LMFDB - Embedded newform 6160.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3080)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{9} +1.00000 q^{11} -0.585786 q^{13} -1.41421 q^{15} -6.24264 q^{17} +5.65685 q^{19} +1.41421 q^{21} +2.00000 q^{23} +1.00000 q^{25} +5.65685 q^{27} -8.82843 q^{29} +7.41421 q^{31} -1.41421 q^{33} -1.00000 q^{35} +2.82843 q^{37} +0.828427 q^{39} -4.24264 q^{41} +2.00000 q^{43} -1.00000 q^{45} -0.242641 q^{47} +1.00000 q^{49} +8.82843 q^{51} +12.4853 q^{53} +1.00000 q^{55} -8.00000 q^{57} -7.41421 q^{59} -3.75736 q^{61} +1.00000 q^{63} -0.585786 q^{65} +13.3137 q^{67} -2.82843 q^{69} -12.4853 q^{71} -1.75736 q^{73} -1.41421 q^{75} -1.00000 q^{77} -12.8284 q^{79} -5.00000 q^{81} -6.34315 q^{83} -6.24264 q^{85} +12.4853 q^{87} -8.82843 q^{89} +0.585786 q^{91} -10.4853 q^{93} +5.65685 q^{95} +13.3137 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^9	$ 2 q + 2 q^{5} - 2 q^{7} - 2 q^{9} + 2 q^{11} - 4 q^{13} - 4 q^{17} + 4 q^{23} + 2 q^{25} - 12 q^{29} + 12 q^{31} - 2 q^{35} - 4 q^{39} + 4 q^{43} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 12 q^{51} + 8 q^{53} + 2 q^{55} - 16 q^{57} - 12 q^{59} - 16 q^{61} + 2 q^{63} - 4 q^{65} + 4 q^{67} - 8 q^{71} - 12 q^{73} - 2 q^{77} - 20 q^{79} - 10 q^{81} - 24 q^{83} - 4 q^{85} + 8 q^{87} - 12 q^{89} + 4 q^{91} - 4 q^{93} + 4 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^9 + 2 * q^11 - 4 * q^13 - 4 * q^17 + 4 * q^23 + 2 * q^25 - 12 * q^29 + 12 * q^31 - 2 * q^35 - 4 * q^39 + 4 * q^43 - 2 * q^45 + 8 * q^47 + 2 * q^49 + 12 * q^51 + 8 * q^53 + 2 * q^55 - 16 * q^57 - 12 * q^59 - 16 * q^61 + 2 * q^63 - 4 * q^65 + 4 * q^67 - 8 * q^71 - 12 * q^73 - 2 * q^77 - 20 * q^79 - 10 * q^81 - 24 * q^83 - 4 * q^85 + 8 * q^87 - 12 * q^89 + 4 * q^91 - 4 * q^93 + 4 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.585786	−0.162468	−0.0812340	−	0.996695i	$-0.525886\pi$
$13$	−0.585786	−0.162468	−0.0812340	+	0.996695i	$0.525886\pi$
$14$	0	0
$15$	−1.41421	−0.365148
$16$	0	0
$17$	−6.24264	−1.51406	−0.757031	−	0.653379i	$-0.773351\pi$
$17$	−6.24264	−1.51406	−0.757031	+	0.653379i	$0.773351\pi$
$18$	0	0
$19$	5.65685	1.29777	0.648886	−	0.760886i	$-0.275235\pi$
$19$	5.65685	1.29777	0.648886	+	0.760886i	$0.275235\pi$
$20$	0	0
$21$	1.41421	0.308607
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−8.82843	−1.63940	−0.819699	−	0.572795i	$-0.805859\pi$
$29$	−8.82843	−1.63940	−0.819699	+	0.572795i	$0.805859\pi$
$30$	0	0
$31$	7.41421	1.33163	0.665816	−	0.746116i	$-0.268084\pi$
$31$	7.41421	1.33163	0.665816	+	0.746116i	$0.268084\pi$
$32$	0	0
$33$	−1.41421	−0.246183
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	2.82843	0.464991	0.232495	−	0.972598i	$-0.425311\pi$
$37$	2.82843	0.464991	0.232495	+	0.972598i	$0.425311\pi$
$38$	0	0
$39$	0.828427	0.132655
$40$	0	0
$41$	−4.24264	−0.662589	−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264	−0.662589	−0.331295	+	0.943527i	$0.607485\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−0.242641	−0.0353928	−0.0176964	−	0.999843i	$-0.505633\pi$
$47$	−0.242641	−0.0353928	−0.0176964	+	0.999843i	$0.505633\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.82843	1.23623
$52$	0	0
$53$	12.4853	1.71499	0.857493	−	0.514496i	$-0.172021\pi$
$53$	12.4853	1.71499	0.857493	+	0.514496i	$0.172021\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	−7.41421	−0.965248	−0.482624	−	0.875828i	$-0.660316\pi$
$59$	−7.41421	−0.965248	−0.482624	+	0.875828i	$0.660316\pi$
$60$	0	0
$61$	−3.75736	−0.481081	−0.240540	−	0.970639i	$-0.577325\pi$
$61$	−3.75736	−0.481081	−0.240540	+	0.970639i	$0.577325\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−0.585786	−0.0726579
$66$	0	0
$67$	13.3137	1.62653	0.813264	−	0.581895i	$-0.197688\pi$
$67$	13.3137	1.62653	0.813264	+	0.581895i	$0.197688\pi$
$68$	0	0
$69$	−2.82843	−0.340503
$70$	0	0
$71$	−12.4853	−1.48173	−0.740865	−	0.671654i	$-0.765584\pi$
$71$	−12.4853	−1.48173	−0.740865	+	0.671654i	$0.765584\pi$
$72$	0	0
$73$	−1.75736	−0.205683	−0.102842	−	0.994698i	$-0.532794\pi$
$73$	−1.75736	−0.205683	−0.102842	+	0.994698i	$0.532794\pi$
$74$	0	0
$75$	−1.41421	−0.163299
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−12.8284	−1.44331	−0.721655	−	0.692252i	$-0.756618\pi$
$79$	−12.8284	−1.44331	−0.721655	+	0.692252i	$0.756618\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−6.34315	−0.696251	−0.348125	−	0.937448i	$-0.613182\pi$
$83$	−6.34315	−0.696251	−0.348125	+	0.937448i	$0.613182\pi$
$84$	0	0
$85$	−6.24264	−0.677109
$86$	0	0
$87$	12.4853	1.33856
$88$	0	0
$89$	−8.82843	−0.935811	−0.467906	−	0.883778i	$-0.654991\pi$
$89$	−8.82843	−0.935811	−0.467906	+	0.883778i	$0.654991\pi$
$90$	0	0
$91$	0.585786	0.0614071
$92$	0	0
$93$	−10.4853	−1.08727
$94$	0	0
$95$	5.65685	0.580381
$96$	0	0
$97$	13.3137	1.35180	0.675901	−	0.736992i	$-0.263755\pi$
$97$	13.3137	1.35180	0.675901	+	0.736992i	$0.263755\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−1.41421	−0.140720	−0.0703598	−	0.997522i	$-0.522415\pi$
$101$	−1.41421	−0.140720	−0.0703598	+	0.997522i	$0.522415\pi$
$102$	0	0
$103$	10.5858	1.04305	0.521524	−	0.853236i	$-0.325364\pi$
$103$	10.5858	1.04305	0.521524	+	0.853236i	$0.325364\pi$
$104$	0	0
$105$	1.41421	0.138013
$106$	0	0
$107$	−11.3137	−1.09374	−0.546869	−	0.837218i	$-0.684180\pi$
$107$	−11.3137	−1.09374	−0.546869	+	0.837218i	$0.684180\pi$
$108$	0	0
$109$	−7.17157	−0.686912	−0.343456	−	0.939169i	$-0.611598\pi$
$109$	−7.17157	−0.686912	−0.343456	+	0.939169i	$0.611598\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	13.3137	1.25245	0.626224	−	0.779643i	$-0.284599\pi$
$113$	13.3137	1.25245	0.626224	+	0.779643i	$0.284599\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	0	0
$117$	0.585786	0.0541560
$118$	0	0
$119$	6.24264	0.572262
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.65685	0.147022	0.0735110	−	0.997294i	$-0.476580\pi$
$127$	1.65685	0.147022	0.0735110	+	0.997294i	$0.476580\pi$
$128$	0	0
$129$	−2.82843	−0.249029
$130$	0	0
$131$	−15.3137	−1.33796	−0.668982	−	0.743278i	$-0.733270\pi$
$131$	−15.3137	−1.33796	−0.668982	+	0.743278i	$0.733270\pi$
$132$	0	0
$133$	−5.65685	−0.490511
$134$	0	0
$135$	5.65685	0.486864
$136$	0	0
$137$	−20.4853	−1.75018	−0.875088	−	0.483964i	$-0.839197\pi$
$137$	−20.4853	−1.75018	−0.875088	+	0.483964i	$0.839197\pi$
$138$	0	0
$139$	8.48528	0.719712	0.359856	−	0.933008i	$-0.382826\pi$
$139$	8.48528	0.719712	0.359856	+	0.933008i	$0.382826\pi$
$140$	0	0
$141$	0.343146	0.0288981
$142$	0	0
$143$	−0.585786	−0.0489859
$144$	0	0
$145$	−8.82843	−0.733161
$146$	0	0
$147$	−1.41421	−0.116642
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−7.17157	−0.583614	−0.291807	−	0.956477i	$-0.594257\pi$
$151$	−7.17157	−0.583614	−0.291807	+	0.956477i	$0.594257\pi$
$152$	0	0
$153$	6.24264	0.504688
$154$	0	0
$155$	7.41421	0.595524
$156$	0	0
$157$	12.1421	0.969048	0.484524	−	0.874778i	$-0.338993\pi$
$157$	12.1421	0.969048	0.484524	+	0.874778i	$0.338993\pi$
$158$	0	0
$159$	−17.6569	−1.40028
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−21.3137	−1.66942	−0.834709	−	0.550691i	$-0.814364\pi$
$163$	−21.3137	−1.66942	−0.834709	+	0.550691i	$0.814364\pi$
$164$	0	0
$165$	−1.41421	−0.110096
$166$	0	0
$167$	−6.14214	−0.475293	−0.237646	−	0.971352i	$-0.576376\pi$
$167$	−6.14214	−0.475293	−0.237646	+	0.971352i	$0.576376\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	−5.65685	−0.432590
$172$	0	0
$173$	−11.4142	−0.867807	−0.433903	−	0.900959i	$-0.642864\pi$
$173$	−11.4142	−0.867807	−0.433903	+	0.900959i	$0.642864\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	10.4853	0.788122
$178$	0	0
$179$	11.3137	0.845626	0.422813	−	0.906217i	$-0.361043\pi$
$179$	11.3137	0.845626	0.422813	+	0.906217i	$0.361043\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	5.31371	0.392801
$184$	0	0
$185$	2.82843	0.207950
$186$	0	0
$187$	−6.24264	−0.456507
$188$	0	0
$189$	−5.65685	−0.411476
$190$	0	0
$191$	9.65685	0.698745	0.349373	−	0.936984i	$-0.386395\pi$
$191$	9.65685	0.698745	0.349373	+	0.936984i	$0.386395\pi$
$192$	0	0
$193$	4.48528	0.322858	0.161429	−	0.986884i	$-0.448390\pi$
$193$	4.48528	0.322858	0.161429	+	0.986884i	$0.448390\pi$
$194$	0	0
$195$	0.828427	0.0593249
$196$	0	0
$197$	−13.1716	−0.938436	−0.469218	−	0.883082i	$-0.655464\pi$
$197$	−13.1716	−0.938436	−0.469218	+	0.883082i	$0.655464\pi$
$198$	0	0
$199$	−9.55635	−0.677432	−0.338716	−	0.940889i	$-0.609993\pi$
$199$	−9.55635	−0.677432	−0.338716	+	0.940889i	$0.609993\pi$
$200$	0	0
$201$	−18.8284	−1.32805
$202$	0	0
$203$	8.82843	0.619634
$204$	0	0
$205$	−4.24264	−0.296319
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	5.65685	0.391293
$210$	0	0
$211$	23.3137	1.60498	0.802491	−	0.596664i	$-0.203508\pi$
$211$	23.3137	1.60498	0.802491	+	0.596664i	$0.203508\pi$
$212$	0	0
$213$	17.6569	1.20983
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	−7.41421	−0.503310
$218$	0	0
$219$	2.48528	0.167940
$220$	0	0
$221$	3.65685	0.245987
$222$	0	0
$223$	4.24264	0.284108	0.142054	−	0.989859i	$-0.454629\pi$
$223$	4.24264	0.284108	0.142054	+	0.989859i	$0.454629\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	−12.9706	−0.860886	−0.430443	−	0.902618i	$-0.641643\pi$
$227$	−12.9706	−0.860886	−0.430443	+	0.902618i	$0.641643\pi$
$228$	0	0
$229$	−4.82843	−0.319071	−0.159536	−	0.987192i	$-0.551000\pi$
$229$	−4.82843	−0.319071	−0.159536	+	0.987192i	$0.551000\pi$
$230$	0	0
$231$	1.41421	0.0930484
$232$	0	0
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	−0.242641	−0.0158281
$236$	0	0
$237$	18.1421	1.17846
$238$	0	0
$239$	−5.65685	−0.365911	−0.182956	−	0.983121i	$-0.558567\pi$
$239$	−5.65685	−0.365911	−0.182956	+	0.983121i	$0.558567\pi$
$240$	0	0
$241$	−3.27208	−0.210773	−0.105387	−	0.994431i	$-0.533608\pi$
$241$	−3.27208	−0.210773	−0.105387	+	0.994431i	$0.533608\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−3.31371	−0.210846
$248$	0	0
$249$	8.97056	0.568486
$250$	0	0
$251$	9.07107	0.572561	0.286280	−	0.958146i	$-0.407581\pi$
$251$	9.07107	0.572561	0.286280	+	0.958146i	$0.407581\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	8.82843	0.552858
$256$	0	0
$257$	−4.82843	−0.301189	−0.150595	−	0.988596i	$-0.548119\pi$
$257$	−4.82843	−0.301189	−0.150595	+	0.988596i	$0.548119\pi$
$258$	0	0
$259$	−2.82843	−0.175750
$260$	0	0
$261$	8.82843	0.546466
$262$	0	0
$263$	−1.65685	−0.102166	−0.0510830	−	0.998694i	$-0.516267\pi$
$263$	−1.65685	−0.102166	−0.0510830	+	0.998694i	$0.516267\pi$
$264$	0	0
$265$	12.4853	0.766965
$266$	0	0
$267$	12.4853	0.764087
$268$	0	0
$269$	−9.31371	−0.567867	−0.283933	−	0.958844i	$-0.591639\pi$
$269$	−9.31371	−0.567867	−0.283933	+	0.958844i	$0.591639\pi$
$270$	0	0
$271$	−6.82843	−0.414797	−0.207399	−	0.978256i	$-0.566500\pi$
$271$	−6.82843	−0.414797	−0.207399	+	0.978256i	$0.566500\pi$
$272$	0	0
$273$	−0.828427	−0.0501387
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−31.7990	−1.91062	−0.955308	−	0.295612i	$-0.904476\pi$
$277$	−31.7990	−1.91062	−0.955308	+	0.295612i	$0.904476\pi$
$278$	0	0
$279$	−7.41421	−0.443877
$280$	0	0
$281$	−14.4853	−0.864119	−0.432060	−	0.901845i	$-0.642213\pi$
$281$	−14.4853	−0.864119	−0.432060	+	0.901845i	$0.642213\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	4.24264	0.250435
$288$	0	0
$289$	21.9706	1.29239
$290$	0	0
$291$	−18.8284	−1.10374
$292$	0	0
$293$	−3.89949	−0.227811	−0.113905	−	0.993492i	$-0.536336\pi$
$293$	−3.89949	−0.227811	−0.113905	+	0.993492i	$0.536336\pi$
$294$	0	0
$295$	−7.41421	−0.431672
$296$	0	0
$297$	5.65685	0.328244
$298$	0	0
$299$	−1.17157	−0.0677538
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	2.00000	0.114897
$304$	0	0
$305$	−3.75736	−0.215146
$306$	0	0
$307$	18.8284	1.07460	0.537298	−	0.843393i	$-0.319445\pi$
$307$	18.8284	1.07460	0.537298	+	0.843393i	$0.319445\pi$
$308$	0	0
$309$	−14.9706	−0.851646
$310$	0	0
$311$	−11.2132	−0.635842	−0.317921	−	0.948117i	$-0.602985\pi$
$311$	−11.2132	−0.635842	−0.317921	+	0.948117i	$0.602985\pi$
$312$	0	0
$313$	13.3137	0.752535	0.376268	−	0.926511i	$-0.377207\pi$
$313$	13.3137	0.752535	0.376268	+	0.926511i	$0.377207\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−18.9706	−1.06549	−0.532746	−	0.846275i	$-0.678840\pi$
$317$	−18.9706	−1.06549	−0.532746	+	0.846275i	$0.678840\pi$
$318$	0	0
$319$	−8.82843	−0.494297
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	−35.3137	−1.96491
$324$	0	0
$325$	−0.585786	−0.0324936
$326$	0	0
$327$	10.1421	0.560861
$328$	0	0
$329$	0.242641	0.0133772
$330$	0	0
$331$	−23.3137	−1.28144	−0.640719	−	0.767776i	$-0.721363\pi$
$331$	−23.3137	−1.28144	−0.640719	+	0.767776i	$0.721363\pi$
$332$	0	0
$333$	−2.82843	−0.154997
$334$	0	0
$335$	13.3137	0.727406
$336$	0	0
$337$	−22.6274	−1.23259	−0.616297	−	0.787514i	$-0.711368\pi$
$337$	−22.6274	−1.23259	−0.616297	+	0.787514i	$0.711368\pi$
$338$	0	0
$339$	−18.8284	−1.02262
$340$	0	0
$341$	7.41421	0.401502
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.82843	−0.152277
$346$	0	0
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	0	0
$349$	−18.3848	−0.984115	−0.492057	−	0.870563i	$-0.663755\pi$
$349$	−18.3848	−0.984115	−0.492057	+	0.870563i	$0.663755\pi$
$350$	0	0
$351$	−3.31371	−0.176873
$352$	0	0
$353$	12.6274	0.672090	0.336045	−	0.941846i	$-0.390911\pi$
$353$	12.6274	0.672090	0.336045	+	0.941846i	$0.390911\pi$
$354$	0	0
$355$	−12.4853	−0.662650
$356$	0	0
$357$	−8.82843	−0.467250
$358$	0	0
$359$	16.1421	0.851949	0.425975	−	0.904735i	$-0.359931\pi$
$359$	16.1421	0.851949	0.425975	+	0.904735i	$0.359931\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	−1.41421	−0.0742270
$364$	0	0
$365$	−1.75736	−0.0919844
$366$	0	0
$367$	32.7279	1.70838	0.854192	−	0.519958i	$-0.174052\pi$
$367$	32.7279	1.70838	0.854192	+	0.519958i	$0.174052\pi$
$368$	0	0
$369$	4.24264	0.220863
$370$	0	0
$371$	−12.4853	−0.648204
$372$	0	0
$373$	−17.1716	−0.889110	−0.444555	−	0.895751i	$-0.646638\pi$
$373$	−17.1716	−0.889110	−0.444555	+	0.895751i	$0.646638\pi$
$374$	0	0
$375$	−1.41421	−0.0730297
$376$	0	0
$377$	5.17157	0.266350
$378$	0	0
$379$	−38.1421	−1.95923	−0.979615	−	0.200884i	$-0.935618\pi$
$379$	−38.1421	−1.95923	−0.979615	+	0.200884i	$0.935618\pi$
$380$	0	0
$381$	−2.34315	−0.120043
$382$	0	0
$383$	20.7279	1.05915	0.529574	−	0.848264i	$-0.322352\pi$
$383$	20.7279	1.05915	0.529574	+	0.848264i	$0.322352\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	−26.2843	−1.33267	−0.666333	−	0.745655i	$-0.732137\pi$
$389$	−26.2843	−1.33267	−0.666333	+	0.745655i	$0.732137\pi$
$390$	0	0
$391$	−12.4853	−0.631408
$392$	0	0
$393$	21.6569	1.09244
$394$	0	0
$395$	−12.8284	−0.645468
$396$	0	0
$397$	−32.1421	−1.61317	−0.806584	−	0.591120i	$-0.798686\pi$
$397$	−32.1421	−1.61317	−0.806584	+	0.591120i	$0.798686\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−1.31371	−0.0656035	−0.0328017	−	0.999462i	$-0.510443\pi$
$401$	−1.31371	−0.0656035	−0.0328017	+	0.999462i	$0.510443\pi$
$402$	0	0
$403$	−4.34315	−0.216347
$404$	0	0
$405$	−5.00000	−0.248452
$406$	0	0
$407$	2.82843	0.140200
$408$	0	0
$409$	−20.2426	−1.00093	−0.500467	−	0.865756i	$-0.666838\pi$
$409$	−20.2426	−1.00093	−0.500467	+	0.865756i	$0.666838\pi$
$410$	0	0
$411$	28.9706	1.42901
$412$	0	0
$413$	7.41421	0.364830
$414$	0	0
$415$	−6.34315	−0.311373
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	4.38478	0.214210	0.107105	−	0.994248i	$-0.465842\pi$
$419$	4.38478	0.214210	0.107105	+	0.994248i	$0.465842\pi$
$420$	0	0
$421$	−10.6274	−0.517949	−0.258974	−	0.965884i	$-0.583384\pi$
$421$	−10.6274	−0.517949	−0.258974	+	0.965884i	$0.583384\pi$
$422$	0	0
$423$	0.242641	0.0117976
$424$	0	0
$425$	−6.24264	−0.302813
$426$	0	0
$427$	3.75736	0.181831
$428$	0	0
$429$	0.828427	0.0399968
$430$	0	0
$431$	20.1421	0.970213	0.485106	−	0.874455i	$-0.338781\pi$
$431$	20.1421	0.970213	0.485106	+	0.874455i	$0.338781\pi$
$432$	0	0
$433$	−23.4558	−1.12722	−0.563608	−	0.826042i	$-0.690587\pi$
$433$	−23.4558	−1.12722	−0.563608	+	0.826042i	$0.690587\pi$
$434$	0	0
$435$	12.4853	0.598623
$436$	0	0
$437$	11.3137	0.541208
$438$	0	0
$439$	3.79899	0.181316	0.0906579	−	0.995882i	$-0.471103\pi$
$439$	3.79899	0.181316	0.0906579	+	0.995882i	$0.471103\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	−33.1127	−1.57323	−0.786616	−	0.617443i	$-0.788169\pi$
$443$	−33.1127	−1.57323	−0.786616	+	0.617443i	$0.788169\pi$
$444$	0	0
$445$	−8.82843	−0.418508
$446$	0	0
$447$	−8.48528	−0.401340
$448$	0	0
$449$	29.6569	1.39959	0.699797	−	0.714342i	$-0.253274\pi$
$449$	29.6569	1.39959	0.699797	+	0.714342i	$0.253274\pi$
$450$	0	0
$451$	−4.24264	−0.199778
$452$	0	0
$453$	10.1421	0.476519
$454$	0	0
$455$	0.585786	0.0274621
$456$	0	0
$457$	−20.4853	−0.958261	−0.479131	−	0.877744i	$-0.659048\pi$
$457$	−20.4853	−0.958261	−0.479131	+	0.877744i	$0.659048\pi$
$458$	0	0
$459$	−35.3137	−1.64830
$460$	0	0
$461$	32.7279	1.52429	0.762146	−	0.647406i	$-0.224146\pi$
$461$	32.7279	1.52429	0.762146	+	0.647406i	$0.224146\pi$
$462$	0	0
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	0	0
$465$	−10.4853	−0.486243
$466$	0	0
$467$	−38.8701	−1.79869	−0.899346	−	0.437238i	$-0.855957\pi$
$467$	−38.8701	−1.79869	−0.899346	+	0.437238i	$0.855957\pi$
$468$	0	0
$469$	−13.3137	−0.614770
$470$	0	0
$471$	−17.1716	−0.791224
$472$	0	0
$473$	2.00000	0.0919601
$474$	0	0
$475$	5.65685	0.259554
$476$	0	0
$477$	−12.4853	−0.571662
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−1.65685	−0.0755461
$482$	0	0
$483$	2.82843	0.128698
$484$	0	0
$485$	13.3137	0.604544
$486$	0	0
$487$	−26.2843	−1.19105	−0.595527	−	0.803335i	$-0.703057\pi$
$487$	−26.2843	−1.19105	−0.595527	+	0.803335i	$0.703057\pi$
$488$	0	0
$489$	30.1421	1.36307
$490$	0	0
$491$	−17.5147	−0.790428	−0.395214	−	0.918589i	$-0.629330\pi$
$491$	−17.5147	−0.790428	−0.395214	+	0.918589i	$0.629330\pi$
$492$	0	0
$493$	55.1127	2.48215
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	12.4853	0.560041
$498$	0	0
$499$	−39.7990	−1.78165	−0.890824	−	0.454349i	$-0.849872\pi$
$499$	−39.7990	−1.78165	−0.890824	+	0.454349i	$0.849872\pi$
$500$	0	0
$501$	8.68629	0.388075
$502$	0	0
$503$	−6.34315	−0.282827	−0.141413	−	0.989951i	$-0.545165\pi$
$503$	−6.34315	−0.282827	−0.141413	+	0.989951i	$0.545165\pi$
$504$	0	0
$505$	−1.41421	−0.0629317
$506$	0	0
$507$	17.8995	0.794944
$508$	0	0
$509$	−1.51472	−0.0671387	−0.0335694	−	0.999436i	$-0.510687\pi$
$509$	−1.51472	−0.0671387	−0.0335694	+	0.999436i	$0.510687\pi$
$510$	0	0
$511$	1.75736	0.0777410
$512$	0	0
$513$	32.0000	1.41283
$514$	0	0
$515$	10.5858	0.466465
$516$	0	0
$517$	−0.242641	−0.0106713
$518$	0	0
$519$	16.1421	0.708561
$520$	0	0
$521$	14.6863	0.643418	0.321709	−	0.946839i	$-0.395743\pi$
$521$	14.6863	0.643418	0.321709	+	0.946839i	$0.395743\pi$
$522$	0	0
$523$	32.7696	1.43291	0.716456	−	0.697632i	$-0.245763\pi$
$523$	32.7696	1.43291	0.716456	+	0.697632i	$0.245763\pi$
$524$	0	0
$525$	1.41421	0.0617213
$526$	0	0
$527$	−46.2843	−2.01617
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	7.41421	0.321749
$532$	0	0
$533$	2.48528	0.107649
$534$	0	0
$535$	−11.3137	−0.489134
$536$	0	0
$537$	−16.0000	−0.690451
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−21.3137	−0.916348	−0.458174	−	0.888863i	$-0.651496\pi$
$541$	−21.3137	−0.916348	−0.458174	+	0.888863i	$0.651496\pi$
$542$	0	0
$543$	8.48528	0.364138
$544$	0	0
$545$	−7.17157	−0.307196
$546$	0	0
$547$	−7.31371	−0.312712	−0.156356	−	0.987701i	$-0.549975\pi$
$547$	−7.31371	−0.312712	−0.156356	+	0.987701i	$0.549975\pi$
$548$	0	0
$549$	3.75736	0.160360
$550$	0	0
$551$	−49.9411	−2.12756
$552$	0	0
$553$	12.8284	0.545520
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	−35.7990	−1.51685	−0.758426	−	0.651759i	$-0.774031\pi$
$557$	−35.7990	−1.51685	−0.758426	+	0.651759i	$0.774031\pi$
$558$	0	0
$559$	−1.17157	−0.0495523
$560$	0	0
$561$	8.82843	0.372736
$562$	0	0
$563$	36.4853	1.53767	0.768836	−	0.639446i	$-0.220836\pi$
$563$	36.4853	1.53767	0.768836	+	0.639446i	$0.220836\pi$
$564$	0	0
$565$	13.3137	0.560112
$566$	0	0
$567$	5.00000	0.209980
$568$	0	0
$569$	39.1716	1.64216	0.821079	−	0.570815i	$-0.193373\pi$
$569$	39.1716	1.64216	0.821079	+	0.570815i	$0.193373\pi$
$570$	0	0
$571$	−18.6274	−0.779533	−0.389767	−	0.920914i	$-0.627444\pi$
$571$	−18.6274	−0.779533	−0.389767	+	0.920914i	$0.627444\pi$
$572$	0	0
$573$	−13.6569	−0.570523
$574$	0	0
$575$	2.00000	0.0834058
$576$	0	0
$577$	−44.6274	−1.85786	−0.928932	−	0.370251i	$-0.879272\pi$
$577$	−44.6274	−1.85786	−0.928932	+	0.370251i	$0.879272\pi$
$578$	0	0
$579$	−6.34315	−0.263612
$580$	0	0
$581$	6.34315	0.263158
$582$	0	0
$583$	12.4853	0.517088
$584$	0	0
$585$	0.585786	0.0242193
$586$	0	0
$587$	12.0416	0.497011	0.248506	−	0.968630i	$-0.420061\pi$
$587$	12.0416	0.497011	0.248506	+	0.968630i	$0.420061\pi$
$588$	0	0
$589$	41.9411	1.72815
$590$	0	0
$591$	18.6274	0.766230
$592$	0	0
$593$	−3.89949	−0.160133	−0.0800665	−	0.996790i	$-0.525513\pi$
$593$	−3.89949	−0.160133	−0.0800665	+	0.996790i	$0.525513\pi$
$594$	0	0
$595$	6.24264	0.255923
$596$	0	0
$597$	13.5147	0.553121
$598$	0	0
$599$	11.7990	0.482094	0.241047	−	0.970513i	$-0.422509\pi$
$599$	11.7990	0.482094	0.241047	+	0.970513i	$0.422509\pi$
$600$	0	0
$601$	8.04163	0.328025	0.164012	−	0.986458i	$-0.447556\pi$
$601$	8.04163	0.328025	0.164012	+	0.986458i	$0.447556\pi$
$602$	0	0
$603$	−13.3137	−0.542176
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	1.85786	0.0754084	0.0377042	−	0.999289i	$-0.487996\pi$
$607$	1.85786	0.0754084	0.0377042	+	0.999289i	$0.487996\pi$
$608$	0	0
$609$	−12.4853	−0.505929
$610$	0	0
$611$	0.142136	0.00575019
$612$	0	0
$613$	19.3137	0.780073	0.390037	−	0.920799i	$-0.372462\pi$
$613$	19.3137	0.780073	0.390037	+	0.920799i	$0.372462\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	−20.4853	−0.824706	−0.412353	−	0.911024i	$-0.635293\pi$
$617$	−20.4853	−0.824706	−0.412353	+	0.911024i	$0.635293\pi$
$618$	0	0
$619$	−41.0711	−1.65079	−0.825393	−	0.564559i	$-0.809046\pi$
$619$	−41.0711	−1.65079	−0.825393	+	0.564559i	$0.809046\pi$
$620$	0	0
$621$	11.3137	0.454003
$622$	0	0
$623$	8.82843	0.353703
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	−17.6569	−0.704025
$630$	0	0
$631$	35.3137	1.40582	0.702908	−	0.711281i	$-0.251884\pi$
$631$	35.3137	1.40582	0.702908	+	0.711281i	$0.251884\pi$
$632$	0	0
$633$	−32.9706	−1.31046
$634$	0	0
$635$	1.65685	0.0657503
$636$	0	0
$637$	−0.585786	−0.0232097
$638$	0	0
$639$	12.4853	0.493910
$640$	0	0
$641$	−31.3137	−1.23682	−0.618409	−	0.785857i	$-0.712222\pi$
$641$	−31.3137	−1.23682	−0.618409	+	0.785857i	$0.712222\pi$
$642$	0	0
$643$	−18.1005	−0.713814	−0.356907	−	0.934140i	$-0.616169\pi$
$643$	−18.1005	−0.713814	−0.356907	+	0.934140i	$0.616169\pi$
$644$	0	0
$645$	−2.82843	−0.111369
$646$	0	0
$647$	38.8701	1.52814	0.764070	−	0.645134i	$-0.223198\pi$
$647$	38.8701	1.52814	0.764070	+	0.645134i	$0.223198\pi$
$648$	0	0
$649$	−7.41421	−0.291033
$650$	0	0
$651$	10.4853	0.410951
$652$	0	0
$653$	17.3137	0.677538	0.338769	−	0.940870i	$-0.389990\pi$
$653$	17.3137	0.677538	0.338769	+	0.940870i	$0.389990\pi$
$654$	0	0
$655$	−15.3137	−0.598356
$656$	0	0
$657$	1.75736	0.0685611
$658$	0	0
$659$	−9.51472	−0.370641	−0.185320	−	0.982678i	$-0.559332\pi$
$659$	−9.51472	−0.370641	−0.185320	+	0.982678i	$0.559332\pi$
$660$	0	0
$661$	−26.4853	−1.03016	−0.515079	−	0.857143i	$-0.672237\pi$
$661$	−26.4853	−1.03016	−0.515079	+	0.857143i	$0.672237\pi$
$662$	0	0
$663$	−5.17157	−0.200847
$664$	0	0
$665$	−5.65685	−0.219363
$666$	0	0
$667$	−17.6569	−0.683676
$668$	0	0
$669$	−6.00000	−0.231973
$670$	0	0
$671$	−3.75736	−0.145051
$672$	0	0
$673$	35.3137	1.36124	0.680622	−	0.732635i	$-0.261710\pi$
$673$	35.3137	1.36124	0.680622	+	0.732635i	$0.261710\pi$
$674$	0	0
$675$	5.65685	0.217732
$676$	0	0
$677$	0.384776	0.0147882	0.00739408	−	0.999973i	$-0.497646\pi$
$677$	0.384776	0.0147882	0.00739408	+	0.999973i	$0.497646\pi$
$678$	0	0
$679$	−13.3137	−0.510933
$680$	0	0
$681$	18.3431	0.702911
$682$	0	0
$683$	11.4558	0.438346	0.219173	−	0.975686i	$-0.429664\pi$
$683$	11.4558	0.438346	0.219173	+	0.975686i	$0.429664\pi$
$684$	0	0
$685$	−20.4853	−0.782702
$686$	0	0
$687$	6.82843	0.260521
$688$	0	0
$689$	−7.31371	−0.278630
$690$	0	0
$691$	5.75736	0.219020	0.109510	−	0.993986i	$-0.465072\pi$
$691$	5.75736	0.219020	0.109510	+	0.993986i	$0.465072\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	8.48528	0.321865
$696$	0	0
$697$	26.4853	1.00320
$698$	0	0
$699$	−5.65685	−0.213962
$700$	0	0
$701$	−23.4558	−0.885915	−0.442958	−	0.896543i	$-0.646071\pi$
$701$	−23.4558	−0.885915	−0.442958	+	0.896543i	$0.646071\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	0	0
$705$	0.343146	0.0129236
$706$	0	0
$707$	1.41421	0.0531870
$708$	0	0
$709$	2.68629	0.100886	0.0504429	−	0.998727i	$-0.483937\pi$
$709$	2.68629	0.100886	0.0504429	+	0.998727i	$0.483937\pi$
$710$	0	0
$711$	12.8284	0.481104
$712$	0	0
$713$	14.8284	0.555329
$714$	0	0
$715$	−0.585786	−0.0219072
$716$	0	0
$717$	8.00000	0.298765
$718$	0	0
$719$	−36.3848	−1.35692	−0.678462	−	0.734636i	$-0.737353\pi$
$719$	−36.3848	−1.35692	−0.678462	+	0.734636i	$0.737353\pi$
$720$	0	0
$721$	−10.5858	−0.394235
$722$	0	0
$723$	4.62742	0.172095
$724$	0	0
$725$	−8.82843	−0.327880
$726$	0	0
$727$	21.8995	0.812207	0.406104	−	0.913827i	$-0.366887\pi$
$727$	21.8995	0.812207	0.406104	+	0.913827i	$0.366887\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−12.4853	−0.461785
$732$	0	0
$733$	−43.2132	−1.59612	−0.798058	−	0.602581i	$-0.794139\pi$
$733$	−43.2132	−1.59612	−0.798058	+	0.602581i	$0.794139\pi$
$734$	0	0
$735$	−1.41421	−0.0521641
$736$	0	0
$737$	13.3137	0.490417
$738$	0	0
$739$	1.65685	0.0609484	0.0304742	−	0.999536i	$-0.490298\pi$
$739$	1.65685	0.0609484	0.0304742	+	0.999536i	$0.490298\pi$
$740$	0	0
$741$	4.68629	0.172155
$742$	0	0
$743$	−20.9706	−0.769335	−0.384668	−	0.923055i	$-0.625684\pi$
$743$	−20.9706	−0.769335	−0.384668	+	0.923055i	$0.625684\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	6.34315	0.232084
$748$	0	0
$749$	11.3137	0.413394
$750$	0	0
$751$	−41.4558	−1.51275	−0.756373	−	0.654141i	$-0.773030\pi$
$751$	−41.4558	−1.51275	−0.756373	+	0.654141i	$0.773030\pi$
$752$	0	0
$753$	−12.8284	−0.467494
$754$	0	0
$755$	−7.17157	−0.261000
$756$	0	0
$757$	47.6569	1.73212	0.866059	−	0.499942i	$-0.166645\pi$
$757$	47.6569	1.73212	0.866059	+	0.499942i	$0.166645\pi$
$758$	0	0
$759$	−2.82843	−0.102665
$760$	0	0
$761$	26.3848	0.956447	0.478224	−	0.878238i	$-0.341281\pi$
$761$	26.3848	0.956447	0.478224	+	0.878238i	$0.341281\pi$
$762$	0	0
$763$	7.17157	0.259628
$764$	0	0
$765$	6.24264	0.225703
$766$	0	0
$767$	4.34315	0.156822
$768$	0	0
$769$	−15.7574	−0.568225	−0.284112	−	0.958791i	$-0.591699\pi$
$769$	−15.7574	−0.568225	−0.284112	+	0.958791i	$0.591699\pi$
$770$	0	0
$771$	6.82843	0.245920
$772$	0	0
$773$	−10.9706	−0.394584	−0.197292	−	0.980345i	$-0.563215\pi$
$773$	−10.9706	−0.394584	−0.197292	+	0.980345i	$0.563215\pi$
$774$	0	0
$775$	7.41421	0.266326
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−12.4853	−0.446758
$782$	0	0
$783$	−49.9411	−1.78475
$784$	0	0
$785$	12.1421	0.433371
$786$	0	0
$787$	37.1716	1.32502	0.662512	−	0.749052i	$-0.269490\pi$
$787$	37.1716	1.32502	0.662512	+	0.749052i	$0.269490\pi$
$788$	0	0
$789$	2.34315	0.0834182
$790$	0	0
$791$	−13.3137	−0.473381
$792$	0	0
$793$	2.20101	0.0781602
$794$	0	0
$795$	−17.6569	−0.626224
$796$	0	0
$797$	47.4558	1.68097	0.840486	−	0.541833i	$-0.182270\pi$
$797$	47.4558	1.68097	0.840486	+	0.541833i	$0.182270\pi$
$798$	0	0
$799$	1.51472	0.0535869
$800$	0	0
$801$	8.82843	0.311937
$802$	0	0
$803$	−1.75736	−0.0620159
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	13.1716	0.463661
$808$	0	0
$809$	−3.37258	−0.118574	−0.0592869	−	0.998241i	$-0.518883\pi$
$809$	−3.37258	−0.118574	−0.0592869	+	0.998241i	$0.518883\pi$
$810$	0	0
$811$	−6.14214	−0.215680	−0.107840	−	0.994168i	$-0.534393\pi$
$811$	−6.14214	−0.215680	−0.107840	+	0.994168i	$0.534393\pi$
$812$	0	0
$813$	9.65685	0.338681
$814$	0	0
$815$	−21.3137	−0.746587
$816$	0	0
$817$	11.3137	0.395817
$818$	0	0
$819$	−0.585786	−0.0204690
$820$	0	0
$821$	33.5980	1.17258	0.586289	−	0.810102i	$-0.300588\pi$
$821$	33.5980	1.17258	0.586289	+	0.810102i	$0.300588\pi$
$822$	0	0
$823$	−19.1716	−0.668279	−0.334140	−	0.942524i	$-0.608446\pi$
$823$	−19.1716	−0.668279	−0.334140	+	0.942524i	$0.608446\pi$
$824$	0	0
$825$	−1.41421	−0.0492366
$826$	0	0
$827$	−9.02944	−0.313984	−0.156992	−	0.987600i	$-0.550180\pi$
$827$	−9.02944	−0.313984	−0.156992	+	0.987600i	$0.550180\pi$
$828$	0	0
$829$	11.6569	0.404859	0.202430	−	0.979297i	$-0.435116\pi$
$829$	11.6569	0.404859	0.202430	+	0.979297i	$0.435116\pi$
$830$	0	0
$831$	44.9706	1.56001
$832$	0	0
$833$	−6.24264	−0.216295
$834$	0	0
$835$	−6.14214	−0.212557
$836$	0	0
$837$	41.9411	1.44970
$838$	0	0
$839$	−31.0122	−1.07066	−0.535330	−	0.844643i	$-0.679813\pi$
$839$	−31.0122	−1.07066	−0.535330	+	0.844643i	$0.679813\pi$
$840$	0	0
$841$	48.9411	1.68763
$842$	0	0
$843$	20.4853	0.705551
$844$	0	0
$845$	−12.6569	−0.435409
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−5.65685	−0.194143
$850$	0	0
$851$	5.65685	0.193914
$852$	0	0
$853$	49.0711	1.68016	0.840081	−	0.542461i	$-0.182508\pi$
$853$	49.0711	1.68016	0.840081	+	0.542461i	$0.182508\pi$
$854$	0	0
$855$	−5.65685	−0.193460
$856$	0	0
$857$	−47.4142	−1.61964	−0.809819	−	0.586679i	$-0.800435\pi$
$857$	−47.4142	−1.61964	−0.809819	+	0.586679i	$0.800435\pi$
$858$	0	0
$859$	18.4437	0.629289	0.314645	−	0.949210i	$-0.398115\pi$
$859$	18.4437	0.629289	0.314645	+	0.949210i	$0.398115\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	3.45584	0.117638	0.0588192	−	0.998269i	$-0.481266\pi$
$863$	3.45584	0.117638	0.0588192	+	0.998269i	$0.481266\pi$
$864$	0	0
$865$	−11.4142	−0.388095
$866$	0	0
$867$	−31.0711	−1.05523
$868$	0	0
$869$	−12.8284	−0.435175
$870$	0	0
$871$	−7.79899	−0.264259
$872$	0	0
$873$	−13.3137	−0.450601
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−27.3137	−0.922318	−0.461159	−	0.887317i	$-0.652566\pi$
$877$	−27.3137	−0.922318	−0.461159	+	0.887317i	$0.652566\pi$
$878$	0	0
$879$	5.51472	0.186007
$880$	0	0
$881$	6.68629	0.225267	0.112633	−	0.993637i	$-0.464071\pi$
$881$	6.68629	0.225267	0.112633	+	0.993637i	$0.464071\pi$
$882$	0	0
$883$	−11.4558	−0.385520	−0.192760	−	0.981246i	$-0.561744\pi$
$883$	−11.4558	−0.385520	−0.192760	+	0.981246i	$0.561744\pi$
$884$	0	0
$885$	10.4853	0.352459
$886$	0	0
$887$	31.7990	1.06771	0.533853	−	0.845577i	$-0.320744\pi$
$887$	31.7990	1.06771	0.533853	+	0.845577i	$0.320744\pi$
$888$	0	0
$889$	−1.65685	−0.0555691
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	−1.37258	−0.0459317
$894$	0	0
$895$	11.3137	0.378176
$896$	0	0
$897$	1.65685	0.0553208
$898$	0	0
$899$	−65.4558	−2.18307
$900$	0	0
$901$	−77.9411	−2.59660
$902$	0	0
$903$	2.82843	0.0941242
$904$	0	0
$905$	−6.00000	−0.199447
$906$	0	0
$907$	39.9411	1.32622	0.663112	−	0.748520i	$-0.269235\pi$
$907$	39.9411	1.32622	0.663112	+	0.748520i	$0.269235\pi$
$908$	0	0
$909$	1.41421	0.0469065
$910$	0	0
$911$	−50.1421	−1.66128	−0.830642	−	0.556808i	$-0.812026\pi$
$911$	−50.1421	−1.66128	−0.830642	+	0.556808i	$0.812026\pi$
$912$	0	0
$913$	−6.34315	−0.209927
$914$	0	0
$915$	5.31371	0.175666
$916$	0	0
$917$	15.3137	0.505703
$918$	0	0
$919$	−32.9706	−1.08760	−0.543799	−	0.839215i	$-0.683015\pi$
$919$	−32.9706	−1.08760	−0.543799	+	0.839215i	$0.683015\pi$
$920$	0	0
$921$	−26.6274	−0.877403
$922$	0	0
$923$	7.31371	0.240734
$924$	0	0
$925$	2.82843	0.0929981
$926$	0	0
$927$	−10.5858	−0.347683
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	5.65685	0.185396
$932$	0	0
$933$	15.8579	0.519163
$934$	0	0
$935$	−6.24264	−0.204156
$936$	0	0
$937$	−21.5563	−0.704215	−0.352108	−	0.935960i	$-0.614535\pi$
$937$	−21.5563	−0.704215	−0.352108	+	0.935960i	$0.614535\pi$
$938$	0	0
$939$	−18.8284	−0.614442
$940$	0	0
$941$	−37.0122	−1.20656	−0.603282	−	0.797528i	$-0.706140\pi$
$941$	−37.0122	−1.20656	−0.603282	+	0.797528i	$0.706140\pi$
$942$	0	0
$943$	−8.48528	−0.276319
$944$	0	0
$945$	−5.65685	−0.184017
$946$	0	0
$947$	−38.4853	−1.25060	−0.625302	−	0.780383i	$-0.715024\pi$
$947$	−38.4853	−1.25060	−0.625302	+	0.780383i	$0.715024\pi$
$948$	0	0
$949$	1.02944	0.0334169
$950$	0	0
$951$	26.8284	0.869971
$952$	0	0
$953$	40.4853	1.31145	0.655723	−	0.755001i	$-0.272364\pi$
$953$	40.4853	1.31145	0.655723	+	0.755001i	$0.272364\pi$
$954$	0	0
$955$	9.65685	0.312488
$956$	0	0
$957$	12.4853	0.403592
$958$	0	0
$959$	20.4853	0.661504
$960$	0	0
$961$	23.9706	0.773244
$962$	0	0
$963$	11.3137	0.364579
$964$	0	0
$965$	4.48528	0.144386
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	49.9411	1.60434
$970$	0	0
$971$	−38.0416	−1.22081	−0.610407	−	0.792088i	$-0.708994\pi$
$971$	−38.0416	−1.22081	−0.610407	+	0.792088i	$0.708994\pi$
$972$	0	0
$973$	−8.48528	−0.272026
$974$	0	0
$975$	0.828427	0.0265309
$976$	0	0
$977$	30.2843	0.968880	0.484440	−	0.874825i	$-0.339023\pi$
$977$	30.2843	0.968880	0.484440	+	0.874825i	$0.339023\pi$
$978$	0	0
$979$	−8.82843	−0.282158
$980$	0	0
$981$	7.17157	0.228971
$982$	0	0
$983$	39.7574	1.26806	0.634031	−	0.773307i	$-0.281399\pi$
$983$	39.7574	1.26806	0.634031	+	0.773307i	$0.281399\pi$
$984$	0	0
$985$	−13.1716	−0.419681
$986$	0	0
$987$	−0.343146	−0.0109224
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	40.7696	1.29509	0.647544	−	0.762028i	$-0.275796\pi$
$991$	40.7696	1.29509	0.647544	+	0.762028i	$0.275796\pi$
$992$	0	0
$993$	32.9706	1.04629
$994$	0	0
$995$	−9.55635	−0.302957
$996$	0	0
$997$	49.8406	1.57847	0.789234	−	0.614092i	$-0.210478\pi$
$997$	49.8406	1.57847	0.789234	+	0.614092i	$0.210478\pi$
$998$	0	0
$999$	16.0000	0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bb.1.1		2
4.3	odd	2		3080.2.a.i.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.i.1.2	✓	2	4.3	odd	2
6160.2.a.bb.1.1		2	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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{9} +1.00000 q^{11} -0.585786 q^{13} -1.41421 q^{15} -6.24264 q^{17} +5.65685 q^{19} +1.41421 q^{21} +2.00000 q^{23} +1.00000 q^{25} +5.65685 q^{27} -8.82843 q^{29} +7.41421 q^{31} -1.41421 q^{33} -1.00000 q^{35} +2.82843 q^{37} +0.828427 q^{39} -4.24264 q^{41} +2.00000 q^{43} -1.00000 q^{45} -0.242641 q^{47} +1.00000 q^{49} +8.82843 q^{51} +12.4853 q^{53} +1.00000 q^{55} -8.00000 q^{57} -7.41421 q^{59} -3.75736 q^{61} +1.00000 q^{63} -0.585786 q^{65} +13.3137 q^{67} -2.82843 q^{69} -12.4853 q^{71} -1.75736 q^{73} -1.41421 q^{75} -1.00000 q^{77} -12.8284 q^{79} -5.00000 q^{81} -6.34315 q^{83} -6.24264 q^{85} +12.4853 q^{87} -8.82843 q^{89} +0.585786 q^{91} -10.4853 q^{93} +5.65685 q^{95} +13.3137 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^9	\( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{9} + 2 q^{11} - 4 q^{13} - 4 q^{17} + 4 q^{23} + 2 q^{25} - 12 q^{29} + 12 q^{31} - 2 q^{35} - 4 q^{39} + 4 q^{43} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 12 q^{51} + 8 q^{53} + 2 q^{55} - 16 q^{57} - 12 q^{59} - 16 q^{61} + 2 q^{63} - 4 q^{65} + 4 q^{67} - 8 q^{71} - 12 q^{73} - 2 q^{77} - 20 q^{79} - 10 q^{81} - 24 q^{83} - 4 q^{85} + 8 q^{87} - 12 q^{89} + 4 q^{91} - 4 q^{93} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^9 + 2 * q^11 - 4 * q^13 - 4 * q^17 + 4 * q^23 + 2 * q^25 - 12 * q^29 + 12 * q^31 - 2 * q^35 - 4 * q^39 + 4 * q^43 - 2 * q^45 + 8 * q^47 + 2 * q^49 + 12 * q^51 + 8 * q^53 + 2 * q^55 - 16 * q^57 - 12 * q^59 - 16 * q^61 + 2 * q^63 - 4 * q^65 + 4 * q^67 - 8 * q^71 - 12 * q^73 - 2 * q^77 - 20 * q^79 - 10 * q^81 - 24 * q^83 - 4 * q^85 + 8 * q^87 - 12 * q^89 + 4 * q^91 - 4 * q^93 + 4 * q^97 - 2 * q^99

Introduction

L-functions

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