LMFDB - Embedded newform 6160.2.a.bc.1.2

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			1.2
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)

$f(q)$	$=$	$q+1.41421 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.41421 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} -1.00000 q^{11} -0.585786 q^{13} +1.41421 q^{15} -0.585786 q^{17} +1.41421 q^{21} -3.65685 q^{23} +1.00000 q^{25} -5.65685 q^{27} -0.828427 q^{29} -3.41421 q^{31} -1.41421 q^{33} +1.00000 q^{35} -10.8284 q^{37} -0.828427 q^{39} -8.24264 q^{41} -7.65685 q^{43} -1.00000 q^{45} +2.58579 q^{47} +1.00000 q^{49} -0.828427 q^{51} -1.17157 q^{53} -1.00000 q^{55} -10.2426 q^{59} -5.41421 q^{61} -1.00000 q^{63} -0.585786 q^{65} +15.6569 q^{67} -5.17157 q^{69} +12.4853 q^{71} +0.585786 q^{73} +1.41421 q^{75} -1.00000 q^{77} -14.4853 q^{79} -5.00000 q^{81} +15.3137 q^{83} -0.585786 q^{85} -1.17157 q^{87} +4.82843 q^{89} -0.585786 q^{91} -4.82843 q^{93} -0.343146 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{29} - 4 q^{31} + 2 q^{35} - 16 q^{37} + 4 q^{39} - 8 q^{41} - 4 q^{43} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 4 q^{51} - 8 q^{53} - 2 q^{55} - 12 q^{59} - 8 q^{61} - 2 q^{63} - 4 q^{65} + 20 q^{67} - 16 q^{69} + 8 q^{71} + 4 q^{73} - 2 q^{77} - 12 q^{79} - 10 q^{81} + 8 q^{83} - 4 q^{85} - 8 q^{87} + 4 q^{89} - 4 q^{91} - 4 q^{93} - 12 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 + 4 * q^29 - 4 * q^31 + 2 * q^35 - 16 * q^37 + 4 * q^39 - 8 * q^41 - 4 * q^43 - 2 * q^45 + 8 * q^47 + 2 * q^49 + 4 * q^51 - 8 * q^53 - 2 * q^55 - 12 * q^59 - 8 * q^61 - 2 * q^63 - 4 * q^65 + 20 * q^67 - 16 * q^69 + 8 * q^71 + 4 * q^73 - 2 * q^77 - 12 * q^79 - 10 * q^81 + 8 * q^83 - 4 * q^85 - 8 * q^87 + 4 * q^89 - 4 * q^91 - 4 * q^93 - 12 * 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	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	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
$17$	−0.585786	−0.142074	−0.0710370	−	0.997474i	$-0.522631\pi$
$17$	−0.585786	−0.142074	−0.0710370	+	0.997474i	$0.522631\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	1.41421	0.308607
$22$	0	0
$23$	−3.65685	−0.762507	−0.381253	−	0.924471i	$-0.624507\pi$
$23$	−3.65685	−0.762507	−0.381253	+	0.924471i	$0.624507\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−0.828427	−0.153835	−0.0769175	−	0.997037i	$-0.524508\pi$
$29$	−0.828427	−0.153835	−0.0769175	+	0.997037i	$0.524508\pi$
$30$	0	0
$31$	−3.41421	−0.613211	−0.306605	−	0.951837i	$-0.599193\pi$
$31$	−3.41421	−0.613211	−0.306605	+	0.951837i	$0.599193\pi$
$32$	0	0
$33$	−1.41421	−0.246183
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−10.8284	−1.78018	−0.890091	−	0.455782i	$-0.849360\pi$
$37$	−10.8284	−1.78018	−0.890091	+	0.455782i	$0.849360\pi$
$38$	0	0
$39$	−0.828427	−0.132655
$40$	0	0
$41$	−8.24264	−1.28728	−0.643642	−	0.765327i	$-0.722577\pi$
$41$	−8.24264	−1.28728	−0.643642	+	0.765327i	$0.722577\pi$
$42$	0	0
$43$	−7.65685	−1.16766	−0.583830	−	0.811876i	$-0.698446\pi$
$43$	−7.65685	−1.16766	−0.583830	+	0.811876i	$0.698446\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	2.58579	0.377176	0.188588	−	0.982056i	$-0.439609\pi$
$47$	2.58579	0.377176	0.188588	+	0.982056i	$0.439609\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.828427	−0.116003
$52$	0	0
$53$	−1.17157	−0.160928	−0.0804640	−	0.996758i	$-0.525640\pi$
$53$	−1.17157	−0.160928	−0.0804640	+	0.996758i	$0.525640\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.2426	−1.33348	−0.666739	−	0.745291i	$-0.732310\pi$
$59$	−10.2426	−1.33348	−0.666739	+	0.745291i	$0.732310\pi$
$60$	0	0
$61$	−5.41421	−0.693219	−0.346610	−	0.938010i	$-0.612667\pi$
$61$	−5.41421	−0.693219	−0.346610	+	0.938010i	$0.612667\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−0.585786	−0.0726579
$66$	0	0
$67$	15.6569	1.91279	0.956395	−	0.292078i	$-0.0943466\pi$
$67$	15.6569	1.91279	0.956395	+	0.292078i	$0.0943466\pi$
$68$	0	0
$69$	−5.17157	−0.622584
$70$	0	0
$71$	12.4853	1.48173	0.740865	−	0.671654i	$-0.234416\pi$
$71$	12.4853	1.48173	0.740865	+	0.671654i	$0.234416\pi$
$72$	0	0
$73$	0.585786	0.0685611	0.0342806	−	0.999412i	$-0.489086\pi$
$73$	0.585786	0.0685611	0.0342806	+	0.999412i	$0.489086\pi$
$74$	0	0
$75$	1.41421	0.163299
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−14.4853	−1.62972	−0.814861	−	0.579657i	$-0.803187\pi$
$79$	−14.4853	−1.62972	−0.814861	+	0.579657i	$0.803187\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	15.3137	1.68090	0.840449	−	0.541891i	$-0.182291\pi$
$83$	15.3137	1.68090	0.840449	+	0.541891i	$0.182291\pi$
$84$	0	0
$85$	−0.585786	−0.0635375
$86$	0	0
$87$	−1.17157	−0.125606
$88$	0	0
$89$	4.82843	0.511812	0.255906	−	0.966702i	$-0.417626\pi$
$89$	4.82843	0.511812	0.255906	+	0.966702i	$0.417626\pi$
$90$	0	0
$91$	−0.585786	−0.0614071
$92$	0	0
$93$	−4.82843	−0.500685
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.343146	−0.0348412	−0.0174206	−	0.999848i	$-0.505545\pi$
$97$	−0.343146	−0.0348412	−0.0174206	+	0.999848i	$0.505545\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	0.242641	0.0241437	0.0120718	−	0.999927i	$-0.496157\pi$
$101$	0.242641	0.0241437	0.0120718	+	0.999927i	$0.496157\pi$
$102$	0	0
$103$	3.07107	0.302601	0.151301	−	0.988488i	$-0.451654\pi$
$103$	3.07107	0.302601	0.151301	+	0.988488i	$0.451654\pi$
$104$	0	0
$105$	1.41421	0.138013
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0.828427	0.0793489	0.0396745	−	0.999213i	$-0.487368\pi$
$109$	0.828427	0.0793489	0.0396745	+	0.999213i	$0.487368\pi$
$110$	0	0
$111$	−15.3137	−1.45351
$112$	0	0
$113$	−17.3137	−1.62874	−0.814368	−	0.580348i	$-0.802916\pi$
$113$	−17.3137	−1.62874	−0.814368	+	0.580348i	$0.802916\pi$
$114$	0	0
$115$	−3.65685	−0.341003
$116$	0	0
$117$	0.585786	0.0541560
$118$	0	0
$119$	−0.585786	−0.0536990
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−11.6569	−1.05106
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.65685	−0.856907	−0.428454	−	0.903564i	$-0.640941\pi$
$127$	−9.65685	−0.856907	−0.428454	+	0.903564i	$0.640941\pi$
$128$	0	0
$129$	−10.8284	−0.953390
$130$	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
$133$	0	0
$134$	0	0
$135$	−5.65685	−0.486864
$136$	0	0
$137$	−14.8284	−1.26688	−0.633439	−	0.773793i	$-0.718357\pi$
$137$	−14.8284	−1.26688	−0.633439	+	0.773793i	$0.718357\pi$
$138$	0	0
$139$	14.1421	1.19952	0.599760	−	0.800180i	$-0.295263\pi$
$139$	14.1421	1.19952	0.599760	+	0.800180i	$0.295263\pi$
$140$	0	0
$141$	3.65685	0.307963
$142$	0	0
$143$	0.585786	0.0489859
$144$	0	0
$145$	−0.828427	−0.0687971
$146$	0	0
$147$	1.41421	0.116642
$148$	0	0
$149$	9.31371	0.763009	0.381504	−	0.924367i	$-0.375406\pi$
$149$	9.31371	0.763009	0.381504	+	0.924367i	$0.375406\pi$
$150$	0	0
$151$	21.7990	1.77398	0.886988	−	0.461792i	$-0.152793\pi$
$151$	21.7990	1.77398	0.886988	+	0.461792i	$0.152793\pi$
$152$	0	0
$153$	0.585786	0.0473580
$154$	0	0
$155$	−3.41421	−0.274236
$156$	0	0
$157$	−1.51472	−0.120888	−0.0604439	−	0.998172i	$-0.519252\pi$
$157$	−1.51472	−0.120888	−0.0604439	+	0.998172i	$0.519252\pi$
$158$	0	0
$159$	−1.65685	−0.131397
$160$	0	0
$161$	−3.65685	−0.288200
$162$	0	0
$163$	11.6569	0.913035	0.456518	−	0.889714i	$-0.349097\pi$
$163$	11.6569	0.913035	0.456518	+	0.889714i	$0.349097\pi$
$164$	0	0
$165$	−1.41421	−0.110096
$166$	0	0
$167$	16.4853	1.27567	0.637835	−	0.770173i	$-0.279830\pi$
$167$	16.4853	1.27567	0.637835	+	0.770173i	$0.279830\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.7279	−1.11974	−0.559872	−	0.828579i	$-0.689150\pi$
$173$	−14.7279	−1.11974	−0.559872	+	0.828579i	$0.689150\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−14.4853	−1.08878
$178$	0	0
$179$	−19.3137	−1.44357	−0.721787	−	0.692115i	$-0.756679\pi$
$179$	−19.3137	−1.44357	−0.721787	+	0.692115i	$0.756679\pi$
$180$	0	0
$181$	−11.6569	−0.866447	−0.433224	−	0.901286i	$-0.642624\pi$
$181$	−11.6569	−0.866447	−0.433224	+	0.901286i	$0.642624\pi$
$182$	0	0
$183$	−7.65685	−0.566011
$184$	0	0
$185$	−10.8284	−0.796122
$186$	0	0
$187$	0.585786	0.0428369
$188$	0	0
$189$	−5.65685	−0.411476
$190$	0	0
$191$	17.6569	1.27761	0.638803	−	0.769371i	$-0.279430\pi$
$191$	17.6569	1.27761	0.638803	+	0.769371i	$0.279430\pi$
$192$	0	0
$193$	−0.485281	−0.0349313	−0.0174657	−	0.999847i	$-0.505560\pi$
$193$	−0.485281	−0.0349313	−0.0174657	+	0.999847i	$0.505560\pi$
$194$	0	0
$195$	−0.828427	−0.0593249
$196$	0	0
$197$	−18.1421	−1.29257	−0.646287	−	0.763095i	$-0.723679\pi$
$197$	−18.1421	−1.29257	−0.646287	+	0.763095i	$0.723679\pi$
$198$	0	0
$199$	2.24264	0.158977	0.0794883	−	0.996836i	$-0.474671\pi$
$199$	2.24264	0.158977	0.0794883	+	0.996836i	$0.474671\pi$
$200$	0	0
$201$	22.1421	1.56179
$202$	0	0
$203$	−0.828427	−0.0581442
$204$	0	0
$205$	−8.24264	−0.575691
$206$	0	0
$207$	3.65685	0.254169
$208$	0	0
$209$	0	0
$210$	0	0
$211$	7.31371	0.503496	0.251748	−	0.967793i	$-0.418995\pi$
$211$	7.31371	0.503496	0.251748	+	0.967793i	$0.418995\pi$
$212$	0	0
$213$	17.6569	1.20983
$214$	0	0
$215$	−7.65685	−0.522193
$216$	0	0
$217$	−3.41421	−0.231772
$218$	0	0
$219$	0.828427	0.0559799
$220$	0	0
$221$	0.343146	0.0230825
$222$	0	0
$223$	−9.89949	−0.662919	−0.331460	−	0.943469i	$-0.607541\pi$
$223$	−9.89949	−0.662919	−0.331460	+	0.943469i	$0.607541\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	−17.6569	−1.17193	−0.585963	−	0.810338i	$-0.699284\pi$
$227$	−17.6569	−1.17193	−0.585963	+	0.810338i	$0.699284\pi$
$228$	0	0
$229$	14.4853	0.957214	0.478607	−	0.878029i	$-0.341142\pi$
$229$	14.4853	0.957214	0.478607	+	0.878029i	$0.341142\pi$
$230$	0	0
$231$	−1.41421	−0.0930484
$232$	0	0
$233$	−16.9706	−1.11178	−0.555889	−	0.831256i	$-0.687622\pi$
$233$	−16.9706	−1.11178	−0.555889	+	0.831256i	$0.687622\pi$
$234$	0	0
$235$	2.58579	0.168678
$236$	0	0
$237$	−20.4853	−1.33066
$238$	0	0
$239$	−8.97056	−0.580257	−0.290129	−	0.956988i	$-0.593698\pi$
$239$	−8.97056	−0.580257	−0.290129	+	0.956988i	$0.593698\pi$
$240$	0	0
$241$	13.4142	0.864085	0.432043	−	0.901853i	$-0.357793\pi$
$241$	13.4142	0.864085	0.432043	+	0.901853i	$0.357793\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	21.6569	1.37245
$250$	0	0
$251$	23.2132	1.46520	0.732602	−	0.680657i	$-0.238306\pi$
$251$	23.2132	1.46520	0.732602	+	0.680657i	$0.238306\pi$
$252$	0	0
$253$	3.65685	0.229904
$254$	0	0
$255$	−0.828427	−0.0518781
$256$	0	0
$257$	4.14214	0.258379	0.129190	−	0.991620i	$-0.458762\pi$
$257$	4.14214	0.258379	0.129190	+	0.991620i	$0.458762\pi$
$258$	0	0
$259$	−10.8284	−0.672846
$260$	0	0
$261$	0.828427	0.0512784
$262$	0	0
$263$	20.9706	1.29310	0.646550	−	0.762871i	$-0.276211\pi$
$263$	20.9706	1.29310	0.646550	+	0.762871i	$0.276211\pi$
$264$	0	0
$265$	−1.17157	−0.0719691
$266$	0	0
$267$	6.82843	0.417893
$268$	0	0
$269$	2.97056	0.181118	0.0905592	−	0.995891i	$-0.471135\pi$
$269$	2.97056	0.181118	0.0905592	+	0.995891i	$0.471135\pi$
$270$	0	0
$271$	14.8284	0.900763	0.450381	−	0.892836i	$-0.351288\pi$
$271$	14.8284	0.900763	0.450381	+	0.892836i	$0.351288\pi$
$272$	0	0
$273$	−0.828427	−0.0501387
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−2.82843	−0.169944	−0.0849719	−	0.996383i	$-0.527080\pi$
$277$	−2.82843	−0.169944	−0.0849719	+	0.996383i	$0.527080\pi$
$278$	0	0
$279$	3.41421	0.204404
$280$	0	0
$281$	4.82843	0.288040	0.144020	−	0.989575i	$-0.453997\pi$
$281$	4.82843	0.288040	0.144020	+	0.989575i	$0.453997\pi$
$282$	0	0
$283$	−7.31371	−0.434755	−0.217377	−	0.976088i	$-0.569750\pi$
$283$	−7.31371	−0.434755	−0.217377	+	0.976088i	$0.569750\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.24264	−0.486548
$288$	0	0
$289$	−16.6569	−0.979815
$290$	0	0
$291$	−0.485281	−0.0284477
$292$	0	0
$293$	−11.8995	−0.695176	−0.347588	−	0.937647i	$-0.612999\pi$
$293$	−11.8995	−0.695176	−0.347588	+	0.937647i	$0.612999\pi$
$294$	0	0
$295$	−10.2426	−0.596350
$296$	0	0
$297$	5.65685	0.328244
$298$	0	0
$299$	2.14214	0.123883
$300$	0	0
$301$	−7.65685	−0.441334
$302$	0	0
$303$	0.343146	0.0197132
$304$	0	0
$305$	−5.41421	−0.310017
$306$	0	0
$307$	−0.485281	−0.0276965	−0.0138482	−	0.999904i	$-0.504408\pi$
$307$	−0.485281	−0.0276965	−0.0138482	+	0.999904i	$0.504408\pi$
$308$	0	0
$309$	4.34315	0.247073
$310$	0	0
$311$	−15.4142	−0.874060	−0.437030	−	0.899447i	$-0.643970\pi$
$311$	−15.4142	−0.874060	−0.437030	+	0.899447i	$0.643970\pi$
$312$	0	0
$313$	−20.6274	−1.16593	−0.582965	−	0.812497i	$-0.698108\pi$
$313$	−20.6274	−1.16593	−0.582965	+	0.812497i	$0.698108\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	−1.02944	−0.0578190	−0.0289095	−	0.999582i	$-0.509203\pi$
$317$	−1.02944	−0.0578190	−0.0289095	+	0.999582i	$0.509203\pi$
$318$	0	0
$319$	0.828427	0.0463830
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−0.585786	−0.0324936
$326$	0	0
$327$	1.17157	0.0647881
$328$	0	0
$329$	2.58579	0.142559
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	10.8284	0.593394
$334$	0	0
$335$	15.6569	0.855425
$336$	0	0
$337$	−9.65685	−0.526042	−0.263021	−	0.964790i	$-0.584719\pi$
$337$	−9.65685	−0.526042	−0.263021	+	0.964790i	$0.584719\pi$
$338$	0	0
$339$	−24.4853	−1.32986
$340$	0	0
$341$	3.41421	0.184890
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−5.17157	−0.278428
$346$	0	0
$347$	6.97056	0.374199	0.187100	−	0.982341i	$-0.440091\pi$
$347$	6.97056	0.374199	0.187100	+	0.982341i	$0.440091\pi$
$348$	0	0
$349$	9.21320	0.493171	0.246586	−	0.969121i	$-0.420691\pi$
$349$	9.21320	0.493171	0.246586	+	0.969121i	$0.420691\pi$
$350$	0	0
$351$	3.31371	0.176873
$352$	0	0
$353$	28.6274	1.52368	0.761842	−	0.647763i	$-0.224295\pi$
$353$	28.6274	1.52368	0.761842	+	0.647763i	$0.224295\pi$
$354$	0	0
$355$	12.4853	0.662650
$356$	0	0
$357$	−0.828427	−0.0438450
$358$	0	0
$359$	17.7990	0.939395	0.469697	−	0.882827i	$-0.344363\pi$
$359$	17.7990	0.939395	0.469697	+	0.882827i	$0.344363\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	1.41421	0.0742270
$364$	0	0
$365$	0.585786	0.0306615
$366$	0	0
$367$	30.8701	1.61140	0.805702	−	0.592321i	$-0.201788\pi$
$367$	30.8701	1.61140	0.805702	+	0.592321i	$0.201788\pi$
$368$	0	0
$369$	8.24264	0.429095
$370$	0	0
$371$	−1.17157	−0.0608250
$372$	0	0
$373$	27.7990	1.43938	0.719689	−	0.694297i	$-0.244285\pi$
$373$	27.7990	1.43938	0.719689	+	0.694297i	$0.244285\pi$
$374$	0	0
$375$	1.41421	0.0730297
$376$	0	0
$377$	0.485281	0.0249933
$378$	0	0
$379$	−5.17157	−0.265646	−0.132823	−	0.991140i	$-0.542404\pi$
$379$	−5.17157	−0.265646	−0.132823	+	0.991140i	$0.542404\pi$
$380$	0	0
$381$	−13.6569	−0.699662
$382$	0	0
$383$	−26.3848	−1.34820	−0.674100	−	0.738641i	$-0.735468\pi$
$383$	−26.3848	−1.34820	−0.674100	+	0.738641i	$0.735468\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	7.65685	0.389220
$388$	0	0
$389$	7.65685	0.388218	0.194109	−	0.980980i	$-0.437818\pi$
$389$	7.65685	0.388218	0.194109	+	0.980980i	$0.437818\pi$
$390$	0	0
$391$	2.14214	0.108332
$392$	0	0
$393$	−21.6569	−1.09244
$394$	0	0
$395$	−14.4853	−0.728834
$396$	0	0
$397$	8.82843	0.443086	0.221543	−	0.975151i	$-0.428891\pi$
$397$	8.82843	0.443086	0.221543	+	0.975151i	$0.428891\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	0	0
$405$	−5.00000	−0.248452
$406$	0	0
$407$	10.8284	0.536745
$408$	0	0
$409$	−0.242641	−0.0119978	−0.00599890	−	0.999982i	$-0.501910\pi$
$409$	−0.242641	−0.0119978	−0.00599890	+	0.999982i	$0.501910\pi$
$410$	0	0
$411$	−20.9706	−1.03440
$412$	0	0
$413$	−10.2426	−0.504007
$414$	0	0
$415$	15.3137	0.751720
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	−31.4142	−1.53468	−0.767342	−	0.641238i	$-0.778421\pi$
$419$	−31.4142	−1.53468	−0.767342	+	0.641238i	$0.778421\pi$
$420$	0	0
$421$	−12.0000	−0.584844	−0.292422	−	0.956289i	$-0.594461\pi$
$421$	−12.0000	−0.584844	−0.292422	+	0.956289i	$0.594461\pi$
$422$	0	0
$423$	−2.58579	−0.125725
$424$	0	0
$425$	−0.585786	−0.0284148
$426$	0	0
$427$	−5.41421	−0.262012
$428$	0	0
$429$	0.828427	0.0399968
$430$	0	0
$431$	−24.8284	−1.19594	−0.597972	−	0.801517i	$-0.704026\pi$
$431$	−24.8284	−1.19594	−0.597972	+	0.801517i	$0.704026\pi$
$432$	0	0
$433$	−39.4558	−1.89613	−0.948063	−	0.318081i	$-0.896961\pi$
$433$	−39.4558	−1.89613	−0.948063	+	0.318081i	$0.896961\pi$
$434$	0	0
$435$	−1.17157	−0.0561726
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−27.7990	−1.32677	−0.663387	−	0.748277i	$-0.730882\pi$
$439$	−27.7990	−1.32677	−0.663387	+	0.748277i	$0.730882\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	−29.7990	−1.41579	−0.707896	−	0.706316i	$-0.750356\pi$
$443$	−29.7990	−1.41579	−0.707896	+	0.706316i	$0.750356\pi$
$444$	0	0
$445$	4.82843	0.228889
$446$	0	0
$447$	13.1716	0.622994
$448$	0	0
$449$	−13.6569	−0.644507	−0.322253	−	0.946653i	$-0.604440\pi$
$449$	−13.6569	−0.644507	−0.322253	+	0.946653i	$0.604440\pi$
$450$	0	0
$451$	8.24264	0.388131
$452$	0	0
$453$	30.8284	1.44845
$454$	0	0
$455$	−0.585786	−0.0274621
$456$	0	0
$457$	11.7990	0.551933	0.275967	−	0.961167i	$-0.411002\pi$
$457$	11.7990	0.551933	0.275967	+	0.961167i	$0.411002\pi$
$458$	0	0
$459$	3.31371	0.154671
$460$	0	0
$461$	37.6985	1.75579	0.877897	−	0.478850i	$-0.158946\pi$
$461$	37.6985	1.75579	0.877897	+	0.478850i	$0.158946\pi$
$462$	0	0
$463$	22.9706	1.06753	0.533766	−	0.845632i	$-0.320776\pi$
$463$	22.9706	1.06753	0.533766	+	0.845632i	$0.320776\pi$
$464$	0	0
$465$	−4.82843	−0.223913
$466$	0	0
$467$	8.24264	0.381424	0.190712	−	0.981646i	$-0.438920\pi$
$467$	8.24264	0.381424	0.190712	+	0.981646i	$0.438920\pi$
$468$	0	0
$469$	15.6569	0.722966
$470$	0	0
$471$	−2.14214	−0.0987044
$472$	0	0
$473$	7.65685	0.352063
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.17157	0.0536426
$478$	0	0
$479$	5.65685	0.258468	0.129234	−	0.991614i	$-0.458748\pi$
$479$	5.65685	0.258468	0.129234	+	0.991614i	$0.458748\pi$
$480$	0	0
$481$	6.34315	0.289223
$482$	0	0
$483$	−5.17157	−0.235315
$484$	0	0
$485$	−0.343146	−0.0155814
$486$	0	0
$487$	−12.6274	−0.572203	−0.286101	−	0.958199i	$-0.592359\pi$
$487$	−12.6274	−0.572203	−0.286101	+	0.958199i	$0.592359\pi$
$488$	0	0
$489$	16.4853	0.745490
$490$	0	0
$491$	8.14214	0.367449	0.183725	−	0.982978i	$-0.441185\pi$
$491$	8.14214	0.367449	0.183725	+	0.982978i	$0.441185\pi$
$492$	0	0
$493$	0.485281	0.0218560
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	12.4853	0.560041
$498$	0	0
$499$	−37.4558	−1.67675	−0.838377	−	0.545091i	$-0.816495\pi$
$499$	−37.4558	−1.67675	−0.838377	+	0.545091i	$0.816495\pi$
$500$	0	0
$501$	23.3137	1.04158
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	0.242641	0.0107974
$506$	0	0
$507$	−17.8995	−0.794944
$508$	0	0
$509$	−4.82843	−0.214016	−0.107008	−	0.994258i	$-0.534127\pi$
$509$	−4.82843	−0.214016	−0.107008	+	0.994258i	$0.534127\pi$
$510$	0	0
$511$	0.585786	0.0259137
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.07107	0.135327
$516$	0	0
$517$	−2.58579	−0.113723
$518$	0	0
$519$	−20.8284	−0.914266
$520$	0	0
$521$	−6.97056	−0.305386	−0.152693	−	0.988274i	$-0.548795\pi$
$521$	−6.97056	−0.305386	−0.152693	+	0.988274i	$0.548795\pi$
$522$	0	0
$523$	−1.17157	−0.0512293	−0.0256147	−	0.999672i	$-0.508154\pi$
$523$	−1.17157	−0.0512293	−0.0256147	+	0.999672i	$0.508154\pi$
$524$	0	0
$525$	1.41421	0.0617213
$526$	0	0
$527$	2.00000	0.0871214
$528$	0	0
$529$	−9.62742	−0.418583
$530$	0	0
$531$	10.2426	0.444493
$532$	0	0
$533$	4.82843	0.209142
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−27.3137	−1.17867
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	9.31371	0.400428	0.200214	−	0.979752i	$-0.435836\pi$
$541$	9.31371	0.400428	0.200214	+	0.979752i	$0.435836\pi$
$542$	0	0
$543$	−16.4853	−0.707451
$544$	0	0
$545$	0.828427	0.0354859
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	0	0
$549$	5.41421	0.231073
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−14.4853	−0.615977
$554$	0	0
$555$	−15.3137	−0.650031
$556$	0	0
$557$	1.17157	0.0496411	0.0248206	−	0.999692i	$-0.492099\pi$
$557$	1.17157	0.0496411	0.0248206	+	0.999692i	$0.492099\pi$
$558$	0	0
$559$	4.48528	0.189707
$560$	0	0
$561$	0.828427	0.0349762
$562$	0	0
$563$	−44.4853	−1.87483	−0.937416	−	0.348213i	$-0.886789\pi$
$563$	−44.4853	−1.87483	−0.937416	+	0.348213i	$0.886789\pi$
$564$	0	0
$565$	−17.3137	−0.728393
$566$	0	0
$567$	−5.00000	−0.209980
$568$	0	0
$569$	−28.1421	−1.17978	−0.589890	−	0.807484i	$-0.700829\pi$
$569$	−28.1421	−1.17978	−0.589890	+	0.807484i	$0.700829\pi$
$570$	0	0
$571$	0.686292	0.0287204	0.0143602	−	0.999897i	$-0.495429\pi$
$571$	0.686292	0.0287204	0.0143602	+	0.999897i	$0.495429\pi$
$572$	0	0
$573$	24.9706	1.04316
$574$	0	0
$575$	−3.65685	−0.152501
$576$	0	0
$577$	23.6569	0.984848	0.492424	−	0.870356i	$-0.336111\pi$
$577$	23.6569	0.984848	0.492424	+	0.870356i	$0.336111\pi$
$578$	0	0
$579$	−0.686292	−0.0285213
$580$	0	0
$581$	15.3137	0.635320
$582$	0	0
$583$	1.17157	0.0485216
$584$	0	0
$585$	0.585786	0.0242193
$586$	0	0
$587$	−18.1005	−0.747088	−0.373544	−	0.927613i	$-0.621857\pi$
$587$	−18.1005	−0.747088	−0.373544	+	0.927613i	$0.621857\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−25.6569	−1.05538
$592$	0	0
$593$	37.0711	1.52233	0.761163	−	0.648560i	$-0.224629\pi$
$593$	37.0711	1.52233	0.761163	+	0.648560i	$0.224629\pi$
$594$	0	0
$595$	−0.585786	−0.0240149
$596$	0	0
$597$	3.17157	0.129804
$598$	0	0
$599$	−5.17157	−0.211305	−0.105652	−	0.994403i	$-0.533693\pi$
$599$	−5.17157	−0.211305	−0.105652	+	0.994403i	$0.533693\pi$
$600$	0	0
$601$	31.3553	1.27901	0.639505	−	0.768787i	$-0.279139\pi$
$601$	31.3553	1.27901	0.639505	+	0.768787i	$0.279139\pi$
$602$	0	0
$603$	−15.6569	−0.637596
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−2.82843	−0.114802	−0.0574012	−	0.998351i	$-0.518281\pi$
$607$	−2.82843	−0.114802	−0.0574012	+	0.998351i	$0.518281\pi$
$608$	0	0
$609$	−1.17157	−0.0474745
$610$	0	0
$611$	−1.51472	−0.0612790
$612$	0	0
$613$	6.34315	0.256197	0.128099	−	0.991761i	$-0.459113\pi$
$613$	6.34315	0.256197	0.128099	+	0.991761i	$0.459113\pi$
$614$	0	0
$615$	−11.6569	−0.470050
$616$	0	0
$617$	−14.8284	−0.596970	−0.298485	−	0.954414i	$-0.596481\pi$
$617$	−14.8284	−0.596970	−0.298485	+	0.954414i	$0.596481\pi$
$618$	0	0
$619$	−39.2132	−1.57611	−0.788056	−	0.615604i	$-0.788912\pi$
$619$	−39.2132	−1.57611	−0.788056	+	0.615604i	$0.788912\pi$
$620$	0	0
$621$	20.6863	0.830112
$622$	0	0
$623$	4.82843	0.193447
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.34315	0.252918
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	10.3431	0.411103
$634$	0	0
$635$	−9.65685	−0.383221
$636$	0	0
$637$	−0.585786	−0.0232097
$638$	0	0
$639$	−12.4853	−0.493910
$640$	0	0
$641$	23.3137	0.920836	0.460418	−	0.887702i	$-0.347700\pi$
$641$	23.3137	0.920836	0.460418	+	0.887702i	$0.347700\pi$
$642$	0	0
$643$	12.0416	0.474876	0.237438	−	0.971403i	$-0.423692\pi$
$643$	12.0416	0.474876	0.237438	+	0.971403i	$0.423692\pi$
$644$	0	0
$645$	−10.8284	−0.426369
$646$	0	0
$647$	24.7279	0.972155	0.486077	−	0.873916i	$-0.338427\pi$
$647$	24.7279	0.972155	0.486077	+	0.873916i	$0.338427\pi$
$648$	0	0
$649$	10.2426	0.402059
$650$	0	0
$651$	−4.82843	−0.189241
$652$	0	0
$653$	−47.2548	−1.84922	−0.924612	−	0.380910i	$-0.875611\pi$
$653$	−47.2548	−1.84922	−0.924612	+	0.380910i	$0.875611\pi$
$654$	0	0
$655$	−15.3137	−0.598356
$656$	0	0
$657$	−0.585786	−0.0228537
$658$	0	0
$659$	−46.4853	−1.81081	−0.905405	−	0.424549i	$-0.860432\pi$
$659$	−46.4853	−1.81081	−0.905405	+	0.424549i	$0.860432\pi$
$660$	0	0
$661$	19.1716	0.745688	0.372844	−	0.927894i	$-0.378383\pi$
$661$	19.1716	0.745688	0.372844	+	0.927894i	$0.378383\pi$
$662$	0	0
$663$	0.485281	0.0188468
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.02944	0.117300
$668$	0	0
$669$	−14.0000	−0.541271
$670$	0	0
$671$	5.41421	0.209013
$672$	0	0
$673$	4.97056	0.191601	0.0958006	−	0.995401i	$-0.469459\pi$
$673$	4.97056	0.191601	0.0958006	+	0.995401i	$0.469459\pi$
$674$	0	0
$675$	−5.65685	−0.217732
$676$	0	0
$677$	16.3848	0.629718	0.314859	−	0.949138i	$-0.398043\pi$
$677$	16.3848	0.629718	0.314859	+	0.949138i	$0.398043\pi$
$678$	0	0
$679$	−0.343146	−0.0131687
$680$	0	0
$681$	−24.9706	−0.956874
$682$	0	0
$683$	−11.1716	−0.427468	−0.213734	−	0.976892i	$-0.568563\pi$
$683$	−11.1716	−0.427468	−0.213734	+	0.976892i	$0.568563\pi$
$684$	0	0
$685$	−14.8284	−0.566565
$686$	0	0
$687$	20.4853	0.781562
$688$	0	0
$689$	0.686292	0.0261456
$690$	0	0
$691$	−2.72792	−0.103775	−0.0518875	−	0.998653i	$-0.516524\pi$
$691$	−2.72792	−0.103775	−0.0518875	+	0.998653i	$0.516524\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	14.1421	0.536442
$696$	0	0
$697$	4.82843	0.182890
$698$	0	0
$699$	−24.0000	−0.907763
$700$	0	0
$701$	−12.1421	−0.458602	−0.229301	−	0.973356i	$-0.573644\pi$
$701$	−12.1421	−0.458602	−0.229301	+	0.973356i	$0.573644\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	3.65685	0.137725
$706$	0	0
$707$	0.242641	0.00912544
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	14.4853	0.543240
$712$	0	0
$713$	12.4853	0.467577
$714$	0	0
$715$	0.585786	0.0219072
$716$	0	0
$717$	−12.6863	−0.473778
$718$	0	0
$719$	−25.5563	−0.953091	−0.476545	−	0.879150i	$-0.658111\pi$
$719$	−25.5563	−0.953091	−0.476545	+	0.879150i	$0.658111\pi$
$720$	0	0
$721$	3.07107	0.114373
$722$	0	0
$723$	18.9706	0.705523
$724$	0	0
$725$	−0.828427	−0.0307670
$726$	0	0
$727$	30.3848	1.12691	0.563454	−	0.826147i	$-0.309472\pi$
$727$	30.3848	1.12691	0.563454	+	0.826147i	$0.309472\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	4.48528	0.165894
$732$	0	0
$733$	45.3553	1.67524	0.837619	−	0.546255i	$-0.183947\pi$
$733$	45.3553	1.67524	0.837619	+	0.546255i	$0.183947\pi$
$734$	0	0
$735$	1.41421	0.0521641
$736$	0	0
$737$	−15.6569	−0.576728
$738$	0	0
$739$	51.5980	1.89806	0.949031	−	0.315182i	$-0.102066\pi$
$739$	51.5980	1.89806	0.949031	+	0.315182i	$0.102066\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11.0294	0.404631	0.202315	−	0.979320i	$-0.435153\pi$
$743$	11.0294	0.404631	0.202315	+	0.979320i	$0.435153\pi$
$744$	0	0
$745$	9.31371	0.341228
$746$	0	0
$747$	−15.3137	−0.560299
$748$	0	0
$749$	0	0
$750$	0	0
$751$	26.8284	0.978983	0.489492	−	0.872008i	$-0.337182\pi$
$751$	26.8284	0.978983	0.489492	+	0.872008i	$0.337182\pi$
$752$	0	0
$753$	32.8284	1.19633
$754$	0	0
$755$	21.7990	0.793346
$756$	0	0
$757$	−11.6569	−0.423676	−0.211838	−	0.977305i	$-0.567945\pi$
$757$	−11.6569	−0.423676	−0.211838	+	0.977305i	$0.567945\pi$
$758$	0	0
$759$	5.17157	0.187716
$760$	0	0
$761$	12.4437	0.451082	0.225541	−	0.974234i	$-0.427585\pi$
$761$	12.4437	0.451082	0.225541	+	0.974234i	$0.427585\pi$
$762$	0	0
$763$	0.828427	0.0299911
$764$	0	0
$765$	0.585786	0.0211792
$766$	0	0
$767$	6.00000	0.216647
$768$	0	0
$769$	−39.0711	−1.40894	−0.704469	−	0.709734i	$-0.748815\pi$
$769$	−39.0711	−1.40894	−0.704469	+	0.709734i	$0.748815\pi$
$770$	0	0
$771$	5.85786	0.210966
$772$	0	0
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	0	0
$775$	−3.41421	−0.122642
$776$	0	0
$777$	−15.3137	−0.549376
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−12.4853	−0.446758
$782$	0	0
$783$	4.68629	0.167474
$784$	0	0
$785$	−1.51472	−0.0540626
$786$	0	0
$787$	−17.8579	−0.636564	−0.318282	−	0.947996i	$-0.603106\pi$
$787$	−17.8579	−0.636564	−0.318282	+	0.947996i	$0.603106\pi$
$788$	0	0
$789$	29.6569	1.05581
$790$	0	0
$791$	−17.3137	−0.615605
$792$	0	0
$793$	3.17157	0.112626
$794$	0	0
$795$	−1.65685	−0.0587626
$796$	0	0
$797$	−32.1421	−1.13853	−0.569266	−	0.822153i	$-0.692773\pi$
$797$	−32.1421	−1.13853	−0.569266	+	0.822153i	$0.692773\pi$
$798$	0	0
$799$	−1.51472	−0.0535869
$800$	0	0
$801$	−4.82843	−0.170604
$802$	0	0
$803$	−0.585786	−0.0206720
$804$	0	0
$805$	−3.65685	−0.128887
$806$	0	0
$807$	4.20101	0.147883
$808$	0	0
$809$	46.0000	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$810$	0	0
$811$	−2.82843	−0.0993195	−0.0496598	−	0.998766i	$-0.515814\pi$
$811$	−2.82843	−0.0993195	−0.0496598	+	0.998766i	$0.515814\pi$
$812$	0	0
$813$	20.9706	0.735470
$814$	0	0
$815$	11.6569	0.408322
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0.585786	0.0204690
$820$	0	0
$821$	42.9706	1.49968	0.749841	−	0.661618i	$-0.230130\pi$
$821$	42.9706	1.49968	0.749841	+	0.661618i	$0.230130\pi$
$822$	0	0
$823$	33.5147	1.16825	0.584125	−	0.811664i	$-0.301438\pi$
$823$	33.5147	1.16825	0.584125	+	0.811664i	$0.301438\pi$
$824$	0	0
$825$	−1.41421	−0.0492366
$826$	0	0
$827$	−14.0000	−0.486828	−0.243414	−	0.969923i	$-0.578267\pi$
$827$	−14.0000	−0.486828	−0.243414	+	0.969923i	$0.578267\pi$
$828$	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
$831$	−4.00000	−0.138758
$832$	0	0
$833$	−0.585786	−0.0202963
$834$	0	0
$835$	16.4853	0.570497
$836$	0	0
$837$	19.3137	0.667579
$838$	0	0
$839$	−22.9289	−0.791595	−0.395797	−	0.918338i	$-0.629532\pi$
$839$	−22.9289	−0.791595	−0.395797	+	0.918338i	$0.629532\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	0	0
$843$	6.82843	0.235184
$844$	0	0
$845$	−12.6569	−0.435409
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−10.3431	−0.354976
$850$	0	0
$851$	39.5980	1.35740
$852$	0	0
$853$	13.7574	0.471043	0.235522	−	0.971869i	$-0.424320\pi$
$853$	13.7574	0.471043	0.235522	+	0.971869i	$0.424320\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.5563	1.41954	0.709769	−	0.704435i	$-0.248799\pi$
$857$	41.5563	1.41954	0.709769	+	0.704435i	$0.248799\pi$
$858$	0	0
$859$	−31.4142	−1.07184	−0.535920	−	0.844269i	$-0.680035\pi$
$859$	−31.4142	−1.07184	−0.535920	+	0.844269i	$0.680035\pi$
$860$	0	0
$861$	−11.6569	−0.397265
$862$	0	0
$863$	9.51472	0.323885	0.161942	−	0.986800i	$-0.448224\pi$
$863$	9.51472	0.323885	0.161942	+	0.986800i	$0.448224\pi$
$864$	0	0
$865$	−14.7279	−0.500764
$866$	0	0
$867$	−23.5563	−0.800016
$868$	0	0
$869$	14.4853	0.491380
$870$	0	0
$871$	−9.17157	−0.310767
$872$	0	0
$873$	0.343146	0.0116137
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−51.5980	−1.74234	−0.871170	−	0.490982i	$-0.836638\pi$
$877$	−51.5980	−1.74234	−0.871170	+	0.490982i	$0.836638\pi$
$878$	0	0
$879$	−16.8284	−0.567609
$880$	0	0
$881$	−14.9706	−0.504371	−0.252186	−	0.967679i	$-0.581149\pi$
$881$	−14.9706	−0.504371	−0.252186	+	0.967679i	$0.581149\pi$
$882$	0	0
$883$	−50.0833	−1.68544	−0.842718	−	0.538355i	$-0.819046\pi$
$883$	−50.0833	−1.68544	−0.842718	+	0.538355i	$0.819046\pi$
$884$	0	0
$885$	−14.4853	−0.486917
$886$	0	0
$887$	45.4558	1.52626	0.763129	−	0.646246i	$-0.223662\pi$
$887$	45.4558	1.52626	0.763129	+	0.646246i	$0.223662\pi$
$888$	0	0
$889$	−9.65685	−0.323880
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−19.3137	−0.645586
$896$	0	0
$897$	3.02944	0.101150
$898$	0	0
$899$	2.82843	0.0943333
$900$	0	0
$901$	0.686292	0.0228637
$902$	0	0
$903$	−10.8284	−0.360347
$904$	0	0
$905$	−11.6569	−0.387487
$906$	0	0
$907$	24.3431	0.808301	0.404150	−	0.914693i	$-0.367567\pi$
$907$	24.3431	0.808301	0.404150	+	0.914693i	$0.367567\pi$
$908$	0	0
$909$	−0.242641	−0.00804788
$910$	0	0
$911$	26.1421	0.866128	0.433064	−	0.901363i	$-0.357432\pi$
$911$	26.1421	0.866128	0.433064	+	0.901363i	$0.357432\pi$
$912$	0	0
$913$	−15.3137	−0.506810
$914$	0	0
$915$	−7.65685	−0.253128
$916$	0	0
$917$	−15.3137	−0.505703
$918$	0	0
$919$	48.9706	1.61539	0.807695	−	0.589601i	$-0.200715\pi$
$919$	48.9706	1.61539	0.807695	+	0.589601i	$0.200715\pi$
$920$	0	0
$921$	−0.686292	−0.0226141
$922$	0	0
$923$	−7.31371	−0.240734
$924$	0	0
$925$	−10.8284	−0.356036
$926$	0	0
$927$	−3.07107	−0.100867
$928$	0	0
$929$	−57.5980	−1.88973	−0.944864	−	0.327462i	$-0.893807\pi$
$929$	−57.5980	−1.88973	−0.944864	+	0.327462i	$0.893807\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−21.7990	−0.713667
$934$	0	0
$935$	0.585786	0.0191573
$936$	0	0
$937$	−4.58579	−0.149811	−0.0749056	−	0.997191i	$-0.523866\pi$
$937$	−4.58579	−0.149811	−0.0749056	+	0.997191i	$0.523866\pi$
$938$	0	0
$939$	−29.1716	−0.951978
$940$	0	0
$941$	−6.10051	−0.198871	−0.0994354	−	0.995044i	$-0.531704\pi$
$941$	−6.10051	−0.198871	−0.0994354	+	0.995044i	$0.531704\pi$
$942$	0	0
$943$	30.1421	0.981563
$944$	0	0
$945$	−5.65685	−0.184017
$946$	0	0
$947$	−1.79899	−0.0584593	−0.0292297	−	0.999573i	$-0.509305\pi$
$947$	−1.79899	−0.0584593	−0.0292297	+	0.999573i	$0.509305\pi$
$948$	0	0
$949$	−0.343146	−0.0111390
$950$	0	0
$951$	−1.45584	−0.0472090
$952$	0	0
$953$	37.4558	1.21331	0.606657	−	0.794964i	$-0.292510\pi$
$953$	37.4558	1.21331	0.606657	+	0.794964i	$0.292510\pi$
$954$	0	0
$955$	17.6569	0.571362
$956$	0	0
$957$	1.17157	0.0378716
$958$	0	0
$959$	−14.8284	−0.478835
$960$	0	0
$961$	−19.3431	−0.623972
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.485281	−0.0156218
$966$	0	0
$967$	1.02944	0.0331045	0.0165522	−	0.999863i	$-0.494731\pi$
$967$	1.02944	0.0331045	0.0165522	+	0.999863i	$0.494731\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.92893	−0.222360	−0.111180	−	0.993800i	$-0.535463\pi$
$971$	−6.92893	−0.222360	−0.111180	+	0.993800i	$0.535463\pi$
$972$	0	0
$973$	14.1421	0.453376
$974$	0	0
$975$	−0.828427	−0.0265309
$976$	0	0
$977$	54.2843	1.73671	0.868354	−	0.495945i	$-0.165178\pi$
$977$	54.2843	1.73671	0.868354	+	0.495945i	$0.165178\pi$
$978$	0	0
$979$	−4.82843	−0.154317
$980$	0	0
$981$	−0.828427	−0.0264496
$982$	0	0
$983$	−32.7279	−1.04386	−0.521929	−	0.852989i	$-0.674788\pi$
$983$	−32.7279	−1.04386	−0.521929	+	0.852989i	$0.674788\pi$
$984$	0	0
$985$	−18.1421	−0.578057
$986$	0	0
$987$	3.65685	0.116399
$988$	0	0
$989$	28.0000	0.890348
$990$	0	0
$991$	−18.1421	−0.576304	−0.288152	−	0.957585i	$-0.593041\pi$
$991$	−18.1421	−0.576304	−0.288152	+	0.957585i	$0.593041\pi$
$992$	0	0
$993$	5.65685	0.179515
$994$	0	0
$995$	2.24264	0.0710965
$996$	0	0
$997$	−37.3553	−1.18306	−0.591528	−	0.806285i	$-0.701475\pi$
$997$	−37.3553	−1.18306	−0.591528	+	0.806285i	$0.701475\pi$
$998$	0	0
$999$	61.2548	1.93802

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.h.1.1	✓	2	4.3	odd	2
6160.2.a.bc.1.2		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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} -1.00000 q^{11} -0.585786 q^{13} +1.41421 q^{15} -0.585786 q^{17} +1.41421 q^{21} -3.65685 q^{23} +1.00000 q^{25} -5.65685 q^{27} -0.828427 q^{29} -3.41421 q^{31} -1.41421 q^{33} +1.00000 q^{35} -10.8284 q^{37} -0.828427 q^{39} -8.24264 q^{41} -7.65685 q^{43} -1.00000 q^{45} +2.58579 q^{47} +1.00000 q^{49} -0.828427 q^{51} -1.17157 q^{53} -1.00000 q^{55} -10.2426 q^{59} -5.41421 q^{61} -1.00000 q^{63} -0.585786 q^{65} +15.6569 q^{67} -5.17157 q^{69} +12.4853 q^{71} +0.585786 q^{73} +1.41421 q^{75} -1.00000 q^{77} -14.4853 q^{79} -5.00000 q^{81} +15.3137 q^{83} -0.585786 q^{85} -1.17157 q^{87} +4.82843 q^{89} -0.585786 q^{91} -4.82843 q^{93} -0.343146 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 + 2 * q^7 - 2 * q^9	\( 2 q + 2 q^{5} + 2 q^{7} - 2 q^{9} - 2 q^{11} - 4 q^{13} - 4 q^{17} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 4 q^{31} + 2 q^{35} - 16 q^{37} + 4 q^{39} - 8 q^{41} - 4 q^{43} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 4 q^{51} - 8 q^{53} - 2 q^{55} - 12 q^{59} - 8 q^{61} - 2 q^{63} - 4 q^{65} + 20 q^{67} - 16 q^{69} + 8 q^{71} + 4 q^{73} - 2 q^{77} - 12 q^{79} - 10 q^{81} + 8 q^{83} - 4 q^{85} - 8 q^{87} + 4 q^{89} - 4 q^{91} - 4 q^{93} - 12 q^{97} + 2 q^{99}+O(q^{100}) \) 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 + 4 * q^29 - 4 * q^31 + 2 * q^35 - 16 * q^37 + 4 * q^39 - 8 * q^41 - 4 * q^43 - 2 * q^45 + 8 * q^47 + 2 * q^49 + 4 * q^51 - 8 * q^53 - 2 * q^55 - 12 * q^59 - 8 * q^61 - 2 * q^63 - 4 * q^65 + 20 * q^67 - 16 * q^69 + 8 * q^71 + 4 * q^73 - 2 * q^77 - 12 * q^79 - 10 * q^81 + 8 * q^83 - 4 * q^85 - 8 * q^87 + 4 * q^89 - 4 * q^91 - 4 * q^93 - 12 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more