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Properties

Label 349.8.a.b
Level $349$
Weight $8$
Character orbit 349.a
Self dual yes
Analytic conductor $109.022$
Analytic rank $0$
Dimension $105$
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] = [349,8,Mod(1,349)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(349, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("349.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 349 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 349.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: \(109.022373894\)
Analytic rank: \(0\)
Dimension: \(105\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 105 q + 55 q^{2} + 242 q^{3} + 7077 q^{4} + 1179 q^{5} + 952 q^{6} + 2389 q^{7} + 9825 q^{8} + 86775 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 105 q + 55 q^{2} + 242 q^{3} + 7077 q^{4} + 1179 q^{5} + 952 q^{6} + 2389 q^{7} + 9825 q^{8} + 86775 q^{9} + 1996 q^{10} + 56501 q^{11} + 36525 q^{12} + 13378 q^{13} + 76334 q^{14} + 75148 q^{15} + 506877 q^{16} + 56033 q^{17} + 125455 q^{18} + 141251 q^{19} + 257432 q^{20} + 177761 q^{21} + 153469 q^{22} + 311568 q^{23} + 200436 q^{24} + 1873896 q^{25} + 387394 q^{26} + 550325 q^{27} + 413914 q^{28} + 549499 q^{29} + 89591 q^{30} + 603934 q^{31} + 1486975 q^{32} + 593863 q^{33} + 495599 q^{34} + 1476524 q^{35} + 7189283 q^{36} + 1161863 q^{37} + 1396931 q^{38} + 1958568 q^{39} - 619765 q^{40} + 2265531 q^{41} + 1541060 q^{42} + 2291535 q^{43} + 7329660 q^{44} + 2062005 q^{45} + 882657 q^{46} + 2728408 q^{47} + 4641642 q^{48} + 14390722 q^{49} + 4354809 q^{50} + 9476017 q^{51} + 5014300 q^{52} + 6386991 q^{53} + 3061138 q^{54} + 3051280 q^{55} + 14229935 q^{56} + 3915187 q^{57} + 5244954 q^{58} + 21673100 q^{59} + 9660389 q^{60} + 5340602 q^{61} + 4271484 q^{62} + 8421595 q^{63} + 39916269 q^{64} + 12186585 q^{65} + 15032438 q^{66} + 13263389 q^{67} + 11532600 q^{68} + 8611329 q^{69} + 11036229 q^{70} + 34770654 q^{71} + 10793456 q^{72} - 1441674 q^{73} + 21893250 q^{74} + 26235834 q^{75} + 9416539 q^{76} + 14371542 q^{77} + 11182233 q^{78} + 19942931 q^{79} + 31141374 q^{80} + 91476277 q^{81} - 2972333 q^{82} + 42250631 q^{83} + 45047987 q^{84} + 6652035 q^{85} + 45381333 q^{86} + 29092375 q^{87} - 25128180 q^{88} + 21238349 q^{89} - 146610632 q^{90} + 3848314 q^{91} - 15626873 q^{92} - 16511111 q^{93} - 73196885 q^{94} + 29561501 q^{95} - 103223896 q^{96} - 23655978 q^{97} - 8520520 q^{98} + 85212951 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −22.5965 44.2174 382.600 −316.710 −999.156 −190.257 −5753.05 −231.823 7156.52
1.2 −21.4650 87.5560 332.745 350.487 −1879.39 −1269.05 −4394.84 5479.06 −7523.18
1.3 −21.3402 −46.3237 327.403 192.630 988.555 975.258 −4255.30 −41.1190 −4110.77
1.4 −21.1164 62.6298 317.900 182.827 −1322.51 1403.93 −4010.00 1735.49 −3860.64
1.5 −20.4004 29.1124 288.177 −108.273 −593.904 −783.465 −3267.67 −1339.47 2208.82
1.6 −20.1630 53.4843 278.545 −61.4289 −1078.40 −1344.13 −3035.43 673.568 1238.59
1.7 −19.9668 −79.4547 270.671 −201.131 1586.45 −1015.62 −2848.68 4126.04 4015.94
1.8 −19.6485 −75.0519 258.063 322.973 1474.66 998.059 −2555.54 3445.79 −6345.93
1.9 −19.3536 −84.7860 246.560 521.263 1640.91 −1071.34 −2294.56 5001.67 −10088.3
1.10 −19.0738 −30.1292 235.811 −232.744 574.680 −1466.34 −2056.37 −1279.23 4439.32
1.11 −19.0435 −42.2096 234.654 −258.296 803.818 1219.82 −2031.06 −405.347 4918.85
1.12 −18.8765 −22.3284 228.322 −410.727 421.482 −166.833 −1893.73 −1688.44 7753.08
1.13 −18.7677 4.49146 224.225 −262.136 −84.2943 1095.70 −1805.92 −2166.83 4919.68
1.14 −18.5232 −46.5003 215.109 151.317 861.335 464.698 −1613.54 −24.7215 −2802.88
1.15 −18.0935 62.3902 199.374 −184.184 −1128.86 −647.619 −1291.41 1705.54 3332.53
1.16 −17.6422 80.4555 183.247 275.435 −1419.41 1377.51 −974.685 4286.08 −4859.28
1.17 −17.0234 26.4784 161.797 268.561 −450.754 −3.34475 −575.349 −1485.89 −4571.84
1.18 −16.1723 −43.0772 133.544 467.796 696.659 920.615 −89.6618 −331.355 −7565.34
1.19 −15.4329 −2.55578 110.174 84.7679 39.4430 231.456 275.113 −2180.47 −1308.21
1.20 −15.0374 46.1284 98.1237 386.440 −693.651 −1639.85 449.262 −59.1727 −5811.06
See next 80 embeddings (of 105 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.105
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(349\) \(1\)

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 349.8.a.b ✓ 105
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
349.8.a.b ✓ 105 1.a even 1 1 trivial