Properties

Label 315.2.i.f
Level $315$
Weight $2$
Character orbit 315.i
Analytic conductor $2.515$
Analytic rank $0$
Dimension $16$
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] = [315,2,Mod(106,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.i (of order \(3\), degree \(2\), 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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 8 x^{14} - 10 x^{13} + 40 x^{12} - 45 x^{11} + 159 x^{10} - 180 x^{9} + 576 x^{8} - 540 x^{7} + 1431 x^{6} - 1215 x^{5} + 3240 x^{4} - 2430 x^{3} + 5832 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{13} + \beta_{2}) q^{2} + \beta_{5} q^{3} + (\beta_{12} - \beta_{10} + \beta_{7} - 2 \beta_{3} - 2) q^{4} + ( - \beta_{3} - 1) q^{5} + ( - \beta_{7} + \beta_{4} - \beta_{3}) q^{6} - \beta_{3} q^{7} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{8} + ( - \beta_{13} - \beta_{12} + \beta_{6} + \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{13} + \beta_{2}) q^{2} + \beta_{5} q^{3} + (\beta_{12} - \beta_{10} + \beta_{7} - 2 \beta_{3} - 2) q^{4} + ( - \beta_{3} - 1) q^{5} + ( - \beta_{7} + \beta_{4} - \beta_{3}) q^{6} - \beta_{3} q^{7} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{8} + ( - \beta_{13} - \beta_{12} + \beta_{6} + \beta_{3} + 1) q^{9} - \beta_{2} q^{10} + ( - \beta_{15} + \beta_{13} - \beta_{12} + \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} + \cdots - 1) q^{11}+ \cdots + ( - \beta_{15} - 2 \beta_{14} - 4 \beta_{13} + 3 \beta_{11} + 2 \beta_{10} + \beta_{9} + \beta_{8} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{2} + q^{3} - 11 q^{4} - 8 q^{5} + 8 q^{6} + 8 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{2} + q^{3} - 11 q^{4} - 8 q^{5} + 8 q^{6} + 8 q^{7} - 6 q^{8} + 3 q^{9} - 2 q^{10} - 4 q^{11} - 3 q^{12} - 5 q^{13} - q^{14} - 2 q^{15} - 21 q^{16} + 8 q^{17} + 26 q^{18} + 6 q^{19} - 11 q^{20} - q^{21} - 23 q^{22} + 8 q^{23} - 38 q^{24} - 8 q^{25} - 6 q^{26} + 10 q^{27} - 22 q^{28} - 19 q^{29} - 13 q^{30} + 12 q^{32} - 21 q^{33} - 9 q^{34} - 16 q^{35} + 70 q^{36} + 42 q^{37} + 28 q^{38} + 32 q^{39} + 3 q^{40} - 20 q^{41} - 5 q^{42} - 13 q^{43} + 30 q^{44} + 9 q^{45} + 34 q^{46} + 11 q^{47} - 18 q^{48} - 8 q^{49} + q^{50} - 14 q^{51} - 13 q^{52} - 16 q^{53} + 8 q^{55} - 3 q^{56} + 8 q^{57} - 37 q^{58} - 7 q^{59} + 9 q^{60} - 24 q^{61} - 30 q^{62} + 12 q^{63} + 110 q^{64} - 5 q^{65} - 11 q^{66} - 16 q^{67} - 5 q^{68} + 21 q^{69} - q^{70} - 10 q^{71} + 17 q^{72} + 20 q^{73} - 21 q^{74} + q^{75} - 25 q^{76} + 4 q^{77} - 61 q^{78} - 27 q^{79} + 42 q^{80} + 11 q^{81} + 72 q^{82} - 5 q^{83} + 6 q^{84} - 4 q^{85} + 27 q^{86} - 46 q^{87} - 67 q^{88} + 54 q^{89} - 7 q^{90} - 10 q^{91} + 93 q^{92} - 9 q^{93} + 17 q^{94} - 3 q^{95} - 98 q^{96} - 27 q^{97} - 2 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 8 x^{14} - 10 x^{13} + 40 x^{12} - 45 x^{11} + 159 x^{10} - 180 x^{9} + 576 x^{8} - 540 x^{7} + 1431 x^{6} - 1215 x^{5} + 3240 x^{4} - 2430 x^{3} + 5832 x^{2} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{15} - 54 \nu^{14} - 13 \nu^{13} - 285 \nu^{12} - 92 \nu^{11} - 1376 \nu^{10} - 564 \nu^{9} - 5133 \nu^{8} - 729 \nu^{7} - 15138 \nu^{6} - 5130 \nu^{5} - 35586 \nu^{4} - 6399 \nu^{3} + \cdots - 105705 ) / 5832 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 331 \nu^{15} - 384 \nu^{14} - 1549 \nu^{13} - 3459 \nu^{12} - 7034 \nu^{11} - 20030 \nu^{10} - 31770 \nu^{9} - 69753 \nu^{8} - 77643 \nu^{7} - 230976 \nu^{6} + \cdots - 2009853 ) / 763992 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23 \nu^{15} + 98 \nu^{14} + 139 \nu^{13} + 571 \nu^{12} + 668 \nu^{11} + 2808 \nu^{10} + 3036 \nu^{9} + 9171 \nu^{8} + 7335 \nu^{7} + 29214 \nu^{6} + 21006 \nu^{5} + 66582 \nu^{4} + \cdots + 185895 ) / 17496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1046 \nu^{15} - 1099 \nu^{14} + 6769 \nu^{13} - 6206 \nu^{12} + 34925 \nu^{11} - 20859 \nu^{10} + 129333 \nu^{9} - 113013 \nu^{8} + 409716 \nu^{7} - 254097 \nu^{6} + \cdots - 2171691 ) / 763992 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3181 \nu^{15} + 19087 \nu^{14} + 19856 \nu^{13} + 110357 \nu^{12} + 111133 \nu^{11} + 545253 \nu^{10} + 450105 \nu^{9} + 1921554 \nu^{8} + 1030491 \nu^{7} + \cdots + 38027556 ) / 2291976 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4417 \nu^{15} - 21707 \nu^{14} + 18134 \nu^{13} - 111667 \nu^{12} + 85891 \nu^{11} - 556257 \nu^{10} + 222711 \nu^{9} - 2202156 \nu^{8} + 1032759 \nu^{7} + \cdots - 48914442 ) / 2291976 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5084 \nu^{15} - 28999 \nu^{14} + 24811 \nu^{13} - 145040 \nu^{12} + 102185 \nu^{11} - 728211 \nu^{10} + 259437 \nu^{9} - 2833947 \nu^{8} + 1367514 \nu^{7} + \cdots - 64820493 ) / 2291976 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7037 \nu^{15} - 4493 \nu^{14} - 37522 \nu^{13} - 33481 \nu^{12} - 196979 \nu^{11} - 182583 \nu^{10} - 785487 \nu^{9} - 500112 \nu^{8} - 2386251 \nu^{7} + \cdots - 11822922 ) / 2291976 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7210 \nu^{15} - 8453 \nu^{14} + 41831 \nu^{13} - 44614 \nu^{12} + 178159 \nu^{11} - 172845 \nu^{10} + 627531 \nu^{9} - 856971 \nu^{8} + 1982160 \nu^{7} + \cdots - 19925757 ) / 2291976 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9662 \nu^{15} + 17977 \nu^{14} - 46603 \nu^{13} + 104798 \nu^{12} - 193703 \nu^{11} + 470901 \nu^{10} - 641559 \nu^{9} + 2068767 \nu^{8} - 2112984 \nu^{7} + \cdots + 46493433 ) / 2291976 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3765 \nu^{15} - 343 \nu^{14} + 21146 \nu^{13} - 1619 \nu^{12} + 99331 \nu^{11} + 9839 \nu^{10} + 355491 \nu^{9} - 51720 \nu^{8} + 1071351 \nu^{7} + 70947 \nu^{6} + \cdots - 1106622 ) / 763992 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12086 \nu^{15} + 23761 \nu^{14} - 60991 \nu^{13} + 125582 \nu^{12} - 286595 \nu^{11} + 566589 \nu^{10} - 937479 \nu^{9} + 2445795 \nu^{8} - 3348720 \nu^{7} + \cdots + 58198257 ) / 2291976 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5017 \nu^{15} + 6145 \nu^{14} - 26580 \nu^{13} + 31991 \nu^{12} - 129525 \nu^{11} + 144563 \nu^{10} - 454785 \nu^{9} + 650802 \nu^{8} - 1497951 \nu^{7} + \cdots + 14521680 ) / 763992 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 16105 \nu^{15} + 1514 \nu^{14} - 86681 \nu^{13} + 4519 \nu^{12} - 420040 \nu^{11} - 14028 \nu^{10} - 1510836 \nu^{9} + 340047 \nu^{8} - 4533309 \nu^{7} + \cdots + 12984219 ) / 2291976 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{12} + \beta_{7} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{10} + \beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{14} + 2 \beta_{13} - 3 \beta_{12} - \beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{14} + \beta_{13} - 3 \beta_{11} - 4 \beta_{9} + 3 \beta_{8} + 2 \beta_{6} + 3 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 5 \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 7 \beta_{15} + 10 \beta_{14} - \beta_{13} + 7 \beta_{12} + 2 \beta_{11} - 7 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - 9 \beta_{6} + 4 \beta_{5} + 3 \beta_{4} + 2 \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4 \beta_{15} - 5 \beta_{14} - 11 \beta_{13} - 4 \beta_{12} + 9 \beta_{11} + 10 \beta_{10} - \beta_{9} - 6 \beta_{8} - \beta_{7} - 4 \beta_{6} - 15 \beta_{5} + 5 \beta_{4} + 12 \beta_{3} - 2 \beta_{2} + \beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 11 \beta_{15} - 5 \beta_{14} - 12 \beta_{13} - 13 \beta_{12} + 12 \beta_{11} + 5 \beta_{10} + 14 \beta_{9} - 9 \beta_{7} - 12 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} - 37 \beta_{3} - 15 \beta_{2} + 25 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 7 \beta_{15} - 6 \beta_{14} + 23 \beta_{13} + 13 \beta_{12} + 2 \beta_{11} - 17 \beta_{10} + 17 \beta_{9} - 7 \beta_{8} + 23 \beta_{7} - 36 \beta_{6} + 28 \beta_{5} + 28 \beta_{4} - 64 \beta_{3} - 6 \beta_{2} + 23 \beta _1 - 117 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 79 \beta_{15} - 26 \beta_{14} - 48 \beta_{13} - 19 \beta_{12} - 22 \beta_{11} + 15 \beta_{10} + 34 \beta_{9} - 22 \beta_{8} + 21 \beta_{7} + 65 \beta_{6} + 40 \beta_{5} + 22 \beta_{4} - 45 \beta_{3} - 25 \beta_{2} - 93 \beta _1 + 46 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 20 \beta_{15} - 58 \beta_{14} + 34 \beta_{13} - 94 \beta_{12} - 21 \beta_{11} + 32 \beta_{10} + 10 \beta_{9} - 45 \beta_{8} - 3 \beta_{7} + 194 \beta_{6} + 9 \beta_{5} - 104 \beta_{4} + 127 \beta_{3} + 126 \beta_{2} + 5 \beta _1 + 165 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 8 \beta_{15} + 109 \beta_{14} + 232 \beta_{13} + 104 \beta_{12} - 68 \beta_{11} - 15 \beta_{10} - 189 \beta_{9} + 193 \beta_{8} + 119 \beta_{7} + 87 \beta_{6} - 22 \beta_{5} - 62 \beta_{4} + 275 \beta_{3} - 19 \beta_{2} + 137 \beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 124 \beta_{15} + 458 \beta_{14} + \beta_{13} + 508 \beta_{12} + 65 \beta_{11} - 137 \beta_{10} + 45 \beta_{9} + 419 \beta_{8} - 181 \beta_{7} - 425 \beta_{6} + 163 \beta_{5} + 192 \beta_{4} + 51 \beta_{3} - 216 \beta_{2} + 124 \beta _1 + 111 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 112 \beta_{15} - 393 \beta_{14} - 134 \beta_{13} - 71 \beta_{12} + 48 \beta_{11} + 4 \beta_{10} + 96 \beta_{9} - 501 \beta_{8} - 276 \beta_{7} - 424 \beta_{6} - 531 \beta_{5} + 292 \beta_{4} + 1464 \beta_{3} + 180 \beta_{2} + \cdots + 438 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 522 \beta_{15} - 734 \beta_{14} - 498 \beta_{13} - 528 \beta_{12} - 438 \beta_{11} + 82 \beta_{10} - 346 \beta_{9} - 744 \beta_{8} + 232 \beta_{7} + 378 \beta_{6} - 1872 \beta_{5} - 514 \beta_{4} - 1208 \beta_{3} - 298 \beta_{2} + \cdots - 113 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1
1.06614 1.36504i
−1.41596 0.997525i
−1.26474 + 1.18340i
1.40613 + 1.01134i
−0.216795 1.71843i
0.803782 + 1.53425i
1.14603 1.29869i
−0.524589 + 1.65070i
1.06614 + 1.36504i
−1.41596 + 0.997525i
−1.26474 1.18340i
1.40613 1.01134i
−0.216795 + 1.71843i
0.803782 1.53425i
1.14603 + 1.29869i
−0.524589 1.65070i
−1.32864 + 2.30128i −0.649093 1.60583i −2.53058 4.38310i −0.500000 0.866025i 4.55786 + 0.639826i 0.500000 0.866025i 8.13440 −2.15736 + 2.08466i 2.65729
106.2 −0.978244 + 1.69437i −1.57186 + 0.727495i −0.913922 1.58296i −0.500000 0.866025i 0.305020 3.37498i 0.500000 0.866025i −0.336819 1.94150 2.28704i 1.95649
106.3 −0.522039 + 0.904198i 0.392487 + 1.68700i 0.454951 + 0.787999i −0.500000 0.866025i −1.73027 0.525791i 0.500000 0.866025i −3.03816 −2.69191 + 1.32425i 1.04408
106.4 −0.219523 + 0.380224i 1.57891 0.712071i 0.903620 + 1.56512i −0.500000 0.866025i −0.0758596 + 0.756655i 0.500000 0.866025i −1.67155 1.98591 2.24859i 0.439045
106.5 0.441371 0.764477i −1.59660 0.671465i 0.610383 + 1.05721i −0.500000 0.866025i −1.21801 + 0.924200i 0.500000 0.866025i 2.84311 2.09827 + 2.14412i −0.882742
106.6 0.627726 1.08725i 1.73059 + 0.0710311i 0.211920 + 0.367057i −0.500000 0.866025i 1.16357 1.83701i 0.500000 0.866025i 3.04302 2.98991 + 0.245852i −1.25545
106.7 1.09035 1.88855i −0.551686 1.64184i −1.37775 2.38633i −0.500000 0.866025i −3.70223 0.748302i 0.500000 0.866025i −1.64751 −2.39128 + 1.81156i −2.18071
106.8 1.38900 2.40581i 1.16725 + 1.27966i −2.85862 4.95128i −0.500000 0.866025i 4.69992 1.03075i 0.500000 0.866025i −10.3265 −0.275041 + 2.98737i −2.77799
211.1 −1.32864 2.30128i −0.649093 + 1.60583i −2.53058 + 4.38310i −0.500000 + 0.866025i 4.55786 0.639826i 0.500000 + 0.866025i 8.13440 −2.15736 2.08466i 2.65729
211.2 −0.978244 1.69437i −1.57186 0.727495i −0.913922 + 1.58296i −0.500000 + 0.866025i 0.305020 + 3.37498i 0.500000 + 0.866025i −0.336819 1.94150 + 2.28704i 1.95649
211.3 −0.522039 0.904198i 0.392487 1.68700i 0.454951 0.787999i −0.500000 + 0.866025i −1.73027 + 0.525791i 0.500000 + 0.866025i −3.03816 −2.69191 1.32425i 1.04408
211.4 −0.219523 0.380224i 1.57891 + 0.712071i 0.903620 1.56512i −0.500000 + 0.866025i −0.0758596 0.756655i 0.500000 + 0.866025i −1.67155 1.98591 + 2.24859i 0.439045
211.5 0.441371 + 0.764477i −1.59660 + 0.671465i 0.610383 1.05721i −0.500000 + 0.866025i −1.21801 0.924200i 0.500000 + 0.866025i 2.84311 2.09827 2.14412i −0.882742
211.6 0.627726 + 1.08725i 1.73059 0.0710311i 0.211920 0.367057i −0.500000 + 0.866025i 1.16357 + 1.83701i 0.500000 + 0.866025i 3.04302 2.98991 0.245852i −1.25545
211.7 1.09035 + 1.88855i −0.551686 + 1.64184i −1.37775 + 2.38633i −0.500000 + 0.866025i −3.70223 + 0.748302i 0.500000 + 0.866025i −1.64751 −2.39128 1.81156i −2.18071
211.8 1.38900 + 2.40581i 1.16725 1.27966i −2.85862 + 4.95128i −0.500000 + 0.866025i 4.69992 + 1.03075i 0.500000 + 0.866025i −10.3265 −0.275041 2.98737i −2.77799
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.i.f 16
3.b odd 2 1 945.2.i.f 16
9.c even 3 1 inner 315.2.i.f 16
9.c even 3 1 2835.2.a.x 8
9.d odd 6 1 945.2.i.f 16
9.d odd 6 1 2835.2.a.y 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.i.f 16 1.a even 1 1 trivial
315.2.i.f 16 9.c even 3 1 inner
945.2.i.f 16 3.b odd 2 1
945.2.i.f 16 9.d odd 6 1
2835.2.a.x 8 9.c even 3 1
2835.2.a.y 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - T_{2}^{15} + 14 T_{2}^{14} - 9 T_{2}^{13} + 131 T_{2}^{12} - 77 T_{2}^{11} + 616 T_{2}^{10} - 216 T_{2}^{9} + 1978 T_{2}^{8} - 660 T_{2}^{7} + 3061 T_{2}^{6} - 220 T_{2}^{5} + 2945 T_{2}^{4} - 186 T_{2}^{3} + 1364 T_{2}^{2} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{15} + 14 T^{14} - 9 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} - T^{15} - T^{14} - 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + 4 T^{15} + 80 T^{14} + \cdots + 241367296 \) Copy content Toggle raw display
$13$ \( T^{16} + 5 T^{15} + 80 T^{14} + \cdots + 3118756 \) Copy content Toggle raw display
$17$ \( (T^{8} - 4 T^{7} - 88 T^{6} + 416 T^{5} + \cdots - 13682)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 3 T^{7} - 101 T^{6} + 243 T^{5} + \cdots + 96256)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 8 T^{15} + 128 T^{14} + \cdots + 358875136 \) Copy content Toggle raw display
$29$ \( T^{16} + 19 T^{15} + \cdots + 181104718096 \) Copy content Toggle raw display
$31$ \( T^{16} + 108 T^{14} + \cdots + 6879707136 \) Copy content Toggle raw display
$37$ \( (T^{8} - 21 T^{7} + 105 T^{6} + \cdots - 82944)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 20 T^{15} + \cdots + 836829184 \) Copy content Toggle raw display
$43$ \( T^{16} + 13 T^{15} + \cdots + 345330171904 \) Copy content Toggle raw display
$47$ \( T^{16} - 11 T^{15} + 192 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( (T^{8} + 8 T^{7} - 118 T^{6} + \cdots - 165056)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 7 T^{15} + \cdots + 63471748096 \) Copy content Toggle raw display
$61$ \( T^{16} + 24 T^{15} + \cdots + 27518828544 \) Copy content Toggle raw display
$67$ \( T^{16} + 16 T^{15} + 233 T^{14} + \cdots + 5456896 \) Copy content Toggle raw display
$71$ \( (T^{8} + 5 T^{7} - 290 T^{6} + \cdots - 1249607)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 10 T^{7} - 88 T^{6} + \cdots + 124561)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + 27 T^{15} + 497 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$83$ \( T^{16} + 5 T^{15} + \cdots + 6791950062769 \) Copy content Toggle raw display
$89$ \( (T^{8} - 27 T^{7} - 53 T^{6} + \cdots - 6305216)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 27 T^{15} + \cdots + 4365349992964 \) Copy content Toggle raw display
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