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Properties

Label 25.6.d.a.11.9
Level $25$
Weight $6$
Character 25.11
Analytic conductor $4.010$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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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] = [25,6,Mod(6,25)] 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("25.6"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 25.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.00959549532\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 11.9
Character \(\chi\) \(=\) 25.11
Dual form 25.6.d.a.16.9

$q$-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+(5.56327 - 4.04195i) q^{2} +(-7.98941 + 24.5889i) q^{3} +(4.72404 - 14.5391i) q^{4} +(52.9093 + 18.0446i) q^{5} +(54.9398 + 169.087i) q^{6} -5.99482 q^{7} +(35.5141 + 109.301i) q^{8} +(-344.191 - 250.069i) q^{9} +(367.284 - 113.470i) q^{10} +(460.750 - 334.754i) q^{11} +(319.758 + 232.317i) q^{12} +(-694.810 - 504.809i) q^{13} +(-33.3508 + 24.2308i) q^{14} +(-866.409 + 1156.81i) q^{15} +(1035.13 + 752.067i) q^{16} +(282.676 + 869.988i) q^{17} -2925.59 q^{18} +(-482.215 - 1484.11i) q^{19} +(512.297 - 684.010i) q^{20} +(47.8951 - 147.406i) q^{21} +(1210.21 - 3724.66i) q^{22} +(2350.86 - 1708.00i) q^{23} -2971.33 q^{24} +(2473.79 + 1909.45i) q^{25} -5905.82 q^{26} +(3816.09 - 2772.55i) q^{27} +(-28.3198 + 87.1592i) q^{28} +(-308.658 + 949.952i) q^{29} +(-144.279 + 9937.65i) q^{30} +(-1593.48 - 4904.24i) q^{31} +5120.89 q^{32} +(4550.11 + 14003.8i) q^{33} +(5089.05 + 3697.41i) q^{34} +(-317.182 - 108.174i) q^{35} +(-5261.75 + 3822.89i) q^{36} +(-477.227 - 346.726i) q^{37} +(-8681.38 - 6307.39i) q^{38} +(17963.8 - 13051.5i) q^{39} +(-93.2649 + 6423.89i) q^{40} +(-4422.90 - 3213.42i) q^{41} +(-329.354 - 1013.65i) q^{42} +9774.65 q^{43} +(-2690.42 - 8280.27i) q^{44} +(-13698.5 - 19441.8i) q^{45} +(6174.81 - 19004.1i) q^{46} +(-6146.94 + 18918.3i) q^{47} +(-26762.6 + 19444.1i) q^{48} -16771.1 q^{49} +(21480.2 + 623.851i) q^{50} -23650.5 q^{51} +(-10621.8 + 7717.16i) q^{52} +(-1960.26 + 6033.06i) q^{53} +(10023.4 - 30848.9i) q^{54} +(30418.4 - 9397.59i) q^{55} +(-212.901 - 655.241i) q^{56} +40345.1 q^{57} +(2122.51 + 6532.42i) q^{58} +(-13944.1 - 10131.0i) q^{59} +(12726.1 + 18061.6i) q^{60} +(-36671.3 + 26643.2i) q^{61} +(-28687.7 - 20842.8i) q^{62} +(2063.36 + 1499.12i) q^{63} +(-4635.32 + 3367.75i) q^{64} +(-27652.8 - 39246.6i) q^{65} +(81916.2 + 59515.6i) q^{66} +(11796.6 + 36306.2i) q^{67} +13984.2 q^{68} +(23215.8 + 71450.9i) q^{69} +(-2201.80 + 680.233i) q^{70} +(24875.0 - 76557.5i) q^{71} +(15109.3 - 46501.5i) q^{72} +(-26282.9 + 19095.7i) q^{73} -4056.39 q^{74} +(-66715.3 + 45572.3i) q^{75} -23855.6 q^{76} +(-2762.11 + 2006.79i) q^{77} +(47184.0 - 145218. i) q^{78} +(-12070.0 + 37147.8i) q^{79} +(41197.3 + 58469.8i) q^{80} +(5738.55 + 17661.4i) q^{81} -37594.3 q^{82} +(-22793.3 - 70150.6i) q^{83} +(-1916.89 - 1392.70i) q^{84} +(-742.346 + 51131.2i) q^{85} +(54379.0 - 39508.6i) q^{86} +(-20892.3 - 15179.1i) q^{87} +(52952.2 + 38472.0i) q^{88} +(-14597.8 + 10605.9i) q^{89} +(-154791. - 52791.0i) q^{90} +(4165.26 + 3026.24i) q^{91} +(-13727.2 - 42248.0i) q^{92} +133321. q^{93} +(42269.9 + 130093. i) q^{94} +(1266.36 - 87224.4i) q^{95} +(-40912.9 + 125917. i) q^{96} +(48208.6 - 148371. i) q^{97} +(-93301.9 + 67787.8i) q^{98} -242298. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9} - 55 q^{10} + 718 q^{11} - 437 q^{12} - 291 q^{13} - 689 q^{14} - 1125 q^{15} + 2609 q^{16} + 718 q^{17} + 8544 q^{18}+ \cdots - 680776 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.56327 − 4.04195i 0.983456 − 0.714523i 0.0249777 − 0.999688i \(-0.492049\pi\)
0.958478 + 0.285165i \(0.0920485\pi\)
\(3\) −7.98941 + 24.5889i −0.512521 + 1.57738i 0.275227 + 0.961379i \(0.411247\pi\)
−0.787748 + 0.615998i \(0.788753\pi\)
\(4\) 4.72404 − 14.5391i 0.147626 − 0.454347i
\(5\) 52.9093 + 18.0446i 0.946470 + 0.322791i
\(6\) 54.9398 + 169.087i 0.623030 + 1.91749i
\(7\) −5.99482 −0.0462414 −0.0231207 − 0.999733i \(-0.507360\pi\)
−0.0231207 + 0.999733i \(0.507360\pi\)
\(8\) 35.5141 + 109.301i 0.196190 + 0.603810i
\(9\) −344.191 − 250.069i −1.41642 − 1.02909i
\(10\) 367.284 − 113.470i 1.16145 − 0.358824i
\(11\) 460.750 − 334.754i 1.14811 − 0.834151i 0.159881 − 0.987136i \(-0.448889\pi\)
0.988228 + 0.152986i \(0.0488888\pi\)
\(12\) 319.758 + 232.317i 0.641014 + 0.465724i
\(13\) −694.810 − 504.809i −1.14027 − 0.828454i −0.153113 − 0.988209i \(-0.548930\pi\)
−0.987157 + 0.159754i \(0.948930\pi\)
\(14\) −33.3508 + 24.2308i −0.0454764 + 0.0330405i
\(15\) −866.409 + 1156.81i −0.994249 + 1.32750i
\(16\) 1035.13 + 752.067i 1.01087 + 0.734440i
\(17\) 282.676 + 869.988i 0.237229 + 0.730115i 0.996818 + 0.0797117i \(0.0254000\pi\)
−0.759589 + 0.650403i \(0.774600\pi\)
\(18\) −2925.59 −2.12830
\(19\) −482.215 − 1484.11i −0.306448 − 0.943150i −0.979133 − 0.203221i \(-0.934859\pi\)
0.672685 − 0.739929i \(-0.265141\pi\)
\(20\) 512.297 − 684.010i 0.286383 − 0.382373i
\(21\) 47.8951 − 147.406i 0.0236997 − 0.0729401i
\(22\) 1210.21 − 3724.66i 0.533096 − 1.64070i
\(23\) 2350.86 − 1708.00i 0.926632 − 0.673237i −0.0185341 − 0.999828i \(-0.505900\pi\)
0.945166 + 0.326591i \(0.105900\pi\)
\(24\) −2971.33 −1.05299
\(25\) 2473.79 + 1909.45i 0.791612 + 0.611024i
\(26\) −5905.82 −1.71335
\(27\) 3816.09 − 2772.55i 1.00742 − 0.731930i
\(28\) −28.3198 + 87.1592i −0.00682644 + 0.0210096i
\(29\) −308.658 + 949.952i −0.0681527 + 0.209752i −0.979333 − 0.202256i \(-0.935173\pi\)
0.911180 + 0.412009i \(0.135173\pi\)
\(30\) −144.279 + 9937.65i −0.0292685 + 2.01595i
\(31\) −1593.48 − 4904.24i −0.297813 − 0.916573i −0.982262 − 0.187513i \(-0.939957\pi\)
0.684449 − 0.729060i \(-0.260043\pi\)
\(32\) 5120.89 0.884037
\(33\) 4550.11 + 14003.8i 0.727340 + 2.23852i
\(34\) 5089.05 + 3697.41i 0.754987 + 0.548530i
\(35\) −317.182 − 108.174i −0.0437661 − 0.0149263i
\(36\) −5261.75 + 3822.89i −0.676665 + 0.491626i
\(37\) −477.227 − 346.726i −0.0573087 − 0.0416372i 0.558762 − 0.829328i \(-0.311276\pi\)
−0.616071 + 0.787691i \(0.711276\pi\)
\(38\) −8681.38 − 6307.39i −0.975281 − 0.708583i
\(39\) 17963.8 − 13051.5i 1.89120 − 1.37403i
\(40\) −93.2649 + 6423.89i −0.00921655 + 0.634816i
\(41\) −4422.90 − 3213.42i −0.410911 − 0.298544i 0.363060 − 0.931766i \(-0.381732\pi\)
−0.773970 + 0.633222i \(0.781732\pi\)
\(42\) −329.354 − 1013.65i −0.0288098 − 0.0886674i
\(43\) 9774.65 0.806176 0.403088 − 0.915161i \(-0.367937\pi\)
0.403088 + 0.915161i \(0.367937\pi\)
\(44\) −2690.42 − 8280.27i −0.209502 − 0.644782i
\(45\) −13698.5 − 19441.8i −1.00842 − 1.43121i
\(46\) 6174.81 − 19004.1i 0.430258 − 1.32420i
\(47\) −6146.94 + 18918.3i −0.405895 + 1.24922i 0.514250 + 0.857640i \(0.328070\pi\)
−0.920145 + 0.391577i \(0.871930\pi\)
\(48\) −26762.6 + 19444.1i −1.67658 + 1.21811i
\(49\) −16771.1 −0.997862
\(50\) 21480.2 + 623.851i 1.21511 + 0.0352904i
\(51\) −23650.5 −1.27325
\(52\) −10621.8 + 7717.16i −0.544739 + 0.395776i
\(53\) −1960.26 + 6033.06i −0.0958570 + 0.295017i −0.987476 − 0.157769i \(-0.949570\pi\)
0.891619 + 0.452786i \(0.149570\pi\)
\(54\) 10023.4 − 30848.9i 0.467768 − 1.43964i
\(55\) 30418.4 − 9397.59i 1.35591 − 0.418899i
\(56\) −212.901 − 655.241i −0.00907209 − 0.0279210i
\(57\) 40345.1 1.64476
\(58\) 2122.51 + 6532.42i 0.0828476 + 0.254979i
\(59\) −13944.1 − 10131.0i −0.521509 − 0.378898i 0.295663 − 0.955292i \(-0.404459\pi\)
−0.817172 + 0.576394i \(0.804459\pi\)
\(60\) 12726.1 + 18061.6i 0.456370 + 0.647708i
\(61\) −36671.3 + 26643.2i −1.26183 + 0.916774i −0.998846 − 0.0480262i \(-0.984707\pi\)
−0.262985 + 0.964800i \(0.584707\pi\)
\(62\) −28687.7 − 20842.8i −0.947798 − 0.688616i
\(63\) 2063.36 + 1499.12i 0.0654974 + 0.0475867i
\(64\) −4635.32 + 3367.75i −0.141459 + 0.102776i
\(65\) −27652.8 − 39246.6i −0.811814 − 1.15218i
\(66\) 81916.2 + 59515.6i 2.31478 + 1.68179i
\(67\) 11796.6 + 36306.2i 0.321047 + 0.988083i 0.973193 + 0.229989i \(0.0738690\pi\)
−0.652146 + 0.758094i \(0.726131\pi\)
\(68\) 13984.2 0.366746
\(69\) 23215.8 + 71450.9i 0.587031 + 1.80670i
\(70\) −2201.80 + 680.233i −0.0537072 + 0.0165925i
\(71\) 24875.0 − 76557.5i 0.585623 − 1.80236i −0.0111320 − 0.999938i \(-0.503543\pi\)
0.596755 − 0.802424i \(-0.296457\pi\)
\(72\) 15109.3 − 46501.5i 0.343488 − 1.05715i
\(73\) −26282.9 + 19095.7i −0.577254 + 0.419400i −0.837733 − 0.546080i \(-0.816119\pi\)
0.260479 + 0.965479i \(0.416119\pi\)
\(74\) −4056.39 −0.0861113
\(75\) −66715.3 + 45572.3i −1.36953 + 0.935508i
\(76\) −23855.6 −0.473757
\(77\) −2762.11 + 2006.79i −0.0530902 + 0.0385723i
\(78\) 47184.0 − 145218.i 0.878130 − 2.70261i
\(79\) −12070.0 + 37147.8i −0.217591 + 0.669677i 0.781368 + 0.624070i \(0.214522\pi\)
−0.998959 + 0.0456065i \(0.985478\pi\)
\(80\) 41197.3 + 58469.8i 0.719688 + 1.02143i
\(81\) 5738.55 + 17661.4i 0.0971828 + 0.299098i
\(82\) −37594.3 −0.617429
\(83\) −22793.3 − 70150.6i −0.363172 − 1.11773i −0.951118 − 0.308827i \(-0.900063\pi\)
0.587946 − 0.808900i \(-0.299937\pi\)
\(84\) −1916.89 − 1392.70i −0.0296414 − 0.0215357i
\(85\) −742.346 + 51131.2i −0.0111445 + 0.767607i
\(86\) 54379.0 − 39508.6i 0.792839 − 0.576031i
\(87\) −20892.3 − 15179.1i −0.295929 − 0.215005i
\(88\) 52952.2 + 38472.0i 0.728916 + 0.529588i
\(89\) −14597.8 + 10605.9i −0.195350 + 0.141930i −0.681160 − 0.732134i \(-0.738524\pi\)
0.485811 + 0.874064i \(0.338524\pi\)
\(90\) −154791. − 52791.0i −2.01437 − 0.686996i
\(91\) 4165.26 + 3026.24i 0.0527277 + 0.0383089i
\(92\) −13727.2 − 42248.0i −0.169088 − 0.520399i
\(93\) 133321. 1.59842
\(94\) 42269.9 + 130093.i 0.493414 + 1.51857i
\(95\) 1266.36 − 87224.4i 0.0143962 − 0.991583i
\(96\) −40912.9 + 125917.i −0.453087 + 1.39446i
\(97\) 48208.6 − 148371.i 0.520230 − 1.60110i −0.253331 − 0.967380i \(-0.581526\pi\)
0.773560 − 0.633723i \(-0.218474\pi\)
\(98\) −93301.9 + 67787.8i −0.981353 + 0.712995i
\(99\) −242298. −2.48463
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 25.6.d.a.11.9 ✓ 44
25.4 even 10 625.6.a.c.1.18 22
25.16 even 5 inner 25.6.d.a.16.9 yes 44
25.21 even 5 625.6.a.d.1.5 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.6.d.a.11.9 ✓ 44 1.1 even 1 trivial
25.6.d.a.16.9 yes 44 25.16 even 5 inner
625.6.a.c.1.18 22 25.4 even 10
625.6.a.d.1.5 22 25.21 even 5