LMFDB - Embedded newform 3549.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3549,2,Mod(1,3549)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3549, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3549.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3549 = 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3549.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:	$28.3389076774$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 13x^{6} + 22x^{5} + 57x^{4} - 72x^{3} - 96x^{2} + 64x + 61 $ x^8 - 2x^7 - 13x^6 + 22x^5 + 57x^4 - 72x^3 - 96x^2 + 64*x + 61
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.72116$ of defining polynomial
Character	$\chi$	$=$	3549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72116 q^{2} -1.00000 q^{3} +5.40472 q^{4} -1.58278 q^{5} +2.72116 q^{6} +1.00000 q^{7} -9.26480 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.72116 q^{2} -1.00000 q^{3} +5.40472 q^{4} -1.58278 q^{5} +2.72116 q^{6} +1.00000 q^{7} -9.26480 q^{8} +1.00000 q^{9} +4.30699 q^{10} -5.54221 q^{11} -5.40472 q^{12} -2.72116 q^{14} +1.58278 q^{15} +14.4016 q^{16} +4.80577 q^{17} -2.72116 q^{18} +0.892533 q^{19} -8.55447 q^{20} -1.00000 q^{21} +15.0812 q^{22} -4.63015 q^{23} +9.26480 q^{24} -2.49482 q^{25} -1.00000 q^{27} +5.40472 q^{28} -1.88217 q^{29} -4.30699 q^{30} +1.47545 q^{31} -20.6594 q^{32} +5.54221 q^{33} -13.0773 q^{34} -1.58278 q^{35} +5.40472 q^{36} +4.99293 q^{37} -2.42873 q^{38} +14.6641 q^{40} +9.75924 q^{41} +2.72116 q^{42} +1.01270 q^{43} -29.9541 q^{44} -1.58278 q^{45} +12.5994 q^{46} +12.5703 q^{47} -14.4016 q^{48} +1.00000 q^{49} +6.78880 q^{50} -4.80577 q^{51} +6.25784 q^{53} +2.72116 q^{54} +8.77208 q^{55} -9.26480 q^{56} -0.892533 q^{57} +5.12170 q^{58} -3.44690 q^{59} +8.55447 q^{60} -3.58549 q^{61} -4.01493 q^{62} +1.00000 q^{63} +27.4144 q^{64} -15.0812 q^{66} -14.7401 q^{67} +25.9738 q^{68} +4.63015 q^{69} +4.30699 q^{70} -13.1398 q^{71} -9.26480 q^{72} -5.33654 q^{73} -13.5866 q^{74} +2.49482 q^{75} +4.82389 q^{76} -5.54221 q^{77} +0.779028 q^{79} -22.7945 q^{80} +1.00000 q^{81} -26.5565 q^{82} +1.47136 q^{83} -5.40472 q^{84} -7.60646 q^{85} -2.75571 q^{86} +1.88217 q^{87} +51.3474 q^{88} +7.65950 q^{89} +4.30699 q^{90} -25.0247 q^{92} -1.47545 q^{93} -34.2058 q^{94} -1.41268 q^{95} +20.6594 q^{96} -0.758679 q^{97} -2.72116 q^{98} -5.54221 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 8 q^{3} + 14 q^{4} - 6 q^{5} + 2 q^{6} + 8 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 2 * q^2 - 8 * q^3 + 14 * q^4 - 6 * q^5 + 2 * q^6 + 8 * q^7 - 12 * q^8 + 8 * q^9	\( 8 q - 2 q^{2} - 8 q^{3} + 14 q^{4} - 6 q^{5} + 2 q^{6} + 8 q^{7} - 12 q^{8} + 8 q^{9} - 4 q^{10} - 16 q^{11} - 14 q^{12} - 2 q^{14} + 6 q^{15} + 10 q^{16} - 2 q^{17} - 2 q^{18} - 14 q^{19} + 10 q^{20} - 8 q^{21} + 18 q^{22} - 6 q^{23} + 12 q^{24} + 10 q^{25} - 8 q^{27} + 14 q^{28} + 12 q^{29} + 4 q^{30} - 16 q^{31} - 26 q^{32} + 16 q^{33} - 32 q^{34} - 6 q^{35} + 14 q^{36} + 8 q^{37} + 12 q^{38} - 14 q^{40} + 4 q^{41} + 2 q^{42} + 14 q^{43} - 68 q^{44} - 6 q^{45} + 28 q^{46} - 6 q^{47} - 10 q^{48} + 8 q^{49} - 32 q^{50} + 2 q^{51} + 14 q^{53} + 2 q^{54} - 2 q^{55} - 12 q^{56} + 14 q^{57} - 24 q^{58} - 50 q^{59} - 10 q^{60} - 2 q^{61} - 20 q^{62} + 8 q^{63} + 24 q^{64} - 18 q^{66} + 16 q^{67} - 36 q^{68} + 6 q^{69} - 4 q^{70} - 34 q^{71} - 12 q^{72} + 8 q^{73} - 6 q^{74} - 10 q^{75} + 4 q^{76} - 16 q^{77} + 46 q^{79} + 24 q^{80} + 8 q^{81} - 42 q^{82} - 42 q^{83} - 14 q^{84} - 20 q^{85} - 50 q^{86} - 12 q^{87} + 62 q^{88} - 28 q^{89} - 4 q^{90} - 58 q^{92} + 16 q^{93} + 24 q^{94} - 24 q^{95} + 26 q^{96} - 16 q^{97} - 2 q^{98} - 16 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 - 8 * q^3 + 14 * q^4 - 6 * q^5 + 2 * q^6 + 8 * q^7 - 12 * q^8 + 8 * q^9 - 4 * q^10 - 16 * q^11 - 14 * q^12 - 2 * q^14 + 6 * q^15 + 10 * q^16 - 2 * q^17 - 2 * q^18 - 14 * q^19 + 10 * q^20 - 8 * q^21 + 18 * q^22 - 6 * q^23 + 12 * q^24 + 10 * q^25 - 8 * q^27 + 14 * q^28 + 12 * q^29 + 4 * q^30 - 16 * q^31 - 26 * q^32 + 16 * q^33 - 32 * q^34 - 6 * q^35 + 14 * q^36 + 8 * q^37 + 12 * q^38 - 14 * q^40 + 4 * q^41 + 2 * q^42 + 14 * q^43 - 68 * q^44 - 6 * q^45 + 28 * q^46 - 6 * q^47 - 10 * q^48 + 8 * q^49 - 32 * q^50 + 2 * q^51 + 14 * q^53 + 2 * q^54 - 2 * q^55 - 12 * q^56 + 14 * q^57 - 24 * q^58 - 50 * q^59 - 10 * q^60 - 2 * q^61 - 20 * q^62 + 8 * q^63 + 24 * q^64 - 18 * q^66 + 16 * q^67 - 36 * q^68 + 6 * q^69 - 4 * q^70 - 34 * q^71 - 12 * q^72 + 8 * q^73 - 6 * q^74 - 10 * q^75 + 4 * q^76 - 16 * q^77 + 46 * q^79 + 24 * q^80 + 8 * q^81 - 42 * q^82 - 42 * q^83 - 14 * q^84 - 20 * q^85 - 50 * q^86 - 12 * q^87 + 62 * q^88 - 28 * q^89 - 4 * q^90 - 58 * q^92 + 16 * q^93 + 24 * q^94 - 24 * q^95 + 26 * q^96 - 16 * q^97 - 2 * q^98 - 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.72116	−1.92415	−0.962076	−	0.272782i	$-0.912056\pi$
$2$	−2.72116	−1.92415	−0.962076	+	0.272782i	$0.912056\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.40472	2.70236
$5$	−1.58278	−0.707839	−0.353920	−	0.935276i	$-0.615151\pi$
$5$	−1.58278	−0.707839	−0.353920	+	0.935276i	$0.615151\pi$
$6$	2.72116	1.11091
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−9.26480	−3.27560
$9$	1.00000	0.333333
$10$	4.30699	1.36199
$11$	−5.54221	−1.67104	−0.835519	−	0.549461i	$-0.814833\pi$
$11$	−5.54221	−1.67104	−0.835519	+	0.549461i	$0.814833\pi$
$12$	−5.40472	−1.56021
$13$	0	0
$13$	0	0
$14$	−2.72116	−0.727261
$15$	1.58278	0.408671
$16$	14.4016	3.60039
$17$	4.80577	1.16557	0.582785	−	0.812627i	$-0.301963\pi$
$17$	4.80577	1.16557	0.582785	+	0.812627i	$0.301963\pi$
$18$	−2.72116	−0.641384
$19$	0.892533	0.204761	0.102381	−	0.994745i	$-0.467354\pi$
$19$	0.892533	0.204761	0.102381	+	0.994745i	$0.467354\pi$
$20$	−8.55447	−1.91284
$21$	−1.00000	−0.218218
$22$	15.0812	3.21533
$23$	−4.63015	−0.965453	−0.482726	−	0.875771i	$-0.660353\pi$
$23$	−4.63015	−0.965453	−0.482726	+	0.875771i	$0.660353\pi$
$24$	9.26480	1.89117
$25$	−2.49482	−0.498963
$26$	0	0
$27$	−1.00000	−0.192450
$28$	5.40472	1.02140
$29$	−1.88217	−0.349511	−0.174755	−	0.984612i	$-0.555913\pi$
$29$	−1.88217	−0.349511	−0.174755	+	0.984612i	$0.555913\pi$
$30$	−4.30699	−0.786346
$31$	1.47545	0.264998	0.132499	−	0.991183i	$-0.457700\pi$
$31$	1.47545	0.264998	0.132499	+	0.991183i	$0.457700\pi$
$32$	−20.6594	−3.65210
$33$	5.54221	0.964775
$34$	−13.0773	−2.24273
$35$	−1.58278	−0.267538
$36$	5.40472	0.900787
$37$	4.99293	0.820832	0.410416	−	0.911898i	$-0.365383\pi$
$37$	4.99293	0.820832	0.410416	+	0.911898i	$0.365383\pi$
$38$	−2.42873	−0.393992
$39$	0	0
$40$	14.6641	2.31860
$41$	9.75924	1.52414	0.762069	−	0.647496i	$-0.224184\pi$
$41$	9.75924	1.52414	0.762069	+	0.647496i	$0.224184\pi$
$42$	2.72116	0.419884
$43$	1.01270	0.154435	0.0772174	−	0.997014i	$-0.475396\pi$
$43$	1.01270	0.154435	0.0772174	+	0.997014i	$0.475396\pi$
$44$	−29.9541	−4.51575
$45$	−1.58278	−0.235946
$46$	12.5994	1.85768
$47$	12.5703	1.83357	0.916783	−	0.399385i	$-0.130776\pi$
$47$	12.5703	1.83357	0.916783	+	0.399385i	$0.130776\pi$
$48$	−14.4016	−2.07869
$49$	1.00000	0.142857
$50$	6.78880	0.960081
$51$	−4.80577	−0.672942
$52$	0	0
$53$	6.25784	0.859581	0.429790	−	0.902929i	$-0.358587\pi$
$53$	6.25784	0.859581	0.429790	+	0.902929i	$0.358587\pi$
$54$	2.72116	0.370303
$55$	8.77208	1.18283
$56$	−9.26480	−1.23806
$57$	−0.892533	−0.118219
$58$	5.12170	0.672512
$59$	−3.44690	−0.448749	−0.224374	−	0.974503i	$-0.572034\pi$
$59$	−3.44690	−0.448749	−0.224374	+	0.974503i	$0.572034\pi$
$60$	8.55447	1.10438
$61$	−3.58549	−0.459075	−0.229538	−	0.973300i	$-0.573721\pi$
$61$	−3.58549	−0.459075	−0.229538	+	0.973300i	$0.573721\pi$
$62$	−4.01493	−0.509896
$63$	1.00000	0.125988
$64$	27.4144	3.42681
$65$	0	0
$66$	−15.0812	−1.85637
$67$	−14.7401	−1.80078	−0.900392	−	0.435079i	$-0.856720\pi$
$67$	−14.7401	−1.80078	−0.900392	+	0.435079i	$0.856720\pi$
$68$	25.9738	3.14979
$69$	4.63015	0.557404
$70$	4.30699	0.514784
$71$	−13.1398	−1.55940	−0.779702	−	0.626151i	$-0.784629\pi$
$71$	−13.1398	−1.55940	−0.779702	+	0.626151i	$0.784629\pi$
$72$	−9.26480	−1.09187
$73$	−5.33654	−0.624595	−0.312298	−	0.949984i	$-0.601099\pi$
$73$	−5.33654	−0.624595	−0.312298	+	0.949984i	$0.601099\pi$
$74$	−13.5866	−1.57941
$75$	2.49482	0.288077
$76$	4.82389	0.553339
$77$	−5.54221	−0.631593
$78$	0	0
$79$	0.779028	0.0876475	0.0438237	−	0.999039i	$-0.486046\pi$
$79$	0.779028	0.0876475	0.0438237	+	0.999039i	$0.486046\pi$
$80$	−22.7945	−2.54850
$81$	1.00000	0.111111
$82$	−26.5565	−2.93267
$83$	1.47136	0.161503	0.0807513	−	0.996734i	$-0.474268\pi$
$83$	1.47136	0.161503	0.0807513	+	0.996734i	$0.474268\pi$
$84$	−5.40472	−0.589703
$85$	−7.60646	−0.825036
$86$	−2.75571	−0.297156
$87$	1.88217	0.201790
$88$	51.3474	5.47365
$89$	7.65950	0.811905	0.405953	−	0.913894i	$-0.366940\pi$
$89$	7.65950	0.811905	0.405953	+	0.913894i	$0.366940\pi$
$90$	4.30699	0.453997
$91$	0	0
$92$	−25.0247	−2.60900
$93$	−1.47545	−0.152997
$94$	−34.2058	−3.52806
$95$	−1.41268	−0.144938
$96$	20.6594	2.10854
$97$	−0.758679	−0.0770322	−0.0385161	−	0.999258i	$-0.512263\pi$
$97$	−0.758679	−0.0770322	−0.0385161	+	0.999258i	$0.512263\pi$
$98$	−2.72116	−0.274879
$99$	−5.54221	−0.557013
$100$	−13.4838	−1.34838
$101$	−11.7740	−1.17156	−0.585779	−	0.810471i	$-0.699211\pi$
$101$	−11.7740	−1.17156	−0.585779	+	0.810471i	$0.699211\pi$
$102$	13.0773	1.29484
$103$	9.61981	0.947868	0.473934	−	0.880560i	$-0.342833\pi$
$103$	9.61981	0.947868	0.473934	+	0.880560i	$0.342833\pi$
$104$	0	0
$105$	1.58278	0.154463
$106$	−17.0286	−1.65396
$107$	3.68918	0.356647	0.178323	−	0.983972i	$-0.442933\pi$
$107$	3.68918	0.356647	0.178323	+	0.983972i	$0.442933\pi$
$108$	−5.40472	−0.520070
$109$	−8.92215	−0.854587	−0.427293	−	0.904113i	$-0.640533\pi$
$109$	−8.92215	−0.854587	−0.427293	+	0.904113i	$0.640533\pi$
$110$	−23.8702	−2.27594
$111$	−4.99293	−0.473908
$112$	14.4016	1.36082
$113$	16.1353	1.51788	0.758941	−	0.651159i	$-0.225717\pi$
$113$	16.1353	1.51788	0.758941	+	0.651159i	$0.225717\pi$
$114$	2.42873	0.227471
$115$	7.32849	0.683385
$116$	−10.1726	−0.944504
$117$	0	0
$118$	9.37958	0.863460
$119$	4.80577	0.440544
$120$	−14.6641	−1.33864
$121$	19.7161	1.79237
$122$	9.75670	0.883330
$123$	−9.75924	−0.879961
$124$	7.97437	0.716120
$125$	11.8626	1.06103
$126$	−2.72116	−0.242420
$127$	−4.62457	−0.410364	−0.205182	−	0.978724i	$-0.565779\pi$
$127$	−4.62457	−0.410364	−0.205182	+	0.978724i	$0.565779\pi$
$128$	−33.2803	−2.94159
$129$	−1.01270	−0.0891630
$130$	0	0
$131$	−7.78183	−0.679902	−0.339951	−	0.940443i	$-0.610410\pi$
$131$	−7.78183	−0.679902	−0.339951	+	0.940443i	$0.610410\pi$
$132$	29.9541	2.60717
$133$	0.892533	0.0773925
$134$	40.1101	3.46498
$135$	1.58278	0.136224
$136$	−44.5245	−3.81794
$137$	−0.276352	−0.0236104	−0.0118052	−	0.999930i	$-0.503758\pi$
$137$	−0.276352	−0.0236104	−0.0118052	+	0.999930i	$0.503758\pi$
$138$	−12.5994	−1.07253
$139$	3.96065	0.335938	0.167969	−	0.985792i	$-0.446279\pi$
$139$	3.96065	0.335938	0.167969	+	0.985792i	$0.446279\pi$
$140$	−8.55447	−0.722984
$141$	−12.5703	−1.05861
$142$	35.7554	3.00053
$143$	0	0
$144$	14.4016	1.20013
$145$	2.97906	0.247398
$146$	14.5216	1.20182
$147$	−1.00000	−0.0824786
$148$	26.9854	2.21818
$149$	21.6897	1.77689	0.888446	−	0.458982i	$-0.151786\pi$
$149$	21.6897	1.77689	0.888446	+	0.458982i	$0.151786\pi$
$150$	−6.78880	−0.554303
$151$	5.66673	0.461152	0.230576	−	0.973054i	$-0.425939\pi$
$151$	5.66673	0.461152	0.230576	+	0.973054i	$0.425939\pi$
$152$	−8.26914	−0.670716
$153$	4.80577	0.388523
$154$	15.0812	1.21528
$155$	−2.33530	−0.187576
$156$	0	0
$157$	1.06406	0.0849216	0.0424608	−	0.999098i	$-0.486480\pi$
$157$	1.06406	0.0849216	0.0424608	+	0.999098i	$0.486480\pi$
$158$	−2.11986	−0.168647
$159$	−6.25784	−0.496279
$160$	32.6992	2.58510
$161$	−4.63015	−0.364907
$162$	−2.72116	−0.213795
$163$	7.20618	0.564432	0.282216	−	0.959351i	$-0.408931\pi$
$163$	7.20618	0.564432	0.282216	+	0.959351i	$0.408931\pi$
$164$	52.7460	4.11877
$165$	−8.77208	−0.682905
$166$	−4.00380	−0.310756
$167$	−0.468497	−0.0362534	−0.0181267	−	0.999836i	$-0.505770\pi$
$167$	−0.468497	−0.0362534	−0.0181267	+	0.999836i	$0.505770\pi$
$168$	9.26480	0.714795
$169$	0	0
$170$	20.6984	1.58749
$171$	0.892533	0.0682537
$172$	5.47335	0.417339
$173$	−10.6101	−0.806670	−0.403335	−	0.915052i	$-0.632149\pi$
$173$	−10.6101	−0.806670	−0.403335	+	0.915052i	$0.632149\pi$
$174$	−5.12170	−0.388275
$175$	−2.49482	−0.188590
$176$	−79.8165	−6.01639
$177$	3.44690	0.259085
$178$	−20.8427	−1.56223
$179$	−19.3674	−1.44759	−0.723793	−	0.690017i	$-0.757603\pi$
$179$	−19.3674	−1.44759	−0.723793	+	0.690017i	$0.757603\pi$
$180$	−8.55447	−0.637612
$181$	−16.0004	−1.18930	−0.594649	−	0.803986i	$-0.702709\pi$
$181$	−16.0004	−1.18930	−0.594649	+	0.803986i	$0.702709\pi$
$182$	0	0
$183$	3.58549	0.265047
$184$	42.8974	3.16244
$185$	−7.90269	−0.581017
$186$	4.01493	0.294389
$187$	−26.6346	−1.94771
$188$	67.9390	4.95496
$189$	−1.00000	−0.0727393
$190$	3.84413	0.278883
$191$	3.14831	0.227804	0.113902	−	0.993492i	$-0.463665\pi$
$191$	3.14831	0.227804	0.113902	+	0.993492i	$0.463665\pi$
$192$	−27.4144	−1.97847
$193$	23.2821	1.67588	0.837942	−	0.545759i	$-0.183759\pi$
$193$	23.2821	1.67588	0.837942	+	0.545759i	$0.183759\pi$
$194$	2.06449	0.148222
$195$	0	0
$196$	5.40472	0.386052
$197$	17.6425	1.25697	0.628487	−	0.777820i	$-0.283675\pi$
$197$	17.6425	1.25697	0.628487	+	0.777820i	$0.283675\pi$
$198$	15.0812	1.07178
$199$	11.9251	0.845348	0.422674	−	0.906282i	$-0.361091\pi$
$199$	11.9251	0.845348	0.422674	+	0.906282i	$0.361091\pi$
$200$	23.1140	1.63440
$201$	14.7401	1.03968
$202$	32.0390	2.25426
$203$	−1.88217	−0.132103
$204$	−25.9738	−1.81853
$205$	−15.4467	−1.07884
$206$	−26.1771	−1.82384
$207$	−4.63015	−0.321818
$208$	0	0
$209$	−4.94661	−0.342164
$210$	−4.30699	−0.297211
$211$	−15.1302	−1.04161	−0.520803	−	0.853677i	$-0.674367\pi$
$211$	−15.1302	−1.04161	−0.520803	+	0.853677i	$0.674367\pi$
$212$	33.8219	2.32290
$213$	13.1398	0.900322
$214$	−10.0389	−0.686243
$215$	−1.60287	−0.109315
$216$	9.26480	0.630390
$217$	1.47545	0.100160
$218$	24.2786	1.64435
$219$	5.33654	0.360610
$220$	47.4106	3.19642
$221$	0	0
$222$	13.5866	0.911870
$223$	19.1562	1.28280	0.641398	−	0.767208i	$-0.278355\pi$
$223$	19.1562	1.28280	0.641398	+	0.767208i	$0.278355\pi$
$224$	−20.6594	−1.38036
$225$	−2.49482	−0.166321
$226$	−43.9068	−2.92064
$227$	−17.4419	−1.15766	−0.578831	−	0.815447i	$-0.696491\pi$
$227$	−17.4419	−1.15766	−0.578831	+	0.815447i	$0.696491\pi$
$228$	−4.82389	−0.319470
$229$	−10.5075	−0.694354	−0.347177	−	0.937800i	$-0.612860\pi$
$229$	−10.5075	−0.694354	−0.347177	+	0.937800i	$0.612860\pi$
$230$	−19.9420	−1.31494
$231$	5.54221	0.364650
$232$	17.4380	1.14486
$233$	−9.69049	−0.634845	−0.317422	−	0.948284i	$-0.602817\pi$
$233$	−9.69049	−0.634845	−0.317422	+	0.948284i	$0.602817\pi$
$234$	0	0
$235$	−19.8960	−1.29787
$236$	−18.6296	−1.21268
$237$	−0.779028	−0.0506033
$238$	−13.0773	−0.847673
$239$	−18.4302	−1.19215	−0.596074	−	0.802929i	$-0.703274\pi$
$239$	−18.4302	−1.19215	−0.596074	+	0.802929i	$0.703274\pi$
$240$	22.7945	1.47138
$241$	−15.3530	−0.988977	−0.494488	−	0.869184i	$-0.664645\pi$
$241$	−15.3530	−0.988977	−0.494488	+	0.869184i	$0.664645\pi$
$242$	−53.6506	−3.44879
$243$	−1.00000	−0.0641500
$244$	−19.3786	−1.24059
$245$	−1.58278	−0.101120
$246$	26.5565	1.69318
$247$	0	0
$248$	−13.6697	−0.868027
$249$	−1.47136	−0.0932436
$250$	−32.2801	−2.04157
$251$	8.62900	0.544658	0.272329	−	0.962204i	$-0.412206\pi$
$251$	8.62900	0.544658	0.272329	+	0.962204i	$0.412206\pi$
$252$	5.40472	0.340465
$253$	25.6612	1.61331
$254$	12.5842	0.789602
$255$	7.60646	0.476335
$256$	35.7323	2.23327
$257$	2.72949	0.170261	0.0851305	−	0.996370i	$-0.472869\pi$
$257$	2.72949	0.170261	0.0851305	+	0.996370i	$0.472869\pi$
$258$	2.75571	0.171563
$259$	4.99293	0.310245
$260$	0	0
$261$	−1.88217	−0.116504
$262$	21.1756	1.30823
$263$	−5.61419	−0.346186	−0.173093	−	0.984906i	$-0.555376\pi$
$263$	−5.61419	−0.346186	−0.173093	+	0.984906i	$0.555376\pi$
$264$	−51.3474	−3.16022
$265$	−9.90477	−0.608445
$266$	−2.42873	−0.148915
$267$	−7.65950	−0.468754
$268$	−79.6659	−4.86637
$269$	−0.182236	−0.0111111	−0.00555556	−	0.999985i	$-0.501768\pi$
$269$	−0.182236	−0.0111111	−0.00555556	+	0.999985i	$0.501768\pi$
$270$	−4.30699	−0.262115
$271$	−29.3326	−1.78183	−0.890914	−	0.454173i	$-0.849935\pi$
$271$	−29.3326	−1.78183	−0.890914	+	0.454173i	$0.849935\pi$
$272$	69.2106	4.19651
$273$	0	0
$274$	0.752000	0.0454300
$275$	13.8268	0.833787
$276$	25.0247	1.50631
$277$	−16.1748	−0.971849	−0.485924	−	0.874001i	$-0.661517\pi$
$277$	−16.1748	−0.971849	−0.485924	+	0.874001i	$0.661517\pi$
$278$	−10.7776	−0.646395
$279$	1.47545	0.0883326
$280$	14.6641	0.876348
$281$	3.71644	0.221704	0.110852	−	0.993837i	$-0.464642\pi$
$281$	3.71644	0.221704	0.110852	+	0.993837i	$0.464642\pi$
$282$	34.2058	2.03693
$283$	3.77370	0.224323	0.112162	−	0.993690i	$-0.464223\pi$
$283$	3.77370	0.224323	0.112162	+	0.993690i	$0.464223\pi$
$284$	−71.0168	−4.21407
$285$	1.41268	0.0836800
$286$	0	0
$287$	9.75924	0.576070
$288$	−20.6594	−1.21737
$289$	6.09539	0.358553
$290$	−8.10651	−0.476030
$291$	0.758679	0.0444746
$292$	−28.8425	−1.68788
$293$	−15.4163	−0.900631	−0.450316	−	0.892869i	$-0.648689\pi$
$293$	−15.4163	−0.900631	−0.450316	+	0.892869i	$0.648689\pi$
$294$	2.72116	0.158701
$295$	5.45568	0.317642
$296$	−46.2585	−2.68872
$297$	5.54221	0.321592
$298$	−59.0212	−3.41901
$299$	0	0
$300$	13.4838	0.778487
$301$	1.01270	0.0583709
$302$	−15.4201	−0.887326
$303$	11.7740	0.676400
$304$	12.8539	0.737221
$305$	5.67503	0.324952
$306$	−13.0773	−0.747578
$307$	33.2617	1.89834	0.949172	−	0.314758i	$-0.101923\pi$
$307$	33.2617	1.89834	0.949172	+	0.314758i	$0.101923\pi$
$308$	−29.9541	−1.70679
$309$	−9.61981	−0.547252
$310$	6.35473	0.360925
$311$	−5.28263	−0.299551	−0.149775	−	0.988720i	$-0.547855\pi$
$311$	−5.28263	−0.299551	−0.149775	+	0.988720i	$0.547855\pi$
$312$	0	0
$313$	−17.4853	−0.988329	−0.494165	−	0.869368i	$-0.664526\pi$
$313$	−17.4853	−0.988329	−0.494165	+	0.869368i	$0.664526\pi$
$314$	−2.89549	−0.163402
$315$	−1.58278	−0.0891794
$316$	4.21043	0.236855
$317$	−6.89451	−0.387234	−0.193617	−	0.981077i	$-0.562022\pi$
$317$	−6.89451	−0.387234	−0.193617	+	0.981077i	$0.562022\pi$
$318$	17.0286	0.954917
$319$	10.4314	0.584046
$320$	−43.3909	−2.42563
$321$	−3.68918	−0.205910
$322$	12.5994	0.702136
$323$	4.28931	0.238663
$324$	5.40472	0.300262
$325$	0	0
$326$	−19.6092	−1.08605
$327$	8.92215	0.493396
$328$	−90.4174	−4.99247
$329$	12.5703	0.693023
$330$	23.8702	1.31401
$331$	−3.00614	−0.165233	−0.0826163	−	0.996581i	$-0.526328\pi$
$331$	−3.00614	−0.165233	−0.0826163	+	0.996581i	$0.526328\pi$
$332$	7.95228	0.436438
$333$	4.99293	0.273611
$334$	1.27486	0.0697571
$335$	23.3302	1.27467
$336$	−14.4016	−0.785670
$337$	−7.21913	−0.393251	−0.196626	−	0.980479i	$-0.562998\pi$
$337$	−7.21913	−0.393251	−0.196626	+	0.980479i	$0.562998\pi$
$338$	0	0
$339$	−16.1353	−0.876350
$340$	−41.1108	−2.22954
$341$	−8.17723	−0.442822
$342$	−2.42873	−0.131331
$343$	1.00000	0.0539949
$344$	−9.38243	−0.505867
$345$	−7.32849	−0.394553
$346$	28.8718	1.55216
$347$	1.92476	0.103327	0.0516633	−	0.998665i	$-0.483548\pi$
$347$	1.92476	0.103327	0.0516633	+	0.998665i	$0.483548\pi$
$348$	10.1726	0.545310
$349$	3.25971	0.174488	0.0872441	−	0.996187i	$-0.472194\pi$
$349$	3.25971	0.174488	0.0872441	+	0.996187i	$0.472194\pi$
$350$	6.78880	0.362877
$351$	0	0
$352$	114.499	6.10280
$353$	−8.91285	−0.474383	−0.237192	−	0.971463i	$-0.576227\pi$
$353$	−8.91285	−0.474383	−0.237192	+	0.971463i	$0.576227\pi$
$354$	−9.37958	−0.498519
$355$	20.7973	1.10381
$356$	41.3974	2.19406
$357$	−4.80577	−0.254348
$358$	52.7018	2.78537
$359$	−20.1320	−1.06253	−0.531263	−	0.847207i	$-0.678282\pi$
$359$	−20.1320	−1.06253	−0.531263	+	0.847207i	$0.678282\pi$
$360$	14.6641	0.772866
$361$	−18.2034	−0.958073
$362$	43.5396	2.28839
$363$	−19.7161	−1.03483
$364$	0	0
$365$	8.44656	0.442113
$366$	−9.75670	−0.509991
$367$	−31.3418	−1.63603	−0.818013	−	0.575200i	$-0.804924\pi$
$367$	−31.3418	−1.63603	−0.818013	+	0.575200i	$0.804924\pi$
$368$	−66.6814	−3.47601
$369$	9.75924	0.508046
$370$	21.5045	1.11797
$371$	6.25784	0.324891
$372$	−7.97437	−0.413452
$373$	17.9434	0.929077	0.464538	−	0.885553i	$-0.346220\pi$
$373$	17.9434	0.929077	0.464538	+	0.885553i	$0.346220\pi$
$374$	72.4769	3.74769
$375$	−11.8626	−0.612583
$376$	−116.461	−6.00603
$377$	0	0
$378$	2.72116	0.139961
$379$	−20.1477	−1.03492	−0.517458	−	0.855709i	$-0.673122\pi$
$379$	−20.1477	−1.03492	−0.517458	+	0.855709i	$0.673122\pi$
$380$	−7.63515	−0.391675
$381$	4.62457	0.236924
$382$	−8.56705	−0.438329
$383$	−37.5129	−1.91682	−0.958410	−	0.285393i	$-0.907876\pi$
$383$	−37.5129	−1.91682	−0.958410	+	0.285393i	$0.907876\pi$
$384$	33.2803	1.69833
$385$	8.77208	0.447067
$386$	−63.3544	−3.22466
$387$	1.01270	0.0514783
$388$	−4.10045	−0.208169
$389$	−23.1946	−1.17601	−0.588006	−	0.808857i	$-0.700087\pi$
$389$	−23.1946	−1.17601	−0.588006	+	0.808857i	$0.700087\pi$
$390$	0	0
$391$	−22.2514	−1.12530
$392$	−9.26480	−0.467943
$393$	7.78183	0.392541
$394$	−48.0080	−2.41861
$395$	−1.23303	−0.0620403
$396$	−29.9541	−1.50525
$397$	38.1606	1.91522	0.957612	−	0.288060i	$-0.0930103\pi$
$397$	38.1606	1.91522	0.957612	+	0.288060i	$0.0930103\pi$
$398$	−32.4501	−1.62658
$399$	−0.892533	−0.0446826
$400$	−35.9293	−1.79646
$401$	8.28812	0.413889	0.206944	−	0.978353i	$-0.433648\pi$
$401$	8.28812	0.413889	0.206944	+	0.978353i	$0.433648\pi$
$402$	−40.1101	−2.00051
$403$	0	0
$404$	−63.6353	−3.16597
$405$	−1.58278	−0.0786488
$406$	5.12170	0.254186
$407$	−27.6718	−1.37164
$408$	44.5245	2.20429
$409$	−39.1125	−1.93399	−0.966994	−	0.254800i	$-0.917990\pi$
$409$	−39.1125	−1.93399	−0.966994	+	0.254800i	$0.917990\pi$
$410$	42.0330	2.07586
$411$	0.276352	0.0136315
$412$	51.9924	2.56148
$413$	−3.44690	−0.169611
$414$	12.5994	0.619226
$415$	−2.32883	−0.114318
$416$	0	0
$417$	−3.96065	−0.193954
$418$	13.4605	0.658375
$419$	27.5466	1.34574	0.672869	−	0.739762i	$-0.265062\pi$
$419$	27.5466	1.34574	0.672869	+	0.739762i	$0.265062\pi$
$420$	8.55447	0.417415
$421$	−34.1236	−1.66308	−0.831540	−	0.555465i	$-0.812540\pi$
$421$	−34.1236	−1.66308	−0.831540	+	0.555465i	$0.812540\pi$
$422$	41.1717	2.00421
$423$	12.5703	0.611189
$424$	−57.9776	−2.81564
$425$	−11.9895	−0.581577
$426$	−35.7554	−1.73236
$427$	−3.58549	−0.173514
$428$	19.9390	0.963788
$429$	0	0
$430$	4.36168	0.210339
$431$	5.00089	0.240884	0.120442	−	0.992720i	$-0.461569\pi$
$431$	5.00089	0.240884	0.120442	+	0.992720i	$0.461569\pi$
$432$	−14.4016	−0.692896
$433$	14.9136	0.716701	0.358351	−	0.933587i	$-0.383339\pi$
$433$	14.9136	0.716701	0.358351	+	0.933587i	$0.383339\pi$
$434$	−4.01493	−0.192723
$435$	−2.97906	−0.142835
$436$	−48.2217	−2.30940
$437$	−4.13256	−0.197687
$438$	−14.5216	−0.693869
$439$	−7.10276	−0.338996	−0.169498	−	0.985531i	$-0.554215\pi$
$439$	−7.10276	−0.338996	−0.169498	+	0.985531i	$0.554215\pi$
$440$	−81.2715	−3.87447
$441$	1.00000	0.0476190
$442$	0	0
$443$	−4.63166	−0.220057	−0.110028	−	0.993928i	$-0.535094\pi$
$443$	−4.63166	−0.220057	−0.110028	+	0.993928i	$0.535094\pi$
$444$	−26.9854	−1.28067
$445$	−12.1233	−0.574698
$446$	−52.1272	−2.46830
$447$	−21.6897	−1.02589
$448$	27.4144	1.29521
$449$	22.5728	1.06528	0.532639	−	0.846343i	$-0.321200\pi$
$449$	22.5728	1.06528	0.532639	+	0.846343i	$0.321200\pi$
$450$	6.78880	0.320027
$451$	−54.0878	−2.54689
$452$	87.2069	4.10187
$453$	−5.66673	−0.266246
$454$	47.4624	2.22752
$455$	0	0
$456$	8.26914	0.387238
$457$	−29.8125	−1.39457	−0.697286	−	0.716793i	$-0.745609\pi$
$457$	−29.8125	−1.39457	−0.697286	+	0.716793i	$0.745609\pi$
$458$	28.5926	1.33604
$459$	−4.80577	−0.224314
$460$	39.6085	1.84675
$461$	−15.3366	−0.714299	−0.357149	−	0.934047i	$-0.616251\pi$
$461$	−15.3366	−0.714299	−0.357149	+	0.934047i	$0.616251\pi$
$462$	−15.0812	−0.701643
$463$	13.7066	0.636998	0.318499	−	0.947923i	$-0.396821\pi$
$463$	13.7066	0.636998	0.318499	+	0.947923i	$0.396821\pi$
$464$	−27.1063	−1.25838
$465$	2.33530	0.108297
$466$	26.3694	1.22154
$467$	−41.2833	−1.91036	−0.955182	−	0.296020i	$-0.904341\pi$
$467$	−41.2833	−1.91036	−0.955182	+	0.296020i	$0.904341\pi$
$468$	0	0
$469$	−14.7401	−0.680633
$470$	54.1402	2.49730
$471$	−1.06406	−0.0490295
$472$	31.9349	1.46992
$473$	−5.61258	−0.258067
$474$	2.11986	0.0973684
$475$	−2.22671	−0.102168
$476$	25.9738	1.19051
$477$	6.25784	0.286527
$478$	50.1514	2.29387
$479$	−3.06705	−0.140137	−0.0700685	−	0.997542i	$-0.522322\pi$
$479$	−3.06705	−0.140137	−0.0700685	+	0.997542i	$0.522322\pi$
$480$	−32.6992	−1.49251
$481$	0	0
$482$	41.7781	1.90294
$483$	4.63015	0.210679
$484$	106.560	4.84363
$485$	1.20082	0.0545264
$486$	2.72116	0.123434
$487$	8.27973	0.375190	0.187595	−	0.982246i	$-0.439931\pi$
$487$	8.27973	0.375190	0.187595	+	0.982246i	$0.439931\pi$
$488$	33.2189	1.50375
$489$	−7.20618	−0.325875
$490$	4.30699	0.194570
$491$	−4.97954	−0.224724	−0.112362	−	0.993667i	$-0.535842\pi$
$491$	−4.97954	−0.224724	−0.112362	+	0.993667i	$0.535842\pi$
$492$	−52.7460	−2.37797
$493$	−9.04529	−0.407379
$494$	0	0
$495$	8.77208	0.394276
$496$	21.2487	0.954096
$497$	−13.1398	−0.589399
$498$	4.00380	0.179415
$499$	−0.00121934	−5.45852e−5 0	−2.72926e−5	−	1.00000i	$-0.500009\pi$
$499$	−0.00121934	−5.45852e−5 0	−2.72926e−5		1.00000i	$0.500009\pi$
$500$	64.1142	2.86727
$501$	0.468497	0.0209309
$502$	−23.4809	−1.04800
$503$	−13.3194	−0.593881	−0.296941	−	0.954896i	$-0.595966\pi$
$503$	−13.3194	−0.593881	−0.296941	+	0.954896i	$0.595966\pi$
$504$	−9.26480	−0.412687
$505$	18.6356	0.829275
$506$	−69.8284	−3.10425
$507$	0	0
$508$	−24.9945	−1.10895
$509$	−24.9832	−1.10736	−0.553682	−	0.832729i	$-0.686777\pi$
$509$	−24.9832	−1.10736	−0.553682	+	0.832729i	$0.686777\pi$
$510$	−20.6984	−0.916541
$511$	−5.33654	−0.236075
$512$	−30.6726	−1.35555
$513$	−0.892533	−0.0394063
$514$	−7.42738	−0.327608
$515$	−15.2260	−0.670939
$516$	−5.47335	−0.240951
$517$	−69.6672	−3.06396
$518$	−13.5866	−0.596959
$519$	10.6101	0.465731
$520$	0	0
$521$	29.3265	1.28482	0.642409	−	0.766362i	$-0.277935\pi$
$521$	29.3265	1.28482	0.642409	+	0.766362i	$0.277935\pi$
$522$	5.12170	0.224171
$523$	11.4512	0.500726	0.250363	−	0.968152i	$-0.419450\pi$
$523$	11.4512	0.500726	0.250363	+	0.968152i	$0.419450\pi$
$524$	−42.0586	−1.83734
$525$	2.49482	0.108883
$526$	15.2771	0.666114
$527$	7.09065	0.308874
$528$	79.8165	3.47357
$529$	−1.56172	−0.0679010
$530$	26.9525	1.17074
$531$	−3.44690	−0.149583
$532$	4.82389	0.209142
$533$	0	0
$534$	20.8427	0.901953
$535$	−5.83915	−0.252449
$536$	136.564	5.89865
$537$	19.3674	0.835764
$538$	0.495894	0.0213795
$539$	−5.54221	−0.238720
$540$	8.55447	0.368126
$541$	7.24411	0.311449	0.155724	−	0.987801i	$-0.450229\pi$
$541$	7.24411	0.311449	0.155724	+	0.987801i	$0.450229\pi$
$542$	79.8187	3.42851
$543$	16.0004	0.686641
$544$	−99.2843	−4.25678
$545$	14.1218	0.604910
$546$	0	0
$547$	24.3579	1.04147	0.520735	−	0.853718i	$-0.325658\pi$
$547$	24.3579	1.04147	0.520735	+	0.853718i	$0.325658\pi$
$548$	−1.49361	−0.0638038
$549$	−3.58549	−0.153025
$550$	−37.6249	−1.60433
$551$	−1.67990	−0.0715663
$552$	−42.8974	−1.82583
$553$	0.779028	0.0331276
$554$	44.0142	1.86998
$555$	7.90269	0.335450
$556$	21.4062	0.907825
$557$	4.14744	0.175733	0.0878664	−	0.996132i	$-0.471995\pi$
$557$	4.14744	0.175733	0.0878664	+	0.996132i	$0.471995\pi$
$558$	−4.01493	−0.169965
$559$	0	0
$560$	−22.7945	−0.963242
$561$	26.6346	1.12451
$562$	−10.1130	−0.426593
$563$	−0.967864	−0.0407906	−0.0203953	−	0.999792i	$-0.506492\pi$
$563$	−0.967864	−0.0407906	−0.0203953	+	0.999792i	$0.506492\pi$
$564$	−67.9390	−2.86075
$565$	−25.5386	−1.07442
$566$	−10.2688	−0.431632
$567$	1.00000	0.0419961
$568$	121.737	5.10798
$569$	−12.1746	−0.510386	−0.255193	−	0.966890i	$-0.582139\pi$
$569$	−12.1746	−0.510386	−0.255193	+	0.966890i	$0.582139\pi$
$570$	−3.84413	−0.161013
$571$	28.8995	1.20941	0.604703	−	0.796451i	$-0.293292\pi$
$571$	28.8995	1.20941	0.604703	+	0.796451i	$0.293292\pi$
$572$	0	0
$573$	−3.14831	−0.131522
$574$	−26.5565	−1.10845
$575$	11.5514	0.481726
$576$	27.4144	1.14227
$577$	−14.5003	−0.603657	−0.301828	−	0.953362i	$-0.597597\pi$
$577$	−14.5003	−0.603657	−0.301828	+	0.953362i	$0.597597\pi$
$578$	−16.5866	−0.689910
$579$	−23.2821	−0.967572
$580$	16.1010	0.668557
$581$	1.47136	0.0610422
$582$	−2.06449	−0.0855758
$583$	−34.6823	−1.43639
$584$	49.4420	2.04592
$585$	0	0
$586$	41.9503	1.73295
$587$	−27.7697	−1.14618	−0.573090	−	0.819493i	$-0.694255\pi$
$587$	−27.7697	−1.14618	−0.573090	+	0.819493i	$0.694255\pi$
$588$	−5.40472	−0.222887
$589$	1.31688	0.0542613
$590$	−14.8458	−0.611191
$591$	−17.6425	−0.725714
$592$	71.9060	2.95532
$593$	−8.09800	−0.332545	−0.166273	−	0.986080i	$-0.553173\pi$
$593$	−8.09800	−0.332545	−0.166273	+	0.986080i	$0.553173\pi$
$594$	−15.0812	−0.618791
$595$	−7.60646	−0.311834
$596$	117.227	4.80180
$597$	−11.9251	−0.488062
$598$	0	0
$599$	12.8054	0.523216	0.261608	−	0.965174i	$-0.415747\pi$
$599$	12.8054	0.523216	0.261608	+	0.965174i	$0.415747\pi$
$600$	−23.1140	−0.943624
$601$	−17.8049	−0.726277	−0.363138	−	0.931735i	$-0.618295\pi$
$601$	−17.8049	−0.726277	−0.363138	+	0.931735i	$0.618295\pi$
$602$	−2.75571	−0.112314
$603$	−14.7401	−0.600262
$604$	30.6271	1.24620
$605$	−31.2061	−1.26871
$606$	−32.0390	−1.30150
$607$	6.10155	0.247654	0.123827	−	0.992304i	$-0.460483\pi$
$607$	6.10155	0.247654	0.123827	+	0.992304i	$0.460483\pi$
$608$	−18.4392	−0.747809
$609$	1.88217	0.0762695
$610$	−15.4427	−0.625256
$611$	0	0
$612$	25.9738	1.04993
$613$	17.0386	0.688184	0.344092	−	0.938936i	$-0.388187\pi$
$613$	17.0386	0.688184	0.344092	+	0.938936i	$0.388187\pi$
$614$	−90.5104	−3.65270
$615$	15.4467	0.622871
$616$	51.3474	2.06885
$617$	−25.3413	−1.02020	−0.510102	−	0.860114i	$-0.670392\pi$
$617$	−25.3413	−1.02020	−0.510102	+	0.860114i	$0.670392\pi$
$618$	26.1771	1.05300
$619$	−45.5457	−1.83064	−0.915319	−	0.402731i	$-0.868061\pi$
$619$	−45.5457	−1.83064	−0.915319	+	0.402731i	$0.868061\pi$
$620$	−12.6217	−0.506898
$621$	4.63015	0.185801
$622$	14.3749	0.576381
$623$	7.65950	0.306871
$624$	0	0
$625$	−6.30180	−0.252072
$626$	47.5804	1.90170
$627$	4.94661	0.197548
$628$	5.75097	0.229489
$629$	23.9948	0.956737
$630$	4.30699	0.171595
$631$	−1.98817	−0.0791477	−0.0395739	−	0.999217i	$-0.512600\pi$
$631$	−1.98817	−0.0791477	−0.0395739	+	0.999217i	$0.512600\pi$
$632$	−7.21754	−0.287098
$633$	15.1302	0.601372
$634$	18.7611	0.745097
$635$	7.31966	0.290472
$636$	−33.8219	−1.34113
$637$	0	0
$638$	−28.3855	−1.12379
$639$	−13.1398	−0.519801
$640$	52.6753	2.08218
$641$	−13.6949	−0.540917	−0.270459	−	0.962732i	$-0.587175\pi$
$641$	−13.6949	−0.540917	−0.270459	+	0.962732i	$0.587175\pi$
$642$	10.0389	0.396202
$643$	19.3516	0.763153	0.381576	−	0.924337i	$-0.375381\pi$
$643$	19.3516	0.763153	0.381576	+	0.924337i	$0.375381\pi$
$644$	−25.0247	−0.986110
$645$	1.60287	0.0631131
$646$	−11.6719	−0.459225
$647$	−36.6708	−1.44168	−0.720839	−	0.693103i	$-0.756243\pi$
$647$	−36.6708	−1.44168	−0.720839	+	0.693103i	$0.756243\pi$
$648$	−9.26480	−0.363956
$649$	19.1035	0.749876
$650$	0	0
$651$	−1.47545	−0.0578273
$652$	38.9474	1.52530
$653$	−15.5962	−0.610325	−0.305163	−	0.952300i	$-0.598711\pi$
$653$	−15.5962	−0.610325	−0.305163	+	0.952300i	$0.598711\pi$
$654$	−24.2786	−0.949368
$655$	12.3169	0.481261
$656$	140.548	5.48749
$657$	−5.33654	−0.208198
$658$	−34.2058	−1.33348
$659$	−27.4628	−1.06980	−0.534900	−	0.844915i	$-0.679651\pi$
$659$	−27.4628	−1.06980	−0.534900	+	0.844915i	$0.679651\pi$
$660$	−47.4106	−1.84546
$661$	−15.2896	−0.594697	−0.297348	−	0.954769i	$-0.596102\pi$
$661$	−15.2896	−0.594697	−0.297348	+	0.954769i	$0.596102\pi$
$662$	8.18021	0.317933
$663$	0	0
$664$	−13.6318	−0.529018
$665$	−1.41268	−0.0547814
$666$	−13.5866	−0.526469
$667$	8.71474	0.337436
$668$	−2.53210	−0.0979698
$669$	−19.1562	−0.740623
$670$	−63.4853	−2.45265
$671$	19.8715	0.767132
$672$	20.6594	0.796954
$673$	−24.8576	−0.958188	−0.479094	−	0.877764i	$-0.659035\pi$
$673$	−24.8576	−0.958188	−0.479094	+	0.877764i	$0.659035\pi$
$674$	19.6444	0.756675
$675$	2.49482	0.0960256
$676$	0	0
$677$	−4.31671	−0.165905	−0.0829523	−	0.996554i	$-0.526435\pi$
$677$	−4.31671	−0.165905	−0.0829523	+	0.996554i	$0.526435\pi$
$678$	43.9068	1.68623
$679$	−0.758679	−0.0291154
$680$	70.4723	2.70249
$681$	17.4419	0.668377
$682$	22.2516	0.852056
$683$	−37.6467	−1.44051	−0.720255	−	0.693709i	$-0.755975\pi$
$683$	−37.6467	−1.44051	−0.720255	+	0.693709i	$0.755975\pi$
$684$	4.82389	0.184446
$685$	0.437404	0.0167124
$686$	−2.72116	−0.103894
$687$	10.5075	0.400886
$688$	14.5844	0.556026
$689$	0	0
$690$	19.9420	0.759179
$691$	−50.2964	−1.91336	−0.956682	−	0.291135i	$-0.905967\pi$
$691$	−50.2964	−1.91336	−0.956682	+	0.291135i	$0.905967\pi$
$692$	−57.3446	−2.17991
$693$	−5.54221	−0.210531
$694$	−5.23758	−0.198816
$695$	−6.26882	−0.237790
$696$	−17.4380	−0.660984
$697$	46.9007	1.77649
$698$	−8.87020	−0.335742
$699$	9.69049	0.366528
$700$	−13.4838	−0.509639
$701$	−12.0336	−0.454504	−0.227252	−	0.973836i	$-0.572974\pi$
$701$	−12.0336	−0.454504	−0.227252	+	0.973836i	$0.572974\pi$
$702$	0	0
$703$	4.45635	0.168075
$704$	−151.937	−5.72632
$705$	19.8960	0.749326
$706$	24.2533	0.912786
$707$	−11.7740	−0.442808
$708$	18.6296	0.700141
$709$	43.2095	1.62277	0.811383	−	0.584515i	$-0.198715\pi$
$709$	43.2095	1.62277	0.811383	+	0.584515i	$0.198715\pi$
$710$	−56.5929	−2.12389
$711$	0.779028	0.0292158
$712$	−70.9637	−2.65948
$713$	−6.83153	−0.255843
$714$	13.0773	0.489404
$715$	0	0
$716$	−104.675	−3.91190
$717$	18.4302	0.688287
$718$	54.7824	2.04446
$719$	40.9527	1.52728	0.763639	−	0.645644i	$-0.223411\pi$
$719$	40.9527	1.52728	0.763639	+	0.645644i	$0.223411\pi$
$720$	−22.7945	−0.849500
$721$	9.61981	0.358261
$722$	49.5344	1.84348
$723$	15.3530	0.570986
$724$	−86.4775	−3.21391
$725$	4.69568	0.174393
$726$	53.6506	1.99116
$727$	−24.7078	−0.916360	−0.458180	−	0.888859i	$-0.651499\pi$
$727$	−24.7078	−0.916360	−0.458180	+	0.888859i	$0.651499\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−22.9845	−0.850693
$731$	4.86679	0.180005
$732$	19.3786	0.716253
$733$	−8.10694	−0.299437	−0.149718	−	0.988729i	$-0.547837\pi$
$733$	−8.10694	−0.299437	−0.149718	+	0.988729i	$0.547837\pi$
$734$	85.2860	3.14796
$735$	1.58278	0.0583816
$736$	95.6561	3.52593
$737$	81.6925	3.00918
$738$	−26.5565	−0.977558
$739$	−36.8696	−1.35627	−0.678135	−	0.734938i	$-0.737211\pi$
$739$	−36.8696	−1.35627	−0.678135	+	0.734938i	$0.737211\pi$
$740$	−42.7118	−1.57012
$741$	0	0
$742$	−17.0286	−0.625140
$743$	7.62191	0.279621	0.139810	−	0.990178i	$-0.455351\pi$
$743$	7.62191	0.279621	0.139810	+	0.990178i	$0.455351\pi$
$744$	13.6697	0.501156
$745$	−34.3300	−1.25775
$746$	−48.8270	−1.78768
$747$	1.47136	0.0538342
$748$	−143.952	−5.26342
$749$	3.68918	0.134800
$750$	32.2801	1.17870
$751$	20.6711	0.754300	0.377150	−	0.926152i	$-0.376904\pi$
$751$	20.6711	0.754300	0.377150	+	0.926152i	$0.376904\pi$
$752$	181.032	6.60156
$753$	−8.62900	−0.314458
$754$	0	0
$755$	−8.96916	−0.326421
$756$	−5.40472	−0.196568
$757$	−2.18816	−0.0795299	−0.0397650	−	0.999209i	$-0.512661\pi$
$757$	−2.18816	−0.0795299	−0.0397650	+	0.999209i	$0.512661\pi$
$758$	54.8250	1.99133
$759$	−25.6612	−0.931444
$760$	13.0882	0.474759
$761$	−31.3762	−1.13739	−0.568694	−	0.822549i	$-0.692551\pi$
$761$	−31.3762	−1.13739	−0.568694	+	0.822549i	$0.692551\pi$
$762$	−12.5842	−0.455877
$763$	−8.92215	−0.323003
$764$	17.0157	0.615607
$765$	−7.60646	−0.275012
$766$	102.079	3.68825
$767$	0	0
$768$	−35.7323	−1.28938
$769$	−8.90840	−0.321245	−0.160623	−	0.987016i	$-0.551350\pi$
$769$	−8.90840	−0.321245	−0.160623	+	0.987016i	$0.551350\pi$
$770$	−23.8702	−0.860224
$771$	−2.72949	−0.0983002
$772$	125.833	4.52884
$773$	3.53796	0.127252	0.0636258	−	0.997974i	$-0.479734\pi$
$773$	3.53796	0.127252	0.0636258	+	0.997974i	$0.479734\pi$
$774$	−2.75571	−0.0990521
$775$	−3.68097	−0.132224
$776$	7.02901	0.252327
$777$	−4.99293	−0.179120
$778$	63.1162	2.26282
$779$	8.71045	0.312084
$780$	0	0
$781$	72.8233	2.60582
$782$	60.5497	2.16525
$783$	1.88217	0.0672634
$784$	14.4016	0.514342
$785$	−1.68418	−0.0601108
$786$	−21.1756	−0.755309
$787$	20.5207	0.731485	0.365742	−	0.930716i	$-0.380815\pi$
$787$	20.5207	0.731485	0.365742	+	0.930716i	$0.380815\pi$
$788$	95.3526	3.39680
$789$	5.61419	0.199870
$790$	3.35527	0.119375
$791$	16.1353	0.573706
$792$	51.3474	1.82455
$793$	0	0
$794$	−103.841	−3.68518
$795$	9.90477	0.351286
$796$	64.4519	2.28444
$797$	−39.8650	−1.41209	−0.706045	−	0.708167i	$-0.749522\pi$
$797$	−39.8650	−1.41209	−0.706045	+	0.708167i	$0.749522\pi$
$798$	2.42873	0.0859760
$799$	60.4099	2.13715
$800$	51.5414	1.82227
$801$	7.65950	0.270635
$802$	−22.5533	−0.796385
$803$	29.5762	1.04372
$804$	79.6659	2.80960
$805$	7.32849	0.258295
$806$	0	0
$807$	0.182236	0.00641501
$808$	109.084	3.83756
$809$	41.6694	1.46502	0.732509	−	0.680757i	$-0.238349\pi$
$809$	41.6694	1.46502	0.732509	+	0.680757i	$0.238349\pi$
$810$	4.30699	0.151332
$811$	20.6446	0.724931	0.362465	−	0.931997i	$-0.381935\pi$
$811$	20.6446	0.724931	0.362465	+	0.931997i	$0.381935\pi$
$812$	−10.1726	−0.356989
$813$	29.3326	1.02874
$814$	75.2995	2.63925
$815$	−11.4058	−0.399527
$816$	−69.2106	−2.42286
$817$	0.903866	0.0316223
$818$	106.431	3.72129
$819$	0	0
$820$	−83.4851	−2.91543
$821$	14.1363	0.493359	0.246679	−	0.969097i	$-0.420661\pi$
$821$	14.1363	0.493359	0.246679	+	0.969097i	$0.420661\pi$
$822$	−0.752000	−0.0262290
$823$	1.64597	0.0573749	0.0286874	−	0.999588i	$-0.490867\pi$
$823$	1.64597	0.0573749	0.0286874	+	0.999588i	$0.490867\pi$
$824$	−89.1256	−3.10484
$825$	−13.8268	−0.481387
$826$	9.37958	0.326357
$827$	10.1980	0.354620	0.177310	−	0.984155i	$-0.443260\pi$
$827$	10.1980	0.354620	0.177310	+	0.984155i	$0.443260\pi$
$828$	−25.0247	−0.869667
$829$	−1.67774	−0.0582703	−0.0291351	−	0.999575i	$-0.509275\pi$
$829$	−1.67774	−0.0582703	−0.0291351	+	0.999575i	$0.509275\pi$
$830$	6.33713	0.219965
$831$	16.1748	0.561097
$832$	0	0
$833$	4.80577	0.166510
$834$	10.7776	0.373197
$835$	0.741527	0.0256616
$836$	−26.7350	−0.924650
$837$	−1.47545	−0.0509989
$838$	−74.9587	−2.58940
$839$	10.7213	0.370140	0.185070	−	0.982725i	$-0.440749\pi$
$839$	10.7213	0.370140	0.185070	+	0.982725i	$0.440749\pi$
$840$	−14.6641	−0.505960
$841$	−25.4574	−0.877842
$842$	92.8557	3.20002
$843$	−3.71644	−0.128001
$844$	−81.7745	−2.81480
$845$	0	0
$846$	−34.2058	−1.17602
$847$	19.7161	0.677452
$848$	90.1228	3.09483
$849$	−3.77370	−0.129513
$850$	32.6254	1.11904
$851$	−23.1180	−0.792475
$852$	71.0168	2.43299
$853$	5.01741	0.171793	0.0858964	−	0.996304i	$-0.472625\pi$
$853$	5.01741	0.171793	0.0858964	+	0.996304i	$0.472625\pi$
$854$	9.75670	0.333868
$855$	−1.41268	−0.0483127
$856$	−34.1795	−1.16823
$857$	25.0009	0.854015	0.427008	−	0.904248i	$-0.359568\pi$
$857$	25.0009	0.854015	0.427008	+	0.904248i	$0.359568\pi$
$858$	0	0
$859$	17.9200	0.611424	0.305712	−	0.952124i	$-0.401106\pi$
$859$	17.9200	0.611424	0.305712	+	0.952124i	$0.401106\pi$
$860$	−8.66309	−0.295409
$861$	−9.75924	−0.332594
$862$	−13.6082	−0.463498
$863$	−7.10038	−0.241700	−0.120850	−	0.992671i	$-0.538562\pi$
$863$	−7.10038	−0.241700	−0.120850	+	0.992671i	$0.538562\pi$
$864$	20.6594	0.702847
$865$	16.7934	0.570993
$866$	−40.5823	−1.37904
$867$	−6.09539	−0.207010
$868$	7.97437	0.270668
$869$	−4.31753	−0.146462
$870$	8.10651	0.274836
$871$	0	0
$872$	82.6619	2.79928
$873$	−0.758679	−0.0256774
$874$	11.2454	0.380380
$875$	11.8626	0.401030
$876$	28.8425	0.974499
$877$	26.7489	0.903248	0.451624	−	0.892208i	$-0.350845\pi$
$877$	26.7489	0.903248	0.451624	+	0.892208i	$0.350845\pi$
$878$	19.3278	0.652280
$879$	15.4163	0.519980
$880$	126.332	4.25864
$881$	10.0293	0.337897	0.168949	−	0.985625i	$-0.445963\pi$
$881$	10.0293	0.337897	0.168949	+	0.985625i	$0.445963\pi$
$882$	−2.72116	−0.0916263
$883$	−1.88319	−0.0633743	−0.0316871	−	0.999498i	$-0.510088\pi$
$883$	−1.88319	−0.0633743	−0.0316871	+	0.999498i	$0.510088\pi$
$884$	0	0
$885$	−5.45568	−0.183391
$886$	12.6035	0.423422
$887$	−38.5774	−1.29530	−0.647652	−	0.761936i	$-0.724249\pi$
$887$	−38.5774	−1.29530	−0.647652	+	0.761936i	$0.724249\pi$
$888$	46.2585	1.55233
$889$	−4.62457	−0.155103
$890$	32.9894	1.10581
$891$	−5.54221	−0.185671
$892$	103.534	3.46658
$893$	11.2194	0.375443
$894$	59.0212	1.97397
$895$	30.6542	1.02466
$896$	−33.2803	−1.11182
$897$	0	0
$898$	−61.4243	−2.04976
$899$	−2.77705	−0.0926197
$900$	−13.4838	−0.449460
$901$	30.0737	1.00190
$902$	147.182	4.90061
$903$	−1.01270	−0.0337005
$904$	−149.490	−4.97198
$905$	25.3250	0.841831
$906$	15.4201	0.512298
$907$	36.3170	1.20589	0.602944	−	0.797784i	$-0.293994\pi$
$907$	36.3170	1.20589	0.602944	+	0.797784i	$0.293994\pi$
$908$	−94.2689	−3.12842
$909$	−11.7740	−0.390520
$910$	0	0
$911$	5.42183	0.179633	0.0898167	−	0.995958i	$-0.471372\pi$
$911$	5.42183	0.179633	0.0898167	+	0.995958i	$0.471372\pi$
$912$	−12.8539	−0.425635
$913$	−8.15458	−0.269877
$914$	81.1248	2.68337
$915$	−5.67503	−0.187611
$916$	−56.7900	−1.87640
$917$	−7.78183	−0.256979
$918$	13.0773	0.431614
$919$	15.8355	0.522364	0.261182	−	0.965290i	$-0.415888\pi$
$919$	15.8355	0.522364	0.261182	+	0.965290i	$0.415888\pi$
$920$	−67.8970	−2.23850
$921$	−33.2617	−1.09601
$922$	41.7335	1.37442
$923$	0	0
$924$	29.9541	0.985417
$925$	−12.4564	−0.409565
$926$	−37.2978	−1.22568
$927$	9.61981	0.315956
$928$	38.8846	1.27645
$929$	−23.0269	−0.755488	−0.377744	−	0.925910i	$-0.623300\pi$
$929$	−23.0269	−0.755488	−0.377744	+	0.925910i	$0.623300\pi$
$930$	−6.35473	−0.208380
$931$	0.892533	0.0292516
$932$	−52.3744	−1.71558
$933$	5.28263	0.172946
$934$	112.339	3.67583
$935$	42.1566	1.37867
$936$	0	0
$937$	−48.9751	−1.59995	−0.799974	−	0.600035i	$-0.795153\pi$
$937$	−48.9751	−1.59995	−0.799974	+	0.600035i	$0.795153\pi$
$938$	40.1101	1.30964
$939$	17.4853	0.570612
$940$	−107.532	−3.50731
$941$	31.1637	1.01591	0.507954	−	0.861384i	$-0.330402\pi$
$941$	31.1637	1.01591	0.507954	+	0.861384i	$0.330402\pi$
$942$	2.89549	0.0943402
$943$	−45.1868	−1.47148
$944$	−49.6408	−1.61567
$945$	1.58278	0.0514877
$946$	15.2727	0.496559
$947$	45.0524	1.46401	0.732003	−	0.681302i	$-0.238586\pi$
$947$	45.0524	1.46401	0.732003	+	0.681302i	$0.238586\pi$
$948$	−4.21043	−0.136748
$949$	0	0
$950$	6.05923	0.196587
$951$	6.89451	0.223570
$952$	−44.5245	−1.44305
$953$	8.73966	0.283105	0.141553	−	0.989931i	$-0.454791\pi$
$953$	8.73966	0.283105	0.141553	+	0.989931i	$0.454791\pi$
$954$	−17.0286	−0.551321
$955$	−4.98307	−0.161248
$956$	−99.6099	−3.22161
$957$	−10.4314	−0.337199
$958$	8.34593	0.269645
$959$	−0.276352	−0.00892389
$960$	43.3909	1.40044
$961$	−28.8231	−0.929776
$962$	0	0
$963$	3.68918	0.118882
$964$	−82.9789	−2.67257
$965$	−36.8504	−1.18626
$966$	−12.5994	−0.405379
$967$	5.92797	0.190631	0.0953154	−	0.995447i	$-0.469614\pi$
$967$	5.92797	0.190631	0.0953154	+	0.995447i	$0.469614\pi$
$968$	−182.665	−5.87109
$969$	−4.28931	−0.137792
$970$	−3.26762	−0.104917
$971$	−31.4172	−1.00823	−0.504114	−	0.863637i	$-0.668181\pi$
$971$	−31.4172	−1.00823	−0.504114	+	0.863637i	$0.668181\pi$
$972$	−5.40472	−0.173357
$973$	3.96065	0.126973
$974$	−22.5305	−0.721923
$975$	0	0
$976$	−51.6367	−1.65285
$977$	38.2801	1.22469	0.612344	−	0.790591i	$-0.290227\pi$
$977$	38.2801	1.22469	0.612344	+	0.790591i	$0.290227\pi$
$978$	19.6092	0.627032
$979$	−42.4505	−1.35672
$980$	−8.55447	−0.273262
$981$	−8.92215	−0.284862
$982$	13.5501	0.432402
$983$	30.9632	0.987572	0.493786	−	0.869583i	$-0.335613\pi$
$983$	30.9632	0.987572	0.493786	+	0.869583i	$0.335613\pi$
$984$	90.4174	2.88240
$985$	−27.9241	−0.889736
$986$	24.6137	0.783860
$987$	−12.5703	−0.400117
$988$	0	0
$989$	−4.68894	−0.149100
$990$	−23.8702	−0.758646
$991$	−11.1295	−0.353540	−0.176770	−	0.984252i	$-0.556565\pi$
$991$	−11.1295	−0.353540	−0.176770	+	0.984252i	$0.556565\pi$
$992$	−30.4818	−0.967799
$993$	3.00614	0.0953971
$994$	35.7554	1.13409
$995$	−18.8748	−0.598371
$996$	−7.95228	−0.251978
$997$	14.2980	0.452823	0.226412	−	0.974032i	$-0.427301\pi$
$997$	14.2980	0.452823	0.226412	+	0.974032i	$0.427301\pi$
$998$	0.00331802	0.000105030 0
$999$	−4.99293	−0.157969

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3549.2.a.ba.1.1		8
13.6	odd	12		273.2.bd.b.127.8	yes	16
13.11	odd	12		273.2.bd.b.43.8	✓	16
13.12	even	2		3549.2.a.bc.1.8		8
39.11	even	12		819.2.ct.c.316.1		16
39.32	even	12		819.2.ct.c.127.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bd.b.43.8	✓	16	13.11	odd	12
273.2.bd.b.127.8	yes	16	13.6	odd	12
819.2.ct.c.127.1		16	39.32	even	12
819.2.ct.c.316.1		16	39.11	even	12
3549.2.a.ba.1.1		8	1.1	even	1	trivial
3549.2.a.bc.1.8		8	13.12	even	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:	\( N \)	\(=\)	\( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3549.a (trivial)

\(f(q)\)	\(=\)	\(q-2.72116 q^{2} -1.00000 q^{3} +5.40472 q^{4} -1.58278 q^{5} +2.72116 q^{6} +1.00000 q^{7} -9.26480 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.72116 q^{2} -1.00000 q^{3} +5.40472 q^{4} -1.58278 q^{5} +2.72116 q^{6} +1.00000 q^{7} -9.26480 q^{8} +1.00000 q^{9} +4.30699 q^{10} -5.54221 q^{11} -5.40472 q^{12} -2.72116 q^{14} +1.58278 q^{15} +14.4016 q^{16} +4.80577 q^{17} -2.72116 q^{18} +0.892533 q^{19} -8.55447 q^{20} -1.00000 q^{21} +15.0812 q^{22} -4.63015 q^{23} +9.26480 q^{24} -2.49482 q^{25} -1.00000 q^{27} +5.40472 q^{28} -1.88217 q^{29} -4.30699 q^{30} +1.47545 q^{31} -20.6594 q^{32} +5.54221 q^{33} -13.0773 q^{34} -1.58278 q^{35} +5.40472 q^{36} +4.99293 q^{37} -2.42873 q^{38} +14.6641 q^{40} +9.75924 q^{41} +2.72116 q^{42} +1.01270 q^{43} -29.9541 q^{44} -1.58278 q^{45} +12.5994 q^{46} +12.5703 q^{47} -14.4016 q^{48} +1.00000 q^{49} +6.78880 q^{50} -4.80577 q^{51} +6.25784 q^{53} +2.72116 q^{54} +8.77208 q^{55} -9.26480 q^{56} -0.892533 q^{57} +5.12170 q^{58} -3.44690 q^{59} +8.55447 q^{60} -3.58549 q^{61} -4.01493 q^{62} +1.00000 q^{63} +27.4144 q^{64} -15.0812 q^{66} -14.7401 q^{67} +25.9738 q^{68} +4.63015 q^{69} +4.30699 q^{70} -13.1398 q^{71} -9.26480 q^{72} -5.33654 q^{73} -13.5866 q^{74} +2.49482 q^{75} +4.82389 q^{76} -5.54221 q^{77} +0.779028 q^{79} -22.7945 q^{80} +1.00000 q^{81} -26.5565 q^{82} +1.47136 q^{83} -5.40472 q^{84} -7.60646 q^{85} -2.75571 q^{86} +1.88217 q^{87} +51.3474 q^{88} +7.65950 q^{89} +4.30699 q^{90} -25.0247 q^{92} -1.47545 q^{93} -34.2058 q^{94} -1.41268 q^{95} +20.6594 q^{96} -0.758679 q^{97} -2.72116 q^{98} -5.54221 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 8 q^{3} + 14 q^{4} - 6 q^{5} + 2 q^{6} + 8 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 2 * q^2 - 8 * q^3 + 14 * q^4 - 6 * q^5 + 2 * q^6 + 8 * q^7 - 12 * q^8 + 8 * q^9	\( 8 q - 2 q^{2} - 8 q^{3} + 14 q^{4} - 6 q^{5} + 2 q^{6} + 8 q^{7} - 12 q^{8} + 8 q^{9} - 4 q^{10} - 16 q^{11} - 14 q^{12} - 2 q^{14} + 6 q^{15} + 10 q^{16} - 2 q^{17} - 2 q^{18} - 14 q^{19} + 10 q^{20} - 8 q^{21} + 18 q^{22} - 6 q^{23} + 12 q^{24} + 10 q^{25} - 8 q^{27} + 14 q^{28} + 12 q^{29} + 4 q^{30} - 16 q^{31} - 26 q^{32} + 16 q^{33} - 32 q^{34} - 6 q^{35} + 14 q^{36} + 8 q^{37} + 12 q^{38} - 14 q^{40} + 4 q^{41} + 2 q^{42} + 14 q^{43} - 68 q^{44} - 6 q^{45} + 28 q^{46} - 6 q^{47} - 10 q^{48} + 8 q^{49} - 32 q^{50} + 2 q^{51} + 14 q^{53} + 2 q^{54} - 2 q^{55} - 12 q^{56} + 14 q^{57} - 24 q^{58} - 50 q^{59} - 10 q^{60} - 2 q^{61} - 20 q^{62} + 8 q^{63} + 24 q^{64} - 18 q^{66} + 16 q^{67} - 36 q^{68} + 6 q^{69} - 4 q^{70} - 34 q^{71} - 12 q^{72} + 8 q^{73} - 6 q^{74} - 10 q^{75} + 4 q^{76} - 16 q^{77} + 46 q^{79} + 24 q^{80} + 8 q^{81} - 42 q^{82} - 42 q^{83} - 14 q^{84} - 20 q^{85} - 50 q^{86} - 12 q^{87} + 62 q^{88} - 28 q^{89} - 4 q^{90} - 58 q^{92} + 16 q^{93} + 24 q^{94} - 24 q^{95} + 26 q^{96} - 16 q^{97} - 2 q^{98} - 16 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 - 8 * q^3 + 14 * q^4 - 6 * q^5 + 2 * q^6 + 8 * q^7 - 12 * q^8 + 8 * q^9 - 4 * q^10 - 16 * q^11 - 14 * q^12 - 2 * q^14 + 6 * q^15 + 10 * q^16 - 2 * q^17 - 2 * q^18 - 14 * q^19 + 10 * q^20 - 8 * q^21 + 18 * q^22 - 6 * q^23 + 12 * q^24 + 10 * q^25 - 8 * q^27 + 14 * q^28 + 12 * q^29 + 4 * q^30 - 16 * q^31 - 26 * q^32 + 16 * q^33 - 32 * q^34 - 6 * q^35 + 14 * q^36 + 8 * q^37 + 12 * q^38 - 14 * q^40 + 4 * q^41 + 2 * q^42 + 14 * q^43 - 68 * q^44 - 6 * q^45 + 28 * q^46 - 6 * q^47 - 10 * q^48 + 8 * q^49 - 32 * q^50 + 2 * q^51 + 14 * q^53 + 2 * q^54 - 2 * q^55 - 12 * q^56 + 14 * q^57 - 24 * q^58 - 50 * q^59 - 10 * q^60 - 2 * q^61 - 20 * q^62 + 8 * q^63 + 24 * q^64 - 18 * q^66 + 16 * q^67 - 36 * q^68 + 6 * q^69 - 4 * q^70 - 34 * q^71 - 12 * q^72 + 8 * q^73 - 6 * q^74 - 10 * q^75 + 4 * q^76 - 16 * q^77 + 46 * q^79 + 24 * q^80 + 8 * q^81 - 42 * q^82 - 42 * q^83 - 14 * q^84 - 20 * q^85 - 50 * q^86 - 12 * q^87 + 62 * q^88 - 28 * q^89 - 4 * q^90 - 58 * q^92 + 16 * q^93 + 24 * q^94 - 24 * q^95 + 26 * q^96 - 16 * q^97 - 2 * q^98 - 16 * q^99

Introduction

L-functions

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