LMFDB - Embedded newform 6160.2.a.y.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 385)
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)

$f(q)$	$=$	$q+1.41421 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.41421 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} -1.00000 q^{11} +3.41421 q^{13} -1.41421 q^{15} -0.585786 q^{17} +1.41421 q^{21} -8.82843 q^{23} +1.00000 q^{25} -5.65685 q^{27} -0.828427 q^{29} +1.75736 q^{31} -1.41421 q^{33} -1.00000 q^{35} +1.17157 q^{37} +4.82843 q^{39} +4.24264 q^{41} -6.00000 q^{43} +1.00000 q^{45} -1.41421 q^{47} +1.00000 q^{49} -0.828427 q^{51} +2.82843 q^{53} +1.00000 q^{55} +7.89949 q^{59} -13.8995 q^{61} -1.00000 q^{63} -3.41421 q^{65} -6.48528 q^{67} -12.4853 q^{69} -9.17157 q^{71} -5.07107 q^{73} +1.41421 q^{75} -1.00000 q^{77} -6.48528 q^{79} -5.00000 q^{81} -12.0000 q^{83} +0.585786 q^{85} -1.17157 q^{87} +0.828427 q^{89} +3.41421 q^{91} +2.48528 q^{93} +9.31371 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 12 q^{23} + 2 q^{25} + 4 q^{29} + 12 q^{31} - 2 q^{35} + 8 q^{37} + 4 q^{39} - 12 q^{43} + 2 q^{45} + 2 q^{49} + 4 q^{51} + 2 q^{55} - 4 q^{59} - 8 q^{61} - 2 q^{63} - 4 q^{65} + 4 q^{67} - 8 q^{69} - 24 q^{71} + 4 q^{73} - 2 q^{77} + 4 q^{79} - 10 q^{81} - 24 q^{83} + 4 q^{85} - 8 q^{87} - 4 q^{89} + 4 q^{91} - 12 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 - 12 * q^23 + 2 * q^25 + 4 * q^29 + 12 * q^31 - 2 * q^35 + 8 * q^37 + 4 * q^39 - 12 * q^43 + 2 * q^45 + 2 * q^49 + 4 * q^51 + 2 * q^55 - 4 * q^59 - 8 * q^61 - 2 * q^63 - 4 * q^65 + 4 * q^67 - 8 * q^69 - 24 * q^71 + 4 * q^73 - 2 * q^77 + 4 * q^79 - 10 * q^81 - 24 * q^83 + 4 * q^85 - 8 * q^87 - 4 * q^89 + 4 * q^91 - 12 * 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	$-0.366140\pi$
$3$	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
$13$	3.41421	0.946932	0.473466	−	0.880812i	$-0.343003\pi$
$13$	3.41421	0.946932	0.473466	+	0.880812i	$0.343003\pi$
$14$	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
$23$	−8.82843	−1.84085	−0.920427	−	0.390914i	$-0.872159\pi$
$23$	−8.82843	−1.84085	−0.920427	+	0.390914i	$0.872159\pi$
$24$	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
$31$	1.75736	0.315631	0.157816	−	0.987469i	$-0.449555\pi$
$31$	1.75736	0.315631	0.157816	+	0.987469i	$0.449555\pi$
$32$	0	0
$33$	−1.41421	−0.246183
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	1.17157	0.192605	0.0963027	−	0.995352i	$-0.469298\pi$
$37$	1.17157	0.192605	0.0963027	+	0.995352i	$0.469298\pi$
$38$	0	0
$39$	4.82843	0.773167
$40$	0	0
$41$	4.24264	0.662589	0.331295	−	0.943527i	$-0.392515\pi$
$41$	4.24264	0.662589	0.331295	+	0.943527i	$0.392515\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−1.41421	−0.206284	−0.103142	−	0.994667i	$-0.532890\pi$
$47$	−1.41421	−0.206284	−0.103142	+	0.994667i	$0.532890\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.828427	−0.116003
$52$	0	0
$53$	2.82843	0.388514	0.194257	−	0.980951i	$-0.437770\pi$
$53$	2.82843	0.388514	0.194257	+	0.980951i	$0.437770\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.89949	1.02843	0.514213	−	0.857662i	$-0.328084\pi$
$59$	7.89949	1.02843	0.514213	+	0.857662i	$0.328084\pi$
$60$	0	0
$61$	−13.8995	−1.77965	−0.889824	−	0.456304i	$-0.849173\pi$
$61$	−13.8995	−1.77965	−0.889824	+	0.456304i	$0.849173\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−3.41421	−0.423481
$66$	0	0
$67$	−6.48528	−0.792303	−0.396152	−	0.918185i	$-0.629655\pi$
$67$	−6.48528	−0.792303	−0.396152	+	0.918185i	$0.629655\pi$
$68$	0	0
$69$	−12.4853	−1.50305
$70$	0	0
$71$	−9.17157	−1.08847	−0.544233	−	0.838934i	$-0.683179\pi$
$71$	−9.17157	−1.08847	−0.544233	+	0.838934i	$0.683179\pi$
$72$	0	0
$73$	−5.07107	−0.593524	−0.296762	−	0.954952i	$-0.595907\pi$
$73$	−5.07107	−0.593524	−0.296762	+	0.954952i	$0.595907\pi$
$74$	0	0
$75$	1.41421	0.163299
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−6.48528	−0.729651	−0.364826	−	0.931076i	$-0.618871\pi$
$79$	−6.48528	−0.729651	−0.364826	+	0.931076i	$0.618871\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	0.585786	0.0635375
$86$	0	0
$87$	−1.17157	−0.125606
$88$	0	0
$89$	0.828427	0.0878131	0.0439065	−	0.999036i	$-0.486020\pi$
$89$	0.828427	0.0878131	0.0439065	+	0.999036i	$0.486020\pi$
$90$	0	0
$91$	3.41421	0.357907
$92$	0	0
$93$	2.48528	0.257712
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.31371	0.945664	0.472832	−	0.881153i	$-0.343232\pi$
$97$	9.31371	0.945664	0.472832	+	0.881153i	$0.343232\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	5.41421	0.538734	0.269367	−	0.963038i	$-0.413185\pi$
$101$	5.41421	0.538734	0.269367	+	0.963038i	$0.413185\pi$
$102$	0	0
$103$	15.0711	1.48500	0.742498	−	0.669848i	$-0.233641\pi$
$103$	15.0711	1.48500	0.742498	+	0.669848i	$0.233641\pi$
$104$	0	0
$105$	−1.41421	−0.138013
$106$	0	0
$107$	−19.3137	−1.86713	−0.933563	−	0.358412i	$-0.883318\pi$
$107$	−19.3137	−1.86713	−0.933563	+	0.358412i	$0.883318\pi$
$108$	0	0
$109$	−4.82843	−0.462479	−0.231240	−	0.972897i	$-0.574278\pi$
$109$	−4.82843	−0.462479	−0.231240	+	0.972897i	$0.574278\pi$
$110$	0	0
$111$	1.65685	0.157262
$112$	0	0
$113$	−3.65685	−0.344008	−0.172004	−	0.985096i	$-0.555024\pi$
$113$	−3.65685	−0.344008	−0.172004	+	0.985096i	$0.555024\pi$
$114$	0	0
$115$	8.82843	0.823255
$116$	0	0
$117$	−3.41421	−0.315644
$118$	0	0
$119$	−0.585786	−0.0536990
$120$	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
$127$	7.31371	0.648987	0.324493	−	0.945888i	$-0.394806\pi$
$127$	7.31371	0.648987	0.324493	+	0.945888i	$0.394806\pi$
$128$	0	0
$129$	−8.48528	−0.747087
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.65685	0.486864
$136$	0	0
$137$	8.48528	0.724947	0.362473	−	0.931994i	$-0.381932\pi$
$137$	8.48528	0.724947	0.362473	+	0.931994i	$0.381932\pi$
$138$	0	0
$139$	12.4853	1.05899	0.529494	−	0.848314i	$-0.322382\pi$
$139$	12.4853	1.05899	0.529494	+	0.848314i	$0.322382\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	−3.41421	−0.285511
$144$	0	0
$145$	0.828427	0.0687971
$146$	0	0
$147$	1.41421	0.116642
$148$	0	0
$149$	11.6569	0.954967	0.477483	−	0.878641i	$-0.341549\pi$
$149$	11.6569	0.954967	0.477483	+	0.878641i	$0.341549\pi$
$150$	0	0
$151$	−14.4853	−1.17880	−0.589398	−	0.807843i	$-0.700635\pi$
$151$	−14.4853	−1.17880	−0.589398	+	0.807843i	$0.700635\pi$
$152$	0	0
$153$	0.585786	0.0473580
$154$	0	0
$155$	−1.75736	−0.141154
$156$	0	0
$157$	6.48528	0.517582	0.258791	−	0.965933i	$-0.416676\pi$
$157$	6.48528	0.517582	0.258791	+	0.965933i	$0.416676\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	−8.82843	−0.695778
$162$	0	0
$163$	6.48528	0.507966	0.253983	−	0.967209i	$-0.418259\pi$
$163$	6.48528	0.507966	0.253983	+	0.967209i	$0.418259\pi$
$164$	0	0
$165$	1.41421	0.110096
$166$	0	0
$167$	−20.4853	−1.58520	−0.792599	−	0.609743i	$-0.791273\pi$
$167$	−20.4853	−1.58520	−0.792599	+	0.609743i	$0.791273\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.92893	0.222683	0.111341	−	0.993782i	$-0.464485\pi$
$173$	2.92893	0.222683	0.111341	+	0.993782i	$0.464485\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	11.1716	0.839707
$178$	0	0
$179$	−3.31371	−0.247678	−0.123839	−	0.992302i	$-0.539521\pi$
$179$	−3.31371	−0.247678	−0.123839	+	0.992302i	$0.539521\pi$
$180$	0	0
$181$	−25.3137	−1.88155	−0.940777	−	0.339027i	$-0.889902\pi$
$181$	−25.3137	−1.88155	−0.940777	+	0.339027i	$0.889902\pi$
$182$	0	0
$183$	−19.6569	−1.45308
$184$	0	0
$185$	−1.17157	−0.0861358
$186$	0	0
$187$	0.585786	0.0428369
$188$	0	0
$189$	−5.65685	−0.411476
$190$	0	0
$191$	−26.6274	−1.92669	−0.963346	−	0.268261i	$-0.913551\pi$
$191$	−26.6274	−1.92669	−0.963346	+	0.268261i	$0.913551\pi$
$192$	0	0
$193$	13.6569	0.983042	0.491521	−	0.870866i	$-0.336441\pi$
$193$	13.6569	0.983042	0.491521	+	0.870866i	$0.336441\pi$
$194$	0	0
$195$	−4.82843	−0.345771
$196$	0	0
$197$	−18.6274	−1.32715	−0.663574	−	0.748110i	$-0.730961\pi$
$197$	−18.6274	−1.32715	−0.663574	+	0.748110i	$0.730961\pi$
$198$	0	0
$199$	−19.8995	−1.41064	−0.705319	−	0.708890i	$-0.749196\pi$
$199$	−19.8995	−1.41064	−0.705319	+	0.708890i	$0.749196\pi$
$200$	0	0
$201$	−9.17157	−0.646913
$202$	0	0
$203$	−0.828427	−0.0581442
$204$	0	0
$205$	−4.24264	−0.296319
$206$	0	0
$207$	8.82843	0.613618
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.9706	−0.892930	−0.446465	−	0.894801i	$-0.647317\pi$
$211$	−12.9706	−0.892930	−0.446465	+	0.894801i	$0.647317\pi$
$212$	0	0
$213$	−12.9706	−0.888728
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	1.75736	0.119297
$218$	0	0
$219$	−7.17157	−0.484610
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	14.3848	0.963276	0.481638	−	0.876370i	$-0.340042\pi$
$223$	14.3848	0.963276	0.481638	+	0.876370i	$0.340042\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	3.31371	0.219939	0.109969	−	0.993935i	$-0.464925\pi$
$227$	3.31371	0.219939	0.109969	+	0.993935i	$0.464925\pi$
$228$	0	0
$229$	16.1421	1.06670	0.533351	−	0.845894i	$-0.320932\pi$
$229$	16.1421	1.06670	0.533351	+	0.845894i	$0.320932\pi$
$230$	0	0
$231$	−1.41421	−0.0930484
$232$	0	0
$233$	−1.85786	−0.121713	−0.0608564	−	0.998147i	$-0.519383\pi$
$233$	−1.85786	−0.121713	−0.0608564	+	0.998147i	$0.519383\pi$
$234$	0	0
$235$	1.41421	0.0922531
$236$	0	0
$237$	−9.17157	−0.595758
$238$	0	0
$239$	−18.3431	−1.18652	−0.593260	−	0.805011i	$-0.702159\pi$
$239$	−18.3431	−1.18652	−0.593260	+	0.805011i	$0.702159\pi$
$240$	0	0
$241$	21.2132	1.36646	0.683231	−	0.730202i	$-0.260574\pi$
$241$	21.2132	1.36646	0.683231	+	0.730202i	$0.260574\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−16.9706	−1.07547
$250$	0	0
$251$	18.7279	1.18210	0.591048	−	0.806636i	$-0.298714\pi$
$251$	18.7279	1.18210	0.591048	+	0.806636i	$0.298714\pi$
$252$	0	0
$253$	8.82843	0.555038
$254$	0	0
$255$	0.828427	0.0518781
$256$	0	0
$257$	−12.1421	−0.757406	−0.378703	−	0.925518i	$-0.623630\pi$
$257$	−12.1421	−0.757406	−0.378703	+	0.925518i	$0.623630\pi$
$258$	0	0
$259$	1.17157	0.0727980
$260$	0	0
$261$	0.828427	0.0512784
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−2.82843	−0.173749
$266$	0	0
$267$	1.17157	0.0716991
$268$	0	0
$269$	−17.3137	−1.05564	−0.527818	−	0.849358i	$-0.676990\pi$
$269$	−17.3137	−1.05564	−0.527818	+	0.849358i	$0.676990\pi$
$270$	0	0
$271$	23.7990	1.44569	0.722843	−	0.691012i	$-0.242835\pi$
$271$	23.7990	1.44569	0.722843	+	0.691012i	$0.242835\pi$
$272$	0	0
$273$	4.82843	0.292230
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−2.34315	−0.140786	−0.0703930	−	0.997519i	$-0.522425\pi$
$277$	−2.34315	−0.140786	−0.0703930	+	0.997519i	$0.522425\pi$
$278$	0	0
$279$	−1.75736	−0.105210
$280$	0	0
$281$	−32.8284	−1.95838	−0.979190	−	0.202946i	$-0.934948\pi$
$281$	−32.8284	−1.95838	−0.979190	+	0.202946i	$0.934948\pi$
$282$	0	0
$283$	13.6569	0.811816	0.405908	−	0.913914i	$-0.366955\pi$
$283$	13.6569	0.811816	0.405908	+	0.913914i	$0.366955\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.24264	0.250435
$288$	0	0
$289$	−16.6569	−0.979815
$290$	0	0
$291$	13.1716	0.772131
$292$	0	0
$293$	12.3848	0.723526	0.361763	−	0.932270i	$-0.382175\pi$
$293$	12.3848	0.723526	0.361763	+	0.932270i	$0.382175\pi$
$294$	0	0
$295$	−7.89949	−0.459926
$296$	0	0
$297$	5.65685	0.328244
$298$	0	0
$299$	−30.1421	−1.74316
$300$	0	0
$301$	−6.00000	−0.345834
$302$	0	0
$303$	7.65685	0.439875
$304$	0	0
$305$	13.8995	0.795883
$306$	0	0
$307$	6.14214	0.350550	0.175275	−	0.984519i	$-0.443919\pi$
$307$	6.14214	0.350550	0.175275	+	0.984519i	$0.443919\pi$
$308$	0	0
$309$	21.3137	1.21249
$310$	0	0
$311$	−20.5858	−1.16731	−0.583656	−	0.812001i	$-0.698378\pi$
$311$	−20.5858	−1.16731	−0.583656	+	0.812001i	$0.698378\pi$
$312$	0	0
$313$	−19.6569	−1.11107	−0.555536	−	0.831493i	$-0.687487\pi$
$313$	−19.6569	−1.11107	−0.555536	+	0.831493i	$0.687487\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−24.6274	−1.38321	−0.691607	−	0.722274i	$-0.743097\pi$
$317$	−24.6274	−1.38321	−0.691607	+	0.722274i	$0.743097\pi$
$318$	0	0
$319$	0.828427	0.0463830
$320$	0	0
$321$	−27.3137	−1.52450
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.41421	0.189386
$326$	0	0
$327$	−6.82843	−0.377613
$328$	0	0
$329$	−1.41421	−0.0779681
$330$	0	0
$331$	25.6569	1.41023	0.705114	−	0.709094i	$-0.250896\pi$
$331$	25.6569	1.41023	0.705114	+	0.709094i	$0.250896\pi$
$332$	0	0
$333$	−1.17157	−0.0642018
$334$	0	0
$335$	6.48528	0.354329
$336$	0	0
$337$	20.4853	1.11590	0.557952	−	0.829873i	$-0.311587\pi$
$337$	20.4853	1.11590	0.557952	+	0.829873i	$0.311587\pi$
$338$	0	0
$339$	−5.17157	−0.280881
$340$	0	0
$341$	−1.75736	−0.0951663
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	12.4853	0.672185
$346$	0	0
$347$	10.9706	0.588931	0.294465	−	0.955662i	$-0.404858\pi$
$347$	10.9706	0.588931	0.294465	+	0.955662i	$0.404858\pi$
$348$	0	0
$349$	16.7279	0.895425	0.447713	−	0.894178i	$-0.352239\pi$
$349$	16.7279	0.895425	0.447713	+	0.894178i	$0.352239\pi$
$350$	0	0
$351$	−19.3137	−1.03089
$352$	0	0
$353$	−20.3431	−1.08276	−0.541378	−	0.840779i	$-0.682097\pi$
$353$	−20.3431	−1.08276	−0.541378	+	0.840779i	$0.682097\pi$
$354$	0	0
$355$	9.17157	0.486777
$356$	0	0
$357$	−0.828427	−0.0438450
$358$	0	0
$359$	27.1716	1.43406	0.717030	−	0.697042i	$-0.245501\pi$
$359$	27.1716	1.43406	0.717030	+	0.697042i	$0.245501\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	1.41421	0.0742270
$364$	0	0
$365$	5.07107	0.265432
$366$	0	0
$367$	−29.6985	−1.55025	−0.775124	−	0.631809i	$-0.782313\pi$
$367$	−29.6985	−1.55025	−0.775124	+	0.631809i	$0.782313\pi$
$368$	0	0
$369$	−4.24264	−0.220863
$370$	0	0
$371$	2.82843	0.146845
$372$	0	0
$373$	−19.3137	−1.00003	−0.500013	−	0.866018i	$-0.666671\pi$
$373$	−19.3137	−1.00003	−0.500013	+	0.866018i	$0.666671\pi$
$374$	0	0
$375$	−1.41421	−0.0730297
$376$	0	0
$377$	−2.82843	−0.145671
$378$	0	0
$379$	29.1716	1.49844	0.749222	−	0.662319i	$-0.230428\pi$
$379$	29.1716	1.49844	0.749222	+	0.662319i	$0.230428\pi$
$380$	0	0
$381$	10.3431	0.529895
$382$	0	0
$383$	5.89949	0.301450	0.150725	−	0.988576i	$-0.451839\pi$
$383$	5.89949	0.301450	0.150725	+	0.988576i	$0.451839\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	6.00000	0.304997
$388$	0	0
$389$	38.2843	1.94109	0.970545	−	0.240921i	$-0.0774494\pi$
$389$	38.2843	1.94109	0.970545	+	0.240921i	$0.0774494\pi$
$390$	0	0
$391$	5.17157	0.261538
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	6.48528	0.326310
$396$	0	0
$397$	−31.1716	−1.56446	−0.782228	−	0.622992i	$-0.785917\pi$
$397$	−31.1716	−1.56446	−0.782228	+	0.622992i	$0.785917\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.31371	0.265354	0.132677	−	0.991159i	$-0.457643\pi$
$401$	5.31371	0.265354	0.132677	+	0.991159i	$0.457643\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	0	0
$405$	5.00000	0.248452
$406$	0	0
$407$	−1.17157	−0.0580727
$408$	0	0
$409$	21.2132	1.04893	0.524463	−	0.851433i	$-0.324266\pi$
$409$	21.2132	1.04893	0.524463	+	0.851433i	$0.324266\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	7.89949	0.388709
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	17.6569	0.864660
$418$	0	0
$419$	4.10051	0.200323	0.100161	−	0.994971i	$-0.468064\pi$
$419$	4.10051	0.200323	0.100161	+	0.994971i	$0.468064\pi$
$420$	0	0
$421$	−28.9706	−1.41194	−0.705969	−	0.708242i	$-0.749488\pi$
$421$	−28.9706	−1.41194	−0.705969	+	0.708242i	$0.749488\pi$
$422$	0	0
$423$	1.41421	0.0687614
$424$	0	0
$425$	−0.585786	−0.0284148
$426$	0	0
$427$	−13.8995	−0.672644
$428$	0	0
$429$	−4.82843	−0.233119
$430$	0	0
$431$	25.1127	1.20964	0.604818	−	0.796364i	$-0.293246\pi$
$431$	25.1127	1.20964	0.604818	+	0.796364i	$0.293246\pi$
$432$	0	0
$433$	0.142136	0.00683060	0.00341530	−	0.999994i	$-0.498913\pi$
$433$	0.142136	0.00683060	0.00341530	+	0.999994i	$0.498913\pi$
$434$	0	0
$435$	1.17157	0.0561726
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−33.4558	−1.59676	−0.798380	−	0.602154i	$-0.794309\pi$
$439$	−33.4558	−1.59676	−0.798380	+	0.602154i	$0.794309\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	28.6274	1.36013	0.680065	−	0.733152i	$-0.261952\pi$
$443$	28.6274	1.36013	0.680065	+	0.733152i	$0.261952\pi$
$444$	0	0
$445$	−0.828427	−0.0392712
$446$	0	0
$447$	16.4853	0.779727
$448$	0	0
$449$	18.3431	0.865667	0.432833	−	0.901474i	$-0.357514\pi$
$449$	18.3431	0.865667	0.432833	+	0.901474i	$0.357514\pi$
$450$	0	0
$451$	−4.24264	−0.199778
$452$	0	0
$453$	−20.4853	−0.962482
$454$	0	0
$455$	−3.41421	−0.160061
$456$	0	0
$457$	32.9706	1.54230	0.771149	−	0.636655i	$-0.219682\pi$
$457$	32.9706	1.54230	0.771149	+	0.636655i	$0.219682\pi$
$458$	0	0
$459$	3.31371	0.154671
$460$	0	0
$461$	6.58579	0.306731	0.153365	−	0.988170i	$-0.450989\pi$
$461$	6.58579	0.306731	0.153365	+	0.988170i	$0.450989\pi$
$462$	0	0
$463$	−40.1421	−1.86556	−0.932782	−	0.360442i	$-0.882626\pi$
$463$	−40.1421	−1.86556	−0.932782	+	0.360442i	$0.882626\pi$
$464$	0	0
$465$	−2.48528	−0.115252
$466$	0	0
$467$	−12.0416	−0.557220	−0.278610	−	0.960404i	$-0.589874\pi$
$467$	−12.0416	−0.557220	−0.278610	+	0.960404i	$0.589874\pi$
$468$	0	0
$469$	−6.48528	−0.299462
$470$	0	0
$471$	9.17157	0.422604
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.82843	−0.129505
$478$	0	0
$479$	4.97056	0.227111	0.113555	−	0.993532i	$-0.463776\pi$
$479$	4.97056	0.227111	0.113555	+	0.993532i	$0.463776\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	−12.4853	−0.568100
$484$	0	0
$485$	−9.31371	−0.422914
$486$	0	0
$487$	−15.4558	−0.700371	−0.350186	−	0.936680i	$-0.613881\pi$
$487$	−15.4558	−0.700371	−0.350186	+	0.936680i	$0.613881\pi$
$488$	0	0
$489$	9.17157	0.414753
$490$	0	0
$491$	−20.1421	−0.909002	−0.454501	−	0.890746i	$-0.650182\pi$
$491$	−20.1421	−0.909002	−0.454501	+	0.890746i	$0.650182\pi$
$492$	0	0
$493$	0.485281	0.0218560
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	−9.17157	−0.411401
$498$	0	0
$499$	−6.82843	−0.305682	−0.152841	−	0.988251i	$-0.548842\pi$
$499$	−6.82843	−0.305682	−0.152841	+	0.988251i	$0.548842\pi$
$500$	0	0
$501$	−28.9706	−1.29431
$502$	0	0
$503$	−18.3431	−0.817880	−0.408940	−	0.912561i	$-0.634102\pi$
$503$	−18.3431	−0.817880	−0.408940	+	0.912561i	$0.634102\pi$
$504$	0	0
$505$	−5.41421	−0.240929
$506$	0	0
$507$	−1.89949	−0.0843595
$508$	0	0
$509$	12.8284	0.568610	0.284305	−	0.958734i	$-0.408237\pi$
$509$	12.8284	0.568610	0.284305	+	0.958734i	$0.408237\pi$
$510$	0	0
$511$	−5.07107	−0.224331
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−15.0711	−0.664111
$516$	0	0
$517$	1.41421	0.0621970
$518$	0	0
$519$	4.14214	0.181820
$520$	0	0
$521$	2.68629	0.117689	0.0588443	−	0.998267i	$-0.481258\pi$
$521$	2.68629	0.117689	0.0588443	+	0.998267i	$0.481258\pi$
$522$	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
$527$	−1.02944	−0.0448430
$528$	0	0
$529$	54.9411	2.38874
$530$	0	0
$531$	−7.89949	−0.342809
$532$	0	0
$533$	14.4853	0.627427
$534$	0	0
$535$	19.3137	0.835004
$536$	0	0
$537$	−4.68629	−0.202228
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	−35.7990	−1.53628
$544$	0	0
$545$	4.82843	0.206827
$546$	0	0
$547$	−3.02944	−0.129529	−0.0647647	−	0.997901i	$-0.520630\pi$
$547$	−3.02944	−0.129529	−0.0647647	+	0.997901i	$0.520630\pi$
$548$	0	0
$549$	13.8995	0.593216
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−6.48528	−0.275782
$554$	0	0
$555$	−1.65685	−0.0703295
$556$	0	0
$557$	17.6569	0.748145	0.374072	−	0.927399i	$-0.377961\pi$
$557$	17.6569	0.748145	0.374072	+	0.927399i	$0.377961\pi$
$558$	0	0
$559$	−20.4853	−0.866435
$560$	0	0
$561$	0.828427	0.0349762
$562$	0	0
$563$	19.7990	0.834428	0.417214	−	0.908808i	$-0.363007\pi$
$563$	19.7990	0.834428	0.417214	+	0.908808i	$0.363007\pi$
$564$	0	0
$565$	3.65685	0.153845
$566$	0	0
$567$	−5.00000	−0.209980
$568$	0	0
$569$	−12.1421	−0.509025	−0.254512	−	0.967070i	$-0.581915\pi$
$569$	−12.1421	−0.509025	−0.254512	+	0.967070i	$0.581915\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	−37.6569	−1.57314
$574$	0	0
$575$	−8.82843	−0.368171
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	0	0
$579$	19.3137	0.802650
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−2.82843	−0.117141
$584$	0	0
$585$	3.41421	0.141160
$586$	0	0
$587$	27.5563	1.13737	0.568686	−	0.822555i	$-0.307452\pi$
$587$	27.5563	1.13737	0.568686	+	0.822555i	$0.307452\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−26.3431	−1.08361
$592$	0	0
$593$	18.7279	0.769064	0.384532	−	0.923112i	$-0.374363\pi$
$593$	18.7279	0.769064	0.384532	+	0.923112i	$0.374363\pi$
$594$	0	0
$595$	0.585786	0.0240149
$596$	0	0
$597$	−28.1421	−1.15178
$598$	0	0
$599$	9.85786	0.402781	0.201391	−	0.979511i	$-0.435454\pi$
$599$	9.85786	0.402781	0.201391	+	0.979511i	$0.435454\pi$
$600$	0	0
$601$	19.2721	0.786124	0.393062	−	0.919512i	$-0.371416\pi$
$601$	19.2721	0.786124	0.393062	+	0.919512i	$0.371416\pi$
$602$	0	0
$603$	6.48528	0.264101
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−25.1716	−1.02168	−0.510841	−	0.859675i	$-0.670666\pi$
$607$	−25.1716	−1.02168	−0.510841	+	0.859675i	$0.670666\pi$
$608$	0	0
$609$	−1.17157	−0.0474745
$610$	0	0
$611$	−4.82843	−0.195337
$612$	0	0
$613$	−20.4853	−0.827393	−0.413696	−	0.910415i	$-0.635762\pi$
$613$	−20.4853	−0.827393	−0.413696	+	0.910415i	$0.635762\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−2.82843	−0.113868	−0.0569341	−	0.998378i	$-0.518132\pi$
$617$	−2.82843	−0.113868	−0.0569341	+	0.998378i	$0.518132\pi$
$618$	0	0
$619$	−12.1005	−0.486360	−0.243180	−	0.969981i	$-0.578191\pi$
$619$	−12.1005	−0.486360	−0.243180	+	0.969981i	$0.578191\pi$
$620$	0	0
$621$	49.9411	2.00407
$622$	0	0
$623$	0.828427	0.0331902
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.686292	−0.0273642
$630$	0	0
$631$	−7.02944	−0.279837	−0.139919	−	0.990163i	$-0.544684\pi$
$631$	−7.02944	−0.279837	−0.139919	+	0.990163i	$0.544684\pi$
$632$	0	0
$633$	−18.3431	−0.729075
$634$	0	0
$635$	−7.31371	−0.290236
$636$	0	0
$637$	3.41421	0.135276
$638$	0	0
$639$	9.17157	0.362822
$640$	0	0
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	0	0
$643$	−16.2426	−0.640547	−0.320274	−	0.947325i	$-0.603775\pi$
$643$	−16.2426	−0.640547	−0.320274	+	0.947325i	$0.603775\pi$
$644$	0	0
$645$	8.48528	0.334108
$646$	0	0
$647$	−30.5858	−1.20245	−0.601226	−	0.799079i	$-0.705321\pi$
$647$	−30.5858	−1.20245	−0.601226	+	0.799079i	$0.705321\pi$
$648$	0	0
$649$	−7.89949	−0.310082
$650$	0	0
$651$	2.48528	0.0974059
$652$	0	0
$653$	−6.68629	−0.261655	−0.130827	−	0.991405i	$-0.541763\pi$
$653$	−6.68629	−0.261655	−0.130827	+	0.991405i	$0.541763\pi$
$654$	0	0
$655$	11.3137	0.442063
$656$	0	0
$657$	5.07107	0.197841
$658$	0	0
$659$	43.4558	1.69280	0.846400	−	0.532548i	$-0.178765\pi$
$659$	43.4558	1.69280	0.846400	+	0.532548i	$0.178765\pi$
$660$	0	0
$661$	17.7990	0.692300	0.346150	−	0.938179i	$-0.387489\pi$
$661$	17.7990	0.692300	0.346150	+	0.938179i	$0.387489\pi$
$662$	0	0
$663$	−2.82843	−0.109847
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.31371	0.283188
$668$	0	0
$669$	20.3431	0.786511
$670$	0	0
$671$	13.8995	0.536584
$672$	0	0
$673$	11.5147	0.443860	0.221930	−	0.975063i	$-0.428764\pi$
$673$	11.5147	0.443860	0.221930	+	0.975063i	$0.428764\pi$
$674$	0	0
$675$	−5.65685	−0.217732
$676$	0	0
$677$	10.4437	0.401382	0.200691	−	0.979655i	$-0.435681\pi$
$677$	10.4437	0.401382	0.200691	+	0.979655i	$0.435681\pi$
$678$	0	0
$679$	9.31371	0.357427
$680$	0	0
$681$	4.68629	0.179579
$682$	0	0
$683$	−17.3137	−0.662491	−0.331245	−	0.943545i	$-0.607469\pi$
$683$	−17.3137	−0.662491	−0.331245	+	0.943545i	$0.607469\pi$
$684$	0	0
$685$	−8.48528	−0.324206
$686$	0	0
$687$	22.8284	0.870959
$688$	0	0
$689$	9.65685	0.367897
$690$	0	0
$691$	−24.5858	−0.935287	−0.467644	−	0.883917i	$-0.654897\pi$
$691$	−24.5858	−0.935287	−0.467644	+	0.883917i	$0.654897\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	−12.4853	−0.473594
$696$	0	0
$697$	−2.48528	−0.0941367
$698$	0	0
$699$	−2.62742	−0.0993780
$700$	0	0
$701$	−50.7696	−1.91754	−0.958770	−	0.284184i	$-0.908277\pi$
$701$	−50.7696	−1.91754	−0.958770	+	0.284184i	$0.908277\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	5.41421	0.203622
$708$	0	0
$709$	−32.6274	−1.22535	−0.612674	−	0.790336i	$-0.709906\pi$
$709$	−32.6274	−1.22535	−0.612674	+	0.790336i	$0.709906\pi$
$710$	0	0
$711$	6.48528	0.243217
$712$	0	0
$713$	−15.5147	−0.581031
$714$	0	0
$715$	3.41421	0.127684
$716$	0	0
$717$	−25.9411	−0.968789
$718$	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
$721$	15.0711	0.561276
$722$	0	0
$723$	30.0000	1.11571
$724$	0	0
$725$	−0.828427	−0.0307670
$726$	0	0
$727$	45.6985	1.69486	0.847431	−	0.530905i	$-0.178148\pi$
$727$	45.6985	1.69486	0.847431	+	0.530905i	$0.178148\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	3.51472	0.129997
$732$	0	0
$733$	50.7279	1.87368	0.936839	−	0.349760i	$-0.113737\pi$
$733$	50.7279	1.87368	0.936839	+	0.349760i	$0.113737\pi$
$734$	0	0
$735$	−1.41421	−0.0521641
$736$	0	0
$737$	6.48528	0.238888
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.3137	0.561805	0.280903	−	0.959736i	$-0.409366\pi$
$743$	15.3137	0.561805	0.280903	+	0.959736i	$0.409366\pi$
$744$	0	0
$745$	−11.6569	−0.427074
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	−19.3137	−0.705708
$750$	0	0
$751$	−33.4558	−1.22082	−0.610411	−	0.792085i	$-0.708996\pi$
$751$	−33.4558	−1.22082	−0.610411	+	0.792085i	$0.708996\pi$
$752$	0	0
$753$	26.4853	0.965177
$754$	0	0
$755$	14.4853	0.527173
$756$	0	0
$757$	24.6274	0.895099	0.447549	−	0.894259i	$-0.352297\pi$
$757$	24.6274	0.895099	0.447549	+	0.894259i	$0.352297\pi$
$758$	0	0
$759$	12.4853	0.453187
$760$	0	0
$761$	−2.78680	−0.101021	−0.0505106	−	0.998724i	$-0.516085\pi$
$761$	−2.78680	−0.101021	−0.0505106	+	0.998724i	$0.516085\pi$
$762$	0	0
$763$	−4.82843	−0.174801
$764$	0	0
$765$	−0.585786	−0.0211792
$766$	0	0
$767$	26.9706	0.973851
$768$	0	0
$769$	−0.242641	−0.00874985	−0.00437492	−	0.999990i	$-0.501393\pi$
$769$	−0.242641	−0.00874985	−0.00437492	+	0.999990i	$0.501393\pi$
$770$	0	0
$771$	−17.1716	−0.618419
$772$	0	0
$773$	−13.3137	−0.478861	−0.239430	−	0.970914i	$-0.576961\pi$
$773$	−13.3137	−0.478861	−0.239430	+	0.970914i	$0.576961\pi$
$774$	0	0
$775$	1.75736	0.0631262
$776$	0	0
$777$	1.65685	0.0594393
$778$	0	0
$779$	0	0
$780$	0	0
$781$	9.17157	0.328185
$782$	0	0
$783$	4.68629	0.167474
$784$	0	0
$785$	−6.48528	−0.231470
$786$	0	0
$787$	−45.4558	−1.62033	−0.810163	−	0.586205i	$-0.800621\pi$
$787$	−45.4558	−1.62033	−0.810163	+	0.586205i	$0.800621\pi$
$788$	0	0
$789$	−16.9706	−0.604168
$790$	0	0
$791$	−3.65685	−0.130023
$792$	0	0
$793$	−47.4558	−1.68521
$794$	0	0
$795$	−4.00000	−0.141865
$796$	0	0
$797$	−40.1421	−1.42191	−0.710954	−	0.703239i	$-0.751736\pi$
$797$	−40.1421	−1.42191	−0.710954	+	0.703239i	$0.751736\pi$
$798$	0	0
$799$	0.828427	0.0293076
$800$	0	0
$801$	−0.828427	−0.0292710
$802$	0	0
$803$	5.07107	0.178954
$804$	0	0
$805$	8.82843	0.311161
$806$	0	0
$807$	−24.4853	−0.861923
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	41.4558	1.45571	0.727856	−	0.685730i	$-0.240517\pi$
$811$	41.4558	1.45571	0.727856	+	0.685730i	$0.240517\pi$
$812$	0	0
$813$	33.6569	1.18040
$814$	0	0
$815$	−6.48528	−0.227169
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−3.41421	−0.119302
$820$	0	0
$821$	12.3431	0.430779	0.215389	−	0.976528i	$-0.430898\pi$
$821$	12.3431	0.430779	0.215389	+	0.976528i	$0.430898\pi$
$822$	0	0
$823$	35.9411	1.25283	0.626414	−	0.779490i	$-0.284522\pi$
$823$	35.9411	1.25283	0.626414	+	0.779490i	$0.284522\pi$
$824$	0	0
$825$	−1.41421	−0.0492366
$826$	0	0
$827$	−40.6274	−1.41275	−0.706377	−	0.707836i	$-0.749672\pi$
$827$	−40.6274	−1.41275	−0.706377	+	0.707836i	$0.749672\pi$
$828$	0	0
$829$	1.02944	0.0357538	0.0178769	−	0.999840i	$-0.494309\pi$
$829$	1.02944	0.0357538	0.0178769	+	0.999840i	$0.494309\pi$
$830$	0	0
$831$	−3.31371	−0.114951
$832$	0	0
$833$	−0.585786	−0.0202963
$834$	0	0
$835$	20.4853	0.708922
$836$	0	0
$837$	−9.94113	−0.343616
$838$	0	0
$839$	32.5858	1.12499	0.562493	−	0.826802i	$-0.309842\pi$
$839$	32.5858	1.12499	0.562493	+	0.826802i	$0.309842\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	0	0
$843$	−46.4264	−1.59901
$844$	0	0
$845$	1.34315	0.0462056
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	19.3137	0.662845
$850$	0	0
$851$	−10.3431	−0.354558
$852$	0	0
$853$	52.6690	1.80335	0.901677	−	0.432410i	$-0.142337\pi$
$853$	52.6690	1.80335	0.901677	+	0.432410i	$0.142337\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−30.0416	−1.02620	−0.513101	−	0.858328i	$-0.671503\pi$
$857$	−30.0416	−1.02620	−0.513101	+	0.858328i	$0.671503\pi$
$858$	0	0
$859$	46.0416	1.57092	0.785460	−	0.618912i	$-0.212426\pi$
$859$	46.0416	1.57092	0.785460	+	0.618912i	$0.212426\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	0	0
$863$	−35.6569	−1.21377	−0.606887	−	0.794788i	$-0.707582\pi$
$863$	−35.6569	−1.21377	−0.606887	+	0.794788i	$0.707582\pi$
$864$	0	0
$865$	−2.92893	−0.0995867
$866$	0	0
$867$	−23.5563	−0.800016
$868$	0	0
$869$	6.48528	0.219998
$870$	0	0
$871$	−22.1421	−0.750258
$872$	0	0
$873$	−9.31371	−0.315221
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	20.4853	0.691739	0.345869	−	0.938283i	$-0.387584\pi$
$877$	20.4853	0.691739	0.345869	+	0.938283i	$0.387584\pi$
$878$	0	0
$879$	17.5147	0.590757
$880$	0	0
$881$	−16.6274	−0.560192	−0.280096	−	0.959972i	$-0.590366\pi$
$881$	−16.6274	−0.560192	−0.280096	+	0.959972i	$0.590366\pi$
$882$	0	0
$883$	−21.3137	−0.717263	−0.358632	−	0.933479i	$-0.616757\pi$
$883$	−21.3137	−0.717263	−0.358632	+	0.933479i	$0.616757\pi$
$884$	0	0
$885$	−11.1716	−0.375528
$886$	0	0
$887$	22.8284	0.766504	0.383252	−	0.923644i	$-0.374804\pi$
$887$	22.8284	0.766504	0.383252	+	0.923644i	$0.374804\pi$
$888$	0	0
$889$	7.31371	0.245294
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	0	0
$894$	0	0
$895$	3.31371	0.110765
$896$	0	0
$897$	−42.6274	−1.42329
$898$	0	0
$899$	−1.45584	−0.0485551
$900$	0	0
$901$	−1.65685	−0.0551978
$902$	0	0
$903$	−8.48528	−0.282372
$904$	0	0
$905$	25.3137	0.841456
$906$	0	0
$907$	48.4264	1.60797	0.803986	−	0.594648i	$-0.202709\pi$
$907$	48.4264	1.60797	0.803986	+	0.594648i	$0.202709\pi$
$908$	0	0
$909$	−5.41421	−0.179578
$910$	0	0
$911$	20.4853	0.678708	0.339354	−	0.940659i	$-0.389792\pi$
$911$	20.4853	0.678708	0.339354	+	0.940659i	$0.389792\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	19.6569	0.649836
$916$	0	0
$917$	−11.3137	−0.373612
$918$	0	0
$919$	41.9411	1.38351	0.691755	−	0.722132i	$-0.256838\pi$
$919$	41.9411	1.38351	0.691755	+	0.722132i	$0.256838\pi$
$920$	0	0
$921$	8.68629	0.286223
$922$	0	0
$923$	−31.3137	−1.03070
$924$	0	0
$925$	1.17157	0.0385211
$926$	0	0
$927$	−15.0711	−0.494999
$928$	0	0
$929$	6.68629	0.219370	0.109685	−	0.993966i	$-0.465016\pi$
$929$	6.68629	0.219370	0.109685	+	0.993966i	$0.465016\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−29.1127	−0.953107
$934$	0	0
$935$	−0.585786	−0.0191573
$936$	0	0
$937$	−17.2721	−0.564254	−0.282127	−	0.959377i	$-0.591040\pi$
$937$	−17.2721	−0.564254	−0.282127	+	0.959377i	$0.591040\pi$
$938$	0	0
$939$	−27.7990	−0.907186
$940$	0	0
$941$	−5.61522	−0.183051	−0.0915255	−	0.995803i	$-0.529174\pi$
$941$	−5.61522	−0.183051	−0.0915255	+	0.995803i	$0.529174\pi$
$942$	0	0
$943$	−37.4558	−1.21973
$944$	0	0
$945$	5.65685	0.184017
$946$	0	0
$947$	25.5980	0.831823	0.415911	−	0.909405i	$-0.363463\pi$
$947$	25.5980	0.831823	0.415911	+	0.909405i	$0.363463\pi$
$948$	0	0
$949$	−17.3137	−0.562027
$950$	0	0
$951$	−34.8284	−1.12939
$952$	0	0
$953$	41.2548	1.33638	0.668188	−	0.743993i	$-0.267070\pi$
$953$	41.2548	1.33638	0.668188	+	0.743993i	$0.267070\pi$
$954$	0	0
$955$	26.6274	0.861643
$956$	0	0
$957$	1.17157	0.0378716
$958$	0	0
$959$	8.48528	0.274004
$960$	0	0
$961$	−27.9117	−0.900377
$962$	0	0
$963$	19.3137	0.622376
$964$	0	0
$965$	−13.6569	−0.439630
$966$	0	0
$967$	5.02944	0.161736	0.0808679	−	0.996725i	$-0.474231\pi$
$967$	5.02944	0.161736	0.0808679	+	0.996725i	$0.474231\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.5858	0.403897	0.201949	−	0.979396i	$-0.435273\pi$
$971$	12.5858	0.403897	0.201949	+	0.979396i	$0.435273\pi$
$972$	0	0
$973$	12.4853	0.400260
$974$	0	0
$975$	4.82843	0.154633
$976$	0	0
$977$	−9.31371	−0.297972	−0.148986	−	0.988839i	$-0.547601\pi$
$977$	−9.31371	−0.297972	−0.148986	+	0.988839i	$0.547601\pi$
$978$	0	0
$979$	−0.828427	−0.0264766
$980$	0	0
$981$	4.82843	0.154160
$982$	0	0
$983$	−14.1005	−0.449736	−0.224868	−	0.974389i	$-0.572195\pi$
$983$	−14.1005	−0.449736	−0.224868	+	0.974389i	$0.572195\pi$
$984$	0	0
$985$	18.6274	0.593519
$986$	0	0
$987$	−2.00000	−0.0636607
$988$	0	0
$989$	52.9706	1.68437
$990$	0	0
$991$	37.4558	1.18982	0.594912	−	0.803791i	$-0.297187\pi$
$991$	37.4558	1.18982	0.594912	+	0.803791i	$0.297187\pi$
$992$	0	0
$993$	36.2843	1.15145
$994$	0	0
$995$	19.8995	0.630856
$996$	0	0
$997$	30.2426	0.957794	0.478897	−	0.877871i	$-0.341037\pi$
$997$	30.2426	0.957794	0.478897	+	0.877871i	$0.341037\pi$
$998$	0	0
$999$	−6.62742	−0.209682

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.y.1.2		2
4.3	odd	2		385.2.a.d.1.1	✓	2
12.11	even	2		3465.2.a.u.1.2		2
20.3	even	4		1925.2.b.j.1849.3		4
20.7	even	4		1925.2.b.j.1849.2		4
20.19	odd	2		1925.2.a.n.1.2		2
28.27	even	2		2695.2.a.e.1.1		2
44.43	even	2		4235.2.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.a.d.1.1	✓	2	4.3	odd	2
1925.2.a.n.1.2		2	20.19	odd	2
1925.2.b.j.1849.2		4	20.7	even	4
1925.2.b.j.1849.3		4	20.3	even	4
2695.2.a.e.1.1		2	28.27	even	2
3465.2.a.u.1.2		2	12.11	even	2
4235.2.a.h.1.2		2	44.43	even	2
6160.2.a.y.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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{9} -1.00000 q^{11} +3.41421 q^{13} -1.41421 q^{15} -0.585786 q^{17} +1.41421 q^{21} -8.82843 q^{23} +1.00000 q^{25} -5.65685 q^{27} -0.828427 q^{29} +1.75736 q^{31} -1.41421 q^{33} -1.00000 q^{35} +1.17157 q^{37} +4.82843 q^{39} +4.24264 q^{41} -6.00000 q^{43} +1.00000 q^{45} -1.41421 q^{47} +1.00000 q^{49} -0.828427 q^{51} +2.82843 q^{53} +1.00000 q^{55} +7.89949 q^{59} -13.8995 q^{61} -1.00000 q^{63} -3.41421 q^{65} -6.48528 q^{67} -12.4853 q^{69} -9.17157 q^{71} -5.07107 q^{73} +1.41421 q^{75} -1.00000 q^{77} -6.48528 q^{79} -5.00000 q^{81} -12.0000 q^{83} +0.585786 q^{85} -1.17157 q^{87} +0.828427 q^{89} +3.41421 q^{91} +2.48528 q^{93} +9.31371 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7 - 2 * q^9	\( 2 q - 2 q^{5} + 2 q^{7} - 2 q^{9} - 2 q^{11} + 4 q^{13} - 4 q^{17} - 12 q^{23} + 2 q^{25} + 4 q^{29} + 12 q^{31} - 2 q^{35} + 8 q^{37} + 4 q^{39} - 12 q^{43} + 2 q^{45} + 2 q^{49} + 4 q^{51} + 2 q^{55} - 4 q^{59} - 8 q^{61} - 2 q^{63} - 4 q^{65} + 4 q^{67} - 8 q^{69} - 24 q^{71} + 4 q^{73} - 2 q^{77} + 4 q^{79} - 10 q^{81} - 24 q^{83} + 4 q^{85} - 8 q^{87} - 4 q^{89} + 4 q^{91} - 12 q^{93} - 4 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 - 2 * q^9 - 2 * q^11 + 4 * q^13 - 4 * q^17 - 12 * q^23 + 2 * q^25 + 4 * q^29 + 12 * q^31 - 2 * q^35 + 8 * q^37 + 4 * q^39 - 12 * q^43 + 2 * q^45 + 2 * q^49 + 4 * q^51 + 2 * q^55 - 4 * q^59 - 8 * q^61 - 2 * q^63 - 4 * q^65 + 4 * q^67 - 8 * q^69 - 24 * q^71 + 4 * q^73 - 2 * q^77 + 4 * q^79 - 10 * q^81 - 24 * q^83 + 4 * q^85 - 8 * q^87 - 4 * q^89 + 4 * q^91 - 12 * q^93 - 4 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more