LMFDB - Embedded newform 1968.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1968,2,Mod(1,1968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1968, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1968.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1968 = 2^{4} \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1968.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:	$15.7145591178$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 492)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	1968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.449490 q^{5} -4.44949 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.449490 q^{5} -4.44949 q^{7} +1.00000 q^{9} +3.44949 q^{11} -0.449490 q^{13} +0.449490 q^{15} -1.44949 q^{17} -6.44949 q^{19} -4.44949 q^{21} -6.44949 q^{23} -4.79796 q^{25} +1.00000 q^{27} +8.34847 q^{29} -7.00000 q^{31} +3.44949 q^{33} -2.00000 q^{35} +3.89898 q^{37} -0.449490 q^{39} -1.00000 q^{41} -1.00000 q^{43} +0.449490 q^{45} -4.55051 q^{47} +12.7980 q^{49} -1.44949 q^{51} -12.8990 q^{53} +1.55051 q^{55} -6.44949 q^{57} +14.6969 q^{59} -12.7980 q^{61} -4.44949 q^{63} -0.202041 q^{65} +6.89898 q^{67} -6.44949 q^{69} -4.34847 q^{71} +1.00000 q^{73} -4.79796 q^{75} -15.3485 q^{77} -10.0000 q^{79} +1.00000 q^{81} -7.55051 q^{83} -0.651531 q^{85} +8.34847 q^{87} -15.7980 q^{89} +2.00000 q^{91} -7.00000 q^{93} -2.89898 q^{95} -3.34847 q^{97} +3.44949 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^5 - 4 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{5} - 4 q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} - 4 q^{15} + 2 q^{17} - 8 q^{19} - 4 q^{21} - 8 q^{23} + 10 q^{25} + 2 q^{27} + 2 q^{29} - 14 q^{31} + 2 q^{33} - 4 q^{35} - 2 q^{37} + 4 q^{39} - 2 q^{41} - 2 q^{43} - 4 q^{45} - 14 q^{47} + 6 q^{49} + 2 q^{51} - 16 q^{53} + 8 q^{55} - 8 q^{57} - 6 q^{61} - 4 q^{63} - 20 q^{65} + 4 q^{67} - 8 q^{69} + 6 q^{71} + 2 q^{73} + 10 q^{75} - 16 q^{77} - 20 q^{79} + 2 q^{81} - 20 q^{83} - 16 q^{85} + 2 q^{87} - 12 q^{89} + 4 q^{91} - 14 q^{93} + 4 q^{95} + 8 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^5 - 4 * q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 - 4 * q^15 + 2 * q^17 - 8 * q^19 - 4 * q^21 - 8 * q^23 + 10 * q^25 + 2 * q^27 + 2 * q^29 - 14 * q^31 + 2 * q^33 - 4 * q^35 - 2 * q^37 + 4 * q^39 - 2 * q^41 - 2 * q^43 - 4 * q^45 - 14 * q^47 + 6 * q^49 + 2 * q^51 - 16 * q^53 + 8 * q^55 - 8 * q^57 - 6 * q^61 - 4 * q^63 - 20 * q^65 + 4 * q^67 - 8 * q^69 + 6 * q^71 + 2 * q^73 + 10 * q^75 - 16 * q^77 - 20 * q^79 + 2 * q^81 - 20 * q^83 - 16 * q^85 + 2 * q^87 - 12 * q^89 + 4 * q^91 - 14 * q^93 + 4 * q^95 + 8 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.449490	0.201018	0.100509	−	0.994936i	$-0.467953\pi$
$5$	0.449490	0.201018	0.100509	+	0.994936i	$0.467953\pi$
$6$	0	0
$7$	−4.44949	−1.68175	−0.840875	−	0.541230i	$-0.817959\pi$
$7$	−4.44949	−1.68175	−0.840875	+	0.541230i	$0.817959\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.44949	1.04006	0.520030	−	0.854148i	$-0.325921\pi$
$11$	3.44949	1.04006	0.520030	+	0.854148i	$0.325921\pi$
$12$	0	0
$13$	−0.449490	−0.124666	−0.0623330	−	0.998055i	$-0.519854\pi$
$13$	−0.449490	−0.124666	−0.0623330	+	0.998055i	$0.519854\pi$
$14$	0	0
$15$	0.449490	0.116058
$16$	0	0
$17$	−1.44949	−0.351553	−0.175776	−	0.984430i	$-0.556244\pi$
$17$	−1.44949	−0.351553	−0.175776	+	0.984430i	$0.556244\pi$
$18$	0	0
$19$	−6.44949	−1.47961	−0.739807	−	0.672819i	$-0.765083\pi$
$19$	−6.44949	−1.47961	−0.739807	+	0.672819i	$0.765083\pi$
$20$	0	0
$21$	−4.44949	−0.970958
$22$	0	0
$23$	−6.44949	−1.34481	−0.672406	−	0.740183i	$-0.734739\pi$
$23$	−6.44949	−1.34481	−0.672406	+	0.740183i	$0.734739\pi$
$24$	0	0
$25$	−4.79796	−0.959592
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.34847	1.55027	0.775136	−	0.631795i	$-0.217681\pi$
$29$	8.34847	1.55027	0.775136	+	0.631795i	$0.217681\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	0	0
$33$	3.44949	0.600479
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	3.89898	0.640988	0.320494	−	0.947250i	$-0.396151\pi$
$37$	3.89898	0.640988	0.320494	+	0.947250i	$0.396151\pi$
$38$	0	0
$39$	−0.449490	−0.0719760
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	0.449490	0.0670060
$46$	0	0
$47$	−4.55051	−0.663760	−0.331880	−	0.943322i	$-0.607683\pi$
$47$	−4.55051	−0.663760	−0.331880	+	0.943322i	$0.607683\pi$
$48$	0	0
$49$	12.7980	1.82828
$50$	0	0
$51$	−1.44949	−0.202969
$52$	0	0
$53$	−12.8990	−1.77181	−0.885906	−	0.463866i	$-0.846462\pi$
$53$	−12.8990	−1.77181	−0.885906	+	0.463866i	$0.846462\pi$
$54$	0	0
$55$	1.55051	0.209071
$56$	0	0
$57$	−6.44949	−0.854256
$58$	0	0
$59$	14.6969	1.91338	0.956689	−	0.291111i	$-0.0940250\pi$
$59$	14.6969	1.91338	0.956689	+	0.291111i	$0.0940250\pi$
$60$	0	0
$61$	−12.7980	−1.63861	−0.819305	−	0.573357i	$-0.805641\pi$
$61$	−12.7980	−1.63861	−0.819305	+	0.573357i	$0.805641\pi$
$62$	0	0
$63$	−4.44949	−0.560583
$64$	0	0
$65$	−0.202041	−0.0250601
$66$	0	0
$67$	6.89898	0.842844	0.421422	−	0.906865i	$-0.361531\pi$
$67$	6.89898	0.842844	0.421422	+	0.906865i	$0.361531\pi$
$68$	0	0
$69$	−6.44949	−0.776427
$70$	0	0
$71$	−4.34847	−0.516068	−0.258034	−	0.966136i	$-0.583075\pi$
$71$	−4.34847	−0.516068	−0.258034	+	0.966136i	$0.583075\pi$
$72$	0	0
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	0	0
$75$	−4.79796	−0.554021
$76$	0	0
$77$	−15.3485	−1.74912
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.55051	−0.828776	−0.414388	−	0.910100i	$-0.636004\pi$
$83$	−7.55051	−0.828776	−0.414388	+	0.910100i	$0.636004\pi$
$84$	0	0
$85$	−0.651531	−0.0706684
$86$	0	0
$87$	8.34847	0.895050
$88$	0	0
$89$	−15.7980	−1.67458	−0.837290	−	0.546759i	$-0.815861\pi$
$89$	−15.7980	−1.67458	−0.837290	+	0.546759i	$0.815861\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	−7.00000	−0.725866
$94$	0	0
$95$	−2.89898	−0.297429
$96$	0	0
$97$	−3.34847	−0.339986	−0.169993	−	0.985445i	$-0.554374\pi$
$97$	−3.34847	−0.339986	−0.169993	+	0.985445i	$0.554374\pi$
$98$	0	0
$99$	3.44949	0.346687
$100$	0	0
$101$	4.34847	0.432689	0.216344	−	0.976317i	$-0.430587\pi$
$101$	4.34847	0.432689	0.216344	+	0.976317i	$0.430587\pi$
$102$	0	0
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	0	0
$109$	5.34847	0.512290	0.256145	−	0.966638i	$-0.417547\pi$
$109$	5.34847	0.512290	0.256145	+	0.966638i	$0.417547\pi$
$110$	0	0
$111$	3.89898	0.370075
$112$	0	0
$113$	−1.34847	−0.126853	−0.0634267	−	0.997987i	$-0.520203\pi$
$113$	−1.34847	−0.126853	−0.0634267	+	0.997987i	$0.520203\pi$
$114$	0	0
$115$	−2.89898	−0.270331
$116$	0	0
$117$	−0.449490	−0.0415553
$118$	0	0
$119$	6.44949	0.591224
$120$	0	0
$121$	0.898979	0.0817254
$122$	0	0
$123$	−1.00000	−0.0901670
$124$	0	0
$125$	−4.40408	−0.393913
$126$	0	0
$127$	18.0000	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	12.2474	1.07006	0.535032	−	0.844832i	$-0.320299\pi$
$131$	12.2474	1.07006	0.535032	+	0.844832i	$0.320299\pi$
$132$	0	0
$133$	28.6969	2.48834
$134$	0	0
$135$	0.449490	0.0386859
$136$	0	0
$137$	−3.44949	−0.294710	−0.147355	−	0.989084i	$-0.547076\pi$
$137$	−3.44949	−0.294710	−0.147355	+	0.989084i	$0.547076\pi$
$138$	0	0
$139$	0.202041	0.0171369	0.00856845	−	0.999963i	$-0.497273\pi$
$139$	0.202041	0.0171369	0.00856845	+	0.999963i	$0.497273\pi$
$140$	0	0
$141$	−4.55051	−0.383222
$142$	0	0
$143$	−1.55051	−0.129660
$144$	0	0
$145$	3.75255	0.311632
$146$	0	0
$147$	12.7980	1.05556
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	19.5959	1.59469	0.797347	−	0.603522i	$-0.206236\pi$
$151$	19.5959	1.59469	0.797347	+	0.603522i	$0.206236\pi$
$152$	0	0
$153$	−1.44949	−0.117184
$154$	0	0
$155$	−3.14643	−0.252727
$156$	0	0
$157$	−7.79796	−0.622345	−0.311172	−	0.950353i	$-0.600722\pi$
$157$	−7.79796	−0.622345	−0.311172	+	0.950353i	$0.600722\pi$
$158$	0	0
$159$	−12.8990	−1.02296
$160$	0	0
$161$	28.6969	2.26164
$162$	0	0
$163$	7.89898	0.618696	0.309348	−	0.950949i	$-0.399889\pi$
$163$	7.89898	0.618696	0.309348	+	0.950949i	$0.399889\pi$
$164$	0	0
$165$	1.55051	0.120707
$166$	0	0
$167$	−13.5959	−1.05208	−0.526042	−	0.850459i	$-0.676324\pi$
$167$	−13.5959	−1.05208	−0.526042	+	0.850459i	$0.676324\pi$
$168$	0	0
$169$	−12.7980	−0.984458
$170$	0	0
$171$	−6.44949	−0.493205
$172$	0	0
$173$	7.55051	0.574055	0.287027	−	0.957922i	$-0.407333\pi$
$173$	7.55051	0.574055	0.287027	+	0.957922i	$0.407333\pi$
$174$	0	0
$175$	21.3485	1.61379
$176$	0	0
$177$	14.6969	1.10469
$178$	0	0
$179$	−6.34847	−0.474507	−0.237253	−	0.971448i	$-0.576247\pi$
$179$	−6.34847	−0.474507	−0.237253	+	0.971448i	$0.576247\pi$
$180$	0	0
$181$	13.7980	1.02559	0.512797	−	0.858510i	$-0.328609\pi$
$181$	13.7980	1.02559	0.512797	+	0.858510i	$0.328609\pi$
$182$	0	0
$183$	−12.7980	−0.946052
$184$	0	0
$185$	1.75255	0.128850
$186$	0	0
$187$	−5.00000	−0.365636
$188$	0	0
$189$	−4.44949	−0.323653
$190$	0	0
$191$	−3.10102	−0.224382	−0.112191	−	0.993687i	$-0.535787\pi$
$191$	−3.10102	−0.224382	−0.112191	+	0.993687i	$0.535787\pi$
$192$	0	0
$193$	7.55051	0.543498	0.271749	−	0.962368i	$-0.412398\pi$
$193$	7.55051	0.543498	0.271749	+	0.962368i	$0.412398\pi$
$194$	0	0
$195$	−0.202041	−0.0144685
$196$	0	0
$197$	−24.0454	−1.71316	−0.856582	−	0.516011i	$-0.827416\pi$
$197$	−24.0454	−1.71316	−0.856582	+	0.516011i	$0.827416\pi$
$198$	0	0
$199$	14.4495	1.02430	0.512149	−	0.858897i	$-0.328850\pi$
$199$	14.4495	1.02430	0.512149	+	0.858897i	$0.328850\pi$
$200$	0	0
$201$	6.89898	0.486616
$202$	0	0
$203$	−37.1464	−2.60717
$204$	0	0
$205$	−0.449490	−0.0313937
$206$	0	0
$207$	−6.44949	−0.448271
$208$	0	0
$209$	−22.2474	−1.53889
$210$	0	0
$211$	4.89898	0.337260	0.168630	−	0.985679i	$-0.446066\pi$
$211$	4.89898	0.337260	0.168630	+	0.985679i	$0.446066\pi$
$212$	0	0
$213$	−4.34847	−0.297952
$214$	0	0
$215$	−0.449490	−0.0306549
$216$	0	0
$217$	31.1464	2.11436
$218$	0	0
$219$	1.00000	0.0675737
$220$	0	0
$221$	0.651531	0.0438267
$222$	0	0
$223$	1.79796	0.120400	0.0602001	−	0.998186i	$-0.480826\pi$
$223$	1.79796	0.120400	0.0602001	+	0.998186i	$0.480826\pi$
$224$	0	0
$225$	−4.79796	−0.319864
$226$	0	0
$227$	−18.5505	−1.23124	−0.615620	−	0.788043i	$-0.711094\pi$
$227$	−18.5505	−1.23124	−0.615620	+	0.788043i	$0.711094\pi$
$228$	0	0
$229$	−9.55051	−0.631115	−0.315558	−	0.948906i	$-0.602192\pi$
$229$	−9.55051	−0.631115	−0.315558	+	0.948906i	$0.602192\pi$
$230$	0	0
$231$	−15.3485	−1.00986
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−2.04541	−0.133428
$236$	0	0
$237$	−10.0000	−0.649570
$238$	0	0
$239$	8.89898	0.575627	0.287814	−	0.957686i	$-0.407072\pi$
$239$	8.89898	0.575627	0.287814	+	0.957686i	$0.407072\pi$
$240$	0	0
$241$	3.69694	0.238141	0.119070	−	0.992886i	$-0.462009\pi$
$241$	3.69694	0.238141	0.119070	+	0.992886i	$0.462009\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.75255	0.367517
$246$	0	0
$247$	2.89898	0.184458
$248$	0	0
$249$	−7.55051	−0.478494
$250$	0	0
$251$	18.8990	1.19289	0.596447	−	0.802653i	$-0.296579\pi$
$251$	18.8990	1.19289	0.596447	+	0.802653i	$0.296579\pi$
$252$	0	0
$253$	−22.2474	−1.39869
$254$	0	0
$255$	−0.651531	−0.0408004
$256$	0	0
$257$	26.1464	1.63097	0.815485	−	0.578779i	$-0.196470\pi$
$257$	26.1464	1.63097	0.815485	+	0.578779i	$0.196470\pi$
$258$	0	0
$259$	−17.3485	−1.07798
$260$	0	0
$261$	8.34847	0.516757
$262$	0	0
$263$	17.2474	1.06352	0.531762	−	0.846894i	$-0.321530\pi$
$263$	17.2474	1.06352	0.531762	+	0.846894i	$0.321530\pi$
$264$	0	0
$265$	−5.79796	−0.356166
$266$	0	0
$267$	−15.7980	−0.966819
$268$	0	0
$269$	−12.8990	−0.786465	−0.393232	−	0.919439i	$-0.628643\pi$
$269$	−12.8990	−0.786465	−0.393232	+	0.919439i	$0.628643\pi$
$270$	0	0
$271$	17.6969	1.07501	0.537506	−	0.843260i	$-0.319366\pi$
$271$	17.6969	1.07501	0.537506	+	0.843260i	$0.319366\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	−16.5505	−0.998033
$276$	0	0
$277$	10.7980	0.648786	0.324393	−	0.945922i	$-0.394840\pi$
$277$	10.7980	0.648786	0.324393	+	0.945922i	$0.394840\pi$
$278$	0	0
$279$	−7.00000	−0.419079
$280$	0	0
$281$	29.4495	1.75681	0.878405	−	0.477918i	$-0.158608\pi$
$281$	29.4495	1.75681	0.878405	+	0.477918i	$0.158608\pi$
$282$	0	0
$283$	−9.89898	−0.588433	−0.294217	−	0.955739i	$-0.595059\pi$
$283$	−9.89898	−0.588433	−0.294217	+	0.955739i	$0.595059\pi$
$284$	0	0
$285$	−2.89898	−0.171721
$286$	0	0
$287$	4.44949	0.262645
$288$	0	0
$289$	−14.8990	−0.876411
$290$	0	0
$291$	−3.34847	−0.196291
$292$	0	0
$293$	14.3485	0.838247	0.419123	−	0.907929i	$-0.362338\pi$
$293$	14.3485	0.838247	0.419123	+	0.907929i	$0.362338\pi$
$294$	0	0
$295$	6.60612	0.384623
$296$	0	0
$297$	3.44949	0.200160
$298$	0	0
$299$	2.89898	0.167652
$300$	0	0
$301$	4.44949	0.256464
$302$	0	0
$303$	4.34847	0.249813
$304$	0	0
$305$	−5.75255	−0.329390
$306$	0	0
$307$	−24.5959	−1.40376	−0.701882	−	0.712294i	$-0.747656\pi$
$307$	−24.5959	−1.40376	−0.701882	+	0.712294i	$0.747656\pi$
$308$	0	0
$309$	−13.0000	−0.739544
$310$	0	0
$311$	−30.6969	−1.74066	−0.870332	−	0.492466i	$-0.836096\pi$
$311$	−30.6969	−1.74066	−0.870332	+	0.492466i	$0.836096\pi$
$312$	0	0
$313$	−9.34847	−0.528407	−0.264203	−	0.964467i	$-0.585109\pi$
$313$	−9.34847	−0.528407	−0.264203	+	0.964467i	$0.585109\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−9.24745	−0.519388	−0.259694	−	0.965691i	$-0.583622\pi$
$317$	−9.24745	−0.519388	−0.259694	+	0.965691i	$0.583622\pi$
$318$	0	0
$319$	28.7980	1.61238
$320$	0	0
$321$	2.00000	0.111629
$322$	0	0
$323$	9.34847	0.520163
$324$	0	0
$325$	2.15663	0.119628
$326$	0	0
$327$	5.34847	0.295771
$328$	0	0
$329$	20.2474	1.11628
$330$	0	0
$331$	25.5505	1.40438	0.702192	−	0.711988i	$-0.252205\pi$
$331$	25.5505	1.40438	0.702192	+	0.711988i	$0.252205\pi$
$332$	0	0
$333$	3.89898	0.213663
$334$	0	0
$335$	3.10102	0.169427
$336$	0	0
$337$	0.797959	0.0434676	0.0217338	−	0.999764i	$-0.493081\pi$
$337$	0.797959	0.0434676	0.0217338	+	0.999764i	$0.493081\pi$
$338$	0	0
$339$	−1.34847	−0.0732388
$340$	0	0
$341$	−24.1464	−1.30760
$342$	0	0
$343$	−25.7980	−1.39296
$344$	0	0
$345$	−2.89898	−0.156076
$346$	0	0
$347$	−13.6515	−0.732853	−0.366426	−	0.930447i	$-0.619419\pi$
$347$	−13.6515	−0.732853	−0.366426	+	0.930447i	$0.619419\pi$
$348$	0	0
$349$	11.8990	0.636938	0.318469	−	0.947933i	$-0.396831\pi$
$349$	11.8990	0.636938	0.318469	+	0.947933i	$0.396831\pi$
$350$	0	0
$351$	−0.449490	−0.0239920
$352$	0	0
$353$	15.7980	0.840841	0.420420	−	0.907329i	$-0.361883\pi$
$353$	15.7980	0.840841	0.420420	+	0.907329i	$0.361883\pi$
$354$	0	0
$355$	−1.95459	−0.103739
$356$	0	0
$357$	6.44949	0.341343
$358$	0	0
$359$	−16.6515	−0.878834	−0.439417	−	0.898283i	$-0.644815\pi$
$359$	−16.6515	−0.878834	−0.439417	+	0.898283i	$0.644815\pi$
$360$	0	0
$361$	22.5959	1.18926
$362$	0	0
$363$	0.898979	0.0471842
$364$	0	0
$365$	0.449490	0.0235274
$366$	0	0
$367$	−2.79796	−0.146052	−0.0730261	−	0.997330i	$-0.523266\pi$
$367$	−2.79796	−0.146052	−0.0730261	+	0.997330i	$0.523266\pi$
$368$	0	0
$369$	−1.00000	−0.0520579
$370$	0	0
$371$	57.3939	2.97974
$372$	0	0
$373$	23.4949	1.21652	0.608260	−	0.793738i	$-0.291868\pi$
$373$	23.4949	1.21652	0.608260	+	0.793738i	$0.291868\pi$
$374$	0	0
$375$	−4.40408	−0.227426
$376$	0	0
$377$	−3.75255	−0.193266
$378$	0	0
$379$	27.3939	1.40713	0.703564	−	0.710631i	$-0.251591\pi$
$379$	27.3939	1.40713	0.703564	+	0.710631i	$0.251591\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	0	0
$383$	−38.1464	−1.94919	−0.974596	−	0.223971i	$-0.928098\pi$
$383$	−38.1464	−1.94919	−0.974596	+	0.223971i	$0.928098\pi$
$384$	0	0
$385$	−6.89898	−0.351605
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	0	0
$389$	6.24745	0.316758	0.158379	−	0.987378i	$-0.449373\pi$
$389$	6.24745	0.316758	0.158379	+	0.987378i	$0.449373\pi$
$390$	0	0
$391$	9.34847	0.472772
$392$	0	0
$393$	12.2474	0.617802
$394$	0	0
$395$	−4.49490	−0.226163
$396$	0	0
$397$	−0.853572	−0.0428395	−0.0214198	−	0.999771i	$-0.506819\pi$
$397$	−0.853572	−0.0428395	−0.0214198	+	0.999771i	$0.506819\pi$
$398$	0	0
$399$	28.6969	1.43664
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	3.14643	0.156735
$404$	0	0
$405$	0.449490	0.0223353
$406$	0	0
$407$	13.4495	0.666666
$408$	0	0
$409$	−9.89898	−0.489473	−0.244737	−	0.969590i	$-0.578701\pi$
$409$	−9.89898	−0.489473	−0.244737	+	0.969590i	$0.578701\pi$
$410$	0	0
$411$	−3.44949	−0.170151
$412$	0	0
$413$	−65.3939	−3.21782
$414$	0	0
$415$	−3.39388	−0.166599
$416$	0	0
$417$	0.202041	0.00989399
$418$	0	0
$419$	16.2474	0.793740	0.396870	−	0.917875i	$-0.370096\pi$
$419$	16.2474	0.793740	0.396870	+	0.917875i	$0.370096\pi$
$420$	0	0
$421$	−21.3939	−1.04267	−0.521337	−	0.853351i	$-0.674566\pi$
$421$	−21.3939	−1.04267	−0.521337	+	0.853351i	$0.674566\pi$
$422$	0	0
$423$	−4.55051	−0.221253
$424$	0	0
$425$	6.95459	0.337347
$426$	0	0
$427$	56.9444	2.75573
$428$	0	0
$429$	−1.55051	−0.0748593
$430$	0	0
$431$	37.1464	1.78928	0.894640	−	0.446787i	$-0.147432\pi$
$431$	37.1464	1.78928	0.894640	+	0.446787i	$0.147432\pi$
$432$	0	0
$433$	−23.4949	−1.12909	−0.564546	−	0.825401i	$-0.690949\pi$
$433$	−23.4949	−1.12909	−0.564546	+	0.825401i	$0.690949\pi$
$434$	0	0
$435$	3.75255	0.179921
$436$	0	0
$437$	41.5959	1.98980
$438$	0	0
$439$	34.0000	1.62273	0.811366	−	0.584539i	$-0.198725\pi$
$439$	34.0000	1.62273	0.811366	+	0.584539i	$0.198725\pi$
$440$	0	0
$441$	12.7980	0.609427
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	−7.10102	−0.336621
$446$	0	0
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−5.30306	−0.250267	−0.125133	−	0.992140i	$-0.539936\pi$
$449$	−5.30306	−0.250267	−0.125133	+	0.992140i	$0.539936\pi$
$450$	0	0
$451$	−3.44949	−0.162430
$452$	0	0
$453$	19.5959	0.920697
$454$	0	0
$455$	0.898979	0.0421448
$456$	0	0
$457$	23.5959	1.10377	0.551885	−	0.833920i	$-0.313909\pi$
$457$	23.5959	1.10377	0.551885	+	0.833920i	$0.313909\pi$
$458$	0	0
$459$	−1.44949	−0.0676564
$460$	0	0
$461$	35.3939	1.64846	0.824229	−	0.566257i	$-0.191609\pi$
$461$	35.3939	1.64846	0.824229	+	0.566257i	$0.191609\pi$
$462$	0	0
$463$	−11.1464	−0.518018	−0.259009	−	0.965875i	$-0.583396\pi$
$463$	−11.1464	−0.518018	−0.259009	+	0.965875i	$0.583396\pi$
$464$	0	0
$465$	−3.14643	−0.145912
$466$	0	0
$467$	−25.5959	−1.18444	−0.592219	−	0.805777i	$-0.701748\pi$
$467$	−25.5959	−1.18444	−0.592219	+	0.805777i	$0.701748\pi$
$468$	0	0
$469$	−30.6969	−1.41745
$470$	0	0
$471$	−7.79796	−0.359311
$472$	0	0
$473$	−3.44949	−0.158608
$474$	0	0
$475$	30.9444	1.41983
$476$	0	0
$477$	−12.8990	−0.590604
$478$	0	0
$479$	4.75255	0.217150	0.108575	−	0.994088i	$-0.465371\pi$
$479$	4.75255	0.217150	0.108575	+	0.994088i	$0.465371\pi$
$480$	0	0
$481$	−1.75255	−0.0799095
$482$	0	0
$483$	28.6969	1.30576
$484$	0	0
$485$	−1.50510	−0.0683432
$486$	0	0
$487$	38.7980	1.75810	0.879052	−	0.476727i	$-0.158177\pi$
$487$	38.7980	1.75810	0.879052	+	0.476727i	$0.158177\pi$
$488$	0	0
$489$	7.89898	0.357204
$490$	0	0
$491$	11.5505	0.521267	0.260634	−	0.965438i	$-0.416069\pi$
$491$	11.5505	0.521267	0.260634	+	0.965438i	$0.416069\pi$
$492$	0	0
$493$	−12.1010	−0.545003
$494$	0	0
$495$	1.55051	0.0696903
$496$	0	0
$497$	19.3485	0.867897
$498$	0	0
$499$	−28.4949	−1.27561	−0.637803	−	0.770199i	$-0.720157\pi$
$499$	−28.4949	−1.27561	−0.637803	+	0.770199i	$0.720157\pi$
$500$	0	0
$501$	−13.5959	−0.607421
$502$	0	0
$503$	−14.1464	−0.630758	−0.315379	−	0.948966i	$-0.602132\pi$
$503$	−14.1464	−0.630758	−0.315379	+	0.948966i	$0.602132\pi$
$504$	0	0
$505$	1.95459	0.0869782
$506$	0	0
$507$	−12.7980	−0.568377
$508$	0	0
$509$	−35.2474	−1.56232	−0.781158	−	0.624334i	$-0.785371\pi$
$509$	−35.2474	−1.56232	−0.781158	+	0.624334i	$0.785371\pi$
$510$	0	0
$511$	−4.44949	−0.196834
$512$	0	0
$513$	−6.44949	−0.284752
$514$	0	0
$515$	−5.84337	−0.257489
$516$	0	0
$517$	−15.6969	−0.690351
$518$	0	0
$519$	7.55051	0.331431
$520$	0	0
$521$	12.5505	0.549848	0.274924	−	0.961466i	$-0.411347\pi$
$521$	12.5505	0.549848	0.274924	+	0.961466i	$0.411347\pi$
$522$	0	0
$523$	−37.5959	−1.64395	−0.821977	−	0.569520i	$-0.807129\pi$
$523$	−37.5959	−1.64395	−0.821977	+	0.569520i	$0.807129\pi$
$524$	0	0
$525$	21.3485	0.931724
$526$	0	0
$527$	10.1464	0.441985
$528$	0	0
$529$	18.5959	0.808518
$530$	0	0
$531$	14.6969	0.637793
$532$	0	0
$533$	0.449490	0.0194696
$534$	0	0
$535$	0.898979	0.0388663
$536$	0	0
$537$	−6.34847	−0.273957
$538$	0	0
$539$	44.1464	1.90152
$540$	0	0
$541$	−27.3939	−1.17775	−0.588877	−	0.808222i	$-0.700430\pi$
$541$	−27.3939	−1.17775	−0.588877	+	0.808222i	$0.700430\pi$
$542$	0	0
$543$	13.7980	0.592127
$544$	0	0
$545$	2.40408	0.102980
$546$	0	0
$547$	−12.8990	−0.551521	−0.275760	−	0.961226i	$-0.588930\pi$
$547$	−12.8990	−0.551521	−0.275760	+	0.961226i	$0.588930\pi$
$548$	0	0
$549$	−12.7980	−0.546204
$550$	0	0
$551$	−53.8434	−2.29380
$552$	0	0
$553$	44.4949	1.89212
$554$	0	0
$555$	1.75255	0.0743917
$556$	0	0
$557$	−11.0454	−0.468009	−0.234004	−	0.972236i	$-0.575183\pi$
$557$	−11.0454	−0.468009	−0.234004	+	0.972236i	$0.575183\pi$
$558$	0	0
$559$	0.449490	0.0190114
$560$	0	0
$561$	−5.00000	−0.211100
$562$	0	0
$563$	−31.6515	−1.33395	−0.666976	−	0.745079i	$-0.732412\pi$
$563$	−31.6515	−1.33395	−0.666976	+	0.745079i	$0.732412\pi$
$564$	0	0
$565$	−0.606123	−0.0254998
$566$	0	0
$567$	−4.44949	−0.186861
$568$	0	0
$569$	−26.4495	−1.10882	−0.554410	−	0.832244i	$-0.687056\pi$
$569$	−26.4495	−1.10882	−0.554410	+	0.832244i	$0.687056\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	−3.10102	−0.129547
$574$	0	0
$575$	30.9444	1.29047
$576$	0	0
$577$	−12.4949	−0.520169	−0.260085	−	0.965586i	$-0.583750\pi$
$577$	−12.4949	−0.520169	−0.260085	+	0.965586i	$0.583750\pi$
$578$	0	0
$579$	7.55051	0.313788
$580$	0	0
$581$	33.5959	1.39379
$582$	0	0
$583$	−44.4949	−1.84279
$584$	0	0
$585$	−0.202041	−0.00835337
$586$	0	0
$587$	6.55051	0.270368	0.135184	−	0.990820i	$-0.456837\pi$
$587$	6.55051	0.270368	0.135184	+	0.990820i	$0.456837\pi$
$588$	0	0
$589$	45.1464	1.86023
$590$	0	0
$591$	−24.0454	−0.989096
$592$	0	0
$593$	−19.0454	−0.782101	−0.391051	−	0.920369i	$-0.627888\pi$
$593$	−19.0454	−0.782101	−0.391051	+	0.920369i	$0.627888\pi$
$594$	0	0
$595$	2.89898	0.118847
$596$	0	0
$597$	14.4495	0.591378
$598$	0	0
$599$	−27.8434	−1.13765	−0.568825	−	0.822459i	$-0.692602\pi$
$599$	−27.8434	−1.13765	−0.568825	+	0.822459i	$0.692602\pi$
$600$	0	0
$601$	34.8990	1.42356	0.711780	−	0.702403i	$-0.247890\pi$
$601$	34.8990	1.42356	0.711780	+	0.702403i	$0.247890\pi$
$602$	0	0
$603$	6.89898	0.280948
$604$	0	0
$605$	0.404082	0.0164283
$606$	0	0
$607$	14.8990	0.604731	0.302365	−	0.953192i	$-0.402224\pi$
$607$	14.8990	0.604731	0.302365	+	0.953192i	$0.402224\pi$
$608$	0	0
$609$	−37.1464	−1.50525
$610$	0	0
$611$	2.04541	0.0827483
$612$	0	0
$613$	−43.1918	−1.74450	−0.872251	−	0.489059i	$-0.837340\pi$
$613$	−43.1918	−1.74450	−0.872251	+	0.489059i	$0.837340\pi$
$614$	0	0
$615$	−0.449490	−0.0181252
$616$	0	0
$617$	−37.3485	−1.50359	−0.751796	−	0.659395i	$-0.770812\pi$
$617$	−37.3485	−1.50359	−0.751796	+	0.659395i	$0.770812\pi$
$618$	0	0
$619$	12.5959	0.506273	0.253136	−	0.967431i	$-0.418538\pi$
$619$	12.5959	0.506273	0.253136	+	0.967431i	$0.418538\pi$
$620$	0	0
$621$	−6.44949	−0.258809
$622$	0	0
$623$	70.2929	2.81622
$624$	0	0
$625$	22.0102	0.880408
$626$	0	0
$627$	−22.2474	−0.888478
$628$	0	0
$629$	−5.65153	−0.225341
$630$	0	0
$631$	12.1010	0.481734	0.240867	−	0.970558i	$-0.422568\pi$
$631$	12.1010	0.481734	0.240867	+	0.970558i	$0.422568\pi$
$632$	0	0
$633$	4.89898	0.194717
$634$	0	0
$635$	8.09082	0.321074
$636$	0	0
$637$	−5.75255	−0.227924
$638$	0	0
$639$	−4.34847	−0.172023
$640$	0	0
$641$	−38.1464	−1.50669	−0.753347	−	0.657624i	$-0.771562\pi$
$641$	−38.1464	−1.50669	−0.753347	+	0.657624i	$0.771562\pi$
$642$	0	0
$643$	−32.8990	−1.29741	−0.648705	−	0.761040i	$-0.724689\pi$
$643$	−32.8990	−1.29741	−0.648705	+	0.761040i	$0.724689\pi$
$644$	0	0
$645$	−0.449490	−0.0176986
$646$	0	0
$647$	34.0454	1.33846	0.669232	−	0.743054i	$-0.266623\pi$
$647$	34.0454	1.33846	0.669232	+	0.743054i	$0.266623\pi$
$648$	0	0
$649$	50.6969	1.99003
$650$	0	0
$651$	31.1464	1.22072
$652$	0	0
$653$	9.85357	0.385600	0.192800	−	0.981238i	$-0.438243\pi$
$653$	9.85357	0.385600	0.192800	+	0.981238i	$0.438243\pi$
$654$	0	0
$655$	5.50510	0.215102
$656$	0	0
$657$	1.00000	0.0390137
$658$	0	0
$659$	19.1010	0.744070	0.372035	−	0.928219i	$-0.378660\pi$
$659$	19.1010	0.744070	0.372035	+	0.928219i	$0.378660\pi$
$660$	0	0
$661$	1.30306	0.0506832	0.0253416	−	0.999679i	$-0.491933\pi$
$661$	1.30306	0.0506832	0.0253416	+	0.999679i	$0.491933\pi$
$662$	0	0
$663$	0.651531	0.0253034
$664$	0	0
$665$	12.8990	0.500201
$666$	0	0
$667$	−53.8434	−2.08482
$668$	0	0
$669$	1.79796	0.0695131
$670$	0	0
$671$	−44.1464	−1.70425
$672$	0	0
$673$	10.4949	0.404549	0.202274	−	0.979329i	$-0.435167\pi$
$673$	10.4949	0.404549	0.202274	+	0.979329i	$0.435167\pi$
$674$	0	0
$675$	−4.79796	−0.184674
$676$	0	0
$677$	39.1464	1.50452	0.752260	−	0.658867i	$-0.228964\pi$
$677$	39.1464	1.50452	0.752260	+	0.658867i	$0.228964\pi$
$678$	0	0
$679$	14.8990	0.571770
$680$	0	0
$681$	−18.5505	−0.710857
$682$	0	0
$683$	−41.5959	−1.59162	−0.795812	−	0.605544i	$-0.792956\pi$
$683$	−41.5959	−1.59162	−0.795812	+	0.605544i	$0.792956\pi$
$684$	0	0
$685$	−1.55051	−0.0592420
$686$	0	0
$687$	−9.55051	−0.364375
$688$	0	0
$689$	5.79796	0.220885
$690$	0	0
$691$	−30.7423	−1.16949	−0.584747	−	0.811216i	$-0.698806\pi$
$691$	−30.7423	−1.16949	−0.584747	+	0.811216i	$0.698806\pi$
$692$	0	0
$693$	−15.3485	−0.583040
$694$	0	0
$695$	0.0908154	0.00344482
$696$	0	0
$697$	1.44949	0.0549033
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	26.8990	1.01596	0.507980	−	0.861369i	$-0.330392\pi$
$701$	26.8990	1.01596	0.507980	+	0.861369i	$0.330392\pi$
$702$	0	0
$703$	−25.1464	−0.948416
$704$	0	0
$705$	−2.04541	−0.0770345
$706$	0	0
$707$	−19.3485	−0.727674
$708$	0	0
$709$	37.1010	1.39336	0.696679	−	0.717383i	$-0.254660\pi$
$709$	37.1010	1.39336	0.696679	+	0.717383i	$0.254660\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	45.1464	1.69075
$714$	0	0
$715$	−0.696938	−0.0260640
$716$	0	0
$717$	8.89898	0.332338
$718$	0	0
$719$	−9.04541	−0.337337	−0.168668	−	0.985673i	$-0.553947\pi$
$719$	−9.04541	−0.337337	−0.168668	+	0.985673i	$0.553947\pi$
$720$	0	0
$721$	57.8434	2.15420
$722$	0	0
$723$	3.69694	0.137491
$724$	0	0
$725$	−40.0556	−1.48763
$726$	0	0
$727$	29.1464	1.08098	0.540491	−	0.841350i	$-0.318239\pi$
$727$	29.1464	1.08098	0.540491	+	0.841350i	$0.318239\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.44949	0.0536113
$732$	0	0
$733$	−22.3031	−0.823782	−0.411891	−	0.911233i	$-0.635132\pi$
$733$	−22.3031	−0.823782	−0.411891	+	0.911233i	$0.635132\pi$
$734$	0	0
$735$	5.75255	0.212186
$736$	0	0
$737$	23.7980	0.876609
$738$	0	0
$739$	−44.7980	−1.64792	−0.823960	−	0.566648i	$-0.808240\pi$
$739$	−44.7980	−1.64792	−0.823960	+	0.566648i	$0.808240\pi$
$740$	0	0
$741$	2.89898	0.106497
$742$	0	0
$743$	−4.20204	−0.154158	−0.0770790	−	0.997025i	$-0.524559\pi$
$743$	−4.20204	−0.154158	−0.0770790	+	0.997025i	$0.524559\pi$
$744$	0	0
$745$	−4.49490	−0.164680
$746$	0	0
$747$	−7.55051	−0.276259
$748$	0	0
$749$	−8.89898	−0.325162
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	0	0
$753$	18.8990	0.688717
$754$	0	0
$755$	8.80816	0.320562
$756$	0	0
$757$	−8.24745	−0.299759	−0.149879	−	0.988704i	$-0.547889\pi$
$757$	−8.24745	−0.299759	−0.149879	+	0.988704i	$0.547889\pi$
$758$	0	0
$759$	−22.2474	−0.807531
$760$	0	0
$761$	−52.4949	−1.90294	−0.951469	−	0.307744i	$-0.900426\pi$
$761$	−52.4949	−1.90294	−0.951469	+	0.307744i	$0.900426\pi$
$762$	0	0
$763$	−23.7980	−0.861544
$764$	0	0
$765$	−0.651531	−0.0235561
$766$	0	0
$767$	−6.60612	−0.238533
$768$	0	0
$769$	−28.1010	−1.01335	−0.506674	−	0.862138i	$-0.669125\pi$
$769$	−28.1010	−1.01335	−0.506674	+	0.862138i	$0.669125\pi$
$770$	0	0
$771$	26.1464	0.941641
$772$	0	0
$773$	15.0454	0.541146	0.270573	−	0.962700i	$-0.412787\pi$
$773$	15.0454	0.541146	0.270573	+	0.962700i	$0.412787\pi$
$774$	0	0
$775$	33.5857	1.20643
$776$	0	0
$777$	−17.3485	−0.622373
$778$	0	0
$779$	6.44949	0.231077
$780$	0	0
$781$	−15.0000	−0.536742
$782$	0	0
$783$	8.34847	0.298350
$784$	0	0
$785$	−3.50510	−0.125102
$786$	0	0
$787$	13.2020	0.470602	0.235301	−	0.971923i	$-0.424392\pi$
$787$	13.2020	0.470602	0.235301	+	0.971923i	$0.424392\pi$
$788$	0	0
$789$	17.2474	0.614025
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	5.75255	0.204279
$794$	0	0
$795$	−5.79796	−0.205632
$796$	0	0
$797$	18.4949	0.655123	0.327561	−	0.944830i	$-0.393773\pi$
$797$	18.4949	0.655123	0.327561	+	0.944830i	$0.393773\pi$
$798$	0	0
$799$	6.59592	0.233347
$800$	0	0
$801$	−15.7980	−0.558193
$802$	0	0
$803$	3.44949	0.121730
$804$	0	0
$805$	12.8990	0.454629
$806$	0	0
$807$	−12.8990	−0.454066
$808$	0	0
$809$	−2.89898	−0.101923	−0.0509613	−	0.998701i	$-0.516229\pi$
$809$	−2.89898	−0.101923	−0.0509613	+	0.998701i	$0.516229\pi$
$810$	0	0
$811$	47.9898	1.68515	0.842575	−	0.538579i	$-0.181039\pi$
$811$	47.9898	1.68515	0.842575	+	0.538579i	$0.181039\pi$
$812$	0	0
$813$	17.6969	0.620659
$814$	0	0
$815$	3.55051	0.124369
$816$	0	0
$817$	6.44949	0.225639
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	18.4495	0.643892	0.321946	−	0.946758i	$-0.395663\pi$
$821$	18.4495	0.643892	0.321946	+	0.946758i	$0.395663\pi$
$822$	0	0
$823$	20.6969	0.721450	0.360725	−	0.932672i	$-0.382529\pi$
$823$	20.6969	0.721450	0.360725	+	0.932672i	$0.382529\pi$
$824$	0	0
$825$	−16.5505	−0.576215
$826$	0	0
$827$	36.1464	1.25693	0.628467	−	0.777836i	$-0.283683\pi$
$827$	36.1464	1.25693	0.628467	+	0.777836i	$0.283683\pi$
$828$	0	0
$829$	32.5959	1.13210	0.566052	−	0.824370i	$-0.308470\pi$
$829$	32.5959	1.13210	0.566052	+	0.824370i	$0.308470\pi$
$830$	0	0
$831$	10.7980	0.374577
$832$	0	0
$833$	−18.5505	−0.642737
$834$	0	0
$835$	−6.11123	−0.211488
$836$	0	0
$837$	−7.00000	−0.241955
$838$	0	0
$839$	−40.8990	−1.41199	−0.705995	−	0.708217i	$-0.749500\pi$
$839$	−40.8990	−1.41199	−0.705995	+	0.708217i	$0.749500\pi$
$840$	0	0
$841$	40.6969	1.40334
$842$	0	0
$843$	29.4495	1.01429
$844$	0	0
$845$	−5.75255	−0.197894
$846$	0	0
$847$	−4.00000	−0.137442
$848$	0	0
$849$	−9.89898	−0.339732
$850$	0	0
$851$	−25.1464	−0.862008
$852$	0	0
$853$	12.0000	0.410872	0.205436	−	0.978671i	$-0.434139\pi$
$853$	12.0000	0.410872	0.205436	+	0.978671i	$0.434139\pi$
$854$	0	0
$855$	−2.89898	−0.0991430
$856$	0	0
$857$	4.89898	0.167346	0.0836730	−	0.996493i	$-0.473335\pi$
$857$	4.89898	0.167346	0.0836730	+	0.996493i	$0.473335\pi$
$858$	0	0
$859$	−23.2020	−0.791643	−0.395822	−	0.918327i	$-0.629540\pi$
$859$	−23.2020	−0.791643	−0.395822	+	0.918327i	$0.629540\pi$
$860$	0	0
$861$	4.44949	0.151638
$862$	0	0
$863$	32.4495	1.10459	0.552297	−	0.833648i	$-0.313752\pi$
$863$	32.4495	1.10459	0.552297	+	0.833648i	$0.313752\pi$
$864$	0	0
$865$	3.39388	0.115395
$866$	0	0
$867$	−14.8990	−0.505996
$868$	0	0
$869$	−34.4949	−1.17016
$870$	0	0
$871$	−3.10102	−0.105074
$872$	0	0
$873$	−3.34847	−0.113329
$874$	0	0
$875$	19.5959	0.662463
$876$	0	0
$877$	−37.8990	−1.27976	−0.639879	−	0.768476i	$-0.721015\pi$
$877$	−37.8990	−1.27976	−0.639879	+	0.768476i	$0.721015\pi$
$878$	0	0
$879$	14.3485	0.483962
$880$	0	0
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	0	0
$883$	36.7423	1.23648	0.618239	−	0.785990i	$-0.287846\pi$
$883$	36.7423	1.23648	0.618239	+	0.785990i	$0.287846\pi$
$884$	0	0
$885$	6.60612	0.222062
$886$	0	0
$887$	−28.3485	−0.951848	−0.475924	−	0.879486i	$-0.657886\pi$
$887$	−28.3485	−0.951848	−0.475924	+	0.879486i	$0.657886\pi$
$888$	0	0
$889$	−80.0908	−2.68616
$890$	0	0
$891$	3.44949	0.115562
$892$	0	0
$893$	29.3485	0.982109
$894$	0	0
$895$	−2.85357	−0.0953844
$896$	0	0
$897$	2.89898	0.0967941
$898$	0	0
$899$	−58.4393	−1.94906
$900$	0	0
$901$	18.6969	0.622885
$902$	0	0
$903$	4.44949	0.148070
$904$	0	0
$905$	6.20204	0.206163
$906$	0	0
$907$	31.7980	1.05583	0.527917	−	0.849296i	$-0.322973\pi$
$907$	31.7980	1.05583	0.527917	+	0.849296i	$0.322973\pi$
$908$	0	0
$909$	4.34847	0.144230
$910$	0	0
$911$	−10.9444	−0.362604	−0.181302	−	0.983427i	$-0.558031\pi$
$911$	−10.9444	−0.362604	−0.181302	+	0.983427i	$0.558031\pi$
$912$	0	0
$913$	−26.0454	−0.861977
$914$	0	0
$915$	−5.75255	−0.190173
$916$	0	0
$917$	−54.4949	−1.79958
$918$	0	0
$919$	−26.2474	−0.865823	−0.432912	−	0.901436i	$-0.642514\pi$
$919$	−26.2474	−0.865823	−0.432912	+	0.901436i	$0.642514\pi$
$920$	0	0
$921$	−24.5959	−0.810463
$922$	0	0
$923$	1.95459	0.0643362
$924$	0	0
$925$	−18.7071	−0.615087
$926$	0	0
$927$	−13.0000	−0.426976
$928$	0	0
$929$	−2.14643	−0.0704220	−0.0352110	−	0.999380i	$-0.511210\pi$
$929$	−2.14643	−0.0704220	−0.0352110	+	0.999380i	$0.511210\pi$
$930$	0	0
$931$	−82.5403	−2.70515
$932$	0	0
$933$	−30.6969	−1.00497
$934$	0	0
$935$	−2.24745	−0.0734994
$936$	0	0
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	0	0
$939$	−9.34847	−0.305076
$940$	0	0
$941$	24.2020	0.788964	0.394482	−	0.918904i	$-0.370924\pi$
$941$	24.2020	0.788964	0.394482	+	0.918904i	$0.370924\pi$
$942$	0	0
$943$	6.44949	0.210024
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	−0.202041	−0.00656545	−0.00328273	−	0.999995i	$-0.501045\pi$
$947$	−0.202041	−0.00656545	−0.00328273	+	0.999995i	$0.501045\pi$
$948$	0	0
$949$	−0.449490	−0.0145911
$950$	0	0
$951$	−9.24745	−0.299869
$952$	0	0
$953$	−14.0454	−0.454975	−0.227488	−	0.973781i	$-0.573051\pi$
$953$	−14.0454	−0.454975	−0.227488	+	0.973781i	$0.573051\pi$
$954$	0	0
$955$	−1.39388	−0.0451048
$956$	0	0
$957$	28.7980	0.930906
$958$	0	0
$959$	15.3485	0.495628
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	2.00000	0.0644491
$964$	0	0
$965$	3.39388	0.109253
$966$	0	0
$967$	−34.0000	−1.09337	−0.546683	−	0.837340i	$-0.684110\pi$
$967$	−34.0000	−1.09337	−0.546683	+	0.837340i	$0.684110\pi$
$968$	0	0
$969$	9.34847	0.300316
$970$	0	0
$971$	34.5505	1.10878	0.554389	−	0.832257i	$-0.312952\pi$
$971$	34.5505	1.10878	0.554389	+	0.832257i	$0.312952\pi$
$972$	0	0
$973$	−0.898979	−0.0288200
$974$	0	0
$975$	2.15663	0.0690675
$976$	0	0
$977$	−21.0454	−0.673302	−0.336651	−	0.941629i	$-0.609294\pi$
$977$	−21.0454	−0.673302	−0.336651	+	0.941629i	$0.609294\pi$
$978$	0	0
$979$	−54.4949	−1.74166
$980$	0	0
$981$	5.34847	0.170763
$982$	0	0
$983$	−35.1464	−1.12100	−0.560498	−	0.828155i	$-0.689390\pi$
$983$	−35.1464	−1.12100	−0.560498	+	0.828155i	$0.689390\pi$
$984$	0	0
$985$	−10.8082	−0.344377
$986$	0	0
$987$	20.2474	0.644483
$988$	0	0
$989$	6.44949	0.205082
$990$	0	0
$991$	−49.8888	−1.58477	−0.792385	−	0.610022i	$-0.791161\pi$
$991$	−49.8888	−1.58477	−0.792385	+	0.610022i	$0.791161\pi$
$992$	0	0
$993$	25.5505	0.810821
$994$	0	0
$995$	6.49490	0.205902
$996$	0	0
$997$	6.65153	0.210656	0.105328	−	0.994438i	$-0.466411\pi$
$997$	6.65153	0.210656	0.105328	+	0.994438i	$0.466411\pi$
$998$	0	0
$999$	3.89898	0.123358

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1968.2.a.s.1.2		2
3.2	odd	2		5904.2.a.ba.1.1		2
4.3	odd	2		492.2.a.c.1.2	✓	2
8.3	odd	2		7872.2.a.bq.1.1		2
8.5	even	2		7872.2.a.bm.1.1		2
12.11	even	2		1476.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
492.2.a.c.1.2	✓	2	4.3	odd	2
1476.2.a.d.1.1		2	12.11	even	2
1968.2.a.s.1.2		2	1.1	even	1	trivial
5904.2.a.ba.1.1		2	3.2	odd	2
7872.2.a.bm.1.1		2	8.5	even	2
7872.2.a.bq.1.1		2	8.3	odd	2

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:	\( N \)	\(=\)	\( 1968 = 2^{4} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1968.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.449490 q^{5} -4.44949 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.449490 q^{5} -4.44949 q^{7} +1.00000 q^{9} +3.44949 q^{11} -0.449490 q^{13} +0.449490 q^{15} -1.44949 q^{17} -6.44949 q^{19} -4.44949 q^{21} -6.44949 q^{23} -4.79796 q^{25} +1.00000 q^{27} +8.34847 q^{29} -7.00000 q^{31} +3.44949 q^{33} -2.00000 q^{35} +3.89898 q^{37} -0.449490 q^{39} -1.00000 q^{41} -1.00000 q^{43} +0.449490 q^{45} -4.55051 q^{47} +12.7980 q^{49} -1.44949 q^{51} -12.8990 q^{53} +1.55051 q^{55} -6.44949 q^{57} +14.6969 q^{59} -12.7980 q^{61} -4.44949 q^{63} -0.202041 q^{65} +6.89898 q^{67} -6.44949 q^{69} -4.34847 q^{71} +1.00000 q^{73} -4.79796 q^{75} -15.3485 q^{77} -10.0000 q^{79} +1.00000 q^{81} -7.55051 q^{83} -0.651531 q^{85} +8.34847 q^{87} -15.7980 q^{89} +2.00000 q^{91} -7.00000 q^{93} -2.89898 q^{95} -3.34847 q^{97} +3.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^5 - 4 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{5} - 4 q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} - 4 q^{15} + 2 q^{17} - 8 q^{19} - 4 q^{21} - 8 q^{23} + 10 q^{25} + 2 q^{27} + 2 q^{29} - 14 q^{31} + 2 q^{33} - 4 q^{35} - 2 q^{37} + 4 q^{39} - 2 q^{41} - 2 q^{43} - 4 q^{45} - 14 q^{47} + 6 q^{49} + 2 q^{51} - 16 q^{53} + 8 q^{55} - 8 q^{57} - 6 q^{61} - 4 q^{63} - 20 q^{65} + 4 q^{67} - 8 q^{69} + 6 q^{71} + 2 q^{73} + 10 q^{75} - 16 q^{77} - 20 q^{79} + 2 q^{81} - 20 q^{83} - 16 q^{85} + 2 q^{87} - 12 q^{89} + 4 q^{91} - 14 q^{93} + 4 q^{95} + 8 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^5 - 4 * q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 - 4 * q^15 + 2 * q^17 - 8 * q^19 - 4 * q^21 - 8 * q^23 + 10 * q^25 + 2 * q^27 + 2 * q^29 - 14 * q^31 + 2 * q^33 - 4 * q^35 - 2 * q^37 + 4 * q^39 - 2 * q^41 - 2 * q^43 - 4 * q^45 - 14 * q^47 + 6 * q^49 + 2 * q^51 - 16 * q^53 + 8 * q^55 - 8 * q^57 - 6 * q^61 - 4 * q^63 - 20 * q^65 + 4 * q^67 - 8 * q^69 + 6 * q^71 + 2 * q^73 + 10 * q^75 - 16 * q^77 - 20 * q^79 + 2 * q^81 - 20 * q^83 - 16 * q^85 + 2 * q^87 - 12 * q^89 + 4 * q^91 - 14 * q^93 + 4 * q^95 + 8 * q^97 + 2 * q^99

Introduction

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