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Properties

Label 8325.2.a.ch.1.5
Level $8325$
Weight $2$
Character 8325.1
Self dual yes
Analytic conductor $66.475$
Analytic rank $0$
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] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,2,0,10,0,0,-11,6,0,0,5,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.973904.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 185)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(3.29298\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.72362 q^{2} +5.41809 q^{4} -3.82710 q^{7} +9.30957 q^{8} +4.41809 q^{11} +3.67583 q^{13} -10.4236 q^{14} +14.5195 q^{16} -2.28688 q^{17} -2.39037 q^{19} +12.0332 q^{22} -0.265251 q^{23} +10.0116 q^{26} -20.7356 q^{28} +6.58595 q^{29} +2.34076 q^{31} +20.9265 q^{32} -6.22860 q^{34} -1.00000 q^{37} -6.51044 q^{38} +4.41809 q^{41} -7.71249 q^{43} +23.9376 q^{44} -0.722443 q^{46} +10.9285 q^{47} +7.64669 q^{49} +19.9160 q^{52} -0.109574 q^{53} -35.6286 q^{56} +17.9376 q^{58} +2.00504 q^{59} +3.96271 q^{61} +6.37534 q^{62} +27.9567 q^{64} -6.80664 q^{67} -12.3905 q^{68} +5.79485 q^{71} +0.140654 q^{73} -2.72362 q^{74} -12.9512 q^{76} -16.9085 q^{77} -6.62418 q^{79} +12.0332 q^{82} +13.9904 q^{83} -21.0059 q^{86} +41.1305 q^{88} -14.8139 q^{89} -14.0678 q^{91} -1.43716 q^{92} +29.7651 q^{94} +8.94394 q^{97} +20.8266 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 10 q^{4} - 11 q^{7} + 6 q^{8} + 5 q^{11} - 4 q^{13} + 8 q^{14} + 16 q^{16} - 4 q^{19} + 8 q^{22} + 4 q^{23} + 4 q^{26} - 28 q^{28} + 4 q^{29} + 8 q^{31} + 14 q^{32} - 32 q^{34} - 5 q^{37}+ \cdots - 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.72362 1.92589 0.962944 − 0.269701i \(-0.0869249\pi\)
0.962944 + 0.269701i \(0.0869249\pi\)
\(3\) 0 0
\(4\) 5.41809 2.70905
\(5\) 0 0
\(6\) 0 0
\(7\) −3.82710 −1.44651 −0.723254 − 0.690582i \(-0.757354\pi\)
−0.723254 + 0.690582i \(0.757354\pi\)
\(8\) 9.30957 3.29143
\(9\) 0 0
\(10\) 0 0
\(11\) 4.41809 1.33210 0.666052 − 0.745905i \(-0.267983\pi\)
0.666052 + 0.745905i \(0.267983\pi\)
\(12\) 0 0
\(13\) 3.67583 1.01949 0.509746 − 0.860325i \(-0.329739\pi\)
0.509746 + 0.860325i \(0.329739\pi\)
\(14\) −10.4236 −2.78581
\(15\) 0 0
\(16\) 14.5195 3.62988
\(17\) −2.28688 −0.554651 −0.277325 − 0.960776i \(-0.589448\pi\)
−0.277325 + 0.960776i \(0.589448\pi\)
\(18\) 0 0
\(19\) −2.39037 −0.548387 −0.274194 − 0.961674i \(-0.588411\pi\)
−0.274194 + 0.961674i \(0.588411\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 12.0332 2.56548
\(23\) −0.265251 −0.0553087 −0.0276544 − 0.999618i \(-0.508804\pi\)
−0.0276544 + 0.999618i \(0.508804\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 10.0116 1.96343
\(27\) 0 0
\(28\) −20.7356 −3.91865
\(29\) 6.58595 1.22298 0.611490 − 0.791252i \(-0.290570\pi\)
0.611490 + 0.791252i \(0.290570\pi\)
\(30\) 0 0
\(31\) 2.34076 0.420413 0.210207 − 0.977657i \(-0.432586\pi\)
0.210207 + 0.977657i \(0.432586\pi\)
\(32\) 20.9265 3.69931
\(33\) 0 0
\(34\) −6.22860 −1.06820
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −6.51044 −1.05613
\(39\) 0 0
\(40\) 0 0
\(41\) 4.41809 0.689990 0.344995 − 0.938605i \(-0.387881\pi\)
0.344995 + 0.938605i \(0.387881\pi\)
\(42\) 0 0
\(43\) −7.71249 −1.17614 −0.588072 − 0.808809i \(-0.700113\pi\)
−0.588072 + 0.808809i \(0.700113\pi\)
\(44\) 23.9376 3.60873
\(45\) 0 0
\(46\) −0.722443 −0.106518
\(47\) 10.9285 1.59409 0.797045 − 0.603920i \(-0.206395\pi\)
0.797045 + 0.603920i \(0.206395\pi\)
\(48\) 0 0
\(49\) 7.64669 1.09238
\(50\) 0 0
\(51\) 0 0
\(52\) 19.9160 2.76185
\(53\) −0.109574 −0.0150512 −0.00752559 − 0.999972i \(-0.502395\pi\)
−0.00752559 + 0.999972i \(0.502395\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −35.6286 −4.76108
\(57\) 0 0
\(58\) 17.9376 2.35532
\(59\) 2.00504 0.261034 0.130517 − 0.991446i \(-0.458336\pi\)
0.130517 + 0.991446i \(0.458336\pi\)
\(60\) 0 0
\(61\) 3.96271 0.507374 0.253687 − 0.967286i \(-0.418357\pi\)
0.253687 + 0.967286i \(0.418357\pi\)
\(62\) 6.37534 0.809669
\(63\) 0 0
\(64\) 27.9567 3.49458
\(65\) 0 0
\(66\) 0 0
\(67\) −6.80664 −0.831563 −0.415782 − 0.909464i \(-0.636492\pi\)
−0.415782 + 0.909464i \(0.636492\pi\)
\(68\) −12.3905 −1.50257
\(69\) 0 0
\(70\) 0 0
\(71\) 5.79485 0.687722 0.343861 − 0.939020i \(-0.388265\pi\)
0.343861 + 0.939020i \(0.388265\pi\)
\(72\) 0 0
\(73\) 0.140654 0.0164623 0.00823116 − 0.999966i \(-0.497380\pi\)
0.00823116 + 0.999966i \(0.497380\pi\)
\(74\) −2.72362 −0.316614
\(75\) 0 0
\(76\) −12.9512 −1.48561
\(77\) −16.9085 −1.92690
\(78\) 0 0
\(79\) −6.62418 −0.745278 −0.372639 − 0.927976i \(-0.621547\pi\)
−0.372639 + 0.927976i \(0.621547\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 12.0332 1.32884
\(83\) 13.9904 1.53565 0.767825 − 0.640660i \(-0.221339\pi\)
0.767825 + 0.640660i \(0.221339\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −21.0059 −2.26512
\(87\) 0 0
\(88\) 41.1305 4.38453
\(89\) −14.8139 −1.57027 −0.785136 − 0.619323i \(-0.787407\pi\)
−0.785136 + 0.619323i \(0.787407\pi\)
\(90\) 0 0
\(91\) −14.0678 −1.47470
\(92\) −1.43716 −0.149834
\(93\) 0 0
\(94\) 29.7651 3.07004
\(95\) 0 0
\(96\) 0 0
\(97\) 8.94394 0.908119 0.454060 − 0.890971i \(-0.349975\pi\)
0.454060 + 0.890971i \(0.349975\pi\)
\(98\) 20.8266 2.10381
\(99\) 0 0
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 8325.2.a.ch.1.5 5
3.2 odd 2 925.2.a.f.1.1 5
5.4 even 2 1665.2.a.p.1.1 5
15.2 even 4 925.2.b.f.149.1 10
15.8 even 4 925.2.b.f.149.10 10
15.14 odd 2 185.2.a.e.1.5 ✓ 5
60.59 even 2 2960.2.a.w.1.5 5
105.104 even 2 9065.2.a.k.1.5 5
555.554 odd 2 6845.2.a.f.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.5 ✓ 5 15.14 odd 2
925.2.a.f.1.1 5 3.2 odd 2
925.2.b.f.149.1 10 15.2 even 4
925.2.b.f.149.10 10 15.8 even 4
1665.2.a.p.1.1 5 5.4 even 2
2960.2.a.w.1.5 5 60.59 even 2
6845.2.a.f.1.1 5 555.554 odd 2
8325.2.a.ch.1.5 5 1.1 even 1 trivial
9065.2.a.k.1.5 5 105.104 even 2