LMFDB - Embedded newform 6422.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, 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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x - 1 $ x^3 - x^2 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.65109$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.65109 q^{3} +1.00000 q^{4} -1.27389 q^{5} -1.65109 q^{6} +2.65109 q^{7} -1.00000 q^{8} -0.273891 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.65109 q^{3} +1.00000 q^{4} -1.27389 q^{5} -1.65109 q^{6} +2.65109 q^{7} -1.00000 q^{8} -0.273891 q^{9} +1.27389 q^{10} +3.27389 q^{11} +1.65109 q^{12} -2.65109 q^{14} -2.10331 q^{15} +1.00000 q^{16} -5.30219 q^{17} +0.273891 q^{18} -1.00000 q^{19} -1.27389 q^{20} +4.37720 q^{21} -3.27389 q^{22} +6.40550 q^{23} -1.65109 q^{24} -3.37720 q^{25} -5.40550 q^{27} +2.65109 q^{28} -7.48052 q^{29} +2.10331 q^{30} -2.47277 q^{31} -1.00000 q^{32} +5.40550 q^{33} +5.30219 q^{34} -3.37720 q^{35} -0.273891 q^{36} -2.20662 q^{37} +1.00000 q^{38} +1.27389 q^{40} -3.89669 q^{41} -4.37720 q^{42} -6.30219 q^{43} +3.27389 q^{44} +0.348907 q^{45} -6.40550 q^{46} +12.3305 q^{47} +1.65109 q^{48} +0.0282963 q^{49} +3.37720 q^{50} -8.75441 q^{51} -2.47277 q^{53} +5.40550 q^{54} -4.17058 q^{55} -2.65109 q^{56} -1.65109 q^{57} +7.48052 q^{58} -9.53711 q^{59} -2.10331 q^{60} -5.67939 q^{61} +2.47277 q^{62} -0.726109 q^{63} +1.00000 q^{64} -5.40550 q^{66} +0.206625 q^{67} -5.30219 q^{68} +10.5761 q^{69} +3.37720 q^{70} +3.00000 q^{71} +0.273891 q^{72} -9.41325 q^{73} +2.20662 q^{74} -5.57608 q^{75} -1.00000 q^{76} +8.67939 q^{77} -12.2555 q^{79} -1.27389 q^{80} -8.10331 q^{81} +3.89669 q^{82} -7.79045 q^{83} +4.37720 q^{84} +6.75441 q^{85} +6.30219 q^{86} -12.3510 q^{87} -3.27389 q^{88} +14.1316 q^{89} -0.348907 q^{90} +6.40550 q^{92} -4.08277 q^{93} -12.3305 q^{94} +1.27389 q^{95} -1.65109 q^{96} +11.3022 q^{97} -0.0282963 q^{98} -0.896688 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 2 q^{6} + q^{7} - 3 q^{8} + q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 2 * q^6 + q^7 - 3 * q^8 + q^9	$ 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 2 q^{6} + q^{7} - 3 q^{8} + q^{9} + 2 q^{10} + 8 q^{11} - 2 q^{12} - q^{14} - 3 q^{15} + 3 q^{16} - 2 q^{17} - q^{18} - 3 q^{19} - 2 q^{20} + 8 q^{21} - 8 q^{22} + 2 q^{23} + 2 q^{24} - 5 q^{25} + q^{27} + q^{28} - 14 q^{29} + 3 q^{30} + 5 q^{31} - 3 q^{32} - q^{33} + 2 q^{34} - 5 q^{35} + q^{36} + 3 q^{38} + 2 q^{40} - 15 q^{41} - 8 q^{42} - 5 q^{43} + 8 q^{44} + 8 q^{45} - 2 q^{46} + 11 q^{47} - 2 q^{48} - 12 q^{49} + 5 q^{50} - 16 q^{51} + 5 q^{53} - q^{54} - 14 q^{55} - q^{56} + 2 q^{57} + 14 q^{58} + 4 q^{59} - 3 q^{60} + 2 q^{61} - 5 q^{62} - 4 q^{63} + 3 q^{64} + q^{66} - 6 q^{67} - 2 q^{68} + 16 q^{69} + 5 q^{70} + 9 q^{71} - q^{72} - 15 q^{73} - q^{75} - 3 q^{76} + 7 q^{77} - 2 q^{79} - 2 q^{80} - 21 q^{81} + 15 q^{82} - 5 q^{83} + 8 q^{84} + 10 q^{85} + 5 q^{86} + 5 q^{87} - 8 q^{88} + 27 q^{89} - 8 q^{90} + 2 q^{92} - 25 q^{93} - 11 q^{94} + 2 q^{95} + 2 q^{96} + 20 q^{97} + 12 q^{98} - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 2 * q^6 + q^7 - 3 * q^8 + q^9 + 2 * q^10 + 8 * q^11 - 2 * q^12 - q^14 - 3 * q^15 + 3 * q^16 - 2 * q^17 - q^18 - 3 * q^19 - 2 * q^20 + 8 * q^21 - 8 * q^22 + 2 * q^23 + 2 * q^24 - 5 * q^25 + q^27 + q^28 - 14 * q^29 + 3 * q^30 + 5 * q^31 - 3 * q^32 - q^33 + 2 * q^34 - 5 * q^35 + q^36 + 3 * q^38 + 2 * q^40 - 15 * q^41 - 8 * q^42 - 5 * q^43 + 8 * q^44 + 8 * q^45 - 2 * q^46 + 11 * q^47 - 2 * q^48 - 12 * q^49 + 5 * q^50 - 16 * q^51 + 5 * q^53 - q^54 - 14 * q^55 - q^56 + 2 * q^57 + 14 * q^58 + 4 * q^59 - 3 * q^60 + 2 * q^61 - 5 * q^62 - 4 * q^63 + 3 * q^64 + q^66 - 6 * q^67 - 2 * q^68 + 16 * q^69 + 5 * q^70 + 9 * q^71 - q^72 - 15 * q^73 - q^75 - 3 * q^76 + 7 * q^77 - 2 * q^79 - 2 * q^80 - 21 * q^81 + 15 * q^82 - 5 * q^83 + 8 * q^84 + 10 * q^85 + 5 * q^86 + 5 * q^87 - 8 * q^88 + 27 * q^89 - 8 * q^90 + 2 * q^92 - 25 * q^93 - 11 * q^94 + 2 * q^95 + 2 * q^96 + 20 * q^97 + 12 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.65109	0.953259	0.476630	−	0.879104i	$-0.341858\pi$
$3$	1.65109	0.953259	0.476630	+	0.879104i	$0.341858\pi$
$4$	1.00000	0.500000
$5$	−1.27389	−0.569701	−0.284851	−	0.958572i	$-0.591944\pi$
$5$	−1.27389	−0.569701	−0.284851	+	0.958572i	$0.591944\pi$
$6$	−1.65109	−0.674056
$7$	2.65109	1.00202	0.501010	−	0.865442i	$-0.332962\pi$
$7$	2.65109	1.00202	0.501010	+	0.865442i	$0.332962\pi$
$8$	−1.00000	−0.353553
$9$	−0.273891	−0.0912969
$10$	1.27389	0.402840
$11$	3.27389	0.987115	0.493558	−	0.869713i	$-0.335696\pi$
$11$	3.27389	0.987115	0.493558	+	0.869713i	$0.335696\pi$
$12$	1.65109	0.476630
$13$	0	0
$13$	0	0
$14$	−2.65109	−0.708535
$15$	−2.10331	−0.543073
$16$	1.00000	0.250000
$17$	−5.30219	−1.28597	−0.642985	−	0.765879i	$-0.722304\pi$
$17$	−5.30219	−1.28597	−0.642985	+	0.765879i	$0.722304\pi$
$18$	0.273891	0.0645566
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.27389	−0.284851
$21$	4.37720	0.955184
$22$	−3.27389	−0.697996
$23$	6.40550	1.33564	0.667819	−	0.744323i	$-0.267228\pi$
$23$	6.40550	1.33564	0.667819	+	0.744323i	$0.267228\pi$
$24$	−1.65109	−0.337028
$25$	−3.37720	−0.675441
$26$	0	0
$27$	−5.40550	−1.04029
$28$	2.65109	0.501010
$29$	−7.48052	−1.38910	−0.694548	−	0.719446i	$-0.744396\pi$
$29$	−7.48052	−1.38910	−0.694548	+	0.719446i	$0.744396\pi$
$30$	2.10331	0.384011
$31$	−2.47277	−0.444122	−0.222061	−	0.975033i	$-0.571278\pi$
$31$	−2.47277	−0.444122	−0.222061	+	0.975033i	$0.571278\pi$
$32$	−1.00000	−0.176777
$33$	5.40550	0.940977
$34$	5.30219	0.909318
$35$	−3.37720	−0.570851
$36$	−0.273891	−0.0456484
$37$	−2.20662	−0.362767	−0.181383	−	0.983412i	$-0.558058\pi$
$37$	−2.20662	−0.362767	−0.181383	+	0.983412i	$0.558058\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	1.27389	0.201420
$41$	−3.89669	−0.608560	−0.304280	−	0.952583i	$-0.598416\pi$
$41$	−3.89669	−0.608560	−0.304280	+	0.952583i	$0.598416\pi$
$42$	−4.37720	−0.675417
$43$	−6.30219	−0.961074	−0.480537	−	0.876974i	$-0.659558\pi$
$43$	−6.30219	−0.961074	−0.480537	+	0.876974i	$0.659558\pi$
$44$	3.27389	0.493558
$45$	0.348907	0.0520119
$46$	−6.40550	−0.944439
$47$	12.3305	1.79859	0.899293	−	0.437347i	$-0.144082\pi$
$47$	12.3305	1.79859	0.899293	+	0.437347i	$0.144082\pi$
$48$	1.65109	0.238315
$49$	0.0282963	0.00404232
$50$	3.37720	0.477609
$51$	−8.75441	−1.22586
$52$	0	0
$53$	−2.47277	−0.339660	−0.169830	−	0.985473i	$-0.554322\pi$
$53$	−2.47277	−0.339660	−0.169830	+	0.985473i	$0.554322\pi$
$54$	5.40550	0.735595
$55$	−4.17058	−0.562361
$56$	−2.65109	−0.354267
$57$	−1.65109	−0.218693
$58$	7.48052	0.982240
$59$	−9.53711	−1.24163	−0.620813	−	0.783959i	$-0.713197\pi$
$59$	−9.53711	−1.24163	−0.620813	+	0.783959i	$0.713197\pi$
$60$	−2.10331	−0.271536
$61$	−5.67939	−0.727171	−0.363586	−	0.931561i	$-0.618448\pi$
$61$	−5.67939	−0.727171	−0.363586	+	0.931561i	$0.618448\pi$
$62$	2.47277	0.314041
$63$	−0.726109	−0.0914812
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.40550	−0.665371
$67$	0.206625	0.0252432	0.0126216	−	0.999920i	$-0.495982\pi$
$67$	0.206625	0.0252432	0.0126216	+	0.999920i	$0.495982\pi$
$68$	−5.30219	−0.642985
$69$	10.5761	1.27321
$70$	3.37720	0.403653
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0.273891	0.0322783
$73$	−9.41325	−1.10174	−0.550869	−	0.834592i	$-0.685704\pi$
$73$	−9.41325	−1.10174	−0.550869	+	0.834592i	$0.685704\pi$
$74$	2.20662	0.256515
$75$	−5.57608	−0.643870
$76$	−1.00000	−0.114708
$77$	8.67939	0.989108
$78$	0	0
$79$	−12.2555	−1.37885	−0.689424	−	0.724358i	$-0.742136\pi$
$79$	−12.2555	−1.37885	−0.689424	+	0.724358i	$0.742136\pi$
$80$	−1.27389	−0.142425
$81$	−8.10331	−0.900368
$82$	3.89669	0.430317
$83$	−7.79045	−0.855113	−0.427557	−	0.903989i	$-0.640626\pi$
$83$	−7.79045	−0.855113	−0.427557	+	0.903989i	$0.640626\pi$
$84$	4.37720	0.477592
$85$	6.75441	0.732618
$86$	6.30219	0.679582
$87$	−12.3510	−1.32417
$88$	−3.27389	−0.348998
$89$	14.1316	1.49795	0.748974	−	0.662600i	$-0.230547\pi$
$89$	14.1316	1.49795	0.748974	+	0.662600i	$0.230547\pi$
$90$	−0.348907	−0.0367780
$91$	0	0
$92$	6.40550	0.667819
$93$	−4.08277	−0.423363
$94$	−12.3305	−1.27179
$95$	1.27389	0.130698
$96$	−1.65109	−0.168514
$97$	11.3022	1.14756	0.573782	−	0.819008i	$-0.305476\pi$
$97$	11.3022	1.14756	0.573782	+	0.819008i	$0.305476\pi$
$98$	−0.0282963	−0.00285835
$99$	−0.896688	−0.0901205
$100$	−3.37720	−0.337720
$101$	−13.8110	−1.37425	−0.687123	−	0.726541i	$-0.741127\pi$
$101$	−13.8110	−1.37425	−0.687123	+	0.726541i	$0.741127\pi$
$102$	8.75441	0.866815
$103$	15.4904	1.52631	0.763157	−	0.646214i	$-0.223648\pi$
$103$	15.4904	1.52631	0.763157	+	0.646214i	$0.223648\pi$
$104$	0	0
$105$	−5.57608	−0.544169
$106$	2.47277	0.240176
$107$	−1.89669	−0.183360	−0.0916799	−	0.995789i	$-0.529224\pi$
$107$	−1.89669	−0.183360	−0.0916799	+	0.995789i	$0.529224\pi$
$108$	−5.40550	−0.520144
$109$	−12.9066	−1.23622	−0.618112	−	0.786090i	$-0.712102\pi$
$109$	−12.9066	−1.23622	−0.618112	+	0.786090i	$0.712102\pi$
$110$	4.17058	0.397649
$111$	−3.64334	−0.345811
$112$	2.65109	0.250505
$113$	20.1522	1.89576	0.947878	−	0.318635i	$-0.103224\pi$
$113$	20.1522	1.89576	0.947878	+	0.318635i	$0.103224\pi$
$114$	1.65109	0.154639
$115$	−8.15990	−0.760915
$116$	−7.48052	−0.694548
$117$	0	0
$118$	9.53711	0.877962
$119$	−14.0566	−1.28857
$120$	2.10331	0.192005
$121$	−0.281641	−0.0256037
$122$	5.67939	0.514188
$123$	−6.43380	−0.580116
$124$	−2.47277	−0.222061
$125$	10.6716	0.954500
$126$	0.726109	0.0646870
$127$	−7.71836	−0.684894	−0.342447	−	0.939537i	$-0.611256\pi$
$127$	−7.71836	−0.684894	−0.342447	+	0.939537i	$0.611256\pi$
$128$	−1.00000	−0.0883883
$129$	−10.4055	−0.916153
$130$	0	0
$131$	−8.97170	−0.783861	−0.391931	−	0.919995i	$-0.628193\pi$
$131$	−8.97170	−0.783861	−0.391931	+	0.919995i	$0.628193\pi$
$132$	5.40550	0.470488
$133$	−2.65109	−0.229879
$134$	−0.206625	−0.0178496
$135$	6.88601	0.592654
$136$	5.30219	0.454659
$137$	−13.9143	−1.18878	−0.594390	−	0.804177i	$-0.702606\pi$
$137$	−13.9143	−1.18878	−0.594390	+	0.804177i	$0.702606\pi$
$138$	−10.5761	−0.900295
$139$	−15.4338	−1.30908	−0.654539	−	0.756028i	$-0.727137\pi$
$139$	−15.4338	−1.30908	−0.654539	+	0.756028i	$0.727137\pi$
$140$	−3.37720	−0.285426
$141$	20.3588	1.71452
$142$	−3.00000	−0.251754
$143$	0	0
$144$	−0.273891	−0.0228242
$145$	9.52936	0.791370
$146$	9.41325	0.779046
$147$	0.0467198	0.00385338
$148$	−2.20662	−0.181383
$149$	20.4055	1.67168	0.835842	−	0.548970i	$-0.184980\pi$
$149$	20.4055	1.67168	0.835842	+	0.548970i	$0.184980\pi$
$150$	5.57608	0.455285
$151$	3.04672	0.247939	0.123969	−	0.992286i	$-0.460438\pi$
$151$	3.04672	0.247939	0.123969	+	0.992286i	$0.460438\pi$
$152$	1.00000	0.0811107
$153$	1.45222	0.117405
$154$	−8.67939	−0.699405
$155$	3.15003	0.253017
$156$	0	0
$157$	−2.17833	−0.173850	−0.0869248	−	0.996215i	$-0.527704\pi$
$157$	−2.17833	−0.173850	−0.0869248	+	0.996215i	$0.527704\pi$
$158$	12.2555	0.974993
$159$	−4.08277	−0.323784
$160$	1.27389	0.100710
$161$	16.9816	1.33834
$162$	8.10331	0.636656
$163$	−0.273891	−0.0214528	−0.0107264	−	0.999942i	$-0.503414\pi$
$163$	−0.273891	−0.0214528	−0.0107264	+	0.999942i	$0.503414\pi$
$164$	−3.89669	−0.304280
$165$	−6.88601	−0.536075
$166$	7.79045	0.604656
$167$	−2.12386	−0.164349	−0.0821746	−	0.996618i	$-0.526187\pi$
$167$	−2.12386	−0.164349	−0.0821746	+	0.996618i	$0.526187\pi$
$168$	−4.37720	−0.337709
$169$	0	0
$170$	−6.75441	−0.518039
$171$	0.273891	0.0209449
$172$	−6.30219	−0.480537
$173$	24.3121	1.84841	0.924206	−	0.381895i	$-0.124728\pi$
$173$	24.3121	1.84841	0.924206	+	0.381895i	$0.124728\pi$
$174$	12.3510	0.936329
$175$	−8.95328	−0.676804
$176$	3.27389	0.246779
$177$	−15.7467	−1.18359
$178$	−14.1316	−1.05921
$179$	10.2661	0.767327	0.383664	−	0.923473i	$-0.374662\pi$
$179$	10.2661	0.767327	0.383664	+	0.923473i	$0.374662\pi$
$180$	0.348907	0.0260060
$181$	−25.6893	−1.90947	−0.954734	−	0.297461i	$-0.903860\pi$
$181$	−25.6893	−1.90947	−0.954734	+	0.297461i	$0.903860\pi$
$182$	0	0
$183$	−9.37720	−0.693183
$184$	−6.40550	−0.472220
$185$	2.81100	0.206669
$186$	4.08277	0.299363
$187$	−17.3588	−1.26940
$188$	12.3305	0.899293
$189$	−14.3305	−1.04239
$190$	−1.27389	−0.0924177
$191$	−0.857718	−0.0620623	−0.0310311	−	0.999518i	$-0.509879\pi$
$191$	−0.857718	−0.0620623	−0.0310311	+	0.999518i	$0.509879\pi$
$192$	1.65109	0.119157
$193$	10.0926	0.726484	0.363242	−	0.931695i	$-0.381670\pi$
$193$	10.0926	0.726484	0.363242	+	0.931695i	$0.381670\pi$
$194$	−11.3022	−0.811450
$195$	0	0
$196$	0.0282963	0.00202116
$197$	−9.69006	−0.690388	−0.345194	−	0.938531i	$-0.612187\pi$
$197$	−9.69006	−0.690388	−0.345194	+	0.938531i	$0.612187\pi$
$198$	0.896688	0.0637248
$199$	24.1960	1.71521	0.857603	−	0.514313i	$-0.171953\pi$
$199$	24.1960	1.71521	0.857603	+	0.514313i	$0.171953\pi$
$200$	3.37720	0.238804
$201$	0.341157	0.0240633
$202$	13.8110	0.971738
$203$	−19.8315	−1.39190
$204$	−8.75441	−0.612931
$205$	4.96395	0.346698
$206$	−15.4904	−1.07927
$207$	−1.75441	−0.121940
$208$	0	0
$209$	−3.27389	−0.226460
$210$	5.57608	0.384786
$211$	6.72823	0.463191	0.231595	−	0.972812i	$-0.425605\pi$
$211$	6.72823	0.463191	0.231595	+	0.972812i	$0.425605\pi$
$212$	−2.47277	−0.169830
$213$	4.95328	0.339393
$214$	1.89669	0.129655
$215$	8.02830	0.547525
$216$	5.40550	0.367798
$217$	−6.55553	−0.445018
$218$	12.9066	0.874143
$219$	−15.5422	−1.05024
$220$	−4.17058	−0.281180
$221$	0	0
$222$	3.64334	0.244525
$223$	15.0566	1.00826	0.504132	−	0.863627i	$-0.331813\pi$
$223$	15.0566	1.00826	0.504132	+	0.863627i	$0.331813\pi$
$224$	−2.65109	−0.177134
$225$	0.924984	0.0616656
$226$	−20.1522	−1.34050
$227$	0.765079	0.0507801	0.0253900	−	0.999678i	$-0.491917\pi$
$227$	0.765079	0.0507801	0.0253900	+	0.999678i	$0.491917\pi$
$228$	−1.65109	−0.109346
$229$	−10.8422	−0.716474	−0.358237	−	0.933631i	$-0.616622\pi$
$229$	−10.8422	−0.716474	−0.358237	+	0.933631i	$0.616622\pi$
$230$	8.15990	0.538048
$231$	14.3305	0.942877
$232$	7.48052	0.491120
$233$	−15.9816	−1.04699	−0.523494	−	0.852029i	$-0.675372\pi$
$233$	−15.9816	−1.04699	−0.523494	+	0.852029i	$0.675372\pi$
$234$	0	0
$235$	−15.7077	−1.02466
$236$	−9.53711	−0.620813
$237$	−20.2349	−1.31440
$238$	14.0566	0.911154
$239$	0.118812	0.00768534	0.00384267	−	0.999993i	$-0.498777\pi$
$239$	0.118812	0.00768534	0.00384267	+	0.999993i	$0.498777\pi$
$240$	−2.10331	−0.135768
$241$	−9.02830	−0.581564	−0.290782	−	0.956789i	$-0.593915\pi$
$241$	−9.02830	−0.581564	−0.290782	+	0.956789i	$0.593915\pi$
$242$	0.281641	0.0181045
$243$	2.83717	0.182005
$244$	−5.67939	−0.363586
$245$	−0.0360463	−0.00230292
$246$	6.43380	0.410204
$247$	0	0
$248$	2.47277	0.157021
$249$	−12.8628	−0.815145
$250$	−10.6716	−0.674934
$251$	3.40550	0.214953	0.107477	−	0.994208i	$-0.465723\pi$
$251$	3.40550	0.214953	0.107477	+	0.994208i	$0.465723\pi$
$252$	−0.726109	−0.0457406
$253$	20.9709	1.31843
$254$	7.71836	0.484293
$255$	11.1522	0.698375
$256$	1.00000	0.0625000
$257$	11.3227	0.706293	0.353146	−	0.935568i	$-0.385112\pi$
$257$	11.3227	0.706293	0.353146	+	0.935568i	$0.385112\pi$
$258$	10.4055	0.647818
$259$	−5.84997	−0.363499
$260$	0	0
$261$	2.04884	0.126820
$262$	8.97170	0.554274
$263$	−29.5294	−1.82086	−0.910429	−	0.413665i	$-0.864248\pi$
$263$	−29.5294	−1.82086	−0.910429	+	0.413665i	$0.864248\pi$
$264$	−5.40550	−0.332685
$265$	3.15003	0.193505
$266$	2.65109	0.162549
$267$	23.3326	1.42793
$268$	0.206625	0.0126216
$269$	15.8187	0.964486	0.482243	−	0.876037i	$-0.339822\pi$
$269$	15.8187	0.964486	0.482243	+	0.876037i	$0.339822\pi$
$270$	−6.88601	−0.419069
$271$	−15.5937	−0.947250	−0.473625	−	0.880727i	$-0.657055\pi$
$271$	−15.5937	−0.947250	−0.473625	+	0.880727i	$0.657055\pi$
$272$	−5.30219	−0.321492
$273$	0	0
$274$	13.9143	0.840594
$275$	−11.0566	−0.666738
$276$	10.5761	0.636605
$277$	−8.25334	−0.495895	−0.247948	−	0.968773i	$-0.579756\pi$
$277$	−8.25334	−0.495895	−0.247948	+	0.968773i	$0.579756\pi$
$278$	15.4338	0.925658
$279$	0.677267	0.0405469
$280$	3.37720	0.201826
$281$	1.38708	0.0827460	0.0413730	−	0.999144i	$-0.486827\pi$
$281$	1.38708	0.0827460	0.0413730	+	0.999144i	$0.486827\pi$
$282$	−20.3588	−1.21235
$283$	−23.8988	−1.42064	−0.710318	−	0.703881i	$-0.751449\pi$
$283$	−23.8988	−1.42064	−0.710318	+	0.703881i	$0.751449\pi$
$284$	3.00000	0.178017
$285$	2.10331	0.124589
$286$	0	0
$287$	−10.3305	−0.609789
$288$	0.273891	0.0161392
$289$	11.1132	0.653717
$290$	−9.52936	−0.559583
$291$	18.6610	1.09393
$292$	−9.41325	−0.550869
$293$	0.798202	0.0466315	0.0233157	−	0.999728i	$-0.492578\pi$
$293$	0.798202	0.0466315	0.0233157	+	0.999728i	$0.492578\pi$
$294$	−0.0467198	−0.00272475
$295$	12.1492	0.707356
$296$	2.20662	0.128257
$297$	−17.6970	−1.02688
$298$	−20.4055	−1.18206
$299$	0	0
$300$	−5.57608	−0.321935
$301$	−16.7077	−0.963015
$302$	−3.04672	−0.175319
$303$	−22.8032	−1.31001
$304$	−1.00000	−0.0573539
$305$	7.23492	0.414270
$306$	−1.45222	−0.0830178
$307$	−18.1522	−1.03600	−0.517999	−	0.855381i	$-0.673323\pi$
$307$	−18.1522	−1.03600	−0.517999	+	0.855381i	$0.673323\pi$
$308$	8.67939	0.494554
$309$	25.5761	1.45497
$310$	−3.15003	−0.178910
$311$	15.5011	0.878985	0.439492	−	0.898246i	$-0.355158\pi$
$311$	15.5011	0.878985	0.439492	+	0.898246i	$0.355158\pi$
$312$	0	0
$313$	0.803248	0.0454023	0.0227011	−	0.999742i	$-0.492773\pi$
$313$	0.803248	0.0454023	0.0227011	+	0.999742i	$0.492773\pi$
$314$	2.17833	0.122930
$315$	0.924984	0.0521169
$316$	−12.2555	−0.689424
$317$	−23.0820	−1.29641	−0.648206	−	0.761465i	$-0.724481\pi$
$317$	−23.0820	−1.29641	−0.648206	+	0.761465i	$0.724481\pi$
$318$	4.08277	0.228950
$319$	−24.4904	−1.37120
$320$	−1.27389	−0.0712126
$321$	−3.13161	−0.174789
$322$	−16.9816	−0.946346
$323$	5.30219	0.295022
$324$	−8.10331	−0.450184
$325$	0	0
$326$	0.273891	0.0151694
$327$	−21.3099	−1.17844
$328$	3.89669	0.215159
$329$	32.6893	1.80222
$330$	6.88601	0.379063
$331$	−7.67164	−0.421671	−0.210836	−	0.977522i	$-0.567619\pi$
$331$	−7.67164	−0.421671	−0.210836	+	0.977522i	$0.567619\pi$
$332$	−7.79045	−0.427557
$333$	0.604374	0.0331195
$334$	2.12386	0.116212
$335$	−0.263217	−0.0143811
$336$	4.37720	0.238796
$337$	4.19887	0.228727	0.114364	−	0.993439i	$-0.463517\pi$
$337$	4.19887	0.228727	0.114364	+	0.993439i	$0.463517\pi$
$338$	0	0
$339$	33.2731	1.80715
$340$	6.75441	0.366309
$341$	−8.09556	−0.438399
$342$	−0.273891	−0.0148103
$343$	−18.4826	−0.997969
$344$	6.30219	0.339791
$345$	−13.4728	−0.725349
$346$	−24.3121	−1.30702
$347$	−7.94048	−0.426268	−0.213134	−	0.977023i	$-0.568367\pi$
$347$	−7.94048	−0.426268	−0.213134	+	0.977023i	$0.568367\pi$
$348$	−12.3510	−0.662085
$349$	21.9194	1.17332	0.586658	−	0.809835i	$-0.300443\pi$
$349$	21.9194	1.17332	0.586658	+	0.809835i	$0.300443\pi$
$350$	8.95328	0.478573
$351$	0	0
$352$	−3.27389	−0.174499
$353$	−25.5966	−1.36237	−0.681185	−	0.732111i	$-0.738535\pi$
$353$	−25.5966	−1.36237	−0.681185	+	0.732111i	$0.738535\pi$
$354$	15.7467	0.836925
$355$	−3.82167	−0.202833
$356$	14.1316	0.748974
$357$	−23.2087	−1.22834
$358$	−10.2661	−0.542582
$359$	9.51948	0.502419	0.251210	−	0.967933i	$-0.419172\pi$
$359$	9.51948	0.502419	0.251210	+	0.967933i	$0.419172\pi$
$360$	−0.348907	−0.0183890
$361$	1.00000	0.0526316
$362$	25.6893	1.35020
$363$	−0.465015	−0.0244070
$364$	0	0
$365$	11.9914	0.627661
$366$	9.37720	0.490154
$367$	−2.64122	−0.137871	−0.0689353	−	0.997621i	$-0.521960\pi$
$367$	−2.64122	−0.137871	−0.0689353	+	0.997621i	$0.521960\pi$
$368$	6.40550	0.333910
$369$	1.06727	0.0555596
$370$	−2.81100	−0.146137
$371$	−6.55553	−0.340346
$372$	−4.08277	−0.211682
$373$	24.8628	1.28735	0.643673	−	0.765301i	$-0.277410\pi$
$373$	24.8628	1.28735	0.643673	+	0.765301i	$0.277410\pi$
$374$	17.3588	0.897601
$375$	17.6199	0.909886
$376$	−12.3305	−0.635896
$377$	0	0
$378$	14.3305	0.737081
$379$	−32.1591	−1.65190	−0.825951	−	0.563742i	$-0.809361\pi$
$379$	−32.1591	−1.65190	−0.825951	+	0.563742i	$0.809361\pi$
$380$	1.27389	0.0653492
$381$	−12.7437	−0.652881
$382$	0.857718	0.0438847
$383$	18.7381	0.957472	0.478736	−	0.877959i	$-0.341095\pi$
$383$	18.7381	0.957472	0.478736	+	0.877959i	$0.341095\pi$
$384$	−1.65109	−0.0842570
$385$	−11.0566	−0.563496
$386$	−10.0926	−0.513702
$387$	1.72611	0.0877431
$388$	11.3022	0.573782
$389$	34.9426	1.77166	0.885830	−	0.464009i	$-0.153590\pi$
$389$	34.9426	1.77166	0.885830	+	0.464009i	$0.153590\pi$
$390$	0	0
$391$	−33.9632	−1.71759
$392$	−0.0282963	−0.00142918
$393$	−14.8131	−0.747223
$394$	9.69006	0.488178
$395$	15.6121	0.785531
$396$	−0.896688	−0.0450603
$397$	25.2547	1.26750	0.633748	−	0.773540i	$-0.281516\pi$
$397$	25.2547	1.26750	0.633748	+	0.773540i	$0.281516\pi$
$398$	−24.1960	−1.21283
$399$	−4.37720	−0.219134
$400$	−3.37720	−0.168860
$401$	−0.943407	−0.0471115	−0.0235558	−	0.999723i	$-0.507499\pi$
$401$	−0.943407	−0.0471115	−0.0235558	+	0.999723i	$0.507499\pi$
$402$	−0.341157	−0.0170153
$403$	0	0
$404$	−13.8110	−0.687123
$405$	10.3227	0.512941
$406$	19.8315	0.984223
$407$	−7.22425	−0.358093
$408$	8.75441	0.433408
$409$	−2.87322	−0.142071	−0.0710357	−	0.997474i	$-0.522630\pi$
$409$	−2.87322	−0.142071	−0.0710357	+	0.997474i	$0.522630\pi$
$410$	−4.96395	−0.245152
$411$	−22.9738	−1.13322
$412$	15.4904	0.763157
$413$	−25.2838	−1.24413
$414$	1.75441	0.0862243
$415$	9.92418	0.487159
$416$	0	0
$417$	−25.4826	−1.24789
$418$	3.27389	0.160131
$419$	−27.7955	−1.35790	−0.678949	−	0.734185i	$-0.737564\pi$
$419$	−27.7955	−1.35790	−0.678949	+	0.734185i	$0.737564\pi$
$420$	−5.57608	−0.272085
$421$	−24.1599	−1.17748	−0.588741	−	0.808322i	$-0.700376\pi$
$421$	−24.1599	−1.17748	−0.588741	+	0.808322i	$0.700376\pi$
$422$	−6.72823	−0.327525
$423$	−3.37720	−0.164205
$424$	2.47277	0.120088
$425$	17.9066	0.868596
$426$	−4.95328	−0.239987
$427$	−15.0566	−0.728640
$428$	−1.89669	−0.0916799
$429$	0	0
$430$	−8.02830	−0.387159
$431$	10.0926	0.486145	0.243073	−	0.970008i	$-0.421845\pi$
$431$	10.0926	0.486145	0.243073	+	0.970008i	$0.421845\pi$
$432$	−5.40550	−0.260072
$433$	4.68926	0.225352	0.112676	−	0.993632i	$-0.464058\pi$
$433$	4.68926	0.225352	0.112676	+	0.993632i	$0.464058\pi$
$434$	6.55553	0.314676
$435$	15.7339	0.754381
$436$	−12.9066	−0.618112
$437$	−6.40550	−0.306417
$438$	15.5422	0.742633
$439$	−36.2370	−1.72950	−0.864750	−	0.502203i	$-0.832523\pi$
$439$	−36.2370	−1.72950	−0.864750	+	0.502203i	$0.832523\pi$
$440$	4.17058	0.198825
$441$	−0.00775008	−0.000369051 0
$442$	0	0
$443$	1.13373	0.0538652	0.0269326	−	0.999637i	$-0.491426\pi$
$443$	1.13373	0.0538652	0.0269326	+	0.999637i	$0.491426\pi$
$444$	−3.64334	−0.172905
$445$	−18.0021	−0.853382
$446$	−15.0566	−0.712950
$447$	33.6914	1.59355
$448$	2.65109	0.125252
$449$	21.9944	1.03798	0.518989	−	0.854781i	$-0.326308\pi$
$449$	21.9944	1.03798	0.518989	+	0.854781i	$0.326308\pi$
$450$	−0.924984	−0.0436042
$451$	−12.7573	−0.600719
$452$	20.1522	0.947878
$453$	5.03042	0.236350
$454$	−0.765079	−0.0359069
$455$	0	0
$456$	1.65109	0.0773195
$457$	−16.7771	−0.784798	−0.392399	−	0.919795i	$-0.628355\pi$
$457$	−16.7771	−0.784798	−0.392399	+	0.919795i	$0.628355\pi$
$458$	10.8422	0.506624
$459$	28.6610	1.33778
$460$	−8.15990	−0.380458
$461$	−34.8081	−1.62117	−0.810587	−	0.585618i	$-0.800852\pi$
$461$	−34.8081	−1.62117	−0.810587	+	0.585618i	$0.800852\pi$
$462$	−14.3305	−0.666714
$463$	−0.842218	−0.0391412	−0.0195706	−	0.999808i	$-0.506230\pi$
$463$	−0.842218	−0.0391412	−0.0195706	+	0.999808i	$0.506230\pi$
$464$	−7.48052	−0.347274
$465$	5.20100	0.241190
$466$	15.9816	0.740332
$467$	−30.5003	−1.41138	−0.705692	−	0.708519i	$-0.749364\pi$
$467$	−30.5003	−1.41138	−0.705692	+	0.708519i	$0.749364\pi$
$468$	0	0
$469$	0.547781	0.0252942
$470$	15.7077	0.724542
$471$	−3.59662	−0.165724
$472$	9.53711	0.438981
$473$	−20.6327	−0.948691
$474$	20.2349	0.929421
$475$	3.37720	0.154957
$476$	−14.0566	−0.644283
$477$	0.677267	0.0310099
$478$	−0.118812	−0.00543436
$479$	17.8500	0.815586	0.407793	−	0.913074i	$-0.366299\pi$
$479$	17.8500	0.815586	0.407793	+	0.913074i	$0.366299\pi$
$480$	2.10331	0.0960026
$481$	0	0
$482$	9.02830	0.411228
$483$	28.0382	1.27578
$484$	−0.281641	−0.0128018
$485$	−14.3977	−0.653768
$486$	−2.83717	−0.128697
$487$	23.8676	1.08154	0.540772	−	0.841169i	$-0.318132\pi$
$487$	23.8676	1.08154	0.540772	+	0.841169i	$0.318132\pi$
$488$	5.67939	0.257094
$489$	−0.452219	−0.0204500
$490$	0.0360463	0.00162841
$491$	3.87322	0.174796	0.0873979	−	0.996173i	$-0.472145\pi$
$491$	3.87322	0.174796	0.0873979	+	0.996173i	$0.472145\pi$
$492$	−6.43380	−0.290058
$493$	39.6631	1.78634
$494$	0	0
$495$	1.14228	0.0513418
$496$	−2.47277	−0.111030
$497$	7.95328	0.356753
$498$	12.8628	0.576394
$499$	−6.30219	−0.282125	−0.141062	−	0.990001i	$-0.545052\pi$
$499$	−6.30219	−0.282125	−0.141062	+	0.990001i	$0.545052\pi$
$500$	10.6716	0.477250
$501$	−3.50669	−0.156667
$502$	−3.40550	−0.151995
$503$	32.7587	1.46064	0.730318	−	0.683107i	$-0.239372\pi$
$503$	32.7587	1.46064	0.730318	+	0.683107i	$0.239372\pi$
$504$	0.726109	0.0323435
$505$	17.5937	0.782909
$506$	−20.9709	−0.932270
$507$	0	0
$508$	−7.71836	−0.342447
$509$	−24.6991	−1.09477	−0.547385	−	0.836881i	$-0.684377\pi$
$509$	−24.6991	−1.09477	−0.547385	+	0.836881i	$0.684377\pi$
$510$	−11.1522	−0.493826
$511$	−24.9554	−1.10396
$512$	−1.00000	−0.0441942
$513$	5.40550	0.238659
$514$	−11.3227	−0.499424
$515$	−19.7331	−0.869542
$516$	−10.4055	−0.458077
$517$	40.3687	1.77541
$518$	5.84997	0.257033
$519$	40.1415	1.76202
$520$	0	0
$521$	−5.82942	−0.255392	−0.127696	−	0.991813i	$-0.540758\pi$
$521$	−5.82942	−0.255392	−0.127696	+	0.991813i	$0.540758\pi$
$522$	−2.04884	−0.0896754
$523$	22.8054	0.997209	0.498605	−	0.866830i	$-0.333846\pi$
$523$	22.8054	0.997209	0.498605	+	0.866830i	$0.333846\pi$
$524$	−8.97170	−0.391931
$525$	−14.7827	−0.645170
$526$	29.5294	1.28754
$527$	13.1111	0.571127
$528$	5.40550	0.235244
$529$	18.0304	0.783931
$530$	−3.15003	−0.136829
$531$	2.61212	0.113357
$532$	−2.65109	−0.114939
$533$	0	0
$534$	−23.3326	−1.00970
$535$	2.41617	0.104460
$536$	−0.206625	−0.00892482
$537$	16.9504	0.731462
$538$	−15.8187	−0.681995
$539$	0.0926389	0.00399024
$540$	6.88601	0.296327
$541$	8.16203	0.350913	0.175456	−	0.984487i	$-0.443860\pi$
$541$	8.16203	0.350913	0.175456	+	0.984487i	$0.443860\pi$
$542$	15.5937	0.669807
$543$	−42.4154	−1.82022
$544$	5.30219	0.227329
$545$	16.4415	0.704278
$546$	0	0
$547$	−32.9378	−1.40832	−0.704159	−	0.710042i	$-0.748676\pi$
$547$	−32.9378	−1.40832	−0.704159	+	0.710042i	$0.748676\pi$
$548$	−13.9143	−0.594390
$549$	1.55553	0.0663885
$550$	11.0566	0.471455
$551$	7.48052	0.318681
$552$	−10.5761	−0.450148
$553$	−32.4904	−1.38163
$554$	8.25334	0.350651
$555$	4.64122	0.197009
$556$	−15.4338	−0.654539
$557$	−12.6901	−0.537695	−0.268848	−	0.963183i	$-0.586643\pi$
$557$	−12.6901	−0.537695	−0.268848	+	0.963183i	$0.586643\pi$
$558$	−0.677267	−0.0286710
$559$	0	0
$560$	−3.37720	−0.142713
$561$	−28.6610	−1.21007
$562$	−1.38708	−0.0585103
$563$	14.2039	0.598624	0.299312	−	0.954155i	$-0.403243\pi$
$563$	14.2039	0.598624	0.299312	+	0.954155i	$0.403243\pi$
$564$	20.3588	0.857259
$565$	−25.6716	−1.08001
$566$	23.8988	1.00454
$567$	−21.4826	−0.902186
$568$	−3.00000	−0.125877
$569$	14.6425	0.613847	0.306924	−	0.951734i	$-0.400700\pi$
$569$	14.6425	0.613847	0.306924	+	0.951734i	$0.400700\pi$
$570$	−2.10331	−0.0880981
$571$	4.61505	0.193134	0.0965669	−	0.995327i	$-0.469214\pi$
$571$	4.61505	0.193134	0.0965669	+	0.995327i	$0.469214\pi$
$572$	0	0
$573$	−1.41617	−0.0591615
$574$	10.3305	0.431186
$575$	−21.6327	−0.902145
$576$	−0.273891	−0.0114121
$577$	42.8102	1.78221	0.891106	−	0.453795i	$-0.149930\pi$
$577$	42.8102	1.78221	0.891106	+	0.453795i	$0.149930\pi$
$578$	−11.1132	−0.462248
$579$	16.6639	0.692528
$580$	9.52936	0.395685
$581$	−20.6532	−0.856840
$582$	−18.6610	−0.773522
$583$	−8.09556	−0.335284
$584$	9.41325	0.389523
$585$	0	0
$586$	−0.798202	−0.0329734
$587$	25.3142	1.04483	0.522414	−	0.852692i	$-0.325032\pi$
$587$	25.3142	1.04483	0.522414	+	0.852692i	$0.325032\pi$
$588$	0.0467198	0.00192669
$589$	2.47277	0.101889
$590$	−12.1492	−0.500176
$591$	−15.9992	−0.658119
$592$	−2.20662	−0.0906917
$593$	−1.13241	−0.0465025	−0.0232512	−	0.999730i	$-0.507402\pi$
$593$	−1.13241	−0.0465025	−0.0232512	+	0.999730i	$0.507402\pi$
$594$	17.6970	0.726117
$595$	17.9066	0.734097
$596$	20.4055	0.835842
$597$	39.9498	1.63504
$598$	0	0
$599$	−5.91994	−0.241882	−0.120941	−	0.992660i	$-0.538591\pi$
$599$	−5.91994	−0.241882	−0.120941	+	0.992660i	$0.538591\pi$
$600$	5.57608	0.227642
$601$	18.8550	0.769112	0.384556	−	0.923102i	$-0.374355\pi$
$601$	18.8550	0.769112	0.384556	+	0.923102i	$0.374355\pi$
$602$	16.7077	0.680954
$603$	−0.0565925	−0.00230463
$604$	3.04672	0.123969
$605$	0.358779	0.0145865
$606$	22.8032	0.926319
$607$	−21.0750	−0.855409	−0.427704	−	0.903919i	$-0.640678\pi$
$607$	−21.0750	−0.855409	−0.427704	+	0.903919i	$0.640678\pi$
$608$	1.00000	0.0405554
$609$	−32.7437	−1.32684
$610$	−7.23492	−0.292933
$611$	0	0
$612$	1.45222	0.0587025
$613$	24.0899	0.972983	0.486492	−	0.873685i	$-0.338276\pi$
$613$	24.0899	0.972983	0.486492	+	0.873685i	$0.338276\pi$
$614$	18.1522	0.732561
$615$	8.19595	0.330493
$616$	−8.67939	−0.349703
$617$	−6.20100	−0.249643	−0.124821	−	0.992179i	$-0.539836\pi$
$617$	−6.20100	−0.249643	−0.124821	+	0.992179i	$0.539836\pi$
$618$	−25.5761	−1.02882
$619$	43.7437	1.75821	0.879105	−	0.476629i	$-0.158141\pi$
$619$	43.7437	1.75821	0.879105	+	0.476629i	$0.158141\pi$
$620$	3.15003	0.126508
$621$	−34.6249	−1.38945
$622$	−15.5011	−0.621536
$623$	37.4642	1.50097
$624$	0	0
$625$	3.29151	0.131661
$626$	−0.803248	−0.0321043
$627$	−5.40550	−0.215875
$628$	−2.17833	−0.0869248
$629$	11.6999	0.466507
$630$	−0.924984	−0.0368522
$631$	−14.9434	−0.594888	−0.297444	−	0.954739i	$-0.596134\pi$
$631$	−14.9434	−0.594888	−0.297444	+	0.954739i	$0.596134\pi$
$632$	12.2555	0.487496
$633$	11.1089	0.441541
$634$	23.0820	0.916702
$635$	9.83235	0.390185
$636$	−4.08277	−0.161892
$637$	0	0
$638$	24.4904	0.969584
$639$	−0.821672	−0.0325048
$640$	1.27389	0.0503549
$641$	−25.1599	−0.993756	−0.496878	−	0.867820i	$-0.665520\pi$
$641$	−25.1599	−0.993756	−0.496878	+	0.867820i	$0.665520\pi$
$642$	3.13161	0.123595
$643$	−44.5294	−1.75607	−0.878033	−	0.478600i	$-0.841145\pi$
$643$	−44.5294	−1.75607	−0.878033	+	0.478600i	$0.841145\pi$
$644$	16.9816	0.669168
$645$	13.2555	0.521934
$646$	−5.30219	−0.208612
$647$	−5.70286	−0.224203	−0.112101	−	0.993697i	$-0.535758\pi$
$647$	−5.70286	−0.224203	−0.112101	+	0.993697i	$0.535758\pi$
$648$	8.10331	0.318328
$649$	−31.2234	−1.22563
$650$	0	0
$651$	−10.8238	−0.424218
$652$	−0.273891	−0.0107264
$653$	8.87322	0.347236	0.173618	−	0.984813i	$-0.444454\pi$
$653$	8.87322	0.347236	0.173618	+	0.984813i	$0.444454\pi$
$654$	21.3099	0.833284
$655$	11.4290	0.446567
$656$	−3.89669	−0.152140
$657$	2.57820	0.100585
$658$	−32.6893	−1.27436
$659$	25.4749	0.992361	0.496180	−	0.868219i	$-0.334735\pi$
$659$	25.4749	0.992361	0.496180	+	0.868219i	$0.334735\pi$
$660$	−6.88601	−0.268038
$661$	−26.7595	−1.04082	−0.520411	−	0.853916i	$-0.674221\pi$
$661$	−26.7595	−1.04082	−0.520411	+	0.853916i	$0.674221\pi$
$662$	7.67164	0.298167
$663$	0	0
$664$	7.79045	0.302328
$665$	3.37720	0.130962
$666$	−0.604374	−0.0234190
$667$	−47.9164	−1.85533
$668$	−2.12386	−0.0821746
$669$	24.8598	0.961137
$670$	0.263217	0.0101690
$671$	−18.5937	−0.717802
$672$	−4.37720	−0.168854
$673$	23.6199	0.910479	0.455240	−	0.890369i	$-0.349554\pi$
$673$	23.6199	0.910479	0.455240	+	0.890369i	$0.349554\pi$
$674$	−4.19887	−0.161735
$675$	18.2555	0.702653
$676$	0	0
$677$	0.317687	0.0122097	0.00610485	−	0.999981i	$-0.498057\pi$
$677$	0.317687	0.0122097	0.00610485	+	0.999981i	$0.498057\pi$
$678$	−33.2731	−1.27785
$679$	29.9632	1.14988
$680$	−6.75441	−0.259020
$681$	1.26322	0.0484066
$682$	8.09556	0.309995
$683$	19.4989	0.746106	0.373053	−	0.927810i	$-0.378311\pi$
$683$	19.4989	0.746106	0.373053	+	0.927810i	$0.378311\pi$
$684$	0.273891	0.0104725
$685$	17.7253	0.677249
$686$	18.4826	0.705670
$687$	−17.9015	−0.682985
$688$	−6.30219	−0.240269
$689$	0	0
$690$	13.4728	0.512899
$691$	−3.69006	−0.140377	−0.0701883	−	0.997534i	$-0.522360\pi$
$691$	−3.69006	−0.140377	−0.0701883	+	0.997534i	$0.522360\pi$
$692$	24.3121	0.924206
$693$	−2.37720	−0.0903025
$694$	7.94048	0.301417
$695$	19.6610	0.745783
$696$	12.3510	0.468165
$697$	20.6610	0.782590
$698$	−21.9194	−0.829660
$699$	−26.3871	−0.998051
$700$	−8.95328	−0.338402
$701$	−13.7381	−0.518881	−0.259441	−	0.965759i	$-0.583538\pi$
$701$	−13.7381	−0.518881	−0.259441	+	0.965759i	$0.583538\pi$
$702$	0	0
$703$	2.20662	0.0832244
$704$	3.27389	0.123389
$705$	−25.9349	−0.976763
$706$	25.5966	0.963342
$707$	−36.6142	−1.37702
$708$	−15.7467	−0.591796
$709$	16.1960	0.608252	0.304126	−	0.952632i	$-0.401636\pi$
$709$	16.1960	0.608252	0.304126	+	0.952632i	$0.401636\pi$
$710$	3.82167	0.143425
$711$	3.35666	0.125884
$712$	−14.1316	−0.529604
$713$	−15.8393	−0.593186
$714$	23.2087	0.868566
$715$	0	0
$716$	10.2661	0.383664
$717$	0.196170	0.00732612
$718$	−9.51948	−0.355264
$719$	−49.7040	−1.85364	−0.926822	−	0.375500i	$-0.877471\pi$
$719$	−49.7040	−1.85364	−0.926822	+	0.375500i	$0.877471\pi$
$720$	0.348907	0.0130030
$721$	41.0665	1.52940
$722$	−1.00000	−0.0372161
$723$	−14.9066	−0.554381
$724$	−25.6893	−0.954734
$725$	25.2632	0.938252
$726$	0.465015	0.0172583
$727$	−42.5470	−1.57798	−0.788990	−	0.614406i	$-0.789396\pi$
$727$	−42.5470	−1.57798	−0.788990	+	0.614406i	$0.789396\pi$
$728$	0	0
$729$	28.9944	1.07387
$730$	−11.9914	−0.443823
$731$	33.4154	1.23591
$732$	−9.37720	−0.346591
$733$	−6.69569	−0.247311	−0.123655	−	0.992325i	$-0.539462\pi$
$733$	−6.69569	−0.247311	−0.123655	+	0.992325i	$0.539462\pi$
$734$	2.64122	0.0974892
$735$	−0.0595159	−0.00219528
$736$	−6.40550	−0.236110
$737$	0.676466	0.0249180
$738$	−1.06727	−0.0392866
$739$	27.5080	1.01190	0.505949	−	0.862563i	$-0.331142\pi$
$739$	27.5080	1.01190	0.505949	+	0.862563i	$0.331142\pi$
$740$	2.81100	0.103334
$741$	0	0
$742$	6.55553	0.240661
$743$	2.11319	0.0775252	0.0387626	−	0.999248i	$-0.487658\pi$
$743$	2.11319	0.0775252	0.0387626	+	0.999248i	$0.487658\pi$
$744$	4.08277	0.149681
$745$	−25.9944	−0.952360
$746$	−24.8628	−0.910290
$747$	2.13373	0.0780691
$748$	−17.3588	−0.634700
$749$	−5.02830	−0.183730
$750$	−17.6199	−0.643387
$751$	2.72531	0.0994479	0.0497240	−	0.998763i	$-0.484166\pi$
$751$	2.72531	0.0994479	0.0497240	+	0.998763i	$0.484166\pi$
$752$	12.3305	0.449646
$753$	5.62280	0.204906
$754$	0	0
$755$	−3.88119	−0.141251
$756$	−14.3305	−0.521195
$757$	−24.6228	−0.894931	−0.447465	−	0.894301i	$-0.647673\pi$
$757$	−24.6228	−0.894931	−0.447465	+	0.894301i	$0.647673\pi$
$758$	32.1591	1.16807
$759$	34.6249	1.25680
$760$	−1.27389	−0.0462089
$761$	42.7488	1.54964	0.774821	−	0.632181i	$-0.217840\pi$
$761$	42.7488	1.54964	0.774821	+	0.632181i	$0.217840\pi$
$762$	12.7437	0.461657
$763$	−34.2165	−1.23872
$764$	−0.857718	−0.0310311
$765$	−1.84997	−0.0668857
$766$	−18.7381	−0.677035
$767$	0	0
$768$	1.65109	0.0595787
$769$	−21.8473	−0.787832	−0.393916	−	0.919146i	$-0.628880\pi$
$769$	−21.8473	−0.787832	−0.393916	+	0.919146i	$0.628880\pi$
$770$	11.0566	0.398452
$771$	18.6949	0.673280
$772$	10.0926	0.363242
$773$	18.0587	0.649527	0.324763	−	0.945795i	$-0.394715\pi$
$773$	18.0587	0.649527	0.324763	+	0.945795i	$0.394715\pi$
$774$	−1.72611	−0.0620437
$775$	8.35103	0.299978
$776$	−11.3022	−0.405725
$777$	−9.65884	−0.346509
$778$	−34.9426	−1.25275
$779$	3.89669	0.139613
$780$	0	0
$781$	9.82167	0.351447
$782$	33.9632	1.21452
$783$	40.4359	1.44506
$784$	0.0282963	0.00101058
$785$	2.77495	0.0990423
$786$	14.8131	0.528367
$787$	−38.3091	−1.36557	−0.682787	−	0.730618i	$-0.739232\pi$
$787$	−38.3091	−1.36557	−0.682787	+	0.730618i	$0.739232\pi$
$788$	−9.69006	−0.345194
$789$	−48.7557	−1.73575
$790$	−15.6121	−0.555454
$791$	53.4252	1.89958
$792$	0.896688	0.0318624
$793$	0	0
$794$	−25.2547	−0.896255
$795$	5.20100	0.184460
$796$	24.1960	0.857603
$797$	−27.1465	−0.961579	−0.480790	−	0.876836i	$-0.659650\pi$
$797$	−27.1465	−0.961579	−0.480790	+	0.876836i	$0.659650\pi$
$798$	4.37720	0.154951
$799$	−65.3785	−2.31293
$800$	3.37720	0.119402
$801$	−3.87051	−0.136758
$802$	0.943407	0.0333129
$803$	−30.8179	−1.08754
$804$	0.341157	0.0120317
$805$	−21.6327	−0.762451
$806$	0	0
$807$	26.1182	0.919405
$808$	13.8110	0.485869
$809$	7.17833	0.252377	0.126188	−	0.992006i	$-0.459726\pi$
$809$	7.17833	0.252377	0.126188	+	0.992006i	$0.459726\pi$
$810$	−10.3227	−0.362704
$811$	−6.01842	−0.211335	−0.105668	−	0.994401i	$-0.533698\pi$
$811$	−6.01842	−0.211335	−0.105668	+	0.994401i	$0.533698\pi$
$812$	−19.8315	−0.695951
$813$	−25.7467	−0.902975
$814$	7.22425	0.253210
$815$	0.348907	0.0122217
$816$	−8.75441	−0.306465
$817$	6.30219	0.220486
$818$	2.87322	0.100460
$819$	0	0
$820$	4.96395	0.173349
$821$	21.0069	0.733148	0.366574	−	0.930389i	$-0.380531\pi$
$821$	21.0069	0.733148	0.366574	+	0.930389i	$0.380531\pi$
$822$	22.9738	0.801304
$823$	−20.6023	−0.718149	−0.359075	−	0.933309i	$-0.616908\pi$
$823$	−20.6023	−0.718149	−0.359075	+	0.933309i	$0.616908\pi$
$824$	−15.4904	−0.539633
$825$	−18.2555	−0.635574
$826$	25.2838	0.879735
$827$	27.0643	0.941119	0.470560	−	0.882368i	$-0.344052\pi$
$827$	27.0643	0.941119	0.470560	+	0.882368i	$0.344052\pi$
$828$	−1.75441	−0.0609698
$829$	14.1805	0.492507	0.246254	−	0.969205i	$-0.420800\pi$
$829$	14.1805	0.492507	0.246254	+	0.969205i	$0.420800\pi$
$830$	−9.92418	−0.344473
$831$	−13.6270	−0.472717
$832$	0	0
$833$	−0.150032	−0.00519830
$834$	25.4826	0.882392
$835$	2.70556	0.0936299
$836$	−3.27389	−0.113230
$837$	13.3665	0.462015
$838$	27.7955	0.960180
$839$	−41.2341	−1.42356	−0.711780	−	0.702403i	$-0.752111\pi$
$839$	−41.2341	−1.42356	−0.711780	+	0.702403i	$0.752111\pi$
$840$	5.57608	0.192393
$841$	26.9581	0.929590
$842$	24.1599	0.832605
$843$	2.29019	0.0788784
$844$	6.72823	0.231595
$845$	0	0
$846$	3.37720	0.116111
$847$	−0.746656	−0.0256554
$848$	−2.47277	−0.0849151
$849$	−39.4592	−1.35424
$850$	−17.9066	−0.614190
$851$	−14.1345	−0.484526
$852$	4.95328	0.169697
$853$	5.96900	0.204375	0.102187	−	0.994765i	$-0.467416\pi$
$853$	5.96900	0.204375	0.102187	+	0.994765i	$0.467416\pi$
$854$	15.0566	0.515226
$855$	−0.348907	−0.0119324
$856$	1.89669	0.0648275
$857$	−39.2285	−1.34002	−0.670010	−	0.742352i	$-0.733710\pi$
$857$	−39.2285	−1.34002	−0.670010	+	0.742352i	$0.733710\pi$
$858$	0	0
$859$	6.83447	0.233189	0.116595	−	0.993180i	$-0.462802\pi$
$859$	6.83447	0.233189	0.116595	+	0.993180i	$0.462802\pi$
$860$	8.02830	0.273763
$861$	−17.0566	−0.581287
$862$	−10.0926	−0.343757
$863$	−38.2760	−1.30293	−0.651465	−	0.758678i	$-0.725845\pi$
$863$	−38.2760	−1.30293	−0.651465	+	0.758678i	$0.725845\pi$
$864$	5.40550	0.183899
$865$	−30.9709	−1.05304
$866$	−4.68926	−0.159348
$867$	18.3489	0.623162
$868$	−6.55553	−0.222509
$869$	−40.1231	−1.36108
$870$	−15.7339	−0.533428
$871$	0	0
$872$	12.9066	0.437071
$873$	−3.09556	−0.104769
$874$	6.40550	0.216669
$875$	28.2915	0.956428
$876$	−15.5422	−0.525121
$877$	10.0480	0.339298	0.169649	−	0.985505i	$-0.445737\pi$
$877$	10.0480	0.339298	0.169649	+	0.985505i	$0.445737\pi$
$878$	36.2370	1.22294
$879$	1.31791	0.0444519
$880$	−4.17058	−0.140590
$881$	−36.5569	−1.23163	−0.615816	−	0.787890i	$-0.711173\pi$
$881$	−36.5569	−1.23163	−0.615816	+	0.787890i	$0.711173\pi$
$882$	0.00775008	0.000260959 0
$883$	−10.0926	−0.339644	−0.169822	−	0.985475i	$-0.554319\pi$
$883$	−10.0926	−0.339644	−0.169822	+	0.985475i	$0.554319\pi$
$884$	0	0
$885$	20.0595	0.674293
$886$	−1.13373	−0.0380884
$887$	14.3596	0.482148	0.241074	−	0.970507i	$-0.422500\pi$
$887$	14.3596	0.482148	0.241074	+	0.970507i	$0.422500\pi$
$888$	3.64334	0.122263
$889$	−20.4621	−0.686277
$890$	18.0021	0.603433
$891$	−26.5294	−0.888767
$892$	15.0566	0.504132
$893$	−12.3305	−0.412624
$894$	−33.6914	−1.12681
$895$	−13.0779	−0.437147
$896$	−2.65109	−0.0885668
$897$	0	0
$898$	−21.9944	−0.733962
$899$	18.4976	0.616928
$900$	0.924984	0.0308328
$901$	13.1111	0.436793
$902$	12.7573	0.424773
$903$	−27.5860	−0.918003
$904$	−20.1522	−0.670251
$905$	32.7253	1.08783
$906$	−5.03042	−0.167124
$907$	−6.26109	−0.207896	−0.103948	−	0.994583i	$-0.533148\pi$
$907$	−6.26109	−0.207896	−0.103948	+	0.994583i	$0.533148\pi$
$908$	0.765079	0.0253900
$909$	3.78270	0.125464
$910$	0	0
$911$	−34.5053	−1.14321	−0.571606	−	0.820528i	$-0.693679\pi$
$911$	−34.5053	−1.14321	−0.571606	+	0.820528i	$0.693679\pi$
$912$	−1.65109	−0.0546732
$913$	−25.5051	−0.844095
$914$	16.7771	0.554936
$915$	11.9455	0.394907
$916$	−10.8422	−0.358237
$917$	−23.7848	−0.785444
$918$	−28.6610	−0.945953
$919$	−56.3374	−1.85840	−0.929200	−	0.369577i	$-0.879503\pi$
$919$	−56.3374	−1.85840	−0.929200	+	0.369577i	$0.879503\pi$
$920$	8.15990	0.269024
$921$	−29.9709	−0.987575
$922$	34.8081	1.14634
$923$	0	0
$924$	14.3305	0.471438
$925$	7.45222	0.245027
$926$	0.842218	0.0276770
$927$	−4.24267	−0.139348
$928$	7.48052	0.245560
$929$	−17.9602	−0.589256	−0.294628	−	0.955612i	$-0.595196\pi$
$929$	−17.9602	−0.589256	−0.294628	+	0.955612i	$0.595196\pi$
$930$	−5.20100	−0.170547
$931$	−0.0282963	−0.000927373 0
$932$	−15.9816	−0.523494
$933$	25.5937	0.837900
$934$	30.5003	0.997999
$935$	22.1132	0.723178
$936$	0	0
$937$	−25.2886	−0.826142	−0.413071	−	0.910699i	$-0.635544\pi$
$937$	−25.2886	−0.826142	−0.413071	+	0.910699i	$0.635544\pi$
$938$	−0.547781	−0.0178857
$939$	1.32624	0.0432801
$940$	−15.7077	−0.512328
$941$	−23.2095	−0.756610	−0.378305	−	0.925681i	$-0.623493\pi$
$941$	−23.2095	−0.756610	−0.378305	+	0.925681i	$0.623493\pi$
$942$	3.59662	0.117184
$943$	−24.9602	−0.812817
$944$	−9.53711	−0.310406
$945$	18.2555	0.593850
$946$	20.6327	0.670826
$947$	55.2675	1.79595	0.897976	−	0.440045i	$-0.145038\pi$
$947$	55.2675	1.79595	0.897976	+	0.440045i	$0.145038\pi$
$948$	−20.2349	−0.657200
$949$	0	0
$950$	−3.37720	−0.109571
$951$	−38.1105	−1.23582
$952$	14.0566	0.455577
$953$	−3.72691	−0.120726	−0.0603632	−	0.998176i	$-0.519226\pi$
$953$	−3.72691	−0.120726	−0.0603632	+	0.998176i	$0.519226\pi$
$954$	−0.677267	−0.0219273
$955$	1.09264	0.0353570
$956$	0.118812	0.00384267
$957$	−40.4359	−1.30711
$958$	−17.8500	−0.576706
$959$	−36.8881	−1.19118
$960$	−2.10331	−0.0678841
$961$	−24.8854	−0.802756
$962$	0	0
$963$	0.519485	0.0167402
$964$	−9.02830	−0.290782
$965$	−12.8569	−0.413879
$966$	−28.0382	−0.902113
$967$	−8.11399	−0.260928	−0.130464	−	0.991453i	$-0.541647\pi$
$967$	−8.11399	−0.260928	−0.130464	+	0.991453i	$0.541647\pi$
$968$	0.281641	0.00905227
$969$	8.75441	0.281232
$970$	14.3977	0.462284
$971$	6.74453	0.216442	0.108221	−	0.994127i	$-0.465485\pi$
$971$	6.74453	0.216442	0.108221	+	0.994127i	$0.465485\pi$
$972$	2.83717	0.0910023
$973$	−40.9164	−1.31172
$974$	−23.8676	−0.764767
$975$	0	0
$976$	−5.67939	−0.181793
$977$	−46.2517	−1.47972	−0.739862	−	0.672758i	$-0.765109\pi$
$977$	−46.2517	−1.47972	−0.739862	+	0.672758i	$0.765109\pi$
$978$	0.452219	0.0144604
$979$	46.2653	1.47865
$980$	−0.0360463	−0.00115146
$981$	3.53499	0.112863
$982$	−3.87322	−0.123599
$983$	26.7488	0.853154	0.426577	−	0.904451i	$-0.359719\pi$
$983$	26.7488	0.853154	0.426577	+	0.904451i	$0.359719\pi$
$984$	6.43380	0.205102
$985$	12.3441	0.393315
$986$	−39.6631	−1.26313
$987$	53.9730	1.71798
$988$	0	0
$989$	−40.3687	−1.28365
$990$	−1.14228	−0.0363041
$991$	−18.1004	−0.574978	−0.287489	−	0.957784i	$-0.592820\pi$
$991$	−18.1004	−0.574978	−0.287489	+	0.957784i	$0.592820\pi$
$992$	2.47277	0.0785104
$993$	−12.6666	−0.401962
$994$	−7.95328	−0.252263
$995$	−30.8230	−0.977155
$996$	−12.8628	−0.407572
$997$	−12.6023	−0.399117	−0.199559	−	0.979886i	$-0.563951\pi$
$997$	−12.6023	−0.399117	−0.199559	+	0.979886i	$0.563951\pi$
$998$	6.30219	0.199492
$999$	11.9279	0.377382

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.m.1.3		3
13.4	even	6		494.2.g.b.419.1	yes	6
13.10	even	6		494.2.g.b.191.1	✓	6
13.12	even	2		6422.2.a.u.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.b.191.1	✓	6	13.10	even	6
494.2.g.b.419.1	yes	6	13.4	even	6
6422.2.a.m.1.3		3	1.1	even	1	trivial
6422.2.a.u.1.3		3	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.65109 q^{3} +1.00000 q^{4} -1.27389 q^{5} -1.65109 q^{6} +2.65109 q^{7} -1.00000 q^{8} -0.273891 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.65109 q^{3} +1.00000 q^{4} -1.27389 q^{5} -1.65109 q^{6} +2.65109 q^{7} -1.00000 q^{8} -0.273891 q^{9} +1.27389 q^{10} +3.27389 q^{11} +1.65109 q^{12} -2.65109 q^{14} -2.10331 q^{15} +1.00000 q^{16} -5.30219 q^{17} +0.273891 q^{18} -1.00000 q^{19} -1.27389 q^{20} +4.37720 q^{21} -3.27389 q^{22} +6.40550 q^{23} -1.65109 q^{24} -3.37720 q^{25} -5.40550 q^{27} +2.65109 q^{28} -7.48052 q^{29} +2.10331 q^{30} -2.47277 q^{31} -1.00000 q^{32} +5.40550 q^{33} +5.30219 q^{34} -3.37720 q^{35} -0.273891 q^{36} -2.20662 q^{37} +1.00000 q^{38} +1.27389 q^{40} -3.89669 q^{41} -4.37720 q^{42} -6.30219 q^{43} +3.27389 q^{44} +0.348907 q^{45} -6.40550 q^{46} +12.3305 q^{47} +1.65109 q^{48} +0.0282963 q^{49} +3.37720 q^{50} -8.75441 q^{51} -2.47277 q^{53} +5.40550 q^{54} -4.17058 q^{55} -2.65109 q^{56} -1.65109 q^{57} +7.48052 q^{58} -9.53711 q^{59} -2.10331 q^{60} -5.67939 q^{61} +2.47277 q^{62} -0.726109 q^{63} +1.00000 q^{64} -5.40550 q^{66} +0.206625 q^{67} -5.30219 q^{68} +10.5761 q^{69} +3.37720 q^{70} +3.00000 q^{71} +0.273891 q^{72} -9.41325 q^{73} +2.20662 q^{74} -5.57608 q^{75} -1.00000 q^{76} +8.67939 q^{77} -12.2555 q^{79} -1.27389 q^{80} -8.10331 q^{81} +3.89669 q^{82} -7.79045 q^{83} +4.37720 q^{84} +6.75441 q^{85} +6.30219 q^{86} -12.3510 q^{87} -3.27389 q^{88} +14.1316 q^{89} -0.348907 q^{90} +6.40550 q^{92} -4.08277 q^{93} -12.3305 q^{94} +1.27389 q^{95} -1.65109 q^{96} +11.3022 q^{97} -0.0282963 q^{98} -0.896688 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 2 q^{6} + q^{7} - 3 q^{8} + q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 2 * q^6 + q^7 - 3 * q^8 + q^9	\( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} + 2 q^{6} + q^{7} - 3 q^{8} + q^{9} + 2 q^{10} + 8 q^{11} - 2 q^{12} - q^{14} - 3 q^{15} + 3 q^{16} - 2 q^{17} - q^{18} - 3 q^{19} - 2 q^{20} + 8 q^{21} - 8 q^{22} + 2 q^{23} + 2 q^{24} - 5 q^{25} + q^{27} + q^{28} - 14 q^{29} + 3 q^{30} + 5 q^{31} - 3 q^{32} - q^{33} + 2 q^{34} - 5 q^{35} + q^{36} + 3 q^{38} + 2 q^{40} - 15 q^{41} - 8 q^{42} - 5 q^{43} + 8 q^{44} + 8 q^{45} - 2 q^{46} + 11 q^{47} - 2 q^{48} - 12 q^{49} + 5 q^{50} - 16 q^{51} + 5 q^{53} - q^{54} - 14 q^{55} - q^{56} + 2 q^{57} + 14 q^{58} + 4 q^{59} - 3 q^{60} + 2 q^{61} - 5 q^{62} - 4 q^{63} + 3 q^{64} + q^{66} - 6 q^{67} - 2 q^{68} + 16 q^{69} + 5 q^{70} + 9 q^{71} - q^{72} - 15 q^{73} - q^{75} - 3 q^{76} + 7 q^{77} - 2 q^{79} - 2 q^{80} - 21 q^{81} + 15 q^{82} - 5 q^{83} + 8 q^{84} + 10 q^{85} + 5 q^{86} + 5 q^{87} - 8 q^{88} + 27 q^{89} - 8 q^{90} + 2 q^{92} - 25 q^{93} - 11 q^{94} + 2 q^{95} + 2 q^{96} + 20 q^{97} + 12 q^{98} - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 2 * q^5 + 2 * q^6 + q^7 - 3 * q^8 + q^9 + 2 * q^10 + 8 * q^11 - 2 * q^12 - q^14 - 3 * q^15 + 3 * q^16 - 2 * q^17 - q^18 - 3 * q^19 - 2 * q^20 + 8 * q^21 - 8 * q^22 + 2 * q^23 + 2 * q^24 - 5 * q^25 + q^27 + q^28 - 14 * q^29 + 3 * q^30 + 5 * q^31 - 3 * q^32 - q^33 + 2 * q^34 - 5 * q^35 + q^36 + 3 * q^38 + 2 * q^40 - 15 * q^41 - 8 * q^42 - 5 * q^43 + 8 * q^44 + 8 * q^45 - 2 * q^46 + 11 * q^47 - 2 * q^48 - 12 * q^49 + 5 * q^50 - 16 * q^51 + 5 * q^53 - q^54 - 14 * q^55 - q^56 + 2 * q^57 + 14 * q^58 + 4 * q^59 - 3 * q^60 + 2 * q^61 - 5 * q^62 - 4 * q^63 + 3 * q^64 + q^66 - 6 * q^67 - 2 * q^68 + 16 * q^69 + 5 * q^70 + 9 * q^71 - q^72 - 15 * q^73 - q^75 - 3 * q^76 + 7 * q^77 - 2 * q^79 - 2 * q^80 - 21 * q^81 + 15 * q^82 - 5 * q^83 + 8 * q^84 + 10 * q^85 + 5 * q^86 + 5 * q^87 - 8 * q^88 + 27 * q^89 - 8 * q^90 + 2 * q^92 - 25 * q^93 - 11 * q^94 + 2 * q^95 + 2 * q^96 + 20 * q^97 + 12 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more