↑

Properties

Label 325.6.b.i.274.9
Level $325$
Weight $6$
Character 325.274
Analytic conductor $52.125$
Analytic rank $0$
Dimension $22$
Inner twists $2$

Related objects

Downloads

Learn more

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(274,325)] 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("325.274"); 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([1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,0,-374,0,702] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(52.1247414392\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.9
Character \(\chi\) \(=\) 325.274
Dual form 325.6.b.i.274.14

$q$-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)
 
\(f(q)\) \(=\) \(q-2.89241i q^{2} -3.45730i q^{3} +23.6340 q^{4} -9.99993 q^{6} +148.288i q^{7} -160.916i q^{8} +231.047 q^{9} -712.658 q^{11} -81.7096i q^{12} +169.000i q^{13} +428.910 q^{14} +290.850 q^{16} +1131.92i q^{17} -668.283i q^{18} -1400.39 q^{19} +512.676 q^{21} +2061.30i q^{22} +897.571i q^{23} -556.336 q^{24} +488.818 q^{26} -1638.92i q^{27} +3504.63i q^{28} -3238.07 q^{29} -7976.53 q^{31} -5990.58i q^{32} +2463.87i q^{33} +3273.98 q^{34} +5460.56 q^{36} -4978.35i q^{37} +4050.50i q^{38} +584.284 q^{39} -15559.9 q^{41} -1482.87i q^{42} -2308.61i q^{43} -16842.9 q^{44} +2596.15 q^{46} +7602.52i q^{47} -1005.56i q^{48} -5182.35 q^{49} +3913.39 q^{51} +3994.14i q^{52} -14138.1i q^{53} -4740.44 q^{54} +23862.0 q^{56} +4841.56i q^{57} +9365.83i q^{58} -49808.7 q^{59} -2516.69 q^{61} +23071.4i q^{62} +34261.5i q^{63} -8020.03 q^{64} +7126.54 q^{66} +38549.3i q^{67} +26751.8i q^{68} +3103.17 q^{69} +68008.5 q^{71} -37179.2i q^{72} -21305.2i q^{73} -14399.4 q^{74} -33096.7 q^{76} -105679. i q^{77} -1689.99i q^{78} -3984.60 q^{79} +50478.2 q^{81} +45005.8i q^{82} -13876.6i q^{83} +12116.6 q^{84} -6677.44 q^{86} +11195.0i q^{87} +114678. i q^{88} -89289.8 q^{89} -25060.7 q^{91} +21213.1i q^{92} +27577.2i q^{93} +21989.6 q^{94} -20711.2 q^{96} +147549. i q^{97} +14989.5i q^{98} -164658. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36}+ \cdots - 853396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.89241i − 0.511311i −0.966768 − 0.255656i \(-0.917709\pi\)
0.966768 − 0.255656i \(-0.0822913\pi\)
\(3\) − 3.45730i − 0.221786i −0.993832 − 0.110893i \(-0.964629\pi\)
0.993832 − 0.110893i \(-0.0353711\pi\)
\(4\) 23.6340 0.738561
\(5\) 0 0
\(6\) −9.99993 −0.113402
\(7\) 148.288i 1.14383i 0.820313 + 0.571914i \(0.193799\pi\)
−0.820313 + 0.571914i \(0.806201\pi\)
\(8\) − 160.916i − 0.888945i
\(9\) 231.047 0.950811
\(10\) 0 0
\(11\) −712.658 −1.77582 −0.887912 − 0.460014i \(-0.847844\pi\)
−0.887912 + 0.460014i \(0.847844\pi\)
\(12\) − 81.7096i − 0.163802i
\(13\) 169.000i 0.277350i
\(14\) 428.910 0.584852
\(15\) 0 0
\(16\) 290.850 0.284033
\(17\) 1131.92i 0.949935i 0.880003 + 0.474968i \(0.157540\pi\)
−0.880003 + 0.474968i \(0.842460\pi\)
\(18\) − 668.283i − 0.486160i
\(19\) −1400.39 −0.889947 −0.444974 − 0.895544i \(-0.646787\pi\)
−0.444974 + 0.895544i \(0.646787\pi\)
\(20\) 0 0
\(21\) 512.676 0.253685
\(22\) 2061.30i 0.907998i
\(23\) 897.571i 0.353793i 0.984229 + 0.176896i \(0.0566058\pi\)
−0.984229 + 0.176896i \(0.943394\pi\)
\(24\) −556.336 −0.197156
\(25\) 0 0
\(26\) 488.818 0.141812
\(27\) − 1638.92i − 0.432662i
\(28\) 3504.63i 0.844787i
\(29\) −3238.07 −0.714975 −0.357488 − 0.933918i \(-0.616367\pi\)
−0.357488 + 0.933918i \(0.616367\pi\)
\(30\) 0 0
\(31\) −7976.53 −1.49077 −0.745383 − 0.666636i \(-0.767733\pi\)
−0.745383 + 0.666636i \(0.767733\pi\)
\(32\) − 5990.58i − 1.03417i
\(33\) 2463.87i 0.393852i
\(34\) 3273.98 0.485712
\(35\) 0 0
\(36\) 5460.56 0.702232
\(37\) − 4978.35i − 0.597835i −0.954279 − 0.298917i \(-0.903374\pi\)
0.954279 − 0.298917i \(-0.0966255\pi\)
\(38\) 4050.50i 0.455040i
\(39\) 584.284 0.0615123
\(40\) 0 0
\(41\) −15559.9 −1.44560 −0.722800 − 0.691057i \(-0.757145\pi\)
−0.722800 + 0.691057i \(0.757145\pi\)
\(42\) − 1482.87i − 0.129712i
\(43\) − 2308.61i − 0.190405i −0.995458 − 0.0952026i \(-0.969650\pi\)
0.995458 − 0.0952026i \(-0.0303499\pi\)
\(44\) −16842.9 −1.31155
\(45\) 0 0
\(46\) 2596.15 0.180898
\(47\) 7602.52i 0.502011i 0.967986 + 0.251005i \(0.0807612\pi\)
−0.967986 + 0.251005i \(0.919239\pi\)
\(48\) − 1005.56i − 0.0629946i
\(49\) −5182.35 −0.308344
\(50\) 0 0
\(51\) 3913.39 0.210682
\(52\) 3994.14i 0.204840i
\(53\) − 14138.1i − 0.691357i −0.938353 − 0.345678i \(-0.887649\pi\)
0.938353 − 0.345678i \(-0.112351\pi\)
\(54\) −4740.44 −0.221225
\(55\) 0 0
\(56\) 23862.0 1.01680
\(57\) 4841.56i 0.197378i
\(58\) 9365.83i 0.365575i
\(59\) −49808.7 −1.86284 −0.931419 − 0.363948i \(-0.881428\pi\)
−0.931419 + 0.363948i \(0.881428\pi\)
\(60\) 0 0
\(61\) −2516.69 −0.0865973 −0.0432986 − 0.999062i \(-0.513787\pi\)
−0.0432986 + 0.999062i \(0.513787\pi\)
\(62\) 23071.4i 0.762245i
\(63\) 34261.5i 1.08757i
\(64\) −8020.03 −0.244752
\(65\) 0 0
\(66\) 7126.54 0.201381
\(67\) 38549.3i 1.04913i 0.851370 + 0.524566i \(0.175772\pi\)
−0.851370 + 0.524566i \(0.824228\pi\)
\(68\) 26751.8i 0.701585i
\(69\) 3103.17 0.0784663
\(70\) 0 0
\(71\) 68008.5 1.60110 0.800548 − 0.599269i \(-0.204542\pi\)
0.800548 + 0.599269i \(0.204542\pi\)
\(72\) − 37179.2i − 0.845219i
\(73\) − 21305.2i − 0.467927i −0.972245 − 0.233963i \(-0.924830\pi\)
0.972245 − 0.233963i \(-0.0751696\pi\)
\(74\) −14399.4 −0.305680
\(75\) 0 0
\(76\) −33096.7 −0.657280
\(77\) − 105679.i − 2.03124i
\(78\) − 1689.99i − 0.0314519i
\(79\) −3984.60 −0.0718318 −0.0359159 − 0.999355i \(-0.511435\pi\)
−0.0359159 + 0.999355i \(0.511435\pi\)
\(80\) 0 0
\(81\) 50478.2 0.854853
\(82\) 45005.8i 0.739152i
\(83\) − 13876.6i − 0.221100i −0.993871 − 0.110550i \(-0.964739\pi\)
0.993871 − 0.110550i \(-0.0352612\pi\)
\(84\) 12116.6 0.187362
\(85\) 0 0
\(86\) −6677.44 −0.0973563
\(87\) 11195.0i 0.158571i
\(88\) 114678.i 1.57861i
\(89\) −89289.8 −1.19489 −0.597443 − 0.801911i \(-0.703817\pi\)
−0.597443 + 0.801911i \(0.703817\pi\)
\(90\) 0 0
\(91\) −25060.7 −0.317241
\(92\) 21213.1i 0.261298i
\(93\) 27577.2i 0.330631i
\(94\) 21989.6 0.256684
\(95\) 0 0
\(96\) −20711.2 −0.229365
\(97\) 147549.i 1.59223i 0.605145 + 0.796116i \(0.293115\pi\)
−0.605145 + 0.796116i \(0.706885\pi\)
\(98\) 14989.5i 0.157660i
\(99\) −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.b.i.274.9 22
5.2 odd 4 325.6.a.k.1.7 yes 11
5.3 odd 4 325.6.a.j.1.5 ✓ 11
5.4 even 2 inner 325.6.b.i.274.14 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
325.6.a.j.1.5 ✓ 11 5.3 odd 4
325.6.a.k.1.7 yes 11 5.2 odd 4
325.6.b.i.274.9 22 1.1 even 1 trivial
325.6.b.i.274.14 22 5.4 even 2 inner