LMFDB - Embedded newform 624.6.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,6,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("624.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	624.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:	$100.079503563$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 14 $ x^2 - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.74166$ of defining polynomial
Character	$\chi$	$=$	624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +13.4166 q^{5} -118.316 q^{7} +81.0000 q^{9} +612.299 q^{11} -169.000 q^{13} +120.749 q^{15} -927.231 q^{17} -1110.28 q^{19} -1064.85 q^{21} -373.197 q^{23} -2945.00 q^{25} +729.000 q^{27} -4434.60 q^{29} +8193.87 q^{31} +5510.69 q^{33} -1587.40 q^{35} +3181.38 q^{37} -1521.00 q^{39} -10101.2 q^{41} +20769.8 q^{43} +1086.74 q^{45} -7139.89 q^{47} -2808.21 q^{49} -8345.08 q^{51} +6726.59 q^{53} +8214.96 q^{55} -9992.54 q^{57} -43325.0 q^{59} -30637.1 q^{61} -9583.63 q^{63} -2267.40 q^{65} -64991.9 q^{67} -3358.77 q^{69} +51374.4 q^{71} +33468.4 q^{73} -26505.0 q^{75} -72445.1 q^{77} +51337.9 q^{79} +6561.00 q^{81} -107064. q^{83} -12440.3 q^{85} -39911.4 q^{87} -121427. q^{89} +19995.5 q^{91} +73744.8 q^{93} -14896.2 q^{95} -81426.5 q^{97} +49596.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} - 48 q^{5} - 72 q^{7} + 162 q^{9} + 596 q^{11} - 338 q^{13} - 432 q^{15} - 268 q^{17} - 1128 q^{19} - 648 q^{21} + 1768 q^{23} - 2298 q^{25} + 1458 q^{27} - 7612 q^{29} + 4160 q^{31} + 5364 q^{33}+ \cdots + 48276 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 - 48 * q^5 - 72 * q^7 + 162 * q^9 + 596 * q^11 - 338 * q^13 - 432 * q^15 - 268 * q^17 - 1128 * q^19 - 648 * q^21 + 1768 * q^23 - 2298 * q^25 + 1458 * q^27 - 7612 * q^29 + 4160 * q^31 + 5364 * q^33 - 4432 * q^35 + 17468 * q^37 - 3042 * q^39 - 28000 * q^41 + 24328 * q^43 - 3888 * q^45 + 18108 * q^47 - 17470 * q^49 - 2412 * q^51 + 1420 * q^53 + 9216 * q^55 - 10152 * q^57 - 6788 * q^59 - 37148 * q^61 - 5832 * q^63 + 8112 * q^65 - 106112 * q^67 + 15912 * q^69 + 30460 * q^71 - 37620 * q^73 - 20682 * q^75 - 73200 * q^77 + 2160 * q^79 + 13122 * q^81 - 207004 * q^83 - 52928 * q^85 - 68508 * q^87 - 74136 * q^89 + 12168 * q^91 + 37440 * q^93 - 13808 * q^95 - 121156 * q^97 + 48276 * 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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	13.4166	0.240003	0.120001	−	0.992774i	$-0.461710\pi$
$5$	13.4166	0.240003	0.120001	+	0.992774i	$0.461710\pi$
$6$	0	0
$7$	−118.316	−0.912641	−0.456321	−	0.889815i	$-0.650833\pi$
$7$	−118.316	−0.912641	−0.456321	+	0.889815i	$0.650833\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	612.299	1.52575	0.762873	−	0.646549i	$-0.223788\pi$
$11$	612.299	1.52575	0.762873	+	0.646549i	$0.223788\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	0	0
$15$	120.749	0.138566
$16$	0	0
$17$	−927.231	−0.778154	−0.389077	−	0.921205i	$-0.627206\pi$
$17$	−927.231	−0.778154	−0.389077	+	0.921205i	$0.627206\pi$
$18$	0	0
$19$	−1110.28	−0.705585	−0.352792	−	0.935702i	$-0.614768\pi$
$19$	−1110.28	−0.705585	−0.352792	+	0.935702i	$0.614768\pi$
$20$	0	0
$21$	−1064.85	−0.526914
$22$	0	0
$23$	−373.197	−0.147102	−0.0735510	−	0.997291i	$-0.523433\pi$
$23$	−373.197	−0.147102	−0.0735510	+	0.997291i	$0.523433\pi$
$24$	0	0
$25$	−2945.00	−0.942399
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−4434.60	−0.979173	−0.489586	−	0.871955i	$-0.662852\pi$
$29$	−4434.60	−0.979173	−0.489586	+	0.871955i	$0.662852\pi$
$30$	0	0
$31$	8193.87	1.53139	0.765693	−	0.643206i	$-0.222396\pi$
$31$	8193.87	1.53139	0.765693	+	0.643206i	$0.222396\pi$
$32$	0	0
$33$	5510.69	0.880889
$34$	0	0
$35$	−1587.40	−0.219037
$36$	0	0
$37$	3181.38	0.382042	0.191021	−	0.981586i	$-0.438820\pi$
$37$	3181.38	0.382042	0.191021	+	0.981586i	$0.438820\pi$
$38$	0	0
$39$	−1521.00	−0.160128
$40$	0	0
$41$	−10101.2	−0.938454	−0.469227	−	0.883078i	$-0.655467\pi$
$41$	−10101.2	−0.938454	−0.469227	+	0.883078i	$0.655467\pi$
$42$	0	0
$43$	20769.8	1.71302	0.856508	−	0.516134i	$-0.172629\pi$
$43$	20769.8	1.71302	0.856508	+	0.516134i	$0.172629\pi$
$44$	0	0
$45$	1086.74	0.0800010
$46$	0	0
$47$	−7139.89	−0.471462	−0.235731	−	0.971818i	$-0.575748\pi$
$47$	−7139.89	−0.471462	−0.235731	+	0.971818i	$0.575748\pi$
$48$	0	0
$49$	−2808.21	−0.167086
$50$	0	0
$51$	−8345.08	−0.449268
$52$	0	0
$53$	6726.59	0.328931	0.164466	−	0.986383i	$-0.447410\pi$
$53$	6726.59	0.328931	0.164466	+	0.986383i	$0.447410\pi$
$54$	0	0
$55$	8214.96	0.366183
$56$	0	0
$57$	−9992.54	−0.407370
$58$	0	0
$59$	−43325.0	−1.62035	−0.810174	−	0.586190i	$-0.800627\pi$
$59$	−43325.0	−1.62035	−0.810174	+	0.586190i	$0.800627\pi$
$60$	0	0
$61$	−30637.1	−1.05420	−0.527100	−	0.849803i	$-0.676721\pi$
$61$	−30637.1	−1.05420	−0.527100	+	0.849803i	$0.676721\pi$
$62$	0	0
$63$	−9583.63	−0.304214
$64$	0	0
$65$	−2267.40	−0.0665648
$66$	0	0
$67$	−64991.9	−1.76877	−0.884387	−	0.466755i	$-0.845423\pi$
$67$	−64991.9	−1.76877	−0.884387	+	0.466755i	$0.845423\pi$
$68$	0	0
$69$	−3358.77	−0.0849293
$70$	0	0
$71$	51374.4	1.20949	0.604743	−	0.796421i	$-0.293276\pi$
$71$	51374.4	1.20949	0.604743	+	0.796421i	$0.293276\pi$
$72$	0	0
$73$	33468.4	0.735069	0.367535	−	0.930010i	$-0.380202\pi$
$73$	33468.4	0.735069	0.367535	+	0.930010i	$0.380202\pi$
$74$	0	0
$75$	−26505.0	−0.544094
$76$	0	0
$77$	−72445.1	−1.39246
$78$	0	0
$79$	51337.9	0.925488	0.462744	−	0.886492i	$-0.346865\pi$
$79$	51337.9	0.925488	0.462744	+	0.886492i	$0.346865\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−107064.	−1.70588	−0.852940	−	0.522009i	$-0.825183\pi$
$83$	−107064.	−1.70588	−0.852940	+	0.522009i	$0.825183\pi$
$84$	0	0
$85$	−12440.3	−0.186759
$86$	0	0
$87$	−39911.4	−0.565326
$88$	0	0
$89$	−121427.	−1.62496	−0.812478	−	0.582992i	$-0.801882\pi$
$89$	−121427.	−1.62496	−0.812478	+	0.582992i	$0.801882\pi$
$90$	0	0
$91$	19995.5	0.253121
$92$	0	0
$93$	73744.8	0.884146
$94$	0	0
$95$	−14896.2	−0.169342
$96$	0	0
$97$	−81426.5	−0.878692	−0.439346	−	0.898318i	$-0.644790\pi$
$97$	−81426.5	−0.878692	−0.439346	+	0.898318i	$0.644790\pi$
$98$	0	0
$99$	49596.2	0.508582
$100$	0	0
$101$	166986.	1.62884	0.814418	−	0.580279i	$-0.197056\pi$
$101$	166986.	1.62884	0.814418	+	0.580279i	$0.197056\pi$
$102$	0	0
$103$	38605.2	0.358553	0.179276	−	0.983799i	$-0.442624\pi$
$103$	38605.2	0.358553	0.179276	+	0.983799i	$0.442624\pi$
$104$	0	0
$105$	−14286.6	−0.126461
$106$	0	0
$107$	73451.0	0.620210	0.310105	−	0.950702i	$-0.399636\pi$
$107$	73451.0	0.620210	0.310105	+	0.950702i	$0.399636\pi$
$108$	0	0
$109$	−102802.	−0.828775	−0.414388	−	0.910100i	$-0.636004\pi$
$109$	−102802.	−0.828775	−0.414388	+	0.910100i	$0.636004\pi$
$110$	0	0
$111$	28632.4	0.220572
$112$	0	0
$113$	−58292.4	−0.429454	−0.214727	−	0.976674i	$-0.568886\pi$
$113$	−58292.4	−0.429454	−0.214727	+	0.976674i	$0.568886\pi$
$114$	0	0
$115$	−5007.02	−0.0353049
$116$	0	0
$117$	−13689.0	−0.0924500
$118$	0	0
$119$	109707.	0.710176
$120$	0	0
$121$	213859.	1.32790
$122$	0	0
$123$	−90910.7	−0.541817
$124$	0	0
$125$	−81438.5	−0.466181
$126$	0	0
$127$	−84798.0	−0.466527	−0.233263	−	0.972414i	$-0.574940\pi$
$127$	−84798.0	−0.466527	−0.233263	+	0.972414i	$0.574940\pi$
$128$	0	0
$129$	186928.	0.989010
$130$	0	0
$131$	−130223.	−0.662994	−0.331497	−	0.943456i	$-0.607554\pi$
$131$	−130223.	−0.662994	−0.331497	+	0.943456i	$0.607554\pi$
$132$	0	0
$133$	131365.	0.643946
$134$	0	0
$135$	9780.68	0.0461886
$136$	0	0
$137$	−100697.	−0.458371	−0.229185	−	0.973383i	$-0.573606\pi$
$137$	−100697.	−0.458371	−0.229185	+	0.973383i	$0.573606\pi$
$138$	0	0
$139$	−336666.	−1.47796	−0.738978	−	0.673729i	$-0.764691\pi$
$139$	−336666.	−1.47796	−0.738978	+	0.673729i	$0.764691\pi$
$140$	0	0
$141$	−64259.0	−0.272199
$142$	0	0
$143$	−103479.	−0.423166
$144$	0	0
$145$	−59497.1	−0.235004
$146$	0	0
$147$	−25273.9	−0.0964672
$148$	0	0
$149$	110168.	0.406528	0.203264	−	0.979124i	$-0.434845\pi$
$149$	110168.	0.406528	0.203264	+	0.979124i	$0.434845\pi$
$150$	0	0
$151$	−402104.	−1.43515	−0.717573	−	0.696484i	$-0.754747\pi$
$151$	−402104.	−1.43515	−0.717573	+	0.696484i	$0.754747\pi$
$152$	0	0
$153$	−75105.7	−0.259385
$154$	0	0
$155$	109934.	0.367537
$156$	0	0
$157$	−337233.	−1.09190	−0.545948	−	0.837819i	$-0.683830\pi$
$157$	−337233.	−1.09190	−0.545948	+	0.837819i	$0.683830\pi$
$158$	0	0
$159$	60539.3	0.189908
$160$	0	0
$161$	44155.3	0.134251
$162$	0	0
$163$	−416815.	−1.22878	−0.614390	−	0.789002i	$-0.710598\pi$
$163$	−416815.	−1.22878	−0.614390	+	0.789002i	$0.710598\pi$
$164$	0	0
$165$	73934.6	0.211416
$166$	0	0
$167$	219728.	0.609669	0.304835	−	0.952405i	$-0.401399\pi$
$167$	219728.	0.609669	0.304835	+	0.952405i	$0.401399\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	−89932.8	−0.235195
$172$	0	0
$173$	−438612.	−1.11421	−0.557103	−	0.830444i	$-0.688087\pi$
$173$	−438612.	−1.11421	−0.557103	+	0.830444i	$0.688087\pi$
$174$	0	0
$175$	348441.	0.860072
$176$	0	0
$177$	−389925.	−0.935508
$178$	0	0
$179$	489393.	1.14163	0.570815	−	0.821079i	$-0.306628\pi$
$179$	489393.	1.14163	0.570815	+	0.821079i	$0.306628\pi$
$180$	0	0
$181$	−143753.	−0.326152	−0.163076	−	0.986614i	$-0.552142\pi$
$181$	−143753.	−0.326152	−0.163076	+	0.986614i	$0.552142\pi$
$182$	0	0
$183$	−275734.	−0.608643
$184$	0	0
$185$	42683.2	0.0916913
$186$	0	0
$187$	−567743.	−1.18727
$188$	0	0
$189$	−86252.7	−0.175638
$190$	0	0
$191$	−673141.	−1.33513	−0.667563	−	0.744553i	$-0.732663\pi$
$191$	−673141.	−1.33513	−0.667563	+	0.744553i	$0.732663\pi$
$192$	0	0
$193$	719060.	1.38954	0.694772	−	0.719230i	$-0.255505\pi$
$193$	719060.	1.38954	0.694772	+	0.719230i	$0.255505\pi$
$194$	0	0
$195$	−20406.6	−0.0384312
$196$	0	0
$197$	−862345.	−1.58313	−0.791563	−	0.611088i	$-0.790732\pi$
$197$	−862345.	−1.58313	−0.791563	+	0.611088i	$0.790732\pi$
$198$	0	0
$199$	−709292.	−1.26967	−0.634837	−	0.772646i	$-0.718933\pi$
$199$	−709292.	−1.26967	−0.634837	+	0.772646i	$0.718933\pi$
$200$	0	0
$201$	−584927.	−1.02120
$202$	0	0
$203$	524686.	0.893633
$204$	0	0
$205$	−135523.	−0.225232
$206$	0	0
$207$	−30228.9	−0.0490340
$208$	0	0
$209$	−679825.	−1.07654
$210$	0	0
$211$	611270.	0.945206	0.472603	−	0.881275i	$-0.343314\pi$
$211$	611270.	0.945206	0.472603	+	0.881275i	$0.343314\pi$
$212$	0	0
$213$	462370.	0.698297
$214$	0	0
$215$	278660.	0.411129
$216$	0	0
$217$	−969469.	−1.39761
$218$	0	0
$219$	301216.	0.424392
$220$	0	0
$221$	156702.	0.215821
$222$	0	0
$223$	1.24428e6	1.67555	0.837773	−	0.546019i	$-0.183857\pi$
$223$	1.24428e6	1.67555	0.837773	+	0.546019i	$0.183857\pi$
$224$	0	0
$225$	−238545.	−0.314133
$226$	0	0
$227$	671174.	0.864511	0.432255	−	0.901751i	$-0.357718\pi$
$227$	671174.	0.864511	0.432255	+	0.901751i	$0.357718\pi$
$228$	0	0
$229$	−201277.	−0.253632	−0.126816	−	0.991926i	$-0.540476\pi$
$229$	−201277.	−0.253632	−0.126816	+	0.991926i	$0.540476\pi$
$230$	0	0
$231$	−652006.	−0.803936
$232$	0	0
$233$	371265.	0.448016	0.224008	−	0.974587i	$-0.428086\pi$
$233$	371265.	0.448016	0.224008	+	0.974587i	$0.428086\pi$
$234$	0	0
$235$	−95792.9	−0.113152
$236$	0	0
$237$	462041.	0.534331
$238$	0	0
$239$	−139866.	−0.158387	−0.0791934	−	0.996859i	$-0.525234\pi$
$239$	−139866.	−0.158387	−0.0791934	+	0.996859i	$0.525234\pi$
$240$	0	0
$241$	−208957.	−0.231747	−0.115873	−	0.993264i	$-0.536967\pi$
$241$	−208957.	−0.231747	−0.115873	+	0.993264i	$0.536967\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−37676.6	−0.0401011
$246$	0	0
$247$	187638.	0.195694
$248$	0	0
$249$	−963577.	−0.984890
$250$	0	0
$251$	−1.57949e6	−1.58246	−0.791229	−	0.611521i	$-0.790558\pi$
$251$	−1.57949e6	−1.58246	−0.791229	+	0.611521i	$0.790558\pi$
$252$	0	0
$253$	−228508.	−0.224440
$254$	0	0
$255$	−111962.	−0.107826
$256$	0	0
$257$	1.42319e6	1.34409	0.672046	−	0.740510i	$-0.265416\pi$
$257$	1.42319e6	1.34409	0.672046	+	0.740510i	$0.265416\pi$
$258$	0	0
$259$	−376410.	−0.348667
$260$	0	0
$261$	−359202.	−0.326391
$262$	0	0
$263$	−377447.	−0.336485	−0.168243	−	0.985746i	$-0.553809\pi$
$263$	−377447.	−0.336485	−0.168243	+	0.985746i	$0.553809\pi$
$264$	0	0
$265$	90247.7	0.0789444
$266$	0	0
$267$	−1.09285e6	−0.938169
$268$	0	0
$269$	−186022.	−0.156741	−0.0783705	−	0.996924i	$-0.524972\pi$
$269$	−186022.	−0.156741	−0.0783705	+	0.996924i	$0.524972\pi$
$270$	0	0
$271$	−394501.	−0.326306	−0.163153	−	0.986601i	$-0.552166\pi$
$271$	−394501.	−0.326306	−0.163153	+	0.986601i	$0.552166\pi$
$272$	0	0
$273$	179959.	0.146140
$274$	0	0
$275$	−1.80322e6	−1.43786
$276$	0	0
$277$	801681.	0.627773	0.313886	−	0.949461i	$-0.398369\pi$
$277$	801681.	0.627773	0.313886	+	0.949461i	$0.398369\pi$
$278$	0	0
$279$	663703.	0.510462
$280$	0	0
$281$	−258921.	−0.195615	−0.0978073	−	0.995205i	$-0.531183\pi$
$281$	−258921.	−0.195615	−0.0978073	+	0.995205i	$0.531183\pi$
$282$	0	0
$283$	−1.39529e6	−1.03562	−0.517809	−	0.855496i	$-0.673252\pi$
$283$	−1.39529e6	−1.03562	−0.517809	+	0.855496i	$0.673252\pi$
$284$	0	0
$285$	−134066.	−0.0977699
$286$	0	0
$287$	1.19514e6	0.856472
$288$	0	0
$289$	−560099.	−0.394476
$290$	0	0
$291$	−732839.	−0.507313
$292$	0	0
$293$	1.67750e6	1.14155	0.570774	−	0.821107i	$-0.306643\pi$
$293$	1.67750e6	1.14155	0.570774	+	0.821107i	$0.306643\pi$
$294$	0	0
$295$	−581273.	−0.388888
$296$	0	0
$297$	446366.	0.293630
$298$	0	0
$299$	63070.3	0.0407987
$300$	0	0
$301$	−2.45741e6	−1.56337
$302$	0	0
$303$	1.50288e6	0.940409
$304$	0	0
$305$	−411045.	−0.253011
$306$	0	0
$307$	−1.03523e6	−0.626889	−0.313444	−	0.949607i	$-0.601483\pi$
$307$	−1.03523e6	−0.626889	−0.313444	+	0.949607i	$0.601483\pi$
$308$	0	0
$309$	347447.	0.207010
$310$	0	0
$311$	405024.	0.237454	0.118727	−	0.992927i	$-0.462119\pi$
$311$	405024.	0.237454	0.118727	+	0.992927i	$0.462119\pi$
$312$	0	0
$313$	−1.87084e6	−1.07939	−0.539693	−	0.841862i	$-0.681460\pi$
$313$	−1.87084e6	−1.07939	−0.539693	+	0.841862i	$0.681460\pi$
$314$	0	0
$315$	−128580.	−0.0730122
$316$	0	0
$317$	65497.1	0.0366078	0.0183039	−	0.999832i	$-0.494173\pi$
$317$	65497.1	0.0366078	0.0183039	+	0.999832i	$0.494173\pi$
$318$	0	0
$319$	−2.71530e6	−1.49397
$320$	0	0
$321$	661059.	0.358078
$322$	0	0
$323$	1.02949e6	0.549054
$324$	0	0
$325$	497704.	0.261374
$326$	0	0
$327$	−925221.	−0.478494
$328$	0	0
$329$	844767.	0.430276
$330$	0	0
$331$	1.39675e6	0.700726	0.350363	−	0.936614i	$-0.386058\pi$
$331$	1.39675e6	0.700726	0.350363	+	0.936614i	$0.386058\pi$
$332$	0	0
$333$	257692.	0.127347
$334$	0	0
$335$	−871968.	−0.424511
$336$	0	0
$337$	−1.31579e6	−0.631121	−0.315561	−	0.948905i	$-0.602193\pi$
$337$	−1.31579e6	−0.631121	−0.315561	+	0.948905i	$0.602193\pi$
$338$	0	0
$339$	−524632.	−0.247945
$340$	0	0
$341$	5.01710e6	2.33651
$342$	0	0
$343$	2.32080e6	1.06513
$344$	0	0
$345$	−45063.2	−0.0203833
$346$	0	0
$347$	−1.12101e6	−0.499790	−0.249895	−	0.968273i	$-0.580396\pi$
$347$	−1.12101e6	−0.499790	−0.249895	+	0.968273i	$0.580396\pi$
$348$	0	0
$349$	−733696.	−0.322443	−0.161221	−	0.986918i	$-0.551543\pi$
$349$	−733696.	−0.322443	−0.161221	+	0.986918i	$0.551543\pi$
$350$	0	0
$351$	−123201.	−0.0533761
$352$	0	0
$353$	−190812.	−0.0815021	−0.0407510	−	0.999169i	$-0.512975\pi$
$353$	−190812.	−0.0815021	−0.0407510	+	0.999169i	$0.512975\pi$
$354$	0	0
$355$	689269.	0.290280
$356$	0	0
$357$	987361.	0.410020
$358$	0	0
$359$	−530326.	−0.217174	−0.108587	−	0.994087i	$-0.534633\pi$
$359$	−530326.	−0.217174	−0.108587	+	0.994087i	$0.534633\pi$
$360$	0	0
$361$	−1.24337e6	−0.502150
$362$	0	0
$363$	1.92473e6	0.766662
$364$	0	0
$365$	449032.	0.176419
$366$	0	0
$367$	1.87975e6	0.728509	0.364255	−	0.931299i	$-0.381324\pi$
$367$	1.87975e6	0.728509	0.364255	+	0.931299i	$0.381324\pi$
$368$	0	0
$369$	−818197.	−0.312818
$370$	0	0
$371$	−795866.	−0.300196
$372$	0	0
$373$	2.61492e6	0.973165	0.486583	−	0.873635i	$-0.338243\pi$
$373$	2.61492e6	0.973165	0.486583	+	0.873635i	$0.338243\pi$
$374$	0	0
$375$	−732947.	−0.269150
$376$	0	0
$377$	749447.	0.271574
$378$	0	0
$379$	3.76631e6	1.34684	0.673422	−	0.739258i	$-0.264824\pi$
$379$	3.76631e6	1.34684	0.673422	+	0.739258i	$0.264824\pi$
$380$	0	0
$381$	−763182.	−0.269349
$382$	0	0
$383$	1.42058e6	0.494845	0.247422	−	0.968908i	$-0.420416\pi$
$383$	1.42058e6	0.494845	0.247422	+	0.968908i	$0.420416\pi$
$384$	0	0
$385$	−971965.	−0.334194
$386$	0	0
$387$	1.68235e6	0.571005
$388$	0	0
$389$	−4.90670e6	−1.64405	−0.822026	−	0.569450i	$-0.807156\pi$
$389$	−4.90670e6	−1.64405	−0.822026	+	0.569450i	$0.807156\pi$
$390$	0	0
$391$	346040.	0.114468
$392$	0	0
$393$	−1.17201e6	−0.382780
$394$	0	0
$395$	688779.	0.222120
$396$	0	0
$397$	1.52253e6	0.484831	0.242416	−	0.970173i	$-0.422060\pi$
$397$	1.52253e6	0.484831	0.242416	+	0.970173i	$0.422060\pi$
$398$	0	0
$399$	1.18228e6	0.371782
$400$	0	0
$401$	−388493.	−0.120649	−0.0603243	−	0.998179i	$-0.519213\pi$
$401$	−388493.	−0.120649	−0.0603243	+	0.998179i	$0.519213\pi$
$402$	0	0
$403$	−1.38476e6	−0.424730
$404$	0	0
$405$	88026.1	0.0266670
$406$	0	0
$407$	1.94796e6	0.582899
$408$	0	0
$409$	1.28523e6	0.379902	0.189951	−	0.981794i	$-0.439167\pi$
$409$	1.28523e6	0.379902	0.189951	+	0.981794i	$0.439167\pi$
$410$	0	0
$411$	−906276.	−0.264640
$412$	0	0
$413$	5.12606e6	1.47880
$414$	0	0
$415$	−1.43643e6	−0.409416
$416$	0	0
$417$	−3.02999e6	−0.853299
$418$	0	0
$419$	4.49606e6	1.25111	0.625557	−	0.780179i	$-0.284872\pi$
$419$	4.49606e6	1.25111	0.625557	+	0.780179i	$0.284872\pi$
$420$	0	0
$421$	5.20144e6	1.43027	0.715135	−	0.698986i	$-0.246365\pi$
$421$	5.20144e6	1.43027	0.715135	+	0.698986i	$0.246365\pi$
$422$	0	0
$423$	−578331.	−0.157154
$424$	0	0
$425$	2.73069e6	0.733332
$426$	0	0
$427$	3.62487e6	0.962107
$428$	0	0
$429$	−931307.	−0.244315
$430$	0	0
$431$	6.97009e6	1.80736	0.903681	−	0.428205i	$-0.140854\pi$
$431$	6.97009e6	1.80736	0.903681	+	0.428205i	$0.140854\pi$
$432$	0	0
$433$	−2.91882e6	−0.748149	−0.374074	−	0.927399i	$-0.622040\pi$
$433$	−2.91882e6	−0.748149	−0.374074	+	0.927399i	$0.622040\pi$
$434$	0	0
$435$	−535474.	−0.135680
$436$	0	0
$437$	414354.	0.103793
$438$	0	0
$439$	−2.54724e6	−0.630825	−0.315413	−	0.948955i	$-0.602143\pi$
$439$	−2.54724e6	−0.630825	−0.315413	+	0.948955i	$0.602143\pi$
$440$	0	0
$441$	−227465.	−0.0556953
$442$	0	0
$443$	−473306.	−0.114586	−0.0572931	−	0.998357i	$-0.518247\pi$
$443$	−473306.	−0.114586	−0.0572931	+	0.998357i	$0.518247\pi$
$444$	0	0
$445$	−1.62914e6	−0.389994
$446$	0	0
$447$	991514.	0.234709
$448$	0	0
$449$	3.22203e6	0.754248	0.377124	−	0.926163i	$-0.376913\pi$
$449$	3.22203e6	0.754248	0.377124	+	0.926163i	$0.376913\pi$
$450$	0	0
$451$	−6.18495e6	−1.43184
$452$	0	0
$453$	−3.61893e6	−0.828582
$454$	0	0
$455$	268271.	0.0607498
$456$	0	0
$457$	2.72127e6	0.609511	0.304756	−	0.952431i	$-0.401425\pi$
$457$	2.72127e6	0.609511	0.304756	+	0.952431i	$0.401425\pi$
$458$	0	0
$459$	−675952.	−0.149756
$460$	0	0
$461$	1.09863e6	0.240769	0.120385	−	0.992727i	$-0.461587\pi$
$461$	1.09863e6	0.240769	0.120385	+	0.992727i	$0.461587\pi$
$462$	0	0
$463$	2.65063e6	0.574642	0.287321	−	0.957834i	$-0.407235\pi$
$463$	2.65063e6	0.574642	0.287321	+	0.957834i	$0.407235\pi$
$464$	0	0
$465$	989403.	0.212198
$466$	0	0
$467$	2.26844e6	0.481322	0.240661	−	0.970609i	$-0.422636\pi$
$467$	2.26844e6	0.481322	0.240661	+	0.970609i	$0.422636\pi$
$468$	0	0
$469$	7.68961e6	1.61426
$470$	0	0
$471$	−3.03510e6	−0.630406
$472$	0	0
$473$	1.27173e7	2.61363
$474$	0	0
$475$	3.26978e6	0.664942
$476$	0	0
$477$	544853.	0.109644
$478$	0	0
$479$	−1.59887e6	−0.318401	−0.159200	−	0.987246i	$-0.550892\pi$
$479$	−1.59887e6	−0.318401	−0.159200	+	0.987246i	$0.550892\pi$
$480$	0	0
$481$	−537653.	−0.105959
$482$	0	0
$483$	397398.	0.0775100
$484$	0	0
$485$	−1.09246e6	−0.210889
$486$	0	0
$487$	7.79908e6	1.49012	0.745059	−	0.666998i	$-0.232421\pi$
$487$	7.79908e6	1.49012	0.745059	+	0.666998i	$0.232421\pi$
$488$	0	0
$489$	−3.75133e6	−0.709437
$490$	0	0
$491$	9.05475e6	1.69501	0.847506	−	0.530786i	$-0.178103\pi$
$491$	9.05475e6	1.69501	0.847506	+	0.530786i	$0.178103\pi$
$492$	0	0
$493$	4.11190e6	0.761948
$494$	0	0
$495$	665412.	0.122061
$496$	0	0
$497$	−6.07844e6	−1.10383
$498$	0	0
$499$	738198.	0.132715	0.0663577	−	0.997796i	$-0.478862\pi$
$499$	738198.	0.132715	0.0663577	+	0.997796i	$0.478862\pi$
$500$	0	0
$501$	1.97755e6	0.351993
$502$	0	0
$503$	6.10876e6	1.07655	0.538274	−	0.842770i	$-0.319077\pi$
$503$	6.10876e6	1.07655	0.538274	+	0.842770i	$0.319077\pi$
$504$	0	0
$505$	2.24038e6	0.390925
$506$	0	0
$507$	257049.	0.0444116
$508$	0	0
$509$	−2.30922e6	−0.395067	−0.197534	−	0.980296i	$-0.563293\pi$
$509$	−2.30922e6	−0.395067	−0.197534	+	0.980296i	$0.563293\pi$
$510$	0	0
$511$	−3.95987e6	−0.670855
$512$	0	0
$513$	−809396.	−0.135790
$514$	0	0
$515$	517950.	0.0860537
$516$	0	0
$517$	−4.37175e6	−0.719331
$518$	0	0
$519$	−3.94751e6	−0.643287
$520$	0	0
$521$	−2.27443e6	−0.367094	−0.183547	−	0.983011i	$-0.558758\pi$
$521$	−2.27443e6	−0.367094	−0.183547	+	0.983011i	$0.558758\pi$
$522$	0	0
$523$	−4.26373e6	−0.681609	−0.340805	−	0.940134i	$-0.610700\pi$
$523$	−4.26373e6	−0.681609	−0.340805	+	0.940134i	$0.610700\pi$
$524$	0	0
$525$	3.13597e6	0.496563
$526$	0	0
$527$	−7.59761e6	−1.19166
$528$	0	0
$529$	−6.29707e6	−0.978361
$530$	0	0
$531$	−3.50932e6	−0.540116
$532$	0	0
$533$	1.70710e6	0.260280
$534$	0	0
$535$	985461.	0.148852
$536$	0	0
$537$	4.40454e6	0.659120
$538$	0	0
$539$	−1.71947e6	−0.254931
$540$	0	0
$541$	1.16109e7	1.70558	0.852791	−	0.522252i	$-0.174908\pi$
$541$	1.16109e7	1.70558	0.852791	+	0.522252i	$0.174908\pi$
$542$	0	0
$543$	−1.29378e6	−0.188304
$544$	0	0
$545$	−1.37926e6	−0.198909
$546$	0	0
$547$	−8.74492e6	−1.24965	−0.624824	−	0.780766i	$-0.714829\pi$
$547$	−8.74492e6	−1.24965	−0.624824	+	0.780766i	$0.714829\pi$
$548$	0	0
$549$	−2.48161e6	−0.351400
$550$	0	0
$551$	4.92365e6	0.690890
$552$	0	0
$553$	−6.07412e6	−0.844638
$554$	0	0
$555$	384149.	0.0529380
$556$	0	0
$557$	1.26430e7	1.72669	0.863343	−	0.504617i	$-0.168366\pi$
$557$	1.26430e7	1.72669	0.863343	+	0.504617i	$0.168366\pi$
$558$	0	0
$559$	−3.51010e6	−0.475105
$560$	0	0
$561$	−5.10969e6	−0.685468
$562$	0	0
$563$	1.00809e7	1.34038	0.670190	−	0.742189i	$-0.266213\pi$
$563$	1.00809e7	1.34038	0.670190	+	0.742189i	$0.266213\pi$
$564$	0	0
$565$	−782085.	−0.103070
$566$	0	0
$567$	−776274.	−0.101405
$568$	0	0
$569$	−9.17539e6	−1.18808	−0.594038	−	0.804437i	$-0.702467\pi$
$569$	−9.17539e6	−1.18808	−0.594038	+	0.804437i	$0.702467\pi$
$570$	0	0
$571$	−7.99125e6	−1.02571	−0.512855	−	0.858475i	$-0.671412\pi$
$571$	−7.99125e6	−1.02571	−0.512855	+	0.858475i	$0.671412\pi$
$572$	0	0
$573$	−6.05827e6	−0.770836
$574$	0	0
$575$	1.09906e6	0.138629
$576$	0	0
$577$	−1.21730e7	−1.52215	−0.761077	−	0.648661i	$-0.775329\pi$
$577$	−1.21730e7	−1.52215	−0.761077	+	0.648661i	$0.775329\pi$
$578$	0	0
$579$	6.47154e6	0.802253
$580$	0	0
$581$	1.26674e7	1.55686
$582$	0	0
$583$	4.11868e6	0.501865
$584$	0	0
$585$	−183659.	−0.0221883
$586$	0	0
$587$	7.54508e6	0.903792	0.451896	−	0.892071i	$-0.350748\pi$
$587$	7.54508e6	0.903792	0.451896	+	0.892071i	$0.350748\pi$
$588$	0	0
$589$	−9.09750e6	−1.08052
$590$	0	0
$591$	−7.76110e6	−0.914018
$592$	0	0
$593$	−2.02354e6	−0.236307	−0.118153	−	0.992995i	$-0.537697\pi$
$593$	−2.02354e6	−0.236307	−0.118153	+	0.992995i	$0.537697\pi$
$594$	0	0
$595$	1.47189e6	0.170444
$596$	0	0
$597$	−6.38363e6	−0.733047
$598$	0	0
$599$	−5.94968e6	−0.677527	−0.338763	−	0.940872i	$-0.610009\pi$
$599$	−5.94968e6	−0.677527	−0.338763	+	0.940872i	$0.610009\pi$
$600$	0	0
$601$	−7.08696e6	−0.800339	−0.400169	−	0.916441i	$-0.631049\pi$
$601$	−7.08696e6	−0.800339	−0.400169	+	0.916441i	$0.631049\pi$
$602$	0	0
$603$	−5.26434e6	−0.589591
$604$	0	0
$605$	2.86926e6	0.318700
$606$	0	0
$607$	3.40921e6	0.375562	0.187781	−	0.982211i	$-0.439870\pi$
$607$	3.40921e6	0.375562	0.187781	+	0.982211i	$0.439870\pi$
$608$	0	0
$609$	4.72217e6	0.515939
$610$	0	0
$611$	1.20664e6	0.130760
$612$	0	0
$613$	1.62737e7	1.74918	0.874589	−	0.484865i	$-0.161131\pi$
$613$	1.62737e7	1.74918	0.874589	+	0.484865i	$0.161131\pi$
$614$	0	0
$615$	−1.21971e6	−0.130038
$616$	0	0
$617$	1.29932e7	1.37405	0.687027	−	0.726632i	$-0.258915\pi$
$617$	1.29932e7	1.37405	0.687027	+	0.726632i	$0.258915\pi$
$618$	0	0
$619$	−724313.	−0.0759801	−0.0379900	−	0.999278i	$-0.512096\pi$
$619$	−724313.	−0.0759801	−0.0379900	+	0.999278i	$0.512096\pi$
$620$	0	0
$621$	−272061.	−0.0283098
$622$	0	0
$623$	1.43669e7	1.48300
$624$	0	0
$625$	8.11048e6	0.830514
$626$	0	0
$627$	−6.11842e6	−0.621542
$628$	0	0
$629$	−2.94988e6	−0.297288
$630$	0	0
$631$	−1.68295e7	−1.68267	−0.841333	−	0.540518i	$-0.818228\pi$
$631$	−1.68295e7	−1.68267	−0.841333	+	0.540518i	$0.818228\pi$
$632$	0	0
$633$	5.50143e6	0.545715
$634$	0	0
$635$	−1.13770e6	−0.111968
$636$	0	0
$637$	474588.	0.0463413
$638$	0	0
$639$	4.16133e6	0.403162
$640$	0	0
$641$	−3.52575e6	−0.338927	−0.169464	−	0.985536i	$-0.554204\pi$
$641$	−3.52575e6	−0.338927	−0.169464	+	0.985536i	$0.554204\pi$
$642$	0	0
$643$	−8.25422e6	−0.787315	−0.393657	−	0.919257i	$-0.628790\pi$
$643$	−8.25422e6	−0.787315	−0.393657	+	0.919257i	$0.628790\pi$
$644$	0	0
$645$	2.50794e6	0.237365
$646$	0	0
$647$	1.55380e6	0.145926	0.0729631	−	0.997335i	$-0.476754\pi$
$647$	1.55380e6	0.145926	0.0729631	+	0.997335i	$0.476754\pi$
$648$	0	0
$649$	−2.65278e7	−2.47224
$650$	0	0
$651$	−8.72523e6	−0.806908
$652$	0	0
$653$	1.52484e6	0.139940	0.0699699	−	0.997549i	$-0.477710\pi$
$653$	1.52484e6	0.139940	0.0699699	+	0.997549i	$0.477710\pi$
$654$	0	0
$655$	−1.74715e6	−0.159121
$656$	0	0
$657$	2.71094e6	0.245023
$658$	0	0
$659$	1.70633e7	1.53056	0.765279	−	0.643699i	$-0.222601\pi$
$659$	1.70633e7	1.53056	0.765279	+	0.643699i	$0.222601\pi$
$660$	0	0
$661$	−9.27424e6	−0.825610	−0.412805	−	0.910819i	$-0.635451\pi$
$661$	−9.27424e6	−0.825610	−0.412805	+	0.910819i	$0.635451\pi$
$662$	0	0
$663$	1.41032e6	0.124604
$664$	0	0
$665$	1.76246e6	0.154549
$666$	0	0
$667$	1.65498e6	0.144038
$668$	0	0
$669$	1.11985e7	0.967377
$670$	0	0
$671$	−1.87591e7	−1.60844
$672$	0	0
$673$	1.49675e7	1.27383	0.636916	−	0.770933i	$-0.280210\pi$
$673$	1.49675e7	1.27383	0.636916	+	0.770933i	$0.280210\pi$
$674$	0	0
$675$	−2.14690e6	−0.181365
$676$	0	0
$677$	3.65749e6	0.306698	0.153349	−	0.988172i	$-0.450994\pi$
$677$	3.65749e6	0.306698	0.153349	+	0.988172i	$0.450994\pi$
$678$	0	0
$679$	9.63410e6	0.801930
$680$	0	0
$681$	6.04056e6	0.499125
$682$	0	0
$683$	1.46597e7	1.20247	0.601233	−	0.799074i	$-0.294676\pi$
$683$	1.46597e7	1.20247	0.601233	+	0.799074i	$0.294676\pi$
$684$	0	0
$685$	−1.35101e6	−0.110010
$686$	0	0
$687$	−1.81149e6	−0.146435
$688$	0	0
$689$	−1.13679e6	−0.0912291
$690$	0	0
$691$	2.45762e7	1.95803	0.979015	−	0.203789i	$-0.0653256\pi$
$691$	2.45762e7	1.95803	0.979015	+	0.203789i	$0.0653256\pi$
$692$	0	0
$693$	−5.86805e6	−0.464153
$694$	0	0
$695$	−4.51690e6	−0.354714
$696$	0	0
$697$	9.36614e6	0.730262
$698$	0	0
$699$	3.34138e6	0.258662
$700$	0	0
$701$	1.83004e7	1.40658	0.703292	−	0.710901i	$-0.251713\pi$
$701$	1.83004e7	1.40658	0.703292	+	0.710901i	$0.251713\pi$
$702$	0	0
$703$	−3.53223e6	−0.269563
$704$	0	0
$705$	−862136.	−0.0653286
$706$	0	0
$707$	−1.97572e7	−1.48654
$708$	0	0
$709$	−1.28635e7	−0.961047	−0.480523	−	0.876982i	$-0.659553\pi$
$709$	−1.28635e7	−0.961047	−0.480523	+	0.876982i	$0.659553\pi$
$710$	0	0
$711$	4.15837e6	0.308496
$712$	0	0
$713$	−3.05793e6	−0.225270
$714$	0	0
$715$	−1.38833e6	−0.101561
$716$	0	0
$717$	−1.25880e6	−0.0914446
$718$	0	0
$719$	1.15140e7	0.830626	0.415313	−	0.909679i	$-0.363672\pi$
$719$	1.15140e7	0.830626	0.415313	+	0.909679i	$0.363672\pi$
$720$	0	0
$721$	−4.56763e6	−0.327230
$722$	0	0
$723$	−1.88061e6	−0.133799
$724$	0	0
$725$	1.30599e7	0.922771
$726$	0	0
$727$	8.18615e6	0.574439	0.287219	−	0.957865i	$-0.407269\pi$
$727$	8.18615e6	0.574439	0.287219	+	0.957865i	$0.407269\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.92584e7	−1.33299
$732$	0	0
$733$	−5.14729e6	−0.353849	−0.176925	−	0.984224i	$-0.556615\pi$
$733$	−5.14729e6	−0.353849	−0.176925	+	0.984224i	$0.556615\pi$
$734$	0	0
$735$	−339090.	−0.0231524
$736$	0	0
$737$	−3.97945e7	−2.69870
$738$	0	0
$739$	−8.80802e6	−0.593290	−0.296645	−	0.954988i	$-0.595868\pi$
$739$	−8.80802e6	−0.593290	−0.296645	+	0.954988i	$0.595868\pi$
$740$	0	0
$741$	1.68874e6	0.112984
$742$	0	0
$743$	−1.29105e7	−0.857971	−0.428985	−	0.903311i	$-0.641129\pi$
$743$	−1.29105e7	−0.857971	−0.428985	+	0.903311i	$0.641129\pi$
$744$	0	0
$745$	1.47808e6	0.0975680
$746$	0	0
$747$	−8.67219e6	−0.568627
$748$	0	0
$749$	−8.69047e6	−0.566029
$750$	0	0
$751$	−2.07147e7	−1.34023	−0.670116	−	0.742257i	$-0.733756\pi$
$751$	−2.07147e7	−1.34023	−0.670116	+	0.742257i	$0.733756\pi$
$752$	0	0
$753$	−1.42154e7	−0.913632
$754$	0	0
$755$	−5.39486e6	−0.344439
$756$	0	0
$757$	−2.41483e7	−1.53161	−0.765804	−	0.643074i	$-0.777659\pi$
$757$	−2.41483e7	−1.53161	−0.765804	+	0.643074i	$0.777659\pi$
$758$	0	0
$759$	−2.05657e6	−0.129581
$760$	0	0
$761$	−1.83678e7	−1.14973	−0.574864	−	0.818249i	$-0.694945\pi$
$761$	−1.83678e7	−1.14973	−0.574864	+	0.818249i	$0.694945\pi$
$762$	0	0
$763$	1.21632e7	0.756375
$764$	0	0
$765$	−1.00766e6	−0.0622531
$766$	0	0
$767$	7.32192e6	0.449404
$768$	0	0
$769$	−7.53946e6	−0.459753	−0.229876	−	0.973220i	$-0.573832\pi$
$769$	−7.53946e6	−0.459753	−0.229876	+	0.973220i	$0.573832\pi$
$770$	0	0
$771$	1.28087e7	0.776011
$772$	0	0
$773$	−2.00447e7	−1.20656	−0.603282	−	0.797528i	$-0.706141\pi$
$773$	−2.00447e7	−1.20656	−0.603282	+	0.797528i	$0.706141\pi$
$774$	0	0
$775$	−2.41309e7	−1.44318
$776$	0	0
$777$	−3.38769e6	−0.201303
$778$	0	0
$779$	1.12152e7	0.662159
$780$	0	0
$781$	3.14565e7	1.84537
$782$	0	0
$783$	−3.23282e6	−0.188442
$784$	0	0
$785$	−4.52451e6	−0.262058
$786$	0	0
$787$	−5.74414e6	−0.330589	−0.165294	−	0.986244i	$-0.552857\pi$
$787$	−5.74414e6	−0.330589	−0.165294	+	0.986244i	$0.552857\pi$
$788$	0	0
$789$	−3.39702e6	−0.194270
$790$	0	0
$791$	6.89696e6	0.391937
$792$	0	0
$793$	5.17767e6	0.292383
$794$	0	0
$795$	812230.	0.0455786
$796$	0	0
$797$	1.41069e7	0.786656	0.393328	−	0.919398i	$-0.371324\pi$
$797$	1.41069e7	0.786656	0.393328	+	0.919398i	$0.371324\pi$
$798$	0	0
$799$	6.62033e6	0.366871
$800$	0	0
$801$	−9.83562e6	−0.541652
$802$	0	0
$803$	2.04927e7	1.12153
$804$	0	0
$805$	592413.	0.0322207
$806$	0	0
$807$	−1.67419e6	−0.0904944
$808$	0	0
$809$	1.08102e7	0.580716	0.290358	−	0.956918i	$-0.406226\pi$
$809$	1.08102e7	0.580716	0.290358	+	0.956918i	$0.406226\pi$
$810$	0	0
$811$	1.38874e7	0.741430	0.370715	−	0.928747i	$-0.379113\pi$
$811$	1.38874e7	0.741430	0.370715	+	0.928747i	$0.379113\pi$
$812$	0	0
$813$	−3.55051e6	−0.188393
$814$	0	0
$815$	−5.59223e6	−0.294911
$816$	0	0
$817$	−2.30603e7	−1.20868
$818$	0	0
$819$	1.61963e6	0.0843737
$820$	0	0
$821$	3.41545e6	0.176844	0.0884218	−	0.996083i	$-0.471818\pi$
$821$	3.41545e6	0.176844	0.0884218	+	0.996083i	$0.471818\pi$
$822$	0	0
$823$	1.17308e7	0.603712	0.301856	−	0.953354i	$-0.402394\pi$
$823$	1.17308e7	0.603712	0.301856	+	0.953354i	$0.402394\pi$
$824$	0	0
$825$	−1.62290e7	−0.830149
$826$	0	0
$827$	2.06476e7	1.04980	0.524899	−	0.851164i	$-0.324103\pi$
$827$	2.06476e7	1.04980	0.524899	+	0.851164i	$0.324103\pi$
$828$	0	0
$829$	−1.82141e7	−0.920496	−0.460248	−	0.887790i	$-0.652239\pi$
$829$	−1.82141e7	−0.920496	−0.460248	+	0.887790i	$0.652239\pi$
$830$	0	0
$831$	7.21513e6	0.362445
$832$	0	0
$833$	2.60386e6	0.130019
$834$	0	0
$835$	2.94800e6	0.146322
$836$	0	0
$837$	5.97333e6	0.294715
$838$	0	0
$839$	−1.01638e7	−0.498483	−0.249242	−	0.968441i	$-0.580181\pi$
$839$	−1.01638e7	−0.498483	−0.249242	+	0.968441i	$0.580181\pi$
$840$	0	0
$841$	−845486.	−0.0412208
$842$	0	0
$843$	−2.33029e6	−0.112938
$844$	0	0
$845$	383191.	0.0184618
$846$	0	0
$847$	−2.53031e7	−1.21189
$848$	0	0
$849$	−1.25576e7	−0.597914
$850$	0	0
$851$	−1.18728e6	−0.0561991
$852$	0	0
$853$	2.04589e7	0.962740	0.481370	−	0.876518i	$-0.340139\pi$
$853$	2.04589e7	0.962740	0.481370	+	0.876518i	$0.340139\pi$
$854$	0	0
$855$	−1.20659e6	−0.0564475
$856$	0	0
$857$	−7.05028e6	−0.327910	−0.163955	−	0.986468i	$-0.552425\pi$
$857$	−7.05028e6	−0.327910	−0.163955	+	0.986468i	$0.552425\pi$
$858$	0	0
$859$	−2.38609e7	−1.10332	−0.551662	−	0.834068i	$-0.686006\pi$
$859$	−2.38609e7	−1.10332	−0.551662	+	0.834068i	$0.686006\pi$
$860$	0	0
$861$	1.07562e7	0.494484
$862$	0	0
$863$	−2.60533e7	−1.19079	−0.595396	−	0.803432i	$-0.703005\pi$
$863$	−2.60533e7	−1.19079	−0.595396	+	0.803432i	$0.703005\pi$
$864$	0	0
$865$	−5.88467e6	−0.267413
$866$	0	0
$867$	−5.04089e6	−0.227751
$868$	0	0
$869$	3.14342e7	1.41206
$870$	0	0
$871$	1.09836e7	0.490569
$872$	0	0
$873$	−6.59555e6	−0.292897
$874$	0	0
$875$	9.63552e6	0.425456
$876$	0	0
$877$	3.93550e7	1.72783	0.863914	−	0.503639i	$-0.168006\pi$
$877$	3.93550e7	1.72783	0.863914	+	0.503639i	$0.168006\pi$
$878$	0	0
$879$	1.50975e7	0.659074
$880$	0	0
$881$	−8.81546e6	−0.382653	−0.191326	−	0.981526i	$-0.561279\pi$
$881$	−8.81546e6	−0.382653	−0.191326	+	0.981526i	$0.561279\pi$
$882$	0	0
$883$	−3.32092e7	−1.43336	−0.716681	−	0.697401i	$-0.754340\pi$
$883$	−3.32092e7	−1.43336	−0.716681	+	0.697401i	$0.754340\pi$
$884$	0	0
$885$	−5.23145e6	−0.224525
$886$	0	0
$887$	−3.09033e6	−0.131885	−0.0659426	−	0.997823i	$-0.521005\pi$
$887$	−3.09033e6	−0.131885	−0.0659426	+	0.997823i	$0.521005\pi$
$888$	0	0
$889$	1.00330e7	0.425772
$890$	0	0
$891$	4.01730e6	0.169527
$892$	0	0
$893$	7.92729e6	0.332657
$894$	0	0
$895$	6.56598e6	0.273994
$896$	0	0
$897$	567632.	0.0235552
$898$	0	0
$899$	−3.63365e7	−1.49949
$900$	0	0
$901$	−6.23710e6	−0.255959
$902$	0	0
$903$	−2.21167e7	−0.902612
$904$	0	0
$905$	−1.92867e6	−0.0782774
$906$	0	0
$907$	1.51978e7	0.613425	0.306712	−	0.951802i	$-0.400771\pi$
$907$	1.51978e7	0.613425	0.306712	+	0.951802i	$0.400771\pi$
$908$	0	0
$909$	1.35259e7	0.542945
$910$	0	0
$911$	8.92224e6	0.356187	0.178094	−	0.984014i	$-0.443007\pi$
$911$	8.92224e6	0.356187	0.178094	+	0.984014i	$0.443007\pi$
$912$	0	0
$913$	−6.55552e7	−2.60274
$914$	0	0
$915$	−3.69940e6	−0.146076
$916$	0	0
$917$	1.54075e7	0.605076
$918$	0	0
$919$	−2.94149e7	−1.14889	−0.574446	−	0.818542i	$-0.694783\pi$
$919$	−2.94149e7	−1.14889	−0.574446	+	0.818542i	$0.694783\pi$
$920$	0	0
$921$	−9.31706e6	−0.361934
$922$	0	0
$923$	−8.68228e6	−0.335451
$924$	0	0
$925$	−9.36915e6	−0.360036
$926$	0	0
$927$	3.12702e6	0.119518
$928$	0	0
$929$	1.64015e7	0.623510	0.311755	−	0.950163i	$-0.399083\pi$
$929$	1.64015e7	0.623510	0.311755	+	0.950163i	$0.399083\pi$
$930$	0	0
$931$	3.11791e6	0.117893
$932$	0	0
$933$	3.64521e6	0.137094
$934$	0	0
$935$	−7.61717e6	−0.284947
$936$	0	0
$937$	3.89038e7	1.44758	0.723791	−	0.690019i	$-0.242398\pi$
$937$	3.89038e7	1.44758	0.723791	+	0.690019i	$0.242398\pi$
$938$	0	0
$939$	−1.68376e7	−0.623184
$940$	0	0
$941$	2.82311e7	1.03933	0.519666	−	0.854369i	$-0.326056\pi$
$941$	2.82311e7	1.03933	0.519666	+	0.854369i	$0.326056\pi$
$942$	0	0
$943$	3.76973e6	0.138048
$944$	0	0
$945$	−1.15722e6	−0.0421536
$946$	0	0
$947$	3.19191e7	1.15658	0.578291	−	0.815831i	$-0.303720\pi$
$947$	3.19191e7	1.15658	0.578291	+	0.815831i	$0.303720\pi$
$948$	0	0
$949$	−5.65617e6	−0.203872
$950$	0	0
$951$	589474.	0.0211356
$952$	0	0
$953$	−2.27842e7	−0.812645	−0.406323	−	0.913730i	$-0.633189\pi$
$953$	−2.27842e7	−0.812645	−0.406323	+	0.913730i	$0.633189\pi$
$954$	0	0
$955$	−9.03124e6	−0.320434
$956$	0	0
$957$	−2.44377e7	−0.862543
$958$	0	0
$959$	1.19142e7	0.418328
$960$	0	0
$961$	3.85103e7	1.34514
$962$	0	0
$963$	5.94953e6	0.206737
$964$	0	0
$965$	9.64733e6	0.333494
$966$	0	0
$967$	1.68036e7	0.577878	0.288939	−	0.957347i	$-0.406697\pi$
$967$	1.68036e7	0.577878	0.288939	+	0.957347i	$0.406697\pi$
$968$	0	0
$969$	9.26539e6	0.316997
$970$	0	0
$971$	−5.22521e7	−1.77851	−0.889254	−	0.457413i	$-0.848776\pi$
$971$	−5.22521e7	−1.77851	−0.889254	+	0.457413i	$0.848776\pi$
$972$	0	0
$973$	3.98331e7	1.34884
$974$	0	0
$975$	4.47934e6	0.150905
$976$	0	0
$977$	−2.90452e7	−0.973503	−0.486752	−	0.873540i	$-0.661818\pi$
$977$	−2.90452e7	−0.973503	−0.486752	+	0.873540i	$0.661818\pi$
$978$	0	0
$979$	−7.43499e7	−2.47927
$980$	0	0
$981$	−8.32699e6	−0.276258
$982$	0	0
$983$	−2.88765e7	−0.953150	−0.476575	−	0.879134i	$-0.658122\pi$
$983$	−2.88765e7	−0.953150	−0.476575	+	0.879134i	$0.658122\pi$
$984$	0	0
$985$	−1.15697e7	−0.379955
$986$	0	0
$987$	7.60290e6	0.248420
$988$	0	0
$989$	−7.75123e6	−0.251988
$990$	0	0
$991$	−8.85182e6	−0.286318	−0.143159	−	0.989700i	$-0.545726\pi$
$991$	−8.85182e6	−0.286318	−0.143159	+	0.989700i	$0.545726\pi$
$992$	0	0
$993$	1.25707e7	0.404564
$994$	0	0
$995$	−9.51627e6	−0.304726
$996$	0	0
$997$	−4.58013e7	−1.45928	−0.729642	−	0.683829i	$-0.760313\pi$
$997$	−4.58013e7	−1.45928	−0.729642	+	0.683829i	$0.760313\pi$
$998$	0	0
$999$	2.31923e6	0.0735240

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.6.a.k.1.2		2
4.3	odd	2		39.6.a.b.1.1	✓	2
12.11	even	2		117.6.a.b.1.2		2
20.19	odd	2		975.6.a.c.1.2		2
52.51	odd	2		507.6.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.6.a.b.1.1	✓	2	4.3	odd	2
117.6.a.b.1.2		2	12.11	even	2
507.6.a.c.1.2		2	52.51	odd	2
624.6.a.k.1.2		2	1.1	even	1	trivial
975.6.a.c.1.2		2	20.19	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +13.4166 q^{5} -118.316 q^{7} +81.0000 q^{9} +612.299 q^{11} -169.000 q^{13} +120.749 q^{15} -927.231 q^{17} -1110.28 q^{19} -1064.85 q^{21} -373.197 q^{23} -2945.00 q^{25} +729.000 q^{27} -4434.60 q^{29} +8193.87 q^{31} +5510.69 q^{33} -1587.40 q^{35} +3181.38 q^{37} -1521.00 q^{39} -10101.2 q^{41} +20769.8 q^{43} +1086.74 q^{45} -7139.89 q^{47} -2808.21 q^{49} -8345.08 q^{51} +6726.59 q^{53} +8214.96 q^{55} -9992.54 q^{57} -43325.0 q^{59} -30637.1 q^{61} -9583.63 q^{63} -2267.40 q^{65} -64991.9 q^{67} -3358.77 q^{69} +51374.4 q^{71} +33468.4 q^{73} -26505.0 q^{75} -72445.1 q^{77} +51337.9 q^{79} +6561.00 q^{81} -107064. q^{83} -12440.3 q^{85} -39911.4 q^{87} -121427. q^{89} +19995.5 q^{91} +73744.8 q^{93} -14896.2 q^{95} -81426.5 q^{97} +49596.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} - 48 q^{5} - 72 q^{7} + 162 q^{9} + 596 q^{11} - 338 q^{13} - 432 q^{15} - 268 q^{17} - 1128 q^{19} - 648 q^{21} + 1768 q^{23} - 2298 q^{25} + 1458 q^{27} - 7612 q^{29} + 4160 q^{31} + 5364 q^{33}+ \cdots + 48276 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 - 48 * q^5 - 72 * q^7 + 162 * q^9 + 596 * q^11 - 338 * q^13 - 432 * q^15 - 268 * q^17 - 1128 * q^19 - 648 * q^21 + 1768 * q^23 - 2298 * q^25 + 1458 * q^27 - 7612 * q^29 + 4160 * q^31 + 5364 * q^33 - 4432 * q^35 + 17468 * q^37 - 3042 * q^39 - 28000 * q^41 + 24328 * q^43 - 3888 * q^45 + 18108 * q^47 - 17470 * q^49 - 2412 * q^51 + 1420 * q^53 + 9216 * q^55 - 10152 * q^57 - 6788 * q^59 - 37148 * q^61 - 5832 * q^63 + 8112 * q^65 - 106112 * q^67 + 15912 * q^69 + 30460 * q^71 - 37620 * q^73 - 20682 * q^75 - 73200 * q^77 + 2160 * q^79 + 13122 * q^81 - 207004 * q^83 - 52928 * q^85 - 68508 * q^87 - 74136 * q^89 + 12168 * q^91 + 37440 * q^93 - 13808 * q^95 - 121156 * q^97 + 48276 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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