LMFDB - Embedded newform 3450.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(1,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, 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("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 690)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.622216 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.622216 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.42864 q^{11} +1.00000 q^{12} -1.37778 q^{13} +0.622216 q^{14} +1.00000 q^{16} -6.42864 q^{17} -1.00000 q^{18} -1.37778 q^{19} -0.622216 q^{21} -4.42864 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.37778 q^{26} +1.00000 q^{27} -0.622216 q^{28} -4.23506 q^{29} -4.00000 q^{31} -1.00000 q^{32} +4.42864 q^{33} +6.42864 q^{34} +1.00000 q^{36} -11.8064 q^{37} +1.37778 q^{38} -1.37778 q^{39} -10.8573 q^{41} +0.622216 q^{42} +1.05086 q^{43} +4.42864 q^{44} +1.00000 q^{46} +11.6128 q^{47} +1.00000 q^{48} -6.61285 q^{49} -6.42864 q^{51} -1.37778 q^{52} -3.37778 q^{53} -1.00000 q^{54} +0.622216 q^{56} -1.37778 q^{57} +4.23506 q^{58} +9.61285 q^{59} +8.66370 q^{61} +4.00000 q^{62} -0.622216 q^{63} +1.00000 q^{64} -4.42864 q^{66} -11.8064 q^{67} -6.42864 q^{68} -1.00000 q^{69} -6.99063 q^{71} -1.00000 q^{72} -2.75557 q^{73} +11.8064 q^{74} -1.37778 q^{76} -2.75557 q^{77} +1.37778 q^{78} +12.0415 q^{79} +1.00000 q^{81} +10.8573 q^{82} -2.19358 q^{83} -0.622216 q^{84} -1.05086 q^{86} -4.23506 q^{87} -4.42864 q^{88} +2.75557 q^{89} +0.857279 q^{91} -1.00000 q^{92} -4.00000 q^{93} -11.6128 q^{94} -1.00000 q^{96} +2.13335 q^{97} +6.61285 q^{98} +4.42864 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{12} - 4 q^{13} + 2 q^{14} + 3 q^{16} - 6 q^{17} - 3 q^{18} - 4 q^{19} - 2 q^{21} - 3 q^{23} - 3 q^{24} + 4 q^{26} + 3 q^{27} - 2 q^{28} + 14 q^{29} - 12 q^{31} - 3 q^{32} + 6 q^{34} + 3 q^{36} - 22 q^{37} + 4 q^{38} - 4 q^{39} - 6 q^{41} + 2 q^{42} - 10 q^{43} + 3 q^{46} + 8 q^{47} + 3 q^{48} + 7 q^{49} - 6 q^{51} - 4 q^{52} - 10 q^{53} - 3 q^{54} + 2 q^{56} - 4 q^{57} - 14 q^{58} + 2 q^{59} - 14 q^{61} + 12 q^{62} - 2 q^{63} + 3 q^{64} - 22 q^{67} - 6 q^{68} - 3 q^{69} + 6 q^{71} - 3 q^{72} - 8 q^{73} + 22 q^{74} - 4 q^{76} - 8 q^{77} + 4 q^{78} - 4 q^{79} + 3 q^{81} + 6 q^{82} - 20 q^{83} - 2 q^{84} + 10 q^{86} + 14 q^{87} + 8 q^{89} - 24 q^{91} - 3 q^{92} - 12 q^{93} - 8 q^{94} - 3 q^{96} + 6 q^{97} - 7 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^12 - 4 * q^13 + 2 * q^14 + 3 * q^16 - 6 * q^17 - 3 * q^18 - 4 * q^19 - 2 * q^21 - 3 * q^23 - 3 * q^24 + 4 * q^26 + 3 * q^27 - 2 * q^28 + 14 * q^29 - 12 * q^31 - 3 * q^32 + 6 * q^34 + 3 * q^36 - 22 * q^37 + 4 * q^38 - 4 * q^39 - 6 * q^41 + 2 * q^42 - 10 * q^43 + 3 * q^46 + 8 * q^47 + 3 * q^48 + 7 * q^49 - 6 * q^51 - 4 * q^52 - 10 * q^53 - 3 * q^54 + 2 * q^56 - 4 * q^57 - 14 * q^58 + 2 * q^59 - 14 * q^61 + 12 * q^62 - 2 * q^63 + 3 * q^64 - 22 * q^67 - 6 * q^68 - 3 * q^69 + 6 * q^71 - 3 * q^72 - 8 * q^73 + 22 * q^74 - 4 * q^76 - 8 * q^77 + 4 * q^78 - 4 * q^79 + 3 * q^81 + 6 * q^82 - 20 * q^83 - 2 * q^84 + 10 * q^86 + 14 * q^87 + 8 * q^89 - 24 * q^91 - 3 * q^92 - 12 * q^93 - 8 * q^94 - 3 * q^96 + 6 * q^97 - 7 * q^98

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
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−0.622216	−0.235175	−0.117588	−	0.993063i	$-0.537516\pi$
$7$	−0.622216	−0.235175	−0.117588	+	0.993063i	$0.537516\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	4.42864	1.33529	0.667643	−	0.744482i	$-0.267303\pi$
$11$	4.42864	1.33529	0.667643	+	0.744482i	$0.267303\pi$
$12$	1.00000	0.288675
$13$	−1.37778	−0.382129	−0.191064	−	0.981578i	$-0.561194\pi$
$13$	−1.37778	−0.382129	−0.191064	+	0.981578i	$0.561194\pi$
$14$	0.622216	0.166294
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.42864	−1.55917	−0.779587	−	0.626294i	$-0.784571\pi$
$17$	−6.42864	−1.55917	−0.779587	+	0.626294i	$0.784571\pi$
$18$	−1.00000	−0.235702
$19$	−1.37778	−0.316085	−0.158043	−	0.987432i	$-0.550518\pi$
$19$	−1.37778	−0.316085	−0.158043	+	0.987432i	$0.550518\pi$
$20$	0	0
$21$	−0.622216	−0.135779
$22$	−4.42864	−0.944189
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	0	0
$26$	1.37778	0.270206
$27$	1.00000	0.192450
$28$	−0.622216	−0.117588
$29$	−4.23506	−0.786432	−0.393216	−	0.919446i	$-0.628637\pi$
$29$	−4.23506	−0.786432	−0.393216	+	0.919446i	$0.628637\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	4.42864	0.770927
$34$	6.42864	1.10250
$35$	0	0
$36$	1.00000	0.166667
$37$	−11.8064	−1.94096	−0.970482	−	0.241173i	$-0.922468\pi$
$37$	−11.8064	−1.94096	−0.970482	+	0.241173i	$0.922468\pi$
$38$	1.37778	0.223506
$39$	−1.37778	−0.220622
$40$	0	0
$41$	−10.8573	−1.69562	−0.847811	−	0.530298i	$-0.822080\pi$
$41$	−10.8573	−1.69562	−0.847811	+	0.530298i	$0.822080\pi$
$42$	0.622216	0.0960100
$43$	1.05086	0.160254	0.0801270	−	0.996785i	$-0.474467\pi$
$43$	1.05086	0.160254	0.0801270	+	0.996785i	$0.474467\pi$
$44$	4.42864	0.667643
$45$	0	0
$46$	1.00000	0.147442
$47$	11.6128	1.69391	0.846954	−	0.531666i	$-0.178434\pi$
$47$	11.6128	1.69391	0.846954	+	0.531666i	$0.178434\pi$
$48$	1.00000	0.144338
$49$	−6.61285	−0.944693
$50$	0	0
$51$	−6.42864	−0.900190
$52$	−1.37778	−0.191064
$53$	−3.37778	−0.463974	−0.231987	−	0.972719i	$-0.574523\pi$
$53$	−3.37778	−0.463974	−0.231987	+	0.972719i	$0.574523\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0.622216	0.0831471
$57$	−1.37778	−0.182492
$58$	4.23506	0.556091
$59$	9.61285	1.25149	0.625743	−	0.780029i	$-0.284796\pi$
$59$	9.61285	1.25149	0.625743	+	0.780029i	$0.284796\pi$
$60$	0	0
$61$	8.66370	1.10927	0.554637	−	0.832093i	$-0.312857\pi$
$61$	8.66370	1.10927	0.554637	+	0.832093i	$0.312857\pi$
$62$	4.00000	0.508001
$63$	−0.622216	−0.0783918
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.42864	−0.545128
$67$	−11.8064	−1.44238	−0.721192	−	0.692735i	$-0.756406\pi$
$67$	−11.8064	−1.44238	−0.721192	+	0.692735i	$0.756406\pi$
$68$	−6.42864	−0.779587
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−6.99063	−0.829635	−0.414818	−	0.909905i	$-0.636155\pi$
$71$	−6.99063	−0.829635	−0.414818	+	0.909905i	$0.636155\pi$
$72$	−1.00000	−0.117851
$73$	−2.75557	−0.322515	−0.161257	−	0.986912i	$-0.551555\pi$
$73$	−2.75557	−0.322515	−0.161257	+	0.986912i	$0.551555\pi$
$74$	11.8064	1.37247
$75$	0	0
$76$	−1.37778	−0.158043
$77$	−2.75557	−0.314026
$78$	1.37778	0.156003
$79$	12.0415	1.35477	0.677387	−	0.735627i	$-0.263112\pi$
$79$	12.0415	1.35477	0.677387	+	0.735627i	$0.263112\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.8573	1.19899
$83$	−2.19358	−0.240776	−0.120388	−	0.992727i	$-0.538414\pi$
$83$	−2.19358	−0.240776	−0.120388	+	0.992727i	$0.538414\pi$
$84$	−0.622216	−0.0678893
$85$	0	0
$86$	−1.05086	−0.113317
$87$	−4.23506	−0.454046
$88$	−4.42864	−0.472095
$89$	2.75557	0.292090	0.146045	−	0.989278i	$-0.453346\pi$
$89$	2.75557	0.292090	0.146045	+	0.989278i	$0.453346\pi$
$90$	0	0
$91$	0.857279	0.0898673
$92$	−1.00000	−0.104257
$93$	−4.00000	−0.414781
$94$	−11.6128	−1.19777
$95$	0	0
$96$	−1.00000	−0.102062
$97$	2.13335	0.216609	0.108305	−	0.994118i	$-0.465458\pi$
$97$	2.13335	0.216609	0.108305	+	0.994118i	$0.465458\pi$
$98$	6.61285	0.667998
$99$	4.42864	0.445095
$100$	0	0
$101$	16.2351	1.61545	0.807725	−	0.589560i	$-0.200699\pi$
$101$	16.2351	1.61545	0.807725	+	0.589560i	$0.200699\pi$
$102$	6.42864	0.636530
$103$	−12.2351	−1.20556	−0.602778	−	0.797909i	$-0.705940\pi$
$103$	−12.2351	−1.20556	−0.602778	+	0.797909i	$0.705940\pi$
$104$	1.37778	0.135103
$105$	0	0
$106$	3.37778	0.328079
$107$	10.6637	1.03090	0.515450	−	0.856920i	$-0.327625\pi$
$107$	10.6637	1.03090	0.515450	+	0.856920i	$0.327625\pi$
$108$	1.00000	0.0962250
$109$	−15.1526	−1.45135	−0.725676	−	0.688036i	$-0.758473\pi$
$109$	−15.1526	−1.45135	−0.725676	+	0.688036i	$0.758473\pi$
$110$	0	0
$111$	−11.8064	−1.12062
$112$	−0.622216	−0.0587939
$113$	−14.4286	−1.35733	−0.678666	−	0.734447i	$-0.737442\pi$
$113$	−14.4286	−1.35733	−0.678666	+	0.734447i	$0.737442\pi$
$114$	1.37778	0.129041
$115$	0	0
$116$	−4.23506	−0.393216
$117$	−1.37778	−0.127376
$118$	−9.61285	−0.884934
$119$	4.00000	0.366679
$120$	0	0
$121$	8.61285	0.782986
$122$	−8.66370	−0.784375
$123$	−10.8573	−0.978968
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0.622216	0.0554314
$127$	4.99063	0.442847	0.221423	−	0.975178i	$-0.428930\pi$
$127$	4.99063	0.442847	0.221423	+	0.975178i	$0.428930\pi$
$128$	−1.00000	−0.0883883
$129$	1.05086	0.0925226
$130$	0	0
$131$	0.755569	0.0660143	0.0330072	−	0.999455i	$-0.489492\pi$
$131$	0.755569	0.0660143	0.0330072	+	0.999455i	$0.489492\pi$
$132$	4.42864	0.385464
$133$	0.857279	0.0743355
$134$	11.8064	1.01992
$135$	0	0
$136$	6.42864	0.551251
$137$	−12.9175	−1.10362	−0.551808	−	0.833971i	$-0.686062\pi$
$137$	−12.9175	−1.10362	−0.551808	+	0.833971i	$0.686062\pi$
$138$	1.00000	0.0851257
$139$	−14.1017	−1.19609	−0.598046	−	0.801462i	$-0.704056\pi$
$139$	−14.1017	−1.19609	−0.598046	+	0.801462i	$0.704056\pi$
$140$	0	0
$141$	11.6128	0.977978
$142$	6.99063	0.586641
$143$	−6.10171	−0.510251
$144$	1.00000	0.0833333
$145$	0	0
$146$	2.75557	0.228052
$147$	−6.61285	−0.545418
$148$	−11.8064	−0.970482
$149$	12.4286	1.01819	0.509097	−	0.860709i	$-0.329979\pi$
$149$	12.4286	1.01819	0.509097	+	0.860709i	$0.329979\pi$
$150$	0	0
$151$	−4.85728	−0.395280	−0.197640	−	0.980275i	$-0.563328\pi$
$151$	−4.85728	−0.395280	−0.197640	+	0.980275i	$0.563328\pi$
$152$	1.37778	0.111753
$153$	−6.42864	−0.519725
$154$	2.75557	0.222050
$155$	0	0
$156$	−1.37778	−0.110311
$157$	9.90813	0.790755	0.395378	−	0.918519i	$-0.370614\pi$
$157$	9.90813	0.790755	0.395378	+	0.918519i	$0.370614\pi$
$158$	−12.0415	−0.957969
$159$	−3.37778	−0.267876
$160$	0	0
$161$	0.622216	0.0490375
$162$	−1.00000	−0.0785674
$163$	−10.1017	−0.791227	−0.395614	−	0.918417i	$-0.629468\pi$
$163$	−10.1017	−0.791227	−0.395614	+	0.918417i	$0.629468\pi$
$164$	−10.8573	−0.847811
$165$	0	0
$166$	2.19358	0.170255
$167$	−1.89829	−0.146894	−0.0734470	−	0.997299i	$-0.523400\pi$
$167$	−1.89829	−0.146894	−0.0734470	+	0.997299i	$0.523400\pi$
$168$	0.622216	0.0480050
$169$	−11.1017	−0.853978
$170$	0	0
$171$	−1.37778	−0.105362
$172$	1.05086	0.0801270
$173$	−21.2257	−1.61376	−0.806880	−	0.590716i	$-0.798846\pi$
$173$	−21.2257	−1.61376	−0.806880	+	0.590716i	$0.798846\pi$
$174$	4.23506	0.321059
$175$	0	0
$176$	4.42864	0.333821
$177$	9.61285	0.722546
$178$	−2.75557	−0.206539
$179$	0.488863	0.0365393	0.0182697	−	0.999833i	$-0.494184\pi$
$179$	0.488863	0.0365393	0.0182697	+	0.999833i	$0.494184\pi$
$180$	0	0
$181$	−20.9304	−1.55575	−0.777873	−	0.628422i	$-0.783701\pi$
$181$	−20.9304	−1.55575	−0.777873	+	0.628422i	$0.783701\pi$
$182$	−0.857279	−0.0635457
$183$	8.66370	0.640439
$184$	1.00000	0.0737210
$185$	0	0
$186$	4.00000	0.293294
$187$	−28.4701	−2.08194
$188$	11.6128	0.846954
$189$	−0.622216	−0.0452595
$190$	0	0
$191$	18.9590	1.37182	0.685912	−	0.727684i	$-0.259403\pi$
$191$	18.9590	1.37182	0.685912	+	0.727684i	$0.259403\pi$
$192$	1.00000	0.0721688
$193$	1.24443	0.0895761	0.0447881	−	0.998997i	$-0.485739\pi$
$193$	1.24443	0.0895761	0.0447881	+	0.998997i	$0.485739\pi$
$194$	−2.13335	−0.153166
$195$	0	0
$196$	−6.61285	−0.472346
$197$	15.2444	1.08612	0.543060	−	0.839694i	$-0.317265\pi$
$197$	15.2444	1.08612	0.543060	+	0.839694i	$0.317265\pi$
$198$	−4.42864	−0.314730
$199$	−14.5303	−1.03003	−0.515015	−	0.857181i	$-0.672214\pi$
$199$	−14.5303	−1.03003	−0.515015	+	0.857181i	$0.672214\pi$
$200$	0	0
$201$	−11.8064	−0.832761
$202$	−16.2351	−1.14230
$203$	2.63512	0.184949
$204$	−6.42864	−0.450095
$205$	0	0
$206$	12.2351	0.852457
$207$	−1.00000	−0.0695048
$208$	−1.37778	−0.0955322
$209$	−6.10171	−0.422064
$210$	0	0
$211$	0.266706	0.0183608	0.00918041	−	0.999958i	$-0.497078\pi$
$211$	0.266706	0.0183608	0.00918041	+	0.999958i	$0.497078\pi$
$212$	−3.37778	−0.231987
$213$	−6.99063	−0.478990
$214$	−10.6637	−0.728956
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	2.48886	0.168955
$218$	15.1526	1.02626
$219$	−2.75557	−0.186204
$220$	0	0
$221$	8.85728	0.595805
$222$	11.8064	0.792395
$223$	0.990632	0.0663376	0.0331688	−	0.999450i	$-0.489440\pi$
$223$	0.990632	0.0663376	0.0331688	+	0.999450i	$0.489440\pi$
$224$	0.622216	0.0415735
$225$	0	0
$226$	14.4286	0.959779
$227$	17.1526	1.13846	0.569228	−	0.822180i	$-0.307242\pi$
$227$	17.1526	1.13846	0.569228	+	0.822180i	$0.307242\pi$
$228$	−1.37778	−0.0912460
$229$	7.15257	0.472655	0.236327	−	0.971673i	$-0.424056\pi$
$229$	7.15257	0.472655	0.236327	+	0.971673i	$0.424056\pi$
$230$	0	0
$231$	−2.75557	−0.181303
$232$	4.23506	0.278046
$233$	3.71456	0.243349	0.121674	−	0.992570i	$-0.461174\pi$
$233$	3.71456	0.243349	0.121674	+	0.992570i	$0.461174\pi$
$234$	1.37778	0.0900686
$235$	0	0
$236$	9.61285	0.625743
$237$	12.0415	0.782179
$238$	−4.00000	−0.259281
$239$	27.8479	1.80133	0.900666	−	0.434512i	$-0.143079\pi$
$239$	27.8479	1.80133	0.900666	+	0.434512i	$0.143079\pi$
$240$	0	0
$241$	−7.12399	−0.458896	−0.229448	−	0.973321i	$-0.573692\pi$
$241$	−7.12399	−0.458896	−0.229448	+	0.973321i	$0.573692\pi$
$242$	−8.61285	−0.553655
$243$	1.00000	0.0641500
$244$	8.66370	0.554637
$245$	0	0
$246$	10.8573	0.692235
$247$	1.89829	0.120785
$248$	4.00000	0.254000
$249$	−2.19358	−0.139012
$250$	0	0
$251$	−22.4099	−1.41450	−0.707250	−	0.706963i	$-0.750065\pi$
$251$	−22.4099	−1.41450	−0.707250	+	0.706963i	$0.750065\pi$
$252$	−0.622216	−0.0391959
$253$	−4.42864	−0.278426
$254$	−4.99063	−0.313140
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.7146	−0.730734	−0.365367	−	0.930864i	$-0.619057\pi$
$257$	−11.7146	−0.730734	−0.365367	+	0.930864i	$0.619057\pi$
$258$	−1.05086	−0.0654234
$259$	7.34614	0.456467
$260$	0	0
$261$	−4.23506	−0.262144
$262$	−0.755569	−0.0466792
$263$	−8.47013	−0.522290	−0.261145	−	0.965300i	$-0.584100\pi$
$263$	−8.47013	−0.522290	−0.261145	+	0.965300i	$0.584100\pi$
$264$	−4.42864	−0.272564
$265$	0	0
$266$	−0.857279	−0.0525631
$267$	2.75557	0.168638
$268$	−11.8064	−0.721192
$269$	28.8256	1.75753	0.878765	−	0.477255i	$-0.158368\pi$
$269$	28.8256	1.75753	0.878765	+	0.477255i	$0.158368\pi$
$270$	0	0
$271$	−13.8350	−0.840417	−0.420208	−	0.907428i	$-0.638043\pi$
$271$	−13.8350	−0.840417	−0.420208	+	0.907428i	$0.638043\pi$
$272$	−6.42864	−0.389794
$273$	0.857279	0.0518849
$274$	12.9175	0.780375
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	2.88892	0.173578	0.0867892	−	0.996227i	$-0.472339\pi$
$277$	2.88892	0.173578	0.0867892	+	0.996227i	$0.472339\pi$
$278$	14.1017	0.845764
$279$	−4.00000	−0.239474
$280$	0	0
$281$	5.51114	0.328767	0.164383	−	0.986397i	$-0.447437\pi$
$281$	5.51114	0.328767	0.164383	+	0.986397i	$0.447437\pi$
$282$	−11.6128	−0.691535
$283$	8.19358	0.487058	0.243529	−	0.969894i	$-0.421695\pi$
$283$	8.19358	0.487058	0.243529	+	0.969894i	$0.421695\pi$
$284$	−6.99063	−0.414818
$285$	0	0
$286$	6.10171	0.360802
$287$	6.75557	0.398769
$288$	−1.00000	−0.0589256
$289$	24.3274	1.43102
$290$	0	0
$291$	2.13335	0.125059
$292$	−2.75557	−0.161257
$293$	12.6222	0.737398	0.368699	−	0.929549i	$-0.379803\pi$
$293$	12.6222	0.737398	0.368699	+	0.929549i	$0.379803\pi$
$294$	6.61285	0.385669
$295$	0	0
$296$	11.8064	0.686234
$297$	4.42864	0.256976
$298$	−12.4286	−0.719972
$299$	1.37778	0.0796793
$300$	0	0
$301$	−0.653858	−0.0376878
$302$	4.85728	0.279505
$303$	16.2351	0.932680
$304$	−1.37778	−0.0790214
$305$	0	0
$306$	6.42864	0.367501
$307$	−0.653858	−0.0373177	−0.0186588	−	0.999826i	$-0.505940\pi$
$307$	−0.653858	−0.0373177	−0.0186588	+	0.999826i	$0.505940\pi$
$308$	−2.75557	−0.157013
$309$	−12.2351	−0.696028
$310$	0	0
$311$	−13.0923	−0.742399	−0.371199	−	0.928553i	$-0.621053\pi$
$311$	−13.0923	−0.742399	−0.371199	+	0.928553i	$0.621053\pi$
$312$	1.37778	0.0780017
$313$	11.2573	0.636302	0.318151	−	0.948040i	$-0.396938\pi$
$313$	11.2573	0.636302	0.318151	+	0.948040i	$0.396938\pi$
$314$	−9.90813	−0.559148
$315$	0	0
$316$	12.0415	0.677387
$317$	−22.4701	−1.26205	−0.631024	−	0.775763i	$-0.717365\pi$
$317$	−22.4701	−1.26205	−0.631024	+	0.775763i	$0.717365\pi$
$318$	3.37778	0.189417
$319$	−18.7556	−1.05011
$320$	0	0
$321$	10.6637	0.595190
$322$	−0.622216	−0.0346747
$323$	8.85728	0.492832
$324$	1.00000	0.0555556
$325$	0	0
$326$	10.1017	0.559482
$327$	−15.1526	−0.837939
$328$	10.8573	0.599493
$329$	−7.22570	−0.398365
$330$	0	0
$331$	−29.4479	−1.61860	−0.809300	−	0.587395i	$-0.800153\pi$
$331$	−29.4479	−1.61860	−0.809300	+	0.587395i	$0.800153\pi$
$332$	−2.19358	−0.120388
$333$	−11.8064	−0.646988
$334$	1.89829	0.103870
$335$	0	0
$336$	−0.622216	−0.0339446
$337$	−24.6222	−1.34126	−0.670629	−	0.741793i	$-0.733976\pi$
$337$	−24.6222	−1.34126	−0.670629	+	0.741793i	$0.733976\pi$
$338$	11.1017	0.603853
$339$	−14.4286	−0.783656
$340$	0	0
$341$	−17.7146	−0.959297
$342$	1.37778	0.0745020
$343$	8.47013	0.457344
$344$	−1.05086	−0.0566583
$345$	0	0
$346$	21.2257	1.14110
$347$	26.1017	1.40121	0.700607	−	0.713548i	$-0.252913\pi$
$347$	26.1017	1.40121	0.700607	+	0.713548i	$0.252913\pi$
$348$	−4.23506	−0.227023
$349$	−21.2257	−1.13619	−0.568093	−	0.822965i	$-0.692319\pi$
$349$	−21.2257	−1.13619	−0.568093	+	0.822965i	$0.692319\pi$
$350$	0	0
$351$	−1.37778	−0.0735407
$352$	−4.42864	−0.236047
$353$	−4.28544	−0.228091	−0.114046	−	0.993476i	$-0.536381\pi$
$353$	−4.28544	−0.228091	−0.114046	+	0.993476i	$0.536381\pi$
$354$	−9.61285	−0.510917
$355$	0	0
$356$	2.75557	0.146045
$357$	4.00000	0.211702
$358$	−0.488863	−0.0258372
$359$	−4.65386	−0.245621	−0.122811	−	0.992430i	$-0.539191\pi$
$359$	−4.65386	−0.245621	−0.122811	+	0.992430i	$0.539191\pi$
$360$	0	0
$361$	−17.1017	−0.900090
$362$	20.9304	1.10008
$363$	8.61285	0.452057
$364$	0.857279	0.0449336
$365$	0	0
$366$	−8.66370	−0.452859
$367$	4.88892	0.255200	0.127600	−	0.991826i	$-0.459273\pi$
$367$	4.88892	0.255200	0.127600	+	0.991826i	$0.459273\pi$
$368$	−1.00000	−0.0521286
$369$	−10.8573	−0.565207
$370$	0	0
$371$	2.10171	0.109115
$372$	−4.00000	−0.207390
$373$	23.4193	1.21260	0.606302	−	0.795235i	$-0.292652\pi$
$373$	23.4193	1.21260	0.606302	+	0.795235i	$0.292652\pi$
$374$	28.4701	1.47216
$375$	0	0
$376$	−11.6128	−0.598887
$377$	5.83500	0.300518
$378$	0.622216	0.0320033
$379$	−4.90766	−0.252089	−0.126045	−	0.992025i	$-0.540228\pi$
$379$	−4.90766	−0.252089	−0.126045	+	0.992025i	$0.540228\pi$
$380$	0	0
$381$	4.99063	0.255678
$382$	−18.9590	−0.970026
$383$	−19.8796	−1.01580	−0.507899	−	0.861417i	$-0.669578\pi$
$383$	−19.8796	−1.01580	−0.507899	+	0.861417i	$0.669578\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−1.24443	−0.0633399
$387$	1.05086	0.0534180
$388$	2.13335	0.108305
$389$	−0.161933	−0.00821034	−0.00410517	−	0.999992i	$-0.501307\pi$
$389$	−0.161933	−0.00821034	−0.00410517	+	0.999992i	$0.501307\pi$
$390$	0	0
$391$	6.42864	0.325110
$392$	6.61285	0.333999
$393$	0.755569	0.0381134
$394$	−15.2444	−0.768003
$395$	0	0
$396$	4.42864	0.222548
$397$	1.76494	0.0885796	0.0442898	−	0.999019i	$-0.485898\pi$
$397$	1.76494	0.0885796	0.0442898	+	0.999019i	$0.485898\pi$
$398$	14.5303	0.728341
$399$	0.857279	0.0429176
$400$	0	0
$401$	−31.3461	−1.56535	−0.782676	−	0.622430i	$-0.786146\pi$
$401$	−31.3461	−1.56535	−0.782676	+	0.622430i	$0.786146\pi$
$402$	11.8064	0.588851
$403$	5.51114	0.274529
$404$	16.2351	0.807725
$405$	0	0
$406$	−2.63512	−0.130779
$407$	−52.2864	−2.59174
$408$	6.42864	0.318265
$409$	−3.51114	−0.173615	−0.0868073	−	0.996225i	$-0.527666\pi$
$409$	−3.51114	−0.173615	−0.0868073	+	0.996225i	$0.527666\pi$
$410$	0	0
$411$	−12.9175	−0.637173
$412$	−12.2351	−0.602778
$413$	−5.98126	−0.294319
$414$	1.00000	0.0491473
$415$	0	0
$416$	1.37778	0.0675514
$417$	−14.1017	−0.690564
$418$	6.10171	0.298444
$419$	−19.7748	−0.966061	−0.483031	−	0.875603i	$-0.660464\pi$
$419$	−19.7748	−0.966061	−0.483031	+	0.875603i	$0.660464\pi$
$420$	0	0
$421$	11.8064	0.575410	0.287705	−	0.957719i	$-0.407108\pi$
$421$	11.8064	0.575410	0.287705	+	0.957719i	$0.407108\pi$
$422$	−0.266706	−0.0129831
$423$	11.6128	0.564636
$424$	3.37778	0.164040
$425$	0	0
$426$	6.99063	0.338697
$427$	−5.39069	−0.260874
$428$	10.6637	0.515450
$429$	−6.10171	−0.294593
$430$	0	0
$431$	24.9403	1.20133	0.600665	−	0.799501i	$-0.294903\pi$
$431$	24.9403	1.20133	0.600665	+	0.799501i	$0.294903\pi$
$432$	1.00000	0.0481125
$433$	−32.0513	−1.54029	−0.770144	−	0.637870i	$-0.779816\pi$
$433$	−32.0513	−1.54029	−0.770144	+	0.637870i	$0.779816\pi$
$434$	−2.48886	−0.119469
$435$	0	0
$436$	−15.1526	−0.725676
$437$	1.37778	0.0659084
$438$	2.75557	0.131666
$439$	0.470127	0.0224379	0.0112190	−	0.999937i	$-0.496429\pi$
$439$	0.470127	0.0224379	0.0112190	+	0.999937i	$0.496429\pi$
$440$	0	0
$441$	−6.61285	−0.314898
$442$	−8.85728	−0.421298
$443$	−0.653858	−0.0310658	−0.0155329	−	0.999879i	$-0.504944\pi$
$443$	−0.653858	−0.0310658	−0.0155329	+	0.999879i	$0.504944\pi$
$444$	−11.8064	−0.560308
$445$	0	0
$446$	−0.990632	−0.0469078
$447$	12.4286	0.587854
$448$	−0.622216	−0.0293969
$449$	17.1427	0.809015	0.404508	−	0.914535i	$-0.367443\pi$
$449$	17.1427	0.809015	0.404508	+	0.914535i	$0.367443\pi$
$450$	0	0
$451$	−48.0830	−2.26414
$452$	−14.4286	−0.678666
$453$	−4.85728	−0.228215
$454$	−17.1526	−0.805010
$455$	0	0
$456$	1.37778	0.0645207
$457$	1.00937	0.0472162	0.0236081	−	0.999721i	$-0.492485\pi$
$457$	1.00937	0.0472162	0.0236081	+	0.999721i	$0.492485\pi$
$458$	−7.15257	−0.334217
$459$	−6.42864	−0.300063
$460$	0	0
$461$	20.3180	0.946305	0.473153	−	0.880980i	$-0.343116\pi$
$461$	20.3180	0.946305	0.473153	+	0.880980i	$0.343116\pi$
$462$	2.75557	0.128201
$463$	12.4572	0.578936	0.289468	−	0.957188i	$-0.406522\pi$
$463$	12.4572	0.578936	0.289468	+	0.957188i	$0.406522\pi$
$464$	−4.23506	−0.196608
$465$	0	0
$466$	−3.71456	−0.172074
$467$	−15.7877	−0.730567	−0.365284	−	0.930896i	$-0.619028\pi$
$467$	−15.7877	−0.730567	−0.365284	+	0.930896i	$0.619028\pi$
$468$	−1.37778	−0.0636881
$469$	7.34614	0.339213
$470$	0	0
$471$	9.90813	0.456543
$472$	−9.61285	−0.442467
$473$	4.65386	0.213985
$474$	−12.0415	−0.553084
$475$	0	0
$476$	4.00000	0.183340
$477$	−3.37778	−0.154658
$478$	−27.8479	−1.27373
$479$	5.12399	0.234121	0.117060	−	0.993125i	$-0.462653\pi$
$479$	5.12399	0.234121	0.117060	+	0.993125i	$0.462653\pi$
$480$	0	0
$481$	16.2667	0.741698
$482$	7.12399	0.324489
$483$	0.622216	0.0283118
$484$	8.61285	0.391493
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−23.2128	−1.05187	−0.525936	−	0.850524i	$-0.676285\pi$
$487$	−23.2128	−1.05187	−0.525936	+	0.850524i	$0.676285\pi$
$488$	−8.66370	−0.392187
$489$	−10.1017	−0.456815
$490$	0	0
$491$	−40.1847	−1.81351	−0.906755	−	0.421658i	$-0.861448\pi$
$491$	−40.1847	−1.81351	−0.906755	+	0.421658i	$0.861448\pi$
$492$	−10.8573	−0.489484
$493$	27.2257	1.22618
$494$	−1.89829	−0.0854081
$495$	0	0
$496$	−4.00000	−0.179605
$497$	4.34968	0.195110
$498$	2.19358	0.0982965
$499$	30.5718	1.36858	0.684292	−	0.729208i	$-0.260112\pi$
$499$	30.5718	1.36858	0.684292	+	0.729208i	$0.260112\pi$
$500$	0	0
$501$	−1.89829	−0.0848093
$502$	22.4099	1.00020
$503$	−23.4291	−1.04465	−0.522326	−	0.852746i	$-0.674936\pi$
$503$	−23.4291	−1.04465	−0.522326	+	0.852746i	$0.674936\pi$
$504$	0.622216	0.0277157
$505$	0	0
$506$	4.42864	0.196877
$507$	−11.1017	−0.493044
$508$	4.99063	0.221423
$509$	−17.2128	−0.762943	−0.381472	−	0.924381i	$-0.624583\pi$
$509$	−17.2128	−0.762943	−0.381472	+	0.924381i	$0.624583\pi$
$510$	0	0
$511$	1.71456	0.0758476
$512$	−1.00000	−0.0441942
$513$	−1.37778	−0.0608307
$514$	11.7146	0.516707
$515$	0	0
$516$	1.05086	0.0462613
$517$	51.4291	2.26185
$518$	−7.34614	−0.322771
$519$	−21.2257	−0.931705
$520$	0	0
$521$	22.3684	0.979978	0.489989	−	0.871729i	$-0.337001\pi$
$521$	22.3684	0.979978	0.489989	+	0.871729i	$0.337001\pi$
$522$	4.23506	0.185364
$523$	30.2953	1.32472	0.662360	−	0.749186i	$-0.269555\pi$
$523$	30.2953	1.32472	0.662360	+	0.749186i	$0.269555\pi$
$524$	0.755569	0.0330072
$525$	0	0
$526$	8.47013	0.369315
$527$	25.7146	1.12014
$528$	4.42864	0.192732
$529$	1.00000	0.0434783
$530$	0	0
$531$	9.61285	0.417162
$532$	0.857279	0.0371678
$533$	14.9590	0.647946
$534$	−2.75557	−0.119245
$535$	0	0
$536$	11.8064	0.509960
$537$	0.488863	0.0210960
$538$	−28.8256	−1.24276
$539$	−29.2859	−1.26143
$540$	0	0
$541$	31.4479	1.35205	0.676024	−	0.736879i	$-0.263701\pi$
$541$	31.4479	1.35205	0.676024	+	0.736879i	$0.263701\pi$
$542$	13.8350	0.594264
$543$	−20.9304	−0.898210
$544$	6.42864	0.275626
$545$	0	0
$546$	−0.857279	−0.0366882
$547$	20.8573	0.891793	0.445896	−	0.895085i	$-0.352885\pi$
$547$	20.8573	0.891793	0.445896	+	0.895085i	$0.352885\pi$
$548$	−12.9175	−0.551808
$549$	8.66370	0.369758
$550$	0	0
$551$	5.83500	0.248580
$552$	1.00000	0.0425628
$553$	−7.49240	−0.318609
$554$	−2.88892	−0.122739
$555$	0	0
$556$	−14.1017	−0.598046
$557$	36.3180	1.53884	0.769422	−	0.638740i	$-0.220544\pi$
$557$	36.3180	1.53884	0.769422	+	0.638740i	$0.220544\pi$
$558$	4.00000	0.169334
$559$	−1.44785	−0.0612376
$560$	0	0
$561$	−28.4701	−1.20201
$562$	−5.51114	−0.232473
$563$	2.58073	0.108765	0.0543824	−	0.998520i	$-0.482681\pi$
$563$	2.58073	0.108765	0.0543824	+	0.998520i	$0.482681\pi$
$564$	11.6128	0.488989
$565$	0	0
$566$	−8.19358	−0.344402
$567$	−0.622216	−0.0261306
$568$	6.99063	0.293320
$569$	−36.3497	−1.52386	−0.761929	−	0.647661i	$-0.775747\pi$
$569$	−36.3497	−1.52386	−0.761929	+	0.647661i	$0.775747\pi$
$570$	0	0
$571$	−45.9309	−1.92215	−0.961074	−	0.276292i	$-0.910894\pi$
$571$	−45.9309	−1.92215	−0.961074	+	0.276292i	$0.910894\pi$
$572$	−6.10171	−0.255125
$573$	18.9590	0.792023
$574$	−6.75557	−0.281972
$575$	0	0
$576$	1.00000	0.0416667
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	−24.3274	−1.01189
$579$	1.24443	0.0517168
$580$	0	0
$581$	1.36488	0.0566247
$582$	−2.13335	−0.0884303
$583$	−14.9590	−0.619538
$584$	2.75557	0.114026
$585$	0	0
$586$	−12.6222	−0.521419
$587$	42.1847	1.74115	0.870574	−	0.492037i	$-0.163748\pi$
$587$	42.1847	1.74115	0.870574	+	0.492037i	$0.163748\pi$
$588$	−6.61285	−0.272709
$589$	5.51114	0.227082
$590$	0	0
$591$	15.2444	0.627072
$592$	−11.8064	−0.485241
$593$	18.7368	0.769430	0.384715	−	0.923036i	$-0.374300\pi$
$593$	18.7368	0.769430	0.384715	+	0.923036i	$0.374300\pi$
$594$	−4.42864	−0.181709
$595$	0	0
$596$	12.4286	0.509097
$597$	−14.5303	−0.594688
$598$	−1.37778	−0.0563418
$599$	−41.5625	−1.69820	−0.849098	−	0.528235i	$-0.822854\pi$
$599$	−41.5625	−1.69820	−0.849098	+	0.528235i	$0.822854\pi$
$600$	0	0
$601$	23.7146	0.967337	0.483668	−	0.875251i	$-0.339304\pi$
$601$	23.7146	0.967337	0.483668	+	0.875251i	$0.339304\pi$
$602$	0.653858	0.0266493
$603$	−11.8064	−0.480795
$604$	−4.85728	−0.197640
$605$	0	0
$606$	−16.2351	−0.659504
$607$	2.74266	0.111321	0.0556606	−	0.998450i	$-0.482274\pi$
$607$	2.74266	0.111321	0.0556606	+	0.998450i	$0.482274\pi$
$608$	1.37778	0.0558765
$609$	2.63512	0.106781
$610$	0	0
$611$	−16.0000	−0.647291
$612$	−6.42864	−0.259862
$613$	−24.0731	−0.972305	−0.486152	−	0.873874i	$-0.661600\pi$
$613$	−24.0731	−0.972305	−0.486152	+	0.873874i	$0.661600\pi$
$614$	0.653858	0.0263876
$615$	0	0
$616$	2.75557	0.111025
$617$	−39.4893	−1.58978	−0.794890	−	0.606753i	$-0.792472\pi$
$617$	−39.4893	−1.58978	−0.794890	+	0.606753i	$0.792472\pi$
$618$	12.2351	0.492166
$619$	−22.6222	−0.909264	−0.454632	−	0.890679i	$-0.650229\pi$
$619$	−22.6222	−0.909264	−0.454632	+	0.890679i	$0.650229\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	13.0923	0.524955
$623$	−1.71456	−0.0686923
$624$	−1.37778	−0.0551555
$625$	0	0
$626$	−11.2573	−0.449934
$627$	−6.10171	−0.243679
$628$	9.90813	0.395378
$629$	75.8992	3.02630
$630$	0	0
$631$	5.93978	0.236459	0.118229	−	0.992986i	$-0.462278\pi$
$631$	5.93978	0.236459	0.118229	+	0.992986i	$0.462278\pi$
$632$	−12.0415	−0.478985
$633$	0.266706	0.0106006
$634$	22.4701	0.892403
$635$	0	0
$636$	−3.37778	−0.133938
$637$	9.11108	0.360994
$638$	18.7556	0.742540
$639$	−6.99063	−0.276545
$640$	0	0
$641$	19.8163	0.782696	0.391348	−	0.920243i	$-0.372009\pi$
$641$	19.8163	0.782696	0.391348	+	0.920243i	$0.372009\pi$
$642$	−10.6637	−0.420863
$643$	44.7467	1.76464	0.882318	−	0.470653i	$-0.155982\pi$
$643$	44.7467	1.76464	0.882318	+	0.470653i	$0.155982\pi$
$644$	0.622216	0.0245187
$645$	0	0
$646$	−8.85728	−0.348485
$647$	32.9403	1.29501	0.647507	−	0.762059i	$-0.275811\pi$
$647$	32.9403	1.29501	0.647507	+	0.762059i	$0.275811\pi$
$648$	−1.00000	−0.0392837
$649$	42.5718	1.67109
$650$	0	0
$651$	2.48886	0.0975462
$652$	−10.1017	−0.395614
$653$	37.2257	1.45675	0.728377	−	0.685177i	$-0.240275\pi$
$653$	37.2257	1.45675	0.728377	+	0.685177i	$0.240275\pi$
$654$	15.1526	0.592512
$655$	0	0
$656$	−10.8573	−0.423906
$657$	−2.75557	−0.107505
$658$	7.22570	0.281687
$659$	24.6953	0.961994	0.480997	−	0.876722i	$-0.340275\pi$
$659$	24.6953	0.961994	0.480997	+	0.876722i	$0.340275\pi$
$660$	0	0
$661$	−14.8287	−0.576770	−0.288385	−	0.957515i	$-0.593118\pi$
$661$	−14.8287	−0.576770	−0.288385	+	0.957515i	$0.593118\pi$
$662$	29.4479	1.14452
$663$	8.85728	0.343988
$664$	2.19358	0.0851273
$665$	0	0
$666$	11.8064	0.457490
$667$	4.23506	0.163982
$668$	−1.89829	−0.0734470
$669$	0.990632	0.0383000
$670$	0	0
$671$	38.3684	1.48120
$672$	0.622216	0.0240025
$673$	8.77430	0.338225	0.169112	−	0.985597i	$-0.445910\pi$
$673$	8.77430	0.338225	0.169112	+	0.985597i	$0.445910\pi$
$674$	24.6222	0.948412
$675$	0	0
$676$	−11.1017	−0.426989
$677$	−12.8256	−0.492929	−0.246465	−	0.969152i	$-0.579269\pi$
$677$	−12.8256	−0.492929	−0.246465	+	0.969152i	$0.579269\pi$
$678$	14.4286	0.554129
$679$	−1.32741	−0.0509412
$680$	0	0
$681$	17.1526	0.657288
$682$	17.7146	0.678325
$683$	8.47013	0.324100	0.162050	−	0.986783i	$-0.448189\pi$
$683$	8.47013	0.324100	0.162050	+	0.986783i	$0.448189\pi$
$684$	−1.37778	−0.0526809
$685$	0	0
$686$	−8.47013	−0.323391
$687$	7.15257	0.272887
$688$	1.05086	0.0400635
$689$	4.65386	0.177298
$690$	0	0
$691$	25.7975	0.981384	0.490692	−	0.871333i	$-0.336744\pi$
$691$	25.7975	0.981384	0.490692	+	0.871333i	$0.336744\pi$
$692$	−21.2257	−0.806880
$693$	−2.75557	−0.104675
$694$	−26.1017	−0.990807
$695$	0	0
$696$	4.23506	0.160530
$697$	69.7975	2.64377
$698$	21.2257	0.803404
$699$	3.71456	0.140497
$700$	0	0
$701$	7.45091	0.281417	0.140709	−	0.990051i	$-0.455062\pi$
$701$	7.45091	0.281417	0.140709	+	0.990051i	$0.455062\pi$
$702$	1.37778	0.0520011
$703$	16.2667	0.613510
$704$	4.42864	0.166911
$705$	0	0
$706$	4.28544	0.161285
$707$	−10.1017	−0.379914
$708$	9.61285	0.361273
$709$	13.2543	0.497775	0.248887	−	0.968532i	$-0.419935\pi$
$709$	13.2543	0.497775	0.248887	+	0.968532i	$0.419935\pi$
$710$	0	0
$711$	12.0415	0.451591
$712$	−2.75557	−0.103269
$713$	4.00000	0.149801
$714$	−4.00000	−0.149696
$715$	0	0
$716$	0.488863	0.0182697
$717$	27.8479	1.04000
$718$	4.65386	0.173680
$719$	−2.33677	−0.0871469	−0.0435735	−	0.999050i	$-0.513874\pi$
$719$	−2.33677	−0.0871469	−0.0435735	+	0.999050i	$0.513874\pi$
$720$	0	0
$721$	7.61285	0.283517
$722$	17.1017	0.636460
$723$	−7.12399	−0.264944
$724$	−20.9304	−0.777873
$725$	0	0
$726$	−8.61285	−0.319653
$727$	−3.37778	−0.125275	−0.0626375	−	0.998036i	$-0.519951\pi$
$727$	−3.37778	−0.125275	−0.0626375	+	0.998036i	$0.519951\pi$
$728$	−0.857279	−0.0317729
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.75557	−0.249864
$732$	8.66370	0.320220
$733$	21.0509	0.777531	0.388766	−	0.921337i	$-0.372902\pi$
$733$	21.0509	0.777531	0.388766	+	0.921337i	$0.372902\pi$
$734$	−4.88892	−0.180453
$735$	0	0
$736$	1.00000	0.0368605
$737$	−52.2864	−1.92599
$738$	10.8573	0.399662
$739$	−22.6351	−0.832646	−0.416323	−	0.909217i	$-0.636682\pi$
$739$	−22.6351	−0.832646	−0.416323	+	0.909217i	$0.636682\pi$
$740$	0	0
$741$	1.89829	0.0697354
$742$	−2.10171	−0.0771562
$743$	−3.87955	−0.142327	−0.0711635	−	0.997465i	$-0.522671\pi$
$743$	−3.87955	−0.142327	−0.0711635	+	0.997465i	$0.522671\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	−23.4193	−0.857440
$747$	−2.19358	−0.0802588
$748$	−28.4701	−1.04097
$749$	−6.63512	−0.242442
$750$	0	0
$751$	19.7748	0.721592	0.360796	−	0.932645i	$-0.382505\pi$
$751$	19.7748	0.721592	0.360796	+	0.932645i	$0.382505\pi$
$752$	11.6128	0.423477
$753$	−22.4099	−0.816662
$754$	−5.83500	−0.212498
$755$	0	0
$756$	−0.622216	−0.0226298
$757$	8.19358	0.297801	0.148900	−	0.988852i	$-0.452427\pi$
$757$	8.19358	0.297801	0.148900	+	0.988852i	$0.452427\pi$
$758$	4.90766	0.178254
$759$	−4.42864	−0.160749
$760$	0	0
$761$	−45.1624	−1.63714	−0.818568	−	0.574410i	$-0.805232\pi$
$761$	−45.1624	−1.63714	−0.818568	+	0.574410i	$0.805232\pi$
$762$	−4.99063	−0.180792
$763$	9.42816	0.341322
$764$	18.9590	0.685912
$765$	0	0
$766$	19.8796	0.718277
$767$	−13.2444	−0.478229
$768$	1.00000	0.0360844
$769$	29.4924	1.06352	0.531762	−	0.846894i	$-0.321530\pi$
$769$	29.4924	1.06352	0.531762	+	0.846894i	$0.321530\pi$
$770$	0	0
$771$	−11.7146	−0.421889
$772$	1.24443	0.0447881
$773$	−41.0923	−1.47799	−0.738994	−	0.673712i	$-0.764699\pi$
$773$	−41.0923	−1.47799	−0.738994	+	0.673712i	$0.764699\pi$
$774$	−1.05086	−0.0377722
$775$	0	0
$776$	−2.13335	−0.0765829
$777$	7.34614	0.263541
$778$	0.161933	0.00580559
$779$	14.9590	0.535961
$780$	0	0
$781$	−30.9590	−1.10780
$782$	−6.42864	−0.229888
$783$	−4.23506	−0.151349
$784$	−6.61285	−0.236173
$785$	0	0
$786$	−0.755569	−0.0269502
$787$	10.0919	0.359736	0.179868	−	0.983691i	$-0.442433\pi$
$787$	10.0919	0.359736	0.179868	+	0.983691i	$0.442433\pi$
$788$	15.2444	0.543060
$789$	−8.47013	−0.301544
$790$	0	0
$791$	8.97773	0.319211
$792$	−4.42864	−0.157365
$793$	−11.9367	−0.423885
$794$	−1.76494	−0.0626353
$795$	0	0
$796$	−14.5303	−0.515015
$797$	−25.2958	−0.896022	−0.448011	−	0.894028i	$-0.647867\pi$
$797$	−25.2958	−0.896022	−0.448011	+	0.894028i	$0.647867\pi$
$798$	−0.857279	−0.0303473
$799$	−74.6548	−2.64110
$800$	0	0
$801$	2.75557	0.0973632
$802$	31.3461	1.10687
$803$	−12.2034	−0.430649
$804$	−11.8064	−0.416380
$805$	0	0
$806$	−5.51114	−0.194122
$807$	28.8256	1.01471
$808$	−16.2351	−0.571148
$809$	43.4479	1.52755	0.763773	−	0.645485i	$-0.223345\pi$
$809$	43.4479	1.52755	0.763773	+	0.645485i	$0.223345\pi$
$810$	0	0
$811$	−8.65386	−0.303878	−0.151939	−	0.988390i	$-0.548552\pi$
$811$	−8.65386	−0.303878	−0.151939	+	0.988390i	$0.548552\pi$
$812$	2.63512	0.0924747
$813$	−13.8350	−0.485215
$814$	52.2864	1.83264
$815$	0	0
$816$	−6.42864	−0.225047
$817$	−1.44785	−0.0506539
$818$	3.51114	0.122764
$819$	0.857279	0.0299558
$820$	0	0
$821$	2.78721	0.0972744	0.0486372	−	0.998817i	$-0.484512\pi$
$821$	2.78721	0.0972744	0.0486372	+	0.998817i	$0.484512\pi$
$822$	12.9175	0.450550
$823$	29.9684	1.04463	0.522316	−	0.852752i	$-0.325068\pi$
$823$	29.9684	1.04463	0.522316	+	0.852752i	$0.325068\pi$
$824$	12.2351	0.426229
$825$	0	0
$826$	5.98126	0.208115
$827$	47.2543	1.64319	0.821596	−	0.570070i	$-0.193084\pi$
$827$	47.2543	1.64319	0.821596	+	0.570070i	$0.193084\pi$
$828$	−1.00000	−0.0347524
$829$	−53.1624	−1.84641	−0.923203	−	0.384312i	$-0.874439\pi$
$829$	−53.1624	−1.84641	−0.923203	+	0.384312i	$0.874439\pi$
$830$	0	0
$831$	2.88892	0.100216
$832$	−1.37778	−0.0477661
$833$	42.5116	1.47294
$834$	14.1017	0.488302
$835$	0	0
$836$	−6.10171	−0.211032
$837$	−4.00000	−0.138260
$838$	19.7748	0.683108
$839$	−17.3274	−0.598208	−0.299104	−	0.954220i	$-0.596688\pi$
$839$	−17.3274	−0.598208	−0.299104	+	0.954220i	$0.596688\pi$
$840$	0	0
$841$	−11.0642	−0.381525
$842$	−11.8064	−0.406876
$843$	5.51114	0.189814
$844$	0.266706	0.00918041
$845$	0	0
$846$	−11.6128	−0.399258
$847$	−5.35905	−0.184139
$848$	−3.37778	−0.115994
$849$	8.19358	0.281203
$850$	0	0
$851$	11.8064	0.404719
$852$	−6.99063	−0.239495
$853$	17.7649	0.608260	0.304130	−	0.952631i	$-0.401634\pi$
$853$	17.7649	0.608260	0.304130	+	0.952631i	$0.401634\pi$
$854$	5.39069	0.184466
$855$	0	0
$856$	−10.6637	−0.364478
$857$	10.0830	0.344428	0.172214	−	0.985060i	$-0.444908\pi$
$857$	10.0830	0.344428	0.172214	+	0.985060i	$0.444908\pi$
$858$	6.10171	0.208309
$859$	5.12399	0.174828	0.0874141	−	0.996172i	$-0.472140\pi$
$859$	5.12399	0.174828	0.0874141	+	0.996172i	$0.472140\pi$
$860$	0	0
$861$	6.75557	0.230229
$862$	−24.9403	−0.849468
$863$	49.2070	1.67502	0.837512	−	0.546419i	$-0.184009\pi$
$863$	49.2070	1.67502	0.837512	+	0.546419i	$0.184009\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	32.0513	1.08915
$867$	24.3274	0.826202
$868$	2.48886	0.0844775
$869$	53.3274	1.80901
$870$	0	0
$871$	16.2667	0.551176
$872$	15.1526	0.513131
$873$	2.13335	0.0722031
$874$	−1.37778	−0.0466043
$875$	0	0
$876$	−2.75557	−0.0931020
$877$	−9.11108	−0.307659	−0.153830	−	0.988097i	$-0.549161\pi$
$877$	−9.11108	−0.307659	−0.153830	+	0.988097i	$0.549161\pi$
$878$	−0.470127	−0.0158660
$879$	12.6222	0.425737
$880$	0	0
$881$	−9.71456	−0.327292	−0.163646	−	0.986519i	$-0.552325\pi$
$881$	−9.71456	−0.327292	−0.163646	+	0.986519i	$0.552325\pi$
$882$	6.61285	0.222666
$883$	33.3274	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$883$	33.3274	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$884$	8.85728	0.297903
$885$	0	0
$886$	0.653858	0.0219668
$887$	53.5941	1.79951	0.899757	−	0.436391i	$-0.143744\pi$
$887$	53.5941	1.79951	0.899757	+	0.436391i	$0.143744\pi$
$888$	11.8064	0.396198
$889$	−3.10525	−0.104147
$890$	0	0
$891$	4.42864	0.148365
$892$	0.990632	0.0331688
$893$	−16.0000	−0.535420
$894$	−12.4286	−0.415676
$895$	0	0
$896$	0.622216	0.0207868
$897$	1.37778	0.0460029
$898$	−17.1427	−0.572060
$899$	16.9403	0.564989
$900$	0	0
$901$	21.7146	0.723417
$902$	48.0830	1.60099
$903$	−0.653858	−0.0217590
$904$	14.4286	0.479889
$905$	0	0
$906$	4.85728	0.161372
$907$	−2.56199	−0.0850696	−0.0425348	−	0.999095i	$-0.513543\pi$
$907$	−2.56199	−0.0850696	−0.0425348	+	0.999095i	$0.513543\pi$
$908$	17.1526	0.569228
$909$	16.2351	0.538483
$910$	0	0
$911$	8.47013	0.280628	0.140314	−	0.990107i	$-0.455189\pi$
$911$	8.47013	0.280628	0.140314	+	0.990107i	$0.455189\pi$
$912$	−1.37778	−0.0456230
$913$	−9.71456	−0.321505
$914$	−1.00937	−0.0333869
$915$	0	0
$916$	7.15257	0.236327
$917$	−0.470127	−0.0155250
$918$	6.42864	0.212177
$919$	−36.8988	−1.21718	−0.608589	−	0.793486i	$-0.708264\pi$
$919$	−36.8988	−1.21718	−0.608589	+	0.793486i	$0.708264\pi$
$920$	0	0
$921$	−0.653858	−0.0215454
$922$	−20.3180	−0.669139
$923$	9.63158	0.317027
$924$	−2.75557	−0.0906516
$925$	0	0
$926$	−12.4572	−0.409370
$927$	−12.2351	−0.401852
$928$	4.23506	0.139023
$929$	−0.164996	−0.00541334	−0.00270667	−	0.999996i	$-0.500862\pi$
$929$	−0.164996	−0.00541334	−0.00270667	+	0.999996i	$0.500862\pi$
$930$	0	0
$931$	9.11108	0.298604
$932$	3.71456	0.121674
$933$	−13.0923	−0.428624
$934$	15.7877	0.516589
$935$	0	0
$936$	1.37778	0.0450343
$937$	40.3180	1.31713	0.658566	−	0.752523i	$-0.271163\pi$
$937$	40.3180	1.31713	0.658566	+	0.752523i	$0.271163\pi$
$938$	−7.34614	−0.239860
$939$	11.2573	0.367369
$940$	0	0
$941$	53.3689	1.73978	0.869888	−	0.493249i	$-0.164191\pi$
$941$	53.3689	1.73978	0.869888	+	0.493249i	$0.164191\pi$
$942$	−9.90813	−0.322824
$943$	10.8573	0.353562
$944$	9.61285	0.312872
$945$	0	0
$946$	−4.65386	−0.151310
$947$	−44.6735	−1.45170	−0.725848	−	0.687856i	$-0.758552\pi$
$947$	−44.6735	−1.45170	−0.725848	+	0.687856i	$0.758552\pi$
$948$	12.0415	0.391089
$949$	3.79658	0.123242
$950$	0	0
$951$	−22.4701	−0.728644
$952$	−4.00000	−0.129641
$953$	21.2672	0.688912	0.344456	−	0.938803i	$-0.388063\pi$
$953$	21.2672	0.688912	0.344456	+	0.938803i	$0.388063\pi$
$954$	3.37778	0.109360
$955$	0	0
$956$	27.8479	0.900666
$957$	−18.7556	−0.606281
$958$	−5.12399	−0.165548
$959$	8.03747	0.259543
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−16.2667	−0.524460
$963$	10.6637	0.343633
$964$	−7.12399	−0.229448
$965$	0	0
$966$	−0.622216	−0.0200195
$967$	−10.9461	−0.352002	−0.176001	−	0.984390i	$-0.556316\pi$
$967$	−10.9461	−0.352002	−0.176001	+	0.984390i	$0.556316\pi$
$968$	−8.61285	−0.276827
$969$	8.85728	0.284537
$970$	0	0
$971$	−16.8988	−0.542307	−0.271154	−	0.962536i	$-0.587405\pi$
$971$	−16.8988	−0.542307	−0.271154	+	0.962536i	$0.587405\pi$
$972$	1.00000	0.0320750
$973$	8.77430	0.281291
$974$	23.2128	0.743786
$975$	0	0
$976$	8.66370	0.277318
$977$	−59.2484	−1.89553	−0.947763	−	0.318976i	$-0.896661\pi$
$977$	−59.2484	−1.89553	−0.947763	+	0.318976i	$0.896661\pi$
$978$	10.1017	0.323017
$979$	12.2034	0.390023
$980$	0	0
$981$	−15.1526	−0.483784
$982$	40.1847	1.28234
$983$	23.8163	0.759621	0.379810	−	0.925064i	$-0.375989\pi$
$983$	23.8163	0.759621	0.379810	+	0.925064i	$0.375989\pi$
$984$	10.8573	0.346117
$985$	0	0
$986$	−27.2257	−0.867043
$987$	−7.22570	−0.229996
$988$	1.89829	0.0603926
$989$	−1.05086	−0.0334152
$990$	0	0
$991$	29.7146	0.943914	0.471957	−	0.881622i	$-0.343548\pi$
$991$	29.7146	0.943914	0.471957	+	0.881622i	$0.343548\pi$
$992$	4.00000	0.127000
$993$	−29.4479	−0.934499
$994$	−4.34968	−0.137963
$995$	0	0
$996$	−2.19358	−0.0695061
$997$	−22.7052	−0.719081	−0.359540	−	0.933130i	$-0.617066\pi$
$997$	−22.7052	−0.719081	−0.359540	+	0.933130i	$0.617066\pi$
$998$	−30.5718	−0.967735
$999$	−11.8064	−0.373539

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.bo.1.2		3
5.2	odd	4		690.2.d.c.139.3	✓	6
5.3	odd	4		690.2.d.c.139.6	yes	6
5.4	even	2		3450.2.a.bt.1.2		3
15.2	even	4		2070.2.d.e.829.4		6
15.8	even	4		2070.2.d.e.829.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.d.c.139.3	✓	6	5.2	odd	4
690.2.d.c.139.6	yes	6	5.3	odd	4
2070.2.d.e.829.1		6	15.8	even	4
2070.2.d.e.829.4		6	15.2	even	4
3450.2.a.bo.1.2		3	1.1	even	1	trivial
3450.2.a.bt.1.2		3	5.4	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.622216 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.622216 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.42864 q^{11} +1.00000 q^{12} -1.37778 q^{13} +0.622216 q^{14} +1.00000 q^{16} -6.42864 q^{17} -1.00000 q^{18} -1.37778 q^{19} -0.622216 q^{21} -4.42864 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.37778 q^{26} +1.00000 q^{27} -0.622216 q^{28} -4.23506 q^{29} -4.00000 q^{31} -1.00000 q^{32} +4.42864 q^{33} +6.42864 q^{34} +1.00000 q^{36} -11.8064 q^{37} +1.37778 q^{38} -1.37778 q^{39} -10.8573 q^{41} +0.622216 q^{42} +1.05086 q^{43} +4.42864 q^{44} +1.00000 q^{46} +11.6128 q^{47} +1.00000 q^{48} -6.61285 q^{49} -6.42864 q^{51} -1.37778 q^{52} -3.37778 q^{53} -1.00000 q^{54} +0.622216 q^{56} -1.37778 q^{57} +4.23506 q^{58} +9.61285 q^{59} +8.66370 q^{61} +4.00000 q^{62} -0.622216 q^{63} +1.00000 q^{64} -4.42864 q^{66} -11.8064 q^{67} -6.42864 q^{68} -1.00000 q^{69} -6.99063 q^{71} -1.00000 q^{72} -2.75557 q^{73} +11.8064 q^{74} -1.37778 q^{76} -2.75557 q^{77} +1.37778 q^{78} +12.0415 q^{79} +1.00000 q^{81} +10.8573 q^{82} -2.19358 q^{83} -0.622216 q^{84} -1.05086 q^{86} -4.23506 q^{87} -4.42864 q^{88} +2.75557 q^{89} +0.857279 q^{91} -1.00000 q^{92} -4.00000 q^{93} -11.6128 q^{94} -1.00000 q^{96} +2.13335 q^{97} +6.61285 q^{98} +4.42864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{12} - 4 q^{13} + 2 q^{14} + 3 q^{16} - 6 q^{17} - 3 q^{18} - 4 q^{19} - 2 q^{21} - 3 q^{23} - 3 q^{24} + 4 q^{26} + 3 q^{27} - 2 q^{28} + 14 q^{29} - 12 q^{31} - 3 q^{32} + 6 q^{34} + 3 q^{36} - 22 q^{37} + 4 q^{38} - 4 q^{39} - 6 q^{41} + 2 q^{42} - 10 q^{43} + 3 q^{46} + 8 q^{47} + 3 q^{48} + 7 q^{49} - 6 q^{51} - 4 q^{52} - 10 q^{53} - 3 q^{54} + 2 q^{56} - 4 q^{57} - 14 q^{58} + 2 q^{59} - 14 q^{61} + 12 q^{62} - 2 q^{63} + 3 q^{64} - 22 q^{67} - 6 q^{68} - 3 q^{69} + 6 q^{71} - 3 q^{72} - 8 q^{73} + 22 q^{74} - 4 q^{76} - 8 q^{77} + 4 q^{78} - 4 q^{79} + 3 q^{81} + 6 q^{82} - 20 q^{83} - 2 q^{84} + 10 q^{86} + 14 q^{87} + 8 q^{89} - 24 q^{91} - 3 q^{92} - 12 q^{93} - 8 q^{94} - 3 q^{96} + 6 q^{97} - 7 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^12 - 4 * q^13 + 2 * q^14 + 3 * q^16 - 6 * q^17 - 3 * q^18 - 4 * q^19 - 2 * q^21 - 3 * q^23 - 3 * q^24 + 4 * q^26 + 3 * q^27 - 2 * q^28 + 14 * q^29 - 12 * q^31 - 3 * q^32 + 6 * q^34 + 3 * q^36 - 22 * q^37 + 4 * q^38 - 4 * q^39 - 6 * q^41 + 2 * q^42 - 10 * q^43 + 3 * q^46 + 8 * q^47 + 3 * q^48 + 7 * q^49 - 6 * q^51 - 4 * q^52 - 10 * q^53 - 3 * q^54 + 2 * q^56 - 4 * q^57 - 14 * q^58 + 2 * q^59 - 14 * q^61 + 12 * q^62 - 2 * q^63 + 3 * q^64 - 22 * q^67 - 6 * q^68 - 3 * q^69 + 6 * q^71 - 3 * q^72 - 8 * q^73 + 22 * q^74 - 4 * q^76 - 8 * q^77 + 4 * q^78 - 4 * q^79 + 3 * q^81 + 6 * q^82 - 20 * q^83 - 2 * q^84 + 10 * q^86 + 14 * q^87 + 8 * q^89 - 24 * q^91 - 3 * q^92 - 12 * q^93 - 8 * q^94 - 3 * q^96 + 6 * q^97 - 7 * q^98

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