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Properties

Label 325.6.a.j.1.11
Level $325$
Weight $6$
Character 325.1
Self dual yes
Analytic conductor $52.125$
Analytic rank $0$
Dimension $11$
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] = [325,6,Mod(1,325)] 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("325.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [11,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(52.1247414392\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5 x^{10} - 257 x^{9} + 1165 x^{8} + 22234 x^{7} - 90282 x^{6} - 751180 x^{5} + 2564400 x^{4} + \cdots + 44115200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-10.1023\) of defining polynomial
Character \(\chi\) \(=\) 325.1

$q$-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)
 
\(f(q)\) \(=\) \(q+10.1023 q^{2} +24.5176 q^{3} +70.0572 q^{4} +247.685 q^{6} +72.1186 q^{7} +384.466 q^{8} +358.110 q^{9} +127.428 q^{11} +1717.63 q^{12} -169.000 q^{13} +728.566 q^{14} +1642.18 q^{16} -2152.48 q^{17} +3617.75 q^{18} +2726.20 q^{19} +1768.17 q^{21} +1287.32 q^{22} -2658.77 q^{23} +9426.17 q^{24} -1707.29 q^{26} +2822.22 q^{27} +5052.43 q^{28} -5885.60 q^{29} +1366.14 q^{31} +4286.91 q^{32} +3124.23 q^{33} -21745.0 q^{34} +25088.2 q^{36} +481.670 q^{37} +27541.0 q^{38} -4143.47 q^{39} +7385.44 q^{41} +17862.7 q^{42} -10167.0 q^{43} +8927.26 q^{44} -26859.8 q^{46} +16361.7 q^{47} +40262.2 q^{48} -11605.9 q^{49} -52773.4 q^{51} -11839.7 q^{52} +9061.97 q^{53} +28511.0 q^{54} +27727.2 q^{56} +66839.8 q^{57} -59458.3 q^{58} +29585.4 q^{59} +16410.0 q^{61} +13801.2 q^{62} +25826.4 q^{63} -9241.89 q^{64} +31562.0 q^{66} +61557.7 q^{67} -150796. q^{68} -65186.5 q^{69} -5229.46 q^{71} +137681. q^{72} -67851.0 q^{73} +4865.99 q^{74} +190990. q^{76} +9189.95 q^{77} -41858.7 q^{78} -89505.0 q^{79} -17826.8 q^{81} +74610.2 q^{82} -78989.9 q^{83} +123873. q^{84} -102711. q^{86} -144300. q^{87} +48991.9 q^{88} +111252. q^{89} -12188.0 q^{91} -186266. q^{92} +33494.5 q^{93} +165292. q^{94} +105105. q^{96} +71553.7 q^{97} -117247. q^{98} +45633.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 5 q^{2} - 11 q^{3} + 187 q^{4} + 351 q^{6} - 208 q^{7} - 165 q^{8} + 1372 q^{9} + 1276 q^{11} - 1533 q^{12} - 1859 q^{13} + 578 q^{14} + 5707 q^{16} - 2218 q^{17} + 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.1023 1.78586 0.892929 − 0.450198i \(-0.148647\pi\)
0.892929 + 0.450198i \(0.148647\pi\)
\(3\) 24.5176 1.57280 0.786401 − 0.617717i \(-0.211942\pi\)
0.786401 + 0.617717i \(0.211942\pi\)
\(4\) 70.0572 2.18929
\(5\) 0 0
\(6\) 247.685 2.80880
\(7\) 72.1186 0.556291 0.278146 − 0.960539i \(-0.410280\pi\)
0.278146 + 0.960539i \(0.410280\pi\)
\(8\) 384.466 2.12390
\(9\) 358.110 1.47370
\(10\) 0 0
\(11\) 127.428 0.317529 0.158765 − 0.987316i \(-0.449249\pi\)
0.158765 + 0.987316i \(0.449249\pi\)
\(12\) 1717.63 3.44331
\(13\) −169.000 −0.277350
\(14\) 728.566 0.993457
\(15\) 0 0
\(16\) 1642.18 1.60369
\(17\) −2152.48 −1.80641 −0.903204 − 0.429211i \(-0.858792\pi\)
−0.903204 + 0.429211i \(0.858792\pi\)
\(18\) 3617.75 2.63183
\(19\) 2726.20 1.73250 0.866252 − 0.499608i \(-0.166523\pi\)
0.866252 + 0.499608i \(0.166523\pi\)
\(20\) 0 0
\(21\) 1768.17 0.874936
\(22\) 1287.32 0.567062
\(23\) −2658.77 −1.04800 −0.524000 − 0.851719i \(-0.675561\pi\)
−0.524000 + 0.851719i \(0.675561\pi\)
\(24\) 9426.17 3.34047
\(25\) 0 0
\(26\) −1707.29 −0.495308
\(27\) 2822.22 0.745044
\(28\) 5052.43 1.21788
\(29\) −5885.60 −1.29956 −0.649779 − 0.760123i \(-0.725139\pi\)
−0.649779 + 0.760123i \(0.725139\pi\)
\(30\) 0 0
\(31\) 1366.14 0.255324 0.127662 − 0.991818i \(-0.459253\pi\)
0.127662 + 0.991818i \(0.459253\pi\)
\(32\) 4286.91 0.740064
\(33\) 3124.23 0.499411
\(34\) −21745.0 −3.22599
\(35\) 0 0
\(36\) 25088.2 3.22636
\(37\) 481.670 0.0578423 0.0289211 − 0.999582i \(-0.490793\pi\)
0.0289211 + 0.999582i \(0.490793\pi\)
\(38\) 27541.0 3.09400
\(39\) −4143.47 −0.436217
\(40\) 0 0
\(41\) 7385.44 0.686146 0.343073 − 0.939309i \(-0.388532\pi\)
0.343073 + 0.939309i \(0.388532\pi\)
\(42\) 17862.7 1.56251
\(43\) −10167.0 −0.838538 −0.419269 − 0.907862i \(-0.637714\pi\)
−0.419269 + 0.907862i \(0.637714\pi\)
\(44\) 8927.26 0.695163
\(45\) 0 0
\(46\) −26859.8 −1.87158
\(47\) 16361.7 1.08040 0.540200 − 0.841537i \(-0.318349\pi\)
0.540200 + 0.841537i \(0.318349\pi\)
\(48\) 40262.2 2.52229
\(49\) −11605.9 −0.690540
\(50\) 0 0
\(51\) −52773.4 −2.84112
\(52\) −11839.7 −0.607199
\(53\) 9061.97 0.443132 0.221566 − 0.975145i \(-0.428883\pi\)
0.221566 + 0.975145i \(0.428883\pi\)
\(54\) 28511.0 1.33054
\(55\) 0 0
\(56\) 27727.2 1.18151
\(57\) 66839.8 2.72488
\(58\) −59458.3 −2.32082
\(59\) 29585.4 1.10649 0.553245 − 0.833019i \(-0.313389\pi\)
0.553245 + 0.833019i \(0.313389\pi\)
\(60\) 0 0
\(61\) 16410.0 0.564656 0.282328 − 0.959318i \(-0.408893\pi\)
0.282328 + 0.959318i \(0.408893\pi\)
\(62\) 13801.2 0.455972
\(63\) 25826.4 0.819809
\(64\) −9241.89 −0.282040
\(65\) 0 0
\(66\) 31562.0 0.891876
\(67\) 61557.7 1.67531 0.837656 − 0.546198i \(-0.183926\pi\)
0.837656 + 0.546198i \(0.183926\pi\)
\(68\) −150796. −3.95475
\(69\) −65186.5 −1.64829
\(70\) 0 0
\(71\) −5229.46 −0.123115 −0.0615575 − 0.998104i \(-0.519607\pi\)
−0.0615575 + 0.998104i \(0.519607\pi\)
\(72\) 137681. 3.13000
\(73\) −67851.0 −1.49021 −0.745107 − 0.666944i \(-0.767602\pi\)
−0.745107 + 0.666944i \(0.767602\pi\)
\(74\) 4865.99 0.103298
\(75\) 0 0
\(76\) 190990. 3.79295
\(77\) 9189.95 0.176639
\(78\) −41858.7 −0.779021
\(79\) −89505.0 −1.61354 −0.806769 − 0.590867i \(-0.798786\pi\)
−0.806769 + 0.590867i \(0.798786\pi\)
\(80\) 0 0
\(81\) −17826.8 −0.301899
\(82\) 74610.2 1.22536
\(83\) −78989.9 −1.25857 −0.629283 − 0.777176i \(-0.716652\pi\)
−0.629283 + 0.777176i \(0.716652\pi\)
\(84\) 123873. 1.91549
\(85\) 0 0
\(86\) −102711. −1.49751
\(87\) −144300. −2.04395
\(88\) 48991.9 0.674399
\(89\) 111252. 1.48879 0.744394 − 0.667741i \(-0.232738\pi\)
0.744394 + 0.667741i \(0.232738\pi\)
\(90\) 0 0
\(91\) −12188.0 −0.154287
\(92\) −186266. −2.29437
\(93\) 33494.5 0.401574
\(94\) 165292. 1.92944
\(95\) 0 0
\(96\) 105105. 1.16397
\(97\) 71553.7 0.772151 0.386076 − 0.922467i \(-0.373830\pi\)
0.386076 + 0.922467i \(0.373830\pi\)
\(98\) −117247. −1.23321
\(99\) 45633.3 0.467945
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 325.6.a.j.1.11 ✓ 11
5.2 odd 4 325.6.b.i.274.21 22
5.3 odd 4 325.6.b.i.274.2 22
5.4 even 2 325.6.a.k.1.1 yes 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
325.6.a.j.1.11 ✓ 11 1.1 even 1 trivial
325.6.a.k.1.1 yes 11 5.4 even 2
325.6.b.i.274.2 22 5.3 odd 4
325.6.b.i.274.21 22 5.2 odd 4