LMFDB - Embedded newform 768.6.a.bd.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,6,Mod(1,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("768.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	768.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,45,0,50,0,98,0,405,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$123.174773616$
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} - x^{4} - 65x^{3} + 85x^{2} + 856x - 1692 $ x^5 - x^4 - 65x^3 + 85x^2 + 856*x - 1692
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$6.79075$ of defining polynomial
Character	$\chi$	$=$	768.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +100.204 q^{5} -154.395 q^{7} +81.0000 q^{9} -85.0809 q^{11} +812.927 q^{13} +901.832 q^{15} +1275.57 q^{17} -393.875 q^{19} -1389.55 q^{21} -2533.66 q^{23} +6915.74 q^{25} +729.000 q^{27} -254.097 q^{29} +1801.33 q^{31} -765.728 q^{33} -15470.9 q^{35} +686.112 q^{37} +7316.35 q^{39} +13255.9 q^{41} -13936.2 q^{43} +8116.48 q^{45} +11557.4 q^{47} +7030.74 q^{49} +11480.1 q^{51} +8023.45 q^{53} -8525.40 q^{55} -3544.88 q^{57} +7967.95 q^{59} -15828.6 q^{61} -12506.0 q^{63} +81458.2 q^{65} +33836.9 q^{67} -22803.0 q^{69} +72786.0 q^{71} +717.727 q^{73} +62241.7 q^{75} +13136.0 q^{77} +32844.2 q^{79} +6561.00 q^{81} +46129.8 q^{83} +127816. q^{85} -2286.88 q^{87} +15255.6 q^{89} -125512. q^{91} +16211.9 q^{93} -39467.7 q^{95} -173584. q^{97} -6891.55 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 45 q^{3} + 50 q^{5} + 98 q^{7} + 405 q^{9} + 676 q^{13} + 450 q^{15} - 202 q^{17} + 716 q^{19} + 882 q^{21} - 836 q^{23} + 4683 q^{25} + 3645 q^{27} + 5046 q^{29} + 9346 q^{31} + 436 q^{35} + 10952 q^{37}+ \cdots + 29686 q^{97}+O(q^{100}) $ 5 * q + 45 * q^3 + 50 * q^5 + 98 * q^7 + 405 * q^9 + 676 * q^13 + 450 * q^15 - 202 * q^17 + 716 * q^19 + 882 * q^21 - 836 * q^23 + 4683 * q^25 + 3645 * q^27 + 5046 * q^29 + 9346 * q^31 + 436 * q^35 + 10952 * q^37 + 6084 * q^39 + 1238 * q^41 + 1676 * q^43 + 4050 * q^45 + 32628 * q^47 + 4689 * q^49 - 1818 * q^51 - 8490 * q^53 + 51712 * q^55 + 6444 * q^57 - 15884 * q^59 - 13824 * q^61 + 7938 * q^63 + 25768 * q^65 + 26652 * q^67 - 7524 * q^69 + 136116 * q^71 + 13694 * q^73 + 42147 * q^75 + 89856 * q^77 + 126154 * q^79 + 32805 * q^81 - 64720 * q^83 + 164636 * q^85 + 45414 * q^87 - 87790 * q^89 - 179992 * q^91 + 84114 * q^93 + 121592 * q^95 + 29686 * q^97

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
$5$	100.204	1.79249	0.896247	−	0.443554i	$-0.146283\pi$
$5$	100.204	1.79249	0.896247	+	0.443554i	$0.146283\pi$
$6$	0	0
$7$	−154.395	−1.19093	−0.595467	−	0.803380i	$-0.703033\pi$
$7$	−154.395	−1.19093	−0.595467	+	0.803380i	$0.703033\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−85.0809	−0.212007	−0.106003	−	0.994366i	$-0.533805\pi$
$11$	−85.0809	−0.212007	−0.106003	+	0.994366i	$0.533805\pi$
$12$	0	0
$13$	812.927	1.33412	0.667058	−	0.745006i	$-0.267553\pi$
$13$	812.927	1.33412	0.667058	+	0.745006i	$0.267553\pi$
$14$	0	0
$15$	901.832	1.03490
$16$	0	0
$17$	1275.57	1.07049	0.535244	−	0.844698i	$-0.320220\pi$
$17$	1275.57	1.07049	0.535244	+	0.844698i	$0.320220\pi$
$18$	0	0
$19$	−393.875	−0.250308	−0.125154	−	0.992137i	$-0.539943\pi$
$19$	−393.875	−0.250308	−0.125154	+	0.992137i	$0.539943\pi$
$20$	0	0
$21$	−1389.55	−0.687586
$22$	0	0
$23$	−2533.66	−0.998686	−0.499343	−	0.866404i	$-0.666425\pi$
$23$	−2533.66	−0.998686	−0.499343	+	0.866404i	$0.666425\pi$
$24$	0	0
$25$	6915.74	2.21304
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−254.097	−0.0561055	−0.0280527	−	0.999606i	$-0.508931\pi$
$29$	−254.097	−0.0561055	−0.0280527	+	0.999606i	$0.508931\pi$
$30$	0	0
$31$	1801.33	0.336658	0.168329	−	0.985731i	$-0.446163\pi$
$31$	1801.33	0.336658	0.168329	+	0.985731i	$0.446163\pi$
$32$	0	0
$33$	−765.728	−0.122402
$34$	0	0
$35$	−15470.9	−2.13474
$36$	0	0
$37$	686.112	0.0823930	0.0411965	−	0.999151i	$-0.486883\pi$
$37$	686.112	0.0823930	0.0411965	+	0.999151i	$0.486883\pi$
$38$	0	0
$39$	7316.35	0.770252
$40$	0	0
$41$	13255.9	1.23154	0.615772	−	0.787924i	$-0.288844\pi$
$41$	13255.9	1.23154	0.615772	+	0.787924i	$0.288844\pi$
$42$	0	0
$43$	−13936.2	−1.14940	−0.574701	−	0.818363i	$-0.694882\pi$
$43$	−13936.2	−1.14940	−0.574701	+	0.818363i	$0.694882\pi$
$44$	0	0
$45$	8116.48	0.597498
$46$	0	0
$47$	11557.4	0.763158	0.381579	−	0.924336i	$-0.375380\pi$
$47$	11557.4	0.763158	0.381579	+	0.924336i	$0.375380\pi$
$48$	0	0
$49$	7030.74	0.418322
$50$	0	0
$51$	11480.1	0.618046
$52$	0	0
$53$	8023.45	0.392348	0.196174	−	0.980569i	$-0.437148\pi$
$53$	8023.45	0.392348	0.196174	+	0.980569i	$0.437148\pi$
$54$	0	0
$55$	−8525.40	−0.380021
$56$	0	0
$57$	−3544.88	−0.144515
$58$	0	0
$59$	7967.95	0.298000	0.149000	−	0.988837i	$-0.452395\pi$
$59$	7967.95	0.298000	0.149000	+	0.988837i	$0.452395\pi$
$60$	0	0
$61$	−15828.6	−0.544651	−0.272326	−	0.962205i	$-0.587793\pi$
$61$	−15828.6	−0.544651	−0.272326	+	0.962205i	$0.587793\pi$
$62$	0	0
$63$	−12506.0	−0.396978
$64$	0	0
$65$	81458.2	2.39140
$66$	0	0
$67$	33836.9	0.920881	0.460440	−	0.887691i	$-0.347691\pi$
$67$	33836.9	0.920881	0.460440	+	0.887691i	$0.347691\pi$
$68$	0	0
$69$	−22803.0	−0.576592
$70$	0	0
$71$	72786.0	1.71357	0.856786	−	0.515672i	$-0.172458\pi$
$71$	72786.0	1.71357	0.856786	+	0.515672i	$0.172458\pi$
$72$	0	0
$73$	717.727	0.0157635	0.00788174	−	0.999969i	$-0.497491\pi$
$73$	717.727	0.0157635	0.00788174	+	0.999969i	$0.497491\pi$
$74$	0	0
$75$	62241.7	1.27770
$76$	0	0
$77$	13136.0	0.252486
$78$	0	0
$79$	32844.2	0.592094	0.296047	−	0.955173i	$-0.404331\pi$
$79$	32844.2	0.592094	0.296047	+	0.955173i	$0.404331\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	46129.8	0.734998	0.367499	−	0.930024i	$-0.380214\pi$
$83$	46129.8	0.734998	0.367499	+	0.930024i	$0.380214\pi$
$84$	0	0
$85$	127816.	1.91884
$86$	0	0
$87$	−2286.88	−0.0323925
$88$	0	0
$89$	15255.6	0.204152	0.102076	−	0.994777i	$-0.467451\pi$
$89$	15255.6	0.204152	0.102076	+	0.994777i	$0.467451\pi$
$90$	0	0
$91$	−125512.	−1.58884
$92$	0	0
$93$	16211.9	0.194369
$94$	0	0
$95$	−39467.7	−0.448676
$96$	0	0
$97$	−173584.	−1.87318	−0.936591	−	0.350426i	$-0.886037\pi$
$97$	−173584.	−1.87318	−0.936591	+	0.350426i	$0.886037\pi$
$98$	0	0
$99$	−6891.55	−0.0706690
$100$	0	0
$101$	−76270.0	−0.743961	−0.371981	−	0.928240i	$-0.621321\pi$
$101$	−76270.0	−0.743961	−0.371981	+	0.928240i	$0.621321\pi$
$102$	0	0
$103$	71216.8	0.661439	0.330719	−	0.943729i	$-0.392709\pi$
$103$	71216.8	0.661439	0.330719	+	0.943729i	$0.392709\pi$
$104$	0	0
$105$	−139238.	−1.23249
$106$	0	0
$107$	144823.	1.22286	0.611432	−	0.791297i	$-0.290594\pi$
$107$	144823.	1.22286	0.611432	+	0.791297i	$0.290594\pi$
$108$	0	0
$109$	−16321.4	−0.131580	−0.0657900	−	0.997833i	$-0.520957\pi$
$109$	−16321.4	−0.131580	−0.0657900	+	0.997833i	$0.520957\pi$
$110$	0	0
$111$	6175.01	0.0475696
$112$	0	0
$113$	−167326.	−1.23273	−0.616366	−	0.787460i	$-0.711396\pi$
$113$	−167326.	−1.23273	−0.616366	+	0.787460i	$0.711396\pi$
$114$	0	0
$115$	−253882.	−1.79014
$116$	0	0
$117$	65847.1	0.444705
$118$	0	0
$119$	−196941.	−1.27488
$120$	0	0
$121$	−153812.	−0.955053
$122$	0	0
$123$	119303.	0.711032
$124$	0	0
$125$	379846.	2.17436
$126$	0	0
$127$	307665.	1.69265	0.846327	−	0.532663i	$-0.178809\pi$
$127$	307665.	1.69265	0.846327	+	0.532663i	$0.178809\pi$
$128$	0	0
$129$	−125425.	−0.663608
$130$	0	0
$131$	197124.	1.00360	0.501802	−	0.864983i	$-0.332671\pi$
$131$	197124.	1.00360	0.501802	+	0.864983i	$0.332671\pi$
$132$	0	0
$133$	60812.3	0.298100
$134$	0	0
$135$	73048.4	0.344966
$136$	0	0
$137$	41854.6	0.190521	0.0952603	−	0.995452i	$-0.469632\pi$
$137$	41854.6	0.190521	0.0952603	+	0.995452i	$0.469632\pi$
$138$	0	0
$139$	−447375.	−1.96397	−0.981984	−	0.188965i	$-0.939487\pi$
$139$	−447375.	−1.96397	−0.981984	+	0.188965i	$0.939487\pi$
$140$	0	0
$141$	104016.	0.440610
$142$	0	0
$143$	−69164.6	−0.282842
$144$	0	0
$145$	−25461.5	−0.100569
$146$	0	0
$147$	63276.6	0.241518
$148$	0	0
$149$	367279.	1.35529	0.677643	−	0.735391i	$-0.263001\pi$
$149$	367279.	1.35529	0.677643	+	0.735391i	$0.263001\pi$
$150$	0	0
$151$	−357113.	−1.27457	−0.637285	−	0.770628i	$-0.719942\pi$
$151$	−357113.	−1.27457	−0.637285	+	0.770628i	$0.719942\pi$
$152$	0	0
$153$	103321.	0.356829
$154$	0	0
$155$	180499.	0.603457
$156$	0	0
$157$	256597.	0.830810	0.415405	−	0.909637i	$-0.363640\pi$
$157$	256597.	0.830810	0.415405	+	0.909637i	$0.363640\pi$
$158$	0	0
$159$	72211.1	0.226522
$160$	0	0
$161$	391184.	1.18937
$162$	0	0
$163$	−560618.	−1.65272	−0.826358	−	0.563145i	$-0.809591\pi$
$163$	−560618.	−1.65272	−0.826358	+	0.563145i	$0.809591\pi$
$164$	0	0
$165$	−76728.6	−0.219405
$166$	0	0
$167$	410936.	1.14021	0.570103	−	0.821573i	$-0.306903\pi$
$167$	410936.	1.14021	0.570103	+	0.821573i	$0.306903\pi$
$168$	0	0
$169$	289558.	0.779864
$170$	0	0
$171$	−31903.9	−0.0834360
$172$	0	0
$173$	−407557.	−1.03532	−0.517658	−	0.855587i	$-0.673196\pi$
$173$	−407557.	−1.03532	−0.517658	+	0.855587i	$0.673196\pi$
$174$	0	0
$175$	−1.06775e6	−2.63558
$176$	0	0
$177$	71711.5	0.172050
$178$	0	0
$179$	73705.9	0.171937	0.0859686	−	0.996298i	$-0.472602\pi$
$179$	73705.9	0.171937	0.0859686	+	0.996298i	$0.472602\pi$
$180$	0	0
$181$	194327.	0.440896	0.220448	−	0.975399i	$-0.429248\pi$
$181$	194327.	0.440896	0.220448	+	0.975399i	$0.429248\pi$
$182$	0	0
$183$	−142458.	−0.314455
$184$	0	0
$185$	68750.8	0.147689
$186$	0	0
$187$	−108526.	−0.226951
$188$	0	0
$189$	−112554.	−0.229195
$190$	0	0
$191$	171483.	0.340125	0.170063	−	0.985433i	$-0.445603\pi$
$191$	171483.	0.340125	0.170063	+	0.985433i	$0.445603\pi$
$192$	0	0
$193$	774459.	1.49660	0.748299	−	0.663362i	$-0.230871\pi$
$193$	774459.	1.49660	0.748299	+	0.663362i	$0.230871\pi$
$194$	0	0
$195$	733124.	1.38067
$196$	0	0
$197$	864723.	1.58749	0.793746	−	0.608250i	$-0.208128\pi$
$197$	864723.	1.58749	0.793746	+	0.608250i	$0.208128\pi$
$198$	0	0
$199$	752715.	1.34740	0.673702	−	0.739003i	$-0.264703\pi$
$199$	752715.	1.34740	0.673702	+	0.739003i	$0.264703\pi$
$200$	0	0
$201$	304532.	0.531671
$202$	0	0
$203$	39231.3	0.0668179
$204$	0	0
$205$	1.32829e6	2.20754
$206$	0	0
$207$	−205227.	−0.332895
$208$	0	0
$209$	33511.3	0.0530671
$210$	0	0
$211$	265517.	0.410569	0.205285	−	0.978702i	$-0.434188\pi$
$211$	265517.	0.410569	0.205285	+	0.978702i	$0.434188\pi$
$212$	0	0
$213$	655074.	0.989331
$214$	0	0
$215$	−1.39645e6	−2.06030
$216$	0	0
$217$	−278115.	−0.400937
$218$	0	0
$219$	6459.54	0.00910105
$220$	0	0
$221$	1.03694e6	1.42815
$222$	0	0
$223$	209136.	0.281623	0.140811	−	0.990036i	$-0.455029\pi$
$223$	209136.	0.281623	0.140811	+	0.990036i	$0.455029\pi$
$224$	0	0
$225$	560175.	0.737679
$226$	0	0
$227$	120120.	0.154722	0.0773610	−	0.997003i	$-0.475351\pi$
$227$	120120.	0.154722	0.0773610	+	0.997003i	$0.475351\pi$
$228$	0	0
$229$	440615.	0.555227	0.277613	−	0.960693i	$-0.410457\pi$
$229$	440615.	0.555227	0.277613	+	0.960693i	$0.410457\pi$
$230$	0	0
$231$	118224.	0.145773
$232$	0	0
$233$	−167121.	−0.201670	−0.100835	−	0.994903i	$-0.532151\pi$
$233$	−167121.	−0.201670	−0.100835	+	0.994903i	$0.532151\pi$
$234$	0	0
$235$	1.15809e6	1.36796
$236$	0	0
$237$	295598.	0.341846
$238$	0	0
$239$	−445757.	−0.504781	−0.252391	−	0.967625i	$-0.581217\pi$
$239$	−445757.	−0.504781	−0.252391	+	0.967625i	$0.581217\pi$
$240$	0	0
$241$	153018.	0.169708	0.0848538	−	0.996393i	$-0.472958\pi$
$241$	153018.	0.169708	0.0848538	+	0.996393i	$0.472958\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	704505.	0.749840
$246$	0	0
$247$	−320192.	−0.333940
$248$	0	0
$249$	415168.	0.424351
$250$	0	0
$251$	−693358.	−0.694662	−0.347331	−	0.937743i	$-0.612912\pi$
$251$	−693358.	−0.694662	−0.347331	+	0.937743i	$0.612912\pi$
$252$	0	0
$253$	215566.	0.211728
$254$	0	0
$255$	1.15035e6	1.10784
$256$	0	0
$257$	−1.71886e6	−1.62334	−0.811668	−	0.584119i	$-0.801440\pi$
$257$	−1.71886e6	−1.62334	−0.811668	+	0.584119i	$0.801440\pi$
$258$	0	0
$259$	−105932.	−0.0981246
$260$	0	0
$261$	−20581.9	−0.0187018
$262$	0	0
$263$	156631.	0.139633	0.0698167	−	0.997560i	$-0.477759\pi$
$263$	156631.	0.139633	0.0698167	+	0.997560i	$0.477759\pi$
$264$	0	0
$265$	803978.	0.703282
$266$	0	0
$267$	137300.	0.117867
$268$	0	0
$269$	800001.	0.674078	0.337039	−	0.941491i	$-0.390575\pi$
$269$	800001.	0.674078	0.337039	+	0.941491i	$0.390575\pi$
$270$	0	0
$271$	1.89597e6	1.56823	0.784113	−	0.620618i	$-0.213118\pi$
$271$	1.89597e6	1.56823	0.784113	+	0.620618i	$0.213118\pi$
$272$	0	0
$273$	−1.12961e6	−0.917319
$274$	0	0
$275$	−588397.	−0.469180
$276$	0	0
$277$	−279803.	−0.219105	−0.109553	−	0.993981i	$-0.534942\pi$
$277$	−279803.	−0.219105	−0.109553	+	0.993981i	$0.534942\pi$
$278$	0	0
$279$	145907.	0.112219
$280$	0	0
$281$	−554219.	−0.418712	−0.209356	−	0.977840i	$-0.567137\pi$
$281$	−554219.	−0.418712	−0.209356	+	0.977840i	$0.567137\pi$
$282$	0	0
$283$	1.57678e6	1.17032	0.585161	−	0.810917i	$-0.301031\pi$
$283$	1.57678e6	1.17032	0.585161	+	0.810917i	$0.301031\pi$
$284$	0	0
$285$	−355209.	−0.259043
$286$	0	0
$287$	−2.04664e6	−1.46669
$288$	0	0
$289$	207218.	0.145943
$290$	0	0
$291$	−1.56225e6	−1.08148
$292$	0	0
$293$	−184563.	−0.125596	−0.0627979	−	0.998026i	$-0.520002\pi$
$293$	−184563.	−0.125596	−0.0627979	+	0.998026i	$0.520002\pi$
$294$	0	0
$295$	798416.	0.534164
$296$	0	0
$297$	−62023.9	−0.0408008
$298$	0	0
$299$	−2.05968e6	−1.33236
$300$	0	0
$301$	2.15167e6	1.36886
$302$	0	0
$303$	−686430.	−0.429526
$304$	0	0
$305$	−1.58608e6	−0.976285
$306$	0	0
$307$	−405878.	−0.245782	−0.122891	−	0.992420i	$-0.539217\pi$
$307$	−405878.	−0.245782	−0.122891	+	0.992420i	$0.539217\pi$
$308$	0	0
$309$	640952.	0.381882
$310$	0	0
$311$	−254438.	−0.149170	−0.0745849	−	0.997215i	$-0.523763\pi$
$311$	−254438.	−0.149170	−0.0745849	+	0.997215i	$0.523763\pi$
$312$	0	0
$313$	−1.03862e6	−0.599234	−0.299617	−	0.954060i	$-0.596859\pi$
$313$	−1.03862e6	−0.599234	−0.299617	+	0.954060i	$0.596859\pi$
$314$	0	0
$315$	−1.25314e6	−0.711581
$316$	0	0
$317$	1.39910e6	0.781990	0.390995	−	0.920393i	$-0.372131\pi$
$317$	1.39910e6	0.781990	0.390995	+	0.920393i	$0.372131\pi$
$318$	0	0
$319$	21618.8	0.0118948
$320$	0	0
$321$	1.30341e6	0.706021
$322$	0	0
$323$	−502415.	−0.267952
$324$	0	0
$325$	5.62200e6	2.95245
$326$	0	0
$327$	−146892.	−0.0759678
$328$	0	0
$329$	−1.78440e6	−0.908870
$330$	0	0
$331$	−2.04667e6	−1.02678	−0.513390	−	0.858156i	$-0.671610\pi$
$331$	−2.04667e6	−1.02678	−0.513390	+	0.858156i	$0.671610\pi$
$332$	0	0
$333$	55575.0	0.0274643
$334$	0	0
$335$	3.39058e6	1.65067
$336$	0	0
$337$	−3.22090e6	−1.54491	−0.772453	−	0.635072i	$-0.780970\pi$
$337$	−3.22090e6	−1.54491	−0.772453	+	0.635072i	$0.780970\pi$
$338$	0	0
$339$	−1.50594e6	−0.711718
$340$	0	0
$341$	−153258.	−0.0713737
$342$	0	0
$343$	1.50940e6	0.692740
$344$	0	0
$345$	−2.28494e6	−1.03354
$346$	0	0
$347$	1.80278e6	0.803748	0.401874	−	0.915695i	$-0.368359\pi$
$347$	1.80278e6	0.803748	0.401874	+	0.915695i	$0.368359\pi$
$348$	0	0
$349$	−3.09976e6	−1.36228	−0.681138	−	0.732155i	$-0.738515\pi$
$349$	−3.09976e6	−1.36228	−0.681138	+	0.732155i	$0.738515\pi$
$350$	0	0
$351$	592624.	0.256751
$352$	0	0
$353$	2.54655e6	1.08772	0.543859	−	0.839177i	$-0.316963\pi$
$353$	2.54655e6	1.08772	0.543859	+	0.839177i	$0.316963\pi$
$354$	0	0
$355$	7.29342e6	3.07157
$356$	0	0
$357$	−1.77247e6	−0.736052
$358$	0	0
$359$	790598.	0.323757	0.161879	−	0.986811i	$-0.448245\pi$
$359$	790598.	0.323757	0.161879	+	0.986811i	$0.448245\pi$
$360$	0	0
$361$	−2.32096e6	−0.937346
$362$	0	0
$363$	−1.38431e6	−0.551400
$364$	0	0
$365$	71918.7	0.0282560
$366$	0	0
$367$	−1.30401e6	−0.505376	−0.252688	−	0.967548i	$-0.581315\pi$
$367$	−1.30401e6	−0.505376	−0.252688	+	0.967548i	$0.581315\pi$
$368$	0	0
$369$	1.07373e6	0.410515
$370$	0	0
$371$	−1.23878e6	−0.467260
$372$	0	0
$373$	−4.94317e6	−1.83964	−0.919822	−	0.392337i	$-0.871667\pi$
$373$	−4.94317e6	−1.83964	−0.919822	+	0.392337i	$0.871667\pi$
$374$	0	0
$375$	3.41861e6	1.25537
$376$	0	0
$377$	−206563.	−0.0748512
$378$	0	0
$379$	1.92614e6	0.688795	0.344397	−	0.938824i	$-0.388083\pi$
$379$	1.92614e6	0.688795	0.344397	+	0.938824i	$0.388083\pi$
$380$	0	0
$381$	2.76898e6	0.977255
$382$	0	0
$383$	−4.69380e6	−1.63504	−0.817519	−	0.575902i	$-0.804651\pi$
$383$	−4.69380e6	−1.63504	−0.817519	+	0.575902i	$0.804651\pi$
$384$	0	0
$385$	1.31628e6	0.452580
$386$	0	0
$387$	−1.12883e6	−0.383134
$388$	0	0
$389$	2.35028e6	0.787492	0.393746	−	0.919219i	$-0.371179\pi$
$389$	2.35028e6	0.787492	0.393746	+	0.919219i	$0.371179\pi$
$390$	0	0
$391$	−3.23186e6	−1.06908
$392$	0	0
$393$	1.77412e6	0.579430
$394$	0	0
$395$	3.29110e6	1.06133
$396$	0	0
$397$	−1.84236e6	−0.586675	−0.293338	−	0.956009i	$-0.594766\pi$
$397$	−1.84236e6	−0.586675	−0.293338	+	0.956009i	$0.594766\pi$
$398$	0	0
$399$	547311.	0.172108
$400$	0	0
$401$	4.73319e6	1.46992	0.734959	−	0.678112i	$-0.237201\pi$
$401$	4.73319e6	1.46992	0.734959	+	0.678112i	$0.237201\pi$
$402$	0	0
$403$	1.46435e6	0.449140
$404$	0	0
$405$	657435.	0.199166
$406$	0	0
$407$	−58375.0	−0.0174679
$408$	0	0
$409$	6.34620e6	1.87588	0.937941	−	0.346796i	$-0.112730\pi$
$409$	6.34620e6	1.87588	0.937941	+	0.346796i	$0.112730\pi$
$410$	0	0
$411$	376692.	0.109997
$412$	0	0
$413$	−1.23021e6	−0.354898
$414$	0	0
$415$	4.62237e6	1.31748
$416$	0	0
$417$	−4.02637e6	−1.13390
$418$	0	0
$419$	1.21837e6	0.339034	0.169517	−	0.985527i	$-0.445779\pi$
$419$	1.21837e6	0.339034	0.169517	+	0.985527i	$0.445779\pi$
$420$	0	0
$421$	3.51877e6	0.967579	0.483789	−	0.875185i	$-0.339260\pi$
$421$	3.51877e6	0.967579	0.483789	+	0.875185i	$0.339260\pi$
$422$	0	0
$423$	936147.	0.254386
$424$	0	0
$425$	8.82151e6	2.36903
$426$	0	0
$427$	2.44386e6	0.648643
$428$	0	0
$429$	−622481.	−0.163299
$430$	0	0
$431$	−3.01231e6	−0.781099	−0.390549	−	0.920582i	$-0.627715\pi$
$431$	−3.01231e6	−0.781099	−0.390549	+	0.920582i	$0.627715\pi$
$432$	0	0
$433$	301545.	0.0772917	0.0386459	−	0.999253i	$-0.487696\pi$
$433$	301545.	0.0772917	0.0386459	+	0.999253i	$0.487696\pi$
$434$	0	0
$435$	−229153.	−0.0580634
$436$	0	0
$437$	997947.	0.249979
$438$	0	0
$439$	4.77660e6	1.18293	0.591464	−	0.806332i	$-0.298550\pi$
$439$	4.77660e6	1.18293	0.591464	+	0.806332i	$0.298550\pi$
$440$	0	0
$441$	569490.	0.139441
$442$	0	0
$443$	−2.87476e6	−0.695973	−0.347986	−	0.937500i	$-0.613134\pi$
$443$	−2.87476e6	−0.695973	−0.347986	+	0.937500i	$0.613134\pi$
$444$	0	0
$445$	1.52867e6	0.365942
$446$	0	0
$447$	3.30552e6	0.782475
$448$	0	0
$449$	5.95349e6	1.39366	0.696828	−	0.717238i	$-0.254594\pi$
$449$	5.95349e6	1.39366	0.696828	+	0.717238i	$0.254594\pi$
$450$	0	0
$451$	−1.12782e6	−0.261096
$452$	0	0
$453$	−3.21402e6	−0.735873
$454$	0	0
$455$	−1.25767e7	−2.84799
$456$	0	0
$457$	2.01722e6	0.451818	0.225909	−	0.974148i	$-0.427465\pi$
$457$	2.01722e6	0.451818	0.225909	+	0.974148i	$0.427465\pi$
$458$	0	0
$459$	929889.	0.206015
$460$	0	0
$461$	−2.25196e6	−0.493525	−0.246763	−	0.969076i	$-0.579367\pi$
$461$	−2.25196e6	−0.493525	−0.246763	+	0.969076i	$0.579367\pi$
$462$	0	0
$463$	3.74465e6	0.811819	0.405909	−	0.913913i	$-0.366955\pi$
$463$	3.74465e6	0.811819	0.405909	+	0.913913i	$0.366955\pi$
$464$	0	0
$465$	1.62449e6	0.348406
$466$	0	0
$467$	−8.62528e6	−1.83013	−0.915063	−	0.403311i	$-0.867859\pi$
$467$	−8.62528e6	−1.83013	−0.915063	+	0.403311i	$0.867859\pi$
$468$	0	0
$469$	−5.22424e6	−1.09671
$470$	0	0
$471$	2.30937e6	0.479668
$472$	0	0
$473$	1.18570e6	0.243681
$474$	0	0
$475$	−2.72394e6	−0.553941
$476$	0	0
$477$	649900.	0.130783
$478$	0	0
$479$	−6.32087e6	−1.25875	−0.629373	−	0.777103i	$-0.716688\pi$
$479$	−6.32087e6	−1.25875	−0.629373	+	0.777103i	$0.716688\pi$
$480$	0	0
$481$	557759.	0.109922
$482$	0	0
$483$	3.52066e6	0.686682
$484$	0	0
$485$	−1.73937e7	−3.35767
$486$	0	0
$487$	−4.53590e6	−0.866644	−0.433322	−	0.901239i	$-0.642659\pi$
$487$	−4.53590e6	−0.866644	−0.433322	+	0.901239i	$0.642659\pi$
$488$	0	0
$489$	−5.04556e6	−0.954196
$490$	0	0
$491$	−2.68402e6	−0.502438	−0.251219	−	0.967930i	$-0.580831\pi$
$491$	−2.68402e6	−0.502438	−0.251219	+	0.967930i	$0.580831\pi$
$492$	0	0
$493$	−324119.	−0.0600602
$494$	0	0
$495$	−690557.	−0.126674
$496$	0	0
$497$	−1.12378e7	−2.04075
$498$	0	0
$499$	−2.57253e6	−0.462496	−0.231248	−	0.972895i	$-0.574281\pi$
$499$	−2.57253e6	−0.462496	−0.231248	+	0.972895i	$0.574281\pi$
$500$	0	0
$501$	3.69843e6	0.658298
$502$	0	0
$503$	−4.22685e6	−0.744898	−0.372449	−	0.928053i	$-0.621482\pi$
$503$	−4.22685e6	−0.744898	−0.372449	+	0.928053i	$0.621482\pi$
$504$	0	0
$505$	−7.64252e6	−1.33355
$506$	0	0
$507$	2.60602e6	0.450255
$508$	0	0
$509$	−2.93969e6	−0.502929	−0.251465	−	0.967866i	$-0.580912\pi$
$509$	−2.93969e6	−0.502929	−0.251465	+	0.967866i	$0.580912\pi$
$510$	0	0
$511$	−110813.	−0.0187732
$512$	0	0
$513$	−287135.	−0.0481718
$514$	0	0
$515$	7.13618e6	1.18563
$516$	0	0
$517$	−983311.	−0.161795
$518$	0	0
$519$	−3.66801e6	−0.597740
$520$	0	0
$521$	−5.13499e6	−0.828792	−0.414396	−	0.910097i	$-0.636007\pi$
$521$	−5.13499e6	−0.828792	−0.414396	+	0.910097i	$0.636007\pi$
$522$	0	0
$523$	−479078.	−0.0765865	−0.0382932	−	0.999267i	$-0.512192\pi$
$523$	−479078.	−0.0765865	−0.0382932	+	0.999267i	$0.512192\pi$
$524$	0	0
$525$	−9.60979e6	−1.52165
$526$	0	0
$527$	2.29772e6	0.360388
$528$	0	0
$529$	−16898.0	−0.00262540
$530$	0	0
$531$	645404.	0.0993334
$532$	0	0
$533$	1.07761e7	1.64302
$534$	0	0
$535$	1.45118e7	2.19198
$536$	0	0
$537$	663353.	0.0992680
$538$	0	0
$539$	−598181.	−0.0886872
$540$	0	0
$541$	−1.03133e7	−1.51497	−0.757486	−	0.652852i	$-0.773572\pi$
$541$	−1.03133e7	−1.51497	−0.757486	+	0.652852i	$0.773572\pi$
$542$	0	0
$543$	1.74894e6	0.254552
$544$	0	0
$545$	−1.63546e6	−0.235857
$546$	0	0
$547$	−1.10504e7	−1.57910	−0.789552	−	0.613684i	$-0.789687\pi$
$547$	−1.10504e7	−1.57910	−0.789552	+	0.613684i	$0.789687\pi$
$548$	0	0
$549$	−1.28212e6	−0.181550
$550$	0	0
$551$	100083.	0.0140437
$552$	0	0
$553$	−5.07097e6	−0.705145
$554$	0	0
$555$	618757.	0.0852683
$556$	0	0
$557$	−1.09232e6	−0.149181	−0.0745904	−	0.997214i	$-0.523765\pi$
$557$	−1.09232e6	−0.149181	−0.0745904	+	0.997214i	$0.523765\pi$
$558$	0	0
$559$	−1.13291e7	−1.53344
$560$	0	0
$561$	−976738.	−0.131030
$562$	0	0
$563$	8.03622e6	1.06852	0.534258	−	0.845322i	$-0.320591\pi$
$563$	8.03622e6	1.06852	0.534258	+	0.845322i	$0.320591\pi$
$564$	0	0
$565$	−1.67667e7	−2.20967
$566$	0	0
$567$	−1.01298e6	−0.132326
$568$	0	0
$569$	−4.89482e6	−0.633806	−0.316903	−	0.948458i	$-0.602643\pi$
$569$	−4.89482e6	−0.633806	−0.316903	+	0.948458i	$0.602643\pi$
$570$	0	0
$571$	−629963.	−0.0808583	−0.0404291	−	0.999182i	$-0.512873\pi$
$571$	−629963.	−0.0808583	−0.0404291	+	0.999182i	$0.512873\pi$
$572$	0	0
$573$	1.54335e6	0.196371
$574$	0	0
$575$	−1.75222e7	−2.21013
$576$	0	0
$577$	−6.90487e6	−0.863408	−0.431704	−	0.902015i	$-0.642087\pi$
$577$	−6.90487e6	−0.863408	−0.431704	+	0.902015i	$0.642087\pi$
$578$	0	0
$579$	6.97013e6	0.864061
$580$	0	0
$581$	−7.12220e6	−0.875333
$582$	0	0
$583$	−682642.	−0.0831805
$584$	0	0
$585$	6.59811e6	0.797132
$586$	0	0
$587$	−1.36076e7	−1.63000	−0.814998	−	0.579463i	$-0.803262\pi$
$587$	−1.36076e7	−1.63000	−0.814998	+	0.579463i	$0.803262\pi$
$588$	0	0
$589$	−709498.	−0.0842681
$590$	0	0
$591$	7.78250e6	0.916538
$592$	0	0
$593$	2.10052e6	0.245296	0.122648	−	0.992450i	$-0.460861\pi$
$593$	2.10052e6	0.245296	0.122648	+	0.992450i	$0.460861\pi$
$594$	0	0
$595$	−1.97342e7	−2.28521
$596$	0	0
$597$	6.77444e6	0.777924
$598$	0	0
$599$	1.62682e6	0.185256	0.0926278	−	0.995701i	$-0.470473\pi$
$599$	1.62682e6	0.185256	0.0926278	+	0.995701i	$0.470473\pi$
$600$	0	0
$601$	−4.49767e6	−0.507928	−0.253964	−	0.967214i	$-0.581734\pi$
$601$	−4.49767e6	−0.507928	−0.253964	+	0.967214i	$0.581734\pi$
$602$	0	0
$603$	2.74079e6	0.306960
$604$	0	0
$605$	−1.54125e7	−1.71193
$606$	0	0
$607$	5.96865e6	0.657513	0.328756	−	0.944415i	$-0.393370\pi$
$607$	5.96865e6	0.657513	0.328756	+	0.944415i	$0.393370\pi$
$608$	0	0
$609$	353082.	0.0385773
$610$	0	0
$611$	9.39531e6	1.01814
$612$	0	0
$613$	3.34021e6	0.359023	0.179511	−	0.983756i	$-0.442548\pi$
$613$	3.34021e6	0.359023	0.179511	+	0.983756i	$0.442548\pi$
$614$	0	0
$615$	1.19546e7	1.27452
$616$	0	0
$617$	−7.12648e6	−0.753637	−0.376818	−	0.926287i	$-0.622982\pi$
$617$	−7.12648e6	−0.753637	−0.376818	+	0.926287i	$0.622982\pi$
$618$	0	0
$619$	−1.20586e7	−1.26494	−0.632469	−	0.774586i	$-0.717958\pi$
$619$	−1.20586e7	−1.26494	−0.632469	+	0.774586i	$0.717958\pi$
$620$	0	0
$621$	−1.84704e6	−0.192197
$622$	0	0
$623$	−2.35539e6	−0.243132
$624$	0	0
$625$	1.64502e7	1.68450
$626$	0	0
$627$	301601.	0.0306383
$628$	0	0
$629$	875182.	0.0882007
$630$	0	0
$631$	−7.68926e6	−0.768796	−0.384398	−	0.923168i	$-0.625591\pi$
$631$	−7.68926e6	−0.768796	−0.384398	+	0.923168i	$0.625591\pi$
$632$	0	0
$633$	2.38965e6	0.237042
$634$	0	0
$635$	3.08291e7	3.03408
$636$	0	0
$637$	5.71548e6	0.558090
$638$	0	0
$639$	5.89567e6	0.571191
$640$	0	0
$641$	−1.55180e7	−1.49174	−0.745868	−	0.666094i	$-0.767965\pi$
$641$	−1.55180e7	−1.49174	−0.745868	+	0.666094i	$0.767965\pi$
$642$	0	0
$643$	1.57536e7	1.50263	0.751314	−	0.659945i	$-0.229420\pi$
$643$	1.57536e7	1.50263	0.751314	+	0.659945i	$0.229420\pi$
$644$	0	0
$645$	−1.25681e7	−1.18951
$646$	0	0
$647$	−1.36896e7	−1.28567	−0.642834	−	0.766006i	$-0.722241\pi$
$647$	−1.36896e7	−1.28567	−0.642834	+	0.766006i	$0.722241\pi$
$648$	0	0
$649$	−677920.	−0.0631781
$650$	0	0
$651$	−2.50304e6	−0.231481
$652$	0	0
$653$	−4.41496e6	−0.405176	−0.202588	−	0.979264i	$-0.564935\pi$
$653$	−4.41496e6	−0.405176	−0.202588	+	0.979264i	$0.564935\pi$
$654$	0	0
$655$	1.97526e7	1.79895
$656$	0	0
$657$	58135.9	0.00525449
$658$	0	0
$659$	−1.73108e6	−0.155276	−0.0776381	−	0.996982i	$-0.524738\pi$
$659$	−1.73108e6	−0.155276	−0.0776381	+	0.996982i	$0.524738\pi$
$660$	0	0
$661$	−8.30967e6	−0.739742	−0.369871	−	0.929083i	$-0.620598\pi$
$661$	−8.30967e6	−0.739742	−0.369871	+	0.929083i	$0.620598\pi$
$662$	0	0
$663$	9.33250e6	0.824545
$664$	0	0
$665$	6.09361e6	0.534343
$666$	0	0
$667$	643797.	0.0560318
$668$	0	0
$669$	1.88223e6	0.162595
$670$	0	0
$671$	1.34671e6	0.115470
$672$	0	0
$673$	−1.56921e6	−0.133550	−0.0667751	−	0.997768i	$-0.521271\pi$
$673$	−1.56921e6	−0.133550	−0.0667751	+	0.997768i	$0.521271\pi$
$674$	0	0
$675$	5.04158e6	0.425899
$676$	0	0
$677$	−1.17260e7	−0.983282	−0.491641	−	0.870798i	$-0.663603\pi$
$677$	−1.17260e7	−0.983282	−0.491641	+	0.870798i	$0.663603\pi$
$678$	0	0
$679$	2.68004e7	2.23083
$680$	0	0
$681$	1.08108e6	0.0893288
$682$	0	0
$683$	2.11345e7	1.73356	0.866782	−	0.498687i	$-0.166184\pi$
$683$	2.11345e7	1.73356	0.866782	+	0.498687i	$0.166184\pi$
$684$	0	0
$685$	4.19398e6	0.341507
$686$	0	0
$687$	3.96553e6	0.320560
$688$	0	0
$689$	6.52248e6	0.523438
$690$	0	0
$691$	−1.84062e7	−1.46646	−0.733230	−	0.679981i	$-0.761988\pi$
$691$	−1.84062e7	−1.46646	−0.733230	+	0.679981i	$0.761988\pi$
$692$	0	0
$693$	1.06402e6	0.0841621
$694$	0	0
$695$	−4.48285e7	−3.52040
$696$	0	0
$697$	1.69088e7	1.31835
$698$	0	0
$699$	−1.50409e6	−0.116434
$700$	0	0
$701$	9.57182e6	0.735698	0.367849	−	0.929886i	$-0.380094\pi$
$701$	9.57182e6	0.735698	0.367849	+	0.929886i	$0.380094\pi$
$702$	0	0
$703$	−270243.	−0.0206236
$704$	0	0
$705$	1.04228e7	0.789790
$706$	0	0
$707$	1.17757e7	0.886008
$708$	0	0
$709$	5.35207e6	0.399859	0.199929	−	0.979810i	$-0.435929\pi$
$709$	5.35207e6	0.399859	0.199929	+	0.979810i	$0.435929\pi$
$710$	0	0
$711$	2.66038e6	0.197365
$712$	0	0
$713$	−4.56395e6	−0.336215
$714$	0	0
$715$	−6.93053e6	−0.506992
$716$	0	0
$717$	−4.01181e6	−0.291436
$718$	0	0
$719$	−1.13450e7	−0.818435	−0.409217	−	0.912437i	$-0.634198\pi$
$719$	−1.13450e7	−0.818435	−0.409217	+	0.912437i	$0.634198\pi$
$720$	0	0
$721$	−1.09955e7	−0.787730
$722$	0	0
$723$	1.37717e6	0.0979807
$724$	0	0
$725$	−1.75727e6	−0.124164
$726$	0	0
$727$	−5.36024e6	−0.376139	−0.188069	−	0.982156i	$-0.560223\pi$
$727$	−5.36024e6	−0.376139	−0.188069	+	0.982156i	$0.560223\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.77765e7	−1.23042
$732$	0	0
$733$	−2.07596e7	−1.42712	−0.713558	−	0.700596i	$-0.752918\pi$
$733$	−2.07596e7	−1.42712	−0.713558	+	0.700596i	$0.752918\pi$
$734$	0	0
$735$	6.34054e6	0.432920
$736$	0	0
$737$	−2.87887e6	−0.195233
$738$	0	0
$739$	2.02904e7	1.36672	0.683359	−	0.730082i	$-0.260518\pi$
$739$	2.02904e7	1.36672	0.683359	+	0.730082i	$0.260518\pi$
$740$	0	0
$741$	−2.88173e6	−0.192800
$742$	0	0
$743$	2.10215e7	1.39699	0.698493	−	0.715617i	$-0.253855\pi$
$743$	2.10215e7	1.39699	0.698493	+	0.715617i	$0.253855\pi$
$744$	0	0
$745$	3.68027e7	2.42934
$746$	0	0
$747$	3.73651e6	0.244999
$748$	0	0
$749$	−2.23599e7	−1.45635
$750$	0	0
$751$	3.49173e6	0.225913	0.112956	−	0.993600i	$-0.463968\pi$
$751$	3.49173e6	0.225913	0.112956	+	0.993600i	$0.463968\pi$
$752$	0	0
$753$	−6.24023e6	−0.401063
$754$	0	0
$755$	−3.57840e7	−2.28466
$756$	0	0
$757$	−2.15165e7	−1.36468	−0.682342	−	0.731033i	$-0.739039\pi$
$757$	−2.15165e7	−1.36468	−0.682342	+	0.731033i	$0.739039\pi$
$758$	0	0
$759$	1.94010e6	0.122242
$760$	0	0
$761$	1.62464e7	1.01694	0.508469	−	0.861080i	$-0.330212\pi$
$761$	1.62464e7	1.01694	0.508469	+	0.861080i	$0.330212\pi$
$762$	0	0
$763$	2.51993e6	0.156703
$764$	0	0
$765$	1.03531e7	0.639614
$766$	0	0
$767$	6.47736e6	0.397567
$768$	0	0
$769$	1.89849e7	1.15769	0.578846	−	0.815437i	$-0.303503\pi$
$769$	1.89849e7	1.15769	0.578846	+	0.815437i	$0.303503\pi$
$770$	0	0
$771$	−1.54698e7	−0.937233
$772$	0	0
$773$	9.61884e6	0.578994	0.289497	−	0.957179i	$-0.406512\pi$
$773$	9.61884e6	0.578994	0.289497	+	0.957179i	$0.406512\pi$
$774$	0	0
$775$	1.24575e7	0.745036
$776$	0	0
$777$	−953388.	−0.0566523
$778$	0	0
$779$	−5.22118e6	−0.308265
$780$	0	0
$781$	−6.19270e6	−0.363289
$782$	0	0
$783$	−185237.	−0.0107975
$784$	0	0
$785$	2.57119e7	1.48922
$786$	0	0
$787$	−9.93586e6	−0.571832	−0.285916	−	0.958255i	$-0.592298\pi$
$787$	−9.93586e6	−0.571832	−0.285916	+	0.958255i	$0.592298\pi$
$788$	0	0
$789$	1.40968e6	0.0806174
$790$	0	0
$791$	2.58343e7	1.46810
$792$	0	0
$793$	−1.28675e7	−0.726628
$794$	0	0
$795$	7.23580e6	0.406040
$796$	0	0
$797$	2.77653e7	1.54831	0.774154	−	0.632998i	$-0.218176\pi$
$797$	2.77653e7	1.54831	0.774154	+	0.632998i	$0.218176\pi$
$798$	0	0
$799$	1.47422e7	0.816951
$800$	0	0
$801$	1.23570e6	0.0680508
$802$	0	0
$803$	−61064.8	−0.00334197
$804$	0	0
$805$	3.91980e7	2.13194
$806$	0	0
$807$	7.20001e6	0.389179
$808$	0	0
$809$	−2.70847e7	−1.45497	−0.727484	−	0.686125i	$-0.759311\pi$
$809$	−2.70847e7	−1.45497	−0.727484	+	0.686125i	$0.759311\pi$
$810$	0	0
$811$	−2.00386e7	−1.06983	−0.534917	−	0.844905i	$-0.679657\pi$
$811$	−2.00386e7	−1.06983	−0.534917	+	0.844905i	$0.679657\pi$
$812$	0	0
$813$	1.70637e7	0.905416
$814$	0	0
$815$	−5.61759e7	−2.96249
$816$	0	0
$817$	5.48911e6	0.287705
$818$	0	0
$819$	−1.01665e7	−0.529614
$820$	0	0
$821$	3.68572e6	0.190838	0.0954188	−	0.995437i	$-0.469581\pi$
$821$	3.68572e6	0.190838	0.0954188	+	0.995437i	$0.469581\pi$
$822$	0	0
$823$	−2.22928e7	−1.14727	−0.573633	−	0.819112i	$-0.694467\pi$
$823$	−2.22928e7	−1.14727	−0.573633	+	0.819112i	$0.694467\pi$
$824$	0	0
$825$	−5.29558e6	−0.270881
$826$	0	0
$827$	−1.98390e7	−1.00869	−0.504343	−	0.863503i	$-0.668265\pi$
$827$	−1.98390e7	−1.00869	−0.504343	+	0.863503i	$0.668265\pi$
$828$	0	0
$829$	−4.83084e6	−0.244139	−0.122069	−	0.992522i	$-0.538953\pi$
$829$	−4.83084e6	−0.244139	−0.122069	+	0.992522i	$0.538953\pi$
$830$	0	0
$831$	−2.51823e6	−0.126500
$832$	0	0
$833$	8.96819e6	0.447808
$834$	0	0
$835$	4.11773e7	2.04381
$836$	0	0
$837$	1.31317e6	0.0647898
$838$	0	0
$839$	−2.24080e7	−1.09900	−0.549501	−	0.835493i	$-0.685182\pi$
$839$	−2.24080e7	−1.09900	−0.549501	+	0.835493i	$0.685182\pi$
$840$	0	0
$841$	−2.04466e7	−0.996852
$842$	0	0
$843$	−4.98797e6	−0.241743
$844$	0	0
$845$	2.90147e7	1.39790
$846$	0	0
$847$	2.37478e7	1.13740
$848$	0	0
$849$	1.41910e7	0.675686
$850$	0	0
$851$	−1.73838e6	−0.0822848
$852$	0	0
$853$	2.55770e7	1.20358	0.601792	−	0.798653i	$-0.294453\pi$
$853$	2.55770e7	1.20358	0.601792	+	0.798653i	$0.294453\pi$
$854$	0	0
$855$	−3.19688e6	−0.149559
$856$	0	0
$857$	1.94781e7	0.905929	0.452964	−	0.891529i	$-0.350366\pi$
$857$	1.94781e7	0.905929	0.452964	+	0.891529i	$0.350366\pi$
$858$	0	0
$859$	2.93437e7	1.35685	0.678424	−	0.734670i	$-0.262663\pi$
$859$	2.93437e7	1.35685	0.678424	+	0.734670i	$0.262663\pi$
$860$	0	0
$861$	−1.84198e7	−0.846792
$862$	0	0
$863$	9.84655e6	0.450046	0.225023	−	0.974353i	$-0.427754\pi$
$863$	9.84655e6	0.450046	0.225023	+	0.974353i	$0.427754\pi$
$864$	0	0
$865$	−4.08387e7	−1.85580
$866$	0	0
$867$	1.86496e6	0.0842601
$868$	0	0
$869$	−2.79441e6	−0.125528
$870$	0	0
$871$	2.75069e7	1.22856
$872$	0	0
$873$	−1.40603e7	−0.624394
$874$	0	0
$875$	−5.86462e7	−2.58952
$876$	0	0
$877$	1.09591e7	0.481147	0.240573	−	0.970631i	$-0.422665\pi$
$877$	1.09591e7	0.481147	0.240573	+	0.970631i	$0.422665\pi$
$878$	0	0
$879$	−1.66107e6	−0.0725128
$880$	0	0
$881$	−4.72688e6	−0.205180	−0.102590	−	0.994724i	$-0.532713\pi$
$881$	−4.72688e6	−0.205180	−0.102590	+	0.994724i	$0.532713\pi$
$882$	0	0
$883$	3.23858e7	1.39782	0.698912	−	0.715207i	$-0.253668\pi$
$883$	3.23858e7	1.39782	0.698912	+	0.715207i	$0.253668\pi$
$884$	0	0
$885$	7.18575e6	0.308400
$886$	0	0
$887$	−4.35277e7	−1.85762	−0.928809	−	0.370559i	$-0.879166\pi$
$887$	−4.35277e7	−1.85762	−0.928809	+	0.370559i	$0.879166\pi$
$888$	0	0
$889$	−4.75018e7	−2.01584
$890$	0	0
$891$	−558216.	−0.0235563
$892$	0	0
$893$	−4.55217e6	−0.191025
$894$	0	0
$895$	7.38559e6	0.308197
$896$	0	0
$897$	−1.85372e7	−0.769240
$898$	0	0
$899$	−457713.	−0.0188883
$900$	0	0
$901$	1.02345e7	0.420004
$902$	0	0
$903$	1.93650e7	0.790312
$904$	0	0
$905$	1.94722e7	0.790304
$906$	0	0
$907$	4.49574e7	1.81461	0.907303	−	0.420477i	$-0.138137\pi$
$907$	4.49574e7	1.81461	0.907303	+	0.420477i	$0.138137\pi$
$908$	0	0
$909$	−6.17787e6	−0.247987
$910$	0	0
$911$	−3.43483e7	−1.37123	−0.685614	−	0.727965i	$-0.740466\pi$
$911$	−3.43483e7	−1.37123	−0.685614	+	0.727965i	$0.740466\pi$
$912$	0	0
$913$	−3.92476e6	−0.155825
$914$	0	0
$915$	−1.42748e7	−0.563658
$916$	0	0
$917$	−3.04350e7	−1.19522
$918$	0	0
$919$	1.51792e7	0.592871	0.296435	−	0.955053i	$-0.404202\pi$
$919$	1.51792e7	0.592871	0.296435	+	0.955053i	$0.404202\pi$
$920$	0	0
$921$	−3.65290e6	−0.141902
$922$	0	0
$923$	5.91698e7	2.28610
$924$	0	0
$925$	4.74497e6	0.182339
$926$	0	0
$927$	5.76856e6	0.220480
$928$	0	0
$929$	−3.12691e7	−1.18871	−0.594356	−	0.804202i	$-0.702593\pi$
$929$	−3.12691e7	−1.18871	−0.594356	+	0.804202i	$0.702593\pi$
$930$	0	0
$931$	−2.76923e6	−0.104709
$932$	0	0
$933$	−2.28994e6	−0.0861232
$934$	0	0
$935$	−1.08747e7	−0.406808
$936$	0	0
$937$	−2.67948e7	−0.997015	−0.498507	−	0.866885i	$-0.666118\pi$
$937$	−2.67948e7	−0.997015	−0.498507	+	0.866885i	$0.666118\pi$
$938$	0	0
$939$	−9.34759e6	−0.345968
$940$	0	0
$941$	8.36072e6	0.307801	0.153900	−	0.988086i	$-0.450817\pi$
$941$	8.36072e6	0.307801	0.153900	+	0.988086i	$0.450817\pi$
$942$	0	0
$943$	−3.35860e7	−1.22993
$944$	0	0
$945$	−1.12783e7	−0.410831
$946$	0	0
$947$	−4.40994e7	−1.59793	−0.798965	−	0.601377i	$-0.794619\pi$
$947$	−4.40994e7	−1.59793	−0.798965	+	0.601377i	$0.794619\pi$
$948$	0	0
$949$	583460.	0.0210303
$950$	0	0
$951$	1.25919e7	0.451482
$952$	0	0
$953$	−1.99247e7	−0.710658	−0.355329	−	0.934741i	$-0.615631\pi$
$953$	−1.99247e7	−0.710658	−0.355329	+	0.934741i	$0.615631\pi$
$954$	0	0
$955$	1.71832e7	0.609673
$956$	0	0
$957$	194569.	0.00686744
$958$	0	0
$959$	−6.46213e6	−0.226897
$960$	0	0
$961$	−2.53844e7	−0.886662
$962$	0	0
$963$	1.17307e7	0.407621
$964$	0	0
$965$	7.76035e7	2.68264
$966$	0	0
$967$	−4.56359e7	−1.56942	−0.784712	−	0.619860i	$-0.787189\pi$
$967$	−4.56359e7	−1.56942	−0.784712	+	0.619860i	$0.787189\pi$
$968$	0	0
$969$	−4.52173e6	−0.154702
$970$	0	0
$971$	2.09777e7	0.714020	0.357010	−	0.934101i	$-0.383796\pi$
$971$	2.09777e7	0.714020	0.357010	+	0.934101i	$0.383796\pi$
$972$	0	0
$973$	6.90723e7	2.33895
$974$	0	0
$975$	5.05980e7	1.70460
$976$	0	0
$977$	−3.03128e7	−1.01599	−0.507995	−	0.861360i	$-0.669613\pi$
$977$	−3.03128e7	−1.01599	−0.507995	+	0.861360i	$0.669613\pi$
$978$	0	0
$979$	−1.29796e6	−0.0432817
$980$	0	0
$981$	−1.32203e6	−0.0438600
$982$	0	0
$983$	−2.04558e7	−0.675200	−0.337600	−	0.941290i	$-0.609615\pi$
$983$	−2.04558e7	−0.675200	−0.337600	+	0.941290i	$0.609615\pi$
$984$	0	0
$985$	8.66483e7	2.84557
$986$	0	0
$987$	−1.60596e7	−0.524737
$988$	0	0
$989$	3.53095e7	1.14789
$990$	0	0
$991$	−1.72449e7	−0.557797	−0.278898	−	0.960321i	$-0.589969\pi$
$991$	−1.72449e7	−0.557797	−0.278898	+	0.960321i	$0.589969\pi$
$992$	0	0
$993$	−1.84200e7	−0.592811
$994$	0	0
$995$	7.54247e7	2.41522
$996$	0	0
$997$	−7.27445e6	−0.231773	−0.115886	−	0.993262i	$-0.536971\pi$
$997$	−7.27445e6	−0.231773	−0.115886	+	0.993262i	$0.536971\pi$
$998$	0	0
$999$	500175.	0.0158565

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.6.a.bd.1.5		5
4.3	odd	2		768.6.a.bb.1.5		5
8.3	odd	2		768.6.a.bc.1.1		5
8.5	even	2		768.6.a.ba.1.1		5
16.3	odd	4		96.6.d.a.49.1		10
16.5	even	4		24.6.d.a.13.8	yes	10
16.11	odd	4		96.6.d.a.49.10		10
16.13	even	4		24.6.d.a.13.7	✓	10
48.5	odd	4		72.6.d.d.37.3		10
48.11	even	4		288.6.d.d.145.1		10
48.29	odd	4		72.6.d.d.37.4		10
48.35	even	4		288.6.d.d.145.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.6.d.a.13.7	✓	10	16.13	even	4
24.6.d.a.13.8	yes	10	16.5	even	4
72.6.d.d.37.3		10	48.5	odd	4
72.6.d.d.37.4		10	48.29	odd	4
96.6.d.a.49.1		10	16.3	odd	4
96.6.d.a.49.10		10	16.11	odd	4
288.6.d.d.145.1		10	48.11	even	4
288.6.d.d.145.10		10	48.35	even	4
768.6.a.ba.1.1		5	8.5	even	2
768.6.a.bb.1.5		5	4.3	odd	2
768.6.a.bc.1.1		5	8.3	odd	2
768.6.a.bd.1.5		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +100.204 q^{5} -154.395 q^{7} +81.0000 q^{9} -85.0809 q^{11} +812.927 q^{13} +901.832 q^{15} +1275.57 q^{17} -393.875 q^{19} -1389.55 q^{21} -2533.66 q^{23} +6915.74 q^{25} +729.000 q^{27} -254.097 q^{29} +1801.33 q^{31} -765.728 q^{33} -15470.9 q^{35} +686.112 q^{37} +7316.35 q^{39} +13255.9 q^{41} -13936.2 q^{43} +8116.48 q^{45} +11557.4 q^{47} +7030.74 q^{49} +11480.1 q^{51} +8023.45 q^{53} -8525.40 q^{55} -3544.88 q^{57} +7967.95 q^{59} -15828.6 q^{61} -12506.0 q^{63} +81458.2 q^{65} +33836.9 q^{67} -22803.0 q^{69} +72786.0 q^{71} +717.727 q^{73} +62241.7 q^{75} +13136.0 q^{77} +32844.2 q^{79} +6561.00 q^{81} +46129.8 q^{83} +127816. q^{85} -2286.88 q^{87} +15255.6 q^{89} -125512. q^{91} +16211.9 q^{93} -39467.7 q^{95} -173584. q^{97} -6891.55 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 45 q^{3} + 50 q^{5} + 98 q^{7} + 405 q^{9} + 676 q^{13} + 450 q^{15} - 202 q^{17} + 716 q^{19} + 882 q^{21} - 836 q^{23} + 4683 q^{25} + 3645 q^{27} + 5046 q^{29} + 9346 q^{31} + 436 q^{35} + 10952 q^{37}+ \cdots + 29686 q^{97}+O(q^{100}) \) 5 * q + 45 * q^3 + 50 * q^5 + 98 * q^7 + 405 * q^9 + 676 * q^13 + 450 * q^15 - 202 * q^17 + 716 * q^19 + 882 * q^21 - 836 * q^23 + 4683 * q^25 + 3645 * q^27 + 5046 * q^29 + 9346 * q^31 + 436 * q^35 + 10952 * q^37 + 6084 * q^39 + 1238 * q^41 + 1676 * q^43 + 4050 * q^45 + 32628 * q^47 + 4689 * q^49 - 1818 * q^51 - 8490 * q^53 + 51712 * q^55 + 6444 * q^57 - 15884 * q^59 - 13824 * q^61 + 7938 * q^63 + 25768 * q^65 + 26652 * q^67 - 7524 * q^69 + 136116 * q^71 + 13694 * q^73 + 42147 * q^75 + 89856 * q^77 + 126154 * q^79 + 32805 * q^81 - 64720 * q^83 + 164636 * q^85 + 45414 * q^87 - 87790 * q^89 - 179992 * q^91 + 84114 * q^93 + 121592 * q^95 + 29686 * q^97

Introduction

L-functions

Modular forms

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Database

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