LMFDB - Embedded newform 1045.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, 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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8 $ x^8 - 2x^7 - 9x^6 + 12x^5 + 28x^4 - 17x^3 - 28x^2 + 6*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.82030$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82030 q^{2} +0.978567 q^{3} +5.95409 q^{4} +1.00000 q^{5} -2.75985 q^{6} -4.31627 q^{7} -11.1517 q^{8} -2.04241 q^{9} +O(q^{10})$	\(q-2.82030 q^{2} +0.978567 q^{3} +5.95409 q^{4} +1.00000 q^{5} -2.75985 q^{6} -4.31627 q^{7} -11.1517 q^{8} -2.04241 q^{9} -2.82030 q^{10} +1.00000 q^{11} +5.82647 q^{12} +4.45224 q^{13} +12.1732 q^{14} +0.978567 q^{15} +19.5430 q^{16} -0.890509 q^{17} +5.76020 q^{18} -1.00000 q^{19} +5.95409 q^{20} -4.22376 q^{21} -2.82030 q^{22} +0.718161 q^{23} -10.9127 q^{24} +1.00000 q^{25} -12.5567 q^{26} -4.93433 q^{27} -25.6995 q^{28} -3.73514 q^{29} -2.75985 q^{30} +6.92133 q^{31} -32.8137 q^{32} +0.978567 q^{33} +2.51150 q^{34} -4.31627 q^{35} -12.1607 q^{36} -9.38577 q^{37} +2.82030 q^{38} +4.35682 q^{39} -11.1517 q^{40} +0.562568 q^{41} +11.9123 q^{42} +2.45537 q^{43} +5.95409 q^{44} -2.04241 q^{45} -2.02543 q^{46} -6.12922 q^{47} +19.1241 q^{48} +11.6302 q^{49} -2.82030 q^{50} -0.871423 q^{51} +26.5091 q^{52} -4.81164 q^{53} +13.9163 q^{54} +1.00000 q^{55} +48.1339 q^{56} -0.978567 q^{57} +10.5342 q^{58} -13.6109 q^{59} +5.82647 q^{60} -12.2483 q^{61} -19.5202 q^{62} +8.81559 q^{63} +53.4585 q^{64} +4.45224 q^{65} -2.75985 q^{66} -8.70936 q^{67} -5.30217 q^{68} +0.702768 q^{69} +12.1732 q^{70} -13.1843 q^{71} +22.7764 q^{72} +4.19184 q^{73} +26.4707 q^{74} +0.978567 q^{75} -5.95409 q^{76} -4.31627 q^{77} -12.2875 q^{78} -0.175970 q^{79} +19.5430 q^{80} +1.29865 q^{81} -1.58661 q^{82} -7.01026 q^{83} -25.1487 q^{84} -0.890509 q^{85} -6.92487 q^{86} -3.65508 q^{87} -11.1517 q^{88} -9.77035 q^{89} +5.76020 q^{90} -19.2171 q^{91} +4.27600 q^{92} +6.77298 q^{93} +17.2862 q^{94} -1.00000 q^{95} -32.1104 q^{96} -0.0271785 q^{97} -32.8007 q^{98} -2.04241 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) $ 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 17 q^{13} + 12 q^{14} - 7 q^{15} + 18 q^{16} - 9 q^{17} - 2 q^{18} - 8 q^{19} + 10 q^{20} + q^{21} - 6 q^{22} - 8 q^{23} + q^{24} + 8 q^{25} + 10 q^{26} - 34 q^{27} - 22 q^{28} - 3 q^{29} - q^{31} - 37 q^{32} - 7 q^{33} - 8 q^{34} - 11 q^{35} + 30 q^{36} - 17 q^{37} + 6 q^{38} + 14 q^{39} - 18 q^{40} - 5 q^{41} + 15 q^{42} - 21 q^{43} + 10 q^{44} + 11 q^{45} - 2 q^{46} - 8 q^{47} + 10 q^{48} + 19 q^{49} - 6 q^{50} - 16 q^{51} + 9 q^{52} - 19 q^{53} - 3 q^{54} + 8 q^{55} + 24 q^{56} + 7 q^{57} + 37 q^{58} - 33 q^{59} - 7 q^{60} - q^{61} - 42 q^{62} - 20 q^{63} + 48 q^{64} - 17 q^{65} - 18 q^{67} - 37 q^{68} + 16 q^{69} + 12 q^{70} - 18 q^{71} + 13 q^{72} - 18 q^{73} + 15 q^{74} - 7 q^{75} - 10 q^{76} - 11 q^{77} - 51 q^{78} - 5 q^{79} + 18 q^{80} + 32 q^{81} + 12 q^{82} - 33 q^{83} - 51 q^{84} - 9 q^{85} - 16 q^{86} - 26 q^{87} - 18 q^{88} - 20 q^{89} - 2 q^{90} + 6 q^{91} - 3 q^{92} + 18 q^{93} + 30 q^{94} - 8 q^{95} + 21 q^{96} - 69 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 17 * q^13 + 12 * q^14 - 7 * q^15 + 18 * q^16 - 9 * q^17 - 2 * q^18 - 8 * q^19 + 10 * q^20 + q^21 - 6 * q^22 - 8 * q^23 + q^24 + 8 * q^25 + 10 * q^26 - 34 * q^27 - 22 * q^28 - 3 * q^29 - q^31 - 37 * q^32 - 7 * q^33 - 8 * q^34 - 11 * q^35 + 30 * q^36 - 17 * q^37 + 6 * q^38 + 14 * q^39 - 18 * q^40 - 5 * q^41 + 15 * q^42 - 21 * q^43 + 10 * q^44 + 11 * q^45 - 2 * q^46 - 8 * q^47 + 10 * q^48 + 19 * q^49 - 6 * q^50 - 16 * q^51 + 9 * q^52 - 19 * q^53 - 3 * q^54 + 8 * q^55 + 24 * q^56 + 7 * q^57 + 37 * q^58 - 33 * q^59 - 7 * q^60 - q^61 - 42 * q^62 - 20 * q^63 + 48 * q^64 - 17 * q^65 - 18 * q^67 - 37 * q^68 + 16 * q^69 + 12 * q^70 - 18 * q^71 + 13 * q^72 - 18 * q^73 + 15 * q^74 - 7 * q^75 - 10 * q^76 - 11 * q^77 - 51 * q^78 - 5 * q^79 + 18 * q^80 + 32 * q^81 + 12 * q^82 - 33 * q^83 - 51 * q^84 - 9 * q^85 - 16 * q^86 - 26 * q^87 - 18 * q^88 - 20 * q^89 - 2 * q^90 + 6 * q^91 - 3 * q^92 + 18 * q^93 + 30 * q^94 - 8 * q^95 + 21 * q^96 - 69 * q^98 + 11 * 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 $

$2$	−2.82030	−1.99425	−0.997127	−	0.0757531i	$-0.975864\pi$
$2$	−2.82030	−1.99425	−0.997127	+	0.0757531i	$0.975864\pi$
$3$	0.978567	0.564976	0.282488	−	0.959271i	$-0.408840\pi$
$3$	0.978567	0.564976	0.282488	+	0.959271i	$0.408840\pi$
$4$	5.95409	2.97705
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.75985	−1.12670
$7$	−4.31627	−1.63140	−0.815699	−	0.578476i	$-0.803648\pi$
$7$	−4.31627	−1.63140	−0.815699	+	0.578476i	$0.803648\pi$
$8$	−11.1517	−3.94273
$9$	−2.04241	−0.680803
$10$	−2.82030	−0.891857
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	5.82647	1.68196
$13$	4.45224	1.23483	0.617415	−	0.786637i	$-0.288180\pi$
$13$	4.45224	1.23483	0.617415	+	0.786637i	$0.288180\pi$
$14$	12.1732	3.25342
$15$	0.978567	0.252665
$16$	19.5430	4.88576
$17$	−0.890509	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$17$	−0.890509	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$18$	5.76020	1.35769
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	5.95409	1.33138
$21$	−4.22376	−0.921700
$22$	−2.82030	−0.601290
$23$	0.718161	0.149747	0.0748735	−	0.997193i	$-0.476145\pi$
$23$	0.718161	0.149747	0.0748735	+	0.997193i	$0.476145\pi$
$24$	−10.9127	−2.22755
$25$	1.00000	0.200000
$26$	−12.5567	−2.46256
$27$	−4.93433	−0.949612
$28$	−25.6995	−4.85675
$29$	−3.73514	−0.693598	−0.346799	−	0.937939i	$-0.612731\pi$
$29$	−3.73514	−0.693598	−0.346799	+	0.937939i	$0.612731\pi$
$30$	−2.75985	−0.503878
$31$	6.92133	1.24311	0.621554	−	0.783372i	$-0.286502\pi$
$31$	6.92133	1.24311	0.621554	+	0.783372i	$0.286502\pi$
$32$	−32.8137	−5.80070
$33$	0.978567	0.170347
$34$	2.51150	0.430719
$35$	−4.31627	−0.729584
$36$	−12.1607	−2.02678
$37$	−9.38577	−1.54301	−0.771505	−	0.636223i	$-0.780496\pi$
$37$	−9.38577	−1.54301	−0.771505	+	0.636223i	$0.780496\pi$
$38$	2.82030	0.457513
$39$	4.35682	0.697649
$40$	−11.1517	−1.76324
$41$	0.562568	0.0878583	0.0439292	−	0.999035i	$-0.486012\pi$
$41$	0.562568	0.0878583	0.0439292	+	0.999035i	$0.486012\pi$
$42$	11.9123	1.83810
$43$	2.45537	0.374440	0.187220	−	0.982318i	$-0.440052\pi$
$43$	2.45537	0.374440	0.187220	+	0.982318i	$0.440052\pi$
$44$	5.95409	0.897613
$45$	−2.04241	−0.304464
$46$	−2.02543	−0.298633
$47$	−6.12922	−0.894039	−0.447019	−	0.894524i	$-0.647515\pi$
$47$	−6.12922	−0.894039	−0.447019	+	0.894524i	$0.647515\pi$
$48$	19.1241	2.76033
$49$	11.6302	1.66146
$50$	−2.82030	−0.398851
$51$	−0.871423	−0.122024
$52$	26.5091	3.67615
$53$	−4.81164	−0.660930	−0.330465	−	0.943818i	$-0.607206\pi$
$53$	−4.81164	−0.660930	−0.330465	+	0.943818i	$0.607206\pi$
$54$	13.9163	1.89377
$55$	1.00000	0.134840
$56$	48.1339	6.43216
$57$	−0.978567	−0.129614
$58$	10.5342	1.38321
$59$	−13.6109	−1.77199	−0.885996	−	0.463693i	$-0.846524\pi$
$59$	−13.6109	−1.77199	−0.885996	+	0.463693i	$0.846524\pi$
$60$	5.82647	0.752195
$61$	−12.2483	−1.56824	−0.784120	−	0.620609i	$-0.786885\pi$
$61$	−12.2483	−1.56824	−0.784120	+	0.620609i	$0.786885\pi$
$62$	−19.5202	−2.47907
$63$	8.81559	1.11066
$64$	53.4585	6.68232
$65$	4.45224	0.552233
$66$	−2.75985	−0.339714
$67$	−8.70936	−1.06402	−0.532009	−	0.846739i	$-0.678563\pi$
$67$	−8.70936	−1.06402	−0.532009	+	0.846739i	$0.678563\pi$
$68$	−5.30217	−0.642983
$69$	0.702768	0.0846034
$70$	12.1732	1.45497
$71$	−13.1843	−1.56469	−0.782347	−	0.622843i	$-0.785978\pi$
$71$	−13.1843	−1.56469	−0.782347	+	0.622843i	$0.785978\pi$
$72$	22.7764	2.68422
$73$	4.19184	0.490618	0.245309	−	0.969445i	$-0.421111\pi$
$73$	4.19184	0.490618	0.245309	+	0.969445i	$0.421111\pi$
$74$	26.4707	3.07715
$75$	0.978567	0.112995
$76$	−5.95409	−0.682981
$77$	−4.31627	−0.491885
$78$	−12.2875	−1.39129
$79$	−0.175970	−0.0197982	−0.00989910	−	0.999951i	$-0.503151\pi$
$79$	−0.175970	−0.0197982	−0.00989910	+	0.999951i	$0.503151\pi$
$80$	19.5430	2.18498
$81$	1.29865	0.144295
$82$	−1.58661	−0.175212
$83$	−7.01026	−0.769476	−0.384738	−	0.923026i	$-0.625708\pi$
$83$	−7.01026	−0.769476	−0.384738	+	0.923026i	$0.625708\pi$
$84$	−25.1487	−2.74394
$85$	−0.890509	−0.0965893
$86$	−6.92487	−0.746728
$87$	−3.65508	−0.391866
$88$	−11.1517	−1.18878
$89$	−9.77035	−1.03565	−0.517827	−	0.855485i	$-0.673259\pi$
$89$	−9.77035	−1.03565	−0.517827	+	0.855485i	$0.673259\pi$
$90$	5.76020	0.607179
$91$	−19.2171	−2.01450
$92$	4.27600	0.445804
$93$	6.77298	0.702325
$94$	17.2862	1.78294
$95$	−1.00000	−0.102598
$96$	−32.1104	−3.27726
$97$	−0.0271785	−0.00275956	−0.00137978	−	0.999999i	$-0.500439\pi$
$97$	−0.0271785	−0.00275956	−0.00137978	+	0.999999i	$0.500439\pi$
$98$	−32.8007	−3.31337
$99$	−2.04241	−0.205270
$100$	5.95409	0.595409
$101$	14.4434	1.43717	0.718584	−	0.695440i	$-0.244791\pi$
$101$	14.4434	1.43717	0.718584	+	0.695440i	$0.244791\pi$
$102$	2.45767	0.243346
$103$	9.59893	0.945811	0.472905	−	0.881113i	$-0.343205\pi$
$103$	9.59893	0.945811	0.472905	+	0.881113i	$0.343205\pi$
$104$	−49.6502	−4.86860
$105$	−4.22376	−0.412197
$106$	13.5703	1.31806
$107$	−0.707954	−0.0684405	−0.0342202	−	0.999414i	$-0.510895\pi$
$107$	−0.707954	−0.0684405	−0.0342202	+	0.999414i	$0.510895\pi$
$108$	−29.3795	−2.82704
$109$	8.64746	0.828277	0.414138	−	0.910214i	$-0.364083\pi$
$109$	8.64746	0.828277	0.414138	+	0.910214i	$0.364083\pi$
$110$	−2.82030	−0.268905
$111$	−9.18460	−0.871763
$112$	−84.3531	−7.97061
$113$	−10.0704	−0.947340	−0.473670	−	0.880702i	$-0.657071\pi$
$113$	−10.0704	−0.947340	−0.473670	+	0.880702i	$0.657071\pi$
$114$	2.75985	0.258484
$115$	0.718161	0.0669689
$116$	−22.2394	−2.06487
$117$	−9.09330	−0.840676
$118$	38.3869	3.53380
$119$	3.84368	0.352350
$120$	−10.9127	−0.996189
$121$	1.00000	0.0909091
$122$	34.5440	3.12747
$123$	0.550510	0.0496378
$124$	41.2102	3.70079
$125$	1.00000	0.0894427
$126$	−24.8626	−2.21494
$127$	−6.89115	−0.611491	−0.305745	−	0.952113i	$-0.598906\pi$
$127$	−6.89115	−0.611491	−0.305745	+	0.952113i	$0.598906\pi$
$128$	−85.1416	−7.52553
$129$	2.40274	0.211549
$130$	−12.5567	−1.10129
$131$	−5.88201	−0.513914	−0.256957	−	0.966423i	$-0.582720\pi$
$131$	−5.88201	−0.513914	−0.256957	+	0.966423i	$0.582720\pi$
$132$	5.82647	0.507130
$133$	4.31627	0.374268
$134$	24.5630	2.12192
$135$	−4.93433	−0.424680
$136$	9.93071	0.851552
$137$	16.8441	1.43909	0.719543	−	0.694448i	$-0.244351\pi$
$137$	16.8441	1.43909	0.719543	+	0.694448i	$0.244351\pi$
$138$	−1.98202	−0.168721
$139$	−19.2183	−1.63007	−0.815037	−	0.579408i	$-0.803284\pi$
$139$	−19.2183	−1.63007	−0.815037	+	0.579408i	$0.803284\pi$
$140$	−25.6995	−2.17200
$141$	−5.99785	−0.505110
$142$	37.1838	3.12040
$143$	4.45224	0.372315
$144$	−39.9148	−3.32624
$145$	−3.73514	−0.310186
$146$	−11.8222	−0.978416
$147$	11.3809	0.938685
$148$	−55.8837	−4.59361
$149$	−6.05061	−0.495685	−0.247842	−	0.968800i	$-0.579722\pi$
$149$	−6.05061	−0.495685	−0.247842	+	0.968800i	$0.579722\pi$
$150$	−2.75985	−0.225341
$151$	−10.0530	−0.818102	−0.409051	−	0.912512i	$-0.634140\pi$
$151$	−10.0530	−0.818102	−0.409051	+	0.912512i	$0.634140\pi$
$152$	11.1517	0.904524
$153$	1.81878	0.147040
$154$	12.1732	0.980943
$155$	6.92133	0.555934
$156$	25.9409	2.07693
$157$	19.9239	1.59010	0.795051	−	0.606542i	$-0.207444\pi$
$157$	19.9239	1.59010	0.795051	+	0.606542i	$0.207444\pi$
$158$	0.496289	0.0394826
$159$	−4.70851	−0.373409
$160$	−32.8137	−2.59415
$161$	−3.09978	−0.244297
$162$	−3.66259	−0.287760
$163$	7.21157	0.564854	0.282427	−	0.959289i	$-0.408861\pi$
$163$	7.21157	0.564854	0.282427	+	0.959289i	$0.408861\pi$
$164$	3.34958	0.261558
$165$	0.978567	0.0761813
$166$	19.7710	1.53453
$167$	10.3106	0.797859	0.398930	−	0.916982i	$-0.369382\pi$
$167$	10.3106	0.797859	0.398930	+	0.916982i	$0.369382\pi$
$168$	47.1022	3.63402
$169$	6.82248	0.524806
$170$	2.51150	0.192624
$171$	2.04241	0.156187
$172$	14.6195	1.11472
$173$	7.74527	0.588862	0.294431	−	0.955673i	$-0.404870\pi$
$173$	7.74527	0.588862	0.294431	+	0.955673i	$0.404870\pi$
$174$	10.3084	0.781480
$175$	−4.31627	−0.326280
$176$	19.5430	1.47311
$177$	−13.3192	−1.00113
$178$	27.5553	2.06536
$179$	−11.1750	−0.835258	−0.417629	−	0.908618i	$-0.637139\pi$
$179$	−11.1750	−0.835258	−0.417629	+	0.908618i	$0.637139\pi$
$180$	−12.1607	−0.906404
$181$	−4.54606	−0.337906	−0.168953	−	0.985624i	$-0.554039\pi$
$181$	−4.54606	−0.337906	−0.168953	+	0.985624i	$0.554039\pi$
$182$	54.1980	4.01742
$183$	−11.9858	−0.886017
$184$	−8.00873	−0.590412
$185$	−9.38577	−0.690055
$186$	−19.1018	−1.40061
$187$	−0.890509	−0.0651205
$188$	−36.4940	−2.66160
$189$	21.2979	1.54920
$190$	2.82030	0.204606
$191$	−10.6372	−0.769678	−0.384839	−	0.922984i	$-0.625743\pi$
$191$	−10.6372	−0.769678	−0.384839	+	0.922984i	$0.625743\pi$
$192$	52.3127	3.77535
$193$	−22.2436	−1.60113	−0.800564	−	0.599248i	$-0.795466\pi$
$193$	−22.2436	−1.60113	−0.800564	+	0.599248i	$0.795466\pi$
$194$	0.0766516	0.00550326
$195$	4.35682	0.311998
$196$	69.2474	4.94624
$197$	5.93859	0.423107	0.211554	−	0.977366i	$-0.432148\pi$
$197$	5.93859	0.423107	0.211554	+	0.977366i	$0.432148\pi$
$198$	5.76020	0.409360
$199$	−0.932308	−0.0660896	−0.0330448	−	0.999454i	$-0.510520\pi$
$199$	−0.932308	−0.0660896	−0.0330448	+	0.999454i	$0.510520\pi$
$200$	−11.1517	−0.788546
$201$	−8.52269	−0.601144
$202$	−40.7346	−2.86608
$203$	16.1219	1.13153
$204$	−5.18853	−0.363270
$205$	0.562568	0.0392914
$206$	−27.0719	−1.88619
$207$	−1.46678	−0.101948
$208$	87.0103	6.03308
$209$	−1.00000	−0.0691714
$210$	11.9123	0.822025
$211$	7.29715	0.502357	0.251178	−	0.967941i	$-0.419182\pi$
$211$	7.29715	0.502357	0.251178	+	0.967941i	$0.419182\pi$
$212$	−28.6490	−1.96762
$213$	−12.9018	−0.884014
$214$	1.99664	0.136488
$215$	2.45537	0.167455
$216$	55.0263	3.74407
$217$	−29.8744	−2.02800
$218$	−24.3884	−1.65179
$219$	4.10199	0.277187
$220$	5.95409	0.401425
$221$	−3.96477	−0.266699
$222$	25.9033	1.73852
$223$	−26.8259	−1.79640	−0.898199	−	0.439590i	$-0.855124\pi$
$223$	−26.8259	−1.79640	−0.898199	+	0.439590i	$0.855124\pi$
$224$	141.633	9.46326
$225$	−2.04241	−0.136161
$226$	28.4014	1.88924
$227$	−8.16934	−0.542218	−0.271109	−	0.962549i	$-0.587390\pi$
$227$	−8.16934	−0.542218	−0.271109	+	0.962549i	$0.587390\pi$
$228$	−5.82647	−0.385868
$229$	9.65742	0.638180	0.319090	−	0.947724i	$-0.396623\pi$
$229$	9.65742	0.638180	0.319090	+	0.947724i	$0.396623\pi$
$230$	−2.02543	−0.133553
$231$	−4.22376	−0.277903
$232$	41.6532	2.73467
$233$	7.99449	0.523736	0.261868	−	0.965104i	$-0.415661\pi$
$233$	7.99449	0.523736	0.261868	+	0.965104i	$0.415661\pi$
$234$	25.6458	1.67652
$235$	−6.12922	−0.399826
$236$	−81.0407	−5.27530
$237$	−0.172199	−0.0111855
$238$	−10.8403	−0.702675
$239$	−13.1696	−0.851873	−0.425936	−	0.904753i	$-0.640055\pi$
$239$	−13.1696	−0.851873	−0.425936	+	0.904753i	$0.640055\pi$
$240$	19.1241	1.23446
$241$	21.5776	1.38994	0.694969	−	0.719040i	$-0.255418\pi$
$241$	21.5776	1.38994	0.694969	+	0.719040i	$0.255418\pi$
$242$	−2.82030	−0.181296
$243$	16.0738	1.03114
$244$	−72.9278	−4.66872
$245$	11.6302	0.743028
$246$	−1.55260	−0.0989903
$247$	−4.45224	−0.283290
$248$	−77.1848	−4.90124
$249$	−6.86001	−0.434735
$250$	−2.82030	−0.178371
$251$	4.31149	0.272139	0.136070	−	0.990699i	$-0.456553\pi$
$251$	4.31149	0.272139	0.136070	+	0.990699i	$0.456553\pi$
$252$	52.4888	3.30649
$253$	0.718161	0.0451504
$254$	19.4351	1.21947
$255$	−0.871423	−0.0545706
$256$	133.208	8.32549
$257$	2.58986	0.161551	0.0807756	−	0.996732i	$-0.474260\pi$
$257$	2.58986	0.161551	0.0807756	+	0.996732i	$0.474260\pi$
$258$	−6.77644	−0.421883
$259$	40.5115	2.51726
$260$	26.5091	1.64402
$261$	7.62868	0.472203
$262$	16.5890	1.02487
$263$	10.1689	0.627044	0.313522	−	0.949581i	$-0.398491\pi$
$263$	10.1689	0.627044	0.313522	+	0.949581i	$0.398491\pi$
$264$	−10.9127	−0.671631
$265$	−4.81164	−0.295577
$266$	−12.1732	−0.746386
$267$	−9.56093	−0.585120
$268$	−51.8563	−3.16763
$269$	−5.18151	−0.315922	−0.157961	−	0.987445i	$-0.550492\pi$
$269$	−5.18151	−0.315922	−0.157961	+	0.987445i	$0.550492\pi$
$270$	13.9163	0.846919
$271$	−11.8406	−0.719266	−0.359633	−	0.933094i	$-0.617098\pi$
$271$	−11.8406	−0.719266	−0.359633	+	0.933094i	$0.617098\pi$
$272$	−17.4032	−1.05523
$273$	−18.8052	−1.13814
$274$	−47.5053	−2.86990
$275$	1.00000	0.0603023
$276$	4.18435	0.251868
$277$	−19.4227	−1.16700	−0.583499	−	0.812114i	$-0.698317\pi$
$277$	−19.4227	−1.16700	−0.583499	+	0.812114i	$0.698317\pi$
$278$	54.2014	3.25078
$279$	−14.1362	−0.846311
$280$	48.1339	2.87655
$281$	29.9204	1.78490	0.892452	−	0.451142i	$-0.148983\pi$
$281$	29.9204	1.78490	0.892452	+	0.451142i	$0.148983\pi$
$282$	16.9157	1.00732
$283$	0.760147	0.0451860	0.0225930	−	0.999745i	$-0.492808\pi$
$283$	0.760147	0.0451860	0.0225930	+	0.999745i	$0.492808\pi$
$284$	−78.5008	−4.65816
$285$	−0.978567	−0.0579653
$286$	−12.5567	−0.742491
$287$	−2.42820	−0.143332
$288$	67.0190	3.94913
$289$	−16.2070	−0.953353
$290$	10.5342	0.618590
$291$	−0.0265960	−0.00155908
$292$	24.9586	1.46059
$293$	−18.6349	−1.08866	−0.544332	−	0.838870i	$-0.683217\pi$
$293$	−18.6349	−1.08866	−0.544332	+	0.838870i	$0.683217\pi$
$294$	−32.0977	−1.87198
$295$	−13.6109	−0.792459
$296$	104.667	6.08367
$297$	−4.93433	−0.286319
$298$	17.0645	0.988521
$299$	3.19743	0.184912
$300$	5.82647	0.336392
$301$	−10.5980	−0.610860
$302$	28.3525	1.63150
$303$	14.1338	0.811965
$304$	−19.5430	−1.12087
$305$	−12.2483	−0.701338
$306$	−5.12951	−0.293235
$307$	10.2710	0.586197	0.293098	−	0.956082i	$-0.405314\pi$
$307$	10.2710	0.586197	0.293098	+	0.956082i	$0.405314\pi$
$308$	−25.6995	−1.46436
$309$	9.39319	0.534360
$310$	−19.5202	−1.10867
$311$	19.7906	1.12222	0.561112	−	0.827740i	$-0.310374\pi$
$311$	19.7906	1.12222	0.561112	+	0.827740i	$0.310374\pi$
$312$	−48.5860	−2.75064
$313$	10.5256	0.594943	0.297471	−	0.954731i	$-0.403857\pi$
$313$	10.5256	0.594943	0.297471	+	0.954731i	$0.403857\pi$
$314$	−56.1915	−3.17107
$315$	8.81559	0.496702
$316$	−1.04774	−0.0589401
$317$	22.2369	1.24895	0.624473	−	0.781046i	$-0.285314\pi$
$317$	22.2369	1.24895	0.624473	+	0.781046i	$0.285314\pi$
$318$	13.2794	0.744673
$319$	−3.73514	−0.209128
$320$	53.4585	2.98842
$321$	−0.692780	−0.0386672
$322$	8.74231	0.487190
$323$	0.890509	0.0495493
$324$	7.73229	0.429572
$325$	4.45224	0.246966
$326$	−20.3388	−1.12646
$327$	8.46212	0.467956
$328$	−6.27360	−0.346402
$329$	26.4554	1.45853
$330$	−2.75985	−0.151925
$331$	8.39778	0.461584	0.230792	−	0.973003i	$-0.425868\pi$
$331$	8.39778	0.461584	0.230792	+	0.973003i	$0.425868\pi$
$332$	−41.7397	−2.29077
$333$	19.1696	1.05049
$334$	−29.0790	−1.59113
$335$	−8.70936	−0.475843
$336$	−82.5451	−4.50320
$337$	5.82341	0.317222	0.158611	−	0.987341i	$-0.449298\pi$
$337$	5.82341	0.317222	0.158611	+	0.987341i	$0.449298\pi$
$338$	−19.2415	−1.04660
$339$	−9.85452	−0.535224
$340$	−5.30217	−0.287551
$341$	6.92133	0.374811
$342$	−5.76020	−0.311476
$343$	−19.9853	−1.07911
$344$	−27.3816	−1.47631
$345$	0.702768	0.0378358
$346$	−21.8440	−1.17434
$347$	14.1063	0.757267	0.378634	−	0.925547i	$-0.376394\pi$
$347$	14.1063	0.757267	0.378634	+	0.925547i	$0.376394\pi$
$348$	−21.7627	−1.16660
$349$	20.1021	1.07604	0.538020	−	0.842932i	$-0.319172\pi$
$349$	20.1021	1.07604	0.538020	+	0.842932i	$0.319172\pi$
$350$	12.1732	0.650684
$351$	−21.9689	−1.17261
$352$	−32.8137	−1.74898
$353$	3.81045	0.202810	0.101405	−	0.994845i	$-0.467666\pi$
$353$	3.81045	0.202810	0.101405	+	0.994845i	$0.467666\pi$
$354$	37.5641	1.99651
$355$	−13.1843	−0.699752
$356$	−58.1735	−3.08319
$357$	3.76130	0.199069
$358$	31.5168	1.66572
$359$	14.3721	0.758529	0.379265	−	0.925288i	$-0.376177\pi$
$359$	14.3721	0.758529	0.379265	+	0.925288i	$0.376177\pi$
$360$	22.7764	1.20042
$361$	1.00000	0.0526316
$362$	12.8213	0.673870
$363$	0.978567	0.0513614
$364$	−114.420	−5.99726
$365$	4.19184	0.219411
$366$	33.8036	1.76694
$367$	30.7446	1.60486	0.802428	−	0.596748i	$-0.203541\pi$
$367$	30.7446	1.60486	0.802428	+	0.596748i	$0.203541\pi$
$368$	14.0350	0.731627
$369$	−1.14899	−0.0598142
$370$	26.4707	1.37614
$371$	20.7684	1.07824
$372$	40.3269	2.09085
$373$	24.4754	1.26729	0.633645	−	0.773624i	$-0.281558\pi$
$373$	24.4754	1.26729	0.633645	+	0.773624i	$0.281558\pi$
$374$	2.51150	0.129867
$375$	0.978567	0.0505330
$376$	68.3514	3.52495
$377$	−16.6298	−0.856476
$378$	−60.0665	−3.08949
$379$	−23.9470	−1.23008	−0.615038	−	0.788498i	$-0.710859\pi$
$379$	−23.9470	−1.23008	−0.615038	+	0.788498i	$0.710859\pi$
$380$	−5.95409	−0.305438
$381$	−6.74345	−0.345477
$382$	30.0000	1.53493
$383$	4.95662	0.253271	0.126636	−	0.991949i	$-0.459582\pi$
$383$	4.95662	0.253271	0.126636	+	0.991949i	$0.459582\pi$
$384$	−83.3168	−4.25174
$385$	−4.31627	−0.219978
$386$	62.7335	3.19305
$387$	−5.01486	−0.254919
$388$	−0.161823	−0.00821534
$389$	4.88198	0.247526	0.123763	−	0.992312i	$-0.460504\pi$
$389$	4.88198	0.247526	0.123763	+	0.992312i	$0.460504\pi$
$390$	−12.2875	−0.622203
$391$	−0.639529	−0.0323424
$392$	−129.697	−6.55069
$393$	−5.75594	−0.290349
$394$	−16.7486	−0.843783
$395$	−0.175970	−0.00885402
$396$	−12.1607	−0.611097
$397$	4.68250	0.235008	0.117504	−	0.993072i	$-0.462511\pi$
$397$	4.68250	0.235008	0.117504	+	0.993072i	$0.462511\pi$
$398$	2.62939	0.131799
$399$	4.22376	0.211453
$400$	19.5430	0.977151
$401$	−6.64076	−0.331624	−0.165812	−	0.986157i	$-0.553024\pi$
$401$	−6.64076	−0.331624	−0.165812	+	0.986157i	$0.553024\pi$
$402$	24.0365	1.19883
$403$	30.8154	1.53503
$404$	85.9971	4.27852
$405$	1.29865	0.0645305
$406$	−45.4686	−2.25657
$407$	−9.38577	−0.465235
$408$	9.71787	0.481106
$409$	−5.28656	−0.261404	−0.130702	−	0.991422i	$-0.541723\pi$
$409$	−5.28656	−0.261404	−0.130702	+	0.991422i	$0.541723\pi$
$410$	−1.58661	−0.0783571
$411$	16.4830	0.813049
$412$	57.1529	2.81572
$413$	58.7485	2.89082
$414$	4.13675	0.203310
$415$	−7.01026	−0.344120
$416$	−146.095	−7.16289
$417$	−18.8064	−0.920952
$418$	2.82030	0.137945
$419$	−5.94182	−0.290277	−0.145138	−	0.989411i	$-0.546363\pi$
$419$	−5.94182	−0.290277	−0.145138	+	0.989411i	$0.546363\pi$
$420$	−25.1487	−1.22713
$421$	−35.7120	−1.74049	−0.870247	−	0.492615i	$-0.836041\pi$
$421$	−35.7120	−1.74049	−0.870247	+	0.492615i	$0.836041\pi$
$422$	−20.5802	−1.00183
$423$	12.5184	0.608664
$424$	53.6581	2.60587
$425$	−0.890509	−0.0431960
$426$	36.3868	1.76295
$427$	52.8672	2.55842
$428$	−4.21522	−0.203750
$429$	4.35682	0.210349
$430$	−6.92487	−0.333947
$431$	−15.0516	−0.725010	−0.362505	−	0.931982i	$-0.618078\pi$
$431$	−15.0516	−0.725010	−0.362505	+	0.931982i	$0.618078\pi$
$432$	−96.4318	−4.63958
$433$	−12.8687	−0.618432	−0.309216	−	0.950992i	$-0.600067\pi$
$433$	−12.8687	−0.618432	−0.309216	+	0.950992i	$0.600067\pi$
$434$	84.2546	4.04435
$435$	−3.65508	−0.175248
$436$	51.4878	2.46582
$437$	−0.718161	−0.0343543
$438$	−11.5689	−0.552781
$439$	2.23667	0.106750	0.0533752	−	0.998575i	$-0.483002\pi$
$439$	2.23667	0.106750	0.0533752	+	0.998575i	$0.483002\pi$
$440$	−11.1517	−0.531638
$441$	−23.7537	−1.13113
$442$	11.1818	0.531865
$443$	−32.5723	−1.54756	−0.773778	−	0.633457i	$-0.781636\pi$
$443$	−32.5723	−1.54756	−0.773778	+	0.633457i	$0.781636\pi$
$444$	−54.6859	−2.59528
$445$	−9.77035	−0.463159
$446$	75.6572	3.58247
$447$	−5.92092	−0.280050
$448$	−230.742	−10.9015
$449$	29.0329	1.37015	0.685074	−	0.728474i	$-0.259770\pi$
$449$	29.0329	1.37015	0.685074	+	0.728474i	$0.259770\pi$
$450$	5.76020	0.271539
$451$	0.562568	0.0264903
$452$	−59.9599	−2.82027
$453$	−9.83753	−0.462208
$454$	23.0400	1.08132
$455$	−19.2171	−0.900912
$456$	10.9127	0.511034
$457$	−30.2618	−1.41559	−0.707793	−	0.706420i	$-0.750309\pi$
$457$	−30.2618	−1.41559	−0.707793	+	0.706420i	$0.750309\pi$
$458$	−27.2368	−1.27269
$459$	4.39407	0.205098
$460$	4.27600	0.199369
$461$	−9.91813	−0.461933	−0.230967	−	0.972962i	$-0.574189\pi$
$461$	−9.91813	−0.461933	−0.230967	+	0.972962i	$0.574189\pi$
$462$	11.9123	0.554209
$463$	−36.2007	−1.68239	−0.841196	−	0.540731i	$-0.818148\pi$
$463$	−36.2007	−1.68239	−0.841196	+	0.540731i	$0.818148\pi$
$464$	−72.9959	−3.38875
$465$	6.77298	0.314089
$466$	−22.5468	−1.04446
$467$	21.3789	0.989296	0.494648	−	0.869093i	$-0.335297\pi$
$467$	21.3789	0.989296	0.494648	+	0.869093i	$0.335297\pi$
$468$	−54.1423	−2.50273
$469$	37.5920	1.73584
$470$	17.2862	0.797355
$471$	19.4969	0.898369
$472$	151.785	6.98649
$473$	2.45537	0.112898
$474$	0.485652	0.0223067
$475$	−1.00000	−0.0458831
$476$	22.8856	1.04896
$477$	9.82734	0.449963
$478$	37.1423	1.69885
$479$	−0.0525247	−0.00239992	−0.00119996	−	0.999999i	$-0.500382\pi$
$479$	−0.0525247	−0.00239992	−0.00119996	+	0.999999i	$0.500382\pi$
$480$	−32.1104	−1.46563
$481$	−41.7877	−1.90536
$482$	−60.8554	−2.77189
$483$	−3.03334	−0.138022
$484$	5.95409	0.270641
$485$	−0.0271785	−0.00123411
$486$	−45.3330	−2.05635
$487$	2.09528	0.0949464	0.0474732	−	0.998873i	$-0.484883\pi$
$487$	2.09528	0.0949464	0.0474732	+	0.998873i	$0.484883\pi$
$488$	136.590	6.18315
$489$	7.05700	0.319129
$490$	−32.8007	−1.48179
$491$	−29.7033	−1.34049	−0.670245	−	0.742140i	$-0.733811\pi$
$491$	−29.7033	−1.34049	−0.670245	+	0.742140i	$0.733811\pi$
$492$	3.27779	0.147774
$493$	3.32618	0.149803
$494$	12.5567	0.564951
$495$	−2.04241	−0.0917994
$496$	135.264	6.07352
$497$	56.9072	2.55264
$498$	19.3473	0.866972
$499$	−2.93723	−0.131489	−0.0657443	−	0.997837i	$-0.520942\pi$
$499$	−2.93723	−0.131489	−0.0657443	+	0.997837i	$0.520942\pi$
$500$	5.95409	0.266275
$501$	10.0896	0.450771
$502$	−12.1597	−0.542714
$503$	−7.12456	−0.317668	−0.158834	−	0.987305i	$-0.550774\pi$
$503$	−7.12456	−0.317668	−0.158834	+	0.987305i	$0.550774\pi$
$504$	−98.3090	−4.37903
$505$	14.4434	0.642721
$506$	−2.02543	−0.0900413
$507$	6.67625	0.296503
$508$	−41.0305	−1.82044
$509$	14.5857	0.646499	0.323250	−	0.946314i	$-0.395225\pi$
$509$	14.5857	0.646499	0.323250	+	0.946314i	$0.395225\pi$
$510$	2.45767	0.108828
$511$	−18.0931	−0.800393
$512$	−205.403	−9.07761
$513$	4.93433	0.217856
$514$	−7.30419	−0.322174
$515$	9.59893	0.422979
$516$	14.3061	0.629792
$517$	−6.12922	−0.269563
$518$	−114.255	−5.02006
$519$	7.57926	0.332693
$520$	−49.6502	−2.17731
$521$	−17.9531	−0.786539	−0.393270	−	0.919423i	$-0.628656\pi$
$521$	−17.9531	−0.786539	−0.393270	+	0.919423i	$0.628656\pi$
$522$	−21.5152	−0.941693
$523$	−16.8919	−0.738630	−0.369315	−	0.929304i	$-0.620408\pi$
$523$	−16.8919	−0.738630	−0.369315	+	0.929304i	$0.620408\pi$
$524$	−35.0220	−1.52994
$525$	−4.22376	−0.184340
$526$	−28.6795	−1.25048
$527$	−6.16351	−0.268487
$528$	19.1241	0.832272
$529$	−22.4842	−0.977576
$530$	13.5703	0.589455
$531$	27.7991	1.20638
$532$	25.6995	1.11421
$533$	2.50469	0.108490
$534$	26.9647	1.16688
$535$	−0.707954	−0.0306075
$536$	97.1244	4.19513
$537$	−10.9355	−0.471900
$538$	14.6134	0.630028
$539$	11.6302	0.500949
$540$	−29.3795	−1.26429
$541$	7.89924	0.339615	0.169807	−	0.985477i	$-0.445685\pi$
$541$	7.89924	0.339615	0.169807	+	0.985477i	$0.445685\pi$
$542$	33.3941	1.43440
$543$	−4.44862	−0.190909
$544$	29.2209	1.25284
$545$	8.64746	0.370417
$546$	53.0364	2.26975
$547$	−1.24065	−0.0530464	−0.0265232	−	0.999648i	$-0.508444\pi$
$547$	−1.24065	−0.0530464	−0.0265232	+	0.999648i	$0.508444\pi$
$548$	100.291	4.28423
$549$	25.0161	1.06766
$550$	−2.82030	−0.120258
$551$	3.73514	0.159122
$552$	−7.83708	−0.333568
$553$	0.759536	0.0322987
$554$	54.7779	2.32729
$555$	−9.18460	−0.389864
$556$	−114.427	−4.85281
$557$	15.5671	0.659598	0.329799	−	0.944051i	$-0.393019\pi$
$557$	15.5671	0.659598	0.329799	+	0.944051i	$0.393019\pi$
$558$	39.8683	1.68776
$559$	10.9319	0.462370
$560$	−84.3531	−3.56457
$561$	−0.871423	−0.0367915
$562$	−84.3846	−3.55955
$563$	4.34507	0.183123	0.0915614	−	0.995799i	$-0.470814\pi$
$563$	4.34507	0.183123	0.0915614	+	0.995799i	$0.470814\pi$
$564$	−35.7118	−1.50374
$565$	−10.0704	−0.423663
$566$	−2.14384	−0.0901124
$567$	−5.60534	−0.235402
$568$	147.028	6.16916
$569$	14.3304	0.600763	0.300381	−	0.953819i	$-0.402886\pi$
$569$	14.3304	0.600763	0.300381	+	0.953819i	$0.402886\pi$
$570$	2.75985	0.115597
$571$	−19.8499	−0.830690	−0.415345	−	0.909664i	$-0.636339\pi$
$571$	−19.8499	−0.830690	−0.415345	+	0.909664i	$0.636339\pi$
$572$	26.5091	1.10840
$573$	−10.4092	−0.434849
$574$	6.84824	0.285840
$575$	0.718161	0.0299494
$576$	−109.184	−4.54934
$577$	−0.865788	−0.0360432	−0.0180216	−	0.999838i	$-0.505737\pi$
$577$	−0.865788	−0.0360432	−0.0180216	+	0.999838i	$0.505737\pi$
$578$	45.7086	1.90123
$579$	−21.7668	−0.904598
$580$	−22.2394	−0.923439
$581$	30.2582	1.25532
$582$	0.0750087	0.00310921
$583$	−4.81164	−0.199278
$584$	−46.7462	−1.93437
$585$	−9.09330	−0.375962
$586$	52.5561	2.17107
$587$	33.7761	1.39409	0.697044	−	0.717028i	$-0.254498\pi$
$587$	33.7761	1.39409	0.697044	+	0.717028i	$0.254498\pi$
$588$	67.7632	2.79451
$589$	−6.92133	−0.285188
$590$	38.3869	1.58036
$591$	5.81131	0.239045
$592$	−183.426	−7.53877
$593$	24.7537	1.01651	0.508255	−	0.861206i	$-0.330291\pi$
$593$	24.7537	1.01651	0.508255	+	0.861206i	$0.330291\pi$
$594$	13.9163	0.570992
$595$	3.84368	0.157576
$596$	−36.0259	−1.47568
$597$	−0.912325	−0.0373390
$598$	−9.01771	−0.368762
$599$	33.8794	1.38428	0.692138	−	0.721765i	$-0.256669\pi$
$599$	33.8794	1.38428	0.692138	+	0.721765i	$0.256669\pi$
$600$	−10.9127	−0.445509
$601$	−20.1635	−0.822486	−0.411243	−	0.911526i	$-0.634905\pi$
$601$	−20.1635	−0.822486	−0.411243	+	0.911526i	$0.634905\pi$
$602$	29.8896	1.21821
$603$	17.7881	0.724386
$604$	−59.8565	−2.43553
$605$	1.00000	0.0406558
$606$	−39.8615	−1.61926
$607$	20.0673	0.814506	0.407253	−	0.913315i	$-0.366487\pi$
$607$	20.0673	0.814506	0.407253	+	0.913315i	$0.366487\pi$
$608$	32.8137	1.33077
$609$	15.7763	0.639289
$610$	34.5440	1.39865
$611$	−27.2888	−1.10399
$612$	10.8292	0.437745
$613$	−20.9089	−0.844502	−0.422251	−	0.906479i	$-0.638760\pi$
$613$	−20.9089	−0.844502	−0.422251	+	0.906479i	$0.638760\pi$
$614$	−28.9673	−1.16902
$615$	0.550510	0.0221987
$616$	48.1339	1.93937
$617$	6.46980	0.260464	0.130232	−	0.991484i	$-0.458428\pi$
$617$	6.46980	0.260464	0.130232	+	0.991484i	$0.458428\pi$
$618$	−26.4916	−1.06565
$619$	−7.83015	−0.314720	−0.157360	−	0.987541i	$-0.550298\pi$
$619$	−7.83015	−0.314720	−0.157360	+	0.987541i	$0.550298\pi$
$620$	41.2102	1.65504
$621$	−3.54364	−0.142202
$622$	−55.8155	−2.23800
$623$	42.1715	1.68957
$624$	85.1454	3.40854
$625$	1.00000	0.0400000
$626$	−29.6854	−1.18647
$627$	−0.978567	−0.0390802
$628$	118.629	4.73381
$629$	8.35811	0.333260
$630$	−24.8626	−0.990550
$631$	33.3315	1.32691	0.663453	−	0.748218i	$-0.269090\pi$
$631$	33.3315	1.32691	0.663453	+	0.748218i	$0.269090\pi$
$632$	1.96237	0.0780589
$633$	7.14075	0.283819
$634$	−62.7146	−2.49072
$635$	−6.89115	−0.273467
$636$	−28.0349	−1.11166
$637$	51.7806	2.05162
$638$	10.5342	0.417053
$639$	26.9278	1.06525
$640$	−85.1416	−3.36552
$641$	−5.99693	−0.236865	−0.118432	−	0.992962i	$-0.537787\pi$
$641$	−5.99693	−0.236865	−0.118432	+	0.992962i	$0.537787\pi$
$642$	1.95385	0.0771122
$643$	22.6462	0.893081	0.446540	−	0.894764i	$-0.352656\pi$
$643$	22.6462	0.893081	0.446540	+	0.894764i	$0.352656\pi$
$644$	−18.4564	−0.727283
$645$	2.40274	0.0946077
$646$	−2.51150	−0.0988138
$647$	−6.34939	−0.249620	−0.124810	−	0.992181i	$-0.539832\pi$
$647$	−6.34939	−0.249620	−0.124810	+	0.992181i	$0.539832\pi$
$648$	−14.4822	−0.568915
$649$	−13.6109	−0.534276
$650$	−12.5567	−0.492513
$651$	−29.2340	−1.14577
$652$	42.9384	1.68160
$653$	1.13832	0.0445460	0.0222730	−	0.999752i	$-0.492910\pi$
$653$	1.13832	0.0445460	0.0222730	+	0.999752i	$0.492910\pi$
$654$	−23.8657	−0.933223
$655$	−5.88201	−0.229829
$656$	10.9943	0.429254
$657$	−8.56144	−0.334014
$658$	−74.6122	−2.90869
$659$	−40.3095	−1.57024	−0.785118	−	0.619346i	$-0.787398\pi$
$659$	−40.3095	−1.57024	−0.785118	+	0.619346i	$0.787398\pi$
$660$	5.82647	0.226795
$661$	−18.5459	−0.721351	−0.360675	−	0.932691i	$-0.617454\pi$
$661$	−18.5459	−0.721351	−0.360675	+	0.932691i	$0.617454\pi$
$662$	−23.6842	−0.920514
$663$	−3.87979	−0.150678
$664$	78.1765	3.03384
$665$	4.31627	0.167378
$666$	−54.0639	−2.09493
$667$	−2.68243	−0.103864
$668$	61.3903	2.37526
$669$	−26.2510	−1.01492
$670$	24.5630	0.948952
$671$	−12.2483	−0.472842
$672$	138.597	5.34651
$673$	−4.19050	−0.161532	−0.0807660	−	0.996733i	$-0.525737\pi$
$673$	−4.19050	−0.161532	−0.0807660	+	0.996733i	$0.525737\pi$
$674$	−16.4238	−0.632620
$675$	−4.93433	−0.189922
$676$	40.6217	1.56237
$677$	45.4150	1.74544	0.872720	−	0.488221i	$-0.162354\pi$
$677$	45.4150	1.74544	0.872720	+	0.488221i	$0.162354\pi$
$678$	27.7927	1.06737
$679$	0.117310	0.00450194
$680$	9.93071	0.380826
$681$	−7.99425	−0.306340
$682$	−19.5202	−0.747468
$683$	−39.2180	−1.50063	−0.750317	−	0.661078i	$-0.770099\pi$
$683$	−39.2180	−1.50063	−0.750317	+	0.661078i	$0.770099\pi$
$684$	12.1607	0.464975
$685$	16.8441	0.643579
$686$	56.3646	2.15201
$687$	9.45043	0.360556
$688$	47.9853	1.82942
$689$	−21.4226	−0.816137
$690$	−1.98202	−0.0754541
$691$	−3.19600	−0.121581	−0.0607907	−	0.998151i	$-0.519362\pi$
$691$	−3.19600	−0.121581	−0.0607907	+	0.998151i	$0.519362\pi$
$692$	46.1160	1.75307
$693$	8.81559	0.334877
$694$	−39.7841	−1.51018
$695$	−19.2183	−0.728992
$696$	40.7605	1.54502
$697$	−0.500972	−0.0189757
$698$	−56.6939	−2.14590
$699$	7.82314	0.295898
$700$	−25.6995	−0.971350
$701$	17.5041	0.661119	0.330559	−	0.943785i	$-0.392763\pi$
$701$	17.5041	0.661119	0.330559	+	0.943785i	$0.392763\pi$
$702$	61.9587	2.33848
$703$	9.38577	0.353991
$704$	53.4585	2.01479
$705$	−5.99785	−0.225892
$706$	−10.7466	−0.404454
$707$	−62.3415	−2.34459
$708$	−79.3037	−2.98042
$709$	−31.0584	−1.16642	−0.583212	−	0.812320i	$-0.698204\pi$
$709$	−31.0584	−1.16642	−0.583212	+	0.812320i	$0.698204\pi$
$710$	37.1838	1.39548
$711$	0.359403	0.0134787
$712$	108.956	4.08331
$713$	4.97063	0.186152
$714$	−10.6080	−0.396994
$715$	4.45224	0.166505
$716$	−66.5369	−2.48660
$717$	−12.8874	−0.481287
$718$	−40.5336	−1.51270
$719$	−0.323259	−0.0120555	−0.00602776	−	0.999982i	$-0.501919\pi$
$719$	−0.323259	−0.0120555	−0.00602776	+	0.999982i	$0.501919\pi$
$720$	−39.9148	−1.48754
$721$	−41.4316	−1.54299
$722$	−2.82030	−0.104961
$723$	21.1151	0.785281
$724$	−27.0677	−1.00596
$725$	−3.73514	−0.138720
$726$	−2.75985	−0.102428
$727$	24.5268	0.909649	0.454824	−	0.890581i	$-0.349702\pi$
$727$	24.5268	0.909649	0.454824	+	0.890581i	$0.349702\pi$
$728$	214.304	7.94263
$729$	11.8333	0.438272
$730$	−11.8222	−0.437561
$731$	−2.18653	−0.0808716
$732$	−71.3647	−2.63771
$733$	41.9011	1.54765	0.773826	−	0.633398i	$-0.218341\pi$
$733$	41.9011	1.54765	0.773826	+	0.633398i	$0.218341\pi$
$734$	−86.7091	−3.20049
$735$	11.3809	0.419793
$736$	−23.5656	−0.868638
$737$	−8.70936	−0.320813
$738$	3.24050	0.119285
$739$	3.51483	0.129295	0.0646475	−	0.997908i	$-0.479408\pi$
$739$	3.51483	0.129295	0.0646475	+	0.997908i	$0.479408\pi$
$740$	−55.8837	−2.05433
$741$	−4.35682	−0.160052
$742$	−58.5731	−2.15028
$743$	38.0482	1.39585	0.697926	−	0.716170i	$-0.254107\pi$
$743$	38.0482	1.39585	0.697926	+	0.716170i	$0.254107\pi$
$744$	−75.5304	−2.76908
$745$	−6.05061	−0.221677
$746$	−69.0281	−2.52730
$747$	14.3178	0.523861
$748$	−5.30217	−0.193867
$749$	3.05572	0.111654
$750$	−2.75985	−0.100776
$751$	48.4780	1.76899	0.884493	−	0.466553i	$-0.154504\pi$
$751$	48.4780	1.76899	0.884493	+	0.466553i	$0.154504\pi$
$752$	−119.784	−4.36806
$753$	4.21908	0.153752
$754$	46.9009	1.70803
$755$	−10.0530	−0.365866
$756$	126.810	4.61203
$757$	−18.7010	−0.679701	−0.339850	−	0.940480i	$-0.610376\pi$
$757$	−18.7010	−0.679701	−0.339850	+	0.940480i	$0.610376\pi$
$758$	67.5378	2.45308
$759$	0.702768	0.0255089
$760$	11.1517	0.404516
$761$	−32.3991	−1.17447	−0.587233	−	0.809418i	$-0.699783\pi$
$761$	−32.3991	−1.17447	−0.587233	+	0.809418i	$0.699783\pi$
$762$	19.0185	0.688969
$763$	−37.3248	−1.35125
$764$	−63.3346	−2.29137
$765$	1.81878	0.0657582
$766$	−13.9791	−0.505087
$767$	−60.5992	−2.18811
$768$	130.353	4.70370
$769$	29.2336	1.05419	0.527096	−	0.849806i	$-0.323281\pi$
$769$	29.2336	1.05419	0.527096	+	0.849806i	$0.323281\pi$
$770$	12.1732	0.438691
$771$	2.53435	0.0912725
$772$	−132.440	−4.76663
$773$	1.06547	0.0383222	0.0191611	−	0.999816i	$-0.493900\pi$
$773$	1.06547	0.0383222	0.0191611	+	0.999816i	$0.493900\pi$
$774$	14.1434	0.508374
$775$	6.92133	0.248621
$776$	0.303087	0.0108802
$777$	39.6432	1.42219
$778$	−13.7686	−0.493630
$779$	−0.562568	−0.0201561
$780$	25.9409	0.928833
$781$	−13.1843	−0.471773
$782$	1.80366	0.0644989
$783$	18.4304	0.658649
$784$	227.290	8.11749
$785$	19.9239	0.711116
$786$	16.2335	0.579029
$787$	8.25293	0.294185	0.147093	−	0.989123i	$-0.453008\pi$
$787$	8.25293	0.294185	0.147093	+	0.989123i	$0.453008\pi$
$788$	35.3589	1.25961
$789$	9.95098	0.354264
$790$	0.496289	0.0176572
$791$	43.4664	1.54549
$792$	22.7764	0.809323
$793$	−54.5326	−1.93651
$794$	−13.2061	−0.468665
$795$	−4.70851	−0.166994
$796$	−5.55105	−0.196752
$797$	47.0444	1.66640	0.833200	−	0.552972i	$-0.186506\pi$
$797$	47.0444	1.66640	0.833200	+	0.552972i	$0.186506\pi$
$798$	−11.9123	−0.421690
$799$	5.45813	0.193095
$800$	−32.8137	−1.16014
$801$	19.9550	0.705076
$802$	18.7289	0.661342
$803$	4.19184	0.147927
$804$	−50.7449	−1.78963
$805$	−3.09978	−0.109253
$806$	−86.9088	−3.06123
$807$	−5.07045	−0.178488
$808$	−161.068	−5.66637
$809$	49.4389	1.73818	0.869089	−	0.494655i	$-0.164706\pi$
$809$	49.4389	1.73818	0.869089	+	0.494655i	$0.164706\pi$
$810$	−3.66259	−0.128690
$811$	4.56930	0.160450	0.0802249	−	0.996777i	$-0.474436\pi$
$811$	4.56930	0.160450	0.0802249	+	0.996777i	$0.474436\pi$
$812$	95.9912	3.36863
$813$	−11.5868	−0.406368
$814$	26.4707	0.927797
$815$	7.21157	0.252610
$816$	−17.0302	−0.596177
$817$	−2.45537	−0.0859024
$818$	14.9097	0.521305
$819$	39.2492	1.37148
$820$	3.34958	0.116972
$821$	22.3891	0.781384	0.390692	−	0.920521i	$-0.372236\pi$
$821$	22.3891	0.781384	0.390692	+	0.920521i	$0.372236\pi$
$822$	−46.4871	−1.62143
$823$	0.409630	0.0142788	0.00713940	−	0.999975i	$-0.497727\pi$
$823$	0.409630	0.0142788	0.00713940	+	0.999975i	$0.497727\pi$
$824$	−107.045	−3.72908
$825$	0.978567	0.0340693
$826$	−165.688	−5.76504
$827$	−39.3486	−1.36829	−0.684143	−	0.729348i	$-0.739823\pi$
$827$	−39.3486	−1.36829	−0.684143	+	0.729348i	$0.739823\pi$
$828$	−8.73333	−0.303504
$829$	25.0834	0.871182	0.435591	−	0.900145i	$-0.356539\pi$
$829$	25.0834	0.871182	0.435591	+	0.900145i	$0.356539\pi$
$830$	19.7710	0.686263
$831$	−19.0064	−0.659325
$832$	238.011	8.25153
$833$	−10.3568	−0.358843
$834$	53.0396	1.83661
$835$	10.3106	0.356813
$836$	−5.95409	−0.205927
$837$	−34.1521	−1.18047
$838$	16.7577	0.578885
$839$	16.8253	0.580873	0.290437	−	0.956894i	$-0.406199\pi$
$839$	16.8253	0.580873	0.290437	+	0.956894i	$0.406199\pi$
$840$	47.1022	1.62518
$841$	−15.0487	−0.518922
$842$	100.718	3.47099
$843$	29.2791	1.00843
$844$	43.4479	1.49554
$845$	6.82248	0.234701
$846$	−35.3056	−1.21383
$847$	−4.31627	−0.148309
$848$	−94.0341	−3.22914
$849$	0.743854	0.0255290
$850$	2.51150	0.0861439
$851$	−6.74049	−0.231061
$852$	−76.8182	−2.63175
$853$	−9.95648	−0.340903	−0.170452	−	0.985366i	$-0.554523\pi$
$853$	−9.95648	−0.340903	−0.170452	+	0.985366i	$0.554523\pi$
$854$	−149.101	−5.10215
$855$	2.04241	0.0698489
$856$	7.89490	0.269842
$857$	−44.0130	−1.50346	−0.751728	−	0.659473i	$-0.770779\pi$
$857$	−44.0130	−1.50346	−0.751728	+	0.659473i	$0.770779\pi$
$858$	−12.2875	−0.419489
$859$	−22.1119	−0.754448	−0.377224	−	0.926122i	$-0.623121\pi$
$859$	−22.1119	−0.754448	−0.377224	+	0.926122i	$0.623121\pi$
$860$	14.6195	0.498520
$861$	−2.37615	−0.0809790
$862$	42.4500	1.44585
$863$	25.0369	0.852266	0.426133	−	0.904661i	$-0.359876\pi$
$863$	25.0369	0.852266	0.426133	+	0.904661i	$0.359876\pi$
$864$	161.914	5.50842
$865$	7.74527	0.263347
$866$	36.2937	1.23331
$867$	−15.8596	−0.538621
$868$	−177.875	−6.03746
$869$	−0.175970	−0.00596938
$870$	10.3084	0.349488
$871$	−38.7762	−1.31388
$872$	−96.4341	−3.26567
$873$	0.0555096	0.00187872
$874$	2.02543	0.0685112
$875$	−4.31627	−0.145917
$876$	24.4236	0.825199
$877$	10.4254	0.352040	0.176020	−	0.984387i	$-0.443678\pi$
$877$	10.4254	0.352040	0.176020	+	0.984387i	$0.443678\pi$
$878$	−6.30807	−0.212887
$879$	−18.2355	−0.615069
$880$	19.5430	0.658795
$881$	13.7326	0.462663	0.231331	−	0.972875i	$-0.425692\pi$
$881$	13.7326	0.462663	0.231331	+	0.972875i	$0.425692\pi$
$882$	66.9924	2.25575
$883$	−17.9267	−0.603283	−0.301641	−	0.953421i	$-0.597535\pi$
$883$	−17.9267	−0.603283	−0.301641	+	0.953421i	$0.597535\pi$
$884$	−23.6066	−0.793975
$885$	−13.3192	−0.447720
$886$	91.8636	3.08622
$887$	−8.02368	−0.269409	−0.134704	−	0.990886i	$-0.543009\pi$
$887$	−8.02368	−0.269409	−0.134704	+	0.990886i	$0.543009\pi$
$888$	102.424	3.43713
$889$	29.7441	0.997585
$890$	27.5553	0.923656
$891$	1.29865	0.0435065
$892$	−159.724	−5.34796
$893$	6.12922	0.205107
$894$	16.6988	0.558490
$895$	−11.1750	−0.373539
$896$	367.495	12.2771
$897$	3.12890	0.104471
$898$	−81.8815	−2.73242
$899$	−25.8521	−0.862217
$900$	−12.1607	−0.405356
$901$	4.28481	0.142748
$902$	−1.58661	−0.0528283
$903$	−10.3709	−0.345121
$904$	112.302	3.73511
$905$	−4.54606	−0.151116
$906$	27.7448	0.921759
$907$	−4.52612	−0.150287	−0.0751437	−	0.997173i	$-0.523942\pi$
$907$	−4.52612	−0.150287	−0.0751437	+	0.997173i	$0.523942\pi$
$908$	−48.6410	−1.61421
$909$	−29.4992	−0.978428
$910$	54.1980	1.79665
$911$	39.4569	1.30727	0.653633	−	0.756812i	$-0.273244\pi$
$911$	39.4569	1.30727	0.653633	+	0.756812i	$0.273244\pi$
$912$	−19.1241	−0.633264
$913$	−7.01026	−0.232006
$914$	85.3473	2.82304
$915$	−11.9858	−0.396239
$916$	57.5012	1.89989
$917$	25.3884	0.838398
$918$	−12.3926	−0.409016
$919$	−35.7313	−1.17867	−0.589333	−	0.807890i	$-0.700609\pi$
$919$	−35.7313	−1.17867	−0.589333	+	0.807890i	$0.700609\pi$
$920$	−8.00873	−0.264040
$921$	10.0509	0.331187
$922$	27.9721	0.921212
$923$	−58.6999	−1.93213
$924$	−25.1487	−0.827330
$925$	−9.38577	−0.308602
$926$	102.097	3.35511
$927$	−19.6049	−0.643910
$928$	122.564	4.02336
$929$	−30.5358	−1.00185	−0.500923	−	0.865492i	$-0.667006\pi$
$929$	−30.5358	−1.00185	−0.500923	+	0.865492i	$0.667006\pi$
$930$	−19.1018	−0.626374
$931$	−11.6302	−0.381165
$932$	47.5999	1.55919
$933$	19.3665	0.634029
$934$	−60.2948	−1.97291
$935$	−0.890509	−0.0291228
$936$	101.406	3.31456
$937$	−23.9640	−0.782869	−0.391434	−	0.920206i	$-0.628021\pi$
$937$	−23.9640	−0.782869	−0.391434	+	0.920206i	$0.628021\pi$
$938$	−106.021	−3.46170
$939$	10.3000	0.336128
$940$	−36.4940	−1.19030
$941$	−1.57323	−0.0512857	−0.0256428	−	0.999671i	$-0.508163\pi$
$941$	−1.57323	−0.0512857	−0.0256428	+	0.999671i	$0.508163\pi$
$942$	−54.9871	−1.79158
$943$	0.404014	0.0131565
$944$	−265.999	−8.65752
$945$	21.2979	0.692822
$946$	−6.92487	−0.225147
$947$	10.6631	0.346503	0.173252	−	0.984878i	$-0.444573\pi$
$947$	10.6631	0.346503	0.173252	+	0.984878i	$0.444573\pi$
$948$	−1.02529	−0.0332997
$949$	18.6631	0.605830
$950$	2.82030	0.0915026
$951$	21.7603	0.705625
$952$	−42.8637	−1.38922
$953$	22.7824	0.737995	0.368998	−	0.929430i	$-0.379701\pi$
$953$	22.7824	0.737995	0.368998	+	0.929430i	$0.379701\pi$
$954$	−27.7160	−0.897340
$955$	−10.6372	−0.344210
$956$	−78.4132	−2.53606
$957$	−3.65508	−0.118152
$958$	0.148135	0.00478604
$959$	−72.7037	−2.34772
$960$	52.3127	1.68839
$961$	16.9048	0.545316
$962$	117.854	3.79976
$963$	1.44593	0.0465944
$964$	128.475	4.13791
$965$	−22.2436	−0.716046
$966$	8.55493	0.275250
$967$	−34.2273	−1.10067	−0.550337	−	0.834942i	$-0.685501\pi$
$967$	−34.2273	−1.10067	−0.550337	+	0.834942i	$0.685501\pi$
$968$	−11.1517	−0.358430
$969$	0.871423	0.0279941
$970$	0.0766516	0.00246113
$971$	46.9432	1.50648	0.753239	−	0.657747i	$-0.228490\pi$
$971$	46.9432	1.50648	0.753239	+	0.657747i	$0.228490\pi$
$972$	95.7049	3.06974
$973$	82.9514	2.65930
$974$	−5.90933	−0.189347
$975$	4.35682	0.139530
$976$	−239.370	−7.66204
$977$	27.7993	0.889380	0.444690	−	0.895685i	$-0.353314\pi$
$977$	27.7993	0.889380	0.444690	+	0.895685i	$0.353314\pi$
$978$	−19.9029	−0.636423
$979$	−9.77035	−0.312262
$980$	69.2474	2.21203
$981$	−17.6616	−0.563893
$982$	83.7722	2.67328
$983$	−55.2447	−1.76203	−0.881017	−	0.473085i	$-0.843140\pi$
$983$	−55.2447	−1.76203	−0.881017	+	0.473085i	$0.843140\pi$
$984$	−6.13913	−0.195708
$985$	5.93859	0.189219
$986$	−9.38081	−0.298746
$987$	25.8884	0.824036
$988$	−26.5091	−0.843366
$989$	1.76335	0.0560712
$990$	5.76020	0.183071
$991$	−17.1535	−0.544900	−0.272450	−	0.962170i	$-0.587834\pi$
$991$	−17.1535	−0.544900	−0.272450	+	0.962170i	$0.587834\pi$
$992$	−227.115	−7.21090
$993$	8.21778	0.260783
$994$	−160.495	−5.09061
$995$	−0.932308	−0.0295561
$996$	−40.8451	−1.29423
$997$	29.7599	0.942504	0.471252	−	0.881999i	$-0.343802\pi$
$997$	29.7599	0.942504	0.471252	+	0.881999i	$0.343802\pi$
$998$	8.28388	0.262222
$999$	46.3125	1.46526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.i.1.1	✓	8
3.2	odd	2		9405.2.a.bf.1.8		8
5.4	even	2		5225.2.a.o.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.i.1.1	✓	8	1.1	even	1	trivial
5225.2.a.o.1.8		8	5.4	even	2
9405.2.a.bf.1.8		8	3.2	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82030 q^{2} +0.978567 q^{3} +5.95409 q^{4} +1.00000 q^{5} -2.75985 q^{6} -4.31627 q^{7} -11.1517 q^{8} -2.04241 q^{9} +O(q^{10})\)	\(q-2.82030 q^{2} +0.978567 q^{3} +5.95409 q^{4} +1.00000 q^{5} -2.75985 q^{6} -4.31627 q^{7} -11.1517 q^{8} -2.04241 q^{9} -2.82030 q^{10} +1.00000 q^{11} +5.82647 q^{12} +4.45224 q^{13} +12.1732 q^{14} +0.978567 q^{15} +19.5430 q^{16} -0.890509 q^{17} +5.76020 q^{18} -1.00000 q^{19} +5.95409 q^{20} -4.22376 q^{21} -2.82030 q^{22} +0.718161 q^{23} -10.9127 q^{24} +1.00000 q^{25} -12.5567 q^{26} -4.93433 q^{27} -25.6995 q^{28} -3.73514 q^{29} -2.75985 q^{30} +6.92133 q^{31} -32.8137 q^{32} +0.978567 q^{33} +2.51150 q^{34} -4.31627 q^{35} -12.1607 q^{36} -9.38577 q^{37} +2.82030 q^{38} +4.35682 q^{39} -11.1517 q^{40} +0.562568 q^{41} +11.9123 q^{42} +2.45537 q^{43} +5.95409 q^{44} -2.04241 q^{45} -2.02543 q^{46} -6.12922 q^{47} +19.1241 q^{48} +11.6302 q^{49} -2.82030 q^{50} -0.871423 q^{51} +26.5091 q^{52} -4.81164 q^{53} +13.9163 q^{54} +1.00000 q^{55} +48.1339 q^{56} -0.978567 q^{57} +10.5342 q^{58} -13.6109 q^{59} +5.82647 q^{60} -12.2483 q^{61} -19.5202 q^{62} +8.81559 q^{63} +53.4585 q^{64} +4.45224 q^{65} -2.75985 q^{66} -8.70936 q^{67} -5.30217 q^{68} +0.702768 q^{69} +12.1732 q^{70} -13.1843 q^{71} +22.7764 q^{72} +4.19184 q^{73} +26.4707 q^{74} +0.978567 q^{75} -5.95409 q^{76} -4.31627 q^{77} -12.2875 q^{78} -0.175970 q^{79} +19.5430 q^{80} +1.29865 q^{81} -1.58661 q^{82} -7.01026 q^{83} -25.1487 q^{84} -0.890509 q^{85} -6.92487 q^{86} -3.65508 q^{87} -11.1517 q^{88} -9.77035 q^{89} +5.76020 q^{90} -19.2171 q^{91} +4.27600 q^{92} +6.77298 q^{93} +17.2862 q^{94} -1.00000 q^{95} -32.1104 q^{96} -0.0271785 q^{97} -32.8007 q^{98} -2.04241 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 17 q^{13} + 12 q^{14} - 7 q^{15} + 18 q^{16} - 9 q^{17} - 2 q^{18} - 8 q^{19} + 10 q^{20} + q^{21} - 6 q^{22} - 8 q^{23} + q^{24} + 8 q^{25} + 10 q^{26} - 34 q^{27} - 22 q^{28} - 3 q^{29} - q^{31} - 37 q^{32} - 7 q^{33} - 8 q^{34} - 11 q^{35} + 30 q^{36} - 17 q^{37} + 6 q^{38} + 14 q^{39} - 18 q^{40} - 5 q^{41} + 15 q^{42} - 21 q^{43} + 10 q^{44} + 11 q^{45} - 2 q^{46} - 8 q^{47} + 10 q^{48} + 19 q^{49} - 6 q^{50} - 16 q^{51} + 9 q^{52} - 19 q^{53} - 3 q^{54} + 8 q^{55} + 24 q^{56} + 7 q^{57} + 37 q^{58} - 33 q^{59} - 7 q^{60} - q^{61} - 42 q^{62} - 20 q^{63} + 48 q^{64} - 17 q^{65} - 18 q^{67} - 37 q^{68} + 16 q^{69} + 12 q^{70} - 18 q^{71} + 13 q^{72} - 18 q^{73} + 15 q^{74} - 7 q^{75} - 10 q^{76} - 11 q^{77} - 51 q^{78} - 5 q^{79} + 18 q^{80} + 32 q^{81} + 12 q^{82} - 33 q^{83} - 51 q^{84} - 9 q^{85} - 16 q^{86} - 26 q^{87} - 18 q^{88} - 20 q^{89} - 2 q^{90} + 6 q^{91} - 3 q^{92} + 18 q^{93} + 30 q^{94} - 8 q^{95} + 21 q^{96} - 69 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 17 * q^13 + 12 * q^14 - 7 * q^15 + 18 * q^16 - 9 * q^17 - 2 * q^18 - 8 * q^19 + 10 * q^20 + q^21 - 6 * q^22 - 8 * q^23 + q^24 + 8 * q^25 + 10 * q^26 - 34 * q^27 - 22 * q^28 - 3 * q^29 - q^31 - 37 * q^32 - 7 * q^33 - 8 * q^34 - 11 * q^35 + 30 * q^36 - 17 * q^37 + 6 * q^38 + 14 * q^39 - 18 * q^40 - 5 * q^41 + 15 * q^42 - 21 * q^43 + 10 * q^44 + 11 * q^45 - 2 * q^46 - 8 * q^47 + 10 * q^48 + 19 * q^49 - 6 * q^50 - 16 * q^51 + 9 * q^52 - 19 * q^53 - 3 * q^54 + 8 * q^55 + 24 * q^56 + 7 * q^57 + 37 * q^58 - 33 * q^59 - 7 * q^60 - q^61 - 42 * q^62 - 20 * q^63 + 48 * q^64 - 17 * q^65 - 18 * q^67 - 37 * q^68 + 16 * q^69 + 12 * q^70 - 18 * q^71 + 13 * q^72 - 18 * q^73 + 15 * q^74 - 7 * q^75 - 10 * q^76 - 11 * q^77 - 51 * q^78 - 5 * q^79 + 18 * q^80 + 32 * q^81 + 12 * q^82 - 33 * q^83 - 51 * q^84 - 9 * q^85 - 16 * q^86 - 26 * q^87 - 18 * q^88 - 20 * q^89 - 2 * q^90 + 6 * q^91 - 3 * q^92 + 18 * q^93 + 30 * q^94 - 8 * q^95 + 21 * q^96 - 69 * q^98 + 11 * q^99

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