LMFDB - Embedded newform 392.6.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,6,Mod(1,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, 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("392.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	392.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:	$62.8704573667$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 200x^{3} - 99x^{2} + 5803x - 3615 $ x^5 - 200x^3 - 99x^2 + 5803*x - 3615
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 7 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.14333$ of defining polynomial
Character	$\chi$	$=$	392.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.2867 q^{3} -26.0667 q^{5} -115.611 q^{9} -434.865 q^{11} -737.247 q^{13} +294.206 q^{15} +1413.04 q^{17} -1737.76 q^{19} -929.808 q^{23} -2445.53 q^{25} +4047.52 q^{27} -1630.63 q^{29} +1872.16 q^{31} +4908.17 q^{33} -7931.14 q^{37} +8321.06 q^{39} -6325.76 q^{41} -20075.7 q^{43} +3013.61 q^{45} -10265.6 q^{47} -15948.5 q^{51} +11967.4 q^{53} +11335.5 q^{55} +19613.5 q^{57} +21184.5 q^{59} +350.115 q^{61} +19217.6 q^{65} -16206.0 q^{67} +10494.4 q^{69} +72282.7 q^{71} +83700.7 q^{73} +27601.8 q^{75} -47131.0 q^{79} -17589.4 q^{81} -843.116 q^{83} -36833.3 q^{85} +18404.3 q^{87} +41349.9 q^{89} -21130.4 q^{93} +45297.7 q^{95} +155369. q^{97} +50275.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 81 q^{5} + 390 q^{9} + 361 q^{11} + 342 q^{13} - 1049 q^{15} + 1809 q^{17} + 1277 q^{19} + 911 q^{23} + 3940 q^{25} - 4751 q^{27} + 5442 q^{29} + 2187 q^{31} - 5553 q^{33} - 8181 q^{37}+ \cdots + 249798 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 81 * q^5 + 390 * q^9 + 361 * q^11 + 342 * q^13 - 1049 * q^15 + 1809 * q^17 + 1277 * q^19 + 911 * q^23 + 3940 * q^25 - 4751 * q^27 + 5442 * q^29 + 2187 * q^31 - 5553 * q^33 - 8181 * q^37 - 3422 * q^39 + 16578 * q^41 + 6332 * q^43 + 41310 * q^45 + 16101 * q^47 - 67865 * q^51 + 16047 * q^53 + 45629 * q^55 + 22347 * q^57 + 71027 * q^59 + 31093 * q^61 - 64370 * q^65 + 47981 * q^67 + 137249 * q^69 + 22512 * q^71 + 123333 * q^73 - 45460 * q^75 - 212481 * q^79 + 52917 * q^81 + 87460 * q^83 + 222141 * q^85 + 318070 * q^87 + 129045 * q^89 - 252835 * q^93 + 300417 * q^95 + 328274 * q^97 + 249798 * 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
$3$	−11.2867	−0.724039	−0.362020	−	0.932170i	$-0.617913\pi$
$3$	−11.2867	−0.724039	−0.362020	+	0.932170i	$0.617913\pi$
$4$	0	0
$5$	−26.0667	−0.466295	−0.233148	−	0.972441i	$-0.574903\pi$
$5$	−26.0667	−0.466295	−0.233148	+	0.972441i	$0.574903\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−115.611	−0.475767
$10$	0	0
$11$	−434.865	−1.08361	−0.541805	−	0.840505i	$-0.682259\pi$
$11$	−434.865	−1.08361	−0.541805	+	0.840505i	$0.682259\pi$
$12$	0	0
$13$	−737.247	−1.20992	−0.604958	−	0.796258i	$-0.706810\pi$
$13$	−737.247	−1.20992	−0.604958	+	0.796258i	$0.706810\pi$
$14$	0	0
$15$	294.206	0.337616
$16$	0	0
$17$	1413.04	1.18586	0.592929	−	0.805255i	$-0.297972\pi$
$17$	1413.04	1.18586	0.592929	+	0.805255i	$0.297972\pi$
$18$	0	0
$19$	−1737.76	−1.10435	−0.552173	−	0.833729i	$-0.686201\pi$
$19$	−1737.76	−1.10435	−0.552173	+	0.833729i	$0.686201\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−929.808	−0.366500	−0.183250	−	0.983066i	$-0.558662\pi$
$23$	−929.808	−0.366500	−0.183250	+	0.983066i	$0.558662\pi$
$24$	0	0
$25$	−2445.53	−0.782569
$26$	0	0
$27$	4047.52	1.06851
$28$	0	0
$29$	−1630.63	−0.360048	−0.180024	−	0.983662i	$-0.557617\pi$
$29$	−1630.63	−0.360048	−0.180024	+	0.983662i	$0.557617\pi$
$30$	0	0
$31$	1872.16	0.349895	0.174948	−	0.984578i	$-0.444024\pi$
$31$	1872.16	0.349895	0.174948	+	0.984578i	$0.444024\pi$
$32$	0	0
$33$	4908.17	0.784575
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7931.14	−0.952426	−0.476213	−	0.879330i	$-0.657991\pi$
$37$	−7931.14	−0.952426	−0.476213	+	0.879330i	$0.657991\pi$
$38$	0	0
$39$	8321.06	0.876026
$40$	0	0
$41$	−6325.76	−0.587697	−0.293848	−	0.955852i	$-0.594936\pi$
$41$	−6325.76	−0.587697	−0.293848	+	0.955852i	$0.594936\pi$
$42$	0	0
$43$	−20075.7	−1.65577	−0.827885	−	0.560898i	$-0.810456\pi$
$43$	−20075.7	−1.65577	−0.827885	+	0.560898i	$0.810456\pi$
$44$	0	0
$45$	3013.61	0.221848
$46$	0	0
$47$	−10265.6	−0.677857	−0.338928	−	0.940812i	$-0.610064\pi$
$47$	−10265.6	−0.677857	−0.338928	+	0.940812i	$0.610064\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−15948.5	−0.858608
$52$	0	0
$53$	11967.4	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$53$	11967.4	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$54$	0	0
$55$	11335.5	0.505282
$56$	0	0
$57$	19613.5	0.799590
$58$	0	0
$59$	21184.5	0.792296	0.396148	−	0.918187i	$-0.370347\pi$
$59$	21184.5	0.792296	0.396148	+	0.918187i	$0.370347\pi$
$60$	0	0
$61$	350.115	0.0120472	0.00602361	−	0.999982i	$-0.498083\pi$
$61$	350.115	0.0120472	0.00602361	+	0.999982i	$0.498083\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	19217.6	0.564178
$66$	0	0
$67$	−16206.0	−0.441052	−0.220526	−	0.975381i	$-0.570777\pi$
$67$	−16206.0	−0.441052	−0.220526	+	0.975381i	$0.570777\pi$
$68$	0	0
$69$	10494.4	0.265360
$70$	0	0
$71$	72282.7	1.70172	0.850861	−	0.525392i	$-0.176081\pi$
$71$	72282.7	1.70172	0.850861	+	0.525392i	$0.176081\pi$
$72$	0	0
$73$	83700.7	1.83832	0.919161	−	0.393881i	$-0.128868\pi$
$73$	83700.7	1.83832	0.919161	+	0.393881i	$0.128868\pi$
$74$	0	0
$75$	27601.8	0.566610
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−47131.0	−0.849648	−0.424824	−	0.905276i	$-0.639664\pi$
$79$	−47131.0	−0.849648	−0.424824	+	0.905276i	$0.639664\pi$
$80$	0	0
$81$	−17589.4	−0.297878
$82$	0	0
$83$	−843.116	−0.0134336	−0.00671679	−	0.999977i	$-0.502138\pi$
$83$	−843.116	−0.0134336	−0.00671679	+	0.999977i	$0.502138\pi$
$84$	0	0
$85$	−36833.3	−0.552960
$86$	0	0
$87$	18404.3	0.260689
$88$	0	0
$89$	41349.9	0.553349	0.276675	−	0.960964i	$-0.410768\pi$
$89$	41349.9	0.553349	0.276675	+	0.960964i	$0.410768\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−21130.4	−0.253338
$94$	0	0
$95$	45297.7	0.514952
$96$	0	0
$97$	155369.	1.67662	0.838312	−	0.545191i	$-0.183543\pi$
$97$	155369.	1.67662	0.838312	+	0.545191i	$0.183543\pi$
$98$	0	0
$99$	50275.4	0.515546
$100$	0	0
$101$	−31423.7	−0.306517	−0.153258	−	0.988186i	$-0.548977\pi$
$101$	−31423.7	−0.306517	−0.153258	+	0.988186i	$0.548977\pi$
$102$	0	0
$103$	−36353.3	−0.337637	−0.168819	−	0.985647i	$-0.553995\pi$
$103$	−36353.3	−0.337637	−0.168819	+	0.985647i	$0.553995\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−210635.	−1.77857	−0.889287	−	0.457350i	$-0.848799\pi$
$107$	−210635.	−1.77857	−0.889287	+	0.457350i	$0.848799\pi$
$108$	0	0
$109$	228714.	1.84385	0.921927	−	0.387364i	$-0.126614\pi$
$109$	228714.	1.84385	0.921927	+	0.387364i	$0.126614\pi$
$110$	0	0
$111$	89516.0	0.689594
$112$	0	0
$113$	240902.	1.77478	0.887390	−	0.461019i	$-0.152516\pi$
$113$	240902.	1.77478	0.887390	+	0.461019i	$0.152516\pi$
$114$	0	0
$115$	24237.0	0.170897
$116$	0	0
$117$	85234.2	0.575638
$118$	0	0
$119$	0	0
$120$	0	0
$121$	28056.5	0.174209
$122$	0	0
$123$	71396.7	0.425515
$124$	0	0
$125$	145205.	0.831204
$126$	0	0
$127$	−134747.	−0.741328	−0.370664	−	0.928767i	$-0.620870\pi$
$127$	−134747.	−0.741328	−0.370664	+	0.928767i	$0.620870\pi$
$128$	0	0
$129$	226588.	1.19884
$130$	0	0
$131$	44321.1	0.225648	0.112824	−	0.993615i	$-0.464010\pi$
$131$	44321.1	0.225648	0.112824	+	0.993615i	$0.464010\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−105506.	−0.498243
$136$	0	0
$137$	−365483.	−1.66366	−0.831832	−	0.555027i	$-0.812708\pi$
$137$	−365483.	−1.66366	−0.831832	+	0.555027i	$0.812708\pi$
$138$	0	0
$139$	134487.	0.590397	0.295198	−	0.955436i	$-0.404614\pi$
$139$	134487.	0.590397	0.295198	+	0.955436i	$0.404614\pi$
$140$	0	0
$141$	115864.	0.490795
$142$	0	0
$143$	320603.	1.31107
$144$	0	0
$145$	42505.1	0.167889
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−330148.	−1.21827	−0.609134	−	0.793067i	$-0.708483\pi$
$149$	−330148.	−1.21827	−0.609134	+	0.793067i	$0.708483\pi$
$150$	0	0
$151$	215976.	0.770838	0.385419	−	0.922742i	$-0.374057\pi$
$151$	215976.	0.770838	0.385419	+	0.922742i	$0.374057\pi$
$152$	0	0
$153$	−163364.	−0.564192
$154$	0	0
$155$	−48800.9	−0.163154
$156$	0	0
$157$	415250.	1.34450	0.672249	−	0.740325i	$-0.265328\pi$
$157$	415250.	1.34450	0.672249	+	0.740325i	$0.265328\pi$
$158$	0	0
$159$	−135071.	−0.423712
$160$	0	0
$161$	0	0
$162$	0	0
$163$	243823.	0.718797	0.359399	−	0.933184i	$-0.382982\pi$
$163$	243823.	0.718797	0.359399	+	0.933184i	$0.382982\pi$
$164$	0	0
$165$	−127940.	−0.365844
$166$	0	0
$167$	−622368.	−1.72685	−0.863427	−	0.504473i	$-0.831687\pi$
$167$	−622368.	−1.72685	−0.863427	+	0.504473i	$0.831687\pi$
$168$	0	0
$169$	172241.	0.463894
$170$	0	0
$171$	200905.	0.525412
$172$	0	0
$173$	195614.	0.496919	0.248459	−	0.968642i	$-0.420076\pi$
$173$	195614.	0.496919	0.248459	+	0.968642i	$0.420076\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−239102.	−0.573653
$178$	0	0
$179$	508181.	1.18546	0.592729	−	0.805402i	$-0.298051\pi$
$179$	508181.	1.18546	0.592729	+	0.805402i	$0.298051\pi$
$180$	0	0
$181$	815807.	1.85093	0.925467	−	0.378827i	$-0.123673\pi$
$181$	815807.	1.85093	0.925467	+	0.378827i	$0.123673\pi$
$182$	0	0
$183$	−3951.63	−0.00872265
$184$	0	0
$185$	206739.	0.444112
$186$	0	0
$187$	−614482.	−1.28501
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−289788.	−0.574774	−0.287387	−	0.957815i	$-0.592787\pi$
$191$	−289788.	−0.574774	−0.287387	+	0.957815i	$0.592787\pi$
$192$	0	0
$193$	−83053.9	−0.160497	−0.0802485	−	0.996775i	$-0.525571\pi$
$193$	−83053.9	−0.160497	−0.0802485	+	0.996775i	$0.525571\pi$
$194$	0	0
$195$	−216903.	−0.408487
$196$	0	0
$197$	−758761.	−1.39296	−0.696481	−	0.717575i	$-0.745252\pi$
$197$	−758761.	−1.39296	−0.696481	+	0.717575i	$0.745252\pi$
$198$	0	0
$199$	−352538.	−0.631063	−0.315532	−	0.948915i	$-0.602183\pi$
$199$	−352538.	−0.631063	−0.315532	+	0.948915i	$0.602183\pi$
$200$	0	0
$201$	182912.	0.319339
$202$	0	0
$203$	0	0
$204$	0	0
$205$	164892.	0.274040
$206$	0	0
$207$	107496.	0.174369
$208$	0	0
$209$	755690.	1.19668
$210$	0	0
$211$	−74904.6	−0.115825	−0.0579125	−	0.998322i	$-0.518444\pi$
$211$	−74904.6	−0.115825	−0.0579125	+	0.998322i	$0.518444\pi$
$212$	0	0
$213$	−815830.	−1.23211
$214$	0	0
$215$	523308.	0.772078
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−944700.	−1.33102
$220$	0	0
$221$	−1.04176e6	−1.43479
$222$	0	0
$223$	−658048.	−0.886125	−0.443063	−	0.896491i	$-0.646108\pi$
$223$	−658048.	−0.886125	−0.443063	+	0.896491i	$0.646108\pi$
$224$	0	0
$225$	282731.	0.372320
$226$	0	0
$227$	−516436.	−0.665200	−0.332600	−	0.943068i	$-0.607926\pi$
$227$	−516436.	−0.665200	−0.332600	+	0.943068i	$0.607926\pi$
$228$	0	0
$229$	322948.	0.406953	0.203476	−	0.979080i	$-0.434776\pi$
$229$	322948.	0.406953	0.203476	+	0.979080i	$0.434776\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.48011e6	1.78610	0.893050	−	0.449958i	$-0.148561\pi$
$233$	1.48011e6	1.78610	0.893050	+	0.449958i	$0.148561\pi$
$234$	0	0
$235$	267589.	0.316082
$236$	0	0
$237$	531952.	0.615179
$238$	0	0
$239$	209299.	0.237014	0.118507	−	0.992953i	$-0.462189\pi$
$239$	209299.	0.237014	0.118507	+	0.992953i	$0.462189\pi$
$240$	0	0
$241$	−1.21271e6	−1.34497	−0.672487	−	0.740109i	$-0.734774\pi$
$241$	−1.21271e6	−1.34497	−0.672487	+	0.740109i	$0.734774\pi$
$242$	0	0
$243$	−785023.	−0.852838
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.28116e6	1.33617
$248$	0	0
$249$	9515.96	0.00972644
$250$	0	0
$251$	−1.64691e6	−1.65001	−0.825004	−	0.565126i	$-0.808827\pi$
$251$	−1.64691e6	−1.65001	−0.825004	+	0.565126i	$0.808827\pi$
$252$	0	0
$253$	404341.	0.397143
$254$	0	0
$255$	415725.	0.400365
$256$	0	0
$257$	−18894.0	−0.0178440	−0.00892198	−	0.999960i	$-0.502840\pi$
$257$	−18894.0	−0.0178440	−0.00892198	+	0.999960i	$0.502840\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	188519.	0.171299
$262$	0	0
$263$	1.25353e6	1.11749	0.558747	−	0.829338i	$-0.311282\pi$
$263$	1.25353e6	1.11749	0.558747	+	0.829338i	$0.311282\pi$
$264$	0	0
$265$	−311950.	−0.272879
$266$	0	0
$267$	−466702.	−0.400647
$268$	0	0
$269$	954812.	0.804521	0.402260	−	0.915525i	$-0.368225\pi$
$269$	954812.	0.804521	0.402260	+	0.915525i	$0.368225\pi$
$270$	0	0
$271$	−450312.	−0.372469	−0.186235	−	0.982505i	$-0.559628\pi$
$271$	−450312.	−0.372469	−0.186235	+	0.982505i	$0.559628\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.06347e6	0.847998
$276$	0	0
$277$	−1.72355e6	−1.34966	−0.674830	−	0.737973i	$-0.735783\pi$
$277$	−1.72355e6	−1.34966	−0.674830	+	0.737973i	$0.735783\pi$
$278$	0	0
$279$	−216443.	−0.166469
$280$	0	0
$281$	503056.	0.380059	0.190029	−	0.981778i	$-0.439142\pi$
$281$	503056.	0.380059	0.190029	+	0.981778i	$0.439142\pi$
$282$	0	0
$283$	−1.59134e6	−1.18113	−0.590565	−	0.806990i	$-0.701095\pi$
$283$	−1.59134e6	−1.18113	−0.590565	+	0.806990i	$0.701095\pi$
$284$	0	0
$285$	−511259.	−0.372845
$286$	0	0
$287$	0	0
$288$	0	0
$289$	576829.	0.406259
$290$	0	0
$291$	−1.75360e6	−1.21394
$292$	0	0
$293$	−1.73373e6	−1.17981	−0.589907	−	0.807471i	$-0.700836\pi$
$293$	−1.73373e6	−1.17981	−0.589907	+	0.807471i	$0.700836\pi$
$294$	0	0
$295$	−552209.	−0.369444
$296$	0	0
$297$	−1.76013e6	−1.15785
$298$	0	0
$299$	685499.	0.443434
$300$	0	0
$301$	0	0
$302$	0	0
$303$	354669.	0.221930
$304$	0	0
$305$	−9126.35	−0.00561756
$306$	0	0
$307$	−955767.	−0.578770	−0.289385	−	0.957213i	$-0.593451\pi$
$307$	−955767.	−0.578770	−0.289385	+	0.957213i	$0.593451\pi$
$308$	0	0
$309$	410307.	0.244463
$310$	0	0
$311$	−2.87849e6	−1.68758	−0.843789	−	0.536674i	$-0.819680\pi$
$311$	−2.87849e6	−1.68758	−0.843789	+	0.536674i	$0.819680\pi$
$312$	0	0
$313$	2.72591e6	1.57272	0.786358	−	0.617771i	$-0.211964\pi$
$313$	2.72591e6	1.57272	0.786358	+	0.617771i	$0.211964\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.86964e6	1.60391	0.801954	−	0.597385i	$-0.203794\pi$
$317$	2.86964e6	1.60391	0.801954	+	0.597385i	$0.203794\pi$
$318$	0	0
$319$	709103.	0.390151
$320$	0	0
$321$	2.37737e6	1.28776
$322$	0	0
$323$	−2.45553e6	−1.30960
$324$	0	0
$325$	1.80296e6	0.946841
$326$	0	0
$327$	−2.58142e6	−1.33502
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−336903.	−0.169019	−0.0845093	−	0.996423i	$-0.526932\pi$
$331$	−336903.	−0.169019	−0.0845093	+	0.996423i	$0.526932\pi$
$332$	0	0
$333$	916931.	0.453133
$334$	0	0
$335$	422438.	0.205660
$336$	0	0
$337$	−16431.9	−0.00788156	−0.00394078	−	0.999992i	$-0.501254\pi$
$337$	−16431.9	−0.00788156	−0.00394078	+	0.999992i	$0.501254\pi$
$338$	0	0
$339$	−2.71898e6	−1.28501
$340$	0	0
$341$	−814135.	−0.379149
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−273555.	−0.123736
$346$	0	0
$347$	639206.	0.284982	0.142491	−	0.989796i	$-0.454489\pi$
$347$	639206.	0.284982	0.142491	+	0.989796i	$0.454489\pi$
$348$	0	0
$349$	−3.22585e6	−1.41769	−0.708843	−	0.705366i	$-0.750783\pi$
$349$	−3.22585e6	−1.41769	−0.708843	+	0.705366i	$0.750783\pi$
$350$	0	0
$351$	−2.98403e6	−1.29281
$352$	0	0
$353$	−2.00724e6	−0.857359	−0.428679	−	0.903457i	$-0.641021\pi$
$353$	−2.00724e6	−0.857359	−0.428679	+	0.903457i	$0.641021\pi$
$354$	0	0
$355$	−1.88417e6	−0.793505
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.40344e6	−1.80325	−0.901626	−	0.432517i	$-0.857626\pi$
$359$	−4.40344e6	−1.80325	−0.901626	+	0.432517i	$0.857626\pi$
$360$	0	0
$361$	543706.	0.219582
$362$	0	0
$363$	−316664.	−0.126134
$364$	0	0
$365$	−2.18180e6	−0.857202
$366$	0	0
$367$	4.19731e6	1.62669	0.813347	−	0.581779i	$-0.197643\pi$
$367$	4.19731e6	1.62669	0.813347	+	0.581779i	$0.197643\pi$
$368$	0	0
$369$	731331.	0.279607
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.58823e6	−0.591072	−0.295536	−	0.955332i	$-0.595498\pi$
$373$	−1.58823e6	−0.591072	−0.295536	+	0.955332i	$0.595498\pi$
$374$	0	0
$375$	−1.63888e6	−0.601824
$376$	0	0
$377$	1.20218e6	0.435627
$378$	0	0
$379$	2.23384e6	0.798830	0.399415	−	0.916770i	$-0.369213\pi$
$379$	2.23384e6	0.798830	0.399415	+	0.916770i	$0.369213\pi$
$380$	0	0
$381$	1.52085e6	0.536751
$382$	0	0
$383$	−884438.	−0.308085	−0.154042	−	0.988064i	$-0.549229\pi$
$383$	−884438.	−0.308085	−0.154042	+	0.988064i	$0.549229\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.32098e6	0.787761
$388$	0	0
$389$	−1.58074e6	−0.529648	−0.264824	−	0.964297i	$-0.585314\pi$
$389$	−1.58074e6	−0.529648	−0.264824	+	0.964297i	$0.585314\pi$
$390$	0	0
$391$	−1.31386e6	−0.434617
$392$	0	0
$393$	−500237.	−0.163378
$394$	0	0
$395$	1.22855e6	0.396187
$396$	0	0
$397$	−978509.	−0.311594	−0.155797	−	0.987789i	$-0.549795\pi$
$397$	−978509.	−0.311594	−0.155797	+	0.987789i	$0.549795\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.39959e6	1.05576	0.527880	−	0.849319i	$-0.322987\pi$
$401$	3.39959e6	1.05576	0.527880	+	0.849319i	$0.322987\pi$
$402$	0	0
$403$	−1.38024e6	−0.423343
$404$	0	0
$405$	458498.	0.138899
$406$	0	0
$407$	3.44897e6	1.03206
$408$	0	0
$409$	−356468.	−0.105369	−0.0526844	−	0.998611i	$-0.516778\pi$
$409$	−356468.	−0.105369	−0.0526844	+	0.998611i	$0.516778\pi$
$410$	0	0
$411$	4.12508e6	1.20456
$412$	0	0
$413$	0	0
$414$	0	0
$415$	21977.3	0.00626402
$416$	0	0
$417$	−1.51791e6	−0.427470
$418$	0	0
$419$	−783777.	−0.218101	−0.109050	−	0.994036i	$-0.534781\pi$
$419$	−783777.	−0.218101	−0.109050	+	0.994036i	$0.534781\pi$
$420$	0	0
$421$	−543516.	−0.149454	−0.0747270	−	0.997204i	$-0.523809\pi$
$421$	−543516.	−0.149454	−0.0747270	+	0.997204i	$0.523809\pi$
$422$	0	0
$423$	1.18682e6	0.322502
$424$	0	0
$425$	−3.45563e6	−0.928015
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.61854e6	−0.949270
$430$	0	0
$431$	−809955.	−0.210023	−0.105012	−	0.994471i	$-0.533488\pi$
$431$	−809955.	−0.210023	−0.105012	+	0.994471i	$0.533488\pi$
$432$	0	0
$433$	894096.	0.229174	0.114587	−	0.993413i	$-0.463446\pi$
$433$	894096.	0.229174	0.114587	+	0.993413i	$0.463446\pi$
$434$	0	0
$435$	−479740.	−0.121558
$436$	0	0
$437$	1.61578e6	0.404743
$438$	0	0
$439$	6.97626e6	1.72767	0.863836	−	0.503773i	$-0.168055\pi$
$439$	6.97626e6	1.72767	0.863836	+	0.503773i	$0.168055\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−466594.	−0.112961	−0.0564807	−	0.998404i	$-0.517988\pi$
$443$	−466594.	−0.112961	−0.0564807	+	0.998404i	$0.517988\pi$
$444$	0	0
$445$	−1.07786e6	−0.258024
$446$	0	0
$447$	3.72627e6	0.882074
$448$	0	0
$449$	2.68786e6	0.629203	0.314602	−	0.949224i	$-0.398129\pi$
$449$	2.68786e6	0.629203	0.314602	+	0.949224i	$0.398129\pi$
$450$	0	0
$451$	2.75085e6	0.636833
$452$	0	0
$453$	−2.43765e6	−0.558117
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−166961.	−0.0373959	−0.0186979	−	0.999825i	$-0.505952\pi$
$457$	−166961.	−0.0373959	−0.0186979	+	0.999825i	$0.505952\pi$
$458$	0	0
$459$	5.71932e6	1.26710
$460$	0	0
$461$	1.53425e6	0.336236	0.168118	−	0.985767i	$-0.446231\pi$
$461$	1.53425e6	0.336236	0.168118	+	0.985767i	$0.446231\pi$
$462$	0	0
$463$	7.11226e6	1.54190	0.770949	−	0.636897i	$-0.219783\pi$
$463$	7.11226e6	1.54190	0.770949	+	0.636897i	$0.219783\pi$
$464$	0	0
$465$	550799.	0.118130
$466$	0	0
$467$	−5.88723e6	−1.24916	−0.624581	−	0.780960i	$-0.714730\pi$
$467$	−5.88723e6	−1.24916	−0.624581	+	0.780960i	$0.714730\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−4.68678e6	−0.973469
$472$	0	0
$473$	8.73022e6	1.79421
$474$	0	0
$475$	4.24974e6	0.864227
$476$	0	0
$477$	−1.38356e6	−0.278422
$478$	0	0
$479$	233079.	0.0464156	0.0232078	−	0.999731i	$-0.492612\pi$
$479$	233079.	0.0464156	0.0232078	+	0.999731i	$0.492612\pi$
$480$	0	0
$481$	5.84721e6	1.15235
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.04996e6	−0.781802
$486$	0	0
$487$	2.29839e6	0.439139	0.219569	−	0.975597i	$-0.429535\pi$
$487$	2.29839e6	0.439139	0.219569	+	0.975597i	$0.429535\pi$
$488$	0	0
$489$	−2.75195e6	−0.520437
$490$	0	0
$491$	268580.	0.0502771	0.0251386	−	0.999684i	$-0.491997\pi$
$491$	268580.	0.0502771	0.0251386	+	0.999684i	$0.491997\pi$
$492$	0	0
$493$	−2.30415e6	−0.426965
$494$	0	0
$495$	−1.31051e6	−0.240397
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.07781e6	1.45225	0.726127	−	0.687561i	$-0.241319\pi$
$499$	8.07781e6	1.45225	0.726127	+	0.687561i	$0.241319\pi$
$500$	0	0
$501$	7.02445e6	1.25031
$502$	0	0
$503$	6.46801e6	1.13986	0.569929	−	0.821694i	$-0.306971\pi$
$503$	6.46801e6	1.13986	0.569929	+	0.821694i	$0.306971\pi$
$504$	0	0
$505$	819113.	0.142927
$506$	0	0
$507$	−1.94402e6	−0.335878
$508$	0	0
$509$	6.61907e6	1.13241	0.566203	−	0.824266i	$-0.308412\pi$
$509$	6.61907e6	1.13241	0.566203	+	0.824266i	$0.308412\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−7.03362e6	−1.18001
$514$	0	0
$515$	947610.	0.157439
$516$	0	0
$517$	4.46413e6	0.734532
$518$	0	0
$519$	−2.20783e6	−0.359789
$520$	0	0
$521$	5.74672e6	0.927525	0.463763	−	0.885959i	$-0.346499\pi$
$521$	5.74672e6	0.927525	0.463763	+	0.885959i	$0.346499\pi$
$522$	0	0
$523$	7.63623e6	1.22074	0.610372	−	0.792115i	$-0.291020\pi$
$523$	7.63623e6	1.22074	0.610372	+	0.792115i	$0.291020\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.64543e6	0.414926
$528$	0	0
$529$	−5.57180e6	−0.865678
$530$	0	0
$531$	−2.44917e6	−0.376948
$532$	0	0
$533$	4.66365e6	0.711063
$534$	0	0
$535$	5.49057e6	0.829341
$536$	0	0
$537$	−5.73566e6	−0.858317
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.46588e6	−0.656015	−0.328008	−	0.944675i	$-0.606377\pi$
$541$	−4.46588e6	−0.656015	−0.328008	+	0.944675i	$0.606377\pi$
$542$	0	0
$543$	−9.20773e6	−1.34015
$544$	0	0
$545$	−5.96182e6	−0.859781
$546$	0	0
$547$	53981.3	0.00771391	0.00385696	−	0.999993i	$-0.498772\pi$
$547$	53981.3	0.00771391	0.00385696	+	0.999993i	$0.498772\pi$
$548$	0	0
$549$	−40477.3	−0.00573167
$550$	0	0
$551$	2.83364e6	0.397617
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.33339e6	−0.321555
$556$	0	0
$557$	1.19677e6	0.163446	0.0817228	−	0.996655i	$-0.473958\pi$
$557$	1.19677e6	0.163446	0.0817228	+	0.996655i	$0.473958\pi$
$558$	0	0
$559$	1.48008e7	2.00334
$560$	0	0
$561$	6.93545e6	0.930395
$562$	0	0
$563$	9.98924e6	1.32819	0.664097	−	0.747646i	$-0.268816\pi$
$563$	9.98924e6	1.32819	0.664097	+	0.747646i	$0.268816\pi$
$564$	0	0
$565$	−6.27953e6	−0.827572
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.19320e7	−1.54502	−0.772509	−	0.635003i	$-0.780999\pi$
$569$	−1.19320e7	−1.54502	−0.772509	+	0.635003i	$0.780999\pi$
$570$	0	0
$571$	−1.00650e7	−1.29189	−0.645943	−	0.763386i	$-0.723535\pi$
$571$	−1.00650e7	−1.29189	−0.645943	+	0.763386i	$0.723535\pi$
$572$	0	0
$573$	3.27074e6	0.416159
$574$	0	0
$575$	2.27387e6	0.286811
$576$	0	0
$577$	9.21284e6	1.15200	0.576002	−	0.817448i	$-0.304612\pi$
$577$	9.21284e6	1.15200	0.576002	+	0.817448i	$0.304612\pi$
$578$	0	0
$579$	937401.	0.116206
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.20418e6	−0.634134
$584$	0	0
$585$	−2.22178e6	−0.268417
$586$	0	0
$587$	−4.95647e6	−0.593714	−0.296857	−	0.954922i	$-0.595938\pi$
$587$	−4.95647e6	−0.593714	−0.296857	+	0.954922i	$0.595938\pi$
$588$	0	0
$589$	−3.25336e6	−0.386405
$590$	0	0
$591$	8.56387e6	1.00856
$592$	0	0
$593$	−9.24149e6	−1.07921	−0.539604	−	0.841919i	$-0.681426\pi$
$593$	−9.24149e6	−1.07921	−0.539604	+	0.841919i	$0.681426\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.97897e6	0.456914
$598$	0	0
$599$	1.07648e7	1.22586	0.612929	−	0.790138i	$-0.289991\pi$
$599$	1.07648e7	1.22586	0.612929	+	0.790138i	$0.289991\pi$
$600$	0	0
$601$	−4.54962e6	−0.513794	−0.256897	−	0.966439i	$-0.582700\pi$
$601$	−4.54962e6	−0.513794	−0.256897	+	0.966439i	$0.582700\pi$
$602$	0	0
$603$	1.87360e6	0.209838
$604$	0	0
$605$	−731340.	−0.0812327
$606$	0	0
$607$	729167.	0.0803258	0.0401629	−	0.999193i	$-0.487212\pi$
$607$	729167.	0.0803258	0.0401629	+	0.999193i	$0.487212\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.56826e6	0.820149
$612$	0	0
$613$	1.07089e7	1.15105	0.575526	−	0.817784i	$-0.304797\pi$
$613$	1.07089e7	1.15105	0.575526	+	0.817784i	$0.304797\pi$
$614$	0	0
$615$	−1.86108e6	−0.198416
$616$	0	0
$617$	−1.13441e7	−1.19966	−0.599828	−	0.800129i	$-0.704764\pi$
$617$	−1.13441e7	−1.19966	−0.599828	+	0.800129i	$0.704764\pi$
$618$	0	0
$619$	−1.60974e7	−1.68861	−0.844303	−	0.535866i	$-0.819985\pi$
$619$	−1.60974e7	−1.68861	−0.844303	+	0.535866i	$0.819985\pi$
$620$	0	0
$621$	−3.76342e6	−0.391610
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3.85725e6	0.394982
$626$	0	0
$627$	−8.52921e6	−0.866443
$628$	0	0
$629$	−1.12070e7	−1.12944
$630$	0	0
$631$	4.09127e6	0.409058	0.204529	−	0.978861i	$-0.434434\pi$
$631$	4.09127e6	0.409058	0.204529	+	0.978861i	$0.434434\pi$
$632$	0	0
$633$	845422.	0.0838618
$634$	0	0
$635$	3.51242e6	0.345678
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.35671e6	−0.809623
$640$	0	0
$641$	6.46808e6	0.621771	0.310885	−	0.950447i	$-0.399375\pi$
$641$	6.46808e6	0.621771	0.310885	+	0.950447i	$0.399375\pi$
$642$	0	0
$643$	1.35591e7	1.29331	0.646655	−	0.762783i	$-0.276168\pi$
$643$	1.35591e7	1.29331	0.646655	+	0.762783i	$0.276168\pi$
$644$	0	0
$645$	−5.90639e6	−0.559015
$646$	0	0
$647$	1.04550e7	0.981890	0.490945	−	0.871190i	$-0.336652\pi$
$647$	1.04550e7	0.981890	0.490945	+	0.871190i	$0.336652\pi$
$648$	0	0
$649$	−9.21238e6	−0.858539
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.64259e6	−0.609613	−0.304807	−	0.952414i	$-0.598592\pi$
$653$	−6.64259e6	−0.609613	−0.304807	+	0.952414i	$0.598592\pi$
$654$	0	0
$655$	−1.15530e6	−0.105219
$656$	0	0
$657$	−9.67675e6	−0.874614
$658$	0	0
$659$	−3.80579e6	−0.341375	−0.170687	−	0.985325i	$-0.554599\pi$
$659$	−3.80579e6	−0.341375	−0.170687	+	0.985325i	$0.554599\pi$
$660$	0	0
$661$	−1.47727e7	−1.31510	−0.657548	−	0.753413i	$-0.728406\pi$
$661$	−1.47727e7	−1.31510	−0.657548	+	0.753413i	$0.728406\pi$
$662$	0	0
$663$	1.17580e7	1.03884
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.51617e6	0.131957
$668$	0	0
$669$	7.42715e6	0.641589
$670$	0	0
$671$	−152253.	−0.0130545
$672$	0	0
$673$	−1.25733e7	−1.07007	−0.535033	−	0.844831i	$-0.679701\pi$
$673$	−1.25733e7	−1.07007	−0.535033	+	0.844831i	$0.679701\pi$
$674$	0	0
$675$	−9.89833e6	−0.836185
$676$	0	0
$677$	1.64300e7	1.37774	0.688868	−	0.724887i	$-0.258108\pi$
$677$	1.64300e7	1.37774	0.688868	+	0.724887i	$0.258108\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5.82884e6	0.481631
$682$	0	0
$683$	−9.16788e6	−0.751999	−0.376000	−	0.926620i	$-0.622701\pi$
$683$	−9.16788e6	−0.751999	−0.376000	+	0.926620i	$0.622701\pi$
$684$	0	0
$685$	9.52694e6	0.775759
$686$	0	0
$687$	−3.64501e6	−0.294650
$688$	0	0
$689$	−8.82291e6	−0.708049
$690$	0	0
$691$	−2.33063e6	−0.185686	−0.0928429	−	0.995681i	$-0.529595\pi$
$691$	−2.33063e6	−0.185686	−0.0928429	+	0.995681i	$0.529595\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.50564e6	−0.275299
$696$	0	0
$697$	−8.93857e6	−0.696925
$698$	0	0
$699$	−1.67055e7	−1.29321
$700$	0	0
$701$	−6.64254e6	−0.510551	−0.255276	−	0.966868i	$-0.582166\pi$
$701$	−6.64254e6	−0.510551	−0.255276	+	0.966868i	$0.582166\pi$
$702$	0	0
$703$	1.37824e7	1.05181
$704$	0	0
$705$	−3.02019e6	−0.228855
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.41760e7	1.80621	0.903105	−	0.429419i	$-0.141282\pi$
$709$	2.41760e7	1.80621	0.903105	+	0.429419i	$0.141282\pi$
$710$	0	0
$711$	5.44889e6	0.404235
$712$	0	0
$713$	−1.74075e6	−0.128236
$714$	0	0
$715$	−8.35707e6	−0.611348
$716$	0	0
$717$	−2.36229e6	−0.171607
$718$	0	0
$719$	−9.33905e6	−0.673721	−0.336861	−	0.941554i	$-0.609365\pi$
$719$	−9.33905e6	−0.673721	−0.336861	+	0.941554i	$0.609365\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.36874e7	0.973814
$724$	0	0
$725$	3.98774e6	0.281762
$726$	0	0
$727$	1.37712e7	0.966352	0.483176	−	0.875523i	$-0.339483\pi$
$727$	1.37712e7	0.966352	0.483176	+	0.875523i	$0.339483\pi$
$728$	0	0
$729$	1.31345e7	0.915366
$730$	0	0
$731$	−2.83678e7	−1.96351
$732$	0	0
$733$	−5.76967e6	−0.396635	−0.198318	−	0.980138i	$-0.563548\pi$
$733$	−5.76967e6	−0.396635	−0.198318	+	0.980138i	$0.563548\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.04743e6	0.477928
$738$	0	0
$739$	−1.30308e7	−0.877730	−0.438865	−	0.898553i	$-0.644619\pi$
$739$	−1.30308e7	−0.877730	−0.438865	+	0.898553i	$0.644619\pi$
$740$	0	0
$741$	−1.44600e7	−0.967436
$742$	0	0
$743$	2.50890e7	1.66729	0.833645	−	0.552300i	$-0.186250\pi$
$743$	2.50890e7	1.66729	0.833645	+	0.552300i	$0.186250\pi$
$744$	0	0
$745$	8.60587e6	0.568073
$746$	0	0
$747$	97473.8	0.00639126
$748$	0	0
$749$	0	0
$750$	0	0
$751$	9.62435e6	0.622689	0.311345	−	0.950297i	$-0.399221\pi$
$751$	9.62435e6	0.622689	0.311345	+	0.950297i	$0.399221\pi$
$752$	0	0
$753$	1.85881e7	1.19467
$754$	0	0
$755$	−5.62978e6	−0.359438
$756$	0	0
$757$	1.81324e7	1.15005	0.575023	−	0.818137i	$-0.304993\pi$
$757$	1.81324e7	1.15005	0.575023	+	0.818137i	$0.304993\pi$
$758$	0	0
$759$	−4.56366e6	−0.287547
$760$	0	0
$761$	7.88507e6	0.493564	0.246782	−	0.969071i	$-0.420627\pi$
$761$	7.88507e6	0.493564	0.246782	+	0.969071i	$0.420627\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4.25836e6	0.263080
$766$	0	0
$767$	−1.56182e7	−0.958610
$768$	0	0
$769$	−2.50529e6	−0.152771	−0.0763857	−	0.997078i	$-0.524338\pi$
$769$	−2.50529e6	−0.152771	−0.0763857	+	0.997078i	$0.524338\pi$
$770$	0	0
$771$	213250.	0.0129197
$772$	0	0
$773$	−1.37371e7	−0.826885	−0.413443	−	0.910530i	$-0.635674\pi$
$773$	−1.37371e7	−0.826885	−0.413443	+	0.910530i	$0.635674\pi$
$774$	0	0
$775$	−4.57841e6	−0.273817
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.09926e7	0.649021
$780$	0	0
$781$	−3.14332e7	−1.84400
$782$	0	0
$783$	−6.60001e6	−0.384716
$784$	0	0
$785$	−1.08242e7	−0.626934
$786$	0	0
$787$	−2.44461e7	−1.40693	−0.703465	−	0.710730i	$-0.748365\pi$
$787$	−2.44461e7	−1.40693	−0.703465	+	0.710730i	$0.748365\pi$
$788$	0	0
$789$	−1.41481e7	−0.809109
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−258122.	−0.0145761
$794$	0	0
$795$	3.52087e6	0.197575
$796$	0	0
$797$	−3.02154e7	−1.68493	−0.842466	−	0.538749i	$-0.818897\pi$
$797$	−3.02154e7	−1.68493	−0.842466	+	0.538749i	$0.818897\pi$
$798$	0	0
$799$	−1.45057e7	−0.803842
$800$	0	0
$801$	−4.78052e6	−0.263266
$802$	0	0
$803$	−3.63985e7	−1.99202
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.07766e7	−0.582505
$808$	0	0
$809$	−2.33535e7	−1.25453	−0.627266	−	0.778805i	$-0.715826\pi$
$809$	−2.33535e7	−1.25453	−0.627266	+	0.778805i	$0.715826\pi$
$810$	0	0
$811$	−271769.	−0.0145093	−0.00725466	−	0.999974i	$-0.502309\pi$
$811$	−271769.	−0.0145093	−0.00725466	+	0.999974i	$0.502309\pi$
$812$	0	0
$813$	5.08252e6	0.269682
$814$	0	0
$815$	−6.35568e6	−0.335172
$816$	0	0
$817$	3.48867e7	1.82854
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.26647e7	−0.655747	−0.327874	−	0.944722i	$-0.606332\pi$
$821$	−1.26647e7	−0.655747	−0.327874	+	0.944722i	$0.606332\pi$
$822$	0	0
$823$	1.66922e7	0.859041	0.429520	−	0.903057i	$-0.358683\pi$
$823$	1.66922e7	0.859041	0.429520	+	0.903057i	$0.358683\pi$
$824$	0	0
$825$	−1.20031e7	−0.613984
$826$	0	0
$827$	−2.52342e7	−1.28300	−0.641500	−	0.767123i	$-0.721687\pi$
$827$	−2.52342e7	−1.28300	−0.641500	+	0.767123i	$0.721687\pi$
$828$	0	0
$829$	839396.	0.0424210	0.0212105	−	0.999775i	$-0.493248\pi$
$829$	839396.	0.0424210	0.0212105	+	0.999775i	$0.493248\pi$
$830$	0	0
$831$	1.94531e7	0.977207
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.62231e7	0.805224
$836$	0	0
$837$	7.57759e6	0.373868
$838$	0	0
$839$	2.27745e7	1.11698	0.558488	−	0.829512i	$-0.311382\pi$
$839$	2.27745e7	1.11698	0.558488	+	0.829512i	$0.311382\pi$
$840$	0	0
$841$	−1.78522e7	−0.870366
$842$	0	0
$843$	−5.67782e6	−0.275177
$844$	0	0
$845$	−4.48975e6	−0.216312
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.79610e7	0.855185
$850$	0	0
$851$	7.37444e6	0.349064
$852$	0	0
$853$	2.56919e7	1.20899	0.604497	−	0.796607i	$-0.293374\pi$
$853$	2.56919e7	1.20899	0.604497	+	0.796607i	$0.293374\pi$
$854$	0	0
$855$	−5.23693e6	−0.244997
$856$	0	0
$857$	−3.17425e7	−1.47635	−0.738174	−	0.674610i	$-0.764312\pi$
$857$	−3.17425e7	−1.47635	−0.738174	+	0.674610i	$0.764312\pi$
$858$	0	0
$859$	1.46054e7	0.675354	0.337677	−	0.941262i	$-0.390359\pi$
$859$	1.46054e7	0.675354	0.337677	+	0.941262i	$0.390359\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.51398e6	0.343434	0.171717	−	0.985146i	$-0.445069\pi$
$863$	7.51398e6	0.343434	0.171717	+	0.985146i	$0.445069\pi$
$864$	0	0
$865$	−5.09902e6	−0.231711
$866$	0	0
$867$	−6.51047e6	−0.294147
$868$	0	0
$869$	2.04956e7	0.920687
$870$	0	0
$871$	1.19479e7	0.533635
$872$	0	0
$873$	−1.79625e7	−0.797683
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.73823e6	0.295833	0.147917	−	0.989000i	$-0.452743\pi$
$877$	6.73823e6	0.295833	0.147917	+	0.989000i	$0.452743\pi$
$878$	0	0
$879$	1.95681e7	0.854231
$880$	0	0
$881$	2.26520e7	0.983256	0.491628	−	0.870805i	$-0.336402\pi$
$881$	2.26520e7	0.983256	0.491628	+	0.870805i	$0.336402\pi$
$882$	0	0
$883$	7.92988e6	0.342267	0.171133	−	0.985248i	$-0.445257\pi$
$883$	7.92988e6	0.342267	0.171133	+	0.985248i	$0.445257\pi$
$884$	0	0
$885$	6.23259e6	0.267492
$886$	0	0
$887$	347573.	0.0148333	0.00741663	−	0.999972i	$-0.497639\pi$
$887$	347573.	0.0148333	0.00741663	+	0.999972i	$0.497639\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	7.64902e6	0.322784
$892$	0	0
$893$	1.78391e7	0.748589
$894$	0	0
$895$	−1.32466e7	−0.552773
$896$	0	0
$897$	−7.73699e6	−0.321063
$898$	0	0
$899$	−3.05279e6	−0.125979
$900$	0	0
$901$	1.69104e7	0.693971
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.12654e7	−0.863083
$906$	0	0
$907$	−1.67448e7	−0.675869	−0.337935	−	0.941170i	$-0.609728\pi$
$907$	−1.67448e7	−0.675869	−0.337935	+	0.941170i	$0.609728\pi$
$908$	0	0
$909$	3.63294e6	0.145831
$910$	0	0
$911$	−8.36320e6	−0.333869	−0.166935	−	0.985968i	$-0.553387\pi$
$911$	−8.36320e6	−0.333869	−0.166935	+	0.985968i	$0.553387\pi$
$912$	0	0
$913$	366641.	0.0145568
$914$	0	0
$915$	103006.	0.00406733
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−2.32444e7	−0.907883	−0.453942	−	0.891031i	$-0.649983\pi$
$919$	−2.32444e7	−0.907883	−0.453942	+	0.891031i	$0.649983\pi$
$920$	0	0
$921$	1.07874e7	0.419052
$922$	0	0
$923$	−5.32902e7	−2.05894
$924$	0	0
$925$	1.93958e7	0.745339
$926$	0	0
$927$	4.20285e6	0.160637
$928$	0	0
$929$	−2.65468e7	−1.00919	−0.504595	−	0.863356i	$-0.668358\pi$
$929$	−2.65468e7	−1.00919	−0.504595	+	0.863356i	$0.668358\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.24885e7	1.22187
$934$	0	0
$935$	1.60175e7	0.599193
$936$	0	0
$937$	4.39209e7	1.63426	0.817132	−	0.576451i	$-0.195563\pi$
$937$	4.39209e7	1.63426	0.817132	+	0.576451i	$0.195563\pi$
$938$	0	0
$939$	−3.07664e7	−1.13871
$940$	0	0
$941$	2.05536e7	0.756684	0.378342	−	0.925666i	$-0.376494\pi$
$941$	2.05536e7	0.756684	0.378342	+	0.925666i	$0.376494\pi$
$942$	0	0
$943$	5.88175e6	0.215391
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.11519e7	−1.12878	−0.564391	−	0.825507i	$-0.690889\pi$
$947$	−3.11519e7	−1.12878	−0.564391	+	0.825507i	$0.690889\pi$
$948$	0	0
$949$	−6.17081e7	−2.22421
$950$	0	0
$951$	−3.23887e7	−1.16129
$952$	0	0
$953$	−4.15913e7	−1.48344	−0.741721	−	0.670709i	$-0.765990\pi$
$953$	−4.15913e7	−1.48344	−0.741721	+	0.670709i	$0.765990\pi$
$954$	0	0
$955$	7.55383e6	0.268015
$956$	0	0
$957$	−8.00340e6	−0.282485
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.51242e7	−0.877573
$962$	0	0
$963$	2.43519e7	0.846187
$964$	0	0
$965$	2.16494e6	0.0748390
$966$	0	0
$967$	4.38061e7	1.50650	0.753249	−	0.657735i	$-0.228485\pi$
$967$	4.38061e7	1.50650	0.753249	+	0.657735i	$0.228485\pi$
$968$	0	0
$969$	2.77147e7	0.948200
$970$	0	0
$971$	−3.71158e7	−1.26331	−0.631656	−	0.775249i	$-0.717624\pi$
$971$	−3.71158e7	−1.26331	−0.631656	+	0.775249i	$0.717624\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.03494e7	−0.685550
$976$	0	0
$977$	1.74795e7	0.585860	0.292930	−	0.956134i	$-0.405370\pi$
$977$	1.74795e7	0.585860	0.292930	+	0.956134i	$0.405370\pi$
$978$	0	0
$979$	−1.79816e7	−0.599614
$980$	0	0
$981$	−2.64420e7	−0.877245
$982$	0	0
$983$	5.36400e7	1.77054	0.885268	−	0.465081i	$-0.153975\pi$
$983$	5.36400e7	1.77054	0.885268	+	0.465081i	$0.153975\pi$
$984$	0	0
$985$	1.97784e7	0.649532
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.86666e7	0.606839
$990$	0	0
$991$	−2.59845e7	−0.840485	−0.420242	−	0.907412i	$-0.638055\pi$
$991$	−2.59845e7	−0.840485	−0.420242	+	0.907412i	$0.638055\pi$
$992$	0	0
$993$	3.80250e6	0.122376
$994$	0	0
$995$	9.18950e6	0.294262
$996$	0	0
$997$	9.23388e6	0.294203	0.147101	−	0.989121i	$-0.453006\pi$
$997$	9.23388e6	0.294203	0.147101	+	0.989121i	$0.453006\pi$
$998$	0	0
$999$	−3.21015e7	−1.01768

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.6.a.j.1.2		5
4.3	odd	2		784.6.a.bl.1.4		5
7.2	even	3		392.6.i.o.361.4		10
7.3	odd	6		56.6.i.b.9.2	✓	10
7.4	even	3		392.6.i.o.177.4		10
7.5	odd	6		56.6.i.b.25.2	yes	10
7.6	odd	2		392.6.a.k.1.4		5
21.5	even	6		504.6.s.b.361.4		10
21.17	even	6		504.6.s.b.289.4		10
28.3	even	6		112.6.i.f.65.4		10
28.19	even	6		112.6.i.f.81.4		10
28.27	even	2		784.6.a.bk.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.i.b.9.2	✓	10	7.3	odd	6
56.6.i.b.25.2	yes	10	7.5	odd	6
112.6.i.f.65.4		10	28.3	even	6
112.6.i.f.81.4		10	28.19	even	6
392.6.a.j.1.2		5	1.1	even	1	trivial
392.6.a.k.1.4		5	7.6	odd	2
392.6.i.o.177.4		10	7.4	even	3
392.6.i.o.361.4		10	7.2	even	3
504.6.s.b.289.4		10	21.17	even	6
504.6.s.b.361.4		10	21.5	even	6
784.6.a.bk.1.2		5	28.27	even	2
784.6.a.bl.1.4		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

\(f(q)\)	\(=\)	\(q-11.2867 q^{3} -26.0667 q^{5} -115.611 q^{9} -434.865 q^{11} -737.247 q^{13} +294.206 q^{15} +1413.04 q^{17} -1737.76 q^{19} -929.808 q^{23} -2445.53 q^{25} +4047.52 q^{27} -1630.63 q^{29} +1872.16 q^{31} +4908.17 q^{33} -7931.14 q^{37} +8321.06 q^{39} -6325.76 q^{41} -20075.7 q^{43} +3013.61 q^{45} -10265.6 q^{47} -15948.5 q^{51} +11967.4 q^{53} +11335.5 q^{55} +19613.5 q^{57} +21184.5 q^{59} +350.115 q^{61} +19217.6 q^{65} -16206.0 q^{67} +10494.4 q^{69} +72282.7 q^{71} +83700.7 q^{73} +27601.8 q^{75} -47131.0 q^{79} -17589.4 q^{81} -843.116 q^{83} -36833.3 q^{85} +18404.3 q^{87} +41349.9 q^{89} -21130.4 q^{93} +45297.7 q^{95} +155369. q^{97} +50275.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 81 q^{5} + 390 q^{9} + 361 q^{11} + 342 q^{13} - 1049 q^{15} + 1809 q^{17} + 1277 q^{19} + 911 q^{23} + 3940 q^{25} - 4751 q^{27} + 5442 q^{29} + 2187 q^{31} - 5553 q^{33} - 8181 q^{37}+ \cdots + 249798 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 81 * q^5 + 390 * q^9 + 361 * q^11 + 342 * q^13 - 1049 * q^15 + 1809 * q^17 + 1277 * q^19 + 911 * q^23 + 3940 * q^25 - 4751 * q^27 + 5442 * q^29 + 2187 * q^31 - 5553 * q^33 - 8181 * q^37 - 3422 * q^39 + 16578 * q^41 + 6332 * q^43 + 41310 * q^45 + 16101 * q^47 - 67865 * q^51 + 16047 * q^53 + 45629 * q^55 + 22347 * q^57 + 71027 * q^59 + 31093 * q^61 - 64370 * q^65 + 47981 * q^67 + 137249 * q^69 + 22512 * q^71 + 123333 * q^73 - 45460 * q^75 - 212481 * q^79 + 52917 * q^81 + 87460 * q^83 + 222141 * q^85 + 318070 * q^87 + 129045 * q^89 - 252835 * q^93 + 300417 * q^95 + 328274 * q^97 + 249798 * q^99

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