LMFDB - Embedded newform 400.6.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.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:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{129}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 32 $ x^2 - x - 32
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.17891$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.7156 q^{3} +94.1469 q^{7} +36.4124 q^{9} +O(q^{10})$	$q+16.7156 q^{3} +94.1469 q^{7} +36.4124 q^{9} -143.706 q^{11} -421.412 q^{13} -1982.11 q^{17} +1317.76 q^{19} +1573.73 q^{21} -4020.02 q^{23} -3453.24 q^{27} -6417.28 q^{29} +2350.64 q^{31} -2402.14 q^{33} +7876.58 q^{37} -7044.18 q^{39} +15081.6 q^{41} +1141.40 q^{43} -21557.3 q^{47} -7943.36 q^{49} -33132.3 q^{51} -9560.44 q^{53} +22027.2 q^{57} -42740.7 q^{59} +32132.1 q^{61} +3428.11 q^{63} -30371.4 q^{67} -67197.2 q^{69} -36006.7 q^{71} +63438.0 q^{73} -13529.5 q^{77} +89922.8 q^{79} -66571.4 q^{81} +38211.2 q^{83} -107269. q^{87} +5745.69 q^{89} -39674.7 q^{91} +39292.5 q^{93} -178780. q^{97} -5232.69 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{3} + 52 q^{7} + 618 q^{9}+O(q^{10}) $ 2 * q - 12 * q^3 + 52 * q^7 + 618 * q^9	$ 2 q - 12 q^{3} + 52 q^{7} + 618 q^{9} - 560 q^{11} - 1388 q^{13} - 148 q^{17} + 1000 q^{19} + 2784 q^{21} - 2452 q^{23} - 13176 q^{27} + 1340 q^{29} + 2248 q^{31} + 9552 q^{33} + 5940 q^{37} + 20712 q^{39} + 23076 q^{41} + 17684 q^{43} - 2908 q^{47} - 22974 q^{49} - 85800 q^{51} + 5412 q^{53} + 31152 q^{57} - 62584 q^{59} + 14108 q^{61} - 21084 q^{63} - 85412 q^{67} - 112224 q^{69} - 47208 q^{71} + 67452 q^{73} + 4016 q^{77} + 65904 q^{79} + 71298 q^{81} + 108724 q^{83} - 330024 q^{87} - 55020 q^{89} + 1064 q^{91} + 42240 q^{93} - 147668 q^{97} - 247344 q^{99}+O(q^{100}) $ 2 * q - 12 * q^3 + 52 * q^7 + 618 * q^9 - 560 * q^11 - 1388 * q^13 - 148 * q^17 + 1000 * q^19 + 2784 * q^21 - 2452 * q^23 - 13176 * q^27 + 1340 * q^29 + 2248 * q^31 + 9552 * q^33 + 5940 * q^37 + 20712 * q^39 + 23076 * q^41 + 17684 * q^43 - 2908 * q^47 - 22974 * q^49 - 85800 * q^51 + 5412 * q^53 + 31152 * q^57 - 62584 * q^59 + 14108 * q^61 - 21084 * q^63 - 85412 * q^67 - 112224 * q^69 - 47208 * q^71 + 67452 * q^73 + 4016 * q^77 + 65904 * q^79 + 71298 * q^81 + 108724 * q^83 - 330024 * q^87 - 55020 * q^89 + 1064 * q^91 + 42240 * q^93 - 147668 * q^97 - 247344 * 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
$3$	16.7156	1.07231	0.536154	−	0.844120i	$-0.319877\pi$
$3$	16.7156	1.07231	0.536154	+	0.844120i	$0.319877\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	94.1469	0.726208	0.363104	−	0.931749i	$-0.381717\pi$
$7$	94.1469	0.726208	0.363104	+	0.931749i	$0.381717\pi$
$8$	0	0
$9$	36.4124	0.149845
$10$	0	0
$11$	−143.706	−0.358091	−0.179046	−	0.983841i	$-0.557301\pi$
$11$	−143.706	−0.358091	−0.179046	+	0.983841i	$0.557301\pi$
$12$	0	0
$13$	−421.412	−0.691590	−0.345795	−	0.938310i	$-0.612391\pi$
$13$	−421.412	−0.691590	−0.345795	+	0.938310i	$0.612391\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1982.11	−1.66344	−0.831718	−	0.555198i	$-0.812642\pi$
$17$	−1982.11	−1.66344	−0.831718	+	0.555198i	$0.812642\pi$
$18$	0	0
$19$	1317.76	0.837439	0.418720	−	0.908116i	$-0.362479\pi$
$19$	1317.76	0.837439	0.418720	+	0.908116i	$0.362479\pi$
$20$	0	0
$21$	1573.73	0.778719
$22$	0	0
$23$	−4020.02	−1.58456	−0.792280	−	0.610157i	$-0.791106\pi$
$23$	−4020.02	−1.58456	−0.792280	+	0.610157i	$0.791106\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3453.24	−0.911628
$28$	0	0
$29$	−6417.28	−1.41695	−0.708477	−	0.705734i	$-0.750617\pi$
$29$	−6417.28	−1.41695	−0.708477	+	0.705734i	$0.750617\pi$
$30$	0	0
$31$	2350.64	0.439322	0.219661	−	0.975576i	$-0.429505\pi$
$31$	2350.64	0.439322	0.219661	+	0.975576i	$0.429505\pi$
$32$	0	0
$33$	−2402.14	−0.383984
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7876.58	0.945874	0.472937	−	0.881096i	$-0.343194\pi$
$37$	7876.58	0.945874	0.472937	+	0.881096i	$0.343194\pi$
$38$	0	0
$39$	−7044.18	−0.741598
$40$	0	0
$41$	15081.6	1.40116	0.700582	−	0.713572i	$-0.252924\pi$
$41$	15081.6	1.40116	0.700582	+	0.713572i	$0.252924\pi$
$42$	0	0
$43$	1141.40	0.0941384	0.0470692	−	0.998892i	$-0.485012\pi$
$43$	1141.40	0.0941384	0.0470692	+	0.998892i	$0.485012\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−21557.3	−1.42348	−0.711738	−	0.702445i	$-0.752092\pi$
$47$	−21557.3	−1.42348	−0.711738	+	0.702445i	$0.752092\pi$
$48$	0	0
$49$	−7943.36	−0.472622
$50$	0	0
$51$	−33132.3	−1.78372
$52$	0	0
$53$	−9560.44	−0.467507	−0.233754	−	0.972296i	$-0.575101\pi$
$53$	−9560.44	−0.467507	−0.233754	+	0.972296i	$0.575101\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	22027.2	0.897993
$58$	0	0
$59$	−42740.7	−1.59850	−0.799248	−	0.601002i	$-0.794768\pi$
$59$	−42740.7	−1.59850	−0.799248	+	0.601002i	$0.794768\pi$
$60$	0	0
$61$	32132.1	1.10564	0.552820	−	0.833301i	$-0.313552\pi$
$61$	32132.1	1.10564	0.552820	+	0.833301i	$0.313552\pi$
$62$	0	0
$63$	3428.11	0.108819
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−30371.4	−0.826567	−0.413283	−	0.910602i	$-0.635618\pi$
$67$	−30371.4	−0.826567	−0.413283	+	0.910602i	$0.635618\pi$
$68$	0	0
$69$	−67197.2	−1.69914
$70$	0	0
$71$	−36006.7	−0.847692	−0.423846	−	0.905734i	$-0.639320\pi$
$71$	−36006.7	−0.847692	−0.423846	+	0.905734i	$0.639320\pi$
$72$	0	0
$73$	63438.0	1.39329	0.696647	−	0.717414i	$-0.254674\pi$
$73$	63438.0	1.39329	0.696647	+	0.717414i	$0.254674\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13529.5	−0.260049
$78$	0	0
$79$	89922.8	1.62107	0.810536	−	0.585689i	$-0.199176\pi$
$79$	89922.8	1.62107	0.810536	+	0.585689i	$0.199176\pi$
$80$	0	0
$81$	−66571.4	−1.12739
$82$	0	0
$83$	38211.2	0.608829	0.304414	−	0.952540i	$-0.401539\pi$
$83$	38211.2	0.608829	0.304414	+	0.952540i	$0.401539\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−107269.	−1.51941
$88$	0	0
$89$	5745.69	0.0768895	0.0384447	−	0.999261i	$-0.487760\pi$
$89$	5745.69	0.0768895	0.0384447	+	0.999261i	$0.487760\pi$
$90$	0	0
$91$	−39674.7	−0.502238
$92$	0	0
$93$	39292.5	0.471088
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−178780.	−1.92926	−0.964629	−	0.263613i	$-0.915086\pi$
$97$	−178780.	−1.92926	−0.964629	+	0.263613i	$0.915086\pi$
$98$	0	0
$99$	−5232.69	−0.0536583
$100$	0	0
$101$	152223.	1.48483	0.742417	−	0.669938i	$-0.233679\pi$
$101$	152223.	1.48483	0.742417	+	0.669938i	$0.233679\pi$
$102$	0	0
$103$	−35830.1	−0.332778	−0.166389	−	0.986060i	$-0.553211\pi$
$103$	−35830.1	−0.332778	−0.166389	+	0.986060i	$0.553211\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	70030.2	0.591324	0.295662	−	0.955293i	$-0.404460\pi$
$107$	70030.2	0.591324	0.295662	+	0.955293i	$0.404460\pi$
$108$	0	0
$109$	38466.9	0.310114	0.155057	−	0.987906i	$-0.450444\pi$
$109$	38466.9	0.310114	0.155057	+	0.987906i	$0.450444\pi$
$110$	0	0
$111$	131662.	1.01427
$112$	0	0
$113$	−39951.6	−0.294333	−0.147166	−	0.989112i	$-0.547015\pi$
$113$	−39951.6	−0.294333	−0.147166	+	0.989112i	$0.547015\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−15344.6	−0.103632
$118$	0	0
$119$	−186610.	−1.20800
$120$	0	0
$121$	−140400.	−0.871771
$122$	0	0
$123$	252099.	1.50248
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−233239.	−1.28319	−0.641596	−	0.767043i	$-0.721727\pi$
$127$	−233239.	−1.28319	−0.641596	+	0.767043i	$0.721727\pi$
$128$	0	0
$129$	19079.2	0.100945
$130$	0	0
$131$	−55237.0	−0.281224	−0.140612	−	0.990065i	$-0.544907\pi$
$131$	−55237.0	−0.281224	−0.140612	+	0.990065i	$0.544907\pi$
$132$	0	0
$133$	124063.	0.608155
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−261520.	−1.19043	−0.595215	−	0.803566i	$-0.702933\pi$
$137$	−261520.	−1.19043	−0.595215	+	0.803566i	$0.702933\pi$
$138$	0	0
$139$	−293366.	−1.28787	−0.643935	−	0.765080i	$-0.722699\pi$
$139$	−293366.	−1.28787	−0.643935	+	0.765080i	$0.722699\pi$
$140$	0	0
$141$	−360345.	−1.52641
$142$	0	0
$143$	60559.6	0.247653
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−132778.	−0.506797
$148$	0	0
$149$	−304505.	−1.12364	−0.561822	−	0.827258i	$-0.689899\pi$
$149$	−304505.	−1.12364	−0.561822	+	0.827258i	$0.689899\pi$
$150$	0	0
$151$	337909.	1.20603	0.603015	−	0.797730i	$-0.293966\pi$
$151$	337909.	1.20603	0.603015	+	0.797730i	$0.293966\pi$
$152$	0	0
$153$	−72173.5	−0.249258
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−68385.9	−0.221420	−0.110710	−	0.993853i	$-0.535313\pi$
$157$	−68385.9	−0.221420	−0.110710	+	0.993853i	$0.535313\pi$
$158$	0	0
$159$	−159809.	−0.501312
$160$	0	0
$161$	−378473.	−1.15072
$162$	0	0
$163$	−404471.	−1.19239	−0.596195	−	0.802840i	$-0.703322\pi$
$163$	−404471.	−1.19239	−0.596195	+	0.802840i	$0.703322\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	411733.	1.14242	0.571209	−	0.820805i	$-0.306475\pi$
$167$	411733.	1.14242	0.571209	+	0.820805i	$0.306475\pi$
$168$	0	0
$169$	−193705.	−0.521703
$170$	0	0
$171$	47982.9	0.125486
$172$	0	0
$173$	162247.	0.412155	0.206077	−	0.978536i	$-0.433930\pi$
$173$	162247.	0.412155	0.206077	+	0.978536i	$0.433930\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−714438.	−1.71408
$178$	0	0
$179$	−384396.	−0.896698	−0.448349	−	0.893859i	$-0.647988\pi$
$179$	−384396.	−0.896698	−0.448349	+	0.893859i	$0.647988\pi$
$180$	0	0
$181$	579219.	1.31415	0.657077	−	0.753823i	$-0.271792\pi$
$181$	579219.	1.31415	0.657077	+	0.753823i	$0.271792\pi$
$182$	0	0
$183$	537108.	1.18559
$184$	0	0
$185$	0	0
$186$	0	0
$187$	284842.	0.595662
$188$	0	0
$189$	−325112.	−0.662031
$190$	0	0
$191$	−142455.	−0.282550	−0.141275	−	0.989970i	$-0.545120\pi$
$191$	−142455.	−0.282550	−0.141275	+	0.989970i	$0.545120\pi$
$192$	0	0
$193$	−267272.	−0.516487	−0.258244	−	0.966080i	$-0.583144\pi$
$193$	−267272.	−0.516487	−0.258244	+	0.966080i	$0.583144\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14189.5	−0.0260496	−0.0130248	−	0.999915i	$-0.504146\pi$
$197$	−14189.5	−0.0260496	−0.0130248	+	0.999915i	$0.504146\pi$
$198$	0	0
$199$	−169198.	−0.302875	−0.151437	−	0.988467i	$-0.548390\pi$
$199$	−169198.	−0.302875	−0.151437	+	0.988467i	$0.548390\pi$
$200$	0	0
$201$	−507677.	−0.886335
$202$	0	0
$203$	−604167.	−1.02900
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−146379.	−0.237439
$208$	0	0
$209$	−189371.	−0.299880
$210$	0	0
$211$	407591.	0.630259	0.315129	−	0.949049i	$-0.397952\pi$
$211$	407591.	0.630259	0.315129	+	0.949049i	$0.397952\pi$
$212$	0	0
$213$	−601875.	−0.908987
$214$	0	0
$215$	0	0
$216$	0	0
$217$	221306.	0.319039
$218$	0	0
$219$	1.06041e6	1.49404
$220$	0	0
$221$	835287.	1.15042
$222$	0	0
$223$	−103743.	−0.139700	−0.0698500	−	0.997558i	$-0.522252\pi$
$223$	−103743.	−0.139700	−0.0698500	+	0.997558i	$0.522252\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−803726.	−1.03524	−0.517622	−	0.855609i	$-0.673183\pi$
$227$	−803726.	−1.03524	−0.517622	+	0.855609i	$0.673183\pi$
$228$	0	0
$229$	1.35955e6	1.71319	0.856596	−	0.515988i	$-0.172575\pi$
$229$	1.35955e6	1.71319	0.856596	+	0.515988i	$0.172575\pi$
$230$	0	0
$231$	−226154.	−0.278852
$232$	0	0
$233$	622827.	0.751584	0.375792	−	0.926704i	$-0.377371\pi$
$233$	622827.	0.751584	0.375792	+	0.926704i	$0.377371\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.50312e6	1.73829
$238$	0	0
$239$	642843.	0.727964	0.363982	−	0.931406i	$-0.381417\pi$
$239$	642843.	0.727964	0.363982	+	0.931406i	$0.381417\pi$
$240$	0	0
$241$	1.11814e6	1.24009	0.620046	−	0.784566i	$-0.287114\pi$
$241$	1.11814e6	1.24009	0.620046	+	0.784566i	$0.287114\pi$
$242$	0	0
$243$	−273644.	−0.297283
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−555322.	−0.579165
$248$	0	0
$249$	638724.	0.652852
$250$	0	0
$251$	−731067.	−0.732442	−0.366221	−	0.930528i	$-0.619349\pi$
$251$	−731067.	−0.732442	−0.366221	+	0.930528i	$0.619349\pi$
$252$	0	0
$253$	577702.	0.567418
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.76622e6	1.66806	0.834030	−	0.551720i	$-0.186028\pi$
$257$	1.76622e6	1.66806	0.834030	+	0.551720i	$0.186028\pi$
$258$	0	0
$259$	741555.	0.686901
$260$	0	0
$261$	−233668.	−0.212324
$262$	0	0
$263$	1.00926e6	0.899735	0.449868	−	0.893095i	$-0.351471\pi$
$263$	1.00926e6	0.899735	0.449868	+	0.893095i	$0.351471\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	96042.8	0.0824492
$268$	0	0
$269$	−775681.	−0.653585	−0.326793	−	0.945096i	$-0.605968\pi$
$269$	−775681.	−0.653585	−0.326793	+	0.945096i	$0.605968\pi$
$270$	0	0
$271$	−1.21395e6	−1.00410	−0.502052	−	0.864837i	$-0.667422\pi$
$271$	−1.21395e6	−1.00410	−0.502052	+	0.864837i	$0.667422\pi$
$272$	0	0
$273$	−663187.	−0.538554
$274$	0	0
$275$	0	0
$276$	0	0
$277$	218505.	0.171104	0.0855522	−	0.996334i	$-0.472735\pi$
$277$	218505.	0.171104	0.0855522	+	0.996334i	$0.472735\pi$
$278$	0	0
$279$	85592.6	0.0658303
$280$	0	0
$281$	−316219.	−0.238903	−0.119452	−	0.992840i	$-0.538114\pi$
$281$	−316219.	−0.238903	−0.119452	+	0.992840i	$0.538114\pi$
$282$	0	0
$283$	−927934.	−0.688733	−0.344366	−	0.938835i	$-0.611906\pi$
$283$	−927934.	−0.688733	−0.344366	+	0.938835i	$0.611906\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.41989e6	1.01754
$288$	0	0
$289$	2.50892e6	1.76702
$290$	0	0
$291$	−2.98842e6	−2.06876
$292$	0	0
$293$	−262992.	−0.178967	−0.0894835	−	0.995988i	$-0.528522\pi$
$293$	−262992.	−0.178967	−0.0894835	+	0.995988i	$0.528522\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	496252.	0.326446
$298$	0	0
$299$	1.69409e6	1.09587
$300$	0	0
$301$	107459.	0.0683640
$302$	0	0
$303$	2.54451e6	1.59220
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.15704e6	1.30621	0.653104	−	0.757269i	$-0.273467\pi$
$307$	2.15704e6	1.30621	0.653104	+	0.757269i	$0.273467\pi$
$308$	0	0
$309$	−598923.	−0.356841
$310$	0	0
$311$	2.33078e6	1.36647	0.683235	−	0.730199i	$-0.260573\pi$
$311$	2.33078e6	1.36647	0.683235	+	0.730199i	$0.260573\pi$
$312$	0	0
$313$	1.12235e6	0.647539	0.323769	−	0.946136i	$-0.395050\pi$
$313$	1.12235e6	0.647539	0.323769	+	0.946136i	$0.395050\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.27955e6	0.715169	0.357584	−	0.933881i	$-0.383600\pi$
$317$	1.27955e6	0.715169	0.357584	+	0.933881i	$0.383600\pi$
$318$	0	0
$319$	922203.	0.507399
$320$	0	0
$321$	1.17060e6	0.634082
$322$	0	0
$323$	−2.61196e6	−1.39303
$324$	0	0
$325$	0	0
$326$	0	0
$327$	642998.	0.332537
$328$	0	0
$329$	−2.02956e6	−1.03374
$330$	0	0
$331$	2.09571e6	1.05138	0.525691	−	0.850675i	$-0.323807\pi$
$331$	2.09571e6	1.05138	0.525691	+	0.850675i	$0.323807\pi$
$332$	0	0
$333$	286805.	0.141735
$334$	0	0
$335$	0	0
$336$	0	0
$337$	571289.	0.274019	0.137010	−	0.990570i	$-0.456251\pi$
$337$	571289.	0.274019	0.137010	+	0.990570i	$0.456251\pi$
$338$	0	0
$339$	−667817.	−0.315615
$340$	0	0
$341$	−337802.	−0.157317
$342$	0	0
$343$	−2.33017e6	−1.06943
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.24067e6	0.998973	0.499486	−	0.866322i	$-0.333522\pi$
$347$	2.24067e6	0.998973	0.499486	+	0.866322i	$0.333522\pi$
$348$	0	0
$349$	130448.	0.0573290	0.0286645	−	0.999589i	$-0.490875\pi$
$349$	130448.	0.0573290	0.0286645	+	0.999589i	$0.490875\pi$
$350$	0	0
$351$	1.45524e6	0.630473
$352$	0	0
$353$	−1.95452e6	−0.834839	−0.417420	−	0.908714i	$-0.637065\pi$
$353$	−1.95452e6	−0.834839	−0.417420	+	0.908714i	$0.637065\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.11930e6	−1.29535
$358$	0	0
$359$	850543.	0.348305	0.174153	−	0.984719i	$-0.444281\pi$
$359$	850543.	0.348305	0.174153	+	0.984719i	$0.444281\pi$
$360$	0	0
$361$	−739600.	−0.298696
$362$	0	0
$363$	−2.34687e6	−0.934807
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.50673e6	−0.583941	−0.291970	−	0.956427i	$-0.594311\pi$
$367$	−1.50673e6	−0.583941	−0.291970	+	0.956427i	$0.594311\pi$
$368$	0	0
$369$	549159.	0.209958
$370$	0	0
$371$	−900086.	−0.339507
$372$	0	0
$373$	−2.90602e6	−1.08150	−0.540749	−	0.841184i	$-0.681859\pi$
$373$	−2.90602e6	−1.08150	−0.540749	+	0.841184i	$0.681859\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.70432e6	0.979952
$378$	0	0
$379$	−5.16710e6	−1.84777	−0.923887	−	0.382665i	$-0.875006\pi$
$379$	−5.16710e6	−1.84777	−0.923887	+	0.382665i	$0.875006\pi$
$380$	0	0
$381$	−3.89874e6	−1.37598
$382$	0	0
$383$	3.85088e6	1.34142	0.670708	−	0.741721i	$-0.265990\pi$
$383$	3.85088e6	1.34142	0.670708	+	0.741721i	$0.265990\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	41561.1	0.0141062
$388$	0	0
$389$	5.49855e6	1.84236	0.921179	−	0.389138i	$-0.127227\pi$
$389$	5.49855e6	1.84236	0.921179	+	0.389138i	$0.127227\pi$
$390$	0	0
$391$	7.96814e6	2.63582
$392$	0	0
$393$	−923321.	−0.301558
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.71425e6	−0.545881	−0.272941	−	0.962031i	$-0.587996\pi$
$397$	−1.71425e6	−0.545881	−0.272941	+	0.962031i	$0.587996\pi$
$398$	0	0
$399$	2.07380e6	0.652130
$400$	0	0
$401$	3.63329e6	1.12834	0.564169	−	0.825660i	$-0.309197\pi$
$401$	3.63329e6	1.12834	0.564169	+	0.825660i	$0.309197\pi$
$402$	0	0
$403$	−990591.	−0.303831
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.13191e6	−0.338709
$408$	0	0
$409$	−1.38246e6	−0.408642	−0.204321	−	0.978904i	$-0.565499\pi$
$409$	−1.38246e6	−0.408642	−0.204321	+	0.978904i	$0.565499\pi$
$410$	0	0
$411$	−4.37148e6	−1.27651
$412$	0	0
$413$	−4.02390e6	−1.16084
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.90379e6	−1.38099
$418$	0	0
$419$	−2.55435e6	−0.710795	−0.355398	−	0.934715i	$-0.615655\pi$
$419$	−2.55435e6	−0.710795	−0.355398	+	0.934715i	$0.615655\pi$
$420$	0	0
$421$	−1.68110e6	−0.462263	−0.231132	−	0.972923i	$-0.574243\pi$
$421$	−1.68110e6	−0.462263	−0.231132	+	0.972923i	$0.574243\pi$
$422$	0	0
$423$	−784954.	−0.213301
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.02513e6	0.802925
$428$	0	0
$429$	1.01229e6	0.265560
$430$	0	0
$431$	2.32369e6	0.602539	0.301269	−	0.953539i	$-0.402590\pi$
$431$	2.32369e6	0.602539	0.301269	+	0.953539i	$0.402590\pi$
$432$	0	0
$433$	4.06439e6	1.04178	0.520890	−	0.853624i	$-0.325600\pi$
$433$	4.06439e6	1.04178	0.520890	+	0.853624i	$0.325600\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.29744e6	−1.32697
$438$	0	0
$439$	−6.39272e6	−1.58316	−0.791579	−	0.611066i	$-0.790741\pi$
$439$	−6.39272e6	−1.58316	−0.791579	+	0.611066i	$0.790741\pi$
$440$	0	0
$441$	−289237.	−0.0708202
$442$	0	0
$443$	3.23515e6	0.783222	0.391611	−	0.920131i	$-0.371918\pi$
$443$	3.23515e6	0.783222	0.391611	+	0.920131i	$0.371918\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5.08999e6	−1.20489
$448$	0	0
$449$	−2.74812e6	−0.643310	−0.321655	−	0.946857i	$-0.604239\pi$
$449$	−2.74812e6	−0.643310	−0.321655	+	0.946857i	$0.604239\pi$
$450$	0	0
$451$	−2.16732e6	−0.501745
$452$	0	0
$453$	5.64837e6	1.29324
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.37326e6	1.20350	0.601751	−	0.798684i	$-0.294470\pi$
$457$	5.37326e6	1.20350	0.601751	+	0.798684i	$0.294470\pi$
$458$	0	0
$459$	6.84472e6	1.51644
$460$	0	0
$461$	−8.19928e6	−1.79690	−0.898449	−	0.439078i	$-0.855305\pi$
$461$	−8.19928e6	−1.79690	−0.898449	+	0.439078i	$0.855305\pi$
$462$	0	0
$463$	3.76963e6	0.817233	0.408617	−	0.912706i	$-0.366011\pi$
$463$	3.76963e6	0.817233	0.408617	+	0.912706i	$0.366011\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.03412e6	1.28033	0.640165	−	0.768237i	$-0.278866\pi$
$467$	6.03412e6	1.28033	0.640165	+	0.768237i	$0.278866\pi$
$468$	0	0
$469$	−2.85937e6	−0.600259
$470$	0	0
$471$	−1.14311e6	−0.237431
$472$	0	0
$473$	−164026.	−0.0337101
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−348119.	−0.0700537
$478$	0	0
$479$	5.26406e6	1.04829	0.524146	−	0.851629i	$-0.324385\pi$
$479$	5.26406e6	1.04829	0.524146	+	0.851629i	$0.324385\pi$
$480$	0	0
$481$	−3.31929e6	−0.654157
$482$	0	0
$483$	−6.32641e6	−1.23393
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.19979e6	−0.420300	−0.210150	−	0.977669i	$-0.567395\pi$
$487$	−2.19979e6	−0.420300	−0.210150	+	0.977669i	$0.567395\pi$
$488$	0	0
$489$	−6.76099e6	−1.27861
$490$	0	0
$491$	−6.47314e6	−1.21174	−0.605872	−	0.795562i	$-0.707176\pi$
$491$	−6.47314e6	−1.21174	−0.605872	+	0.795562i	$0.707176\pi$
$492$	0	0
$493$	1.27198e7	2.35701
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.38992e6	−0.615600
$498$	0	0
$499$	−8.62090e6	−1.54989	−0.774946	−	0.632028i	$-0.782223\pi$
$499$	−8.62090e6	−1.54989	−0.774946	+	0.632028i	$0.782223\pi$
$500$	0	0
$501$	6.88238e6	1.22502
$502$	0	0
$503$	8.10557e6	1.42845	0.714223	−	0.699918i	$-0.246780\pi$
$503$	8.10557e6	1.42845	0.714223	+	0.699918i	$0.246780\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.23789e6	−0.559426
$508$	0	0
$509$	−6.39610e6	−1.09426	−0.547131	−	0.837047i	$-0.684280\pi$
$509$	−6.39610e6	−1.09426	−0.547131	+	0.837047i	$0.684280\pi$
$510$	0	0
$511$	5.97250e6	1.01182
$512$	0	0
$513$	−4.55055e6	−0.763433
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.09792e6	0.509735
$518$	0	0
$519$	2.71205e6	0.441957
$520$	0	0
$521$	7.53647e6	1.21639	0.608197	−	0.793786i	$-0.291893\pi$
$521$	7.53647e6	1.21639	0.608197	+	0.793786i	$0.291893\pi$
$522$	0	0
$523$	1.87780e6	0.300189	0.150095	−	0.988672i	$-0.452042\pi$
$523$	1.87780e6	0.300189	0.150095	+	0.988672i	$0.452042\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.65924e6	−0.730784
$528$	0	0
$529$	9.72424e6	1.51083
$530$	0	0
$531$	−1.55629e6	−0.239527
$532$	0	0
$533$	−6.35559e6	−0.969031
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−6.42542e6	−0.961537
$538$	0	0
$539$	1.14151e6	0.169242
$540$	0	0
$541$	2.44482e6	0.359131	0.179566	−	0.983746i	$-0.442531\pi$
$541$	2.44482e6	0.359131	0.179566	+	0.983746i	$0.442531\pi$
$542$	0	0
$543$	9.68202e6	1.40918
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.09485e7	−1.56453	−0.782266	−	0.622944i	$-0.785936\pi$
$547$	−1.09485e7	−1.56453	−0.782266	+	0.622944i	$0.785936\pi$
$548$	0	0
$549$	1.17001e6	0.165675
$550$	0	0
$551$	−8.45645e6	−1.18661
$552$	0	0
$553$	8.46595e6	1.17723
$554$	0	0
$555$	0	0
$556$	0	0
$557$	768938.	0.105015	0.0525077	−	0.998621i	$-0.483279\pi$
$557$	768938.	0.105015	0.0525077	+	0.998621i	$0.483279\pi$
$558$	0	0
$559$	−481000.	−0.0651052
$560$	0	0
$561$	4.76131e6	0.638733
$562$	0	0
$563$	−1.47386e7	−1.95968	−0.979838	−	0.199792i	$-0.935974\pi$
$563$	−1.47386e7	−1.95968	−0.979838	+	0.199792i	$0.935974\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−6.26749e6	−0.818721
$568$	0	0
$569$	−3.33696e6	−0.432086	−0.216043	−	0.976384i	$-0.569315\pi$
$569$	−3.33696e6	−0.432086	−0.216043	+	0.976384i	$0.569315\pi$
$570$	0	0
$571$	−1.54413e7	−1.98195	−0.990974	−	0.134051i	$-0.957201\pi$
$571$	−1.54413e7	−1.98195	−0.990974	+	0.134051i	$0.957201\pi$
$572$	0	0
$573$	−2.38123e6	−0.302981
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.05636e6	0.507221	0.253610	−	0.967306i	$-0.418382\pi$
$577$	4.05636e6	0.507221	0.253610	+	0.967306i	$0.418382\pi$
$578$	0	0
$579$	−4.46762e6	−0.553834
$580$	0	0
$581$	3.59746e6	0.442136
$582$	0	0
$583$	1.37389e6	0.167410
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.22567e7	−1.46818	−0.734090	−	0.679052i	$-0.762391\pi$
$587$	−1.22567e7	−1.46818	−0.734090	+	0.679052i	$0.762391\pi$
$588$	0	0
$589$	3.09759e6	0.367905
$590$	0	0
$591$	−237186.	−0.0279332
$592$	0	0
$593$	−8.58993e6	−1.00312	−0.501560	−	0.865123i	$-0.667240\pi$
$593$	−8.58993e6	−1.00312	−0.501560	+	0.865123i	$0.667240\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.82825e6	−0.324775
$598$	0	0
$599$	−8.85849e6	−1.00877	−0.504386	−	0.863479i	$-0.668281\pi$
$599$	−8.85849e6	−1.00877	−0.504386	+	0.863479i	$0.668281\pi$
$600$	0	0
$601$	−8.50579e6	−0.960569	−0.480284	−	0.877113i	$-0.659467\pi$
$601$	−8.50579e6	−0.960569	−0.480284	+	0.877113i	$0.659467\pi$
$602$	0	0
$603$	−1.10590e6	−0.123857
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.15481e6	−0.347537	−0.173768	−	0.984787i	$-0.555594\pi$
$607$	−3.15481e6	−0.347537	−0.173768	+	0.984787i	$0.555594\pi$
$608$	0	0
$609$	−1.00990e7	−1.10341
$610$	0	0
$611$	9.08453e6	0.984463
$612$	0	0
$613$	−3.20689e6	−0.344694	−0.172347	−	0.985036i	$-0.555135\pi$
$613$	−3.20689e6	−0.344694	−0.172347	+	0.985036i	$0.555135\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.31916e6	0.139504	0.0697519	−	0.997564i	$-0.477779\pi$
$617$	1.31916e6	0.139504	0.0697519	+	0.997564i	$0.477779\pi$
$618$	0	0
$619$	3.36061e6	0.352526	0.176263	−	0.984343i	$-0.443599\pi$
$619$	3.36061e6	0.352526	0.176263	+	0.984343i	$0.443599\pi$
$620$	0	0
$621$	1.38821e7	1.44453
$622$	0	0
$623$	540939.	0.0558377
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.16545e6	−0.321563
$628$	0	0
$629$	−1.56123e7	−1.57340
$630$	0	0
$631$	1.46547e7	1.46522	0.732610	−	0.680648i	$-0.238302\pi$
$631$	1.46547e7	1.46522	0.732610	+	0.680648i	$0.238302\pi$
$632$	0	0
$633$	6.81315e6	0.675832
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.34743e6	0.326861
$638$	0	0
$639$	−1.31109e6	−0.127023
$640$	0	0
$641$	−5.67545e6	−0.545576	−0.272788	−	0.962074i	$-0.587946\pi$
$641$	−5.67545e6	−0.545576	−0.272788	+	0.962074i	$0.587946\pi$
$642$	0	0
$643$	−1.81422e7	−1.73047	−0.865233	−	0.501370i	$-0.832830\pi$
$643$	−1.81422e7	−1.73047	−0.865233	+	0.501370i	$0.832830\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	472390.	0.0443649	0.0221825	−	0.999754i	$-0.492939\pi$
$647$	472390.	0.0443649	0.0221825	+	0.999754i	$0.492939\pi$
$648$	0	0
$649$	6.14210e6	0.572407
$650$	0	0
$651$	3.69927e6	0.342108
$652$	0	0
$653$	−9.79442e6	−0.898867	−0.449434	−	0.893314i	$-0.648374\pi$
$653$	−9.79442e6	−0.898867	−0.449434	+	0.893314i	$0.648374\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.30993e6	0.208778
$658$	0	0
$659$	9.83251e6	0.881964	0.440982	−	0.897516i	$-0.354630\pi$
$659$	9.83251e6	0.881964	0.440982	+	0.897516i	$0.354630\pi$
$660$	0	0
$661$	1.84120e6	0.163907	0.0819535	−	0.996636i	$-0.473884\pi$
$661$	1.84120e6	0.163907	0.0819535	+	0.996636i	$0.473884\pi$
$662$	0	0
$663$	1.39624e7	1.23360
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.57976e7	2.24525
$668$	0	0
$669$	−1.73413e6	−0.149802
$670$	0	0
$671$	−4.61758e6	−0.395920
$672$	0	0
$673$	1.02990e7	0.876510	0.438255	−	0.898851i	$-0.355597\pi$
$673$	1.02990e7	0.876510	0.438255	+	0.898851i	$0.355597\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.63461e6	0.304779	0.152390	−	0.988320i	$-0.451303\pi$
$677$	3.63461e6	0.304779	0.152390	+	0.988320i	$0.451303\pi$
$678$	0	0
$679$	−1.68316e7	−1.40104
$680$	0	0
$681$	−1.34348e7	−1.11010
$682$	0	0
$683$	−9.78443e6	−0.802571	−0.401286	−	0.915953i	$-0.631437\pi$
$683$	−9.78443e6	−0.802571	−0.401286	+	0.915953i	$0.631437\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.27257e7	1.83707
$688$	0	0
$689$	4.02889e6	0.323323
$690$	0	0
$691$	−4.67949e6	−0.372824	−0.186412	−	0.982472i	$-0.559686\pi$
$691$	−4.67949e6	−0.372824	−0.186412	+	0.982472i	$0.559686\pi$
$692$	0	0
$693$	−492641.	−0.0389671
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2.98935e7	−2.33075
$698$	0	0
$699$	1.04110e7	0.805930
$700$	0	0
$701$	−565147.	−0.0434376	−0.0217188	−	0.999764i	$-0.506914\pi$
$701$	−565147.	−0.0434376	−0.0217188	+	0.999764i	$0.506914\pi$
$702$	0	0
$703$	1.03795e7	0.792112
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.43314e7	1.07830
$708$	0	0
$709$	−1.00907e6	−0.0753889	−0.0376944	−	0.999289i	$-0.512001\pi$
$709$	−1.00907e6	−0.0753889	−0.0376944	+	0.999289i	$0.512001\pi$
$710$	0	0
$711$	3.27431e6	0.242910
$712$	0	0
$713$	−9.44964e6	−0.696132
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.07455e7	0.780602
$718$	0	0
$719$	6.02329e6	0.434522	0.217261	−	0.976114i	$-0.430288\pi$
$719$	6.02329e6	0.434522	0.217261	+	0.976114i	$0.430288\pi$
$720$	0	0
$721$	−3.37329e6	−0.241666
$722$	0	0
$723$	1.86904e7	1.32976
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.30828e6	−0.161977	−0.0809883	−	0.996715i	$-0.525808\pi$
$727$	−2.30828e6	−0.161977	−0.0809883	+	0.996715i	$0.525808\pi$
$728$	0	0
$729$	1.16027e7	0.808612
$730$	0	0
$731$	−2.26238e6	−0.156593
$732$	0	0
$733$	1.65224e7	1.13583	0.567914	−	0.823088i	$-0.307751\pi$
$733$	1.65224e7	1.13583	0.567914	+	0.823088i	$0.307751\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.36456e6	0.295986
$738$	0	0
$739$	−2.41051e7	−1.62367	−0.811834	−	0.583888i	$-0.801531\pi$
$739$	−2.41051e7	−1.62367	−0.811834	+	0.583888i	$0.801531\pi$
$740$	0	0
$741$	−9.28255e6	−0.621043
$742$	0	0
$743$	−2.54593e7	−1.69190	−0.845950	−	0.533262i	$-0.820966\pi$
$743$	−2.54593e7	−1.69190	−0.845950	+	0.533262i	$0.820966\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.39136e6	0.0912301
$748$	0	0
$749$	6.59312e6	0.429424
$750$	0	0
$751$	1.25073e7	0.809215	0.404608	−	0.914490i	$-0.367408\pi$
$751$	1.25073e7	0.809215	0.404608	+	0.914490i	$0.367408\pi$
$752$	0	0
$753$	−1.22203e7	−0.785404
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.34914e6	0.339269	0.169635	−	0.985507i	$-0.445741\pi$
$757$	5.34914e6	0.339269	0.169635	+	0.985507i	$0.445741\pi$
$758$	0	0
$759$	9.65666e6	0.608447
$760$	0	0
$761$	−1.80369e7	−1.12901	−0.564507	−	0.825428i	$-0.690934\pi$
$761$	−1.80369e7	−1.12901	−0.564507	+	0.825428i	$0.690934\pi$
$762$	0	0
$763$	3.62154e6	0.225207
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.80115e7	1.10550
$768$	0	0
$769$	2.41928e7	1.47526	0.737632	−	0.675203i	$-0.235944\pi$
$769$	2.41928e7	1.47526	0.737632	+	0.675203i	$0.235944\pi$
$770$	0	0
$771$	2.95234e7	1.78867
$772$	0	0
$773$	3.22531e6	0.194143	0.0970716	−	0.995277i	$-0.469052\pi$
$773$	3.22531e6	0.194143	0.0970716	+	0.995277i	$0.469052\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.23956e7	0.736570
$778$	0	0
$779$	1.98740e7	1.17339
$780$	0	0
$781$	5.17439e6	0.303551
$782$	0	0
$783$	2.21604e7	1.29174
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.12227e6	−0.0645896	−0.0322948	−	0.999478i	$-0.510282\pi$
$787$	−1.12227e6	−0.0645896	−0.0322948	+	0.999478i	$0.510282\pi$
$788$	0	0
$789$	1.68705e7	0.964793
$790$	0	0
$791$	−3.76132e6	−0.213747
$792$	0	0
$793$	−1.35408e7	−0.764650
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.48763e7	0.829560	0.414780	−	0.909922i	$-0.363859\pi$
$797$	1.48763e7	0.829560	0.414780	+	0.909922i	$0.363859\pi$
$798$	0	0
$799$	4.27291e7	2.36786
$800$	0	0
$801$	209214.	0.0115215
$802$	0	0
$803$	−9.11644e6	−0.498926
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.29660e7	−0.700845
$808$	0	0
$809$	1.86436e7	1.00152	0.500759	−	0.865587i	$-0.333054\pi$
$809$	1.86436e7	1.00152	0.500759	+	0.865587i	$0.333054\pi$
$810$	0	0
$811$	−4.50701e6	−0.240623	−0.120311	−	0.992736i	$-0.538389\pi$
$811$	−4.50701e6	−0.240623	−0.120311	+	0.992736i	$0.538389\pi$
$812$	0	0
$813$	−2.02920e7	−1.07671
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.50409e6	0.0788352
$818$	0	0
$819$	−1.44465e6	−0.0752580
$820$	0	0
$821$	−2.26560e7	−1.17307	−0.586537	−	0.809922i	$-0.699509\pi$
$821$	−2.26560e7	−1.17307	−0.586537	+	0.809922i	$0.699509\pi$
$822$	0	0
$823$	−2.76806e7	−1.42454	−0.712272	−	0.701903i	$-0.752334\pi$
$823$	−2.76806e7	−1.42454	−0.712272	+	0.701903i	$0.752334\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.42673e6	−0.377602	−0.188801	−	0.982015i	$-0.560460\pi$
$827$	−7.42673e6	−0.377602	−0.188801	+	0.982015i	$0.560460\pi$
$828$	0	0
$829$	3.41770e7	1.72722	0.863610	−	0.504160i	$-0.168198\pi$
$829$	3.41770e7	1.72722	0.863610	+	0.504160i	$0.168198\pi$
$830$	0	0
$831$	3.65244e6	0.183477
$832$	0	0
$833$	1.57446e7	0.786177
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.11734e6	−0.400498
$838$	0	0
$839$	−155990.	−0.00765051	−0.00382526	−	0.999993i	$-0.501218\pi$
$839$	−155990.	−0.00765051	−0.00382526	+	0.999993i	$0.501218\pi$
$840$	0	0
$841$	2.06703e7	1.00776
$842$	0	0
$843$	−5.28580e6	−0.256178
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.32182e7	−0.633087
$848$	0	0
$849$	−1.55110e7	−0.738534
$850$	0	0
$851$	−3.16640e7	−1.49879
$852$	0	0
$853$	2.52778e7	1.18951	0.594754	−	0.803908i	$-0.297250\pi$
$853$	2.52778e7	1.18951	0.594754	+	0.803908i	$0.297250\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.40854e6	−0.0655112	−0.0327556	−	0.999463i	$-0.510428\pi$
$857$	−1.40854e6	−0.0655112	−0.0327556	+	0.999463i	$0.510428\pi$
$858$	0	0
$859$	246158.	0.0113823	0.00569116	−	0.999984i	$-0.498188\pi$
$859$	246158.	0.0113823	0.00569116	+	0.999984i	$0.498188\pi$
$860$	0	0
$861$	2.37344e7	1.09111
$862$	0	0
$863$	1.17019e7	0.534849	0.267424	−	0.963579i	$-0.413827\pi$
$863$	1.17019e7	0.534849	0.267424	+	0.963579i	$0.413827\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.19381e7	1.89479
$868$	0	0
$869$	−1.29225e7	−0.580492
$870$	0	0
$871$	1.27989e7	0.571646
$872$	0	0
$873$	−6.50982e6	−0.289090
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.85174e6	0.344720	0.172360	−	0.985034i	$-0.444861\pi$
$877$	7.85174e6	0.344720	0.172360	+	0.985034i	$0.444861\pi$
$878$	0	0
$879$	−4.39607e6	−0.191908
$880$	0	0
$881$	105461.	0.00457777	0.00228888	−	0.999997i	$-0.499271\pi$
$881$	105461.	0.00457777	0.00228888	+	0.999997i	$0.499271\pi$
$882$	0	0
$883$	1.43760e7	0.620491	0.310245	−	0.950656i	$-0.399589\pi$
$883$	1.43760e7	0.620491	0.310245	+	0.950656i	$0.399589\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.71725e7	1.15963	0.579816	−	0.814748i	$-0.303125\pi$
$887$	2.71725e7	1.15963	0.579816	+	0.814748i	$0.303125\pi$
$888$	0	0
$889$	−2.19587e7	−0.931864
$890$	0	0
$891$	9.56672e6	0.403709
$892$	0	0
$893$	−2.84075e7	−1.19208
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.83177e7	1.17511
$898$	0	0
$899$	−1.50847e7	−0.622499
$900$	0	0
$901$	1.89499e7	0.777668
$902$	0	0
$903$	1.79625e6	0.0733073
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−5.25290e6	−0.212022	−0.106011	−	0.994365i	$-0.533808\pi$
$907$	−5.25290e6	−0.212022	−0.106011	+	0.994365i	$0.533808\pi$
$908$	0	0
$909$	5.54282e6	0.222495
$910$	0	0
$911$	−2.68057e7	−1.07012	−0.535059	−	0.844815i	$-0.679711\pi$
$911$	−2.68057e7	−1.07012	−0.535059	+	0.844815i	$0.679711\pi$
$912$	0	0
$913$	−5.49118e6	−0.218016
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.20039e6	−0.204227
$918$	0	0
$919$	1.25954e7	0.491951	0.245976	−	0.969276i	$-0.420892\pi$
$919$	1.25954e7	0.491951	0.245976	+	0.969276i	$0.420892\pi$
$920$	0	0
$921$	3.60563e7	1.40066
$922$	0	0
$923$	1.51737e7	0.586255
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.30466e6	−0.0498652
$928$	0	0
$929$	−3.23322e7	−1.22912	−0.614562	−	0.788868i	$-0.710667\pi$
$929$	−3.23322e7	−1.22912	−0.614562	+	0.788868i	$0.710667\pi$
$930$	0	0
$931$	−1.04675e7	−0.395792
$932$	0	0
$933$	3.89604e7	1.46528
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.04882e7	−0.762350	−0.381175	−	0.924503i	$-0.624481\pi$
$937$	−2.04882e7	−0.762350	−0.381175	+	0.924503i	$0.624481\pi$
$938$	0	0
$939$	1.87607e7	0.694361
$940$	0	0
$941$	−5.87135e6	−0.216154	−0.108077	−	0.994143i	$-0.534469\pi$
$941$	−5.87135e6	−0.216154	−0.108077	+	0.994143i	$0.534469\pi$
$942$	0	0
$943$	−6.06285e7	−2.22023
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.86718e7	−0.676567	−0.338284	−	0.941044i	$-0.609846\pi$
$947$	−1.86718e7	−0.676567	−0.338284	+	0.941044i	$0.609846\pi$
$948$	0	0
$949$	−2.67336e7	−0.963588
$950$	0	0
$951$	2.13885e7	0.766882
$952$	0	0
$953$	−1.55896e7	−0.556037	−0.278018	−	0.960576i	$-0.589678\pi$
$953$	−1.55896e7	−0.556037	−0.278018	+	0.960576i	$0.589678\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.54152e7	0.544088
$958$	0	0
$959$	−2.46213e7	−0.864500
$960$	0	0
$961$	−2.31036e7	−0.806996
$962$	0	0
$963$	2.54997e6	0.0886071
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−3.24491e7	−1.11593	−0.557964	−	0.829865i	$-0.688417\pi$
$967$	−3.24491e7	−1.11593	−0.557964	+	0.829865i	$0.688417\pi$
$968$	0	0
$969$	−4.36605e7	−1.49375
$970$	0	0
$971$	−3.73661e7	−1.27183	−0.635916	−	0.771758i	$-0.719378\pi$
$971$	−3.73661e7	−1.27183	−0.635916	+	0.771758i	$0.719378\pi$
$972$	0	0
$973$	−2.76195e7	−0.935261
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.23874e7	1.75586	0.877931	−	0.478787i	$-0.158923\pi$
$977$	5.23874e7	1.75586	0.877931	+	0.478787i	$0.158923\pi$
$978$	0	0
$979$	−825691.	−0.0275335
$980$	0	0
$981$	1.40067e6	0.0464690
$982$	0	0
$983$	−4.75140e7	−1.56833	−0.784166	−	0.620551i	$-0.786909\pi$
$983$	−4.75140e7	−1.56833	−0.784166	+	0.620551i	$0.786909\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.39253e7	−1.10849
$988$	0	0
$989$	−4.58846e6	−0.149168
$990$	0	0
$991$	−2.47278e7	−0.799835	−0.399918	−	0.916551i	$-0.630961\pi$
$991$	−2.47278e7	−0.799835	−0.399918	+	0.916551i	$0.630961\pi$
$992$	0	0
$993$	3.50311e7	1.12741
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.25139e7	−0.398709	−0.199355	−	0.979927i	$-0.563885\pi$
$997$	−1.25139e7	−0.398709	−0.199355	+	0.979927i	$0.563885\pi$
$998$	0	0
$999$	−2.71997e7	−0.862285

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.q.1.2		2
4.3	odd	2		200.6.a.g.1.1		2
5.2	odd	4		400.6.c.l.49.2		4
5.3	odd	4		400.6.c.l.49.3		4
5.4	even	2		80.6.a.i.1.1		2
15.14	odd	2		720.6.a.z.1.1		2
20.3	even	4		200.6.c.e.49.2		4
20.7	even	4		200.6.c.e.49.3		4
20.19	odd	2		40.6.a.d.1.2	✓	2
40.19	odd	2		320.6.a.w.1.1		2
40.29	even	2		320.6.a.q.1.2		2
60.59	even	2		360.6.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.a.d.1.2	✓	2	20.19	odd	2
80.6.a.i.1.1		2	5.4	even	2
200.6.a.g.1.1		2	4.3	odd	2
200.6.c.e.49.2		4	20.3	even	4
200.6.c.e.49.3		4	20.7	even	4
320.6.a.q.1.2		2	40.29	even	2
320.6.a.w.1.1		2	40.19	odd	2
360.6.a.l.1.2		2	60.59	even	2
400.6.a.q.1.2		2	1.1	even	1	trivial
400.6.c.l.49.2		4	5.2	odd	4
400.6.c.l.49.3		4	5.3	odd	4
720.6.a.z.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q+16.7156 q^{3} +94.1469 q^{7} +36.4124 q^{9} +O(q^{10})\)	\(q+16.7156 q^{3} +94.1469 q^{7} +36.4124 q^{9} -143.706 q^{11} -421.412 q^{13} -1982.11 q^{17} +1317.76 q^{19} +1573.73 q^{21} -4020.02 q^{23} -3453.24 q^{27} -6417.28 q^{29} +2350.64 q^{31} -2402.14 q^{33} +7876.58 q^{37} -7044.18 q^{39} +15081.6 q^{41} +1141.40 q^{43} -21557.3 q^{47} -7943.36 q^{49} -33132.3 q^{51} -9560.44 q^{53} +22027.2 q^{57} -42740.7 q^{59} +32132.1 q^{61} +3428.11 q^{63} -30371.4 q^{67} -67197.2 q^{69} -36006.7 q^{71} +63438.0 q^{73} -13529.5 q^{77} +89922.8 q^{79} -66571.4 q^{81} +38211.2 q^{83} -107269. q^{87} +5745.69 q^{89} -39674.7 q^{91} +39292.5 q^{93} -178780. q^{97} -5232.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{3} + 52 q^{7} + 618 q^{9}+O(q^{10}) \) 2 * q - 12 * q^3 + 52 * q^7 + 618 * q^9	\( 2 q - 12 q^{3} + 52 q^{7} + 618 q^{9} - 560 q^{11} - 1388 q^{13} - 148 q^{17} + 1000 q^{19} + 2784 q^{21} - 2452 q^{23} - 13176 q^{27} + 1340 q^{29} + 2248 q^{31} + 9552 q^{33} + 5940 q^{37} + 20712 q^{39} + 23076 q^{41} + 17684 q^{43} - 2908 q^{47} - 22974 q^{49} - 85800 q^{51} + 5412 q^{53} + 31152 q^{57} - 62584 q^{59} + 14108 q^{61} - 21084 q^{63} - 85412 q^{67} - 112224 q^{69} - 47208 q^{71} + 67452 q^{73} + 4016 q^{77} + 65904 q^{79} + 71298 q^{81} + 108724 q^{83} - 330024 q^{87} - 55020 q^{89} + 1064 q^{91} + 42240 q^{93} - 147668 q^{97} - 247344 q^{99}+O(q^{100}) \) 2 * q - 12 * q^3 + 52 * q^7 + 618 * q^9 - 560 * q^11 - 1388 * q^13 - 148 * q^17 + 1000 * q^19 + 2784 * q^21 - 2452 * q^23 - 13176 * q^27 + 1340 * q^29 + 2248 * q^31 + 9552 * q^33 + 5940 * q^37 + 20712 * q^39 + 23076 * q^41 + 17684 * q^43 - 2908 * q^47 - 22974 * q^49 - 85800 * q^51 + 5412 * q^53 + 31152 * q^57 - 62584 * q^59 + 14108 * q^61 - 21084 * q^63 - 85412 * q^67 - 112224 * q^69 - 47208 * q^71 + 67452 * q^73 + 4016 * q^77 + 65904 * q^79 + 71298 * q^81 + 108724 * q^83 - 330024 * q^87 - 55020 * q^89 + 1064 * q^91 + 42240 * q^93 - 147668 * q^97 - 247344 * q^99

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