LMFDB - Embedded newform 7942.2.a.cd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, 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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.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:	$63.4171892853$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} - 14 x^{14} + 128 x^{13} + 39 x^{12} - 1110 x^{11} + 348 x^{10} + 4996 x^{9} + \cdots - 359 $ x^16 - 6x^15 - 14x^14 + 128x^13 + 39x^12 - 1110x^11 + 348x^10 + 4996x^9 - 2932x^8 - 12256x^7 + 8993x^6 + 15430x^5 - 12896x^4 - 7688x^3 + 7371x^2 - 64*x - 359
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.11007$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.11007 q^{3} +1.00000 q^{4} -0.774003 q^{5} -1.11007 q^{6} +4.10393 q^{7} +1.00000 q^{8} -1.76773 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.11007 q^{3} +1.00000 q^{4} -0.774003 q^{5} -1.11007 q^{6} +4.10393 q^{7} +1.00000 q^{8} -1.76773 q^{9} -0.774003 q^{10} -1.00000 q^{11} -1.11007 q^{12} +5.19569 q^{13} +4.10393 q^{14} +0.859202 q^{15} +1.00000 q^{16} +1.38233 q^{17} -1.76773 q^{18} -0.774003 q^{20} -4.55567 q^{21} -1.00000 q^{22} +3.46287 q^{23} -1.11007 q^{24} -4.40092 q^{25} +5.19569 q^{26} +5.29254 q^{27} +4.10393 q^{28} +2.92082 q^{29} +0.859202 q^{30} -3.59274 q^{31} +1.00000 q^{32} +1.11007 q^{33} +1.38233 q^{34} -3.17645 q^{35} -1.76773 q^{36} -2.47859 q^{37} -5.76760 q^{39} -0.774003 q^{40} -1.45242 q^{41} -4.55567 q^{42} +2.15057 q^{43} -1.00000 q^{44} +1.36823 q^{45} +3.46287 q^{46} +6.48395 q^{47} -1.11007 q^{48} +9.84224 q^{49} -4.40092 q^{50} -1.53449 q^{51} +5.19569 q^{52} +8.22624 q^{53} +5.29254 q^{54} +0.774003 q^{55} +4.10393 q^{56} +2.92082 q^{58} +2.94920 q^{59} +0.859202 q^{60} +3.59649 q^{61} -3.59274 q^{62} -7.25465 q^{63} +1.00000 q^{64} -4.02148 q^{65} +1.11007 q^{66} +11.1660 q^{67} +1.38233 q^{68} -3.84405 q^{69} -3.17645 q^{70} -5.31877 q^{71} -1.76773 q^{72} -5.61146 q^{73} -2.47859 q^{74} +4.88535 q^{75} -4.10393 q^{77} -5.76760 q^{78} -17.1631 q^{79} -0.774003 q^{80} -0.571917 q^{81} -1.45242 q^{82} -9.74617 q^{83} -4.55567 q^{84} -1.06992 q^{85} +2.15057 q^{86} -3.24233 q^{87} -1.00000 q^{88} +8.13158 q^{89} +1.36823 q^{90} +21.3227 q^{91} +3.46287 q^{92} +3.98821 q^{93} +6.48395 q^{94} -1.11007 q^{96} -0.660421 q^{97} +9.84224 q^{98} +1.76773 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 10 q^{3} + 16 q^{4} + 10 q^{6} + 6 q^{7} + 16 q^{8} + 20 q^{9}+O(q^{10}) $ 16 * q + 16 * q^2 + 10 * q^3 + 16 * q^4 + 10 * q^6 + 6 * q^7 + 16 * q^8 + 20 * q^9	$ 16 q + 16 q^{2} + 10 q^{3} + 16 q^{4} + 10 q^{6} + 6 q^{7} + 16 q^{8} + 20 q^{9} - 16 q^{11} + 10 q^{12} - 2 q^{13} + 6 q^{14} - 8 q^{15} + 16 q^{16} + 20 q^{18} + 6 q^{21} - 16 q^{22} - 6 q^{23} + 10 q^{24} + 18 q^{25} - 2 q^{26} + 46 q^{27} + 6 q^{28} + 28 q^{29} - 8 q^{30} + 8 q^{31} + 16 q^{32} - 10 q^{33} + 14 q^{35} + 20 q^{36} + 32 q^{37} + 24 q^{39} + 24 q^{41} + 6 q^{42} + 28 q^{43} - 16 q^{44} - 16 q^{45} - 6 q^{46} + 32 q^{47} + 10 q^{48} + 16 q^{49} + 18 q^{50} + 14 q^{51} - 2 q^{52} + 16 q^{53} + 46 q^{54} + 6 q^{56} + 28 q^{58} + 30 q^{59} - 8 q^{60} - 8 q^{61} + 8 q^{62} - 8 q^{63} + 16 q^{64} + 26 q^{65} - 10 q^{66} + 44 q^{67} + 16 q^{69} + 14 q^{70} + 36 q^{71} + 20 q^{72} + 10 q^{73} + 32 q^{74} + 76 q^{75} - 6 q^{77} + 24 q^{78} - 16 q^{79} + 48 q^{81} + 24 q^{82} + 12 q^{83} + 6 q^{84} - 4 q^{85} + 28 q^{86} + 24 q^{87} - 16 q^{88} + 32 q^{89} - 16 q^{90} - 24 q^{91} - 6 q^{92} - 14 q^{93} + 32 q^{94} + 10 q^{96} + 52 q^{97} + 16 q^{98} - 20 q^{99}+O(q^{100}) $ 16 * q + 16 * q^2 + 10 * q^3 + 16 * q^4 + 10 * q^6 + 6 * q^7 + 16 * q^8 + 20 * q^9 - 16 * q^11 + 10 * q^12 - 2 * q^13 + 6 * q^14 - 8 * q^15 + 16 * q^16 + 20 * q^18 + 6 * q^21 - 16 * q^22 - 6 * q^23 + 10 * q^24 + 18 * q^25 - 2 * q^26 + 46 * q^27 + 6 * q^28 + 28 * q^29 - 8 * q^30 + 8 * q^31 + 16 * q^32 - 10 * q^33 + 14 * q^35 + 20 * q^36 + 32 * q^37 + 24 * q^39 + 24 * q^41 + 6 * q^42 + 28 * q^43 - 16 * q^44 - 16 * q^45 - 6 * q^46 + 32 * q^47 + 10 * q^48 + 16 * q^49 + 18 * q^50 + 14 * q^51 - 2 * q^52 + 16 * q^53 + 46 * q^54 + 6 * q^56 + 28 * q^58 + 30 * q^59 - 8 * q^60 - 8 * q^61 + 8 * q^62 - 8 * q^63 + 16 * q^64 + 26 * q^65 - 10 * q^66 + 44 * q^67 + 16 * q^69 + 14 * q^70 + 36 * q^71 + 20 * q^72 + 10 * q^73 + 32 * q^74 + 76 * q^75 - 6 * q^77 + 24 * q^78 - 16 * q^79 + 48 * q^81 + 24 * q^82 + 12 * q^83 + 6 * q^84 - 4 * q^85 + 28 * q^86 + 24 * q^87 - 16 * q^88 + 32 * q^89 - 16 * q^90 - 24 * q^91 - 6 * q^92 - 14 * q^93 + 32 * q^94 + 10 * q^96 + 52 * q^97 + 16 * q^98 - 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.11007	−0.640902	−0.320451	−	0.947265i	$-0.603834\pi$
$3$	−1.11007	−0.640902	−0.320451	+	0.947265i	$0.603834\pi$
$4$	1.00000	0.500000
$5$	−0.774003	−0.346145	−0.173072	−	0.984909i	$-0.555369\pi$
$5$	−0.774003	−0.346145	−0.173072	+	0.984909i	$0.555369\pi$
$6$	−1.11007	−0.453186
$7$	4.10393	1.55114	0.775570	−	0.631262i	$-0.217463\pi$
$7$	4.10393	1.55114	0.775570	+	0.631262i	$0.217463\pi$
$8$	1.00000	0.353553
$9$	−1.76773	−0.589245
$10$	−0.774003	−0.244761
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.11007	−0.320451
$13$	5.19569	1.44102	0.720512	−	0.693443i	$-0.243907\pi$
$13$	5.19569	1.44102	0.720512	+	0.693443i	$0.243907\pi$
$14$	4.10393	1.09682
$15$	0.859202	0.221845
$16$	1.00000	0.250000
$17$	1.38233	0.335263	0.167632	−	0.985850i	$-0.446388\pi$
$17$	1.38233	0.335263	0.167632	+	0.985850i	$0.446388\pi$
$18$	−1.76773	−0.416659
$19$	0	0
$19$	0	0
$20$	−0.774003	−0.173072
$21$	−4.55567	−0.994128
$22$	−1.00000	−0.213201
$23$	3.46287	0.722059	0.361029	−	0.932554i	$-0.382425\pi$
$23$	3.46287	0.722059	0.361029	+	0.932554i	$0.382425\pi$
$24$	−1.11007	−0.226593
$25$	−4.40092	−0.880184
$26$	5.19569	1.01896
$27$	5.29254	1.01855
$28$	4.10393	0.775570
$29$	2.92082	0.542382	0.271191	−	0.962526i	$-0.412582\pi$
$29$	2.92082	0.542382	0.271191	+	0.962526i	$0.412582\pi$
$30$	0.859202	0.156868
$31$	−3.59274	−0.645275	−0.322638	−	0.946523i	$-0.604570\pi$
$31$	−3.59274	−0.645275	−0.322638	+	0.946523i	$0.604570\pi$
$32$	1.00000	0.176777
$33$	1.11007	0.193239
$34$	1.38233	0.237067
$35$	−3.17645	−0.536919
$36$	−1.76773	−0.294622
$37$	−2.47859	−0.407478	−0.203739	−	0.979025i	$-0.565309\pi$
$37$	−2.47859	−0.407478	−0.203739	+	0.979025i	$0.565309\pi$
$38$	0	0
$39$	−5.76760	−0.923555
$40$	−0.774003	−0.122381
$41$	−1.45242	−0.226830	−0.113415	−	0.993548i	$-0.536179\pi$
$41$	−1.45242	−0.226830	−0.113415	+	0.993548i	$0.536179\pi$
$42$	−4.55567	−0.702955
$43$	2.15057	0.327959	0.163980	−	0.986464i	$-0.447567\pi$
$43$	2.15057	0.327959	0.163980	+	0.986464i	$0.447567\pi$
$44$	−1.00000	−0.150756
$45$	1.36823	0.203964
$46$	3.46287	0.510572
$47$	6.48395	0.945781	0.472891	−	0.881121i	$-0.343211\pi$
$47$	6.48395	0.945781	0.472891	+	0.881121i	$0.343211\pi$
$48$	−1.11007	−0.160226
$49$	9.84224	1.40603
$50$	−4.40092	−0.622384
$51$	−1.53449	−0.214871
$52$	5.19569	0.720512
$53$	8.22624	1.12996	0.564980	−	0.825104i	$-0.308884\pi$
$53$	8.22624	1.12996	0.564980	+	0.825104i	$0.308884\pi$
$54$	5.29254	0.720224
$55$	0.774003	0.104367
$56$	4.10393	0.548411
$57$	0	0
$58$	2.92082	0.383522
$59$	2.94920	0.383953	0.191977	−	0.981399i	$-0.438510\pi$
$59$	2.94920	0.383953	0.191977	+	0.981399i	$0.438510\pi$
$60$	0.859202	0.110922
$61$	3.59649	0.460483	0.230242	−	0.973134i	$-0.426048\pi$
$61$	3.59649	0.460483	0.230242	+	0.973134i	$0.426048\pi$
$62$	−3.59274	−0.456279
$63$	−7.25465	−0.914000
$64$	1.00000	0.125000
$65$	−4.02148	−0.498803
$66$	1.11007	0.136641
$67$	11.1660	1.36415	0.682073	−	0.731284i	$-0.261079\pi$
$67$	11.1660	1.36415	0.682073	+	0.731284i	$0.261079\pi$
$68$	1.38233	0.167632
$69$	−3.84405	−0.462769
$70$	−3.17645	−0.379659
$71$	−5.31877	−0.631222	−0.315611	−	0.948889i	$-0.602209\pi$
$71$	−5.31877	−0.631222	−0.315611	+	0.948889i	$0.602209\pi$
$72$	−1.76773	−0.208329
$73$	−5.61146	−0.656772	−0.328386	−	0.944544i	$-0.606505\pi$
$73$	−5.61146	−0.656772	−0.328386	+	0.944544i	$0.606505\pi$
$74$	−2.47859	−0.288131
$75$	4.88535	0.564112
$76$	0	0
$77$	−4.10393	−0.467686
$78$	−5.76760	−0.653052
$79$	−17.1631	−1.93100	−0.965502	−	0.260395i	$-0.916147\pi$
$79$	−17.1631	−1.93100	−0.965502	+	0.260395i	$0.916147\pi$
$80$	−0.774003	−0.0865362
$81$	−0.571917	−0.0635463
$82$	−1.45242	−0.160393
$83$	−9.74617	−1.06978	−0.534890	−	0.844921i	$-0.679647\pi$
$83$	−9.74617	−1.06978	−0.534890	+	0.844921i	$0.679647\pi$
$84$	−4.55567	−0.497064
$85$	−1.06992	−0.116050
$86$	2.15057	0.231902
$87$	−3.24233	−0.347614
$88$	−1.00000	−0.106600
$89$	8.13158	0.861946	0.430973	−	0.902365i	$-0.358170\pi$
$89$	8.13158	0.861946	0.430973	+	0.902365i	$0.358170\pi$
$90$	1.36823	0.144224
$91$	21.3227	2.23523
$92$	3.46287	0.361029
$93$	3.98821	0.413558
$94$	6.48395	0.668768
$95$	0	0
$96$	−1.11007	−0.113297
$97$	−0.660421	−0.0670556	−0.0335278	−	0.999438i	$-0.510674\pi$
$97$	−0.660421	−0.0670556	−0.0335278	+	0.999438i	$0.510674\pi$
$98$	9.84224	0.994216
$99$	1.76773	0.177664
$100$	−4.40092	−0.440092
$101$	−1.47039	−0.146310	−0.0731548	−	0.997321i	$-0.523307\pi$
$101$	−1.47039	−0.146310	−0.0731548	+	0.997321i	$0.523307\pi$
$102$	−1.53449	−0.151937
$103$	−2.66019	−0.262116	−0.131058	−	0.991375i	$-0.541837\pi$
$103$	−2.66019	−0.262116	−0.131058	+	0.991375i	$0.541837\pi$
$104$	5.19569	0.509479
$105$	3.52610	0.344112
$106$	8.22624	0.799003
$107$	15.2497	1.47425	0.737124	−	0.675758i	$-0.236184\pi$
$107$	15.2497	1.47425	0.737124	+	0.675758i	$0.236184\pi$
$108$	5.29254	0.509275
$109$	2.73598	0.262059	0.131030	−	0.991378i	$-0.458172\pi$
$109$	2.73598	0.262059	0.131030	+	0.991378i	$0.458172\pi$
$110$	0.774003	0.0737983
$111$	2.75142	0.261154
$112$	4.10393	0.387785
$113$	9.55349	0.898716	0.449358	−	0.893352i	$-0.351653\pi$
$113$	9.55349	0.898716	0.449358	+	0.893352i	$0.351653\pi$
$114$	0	0
$115$	−2.68027	−0.249937
$116$	2.92082	0.271191
$117$	−9.18459	−0.849115
$118$	2.94920	0.271496
$119$	5.67297	0.520040
$120$	0.859202	0.0784340
$121$	1.00000	0.0909091
$122$	3.59649	0.325611
$123$	1.61230	0.145376
$124$	−3.59274	−0.322638
$125$	7.27634	0.650816
$126$	−7.25465	−0.646296
$127$	−12.3443	−1.09538	−0.547689	−	0.836682i	$-0.684492\pi$
$127$	−12.3443	−1.09538	−0.547689	+	0.836682i	$0.684492\pi$
$128$	1.00000	0.0883883
$129$	−2.38730	−0.210190
$130$	−4.02148	−0.352707
$131$	−7.50742	−0.655927	−0.327963	−	0.944690i	$-0.606362\pi$
$131$	−7.50742	−0.655927	−0.327963	+	0.944690i	$0.606362\pi$
$132$	1.11007	0.0966196
$133$	0	0
$134$	11.1660	0.964596
$135$	−4.09644	−0.352566
$136$	1.38233	0.118533
$137$	6.86830	0.586799	0.293399	−	0.955990i	$-0.405213\pi$
$137$	6.86830	0.586799	0.293399	+	0.955990i	$0.405213\pi$
$138$	−3.84405	−0.327227
$139$	−10.1137	−0.857835	−0.428918	−	0.903344i	$-0.641105\pi$
$139$	−10.1137	−0.857835	−0.428918	+	0.903344i	$0.641105\pi$
$140$	−3.17645	−0.268459
$141$	−7.19767	−0.606153
$142$	−5.31877	−0.446342
$143$	−5.19569	−0.434485
$144$	−1.76773	−0.147311
$145$	−2.26072	−0.187743
$146$	−5.61146	−0.464408
$147$	−10.9256	−0.901130
$148$	−2.47859	−0.203739
$149$	−15.7180	−1.28767	−0.643833	−	0.765166i	$-0.722657\pi$
$149$	−15.7180	−1.28767	−0.643833	+	0.765166i	$0.722657\pi$
$150$	4.88535	0.398887
$151$	17.1407	1.39489	0.697444	−	0.716640i	$-0.254321\pi$
$151$	17.1407	1.39489	0.697444	+	0.716640i	$0.254321\pi$
$152$	0	0
$153$	−2.44358	−0.197552
$154$	−4.10393	−0.330704
$155$	2.78079	0.223359
$156$	−5.76760	−0.461778
$157$	−22.0313	−1.75829	−0.879143	−	0.476558i	$-0.841884\pi$
$157$	−22.0313	−1.75829	−0.879143	+	0.476558i	$0.841884\pi$
$158$	−17.1631	−1.36543
$159$	−9.13174	−0.724194
$160$	−0.774003	−0.0611903
$161$	14.2114	1.12001
$162$	−0.571917	−0.0449340
$163$	−9.18227	−0.719211	−0.359605	−	0.933104i	$-0.617089\pi$
$163$	−9.18227	−0.719211	−0.359605	+	0.933104i	$0.617089\pi$
$164$	−1.45242	−0.113415
$165$	−0.859202	−0.0668888
$166$	−9.74617	−0.756449
$167$	24.8852	1.92567	0.962836	−	0.270087i	$-0.0870524\pi$
$167$	24.8852	1.92567	0.962836	+	0.270087i	$0.0870524\pi$
$168$	−4.55567	−0.351478
$169$	13.9951	1.07655
$170$	−1.06992	−0.0820595
$171$	0	0
$172$	2.15057	0.163980
$173$	5.92707	0.450626	0.225313	−	0.974286i	$-0.427659\pi$
$173$	5.92707	0.450626	0.225313	+	0.974286i	$0.427659\pi$
$174$	−3.24233	−0.245800
$175$	−18.0611	−1.36529
$176$	−1.00000	−0.0753778
$177$	−3.27384	−0.246077
$178$	8.13158	0.609488
$179$	13.3633	0.998819	0.499410	−	0.866366i	$-0.333550\pi$
$179$	13.3633	0.998819	0.499410	+	0.866366i	$0.333550\pi$
$180$	1.36823	0.101982
$181$	6.02646	0.447943	0.223972	−	0.974596i	$-0.428098\pi$
$181$	6.02646	0.447943	0.223972	+	0.974596i	$0.428098\pi$
$182$	21.3227	1.58055
$183$	−3.99237	−0.295125
$184$	3.46287	0.255286
$185$	1.91844	0.141046
$186$	3.98821	0.292430
$187$	−1.38233	−0.101086
$188$	6.48395	0.472891
$189$	21.7202	1.57991
$190$	0	0
$191$	−14.8933	−1.07764	−0.538819	−	0.842421i	$-0.681129\pi$
$191$	−14.8933	−1.07764	−0.538819	+	0.842421i	$0.681129\pi$
$192$	−1.11007	−0.0801128
$193$	5.42938	0.390815	0.195407	−	0.980722i	$-0.437397\pi$
$193$	5.42938	0.390815	0.195407	+	0.980722i	$0.437397\pi$
$194$	−0.660421	−0.0474155
$195$	4.46414	0.319684
$196$	9.84224	0.703017
$197$	−18.9028	−1.34677	−0.673384	−	0.739293i	$-0.735160\pi$
$197$	−18.9028	−1.34677	−0.673384	+	0.739293i	$0.735160\pi$
$198$	1.76773	0.125627
$199$	12.4997	0.886083	0.443042	−	0.896501i	$-0.353899\pi$
$199$	12.4997	0.886083	0.443042	+	0.896501i	$0.353899\pi$
$200$	−4.40092	−0.311192
$201$	−12.3951	−0.874283
$202$	−1.47039	−0.103457
$203$	11.9868	0.841310
$204$	−1.53449	−0.107435
$205$	1.12418	0.0785160
$206$	−2.66019	−0.185344
$207$	−6.12143	−0.425469
$208$	5.19569	0.360256
$209$	0	0
$210$	3.52610	0.243324
$211$	28.3972	1.95494	0.977471	−	0.211071i	$-0.0676950\pi$
$211$	28.3972	1.95494	0.977471	+	0.211071i	$0.0676950\pi$
$212$	8.22624	0.564980
$213$	5.90424	0.404552
$214$	15.2497	1.04245
$215$	−1.66455	−0.113521
$216$	5.29254	0.360112
$217$	−14.7444	−1.00091
$218$	2.73598	0.185304
$219$	6.22914	0.420927
$220$	0.774003	0.0521833
$221$	7.18213	0.483122
$222$	2.75142	0.184663
$223$	−3.26943	−0.218937	−0.109469	−	0.993990i	$-0.534915\pi$
$223$	−3.26943	−0.218937	−0.109469	+	0.993990i	$0.534915\pi$
$224$	4.10393	0.274205
$225$	7.77965	0.518644
$226$	9.55349	0.635488
$227$	−7.37560	−0.489536	−0.244768	−	0.969582i	$-0.578712\pi$
$227$	−7.37560	−0.489536	−0.244768	+	0.969582i	$0.578712\pi$
$228$	0	0
$229$	28.6009	1.89000	0.945002	−	0.327065i	$-0.106060\pi$
$229$	28.6009	1.89000	0.945002	+	0.327065i	$0.106060\pi$
$230$	−2.68027	−0.176732
$231$	4.55567	0.299741
$232$	2.92082	0.191761
$233$	26.8013	1.75581	0.877905	−	0.478835i	$-0.158941\pi$
$233$	26.8013	1.75581	0.877905	+	0.478835i	$0.158941\pi$
$234$	−9.18459	−0.600415
$235$	−5.01860	−0.327377
$236$	2.94920	0.191977
$237$	19.0524	1.23758
$238$	5.67297	0.367724
$239$	29.7199	1.92242	0.961212	−	0.275812i	$-0.0889468\pi$
$239$	29.7199	1.92242	0.961212	+	0.275812i	$0.0889468\pi$
$240$	0.859202	0.0554612
$241$	24.5878	1.58384	0.791920	−	0.610624i	$-0.209081\pi$
$241$	24.5878	1.58384	0.791920	+	0.610624i	$0.209081\pi$
$242$	1.00000	0.0642824
$243$	−15.2428	−0.977823
$244$	3.59649	0.230242
$245$	−7.61792	−0.486691
$246$	1.61230	0.102796
$247$	0	0
$248$	−3.59274	−0.228139
$249$	10.8190	0.685625
$250$	7.27634	0.460196
$251$	−16.7711	−1.05858	−0.529291	−	0.848440i	$-0.677542\pi$
$251$	−16.7711	−1.05858	−0.529291	+	0.848440i	$0.677542\pi$
$252$	−7.25465	−0.457000
$253$	−3.46287	−0.217709
$254$	−12.3443	−0.774550
$255$	1.18770	0.0743764
$256$	1.00000	0.0625000
$257$	−15.1683	−0.946174	−0.473087	−	0.881016i	$-0.656860\pi$
$257$	−15.1683	−0.946174	−0.473087	+	0.881016i	$0.656860\pi$
$258$	−2.38730	−0.148627
$259$	−10.1720	−0.632056
$260$	−4.02148	−0.249401
$261$	−5.16323	−0.319596
$262$	−7.50742	−0.463810
$263$	−13.4736	−0.830818	−0.415409	−	0.909635i	$-0.636362\pi$
$263$	−13.4736	−0.830818	−0.415409	+	0.909635i	$0.636362\pi$
$264$	1.11007	0.0683204
$265$	−6.36714	−0.391130
$266$	0	0
$267$	−9.02666	−0.552423
$268$	11.1660	0.682073
$269$	3.49743	0.213242	0.106621	−	0.994300i	$-0.465997\pi$
$269$	3.49743	0.213242	0.106621	+	0.994300i	$0.465997\pi$
$270$	−4.09644	−0.249302
$271$	24.8015	1.50659	0.753293	−	0.657685i	$-0.228464\pi$
$271$	24.8015	1.50659	0.753293	+	0.657685i	$0.228464\pi$
$272$	1.38233	0.0838158
$273$	−23.6698	−1.43256
$274$	6.86830	0.414929
$275$	4.40092	0.265385
$276$	−3.84405	−0.231384
$277$	31.5106	1.89329	0.946645	−	0.322279i	$-0.104449\pi$
$277$	31.5106	1.89329	0.946645	+	0.322279i	$0.104449\pi$
$278$	−10.1137	−0.606581
$279$	6.35101	0.380225
$280$	−3.17645	−0.189829
$281$	−11.9153	−0.710808	−0.355404	−	0.934713i	$-0.615657\pi$
$281$	−11.9153	−0.710808	−0.355404	+	0.934713i	$0.615657\pi$
$282$	−7.19767	−0.428615
$283$	4.42445	0.263006	0.131503	−	0.991316i	$-0.458020\pi$
$283$	4.42445	0.263006	0.131503	+	0.991316i	$0.458020\pi$
$284$	−5.31877	−0.315611
$285$	0	0
$286$	−5.19569	−0.307227
$287$	−5.96063	−0.351845
$288$	−1.76773	−0.104165
$289$	−15.0892	−0.887599
$290$	−2.26072	−0.132754
$291$	0.733117	0.0429761
$292$	−5.61146	−0.328386
$293$	−9.41980	−0.550311	−0.275155	−	0.961400i	$-0.588729\pi$
$293$	−9.41980	−0.550311	−0.275155	+	0.961400i	$0.588729\pi$
$294$	−10.9256	−0.637195
$295$	−2.28269	−0.132903
$296$	−2.47859	−0.144065
$297$	−5.29254	−0.307104
$298$	−15.7180	−0.910518
$299$	17.9920	1.04050
$300$	4.88535	0.282056
$301$	8.82579	0.508710
$302$	17.1407	0.986334
$303$	1.63225	0.0937702
$304$	0	0
$305$	−2.78369	−0.159394
$306$	−2.44358	−0.139690
$307$	3.54698	0.202437	0.101218	−	0.994864i	$-0.467726\pi$
$307$	3.54698	0.202437	0.101218	+	0.994864i	$0.467726\pi$
$308$	−4.10393	−0.233843
$309$	2.95301	0.167991
$310$	2.78079	0.157938
$311$	−25.9239	−1.47001	−0.735004	−	0.678063i	$-0.762820\pi$
$311$	−25.9239	−1.47001	−0.735004	+	0.678063i	$0.762820\pi$
$312$	−5.76760	−0.326526
$313$	−13.7951	−0.779745	−0.389872	−	0.920869i	$-0.627481\pi$
$313$	−13.7951	−0.779745	−0.389872	+	0.920869i	$0.627481\pi$
$314$	−22.0313	−1.24330
$315$	5.61513	0.316376
$316$	−17.1631	−0.965502
$317$	35.2559	1.98017	0.990085	−	0.140471i	$-0.0448615\pi$
$317$	35.2559	1.98017	0.990085	+	0.140471i	$0.0448615\pi$
$318$	−9.13174	−0.512083
$319$	−2.92082	−0.163534
$320$	−0.774003	−0.0432681
$321$	−16.9283	−0.944848
$322$	14.2114	0.791969
$323$	0	0
$324$	−0.571917	−0.0317732
$325$	−22.8658	−1.26837
$326$	−9.18227	−0.508559
$327$	−3.03714	−0.167954
$328$	−1.45242	−0.0801965
$329$	26.6097	1.46704
$330$	−0.859202	−0.0472975
$331$	19.2912	1.06034	0.530170	−	0.847892i	$-0.322128\pi$
$331$	19.2912	1.06034	0.530170	+	0.847892i	$0.322128\pi$
$332$	−9.74617	−0.534890
$333$	4.38149	0.240104
$334$	24.8852	1.36166
$335$	−8.64253	−0.472192
$336$	−4.55567	−0.248532
$337$	0.653866	0.0356184	0.0178092	−	0.999841i	$-0.494331\pi$
$337$	0.653866	0.0356184	0.0178092	+	0.999841i	$0.494331\pi$
$338$	13.9951	0.761236
$339$	−10.6051	−0.575989
$340$	−1.06992	−0.0580248
$341$	3.59274	0.194558
$342$	0	0
$343$	11.6643	0.629815
$344$	2.15057	0.115951
$345$	2.97530	0.160185
$346$	5.92707	0.318641
$347$	−0.714978	−0.0383820	−0.0191910	−	0.999816i	$-0.506109\pi$
$347$	−0.714978	−0.0383820	−0.0191910	+	0.999816i	$0.506109\pi$
$348$	−3.24233	−0.173807
$349$	5.79508	0.310203	0.155102	−	0.987899i	$-0.450429\pi$
$349$	5.79508	0.310203	0.155102	+	0.987899i	$0.450429\pi$
$350$	−18.0611	−0.965404
$351$	27.4984	1.46776
$352$	−1.00000	−0.0533002
$353$	7.06391	0.375974	0.187987	−	0.982171i	$-0.439804\pi$
$353$	7.06391	0.375974	0.187987	+	0.982171i	$0.439804\pi$
$354$	−3.27384	−0.174002
$355$	4.11675	0.218494
$356$	8.13158	0.430973
$357$	−6.29742	−0.333295
$358$	13.3633	0.706272
$359$	12.8776	0.679654	0.339827	−	0.940488i	$-0.389631\pi$
$359$	12.8776	0.679654	0.339827	+	0.940488i	$0.389631\pi$
$360$	1.36823	0.0721121
$361$	0	0
$362$	6.02646	0.316744
$363$	−1.11007	−0.0582638
$364$	21.3227	1.11761
$365$	4.34329	0.227338
$366$	−3.99237	−0.208685
$367$	−35.9675	−1.87749	−0.938745	−	0.344613i	$-0.888010\pi$
$367$	−35.9675	−1.87749	−0.938745	+	0.344613i	$0.888010\pi$
$368$	3.46287	0.180515
$369$	2.56749	0.133658
$370$	1.91844	0.0997349
$371$	33.7599	1.75273
$372$	3.98821	0.206779
$373$	−11.0114	−0.570147	−0.285074	−	0.958506i	$-0.592018\pi$
$373$	−11.0114	−0.570147	−0.285074	+	0.958506i	$0.592018\pi$
$374$	−1.38233	−0.0714784
$375$	−8.07728	−0.417109
$376$	6.48395	0.334384
$377$	15.1756	0.781585
$378$	21.7202	1.11717
$379$	−33.8016	−1.73627	−0.868135	−	0.496328i	$-0.834681\pi$
$379$	−33.8016	−1.73627	−0.868135	+	0.496328i	$0.834681\pi$
$380$	0	0
$381$	13.7031	0.702030
$382$	−14.8933	−0.762006
$383$	28.9498	1.47927	0.739633	−	0.673010i	$-0.234999\pi$
$383$	28.9498	1.47927	0.739633	+	0.673010i	$0.234999\pi$
$384$	−1.11007	−0.0566483
$385$	3.17645	0.161887
$386$	5.42938	0.276348
$387$	−3.80164	−0.193248
$388$	−0.660421	−0.0335278
$389$	25.2740	1.28144	0.640720	−	0.767774i	$-0.278636\pi$
$389$	25.2740	1.28144	0.640720	+	0.767774i	$0.278636\pi$
$390$	4.46414	0.226051
$391$	4.78682	0.242080
$392$	9.84224	0.497108
$393$	8.33380	0.420385
$394$	−18.9028	−0.952309
$395$	13.2843	0.668407
$396$	1.76773	0.0888320
$397$	24.2779	1.21847	0.609235	−	0.792989i	$-0.291476\pi$
$397$	24.2779	1.21847	0.609235	+	0.792989i	$0.291476\pi$
$398$	12.4997	0.626555
$399$	0	0
$400$	−4.40092	−0.220046
$401$	−10.5621	−0.527447	−0.263724	−	0.964598i	$-0.584951\pi$
$401$	−10.5621	−0.527447	−0.263724	+	0.964598i	$0.584951\pi$
$402$	−12.3951	−0.618212
$403$	−18.6668	−0.929857
$404$	−1.47039	−0.0731548
$405$	0.442665	0.0219962
$406$	11.9868	0.594896
$407$	2.47859	0.122859
$408$	−1.53449	−0.0759683
$409$	20.9834	1.03756	0.518782	−	0.854906i	$-0.326386\pi$
$409$	20.9834	1.03756	0.518782	+	0.854906i	$0.326386\pi$
$410$	1.12418	0.0555192
$411$	−7.62433	−0.376081
$412$	−2.66019	−0.131058
$413$	12.1033	0.595565
$414$	−6.12143	−0.300852
$415$	7.54356	0.370299
$416$	5.19569	0.254739
$417$	11.2270	0.549788
$418$	0	0
$419$	4.08683	0.199655	0.0998274	−	0.995005i	$-0.468171\pi$
$419$	4.08683	0.199655	0.0998274	+	0.995005i	$0.468171\pi$
$420$	3.52610	0.172056
$421$	−31.7099	−1.54545	−0.772724	−	0.634743i	$-0.781106\pi$
$421$	−31.7099	−1.54545	−0.772724	+	0.634743i	$0.781106\pi$
$422$	28.3972	1.38235
$423$	−11.4619	−0.557296
$424$	8.22624	0.399501
$425$	−6.08350	−0.295093
$426$	5.90424	0.286061
$427$	14.7597	0.714273
$428$	15.2497	0.737124
$429$	5.76760	0.278462
$430$	−1.66455	−0.0802717
$431$	29.6640	1.42887	0.714433	−	0.699703i	$-0.246685\pi$
$431$	29.6640	1.42887	0.714433	+	0.699703i	$0.246685\pi$
$432$	5.29254	0.254638
$433$	−36.6044	−1.75909	−0.879547	−	0.475812i	$-0.842154\pi$
$433$	−36.6044	−1.75909	−0.879547	+	0.475812i	$0.842154\pi$
$434$	−14.7444	−0.707752
$435$	2.50957	0.120325
$436$	2.73598	0.131030
$437$	0	0
$438$	6.22914	0.297640
$439$	−7.37392	−0.351938	−0.175969	−	0.984396i	$-0.556306\pi$
$439$	−7.37392	−0.351938	−0.175969	+	0.984396i	$0.556306\pi$
$440$	0.774003	0.0368992
$441$	−17.3985	−0.828498
$442$	7.18213	0.341619
$443$	−1.65116	−0.0784492	−0.0392246	−	0.999230i	$-0.512489\pi$
$443$	−1.65116	−0.0784492	−0.0392246	+	0.999230i	$0.512489\pi$
$444$	2.75142	0.130577
$445$	−6.29387	−0.298358
$446$	−3.26943	−0.154812
$447$	17.4481	0.825268
$448$	4.10393	0.193892
$449$	15.2799	0.721102	0.360551	−	0.932740i	$-0.382589\pi$
$449$	15.2799	0.721102	0.360551	+	0.932740i	$0.382589\pi$
$450$	7.77965	0.366736
$451$	1.45242	0.0683918
$452$	9.55349	0.449358
$453$	−19.0274	−0.893986
$454$	−7.37560	−0.346154
$455$	−16.5039	−0.773713
$456$	0	0
$457$	−14.2014	−0.664316	−0.332158	−	0.943224i	$-0.607777\pi$
$457$	−14.2014	−0.664316	−0.332158	+	0.943224i	$0.607777\pi$
$458$	28.6009	1.33643
$459$	7.31602	0.341482
$460$	−2.68027	−0.124968
$461$	−34.6919	−1.61576	−0.807881	−	0.589346i	$-0.799386\pi$
$461$	−34.6919	−1.61576	−0.807881	+	0.589346i	$0.799386\pi$
$462$	4.55567	0.211949
$463$	14.6523	0.680950	0.340475	−	0.940253i	$-0.389412\pi$
$463$	14.6523	0.680950	0.340475	+	0.940253i	$0.389412\pi$
$464$	2.92082	0.135595
$465$	−3.08689	−0.143151
$466$	26.8013	1.24155
$467$	4.44379	0.205634	0.102817	−	0.994700i	$-0.467214\pi$
$467$	4.44379	0.205634	0.102817	+	0.994700i	$0.467214\pi$
$468$	−9.18459	−0.424558
$469$	45.8245	2.11598
$470$	−5.01860	−0.231491
$471$	24.4564	1.12689
$472$	2.94920	0.135748
$473$	−2.15057	−0.0988834
$474$	19.0524	0.875104
$475$	0	0
$476$	5.67297	0.260020
$477$	−14.5418	−0.665823
$478$	29.7199	1.35936
$479$	41.8374	1.91160	0.955800	−	0.294019i	$-0.0949930\pi$
$479$	41.8374	1.91160	0.955800	+	0.294019i	$0.0949930\pi$
$480$	0.859202	0.0392170
$481$	−12.8780	−0.587186
$482$	24.5878	1.11994
$483$	−15.7757	−0.717819
$484$	1.00000	0.0454545
$485$	0.511168	0.0232109
$486$	−15.2428	−0.691425
$487$	15.9096	0.720933	0.360467	−	0.932772i	$-0.382617\pi$
$487$	15.9096	0.720933	0.360467	+	0.932772i	$0.382617\pi$
$488$	3.59649	0.162805
$489$	10.1930	0.460944
$490$	−7.61792	−0.344143
$491$	−31.1476	−1.40567	−0.702836	−	0.711352i	$-0.748083\pi$
$491$	−31.1476	−1.40567	−0.702836	+	0.711352i	$0.748083\pi$
$492$	1.61230	0.0726879
$493$	4.03752	0.181841
$494$	0	0
$495$	−1.36823	−0.0614974
$496$	−3.59274	−0.161319
$497$	−21.8279	−0.979114
$498$	10.8190	0.484810
$499$	−33.8679	−1.51613	−0.758067	−	0.652177i	$-0.773856\pi$
$499$	−33.8679	−1.51613	−0.758067	+	0.652177i	$0.773856\pi$
$500$	7.27634	0.325408
$501$	−27.6244	−1.23417
$502$	−16.7711	−0.748531
$503$	−11.6852	−0.521019	−0.260509	−	0.965471i	$-0.583891\pi$
$503$	−11.6852	−0.521019	−0.260509	+	0.965471i	$0.583891\pi$
$504$	−7.25465	−0.323148
$505$	1.13809	0.0506443
$506$	−3.46287	−0.153943
$507$	−15.5357	−0.689963
$508$	−12.3443	−0.547689
$509$	31.8808	1.41309	0.706545	−	0.707668i	$-0.250253\pi$
$509$	31.8808	1.41309	0.706545	+	0.707668i	$0.250253\pi$
$510$	1.18770	0.0525921
$511$	−23.0290	−1.01874
$512$	1.00000	0.0441942
$513$	0	0
$514$	−15.1683	−0.669046
$515$	2.05899	0.0907301
$516$	−2.38730	−0.105095
$517$	−6.48395	−0.285164
$518$	−10.1720	−0.446931
$519$	−6.57949	−0.288807
$520$	−4.02148	−0.176353
$521$	38.5303	1.68805	0.844023	−	0.536308i	$-0.180181\pi$
$521$	38.5303	1.68805	0.844023	+	0.536308i	$0.180181\pi$
$522$	−5.16323	−0.225988
$523$	−5.38542	−0.235488	−0.117744	−	0.993044i	$-0.537566\pi$
$523$	−5.38542	−0.235488	−0.117744	+	0.993044i	$0.537566\pi$
$524$	−7.50742	−0.327963
$525$	20.0491	0.875016
$526$	−13.4736	−0.587477
$527$	−4.96634	−0.216337
$528$	1.11007	0.0483098
$529$	−11.0085	−0.478631
$530$	−6.36714	−0.276571
$531$	−5.21340	−0.226242
$532$	0	0
$533$	−7.54632	−0.326867
$534$	−9.02666	−0.390622
$535$	−11.8033	−0.510303
$536$	11.1660	0.482298
$537$	−14.8343	−0.640145
$538$	3.49743	0.150785
$539$	−9.84224	−0.423935
$540$	−4.09644	−0.176283
$541$	4.42055	0.190054	0.0950272	−	0.995475i	$-0.469706\pi$
$541$	4.42055	0.190054	0.0950272	+	0.995475i	$0.469706\pi$
$542$	24.8015	1.06532
$543$	−6.68982	−0.287088
$544$	1.38233	0.0592667
$545$	−2.11766	−0.0907105
$546$	−23.6698	−1.01297
$547$	−21.3849	−0.914354	−0.457177	−	0.889376i	$-0.651139\pi$
$547$	−21.3849	−0.914354	−0.457177	+	0.889376i	$0.651139\pi$
$548$	6.86830	0.293399
$549$	−6.35763	−0.271337
$550$	4.40092	0.187656
$551$	0	0
$552$	−3.84405	−0.163613
$553$	−70.4363	−2.99526
$554$	31.5106	1.33876
$555$	−2.12961	−0.0903970
$556$	−10.1137	−0.428918
$557$	29.0899	1.23258	0.616288	−	0.787520i	$-0.288636\pi$
$557$	29.0899	1.23258	0.616288	+	0.787520i	$0.288636\pi$
$558$	6.35101	0.268860
$559$	11.1737	0.472597
$560$	−3.17645	−0.134230
$561$	1.53449	0.0647860
$562$	−11.9153	−0.502617
$563$	13.2076	0.556633	0.278316	−	0.960489i	$-0.410224\pi$
$563$	13.2076	0.556633	0.278316	+	0.960489i	$0.410224\pi$
$564$	−7.19767	−0.303077
$565$	−7.39443	−0.311086
$566$	4.42445	0.185973
$567$	−2.34711	−0.0985692
$568$	−5.31877	−0.223171
$569$	−39.2120	−1.64385	−0.821927	−	0.569593i	$-0.807101\pi$
$569$	−39.2120	−1.64385	−0.821927	+	0.569593i	$0.807101\pi$
$570$	0	0
$571$	20.1551	0.843463	0.421731	−	0.906721i	$-0.361423\pi$
$571$	20.1551	0.843463	0.421731	+	0.906721i	$0.361423\pi$
$572$	−5.19569	−0.217243
$573$	16.5326	0.690661
$574$	−5.96063	−0.248792
$575$	−15.2398	−0.635544
$576$	−1.76773	−0.0736556
$577$	36.1883	1.50654	0.753269	−	0.657713i	$-0.228476\pi$
$577$	36.1883	1.50654	0.753269	+	0.657713i	$0.228476\pi$
$578$	−15.0892	−0.627627
$579$	−6.02701	−0.250474
$580$	−2.26072	−0.0938713
$581$	−39.9976	−1.65938
$582$	0.733117	0.0303887
$583$	−8.22624	−0.340696
$584$	−5.61146	−0.232204
$585$	7.10890	0.293917
$586$	−9.41980	−0.389128
$587$	2.36877	0.0977697	0.0488849	−	0.998804i	$-0.484433\pi$
$587$	2.36877	0.0977697	0.0488849	+	0.998804i	$0.484433\pi$
$588$	−10.9256	−0.450565
$589$	0	0
$590$	−2.28269	−0.0939770
$591$	20.9835	0.863146
$592$	−2.47859	−0.101870
$593$	−14.1167	−0.579704	−0.289852	−	0.957071i	$-0.593606\pi$
$593$	−14.1167	−0.579704	−0.289852	+	0.957071i	$0.593606\pi$
$594$	−5.29254	−0.217156
$595$	−4.39089	−0.180009
$596$	−15.7180	−0.643833
$597$	−13.8756	−0.567892
$598$	17.9920	0.735747
$599$	27.8599	1.13832	0.569162	−	0.822225i	$-0.307268\pi$
$599$	27.8599	1.13832	0.569162	+	0.822225i	$0.307268\pi$
$600$	4.88535	0.199444
$601$	−5.06763	−0.206713	−0.103356	−	0.994644i	$-0.532958\pi$
$601$	−5.06763	−0.206713	−0.103356	+	0.994644i	$0.532958\pi$
$602$	8.82579	0.359712
$603$	−19.7385	−0.803815
$604$	17.1407	0.697444
$605$	−0.774003	−0.0314677
$606$	1.63225	0.0663055
$607$	0.906200	0.0367815	0.0183908	−	0.999831i	$-0.494146\pi$
$607$	0.906200	0.0367815	0.0183908	+	0.999831i	$0.494146\pi$
$608$	0	0
$609$	−13.3063	−0.539197
$610$	−2.78369	−0.112708
$611$	33.6886	1.36289
$612$	−2.44358	−0.0987760
$613$	12.3869	0.500303	0.250151	−	0.968207i	$-0.419520\pi$
$613$	12.3869	0.500303	0.250151	+	0.968207i	$0.419520\pi$
$614$	3.54698	0.143144
$615$	−1.24792	−0.0503211
$616$	−4.10393	−0.165352
$617$	−23.6809	−0.953358	−0.476679	−	0.879077i	$-0.658160\pi$
$617$	−23.6809	−0.953358	−0.476679	+	0.879077i	$0.658160\pi$
$618$	2.95301	0.118787
$619$	22.7019	0.912466	0.456233	−	0.889860i	$-0.349198\pi$
$619$	22.7019	0.912466	0.456233	+	0.889860i	$0.349198\pi$
$620$	2.78079	0.111679
$621$	18.3274	0.735453
$622$	−25.9239	−1.03945
$623$	33.3714	1.33700
$624$	−5.76760	−0.230889
$625$	16.3727	0.654907
$626$	−13.7951	−0.551363
$627$	0	0
$628$	−22.0313	−0.879143
$629$	−3.42622	−0.136612
$630$	5.61513	0.223712
$631$	21.6232	0.860807	0.430404	−	0.902637i	$-0.358371\pi$
$631$	21.6232	0.860807	0.430404	+	0.902637i	$0.358371\pi$
$632$	−17.1631	−0.682713
$633$	−31.5230	−1.25293
$634$	35.2559	1.40019
$635$	9.55452	0.379160
$636$	−9.13174	−0.362097
$637$	51.1372	2.02613
$638$	−2.92082	−0.115636
$639$	9.40217	0.371944
$640$	−0.774003	−0.0305952
$641$	31.6629	1.25061	0.625304	−	0.780381i	$-0.284975\pi$
$641$	31.6629	1.25061	0.625304	+	0.780381i	$0.284975\pi$
$642$	−16.9283	−0.668109
$643$	−30.7888	−1.21419	−0.607096	−	0.794628i	$-0.707666\pi$
$643$	−30.7888	−1.21419	−0.607096	+	0.794628i	$0.707666\pi$
$644$	14.2114	0.560007
$645$	1.84777	0.0727561
$646$	0	0
$647$	8.82313	0.346873	0.173437	−	0.984845i	$-0.444513\pi$
$647$	8.82313	0.346873	0.173437	+	0.984845i	$0.444513\pi$
$648$	−0.571917	−0.0224670
$649$	−2.94920	−0.115766
$650$	−22.8658	−0.896870
$651$	16.3673	0.641487
$652$	−9.18227	−0.359605
$653$	15.9441	0.623939	0.311970	−	0.950092i	$-0.399011\pi$
$653$	15.9441	0.623939	0.311970	+	0.950092i	$0.399011\pi$
$654$	−3.03714	−0.118762
$655$	5.81077	0.227046
$656$	−1.45242	−0.0567075
$657$	9.91957	0.386999
$658$	26.6097	1.03735
$659$	−12.6026	−0.490929	−0.245465	−	0.969406i	$-0.578941\pi$
$659$	−12.6026	−0.490929	−0.245465	+	0.969406i	$0.578941\pi$
$660$	−0.859202	−0.0334444
$661$	−43.7314	−1.70095	−0.850477	−	0.526013i	$-0.823686\pi$
$661$	−43.7314	−1.70095	−0.850477	+	0.526013i	$0.823686\pi$
$662$	19.2912	0.749773
$663$	−7.97270	−0.309634
$664$	−9.74617	−0.378225
$665$	0	0
$666$	4.38149	0.169779
$667$	10.1144	0.391632
$668$	24.8852	0.962836
$669$	3.62931	0.140317
$670$	−8.64253	−0.333890
$671$	−3.59649	−0.138841
$672$	−4.55567	−0.175739
$673$	−34.0098	−1.31098	−0.655491	−	0.755203i	$-0.727538\pi$
$673$	−34.0098	−1.31098	−0.655491	+	0.755203i	$0.727538\pi$
$674$	0.653866	0.0251860
$675$	−23.2920	−0.896511
$676$	13.9951	0.538275
$677$	−30.2431	−1.16234	−0.581169	−	0.813783i	$-0.697404\pi$
$677$	−30.2431	−1.16234	−0.581169	+	0.813783i	$0.697404\pi$
$678$	−10.6051	−0.407286
$679$	−2.71032	−0.104013
$680$	−1.06992	−0.0410297
$681$	8.18746	0.313744
$682$	3.59274	0.137573
$683$	−20.7998	−0.795882	−0.397941	−	0.917411i	$-0.630275\pi$
$683$	−20.7998	−0.795882	−0.397941	+	0.917411i	$0.630275\pi$
$684$	0	0
$685$	−5.31609	−0.203117
$686$	11.6643	0.445346
$687$	−31.7492	−1.21131
$688$	2.15057	0.0819898
$689$	42.7410	1.62830
$690$	2.97530	0.113268
$691$	−36.5565	−1.39067	−0.695337	−	0.718684i	$-0.744745\pi$
$691$	−36.5565	−1.39067	−0.695337	+	0.718684i	$0.744745\pi$
$692$	5.92707	0.225313
$693$	7.25465	0.275582
$694$	−0.714978	−0.0271402
$695$	7.82806	0.296935
$696$	−3.24233	−0.122900
$697$	−2.00772	−0.0760477
$698$	5.79508	0.219347
$699$	−29.7514	−1.12530
$700$	−18.0611	−0.682644
$701$	−1.32953	−0.0502156	−0.0251078	−	0.999685i	$-0.507993\pi$
$701$	−1.32953	−0.0502156	−0.0251078	+	0.999685i	$0.507993\pi$
$702$	27.4984	1.03786
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	5.57102	0.209817
$706$	7.06391	0.265854
$707$	−6.03439	−0.226947
$708$	−3.27384	−0.123038
$709$	−36.1810	−1.35881	−0.679403	−	0.733766i	$-0.737761\pi$
$709$	−36.1810	−1.35881	−0.679403	+	0.733766i	$0.737761\pi$
$710$	4.11675	0.154499
$711$	30.3399	1.13783
$712$	8.13158	0.304744
$713$	−12.4412	−0.465927
$714$	−6.29742	−0.235675
$715$	4.02148	0.150395
$716$	13.3633	0.499410
$717$	−32.9914	−1.23208
$718$	12.8776	0.480588
$719$	39.0724	1.45715	0.728577	−	0.684964i	$-0.240182\pi$
$719$	39.0724	1.45715	0.728577	+	0.684964i	$0.240182\pi$
$720$	1.36823	0.0509910
$721$	−10.9172	−0.406578
$722$	0	0
$723$	−27.2943	−1.01509
$724$	6.02646	0.223972
$725$	−12.8543	−0.477396
$726$	−1.11007	−0.0411987
$727$	4.26604	0.158219	0.0791093	−	0.996866i	$-0.474792\pi$
$727$	4.26604	0.158219	0.0791093	+	0.996866i	$0.474792\pi$
$728$	21.3227	0.790273
$729$	18.6363	0.690235
$730$	4.34329	0.160752
$731$	2.97279	0.109953
$732$	−3.99237	−0.147562
$733$	−40.2126	−1.48529	−0.742643	−	0.669688i	$-0.766428\pi$
$733$	−40.2126	−1.48529	−0.742643	+	0.669688i	$0.766428\pi$
$734$	−35.9675	−1.32759
$735$	8.45647	0.311921
$736$	3.46287	0.127643
$737$	−11.1660	−0.411305
$738$	2.56749	0.0945107
$739$	−8.78345	−0.323104	−0.161552	−	0.986864i	$-0.551650\pi$
$739$	−8.78345	−0.323104	−0.161552	+	0.986864i	$0.551650\pi$
$740$	1.91844	0.0705232
$741$	0	0
$742$	33.7599	1.23937
$743$	26.7351	0.980815	0.490408	−	0.871493i	$-0.336848\pi$
$743$	26.7351	0.980815	0.490408	+	0.871493i	$0.336848\pi$
$744$	3.98821	0.146215
$745$	12.1658	0.445719
$746$	−11.0114	−0.403155
$747$	17.2286	0.630363
$748$	−1.38233	−0.0505428
$749$	62.5838	2.28676
$750$	−8.07728	−0.294941
$751$	−4.77164	−0.174119	−0.0870597	−	0.996203i	$-0.527747\pi$
$751$	−4.77164	−0.174119	−0.0870597	+	0.996203i	$0.527747\pi$
$752$	6.48395	0.236445
$753$	18.6172	0.678447
$754$	15.1756	0.552664
$755$	−13.2669	−0.482833
$756$	21.7202	0.789957
$757$	−31.1836	−1.13339	−0.566694	−	0.823929i	$-0.691778\pi$
$757$	−31.1836	−1.13339	−0.566694	+	0.823929i	$0.691778\pi$
$758$	−33.8016	−1.22773
$759$	3.84405	0.139530
$760$	0	0
$761$	36.7107	1.33076	0.665381	−	0.746504i	$-0.268269\pi$
$761$	36.7107	1.33076	0.665381	+	0.746504i	$0.268269\pi$
$762$	13.7031	0.496411
$763$	11.2283	0.406491
$764$	−14.8933	−0.538819
$765$	1.89134	0.0683816
$766$	28.9498	1.04600
$767$	15.3231	0.553286
$768$	−1.11007	−0.0400564
$769$	−10.9231	−0.393896	−0.196948	−	0.980414i	$-0.563103\pi$
$769$	−10.9231	−0.393896	−0.196948	+	0.980414i	$0.563103\pi$
$770$	3.17645	0.114471
$771$	16.8380	0.606405
$772$	5.42938	0.195407
$773$	26.8395	0.965349	0.482675	−	0.875800i	$-0.339665\pi$
$773$	26.8395	0.965349	0.482675	+	0.875800i	$0.339665\pi$
$774$	−3.80164	−0.136647
$775$	15.8114	0.567961
$776$	−0.660421	−0.0237077
$777$	11.2916	0.405086
$778$	25.2740	0.906116
$779$	0	0
$780$	4.46414	0.159842
$781$	5.31877	0.190321
$782$	4.78682	0.171176
$783$	15.4585	0.552443
$784$	9.84224	0.351508
$785$	17.0523	0.608622
$786$	8.33380	0.297257
$787$	−26.3438	−0.939057	−0.469528	−	0.882917i	$-0.655576\pi$
$787$	−26.3438	−0.939057	−0.469528	+	0.882917i	$0.655576\pi$
$788$	−18.9028	−0.673384
$789$	14.9567	0.532473
$790$	13.2843	0.472635
$791$	39.2068	1.39403
$792$	1.76773	0.0628137
$793$	18.6862	0.663567
$794$	24.2779	0.861589
$795$	7.06800	0.250676
$796$	12.4997	0.443042
$797$	5.30691	0.187981	0.0939903	−	0.995573i	$-0.470038\pi$
$797$	5.30691	0.187981	0.0939903	+	0.995573i	$0.470038\pi$
$798$	0	0
$799$	8.96293	0.317086
$800$	−4.40092	−0.155596
$801$	−14.3745	−0.507897
$802$	−10.5621	−0.372961
$803$	5.61146	0.198024
$804$	−12.3951	−0.437142
$805$	−10.9997	−0.387687
$806$	−18.6668	−0.657508
$807$	−3.88241	−0.136667
$808$	−1.47039	−0.0517283
$809$	−23.8209	−0.837498	−0.418749	−	0.908102i	$-0.637531\pi$
$809$	−23.8209	−0.837498	−0.418749	+	0.908102i	$0.637531\pi$
$810$	0.442665	0.0155537
$811$	37.7162	1.32439	0.662197	−	0.749330i	$-0.269624\pi$
$811$	37.7162	1.32439	0.662197	+	0.749330i	$0.269624\pi$
$812$	11.9868	0.420655
$813$	−27.5316	−0.965575
$814$	2.47859	0.0868746
$815$	7.10711	0.248951
$816$	−1.53449	−0.0537177
$817$	0	0
$818$	20.9834	0.733669
$819$	−37.6929	−1.31710
$820$	1.12418	0.0392580
$821$	−14.7564	−0.515002	−0.257501	−	0.966278i	$-0.582899\pi$
$821$	−14.7564	−0.515002	−0.257501	+	0.966278i	$0.582899\pi$
$822$	−7.62433	−0.265929
$823$	−23.7657	−0.828422	−0.414211	−	0.910181i	$-0.635942\pi$
$823$	−23.7657	−0.828422	−0.414211	+	0.910181i	$0.635942\pi$
$824$	−2.66019	−0.0926720
$825$	−4.88535	−0.170086
$826$	12.1033	0.421128
$827$	−2.13232	−0.0741482	−0.0370741	−	0.999313i	$-0.511804\pi$
$827$	−2.13232	−0.0741482	−0.0370741	+	0.999313i	$0.511804\pi$
$828$	−6.12143	−0.212735
$829$	−39.3546	−1.36684	−0.683421	−	0.730024i	$-0.739509\pi$
$829$	−39.3546	−1.36684	−0.683421	+	0.730024i	$0.739509\pi$
$830$	7.54356	0.261841
$831$	−34.9791	−1.21341
$832$	5.19569	0.180128
$833$	13.6052	0.471391
$834$	11.2270	0.388759
$835$	−19.2612	−0.666561
$836$	0	0
$837$	−19.0147	−0.657245
$838$	4.08683	0.141177
$839$	29.9560	1.03419	0.517097	−	0.855927i	$-0.327013\pi$
$839$	29.9560	1.03419	0.517097	+	0.855927i	$0.327013\pi$
$840$	3.52610	0.121662
$841$	−20.4688	−0.705822
$842$	−31.7099	−1.09280
$843$	13.2269	0.455558
$844$	28.3972	0.977471
$845$	−10.8323	−0.372642
$846$	−11.4619	−0.394068
$847$	4.10393	0.141013
$848$	8.22624	0.282490
$849$	−4.91147	−0.168561
$850$	−6.08350	−0.208662
$851$	−8.58305	−0.294223
$852$	5.90424	0.202276
$853$	39.9783	1.36883	0.684415	−	0.729093i	$-0.260058\pi$
$853$	39.9783	1.36883	0.684415	+	0.729093i	$0.260058\pi$
$854$	14.7597	0.505068
$855$	0	0
$856$	15.2497	0.521225
$857$	32.8138	1.12090	0.560449	−	0.828189i	$-0.310628\pi$
$857$	32.8138	1.12090	0.560449	+	0.828189i	$0.310628\pi$
$858$	5.76760	0.196903
$859$	26.2698	0.896312	0.448156	−	0.893955i	$-0.352081\pi$
$859$	26.2698	0.896312	0.448156	+	0.893955i	$0.352081\pi$
$860$	−1.66455	−0.0567607
$861$	6.61675	0.225498
$862$	29.6640	1.01036
$863$	13.8689	0.472104	0.236052	−	0.971740i	$-0.424146\pi$
$863$	13.8689	0.472104	0.236052	+	0.971740i	$0.424146\pi$
$864$	5.29254	0.180056
$865$	−4.58757	−0.155982
$866$	−36.6044	−1.24387
$867$	16.7501	0.568864
$868$	−14.7444	−0.500456
$869$	17.1631	0.582220
$870$	2.50957	0.0850824
$871$	58.0151	1.96577
$872$	2.73598	0.0926520
$873$	1.16745	0.0395121
$874$	0	0
$875$	29.8616	1.00951
$876$	6.22914	0.210463
$877$	−37.9843	−1.28264	−0.641320	−	0.767274i	$-0.721613\pi$
$877$	−37.9843	−1.28264	−0.641320	+	0.767274i	$0.721613\pi$
$878$	−7.37392	−0.248858
$879$	10.4567	0.352695
$880$	0.774003	0.0260916
$881$	−4.96872	−0.167400	−0.0837001	−	0.996491i	$-0.526674\pi$
$881$	−4.96872	−0.167400	−0.0837001	+	0.996491i	$0.526674\pi$
$882$	−17.3985	−0.585836
$883$	7.70851	0.259412	0.129706	−	0.991553i	$-0.458597\pi$
$883$	7.70851	0.259412	0.129706	+	0.991553i	$0.458597\pi$
$884$	7.18213	0.241561
$885$	2.53396	0.0851781
$886$	−1.65116	−0.0554720
$887$	10.0843	0.338598	0.169299	−	0.985565i	$-0.445850\pi$
$887$	10.0843	0.338598	0.169299	+	0.985565i	$0.445850\pi$
$888$	2.75142	0.0923317
$889$	−50.6601	−1.69909
$890$	−6.29387	−0.210971
$891$	0.571917	0.0191599
$892$	−3.26943	−0.109469
$893$	0	0
$894$	17.4481	0.583553
$895$	−10.3432	−0.345736
$896$	4.10393	0.137103
$897$	−19.9725	−0.666861
$898$	15.2799	0.509896
$899$	−10.4937	−0.349986
$900$	7.77965	0.259322
$901$	11.3713	0.378834
$902$	1.45242	0.0483603
$903$	−9.79729	−0.326033
$904$	9.55349	0.317744
$905$	−4.66450	−0.155053
$906$	−19.0274	−0.632144
$907$	−13.4455	−0.446451	−0.223225	−	0.974767i	$-0.571659\pi$
$907$	−13.4455	−0.446451	−0.223225	+	0.974767i	$0.571659\pi$
$908$	−7.37560	−0.244768
$909$	2.59927	0.0862122
$910$	−16.5039	−0.547098
$911$	10.6301	0.352190	0.176095	−	0.984373i	$-0.443653\pi$
$911$	10.6301	0.352190	0.176095	+	0.984373i	$0.443653\pi$
$912$	0	0
$913$	9.74617	0.322551
$914$	−14.2014	−0.469742
$915$	3.09011	0.102156
$916$	28.6009	0.945002
$917$	−30.8099	−1.01743
$918$	7.31602	0.241465
$919$	−37.2895	−1.23007	−0.615034	−	0.788501i	$-0.710858\pi$
$919$	−37.2895	−1.23007	−0.615034	+	0.788501i	$0.710858\pi$
$920$	−2.68027	−0.0883660
$921$	−3.93741	−0.129742
$922$	−34.6919	−1.14252
$923$	−27.6347	−0.909606
$924$	4.55567	0.149871
$925$	10.9081	0.358656
$926$	14.6523	0.481505
$927$	4.70250	0.154450
$928$	2.92082	0.0958805
$929$	−5.35066	−0.175549	−0.0877747	−	0.996140i	$-0.527976\pi$
$929$	−5.35066	−0.175549	−0.0877747	+	0.996140i	$0.527976\pi$
$930$	−3.08689	−0.101223
$931$	0	0
$932$	26.8013	0.877905
$933$	28.7774	0.942131
$934$	4.44379	0.145405
$935$	1.06992	0.0349903
$936$	−9.18459	−0.300208
$937$	3.47661	0.113576	0.0567879	−	0.998386i	$-0.481914\pi$
$937$	3.47661	0.113576	0.0567879	+	0.998386i	$0.481914\pi$
$938$	45.8245	1.49622
$939$	15.3136	0.499740
$940$	−5.01860	−0.163689
$941$	−45.3731	−1.47912	−0.739561	−	0.673090i	$-0.764967\pi$
$941$	−45.3731	−1.47912	−0.739561	+	0.673090i	$0.764967\pi$
$942$	24.4564	0.796831
$943$	−5.02955	−0.163785
$944$	2.94920	0.0959884
$945$	−16.8115	−0.546879
$946$	−2.15057	−0.0699211
$947$	−44.1295	−1.43402	−0.717008	−	0.697065i	$-0.754489\pi$
$947$	−44.1295	−1.43402	−0.717008	+	0.697065i	$0.754489\pi$
$948$	19.0524	0.618792
$949$	−29.1554	−0.946424
$950$	0	0
$951$	−39.1367	−1.26909
$952$	5.67297	0.183862
$953$	12.5953	0.408002	0.204001	−	0.978971i	$-0.434605\pi$
$953$	12.5953	0.408002	0.204001	+	0.978971i	$0.434605\pi$
$954$	−14.5418	−0.470808
$955$	11.5274	0.373019
$956$	29.7199	0.961212
$957$	3.24233	0.104809
$958$	41.8374	1.35170
$959$	28.1870	0.910207
$960$	0.859202	0.0277306
$961$	−18.0922	−0.583620
$962$	−12.8780	−0.415203
$963$	−26.9575	−0.868692
$964$	24.5878	0.791920
$965$	−4.20235	−0.135279
$966$	−15.7757	−0.507575
$967$	−44.4677	−1.42999	−0.714993	−	0.699132i	$-0.753570\pi$
$967$	−44.4677	−1.42999	−0.714993	+	0.699132i	$0.753570\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	0.511168	0.0164126
$971$	−12.4354	−0.399070	−0.199535	−	0.979891i	$-0.563943\pi$
$971$	−12.4354	−0.399070	−0.199535	+	0.979891i	$0.563943\pi$
$972$	−15.2428	−0.488912
$973$	−41.5060	−1.33062
$974$	15.9096	0.509777
$975$	25.3827	0.812898
$976$	3.59649	0.115121
$977$	12.8142	0.409963	0.204982	−	0.978766i	$-0.434287\pi$
$977$	12.8142	0.409963	0.204982	+	0.978766i	$0.434287\pi$
$978$	10.1930	0.325936
$979$	−8.13158	−0.259886
$980$	−7.61792	−0.243346
$981$	−4.83649	−0.154417
$982$	−31.1476	−0.993960
$983$	45.1194	1.43909	0.719543	−	0.694448i	$-0.244351\pi$
$983$	45.1194	1.43909	0.719543	+	0.694448i	$0.244351\pi$
$984$	1.61230	0.0513981
$985$	14.6308	0.466177
$986$	4.03752	0.128581
$987$	−29.5387	−0.940228
$988$	0	0
$989$	7.44715	0.236806
$990$	−1.36823	−0.0434853
$991$	−12.5332	−0.398130	−0.199065	−	0.979986i	$-0.563791\pi$
$991$	−12.5332	−0.398130	−0.199065	+	0.979986i	$0.563791\pi$
$992$	−3.59274	−0.114070
$993$	−21.4147	−0.679574
$994$	−21.8279	−0.692338
$995$	−9.67484	−0.306713
$996$	10.8190	0.342812
$997$	43.1939	1.36796	0.683982	−	0.729499i	$-0.260247\pi$
$997$	43.1939	1.36796	0.683982	+	0.729499i	$0.260247\pi$
$998$	−33.8679	−1.07207
$999$	−13.1181	−0.415037

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.cd.1.4	yes	16
19.18	odd	2		7942.2.a.cc.1.13	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7942.2.a.cc.1.13	✓	16	19.18	odd	2
7942.2.a.cd.1.4	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.11007 q^{3} +1.00000 q^{4} -0.774003 q^{5} -1.11007 q^{6} +4.10393 q^{7} +1.00000 q^{8} -1.76773 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.11007 q^{3} +1.00000 q^{4} -0.774003 q^{5} -1.11007 q^{6} +4.10393 q^{7} +1.00000 q^{8} -1.76773 q^{9} -0.774003 q^{10} -1.00000 q^{11} -1.11007 q^{12} +5.19569 q^{13} +4.10393 q^{14} +0.859202 q^{15} +1.00000 q^{16} +1.38233 q^{17} -1.76773 q^{18} -0.774003 q^{20} -4.55567 q^{21} -1.00000 q^{22} +3.46287 q^{23} -1.11007 q^{24} -4.40092 q^{25} +5.19569 q^{26} +5.29254 q^{27} +4.10393 q^{28} +2.92082 q^{29} +0.859202 q^{30} -3.59274 q^{31} +1.00000 q^{32} +1.11007 q^{33} +1.38233 q^{34} -3.17645 q^{35} -1.76773 q^{36} -2.47859 q^{37} -5.76760 q^{39} -0.774003 q^{40} -1.45242 q^{41} -4.55567 q^{42} +2.15057 q^{43} -1.00000 q^{44} +1.36823 q^{45} +3.46287 q^{46} +6.48395 q^{47} -1.11007 q^{48} +9.84224 q^{49} -4.40092 q^{50} -1.53449 q^{51} +5.19569 q^{52} +8.22624 q^{53} +5.29254 q^{54} +0.774003 q^{55} +4.10393 q^{56} +2.92082 q^{58} +2.94920 q^{59} +0.859202 q^{60} +3.59649 q^{61} -3.59274 q^{62} -7.25465 q^{63} +1.00000 q^{64} -4.02148 q^{65} +1.11007 q^{66} +11.1660 q^{67} +1.38233 q^{68} -3.84405 q^{69} -3.17645 q^{70} -5.31877 q^{71} -1.76773 q^{72} -5.61146 q^{73} -2.47859 q^{74} +4.88535 q^{75} -4.10393 q^{77} -5.76760 q^{78} -17.1631 q^{79} -0.774003 q^{80} -0.571917 q^{81} -1.45242 q^{82} -9.74617 q^{83} -4.55567 q^{84} -1.06992 q^{85} +2.15057 q^{86} -3.24233 q^{87} -1.00000 q^{88} +8.13158 q^{89} +1.36823 q^{90} +21.3227 q^{91} +3.46287 q^{92} +3.98821 q^{93} +6.48395 q^{94} -1.11007 q^{96} -0.660421 q^{97} +9.84224 q^{98} +1.76773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 10 q^{3} + 16 q^{4} + 10 q^{6} + 6 q^{7} + 16 q^{8} + 20 q^{9}+O(q^{10}) \) 16 * q + 16 * q^2 + 10 * q^3 + 16 * q^4 + 10 * q^6 + 6 * q^7 + 16 * q^8 + 20 * q^9	\( 16 q + 16 q^{2} + 10 q^{3} + 16 q^{4} + 10 q^{6} + 6 q^{7} + 16 q^{8} + 20 q^{9} - 16 q^{11} + 10 q^{12} - 2 q^{13} + 6 q^{14} - 8 q^{15} + 16 q^{16} + 20 q^{18} + 6 q^{21} - 16 q^{22} - 6 q^{23} + 10 q^{24} + 18 q^{25} - 2 q^{26} + 46 q^{27} + 6 q^{28} + 28 q^{29} - 8 q^{30} + 8 q^{31} + 16 q^{32} - 10 q^{33} + 14 q^{35} + 20 q^{36} + 32 q^{37} + 24 q^{39} + 24 q^{41} + 6 q^{42} + 28 q^{43} - 16 q^{44} - 16 q^{45} - 6 q^{46} + 32 q^{47} + 10 q^{48} + 16 q^{49} + 18 q^{50} + 14 q^{51} - 2 q^{52} + 16 q^{53} + 46 q^{54} + 6 q^{56} + 28 q^{58} + 30 q^{59} - 8 q^{60} - 8 q^{61} + 8 q^{62} - 8 q^{63} + 16 q^{64} + 26 q^{65} - 10 q^{66} + 44 q^{67} + 16 q^{69} + 14 q^{70} + 36 q^{71} + 20 q^{72} + 10 q^{73} + 32 q^{74} + 76 q^{75} - 6 q^{77} + 24 q^{78} - 16 q^{79} + 48 q^{81} + 24 q^{82} + 12 q^{83} + 6 q^{84} - 4 q^{85} + 28 q^{86} + 24 q^{87} - 16 q^{88} + 32 q^{89} - 16 q^{90} - 24 q^{91} - 6 q^{92} - 14 q^{93} + 32 q^{94} + 10 q^{96} + 52 q^{97} + 16 q^{98} - 20 q^{99}+O(q^{100}) \) 16 * q + 16 * q^2 + 10 * q^3 + 16 * q^4 + 10 * q^6 + 6 * q^7 + 16 * q^8 + 20 * q^9 - 16 * q^11 + 10 * q^12 - 2 * q^13 + 6 * q^14 - 8 * q^15 + 16 * q^16 + 20 * q^18 + 6 * q^21 - 16 * q^22 - 6 * q^23 + 10 * q^24 + 18 * q^25 - 2 * q^26 + 46 * q^27 + 6 * q^28 + 28 * q^29 - 8 * q^30 + 8 * q^31 + 16 * q^32 - 10 * q^33 + 14 * q^35 + 20 * q^36 + 32 * q^37 + 24 * q^39 + 24 * q^41 + 6 * q^42 + 28 * q^43 - 16 * q^44 - 16 * q^45 - 6 * q^46 + 32 * q^47 + 10 * q^48 + 16 * q^49 + 18 * q^50 + 14 * q^51 - 2 * q^52 + 16 * q^53 + 46 * q^54 + 6 * q^56 + 28 * q^58 + 30 * q^59 - 8 * q^60 - 8 * q^61 + 8 * q^62 - 8 * q^63 + 16 * q^64 + 26 * q^65 - 10 * q^66 + 44 * q^67 + 16 * q^69 + 14 * q^70 + 36 * q^71 + 20 * q^72 + 10 * q^73 + 32 * q^74 + 76 * q^75 - 6 * q^77 + 24 * q^78 - 16 * q^79 + 48 * q^81 + 24 * q^82 + 12 * q^83 + 6 * q^84 - 4 * q^85 + 28 * q^86 + 24 * q^87 - 16 * q^88 + 32 * q^89 - 16 * q^90 - 24 * q^91 - 6 * q^92 - 14 * q^93 + 32 * q^94 + 10 * q^96 + 52 * q^97 + 16 * q^98 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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