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Properties

Label 48.3.i.b
Level $48$
Weight $3$
Character orbit 48.i
Analytic conductor $1.308$
Analytic rank $0$
Dimension $20$
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] = [48,3,Mod(5,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 48.i (of order \(4\), 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: \(1.30790526893\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{18} + 6 x^{16} - 24 x^{14} - 24 x^{12} + 1216 x^{10} - 384 x^{8} - 6144 x^{6} + 24576 x^{4} - 131072 x^{2} + 1048576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{15} q^{2} + \beta_{4} q^{3} + ( - \beta_{12} + \beta_{8} + \beta_1) q^{4} + ( - \beta_{18} + \beta_{15} - \beta_{14} + \beta_{12} + \beta_{10}) q^{5} + (\beta_{18} - \beta_{11} - \beta_{10} - \beta_{3} - \beta_1 - 1) q^{6} + (\beta_{14} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} + 1) q^{7} + ( - \beta_{18} - \beta_{15} - \beta_{13} + \beta_{9}) q^{8} + ( - \beta_{19} - \beta_{14} + \beta_{12} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{15} q^{2} + \beta_{4} q^{3} + ( - \beta_{12} + \beta_{8} + \beta_1) q^{4} + ( - \beta_{18} + \beta_{15} - \beta_{14} + \beta_{12} + \beta_{10}) q^{5} + (\beta_{18} - \beta_{11} - \beta_{10} - \beta_{3} - \beta_1 - 1) q^{6} + (\beta_{14} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} + 1) q^{7} + ( - \beta_{18} - \beta_{15} - \beta_{13} + \beta_{9}) q^{8} + ( - \beta_{19} - \beta_{14} + \beta_{12} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_1) q^{9} + (\beta_{13} + \beta_{12} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{4} - 3 \beta_1 + 1) q^{10} + (2 \beta_{19} - \beta_{17} + \beta_{16} + 2 \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots + \beta_{2}) q^{11}+ \cdots + (4 \beta_{19} - 7 \beta_{17} + 5 \beta_{16} + 36 \beta_{15} - 3 \beta_{14} - 8 \beta_{13} + \cdots - 24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 6 q^{3} + 4 q^{4} - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 6 q^{3} + 4 q^{4} - 12 q^{6} + 32 q^{10} - 88 q^{12} + 92 q^{13} - 116 q^{15} - 16 q^{16} + 4 q^{18} - 52 q^{19} + 48 q^{21} + 24 q^{22} - 8 q^{24} + 18 q^{27} + 56 q^{28} + 28 q^{30} - 80 q^{31} + 60 q^{33} + 104 q^{34} + 92 q^{36} - 116 q^{37} + 88 q^{40} + 304 q^{42} + 172 q^{43} + 60 q^{45} - 424 q^{46} + 176 q^{48} - 364 q^{49} + 128 q^{51} - 208 q^{52} + 40 q^{54} - 512 q^{58} - 240 q^{60} - 244 q^{61} + 296 q^{63} + 88 q^{64} - 492 q^{66} + 356 q^{67} - 20 q^{69} + 200 q^{70} - 472 q^{72} - 146 q^{75} + 328 q^{76} + 84 q^{78} + 384 q^{79} - 188 q^{81} + 560 q^{82} + 816 q^{84} + 48 q^{85} + 416 q^{88} + 616 q^{90} + 136 q^{91} - 132 q^{93} + 32 q^{94} - 24 q^{96} + 472 q^{97} - 452 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{18} + 6 x^{16} - 24 x^{14} - 24 x^{12} + 1216 x^{10} - 384 x^{8} - 6144 x^{6} + 24576 x^{4} - 131072 x^{2} + 1048576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 33 \nu^{18} + 42 \nu^{16} - 1090 \nu^{14} + 528 \nu^{12} + 11816 \nu^{10} - 6496 \nu^{8} + 64512 \nu^{6} - 1111040 \nu^{4} - 1482752 \nu^{2} + 16842752 ) / 10158080 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 25 \nu^{19} + 68 \nu^{18} - 2270 \nu^{17} - 1448 \nu^{16} + 7850 \nu^{15} - 10920 \nu^{14} + 2360 \nu^{13} + 33248 \nu^{12} - 93800 \nu^{11} + 41376 \nu^{10} + \cdots + 120061952 ) / 40632320 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25 \nu^{19} - 68 \nu^{18} - 2270 \nu^{17} + 1448 \nu^{16} + 7850 \nu^{15} + 10920 \nu^{14} + 2360 \nu^{13} - 33248 \nu^{12} - 93800 \nu^{11} - 41376 \nu^{10} + \cdots - 120061952 ) / 40632320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4 \nu^{19} + 112 \nu^{18} - 101 \nu^{17} - 542 \nu^{16} - 410 \nu^{15} + 980 \nu^{14} + 1946 \nu^{13} + 5372 \nu^{12} - 368 \nu^{11} - 13216 \nu^{10} + 24248 \nu^{9} + \cdots - 18481152 ) / 10158080 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4 \nu^{19} + 112 \nu^{18} + 101 \nu^{17} - 542 \nu^{16} + 410 \nu^{15} + 980 \nu^{14} - 1946 \nu^{13} + 5372 \nu^{12} + 368 \nu^{11} - 13216 \nu^{10} - 24248 \nu^{9} + 64976 \nu^{8} + \cdots - 18481152 ) / 10158080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4 \nu^{19} - 149 \nu^{18} + 101 \nu^{17} - 836 \nu^{16} + 410 \nu^{15} + 790 \nu^{14} - 1946 \nu^{13} - 12204 \nu^{12} + 368 \nu^{11} - 36008 \nu^{10} - 24248 \nu^{9} + 8848 \nu^{8} + \cdots - 63307776 ) / 10158080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4 \nu^{19} + 145 \nu^{18} + 101 \nu^{17} - 500 \nu^{16} + 410 \nu^{15} - 110 \nu^{14} - 1946 \nu^{13} + 5900 \nu^{12} + 368 \nu^{11} - 1400 \nu^{10} - 24248 \nu^{9} + 58480 \nu^{8} + \cdots - 1638400 ) / 10158080 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 79 \nu^{19} - 160 \nu^{18} + 34 \nu^{17} - 2040 \nu^{16} + 2030 \nu^{15} + 2960 \nu^{14} + 4896 \nu^{13} + 31600 \nu^{12} + 12392 \nu^{11} - 109440 \nu^{10} - 68512 \nu^{9} + \cdots - 82575360 ) / 40632320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 51 \nu^{19} - 448 \nu^{18} - 1606 \nu^{17} - 32 \nu^{16} + 2450 \nu^{15} + 12480 \nu^{14} + 56 \nu^{13} + 17472 \nu^{12} + 24312 \nu^{11} + 2304 \nu^{10} - 2752 \nu^{9} + \cdots - 70254592 ) / 40632320 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 33 \nu^{19} - 42 \nu^{17} + 1090 \nu^{15} - 528 \nu^{13} - 11816 \nu^{11} + 6496 \nu^{9} - 64512 \nu^{7} + 1111040 \nu^{5} + 1482752 \nu^{3} - 6684672 \nu ) / 10158080 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 33 \nu^{19} + 42 \nu^{17} - 1090 \nu^{15} + 528 \nu^{13} + 11816 \nu^{11} - 6496 \nu^{9} + 64512 \nu^{7} - 1111040 \nu^{5} - 1482752 \nu^{3} + 27000832 \nu ) / 10158080 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 79 \nu^{19} + 592 \nu^{18} + 34 \nu^{17} - 3112 \nu^{16} + 2030 \nu^{15} + 2320 \nu^{14} + 4896 \nu^{13} + 18832 \nu^{12} + 12392 \nu^{11} - 77056 \nu^{10} - 68512 \nu^{9} + \cdots - 96468992 ) / 40632320 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 51 \nu^{19} - 448 \nu^{18} + 1606 \nu^{17} - 32 \nu^{16} - 2450 \nu^{15} + 12480 \nu^{14} - 56 \nu^{13} + 17472 \nu^{12} - 24312 \nu^{11} + 2304 \nu^{10} + 2752 \nu^{9} + \cdots - 70254592 ) / 40632320 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 79 \nu^{19} + 592 \nu^{18} - 34 \nu^{17} - 3112 \nu^{16} - 2030 \nu^{15} + 2320 \nu^{14} - 4896 \nu^{13} + 18832 \nu^{12} - 12392 \nu^{11} - 77056 \nu^{10} + 68512 \nu^{9} + \cdots - 96468992 ) / 40632320 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{19} - 2 \nu^{17} + 6 \nu^{15} - 24 \nu^{13} - 24 \nu^{11} + 1216 \nu^{9} - 384 \nu^{7} - 6144 \nu^{5} + 24576 \nu^{3} - 131072 \nu ) / 262144 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 211 \nu^{19} - 468 \nu^{18} + 334 \nu^{17} + 1008 \nu^{16} - 2470 \nu^{15} + 2520 \nu^{14} + 4496 \nu^{13} - 14768 \nu^{12} - 57928 \nu^{11} - 136096 \nu^{10} + 99168 \nu^{9} + \cdots - 122683392 ) / 40632320 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 211 \nu^{19} + 468 \nu^{18} + 334 \nu^{17} - 1008 \nu^{16} - 2470 \nu^{15} - 2520 \nu^{14} + 4496 \nu^{13} + 14768 \nu^{12} - 57928 \nu^{11} + 136096 \nu^{10} + 99168 \nu^{9} + \cdots + 122683392 ) / 40632320 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 257 \nu^{19} - 1042 \nu^{17} + 870 \nu^{15} + 11272 \nu^{13} - 14616 \nu^{11} + 123456 \nu^{9} + 5248 \nu^{7} - 2611200 \nu^{5} + 24092672 \nu^{3} - 9961472 \nu ) / 40632320 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 157 \nu^{19} - 140 \nu^{18} + 982 \nu^{17} + 240 \nu^{16} - 1110 \nu^{15} + 2600 \nu^{14} - 8192 \nu^{13} + 2160 \nu^{12} + 27896 \nu^{11} + 36000 \nu^{10} - 116576 \nu^{9} + \cdots + 14745600 ) / 10158080 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{5} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + 2\beta_{18} + \beta_{16} - \beta_{13} - \beta_{11} + \beta_{7} - \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} - \beta_{16} - \beta_{14} - \beta_{13} - 2\beta_{12} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} - 7\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6 \beta_{18} - \beta_{17} - \beta_{16} - 6 \beta_{15} + 4 \beta_{14} - \beta_{13} - 4 \beta_{12} - 7 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{17} + 2 \beta_{16} - 10 \beta_{14} - 4 \beta_{12} - 6 \beta_{8} + 6 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + 12 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 4 \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10 \beta_{19} + 28 \beta_{18} - 12 \beta_{17} - 2 \beta_{16} + 8 \beta_{15} + 4 \beta_{14} + 6 \beta_{13} - 4 \beta_{12} + 12 \beta_{11} - 2 \beta_{10} - 16 \beta_{9} + 10 \beta_{7} - 10 \beta_{5} + 10 \beta_{4} + 8 \beta_{3} + 8 \beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2 \beta_{17} + 2 \beta_{16} - 10 \beta_{14} + 2 \beta_{13} + 44 \beta_{12} + 2 \beta_{9} - 54 \beta_{8} - 10 \beta_{7} - 8 \beta_{6} - 18 \beta_{4} - 24 \beta_{3} + 24 \beta_{2} - 146 \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4 \beta_{19} - 44 \beta_{18} - 2 \beta_{17} + 2 \beta_{16} + 180 \beta_{15} - 24 \beta_{14} + 2 \beta_{13} + 24 \beta_{12} - 2 \beta_{11} + 58 \beta_{10} - 6 \beta_{9} + 4 \beta_{7} - 82 \beta_{5} + 82 \beta_{4} - 12 \beta_{3} - 12 \beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 80 \beta_{17} - 80 \beta_{16} - 64 \beta_{14} + 76 \beta_{13} + 16 \beta_{12} + 76 \beta_{9} - 80 \beta_{8} + 8 \beta_{7} + 52 \beta_{6} - 32 \beta_{5} + 28 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} + 276 \beta _1 - 660 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 68 \beta_{19} + 128 \beta_{18} - 116 \beta_{17} - 48 \beta_{16} + 296 \beta_{15} - 200 \beta_{14} - 112 \beta_{13} + 200 \beta_{12} - 204 \beta_{11} - 360 \beta_{10} + 44 \beta_{9} + 68 \beta_{7} - 72 \beta_{5} + 72 \beta_{4} - 56 \beta_{3} - 56 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 44 \beta_{17} + 44 \beta_{16} - 76 \beta_{14} + 52 \beta_{13} - 312 \beta_{12} + 52 \beta_{9} + 236 \beta_{8} - 524 \beta_{7} - 312 \beta_{6} + 928 \beta_{5} + 92 \beta_{4} - 232 \beta_{3} + 232 \beta_{2} - 116 \beta _1 - 1348 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1120 \beta_{19} - 328 \beta_{18} + 236 \beta_{17} - 884 \beta_{16} - 1720 \beta_{15} - 832 \beta_{14} + 972 \beta_{13} + 832 \beta_{12} - 660 \beta_{11} - 132 \beta_{10} + 148 \beta_{9} - 1120 \beta_{7} + 580 \beta_{5} + \cdots - 536 \beta_{2} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 344 \beta_{17} + 344 \beta_{16} - 56 \beta_{14} + 1424 \beta_{13} + 1648 \beta_{12} + 1424 \beta_{9} - 1704 \beta_{8} - 2024 \beta_{7} - 56 \beta_{6} + 2272 \beta_{5} + 192 \beta_{4} + 472 \beta_{3} - 472 \beta_{2} + \cdots + 7840 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1928 \beta_{19} - 4240 \beta_{18} - 1928 \beta_{16} + 6592 \beta_{15} - 5808 \beta_{14} + 2616 \beta_{13} + 5808 \beta_{12} + 7248 \beta_{11} + 5096 \beta_{10} - 688 \beta_{9} - 1928 \beta_{7} + 3928 \beta_{5} + \cdots + 576 \beta_{2} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3928 \beta_{17} + 3928 \beta_{16} + 2152 \beta_{14} - 72 \beta_{13} + 6608 \beta_{12} - 72 \beta_{9} - 4456 \beta_{8} + 2376 \beta_{7} - 7808 \beta_{6} - 4608 \beta_{5} - 10040 \beta_{4} - 13304 \beta _1 - 17656 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1424 \beta_{19} - 7568 \beta_{18} + 9960 \beta_{17} + 8536 \beta_{16} + 6512 \beta_{15} - 15328 \beta_{14} - 680 \beta_{13} + 15328 \beta_{12} - 22200 \beta_{11} - 5736 \beta_{10} + 2104 \beta_{9} + \cdots - 7760 \beta_{2} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2880 \beta_{17} - 2880 \beta_{16} + 3456 \beta_{14} - 14192 \beta_{13} - 5568 \beta_{12} - 14192 \beta_{9} + 9024 \beta_{8} + 544 \beta_{7} - 11024 \beta_{6} + 24768 \beta_{5} + 14288 \beta_{4} + 19920 \beta_{3} + \cdots - 11376 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 42128 \beta_{19} + 57472 \beta_{18} + 14480 \beta_{17} - 27648 \beta_{16} - 51232 \beta_{15} + 34720 \beta_{14} + 21888 \beta_{13} - 34720 \beta_{12} - 54480 \beta_{11} - 53472 \beta_{10} + \cdots - 27808 \beta_{2} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−1.96139 − 0.391068i
−1.85381 − 0.750590i
−1.28499 + 1.53258i
−1.21144 − 1.59136i
−0.312316 + 1.97546i
0.312316 − 1.97546i
1.21144 + 1.59136i
1.28499 − 1.53258i
1.85381 + 0.750590i
1.96139 + 0.391068i
−1.96139 + 0.391068i
−1.85381 + 0.750590i
−1.28499 − 1.53258i
−1.21144 + 1.59136i
−0.312316 − 1.97546i
0.312316 + 1.97546i
1.21144 − 1.59136i
1.28499 + 1.53258i
1.85381 − 0.750590i
1.96139 − 0.391068i
−1.96139 + 0.391068i −2.99548 + 0.164573i 3.69413 − 1.53408i 3.61305 − 3.61305i 5.81096 − 1.49423i − 12.2792i −6.64572 + 4.45358i 8.94583 − 0.985948i −5.67366 + 8.49955i
5.2 −1.85381 + 0.750590i 1.50491 + 2.59524i 2.87323 − 2.78290i −2.59897 + 2.59897i −4.73777 − 3.68151i 7.30027i −3.23761 + 7.31559i −4.47050 + 7.81118i 2.86723 − 6.76875i
5.3 −1.28499 − 1.53258i −2.06336 + 2.17774i −0.697601 + 3.93870i −3.17955 + 3.17955i 5.98896 + 0.363879i 6.03979i 6.93278 − 3.99206i −0.485128 − 8.98692i 8.95859 + 0.787223i
5.4 −1.21144 + 1.59136i 1.14944 − 2.77106i −1.06484 − 3.85566i 4.80434 − 4.80434i 3.01728 + 5.18614i 7.36187i 7.42573 + 2.97634i −6.35757 − 6.37035i 1.82527 + 13.4656i
5.5 −0.312316 − 1.97546i −1.18505 − 2.75602i −3.80492 + 1.23394i −0.00985921 + 0.00985921i −5.07432 + 3.20176i − 6.42277i 3.62594 + 7.13110i −6.19134 + 6.53203i 0.0225557 + 0.0163973i
5.6 0.312316 + 1.97546i 2.75602 + 1.18505i −3.80492 + 1.23394i 0.00985921 − 0.00985921i −1.48026 + 5.81454i − 6.42277i −3.62594 − 7.13110i 6.19134 + 6.53203i 0.0225557 + 0.0163973i
5.7 1.21144 − 1.59136i 2.77106 − 1.14944i −1.06484 − 3.85566i −4.80434 + 4.80434i 1.52779 − 5.80223i 7.36187i −7.42573 − 2.97634i 6.35757 − 6.37035i 1.82527 + 13.4656i
5.8 1.28499 + 1.53258i −2.17774 + 2.06336i −0.697601 + 3.93870i 3.17955 − 3.17955i −5.96063 − 0.686173i 6.03979i −6.93278 + 3.99206i 0.485128 − 8.98692i 8.95859 + 0.787223i
5.9 1.85381 − 0.750590i −2.59524 − 1.50491i 2.87323 − 2.78290i 2.59897 − 2.59897i −5.94065 − 0.841858i 7.30027i 3.23761 − 7.31559i 4.47050 + 7.81118i 2.86723 − 6.76875i
5.10 1.96139 − 0.391068i −0.164573 + 2.99548i 3.69413 − 1.53408i −3.61305 + 3.61305i 0.848646 + 5.93968i − 12.2792i 6.64572 − 4.45358i −8.94583 − 0.985948i −5.67366 + 8.49955i
29.1 −1.96139 − 0.391068i −2.99548 − 0.164573i 3.69413 + 1.53408i 3.61305 + 3.61305i 5.81096 + 1.49423i 12.2792i −6.64572 − 4.45358i 8.94583 + 0.985948i −5.67366 − 8.49955i
29.2 −1.85381 − 0.750590i 1.50491 − 2.59524i 2.87323 + 2.78290i −2.59897 − 2.59897i −4.73777 + 3.68151i − 7.30027i −3.23761 − 7.31559i −4.47050 − 7.81118i 2.86723 + 6.76875i
29.3 −1.28499 + 1.53258i −2.06336 − 2.17774i −0.697601 − 3.93870i −3.17955 − 3.17955i 5.98896 − 0.363879i − 6.03979i 6.93278 + 3.99206i −0.485128 + 8.98692i 8.95859 − 0.787223i
29.4 −1.21144 − 1.59136i 1.14944 + 2.77106i −1.06484 + 3.85566i 4.80434 + 4.80434i 3.01728 − 5.18614i − 7.36187i 7.42573 − 2.97634i −6.35757 + 6.37035i 1.82527 − 13.4656i
29.5 −0.312316 + 1.97546i −1.18505 + 2.75602i −3.80492 − 1.23394i −0.00985921 − 0.00985921i −5.07432 − 3.20176i 6.42277i 3.62594 − 7.13110i −6.19134 − 6.53203i 0.0225557 − 0.0163973i
29.6 0.312316 − 1.97546i 2.75602 − 1.18505i −3.80492 − 1.23394i 0.00985921 + 0.00985921i −1.48026 − 5.81454i 6.42277i −3.62594 + 7.13110i 6.19134 − 6.53203i 0.0225557 − 0.0163973i
29.7 1.21144 + 1.59136i 2.77106 + 1.14944i −1.06484 + 3.85566i −4.80434 − 4.80434i 1.52779 + 5.80223i − 7.36187i −7.42573 + 2.97634i 6.35757 + 6.37035i 1.82527 − 13.4656i
29.8 1.28499 − 1.53258i −2.17774 − 2.06336i −0.697601 − 3.93870i 3.17955 + 3.17955i −5.96063 + 0.686173i − 6.03979i −6.93278 − 3.99206i 0.485128 + 8.98692i 8.95859 − 0.787223i
29.9 1.85381 + 0.750590i −2.59524 + 1.50491i 2.87323 + 2.78290i 2.59897 + 2.59897i −5.94065 + 0.841858i − 7.30027i 3.23761 + 7.31559i 4.47050 − 7.81118i 2.86723 + 6.76875i
29.10 1.96139 + 0.391068i −0.164573 − 2.99548i 3.69413 + 1.53408i −3.61305 − 3.61305i 0.848646 − 5.93968i 12.2792i 6.64572 + 4.45358i −8.94583 + 0.985948i −5.67366 − 8.49955i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.3.i.b ✓ 20
3.b odd 2 1 inner 48.3.i.b ✓ 20
4.b odd 2 1 192.3.i.b 20
8.b even 2 1 384.3.i.d 20
8.d odd 2 1 384.3.i.c 20
12.b even 2 1 192.3.i.b 20
16.e even 4 1 inner 48.3.i.b ✓ 20
16.e even 4 1 384.3.i.d 20
16.f odd 4 1 192.3.i.b 20
16.f odd 4 1 384.3.i.c 20
24.f even 2 1 384.3.i.c 20
24.h odd 2 1 384.3.i.d 20
48.i odd 4 1 inner 48.3.i.b ✓ 20
48.i odd 4 1 384.3.i.d 20
48.k even 4 1 192.3.i.b 20
48.k even 4 1 384.3.i.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.i.b ✓ 20 1.a even 1 1 trivial
48.3.i.b ✓ 20 3.b odd 2 1 inner
48.3.i.b ✓ 20 16.e even 4 1 inner
48.3.i.b ✓ 20 48.i odd 4 1 inner
192.3.i.b 20 4.b odd 2 1
192.3.i.b 20 12.b even 2 1
192.3.i.b 20 16.f odd 4 1
192.3.i.b 20 48.k even 4 1
384.3.i.c 20 8.d odd 2 1
384.3.i.c 20 16.f odd 4 1
384.3.i.c 20 24.f even 2 1
384.3.i.c 20 48.k even 4 1
384.3.i.d 20 8.b even 2 1
384.3.i.d 20 16.e even 4 1
384.3.i.d 20 24.h odd 2 1
384.3.i.d 20 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 3404T_{5}^{16} + 3190384T_{5}^{12} + 1068787520T_{5}^{8} + 108375444480T_{5}^{4} + 4096 \) acting on \(S_{3}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 2 T^{18} + 6 T^{16} + \cdots + 1048576 \) Copy content Toggle raw display
$3$ \( T^{20} + 6 T^{19} + \cdots + 3486784401 \) Copy content Toggle raw display
$5$ \( T^{20} + 3404 T^{16} + 3190384 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( (T^{10} + 336 T^{8} + 40676 T^{6} + \cdots + 655360000)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + 207308 T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} - 46 T^{9} + 1058 T^{8} + \cdots + 33620000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + 952 T^{8} + \cdots + 15510536192)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + 26 T^{9} + 338 T^{8} + \cdots + 23975244288)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} - 2236 T^{8} + \cdots - 2157878476800)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + 8250700 T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + 20 T^{4} - 2750 T^{3} + \cdots + 6473680)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + 58 T^{9} + \cdots + 93878430976800)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} - 8644 T^{8} + \cdots - 89172136396800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} - 86 T^{9} + \cdots + 398518394892800)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + 4944 T^{8} + \cdots + 2199023255552)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + 33387084 T^{16} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + 96029644 T^{16} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + 122 T^{9} + \cdots + 79\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 178 T^{9} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 12876 T^{8} + \cdots - 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 16160 T^{8} + \cdots + 900192010240000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 96 T^{4} - 3534 T^{3} + \cdots - 147403248)^{4} \) Copy content Toggle raw display
$83$ \( T^{20} + 433330892 T^{16} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{10} - 49740 T^{8} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} - 118 T^{4} - 11780 T^{3} + \cdots - 2657552000)^{4} \) Copy content Toggle raw display
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