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Properties

Label 45.14.a.h.1.4
Level $45$
Weight $14$
Character 45.1
Self dual yes
Analytic conductor $48.254$
Analytic rank $1$
Dimension $4$
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] = [45,14,Mod(1,45)] 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("45.1"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,15] 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: \(48.2539180284\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 24774x^{2} - 86616x + 52534656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{4}\cdot 5^{3} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-146.991\) of defining polynomial
Character \(\chi\) \(=\) 45.1

$q$-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)
 
\(f(q)\) \(=\) \(q+150.991 q^{2} +14606.2 q^{4} -15625.0 q^{5} -242904. q^{7} +968489. q^{8} -2.35923e6 q^{10} -6.86310e6 q^{11} +2.01315e7 q^{13} -3.66762e7 q^{14} +2.65787e7 q^{16} -1.41199e8 q^{17} -2.08405e8 q^{19} -2.28222e8 q^{20} -1.03626e9 q^{22} -2.35142e8 q^{23} +2.44141e8 q^{25} +3.03967e9 q^{26} -3.54790e9 q^{28} +2.00980e9 q^{29} -3.36358e9 q^{31} -3.92072e9 q^{32} -2.13197e10 q^{34} +3.79537e9 q^{35} -9.62526e9 q^{37} -3.14672e10 q^{38} -1.51326e10 q^{40} +2.34132e10 q^{41} +8.82167e9 q^{43} -1.00244e11 q^{44} -3.55042e10 q^{46} -1.04323e11 q^{47} -3.78868e10 q^{49} +3.68630e10 q^{50} +2.94045e11 q^{52} +1.44937e11 q^{53} +1.07236e11 q^{55} -2.35249e11 q^{56} +3.03461e11 q^{58} +1.25803e11 q^{59} +7.31437e11 q^{61} -5.07869e11 q^{62} -8.09725e11 q^{64} -3.14554e11 q^{65} +1.13018e12 q^{67} -2.06238e12 q^{68} +5.73066e11 q^{70} -2.11613e12 q^{71} -2.39041e12 q^{73} -1.45333e12 q^{74} -3.04401e12 q^{76} +1.66707e12 q^{77} -1.88805e12 q^{79} -4.15292e11 q^{80} +3.53517e12 q^{82} +6.14850e11 q^{83} +2.20623e12 q^{85} +1.33199e12 q^{86} -6.64683e12 q^{88} +2.90766e12 q^{89} -4.89001e12 q^{91} -3.43453e12 q^{92} -1.57518e13 q^{94} +3.25633e12 q^{95} +1.60768e13 q^{97} -5.72056e12 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 15 q^{2} + 16837 q^{4} - 62500 q^{5} + 343040 q^{7} + 14865 q^{8} - 234375 q^{10} - 12697800 q^{11} + 34336040 q^{13} - 26944650 q^{14} + 66562801 q^{16} - 84377280 q^{17} - 131821144 q^{19} - 263078125 q^{20}+ \cdots + 17650752985395 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 150.991 1.66823 0.834114 − 0.551592i \(-0.185979\pi\)
0.834114 + 0.551592i \(0.185979\pi\)
\(3\) 0 0
\(4\) 14606.2 1.78299
\(5\) −15625.0 −0.447214
\(6\) 0 0
\(7\) −242904. −0.780363 −0.390182 − 0.920738i \(-0.627588\pi\)
−0.390182 + 0.920738i \(0.627588\pi\)
\(8\) 968489. 1.30620
\(9\) 0 0
\(10\) −2.35923e6 −0.746054
\(11\) −6.86310e6 −1.16807 −0.584034 − 0.811730i \(-0.698526\pi\)
−0.584034 + 0.811730i \(0.698526\pi\)
\(12\) 0 0
\(13\) 2.01315e7 1.15676 0.578380 − 0.815767i \(-0.303685\pi\)
0.578380 + 0.815767i \(0.303685\pi\)
\(14\) −3.66762e7 −1.30182
\(15\) 0 0
\(16\) 2.65787e7 0.396054
\(17\) −1.41199e8 −1.41877 −0.709387 − 0.704819i \(-0.751028\pi\)
−0.709387 + 0.704819i \(0.751028\pi\)
\(18\) 0 0
\(19\) −2.08405e8 −1.01627 −0.508135 − 0.861277i \(-0.669665\pi\)
−0.508135 + 0.861277i \(0.669665\pi\)
\(20\) −2.28222e8 −0.797376
\(21\) 0 0
\(22\) −1.03626e9 −1.94860
\(23\) −2.35142e8 −0.331206 −0.165603 − 0.986192i \(-0.552957\pi\)
−0.165603 + 0.986192i \(0.552957\pi\)
\(24\) 0 0
\(25\) 2.44141e8 0.200000
\(26\) 3.03967e9 1.92974
\(27\) 0 0
\(28\) −3.54790e9 −1.39138
\(29\) 2.00980e9 0.627430 0.313715 − 0.949517i \(-0.398426\pi\)
0.313715 + 0.949517i \(0.398426\pi\)
\(30\) 0 0
\(31\) −3.36358e9 −0.680692 −0.340346 − 0.940300i \(-0.610544\pi\)
−0.340346 + 0.940300i \(0.610544\pi\)
\(32\) −3.92072e9 −0.645492
\(33\) 0 0
\(34\) −2.13197e10 −2.36684
\(35\) 3.79537e9 0.348989
\(36\) 0 0
\(37\) −9.62526e9 −0.616739 −0.308369 − 0.951267i \(-0.599783\pi\)
−0.308369 + 0.951267i \(0.599783\pi\)
\(38\) −3.14672e10 −1.69537
\(39\) 0 0
\(40\) −1.51326e10 −0.584150
\(41\) 2.34132e10 0.769777 0.384888 − 0.922963i \(-0.374240\pi\)
0.384888 + 0.922963i \(0.374240\pi\)
\(42\) 0 0
\(43\) 8.82167e9 0.212817 0.106408 − 0.994323i \(-0.466065\pi\)
0.106408 + 0.994323i \(0.466065\pi\)
\(44\) −1.00244e11 −2.08265
\(45\) 0 0
\(46\) −3.55042e10 −0.552527
\(47\) −1.04323e11 −1.41170 −0.705851 − 0.708360i \(-0.749435\pi\)
−0.705851 + 0.708360i \(0.749435\pi\)
\(48\) 0 0
\(49\) −3.78868e10 −0.391033
\(50\) 3.68630e10 0.333646
\(51\) 0 0
\(52\) 2.94045e11 2.06249
\(53\) 1.44937e11 0.898225 0.449112 − 0.893475i \(-0.351740\pi\)
0.449112 + 0.893475i \(0.351740\pi\)
\(54\) 0 0
\(55\) 1.07236e11 0.522375
\(56\) −2.35249e11 −1.01931
\(57\) 0 0
\(58\) 3.03461e11 1.04670
\(59\) 1.25803e11 0.388286 0.194143 − 0.980973i \(-0.437807\pi\)
0.194143 + 0.980973i \(0.437807\pi\)
\(60\) 0 0
\(61\) 7.31437e11 1.81775 0.908873 − 0.417073i \(-0.136944\pi\)
0.908873 + 0.417073i \(0.136944\pi\)
\(62\) −5.07869e11 −1.13555
\(63\) 0 0
\(64\) −8.09725e11 −1.47288
\(65\) −3.14554e11 −0.517319
\(66\) 0 0
\(67\) 1.13018e12 1.52637 0.763186 − 0.646179i \(-0.223634\pi\)
0.763186 + 0.646179i \(0.223634\pi\)
\(68\) −2.06238e12 −2.52966
\(69\) 0 0
\(70\) 5.73066e11 0.582193
\(71\) −2.11613e12 −1.96049 −0.980244 − 0.197793i \(-0.936623\pi\)
−0.980244 + 0.197793i \(0.936623\pi\)
\(72\) 0 0
\(73\) −2.39041e12 −1.84873 −0.924365 − 0.381509i \(-0.875405\pi\)
−0.924365 + 0.381509i \(0.875405\pi\)
\(74\) −1.45333e12 −1.02886
\(75\) 0 0
\(76\) −3.04401e12 −1.81200
\(77\) 1.66707e12 0.911516
\(78\) 0 0
\(79\) −1.88805e12 −0.873850 −0.436925 − 0.899498i \(-0.643932\pi\)
−0.436925 + 0.899498i \(0.643932\pi\)
\(80\) −4.15292e11 −0.177121
\(81\) 0 0
\(82\) 3.53517e12 1.28416
\(83\) 6.14850e11 0.206425 0.103212 − 0.994659i \(-0.467088\pi\)
0.103212 + 0.994659i \(0.467088\pi\)
\(84\) 0 0
\(85\) 2.20623e12 0.634495
\(86\) 1.33199e12 0.355027
\(87\) 0 0
\(88\) −6.64683e12 −1.52573
\(89\) 2.90766e12 0.620167 0.310084 − 0.950709i \(-0.399643\pi\)
0.310084 + 0.950709i \(0.399643\pi\)
\(90\) 0 0
\(91\) −4.89001e12 −0.902693
\(92\) −3.43453e12 −0.590536
\(93\) 0 0
\(94\) −1.57518e13 −2.35504
\(95\) 3.25633e12 0.454490
\(96\) 0 0
\(97\) 1.60768e13 1.95967 0.979836 − 0.199804i \(-0.0640306\pi\)
0.979836 + 0.199804i \(0.0640306\pi\)
\(98\) −5.72056e12 −0.652333
\(99\) 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 45.14.a.h.1.4 yes 4
3.2 odd 2 45.14.a.g.1.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.14.a.g.1.1 ✓ 4 3.2 odd 2
45.14.a.h.1.4 yes 4 1.1 even 1 trivial