↑

Properties

Label 35.5.l.a.23.3
Level $35$
Weight $5$
Character 35.23
Analytic conductor $3.618$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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] = [35,5,Mod(2,35)] 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("35.2"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 35.l (of order \(12\), 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: \(3.61794870793\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 23.3
Character \(\chi\) \(=\) 35.23
Dual form 35.5.l.a.32.3

$q$-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+(-1.49563 - 5.58177i) q^{2} +(-3.00826 + 11.2270i) q^{3} +(-15.0628 + 8.69652i) q^{4} +(21.2360 + 13.1921i) q^{5} +67.1656 q^{6} +(30.0834 + 38.6780i) q^{7} +(5.69217 + 5.69217i) q^{8} +(-46.8474 - 27.0474i) q^{9} +(41.8741 - 138.265i) q^{10} +(-7.11215 - 12.3186i) q^{11} +(-52.3228 - 195.271i) q^{12} +(195.546 + 195.546i) q^{13} +(170.898 - 225.766i) q^{14} +(-211.991 + 198.731i) q^{15} +(-115.885 + 200.719i) q^{16} +(39.0789 + 10.4712i) q^{17} +(-80.9057 + 301.944i) q^{18} +(-222.686 - 128.568i) q^{19} +(-434.600 - 14.0308i) q^{20} +(-524.736 + 221.392i) q^{21} +(-58.1225 + 58.1225i) q^{22} +(83.9010 - 22.4812i) q^{23} +(-81.0294 + 46.7823i) q^{24} +(276.937 + 560.296i) q^{25} +(799.027 - 1383.96i) q^{26} +(-221.127 + 221.127i) q^{27} +(-789.505 - 320.980i) q^{28} -1116.77i q^{29} +(1426.33 + 886.056i) q^{30} +(-890.180 - 1541.84i) q^{31} +(1418.10 + 379.979i) q^{32} +(159.696 - 42.7904i) q^{33} -233.790i q^{34} +(128.606 + 1218.23i) q^{35} +940.872 q^{36} +(-215.806 - 805.400i) q^{37} +(-384.580 + 1435.27i) q^{38} +(-2783.64 + 1607.14i) q^{39} +(45.7873 + 195.971i) q^{40} +2737.91 q^{41} +(2020.57 + 2597.84i) q^{42} +(-209.828 - 209.828i) q^{43} +(214.258 + 123.702i) q^{44} +(-638.040 - 1192.39i) q^{45} +(-250.970 - 434.692i) q^{46} +(140.190 + 523.197i) q^{47} +(-1904.86 - 1904.86i) q^{48} +(-590.982 + 2327.13i) q^{49} +(2713.24 - 2383.79i) q^{50} +(-235.119 + 407.238i) q^{51} +(-4646.04 - 1244.90i) q^{52} +(871.725 - 3253.32i) q^{53} +(1565.00 + 903.555i) q^{54} +(11.4746 - 355.422i) q^{55} +(-48.9223 + 391.401i) q^{56} +(2113.33 - 2113.33i) q^{57} +(-6233.55 + 1670.27i) q^{58} +(1429.00 - 825.035i) q^{59} +(1464.91 - 4837.03i) q^{60} +(-438.372 + 759.283i) q^{61} +(-7274.80 + 7274.80i) q^{62} +(-363.189 - 2625.64i) q^{63} -4775.49i q^{64} +(1572.95 + 6732.28i) q^{65} +(-477.692 - 827.387i) q^{66} +(1364.24 + 365.546i) q^{67} +(-679.701 + 182.125i) q^{68} +1009.58i q^{69} +(6607.53 - 2539.87i) q^{70} +1621.68 q^{71} +(-112.705 - 420.621i) q^{72} +(-1341.54 + 5006.69i) q^{73} +(-4172.79 + 2409.16i) q^{74} +(-7123.52 + 1423.65i) q^{75} +4472.38 q^{76} +(262.502 - 645.669i) q^{77} +(13134.0 + 13134.0i) q^{78} +(890.409 + 514.078i) q^{79} +(-5108.85 + 2733.71i) q^{80} +(-4008.22 - 6942.44i) q^{81} +(-4094.90 - 15282.4i) q^{82} +(-6269.86 - 6269.86i) q^{83} +(5978.67 - 7898.16i) q^{84} +(691.744 + 737.899i) q^{85} +(-857.387 + 1485.04i) q^{86} +(12537.9 + 3359.53i) q^{87} +(29.6360 - 110.603i) q^{88} +(5777.55 + 3335.67i) q^{89} +(-5701.39 + 5344.77i) q^{90} +(-1680.65 + 13446.0i) q^{91} +(-1068.28 + 1068.28i) q^{92} +(19988.1 - 5355.79i) q^{93} +(2710.69 - 1565.02i) q^{94} +(-3032.89 - 5667.97i) q^{95} +(-8532.03 + 14777.9i) q^{96} +(12264.2 - 12264.2i) q^{97} +(13873.4 - 181.805i) q^{98} +769.460i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8} - 66 q^{10} + 296 q^{11} - 358 q^{12} - 8 q^{13} - 68 q^{15} + 468 q^{16} + 28 q^{17} - 868 q^{18} - 1032 q^{20} + 1280 q^{21} + 56 q^{22}+ \cdots - 78606 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.49563 − 5.58177i −0.373908 − 1.39544i −0.854934 − 0.518736i \(-0.826403\pi\)
0.481027 − 0.876706i \(-0.340264\pi\)
\(3\) −3.00826 + 11.2270i −0.334251 + 1.24744i 0.570428 + 0.821348i \(0.306778\pi\)
−0.904679 + 0.426094i \(0.859889\pi\)
\(4\) −15.0628 + 8.69652i −0.941426 + 0.543533i
\(5\) 21.2360 + 13.1921i 0.849441 + 0.527684i
\(6\) 67.1656 1.86571
\(7\) 30.0834 + 38.6780i 0.613946 + 0.789348i
\(8\) 5.69217 + 5.69217i 0.0889401 + 0.0889401i
\(9\) −46.8474 − 27.0474i −0.578363 − 0.333918i
\(10\) 41.8741 − 138.265i 0.418741 − 1.38265i
\(11\) −7.11215 − 12.3186i −0.0587781 − 0.101807i 0.835139 − 0.550039i \(-0.185387\pi\)
−0.893917 + 0.448232i \(0.852054\pi\)
\(12\) −52.3228 − 195.271i −0.363353 − 1.35605i
\(13\) 195.546 + 195.546i 1.15708 + 1.15708i 0.985101 + 0.171975i \(0.0550148\pi\)
0.171975 + 0.985101i \(0.444985\pi\)
\(14\) 170.898 − 225.766i 0.871930 − 1.15187i
\(15\) −211.991 + 198.731i −0.942182 + 0.883249i
\(16\) −115.885 + 200.719i −0.452677 + 0.784060i
\(17\) 39.0789 + 10.4712i 0.135221 + 0.0362324i 0.325795 − 0.945440i \(-0.394368\pi\)
−0.190574 + 0.981673i \(0.561035\pi\)
\(18\) −80.9057 + 301.944i −0.249709 + 0.931926i
\(19\) −222.686 − 128.568i −0.616859 − 0.356144i 0.158786 − 0.987313i \(-0.449242\pi\)
−0.775645 + 0.631169i \(0.782575\pi\)
\(20\) −434.600 − 14.0308i −1.08650 − 0.0350771i
\(21\) −524.736 + 221.392i −1.18988 + 0.502022i
\(22\) −58.1225 + 58.1225i −0.120088 + 0.120088i
\(23\) 83.9010 − 22.4812i 0.158603 − 0.0424975i −0.178644 − 0.983914i \(-0.557171\pi\)
0.337247 + 0.941416i \(0.390504\pi\)
\(24\) −81.0294 + 46.7823i −0.140676 + 0.0812193i
\(25\) 276.937 + 560.296i 0.443099 + 0.896473i
\(26\) 799.027 − 1383.96i 1.18199 − 2.04727i
\(27\) −221.127 + 221.127i −0.303329 + 0.303329i
\(28\) −789.505 − 320.980i −1.00702 − 0.409413i
\(29\) − 1116.77i − 1.32791i −0.747774 − 0.663953i \(-0.768877\pi\)
0.747774 − 0.663953i \(-0.231123\pi\)
\(30\) 1426.33 + 886.056i 1.58481 + 0.984507i
\(31\) −890.180 − 1541.84i −0.926306 − 1.60441i −0.789447 − 0.613819i \(-0.789632\pi\)
−0.136860 − 0.990590i \(-0.543701\pi\)
\(32\) 1418.10 + 379.979i 1.38486 + 0.371073i
\(33\) 159.696 − 42.7904i 0.146645 − 0.0392933i
\(34\) − 233.790i − 0.202241i
\(35\) 128.606 + 1218.23i 0.104985 + 0.994474i
\(36\) 940.872 0.725981
\(37\) −215.806 − 805.400i −0.157638 − 0.588313i −0.998865 − 0.0476309i \(-0.984833\pi\)
0.841227 − 0.540682i \(-0.181834\pi\)
\(38\) −384.580 + 1435.27i −0.266330 + 0.993956i
\(39\) −2783.64 + 1607.14i −1.83014 + 1.05663i
\(40\) 45.7873 + 195.971i 0.0286170 + 0.122482i
\(41\) 2737.91 1.62874 0.814370 − 0.580346i \(-0.197083\pi\)
0.814370 + 0.580346i \(0.197083\pi\)
\(42\) 2020.57 + 2597.84i 1.14545 + 1.47270i
\(43\) −209.828 − 209.828i −0.113482 − 0.113482i 0.648086 − 0.761568i \(-0.275570\pi\)
−0.761568 + 0.648086i \(0.775570\pi\)
\(44\) 214.258 + 123.702i 0.110670 + 0.0638956i
\(45\) −638.040 − 1192.39i −0.315082 − 0.588836i
\(46\) −250.970 − 434.692i −0.118606 − 0.205431i
\(47\) 140.190 + 523.197i 0.0634632 + 0.236848i 0.990370 − 0.138443i \(-0.0442096\pi\)
−0.926907 + 0.375290i \(0.877543\pi\)
\(48\) −1904.86 − 1904.86i −0.826762 − 0.826762i
\(49\) −590.982 + 2327.13i −0.246140 + 0.969234i
\(50\) 2713.24 − 2383.79i 1.08530 − 0.953516i
\(51\) −235.119 + 407.238i −0.0903957 + 0.156570i
\(52\) −4646.04 − 1244.90i −1.71821 − 0.460393i
\(53\) 871.725 − 3253.32i 0.310333 − 1.15818i −0.617924 − 0.786238i \(-0.712026\pi\)
0.928257 − 0.371940i \(-0.121307\pi\)
\(54\) 1565.00 + 903.555i 0.536695 + 0.309861i
\(55\) 11.4746 − 355.422i 0.00379327 − 0.117495i
\(56\) −48.9223 + 391.401i −0.0156002 + 0.124809i
\(57\) 2113.33 − 2113.33i 0.650455 − 0.650455i
\(58\) −6233.55 + 1670.27i −1.85302 + 0.496514i
\(59\) 1429.00 − 825.035i 0.410515 − 0.237011i −0.280496 − 0.959855i \(-0.590499\pi\)
0.691011 + 0.722844i \(0.257166\pi\)
\(60\) 1464.91 − 4837.03i 0.406920 − 1.34362i
\(61\) −438.372 + 759.283i −0.117810 + 0.204053i −0.918900 − 0.394491i \(-0.870921\pi\)
0.801089 + 0.598545i \(0.204254\pi\)
\(62\) −7274.80 + 7274.80i −1.89251 + 1.89251i
\(63\) −363.189 − 2625.64i −0.0915064 − 0.661537i
\(64\) − 4775.49i − 1.16589i
\(65\) 1572.95 + 6732.28i 0.372297 + 1.59344i
\(66\) −477.692 − 827.387i −0.109663 − 0.189942i
\(67\) 1364.24 + 365.546i 0.303907 + 0.0814316i 0.407550 − 0.913183i \(-0.366383\pi\)
−0.103643 + 0.994615i \(0.533050\pi\)
\(68\) −679.701 + 182.125i −0.146994 + 0.0393870i
\(69\) 1009.58i 0.212053i
\(70\) 6607.53 − 2539.87i 1.34848 − 0.518341i
\(71\) 1621.68 0.321698 0.160849 − 0.986979i \(-0.448577\pi\)
0.160849 + 0.986979i \(0.448577\pi\)
\(72\) −112.705 − 420.621i −0.0217410 − 0.0811384i
\(73\) −1341.54 + 5006.69i −0.251743 + 0.939517i 0.718131 + 0.695908i \(0.244998\pi\)
−0.969874 + 0.243609i \(0.921669\pi\)
\(74\) −4172.79 + 2409.16i −0.762015 + 0.439949i
\(75\) −7123.52 + 1423.65i −1.26640 + 0.253093i
\(76\) 4472.38 0.774303
\(77\) 262.502 − 645.669i 0.0442743 − 0.108900i
\(78\) 13134.0 + 13134.0i 2.15877 + 2.15877i
\(79\) 890.409 + 514.078i 0.142671 + 0.0823711i 0.569636 − 0.821897i \(-0.307084\pi\)
−0.426965 + 0.904268i \(0.640417\pi\)
\(80\) −5108.85 + 2733.71i −0.798259 + 0.427142i
\(81\) −4008.22 − 6942.44i −0.610916 − 1.05814i
\(82\) −4094.90 − 15282.4i −0.608998 − 2.27281i
\(83\) −6269.86 − 6269.86i −0.910127 − 0.910127i 0.0861548 − 0.996282i \(-0.472542\pi\)
−0.996282 + 0.0861548i \(0.972542\pi\)
\(84\) 5978.67 − 7898.16i 0.847317 − 1.11935i
\(85\) 691.744 + 737.899i 0.0957431 + 0.102131i
\(86\) −857.387 + 1485.04i −0.115926 + 0.200789i
\(87\) 12537.9 + 3359.53i 1.65649 + 0.443854i
\(88\) 29.6360 − 110.603i 0.00382697 − 0.0142824i
\(89\) 5777.55 + 3335.67i 0.729396 + 0.421117i 0.818201 − 0.574932i \(-0.194971\pi\)
−0.0888049 + 0.996049i \(0.528305\pi\)
\(90\) −5701.39 + 5344.77i −0.703876 + 0.659849i
\(91\) −1680.65 + 13446.0i −0.202953 + 1.62372i
\(92\) −1068.28 + 1068.28i −0.126214 + 0.126214i
\(93\) 19988.1 − 5355.79i 2.31103 − 0.619238i
\(94\) 2710.69 − 1565.02i 0.306778 − 0.177118i
\(95\) −3032.89 − 5667.97i −0.336054 − 0.628030i
\(96\) −8532.03 + 14777.9i −0.925785 + 1.60351i
\(97\) 12264.2 − 12264.2i 1.30345 − 1.30345i 0.377399 − 0.926051i \(-0.376819\pi\)
0.926051 − 0.377399i \(-0.123181\pi\)
\(98\) 13873.4 − 181.805i 1.44454 − 0.0189302i
\(99\) 769.460i 0.0785083i
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 35.5.l.a.23.3 yes 56
5.2 odd 4 inner 35.5.l.a.2.12 ✓ 56
7.4 even 3 inner 35.5.l.a.18.12 yes 56
35.32 odd 12 inner 35.5.l.a.32.3 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.12 ✓ 56 5.2 odd 4 inner
35.5.l.a.18.12 yes 56 7.4 even 3 inner
35.5.l.a.23.3 yes 56 1.1 even 1 trivial
35.5.l.a.32.3 yes 56 35.32 odd 12 inner