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Properties

Label 349.6.a.a
Level $349$
Weight $6$
Character orbit 349.a
Self dual yes
Analytic conductor $55.974$
Analytic rank $1$
Dimension $69$
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] = [349,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("349.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 349 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(55.9739531147\)
Analytic rank: \(1\)
Dimension: \(69\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 69 q - 25 q^{2} - 82 q^{3} + 981 q^{4} - 279 q^{5} - 212 q^{6} - 283 q^{7} - 1095 q^{8} + 4563 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 69 q - 25 q^{2} - 82 q^{3} + 981 q^{4} - 279 q^{5} - 212 q^{6} - 283 q^{7} - 1095 q^{8} + 4563 q^{9} - 364 q^{10} - 4883 q^{11} - 2619 q^{12} - 946 q^{13} - 3814 q^{14} - 6140 q^{15} + 11325 q^{16} - 4813 q^{17} - 5381 q^{18} - 8285 q^{19} - 14464 q^{20} - 10837 q^{21} - 7387 q^{22} - 13344 q^{23} - 17124 q^{24} + 36798 q^{25} - 21922 q^{26} - 25135 q^{27} - 16126 q^{28} - 21647 q^{29} - 17545 q^{30} - 15706 q^{31} - 41185 q^{32} - 7367 q^{33} - 23045 q^{34} - 47068 q^{35} + 21467 q^{36} - 39395 q^{37} - 47625 q^{38} - 35664 q^{39} - 8117 q^{40} - 50359 q^{41} - 18076 q^{42} - 42561 q^{43} - 132020 q^{44} - 45513 q^{45} - 54591 q^{46} - 46008 q^{47} - 88134 q^{48} + 95516 q^{49} - 134743 q^{50} - 129035 q^{51} - 61804 q^{52} - 127891 q^{53} - 41150 q^{54} - 50872 q^{55} - 192017 q^{56} - 93347 q^{57} - 64662 q^{58} - 304312 q^{59} - 308491 q^{60} - 69410 q^{61} - 116844 q^{62} - 147581 q^{63} + 31533 q^{64} - 161985 q^{65} - 128842 q^{66} - 185243 q^{67} - 191856 q^{68} - 200157 q^{69} - 193707 q^{70} - 471018 q^{71} - 241480 q^{72} - 14946 q^{73} - 305102 q^{74} - 323802 q^{75} - 216541 q^{76} - 200070 q^{77} - 50703 q^{78} - 359837 q^{79} - 461778 q^{80} + 54889 q^{81} - 41785 q^{82} - 566313 q^{83} - 677749 q^{84} - 13167 q^{85} - 666451 q^{86} - 256649 q^{87} - 474516 q^{88} - 592577 q^{89} - 1458380 q^{90} - 910030 q^{91} - 1229521 q^{92} - 822221 q^{93} - 1102801 q^{94} - 1073323 q^{95} - 2081128 q^{96} - 487026 q^{97} - 1236400 q^{98} - 1640457 q^{99}+O(q^{100}) \) 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 −11.1867 22.9118 93.1414 31.0196 −256.306 62.4090 −683.968 281.950 −347.006
1.2 −10.9485 3.01009 87.8686 −82.1234 −32.9558 −213.251 −611.675 −233.939 899.125
1.3 −10.5373 0.154512 79.0339 −8.82886 −1.62813 −86.9287 −495.608 −242.976 93.0320
1.4 −10.4706 −11.6981 77.6333 −41.2166 122.486 172.870 −477.808 −106.155 431.563
1.5 −10.0511 10.1236 69.0252 62.3208 −101.753 −128.541 −372.145 −140.513 −626.395
1.6 −9.97202 12.4034 67.4412 −100.037 −123.687 192.357 −353.420 −89.1546 997.569
1.7 −9.85168 −27.9086 65.0555 −18.4372 274.946 −66.5317 −325.652 535.890 181.637
1.8 −9.37993 −26.8531 55.9832 6.62032 251.881 253.525 −224.960 478.091 −62.0982
1.9 −9.20187 19.7510 52.6743 50.0626 −181.746 −203.519 −190.242 147.103 −460.669
1.10 −8.81130 −27.5402 45.6391 84.1763 242.665 −7.81426 −120.178 515.460 −741.703
1.11 −8.26630 16.0390 36.3318 −2.15746 −132.583 24.7625 −35.8077 14.2483 17.8342
1.12 −8.23553 −13.6565 35.8239 −28.3609 112.468 −246.150 −31.4920 −56.5001 233.567
1.13 −7.68312 −12.9535 27.0303 52.5524 99.5229 25.9390 38.1825 −75.2081 −403.766
1.14 −7.44815 −17.4326 23.4749 −98.1529 129.841 −39.8687 63.4960 60.8959 731.057
1.15 −7.17821 −15.7011 19.5267 9.19092 112.706 166.694 89.5363 3.52593 −65.9744
1.16 −7.07131 6.33238 18.0035 52.3120 −44.7782 185.730 98.9739 −202.901 −369.915
1.17 −6.74086 26.9814 13.4392 −50.1156 −181.878 200.373 125.116 484.997 337.823
1.18 −6.54900 19.2735 10.8894 24.8478 −126.222 77.9832 138.254 128.468 −162.728
1.19 −6.38864 −3.77266 8.81472 −98.5503 24.1022 182.515 148.122 −228.767 629.602
1.20 −6.32566 7.61116 8.01394 −64.2728 −48.1456 −61.7164 151.728 −185.070 406.568
See all 69 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.69
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 349.6.a.a ✓ 69
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
349.6.a.a ✓ 69 1.a even 1 1 trivial