.1 −2.25487 0.604191i 0 2.98735 + 1.72474i 0 0 1.04649 3.90555i −2.39264 2.39264i 0 0
557.2 −0.716682 0.192034i 0 −1.25529 0.724745i 0 0 0.332613 1.24133i 1.80977 + 1.80977i 0 0
557.3 0.716682 + 0.192034i 0 −1.25529 0.724745i 0 0 −0.332613 + 1.24133i −1.80977 1.80977i 0 0
557.4 2.25487 + 0.604191i 0 2.98735 + 1.72474i 0 0 −1.04649 + 3.90555i 2.39264 + 2.39264i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
9.d odd 6 1 inner
45.h odd 6 1 inner
45.l even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.q.b 16
3.b odd 2 1 225.2.p.a 16
5.b even 2 1 inner 675.2.q.b 16
5.c odd 4 2 inner 675.2.q.b 16
9.c even 3 1 225.2.p.a 16
9.d odd 6 1 inner 675.2.q.b 16
15.d odd 2 1 225.2.p.a 16
15.e even 4 2 225.2.p.a 16
45.h odd 6 1 inner 675.2.q.b 16
45.j even 6 1 225.2.p.a 16
45.k odd 12 2 225.2.p.a 16
45.l even 12 2 inner 675.2.q.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.p.a 16 3.b odd 2 1
225.2.p.a 16 9.c even 3 1
225.2.p.a 16 15.d odd 2 1
225.2.p.a 16 15.e even 4 2
225.2.p.a 16 45.j even 6 1
225.2.p.a 16 45.k odd 12 2
675.2.q.b 16 1.a even 1 1 trivial
675.2.q.b 16 5.b even 2 1 inner
675.2.q.b 16 5.c odd 4 2 inner
675.2.q.b 16 9.d odd 6 1 inner
675.2.q.b 16 45.h odd 6 1 inner
675.2.q.b 16 45.l even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 30T_{2}^{12} + 891T_{2}^{8} - 270T_{2}^{4} + 81 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 30 T^{12} + 891 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 270 T^{12} + 72171 T^{8} + \cdots + 531441 \) Copy content Toggle raw display
$11$ \( (T^{4} - 18 T^{2} + 324)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} - 1080 T^{12} + \cdots + 136048896 \) Copy content Toggle raw display
$17$ \( (T^{8} + 120 T^{4} + 144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 20 T^{2} + 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} - 2910 T^{12} + 8468091 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( (T^{4} + 27 T^{2} + 729)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + 18 T^{2} + 8 T + 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 6264 T^{4} + 7290000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 18 T^{3} + 117 T^{2} - 162 T + 81)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} - 1080 T^{12} + \cdots + 136048896 \) Copy content Toggle raw display
$47$ \( T^{16} - 270 T^{12} + 72171 T^{8} + \cdots + 531441 \) Copy content Toggle raw display
$53$ \( (T^{8} + 2784 T^{4} + 1440000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 72 T^{2} + 5184)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 10 T^{3} + 81 T^{2} + 190 T + 361)^{4} \) Copy content Toggle raw display
$67$ \( T^{16} - 2430 T^{12} + \cdots + 3486784401 \) Copy content Toggle raw display
$71$ \( (T^{4} + 252 T^{2} + 8100)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( (T^{8} - 200 T^{6} + 31536 T^{4} + \cdots + 71639296)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 7230 T^{12} + \cdots + 6343189807761 \) Copy content Toggle raw display
$89$ \( (T^{4} - 198 T^{2} + 2025)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} - 25056 T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
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