LMFDB - Embedded newform 1568.3.c.g.97.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(97,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1568.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26 x^{14} - 16 x^{13} + 469 x^{12} + 144 x^{11} - 4526 x^{10} + 4440 x^{9} + 32608 x^{8} + \cdots + 208849 $ x^16 - 26x^14 - 16x^13 + 469x^12 + 144x^11 - 4526x^10 + 4440x^9 + 32608x^8 - 33728x^7 - 49760x^6 + 203528x^5 + 27401x^4 - 156928x^3 + 114964x^2 - 248608x + 208849
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.12
Root			$-1.90990 + 0.286185i$ of defining polynomial
Character	$\chi$	$=$	1568.97
Dual form			1568.3.c.g.97.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.54150i q^{3} +4.11878i q^{5} +2.54076 q^{9} +O(q^{10})$	$q+2.54150i q^{3} +4.11878i q^{5} +2.54076 q^{9} +3.26783 q^{11} +5.88759i q^{13} -10.4679 q^{15} +13.8800i q^{17} -15.8261i q^{19} +36.5036 q^{23} +8.03567 q^{25} +29.3309i q^{27} -28.4655 q^{29} +41.8017i q^{31} +8.30521i q^{33} +14.2857 q^{37} -14.9633 q^{39} +21.3515i q^{41} +55.3992 q^{43} +10.4648i q^{45} +33.8533i q^{47} -35.2760 q^{51} -84.8542 q^{53} +13.4595i q^{55} +40.2220 q^{57} -67.5572i q^{59} -29.5759i q^{61} -24.2497 q^{65} -54.9578 q^{67} +92.7741i q^{69} +83.8102 q^{71} +125.613i q^{73} +20.4227i q^{75} +70.3910 q^{79} -51.6777 q^{81} +27.1264i q^{83} -57.1685 q^{85} -72.3451i q^{87} -146.131i q^{89} -106.239 q^{93} +65.1841 q^{95} -11.3574i q^{97} +8.30278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 80 q^{9}+O(q^{10}) $ 16 * q - 80 * q^9	$ 16 q - 80 q^{9} - 160 q^{25} - 16 q^{29} - 144 q^{37} - 80 q^{53} + 368 q^{57} + 336 q^{65} + 768 q^{81} - 1072 q^{85} + 336 q^{93}+O(q^{100}) $ 16 * q - 80 * q^9 - 160 * q^25 - 16 * q^29 - 144 * q^37 - 80 * q^53 + 368 * q^57 + 336 * q^65 + 768 * q^81 - 1072 * q^85 + 336 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.54150i	0.847168i	0.905857	+	0.423584i	$0.139228\pi$
$3$	2.54150i	0.847168i	−0.905857	+	0.423584i	$0.860772\pi$
$4$	0	0
$5$	4.11878i	0.823756i	0.911239	+	0.411878i	$0.135127\pi$
$5$	4.11878i	0.823756i	−0.911239	+	0.411878i	$0.864873\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.54076	0.282306
$10$	0	0
$11$	3.26783	0.297076	0.148538	−	0.988907i	$-0.452543\pi$
$11$	3.26783	0.297076	0.148538	+	0.988907i	$0.452543\pi$
$12$	0	0
$13$	5.88759i	0.452892i	0.974024	+	0.226446i	$0.0727107\pi$
$13$	5.88759i	0.452892i	−0.974024	+	0.226446i	$0.927289\pi$
$14$	0	0
$15$	−10.4679	−0.697859
$16$	0	0
$17$	13.8800i	0.816469i	0.912877	+	0.408234i	$0.133855\pi$
$17$	13.8800i	0.816469i	−0.912877	+	0.408234i	$0.866145\pi$
$18$	0	0
$19$	− 15.8261i	− 0.832951i	−0.909147	−	0.416476i	$-0.863265\pi$
$19$	− 15.8261i	− 0.832951i	0.909147	−	0.416476i	$-0.136735\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	36.5036	1.58711	0.793557	−	0.608496i	$-0.208227\pi$
$23$	36.5036	1.58711	0.793557	+	0.608496i	$0.208227\pi$
$24$	0	0
$25$	8.03567	0.321427
$26$	0	0
$27$	29.3309i	1.08633i
$28$	0	0
$29$	−28.4655	−0.981568	−0.490784	−	0.871281i	$-0.663290\pi$
$29$	−28.4655	−0.981568	−0.490784	+	0.871281i	$0.663290\pi$
$30$	0	0
$31$	41.8017i	1.34844i	0.738529	+	0.674222i	$0.235521\pi$
$31$	41.8017i	1.34844i	−0.738529	+	0.674222i	$0.764479\pi$
$32$	0	0
$33$	8.30521i	0.251673i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	14.2857	0.386100	0.193050	−	0.981189i	$-0.438162\pi$
$37$	14.2857	0.386100	0.193050	+	0.981189i	$0.438162\pi$
$38$	0	0
$39$	−14.9633	−0.383675
$40$	0	0
$41$	21.3515i	0.520769i	0.965505	+	0.260385i	$0.0838493\pi$
$41$	21.3515i	0.520769i	−0.965505	+	0.260385i	$0.916151\pi$
$42$	0	0
$43$	55.3992	1.28835	0.644177	−	0.764877i	$-0.277200\pi$
$43$	55.3992	1.28835	0.644177	+	0.764877i	$0.277200\pi$
$44$	0	0
$45$	10.4648i	0.232551i
$46$	0	0
$47$	33.8533i	0.720282i	0.932898	+	0.360141i	$0.117271\pi$
$47$	33.8533i	0.720282i	−0.932898	+	0.360141i	$0.882729\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−35.2760	−0.691686
$52$	0	0
$53$	−84.8542	−1.60102	−0.800512	−	0.599317i	$-0.795439\pi$
$53$	−84.8542	−1.60102	−0.800512	+	0.599317i	$0.795439\pi$
$54$	0	0
$55$	13.4595i	0.244718i
$56$	0	0
$57$	40.2220	0.705649
$58$	0	0
$59$	− 67.5572i	− 1.14504i	−0.819892	−	0.572518i	$-0.805967\pi$
$59$	− 67.5572i	− 1.14504i	0.819892	−	0.572518i	$-0.194033\pi$
$60$	0	0
$61$	− 29.5759i	− 0.484851i	−0.970170	−	0.242425i	$-0.922057\pi$
$61$	− 29.5759i	− 0.484851i	0.970170	−	0.242425i	$-0.0779429\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−24.2497	−0.373072
$66$	0	0
$67$	−54.9578	−0.820266	−0.410133	−	0.912026i	$-0.634518\pi$
$67$	−54.9578	−0.820266	−0.410133	+	0.912026i	$0.634518\pi$
$68$	0	0
$69$	92.7741i	1.34455i
$70$	0	0
$71$	83.8102	1.18043	0.590213	−	0.807248i	$-0.299044\pi$
$71$	83.8102	1.18043	0.590213	+	0.807248i	$0.299044\pi$
$72$	0	0
$73$	125.613i	1.72073i	0.509682	+	0.860363i	$0.329763\pi$
$73$	125.613i	1.72073i	−0.509682	+	0.860363i	$0.670237\pi$
$74$	0	0
$75$	20.4227i	0.272303i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	70.3910	0.891025	0.445512	−	0.895276i	$-0.353022\pi$
$79$	70.3910	0.891025	0.445512	+	0.895276i	$0.353022\pi$
$80$	0	0
$81$	−51.6777	−0.637997
$82$	0	0
$83$	27.1264i	0.326824i	0.986558	+	0.163412i	$0.0522500\pi$
$83$	27.1264i	0.326824i	−0.986558	+	0.163412i	$0.947750\pi$
$84$	0	0
$85$	−57.1685	−0.672571
$86$	0	0
$87$	− 72.3451i	− 0.831553i
$88$	0	0
$89$	− 146.131i	− 1.64192i	−0.570987	−	0.820959i	$-0.693439\pi$
$89$	− 146.131i	− 1.64192i	0.570987	−	0.820959i	$-0.306561\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−106.239	−1.14236
$94$	0	0
$95$	65.1841	0.686148
$96$	0	0
$97$	− 11.3574i	− 0.117086i	−0.998285	−	0.0585431i	$-0.981354\pi$
$97$	− 11.3574i	− 0.117086i	0.998285	−	0.0585431i	$-0.0186455\pi$
$98$	0	0
$99$	8.30278	0.0838664
$100$	0	0
$101$	41.8984i	0.414836i	0.978252	+	0.207418i	$0.0665060\pi$
$101$	41.8984i	0.414836i	−0.978252	+	0.207418i	$0.933494\pi$
$102$	0	0
$103$	97.1798i	0.943493i	0.881734	+	0.471746i	$0.156376\pi$
$103$	97.1798i	0.943493i	−0.881734	+	0.471746i	$0.843624\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−83.4437	−0.779847	−0.389924	−	0.920847i	$-0.627499\pi$
$107$	−83.4437	−0.779847	−0.389924	+	0.920847i	$0.627499\pi$
$108$	0	0
$109$	−161.993	−1.48618	−0.743088	−	0.669194i	$-0.766639\pi$
$109$	−161.993	−1.48618	−0.743088	+	0.669194i	$0.766639\pi$
$110$	0	0
$111$	36.3072i	0.327092i
$112$	0	0
$113$	27.2503	0.241153	0.120576	−	0.992704i	$-0.461526\pi$
$113$	27.2503	0.241153	0.120576	+	0.992704i	$0.461526\pi$
$114$	0	0
$115$	150.350i	1.30739i
$116$	0	0
$117$	14.9590i	0.127854i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−110.321	−0.911746
$122$	0	0
$123$	−54.2650	−0.441179
$124$	0	0
$125$	136.067i	1.08853i
$126$	0	0
$127$	−232.457	−1.83037	−0.915183	−	0.403038i	$-0.867954\pi$
$127$	−232.457	−1.83037	−0.915183	+	0.403038i	$0.867954\pi$
$128$	0	0
$129$	140.797i	1.09145i
$130$	0	0
$131$	− 176.102i	− 1.34429i	−0.740418	−	0.672146i	$-0.765373\pi$
$131$	− 176.102i	− 1.34429i	0.740418	−	0.672146i	$-0.234627\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−120.807	−0.894869
$136$	0	0
$137$	−264.925	−1.93376	−0.966879	−	0.255234i	$-0.917847\pi$
$137$	−264.925	−1.93376	−0.966879	+	0.255234i	$0.917847\pi$
$138$	0	0
$139$	− 267.680i	− 1.92576i	−0.269935	−	0.962878i	$-0.587002\pi$
$139$	− 267.680i	− 1.92576i	0.269935	−	0.962878i	$-0.412998\pi$
$140$	0	0
$141$	−86.0382	−0.610200
$142$	0	0
$143$	19.2397i	0.134543i
$144$	0	0
$145$	− 117.243i	− 0.808572i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.94540	−0.0130564	−0.00652820	−	0.999979i	$-0.502078\pi$
$149$	−1.94540	−0.0130564	−0.00652820	+	0.999979i	$0.502078\pi$
$150$	0	0
$151$	100.305	0.664270	0.332135	−	0.943232i	$-0.392231\pi$
$151$	100.305	0.664270	0.332135	+	0.943232i	$0.392231\pi$
$152$	0	0
$153$	35.2656i	0.230494i
$154$	0	0
$155$	−172.172	−1.11079
$156$	0	0
$157$	135.277i	0.861638i	0.902438	+	0.430819i	$0.141775\pi$
$157$	135.277i	0.861638i	−0.902438	+	0.430819i	$0.858225\pi$
$158$	0	0
$159$	− 215.657i	− 1.35634i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	254.706	1.56261	0.781307	−	0.624146i	$-0.214553\pi$
$163$	254.706	1.56261	0.781307	+	0.624146i	$0.214553\pi$
$164$	0	0
$165$	−34.2073	−0.207317
$166$	0	0
$167$	50.9246i	0.304938i	0.988308	+	0.152469i	$0.0487224\pi$
$167$	50.9246i	0.304938i	−0.988308	+	0.152469i	$0.951278\pi$
$168$	0	0
$169$	134.336	0.794889
$170$	0	0
$171$	− 40.2102i	− 0.235147i
$172$	0	0
$173$	70.4200i	0.407052i	0.979070	+	0.203526i	$0.0652401\pi$
$173$	70.4200i	0.407052i	−0.979070	+	0.203526i	$0.934760\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	171.697	0.970038
$178$	0	0
$179$	147.440	0.823689	0.411845	−	0.911254i	$-0.364885\pi$
$179$	147.440	0.823689	0.411845	+	0.911254i	$0.364885\pi$
$180$	0	0
$181$	294.491i	1.62702i	0.581550	+	0.813511i	$0.302447\pi$
$181$	294.491i	1.62702i	−0.581550	+	0.813511i	$0.697553\pi$
$182$	0	0
$183$	75.1672	0.410750
$184$	0	0
$185$	58.8396i	0.318052i
$186$	0	0
$187$	45.3574i	0.242553i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−112.519	−0.589105	−0.294553	−	0.955635i	$-0.595171\pi$
$191$	−112.519	−0.589105	−0.294553	+	0.955635i	$0.595171\pi$
$192$	0	0
$193$	127.487	0.660554	0.330277	−	0.943884i	$-0.392858\pi$
$193$	127.487	0.660554	0.330277	+	0.943884i	$0.392858\pi$
$194$	0	0
$195$	− 61.6307i	− 0.316055i
$196$	0	0
$197$	−293.140	−1.48802	−0.744011	−	0.668167i	$-0.767079\pi$
$197$	−293.140	−1.48802	−0.744011	+	0.668167i	$0.767079\pi$
$198$	0	0
$199$	9.74194i	0.0489545i	0.999700	+	0.0244772i	$0.00779212\pi$
$199$	9.74194i	0.0489545i	−0.999700	+	0.0244772i	$0.992208\pi$
$200$	0	0
$201$	− 139.676i	− 0.694903i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−87.9422	−0.428986
$206$	0	0
$207$	92.7469	0.448052
$208$	0	0
$209$	− 51.7170i	− 0.247450i
$210$	0	0
$211$	−139.516	−0.661214	−0.330607	−	0.943769i	$-0.607253\pi$
$211$	−139.516	−0.661214	−0.330607	+	0.943769i	$0.607253\pi$
$212$	0	0
$213$	213.004i	1.00002i
$214$	0	0
$215$	228.177i	1.06129i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−319.246	−1.45774
$220$	0	0
$221$	−81.7196	−0.369772
$222$	0	0
$223$	− 273.426i	− 1.22612i	−0.790035	−	0.613062i	$-0.789938\pi$
$223$	− 273.426i	− 1.22612i	0.790035	−	0.613062i	$-0.210062\pi$
$224$	0	0
$225$	20.4167	0.0907409
$226$	0	0
$227$	− 67.5444i	− 0.297552i	−0.988871	−	0.148776i	$-0.952467\pi$
$227$	− 67.5444i	− 0.297552i	0.988871	−	0.148776i	$-0.0475334\pi$
$228$	0	0
$229$	192.563i	0.840888i	0.907319	+	0.420444i	$0.138126\pi$
$229$	192.563i	0.840888i	−0.907319	+	0.420444i	$0.861874\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	81.4048	0.349377	0.174689	−	0.984624i	$-0.444108\pi$
$233$	81.4048	0.349377	0.174689	+	0.984624i	$0.444108\pi$
$234$	0	0
$235$	−139.434	−0.593336
$236$	0	0
$237$	178.899i	0.754848i
$238$	0	0
$239$	22.6152	0.0946244	0.0473122	−	0.998880i	$-0.484934\pi$
$239$	22.6152	0.0946244	0.0473122	+	0.998880i	$0.484934\pi$
$240$	0	0
$241$	94.1663i	0.390732i	0.980730	+	0.195366i	$0.0625894\pi$
$241$	94.1663i	0.390732i	−0.980730	+	0.195366i	$0.937411\pi$
$242$	0	0
$243$	132.639i	0.545839i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	93.1775	0.377237
$248$	0	0
$249$	−68.9419	−0.276875
$250$	0	0
$251$	316.694i	1.26173i	0.775892	+	0.630865i	$0.217300\pi$
$251$	316.694i	1.26173i	−0.775892	+	0.630865i	$0.782700\pi$
$252$	0	0
$253$	119.288	0.471493
$254$	0	0
$255$	− 145.294i	− 0.569780i
$256$	0	0
$257$	47.0598i	0.183112i	0.995800	+	0.0915560i	$0.0291840\pi$
$257$	47.0598i	0.183112i	−0.995800	+	0.0915560i	$0.970816\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−72.3239	−0.277103
$262$	0	0
$263$	−406.504	−1.54564	−0.772822	−	0.634623i	$-0.781155\pi$
$263$	−406.504	−1.54564	−0.772822	+	0.634623i	$0.781155\pi$
$264$	0	0
$265$	− 349.496i	− 1.31885i
$266$	0	0
$267$	371.392	1.39098
$268$	0	0
$269$	− 332.504i	− 1.23608i	−0.786148	−	0.618038i	$-0.787928\pi$
$269$	− 332.504i	− 1.23608i	0.786148	−	0.618038i	$-0.212072\pi$
$270$	0	0
$271$	− 71.3242i	− 0.263189i	−0.991304	−	0.131595i	$-0.957990\pi$
$271$	− 71.3242i	− 0.263189i	0.991304	−	0.131595i	$-0.0420097\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	26.2592	0.0954881
$276$	0	0
$277$	−29.7147	−0.107273	−0.0536367	−	0.998561i	$-0.517081\pi$
$277$	−29.7147	−0.107273	−0.0536367	+	0.998561i	$0.517081\pi$
$278$	0	0
$279$	106.208i	0.380674i
$280$	0	0
$281$	9.06447	0.0322579	0.0161289	−	0.999870i	$-0.494866\pi$
$281$	9.06447	0.0322579	0.0161289	+	0.999870i	$0.494866\pi$
$282$	0	0
$283$	− 184.738i	− 0.652784i	−0.945235	−	0.326392i	$-0.894167\pi$
$283$	− 184.738i	− 0.652784i	0.945235	−	0.326392i	$-0.105833\pi$
$284$	0	0
$285$	165.666i	0.581283i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	96.3464	0.333379
$290$	0	0
$291$	28.8648	0.0991917
$292$	0	0
$293$	300.389i	1.02522i	0.858622	+	0.512609i	$0.171321\pi$
$293$	300.389i	1.02522i	−0.858622	+	0.512609i	$0.828679\pi$
$294$	0	0
$295$	278.253	0.943230
$296$	0	0
$297$	95.8485i	0.322722i
$298$	0	0
$299$	214.918i	0.718791i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−106.485	−0.351436
$304$	0	0
$305$	121.817	0.399398
$306$	0	0
$307$	390.385i	1.27161i	0.771848	+	0.635807i	$0.219333\pi$
$307$	390.385i	1.27161i	−0.771848	+	0.635807i	$0.780667\pi$
$308$	0	0
$309$	−246.983	−0.799297
$310$	0	0
$311$	96.0768i	0.308929i	0.987998	+	0.154464i	$0.0493652\pi$
$311$	96.0768i	0.308929i	−0.987998	+	0.154464i	$0.950635\pi$
$312$	0	0
$313$	− 484.883i	− 1.54915i	−0.632484	−	0.774574i	$-0.717964\pi$
$313$	− 484.883i	− 1.54915i	0.632484	−	0.774574i	$-0.282036\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	67.9427	0.214330	0.107165	−	0.994241i	$-0.465823\pi$
$317$	67.9427	0.214330	0.107165	+	0.994241i	$0.465823\pi$
$318$	0	0
$319$	−93.0205	−0.291600
$320$	0	0
$321$	− 212.072i	− 0.660662i
$322$	0	0
$323$	219.665	0.680079
$324$	0	0
$325$	47.3108i	0.145572i
$326$	0	0
$327$	− 411.706i	− 1.25904i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	265.268	0.801415	0.400707	−	0.916206i	$-0.368764\pi$
$331$	265.268	0.801415	0.400707	+	0.916206i	$0.368764\pi$
$332$	0	0
$333$	36.2965	0.108999
$334$	0	0
$335$	− 226.359i	− 0.675699i
$336$	0	0
$337$	549.980	1.63199	0.815993	−	0.578061i	$-0.196191\pi$
$337$	549.980	1.63199	0.815993	+	0.578061i	$0.196191\pi$
$338$	0	0
$339$	69.2566i	0.204297i
$340$	0	0
$341$	136.601i	0.400590i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−382.116	−1.10758
$346$	0	0
$347$	678.750	1.95605	0.978026	−	0.208481i	$-0.0668520\pi$
$347$	678.750	1.95605	0.978026	+	0.208481i	$0.0668520\pi$
$348$	0	0
$349$	− 170.081i	− 0.487339i	−0.969858	−	0.243669i	$-0.921649\pi$
$349$	− 170.081i	− 0.487339i	0.969858	−	0.243669i	$-0.0783511\pi$
$350$	0	0
$351$	−172.688	−0.491990
$352$	0	0
$353$	− 236.047i	− 0.668689i	−0.942451	−	0.334345i	$-0.891485\pi$
$353$	− 236.047i	− 0.668689i	0.942451	−	0.334345i	$-0.108515\pi$
$354$	0	0
$355$	345.196i	0.972382i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	458.116	1.27609	0.638044	−	0.770000i	$-0.279744\pi$
$359$	458.116	1.27609	0.638044	+	0.770000i	$0.279744\pi$
$360$	0	0
$361$	110.535	0.306193
$362$	0	0
$363$	− 280.382i	− 0.772402i
$364$	0	0
$365$	−517.372	−1.41746
$366$	0	0
$367$	411.437i	1.12108i	0.828127	+	0.560541i	$0.189407\pi$
$367$	411.437i	1.12108i	−0.828127	+	0.560541i	$0.810593\pi$
$368$	0	0
$369$	54.2491i	0.147016i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	15.7971	0.0423514	0.0211757	−	0.999776i	$-0.493259\pi$
$373$	15.7971	0.0423514	0.0211757	+	0.999776i	$0.493259\pi$
$374$	0	0
$375$	−345.814	−0.922170
$376$	0	0
$377$	− 167.593i	− 0.444544i
$378$	0	0
$379$	−455.384	−1.20154	−0.600770	−	0.799422i	$-0.705139\pi$
$379$	−455.384	−1.20154	−0.600770	+	0.799422i	$0.705139\pi$
$380$	0	0
$381$	− 590.789i	− 1.55063i
$382$	0	0
$383$	183.288i	0.478558i	0.970951	+	0.239279i	$0.0769111\pi$
$383$	183.288i	0.478558i	−0.970951	+	0.239279i	$0.923089\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	140.756	0.363710
$388$	0	0
$389$	−185.494	−0.476849	−0.238424	−	0.971161i	$-0.576631\pi$
$389$	−185.494	−0.476849	−0.238424	+	0.971161i	$0.576631\pi$
$390$	0	0
$391$	506.669i	1.29583i
$392$	0	0
$393$	447.565	1.13884
$394$	0	0
$395$	289.925i	0.733987i
$396$	0	0
$397$	− 45.6707i	− 0.115039i	−0.998344	−	0.0575197i	$-0.981681\pi$
$397$	− 45.6707i	− 0.115039i	0.998344	−	0.0575197i	$-0.0183192\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	38.4623	0.0959160	0.0479580	−	0.998849i	$-0.484729\pi$
$401$	38.4623	0.0959160	0.0479580	+	0.998849i	$0.484729\pi$
$402$	0	0
$403$	−246.112	−0.610699
$404$	0	0
$405$	− 212.849i	− 0.525553i
$406$	0	0
$407$	46.6833	0.114701
$408$	0	0
$409$	167.737i	0.410114i	0.978750	+	0.205057i	$0.0657380\pi$
$409$	167.737i	0.410114i	−0.978750	+	0.205057i	$0.934262\pi$
$410$	0	0
$411$	− 673.308i	− 1.63822i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−111.728	−0.269223
$416$	0	0
$417$	680.310	1.63144
$418$	0	0
$419$	300.318i	0.716751i	0.933578	+	0.358375i	$0.116669\pi$
$419$	300.318i	0.716751i	−0.933578	+	0.358375i	$0.883331\pi$
$420$	0	0
$421$	280.567	0.666430	0.333215	−	0.942851i	$-0.391867\pi$
$421$	280.567	0.666430	0.333215	+	0.942851i	$0.391867\pi$
$422$	0	0
$423$	86.0129i	0.203340i
$424$	0	0
$425$	111.535i	0.262435i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−48.8977	−0.113981
$430$	0	0
$431$	224.764	0.521495	0.260747	−	0.965407i	$-0.416031\pi$
$431$	224.764	0.521495	0.260747	+	0.965407i	$0.416031\pi$
$432$	0	0
$433$	− 731.236i	− 1.68877i	−0.535739	−	0.844383i	$-0.679967\pi$
$433$	− 731.236i	− 1.68877i	0.535739	−	0.844383i	$-0.320033\pi$
$434$	0	0
$435$	297.973	0.684996
$436$	0	0
$437$	− 577.709i	− 1.32199i
$438$	0	0
$439$	− 299.439i	− 0.682094i	−0.940046	−	0.341047i	$-0.889218\pi$
$439$	− 299.439i	− 0.682094i	0.940046	−	0.341047i	$-0.110782\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	396.155	0.894255	0.447127	−	0.894470i	$-0.352447\pi$
$443$	396.155	0.894255	0.447127	+	0.894470i	$0.352447\pi$
$444$	0	0
$445$	601.880	1.35254
$446$	0	0
$447$	− 4.94425i	− 0.0110610i
$448$	0	0
$449$	128.183	0.285486	0.142743	−	0.989760i	$-0.454408\pi$
$449$	128.183	0.285486	0.142743	+	0.989760i	$0.454408\pi$
$450$	0	0
$451$	69.7733i	0.154708i
$452$	0	0
$453$	254.925i	0.562748i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	405.147	0.886537	0.443268	−	0.896389i	$-0.353819\pi$
$457$	405.147	0.886537	0.443268	+	0.896389i	$0.353819\pi$
$458$	0	0
$459$	−407.112	−0.886954
$460$	0	0
$461$	− 312.620i	− 0.678134i	−0.940762	−	0.339067i	$-0.889889\pi$
$461$	− 312.620i	− 0.678134i	0.940762	−	0.339067i	$-0.110111\pi$
$462$	0	0
$463$	246.396	0.532173	0.266087	−	0.963949i	$-0.414269\pi$
$463$	246.396	0.532173	0.266087	+	0.963949i	$0.414269\pi$
$464$	0	0
$465$	− 437.576i	− 0.941024i
$466$	0	0
$467$	21.7706i	0.0466180i	0.999728	+	0.0233090i	$0.00742015\pi$
$467$	21.7706i	0.0466180i	−0.999728	+	0.0233090i	$0.992580\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−343.807	−0.729952
$472$	0	0
$473$	181.035	0.382739
$474$	0	0
$475$	− 127.173i	− 0.267733i
$476$	0	0
$477$	−215.594	−0.451979
$478$	0	0
$479$	− 522.490i	− 1.09079i	−0.838178	−	0.545397i	$-0.816379\pi$
$479$	− 522.490i	− 1.09079i	0.838178	−	0.545397i	$-0.183621\pi$
$480$	0	0
$481$	84.1084i	0.174862i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	46.7785	0.0964505
$486$	0	0
$487$	818.134	1.67995	0.839974	−	0.542627i	$-0.182570\pi$
$487$	818.134	1.67995	0.839974	+	0.542627i	$0.182570\pi$
$488$	0	0
$489$	647.337i	1.32380i
$490$	0	0
$491$	−762.002	−1.55194	−0.775969	−	0.630771i	$-0.782739\pi$
$491$	−762.002	−1.55194	−0.775969	+	0.630771i	$0.782739\pi$
$492$	0	0
$493$	− 395.100i	− 0.801420i
$494$	0	0
$495$	34.1973i	0.0690854i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−50.9869	−0.102178	−0.0510891	−	0.998694i	$-0.516269\pi$
$499$	−50.9869	−0.102178	−0.0510891	+	0.998694i	$0.516269\pi$
$500$	0	0
$501$	−129.425	−0.258334
$502$	0	0
$503$	− 305.233i	− 0.606825i	−0.952859	−	0.303413i	$-0.901874\pi$
$503$	− 305.233i	− 0.606825i	0.952859	−	0.303413i	$-0.0981260\pi$
$504$	0	0
$505$	−172.570	−0.341723
$506$	0	0
$507$	341.416i	0.673404i
$508$	0	0
$509$	180.752i	0.355112i	0.984111	+	0.177556i	$0.0568191\pi$
$509$	180.752i	0.355112i	−0.984111	+	0.177556i	$0.943181\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	464.193	0.904859
$514$	0	0
$515$	−400.262	−0.777207
$516$	0	0
$517$	110.627i	0.213978i
$518$	0	0
$519$	−178.973	−0.344841
$520$	0	0
$521$	361.152i	0.693190i	0.938015	+	0.346595i	$0.112662\pi$
$521$	361.152i	0.693190i	−0.938015	+	0.346595i	$0.887338\pi$
$522$	0	0
$523$	654.144i	1.25075i	0.780323	+	0.625377i	$0.215055\pi$
$523$	654.144i	1.25075i	−0.780323	+	0.625377i	$0.784945\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−580.207	−1.10096
$528$	0	0
$529$	803.514	1.51893
$530$	0	0
$531$	− 171.646i	− 0.323251i
$532$	0	0
$533$	−125.709	−0.235852
$534$	0	0
$535$	− 343.686i	− 0.642404i
$536$	0	0
$537$	374.720i	0.697803i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	603.284	1.11513	0.557564	−	0.830134i	$-0.311736\pi$
$541$	603.284	1.11513	0.557564	+	0.830134i	$0.311736\pi$
$542$	0	0
$543$	−748.450	−1.37836
$544$	0	0
$545$	− 667.214i	− 1.22424i
$546$	0	0
$547$	686.167	1.25442	0.627209	−	0.778851i	$-0.284197\pi$
$547$	686.167	1.25442	0.627209	+	0.778851i	$0.284197\pi$
$548$	0	0
$549$	− 75.1452i	− 0.136876i
$550$	0	0
$551$	450.497i	0.817598i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−149.541	−0.269443
$556$	0	0
$557$	785.321	1.40991	0.704956	−	0.709251i	$-0.250967\pi$
$557$	785.321	1.40991	0.704956	+	0.709251i	$0.250967\pi$
$558$	0	0
$559$	326.168i	0.583485i
$560$	0	0
$561$	−115.276	−0.205483
$562$	0	0
$563$	− 454.058i	− 0.806498i	−0.915090	−	0.403249i	$-0.867881\pi$
$563$	− 454.058i	− 0.806498i	0.915090	−	0.403249i	$-0.132119\pi$
$564$	0	0
$565$	112.238i	0.198651i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−71.4274	−0.125532	−0.0627658	−	0.998028i	$-0.519992\pi$
$569$	−71.4274	−0.125532	−0.0627658	+	0.998028i	$0.519992\pi$
$570$	0	0
$571$	718.067	1.25756	0.628780	−	0.777583i	$-0.283555\pi$
$571$	718.067	1.25756	0.628780	+	0.777583i	$0.283555\pi$
$572$	0	0
$573$	− 285.968i	− 0.499071i
$574$	0	0
$575$	293.331	0.510141
$576$	0	0
$577$	− 907.985i	− 1.57363i	−0.617188	−	0.786815i	$-0.711728\pi$
$577$	− 907.985i	− 1.57363i	0.617188	−	0.786815i	$-0.288272\pi$
$578$	0	0
$579$	324.009i	0.559600i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−277.290	−0.475625
$584$	0	0
$585$	−61.6126	−0.105321
$586$	0	0
$587$	− 862.870i	− 1.46997i	−0.678086	−	0.734983i	$-0.737190\pi$
$587$	− 862.870i	− 1.46997i	0.678086	−	0.734983i	$-0.262810\pi$
$588$	0	0
$589$	661.557	1.12319
$590$	0	0
$591$	− 745.017i	− 1.26060i
$592$	0	0
$593$	542.856i	0.915440i	0.889096	+	0.457720i	$0.151334\pi$
$593$	542.856i	0.915440i	−0.889096	+	0.457720i	$0.848666\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−24.7592	−0.0414727
$598$	0	0
$599$	78.6147	0.131243	0.0656216	−	0.997845i	$-0.479097\pi$
$599$	78.6147	0.131243	0.0656216	+	0.997845i	$0.479097\pi$
$600$	0	0
$601$	− 851.603i	− 1.41698i	−0.705722	−	0.708489i	$-0.749377\pi$
$601$	− 851.603i	− 1.41698i	0.705722	−	0.708489i	$-0.250623\pi$
$602$	0	0
$603$	−139.635	−0.231566
$604$	0	0
$605$	− 454.389i	− 0.751056i
$606$	0	0
$607$	233.575i	0.384802i	0.981316	+	0.192401i	$0.0616275\pi$
$607$	233.575i	0.384802i	−0.981316	+	0.192401i	$0.938373\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−199.314	−0.326210
$612$	0	0
$613$	81.1241	0.132339	0.0661697	−	0.997808i	$-0.478922\pi$
$613$	81.1241	0.132339	0.0661697	+	0.997808i	$0.478922\pi$
$614$	0	0
$615$	− 223.505i	− 0.363424i
$616$	0	0
$617$	47.2962	0.0766552	0.0383276	−	0.999265i	$-0.487797\pi$
$617$	47.2962	0.0766552	0.0383276	+	0.999265i	$0.487797\pi$
$618$	0	0
$619$	− 657.406i	− 1.06205i	−0.847357	−	0.531023i	$-0.821808\pi$
$619$	− 657.406i	− 1.06205i	0.847357	−	0.531023i	$-0.178192\pi$
$620$	0	0
$621$	1070.68i	1.72413i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−359.536	−0.575258
$626$	0	0
$627$	131.439	0.209631
$628$	0	0
$629$	198.285i	0.315239i
$630$	0	0
$631$	−270.276	−0.428330	−0.214165	−	0.976797i	$-0.568703\pi$
$631$	−270.276	−0.428330	−0.214165	+	0.976797i	$0.568703\pi$
$632$	0	0
$633$	− 354.581i	− 0.560159i
$634$	0	0
$635$	− 957.437i	− 1.50777i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	212.941	0.333242
$640$	0	0
$641$	244.610	0.381606	0.190803	−	0.981628i	$-0.438891\pi$
$641$	244.610	0.381606	0.190803	+	0.981628i	$0.438891\pi$
$642$	0	0
$643$	− 358.233i	− 0.557128i	−0.960418	−	0.278564i	$-0.910142\pi$
$643$	− 358.233i	− 0.557128i	0.960418	−	0.278564i	$-0.0898584\pi$
$644$	0	0
$645$	−579.913	−0.899089
$646$	0	0
$647$	1182.75i	1.82806i	0.405647	+	0.914030i	$0.367046\pi$
$647$	1182.75i	1.82806i	−0.405647	+	0.914030i	$0.632954\pi$
$648$	0	0
$649$	− 220.766i	− 0.340163i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	215.295	0.329701	0.164850	−	0.986319i	$-0.447286\pi$
$653$	215.295	0.329701	0.164850	+	0.986319i	$0.447286\pi$
$654$	0	0
$655$	725.327	1.10737
$656$	0	0
$657$	319.152i	0.485772i
$658$	0	0
$659$	−254.983	−0.386925	−0.193462	−	0.981108i	$-0.561972\pi$
$659$	−254.983	−0.386925	−0.193462	+	0.981108i	$0.561972\pi$
$660$	0	0
$661$	1243.66i	1.88148i	0.339126	+	0.940741i	$0.389869\pi$
$661$	1243.66i	1.88148i	−0.339126	+	0.940741i	$0.610131\pi$
$662$	0	0
$663$	− 207.691i	− 0.313259i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1039.09	−1.55786
$668$	0	0
$669$	694.912	1.03873
$670$	0	0
$671$	− 96.6491i	− 0.144037i
$672$	0	0
$673$	−63.0354	−0.0936633	−0.0468317	−	0.998903i	$-0.514912\pi$
$673$	−63.0354	−0.0936633	−0.0468317	+	0.998903i	$0.514912\pi$
$674$	0	0
$675$	235.693i	0.349175i
$676$	0	0
$677$	− 987.456i	− 1.45858i	−0.684207	−	0.729288i	$-0.739851\pi$
$677$	− 987.456i	− 1.45858i	0.684207	−	0.729288i	$-0.260149\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	171.664	0.252077
$682$	0	0
$683$	−934.895	−1.36881	−0.684403	−	0.729104i	$-0.739937\pi$
$683$	−934.895	−1.36881	−0.684403	+	0.729104i	$0.739937\pi$
$684$	0	0
$685$	− 1091.17i	− 1.59294i
$686$	0	0
$687$	−489.400	−0.712373
$688$	0	0
$689$	− 499.587i	− 0.725090i
$690$	0	0
$691$	− 972.480i	− 1.40735i	−0.710521	−	0.703676i	$-0.751541\pi$
$691$	− 972.480i	− 1.40735i	0.710521	−	0.703676i	$-0.248459\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1102.52	1.58635
$696$	0	0
$697$	−296.359	−0.425192
$698$	0	0
$699$	206.891i	0.295981i
$700$	0	0
$701$	695.549	0.992224	0.496112	−	0.868259i	$-0.334761\pi$
$701$	695.549	0.992224	0.496112	+	0.868259i	$0.334761\pi$
$702$	0	0
$703$	− 226.086i	− 0.321602i
$704$	0	0
$705$	− 354.372i	− 0.502656i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−157.264	−0.221811	−0.110905	−	0.993831i	$-0.535375\pi$
$709$	−157.264	−0.221811	−0.110905	+	0.993831i	$0.535375\pi$
$710$	0	0
$711$	178.846	0.251542
$712$	0	0
$713$	1525.91i	2.14013i
$714$	0	0
$715$	−79.2440	−0.110831
$716$	0	0
$717$	57.4767i	0.0801628i
$718$	0	0
$719$	211.949i	0.294783i	0.989078	+	0.147392i	$0.0470878\pi$
$719$	211.949i	0.294783i	−0.989078	+	0.147392i	$0.952912\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−239.324	−0.331015
$724$	0	0
$725$	−228.739	−0.315502
$726$	0	0
$727$	271.507i	0.373462i	0.982411	+	0.186731i	$0.0597893\pi$
$727$	271.507i	0.373462i	−0.982411	+	0.186731i	$0.940211\pi$
$728$	0	0
$729$	−802.202	−1.10041
$730$	0	0
$731$	768.939i	1.05190i
$732$	0	0
$733$	− 510.429i	− 0.696357i	−0.937428	−	0.348178i	$-0.886800\pi$
$733$	− 510.429i	− 0.696357i	0.937428	−	0.348178i	$-0.113200\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−179.593	−0.243681
$738$	0	0
$739$	−374.168	−0.506317	−0.253158	−	0.967425i	$-0.581469\pi$
$739$	−374.168	−0.506317	−0.253158	+	0.967425i	$0.581469\pi$
$740$	0	0
$741$	236.811i	0.319583i
$742$	0	0
$743$	−792.307	−1.06636	−0.533181	−	0.846001i	$-0.679004\pi$
$743$	−792.307	−1.06636	−0.533181	+	0.846001i	$0.679004\pi$
$744$	0	0
$745$	− 8.01268i	− 0.0107553i
$746$	0	0
$747$	68.9217i	0.0922646i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−26.2964	−0.0350152	−0.0175076	−	0.999847i	$-0.505573\pi$
$751$	−26.2964	−0.0350152	−0.0175076	+	0.999847i	$0.505573\pi$
$752$	0	0
$753$	−804.880	−1.06890
$754$	0	0
$755$	413.133i	0.547196i
$756$	0	0
$757$	1287.30	1.70053	0.850264	−	0.526356i	$-0.176442\pi$
$757$	1287.30	1.70053	0.850264	+	0.526356i	$0.176442\pi$
$758$	0	0
$759$	303.170i	0.399434i
$760$	0	0
$761$	969.451i	1.27392i	0.770898	+	0.636959i	$0.219808\pi$
$761$	969.451i	1.27392i	−0.770898	+	0.636959i	$0.780192\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−145.251	−0.189871
$766$	0	0
$767$	397.749	0.518578
$768$	0	0
$769$	− 499.204i	− 0.649160i	−0.945858	−	0.324580i	$-0.894777\pi$
$769$	− 499.204i	− 0.649160i	0.945858	−	0.324580i	$-0.105223\pi$
$770$	0	0
$771$	−119.603	−0.155127
$772$	0	0
$773$	145.343i	0.188024i	0.995571	+	0.0940120i	$0.0299692\pi$
$773$	145.343i	0.188024i	−0.995571	+	0.0940120i	$0.970031\pi$
$774$	0	0
$775$	335.905i	0.433426i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	337.911	0.433775
$780$	0	0
$781$	273.878	0.350676
$782$	0	0
$783$	− 834.918i	− 1.06631i
$784$	0	0
$785$	−557.176	−0.709779
$786$	0	0
$787$	− 384.668i	− 0.488778i	−0.969677	−	0.244389i	$-0.921413\pi$
$787$	− 384.668i	− 0.488778i	0.969677	−	0.244389i	$-0.0785873\pi$
$788$	0	0
$789$	− 1033.13i	− 1.30942i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	174.131	0.219585
$794$	0	0
$795$	888.245	1.11729
$796$	0	0
$797$	9.29571i	0.0116634i	0.999983	+	0.00583169i	$0.00185629\pi$
$797$	9.29571i	0.0116634i	−0.999983	+	0.00583169i	$0.998144\pi$
$798$	0	0
$799$	−469.882	−0.588088
$800$	0	0
$801$	− 371.283i	− 0.463524i
$802$	0	0
$803$	410.482i	0.511186i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	845.061	1.04716
$808$	0	0
$809$	−815.391	−1.00790	−0.503950	−	0.863733i	$-0.668120\pi$
$809$	−815.391	−1.00790	−0.503950	+	0.863733i	$0.668120\pi$
$810$	0	0
$811$	762.398i	0.940071i	0.882647	+	0.470036i	$0.155759\pi$
$811$	762.398i	0.940071i	−0.882647	+	0.470036i	$0.844241\pi$
$812$	0	0
$813$	181.271	0.222965
$814$	0	0
$815$	1049.08i	1.28721i
$816$	0	0
$817$	− 876.751i	− 1.07314i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1123.94	−1.36899	−0.684493	−	0.729020i	$-0.739976\pi$
$821$	−1123.94	−1.36899	−0.684493	+	0.729020i	$0.739976\pi$
$822$	0	0
$823$	−332.822	−0.404401	−0.202200	−	0.979344i	$-0.564809\pi$
$823$	−332.822	−0.404401	−0.202200	+	0.979344i	$0.564809\pi$
$824$	0	0
$825$	66.7380i	0.0808945i
$826$	0	0
$827$	451.741	0.546241	0.273120	−	0.961980i	$-0.411944\pi$
$827$	451.741	0.546241	0.273120	+	0.961980i	$0.411944\pi$
$828$	0	0
$829$	− 457.296i	− 0.551624i	−0.961212	−	0.275812i	$-0.911053\pi$
$829$	− 457.296i	− 0.551624i	0.961212	−	0.275812i	$-0.0889467\pi$
$830$	0	0
$831$	− 75.5202i	− 0.0908786i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−209.747	−0.251194
$836$	0	0
$837$	−1226.08	−1.46485
$838$	0	0
$839$	− 97.6089i	− 0.116340i	−0.998307	−	0.0581698i	$-0.981474\pi$
$839$	− 97.6089i	− 0.116340i	0.998307	−	0.0581698i	$-0.0185265\pi$
$840$	0	0
$841$	−30.7167	−0.0365240
$842$	0	0
$843$	23.0374i	0.0273278i
$844$	0	0
$845$	553.301i	0.654794i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	469.512	0.553017
$850$	0	0
$851$	521.480	0.612785
$852$	0	0
$853$	− 832.329i	− 0.975766i	−0.872909	−	0.487883i	$-0.837769\pi$
$853$	− 832.329i	− 0.975766i	0.872909	−	0.487883i	$-0.162231\pi$
$854$	0	0
$855$	165.617	0.193704
$856$	0	0
$857$	976.964i	1.13998i	0.821651	+	0.569991i	$0.193053\pi$
$857$	976.964i	1.13998i	−0.821651	+	0.569991i	$0.806947\pi$
$858$	0	0
$859$	− 697.938i	− 0.812500i	−0.913762	−	0.406250i	$-0.866836\pi$
$859$	− 697.938i	− 0.812500i	0.913762	−	0.406250i	$-0.133164\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	843.036	0.976867	0.488434	−	0.872601i	$-0.337568\pi$
$863$	843.036	0.976867	0.488434	+	0.872601i	$0.337568\pi$
$864$	0	0
$865$	−290.044	−0.335311
$866$	0	0
$867$	244.865i	0.282428i
$868$	0	0
$869$	230.026	0.264702
$870$	0	0
$871$	− 323.569i	− 0.371492i
$872$	0	0
$873$	− 28.8563i	− 0.0330542i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	58.1145	0.0662651	0.0331326	−	0.999451i	$-0.489452\pi$
$877$	58.1145	0.0662651	0.0331326	+	0.999451i	$0.489452\pi$
$878$	0	0
$879$	−763.440	−0.868532
$880$	0	0
$881$	1034.35i	1.17406i	0.809565	+	0.587030i	$0.199703\pi$
$881$	1034.35i	1.17406i	−0.809565	+	0.587030i	$0.800297\pi$
$882$	0	0
$883$	483.067	0.547075	0.273538	−	0.961861i	$-0.411806\pi$
$883$	483.067	0.547075	0.273538	+	0.961861i	$0.411806\pi$
$884$	0	0
$885$	707.181i	0.799075i
$886$	0	0
$887$	− 440.647i	− 0.496783i	−0.968660	−	0.248392i	$-0.920098\pi$
$887$	− 440.647i	− 0.496783i	0.968660	−	0.248392i	$-0.0799020\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−168.874	−0.189533
$892$	0	0
$893$	535.764	0.599960
$894$	0	0
$895$	607.274i	0.678519i
$896$	0	0
$897$	−546.216	−0.608937
$898$	0	0
$899$	− 1189.91i	− 1.32359i
$900$	0	0
$901$	− 1177.77i	− 1.30719i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1212.94	−1.34027
$906$	0	0
$907$	−459.549	−0.506669	−0.253334	−	0.967379i	$-0.581527\pi$
$907$	−459.549	−0.506669	−0.253334	+	0.967379i	$0.581527\pi$
$908$	0	0
$909$	106.454i	0.117111i
$910$	0	0
$911$	1515.03	1.66304	0.831518	−	0.555498i	$-0.187472\pi$
$911$	1515.03	1.66304	0.831518	+	0.555498i	$0.187472\pi$
$912$	0	0
$913$	88.6447i	0.0970916i
$914$	0	0
$915$	309.597i	0.338358i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−103.364	−0.112475	−0.0562373	−	0.998417i	$-0.517910\pi$
$919$	−103.364	−0.112475	−0.0562373	+	0.998417i	$0.517910\pi$
$920$	0	0
$921$	−992.166	−1.07727
$922$	0	0
$923$	493.441i	0.534605i
$924$	0	0
$925$	114.795	0.124103
$926$	0	0
$927$	246.910i	0.266354i
$928$	0	0
$929$	1353.04i	1.45645i	0.685338	+	0.728225i	$0.259654\pi$
$929$	1353.04i	1.45645i	−0.685338	+	0.728225i	$0.740346\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−244.180	−0.261715
$934$	0	0
$935$	−186.817	−0.199805
$936$	0	0
$937$	1443.67i	1.54074i	0.637600	+	0.770368i	$0.279927\pi$
$937$	1443.67i	1.54074i	−0.637600	+	0.770368i	$0.720073\pi$
$938$	0	0
$939$	1232.33	1.31239
$940$	0	0
$941$	1267.74i	1.34722i	0.739085	+	0.673612i	$0.235258\pi$
$941$	1267.74i	1.34722i	−0.739085	+	0.673612i	$0.764742\pi$
$942$	0	0
$943$	779.408i	0.826520i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−401.424	−0.423890	−0.211945	−	0.977282i	$-0.567980\pi$
$947$	−401.424	−0.423890	−0.211945	+	0.977282i	$0.567980\pi$
$948$	0	0
$949$	−739.558	−0.779303
$950$	0	0
$951$	172.677i	0.181574i
$952$	0	0
$953$	−1544.95	−1.62115	−0.810573	−	0.585637i	$-0.800844\pi$
$953$	−1544.95	−1.62115	−0.810573	+	0.585637i	$0.800844\pi$
$954$	0	0
$955$	− 463.441i	− 0.485279i
$956$	0	0
$957$	− 236.412i	− 0.247034i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−786.386	−0.818299
$962$	0	0
$963$	−212.010	−0.220156
$964$	0	0
$965$	525.090i	0.544135i
$966$	0	0
$967$	1272.03	1.31544	0.657718	−	0.753264i	$-0.271522\pi$
$967$	1272.03	1.31544	0.657718	+	0.753264i	$0.271522\pi$
$968$	0	0
$969$	558.280i	0.576141i
$970$	0	0
$971$	1057.91i	1.08950i	0.838598	+	0.544751i	$0.183376\pi$
$971$	1057.91i	1.08950i	−0.838598	+	0.544751i	$0.816624\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−120.241	−0.123324
$976$	0	0
$977$	455.290	0.466008	0.233004	−	0.972476i	$-0.425144\pi$
$977$	455.290	0.466008	0.233004	+	0.972476i	$0.425144\pi$
$978$	0	0
$979$	− 477.531i	− 0.487774i
$980$	0	0
$981$	−411.585	−0.419557
$982$	0	0
$983$	− 1083.26i	− 1.10199i	−0.834508	−	0.550995i	$-0.814248\pi$
$983$	− 1083.26i	− 1.10199i	0.834508	−	0.550995i	$-0.185752\pi$
$984$	0	0
$985$	− 1207.38i	− 1.22577i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2022.27	2.04476
$990$	0	0
$991$	−1110.08	−1.12016	−0.560081	−	0.828438i	$-0.689230\pi$
$991$	−1110.08	−1.12016	−0.560081	+	0.828438i	$0.689230\pi$
$992$	0	0
$993$	674.180i	0.678933i
$994$	0	0
$995$	−40.1249	−0.0403265
$996$	0	0
$997$	1538.71i	1.54334i	0.636025	+	0.771668i	$0.280577\pi$
$997$	1538.71i	1.54334i	−0.636025	+	0.771668i	$0.719423\pi$
$998$	0	0
$999$	419.012i	0.419432i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.c.g.97.12		16
4.3	odd	2	inner	1568.3.c.g.97.6		16
7.2	even	3		224.3.s.b.129.3	yes	16
7.3	odd	6		224.3.s.b.33.3	✓	16
7.6	odd	2	inner	1568.3.c.g.97.5		16
28.3	even	6		224.3.s.b.33.6	yes	16
28.23	odd	6		224.3.s.b.129.6	yes	16
28.27	even	2	inner	1568.3.c.g.97.11		16
56.3	even	6		448.3.s.h.257.3		16
56.37	even	6		448.3.s.h.129.6		16
56.45	odd	6		448.3.s.h.257.6		16
56.51	odd	6		448.3.s.h.129.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.s.b.33.3	✓	16	7.3	odd	6
224.3.s.b.33.6	yes	16	28.3	even	6
224.3.s.b.129.3	yes	16	7.2	even	3
224.3.s.b.129.6	yes	16	28.23	odd	6
448.3.s.h.129.3		16	56.51	odd	6
448.3.s.h.129.6		16	56.37	even	6
448.3.s.h.257.3		16	56.3	even	6
448.3.s.h.257.6		16	56.45	odd	6
1568.3.c.g.97.5		16	7.6	odd	2	inner
1568.3.c.g.97.6		16	4.3	odd	2	inner
1568.3.c.g.97.11		16	28.27	even	2	inner
1568.3.c.g.97.12		16	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.54150i q^{3} +4.11878i q^{5} +2.54076 q^{9} +O(q^{10})\)	\(q+2.54150i q^{3} +4.11878i q^{5} +2.54076 q^{9} +3.26783 q^{11} +5.88759i q^{13} -10.4679 q^{15} +13.8800i q^{17} -15.8261i q^{19} +36.5036 q^{23} +8.03567 q^{25} +29.3309i q^{27} -28.4655 q^{29} +41.8017i q^{31} +8.30521i q^{33} +14.2857 q^{37} -14.9633 q^{39} +21.3515i q^{41} +55.3992 q^{43} +10.4648i q^{45} +33.8533i q^{47} -35.2760 q^{51} -84.8542 q^{53} +13.4595i q^{55} +40.2220 q^{57} -67.5572i q^{59} -29.5759i q^{61} -24.2497 q^{65} -54.9578 q^{67} +92.7741i q^{69} +83.8102 q^{71} +125.613i q^{73} +20.4227i q^{75} +70.3910 q^{79} -51.6777 q^{81} +27.1264i q^{83} -57.1685 q^{85} -72.3451i q^{87} -146.131i q^{89} -106.239 q^{93} +65.1841 q^{95} -11.3574i q^{97} +8.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 80 q^{9}+O(q^{10}) \) 16 * q - 80 * q^9	\( 16 q - 80 q^{9} - 160 q^{25} - 16 q^{29} - 144 q^{37} - 80 q^{53} + 368 q^{57} + 336 q^{65} + 768 q^{81} - 1072 q^{85} + 336 q^{93}+O(q^{100}) \) 16 * q - 80 * q^9 - 160 * q^25 - 16 * q^29 - 144 * q^37 - 80 * q^53 + 368 * q^57 + 336 * q^65 + 768 * q^81 - 1072 * q^85 + 336 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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