LMFDB - Embedded newform 6422.2.a.y.1.2

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1129.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 3 $ x^3 - 7*x - 3
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			1.2
Root			$-0.440808$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.440808 q^{3} +1.00000 q^{4} -0.440808 q^{6} +4.92407 q^{7} +1.00000 q^{8} -2.80569 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.440808 q^{3} +1.00000 q^{4} -0.440808 q^{6} +4.92407 q^{7} +1.00000 q^{8} -2.80569 q^{9} -0.440808 q^{12} +4.92407 q^{14} +1.00000 q^{16} -4.44081 q^{17} -2.80569 q^{18} -1.00000 q^{19} -2.17057 q^{21} -6.68730 q^{23} -0.440808 q^{24} -5.00000 q^{25} +2.55919 q^{27} +4.92407 q^{28} +2.48327 q^{29} +8.72976 q^{31} +1.00000 q^{32} -4.44081 q^{34} -2.80569 q^{36} +4.11838 q^{37} -1.00000 q^{38} +10.4930 q^{41} -2.17057 q^{42} +4.88162 q^{43} -6.68730 q^{46} +2.76323 q^{47} -0.440808 q^{48} +17.2465 q^{49} -5.00000 q^{50} +1.95754 q^{51} +8.24650 q^{53} +2.55919 q^{54} +4.92407 q^{56} +0.440808 q^{57} +2.48327 q^{58} +11.3649 q^{59} -6.88162 q^{61} +8.72976 q^{62} -13.8154 q^{63} +1.00000 q^{64} +3.36488 q^{67} -4.44081 q^{68} +2.94782 q^{69} +1.61138 q^{71} -2.80569 q^{72} +9.41707 q^{73} +4.11838 q^{74} +2.20404 q^{75} -1.00000 q^{76} -5.61138 q^{79} +7.28895 q^{81} +10.4930 q^{82} +7.61138 q^{83} -2.17057 q^{84} +4.88162 q^{86} -1.09464 q^{87} +15.4595 q^{89} -6.68730 q^{92} -3.84815 q^{93} +2.76323 q^{94} -0.440808 q^{96} -4.88162 q^{97} +17.2465 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 4 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 4 * q^7 + 3 * q^8 + 5 * q^9	$ 3 q + 3 q^{2} + 3 q^{4} + 4 q^{7} + 3 q^{8} + 5 q^{9} + 4 q^{14} + 3 q^{16} - 12 q^{17} + 5 q^{18} - 3 q^{19} + 19 q^{21} - 4 q^{23} - 15 q^{25} + 9 q^{27} + 4 q^{28} - 2 q^{29} + 2 q^{31} + 3 q^{32} - 12 q^{34} + 5 q^{36} + 15 q^{37} - 3 q^{38} + 2 q^{41} + 19 q^{42} + 12 q^{43} - 4 q^{46} + 3 q^{47} + 37 q^{49} - 15 q^{50} + 14 q^{51} + 10 q^{53} + 9 q^{54} + 4 q^{56} - 2 q^{58} + 22 q^{59} - 18 q^{61} + 2 q^{62} - 8 q^{63} + 3 q^{64} - 2 q^{67} - 12 q^{68} + 37 q^{69} - 22 q^{71} + 5 q^{72} - 12 q^{73} + 15 q^{74} - 3 q^{76} + 10 q^{79} - q^{81} + 2 q^{82} - 4 q^{83} + 19 q^{84} + 12 q^{86} + 33 q^{87} - 2 q^{89} - 4 q^{92} + 10 q^{93} + 3 q^{94} - 12 q^{97} + 37 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 4 * q^7 + 3 * q^8 + 5 * q^9 + 4 * q^14 + 3 * q^16 - 12 * q^17 + 5 * q^18 - 3 * q^19 + 19 * q^21 - 4 * q^23 - 15 * q^25 + 9 * q^27 + 4 * q^28 - 2 * q^29 + 2 * q^31 + 3 * q^32 - 12 * q^34 + 5 * q^36 + 15 * q^37 - 3 * q^38 + 2 * q^41 + 19 * q^42 + 12 * q^43 - 4 * q^46 + 3 * q^47 + 37 * q^49 - 15 * q^50 + 14 * q^51 + 10 * q^53 + 9 * q^54 + 4 * q^56 - 2 * q^58 + 22 * q^59 - 18 * q^61 + 2 * q^62 - 8 * q^63 + 3 * q^64 - 2 * q^67 - 12 * q^68 + 37 * q^69 - 22 * q^71 + 5 * q^72 - 12 * q^73 + 15 * q^74 - 3 * q^76 + 10 * q^79 - q^81 + 2 * q^82 - 4 * q^83 + 19 * q^84 + 12 * q^86 + 33 * q^87 - 2 * q^89 - 4 * q^92 + 10 * q^93 + 3 * q^94 - 12 * q^97 + 37 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−0.440808	−0.254500	−0.127250	−	0.991871i	$-0.540615\pi$
$3$	−0.440808	−0.254500	−0.127250	+	0.991871i	$0.540615\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−0.440808	−0.179959
$7$	4.92407	1.86112	0.930562	−	0.366133i	$-0.119319\pi$
$7$	4.92407	1.86112	0.930562	+	0.366133i	$0.119319\pi$
$8$	1.00000	0.353553
$9$	−2.80569	−0.935230
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−0.440808	−0.127250
$13$	0	0
$13$	0	0
$14$	4.92407	1.31601
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.44081	−1.07705	−0.538527	−	0.842608i	$-0.681019\pi$
$17$	−4.44081	−1.07705	−0.538527	+	0.842608i	$0.681019\pi$
$18$	−2.80569	−0.661307
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.17057	−0.473657
$22$	0	0
$23$	−6.68730	−1.39440	−0.697200	−	0.716877i	$-0.745571\pi$
$23$	−6.68730	−1.39440	−0.697200	+	0.716877i	$0.745571\pi$
$24$	−0.440808	−0.0899795
$25$	−5.00000	−1.00000
$26$	0	0
$27$	2.55919	0.492517
$28$	4.92407	0.930562
$29$	2.48327	0.461131	0.230565	−	0.973057i	$-0.425942\pi$
$29$	2.48327	0.461131	0.230565	+	0.973057i	$0.425942\pi$
$30$	0	0
$31$	8.72976	1.56791	0.783956	−	0.620817i	$-0.213199\pi$
$31$	8.72976	1.56791	0.783956	+	0.620817i	$0.213199\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.44081	−0.761592
$35$	0	0
$36$	−2.80569	−0.467615
$37$	4.11838	0.677058	0.338529	−	0.940956i	$-0.390071\pi$
$37$	4.11838	0.677058	0.338529	+	0.940956i	$0.390071\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	10.4930	1.63873	0.819365	−	0.573272i	$-0.194326\pi$
$41$	10.4930	1.63873	0.819365	+	0.573272i	$0.194326\pi$
$42$	−2.17057	−0.334926
$43$	4.88162	0.744439	0.372220	−	0.928145i	$-0.378597\pi$
$43$	4.88162	0.744439	0.372220	+	0.928145i	$0.378597\pi$
$44$	0	0
$45$	0	0
$46$	−6.68730	−0.985989
$47$	2.76323	0.403059	0.201529	−	0.979482i	$-0.435409\pi$
$47$	2.76323	0.403059	0.201529	+	0.979482i	$0.435409\pi$
$48$	−0.440808	−0.0636251
$49$	17.2465	2.46379
$50$	−5.00000	−0.707107
$51$	1.95754	0.274111
$52$	0	0
$53$	8.24650	1.13274	0.566372	−	0.824150i	$-0.308347\pi$
$53$	8.24650	1.13274	0.566372	+	0.824150i	$0.308347\pi$
$54$	2.55919	0.348262
$55$	0	0
$56$	4.92407	0.658007
$57$	0.440808	0.0583864
$58$	2.48327	0.326069
$59$	11.3649	1.47958	0.739791	−	0.672837i	$-0.234924\pi$
$59$	11.3649	1.47958	0.739791	+	0.672837i	$0.234924\pi$
$60$	0	0
$61$	−6.88162	−0.881101	−0.440550	−	0.897728i	$-0.645217\pi$
$61$	−6.88162	−0.881101	−0.440550	+	0.897728i	$0.645217\pi$
$62$	8.72976	1.10868
$63$	−13.8154	−1.74058
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	3.36488	0.411086	0.205543	−	0.978648i	$-0.434104\pi$
$67$	3.36488	0.411086	0.205543	+	0.978648i	$0.434104\pi$
$68$	−4.44081	−0.538527
$69$	2.94782	0.354875
$70$	0	0
$71$	1.61138	0.191235	0.0956176	−	0.995418i	$-0.469517\pi$
$71$	1.61138	0.191235	0.0956176	+	0.995418i	$0.469517\pi$
$72$	−2.80569	−0.330654
$73$	9.41707	1.10218	0.551092	−	0.834444i	$-0.314211\pi$
$73$	9.41707	1.10218	0.551092	+	0.834444i	$0.314211\pi$
$74$	4.11838	0.478752
$75$	2.20404	0.254500
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	−5.61138	−0.631329	−0.315665	−	0.948871i	$-0.602227\pi$
$79$	−5.61138	−0.631329	−0.315665	+	0.948871i	$0.602227\pi$
$80$	0	0
$81$	7.28895	0.809884
$82$	10.4930	1.15876
$83$	7.61138	0.835457	0.417729	−	0.908572i	$-0.362826\pi$
$83$	7.61138	0.835457	0.417729	+	0.908572i	$0.362826\pi$
$84$	−2.17057	−0.236829
$85$	0	0
$86$	4.88162	0.526398
$87$	−1.09464	−0.117358
$88$	0	0
$89$	15.4595	1.63871	0.819353	−	0.573289i	$-0.194333\pi$
$89$	15.4595	1.63871	0.819353	+	0.573289i	$0.194333\pi$
$90$	0	0
$91$	0	0
$92$	−6.68730	−0.697200
$93$	−3.84815	−0.399034
$94$	2.76323	0.285006
$95$	0	0
$96$	−0.440808	−0.0449897
$97$	−4.88162	−0.495653	−0.247826	−	0.968804i	$-0.579716\pi$
$97$	−4.88162	−0.495653	−0.247826	+	0.968804i	$0.579716\pi$
$98$	17.2465	1.74216
$99$	0	0
$100$	−5.00000	−0.500000
$101$	−10.7298	−1.06765	−0.533826	−	0.845595i	$-0.679246\pi$
$101$	−10.7298	−1.06765	−0.533826	+	0.845595i	$0.679246\pi$
$102$	1.95754	0.193826
$103$	−10.3411	−1.01894	−0.509471	−	0.860488i	$-0.670159\pi$
$103$	−10.3411	−1.01894	−0.509471	+	0.860488i	$0.670159\pi$
$104$	0	0
$105$	0	0
$106$	8.24650	0.800971
$107$	4.44081	0.429309	0.214655	−	0.976690i	$-0.431137\pi$
$107$	4.44081	0.429309	0.214655	+	0.976690i	$0.431137\pi$
$108$	2.55919	0.246258
$109$	6.48327	0.620984	0.310492	−	0.950576i	$-0.399506\pi$
$109$	6.48327	0.620984	0.310492	+	0.950576i	$0.399506\pi$
$110$	0	0
$111$	−1.81542	−0.172312
$112$	4.92407	0.465281
$113$	15.4595	1.45431	0.727155	−	0.686474i	$-0.240842\pi$
$113$	15.4595	1.45431	0.727155	+	0.686474i	$0.240842\pi$
$114$	0.440808	0.0412854
$115$	0	0
$116$	2.48327	0.230565
$117$	0	0
$118$	11.3649	1.04622
$119$	−21.8669	−2.00453
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−6.88162	−0.623032
$123$	−4.62539	−0.417058
$124$	8.72976	0.783956
$125$	0	0
$126$	−13.8154	−1.23078
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	−2.15185	−0.189460
$130$	0	0
$131$	−13.6114	−1.18923	−0.594616	−	0.804010i	$-0.702696\pi$
$131$	−13.6114	−1.18923	−0.594616	+	0.804010i	$0.702696\pi$
$132$	0	0
$133$	−4.92407	−0.426971
$134$	3.36488	0.290681
$135$	0	0
$136$	−4.44081	−0.380796
$137$	−8.00973	−0.684317	−0.342159	−	0.939642i	$-0.611158\pi$
$137$	−8.00973	−0.684317	−0.342159	+	0.939642i	$0.611158\pi$
$138$	2.94782	0.250935
$139$	6.96653	0.590893	0.295447	−	0.955359i	$-0.404532\pi$
$139$	6.96653	0.590893	0.295447	+	0.955359i	$0.404532\pi$
$140$	0	0
$141$	−1.21805	−0.102579
$142$	1.61138	0.135224
$143$	0	0
$144$	−2.80569	−0.233807
$145$	0	0
$146$	9.41707	0.779362
$147$	−7.60239	−0.627034
$148$	4.11838	0.338529
$149$	−13.6114	−1.11509	−0.557544	−	0.830148i	$-0.688256\pi$
$149$	−13.6114	−1.11509	−0.557544	+	0.830148i	$0.688256\pi$
$150$	2.20404	0.179959
$151$	12.8816	1.04829	0.524145	−	0.851629i	$-0.324385\pi$
$151$	12.8816	1.04829	0.524145	+	0.851629i	$0.324385\pi$
$152$	−1.00000	−0.0811107
$153$	12.4595	1.00729
$154$	0	0
$155$	0	0
$156$	0	0
$157$	24.7298	1.97365	0.986825	−	0.161792i	$-0.0517273\pi$
$157$	24.7298	1.97365	0.986825	+	0.161792i	$0.0517273\pi$
$158$	−5.61138	−0.446417
$159$	−3.63512	−0.288284
$160$	0	0
$161$	−32.9288	−2.59515
$162$	7.28895	0.572674
$163$	−0.966531	−0.0757045	−0.0378523	−	0.999283i	$-0.512052\pi$
$163$	−0.966531	−0.0757045	−0.0378523	+	0.999283i	$0.512052\pi$
$164$	10.4930	0.819365
$165$	0	0
$166$	7.61138	0.590757
$167$	−21.4595	−1.66059	−0.830294	−	0.557326i	$-0.811827\pi$
$167$	−21.4595	−1.66059	−0.830294	+	0.557326i	$0.811827\pi$
$168$	−2.17057	−0.167463
$169$	0	0
$170$	0	0
$171$	2.80569	0.214556
$172$	4.88162	0.372220
$173$	−9.22275	−0.701193	−0.350597	−	0.936527i	$-0.614021\pi$
$173$	−9.22275	−0.701193	−0.350597	+	0.936527i	$0.614021\pi$
$174$	−1.09464	−0.0829846
$175$	−24.6204	−1.86112
$176$	0	0
$177$	−5.00973	−0.376554
$178$	15.4595	1.15874
$179$	−9.57791	−0.715886	−0.357943	−	0.933743i	$-0.616522\pi$
$179$	−9.57791	−0.715886	−0.357943	+	0.933743i	$0.616522\pi$
$180$	0	0
$181$	7.76323	0.577036	0.288518	−	0.957474i	$-0.406837\pi$
$181$	7.76323	0.577036	0.288518	+	0.957474i	$0.406837\pi$
$182$	0	0
$183$	3.03347	0.224240
$184$	−6.68730	−0.492995
$185$	0	0
$186$	−3.84815	−0.282160
$187$	0	0
$188$	2.76323	0.201529
$189$	12.6017	0.916635
$190$	0	0
$191$	0.0521848	0.00377596	0.00188798	−	0.999998i	$-0.499399\pi$
$191$	0.0521848	0.00377596	0.00188798	+	0.999998i	$0.499399\pi$
$192$	−0.440808	−0.0318126
$193$	−3.84815	−0.276996	−0.138498	−	0.990363i	$-0.544227\pi$
$193$	−3.84815	−0.276996	−0.138498	+	0.990363i	$0.544227\pi$
$194$	−4.88162	−0.350480
$195$	0	0
$196$	17.2465	1.23189
$197$	−3.35515	−0.239045	−0.119522	−	0.992832i	$-0.538136\pi$
$197$	−3.35515	−0.239045	−0.119522	+	0.992832i	$0.538136\pi$
$198$	0	0
$199$	−3.94782	−0.279853	−0.139927	−	0.990162i	$-0.544687\pi$
$199$	−3.94782	−0.279853	−0.139927	+	0.990162i	$0.544687\pi$
$200$	−5.00000	−0.353553
$201$	−1.48327	−0.104621
$202$	−10.7298	−0.754943
$203$	12.2278	0.858222
$204$	1.95754	0.137055
$205$	0	0
$206$	−10.3411	−0.720501
$207$	18.7625	1.30408
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−24.8244	−1.70898	−0.854491	−	0.519466i	$-0.826131\pi$
$211$	−24.8244	−1.70898	−0.854491	+	0.519466i	$0.826131\pi$
$212$	8.24650	0.566372
$213$	−0.710307	−0.0486695
$214$	4.44081	0.303567
$215$	0	0
$216$	2.55919	0.174131
$217$	42.9860	2.91808
$218$	6.48327	0.439102
$219$	−4.15112	−0.280506
$220$	0	0
$221$	0	0
$222$	−1.81542	−0.121843
$223$	−3.50701	−0.234847	−0.117423	−	0.993082i	$-0.537463\pi$
$223$	−3.50701	−0.234847	−0.117423	+	0.993082i	$0.537463\pi$
$224$	4.92407	0.329003
$225$	14.0284	0.935230
$226$	15.4595	1.02835
$227$	−13.3224	−0.884240	−0.442120	−	0.896956i	$-0.645774\pi$
$227$	−13.3224	−0.884240	−0.442120	+	0.896956i	$0.645774\pi$
$228$	0.440808	0.0291932
$229$	28.1044	1.85719	0.928595	−	0.371096i	$-0.121018\pi$
$229$	28.1044	1.85719	0.928595	+	0.371096i	$0.121018\pi$
$230$	0	0
$231$	0	0
$232$	2.48327	0.163034
$233$	30.2228	1.97996	0.989979	−	0.141213i	$-0.0451003\pi$
$233$	30.2228	1.97996	0.989979	+	0.141213i	$0.0451003\pi$
$234$	0	0
$235$	0	0
$236$	11.3649	0.739791
$237$	2.47354	0.160674
$238$	−21.8669	−1.41742
$239$	−11.0759	−0.716442	−0.358221	−	0.933637i	$-0.616617\pi$
$239$	−11.0759	−0.716442	−0.358221	+	0.933637i	$0.616617\pi$
$240$	0	0
$241$	0.644846	0.0415382	0.0207691	−	0.999784i	$-0.493389\pi$
$241$	0.644846	0.0415382	0.0207691	+	0.999784i	$0.493389\pi$
$242$	−11.0000	−0.707107
$243$	−10.8906	−0.698633
$244$	−6.88162	−0.440550
$245$	0	0
$246$	−4.62539	−0.294904
$247$	0	0
$248$	8.72976	0.554340
$249$	−3.35515	−0.212624
$250$	0	0
$251$	−1.84815	−0.116654	−0.0583270	−	0.998298i	$-0.518577\pi$
$251$	−1.84815	−0.116654	−0.0583270	+	0.998298i	$0.518577\pi$
$252$	−13.8154	−0.870289
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.8147	−1.54790	−0.773949	−	0.633249i	$-0.781721\pi$
$257$	−24.8147	−1.54790	−0.773949	+	0.633249i	$0.781721\pi$
$258$	−2.15185	−0.133969
$259$	20.2792	1.26009
$260$	0	0
$261$	−6.96727	−0.431263
$262$	−13.6114	−0.840914
$263$	−6.22275	−0.383711	−0.191856	−	0.981423i	$-0.561451\pi$
$263$	−6.22275	−0.383711	−0.191856	+	0.981423i	$0.561451\pi$
$264$	0	0
$265$	0	0
$266$	−4.92407	−0.301914
$267$	−6.81468	−0.417051
$268$	3.36488	0.205543
$269$	−9.10437	−0.555103	−0.277552	−	0.960711i	$-0.589523\pi$
$269$	−9.10437	−0.555103	−0.277552	+	0.960711i	$0.589523\pi$
$270$	0	0
$271$	7.17057	0.435581	0.217791	−	0.975996i	$-0.430115\pi$
$271$	7.17057	0.435581	0.217791	+	0.975996i	$0.430115\pi$
$272$	−4.44081	−0.269264
$273$	0	0
$274$	−8.00973	−0.483885
$275$	0	0
$276$	2.94782	0.177438
$277$	−11.9525	−0.718157	−0.359079	−	0.933307i	$-0.616909\pi$
$277$	−11.9525	−0.718157	−0.359079	+	0.933307i	$0.616909\pi$
$278$	6.96653	0.417825
$279$	−24.4930	−1.46636
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	−1.21805	−0.0725340
$283$	24.8341	1.47624	0.738118	−	0.674672i	$-0.235715\pi$
$283$	24.8341	1.47624	0.738118	+	0.674672i	$0.235715\pi$
$284$	1.61138	0.0956176
$285$	0	0
$286$	0	0
$287$	51.6683	3.04988
$288$	−2.80569	−0.165327
$289$	2.72077	0.160045
$290$	0	0
$291$	2.15185	0.126144
$292$	9.41707	0.551092
$293$	28.3836	1.65819	0.829094	−	0.559110i	$-0.188857\pi$
$293$	28.3836	1.65819	0.829094	+	0.559110i	$0.188857\pi$
$294$	−7.60239	−0.443380
$295$	0	0
$296$	4.11838	0.239376
$297$	0	0
$298$	−13.6114	−0.788486
$299$	0	0
$300$	2.20404	0.127250
$301$	24.0374	1.38549
$302$	12.8816	0.741254
$303$	4.72976	0.271718
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	12.4595	0.712264
$307$	30.8676	1.76171	0.880854	−	0.473389i	$-0.156969\pi$
$307$	30.8676	1.76171	0.880854	+	0.473389i	$0.156969\pi$
$308$	0	0
$309$	4.55845	0.259321
$310$	0	0
$311$	4.09464	0.232186	0.116093	−	0.993238i	$-0.462963\pi$
$311$	4.09464	0.232186	0.116093	+	0.993238i	$0.462963\pi$
$312$	0	0
$313$	0.914346	0.0516819	0.0258409	−	0.999666i	$-0.491774\pi$
$313$	0.914346	0.0516819	0.0258409	+	0.999666i	$0.491774\pi$
$314$	24.7298	1.39558
$315$	0	0
$316$	−5.61138	−0.315665
$317$	−15.1706	−0.852064	−0.426032	−	0.904708i	$-0.640089\pi$
$317$	−15.1706	−0.852064	−0.426032	+	0.904708i	$0.640089\pi$
$318$	−3.63512	−0.203847
$319$	0	0
$320$	0	0
$321$	−1.95754	−0.109259
$322$	−32.9288	−1.83505
$323$	4.44081	0.247093
$324$	7.28895	0.404942
$325$	0	0
$326$	−0.966531	−0.0535312
$327$	−2.85787	−0.158041
$328$	10.4930	0.579379
$329$	13.6064	0.750142
$330$	0	0
$331$	−25.1231	−1.38089	−0.690445	−	0.723385i	$-0.742585\pi$
$331$	−25.1231	−1.38089	−0.690445	+	0.723385i	$0.742585\pi$
$332$	7.61138	0.417729
$333$	−11.5549	−0.633205
$334$	−21.4595	−1.17421
$335$	0	0
$336$	−2.17057	−0.118414
$337$	14.3411	0.781212	0.390606	−	0.920558i	$-0.372266\pi$
$337$	14.3411	0.781212	0.390606	+	0.920558i	$0.372266\pi$
$338$	0	0
$339$	−6.81468	−0.370122
$340$	0	0
$341$	0	0
$342$	2.80569	0.151714
$343$	50.4545	2.72429
$344$	4.88162	0.263199
$345$	0	0
$346$	−9.22275	−0.495818
$347$	−21.8007	−1.17032	−0.585160	−	0.810918i	$-0.698968\pi$
$347$	−21.8007	−1.17032	−0.585160	+	0.810918i	$0.698968\pi$
$348$	−1.09464	−0.0586790
$349$	−12.3886	−0.663148	−0.331574	−	0.943429i	$-0.607580\pi$
$349$	−12.3886	−0.663148	−0.331574	+	0.943429i	$0.607580\pi$
$350$	−24.6204	−1.31601
$351$	0	0
$352$	0	0
$353$	−9.80569	−0.521904	−0.260952	−	0.965352i	$-0.584036\pi$
$353$	−9.80569	−0.521904	−0.260952	+	0.965352i	$0.584036\pi$
$354$	−5.00973	−0.266264
$355$	0	0
$356$	15.4595	0.819353
$357$	9.63908	0.510154
$358$	−9.57791	−0.506208
$359$	−4.89134	−0.258155	−0.129078	−	0.991634i	$-0.541202\pi$
$359$	−4.89134	−0.258155	−0.129078	+	0.991634i	$0.541202\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	7.76323	0.408026
$363$	4.84888	0.254500
$364$	0	0
$365$	0	0
$366$	3.03347	0.158562
$367$	8.22275	0.429224	0.214612	−	0.976699i	$-0.431151\pi$
$367$	8.22275	0.429224	0.214612	+	0.976699i	$0.431151\pi$
$368$	−6.68730	−0.348600
$369$	−29.4401	−1.53259
$370$	0	0
$371$	40.6064	2.10818
$372$	−3.84815	−0.199517
$373$	28.9338	1.49814	0.749068	−	0.662493i	$-0.230502\pi$
$373$	28.9338	1.49814	0.749068	+	0.662493i	$0.230502\pi$
$374$	0	0
$375$	0	0
$376$	2.76323	0.142503
$377$	0	0
$378$	12.6017	0.648159
$379$	10.1274	0.520208	0.260104	−	0.965581i	$-0.416243\pi$
$379$	10.1274	0.520208	0.260104	+	0.965581i	$0.416243\pi$
$380$	0	0
$381$	−3.52646	−0.180666
$382$	0.0521848	0.00267001
$383$	9.11838	0.465928	0.232964	−	0.972485i	$-0.425158\pi$
$383$	9.11838	0.465928	0.232964	+	0.972485i	$0.425158\pi$
$384$	−0.440808	−0.0224949
$385$	0	0
$386$	−3.84815	−0.195865
$387$	−13.6963	−0.696222
$388$	−4.88162	−0.247826
$389$	8.64485	0.438311	0.219156	−	0.975690i	$-0.429670\pi$
$389$	8.64485	0.438311	0.219156	+	0.975690i	$0.429670\pi$
$390$	0	0
$391$	29.6970	1.50184
$392$	17.2465	0.871080
$393$	6.00000	0.302660
$394$	−3.35515	−0.169030
$395$	0	0
$396$	0	0
$397$	−32.5779	−1.63504	−0.817519	−	0.575901i	$-0.804651\pi$
$397$	−32.5779	−1.63504	−0.817519	+	0.575901i	$0.804651\pi$
$398$	−3.94782	−0.197886
$399$	2.17057	0.108664
$400$	−5.00000	−0.250000
$401$	−22.7298	−1.13507	−0.567535	−	0.823349i	$-0.692103\pi$
$401$	−22.7298	−1.13507	−0.567535	+	0.823349i	$0.692103\pi$
$402$	−1.48327	−0.0739786
$403$	0	0
$404$	−10.7298	−0.533826
$405$	0	0
$406$	12.2278	0.606855
$407$	0	0
$408$	1.95754	0.0969128
$409$	7.22275	0.357142	0.178571	−	0.983927i	$-0.442853\pi$
$409$	7.22275	0.357142	0.178571	+	0.983927i	$0.442853\pi$
$410$	0	0
$411$	3.53075	0.174159
$412$	−10.3411	−0.509471
$413$	55.9615	2.75369
$414$	18.7625	0.922126
$415$	0	0
$416$	0	0
$417$	−3.07090	−0.150383
$418$	0	0
$419$	−8.64485	−0.422328	−0.211164	−	0.977451i	$-0.567725\pi$
$419$	−8.64485	−0.422328	−0.211164	+	0.977451i	$0.567725\pi$
$420$	0	0
$421$	−18.1468	−0.884422	−0.442211	−	0.896911i	$-0.645806\pi$
$421$	−18.1468	−0.884422	−0.442211	+	0.896911i	$0.645806\pi$
$422$	−24.8244	−1.20843
$423$	−7.75277	−0.376952
$424$	8.24650	0.400485
$425$	22.2040	1.07705
$426$	−0.710307	−0.0344145
$427$	−33.8856	−1.63984
$428$	4.44081	0.214655
$429$	0	0
$430$	0	0
$431$	−11.8676	−0.571642	−0.285821	−	0.958283i	$-0.592266\pi$
$431$	−11.8676	−0.571642	−0.285821	+	0.958283i	$0.592266\pi$
$432$	2.55919	0.123129
$433$	−26.2562	−1.26179	−0.630897	−	0.775867i	$-0.717313\pi$
$433$	−26.2562	−1.26179	−0.630897	+	0.775867i	$0.717313\pi$
$434$	42.9860	2.06339
$435$	0	0
$436$	6.48327	0.310492
$437$	6.68730	0.319897
$438$	−4.15112	−0.198348
$439$	−6.64485	−0.317141	−0.158571	−	0.987348i	$-0.550689\pi$
$439$	−6.64485	−0.317141	−0.158571	+	0.987348i	$0.550689\pi$
$440$	0	0
$441$	−48.3883	−2.30420
$442$	0	0
$443$	−29.5444	−1.40370	−0.701849	−	0.712325i	$-0.747642\pi$
$443$	−29.5444	−1.40370	−0.701849	+	0.712325i	$0.747642\pi$
$444$	−1.81542	−0.0861558
$445$	0	0
$446$	−3.50701	−0.166062
$447$	6.00000	0.283790
$448$	4.92407	0.232641
$449$	−1.61138	−0.0760456	−0.0380228	−	0.999277i	$-0.512106\pi$
$449$	−1.61138	−0.0760456	−0.0380228	+	0.999277i	$0.512106\pi$
$450$	14.0284	0.661307
$451$	0	0
$452$	15.4595	0.727155
$453$	−5.67832	−0.266791
$454$	−13.3224	−0.625252
$455$	0	0
$456$	0.440808	0.0206427
$457$	−21.4268	−1.00230	−0.501152	−	0.865360i	$-0.667090\pi$
$457$	−21.4268	−1.00230	−0.501152	+	0.865360i	$0.667090\pi$
$458$	28.1044	1.31323
$459$	−11.3649	−0.530467
$460$	0	0
$461$	9.93306	0.462629	0.231314	−	0.972879i	$-0.425697\pi$
$461$	9.93306	0.462629	0.231314	+	0.972879i	$0.425697\pi$
$462$	0	0
$463$	−25.7632	−1.19732	−0.598659	−	0.801004i	$-0.704300\pi$
$463$	−25.7632	−1.19732	−0.598659	+	0.801004i	$0.704300\pi$
$464$	2.48327	0.115283
$465$	0	0
$466$	30.2228	1.40004
$467$	−37.7157	−1.74528	−0.872638	−	0.488367i	$-0.837593\pi$
$467$	−37.7157	−1.74528	−0.872638	+	0.488367i	$0.837593\pi$
$468$	0	0
$469$	16.5689	0.765082
$470$	0	0
$471$	−10.9011	−0.502295
$472$	11.3649	0.523111
$473$	0	0
$474$	2.47354	0.113613
$475$	5.00000	0.229416
$476$	−21.8669	−1.00227
$477$	−23.1371	−1.05938
$478$	−11.0759	−0.506601
$479$	5.76323	0.263329	0.131664	−	0.991294i	$-0.457968\pi$
$479$	5.76323	0.263329	0.131664	+	0.991294i	$0.457968\pi$
$480$	0	0
$481$	0	0
$482$	0.644846	0.0293719
$483$	14.5153	0.660467
$484$	−11.0000	−0.500000
$485$	0	0
$486$	−10.8906	−0.494008
$487$	30.1893	1.36801	0.684004	−	0.729479i	$-0.260237\pi$
$487$	30.1893	1.36801	0.684004	+	0.729479i	$0.260237\pi$
$488$	−6.88162	−0.311516
$489$	0.426054	0.0192668
$490$	0	0
$491$	−36.8147	−1.66142	−0.830712	−	0.556703i	$-0.812066\pi$
$491$	−36.8147	−1.66142	−0.830712	+	0.556703i	$0.812066\pi$
$492$	−4.62539	−0.208529
$493$	−11.0277	−0.496663
$494$	0	0
$495$	0	0
$496$	8.72976	0.391978
$497$	7.93454	0.355913
$498$	−3.35515	−0.150348
$499$	20.3886	0.912720	0.456360	−	0.889795i	$-0.349153\pi$
$499$	20.3886	0.912720	0.456360	+	0.889795i	$0.349153\pi$
$500$	0	0
$501$	9.45952	0.422620
$502$	−1.84815	−0.0824868
$503$	−33.0612	−1.47412	−0.737062	−	0.675825i	$-0.763788\pi$
$503$	−33.0612	−1.47412	−0.737062	+	0.675825i	$0.763788\pi$
$504$	−13.8154	−0.615388
$505$	0	0
$506$	0	0
$507$	0	0
$508$	8.00000	0.354943
$509$	15.4595	0.685231	0.342616	−	0.939476i	$-0.388687\pi$
$509$	15.4595	0.685231	0.342616	+	0.939476i	$0.388687\pi$
$510$	0	0
$511$	46.3703	2.05130
$512$	1.00000	0.0441942
$513$	−2.55919	−0.112991
$514$	−24.8147	−1.09453
$515$	0	0
$516$	−2.15185	−0.0947301
$517$	0	0
$518$	20.2792	0.891018
$519$	4.06546	0.178454
$520$	0	0
$521$	−36.6823	−1.60708	−0.803540	−	0.595251i	$-0.797053\pi$
$521$	−36.6823	−1.60708	−0.803540	+	0.595251i	$0.797053\pi$
$522$	−6.96727	−0.304949
$523$	25.6538	1.12176	0.560882	−	0.827896i	$-0.310462\pi$
$523$	25.6538	1.12176	0.560882	+	0.827896i	$0.310462\pi$
$524$	−13.6114	−0.594616
$525$	10.8528	0.473657
$526$	−6.22275	−0.271325
$527$	−38.7672	−1.68873
$528$	0	0
$529$	21.7200	0.944349
$530$	0	0
$531$	−31.8863	−1.38375
$532$	−4.92407	−0.213486
$533$	0	0
$534$	−6.81468	−0.294900
$535$	0	0
$536$	3.36488	0.145341
$537$	4.22202	0.182193
$538$	−9.10437	−0.392517
$539$	0	0
$540$	0	0
$541$	−36.5974	−1.57344	−0.786722	−	0.617308i	$-0.788223\pi$
$541$	−36.5974	−1.57344	−0.786722	+	0.617308i	$0.788223\pi$
$542$	7.17057	0.308002
$543$	−3.42209	−0.146856
$544$	−4.44081	−0.190398
$545$	0	0
$546$	0	0
$547$	26.9190	1.15098	0.575488	−	0.817810i	$-0.304812\pi$
$547$	26.9190	1.15098	0.575488	+	0.817810i	$0.304812\pi$
$548$	−8.00973	−0.342159
$549$	19.3077	0.824031
$550$	0	0
$551$	−2.48327	−0.105791
$552$	2.94782	0.125467
$553$	−27.6308	−1.17498
$554$	−11.9525	−0.507814
$555$	0	0
$556$	6.96653	0.295447
$557$	12.9011	0.546636	0.273318	−	0.961924i	$-0.411879\pi$
$557$	12.9011	0.546636	0.273318	+	0.961924i	$0.411879\pi$
$558$	−24.4930	−1.03687
$559$	0	0
$560$	0	0
$561$	0	0
$562$	12.0000	0.506189
$563$	−21.9331	−0.924368	−0.462184	−	0.886784i	$-0.652934\pi$
$563$	−21.9331	−0.924368	−0.462184	+	0.886784i	$0.652934\pi$
$564$	−1.21805	−0.0512893
$565$	0	0
$566$	24.8341	1.04386
$567$	35.8913	1.50729
$568$	1.61138	0.0676119
$569$	−1.27024	−0.0532512	−0.0266256	−	0.999645i	$-0.508476\pi$
$569$	−1.27024	−0.0532512	−0.0266256	+	0.999645i	$0.508476\pi$
$570$	0	0
$571$	47.4595	1.98612	0.993060	−	0.117612i	$-0.0375239\pi$
$571$	47.4595	1.98612	0.993060	+	0.117612i	$0.0375239\pi$
$572$	0	0
$573$	−0.0230035	−0.000960984 0
$574$	51.6683	2.15659
$575$	33.4365	1.39440
$576$	−2.80569	−0.116904
$577$	2.22704	0.0927130	0.0463565	−	0.998925i	$-0.485239\pi$
$577$	2.22704	0.0927130	0.0463565	+	0.998925i	$0.485239\pi$
$578$	2.72077	0.113169
$579$	1.69629	0.0704955
$580$	0	0
$581$	37.4790	1.55489
$582$	2.15185	0.0891972
$583$	0	0
$584$	9.41707	0.389681
$585$	0	0
$586$	28.3836	1.17252
$587$	−21.4595	−0.885729	−0.442865	−	0.896588i	$-0.646038\pi$
$587$	−21.4595	−0.885729	−0.442865	+	0.896588i	$0.646038\pi$
$588$	−7.60239	−0.313517
$589$	−8.72976	−0.359704
$590$	0	0
$591$	1.47898	0.0608370
$592$	4.11838	0.169265
$593$	34.1558	1.40261	0.701306	−	0.712861i	$-0.252601\pi$
$593$	34.1558	1.40261	0.701306	+	0.712861i	$0.252601\pi$
$594$	0	0
$595$	0	0
$596$	−13.6114	−0.557544
$597$	1.74023	0.0712228
$598$	0	0
$599$	3.22275	0.131678	0.0658391	−	0.997830i	$-0.479028\pi$
$599$	3.22275	0.131678	0.0658391	+	0.997830i	$0.479028\pi$
$600$	2.20404	0.0899795
$601$	24.7298	1.00875	0.504374	−	0.863485i	$-0.331723\pi$
$601$	24.7298	1.00875	0.504374	+	0.863485i	$0.331723\pi$
$602$	24.0374	0.979693
$603$	−9.44081	−0.384459
$604$	12.8816	0.524145
$605$	0	0
$606$	4.72976	0.192133
$607$	−20.3886	−0.827549	−0.413774	−	0.910379i	$-0.635790\pi$
$607$	−20.3886	−0.827549	−0.413774	+	0.910379i	$0.635790\pi$
$608$	−1.00000	−0.0405554
$609$	−5.39010	−0.218418
$610$	0	0
$611$	0	0
$612$	12.4595	0.503646
$613$	−30.3606	−1.22625	−0.613126	−	0.789985i	$-0.710088\pi$
$613$	−30.3606	−1.22625	−0.613126	+	0.789985i	$0.710088\pi$
$614$	30.8676	1.24571
$615$	0	0
$616$	0	0
$617$	−30.4595	−1.22625	−0.613127	−	0.789984i	$-0.710089\pi$
$617$	−30.4595	−1.22625	−0.613127	+	0.789984i	$0.710089\pi$
$618$	4.55845	0.183368
$619$	10.1044	0.406129	0.203064	−	0.979165i	$-0.434910\pi$
$619$	10.1044	0.406129	0.203064	+	0.979165i	$0.434910\pi$
$620$	0	0
$621$	−17.1141	−0.686765
$622$	4.09464	0.164180
$623$	76.1238	3.04984
$624$	0	0
$625$	25.0000	1.00000
$626$	0.914346	0.0365446
$627$	0	0
$628$	24.7298	0.986825
$629$	−18.2890	−0.729228
$630$	0	0
$631$	34.4595	1.37181	0.685906	−	0.727690i	$-0.259406\pi$
$631$	34.4595	1.37181	0.685906	+	0.727690i	$0.259406\pi$
$632$	−5.61138	−0.223209
$633$	10.9428	0.434937
$634$	−15.1706	−0.602500
$635$	0	0
$636$	−3.63512	−0.144142
$637$	0	0
$638$	0	0
$639$	−4.52102	−0.178849
$640$	0	0
$641$	−6.57791	−0.259812	−0.129906	−	0.991526i	$-0.541468\pi$
$641$	−6.57791	−0.259812	−0.129906	+	0.991526i	$0.541468\pi$
$642$	−1.95754	−0.0772580
$643$	−2.47354	−0.0975468	−0.0487734	−	0.998810i	$-0.515531\pi$
$643$	−2.47354	−0.0975468	−0.0487734	+	0.998810i	$0.515531\pi$
$644$	−32.9288	−1.29758
$645$	0	0
$646$	4.44081	0.174721
$647$	40.0946	1.57628	0.788142	−	0.615494i	$-0.211043\pi$
$647$	40.0946	1.57628	0.788142	+	0.615494i	$0.211043\pi$
$648$	7.28895	0.286337
$649$	0	0
$650$	0	0
$651$	−18.9486	−0.742652
$652$	−0.966531	−0.0378523
$653$	−10.7298	−0.419888	−0.209944	−	0.977713i	$-0.567328\pi$
$653$	−10.7298	−0.419888	−0.209944	+	0.977713i	$0.567328\pi$
$654$	−2.85787	−0.111752
$655$	0	0
$656$	10.4930	0.409683
$657$	−26.4214	−1.03080
$658$	13.6064	0.530431
$659$	10.9435	0.426299	0.213150	−	0.977020i	$-0.431628\pi$
$659$	10.9435	0.426299	0.213150	+	0.977020i	$0.431628\pi$
$660$	0	0
$661$	19.9101	0.774412	0.387206	−	0.921993i	$-0.373440\pi$
$661$	19.9101	0.774412	0.387206	+	0.921993i	$0.373440\pi$
$662$	−25.1231	−0.976436
$663$	0	0
$664$	7.61138	0.295379
$665$	0	0
$666$	−11.5549	−0.447743
$667$	−16.6064	−0.643000
$668$	−21.4595	−0.830294
$669$	1.54592	0.0597686
$670$	0	0
$671$	0	0
$672$	−2.17057	−0.0837315
$673$	−14.8341	−0.571814	−0.285907	−	0.958257i	$-0.592295\pi$
$673$	−14.8341	−0.571814	−0.285907	+	0.958257i	$0.592295\pi$
$674$	14.3411	0.552400
$675$	−12.7960	−0.492517
$676$	0	0
$677$	6.86686	0.263915	0.131957	−	0.991255i	$-0.457874\pi$
$677$	6.86686	0.263915	0.131957	+	0.991255i	$0.457874\pi$
$678$	−6.81468	−0.261716
$679$	−24.0374	−0.922472
$680$	0	0
$681$	5.87263	0.225040
$682$	0	0
$683$	3.81468	0.145965	0.0729823	−	0.997333i	$-0.476748\pi$
$683$	3.81468	0.145965	0.0729823	+	0.997333i	$0.476748\pi$
$684$	2.80569	0.107278
$685$	0	0
$686$	50.4545	1.92636
$687$	−12.3886	−0.472655
$688$	4.88162	0.186110
$689$	0	0
$690$	0	0
$691$	39.4120	1.49930	0.749652	−	0.661832i	$-0.230221\pi$
$691$	39.4120	1.49930	0.749652	+	0.661832i	$0.230221\pi$
$692$	−9.22275	−0.350597
$693$	0	0
$694$	−21.8007	−0.827542
$695$	0	0
$696$	−1.09464	−0.0414923
$697$	−46.5974	−1.76500
$698$	−12.3886	−0.468916
$699$	−13.3224	−0.503900
$700$	−24.6204	−0.930562
$701$	32.8996	1.24260	0.621300	−	0.783573i	$-0.286605\pi$
$701$	32.8996	1.24260	0.621300	+	0.783573i	$0.286605\pi$
$702$	0	0
$703$	−4.11838	−0.155328
$704$	0	0
$705$	0	0
$706$	−9.80569	−0.369042
$707$	−52.8341	−1.98703
$708$	−5.00973	−0.188277
$709$	34.3411	1.28971	0.644854	−	0.764306i	$-0.276918\pi$
$709$	34.3411	1.28971	0.644854	+	0.764306i	$0.276918\pi$
$710$	0	0
$711$	15.7438	0.590438
$712$	15.4595	0.579370
$713$	−58.3786	−2.18629
$714$	9.63908	0.360734
$715$	0	0
$716$	−9.57791	−0.357943
$717$	4.88235	0.182335
$718$	−4.89134	−0.182543
$719$	−28.6204	−1.06736	−0.533680	−	0.845687i	$-0.679191\pi$
$719$	−28.6204	−1.06736	−0.533680	+	0.845687i	$0.679191\pi$
$720$	0	0
$721$	−50.9205	−1.89638
$722$	1.00000	0.0372161
$723$	−0.284253	−0.0105715
$724$	7.76323	0.288518
$725$	−12.4163	−0.461131
$726$	4.84888	0.179959
$727$	32.7053	1.21297	0.606486	−	0.795094i	$-0.292579\pi$
$727$	32.7053	1.21297	0.606486	+	0.795094i	$0.292579\pi$
$728$	0	0
$729$	−17.0662	−0.632081
$730$	0	0
$731$	−21.6783	−0.801801
$732$	3.03347	0.112120
$733$	14.1698	0.523375	0.261687	−	0.965153i	$-0.415721\pi$
$733$	14.1698	0.523375	0.261687	+	0.965153i	$0.415721\pi$
$734$	8.22275	0.303507
$735$	0	0
$736$	−6.68730	−0.246497
$737$	0	0
$738$	−29.4401	−1.08370
$739$	34.6823	1.27581	0.637904	−	0.770116i	$-0.279802\pi$
$739$	34.6823	1.27581	0.637904	+	0.770116i	$0.279802\pi$
$740$	0	0
$741$	0	0
$742$	40.6064	1.49071
$743$	37.2422	1.36628	0.683142	−	0.730286i	$-0.260613\pi$
$743$	37.2422	1.36628	0.683142	+	0.730286i	$0.260613\pi$
$744$	−3.84815	−0.141080
$745$	0	0
$746$	28.9338	1.05934
$747$	−21.3552	−0.781344
$748$	0	0
$749$	21.8669	0.798998
$750$	0	0
$751$	−19.1184	−0.697640	−0.348820	−	0.937190i	$-0.613418\pi$
$751$	−19.1184	−0.697640	−0.348820	+	0.937190i	$0.613418\pi$
$752$	2.76323	0.100765
$753$	0.814677	0.0296885
$754$	0	0
$755$	0	0
$756$	12.6017	0.458318
$757$	8.10437	0.294558	0.147279	−	0.989095i	$-0.452948\pi$
$757$	8.10437	0.294558	0.147279	+	0.989095i	$0.452948\pi$
$758$	10.1274	0.367843
$759$	0	0
$760$	0	0
$761$	−26.4782	−0.959835	−0.479918	−	0.877314i	$-0.659333\pi$
$761$	−26.4782	−0.959835	−0.479918	+	0.877314i	$0.659333\pi$
$762$	−3.52646	−0.127750
$763$	31.9241	1.15573
$764$	0.0521848	0.00188798
$765$	0	0
$766$	9.11838	0.329461
$767$	0	0
$768$	−0.440808	−0.0159063
$769$	−45.1850	−1.62941	−0.814706	−	0.579874i	$-0.803102\pi$
$769$	−45.1850	−1.62941	−0.814706	+	0.579874i	$0.803102\pi$
$770$	0	0
$771$	10.9385	0.393940
$772$	−3.84815	−0.138498
$773$	−12.7395	−0.458208	−0.229104	−	0.973402i	$-0.573580\pi$
$773$	−12.7395	−0.458208	−0.229104	+	0.973402i	$0.573580\pi$
$774$	−13.6963	−0.492303
$775$	−43.6488	−1.56791
$776$	−4.88162	−0.175240
$777$	−8.93924	−0.320693
$778$	8.64485	0.309933
$779$	−10.4930	−0.375950
$780$	0	0
$781$	0	0
$782$	29.6970	1.06196
$783$	6.35515	0.227115
$784$	17.2465	0.615946
$785$	0	0
$786$	6.00000	0.214013
$787$	4.36414	0.155565	0.0777824	−	0.996970i	$-0.475216\pi$
$787$	4.36414	0.155565	0.0777824	+	0.996970i	$0.475216\pi$
$788$	−3.35515	−0.119522
$789$	2.74304	0.0976547
$790$	0	0
$791$	76.1238	2.70665
$792$	0	0
$793$	0	0
$794$	−32.5779	−1.15615
$795$	0	0
$796$	−3.94782	−0.139927
$797$	−32.2987	−1.14408	−0.572039	−	0.820226i	$-0.693847\pi$
$797$	−32.2987	−1.14408	−0.572039	+	0.820226i	$0.693847\pi$
$798$	2.17057	0.0768373
$799$	−12.2710	−0.434116
$800$	−5.00000	−0.176777
$801$	−43.3746	−1.53257
$802$	−22.7298	−0.802616
$803$	0	0
$804$	−1.48327	−0.0523107
$805$	0	0
$806$	0	0
$807$	4.01328	0.141274
$808$	−10.7298	−0.377472
$809$	15.9673	0.561379	0.280690	−	0.959799i	$-0.409437\pi$
$809$	15.9673	0.561379	0.280690	+	0.959799i	$0.409437\pi$
$810$	0	0
$811$	−42.8863	−1.50594	−0.752971	−	0.658054i	$-0.771380\pi$
$811$	−42.8863	−1.50594	−0.752971	+	0.658054i	$0.771380\pi$
$812$	12.2278	0.429111
$813$	−3.16084	−0.110856
$814$	0	0
$815$	0	0
$816$	1.95754	0.0685277
$817$	−4.88162	−0.170786
$818$	7.22275	0.252538
$819$	0	0
$820$	0	0
$821$	−13.0335	−0.454871	−0.227436	−	0.973793i	$-0.573034\pi$
$821$	−13.0335	−0.454871	−0.227436	+	0.973793i	$0.573034\pi$
$822$	3.53075	0.123149
$823$	7.00074	0.244030	0.122015	−	0.992528i	$-0.461064\pi$
$823$	7.00074	0.244030	0.122015	+	0.992528i	$0.461064\pi$
$824$	−10.3411	−0.360251
$825$	0	0
$826$	55.9615	1.94715
$827$	−1.05647	−0.0367372	−0.0183686	−	0.999831i	$-0.505847\pi$
$827$	−1.05647	−0.0367372	−0.0183686	+	0.999831i	$0.505847\pi$
$828$	18.7625	0.652042
$829$	47.2177	1.63994	0.819970	−	0.572407i	$-0.193990\pi$
$829$	47.2177	1.63994	0.819970	+	0.572407i	$0.193990\pi$
$830$	0	0
$831$	5.26876	0.182771
$832$	0	0
$833$	−76.5884	−2.65363
$834$	−3.07090	−0.106337
$835$	0	0
$836$	0	0
$837$	22.3411	0.772223
$838$	−8.64485	−0.298631
$839$	11.6308	0.401541	0.200770	−	0.979638i	$-0.435655\pi$
$839$	11.6308	0.401541	0.200770	+	0.979638i	$0.435655\pi$
$840$	0	0
$841$	−22.8334	−0.787358
$842$	−18.1468	−0.625381
$843$	−5.28969	−0.182187
$844$	−24.8244	−0.854491
$845$	0	0
$846$	−7.75277	−0.266546
$847$	−54.1648	−1.86112
$848$	8.24650	0.283186
$849$	−10.9471	−0.375703
$850$	22.2040	0.761592
$851$	−27.5409	−0.944090
$852$	−0.710307	−0.0243347
$853$	−4.32168	−0.147972	−0.0739858	−	0.997259i	$-0.523572\pi$
$853$	−4.32168	−0.147972	−0.0739858	+	0.997259i	$0.523572\pi$
$854$	−33.8856	−1.15954
$855$	0	0
$856$	4.44081	0.151784
$857$	−2.18929	−0.0747846	−0.0373923	−	0.999301i	$-0.511905\pi$
$857$	−2.18929	−0.0747846	−0.0373923	+	0.999301i	$0.511905\pi$
$858$	0	0
$859$	−30.1893	−1.03005	−0.515023	−	0.857177i	$-0.672216\pi$
$859$	−30.1893	−1.03005	−0.515023	+	0.857177i	$0.672216\pi$
$860$	0	0
$861$	−22.7758	−0.776196
$862$	−11.8676	−0.404212
$863$	−14.1893	−0.483009	−0.241504	−	0.970400i	$-0.577641\pi$
$863$	−14.1893	−0.483009	−0.241504	+	0.970400i	$0.577641\pi$
$864$	2.55919	0.0870655
$865$	0	0
$866$	−26.2562	−0.892223
$867$	−1.19934	−0.0407316
$868$	42.9860	1.45904
$869$	0	0
$870$	0	0
$871$	0	0
$872$	6.48327	0.219551
$873$	13.6963	0.463549
$874$	6.68730	0.226201
$875$	0	0
$876$	−4.15112	−0.140253
$877$	2.31343	0.0781191	0.0390596	−	0.999237i	$-0.487564\pi$
$877$	2.31343	0.0781191	0.0390596	+	0.999237i	$0.487564\pi$
$878$	−6.64485	−0.224253
$879$	−12.5117	−0.422009
$880$	0	0
$881$	−11.7772	−0.396785	−0.198393	−	0.980123i	$-0.563572\pi$
$881$	−11.7772	−0.396785	−0.198393	+	0.980123i	$0.563572\pi$
$882$	−48.3883	−1.62932
$883$	−36.4261	−1.22583	−0.612917	−	0.790147i	$-0.710004\pi$
$883$	−36.4261	−1.22583	−0.612917	+	0.790147i	$0.710004\pi$
$884$	0	0
$885$	0	0
$886$	−29.5444	−0.992565
$887$	−7.37461	−0.247615	−0.123808	−	0.992306i	$-0.539511\pi$
$887$	−7.37461	−0.247615	−0.123808	+	0.992306i	$0.539511\pi$
$888$	−1.81542	−0.0609214
$889$	39.3926	1.32118
$890$	0	0
$891$	0	0
$892$	−3.50701	−0.117423
$893$	−2.76323	−0.0924680
$894$	6.00000	0.200670
$895$	0	0
$896$	4.92407	0.164502
$897$	0	0
$898$	−1.61138	−0.0537724
$899$	21.6783	0.723012
$900$	14.0284	0.467615
$901$	−36.6211	−1.22003
$902$	0	0
$903$	−10.5959	−0.352609
$904$	15.4595	0.514176
$905$	0	0
$906$	−5.67832	−0.188649
$907$	−37.6211	−1.24919	−0.624594	−	0.780950i	$-0.714736\pi$
$907$	−37.6211	−1.24919	−0.624594	+	0.780950i	$0.714736\pi$
$908$	−13.3224	−0.442120
$909$	30.1044	0.998499
$910$	0	0
$911$	−53.4120	−1.76962	−0.884810	−	0.465952i	$-0.845712\pi$
$911$	−53.4120	−1.76962	−0.884810	+	0.465952i	$0.845712\pi$
$912$	0.440808	0.0145966
$913$	0	0
$914$	−21.4268	−0.708735
$915$	0	0
$916$	28.1044	0.928595
$917$	−67.0234	−2.21331
$918$	−11.3649	−0.375097
$919$	25.0939	0.827772	0.413886	−	0.910329i	$-0.364171\pi$
$919$	25.0939	0.827772	0.413886	+	0.910329i	$0.364171\pi$
$920$	0	0
$921$	−13.6067	−0.448355
$922$	9.93306	0.327128
$923$	0	0
$924$	0	0
$925$	−20.5919	−0.677058
$926$	−25.7632	−0.846632
$927$	29.0140	0.952945
$928$	2.48327	0.0815172
$929$	34.2332	1.12316	0.561578	−	0.827424i	$-0.310195\pi$
$929$	34.2332	1.12316	0.561578	+	0.827424i	$0.310195\pi$
$930$	0	0
$931$	−17.2465	−0.565231
$932$	30.2228	0.989979
$933$	−1.80495	−0.0590914
$934$	−37.7157	−1.23410
$935$	0	0
$936$	0	0
$937$	−40.3361	−1.31772	−0.658862	−	0.752264i	$-0.728962\pi$
$937$	−40.3361	−1.31772	−0.658862	+	0.752264i	$0.728962\pi$
$938$	16.5689	0.540994
$939$	−0.403051	−0.0131531
$940$	0	0
$941$	−9.93809	−0.323972	−0.161986	−	0.986793i	$-0.551790\pi$
$941$	−9.93809	−0.323972	−0.161986	+	0.986793i	$0.551790\pi$
$942$	−10.9011	−0.355176
$943$	−70.1698	−2.28504
$944$	11.3649	0.369895
$945$	0	0
$946$	0	0
$947$	59.7532	1.94172	0.970859	−	0.239653i	$-0.0770336\pi$
$947$	59.7532	1.94172	0.970859	+	0.239653i	$0.0770336\pi$
$948$	2.47354	0.0803368
$949$	0	0
$950$	5.00000	0.162221
$951$	6.68730	0.216851
$952$	−21.8669	−0.708709
$953$	31.4790	1.01970	0.509852	−	0.860262i	$-0.329700\pi$
$953$	31.4790	1.01970	0.509852	+	0.860262i	$0.329700\pi$
$954$	−23.1371	−0.749091
$955$	0	0
$956$	−11.0759	−0.358221
$957$	0	0
$958$	5.76323	0.186202
$959$	−39.4405	−1.27360
$960$	0	0
$961$	45.2087	1.45835
$962$	0	0
$963$	−12.4595	−0.401503
$964$	0.644846	0.0207691
$965$	0	0
$966$	14.5153	0.467021
$967$	32.3926	1.04168	0.520838	−	0.853656i	$-0.325620\pi$
$967$	32.3926	1.04168	0.520838	+	0.853656i	$0.325620\pi$
$968$	−11.0000	−0.353553
$969$	−1.95754	−0.0628853
$970$	0	0
$971$	−8.89563	−0.285474	−0.142737	−	0.989761i	$-0.545590\pi$
$971$	−8.89563	−0.285474	−0.142737	+	0.989761i	$0.545590\pi$
$972$	−10.8906	−0.349316
$973$	34.3037	1.09973
$974$	30.1893	0.967327
$975$	0	0
$976$	−6.88162	−0.220275
$977$	−8.40808	−0.268998	−0.134499	−	0.990914i	$-0.542943\pi$
$977$	−8.40808	−0.268998	−0.134499	+	0.990914i	$0.542943\pi$
$978$	0.426054	0.0136237
$979$	0	0
$980$	0	0
$981$	−18.1900	−0.580763
$982$	−36.8147	−1.17480
$983$	29.7632	0.949300	0.474650	−	0.880175i	$-0.342575\pi$
$983$	29.7632	0.949300	0.474650	+	0.880175i	$0.342575\pi$
$984$	−4.62539	−0.147452
$985$	0	0
$986$	−11.0277	−0.351194
$987$	−5.99778	−0.190912
$988$	0	0
$989$	−32.6448	−1.03805
$990$	0	0
$991$	18.3606	0.583243	0.291622	−	0.956534i	$-0.405805\pi$
$991$	18.3606	0.583243	0.291622	+	0.956534i	$0.405805\pi$
$992$	8.72976	0.277170
$993$	11.0744	0.351437
$994$	7.93454	0.251668
$995$	0	0
$996$	−3.35515	−0.106312
$997$	−28.2087	−0.893380	−0.446690	−	0.894689i	$-0.647397\pi$
$997$	−28.2087	−0.893380	−0.446690	+	0.894689i	$0.647397\pi$
$998$	20.3886	0.645391
$999$	10.5397	0.333463

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.y.1.2		3
13.4	even	6		494.2.g.c.419.2	yes	6
13.10	even	6		494.2.g.c.191.2	✓	6
13.12	even	2		6422.2.a.o.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.c.191.2	✓	6	13.10	even	6
494.2.g.c.419.2	yes	6	13.4	even	6
6422.2.a.o.1.2		3	13.12	even	2
6422.2.a.y.1.2		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.440808 q^{3} +1.00000 q^{4} -0.440808 q^{6} +4.92407 q^{7} +1.00000 q^{8} -2.80569 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.440808 q^{3} +1.00000 q^{4} -0.440808 q^{6} +4.92407 q^{7} +1.00000 q^{8} -2.80569 q^{9} -0.440808 q^{12} +4.92407 q^{14} +1.00000 q^{16} -4.44081 q^{17} -2.80569 q^{18} -1.00000 q^{19} -2.17057 q^{21} -6.68730 q^{23} -0.440808 q^{24} -5.00000 q^{25} +2.55919 q^{27} +4.92407 q^{28} +2.48327 q^{29} +8.72976 q^{31} +1.00000 q^{32} -4.44081 q^{34} -2.80569 q^{36} +4.11838 q^{37} -1.00000 q^{38} +10.4930 q^{41} -2.17057 q^{42} +4.88162 q^{43} -6.68730 q^{46} +2.76323 q^{47} -0.440808 q^{48} +17.2465 q^{49} -5.00000 q^{50} +1.95754 q^{51} +8.24650 q^{53} +2.55919 q^{54} +4.92407 q^{56} +0.440808 q^{57} +2.48327 q^{58} +11.3649 q^{59} -6.88162 q^{61} +8.72976 q^{62} -13.8154 q^{63} +1.00000 q^{64} +3.36488 q^{67} -4.44081 q^{68} +2.94782 q^{69} +1.61138 q^{71} -2.80569 q^{72} +9.41707 q^{73} +4.11838 q^{74} +2.20404 q^{75} -1.00000 q^{76} -5.61138 q^{79} +7.28895 q^{81} +10.4930 q^{82} +7.61138 q^{83} -2.17057 q^{84} +4.88162 q^{86} -1.09464 q^{87} +15.4595 q^{89} -6.68730 q^{92} -3.84815 q^{93} +2.76323 q^{94} -0.440808 q^{96} -4.88162 q^{97} +17.2465 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 4 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 4 * q^7 + 3 * q^8 + 5 * q^9	\( 3 q + 3 q^{2} + 3 q^{4} + 4 q^{7} + 3 q^{8} + 5 q^{9} + 4 q^{14} + 3 q^{16} - 12 q^{17} + 5 q^{18} - 3 q^{19} + 19 q^{21} - 4 q^{23} - 15 q^{25} + 9 q^{27} + 4 q^{28} - 2 q^{29} + 2 q^{31} + 3 q^{32} - 12 q^{34} + 5 q^{36} + 15 q^{37} - 3 q^{38} + 2 q^{41} + 19 q^{42} + 12 q^{43} - 4 q^{46} + 3 q^{47} + 37 q^{49} - 15 q^{50} + 14 q^{51} + 10 q^{53} + 9 q^{54} + 4 q^{56} - 2 q^{58} + 22 q^{59} - 18 q^{61} + 2 q^{62} - 8 q^{63} + 3 q^{64} - 2 q^{67} - 12 q^{68} + 37 q^{69} - 22 q^{71} + 5 q^{72} - 12 q^{73} + 15 q^{74} - 3 q^{76} + 10 q^{79} - q^{81} + 2 q^{82} - 4 q^{83} + 19 q^{84} + 12 q^{86} + 33 q^{87} - 2 q^{89} - 4 q^{92} + 10 q^{93} + 3 q^{94} - 12 q^{97} + 37 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 4 * q^7 + 3 * q^8 + 5 * q^9 + 4 * q^14 + 3 * q^16 - 12 * q^17 + 5 * q^18 - 3 * q^19 + 19 * q^21 - 4 * q^23 - 15 * q^25 + 9 * q^27 + 4 * q^28 - 2 * q^29 + 2 * q^31 + 3 * q^32 - 12 * q^34 + 5 * q^36 + 15 * q^37 - 3 * q^38 + 2 * q^41 + 19 * q^42 + 12 * q^43 - 4 * q^46 + 3 * q^47 + 37 * q^49 - 15 * q^50 + 14 * q^51 + 10 * q^53 + 9 * q^54 + 4 * q^56 - 2 * q^58 + 22 * q^59 - 18 * q^61 + 2 * q^62 - 8 * q^63 + 3 * q^64 - 2 * q^67 - 12 * q^68 + 37 * q^69 - 22 * q^71 + 5 * q^72 - 12 * q^73 + 15 * q^74 - 3 * q^76 + 10 * q^79 - q^81 + 2 * q^82 - 4 * q^83 + 19 * q^84 + 12 * q^86 + 33 * q^87 - 2 * q^89 - 4 * q^92 + 10 * q^93 + 3 * q^94 - 12 * q^97 + 37 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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