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Properties

Label 325.6.a.j.1.10
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 1.10
Root \(-9.60154\) 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)
 
\(f(q)\) \(=\) \(q+9.60154 q^{2} -25.9222 q^{3} +60.1895 q^{4} -248.893 q^{6} -181.554 q^{7} +270.662 q^{8} +428.963 q^{9} +512.944 q^{11} -1560.25 q^{12} -169.000 q^{13} -1743.19 q^{14} +672.710 q^{16} -1668.15 q^{17} +4118.70 q^{18} -464.071 q^{19} +4706.28 q^{21} +4925.05 q^{22} +2095.04 q^{23} -7016.17 q^{24} -1622.66 q^{26} -4820.57 q^{27} -10927.6 q^{28} +5935.75 q^{29} +8397.76 q^{31} -2202.14 q^{32} -13296.7 q^{33} -16016.8 q^{34} +25819.0 q^{36} +8909.35 q^{37} -4455.79 q^{38} +4380.86 q^{39} +3256.85 q^{41} +45187.5 q^{42} +5951.12 q^{43} +30873.8 q^{44} +20115.6 q^{46} +10741.1 q^{47} -17438.1 q^{48} +16154.7 q^{49} +43242.3 q^{51} -10172.0 q^{52} +39344.1 q^{53} -46284.9 q^{54} -49139.7 q^{56} +12029.8 q^{57} +56992.4 q^{58} -3940.20 q^{59} -41477.5 q^{61} +80631.4 q^{62} -77879.7 q^{63} -42670.7 q^{64} -127668. q^{66} -18589.7 q^{67} -100405. q^{68} -54308.1 q^{69} +70348.8 q^{71} +116104. q^{72} +78602.6 q^{73} +85543.4 q^{74} -27932.2 q^{76} -93126.8 q^{77} +42063.0 q^{78} -71703.3 q^{79} +20722.1 q^{81} +31270.8 q^{82} -208.877 q^{83} +283268. q^{84} +57139.9 q^{86} -153868. q^{87} +138835. q^{88} -43537.5 q^{89} +30682.6 q^{91} +126099. q^{92} -217689. q^{93} +103131. q^{94} +57084.5 q^{96} -15298.0 q^{97} +155110. q^{98} +220034. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 5 q^{2} - 11 q^{3} + 187 q^{4} + 351 q^{6} - 208 q^{7} - 165 q^{8} + 1372 q^{9} + 1276 q^{11} - 1533 q^{12} - 1859 q^{13} + 578 q^{14} + 5707 q^{16} - 2218 q^{17} + 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.60154 1.69733 0.848664 − 0.528933i \(-0.177408\pi\)
0.848664 + 0.528933i \(0.177408\pi\)
\(3\) −25.9222 −1.66291 −0.831456 − 0.555590i \(-0.812492\pi\)
−0.831456 + 0.555590i \(0.812492\pi\)
\(4\) 60.1895 1.88092
\(5\) 0 0
\(6\) −248.893 −2.82251
\(7\) −181.554 −1.40042 −0.700212 − 0.713935i \(-0.746911\pi\)
−0.700212 + 0.713935i \(0.746911\pi\)
\(8\) 270.662 1.49521
\(9\) 428.963 1.76528
\(10\) 0 0
\(11\) 512.944 1.27817 0.639084 − 0.769137i \(-0.279313\pi\)
0.639084 + 0.769137i \(0.279313\pi\)
\(12\) −1560.25 −3.12781
\(13\) −169.000 −0.277350
\(14\) −1743.19 −2.37698
\(15\) 0 0
\(16\) 672.710 0.656943
\(17\) −1668.15 −1.39995 −0.699977 − 0.714165i \(-0.746807\pi\)
−0.699977 + 0.714165i \(0.746807\pi\)
\(18\) 4118.70 2.99626
\(19\) −464.071 −0.294917 −0.147459 − 0.989068i \(-0.547109\pi\)
−0.147459 + 0.989068i \(0.547109\pi\)
\(20\) 0 0
\(21\) 4706.28 2.32878
\(22\) 4925.05 2.16947
\(23\) 2095.04 0.825795 0.412897 − 0.910778i \(-0.364517\pi\)
0.412897 + 0.910778i \(0.364517\pi\)
\(24\) −7016.17 −2.48641
\(25\) 0 0
\(26\) −1622.66 −0.470754
\(27\) −4820.57 −1.27259
\(28\) −10927.6 −2.63409
\(29\) 5935.75 1.31063 0.655316 − 0.755355i \(-0.272535\pi\)
0.655316 + 0.755355i \(0.272535\pi\)
\(30\) 0 0
\(31\) 8397.76 1.56949 0.784746 − 0.619817i \(-0.212793\pi\)
0.784746 + 0.619817i \(0.212793\pi\)
\(32\) −2202.14 −0.380164
\(33\) −13296.7 −2.12548
\(34\) −16016.8 −2.37618
\(35\) 0 0
\(36\) 25819.0 3.32035
\(37\) 8909.35 1.06990 0.534948 − 0.844885i \(-0.320331\pi\)
0.534948 + 0.844885i \(0.320331\pi\)
\(38\) −4455.79 −0.500571
\(39\) 4380.86 0.461209
\(40\) 0 0
\(41\) 3256.85 0.302579 0.151289 − 0.988490i \(-0.451657\pi\)
0.151289 + 0.988490i \(0.451657\pi\)
\(42\) 45187.5 3.95271
\(43\) 5951.12 0.490826 0.245413 − 0.969419i \(-0.421076\pi\)
0.245413 + 0.969419i \(0.421076\pi\)
\(44\) 30873.8 2.40413
\(45\) 0 0
\(46\) 20115.6 1.40164
\(47\) 10741.1 0.709255 0.354628 − 0.935008i \(-0.384608\pi\)
0.354628 + 0.935008i \(0.384608\pi\)
\(48\) −17438.1 −1.09244
\(49\) 16154.7 0.961190
\(50\) 0 0
\(51\) 43242.3 2.32800
\(52\) −10172.0 −0.521674
\(53\) 39344.1 1.92393 0.961967 − 0.273167i \(-0.0880710\pi\)
0.961967 + 0.273167i \(0.0880710\pi\)
\(54\) −46284.9 −2.16001
\(55\) 0 0
\(56\) −49139.7 −2.09393
\(57\) 12029.8 0.490422
\(58\) 56992.4 2.22457
\(59\) −3940.20 −0.147363 −0.0736815 − 0.997282i \(-0.523475\pi\)
−0.0736815 + 0.997282i \(0.523475\pi\)
\(60\) 0 0
\(61\) −41477.5 −1.42721 −0.713605 − 0.700548i \(-0.752939\pi\)
−0.713605 + 0.700548i \(0.752939\pi\)
\(62\) 80631.4 2.66394
\(63\) −77879.7 −2.47214
\(64\) −42670.7 −1.30221
\(65\) 0 0
\(66\) −127668. −3.60764
\(67\) −18589.7 −0.505923 −0.252961 − 0.967476i \(-0.581405\pi\)
−0.252961 + 0.967476i \(0.581405\pi\)
\(68\) −100405. −2.63320
\(69\) −54308.1 −1.37322
\(70\) 0 0
\(71\) 70348.8 1.65619 0.828096 − 0.560586i \(-0.189424\pi\)
0.828096 + 0.560586i \(0.189424\pi\)
\(72\) 116104. 2.63947
\(73\) 78602.6 1.72635 0.863176 − 0.504902i \(-0.168471\pi\)
0.863176 + 0.504902i \(0.168471\pi\)
\(74\) 85543.4 1.81596
\(75\) 0 0
\(76\) −27932.2 −0.554716
\(77\) −93126.8 −1.78998
\(78\) 42063.0 0.782823
\(79\) −71703.3 −1.29262 −0.646310 − 0.763075i \(-0.723689\pi\)
−0.646310 + 0.763075i \(0.723689\pi\)
\(80\) 0 0
\(81\) 20722.1 0.350931
\(82\) 31270.8 0.513575
\(83\) −208.877 −0.00332808 −0.00166404 − 0.999999i \(-0.500530\pi\)
−0.00166404 + 0.999999i \(0.500530\pi\)
\(84\) 283268. 4.38026
\(85\) 0 0
\(86\) 57139.9 0.833093
\(87\) −153868. −2.17947
\(88\) 138835. 1.91113
\(89\) −43537.5 −0.582624 −0.291312 − 0.956628i \(-0.594092\pi\)
−0.291312 + 0.956628i \(0.594092\pi\)
\(90\) 0 0
\(91\) 30682.6 0.388408
\(92\) 126099. 1.55325
\(93\) −217689. −2.60993
\(94\) 103131. 1.20384
\(95\) 0 0
\(96\) 57084.5 0.632179
\(97\) −15298.0 −0.165084 −0.0825419 − 0.996588i \(-0.526304\pi\)
−0.0825419 + 0.996588i \(0.526304\pi\)
\(98\) 155110. 1.63145
\(99\) 220034. 2.25632
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.1.10 ✓ 11
5.2 odd 4 325.6.b.i.274.20 22
5.3 odd 4 325.6.b.i.274.3 22
5.4 even 2 325.6.a.k.1.2 yes 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
325.6.a.j.1.10 ✓ 11 1.1 even 1 trivial
325.6.a.k.1.2 yes 11 5.4 even 2
325.6.b.i.274.3 22 5.3 odd 4
325.6.b.i.274.20 22 5.2 odd 4