Properties

Label 105.8.a.c
Level $105$
Weight $8$
Character orbit 105.a
Self dual yes
Analytic conductor $32.800$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [105,8,Mod(1,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 105.a (trivial)

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: yes
Analytic conductor: \(32.8004276758\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 4\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta - 6) q^{2} + 27 q^{3} + ( - 24 \beta + 36) q^{4} + 125 q^{5} + (54 \beta - 162) q^{6} + 343 q^{7} + ( - 40 \beta - 984) q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta - 6) q^{2} + 27 q^{3} + ( - 24 \beta + 36) q^{4} + 125 q^{5} + (54 \beta - 162) q^{6} + 343 q^{7} + ( - 40 \beta - 984) q^{8} + 729 q^{9} + (250 \beta - 750) q^{10} + (95 \beta - 5488) q^{11} + ( - 648 \beta + 972) q^{12} + ( - 136 \beta - 1398) q^{13} + (686 \beta - 2058) q^{14} + 3375 q^{15} + (1344 \beta - 1264) q^{16} + (939 \beta - 4142) q^{17} + (1458 \beta - 4374) q^{18} + ( - 143 \beta - 4048) q^{19} + ( - 3000 \beta + 4500) q^{20} + 9261 q^{21} + ( - 11546 \beta + 39008) q^{22} + ( - 5307 \beta - 45488) q^{23} + ( - 1080 \beta - 26568) q^{24} + 15625 q^{25} + ( - 1980 \beta - 316) q^{26} + 19683 q^{27} + ( - 8232 \beta + 12348) q^{28} + ( - 8967 \beta - 6266) q^{29} + (6750 \beta - 20250) q^{30} + ( - 39296 \beta - 58980) q^{31} + ( - 5472 \beta + 219552) q^{32} + (2565 \beta - 148176) q^{33} + ( - 13918 \beta + 84948) q^{34} + 42875 q^{35} + ( - 17496 \beta + 26244) q^{36} + ( - 53383 \beta - 87106) q^{37} + ( - 7238 \beta + 15136) q^{38} + ( - 3672 \beta - 37746) q^{39} + ( - 5000 \beta - 123000) q^{40} + (53640 \beta - 246350) q^{41} + (18522 \beta - 55566) q^{42} + (68551 \beta - 330588) q^{43} + (135132 \beta - 270528) q^{44} + 91125 q^{45} + ( - 59134 \beta - 66720) q^{46} + (1819 \beta - 837704) q^{47} + (36288 \beta - 34128) q^{48} + 117649 q^{49} + (31250 \beta - 93750) q^{50} + (25353 \beta - 111834) q^{51} + (28656 \beta + 54120) q^{52} + (317207 \beta - 277718) q^{53} + (39366 \beta - 118098) q^{54} + (11875 \beta - 686000) q^{55} + ( - 13720 \beta - 337512) q^{56} + ( - 3861 \beta - 109296) q^{57} + (41270 \beta - 536292) q^{58} + ( - 154552 \beta - 1560588) q^{59} + ( - 81000 \beta + 121500) q^{60} + ( - 379281 \beta - 255794) q^{61} + (117816 \beta - 2161064) q^{62} + 250047 q^{63} + (299904 \beta - 1505728) q^{64} + ( - 17000 \beta - 174750) q^{65} + ( - 311742 \beta + 1053216) q^{66} + (20769 \beta - 126364) q^{67} + (133212 \beta - 870264) q^{68} + ( - 143289 \beta - 1228176) q^{69} + (85750 \beta - 257250) q^{70} + (102490 \beta - 549668) q^{71} + ( - 29160 \beta - 717336) q^{72} + ( - 20024 \beta + 2506294) q^{73} + (146086 \beta - 2893876) q^{74} + 421875 q^{75} + (92004 \beta - 35904) q^{76} + (32585 \beta - 1882384) q^{77} + ( - 53460 \beta - 8532) q^{78} + (30814 \beta + 41824) q^{79} + (168000 \beta - 158000) q^{80} + 531441 q^{81} + ( - 814540 \beta + 4911060) q^{82} + (943328 \beta - 2202092) q^{83} + ( - 222264 \beta + 333396) q^{84} + (117375 \beta - 517750) q^{85} + ( - 1072482 \beta + 6370792) q^{86} + ( - 242109 \beta - 169182) q^{87} + (126040 \beta + 5278592) q^{88} + (60216 \beta - 2190782) q^{89} + (182250 \beta - 546750) q^{90} + ( - 46648 \beta - 479514) q^{91} + (900660 \beta + 2438208) q^{92} + ( - 1060992 \beta - 1592460) q^{93} + ( - 1686322 \beta + 5142640) q^{94} + ( - 17875 \beta - 506000) q^{95} + ( - 147744 \beta + 5927904) q^{96} + (2341208 \beta - 4269914) q^{97} + (235298 \beta - 705894) q^{98} + (69255 \beta - 4000752) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{2} + 54 q^{3} + 72 q^{4} + 250 q^{5} - 324 q^{6} + 686 q^{7} - 1968 q^{8} + 1458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{2} + 54 q^{3} + 72 q^{4} + 250 q^{5} - 324 q^{6} + 686 q^{7} - 1968 q^{8} + 1458 q^{9} - 1500 q^{10} - 10976 q^{11} + 1944 q^{12} - 2796 q^{13} - 4116 q^{14} + 6750 q^{15} - 2528 q^{16} - 8284 q^{17} - 8748 q^{18} - 8096 q^{19} + 9000 q^{20} + 18522 q^{21} + 78016 q^{22} - 90976 q^{23} - 53136 q^{24} + 31250 q^{25} - 632 q^{26} + 39366 q^{27} + 24696 q^{28} - 12532 q^{29} - 40500 q^{30} - 117960 q^{31} + 439104 q^{32} - 296352 q^{33} + 169896 q^{34} + 85750 q^{35} + 52488 q^{36} - 174212 q^{37} + 30272 q^{38} - 75492 q^{39} - 246000 q^{40} - 492700 q^{41} - 111132 q^{42} - 661176 q^{43} - 541056 q^{44} + 182250 q^{45} - 133440 q^{46} - 1675408 q^{47} - 68256 q^{48} + 235298 q^{49} - 187500 q^{50} - 223668 q^{51} + 108240 q^{52} - 555436 q^{53} - 236196 q^{54} - 1372000 q^{55} - 675024 q^{56} - 218592 q^{57} - 1072584 q^{58} - 3121176 q^{59} + 243000 q^{60} - 511588 q^{61} - 4322128 q^{62} + 500094 q^{63} - 3011456 q^{64} - 349500 q^{65} + 2106432 q^{66} - 252728 q^{67} - 1740528 q^{68} - 2456352 q^{69} - 514500 q^{70} - 1099336 q^{71} - 1434672 q^{72} + 5012588 q^{73} - 5787752 q^{74} + 843750 q^{75} - 71808 q^{76} - 3764768 q^{77} - 17064 q^{78} + 83648 q^{79} - 316000 q^{80} + 1062882 q^{81} + 9822120 q^{82} - 4404184 q^{83} + 666792 q^{84} - 1035500 q^{85} + 12741584 q^{86} - 338364 q^{87} + 10557184 q^{88} - 4381564 q^{89} - 1093500 q^{90} - 959028 q^{91} + 4876416 q^{92} - 3184920 q^{93} + 10285280 q^{94} - 1012000 q^{95} + 11855808 q^{96} - 8539828 q^{97} - 1411788 q^{98} - 8001504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−17.3137 27.0000 171.765 125.000 −467.470 343.000 −757.726 729.000 −2164.21
1.2 5.31371 27.0000 −99.7645 125.000 143.470 343.000 −1210.27 729.000 664.214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.8.a.c 2
3.b odd 2 1 315.8.a.d 2
5.b even 2 1 525.8.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.8.a.c 2 1.a even 1 1 trivial
315.8.a.d 2 3.b odd 2 1
525.8.a.f 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 12T_{2} - 92 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(105))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 12T - 92 \) Copy content Toggle raw display
$3$ \( (T - 27)^{2} \) Copy content Toggle raw display
$5$ \( (T - 125)^{2} \) Copy content Toggle raw display
$7$ \( (T - 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 10976 T + 29829344 \) Copy content Toggle raw display
$13$ \( T^{2} + 2796 T + 1362532 \) Copy content Toggle raw display
$17$ \( T^{2} + 8284 T - 11058908 \) Copy content Toggle raw display
$19$ \( T^{2} + 8096 T + 15731936 \) Copy content Toggle raw display
$23$ \( T^{2} + 90976 T + 1167902176 \) Copy content Toggle raw display
$29$ \( T^{2} + 12532 T - 2533764092 \) Copy content Toggle raw display
$31$ \( T^{2} + 117960 T - 45934979312 \) Copy content Toggle raw display
$37$ \( T^{2} + 174212 T - 83604374812 \) Copy content Toggle raw display
$41$ \( T^{2} + 492700 T - 31383664700 \) Copy content Toggle raw display
$43$ \( T^{2} + 661176 T - 41087241488 \) Copy content Toggle raw display
$47$ \( T^{2} + 1675408 T + 701642111264 \) Copy content Toggle raw display
$53$ \( T^{2} + 555436 T - 3142721699644 \) Copy content Toggle raw display
$59$ \( T^{2} + 3121176 T + 1671072643216 \) Copy content Toggle raw display
$61$ \( T^{2} + 511588 T - 4537899892316 \) Copy content Toggle raw display
$67$ \( T^{2} + 252728 T + 2164616944 \) Copy content Toggle raw display
$71$ \( T^{2} + 1099336 T - 33999492976 \) Copy content Toggle raw display
$73$ \( T^{2} - 5012588 T + 6268678876004 \) Copy content Toggle raw display
$79$ \( T^{2} - 83648 T - 28634836096 \) Copy content Toggle raw display
$83$ \( T^{2} + 4404184 T - 23626557722224 \) Copy content Toggle raw display
$89$ \( T^{2} + 4381564 T + 4683494838532 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 157167991209052 \) Copy content Toggle raw display
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