Properties

Label 25.22.a.d.1.6
Level $25$
Weight $22$
Character 25.1
Self dual yes
Analytic conductor $69.869$
Analytic rank $1$
Dimension $7$
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] = [25,22,Mod(1,25)] 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("25.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,-737] 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: \(69.8693360718\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2 x^{6} - 11176187 x^{5} + 2419003474 x^{4} + 32443001467723 x^{3} + \cdots + 99\!\cdots\!78 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{13}\cdot 3^{3}\cdot 5^{10}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1647.62\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1542.62 q^{2} +110812. q^{3} +282527. q^{4} +1.70941e8 q^{6} +3.07836e8 q^{7} -2.79928e9 q^{8} +1.81900e9 q^{9} +6.74227e9 q^{11} +3.13075e10 q^{12} -7.21023e11 q^{13} +4.74873e11 q^{14} -4.91073e12 q^{16} -6.60745e12 q^{17} +2.80603e12 q^{18} -3.21646e12 q^{19} +3.41119e13 q^{21} +1.04008e13 q^{22} +1.77020e14 q^{23} -3.10194e14 q^{24} -1.11227e15 q^{26} -9.57568e14 q^{27} +8.69719e13 q^{28} -7.62161e14 q^{29} -6.75856e15 q^{31} -1.70488e15 q^{32} +7.47126e14 q^{33} -1.01928e16 q^{34} +5.13917e14 q^{36} +3.27807e16 q^{37} -4.96178e15 q^{38} -7.98982e16 q^{39} -3.82201e16 q^{41} +5.26218e16 q^{42} -2.75494e17 q^{43} +1.90487e15 q^{44} +2.73075e17 q^{46} +5.22162e17 q^{47} -5.44169e17 q^{48} -4.63783e17 q^{49} -7.32187e17 q^{51} -2.03708e17 q^{52} +9.56366e17 q^{53} -1.47716e18 q^{54} -8.61717e17 q^{56} -3.56423e17 q^{57} -1.17573e18 q^{58} +5.30330e18 q^{59} -5.47739e18 q^{61} -1.04259e19 q^{62} +5.59953e17 q^{63} +7.66856e18 q^{64} +1.15253e18 q^{66} -4.98720e18 q^{67} -1.86678e18 q^{68} +1.96160e19 q^{69} +1.90817e19 q^{71} -5.09189e18 q^{72} -2.67876e19 q^{73} +5.05682e19 q^{74} -9.08737e17 q^{76} +2.07551e18 q^{77} -1.23253e20 q^{78} +6.90931e19 q^{79} -1.25138e20 q^{81} -5.89590e19 q^{82} -2.45646e20 q^{83} +9.63755e18 q^{84} -4.24983e20 q^{86} -8.44568e19 q^{87} -1.88735e19 q^{88} -5.05575e20 q^{89} -2.21956e20 q^{91} +5.00129e19 q^{92} -7.48931e20 q^{93} +8.05498e20 q^{94} -1.88921e20 q^{96} -2.51872e20 q^{97} -7.15442e20 q^{98} +1.22642e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 737 q^{2} - 50381 q^{3} + 7749909 q^{4} - 30442931 q^{6} - 817782442 q^{7} + 3231986745 q^{8} + 1856104186 q^{9} - 17026930591 q^{11} - 286884094477 q^{12} + 325366982844 q^{13} + 13403645898 q^{14}+ \cdots + 26\!\cdots\!82 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1542.62 1.06523 0.532616 − 0.846357i \(-0.321209\pi\)
0.532616 + 0.846357i \(0.321209\pi\)
\(3\) 110812. 1.08346 0.541732 − 0.840551i \(-0.317769\pi\)
0.541732 + 0.840551i \(0.317769\pi\)
\(4\) 282527. 0.134719
\(5\) 0 0
\(6\) 1.70941e8 1.15414
\(7\) 3.07836e8 0.411898 0.205949 − 0.978563i \(-0.433972\pi\)
0.205949 + 0.978563i \(0.433972\pi\)
\(8\) −2.79928e9 −0.921725
\(9\) 1.81900e9 0.173895
\(10\) 0 0
\(11\) 6.74227e9 0.0783760 0.0391880 − 0.999232i \(-0.487523\pi\)
0.0391880 + 0.999232i \(0.487523\pi\)
\(12\) 3.13075e10 0.145964
\(13\) −7.21023e11 −1.45059 −0.725294 − 0.688440i \(-0.758296\pi\)
−0.725294 + 0.688440i \(0.758296\pi\)
\(14\) 4.74873e11 0.438767
\(15\) 0 0
\(16\) −4.91073e12 −1.11657
\(17\) −6.60745e12 −0.794914 −0.397457 − 0.917621i \(-0.630107\pi\)
−0.397457 + 0.917621i \(0.630107\pi\)
\(18\) 2.80603e12 0.185238
\(19\) −3.21646e12 −0.120355 −0.0601777 − 0.998188i \(-0.519167\pi\)
−0.0601777 + 0.998188i \(0.519167\pi\)
\(20\) 0 0
\(21\) 3.41119e13 0.446276
\(22\) 1.04008e13 0.0834886
\(23\) 1.77020e14 0.891004 0.445502 − 0.895281i \(-0.353025\pi\)
0.445502 + 0.895281i \(0.353025\pi\)
\(24\) −3.10194e14 −0.998656
\(25\) 0 0
\(26\) −1.11227e15 −1.54521
\(27\) −9.57568e14 −0.895055
\(28\) 8.69719e13 0.0554906
\(29\) −7.62161e14 −0.336409 −0.168205 − 0.985752i \(-0.553797\pi\)
−0.168205 + 0.985752i \(0.553797\pi\)
\(30\) 0 0
\(31\) −6.75856e15 −1.48100 −0.740502 − 0.672054i \(-0.765412\pi\)
−0.740502 + 0.672054i \(0.765412\pi\)
\(32\) −1.70488e15 −0.267682
\(33\) 7.47126e14 0.0849176
\(34\) −1.01928e16 −0.846768
\(35\) 0 0
\(36\) 5.13917e14 0.0234270
\(37\) 3.27807e16 1.12073 0.560364 − 0.828246i \(-0.310661\pi\)
0.560364 + 0.828246i \(0.310661\pi\)
\(38\) −4.96178e15 −0.128206
\(39\) −7.98982e16 −1.57166
\(40\) 0 0
\(41\) −3.82201e16 −0.444693 −0.222347 − 0.974968i \(-0.571372\pi\)
−0.222347 + 0.974968i \(0.571372\pi\)
\(42\) 5.26218e16 0.475388
\(43\) −2.75494e17 −1.94399 −0.971993 − 0.235010i \(-0.924488\pi\)
−0.971993 + 0.235010i \(0.924488\pi\)
\(44\) 1.90487e15 0.0105588
\(45\) 0 0
\(46\) 2.73075e17 0.949126
\(47\) 5.22162e17 1.44803 0.724015 − 0.689784i \(-0.242294\pi\)
0.724015 + 0.689784i \(0.242294\pi\)
\(48\) −5.44169e17 −1.20976
\(49\) −4.63783e17 −0.830340
\(50\) 0 0
\(51\) −7.32187e17 −0.861261
\(52\) −2.03708e17 −0.195422
\(53\) 9.56366e17 0.751151 0.375576 − 0.926792i \(-0.377445\pi\)
0.375576 + 0.926792i \(0.377445\pi\)
\(54\) −1.47716e18 −0.953442
\(55\) 0 0
\(56\) −8.61717e17 −0.379656
\(57\) −3.56423e17 −0.130401
\(58\) −1.17573e18 −0.358354
\(59\) 5.30330e18 1.35083 0.675415 − 0.737438i \(-0.263965\pi\)
0.675415 + 0.737438i \(0.263965\pi\)
\(60\) 0 0
\(61\) −5.47739e18 −0.983128 −0.491564 − 0.870841i \(-0.663575\pi\)
−0.491564 + 0.870841i \(0.663575\pi\)
\(62\) −1.04259e19 −1.57761
\(63\) 5.59953e17 0.0716269
\(64\) 7.66856e18 0.831427
\(65\) 0 0
\(66\) 1.15253e18 0.0904569
\(67\) −4.98720e18 −0.334250 −0.167125 − 0.985936i \(-0.553448\pi\)
−0.167125 + 0.985936i \(0.553448\pi\)
\(68\) −1.86678e18 −0.107090
\(69\) 1.96160e19 0.965371
\(70\) 0 0
\(71\) 1.90817e19 0.695673 0.347836 − 0.937555i \(-0.386917\pi\)
0.347836 + 0.937555i \(0.386917\pi\)
\(72\) −5.09189e18 −0.160283
\(73\) −2.67876e19 −0.729532 −0.364766 − 0.931099i \(-0.618851\pi\)
−0.364766 + 0.931099i \(0.618851\pi\)
\(74\) 5.05682e19 1.19384
\(75\) 0 0
\(76\) −9.08737e17 −0.0162142
\(77\) 2.07551e18 0.0322829
\(78\) −1.23253e20 −1.67418
\(79\) 6.90931e19 0.821014 0.410507 − 0.911858i \(-0.365352\pi\)
0.410507 + 0.911858i \(0.365352\pi\)
\(80\) 0 0
\(81\) −1.25138e20 −1.14366
\(82\) −5.89590e19 −0.473702
\(83\) −2.45646e20 −1.73776 −0.868880 − 0.495022i \(-0.835160\pi\)
−0.868880 + 0.495022i \(0.835160\pi\)
\(84\) 9.63755e18 0.0601221
\(85\) 0 0
\(86\) −4.24983e20 −2.07080
\(87\) −8.44568e19 −0.364487
\(88\) −1.88735e19 −0.0722411
\(89\) −5.05575e20 −1.71866 −0.859331 − 0.511420i \(-0.829120\pi\)
−0.859331 + 0.511420i \(0.829120\pi\)
\(90\) 0 0
\(91\) −2.21956e20 −0.597494
\(92\) 5.00129e19 0.120036
\(93\) −7.48931e20 −1.60462
\(94\) 8.05498e20 1.54249
\(95\) 0 0
\(96\) −1.88921e20 −0.290023
\(97\) −2.51872e20 −0.346798 −0.173399 − 0.984852i \(-0.555475\pi\)
−0.173399 + 0.984852i \(0.555475\pi\)
\(98\) −7.15442e20 −0.884505
\(99\) 1.22642e19 0.0136292
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 25.22.a.d.1.6 ✓ 7
5.2 odd 4 25.22.b.d.24.11 14
5.3 odd 4 25.22.b.d.24.4 14
5.4 even 2 25.22.a.e.1.2 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.22.a.d.1.6 ✓ 7 1.1 even 1 trivial
25.22.a.e.1.2 yes 7 5.4 even 2
25.22.b.d.24.4 14 5.3 odd 4
25.22.b.d.24.11 14 5.2 odd 4