LMFDB - Embedded newform 93.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [93,2,Mod(1,93)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(93, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("93.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 93 = 3 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	93.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:	$0.742608738798$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	93.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966 q^{2} -1.00000 q^{3} -1.85410 q^{4} -4.23607 q^{5} +0.381966 q^{6} +0.236068 q^{7} +1.47214 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.381966 q^{2} -1.00000 q^{3} -1.85410 q^{4} -4.23607 q^{5} +0.381966 q^{6} +0.236068 q^{7} +1.47214 q^{8} +1.00000 q^{9} +1.61803 q^{10} -0.763932 q^{11} +1.85410 q^{12} +1.23607 q^{13} -0.0901699 q^{14} +4.23607 q^{15} +3.14590 q^{16} -6.47214 q^{17} -0.381966 q^{18} -6.23607 q^{19} +7.85410 q^{20} -0.236068 q^{21} +0.291796 q^{22} -1.23607 q^{23} -1.47214 q^{24} +12.9443 q^{25} -0.472136 q^{26} -1.00000 q^{27} -0.437694 q^{28} +3.23607 q^{29} -1.61803 q^{30} -1.00000 q^{31} -4.14590 q^{32} +0.763932 q^{33} +2.47214 q^{34} -1.00000 q^{35} -1.85410 q^{36} -5.70820 q^{37} +2.38197 q^{38} -1.23607 q^{39} -6.23607 q^{40} +6.70820 q^{41} +0.0901699 q^{42} -9.70820 q^{43} +1.41641 q^{44} -4.23607 q^{45} +0.472136 q^{46} -2.47214 q^{47} -3.14590 q^{48} -6.94427 q^{49} -4.94427 q^{50} +6.47214 q^{51} -2.29180 q^{52} +8.94427 q^{53} +0.381966 q^{54} +3.23607 q^{55} +0.347524 q^{56} +6.23607 q^{57} -1.23607 q^{58} -3.00000 q^{59} -7.85410 q^{60} +8.00000 q^{61} +0.381966 q^{62} +0.236068 q^{63} -4.70820 q^{64} -5.23607 q^{65} -0.291796 q^{66} -12.0000 q^{67} +12.0000 q^{68} +1.23607 q^{69} +0.381966 q^{70} +9.00000 q^{71} +1.47214 q^{72} +3.23607 q^{73} +2.18034 q^{74} -12.9443 q^{75} +11.5623 q^{76} -0.180340 q^{77} +0.472136 q^{78} +8.47214 q^{79} -13.3262 q^{80} +1.00000 q^{81} -2.56231 q^{82} -16.4721 q^{83} +0.437694 q^{84} +27.4164 q^{85} +3.70820 q^{86} -3.23607 q^{87} -1.12461 q^{88} +6.94427 q^{89} +1.61803 q^{90} +0.291796 q^{91} +2.29180 q^{92} +1.00000 q^{93} +0.944272 q^{94} +26.4164 q^{95} +4.14590 q^{96} +9.00000 q^{97} +2.65248 q^{98} -0.763932 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} + 3 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 + 3 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9	\( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} + 3 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + q^{10} - 6 q^{11} - 3 q^{12} - 2 q^{13} + 11 q^{14} + 4 q^{15} + 13 q^{16} - 4 q^{17} - 3 q^{18} - 8 q^{19} + 9 q^{20} + 4 q^{21} + 14 q^{22} + 2 q^{23} + 6 q^{24} + 8 q^{25} + 8 q^{26} - 2 q^{27} - 21 q^{28} + 2 q^{29} - q^{30} - 2 q^{31} - 15 q^{32} + 6 q^{33} - 4 q^{34} - 2 q^{35} + 3 q^{36} + 2 q^{37} + 7 q^{38} + 2 q^{39} - 8 q^{40} - 11 q^{42} - 6 q^{43} - 24 q^{44} - 4 q^{45} - 8 q^{46} + 4 q^{47} - 13 q^{48} + 4 q^{49} + 8 q^{50} + 4 q^{51} - 18 q^{52} + 3 q^{54} + 2 q^{55} + 32 q^{56} + 8 q^{57} + 2 q^{58} - 6 q^{59} - 9 q^{60} + 16 q^{61} + 3 q^{62} - 4 q^{63} + 4 q^{64} - 6 q^{65} - 14 q^{66} - 24 q^{67} + 24 q^{68} - 2 q^{69} + 3 q^{70} + 18 q^{71} - 6 q^{72} + 2 q^{73} - 18 q^{74} - 8 q^{75} + 3 q^{76} + 22 q^{77} - 8 q^{78} + 8 q^{79} - 11 q^{80} + 2 q^{81} + 15 q^{82} - 24 q^{83} + 21 q^{84} + 28 q^{85} - 6 q^{86} - 2 q^{87} + 38 q^{88} - 4 q^{89} + q^{90} + 14 q^{91} + 18 q^{92} + 2 q^{93} - 16 q^{94} + 26 q^{95} + 15 q^{96} + 18 q^{97} - 26 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 + 3 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 + q^10 - 6 * q^11 - 3 * q^12 - 2 * q^13 + 11 * q^14 + 4 * q^15 + 13 * q^16 - 4 * q^17 - 3 * q^18 - 8 * q^19 + 9 * q^20 + 4 * q^21 + 14 * q^22 + 2 * q^23 + 6 * q^24 + 8 * q^25 + 8 * q^26 - 2 * q^27 - 21 * q^28 + 2 * q^29 - q^30 - 2 * q^31 - 15 * q^32 + 6 * q^33 - 4 * q^34 - 2 * q^35 + 3 * q^36 + 2 * q^37 + 7 * q^38 + 2 * q^39 - 8 * q^40 - 11 * q^42 - 6 * q^43 - 24 * q^44 - 4 * q^45 - 8 * q^46 + 4 * q^47 - 13 * q^48 + 4 * q^49 + 8 * q^50 + 4 * q^51 - 18 * q^52 + 3 * q^54 + 2 * q^55 + 32 * q^56 + 8 * q^57 + 2 * q^58 - 6 * q^59 - 9 * q^60 + 16 * q^61 + 3 * q^62 - 4 * q^63 + 4 * q^64 - 6 * q^65 - 14 * q^66 - 24 * q^67 + 24 * q^68 - 2 * q^69 + 3 * q^70 + 18 * q^71 - 6 * q^72 + 2 * q^73 - 18 * q^74 - 8 * q^75 + 3 * q^76 + 22 * q^77 - 8 * q^78 + 8 * q^79 - 11 * q^80 + 2 * q^81 + 15 * q^82 - 24 * q^83 + 21 * q^84 + 28 * q^85 - 6 * q^86 - 2 * q^87 + 38 * q^88 - 4 * q^89 + q^90 + 14 * q^91 + 18 * q^92 + 2 * q^93 - 16 * q^94 + 26 * q^95 + 15 * q^96 + 18 * q^97 - 26 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.381966	−0.270091	−0.135045	−	0.990839i	$-0.543118\pi$
$2$	−0.381966	−0.270091	−0.135045	+	0.990839i	$0.543118\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.85410	−0.927051
$5$	−4.23607	−1.89443	−0.947214	−	0.320603i	$-0.896114\pi$
$5$	−4.23607	−1.89443	−0.947214	+	0.320603i	$0.896114\pi$
$6$	0.381966	0.155937
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	1.47214	0.520479
$9$	1.00000	0.333333
$10$	1.61803	0.511667
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	1.85410	0.535233
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	−0.0901699	−0.0240989
$15$	4.23607	1.09375
$16$	3.14590	0.786475
$17$	−6.47214	−1.56972	−0.784862	−	0.619671i	$-0.787266\pi$
$17$	−6.47214	−1.56972	−0.784862	+	0.619671i	$0.787266\pi$
$18$	−0.381966	−0.0900303
$19$	−6.23607	−1.43065	−0.715326	−	0.698791i	$-0.753722\pi$
$19$	−6.23607	−1.43065	−0.715326	+	0.698791i	$0.753722\pi$
$20$	7.85410	1.75623
$21$	−0.236068	−0.0515143
$22$	0.291796	0.0622111
$23$	−1.23607	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$24$	−1.47214	−0.300498
$25$	12.9443	2.58885
$26$	−0.472136	−0.0925935
$27$	−1.00000	−0.192450
$28$	−0.437694	−0.0827164
$29$	3.23607	0.600923	0.300461	−	0.953794i	$-0.402859\pi$
$29$	3.23607	0.600923	0.300461	+	0.953794i	$0.402859\pi$
$30$	−1.61803	−0.295411
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−4.14590	−0.732898
$33$	0.763932	0.132983
$34$	2.47214	0.423968
$35$	−1.00000	−0.169031
$36$	−1.85410	−0.309017
$37$	−5.70820	−0.938423	−0.469211	−	0.883086i	$-0.655462\pi$
$37$	−5.70820	−0.938423	−0.469211	+	0.883086i	$0.655462\pi$
$38$	2.38197	0.386406
$39$	−1.23607	−0.197929
$40$	−6.23607	−0.986009
$41$	6.70820	1.04765	0.523823	−	0.851827i	$-0.324505\pi$
$41$	6.70820	1.04765	0.523823	+	0.851827i	$0.324505\pi$
$42$	0.0901699	0.0139135
$43$	−9.70820	−1.48049	−0.740244	−	0.672339i	$-0.765290\pi$
$43$	−9.70820	−1.48049	−0.740244	+	0.672339i	$0.765290\pi$
$44$	1.41641	0.213532
$45$	−4.23607	−0.631476
$46$	0.472136	0.0696126
$47$	−2.47214	−0.360598	−0.180299	−	0.983612i	$-0.557707\pi$
$47$	−2.47214	−0.360598	−0.180299	+	0.983612i	$0.557707\pi$
$48$	−3.14590	−0.454071
$49$	−6.94427	−0.992039
$50$	−4.94427	−0.699226
$51$	6.47214	0.906280
$52$	−2.29180	−0.317815
$53$	8.94427	1.22859	0.614295	−	0.789076i	$-0.289440\pi$
$53$	8.94427	1.22859	0.614295	+	0.789076i	$0.289440\pi$
$54$	0.381966	0.0519790
$55$	3.23607	0.436351
$56$	0.347524	0.0464399
$57$	6.23607	0.825987
$58$	−1.23607	−0.162304
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	−7.85410	−1.01396
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0.381966	0.0485097
$63$	0.236068	0.0297418
$64$	−4.70820	−0.588525
$65$	−5.23607	−0.649454
$66$	−0.291796	−0.0359176
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	12.0000	1.45521
$69$	1.23607	0.148805
$70$	0.381966	0.0456537
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	1.47214	0.173493
$73$	3.23607	0.378753	0.189377	−	0.981905i	$-0.439353\pi$
$73$	3.23607	0.378753	0.189377	+	0.981905i	$0.439353\pi$
$74$	2.18034	0.253459
$75$	−12.9443	−1.49468
$76$	11.5623	1.32629
$77$	−0.180340	−0.0205516
$78$	0.472136	0.0534589
$79$	8.47214	0.953190	0.476595	−	0.879123i	$-0.341871\pi$
$79$	8.47214	0.953190	0.476595	+	0.879123i	$0.341871\pi$
$80$	−13.3262	−1.48992
$81$	1.00000	0.111111
$82$	−2.56231	−0.282959
$83$	−16.4721	−1.80805	−0.904026	−	0.427478i	$-0.859402\pi$
$83$	−16.4721	−1.80805	−0.904026	+	0.427478i	$0.859402\pi$
$84$	0.437694	0.0477563
$85$	27.4164	2.97373
$86$	3.70820	0.399866
$87$	−3.23607	−0.346943
$88$	−1.12461	−0.119884
$89$	6.94427	0.736091	0.368046	−	0.929808i	$-0.380027\pi$
$89$	6.94427	0.736091	0.368046	+	0.929808i	$0.380027\pi$
$90$	1.61803	0.170556
$91$	0.291796	0.0305885
$92$	2.29180	0.238936
$93$	1.00000	0.103695
$94$	0.944272	0.0973942
$95$	26.4164	2.71027
$96$	4.14590	0.423139
$97$	9.00000	0.913812	0.456906	−	0.889515i	$-0.348958\pi$
$97$	9.00000	0.913812	0.456906	+	0.889515i	$0.348958\pi$
$98$	2.65248	0.267941
$99$	−0.763932	−0.0767781
$100$	−24.0000	−2.40000
$101$	−5.76393	−0.573533	−0.286766	−	0.958001i	$-0.592580\pi$
$101$	−5.76393	−0.573533	−0.286766	+	0.958001i	$0.592580\pi$
$102$	−2.47214	−0.244778
$103$	−13.7639	−1.35620	−0.678100	−	0.734969i	$-0.737196\pi$
$103$	−13.7639	−1.35620	−0.678100	+	0.734969i	$0.737196\pi$
$104$	1.81966	0.178432
$105$	1.00000	0.0975900
$106$	−3.41641	−0.331831
$107$	1.47214	0.142317	0.0711584	−	0.997465i	$-0.477330\pi$
$107$	1.47214	0.142317	0.0711584	+	0.997465i	$0.477330\pi$
$108$	1.85410	0.178411
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	−1.23607	−0.117854
$111$	5.70820	0.541799
$112$	0.742646	0.0701734
$113$	−1.76393	−0.165937	−0.0829684	−	0.996552i	$-0.526440\pi$
$113$	−1.76393	−0.165937	−0.0829684	+	0.996552i	$0.526440\pi$
$114$	−2.38197	−0.223092
$115$	5.23607	0.488266
$116$	−6.00000	−0.557086
$117$	1.23607	0.114275
$118$	1.14590	0.105488
$119$	−1.52786	−0.140059
$120$	6.23607	0.569273
$121$	−10.4164	−0.946946
$122$	−3.05573	−0.276653
$123$	−6.70820	−0.604858
$124$	1.85410	0.166503
$125$	−33.6525	−3.00997
$126$	−0.0901699	−0.00803298
$127$	7.23607	0.642097	0.321049	−	0.947063i	$-0.395965\pi$
$127$	7.23607	0.642097	0.321049	+	0.947063i	$0.395965\pi$
$128$	10.0902	0.891853
$129$	9.70820	0.854760
$130$	2.00000	0.175412
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−1.41641	−0.123282
$133$	−1.47214	−0.127650
$134$	4.58359	0.395962
$135$	4.23607	0.364583
$136$	−9.52786	−0.817008
$137$	−10.1803	−0.869765	−0.434883	−	0.900487i	$-0.643210\pi$
$137$	−10.1803	−0.869765	−0.434883	+	0.900487i	$0.643210\pi$
$138$	−0.472136	−0.0401909
$139$	5.70820	0.484164	0.242082	−	0.970256i	$-0.422170\pi$
$139$	5.70820	0.484164	0.242082	+	0.970256i	$0.422170\pi$
$140$	1.85410	0.156700
$141$	2.47214	0.208191
$142$	−3.43769	−0.288485
$143$	−0.944272	−0.0789640
$144$	3.14590	0.262158
$145$	−13.7082	−1.13840
$146$	−1.23607	−0.102298
$147$	6.94427	0.572754
$148$	10.5836	0.869966
$149$	−13.4164	−1.09911	−0.549557	−	0.835456i	$-0.685204\pi$
$149$	−13.4164	−1.09911	−0.549557	+	0.835456i	$0.685204\pi$
$150$	4.94427	0.403698
$151$	15.2361	1.23989	0.619947	−	0.784644i	$-0.287154\pi$
$151$	15.2361	1.23989	0.619947	+	0.784644i	$0.287154\pi$
$152$	−9.18034	−0.744624
$153$	−6.47214	−0.523241
$154$	0.0688837	0.00555081
$155$	4.23607	0.340249
$156$	2.29180	0.183491
$157$	−4.05573	−0.323682	−0.161841	−	0.986817i	$-0.551743\pi$
$157$	−4.05573	−0.323682	−0.161841	+	0.986817i	$0.551743\pi$
$158$	−3.23607	−0.257448
$159$	−8.94427	−0.709327
$160$	17.5623	1.38842
$161$	−0.291796	−0.0229968
$162$	−0.381966	−0.0300101
$163$	−7.18034	−0.562408	−0.281204	−	0.959648i	$-0.590734\pi$
$163$	−7.18034	−0.562408	−0.281204	+	0.959648i	$0.590734\pi$
$164$	−12.4377	−0.971221
$165$	−3.23607	−0.251928
$166$	6.29180	0.488338
$167$	−16.6525	−1.28861	−0.644304	−	0.764770i	$-0.722853\pi$
$167$	−16.6525	−1.28861	−0.644304	+	0.764770i	$0.722853\pi$
$168$	−0.347524	−0.0268121
$169$	−11.4721	−0.882472
$170$	−10.4721	−0.803176
$171$	−6.23607	−0.476884
$172$	18.0000	1.37249
$173$	6.94427	0.527963	0.263982	−	0.964528i	$-0.414964\pi$
$173$	6.94427	0.527963	0.263982	+	0.964528i	$0.414964\pi$
$174$	1.23607	0.0937061
$175$	3.05573	0.230991
$176$	−2.40325	−0.181152
$177$	3.00000	0.225494
$178$	−2.65248	−0.198811
$179$	19.4164	1.45125	0.725625	−	0.688090i	$-0.241551\pi$
$179$	19.4164	1.45125	0.725625	+	0.688090i	$0.241551\pi$
$180$	7.85410	0.585410
$181$	19.4164	1.44321	0.721605	−	0.692305i	$-0.243405\pi$
$181$	19.4164	1.44321	0.721605	+	0.692305i	$0.243405\pi$
$182$	−0.111456	−0.00826168
$183$	−8.00000	−0.591377
$184$	−1.81966	−0.134147
$185$	24.1803	1.77777
$186$	−0.381966	−0.0280071
$187$	4.94427	0.361561
$188$	4.58359	0.334293
$189$	−0.236068	−0.0171714
$190$	−10.0902	−0.732018
$191$	−13.9443	−1.00897	−0.504486	−	0.863420i	$-0.668318\pi$
$191$	−13.9443	−1.00897	−0.504486	+	0.863420i	$0.668318\pi$
$192$	4.70820	0.339785
$193$	−3.00000	−0.215945	−0.107972	−	0.994154i	$-0.534436\pi$
$193$	−3.00000	−0.215945	−0.107972	+	0.994154i	$0.534436\pi$
$194$	−3.43769	−0.246812
$195$	5.23607	0.374963
$196$	12.8754	0.919671
$197$	5.23607	0.373054	0.186527	−	0.982450i	$-0.440277\pi$
$197$	5.23607	0.373054	0.186527	+	0.982450i	$0.440277\pi$
$198$	0.291796	0.0207370
$199$	−12.1803	−0.863441	−0.431721	−	0.902007i	$-0.642093\pi$
$199$	−12.1803	−0.863441	−0.431721	+	0.902007i	$0.642093\pi$
$200$	19.0557	1.34744
$201$	12.0000	0.846415
$202$	2.20163	0.154906
$203$	0.763932	0.0536175
$204$	−12.0000	−0.840168
$205$	−28.4164	−1.98469
$206$	5.25735	0.366297
$207$	−1.23607	−0.0859127
$208$	3.88854	0.269622
$209$	4.76393	0.329528
$210$	−0.381966	−0.0263582
$211$	0.708204	0.0487548	0.0243774	−	0.999703i	$-0.492240\pi$
$211$	0.708204	0.0487548	0.0243774	+	0.999703i	$0.492240\pi$
$212$	−16.5836	−1.13897
$213$	−9.00000	−0.616670
$214$	−0.562306	−0.0384384
$215$	41.1246	2.80468
$216$	−1.47214	−0.100166
$217$	−0.236068	−0.0160253
$218$	1.90983	0.129350
$219$	−3.23607	−0.218673
$220$	−6.00000	−0.404520
$221$	−8.00000	−0.538138
$222$	−2.18034	−0.146335
$223$	−1.81966	−0.121853	−0.0609267	−	0.998142i	$-0.519406\pi$
$223$	−1.81966	−0.121853	−0.0609267	+	0.998142i	$0.519406\pi$
$224$	−0.978714	−0.0653931
$225$	12.9443	0.862951
$226$	0.673762	0.0448180
$227$	4.94427	0.328163	0.164081	−	0.986447i	$-0.447534\pi$
$227$	4.94427	0.328163	0.164081	+	0.986447i	$0.447534\pi$
$228$	−11.5623	−0.765732
$229$	9.70820	0.641536	0.320768	−	0.947158i	$-0.396059\pi$
$229$	9.70820	0.641536	0.320768	+	0.947158i	$0.396059\pi$
$230$	−2.00000	−0.131876
$231$	0.180340	0.0118655
$232$	4.76393	0.312767
$233$	0.708204	0.0463960	0.0231980	−	0.999731i	$-0.492615\pi$
$233$	0.708204	0.0463960	0.0231980	+	0.999731i	$0.492615\pi$
$234$	−0.472136	−0.0308645
$235$	10.4721	0.683127
$236$	5.56231	0.362075
$237$	−8.47214	−0.550324
$238$	0.583592	0.0378287
$239$	−11.8885	−0.769006	−0.384503	−	0.923124i	$-0.625627\pi$
$239$	−11.8885	−0.769006	−0.384503	+	0.923124i	$0.625627\pi$
$240$	13.3262	0.860205
$241$	−13.7082	−0.883023	−0.441512	−	0.897256i	$-0.645558\pi$
$241$	−13.7082	−0.883023	−0.441512	+	0.897256i	$0.645558\pi$
$242$	3.97871	0.255761
$243$	−1.00000	−0.0641500
$244$	−14.8328	−0.949574
$245$	29.4164	1.87935
$246$	2.56231	0.163367
$247$	−7.70820	−0.490461
$248$	−1.47214	−0.0934807
$249$	16.4721	1.04388
$250$	12.8541	0.812965
$251$	−11.2361	−0.709214	−0.354607	−	0.935015i	$-0.615385\pi$
$251$	−11.2361	−0.709214	−0.354607	+	0.935015i	$0.615385\pi$
$252$	−0.437694	−0.0275721
$253$	0.944272	0.0593659
$254$	−2.76393	−0.173425
$255$	−27.4164	−1.71688
$256$	5.56231	0.347644
$257$	5.29180	0.330093	0.165047	−	0.986286i	$-0.447223\pi$
$257$	5.29180	0.330093	0.165047	+	0.986286i	$0.447223\pi$
$258$	−3.70820	−0.230863
$259$	−1.34752	−0.0837311
$260$	9.70820	0.602077
$261$	3.23607	0.200308
$262$	0	0
$263$	18.4721	1.13904	0.569520	−	0.821977i	$-0.307129\pi$
$263$	18.4721	1.13904	0.569520	+	0.821977i	$0.307129\pi$
$264$	1.12461	0.0692151
$265$	−37.8885	−2.32747
$266$	0.562306	0.0344772
$267$	−6.94427	−0.424983
$268$	22.2492	1.35909
$269$	−12.6525	−0.771435	−0.385718	−	0.922617i	$-0.626046\pi$
$269$	−12.6525	−0.771435	−0.385718	+	0.922617i	$0.626046\pi$
$270$	−1.61803	−0.0984704
$271$	−13.4164	−0.814989	−0.407494	−	0.913208i	$-0.633597\pi$
$271$	−13.4164	−0.814989	−0.407494	+	0.913208i	$0.633597\pi$
$272$	−20.3607	−1.23455
$273$	−0.291796	−0.0176603
$274$	3.88854	0.234916
$275$	−9.88854	−0.596302
$276$	−2.29180	−0.137950
$277$	−8.47214	−0.509041	−0.254521	−	0.967067i	$-0.581918\pi$
$277$	−8.47214	−0.509041	−0.254521	+	0.967067i	$0.581918\pi$
$278$	−2.18034	−0.130768
$279$	−1.00000	−0.0598684
$280$	−1.47214	−0.0879770
$281$	−28.2361	−1.68442	−0.842211	−	0.539148i	$-0.818746\pi$
$281$	−28.2361	−1.68442	−0.842211	+	0.539148i	$0.818746\pi$
$282$	−0.944272	−0.0562306
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	−16.6869	−0.990186
$285$	−26.4164	−1.56477
$286$	0.360680	0.0213274
$287$	1.58359	0.0934765
$288$	−4.14590	−0.244299
$289$	24.8885	1.46403
$290$	5.23607	0.307472
$291$	−9.00000	−0.527589
$292$	−6.00000	−0.351123
$293$	14.3607	0.838960	0.419480	−	0.907765i	$-0.362212\pi$
$293$	14.3607	0.838960	0.419480	+	0.907765i	$0.362212\pi$
$294$	−2.65248	−0.154696
$295$	12.7082	0.739900
$296$	−8.40325	−0.488429
$297$	0.763932	0.0443278
$298$	5.12461	0.296861
$299$	−1.52786	−0.0883587
$300$	24.0000	1.38564
$301$	−2.29180	−0.132097
$302$	−5.81966	−0.334884
$303$	5.76393	0.331129
$304$	−19.6180	−1.12517
$305$	−33.8885	−1.94045
$306$	2.47214	0.141323
$307$	−10.2361	−0.584203	−0.292102	−	0.956387i	$-0.594355\pi$
$307$	−10.2361	−0.584203	−0.292102	+	0.956387i	$0.594355\pi$
$308$	0.334369	0.0190524
$309$	13.7639	0.783003
$310$	−1.61803	−0.0918982
$311$	−5.47214	−0.310296	−0.155148	−	0.987891i	$-0.549586\pi$
$311$	−5.47214	−0.310296	−0.155148	+	0.987891i	$0.549586\pi$
$312$	−1.81966	−0.103018
$313$	−6.47214	−0.365827	−0.182913	−	0.983129i	$-0.558553\pi$
$313$	−6.47214	−0.365827	−0.182913	+	0.983129i	$0.558553\pi$
$314$	1.54915	0.0874236
$315$	−1.00000	−0.0563436
$316$	−15.7082	−0.883656
$317$	32.1246	1.80430	0.902149	−	0.431425i	$-0.141989\pi$
$317$	32.1246	1.80430	0.902149	+	0.431425i	$0.141989\pi$
$318$	3.41641	0.191583
$319$	−2.47214	−0.138413
$320$	19.9443	1.11492
$321$	−1.47214	−0.0821666
$322$	0.111456	0.00621121
$323$	40.3607	2.24573
$324$	−1.85410	−0.103006
$325$	16.0000	0.887520
$326$	2.74265	0.151901
$327$	5.00000	0.276501
$328$	9.87539	0.545277
$329$	−0.583592	−0.0321745
$330$	1.23607	0.0680433
$331$	20.8328	1.14508	0.572538	−	0.819878i	$-0.305959\pi$
$331$	20.8328	1.14508	0.572538	+	0.819878i	$0.305959\pi$
$332$	30.5410	1.67616
$333$	−5.70820	−0.312808
$334$	6.36068	0.348041
$335$	50.8328	2.77729
$336$	−0.742646	−0.0405146
$337$	17.0557	0.929085	0.464542	−	0.885551i	$-0.346219\pi$
$337$	17.0557	0.929085	0.464542	+	0.885551i	$0.346219\pi$
$338$	4.38197	0.238348
$339$	1.76393	0.0958036
$340$	−50.8328	−2.75680
$341$	0.763932	0.0413692
$342$	2.38197	0.128802
$343$	−3.29180	−0.177740
$344$	−14.2918	−0.770562
$345$	−5.23607	−0.281900
$346$	−2.65248	−0.142598
$347$	3.34752	0.179705	0.0898523	−	0.995955i	$-0.471361\pi$
$347$	3.34752	0.179705	0.0898523	+	0.995955i	$0.471361\pi$
$348$	6.00000	0.321634
$349$	27.8885	1.49284	0.746420	−	0.665475i	$-0.231771\pi$
$349$	27.8885	1.49284	0.746420	+	0.665475i	$0.231771\pi$
$350$	−1.16718	−0.0623886
$351$	−1.23607	−0.0659764
$352$	3.16718	0.168811
$353$	−17.2361	−0.917383	−0.458692	−	0.888595i	$-0.651682\pi$
$353$	−17.2361	−0.917383	−0.458692	+	0.888595i	$0.651682\pi$
$354$	−1.14590	−0.0609038
$355$	−38.1246	−2.02344
$356$	−12.8754	−0.682394
$357$	1.52786	0.0808631
$358$	−7.41641	−0.391969
$359$	−8.88854	−0.469119	−0.234560	−	0.972102i	$-0.575365\pi$
$359$	−8.88854	−0.469119	−0.234560	+	0.972102i	$0.575365\pi$
$360$	−6.23607	−0.328670
$361$	19.8885	1.04677
$362$	−7.41641	−0.389798
$363$	10.4164	0.546720
$364$	−0.541020	−0.0283571
$365$	−13.7082	−0.717520
$366$	3.05573	0.159725
$367$	17.0557	0.890302	0.445151	−	0.895456i	$-0.353150\pi$
$367$	17.0557	0.890302	0.445151	+	0.895456i	$0.353150\pi$
$368$	−3.88854	−0.202704
$369$	6.70820	0.349215
$370$	−9.23607	−0.480160
$371$	2.11146	0.109621
$372$	−1.85410	−0.0961307
$373$	7.58359	0.392664	0.196332	−	0.980538i	$-0.437097\pi$
$373$	7.58359	0.392664	0.196332	+	0.980538i	$0.437097\pi$
$374$	−1.88854	−0.0976543
$375$	33.6525	1.73781
$376$	−3.63932	−0.187684
$377$	4.00000	0.206010
$378$	0.0901699	0.00463784
$379$	17.5279	0.900346	0.450173	−	0.892941i	$-0.351362\pi$
$379$	17.5279	0.900346	0.450173	+	0.892941i	$0.351362\pi$
$380$	−48.9787	−2.51256
$381$	−7.23607	−0.370715
$382$	5.32624	0.272514
$383$	2.94427	0.150445	0.0752226	−	0.997167i	$-0.476033\pi$
$383$	2.94427	0.150445	0.0752226	+	0.997167i	$0.476033\pi$
$384$	−10.0902	−0.514912
$385$	0.763932	0.0389336
$386$	1.14590	0.0583247
$387$	−9.70820	−0.493496
$388$	−16.6869	−0.847150
$389$	26.1803	1.32740	0.663698	−	0.748001i	$-0.268986\pi$
$389$	26.1803	1.32740	0.663698	+	0.748001i	$0.268986\pi$
$390$	−2.00000	−0.101274
$391$	8.00000	0.404577
$392$	−10.2229	−0.516335
$393$	0	0
$394$	−2.00000	−0.100759
$395$	−35.8885	−1.80575
$396$	1.41641	0.0711772
$397$	−30.4164	−1.52656	−0.763278	−	0.646070i	$-0.776411\pi$
$397$	−30.4164	−1.52656	−0.763278	+	0.646070i	$0.776411\pi$
$398$	4.65248	0.233208
$399$	1.47214	0.0736990
$400$	40.7214	2.03607
$401$	0.763932	0.0381489	0.0190745	−	0.999818i	$-0.493928\pi$
$401$	0.763932	0.0381489	0.0190745	+	0.999818i	$0.493928\pi$
$402$	−4.58359	−0.228609
$403$	−1.23607	−0.0615729
$404$	10.6869	0.531694
$405$	−4.23607	−0.210492
$406$	−0.291796	−0.0144816
$407$	4.36068	0.216151
$408$	9.52786	0.471700
$409$	23.1246	1.14344	0.571719	−	0.820449i	$-0.306277\pi$
$409$	23.1246	1.14344	0.571719	+	0.820449i	$0.306277\pi$
$410$	10.8541	0.536046
$411$	10.1803	0.502159
$412$	25.5197	1.25727
$413$	−0.708204	−0.0348484
$414$	0.472136	0.0232042
$415$	69.7771	3.42522
$416$	−5.12461	−0.251255
$417$	−5.70820	−0.279532
$418$	−1.81966	−0.0890025
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	−1.85410	−0.0904709
$421$	−14.4164	−0.702613	−0.351306	−	0.936261i	$-0.614262\pi$
$421$	−14.4164	−0.702613	−0.351306	+	0.936261i	$0.614262\pi$
$422$	−0.270510	−0.0131682
$423$	−2.47214	−0.120199
$424$	13.1672	0.639455
$425$	−83.7771	−4.06379
$426$	3.43769	0.166557
$427$	1.88854	0.0913930
$428$	−2.72949	−0.131935
$429$	0.944272	0.0455899
$430$	−15.7082	−0.757517
$431$	−6.47214	−0.311752	−0.155876	−	0.987777i	$-0.549820\pi$
$431$	−6.47214	−0.311752	−0.155876	+	0.987777i	$0.549820\pi$
$432$	−3.14590	−0.151357
$433$	12.4721	0.599373	0.299686	−	0.954038i	$-0.403118\pi$
$433$	12.4721	0.599373	0.299686	+	0.954038i	$0.403118\pi$
$434$	0.0901699	0.00432830
$435$	13.7082	0.657258
$436$	9.27051	0.443977
$437$	7.70820	0.368733
$438$	1.23607	0.0590616
$439$	−10.7082	−0.511075	−0.255537	−	0.966799i	$-0.582252\pi$
$439$	−10.7082	−0.511075	−0.255537	+	0.966799i	$0.582252\pi$
$440$	4.76393	0.227112
$441$	−6.94427	−0.330680
$442$	3.05573	0.145346
$443$	−25.3607	−1.20492	−0.602461	−	0.798148i	$-0.705813\pi$
$443$	−25.3607	−1.20492	−0.602461	+	0.798148i	$0.705813\pi$
$444$	−10.5836	−0.502275
$445$	−29.4164	−1.39447
$446$	0.695048	0.0329115
$447$	13.4164	0.634574
$448$	−1.11146	−0.0525114
$449$	−30.6525	−1.44658	−0.723290	−	0.690545i	$-0.757371\pi$
$449$	−30.6525	−1.44658	−0.723290	+	0.690545i	$0.757371\pi$
$450$	−4.94427	−0.233075
$451$	−5.12461	−0.241309
$452$	3.27051	0.153832
$453$	−15.2361	−0.715853
$454$	−1.88854	−0.0886338
$455$	−1.23607	−0.0579478
$456$	9.18034	0.429909
$457$	11.1246	0.520387	0.260194	−	0.965556i	$-0.416214\pi$
$457$	11.1246	0.520387	0.260194	+	0.965556i	$0.416214\pi$
$458$	−3.70820	−0.173273
$459$	6.47214	0.302093
$460$	−9.70820	−0.452647
$461$	−23.8885	−1.11260	−0.556300	−	0.830981i	$-0.687780\pi$
$461$	−23.8885	−1.11260	−0.556300	+	0.830981i	$0.687780\pi$
$462$	−0.0688837	−0.00320476
$463$	−24.8328	−1.15408	−0.577039	−	0.816716i	$-0.695792\pi$
$463$	−24.8328	−1.15408	−0.577039	+	0.816716i	$0.695792\pi$
$464$	10.1803	0.472610
$465$	−4.23607	−0.196443
$466$	−0.270510	−0.0125311
$467$	20.0557	0.928068	0.464034	−	0.885817i	$-0.346401\pi$
$467$	20.0557	0.928068	0.464034	+	0.885817i	$0.346401\pi$
$468$	−2.29180	−0.105938
$469$	−2.83282	−0.130807
$470$	−4.00000	−0.184506
$471$	4.05573	0.186878
$472$	−4.41641	−0.203282
$473$	7.41641	0.341007
$474$	3.23607	0.148638
$475$	−80.7214	−3.70375
$476$	2.83282	0.129842
$477$	8.94427	0.409530
$478$	4.54102	0.207701
$479$	−0.0557281	−0.00254628	−0.00127314	−	0.999999i	$-0.500405\pi$
$479$	−0.0557281	−0.00254628	−0.00127314	+	0.999999i	$0.500405\pi$
$480$	−17.5623	−0.801606
$481$	−7.05573	−0.321714
$482$	5.23607	0.238496
$483$	0.291796	0.0132772
$484$	19.3131	0.877867
$485$	−38.1246	−1.73115
$486$	0.381966	0.0173263
$487$	−3.70820	−0.168035	−0.0840174	−	0.996464i	$-0.526775\pi$
$487$	−3.70820	−0.168035	−0.0840174	+	0.996464i	$0.526775\pi$
$488$	11.7771	0.533124
$489$	7.18034	0.324706
$490$	−11.2361	−0.507594
$491$	−13.8197	−0.623673	−0.311836	−	0.950136i	$-0.600944\pi$
$491$	−13.8197	−0.623673	−0.311836	+	0.950136i	$0.600944\pi$
$492$	12.4377	0.560735
$493$	−20.9443	−0.943283
$494$	2.94427	0.132469
$495$	3.23607	0.145450
$496$	−3.14590	−0.141255
$497$	2.12461	0.0953019
$498$	−6.29180	−0.281942
$499$	19.8885	0.890333	0.445167	−	0.895448i	$-0.353144\pi$
$499$	19.8885	0.890333	0.445167	+	0.895448i	$0.353144\pi$
$500$	62.3951	2.79039
$501$	16.6525	0.743978
$502$	4.29180	0.191552
$503$	13.3607	0.595723	0.297862	−	0.954609i	$-0.403727\pi$
$503$	13.3607	0.595723	0.297862	+	0.954609i	$0.403727\pi$
$504$	0.347524	0.0154800
$505$	24.4164	1.08652
$506$	−0.360680	−0.0160342
$507$	11.4721	0.509495
$508$	−13.4164	−0.595257
$509$	26.2918	1.16536	0.582682	−	0.812700i	$-0.302003\pi$
$509$	26.2918	1.16536	0.582682	+	0.812700i	$0.302003\pi$
$510$	10.4721	0.463714
$511$	0.763932	0.0337944
$512$	−22.3050	−0.985749
$513$	6.23607	0.275329
$514$	−2.02129	−0.0891551
$515$	58.3050	2.56922
$516$	−18.0000	−0.792406
$517$	1.88854	0.0830581
$518$	0.514708	0.0226150
$519$	−6.94427	−0.304820
$520$	−7.70820	−0.338027
$521$	−27.5279	−1.20602	−0.603009	−	0.797735i	$-0.706032\pi$
$521$	−27.5279	−1.20602	−0.603009	+	0.797735i	$0.706032\pi$
$522$	−1.23607	−0.0541012
$523$	42.5410	1.86019	0.930094	−	0.367320i	$-0.119725\pi$
$523$	42.5410	1.86019	0.930094	+	0.367320i	$0.119725\pi$
$524$	0	0
$525$	−3.05573	−0.133363
$526$	−7.05573	−0.307644
$527$	6.47214	0.281931
$528$	2.40325	0.104588
$529$	−21.4721	−0.933571
$530$	14.4721	0.628629
$531$	−3.00000	−0.130189
$532$	2.72949	0.118338
$533$	8.29180	0.359158
$534$	2.65248	0.114784
$535$	−6.23607	−0.269609
$536$	−17.6656	−0.763039
$537$	−19.4164	−0.837880
$538$	4.83282	0.208357
$539$	5.30495	0.228500
$540$	−7.85410	−0.337987
$541$	−9.00000	−0.386940	−0.193470	−	0.981106i	$-0.561974\pi$
$541$	−9.00000	−0.386940	−0.193470	+	0.981106i	$0.561974\pi$
$542$	5.12461	0.220121
$543$	−19.4164	−0.833238
$544$	26.8328	1.15045
$545$	21.1803	0.907266
$546$	0.111456	0.00476988
$547$	−31.5410	−1.34860	−0.674298	−	0.738459i	$-0.735554\pi$
$547$	−31.5410	−1.34860	−0.674298	+	0.738459i	$0.735554\pi$
$548$	18.8754	0.806317
$549$	8.00000	0.341432
$550$	3.77709	0.161056
$551$	−20.1803	−0.859711
$552$	1.81966	0.0774499
$553$	2.00000	0.0850487
$554$	3.23607	0.137487
$555$	−24.1803	−1.02640
$556$	−10.5836	−0.448844
$557$	24.6525	1.04456	0.522279	−	0.852774i	$-0.325082\pi$
$557$	24.6525	1.04456	0.522279	+	0.852774i	$0.325082\pi$
$558$	0.381966	0.0161699
$559$	−12.0000	−0.507546
$560$	−3.14590	−0.132938
$561$	−4.94427	−0.208747
$562$	10.7852	0.454947
$563$	−9.11146	−0.384002	−0.192001	−	0.981395i	$-0.561498\pi$
$563$	−9.11146	−0.384002	−0.192001	+	0.981395i	$0.561498\pi$
$564$	−4.58359	−0.193004
$565$	7.47214	0.314355
$566$	3.05573	0.128442
$567$	0.236068	0.00991392
$568$	13.2492	0.555925
$569$	29.8885	1.25299	0.626496	−	0.779424i	$-0.284488\pi$
$569$	29.8885	1.25299	0.626496	+	0.779424i	$0.284488\pi$
$570$	10.0902	0.422631
$571$	23.7082	0.992157	0.496079	−	0.868278i	$-0.334773\pi$
$571$	23.7082	0.992157	0.496079	+	0.868278i	$0.334773\pi$
$572$	1.75078	0.0732036
$573$	13.9443	0.582530
$574$	−0.604878	−0.0252471
$575$	−16.0000	−0.667246
$576$	−4.70820	−0.196175
$577$	30.3607	1.26393	0.631966	−	0.774996i	$-0.282248\pi$
$577$	30.3607	1.26393	0.631966	+	0.774996i	$0.282248\pi$
$578$	−9.50658	−0.395422
$579$	3.00000	0.124676
$580$	25.4164	1.05536
$581$	−3.88854	−0.161324
$582$	3.43769	0.142497
$583$	−6.83282	−0.282986
$584$	4.76393	0.197133
$585$	−5.23607	−0.216485
$586$	−5.48529	−0.226595
$587$	−34.4721	−1.42282	−0.711409	−	0.702779i	$-0.751942\pi$
$587$	−34.4721	−1.42282	−0.711409	+	0.702779i	$0.751942\pi$
$588$	−12.8754	−0.530972
$589$	6.23607	0.256953
$590$	−4.85410	−0.199840
$591$	−5.23607	−0.215383
$592$	−17.9574	−0.738046
$593$	−5.29180	−0.217308	−0.108654	−	0.994080i	$-0.534654\pi$
$593$	−5.29180	−0.217308	−0.108654	+	0.994080i	$0.534654\pi$
$594$	−0.291796	−0.0119725
$595$	6.47214	0.265332
$596$	24.8754	1.01894
$597$	12.1803	0.498508
$598$	0.583592	0.0238649
$599$	−37.9443	−1.55036	−0.775180	−	0.631740i	$-0.782341\pi$
$599$	−37.9443	−1.55036	−0.775180	+	0.631740i	$0.782341\pi$
$600$	−19.0557	−0.777947
$601$	−20.5410	−0.837886	−0.418943	−	0.908013i	$-0.637599\pi$
$601$	−20.5410	−0.837886	−0.418943	+	0.908013i	$0.637599\pi$
$602$	0.875388	0.0356782
$603$	−12.0000	−0.488678
$604$	−28.2492	−1.14944
$605$	44.1246	1.79392
$606$	−2.20163	−0.0894349
$607$	26.8328	1.08911	0.544555	−	0.838725i	$-0.316698\pi$
$607$	26.8328	1.08911	0.544555	+	0.838725i	$0.316698\pi$
$608$	25.8541	1.04852
$609$	−0.763932	−0.0309561
$610$	12.9443	0.524098
$611$	−3.05573	−0.123622
$612$	12.0000	0.485071
$613$	−17.4164	−0.703442	−0.351721	−	0.936105i	$-0.614403\pi$
$613$	−17.4164	−0.703442	−0.351721	+	0.936105i	$0.614403\pi$
$614$	3.90983	0.157788
$615$	28.4164	1.14586
$616$	−0.265485	−0.0106967
$617$	13.4164	0.540124	0.270062	−	0.962843i	$-0.412956\pi$
$617$	13.4164	0.540124	0.270062	+	0.962843i	$0.412956\pi$
$618$	−5.25735	−0.211482
$619$	6.00000	0.241160	0.120580	−	0.992704i	$-0.461525\pi$
$619$	6.00000	0.241160	0.120580	+	0.992704i	$0.461525\pi$
$620$	−7.85410	−0.315428
$621$	1.23607	0.0496017
$622$	2.09017	0.0838082
$623$	1.63932	0.0656780
$624$	−3.88854	−0.155666
$625$	77.8328	3.11331
$626$	2.47214	0.0988064
$627$	−4.76393	−0.190253
$628$	7.51973	0.300070
$629$	36.9443	1.47306
$630$	0.381966	0.0152179
$631$	−24.8328	−0.988579	−0.494289	−	0.869297i	$-0.664572\pi$
$631$	−24.8328	−0.988579	−0.494289	+	0.869297i	$0.664572\pi$
$632$	12.4721	0.496115
$633$	−0.708204	−0.0281486
$634$	−12.2705	−0.487324
$635$	−30.6525	−1.21641
$636$	16.5836	0.657582
$637$	−8.58359	−0.340094
$638$	0.944272	0.0373841
$639$	9.00000	0.356034
$640$	−42.7426	−1.68955
$641$	4.58359	0.181041	0.0905205	−	0.995895i	$-0.471147\pi$
$641$	4.58359	0.181041	0.0905205	+	0.995895i	$0.471147\pi$
$642$	0.562306	0.0221924
$643$	−44.4721	−1.75381	−0.876905	−	0.480664i	$-0.840396\pi$
$643$	−44.4721	−1.75381	−0.876905	+	0.480664i	$0.840396\pi$
$644$	0.541020	0.0213192
$645$	−41.1246	−1.61928
$646$	−15.4164	−0.606550
$647$	9.59675	0.377287	0.188644	−	0.982046i	$-0.439591\pi$
$647$	9.59675	0.377287	0.188644	+	0.982046i	$0.439591\pi$
$648$	1.47214	0.0578310
$649$	2.29180	0.0899609
$650$	−6.11146	−0.239711
$651$	0.236068	0.00925223
$652$	13.3131	0.521381
$653$	−23.8885	−0.934831	−0.467415	−	0.884038i	$-0.654815\pi$
$653$	−23.8885	−0.934831	−0.467415	+	0.884038i	$0.654815\pi$
$654$	−1.90983	−0.0746803
$655$	0	0
$656$	21.1033	0.823946
$657$	3.23607	0.126251
$658$	0.222912	0.00869003
$659$	−44.3050	−1.72588	−0.862938	−	0.505310i	$-0.831378\pi$
$659$	−44.3050	−1.72588	−0.862938	+	0.505310i	$0.831378\pi$
$660$	6.00000	0.233550
$661$	9.47214	0.368423	0.184212	−	0.982887i	$-0.441027\pi$
$661$	9.47214	0.368423	0.184212	+	0.982887i	$0.441027\pi$
$662$	−7.95743	−0.309274
$663$	8.00000	0.310694
$664$	−24.2492	−0.941052
$665$	6.23607	0.241824
$666$	2.18034	0.0844865
$667$	−4.00000	−0.154881
$668$	30.8754	1.19460
$669$	1.81966	0.0703521
$670$	−19.4164	−0.750121
$671$	−6.11146	−0.235930
$672$	0.978714	0.0377547
$673$	−33.1246	−1.27686	−0.638430	−	0.769680i	$-0.720416\pi$
$673$	−33.1246	−1.27686	−0.638430	+	0.769680i	$0.720416\pi$
$674$	−6.51471	−0.250937
$675$	−12.9443	−0.498225
$676$	21.2705	0.818097
$677$	−10.6525	−0.409408	−0.204704	−	0.978824i	$-0.565623\pi$
$677$	−10.6525	−0.409408	−0.204704	+	0.978824i	$0.565623\pi$
$678$	−0.673762	−0.0258757
$679$	2.12461	0.0815351
$680$	40.3607	1.54776
$681$	−4.94427	−0.189465
$682$	−0.291796	−0.0111734
$683$	−26.8885	−1.02886	−0.514431	−	0.857532i	$-0.671997\pi$
$683$	−26.8885	−1.02886	−0.514431	+	0.857532i	$0.671997\pi$
$684$	11.5623	0.442096
$685$	43.1246	1.64771
$686$	1.25735	0.0480060
$687$	−9.70820	−0.370391
$688$	−30.5410	−1.16437
$689$	11.0557	0.421190
$690$	2.00000	0.0761387
$691$	−22.7082	−0.863861	−0.431930	−	0.901907i	$-0.642167\pi$
$691$	−22.7082	−0.863861	−0.431930	+	0.901907i	$0.642167\pi$
$692$	−12.8754	−0.489449
$693$	−0.180340	−0.00685055
$694$	−1.27864	−0.0485365
$695$	−24.1803	−0.917213
$696$	−4.76393	−0.180576
$697$	−43.4164	−1.64451
$698$	−10.6525	−0.403202
$699$	−0.708204	−0.0267867
$700$	−5.66563	−0.214141
$701$	−24.7082	−0.933216	−0.466608	−	0.884464i	$-0.654524\pi$
$701$	−24.7082	−0.933216	−0.466608	+	0.884464i	$0.654524\pi$
$702$	0.472136	0.0178196
$703$	35.5967	1.34256
$704$	3.59675	0.135558
$705$	−10.4721	−0.394403
$706$	6.58359	0.247777
$707$	−1.36068	−0.0511736
$708$	−5.56231	−0.209044
$709$	−38.8328	−1.45840	−0.729199	−	0.684302i	$-0.760107\pi$
$709$	−38.8328	−1.45840	−0.729199	+	0.684302i	$0.760107\pi$
$710$	14.5623	0.546514
$711$	8.47214	0.317730
$712$	10.2229	0.383120
$713$	1.23607	0.0462911
$714$	−0.583592	−0.0218404
$715$	4.00000	0.149592
$716$	−36.0000	−1.34538
$717$	11.8885	0.443986
$718$	3.39512	0.126705
$719$	53.1935	1.98378	0.991891	−	0.127089i	$-0.0405634\pi$
$719$	53.1935	1.98378	0.991891	+	0.127089i	$0.0405634\pi$
$720$	−13.3262	−0.496640
$721$	−3.24922	−0.121007
$722$	−7.59675	−0.282722
$723$	13.7082	0.509814
$724$	−36.0000	−1.33793
$725$	41.8885	1.55570
$726$	−3.97871	−0.147664
$727$	24.1246	0.894732	0.447366	−	0.894351i	$-0.352362\pi$
$727$	24.1246	0.894732	0.447366	+	0.894351i	$0.352362\pi$
$728$	0.429563	0.0159207
$729$	1.00000	0.0370370
$730$	5.23607	0.193796
$731$	62.8328	2.32396
$732$	14.8328	0.548237
$733$	−7.00000	−0.258551	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−7.00000	−0.258551	−0.129275	+	0.991609i	$0.541265\pi$
$734$	−6.51471	−0.240462
$735$	−29.4164	−1.08504
$736$	5.12461	0.188896
$737$	9.16718	0.337678
$738$	−2.56231	−0.0943198
$739$	43.2361	1.59046	0.795232	−	0.606305i	$-0.207349\pi$
$739$	43.2361	1.59046	0.795232	+	0.606305i	$0.207349\pi$
$740$	−44.8328	−1.64809
$741$	7.70820	0.283168
$742$	−0.806504	−0.0296077
$743$	−39.2361	−1.43943	−0.719716	−	0.694269i	$-0.755728\pi$
$743$	−39.2361	−1.43943	−0.719716	+	0.694269i	$0.755728\pi$
$744$	1.47214	0.0539711
$745$	56.8328	2.08219
$746$	−2.89667	−0.106055
$747$	−16.4721	−0.602684
$748$	−9.16718	−0.335185
$749$	0.347524	0.0126983
$750$	−12.8541	−0.469365
$751$	0.708204	0.0258427	0.0129214	−	0.999917i	$-0.495887\pi$
$751$	0.708204	0.0258427	0.0129214	+	0.999917i	$0.495887\pi$
$752$	−7.77709	−0.283601
$753$	11.2361	0.409465
$754$	−1.52786	−0.0556415
$755$	−64.5410	−2.34889
$756$	0.437694	0.0159188
$757$	7.59675	0.276108	0.138054	−	0.990425i	$-0.455915\pi$
$757$	7.59675	0.276108	0.138054	+	0.990425i	$0.455915\pi$
$758$	−6.69505	−0.243175
$759$	−0.944272	−0.0342749
$760$	38.8885	1.41064
$761$	−32.7639	−1.18769	−0.593846	−	0.804579i	$-0.702391\pi$
$761$	−32.7639	−1.18769	−0.593846	+	0.804579i	$0.702391\pi$
$762$	2.76393	0.100127
$763$	−1.18034	−0.0427312
$764$	25.8541	0.935369
$765$	27.4164	0.991242
$766$	−1.12461	−0.0406339
$767$	−3.70820	−0.133895
$768$	−5.56231	−0.200712
$769$	25.5836	0.922568	0.461284	−	0.887253i	$-0.347389\pi$
$769$	25.5836	0.922568	0.461284	+	0.887253i	$0.347389\pi$
$770$	−0.291796	−0.0105156
$771$	−5.29180	−0.190579
$772$	5.56231	0.200192
$773$	28.7639	1.03457	0.517283	−	0.855814i	$-0.326943\pi$
$773$	28.7639	1.03457	0.517283	+	0.855814i	$0.326943\pi$
$774$	3.70820	0.133289
$775$	−12.9443	−0.464972
$776$	13.2492	0.475619
$777$	1.34752	0.0483422
$778$	−10.0000	−0.358517
$779$	−41.8328	−1.49882
$780$	−9.70820	−0.347609
$781$	−6.87539	−0.246021
$782$	−3.05573	−0.109273
$783$	−3.23607	−0.115648
$784$	−21.8460	−0.780213
$785$	17.1803	0.613193
$786$	0	0
$787$	−40.8328	−1.45553	−0.727766	−	0.685825i	$-0.759441\pi$
$787$	−40.8328	−1.45553	−0.727766	+	0.685825i	$0.759441\pi$
$788$	−9.70820	−0.345840
$789$	−18.4721	−0.657625
$790$	13.7082	0.487716
$791$	−0.416408	−0.0148058
$792$	−1.12461	−0.0399613
$793$	9.88854	0.351152
$794$	11.6180	0.412309
$795$	37.8885	1.34377
$796$	22.5836	0.800454
$797$	23.7771	0.842228	0.421114	−	0.907008i	$-0.361639\pi$
$797$	23.7771	0.842228	0.421114	+	0.907008i	$0.361639\pi$
$798$	−0.562306	−0.0199054
$799$	16.0000	0.566039
$800$	−53.6656	−1.89737
$801$	6.94427	0.245364
$802$	−0.291796	−0.0103037
$803$	−2.47214	−0.0872398
$804$	−22.2492	−0.784670
$805$	1.23607	0.0435657
$806$	0.472136	0.0166303
$807$	12.6525	0.445388
$808$	−8.48529	−0.298512
$809$	−53.0132	−1.86384	−0.931922	−	0.362660i	$-0.881869\pi$
$809$	−53.0132	−1.86384	−0.931922	+	0.362660i	$0.881869\pi$
$810$	1.61803	0.0568519
$811$	−46.8328	−1.64452	−0.822261	−	0.569110i	$-0.807288\pi$
$811$	−46.8328	−1.64452	−0.822261	+	0.569110i	$0.807288\pi$
$812$	−1.41641	−0.0497062
$813$	13.4164	0.470534
$814$	−1.66563	−0.0583804
$815$	30.4164	1.06544
$816$	20.3607	0.712766
$817$	60.5410	2.11806
$818$	−8.83282	−0.308832
$819$	0.291796	0.0101962
$820$	52.6869	1.83991
$821$	55.3050	1.93016	0.965078	−	0.261962i	$-0.0843697\pi$
$821$	55.3050	1.93016	0.965078	+	0.261962i	$0.0843697\pi$
$822$	−3.88854	−0.135629
$823$	18.8328	0.656471	0.328235	−	0.944596i	$-0.393546\pi$
$823$	18.8328	0.656471	0.328235	+	0.944596i	$0.393546\pi$
$824$	−20.2624	−0.705873
$825$	9.88854	0.344275
$826$	0.270510	0.00941224
$827$	−0.875388	−0.0304402	−0.0152201	−	0.999884i	$-0.504845\pi$
$827$	−0.875388	−0.0304402	−0.0152201	+	0.999884i	$0.504845\pi$
$828$	2.29180	0.0796454
$829$	−5.41641	−0.188120	−0.0940598	−	0.995567i	$-0.529984\pi$
$829$	−5.41641	−0.188120	−0.0940598	+	0.995567i	$0.529984\pi$
$830$	−26.6525	−0.925121
$831$	8.47214	0.293895
$832$	−5.81966	−0.201760
$833$	44.9443	1.55723
$834$	2.18034	0.0754990
$835$	70.5410	2.44117
$836$	−8.83282	−0.305489
$837$	1.00000	0.0345651
$838$	−3.43769	−0.118753
$839$	−20.9443	−0.723077	−0.361538	−	0.932357i	$-0.617748\pi$
$839$	−20.9443	−0.723077	−0.361538	+	0.932357i	$0.617748\pi$
$840$	1.47214	0.0507935
$841$	−18.5279	−0.638892
$842$	5.50658	0.189769
$843$	28.2361	0.972502
$844$	−1.31308	−0.0451982
$845$	48.5967	1.67178
$846$	0.944272	0.0324647
$847$	−2.45898	−0.0844916
$848$	28.1378	0.966255
$849$	8.00000	0.274559
$850$	32.0000	1.09759
$851$	7.05573	0.241867
$852$	16.6869	0.571684
$853$	−23.5279	−0.805579	−0.402789	−	0.915293i	$-0.631959\pi$
$853$	−23.5279	−0.805579	−0.402789	+	0.915293i	$0.631959\pi$
$854$	−0.721360	−0.0246844
$855$	26.4164	0.903422
$856$	2.16718	0.0740728
$857$	−7.30495	−0.249532	−0.124766	−	0.992186i	$-0.539818\pi$
$857$	−7.30495	−0.249532	−0.124766	+	0.992186i	$0.539818\pi$
$858$	−0.360680	−0.0123134
$859$	22.1803	0.756783	0.378392	−	0.925646i	$-0.376477\pi$
$859$	22.1803	0.756783	0.378392	+	0.925646i	$0.376477\pi$
$860$	−76.2492	−2.60008
$861$	−1.58359	−0.0539687
$862$	2.47214	0.0842013
$863$	−11.3050	−0.384825	−0.192413	−	0.981314i	$-0.561631\pi$
$863$	−11.3050	−0.384825	−0.192413	+	0.981314i	$0.561631\pi$
$864$	4.14590	0.141046
$865$	−29.4164	−1.00019
$866$	−4.76393	−0.161885
$867$	−24.8885	−0.845259
$868$	0.437694	0.0148563
$869$	−6.47214	−0.219552
$870$	−5.23607	−0.177519
$871$	−14.8328	−0.502591
$872$	−7.36068	−0.249264
$873$	9.00000	0.304604
$874$	−2.94427	−0.0995915
$875$	−7.94427	−0.268565
$876$	6.00000	0.202721
$877$	−29.8328	−1.00738	−0.503691	−	0.863884i	$-0.668025\pi$
$877$	−29.8328	−1.00738	−0.503691	+	0.863884i	$0.668025\pi$
$878$	4.09017	0.138037
$879$	−14.3607	−0.484374
$880$	10.1803	0.343179
$881$	45.3050	1.52636	0.763181	−	0.646184i	$-0.223636\pi$
$881$	45.3050	1.52636	0.763181	+	0.646184i	$0.223636\pi$
$882$	2.65248	0.0893135
$883$	26.8328	0.902996	0.451498	−	0.892272i	$-0.350890\pi$
$883$	26.8328	0.902996	0.451498	+	0.892272i	$0.350890\pi$
$884$	14.8328	0.498882
$885$	−12.7082	−0.427182
$886$	9.68692	0.325438
$887$	19.5836	0.657553	0.328776	−	0.944408i	$-0.393364\pi$
$887$	19.5836	0.657553	0.328776	+	0.944408i	$0.393364\pi$
$888$	8.40325	0.281995
$889$	1.70820	0.0572913
$890$	11.2361	0.376634
$891$	−0.763932	−0.0255927
$892$	3.37384	0.112964
$893$	15.4164	0.515890
$894$	−5.12461	−0.171393
$895$	−82.2492	−2.74929
$896$	2.38197	0.0795759
$897$	1.52786	0.0510139
$898$	11.7082	0.390708
$899$	−3.23607	−0.107929
$900$	−24.0000	−0.800000
$901$	−57.8885	−1.92855
$902$	1.95743	0.0651752
$903$	2.29180	0.0762662
$904$	−2.59675	−0.0863665
$905$	−82.2492	−2.73406
$906$	5.81966	0.193345
$907$	42.9574	1.42638	0.713189	−	0.700972i	$-0.247250\pi$
$907$	42.9574	1.42638	0.713189	+	0.700972i	$0.247250\pi$
$908$	−9.16718	−0.304224
$909$	−5.76393	−0.191178
$910$	0.472136	0.0156512
$911$	−20.1803	−0.668604	−0.334302	−	0.942466i	$-0.608501\pi$
$911$	−20.1803	−0.668604	−0.334302	+	0.942466i	$0.608501\pi$
$912$	19.6180	0.649618
$913$	12.5836	0.416456
$914$	−4.24922	−0.140552
$915$	33.8885	1.12032
$916$	−18.0000	−0.594737
$917$	0	0
$918$	−2.47214	−0.0815926
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	7.70820	0.254132
$921$	10.2361	0.337290
$922$	9.12461	0.300503
$923$	11.1246	0.366171
$924$	−0.334369	−0.0109999
$925$	−73.8885	−2.42944
$926$	9.48529	0.311706
$927$	−13.7639	−0.452067
$928$	−13.4164	−0.440415
$929$	−17.2361	−0.565497	−0.282749	−	0.959194i	$-0.591246\pi$
$929$	−17.2361	−0.565497	−0.282749	+	0.959194i	$0.591246\pi$
$930$	1.61803	0.0530574
$931$	43.3050	1.41926
$932$	−1.31308	−0.0430114
$933$	5.47214	0.179150
$934$	−7.66061	−0.250663
$935$	−20.9443	−0.684951
$936$	1.81966	0.0594775
$937$	−32.8328	−1.07260	−0.536301	−	0.844027i	$-0.680179\pi$
$937$	−32.8328	−1.07260	−0.536301	+	0.844027i	$0.680179\pi$
$938$	1.08204	0.0353298
$939$	6.47214	0.211210
$940$	−19.4164	−0.633293
$941$	−26.9443	−0.878358	−0.439179	−	0.898400i	$-0.644731\pi$
$941$	−26.9443	−0.878358	−0.439179	+	0.898400i	$0.644731\pi$
$942$	−1.54915	−0.0504740
$943$	−8.29180	−0.270018
$944$	−9.43769	−0.307171
$945$	1.00000	0.0325300
$946$	−2.83282	−0.0921028
$947$	−7.81966	−0.254105	−0.127052	−	0.991896i	$-0.540552\pi$
$947$	−7.81966	−0.254105	−0.127052	+	0.991896i	$0.540552\pi$
$948$	15.7082	0.510179
$949$	4.00000	0.129845
$950$	30.8328	1.00035
$951$	−32.1246	−1.04171
$952$	−2.24922	−0.0728978
$953$	−0.944272	−0.0305880	−0.0152940	−	0.999883i	$-0.504868\pi$
$953$	−0.944272	−0.0305880	−0.0152940	+	0.999883i	$0.504868\pi$
$954$	−3.41641	−0.110610
$955$	59.0689	1.91142
$956$	22.0426	0.712908
$957$	2.47214	0.0799128
$958$	0.0212862	0.000687727 0
$959$	−2.40325	−0.0776051
$960$	−19.9443	−0.643699
$961$	1.00000	0.0322581
$962$	2.69505	0.0868918
$963$	1.47214	0.0474389
$964$	25.4164	0.818607
$965$	12.7082	0.409092
$966$	−0.111456	−0.00358604
$967$	−50.5410	−1.62529	−0.812645	−	0.582759i	$-0.801973\pi$
$967$	−50.5410	−1.62529	−0.812645	+	0.582759i	$0.801973\pi$
$968$	−15.3344	−0.492865
$969$	−40.3607	−1.29657
$970$	14.5623	0.467567
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	1.85410	0.0594703
$973$	1.34752	0.0431996
$974$	1.41641	0.0453846
$975$	−16.0000	−0.512410
$976$	25.1672	0.805582
$977$	11.2918	0.361257	0.180628	−	0.983551i	$-0.442187\pi$
$977$	11.2918	0.361257	0.180628	+	0.983551i	$0.442187\pi$
$978$	−2.74265	−0.0877001
$979$	−5.30495	−0.169547
$980$	−54.5410	−1.74225
$981$	−5.00000	−0.159638
$982$	5.27864	0.168448
$983$	−28.7639	−0.917427	−0.458713	−	0.888584i	$-0.651690\pi$
$983$	−28.7639	−0.917427	−0.458713	+	0.888584i	$0.651690\pi$
$984$	−9.87539	−0.314816
$985$	−22.1803	−0.706724
$986$	8.00000	0.254772
$987$	0.583592	0.0185759
$988$	14.2918	0.454683
$989$	12.0000	0.381578
$990$	−1.23607	−0.0392848
$991$	36.7214	1.16649	0.583246	−	0.812295i	$-0.301782\pi$
$991$	36.7214	1.16649	0.583246	+	0.812295i	$0.301782\pi$
$992$	4.14590	0.131632
$993$	−20.8328	−0.661109
$994$	−0.811529	−0.0257402
$995$	51.5967	1.63573
$996$	−30.5410	−0.967729
$997$	−4.41641	−0.139869	−0.0699345	−	0.997552i	$-0.522279\pi$
$997$	−4.41641	−0.139869	−0.0699345	+	0.997552i	$0.522279\pi$
$998$	−7.59675	−0.240471
$999$	5.70820	0.180600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.a.a.1.2	✓	2
3.2	odd	2		279.2.a.b.1.1		2
4.3	odd	2		1488.2.a.q.1.1		2
5.2	odd	4		2325.2.c.h.1024.2		4
5.3	odd	4		2325.2.c.h.1024.3		4
5.4	even	2		2325.2.a.o.1.1		2
7.6	odd	2		4557.2.a.p.1.2		2
8.3	odd	2		5952.2.a.bo.1.2		2
8.5	even	2		5952.2.a.bv.1.2		2
12.11	even	2		4464.2.a.bn.1.2		2
15.14	odd	2		6975.2.a.t.1.2		2
31.30	odd	2		2883.2.a.a.1.2		2
93.92	even	2		8649.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.a.a.1.2	✓	2	1.1	even	1	trivial
279.2.a.b.1.1		2	3.2	odd	2
1488.2.a.q.1.1		2	4.3	odd	2
2325.2.a.o.1.1		2	5.4	even	2
2325.2.c.h.1024.2		4	5.2	odd	4
2325.2.c.h.1024.3		4	5.3	odd	4
2883.2.a.a.1.2		2	31.30	odd	2
4464.2.a.bn.1.2		2	12.11	even	2
4557.2.a.p.1.2		2	7.6	odd	2
5952.2.a.bo.1.2		2	8.3	odd	2
5952.2.a.bv.1.2		2	8.5	even	2
6975.2.a.t.1.2		2	15.14	odd	2
8649.2.a.m.1.1		2	93.92	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 93 = 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	93.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381966 q^{2} -1.00000 q^{3} -1.85410 q^{4} -4.23607 q^{5} +0.381966 q^{6} +0.236068 q^{7} +1.47214 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.381966 q^{2} -1.00000 q^{3} -1.85410 q^{4} -4.23607 q^{5} +0.381966 q^{6} +0.236068 q^{7} +1.47214 q^{8} +1.00000 q^{9} +1.61803 q^{10} -0.763932 q^{11} +1.85410 q^{12} +1.23607 q^{13} -0.0901699 q^{14} +4.23607 q^{15} +3.14590 q^{16} -6.47214 q^{17} -0.381966 q^{18} -6.23607 q^{19} +7.85410 q^{20} -0.236068 q^{21} +0.291796 q^{22} -1.23607 q^{23} -1.47214 q^{24} +12.9443 q^{25} -0.472136 q^{26} -1.00000 q^{27} -0.437694 q^{28} +3.23607 q^{29} -1.61803 q^{30} -1.00000 q^{31} -4.14590 q^{32} +0.763932 q^{33} +2.47214 q^{34} -1.00000 q^{35} -1.85410 q^{36} -5.70820 q^{37} +2.38197 q^{38} -1.23607 q^{39} -6.23607 q^{40} +6.70820 q^{41} +0.0901699 q^{42} -9.70820 q^{43} +1.41641 q^{44} -4.23607 q^{45} +0.472136 q^{46} -2.47214 q^{47} -3.14590 q^{48} -6.94427 q^{49} -4.94427 q^{50} +6.47214 q^{51} -2.29180 q^{52} +8.94427 q^{53} +0.381966 q^{54} +3.23607 q^{55} +0.347524 q^{56} +6.23607 q^{57} -1.23607 q^{58} -3.00000 q^{59} -7.85410 q^{60} +8.00000 q^{61} +0.381966 q^{62} +0.236068 q^{63} -4.70820 q^{64} -5.23607 q^{65} -0.291796 q^{66} -12.0000 q^{67} +12.0000 q^{68} +1.23607 q^{69} +0.381966 q^{70} +9.00000 q^{71} +1.47214 q^{72} +3.23607 q^{73} +2.18034 q^{74} -12.9443 q^{75} +11.5623 q^{76} -0.180340 q^{77} +0.472136 q^{78} +8.47214 q^{79} -13.3262 q^{80} +1.00000 q^{81} -2.56231 q^{82} -16.4721 q^{83} +0.437694 q^{84} +27.4164 q^{85} +3.70820 q^{86} -3.23607 q^{87} -1.12461 q^{88} +6.94427 q^{89} +1.61803 q^{90} +0.291796 q^{91} +2.29180 q^{92} +1.00000 q^{93} +0.944272 q^{94} +26.4164 q^{95} +4.14590 q^{96} +9.00000 q^{97} +2.65248 q^{98} -0.763932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} + 3 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 + 3 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9	\( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} + 3 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + q^{10} - 6 q^{11} - 3 q^{12} - 2 q^{13} + 11 q^{14} + 4 q^{15} + 13 q^{16} - 4 q^{17} - 3 q^{18} - 8 q^{19} + 9 q^{20} + 4 q^{21} + 14 q^{22} + 2 q^{23} + 6 q^{24} + 8 q^{25} + 8 q^{26} - 2 q^{27} - 21 q^{28} + 2 q^{29} - q^{30} - 2 q^{31} - 15 q^{32} + 6 q^{33} - 4 q^{34} - 2 q^{35} + 3 q^{36} + 2 q^{37} + 7 q^{38} + 2 q^{39} - 8 q^{40} - 11 q^{42} - 6 q^{43} - 24 q^{44} - 4 q^{45} - 8 q^{46} + 4 q^{47} - 13 q^{48} + 4 q^{49} + 8 q^{50} + 4 q^{51} - 18 q^{52} + 3 q^{54} + 2 q^{55} + 32 q^{56} + 8 q^{57} + 2 q^{58} - 6 q^{59} - 9 q^{60} + 16 q^{61} + 3 q^{62} - 4 q^{63} + 4 q^{64} - 6 q^{65} - 14 q^{66} - 24 q^{67} + 24 q^{68} - 2 q^{69} + 3 q^{70} + 18 q^{71} - 6 q^{72} + 2 q^{73} - 18 q^{74} - 8 q^{75} + 3 q^{76} + 22 q^{77} - 8 q^{78} + 8 q^{79} - 11 q^{80} + 2 q^{81} + 15 q^{82} - 24 q^{83} + 21 q^{84} + 28 q^{85} - 6 q^{86} - 2 q^{87} + 38 q^{88} - 4 q^{89} + q^{90} + 14 q^{91} + 18 q^{92} + 2 q^{93} - 16 q^{94} + 26 q^{95} + 15 q^{96} + 18 q^{97} - 26 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 + 3 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 + q^10 - 6 * q^11 - 3 * q^12 - 2 * q^13 + 11 * q^14 + 4 * q^15 + 13 * q^16 - 4 * q^17 - 3 * q^18 - 8 * q^19 + 9 * q^20 + 4 * q^21 + 14 * q^22 + 2 * q^23 + 6 * q^24 + 8 * q^25 + 8 * q^26 - 2 * q^27 - 21 * q^28 + 2 * q^29 - q^30 - 2 * q^31 - 15 * q^32 + 6 * q^33 - 4 * q^34 - 2 * q^35 + 3 * q^36 + 2 * q^37 + 7 * q^38 + 2 * q^39 - 8 * q^40 - 11 * q^42 - 6 * q^43 - 24 * q^44 - 4 * q^45 - 8 * q^46 + 4 * q^47 - 13 * q^48 + 4 * q^49 + 8 * q^50 + 4 * q^51 - 18 * q^52 + 3 * q^54 + 2 * q^55 + 32 * q^56 + 8 * q^57 + 2 * q^58 - 6 * q^59 - 9 * q^60 + 16 * q^61 + 3 * q^62 - 4 * q^63 + 4 * q^64 - 6 * q^65 - 14 * q^66 - 24 * q^67 + 24 * q^68 - 2 * q^69 + 3 * q^70 + 18 * q^71 - 6 * q^72 + 2 * q^73 - 18 * q^74 - 8 * q^75 + 3 * q^76 + 22 * q^77 - 8 * q^78 + 8 * q^79 - 11 * q^80 + 2 * q^81 + 15 * q^82 - 24 * q^83 + 21 * q^84 + 28 * q^85 - 6 * q^86 - 2 * q^87 + 38 * q^88 - 4 * q^89 + q^90 + 14 * q^91 + 18 * q^92 + 2 * q^93 - 16 * q^94 + 26 * q^95 + 15 * q^96 + 18 * q^97 - 26 * q^98 - 6 * q^99

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