LMFDB - Embedded newform 3381.2.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.3176240.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} + 7x^{2} + 19x - 9 $ x^5 - x^4 - 9x^3 + 7x^2 + 19*x - 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.443666$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.443666 q^{2} -1.00000 q^{3} -1.80316 q^{4} +2.13100 q^{5} -0.443666 q^{6} -1.68733 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.443666 q^{2} -1.00000 q^{3} -1.80316 q^{4} +2.13100 q^{5} -0.443666 q^{6} -1.68733 q^{8} +1.00000 q^{9} +0.945453 q^{10} -4.24683 q^{11} +1.80316 q^{12} -2.11583 q^{13} -2.13100 q^{15} +2.85771 q^{16} +3.08620 q^{17} +0.443666 q^{18} +4.66087 q^{19} -3.84253 q^{20} -1.88417 q^{22} +1.00000 q^{23} +1.68733 q^{24} -0.458839 q^{25} -0.938721 q^{26} -1.00000 q^{27} +4.52012 q^{29} -0.945453 q^{30} -1.32784 q^{31} +4.64254 q^{32} +4.24683 q^{33} +1.36924 q^{34} -1.80316 q^{36} +0.106077 q^{37} +2.06787 q^{38} +2.11583 q^{39} -3.59571 q^{40} -2.90251 q^{41} -6.11583 q^{43} +7.65771 q^{44} +2.13100 q^{45} +0.443666 q^{46} -9.37467 q^{47} -2.85771 q^{48} -0.203572 q^{50} -3.08620 q^{51} +3.81517 q^{52} +7.00203 q^{53} -0.443666 q^{54} -9.04999 q^{55} -4.66087 q^{57} +2.00542 q^{58} -10.7767 q^{59} +3.84253 q^{60} -7.02850 q^{61} -0.589118 q^{62} -3.65568 q^{64} -4.50883 q^{65} +1.88417 q^{66} -14.9907 q^{67} -5.56492 q^{68} -1.00000 q^{69} +4.38912 q^{71} -1.68733 q^{72} +14.9975 q^{73} +0.0470629 q^{74} +0.458839 q^{75} -8.40429 q^{76} +0.938721 q^{78} +13.0762 q^{79} +6.08977 q^{80} +1.00000 q^{81} -1.28774 q^{82} -5.97353 q^{83} +6.57670 q^{85} -2.71339 q^{86} -4.52012 q^{87} +7.16581 q^{88} -0.438964 q^{89} +0.945453 q^{90} -1.80316 q^{92} +1.32784 q^{93} -4.15922 q^{94} +9.93231 q^{95} -4.64254 q^{96} -3.31655 q^{97} -4.24683 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} - 5 q^{3} + 9 q^{4} - 2 q^{5} - q^{6} + 3 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q + q^2 - 5 * q^3 + 9 * q^4 - 2 * q^5 - q^6 + 3 * q^8 + 5 * q^9	$ 5 q + q^{2} - 5 q^{3} + 9 q^{4} - 2 q^{5} - q^{6} + 3 q^{8} + 5 q^{9} - 2 q^{11} - 9 q^{12} - 4 q^{13} + 2 q^{15} + q^{16} - 2 q^{17} + q^{18} - 8 q^{19} - 12 q^{20} - 16 q^{22} + 5 q^{23} - 3 q^{24} + 5 q^{25} - 16 q^{26} - 5 q^{27} + 4 q^{29} - 12 q^{31} + 7 q^{32} + 2 q^{33} - 10 q^{34} + 9 q^{36} - 6 q^{37} + 8 q^{38} + 4 q^{39} - 30 q^{40} - 6 q^{41} - 24 q^{43} + 16 q^{44} - 2 q^{45} + q^{46} - 24 q^{47} - q^{48} - 3 q^{50} + 2 q^{51} + 4 q^{52} + 2 q^{53} - q^{54} - 8 q^{55} + 8 q^{57} + 4 q^{58} - 16 q^{59} + 12 q^{60} - 22 q^{61} - 6 q^{62} - 29 q^{64} + 22 q^{65} + 16 q^{66} - 16 q^{67} - 14 q^{68} - 5 q^{69} + 16 q^{71} + 3 q^{72} + 8 q^{74} - 5 q^{75} - 30 q^{76} + 16 q^{78} + 12 q^{79} + 6 q^{80} + 5 q^{81} - 24 q^{82} - 10 q^{83} - 14 q^{85} - 20 q^{86} - 4 q^{87} - 8 q^{88} + 16 q^{89} + 9 q^{92} + 12 q^{93} + 32 q^{94} + 18 q^{95} - 7 q^{96} + 4 q^{97} - 2 q^{99}+O(q^{100}) $ 5 * q + q^2 - 5 * q^3 + 9 * q^4 - 2 * q^5 - q^6 + 3 * q^8 + 5 * q^9 - 2 * q^11 - 9 * q^12 - 4 * q^13 + 2 * q^15 + q^16 - 2 * q^17 + q^18 - 8 * q^19 - 12 * q^20 - 16 * q^22 + 5 * q^23 - 3 * q^24 + 5 * q^25 - 16 * q^26 - 5 * q^27 + 4 * q^29 - 12 * q^31 + 7 * q^32 + 2 * q^33 - 10 * q^34 + 9 * q^36 - 6 * q^37 + 8 * q^38 + 4 * q^39 - 30 * q^40 - 6 * q^41 - 24 * q^43 + 16 * q^44 - 2 * q^45 + q^46 - 24 * q^47 - q^48 - 3 * q^50 + 2 * q^51 + 4 * q^52 + 2 * q^53 - q^54 - 8 * q^55 + 8 * q^57 + 4 * q^58 - 16 * q^59 + 12 * q^60 - 22 * q^61 - 6 * q^62 - 29 * q^64 + 22 * q^65 + 16 * q^66 - 16 * q^67 - 14 * q^68 - 5 * q^69 + 16 * q^71 + 3 * q^72 + 8 * q^74 - 5 * q^75 - 30 * q^76 + 16 * q^78 + 12 * q^79 + 6 * q^80 + 5 * q^81 - 24 * q^82 - 10 * q^83 - 14 * q^85 - 20 * q^86 - 4 * q^87 - 8 * q^88 + 16 * q^89 + 9 * q^92 + 12 * q^93 + 32 * q^94 + 18 * q^95 - 7 * q^96 + 4 * q^97 - 2 * 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 $

$2$	0.443666	0.313719	0.156860	−	0.987621i	$-0.449863\pi$
$2$	0.443666	0.313719	0.156860	+	0.987621i	$0.449863\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.80316	−0.901580
$5$	2.13100	0.953012	0.476506	−	0.879171i	$-0.341903\pi$
$5$	2.13100	0.953012	0.476506	+	0.879171i	$0.341903\pi$
$6$	−0.443666	−0.181126
$7$	0	0
$7$	0	0
$8$	−1.68733	−0.596563
$9$	1.00000	0.333333
$10$	0.945453	0.298978
$11$	−4.24683	−1.28047	−0.640233	−	0.768181i	$-0.721162\pi$
$11$	−4.24683	−1.28047	−0.640233	+	0.768181i	$0.721162\pi$
$12$	1.80316	0.520528
$13$	−2.11583	−0.586825	−0.293412	−	0.955986i	$-0.594791\pi$
$13$	−2.11583	−0.586825	−0.293412	+	0.955986i	$0.594791\pi$
$14$	0	0
$15$	−2.13100	−0.550222
$16$	2.85771	0.714427
$17$	3.08620	0.748514	0.374257	−	0.927325i	$-0.377898\pi$
$17$	3.08620	0.748514	0.374257	+	0.927325i	$0.377898\pi$
$18$	0.443666	0.104573
$19$	4.66087	1.06928	0.534638	−	0.845081i	$-0.320448\pi$
$19$	4.66087	1.06928	0.534638	+	0.845081i	$0.320448\pi$
$20$	−3.84253	−0.859217
$21$	0	0
$22$	−1.88417	−0.401707
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.68733	0.344426
$25$	−0.458839	−0.0917679
$26$	−0.938721	−0.184098
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.52012	0.839365	0.419682	−	0.907671i	$-0.362141\pi$
$29$	4.52012	0.839365	0.419682	+	0.907671i	$0.362141\pi$
$30$	−0.945453	−0.172615
$31$	−1.32784	−0.238487	−0.119244	−	0.992865i	$-0.538047\pi$
$31$	−1.32784	−0.238487	−0.119244	+	0.992865i	$0.538047\pi$
$32$	4.64254	0.820692
$33$	4.24683	0.739278
$34$	1.36924	0.234823
$35$	0	0
$36$	−1.80316	−0.300527
$37$	0.106077	0.0174390	0.00871950	−	0.999962i	$-0.497224\pi$
$37$	0.106077	0.0174390	0.00871950	+	0.999962i	$0.497224\pi$
$38$	2.06787	0.335453
$39$	2.11583	0.338803
$40$	−3.59571	−0.568531
$41$	−2.90251	−0.453295	−0.226648	−	0.973977i	$-0.572777\pi$
$41$	−2.90251	−0.453295	−0.226648	+	0.973977i	$0.572777\pi$
$42$	0	0
$43$	−6.11583	−0.932655	−0.466327	−	0.884612i	$-0.654423\pi$
$43$	−6.11583	−0.932655	−0.466327	+	0.884612i	$0.654423\pi$
$44$	7.65771	1.15444
$45$	2.13100	0.317671
$46$	0.443666	0.0654150
$47$	−9.37467	−1.36744	−0.683718	−	0.729746i	$-0.739638\pi$
$47$	−9.37467	−1.36744	−0.683718	+	0.729746i	$0.739638\pi$
$48$	−2.85771	−0.412475
$49$	0	0
$50$	−0.203572	−0.0287894
$51$	−3.08620	−0.432155
$52$	3.81517	0.529070
$53$	7.00203	0.961803	0.480901	−	0.876775i	$-0.340310\pi$
$53$	7.00203	0.961803	0.480901	+	0.876775i	$0.340310\pi$
$54$	−0.443666	−0.0603753
$55$	−9.04999	−1.22030
$56$	0	0
$57$	−4.66087	−0.617347
$58$	2.00542	0.263325
$59$	−10.7767	−1.40301	−0.701503	−	0.712666i	$-0.747487\pi$
$59$	−10.7767	−1.40301	−0.701503	+	0.712666i	$0.747487\pi$
$60$	3.84253	0.496069
$61$	−7.02850	−0.899907	−0.449953	−	0.893052i	$-0.648559\pi$
$61$	−7.02850	−0.899907	−0.449953	+	0.893052i	$0.648559\pi$
$62$	−0.589118	−0.0748180
$63$	0	0
$64$	−3.65568	−0.456960
$65$	−4.50883	−0.559251
$66$	1.88417	0.231926
$67$	−14.9907	−1.83141	−0.915705	−	0.401851i	$-0.868367\pi$
$67$	−14.9907	−1.83141	−0.915705	+	0.401851i	$0.868367\pi$
$68$	−5.56492	−0.674845
$69$	−1.00000	−0.120386
$70$	0	0
$71$	4.38912	0.520893	0.260446	−	0.965488i	$-0.416130\pi$
$71$	4.38912	0.520893	0.260446	+	0.965488i	$0.416130\pi$
$72$	−1.68733	−0.198854
$73$	14.9975	1.75532	0.877661	−	0.479283i	$-0.159103\pi$
$73$	14.9975	1.75532	0.877661	+	0.479283i	$0.159103\pi$
$74$	0.0470629	0.00547095
$75$	0.458839	0.0529822
$76$	−8.40429	−0.964038
$77$	0	0
$78$	0.938721	0.106289
$79$	13.0762	1.47119	0.735595	−	0.677422i	$-0.236903\pi$
$79$	13.0762	1.47119	0.735595	+	0.677422i	$0.236903\pi$
$80$	6.08977	0.680858
$81$	1.00000	0.111111
$82$	−1.28774	−0.142207
$83$	−5.97353	−0.655681	−0.327840	−	0.944733i	$-0.606321\pi$
$83$	−5.97353	−0.655681	−0.327840	+	0.944733i	$0.606321\pi$
$84$	0	0
$85$	6.57670	0.713343
$86$	−2.71339	−0.292592
$87$	−4.52012	−0.484608
$88$	7.16581	0.763878
$89$	−0.438964	−0.0465301	−0.0232651	−	0.999729i	$-0.507406\pi$
$89$	−0.438964	−0.0465301	−0.0232651	+	0.999729i	$0.507406\pi$
$90$	0.945453	0.0996595
$91$	0	0
$92$	−1.80316	−0.187992
$93$	1.32784	0.137691
$94$	−4.15922	−0.428991
$95$	9.93231	1.01903
$96$	−4.64254	−0.473827
$97$	−3.31655	−0.336744	−0.168372	−	0.985723i	$-0.553851\pi$
$97$	−3.31655	−0.336744	−0.168372	+	0.985723i	$0.553851\pi$
$98$	0	0
$99$	−4.24683	−0.426822
$100$	0.827361	0.0827361
$101$	−15.6048	−1.55273	−0.776367	−	0.630281i	$-0.782939\pi$
$101$	−15.6048	−1.55273	−0.776367	+	0.630281i	$0.782939\pi$
$102$	−1.36924	−0.135575
$103$	−2.71899	−0.267910	−0.133955	−	0.990987i	$-0.542768\pi$
$103$	−2.71899	−0.267910	−0.133955	+	0.990987i	$0.542768\pi$
$104$	3.57011	0.350078
$105$	0	0
$106$	3.10656	0.301736
$107$	−6.79187	−0.656595	−0.328297	−	0.944574i	$-0.606475\pi$
$107$	−6.79187	−0.656595	−0.328297	+	0.944574i	$0.606475\pi$
$108$	1.80316	0.173509
$109$	−4.67889	−0.448157	−0.224078	−	0.974571i	$-0.571937\pi$
$109$	−4.67889	−0.448157	−0.224078	+	0.974571i	$0.571937\pi$
$110$	−4.01517	−0.382832
$111$	−0.106077	−0.0100684
$112$	0	0
$113$	11.4324	1.07547	0.537734	−	0.843115i	$-0.319281\pi$
$113$	11.4324	1.07547	0.537734	+	0.843115i	$0.319281\pi$
$114$	−2.06787	−0.193674
$115$	2.13100	0.198717
$116$	−8.15050	−0.756755
$117$	−2.11583	−0.195608
$118$	−4.78126	−0.440150
$119$	0	0
$120$	3.59571	0.328242
$121$	7.03554	0.639594
$122$	−3.11831	−0.282318
$123$	2.90251	0.261710
$124$	2.39431	0.215015
$125$	−11.6328	−1.04047
$126$	0	0
$127$	−1.56793	−0.139132	−0.0695658	−	0.997577i	$-0.522161\pi$
$127$	−1.56793	−0.139132	−0.0695658	+	0.997577i	$0.522161\pi$
$128$	−10.9070	−0.964049
$129$	6.11583	0.538469
$130$	−2.00041	−0.175448
$131$	−4.56176	−0.398563	−0.199281	−	0.979942i	$-0.563861\pi$
$131$	−4.56176	−0.398563	−0.199281	+	0.979942i	$0.563861\pi$
$132$	−7.65771	−0.666518
$133$	0	0
$134$	−6.65088	−0.574549
$135$	−2.13100	−0.183407
$136$	−5.20745	−0.446535
$137$	−1.13416	−0.0968977	−0.0484489	−	0.998826i	$-0.515428\pi$
$137$	−1.13416	−0.0968977	−0.0484489	+	0.998826i	$0.515428\pi$
$138$	−0.443666	−0.0377674
$139$	−16.5034	−1.39980	−0.699900	−	0.714241i	$-0.746772\pi$
$139$	−16.5034	−1.39980	−0.699900	+	0.714241i	$0.746772\pi$
$140$	0	0
$141$	9.37467	0.789489
$142$	1.94730	0.163414
$143$	8.98555	0.751409
$144$	2.85771	0.238142
$145$	9.63237	0.799925
$146$	6.65387	0.550678
$147$	0	0
$148$	−0.191274	−0.0157226
$149$	−18.7027	−1.53219	−0.766093	−	0.642729i	$-0.777802\pi$
$149$	−18.7027	−1.53219	−0.766093	+	0.642729i	$0.777802\pi$
$150$	0.203572	0.0166215
$151$	−1.08232	−0.0880780	−0.0440390	−	0.999030i	$-0.514023\pi$
$151$	−1.08232	−0.0880780	−0.0440390	+	0.999030i	$0.514023\pi$
$152$	−7.86444	−0.637890
$153$	3.08620	0.249505
$154$	0	0
$155$	−2.82963	−0.227281
$156$	−3.81517	−0.305458
$157$	9.77466	0.780103	0.390052	−	0.920793i	$-0.372457\pi$
$157$	9.77466	0.780103	0.390052	+	0.920793i	$0.372457\pi$
$158$	5.80148	0.461541
$159$	−7.00203	−0.555297
$160$	9.89324	0.782130
$161$	0	0
$162$	0.443666	0.0348577
$163$	11.8048	0.924621	0.462311	−	0.886718i	$-0.347020\pi$
$163$	11.8048	0.924621	0.462311	+	0.886718i	$0.347020\pi$
$164$	5.23368	0.408682
$165$	9.04999	0.704541
$166$	−2.65026	−0.205700
$167$	−16.1744	−1.25161	−0.625806	−	0.779979i	$-0.715230\pi$
$167$	−16.1744	−1.25161	−0.625806	+	0.779979i	$0.715230\pi$
$168$	0	0
$169$	−8.52328	−0.655637
$170$	2.91786	0.223789
$171$	4.66087	0.356425
$172$	11.0278	0.840863
$173$	−17.9165	−1.36216	−0.681081	−	0.732208i	$-0.738490\pi$
$173$	−17.9165	−1.36216	−0.681081	+	0.732208i	$0.738490\pi$
$174$	−2.00542	−0.152031
$175$	0	0
$176$	−12.1362	−0.914800
$177$	10.7767	0.810026
$178$	−0.194754	−0.0145974
$179$	11.4527	0.856018	0.428009	−	0.903775i	$-0.359215\pi$
$179$	11.4527	0.856018	0.428009	+	0.903775i	$0.359215\pi$
$180$	−3.84253	−0.286406
$181$	−7.48489	−0.556348	−0.278174	−	0.960531i	$-0.589729\pi$
$181$	−7.48489	−0.556348	−0.278174	+	0.960531i	$0.589729\pi$
$182$	0	0
$183$	7.02850	0.519561
$184$	−1.68733	−0.124392
$185$	0.226051	0.0166196
$186$	0.589118	0.0431962
$187$	−13.1066	−0.958447
$188$	16.9040	1.23285
$189$	0	0
$190$	4.40663	0.319691
$191$	−2.35160	−0.170155	−0.0850777	−	0.996374i	$-0.527114\pi$
$191$	−2.35160	−0.170155	−0.0850777	+	0.996374i	$0.527114\pi$
$192$	3.65568	0.263826
$193$	−22.8719	−1.64635	−0.823177	−	0.567784i	$-0.807801\pi$
$193$	−22.8719	−1.64635	−0.823177	+	0.567784i	$0.807801\pi$
$194$	−1.47144	−0.105643
$195$	4.50883	0.322884
$196$	0	0
$197$	12.7602	0.909128	0.454564	−	0.890714i	$-0.349795\pi$
$197$	12.7602	0.909128	0.454564	+	0.890714i	$0.349795\pi$
$198$	−1.88417	−0.133902
$199$	−27.3627	−1.93969	−0.969845	−	0.243722i	$-0.921631\pi$
$199$	−27.3627	−1.93969	−0.969845	+	0.243722i	$0.921631\pi$
$200$	0.774215	0.0547453
$201$	14.9907	1.05737
$202$	−6.92331	−0.487123
$203$	0	0
$204$	5.56492	0.389622
$205$	−6.18524	−0.431996
$206$	−1.20632	−0.0840485
$207$	1.00000	0.0695048
$208$	−6.04641	−0.419243
$209$	−19.7939	−1.36917
$210$	0	0
$211$	26.0130	1.79081	0.895403	−	0.445257i	$-0.146888\pi$
$211$	26.0130	1.79081	0.895403	+	0.445257i	$0.146888\pi$
$212$	−12.6258	−0.867142
$213$	−4.38912	−0.300737
$214$	−3.01332	−0.205986
$215$	−13.0328	−0.888831
$216$	1.68733	0.114809
$217$	0	0
$218$	−2.07587	−0.140595
$219$	−14.9975	−1.01344
$220$	16.3186	1.10020
$221$	−6.52987	−0.439246
$222$	−0.0470629	−0.00315865
$223$	−3.90697	−0.261630	−0.130815	−	0.991407i	$-0.541759\pi$
$223$	−3.90697	−0.261630	−0.130815	+	0.991407i	$0.541759\pi$
$224$	0	0
$225$	−0.458839	−0.0305893
$226$	5.07216	0.337395
$227$	8.96540	0.595055	0.297527	−	0.954713i	$-0.403838\pi$
$227$	8.96540	0.595055	0.297527	+	0.954713i	$0.403838\pi$
$228$	8.40429	0.556588
$229$	−1.86403	−0.123178	−0.0615891	−	0.998102i	$-0.519617\pi$
$229$	−1.86403	−0.123178	−0.0615891	+	0.998102i	$0.519617\pi$
$230$	0.945453	0.0623413
$231$	0	0
$232$	−7.62695	−0.500734
$233$	−3.53457	−0.231557	−0.115779	−	0.993275i	$-0.536936\pi$
$233$	−3.53457	−0.231557	−0.115779	+	0.993275i	$0.536936\pi$
$234$	−0.938721	−0.0613661
$235$	−19.9774	−1.30318
$236$	19.4321	1.26492
$237$	−13.0762	−0.849392
$238$	0	0
$239$	−11.8536	−0.766743	−0.383372	−	0.923594i	$-0.625237\pi$
$239$	−11.8536	−0.766743	−0.383372	+	0.923594i	$0.625237\pi$
$240$	−6.08977	−0.393093
$241$	2.69753	0.173763	0.0868817	−	0.996219i	$-0.472310\pi$
$241$	2.69753	0.173763	0.0868817	+	0.996219i	$0.472310\pi$
$242$	3.12143	0.200653
$243$	−1.00000	−0.0641500
$244$	12.6735	0.811338
$245$	0	0
$246$	1.28774	0.0821035
$247$	−9.86159	−0.627478
$248$	2.24051	0.142272
$249$	5.97353	0.378557
$250$	−5.16107	−0.326415
$251$	12.0407	0.760004	0.380002	−	0.924986i	$-0.375923\pi$
$251$	12.0407	0.760004	0.380002	+	0.924986i	$0.375923\pi$
$252$	0	0
$253$	−4.24683	−0.266996
$254$	−0.695639	−0.0436483
$255$	−6.57670	−0.411849
$256$	2.47230	0.154519
$257$	6.96752	0.434622	0.217311	−	0.976102i	$-0.430271\pi$
$257$	6.96752	0.434622	0.217311	+	0.976102i	$0.430271\pi$
$258$	2.71339	0.168928
$259$	0	0
$260$	8.13014	0.504210
$261$	4.52012	0.279788
$262$	−2.02390	−0.125037
$263$	−5.27360	−0.325184	−0.162592	−	0.986693i	$-0.551985\pi$
$263$	−5.27360	−0.325184	−0.162592	+	0.986693i	$0.551985\pi$
$264$	−7.16581	−0.441025
$265$	14.9213	0.916610
$266$	0	0
$267$	0.438964	0.0268642
$268$	27.0307	1.65116
$269$	−13.8453	−0.844162	−0.422081	−	0.906558i	$-0.638700\pi$
$269$	−13.8453	−0.844162	−0.422081	+	0.906558i	$0.638700\pi$
$270$	−0.945453	−0.0575384
$271$	−0.0889114	−0.00540098	−0.00270049	−	0.999996i	$-0.500860\pi$
$271$	−0.0889114	−0.00540098	−0.00270049	+	0.999996i	$0.500860\pi$
$272$	8.81946	0.534758
$273$	0	0
$274$	−0.503188	−0.0303987
$275$	1.94861	0.117506
$276$	1.80316	0.108537
$277$	−6.35249	−0.381684	−0.190842	−	0.981621i	$-0.561122\pi$
$277$	−6.35249	−0.381684	−0.190842	+	0.981621i	$0.561122\pi$
$278$	−7.32200	−0.439144
$279$	−1.32784	−0.0794957
$280$	0	0
$281$	−8.53718	−0.509285	−0.254643	−	0.967035i	$-0.581958\pi$
$281$	−8.53718	−0.509285	−0.254643	+	0.967035i	$0.581958\pi$
$282$	4.15922	0.247678
$283$	−30.0708	−1.78753	−0.893763	−	0.448540i	$-0.851944\pi$
$283$	−30.0708	−1.78753	−0.893763	+	0.448540i	$0.851944\pi$
$284$	−7.91428	−0.469626
$285$	−9.93231	−0.588339
$286$	3.98658	0.235732
$287$	0	0
$288$	4.64254	0.273564
$289$	−7.47536	−0.439727
$290$	4.27356	0.250952
$291$	3.31655	0.194419
$292$	−27.0428	−1.58256
$293$	−6.38456	−0.372990	−0.186495	−	0.982456i	$-0.559713\pi$
$293$	−6.38456	−0.372990	−0.186495	+	0.982456i	$0.559713\pi$
$294$	0	0
$295$	−22.9651	−1.33708
$296$	−0.178988	−0.0104034
$297$	4.24683	0.246426
$298$	−8.29777	−0.480677
$299$	−2.11583	−0.122361
$300$	−0.827361	−0.0477677
$301$	0	0
$302$	−0.480189	−0.0276318
$303$	15.6048	0.896471
$304$	13.3194	0.763920
$305$	−14.9777	−0.857622
$306$	1.36924	0.0782744
$307$	−14.3546	−0.819261	−0.409631	−	0.912252i	$-0.634342\pi$
$307$	−14.3546	−0.819261	−0.409631	+	0.912252i	$0.634342\pi$
$308$	0	0
$309$	2.71899	0.154678
$310$	−1.25541	−0.0713025
$311$	12.3453	0.700035	0.350018	−	0.936743i	$-0.386176\pi$
$311$	12.3453	0.700035	0.350018	+	0.936743i	$0.386176\pi$
$312$	−3.57011	−0.202117
$313$	−7.60776	−0.430016	−0.215008	−	0.976612i	$-0.568978\pi$
$313$	−7.60776	−0.430016	−0.215008	+	0.976612i	$0.568978\pi$
$314$	4.33669	0.244733
$315$	0	0
$316$	−23.5785	−1.32640
$317$	−10.9133	−0.612952	−0.306476	−	0.951878i	$-0.599150\pi$
$317$	−10.9133	−0.612952	−0.306476	+	0.951878i	$0.599150\pi$
$318$	−3.10656	−0.174207
$319$	−19.1962	−1.07478
$320$	−7.79025	−0.435488
$321$	6.79187	0.379085
$322$	0	0
$323$	14.3844	0.800368
$324$	−1.80316	−0.100176
$325$	0.970825	0.0538517
$326$	5.23738	0.290072
$327$	4.67889	0.258743
$328$	4.89750	0.270419
$329$	0	0
$330$	4.01517	0.221028
$331$	0.337516	0.0185516	0.00927578	−	0.999957i	$-0.497047\pi$
$331$	0.337516	0.0185516	0.00927578	+	0.999957i	$0.497047\pi$
$332$	10.7712	0.591149
$333$	0.106077	0.00581300
$334$	−7.17603	−0.392655
$335$	−31.9453	−1.74536
$336$	0	0
$337$	−8.31713	−0.453063	−0.226532	−	0.974004i	$-0.572739\pi$
$337$	−8.31713	−0.453063	−0.226532	+	0.974004i	$0.572739\pi$
$338$	−3.78149	−0.205686
$339$	−11.4324	−0.620921
$340$	−11.8588	−0.643136
$341$	5.63910	0.305375
$342$	2.06787	0.111818
$343$	0	0
$344$	10.3194	0.556387
$345$	−2.13100	−0.114729
$346$	−7.94893	−0.427337
$347$	3.00339	0.161231	0.0806153	−	0.996745i	$-0.474311\pi$
$347$	3.00339	0.161231	0.0806153	+	0.996745i	$0.474311\pi$
$348$	8.15050	0.436913
$349$	18.1636	0.972273	0.486136	−	0.873883i	$-0.338406\pi$
$349$	18.1636	0.972273	0.486136	+	0.873883i	$0.338406\pi$
$350$	0	0
$351$	2.11583	0.112934
$352$	−19.7160	−1.05087
$353$	27.3645	1.45646	0.728231	−	0.685331i	$-0.240343\pi$
$353$	27.3645	1.45646	0.728231	+	0.685331i	$0.240343\pi$
$354$	4.78126	0.254121
$355$	9.35321	0.496417
$356$	0.791523	0.0419506
$357$	0	0
$358$	5.08119	0.268549
$359$	27.5957	1.45644	0.728222	−	0.685342i	$-0.240347\pi$
$359$	27.5957	1.45644	0.728222	+	0.685342i	$0.240347\pi$
$360$	−3.59571	−0.189510
$361$	2.72369	0.143352
$362$	−3.32079	−0.174537
$363$	−7.03554	−0.369270
$364$	0	0
$365$	31.9596	1.67284
$366$	3.11831	0.162996
$367$	16.9624	0.885429	0.442715	−	0.896663i	$-0.354015\pi$
$367$	16.9624	0.885429	0.442715	+	0.896663i	$0.354015\pi$
$368$	2.85771	0.148968
$369$	−2.90251	−0.151098
$370$	0.100291	0.00521388
$371$	0	0
$372$	−2.39431	−0.124139
$373$	6.75950	0.349994	0.174997	−	0.984569i	$-0.444008\pi$
$373$	6.75950	0.349994	0.174997	+	0.984569i	$0.444008\pi$
$374$	−5.81494	−0.300683
$375$	11.6328	0.600715
$376$	15.8182	0.815761
$377$	−9.56379	−0.492560
$378$	0	0
$379$	−38.4027	−1.97261	−0.986307	−	0.164919i	$-0.947264\pi$
$379$	−38.4027	−1.97261	−0.986307	+	0.164919i	$0.947264\pi$
$380$	−17.9095	−0.918740
$381$	1.56793	0.0803277
$382$	−1.04332	−0.0533811
$383$	11.4138	0.583216	0.291608	−	0.956538i	$-0.405810\pi$
$383$	11.4138	0.583216	0.291608	+	0.956538i	$0.405810\pi$
$384$	10.9070	0.556594
$385$	0	0
$386$	−10.1475	−0.516493
$387$	−6.11583	−0.310885
$388$	5.98027	0.303602
$389$	8.97494	0.455047	0.227524	−	0.973773i	$-0.426937\pi$
$389$	8.97494	0.455047	0.227524	+	0.973773i	$0.426937\pi$
$390$	2.00041	0.101295
$391$	3.08620	0.156076
$392$	0	0
$393$	4.56176	0.230110
$394$	5.66128	0.285211
$395$	27.8654	1.40206
$396$	7.65771	0.384814
$397$	18.3385	0.920382	0.460191	−	0.887820i	$-0.347781\pi$
$397$	18.3385	0.920382	0.460191	+	0.887820i	$0.347781\pi$
$398$	−12.1399	−0.608518
$399$	0	0
$400$	−1.31123	−0.0655615
$401$	15.6622	0.782134	0.391067	−	0.920362i	$-0.372106\pi$
$401$	15.6622	0.782134	0.391067	+	0.920362i	$0.372106\pi$
$402$	6.65088	0.331716
$403$	2.80948	0.139950
$404$	28.1379	1.39991
$405$	2.13100	0.105890
$406$	0	0
$407$	−0.450492	−0.0223300
$408$	5.20745	0.257807
$409$	18.2622	0.903008	0.451504	−	0.892269i	$-0.350888\pi$
$409$	18.2622	0.903008	0.451504	+	0.892269i	$0.350888\pi$
$410$	−2.74418	−0.135525
$411$	1.13416	0.0559439
$412$	4.90277	0.241542
$413$	0	0
$414$	0.443666	0.0218050
$415$	−12.7296	−0.624872
$416$	−9.82280	−0.481602
$417$	16.5034	0.808175
$418$	−8.78188	−0.429536
$419$	−0.754074	−0.0368389	−0.0184195	−	0.999830i	$-0.505863\pi$
$419$	−0.754074	−0.0368389	−0.0184195	+	0.999830i	$0.505863\pi$
$420$	0	0
$421$	3.08905	0.150551	0.0752756	−	0.997163i	$-0.476016\pi$
$421$	3.08905	0.150551	0.0752756	+	0.997163i	$0.476016\pi$
$422$	11.5411	0.561810
$423$	−9.37467	−0.455812
$424$	−11.8148	−0.573776
$425$	−1.41607	−0.0686895
$426$	−1.94730	−0.0943472
$427$	0	0
$428$	12.2468	0.591973
$429$	−8.98555	−0.433826
$430$	−5.78222	−0.278844
$431$	−4.10893	−0.197920	−0.0989601	−	0.995091i	$-0.531552\pi$
$431$	−4.10893	−0.197920	−0.0989601	+	0.995091i	$0.531552\pi$
$432$	−2.85771	−0.137492
$433$	−28.0316	−1.34711	−0.673556	−	0.739136i	$-0.735234\pi$
$433$	−28.0316	−1.34711	−0.673556	+	0.739136i	$0.735234\pi$
$434$	0	0
$435$	−9.63237	−0.461837
$436$	8.43679	0.404049
$437$	4.66087	0.222960
$438$	−6.65387	−0.317934
$439$	24.2663	1.15817	0.579083	−	0.815269i	$-0.303411\pi$
$439$	24.2663	1.15817	0.579083	+	0.815269i	$0.303411\pi$
$440$	15.2703	0.727985
$441$	0	0
$442$	−2.89708	−0.137800
$443$	17.3114	0.822487	0.411244	−	0.911525i	$-0.365095\pi$
$443$	17.3114	0.822487	0.411244	+	0.911525i	$0.365095\pi$
$444$	0.191274	0.00907748
$445$	−0.935432	−0.0443438
$446$	−1.73339	−0.0820785
$447$	18.7027	0.884608
$448$	0	0
$449$	10.4219	0.491840	0.245920	−	0.969290i	$-0.420910\pi$
$449$	10.4219	0.491840	0.245920	+	0.969290i	$0.420910\pi$
$450$	−0.203572	−0.00959646
$451$	12.3264	0.580429
$452$	−20.6144	−0.969620
$453$	1.08232	0.0508519
$454$	3.97765	0.186680
$455$	0	0
$456$	7.86444	0.368286
$457$	32.6641	1.52796	0.763980	−	0.645240i	$-0.223242\pi$
$457$	32.6641	1.52796	0.763980	+	0.645240i	$0.223242\pi$
$458$	−0.827005	−0.0386434
$459$	−3.08620	−0.144052
$460$	−3.84253	−0.179159
$461$	25.0653	1.16741	0.583705	−	0.811966i	$-0.301603\pi$
$461$	25.0653	1.16741	0.583705	+	0.811966i	$0.301603\pi$
$462$	0	0
$463$	−12.1025	−0.562453	−0.281227	−	0.959641i	$-0.590741\pi$
$463$	−12.1025	−0.562453	−0.281227	+	0.959641i	$0.590741\pi$
$464$	12.9172	0.599665
$465$	2.82963	0.131221
$466$	−1.56817	−0.0726440
$467$	39.0680	1.80785	0.903925	−	0.427691i	$-0.140673\pi$
$467$	39.0680	1.80785	0.903925	+	0.427691i	$0.140673\pi$
$468$	3.81517	0.176357
$469$	0	0
$470$	−8.86330	−0.408834
$471$	−9.77466	−0.450393
$472$	18.1839	0.836981
$473$	25.9729	1.19423
$474$	−5.80148	−0.266471
$475$	−2.13859	−0.0981252
$476$	0	0
$477$	7.00203	0.320601
$478$	−5.25902	−0.240542
$479$	−26.7100	−1.22041	−0.610207	−	0.792242i	$-0.708914\pi$
$479$	−26.7100	−1.22041	−0.610207	+	0.792242i	$0.708914\pi$
$480$	−9.89324	−0.451563
$481$	−0.224441	−0.0102336
$482$	1.19680	0.0545129
$483$	0	0
$484$	−12.6862	−0.576645
$485$	−7.06756	−0.320921
$486$	−0.443666	−0.0201251
$487$	−34.0490	−1.54291	−0.771454	−	0.636285i	$-0.780470\pi$
$487$	−34.0490	−1.54291	−0.771454	+	0.636285i	$0.780470\pi$
$488$	11.8594	0.536851
$489$	−11.8048	−0.533830
$490$	0	0
$491$	−15.0417	−0.678823	−0.339411	−	0.940638i	$-0.610228\pi$
$491$	−15.0417	−0.678823	−0.339411	+	0.940638i	$0.610228\pi$
$492$	−5.23368	−0.235953
$493$	13.9500	0.628276
$494$	−4.37525	−0.196852
$495$	−9.04999	−0.406767
$496$	−3.79458	−0.170382
$497$	0	0
$498$	2.65026	0.118761
$499$	−30.3156	−1.35711	−0.678556	−	0.734548i	$-0.737394\pi$
$499$	−30.3156	−1.35711	−0.678556	+	0.734548i	$0.737394\pi$
$500$	20.9758	0.938065
$501$	16.1744	0.722619
$502$	5.34206	0.238428
$503$	−2.76265	−0.123180	−0.0615902	−	0.998102i	$-0.519617\pi$
$503$	−2.76265	−0.123180	−0.0615902	+	0.998102i	$0.519617\pi$
$504$	0	0
$505$	−33.2538	−1.47977
$506$	−1.88417	−0.0837617
$507$	8.52328	0.378532
$508$	2.82724	0.125438
$509$	6.72811	0.298218	0.149109	−	0.988821i	$-0.452359\pi$
$509$	6.72811	0.298218	0.149109	+	0.988821i	$0.452359\pi$
$510$	−2.91786	−0.129205
$511$	0	0
$512$	22.9108	1.01252
$513$	−4.66087	−0.205782
$514$	3.09126	0.136349
$515$	−5.79416	−0.255321
$516$	−11.0278	−0.485473
$517$	39.8126	1.75096
$518$	0	0
$519$	17.9165	0.786445
$520$	7.60789	0.333628
$521$	−8.01038	−0.350941	−0.175471	−	0.984485i	$-0.556145\pi$
$521$	−8.01038	−0.350941	−0.175471	+	0.984485i	$0.556145\pi$
$522$	2.00542	0.0877750
$523$	−2.31040	−0.101027	−0.0505134	−	0.998723i	$-0.516086\pi$
$523$	−2.31040	−0.101027	−0.0505134	+	0.998723i	$0.516086\pi$
$524$	8.22558	0.359336
$525$	0	0
$526$	−2.33972	−0.102017
$527$	−4.09798	−0.178511
$528$	12.1362	0.528160
$529$	1.00000	0.0434783
$530$	6.62009	0.287558
$531$	−10.7767	−0.467669
$532$	0	0
$533$	6.14120	0.266005
$534$	0.194754	0.00842781
$535$	−14.4735	−0.625743
$536$	25.2944	1.09255
$537$	−11.4527	−0.494222
$538$	−6.14269	−0.264830
$539$	0	0
$540$	3.84253	0.165356
$541$	39.1503	1.68320	0.841602	−	0.540099i	$-0.181613\pi$
$541$	39.1503	1.68320	0.841602	+	0.540099i	$0.181613\pi$
$542$	−0.0394470	−0.00169439
$543$	7.48489	0.321207
$544$	14.3278	0.614299
$545$	−9.97072	−0.427099
$546$	0	0
$547$	−30.0310	−1.28403	−0.642017	−	0.766691i	$-0.721902\pi$
$547$	−30.0310	−1.28403	−0.642017	+	0.766691i	$0.721902\pi$
$548$	2.04507	0.0873611
$549$	−7.02850	−0.299969
$550$	0.864533	0.0368638
$551$	21.0677	0.897513
$552$	1.68733	0.0718177
$553$	0	0
$554$	−2.81839	−0.119742
$555$	−0.226051	−0.00959531
$556$	29.7583	1.26203
$557$	24.7714	1.04960	0.524800	−	0.851226i	$-0.324140\pi$
$557$	24.7714	1.04960	0.524800	+	0.851226i	$0.324140\pi$
$558$	−0.589118	−0.0249393
$559$	12.9400	0.547305
$560$	0	0
$561$	13.1066	0.553360
$562$	−3.78766	−0.159773
$563$	−32.1575	−1.35528	−0.677638	−	0.735396i	$-0.736996\pi$
$563$	−32.1575	−1.35528	−0.677638	+	0.735396i	$0.736996\pi$
$564$	−16.9040	−0.711788
$565$	24.3624	1.02493
$566$	−13.3414	−0.560781
$567$	0	0
$568$	−7.40591	−0.310745
$569$	−10.9873	−0.460612	−0.230306	−	0.973118i	$-0.573973\pi$
$569$	−10.9873	−0.460612	−0.230306	+	0.973118i	$0.573973\pi$
$570$	−4.40663	−0.184573
$571$	30.8166	1.28963	0.644816	−	0.764337i	$-0.276934\pi$
$571$	30.8166	1.28963	0.644816	+	0.764337i	$0.276934\pi$
$572$	−16.2024	−0.677456
$573$	2.35160	0.0982393
$574$	0	0
$575$	−0.458839	−0.0191349
$576$	−3.65568	−0.152320
$577$	−34.3918	−1.43175	−0.715876	−	0.698228i	$-0.753972\pi$
$577$	−34.3918	−1.43175	−0.715876	+	0.698228i	$0.753972\pi$
$578$	−3.31656	−0.137951
$579$	22.8719	0.950523
$580$	−17.3687	−0.721196
$581$	0	0
$582$	1.47144	0.0609931
$583$	−29.7364	−1.23156
$584$	−25.3057	−1.04716
$585$	−4.50883	−0.186417
$586$	−2.83261	−0.117014
$587$	28.7235	1.18555	0.592773	−	0.805370i	$-0.298033\pi$
$587$	28.7235	1.18555	0.592773	+	0.805370i	$0.298033\pi$
$588$	0	0
$589$	−6.18889	−0.255009
$590$	−10.1889	−0.419468
$591$	−12.7602	−0.524885
$592$	0.303138	0.0124589
$593$	37.8380	1.55382	0.776910	−	0.629612i	$-0.216786\pi$
$593$	37.8380	1.55382	0.776910	+	0.629612i	$0.216786\pi$
$594$	1.88417	0.0773086
$595$	0	0
$596$	33.7240	1.38139
$597$	27.3627	1.11988
$598$	−0.938721	−0.0383871
$599$	36.5860	1.49486	0.747431	−	0.664339i	$-0.231287\pi$
$599$	36.5860	1.49486	0.747431	+	0.664339i	$0.231287\pi$
$600$	−0.774215	−0.0316072
$601$	28.9301	1.18008	0.590042	−	0.807372i	$-0.299111\pi$
$601$	28.9301	1.18008	0.590042	+	0.807372i	$0.299111\pi$
$602$	0	0
$603$	−14.9907	−0.610470
$604$	1.95160	0.0794094
$605$	14.9927	0.609541
$606$	6.92331	0.281240
$607$	−31.4315	−1.27577	−0.637883	−	0.770133i	$-0.720190\pi$
$607$	−31.4315	−1.27577	−0.637883	+	0.770133i	$0.720190\pi$
$608$	21.6382	0.877547
$609$	0	0
$610$	−6.64511	−0.269053
$611$	19.8352	0.802445
$612$	−5.56492	−0.224948
$613$	19.6432	0.793381	0.396691	−	0.917952i	$-0.370159\pi$
$613$	19.6432	0.793381	0.396691	+	0.917952i	$0.370159\pi$
$614$	−6.36866	−0.257018
$615$	6.18524	0.249413
$616$	0	0
$617$	42.8896	1.72667	0.863335	−	0.504632i	$-0.168372\pi$
$617$	42.8896	1.72667	0.863335	+	0.504632i	$0.168372\pi$
$618$	1.20632	0.0485254
$619$	−48.8018	−1.96151	−0.980755	−	0.195243i	$-0.937451\pi$
$619$	−48.8018	−1.96151	−0.980755	+	0.195243i	$0.937451\pi$
$620$	5.10227	0.204912
$621$	−1.00000	−0.0401286
$622$	5.47717	0.219615
$623$	0	0
$624$	6.04641	0.242050
$625$	−22.4953	−0.899811
$626$	−3.37531	−0.134904
$627$	19.7939	0.790492
$628$	−17.6253	−0.703326
$629$	0.327376	0.0130533
$630$	0	0
$631$	−3.30202	−0.131451	−0.0657257	−	0.997838i	$-0.520936\pi$
$631$	−3.30202	−0.131451	−0.0657257	+	0.997838i	$0.520936\pi$
$632$	−22.0639	−0.877656
$633$	−26.0130	−1.03392
$634$	−4.84186	−0.192295
$635$	−3.34127	−0.132594
$636$	12.6258	0.500645
$637$	0	0
$638$	−8.51669	−0.337179
$639$	4.38912	0.173631
$640$	−23.2428	−0.918751
$641$	−43.2501	−1.70828	−0.854138	−	0.520046i	$-0.825915\pi$
$641$	−43.2501	−1.70828	−0.854138	+	0.520046i	$0.825915\pi$
$642$	3.01332	0.118926
$643$	−16.3299	−0.643989	−0.321994	−	0.946742i	$-0.604353\pi$
$643$	−16.3299	−0.643989	−0.321994	+	0.946742i	$0.604353\pi$
$644$	0	0
$645$	13.0328	0.513167
$646$	6.38186	0.251091
$647$	3.18790	0.125329	0.0626646	−	0.998035i	$-0.480040\pi$
$647$	3.18790	0.125329	0.0626646	+	0.998035i	$0.480040\pi$
$648$	−1.68733	−0.0662847
$649$	45.7668	1.79650
$650$	0.430722	0.0168943
$651$	0	0
$652$	−21.2859	−0.833620
$653$	20.2811	0.793661	0.396830	−	0.917892i	$-0.370110\pi$
$653$	20.2811	0.793661	0.396830	+	0.917892i	$0.370110\pi$
$654$	2.07587	0.0811728
$655$	−9.72111	−0.379835
$656$	−8.29451	−0.323846
$657$	14.9975	0.585107
$658$	0	0
$659$	26.4070	1.02867	0.514336	−	0.857589i	$-0.328039\pi$
$659$	26.4070	1.02867	0.514336	+	0.857589i	$0.328039\pi$
$660$	−16.3186	−0.635200
$661$	−27.2046	−1.05814	−0.529069	−	0.848579i	$-0.677459\pi$
$661$	−27.2046	−1.05814	−0.529069	+	0.848579i	$0.677459\pi$
$662$	0.149744	0.00581998
$663$	6.52987	0.253599
$664$	10.0793	0.391154
$665$	0	0
$666$	0.0470629	0.00182365
$667$	4.52012	0.175020
$668$	29.1650	1.12843
$669$	3.90697	0.151052
$670$	−14.1730	−0.547552
$671$	29.8488	1.15230
$672$	0	0
$673$	18.8152	0.725273	0.362637	−	0.931931i	$-0.381877\pi$
$673$	18.8152	0.725273	0.362637	+	0.931931i	$0.381877\pi$
$674$	−3.69003	−0.142135
$675$	0.458839	0.0176607
$676$	15.3688	0.591109
$677$	23.6899	0.910477	0.455239	−	0.890369i	$-0.349554\pi$
$677$	23.6899	0.910477	0.455239	+	0.890369i	$0.349554\pi$
$678$	−5.07216	−0.194795
$679$	0	0
$680$	−11.0971	−0.425554
$681$	−8.96540	−0.343555
$682$	2.50188	0.0958019
$683$	41.6025	1.59188	0.795938	−	0.605378i	$-0.206978\pi$
$683$	41.6025	1.59188	0.795938	+	0.605378i	$0.206978\pi$
$684$	−8.40429	−0.321346
$685$	−2.41689	−0.0923447
$686$	0	0
$687$	1.86403	0.0711170
$688$	−17.4772	−0.666314
$689$	−14.8151	−0.564410
$690$	−0.945453	−0.0359928
$691$	−29.9774	−1.14039	−0.570196	−	0.821509i	$-0.693133\pi$
$691$	−29.9774	−1.14039	−0.570196	+	0.821509i	$0.693133\pi$
$692$	32.3062	1.22810
$693$	0	0
$694$	1.33250	0.0505812
$695$	−35.1687	−1.33403
$696$	7.62695	0.289099
$697$	−8.95772	−0.339298
$698$	8.05855	0.305021
$699$	3.53457	0.133690
$700$	0	0
$701$	5.03608	0.190210	0.0951050	−	0.995467i	$-0.469681\pi$
$701$	5.03608	0.190210	0.0951050	+	0.995467i	$0.469681\pi$
$702$	0.938721	0.0354297
$703$	0.494412	0.0186471
$704$	15.5250	0.585122
$705$	19.9774	0.752393
$706$	12.1407	0.456921
$707$	0	0
$708$	−19.4321	−0.730303
$709$	−6.44620	−0.242092	−0.121046	−	0.992647i	$-0.538625\pi$
$709$	−6.44620	−0.242092	−0.121046	+	0.992647i	$0.538625\pi$
$710$	4.14970	0.155736
$711$	13.0762	0.490396
$712$	0.740679	0.0277581
$713$	−1.32784	−0.0497280
$714$	0	0
$715$	19.1482	0.716102
$716$	−20.6511	−0.771769
$717$	11.8536	0.442679
$718$	12.2433	0.456914
$719$	−20.2867	−0.756567	−0.378284	−	0.925690i	$-0.623486\pi$
$719$	−20.2867	−0.756567	−0.378284	+	0.925690i	$0.623486\pi$
$720$	6.08977	0.226953
$721$	0	0
$722$	1.20841	0.0449723
$723$	−2.69753	−0.100322
$724$	13.4965	0.501592
$725$	−2.07401	−0.0770268
$726$	−3.12143	−0.115847
$727$	36.8673	1.36733	0.683667	−	0.729794i	$-0.260384\pi$
$727$	36.8673	1.36733	0.683667	+	0.729794i	$0.260384\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	14.1794	0.524803
$731$	−18.8747	−0.698105
$732$	−12.6735	−0.468426
$733$	38.4041	1.41849	0.709243	−	0.704964i	$-0.249037\pi$
$733$	38.4041	1.41849	0.709243	+	0.704964i	$0.249037\pi$
$734$	7.52564	0.277776
$735$	0	0
$736$	4.64254	0.171126
$737$	63.6631	2.34506
$738$	−1.28774	−0.0474025
$739$	35.2084	1.29516	0.647581	−	0.761996i	$-0.275781\pi$
$739$	35.2084	1.29516	0.647581	+	0.761996i	$0.275781\pi$
$740$	−0.407605	−0.0149839
$741$	9.86159	0.362275
$742$	0	0
$743$	−35.7365	−1.31105	−0.655523	−	0.755175i	$-0.727552\pi$
$743$	−35.7365	−1.31105	−0.655523	+	0.755175i	$0.727552\pi$
$744$	−2.24051	−0.0821410
$745$	−39.8555	−1.46019
$746$	2.99896	0.109800
$747$	−5.97353	−0.218560
$748$	23.6332	0.864117
$749$	0	0
$750$	5.16107	0.188456
$751$	39.0727	1.42578	0.712891	−	0.701275i	$-0.247386\pi$
$751$	39.0727	1.42578	0.712891	+	0.701275i	$0.247386\pi$
$752$	−26.7901	−0.976933
$753$	−12.0407	−0.438788
$754$	−4.24313	−0.154526
$755$	−2.30643	−0.0839394
$756$	0	0
$757$	−1.24601	−0.0452868	−0.0226434	−	0.999744i	$-0.507208\pi$
$757$	−1.24601	−0.0452868	−0.0226434	+	0.999744i	$0.507208\pi$
$758$	−17.0380	−0.618847
$759$	4.24683	0.154150
$760$	−16.7591	−0.607917
$761$	18.7199	0.678595	0.339298	−	0.940679i	$-0.389811\pi$
$761$	18.7199	0.678595	0.339298	+	0.940679i	$0.389811\pi$
$762$	0.695639	0.0252004
$763$	0	0
$764$	4.24030	0.153409
$765$	6.57670	0.237781
$766$	5.06390	0.182966
$767$	22.8016	0.823319
$768$	−2.47230	−0.0892116
$769$	−23.7683	−0.857109	−0.428554	−	0.903516i	$-0.640977\pi$
$769$	−23.7683	−0.857109	−0.428554	+	0.903516i	$0.640977\pi$
$770$	0	0
$771$	−6.96752	−0.250929
$772$	41.2417	1.48432
$773$	−53.3111	−1.91747	−0.958733	−	0.284308i	$-0.908236\pi$
$773$	−53.3111	−1.91747	−0.958733	+	0.284308i	$0.908236\pi$
$774$	−2.71339	−0.0975306
$775$	0.609265	0.0218855
$776$	5.59612	0.200889
$777$	0	0
$778$	3.98188	0.142757
$779$	−13.5282	−0.484698
$780$	−8.13014	−0.291106
$781$	−18.6398	−0.666985
$782$	1.36924	0.0489640
$783$	−4.52012	−0.161536
$784$	0	0
$785$	20.8298	0.743448
$786$	2.02390	0.0721900
$787$	−51.1120	−1.82195	−0.910973	−	0.412467i	$-0.864667\pi$
$787$	−51.1120	−1.82195	−0.910973	+	0.412467i	$0.864667\pi$
$788$	−23.0087	−0.819651
$789$	5.27360	0.187745
$790$	12.3629	0.439854
$791$	0	0
$792$	7.16581	0.254626
$793$	14.8711	0.528087
$794$	8.13617	0.288742
$795$	−14.9213	−0.529205
$796$	49.3393	1.74879
$797$	10.8882	0.385681	0.192840	−	0.981230i	$-0.438230\pi$
$797$	10.8882	0.385681	0.192840	+	0.981230i	$0.438230\pi$
$798$	0	0
$799$	−28.9321	−1.02354
$800$	−2.13018	−0.0753132
$801$	−0.438964	−0.0155100
$802$	6.94880	0.245371
$803$	−63.6916	−2.24763
$804$	−27.0307	−0.953299
$805$	0	0
$806$	1.24647	0.0439051
$807$	13.8453	0.487377
$808$	26.3305	0.926303
$809$	46.6723	1.64091	0.820456	−	0.571710i	$-0.193720\pi$
$809$	46.6723	1.64091	0.820456	+	0.571710i	$0.193720\pi$
$810$	0.945453	0.0332198
$811$	−0.341433	−0.0119893	−0.00599466	−	0.999982i	$-0.501908\pi$
$811$	−0.341433	−0.0119893	−0.00599466	+	0.999982i	$0.501908\pi$
$812$	0	0
$813$	0.0889114	0.00311826
$814$	−0.199868	−0.00700537
$815$	25.1560	0.881175
$816$	−8.81946	−0.308743
$817$	−28.5051	−0.997266
$818$	8.10233	0.283291
$819$	0	0
$820$	11.1530	0.389479
$821$	3.54610	0.123760	0.0618798	−	0.998084i	$-0.480290\pi$
$821$	3.54610	0.123760	0.0618798	+	0.998084i	$0.480290\pi$
$822$	0.503188	0.0175507
$823$	−37.8726	−1.32016	−0.660078	−	0.751197i	$-0.729477\pi$
$823$	−37.8726	−1.32016	−0.660078	+	0.751197i	$0.729477\pi$
$824$	4.58784	0.159825
$825$	−1.94861	−0.0678419
$826$	0	0
$827$	−12.5848	−0.437616	−0.218808	−	0.975768i	$-0.570217\pi$
$827$	−12.5848	−0.437616	−0.218808	+	0.975768i	$0.570217\pi$
$828$	−1.80316	−0.0626642
$829$	−41.8305	−1.45283	−0.726417	−	0.687254i	$-0.758816\pi$
$829$	−41.8305	−1.45283	−0.726417	+	0.687254i	$0.758816\pi$
$830$	−5.64769	−0.196034
$831$	6.35249	0.220365
$832$	7.73478	0.268155
$833$	0	0
$834$	7.32200	0.253540
$835$	−34.4676	−1.19280
$836$	35.6916	1.23442
$837$	1.32784	0.0458969
$838$	−0.334557	−0.0115571
$839$	44.1860	1.52547	0.762735	−	0.646711i	$-0.223856\pi$
$839$	44.1860	1.52547	0.762735	+	0.646711i	$0.223856\pi$
$840$	0	0
$841$	−8.56853	−0.295466
$842$	1.37051	0.0472308
$843$	8.53718	0.294036
$844$	−46.9055	−1.61455
$845$	−18.1631	−0.624830
$846$	−4.15922	−0.142997
$847$	0	0
$848$	20.0098	0.687138
$849$	30.0708	1.03203
$850$	−0.628263	−0.0215492
$851$	0.106077	0.00363628
$852$	7.91428	0.271139
$853$	−28.6584	−0.981244	−0.490622	−	0.871372i	$-0.663230\pi$
$853$	−28.6584	−0.981244	−0.490622	+	0.871372i	$0.663230\pi$
$854$	0	0
$855$	9.93231	0.339678
$856$	11.4601	0.391700
$857$	0.999871	0.0341549	0.0170775	−	0.999854i	$-0.494564\pi$
$857$	0.999871	0.0341549	0.0170775	+	0.999854i	$0.494564\pi$
$858$	−3.98658	−0.136100
$859$	31.4841	1.07422	0.537112	−	0.843511i	$-0.319515\pi$
$859$	31.4841	1.07422	0.537112	+	0.843511i	$0.319515\pi$
$860$	23.5003	0.801353
$861$	0	0
$862$	−1.82299	−0.0620914
$863$	48.5018	1.65102	0.825511	−	0.564386i	$-0.190887\pi$
$863$	48.5018	1.65102	0.825511	+	0.564386i	$0.190887\pi$
$864$	−4.64254	−0.157942
$865$	−38.1800	−1.29816
$866$	−12.4367	−0.422615
$867$	7.47536	0.253876
$868$	0	0
$869$	−55.5324	−1.88381
$870$	−4.27356	−0.144887
$871$	31.7178	1.07472
$872$	7.89485	0.267353
$873$	−3.31655	−0.112248
$874$	2.06787	0.0699467
$875$	0	0
$876$	27.0428	0.913693
$877$	−30.8693	−1.04238	−0.521191	−	0.853440i	$-0.674512\pi$
$877$	−30.8693	−1.04238	−0.521191	+	0.853440i	$0.674512\pi$
$878$	10.7661	0.363339
$879$	6.38456	0.215346
$880$	−25.8622	−0.871815
$881$	−52.4488	−1.76705	−0.883523	−	0.468387i	$-0.844835\pi$
$881$	−52.4488	−1.76705	−0.883523	+	0.468387i	$0.844835\pi$
$882$	0	0
$883$	18.7213	0.630023	0.315011	−	0.949088i	$-0.397992\pi$
$883$	18.7213	0.630023	0.315011	+	0.949088i	$0.397992\pi$
$884$	11.7744	0.396016
$885$	22.9651	0.771965
$886$	7.68046	0.258030
$887$	−18.0701	−0.606734	−0.303367	−	0.952874i	$-0.598111\pi$
$887$	−18.0701	−0.606734	−0.303367	+	0.952874i	$0.598111\pi$
$888$	0.178988	0.00600643
$889$	0	0
$890$	−0.415020	−0.0139115
$891$	−4.24683	−0.142274
$892$	7.04490	0.235881
$893$	−43.6941	−1.46217
$894$	8.29777	0.277519
$895$	24.4058	0.815795
$896$	0	0
$897$	2.11583	0.0706454
$898$	4.62385	0.154300
$899$	−6.00199	−0.200178
$900$	0.827361	0.0275787
$901$	21.6097	0.719923
$902$	5.46882	0.182092
$903$	0	0
$904$	−19.2902	−0.641583
$905$	−15.9503	−0.530206
$906$	0.480189	0.0159532
$907$	−41.0300	−1.36238	−0.681190	−	0.732107i	$-0.738537\pi$
$907$	−41.0300	−1.36238	−0.681190	+	0.732107i	$0.738537\pi$
$908$	−16.1661	−0.536489
$909$	−15.6048	−0.517578
$910$	0	0
$911$	−32.1141	−1.06399	−0.531993	−	0.846749i	$-0.678557\pi$
$911$	−32.1141	−1.06399	−0.531993	+	0.846749i	$0.678557\pi$
$912$	−13.3194	−0.441049
$913$	25.3686	0.839577
$914$	14.4919	0.479351
$915$	14.9777	0.495148
$916$	3.36114	0.111055
$917$	0	0
$918$	−1.36924	−0.0451918
$919$	−13.3777	−0.441289	−0.220645	−	0.975354i	$-0.570816\pi$
$919$	−13.3777	−0.441289	−0.220645	+	0.975354i	$0.570816\pi$
$920$	−3.59571	−0.118547
$921$	14.3546	0.473001
$922$	11.1206	0.366239
$923$	−9.28661	−0.305673
$924$	0	0
$925$	−0.0486724	−0.00160034
$926$	−5.36949	−0.176452
$927$	−2.71899	−0.0893033
$928$	20.9848	0.688860
$929$	25.5091	0.836928	0.418464	−	0.908233i	$-0.362569\pi$
$929$	25.5091	0.836928	0.418464	+	0.908233i	$0.362569\pi$
$930$	1.25541	0.0411665
$931$	0	0
$932$	6.37340	0.208768
$933$	−12.3453	−0.404166
$934$	17.3331	0.567157
$935$	−27.9301	−0.913411
$936$	3.57011	0.116693
$937$	−6.88538	−0.224935	−0.112468	−	0.993655i	$-0.535875\pi$
$937$	−6.88538	−0.224935	−0.112468	+	0.993655i	$0.535875\pi$
$938$	0	0
$939$	7.60776	0.248270
$940$	36.0225	1.17492
$941$	−23.3389	−0.760826	−0.380413	−	0.924817i	$-0.624218\pi$
$941$	−23.3389	−0.760826	−0.380413	+	0.924817i	$0.624218\pi$
$942$	−4.33669	−0.141297
$943$	−2.90251	−0.0945186
$944$	−30.7966	−1.00235
$945$	0	0
$946$	11.5233	0.374654
$947$	−32.6904	−1.06230	−0.531148	−	0.847279i	$-0.678239\pi$
$947$	−32.6904	−1.06230	−0.531148	+	0.847279i	$0.678239\pi$
$948$	23.5785	0.765795
$949$	−31.7320	−1.03007
$950$	−0.948820	−0.0307838
$951$	10.9133	0.353888
$952$	0	0
$953$	6.46250	0.209341	0.104670	−	0.994507i	$-0.466621\pi$
$953$	6.46250	0.209341	0.104670	+	0.994507i	$0.466621\pi$
$954$	3.10656	0.100579
$955$	−5.01125	−0.162160
$956$	21.3739	0.691280
$957$	19.1962	0.620524
$958$	−11.8503	−0.382867
$959$	0	0
$960$	7.79025	0.251429
$961$	−29.2368	−0.943124
$962$	−0.0995769	−0.00321049
$963$	−6.79187	−0.218865
$964$	−4.86408	−0.156662
$965$	−48.7400	−1.56900
$966$	0	0
$967$	19.4217	0.624560	0.312280	−	0.949990i	$-0.398907\pi$
$967$	19.4217	0.624560	0.312280	+	0.949990i	$0.398907\pi$
$968$	−11.8713	−0.381558
$969$	−14.3844	−0.462093
$970$	−3.13564	−0.100679
$971$	−1.10488	−0.0354571	−0.0177286	−	0.999843i	$-0.505643\pi$
$971$	−1.10488	−0.0354571	−0.0177286	+	0.999843i	$0.505643\pi$
$972$	1.80316	0.0578364
$973$	0	0
$974$	−15.1064	−0.484040
$975$	−0.970825	−0.0310913
$976$	−20.0854	−0.642918
$977$	21.4638	0.686688	0.343344	−	0.939210i	$-0.388440\pi$
$977$	21.4638	0.686688	0.343344	+	0.939210i	$0.388440\pi$
$978$	−5.23738	−0.167473
$979$	1.86420	0.0595802
$980$	0	0
$981$	−4.67889	−0.149386
$982$	−6.67350	−0.212960
$983$	−57.5436	−1.83536	−0.917678	−	0.397324i	$-0.869939\pi$
$983$	−57.5436	−1.83536	−0.917678	+	0.397324i	$0.869939\pi$
$984$	−4.89750	−0.156126
$985$	27.1920	0.866410
$986$	6.18914	0.197102
$987$	0	0
$988$	17.7820	0.565722
$989$	−6.11583	−0.194472
$990$	−4.01517	−0.127611
$991$	−43.1279	−1.37000	−0.685002	−	0.728541i	$-0.740199\pi$
$991$	−43.1279	−1.37000	−0.685002	+	0.728541i	$0.740199\pi$
$992$	−6.16454	−0.195724
$993$	−0.337516	−0.0107107
$994$	0	0
$995$	−58.3099	−1.84855
$996$	−10.7712	−0.341300
$997$	46.2313	1.46416	0.732081	−	0.681218i	$-0.238549\pi$
$997$	46.2313	1.46416	0.732081	+	0.681218i	$0.238549\pi$
$998$	−13.4500	−0.425752
$999$	−0.106077	−0.00335614

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.y.1.3	✓	5
7.6	odd	2		3381.2.a.z.1.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3381.2.a.y.1.3	✓	5	1.1	even	1	trivial
3381.2.a.z.1.3	yes	5	7.6	odd	2

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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q+0.443666 q^{2} -1.00000 q^{3} -1.80316 q^{4} +2.13100 q^{5} -0.443666 q^{6} -1.68733 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.443666 q^{2} -1.00000 q^{3} -1.80316 q^{4} +2.13100 q^{5} -0.443666 q^{6} -1.68733 q^{8} +1.00000 q^{9} +0.945453 q^{10} -4.24683 q^{11} +1.80316 q^{12} -2.11583 q^{13} -2.13100 q^{15} +2.85771 q^{16} +3.08620 q^{17} +0.443666 q^{18} +4.66087 q^{19} -3.84253 q^{20} -1.88417 q^{22} +1.00000 q^{23} +1.68733 q^{24} -0.458839 q^{25} -0.938721 q^{26} -1.00000 q^{27} +4.52012 q^{29} -0.945453 q^{30} -1.32784 q^{31} +4.64254 q^{32} +4.24683 q^{33} +1.36924 q^{34} -1.80316 q^{36} +0.106077 q^{37} +2.06787 q^{38} +2.11583 q^{39} -3.59571 q^{40} -2.90251 q^{41} -6.11583 q^{43} +7.65771 q^{44} +2.13100 q^{45} +0.443666 q^{46} -9.37467 q^{47} -2.85771 q^{48} -0.203572 q^{50} -3.08620 q^{51} +3.81517 q^{52} +7.00203 q^{53} -0.443666 q^{54} -9.04999 q^{55} -4.66087 q^{57} +2.00542 q^{58} -10.7767 q^{59} +3.84253 q^{60} -7.02850 q^{61} -0.589118 q^{62} -3.65568 q^{64} -4.50883 q^{65} +1.88417 q^{66} -14.9907 q^{67} -5.56492 q^{68} -1.00000 q^{69} +4.38912 q^{71} -1.68733 q^{72} +14.9975 q^{73} +0.0470629 q^{74} +0.458839 q^{75} -8.40429 q^{76} +0.938721 q^{78} +13.0762 q^{79} +6.08977 q^{80} +1.00000 q^{81} -1.28774 q^{82} -5.97353 q^{83} +6.57670 q^{85} -2.71339 q^{86} -4.52012 q^{87} +7.16581 q^{88} -0.438964 q^{89} +0.945453 q^{90} -1.80316 q^{92} +1.32784 q^{93} -4.15922 q^{94} +9.93231 q^{95} -4.64254 q^{96} -3.31655 q^{97} -4.24683 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} - 5 q^{3} + 9 q^{4} - 2 q^{5} - q^{6} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q + q^2 - 5 * q^3 + 9 * q^4 - 2 * q^5 - q^6 + 3 * q^8 + 5 * q^9	\( 5 q + q^{2} - 5 q^{3} + 9 q^{4} - 2 q^{5} - q^{6} + 3 q^{8} + 5 q^{9} - 2 q^{11} - 9 q^{12} - 4 q^{13} + 2 q^{15} + q^{16} - 2 q^{17} + q^{18} - 8 q^{19} - 12 q^{20} - 16 q^{22} + 5 q^{23} - 3 q^{24} + 5 q^{25} - 16 q^{26} - 5 q^{27} + 4 q^{29} - 12 q^{31} + 7 q^{32} + 2 q^{33} - 10 q^{34} + 9 q^{36} - 6 q^{37} + 8 q^{38} + 4 q^{39} - 30 q^{40} - 6 q^{41} - 24 q^{43} + 16 q^{44} - 2 q^{45} + q^{46} - 24 q^{47} - q^{48} - 3 q^{50} + 2 q^{51} + 4 q^{52} + 2 q^{53} - q^{54} - 8 q^{55} + 8 q^{57} + 4 q^{58} - 16 q^{59} + 12 q^{60} - 22 q^{61} - 6 q^{62} - 29 q^{64} + 22 q^{65} + 16 q^{66} - 16 q^{67} - 14 q^{68} - 5 q^{69} + 16 q^{71} + 3 q^{72} + 8 q^{74} - 5 q^{75} - 30 q^{76} + 16 q^{78} + 12 q^{79} + 6 q^{80} + 5 q^{81} - 24 q^{82} - 10 q^{83} - 14 q^{85} - 20 q^{86} - 4 q^{87} - 8 q^{88} + 16 q^{89} + 9 q^{92} + 12 q^{93} + 32 q^{94} + 18 q^{95} - 7 q^{96} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 5 * q + q^2 - 5 * q^3 + 9 * q^4 - 2 * q^5 - q^6 + 3 * q^8 + 5 * q^9 - 2 * q^11 - 9 * q^12 - 4 * q^13 + 2 * q^15 + q^16 - 2 * q^17 + q^18 - 8 * q^19 - 12 * q^20 - 16 * q^22 + 5 * q^23 - 3 * q^24 + 5 * q^25 - 16 * q^26 - 5 * q^27 + 4 * q^29 - 12 * q^31 + 7 * q^32 + 2 * q^33 - 10 * q^34 + 9 * q^36 - 6 * q^37 + 8 * q^38 + 4 * q^39 - 30 * q^40 - 6 * q^41 - 24 * q^43 + 16 * q^44 - 2 * q^45 + q^46 - 24 * q^47 - q^48 - 3 * q^50 + 2 * q^51 + 4 * q^52 + 2 * q^53 - q^54 - 8 * q^55 + 8 * q^57 + 4 * q^58 - 16 * q^59 + 12 * q^60 - 22 * q^61 - 6 * q^62 - 29 * q^64 + 22 * q^65 + 16 * q^66 - 16 * q^67 - 14 * q^68 - 5 * q^69 + 16 * q^71 + 3 * q^72 + 8 * q^74 - 5 * q^75 - 30 * q^76 + 16 * q^78 + 12 * q^79 + 6 * q^80 + 5 * q^81 - 24 * q^82 - 10 * q^83 - 14 * q^85 - 20 * q^86 - 4 * q^87 - 8 * q^88 + 16 * q^89 + 9 * q^92 + 12 * q^93 + 32 * q^94 + 18 * q^95 - 7 * q^96 + 4 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more