LMFDB - Embedded newform 1968.2.a.w.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 123)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.34292$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.853635 q^{5} +3.83221 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.853635 q^{5} +3.83221 q^{7} +1.00000 q^{9} +3.34292 q^{11} +3.14637 q^{13} +0.853635 q^{15} +3.63565 q^{17} -1.14637 q^{19} +3.83221 q^{21} +2.85363 q^{23} -4.27131 q^{25} +1.00000 q^{27} -8.02877 q^{29} -9.86098 q^{31} +3.34292 q^{33} +3.27131 q^{35} +8.19656 q^{37} +3.14637 q^{39} +1.00000 q^{41} -11.7606 q^{43} +0.853635 q^{45} -8.32150 q^{47} +7.68585 q^{49} +3.63565 q^{51} +1.60688 q^{53} +2.85363 q^{55} -1.14637 q^{57} +11.6644 q^{59} -4.19656 q^{61} +3.83221 q^{63} +2.68585 q^{65} -6.10038 q^{67} +2.85363 q^{69} +8.65708 q^{71} -10.1537 q^{73} -4.27131 q^{75} +12.8108 q^{77} +3.60688 q^{79} +1.00000 q^{81} +10.1249 q^{83} +3.10352 q^{85} -8.02877 q^{87} -3.37169 q^{89} +12.0575 q^{91} -9.86098 q^{93} -0.978577 q^{95} -7.53948 q^{97} +3.34292 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 4 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 4 * q^5 - 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 4 q^{5} - 2 q^{7} + 3 q^{9} + 4 q^{11} + 8 q^{13} + 4 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{21} + 10 q^{23} + 5 q^{25} + 3 q^{27} - 6 q^{29} + 2 q^{31} + 4 q^{33} - 8 q^{35} + 20 q^{37} + 8 q^{39} + 3 q^{41} - 10 q^{43} + 4 q^{45} - 4 q^{47} + 11 q^{49} + 2 q^{51} + 14 q^{53} + 10 q^{55} - 2 q^{57} + 8 q^{59} - 8 q^{61} - 2 q^{63} - 4 q^{65} - 12 q^{67} + 10 q^{69} + 32 q^{71} + 4 q^{73} + 5 q^{75} + 10 q^{77} + 20 q^{79} + 3 q^{81} + 14 q^{83} - 22 q^{85} - 6 q^{87} + 14 q^{89} + 2 q^{93} + 12 q^{95} - 12 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 4 * q^5 - 2 * q^7 + 3 * q^9 + 4 * q^11 + 8 * q^13 + 4 * q^15 + 2 * q^17 - 2 * q^19 - 2 * q^21 + 10 * q^23 + 5 * q^25 + 3 * q^27 - 6 * q^29 + 2 * q^31 + 4 * q^33 - 8 * q^35 + 20 * q^37 + 8 * q^39 + 3 * q^41 - 10 * q^43 + 4 * q^45 - 4 * q^47 + 11 * q^49 + 2 * q^51 + 14 * q^53 + 10 * q^55 - 2 * q^57 + 8 * q^59 - 8 * q^61 - 2 * q^63 - 4 * q^65 - 12 * q^67 + 10 * q^69 + 32 * q^71 + 4 * q^73 + 5 * q^75 + 10 * q^77 + 20 * q^79 + 3 * q^81 + 14 * q^83 - 22 * q^85 - 6 * q^87 + 14 * q^89 + 2 * q^93 + 12 * q^95 - 12 * q^97 + 4 * 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
$5$	0.853635	0.381757	0.190878	−	0.981614i	$-0.438866\pi$
$5$	0.853635	0.381757	0.190878	+	0.981614i	$0.438866\pi$
$6$	0	0
$7$	3.83221	1.44844	0.724220	−	0.689569i	$-0.242200\pi$
$7$	3.83221	1.44844	0.724220	+	0.689569i	$0.242200\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.34292	1.00793	0.503965	−	0.863724i	$-0.331874\pi$
$11$	3.34292	1.00793	0.503965	+	0.863724i	$0.331874\pi$
$12$	0	0
$13$	3.14637	0.872645	0.436322	−	0.899790i	$-0.356281\pi$
$13$	3.14637	0.872645	0.436322	+	0.899790i	$0.356281\pi$
$14$	0	0
$15$	0.853635	0.220407
$16$	0	0
$17$	3.63565	0.881776	0.440888	−	0.897562i	$-0.354664\pi$
$17$	3.63565	0.881776	0.440888	+	0.897562i	$0.354664\pi$
$18$	0	0
$19$	−1.14637	−0.262994	−0.131497	−	0.991317i	$-0.541978\pi$
$19$	−1.14637	−0.262994	−0.131497	+	0.991317i	$0.541978\pi$
$20$	0	0
$21$	3.83221	0.836257
$22$	0	0
$23$	2.85363	0.595024	0.297512	−	0.954718i	$-0.403843\pi$
$23$	2.85363	0.595024	0.297512	+	0.954718i	$0.403843\pi$
$24$	0	0
$25$	−4.27131	−0.854262
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.02877	−1.49091	−0.745453	−	0.666559i	$-0.767767\pi$
$29$	−8.02877	−1.49091	−0.745453	+	0.666559i	$0.767767\pi$
$30$	0	0
$31$	−9.86098	−1.77108	−0.885542	−	0.464559i	$-0.846213\pi$
$31$	−9.86098	−1.77108	−0.885542	+	0.464559i	$0.846213\pi$
$32$	0	0
$33$	3.34292	0.581928
$34$	0	0
$35$	3.27131	0.552952
$36$	0	0
$37$	8.19656	1.34751	0.673753	−	0.738957i	$-0.264681\pi$
$37$	8.19656	1.34751	0.673753	+	0.738957i	$0.264681\pi$
$38$	0	0
$39$	3.14637	0.503822
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−11.7606	−1.79347	−0.896737	−	0.442564i	$-0.854069\pi$
$43$	−11.7606	−1.79347	−0.896737	+	0.442564i	$0.854069\pi$
$44$	0	0
$45$	0.853635	0.127252
$46$	0	0
$47$	−8.32150	−1.21382	−0.606908	−	0.794772i	$-0.707590\pi$
$47$	−8.32150	−1.21382	−0.606908	+	0.794772i	$0.707590\pi$
$48$	0	0
$49$	7.68585	1.09798
$50$	0	0
$51$	3.63565	0.509093
$52$	0	0
$53$	1.60688	0.220723	0.110361	−	0.993892i	$-0.464799\pi$
$53$	1.60688	0.220723	0.110361	+	0.993892i	$0.464799\pi$
$54$	0	0
$55$	2.85363	0.384784
$56$	0	0
$57$	−1.14637	−0.151840
$58$	0	0
$59$	11.6644	1.51858	0.759289	−	0.650753i	$-0.225547\pi$
$59$	11.6644	1.51858	0.759289	+	0.650753i	$0.225547\pi$
$60$	0	0
$61$	−4.19656	−0.537314	−0.268657	−	0.963236i	$-0.586580\pi$
$61$	−4.19656	−0.537314	−0.268657	+	0.963236i	$0.586580\pi$
$62$	0	0
$63$	3.83221	0.482813
$64$	0	0
$65$	2.68585	0.333138
$66$	0	0
$67$	−6.10038	−0.745281	−0.372640	−	0.927976i	$-0.621547\pi$
$67$	−6.10038	−0.745281	−0.372640	+	0.927976i	$0.621547\pi$
$68$	0	0
$69$	2.85363	0.343537
$70$	0	0
$71$	8.65708	1.02741	0.513703	−	0.857968i	$-0.328273\pi$
$71$	8.65708	1.02741	0.513703	+	0.857968i	$0.328273\pi$
$72$	0	0
$73$	−10.1537	−1.18840	−0.594201	−	0.804317i	$-0.702532\pi$
$73$	−10.1537	−1.18840	−0.594201	+	0.804317i	$0.702532\pi$
$74$	0	0
$75$	−4.27131	−0.493208
$76$	0	0
$77$	12.8108	1.45992
$78$	0	0
$79$	3.60688	0.405806	0.202903	−	0.979199i	$-0.434962\pi$
$79$	3.60688	0.405806	0.202903	+	0.979199i	$0.434962\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.1249	1.11136	0.555678	−	0.831397i	$-0.312459\pi$
$83$	10.1249	1.11136	0.555678	+	0.831397i	$0.312459\pi$
$84$	0	0
$85$	3.10352	0.336624
$86$	0	0
$87$	−8.02877	−0.860774
$88$	0	0
$89$	−3.37169	−0.357399	−0.178699	−	0.983904i	$-0.557189\pi$
$89$	−3.37169	−0.357399	−0.178699	+	0.983904i	$0.557189\pi$
$90$	0	0
$91$	12.0575	1.26397
$92$	0	0
$93$	−9.86098	−1.02254
$94$	0	0
$95$	−0.978577	−0.100400
$96$	0	0
$97$	−7.53948	−0.765518	−0.382759	−	0.923848i	$-0.625026\pi$
$97$	−7.53948	−0.765518	−0.382759	+	0.923848i	$0.625026\pi$
$98$	0	0
$99$	3.34292	0.335976
$100$	0	0
$101$	5.00735	0.498250	0.249125	−	0.968471i	$-0.419857\pi$
$101$	5.00735	0.498250	0.249125	+	0.968471i	$0.419857\pi$
$102$	0	0
$103$	−1.21798	−0.120011	−0.0600056	−	0.998198i	$-0.519112\pi$
$103$	−1.21798	−0.120011	−0.0600056	+	0.998198i	$0.519112\pi$
$104$	0	0
$105$	3.27131	0.319247
$106$	0	0
$107$	−13.7648	−1.33069	−0.665347	−	0.746534i	$-0.731716\pi$
$107$	−13.7648	−1.33069	−0.665347	+	0.746534i	$0.731716\pi$
$108$	0	0
$109$	3.14637	0.301367	0.150684	−	0.988582i	$-0.451853\pi$
$109$	3.14637	0.301367	0.150684	+	0.988582i	$0.451853\pi$
$110$	0	0
$111$	8.19656	0.777983
$112$	0	0
$113$	−7.73183	−0.727349	−0.363675	−	0.931526i	$-0.618478\pi$
$113$	−7.73183	−0.727349	−0.363675	+	0.931526i	$0.618478\pi$
$114$	0	0
$115$	2.43596	0.227155
$116$	0	0
$117$	3.14637	0.290882
$118$	0	0
$119$	13.9326	1.27720
$120$	0	0
$121$	0.175135	0.0159213
$122$	0	0
$123$	1.00000	0.0901670
$124$	0	0
$125$	−7.91431	−0.707877
$126$	0	0
$127$	12.0575	1.06993	0.534967	−	0.844873i	$-0.320324\pi$
$127$	12.0575	1.06993	0.534967	+	0.844873i	$0.320324\pi$
$128$	0	0
$129$	−11.7606	−1.03546
$130$	0	0
$131$	−12.2253	−1.06813	−0.534066	−	0.845443i	$-0.679337\pi$
$131$	−12.2253	−1.06813	−0.534066	+	0.845443i	$0.679337\pi$
$132$	0	0
$133$	−4.39312	−0.380931
$134$	0	0
$135$	0.853635	0.0734692
$136$	0	0
$137$	7.69319	0.657274	0.328637	−	0.944456i	$-0.393411\pi$
$137$	7.69319	0.657274	0.328637	+	0.944456i	$0.393411\pi$
$138$	0	0
$139$	18.0147	1.52799	0.763993	−	0.645224i	$-0.223236\pi$
$139$	18.0147	1.52799	0.763993	+	0.645224i	$0.223236\pi$
$140$	0	0
$141$	−8.32150	−0.700797
$142$	0	0
$143$	10.5181	0.879564
$144$	0	0
$145$	−6.85363	−0.569163
$146$	0	0
$147$	7.68585	0.633918
$148$	0	0
$149$	12.5426	1.02753	0.513766	−	0.857931i	$-0.328250\pi$
$149$	12.5426	1.02753	0.513766	+	0.857931i	$0.328250\pi$
$150$	0	0
$151$	−2.62831	−0.213889	−0.106944	−	0.994265i	$-0.534107\pi$
$151$	−2.62831	−0.213889	−0.106944	+	0.994265i	$0.534107\pi$
$152$	0	0
$153$	3.63565	0.293925
$154$	0	0
$155$	−8.41767	−0.676124
$156$	0	0
$157$	20.1579	1.60878	0.804389	−	0.594103i	$-0.202493\pi$
$157$	20.1579	1.60878	0.804389	+	0.594103i	$0.202493\pi$
$158$	0	0
$159$	1.60688	0.127434
$160$	0	0
$161$	10.9357	0.861856
$162$	0	0
$163$	−18.2541	−1.42977	−0.714886	−	0.699241i	$-0.753521\pi$
$163$	−18.2541	−1.42977	−0.714886	+	0.699241i	$0.753521\pi$
$164$	0	0
$165$	2.85363	0.222155
$166$	0	0
$167$	−1.31415	−0.101692	−0.0508461	−	0.998706i	$-0.516192\pi$
$167$	−1.31415	−0.101692	−0.0508461	+	0.998706i	$0.516192\pi$
$168$	0	0
$169$	−3.10038	−0.238491
$170$	0	0
$171$	−1.14637	−0.0876648
$172$	0	0
$173$	1.18921	0.0904141	0.0452070	−	0.998978i	$-0.485605\pi$
$173$	1.18921	0.0904141	0.0452070	+	0.998978i	$0.485605\pi$
$174$	0	0
$175$	−16.3686	−1.23735
$176$	0	0
$177$	11.6644	0.876752
$178$	0	0
$179$	19.0073	1.42068	0.710338	−	0.703861i	$-0.248542\pi$
$179$	19.0073	1.42068	0.710338	+	0.703861i	$0.248542\pi$
$180$	0	0
$181$	10.3931	0.772514	0.386257	−	0.922391i	$-0.373768\pi$
$181$	10.3931	0.772514	0.386257	+	0.922391i	$0.373768\pi$
$182$	0	0
$183$	−4.19656	−0.310218
$184$	0	0
$185$	6.99686	0.514420
$186$	0	0
$187$	12.1537	0.888767
$188$	0	0
$189$	3.83221	0.278752
$190$	0	0
$191$	6.87819	0.497689	0.248844	−	0.968544i	$-0.419949\pi$
$191$	6.87819	0.497689	0.248844	+	0.968544i	$0.419949\pi$
$192$	0	0
$193$	−19.5970	−1.41062	−0.705312	−	0.708897i	$-0.749193\pi$
$193$	−19.5970	−1.41062	−0.705312	+	0.708897i	$0.749193\pi$
$194$	0	0
$195$	2.68585	0.192337
$196$	0	0
$197$	−24.7679	−1.76464	−0.882321	−	0.470647i	$-0.844020\pi$
$197$	−24.7679	−1.76464	−0.882321	+	0.470647i	$0.844020\pi$
$198$	0	0
$199$	−12.8108	−0.908133	−0.454066	−	0.890968i	$-0.650027\pi$
$199$	−12.8108	−0.908133	−0.454066	+	0.890968i	$0.650027\pi$
$200$	0	0
$201$	−6.10038	−0.430288
$202$	0	0
$203$	−30.7679	−2.15949
$204$	0	0
$205$	0.853635	0.0596204
$206$	0	0
$207$	2.85363	0.198341
$208$	0	0
$209$	−3.83221	−0.265080
$210$	0	0
$211$	−3.66442	−0.252269	−0.126135	−	0.992013i	$-0.540257\pi$
$211$	−3.66442	−0.252269	−0.126135	+	0.992013i	$0.540257\pi$
$212$	0	0
$213$	8.65708	0.593173
$214$	0	0
$215$	−10.0393	−0.684671
$216$	0	0
$217$	−37.7894	−2.56531
$218$	0	0
$219$	−10.1537	−0.686124
$220$	0	0
$221$	11.4391	0.769477
$222$	0	0
$223$	−9.17092	−0.614130	−0.307065	−	0.951688i	$-0.599347\pi$
$223$	−9.17092	−0.614130	−0.307065	+	0.951688i	$0.599347\pi$
$224$	0	0
$225$	−4.27131	−0.284754
$226$	0	0
$227$	−0.263962	−0.0175198	−0.00875988	−	0.999962i	$-0.502788\pi$
$227$	−0.263962	−0.0175198	−0.00875988	+	0.999962i	$0.502788\pi$
$228$	0	0
$229$	7.20390	0.476047	0.238024	−	0.971259i	$-0.423500\pi$
$229$	7.20390	0.476047	0.238024	+	0.971259i	$0.423500\pi$
$230$	0	0
$231$	12.8108	0.842888
$232$	0	0
$233$	16.1579	1.05854	0.529270	−	0.848453i	$-0.322466\pi$
$233$	16.1579	1.05854	0.529270	+	0.848453i	$0.322466\pi$
$234$	0	0
$235$	−7.10352	−0.463383
$236$	0	0
$237$	3.60688	0.234292
$238$	0	0
$239$	15.6644	1.01325	0.506624	−	0.862167i	$-0.330893\pi$
$239$	15.6644	1.01325	0.506624	+	0.862167i	$0.330893\pi$
$240$	0	0
$241$	10.5468	0.679381	0.339690	−	0.940537i	$-0.389678\pi$
$241$	10.5468	0.679381	0.339690	+	0.940537i	$0.389678\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.56090	0.419161
$246$	0	0
$247$	−3.60688	−0.229501
$248$	0	0
$249$	10.1249	0.641642
$250$	0	0
$251$	−21.4292	−1.35260	−0.676301	−	0.736626i	$-0.736418\pi$
$251$	−21.4292	−1.35260	−0.676301	+	0.736626i	$0.736418\pi$
$252$	0	0
$253$	9.53948	0.599742
$254$	0	0
$255$	3.10352	0.194350
$256$	0	0
$257$	−7.10773	−0.443368	−0.221684	−	0.975119i	$-0.571155\pi$
$257$	−7.10773	−0.443368	−0.221684	+	0.975119i	$0.571155\pi$
$258$	0	0
$259$	31.4109	1.95178
$260$	0	0
$261$	−8.02877	−0.496968
$262$	0	0
$263$	9.97123	0.614852	0.307426	−	0.951572i	$-0.400532\pi$
$263$	9.97123	0.614852	0.307426	+	0.951572i	$0.400532\pi$
$264$	0	0
$265$	1.37169	0.0842624
$266$	0	0
$267$	−3.37169	−0.206344
$268$	0	0
$269$	−0.685846	−0.0418168	−0.0209084	−	0.999781i	$-0.506656\pi$
$269$	−0.685846	−0.0418168	−0.0209084	+	0.999781i	$0.506656\pi$
$270$	0	0
$271$	−13.8034	−0.838499	−0.419250	−	0.907871i	$-0.637707\pi$
$271$	−13.8034	−0.838499	−0.419250	+	0.907871i	$0.637707\pi$
$272$	0	0
$273$	12.0575	0.729755
$274$	0	0
$275$	−14.2787	−0.861035
$276$	0	0
$277$	22.2113	1.33454	0.667272	−	0.744814i	$-0.267462\pi$
$277$	22.2113	1.33454	0.667272	+	0.744814i	$0.267462\pi$
$278$	0	0
$279$	−9.86098	−0.590361
$280$	0	0
$281$	4.61423	0.275262	0.137631	−	0.990484i	$-0.456051\pi$
$281$	4.61423	0.275262	0.137631	+	0.990484i	$0.456051\pi$
$282$	0	0
$283$	−9.13229	−0.542858	−0.271429	−	0.962458i	$-0.587496\pi$
$283$	−9.13229	−0.542858	−0.271429	+	0.962458i	$0.587496\pi$
$284$	0	0
$285$	−0.978577	−0.0579659
$286$	0	0
$287$	3.83221	0.226208
$288$	0	0
$289$	−3.78202	−0.222472
$290$	0	0
$291$	−7.53948	−0.441972
$292$	0	0
$293$	23.4433	1.36957	0.684786	−	0.728744i	$-0.259896\pi$
$293$	23.4433	1.36957	0.684786	+	0.728744i	$0.259896\pi$
$294$	0	0
$295$	9.95715	0.579728
$296$	0	0
$297$	3.34292	0.193976
$298$	0	0
$299$	8.97858	0.519245
$300$	0	0
$301$	−45.0691	−2.59774
$302$	0	0
$303$	5.00735	0.287665
$304$	0	0
$305$	−3.58233	−0.205123
$306$	0	0
$307$	−18.2541	−1.04182	−0.520908	−	0.853613i	$-0.674407\pi$
$307$	−18.2541	−1.04182	−0.520908	+	0.853613i	$0.674407\pi$
$308$	0	0
$309$	−1.21798	−0.0692885
$310$	0	0
$311$	−14.8782	−0.843665	−0.421832	−	0.906674i	$-0.638613\pi$
$311$	−14.8782	−0.843665	−0.421832	+	0.906674i	$0.638613\pi$
$312$	0	0
$313$	−26.4752	−1.49647	−0.748234	−	0.663435i	$-0.769098\pi$
$313$	−26.4752	−1.49647	−0.748234	+	0.663435i	$0.769098\pi$
$314$	0	0
$315$	3.27131	0.184317
$316$	0	0
$317$	29.6503	1.66533	0.832665	−	0.553777i	$-0.186814\pi$
$317$	29.6503	1.66533	0.832665	+	0.553777i	$0.186814\pi$
$318$	0	0
$319$	−26.8396	−1.50273
$320$	0	0
$321$	−13.7648	−0.768277
$322$	0	0
$323$	−4.16779	−0.231902
$324$	0	0
$325$	−13.4391	−0.745467
$326$	0	0
$327$	3.14637	0.173994
$328$	0	0
$329$	−31.8898	−1.75814
$330$	0	0
$331$	−2.60375	−0.143115	−0.0715575	−	0.997436i	$-0.522797\pi$
$331$	−2.60375	−0.143115	−0.0715575	+	0.997436i	$0.522797\pi$
$332$	0	0
$333$	8.19656	0.449169
$334$	0	0
$335$	−5.20750	−0.284516
$336$	0	0
$337$	−1.51071	−0.0822937	−0.0411468	−	0.999153i	$-0.513101\pi$
$337$	−1.51071	−0.0822937	−0.0411468	+	0.999153i	$0.513101\pi$
$338$	0	0
$339$	−7.73183	−0.419935
$340$	0	0
$341$	−32.9645	−1.78513
$342$	0	0
$343$	2.62831	0.141915
$344$	0	0
$345$	2.43596	0.131148
$346$	0	0
$347$	14.2787	0.766518	0.383259	−	0.923641i	$-0.374802\pi$
$347$	14.2787	0.766518	0.383259	+	0.923641i	$0.374802\pi$
$348$	0	0
$349$	−16.2541	−0.870062	−0.435031	−	0.900416i	$-0.643263\pi$
$349$	−16.2541	−0.870062	−0.435031	+	0.900416i	$0.643263\pi$
$350$	0	0
$351$	3.14637	0.167941
$352$	0	0
$353$	11.1709	0.594568	0.297284	−	0.954789i	$-0.403919\pi$
$353$	11.1709	0.594568	0.297284	+	0.954789i	$0.403919\pi$
$354$	0	0
$355$	7.38998	0.392219
$356$	0	0
$357$	13.9326	0.737391
$358$	0	0
$359$	16.2253	0.856340	0.428170	−	0.903698i	$-0.359158\pi$
$359$	16.2253	0.856340	0.428170	+	0.903698i	$0.359158\pi$
$360$	0	0
$361$	−17.6858	−0.930834
$362$	0	0
$363$	0.175135	0.00919219
$364$	0	0
$365$	−8.66756	−0.453681
$366$	0	0
$367$	−0.431750	−0.0225372	−0.0112686	−	0.999937i	$-0.503587\pi$
$367$	−0.431750	−0.0225372	−0.0112686	+	0.999937i	$0.503587\pi$
$368$	0	0
$369$	1.00000	0.0520579
$370$	0	0
$371$	6.15792	0.319703
$372$	0	0
$373$	−3.66863	−0.189955	−0.0949773	−	0.995479i	$-0.530278\pi$
$373$	−3.66863	−0.189955	−0.0949773	+	0.995479i	$0.530278\pi$
$374$	0	0
$375$	−7.91431	−0.408693
$376$	0	0
$377$	−25.2614	−1.30103
$378$	0	0
$379$	9.89962	0.508509	0.254255	−	0.967137i	$-0.418170\pi$
$379$	9.89962	0.508509	0.254255	+	0.967137i	$0.418170\pi$
$380$	0	0
$381$	12.0575	0.617726
$382$	0	0
$383$	31.3148	1.60011	0.800055	−	0.599927i	$-0.204804\pi$
$383$	31.3148	1.60011	0.800055	+	0.599927i	$0.204804\pi$
$384$	0	0
$385$	10.9357	0.557336
$386$	0	0
$387$	−11.7606	−0.597825
$388$	0	0
$389$	7.39625	0.375005	0.187502	−	0.982264i	$-0.439961\pi$
$389$	7.39625	0.375005	0.187502	+	0.982264i	$0.439961\pi$
$390$	0	0
$391$	10.3748	0.524678
$392$	0	0
$393$	−12.2253	−0.616686
$394$	0	0
$395$	3.07896	0.154919
$396$	0	0
$397$	11.9326	0.598880	0.299440	−	0.954115i	$-0.403200\pi$
$397$	11.9326	0.598880	0.299440	+	0.954115i	$0.403200\pi$
$398$	0	0
$399$	−4.39312	−0.219931
$400$	0	0
$401$	−15.7648	−0.787257	−0.393628	−	0.919270i	$-0.628780\pi$
$401$	−15.7648	−0.787257	−0.393628	+	0.919270i	$0.628780\pi$
$402$	0	0
$403$	−31.0263	−1.54553
$404$	0	0
$405$	0.853635	0.0424174
$406$	0	0
$407$	27.4005	1.35819
$408$	0	0
$409$	−7.55356	−0.373499	−0.186750	−	0.982408i	$-0.559795\pi$
$409$	−7.55356	−0.373499	−0.186750	+	0.982408i	$0.559795\pi$
$410$	0	0
$411$	7.69319	0.379477
$412$	0	0
$413$	44.7005	2.19957
$414$	0	0
$415$	8.64300	0.424268
$416$	0	0
$417$	18.0147	0.882183
$418$	0	0
$419$	−9.59702	−0.468845	−0.234423	−	0.972135i	$-0.575320\pi$
$419$	−9.59702	−0.468845	−0.234423	+	0.972135i	$0.575320\pi$
$420$	0	0
$421$	12.3503	0.601915	0.300958	−	0.953638i	$-0.402694\pi$
$421$	12.3503	0.601915	0.300958	+	0.953638i	$0.402694\pi$
$422$	0	0
$423$	−8.32150	−0.404605
$424$	0	0
$425$	−15.5290	−0.753267
$426$	0	0
$427$	−16.0821	−0.778267
$428$	0	0
$429$	10.5181	0.507817
$430$	0	0
$431$	−5.14637	−0.247892	−0.123946	−	0.992289i	$-0.539555\pi$
$431$	−5.14637	−0.247892	−0.123946	+	0.992289i	$0.539555\pi$
$432$	0	0
$433$	17.0894	0.821266	0.410633	−	0.911801i	$-0.365308\pi$
$433$	17.0894	0.821266	0.410633	+	0.911801i	$0.365308\pi$
$434$	0	0
$435$	−6.85363	−0.328607
$436$	0	0
$437$	−3.27131	−0.156488
$438$	0	0
$439$	22.3503	1.06672	0.533360	−	0.845888i	$-0.320929\pi$
$439$	22.3503	1.06672	0.533360	+	0.845888i	$0.320929\pi$
$440$	0	0
$441$	7.68585	0.365993
$442$	0	0
$443$	−23.4145	−1.11246	−0.556229	−	0.831029i	$-0.687752\pi$
$443$	−23.4145	−1.11246	−0.556229	+	0.831029i	$0.687752\pi$
$444$	0	0
$445$	−2.87819	−0.136439
$446$	0	0
$447$	12.5426	0.593245
$448$	0	0
$449$	−33.2713	−1.57017	−0.785085	−	0.619388i	$-0.787381\pi$
$449$	−33.2713	−1.57017	−0.785085	+	0.619388i	$0.787381\pi$
$450$	0	0
$451$	3.34292	0.157412
$452$	0	0
$453$	−2.62831	−0.123489
$454$	0	0
$455$	10.2927	0.482531
$456$	0	0
$457$	−26.3931	−1.23462	−0.617309	−	0.786721i	$-0.711777\pi$
$457$	−26.3931	−1.23462	−0.617309	+	0.786721i	$0.711777\pi$
$458$	0	0
$459$	3.63565	0.169698
$460$	0	0
$461$	2.20077	0.102500	0.0512500	−	0.998686i	$-0.483679\pi$
$461$	2.20077	0.102500	0.0512500	+	0.998686i	$0.483679\pi$
$462$	0	0
$463$	−24.5328	−1.14013	−0.570067	−	0.821598i	$-0.693083\pi$
$463$	−24.5328	−1.14013	−0.570067	+	0.821598i	$0.693083\pi$
$464$	0	0
$465$	−8.41767	−0.390360
$466$	0	0
$467$	22.9357	1.06134	0.530670	−	0.847579i	$-0.321941\pi$
$467$	22.9357	1.06134	0.530670	+	0.847579i	$0.321941\pi$
$468$	0	0
$469$	−23.3780	−1.07949
$470$	0	0
$471$	20.1579	0.928828
$472$	0	0
$473$	−39.3148	−1.80770
$474$	0	0
$475$	4.89648	0.224666
$476$	0	0
$477$	1.60688	0.0735742
$478$	0	0
$479$	−41.6075	−1.90110	−0.950548	−	0.310579i	$-0.899477\pi$
$479$	−41.6075	−1.90110	−0.950548	+	0.310579i	$0.899477\pi$
$480$	0	0
$481$	25.7894	1.17589
$482$	0	0
$483$	10.9357	0.497593
$484$	0	0
$485$	−6.43596	−0.292242
$486$	0	0
$487$	−40.4036	−1.83086	−0.915431	−	0.402475i	$-0.868150\pi$
$487$	−40.4036	−1.83086	−0.915431	+	0.402475i	$0.868150\pi$
$488$	0	0
$489$	−18.2541	−0.825479
$490$	0	0
$491$	16.6184	0.749980	0.374990	−	0.927029i	$-0.377646\pi$
$491$	16.6184	0.749980	0.374990	+	0.927029i	$0.377646\pi$
$492$	0	0
$493$	−29.1898	−1.31464
$494$	0	0
$495$	2.85363	0.128261
$496$	0	0
$497$	33.1758	1.48814
$498$	0	0
$499$	0.921039	0.0412314	0.0206157	−	0.999787i	$-0.493437\pi$
$499$	0.921039	0.0412314	0.0206157	+	0.999787i	$0.493437\pi$
$500$	0	0
$501$	−1.31415	−0.0587121
$502$	0	0
$503$	−18.9217	−0.843675	−0.421837	−	0.906671i	$-0.638615\pi$
$503$	−18.9217	−0.843675	−0.421837	+	0.906671i	$0.638615\pi$
$504$	0	0
$505$	4.27444	0.190210
$506$	0	0
$507$	−3.10038	−0.137693
$508$	0	0
$509$	27.8855	1.23600	0.618002	−	0.786176i	$-0.287942\pi$
$509$	27.8855	1.23600	0.618002	+	0.786176i	$0.287942\pi$
$510$	0	0
$511$	−38.9112	−1.72133
$512$	0	0
$513$	−1.14637	−0.0506133
$514$	0	0
$515$	−1.03971	−0.0458151
$516$	0	0
$517$	−27.8181	−1.22344
$518$	0	0
$519$	1.18921	0.0522006
$520$	0	0
$521$	9.93681	0.435339	0.217670	−	0.976022i	$-0.430154\pi$
$521$	9.93681	0.435339	0.217670	+	0.976022i	$0.430154\pi$
$522$	0	0
$523$	35.9718	1.57294	0.786470	−	0.617629i	$-0.211907\pi$
$523$	35.9718	1.57294	0.786470	+	0.617629i	$0.211907\pi$
$524$	0	0
$525$	−16.3686	−0.714382
$526$	0	0
$527$	−35.8511	−1.56170
$528$	0	0
$529$	−14.8568	−0.645947
$530$	0	0
$531$	11.6644	0.506193
$532$	0	0
$533$	3.14637	0.136284
$534$	0	0
$535$	−11.7501	−0.508002
$536$	0	0
$537$	19.0073	0.820228
$538$	0	0
$539$	25.6932	1.10668
$540$	0	0
$541$	9.79923	0.421302	0.210651	−	0.977561i	$-0.432442\pi$
$541$	9.79923	0.421302	0.210651	+	0.977561i	$0.432442\pi$
$542$	0	0
$543$	10.3931	0.446011
$544$	0	0
$545$	2.68585	0.115049
$546$	0	0
$547$	33.8223	1.44614	0.723070	−	0.690775i	$-0.242731\pi$
$547$	33.8223	1.44614	0.723070	+	0.690775i	$0.242731\pi$
$548$	0	0
$549$	−4.19656	−0.179105
$550$	0	0
$551$	9.20390	0.392099
$552$	0	0
$553$	13.8223	0.587786
$554$	0	0
$555$	6.99686	0.297000
$556$	0	0
$557$	−20.9217	−0.886479	−0.443239	−	0.896403i	$-0.646171\pi$
$557$	−20.9217	−0.886479	−0.443239	+	0.896403i	$0.646171\pi$
$558$	0	0
$559$	−37.0031	−1.56507
$560$	0	0
$561$	12.1537	0.513130
$562$	0	0
$563$	28.6289	1.20657	0.603283	−	0.797527i	$-0.293859\pi$
$563$	28.6289	1.20657	0.603283	+	0.797527i	$0.293859\pi$
$564$	0	0
$565$	−6.60015	−0.277671
$566$	0	0
$567$	3.83221	0.160938
$568$	0	0
$569$	−20.9603	−0.878701	−0.439351	−	0.898316i	$-0.644791\pi$
$569$	−20.9603	−0.878701	−0.439351	+	0.898316i	$0.644791\pi$
$570$	0	0
$571$	13.2860	0.556002	0.278001	−	0.960581i	$-0.410328\pi$
$571$	13.2860	0.556002	0.278001	+	0.960581i	$0.410328\pi$
$572$	0	0
$573$	6.87819	0.287341
$574$	0	0
$575$	−12.1888	−0.508306
$576$	0	0
$577$	1.35700	0.0564926	0.0282463	−	0.999601i	$-0.491008\pi$
$577$	1.35700	0.0564926	0.0282463	+	0.999601i	$0.491008\pi$
$578$	0	0
$579$	−19.5970	−0.814424
$580$	0	0
$581$	38.8009	1.60973
$582$	0	0
$583$	5.37169	0.222473
$584$	0	0
$585$	2.68585	0.111046
$586$	0	0
$587$	−29.9431	−1.23588	−0.617942	−	0.786224i	$-0.712033\pi$
$587$	−29.9431	−1.23588	−0.617942	+	0.786224i	$0.712033\pi$
$588$	0	0
$589$	11.3043	0.465785
$590$	0	0
$591$	−24.7679	−1.01882
$592$	0	0
$593$	−6.01408	−0.246969	−0.123484	−	0.992347i	$-0.539407\pi$
$593$	−6.01408	−0.246969	−0.123484	+	0.992347i	$0.539407\pi$
$594$	0	0
$595$	11.8933	0.487580
$596$	0	0
$597$	−12.8108	−0.524311
$598$	0	0
$599$	−30.0477	−1.22771	−0.613857	−	0.789417i	$-0.710383\pi$
$599$	−30.0477	−1.22771	−0.613857	+	0.789417i	$0.710383\pi$
$600$	0	0
$601$	−38.1642	−1.55675	−0.778375	−	0.627800i	$-0.783956\pi$
$601$	−38.1642	−1.55675	−0.778375	+	0.627800i	$0.783956\pi$
$602$	0	0
$603$	−6.10038	−0.248427
$604$	0	0
$605$	0.149501	0.00607808
$606$	0	0
$607$	−35.7220	−1.44991	−0.724955	−	0.688796i	$-0.758139\pi$
$607$	−35.7220	−1.44991	−0.724955	+	0.688796i	$0.758139\pi$
$608$	0	0
$609$	−30.7679	−1.24678
$610$	0	0
$611$	−26.1825	−1.05923
$612$	0	0
$613$	−8.68585	−0.350818	−0.175409	−	0.984496i	$-0.556125\pi$
$613$	−8.68585	−0.350818	−0.175409	+	0.984496i	$0.556125\pi$
$614$	0	0
$615$	0.853635	0.0344219
$616$	0	0
$617$	25.4391	1.02414	0.512070	−	0.858944i	$-0.328879\pi$
$617$	25.4391	1.02414	0.512070	+	0.858944i	$0.328879\pi$
$618$	0	0
$619$	−4.28852	−0.172370	−0.0861851	−	0.996279i	$-0.527468\pi$
$619$	−4.28852	−0.172370	−0.0861851	+	0.996279i	$0.527468\pi$
$620$	0	0
$621$	2.85363	0.114512
$622$	0	0
$623$	−12.9210	−0.517670
$624$	0	0
$625$	14.6006	0.584025
$626$	0	0
$627$	−3.83221	−0.153044
$628$	0	0
$629$	29.7998	1.18820
$630$	0	0
$631$	3.17513	0.126400	0.0632001	−	0.998001i	$-0.479869\pi$
$631$	3.17513	0.126400	0.0632001	+	0.998001i	$0.479869\pi$
$632$	0	0
$633$	−3.66442	−0.145648
$634$	0	0
$635$	10.2927	0.408455
$636$	0	0
$637$	24.1825	0.958145
$638$	0	0
$639$	8.65708	0.342469
$640$	0	0
$641$	−21.0565	−0.831680	−0.415840	−	0.909438i	$-0.636512\pi$
$641$	−21.0565	−0.831680	−0.415840	+	0.909438i	$0.636512\pi$
$642$	0	0
$643$	18.2070	0.718016	0.359008	−	0.933335i	$-0.383115\pi$
$643$	18.2070	0.718016	0.359008	+	0.933335i	$0.383115\pi$
$644$	0	0
$645$	−10.0393	−0.395295
$646$	0	0
$647$	19.6890	0.774054	0.387027	−	0.922068i	$-0.373502\pi$
$647$	19.6890	0.774054	0.387027	+	0.922068i	$0.373502\pi$
$648$	0	0
$649$	38.9933	1.53062
$650$	0	0
$651$	−37.7894	−1.48108
$652$	0	0
$653$	−2.80031	−0.109584	−0.0547922	−	0.998498i	$-0.517450\pi$
$653$	−2.80031	−0.109584	−0.0547922	+	0.998498i	$0.517450\pi$
$654$	0	0
$655$	−10.4360	−0.407767
$656$	0	0
$657$	−10.1537	−0.396134
$658$	0	0
$659$	7.07896	0.275757	0.137879	−	0.990449i	$-0.455972\pi$
$659$	7.07896	0.275757	0.137879	+	0.990449i	$0.455972\pi$
$660$	0	0
$661$	−15.6791	−0.609847	−0.304923	−	0.952377i	$-0.598631\pi$
$661$	−15.6791	−0.609847	−0.304923	+	0.952377i	$0.598631\pi$
$662$	0	0
$663$	11.4391	0.444258
$664$	0	0
$665$	−3.75011	−0.145423
$666$	0	0
$667$	−22.9112	−0.887124
$668$	0	0
$669$	−9.17092	−0.354568
$670$	0	0
$671$	−14.0288	−0.541575
$672$	0	0
$673$	−12.2070	−0.470547	−0.235273	−	0.971929i	$-0.575599\pi$
$673$	−12.2070	−0.470547	−0.235273	+	0.971929i	$0.575599\pi$
$674$	0	0
$675$	−4.27131	−0.164403
$676$	0	0
$677$	−39.7121	−1.52626	−0.763130	−	0.646245i	$-0.776338\pi$
$677$	−39.7121	−1.52626	−0.763130	+	0.646245i	$0.776338\pi$
$678$	0	0
$679$	−28.8929	−1.10881
$680$	0	0
$681$	−0.263962	−0.0101150
$682$	0	0
$683$	−9.22846	−0.353117	−0.176559	−	0.984290i	$-0.556497\pi$
$683$	−9.22846	−0.353117	−0.176559	+	0.984290i	$0.556497\pi$
$684$	0	0
$685$	6.56717	0.250919
$686$	0	0
$687$	7.20390	0.274846
$688$	0	0
$689$	5.05585	0.192612
$690$	0	0
$691$	4.75325	0.180822	0.0904111	−	0.995905i	$-0.471182\pi$
$691$	4.75325	0.180822	0.0904111	+	0.995905i	$0.471182\pi$
$692$	0	0
$693$	12.8108	0.486642
$694$	0	0
$695$	15.3780	0.583319
$696$	0	0
$697$	3.63565	0.137710
$698$	0	0
$699$	16.1579	0.611149
$700$	0	0
$701$	−21.0790	−0.796141	−0.398071	−	0.917355i	$-0.630320\pi$
$701$	−21.0790	−0.796141	−0.398071	+	0.917355i	$0.630320\pi$
$702$	0	0
$703$	−9.39625	−0.354386
$704$	0	0
$705$	−7.10352	−0.267534
$706$	0	0
$707$	19.1892	0.721685
$708$	0	0
$709$	42.7497	1.60550	0.802749	−	0.596318i	$-0.203370\pi$
$709$	42.7497	1.60550	0.802749	+	0.596318i	$0.203370\pi$
$710$	0	0
$711$	3.60688	0.135269
$712$	0	0
$713$	−28.1396	−1.05384
$714$	0	0
$715$	8.97858	0.335780
$716$	0	0
$717$	15.6644	0.584999
$718$	0	0
$719$	7.00735	0.261330	0.130665	−	0.991427i	$-0.458289\pi$
$719$	7.00735	0.261330	0.130665	+	0.991427i	$0.458289\pi$
$720$	0	0
$721$	−4.66756	−0.173829
$722$	0	0
$723$	10.5468	0.392241
$724$	0	0
$725$	34.2933	1.27362
$726$	0	0
$727$	33.3963	1.23860	0.619299	−	0.785155i	$-0.287417\pi$
$727$	33.3963	1.23860	0.619299	+	0.785155i	$0.287417\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−42.7575	−1.58144
$732$	0	0
$733$	13.1176	0.484509	0.242255	−	0.970213i	$-0.422113\pi$
$733$	13.1176	0.484509	0.242255	+	0.970213i	$0.422113\pi$
$734$	0	0
$735$	6.56090	0.242003
$736$	0	0
$737$	−20.3931	−0.751190
$738$	0	0
$739$	47.2902	1.73960	0.869799	−	0.493406i	$-0.164248\pi$
$739$	47.2902	1.73960	0.869799	+	0.493406i	$0.164248\pi$
$740$	0	0
$741$	−3.60688	−0.132502
$742$	0	0
$743$	18.5510	0.680572	0.340286	−	0.940322i	$-0.389476\pi$
$743$	18.5510	0.680572	0.340286	+	0.940322i	$0.389476\pi$
$744$	0	0
$745$	10.7068	0.392267
$746$	0	0
$747$	10.1249	0.370452
$748$	0	0
$749$	−52.7497	−1.92743
$750$	0	0
$751$	13.6791	0.499158	0.249579	−	0.968354i	$-0.419708\pi$
$751$	13.6791	0.499158	0.249579	+	0.968354i	$0.419708\pi$
$752$	0	0
$753$	−21.4292	−0.780925
$754$	0	0
$755$	−2.24361	−0.0816535
$756$	0	0
$757$	−23.0031	−0.836063	−0.418032	−	0.908432i	$-0.637280\pi$
$757$	−23.0031	−0.836063	−0.418032	+	0.908432i	$0.637280\pi$
$758$	0	0
$759$	9.53948	0.346261
$760$	0	0
$761$	20.4851	0.742583	0.371292	−	0.928516i	$-0.378915\pi$
$761$	20.4851	0.742583	0.371292	+	0.928516i	$0.378915\pi$
$762$	0	0
$763$	12.0575	0.436512
$764$	0	0
$765$	3.10352	0.112208
$766$	0	0
$767$	36.7005	1.32518
$768$	0	0
$769$	−17.5107	−0.631452	−0.315726	−	0.948850i	$-0.602248\pi$
$769$	−17.5107	−0.631452	−0.315726	+	0.948850i	$0.602248\pi$
$770$	0	0
$771$	−7.10773	−0.255979
$772$	0	0
$773$	−31.8083	−1.14406	−0.572032	−	0.820231i	$-0.693845\pi$
$773$	−31.8083	−1.14406	−0.572032	+	0.820231i	$0.693845\pi$
$774$	0	0
$775$	42.1193	1.51297
$776$	0	0
$777$	31.4109	1.12686
$778$	0	0
$779$	−1.14637	−0.0410728
$780$	0	0
$781$	28.9399	1.03555
$782$	0	0
$783$	−8.02877	−0.286925
$784$	0	0
$785$	17.2075	0.614162
$786$	0	0
$787$	−14.5040	−0.517011	−0.258506	−	0.966010i	$-0.583230\pi$
$787$	−14.5040	−0.517011	−0.258506	+	0.966010i	$0.583230\pi$
$788$	0	0
$789$	9.97123	0.354985
$790$	0	0
$791$	−29.6300	−1.05352
$792$	0	0
$793$	−13.2039	−0.468884
$794$	0	0
$795$	1.37169	0.0486489
$796$	0	0
$797$	50.3650	1.78402	0.892009	−	0.452017i	$-0.149295\pi$
$797$	50.3650	1.78402	0.892009	+	0.452017i	$0.149295\pi$
$798$	0	0
$799$	−30.2541	−1.07031
$800$	0	0
$801$	−3.37169	−0.119133
$802$	0	0
$803$	−33.9431	−1.19783
$804$	0	0
$805$	9.33512	0.329020
$806$	0	0
$807$	−0.685846	−0.0241429
$808$	0	0
$809$	35.2369	1.23886	0.619431	−	0.785051i	$-0.287363\pi$
$809$	35.2369	1.23886	0.619431	+	0.785051i	$0.287363\pi$
$810$	0	0
$811$	13.8610	0.486725	0.243362	−	0.969935i	$-0.421750\pi$
$811$	13.8610	0.486725	0.243362	+	0.969935i	$0.421750\pi$
$812$	0	0
$813$	−13.8034	−0.484108
$814$	0	0
$815$	−15.5823	−0.545825
$816$	0	0
$817$	13.4819	0.471673
$818$	0	0
$819$	12.0575	0.421324
$820$	0	0
$821$	7.20390	0.251418	0.125709	−	0.992067i	$-0.459879\pi$
$821$	7.20390	0.251418	0.125709	+	0.992067i	$0.459879\pi$
$822$	0	0
$823$	9.81392	0.342092	0.171046	−	0.985263i	$-0.445285\pi$
$823$	9.81392	0.342092	0.171046	+	0.985263i	$0.445285\pi$
$824$	0	0
$825$	−14.2787	−0.497119
$826$	0	0
$827$	−24.8788	−0.865121	−0.432560	−	0.901605i	$-0.642390\pi$
$827$	−24.8788	−0.865121	−0.432560	+	0.901605i	$0.642390\pi$
$828$	0	0
$829$	13.3759	0.464564	0.232282	−	0.972648i	$-0.425381\pi$
$829$	13.3759	0.464564	0.232282	+	0.972648i	$0.425381\pi$
$830$	0	0
$831$	22.2113	0.770500
$832$	0	0
$833$	27.9431	0.968170
$834$	0	0
$835$	−1.12181	−0.0388217
$836$	0	0
$837$	−9.86098	−0.340845
$838$	0	0
$839$	12.9210	0.446084	0.223042	−	0.974809i	$-0.428401\pi$
$839$	12.9210	0.446084	0.223042	+	0.974809i	$0.428401\pi$
$840$	0	0
$841$	35.4611	1.22280
$842$	0	0
$843$	4.61423	0.158923
$844$	0	0
$845$	−2.64659	−0.0910456
$846$	0	0
$847$	0.671153	0.0230611
$848$	0	0
$849$	−9.13229	−0.313419
$850$	0	0
$851$	23.3900	0.801798
$852$	0	0
$853$	16.4851	0.564438	0.282219	−	0.959350i	$-0.408929\pi$
$853$	16.4851	0.564438	0.282219	+	0.959350i	$0.408929\pi$
$854$	0	0
$855$	−0.978577	−0.0334666
$856$	0	0
$857$	−11.5149	−0.393342	−0.196671	−	0.980470i	$-0.563013\pi$
$857$	−11.5149	−0.393342	−0.196671	+	0.980470i	$0.563013\pi$
$858$	0	0
$859$	28.7392	0.980568	0.490284	−	0.871563i	$-0.336893\pi$
$859$	28.7392	0.980568	0.490284	+	0.871563i	$0.336893\pi$
$860$	0	0
$861$	3.83221	0.130601
$862$	0	0
$863$	−51.1611	−1.74154	−0.870771	−	0.491688i	$-0.836380\pi$
$863$	−51.1611	−1.74154	−0.870771	+	0.491688i	$0.836380\pi$
$864$	0	0
$865$	1.01515	0.0345162
$866$	0	0
$867$	−3.78202	−0.128444
$868$	0	0
$869$	12.0575	0.409024
$870$	0	0
$871$	−19.1940	−0.650365
$872$	0	0
$873$	−7.53948	−0.255173
$874$	0	0
$875$	−30.3293	−1.02532
$876$	0	0
$877$	34.3179	1.15883	0.579417	−	0.815031i	$-0.303280\pi$
$877$	34.3179	1.15883	0.579417	+	0.815031i	$0.303280\pi$
$878$	0	0
$879$	23.4433	0.790723
$880$	0	0
$881$	−38.9786	−1.31322	−0.656611	−	0.754230i	$-0.728011\pi$
$881$	−38.9786	−1.31322	−0.656611	+	0.754230i	$0.728011\pi$
$882$	0	0
$883$	28.2253	0.949858	0.474929	−	0.880024i	$-0.342474\pi$
$883$	28.2253	0.949858	0.474929	+	0.880024i	$0.342474\pi$
$884$	0	0
$885$	9.95715	0.334706
$886$	0	0
$887$	30.1438	1.01213	0.506066	−	0.862495i	$-0.331099\pi$
$887$	30.1438	1.01213	0.506066	+	0.862495i	$0.331099\pi$
$888$	0	0
$889$	46.2070	1.54973
$890$	0	0
$891$	3.34292	0.111992
$892$	0	0
$893$	9.53948	0.319227
$894$	0	0
$895$	16.2253	0.542353
$896$	0	0
$897$	8.97858	0.299786
$898$	0	0
$899$	79.1715	2.64052
$900$	0	0
$901$	5.84208	0.194628
$902$	0	0
$903$	−45.0691	−1.49981
$904$	0	0
$905$	8.87192	0.294913
$906$	0	0
$907$	−41.3435	−1.37279	−0.686395	−	0.727229i	$-0.740808\pi$
$907$	−41.3435	−1.37279	−0.686395	+	0.727229i	$0.740808\pi$
$908$	0	0
$909$	5.00735	0.166083
$910$	0	0
$911$	1.23206	0.0408199	0.0204099	−	0.999792i	$-0.493503\pi$
$911$	1.23206	0.0408199	0.0204099	+	0.999792i	$0.493503\pi$
$912$	0	0
$913$	33.8469	1.12017
$914$	0	0
$915$	−3.58233	−0.118428
$916$	0	0
$917$	−46.8500	−1.54712
$918$	0	0
$919$	21.4047	0.706075	0.353037	−	0.935609i	$-0.385149\pi$
$919$	21.4047	0.706075	0.353037	+	0.935609i	$0.385149\pi$
$920$	0	0
$921$	−18.2541	−0.601493
$922$	0	0
$923$	27.2383	0.896560
$924$	0	0
$925$	−35.0100	−1.15112
$926$	0	0
$927$	−1.21798	−0.0400037
$928$	0	0
$929$	32.8150	1.07663	0.538313	−	0.842745i	$-0.319062\pi$
$929$	32.8150	1.07663	0.538313	+	0.842745i	$0.319062\pi$
$930$	0	0
$931$	−8.81079	−0.288762
$932$	0	0
$933$	−14.8782	−0.487090
$934$	0	0
$935$	10.3748	0.339293
$936$	0	0
$937$	27.4783	0.897678	0.448839	−	0.893613i	$-0.351838\pi$
$937$	27.4783	0.897678	0.448839	+	0.893613i	$0.351838\pi$
$938$	0	0
$939$	−26.4752	−0.863986
$940$	0	0
$941$	38.1151	1.24252	0.621258	−	0.783606i	$-0.286622\pi$
$941$	38.1151	1.24252	0.621258	+	0.783606i	$0.286622\pi$
$942$	0	0
$943$	2.85363	0.0929271
$944$	0	0
$945$	3.27131	0.106416
$946$	0	0
$947$	9.85050	0.320098	0.160049	−	0.987109i	$-0.448835\pi$
$947$	9.85050	0.320098	0.160049	+	0.987109i	$0.448835\pi$
$948$	0	0
$949$	−31.9473	−1.03705
$950$	0	0
$951$	29.6503	0.961478
$952$	0	0
$953$	3.92417	0.127116	0.0635582	−	0.997978i	$-0.479755\pi$
$953$	3.92417	0.127116	0.0635582	+	0.997978i	$0.479755\pi$
$954$	0	0
$955$	5.87146	0.189996
$956$	0	0
$957$	−26.8396	−0.867600
$958$	0	0
$959$	29.4819	0.952022
$960$	0	0
$961$	66.2389	2.13674
$962$	0	0
$963$	−13.7648	−0.443565
$964$	0	0
$965$	−16.7287	−0.538516
$966$	0	0
$967$	40.3074	1.29620	0.648100	−	0.761556i	$-0.275564\pi$
$967$	40.3074	1.29620	0.648100	+	0.761556i	$0.275564\pi$
$968$	0	0
$969$	−4.16779	−0.133889
$970$	0	0
$971$	−37.8280	−1.21396	−0.606979	−	0.794718i	$-0.707619\pi$
$971$	−37.8280	−1.21396	−0.606979	+	0.794718i	$0.707619\pi$
$972$	0	0
$973$	69.0361	2.21320
$974$	0	0
$975$	−13.4391	−0.430396
$976$	0	0
$977$	8.89227	0.284489	0.142244	−	0.989832i	$-0.454568\pi$
$977$	8.89227	0.284489	0.142244	+	0.989832i	$0.454568\pi$
$978$	0	0
$979$	−11.2713	−0.360233
$980$	0	0
$981$	3.14637	0.100456
$982$	0	0
$983$	30.6247	0.976777	0.488388	−	0.872626i	$-0.337585\pi$
$983$	30.6247	0.976777	0.488388	+	0.872626i	$0.337585\pi$
$984$	0	0
$985$	−21.1428	−0.673665
$986$	0	0
$987$	−31.8898	−1.01506
$988$	0	0
$989$	−33.5604	−1.06716
$990$	0	0
$991$	53.1512	1.68840	0.844202	−	0.536026i	$-0.180075\pi$
$991$	53.1512	1.68840	0.844202	+	0.536026i	$0.180075\pi$
$992$	0	0
$993$	−2.60375	−0.0826275
$994$	0	0
$995$	−10.9357	−0.346686
$996$	0	0
$997$	−44.5181	−1.40990	−0.704951	−	0.709256i	$-0.749031\pi$
$997$	−44.5181	−1.40990	−0.704951	+	0.709256i	$0.749031\pi$
$998$	0	0
$999$	8.19656	0.259328

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1968.2.a.w.1.2		3
3.2	odd	2		5904.2.a.bd.1.2		3
4.3	odd	2		123.2.a.d.1.3	✓	3
8.3	odd	2		7872.2.a.bx.1.2		3
8.5	even	2		7872.2.a.bs.1.2		3
12.11	even	2		369.2.a.e.1.1		3
20.19	odd	2		3075.2.a.t.1.1		3
28.27	even	2		6027.2.a.s.1.3		3
60.59	even	2		9225.2.a.bx.1.3		3
164.163	odd	2		5043.2.a.n.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.a.d.1.3	✓	3	4.3	odd	2
369.2.a.e.1.1		3	12.11	even	2
1968.2.a.w.1.2		3	1.1	even	1	trivial
3075.2.a.t.1.1		3	20.19	odd	2
5043.2.a.n.1.3		3	164.163	odd	2
5904.2.a.bd.1.2		3	3.2	odd	2
6027.2.a.s.1.3		3	28.27	even	2
7872.2.a.bs.1.2		3	8.5	even	2
7872.2.a.bx.1.2		3	8.3	odd	2
9225.2.a.bx.1.3		3	60.59	even	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.853635 q^{5} +3.83221 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.853635 q^{5} +3.83221 q^{7} +1.00000 q^{9} +3.34292 q^{11} +3.14637 q^{13} +0.853635 q^{15} +3.63565 q^{17} -1.14637 q^{19} +3.83221 q^{21} +2.85363 q^{23} -4.27131 q^{25} +1.00000 q^{27} -8.02877 q^{29} -9.86098 q^{31} +3.34292 q^{33} +3.27131 q^{35} +8.19656 q^{37} +3.14637 q^{39} +1.00000 q^{41} -11.7606 q^{43} +0.853635 q^{45} -8.32150 q^{47} +7.68585 q^{49} +3.63565 q^{51} +1.60688 q^{53} +2.85363 q^{55} -1.14637 q^{57} +11.6644 q^{59} -4.19656 q^{61} +3.83221 q^{63} +2.68585 q^{65} -6.10038 q^{67} +2.85363 q^{69} +8.65708 q^{71} -10.1537 q^{73} -4.27131 q^{75} +12.8108 q^{77} +3.60688 q^{79} +1.00000 q^{81} +10.1249 q^{83} +3.10352 q^{85} -8.02877 q^{87} -3.37169 q^{89} +12.0575 q^{91} -9.86098 q^{93} -0.978577 q^{95} -7.53948 q^{97} +3.34292 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 4 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 4 * q^5 - 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 4 q^{5} - 2 q^{7} + 3 q^{9} + 4 q^{11} + 8 q^{13} + 4 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{21} + 10 q^{23} + 5 q^{25} + 3 q^{27} - 6 q^{29} + 2 q^{31} + 4 q^{33} - 8 q^{35} + 20 q^{37} + 8 q^{39} + 3 q^{41} - 10 q^{43} + 4 q^{45} - 4 q^{47} + 11 q^{49} + 2 q^{51} + 14 q^{53} + 10 q^{55} - 2 q^{57} + 8 q^{59} - 8 q^{61} - 2 q^{63} - 4 q^{65} - 12 q^{67} + 10 q^{69} + 32 q^{71} + 4 q^{73} + 5 q^{75} + 10 q^{77} + 20 q^{79} + 3 q^{81} + 14 q^{83} - 22 q^{85} - 6 q^{87} + 14 q^{89} + 2 q^{93} + 12 q^{95} - 12 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 4 * q^5 - 2 * q^7 + 3 * q^9 + 4 * q^11 + 8 * q^13 + 4 * q^15 + 2 * q^17 - 2 * q^19 - 2 * q^21 + 10 * q^23 + 5 * q^25 + 3 * q^27 - 6 * q^29 + 2 * q^31 + 4 * q^33 - 8 * q^35 + 20 * q^37 + 8 * q^39 + 3 * q^41 - 10 * q^43 + 4 * q^45 - 4 * q^47 + 11 * q^49 + 2 * q^51 + 14 * q^53 + 10 * q^55 - 2 * q^57 + 8 * q^59 - 8 * q^61 - 2 * q^63 - 4 * q^65 - 12 * q^67 + 10 * q^69 + 32 * q^71 + 4 * q^73 + 5 * q^75 + 10 * q^77 + 20 * q^79 + 3 * q^81 + 14 * q^83 - 22 * q^85 - 6 * q^87 + 14 * q^89 + 2 * q^93 + 12 * q^95 - 12 * q^97 + 4 * q^99

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