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Properties

Label 325.6.a.k.1.7
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 1.7
Root \(2.89241\) 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)
 
\(f(q)\) \(=\) \(q+2.89241 q^{2} -3.45730 q^{3} -23.6340 q^{4} -9.99993 q^{6} -148.288 q^{7} -160.916 q^{8} -231.047 q^{9} -712.658 q^{11} +81.7096 q^{12} +169.000 q^{13} -428.910 q^{14} +290.850 q^{16} -1131.92 q^{17} -668.283 q^{18} +1400.39 q^{19} +512.676 q^{21} -2061.30 q^{22} +897.571 q^{23} +556.336 q^{24} +488.818 q^{26} +1638.92 q^{27} +3504.63 q^{28} +3238.07 q^{29} -7976.53 q^{31} +5990.58 q^{32} +2463.87 q^{33} -3273.98 q^{34} +5460.56 q^{36} +4978.35 q^{37} +4050.50 q^{38} -584.284 q^{39} -15559.9 q^{41} +1482.87 q^{42} -2308.61 q^{43} +16842.9 q^{44} +2596.15 q^{46} -7602.52 q^{47} -1005.56 q^{48} +5182.35 q^{49} +3913.39 q^{51} -3994.14 q^{52} -14138.1 q^{53} +4740.44 q^{54} +23862.0 q^{56} -4841.56 q^{57} +9365.83 q^{58} +49808.7 q^{59} -2516.69 q^{61} -23071.4 q^{62} +34261.5 q^{63} +8020.03 q^{64} +7126.54 q^{66} -38549.3 q^{67} +26751.8 q^{68} -3103.17 q^{69} +68008.5 q^{71} +37179.2 q^{72} -21305.2 q^{73} +14399.4 q^{74} -33096.7 q^{76} +105679. q^{77} -1689.99 q^{78} +3984.60 q^{79} +50478.2 q^{81} -45005.8 q^{82} -13876.6 q^{83} -12116.6 q^{84} -6677.44 q^{86} -11195.0 q^{87} +114678. q^{88} +89289.8 q^{89} -25060.7 q^{91} -21213.1 q^{92} +27577.2 q^{93} -21989.6 q^{94} -20711.2 q^{96} -147549. q^{97} +14989.5 q^{98} +164658. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 5 q^{2} + 11 q^{3} + 187 q^{4} + 351 q^{6} + 208 q^{7} + 165 q^{8} + 1372 q^{9} + 1276 q^{11} + 1533 q^{12} + 1859 q^{13} + 578 q^{14} + 5707 q^{16} + 2218 q^{17} - 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.89241 0.511311 0.255656 − 0.966768i \(-0.417709\pi\)
0.255656 + 0.966768i \(0.417709\pi\)
\(3\) −3.45730 −0.221786 −0.110893 − 0.993832i \(-0.535371\pi\)
−0.110893 + 0.993832i \(0.535371\pi\)
\(4\) −23.6340 −0.738561
\(5\) 0 0
\(6\) −9.99993 −0.113402
\(7\) −148.288 −1.14383 −0.571914 − 0.820313i \(-0.693799\pi\)
−0.571914 + 0.820313i \(0.693799\pi\)
\(8\) −160.916 −0.888945
\(9\) −231.047 −0.950811
\(10\) 0 0
\(11\) −712.658 −1.77582 −0.887912 − 0.460014i \(-0.847844\pi\)
−0.887912 + 0.460014i \(0.847844\pi\)
\(12\) 81.7096 0.163802
\(13\) 169.000 0.277350
\(14\) −428.910 −0.584852
\(15\) 0 0
\(16\) 290.850 0.284033
\(17\) −1131.92 −0.949935 −0.474968 − 0.880003i \(-0.657540\pi\)
−0.474968 + 0.880003i \(0.657540\pi\)
\(18\) −668.283 −0.486160
\(19\) 1400.39 0.889947 0.444974 − 0.895544i \(-0.353213\pi\)
0.444974 + 0.895544i \(0.353213\pi\)
\(20\) 0 0
\(21\) 512.676 0.253685
\(22\) −2061.30 −0.907998
\(23\) 897.571 0.353793 0.176896 − 0.984229i \(-0.443394\pi\)
0.176896 + 0.984229i \(0.443394\pi\)
\(24\) 556.336 0.197156
\(25\) 0 0
\(26\) 488.818 0.141812
\(27\) 1638.92 0.432662
\(28\) 3504.63 0.844787
\(29\) 3238.07 0.714975 0.357488 − 0.933918i \(-0.383633\pi\)
0.357488 + 0.933918i \(0.383633\pi\)
\(30\) 0 0
\(31\) −7976.53 −1.49077 −0.745383 − 0.666636i \(-0.767733\pi\)
−0.745383 + 0.666636i \(0.767733\pi\)
\(32\) 5990.58 1.03417
\(33\) 2463.87 0.393852
\(34\) −3273.98 −0.485712
\(35\) 0 0
\(36\) 5460.56 0.702232
\(37\) 4978.35 0.597835 0.298917 − 0.954279i \(-0.403374\pi\)
0.298917 + 0.954279i \(0.403374\pi\)
\(38\) 4050.50 0.455040
\(39\) −584.284 −0.0615123
\(40\) 0 0
\(41\) −15559.9 −1.44560 −0.722800 − 0.691057i \(-0.757145\pi\)
−0.722800 + 0.691057i \(0.757145\pi\)
\(42\) 1482.87 0.129712
\(43\) −2308.61 −0.190405 −0.0952026 − 0.995458i \(-0.530350\pi\)
−0.0952026 + 0.995458i \(0.530350\pi\)
\(44\) 16842.9 1.31155
\(45\) 0 0
\(46\) 2596.15 0.180898
\(47\) −7602.52 −0.502011 −0.251005 − 0.967986i \(-0.580761\pi\)
−0.251005 + 0.967986i \(0.580761\pi\)
\(48\) −1005.56 −0.0629946
\(49\) 5182.35 0.308344
\(50\) 0 0
\(51\) 3913.39 0.210682
\(52\) −3994.14 −0.204840
\(53\) −14138.1 −0.691357 −0.345678 − 0.938353i \(-0.612351\pi\)
−0.345678 + 0.938353i \(0.612351\pi\)
\(54\) 4740.44 0.221225
\(55\) 0 0
\(56\) 23862.0 1.01680
\(57\) −4841.56 −0.197378
\(58\) 9365.83 0.365575
\(59\) 49808.7 1.86284 0.931419 − 0.363948i \(-0.118572\pi\)
0.931419 + 0.363948i \(0.118572\pi\)
\(60\) 0 0
\(61\) −2516.69 −0.0865973 −0.0432986 − 0.999062i \(-0.513787\pi\)
−0.0432986 + 0.999062i \(0.513787\pi\)
\(62\) −23071.4 −0.762245
\(63\) 34261.5 1.08757
\(64\) 8020.03 0.244752
\(65\) 0 0
\(66\) 7126.54 0.201381
\(67\) −38549.3 −1.04913 −0.524566 − 0.851370i \(-0.675772\pi\)
−0.524566 + 0.851370i \(0.675772\pi\)
\(68\) 26751.8 0.701585
\(69\) −3103.17 −0.0784663
\(70\) 0 0
\(71\) 68008.5 1.60110 0.800548 − 0.599269i \(-0.204542\pi\)
0.800548 + 0.599269i \(0.204542\pi\)
\(72\) 37179.2 0.845219
\(73\) −21305.2 −0.467927 −0.233963 − 0.972245i \(-0.575170\pi\)
−0.233963 + 0.972245i \(0.575170\pi\)
\(74\) 14399.4 0.305680
\(75\) 0 0
\(76\) −33096.7 −0.657280
\(77\) 105679. 2.03124
\(78\) −1689.99 −0.0314519
\(79\) 3984.60 0.0718318 0.0359159 − 0.999355i \(-0.488565\pi\)
0.0359159 + 0.999355i \(0.488565\pi\)
\(80\) 0 0
\(81\) 50478.2 0.854853
\(82\) −45005.8 −0.739152
\(83\) −13876.6 −0.221100 −0.110550 − 0.993871i \(-0.535261\pi\)
−0.110550 + 0.993871i \(0.535261\pi\)
\(84\) −12116.6 −0.187362
\(85\) 0 0
\(86\) −6677.44 −0.0973563
\(87\) −11195.0 −0.158571
\(88\) 114678. 1.57861
\(89\) 89289.8 1.19489 0.597443 − 0.801911i \(-0.296183\pi\)
0.597443 + 0.801911i \(0.296183\pi\)
\(90\) 0 0
\(91\) −25060.7 −0.317241
\(92\) −21213.1 −0.261298
\(93\) 27577.2 0.330631
\(94\) −21989.6 −0.256684
\(95\) 0 0
\(96\) −20711.2 −0.229365
\(97\) −147549. −1.59223 −0.796116 − 0.605145i \(-0.793115\pi\)
−0.796116 + 0.605145i \(0.793115\pi\)
\(98\) 14989.5 0.157660
\(99\) 164658. 1.68847
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.k.1.7 yes 11
5.2 odd 4 325.6.b.i.274.14 22
5.3 odd 4 325.6.b.i.274.9 22
5.4 even 2 325.6.a.j.1.5 ✓ 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
325.6.a.j.1.5 ✓ 11 5.4 even 2
325.6.a.k.1.7 yes 11 1.1 even 1 trivial
325.6.b.i.274.9 22 5.3 odd 4
325.6.b.i.274.14 22 5.2 odd 4