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Properties

Label 49.10.a.e.1.2
Level $49$
Weight $10$
Character 49.1
Self dual yes
Analytic conductor $25.237$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,10,Mod(1,49)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("49.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,18,-161] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.2367559720\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 429x^{3} + 184x^{2} + 37472x + 27600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(11.7982\) of defining polynomial
Character \(\chi\) \(=\) 49.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-19.5965 q^{2} -209.955 q^{3} -127.978 q^{4} -1967.58 q^{5} +4114.37 q^{6} +12541.3 q^{8} +24397.9 q^{9} +38557.6 q^{10} +31797.8 q^{11} +26869.6 q^{12} -100188. q^{13} +413103. q^{15} -180241. q^{16} +338258. q^{17} -478113. q^{18} +261517. q^{19} +251808. q^{20} -623124. q^{22} -544675. q^{23} -2.63311e6 q^{24} +1.91825e6 q^{25} +1.96333e6 q^{26} -989914. q^{27} +2.32128e6 q^{29} -8.09535e6 q^{30} -5.07071e6 q^{31} -2.88907e6 q^{32} -6.67608e6 q^{33} -6.62867e6 q^{34} -3.12240e6 q^{36} +5.86469e6 q^{37} -5.12481e6 q^{38} +2.10349e7 q^{39} -2.46761e7 q^{40} -2.81312e7 q^{41} +3.88522e7 q^{43} -4.06942e6 q^{44} -4.80048e7 q^{45} +1.06737e7 q^{46} +2.53481e7 q^{47} +3.78423e7 q^{48} -3.75909e7 q^{50} -7.10189e7 q^{51} +1.28219e7 q^{52} -4.52978e7 q^{53} +1.93988e7 q^{54} -6.25647e7 q^{55} -5.49067e7 q^{57} -4.54890e7 q^{58} +1.88911e7 q^{59} -5.28682e7 q^{60} +1.01651e8 q^{61} +9.93680e7 q^{62} +1.48899e8 q^{64} +1.97128e8 q^{65} +1.30828e8 q^{66} +1.31099e8 q^{67} -4.32897e7 q^{68} +1.14357e8 q^{69} +9.51959e7 q^{71} +3.05982e8 q^{72} +2.38111e8 q^{73} -1.14927e8 q^{74} -4.02745e8 q^{75} -3.34685e7 q^{76} -4.12210e8 q^{78} +1.88520e8 q^{79} +3.54638e8 q^{80} -2.72387e8 q^{81} +5.51273e8 q^{82} -1.76014e8 q^{83} -6.65551e8 q^{85} -7.61366e8 q^{86} -4.87364e8 q^{87} +3.98786e8 q^{88} -4.83034e8 q^{89} +9.40725e8 q^{90} +6.97066e7 q^{92} +1.06462e9 q^{93} -4.96734e8 q^{94} -5.14556e8 q^{95} +6.06574e8 q^{96} -2.28821e7 q^{97} +7.75799e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 18 q^{2} - 161 q^{3} + 940 q^{4} - 1533 q^{5} - 4354 q^{6} + 17136 q^{8} + 35734 q^{9} - 4298 q^{10} - 42213 q^{11} - 135604 q^{12} - 159838 q^{13} + 75697 q^{15} - 322064 q^{16} - 324681 q^{17} + 1012868 q^{18}+ \cdots - 950388590 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.5965 −0.866050 −0.433025 − 0.901382i \(-0.642554\pi\)
−0.433025 + 0.901382i \(0.642554\pi\)
\(3\) −209.955 −1.49651 −0.748255 − 0.663411i \(-0.769108\pi\)
−0.748255 + 0.663411i \(0.769108\pi\)
\(4\) −127.978 −0.249958
\(5\) −1967.58 −1.40789 −0.703943 − 0.710256i \(-0.748579\pi\)
−0.703943 + 0.710256i \(0.748579\pi\)
\(6\) 4114.37 1.29605
\(7\) 0 0
\(8\) 12541.3 1.08253
\(9\) 24397.9 1.23954
\(10\) 38557.6 1.21930
\(11\) 31797.8 0.654831 0.327416 − 0.944880i \(-0.393822\pi\)
0.327416 + 0.944880i \(0.393822\pi\)
\(12\) 26869.6 0.374064
\(13\) −100188. −0.972904 −0.486452 − 0.873707i \(-0.661709\pi\)
−0.486452 + 0.873707i \(0.661709\pi\)
\(14\) 0 0
\(15\) 413103. 2.10692
\(16\) −180241. −0.687564
\(17\) 338258. 0.982264 0.491132 − 0.871085i \(-0.336583\pi\)
0.491132 + 0.871085i \(0.336583\pi\)
\(18\) −478113. −1.07350
\(19\) 261517. 0.460372 0.230186 − 0.973147i \(-0.426067\pi\)
0.230186 + 0.973147i \(0.426067\pi\)
\(20\) 251808. 0.351912
\(21\) 0 0
\(22\) −623124. −0.567117
\(23\) −544675. −0.405847 −0.202923 − 0.979195i \(-0.565044\pi\)
−0.202923 + 0.979195i \(0.565044\pi\)
\(24\) −2.63311e6 −1.62001
\(25\) 1.91825e6 0.982144
\(26\) 1.96333e6 0.842583
\(27\) −989914. −0.358476
\(28\) 0 0
\(29\) 2.32128e6 0.609449 0.304724 − 0.952441i \(-0.401436\pi\)
0.304724 + 0.952441i \(0.401436\pi\)
\(30\) −8.09535e6 −1.82469
\(31\) −5.07071e6 −0.986146 −0.493073 − 0.869988i \(-0.664126\pi\)
−0.493073 + 0.869988i \(0.664126\pi\)
\(32\) −2.88907e6 −0.487061
\(33\) −6.67608e6 −0.979962
\(34\) −6.62867e6 −0.850690
\(35\) 0 0
\(36\) −3.12240e6 −0.309833
\(37\) 5.86469e6 0.514443 0.257222 − 0.966352i \(-0.417193\pi\)
0.257222 + 0.966352i \(0.417193\pi\)
\(38\) −5.12481e6 −0.398705
\(39\) 2.10349e7 1.45596
\(40\) −2.46761e7 −1.52407
\(41\) −2.81312e7 −1.55475 −0.777376 − 0.629036i \(-0.783450\pi\)
−0.777376 + 0.629036i \(0.783450\pi\)
\(42\) 0 0
\(43\) 3.88522e7 1.73304 0.866518 − 0.499147i \(-0.166353\pi\)
0.866518 + 0.499147i \(0.166353\pi\)
\(44\) −4.06942e6 −0.163680
\(45\) −4.80048e7 −1.74513
\(46\) 1.06737e7 0.351484
\(47\) 2.53481e7 0.757714 0.378857 − 0.925455i \(-0.376317\pi\)
0.378857 + 0.925455i \(0.376317\pi\)
\(48\) 3.78423e7 1.02895
\(49\) 0 0
\(50\) −3.75909e7 −0.850586
\(51\) −7.10189e7 −1.46997
\(52\) 1.28219e7 0.243185
\(53\) −4.52978e7 −0.788563 −0.394281 − 0.918990i \(-0.629006\pi\)
−0.394281 + 0.918990i \(0.629006\pi\)
\(54\) 1.93988e7 0.310458
\(55\) −6.25647e7 −0.921928
\(56\) 0 0
\(57\) −5.49067e7 −0.688951
\(58\) −4.54890e7 −0.527813
\(59\) 1.88911e7 0.202966 0.101483 − 0.994837i \(-0.467641\pi\)
0.101483 + 0.994837i \(0.467641\pi\)
\(60\) −5.28682e7 −0.526640
\(61\) 1.01651e8 0.939998 0.469999 − 0.882667i \(-0.344254\pi\)
0.469999 + 0.882667i \(0.344254\pi\)
\(62\) 9.93680e7 0.854051
\(63\) 0 0
\(64\) 1.48899e8 1.10938
\(65\) 1.97128e8 1.36974
\(66\) 1.30828e8 0.848696
\(67\) 1.31099e8 0.794810 0.397405 − 0.917643i \(-0.369911\pi\)
0.397405 + 0.917643i \(0.369911\pi\)
\(68\) −4.32897e7 −0.245524
\(69\) 1.14357e8 0.607354
\(70\) 0 0
\(71\) 9.51959e7 0.444586 0.222293 − 0.974980i \(-0.428646\pi\)
0.222293 + 0.974980i \(0.428646\pi\)
\(72\) 3.05982e8 1.34184
\(73\) 2.38111e8 0.981358 0.490679 − 0.871341i \(-0.336749\pi\)
0.490679 + 0.871341i \(0.336749\pi\)
\(74\) −1.14927e8 −0.445534
\(75\) −4.02745e8 −1.46979
\(76\) −3.34685e7 −0.115074
\(77\) 0 0
\(78\) −4.12210e8 −1.26093
\(79\) 1.88520e8 0.544547 0.272274 − 0.962220i \(-0.412224\pi\)
0.272274 + 0.962220i \(0.412224\pi\)
\(80\) 3.54638e8 0.968011
\(81\) −2.72387e8 −0.703078
\(82\) 5.51273e8 1.34649
\(83\) −1.76014e8 −0.407095 −0.203547 − 0.979065i \(-0.565247\pi\)
−0.203547 + 0.979065i \(0.565247\pi\)
\(84\) 0 0
\(85\) −6.65551e8 −1.38292
\(86\) −7.61366e8 −1.50089
\(87\) −4.87364e8 −0.912046
\(88\) 3.98786e8 0.708872
\(89\) −4.83034e8 −0.816061 −0.408030 − 0.912968i \(-0.633784\pi\)
−0.408030 + 0.912968i \(0.633784\pi\)
\(90\) 9.40725e8 1.51137
\(91\) 0 0
\(92\) 6.97066e7 0.101445
\(93\) 1.06462e9 1.47578
\(94\) −4.96734e8 −0.656218
\(95\) −5.14556e8 −0.648151
\(96\) 6.06574e8 0.728892
\(97\) −2.28821e7 −0.0262436 −0.0131218 − 0.999914i \(-0.504177\pi\)
−0.0131218 + 0.999914i \(0.504177\pi\)
\(98\) 0 0
\(99\) 7.75799e8 0.811691
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 49.10.a.e.1.2 5
7.2 even 3 7.10.c.a.4.4 yes 10
7.3 odd 6 49.10.c.g.30.4 10
7.4 even 3 7.10.c.a.2.4 ✓ 10
7.5 odd 6 49.10.c.g.18.4 10
7.6 odd 2 49.10.a.f.1.2 5
21.2 odd 6 63.10.e.b.46.2 10
21.11 odd 6 63.10.e.b.37.2 10
28.11 odd 6 112.10.i.c.65.2 10
28.23 odd 6 112.10.i.c.81.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.10.c.a.2.4 ✓ 10 7.4 even 3
7.10.c.a.4.4 yes 10 7.2 even 3
49.10.a.e.1.2 5 1.1 even 1 trivial
49.10.a.f.1.2 5 7.6 odd 2
49.10.c.g.18.4 10 7.5 odd 6
49.10.c.g.30.4 10 7.3 odd 6
63.10.e.b.37.2 10 21.11 odd 6
63.10.e.b.46.2 10 21.2 odd 6
112.10.i.c.65.2 10 28.11 odd 6
112.10.i.c.81.2 10 28.23 odd 6