Properties

Label 102.5.g.b
Level $102$
Weight $5$
Character orbit 102.g
Analytic conductor $10.544$
Analytic rank $0$
Dimension $4$
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] = [102,5,Mod(53,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 7]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("102.53");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 102.g (of order \(8\), 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: \(10.5437362346\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{8}^{2} + 2) q^{2} + (6 \zeta_{8}^{3} + 6 \zeta_{8} + 3) q^{3} + 8 \zeta_{8}^{2} q^{4} + ( - 14 \zeta_{8}^{3} + 14 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{5} + (24 \zeta_{8}^{3} + 6 \zeta_{8}^{2} + 6) q^{6} + ( - 33 \zeta_{8}^{3} + 19 \zeta_{8}^{2} + 19 \zeta_{8} - 33) q^{7} + (16 \zeta_{8}^{2} - 16) q^{8} + (36 \zeta_{8}^{3} + 36 \zeta_{8} - 63) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{8}^{2} + 2) q^{2} + (6 \zeta_{8}^{3} + 6 \zeta_{8} + 3) q^{3} + 8 \zeta_{8}^{2} q^{4} + ( - 14 \zeta_{8}^{3} + 14 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{5} + (24 \zeta_{8}^{3} + 6 \zeta_{8}^{2} + 6) q^{6} + ( - 33 \zeta_{8}^{3} + 19 \zeta_{8}^{2} + 19 \zeta_{8} - 33) q^{7} + (16 \zeta_{8}^{2} - 16) q^{8} + (36 \zeta_{8}^{3} + 36 \zeta_{8} - 63) q^{9} + ( - 32 \zeta_{8}^{3} + 24 \zeta_{8}^{2} + 24 \zeta_{8} - 32) q^{10} + (30 \zeta_{8}^{3} + 83 \zeta_{8}^{2} + 83 \zeta_{8} + 30) q^{11} + (48 \zeta_{8}^{3} + 24 \zeta_{8}^{2} - 48 \zeta_{8}) q^{12} + ( - 34 \zeta_{8}^{3} - 76 \zeta_{8}^{2} - 34 \zeta_{8}) q^{13} + ( - 28 \zeta_{8}^{3} - 28 \zeta_{8}^{2} + 104 \zeta_{8} - 104) q^{14} + (30 \zeta_{8}^{3} + 114 \zeta_{8}^{2} - 102 \zeta_{8} + 90) q^{15} - 64 q^{16} + ( - 51 \zeta_{8}^{3} + 102 \zeta_{8}^{2} - 204 \zeta_{8} - 170) q^{17} + (144 \zeta_{8}^{3} - 126 \zeta_{8}^{2} - 126) q^{18} + ( - 260 \zeta_{8}^{3} + 142 \zeta_{8}^{2} - 142) q^{19} + ( - 16 \zeta_{8}^{3} - 16 \zeta_{8}^{2} + 112 \zeta_{8} - 112) q^{20} + ( - 183 \zeta_{8}^{3} + 369 \zeta_{8}^{2} - 255 \zeta_{8} - 15) q^{21} + (226 \zeta_{8}^{3} + 226 \zeta_{8}^{2} + 106 \zeta_{8} - 106) q^{22} + (221 \zeta_{8}^{3} + 71 \zeta_{8}^{2} + 71 \zeta_{8} + 221) q^{23} + (48 \zeta_{8}^{2} - 192 \zeta_{8} - 48) q^{24} + ( - 248 \zeta_{8}^{2} - 225 \zeta_{8} - 248) q^{25} + ( - 136 \zeta_{8}^{3} - 152 \zeta_{8}^{2} + 152) q^{26} + ( - 270 \zeta_{8}^{3} - 270 \zeta_{8} - 621) q^{27} + (152 \zeta_{8}^{3} - 264 \zeta_{8}^{2} + 264 \zeta_{8} - 152) q^{28} + (34 \zeta_{8}^{3} - 34 \zeta_{8}^{2} + 824 \zeta_{8} + 824) q^{29} + ( - 144 \zeta_{8}^{3} + 408 \zeta_{8}^{2} - 264 \zeta_{8} - 48) q^{30} + (85 \zeta_{8}^{3} - 85 \zeta_{8}^{2} + 3 \zeta_{8} + 3) q^{31} + ( - 128 \zeta_{8}^{2} - 128) q^{32} + (768 \zeta_{8}^{3} + 567 \zeta_{8}^{2} - 69 \zeta_{8} - 588) q^{33} + ( - 510 \zeta_{8}^{3} - 136 \zeta_{8}^{2} - 306 \zeta_{8} - 544) q^{34} + (756 \zeta_{8}^{3} - 1000 \zeta_{8}^{2} + 756 \zeta_{8}) q^{35} + (288 \zeta_{8}^{3} - 504 \zeta_{8}^{2} - 288 \zeta_{8}) q^{36} + ( - 118 \zeta_{8}^{3} + 118 \zeta_{8}^{2} + 550 \zeta_{8} + 550) q^{37} + ( - 520 \zeta_{8}^{3} + 520 \zeta_{8} - 568) q^{38} + ( - 558 \zeta_{8}^{3} - 228 \zeta_{8}^{2} + 354 \zeta_{8} + 408) q^{39} + (192 \zeta_{8}^{3} - 256 \zeta_{8}^{2} + 256 \zeta_{8} - 192) q^{40} + ( - 340 \zeta_{8}^{3} + 807 \zeta_{8}^{2} - 807 \zeta_{8} + 340) q^{41} + ( - 876 \zeta_{8}^{3} + 708 \zeta_{8}^{2} - 144 \zeta_{8} - 768) q^{42} + (483 \zeta_{8}^{2} + 2298 \zeta_{8} + 483) q^{43} + (664 \zeta_{8}^{3} + 240 \zeta_{8}^{2} - 240 \zeta_{8} - 664) q^{44} + (1314 \zeta_{8}^{3} - 450 \zeta_{8}^{2} - 450 \zeta_{8} + 702) q^{45} + (584 \zeta_{8}^{3} + 584 \zeta_{8}^{2} - 300 \zeta_{8} + 300) q^{46} + ( - 596 \zeta_{8}^{3} + 596 \zeta_{8} - 550) q^{47} + ( - 384 \zeta_{8}^{3} - 384 \zeta_{8} - 192) q^{48} + (499 \zeta_{8}^{3} - 1982 \zeta_{8}^{2} + 1982) q^{49} + ( - 450 \zeta_{8}^{3} - 992 \zeta_{8}^{2} - 450 \zeta_{8}) q^{50} + ( - 561 \zeta_{8}^{3} - 612 \zeta_{8}^{2} - 2244 \zeta_{8} + 1020) q^{51} + ( - 272 \zeta_{8}^{3} + 272 \zeta_{8} + 608) q^{52} + (1022 \zeta_{8}^{2} - 3182 \zeta_{8} + 1022) q^{53} + ( - 1080 \zeta_{8}^{3} - 1242 \zeta_{8}^{2} - 1242) q^{54} + (516 \zeta_{8}^{3} + 508 \zeta_{8}^{2} + 516 \zeta_{8}) q^{55} + (832 \zeta_{8}^{3} - 832 \zeta_{8}^{2} + 224 \zeta_{8} + 224) q^{56} + ( - 780 \zeta_{8}^{3} + 1986 \zeta_{8}^{2} - 1704 \zeta_{8} + 1134) q^{57} + (1716 \zeta_{8}^{3} + 1580 \zeta_{8}^{2} + 1580 \zeta_{8} + 1716) q^{58} + ( - 3950 \zeta_{8}^{3} + 217 \zeta_{8}^{2} - 217) q^{59} + ( - 816 \zeta_{8}^{3} + 720 \zeta_{8}^{2} - 240 \zeta_{8} - 912) q^{60} + (2590 \zeta_{8}^{3} + 300 \zeta_{8}^{2} + 300 \zeta_{8} + 2590) q^{61} + (176 \zeta_{8}^{3} - 164 \zeta_{8}^{2} - 164 \zeta_{8} + 176) q^{62} + (1575 \zeta_{8}^{3} + 675 \zeta_{8}^{2} - 3069 \zeta_{8} + 2583) q^{63} - 512 \zeta_{8}^{2} q^{64} + ( - 256 \zeta_{8}^{3} - 256 \zeta_{8}^{2} - 520 \zeta_{8} + 520) q^{65} + (1398 \zeta_{8}^{3} - 42 \zeta_{8}^{2} - 1674 \zeta_{8} - 2310) q^{66} + ( - 1405 \zeta_{8}^{3} + 1405 \zeta_{8} + 5766) q^{67} + ( - 1632 \zeta_{8}^{3} - 1360 \zeta_{8}^{2} + 408 \zeta_{8} - 816) q^{68} + (2415 \zeta_{8}^{3} - 687 \zeta_{8}^{2} + 1113 \zeta_{8} - 1089) q^{69} + (3024 \zeta_{8}^{3} - 2000 \zeta_{8}^{2} + 2000) q^{70} + ( - 2017 \zeta_{8}^{3} - 2017 \zeta_{8}^{2} + 2757 \zeta_{8} - 2757) q^{71} + ( - 1008 \zeta_{8}^{2} - 1152 \zeta_{8} + 1008) q^{72} + ( - 2203 \zeta_{8}^{3} - 2203 \zeta_{8}^{2} + 3762 \zeta_{8} - 3762) q^{73} + (864 \zeta_{8}^{3} + 1336 \zeta_{8}^{2} + 1336 \zeta_{8} + 864) q^{74} + ( - 2976 \zeta_{8}^{3} - 2094 \zeta_{8}^{2} - 675 \zeta_{8} + 606) q^{75} + ( - 1136 \zeta_{8}^{2} + 2080 \zeta_{8} - 1136) q^{76} + (1174 \zeta_{8}^{3} + 398 \zeta_{8}^{2} - 398) q^{77} + ( - 408 \zeta_{8}^{3} + 360 \zeta_{8}^{2} + 1824 \zeta_{8} + 1272) q^{78} + ( - 799 \zeta_{8}^{3} - 4727 \zeta_{8}^{2} + 4727 \zeta_{8} + 799) q^{79} + (896 \zeta_{8}^{3} - 896 \zeta_{8}^{2} + 128 \zeta_{8} + 128) q^{80} + ( - 4536 \zeta_{8}^{3} - 4536 \zeta_{8} + 1377) q^{81} + ( - 2294 \zeta_{8}^{3} + 2294 \zeta_{8}^{2} - 934 \zeta_{8} - 934) q^{82} + (3763 \zeta_{8}^{2} - 7242 \zeta_{8} + 3763) q^{83} + ( - 2040 \zeta_{8}^{3} - 120 \zeta_{8}^{2} + 1464 \zeta_{8} - 2952) q^{84} + ( - 578 \zeta_{8}^{3} - 2890 \zeta_{8}^{2} + 2890 \zeta_{8} - 4046) q^{85} + (4596 \zeta_{8}^{3} + 1932 \zeta_{8}^{2} + 4596 \zeta_{8}) q^{86} + (4842 \zeta_{8}^{3} + 4638 \zeta_{8}^{2} + 7620 \zeta_{8} - 2676) q^{87} + (848 \zeta_{8}^{3} - 848 \zeta_{8}^{2} - 1808 \zeta_{8} - 1808) q^{88} + (6545 \zeta_{8}^{3} - 6545 \zeta_{8} - 932) q^{89} + (1728 \zeta_{8}^{3} + 504 \zeta_{8}^{2} - 3528 \zeta_{8} + 2304) q^{90} + ( - 968 \zeta_{8}^{3} + 740 \zeta_{8}^{2} - 740 \zeta_{8} + 968) q^{91} + (568 \zeta_{8}^{3} + 1768 \zeta_{8}^{2} - 1768 \zeta_{8} - 568) q^{92} + ( - 237 \zeta_{8}^{3} - 747 \zeta_{8}^{2} + 537 \zeta_{8} - 519) q^{93} + ( - 1100 \zeta_{8}^{2} + 2384 \zeta_{8} - 1100) q^{94} + (2224 \zeta_{8}^{3} - 5912 \zeta_{8}^{2} + 5912 \zeta_{8} - 2224) q^{95} + ( - 1536 \zeta_{8}^{3} - 384 \zeta_{8}^{2} - 384) q^{96} + (3150 \zeta_{8}^{3} + 3150 \zeta_{8}^{2} - 3855 \zeta_{8} + 3855) q^{97} + (998 \zeta_{8}^{3} - 998 \zeta_{8} + 7928) q^{98} + (2178 \zeta_{8}^{3} - 3321 \zeta_{8}^{2} - 7137 \zeta_{8} - 5958) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 12 q^{3} - 8 q^{5} + 24 q^{6} - 132 q^{7} - 64 q^{8} - 252 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} + 12 q^{3} - 8 q^{5} + 24 q^{6} - 132 q^{7} - 64 q^{8} - 252 q^{9} - 128 q^{10} + 120 q^{11} - 416 q^{14} + 360 q^{15} - 256 q^{16} - 680 q^{17} - 504 q^{18} - 568 q^{19} - 448 q^{20} - 60 q^{21} - 424 q^{22} + 884 q^{23} - 192 q^{24} - 992 q^{25} + 608 q^{26} - 2484 q^{27} - 608 q^{28} + 3296 q^{29} - 192 q^{30} + 12 q^{31} - 512 q^{32} - 2352 q^{33} - 2176 q^{34} + 2200 q^{37} - 2272 q^{38} + 1632 q^{39} - 768 q^{40} + 1360 q^{41} - 3072 q^{42} + 1932 q^{43} - 2656 q^{44} + 2808 q^{45} + 1200 q^{46} - 2200 q^{47} - 768 q^{48} + 7928 q^{49} + 4080 q^{51} + 2432 q^{52} + 4088 q^{53} - 4968 q^{54} + 896 q^{56} + 4536 q^{57} + 6864 q^{58} - 868 q^{59} - 3648 q^{60} + 10360 q^{61} + 704 q^{62} + 10332 q^{63} + 2080 q^{65} - 9240 q^{66} + 23064 q^{67} - 3264 q^{68} - 4356 q^{69} + 8000 q^{70} - 11028 q^{71} + 4032 q^{72} - 15048 q^{73} + 3456 q^{74} + 2424 q^{75} - 4544 q^{76} - 1592 q^{77} + 5088 q^{78} + 3196 q^{79} + 512 q^{80} + 5508 q^{81} - 3736 q^{82} + 15052 q^{83} - 11808 q^{84} - 16184 q^{85} - 10704 q^{87} - 7232 q^{88} - 3728 q^{89} + 9216 q^{90} + 3872 q^{91} - 2272 q^{92} - 2076 q^{93} - 4400 q^{94} - 8896 q^{95} - 1536 q^{96} + 15420 q^{97} + 31712 q^{98} - 23832 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/102\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(-1\) \(\zeta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
2.00000 2.00000i 3.00000 8.48528i 8.00000i 6.48528 2.68629i −10.9706 22.9706i 3.76955 9.10051i −16.0000 16.0000i −63.0000 50.9117i 7.59798 18.3431i
59.1 2.00000 + 2.00000i 3.00000 8.48528i 8.00000i −10.4853 + 25.3137i 22.9706 10.9706i −69.7696 + 28.8995i −16.0000 + 16.0000i −63.0000 50.9117i −71.5980 + 29.6569i
77.1 2.00000 + 2.00000i 3.00000 + 8.48528i 8.00000i 6.48528 + 2.68629i −10.9706 + 22.9706i 3.76955 + 9.10051i −16.0000 + 16.0000i −63.0000 + 50.9117i 7.59798 + 18.3431i
83.1 2.00000 2.00000i 3.00000 + 8.48528i 8.00000i −10.4853 25.3137i 22.9706 + 10.9706i −69.7696 28.8995i −16.0000 16.0000i −63.0000 + 50.9117i −71.5980 29.6569i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
51.g odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 102.5.g.b yes 4
3.b odd 2 1 102.5.g.a 4
17.d even 8 1 102.5.g.a 4
51.g odd 8 1 inner 102.5.g.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.5.g.a 4 3.b odd 2 1
102.5.g.a 4 17.d even 8 1
102.5.g.b yes 4 1.a even 1 1 trivial
102.5.g.b yes 4 51.g odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 8T_{5}^{3} + 528T_{5}^{2} - 8704T_{5} + 36992 \) acting on \(S_{5}^{\mathrm{new}}(102, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 6 T + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 8 T^{3} + 528 T^{2} + \cdots + 36992 \) Copy content Toggle raw display
$7$ \( T^{4} + 132 T^{3} + 4748 T^{2} + \cdots + 553352 \) Copy content Toggle raw display
$11$ \( T^{4} - 120 T^{3} + 29138 T^{2} + \cdots + 2036162 \) Copy content Toggle raw display
$13$ \( T^{4} + 16176 T^{2} + \cdots + 11999296 \) Copy content Toggle raw display
$17$ \( T^{4} + 680 T^{3} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{4} + 568 T^{3} + \cdots + 743761984 \) Copy content Toggle raw display
$23$ \( T^{4} - 884 T^{3} + \cdots + 11304666248 \) Copy content Toggle raw display
$29$ \( T^{4} - 3296 T^{3} + \cdots + 1077077515808 \) Copy content Toggle raw display
$31$ \( T^{4} - 12 T^{3} + 15524 T^{2} + \cdots + 89940872 \) Copy content Toggle raw display
$37$ \( T^{4} - 2200 T^{3} + \cdots + 50419636352 \) Copy content Toggle raw display
$41$ \( T^{4} - 1360 T^{3} + \cdots + 343796642 \) Copy content Toggle raw display
$43$ \( T^{4} - 1932 T^{3} + \cdots + 23176771979076 \) Copy content Toggle raw display
$47$ \( (T^{2} + 1100 T - 407932)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 4088 T^{3} + \cdots + 64579803256336 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 240508051255684 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 133566436820000 \) Copy content Toggle raw display
$67$ \( (T^{2} - 11532 T + 29298706)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 115185174168968 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 105878518455938 \) Copy content Toggle raw display
$79$ \( T^{4} - 3196 T^{3} + \cdots + 17\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 582074781003076 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1864 T - 84805426)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 748688273201250 \) Copy content Toggle raw display
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