LMFDB - Embedded newform 9200.2.a.ce.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9200,2,Mod(1,9200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9200.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,0,0,2,0,10,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$73.4623698596$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2597.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 8 $ x^3 - x^2 - 9*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.878468$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.878468 q^{3} +0.121532 q^{7} -2.22829 q^{9} -2.87847 q^{11} -5.22829 q^{13} +2.22829 q^{17} -1.22829 q^{19} +0.106762 q^{21} +1.00000 q^{23} -4.59289 q^{27} +9.34983 q^{29} +2.12153 q^{31} -2.52864 q^{33} -5.59289 q^{37} -4.59289 q^{39} +8.22829 q^{41} +8.00000 q^{43} -10.4566 q^{47} -6.98523 q^{49} +1.95749 q^{51} +3.59289 q^{53} -1.07902 q^{57} +0.650174 q^{59} -7.33506 q^{61} -0.270809 q^{63} +5.59289 q^{67} +0.878468 q^{69} +13.9852 q^{71} -12.9427 q^{73} -0.349826 q^{77} +3.51387 q^{79} +2.65017 q^{81} -11.1068 q^{83} +8.21352 q^{87} -0.486128 q^{89} -0.635404 q^{91} +1.86370 q^{93} -0.635404 q^{97} +6.41407 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 2 q^{7} + 10 q^{9} - 7 q^{11} + q^{13} - 10 q^{17} + 13 q^{19} - 18 q^{21} + 3 q^{23} - 2 q^{27} + 13 q^{29} + 8 q^{31} - 21 q^{33} - 5 q^{37} - 2 q^{39} + 8 q^{41} + 24 q^{43} + 2 q^{47}+ \cdots - 21 q^{99}+O(q^{100}) $ 3 * q + q^3 + 2 * q^7 + 10 * q^9 - 7 * q^11 + q^13 - 10 * q^17 + 13 * q^19 - 18 * q^21 + 3 * q^23 - 2 * q^27 + 13 * q^29 + 8 * q^31 - 21 * q^33 - 5 * q^37 - 2 * q^39 + 8 * q^41 + 24 * q^43 + 2 * q^47 - q^49 - q^51 - q^53 + 2 * q^57 + 17 * q^59 + 13 * q^61 + 9 * q^63 + 5 * q^67 + q^69 + 22 * q^71 - 12 * q^73 + 14 * q^77 + 4 * q^79 + 23 * q^81 - 15 * q^83 - 12 * q^87 - 8 * q^89 + 3 * q^91 - 16 * q^93 + 3 * q^97 - 21 * q^99

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
$3$	0.878468	0.507184	0.253592	−	0.967311i	$-0.418388\pi$
$3$	0.878468	0.507184	0.253592	+	0.967311i	$0.418388\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.121532	0.0459347	0.0229674	−	0.999736i	$-0.492689\pi$
$7$	0.121532	0.0459347	0.0229674	+	0.999736i	$0.492689\pi$
$8$	0	0
$9$	−2.22829	−0.742765
$10$	0	0
$11$	−2.87847	−0.867891	−0.433945	−	0.900939i	$-0.642879\pi$
$11$	−2.87847	−0.867891	−0.433945	+	0.900939i	$0.642879\pi$
$12$	0	0
$13$	−5.22829	−1.45007	−0.725034	−	0.688713i	$-0.758176\pi$
$13$	−5.22829	−1.45007	−0.725034	+	0.688713i	$0.758176\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.22829	0.540441	0.270220	−	0.962799i	$-0.412903\pi$
$17$	2.22829	0.540441	0.270220	+	0.962799i	$0.412903\pi$
$18$	0	0
$19$	−1.22829	−0.281790	−0.140895	−	0.990025i	$-0.544998\pi$
$19$	−1.22829	−0.281790	−0.140895	+	0.990025i	$0.544998\pi$
$20$	0	0
$21$	0.106762	0.0232974
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4.59289	−0.883902
$28$	0	0
$29$	9.34983	1.73622	0.868110	−	0.496373i	$-0.165335\pi$
$29$	9.34983	1.73622	0.868110	+	0.496373i	$0.165335\pi$
$30$	0	0
$31$	2.12153	0.381038	0.190519	−	0.981683i	$-0.438983\pi$
$31$	2.12153	0.381038	0.190519	+	0.981683i	$0.438983\pi$
$32$	0	0
$33$	−2.52864	−0.440180
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.59289	−0.919465	−0.459733	−	0.888057i	$-0.652055\pi$
$37$	−5.59289	−0.919465	−0.459733	+	0.888057i	$0.652055\pi$
$38$	0	0
$39$	−4.59289	−0.735451
$40$	0	0
$41$	8.22829	1.28504	0.642522	−	0.766267i	$-0.277888\pi$
$41$	8.22829	1.28504	0.642522	+	0.766267i	$0.277888\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.4566	−1.52525	−0.762625	−	0.646841i	$-0.776090\pi$
$47$	−10.4566	−1.52525	−0.762625	+	0.646841i	$0.776090\pi$
$48$	0	0
$49$	−6.98523	−0.997890
$50$	0	0
$51$	1.95749	0.274103
$52$	0	0
$53$	3.59289	0.493521	0.246761	−	0.969076i	$-0.420634\pi$
$53$	3.59289	0.493521	0.246761	+	0.969076i	$0.420634\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.07902	−0.142919
$58$	0	0
$59$	0.650174	0.0846455	0.0423227	−	0.999104i	$-0.486524\pi$
$59$	0.650174	0.0846455	0.0423227	+	0.999104i	$0.486524\pi$
$60$	0	0
$61$	−7.33506	−0.939158	−0.469579	−	0.882891i	$-0.655594\pi$
$61$	−7.33506	−0.939158	−0.469579	+	0.882891i	$0.655594\pi$
$62$	0	0
$63$	−0.270809	−0.0341187
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.59289	0.683280	0.341640	−	0.939831i	$-0.389018\pi$
$67$	5.59289	0.683280	0.341640	+	0.939831i	$0.389018\pi$
$68$	0	0
$69$	0.878468	0.105755
$70$	0	0
$71$	13.9852	1.65974	0.829871	−	0.557956i	$-0.188414\pi$
$71$	13.9852	1.65974	0.829871	+	0.557956i	$0.188414\pi$
$72$	0	0
$73$	−12.9427	−1.51483	−0.757415	−	0.652934i	$-0.773538\pi$
$73$	−12.9427	−1.51483	−0.757415	+	0.652934i	$0.773538\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.349826	−0.0398663
$78$	0	0
$79$	3.51387	0.395342	0.197671	−	0.980268i	$-0.436662\pi$
$79$	3.51387	0.395342	0.197671	+	0.980268i	$0.436662\pi$
$80$	0	0
$81$	2.65017	0.294464
$82$	0	0
$83$	−11.1068	−1.21913	−0.609563	−	0.792738i	$-0.708655\pi$
$83$	−11.1068	−1.21913	−0.609563	+	0.792738i	$0.708655\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.21352	0.880582
$88$	0	0
$89$	−0.486128	−0.0515294	−0.0257647	−	0.999668i	$-0.508202\pi$
$89$	−0.486128	−0.0515294	−0.0257647	+	0.999668i	$0.508202\pi$
$90$	0	0
$91$	−0.635404	−0.0666085
$92$	0	0
$93$	1.86370	0.193256
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.635404	−0.0645155	−0.0322578	−	0.999480i	$-0.510270\pi$
$97$	−0.635404	−0.0645155	−0.0322578	+	0.999480i	$0.510270\pi$
$98$	0	0
$99$	6.41407	0.644639
$100$	0	0
$101$	13.3498	1.32836	0.664179	−	0.747574i	$-0.268781\pi$
$101$	13.3498	1.32836	0.664179	+	0.747574i	$0.268781\pi$
$102$	0	0
$103$	−7.33506	−0.722745	−0.361372	−	0.932422i	$-0.617692\pi$
$103$	−7.33506	−0.722745	−0.361372	+	0.932422i	$0.617692\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0495	1.55156	0.775781	−	0.631003i	$-0.217356\pi$
$107$	16.0495	1.55156	0.775781	+	0.631003i	$0.217356\pi$
$108$	0	0
$109$	1.01477	0.0971973	0.0485987	−	0.998818i	$-0.484524\pi$
$109$	1.01477	0.0971973	0.0485987	+	0.998818i	$0.484524\pi$
$110$	0	0
$111$	−4.91318	−0.466338
$112$	0	0
$113$	−17.3203	−1.62936	−0.814678	−	0.579914i	$-0.803086\pi$
$113$	−17.3203	−1.62936	−0.814678	+	0.579914i	$0.803086\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	11.6502	1.07706
$118$	0	0
$119$	0.270809	0.0248250
$120$	0	0
$121$	−2.71442	−0.246766
$122$	0	0
$123$	7.22829	0.651753
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.75694	0.155903	0.0779514	−	0.996957i	$-0.475162\pi$
$127$	1.75694	0.155903	0.0779514	+	0.996957i	$0.475162\pi$
$128$	0	0
$129$	7.02774	0.618758
$130$	0	0
$131$	20.2135	1.76606	0.883032	−	0.469313i	$-0.155498\pi$
$131$	20.2135	1.76606	0.883032	+	0.469313i	$0.155498\pi$
$132$	0	0
$133$	−0.149277	−0.0129439
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.9279	1.53169	0.765844	−	0.643027i	$-0.222322\pi$
$137$	17.9279	1.53169	0.765844	+	0.643027i	$0.222322\pi$
$138$	0	0
$139$	10.8637	0.921447	0.460723	−	0.887544i	$-0.347590\pi$
$139$	10.8637	0.921447	0.460723	+	0.887544i	$0.347590\pi$
$140$	0	0
$141$	−9.18578	−0.773582
$142$	0	0
$143$	15.0495	1.25850
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.13630	−0.506114
$148$	0	0
$149$	−15.7144	−1.28738	−0.643688	−	0.765288i	$-0.722596\pi$
$149$	−15.7144	−1.28738	−0.643688	+	0.765288i	$0.722596\pi$
$150$	0	0
$151$	8.39234	0.682959	0.341479	−	0.939889i	$-0.389072\pi$
$151$	8.39234	0.682959	0.341479	+	0.939889i	$0.389072\pi$
$152$	0	0
$153$	−4.96529	−0.401420
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19.5633	1.56133	0.780663	−	0.624953i	$-0.214882\pi$
$157$	19.5633	1.56133	0.780663	+	0.624953i	$0.214882\pi$
$158$	0	0
$159$	3.15624	0.250306
$160$	0	0
$161$	0.121532	0.00957805
$162$	0	0
$163$	1.01477	0.0794829	0.0397415	−	0.999210i	$-0.487347\pi$
$163$	1.01477	0.0794829	0.0397415	+	0.999210i	$0.487347\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	14.3351	1.10270
$170$	0	0
$171$	2.73700	0.209304
$172$	0	0
$173$	2.63540	0.200366	0.100183	−	0.994969i	$-0.468057\pi$
$173$	2.63540	0.200366	0.100183	+	0.994969i	$0.468057\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.571157	0.0429308
$178$	0	0
$179$	14.4566	1.08054	0.540268	−	0.841493i	$-0.318323\pi$
$179$	14.4566	1.08054	0.540268	+	0.841493i	$0.318323\pi$
$180$	0	0
$181$	10.7422	0.798459	0.399229	−	0.916851i	$-0.369278\pi$
$181$	10.7422	0.798459	0.399229	+	0.916851i	$0.369278\pi$
$182$	0	0
$183$	−6.44361	−0.476326
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.41407	−0.469043
$188$	0	0
$189$	−0.558182	−0.0406018
$190$	0	0
$191$	22.6701	1.64035	0.820176	−	0.572112i	$-0.193876\pi$
$191$	22.6701	1.64035	0.820176	+	0.572112i	$0.193876\pi$
$192$	0	0
$193$	−2.24306	−0.161459	−0.0807296	−	0.996736i	$-0.525725\pi$
$193$	−2.24306	−0.161459	−0.0807296	+	0.996736i	$0.525725\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.57812	0.539919	0.269959	−	0.962872i	$-0.412990\pi$
$197$	7.57812	0.539919	0.269959	+	0.962872i	$0.412990\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	4.91318	0.346549
$202$	0	0
$203$	1.13630	0.0797528
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.22829	−0.154877
$208$	0	0
$209$	3.53560	0.244563
$210$	0	0
$211$	3.10676	0.213878	0.106939	−	0.994266i	$-0.465895\pi$
$211$	3.10676	0.213878	0.106939	+	0.994266i	$0.465895\pi$
$212$	0	0
$213$	12.2856	0.841794
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.257834	0.0175029
$218$	0	0
$219$	−11.3698	−0.768297
$220$	0	0
$221$	−11.6502	−0.783676
$222$	0	0
$223$	18.2135	1.21967	0.609834	−	0.792529i	$-0.291236\pi$
$223$	18.2135	1.21967	0.609834	+	0.792529i	$0.291236\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.9705	0.927252	0.463626	−	0.886031i	$-0.346548\pi$
$227$	13.9705	0.927252	0.463626	+	0.886031i	$0.346548\pi$
$228$	0	0
$229$	12.2135	0.807092	0.403546	−	0.914959i	$-0.367777\pi$
$229$	12.2135	0.807092	0.403546	+	0.914959i	$0.367777\pi$
$230$	0	0
$231$	−0.307311	−0.0202196
$232$	0	0
$233$	−9.75694	−0.639198	−0.319599	−	0.947553i	$-0.603548\pi$
$233$	−9.75694	−0.639198	−0.319599	+	0.947553i	$0.603548\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.08682	0.200511
$238$	0	0
$239$	−8.62063	−0.557622	−0.278811	−	0.960346i	$-0.589940\pi$
$239$	−8.62063	−0.557622	−0.278811	+	0.960346i	$0.589940\pi$
$240$	0	0
$241$	3.18578	0.205214	0.102607	−	0.994722i	$-0.467282\pi$
$241$	3.18578	0.205214	0.102607	+	0.994722i	$0.467282\pi$
$242$	0	0
$243$	16.1068	1.03325
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.42188	0.408614
$248$	0	0
$249$	−9.75694	−0.618321
$250$	0	0
$251$	25.2283	1.59240	0.796198	−	0.605036i	$-0.206841\pi$
$251$	25.2283	1.59240	0.796198	+	0.605036i	$0.206841\pi$
$252$	0	0
$253$	−2.87847	−0.180968
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.6701	0.665583	0.332792	−	0.943000i	$-0.392009\pi$
$257$	10.6701	0.665583	0.332792	+	0.943000i	$0.392009\pi$
$258$	0	0
$259$	−0.679714	−0.0422354
$260$	0	0
$261$	−20.8342	−1.28960
$262$	0	0
$263$	−18.6849	−1.15216	−0.576080	−	0.817394i	$-0.695418\pi$
$263$	−18.6849	−1.15216	−0.576080	+	0.817394i	$0.695418\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.427048	−0.0261349
$268$	0	0
$269$	−24.8637	−1.51597	−0.757983	−	0.652274i	$-0.773815\pi$
$269$	−24.8637	−1.51597	−0.757983	+	0.652274i	$0.773815\pi$
$270$	0	0
$271$	−0.578119	−0.0351183	−0.0175591	−	0.999846i	$-0.505590\pi$
$271$	−0.578119	−0.0351183	−0.0175591	+	0.999846i	$0.505590\pi$
$272$	0	0
$273$	−0.558182	−0.0337827
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.51387	0.571633	0.285817	−	0.958284i	$-0.407735\pi$
$277$	9.51387	0.571633	0.285817	+	0.958284i	$0.407735\pi$
$278$	0	0
$279$	−4.72740	−0.283022
$280$	0	0
$281$	−13.1562	−0.784835	−0.392418	−	0.919787i	$-0.628361\pi$
$281$	−13.1562	−0.784835	−0.392418	+	0.919787i	$0.628361\pi$
$282$	0	0
$283$	24.5061	1.45673	0.728367	−	0.685187i	$-0.240280\pi$
$283$	24.5061	1.45673	0.728367	+	0.685187i	$0.240280\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.00000	0.0590281
$288$	0	0
$289$	−12.0347	−0.707924
$290$	0	0
$291$	−0.558182	−0.0327212
$292$	0	0
$293$	6.29254	0.367614	0.183807	−	0.982962i	$-0.441158\pi$
$293$	6.29254	0.367614	0.183807	+	0.982962i	$0.441158\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	13.2205	0.767130
$298$	0	0
$299$	−5.22829	−0.302360
$300$	0	0
$301$	0.972255	0.0560398
$302$	0	0
$303$	11.7274	0.673721
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.3351	0.761072	0.380536	−	0.924766i	$-0.375740\pi$
$307$	13.3351	0.761072	0.380536	+	0.924766i	$0.375740\pi$
$308$	0	0
$309$	−6.44361	−0.366564
$310$	0	0
$311$	24.4270	1.38513	0.692565	−	0.721355i	$-0.256480\pi$
$311$	24.4270	1.38513	0.692565	+	0.721355i	$0.256480\pi$
$312$	0	0
$313$	−26.5486	−1.50061	−0.750307	−	0.661089i	$-0.770094\pi$
$313$	−26.5486	−1.50061	−0.750307	+	0.661089i	$0.770094\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.9852	0.953986	0.476993	−	0.878907i	$-0.341727\pi$
$317$	16.9852	0.953986	0.476993	+	0.878907i	$0.341727\pi$
$318$	0	0
$319$	−26.9132	−1.50685
$320$	0	0
$321$	14.0990	0.786927
$322$	0	0
$323$	−2.73700	−0.152291
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.891443	0.0492969
$328$	0	0
$329$	−1.27081	−0.0700620
$330$	0	0
$331$	−17.8914	−0.983403	−0.491701	−	0.870764i	$-0.663625\pi$
$331$	−17.8914	−0.983403	−0.491701	+	0.870764i	$0.663625\pi$
$332$	0	0
$333$	12.4626	0.682946
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.52864	0.246691	0.123345	−	0.992364i	$-0.460638\pi$
$337$	4.52864	0.246691	0.123345	+	0.992364i	$0.460638\pi$
$338$	0	0
$339$	−15.2153	−0.826383
$340$	0	0
$341$	−6.10676	−0.330700
$342$	0	0
$343$	−1.69965	−0.0917725
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−0.0937869	−0.00503475	−0.00251737	−	0.999997i	$-0.500801\pi$
$347$	−0.0937869	−0.00503475	−0.00251737	+	0.999997i	$0.500801\pi$
$348$	0	0
$349$	8.07902	0.432460	0.216230	−	0.976342i	$-0.430624\pi$
$349$	8.07902	0.432460	0.216230	+	0.976342i	$0.430624\pi$
$350$	0	0
$351$	24.0130	1.28172
$352$	0	0
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.237897	0.0125908
$358$	0	0
$359$	−8.69965	−0.459150	−0.229575	−	0.973291i	$-0.573734\pi$
$359$	−8.69965	−0.459150	−0.229575	+	0.973291i	$0.573734\pi$
$360$	0	0
$361$	−17.4913	−0.920594
$362$	0	0
$363$	−2.38453	−0.125156
$364$	0	0
$365$	0	0
$366$	0	0
$367$	31.8064	1.66028	0.830141	−	0.557554i	$-0.188260\pi$
$367$	31.8064	1.66028	0.830141	+	0.557554i	$0.188260\pi$
$368$	0	0
$369$	−18.3351	−0.954485
$370$	0	0
$371$	0.436651	0.0226698
$372$	0	0
$373$	13.4288	0.695319	0.347660	−	0.937621i	$-0.386977\pi$
$373$	13.4288	0.695319	0.347660	+	0.937621i	$0.386977\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−48.8836	−2.51764
$378$	0	0
$379$	−31.8212	−1.63454	−0.817272	−	0.576252i	$-0.804515\pi$
$379$	−31.8212	−1.63454	−0.817272	+	0.576252i	$0.804515\pi$
$380$	0	0
$381$	1.54341	0.0790714
$382$	0	0
$383$	34.3480	1.75510	0.877551	−	0.479483i	$-0.159176\pi$
$383$	34.3480	1.75510	0.877551	+	0.479483i	$0.159176\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−17.8264	−0.906164
$388$	0	0
$389$	8.14928	0.413185	0.206592	−	0.978427i	$-0.433763\pi$
$389$	8.14928	0.413185	0.206592	+	0.978427i	$0.433763\pi$
$390$	0	0
$391$	2.22829	0.112690
$392$	0	0
$393$	17.7569	0.895719
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−19.0625	−0.956717	−0.478359	−	0.878165i	$-0.658768\pi$
$397$	−19.0625	−0.956717	−0.478359	+	0.878165i	$0.658768\pi$
$398$	0	0
$399$	−0.131135	−0.00656496
$400$	0	0
$401$	19.9705	0.997277	0.498639	−	0.866810i	$-0.333833\pi$
$401$	19.9705	0.997277	0.498639	+	0.866810i	$0.333833\pi$
$402$	0	0
$403$	−11.0920	−0.552531
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.0990	0.797996
$408$	0	0
$409$	−10.3351	−0.511036	−0.255518	−	0.966804i	$-0.582246\pi$
$409$	−10.3351	−0.511036	−0.255518	+	0.966804i	$0.582246\pi$
$410$	0	0
$411$	15.7491	0.776847
$412$	0	0
$413$	0.0790169	0.00388817
$414$	0	0
$415$	0	0
$416$	0	0
$417$	9.54341	0.467343
$418$	0	0
$419$	−5.72740	−0.279802	−0.139901	−	0.990166i	$-0.544678\pi$
$419$	−5.72740	−0.279802	−0.139901	+	0.990166i	$0.544678\pi$
$420$	0	0
$421$	−8.44361	−0.411516	−0.205758	−	0.978603i	$-0.565966\pi$
$421$	−8.44361	−0.411516	−0.205758	+	0.978603i	$0.565966\pi$
$422$	0	0
$423$	23.3003	1.13290
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.891443	−0.0431400
$428$	0	0
$429$	13.2205	0.638291
$430$	0	0
$431$	−35.3698	−1.70370	−0.851851	−	0.523785i	$-0.824520\pi$
$431$	−35.3698	−1.70370	−0.851851	+	0.523785i	$0.824520\pi$
$432$	0	0
$433$	−13.3923	−0.643595	−0.321797	−	0.946809i	$-0.604287\pi$
$433$	−13.3923	−0.643595	−0.321797	+	0.946809i	$0.604287\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.22829	−0.0587573
$438$	0	0
$439$	21.1137	1.00770	0.503852	−	0.863790i	$-0.331916\pi$
$439$	21.1137	1.00770	0.503852	+	0.863790i	$0.331916\pi$
$440$	0	0
$441$	15.5651	0.741197
$442$	0	0
$443$	−31.1692	−1.48089	−0.740447	−	0.672115i	$-0.765386\pi$
$443$	−31.1692	−1.48089	−0.740447	+	0.672115i	$0.765386\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−13.8046	−0.652936
$448$	0	0
$449$	−6.09199	−0.287499	−0.143749	−	0.989614i	$-0.545916\pi$
$449$	−6.09199	−0.287499	−0.143749	+	0.989614i	$0.545916\pi$
$450$	0	0
$451$	−23.6849	−1.11528
$452$	0	0
$453$	7.37240	0.346386
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.1345	−1.03541	−0.517704	−	0.855560i	$-0.673213\pi$
$457$	−22.1345	−1.03541	−0.517704	+	0.855560i	$0.673213\pi$
$458$	0	0
$459$	−10.2343	−0.477697
$460$	0	0
$461$	−11.5434	−0.537630	−0.268815	−	0.963192i	$-0.586632\pi$
$461$	−11.5434	−0.537630	−0.268815	+	0.963192i	$0.586632\pi$
$462$	0	0
$463$	30.4270	1.41406	0.707032	−	0.707181i	$-0.250033\pi$
$463$	30.4270	1.41406	0.707032	+	0.707181i	$0.250033\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.3776	0.480217	0.240108	−	0.970746i	$-0.422817\pi$
$467$	10.3776	0.480217	0.240108	+	0.970746i	$0.422817\pi$
$468$	0	0
$469$	0.679714	0.0313863
$470$	0	0
$471$	17.1858	0.791879
$472$	0	0
$473$	−23.0277	−1.05882
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−8.00601	−0.366570
$478$	0	0
$479$	29.6128	1.35304	0.676522	−	0.736422i	$-0.263486\pi$
$479$	29.6128	1.35304	0.676522	+	0.736422i	$0.263486\pi$
$480$	0	0
$481$	29.2413	1.33329
$482$	0	0
$483$	0.106762	0.00485783
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11.1562	0.505537	0.252769	−	0.967527i	$-0.418659\pi$
$487$	11.1562	0.505537	0.252769	+	0.967527i	$0.418659\pi$
$488$	0	0
$489$	0.891443	0.0403125
$490$	0	0
$491$	−38.2630	−1.72679	−0.863393	−	0.504533i	$-0.831665\pi$
$491$	−38.2630	−1.72679	−0.863393	+	0.504533i	$0.831665\pi$
$492$	0	0
$493$	20.8342	0.938323
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.69965	0.0762398
$498$	0	0
$499$	−6.40711	−0.286822	−0.143411	−	0.989663i	$-0.545807\pi$
$499$	−6.40711	−0.286822	−0.143411	+	0.989663i	$0.545807\pi$
$500$	0	0
$501$	−7.02774	−0.313976
$502$	0	0
$503$	−18.0920	−0.806682	−0.403341	−	0.915050i	$-0.632151\pi$
$503$	−18.0920	−0.806682	−0.403341	+	0.915050i	$0.632151\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.5929	0.559270
$508$	0	0
$509$	−34.1285	−1.51272	−0.756359	−	0.654156i	$-0.773024\pi$
$509$	−34.1285	−1.51272	−0.756359	+	0.654156i	$0.773024\pi$
$510$	0	0
$511$	−1.57295	−0.0695833
$512$	0	0
$513$	5.64142	0.249075
$514$	0	0
$515$	0	0
$516$	0	0
$517$	30.0990	1.32375
$518$	0	0
$519$	2.31512	0.101622
$520$	0	0
$521$	35.8854	1.57217	0.786085	−	0.618119i	$-0.212105\pi$
$521$	35.8854	1.57217	0.786085	+	0.618119i	$0.212105\pi$
$522$	0	0
$523$	11.7865	0.515387	0.257693	−	0.966227i	$-0.417038\pi$
$523$	11.7865	0.515387	0.257693	+	0.966227i	$0.417038\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.72740	0.205929
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−1.44878	−0.0628717
$532$	0	0
$533$	−43.0199	−1.86340
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.6997	0.548030
$538$	0	0
$539$	20.1068	0.866060
$540$	0	0
$541$	−17.2413	−0.741260	−0.370630	−	0.928781i	$-0.620858\pi$
$541$	−17.2413	−0.741260	−0.370630	+	0.928781i	$0.620858\pi$
$542$	0	0
$543$	9.43665	0.404965
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.6571	0.626694	0.313347	−	0.949639i	$-0.398550\pi$
$547$	14.6571	0.626694	0.313347	+	0.949639i	$0.398550\pi$
$548$	0	0
$549$	16.3447	0.697573
$550$	0	0
$551$	−11.4843	−0.489249
$552$	0	0
$553$	0.427048	0.0181599
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.07902	−0.0880908	−0.0440454	−	0.999030i	$-0.514025\pi$
$557$	−2.07902	−0.0880908	−0.0440454	+	0.999030i	$0.514025\pi$
$558$	0	0
$559$	−41.8264	−1.76907
$560$	0	0
$561$	−5.63456	−0.237891
$562$	0	0
$563$	−5.10676	−0.215224	−0.107612	−	0.994193i	$-0.534320\pi$
$563$	−5.10676	−0.215224	−0.107612	+	0.994193i	$0.534320\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.322081	0.0135261
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−10.0642	−0.421176	−0.210588	−	0.977575i	$-0.567538\pi$
$571$	−10.0642	−0.421176	−0.210588	+	0.977575i	$0.567538\pi$
$572$	0	0
$573$	19.9150	0.831960
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.7274	0.654740	0.327370	−	0.944896i	$-0.393838\pi$
$577$	15.7274	0.654740	0.327370	+	0.944896i	$0.393838\pi$
$578$	0	0
$579$	−1.97046	−0.0818895
$580$	0	0
$581$	−1.34983	−0.0560002
$582$	0	0
$583$	−10.3420	−0.428323
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.7344	−0.773250	−0.386625	−	0.922237i	$-0.626359\pi$
$587$	−18.7344	−0.773250	−0.386625	+	0.922237i	$0.626359\pi$
$588$	0	0
$589$	−2.60586	−0.107373
$590$	0	0
$591$	6.65714	0.273838
$592$	0	0
$593$	3.21532	0.132037	0.0660187	−	0.997818i	$-0.478970\pi$
$593$	3.21532	0.132037	0.0660187	+	0.997818i	$0.478970\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.2986	0.503346
$598$	0	0
$599$	−22.7639	−0.930108	−0.465054	−	0.885282i	$-0.653965\pi$
$599$	−22.7639	−0.930108	−0.465054	+	0.885282i	$0.653965\pi$
$600$	0	0
$601$	4.30552	0.175626	0.0878128	−	0.996137i	$-0.472012\pi$
$601$	4.30552	0.175626	0.0878128	+	0.996137i	$0.472012\pi$
$602$	0	0
$603$	−12.4626	−0.507516
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11.0868	0.450000	0.225000	−	0.974359i	$-0.427762\pi$
$607$	11.0868	0.450000	0.225000	+	0.974359i	$0.427762\pi$
$608$	0	0
$609$	0.998205	0.0404493
$610$	0	0
$611$	54.6701	2.21172
$612$	0	0
$613$	−29.9409	−1.20930	−0.604651	−	0.796490i	$-0.706687\pi$
$613$	−29.9409	−1.20930	−0.604651	+	0.796490i	$0.706687\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.0130	−1.40957	−0.704785	−	0.709421i	$-0.748956\pi$
$617$	−35.0130	−1.40957	−0.704785	+	0.709421i	$0.748956\pi$
$618$	0	0
$619$	−4.98523	−0.200373	−0.100187	−	0.994969i	$-0.531944\pi$
$619$	−4.98523	−0.200373	−0.100187	+	0.994969i	$0.531944\pi$
$620$	0	0
$621$	−4.59289	−0.184306
$622$	0	0
$623$	−0.0590800	−0.00236699
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.10592	0.124038
$628$	0	0
$629$	−12.4626	−0.496916
$630$	0	0
$631$	3.64237	0.145000	0.0725002	−	0.997368i	$-0.476902\pi$
$631$	3.64237	0.145000	0.0725002	+	0.997368i	$0.476902\pi$
$632$	0	0
$633$	2.72919	0.108476
$634$	0	0
$635$	0	0
$636$	0	0
$637$	36.5208	1.44701
$638$	0	0
$639$	−31.1632	−1.23280
$640$	0	0
$641$	38.6997	1.52854	0.764272	−	0.644894i	$-0.223098\pi$
$641$	38.6997	1.52854	0.764272	+	0.644894i	$0.223098\pi$
$642$	0	0
$643$	−28.8637	−1.13827	−0.569137	−	0.822243i	$-0.692722\pi$
$643$	−28.8637	−1.13827	−0.569137	+	0.822243i	$0.692722\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.8854	−0.781777	−0.390888	−	0.920438i	$-0.627832\pi$
$647$	−19.8854	−0.781777	−0.390888	+	0.920438i	$0.627832\pi$
$648$	0	0
$649$	−1.87151	−0.0734630
$650$	0	0
$651$	0.226499	0.00887719
$652$	0	0
$653$	27.1988	1.06437	0.532185	−	0.846628i	$-0.321371\pi$
$653$	27.1988	1.06437	0.532185	+	0.846628i	$0.321371\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	28.8402	1.12516
$658$	0	0
$659$	17.7274	0.690561	0.345281	−	0.938499i	$-0.387784\pi$
$659$	17.7274	0.690561	0.345281	+	0.938499i	$0.387784\pi$
$660$	0	0
$661$	40.7048	1.58323	0.791617	−	0.611018i	$-0.209240\pi$
$661$	40.7048	1.58323	0.791617	+	0.611018i	$0.209240\pi$
$662$	0	0
$663$	−10.2343	−0.397468
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.34983	0.362027
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	21.1137	0.815086
$672$	0	0
$673$	−12.2135	−0.470797	−0.235398	−	0.971899i	$-0.575639\pi$
$673$	−12.2135	−0.470797	−0.235398	+	0.971899i	$0.575639\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.5061	−0.634380	−0.317190	−	0.948362i	$-0.602739\pi$
$677$	−16.5061	−0.634380	−0.317190	+	0.948362i	$0.602739\pi$
$678$	0	0
$679$	−0.0772219	−0.00296350
$680$	0	0
$681$	12.2726	0.470287
$682$	0	0
$683$	−29.1988	−1.11726	−0.558630	−	0.829417i	$-0.688673\pi$
$683$	−29.1988	−1.11726	−0.558630	+	0.829417i	$0.688673\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.7292	0.409344
$688$	0	0
$689$	−18.7847	−0.715639
$690$	0	0
$691$	15.9150	0.605434	0.302717	−	0.953080i	$-0.402106\pi$
$691$	15.9150	0.605434	0.302717	+	0.953080i	$0.402106\pi$
$692$	0	0
$693$	0.779514	0.0296113
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.3351	0.694490
$698$	0	0
$699$	−8.57116	−0.324191
$700$	0	0
$701$	34.5981	1.30675	0.653375	−	0.757034i	$-0.273352\pi$
$701$	34.5981	1.30675	0.653375	+	0.757034i	$0.273352\pi$
$702$	0	0
$703$	6.86971	0.259096
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.62243	0.0610177
$708$	0	0
$709$	35.8212	1.34529	0.672646	−	0.739964i	$-0.265158\pi$
$709$	35.8212	1.34529	0.672646	+	0.739964i	$0.265158\pi$
$710$	0	0
$711$	−7.82994	−0.293646
$712$	0	0
$713$	2.12153	0.0794520
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.57295	−0.282817
$718$	0	0
$719$	32.5781	1.21496	0.607479	−	0.794335i	$-0.292181\pi$
$719$	32.5781	1.21496	0.607479	+	0.794335i	$0.292181\pi$
$720$	0	0
$721$	−0.891443	−0.0331991
$722$	0	0
$723$	2.79861	0.104081
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−31.9557	−1.18517	−0.592585	−	0.805508i	$-0.701893\pi$
$727$	−31.9557	−1.18517	−0.592585	+	0.805508i	$0.701893\pi$
$728$	0	0
$729$	6.19875	0.229583
$730$	0	0
$731$	17.8264	0.659331
$732$	0	0
$733$	2.19359	0.0810220	0.0405110	−	0.999179i	$-0.487101\pi$
$733$	2.19359	0.0810220	0.0405110	+	0.999179i	$0.487101\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.0990	−0.593013
$738$	0	0
$739$	51.7473	1.90356	0.951778	−	0.306787i	$-0.0992539\pi$
$739$	51.7473	1.90356	0.951778	+	0.306787i	$0.0992539\pi$
$740$	0	0
$741$	5.64142	0.207243
$742$	0	0
$743$	−14.4436	−0.529885	−0.264942	−	0.964264i	$-0.585353\pi$
$743$	−14.4436	−0.529885	−0.264942	+	0.964264i	$0.585353\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	24.7491	0.905523
$748$	0	0
$749$	1.95052	0.0712706
$750$	0	0
$751$	−17.4288	−0.635987	−0.317994	−	0.948093i	$-0.603009\pi$
$751$	−17.4288	−0.635987	−0.317994	+	0.948093i	$0.603009\pi$
$752$	0	0
$753$	22.1623	0.807637
$754$	0	0
$755$	0	0
$756$	0	0
$757$	34.9331	1.26967	0.634833	−	0.772650i	$-0.281069\pi$
$757$	34.9331	1.26967	0.634833	+	0.772650i	$0.281069\pi$
$758$	0	0
$759$	−2.52864	−0.0917839
$760$	0	0
$761$	−23.2482	−0.842748	−0.421374	−	0.906887i	$-0.638452\pi$
$761$	−23.2482	−0.842748	−0.421374	+	0.906887i	$0.638452\pi$
$762$	0	0
$763$	0.123327	0.00446473
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.39930	−0.122742
$768$	0	0
$769$	−48.6997	−1.75615	−0.878077	−	0.478519i	$-0.841174\pi$
$769$	−48.6997	−1.75615	−0.878077	+	0.478519i	$0.841174\pi$
$770$	0	0
$771$	9.37335	0.337573
$772$	0	0
$773$	15.6128	0.561554	0.280777	−	0.959773i	$-0.409408\pi$
$773$	15.6128	0.561554	0.280777	+	0.959773i	$0.409408\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.597107	−0.0214211
$778$	0	0
$779$	−10.1068	−0.362112
$780$	0	0
$781$	−40.2560	−1.44047
$782$	0	0
$783$	−42.9427	−1.53465
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24.2630	0.864883	0.432441	−	0.901662i	$-0.357652\pi$
$787$	24.2630	0.864883	0.432441	+	0.901662i	$0.357652\pi$
$788$	0	0
$789$	−16.4141	−0.584356
$790$	0	0
$791$	−2.10497	−0.0748440
$792$	0	0
$793$	38.3498	1.36184
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−44.5911	−1.57950	−0.789749	−	0.613430i	$-0.789789\pi$
$797$	−44.5911	−1.57950	−0.789749	+	0.613430i	$0.789789\pi$
$798$	0	0
$799$	−23.3003	−0.824307
$800$	0	0
$801$	1.08323	0.0382742
$802$	0	0
$803$	37.2552	1.31471
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−21.8420	−0.768874
$808$	0	0
$809$	−44.7144	−1.57208	−0.786038	−	0.618179i	$-0.787871\pi$
$809$	−44.7144	−1.57208	−0.786038	+	0.618179i	$0.787871\pi$
$810$	0	0
$811$	−1.19179	−0.0418495	−0.0209247	−	0.999781i	$-0.506661\pi$
$811$	−1.19179	−0.0418495	−0.0209247	+	0.999781i	$0.506661\pi$
$812$	0	0
$813$	−0.507859	−0.0178114
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−9.82635	−0.343780
$818$	0	0
$819$	1.41587	0.0494744
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	26.7551	0.932626	0.466313	−	0.884620i	$-0.345582\pi$
$823$	26.7551	0.932626	0.466313	+	0.884620i	$0.345582\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−27.2907	−0.948992	−0.474496	−	0.880258i	$-0.657370\pi$
$827$	−27.2907	−0.948992	−0.474496	+	0.880258i	$0.657370\pi$
$828$	0	0
$829$	34.2925	1.19103	0.595515	−	0.803344i	$-0.296948\pi$
$829$	34.2925	1.19103	0.595515	+	0.803344i	$0.296948\pi$
$830$	0	0
$831$	8.35763	0.289923
$832$	0	0
$833$	−15.5651	−0.539300
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9.74396	−0.336801
$838$	0	0
$839$	40.6701	1.40409	0.702044	−	0.712133i	$-0.252271\pi$
$839$	40.6701	1.40409	0.702044	+	0.712133i	$0.252271\pi$
$840$	0	0
$841$	58.4192	2.01446
$842$	0	0
$843$	−11.5573	−0.398056
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.329889	−0.0113351
$848$	0	0
$849$	21.5278	0.738832
$850$	0	0
$851$	−5.59289	−0.191722
$852$	0	0
$853$	34.6276	1.18563	0.592813	−	0.805340i	$-0.298017\pi$
$853$	34.6276	1.18563	0.592813	+	0.805340i	$0.298017\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26.4861	−0.904749	−0.452374	−	0.891828i	$-0.649423\pi$
$857$	−26.4861	−0.904749	−0.452374	+	0.891828i	$0.649423\pi$
$858$	0	0
$859$	−40.9627	−1.39763	−0.698814	−	0.715304i	$-0.746288\pi$
$859$	−40.9627	−1.39763	−0.698814	+	0.715304i	$0.746288\pi$
$860$	0	0
$861$	0.878468	0.0299381
$862$	0	0
$863$	−18.5712	−0.632170	−0.316085	−	0.948731i	$-0.602368\pi$
$863$	−18.5712	−0.632170	−0.316085	+	0.948731i	$0.602368\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−10.5721	−0.359048
$868$	0	0
$869$	−10.1146	−0.343113
$870$	0	0
$871$	−29.2413	−0.990803
$872$	0	0
$873$	1.41587	0.0479199
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28.4913	0.962083	0.481041	−	0.876698i	$-0.340259\pi$
$877$	28.4913	0.962083	0.481041	+	0.876698i	$0.340259\pi$
$878$	0	0
$879$	5.52780	0.186448
$880$	0	0
$881$	−4.91318	−0.165529	−0.0827645	−	0.996569i	$-0.526375\pi$
$881$	−4.91318	−0.165529	−0.0827645	+	0.996569i	$0.526375\pi$
$882$	0	0
$883$	30.1710	1.01534	0.507668	−	0.861553i	$-0.330508\pi$
$883$	30.1710	1.01534	0.507668	+	0.861553i	$0.330508\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.7865	0.865825	0.432913	−	0.901436i	$-0.357486\pi$
$887$	25.7865	0.865825	0.432913	+	0.901436i	$0.357486\pi$
$888$	0	0
$889$	0.213524	0.00716136
$890$	0	0
$891$	−7.62844	−0.255562
$892$	0	0
$893$	12.8438	0.429800
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.59289	−0.153352
$898$	0	0
$899$	19.8360	0.661566
$900$	0	0
$901$	8.00601	0.266719
$902$	0	0
$903$	0.854095	0.0284225
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−44.3776	−1.47353	−0.736767	−	0.676147i	$-0.763648\pi$
$907$	−44.3776	−1.47353	−0.736767	+	0.676147i	$0.763648\pi$
$908$	0	0
$909$	−29.7473	−0.986657
$910$	0	0
$911$	−47.7968	−1.58358	−0.791789	−	0.610794i	$-0.790850\pi$
$911$	−47.7968	−1.58358	−0.791789	+	0.610794i	$0.790850\pi$
$912$	0	0
$913$	31.9705	1.05807
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.45659	0.0811237
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	11.7144	0.386003
$922$	0	0
$923$	−73.1189	−2.40674
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.3447	0.536829
$928$	0	0
$929$	24.0199	0.788069	0.394034	−	0.919096i	$-0.371079\pi$
$929$	24.0199	0.788069	0.394034	+	0.919096i	$0.371079\pi$
$930$	0	0
$931$	8.57991	0.281195
$932$	0	0
$933$	21.4584	0.702516
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.384533	−0.0125621	−0.00628107	−	0.999980i	$-0.501999\pi$
$937$	−0.384533	−0.0125621	−0.00628107	+	0.999980i	$0.501999\pi$
$938$	0	0
$939$	−23.3221	−0.761087
$940$	0	0
$941$	25.0920	0.817976	0.408988	−	0.912540i	$-0.365882\pi$
$941$	25.0920	0.817976	0.408988	+	0.912540i	$0.365882\pi$
$942$	0	0
$943$	8.22829	0.267950
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.7126	1.45297	0.726483	−	0.687185i	$-0.241154\pi$
$947$	44.7126	1.45297	0.726483	+	0.687185i	$0.241154\pi$
$948$	0	0
$949$	67.6683	2.19661
$950$	0	0
$951$	14.9210	0.483846
$952$	0	0
$953$	21.2500	0.688356	0.344178	−	0.938904i	$-0.388158\pi$
$953$	21.2500	0.688356	0.344178	+	0.938904i	$0.388158\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−23.6424	−0.764249
$958$	0	0
$959$	2.17882	0.0703577
$960$	0	0
$961$	−26.4991	−0.854810
$962$	0	0
$963$	−35.7629	−1.15244
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−41.0417	−1.31981	−0.659906	−	0.751349i	$-0.729404\pi$
$967$	−41.0417	−1.31981	−0.659906	+	0.751349i	$0.729404\pi$
$968$	0	0
$969$	−2.40437	−0.0772394
$970$	0	0
$971$	−35.2760	−1.13206	−0.566030	−	0.824385i	$-0.691521\pi$
$971$	−35.2760	−1.13206	−0.566030	+	0.824385i	$0.691521\pi$
$972$	0	0
$973$	1.32029	0.0423264
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.79164	0.217284	0.108642	−	0.994081i	$-0.465350\pi$
$977$	6.79164	0.217284	0.108642	+	0.994081i	$0.465350\pi$
$978$	0	0
$979$	1.39930	0.0447219
$980$	0	0
$981$	−2.26121	−0.0721947
$982$	0	0
$983$	19.5981	0.625081	0.312540	−	0.949904i	$-0.398820\pi$
$983$	19.5981	0.625081	0.312540	+	0.949904i	$0.398820\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.11636	−0.0355343
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	37.2995	1.18486	0.592429	−	0.805623i	$-0.298169\pi$
$991$	37.2995	1.18486	0.592429	+	0.805623i	$0.298169\pi$
$992$	0	0
$993$	−15.7171	−0.498766
$994$	0	0
$995$	0	0
$996$	0	0
$997$	19.7274	0.624773	0.312386	−	0.949955i	$-0.398872\pi$
$997$	19.7274	0.624773	0.312386	+	0.949955i	$0.398872\pi$
$998$	0	0
$999$	25.6875	0.812717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.ce.1.2		3
4.3	odd	2		4600.2.a.x.1.2		3
5.4	even	2		1840.2.a.s.1.2		3
20.3	even	4		4600.2.e.p.4049.3		6
20.7	even	4		4600.2.e.p.4049.4		6
20.19	odd	2		920.2.a.h.1.2	✓	3
40.19	odd	2		7360.2.a.by.1.2		3
40.29	even	2		7360.2.a.cc.1.2		3
60.59	even	2		8280.2.a.bj.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.h.1.2	✓	3	20.19	odd	2
1840.2.a.s.1.2		3	5.4	even	2
4600.2.a.x.1.2		3	4.3	odd	2
4600.2.e.p.4049.3		6	20.3	even	4
4600.2.e.p.4049.4		6	20.7	even	4
7360.2.a.by.1.2		3	40.19	odd	2
7360.2.a.cc.1.2		3	40.29	even	2
8280.2.a.bj.1.2		3	60.59	even	2
9200.2.a.ce.1.2		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+0.878468 q^{3} +0.121532 q^{7} -2.22829 q^{9} -2.87847 q^{11} -5.22829 q^{13} +2.22829 q^{17} -1.22829 q^{19} +0.106762 q^{21} +1.00000 q^{23} -4.59289 q^{27} +9.34983 q^{29} +2.12153 q^{31} -2.52864 q^{33} -5.59289 q^{37} -4.59289 q^{39} +8.22829 q^{41} +8.00000 q^{43} -10.4566 q^{47} -6.98523 q^{49} +1.95749 q^{51} +3.59289 q^{53} -1.07902 q^{57} +0.650174 q^{59} -7.33506 q^{61} -0.270809 q^{63} +5.59289 q^{67} +0.878468 q^{69} +13.9852 q^{71} -12.9427 q^{73} -0.349826 q^{77} +3.51387 q^{79} +2.65017 q^{81} -11.1068 q^{83} +8.21352 q^{87} -0.486128 q^{89} -0.635404 q^{91} +1.86370 q^{93} -0.635404 q^{97} +6.41407 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 2 q^{7} + 10 q^{9} - 7 q^{11} + q^{13} - 10 q^{17} + 13 q^{19} - 18 q^{21} + 3 q^{23} - 2 q^{27} + 13 q^{29} + 8 q^{31} - 21 q^{33} - 5 q^{37} - 2 q^{39} + 8 q^{41} + 24 q^{43} + 2 q^{47}+ \cdots - 21 q^{99}+O(q^{100}) \) 3 * q + q^3 + 2 * q^7 + 10 * q^9 - 7 * q^11 + q^13 - 10 * q^17 + 13 * q^19 - 18 * q^21 + 3 * q^23 - 2 * q^27 + 13 * q^29 + 8 * q^31 - 21 * q^33 - 5 * q^37 - 2 * q^39 + 8 * q^41 + 24 * q^43 + 2 * q^47 - q^49 - q^51 - q^53 + 2 * q^57 + 17 * q^59 + 13 * q^61 + 9 * q^63 + 5 * q^67 + q^69 + 22 * q^71 - 12 * q^73 + 14 * q^77 + 4 * q^79 + 23 * q^81 - 15 * q^83 - 12 * q^87 - 8 * q^89 + 3 * q^91 - 16 * q^93 + 3 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more