LMFDB - Embedded newform 7098.2.a.cn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, 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("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7098.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:	$56.6778153547$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13968.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 4 $ x^4 - 2x^3 - 7x^2 + 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.386509$ of defining polynomial
Character	$\chi$	$=$	7098.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.05596 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.05596 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.05596 q^{10} -3.44247 q^{11} -1.00000 q^{12} +1.00000 q^{14} +3.05596 q^{15} +1.00000 q^{16} -2.82898 q^{17} -1.00000 q^{18} +1.72542 q^{19} -3.05596 q^{20} +1.00000 q^{21} +3.44247 q^{22} +3.06260 q^{23} +1.00000 q^{24} +4.33891 q^{25} -1.00000 q^{27} -1.00000 q^{28} +3.85061 q^{29} -3.05596 q^{30} -0.978370 q^{31} -1.00000 q^{32} +3.44247 q^{33} +2.82898 q^{34} +3.05596 q^{35} +1.00000 q^{36} +5.29308 q^{37} -1.72542 q^{38} +3.05596 q^{40} -9.96254 q^{41} -1.00000 q^{42} -3.02163 q^{43} -3.44247 q^{44} -3.05596 q^{45} -3.06260 q^{46} -8.04447 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.33891 q^{50} +2.82898 q^{51} -8.33405 q^{53} +1.00000 q^{54} +10.5201 q^{55} +1.00000 q^{56} -1.72542 q^{57} -3.85061 q^{58} -9.97753 q^{59} +3.05596 q^{60} -11.5452 q^{61} +0.978370 q^{62} -1.00000 q^{63} +1.00000 q^{64} -3.44247 q^{66} -4.75055 q^{67} -2.82898 q^{68} -3.06260 q^{69} -3.05596 q^{70} -3.66945 q^{71} -1.00000 q^{72} -10.1119 q^{73} -5.29308 q^{74} -4.33891 q^{75} +1.72542 q^{76} +3.44247 q^{77} -4.49843 q^{79} -3.05596 q^{80} +1.00000 q^{81} +9.96254 q^{82} +7.15869 q^{83} +1.00000 q^{84} +8.64526 q^{85} +3.02163 q^{86} -3.85061 q^{87} +3.44247 q^{88} -7.90307 q^{89} +3.05596 q^{90} +3.06260 q^{92} +0.978370 q^{93} +8.04447 q^{94} -5.27281 q^{95} +1.00000 q^{96} -9.09922 q^{97} -1.00000 q^{98} -3.44247 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 6 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9} + 6 q^{10} - 4 q^{11} - 4 q^{12} + 4 q^{14} + 6 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} - 2 q^{19} - 6 q^{20} + 4 q^{21} + 4 q^{22} + 8 q^{23} + 4 q^{24} + 12 q^{25} - 4 q^{27} - 4 q^{28} - 2 q^{29} - 6 q^{30} - 8 q^{31} - 4 q^{32} + 4 q^{33} - 2 q^{34} + 6 q^{35} + 4 q^{36} - 6 q^{37} + 2 q^{38} + 6 q^{40} - 10 q^{41} - 4 q^{42} - 8 q^{43} - 4 q^{44} - 6 q^{45} - 8 q^{46} - 2 q^{47} - 4 q^{48} + 4 q^{49} - 12 q^{50} - 2 q^{51} - 6 q^{53} + 4 q^{54} + 22 q^{55} + 4 q^{56} + 2 q^{57} + 2 q^{58} - 4 q^{59} + 6 q^{60} + 8 q^{61} + 8 q^{62} - 4 q^{63} + 4 q^{64} - 4 q^{66} + 24 q^{67} + 2 q^{68} - 8 q^{69} - 6 q^{70} - 12 q^{71} - 4 q^{72} - 28 q^{73} + 6 q^{74} - 12 q^{75} - 2 q^{76} + 4 q^{77} - 2 q^{79} - 6 q^{80} + 4 q^{81} + 10 q^{82} - 22 q^{83} + 4 q^{84} + 6 q^{85} + 8 q^{86} + 2 q^{87} + 4 q^{88} - 38 q^{89} + 6 q^{90} + 8 q^{92} + 8 q^{93} + 2 q^{94} - 50 q^{95} + 4 q^{96} - 22 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 6 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 4 * q^9 + 6 * q^10 - 4 * q^11 - 4 * q^12 + 4 * q^14 + 6 * q^15 + 4 * q^16 + 2 * q^17 - 4 * q^18 - 2 * q^19 - 6 * q^20 + 4 * q^21 + 4 * q^22 + 8 * q^23 + 4 * q^24 + 12 * q^25 - 4 * q^27 - 4 * q^28 - 2 * q^29 - 6 * q^30 - 8 * q^31 - 4 * q^32 + 4 * q^33 - 2 * q^34 + 6 * q^35 + 4 * q^36 - 6 * q^37 + 2 * q^38 + 6 * q^40 - 10 * q^41 - 4 * q^42 - 8 * q^43 - 4 * q^44 - 6 * q^45 - 8 * q^46 - 2 * q^47 - 4 * q^48 + 4 * q^49 - 12 * q^50 - 2 * q^51 - 6 * q^53 + 4 * q^54 + 22 * q^55 + 4 * q^56 + 2 * q^57 + 2 * q^58 - 4 * q^59 + 6 * q^60 + 8 * q^61 + 8 * q^62 - 4 * q^63 + 4 * q^64 - 4 * q^66 + 24 * q^67 + 2 * q^68 - 8 * q^69 - 6 * q^70 - 12 * q^71 - 4 * q^72 - 28 * q^73 + 6 * q^74 - 12 * q^75 - 2 * q^76 + 4 * q^77 - 2 * q^79 - 6 * q^80 + 4 * q^81 + 10 * q^82 - 22 * q^83 + 4 * q^84 + 6 * q^85 + 8 * q^86 + 2 * q^87 + 4 * q^88 - 38 * q^89 + 6 * q^90 + 8 * q^92 + 8 * q^93 + 2 * q^94 - 50 * q^95 + 4 * q^96 - 22 * q^97 - 4 * q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−3.05596	−1.36667	−0.683334	−	0.730106i	$-0.739471\pi$
$5$	−3.05596	−1.36667	−0.683334	+	0.730106i	$0.739471\pi$
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.05596	0.966380
$11$	−3.44247	−1.03794	−0.518972	−	0.854791i	$-0.673685\pi$
$11$	−3.44247	−1.03794	−0.518972	+	0.854791i	$0.673685\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	3.05596	0.789046
$16$	1.00000	0.250000
$17$	−2.82898	−0.686129	−0.343064	−	0.939312i	$-0.611465\pi$
$17$	−2.82898	−0.686129	−0.343064	+	0.939312i	$0.611465\pi$
$18$	−1.00000	−0.235702
$19$	1.72542	0.395838	0.197919	−	0.980218i	$-0.436582\pi$
$19$	1.72542	0.395838	0.197919	+	0.980218i	$0.436582\pi$
$20$	−3.05596	−0.683334
$21$	1.00000	0.218218
$22$	3.44247	0.733937
$23$	3.06260	0.638596	0.319298	−	0.947654i	$-0.396553\pi$
$23$	3.06260	0.638596	0.319298	+	0.947654i	$0.396553\pi$
$24$	1.00000	0.204124
$25$	4.33891	0.867781
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	3.85061	0.715040	0.357520	−	0.933905i	$-0.383622\pi$
$29$	3.85061	0.715040	0.357520	+	0.933905i	$0.383622\pi$
$30$	−3.05596	−0.557940
$31$	−0.978370	−0.175720	−0.0878602	−	0.996133i	$-0.528003\pi$
$31$	−0.978370	−0.175720	−0.0878602	+	0.996133i	$0.528003\pi$
$32$	−1.00000	−0.176777
$33$	3.44247	0.599257
$34$	2.82898	0.485166
$35$	3.05596	0.516552
$36$	1.00000	0.166667
$37$	5.29308	0.870177	0.435089	−	0.900388i	$-0.356717\pi$
$37$	5.29308	0.870177	0.435089	+	0.900388i	$0.356717\pi$
$38$	−1.72542	−0.279899
$39$	0	0
$40$	3.05596	0.483190
$41$	−9.96254	−1.55589	−0.777943	−	0.628334i	$-0.783737\pi$
$41$	−9.96254	−1.55589	−0.777943	+	0.628334i	$0.783737\pi$
$42$	−1.00000	−0.154303
$43$	−3.02163	−0.460794	−0.230397	−	0.973097i	$-0.574003\pi$
$43$	−3.02163	−0.460794	−0.230397	+	0.973097i	$0.574003\pi$
$44$	−3.44247	−0.518972
$45$	−3.05596	−0.455556
$46$	−3.06260	−0.451555
$47$	−8.04447	−1.17341	−0.586703	−	0.809802i	$-0.699575\pi$
$47$	−8.04447	−1.17341	−0.586703	+	0.809802i	$0.699575\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−4.33891	−0.613614
$51$	2.82898	0.396137
$52$	0	0
$53$	−8.33405	−1.14477	−0.572385	−	0.819985i	$-0.693982\pi$
$53$	−8.33405	−1.14477	−0.572385	+	0.819985i	$0.693982\pi$
$54$	1.00000	0.136083
$55$	10.5201	1.41853
$56$	1.00000	0.133631
$57$	−1.72542	−0.228537
$58$	−3.85061	−0.505610
$59$	−9.97753	−1.29896	−0.649482	−	0.760377i	$-0.725014\pi$
$59$	−9.97753	−1.29896	−0.649482	+	0.760377i	$0.725014\pi$
$60$	3.05596	0.394523
$61$	−11.5452	−1.47821	−0.739106	−	0.673590i	$-0.764752\pi$
$61$	−11.5452	−1.47821	−0.739106	+	0.673590i	$0.764752\pi$
$62$	0.978370	0.124253
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.44247	−0.423739
$67$	−4.75055	−0.580372	−0.290186	−	0.956970i	$-0.593717\pi$
$67$	−4.75055	−0.580372	−0.290186	+	0.956970i	$0.593717\pi$
$68$	−2.82898	−0.343064
$69$	−3.06260	−0.368693
$70$	−3.05596	−0.365257
$71$	−3.66945	−0.435484	−0.217742	−	0.976006i	$-0.569869\pi$
$71$	−3.66945	−0.435484	−0.217742	+	0.976006i	$0.569869\pi$
$72$	−1.00000	−0.117851
$73$	−10.1119	−1.18351	−0.591756	−	0.806117i	$-0.701565\pi$
$73$	−10.1119	−1.18351	−0.591756	+	0.806117i	$0.701565\pi$
$74$	−5.29308	−0.615308
$75$	−4.33891	−0.501014
$76$	1.72542	0.197919
$77$	3.44247	0.392306
$78$	0	0
$79$	−4.49843	−0.506113	−0.253057	−	0.967451i	$-0.581436\pi$
$79$	−4.49843	−0.506113	−0.253057	+	0.967451i	$0.581436\pi$
$80$	−3.05596	−0.341667
$81$	1.00000	0.111111
$82$	9.96254	1.10018
$83$	7.15869	0.785768	0.392884	−	0.919588i	$-0.371477\pi$
$83$	7.15869	0.785768	0.392884	+	0.919588i	$0.371477\pi$
$84$	1.00000	0.109109
$85$	8.64526	0.937710
$86$	3.02163	0.325831
$87$	−3.85061	−0.412829
$88$	3.44247	0.366969
$89$	−7.90307	−0.837724	−0.418862	−	0.908050i	$-0.637571\pi$
$89$	−7.90307	−0.837724	−0.418862	+	0.908050i	$0.637571\pi$
$90$	3.05596	0.322127
$91$	0	0
$92$	3.06260	0.319298
$93$	0.978370	0.101452
$94$	8.04447	0.829724
$95$	−5.27281	−0.540979
$96$	1.00000	0.102062
$97$	−9.09922	−0.923886	−0.461943	−	0.886910i	$-0.652848\pi$
$97$	−9.09922	−0.923886	−0.461943	+	0.886910i	$0.652848\pi$
$98$	−1.00000	−0.101015
$99$	−3.44247	−0.345981
$100$	4.33891	0.433891
$101$	18.8533	1.87597	0.937985	−	0.346675i	$-0.112689\pi$
$101$	18.8533	1.87597	0.937985	+	0.346675i	$0.112689\pi$
$102$	−2.82898	−0.280111
$103$	13.7079	1.35068	0.675338	−	0.737509i	$-0.263998\pi$
$103$	13.7079	1.35068	0.675338	+	0.737509i	$0.263998\pi$
$104$	0	0
$105$	−3.05596	−0.298231
$106$	8.33405	0.809474
$107$	−3.31471	−0.320445	−0.160223	−	0.987081i	$-0.551221\pi$
$107$	−3.31471	−0.320445	−0.160223	+	0.987081i	$0.551221\pi$
$108$	−1.00000	−0.0962250
$109$	8.67525	0.830938	0.415469	−	0.909607i	$-0.363617\pi$
$109$	8.67525	0.830938	0.415469	+	0.909607i	$0.363617\pi$
$110$	−10.5201	−1.00305
$111$	−5.29308	−0.502397
$112$	−1.00000	−0.0944911
$113$	−11.2111	−1.05466	−0.527328	−	0.849662i	$-0.676806\pi$
$113$	−11.2111	−1.05466	−0.527328	+	0.849662i	$0.676806\pi$
$114$	1.72542	0.161600
$115$	−9.35918	−0.872748
$116$	3.85061	0.357520
$117$	0	0
$118$	9.97753	0.918506
$119$	2.82898	0.259332
$120$	−3.05596	−0.278970
$121$	0.850611	0.0773282
$122$	11.5452	1.04525
$123$	9.96254	0.898292
$124$	−0.978370	−0.0878602
$125$	2.02028	0.180699
$126$	1.00000	0.0890871
$127$	5.11506	0.453888	0.226944	−	0.973908i	$-0.427127\pi$
$127$	5.11506	0.453888	0.226944	+	0.973908i	$0.427127\pi$
$128$	−1.00000	−0.0883883
$129$	3.02163	0.266040
$130$	0	0
$131$	−18.7757	−1.64044	−0.820219	−	0.572049i	$-0.806149\pi$
$131$	−18.7757	−1.64044	−0.820219	+	0.572049i	$0.806149\pi$
$132$	3.44247	0.299629
$133$	−1.72542	−0.149613
$134$	4.75055	0.410385
$135$	3.05596	0.263015
$136$	2.82898	0.242583
$137$	−2.66109	−0.227353	−0.113676	−	0.993518i	$-0.536263\pi$
$137$	−2.66109	−0.227353	−0.113676	+	0.993518i	$0.536263\pi$
$138$	3.06260	0.260706
$139$	−16.7064	−1.41702	−0.708511	−	0.705699i	$-0.750633\pi$
$139$	−16.7064	−1.41702	−0.708511	+	0.705699i	$0.750633\pi$
$140$	3.05596	0.258276
$141$	8.04447	0.677467
$142$	3.66945	0.307934
$143$	0	0
$144$	1.00000	0.0833333
$145$	−11.7673	−0.977223
$146$	10.1119	0.836869
$147$	−1.00000	−0.0824786
$148$	5.29308	0.435089
$149$	2.50157	0.204936	0.102468	−	0.994736i	$-0.467326\pi$
$149$	2.50157	0.204936	0.102468	+	0.994736i	$0.467326\pi$
$150$	4.33891	0.354270
$151$	12.6465	1.02916	0.514578	−	0.857444i	$-0.327949\pi$
$151$	12.6465	1.02916	0.514578	+	0.857444i	$0.327949\pi$
$152$	−1.72542	−0.139950
$153$	−2.82898	−0.228710
$154$	−3.44247	−0.277402
$155$	2.98986	0.240151
$156$	0	0
$157$	17.8131	1.42164	0.710822	−	0.703372i	$-0.248323\pi$
$157$	17.8131	1.42164	0.710822	+	0.703372i	$0.248323\pi$
$158$	4.49843	0.357876
$159$	8.33405	0.660933
$160$	3.05596	0.241595
$161$	−3.06260	−0.241366
$162$	−1.00000	−0.0785674
$163$	8.87831	0.695403	0.347701	−	0.937605i	$-0.386962\pi$
$163$	8.87831	0.695403	0.347701	+	0.937605i	$0.386962\pi$
$164$	−9.96254	−0.777943
$165$	−10.5201	−0.818986
$166$	−7.15869	−0.555622
$167$	−9.76989	−0.756017	−0.378008	−	0.925802i	$-0.623391\pi$
$167$	−9.76989	−0.756017	−0.378008	+	0.925802i	$0.623391\pi$
$168$	−1.00000	−0.0771517
$169$	0	0
$170$	−8.64526	−0.663061
$171$	1.72542	0.131946
$172$	−3.02163	−0.230397
$173$	8.67525	0.659567	0.329783	−	0.944057i	$-0.393024\pi$
$173$	8.67525	0.659567	0.329783	+	0.944057i	$0.393024\pi$
$174$	3.85061	0.291914
$175$	−4.33891	−0.327991
$176$	−3.44247	−0.259486
$177$	9.97753	0.749957
$178$	7.90307	0.592360
$179$	−23.4278	−1.75108	−0.875540	−	0.483146i	$-0.839494\pi$
$179$	−23.4278	−1.75108	−0.875540	+	0.483146i	$0.839494\pi$
$180$	−3.05596	−0.227778
$181$	−7.66632	−0.569833	−0.284917	−	0.958552i	$-0.591966\pi$
$181$	−7.66632	−0.569833	−0.284917	+	0.958552i	$0.591966\pi$
$182$	0	0
$183$	11.5452	0.853446
$184$	−3.06260	−0.225778
$185$	−16.1755	−1.18924
$186$	−0.978370	−0.0717376
$187$	9.73869	0.712163
$188$	−8.04447	−0.586703
$189$	1.00000	0.0727393
$190$	5.27281	0.382530
$191$	25.4945	1.84471	0.922357	−	0.386338i	$-0.126260\pi$
$191$	25.4945	1.84471	0.922357	+	0.386338i	$0.126260\pi$
$192$	−1.00000	−0.0721688
$193$	−8.52586	−0.613705	−0.306852	−	0.951757i	$-0.599276\pi$
$193$	−8.52586	−0.613705	−0.306852	+	0.951757i	$0.599276\pi$
$194$	9.09922	0.653286
$195$	0	0
$196$	1.00000	0.0714286
$197$	−23.6823	−1.68729	−0.843645	−	0.536901i	$-0.819595\pi$
$197$	−23.6823	−1.68729	−0.843645	+	0.536901i	$0.819595\pi$
$198$	3.44247	0.244646
$199$	−24.8466	−1.76133	−0.880666	−	0.473738i	$-0.842904\pi$
$199$	−24.8466	−1.76133	−0.880666	+	0.473738i	$0.842904\pi$
$200$	−4.33891	−0.306807
$201$	4.75055	0.335078
$202$	−18.8533	−1.32651
$203$	−3.85061	−0.270260
$204$	2.82898	0.198068
$205$	30.4451	2.12638
$206$	−13.7079	−0.955072
$207$	3.06260	0.212865
$208$	0	0
$209$	−5.93970	−0.410857
$210$	3.05596	0.210881
$211$	0.773018	0.0532168	0.0266084	−	0.999646i	$-0.491529\pi$
$211$	0.773018	0.0532168	0.0266084	+	0.999646i	$0.491529\pi$
$212$	−8.33405	−0.572385
$213$	3.66945	0.251427
$214$	3.31471	0.226589
$215$	9.23399	0.629753
$216$	1.00000	0.0680414
$217$	0.978370	0.0664161
$218$	−8.67525	−0.587562
$219$	10.1119	0.683301
$220$	10.5201	0.709263
$221$	0	0
$222$	5.29308	0.355248
$223$	21.5354	1.44212	0.721060	−	0.692873i	$-0.243655\pi$
$223$	21.5354	1.44212	0.721060	+	0.692873i	$0.243655\pi$
$224$	1.00000	0.0668153
$225$	4.33891	0.289260
$226$	11.2111	0.745754
$227$	17.5444	1.16446	0.582230	−	0.813024i	$-0.302180\pi$
$227$	17.5444	1.16446	0.582230	+	0.813024i	$0.302180\pi$
$228$	−1.72542	−0.114268
$229$	−10.3222	−0.682109	−0.341055	−	0.940043i	$-0.610784\pi$
$229$	−10.3222	−0.682109	−0.341055	+	0.940043i	$0.610784\pi$
$230$	9.35918	0.617126
$231$	−3.44247	−0.226498
$232$	−3.85061	−0.252805
$233$	17.8290	1.16802	0.584008	−	0.811748i	$-0.301484\pi$
$233$	17.8290	1.16802	0.584008	+	0.811748i	$0.301484\pi$
$234$	0	0
$235$	24.5836	1.60366
$236$	−9.97753	−0.649482
$237$	4.49843	0.292205
$238$	−2.82898	−0.183376
$239$	5.71271	0.369525	0.184762	−	0.982783i	$-0.440848\pi$
$239$	5.71271	0.369525	0.184762	+	0.982783i	$0.440848\pi$
$240$	3.05596	0.197262
$241$	−1.00313	−0.0646174	−0.0323087	−	0.999478i	$-0.510286\pi$
$241$	−1.00313	−0.0646174	−0.0323087	+	0.999478i	$0.510286\pi$
$242$	−0.850611	−0.0546793
$243$	−1.00000	−0.0641500
$244$	−11.5452	−0.739106
$245$	−3.05596	−0.195238
$246$	−9.96254	−0.635188
$247$	0	0
$248$	0.978370	0.0621266
$249$	−7.15869	−0.453663
$250$	−2.02028	−0.127773
$251$	−0.672585	−0.0424532	−0.0212266	−	0.999775i	$-0.506757\pi$
$251$	−0.672585	−0.0424532	−0.0212266	+	0.999775i	$0.506757\pi$
$252$	−1.00000	−0.0629941
$253$	−10.5429	−0.662827
$254$	−5.11506	−0.320947
$255$	−8.64526	−0.541387
$256$	1.00000	0.0625000
$257$	24.3433	1.51850	0.759248	−	0.650801i	$-0.225567\pi$
$257$	24.3433	1.51850	0.759248	+	0.650801i	$0.225567\pi$
$258$	−3.02163	−0.188118
$259$	−5.29308	−0.328896
$260$	0	0
$261$	3.85061	0.238347
$262$	18.7757	1.15997
$263$	−17.0930	−1.05400	−0.526999	−	0.849866i	$-0.676683\pi$
$263$	−17.0930	−1.05400	−0.526999	+	0.849866i	$0.676683\pi$
$264$	−3.44247	−0.211869
$265$	25.4685	1.56452
$266$	1.72542	0.105792
$267$	7.90307	0.483660
$268$	−4.75055	−0.290186
$269$	−26.6594	−1.62545	−0.812727	−	0.582645i	$-0.802018\pi$
$269$	−26.6594	−1.62545	−0.812727	+	0.582645i	$0.802018\pi$
$270$	−3.05596	−0.185980
$271$	24.4495	1.48520	0.742600	−	0.669735i	$-0.233592\pi$
$271$	24.4495	1.48520	0.742600	+	0.669735i	$0.233592\pi$
$272$	−2.82898	−0.171532
$273$	0	0
$274$	2.66109	0.160763
$275$	−14.9366	−0.900709
$276$	−3.06260	−0.184347
$277$	−28.4812	−1.71127	−0.855636	−	0.517578i	$-0.826834\pi$
$277$	−28.4812	−1.71127	−0.855636	+	0.517578i	$0.826834\pi$
$278$	16.7064	1.00199
$279$	−0.978370	−0.0585735
$280$	−3.05596	−0.182629
$281$	−9.76032	−0.582252	−0.291126	−	0.956685i	$-0.594030\pi$
$281$	−9.76032	−0.582252	−0.291126	+	0.956685i	$0.594030\pi$
$282$	−8.04447	−0.479041
$283$	11.6712	0.693783	0.346891	−	0.937905i	$-0.387237\pi$
$283$	11.6712	0.693783	0.346891	+	0.937905i	$0.387237\pi$
$284$	−3.66945	−0.217742
$285$	5.27281	0.312334
$286$	0	0
$287$	9.96254	0.588070
$288$	−1.00000	−0.0589256
$289$	−8.99687	−0.529228
$290$	11.7673	0.691001
$291$	9.09922	0.533406
$292$	−10.1119	−0.591756
$293$	−21.8202	−1.27475	−0.637373	−	0.770555i	$-0.719979\pi$
$293$	−21.8202	−1.27475	−0.637373	+	0.770555i	$0.719979\pi$
$294$	1.00000	0.0583212
$295$	30.4910	1.77525
$296$	−5.29308	−0.307654
$297$	3.44247	0.199752
$298$	−2.50157	−0.144912
$299$	0	0
$300$	−4.33891	−0.250507
$301$	3.02163	0.174164
$302$	−12.6465	−0.727723
$303$	−18.8533	−1.08309
$304$	1.72542	0.0989594
$305$	35.2817	2.02022
$306$	2.82898	0.161722
$307$	4.87046	0.277972	0.138986	−	0.990294i	$-0.455616\pi$
$307$	4.87046	0.277972	0.138986	+	0.990294i	$0.455616\pi$
$308$	3.44247	0.196153
$309$	−13.7079	−0.779813
$310$	−2.98986	−0.169813
$311$	3.90344	0.221344	0.110672	−	0.993857i	$-0.464700\pi$
$311$	3.90344	0.221344	0.110672	+	0.993857i	$0.464700\pi$
$312$	0	0
$313$	26.0528	1.47259	0.736297	−	0.676659i	$-0.236573\pi$
$313$	26.0528	1.47259	0.736297	+	0.676659i	$0.236573\pi$
$314$	−17.8131	−1.00525
$315$	3.05596	0.172184
$316$	−4.49843	−0.253057
$317$	11.1107	0.624040	0.312020	−	0.950076i	$-0.398994\pi$
$317$	11.1107	0.624040	0.312020	+	0.950076i	$0.398994\pi$
$318$	−8.33405	−0.467350
$319$	−13.2556	−0.742172
$320$	−3.05596	−0.170833
$321$	3.31471	0.185009
$322$	3.06260	0.170672
$323$	−4.88117	−0.271595
$324$	1.00000	0.0555556
$325$	0	0
$326$	−8.87831	−0.491724
$327$	−8.67525	−0.479742
$328$	9.96254	0.550089
$329$	8.04447	0.443506
$330$	10.5201	0.579110
$331$	14.6478	0.805117	0.402559	−	0.915394i	$-0.368121\pi$
$331$	14.6478	0.805117	0.402559	+	0.915394i	$0.368121\pi$
$332$	7.15869	0.392884
$333$	5.29308	0.290059
$334$	9.76989	0.534584
$335$	14.5175	0.793176
$336$	1.00000	0.0545545
$337$	27.5456	1.50050	0.750251	−	0.661153i	$-0.229932\pi$
$337$	27.5456	1.50050	0.750251	+	0.661153i	$0.229932\pi$
$338$	0	0
$339$	11.2111	0.608906
$340$	8.64526	0.468855
$341$	3.36801	0.182388
$342$	−1.72542	−0.0932998
$343$	−1.00000	−0.0539949
$344$	3.02163	0.162915
$345$	9.35918	0.503881
$346$	−8.67525	−0.466384
$347$	5.24146	0.281376	0.140688	−	0.990054i	$-0.455068\pi$
$347$	5.24146	0.281376	0.140688	+	0.990054i	$0.455068\pi$
$348$	−3.85061	−0.206414
$349$	−5.22965	−0.279937	−0.139968	−	0.990156i	$-0.544700\pi$
$349$	−5.22965	−0.279937	−0.139968	+	0.990156i	$0.544700\pi$
$350$	4.33891	0.231924
$351$	0	0
$352$	3.44247	0.183484
$353$	−3.76695	−0.200495	−0.100247	−	0.994963i	$-0.531963\pi$
$353$	−3.76695	−0.200495	−0.100247	+	0.994963i	$0.531963\pi$
$354$	−9.97753	−0.530300
$355$	11.2137	0.595162
$356$	−7.90307	−0.418862
$357$	−2.82898	−0.149726
$358$	23.4278	1.23820
$359$	−20.6003	−1.08724	−0.543622	−	0.839330i	$-0.682947\pi$
$359$	−20.6003	−1.08724	−0.543622	+	0.839330i	$0.682947\pi$
$360$	3.05596	0.161063
$361$	−16.0229	−0.843313
$362$	7.66632	0.402933
$363$	−0.850611	−0.0446455
$364$	0	0
$365$	30.9017	1.61747
$366$	−11.5452	−0.603477
$367$	−27.9159	−1.45720	−0.728598	−	0.684941i	$-0.759828\pi$
$367$	−27.9159	−1.45720	−0.728598	+	0.684941i	$0.759828\pi$
$368$	3.06260	0.159649
$369$	−9.96254	−0.518629
$370$	16.1755	0.840922
$371$	8.33405	0.432682
$372$	0.978370	0.0507261
$373$	31.4183	1.62678	0.813388	−	0.581721i	$-0.197620\pi$
$373$	31.4183	1.62678	0.813388	+	0.581721i	$0.197620\pi$
$374$	−9.73869	−0.503575
$375$	−2.02028	−0.104327
$376$	8.04447	0.414862
$377$	0	0
$378$	−1.00000	−0.0514344
$379$	31.8406	1.63554	0.817770	−	0.575545i	$-0.195210\pi$
$379$	31.8406	1.63554	0.817770	+	0.575545i	$0.195210\pi$
$380$	−5.27281	−0.270489
$381$	−5.11506	−0.262052
$382$	−25.4945	−1.30441
$383$	−4.72228	−0.241297	−0.120649	−	0.992695i	$-0.538497\pi$
$383$	−4.72228	−0.241297	−0.120649	+	0.992695i	$0.538497\pi$
$384$	1.00000	0.0510310
$385$	−10.5201	−0.536152
$386$	8.52586	0.433955
$387$	−3.02163	−0.153598
$388$	−9.09922	−0.461943
$389$	30.5902	1.55099	0.775493	−	0.631356i	$-0.217501\pi$
$389$	30.5902	1.55099	0.775493	+	0.631356i	$0.217501\pi$
$390$	0	0
$391$	−8.66403	−0.438159
$392$	−1.00000	−0.0505076
$393$	18.7757	0.947108
$394$	23.6823	1.19309
$395$	13.7470	0.691689
$396$	−3.44247	−0.172991
$397$	−11.6778	−0.586093	−0.293046	−	0.956098i	$-0.594669\pi$
$397$	−11.6778	−0.586093	−0.293046	+	0.956098i	$0.594669\pi$
$398$	24.8466	1.24545
$399$	1.72542	0.0863788
$400$	4.33891	0.216945
$401$	9.25697	0.462271	0.231136	−	0.972922i	$-0.425756\pi$
$401$	9.25697	0.462271	0.231136	+	0.972922i	$0.425756\pi$
$402$	−4.75055	−0.236936
$403$	0	0
$404$	18.8533	0.937985
$405$	−3.05596	−0.151852
$406$	3.85061	0.191103
$407$	−18.2213	−0.903196
$408$	−2.82898	−0.140055
$409$	−25.5989	−1.26578	−0.632891	−	0.774241i	$-0.718132\pi$
$409$	−25.5989	−1.26578	−0.632891	+	0.774241i	$0.718132\pi$
$410$	−30.4451	−1.50358
$411$	2.66109	0.131262
$412$	13.7079	0.675338
$413$	9.97753	0.490962
$414$	−3.06260	−0.150518
$415$	−21.8767	−1.07388
$416$	0	0
$417$	16.7064	0.818118
$418$	5.93970	0.290520
$419$	28.2037	1.37784	0.688920	−	0.724838i	$-0.258085\pi$
$419$	28.2037	1.37784	0.688920	+	0.724838i	$0.258085\pi$
$420$	−3.05596	−0.149116
$421$	5.07012	0.247102	0.123551	−	0.992338i	$-0.460572\pi$
$421$	5.07012	0.247102	0.123551	+	0.992338i	$0.460572\pi$
$422$	−0.773018	−0.0376299
$423$	−8.04447	−0.391136
$424$	8.33405	0.404737
$425$	−12.2747	−0.595410
$426$	−3.66945	−0.177786
$427$	11.5452	0.558711
$428$	−3.31471	−0.160223
$429$	0	0
$430$	−9.23399	−0.445302
$431$	16.1437	0.777614	0.388807	−	0.921319i	$-0.372887\pi$
$431$	16.1437	0.777614	0.388807	+	0.921319i	$0.372887\pi$
$432$	−1.00000	−0.0481125
$433$	−22.6029	−1.08623	−0.543113	−	0.839660i	$-0.682754\pi$
$433$	−22.6029	−1.08623	−0.543113	+	0.839660i	$0.682754\pi$
$434$	−0.978370	−0.0469633
$435$	11.7673	0.564200
$436$	8.67525	0.415469
$437$	5.28425	0.252780
$438$	−10.1119	−0.483166
$439$	28.2971	1.35055	0.675273	−	0.737567i	$-0.264026\pi$
$439$	28.2971	1.35055	0.675273	+	0.737567i	$0.264026\pi$
$440$	−10.5201	−0.501524
$441$	1.00000	0.0476190
$442$	0	0
$443$	−1.38096	−0.0656112	−0.0328056	−	0.999462i	$-0.510444\pi$
$443$	−1.38096	−0.0656112	−0.0328056	+	0.999462i	$0.510444\pi$
$444$	−5.29308	−0.251199
$445$	24.1515	1.14489
$446$	−21.5354	−1.01973
$447$	−2.50157	−0.118320
$448$	−1.00000	−0.0472456
$449$	37.0059	1.74642	0.873208	−	0.487348i	$-0.162036\pi$
$449$	37.0059	1.74642	0.873208	+	0.487348i	$0.162036\pi$
$450$	−4.33891	−0.204538
$451$	34.2957	1.61492
$452$	−11.2111	−0.527328
$453$	−12.6465	−0.594183
$454$	−17.5444	−0.823398
$455$	0	0
$456$	1.72542	0.0808000
$457$	−13.9176	−0.651036	−0.325518	−	0.945536i	$-0.605539\pi$
$457$	−13.9176	−0.651036	−0.325518	+	0.945536i	$0.605539\pi$
$458$	10.3222	0.482324
$459$	2.82898	0.132046
$460$	−9.35918	−0.436374
$461$	4.84795	0.225791	0.112896	−	0.993607i	$-0.463987\pi$
$461$	4.84795	0.225791	0.112896	+	0.993607i	$0.463987\pi$
$462$	3.44247	0.160158
$463$	−24.4377	−1.13571	−0.567857	−	0.823127i	$-0.692227\pi$
$463$	−24.4377	−1.13571	−0.567857	+	0.823127i	$0.692227\pi$
$464$	3.85061	0.178760
$465$	−2.98986	−0.138652
$466$	−17.8290	−0.825912
$467$	36.0027	1.66600	0.833002	−	0.553270i	$-0.186620\pi$
$467$	36.0027	1.66600	0.833002	+	0.553270i	$0.186620\pi$
$468$	0	0
$469$	4.75055	0.219360
$470$	−24.5836	−1.13396
$471$	−17.8131	−0.820786
$472$	9.97753	0.459253
$473$	10.4019	0.478279
$474$	−4.49843	−0.206620
$475$	7.48642	0.343500
$476$	2.82898	0.129666
$477$	−8.33405	−0.381590
$478$	−5.71271	−0.261293
$479$	−9.49288	−0.433741	−0.216870	−	0.976200i	$-0.569585\pi$
$479$	−9.49288	−0.433741	−0.216870	+	0.976200i	$0.569585\pi$
$480$	−3.05596	−0.139485
$481$	0	0
$482$	1.00313	0.0456914
$483$	3.06260	0.139353
$484$	0.850611	0.0386641
$485$	27.8069	1.26265
$486$	1.00000	0.0453609
$487$	−40.9206	−1.85429	−0.927144	−	0.374704i	$-0.877744\pi$
$487$	−40.9206	−1.85429	−0.927144	+	0.374704i	$0.877744\pi$
$488$	11.5452	0.522627
$489$	−8.87831	−0.401491
$490$	3.05596	0.138054
$491$	−5.11506	−0.230839	−0.115420	−	0.993317i	$-0.536821\pi$
$491$	−5.11506	−0.230839	−0.115420	+	0.993317i	$0.536821\pi$
$492$	9.96254	0.449146
$493$	−10.8933	−0.490610
$494$	0	0
$495$	10.5201	0.472842
$496$	−0.978370	−0.0439301
$497$	3.66945	0.164597
$498$	7.15869	0.320788
$499$	19.5827	0.876640	0.438320	−	0.898819i	$-0.355574\pi$
$499$	19.5827	0.876640	0.438320	+	0.898819i	$0.355574\pi$
$500$	2.02028	0.0903495
$501$	9.76989	0.436486
$502$	0.672585	0.0300189
$503$	−29.0858	−1.29687	−0.648436	−	0.761269i	$-0.724577\pi$
$503$	−29.0858	−1.29687	−0.648436	+	0.761269i	$0.724577\pi$
$504$	1.00000	0.0445435
$505$	−57.6149	−2.56383
$506$	10.5429	0.468689
$507$	0	0
$508$	5.11506	0.226944
$509$	5.21983	0.231365	0.115682	−	0.993286i	$-0.463094\pi$
$509$	5.21983	0.231365	0.115682	+	0.993286i	$0.463094\pi$
$510$	8.64526	0.382818
$511$	10.1119	0.447325
$512$	−1.00000	−0.0441942
$513$	−1.72542	−0.0761790
$514$	−24.3433	−1.07374
$515$	−41.8907	−1.84592
$516$	3.02163	0.133020
$517$	27.6929	1.21793
$518$	5.29308	0.232565
$519$	−8.67525	−0.380801
$520$	0	0
$521$	−14.5259	−0.636389	−0.318195	−	0.948025i	$-0.603077\pi$
$521$	−14.5259	−0.636389	−0.318195	+	0.948025i	$0.603077\pi$
$522$	−3.85061	−0.168537
$523$	27.8735	1.21882	0.609410	−	0.792855i	$-0.291406\pi$
$523$	27.8735	1.21882	0.609410	+	0.792855i	$0.291406\pi$
$524$	−18.7757	−0.820219
$525$	4.33891	0.189365
$526$	17.0930	0.745288
$527$	2.76779	0.120567
$528$	3.44247	0.149814
$529$	−13.6205	−0.592196
$530$	−25.4685	−1.10628
$531$	−9.97753	−0.432988
$532$	−1.72542	−0.0748063
$533$	0	0
$534$	−7.90307	−0.341999
$535$	10.1296	0.437942
$536$	4.75055	0.205192
$537$	23.4278	1.01099
$538$	26.6594	1.14937
$539$	−3.44247	−0.148278
$540$	3.05596	0.131508
$541$	1.42341	0.0611970	0.0305985	−	0.999532i	$-0.490259\pi$
$541$	1.42341	0.0611970	0.0305985	+	0.999532i	$0.490259\pi$
$542$	−24.4495	−1.05019
$543$	7.66632	0.328993
$544$	2.82898	0.121292
$545$	−26.5112	−1.13562
$546$	0	0
$547$	−9.33008	−0.398925	−0.199463	−	0.979905i	$-0.563920\pi$
$547$	−9.33008	−0.398925	−0.199463	+	0.979905i	$0.563920\pi$
$548$	−2.66109	−0.113676
$549$	−11.5452	−0.492737
$550$	14.9366	0.636897
$551$	6.64390	0.283040
$552$	3.06260	0.130353
$553$	4.49843	0.191293
$554$	28.4812	1.21005
$555$	16.1755	0.686610
$556$	−16.7064	−0.708511
$557$	10.7857	0.457006	0.228503	−	0.973543i	$-0.426617\pi$
$557$	10.7857	0.457006	0.228503	+	0.973543i	$0.426617\pi$
$558$	0.978370	0.0414177
$559$	0	0
$560$	3.05596	0.129138
$561$	−9.73869	−0.411168
$562$	9.76032	0.411714
$563$	25.5082	1.07504	0.537522	−	0.843250i	$-0.319360\pi$
$563$	25.5082	1.07504	0.537522	+	0.843250i	$0.319360\pi$
$564$	8.04447	0.338733
$565$	34.2608	1.44136
$566$	−11.6712	−0.490578
$567$	−1.00000	−0.0419961
$568$	3.66945	0.153967
$569$	−8.21670	−0.344462	−0.172231	−	0.985057i	$-0.555098\pi$
$569$	−8.21670	−0.344462	−0.172231	+	0.985057i	$0.555098\pi$
$570$	−5.27281	−0.220854
$571$	−2.70500	−0.113201	−0.0566003	−	0.998397i	$-0.518026\pi$
$571$	−2.70500	−0.113201	−0.0566003	+	0.998397i	$0.518026\pi$
$572$	0	0
$573$	−25.4945	−1.06505
$574$	−9.96254	−0.415828
$575$	13.2883	0.554161
$576$	1.00000	0.0416667
$577$	21.4253	0.891946	0.445973	−	0.895046i	$-0.352858\pi$
$577$	21.4253	0.891946	0.445973	+	0.895046i	$0.352858\pi$
$578$	8.99687	0.374220
$579$	8.52586	0.354323
$580$	−11.7673	−0.488611
$581$	−7.15869	−0.296992
$582$	−9.09922	−0.377175
$583$	28.6897	1.18821
$584$	10.1119	0.418434
$585$	0	0
$586$	21.8202	0.901382
$587$	5.73326	0.236637	0.118319	−	0.992976i	$-0.462250\pi$
$587$	5.73326	0.236637	0.118319	+	0.992976i	$0.462250\pi$
$588$	−1.00000	−0.0412393
$589$	−1.68810	−0.0695567
$590$	−30.4910	−1.25529
$591$	23.6823	0.974158
$592$	5.29308	0.217544
$593$	17.3442	0.712240	0.356120	−	0.934440i	$-0.384099\pi$
$593$	17.3442	0.712240	0.356120	+	0.934440i	$0.384099\pi$
$594$	−3.44247	−0.141246
$595$	−8.64526	−0.354421
$596$	2.50157	0.102468
$597$	24.8466	1.01691
$598$	0	0
$599$	−31.5835	−1.29047	−0.645234	−	0.763985i	$-0.723240\pi$
$599$	−31.5835	−1.29047	−0.645234	+	0.763985i	$0.723240\pi$
$600$	4.33891	0.177135
$601$	24.8956	1.01551	0.507757	−	0.861500i	$-0.330475\pi$
$601$	24.8956	1.01551	0.507757	+	0.861500i	$0.330475\pi$
$602$	−3.02163	−0.123152
$603$	−4.75055	−0.193457
$604$	12.6465	0.514578
$605$	−2.59943	−0.105682
$606$	18.8533	0.765862
$607$	15.2590	0.619345	0.309672	−	0.950843i	$-0.399781\pi$
$607$	15.2590	0.619345	0.309672	+	0.950843i	$0.399781\pi$
$608$	−1.72542	−0.0699749
$609$	3.85061	0.156035
$610$	−35.2817	−1.42851
$611$	0	0
$612$	−2.82898	−0.114355
$613$	45.7515	1.84789	0.923943	−	0.382531i	$-0.124948\pi$
$613$	45.7515	1.84789	0.923943	+	0.382531i	$0.124948\pi$
$614$	−4.87046	−0.196556
$615$	−30.4451	−1.22767
$616$	−3.44247	−0.138701
$617$	42.2813	1.70218	0.851090	−	0.525020i	$-0.175942\pi$
$617$	42.2813	1.70218	0.851090	+	0.525020i	$0.175942\pi$
$618$	13.7079	0.551411
$619$	−22.0261	−0.885303	−0.442651	−	0.896694i	$-0.645962\pi$
$619$	−22.0261	−0.885303	−0.442651	+	0.896694i	$0.645962\pi$
$620$	2.98986	0.120076
$621$	−3.06260	−0.122898
$622$	−3.90344	−0.156514
$623$	7.90307	0.316630
$624$	0	0
$625$	−27.8684	−1.11474
$626$	−26.0528	−1.04128
$627$	5.93970	0.237209
$628$	17.8131	0.710822
$629$	−14.9740	−0.597054
$630$	−3.05596	−0.121752
$631$	−20.8679	−0.830738	−0.415369	−	0.909653i	$-0.636348\pi$
$631$	−20.8679	−0.830738	−0.415369	+	0.909653i	$0.636348\pi$
$632$	4.49843	0.178938
$633$	−0.773018	−0.0307247
$634$	−11.1107	−0.441263
$635$	−15.6314	−0.620314
$636$	8.33405	0.330467
$637$	0	0
$638$	13.2556	0.524795
$639$	−3.66945	−0.145161
$640$	3.05596	0.120798
$641$	−20.9052	−0.825707	−0.412853	−	0.910798i	$-0.635468\pi$
$641$	−20.9052	−0.825707	−0.412853	+	0.910798i	$0.635468\pi$
$642$	−3.31471	−0.130821
$643$	39.1100	1.54235	0.771174	−	0.636624i	$-0.219670\pi$
$643$	39.1100	1.54235	0.771174	+	0.636624i	$0.219670\pi$
$644$	−3.06260	−0.120683
$645$	−9.23399	−0.363588
$646$	4.88117	0.192047
$647$	−7.93511	−0.311961	−0.155981	−	0.987760i	$-0.549854\pi$
$647$	−7.93511	−0.311961	−0.155981	+	0.987760i	$0.549854\pi$
$648$	−1.00000	−0.0392837
$649$	34.3474	1.34825
$650$	0	0
$651$	−0.978370	−0.0383453
$652$	8.87831	0.347701
$653$	42.7996	1.67488	0.837439	−	0.546531i	$-0.184052\pi$
$653$	42.7996	1.67488	0.837439	+	0.546531i	$0.184052\pi$
$654$	8.67525	0.339229
$655$	57.3778	2.24194
$656$	−9.96254	−0.388972
$657$	−10.1119	−0.394504
$658$	−8.04447	−0.313606
$659$	0.728548	0.0283802	0.0141901	−	0.999899i	$-0.495483\pi$
$659$	0.728548	0.0283802	0.0141901	+	0.999899i	$0.495483\pi$
$660$	−10.5201	−0.409493
$661$	−2.35965	−0.0917798	−0.0458899	−	0.998947i	$-0.514612\pi$
$661$	−2.35965	−0.0917798	−0.0458899	+	0.998947i	$0.514612\pi$
$662$	−14.6478	−0.569304
$663$	0	0
$664$	−7.15869	−0.277811
$665$	5.27281	0.204471
$666$	−5.29308	−0.205103
$667$	11.7929	0.456622
$668$	−9.76989	−0.378008
$669$	−21.5354	−0.832608
$670$	−14.5175	−0.560860
$671$	39.7440	1.53430
$672$	−1.00000	−0.0385758
$673$	31.2557	1.20482	0.602410	−	0.798187i	$-0.294207\pi$
$673$	31.2557	1.20482	0.602410	+	0.798187i	$0.294207\pi$
$674$	−27.5456	−1.06102
$675$	−4.33891	−0.167005
$676$	0	0
$677$	−41.7294	−1.60379	−0.801895	−	0.597464i	$-0.796175\pi$
$677$	−41.7294	−1.60379	−0.801895	+	0.597464i	$0.796175\pi$
$678$	−11.2111	−0.430562
$679$	9.09922	0.349196
$680$	−8.64526	−0.331531
$681$	−17.5444	−0.672301
$682$	−3.36801	−0.128968
$683$	9.21998	0.352793	0.176396	−	0.984319i	$-0.443556\pi$
$683$	9.21998	0.352793	0.176396	+	0.984319i	$0.443556\pi$
$684$	1.72542	0.0659729
$685$	8.13220	0.310715
$686$	1.00000	0.0381802
$687$	10.3222	0.393816
$688$	−3.02163	−0.115199
$689$	0	0
$690$	−9.35918	−0.356298
$691$	9.20858	0.350311	0.175156	−	0.984541i	$-0.443957\pi$
$691$	9.20858	0.350311	0.175156	+	0.984541i	$0.443957\pi$
$692$	8.67525	0.329783
$693$	3.44247	0.130769
$694$	−5.24146	−0.198963
$695$	51.0543	1.93660
$696$	3.85061	0.145957
$697$	28.1838	1.06754
$698$	5.22965	0.197945
$699$	−17.8290	−0.674354
$700$	−4.33891	−0.163995
$701$	−20.8683	−0.788184	−0.394092	−	0.919071i	$-0.628941\pi$
$701$	−20.8683	−0.788184	−0.394092	+	0.919071i	$0.628941\pi$
$702$	0	0
$703$	9.13277	0.344449
$704$	−3.44247	−0.129743
$705$	−24.5836	−0.925872
$706$	3.76695	0.141771
$707$	−18.8533	−0.709050
$708$	9.97753	0.374979
$709$	14.8645	0.558249	0.279124	−	0.960255i	$-0.409956\pi$
$709$	14.8645	0.558249	0.279124	+	0.960255i	$0.409956\pi$
$710$	−11.2137	−0.420843
$711$	−4.49843	−0.168704
$712$	7.90307	0.296180
$713$	−2.99635	−0.112214
$714$	2.82898	0.105872
$715$	0	0
$716$	−23.4278	−0.875540
$717$	−5.71271	−0.213345
$718$	20.6003	0.768797
$719$	−25.6531	−0.956697	−0.478349	−	0.878170i	$-0.658764\pi$
$719$	−25.6531	−0.956697	−0.478349	+	0.878170i	$0.658764\pi$
$720$	−3.05596	−0.113889
$721$	−13.7079	−0.510507
$722$	16.0229	0.596312
$723$	1.00313	0.0373069
$724$	−7.66632	−0.284917
$725$	16.7074	0.620499
$726$	0.850611	0.0315691
$727$	−41.6568	−1.54497	−0.772483	−	0.635036i	$-0.780985\pi$
$727$	−41.6568	−1.54497	−0.772483	+	0.635036i	$0.780985\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−30.9017	−1.14372
$731$	8.54813	0.316164
$732$	11.5452	0.426723
$733$	−26.4965	−0.978671	−0.489336	−	0.872096i	$-0.662761\pi$
$733$	−26.4965	−0.978671	−0.489336	+	0.872096i	$0.662761\pi$
$734$	27.9159	1.03039
$735$	3.05596	0.112721
$736$	−3.06260	−0.112889
$737$	16.3536	0.602394
$738$	9.96254	0.366726
$739$	20.9011	0.768861	0.384431	−	0.923154i	$-0.374398\pi$
$739$	20.9011	0.768861	0.384431	+	0.923154i	$0.374398\pi$
$740$	−16.1755	−0.594622
$741$	0	0
$742$	−8.33405	−0.305953
$743$	−5.64647	−0.207149	−0.103574	−	0.994622i	$-0.533028\pi$
$743$	−5.64647	−0.207149	−0.103574	+	0.994622i	$0.533028\pi$
$744$	−0.978370	−0.0358688
$745$	−7.64469	−0.280080
$746$	−31.4183	−1.15030
$747$	7.15869	0.261923
$748$	9.73869	0.356082
$749$	3.31471	0.121117
$750$	2.02028	0.0737701
$751$	15.4088	0.562274	0.281137	−	0.959668i	$-0.409288\pi$
$751$	15.4088	0.562274	0.281137	+	0.959668i	$0.409288\pi$
$752$	−8.04447	−0.293352
$753$	0.672585	0.0245104
$754$	0	0
$755$	−38.6471	−1.40651
$756$	1.00000	0.0363696
$757$	−35.5372	−1.29162	−0.645811	−	0.763497i	$-0.723481\pi$
$757$	−35.5372	−1.29162	−0.645811	+	0.763497i	$0.723481\pi$
$758$	−31.8406	−1.15650
$759$	10.5429	0.382683
$760$	5.27281	0.191265
$761$	−41.8247	−1.51615	−0.758073	−	0.652170i	$-0.773859\pi$
$761$	−41.8247	−1.51615	−0.758073	+	0.652170i	$0.773859\pi$
$762$	5.11506	0.185299
$763$	−8.67525	−0.314065
$764$	25.4945	0.922357
$765$	8.64526	0.312570
$766$	4.72228	0.170623
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−17.8527	−0.643785	−0.321893	−	0.946776i	$-0.604319\pi$
$769$	−17.8527	−0.643785	−0.321893	+	0.946776i	$0.604319\pi$
$770$	10.5201	0.379117
$771$	−24.3433	−0.876704
$772$	−8.52586	−0.306852
$773$	12.1119	0.435636	0.217818	−	0.975989i	$-0.430106\pi$
$773$	12.1119	0.435636	0.217818	+	0.975989i	$0.430106\pi$
$774$	3.02163	0.108610
$775$	−4.24506	−0.152487
$776$	9.09922	0.326643
$777$	5.29308	0.189888
$778$	−30.5902	−1.09671
$779$	−17.1895	−0.615878
$780$	0	0
$781$	12.6320	0.452008
$782$	8.66403	0.309825
$783$	−3.85061	−0.137610
$784$	1.00000	0.0357143
$785$	−54.4363	−1.94292
$786$	−18.7757	−0.669706
$787$	−32.0549	−1.14263	−0.571317	−	0.820729i	$-0.693567\pi$
$787$	−32.0549	−1.14263	−0.571317	+	0.820729i	$0.693567\pi$
$788$	−23.6823	−0.843645
$789$	17.0930	0.608525
$790$	−13.7470	−0.489098
$791$	11.2111	0.398623
$792$	3.44247	0.122323
$793$	0	0
$794$	11.6778	0.414430
$795$	−25.4685	−0.903276
$796$	−24.8466	−0.880666
$797$	9.13089	0.323433	0.161716	−	0.986837i	$-0.448297\pi$
$797$	9.13089	0.323433	0.161716	+	0.986837i	$0.448297\pi$
$798$	−1.72542	−0.0610791
$799$	22.7577	0.805108
$800$	−4.33891	−0.153404
$801$	−7.90307	−0.279241
$802$	−9.25697	−0.326875
$803$	34.8100	1.22842
$804$	4.75055	0.167539
$805$	9.35918	0.329868
$806$	0	0
$807$	26.6594	0.938456
$808$	−18.8533	−0.663256
$809$	21.4623	0.754574	0.377287	−	0.926096i	$-0.376857\pi$
$809$	21.4623	0.754574	0.377287	+	0.926096i	$0.376857\pi$
$810$	3.05596	0.107376
$811$	44.2671	1.55443	0.777214	−	0.629236i	$-0.216632\pi$
$811$	44.2671	1.55443	0.777214	+	0.629236i	$0.216632\pi$
$812$	−3.85061	−0.135130
$813$	−24.4495	−0.857481
$814$	18.2213	0.638656
$815$	−27.1318	−0.950385
$816$	2.82898	0.0990341
$817$	−5.21357	−0.182400
$818$	25.5989	0.895043
$819$	0	0
$820$	30.4451	1.06319
$821$	9.31994	0.325268	0.162634	−	0.986686i	$-0.448001\pi$
$821$	9.31994	0.325268	0.162634	+	0.986686i	$0.448001\pi$
$822$	−2.66109	−0.0928163
$823$	29.6852	1.03476	0.517381	−	0.855755i	$-0.326907\pi$
$823$	29.6852	1.03476	0.517381	+	0.855755i	$0.326907\pi$
$824$	−13.7079	−0.477536
$825$	14.9366	0.520024
$826$	−9.97753	−0.347163
$827$	−22.6364	−0.787146	−0.393573	−	0.919293i	$-0.628761\pi$
$827$	−22.6364	−0.787146	−0.393573	+	0.919293i	$0.628761\pi$
$828$	3.06260	0.106433
$829$	−9.64334	−0.334927	−0.167463	−	0.985878i	$-0.553558\pi$
$829$	−9.64334	−0.334927	−0.167463	+	0.985878i	$0.553558\pi$
$830$	21.8767	0.759351
$831$	28.4812	0.988003
$832$	0	0
$833$	−2.82898	−0.0980184
$834$	−16.7064	−0.578497
$835$	29.8564	1.03322
$836$	−5.93970	−0.205429
$837$	0.978370	0.0338174
$838$	−28.2037	−0.974280
$839$	33.4844	1.15601	0.578005	−	0.816034i	$-0.303832\pi$
$839$	33.4844	1.15601	0.578005	+	0.816034i	$0.303832\pi$
$840$	3.05596	0.105441
$841$	−14.1728	−0.488717
$842$	−5.07012	−0.174728
$843$	9.76032	0.336163
$844$	0.773018	0.0266084
$845$	0	0
$846$	8.04447	0.276575
$847$	−0.850611	−0.0292273
$848$	−8.33405	−0.286192
$849$	−11.6712	−0.400556
$850$	12.2747	0.421018
$851$	16.2106	0.555692
$852$	3.66945	0.125713
$853$	23.7246	0.812316	0.406158	−	0.913803i	$-0.366868\pi$
$853$	23.7246	0.812316	0.406158	+	0.913803i	$0.366868\pi$
$854$	−11.5452	−0.395069
$855$	−5.27281	−0.180326
$856$	3.31471	0.113295
$857$	24.4823	0.836299	0.418149	−	0.908378i	$-0.362679\pi$
$857$	24.4823	0.836299	0.418149	+	0.908378i	$0.362679\pi$
$858$	0	0
$859$	−26.7787	−0.913676	−0.456838	−	0.889550i	$-0.651018\pi$
$859$	−26.7787	−0.913676	−0.456838	+	0.889550i	$0.651018\pi$
$860$	9.23399	0.314876
$861$	−9.96254	−0.339522
$862$	−16.1437	−0.549856
$863$	54.8533	1.86723	0.933614	−	0.358282i	$-0.116637\pi$
$863$	54.8533	1.86723	0.933614	+	0.358282i	$0.116637\pi$
$864$	1.00000	0.0340207
$865$	−26.5112	−0.901409
$866$	22.6029	0.768077
$867$	8.99687	0.305550
$868$	0.978370	0.0332080
$869$	15.4857	0.525317
$870$	−11.7673	−0.398950
$871$	0	0
$872$	−8.67525	−0.293781
$873$	−9.09922	−0.307962
$874$	−5.28425	−0.178743
$875$	−2.02028	−0.0682978
$876$	10.1119	0.341650
$877$	5.39968	0.182334	0.0911671	−	0.995836i	$-0.470940\pi$
$877$	5.39968	0.182334	0.0911671	+	0.995836i	$0.470940\pi$
$878$	−28.2971	−0.954981
$879$	21.8202	0.735975
$880$	10.5201	0.354631
$881$	−16.3667	−0.551407	−0.275703	−	0.961243i	$-0.588911\pi$
$881$	−16.3667	−0.551407	−0.275703	+	0.961243i	$0.588911\pi$
$882$	−1.00000	−0.0336718
$883$	−10.8294	−0.364440	−0.182220	−	0.983258i	$-0.558328\pi$
$883$	−10.8294	−0.364440	−0.182220	+	0.983258i	$0.558328\pi$
$884$	0	0
$885$	−30.4910	−1.02494
$886$	1.38096	0.0463941
$887$	−33.1553	−1.11325	−0.556623	−	0.830765i	$-0.687903\pi$
$887$	−33.1553	−1.11325	−0.556623	+	0.830765i	$0.687903\pi$
$888$	5.29308	0.177624
$889$	−5.11506	−0.171553
$890$	−24.1515	−0.809560
$891$	−3.44247	−0.115327
$892$	21.5354	0.721060
$893$	−13.8801	−0.464478
$894$	2.50157	0.0836649
$895$	71.5946	2.39314
$896$	1.00000	0.0334077
$897$	0	0
$898$	−37.0059	−1.23490
$899$	−3.76732	−0.125647
$900$	4.33891	0.144630
$901$	23.5769	0.785459
$902$	−34.2957	−1.14192
$903$	−3.02163	−0.100554
$904$	11.2111	0.372877
$905$	23.4280	0.778773
$906$	12.6465	0.420151
$907$	31.0699	1.03166	0.515829	−	0.856691i	$-0.327484\pi$
$907$	31.0699	1.03166	0.515829	+	0.856691i	$0.327484\pi$
$908$	17.5444	0.582230
$909$	18.8533	0.625324
$910$	0	0
$911$	−7.26309	−0.240637	−0.120318	−	0.992735i	$-0.538392\pi$
$911$	−7.26309	−0.240637	−0.120318	+	0.992735i	$0.538392\pi$
$912$	−1.72542	−0.0571342
$913$	−24.6436	−0.815583
$914$	13.9176	0.460352
$915$	−35.2817	−1.16638
$916$	−10.3222	−0.341055
$917$	18.7757	0.620028
$918$	−2.82898	−0.0933703
$919$	34.1726	1.12725	0.563625	−	0.826031i	$-0.309406\pi$
$919$	34.1726	1.12725	0.563625	+	0.826031i	$0.309406\pi$
$920$	9.35918	0.308563
$921$	−4.87046	−0.160487
$922$	−4.84795	−0.159659
$923$	0	0
$924$	−3.44247	−0.113249
$925$	22.9662	0.755124
$926$	24.4377	0.803071
$927$	13.7079	0.450225
$928$	−3.85061	−0.126402
$929$	−13.6546	−0.447992	−0.223996	−	0.974590i	$-0.571910\pi$
$929$	−13.6546	−0.447992	−0.223996	+	0.974590i	$0.571910\pi$
$930$	2.98986	0.0980414
$931$	1.72542	0.0565482
$932$	17.8290	0.584008
$933$	−3.90344	−0.127793
$934$	−36.0027	−1.17804
$935$	−29.7611	−0.973291
$936$	0	0
$937$	43.5716	1.42342	0.711712	−	0.702472i	$-0.247920\pi$
$937$	43.5716	1.42342	0.711712	+	0.702472i	$0.247920\pi$
$938$	−4.75055	−0.155111
$939$	−26.0528	−0.850202
$940$	24.5836	0.801829
$941$	−55.8282	−1.81995	−0.909973	−	0.414666i	$-0.863898\pi$
$941$	−55.8282	−1.81995	−0.909973	+	0.414666i	$0.863898\pi$
$942$	17.8131	0.580384
$943$	−30.5112	−0.993583
$944$	−9.97753	−0.324741
$945$	−3.05596	−0.0994105
$946$	−10.4019	−0.338194
$947$	52.4896	1.70568	0.852842	−	0.522170i	$-0.174877\pi$
$947$	52.4896	1.70568	0.852842	+	0.522170i	$0.174877\pi$
$948$	4.49843	0.146102
$949$	0	0
$950$	−7.48642	−0.242891
$951$	−11.1107	−0.360290
$952$	−2.82898	−0.0916878
$953$	−7.01971	−0.227391	−0.113695	−	0.993516i	$-0.536269\pi$
$953$	−7.01971	−0.227391	−0.113695	+	0.993516i	$0.536269\pi$
$954$	8.33405	0.269825
$955$	−77.9101	−2.52111
$956$	5.71271	0.184762
$957$	13.2556	0.428493
$958$	9.49288	0.306701
$959$	2.66109	0.0859312
$960$	3.05596	0.0986308
$961$	−30.0428	−0.969122
$962$	0	0
$963$	−3.31471	−0.106815
$964$	−1.00313	−0.0323087
$965$	26.0547	0.838731
$966$	−3.06260	−0.0985375
$967$	−36.8770	−1.18588	−0.592942	−	0.805245i	$-0.702034\pi$
$967$	−36.8770	−1.18588	−0.592942	+	0.805245i	$0.702034\pi$
$968$	−0.850611	−0.0273397
$969$	4.88117	0.156806
$970$	−27.8069	−0.892825
$971$	27.6403	0.887021	0.443511	−	0.896269i	$-0.353733\pi$
$971$	27.6403	0.887021	0.443511	+	0.896269i	$0.353733\pi$
$972$	−1.00000	−0.0320750
$973$	16.7064	0.535584
$974$	40.9206	1.31118
$975$	0	0
$976$	−11.5452	−0.369553
$977$	55.6454	1.78026	0.890128	−	0.455711i	$-0.150615\pi$
$977$	55.6454	1.78026	0.890128	+	0.455711i	$0.150615\pi$
$978$	8.87831	0.283897
$979$	27.2061	0.869511
$980$	−3.05596	−0.0976191
$981$	8.67525	0.276979
$982$	5.11506	0.163228
$983$	25.0022	0.797446	0.398723	−	0.917071i	$-0.369453\pi$
$983$	25.0022	0.797446	0.398723	+	0.917071i	$0.369453\pi$
$984$	−9.96254	−0.317594
$985$	72.3721	2.30597
$986$	10.8933	0.346913
$987$	−8.04447	−0.256058
$988$	0	0
$989$	−9.25404	−0.294261
$990$	−10.5201	−0.334350
$991$	−27.2151	−0.864517	−0.432258	−	0.901750i	$-0.642283\pi$
$991$	−27.2151	−0.864517	−0.432258	+	0.901750i	$0.642283\pi$
$992$	0.978370	0.0310633
$993$	−14.6478	−0.464835
$994$	−3.66945	−0.116388
$995$	75.9304	2.40716
$996$	−7.15869	−0.226832
$997$	−16.8541	−0.533775	−0.266888	−	0.963728i	$-0.585995\pi$
$997$	−16.8541	−0.533775	−0.266888	+	0.963728i	$0.585995\pi$
$998$	−19.5827	−0.619878
$999$	−5.29308	−0.167466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.cn.1.2		4
13.6	odd	12		546.2.s.e.127.4	yes	8
13.11	odd	12		546.2.s.e.43.3	✓	8
13.12	even	2		7098.2.a.co.1.3		4
39.11	even	12		1638.2.bj.f.1135.2		8
39.32	even	12		1638.2.bj.f.127.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.s.e.43.3	✓	8	13.11	odd	12
546.2.s.e.127.4	yes	8	13.6	odd	12
1638.2.bj.f.127.1		8	39.32	even	12
1638.2.bj.f.1135.2		8	39.11	even	12
7098.2.a.cn.1.2		4	1.1	even	1	trivial
7098.2.a.co.1.3		4	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:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.05596 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.05596 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.05596 q^{10} -3.44247 q^{11} -1.00000 q^{12} +1.00000 q^{14} +3.05596 q^{15} +1.00000 q^{16} -2.82898 q^{17} -1.00000 q^{18} +1.72542 q^{19} -3.05596 q^{20} +1.00000 q^{21} +3.44247 q^{22} +3.06260 q^{23} +1.00000 q^{24} +4.33891 q^{25} -1.00000 q^{27} -1.00000 q^{28} +3.85061 q^{29} -3.05596 q^{30} -0.978370 q^{31} -1.00000 q^{32} +3.44247 q^{33} +2.82898 q^{34} +3.05596 q^{35} +1.00000 q^{36} +5.29308 q^{37} -1.72542 q^{38} +3.05596 q^{40} -9.96254 q^{41} -1.00000 q^{42} -3.02163 q^{43} -3.44247 q^{44} -3.05596 q^{45} -3.06260 q^{46} -8.04447 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.33891 q^{50} +2.82898 q^{51} -8.33405 q^{53} +1.00000 q^{54} +10.5201 q^{55} +1.00000 q^{56} -1.72542 q^{57} -3.85061 q^{58} -9.97753 q^{59} +3.05596 q^{60} -11.5452 q^{61} +0.978370 q^{62} -1.00000 q^{63} +1.00000 q^{64} -3.44247 q^{66} -4.75055 q^{67} -2.82898 q^{68} -3.06260 q^{69} -3.05596 q^{70} -3.66945 q^{71} -1.00000 q^{72} -10.1119 q^{73} -5.29308 q^{74} -4.33891 q^{75} +1.72542 q^{76} +3.44247 q^{77} -4.49843 q^{79} -3.05596 q^{80} +1.00000 q^{81} +9.96254 q^{82} +7.15869 q^{83} +1.00000 q^{84} +8.64526 q^{85} +3.02163 q^{86} -3.85061 q^{87} +3.44247 q^{88} -7.90307 q^{89} +3.05596 q^{90} +3.06260 q^{92} +0.978370 q^{93} +8.04447 q^{94} -5.27281 q^{95} +1.00000 q^{96} -9.09922 q^{97} -1.00000 q^{98} -3.44247 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 6 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9} + 6 q^{10} - 4 q^{11} - 4 q^{12} + 4 q^{14} + 6 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} - 2 q^{19} - 6 q^{20} + 4 q^{21} + 4 q^{22} + 8 q^{23} + 4 q^{24} + 12 q^{25} - 4 q^{27} - 4 q^{28} - 2 q^{29} - 6 q^{30} - 8 q^{31} - 4 q^{32} + 4 q^{33} - 2 q^{34} + 6 q^{35} + 4 q^{36} - 6 q^{37} + 2 q^{38} + 6 q^{40} - 10 q^{41} - 4 q^{42} - 8 q^{43} - 4 q^{44} - 6 q^{45} - 8 q^{46} - 2 q^{47} - 4 q^{48} + 4 q^{49} - 12 q^{50} - 2 q^{51} - 6 q^{53} + 4 q^{54} + 22 q^{55} + 4 q^{56} + 2 q^{57} + 2 q^{58} - 4 q^{59} + 6 q^{60} + 8 q^{61} + 8 q^{62} - 4 q^{63} + 4 q^{64} - 4 q^{66} + 24 q^{67} + 2 q^{68} - 8 q^{69} - 6 q^{70} - 12 q^{71} - 4 q^{72} - 28 q^{73} + 6 q^{74} - 12 q^{75} - 2 q^{76} + 4 q^{77} - 2 q^{79} - 6 q^{80} + 4 q^{81} + 10 q^{82} - 22 q^{83} + 4 q^{84} + 6 q^{85} + 8 q^{86} + 2 q^{87} + 4 q^{88} - 38 q^{89} + 6 q^{90} + 8 q^{92} + 8 q^{93} + 2 q^{94} - 50 q^{95} + 4 q^{96} - 22 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 6 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 4 * q^9 + 6 * q^10 - 4 * q^11 - 4 * q^12 + 4 * q^14 + 6 * q^15 + 4 * q^16 + 2 * q^17 - 4 * q^18 - 2 * q^19 - 6 * q^20 + 4 * q^21 + 4 * q^22 + 8 * q^23 + 4 * q^24 + 12 * q^25 - 4 * q^27 - 4 * q^28 - 2 * q^29 - 6 * q^30 - 8 * q^31 - 4 * q^32 + 4 * q^33 - 2 * q^34 + 6 * q^35 + 4 * q^36 - 6 * q^37 + 2 * q^38 + 6 * q^40 - 10 * q^41 - 4 * q^42 - 8 * q^43 - 4 * q^44 - 6 * q^45 - 8 * q^46 - 2 * q^47 - 4 * q^48 + 4 * q^49 - 12 * q^50 - 2 * q^51 - 6 * q^53 + 4 * q^54 + 22 * q^55 + 4 * q^56 + 2 * q^57 + 2 * q^58 - 4 * q^59 + 6 * q^60 + 8 * q^61 + 8 * q^62 - 4 * q^63 + 4 * q^64 - 4 * q^66 + 24 * q^67 + 2 * q^68 - 8 * q^69 - 6 * q^70 - 12 * q^71 - 4 * q^72 - 28 * q^73 + 6 * q^74 - 12 * q^75 - 2 * q^76 + 4 * q^77 - 2 * q^79 - 6 * q^80 + 4 * q^81 + 10 * q^82 - 22 * q^83 + 4 * q^84 + 6 * q^85 + 8 * q^86 + 2 * q^87 + 4 * q^88 - 38 * q^89 + 6 * q^90 + 8 * q^92 + 8 * q^93 + 2 * q^94 - 50 * q^95 + 4 * q^96 - 22 * q^97 - 4 * q^98 - 4 * q^99

Introduction

L-functions

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