LMFDB - Embedded newform 6422.2.a.bh.1.1

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 4x^{6} + 22x^{5} + 3x^{4} - 28x^{3} + 7x^{2} + 6x - 2 $ x^8 - 4x^7 - 4x^6 + 22x^5 + 3x^4 - 28x^3 + 7x^2 + 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.1
Root			$3.36639$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -3.36639 q^{3} +1.00000 q^{4} +0.659053 q^{5} -3.36639 q^{6} -1.55483 q^{7} +1.00000 q^{8} +8.33259 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.36639 q^{3} +1.00000 q^{4} +0.659053 q^{5} -3.36639 q^{6} -1.55483 q^{7} +1.00000 q^{8} +8.33259 q^{9} +0.659053 q^{10} -2.14446 q^{11} -3.36639 q^{12} -1.55483 q^{14} -2.21863 q^{15} +1.00000 q^{16} +2.31516 q^{17} +8.33259 q^{18} -1.00000 q^{19} +0.659053 q^{20} +5.23415 q^{21} -2.14446 q^{22} +0.906597 q^{23} -3.36639 q^{24} -4.56565 q^{25} -17.9516 q^{27} -1.55483 q^{28} +6.14787 q^{29} -2.21863 q^{30} -6.56326 q^{31} +1.00000 q^{32} +7.21909 q^{33} +2.31516 q^{34} -1.02471 q^{35} +8.33259 q^{36} +1.53099 q^{37} -1.00000 q^{38} +0.659053 q^{40} +5.85275 q^{41} +5.23415 q^{42} +0.197219 q^{43} -2.14446 q^{44} +5.49162 q^{45} +0.906597 q^{46} -0.0646766 q^{47} -3.36639 q^{48} -4.58252 q^{49} -4.56565 q^{50} -7.79373 q^{51} -12.5347 q^{53} -17.9516 q^{54} -1.41331 q^{55} -1.55483 q^{56} +3.36639 q^{57} +6.14787 q^{58} +14.3903 q^{59} -2.21863 q^{60} -0.404778 q^{61} -6.56326 q^{62} -12.9557 q^{63} +1.00000 q^{64} +7.21909 q^{66} +9.25247 q^{67} +2.31516 q^{68} -3.05196 q^{69} -1.02471 q^{70} -5.02728 q^{71} +8.33259 q^{72} +6.09824 q^{73} +1.53099 q^{74} +15.3698 q^{75} -1.00000 q^{76} +3.33426 q^{77} -4.22818 q^{79} +0.659053 q^{80} +35.4343 q^{81} +5.85275 q^{82} -4.60991 q^{83} +5.23415 q^{84} +1.52581 q^{85} +0.197219 q^{86} -20.6961 q^{87} -2.14446 q^{88} -14.3869 q^{89} +5.49162 q^{90} +0.906597 q^{92} +22.0945 q^{93} -0.0646766 q^{94} -0.659053 q^{95} -3.36639 q^{96} +15.6508 q^{97} -4.58252 q^{98} -17.8689 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} + 8 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^7 + 8 * q^8	$ 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} + 8 q^{8} - 2 q^{10} - 10 q^{11} - 4 q^{12} - 2 q^{14} + 8 q^{16} + 2 q^{17} - 8 q^{19} - 2 q^{20} + 12 q^{21} - 10 q^{22} - 8 q^{23} - 4 q^{24} - 14 q^{25} - 22 q^{27} - 2 q^{28} + 8 q^{29} - 12 q^{31} + 8 q^{32} + 4 q^{33} + 2 q^{34} - 10 q^{35} - 8 q^{38} - 2 q^{40} + 2 q^{41} + 12 q^{42} - 16 q^{43} - 10 q^{44} - 12 q^{45} - 8 q^{46} + 12 q^{47} - 4 q^{48} - 14 q^{49} - 14 q^{50} - 22 q^{51} - 24 q^{53} - 22 q^{54} - 10 q^{55} - 2 q^{56} + 4 q^{57} + 8 q^{58} - 4 q^{59} - 26 q^{61} - 12 q^{62} - 12 q^{63} + 8 q^{64} + 4 q^{66} + 10 q^{67} + 2 q^{68} + 6 q^{69} - 10 q^{70} - 36 q^{71} + 20 q^{73} + 6 q^{75} - 8 q^{76} - 12 q^{77} - 22 q^{79} - 2 q^{80} + 36 q^{81} + 2 q^{82} + 18 q^{83} + 12 q^{84} + 26 q^{85} - 16 q^{86} - 38 q^{87} - 10 q^{88} - 18 q^{89} - 12 q^{90} - 8 q^{92} + 12 q^{93} + 12 q^{94} + 2 q^{95} - 4 q^{96} - 8 q^{97} - 14 q^{98} - 8 q^{99}+O(q^{100}) $ 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^7 + 8 * q^8 - 2 * q^10 - 10 * q^11 - 4 * q^12 - 2 * q^14 + 8 * q^16 + 2 * q^17 - 8 * q^19 - 2 * q^20 + 12 * q^21 - 10 * q^22 - 8 * q^23 - 4 * q^24 - 14 * q^25 - 22 * q^27 - 2 * q^28 + 8 * q^29 - 12 * q^31 + 8 * q^32 + 4 * q^33 + 2 * q^34 - 10 * q^35 - 8 * q^38 - 2 * q^40 + 2 * q^41 + 12 * q^42 - 16 * q^43 - 10 * q^44 - 12 * q^45 - 8 * q^46 + 12 * q^47 - 4 * q^48 - 14 * q^49 - 14 * q^50 - 22 * q^51 - 24 * q^53 - 22 * q^54 - 10 * q^55 - 2 * q^56 + 4 * q^57 + 8 * q^58 - 4 * q^59 - 26 * q^61 - 12 * q^62 - 12 * q^63 + 8 * q^64 + 4 * q^66 + 10 * q^67 + 2 * q^68 + 6 * q^69 - 10 * q^70 - 36 * q^71 + 20 * q^73 + 6 * q^75 - 8 * q^76 - 12 * q^77 - 22 * q^79 - 2 * q^80 + 36 * q^81 + 2 * q^82 + 18 * q^83 + 12 * q^84 + 26 * q^85 - 16 * q^86 - 38 * q^87 - 10 * q^88 - 18 * q^89 - 12 * q^90 - 8 * q^92 + 12 * q^93 + 12 * q^94 + 2 * q^95 - 4 * q^96 - 8 * q^97 - 14 * q^98 - 8 * 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
$3$	−3.36639	−1.94359	−0.971794	−	0.235833i	$-0.924218\pi$
$3$	−3.36639	−1.94359	−0.971794	+	0.235833i	$0.924218\pi$
$4$	1.00000	0.500000
$5$	0.659053	0.294737	0.147369	−	0.989082i	$-0.452920\pi$
$5$	0.659053	0.294737	0.147369	+	0.989082i	$0.452920\pi$
$6$	−3.36639	−1.37432
$7$	−1.55483	−0.587669	−0.293834	−	0.955856i	$-0.594931\pi$
$7$	−1.55483	−0.587669	−0.293834	+	0.955856i	$0.594931\pi$
$8$	1.00000	0.353553
$9$	8.33259	2.77753
$10$	0.659053	0.208411
$11$	−2.14446	−0.646579	−0.323290	−	0.946300i	$-0.604789\pi$
$11$	−2.14446	−0.646579	−0.323290	+	0.946300i	$0.604789\pi$
$12$	−3.36639	−0.971794
$13$	0	0
$13$	0	0
$14$	−1.55483	−0.415545
$15$	−2.21863	−0.572848
$16$	1.00000	0.250000
$17$	2.31516	0.561508	0.280754	−	0.959780i	$-0.409415\pi$
$17$	2.31516	0.561508	0.280754	+	0.959780i	$0.409415\pi$
$18$	8.33259	1.96401
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.659053	0.147369
$21$	5.23415	1.14219
$22$	−2.14446	−0.457201
$23$	0.906597	0.189039	0.0945193	−	0.995523i	$-0.469869\pi$
$23$	0.906597	0.189039	0.0945193	+	0.995523i	$0.469869\pi$
$24$	−3.36639	−0.687162
$25$	−4.56565	−0.913130
$26$	0	0
$27$	−17.9516	−3.45479
$28$	−1.55483	−0.293834
$29$	6.14787	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$29$	6.14787	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$30$	−2.21863	−0.405064
$31$	−6.56326	−1.17880	−0.589398	−	0.807843i	$-0.700635\pi$
$31$	−6.56326	−1.17880	−0.589398	+	0.807843i	$0.700635\pi$
$32$	1.00000	0.176777
$33$	7.21909	1.25668
$34$	2.31516	0.397046
$35$	−1.02471	−0.173208
$36$	8.33259	1.38877
$37$	1.53099	0.251693	0.125847	−	0.992050i	$-0.459835\pi$
$37$	1.53099	0.251693	0.125847	+	0.992050i	$0.459835\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0.659053	0.104205
$41$	5.85275	0.914046	0.457023	−	0.889455i	$-0.348916\pi$
$41$	5.85275	0.914046	0.457023	+	0.889455i	$0.348916\pi$
$42$	5.23415	0.807647
$43$	0.197219	0.0300757	0.0150378	−	0.999887i	$-0.495213\pi$
$43$	0.197219	0.0300757	0.0150378	+	0.999887i	$0.495213\pi$
$44$	−2.14446	−0.323290
$45$	5.49162	0.818642
$46$	0.906597	0.133670
$47$	−0.0646766	−0.00943405	−0.00471703	−	0.999989i	$-0.501501\pi$
$47$	−0.0646766	−0.00943405	−0.00471703	+	0.999989i	$0.501501\pi$
$48$	−3.36639	−0.485897
$49$	−4.58252	−0.654646
$50$	−4.56565	−0.645680
$51$	−7.79373	−1.09134
$52$	0	0
$53$	−12.5347	−1.72177	−0.860885	−	0.508799i	$-0.830090\pi$
$53$	−12.5347	−1.72177	−0.860885	+	0.508799i	$0.830090\pi$
$54$	−17.9516	−2.44290
$55$	−1.41331	−0.190571
$56$	−1.55483	−0.207772
$57$	3.36639	0.445889
$58$	6.14787	0.807255
$59$	14.3903	1.87345	0.936726	−	0.350064i	$-0.113840\pi$
$59$	14.3903	1.87345	0.936726	+	0.350064i	$0.113840\pi$
$60$	−2.21863	−0.286424
$61$	−0.404778	−0.0518265	−0.0259132	−	0.999664i	$-0.508249\pi$
$61$	−0.404778	−0.0518265	−0.0259132	+	0.999664i	$0.508249\pi$
$62$	−6.56326	−0.833535
$63$	−12.9557	−1.63227
$64$	1.00000	0.125000
$65$	0	0
$66$	7.21909	0.888609
$67$	9.25247	1.13037	0.565185	−	0.824964i	$-0.308805\pi$
$67$	9.25247	1.13037	0.565185	+	0.824964i	$0.308805\pi$
$68$	2.31516	0.280754
$69$	−3.05196	−0.367413
$70$	−1.02471	−0.122476
$71$	−5.02728	−0.596628	−0.298314	−	0.954468i	$-0.596424\pi$
$71$	−5.02728	−0.596628	−0.298314	+	0.954468i	$0.596424\pi$
$72$	8.33259	0.982006
$73$	6.09824	0.713745	0.356872	−	0.934153i	$-0.383843\pi$
$73$	6.09824	0.713745	0.356872	+	0.934153i	$0.383843\pi$
$74$	1.53099	0.177974
$75$	15.3698	1.77475
$76$	−1.00000	−0.114708
$77$	3.33426	0.379974
$78$	0	0
$79$	−4.22818	−0.475707	−0.237854	−	0.971301i	$-0.576444\pi$
$79$	−4.22818	−0.475707	−0.237854	+	0.971301i	$0.576444\pi$
$80$	0.659053	0.0736843
$81$	35.4343	3.93715
$82$	5.85275	0.646328
$83$	−4.60991	−0.506003	−0.253002	−	0.967466i	$-0.581418\pi$
$83$	−4.60991	−0.506003	−0.253002	+	0.967466i	$0.581418\pi$
$84$	5.23415	0.571093
$85$	1.52581	0.165497
$86$	0.197219	0.0212667
$87$	−20.6961	−2.21886
$88$	−2.14446	−0.228600
$89$	−14.3869	−1.52500	−0.762502	−	0.646985i	$-0.776030\pi$
$89$	−14.3869	−1.52500	−0.762502	+	0.646985i	$0.776030\pi$
$90$	5.49162	0.578867
$91$	0	0
$92$	0.906597	0.0945193
$93$	22.0945	2.29109
$94$	−0.0646766	−0.00667088
$95$	−0.659053	−0.0676174
$96$	−3.36639	−0.343581
$97$	15.6508	1.58910	0.794551	−	0.607197i	$-0.207706\pi$
$97$	15.6508	1.58910	0.794551	+	0.607197i	$0.207706\pi$
$98$	−4.58252	−0.462904
$99$	−17.8689	−1.79589
$100$	−4.56565	−0.456565
$101$	6.67794	0.664479	0.332240	−	0.943195i	$-0.392196\pi$
$101$	6.67794	0.664479	0.332240	+	0.943195i	$0.392196\pi$
$102$	−7.79373	−0.771694
$103$	10.3411	1.01893	0.509467	−	0.860490i	$-0.329843\pi$
$103$	10.3411	1.01893	0.509467	+	0.860490i	$0.329843\pi$
$104$	0	0
$105$	3.44958	0.336645
$106$	−12.5347	−1.21748
$107$	−17.5465	−1.69628	−0.848141	−	0.529770i	$-0.822278\pi$
$107$	−17.5465	−1.69628	−0.848141	+	0.529770i	$0.822278\pi$
$108$	−17.9516	−1.72739
$109$	10.1078	0.968155	0.484077	−	0.875025i	$-0.339155\pi$
$109$	10.1078	0.968155	0.484077	+	0.875025i	$0.339155\pi$
$110$	−1.41331	−0.134754
$111$	−5.15391	−0.489187
$112$	−1.55483	−0.146917
$113$	−1.33176	−0.125282	−0.0626408	−	0.998036i	$-0.519952\pi$
$113$	−1.33176	−0.125282	−0.0626408	+	0.998036i	$0.519952\pi$
$114$	3.36639	0.315291
$115$	0.597495	0.0557167
$116$	6.14787	0.570816
$117$	0	0
$118$	14.3903	1.32473
$119$	−3.59967	−0.329981
$120$	−2.21863	−0.202532
$121$	−6.40129	−0.581935
$122$	−0.404778	−0.0366469
$123$	−19.7026	−1.77653
$124$	−6.56326	−0.589398
$125$	−6.30427	−0.563871
$126$	−12.9557	−1.15419
$127$	−0.0377111	−0.00334632	−0.00167316	−	0.999999i	$-0.500533\pi$
$127$	−0.0377111	−0.00334632	−0.00167316	+	0.999999i	$0.500533\pi$
$128$	1.00000	0.0883883
$129$	−0.663918	−0.0584547
$130$	0	0
$131$	−11.1949	−0.978102	−0.489051	−	0.872255i	$-0.662657\pi$
$131$	−11.1949	−0.978102	−0.489051	+	0.872255i	$0.662657\pi$
$132$	7.21909	0.628341
$133$	1.55483	0.134820
$134$	9.25247	0.799292
$135$	−11.8310	−1.01825
$136$	2.31516	0.198523
$137$	−15.5246	−1.32636	−0.663178	−	0.748462i	$-0.730793\pi$
$137$	−15.5246	−1.32636	−0.663178	+	0.748462i	$0.730793\pi$
$138$	−3.05196	−0.259800
$139$	20.6863	1.75459	0.877296	−	0.479950i	$-0.159345\pi$
$139$	20.6863	1.75459	0.877296	+	0.479950i	$0.159345\pi$
$140$	−1.02471	−0.0866040
$141$	0.217727	0.0183359
$142$	−5.02728	−0.421880
$143$	0	0
$144$	8.33259	0.694383
$145$	4.05177	0.336481
$146$	6.09824	0.504694
$147$	15.4266	1.27236
$148$	1.53099	0.125847
$149$	4.50875	0.369371	0.184686	−	0.982798i	$-0.440873\pi$
$149$	4.50875	0.369371	0.184686	+	0.982798i	$0.440873\pi$
$150$	15.3698	1.25494
$151$	−13.7418	−1.11829	−0.559145	−	0.829070i	$-0.688870\pi$
$151$	−13.7418	−1.11829	−0.559145	+	0.829070i	$0.688870\pi$
$152$	−1.00000	−0.0811107
$153$	19.2913	1.55961
$154$	3.33426	0.268682
$155$	−4.32553	−0.347435
$156$	0	0
$157$	−14.5420	−1.16058	−0.580289	−	0.814410i	$-0.697061\pi$
$157$	−14.5420	−1.16058	−0.580289	+	0.814410i	$0.697061\pi$
$158$	−4.22818	−0.336376
$159$	42.1966	3.34641
$160$	0.659053	0.0521027
$161$	−1.40960	−0.111092
$162$	35.4343	2.78398
$163$	−7.41566	−0.580839	−0.290420	−	0.956899i	$-0.593795\pi$
$163$	−7.41566	−0.580839	−0.290420	+	0.956899i	$0.593795\pi$
$164$	5.85275	0.457023
$165$	4.75776	0.370391
$166$	−4.60991	−0.357798
$167$	−5.52469	−0.427513	−0.213757	−	0.976887i	$-0.568570\pi$
$167$	−5.52469	−0.427513	−0.213757	+	0.976887i	$0.568570\pi$
$168$	5.23415	0.403824
$169$	0	0
$170$	1.52581	0.117024
$171$	−8.33259	−0.637209
$172$	0.197219	0.0150378
$173$	22.3664	1.70049	0.850244	−	0.526388i	$-0.176454\pi$
$173$	22.3664	1.70049	0.850244	+	0.526388i	$0.176454\pi$
$174$	−20.6961	−1.56897
$175$	7.09879	0.536618
$176$	−2.14446	−0.161645
$177$	−48.4432	−3.64122
$178$	−14.3869	−1.07834
$179$	−0.851299	−0.0636291	−0.0318145	−	0.999494i	$-0.510129\pi$
$179$	−0.851299	−0.0636291	−0.0318145	+	0.999494i	$0.510129\pi$
$180$	5.49162	0.409321
$181$	5.30763	0.394513	0.197256	−	0.980352i	$-0.436797\pi$
$181$	5.30763	0.394513	0.197256	+	0.980352i	$0.436797\pi$
$182$	0	0
$183$	1.36264	0.100729
$184$	0.906597	0.0668352
$185$	1.00900	0.0741833
$186$	22.0945	1.62005
$187$	−4.96476	−0.363060
$188$	−0.0646766	−0.00471703
$189$	27.9116	2.03027
$190$	−0.659053	−0.0478127
$191$	−20.0068	−1.44764	−0.723821	−	0.689987i	$-0.757616\pi$
$191$	−20.0068	−1.44764	−0.723821	+	0.689987i	$0.757616\pi$
$192$	−3.36639	−0.242948
$193$	21.3002	1.53322	0.766611	−	0.642111i	$-0.221941\pi$
$193$	21.3002	1.53322	0.766611	+	0.642111i	$0.221941\pi$
$194$	15.6508	1.12367
$195$	0	0
$196$	−4.58252	−0.327323
$197$	−19.2090	−1.36858	−0.684291	−	0.729209i	$-0.739888\pi$
$197$	−19.2090	−1.36858	−0.684291	+	0.729209i	$0.739888\pi$
$198$	−17.8689	−1.26989
$199$	13.5279	0.958964	0.479482	−	0.877552i	$-0.340825\pi$
$199$	13.5279	0.958964	0.479482	+	0.877552i	$0.340825\pi$
$200$	−4.56565	−0.322840
$201$	−31.1474	−2.19697
$202$	6.67794	0.469858
$203$	−9.55887	−0.670901
$204$	−7.79373	−0.545670
$205$	3.85727	0.269403
$206$	10.3411	0.720495
$207$	7.55430	0.525060
$208$	0	0
$209$	2.14446	0.148335
$210$	3.44958	0.238044
$211$	−22.2608	−1.53250	−0.766248	−	0.642544i	$-0.777879\pi$
$211$	−22.2608	−1.53250	−0.766248	+	0.642544i	$0.777879\pi$
$212$	−12.5347	−0.860885
$213$	16.9238	1.15960
$214$	−17.5465	−1.19945
$215$	0.129978	0.00886443
$216$	−17.9516	−1.22145
$217$	10.2047	0.692741
$218$	10.1078	0.684589
$219$	−20.5291	−1.38723
$220$	−1.41331	−0.0952855
$221$	0	0
$222$	−5.15391	−0.345908
$223$	−23.7309	−1.58914	−0.794568	−	0.607175i	$-0.792303\pi$
$223$	−23.7309	−1.58914	−0.794568	+	0.607175i	$0.792303\pi$
$224$	−1.55483	−0.103886
$225$	−38.0437	−2.53625
$226$	−1.33176	−0.0885874
$227$	−13.4937	−0.895610	−0.447805	−	0.894131i	$-0.647794\pi$
$227$	−13.4937	−0.895610	−0.447805	+	0.894131i	$0.647794\pi$
$228$	3.36639	0.222945
$229$	17.8510	1.17962	0.589812	−	0.807540i	$-0.299202\pi$
$229$	17.8510	1.17962	0.589812	+	0.807540i	$0.299202\pi$
$230$	0.597495	0.0393977
$231$	−11.2244	−0.738513
$232$	6.14787	0.403628
$233$	−28.6359	−1.87600	−0.938000	−	0.346636i	$-0.887324\pi$
$233$	−28.6359	−1.87600	−0.938000	+	0.346636i	$0.887324\pi$
$234$	0	0
$235$	−0.0426253	−0.00278057
$236$	14.3903	0.936726
$237$	14.2337	0.924579
$238$	−3.59967	−0.233332
$239$	−7.83970	−0.507108	−0.253554	−	0.967321i	$-0.581600\pi$
$239$	−7.83970	−0.507108	−0.253554	+	0.967321i	$0.581600\pi$
$240$	−2.21863	−0.143212
$241$	5.79801	0.373483	0.186741	−	0.982409i	$-0.440207\pi$
$241$	5.79801	0.373483	0.186741	+	0.982409i	$0.440207\pi$
$242$	−6.40129	−0.411490
$243$	−65.4310	−4.19740
$244$	−0.404778	−0.0259132
$245$	−3.02012	−0.192948
$246$	−19.7026	−1.25620
$247$	0	0
$248$	−6.56326	−0.416767
$249$	15.5188	0.983462
$250$	−6.30427	−0.398717
$251$	−13.5008	−0.852163	−0.426082	−	0.904685i	$-0.640106\pi$
$251$	−13.5008	−0.852163	−0.426082	+	0.904685i	$0.640106\pi$
$252$	−12.9557	−0.816134
$253$	−1.94416	−0.122228
$254$	−0.0377111	−0.00236621
$255$	−5.13648	−0.321659
$256$	1.00000	0.0625000
$257$	−9.98458	−0.622821	−0.311410	−	0.950275i	$-0.600801\pi$
$257$	−9.98458	−0.622821	−0.311410	+	0.950275i	$0.600801\pi$
$258$	−0.663918	−0.0413337
$259$	−2.38042	−0.147912
$260$	0	0
$261$	51.2277	3.17092
$262$	−11.1949	−0.691622
$263$	−5.72498	−0.353018	−0.176509	−	0.984299i	$-0.556480\pi$
$263$	−5.72498	−0.353018	−0.176509	+	0.984299i	$0.556480\pi$
$264$	7.21909	0.444305
$265$	−8.26102	−0.507470
$266$	1.55483	0.0953325
$267$	48.4318	2.96398
$268$	9.25247	0.565185
$269$	−2.30255	−0.140389	−0.0701945	−	0.997533i	$-0.522362\pi$
$269$	−2.30255	−0.140389	−0.0701945	+	0.997533i	$0.522362\pi$
$270$	−11.8310	−0.720015
$271$	−24.5648	−1.49220	−0.746102	−	0.665832i	$-0.768077\pi$
$271$	−24.5648	−1.49220	−0.746102	+	0.665832i	$0.768077\pi$
$272$	2.31516	0.140377
$273$	0	0
$274$	−15.5246	−0.937875
$275$	9.79086	0.590411
$276$	−3.05196	−0.183706
$277$	−26.2419	−1.57672	−0.788362	−	0.615211i	$-0.789071\pi$
$277$	−26.2419	−1.57672	−0.788362	+	0.615211i	$0.789071\pi$
$278$	20.6863	1.24068
$279$	−54.6890	−3.27414
$280$	−1.02471	−0.0612382
$281$	5.07557	0.302783	0.151392	−	0.988474i	$-0.451625\pi$
$281$	5.07557	0.302783	0.151392	+	0.988474i	$0.451625\pi$
$282$	0.217727	0.0129654
$283$	−9.07363	−0.539371	−0.269686	−	0.962948i	$-0.586920\pi$
$283$	−9.07363	−0.539371	−0.269686	+	0.962948i	$0.586920\pi$
$284$	−5.02728	−0.298314
$285$	2.21863	0.131420
$286$	0	0
$287$	−9.10000	−0.537156
$288$	8.33259	0.491003
$289$	−11.6400	−0.684709
$290$	4.05177	0.237928
$291$	−52.6869	−3.08856
$292$	6.09824	0.356872
$293$	15.8216	0.924306	0.462153	−	0.886800i	$-0.347077\pi$
$293$	15.8216	0.924306	0.462153	+	0.886800i	$0.347077\pi$
$294$	15.4266	0.899695
$295$	9.48394	0.552176
$296$	1.53099	0.0889869
$297$	38.4965	2.23379
$298$	4.50875	0.261185
$299$	0	0
$300$	15.3698	0.887374
$301$	−0.306642	−0.0176745
$302$	−13.7418	−0.790750
$303$	−22.4805	−1.29147
$304$	−1.00000	−0.0573539
$305$	−0.266770	−0.0152752
$306$	19.2913	1.10281
$307$	9.52079	0.543380	0.271690	−	0.962385i	$-0.412417\pi$
$307$	9.52079	0.543380	0.271690	+	0.962385i	$0.412417\pi$
$308$	3.33426	0.189987
$309$	−34.8120	−1.98039
$310$	−4.32553	−0.245674
$311$	25.2248	1.43037	0.715183	−	0.698938i	$-0.246343\pi$
$311$	25.2248	1.43037	0.715183	+	0.698938i	$0.246343\pi$
$312$	0	0
$313$	−30.7340	−1.73719	−0.868595	−	0.495523i	$-0.834976\pi$
$313$	−30.7340	−1.73719	−0.868595	+	0.495523i	$0.834976\pi$
$314$	−14.5420	−0.820653
$315$	−8.53851	−0.481090
$316$	−4.22818	−0.237854
$317$	2.56431	0.144026	0.0720130	−	0.997404i	$-0.477058\pi$
$317$	2.56431	0.144026	0.0720130	+	0.997404i	$0.477058\pi$
$318$	42.1966	2.36627
$319$	−13.1839	−0.738155
$320$	0.659053	0.0368422
$321$	59.0683	3.29687
$322$	−1.40960	−0.0785539
$323$	−2.31516	−0.128819
$324$	35.4343	1.96857
$325$	0	0
$326$	−7.41566	−0.410716
$327$	−34.0269	−1.88169
$328$	5.85275	0.323164
$329$	0.100561	0.00554410
$330$	4.75776	0.261906
$331$	−4.57933	−0.251703	−0.125851	−	0.992049i	$-0.540166\pi$
$331$	−4.57933	−0.251703	−0.125851	+	0.992049i	$0.540166\pi$
$332$	−4.60991	−0.253002
$333$	12.7571	0.699085
$334$	−5.52469	−0.302297
$335$	6.09787	0.333162
$336$	5.23415	0.285546
$337$	−23.2052	−1.26407	−0.632033	−	0.774942i	$-0.717779\pi$
$337$	−23.2052	−1.26407	−0.632033	+	0.774942i	$0.717779\pi$
$338$	0	0
$339$	4.48323	0.243496
$340$	1.52581	0.0827487
$341$	14.0746	0.762185
$342$	−8.33259	−0.450575
$343$	18.0088	0.972383
$344$	0.197219	0.0106334
$345$	−2.01140	−0.108290
$346$	22.3664	1.20243
$347$	20.7095	1.11174	0.555871	−	0.831269i	$-0.312385\pi$
$347$	20.7095	1.11174	0.555871	+	0.831269i	$0.312385\pi$
$348$	−20.6961	−1.10943
$349$	18.3764	0.983668	0.491834	−	0.870689i	$-0.336327\pi$
$349$	18.3764	0.983668	0.491834	+	0.870689i	$0.336327\pi$
$350$	7.09879	0.379446
$351$	0	0
$352$	−2.14446	−0.114300
$353$	−18.3367	−0.975963	−0.487982	−	0.872854i	$-0.662267\pi$
$353$	−18.3367	−0.975963	−0.487982	+	0.872854i	$0.662267\pi$
$354$	−48.4432	−2.57473
$355$	−3.31324	−0.175849
$356$	−14.3869	−0.762502
$357$	12.1179	0.641346
$358$	−0.851299	−0.0449925
$359$	5.11843	0.270141	0.135070	−	0.990836i	$-0.456874\pi$
$359$	5.11843	0.270141	0.135070	+	0.990836i	$0.456874\pi$
$360$	5.49162	0.289434
$361$	1.00000	0.0526316
$362$	5.30763	0.278963
$363$	21.5492	1.13104
$364$	0	0
$365$	4.01906	0.210367
$366$	1.36264	0.0712264
$367$	−3.19408	−0.166730	−0.0833648	−	0.996519i	$-0.526567\pi$
$367$	−3.19408	−0.166730	−0.0833648	+	0.996519i	$0.526567\pi$
$368$	0.906597	0.0472596
$369$	48.7686	2.53879
$370$	1.00900	0.0524555
$371$	19.4892	1.01183
$372$	22.0945	1.14555
$373$	−21.2217	−1.09882	−0.549409	−	0.835554i	$-0.685147\pi$
$373$	−21.2217	−1.09882	−0.549409	+	0.835554i	$0.685147\pi$
$374$	−4.96476	−0.256722
$375$	21.2226	1.09593
$376$	−0.0646766	−0.00333544
$377$	0	0
$378$	27.9116	1.43562
$379$	11.9023	0.611378	0.305689	−	0.952131i	$-0.401113\pi$
$379$	11.9023	0.611378	0.305689	+	0.952131i	$0.401113\pi$
$380$	−0.659053	−0.0338087
$381$	0.126950	0.00650387
$382$	−20.0068	−1.02364
$383$	−35.6638	−1.82234	−0.911169	−	0.412034i	$-0.864819\pi$
$383$	−35.6638	−1.82234	−0.911169	+	0.412034i	$0.864819\pi$
$384$	−3.36639	−0.171790
$385$	2.19745	0.111993
$386$	21.3002	1.08415
$387$	1.64335	0.0835362
$388$	15.6508	0.794551
$389$	−21.3030	−1.08010	−0.540052	−	0.841632i	$-0.681595\pi$
$389$	−21.3030	−1.08010	−0.540052	+	0.841632i	$0.681595\pi$
$390$	0	0
$391$	2.09891	0.106147
$392$	−4.58252	−0.231452
$393$	37.6864	1.90103
$394$	−19.2090	−0.967734
$395$	−2.78659	−0.140209
$396$	−17.8689	−0.897947
$397$	−32.3064	−1.62141	−0.810707	−	0.585452i	$-0.800917\pi$
$397$	−32.3064	−1.62141	−0.810707	+	0.585452i	$0.800917\pi$
$398$	13.5279	0.678090
$399$	−5.23415	−0.262035
$400$	−4.56565	−0.228282
$401$	14.9787	0.747999	0.374000	−	0.927429i	$-0.377986\pi$
$401$	14.9787	0.747999	0.374000	+	0.927429i	$0.377986\pi$
$402$	−31.1474	−1.55349
$403$	0	0
$404$	6.67794	0.332240
$405$	23.3531	1.16042
$406$	−9.55887	−0.474399
$407$	−3.28315	−0.162739
$408$	−7.79373	−0.385847
$409$	−6.43031	−0.317959	−0.158979	−	0.987282i	$-0.550820\pi$
$409$	−6.43031	−0.317959	−0.158979	+	0.987282i	$0.550820\pi$
$410$	3.85727	0.190497
$411$	52.2619	2.57789
$412$	10.3411	0.509467
$413$	−22.3743	−1.10097
$414$	7.55430	0.371274
$415$	−3.03817	−0.149138
$416$	0	0
$417$	−69.6383	−3.41020
$418$	2.14446	0.104889
$419$	−0.238887	−0.0116704	−0.00583520	−	0.999983i	$-0.501857\pi$
$419$	−0.238887	−0.0116704	−0.00583520	+	0.999983i	$0.501857\pi$
$420$	3.44958	0.168322
$421$	−34.4329	−1.67816	−0.839079	−	0.544010i	$-0.816905\pi$
$421$	−34.4329	−1.67816	−0.839079	+	0.544010i	$0.816905\pi$
$422$	−22.2608	−1.08364
$423$	−0.538924	−0.0262034
$424$	−12.5347	−0.608738
$425$	−10.5702	−0.512730
$426$	16.9238	0.819960
$427$	0.629359	0.0304568
$428$	−17.5465	−0.848141
$429$	0	0
$430$	0.129978	0.00626810
$431$	20.1204	0.969168	0.484584	−	0.874745i	$-0.338971\pi$
$431$	20.1204	0.969168	0.484584	+	0.874745i	$0.338971\pi$
$432$	−17.9516	−0.863697
$433$	−4.23488	−0.203515	−0.101758	−	0.994809i	$-0.532447\pi$
$433$	−4.23488	−0.203515	−0.101758	+	0.994809i	$0.532447\pi$
$434$	10.2047	0.489842
$435$	−13.6399	−0.653981
$436$	10.1078	0.484077
$437$	−0.906597	−0.0433684
$438$	−20.5291	−0.980916
$439$	−18.7969	−0.897128	−0.448564	−	0.893751i	$-0.648064\pi$
$439$	−18.7969	−0.897128	−0.448564	+	0.893751i	$0.648064\pi$
$440$	−1.41331	−0.0673770
$441$	−38.1843	−1.81830
$442$	0	0
$443$	−15.3982	−0.731590	−0.365795	−	0.930695i	$-0.619203\pi$
$443$	−15.3982	−0.731590	−0.365795	+	0.930695i	$0.619203\pi$
$444$	−5.15391	−0.244594
$445$	−9.48170	−0.449476
$446$	−23.7309	−1.12369
$447$	−15.1782	−0.717905
$448$	−1.55483	−0.0734586
$449$	6.74190	0.318170	0.159085	−	0.987265i	$-0.449146\pi$
$449$	6.74190	0.318170	0.159085	+	0.987265i	$0.449146\pi$
$450$	−38.0437	−1.79340
$451$	−12.5510	−0.591003
$452$	−1.33176	−0.0626408
$453$	46.2602	2.17349
$454$	−13.4937	−0.633292
$455$	0	0
$456$	3.36639	0.157646
$457$	−28.2888	−1.32329	−0.661647	−	0.749816i	$-0.730142\pi$
$457$	−28.2888	−1.32329	−0.661647	+	0.749816i	$0.730142\pi$
$458$	17.8510	0.834120
$459$	−41.5608	−1.93989
$460$	0.597495	0.0278584
$461$	−17.3637	−0.808709	−0.404354	−	0.914602i	$-0.632504\pi$
$461$	−17.3637	−0.808709	−0.404354	+	0.914602i	$0.632504\pi$
$462$	−11.2244	−0.522208
$463$	2.55506	0.118744	0.0593719	−	0.998236i	$-0.481090\pi$
$463$	2.55506	0.118744	0.0593719	+	0.998236i	$0.481090\pi$
$464$	6.14787	0.285408
$465$	14.5614	0.675270
$466$	−28.6359	−1.32653
$467$	−13.8566	−0.641207	−0.320603	−	0.947214i	$-0.603886\pi$
$467$	−13.8566	−0.641207	−0.320603	+	0.947214i	$0.603886\pi$
$468$	0	0
$469$	−14.3860	−0.664283
$470$	−0.0426253	−0.00196616
$471$	48.9541	2.25569
$472$	14.3903	0.662365
$473$	−0.422929	−0.0194463
$474$	14.2337	0.653776
$475$	4.56565	0.209486
$476$	−3.59967	−0.164990
$477$	−104.446	−4.78227
$478$	−7.83970	−0.358580
$479$	−21.5179	−0.983176	−0.491588	−	0.870828i	$-0.663583\pi$
$479$	−21.5179	−0.983176	−0.491588	+	0.870828i	$0.663583\pi$
$480$	−2.21863	−0.101266
$481$	0	0
$482$	5.79801	0.264092
$483$	4.74526	0.215917
$484$	−6.40129	−0.290968
$485$	10.3147	0.468368
$486$	−65.4310	−2.96801
$487$	10.7576	0.487474	0.243737	−	0.969841i	$-0.421627\pi$
$487$	10.7576	0.487474	0.243737	+	0.969841i	$0.421627\pi$
$488$	−0.404778	−0.0183234
$489$	24.9640	1.12891
$490$	−3.02012	−0.136435
$491$	38.1963	1.72378	0.861888	−	0.507099i	$-0.169282\pi$
$491$	38.1963	1.72378	0.861888	+	0.507099i	$0.169282\pi$
$492$	−19.7026	−0.888264
$493$	14.2333	0.641035
$494$	0	0
$495$	−11.7766	−0.529317
$496$	−6.56326	−0.294699
$497$	7.81654	0.350620
$498$	15.5188	0.695412
$499$	20.3365	0.910386	0.455193	−	0.890393i	$-0.349570\pi$
$499$	20.3365	0.910386	0.455193	+	0.890393i	$0.349570\pi$
$500$	−6.30427	−0.281935
$501$	18.5983	0.830909
$502$	−13.5008	−0.602570
$503$	15.9668	0.711923	0.355962	−	0.934501i	$-0.384153\pi$
$503$	15.9668	0.711923	0.355962	+	0.934501i	$0.384153\pi$
$504$	−12.9557	−0.577094
$505$	4.40111	0.195847
$506$	−1.94416	−0.0864285
$507$	0	0
$508$	−0.0377111	−0.00167316
$509$	−2.43816	−0.108070	−0.0540348	−	0.998539i	$-0.517208\pi$
$509$	−2.43816	−0.108070	−0.0540348	+	0.998539i	$0.517208\pi$
$510$	−5.13648	−0.227447
$511$	−9.48169	−0.419445
$512$	1.00000	0.0441942
$513$	17.9516	0.792582
$514$	−9.98458	−0.440401
$515$	6.81530	0.300318
$516$	−0.663918	−0.0292274
$517$	0.138696	0.00609986
$518$	−2.38042	−0.104590
$519$	−75.2942	−3.30505
$520$	0	0
$521$	−29.1880	−1.27875	−0.639375	−	0.768895i	$-0.720807\pi$
$521$	−29.1880	−1.27875	−0.639375	+	0.768895i	$0.720807\pi$
$522$	51.2277	2.24218
$523$	−14.4899	−0.633598	−0.316799	−	0.948493i	$-0.602608\pi$
$523$	−14.4899	−0.633598	−0.316799	+	0.948493i	$0.602608\pi$
$524$	−11.1949	−0.489051
$525$	−23.8973	−1.04296
$526$	−5.72498	−0.249621
$527$	−15.1950	−0.661904
$528$	7.21909	0.314171
$529$	−22.1781	−0.964264
$530$	−8.26102	−0.358836
$531$	119.908	5.20357
$532$	1.55483	0.0674102
$533$	0	0
$534$	48.4318	2.09585
$535$	−11.5641	−0.499958
$536$	9.25247	0.399646
$537$	2.86580	0.123669
$538$	−2.30255	−0.0992700
$539$	9.82703	0.423280
$540$	−11.8310	−0.509127
$541$	1.93121	0.0830293	0.0415147	−	0.999138i	$-0.486782\pi$
$541$	1.93121	0.0830293	0.0415147	+	0.999138i	$0.486782\pi$
$542$	−24.5648	−1.05515
$543$	−17.8675	−0.766770
$544$	2.31516	0.0992616
$545$	6.66159	0.285351
$546$	0	0
$547$	−17.5237	−0.749261	−0.374630	−	0.927174i	$-0.622230\pi$
$547$	−17.5237	−0.749261	−0.374630	+	0.927174i	$0.622230\pi$
$548$	−15.5246	−0.663178
$549$	−3.37285	−0.143950
$550$	9.79086	0.417483
$551$	−6.14787	−0.261908
$552$	−3.05196	−0.129900
$553$	6.57408	0.279558
$554$	−26.2419	−1.11491
$555$	−3.39670	−0.144182
$556$	20.6863	0.877296
$557$	−17.4101	−0.737689	−0.368844	−	0.929491i	$-0.620246\pi$
$557$	−17.4101	−0.737689	−0.368844	+	0.929491i	$0.620246\pi$
$558$	−54.6890	−2.31517
$559$	0	0
$560$	−1.02471	−0.0433020
$561$	16.7133	0.705638
$562$	5.07557	0.214100
$563$	10.3029	0.434214	0.217107	−	0.976148i	$-0.430338\pi$
$563$	10.3029	0.434214	0.217107	+	0.976148i	$0.430338\pi$
$564$	0.217727	0.00916795
$565$	−0.877701	−0.0369252
$566$	−9.07363	−0.381393
$567$	−55.0942	−2.31374
$568$	−5.02728	−0.210940
$569$	−16.5450	−0.693602	−0.346801	−	0.937939i	$-0.612732\pi$
$569$	−16.5450	−0.693602	−0.346801	+	0.937939i	$0.612732\pi$
$570$	2.21863	0.0929282
$571$	0.221196	0.00925678	0.00462839	−	0.999989i	$-0.498527\pi$
$571$	0.221196	0.00925678	0.00462839	+	0.999989i	$0.498527\pi$
$572$	0	0
$573$	67.3508	2.81362
$574$	−9.10000	−0.379827
$575$	−4.13920	−0.172617
$576$	8.33259	0.347191
$577$	−10.2652	−0.427344	−0.213672	−	0.976905i	$-0.568542\pi$
$577$	−10.2652	−0.427344	−0.213672	+	0.976905i	$0.568542\pi$
$578$	−11.6400	−0.484162
$579$	−71.7049	−2.97995
$580$	4.05177	0.168241
$581$	7.16760	0.297362
$582$	−52.6869	−2.18394
$583$	26.8801	1.11326
$584$	6.09824	0.252347
$585$	0	0
$586$	15.8216	0.653583
$587$	7.79489	0.321730	0.160865	−	0.986976i	$-0.448572\pi$
$587$	7.79489	0.321730	0.160865	+	0.986976i	$0.448572\pi$
$588$	15.4266	0.636180
$589$	6.56326	0.270434
$590$	9.48394	0.390447
$591$	64.6649	2.65996
$592$	1.53099	0.0629233
$593$	−8.18825	−0.336251	−0.168126	−	0.985766i	$-0.553771\pi$
$593$	−8.18825	−0.336251	−0.168126	+	0.985766i	$0.553771\pi$
$594$	38.4965	1.57953
$595$	−2.37237	−0.0972577
$596$	4.50875	0.184686
$597$	−45.5400	−1.86383
$598$	0	0
$599$	18.5390	0.757484	0.378742	−	0.925502i	$-0.376357\pi$
$599$	18.5390	0.757484	0.378742	+	0.925502i	$0.376357\pi$
$600$	15.3698	0.627468
$601$	7.26286	0.296258	0.148129	−	0.988968i	$-0.452675\pi$
$601$	7.26286	0.296258	0.148129	+	0.988968i	$0.452675\pi$
$602$	−0.306642	−0.0124978
$603$	77.0971	3.13964
$604$	−13.7418	−0.559145
$605$	−4.21879	−0.171518
$606$	−22.4805	−0.913210
$607$	−20.9815	−0.851611	−0.425806	−	0.904815i	$-0.640009\pi$
$607$	−20.9815	−0.851611	−0.425806	+	0.904815i	$0.640009\pi$
$608$	−1.00000	−0.0405554
$609$	32.1789	1.30395
$610$	−0.266770	−0.0108012
$611$	0	0
$612$	19.2913	0.779803
$613$	43.8165	1.76973	0.884866	−	0.465845i	$-0.154250\pi$
$613$	43.8165	1.76973	0.884866	+	0.465845i	$0.154250\pi$
$614$	9.52079	0.384228
$615$	−12.9851	−0.523609
$616$	3.33426	0.134341
$617$	−2.29861	−0.0925385	−0.0462693	−	0.998929i	$-0.514733\pi$
$617$	−2.29861	−0.0925385	−0.0462693	+	0.998929i	$0.514733\pi$
$618$	−34.8120	−1.40035
$619$	40.0864	1.61121	0.805604	−	0.592455i	$-0.201841\pi$
$619$	40.0864	1.61121	0.805604	+	0.592455i	$0.201841\pi$
$620$	−4.32553	−0.173718
$621$	−16.2749	−0.653088
$622$	25.2248	1.01142
$623$	22.3691	0.896198
$624$	0	0
$625$	18.6734	0.746936
$626$	−30.7340	−1.22838
$627$	−7.21909	−0.288303
$628$	−14.5420	−0.580289
$629$	3.54448	0.141328
$630$	−8.53851	−0.340182
$631$	−9.22321	−0.367170	−0.183585	−	0.983004i	$-0.558770\pi$
$631$	−9.22321	−0.367170	−0.183585	+	0.983004i	$0.558770\pi$
$632$	−4.22818	−0.168188
$633$	74.9386	2.97854
$634$	2.56431	0.101842
$635$	−0.0248536	−0.000986286 0
$636$	42.1966	1.67321
$637$	0	0
$638$	−13.1839	−0.521954
$639$	−41.8903	−1.65715
$640$	0.659053	0.0260513
$641$	38.7348	1.52993	0.764967	−	0.644070i	$-0.222755\pi$
$641$	38.7348	1.52993	0.764967	+	0.644070i	$0.222755\pi$
$642$	59.0683	2.33124
$643$	−6.83413	−0.269512	−0.134756	−	0.990879i	$-0.543025\pi$
$643$	−6.83413	−0.269512	−0.134756	+	0.990879i	$0.543025\pi$
$644$	−1.40960	−0.0555460
$645$	−0.437557	−0.0172288
$646$	−2.31516	−0.0910887
$647$	31.7010	1.24630	0.623148	−	0.782104i	$-0.285854\pi$
$647$	31.7010	1.24630	0.623148	+	0.782104i	$0.285854\pi$
$648$	35.4343	1.39199
$649$	−30.8593	−1.21133
$650$	0	0
$651$	−34.3531	−1.34640
$652$	−7.41566	−0.290420
$653$	−46.1273	−1.80510	−0.902551	−	0.430582i	$-0.858308\pi$
$653$	−46.1273	−1.80510	−0.902551	+	0.430582i	$0.858308\pi$
$654$	−34.0269	−1.33056
$655$	−7.37802	−0.288283
$656$	5.85275	0.228511
$657$	50.8141	1.98245
$658$	0.100561	0.00392027
$659$	11.3580	0.442445	0.221222	−	0.975223i	$-0.428995\pi$
$659$	11.3580	0.442445	0.221222	+	0.975223i	$0.428995\pi$
$660$	4.75776	0.185196
$661$	47.2706	1.83861	0.919307	−	0.393541i	$-0.128750\pi$
$661$	47.2706	1.83861	0.919307	+	0.393541i	$0.128750\pi$
$662$	−4.57933	−0.177981
$663$	0	0
$664$	−4.60991	−0.178899
$665$	1.02471	0.0397366
$666$	12.7571	0.494328
$667$	5.57364	0.215812
$668$	−5.52469	−0.213757
$669$	79.8874	3.08862
$670$	6.09787	0.235581
$671$	0.868030	0.0335099
$672$	5.23415	0.201912
$673$	−19.8580	−0.765470	−0.382735	−	0.923858i	$-0.625018\pi$
$673$	−19.8580	−0.765470	−0.382735	+	0.923858i	$0.625018\pi$
$674$	−23.2052	−0.893830
$675$	81.9607	3.15467
$676$	0	0
$677$	29.3639	1.12855	0.564273	−	0.825588i	$-0.309156\pi$
$677$	29.3639	1.12855	0.564273	+	0.825588i	$0.309156\pi$
$678$	4.48323	0.172177
$679$	−24.3343	−0.933866
$680$	1.52581	0.0585122
$681$	45.4252	1.74070
$682$	14.0746	0.538946
$683$	12.1806	0.466078	0.233039	−	0.972467i	$-0.425133\pi$
$683$	12.1806	0.466078	0.233039	+	0.972467i	$0.425133\pi$
$684$	−8.33259	−0.318605
$685$	−10.2315	−0.390926
$686$	18.0088	0.687579
$687$	−60.0933	−2.29270
$688$	0.197219	0.00751892
$689$	0	0
$690$	−2.01140	−0.0765728
$691$	12.5640	0.477958	0.238979	−	0.971025i	$-0.423187\pi$
$691$	12.5640	0.477958	0.238979	+	0.971025i	$0.423187\pi$
$692$	22.3664	0.850244
$693$	27.7830	1.05539
$694$	20.7095	0.786120
$695$	13.6334	0.517144
$696$	−20.6961	−0.784485
$697$	13.5500	0.513244
$698$	18.3764	0.695558
$699$	96.3996	3.64617
$700$	7.09879	0.268309
$701$	46.7967	1.76749	0.883743	−	0.467973i	$-0.155016\pi$
$701$	46.7967	1.76749	0.883743	+	0.467973i	$0.155016\pi$
$702$	0	0
$703$	−1.53099	−0.0577423
$704$	−2.14446	−0.0808224
$705$	0.143493	0.00540427
$706$	−18.3367	−0.690110
$707$	−10.3830	−0.390494
$708$	−48.4432	−1.82061
$709$	−32.9155	−1.23617	−0.618084	−	0.786112i	$-0.712091\pi$
$709$	−32.9155	−1.23617	−0.618084	+	0.786112i	$0.712091\pi$
$710$	−3.31324	−0.124344
$711$	−35.2317	−1.32129
$712$	−14.3869	−0.539171
$713$	−5.95023	−0.222838
$714$	12.1179	0.453500
$715$	0	0
$716$	−0.851299	−0.0318145
$717$	26.3915	0.985609
$718$	5.11843	0.191018
$719$	22.7335	0.847816	0.423908	−	0.905705i	$-0.360658\pi$
$719$	22.7335	0.847816	0.423908	+	0.905705i	$0.360658\pi$
$720$	5.49162	0.204661
$721$	−16.0785	−0.598796
$722$	1.00000	0.0372161
$723$	−19.5184	−0.725896
$724$	5.30763	0.197256
$725$	−28.0690	−1.04246
$726$	21.5492	0.799768
$727$	−49.8150	−1.84753	−0.923767	−	0.382954i	$-0.874907\pi$
$727$	−49.8150	−1.84753	−0.923767	+	0.382954i	$0.874907\pi$
$728$	0	0
$729$	113.964	4.22087
$730$	4.01906	0.148752
$731$	0.456594	0.0168877
$732$	1.36264	0.0503647
$733$	5.01600	0.185270	0.0926351	−	0.995700i	$-0.470471\pi$
$733$	5.01600	0.185270	0.0926351	+	0.995700i	$0.470471\pi$
$734$	−3.19408	−0.117896
$735$	10.1669	0.375012
$736$	0.906597	0.0334176
$737$	−19.8416	−0.730873
$738$	48.7686	1.79520
$739$	−11.1272	−0.409321	−0.204661	−	0.978833i	$-0.565609\pi$
$739$	−11.1272	−0.409321	−0.204661	+	0.978833i	$0.565609\pi$
$740$	1.00900	0.0370917
$741$	0	0
$742$	19.4892	0.715472
$743$	−35.9871	−1.32024	−0.660118	−	0.751161i	$-0.729494\pi$
$743$	−35.9871	−1.32024	−0.660118	+	0.751161i	$0.729494\pi$
$744$	22.0945	0.810023
$745$	2.97151	0.108868
$746$	−21.2217	−0.776982
$747$	−38.4125	−1.40544
$748$	−4.96476	−0.181530
$749$	27.2817	0.996852
$750$	21.2226	0.774941
$751$	18.2229	0.664962	0.332481	−	0.943110i	$-0.392114\pi$
$751$	18.2229	0.664962	0.332481	+	0.943110i	$0.392114\pi$
$752$	−0.0646766	−0.00235851
$753$	45.4490	1.65625
$754$	0	0
$755$	−9.05655	−0.329602
$756$	27.9116	1.01514
$757$	−24.9727	−0.907649	−0.453824	−	0.891091i	$-0.649941\pi$
$757$	−24.9727	−0.907649	−0.453824	+	0.891091i	$0.649941\pi$
$758$	11.9023	0.432310
$759$	6.54481	0.237561
$760$	−0.659053	−0.0239064
$761$	16.5240	0.598994	0.299497	−	0.954097i	$-0.403181\pi$
$761$	16.5240	0.598994	0.299497	+	0.954097i	$0.403181\pi$
$762$	0.126950	0.00459893
$763$	−15.7159	−0.568954
$764$	−20.0068	−0.723821
$765$	12.7140	0.459674
$766$	−35.6638	−1.28859
$767$	0	0
$768$	−3.36639	−0.121474
$769$	3.79823	0.136968	0.0684839	−	0.997652i	$-0.478184\pi$
$769$	3.79823	0.136968	0.0684839	+	0.997652i	$0.478184\pi$
$770$	2.19745	0.0791907
$771$	33.6120	1.21051
$772$	21.3002	0.766611
$773$	34.8182	1.25232	0.626162	−	0.779693i	$-0.284625\pi$
$773$	34.8182	1.25232	0.626162	+	0.779693i	$0.284625\pi$
$774$	1.64335	0.0590690
$775$	29.9655	1.07639
$776$	15.6508	0.561833
$777$	8.01343	0.287480
$778$	−21.3030	−0.763749
$779$	−5.85275	−0.209697
$780$	0	0
$781$	10.7808	0.385767
$782$	2.09891	0.0750570
$783$	−110.364	−3.94409
$784$	−4.58252	−0.163661
$785$	−9.58395	−0.342066
$786$	37.6864	1.34423
$787$	13.1410	0.468426	0.234213	−	0.972185i	$-0.424749\pi$
$787$	13.1410	0.468426	0.234213	+	0.972185i	$0.424749\pi$
$788$	−19.2090	−0.684291
$789$	19.2725	0.686121
$790$	−2.78659	−0.0991426
$791$	2.07066	0.0736241
$792$	−17.8689	−0.634944
$793$	0	0
$794$	−32.3064	−1.14651
$795$	27.8098	0.986312
$796$	13.5279	0.479482
$797$	38.4927	1.36348	0.681741	−	0.731594i	$-0.261223\pi$
$797$	38.4927	1.36348	0.681741	+	0.731594i	$0.261223\pi$
$798$	−5.23415	−0.185287
$799$	−0.149737	−0.00529730
$800$	−4.56565	−0.161420
$801$	−119.880	−4.23575
$802$	14.9787	0.528915
$803$	−13.0774	−0.461492
$804$	−31.1474	−1.09849
$805$	−0.929000	−0.0327430
$806$	0	0
$807$	7.75129	0.272858
$808$	6.67794	0.234929
$809$	−0.282721	−0.00993995	−0.00496998	−	0.999988i	$-0.501582\pi$
$809$	−0.282721	−0.00993995	−0.00496998	+	0.999988i	$0.501582\pi$
$810$	23.3531	0.820544
$811$	−51.1669	−1.79671	−0.898356	−	0.439268i	$-0.855238\pi$
$811$	−51.1669	−1.79671	−0.898356	+	0.439268i	$0.855238\pi$
$812$	−9.55887	−0.335450
$813$	82.6947	2.90023
$814$	−3.28315	−0.115074
$815$	−4.88731	−0.171195
$816$	−7.79373	−0.272835
$817$	−0.197219	−0.00689984
$818$	−6.43031	−0.224831
$819$	0	0
$820$	3.85727	0.134702
$821$	4.43545	0.154798	0.0773991	−	0.997000i	$-0.475338\pi$
$821$	4.43545	0.154798	0.0773991	+	0.997000i	$0.475338\pi$
$822$	52.2619	1.82284
$823$	−45.5433	−1.58754	−0.793770	−	0.608218i	$-0.791885\pi$
$823$	−45.5433	−1.58754	−0.793770	+	0.608218i	$0.791885\pi$
$824$	10.3411	0.360248
$825$	−32.9599	−1.14751
$826$	−22.3743	−0.778503
$827$	40.8431	1.42025	0.710127	−	0.704073i	$-0.248637\pi$
$827$	40.8431	1.42025	0.710127	+	0.704073i	$0.248637\pi$
$828$	7.55430	0.262530
$829$	−3.65715	−0.127018	−0.0635090	−	0.997981i	$-0.520229\pi$
$829$	−3.65715	−0.127018	−0.0635090	+	0.997981i	$0.520229\pi$
$830$	−3.03817	−0.105457
$831$	88.3406	3.06450
$832$	0	0
$833$	−10.6093	−0.367589
$834$	−69.6383	−2.41138
$835$	−3.64106	−0.126004
$836$	2.14446	0.0741677
$837$	117.821	4.07249
$838$	−0.238887	−0.00825222
$839$	23.6861	0.817735	0.408868	−	0.912594i	$-0.365924\pi$
$839$	23.6861	0.817735	0.408868	+	0.912594i	$0.365924\pi$
$840$	3.44958	0.119022
$841$	8.79634	0.303322
$842$	−34.4329	−1.18664
$843$	−17.0864	−0.588486
$844$	−22.2608	−0.766248
$845$	0	0
$846$	−0.538924	−0.0185286
$847$	9.95289	0.341985
$848$	−12.5347	−0.430443
$849$	30.5454	1.04832
$850$	−10.5702	−0.362555
$851$	1.38799	0.0475797
$852$	16.9238	0.579799
$853$	17.2391	0.590254	0.295127	−	0.955458i	$-0.404638\pi$
$853$	17.2391	0.590254	0.295127	+	0.955458i	$0.404638\pi$
$854$	0.629359	0.0215362
$855$	−5.49162	−0.187809
$856$	−17.5465	−0.599726
$857$	−12.5765	−0.429604	−0.214802	−	0.976658i	$-0.568911\pi$
$857$	−12.5765	−0.429604	−0.214802	+	0.976658i	$0.568911\pi$
$858$	0	0
$859$	−7.90850	−0.269835	−0.134917	−	0.990857i	$-0.543077\pi$
$859$	−7.90850	−0.269835	−0.134917	+	0.990857i	$0.543077\pi$
$860$	0.129978	0.00443221
$861$	30.6342	1.04401
$862$	20.1204	0.685305
$863$	45.0795	1.53453	0.767263	−	0.641333i	$-0.221618\pi$
$863$	45.0795	1.53453	0.767263	+	0.641333i	$0.221618\pi$
$864$	−17.9516	−0.610726
$865$	14.7407	0.501198
$866$	−4.23488	−0.143907
$867$	39.1850	1.33079
$868$	10.2047	0.346371
$869$	9.06717	0.307583
$870$	−13.6399	−0.462434
$871$	0	0
$872$	10.1078	0.342294
$873$	130.412	4.41378
$874$	−0.906597	−0.0306661
$875$	9.80203	0.331369
$876$	−20.5291	−0.693613
$877$	−3.09592	−0.104542	−0.0522710	−	0.998633i	$-0.516646\pi$
$877$	−3.09592	−0.104542	−0.0522710	+	0.998633i	$0.516646\pi$
$878$	−18.7969	−0.634365
$879$	−53.2616	−1.79647
$880$	−1.41331	−0.0476428
$881$	−33.6727	−1.13446	−0.567232	−	0.823558i	$-0.691986\pi$
$881$	−33.6727	−1.13446	−0.567232	+	0.823558i	$0.691986\pi$
$882$	−38.1843	−1.28573
$883$	−12.9000	−0.434119	−0.217060	−	0.976158i	$-0.569647\pi$
$883$	−12.9000	−0.434119	−0.217060	+	0.976158i	$0.569647\pi$
$884$	0	0
$885$	−31.9266	−1.07320
$886$	−15.3982	−0.517312
$887$	−28.4001	−0.953582	−0.476791	−	0.879017i	$-0.658200\pi$
$887$	−28.4001	−0.953582	−0.476791	+	0.879017i	$0.658200\pi$
$888$	−5.15391	−0.172954
$889$	0.0586342	0.00196653
$890$	−9.48170	−0.317827
$891$	−75.9875	−2.54568
$892$	−23.7309	−0.794568
$893$	0.0646766	0.00216432
$894$	−15.1782	−0.507636
$895$	−0.561051	−0.0187539
$896$	−1.55483	−0.0519431
$897$	0	0
$898$	6.74190	0.224980
$899$	−40.3501	−1.34575
$900$	−38.0437	−1.26812
$901$	−29.0198	−0.966788
$902$	−12.5510	−0.417902
$903$	1.03228	0.0343520
$904$	−1.33176	−0.0442937
$905$	3.49800	0.116278
$906$	46.2602	1.53689
$907$	31.6558	1.05111	0.525557	−	0.850759i	$-0.323857\pi$
$907$	31.6558	1.05111	0.525557	+	0.850759i	$0.323857\pi$
$908$	−13.4937	−0.447805
$909$	55.6445	1.84561
$910$	0	0
$911$	−18.6057	−0.616434	−0.308217	−	0.951316i	$-0.599732\pi$
$911$	−18.6057	−0.616434	−0.308217	+	0.951316i	$0.599732\pi$
$912$	3.36639	0.111472
$913$	9.88577	0.327171
$914$	−28.2888	−0.935710
$915$	0.898052	0.0296887
$916$	17.8510	0.589812
$917$	17.4061	0.574800
$918$	−41.5608	−1.37171
$919$	41.0076	1.35272	0.676358	−	0.736573i	$-0.263557\pi$
$919$	41.0076	1.35272	0.676358	+	0.736573i	$0.263557\pi$
$920$	0.597495	0.0196988
$921$	−32.0507	−1.05611
$922$	−17.3637	−0.571844
$923$	0	0
$924$	−11.2244	−0.369257
$925$	−6.98996	−0.229828
$926$	2.55506	0.0839646
$927$	86.1678	2.83012
$928$	6.14787	0.201814
$929$	−2.65194	−0.0870072	−0.0435036	−	0.999053i	$-0.513852\pi$
$929$	−2.65194	−0.0870072	−0.0435036	+	0.999053i	$0.513852\pi$
$930$	14.5614	0.477488
$931$	4.58252	0.150186
$932$	−28.6359	−0.938000
$933$	−84.9165	−2.78004
$934$	−13.8566	−0.453402
$935$	−3.27204	−0.107007
$936$	0	0
$937$	14.6898	0.479894	0.239947	−	0.970786i	$-0.422870\pi$
$937$	14.6898	0.479894	0.239947	+	0.970786i	$0.422870\pi$
$938$	−14.3860	−0.469719
$939$	103.463	3.37638
$940$	−0.0426253	−0.00139028
$941$	−29.2022	−0.951965	−0.475982	−	0.879455i	$-0.657907\pi$
$941$	−29.2022	−0.951965	−0.475982	+	0.879455i	$0.657907\pi$
$942$	48.9541	1.59501
$943$	5.30608	0.172790
$944$	14.3903	0.468363
$945$	18.3952	0.598396
$946$	−0.422929	−0.0137506
$947$	−48.7605	−1.58450	−0.792251	−	0.610195i	$-0.791091\pi$
$947$	−48.7605	−1.58450	−0.792251	+	0.610195i	$0.791091\pi$
$948$	14.2337	0.462289
$949$	0	0
$950$	4.56565	0.148129
$951$	−8.63248	−0.279927
$952$	−3.59967	−0.116666
$953$	−23.7242	−0.768502	−0.384251	−	0.923229i	$-0.625540\pi$
$953$	−23.7242	−0.768502	−0.384251	+	0.923229i	$0.625540\pi$
$954$	−104.446	−3.38158
$955$	−13.1855	−0.426674
$956$	−7.83970	−0.253554
$957$	44.3821	1.43467
$958$	−21.5179	−0.695210
$959$	24.1380	0.779458
$960$	−2.21863	−0.0716060
$961$	12.0763	0.389560
$962$	0	0
$963$	−146.208	−4.71148
$964$	5.79801	0.186741
$965$	14.0380	0.451898
$966$	4.74526	0.152676
$967$	−21.4717	−0.690484	−0.345242	−	0.938514i	$-0.612203\pi$
$967$	−21.4717	−0.690484	−0.345242	+	0.938514i	$0.612203\pi$
$968$	−6.40129	−0.205745
$969$	7.79373	0.250371
$970$	10.3147	0.331186
$971$	−45.8967	−1.47289	−0.736447	−	0.676495i	$-0.763498\pi$
$971$	−45.8967	−1.47289	−0.736447	+	0.676495i	$0.763498\pi$
$972$	−65.4310	−2.09870
$973$	−32.1636	−1.03112
$974$	10.7576	0.344696
$975$	0	0
$976$	−0.404778	−0.0129566
$977$	−29.3098	−0.937704	−0.468852	−	0.883277i	$-0.655332\pi$
$977$	−29.3098	−0.937704	−0.468852	+	0.883277i	$0.655332\pi$
$978$	24.9640	0.798261
$979$	30.8521	0.986036
$980$	−3.02012	−0.0964742
$981$	84.2245	2.68908
$982$	38.1963	1.21889
$983$	3.36484	0.107322	0.0536609	−	0.998559i	$-0.482911\pi$
$983$	3.36484	0.107322	0.0536609	+	0.998559i	$0.482911\pi$
$984$	−19.7026	−0.628098
$985$	−12.6597	−0.403372
$986$	14.2333	0.453280
$987$	−0.338527	−0.0107754
$988$	0	0
$989$	0.178799	0.00568546
$990$	−11.7766	−0.374284
$991$	39.4061	1.25177	0.625887	−	0.779913i	$-0.284737\pi$
$991$	39.4061	1.25177	0.625887	+	0.779913i	$0.284737\pi$
$992$	−6.56326	−0.208384
$993$	15.4158	0.489206
$994$	7.81654	0.247926
$995$	8.91557	0.282642
$996$	15.5188	0.491731
$997$	18.4286	0.583638	0.291819	−	0.956474i	$-0.405739\pi$
$997$	18.4286	0.583638	0.291819	+	0.956474i	$0.405739\pi$
$998$	20.3365	0.643740
$999$	−27.4837	−0.869546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bh.1.1		8
13.2	odd	12		494.2.m.a.381.8	yes	16
13.7	odd	12		494.2.m.a.153.8	✓	16
13.12	even	2		6422.2.a.bg.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.m.a.153.8	✓	16	13.7	odd	12
494.2.m.a.381.8	yes	16	13.2	odd	12
6422.2.a.bg.1.1		8	13.12	even	2
6422.2.a.bh.1.1		8	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.36639 q^{3} +1.00000 q^{4} +0.659053 q^{5} -3.36639 q^{6} -1.55483 q^{7} +1.00000 q^{8} +8.33259 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.36639 q^{3} +1.00000 q^{4} +0.659053 q^{5} -3.36639 q^{6} -1.55483 q^{7} +1.00000 q^{8} +8.33259 q^{9} +0.659053 q^{10} -2.14446 q^{11} -3.36639 q^{12} -1.55483 q^{14} -2.21863 q^{15} +1.00000 q^{16} +2.31516 q^{17} +8.33259 q^{18} -1.00000 q^{19} +0.659053 q^{20} +5.23415 q^{21} -2.14446 q^{22} +0.906597 q^{23} -3.36639 q^{24} -4.56565 q^{25} -17.9516 q^{27} -1.55483 q^{28} +6.14787 q^{29} -2.21863 q^{30} -6.56326 q^{31} +1.00000 q^{32} +7.21909 q^{33} +2.31516 q^{34} -1.02471 q^{35} +8.33259 q^{36} +1.53099 q^{37} -1.00000 q^{38} +0.659053 q^{40} +5.85275 q^{41} +5.23415 q^{42} +0.197219 q^{43} -2.14446 q^{44} +5.49162 q^{45} +0.906597 q^{46} -0.0646766 q^{47} -3.36639 q^{48} -4.58252 q^{49} -4.56565 q^{50} -7.79373 q^{51} -12.5347 q^{53} -17.9516 q^{54} -1.41331 q^{55} -1.55483 q^{56} +3.36639 q^{57} +6.14787 q^{58} +14.3903 q^{59} -2.21863 q^{60} -0.404778 q^{61} -6.56326 q^{62} -12.9557 q^{63} +1.00000 q^{64} +7.21909 q^{66} +9.25247 q^{67} +2.31516 q^{68} -3.05196 q^{69} -1.02471 q^{70} -5.02728 q^{71} +8.33259 q^{72} +6.09824 q^{73} +1.53099 q^{74} +15.3698 q^{75} -1.00000 q^{76} +3.33426 q^{77} -4.22818 q^{79} +0.659053 q^{80} +35.4343 q^{81} +5.85275 q^{82} -4.60991 q^{83} +5.23415 q^{84} +1.52581 q^{85} +0.197219 q^{86} -20.6961 q^{87} -2.14446 q^{88} -14.3869 q^{89} +5.49162 q^{90} +0.906597 q^{92} +22.0945 q^{93} -0.0646766 q^{94} -0.659053 q^{95} -3.36639 q^{96} +15.6508 q^{97} -4.58252 q^{98} -17.8689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} + 8 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^7 + 8 * q^8	\( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} + 8 q^{8} - 2 q^{10} - 10 q^{11} - 4 q^{12} - 2 q^{14} + 8 q^{16} + 2 q^{17} - 8 q^{19} - 2 q^{20} + 12 q^{21} - 10 q^{22} - 8 q^{23} - 4 q^{24} - 14 q^{25} - 22 q^{27} - 2 q^{28} + 8 q^{29} - 12 q^{31} + 8 q^{32} + 4 q^{33} + 2 q^{34} - 10 q^{35} - 8 q^{38} - 2 q^{40} + 2 q^{41} + 12 q^{42} - 16 q^{43} - 10 q^{44} - 12 q^{45} - 8 q^{46} + 12 q^{47} - 4 q^{48} - 14 q^{49} - 14 q^{50} - 22 q^{51} - 24 q^{53} - 22 q^{54} - 10 q^{55} - 2 q^{56} + 4 q^{57} + 8 q^{58} - 4 q^{59} - 26 q^{61} - 12 q^{62} - 12 q^{63} + 8 q^{64} + 4 q^{66} + 10 q^{67} + 2 q^{68} + 6 q^{69} - 10 q^{70} - 36 q^{71} + 20 q^{73} + 6 q^{75} - 8 q^{76} - 12 q^{77} - 22 q^{79} - 2 q^{80} + 36 q^{81} + 2 q^{82} + 18 q^{83} + 12 q^{84} + 26 q^{85} - 16 q^{86} - 38 q^{87} - 10 q^{88} - 18 q^{89} - 12 q^{90} - 8 q^{92} + 12 q^{93} + 12 q^{94} + 2 q^{95} - 4 q^{96} - 8 q^{97} - 14 q^{98} - 8 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^7 + 8 * q^8 - 2 * q^10 - 10 * q^11 - 4 * q^12 - 2 * q^14 + 8 * q^16 + 2 * q^17 - 8 * q^19 - 2 * q^20 + 12 * q^21 - 10 * q^22 - 8 * q^23 - 4 * q^24 - 14 * q^25 - 22 * q^27 - 2 * q^28 + 8 * q^29 - 12 * q^31 + 8 * q^32 + 4 * q^33 + 2 * q^34 - 10 * q^35 - 8 * q^38 - 2 * q^40 + 2 * q^41 + 12 * q^42 - 16 * q^43 - 10 * q^44 - 12 * q^45 - 8 * q^46 + 12 * q^47 - 4 * q^48 - 14 * q^49 - 14 * q^50 - 22 * q^51 - 24 * q^53 - 22 * q^54 - 10 * q^55 - 2 * q^56 + 4 * q^57 + 8 * q^58 - 4 * q^59 - 26 * q^61 - 12 * q^62 - 12 * q^63 + 8 * q^64 + 4 * q^66 + 10 * q^67 + 2 * q^68 + 6 * q^69 - 10 * q^70 - 36 * q^71 + 20 * q^73 + 6 * q^75 - 8 * q^76 - 12 * q^77 - 22 * q^79 - 2 * q^80 + 36 * q^81 + 2 * q^82 + 18 * q^83 + 12 * q^84 + 26 * q^85 - 16 * q^86 - 38 * q^87 - 10 * q^88 - 18 * q^89 - 12 * q^90 - 8 * q^92 + 12 * q^93 + 12 * q^94 + 2 * q^95 - 4 * q^96 - 8 * q^97 - 14 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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