LMFDB - Embedded newform 4033.2.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4033,2,Mod(1,4033)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4033, 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("4033.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4033 = 37 \cdot 109 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4033.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:	$32.2036671352$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	4033.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52769 q^{2} +2.28955 q^{3} +4.38920 q^{4} -2.00544 q^{5} -5.78727 q^{6} -4.48140 q^{7} -6.03914 q^{8} +2.24205 q^{9} +O(q^{10})$	\(q-2.52769 q^{2} +2.28955 q^{3} +4.38920 q^{4} -2.00544 q^{5} -5.78727 q^{6} -4.48140 q^{7} -6.03914 q^{8} +2.24205 q^{9} +5.06913 q^{10} -5.32395 q^{11} +10.0493 q^{12} -6.68550 q^{13} +11.3276 q^{14} -4.59157 q^{15} +6.48667 q^{16} -3.61002 q^{17} -5.66721 q^{18} +0.316235 q^{19} -8.80229 q^{20} -10.2604 q^{21} +13.4573 q^{22} +4.72950 q^{23} -13.8269 q^{24} -0.978192 q^{25} +16.8988 q^{26} -1.73536 q^{27} -19.6697 q^{28} +0.0523907 q^{29} +11.6061 q^{30} -3.67688 q^{31} -4.31797 q^{32} -12.1895 q^{33} +9.12500 q^{34} +8.98719 q^{35} +9.84082 q^{36} -1.00000 q^{37} -0.799343 q^{38} -15.3068 q^{39} +12.1112 q^{40} -10.4937 q^{41} +25.9351 q^{42} +4.68390 q^{43} -23.3679 q^{44} -4.49631 q^{45} -11.9547 q^{46} +11.7990 q^{47} +14.8516 q^{48} +13.0829 q^{49} +2.47256 q^{50} -8.26533 q^{51} -29.3440 q^{52} +2.61056 q^{53} +4.38644 q^{54} +10.6769 q^{55} +27.0638 q^{56} +0.724037 q^{57} -0.132427 q^{58} -0.690019 q^{59} -20.1533 q^{60} -13.9320 q^{61} +9.29400 q^{62} -10.0475 q^{63} -2.05886 q^{64} +13.4074 q^{65} +30.8111 q^{66} -9.81308 q^{67} -15.8451 q^{68} +10.8284 q^{69} -22.7168 q^{70} -15.4398 q^{71} -13.5401 q^{72} -11.7646 q^{73} +2.52769 q^{74} -2.23962 q^{75} +1.38802 q^{76} +23.8587 q^{77} +38.6908 q^{78} -0.761520 q^{79} -13.0086 q^{80} -10.6994 q^{81} +26.5248 q^{82} -7.84499 q^{83} -45.0349 q^{84} +7.23969 q^{85} -11.8394 q^{86} +0.119951 q^{87} +32.1521 q^{88} +18.2334 q^{89} +11.3653 q^{90} +29.9604 q^{91} +20.7587 q^{92} -8.41841 q^{93} -29.8241 q^{94} -0.634192 q^{95} -9.88621 q^{96} -6.76945 q^{97} -33.0695 q^{98} -11.9366 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 10 q^{2} + 17 q^{3} + 88 q^{4} + 22 q^{5} + 4 q^{6} + 15 q^{7} + 33 q^{8} + 91 q^{9}+O(q^{10}) $ 82 * q + 10 * q^2 + 17 * q^3 + 88 * q^4 + 22 * q^5 + 4 * q^6 + 15 * q^7 + 33 * q^8 + 91 * q^9	\( 82 q + 10 q^{2} + 17 q^{3} + 88 q^{4} + 22 q^{5} + 4 q^{6} + 15 q^{7} + 33 q^{8} + 91 q^{9} + 13 q^{10} + 2 q^{11} + 36 q^{12} + 23 q^{13} + 24 q^{14} + 35 q^{15} + 104 q^{16} + 43 q^{17} + 42 q^{18} + 15 q^{19} + 59 q^{20} + 9 q^{21} + 6 q^{22} + 70 q^{23} + 15 q^{24} + 98 q^{25} + 10 q^{26} + 65 q^{27} + 37 q^{28} + 27 q^{29} + 17 q^{30} + 59 q^{31} + 46 q^{32} + 16 q^{33} - 16 q^{34} + 80 q^{35} + 88 q^{36} - 82 q^{37} + 82 q^{38} + 13 q^{39} + 14 q^{40} + 3 q^{41} + 62 q^{42} + 7 q^{43} - 11 q^{44} + 42 q^{45} - 11 q^{46} + 123 q^{47} + 45 q^{48} + 105 q^{49} + 27 q^{50} + 3 q^{51} - 30 q^{52} + 82 q^{53} - 27 q^{54} + 37 q^{55} + 66 q^{56} + 29 q^{57} - 34 q^{58} + 60 q^{59} + 94 q^{60} + 9 q^{61} - 2 q^{62} + 106 q^{63} + 93 q^{64} + 9 q^{65} + 63 q^{66} + 113 q^{68} + 48 q^{69} + 47 q^{70} + 59 q^{71} + 63 q^{72} + 21 q^{73} - 10 q^{74} + 77 q^{75} + 22 q^{76} + 30 q^{77} + 29 q^{78} + 55 q^{79} + 88 q^{80} + 42 q^{81} + 4 q^{82} + 92 q^{83} + 43 q^{84} - 7 q^{85} + 17 q^{86} + 147 q^{87} - 13 q^{88} + 100 q^{89} + 91 q^{90} + 28 q^{91} + 127 q^{92} + 3 q^{93} + 30 q^{94} + 48 q^{95} - 31 q^{96} + 53 q^{97} + 101 q^{98} - 20 q^{99}+O(q^{100}) \) 82 * q + 10 * q^2 + 17 * q^3 + 88 * q^4 + 22 * q^5 + 4 * q^6 + 15 * q^7 + 33 * q^8 + 91 * q^9 + 13 * q^10 + 2 * q^11 + 36 * q^12 + 23 * q^13 + 24 * q^14 + 35 * q^15 + 104 * q^16 + 43 * q^17 + 42 * q^18 + 15 * q^19 + 59 * q^20 + 9 * q^21 + 6 * q^22 + 70 * q^23 + 15 * q^24 + 98 * q^25 + 10 * q^26 + 65 * q^27 + 37 * q^28 + 27 * q^29 + 17 * q^30 + 59 * q^31 + 46 * q^32 + 16 * q^33 - 16 * q^34 + 80 * q^35 + 88 * q^36 - 82 * q^37 + 82 * q^38 + 13 * q^39 + 14 * q^40 + 3 * q^41 + 62 * q^42 + 7 * q^43 - 11 * q^44 + 42 * q^45 - 11 * q^46 + 123 * q^47 + 45 * q^48 + 105 * q^49 + 27 * q^50 + 3 * q^51 - 30 * q^52 + 82 * q^53 - 27 * q^54 + 37 * q^55 + 66 * q^56 + 29 * q^57 - 34 * q^58 + 60 * q^59 + 94 * q^60 + 9 * q^61 - 2 * q^62 + 106 * q^63 + 93 * q^64 + 9 * q^65 + 63 * q^66 + 113 * q^68 + 48 * q^69 + 47 * q^70 + 59 * q^71 + 63 * q^72 + 21 * q^73 - 10 * q^74 + 77 * q^75 + 22 * q^76 + 30 * q^77 + 29 * q^78 + 55 * q^79 + 88 * q^80 + 42 * q^81 + 4 * q^82 + 92 * q^83 + 43 * q^84 - 7 * q^85 + 17 * q^86 + 147 * q^87 - 13 * q^88 + 100 * q^89 + 91 * q^90 + 28 * q^91 + 127 * q^92 + 3 * q^93 + 30 * q^94 + 48 * q^95 - 31 * q^96 + 53 * q^97 + 101 * q^98 - 20 * 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 $

$2$	−2.52769	−1.78734	−0.893672	−	0.448721i	$-0.851880\pi$
$2$	−2.52769	−1.78734	−0.893672	+	0.448721i	$0.851880\pi$
$3$	2.28955	1.32187	0.660937	−	0.750441i	$-0.270159\pi$
$3$	2.28955	1.32187	0.660937	+	0.750441i	$0.270159\pi$
$4$	4.38920	2.19460
$5$	−2.00544	−0.896862	−0.448431	−	0.893817i	$-0.648017\pi$
$5$	−2.00544	−0.896862	−0.448431	+	0.893817i	$0.648017\pi$
$6$	−5.78727	−2.36264
$7$	−4.48140	−1.69381	−0.846904	−	0.531745i	$-0.821536\pi$
$7$	−4.48140	−1.69381	−0.846904	+	0.531745i	$0.821536\pi$
$8$	−6.03914	−2.13516
$9$	2.24205	0.747351
$10$	5.06913	1.60300
$11$	−5.32395	−1.60523	−0.802615	−	0.596497i	$-0.796559\pi$
$11$	−5.32395	−1.60523	−0.802615	+	0.596497i	$0.796559\pi$
$12$	10.0493	2.90098
$13$	−6.68550	−1.85422	−0.927111	−	0.374786i	$-0.877716\pi$
$13$	−6.68550	−1.85422	−0.927111	+	0.374786i	$0.877716\pi$
$14$	11.3276	3.02742
$15$	−4.59157	−1.18554
$16$	6.48667	1.62167
$17$	−3.61002	−0.875558	−0.437779	−	0.899083i	$-0.644235\pi$
$17$	−3.61002	−0.875558	−0.437779	+	0.899083i	$0.644235\pi$
$18$	−5.66721	−1.33577
$19$	0.316235	0.0725493	0.0362747	−	0.999342i	$-0.488451\pi$
$19$	0.316235	0.0725493	0.0362747	+	0.999342i	$0.488451\pi$
$20$	−8.80229	−1.96825
$21$	−10.2604	−2.23900
$22$	13.4573	2.86910
$23$	4.72950	0.986168	0.493084	−	0.869982i	$-0.335870\pi$
$23$	4.72950	0.986168	0.493084	+	0.869982i	$0.335870\pi$
$24$	−13.8269	−2.82241
$25$	−0.978192	−0.195638
$26$	16.8988	3.31413
$27$	−1.73536	−0.333970
$28$	−19.6697	−3.71723
$29$	0.0523907	0.00972871	0.00486436	−	0.999988i	$-0.498452\pi$
$29$	0.0523907	0.00972871	0.00486436	+	0.999988i	$0.498452\pi$
$30$	11.6061	2.11897
$31$	−3.67688	−0.660387	−0.330194	−	0.943913i	$-0.607114\pi$
$31$	−3.67688	−0.660387	−0.330194	+	0.943913i	$0.607114\pi$
$32$	−4.31797	−0.763316
$33$	−12.1895	−2.12191
$34$	9.12500	1.56492
$35$	8.98719	1.51911
$36$	9.84082	1.64014
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−0.799343	−0.129671
$39$	−15.3068	−2.45105
$40$	12.1112	1.91494
$41$	−10.4937	−1.63884	−0.819421	−	0.573192i	$-0.805705\pi$
$41$	−10.4937	−1.63884	−0.819421	+	0.573192i	$0.805705\pi$
$42$	25.9351	4.00187
$43$	4.68390	0.714288	0.357144	−	0.934049i	$-0.383751\pi$
$43$	4.68390	0.714288	0.357144	+	0.934049i	$0.383751\pi$
$44$	−23.3679	−3.52284
$45$	−4.49631	−0.670271
$46$	−11.9547	−1.76262
$47$	11.7990	1.72106	0.860530	−	0.509400i	$-0.170133\pi$
$47$	11.7990	1.72106	0.860530	+	0.509400i	$0.170133\pi$
$48$	14.8516	2.14364
$49$	13.0829	1.86899
$50$	2.47256	0.349673
$51$	−8.26533	−1.15738
$52$	−29.3440	−4.06928
$53$	2.61056	0.358589	0.179294	−	0.983795i	$-0.442619\pi$
$53$	2.61056	0.358589	0.179294	+	0.983795i	$0.442619\pi$
$54$	4.38644	0.596919
$55$	10.6769	1.43967
$56$	27.0638	3.61655
$57$	0.724037	0.0959011
$58$	−0.132427	−0.0173886
$59$	−0.690019	−0.0898328	−0.0449164	−	0.998991i	$-0.514302\pi$
$59$	−0.690019	−0.0898328	−0.0449164	+	0.998991i	$0.514302\pi$
$60$	−20.1533	−2.60178
$61$	−13.9320	−1.78381	−0.891905	−	0.452224i	$-0.850631\pi$
$61$	−13.9320	−1.78381	−0.891905	+	0.452224i	$0.850631\pi$
$62$	9.29400	1.18034
$63$	−10.0475	−1.26587
$64$	−2.05886	−0.257358
$65$	13.4074	1.66298
$66$	30.8111	3.79259
$67$	−9.81308	−1.19886	−0.599429	−	0.800428i	$-0.704606\pi$
$67$	−9.81308	−1.19886	−0.599429	+	0.800428i	$0.704606\pi$
$68$	−15.8451	−1.92150
$69$	10.8284	1.30359
$70$	−22.7168	−2.71518
$71$	−15.4398	−1.83237	−0.916183	−	0.400760i	$-0.868746\pi$
$71$	−15.4398	−1.83237	−0.916183	+	0.400760i	$0.868746\pi$
$72$	−13.5401	−1.59571
$73$	−11.7646	−1.37695	−0.688473	−	0.725262i	$-0.741719\pi$
$73$	−11.7646	−1.37695	−0.688473	+	0.725262i	$0.741719\pi$
$74$	2.52769	0.293838
$75$	−2.23962	−0.258609
$76$	1.38802	0.159217
$77$	23.8587	2.71895
$78$	38.6908	4.38087
$79$	−0.761520	−0.0856777	−0.0428388	−	0.999082i	$-0.513640\pi$
$79$	−0.761520	−0.0856777	−0.0428388	+	0.999082i	$0.513640\pi$
$80$	−13.0086	−1.45441
$81$	−10.6994	−1.18882
$82$	26.5248	2.92917
$83$	−7.84499	−0.861100	−0.430550	−	0.902567i	$-0.641680\pi$
$83$	−7.84499	−0.861100	−0.430550	+	0.902567i	$0.641680\pi$
$84$	−45.0349	−4.91371
$85$	7.23969	0.785255
$86$	−11.8394	−1.27668
$87$	0.119951	0.0128601
$88$	32.1521	3.42742
$89$	18.2334	1.93274	0.966368	−	0.257164i	$-0.0827879\pi$
$89$	18.2334	1.93274	0.966368	+	0.257164i	$0.0827879\pi$
$90$	11.3653	1.19800
$91$	29.9604	3.14070
$92$	20.7587	2.16424
$93$	−8.41841	−0.872949
$94$	−29.8241	−3.07613
$95$	−0.634192	−0.0650667
$96$	−9.88621	−1.00901
$97$	−6.76945	−0.687334	−0.343667	−	0.939092i	$-0.611669\pi$
$97$	−6.76945	−0.687334	−0.343667	+	0.939092i	$0.611669\pi$
$98$	−33.0695	−3.34052
$99$	−11.9366	−1.19967
$100$	−4.29348	−0.429348
$101$	5.65537	0.562731	0.281365	−	0.959601i	$-0.409213\pi$
$101$	5.65537	0.562731	0.281365	+	0.959601i	$0.409213\pi$
$102$	20.8922	2.06863
$103$	4.67162	0.460309	0.230154	−	0.973154i	$-0.426077\pi$
$103$	4.67162	0.460309	0.230154	+	0.973154i	$0.426077\pi$
$104$	40.3747	3.95906
$105$	20.5767	2.00808
$106$	−6.59869	−0.640921
$107$	−13.8878	−1.34259	−0.671294	−	0.741191i	$-0.734261\pi$
$107$	−13.8878	−1.34259	−0.671294	+	0.741191i	$0.734261\pi$
$108$	−7.61683	−0.732930
$109$	1.00000	0.0957826
$109$	1.00000	0.0957826
$110$	−26.9878	−2.57319
$111$	−2.28955	−0.217315
$112$	−29.0693	−2.74679
$113$	1.92233	0.180838	0.0904188	−	0.995904i	$-0.471179\pi$
$113$	1.92233	0.180838	0.0904188	+	0.995904i	$0.471179\pi$
$114$	−1.83014	−0.171408
$115$	−9.48474	−0.884457
$116$	0.229953	0.0213506
$117$	−14.9892	−1.38576
$118$	1.74415	0.160562
$119$	16.1779	1.48303
$120$	27.7292	2.53131
$121$	17.3444	1.57677
$122$	35.2157	3.18828
$123$	−24.0259	−2.16634
$124$	−16.1386	−1.44929
$125$	11.9889	1.07232
$126$	25.3970	2.26254
$127$	1.44801	0.128490	0.0642450	−	0.997934i	$-0.479536\pi$
$127$	1.44801	0.128490	0.0642450	+	0.997934i	$0.479536\pi$
$128$	13.8401	1.22330
$129$	10.7240	0.944198
$130$	−33.8897	−2.97232
$131$	11.8103	1.03187	0.515936	−	0.856627i	$-0.327444\pi$
$131$	11.8103	1.03187	0.515936	+	0.856627i	$0.327444\pi$
$132$	−53.5020	−4.65675
$133$	−1.41717	−0.122885
$134$	24.8044	2.14277
$135$	3.48017	0.299525
$136$	21.8014	1.86946
$137$	16.5508	1.41403	0.707013	−	0.707200i	$-0.250042\pi$
$137$	16.5508	1.41403	0.707013	+	0.707200i	$0.250042\pi$
$138$	−27.3709	−2.32996
$139$	−10.4366	−0.885222	−0.442611	−	0.896714i	$-0.645948\pi$
$139$	−10.4366	−0.885222	−0.442611	+	0.896714i	$0.645948\pi$
$140$	39.4466	3.33384
$141$	27.0144	2.27502
$142$	39.0270	3.27507
$143$	35.5932	2.97646
$144$	14.5435	1.21195
$145$	−0.105067	−0.00872531
$146$	29.7373	2.46108
$147$	29.9540	2.47057
$148$	−4.38920	−0.360790
$149$	5.98999	0.490719	0.245359	−	0.969432i	$-0.421094\pi$
$149$	5.98999	0.490719	0.245359	+	0.969432i	$0.421094\pi$
$150$	5.66106	0.462224
$151$	−16.1436	−1.31375	−0.656874	−	0.754000i	$-0.728122\pi$
$151$	−16.1436	−1.31375	−0.656874	+	0.754000i	$0.728122\pi$
$152$	−1.90979	−0.154904
$153$	−8.09386	−0.654349
$154$	−60.3074	−4.85971
$155$	7.37378	0.592276
$156$	−67.1846	−5.37907
$157$	4.99628	0.398747	0.199373	−	0.979924i	$-0.436109\pi$
$157$	4.99628	0.398747	0.199373	+	0.979924i	$0.436109\pi$
$158$	1.92488	0.153136
$159$	5.97702	0.474009
$160$	8.65944	0.684589
$161$	−21.1947	−1.67038
$162$	27.0446	2.12483
$163$	11.0591	0.866213	0.433107	−	0.901343i	$-0.357417\pi$
$163$	11.0591	0.866213	0.433107	+	0.901343i	$0.357417\pi$
$164$	−46.0590	−3.59660
$165$	24.4453	1.90306
$166$	19.8297	1.53908
$167$	1.35446	0.104811	0.0524056	−	0.998626i	$-0.483311\pi$
$167$	1.35446	0.104811	0.0524056	+	0.998626i	$0.483311\pi$
$168$	61.9640	4.78063
$169$	31.6959	2.43814
$170$	−18.2997	−1.40352
$171$	0.709016	0.0542198
$172$	20.5586	1.56757
$173$	10.2789	0.781490	0.390745	−	0.920499i	$-0.372217\pi$
$173$	10.2789	0.781490	0.390745	+	0.920499i	$0.372217\pi$
$174$	−0.303199	−0.0229855
$175$	4.38367	0.331374
$176$	−34.5347	−2.60315
$177$	−1.57984	−0.118748
$178$	−46.0883	−3.45446
$179$	−22.2662	−1.66426	−0.832128	−	0.554584i	$-0.812877\pi$
$179$	−22.2662	−1.66426	−0.832128	+	0.554584i	$0.812877\pi$
$180$	−19.7352	−1.47098
$181$	13.3854	0.994931	0.497466	−	0.867484i	$-0.334264\pi$
$181$	13.3854	0.994931	0.497466	+	0.867484i	$0.334264\pi$
$182$	−75.7304	−5.61351
$183$	−31.8980	−2.35797
$184$	−28.5621	−2.10563
$185$	2.00544	0.147443
$186$	21.2791	1.56026
$187$	19.2196	1.40547
$188$	51.7881	3.77704
$189$	7.77683	0.565681
$190$	1.60304	0.116297
$191$	−25.3950	−1.83752	−0.918760	−	0.394817i	$-0.870808\pi$
$191$	−25.3950	−1.83752	−0.918760	+	0.394817i	$0.870808\pi$
$192$	−4.71388	−0.340195
$193$	−2.32696	−0.167498	−0.0837491	−	0.996487i	$-0.526689\pi$
$193$	−2.32696	−0.167498	−0.0837491	+	0.996487i	$0.526689\pi$
$194$	17.1111	1.22850
$195$	30.6969	2.19825
$196$	57.4235	4.10168
$197$	21.0780	1.50174	0.750871	−	0.660448i	$-0.229634\pi$
$197$	21.0780	1.50174	0.750871	+	0.660448i	$0.229634\pi$
$198$	30.1719	2.14423
$199$	11.6106	0.823054	0.411527	−	0.911397i	$-0.364995\pi$
$199$	11.6106	0.823054	0.411527	+	0.911397i	$0.364995\pi$
$200$	5.90744	0.417719
$201$	−22.4676	−1.58474
$202$	−14.2950	−1.00579
$203$	−0.234784	−0.0164786
$204$	−36.2782	−2.53998
$205$	21.0446	1.46982
$206$	−11.8084	−0.822730
$207$	10.6038	0.737014
$208$	−43.3666	−3.00693
$209$	−1.68362	−0.116458
$210$	−52.0113	−3.58912
$211$	1.01310	0.0697448	0.0348724	−	0.999392i	$-0.488898\pi$
$211$	1.01310	0.0697448	0.0348724	+	0.999392i	$0.488898\pi$
$212$	11.4583	0.786958
$213$	−35.3502	−2.42216
$214$	35.1041	2.39967
$215$	−9.39329	−0.640617
$216$	10.4801	0.713079
$217$	16.4776	1.11857
$218$	−2.52769	−0.171197
$219$	−26.9357	−1.82015
$220$	46.8630	3.15950
$221$	24.1348	1.62348
$222$	5.78727	0.388416
$223$	3.79189	0.253924	0.126962	−	0.991908i	$-0.459477\pi$
$223$	3.79189	0.253924	0.126962	+	0.991908i	$0.459477\pi$
$224$	19.3505	1.29291
$225$	−2.19316	−0.146211
$226$	−4.85905	−0.323219
$227$	14.6220	0.970495	0.485248	−	0.874377i	$-0.338729\pi$
$227$	14.6220	0.970495	0.485248	+	0.874377i	$0.338729\pi$
$228$	3.17794	0.210464
$229$	−21.7466	−1.43706	−0.718528	−	0.695498i	$-0.755184\pi$
$229$	−21.7466	−1.43706	−0.718528	+	0.695498i	$0.755184\pi$
$230$	23.9744	1.58083
$231$	54.6258	3.59411
$232$	−0.316395	−0.0207724
$233$	−17.8829	−1.17154	−0.585772	−	0.810476i	$-0.699209\pi$
$233$	−17.8829	−1.17154	−0.585772	+	0.810476i	$0.699209\pi$
$234$	37.8881	2.47682
$235$	−23.6622	−1.54355
$236$	−3.02863	−0.197147
$237$	−1.74354	−0.113255
$238$	−40.8927	−2.65068
$239$	14.0993	0.912007	0.456004	−	0.889978i	$-0.349280\pi$
$239$	14.0993	0.912007	0.456004	+	0.889978i	$0.349280\pi$
$240$	−29.7840	−1.92255
$241$	−11.0632	−0.712641	−0.356321	−	0.934364i	$-0.615969\pi$
$241$	−11.0632	−0.712641	−0.356321	+	0.934364i	$0.615969\pi$
$242$	−43.8413	−2.81822
$243$	−19.2907	−1.23750
$244$	−61.1503	−3.91475
$245$	−26.2370	−1.67622
$246$	60.7299	3.87200
$247$	−2.11419	−0.134523
$248$	22.2052	1.41003
$249$	−17.9615	−1.13827
$250$	−30.3043	−1.91661
$251$	−17.7111	−1.11791	−0.558956	−	0.829197i	$-0.688798\pi$
$251$	−17.7111	−1.11791	−0.558956	+	0.829197i	$0.688798\pi$
$252$	−44.1006	−2.77808
$253$	−25.1796	−1.58303
$254$	−3.66011	−0.229656
$255$	16.5757	1.03801
$256$	−30.8657	−1.92911
$257$	2.91234	0.181667	0.0908335	−	0.995866i	$-0.471047\pi$
$257$	2.91234	0.181667	0.0908335	+	0.995866i	$0.471047\pi$
$258$	−27.1070	−1.68761
$259$	4.48140	0.278460
$260$	58.8477	3.64958
$261$	0.117463	0.00727076
$262$	−29.8527	−1.84431
$263$	20.7407	1.27893	0.639465	−	0.768820i	$-0.279156\pi$
$263$	20.7407	1.27893	0.639465	+	0.768820i	$0.279156\pi$
$264$	73.6139	4.53062
$265$	−5.23534	−0.321604
$266$	3.58217	0.219637
$267$	41.7463	2.55483
$268$	−43.0716	−2.63101
$269$	−17.1634	−1.04647	−0.523234	−	0.852189i	$-0.675275\pi$
$269$	−17.1634	−1.04647	−0.523234	+	0.852189i	$0.675275\pi$
$270$	−8.79677	−0.535354
$271$	−15.5025	−0.941707	−0.470854	−	0.882211i	$-0.656054\pi$
$271$	−15.5025	−0.941707	−0.470854	+	0.882211i	$0.656054\pi$
$272$	−23.4170	−1.41986
$273$	68.5958	4.15161
$274$	−41.8351	−2.52735
$275$	5.20784	0.314045
$276$	47.5281	2.86086
$277$	27.0110	1.62293	0.811467	−	0.584399i	$-0.198670\pi$
$277$	27.0110	1.62293	0.811467	+	0.584399i	$0.198670\pi$
$278$	26.3805	1.58220
$279$	−8.24376	−0.493541
$280$	−54.2749	−3.24355
$281$	−10.3124	−0.615187	−0.307594	−	0.951518i	$-0.599524\pi$
$281$	−10.3124	−0.615187	−0.307594	+	0.951518i	$0.599524\pi$
$282$	−68.2840	−4.06625
$283$	−24.5378	−1.45862	−0.729310	−	0.684183i	$-0.760159\pi$
$283$	−24.5378	−1.45862	−0.729310	+	0.684183i	$0.760159\pi$
$284$	−67.7683	−4.02131
$285$	−1.45202	−0.0860100
$286$	−89.9685	−5.31995
$287$	47.0265	2.77588
$288$	−9.68111	−0.570465
$289$	−3.96776	−0.233398
$290$	0.265576	0.0155951
$291$	−15.4990	−0.908569
$292$	−51.6373	−3.02184
$293$	25.4770	1.48838	0.744191	−	0.667967i	$-0.232835\pi$
$293$	25.4770	1.48838	0.744191	+	0.667967i	$0.232835\pi$
$294$	−75.7144	−4.41575
$295$	1.38379	0.0805677
$296$	6.03914	0.351018
$297$	9.23896	0.536099
$298$	−15.1408	−0.877083
$299$	−31.6190	−1.82858
$300$	−9.83015	−0.567544
$301$	−20.9904	−1.20987
$302$	40.8060	2.34812
$303$	12.9483	0.743859
$304$	2.05131	0.117651
$305$	27.9398	1.59983
$306$	20.4587	1.16955
$307$	−10.5175	−0.600266	−0.300133	−	0.953897i	$-0.597031\pi$
$307$	−10.5175	−0.600266	−0.300133	+	0.953897i	$0.597031\pi$
$308$	104.721	5.96701
$309$	10.6959	0.608470
$310$	−18.6386	−1.05860
$311$	14.4564	0.819745	0.409872	−	0.912143i	$-0.365573\pi$
$311$	14.4564	0.819745	0.409872	+	0.912143i	$0.365573\pi$
$312$	92.4400	5.23338
$313$	−10.4243	−0.589215	−0.294608	−	0.955618i	$-0.595189\pi$
$313$	−10.4243	−0.589215	−0.294608	+	0.955618i	$0.595189\pi$
$314$	−12.6290	−0.712697
$315$	20.1498	1.13531
$316$	−3.34246	−0.188028
$317$	−27.7612	−1.55922	−0.779611	−	0.626264i	$-0.784583\pi$
$317$	−27.7612	−1.55922	−0.779611	+	0.626264i	$0.784583\pi$
$318$	−15.1080	−0.847217
$319$	−0.278925	−0.0156168
$320$	4.12894	0.230815
$321$	−31.7970	−1.77473
$322$	53.5737	2.98554
$323$	−1.14161	−0.0635211
$324$	−46.9616	−2.60898
$325$	6.53970	0.362757
$326$	−27.9539	−1.54822
$327$	2.28955	0.126613
$328$	63.3730	3.49919
$329$	−52.8759	−2.91514
$330$	−61.7900	−3.40143
$331$	−20.6674	−1.13598	−0.567991	−	0.823035i	$-0.692279\pi$
$331$	−20.6674	−1.13598	−0.567991	+	0.823035i	$0.692279\pi$
$332$	−34.4332	−1.88977
$333$	−2.24205	−0.122864
$334$	−3.42365	−0.187334
$335$	19.6796	1.07521
$336$	−66.5557	−3.63091
$337$	32.1757	1.75272	0.876360	−	0.481656i	$-0.159964\pi$
$337$	32.1757	1.75272	0.876360	+	0.481656i	$0.159964\pi$
$338$	−80.1172	−4.35780
$339$	4.40128	0.239045
$340$	31.7764	1.72332
$341$	19.5755	1.06007
$342$	−1.79217	−0.0969095
$343$	−27.2599	−1.47190
$344$	−28.2867	−1.52512
$345$	−21.7158	−1.16914
$346$	−25.9818	−1.39679
$347$	14.9282	0.801386	0.400693	−	0.916212i	$-0.368769\pi$
$347$	14.9282	0.801386	0.400693	+	0.916212i	$0.368769\pi$
$348$	0.526490	0.0282228
$349$	−11.3989	−0.610167	−0.305084	−	0.952326i	$-0.598684\pi$
$349$	−11.3989	−0.610167	−0.305084	+	0.952326i	$0.598684\pi$
$350$	−11.0805	−0.592279
$351$	11.6017	0.619255
$352$	22.9886	1.22530
$353$	−32.0508	−1.70589	−0.852945	−	0.522000i	$-0.825186\pi$
$353$	−32.0508	−1.70589	−0.852945	+	0.522000i	$0.825186\pi$
$354$	3.99333	0.212243
$355$	30.9637	1.64338
$356$	80.0300	4.24158
$357$	37.0402	1.96038
$358$	56.2820	2.97460
$359$	−33.2891	−1.75693	−0.878465	−	0.477806i	$-0.841432\pi$
$359$	−33.2891	−1.75693	−0.878465	+	0.477806i	$0.841432\pi$
$360$	27.1539	1.43114
$361$	−18.9000	−0.994737
$362$	−33.8342	−1.77828
$363$	39.7110	2.08429
$364$	131.502	6.89257
$365$	23.5933	1.23493
$366$	80.6282	4.21451
$367$	−36.2124	−1.89027	−0.945136	−	0.326677i	$-0.894071\pi$
$367$	−36.2124	−1.89027	−0.945136	+	0.326677i	$0.894071\pi$
$368$	30.6787	1.59924
$369$	−23.5275	−1.22479
$370$	−5.06913	−0.263532
$371$	−11.6990	−0.607380
$372$	−36.9501	−1.91577
$373$	1.91564	0.0991879	0.0495940	−	0.998769i	$-0.484207\pi$
$373$	1.91564	0.0991879	0.0495940	+	0.998769i	$0.484207\pi$
$374$	−48.5810	−2.51206
$375$	27.4493	1.41748
$376$	−71.2558	−3.67474
$377$	−0.350258	−0.0180392
$378$	−19.6574	−1.01107
$379$	−29.4459	−1.51254	−0.756268	−	0.654262i	$-0.772979\pi$
$379$	−29.4459	−1.51254	−0.756268	+	0.654262i	$0.772979\pi$
$380$	−2.78359	−0.142795
$381$	3.31529	0.169848
$382$	64.1906	3.28428
$383$	16.7614	0.856465	0.428233	−	0.903669i	$-0.359136\pi$
$383$	16.7614	0.856465	0.428233	+	0.903669i	$0.359136\pi$
$384$	31.6876	1.61705
$385$	−47.8473	−2.43853
$386$	5.88182	0.299377
$387$	10.5015	0.533824
$388$	−29.7125	−1.50842
$389$	−0.932087	−0.0472587	−0.0236293	−	0.999721i	$-0.507522\pi$
$389$	−0.932087	−0.0472587	−0.0236293	+	0.999721i	$0.507522\pi$
$390$	−77.5922	−3.92903
$391$	−17.0736	−0.863447
$392$	−79.0096	−3.99059
$393$	27.0403	1.36400
$394$	−53.2785	−2.68413
$395$	1.52719	0.0768411
$396$	−52.3920	−2.63280
$397$	−6.40324	−0.321370	−0.160685	−	0.987006i	$-0.551370\pi$
$397$	−6.40324	−0.321370	−0.160685	+	0.987006i	$0.551370\pi$
$398$	−29.3480	−1.47108
$399$	−3.24470	−0.162438
$400$	−6.34520	−0.317260
$401$	27.9489	1.39570	0.697851	−	0.716243i	$-0.254140\pi$
$401$	27.9489	1.39570	0.697851	+	0.716243i	$0.254140\pi$
$402$	56.7910	2.83248
$403$	24.5818	1.22451
$404$	24.8226	1.23497
$405$	21.4570	1.06621
$406$	0.593459	0.0294529
$407$	5.32395	0.263898
$408$	49.9155	2.47119
$409$	26.9898	1.33456	0.667280	−	0.744807i	$-0.267459\pi$
$409$	26.9898	1.33456	0.667280	+	0.744807i	$0.267459\pi$
$410$	−53.1940	−2.62707
$411$	37.8938	1.86916
$412$	20.5047	1.01019
$413$	3.09225	0.152160
$414$	−26.8030	−1.31730
$415$	15.7327	0.772288
$416$	28.8677	1.41536
$417$	−23.8952	−1.17015
$418$	4.25566	0.208151
$419$	−12.7188	−0.621354	−0.310677	−	0.950516i	$-0.600556\pi$
$419$	−12.7188	−0.621354	−0.310677	+	0.950516i	$0.600556\pi$
$420$	90.3150	4.40692
$421$	1.81679	0.0885449	0.0442724	−	0.999019i	$-0.485903\pi$
$421$	1.81679	0.0885449	0.0442724	+	0.999019i	$0.485903\pi$
$422$	−2.56080	−0.124658
$423$	26.4540	1.28624
$424$	−15.7656	−0.765644
$425$	3.53129	0.171293
$426$	89.3543	4.32923
$427$	62.4348	3.02143
$428$	−60.9565	−2.94644
$429$	81.4926	3.93450
$430$	23.7433	1.14500
$431$	20.4114	0.983183	0.491591	−	0.870826i	$-0.336415\pi$
$431$	20.4114	0.983183	0.491591	+	0.870826i	$0.336415\pi$
$432$	−11.2567	−0.541588
$433$	−34.4971	−1.65782	−0.828912	−	0.559378i	$-0.811040\pi$
$433$	−34.4971	−1.65782	−0.828912	+	0.559378i	$0.811040\pi$
$434$	−41.6501	−1.99927
$435$	−0.240556	−0.0115338
$436$	4.38920	0.210204
$437$	1.49563	0.0715458
$438$	68.0851	3.25323
$439$	15.1144	0.721371	0.360685	−	0.932688i	$-0.382543\pi$
$439$	15.1144	0.721371	0.360685	+	0.932688i	$0.382543\pi$
$440$	−64.4792	−3.07393
$441$	29.3326	1.39679
$442$	−61.0051	−2.90172
$443$	12.4903	0.593430	0.296715	−	0.954966i	$-0.404109\pi$
$443$	12.4903	0.593430	0.296715	+	0.954966i	$0.404109\pi$
$444$	−10.0493	−0.476919
$445$	−36.5661	−1.73340
$446$	−9.58470	−0.453849
$447$	13.7144	0.648668
$448$	9.22659	0.435915
$449$	−0.353379	−0.0166770	−0.00833850	−	0.999965i	$-0.502654\pi$
$449$	−0.353379	−0.0166770	−0.00833850	+	0.999965i	$0.502654\pi$
$450$	5.54362	0.261329
$451$	55.8680	2.63072
$452$	8.43749	0.396866
$453$	−36.9616	−1.73661
$454$	−36.9598	−1.73461
$455$	−60.0838	−2.81677
$456$	−4.37256	−0.204764
$457$	−3.99213	−0.186744	−0.0933720	−	0.995631i	$-0.529765\pi$
$457$	−3.99213	−0.186744	−0.0933720	+	0.995631i	$0.529765\pi$
$458$	54.9686	2.56851
$459$	6.26468	0.292410
$460$	−41.6304	−1.94103
$461$	−25.2976	−1.17823	−0.589114	−	0.808050i	$-0.700523\pi$
$461$	−25.2976	−1.17823	−0.589114	+	0.808050i	$0.700523\pi$
$462$	−138.077	−6.42392
$463$	4.72163	0.219433	0.109716	−	0.993963i	$-0.465006\pi$
$463$	4.72163	0.219433	0.109716	+	0.993963i	$0.465006\pi$
$464$	0.339841	0.0157767
$465$	16.8827	0.782915
$466$	45.2022	2.09395
$467$	0.602017	0.0278580	0.0139290	−	0.999903i	$-0.495566\pi$
$467$	0.602017	0.0278580	0.0139290	+	0.999903i	$0.495566\pi$
$468$	−65.7907	−3.04118
$469$	43.9763	2.03064
$470$	59.8107	2.75886
$471$	11.4392	0.527093
$472$	4.16712	0.191807
$473$	−24.9368	−1.14660
$474$	4.40712	0.202426
$475$	−0.309339	−0.0141934
$476$	71.0081	3.25465
$477$	5.85302	0.267992
$478$	−35.6386	−1.63007
$479$	8.18512	0.373987	0.186994	−	0.982361i	$-0.440126\pi$
$479$	8.18512	0.373987	0.186994	+	0.982361i	$0.440126\pi$
$480$	19.8263	0.904941
$481$	6.68550	0.304832
$482$	27.9642	1.27373
$483$	−48.5265	−2.20803
$484$	76.1281	3.46037
$485$	13.5758	0.616444
$486$	48.7608	2.21183
$487$	−0.525144	−0.0237965	−0.0118983	−	0.999929i	$-0.503787\pi$
$487$	−0.525144	−0.0237965	−0.0118983	+	0.999929i	$0.503787\pi$
$488$	84.1373	3.80872
$489$	25.3203	1.14502
$490$	66.3190	2.99599
$491$	−6.46278	−0.291661	−0.145831	−	0.989310i	$-0.546585\pi$
$491$	−6.46278	−0.291661	−0.145831	+	0.989310i	$0.546585\pi$
$492$	−105.454	−4.75425
$493$	−0.189131	−0.00851805
$494$	5.34401	0.240438
$495$	23.9381	1.07594
$496$	−23.8507	−1.07093
$497$	69.1918	3.10368
$498$	45.4011	2.03447
$499$	11.8630	0.531059	0.265530	−	0.964103i	$-0.414453\pi$
$499$	11.8630	0.531059	0.265530	+	0.964103i	$0.414453\pi$
$500$	52.6218	2.35332
$501$	3.10111	0.138547
$502$	44.7680	1.99809
$503$	−2.21979	−0.0989756	−0.0494878	−	0.998775i	$-0.515759\pi$
$503$	−2.21979	−0.0989756	−0.0494878	+	0.998775i	$0.515759\pi$
$504$	60.6785	2.70283
$505$	−11.3415	−0.504692
$506$	63.6461	2.82941
$507$	72.5693	3.22292
$508$	6.35560	0.281984
$509$	17.4838	0.774955	0.387477	−	0.921879i	$-0.373347\pi$
$509$	17.4838	0.774955	0.387477	+	0.921879i	$0.373347\pi$
$510$	−41.8981	−1.85528
$511$	52.7220	2.33228
$512$	50.3386	2.22467
$513$	−0.548781	−0.0242293
$514$	−7.36149	−0.324701
$515$	−9.36868	−0.412833
$516$	47.0699	2.07214
$517$	−62.8172	−2.76270
$518$	−11.3276	−0.497705
$519$	23.5341	1.03303
$520$	−80.9692	−3.55073
$521$	1.10695	0.0484962	0.0242481	−	0.999706i	$-0.492281\pi$
$521$	1.10695	0.0484962	0.0242481	+	0.999706i	$0.492281\pi$
$522$	−0.296909	−0.0129954
$523$	33.5576	1.46737	0.733686	−	0.679489i	$-0.237798\pi$
$523$	33.5576	1.46737	0.733686	+	0.679489i	$0.237798\pi$
$524$	51.8378	2.26454
$525$	10.0366	0.438035
$526$	−52.4261	−2.28589
$527$	13.2736	0.578208
$528$	−79.0690	−3.44103
$529$	−0.631873	−0.0274727
$530$	13.2333	0.574818
$531$	−1.54706	−0.0671367
$532$	−6.22026	−0.269683
$533$	70.1556	3.03878
$534$	−105.522	−4.56637
$535$	27.8513	1.20412
$536$	59.2626	2.55975
$537$	−50.9797	−2.19994
$538$	43.3836	1.87040
$539$	−69.6527	−3.00016
$540$	15.2751	0.657337
$541$	−11.5955	−0.498530	−0.249265	−	0.968435i	$-0.580189\pi$
$541$	−11.5955	−0.498530	−0.249265	+	0.968435i	$0.580189\pi$
$542$	39.1853	1.68315
$543$	30.6467	1.31517
$544$	15.5879	0.668328
$545$	−2.00544	−0.0859038
$546$	−173.389	−7.42035
$547$	−28.4042	−1.21448	−0.607239	−	0.794519i	$-0.707723\pi$
$547$	−28.4042	−1.21448	−0.607239	+	0.794519i	$0.707723\pi$
$548$	72.6445	3.10322
$549$	−31.2363	−1.33313
$550$	−13.1638	−0.561306
$551$	0.0165678	0.000705811 0
$552$	−65.3945	−2.78337
$553$	3.41267	0.145122
$554$	−68.2753	−2.90074
$555$	4.59157	0.194901
$556$	−45.8084	−1.94271
$557$	1.40863	0.0596856	0.0298428	−	0.999555i	$-0.490499\pi$
$557$	1.40863	0.0596856	0.0298428	+	0.999555i	$0.490499\pi$
$558$	20.8376	0.882128
$559$	−31.3142	−1.32445
$560$	58.2969	2.46349
$561$	44.0042	1.85786
$562$	26.0666	1.09955
$563$	26.9885	1.13743	0.568716	−	0.822534i	$-0.307440\pi$
$563$	26.9885	1.13743	0.568716	+	0.822534i	$0.307440\pi$
$564$	118.572	4.99277
$565$	−3.85513	−0.162186
$566$	62.0239	2.60706
$567$	47.9481	2.01363
$568$	93.2431	3.91240
$569$	−35.8200	−1.50165	−0.750827	−	0.660499i	$-0.770345\pi$
$569$	−35.8200	−1.50165	−0.750827	+	0.660499i	$0.770345\pi$
$570$	3.67024	0.153730
$571$	−12.6123	−0.527807	−0.263904	−	0.964549i	$-0.585010\pi$
$571$	−12.6123	−0.527807	−0.263904	+	0.964549i	$0.585010\pi$
$572$	156.226	6.53213
$573$	−58.1433	−2.42897
$574$	−118.868	−4.96146
$575$	−4.62636	−0.192932
$576$	−4.61608	−0.192337
$577$	−2.01768	−0.0839972	−0.0419986	−	0.999118i	$-0.513373\pi$
$577$	−2.01768	−0.0839972	−0.0419986	+	0.999118i	$0.513373\pi$
$578$	10.0293	0.417162
$579$	−5.32770	−0.221411
$580$	−0.461159	−0.0191486
$581$	35.1565	1.45854
$582$	39.1767	1.62392
$583$	−13.8985	−0.575617
$584$	71.0483	2.94000
$585$	30.0601	1.24283
$586$	−64.3978	−2.66025
$587$	−16.9953	−0.701472	−0.350736	−	0.936474i	$-0.614069\pi$
$587$	−16.9953	−0.701472	−0.350736	+	0.936474i	$0.614069\pi$
$588$	131.474	5.42190
$589$	−1.16276	−0.0479106
$590$	−3.49780	−0.144002
$591$	48.2591	1.98511
$592$	−6.48667	−0.266600
$593$	−16.1672	−0.663909	−0.331955	−	0.943295i	$-0.607708\pi$
$593$	−16.1672	−0.663909	−0.331955	+	0.943295i	$0.607708\pi$
$594$	−23.3532	−0.958193
$595$	−32.4439	−1.33007
$596$	26.2912	1.07693
$597$	26.5831	1.08797
$598$	79.9230	3.26829
$599$	35.1204	1.43498	0.717491	−	0.696568i	$-0.245290\pi$
$599$	35.1204	1.43498	0.717491	+	0.696568i	$0.245290\pi$
$600$	13.5254	0.552172
$601$	23.5793	0.961820	0.480910	−	0.876770i	$-0.340306\pi$
$601$	23.5793	0.961820	0.480910	+	0.876770i	$0.340306\pi$
$602$	53.0571	2.16245
$603$	−22.0015	−0.895968
$604$	−70.8575	−2.88315
$605$	−34.7833	−1.41414
$606$	−32.7292	−1.32953
$607$	4.24023	0.172105	0.0860527	−	0.996291i	$-0.472575\pi$
$607$	4.24023	0.172105	0.0860527	+	0.996291i	$0.472575\pi$
$608$	−1.36549	−0.0553780
$609$	−0.537549	−0.0217826
$610$	−70.6232	−2.85945
$611$	−78.8821	−3.19123
$612$	−35.5255	−1.43603
$613$	−11.7957	−0.476424	−0.238212	−	0.971213i	$-0.576561\pi$
$613$	−11.7957	−0.476424	−0.238212	+	0.971213i	$0.576561\pi$
$614$	26.5849	1.07288
$615$	48.1826	1.94291
$616$	−144.086	−5.80540
$617$	−17.7868	−0.716068	−0.358034	−	0.933708i	$-0.616553\pi$
$617$	−17.7868	−0.716068	−0.358034	+	0.933708i	$0.616553\pi$
$618$	−27.0360	−1.08755
$619$	34.9222	1.40364	0.701820	−	0.712354i	$-0.252371\pi$
$619$	34.9222	1.40364	0.701820	+	0.712354i	$0.252371\pi$
$620$	32.3650	1.29981
$621$	−8.20737	−0.329350
$622$	−36.5411	−1.46517
$623$	−81.7111	−3.27368
$624$	−99.2901	−3.97478
$625$	−19.1522	−0.766087
$626$	26.3493	1.05313
$627$	−3.85474	−0.153943
$628$	21.9297	0.875089
$629$	3.61002	0.143941
$630$	−50.9323	−2.02919
$631$	11.2419	0.447532	0.223766	−	0.974643i	$-0.428165\pi$
$631$	11.2419	0.447532	0.223766	+	0.974643i	$0.428165\pi$
$632$	4.59893	0.182936
$633$	2.31955	0.0921938
$634$	70.1715	2.78687
$635$	−2.90390	−0.115238
$636$	26.2343	1.04026
$637$	−87.4657	−3.46552
$638$	0.705036	0.0279126
$639$	−34.6168	−1.36942
$640$	−27.7555	−1.09713
$641$	−28.1495	−1.11184	−0.555920	−	0.831236i	$-0.687634\pi$
$641$	−28.1495	−1.11184	−0.555920	+	0.831236i	$0.687634\pi$
$642$	80.3727	3.17206
$643$	−2.03427	−0.0802238	−0.0401119	−	0.999195i	$-0.512771\pi$
$643$	−2.03427	−0.0802238	−0.0401119	+	0.999195i	$0.512771\pi$
$644$	−93.0279	−3.66581
$645$	−21.5064	−0.846816
$646$	2.88564	0.113534
$647$	−30.9556	−1.21699	−0.608495	−	0.793557i	$-0.708227\pi$
$647$	−30.9556	−1.21699	−0.608495	+	0.793557i	$0.708227\pi$
$648$	64.6150	2.53832
$649$	3.67363	0.144202
$650$	−16.5303	−0.648372
$651$	37.7262	1.47861
$652$	48.5404	1.90099
$653$	9.29517	0.363748	0.181874	−	0.983322i	$-0.441784\pi$
$653$	9.29517	0.363748	0.181874	+	0.983322i	$0.441784\pi$
$654$	−5.78727	−0.226300
$655$	−23.6849	−0.925446
$656$	−68.0692	−2.65765
$657$	−26.3769	−1.02906
$658$	133.654	5.21037
$659$	6.90281	0.268895	0.134448	−	0.990921i	$-0.457074\pi$
$659$	6.90281	0.268895	0.134448	+	0.990921i	$0.457074\pi$
$660$	107.295	4.17646
$661$	4.70001	0.182809	0.0914046	−	0.995814i	$-0.470864\pi$
$661$	4.70001	0.182809	0.0914046	+	0.995814i	$0.470864\pi$
$662$	52.2406	2.03039
$663$	55.2578	2.14604
$664$	47.3770	1.83859
$665$	2.84207	0.110211
$666$	5.66721	0.219600
$667$	0.247782	0.00959414
$668$	5.94499	0.230019
$669$	8.68173	0.335655
$670$	−49.7438	−1.92177
$671$	74.1732	2.86343
$672$	44.3040	1.70907
$673$	−25.1824	−0.970709	−0.485355	−	0.874317i	$-0.661309\pi$
$673$	−25.1824	−0.970709	−0.485355	+	0.874317i	$0.661309\pi$
$674$	−81.3300	−3.13272
$675$	1.69751	0.0653373
$676$	139.119	5.35075
$677$	−38.3140	−1.47253	−0.736263	−	0.676696i	$-0.763411\pi$
$677$	−38.3140	−1.47253	−0.736263	+	0.676696i	$0.763411\pi$
$678$	−11.1251	−0.427255
$679$	30.3366	1.16421
$680$	−43.7215	−1.67664
$681$	33.4778	1.28287
$682$	−49.4808	−1.89472
$683$	−25.5934	−0.979306	−0.489653	−	0.871917i	$-0.662877\pi$
$683$	−25.5934	−0.979306	−0.489653	+	0.871917i	$0.662877\pi$
$684$	3.11201	0.118991
$685$	−33.1916	−1.26819
$686$	68.9045	2.63079
$687$	−49.7900	−1.89961
$688$	30.3829	1.15834
$689$	−17.4529	−0.664903
$690$	54.8908	2.08966
$691$	−18.2257	−0.693338	−0.346669	−	0.937988i	$-0.612687\pi$
$691$	−18.2257	−0.693338	−0.346669	+	0.937988i	$0.612687\pi$
$692$	45.1161	1.71506
$693$	53.4925	2.03201
$694$	−37.7337	−1.43235
$695$	20.9301	0.793922
$696$	−0.724403	−0.0274584
$697$	37.8825	1.43490
$698$	28.8128	1.09058
$699$	−40.9437	−1.54863
$700$	19.2408	0.727233
$701$	−44.6590	−1.68675	−0.843373	−	0.537329i	$-0.819433\pi$
$701$	−44.6590	−1.68675	−0.843373	+	0.537329i	$0.819433\pi$
$702$	−29.3255	−1.10682
$703$	−0.316235	−0.0119270
$704$	10.9613	0.413119
$705$	−54.1759	−2.04038
$706$	81.0143	3.04901
$707$	−25.3440	−0.953158
$708$	−6.93421	−0.260604
$709$	−18.3908	−0.690680	−0.345340	−	0.938478i	$-0.612236\pi$
$709$	−18.3908	−0.690680	−0.345340	+	0.938478i	$0.612236\pi$
$710$	−78.2664	−2.93729
$711$	−1.70737	−0.0640313
$712$	−110.114	−4.12670
$713$	−17.3898	−0.651253
$714$	−93.6260	−3.50387
$715$	−71.3802	−2.66947
$716$	−97.7308	−3.65237
$717$	32.2811	1.20556
$718$	84.1444	3.14024
$719$	24.6636	0.919797	0.459899	−	0.887971i	$-0.347886\pi$
$719$	24.6636	0.919797	0.459899	+	0.887971i	$0.347886\pi$
$720$	−29.1661	−1.08696
$721$	−20.9354	−0.779675
$722$	47.7733	1.77794
$723$	−25.3297	−0.942022
$724$	58.7513	2.18348
$725$	−0.0512482	−0.00190331
$726$	−100.377	−3.72534
$727$	−4.58368	−0.169999	−0.0849997	−	0.996381i	$-0.527089\pi$
$727$	−4.58368	−0.169999	−0.0849997	+	0.996381i	$0.527089\pi$
$728$	−180.935	−6.70589
$729$	−12.0689	−0.446998
$730$	−59.6365	−2.20725
$731$	−16.9090	−0.625400
$732$	−140.007	−5.17480
$733$	1.38738	0.0512441	0.0256221	−	0.999672i	$-0.491843\pi$
$733$	1.38738	0.0512441	0.0256221	+	0.999672i	$0.491843\pi$
$734$	91.5336	3.37857
$735$	−60.0711	−2.21576
$736$	−20.4218	−0.752758
$737$	52.2443	1.92444
$738$	59.4700	2.18912
$739$	−3.07324	−0.113051	−0.0565254	−	0.998401i	$-0.518002\pi$
$739$	−3.07324	−0.113051	−0.0565254	+	0.998401i	$0.518002\pi$
$740$	8.80229	0.323579
$741$	−4.84055	−0.177822
$742$	29.5713	1.08560
$743$	−27.8508	−1.02175	−0.510874	−	0.859656i	$-0.670678\pi$
$743$	−27.8508	−1.02175	−0.510874	+	0.859656i	$0.670678\pi$
$744$	50.8400	1.86389
$745$	−12.0126	−0.440107
$746$	−4.84213	−0.177283
$747$	−17.5889	−0.643544
$748$	84.3584	3.08445
$749$	62.2369	2.27409
$750$	−69.3832	−2.53352
$751$	−17.6792	−0.645122	−0.322561	−	0.946549i	$-0.604544\pi$
$751$	−17.6792	−0.645122	−0.322561	+	0.946549i	$0.604544\pi$
$752$	76.5361	2.79098
$753$	−40.5504	−1.47774
$754$	0.885342	0.0322423
$755$	32.3751	1.17825
$756$	34.1340	1.24144
$757$	−16.2127	−0.589259	−0.294630	−	0.955612i	$-0.595196\pi$
$757$	−16.2127	−0.589259	−0.294630	+	0.955612i	$0.595196\pi$
$758$	74.4301	2.70342
$759$	−57.6500	−2.09256
$760$	3.82998	0.138928
$761$	−18.0127	−0.652960	−0.326480	−	0.945204i	$-0.605863\pi$
$761$	−18.0127	−0.652960	−0.326480	+	0.945204i	$0.605863\pi$
$762$	−8.38002	−0.303576
$763$	−4.48140	−0.162237
$764$	−111.464	−4.03262
$765$	16.2318	0.586861
$766$	−42.3674	−1.53080
$767$	4.61312	0.166570
$768$	−70.6686	−2.55004
$769$	30.2411	1.09052	0.545261	−	0.838266i	$-0.316430\pi$
$769$	30.2411	1.09052	0.545261	+	0.838266i	$0.316430\pi$
$770$	120.943	4.35849
$771$	6.66796	0.240141
$772$	−10.2135	−0.367591
$773$	−14.3832	−0.517328	−0.258664	−	0.965967i	$-0.583282\pi$
$773$	−14.3832	−0.517328	−0.258664	+	0.965967i	$0.583282\pi$
$774$	−26.5446	−0.954127
$775$	3.59670	0.129197
$776$	40.8817	1.46757
$777$	10.2604	0.368090
$778$	2.35602	0.0844676
$779$	−3.31848	−0.118897
$780$	134.735	4.82428
$781$	82.2007	2.94137
$782$	43.1566	1.54328
$783$	−0.0909167	−0.00324910
$784$	84.8644	3.03087
$785$	−10.0198	−0.357621
$786$	−68.3494	−2.43794
$787$	−6.70946	−0.239166	−0.119583	−	0.992824i	$-0.538156\pi$
$787$	−6.70946	−0.239166	−0.119583	+	0.992824i	$0.538156\pi$
$788$	92.5154	3.29572
$789$	47.4870	1.69058
$790$	−3.86025	−0.137341
$791$	−8.61472	−0.306304
$792$	72.0867	2.56149
$793$	93.1423	3.30758
$794$	16.1854	0.574398
$795$	−11.9866	−0.425121
$796$	50.9613	1.80627
$797$	−6.13276	−0.217233	−0.108617	−	0.994084i	$-0.534642\pi$
$797$	−6.13276	−0.217233	−0.108617	+	0.994084i	$0.534642\pi$
$798$	8.20158	0.290333
$799$	−42.5946	−1.50689
$800$	4.22380	0.149334
$801$	40.8802	1.44443
$802$	−70.6461	−2.49460
$803$	62.6343	2.21032
$804$	−98.6146	−3.47787
$805$	42.5049	1.49810
$806$	−62.1350	−2.18861
$807$	−39.2964	−1.38330
$808$	−34.1536	−1.20152
$809$	−13.5950	−0.477975	−0.238988	−	0.971023i	$-0.576816\pi$
$809$	−13.5950	−0.477975	−0.238988	+	0.971023i	$0.576816\pi$
$810$	−54.2365	−1.90568
$811$	9.40773	0.330350	0.165175	−	0.986264i	$-0.447181\pi$
$811$	9.40773	0.330350	0.165175	+	0.986264i	$0.447181\pi$
$812$	−1.03051	−0.0361639
$813$	−35.4937	−1.24482
$814$	−13.4573	−0.471677
$815$	−22.1784	−0.776874
$816$	−53.6144	−1.87688
$817$	1.48121	0.0518211
$818$	−68.2217	−2.38532
$819$	67.1727	2.34720
$820$	92.3687	3.22566
$821$	11.6152	0.405374	0.202687	−	0.979244i	$-0.435033\pi$
$821$	11.6152	0.405374	0.202687	+	0.979244i	$0.435033\pi$
$822$	−95.7837	−3.34084
$823$	14.8533	0.517752	0.258876	−	0.965911i	$-0.416648\pi$
$823$	14.8533	0.517752	0.258876	+	0.965911i	$0.416648\pi$
$824$	−28.2126	−0.982833
$825$	11.9236	0.415128
$826$	−7.81623	−0.271962
$827$	−31.8027	−1.10589	−0.552945	−	0.833218i	$-0.686496\pi$
$827$	−31.8027	−1.10589	−0.552945	+	0.833218i	$0.686496\pi$
$828$	46.5421	1.61745
$829$	0.174994	0.00607778	0.00303889	−	0.999995i	$-0.499033\pi$
$829$	0.174994	0.00607778	0.00303889	+	0.999995i	$0.499033\pi$
$830$	−39.7673	−1.38034
$831$	61.8431	2.14531
$832$	13.7645	0.477199
$833$	−47.2295	−1.63641
$834$	60.3995	2.09146
$835$	−2.71629	−0.0940012
$836$	−7.38974	−0.255579
$837$	6.38071	0.220549
$838$	32.1491	1.11057
$839$	−5.56210	−0.192025	−0.0960126	−	0.995380i	$-0.530609\pi$
$839$	−5.56210	−0.192025	−0.0960126	+	0.995380i	$0.530609\pi$
$840$	−124.265	−4.28756
$841$	−28.9973	−0.999905
$842$	−4.59227	−0.158260
$843$	−23.6108	−0.813200
$844$	4.44670	0.153062
$845$	−63.5643	−2.18668
$846$	−66.8673	−2.29895
$847$	−77.7272	−2.67074
$848$	16.9339	0.581511
$849$	−56.1806	−1.92811
$850$	−8.92600	−0.306159
$851$	−4.72950	−0.162125
$852$	−155.159	−5.31567
$853$	9.89925	0.338944	0.169472	−	0.985535i	$-0.445794\pi$
$853$	9.89925	0.338944	0.169472	+	0.985535i	$0.445794\pi$
$854$	−157.816	−5.40034
$855$	−1.42189	−0.0486277
$856$	83.8707	2.86664
$857$	34.2442	1.16976	0.584881	−	0.811119i	$-0.301141\pi$
$857$	34.2442	1.16976	0.584881	+	0.811119i	$0.301141\pi$
$858$	−205.988	−7.03230
$859$	1.81122	0.0617982	0.0308991	−	0.999523i	$-0.490163\pi$
$859$	1.81122	0.0617982	0.0308991	+	0.999523i	$0.490163\pi$
$860$	−41.2290	−1.40590
$861$	107.670	3.66937
$862$	−51.5936	−1.75729
$863$	30.3843	1.03429	0.517147	−	0.855896i	$-0.326994\pi$
$863$	30.3843	1.03429	0.517147	+	0.855896i	$0.326994\pi$
$864$	7.49322	0.254925
$865$	−20.6138	−0.700889
$866$	87.1978	2.96310
$867$	−9.08440	−0.308523
$868$	72.3233	2.45481
$869$	4.05429	0.137532
$870$	0.608050	0.0206148
$871$	65.6053	2.22295
$872$	−6.03914	−0.204511
$873$	−15.1775	−0.513680
$874$	−3.78049	−0.127877
$875$	−53.7272	−1.81631
$876$	−118.226	−3.99450
$877$	−29.1897	−0.985667	−0.492834	−	0.870124i	$-0.664039\pi$
$877$	−29.1897	−0.985667	−0.492834	+	0.870124i	$0.664039\pi$
$878$	−38.2044	−1.28934
$879$	58.3309	1.96745
$880$	69.2574	2.33467
$881$	−3.30261	−0.111268	−0.0556338	−	0.998451i	$-0.517718\pi$
$881$	−3.30261	−0.111268	−0.0556338	+	0.998451i	$0.517718\pi$
$882$	−74.1436	−2.49654
$883$	−38.8898	−1.30875	−0.654373	−	0.756172i	$-0.727067\pi$
$883$	−38.8898	−1.30875	−0.654373	+	0.756172i	$0.727067\pi$
$884$	105.932	3.56289
$885$	3.16827	0.106500
$886$	−31.5714	−1.06066
$887$	−10.9904	−0.369021	−0.184511	−	0.982831i	$-0.559070\pi$
$887$	−10.9904	−0.369021	−0.184511	+	0.982831i	$0.559070\pi$
$888$	13.8269	0.464002
$889$	−6.48910	−0.217637
$890$	92.4275	3.09818
$891$	56.9628	1.90833
$892$	16.6433	0.557261
$893$	3.73125	0.124862
$894$	−34.6657	−1.15939
$895$	44.6537	1.49261
$896$	−62.0230	−2.07204
$897$	−72.3934	−2.41715
$898$	0.893232	0.0298076
$899$	−0.192634	−0.00642472
$900$	−9.62621	−0.320874
$901$	−9.42418	−0.313965
$902$	−141.217	−4.70200
$903$	−48.0586	−1.59929
$904$	−11.6092	−0.386117
$905$	−26.8437	−0.892316
$906$	93.4274	3.10392
$907$	−11.1027	−0.368661	−0.184330	−	0.982864i	$-0.559012\pi$
$907$	−11.1027	−0.368661	−0.184330	+	0.982864i	$0.559012\pi$
$908$	64.1788	2.12985
$909$	12.6797	0.420557
$910$	151.873	5.03454
$911$	0.365097	0.0120962	0.00604809	−	0.999982i	$-0.498075\pi$
$911$	0.365097	0.0120962	0.00604809	+	0.999982i	$0.498075\pi$
$912$	4.69659	0.155520
$913$	41.7663	1.38226
$914$	10.0909	0.333776
$915$	63.9698	2.11477
$916$	−95.4502	−3.15376
$917$	−52.9266	−1.74779
$918$	−15.8351	−0.522637
$919$	−34.7145	−1.14513	−0.572563	−	0.819861i	$-0.694051\pi$
$919$	−34.7145	−1.14513	−0.572563	+	0.819861i	$0.694051\pi$
$920$	57.2797	1.88846
$921$	−24.0804	−0.793475
$922$	63.9444	2.10590
$923$	103.223	3.39762
$924$	239.763	7.88764
$925$	0.978192	0.0321628
$926$	−11.9348	−0.392202
$927$	10.4740	0.344012
$928$	−0.226221	−0.00742608
$929$	58.9405	1.93377	0.966887	−	0.255205i	$-0.0821429\pi$
$929$	58.9405	1.93377	0.966887	+	0.255205i	$0.0821429\pi$
$930$	−42.6741	−1.39934
$931$	4.13728	0.135594
$932$	−78.4914	−2.57107
$933$	33.0986	1.08360
$934$	−1.52171	−0.0497919
$935$	−38.5437	−1.26052
$936$	90.5222	2.95881
$937$	15.1297	0.494267	0.247133	−	0.968982i	$-0.420511\pi$
$937$	15.1297	0.494267	0.247133	+	0.968982i	$0.420511\pi$
$938$	−111.158	−3.62945
$939$	−23.8670	−0.778869
$940$	−103.858	−3.38748
$941$	28.7266	0.936461	0.468230	−	0.883606i	$-0.344892\pi$
$941$	28.7266	0.936461	0.468230	+	0.883606i	$0.344892\pi$
$942$	−28.9148	−0.942096
$943$	−49.6299	−1.61617
$944$	−4.47592	−0.145679
$945$	−15.5960	−0.507338
$946$	63.0325	2.04936
$947$	3.95712	0.128589	0.0642945	−	0.997931i	$-0.479520\pi$
$947$	3.95712	0.128589	0.0642945	+	0.997931i	$0.479520\pi$
$948$	−7.65274	−0.248550
$949$	78.6524	2.55316
$950$	0.781911	0.0253686
$951$	−63.5607	−2.06110
$952$	−97.7008	−3.16650
$953$	−8.31292	−0.269282	−0.134641	−	0.990894i	$-0.542988\pi$
$953$	−8.31292	−0.269282	−0.134641	+	0.990894i	$0.542988\pi$
$954$	−14.7946	−0.478993
$955$	50.9283	1.64800
$956$	61.8846	2.00149
$957$	−0.638615	−0.0206435
$958$	−20.6894	−0.668444
$959$	−74.1705	−2.39509
$960$	9.45342	0.305108
$961$	−17.4805	−0.563889
$962$	−16.8988	−0.544840
$963$	−31.1373	−1.00339
$964$	−48.5584	−1.56396
$965$	4.66659	0.150223
$966$	122.660	3.94651
$967$	16.5877	0.533424	0.266712	−	0.963776i	$-0.414063\pi$
$967$	16.5877	0.533424	0.266712	+	0.963776i	$0.414063\pi$
$968$	−104.745	−3.36665
$969$	−2.61379	−0.0839670
$970$	−34.3153	−1.10180
$971$	6.21462	0.199437	0.0997184	−	0.995016i	$-0.468206\pi$
$971$	6.21462	0.199437	0.0997184	+	0.995016i	$0.468206\pi$
$972$	−84.6706	−2.71581
$973$	46.7706	1.49940
$974$	1.32740	0.0425326
$975$	14.9730	0.479519
$976$	−90.3722	−2.89274
$977$	−24.0304	−0.768801	−0.384400	−	0.923166i	$-0.625592\pi$
$977$	−24.0304	−0.768801	−0.384400	+	0.923166i	$0.625592\pi$
$978$	−64.0018	−2.04655
$979$	−97.0736	−3.10249
$980$	−115.160	−3.67864
$981$	2.24205	0.0715833
$982$	16.3359	0.521299
$983$	33.2693	1.06113	0.530564	−	0.847645i	$-0.321980\pi$
$983$	33.2693	1.06113	0.530564	+	0.847645i	$0.321980\pi$
$984$	145.096	4.62549
$985$	−42.2707	−1.34686
$986$	0.478065	0.0152247
$987$	−121.062	−3.85345
$988$	−9.27959	−0.295223
$989$	22.1525	0.704407
$990$	−60.5081	−1.92307
$991$	11.2754	0.358174	0.179087	−	0.983833i	$-0.442686\pi$
$991$	11.2754	0.358174	0.179087	+	0.983833i	$0.442686\pi$
$992$	15.8766	0.504084
$993$	−47.3190	−1.50162
$994$	−174.895	−5.54734
$995$	−23.2844	−0.738166
$996$	−78.8367	−2.49804
$997$	17.5737	0.556565	0.278283	−	0.960499i	$-0.410235\pi$
$997$	17.5737	0.556565	0.278283	+	0.960499i	$0.410235\pi$
$998$	−29.9859	−0.949186
$999$	1.73536	0.0549043

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4033.2.a.e.1.6	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4033.2.a.e.1.6	✓	82	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:	\( N \)	\(=\)	\( 4033 = 37 \cdot 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4033.a (trivial)

\(f(q)\)	\(=\)	\(q-2.52769 q^{2} +2.28955 q^{3} +4.38920 q^{4} -2.00544 q^{5} -5.78727 q^{6} -4.48140 q^{7} -6.03914 q^{8} +2.24205 q^{9} +O(q^{10})\)	\(q-2.52769 q^{2} +2.28955 q^{3} +4.38920 q^{4} -2.00544 q^{5} -5.78727 q^{6} -4.48140 q^{7} -6.03914 q^{8} +2.24205 q^{9} +5.06913 q^{10} -5.32395 q^{11} +10.0493 q^{12} -6.68550 q^{13} +11.3276 q^{14} -4.59157 q^{15} +6.48667 q^{16} -3.61002 q^{17} -5.66721 q^{18} +0.316235 q^{19} -8.80229 q^{20} -10.2604 q^{21} +13.4573 q^{22} +4.72950 q^{23} -13.8269 q^{24} -0.978192 q^{25} +16.8988 q^{26} -1.73536 q^{27} -19.6697 q^{28} +0.0523907 q^{29} +11.6061 q^{30} -3.67688 q^{31} -4.31797 q^{32} -12.1895 q^{33} +9.12500 q^{34} +8.98719 q^{35} +9.84082 q^{36} -1.00000 q^{37} -0.799343 q^{38} -15.3068 q^{39} +12.1112 q^{40} -10.4937 q^{41} +25.9351 q^{42} +4.68390 q^{43} -23.3679 q^{44} -4.49631 q^{45} -11.9547 q^{46} +11.7990 q^{47} +14.8516 q^{48} +13.0829 q^{49} +2.47256 q^{50} -8.26533 q^{51} -29.3440 q^{52} +2.61056 q^{53} +4.38644 q^{54} +10.6769 q^{55} +27.0638 q^{56} +0.724037 q^{57} -0.132427 q^{58} -0.690019 q^{59} -20.1533 q^{60} -13.9320 q^{61} +9.29400 q^{62} -10.0475 q^{63} -2.05886 q^{64} +13.4074 q^{65} +30.8111 q^{66} -9.81308 q^{67} -15.8451 q^{68} +10.8284 q^{69} -22.7168 q^{70} -15.4398 q^{71} -13.5401 q^{72} -11.7646 q^{73} +2.52769 q^{74} -2.23962 q^{75} +1.38802 q^{76} +23.8587 q^{77} +38.6908 q^{78} -0.761520 q^{79} -13.0086 q^{80} -10.6994 q^{81} +26.5248 q^{82} -7.84499 q^{83} -45.0349 q^{84} +7.23969 q^{85} -11.8394 q^{86} +0.119951 q^{87} +32.1521 q^{88} +18.2334 q^{89} +11.3653 q^{90} +29.9604 q^{91} +20.7587 q^{92} -8.41841 q^{93} -29.8241 q^{94} -0.634192 q^{95} -9.88621 q^{96} -6.76945 q^{97} -33.0695 q^{98} -11.9366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 10 q^{2} + 17 q^{3} + 88 q^{4} + 22 q^{5} + 4 q^{6} + 15 q^{7} + 33 q^{8} + 91 q^{9}+O(q^{10}) \) 82 * q + 10 * q^2 + 17 * q^3 + 88 * q^4 + 22 * q^5 + 4 * q^6 + 15 * q^7 + 33 * q^8 + 91 * q^9	\( 82 q + 10 q^{2} + 17 q^{3} + 88 q^{4} + 22 q^{5} + 4 q^{6} + 15 q^{7} + 33 q^{8} + 91 q^{9} + 13 q^{10} + 2 q^{11} + 36 q^{12} + 23 q^{13} + 24 q^{14} + 35 q^{15} + 104 q^{16} + 43 q^{17} + 42 q^{18} + 15 q^{19} + 59 q^{20} + 9 q^{21} + 6 q^{22} + 70 q^{23} + 15 q^{24} + 98 q^{25} + 10 q^{26} + 65 q^{27} + 37 q^{28} + 27 q^{29} + 17 q^{30} + 59 q^{31} + 46 q^{32} + 16 q^{33} - 16 q^{34} + 80 q^{35} + 88 q^{36} - 82 q^{37} + 82 q^{38} + 13 q^{39} + 14 q^{40} + 3 q^{41} + 62 q^{42} + 7 q^{43} - 11 q^{44} + 42 q^{45} - 11 q^{46} + 123 q^{47} + 45 q^{48} + 105 q^{49} + 27 q^{50} + 3 q^{51} - 30 q^{52} + 82 q^{53} - 27 q^{54} + 37 q^{55} + 66 q^{56} + 29 q^{57} - 34 q^{58} + 60 q^{59} + 94 q^{60} + 9 q^{61} - 2 q^{62} + 106 q^{63} + 93 q^{64} + 9 q^{65} + 63 q^{66} + 113 q^{68} + 48 q^{69} + 47 q^{70} + 59 q^{71} + 63 q^{72} + 21 q^{73} - 10 q^{74} + 77 q^{75} + 22 q^{76} + 30 q^{77} + 29 q^{78} + 55 q^{79} + 88 q^{80} + 42 q^{81} + 4 q^{82} + 92 q^{83} + 43 q^{84} - 7 q^{85} + 17 q^{86} + 147 q^{87} - 13 q^{88} + 100 q^{89} + 91 q^{90} + 28 q^{91} + 127 q^{92} + 3 q^{93} + 30 q^{94} + 48 q^{95} - 31 q^{96} + 53 q^{97} + 101 q^{98} - 20 q^{99}+O(q^{100}) \) 82 * q + 10 * q^2 + 17 * q^3 + 88 * q^4 + 22 * q^5 + 4 * q^6 + 15 * q^7 + 33 * q^8 + 91 * q^9 + 13 * q^10 + 2 * q^11 + 36 * q^12 + 23 * q^13 + 24 * q^14 + 35 * q^15 + 104 * q^16 + 43 * q^17 + 42 * q^18 + 15 * q^19 + 59 * q^20 + 9 * q^21 + 6 * q^22 + 70 * q^23 + 15 * q^24 + 98 * q^25 + 10 * q^26 + 65 * q^27 + 37 * q^28 + 27 * q^29 + 17 * q^30 + 59 * q^31 + 46 * q^32 + 16 * q^33 - 16 * q^34 + 80 * q^35 + 88 * q^36 - 82 * q^37 + 82 * q^38 + 13 * q^39 + 14 * q^40 + 3 * q^41 + 62 * q^42 + 7 * q^43 - 11 * q^44 + 42 * q^45 - 11 * q^46 + 123 * q^47 + 45 * q^48 + 105 * q^49 + 27 * q^50 + 3 * q^51 - 30 * q^52 + 82 * q^53 - 27 * q^54 + 37 * q^55 + 66 * q^56 + 29 * q^57 - 34 * q^58 + 60 * q^59 + 94 * q^60 + 9 * q^61 - 2 * q^62 + 106 * q^63 + 93 * q^64 + 9 * q^65 + 63 * q^66 + 113 * q^68 + 48 * q^69 + 47 * q^70 + 59 * q^71 + 63 * q^72 + 21 * q^73 - 10 * q^74 + 77 * q^75 + 22 * q^76 + 30 * q^77 + 29 * q^78 + 55 * q^79 + 88 * q^80 + 42 * q^81 + 4 * q^82 + 92 * q^83 + 43 * q^84 - 7 * q^85 + 17 * q^86 + 147 * q^87 - 13 * q^88 + 100 * q^89 + 91 * q^90 + 28 * q^91 + 127 * q^92 + 3 * q^93 + 30 * q^94 + 48 * q^95 - 31 * q^96 + 53 * q^97 + 101 * q^98 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more