Properties

Label 210.8.a.n
Level $210$
Weight $8$
Character orbit 210.a
Self dual yes
Analytic conductor $65.601$
Analytic rank $0$
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] = [210,8,Mod(1,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, 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("210.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 210.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: \(65.6008553517\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{536866}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 536866 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\beta = 4\sqrt{536866}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} - 27 q^{3} + 64 q^{4} + 125 q^{5} - 216 q^{6} - 343 q^{7} + 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} - 27 q^{3} + 64 q^{4} + 125 q^{5} - 216 q^{6} - 343 q^{7} + 512 q^{8} + 729 q^{9} + 1000 q^{10} + (2 \beta + 1368) q^{11} - 1728 q^{12} + ( - 3 \beta - 4046) q^{13} - 2744 q^{14} - 3375 q^{15} + 4096 q^{16} + (\beta + 11642) q^{17} + 5832 q^{18} + (4 \beta + 13568) q^{19} + 8000 q^{20} + 9261 q^{21} + (16 \beta + 10944) q^{22} + ( - 23 \beta - 29232) q^{23} - 13824 q^{24} + 15625 q^{25} + ( - 24 \beta - 32368) q^{26} - 19683 q^{27} - 21952 q^{28} + ( - 29 \beta - 8922) q^{29} - 27000 q^{30} + (83 \beta + 78188) q^{31} + 32768 q^{32} + ( - 54 \beta - 36936) q^{33} + (8 \beta + 93136) q^{34} - 42875 q^{35} + 46656 q^{36} + (25 \beta + 112150) q^{37} + (32 \beta + 108544) q^{38} + (81 \beta + 109242) q^{39} + 64000 q^{40} + (80 \beta + 324674) q^{41} + 74088 q^{42} + ( - 87 \beta + 406700) q^{43} + (128 \beta + 87552) q^{44} + 91125 q^{45} + ( - 184 \beta - 233856) q^{46} + ( - 115 \beta - 375880) q^{47} - 110592 q^{48} + 117649 q^{49} + 125000 q^{50} + ( - 27 \beta - 314334) q^{51} + ( - 192 \beta - 258944) q^{52} + ( - 300 \beta - 169958) q^{53} - 157464 q^{54} + (250 \beta + 171000) q^{55} - 175616 q^{56} + ( - 108 \beta - 366336) q^{57} + ( - 232 \beta - 71376) q^{58} + (518 \beta + 699044) q^{59} - 216000 q^{60} + ( - 43 \beta + 1801918) q^{61} + (664 \beta + 625504) q^{62} - 250047 q^{63} + 262144 q^{64} + ( - 375 \beta - 505750) q^{65} + ( - 432 \beta - 295488) q^{66} + (45 \beta + 264012) q^{67} + (64 \beta + 745088) q^{68} + (621 \beta + 789264) q^{69} - 343000 q^{70} + (167 \beta + 4076980) q^{71} + 373248 q^{72} + (115 \beta + 5345510) q^{73} + (200 \beta + 897200) q^{74} - 421875 q^{75} + (256 \beta + 868352) q^{76} + ( - 686 \beta - 469224) q^{77} + (648 \beta + 873936) q^{78} + ( - 1164 \beta + 1073416) q^{79} + 512000 q^{80} + 531441 q^{81} + (640 \beta + 2597392) q^{82} + (802 \beta + 5238604) q^{83} + 592704 q^{84} + (125 \beta + 1455250) q^{85} + ( - 696 \beta + 3253600) q^{86} + (783 \beta + 240894) q^{87} + (1024 \beta + 700416) q^{88} + ( - 3600 \beta + 1609586) q^{89} + 729000 q^{90} + (1029 \beta + 1387778) q^{91} + ( - 1472 \beta - 1870848) q^{92} + ( - 2241 \beta - 2111076) q^{93} + ( - 920 \beta - 3007040) q^{94} + (500 \beta + 1696000) q^{95} - 884736 q^{96} + ( - 2139 \beta + 6570374) q^{97} + 941192 q^{98} + (1458 \beta + 997272) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{2} - 54 q^{3} + 128 q^{4} + 250 q^{5} - 432 q^{6} - 686 q^{7} + 1024 q^{8} + 1458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{2} - 54 q^{3} + 128 q^{4} + 250 q^{5} - 432 q^{6} - 686 q^{7} + 1024 q^{8} + 1458 q^{9} + 2000 q^{10} + 2736 q^{11} - 3456 q^{12} - 8092 q^{13} - 5488 q^{14} - 6750 q^{15} + 8192 q^{16} + 23284 q^{17} + 11664 q^{18} + 27136 q^{19} + 16000 q^{20} + 18522 q^{21} + 21888 q^{22} - 58464 q^{23} - 27648 q^{24} + 31250 q^{25} - 64736 q^{26} - 39366 q^{27} - 43904 q^{28} - 17844 q^{29} - 54000 q^{30} + 156376 q^{31} + 65536 q^{32} - 73872 q^{33} + 186272 q^{34} - 85750 q^{35} + 93312 q^{36} + 224300 q^{37} + 217088 q^{38} + 218484 q^{39} + 128000 q^{40} + 649348 q^{41} + 148176 q^{42} + 813400 q^{43} + 175104 q^{44} + 182250 q^{45} - 467712 q^{46} - 751760 q^{47} - 221184 q^{48} + 235298 q^{49} + 250000 q^{50} - 628668 q^{51} - 517888 q^{52} - 339916 q^{53} - 314928 q^{54} + 342000 q^{55} - 351232 q^{56} - 732672 q^{57} - 142752 q^{58} + 1398088 q^{59} - 432000 q^{60} + 3603836 q^{61} + 1251008 q^{62} - 500094 q^{63} + 524288 q^{64} - 1011500 q^{65} - 590976 q^{66} + 528024 q^{67} + 1490176 q^{68} + 1578528 q^{69} - 686000 q^{70} + 8153960 q^{71} + 746496 q^{72} + 10691020 q^{73} + 1794400 q^{74} - 843750 q^{75} + 1736704 q^{76} - 938448 q^{77} + 1747872 q^{78} + 2146832 q^{79} + 1024000 q^{80} + 1062882 q^{81} + 5194784 q^{82} + 10477208 q^{83} + 1185408 q^{84} + 2910500 q^{85} + 6507200 q^{86} + 481788 q^{87} + 1400832 q^{88} + 3219172 q^{89} + 1458000 q^{90} + 2775556 q^{91} - 3741696 q^{92} - 4222152 q^{93} - 6014080 q^{94} + 3392000 q^{95} - 1769472 q^{96} + 13140748 q^{97} + 1882384 q^{98} + 1994544 q^{99}+O(q^{100}) \) 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
−732.711
732.711
8.00000 −27.0000 64.0000 125.000 −216.000 −343.000 512.000 729.000 1000.00
1.2 8.00000 −27.0000 64.0000 125.000 −216.000 −343.000 512.000 729.000 1000.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

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 210.8.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.8.a.n 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - 2736T_{11} - 32488000 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(210))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 8)^{2} \) 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
$11$ \( T^{2} - 2736 T - 32488000 \) Copy content Toggle raw display
$13$ \( T^{2} + 8092 T - 60938588 \) Copy content Toggle raw display
$17$ \( T^{2} - 23284 T + 126946308 \) Copy content Toggle raw display
$19$ \( T^{2} - 27136 T + 46652928 \) Copy content Toggle raw display
$23$ \( T^{2} + 58464 T - 3689524000 \) Copy content Toggle raw display
$29$ \( T^{2} + 17844 T - 7144466812 \) Copy content Toggle raw display
$31$ \( T^{2} - 156376 T - 53062154640 \) Copy content Toggle raw display
$37$ \( T^{2} - 224300 T + 7208962500 \) Copy content Toggle raw display
$41$ \( T^{2} - 649348 T + 50438127876 \) Copy content Toggle raw display
$43$ \( T^{2} - 813400 T + 100388269936 \) Copy content Toggle raw display
$47$ \( T^{2} + 751760 T + 27684928800 \) Copy content Toggle raw display
$53$ \( T^{2} + 339916 T - 744201318236 \) Copy content Toggle raw display
$59$ \( T^{2} - 1398088 T - 1816202007408 \) Copy content Toggle raw display
$61$ \( T^{2} - 3603836 T + 3231025834980 \) Copy content Toggle raw display
$67$ \( T^{2} - 528024 T + 52307877744 \) Copy content Toggle raw display
$71$ \( T^{2} - 8153960 T + 16382203426416 \) Copy content Toggle raw display
$73$ \( T^{2} - 10691020 T + 28460876314500 \) Copy content Toggle raw display
$79$ \( T^{2} - 2146832 T - 10486139625920 \) Copy content Toggle raw display
$83$ \( T^{2} - 10477208 T + 21917942130192 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 108733766668604 \) Copy content Toggle raw display
$97$ \( T^{2} - 13140748 T + 3868465956100 \) Copy content Toggle raw display
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