LMFDB - Embedded newform 445.2.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [445,2,Mod(1,445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(445, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 445 = 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	445.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:	$3.55334288995$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 8x^{5} + 6x^{4} + 19x^{3} - 10x^{2} - 12x + 6 $ x^7 - x^6 - 8x^5 + 6x^4 + 19x^3 - 10x^2 - 12*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.89340$ of defining polynomial
Character	$\chi$	$=$	445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.584976 q^{2} +0.172659 q^{3} -1.65780 q^{4} +1.00000 q^{5} +0.101002 q^{6} -2.74591 q^{7} -2.13973 q^{8} -2.97019 q^{9} +O(q^{10})$	\(q+0.584976 q^{2} +0.172659 q^{3} -1.65780 q^{4} +1.00000 q^{5} +0.101002 q^{6} -2.74591 q^{7} -2.13973 q^{8} -2.97019 q^{9} +0.584976 q^{10} -0.429723 q^{11} -0.286235 q^{12} -6.91659 q^{13} -1.60629 q^{14} +0.172659 q^{15} +2.06392 q^{16} +5.29153 q^{17} -1.73749 q^{18} -1.63823 q^{19} -1.65780 q^{20} -0.474108 q^{21} -0.251378 q^{22} -2.26995 q^{23} -0.369444 q^{24} +1.00000 q^{25} -4.04604 q^{26} -1.03081 q^{27} +4.55219 q^{28} +3.93691 q^{29} +0.101002 q^{30} -2.28028 q^{31} +5.48679 q^{32} -0.0741958 q^{33} +3.09541 q^{34} -2.74591 q^{35} +4.92399 q^{36} -7.74257 q^{37} -0.958324 q^{38} -1.19421 q^{39} -2.13973 q^{40} -5.18947 q^{41} -0.277342 q^{42} -6.95847 q^{43} +0.712397 q^{44} -2.97019 q^{45} -1.32787 q^{46} +8.34712 q^{47} +0.356355 q^{48} +0.540044 q^{49} +0.584976 q^{50} +0.913632 q^{51} +11.4664 q^{52} +5.51251 q^{53} -0.602998 q^{54} -0.429723 q^{55} +5.87550 q^{56} -0.282856 q^{57} +2.30300 q^{58} -6.35400 q^{59} -0.286235 q^{60} +9.00292 q^{61} -1.33391 q^{62} +8.15588 q^{63} -0.918199 q^{64} -6.91659 q^{65} -0.0434027 q^{66} -8.39214 q^{67} -8.77231 q^{68} -0.391929 q^{69} -1.60629 q^{70} +12.7822 q^{71} +6.35539 q^{72} +5.08311 q^{73} -4.52921 q^{74} +0.172659 q^{75} +2.71586 q^{76} +1.17998 q^{77} -0.698586 q^{78} -1.33370 q^{79} +2.06392 q^{80} +8.73259 q^{81} -3.03571 q^{82} -6.64852 q^{83} +0.785978 q^{84} +5.29153 q^{85} -4.07053 q^{86} +0.679745 q^{87} +0.919490 q^{88} +1.00000 q^{89} -1.73749 q^{90} +18.9924 q^{91} +3.76314 q^{92} -0.393712 q^{93} +4.88286 q^{94} -1.63823 q^{95} +0.947346 q^{96} +0.828069 q^{97} +0.315913 q^{98} +1.27636 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9}+O(q^{10}) $ 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9} - 4 q^{10} - 10 q^{11} - 11 q^{12} - 7 q^{13} + 3 q^{14} - 8 q^{15} + 10 q^{16} - 13 q^{17} + 4 q^{18} - 7 q^{19} + 8 q^{20} + 16 q^{21} + 2 q^{22} - 13 q^{23} + 4 q^{24} + 7 q^{25} + q^{26} - 23 q^{27} - 21 q^{28} - 4 q^{29} - 2 q^{30} + q^{31} - 13 q^{32} - 6 q^{33} + 10 q^{34} - 16 q^{35} + 20 q^{36} - 5 q^{37} - 40 q^{38} - 13 q^{39} - 12 q^{40} + 5 q^{41} + 30 q^{42} - 31 q^{43} - 21 q^{44} + 11 q^{45} + 16 q^{46} - 14 q^{47} - 7 q^{48} + 19 q^{49} - 4 q^{50} - q^{51} - 13 q^{53} - 17 q^{54} - 10 q^{55} - q^{56} + 21 q^{57} + 17 q^{58} - 14 q^{59} - 11 q^{60} + 3 q^{61} + 26 q^{62} - 54 q^{63} + 14 q^{64} - 7 q^{65} + 36 q^{66} + q^{67} - 35 q^{68} + 31 q^{69} + 3 q^{70} - 8 q^{71} + 53 q^{72} + 9 q^{73} - 35 q^{74} - 8 q^{75} + 40 q^{76} + 42 q^{77} + 46 q^{78} + 9 q^{79} + 10 q^{80} + 35 q^{81} + 29 q^{82} - 42 q^{83} + 55 q^{84} - 13 q^{85} + 35 q^{86} + 6 q^{87} + 30 q^{88} + 7 q^{89} + 4 q^{90} + 31 q^{91} + 19 q^{92} + 24 q^{93} + 37 q^{94} - 7 q^{95} + 44 q^{96} - 7 q^{97} + 9 q^{98} + 5 q^{99}+O(q^{100}) \) 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9 - 4 * q^10 - 10 * q^11 - 11 * q^12 - 7 * q^13 + 3 * q^14 - 8 * q^15 + 10 * q^16 - 13 * q^17 + 4 * q^18 - 7 * q^19 + 8 * q^20 + 16 * q^21 + 2 * q^22 - 13 * q^23 + 4 * q^24 + 7 * q^25 + q^26 - 23 * q^27 - 21 * q^28 - 4 * q^29 - 2 * q^30 + q^31 - 13 * q^32 - 6 * q^33 + 10 * q^34 - 16 * q^35 + 20 * q^36 - 5 * q^37 - 40 * q^38 - 13 * q^39 - 12 * q^40 + 5 * q^41 + 30 * q^42 - 31 * q^43 - 21 * q^44 + 11 * q^45 + 16 * q^46 - 14 * q^47 - 7 * q^48 + 19 * q^49 - 4 * q^50 - q^51 - 13 * q^53 - 17 * q^54 - 10 * q^55 - q^56 + 21 * q^57 + 17 * q^58 - 14 * q^59 - 11 * q^60 + 3 * q^61 + 26 * q^62 - 54 * q^63 + 14 * q^64 - 7 * q^65 + 36 * q^66 + q^67 - 35 * q^68 + 31 * q^69 + 3 * q^70 - 8 * q^71 + 53 * q^72 + 9 * q^73 - 35 * q^74 - 8 * q^75 + 40 * q^76 + 42 * q^77 + 46 * q^78 + 9 * q^79 + 10 * q^80 + 35 * q^81 + 29 * q^82 - 42 * q^83 + 55 * q^84 - 13 * q^85 + 35 * q^86 + 6 * q^87 + 30 * q^88 + 7 * q^89 + 4 * q^90 + 31 * q^91 + 19 * q^92 + 24 * q^93 + 37 * q^94 - 7 * q^95 + 44 * q^96 - 7 * q^97 + 9 * q^98 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.584976	0.413640	0.206820	−	0.978379i	$-0.433689\pi$
$2$	0.584976	0.413640	0.206820	+	0.978379i	$0.433689\pi$
$3$	0.172659	0.0996849	0.0498425	−	0.998757i	$-0.484128\pi$
$3$	0.172659	0.0996849	0.0498425	+	0.998757i	$0.484128\pi$
$4$	−1.65780	−0.828902
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.101002	0.0412337
$7$	−2.74591	−1.03786	−0.518929	−	0.854817i	$-0.673669\pi$
$7$	−2.74591	−1.03786	−0.518929	+	0.854817i	$0.673669\pi$
$8$	−2.13973	−0.756507
$9$	−2.97019	−0.990063
$10$	0.584976	0.184986
$11$	−0.429723	−0.129566	−0.0647832	−	0.997899i	$-0.520636\pi$
$11$	−0.429723	−0.129566	−0.0647832	+	0.997899i	$0.520636\pi$
$12$	−0.286235	−0.0826290
$13$	−6.91659	−1.91832	−0.959159	−	0.282868i	$-0.908714\pi$
$13$	−6.91659	−1.91832	−0.959159	+	0.282868i	$0.908714\pi$
$14$	−1.60629	−0.429300
$15$	0.172659	0.0445805
$16$	2.06392	0.515980
$17$	5.29153	1.28338	0.641692	−	0.766963i	$-0.278233\pi$
$17$	5.29153	1.28338	0.641692	+	0.766963i	$0.278233\pi$
$18$	−1.73749	−0.409530
$19$	−1.63823	−0.375835	−0.187918	−	0.982185i	$-0.560174\pi$
$19$	−1.63823	−0.375835	−0.187918	+	0.982185i	$0.560174\pi$
$20$	−1.65780	−0.370696
$21$	−0.474108	−0.103459
$22$	−0.251378	−0.0535939
$23$	−2.26995	−0.473318	−0.236659	−	0.971593i	$-0.576052\pi$
$23$	−2.26995	−0.473318	−0.236659	+	0.971593i	$0.576052\pi$
$24$	−0.369444	−0.0754124
$25$	1.00000	0.200000
$26$	−4.04604	−0.793493
$27$	−1.03081	−0.198379
$28$	4.55219	0.860282
$29$	3.93691	0.731066	0.365533	−	0.930798i	$-0.380887\pi$
$29$	3.93691	0.731066	0.365533	+	0.930798i	$0.380887\pi$
$30$	0.101002	0.0184403
$31$	−2.28028	−0.409550	−0.204775	−	0.978809i	$-0.565646\pi$
$31$	−2.28028	−0.409550	−0.204775	+	0.978809i	$0.565646\pi$
$32$	5.48679	0.969937
$33$	−0.0741958	−0.0129158
$34$	3.09541	0.530859
$35$	−2.74591	−0.464144
$36$	4.92399	0.820665
$37$	−7.74257	−1.27287	−0.636435	−	0.771330i	$-0.719592\pi$
$37$	−7.74257	−1.27287	−0.636435	+	0.771330i	$0.719592\pi$
$38$	−0.958324	−0.155461
$39$	−1.19421	−0.191227
$40$	−2.13973	−0.338320
$41$	−5.18947	−0.810459	−0.405230	−	0.914215i	$-0.632808\pi$
$41$	−5.18947	−0.810459	−0.405230	+	0.914215i	$0.632808\pi$
$42$	−0.277342	−0.0427947
$43$	−6.95847	−1.06116	−0.530578	−	0.847636i	$-0.678025\pi$
$43$	−6.95847	−1.06116	−0.530578	+	0.847636i	$0.678025\pi$
$44$	0.712397	0.107398
$45$	−2.97019	−0.442770
$46$	−1.32787	−0.195783
$47$	8.34712	1.21755	0.608776	−	0.793342i	$-0.291661\pi$
$47$	8.34712	1.21755	0.608776	+	0.793342i	$0.291661\pi$
$48$	0.356355	0.0514354
$49$	0.540044	0.0771492
$50$	0.584976	0.0827280
$51$	0.913632	0.127934
$52$	11.4664	1.59010
$53$	5.51251	0.757201	0.378600	−	0.925560i	$-0.376405\pi$
$53$	5.51251	0.757201	0.378600	+	0.925560i	$0.376405\pi$
$54$	−0.602998	−0.0820577
$55$	−0.429723	−0.0579439
$56$	5.87550	0.785147
$57$	−0.282856	−0.0374651
$58$	2.30300	0.302398
$59$	−6.35400	−0.827220	−0.413610	−	0.910454i	$-0.635732\pi$
$59$	−6.35400	−0.827220	−0.413610	+	0.910454i	$0.635732\pi$
$60$	−0.286235	−0.0369528
$61$	9.00292	1.15271	0.576353	−	0.817201i	$-0.304475\pi$
$61$	9.00292	1.15271	0.576353	+	0.817201i	$0.304475\pi$
$62$	−1.33391	−0.169406
$63$	8.15588	1.02754
$64$	−0.918199	−0.114775
$65$	−6.91659	−0.857898
$66$	−0.0434027	−0.00534250
$67$	−8.39214	−1.02526	−0.512632	−	0.858609i	$-0.671329\pi$
$67$	−8.39214	−1.02526	−0.512632	+	0.858609i	$0.671329\pi$
$68$	−8.77231	−1.06380
$69$	−0.391929	−0.0471827
$70$	−1.60629	−0.191989
$71$	12.7822	1.51696	0.758482	−	0.651695i	$-0.225942\pi$
$71$	12.7822	1.51696	0.758482	+	0.651695i	$0.225942\pi$
$72$	6.35539	0.748990
$73$	5.08311	0.594933	0.297466	−	0.954732i	$-0.403858\pi$
$73$	5.08311	0.594933	0.297466	+	0.954732i	$0.403858\pi$
$74$	−4.52921	−0.526510
$75$	0.172659	0.0199370
$76$	2.71586	0.311531
$77$	1.17998	0.134472
$78$	−0.698586	−0.0790993
$79$	−1.33370	−0.150053	−0.0750266	−	0.997182i	$-0.523904\pi$
$79$	−1.33370	−0.150053	−0.0750266	+	0.997182i	$0.523904\pi$
$80$	2.06392	0.230753
$81$	8.73259	0.970287
$82$	−3.03571	−0.335238
$83$	−6.64852	−0.729770	−0.364885	−	0.931053i	$-0.618892\pi$
$83$	−6.64852	−0.729770	−0.364885	+	0.931053i	$0.618892\pi$
$84$	0.785978	0.0857572
$85$	5.29153	0.573947
$86$	−4.07053	−0.438937
$87$	0.679745	0.0728763
$88$	0.919490	0.0980180
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	−1.73749	−0.183147
$91$	18.9924	1.99094
$92$	3.76314	0.392334
$93$	−0.393712	−0.0408260
$94$	4.88286	0.503629
$95$	−1.63823	−0.168079
$96$	0.947346	0.0966881
$97$	0.828069	0.0840776	0.0420388	−	0.999116i	$-0.486615\pi$
$97$	0.828069	0.0840776	0.0420388	+	0.999116i	$0.486615\pi$
$98$	0.315913	0.0319120
$99$	1.27636	0.128279
$100$	−1.65780	−0.165780
$101$	−1.43539	−0.142826	−0.0714132	−	0.997447i	$-0.522751\pi$
$101$	−1.43539	−0.142826	−0.0714132	+	0.997447i	$0.522751\pi$
$102$	0.534452	0.0529186
$103$	−6.02978	−0.594132	−0.297066	−	0.954857i	$-0.596008\pi$
$103$	−6.02978	−0.594132	−0.297066	+	0.954857i	$0.596008\pi$
$104$	14.7996	1.45122
$105$	−0.474108	−0.0462682
$106$	3.22468	0.313209
$107$	−11.0707	−1.07024	−0.535121	−	0.844775i	$-0.679734\pi$
$107$	−11.0707	−1.07024	−0.535121	+	0.844775i	$0.679734\pi$
$108$	1.70888	0.164437
$109$	8.99213	0.861290	0.430645	−	0.902521i	$-0.358286\pi$
$109$	8.99213	0.861290	0.430645	+	0.902521i	$0.358286\pi$
$110$	−0.251378	−0.0239679
$111$	−1.33683	−0.126886
$112$	−5.66735	−0.535514
$113$	−5.54942	−0.522045	−0.261023	−	0.965333i	$-0.584060\pi$
$113$	−5.54942	−0.522045	−0.261023	+	0.965333i	$0.584060\pi$
$114$	−0.165464	−0.0154971
$115$	−2.26995	−0.211674
$116$	−6.52663	−0.605982
$117$	20.5436	1.89926
$118$	−3.71694	−0.342172
$119$	−14.5301	−1.33197
$120$	−0.369444	−0.0337254
$121$	−10.8153	−0.983213
$122$	5.26649	0.476806
$123$	−0.896011	−0.0807906
$124$	3.78025	0.339477
$125$	1.00000	0.0894427
$126$	4.77099	0.425034
$127$	−16.6399	−1.47655	−0.738277	−	0.674497i	$-0.764360\pi$
$127$	−16.6399	−1.47655	−0.738277	+	0.674497i	$0.764360\pi$
$128$	−11.5107	−1.01741
$129$	−1.20144	−0.105781
$130$	−4.04604	−0.354861
$131$	0.840072	0.0733974	0.0366987	−	0.999326i	$-0.488316\pi$
$131$	0.840072	0.0733974	0.0366987	+	0.999326i	$0.488316\pi$
$132$	0.123002	0.0107060
$133$	4.49843	0.390064
$134$	−4.90920	−0.424090
$135$	−1.03081	−0.0887179
$136$	−11.3224	−0.970889
$137$	−15.0870	−1.28897	−0.644485	−	0.764617i	$-0.722928\pi$
$137$	−15.0870	−1.28897	−0.644485	+	0.764617i	$0.722928\pi$
$138$	−0.229269	−0.0195167
$139$	−4.93675	−0.418730	−0.209365	−	0.977838i	$-0.567140\pi$
$139$	−4.93675	−0.418730	−0.209365	+	0.977838i	$0.567140\pi$
$140$	4.55219	0.384730
$141$	1.44121	0.121372
$142$	7.47725	0.627477
$143$	2.97222	0.248550
$144$	−6.13023	−0.510853
$145$	3.93691	0.326943
$146$	2.97349	0.246088
$147$	0.0932437	0.00769061
$148$	12.8357	1.05508
$149$	22.4744	1.84117	0.920587	−	0.390537i	$-0.127711\pi$
$149$	22.4744	1.84117	0.920587	+	0.390537i	$0.127711\pi$
$150$	0.101002	0.00824674
$151$	8.83321	0.718837	0.359418	−	0.933177i	$-0.382975\pi$
$151$	8.83321	0.718837	0.359418	+	0.933177i	$0.382975\pi$
$152$	3.50536	0.284322
$153$	−15.7168	−1.27063
$154$	0.690261	0.0556229
$155$	−2.28028	−0.183156
$156$	1.97977	0.158509
$157$	−4.37859	−0.349450	−0.174725	−	0.984617i	$-0.555904\pi$
$157$	−4.37859	−0.349450	−0.174725	+	0.984617i	$0.555904\pi$
$158$	−0.780183	−0.0620681
$159$	0.951786	0.0754815
$160$	5.48679	0.433769
$161$	6.23310	0.491237
$162$	5.10835	0.401350
$163$	−17.3126	−1.35603	−0.678014	−	0.735049i	$-0.737159\pi$
$163$	−17.3126	−1.35603	−0.678014	+	0.735049i	$0.737159\pi$
$164$	8.60312	0.671791
$165$	−0.0741958	−0.00577613
$166$	−3.88922	−0.301862
$167$	4.89427	0.378730	0.189365	−	0.981907i	$-0.439357\pi$
$167$	4.89427	0.378730	0.189365	+	0.981907i	$0.439357\pi$
$168$	1.01446	0.0782673
$169$	34.8393	2.67994
$170$	3.09541	0.237407
$171$	4.86585	0.372101
$172$	11.5358	0.879594
$173$	−2.08921	−0.158840	−0.0794198	−	0.996841i	$-0.525307\pi$
$173$	−2.08921	−0.158840	−0.0794198	+	0.996841i	$0.525307\pi$
$174$	0.397634	0.0301446
$175$	−2.74591	−0.207572
$176$	−0.886914	−0.0668537
$177$	−1.09708	−0.0824614
$178$	0.584976	0.0438458
$179$	−17.4654	−1.30542	−0.652712	−	0.757606i	$-0.726369\pi$
$179$	−17.4654	−1.30542	−0.652712	+	0.757606i	$0.726369\pi$
$180$	4.92399	0.367013
$181$	12.0475	0.895485	0.447743	−	0.894162i	$-0.352228\pi$
$181$	12.0475	0.895485	0.447743	+	0.894162i	$0.352228\pi$
$182$	11.1101	0.823533
$183$	1.55444	0.114907
$184$	4.85708	0.358069
$185$	−7.74257	−0.569245
$186$	−0.230312	−0.0168873
$187$	−2.27389	−0.166283
$188$	−13.8379	−1.00923
$189$	2.83051	0.205890
$190$	−0.958324	−0.0695241
$191$	−7.90966	−0.572323	−0.286161	−	0.958181i	$-0.592379\pi$
$191$	−7.90966	−0.572323	−0.286161	+	0.958181i	$0.592379\pi$
$192$	−0.158536	−0.0114413
$193$	2.39653	0.172506	0.0862530	−	0.996273i	$-0.472511\pi$
$193$	2.39653	0.172506	0.0862530	+	0.996273i	$0.472511\pi$
$194$	0.484400	0.0347779
$195$	−1.19421	−0.0855195
$196$	−0.895288	−0.0639491
$197$	−16.9355	−1.20660	−0.603302	−	0.797513i	$-0.706149\pi$
$197$	−16.9355	−1.20660	−0.603302	+	0.797513i	$0.706149\pi$
$198$	0.746639	0.0530613
$199$	−25.0231	−1.77384	−0.886919	−	0.461926i	$-0.847159\pi$
$199$	−25.0231	−1.77384	−0.886919	+	0.461926i	$0.847159\pi$
$200$	−2.13973	−0.151301
$201$	−1.44898	−0.102203
$202$	−0.839666	−0.0590787
$203$	−10.8104	−0.758743
$204$	−1.51462	−0.106045
$205$	−5.18947	−0.362448
$206$	−3.52728	−0.245757
$207$	6.74219	0.468615
$208$	−14.2753	−0.989814
$209$	0.703985	0.0486957
$210$	−0.277342	−0.0191384
$211$	8.90186	0.612829	0.306414	−	0.951898i	$-0.400871\pi$
$211$	8.90186	0.612829	0.306414	+	0.951898i	$0.400871\pi$
$212$	−9.13865	−0.627645
$213$	2.20696	0.151218
$214$	−6.47607	−0.442695
$215$	−6.95847	−0.474564
$216$	2.20565	0.150075
$217$	6.26145	0.425055
$218$	5.26018	0.356264
$219$	0.877646	0.0593058
$220$	0.712397	0.0480298
$221$	−36.5993	−2.46194
$222$	−0.782011	−0.0524851
$223$	28.1197	1.88303	0.941516	−	0.336968i	$-0.109401\pi$
$223$	28.1197	1.88303	0.941516	+	0.336968i	$0.109401\pi$
$224$	−15.0663	−1.00666
$225$	−2.97019	−0.198013
$226$	−3.24627	−0.215939
$227$	−16.0209	−1.06334	−0.531671	−	0.846951i	$-0.678436\pi$
$227$	−16.0209	−1.06334	−0.531671	+	0.846951i	$0.678436\pi$
$228$	0.468919	0.0310549
$229$	6.71542	0.443767	0.221884	−	0.975073i	$-0.428779\pi$
$229$	6.71542	0.443767	0.221884	+	0.975073i	$0.428779\pi$
$230$	−1.32787	−0.0875570
$231$	0.203735	0.0134048
$232$	−8.42391	−0.553057
$233$	−9.20337	−0.602933	−0.301466	−	0.953477i	$-0.597476\pi$
$233$	−9.20337	−0.602933	−0.301466	+	0.953477i	$0.597476\pi$
$234$	12.0175	0.785608
$235$	8.34712	0.544506
$236$	10.5337	0.685684
$237$	−0.230276	−0.0149580
$238$	−8.49974	−0.550956
$239$	−1.01200	−0.0654610	−0.0327305	−	0.999464i	$-0.510420\pi$
$239$	−1.01200	−0.0654610	−0.0327305	+	0.999464i	$0.510420\pi$
$240$	0.356355	0.0230026
$241$	6.14399	0.395769	0.197885	−	0.980225i	$-0.436593\pi$
$241$	6.14399	0.395769	0.197885	+	0.980225i	$0.436593\pi$
$242$	−6.32671	−0.406696
$243$	4.60019	0.295102
$244$	−14.9251	−0.955480
$245$	0.540044	0.0345022
$246$	−0.524144	−0.0334182
$247$	11.3310	0.720972
$248$	4.87917	0.309828
$249$	−1.14793	−0.0727471
$250$	0.584976	0.0369971
$251$	15.4001	0.972049	0.486024	−	0.873945i	$-0.338447\pi$
$251$	15.4001	0.972049	0.486024	+	0.873945i	$0.338447\pi$
$252$	−13.5209	−0.851734
$253$	0.975453	0.0613262
$254$	−9.73395	−0.610762
$255$	0.913632	0.0572138
$256$	−4.89709	−0.306068
$257$	1.91078	0.119191	0.0595957	−	0.998223i	$-0.481019\pi$
$257$	1.91078	0.119191	0.0595957	+	0.998223i	$0.481019\pi$
$258$	−0.702816	−0.0437554
$259$	21.2604	1.32106
$260$	11.4664	0.711113
$261$	−11.6934	−0.723802
$262$	0.491421	0.0303601
$263$	−21.3509	−1.31655	−0.658277	−	0.752776i	$-0.728714\pi$
$263$	−21.3509	−1.31655	−0.658277	+	0.752776i	$0.728714\pi$
$264$	0.158759	0.00977092
$265$	5.51251	0.338631
$266$	2.63147	0.161346
$267$	0.172659	0.0105666
$268$	13.9125	0.849843
$269$	−3.71865	−0.226730	−0.113365	−	0.993553i	$-0.536163\pi$
$269$	−3.71865	−0.226730	−0.113365	+	0.993553i	$0.536163\pi$
$270$	−0.602998	−0.0366973
$271$	−29.0720	−1.76600	−0.882999	−	0.469375i	$-0.844479\pi$
$271$	−29.0720	−1.76600	−0.882999	+	0.469375i	$0.844479\pi$
$272$	10.9213	0.662200
$273$	3.27921	0.198467
$274$	−8.82553	−0.533170
$275$	−0.429723	−0.0259133
$276$	0.649741	0.0391098
$277$	−22.1766	−1.33246	−0.666230	−	0.745746i	$-0.732093\pi$
$277$	−22.1766	−1.33246	−0.666230	+	0.745746i	$0.732093\pi$
$278$	−2.88788	−0.173203
$279$	6.77286	0.405480
$280$	5.87550	0.351128
$281$	−27.5045	−1.64078	−0.820391	−	0.571803i	$-0.806244\pi$
$281$	−27.5045	−1.64078	−0.820391	+	0.571803i	$0.806244\pi$
$282$	0.843072	0.0502042
$283$	−11.7574	−0.698902	−0.349451	−	0.936955i	$-0.613632\pi$
$283$	−11.7574	−0.698902	−0.349451	+	0.936955i	$0.613632\pi$
$284$	−21.1903	−1.25741
$285$	−0.282856	−0.0167549
$286$	1.73868	0.102810
$287$	14.2498	0.841141
$288$	−16.2968	−0.960299
$289$	11.0002	0.647073
$290$	2.30300	0.135237
$291$	0.142974	0.00838127
$292$	−8.42680	−0.493141
$293$	−12.7263	−0.743476	−0.371738	−	0.928338i	$-0.621238\pi$
$293$	−12.7263	−0.743476	−0.371738	+	0.928338i	$0.621238\pi$
$294$	0.0545453	0.00318115
$295$	−6.35400	−0.369944
$296$	16.5670	0.962935
$297$	0.442963	0.0257033
$298$	13.1470	0.761584
$299$	15.7004	0.907975
$300$	−0.286235	−0.0165258
$301$	19.1074	1.10133
$302$	5.16721	0.297340
$303$	−0.247833	−0.0142376
$304$	−3.38117	−0.193924
$305$	9.00292	0.515506
$306$	−9.19396	−0.525584
$307$	−4.59039	−0.261987	−0.130994	−	0.991383i	$-0.541817\pi$
$307$	−4.59039	−0.261987	−0.130994	+	0.991383i	$0.541817\pi$
$308$	−1.95618	−0.111464
$309$	−1.04110	−0.0592260
$310$	−1.33391	−0.0757609
$311$	−15.6135	−0.885361	−0.442680	−	0.896679i	$-0.645972\pi$
$311$	−15.6135	−0.885361	−0.442680	+	0.896679i	$0.645972\pi$
$312$	2.55529	0.144665
$313$	22.3598	1.26385	0.631925	−	0.775029i	$-0.282265\pi$
$313$	22.3598	1.26385	0.631925	+	0.775029i	$0.282265\pi$
$314$	−2.56137	−0.144547
$315$	8.15588	0.459532
$316$	2.21102	0.124379
$317$	19.3949	1.08932	0.544662	−	0.838655i	$-0.316658\pi$
$317$	19.3949	1.08932	0.544662	+	0.838655i	$0.316658\pi$
$318$	0.556771	0.0312222
$319$	−1.69178	−0.0947217
$320$	−0.918199	−0.0513289
$321$	−1.91146	−0.106687
$322$	3.64621	0.203195
$323$	−8.66873	−0.482341
$324$	−14.4769	−0.804273
$325$	−6.91659	−0.383664
$326$	−10.1274	−0.560908
$327$	1.55258	0.0858576
$328$	11.1040	0.613118
$329$	−22.9205	−1.26365
$330$	−0.0434027	−0.00238924
$331$	25.2472	1.38771	0.693855	−	0.720115i	$-0.255911\pi$
$331$	25.2472	1.38771	0.693855	+	0.720115i	$0.255911\pi$
$332$	11.0219	0.604908
$333$	22.9969	1.26022
$334$	2.86303	0.156658
$335$	−8.39214	−0.458512
$336$	−0.978521	−0.0533827
$337$	4.05916	0.221116	0.110558	−	0.993870i	$-0.464736\pi$
$337$	4.05916	0.221116	0.110558	+	0.993870i	$0.464736\pi$
$338$	20.3801	1.10853
$339$	−0.958159	−0.0520400
$340$	−8.77231	−0.475745
$341$	0.979889	0.0530640
$342$	2.84640	0.153916
$343$	17.7385	0.957788
$344$	14.8892	0.802773
$345$	−0.391929	−0.0211007
$346$	−1.22214	−0.0657025
$347$	25.2031	1.35297	0.676487	−	0.736454i	$-0.263501\pi$
$347$	25.2031	1.35297	0.676487	+	0.736454i	$0.263501\pi$
$348$	−1.12688	−0.0604073
$349$	−11.8501	−0.634321	−0.317161	−	0.948372i	$-0.602729\pi$
$349$	−11.8501	−0.634321	−0.317161	+	0.948372i	$0.602729\pi$
$350$	−1.60629	−0.0858600
$351$	7.12969	0.380555
$352$	−2.35780	−0.125671
$353$	13.9770	0.743921	0.371961	−	0.928249i	$-0.378686\pi$
$353$	13.9770	0.743921	0.371961	+	0.928249i	$0.378686\pi$
$354$	−0.641764	−0.0341094
$355$	12.7822	0.678406
$356$	−1.65780	−0.0878634
$357$	−2.50875	−0.132777
$358$	−10.2168	−0.539976
$359$	12.6401	0.667120	0.333560	−	0.942729i	$-0.391750\pi$
$359$	12.6401	0.667120	0.333560	+	0.942729i	$0.391750\pi$
$360$	6.35539	0.334958
$361$	−16.3162	−0.858748
$362$	7.04751	0.370409
$363$	−1.86737	−0.0980115
$364$	−31.4856	−1.65029
$365$	5.08311	0.266062
$366$	0.909309	0.0475303
$367$	−18.5135	−0.966398	−0.483199	−	0.875510i	$-0.660525\pi$
$367$	−18.5135	−0.966398	−0.483199	+	0.875510i	$0.660525\pi$
$368$	−4.68500	−0.244223
$369$	15.4137	0.802405
$370$	−4.52921	−0.235463
$371$	−15.1369	−0.785867
$372$	0.652696	0.0338407
$373$	29.1116	1.50734	0.753670	−	0.657253i	$-0.228282\pi$
$373$	29.1116	1.50734	0.753670	+	0.657253i	$0.228282\pi$
$374$	−1.33017	−0.0687815
$375$	0.172659	0.00891609
$376$	−17.8605	−0.921087
$377$	−27.2300	−1.40242
$378$	1.65578	0.0851642
$379$	16.6575	0.855637	0.427818	−	0.903865i	$-0.359282\pi$
$379$	16.6575	0.855637	0.427818	+	0.903865i	$0.359282\pi$
$380$	2.71586	0.139321
$381$	−2.87304	−0.147190
$382$	−4.62696	−0.236736
$383$	6.49778	0.332021	0.166010	−	0.986124i	$-0.446911\pi$
$383$	6.49778	0.332021	0.166010	+	0.986124i	$0.446911\pi$
$384$	−1.98743	−0.101421
$385$	1.17998	0.0601375
$386$	1.40191	0.0713554
$387$	20.6680	1.05061
$388$	−1.37277	−0.0696921
$389$	26.9389	1.36585	0.682927	−	0.730486i	$-0.260706\pi$
$389$	26.9389	1.36585	0.682927	+	0.730486i	$0.260706\pi$
$390$	−0.698586	−0.0353743
$391$	−12.0115	−0.607449
$392$	−1.15555	−0.0583639
$393$	0.145046	0.00731662
$394$	−9.90686	−0.499100
$395$	−1.33370	−0.0671058
$396$	−2.11595	−0.106331
$397$	26.0878	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$397$	26.0878	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$398$	−14.6379	−0.733730
$399$	0.776697	0.0388835
$400$	2.06392	0.103196
$401$	37.7689	1.88609	0.943045	−	0.332666i	$-0.107948\pi$
$401$	37.7689	1.88609	0.943045	+	0.332666i	$0.107948\pi$
$402$	−0.847619	−0.0422754
$403$	15.7718	0.785647
$404$	2.37959	0.118389
$405$	8.73259	0.433926
$406$	−6.32383	−0.313847
$407$	3.32716	0.164921
$408$	−1.95492	−0.0967830
$409$	−32.8776	−1.62569	−0.812846	−	0.582479i	$-0.802083\pi$
$409$	−32.8776	−1.62569	−0.812846	+	0.582479i	$0.802083\pi$
$410$	−3.03571	−0.149923
$411$	−2.60491	−0.128491
$412$	9.99620	0.492477
$413$	17.4475	0.858537
$414$	3.94402	0.193838
$415$	−6.64852	−0.326363
$416$	−37.9499	−1.86065
$417$	−0.852377	−0.0417411
$418$	0.411814	0.0201425
$419$	−17.5007	−0.854964	−0.427482	−	0.904024i	$-0.640599\pi$
$419$	−17.5007	−0.854964	−0.427482	+	0.904024i	$0.640599\pi$
$420$	0.785978	0.0383518
$421$	2.78590	0.135777	0.0678883	−	0.997693i	$-0.478374\pi$
$421$	2.78590	0.135777	0.0678883	+	0.997693i	$0.478374\pi$
$422$	5.20737	0.253491
$423$	−24.7925	−1.20545
$424$	−11.7953	−0.572828
$425$	5.29153	0.256677
$426$	1.29102	0.0625500
$427$	−24.7213	−1.19635
$428$	18.3530	0.887126
$429$	0.513182	0.0247767
$430$	−4.07053	−0.196299
$431$	14.3931	0.693290	0.346645	−	0.937996i	$-0.387321\pi$
$431$	14.3931	0.693290	0.346645	+	0.937996i	$0.387321\pi$
$432$	−2.12751	−0.102360
$433$	6.91601	0.332362	0.166181	−	0.986095i	$-0.446856\pi$
$433$	6.91601	0.332362	0.166181	+	0.986095i	$0.446856\pi$
$434$	3.66280	0.175820
$435$	0.679745	0.0325913
$436$	−14.9072	−0.713925
$437$	3.71870	0.177890
$438$	0.513402	0.0245313
$439$	33.3144	1.59001	0.795005	−	0.606602i	$-0.207468\pi$
$439$	33.3144	1.59001	0.795005	+	0.606602i	$0.207468\pi$
$440$	0.919490	0.0438350
$441$	−1.60403	−0.0763826
$442$	−21.4097	−1.01836
$443$	−17.6967	−0.840794	−0.420397	−	0.907340i	$-0.638109\pi$
$443$	−17.6967	−0.840794	−0.420397	+	0.907340i	$0.638109\pi$
$444$	2.21620	0.105176
$445$	1.00000	0.0474045
$446$	16.4493	0.778898
$447$	3.88042	0.183537
$448$	2.52129	0.119120
$449$	42.0230	1.98319	0.991593	−	0.129394i	$-0.0413032\pi$
$449$	42.0230	1.98319	0.991593	+	0.129394i	$0.0413032\pi$
$450$	−1.73749	−0.0819060
$451$	2.23004	0.105008
$452$	9.19984	0.432724
$453$	1.52514	0.0716572
$454$	−9.37182	−0.439841
$455$	18.9924	0.890376
$456$	0.605233	0.0283426
$457$	−16.9157	−0.791283	−0.395641	−	0.918405i	$-0.629478\pi$
$457$	−16.9157	−0.791283	−0.395641	+	0.918405i	$0.629478\pi$
$458$	3.92836	0.183560
$459$	−5.45455	−0.254597
$460$	3.76314	0.175457
$461$	−9.77419	−0.455229	−0.227615	−	0.973751i	$-0.573093\pi$
$461$	−9.77419	−0.455229	−0.227615	+	0.973751i	$0.573093\pi$
$462$	0.119180	0.00554476
$463$	40.4822	1.88137	0.940684	−	0.339283i	$-0.110184\pi$
$463$	40.4822	1.88137	0.940684	+	0.339283i	$0.110184\pi$
$464$	8.12547	0.377216
$465$	−0.393712	−0.0182579
$466$	−5.38375	−0.249397
$467$	−21.7726	−1.00752	−0.503759	−	0.863844i	$-0.668050\pi$
$467$	−21.7726	−1.00752	−0.503759	+	0.863844i	$0.668050\pi$
$468$	−34.0572	−1.57430
$469$	23.0441	1.06408
$470$	4.88286	0.225230
$471$	−0.756005	−0.0348349
$472$	13.5958	0.625798
$473$	2.99022	0.137490
$474$	−0.134706	−0.00618725
$475$	−1.63823	−0.0751671
$476$	24.0880	1.10407
$477$	−16.3732	−0.749677
$478$	−0.591996	−0.0270773
$479$	−14.3976	−0.657842	−0.328921	−	0.944357i	$-0.606685\pi$
$479$	−14.3976	−0.657842	−0.328921	+	0.944357i	$0.606685\pi$
$480$	0.947346	0.0432403
$481$	53.5522	2.44177
$482$	3.59408	0.163706
$483$	1.07620	0.0489689
$484$	17.9297	0.814987
$485$	0.828069	0.0376007
$486$	2.69100	0.122066
$487$	−29.7669	−1.34886	−0.674432	−	0.738337i	$-0.735612\pi$
$487$	−29.7669	−1.34886	−0.674432	+	0.738337i	$0.735612\pi$
$488$	−19.2638	−0.872031
$489$	−2.98918	−0.135176
$490$	0.315913	0.0142715
$491$	−32.2121	−1.45371	−0.726857	−	0.686789i	$-0.759020\pi$
$491$	−32.2121	−1.45371	−0.726857	+	0.686789i	$0.759020\pi$
$492$	1.48541	0.0669674
$493$	20.8323	0.938238
$494$	6.62834	0.298223
$495$	1.27636	0.0573681
$496$	−4.70631	−0.211320
$497$	−35.0987	−1.57439
$498$	−0.671511	−0.0300911
$499$	20.7011	0.926707	0.463353	−	0.886174i	$-0.346646\pi$
$499$	20.7011	0.926707	0.463353	+	0.886174i	$0.346646\pi$
$500$	−1.65780	−0.0741392
$501$	0.845041	0.0377536
$502$	9.00871	0.402078
$503$	39.1454	1.74541	0.872703	−	0.488251i	$-0.162365\pi$
$503$	39.1454	1.74541	0.872703	+	0.488251i	$0.162365\pi$
$504$	−17.4514	−0.777345
$505$	−1.43539	−0.0638739
$506$	0.570616	0.0253670
$507$	6.01533	0.267150
$508$	27.5857	1.22392
$509$	−8.05887	−0.357203	−0.178602	−	0.983921i	$-0.557157\pi$
$509$	−8.05887	−0.357203	−0.178602	+	0.983921i	$0.557157\pi$
$510$	0.534452	0.0236659
$511$	−13.9578	−0.617456
$512$	20.1567	0.890811
$513$	1.68870	0.0745580
$514$	1.11776	0.0493023
$515$	−6.02978	−0.265704
$516$	1.99176	0.0876823
$517$	−3.58695	−0.157754
$518$	12.4368	0.546443
$519$	−0.360722	−0.0158339
$520$	14.7996	0.649006
$521$	5.83455	0.255616	0.127808	−	0.991799i	$-0.459206\pi$
$521$	5.83455	0.255616	0.127808	+	0.991799i	$0.459206\pi$
$522$	−6.84034	−0.299393
$523$	−33.8865	−1.48175	−0.740877	−	0.671640i	$-0.765590\pi$
$523$	−33.8865	−1.48175	−0.740877	+	0.671640i	$0.765590\pi$
$524$	−1.39267	−0.0608392
$525$	−0.474108	−0.0206918
$526$	−12.4898	−0.544580
$527$	−12.0662	−0.525610
$528$	−0.153134	−0.00666431
$529$	−17.8473	−0.775970
$530$	3.22468	0.140071
$531$	18.8726	0.819000
$532$	−7.45752	−0.323325
$533$	35.8935	1.55472
$534$	0.101002	0.00437076
$535$	−11.0707	−0.478627
$536$	17.9569	0.775619
$537$	−3.01556	−0.130131
$538$	−2.17532	−0.0937847
$539$	−0.232070	−0.00999595
$540$	1.70888	0.0735384
$541$	−22.7941	−0.979995	−0.489997	−	0.871724i	$-0.663002\pi$
$541$	−22.7941	−0.979995	−0.489997	+	0.871724i	$0.663002\pi$
$542$	−17.0064	−0.730488
$543$	2.08012	0.0892664
$544$	29.0335	1.24480
$545$	8.99213	0.385181
$546$	1.91826	0.0820939
$547$	21.1319	0.903533	0.451767	−	0.892136i	$-0.350794\pi$
$547$	21.1319	0.903533	0.451767	+	0.892136i	$0.350794\pi$
$548$	25.0113	1.06843
$549$	−26.7404	−1.14125
$550$	−0.251378	−0.0107188
$551$	−6.44956	−0.274761
$552$	0.838621	0.0356941
$553$	3.66223	0.155734
$554$	−12.9727	−0.551159
$555$	−1.33683	−0.0567451
$556$	8.18417	0.347086
$557$	19.8251	0.840017	0.420008	−	0.907520i	$-0.362027\pi$
$557$	19.8251	0.840017	0.420008	+	0.907520i	$0.362027\pi$
$558$	3.96196	0.167723
$559$	48.1289	2.03564
$560$	−5.66735	−0.239489
$561$	−0.392609	−0.0165760
$562$	−16.0895	−0.678693
$563$	−10.7352	−0.452433	−0.226217	−	0.974077i	$-0.572636\pi$
$563$	−10.7352	−0.452433	−0.226217	+	0.974077i	$0.572636\pi$
$564$	−2.38924	−0.100605
$565$	−5.54942	−0.233466
$566$	−6.87776	−0.289094
$567$	−23.9789	−1.00702
$568$	−27.3503	−1.14759
$569$	19.1468	0.802677	0.401338	−	0.915930i	$-0.368545\pi$
$569$	19.1468	0.802677	0.401338	+	0.915930i	$0.368545\pi$
$570$	−0.165464	−0.00693051
$571$	−22.6449	−0.947661	−0.473831	−	0.880616i	$-0.657129\pi$
$571$	−22.6449	−0.947661	−0.473831	+	0.880616i	$0.657129\pi$
$572$	−4.92736	−0.206023
$573$	−1.36568	−0.0570520
$574$	8.33581	0.347930
$575$	−2.26995	−0.0946637
$576$	2.72722	0.113634
$577$	−24.6727	−1.02714	−0.513570	−	0.858048i	$-0.671677\pi$
$577$	−24.6727	−1.02714	−0.513570	+	0.858048i	$0.671677\pi$
$578$	6.43487	0.267655
$579$	0.413783	0.0171963
$580$	−6.52663	−0.271003
$581$	18.2563	0.757398
$582$	0.0836362	0.00346683
$583$	−2.36885	−0.0981078
$584$	−10.8765	−0.450071
$585$	20.5436	0.849373
$586$	−7.44456	−0.307532
$587$	−14.3041	−0.590395	−0.295198	−	0.955436i	$-0.595385\pi$
$587$	−14.3041	−0.590395	−0.295198	+	0.955436i	$0.595385\pi$
$588$	−0.154580	−0.00637476
$589$	3.73562	0.153923
$590$	−3.71694	−0.153024
$591$	−2.92407	−0.120280
$592$	−15.9800	−0.656775
$593$	31.7751	1.30485	0.652423	−	0.757855i	$-0.273752\pi$
$593$	31.7751	1.30485	0.652423	+	0.757855i	$0.273752\pi$
$594$	0.259122	0.0106319
$595$	−14.5301	−0.595675
$596$	−37.2581	−1.52615
$597$	−4.32047	−0.176825
$598$	9.18432	0.375575
$599$	−33.7567	−1.37926	−0.689631	−	0.724161i	$-0.742227\pi$
$599$	−33.7567	−1.37926	−0.689631	+	0.724161i	$0.742227\pi$
$600$	−0.369444	−0.0150825
$601$	−34.6759	−1.41446	−0.707229	−	0.706985i	$-0.750055\pi$
$601$	−34.6759	−1.41446	−0.707229	+	0.706985i	$0.750055\pi$
$602$	11.1773	0.455554
$603$	24.9262	1.01508
$604$	−14.6437	−0.595845
$605$	−10.8153	−0.439706
$606$	−0.144976	−0.00588926
$607$	−12.8661	−0.522217	−0.261109	−	0.965309i	$-0.584088\pi$
$607$	−12.8661	−0.522217	−0.261109	+	0.965309i	$0.584088\pi$
$608$	−8.98862	−0.364537
$609$	−1.86652	−0.0756353
$610$	5.26649	0.213234
$611$	−57.7336	−2.33565
$612$	26.0554	1.05323
$613$	16.6348	0.671872	0.335936	−	0.941885i	$-0.390947\pi$
$613$	16.6348	0.671872	0.335936	+	0.941885i	$0.390947\pi$
$614$	−2.68526	−0.108368
$615$	−0.896011	−0.0361306
$616$	−2.52484	−0.101729
$617$	33.5170	1.34934	0.674671	−	0.738118i	$-0.264285\pi$
$617$	33.5170	1.34934	0.674671	+	0.738118i	$0.264285\pi$
$618$	−0.609017	−0.0244983
$619$	34.1671	1.37329	0.686646	−	0.726992i	$-0.259082\pi$
$619$	34.1671	1.37329	0.686646	+	0.726992i	$0.259082\pi$
$620$	3.78025	0.151819
$621$	2.33989	0.0938966
$622$	−9.13352	−0.366221
$623$	−2.74591	−0.110013
$624$	−2.46476	−0.0986695
$625$	1.00000	0.0400000
$626$	13.0799	0.522779
$627$	0.121550	0.00485422
$628$	7.25885	0.289660
$629$	−40.9700	−1.63358
$630$	4.77099	0.190081
$631$	−2.76256	−0.109976	−0.0549878	−	0.998487i	$-0.517512\pi$
$631$	−2.76256	−0.109976	−0.0549878	+	0.998487i	$0.517512\pi$
$632$	2.85376	0.113516
$633$	1.53699	0.0610898
$634$	11.3455	0.450588
$635$	−16.6399	−0.660335
$636$	−1.57787	−0.0625668
$637$	−3.73527	−0.147997
$638$	−0.989652	−0.0391807
$639$	−37.9654	−1.50189
$640$	−11.5107	−0.455001
$641$	46.4551	1.83487	0.917434	−	0.397889i	$-0.130257\pi$
$641$	46.4551	1.83487	0.917434	+	0.397889i	$0.130257\pi$
$642$	−1.11816	−0.0441301
$643$	−45.9246	−1.81109	−0.905545	−	0.424249i	$-0.860538\pi$
$643$	−45.9246	−1.81109	−0.905545	+	0.424249i	$0.860538\pi$
$644$	−10.3333	−0.407187
$645$	−1.20144	−0.0473068
$646$	−5.07099	−0.199516
$647$	14.4744	0.569048	0.284524	−	0.958669i	$-0.408164\pi$
$647$	14.4744	0.569048	0.284524	+	0.958669i	$0.408164\pi$
$648$	−18.6853	−0.734030
$649$	2.73046	0.107180
$650$	−4.04604	−0.158699
$651$	1.08110	0.0423716
$652$	28.7009	1.12401
$653$	29.4446	1.15226	0.576129	−	0.817359i	$-0.304563\pi$
$653$	29.4446	1.15226	0.576129	+	0.817359i	$0.304563\pi$
$654$	0.908219	0.0355142
$655$	0.840072	0.0328243
$656$	−10.7106	−0.418181
$657$	−15.0978	−0.589021
$658$	−13.4079	−0.522695
$659$	−25.5581	−0.995601	−0.497800	−	0.867292i	$-0.665859\pi$
$659$	−25.5581	−0.995601	−0.497800	+	0.867292i	$0.665859\pi$
$660$	0.123002	0.00478785
$661$	−22.7900	−0.886428	−0.443214	−	0.896416i	$-0.646162\pi$
$661$	−22.7900	−0.886428	−0.443214	+	0.896416i	$0.646162\pi$
$662$	14.7690	0.574013
$663$	−6.31922	−0.245418
$664$	14.2260	0.552076
$665$	4.49843	0.174442
$666$	13.4526	0.521278
$667$	−8.93661	−0.346027
$668$	−8.11373	−0.313930
$669$	4.85512	0.187710
$670$	−4.90920	−0.189659
$671$	−3.86877	−0.149352
$672$	−2.60133	−0.100349
$673$	26.0465	1.00402	0.502009	−	0.864862i	$-0.332594\pi$
$673$	26.0465	1.00402	0.502009	+	0.864862i	$0.332594\pi$
$674$	2.37451	0.0914627
$675$	−1.03081	−0.0396759
$676$	−57.7567	−2.22141
$677$	8.03655	0.308870	0.154435	−	0.988003i	$-0.450644\pi$
$677$	8.03655	0.308870	0.154435	+	0.988003i	$0.450644\pi$
$678$	−0.560500	−0.0215259
$679$	−2.27381	−0.0872606
$680$	−11.3224	−0.434195
$681$	−2.76615	−0.105999
$682$	0.573211	0.0219494
$683$	−8.08860	−0.309502	−0.154751	−	0.987954i	$-0.549457\pi$
$683$	−8.08860	−0.309502	−0.154751	+	0.987954i	$0.549457\pi$
$684$	−8.06662	−0.308435
$685$	−15.0870	−0.576445
$686$	10.3766	0.396180
$687$	1.15948	0.0442369
$688$	−14.3617	−0.547535
$689$	−38.1278	−1.45255
$690$	−0.229269	−0.00872812
$691$	27.2840	1.03793	0.518966	−	0.854795i	$-0.326317\pi$
$691$	27.2840	1.03793	0.518966	+	0.854795i	$0.326317\pi$
$692$	3.46350	0.131662
$693$	−3.50477	−0.133135
$694$	14.7432	0.559645
$695$	−4.93675	−0.187262
$696$	−1.45447	−0.0551315
$697$	−27.4602	−1.04013
$698$	−6.93202	−0.262381
$699$	−1.58905	−0.0601033
$700$	4.55219	0.172056
$701$	−9.03883	−0.341392	−0.170696	−	0.985324i	$-0.554602\pi$
$701$	−9.03883	−0.341392	−0.170696	+	0.985324i	$0.554602\pi$
$702$	4.17069	0.157413
$703$	12.6841	0.478390
$704$	0.394571	0.0148710
$705$	1.44121	0.0542790
$706$	8.17621	0.307716
$707$	3.94145	0.148233
$708$	1.81874	0.0683524
$709$	−46.5488	−1.74818	−0.874088	−	0.485767i	$-0.838540\pi$
$709$	−46.5488	−1.74818	−0.874088	+	0.485767i	$0.838540\pi$
$710$	7.47725	0.280616
$711$	3.96135	0.148562
$712$	−2.13973	−0.0801896
$713$	5.17613	0.193848
$714$	−1.46756	−0.0549220
$715$	2.97222	0.111155
$716$	28.9542	1.08207
$717$	−0.174732	−0.00652547
$718$	7.39416	0.275948
$719$	−49.3092	−1.83892	−0.919462	−	0.393180i	$-0.871375\pi$
$719$	−49.3092	−1.83892	−0.919462	+	0.393180i	$0.871375\pi$
$720$	−6.13023	−0.228460
$721$	16.5573	0.616625
$722$	−9.54458	−0.355213
$723$	1.06082	0.0394522
$724$	−19.9724	−0.742269
$725$	3.93691	0.146213
$726$	−1.09237	−0.0405415
$727$	−39.4930	−1.46471	−0.732357	−	0.680921i	$-0.761580\pi$
$727$	−39.4930	−1.46471	−0.732357	+	0.680921i	$0.761580\pi$
$728$	−40.6385	−1.50616
$729$	−25.4035	−0.940870
$730$	2.97349	0.110054
$731$	−36.8209	−1.36187
$732$	−2.57696	−0.0952470
$733$	−4.32917	−0.159902	−0.0799508	−	0.996799i	$-0.525476\pi$
$733$	−4.32917	−0.159902	−0.0799508	+	0.996799i	$0.525476\pi$
$734$	−10.8300	−0.399741
$735$	0.0932437	0.00343935
$736$	−12.4548	−0.459089
$737$	3.60630	0.132840
$738$	9.01664	0.331907
$739$	2.52150	0.0927549	0.0463775	−	0.998924i	$-0.485232\pi$
$739$	2.52150	0.0927549	0.0463775	+	0.998924i	$0.485232\pi$
$740$	12.8357	0.471848
$741$	1.95640	0.0718700
$742$	−8.85470	−0.325066
$743$	43.3061	1.58875	0.794373	−	0.607431i	$-0.207800\pi$
$743$	43.3061	1.58875	0.794373	+	0.607431i	$0.207800\pi$
$744$	0.842435	0.0308852
$745$	22.4744	0.823398
$746$	17.0296	0.623497
$747$	19.7474	0.722518
$748$	3.76967	0.137833
$749$	30.3991	1.11076
$750$	0.101002	0.00368805
$751$	43.4241	1.58457	0.792284	−	0.610153i	$-0.208892\pi$
$751$	43.4241	1.58457	0.792284	+	0.610153i	$0.208892\pi$
$752$	17.2278	0.628232
$753$	2.65898	0.0968986
$754$	−15.9289	−0.580096
$755$	8.83321	0.321474
$756$	−4.69243	−0.170662
$757$	46.8153	1.70153	0.850765	−	0.525547i	$-0.176139\pi$
$757$	46.8153	1.70153	0.850765	+	0.525547i	$0.176139\pi$
$758$	9.74421	0.353926
$759$	0.168421	0.00611330
$760$	3.50536	0.127153
$761$	−48.9144	−1.77315	−0.886573	−	0.462588i	$-0.846921\pi$
$761$	−48.9144	−1.77315	−0.886573	+	0.462588i	$0.846921\pi$
$762$	−1.68066	−0.0608838
$763$	−24.6916	−0.893896
$764$	13.1127	0.474400
$765$	−15.7168	−0.568243
$766$	3.80104	0.137337
$767$	43.9480	1.58687
$768$	−0.845528	−0.0305104
$769$	12.2118	0.440370	0.220185	−	0.975458i	$-0.429334\pi$
$769$	12.2118	0.440370	0.220185	+	0.975458i	$0.429334\pi$
$770$	0.690261	0.0248753
$771$	0.329914	0.0118816
$772$	−3.97298	−0.142991
$773$	−23.1151	−0.831391	−0.415695	−	0.909504i	$-0.636462\pi$
$773$	−23.1151	−0.831391	−0.415695	+	0.909504i	$0.636462\pi$
$774$	12.0903	0.434575
$775$	−2.28028	−0.0819100
$776$	−1.77184	−0.0636053
$777$	3.67081	0.131690
$778$	15.7586	0.564973
$779$	8.50154	0.304599
$780$	1.97977	0.0708873
$781$	−5.49279	−0.196548
$782$	−7.02645	−0.251265
$783$	−4.05821	−0.145028
$784$	1.11461	0.0398074
$785$	−4.37859	−0.156279
$786$	0.0848485	0.00302645
$787$	−24.5229	−0.874147	−0.437073	−	0.899426i	$-0.643985\pi$
$787$	−24.5229	−0.874147	−0.437073	+	0.899426i	$0.643985\pi$
$788$	28.0757	1.00016
$789$	−3.68644	−0.131241
$790$	−0.780183	−0.0277577
$791$	15.2382	0.541809
$792$	−2.73106	−0.0970440
$793$	−62.2696	−2.21126
$794$	15.2607	0.541582
$795$	0.951786	0.0337564
$796$	41.4833	1.47034
$797$	−46.4290	−1.64460	−0.822299	−	0.569055i	$-0.807309\pi$
$797$	−46.4290	−1.64460	−0.822299	+	0.569055i	$0.807309\pi$
$798$	0.454349	0.0160838
$799$	44.1690	1.56259
$800$	5.48679	0.193987
$801$	−2.97019	−0.104946
$802$	22.0939	0.780163
$803$	−2.18433	−0.0770833
$804$	2.40213	0.0847165
$805$	6.23310	0.219688
$806$	9.22610	0.324975
$807$	−0.642060	−0.0226016
$808$	3.07133	0.108049
$809$	−9.67473	−0.340145	−0.170073	−	0.985432i	$-0.554400\pi$
$809$	−9.67473	−0.340145	−0.170073	+	0.985432i	$0.554400\pi$
$810$	5.10835	0.179489
$811$	27.8332	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$811$	27.8332	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$812$	17.9216	0.628923
$813$	−5.01955	−0.176043
$814$	1.94631	0.0682181
$815$	−17.3126	−0.606434
$816$	1.88566	0.0660114
$817$	11.3996	0.398820
$818$	−19.2326	−0.672451
$819$	−56.4109	−1.97116
$820$	8.60312	0.300434
$821$	−37.2035	−1.29841	−0.649206	−	0.760612i	$-0.724899\pi$
$821$	−37.2035	−1.29841	−0.649206	+	0.760612i	$0.724899\pi$
$822$	−1.52381	−0.0531490
$823$	−25.9987	−0.906257	−0.453129	−	0.891445i	$-0.649692\pi$
$823$	−25.9987	−0.906257	−0.453129	+	0.891445i	$0.649692\pi$
$824$	12.9021	0.449465
$825$	−0.0741958	−0.00258316
$826$	10.2064	0.355126
$827$	10.2724	0.357207	0.178603	−	0.983921i	$-0.442842\pi$
$827$	10.2724	0.357207	0.178603	+	0.983921i	$0.442842\pi$
$828$	−11.1772	−0.388436
$829$	13.8050	0.479466	0.239733	−	0.970839i	$-0.422940\pi$
$829$	13.8050	0.479466	0.239733	+	0.970839i	$0.422940\pi$
$830$	−3.88922	−0.134997
$831$	−3.82899	−0.132826
$832$	6.35081	0.220175
$833$	2.85766	0.0990120
$834$	−0.498619	−0.0172658
$835$	4.89427	0.169373
$836$	−1.16707	−0.0403639
$837$	2.35053	0.0812463
$838$	−10.2375	−0.353647
$839$	36.6994	1.26700	0.633502	−	0.773741i	$-0.281617\pi$
$839$	36.6994	1.26700	0.633502	+	0.773741i	$0.281617\pi$
$840$	1.01446	0.0350022
$841$	−13.5007	−0.465542
$842$	1.62969	0.0561627
$843$	−4.74891	−0.163561
$844$	−14.7575	−0.507975
$845$	34.8393	1.19851
$846$	−14.5030	−0.498624
$847$	29.6980	1.02043
$848$	11.3774	0.390700
$849$	−2.03002	−0.0696700
$850$	3.09541	0.106172
$851$	17.5753	0.602473
$852$	−3.65871	−0.125345
$853$	−42.5986	−1.45855	−0.729275	−	0.684221i	$-0.760142\pi$
$853$	−42.5986	−1.45855	−0.729275	+	0.684221i	$0.760142\pi$
$854$	−14.4613	−0.494857
$855$	4.86585	0.166408
$856$	23.6882	0.809646
$857$	53.8456	1.83933	0.919665	−	0.392704i	$-0.128460\pi$
$857$	53.8456	1.83933	0.919665	+	0.392704i	$0.128460\pi$
$858$	0.300199	0.0102486
$859$	−3.35768	−0.114563	−0.0572813	−	0.998358i	$-0.518243\pi$
$859$	−3.35768	−0.114563	−0.0572813	+	0.998358i	$0.518243\pi$
$860$	11.5358	0.393367
$861$	2.46037	0.0838491
$862$	8.41959	0.286772
$863$	58.0162	1.97489	0.987447	−	0.157949i	$-0.0504881\pi$
$863$	58.0162	1.97489	0.987447	+	0.157949i	$0.0504881\pi$
$864$	−5.65584	−0.192415
$865$	−2.08921	−0.0710353
$866$	4.04570	0.137478
$867$	1.89930	0.0645034
$868$	−10.3803	−0.352329
$869$	0.573123	0.0194419
$870$	0.397634	0.0134811
$871$	58.0450	1.96678
$872$	−19.2407	−0.651572
$873$	−2.45952	−0.0832421
$874$	2.17535	0.0735824
$875$	−2.74591	−0.0928288
$876$	−1.45497	−0.0491587
$877$	−33.8297	−1.14235	−0.571173	−	0.820830i	$-0.693512\pi$
$877$	−33.8297	−1.14235	−0.571173	+	0.820830i	$0.693512\pi$
$878$	19.4881	0.657692
$879$	−2.19731	−0.0741134
$880$	−0.886914	−0.0298979
$881$	17.7123	0.596744	0.298372	−	0.954450i	$-0.403556\pi$
$881$	17.7123	0.596744	0.298372	+	0.954450i	$0.403556\pi$
$882$	−0.938321	−0.0315949
$883$	−46.8318	−1.57602	−0.788008	−	0.615665i	$-0.788888\pi$
$883$	−46.8318	−1.57602	−0.788008	+	0.615665i	$0.788888\pi$
$884$	60.6745	2.04070
$885$	−1.09708	−0.0368779
$886$	−10.3521	−0.347786
$887$	−19.8302	−0.665832	−0.332916	−	0.942957i	$-0.608033\pi$
$887$	−19.8302	−0.665832	−0.332916	+	0.942957i	$0.608033\pi$
$888$	2.86044	0.0959902
$889$	45.6918	1.53245
$890$	0.584976	0.0196084
$891$	−3.75260	−0.125717
$892$	−46.6169	−1.56085
$893$	−13.6745	−0.457599
$894$	2.26995	0.0759184
$895$	−17.4654	−0.583803
$896$	31.6074	1.05593
$897$	2.71081	0.0905114
$898$	24.5824	0.820326
$899$	−8.97726	−0.299408
$900$	4.92399	0.164133
$901$	29.1696	0.971779
$902$	1.30452	0.0434357
$903$	3.29906	0.109786
$904$	11.8742	0.394931
$905$	12.0475	0.400473
$906$	0.892168	0.0296403
$907$	−4.41800	−0.146697	−0.0733487	−	0.997306i	$-0.523369\pi$
$907$	−4.41800	−0.146697	−0.0733487	+	0.997306i	$0.523369\pi$
$908$	26.5595	0.881407
$909$	4.26337	0.141407
$910$	11.1101	0.368295
$911$	43.9432	1.45590	0.727951	−	0.685629i	$-0.240473\pi$
$911$	43.9432	1.45590	0.727951	+	0.685629i	$0.240473\pi$
$912$	−0.583791	−0.0193313
$913$	2.85702	0.0945537
$914$	−9.89526	−0.327306
$915$	1.55444	0.0513882
$916$	−11.1328	−0.367840
$917$	−2.30676	−0.0761761
$918$	−3.19078	−0.105311
$919$	54.2143	1.78836	0.894182	−	0.447704i	$-0.147758\pi$
$919$	54.2143	1.78836	0.894182	+	0.447704i	$0.147758\pi$
$920$	4.85708	0.160133
$921$	−0.792573	−0.0261162
$922$	−5.71766	−0.188301
$923$	−88.4090	−2.91002
$924$	−0.337753	−0.0111113
$925$	−7.74257	−0.254574
$926$	23.6811	0.778210
$927$	17.9096	0.588228
$928$	21.6010	0.709089
$929$	−0.0310742	−0.00101951	−0.000509756	−	1.00000i	$-0.500162\pi$
$929$	−0.0310742	−0.00101951	−0.000509756		1.00000i	$0.500162\pi$
$930$	−0.230312	−0.00755222
$931$	−0.884716	−0.0289954
$932$	15.2574	0.499772
$933$	−2.69582	−0.0882571
$934$	−12.7365	−0.416750
$935$	−2.27389	−0.0743642
$936$	−43.9576	−1.43680
$937$	−18.6118	−0.608020	−0.304010	−	0.952669i	$-0.598326\pi$
$937$	−18.6118	−0.608020	−0.304010	+	0.952669i	$0.598326\pi$
$938$	13.4802	0.440145
$939$	3.86063	0.125987
$940$	−13.8379	−0.451342
$941$	4.95121	0.161405	0.0807025	−	0.996738i	$-0.474284\pi$
$941$	4.95121	0.161405	0.0807025	+	0.996738i	$0.474284\pi$
$942$	−0.442245	−0.0144091
$943$	11.7799	0.383605
$944$	−13.1141	−0.426829
$945$	2.83051	0.0920766
$946$	1.74920	0.0568715
$947$	33.9363	1.10278	0.551391	−	0.834247i	$-0.314097\pi$
$947$	33.9363	1.10278	0.551391	+	0.834247i	$0.314097\pi$
$948$	0.381753	0.0123988
$949$	−35.1578	−1.14127
$950$	−0.958324	−0.0310921
$951$	3.34871	0.108589
$952$	31.0904	1.00764
$953$	−5.44095	−0.176250	−0.0881249	−	0.996109i	$-0.528087\pi$
$953$	−5.44095	−0.176250	−0.0881249	+	0.996109i	$0.528087\pi$
$954$	−9.57791	−0.310096
$955$	−7.90966	−0.255951
$956$	1.67770	0.0542607
$957$	−0.292102	−0.00944232
$958$	−8.42223	−0.272110
$959$	41.4276	1.33777
$960$	−0.158536	−0.00511672
$961$	−25.8003	−0.832269
$962$	31.3267	1.01001
$963$	32.8820	1.05961
$964$	−10.1855	−0.328054
$965$	2.39653	0.0771470
$966$	0.629553	0.0202555
$967$	0.538092	0.0173039	0.00865193	−	0.999963i	$-0.497246\pi$
$967$	0.538092	0.0173039	0.00865193	+	0.999963i	$0.497246\pi$
$968$	23.1419	0.743807
$969$	−1.49674	−0.0480821
$970$	0.484400	0.0155531
$971$	−6.65013	−0.213413	−0.106706	−	0.994291i	$-0.534031\pi$
$971$	−6.65013	−0.213413	−0.106706	+	0.994291i	$0.534031\pi$
$972$	−7.62621	−0.244611
$973$	13.5559	0.434582
$974$	−17.4129	−0.557945
$975$	−1.19421	−0.0382455
$976$	18.5813	0.594773
$977$	−19.8824	−0.636094	−0.318047	−	0.948075i	$-0.603027\pi$
$977$	−19.8824	−0.636094	−0.318047	+	0.948075i	$0.603027\pi$
$978$	−1.74860	−0.0559140
$979$	−0.429723	−0.0137340
$980$	−0.895288	−0.0285989
$981$	−26.7083	−0.852731
$982$	−18.8433	−0.601314
$983$	−13.2974	−0.424121	−0.212060	−	0.977257i	$-0.568017\pi$
$983$	−13.2974	−0.424121	−0.212060	+	0.977257i	$0.568017\pi$
$984$	1.91722	0.0611187
$985$	−16.9355	−0.539610
$986$	12.1864	0.388093
$987$	−3.95743	−0.125966
$988$	−18.7845	−0.597615
$989$	15.7954	0.502265
$990$	0.746639	0.0237297
$991$	−24.2737	−0.771081	−0.385540	−	0.922691i	$-0.625985\pi$
$991$	−24.2737	−0.771081	−0.385540	+	0.922691i	$0.625985\pi$
$992$	−12.5114	−0.397238
$993$	4.35916	0.138334
$994$	−20.5319	−0.651232
$995$	−25.0231	−0.793284
$996$	1.90304	0.0603002
$997$	−4.40285	−0.139440	−0.0697199	−	0.997567i	$-0.522211\pi$
$997$	−4.40285	−0.139440	−0.0697199	+	0.997567i	$0.522211\pi$
$998$	12.1096	0.383323
$999$	7.98111	0.252511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	445.2.a.f.1.5	✓	7
3.2	odd	2		4005.2.a.o.1.3		7
4.3	odd	2		7120.2.a.bj.1.2		7
5.4	even	2		2225.2.a.k.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.f.1.5	✓	7	1.1	even	1	trivial
2225.2.a.k.1.3		7	5.4	even	2
4005.2.a.o.1.3		7	3.2	odd	2
7120.2.a.bj.1.2		7	4.3	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:	\( N \)	\(=\)	\( 445 = 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	445.a (trivial)

\(f(q)\)	\(=\)	\(q+0.584976 q^{2} +0.172659 q^{3} -1.65780 q^{4} +1.00000 q^{5} +0.101002 q^{6} -2.74591 q^{7} -2.13973 q^{8} -2.97019 q^{9} +O(q^{10})\)	\(q+0.584976 q^{2} +0.172659 q^{3} -1.65780 q^{4} +1.00000 q^{5} +0.101002 q^{6} -2.74591 q^{7} -2.13973 q^{8} -2.97019 q^{9} +0.584976 q^{10} -0.429723 q^{11} -0.286235 q^{12} -6.91659 q^{13} -1.60629 q^{14} +0.172659 q^{15} +2.06392 q^{16} +5.29153 q^{17} -1.73749 q^{18} -1.63823 q^{19} -1.65780 q^{20} -0.474108 q^{21} -0.251378 q^{22} -2.26995 q^{23} -0.369444 q^{24} +1.00000 q^{25} -4.04604 q^{26} -1.03081 q^{27} +4.55219 q^{28} +3.93691 q^{29} +0.101002 q^{30} -2.28028 q^{31} +5.48679 q^{32} -0.0741958 q^{33} +3.09541 q^{34} -2.74591 q^{35} +4.92399 q^{36} -7.74257 q^{37} -0.958324 q^{38} -1.19421 q^{39} -2.13973 q^{40} -5.18947 q^{41} -0.277342 q^{42} -6.95847 q^{43} +0.712397 q^{44} -2.97019 q^{45} -1.32787 q^{46} +8.34712 q^{47} +0.356355 q^{48} +0.540044 q^{49} +0.584976 q^{50} +0.913632 q^{51} +11.4664 q^{52} +5.51251 q^{53} -0.602998 q^{54} -0.429723 q^{55} +5.87550 q^{56} -0.282856 q^{57} +2.30300 q^{58} -6.35400 q^{59} -0.286235 q^{60} +9.00292 q^{61} -1.33391 q^{62} +8.15588 q^{63} -0.918199 q^{64} -6.91659 q^{65} -0.0434027 q^{66} -8.39214 q^{67} -8.77231 q^{68} -0.391929 q^{69} -1.60629 q^{70} +12.7822 q^{71} +6.35539 q^{72} +5.08311 q^{73} -4.52921 q^{74} +0.172659 q^{75} +2.71586 q^{76} +1.17998 q^{77} -0.698586 q^{78} -1.33370 q^{79} +2.06392 q^{80} +8.73259 q^{81} -3.03571 q^{82} -6.64852 q^{83} +0.785978 q^{84} +5.29153 q^{85} -4.07053 q^{86} +0.679745 q^{87} +0.919490 q^{88} +1.00000 q^{89} -1.73749 q^{90} +18.9924 q^{91} +3.76314 q^{92} -0.393712 q^{93} +4.88286 q^{94} -1.63823 q^{95} +0.947346 q^{96} +0.828069 q^{97} +0.315913 q^{98} +1.27636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9}+O(q^{10}) \) 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9} - 4 q^{10} - 10 q^{11} - 11 q^{12} - 7 q^{13} + 3 q^{14} - 8 q^{15} + 10 q^{16} - 13 q^{17} + 4 q^{18} - 7 q^{19} + 8 q^{20} + 16 q^{21} + 2 q^{22} - 13 q^{23} + 4 q^{24} + 7 q^{25} + q^{26} - 23 q^{27} - 21 q^{28} - 4 q^{29} - 2 q^{30} + q^{31} - 13 q^{32} - 6 q^{33} + 10 q^{34} - 16 q^{35} + 20 q^{36} - 5 q^{37} - 40 q^{38} - 13 q^{39} - 12 q^{40} + 5 q^{41} + 30 q^{42} - 31 q^{43} - 21 q^{44} + 11 q^{45} + 16 q^{46} - 14 q^{47} - 7 q^{48} + 19 q^{49} - 4 q^{50} - q^{51} - 13 q^{53} - 17 q^{54} - 10 q^{55} - q^{56} + 21 q^{57} + 17 q^{58} - 14 q^{59} - 11 q^{60} + 3 q^{61} + 26 q^{62} - 54 q^{63} + 14 q^{64} - 7 q^{65} + 36 q^{66} + q^{67} - 35 q^{68} + 31 q^{69} + 3 q^{70} - 8 q^{71} + 53 q^{72} + 9 q^{73} - 35 q^{74} - 8 q^{75} + 40 q^{76} + 42 q^{77} + 46 q^{78} + 9 q^{79} + 10 q^{80} + 35 q^{81} + 29 q^{82} - 42 q^{83} + 55 q^{84} - 13 q^{85} + 35 q^{86} + 6 q^{87} + 30 q^{88} + 7 q^{89} + 4 q^{90} + 31 q^{91} + 19 q^{92} + 24 q^{93} + 37 q^{94} - 7 q^{95} + 44 q^{96} - 7 q^{97} + 9 q^{98} + 5 q^{99}+O(q^{100}) \) 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9 - 4 * q^10 - 10 * q^11 - 11 * q^12 - 7 * q^13 + 3 * q^14 - 8 * q^15 + 10 * q^16 - 13 * q^17 + 4 * q^18 - 7 * q^19 + 8 * q^20 + 16 * q^21 + 2 * q^22 - 13 * q^23 + 4 * q^24 + 7 * q^25 + q^26 - 23 * q^27 - 21 * q^28 - 4 * q^29 - 2 * q^30 + q^31 - 13 * q^32 - 6 * q^33 + 10 * q^34 - 16 * q^35 + 20 * q^36 - 5 * q^37 - 40 * q^38 - 13 * q^39 - 12 * q^40 + 5 * q^41 + 30 * q^42 - 31 * q^43 - 21 * q^44 + 11 * q^45 + 16 * q^46 - 14 * q^47 - 7 * q^48 + 19 * q^49 - 4 * q^50 - q^51 - 13 * q^53 - 17 * q^54 - 10 * q^55 - q^56 + 21 * q^57 + 17 * q^58 - 14 * q^59 - 11 * q^60 + 3 * q^61 + 26 * q^62 - 54 * q^63 + 14 * q^64 - 7 * q^65 + 36 * q^66 + q^67 - 35 * q^68 + 31 * q^69 + 3 * q^70 - 8 * q^71 + 53 * q^72 + 9 * q^73 - 35 * q^74 - 8 * q^75 + 40 * q^76 + 42 * q^77 + 46 * q^78 + 9 * q^79 + 10 * q^80 + 35 * q^81 + 29 * q^82 - 42 * q^83 + 55 * q^84 - 13 * q^85 + 35 * q^86 + 6 * q^87 + 30 * q^88 + 7 * q^89 + 4 * q^90 + 31 * q^91 + 19 * q^92 + 24 * q^93 + 37 * q^94 - 7 * q^95 + 44 * q^96 - 7 * q^97 + 9 * q^98 + 5 * q^99

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