LMFDB - Embedded newform 8470.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.61803 q^{6} -1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.61803 q^{6} -1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} -1.00000 q^{10} +1.61803 q^{12} -5.61803 q^{13} +1.00000 q^{14} +1.61803 q^{15} +1.00000 q^{16} -4.61803 q^{17} +0.381966 q^{18} +5.23607 q^{19} +1.00000 q^{20} -1.61803 q^{21} +8.47214 q^{23} -1.61803 q^{24} +1.00000 q^{25} +5.61803 q^{26} -5.47214 q^{27} -1.00000 q^{28} +5.85410 q^{29} -1.61803 q^{30} +5.23607 q^{31} -1.00000 q^{32} +4.61803 q^{34} -1.00000 q^{35} -0.381966 q^{36} -3.70820 q^{37} -5.23607 q^{38} -9.09017 q^{39} -1.00000 q^{40} -8.94427 q^{41} +1.61803 q^{42} -3.23607 q^{43} -0.381966 q^{45} -8.47214 q^{46} -13.0902 q^{47} +1.61803 q^{48} +1.00000 q^{49} -1.00000 q^{50} -7.47214 q^{51} -5.61803 q^{52} +6.47214 q^{53} +5.47214 q^{54} +1.00000 q^{56} +8.47214 q^{57} -5.85410 q^{58} +4.76393 q^{59} +1.61803 q^{60} +11.7082 q^{61} -5.23607 q^{62} +0.381966 q^{63} +1.00000 q^{64} -5.61803 q^{65} -1.23607 q^{67} -4.61803 q^{68} +13.7082 q^{69} +1.00000 q^{70} -7.32624 q^{71} +0.381966 q^{72} -4.32624 q^{73} +3.70820 q^{74} +1.61803 q^{75} +5.23607 q^{76} +9.09017 q^{78} +0.854102 q^{79} +1.00000 q^{80} -7.70820 q^{81} +8.94427 q^{82} -13.6180 q^{83} -1.61803 q^{84} -4.61803 q^{85} +3.23607 q^{86} +9.47214 q^{87} +0.291796 q^{89} +0.381966 q^{90} +5.61803 q^{91} +8.47214 q^{92} +8.47214 q^{93} +13.0902 q^{94} +5.23607 q^{95} -1.61803 q^{96} -3.38197 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 + 2 * q^5 - q^6 - 2 * q^7 - 2 * q^8 - 3 * q^9	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9} - 2 q^{10} + q^{12} - 9 q^{13} + 2 q^{14} + q^{15} + 2 q^{16} - 7 q^{17} + 3 q^{18} + 6 q^{19} + 2 q^{20} - q^{21} + 8 q^{23} - q^{24} + 2 q^{25} + 9 q^{26} - 2 q^{27} - 2 q^{28} + 5 q^{29} - q^{30} + 6 q^{31} - 2 q^{32} + 7 q^{34} - 2 q^{35} - 3 q^{36} + 6 q^{37} - 6 q^{38} - 7 q^{39} - 2 q^{40} + q^{42} - 2 q^{43} - 3 q^{45} - 8 q^{46} - 15 q^{47} + q^{48} + 2 q^{49} - 2 q^{50} - 6 q^{51} - 9 q^{52} + 4 q^{53} + 2 q^{54} + 2 q^{56} + 8 q^{57} - 5 q^{58} + 14 q^{59} + q^{60} + 10 q^{61} - 6 q^{62} + 3 q^{63} + 2 q^{64} - 9 q^{65} + 2 q^{67} - 7 q^{68} + 14 q^{69} + 2 q^{70} + q^{71} + 3 q^{72} + 7 q^{73} - 6 q^{74} + q^{75} + 6 q^{76} + 7 q^{78} - 5 q^{79} + 2 q^{80} - 2 q^{81} - 25 q^{83} - q^{84} - 7 q^{85} + 2 q^{86} + 10 q^{87} + 14 q^{89} + 3 q^{90} + 9 q^{91} + 8 q^{92} + 8 q^{93} + 15 q^{94} + 6 q^{95} - q^{96} - 9 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 + 2 * q^5 - q^6 - 2 * q^7 - 2 * q^8 - 3 * q^9 - 2 * q^10 + q^12 - 9 * q^13 + 2 * q^14 + q^15 + 2 * q^16 - 7 * q^17 + 3 * q^18 + 6 * q^19 + 2 * q^20 - q^21 + 8 * q^23 - q^24 + 2 * q^25 + 9 * q^26 - 2 * q^27 - 2 * q^28 + 5 * q^29 - q^30 + 6 * q^31 - 2 * q^32 + 7 * q^34 - 2 * q^35 - 3 * q^36 + 6 * q^37 - 6 * q^38 - 7 * q^39 - 2 * q^40 + q^42 - 2 * q^43 - 3 * q^45 - 8 * q^46 - 15 * q^47 + q^48 + 2 * q^49 - 2 * q^50 - 6 * q^51 - 9 * q^52 + 4 * q^53 + 2 * q^54 + 2 * q^56 + 8 * q^57 - 5 * q^58 + 14 * q^59 + q^60 + 10 * q^61 - 6 * q^62 + 3 * q^63 + 2 * q^64 - 9 * q^65 + 2 * q^67 - 7 * q^68 + 14 * q^69 + 2 * q^70 + q^71 + 3 * q^72 + 7 * q^73 - 6 * q^74 + q^75 + 6 * q^76 + 7 * q^78 - 5 * q^79 + 2 * q^80 - 2 * q^81 - 25 * q^83 - q^84 - 7 * q^85 + 2 * q^86 + 10 * q^87 + 14 * q^89 + 3 * q^90 + 9 * q^91 + 8 * q^92 + 8 * q^93 + 15 * q^94 + 6 * q^95 - q^96 - 9 * q^97 - 2 * 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
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.61803	−0.660560
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−0.381966	−0.127322
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	1.61803	0.467086
$13$	−5.61803	−1.55816	−0.779081	−	0.626923i	$-0.784314\pi$
$13$	−5.61803	−1.55816	−0.779081	+	0.626923i	$0.784314\pi$
$14$	1.00000	0.267261
$15$	1.61803	0.417775
$16$	1.00000	0.250000
$17$	−4.61803	−1.12004	−0.560019	−	0.828480i	$-0.689206\pi$
$17$	−4.61803	−1.12004	−0.560019	+	0.828480i	$0.689206\pi$
$18$	0.381966	0.0900303
$19$	5.23607	1.20124	0.600618	−	0.799536i	$-0.294921\pi$
$19$	5.23607	1.20124	0.600618	+	0.799536i	$0.294921\pi$
$20$	1.00000	0.223607
$21$	−1.61803	−0.353084
$22$	0	0
$23$	8.47214	1.76656	0.883281	−	0.468844i	$-0.155329\pi$
$23$	8.47214	1.76656	0.883281	+	0.468844i	$0.155329\pi$
$24$	−1.61803	−0.330280
$25$	1.00000	0.200000
$26$	5.61803	1.10179
$27$	−5.47214	−1.05311
$28$	−1.00000	−0.188982
$29$	5.85410	1.08708	0.543540	−	0.839383i	$-0.317084\pi$
$29$	5.85410	1.08708	0.543540	+	0.839383i	$0.317084\pi$
$30$	−1.61803	−0.295411
$31$	5.23607	0.940426	0.470213	−	0.882553i	$-0.344177\pi$
$31$	5.23607	0.940426	0.470213	+	0.882553i	$0.344177\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.61803	0.791986
$35$	−1.00000	−0.169031
$36$	−0.381966	−0.0636610
$37$	−3.70820	−0.609625	−0.304812	−	0.952412i	$-0.598594\pi$
$37$	−3.70820	−0.609625	−0.304812	+	0.952412i	$0.598594\pi$
$38$	−5.23607	−0.849402
$39$	−9.09017	−1.45559
$40$	−1.00000	−0.158114
$41$	−8.94427	−1.39686	−0.698430	−	0.715678i	$-0.746118\pi$
$41$	−8.94427	−1.39686	−0.698430	+	0.715678i	$0.746118\pi$
$42$	1.61803	0.249668
$43$	−3.23607	−0.493496	−0.246748	−	0.969080i	$-0.579362\pi$
$43$	−3.23607	−0.493496	−0.246748	+	0.969080i	$0.579362\pi$
$44$	0	0
$45$	−0.381966	−0.0569401
$46$	−8.47214	−1.24915
$47$	−13.0902	−1.90940	−0.954699	−	0.297574i	$-0.903823\pi$
$47$	−13.0902	−1.90940	−0.954699	+	0.297574i	$0.903823\pi$
$48$	1.61803	0.233543
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−7.47214	−1.04631
$52$	−5.61803	−0.779081
$53$	6.47214	0.889016	0.444508	−	0.895775i	$-0.353378\pi$
$53$	6.47214	0.889016	0.444508	+	0.895775i	$0.353378\pi$
$54$	5.47214	0.744663
$55$	0	0
$56$	1.00000	0.133631
$57$	8.47214	1.12216
$58$	−5.85410	−0.768681
$59$	4.76393	0.620211	0.310106	−	0.950702i	$-0.399636\pi$
$59$	4.76393	0.620211	0.310106	+	0.950702i	$0.399636\pi$
$60$	1.61803	0.208887
$61$	11.7082	1.49908	0.749541	−	0.661958i	$-0.230274\pi$
$61$	11.7082	1.49908	0.749541	+	0.661958i	$0.230274\pi$
$62$	−5.23607	−0.664981
$63$	0.381966	0.0481232
$64$	1.00000	0.125000
$65$	−5.61803	−0.696831
$66$	0	0
$67$	−1.23607	−0.151010	−0.0755049	−	0.997145i	$-0.524057\pi$
$67$	−1.23607	−0.151010	−0.0755049	+	0.997145i	$0.524057\pi$
$68$	−4.61803	−0.560019
$69$	13.7082	1.65027
$70$	1.00000	0.119523
$71$	−7.32624	−0.869464	−0.434732	−	0.900560i	$-0.643157\pi$
$71$	−7.32624	−0.869464	−0.434732	+	0.900560i	$0.643157\pi$
$72$	0.381966	0.0450151
$73$	−4.32624	−0.506348	−0.253174	−	0.967421i	$-0.581474\pi$
$73$	−4.32624	−0.506348	−0.253174	+	0.967421i	$0.581474\pi$
$74$	3.70820	0.431070
$75$	1.61803	0.186834
$76$	5.23607	0.600618
$77$	0	0
$78$	9.09017	1.02926
$79$	0.854102	0.0960940	0.0480470	−	0.998845i	$-0.484700\pi$
$79$	0.854102	0.0960940	0.0480470	+	0.998845i	$0.484700\pi$
$80$	1.00000	0.111803
$81$	−7.70820	−0.856467
$82$	8.94427	0.987730
$83$	−13.6180	−1.49477	−0.747387	−	0.664389i	$-0.768692\pi$
$83$	−13.6180	−1.49477	−0.747387	+	0.664389i	$0.768692\pi$
$84$	−1.61803	−0.176542
$85$	−4.61803	−0.500896
$86$	3.23607	0.348954
$87$	9.47214	1.01552
$88$	0	0
$89$	0.291796	0.0309303	0.0154652	−	0.999880i	$-0.495077\pi$
$89$	0.291796	0.0309303	0.0154652	+	0.999880i	$0.495077\pi$
$90$	0.381966	0.0402628
$91$	5.61803	0.588930
$92$	8.47214	0.883281
$93$	8.47214	0.878520
$94$	13.0902	1.35015
$95$	5.23607	0.537209
$96$	−1.61803	−0.165140
$97$	−3.38197	−0.343387	−0.171693	−	0.985150i	$-0.554924\pi$
$97$	−3.38197	−0.343387	−0.171693	+	0.985150i	$0.554924\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	5.23607	0.521008	0.260504	−	0.965473i	$-0.416111\pi$
$101$	5.23607	0.521008	0.260504	+	0.965473i	$0.416111\pi$
$102$	7.47214	0.739852
$103$	3.56231	0.351004	0.175502	−	0.984479i	$-0.443845\pi$
$103$	3.56231	0.351004	0.175502	+	0.984479i	$0.443845\pi$
$104$	5.61803	0.550894
$105$	−1.61803	−0.157904
$106$	−6.47214	−0.628629
$107$	10.7639	1.04059	0.520294	−	0.853987i	$-0.325822\pi$
$107$	10.7639	1.04059	0.520294	+	0.853987i	$0.325822\pi$
$108$	−5.47214	−0.526557
$109$	−3.52786	−0.337908	−0.168954	−	0.985624i	$-0.554039\pi$
$109$	−3.52786	−0.337908	−0.168954	+	0.985624i	$0.554039\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	−1.00000	−0.0944911
$113$	−1.52786	−0.143729	−0.0718647	−	0.997414i	$-0.522895\pi$
$113$	−1.52786	−0.143729	−0.0718647	+	0.997414i	$0.522895\pi$
$114$	−8.47214	−0.793488
$115$	8.47214	0.790031
$116$	5.85410	0.543540
$117$	2.14590	0.198388
$118$	−4.76393	−0.438555
$119$	4.61803	0.423334
$120$	−1.61803	−0.147706
$121$	0	0
$122$	−11.7082	−1.06001
$123$	−14.4721	−1.30491
$124$	5.23607	0.470213
$125$	1.00000	0.0894427
$126$	−0.381966	−0.0340282
$127$	−20.1803	−1.79072	−0.895358	−	0.445348i	$-0.853080\pi$
$127$	−20.1803	−1.79072	−0.895358	+	0.445348i	$0.853080\pi$
$128$	−1.00000	−0.0883883
$129$	−5.23607	−0.461010
$130$	5.61803	0.492734
$131$	11.4164	0.997456	0.498728	−	0.866758i	$-0.333801\pi$
$131$	11.4164	0.997456	0.498728	+	0.866758i	$0.333801\pi$
$132$	0	0
$133$	−5.23607	−0.454025
$134$	1.23607	0.106780
$135$	−5.47214	−0.470966
$136$	4.61803	0.395993
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	−13.7082	−1.16692
$139$	−8.94427	−0.758643	−0.379322	−	0.925265i	$-0.623843\pi$
$139$	−8.94427	−0.758643	−0.379322	+	0.925265i	$0.623843\pi$
$140$	−1.00000	−0.0845154
$141$	−21.1803	−1.78371
$142$	7.32624	0.614804
$143$	0	0
$144$	−0.381966	−0.0318305
$145$	5.85410	0.486157
$146$	4.32624	0.358042
$147$	1.61803	0.133453
$148$	−3.70820	−0.304812
$149$	−20.8541	−1.70843	−0.854217	−	0.519916i	$-0.825963\pi$
$149$	−20.8541	−1.70843	−0.854217	+	0.519916i	$0.825963\pi$
$150$	−1.61803	−0.132112
$151$	−21.3820	−1.74004	−0.870020	−	0.493017i	$-0.835894\pi$
$151$	−21.3820	−1.74004	−0.870020	+	0.493017i	$0.835894\pi$
$152$	−5.23607	−0.424701
$153$	1.76393	0.142605
$154$	0	0
$155$	5.23607	0.420571
$156$	−9.09017	−0.727796
$157$	−11.0902	−0.885092	−0.442546	−	0.896746i	$-0.645925\pi$
$157$	−11.0902	−0.885092	−0.442546	+	0.896746i	$0.645925\pi$
$158$	−0.854102	−0.0679487
$159$	10.4721	0.830494
$160$	−1.00000	−0.0790569
$161$	−8.47214	−0.667698
$162$	7.70820	0.605614
$163$	−14.6525	−1.14767	−0.573835	−	0.818971i	$-0.694545\pi$
$163$	−14.6525	−1.14767	−0.573835	+	0.818971i	$0.694545\pi$
$164$	−8.94427	−0.698430
$165$	0	0
$166$	13.6180	1.05696
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	1.61803	0.124834
$169$	18.5623	1.42787
$170$	4.61803	0.354187
$171$	−2.00000	−0.152944
$172$	−3.23607	−0.246748
$173$	15.5066	1.17894	0.589472	−	0.807789i	$-0.299336\pi$
$173$	15.5066	1.17894	0.589472	+	0.807789i	$0.299336\pi$
$174$	−9.47214	−0.718081
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	7.70820	0.579384
$178$	−0.291796	−0.0218710
$179$	−19.5623	−1.46216	−0.731078	−	0.682294i	$-0.760982\pi$
$179$	−19.5623	−1.46216	−0.731078	+	0.682294i	$0.760982\pi$
$180$	−0.381966	−0.0284701
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	−5.61803	−0.416436
$183$	18.9443	1.40040
$184$	−8.47214	−0.624574
$185$	−3.70820	−0.272633
$186$	−8.47214	−0.621207
$187$	0	0
$188$	−13.0902	−0.954699
$189$	5.47214	0.398039
$190$	−5.23607	−0.379864
$191$	0.854102	0.0618006	0.0309003	−	0.999522i	$-0.490163\pi$
$191$	0.854102	0.0618006	0.0309003	+	0.999522i	$0.490163\pi$
$192$	1.61803	0.116772
$193$	−12.9443	−0.931749	−0.465875	−	0.884851i	$-0.654260\pi$
$193$	−12.9443	−0.931749	−0.465875	+	0.884851i	$0.654260\pi$
$194$	3.38197	0.242811
$195$	−9.09017	−0.650961
$196$	1.00000	0.0714286
$197$	15.4164	1.09837	0.549187	−	0.835700i	$-0.314938\pi$
$197$	15.4164	1.09837	0.549187	+	0.835700i	$0.314938\pi$
$198$	0	0
$199$	−10.9443	−0.775819	−0.387909	−	0.921697i	$-0.626803\pi$
$199$	−10.9443	−0.775819	−0.387909	+	0.921697i	$0.626803\pi$
$200$	−1.00000	−0.0707107
$201$	−2.00000	−0.141069
$202$	−5.23607	−0.368408
$203$	−5.85410	−0.410877
$204$	−7.47214	−0.523154
$205$	−8.94427	−0.624695
$206$	−3.56231	−0.248198
$207$	−3.23607	−0.224922
$208$	−5.61803	−0.389541
$209$	0	0
$210$	1.61803	0.111655
$211$	−15.3262	−1.05510	−0.527551	−	0.849523i	$-0.676890\pi$
$211$	−15.3262	−1.05510	−0.527551	+	0.849523i	$0.676890\pi$
$212$	6.47214	0.444508
$213$	−11.8541	−0.812230
$214$	−10.7639	−0.735807
$215$	−3.23607	−0.220698
$216$	5.47214	0.372332
$217$	−5.23607	−0.355447
$218$	3.52786	0.238937
$219$	−7.00000	−0.473016
$220$	0	0
$221$	25.9443	1.74520
$222$	6.00000	0.402694
$223$	−18.4721	−1.23699	−0.618493	−	0.785790i	$-0.712256\pi$
$223$	−18.4721	−1.23699	−0.618493	+	0.785790i	$0.712256\pi$
$224$	1.00000	0.0668153
$225$	−0.381966	−0.0254644
$226$	1.52786	0.101632
$227$	−7.14590	−0.474290	−0.237145	−	0.971474i	$-0.576212\pi$
$227$	−7.14590	−0.474290	−0.237145	+	0.971474i	$0.576212\pi$
$228$	8.47214	0.561081
$229$	7.70820	0.509372	0.254686	−	0.967024i	$-0.418028\pi$
$229$	7.70820	0.509372	0.254686	+	0.967024i	$0.418028\pi$
$230$	−8.47214	−0.558636
$231$	0	0
$232$	−5.85410	−0.384341
$233$	16.6525	1.09094	0.545470	−	0.838130i	$-0.316351\pi$
$233$	16.6525	1.09094	0.545470	+	0.838130i	$0.316351\pi$
$234$	−2.14590	−0.140282
$235$	−13.0902	−0.853909
$236$	4.76393	0.310106
$237$	1.38197	0.0897683
$238$	−4.61803	−0.299343
$239$	−8.85410	−0.572724	−0.286362	−	0.958121i	$-0.592446\pi$
$239$	−8.85410	−0.572724	−0.286362	+	0.958121i	$0.592446\pi$
$240$	1.61803	0.104444
$241$	3.70820	0.238866	0.119433	−	0.992842i	$-0.461892\pi$
$241$	3.70820	0.238866	0.119433	+	0.992842i	$0.461892\pi$
$242$	0	0
$243$	3.94427	0.253025
$244$	11.7082	0.749541
$245$	1.00000	0.0638877
$246$	14.4721	0.922710
$247$	−29.4164	−1.87172
$248$	−5.23607	−0.332491
$249$	−22.0344	−1.39638
$250$	−1.00000	−0.0632456
$251$	0.763932	0.0482190	0.0241095	−	0.999709i	$-0.492325\pi$
$251$	0.763932	0.0482190	0.0241095	+	0.999709i	$0.492325\pi$
$252$	0.381966	0.0240616
$253$	0	0
$254$	20.1803	1.26623
$255$	−7.47214	−0.467923
$256$	1.00000	0.0625000
$257$	−13.2705	−0.827792	−0.413896	−	0.910324i	$-0.635832\pi$
$257$	−13.2705	−0.827792	−0.413896	+	0.910324i	$0.635832\pi$
$258$	5.23607	0.325983
$259$	3.70820	0.230417
$260$	−5.61803	−0.348416
$261$	−2.23607	−0.138409
$262$	−11.4164	−0.705308
$263$	−14.2918	−0.881270	−0.440635	−	0.897686i	$-0.645247\pi$
$263$	−14.2918	−0.881270	−0.440635	+	0.897686i	$0.645247\pi$
$264$	0	0
$265$	6.47214	0.397580
$266$	5.23607	0.321044
$267$	0.472136	0.0288943
$268$	−1.23607	−0.0755049
$269$	19.7082	1.20163	0.600815	−	0.799388i	$-0.294843\pi$
$269$	19.7082	1.20163	0.600815	+	0.799388i	$0.294843\pi$
$270$	5.47214	0.333024
$271$	−15.1246	−0.918755	−0.459377	−	0.888241i	$-0.651927\pi$
$271$	−15.1246	−0.918755	−0.459377	+	0.888241i	$0.651927\pi$
$272$	−4.61803	−0.280009
$273$	9.09017	0.550162
$274$	16.0000	0.966595
$275$	0	0
$276$	13.7082	0.825137
$277$	−13.4164	−0.806114	−0.403057	−	0.915175i	$-0.632052\pi$
$277$	−13.4164	−0.806114	−0.403057	+	0.915175i	$0.632052\pi$
$278$	8.94427	0.536442
$279$	−2.00000	−0.119737
$280$	1.00000	0.0597614
$281$	−18.3607	−1.09531	−0.547653	−	0.836705i	$-0.684479\pi$
$281$	−18.3607	−1.09531	−0.547653	+	0.836705i	$0.684479\pi$
$282$	21.1803	1.26127
$283$	14.3262	0.851606	0.425803	−	0.904816i	$-0.359992\pi$
$283$	14.3262	0.851606	0.425803	+	0.904816i	$0.359992\pi$
$284$	−7.32624	−0.434732
$285$	8.47214	0.501846
$286$	0	0
$287$	8.94427	0.527964
$288$	0.381966	0.0225076
$289$	4.32624	0.254485
$290$	−5.85410	−0.343765
$291$	−5.47214	−0.320782
$292$	−4.32624	−0.253174
$293$	25.4164	1.48484	0.742421	−	0.669933i	$-0.233677\pi$
$293$	25.4164	1.48484	0.742421	+	0.669933i	$0.233677\pi$
$294$	−1.61803	−0.0943657
$295$	4.76393	0.277367
$296$	3.70820	0.215535
$297$	0	0
$298$	20.8541	1.20805
$299$	−47.5967	−2.75259
$300$	1.61803	0.0934172
$301$	3.23607	0.186524
$302$	21.3820	1.23039
$303$	8.47214	0.486711
$304$	5.23607	0.300309
$305$	11.7082	0.670410
$306$	−1.76393	−0.100837
$307$	33.4508	1.90914	0.954570	−	0.297985i	$-0.0963147\pi$
$307$	33.4508	1.90914	0.954570	+	0.297985i	$0.0963147\pi$
$308$	0	0
$309$	5.76393	0.327899
$310$	−5.23607	−0.297389
$311$	10.1803	0.577274	0.288637	−	0.957439i	$-0.406798\pi$
$311$	10.1803	0.577274	0.288637	+	0.957439i	$0.406798\pi$
$312$	9.09017	0.514630
$313$	11.8885	0.671980	0.335990	−	0.941866i	$-0.390929\pi$
$313$	11.8885	0.671980	0.335990	+	0.941866i	$0.390929\pi$
$314$	11.0902	0.625854
$315$	0.381966	0.0215213
$316$	0.854102	0.0480470
$317$	17.8885	1.00472	0.502360	−	0.864658i	$-0.332465\pi$
$317$	17.8885	1.00472	0.502360	+	0.864658i	$0.332465\pi$
$318$	−10.4721	−0.587248
$319$	0	0
$320$	1.00000	0.0559017
$321$	17.4164	0.972089
$322$	8.47214	0.472134
$323$	−24.1803	−1.34543
$324$	−7.70820	−0.428234
$325$	−5.61803	−0.311632
$326$	14.6525	0.811526
$327$	−5.70820	−0.315664
$328$	8.94427	0.493865
$329$	13.0902	0.721684
$330$	0	0
$331$	−6.38197	−0.350785	−0.175392	−	0.984499i	$-0.556119\pi$
$331$	−6.38197	−0.350785	−0.175392	+	0.984499i	$0.556119\pi$
$332$	−13.6180	−0.747387
$333$	1.41641	0.0776187
$334$	20.0000	1.09435
$335$	−1.23607	−0.0675336
$336$	−1.61803	−0.0882710
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	−18.5623	−1.00966
$339$	−2.47214	−0.134268
$340$	−4.61803	−0.250448
$341$	0	0
$342$	2.00000	0.108148
$343$	−1.00000	−0.0539949
$344$	3.23607	0.174477
$345$	13.7082	0.738025
$346$	−15.5066	−0.833639
$347$	−25.8885	−1.38977	−0.694885	−	0.719121i	$-0.744545\pi$
$347$	−25.8885	−1.38977	−0.694885	+	0.719121i	$0.744545\pi$
$348$	9.47214	0.507760
$349$	26.9443	1.44229	0.721147	−	0.692782i	$-0.243615\pi$
$349$	26.9443	1.44229	0.721147	+	0.692782i	$0.243615\pi$
$350$	1.00000	0.0534522
$351$	30.7426	1.64092
$352$	0	0
$353$	−24.4508	−1.30139	−0.650694	−	0.759340i	$-0.725522\pi$
$353$	−24.4508	−1.30139	−0.650694	+	0.759340i	$0.725522\pi$
$354$	−7.70820	−0.409686
$355$	−7.32624	−0.388836
$356$	0.291796	0.0154652
$357$	7.47214	0.395467
$358$	19.5623	1.03390
$359$	−17.9787	−0.948880	−0.474440	−	0.880288i	$-0.657349\pi$
$359$	−17.9787	−0.948880	−0.474440	+	0.880288i	$0.657349\pi$
$360$	0.381966	0.0201314
$361$	8.41641	0.442969
$362$	0	0
$363$	0	0
$364$	5.61803	0.294465
$365$	−4.32624	−0.226446
$366$	−18.9443	−0.990233
$367$	35.3262	1.84401	0.922007	−	0.387172i	$-0.126548\pi$
$367$	35.3262	1.84401	0.922007	+	0.387172i	$0.126548\pi$
$368$	8.47214	0.441641
$369$	3.41641	0.177851
$370$	3.70820	0.192780
$371$	−6.47214	−0.336017
$372$	8.47214	0.439260
$373$	2.65248	0.137340	0.0686700	−	0.997639i	$-0.478124\pi$
$373$	2.65248	0.137340	0.0686700	+	0.997639i	$0.478124\pi$
$374$	0	0
$375$	1.61803	0.0835549
$376$	13.0902	0.675074
$377$	−32.8885	−1.69385
$378$	−5.47214	−0.281456
$379$	16.1459	0.829359	0.414680	−	0.909968i	$-0.363894\pi$
$379$	16.1459	0.829359	0.414680	+	0.909968i	$0.363894\pi$
$380$	5.23607	0.268605
$381$	−32.6525	−1.67284
$382$	−0.854102	−0.0436997
$383$	8.85410	0.452423	0.226212	−	0.974078i	$-0.427366\pi$
$383$	8.85410	0.452423	0.226212	+	0.974078i	$0.427366\pi$
$384$	−1.61803	−0.0825700
$385$	0	0
$386$	12.9443	0.658846
$387$	1.23607	0.0628329
$388$	−3.38197	−0.171693
$389$	−20.2705	−1.02776	−0.513878	−	0.857863i	$-0.671792\pi$
$389$	−20.2705	−1.02776	−0.513878	+	0.857863i	$0.671792\pi$
$390$	9.09017	0.460299
$391$	−39.1246	−1.97862
$392$	−1.00000	−0.0505076
$393$	18.4721	0.931796
$394$	−15.4164	−0.776667
$395$	0.854102	0.0429745
$396$	0	0
$397$	−30.5623	−1.53388	−0.766939	−	0.641720i	$-0.778221\pi$
$397$	−30.5623	−1.53388	−0.766939	+	0.641720i	$0.778221\pi$
$398$	10.9443	0.548587
$399$	−8.47214	−0.424137
$400$	1.00000	0.0500000
$401$	−36.6180	−1.82862	−0.914309	−	0.405018i	$-0.867265\pi$
$401$	−36.6180	−1.82862	−0.914309	+	0.405018i	$0.867265\pi$
$402$	2.00000	0.0997509
$403$	−29.4164	−1.46534
$404$	5.23607	0.260504
$405$	−7.70820	−0.383024
$406$	5.85410	0.290534
$407$	0	0
$408$	7.47214	0.369926
$409$	−5.23607	−0.258907	−0.129453	−	0.991586i	$-0.541322\pi$
$409$	−5.23607	−0.258907	−0.129453	+	0.991586i	$0.541322\pi$
$410$	8.94427	0.441726
$411$	−25.8885	−1.27699
$412$	3.56231	0.175502
$413$	−4.76393	−0.234418
$414$	3.23607	0.159044
$415$	−13.6180	−0.668483
$416$	5.61803	0.275447
$417$	−14.4721	−0.708704
$418$	0	0
$419$	6.94427	0.339250	0.169625	−	0.985509i	$-0.445744\pi$
$419$	6.94427	0.339250	0.169625	+	0.985509i	$0.445744\pi$
$420$	−1.61803	−0.0789520
$421$	−11.3262	−0.552007	−0.276004	−	0.961157i	$-0.589010\pi$
$421$	−11.3262	−0.552007	−0.276004	+	0.961157i	$0.589010\pi$
$422$	15.3262	0.746070
$423$	5.00000	0.243108
$424$	−6.47214	−0.314315
$425$	−4.61803	−0.224008
$426$	11.8541	0.574333
$427$	−11.7082	−0.566600
$428$	10.7639	0.520294
$429$	0	0
$430$	3.23607	0.156057
$431$	15.5623	0.749610	0.374805	−	0.927104i	$-0.377710\pi$
$431$	15.5623	0.749610	0.374805	+	0.927104i	$0.377710\pi$
$432$	−5.47214	−0.263278
$433$	4.09017	0.196561	0.0982805	−	0.995159i	$-0.468666\pi$
$433$	4.09017	0.196561	0.0982805	+	0.995159i	$0.468666\pi$
$434$	5.23607	0.251339
$435$	9.47214	0.454154
$436$	−3.52786	−0.168954
$437$	44.3607	2.12206
$438$	7.00000	0.334473
$439$	10.6525	0.508415	0.254207	−	0.967150i	$-0.418185\pi$
$439$	10.6525	0.508415	0.254207	+	0.967150i	$0.418185\pi$
$440$	0	0
$441$	−0.381966	−0.0181889
$442$	−25.9443	−1.23404
$443$	−15.2361	−0.723887	−0.361944	−	0.932200i	$-0.617887\pi$
$443$	−15.2361	−0.723887	−0.361944	+	0.932200i	$0.617887\pi$
$444$	−6.00000	−0.284747
$445$	0.291796	0.0138325
$446$	18.4721	0.874681
$447$	−33.7426	−1.59597
$448$	−1.00000	−0.0472456
$449$	−19.5066	−0.920572	−0.460286	−	0.887771i	$-0.652253\pi$
$449$	−19.5066	−0.920572	−0.460286	+	0.887771i	$0.652253\pi$
$450$	0.381966	0.0180061
$451$	0	0
$452$	−1.52786	−0.0718647
$453$	−34.5967	−1.62550
$454$	7.14590	0.335374
$455$	5.61803	0.263377
$456$	−8.47214	−0.396744
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	−7.70820	−0.360181
$459$	25.2705	1.17953
$460$	8.47214	0.395015
$461$	−25.1246	−1.17017	−0.585085	−	0.810972i	$-0.698939\pi$
$461$	−25.1246	−1.17017	−0.585085	+	0.810972i	$0.698939\pi$
$462$	0	0
$463$	−14.1803	−0.659016	−0.329508	−	0.944153i	$-0.606883\pi$
$463$	−14.1803	−0.659016	−0.329508	+	0.944153i	$0.606883\pi$
$464$	5.85410	0.271770
$465$	8.47214	0.392886
$466$	−16.6525	−0.771411
$467$	21.6180	1.00036	0.500182	−	0.865920i	$-0.333267\pi$
$467$	21.6180	1.00036	0.500182	+	0.865920i	$0.333267\pi$
$468$	2.14590	0.0991942
$469$	1.23607	0.0570763
$470$	13.0902	0.603805
$471$	−17.9443	−0.826828
$472$	−4.76393	−0.219278
$473$	0	0
$474$	−1.38197	−0.0634758
$475$	5.23607	0.240247
$476$	4.61803	0.211667
$477$	−2.47214	−0.113191
$478$	8.85410	0.404977
$479$	35.1246	1.60488	0.802442	−	0.596730i	$-0.203534\pi$
$479$	35.1246	1.60488	0.802442	+	0.596730i	$0.203534\pi$
$480$	−1.61803	−0.0738528
$481$	20.8328	0.949895
$482$	−3.70820	−0.168904
$483$	−13.7082	−0.623745
$484$	0	0
$485$	−3.38197	−0.153567
$486$	−3.94427	−0.178916
$487$	11.8197	0.535600	0.267800	−	0.963475i	$-0.413703\pi$
$487$	11.8197	0.535600	0.267800	+	0.963475i	$0.413703\pi$
$488$	−11.7082	−0.530005
$489$	−23.7082	−1.07212
$490$	−1.00000	−0.0451754
$491$	−0.944272	−0.0426144	−0.0213072	−	0.999773i	$-0.506783\pi$
$491$	−0.944272	−0.0426144	−0.0213072	+	0.999773i	$0.506783\pi$
$492$	−14.4721	−0.652454
$493$	−27.0344	−1.21757
$494$	29.4164	1.32351
$495$	0	0
$496$	5.23607	0.235106
$497$	7.32624	0.328627
$498$	22.0344	0.987387
$499$	−24.7984	−1.11013	−0.555064	−	0.831808i	$-0.687306\pi$
$499$	−24.7984	−1.11013	−0.555064	+	0.831808i	$0.687306\pi$
$500$	1.00000	0.0447214
$501$	−32.3607	−1.44577
$502$	−0.763932	−0.0340960
$503$	−25.9787	−1.15833	−0.579167	−	0.815209i	$-0.696622\pi$
$503$	−25.9787	−1.15833	−0.579167	+	0.815209i	$0.696622\pi$
$504$	−0.381966	−0.0170141
$505$	5.23607	0.233002
$506$	0	0
$507$	30.0344	1.33388
$508$	−20.1803	−0.895358
$509$	−19.8885	−0.881544	−0.440772	−	0.897619i	$-0.645295\pi$
$509$	−19.8885	−0.881544	−0.440772	+	0.897619i	$0.645295\pi$
$510$	7.47214	0.330872
$511$	4.32624	0.191381
$512$	−1.00000	−0.0441942
$513$	−28.6525	−1.26504
$514$	13.2705	0.585337
$515$	3.56231	0.156974
$516$	−5.23607	−0.230505
$517$	0	0
$518$	−3.70820	−0.162929
$519$	25.0902	1.10134
$520$	5.61803	0.246367
$521$	−32.5410	−1.42565	−0.712824	−	0.701343i	$-0.752584\pi$
$521$	−32.5410	−1.42565	−0.712824	+	0.701343i	$0.752584\pi$
$522$	2.23607	0.0978700
$523$	9.74265	0.426016	0.213008	−	0.977050i	$-0.431674\pi$
$523$	9.74265	0.426016	0.213008	+	0.977050i	$0.431674\pi$
$524$	11.4164	0.498728
$525$	−1.61803	−0.0706168
$526$	14.2918	0.623152
$527$	−24.1803	−1.05331
$528$	0	0
$529$	48.7771	2.12074
$530$	−6.47214	−0.281132
$531$	−1.81966	−0.0789665
$532$	−5.23607	−0.227012
$533$	50.2492	2.17654
$534$	−0.472136	−0.0204313
$535$	10.7639	0.465365
$536$	1.23607	0.0533900
$537$	−31.6525	−1.36591
$538$	−19.7082	−0.849681
$539$	0	0
$540$	−5.47214	−0.235483
$541$	8.09017	0.347824	0.173912	−	0.984761i	$-0.444359\pi$
$541$	8.09017	0.347824	0.173912	+	0.984761i	$0.444359\pi$
$542$	15.1246	0.649658
$543$	0	0
$544$	4.61803	0.197997
$545$	−3.52786	−0.151117
$546$	−9.09017	−0.389023
$547$	22.1803	0.948363	0.474181	−	0.880427i	$-0.342744\pi$
$547$	22.1803	0.948363	0.474181	+	0.880427i	$0.342744\pi$
$548$	−16.0000	−0.683486
$549$	−4.47214	−0.190866
$550$	0	0
$551$	30.6525	1.30584
$552$	−13.7082	−0.583460
$553$	−0.854102	−0.0363201
$554$	13.4164	0.570009
$555$	−6.00000	−0.254686
$556$	−8.94427	−0.379322
$557$	45.5967	1.93200	0.965998	−	0.258549i	$-0.0832444\pi$
$557$	45.5967	1.93200	0.965998	+	0.258549i	$0.0832444\pi$
$558$	2.00000	0.0846668
$559$	18.1803	0.768946
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	18.3607	0.774499
$563$	23.5623	0.993033	0.496516	−	0.868027i	$-0.334612\pi$
$563$	23.5623	0.993033	0.496516	+	0.868027i	$0.334612\pi$
$564$	−21.1803	−0.891853
$565$	−1.52786	−0.0642777
$566$	−14.3262	−0.602177
$567$	7.70820	0.323714
$568$	7.32624	0.307402
$569$	−18.7984	−0.788069	−0.394034	−	0.919096i	$-0.628921\pi$
$569$	−18.7984	−0.788069	−0.394034	+	0.919096i	$0.628921\pi$
$570$	−8.47214	−0.354859
$571$	43.8541	1.83524	0.917619	−	0.397462i	$-0.130109\pi$
$571$	43.8541	1.83524	0.917619	+	0.397462i	$0.130109\pi$
$572$	0	0
$573$	1.38197	0.0577325
$574$	−8.94427	−0.373327
$575$	8.47214	0.353312
$576$	−0.381966	−0.0159153
$577$	42.2148	1.75742	0.878712	−	0.477352i	$-0.158403\pi$
$577$	42.2148	1.75742	0.878712	+	0.477352i	$0.158403\pi$
$578$	−4.32624	−0.179948
$579$	−20.9443	−0.870414
$580$	5.85410	0.243078
$581$	13.6180	0.564971
$582$	5.47214	0.226827
$583$	0	0
$584$	4.32624	0.179021
$585$	2.14590	0.0887220
$586$	−25.4164	−1.04994
$587$	19.5066	0.805123	0.402561	−	0.915393i	$-0.368120\pi$
$587$	19.5066	0.805123	0.402561	+	0.915393i	$0.368120\pi$
$588$	1.61803	0.0667266
$589$	27.4164	1.12967
$590$	−4.76393	−0.196128
$591$	24.9443	1.02607
$592$	−3.70820	−0.152406
$593$	−14.7984	−0.607696	−0.303848	−	0.952720i	$-0.598272\pi$
$593$	−14.7984	−0.607696	−0.303848	+	0.952720i	$0.598272\pi$
$594$	0	0
$595$	4.61803	0.189321
$596$	−20.8541	−0.854217
$597$	−17.7082	−0.724749
$598$	47.5967	1.94638
$599$	38.3951	1.56878	0.784391	−	0.620267i	$-0.212976\pi$
$599$	38.3951	1.56878	0.784391	+	0.620267i	$0.212976\pi$
$600$	−1.61803	−0.0660560
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	−3.23607	−0.131892
$603$	0.472136	0.0192269
$604$	−21.3820	−0.870020
$605$	0	0
$606$	−8.47214	−0.344157
$607$	38.6869	1.57025	0.785127	−	0.619335i	$-0.212598\pi$
$607$	38.6869	1.57025	0.785127	+	0.619335i	$0.212598\pi$
$608$	−5.23607	−0.212351
$609$	−9.47214	−0.383830
$610$	−11.7082	−0.474051
$611$	73.5410	2.97515
$612$	1.76393	0.0713027
$613$	22.9443	0.926710	0.463355	−	0.886173i	$-0.346645\pi$
$613$	22.9443	0.926710	0.463355	+	0.886173i	$0.346645\pi$
$614$	−33.4508	−1.34997
$615$	−14.4721	−0.583573
$616$	0	0
$617$	−3.81966	−0.153774	−0.0768869	−	0.997040i	$-0.524498\pi$
$617$	−3.81966	−0.153774	−0.0768869	+	0.997040i	$0.524498\pi$
$618$	−5.76393	−0.231859
$619$	0.583592	0.0234565	0.0117283	−	0.999931i	$-0.496267\pi$
$619$	0.583592	0.0234565	0.0117283	+	0.999931i	$0.496267\pi$
$620$	5.23607	0.210286
$621$	−46.3607	−1.86039
$622$	−10.1803	−0.408194
$623$	−0.291796	−0.0116906
$624$	−9.09017	−0.363898
$625$	1.00000	0.0400000
$626$	−11.8885	−0.475162
$627$	0	0
$628$	−11.0902	−0.442546
$629$	17.1246	0.682803
$630$	−0.381966	−0.0152179
$631$	9.45085	0.376232	0.188116	−	0.982147i	$-0.439762\pi$
$631$	9.45085	0.376232	0.188116	+	0.982147i	$0.439762\pi$
$632$	−0.854102	−0.0339744
$633$	−24.7984	−0.985647
$634$	−17.8885	−0.710445
$635$	−20.1803	−0.800832
$636$	10.4721	0.415247
$637$	−5.61803	−0.222595
$638$	0	0
$639$	2.79837	0.110702
$640$	−1.00000	−0.0395285
$641$	9.79837	0.387012	0.193506	−	0.981099i	$-0.438014\pi$
$641$	9.79837	0.387012	0.193506	+	0.981099i	$0.438014\pi$
$642$	−17.4164	−0.687371
$643$	−15.1459	−0.597296	−0.298648	−	0.954363i	$-0.596536\pi$
$643$	−15.1459	−0.597296	−0.298648	+	0.954363i	$0.596536\pi$
$644$	−8.47214	−0.333849
$645$	−5.23607	−0.206170
$646$	24.1803	0.951363
$647$	−41.7426	−1.64107	−0.820536	−	0.571594i	$-0.806325\pi$
$647$	−41.7426	−1.64107	−0.820536	+	0.571594i	$0.806325\pi$
$648$	7.70820	0.302807
$649$	0	0
$650$	5.61803	0.220357
$651$	−8.47214	−0.332049
$652$	−14.6525	−0.573835
$653$	−14.2918	−0.559281	−0.279641	−	0.960105i	$-0.590215\pi$
$653$	−14.2918	−0.559281	−0.279641	+	0.960105i	$0.590215\pi$
$654$	5.70820	0.223208
$655$	11.4164	0.446076
$656$	−8.94427	−0.349215
$657$	1.65248	0.0644692
$658$	−13.0902	−0.510308
$659$	−5.25735	−0.204797	−0.102399	−	0.994743i	$-0.532652\pi$
$659$	−5.25735	−0.204797	−0.102399	+	0.994743i	$0.532652\pi$
$660$	0	0
$661$	−18.9443	−0.736847	−0.368423	−	0.929658i	$-0.620102\pi$
$661$	−18.9443	−0.736847	−0.368423	+	0.929658i	$0.620102\pi$
$662$	6.38197	0.248042
$663$	41.9787	1.63032
$664$	13.6180	0.528482
$665$	−5.23607	−0.203046
$666$	−1.41641	−0.0548847
$667$	49.5967	1.92039
$668$	−20.0000	−0.773823
$669$	−29.8885	−1.15556
$670$	1.23607	0.0477535
$671$	0	0
$672$	1.61803	0.0624170
$673$	4.58359	0.176684	0.0883422	−	0.996090i	$-0.471843\pi$
$673$	4.58359	0.176684	0.0883422	+	0.996090i	$0.471843\pi$
$674$	8.00000	0.308148
$675$	−5.47214	−0.210623
$676$	18.5623	0.713935
$677$	13.9230	0.535104	0.267552	−	0.963543i	$-0.413785\pi$
$677$	13.9230	0.535104	0.267552	+	0.963543i	$0.413785\pi$
$678$	2.47214	0.0949418
$679$	3.38197	0.129788
$680$	4.61803	0.177094
$681$	−11.5623	−0.443069
$682$	0	0
$683$	−43.5967	−1.66818	−0.834092	−	0.551626i	$-0.814008\pi$
$683$	−43.5967	−1.66818	−0.834092	+	0.551626i	$0.814008\pi$
$684$	−2.00000	−0.0764719
$685$	−16.0000	−0.611329
$686$	1.00000	0.0381802
$687$	12.4721	0.475842
$688$	−3.23607	−0.123374
$689$	−36.3607	−1.38523
$690$	−13.7082	−0.521862
$691$	2.47214	0.0940445	0.0470222	−	0.998894i	$-0.485027\pi$
$691$	2.47214	0.0940445	0.0470222	+	0.998894i	$0.485027\pi$
$692$	15.5066	0.589472
$693$	0	0
$694$	25.8885	0.982716
$695$	−8.94427	−0.339276
$696$	−9.47214	−0.359040
$697$	41.3050	1.56454
$698$	−26.9443	−1.01986
$699$	26.9443	1.01913
$700$	−1.00000	−0.0377964
$701$	−39.5279	−1.49295	−0.746473	−	0.665415i	$-0.768255\pi$
$701$	−39.5279	−1.49295	−0.746473	+	0.665415i	$0.768255\pi$
$702$	−30.7426	−1.16031
$703$	−19.4164	−0.732304
$704$	0	0
$705$	−21.1803	−0.797698
$706$	24.4508	0.920220
$707$	−5.23607	−0.196923
$708$	7.70820	0.289692
$709$	6.96556	0.261597	0.130799	−	0.991409i	$-0.458246\pi$
$709$	6.96556	0.261597	0.130799	+	0.991409i	$0.458246\pi$
$710$	7.32624	0.274949
$711$	−0.326238	−0.0122349
$712$	−0.291796	−0.0109355
$713$	44.3607	1.66132
$714$	−7.47214	−0.279638
$715$	0	0
$716$	−19.5623	−0.731078
$717$	−14.3262	−0.535023
$718$	17.9787	0.670960
$719$	−32.3607	−1.20685	−0.603425	−	0.797420i	$-0.706198\pi$
$719$	−32.3607	−1.20685	−0.603425	+	0.797420i	$0.706198\pi$
$720$	−0.381966	−0.0142350
$721$	−3.56231	−0.132667
$722$	−8.41641	−0.313226
$723$	6.00000	0.223142
$724$	0	0
$725$	5.85410	0.217416
$726$	0	0
$727$	39.3262	1.45853	0.729265	−	0.684232i	$-0.239862\pi$
$727$	39.3262	1.45853	0.729265	+	0.684232i	$0.239862\pi$
$728$	−5.61803	−0.208218
$729$	29.5066	1.09284
$730$	4.32624	0.160121
$731$	14.9443	0.552734
$732$	18.9443	0.700200
$733$	−5.25735	−0.194185	−0.0970924	−	0.995275i	$-0.530954\pi$
$733$	−5.25735	−0.194185	−0.0970924	+	0.995275i	$0.530954\pi$
$734$	−35.3262	−1.30392
$735$	1.61803	0.0596821
$736$	−8.47214	−0.312287
$737$	0	0
$738$	−3.41641	−0.125760
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−3.70820	−0.136316
$741$	−47.5967	−1.74851
$742$	6.47214	0.237600
$743$	19.1246	0.701614	0.350807	−	0.936448i	$-0.385907\pi$
$743$	19.1246	0.701614	0.350807	+	0.936448i	$0.385907\pi$
$744$	−8.47214	−0.310604
$745$	−20.8541	−0.764035
$746$	−2.65248	−0.0971140
$747$	5.20163	0.190318
$748$	0	0
$749$	−10.7639	−0.393306
$750$	−1.61803	−0.0590822
$751$	−15.2705	−0.557229	−0.278614	−	0.960403i	$-0.589875\pi$
$751$	−15.2705	−0.557229	−0.278614	+	0.960403i	$0.589875\pi$
$752$	−13.0902	−0.477349
$753$	1.23607	0.0450448
$754$	32.8885	1.19773
$755$	−21.3820	−0.778169
$756$	5.47214	0.199020
$757$	−12.6525	−0.459862	−0.229931	−	0.973207i	$-0.573850\pi$
$757$	−12.6525	−0.459862	−0.229931	+	0.973207i	$0.573850\pi$
$758$	−16.1459	−0.586445
$759$	0	0
$760$	−5.23607	−0.189932
$761$	−18.5836	−0.673655	−0.336827	−	0.941566i	$-0.609354\pi$
$761$	−18.5836	−0.673655	−0.336827	+	0.941566i	$0.609354\pi$
$762$	32.6525	1.18287
$763$	3.52786	0.127717
$764$	0.854102	0.0309003
$765$	1.76393	0.0637751
$766$	−8.85410	−0.319912
$767$	−26.7639	−0.966390
$768$	1.61803	0.0583858
$769$	21.7082	0.782818	0.391409	−	0.920217i	$-0.371988\pi$
$769$	21.7082	0.782818	0.391409	+	0.920217i	$0.371988\pi$
$770$	0	0
$771$	−21.4721	−0.773300
$772$	−12.9443	−0.465875
$773$	39.5066	1.42095	0.710476	−	0.703721i	$-0.248479\pi$
$773$	39.5066	1.42095	0.710476	+	0.703721i	$0.248479\pi$
$774$	−1.23607	−0.0444295
$775$	5.23607	0.188085
$776$	3.38197	0.121406
$777$	6.00000	0.215249
$778$	20.2705	0.726733
$779$	−46.8328	−1.67796
$780$	−9.09017	−0.325480
$781$	0	0
$782$	39.1246	1.39909
$783$	−32.0344	−1.14482
$784$	1.00000	0.0357143
$785$	−11.0902	−0.395825
$786$	−18.4721	−0.658879
$787$	15.9098	0.567124	0.283562	−	0.958954i	$-0.408484\pi$
$787$	15.9098	0.567124	0.283562	+	0.958954i	$0.408484\pi$
$788$	15.4164	0.549187
$789$	−23.1246	−0.823258
$790$	−0.854102	−0.0303876
$791$	1.52786	0.0543246
$792$	0	0
$793$	−65.7771	−2.33581
$794$	30.5623	1.08462
$795$	10.4721	0.371408
$796$	−10.9443	−0.387909
$797$	−32.6738	−1.15736	−0.578682	−	0.815553i	$-0.696433\pi$
$797$	−32.6738	−1.15736	−0.578682	+	0.815553i	$0.696433\pi$
$798$	8.47214	0.299910
$799$	60.4508	2.13860
$800$	−1.00000	−0.0353553
$801$	−0.111456	−0.00393811
$802$	36.6180	1.29303
$803$	0	0
$804$	−2.00000	−0.0705346
$805$	−8.47214	−0.298604
$806$	29.4164	1.03615
$807$	31.8885	1.12253
$808$	−5.23607	−0.184204
$809$	27.6869	0.973420	0.486710	−	0.873564i	$-0.338197\pi$
$809$	27.6869	0.973420	0.486710	+	0.873564i	$0.338197\pi$
$810$	7.70820	0.270839
$811$	18.1115	0.635979	0.317990	−	0.948094i	$-0.396992\pi$
$811$	18.1115	0.635979	0.317990	+	0.948094i	$0.396992\pi$
$812$	−5.85410	−0.205439
$813$	−24.4721	−0.858275
$814$	0	0
$815$	−14.6525	−0.513254
$816$	−7.47214	−0.261577
$817$	−16.9443	−0.592805
$818$	5.23607	0.183075
$819$	−2.14590	−0.0749837
$820$	−8.94427	−0.312348
$821$	−14.2148	−0.496099	−0.248050	−	0.968747i	$-0.579790\pi$
$821$	−14.2148	−0.496099	−0.248050	+	0.968747i	$0.579790\pi$
$822$	25.8885	0.902967
$823$	−40.3607	−1.40688	−0.703442	−	0.710752i	$-0.748355\pi$
$823$	−40.3607	−1.40688	−0.703442	+	0.710752i	$0.748355\pi$
$824$	−3.56231	−0.124099
$825$	0	0
$826$	4.76393	0.165758
$827$	32.9443	1.14558	0.572792	−	0.819701i	$-0.305860\pi$
$827$	32.9443	1.14558	0.572792	+	0.819701i	$0.305860\pi$
$828$	−3.23607	−0.112461
$829$	29.8885	1.03807	0.519036	−	0.854752i	$-0.326291\pi$
$829$	29.8885	1.03807	0.519036	+	0.854752i	$0.326291\pi$
$830$	13.6180	0.472689
$831$	−21.7082	−0.753049
$832$	−5.61803	−0.194770
$833$	−4.61803	−0.160005
$834$	14.4721	0.501129
$835$	−20.0000	−0.692129
$836$	0	0
$837$	−28.6525	−0.990374
$838$	−6.94427	−0.239886
$839$	−0.360680	−0.0124520	−0.00622602	−	0.999981i	$-0.501982\pi$
$839$	−0.360680	−0.0124520	−0.00622602	+	0.999981i	$0.501982\pi$
$840$	1.61803	0.0558275
$841$	5.27051	0.181742
$842$	11.3262	0.390328
$843$	−29.7082	−1.02320
$844$	−15.3262	−0.527551
$845$	18.5623	0.638563
$846$	−5.00000	−0.171904
$847$	0	0
$848$	6.47214	0.222254
$849$	23.1803	0.795547
$850$	4.61803	0.158397
$851$	−31.4164	−1.07694
$852$	−11.8541	−0.406115
$853$	−8.27051	−0.283177	−0.141588	−	0.989926i	$-0.545221\pi$
$853$	−8.27051	−0.283177	−0.141588	+	0.989926i	$0.545221\pi$
$854$	11.7082	0.400646
$855$	−2.00000	−0.0683986
$856$	−10.7639	−0.367904
$857$	−33.4164	−1.14148	−0.570741	−	0.821130i	$-0.693344\pi$
$857$	−33.4164	−1.14148	−0.570741	+	0.821130i	$0.693344\pi$
$858$	0	0
$859$	55.4164	1.89078	0.945392	−	0.325936i	$-0.105680\pi$
$859$	55.4164	1.89078	0.945392	+	0.325936i	$0.105680\pi$
$860$	−3.23607	−0.110349
$861$	14.4721	0.493209
$862$	−15.5623	−0.530054
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	5.47214	0.186166
$865$	15.5066	0.527239
$866$	−4.09017	−0.138990
$867$	7.00000	0.237732
$868$	−5.23607	−0.177724
$869$	0	0
$870$	−9.47214	−0.321135
$871$	6.94427	0.235298
$872$	3.52786	0.119469
$873$	1.29180	0.0437207
$874$	−44.3607	−1.50052
$875$	−1.00000	−0.0338062
$876$	−7.00000	−0.236508
$877$	−26.5410	−0.896227	−0.448113	−	0.893977i	$-0.647904\pi$
$877$	−26.5410	−0.896227	−0.448113	+	0.893977i	$0.647904\pi$
$878$	−10.6525	−0.359504
$879$	41.1246	1.38710
$880$	0	0
$881$	37.4853	1.26291	0.631456	−	0.775412i	$-0.282458\pi$
$881$	37.4853	1.26291	0.631456	+	0.775412i	$0.282458\pi$
$882$	0.381966	0.0128615
$883$	−43.2361	−1.45501	−0.727505	−	0.686103i	$-0.759320\pi$
$883$	−43.2361	−1.45501	−0.727505	+	0.686103i	$0.759320\pi$
$884$	25.9443	0.872600
$885$	7.70820	0.259108
$886$	15.2361	0.511866
$887$	−35.8541	−1.20386	−0.601931	−	0.798548i	$-0.705602\pi$
$887$	−35.8541	−1.20386	−0.601931	+	0.798548i	$0.705602\pi$
$888$	6.00000	0.201347
$889$	20.1803	0.676827
$890$	−0.291796	−0.00978103
$891$	0	0
$892$	−18.4721	−0.618493
$893$	−68.5410	−2.29364
$894$	33.7426	1.12852
$895$	−19.5623	−0.653896
$896$	1.00000	0.0334077
$897$	−77.0132	−2.57139
$898$	19.5066	0.650943
$899$	30.6525	1.02232
$900$	−0.381966	−0.0127322
$901$	−29.8885	−0.995732
$902$	0	0
$903$	5.23607	0.174245
$904$	1.52786	0.0508160
$905$	0	0
$906$	34.5967	1.14940
$907$	−30.3607	−1.00811	−0.504055	−	0.863672i	$-0.668159\pi$
$907$	−30.3607	−1.00811	−0.504055	+	0.863672i	$0.668159\pi$
$908$	−7.14590	−0.237145
$909$	−2.00000	−0.0663358
$910$	−5.61803	−0.186236
$911$	17.4508	0.578172	0.289086	−	0.957303i	$-0.406649\pi$
$911$	17.4508	0.578172	0.289086	+	0.957303i	$0.406649\pi$
$912$	8.47214	0.280540
$913$	0	0
$914$	−14.0000	−0.463079
$915$	18.9443	0.626278
$916$	7.70820	0.254686
$917$	−11.4164	−0.377003
$918$	−25.2705	−0.834051
$919$	−52.5066	−1.73203	−0.866016	−	0.500016i	$-0.833327\pi$
$919$	−52.5066	−1.73203	−0.866016	+	0.500016i	$0.833327\pi$
$920$	−8.47214	−0.279318
$921$	54.1246	1.78347
$922$	25.1246	0.827435
$923$	41.1591	1.35477
$924$	0	0
$925$	−3.70820	−0.121925
$926$	14.1803	0.465995
$927$	−1.36068	−0.0446906
$928$	−5.85410	−0.192170
$929$	27.7771	0.911337	0.455668	−	0.890150i	$-0.349400\pi$
$929$	27.7771	0.911337	0.455668	+	0.890150i	$0.349400\pi$
$930$	−8.47214	−0.277812
$931$	5.23607	0.171605
$932$	16.6525	0.545470
$933$	16.4721	0.539274
$934$	−21.6180	−0.707364
$935$	0	0
$936$	−2.14590	−0.0701409
$937$	37.3951	1.22165	0.610823	−	0.791767i	$-0.290839\pi$
$937$	37.3951	1.22165	0.610823	+	0.791767i	$0.290839\pi$
$938$	−1.23607	−0.0403591
$939$	19.2361	0.627745
$940$	−13.0902	−0.426954
$941$	−42.6525	−1.39043	−0.695215	−	0.718801i	$-0.744691\pi$
$941$	−42.6525	−1.39043	−0.695215	+	0.718801i	$0.744691\pi$
$942$	17.9443	0.584656
$943$	−75.7771	−2.46764
$944$	4.76393	0.155053
$945$	5.47214	0.178009
$946$	0	0
$947$	32.6525	1.06106	0.530531	−	0.847665i	$-0.321992\pi$
$947$	32.6525	1.06106	0.530531	+	0.847665i	$0.321992\pi$
$948$	1.38197	0.0448842
$949$	24.3050	0.788972
$950$	−5.23607	−0.169880
$951$	28.9443	0.938582
$952$	−4.61803	−0.149671
$953$	−2.58359	−0.0836908	−0.0418454	−	0.999124i	$-0.513324\pi$
$953$	−2.58359	−0.0836908	−0.0418454	+	0.999124i	$0.513324\pi$
$954$	2.47214	0.0800384
$955$	0.854102	0.0276381
$956$	−8.85410	−0.286362
$957$	0	0
$958$	−35.1246	−1.13482
$959$	16.0000	0.516667
$960$	1.61803	0.0522218
$961$	−3.58359	−0.115600
$962$	−20.8328	−0.671677
$963$	−4.11146	−0.132490
$964$	3.70820	0.119433
$965$	−12.9443	−0.416691
$966$	13.7082	0.441054
$967$	4.29180	0.138015	0.0690074	−	0.997616i	$-0.478017\pi$
$967$	4.29180	0.138015	0.0690074	+	0.997616i	$0.478017\pi$
$968$	0	0
$969$	−39.1246	−1.25686
$970$	3.38197	0.108588
$971$	42.3607	1.35942	0.679709	−	0.733481i	$-0.262106\pi$
$971$	42.3607	1.35942	0.679709	+	0.733481i	$0.262106\pi$
$972$	3.94427	0.126513
$973$	8.94427	0.286740
$974$	−11.8197	−0.378726
$975$	−9.09017	−0.291118
$976$	11.7082	0.374770
$977$	52.9017	1.69248	0.846238	−	0.532806i	$-0.178862\pi$
$977$	52.9017	1.69248	0.846238	+	0.532806i	$0.178862\pi$
$978$	23.7082	0.758105
$979$	0	0
$980$	1.00000	0.0319438
$981$	1.34752	0.0430231
$982$	0.944272	0.0301329
$983$	−30.0344	−0.957950	−0.478975	−	0.877829i	$-0.658992\pi$
$983$	−30.0344	−0.957950	−0.478975	+	0.877829i	$0.658992\pi$
$984$	14.4721	0.461355
$985$	15.4164	0.491208
$986$	27.0344	0.860952
$987$	21.1803	0.674178
$988$	−29.4164	−0.935861
$989$	−27.4164	−0.871791
$990$	0	0
$991$	10.9230	0.346980	0.173490	−	0.984836i	$-0.444496\pi$
$991$	10.9230	0.346980	0.173490	+	0.984836i	$0.444496\pi$
$992$	−5.23607	−0.166245
$993$	−10.3262	−0.327693
$994$	−7.32624	−0.232374
$995$	−10.9443	−0.346957
$996$	−22.0344	−0.698188
$997$	−17.5623	−0.556204	−0.278102	−	0.960552i	$-0.589705\pi$
$997$	−17.5623	−0.556204	−0.278102	+	0.960552i	$0.589705\pi$
$998$	24.7984	0.784979
$999$	20.2918	0.642004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.bq.1.2		2
11.7	odd	10		770.2.n.a.71.1	✓	4
11.8	odd	10		770.2.n.a.141.1	yes	4
11.10	odd	2		8470.2.a.cc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.a.71.1	✓	4	11.7	odd	10
770.2.n.a.141.1	yes	4	11.8	odd	10
8470.2.a.bq.1.2		2	1.1	even	1	trivial
8470.2.a.cc.1.2		2	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.61803 q^{6} -1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.61803 q^{6} -1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} -1.00000 q^{10} +1.61803 q^{12} -5.61803 q^{13} +1.00000 q^{14} +1.61803 q^{15} +1.00000 q^{16} -4.61803 q^{17} +0.381966 q^{18} +5.23607 q^{19} +1.00000 q^{20} -1.61803 q^{21} +8.47214 q^{23} -1.61803 q^{24} +1.00000 q^{25} +5.61803 q^{26} -5.47214 q^{27} -1.00000 q^{28} +5.85410 q^{29} -1.61803 q^{30} +5.23607 q^{31} -1.00000 q^{32} +4.61803 q^{34} -1.00000 q^{35} -0.381966 q^{36} -3.70820 q^{37} -5.23607 q^{38} -9.09017 q^{39} -1.00000 q^{40} -8.94427 q^{41} +1.61803 q^{42} -3.23607 q^{43} -0.381966 q^{45} -8.47214 q^{46} -13.0902 q^{47} +1.61803 q^{48} +1.00000 q^{49} -1.00000 q^{50} -7.47214 q^{51} -5.61803 q^{52} +6.47214 q^{53} +5.47214 q^{54} +1.00000 q^{56} +8.47214 q^{57} -5.85410 q^{58} +4.76393 q^{59} +1.61803 q^{60} +11.7082 q^{61} -5.23607 q^{62} +0.381966 q^{63} +1.00000 q^{64} -5.61803 q^{65} -1.23607 q^{67} -4.61803 q^{68} +13.7082 q^{69} +1.00000 q^{70} -7.32624 q^{71} +0.381966 q^{72} -4.32624 q^{73} +3.70820 q^{74} +1.61803 q^{75} +5.23607 q^{76} +9.09017 q^{78} +0.854102 q^{79} +1.00000 q^{80} -7.70820 q^{81} +8.94427 q^{82} -13.6180 q^{83} -1.61803 q^{84} -4.61803 q^{85} +3.23607 q^{86} +9.47214 q^{87} +0.291796 q^{89} +0.381966 q^{90} +5.61803 q^{91} +8.47214 q^{92} +8.47214 q^{93} +13.0902 q^{94} +5.23607 q^{95} -1.61803 q^{96} -3.38197 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 + 2 * q^5 - q^6 - 2 * q^7 - 2 * q^8 - 3 * q^9	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9} - 2 q^{10} + q^{12} - 9 q^{13} + 2 q^{14} + q^{15} + 2 q^{16} - 7 q^{17} + 3 q^{18} + 6 q^{19} + 2 q^{20} - q^{21} + 8 q^{23} - q^{24} + 2 q^{25} + 9 q^{26} - 2 q^{27} - 2 q^{28} + 5 q^{29} - q^{30} + 6 q^{31} - 2 q^{32} + 7 q^{34} - 2 q^{35} - 3 q^{36} + 6 q^{37} - 6 q^{38} - 7 q^{39} - 2 q^{40} + q^{42} - 2 q^{43} - 3 q^{45} - 8 q^{46} - 15 q^{47} + q^{48} + 2 q^{49} - 2 q^{50} - 6 q^{51} - 9 q^{52} + 4 q^{53} + 2 q^{54} + 2 q^{56} + 8 q^{57} - 5 q^{58} + 14 q^{59} + q^{60} + 10 q^{61} - 6 q^{62} + 3 q^{63} + 2 q^{64} - 9 q^{65} + 2 q^{67} - 7 q^{68} + 14 q^{69} + 2 q^{70} + q^{71} + 3 q^{72} + 7 q^{73} - 6 q^{74} + q^{75} + 6 q^{76} + 7 q^{78} - 5 q^{79} + 2 q^{80} - 2 q^{81} - 25 q^{83} - q^{84} - 7 q^{85} + 2 q^{86} + 10 q^{87} + 14 q^{89} + 3 q^{90} + 9 q^{91} + 8 q^{92} + 8 q^{93} + 15 q^{94} + 6 q^{95} - q^{96} - 9 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 + 2 * q^5 - q^6 - 2 * q^7 - 2 * q^8 - 3 * q^9 - 2 * q^10 + q^12 - 9 * q^13 + 2 * q^14 + q^15 + 2 * q^16 - 7 * q^17 + 3 * q^18 + 6 * q^19 + 2 * q^20 - q^21 + 8 * q^23 - q^24 + 2 * q^25 + 9 * q^26 - 2 * q^27 - 2 * q^28 + 5 * q^29 - q^30 + 6 * q^31 - 2 * q^32 + 7 * q^34 - 2 * q^35 - 3 * q^36 + 6 * q^37 - 6 * q^38 - 7 * q^39 - 2 * q^40 + q^42 - 2 * q^43 - 3 * q^45 - 8 * q^46 - 15 * q^47 + q^48 + 2 * q^49 - 2 * q^50 - 6 * q^51 - 9 * q^52 + 4 * q^53 + 2 * q^54 + 2 * q^56 + 8 * q^57 - 5 * q^58 + 14 * q^59 + q^60 + 10 * q^61 - 6 * q^62 + 3 * q^63 + 2 * q^64 - 9 * q^65 + 2 * q^67 - 7 * q^68 + 14 * q^69 + 2 * q^70 + q^71 + 3 * q^72 + 7 * q^73 - 6 * q^74 + q^75 + 6 * q^76 + 7 * q^78 - 5 * q^79 + 2 * q^80 - 2 * q^81 - 25 * q^83 - q^84 - 7 * q^85 + 2 * q^86 + 10 * q^87 + 14 * q^89 + 3 * q^90 + 9 * q^91 + 8 * q^92 + 8 * q^93 + 15 * q^94 + 6 * q^95 - q^96 - 9 * q^97 - 2 * q^98

Introduction

L-functions

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