LMFDB - Embedded newform 4009.2.a.c.1.36

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.36
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.225751 q^{2} +1.49745 q^{3} -1.94904 q^{4} +2.28431 q^{5} -0.338051 q^{6} -2.29657 q^{7} +0.891498 q^{8} -0.757634 q^{9} +O(q^{10})$	\(q-0.225751 q^{2} +1.49745 q^{3} -1.94904 q^{4} +2.28431 q^{5} -0.338051 q^{6} -2.29657 q^{7} +0.891498 q^{8} -0.757634 q^{9} -0.515685 q^{10} -1.57543 q^{11} -2.91859 q^{12} +0.768776 q^{13} +0.518451 q^{14} +3.42065 q^{15} +3.69682 q^{16} +1.66603 q^{17} +0.171036 q^{18} +1.00000 q^{19} -4.45221 q^{20} -3.43900 q^{21} +0.355655 q^{22} -7.57889 q^{23} +1.33498 q^{24} +0.218082 q^{25} -0.173552 q^{26} -5.62688 q^{27} +4.47609 q^{28} +5.28827 q^{29} -0.772214 q^{30} +9.38342 q^{31} -2.61756 q^{32} -2.35914 q^{33} -0.376108 q^{34} -5.24607 q^{35} +1.47666 q^{36} -0.0976433 q^{37} -0.225751 q^{38} +1.15121 q^{39} +2.03646 q^{40} -7.54895 q^{41} +0.776357 q^{42} +11.7715 q^{43} +3.07058 q^{44} -1.73067 q^{45} +1.71094 q^{46} -6.91598 q^{47} +5.53581 q^{48} -1.72579 q^{49} -0.0492321 q^{50} +2.49480 q^{51} -1.49837 q^{52} -10.4319 q^{53} +1.27027 q^{54} -3.59878 q^{55} -2.04738 q^{56} +1.49745 q^{57} -1.19383 q^{58} -11.0971 q^{59} -6.66697 q^{60} +2.95668 q^{61} -2.11831 q^{62} +1.73996 q^{63} -6.80272 q^{64} +1.75613 q^{65} +0.532577 q^{66} -10.8424 q^{67} -3.24715 q^{68} -11.3490 q^{69} +1.18430 q^{70} +5.30306 q^{71} -0.675429 q^{72} -3.40422 q^{73} +0.0220431 q^{74} +0.326567 q^{75} -1.94904 q^{76} +3.61809 q^{77} -0.259886 q^{78} +2.02379 q^{79} +8.44468 q^{80} -6.15309 q^{81} +1.70418 q^{82} -6.72746 q^{83} +6.70274 q^{84} +3.80573 q^{85} -2.65742 q^{86} +7.91893 q^{87} -1.40450 q^{88} -15.5984 q^{89} +0.390701 q^{90} -1.76555 q^{91} +14.7715 q^{92} +14.0512 q^{93} +1.56129 q^{94} +2.28431 q^{95} -3.91967 q^{96} +0.271021 q^{97} +0.389598 q^{98} +1.19360 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.225751	−0.159630	−0.0798149	−	0.996810i	$-0.525433\pi$
$2$	−0.225751	−0.159630	−0.0798149	+	0.996810i	$0.525433\pi$
$3$	1.49745	0.864555	0.432277	−	0.901741i	$-0.357710\pi$
$3$	1.49745	0.864555	0.432277	+	0.901741i	$0.357710\pi$
$4$	−1.94904	−0.974518
$5$	2.28431	1.02158	0.510788	−	0.859707i	$-0.329354\pi$
$5$	2.28431	1.02158	0.510788	+	0.859707i	$0.329354\pi$
$6$	−0.338051	−0.138009
$7$	−2.29657	−0.868020	−0.434010	−	0.900908i	$-0.642902\pi$
$7$	−2.29657	−0.868020	−0.434010	+	0.900908i	$0.642902\pi$
$8$	0.891498	0.315192
$9$	−0.757634	−0.252545
$10$	−0.515685	−0.163074
$11$	−1.57543	−0.475011	−0.237505	−	0.971386i	$-0.576330\pi$
$11$	−1.57543	−0.475011	−0.237505	+	0.971386i	$0.576330\pi$
$12$	−2.91859	−0.842525
$13$	0.768776	0.213220	0.106610	−	0.994301i	$-0.466000\pi$
$13$	0.768776	0.213220	0.106610	+	0.994301i	$0.466000\pi$
$14$	0.518451	0.138562
$15$	3.42065	0.883208
$16$	3.69682	0.924204
$17$	1.66603	0.404072	0.202036	−	0.979378i	$-0.435244\pi$
$17$	1.66603	0.404072	0.202036	+	0.979378i	$0.435244\pi$
$18$	0.171036	0.0403137
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−4.45221	−0.995544
$21$	−3.43900	−0.750451
$22$	0.355655	0.0758259
$23$	−7.57889	−1.58031	−0.790154	−	0.612908i	$-0.790000\pi$
$23$	−7.57889	−1.58031	−0.790154	+	0.612908i	$0.790000\pi$
$24$	1.33498	0.272501
$25$	0.218082	0.0436163
$26$	−0.173552	−0.0340363
$27$	−5.62688	−1.08289
$28$	4.47609	0.845902
$29$	5.28827	0.982007	0.491003	−	0.871158i	$-0.336630\pi$
$29$	5.28827	0.982007	0.491003	+	0.871158i	$0.336630\pi$
$30$	−0.772214	−0.140986
$31$	9.38342	1.68531	0.842656	−	0.538453i	$-0.180991\pi$
$31$	9.38342	1.68531	0.842656	+	0.538453i	$0.180991\pi$
$32$	−2.61756	−0.462723
$33$	−2.35914	−0.410673
$34$	−0.376108	−0.0645019
$35$	−5.24607	−0.886748
$36$	1.47666	0.246109
$37$	−0.0976433	−0.0160525	−0.00802623	−	0.999968i	$-0.502555\pi$
$37$	−0.0976433	−0.0160525	−0.00802623	+	0.999968i	$0.502555\pi$
$38$	−0.225751	−0.0366216
$39$	1.15121	0.184341
$40$	2.03646	0.321993
$41$	−7.54895	−1.17895	−0.589474	−	0.807788i	$-0.700665\pi$
$41$	−7.54895	−1.17895	−0.589474	+	0.807788i	$0.700665\pi$
$42$	0.776357	0.119794
$43$	11.7715	1.79514	0.897568	−	0.440876i	$-0.145332\pi$
$43$	11.7715	1.79514	0.897568	+	0.440876i	$0.145332\pi$
$44$	3.07058	0.462907
$45$	−1.73067	−0.257993
$46$	1.71094	0.252264
$47$	−6.91598	−1.00880	−0.504399	−	0.863470i	$-0.668286\pi$
$47$	−6.91598	−1.00880	−0.504399	+	0.863470i	$0.668286\pi$
$48$	5.53581	0.799025
$49$	−1.72579	−0.246541
$50$	−0.0492321	−0.00696247
$51$	2.49480	0.349342
$52$	−1.49837	−0.207787
$53$	−10.4319	−1.43293	−0.716463	−	0.697626i	$-0.754240\pi$
$53$	−10.4319	−1.43293	−0.716463	+	0.697626i	$0.754240\pi$
$54$	1.27027	0.172862
$55$	−3.59878	−0.485259
$56$	−2.04738	−0.273593
$57$	1.49745	0.198343
$58$	−1.19383	−0.156758
$59$	−11.0971	−1.44472	−0.722358	−	0.691519i	$-0.756942\pi$
$59$	−11.0971	−1.44472	−0.722358	+	0.691519i	$0.756942\pi$
$60$	−6.66697	−0.860702
$61$	2.95668	0.378564	0.189282	−	0.981923i	$-0.439384\pi$
$61$	2.95668	0.378564	0.189282	+	0.981923i	$0.439384\pi$
$62$	−2.11831	−0.269026
$63$	1.73996	0.219214
$64$	−6.80272	−0.850340
$65$	1.75613	0.217821
$66$	0.532577	0.0655557
$67$	−10.8424	−1.32461	−0.662307	−	0.749232i	$-0.730423\pi$
$67$	−10.8424	−1.32461	−0.662307	+	0.749232i	$0.730423\pi$
$68$	−3.24715	−0.393775
$69$	−11.3490	−1.36626
$70$	1.18430	0.141552
$71$	5.30306	0.629358	0.314679	−	0.949198i	$-0.398103\pi$
$71$	5.30306	0.629358	0.314679	+	0.949198i	$0.398103\pi$
$72$	−0.675429	−0.0796001
$73$	−3.40422	−0.398433	−0.199217	−	0.979955i	$-0.563840\pi$
$73$	−3.40422	−0.398433	−0.199217	+	0.979955i	$0.563840\pi$
$74$	0.0220431	0.00256245
$75$	0.326567	0.0377087
$76$	−1.94904	−0.223570
$77$	3.61809	0.412319
$78$	−0.259886	−0.0294263
$79$	2.02379	0.227694	0.113847	−	0.993498i	$-0.463683\pi$
$79$	2.02379	0.227694	0.113847	+	0.993498i	$0.463683\pi$
$80$	8.44468	0.944144
$81$	−6.15309	−0.683676
$82$	1.70418	0.188195
$83$	−6.72746	−0.738435	−0.369218	−	0.929343i	$-0.620374\pi$
$83$	−6.72746	−0.738435	−0.369218	+	0.929343i	$0.620374\pi$
$84$	6.70274	0.731328
$85$	3.80573	0.412790
$86$	−2.65742	−0.286557
$87$	7.91893	0.848999
$88$	−1.40450	−0.149720
$89$	−15.5984	−1.65343	−0.826714	−	0.562622i	$-0.809793\pi$
$89$	−15.5984	−1.65343	−0.826714	+	0.562622i	$0.809793\pi$
$90$	0.390701	0.0411835
$91$	−1.76555	−0.185079
$92$	14.7715	1.54004
$93$	14.0512	1.45704
$94$	1.56129	0.161034
$95$	2.28431	0.234365
$96$	−3.91967	−0.400049
$97$	0.271021	0.0275180	0.0137590	−	0.999905i	$-0.495620\pi$
$97$	0.271021	0.0275180	0.0137590	+	0.999905i	$0.495620\pi$
$98$	0.389598	0.0393553
$99$	1.19360	0.119962
$100$	−0.425049	−0.0425049
$101$	12.8807	1.28168	0.640840	−	0.767674i	$-0.278586\pi$
$101$	12.8807	1.28168	0.640840	+	0.767674i	$0.278586\pi$
$102$	−0.563204	−0.0557655
$103$	6.27812	0.618602	0.309301	−	0.950964i	$-0.399905\pi$
$103$	6.27812	0.618602	0.309301	+	0.950964i	$0.399905\pi$
$104$	0.685363	0.0672053
$105$	−7.85575	−0.766642
$106$	2.35500	0.228738
$107$	2.58781	0.250173	0.125086	−	0.992146i	$-0.460079\pi$
$107$	2.58781	0.250173	0.125086	+	0.992146i	$0.460079\pi$
$108$	10.9670	1.05530
$109$	−6.20940	−0.594752	−0.297376	−	0.954760i	$-0.596112\pi$
$109$	−6.20940	−0.594752	−0.297376	+	0.954760i	$0.596112\pi$
$110$	0.812427	0.0774619
$111$	−0.146216	−0.0138782
$112$	−8.48998	−0.802228
$113$	−6.52137	−0.613479	−0.306739	−	0.951794i	$-0.599238\pi$
$113$	−6.52137	−0.613479	−0.306739	+	0.951794i	$0.599238\pi$
$114$	−0.338051	−0.0316614
$115$	−17.3126	−1.61440
$116$	−10.3070	−0.956983
$117$	−0.582451	−0.0538476
$118$	2.50517	0.230620
$119$	−3.82615	−0.350742
$120$	3.04950	0.278380
$121$	−8.51801	−0.774365
$122$	−0.667472	−0.0604301
$123$	−11.3042	−1.01926
$124$	−18.2886	−1.64237
$125$	−10.9234	−0.977018
$126$	−0.392796	−0.0349931
$127$	1.69962	0.150817	0.0754084	−	0.997153i	$-0.475974\pi$
$127$	1.69962	0.150817	0.0754084	+	0.997153i	$0.475974\pi$
$128$	6.77083	0.598462
$129$	17.6273	1.55199
$130$	−0.396447	−0.0347707
$131$	−19.0687	−1.66604	−0.833018	−	0.553245i	$-0.813389\pi$
$131$	−19.0687	−1.66604	−0.833018	+	0.553245i	$0.813389\pi$
$132$	4.59804	0.400208
$133$	−2.29657	−0.199137
$134$	2.44769	0.211448
$135$	−12.8536	−1.10626
$136$	1.48526	0.127360
$137$	−17.4784	−1.49328	−0.746640	−	0.665229i	$-0.768334\pi$
$137$	−17.4784	−1.49328	−0.746640	+	0.665229i	$0.768334\pi$
$138$	2.56205	0.218096
$139$	−3.69604	−0.313494	−0.156747	−	0.987639i	$-0.550101\pi$
$139$	−3.69604	−0.313494	−0.156747	+	0.987639i	$0.550101\pi$
$140$	10.2248	0.864152
$141$	−10.3564	−0.872162
$142$	−1.19717	−0.100464
$143$	−1.21116	−0.101282
$144$	−2.80083	−0.233403
$145$	12.0801	1.00319
$146$	0.768505	0.0636019
$147$	−2.58428	−0.213148
$148$	0.190310	0.0156434
$149$	−15.1512	−1.24123	−0.620617	−	0.784114i	$-0.713118\pi$
$149$	−15.1512	−1.24123	−0.620617	+	0.784114i	$0.713118\pi$
$150$	−0.0737227	−0.00601944
$151$	−16.4680	−1.34015	−0.670075	−	0.742293i	$-0.733738\pi$
$151$	−16.4680	−1.34015	−0.670075	+	0.742293i	$0.733738\pi$
$152$	0.891498	0.0723100
$153$	−1.26224	−0.102046
$154$	−0.816786	−0.0658185
$155$	21.4347	1.72167
$156$	−2.24374	−0.179643
$157$	21.1047	1.68434	0.842170	−	0.539212i	$-0.181278\pi$
$157$	21.1047	1.68434	0.842170	+	0.539212i	$0.181278\pi$
$158$	−0.456872	−0.0363468
$159$	−15.6212	−1.23884
$160$	−5.97931	−0.472706
$161$	17.4054	1.37174
$162$	1.38906	0.109135
$163$	−11.3287	−0.887336	−0.443668	−	0.896191i	$-0.646323\pi$
$163$	−11.3287	−0.887336	−0.443668	+	0.896191i	$0.646323\pi$
$164$	14.7132	1.14891
$165$	−5.38901	−0.419534
$166$	1.51873	0.117876
$167$	2.32139	0.179634	0.0898171	−	0.995958i	$-0.471372\pi$
$167$	2.32139	0.179634	0.0898171	+	0.995958i	$0.471372\pi$
$168$	−3.06586	−0.236536
$169$	−12.4090	−0.954537
$170$	−0.859147	−0.0658936
$171$	−0.757634	−0.0579377
$172$	−22.9431	−1.74939
$173$	10.8344	0.823728	0.411864	−	0.911245i	$-0.364878\pi$
$173$	10.8344	0.823728	0.411864	+	0.911245i	$0.364878\pi$
$174$	−1.78771	−0.135526
$175$	−0.500839	−0.0378598
$176$	−5.82409	−0.439007
$177$	−16.6174	−1.24904
$178$	3.52135	0.263937
$179$	13.4631	1.00628	0.503139	−	0.864205i	$-0.332178\pi$
$179$	13.4631	1.00628	0.503139	+	0.864205i	$0.332178\pi$
$180$	3.37314	0.251419
$181$	25.6236	1.90459	0.952294	−	0.305181i	$-0.0987171\pi$
$181$	25.6236	1.90459	0.952294	+	0.305181i	$0.0987171\pi$
$182$	0.398573	0.0295442
$183$	4.42749	0.327289
$184$	−6.75657	−0.498101
$185$	−0.223048	−0.0163988
$186$	−3.17208	−0.232588
$187$	−2.62472	−0.191939
$188$	13.4795	0.983093
$189$	12.9225	0.939974
$190$	−0.515685	−0.0374117
$191$	−6.34501	−0.459108	−0.229554	−	0.973296i	$-0.573727\pi$
$191$	−6.34501	−0.459108	−0.229554	+	0.973296i	$0.573727\pi$
$192$	−10.1868	−0.735166
$193$	8.08444	0.581931	0.290965	−	0.956734i	$-0.406024\pi$
$193$	8.08444	0.581931	0.290965	+	0.956734i	$0.406024\pi$
$194$	−0.0611832	−0.00439270
$195$	2.62972	0.188318
$196$	3.36362	0.240259
$197$	8.38767	0.597597	0.298799	−	0.954316i	$-0.403414\pi$
$197$	8.38767	0.597597	0.298799	+	0.954316i	$0.403414\pi$
$198$	−0.269457	−0.0191494
$199$	−3.26382	−0.231366	−0.115683	−	0.993286i	$-0.536906\pi$
$199$	−3.26382	−0.231366	−0.115683	+	0.993286i	$0.536906\pi$
$200$	0.194419	0.0137475
$201$	−16.2360	−1.14520
$202$	−2.90784	−0.204595
$203$	−12.1449	−0.852402
$204$	−4.86246	−0.340440
$205$	−17.2441	−1.20438
$206$	−1.41729	−0.0987473
$207$	5.74203	0.399098
$208$	2.84203	0.197059
$209$	−1.57543	−0.108975
$210$	1.77344	0.122379
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	20.3321	1.39641
$213$	7.94109	0.544115
$214$	−0.584200	−0.0399351
$215$	26.8898	1.83387
$216$	−5.01635	−0.341320
$217$	−21.5496	−1.46288
$218$	1.40178	0.0949403
$219$	−5.09766	−0.344468
$220$	7.01416	0.472894
$221$	1.28081	0.0861563
$222$	0.0330084	0.00221538
$223$	−10.3137	−0.690657	−0.345329	−	0.938482i	$-0.612233\pi$
$223$	−10.3137	−0.690657	−0.345329	+	0.938482i	$0.612233\pi$
$224$	6.01139	0.401653
$225$	−0.165226	−0.0110151
$226$	1.47220	0.0979295
$227$	−14.6299	−0.971021	−0.485511	−	0.874231i	$-0.661366\pi$
$227$	−14.6299	−0.971021	−0.485511	+	0.874231i	$0.661366\pi$
$228$	−2.91859	−0.193288
$229$	−13.1082	−0.866211	−0.433106	−	0.901343i	$-0.642582\pi$
$229$	−13.1082	−0.866211	−0.433106	+	0.901343i	$0.642582\pi$
$230$	3.90832	0.257707
$231$	5.41791	0.356473
$232$	4.71448	0.309521
$233$	17.0456	1.11670	0.558348	−	0.829607i	$-0.311435\pi$
$233$	17.0456	1.11670	0.558348	+	0.829607i	$0.311435\pi$
$234$	0.131489	0.00859569
$235$	−15.7983	−1.03056
$236$	21.6286	1.40790
$237$	3.03053	0.196854
$238$	0.863756	0.0559890
$239$	−16.1964	−1.04765	−0.523827	−	0.851824i	$-0.675496\pi$
$239$	−16.1964	−1.04765	−0.523827	+	0.851824i	$0.675496\pi$
$240$	12.6455	0.816265
$241$	15.5667	1.00274	0.501370	−	0.865233i	$-0.332829\pi$
$241$	15.5667	1.00274	0.501370	+	0.865233i	$0.332829\pi$
$242$	1.92295	0.123612
$243$	7.66668	0.491818
$244$	−5.76267	−0.368917
$245$	−3.94224	−0.251860
$246$	2.55193	0.162705
$247$	0.768776	0.0489161
$248$	8.36530	0.531197
$249$	−10.0741	−0.638418
$250$	2.46596	0.155961
$251$	−0.343149	−0.0216593	−0.0108297	−	0.999941i	$-0.503447\pi$
$251$	−0.343149	−0.0216593	−0.0108297	+	0.999941i	$0.503447\pi$
$252$	−3.39124	−0.213628
$253$	11.9400	0.750664
$254$	−0.383690	−0.0240749
$255$	5.69891	0.356879
$256$	12.0769	0.754807
$257$	24.2911	1.51524	0.757619	−	0.652697i	$-0.226363\pi$
$257$	24.2911	1.51524	0.757619	+	0.652697i	$0.226363\pi$
$258$	−3.97937	−0.247745
$259$	0.224244	0.0139339
$260$	−3.42275	−0.212270
$261$	−4.00657	−0.248001
$262$	4.30476	0.265949
$263$	2.24957	0.138715	0.0693573	−	0.997592i	$-0.477905\pi$
$263$	2.24957	0.138715	0.0693573	+	0.997592i	$0.477905\pi$
$264$	−2.10317	−0.129441
$265$	−23.8296	−1.46384
$266$	0.518451	0.0317883
$267$	−23.3579	−1.42948
$268$	21.1323	1.29086
$269$	−1.69109	−0.103108	−0.0515539	−	0.998670i	$-0.516417\pi$
$269$	−1.69109	−0.103108	−0.0515539	+	0.998670i	$0.516417\pi$
$270$	2.90170	0.176592
$271$	−12.0716	−0.733297	−0.366648	−	0.930360i	$-0.619495\pi$
$271$	−12.0716	−0.733297	−0.366648	+	0.930360i	$0.619495\pi$
$272$	6.15901	0.373445
$273$	−2.64382	−0.160011
$274$	3.94576	0.238372
$275$	−0.343573	−0.0207182
$276$	22.1197	1.33145
$277$	−0.141471	−0.00850019	−0.00425010	−	0.999991i	$-0.501353\pi$
$277$	−0.141471	−0.00850019	−0.00425010	+	0.999991i	$0.501353\pi$
$278$	0.834383	0.0500430
$279$	−7.10920	−0.425617
$280$	−4.67686	−0.279496
$281$	18.0149	1.07468	0.537339	−	0.843366i	$-0.319430\pi$
$281$	18.0149	1.07468	0.537339	+	0.843366i	$0.319430\pi$
$282$	2.33795	0.139223
$283$	−23.1446	−1.37580	−0.687902	−	0.725804i	$-0.741468\pi$
$283$	−23.1446	−1.37580	−0.687902	+	0.725804i	$0.741468\pi$
$284$	−10.3359	−0.613321
$285$	3.42065	0.202622
$286$	0.273419	0.0161676
$287$	17.3367	1.02335
$288$	1.98315	0.116858
$289$	−14.2243	−0.836726
$290$	−2.72708	−0.160140
$291$	0.405841	0.0237908
$292$	6.63494	0.388281
$293$	12.4418	0.726856	0.363428	−	0.931622i	$-0.381606\pi$
$293$	12.4418	0.726856	0.363428	+	0.931622i	$0.381606\pi$
$294$	0.583404	0.0340248
$295$	−25.3492	−1.47589
$296$	−0.0870488	−0.00505961
$297$	8.86477	0.514386
$298$	3.42039	0.198138
$299$	−5.82647	−0.336954
$300$	−0.636491	−0.0367478
$301$	−27.0340	−1.55821
$302$	3.71767	0.213928
$303$	19.2883	1.10808
$304$	3.69682	0.212027
$305$	6.75398	0.386732
$306$	0.284952	0.0162896
$307$	−5.14310	−0.293532	−0.146766	−	0.989171i	$-0.546886\pi$
$307$	−5.14310	−0.293532	−0.146766	+	0.989171i	$0.546886\pi$
$308$	−7.05178	−0.401812
$309$	9.40119	0.534815
$310$	−4.83889	−0.274830
$311$	−14.3109	−0.811494	−0.405747	−	0.913985i	$-0.632989\pi$
$311$	−14.3109	−0.811494	−0.405747	+	0.913985i	$0.632989\pi$
$312$	1.02630	0.0581027
$313$	17.4427	0.985918	0.492959	−	0.870052i	$-0.335915\pi$
$313$	17.4427	0.985918	0.492959	+	0.870052i	$0.335915\pi$
$314$	−4.76441	−0.268871
$315$	3.97460	0.223944
$316$	−3.94444	−0.221892
$317$	−3.03095	−0.170235	−0.0851177	−	0.996371i	$-0.527127\pi$
$317$	−3.03095	−0.170235	−0.0851177	+	0.996371i	$0.527127\pi$
$318$	3.52650	0.197756
$319$	−8.33131	−0.466464
$320$	−15.5395	−0.868686
$321$	3.87512	0.216288
$322$	−3.92929	−0.218971
$323$	1.66603	0.0927004
$324$	11.9926	0.666255
$325$	0.167656	0.00929988
$326$	2.55747	0.141645
$327$	−9.29828	−0.514196
$328$	−6.72987	−0.371595
$329$	15.8830	0.875658
$330$	1.21657	0.0669701
$331$	31.6013	1.73697	0.868483	−	0.495719i	$-0.165096\pi$
$331$	31.6013	1.73697	0.868483	+	0.495719i	$0.165096\pi$
$332$	13.1121	0.719619
$333$	0.0739779	0.00405397
$334$	−0.524055	−0.0286750
$335$	−24.7675	−1.35319
$336$	−12.7134	−0.693570
$337$	7.99863	0.435713	0.217857	−	0.975981i	$-0.430093\pi$
$337$	7.99863	0.435713	0.217857	+	0.975981i	$0.430093\pi$
$338$	2.80134	0.152373
$339$	−9.76544	−0.530386
$340$	−7.41751	−0.402271
$341$	−14.7829	−0.800541
$342$	0.171036	0.00924859
$343$	20.0393	1.08202
$344$	10.4943	0.565813
$345$	−25.9247	−1.39574
$346$	−2.44588	−0.131492
$347$	29.9506	1.60783	0.803917	−	0.594742i	$-0.202746\pi$
$347$	29.9506	1.60783	0.803917	+	0.594742i	$0.202746\pi$
$348$	−15.4343	−0.827365
$349$	1.23765	0.0662500	0.0331250	−	0.999451i	$-0.489454\pi$
$349$	1.23765	0.0662500	0.0331250	+	0.999451i	$0.489454\pi$
$350$	0.113065	0.00604356
$351$	−4.32581	−0.230895
$352$	4.12378	0.219798
$353$	−13.0173	−0.692842	−0.346421	−	0.938079i	$-0.612603\pi$
$353$	−13.0173	−0.692842	−0.346421	+	0.938079i	$0.612603\pi$
$354$	3.75138	0.199384
$355$	12.1139	0.642937
$356$	30.4019	1.61130
$357$	−5.72948	−0.303236
$358$	−3.03930	−0.160632
$359$	−5.35397	−0.282571	−0.141286	−	0.989969i	$-0.545124\pi$
$359$	−5.35397	−0.282571	−0.141286	+	0.989969i	$0.545124\pi$
$360$	−1.54289	−0.0813175
$361$	1.00000	0.0526316
$362$	−5.78455	−0.304029
$363$	−12.7553	−0.669481
$364$	3.44111	0.180363
$365$	−7.77629	−0.407030
$366$	−0.999509	−0.0522452
$367$	12.8245	0.669435	0.334718	−	0.942319i	$-0.391359\pi$
$367$	12.8245	0.669435	0.334718	+	0.942319i	$0.391359\pi$
$368$	−28.0178	−1.46053
$369$	5.71934	0.297737
$370$	0.0503532	0.00261774
$371$	23.9574	1.24381
$372$	−27.3864	−1.41992
$373$	−14.5413	−0.752918	−0.376459	−	0.926433i	$-0.622858\pi$
$373$	−14.5413	−0.752918	−0.376459	+	0.926433i	$0.622858\pi$
$374$	0.592532	0.0306391
$375$	−16.3573	−0.844686
$376$	−6.16558	−0.317966
$377$	4.06550	0.209384
$378$	−2.91726	−0.150048
$379$	−20.1799	−1.03657	−0.518287	−	0.855207i	$-0.673430\pi$
$379$	−20.1799	−1.03657	−0.518287	+	0.855207i	$0.673430\pi$
$380$	−4.45221	−0.228393
$381$	2.54510	0.130389
$382$	1.43239	0.0732874
$383$	−27.4429	−1.40227	−0.701133	−	0.713030i	$-0.747322\pi$
$383$	−27.4429	−1.40227	−0.701133	+	0.713030i	$0.747322\pi$
$384$	10.1390	0.517404
$385$	8.26484	0.421215
$386$	−1.82507	−0.0928936
$387$	−8.91849	−0.453352
$388$	−0.528230	−0.0268168
$389$	−14.2869	−0.724372	−0.362186	−	0.932106i	$-0.617970\pi$
$389$	−14.2869	−0.724372	−0.362186	+	0.932106i	$0.617970\pi$
$390$	−0.593660	−0.0300612
$391$	−12.6267	−0.638558
$392$	−1.53854	−0.0777078
$393$	−28.5544	−1.44038
$394$	−1.89352	−0.0953944
$395$	4.62297	0.232607
$396$	−2.32637	−0.116905
$397$	10.0602	0.504906	0.252453	−	0.967609i	$-0.418763\pi$
$397$	10.0602	0.504906	0.252453	+	0.967609i	$0.418763\pi$
$398$	0.736810	0.0369330
$399$	−3.43900	−0.172165
$400$	0.806208	0.0403104
$401$	1.18402	0.0591269	0.0295635	−	0.999563i	$-0.490588\pi$
$401$	1.18402	0.0591269	0.0295635	+	0.999563i	$0.490588\pi$
$402$	3.66530	0.182809
$403$	7.21375	0.359343
$404$	−25.1050	−1.24902
$405$	−14.0556	−0.698427
$406$	2.74171	0.136069
$407$	0.153831	0.00762510
$408$	2.22411	0.110110
$409$	11.8681	0.586840	0.293420	−	0.955984i	$-0.405207\pi$
$409$	11.8681	0.586840	0.293420	+	0.955984i	$0.405207\pi$
$410$	3.89288	0.192256
$411$	−26.1731	−1.29102
$412$	−12.2363	−0.602839
$413$	25.4852	1.25404
$414$	−1.29627	−0.0637080
$415$	−15.3676	−0.754367
$416$	−2.01231	−0.0986619
$417$	−5.53464	−0.271033
$418$	0.355655	0.0173957
$419$	21.0041	1.02612	0.513059	−	0.858353i	$-0.328512\pi$
$419$	21.0041	1.02612	0.513059	+	0.858353i	$0.328512\pi$
$420$	15.3111	0.747107
$421$	−24.3609	−1.18728	−0.593640	−	0.804731i	$-0.702309\pi$
$421$	−24.3609	−1.18728	−0.593640	+	0.804731i	$0.702309\pi$
$422$	−0.225751	−0.0109894
$423$	5.23978	0.254767
$424$	−9.29998	−0.451647
$425$	0.363331	0.0176241
$426$	−1.79271	−0.0868570
$427$	−6.79021	−0.328601
$428$	−5.04373	−0.243798
$429$	−1.81365	−0.0875638
$430$	−6.07039	−0.292740
$431$	−8.22374	−0.396124	−0.198062	−	0.980190i	$-0.563465\pi$
$431$	−8.22374	−0.396124	−0.198062	+	0.980190i	$0.563465\pi$
$432$	−20.8015	−1.00081
$433$	34.8071	1.67272	0.836362	−	0.548178i	$-0.184678\pi$
$433$	34.8071	1.67272	0.836362	+	0.548178i	$0.184678\pi$
$434$	4.86485	0.233520
$435$	18.0893	0.867316
$436$	12.1023	0.579597
$437$	−7.57889	−0.362548
$438$	1.15080	0.0549873
$439$	−16.7210	−0.798049	−0.399024	−	0.916940i	$-0.630651\pi$
$439$	−16.7210	−0.798049	−0.399024	+	0.916940i	$0.630651\pi$
$440$	−3.20831	−0.152950
$441$	1.30751	0.0622626
$442$	−0.289143	−0.0137531
$443$	−32.3971	−1.53923	−0.769616	−	0.638507i	$-0.779552\pi$
$443$	−32.3971	−1.53923	−0.769616	+	0.638507i	$0.779552\pi$
$444$	0.284981	0.0135246
$445$	−35.6316	−1.68910
$446$	2.32833	0.110250
$447$	−22.6882	−1.07312
$448$	15.6229	0.738112
$449$	−12.8039	−0.604255	−0.302128	−	0.953267i	$-0.597697\pi$
$449$	−12.8039	−0.604255	−0.302128	+	0.953267i	$0.597697\pi$
$450$	0.0372999	0.00175833
$451$	11.8929	0.560013
$452$	12.7104	0.597846
$453$	−24.6601	−1.15863
$454$	3.30271	0.155004
$455$	−4.03306	−0.189073
$456$	1.33498	0.0625160
$457$	−1.31915	−0.0617073	−0.0308536	−	0.999524i	$-0.509823\pi$
$457$	−1.31915	−0.0617073	−0.0308536	+	0.999524i	$0.509823\pi$
$458$	2.95918	0.138273
$459$	−9.37456	−0.437567
$460$	33.7428	1.57327
$461$	17.7309	0.825812	0.412906	−	0.910774i	$-0.364514\pi$
$461$	17.7309	0.825812	0.412906	+	0.910774i	$0.364514\pi$
$462$	−1.22310	−0.0569037
$463$	−21.3679	−0.993050	−0.496525	−	0.868022i	$-0.665391\pi$
$463$	−21.3679	−0.993050	−0.496525	+	0.868022i	$0.665391\pi$
$464$	19.5498	0.907575
$465$	32.0974	1.48848
$466$	−3.84806	−0.178258
$467$	4.29755	0.198867	0.0994335	−	0.995044i	$-0.468297\pi$
$467$	4.29755	0.198867	0.0994335	+	0.995044i	$0.468297\pi$
$468$	1.13522	0.0524755
$469$	24.9004	1.14979
$470$	3.56647	0.164509
$471$	31.6033	1.45620
$472$	−9.89302	−0.455363
$473$	−18.5452	−0.852709
$474$	−0.684144	−0.0314238
$475$	0.218082	0.0100063
$476$	7.45730	0.341805
$477$	7.90353	0.361878
$478$	3.65634	0.167237
$479$	12.5089	0.571548	0.285774	−	0.958297i	$-0.407749\pi$
$479$	12.5089	0.571548	0.285774	+	0.958297i	$0.407749\pi$
$480$	−8.95374	−0.408680
$481$	−0.0750659	−0.00342271
$482$	−3.51420	−0.160067
$483$	26.0638	1.18594
$484$	16.6019	0.754632
$485$	0.619096	0.0281117
$486$	−1.73076	−0.0785088
$487$	−23.5515	−1.06722	−0.533609	−	0.845731i	$-0.679165\pi$
$487$	−23.5515	−1.06722	−0.533609	+	0.845731i	$0.679165\pi$
$488$	2.63587	0.119320
$489$	−16.9643	−0.767150
$490$	0.889963	0.0402044
$491$	−7.30373	−0.329613	−0.164806	−	0.986326i	$-0.552700\pi$
$491$	−7.30373	−0.329613	−0.164806	+	0.986326i	$0.552700\pi$
$492$	22.0323	0.993292
$493$	8.81042	0.396801
$494$	−0.173552	−0.00780847
$495$	2.72656	0.122550
$496$	34.6888	1.55757
$497$	−12.1788	−0.546295
$498$	2.27423	0.101911
$499$	15.7200	0.703726	0.351863	−	0.936052i	$-0.385548\pi$
$499$	15.7200	0.703726	0.351863	+	0.936052i	$0.385548\pi$
$500$	21.2901	0.952122
$501$	3.47617	0.155304
$502$	0.0774661	0.00345748
$503$	−8.06870	−0.359765	−0.179883	−	0.983688i	$-0.557572\pi$
$503$	−8.06870	−0.359765	−0.179883	+	0.983688i	$0.557572\pi$
$504$	1.55117	0.0690945
$505$	29.4236	1.30933
$506$	−2.69547	−0.119828
$507$	−18.5819	−0.825250
$508$	−3.31262	−0.146974
$509$	1.24272	0.0550826	0.0275413	−	0.999621i	$-0.491232\pi$
$509$	1.24272	0.0550826	0.0275413	+	0.999621i	$0.491232\pi$
$510$	−1.28653	−0.0569686
$511$	7.81801	0.345848
$512$	−16.2680	−0.718952
$513$	−5.62688	−0.248433
$514$	−5.48374	−0.241877
$515$	14.3412	0.631948
$516$	−34.3562	−1.51245
$517$	10.8957	0.479191
$518$	−0.0506233	−0.00222426
$519$	16.2241	0.712158
$520$	1.56558	0.0686553
$521$	−28.5862	−1.25238	−0.626191	−	0.779670i	$-0.715387\pi$
$521$	−28.5862	−1.25238	−0.626191	+	0.779670i	$0.715387\pi$
$522$	0.904487	0.0395883
$523$	7.85270	0.343374	0.171687	−	0.985152i	$-0.445078\pi$
$523$	7.85270	0.343374	0.171687	+	0.985152i	$0.445078\pi$
$524$	37.1655	1.62358
$525$	−0.749982	−0.0327319
$526$	−0.507843	−0.0221430
$527$	15.6331	0.680987
$528$	−8.72130	−0.379546
$529$	34.4396	1.49737
$530$	5.37955	0.233673
$531$	8.40752	0.364855
$532$	4.47609	0.194063
$533$	−5.80345	−0.251375
$534$	5.27306	0.228188
$535$	5.91136	0.255570
$536$	−9.66601	−0.417508
$537$	20.1603	0.869983
$538$	0.381766	0.0164591
$539$	2.71886	0.117110
$540$	25.0520	1.07807
$541$	20.7417	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$541$	20.7417	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$542$	2.72517	0.117056
$543$	38.3702	1.64662
$544$	−4.36093	−0.186973
$545$	−14.1842	−0.607584
$546$	0.596845	0.0255426
$547$	−11.2094	−0.479280	−0.239640	−	0.970862i	$-0.577029\pi$
$547$	−11.2094	−0.479280	−0.239640	+	0.970862i	$0.577029\pi$
$548$	34.0660	1.45523
$549$	−2.24008	−0.0956043
$550$	0.0775618	0.00330725
$551$	5.28827	0.225288
$552$	−10.1176	−0.430635
$553$	−4.64777	−0.197643
$554$	0.0319373	0.00135688
$555$	−0.334004	−0.0141777
$556$	7.20371	0.305505
$557$	−7.91889	−0.335534	−0.167767	−	0.985827i	$-0.553656\pi$
$557$	−7.91889	−0.335534	−0.167767	+	0.985827i	$0.553656\pi$
$558$	1.60491	0.0679411
$559$	9.04965	0.382759
$560$	−19.3938	−0.819536
$561$	−3.93039	−0.165941
$562$	−4.06687	−0.171551
$563$	−17.2233	−0.725875	−0.362937	−	0.931814i	$-0.618226\pi$
$563$	−17.2233	−0.725875	−0.362937	+	0.931814i	$0.618226\pi$
$564$	20.1849	0.849938
$565$	−14.8968	−0.626715
$566$	5.22491	0.219619
$567$	14.1310	0.593445
$568$	4.72767	0.198369
$569$	−7.71802	−0.323556	−0.161778	−	0.986827i	$-0.551723\pi$
$569$	−7.71802	−0.323556	−0.161778	+	0.986827i	$0.551723\pi$
$570$	−0.772214	−0.0323445
$571$	4.49547	0.188129	0.0940647	−	0.995566i	$-0.470014\pi$
$571$	4.49547	0.188129	0.0940647	+	0.995566i	$0.470014\pi$
$572$	2.36059	0.0987011
$573$	−9.50135	−0.396925
$574$	−3.91376	−0.163357
$575$	−1.65282	−0.0689272
$576$	5.15397	0.214749
$577$	−23.1086	−0.962024	−0.481012	−	0.876714i	$-0.659731\pi$
$577$	−23.1086	−0.962024	−0.481012	+	0.876714i	$0.659731\pi$
$578$	3.21116	0.133566
$579$	12.1061	0.503111
$580$	−23.5445	−0.977631
$581$	15.4501	0.640977
$582$	−0.0916189	−0.00379773
$583$	16.4347	0.680655
$584$	−3.03485	−0.125583
$585$	−1.33050	−0.0550094
$586$	−2.80874	−0.116028
$587$	−12.0423	−0.497041	−0.248520	−	0.968627i	$-0.579944\pi$
$587$	−12.0423	−0.497041	−0.248520	+	0.968627i	$0.579944\pi$
$588$	5.03687	0.207717
$589$	9.38342	0.386637
$590$	5.72260	0.235596
$591$	12.5601	0.516656
$592$	−0.360970	−0.0148358
$593$	8.12137	0.333505	0.166752	−	0.985999i	$-0.446672\pi$
$593$	8.12137	0.333505	0.166752	+	0.985999i	$0.446672\pi$
$594$	−2.00123	−0.0821114
$595$	−8.74012	−0.358310
$596$	29.5302	1.20961
$597$	−4.88742	−0.200029
$598$	1.31533	0.0537879
$599$	42.4768	1.73555	0.867777	−	0.496953i	$-0.165548\pi$
$599$	42.4768	1.73555	0.867777	+	0.496953i	$0.165548\pi$
$600$	0.291134	0.0118855
$601$	21.1024	0.860784	0.430392	−	0.902642i	$-0.358375\pi$
$601$	21.1024	0.860784	0.430392	+	0.902642i	$0.358375\pi$
$602$	6.10295	0.248738
$603$	8.21460	0.334524
$604$	32.0968	1.30600
$605$	−19.4578	−0.791072
$606$	−4.35435	−0.176883
$607$	−8.89882	−0.361192	−0.180596	−	0.983557i	$-0.557803\pi$
$607$	−8.89882	−0.361192	−0.180596	+	0.983557i	$0.557803\pi$
$608$	−2.61756	−0.106156
$609$	−18.1863	−0.736948
$610$	−1.52472	−0.0617339
$611$	−5.31684	−0.215096
$612$	2.46016	0.0994459
$613$	10.2937	0.415757	0.207878	−	0.978155i	$-0.433344\pi$
$613$	10.2937	0.415757	0.207878	+	0.978155i	$0.433344\pi$
$614$	1.16106	0.0468565
$615$	−25.8223	−1.04126
$616$	3.22552	0.129960
$617$	41.6812	1.67802	0.839011	−	0.544115i	$-0.183135\pi$
$617$	41.6812	1.67802	0.839011	+	0.544115i	$0.183135\pi$
$618$	−2.12233	−0.0853725
$619$	−23.6513	−0.950626	−0.475313	−	0.879817i	$-0.657665\pi$
$619$	−23.6513	−0.950626	−0.475313	+	0.879817i	$0.657665\pi$
$620$	−41.7769	−1.67780
$621$	42.6455	1.71131
$622$	3.23069	0.129539
$623$	35.8228	1.43521
$624$	4.25580	0.170368
$625$	−26.0428	−1.04171
$626$	−3.93770	−0.157382
$627$	−2.35914	−0.0942149
$628$	−41.1339	−1.64142
$629$	−0.162677	−0.00648635
$630$	−0.897270	−0.0357481
$631$	−14.4345	−0.574627	−0.287313	−	0.957837i	$-0.592762\pi$
$631$	−14.4345	−0.574627	−0.287313	+	0.957837i	$0.592762\pi$
$632$	1.80420	0.0717674
$633$	1.49745	0.0595184
$634$	0.684240	0.0271747
$635$	3.88246	0.154071
$636$	30.4463	1.20727
$637$	−1.32674	−0.0525675
$638$	1.88080	0.0744616
$639$	−4.01778	−0.158941
$640$	15.4667	0.611374
$641$	34.9813	1.38168	0.690839	−	0.723008i	$-0.257241\pi$
$641$	34.9813	1.38168	0.690839	+	0.723008i	$0.257241\pi$
$642$	−0.874812	−0.0345261
$643$	−4.11707	−0.162361	−0.0811806	−	0.996699i	$-0.525869\pi$
$643$	−4.11707	−0.162361	−0.0811806	+	0.996699i	$0.525869\pi$
$644$	−33.9238	−1.33679
$645$	40.2662	1.58548
$646$	−0.376108	−0.0147978
$647$	−31.1524	−1.22473	−0.612364	−	0.790576i	$-0.709781\pi$
$647$	−31.1524	−1.22473	−0.612364	+	0.790576i	$0.709781\pi$
$648$	−5.48547	−0.215489
$649$	17.4827	0.686256
$650$	−0.0378485	−0.00148454
$651$	−32.2696	−1.26474
$652$	22.0801	0.864725
$653$	25.5998	1.00180	0.500898	−	0.865506i	$-0.333003\pi$
$653$	25.5998	1.00180	0.500898	+	0.865506i	$0.333003\pi$
$654$	2.09909	0.0820811
$655$	−43.5588	−1.70198
$656$	−27.9071	−1.08959
$657$	2.57915	0.100622
$658$	−3.58560	−0.139781
$659$	12.2662	0.477823	0.238911	−	0.971041i	$-0.423209\pi$
$659$	12.2662	0.477823	0.238911	+	0.971041i	$0.423209\pi$
$660$	10.5034	0.408843
$661$	−18.7004	−0.727361	−0.363680	−	0.931524i	$-0.618480\pi$
$661$	−18.7004	−0.727361	−0.363680	+	0.931524i	$0.618480\pi$
$662$	−7.13402	−0.277272
$663$	1.91795	0.0744868
$664$	−5.99752	−0.232749
$665$	−5.24607	−0.203434
$666$	−0.0167006	−0.000647134 0
$667$	−40.0792	−1.55187
$668$	−4.52447	−0.175057
$669$	−15.4443	−0.597111
$670$	5.59129	0.216010
$671$	−4.65805	−0.179822
$672$	9.00177	0.347251
$673$	36.1737	1.39440	0.697198	−	0.716879i	$-0.254430\pi$
$673$	36.1737	1.39440	0.697198	+	0.716879i	$0.254430\pi$
$674$	−1.80570	−0.0695529
$675$	−1.22712	−0.0472318
$676$	24.1856	0.930214
$677$	−22.3634	−0.859495	−0.429747	−	0.902949i	$-0.641397\pi$
$677$	−22.3634	−0.859495	−0.429747	+	0.902949i	$0.641397\pi$
$678$	2.20456	0.0846655
$679$	−0.622417	−0.0238862
$680$	3.39280	0.130108
$681$	−21.9076	−0.839501
$682$	3.33726	0.127790
$683$	−3.70536	−0.141782	−0.0708908	−	0.997484i	$-0.522584\pi$
$683$	−3.70536	−0.141782	−0.0708908	+	0.997484i	$0.522584\pi$
$684$	1.47666	0.0564614
$685$	−39.9261	−1.52550
$686$	−4.52390	−0.172723
$687$	−19.6288	−0.748887
$688$	43.5171	1.65907
$689$	−8.01976	−0.305529
$690$	5.85253	0.222802
$691$	−15.6188	−0.594166	−0.297083	−	0.954852i	$-0.596014\pi$
$691$	−15.6188	−0.594166	−0.297083	+	0.954852i	$0.596014\pi$
$692$	−21.1167	−0.802738
$693$	−2.74118	−0.104129
$694$	−6.76137	−0.256658
$695$	−8.44290	−0.320257
$696$	7.05971	0.267598
$697$	−12.5768	−0.476379
$698$	−0.279401	−0.0105755
$699$	25.5250	0.965445
$700$	0.976153	0.0368951
$701$	−33.1195	−1.25090	−0.625452	−	0.780262i	$-0.715086\pi$
$701$	−33.1195	−1.25090	−0.625452	+	0.780262i	$0.715086\pi$
$702$	0.976556	0.0368577
$703$	−0.0976433	−0.00368269
$704$	10.7172	0.403921
$705$	−23.6571	−0.890979
$706$	2.93867	0.110598
$707$	−29.5814	−1.11252
$708$	32.3878	1.21721
$709$	16.4062	0.616149	0.308074	−	0.951362i	$-0.400315\pi$
$709$	16.4062	0.616149	0.308074	+	0.951362i	$0.400315\pi$
$710$	−2.73471	−0.102632
$711$	−1.53329	−0.0575029
$712$	−13.9060	−0.521148
$713$	−71.1159	−2.66331
$714$	1.29343	0.0484056
$715$	−2.76666	−0.103467
$716$	−26.2401	−0.980637
$717$	−24.2533	−0.905755
$718$	1.20866	0.0451069
$719$	−35.6540	−1.32967	−0.664836	−	0.746990i	$-0.731498\pi$
$719$	−35.6540	−1.32967	−0.664836	+	0.746990i	$0.731498\pi$
$720$	−6.39798	−0.238439
$721$	−14.4181	−0.536959
$722$	−0.225751	−0.00840157
$723$	23.3104	0.866924
$724$	−49.9414	−1.85606
$725$	1.15327	0.0428315
$726$	2.87952	0.106869
$727$	−3.21317	−0.119170	−0.0595850	−	0.998223i	$-0.518978\pi$
$727$	−3.21317	−0.119170	−0.0595850	+	0.998223i	$0.518978\pi$
$728$	−1.57398	−0.0583356
$729$	29.9398	1.10888
$730$	1.75550	0.0649741
$731$	19.6117	0.725364
$732$	−8.62933	−0.318949
$733$	47.1009	1.73971	0.869856	−	0.493306i	$-0.164212\pi$
$733$	47.1009	1.73971	0.869856	+	0.493306i	$0.164212\pi$
$734$	−2.89515	−0.106862
$735$	−5.90331	−0.217747
$736$	19.8382	0.731245
$737$	17.0815	0.629207
$738$	−1.29115	−0.0475277
$739$	−7.53605	−0.277218	−0.138609	−	0.990347i	$-0.544263\pi$
$739$	−7.53605	−0.277218	−0.138609	+	0.990347i	$0.544263\pi$
$740$	0.434728	0.0159809
$741$	1.15121	0.0422906
$742$	−5.40841	−0.198549
$743$	28.4484	1.04367	0.521836	−	0.853046i	$-0.325247\pi$
$743$	28.4484	1.04367	0.521836	+	0.853046i	$0.325247\pi$
$744$	12.5266	0.459249
$745$	−34.6101	−1.26801
$746$	3.28270	0.120188
$747$	5.09696	0.186488
$748$	5.11567	0.187048
$749$	−5.94307	−0.217155
$750$	3.69267	0.134837
$751$	25.8585	0.943591	0.471796	−	0.881708i	$-0.343606\pi$
$751$	25.8585	0.943591	0.471796	+	0.881708i	$0.343606\pi$
$752$	−25.5671	−0.932336
$753$	−0.513849	−0.0187257
$754$	−0.917789	−0.0334239
$755$	−37.6181	−1.36906
$756$	−25.1864	−0.916021
$757$	−12.2394	−0.444850	−0.222425	−	0.974950i	$-0.571397\pi$
$757$	−12.2394	−0.444850	−0.222425	+	0.974950i	$0.571397\pi$
$758$	4.55563	0.165468
$759$	17.8796	0.648990
$760$	2.03646	0.0738702
$761$	34.1282	1.23715	0.618573	−	0.785727i	$-0.287711\pi$
$761$	34.1282	1.23715	0.618573	+	0.785727i	$0.287711\pi$
$762$	−0.574558	−0.0208141
$763$	14.2603	0.516257
$764$	12.3666	0.447410
$765$	−2.88335	−0.104248
$766$	6.19525	0.223844
$767$	−8.53117	−0.308043
$768$	18.0846	0.652572
$769$	46.0799	1.66168	0.830842	−	0.556508i	$-0.187859\pi$
$769$	46.0799	1.66168	0.830842	+	0.556508i	$0.187859\pi$
$770$	−1.86579	−0.0672385
$771$	36.3748	1.31001
$772$	−15.7569	−0.567102
$773$	12.7472	0.458484	0.229242	−	0.973369i	$-0.426375\pi$
$773$	12.7472	0.458484	0.229242	+	0.973369i	$0.426375\pi$
$774$	2.01335	0.0723686
$775$	2.04635	0.0735071
$776$	0.241615	0.00867346
$777$	0.335795	0.0120466
$778$	3.22527	0.115631
$779$	−7.54895	−0.270469
$780$	−5.12541	−0.183519
$781$	−8.35462	−0.298952
$782$	2.85048	0.101933
$783$	−29.7565	−1.06341
$784$	−6.37992	−0.227854
$785$	48.2098	1.72068
$786$	6.44618	0.229928
$787$	26.9652	0.961204	0.480602	−	0.876939i	$-0.340418\pi$
$787$	26.9652	0.961204	0.480602	+	0.876939i	$0.340418\pi$
$788$	−16.3479	−0.582369
$789$	3.36863	0.119926
$790$	−1.04364	−0.0371310
$791$	14.9767	0.532512
$792$	1.06409	0.0378109
$793$	2.27302	0.0807175
$794$	−2.27110	−0.0805982
$795$	−35.6837	−1.26557
$796$	6.36131	0.225471
$797$	42.3418	1.49982	0.749911	−	0.661538i	$-0.230096\pi$
$797$	42.3418	1.49982	0.749911	+	0.661538i	$0.230096\pi$
$798$	0.776357	0.0274827
$799$	−11.5222	−0.407627
$800$	−0.570841	−0.0201823
$801$	11.8179	0.417565
$802$	−0.267292	−0.00943842
$803$	5.36312	0.189260
$804$	31.6446	1.11602
$805$	39.7594	1.40134
$806$	−1.62851	−0.0573618
$807$	−2.53233	−0.0891424
$808$	11.4831	0.403976
$809$	−22.1157	−0.777547	−0.388774	−	0.921333i	$-0.627101\pi$
$809$	−22.1157	−0.777547	−0.388774	+	0.921333i	$0.627101\pi$
$810$	3.17306	0.111490
$811$	−2.08542	−0.0732290	−0.0366145	−	0.999329i	$-0.511657\pi$
$811$	−2.08542	−0.0732290	−0.0366145	+	0.999329i	$0.511657\pi$
$812$	23.6708	0.830681
$813$	−18.0766	−0.633975
$814$	−0.0347274	−0.00121719
$815$	−25.8784	−0.906480
$816$	9.22283	0.322864
$817$	11.7715	0.411832
$818$	−2.67923	−0.0936772
$819$	1.33764	0.0467408
$820$	33.6095	1.17369
$821$	−51.3781	−1.79311	−0.896555	−	0.442933i	$-0.853938\pi$
$821$	−51.3781	−1.79311	−0.896555	+	0.442933i	$0.853938\pi$
$822$	5.90859	0.206086
$823$	−18.5839	−0.647795	−0.323898	−	0.946092i	$-0.604993\pi$
$823$	−18.5839	−0.647795	−0.323898	+	0.946092i	$0.604993\pi$
$824$	5.59693	0.194978
$825$	−0.514484	−0.0179120
$826$	−5.75329	−0.200183
$827$	12.7876	0.444669	0.222334	−	0.974971i	$-0.428632\pi$
$827$	12.7876	0.444669	0.222334	+	0.974971i	$0.428632\pi$
$828$	−11.1914	−0.388929
$829$	12.4565	0.432632	0.216316	−	0.976323i	$-0.430596\pi$
$829$	12.4565	0.432632	0.216316	+	0.976323i	$0.430596\pi$
$830$	3.46925	0.120420
$831$	−0.211847	−0.00734888
$832$	−5.22977	−0.181310
$833$	−2.87521	−0.0996202
$834$	1.24945	0.0432649
$835$	5.30277	0.183510
$836$	3.07058	0.106198
$837$	−52.7994	−1.82501
$838$	−4.74170	−0.163799
$839$	39.8507	1.37580	0.687899	−	0.725806i	$-0.258533\pi$
$839$	39.8507	1.37580	0.687899	+	0.725806i	$0.258533\pi$
$840$	−7.00338	−0.241640
$841$	−1.03422	−0.0356629
$842$	5.49950	0.189525
$843$	26.9765	0.929118
$844$	−1.94904	−0.0670886
$845$	−28.3460	−0.975132
$846$	−1.18288	−0.0406684
$847$	19.5622	0.672164
$848$	−38.5646	−1.32432
$849$	−34.6580	−1.18946
$850$	−0.0820222	−0.00281334
$851$	0.740028	0.0253678
$852$	−15.4775	−0.530250
$853$	51.0517	1.74798	0.873988	−	0.485947i	$-0.161525\pi$
$853$	51.0517	1.74798	0.873988	+	0.485947i	$0.161525\pi$
$854$	1.53289	0.0524546
$855$	−1.73067	−0.0591878
$856$	2.30703	0.0788525
$857$	35.7468	1.22109	0.610544	−	0.791982i	$-0.290951\pi$
$857$	35.7468	1.22109	0.610544	+	0.791982i	$0.290951\pi$
$858$	0.409433	0.0139778
$859$	−45.3917	−1.54875	−0.774373	−	0.632730i	$-0.781934\pi$
$859$	−45.3917	−1.54875	−0.774373	+	0.632730i	$0.781934\pi$
$860$	−52.4091	−1.78714
$861$	25.9608	0.884742
$862$	1.85652	0.0632332
$863$	46.3696	1.57844	0.789220	−	0.614111i	$-0.210485\pi$
$863$	46.3696	1.57844	0.789220	+	0.614111i	$0.210485\pi$
$864$	14.7287	0.501080
$865$	24.7493	0.841500
$866$	−7.85773	−0.267017
$867$	−21.3003	−0.723396
$868$	42.0010	1.42561
$869$	−3.18834	−0.108157
$870$	−4.08368	−0.138450
$871$	−8.33541	−0.282435
$872$	−5.53567	−0.187461
$873$	−0.205335	−0.00694952
$874$	1.71094	0.0578734
$875$	25.0863	0.848071
$876$	9.93552	0.335690
$877$	−11.4296	−0.385950	−0.192975	−	0.981204i	$-0.561814\pi$
$877$	−11.4296	−0.385950	−0.192975	+	0.981204i	$0.561814\pi$
$878$	3.77477	0.127392
$879$	18.6310	0.628407
$880$	−13.3040	−0.448479
$881$	−19.8824	−0.669856	−0.334928	−	0.942244i	$-0.608712\pi$
$881$	−19.8824	−0.669856	−0.334928	+	0.942244i	$0.608712\pi$
$882$	−0.295172	−0.00993898
$883$	−10.6010	−0.356753	−0.178377	−	0.983962i	$-0.557085\pi$
$883$	−10.6010	−0.356753	−0.178377	+	0.983962i	$0.557085\pi$
$884$	−2.49634	−0.0839609
$885$	−37.9592	−1.27599
$886$	7.31367	0.245707
$887$	36.9949	1.24217	0.621084	−	0.783744i	$-0.286692\pi$
$887$	36.9949	1.24217	0.621084	+	0.783744i	$0.286692\pi$
$888$	−0.130352	−0.00437431
$889$	−3.90329	−0.130912
$890$	8.04387	0.269631
$891$	9.69378	0.324754
$892$	20.1018	0.673058
$893$	−6.91598	−0.231434
$894$	5.12188	0.171301
$895$	30.7539	1.02799
$896$	−15.5497	−0.519477
$897$	−8.72487	−0.291315
$898$	2.89050	0.0964572
$899$	49.6220	1.65499
$900$	0.322032	0.0107344
$901$	−17.3798	−0.579005
$902$	−2.68482	−0.0893948
$903$	−40.4822	−1.34716
$904$	−5.81379	−0.193364
$905$	58.5323	1.94568
$906$	5.56704	0.184953
$907$	49.6598	1.64893	0.824463	−	0.565916i	$-0.191477\pi$
$907$	49.6598	1.64893	0.824463	+	0.565916i	$0.191477\pi$
$908$	28.5142	0.946278
$909$	−9.75888	−0.323682
$910$	0.910466	0.0301816
$911$	23.7799	0.787863	0.393932	−	0.919140i	$-0.371115\pi$
$911$	23.7799	0.787863	0.393932	+	0.919140i	$0.371115\pi$
$912$	5.53581	0.183309
$913$	10.5987	0.350765
$914$	0.297799	0.00985033
$915$	10.1138	0.334351
$916$	25.5483	0.844138
$917$	43.7924	1.44615
$918$	2.11631	0.0698487
$919$	9.20794	0.303742	0.151871	−	0.988400i	$-0.451470\pi$
$919$	9.20794	0.303742	0.151871	+	0.988400i	$0.451470\pi$
$920$	−15.4341	−0.508847
$921$	−7.70155	−0.253775
$922$	−4.00277	−0.131824
$923$	4.07687	0.134192
$924$	−10.5597	−0.347389
$925$	−0.0212942	−0.000700149 0
$926$	4.82382	0.158521
$927$	−4.75652	−0.156225
$928$	−13.8423	−0.454397
$929$	26.8402	0.880599	0.440300	−	0.897851i	$-0.354872\pi$
$929$	26.8402	0.880599	0.440300	+	0.897851i	$0.354872\pi$
$930$	−7.24601	−0.237606
$931$	−1.72579	−0.0565604
$932$	−33.2225	−1.08824
$933$	−21.4298	−0.701581
$934$	−0.970175	−0.0317451
$935$	−5.99568	−0.196080
$936$	−0.519254	−0.0169724
$937$	−43.3453	−1.41603	−0.708014	−	0.706198i	$-0.750409\pi$
$937$	−43.3453	−1.41603	−0.708014	+	0.706198i	$0.750409\pi$
$938$	−5.62128	−0.183541
$939$	26.1196	0.852380
$940$	30.7914	1.00430
$941$	−43.0289	−1.40270	−0.701351	−	0.712817i	$-0.747419\pi$
$941$	−43.0289	−1.40270	−0.701351	+	0.712817i	$0.747419\pi$
$942$	−7.13447	−0.232454
$943$	57.2126	1.86310
$944$	−41.0239	−1.33521
$945$	29.5190	0.960254
$946$	4.18659	0.136118
$947$	−18.4628	−0.599960	−0.299980	−	0.953946i	$-0.596980\pi$
$947$	−18.4628	−0.599960	−0.299980	+	0.953946i	$0.596980\pi$
$948$	−5.90661	−0.191838
$949$	−2.61708	−0.0849541
$950$	−0.0492321	−0.00159730
$951$	−4.53871	−0.147178
$952$	−3.41100	−0.110551
$953$	−34.6165	−1.12134	−0.560669	−	0.828040i	$-0.689456\pi$
$953$	−34.6165	−1.12134	−0.560669	+	0.828040i	$0.689456\pi$
$954$	−1.78423	−0.0577665
$955$	−14.4940	−0.469014
$956$	31.5673	1.02096
$957$	−12.4757	−0.403284
$958$	−2.82390	−0.0912361
$959$	40.1403	1.29620
$960$	−23.2697	−0.751027
$961$	57.0485	1.84027
$962$	0.0169462	0.000546367 0
$963$	−1.96061	−0.0631798
$964$	−30.3401	−0.977189
$965$	18.4674	0.594486
$966$	−5.88392	−0.189312
$967$	4.48179	0.144125	0.0720624	−	0.997400i	$-0.477042\pi$
$967$	4.48179	0.144125	0.0720624	+	0.997400i	$0.477042\pi$
$968$	−7.59379	−0.244074
$969$	2.49480	0.0801446
$970$	−0.139761	−0.00448747
$971$	−14.0223	−0.449996	−0.224998	−	0.974359i	$-0.572238\pi$
$971$	−14.0223	−0.449996	−0.224998	+	0.974359i	$0.572238\pi$
$972$	−14.9426	−0.479286
$973$	8.48819	0.272119
$974$	5.31676	0.170360
$975$	0.251057	0.00804026
$976$	10.9303	0.349870
$977$	−52.2216	−1.67072	−0.835359	−	0.549705i	$-0.814740\pi$
$977$	−52.2216	−1.67072	−0.835359	+	0.549705i	$0.814740\pi$
$978$	3.82969	0.122460
$979$	24.5743	0.785397
$980$	7.68356	0.245442
$981$	4.70445	0.150202
$982$	1.64882	0.0526160
$983$	6.72371	0.214453	0.107226	−	0.994235i	$-0.465803\pi$
$983$	6.72371	0.214453	0.107226	+	0.994235i	$0.465803\pi$
$984$	−10.0777	−0.321264
$985$	19.1601	0.610490
$986$	−1.98896	−0.0633413
$987$	23.7840	0.757054
$988$	−1.49837	−0.0476696
$989$	−89.2149	−2.83687
$990$	−0.615523	−0.0195626
$991$	27.0379	0.858887	0.429443	−	0.903094i	$-0.358710\pi$
$991$	27.0379	0.858887	0.429443	+	0.903094i	$0.358710\pi$
$992$	−24.5616	−0.779832
$993$	47.3215	1.50170
$994$	2.74938	0.0872051
$995$	−7.45559	−0.236358
$996$	19.6347	0.622150
$997$	20.8753	0.661126	0.330563	−	0.943784i	$-0.392761\pi$
$997$	20.8753	0.661126	0.330563	+	0.943784i	$0.392761\pi$
$998$	−3.54881	−0.112336
$999$	0.549427	0.0173831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.36	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.36	✓	71	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-0.225751 q^{2} +1.49745 q^{3} -1.94904 q^{4} +2.28431 q^{5} -0.338051 q^{6} -2.29657 q^{7} +0.891498 q^{8} -0.757634 q^{9} +O(q^{10})\)	\(q-0.225751 q^{2} +1.49745 q^{3} -1.94904 q^{4} +2.28431 q^{5} -0.338051 q^{6} -2.29657 q^{7} +0.891498 q^{8} -0.757634 q^{9} -0.515685 q^{10} -1.57543 q^{11} -2.91859 q^{12} +0.768776 q^{13} +0.518451 q^{14} +3.42065 q^{15} +3.69682 q^{16} +1.66603 q^{17} +0.171036 q^{18} +1.00000 q^{19} -4.45221 q^{20} -3.43900 q^{21} +0.355655 q^{22} -7.57889 q^{23} +1.33498 q^{24} +0.218082 q^{25} -0.173552 q^{26} -5.62688 q^{27} +4.47609 q^{28} +5.28827 q^{29} -0.772214 q^{30} +9.38342 q^{31} -2.61756 q^{32} -2.35914 q^{33} -0.376108 q^{34} -5.24607 q^{35} +1.47666 q^{36} -0.0976433 q^{37} -0.225751 q^{38} +1.15121 q^{39} +2.03646 q^{40} -7.54895 q^{41} +0.776357 q^{42} +11.7715 q^{43} +3.07058 q^{44} -1.73067 q^{45} +1.71094 q^{46} -6.91598 q^{47} +5.53581 q^{48} -1.72579 q^{49} -0.0492321 q^{50} +2.49480 q^{51} -1.49837 q^{52} -10.4319 q^{53} +1.27027 q^{54} -3.59878 q^{55} -2.04738 q^{56} +1.49745 q^{57} -1.19383 q^{58} -11.0971 q^{59} -6.66697 q^{60} +2.95668 q^{61} -2.11831 q^{62} +1.73996 q^{63} -6.80272 q^{64} +1.75613 q^{65} +0.532577 q^{66} -10.8424 q^{67} -3.24715 q^{68} -11.3490 q^{69} +1.18430 q^{70} +5.30306 q^{71} -0.675429 q^{72} -3.40422 q^{73} +0.0220431 q^{74} +0.326567 q^{75} -1.94904 q^{76} +3.61809 q^{77} -0.259886 q^{78} +2.02379 q^{79} +8.44468 q^{80} -6.15309 q^{81} +1.70418 q^{82} -6.72746 q^{83} +6.70274 q^{84} +3.80573 q^{85} -2.65742 q^{86} +7.91893 q^{87} -1.40450 q^{88} -15.5984 q^{89} +0.390701 q^{90} -1.76555 q^{91} +14.7715 q^{92} +14.0512 q^{93} +1.56129 q^{94} +2.28431 q^{95} -3.91967 q^{96} +0.271021 q^{97} +0.389598 q^{98} +1.19360 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more