LMFDB - Embedded newform 6045.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.230224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 3x - 2 $ x^5 - 2x^4 - 4x^3 + 6x^2 + 3x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.49592$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.25814 q^{2} +1.00000 q^{3} -0.417093 q^{4} +1.00000 q^{5} -1.25814 q^{6} -1.35765 q^{7} +3.04103 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.25814 q^{2} +1.00000 q^{3} -0.417093 q^{4} +1.00000 q^{5} -1.25814 q^{6} -1.35765 q^{7} +3.04103 q^{8} +1.00000 q^{9} -1.25814 q^{10} -0.345120 q^{11} -0.417093 q^{12} -1.00000 q^{13} +1.70811 q^{14} +1.00000 q^{15} -2.99185 q^{16} -6.69591 q^{17} -1.25814 q^{18} +1.57605 q^{19} -0.417093 q^{20} -1.35765 q^{21} +0.434207 q^{22} +5.40273 q^{23} +3.04103 q^{24} +1.00000 q^{25} +1.25814 q^{26} +1.00000 q^{27} +0.566267 q^{28} +0.119863 q^{29} -1.25814 q^{30} -1.00000 q^{31} -2.31791 q^{32} -0.345120 q^{33} +8.42437 q^{34} -1.35765 q^{35} -0.417093 q^{36} -3.44803 q^{37} -1.98289 q^{38} -1.00000 q^{39} +3.04103 q^{40} +1.58420 q^{41} +1.70811 q^{42} +7.03352 q^{43} +0.143947 q^{44} +1.00000 q^{45} -6.79737 q^{46} +2.35846 q^{47} -2.99185 q^{48} -5.15679 q^{49} -1.25814 q^{50} -6.69591 q^{51} +0.417093 q^{52} -2.33098 q^{53} -1.25814 q^{54} -0.345120 q^{55} -4.12866 q^{56} +1.57605 q^{57} -0.150804 q^{58} +11.2766 q^{59} -0.417093 q^{60} -10.6280 q^{61} +1.25814 q^{62} -1.35765 q^{63} +8.89995 q^{64} -1.00000 q^{65} +0.434207 q^{66} +1.27719 q^{67} +2.79282 q^{68} +5.40273 q^{69} +1.70811 q^{70} +2.26693 q^{71} +3.04103 q^{72} -9.90714 q^{73} +4.33810 q^{74} +1.00000 q^{75} -0.657360 q^{76} +0.468552 q^{77} +1.25814 q^{78} +3.65244 q^{79} -2.99185 q^{80} +1.00000 q^{81} -1.99314 q^{82} -11.9725 q^{83} +0.566267 q^{84} -6.69591 q^{85} -8.84913 q^{86} +0.119863 q^{87} -1.04952 q^{88} -14.5357 q^{89} -1.25814 q^{90} +1.35765 q^{91} -2.25344 q^{92} -1.00000 q^{93} -2.96726 q^{94} +1.57605 q^{95} -2.31791 q^{96} -15.1206 q^{97} +6.48794 q^{98} -0.345120 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 5 * q^9	$ 5 q + 5 q^{3} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 5 q^{9} - 9 q^{11} + 6 q^{12} - 5 q^{13} - 26 q^{14} + 5 q^{15} - 4 q^{16} - 17 q^{17} - 2 q^{19} + 6 q^{20} - q^{21} + 8 q^{22} - 17 q^{23} + 6 q^{24} + 5 q^{25} + 5 q^{27} - 16 q^{28} - 6 q^{29} - 5 q^{31} - 8 q^{32} - 9 q^{33} - 16 q^{34} - q^{35} + 6 q^{36} - 3 q^{37} + 4 q^{38} - 5 q^{39} + 6 q^{40} + 9 q^{41} - 26 q^{42} + 8 q^{43} - 30 q^{44} + 5 q^{45} - 4 q^{46} - 2 q^{47} - 4 q^{48} + 30 q^{49} - 17 q^{51} - 6 q^{52} - 51 q^{53} - 9 q^{55} - 18 q^{56} - 2 q^{57} + 12 q^{58} - 6 q^{59} + 6 q^{60} - 23 q^{61} - q^{63} - 12 q^{64} - 5 q^{65} + 8 q^{66} + 2 q^{68} - 17 q^{69} - 26 q^{70} + 9 q^{71} + 6 q^{72} - 16 q^{73} + 12 q^{74} + 5 q^{75} - 38 q^{76} - 13 q^{77} + 21 q^{79} - 4 q^{80} + 5 q^{81} + 8 q^{82} - 44 q^{83} - 16 q^{84} - 17 q^{85} + 12 q^{86} - 6 q^{87} - 12 q^{88} - 65 q^{89} + q^{91} - 34 q^{92} - 5 q^{93} - 24 q^{94} - 2 q^{95} - 8 q^{96} + 11 q^{97} + 22 q^{98} - 9 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 5 * q^9 - 9 * q^11 + 6 * q^12 - 5 * q^13 - 26 * q^14 + 5 * q^15 - 4 * q^16 - 17 * q^17 - 2 * q^19 + 6 * q^20 - q^21 + 8 * q^22 - 17 * q^23 + 6 * q^24 + 5 * q^25 + 5 * q^27 - 16 * q^28 - 6 * q^29 - 5 * q^31 - 8 * q^32 - 9 * q^33 - 16 * q^34 - q^35 + 6 * q^36 - 3 * q^37 + 4 * q^38 - 5 * q^39 + 6 * q^40 + 9 * q^41 - 26 * q^42 + 8 * q^43 - 30 * q^44 + 5 * q^45 - 4 * q^46 - 2 * q^47 - 4 * q^48 + 30 * q^49 - 17 * q^51 - 6 * q^52 - 51 * q^53 - 9 * q^55 - 18 * q^56 - 2 * q^57 + 12 * q^58 - 6 * q^59 + 6 * q^60 - 23 * q^61 - q^63 - 12 * q^64 - 5 * q^65 + 8 * q^66 + 2 * q^68 - 17 * q^69 - 26 * q^70 + 9 * q^71 + 6 * q^72 - 16 * q^73 + 12 * q^74 + 5 * q^75 - 38 * q^76 - 13 * q^77 + 21 * q^79 - 4 * q^80 + 5 * q^81 + 8 * q^82 - 44 * q^83 - 16 * q^84 - 17 * q^85 + 12 * q^86 - 6 * q^87 - 12 * q^88 - 65 * q^89 + q^91 - 34 * q^92 - 5 * q^93 - 24 * q^94 - 2 * q^95 - 8 * q^96 + 11 * q^97 + 22 * q^98 - 9 * 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 $

$2$	−1.25814	−0.889637	−0.444818	−	0.895621i	$-0.646732\pi$
$2$	−1.25814	−0.889637	−0.444818	+	0.895621i	$0.646732\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.417093	−0.208547
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.25814	−0.513632
$7$	−1.35765	−0.513144	−0.256572	−	0.966525i	$-0.582593\pi$
$7$	−1.35765	−0.513144	−0.256572	+	0.966525i	$0.582593\pi$
$8$	3.04103	1.07517
$9$	1.00000	0.333333
$10$	−1.25814	−0.397858
$11$	−0.345120	−0.104057	−0.0520287	−	0.998646i	$-0.516569\pi$
$11$	−0.345120	−0.104057	−0.0520287	+	0.998646i	$0.516569\pi$
$12$	−0.417093	−0.120404
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.70811	0.456511
$15$	1.00000	0.258199
$16$	−2.99185	−0.747962
$17$	−6.69591	−1.62400	−0.811999	−	0.583659i	$-0.801620\pi$
$17$	−6.69591	−1.62400	−0.811999	+	0.583659i	$0.801620\pi$
$18$	−1.25814	−0.296546
$19$	1.57605	0.361571	0.180785	−	0.983523i	$-0.442136\pi$
$19$	1.57605	0.361571	0.180785	+	0.983523i	$0.442136\pi$
$20$	−0.417093	−0.0932649
$21$	−1.35765	−0.296264
$22$	0.434207	0.0925733
$23$	5.40273	1.12655	0.563273	−	0.826271i	$-0.309542\pi$
$23$	5.40273	1.12655	0.563273	+	0.826271i	$0.309542\pi$
$24$	3.04103	0.620748
$25$	1.00000	0.200000
$26$	1.25814	0.246741
$27$	1.00000	0.192450
$28$	0.566267	0.107014
$29$	0.119863	0.0222580	0.0111290	−	0.999938i	$-0.496457\pi$
$29$	0.119863	0.0222580	0.0111290	+	0.999938i	$0.496457\pi$
$30$	−1.25814	−0.229703
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−2.31791	−0.409753
$33$	−0.345120	−0.0600776
$34$	8.42437	1.44477
$35$	−1.35765	−0.229485
$36$	−0.417093	−0.0695155
$37$	−3.44803	−0.566853	−0.283427	−	0.958994i	$-0.591471\pi$
$37$	−3.44803	−0.566853	−0.283427	+	0.958994i	$0.591471\pi$
$38$	−1.98289	−0.321667
$39$	−1.00000	−0.160128
$40$	3.04103	0.480829
$41$	1.58420	0.247411	0.123705	−	0.992319i	$-0.460522\pi$
$41$	1.58420	0.247411	0.123705	+	0.992319i	$0.460522\pi$
$42$	1.70811	0.263567
$43$	7.03352	1.07260	0.536301	−	0.844027i	$-0.319821\pi$
$43$	7.03352	1.07260	0.536301	+	0.844027i	$0.319821\pi$
$44$	0.143947	0.0217008
$45$	1.00000	0.149071
$46$	−6.79737	−1.00222
$47$	2.35846	0.344017	0.172008	−	0.985096i	$-0.444974\pi$
$47$	2.35846	0.344017	0.172008	+	0.985096i	$0.444974\pi$
$48$	−2.99185	−0.431836
$49$	−5.15679	−0.736684
$50$	−1.25814	−0.177927
$51$	−6.69591	−0.937615
$52$	0.417093	0.0578404
$53$	−2.33098	−0.320185	−0.160093	−	0.987102i	$-0.551179\pi$
$53$	−2.33098	−0.320185	−0.160093	+	0.987102i	$0.551179\pi$
$54$	−1.25814	−0.171211
$55$	−0.345120	−0.0465359
$56$	−4.12866	−0.551715
$57$	1.57605	0.208753
$58$	−0.150804	−0.0198015
$59$	11.2766	1.46809	0.734047	−	0.679098i	$-0.237629\pi$
$59$	11.2766	1.46809	0.734047	+	0.679098i	$0.237629\pi$
$60$	−0.417093	−0.0538465
$61$	−10.6280	−1.36077	−0.680387	−	0.732853i	$-0.738188\pi$
$61$	−10.6280	−1.36077	−0.680387	+	0.732853i	$0.738188\pi$
$62$	1.25814	0.159783
$63$	−1.35765	−0.171048
$64$	8.89995	1.11249
$65$	−1.00000	−0.124035
$66$	0.434207	0.0534472
$67$	1.27719	0.156034	0.0780168	−	0.996952i	$-0.475141\pi$
$67$	1.27719	0.156034	0.0780168	+	0.996952i	$0.475141\pi$
$68$	2.79282	0.338679
$69$	5.40273	0.650412
$70$	1.70811	0.204158
$71$	2.26693	0.269035	0.134518	−	0.990911i	$-0.457052\pi$
$71$	2.26693	0.269035	0.134518	+	0.990911i	$0.457052\pi$
$72$	3.04103	0.358389
$73$	−9.90714	−1.15954	−0.579771	−	0.814779i	$-0.696858\pi$
$73$	−9.90714	−1.15954	−0.579771	+	0.814779i	$0.696858\pi$
$74$	4.33810	0.504294
$75$	1.00000	0.115470
$76$	−0.657360	−0.0754043
$77$	0.468552	0.0533964
$78$	1.25814	0.142456
$79$	3.65244	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$79$	3.65244	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$80$	−2.99185	−0.334499
$81$	1.00000	0.111111
$82$	−1.99314	−0.220106
$83$	−11.9725	−1.31415	−0.657074	−	0.753826i	$-0.728206\pi$
$83$	−11.9725	−1.31415	−0.657074	+	0.753826i	$0.728206\pi$
$84$	0.566267	0.0617847
$85$	−6.69591	−0.726274
$86$	−8.84913	−0.954226
$87$	0.119863	0.0128507
$88$	−1.04952	−0.111879
$89$	−14.5357	−1.54078	−0.770388	−	0.637575i	$-0.779938\pi$
$89$	−14.5357	−1.54078	−0.770388	+	0.637575i	$0.779938\pi$
$90$	−1.25814	−0.132619
$91$	1.35765	0.142320
$92$	−2.25344	−0.234937
$93$	−1.00000	−0.103695
$94$	−2.96726	−0.306050
$95$	1.57605	0.161699
$96$	−2.31791	−0.236571
$97$	−15.1206	−1.53526	−0.767632	−	0.640891i	$-0.778565\pi$
$97$	−15.1206	−1.53526	−0.767632	+	0.640891i	$0.778565\pi$
$98$	6.48794	0.655381
$99$	−0.345120	−0.0346858
$100$	−0.417093	−0.0417093
$101$	4.71626	0.469286	0.234643	−	0.972082i	$-0.424608\pi$
$101$	4.71626	0.469286	0.234643	+	0.972082i	$0.424608\pi$
$102$	8.42437	0.834137
$103$	−4.28728	−0.422438	−0.211219	−	0.977439i	$-0.567743\pi$
$103$	−4.28728	−0.422438	−0.211219	+	0.977439i	$0.567743\pi$
$104$	−3.04103	−0.298198
$105$	−1.35765	−0.132493
$106$	2.93270	0.284849
$107$	13.9235	1.34604	0.673019	−	0.739625i	$-0.264997\pi$
$107$	13.9235	1.34604	0.673019	+	0.739625i	$0.264997\pi$
$108$	−0.417093	−0.0401348
$109$	11.2620	1.07871	0.539353	−	0.842080i	$-0.318669\pi$
$109$	11.2620	1.07871	0.539353	+	0.842080i	$0.318669\pi$
$110$	0.434207	0.0414001
$111$	−3.44803	−0.327273
$112$	4.06188	0.383812
$113$	−3.23351	−0.304183	−0.152092	−	0.988366i	$-0.548601\pi$
$113$	−3.23351	−0.304183	−0.152092	+	0.988366i	$0.548601\pi$
$114$	−1.98289	−0.185714
$115$	5.40273	0.503807
$116$	−0.0499940	−0.00464183
$117$	−1.00000	−0.0924500
$118$	−14.1876	−1.30607
$119$	9.09071	0.833344
$120$	3.04103	0.277607
$121$	−10.8809	−0.989172
$122$	13.3715	1.21059
$123$	1.58420	0.142843
$124$	0.417093	0.0374561
$125$	1.00000	0.0894427
$126$	1.70811	0.152170
$127$	−14.1644	−1.25689	−0.628445	−	0.777854i	$-0.716308\pi$
$127$	−14.1644	−1.25689	−0.628445	+	0.777854i	$0.716308\pi$
$128$	−6.56152	−0.579962
$129$	7.03352	0.619267
$130$	1.25814	0.110346
$131$	−5.81171	−0.507771	−0.253886	−	0.967234i	$-0.581709\pi$
$131$	−5.81171	−0.507771	−0.253886	+	0.967234i	$0.581709\pi$
$132$	0.143947	0.0125290
$133$	−2.13972	−0.185538
$134$	−1.60688	−0.138813
$135$	1.00000	0.0860663
$136$	−20.3625	−1.74607
$137$	−14.5828	−1.24590	−0.622948	−	0.782263i	$-0.714065\pi$
$137$	−14.5828	−1.24590	−0.622948	+	0.782263i	$0.714065\pi$
$138$	−6.79737	−0.578630
$139$	11.7019	0.992542	0.496271	−	0.868168i	$-0.334702\pi$
$139$	11.7019	0.992542	0.496271	+	0.868168i	$0.334702\pi$
$140$	0.566267	0.0478583
$141$	2.35846	0.198618
$142$	−2.85211	−0.239344
$143$	0.345120	0.0288604
$144$	−2.99185	−0.249321
$145$	0.119863	0.00995408
$146$	12.4645	1.03157
$147$	−5.15679	−0.425325
$148$	1.43815	0.118215
$149$	−7.58040	−0.621011	−0.310505	−	0.950572i	$-0.600498\pi$
$149$	−7.58040	−0.621011	−0.310505	+	0.950572i	$0.600498\pi$
$150$	−1.25814	−0.102726
$151$	4.13400	0.336420	0.168210	−	0.985751i	$-0.446201\pi$
$151$	4.13400	0.336420	0.168210	+	0.985751i	$0.446201\pi$
$152$	4.79282	0.388749
$153$	−6.69591	−0.541332
$154$	−0.589502	−0.0475034
$155$	−1.00000	−0.0803219
$156$	0.417093	0.0333942
$157$	12.6022	1.00577	0.502883	−	0.864354i	$-0.332273\pi$
$157$	12.6022	1.00577	0.502883	+	0.864354i	$0.332273\pi$
$158$	−4.59527	−0.365580
$159$	−2.33098	−0.184859
$160$	−2.31791	−0.183247
$161$	−7.33501	−0.578080
$162$	−1.25814	−0.0988485
$163$	9.48867	0.743210	0.371605	−	0.928391i	$-0.378808\pi$
$163$	9.48867	0.743210	0.371605	+	0.928391i	$0.378808\pi$
$164$	−0.660760	−0.0515967
$165$	−0.345120	−0.0268675
$166$	15.0630	1.16911
$167$	−19.3453	−1.49698	−0.748492	−	0.663143i	$-0.769222\pi$
$167$	−19.3453	−1.49698	−0.748492	+	0.663143i	$0.769222\pi$
$168$	−4.12866	−0.318533
$169$	1.00000	0.0769231
$170$	8.42437	0.646120
$171$	1.57605	0.120524
$172$	−2.93363	−0.223687
$173$	−2.78667	−0.211867	−0.105933	−	0.994373i	$-0.533783\pi$
$173$	−2.78667	−0.211867	−0.105933	+	0.994373i	$0.533783\pi$
$174$	−0.150804	−0.0114324
$175$	−1.35765	−0.102629
$176$	1.03255	0.0778310
$177$	11.2766	0.847605
$178$	18.2878	1.37073
$179$	8.16315	0.610143	0.305071	−	0.952329i	$-0.401320\pi$
$179$	8.16315	0.610143	0.305071	+	0.952329i	$0.401320\pi$
$180$	−0.417093	−0.0310883
$181$	−12.3242	−0.916051	−0.458025	−	0.888939i	$-0.651443\pi$
$181$	−12.3242	−0.916051	−0.458025	+	0.888939i	$0.651443\pi$
$182$	−1.70811	−0.126613
$183$	−10.6280	−0.785643
$184$	16.4299	1.21123
$185$	−3.44803	−0.253505
$186$	1.25814	0.0922510
$187$	2.31089	0.168989
$188$	−0.983697	−0.0717435
$189$	−1.35765	−0.0987545
$190$	−1.98289	−0.143854
$191$	−10.8585	−0.785695	−0.392847	−	0.919604i	$-0.628510\pi$
$191$	−10.8585	−0.785695	−0.392847	+	0.919604i	$0.628510\pi$
$192$	8.89995	0.642298
$193$	16.7713	1.20722	0.603611	−	0.797279i	$-0.293728\pi$
$193$	16.7713	1.20722	0.603611	+	0.797279i	$0.293728\pi$
$194$	19.0238	1.36583
$195$	−1.00000	−0.0716115
$196$	2.15086	0.153633
$197$	22.2985	1.58870	0.794350	−	0.607461i	$-0.207812\pi$
$197$	22.2985	1.58870	0.794350	+	0.607461i	$0.207812\pi$
$198$	0.434207	0.0308578
$199$	−25.9324	−1.83830	−0.919150	−	0.393907i	$-0.871123\pi$
$199$	−25.9324	−1.83830	−0.919150	+	0.393907i	$0.871123\pi$
$200$	3.04103	0.215033
$201$	1.27719	0.0900860
$202$	−5.93370	−0.417494
$203$	−0.162732	−0.0114215
$204$	2.79282	0.195536
$205$	1.58420	0.110646
$206$	5.39398	0.375817
$207$	5.40273	0.375515
$208$	2.99185	0.207447
$209$	−0.543926	−0.0376241
$210$	1.70811	0.117871
$211$	24.4100	1.68045	0.840227	−	0.542235i	$-0.182422\pi$
$211$	24.4100	1.68045	0.840227	+	0.542235i	$0.182422\pi$
$212$	0.972238	0.0667736
$213$	2.26693	0.155328
$214$	−17.5177	−1.19749
$215$	7.03352	0.479682
$216$	3.04103	0.206916
$217$	1.35765	0.0921633
$218$	−14.1691	−0.959656
$219$	−9.90714	−0.669462
$220$	0.143947	0.00970491
$221$	6.69591	0.450416
$222$	4.33810	0.291154
$223$	−4.08272	−0.273399	−0.136700	−	0.990613i	$-0.543650\pi$
$223$	−4.08272	−0.273399	−0.136700	+	0.990613i	$0.543650\pi$
$224$	3.14692	0.210262
$225$	1.00000	0.0666667
$226$	4.06820	0.270613
$227$	12.0446	0.799427	0.399713	−	0.916640i	$-0.369110\pi$
$227$	12.0446	0.799427	0.399713	+	0.916640i	$0.369110\pi$
$228$	−0.657360	−0.0435347
$229$	16.6470	1.10006	0.550032	−	0.835144i	$-0.314616\pi$
$229$	16.6470	1.10006	0.550032	+	0.835144i	$0.314616\pi$
$230$	−6.79737	−0.448205
$231$	0.468552	0.0308284
$232$	0.364507	0.0239311
$233$	21.3869	1.40110	0.700551	−	0.713603i	$-0.252938\pi$
$233$	21.3869	1.40110	0.700551	+	0.713603i	$0.252938\pi$
$234$	1.25814	0.0822469
$235$	2.35846	0.153849
$236$	−4.70341	−0.306166
$237$	3.65244	0.237252
$238$	−11.4373	−0.741373
$239$	−23.9668	−1.55028	−0.775142	−	0.631787i	$-0.782322\pi$
$239$	−23.9668	−1.55028	−0.775142	+	0.631787i	$0.782322\pi$
$240$	−2.99185	−0.193123
$241$	−20.8191	−1.34108	−0.670538	−	0.741876i	$-0.733937\pi$
$241$	−20.8191	−1.34108	−0.670538	+	0.741876i	$0.733937\pi$
$242$	13.6896	0.880004
$243$	1.00000	0.0641500
$244$	4.43286	0.283785
$245$	−5.15679	−0.329455
$246$	−1.99314	−0.127078
$247$	−1.57605	−0.100282
$248$	−3.04103	−0.193106
$249$	−11.9725	−0.758724
$250$	−1.25814	−0.0795715
$251$	−21.9196	−1.38355	−0.691777	−	0.722111i	$-0.743172\pi$
$251$	−21.9196	−1.38355	−0.691777	+	0.722111i	$0.743172\pi$
$252$	0.566267	0.0356714
$253$	−1.86459	−0.117226
$254$	17.8208	1.11818
$255$	−6.69591	−0.419314
$256$	−9.54461	−0.596538
$257$	−26.8534	−1.67507	−0.837535	−	0.546383i	$-0.816004\pi$
$257$	−26.8534	−1.67507	−0.837535	+	0.546383i	$0.816004\pi$
$258$	−8.84913	−0.550923
$259$	4.68122	0.290877
$260$	0.417093	0.0258670
$261$	0.119863	0.00741933
$262$	7.31192	0.451732
$263$	−15.0308	−0.926837	−0.463419	−	0.886139i	$-0.653377\pi$
$263$	−15.0308	−0.926837	−0.463419	+	0.886139i	$0.653377\pi$
$264$	−1.04952	−0.0645935
$265$	−2.33098	−0.143191
$266$	2.69207	0.165061
$267$	−14.5357	−0.889568
$268$	−0.532707	−0.0325403
$269$	−25.4175	−1.54973	−0.774866	−	0.632125i	$-0.782183\pi$
$269$	−25.4175	−1.54973	−0.774866	+	0.632125i	$0.782183\pi$
$270$	−1.25814	−0.0765677
$271$	12.6156	0.766341	0.383170	−	0.923678i	$-0.374832\pi$
$271$	12.6156	0.766341	0.383170	+	0.923678i	$0.374832\pi$
$272$	20.0331	1.21469
$273$	1.35765	0.0821687
$274$	18.3472	1.10839
$275$	−0.345120	−0.0208115
$276$	−2.25344	−0.135641
$277$	−20.9698	−1.25995	−0.629977	−	0.776614i	$-0.716936\pi$
$277$	−20.9698	−1.25995	−0.629977	+	0.776614i	$0.716936\pi$
$278$	−14.7226	−0.883002
$279$	−1.00000	−0.0598684
$280$	−4.12866	−0.246735
$281$	−9.09780	−0.542729	−0.271365	−	0.962477i	$-0.587475\pi$
$281$	−9.09780	−0.542729	−0.271365	+	0.962477i	$0.587475\pi$
$282$	−2.96726	−0.176698
$283$	−4.22640	−0.251233	−0.125617	−	0.992079i	$-0.540091\pi$
$283$	−4.22640	−0.251233	−0.125617	+	0.992079i	$0.540091\pi$
$284$	−0.945522	−0.0561064
$285$	1.57605	0.0933571
$286$	−0.434207	−0.0256752
$287$	−2.15079	−0.126957
$288$	−2.31791	−0.136584
$289$	27.8352	1.63737
$290$	−0.150804	−0.00885551
$291$	−15.1206	−0.886385
$292$	4.13220	0.241819
$293$	−2.96980	−0.173497	−0.0867487	−	0.996230i	$-0.527648\pi$
$293$	−2.96980	−0.173497	−0.0867487	+	0.996230i	$0.527648\pi$
$294$	6.48794	0.378384
$295$	11.2766	0.656552
$296$	−10.4856	−0.609462
$297$	−0.345120	−0.0200259
$298$	9.53718	0.552474
$299$	−5.40273	−0.312448
$300$	−0.417093	−0.0240809
$301$	−9.54906	−0.550399
$302$	−5.20113	−0.299292
$303$	4.71626	0.270942
$304$	−4.71530	−0.270441
$305$	−10.6280	−0.608557
$306$	8.42437	0.481589
$307$	−32.2706	−1.84178	−0.920890	−	0.389822i	$-0.872537\pi$
$307$	−32.2706	−1.84178	−0.920890	+	0.389822i	$0.872537\pi$
$308$	−0.195430	−0.0111356
$309$	−4.28728	−0.243895
$310$	1.25814	0.0714573
$311$	−18.6126	−1.05543	−0.527713	−	0.849423i	$-0.676950\pi$
$311$	−18.6126	−1.05543	−0.527713	+	0.849423i	$0.676950\pi$
$312$	−3.04103	−0.172165
$313$	23.4253	1.32408	0.662038	−	0.749471i	$-0.269692\pi$
$313$	23.4253	1.32408	0.662038	+	0.749471i	$0.269692\pi$
$314$	−15.8553	−0.894767
$315$	−1.35765	−0.0764949
$316$	−1.52341	−0.0856984
$317$	−10.4490	−0.586875	−0.293437	−	0.955978i	$-0.594799\pi$
$317$	−10.4490	−0.586875	−0.293437	+	0.955978i	$0.594799\pi$
$318$	2.93270	0.164457
$319$	−0.0413671	−0.00231611
$320$	8.89995	0.497522
$321$	13.9235	0.777136
$322$	9.22844	0.514281
$323$	−10.5531	−0.587190
$324$	−0.417093	−0.0231718
$325$	−1.00000	−0.0554700
$326$	−11.9380	−0.661187
$327$	11.2620	0.622791
$328$	4.81761	0.266008
$329$	−3.20196	−0.176530
$330$	0.434207	0.0239023
$331$	19.2152	1.05616	0.528081	−	0.849194i	$-0.322912\pi$
$331$	19.2152	1.05616	0.528081	+	0.849194i	$0.322912\pi$
$332$	4.99363	0.274061
$333$	−3.44803	−0.188951
$334$	24.3390	1.33177
$335$	1.27719	0.0697803
$336$	4.06188	0.221594
$337$	6.06317	0.330282	0.165141	−	0.986270i	$-0.447192\pi$
$337$	6.06317	0.330282	0.165141	+	0.986270i	$0.447192\pi$
$338$	−1.25814	−0.0684336
$339$	−3.23351	−0.175620
$340$	2.79282	0.151462
$341$	0.345120	0.0186893
$342$	−1.98289	−0.107222
$343$	16.5047	0.891168
$344$	21.3892	1.15323
$345$	5.40273	0.290873
$346$	3.50601	0.188484
$347$	−6.30198	−0.338308	−0.169154	−	0.985590i	$-0.554104\pi$
$347$	−6.30198	−0.338308	−0.169154	+	0.985590i	$0.554104\pi$
$348$	−0.0499940	−0.00267996
$349$	19.2641	1.03118	0.515591	−	0.856835i	$-0.327573\pi$
$349$	19.2641	1.03118	0.515591	+	0.856835i	$0.327573\pi$
$350$	1.70811	0.0913023
$351$	−1.00000	−0.0533761
$352$	0.799957	0.0426379
$353$	−28.9493	−1.54082	−0.770408	−	0.637552i	$-0.779947\pi$
$353$	−28.9493	−1.54082	−0.770408	+	0.637552i	$0.779947\pi$
$354$	−14.1876	−0.754060
$355$	2.26693	0.120316
$356$	6.06272	0.321324
$357$	9.09071	0.481131
$358$	−10.2704	−0.542805
$359$	−3.83008	−0.202144	−0.101072	−	0.994879i	$-0.532227\pi$
$359$	−3.83008	−0.202144	−0.101072	+	0.994879i	$0.532227\pi$
$360$	3.04103	0.160276
$361$	−16.5161	−0.869267
$362$	15.5055	0.814952
$363$	−10.8809	−0.571099
$364$	−0.566267	−0.0296804
$365$	−9.90714	−0.518563
$366$	13.3715	0.698937
$367$	24.1046	1.25825	0.629126	−	0.777304i	$-0.283413\pi$
$367$	24.1046	1.25825	0.629126	+	0.777304i	$0.283413\pi$
$368$	−16.1641	−0.842614
$369$	1.58420	0.0824703
$370$	4.33810	0.225527
$371$	3.16466	0.164301
$372$	0.417093	0.0216253
$373$	−1.92053	−0.0994411	−0.0497206	−	0.998763i	$-0.515833\pi$
$373$	−1.92053	−0.0994411	−0.0497206	+	0.998763i	$0.515833\pi$
$374$	−2.90742	−0.150339
$375$	1.00000	0.0516398
$376$	7.17215	0.369875
$377$	−0.119863	−0.00617326
$378$	1.70811	0.0878556
$379$	−30.8098	−1.58259	−0.791295	−	0.611434i	$-0.790593\pi$
$379$	−30.8098	−1.58259	−0.791295	+	0.611434i	$0.790593\pi$
$380$	−0.657360	−0.0337218
$381$	−14.1644	−0.725666
$382$	13.6615	0.698983
$383$	−24.3233	−1.24286	−0.621432	−	0.783468i	$-0.713449\pi$
$383$	−24.3233	−1.24286	−0.621432	+	0.783468i	$0.713449\pi$
$384$	−6.56152	−0.334841
$385$	0.468552	0.0238796
$386$	−21.1006	−1.07399
$387$	7.03352	0.357534
$388$	6.30669	0.320174
$389$	−36.2246	−1.83666	−0.918329	−	0.395817i	$-0.870461\pi$
$389$	−36.2246	−1.83666	−0.918329	+	0.395817i	$0.870461\pi$
$390$	1.25814	0.0637082
$391$	−36.1762	−1.82951
$392$	−15.6820	−0.792058
$393$	−5.81171	−0.293162
$394$	−28.0545	−1.41337
$395$	3.65244	0.183774
$396$	0.143947	0.00723361
$397$	−10.2799	−0.515933	−0.257967	−	0.966154i	$-0.583052\pi$
$397$	−10.2799	−0.515933	−0.257967	+	0.966154i	$0.583052\pi$
$398$	32.6265	1.63542
$399$	−2.13972	−0.107120
$400$	−2.99185	−0.149592
$401$	11.2253	0.560566	0.280283	−	0.959917i	$-0.409572\pi$
$401$	11.2253	0.560566	0.280283	+	0.959917i	$0.409572\pi$
$402$	−1.60688	−0.0801438
$403$	1.00000	0.0498135
$404$	−1.96712	−0.0978679
$405$	1.00000	0.0496904
$406$	0.204739	0.0101610
$407$	1.18998	0.0589853
$408$	−20.3625	−1.00809
$409$	−3.55715	−0.175890	−0.0879449	−	0.996125i	$-0.528030\pi$
$409$	−3.55715	−0.175890	−0.0879449	+	0.996125i	$0.528030\pi$
$410$	−1.99314	−0.0984343
$411$	−14.5828	−0.719318
$412$	1.78820	0.0880981
$413$	−15.3097	−0.753343
$414$	−6.79737	−0.334072
$415$	−11.9725	−0.587705
$416$	2.31791	0.113645
$417$	11.7019	0.573044
$418$	0.684333	0.0334718
$419$	18.0960	0.884045	0.442023	−	0.897004i	$-0.354261\pi$
$419$	18.0960	0.884045	0.442023	+	0.897004i	$0.354261\pi$
$420$	0.566267	0.0276310
$421$	20.0283	0.976118	0.488059	−	0.872811i	$-0.337705\pi$
$421$	20.0283	0.976118	0.488059	+	0.872811i	$0.337705\pi$
$422$	−30.7111	−1.49499
$423$	2.35846	0.114672
$424$	−7.08860	−0.344253
$425$	−6.69591	−0.324799
$426$	−2.85211	−0.138185
$427$	14.4291	0.698272
$428$	−5.80741	−0.280712
$429$	0.345120	0.0166625
$430$	−8.84913	−0.426743
$431$	36.5616	1.76111	0.880554	−	0.473946i	$-0.157171\pi$
$431$	36.5616	1.76111	0.880554	+	0.473946i	$0.157171\pi$
$432$	−2.99185	−0.143945
$433$	3.54204	0.170220	0.0851098	−	0.996372i	$-0.472876\pi$
$433$	3.54204	0.170220	0.0851098	+	0.996372i	$0.472876\pi$
$434$	−1.70811	−0.0819919
$435$	0.119863	0.00574699
$436$	−4.69731	−0.224960
$437$	8.51497	0.407326
$438$	12.4645	0.595578
$439$	6.32219	0.301742	0.150871	−	0.988553i	$-0.451792\pi$
$439$	6.32219	0.301742	0.150871	+	0.988553i	$0.451792\pi$
$440$	−1.04952	−0.0500339
$441$	−5.15679	−0.245561
$442$	−8.42437	−0.400706
$443$	−22.3162	−1.06028	−0.530138	−	0.847911i	$-0.677860\pi$
$443$	−22.3162	−1.06028	−0.530138	+	0.847911i	$0.677860\pi$
$444$	1.43815	0.0682516
$445$	−14.5357	−0.689056
$446$	5.13662	0.243226
$447$	−7.58040	−0.358541
$448$	−12.0830	−0.570869
$449$	−3.01840	−0.142447	−0.0712235	−	0.997460i	$-0.522690\pi$
$449$	−3.01840	−0.142447	−0.0712235	+	0.997460i	$0.522690\pi$
$450$	−1.25814	−0.0593091
$451$	−0.546739	−0.0257450
$452$	1.34868	0.0634364
$453$	4.13400	0.194232
$454$	−15.1537	−0.711199
$455$	1.35765	0.0636476
$456$	4.79282	0.224444
$457$	10.0796	0.471506	0.235753	−	0.971813i	$-0.424244\pi$
$457$	10.0796	0.471506	0.235753	+	0.971813i	$0.424244\pi$
$458$	−20.9442	−0.978657
$459$	−6.69591	−0.312538
$460$	−2.25344	−0.105067
$461$	11.1471	0.519172	0.259586	−	0.965720i	$-0.416414\pi$
$461$	11.1471	0.519172	0.259586	+	0.965720i	$0.416414\pi$
$462$	−0.589502	−0.0274261
$463$	−31.1483	−1.44758	−0.723792	−	0.690018i	$-0.757602\pi$
$463$	−31.1483	−1.44758	−0.723792	+	0.690018i	$0.757602\pi$
$464$	−0.358612	−0.0166481
$465$	−1.00000	−0.0463739
$466$	−26.9076	−1.24647
$467$	5.86653	0.271470	0.135735	−	0.990745i	$-0.456660\pi$
$467$	5.86653	0.271470	0.135735	+	0.990745i	$0.456660\pi$
$468$	0.417093	0.0192801
$469$	−1.73398	−0.0800676
$470$	−2.96726	−0.136870
$471$	12.6022	0.580679
$472$	34.2927	1.57845
$473$	−2.42741	−0.111612
$474$	−4.59527	−0.211068
$475$	1.57605	0.0723141
$476$	−3.79167	−0.173791
$477$	−2.33098	−0.106728
$478$	30.1535	1.37919
$479$	29.4485	1.34554	0.672769	−	0.739852i	$-0.265105\pi$
$479$	29.4485	1.34554	0.672769	+	0.739852i	$0.265105\pi$
$480$	−2.31791	−0.105798
$481$	3.44803	0.157217
$482$	26.1932	1.19307
$483$	−7.33501	−0.333755
$484$	4.53835	0.206288
$485$	−15.1206	−0.686591
$486$	−1.25814	−0.0570702
$487$	−4.21745	−0.191111	−0.0955555	−	0.995424i	$-0.530463\pi$
$487$	−4.21745	−0.191111	−0.0955555	+	0.995424i	$0.530463\pi$
$488$	−32.3200	−1.46306
$489$	9.48867	0.429092
$490$	6.48794	0.293095
$491$	−2.32957	−0.105132	−0.0525660	−	0.998617i	$-0.516740\pi$
$491$	−2.32957	−0.105132	−0.0525660	+	0.998617i	$0.516740\pi$
$492$	−0.660760	−0.0297894
$493$	−0.802592	−0.0361469
$494$	1.98289	0.0892142
$495$	−0.345120	−0.0155120
$496$	2.99185	0.134338
$497$	−3.07770	−0.138054
$498$	15.0630	0.674989
$499$	18.5144	0.828819	0.414409	−	0.910091i	$-0.363988\pi$
$499$	18.5144	0.828819	0.414409	+	0.910091i	$0.363988\pi$
$500$	−0.417093	−0.0186530
$501$	−19.3453	−0.864285
$502$	27.5779	1.23086
$503$	−21.1174	−0.941576	−0.470788	−	0.882246i	$-0.656030\pi$
$503$	−21.1174	−0.941576	−0.470788	+	0.882246i	$0.656030\pi$
$504$	−4.12866	−0.183905
$505$	4.71626	0.209871
$506$	2.34590	0.104288
$507$	1.00000	0.0444116
$508$	5.90789	0.262120
$509$	13.5021	0.598468	0.299234	−	0.954180i	$-0.403269\pi$
$509$	13.5021	0.598468	0.299234	+	0.954180i	$0.403269\pi$
$510$	8.42437	0.373037
$511$	13.4504	0.595012
$512$	25.1315	1.11066
$513$	1.57605	0.0695843
$514$	33.7853	1.49020
$515$	−4.28728	−0.188920
$516$	−2.93363	−0.129146
$517$	−0.813950	−0.0357975
$518$	−5.88962	−0.258775
$519$	−2.78667	−0.122321
$520$	−3.04103	−0.133358
$521$	−11.3628	−0.497814	−0.248907	−	0.968527i	$-0.580071\pi$
$521$	−11.3628	−0.497814	−0.248907	+	0.968527i	$0.580071\pi$
$522$	−0.150804	−0.00660051
$523$	−25.9377	−1.13418	−0.567088	−	0.823657i	$-0.691930\pi$
$523$	−25.9377	−1.13418	−0.567088	+	0.823657i	$0.691930\pi$
$524$	2.42402	0.105894
$525$	−1.35765	−0.0592527
$526$	18.9108	0.824548
$527$	6.69591	0.291679
$528$	1.03255	0.0449358
$529$	6.18945	0.269106
$530$	2.93270	0.127388
$531$	11.2766	0.489365
$532$	0.892464	0.0386932
$533$	−1.58420	−0.0686194
$534$	18.2878	0.791392
$535$	13.9235	0.601967
$536$	3.88397	0.167762
$537$	8.16315	0.352266
$538$	31.9787	1.37870
$539$	1.77971	0.0766574
$540$	−0.417093	−0.0179488
$541$	2.01569	0.0866613	0.0433306	−	0.999061i	$-0.486203\pi$
$541$	2.01569	0.0866613	0.0433306	+	0.999061i	$0.486203\pi$
$542$	−15.8721	−0.681765
$543$	−12.3242	−0.528882
$544$	15.5205	0.665438
$545$	11.2620	0.482412
$546$	−1.70811	−0.0731003
$547$	22.9404	0.980859	0.490429	−	0.871481i	$-0.336840\pi$
$547$	22.9404	0.980859	0.490429	+	0.871481i	$0.336840\pi$
$548$	6.08240	0.259827
$549$	−10.6280	−0.453591
$550$	0.434207	0.0185147
$551$	0.188910	0.00804784
$552$	16.4299	0.699302
$553$	−4.95874	−0.210867
$554$	26.3829	1.12090
$555$	−3.44803	−0.146361
$556$	−4.88078	−0.206991
$557$	−24.1611	−1.02374	−0.511869	−	0.859064i	$-0.671047\pi$
$557$	−24.1611	−1.02374	−0.511869	+	0.859064i	$0.671047\pi$
$558$	1.25814	0.0532612
$559$	−7.03352	−0.297486
$560$	4.06188	0.171646
$561$	2.31089	0.0975659
$562$	11.4463	0.482832
$563$	−38.4214	−1.61927	−0.809634	−	0.586935i	$-0.800334\pi$
$563$	−38.4214	−1.61927	−0.809634	+	0.586935i	$0.800334\pi$
$564$	−0.983697	−0.0414211
$565$	−3.23351	−0.136035
$566$	5.31739	0.223506
$567$	−1.35765	−0.0570160
$568$	6.89381	0.289258
$569$	3.47076	0.145502	0.0727509	−	0.997350i	$-0.476822\pi$
$569$	3.47076	0.145502	0.0727509	+	0.997350i	$0.476822\pi$
$570$	−1.98289	−0.0830539
$571$	0.0100158	0.000419146 0	0.000209573	−	1.00000i	$-0.499933\pi$
$571$	0.0100158	0.000419146 0	0.000209573		1.00000i	$0.499933\pi$
$572$	−0.143947	−0.00601873
$573$	−10.8585	−0.453621
$574$	2.70599	0.112946
$575$	5.40273	0.225309
$576$	8.89995	0.370831
$577$	−25.3922	−1.05709	−0.528546	−	0.848905i	$-0.677263\pi$
$577$	−25.3922	−1.05709	−0.528546	+	0.848905i	$0.677263\pi$
$578$	−35.0205	−1.45666
$579$	16.7713	0.696990
$580$	−0.0499940	−0.00207589
$581$	16.2544	0.674347
$582$	19.0238	0.788560
$583$	0.804469	0.0333177
$584$	−30.1279	−1.24670
$585$	−1.00000	−0.0413449
$586$	3.73641	0.154350
$587$	−39.5759	−1.63347	−0.816737	−	0.577010i	$-0.804219\pi$
$587$	−39.5759	−1.63347	−0.816737	+	0.577010i	$0.804219\pi$
$588$	2.15086	0.0887000
$589$	−1.57605	−0.0649400
$590$	−14.1876	−0.584093
$591$	22.2985	0.917236
$592$	10.3160	0.423985
$593$	−30.3121	−1.24477	−0.622384	−	0.782712i	$-0.713836\pi$
$593$	−30.3121	−1.24477	−0.622384	+	0.782712i	$0.713836\pi$
$594$	0.434207	0.0178157
$595$	9.09071	0.372683
$596$	3.16173	0.129510
$597$	−25.9324	−1.06134
$598$	6.79737	0.277965
$599$	−27.4673	−1.12228	−0.561142	−	0.827719i	$-0.689638\pi$
$599$	−27.4673	−1.12228	−0.561142	+	0.827719i	$0.689638\pi$
$600$	3.04103	0.124150
$601$	33.1017	1.35025	0.675124	−	0.737704i	$-0.264090\pi$
$601$	33.1017	1.35025	0.675124	+	0.737704i	$0.264090\pi$
$602$	12.0140	0.489655
$603$	1.27719	0.0520112
$604$	−1.72426	−0.0701592
$605$	−10.8809	−0.442371
$606$	−5.93370	−0.241040
$607$	−43.1621	−1.75190	−0.875949	−	0.482404i	$-0.839764\pi$
$607$	−43.1621	−1.75190	−0.875949	+	0.482404i	$0.839764\pi$
$608$	−3.65315	−0.148155
$609$	−0.162732	−0.00659423
$610$	13.3715	0.541394
$611$	−2.35846	−0.0954130
$612$	2.79282	0.112893
$613$	9.91320	0.400390	0.200195	−	0.979756i	$-0.435842\pi$
$613$	9.91320	0.400390	0.200195	+	0.979756i	$0.435842\pi$
$614$	40.6008	1.63852
$615$	1.58420	0.0638812
$616$	1.42488	0.0574101
$617$	34.3745	1.38387	0.691933	−	0.721962i	$-0.256759\pi$
$617$	34.3745	1.38387	0.691933	+	0.721962i	$0.256759\pi$
$618$	5.39398	0.216978
$619$	−11.0328	−0.443447	−0.221723	−	0.975110i	$-0.571168\pi$
$619$	−11.0328	−0.443447	−0.221723	+	0.975110i	$0.571168\pi$
$620$	0.417093	0.0167509
$621$	5.40273	0.216804
$622$	23.4172	0.938945
$623$	19.7343	0.790640
$624$	2.99185	0.119770
$625$	1.00000	0.0400000
$626$	−29.4722	−1.17795
$627$	−0.543926	−0.0217223
$628$	−5.25630	−0.209749
$629$	23.0877	0.920568
$630$	1.70811	0.0680527
$631$	26.7069	1.06319	0.531593	−	0.847000i	$-0.321594\pi$
$631$	26.7069	1.06319	0.531593	+	0.847000i	$0.321594\pi$
$632$	11.1072	0.441820
$633$	24.4100	0.970211
$634$	13.1463	0.522105
$635$	−14.1644	−0.562099
$636$	0.972238	0.0385517
$637$	5.15679	0.204319
$638$	0.0520454	0.00206050
$639$	2.26693	0.0896784
$640$	−6.56152	−0.259367
$641$	−37.5115	−1.48162	−0.740808	−	0.671717i	$-0.765557\pi$
$641$	−37.5115	−1.48162	−0.740808	+	0.671717i	$0.765557\pi$
$642$	−17.5177	−0.691368
$643$	−13.2272	−0.521629	−0.260815	−	0.965389i	$-0.583991\pi$
$643$	−13.2272	−0.521629	−0.260815	+	0.965389i	$0.583991\pi$
$644$	3.05938	0.120557
$645$	7.03352	0.276945
$646$	13.2772	0.522386
$647$	−38.8116	−1.52584	−0.762921	−	0.646492i	$-0.776235\pi$
$647$	−38.8116	−1.52584	−0.762921	+	0.646492i	$0.776235\pi$
$648$	3.04103	0.119463
$649$	−3.89179	−0.152766
$650$	1.25814	0.0493482
$651$	1.35765	0.0532105
$652$	−3.95766	−0.154994
$653$	−0.763484	−0.0298775	−0.0149387	−	0.999888i	$-0.504755\pi$
$653$	−0.763484	−0.0298775	−0.0149387	+	0.999888i	$0.504755\pi$
$654$	−14.1691	−0.554058
$655$	−5.81171	−0.227082
$656$	−4.73969	−0.185054
$657$	−9.90714	−0.386514
$658$	4.02850	0.157047
$659$	−4.27979	−0.166717	−0.0833586	−	0.996520i	$-0.526565\pi$
$659$	−4.27979	−0.166717	−0.0833586	+	0.996520i	$0.526565\pi$
$660$	0.143947	0.00560313
$661$	−13.8127	−0.537251	−0.268625	−	0.963245i	$-0.586569\pi$
$661$	−13.8127	−0.537251	−0.268625	+	0.963245i	$0.586569\pi$
$662$	−24.1753	−0.939601
$663$	6.69591	0.260048
$664$	−36.4086	−1.41293
$665$	−2.13972	−0.0829750
$666$	4.33810	0.168098
$667$	0.647587	0.0250747
$668$	8.06879	0.312191
$669$	−4.08272	−0.157847
$670$	−1.60688	−0.0620791
$671$	3.66793	0.141599
$672$	3.14692	0.121395
$673$	19.9255	0.768071	0.384035	−	0.923318i	$-0.374534\pi$
$673$	19.9255	0.768071	0.384035	+	0.923318i	$0.374534\pi$
$674$	−7.62829	−0.293831
$675$	1.00000	0.0384900
$676$	−0.417093	−0.0160420
$677$	32.3534	1.24344	0.621720	−	0.783239i	$-0.286434\pi$
$677$	32.3534	1.24344	0.621720	+	0.783239i	$0.286434\pi$
$678$	4.06820	0.156238
$679$	20.5285	0.787811
$680$	−20.3625	−0.780866
$681$	12.0446	0.461549
$682$	−0.434207	−0.0166267
$683$	−37.1181	−1.42029	−0.710143	−	0.704057i	$-0.751370\pi$
$683$	−37.1181	−1.42029	−0.710143	+	0.704057i	$0.751370\pi$
$684$	−0.657360	−0.0251348
$685$	−14.5828	−0.557182
$686$	−20.7651	−0.792816
$687$	16.6470	0.635122
$688$	−21.0432	−0.802265
$689$	2.33098	0.0888035
$690$	−6.79737	−0.258771
$691$	22.3455	0.850061	0.425030	−	0.905179i	$-0.360263\pi$
$691$	22.3455	0.850061	0.425030	+	0.905179i	$0.360263\pi$
$692$	1.16230	0.0441841
$693$	0.468552	0.0177988
$694$	7.92875	0.300971
$695$	11.7019	0.443878
$696$	0.364507	0.0138166
$697$	−10.6077	−0.401795
$698$	−24.2368	−0.917376
$699$	21.3869	0.808926
$700$	0.566267	0.0214029
$701$	−8.91295	−0.336637	−0.168319	−	0.985733i	$-0.553834\pi$
$701$	−8.91295	−0.336637	−0.168319	+	0.985733i	$0.553834\pi$
$702$	1.25814	0.0474853
$703$	−5.43427	−0.204958
$704$	−3.07155	−0.115763
$705$	2.35846	0.0888247
$706$	36.4221	1.37077
$707$	−6.40303	−0.240811
$708$	−4.70341	−0.176765
$709$	23.4636	0.881192	0.440596	−	0.897705i	$-0.354767\pi$
$709$	23.4636	0.881192	0.440596	+	0.897705i	$0.354767\pi$
$710$	−2.85211	−0.107038
$711$	3.65244	0.136977
$712$	−44.2034	−1.65659
$713$	−5.40273	−0.202334
$714$	−11.4373	−0.428032
$715$	0.345120	0.0129067
$716$	−3.40479	−0.127243
$717$	−23.9668	−0.895057
$718$	4.81876	0.179835
$719$	32.4728	1.21103	0.605516	−	0.795833i	$-0.292967\pi$
$719$	32.4728	1.21103	0.605516	+	0.795833i	$0.292967\pi$
$720$	−2.99185	−0.111500
$721$	5.82063	0.216772
$722$	20.7795	0.773332
$723$	−20.8191	−0.774270
$724$	5.14034	0.191039
$725$	0.119863	0.00445160
$726$	13.6896	0.508070
$727$	−53.6867	−1.99113	−0.995565	−	0.0940733i	$-0.970011\pi$
$727$	−53.6867	−1.99113	−0.995565	+	0.0940733i	$0.970011\pi$
$728$	4.12866	0.153018
$729$	1.00000	0.0370370
$730$	12.4645	0.461333
$731$	−47.0959	−1.74190
$732$	4.43286	0.163843
$733$	−27.7686	−1.02566	−0.512828	−	0.858491i	$-0.671402\pi$
$733$	−27.7686	−1.02566	−0.512828	+	0.858491i	$0.671402\pi$
$734$	−30.3269	−1.11939
$735$	−5.15679	−0.190211
$736$	−12.5231	−0.461606
$737$	−0.440783	−0.0162365
$738$	−1.99314	−0.0733686
$739$	−14.6735	−0.539772	−0.269886	−	0.962892i	$-0.586986\pi$
$739$	−14.6735	−0.539772	−0.269886	+	0.962892i	$0.586986\pi$
$740$	1.43815	0.0528675
$741$	−1.57605	−0.0578976
$742$	−3.98158	−0.146168
$743$	−29.4556	−1.08062	−0.540310	−	0.841466i	$-0.681693\pi$
$743$	−29.4556	−1.08062	−0.540310	+	0.841466i	$0.681693\pi$
$744$	−3.04103	−0.111490
$745$	−7.58040	−0.277724
$746$	2.41628	0.0884665
$747$	−11.9725	−0.438049
$748$	−0.963857	−0.0352421
$749$	−18.9033	−0.690711
$750$	−1.25814	−0.0459406
$751$	0.879829	0.0321054	0.0160527	−	0.999871i	$-0.494890\pi$
$751$	0.879829	0.0321054	0.0160527	+	0.999871i	$0.494890\pi$
$752$	−7.05615	−0.257311
$753$	−21.9196	−0.798796
$754$	0.150804	0.00549196
$755$	4.13400	0.150452
$756$	0.566267	0.0205949
$757$	30.4962	1.10840	0.554201	−	0.832383i	$-0.313024\pi$
$757$	30.4962	1.10840	0.554201	+	0.832383i	$0.313024\pi$
$758$	38.7629	1.40793
$759$	−1.86459	−0.0676802
$760$	4.79282	0.173854
$761$	4.52286	0.163954	0.0819768	−	0.996634i	$-0.473877\pi$
$761$	4.52286	0.163954	0.0819768	+	0.996634i	$0.473877\pi$
$762$	17.8208	0.645579
$763$	−15.2899	−0.553531
$764$	4.52901	0.163854
$765$	−6.69591	−0.242091
$766$	30.6021	1.10570
$767$	−11.2766	−0.407176
$768$	−9.54461	−0.344411
$769$	27.1955	0.980695	0.490347	−	0.871527i	$-0.336870\pi$
$769$	27.1955	0.980695	0.490347	+	0.871527i	$0.336870\pi$
$770$	−0.589502	−0.0212442
$771$	−26.8534	−0.967103
$772$	−6.99518	−0.251762
$773$	11.2066	0.403074	0.201537	−	0.979481i	$-0.435406\pi$
$773$	11.2066	0.403074	0.201537	+	0.979481i	$0.435406\pi$
$774$	−8.84913	−0.318075
$775$	−1.00000	−0.0359211
$776$	−45.9822	−1.65067
$777$	4.68122	0.167938
$778$	45.5754	1.63396
$779$	2.49678	0.0894565
$780$	0.417093	0.0149343
$781$	−0.782363	−0.0279951
$782$	45.5146	1.62760
$783$	0.119863	0.00428355
$784$	15.4283	0.551011
$785$	12.6022	0.449792
$786$	7.31192	0.260807
$787$	10.0338	0.357666	0.178833	−	0.983879i	$-0.442768\pi$
$787$	10.0338	0.357666	0.178833	+	0.983879i	$0.442768\pi$
$788$	−9.30053	−0.331318
$789$	−15.0308	−0.535110
$790$	−4.59527	−0.163492
$791$	4.38998	0.156090
$792$	−1.04952	−0.0372931
$793$	10.6280	0.377411
$794$	12.9335	0.458993
$795$	−2.33098	−0.0826715
$796$	10.8162	0.383371
$797$	−43.4123	−1.53774	−0.768871	−	0.639404i	$-0.779181\pi$
$797$	−43.4123	−1.53774	−0.768871	+	0.639404i	$0.779181\pi$
$798$	2.69207	0.0952981
$799$	−15.7920	−0.558682
$800$	−2.31791	−0.0819506
$801$	−14.5357	−0.513592
$802$	−14.1230	−0.498700
$803$	3.41915	0.120659
$804$	−0.532707	−0.0187871
$805$	−7.33501	−0.258525
$806$	−1.25814	−0.0443160
$807$	−25.4175	−0.894738
$808$	14.3423	0.504561
$809$	−5.51418	−0.193868	−0.0969342	−	0.995291i	$-0.530904\pi$
$809$	−5.51418	−0.193868	−0.0969342	+	0.995291i	$0.530904\pi$
$810$	−1.25814	−0.0442064
$811$	−15.8646	−0.557082	−0.278541	−	0.960424i	$-0.589851\pi$
$811$	−15.8646	−0.557082	−0.278541	+	0.960424i	$0.589851\pi$
$812$	0.0678744	0.00238192
$813$	12.6156	0.442447
$814$	−1.49716	−0.0524755
$815$	9.48867	0.332374
$816$	20.0331	0.701300
$817$	11.0852	0.387821
$818$	4.47538	0.156478
$819$	1.35765	0.0474401
$820$	−0.660760	−0.0230747
$821$	−13.6626	−0.476828	−0.238414	−	0.971164i	$-0.576628\pi$
$821$	−13.6626	−0.476828	−0.238414	+	0.971164i	$0.576628\pi$
$822$	18.3472	0.639932
$823$	−10.0692	−0.350990	−0.175495	−	0.984480i	$-0.556153\pi$
$823$	−10.0692	−0.350990	−0.175495	+	0.984480i	$0.556153\pi$
$824$	−13.0378	−0.454192
$825$	−0.345120	−0.0120155
$826$	19.2617	0.670202
$827$	−35.6393	−1.23930	−0.619649	−	0.784879i	$-0.712725\pi$
$827$	−35.6393	−1.23930	−0.619649	+	0.784879i	$0.712725\pi$
$828$	−2.25344	−0.0783124
$829$	−1.86268	−0.0646936	−0.0323468	−	0.999477i	$-0.510298\pi$
$829$	−1.86268	−0.0646936	−0.0323468	+	0.999477i	$0.510298\pi$
$830$	15.0630	0.522844
$831$	−20.9698	−0.727435
$832$	−8.89995	−0.308550
$833$	34.5294	1.19637
$834$	−14.7226	−0.509801
$835$	−19.3453	−0.669472
$836$	0.226868	0.00784638
$837$	−1.00000	−0.0345651
$838$	−22.7672	−0.786479
$839$	33.4730	1.15562	0.577808	−	0.816173i	$-0.303908\pi$
$839$	33.4730	1.15562	0.577808	+	0.816173i	$0.303908\pi$
$840$	−4.12866	−0.142452
$841$	−28.9856	−0.999505
$842$	−25.1983	−0.868391
$843$	−9.09780	−0.313345
$844$	−10.1812	−0.350453
$845$	1.00000	0.0344010
$846$	−2.96726	−0.102017
$847$	14.7724	0.507587
$848$	6.97395	0.239486
$849$	−4.22640	−0.145050
$850$	8.42437	0.288954
$851$	−18.6288	−0.638586
$852$	−0.945522	−0.0323930
$853$	17.3115	0.592736	0.296368	−	0.955074i	$-0.404225\pi$
$853$	17.3115	0.592736	0.296368	+	0.955074i	$0.404225\pi$
$854$	−18.1538	−0.621209
$855$	1.57605	0.0538998
$856$	42.3419	1.44722
$857$	8.49247	0.290097	0.145049	−	0.989425i	$-0.453666\pi$
$857$	8.49247	0.290097	0.145049	+	0.989425i	$0.453666\pi$
$858$	−0.434207	−0.0148236
$859$	9.71667	0.331528	0.165764	−	0.986165i	$-0.446991\pi$
$859$	9.71667	0.331528	0.165764	+	0.986165i	$0.446991\pi$
$860$	−2.93363	−0.100036
$861$	−2.15079	−0.0732988
$862$	−45.9994	−1.56675
$863$	12.7606	0.434375	0.217187	−	0.976130i	$-0.430312\pi$
$863$	12.7606	0.434375	0.217187	+	0.976130i	$0.430312\pi$
$864$	−2.31791	−0.0788570
$865$	−2.78667	−0.0947497
$866$	−4.45637	−0.151434
$867$	27.8352	0.945335
$868$	−0.566267	−0.0192203
$869$	−1.26053	−0.0427605
$870$	−0.150804	−0.00511273
$871$	−1.27719	−0.0432759
$872$	34.2481	1.15979
$873$	−15.1206	−0.511755
$874$	−10.7130	−0.362372
$875$	−1.35765	−0.0458970
$876$	4.13220	0.139614
$877$	45.8461	1.54811	0.774056	−	0.633117i	$-0.218225\pi$
$877$	45.8461	1.54811	0.774056	+	0.633117i	$0.218225\pi$
$878$	−7.95418	−0.268440
$879$	−2.96980	−0.100169
$880$	1.03255	0.0348071
$881$	1.67828	0.0565428	0.0282714	−	0.999600i	$-0.491000\pi$
$881$	1.67828	0.0565428	0.0282714	+	0.999600i	$0.491000\pi$
$882$	6.48794	0.218460
$883$	−29.3642	−0.988186	−0.494093	−	0.869409i	$-0.664500\pi$
$883$	−29.3642	−0.988186	−0.494093	+	0.869409i	$0.664500\pi$
$884$	−2.79282	−0.0939327
$885$	11.2766	0.379060
$886$	28.0769	0.943261
$887$	−5.04351	−0.169345	−0.0846723	−	0.996409i	$-0.526984\pi$
$887$	−5.04351	−0.169345	−0.0846723	+	0.996409i	$0.526984\pi$
$888$	−10.4856	−0.351873
$889$	19.2304	0.644965
$890$	18.2878	0.613010
$891$	−0.345120	−0.0115619
$892$	1.70287	0.0570165
$893$	3.71705	0.124386
$894$	9.53718	0.318971
$895$	8.16315	0.272864
$896$	8.90825	0.297604
$897$	−5.40273	−0.180392
$898$	3.79756	0.126726
$899$	−0.119863	−0.00399765
$900$	−0.417093	−0.0139031
$901$	15.6081	0.519980
$902$	0.687873	0.0229037
$903$	−9.54906	−0.317773
$904$	−9.83321	−0.327048
$905$	−12.3242	−0.409670
$906$	−5.20113	−0.172796
$907$	48.8242	1.62118	0.810591	−	0.585613i	$-0.199146\pi$
$907$	48.8242	1.62118	0.810591	+	0.585613i	$0.199146\pi$
$908$	−5.02371	−0.166718
$909$	4.71626	0.156429
$910$	−1.70811	−0.0566233
$911$	51.2085	1.69661	0.848306	−	0.529506i	$-0.177623\pi$
$911$	51.2085	1.69661	0.848306	+	0.529506i	$0.177623\pi$
$912$	−4.71530	−0.156139
$913$	4.13193	0.136747
$914$	−12.6816	−0.419469
$915$	−10.6280	−0.351350
$916$	−6.94334	−0.229414
$917$	7.89026	0.260559
$918$	8.42437	0.278046
$919$	−14.8367	−0.489417	−0.244708	−	0.969597i	$-0.578692\pi$
$919$	−14.8367	−0.489417	−0.244708	+	0.969597i	$0.578692\pi$
$920$	16.4299	0.541677
$921$	−32.2706	−1.06335
$922$	−14.0246	−0.461874
$923$	−2.26693	−0.0746170
$924$	−0.195430	−0.00642916
$925$	−3.44803	−0.113371
$926$	39.1888	1.28782
$927$	−4.28728	−0.140813
$928$	−0.277832	−0.00912028
$929$	12.2951	0.403391	0.201695	−	0.979448i	$-0.435355\pi$
$929$	12.2951	0.403391	0.201695	+	0.979448i	$0.435355\pi$
$930$	1.25814	0.0412559
$931$	−8.12735	−0.266363
$932$	−8.92032	−0.292195
$933$	−18.6126	−0.609350
$934$	−7.38089	−0.241510
$935$	2.31089	0.0755742
$936$	−3.04103	−0.0993993
$937$	8.30662	0.271365	0.135683	−	0.990752i	$-0.456677\pi$
$937$	8.30662	0.271365	0.135683	+	0.990752i	$0.456677\pi$
$938$	2.18158	0.0712311
$939$	23.4253	0.764455
$940$	−0.983697	−0.0320847
$941$	38.5489	1.25666	0.628328	−	0.777948i	$-0.283739\pi$
$941$	38.5489	1.25666	0.628328	+	0.777948i	$0.283739\pi$
$942$	−15.8553	−0.516594
$943$	8.55901	0.278720
$944$	−33.7380	−1.09808
$945$	−1.35765	−0.0441644
$946$	3.05401	0.0992944
$947$	50.6677	1.64648	0.823239	−	0.567695i	$-0.192165\pi$
$947$	50.6677	1.64648	0.823239	+	0.567695i	$0.192165\pi$
$948$	−1.52341	−0.0494780
$949$	9.90714	0.321599
$950$	−1.98289	−0.0643333
$951$	−10.4490	−0.338832
$952$	27.6451	0.895984
$953$	−40.9239	−1.32566	−0.662828	−	0.748772i	$-0.730644\pi$
$953$	−40.9239	−1.32566	−0.662828	+	0.748772i	$0.730644\pi$
$954$	2.93270	0.0949496
$955$	−10.8585	−0.351373
$956$	9.99640	0.323307
$957$	−0.0413671	−0.00133721
$958$	−37.0503	−1.19704
$959$	19.7984	0.639324
$960$	8.89995	0.287245
$961$	1.00000	0.0322581
$962$	−4.33810	−0.139866
$963$	13.9235	0.448679
$964$	8.68350	0.279677
$965$	16.7713	0.539886
$966$	9.22844	0.296920
$967$	−6.91097	−0.222242	−0.111121	−	0.993807i	$-0.535444\pi$
$967$	−6.91097	−0.222242	−0.111121	+	0.993807i	$0.535444\pi$
$968$	−33.0891	−1.06353
$969$	−10.5531	−0.339014
$970$	19.0238	0.610816
$971$	5.39277	0.173062	0.0865312	−	0.996249i	$-0.472422\pi$
$971$	5.39277	0.173062	0.0865312	+	0.996249i	$0.472422\pi$
$972$	−0.417093	−0.0133783
$973$	−15.8871	−0.509316
$974$	5.30613	0.170019
$975$	−1.00000	−0.0320256
$976$	31.7973	1.01781
$977$	5.51200	0.176344	0.0881722	−	0.996105i	$-0.471897\pi$
$977$	5.51200	0.176344	0.0881722	+	0.996105i	$0.471897\pi$
$978$	−11.9380	−0.381736
$979$	5.01654	0.160329
$980$	2.15086	0.0687067
$981$	11.2620	0.359568
$982$	2.93092	0.0935293
$983$	7.64599	0.243869	0.121935	−	0.992538i	$-0.461090\pi$
$983$	7.64599	0.243869	0.121935	+	0.992538i	$0.461090\pi$
$984$	4.81761	0.153580
$985$	22.2985	0.710488
$986$	1.00977	0.0321576
$987$	−3.20196	−0.101920
$988$	0.657360	0.0209134
$989$	38.0002	1.20834
$990$	0.434207	0.0138000
$991$	−41.5901	−1.32115	−0.660576	−	0.750759i	$-0.729688\pi$
$991$	−41.5901	−1.32115	−0.660576	+	0.750759i	$0.729688\pi$
$992$	2.31791	0.0735938
$993$	19.2152	0.609776
$994$	3.87217	0.122818
$995$	−25.9324	−0.822113
$996$	4.99363	0.158229
$997$	56.8328	1.79991	0.899956	−	0.435981i	$-0.143599\pi$
$997$	56.8328	1.79991	0.899956	+	0.435981i	$0.143599\pi$
$998$	−23.2937	−0.737348
$999$	−3.44803	−0.109091

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.s.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.s.1.2	✓	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-1.25814 q^{2} +1.00000 q^{3} -0.417093 q^{4} +1.00000 q^{5} -1.25814 q^{6} -1.35765 q^{7} +3.04103 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.25814 q^{2} +1.00000 q^{3} -0.417093 q^{4} +1.00000 q^{5} -1.25814 q^{6} -1.35765 q^{7} +3.04103 q^{8} +1.00000 q^{9} -1.25814 q^{10} -0.345120 q^{11} -0.417093 q^{12} -1.00000 q^{13} +1.70811 q^{14} +1.00000 q^{15} -2.99185 q^{16} -6.69591 q^{17} -1.25814 q^{18} +1.57605 q^{19} -0.417093 q^{20} -1.35765 q^{21} +0.434207 q^{22} +5.40273 q^{23} +3.04103 q^{24} +1.00000 q^{25} +1.25814 q^{26} +1.00000 q^{27} +0.566267 q^{28} +0.119863 q^{29} -1.25814 q^{30} -1.00000 q^{31} -2.31791 q^{32} -0.345120 q^{33} +8.42437 q^{34} -1.35765 q^{35} -0.417093 q^{36} -3.44803 q^{37} -1.98289 q^{38} -1.00000 q^{39} +3.04103 q^{40} +1.58420 q^{41} +1.70811 q^{42} +7.03352 q^{43} +0.143947 q^{44} +1.00000 q^{45} -6.79737 q^{46} +2.35846 q^{47} -2.99185 q^{48} -5.15679 q^{49} -1.25814 q^{50} -6.69591 q^{51} +0.417093 q^{52} -2.33098 q^{53} -1.25814 q^{54} -0.345120 q^{55} -4.12866 q^{56} +1.57605 q^{57} -0.150804 q^{58} +11.2766 q^{59} -0.417093 q^{60} -10.6280 q^{61} +1.25814 q^{62} -1.35765 q^{63} +8.89995 q^{64} -1.00000 q^{65} +0.434207 q^{66} +1.27719 q^{67} +2.79282 q^{68} +5.40273 q^{69} +1.70811 q^{70} +2.26693 q^{71} +3.04103 q^{72} -9.90714 q^{73} +4.33810 q^{74} +1.00000 q^{75} -0.657360 q^{76} +0.468552 q^{77} +1.25814 q^{78} +3.65244 q^{79} -2.99185 q^{80} +1.00000 q^{81} -1.99314 q^{82} -11.9725 q^{83} +0.566267 q^{84} -6.69591 q^{85} -8.84913 q^{86} +0.119863 q^{87} -1.04952 q^{88} -14.5357 q^{89} -1.25814 q^{90} +1.35765 q^{91} -2.25344 q^{92} -1.00000 q^{93} -2.96726 q^{94} +1.57605 q^{95} -2.31791 q^{96} -15.1206 q^{97} +6.48794 q^{98} -0.345120 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 5 * q^9	\( 5 q + 5 q^{3} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 5 q^{9} - 9 q^{11} + 6 q^{12} - 5 q^{13} - 26 q^{14} + 5 q^{15} - 4 q^{16} - 17 q^{17} - 2 q^{19} + 6 q^{20} - q^{21} + 8 q^{22} - 17 q^{23} + 6 q^{24} + 5 q^{25} + 5 q^{27} - 16 q^{28} - 6 q^{29} - 5 q^{31} - 8 q^{32} - 9 q^{33} - 16 q^{34} - q^{35} + 6 q^{36} - 3 q^{37} + 4 q^{38} - 5 q^{39} + 6 q^{40} + 9 q^{41} - 26 q^{42} + 8 q^{43} - 30 q^{44} + 5 q^{45} - 4 q^{46} - 2 q^{47} - 4 q^{48} + 30 q^{49} - 17 q^{51} - 6 q^{52} - 51 q^{53} - 9 q^{55} - 18 q^{56} - 2 q^{57} + 12 q^{58} - 6 q^{59} + 6 q^{60} - 23 q^{61} - q^{63} - 12 q^{64} - 5 q^{65} + 8 q^{66} + 2 q^{68} - 17 q^{69} - 26 q^{70} + 9 q^{71} + 6 q^{72} - 16 q^{73} + 12 q^{74} + 5 q^{75} - 38 q^{76} - 13 q^{77} + 21 q^{79} - 4 q^{80} + 5 q^{81} + 8 q^{82} - 44 q^{83} - 16 q^{84} - 17 q^{85} + 12 q^{86} - 6 q^{87} - 12 q^{88} - 65 q^{89} + q^{91} - 34 q^{92} - 5 q^{93} - 24 q^{94} - 2 q^{95} - 8 q^{96} + 11 q^{97} + 22 q^{98} - 9 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 5 * q^9 - 9 * q^11 + 6 * q^12 - 5 * q^13 - 26 * q^14 + 5 * q^15 - 4 * q^16 - 17 * q^17 - 2 * q^19 + 6 * q^20 - q^21 + 8 * q^22 - 17 * q^23 + 6 * q^24 + 5 * q^25 + 5 * q^27 - 16 * q^28 - 6 * q^29 - 5 * q^31 - 8 * q^32 - 9 * q^33 - 16 * q^34 - q^35 + 6 * q^36 - 3 * q^37 + 4 * q^38 - 5 * q^39 + 6 * q^40 + 9 * q^41 - 26 * q^42 + 8 * q^43 - 30 * q^44 + 5 * q^45 - 4 * q^46 - 2 * q^47 - 4 * q^48 + 30 * q^49 - 17 * q^51 - 6 * q^52 - 51 * q^53 - 9 * q^55 - 18 * q^56 - 2 * q^57 + 12 * q^58 - 6 * q^59 + 6 * q^60 - 23 * q^61 - q^63 - 12 * q^64 - 5 * q^65 + 8 * q^66 + 2 * q^68 - 17 * q^69 - 26 * q^70 + 9 * q^71 + 6 * q^72 - 16 * q^73 + 12 * q^74 + 5 * q^75 - 38 * q^76 - 13 * q^77 + 21 * q^79 - 4 * q^80 + 5 * q^81 + 8 * q^82 - 44 * q^83 - 16 * q^84 - 17 * q^85 + 12 * q^86 - 6 * q^87 - 12 * q^88 - 65 * q^89 + q^91 - 34 * q^92 - 5 * q^93 - 24 * q^94 - 2 * q^95 - 8 * q^96 + 11 * q^97 + 22 * q^98 - 9 * q^99

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