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Properties

Label 349.10.a.a
Level $349$
Weight $10$
Character orbit 349.a
Self dual yes
Analytic conductor $179.748$
Analytic rank $1$
Dimension $127$
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] = [349,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("349.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 349 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(179.747506821\)
Analytic rank: \(1\)
Dimension: \(127\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 127 q - 113 q^{2} - 649 q^{3} + 30805 q^{4} - 5454 q^{5} - 9296 q^{6} - 16883 q^{7} - 90711 q^{8} + 711372 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 127 q - 113 q^{2} - 649 q^{3} + 30805 q^{4} - 5454 q^{5} - 9296 q^{6} - 16883 q^{7} - 90711 q^{8} + 711372 q^{9} - 8644 q^{10} - 604048 q^{11} - 381819 q^{12} - 166417 q^{13} - 968242 q^{14} - 1089110 q^{15} + 7191901 q^{16} - 850141 q^{17} - 1529633 q^{18} - 2757449 q^{19} - 4289584 q^{20} - 3548344 q^{21} - 3152371 q^{22} - 8537958 q^{23} - 5337396 q^{24} + 42856883 q^{25} - 9353918 q^{26} - 12954418 q^{27} - 10463102 q^{28} - 23513090 q^{29} - 2361145 q^{30} - 24560734 q^{31} - 53784065 q^{32} - 18132830 q^{33} - 30348441 q^{34} - 72705388 q^{35} + 98121179 q^{36} - 23713320 q^{37} - 41921341 q^{38} - 80830134 q^{39} + 2772683 q^{40} - 92485521 q^{41} - 88230868 q^{42} - 122513944 q^{43} - 321871572 q^{44} - 79054728 q^{45} - 114434351 q^{46} - 182580641 q^{47} - 302470086 q^{48} + 628726892 q^{49} - 193666143 q^{50} - 422181056 q^{51} - 236372172 q^{52} - 297882939 q^{53} - 252040310 q^{54} - 62446622 q^{55} - 912219793 q^{56} - 366162620 q^{57} - 294998126 q^{58} - 1137790108 q^{59} - 750381931 q^{60} - 286392259 q^{61} - 221795604 q^{62} - 326510501 q^{63} + 1508718541 q^{64} - 603802770 q^{65} - 516666298 q^{66} - 1304719355 q^{67} - 1075437744 q^{68} - 394540584 q^{69} - 444957387 q^{70} - 2161015251 q^{71} - 819252856 q^{72} - 351543645 q^{73} - 1385001582 q^{74} - 1347830007 q^{75} - 1074081853 q^{76} - 1087879488 q^{77} - 949494903 q^{78} - 956383587 q^{79} - 1939195698 q^{80} + 1911750271 q^{81} - 823220733 q^{82} - 2947168459 q^{83} - 3015449701 q^{84} - 1314493792 q^{85} - 4999885155 q^{86} - 1496635184 q^{87} - 6179137460 q^{88} - 3850368406 q^{89} - 16817058380 q^{90} - 7660329398 q^{91} - 11198408993 q^{92} - 5103913382 q^{93} - 9272885877 q^{94} - 4596108328 q^{95} - 6837801832 q^{96} - 1552261742 q^{97} - 7832752468 q^{98} - 10894547238 q^{99}+O(q^{100}) \) 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 −44.0889 −213.701 1431.83 2455.95 9421.84 −2651.96 −40554.4 25985.0 −108280.
1.2 −44.0656 −32.2954 1429.77 1556.61 1423.11 12056.5 −40442.2 −18640.0 −68592.8
1.3 −44.0055 −37.7159 1424.49 −2467.41 1659.71 11365.0 −40154.5 −18260.5 108580.
1.4 −42.7048 67.5072 1311.70 −2227.39 −2882.88 947.682 −34150.9 −15125.8 95120.3
1.5 −42.5752 230.770 1300.65 −317.854 −9825.10 5635.84 −33577.0 33571.9 13532.7
1.6 −41.9831 −116.645 1250.58 114.004 4897.14 −723.794 −31008.0 −6076.86 −4786.23
1.7 −41.5484 203.579 1214.27 −2056.57 −8458.39 −8943.85 −29178.1 21761.5 85447.3
1.8 −41.2257 −171.625 1187.56 1580.19 7075.38 351.897 −27850.5 9772.23 −65144.3
1.9 −41.1838 −182.357 1184.11 −1383.52 7510.18 6543.23 −27679.9 13571.2 56978.8
1.10 −40.9353 266.797 1163.70 562.794 −10921.4 −1964.27 −26677.4 51497.7 −23038.1
1.11 −39.5341 −126.410 1050.95 −1337.98 4997.52 5495.38 −21306.8 −3703.41 52895.8
1.12 −38.8884 121.505 1000.31 2003.34 −4725.13 4120.99 −18989.5 −4919.57 −77906.6
1.13 −37.6980 −177.880 909.137 393.427 6705.72 −10315.9 −14971.3 11958.3 −14831.4
1.14 −36.9335 −42.9512 852.084 −1750.64 1586.34 −7640.54 −12560.5 −17838.2 64657.3
1.15 −36.5426 101.174 823.360 −877.563 −3697.17 −1114.09 −11377.9 −9446.77 32068.4
1.16 −36.3641 −26.6322 810.344 1845.01 968.453 3396.47 −10849.0 −18973.7 −67092.1
1.17 −36.2114 −63.8105 799.265 2504.43 2310.67 −4870.43 −10402.3 −15611.2 −90688.9
1.18 −35.6295 −30.8843 757.461 62.0064 1100.39 −3765.21 −8745.65 −18729.2 −2209.26
1.19 −34.9564 −265.383 709.950 323.634 9276.83 −9189.74 −6919.63 50745.0 −11313.1
1.20 −34.9527 189.406 709.692 1683.21 −6620.24 8702.35 −6909.86 16191.5 −58832.8
See next 80 embeddings (of 127 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.127
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 349.10.a.a ✓ 127
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
349.10.a.a ✓ 127 1.a even 1 1 trivial