LMFDB - Embedded newform 768.4.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.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:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 384)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} +18.4222 q^{5} +22.4222 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +18.4222 q^{5} +22.4222 q^{7} +9.00000 q^{9} +53.6888 q^{11} -7.15559 q^{13} -55.2666 q^{15} +39.6888 q^{17} +125.689 q^{19} -67.2666 q^{21} +99.1556 q^{23} +214.378 q^{25} -27.0000 q^{27} -205.800 q^{29} +147.489 q^{31} -161.066 q^{33} +413.066 q^{35} -125.689 q^{37} +21.4668 q^{39} -506.444 q^{41} -413.689 q^{43} +165.800 q^{45} -313.911 q^{47} +159.755 q^{49} -119.066 q^{51} +44.3331 q^{53} +989.066 q^{55} -377.066 q^{57} -324.000 q^{59} -324.000 q^{61} +201.800 q^{63} -131.822 q^{65} -464.266 q^{67} -297.467 q^{69} +1052.84 q^{71} +1022.27 q^{73} -643.133 q^{75} +1203.82 q^{77} -602.910 q^{79} +81.0000 q^{81} -15.8217 q^{83} +731.156 q^{85} +617.400 q^{87} +381.378 q^{89} -160.444 q^{91} -442.466 q^{93} +2315.47 q^{95} +659.154 q^{97} +483.199 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 8 q^{5} + 16 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 8 * q^5 + 16 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} + 8 q^{5} + 16 q^{7} + 18 q^{9} - 8 q^{11} - 72 q^{13} - 24 q^{15} - 36 q^{17} + 136 q^{19} - 48 q^{21} + 256 q^{23} + 198 q^{25} - 54 q^{27} - 152 q^{29} - 80 q^{31} + 24 q^{33} + 480 q^{35} - 136 q^{37} + 216 q^{39} - 436 q^{41} - 712 q^{43} + 72 q^{45} - 224 q^{47} - 142 q^{49} + 108 q^{51} - 344 q^{53} + 1632 q^{55} - 408 q^{57} - 648 q^{59} - 648 q^{61} + 144 q^{63} + 544 q^{65} + 456 q^{67} - 768 q^{69} + 2048 q^{71} + 660 q^{73} - 594 q^{75} + 1600 q^{77} + 496 q^{79} + 162 q^{81} + 776 q^{83} + 1520 q^{85} + 456 q^{87} + 532 q^{89} + 256 q^{91} + 240 q^{93} + 2208 q^{95} - 1220 q^{97} - 72 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 8 * q^5 + 16 * q^7 + 18 * q^9 - 8 * q^11 - 72 * q^13 - 24 * q^15 - 36 * q^17 + 136 * q^19 - 48 * q^21 + 256 * q^23 + 198 * q^25 - 54 * q^27 - 152 * q^29 - 80 * q^31 + 24 * q^33 + 480 * q^35 - 136 * q^37 + 216 * q^39 - 436 * q^41 - 712 * q^43 + 72 * q^45 - 224 * q^47 - 142 * q^49 + 108 * q^51 - 344 * q^53 + 1632 * q^55 - 408 * q^57 - 648 * q^59 - 648 * q^61 + 144 * q^63 + 544 * q^65 + 456 * q^67 - 768 * q^69 + 2048 * q^71 + 660 * q^73 - 594 * q^75 + 1600 * q^77 + 496 * q^79 + 162 * q^81 + 776 * q^83 + 1520 * q^85 + 456 * q^87 + 532 * q^89 + 256 * q^91 + 240 * q^93 + 2208 * q^95 - 1220 * q^97 - 72 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	18.4222	1.64773	0.823866	−	0.566785i	$-0.191813\pi$
$5$	18.4222	1.64773	0.823866	+	0.566785i	$0.191813\pi$
$6$	0	0
$7$	22.4222	1.21069	0.605343	−	0.795965i	$-0.293036\pi$
$7$	22.4222	1.21069	0.605343	+	0.795965i	$0.293036\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	53.6888	1.47162	0.735809	−	0.677190i	$-0.236802\pi$
$11$	53.6888	1.47162	0.735809	+	0.677190i	$0.236802\pi$
$12$	0	0
$13$	−7.15559	−0.152662	−0.0763309	−	0.997083i	$-0.524321\pi$
$13$	−7.15559	−0.152662	−0.0763309	+	0.997083i	$0.524321\pi$
$14$	0	0
$15$	−55.2666	−0.951319
$16$	0	0
$17$	39.6888	0.566233	0.283116	−	0.959086i	$-0.408632\pi$
$17$	39.6888	0.566233	0.283116	+	0.959086i	$0.408632\pi$
$18$	0	0
$19$	125.689	1.51763	0.758816	−	0.651306i	$-0.225778\pi$
$19$	125.689	1.51763	0.758816	+	0.651306i	$0.225778\pi$
$20$	0	0
$21$	−67.2666	−0.698989
$22$	0	0
$23$	99.1556	0.898929	0.449465	−	0.893298i	$-0.351615\pi$
$23$	99.1556	0.898929	0.449465	+	0.893298i	$0.351615\pi$
$24$	0	0
$25$	214.378	1.71502
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−205.800	−1.31780	−0.658898	−	0.752232i	$-0.728977\pi$
$29$	−205.800	−1.31780	−0.658898	+	0.752232i	$0.728977\pi$
$30$	0	0
$31$	147.489	0.854508	0.427254	−	0.904132i	$-0.359481\pi$
$31$	147.489	0.854508	0.427254	+	0.904132i	$0.359481\pi$
$32$	0	0
$33$	−161.066	−0.849639
$34$	0	0
$35$	413.066	1.99489
$36$	0	0
$37$	−125.689	−0.558463	−0.279231	−	0.960224i	$-0.590080\pi$
$37$	−125.689	−0.558463	−0.279231	+	0.960224i	$0.590080\pi$
$38$	0	0
$39$	21.4668	0.0881393
$40$	0	0
$41$	−506.444	−1.92910	−0.964552	−	0.263892i	$-0.914994\pi$
$41$	−506.444	−1.92910	−0.964552	+	0.263892i	$0.914994\pi$
$42$	0	0
$43$	−413.689	−1.46714	−0.733569	−	0.679615i	$-0.762147\pi$
$43$	−413.689	−1.46714	−0.733569	+	0.679615i	$0.762147\pi$
$44$	0	0
$45$	165.800	0.549244
$46$	0	0
$47$	−313.911	−0.974226	−0.487113	−	0.873339i	$-0.661950\pi$
$47$	−313.911	−0.974226	−0.487113	+	0.873339i	$0.661950\pi$
$48$	0	0
$49$	159.755	0.465759
$50$	0	0
$51$	−119.066	−0.326914
$52$	0	0
$53$	44.3331	0.114898	0.0574492	−	0.998348i	$-0.481703\pi$
$53$	44.3331	0.114898	0.0574492	+	0.998348i	$0.481703\pi$
$54$	0	0
$55$	989.066	2.42483
$56$	0	0
$57$	−377.066	−0.876205
$58$	0	0
$59$	−324.000	−0.714936	−0.357468	−	0.933925i	$-0.616360\pi$
$59$	−324.000	−0.714936	−0.357468	+	0.933925i	$0.616360\pi$
$60$	0	0
$61$	−324.000	−0.680065	−0.340032	−	0.940414i	$-0.610438\pi$
$61$	−324.000	−0.680065	−0.340032	+	0.940414i	$0.610438\pi$
$62$	0	0
$63$	201.800	0.403562
$64$	0	0
$65$	−131.822	−0.251546
$66$	0	0
$67$	−464.266	−0.846554	−0.423277	−	0.906000i	$-0.639120\pi$
$67$	−464.266	−0.846554	−0.423277	+	0.906000i	$0.639120\pi$
$68$	0	0
$69$	−297.467	−0.518997
$70$	0	0
$71$	1052.84	1.75985	0.879927	−	0.475109i	$-0.157591\pi$
$71$	1052.84	1.75985	0.879927	+	0.475109i	$0.157591\pi$
$72$	0	0
$73$	1022.27	1.63900	0.819501	−	0.573078i	$-0.194251\pi$
$73$	1022.27	1.63900	0.819501	+	0.573078i	$0.194251\pi$
$74$	0	0
$75$	−643.133	−0.990168
$76$	0	0
$77$	1203.82	1.78167
$78$	0	0
$79$	−602.910	−0.858642	−0.429321	−	0.903152i	$-0.641247\pi$
$79$	−602.910	−0.858642	−0.429321	+	0.903152i	$0.641247\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−15.8217	−0.0209236	−0.0104618	−	0.999945i	$-0.503330\pi$
$83$	−15.8217	−0.0209236	−0.0104618	+	0.999945i	$0.503330\pi$
$84$	0	0
$85$	731.156	0.933000
$86$	0	0
$87$	617.400	0.760830
$88$	0	0
$89$	381.378	0.454224	0.227112	−	0.973869i	$-0.427072\pi$
$89$	381.378	0.454224	0.227112	+	0.973869i	$0.427072\pi$
$90$	0	0
$91$	−160.444	−0.184825
$92$	0	0
$93$	−442.466	−0.493350
$94$	0	0
$95$	2315.47	2.50065
$96$	0	0
$97$	659.154	0.689969	0.344984	−	0.938608i	$-0.387884\pi$
$97$	659.154	0.689969	0.344984	+	0.938608i	$0.387884\pi$
$98$	0	0
$99$	483.199	0.490539
$100$	0	0
$101$	−498.510	−0.491125	−0.245562	−	0.969381i	$-0.578973\pi$
$101$	−498.510	−0.491125	−0.245562	+	0.969381i	$0.578973\pi$
$102$	0	0
$103$	−196.821	−0.188285	−0.0941425	−	0.995559i	$-0.530011\pi$
$103$	−196.821	−0.188285	−0.0941425	+	0.995559i	$0.530011\pi$
$104$	0	0
$105$	−1239.20	−1.15175
$106$	0	0
$107$	−359.378	−0.324695	−0.162347	−	0.986734i	$-0.551907\pi$
$107$	−359.378	−0.324695	−0.162347	+	0.986734i	$0.551907\pi$
$108$	0	0
$109$	−1969.73	−1.73088	−0.865441	−	0.501011i	$-0.832962\pi$
$109$	−1969.73	−1.73088	−0.865441	+	0.501011i	$0.832962\pi$
$110$	0	0
$111$	377.066	0.322429
$112$	0	0
$113$	−693.643	−0.577456	−0.288728	−	0.957411i	$-0.593232\pi$
$113$	−693.643	−0.577456	−0.288728	+	0.957411i	$0.593232\pi$
$114$	0	0
$115$	1826.66	1.48119
$116$	0	0
$117$	−64.4003	−0.0508873
$118$	0	0
$119$	889.911	0.685529
$120$	0	0
$121$	1551.49	1.16566
$122$	0	0
$123$	1519.33	1.11377
$124$	0	0
$125$	1646.53	1.17816
$126$	0	0
$127$	2656.78	1.85631	0.928153	−	0.372199i	$-0.121396\pi$
$127$	2656.78	1.85631	0.928153	+	0.372199i	$0.121396\pi$
$128$	0	0
$129$	1241.07	0.847053
$130$	0	0
$131$	−615.734	−0.410664	−0.205332	−	0.978692i	$-0.565827\pi$
$131$	−615.734	−0.410664	−0.205332	+	0.978692i	$0.565827\pi$
$132$	0	0
$133$	2818.22	1.83737
$134$	0	0
$135$	−497.400	−0.317106
$136$	0	0
$137$	−613.290	−0.382459	−0.191230	−	0.981545i	$-0.561247\pi$
$137$	−613.290	−0.382459	−0.191230	+	0.981545i	$0.561247\pi$
$138$	0	0
$139$	1899.29	1.15896	0.579480	−	0.814987i	$-0.303256\pi$
$139$	1899.29	1.15896	0.579480	+	0.814987i	$0.303256\pi$
$140$	0	0
$141$	941.733	0.562469
$142$	0	0
$143$	−384.175	−0.224660
$144$	0	0
$145$	−3791.29	−2.17137
$146$	0	0
$147$	−479.266	−0.268906
$148$	0	0
$149$	976.377	0.536832	0.268416	−	0.963303i	$-0.413500\pi$
$149$	976.377	0.536832	0.268416	+	0.963303i	$0.413500\pi$
$150$	0	0
$151$	683.132	0.368162	0.184081	−	0.982911i	$-0.441069\pi$
$151$	683.132	0.368162	0.184081	+	0.982911i	$0.441069\pi$
$152$	0	0
$153$	357.199	0.188744
$154$	0	0
$155$	2717.07	1.40800
$156$	0	0
$157$	511.109	0.259815	0.129907	−	0.991526i	$-0.458532\pi$
$157$	511.109	0.259815	0.129907	+	0.991526i	$0.458532\pi$
$158$	0	0
$159$	−132.999	−0.0663366
$160$	0	0
$161$	2223.29	1.08832
$162$	0	0
$163$	−2425.95	−1.16574	−0.582869	−	0.812566i	$-0.698070\pi$
$163$	−2425.95	−1.16574	−0.582869	+	0.812566i	$0.698070\pi$
$164$	0	0
$165$	−2967.20	−1.39998
$166$	0	0
$167$	337.332	0.156309	0.0781544	−	0.996941i	$-0.475097\pi$
$167$	337.332	0.156309	0.0781544	+	0.996941i	$0.475097\pi$
$168$	0	0
$169$	−2145.80	−0.976694
$170$	0	0
$171$	1131.20	0.505877
$172$	0	0
$173$	2648.29	1.16385	0.581924	−	0.813243i	$-0.302300\pi$
$173$	2648.29	1.16385	0.581924	+	0.813243i	$0.302300\pi$
$174$	0	0
$175$	4806.82	2.07635
$176$	0	0
$177$	972.000	0.412768
$178$	0	0
$179$	−2907.29	−1.21397	−0.606986	−	0.794713i	$-0.707621\pi$
$179$	−2907.29	−1.21397	−0.606986	+	0.794713i	$0.707621\pi$
$180$	0	0
$181$	−3682.80	−1.51238	−0.756188	−	0.654354i	$-0.772940\pi$
$181$	−3682.80	−1.51238	−0.756188	+	0.654354i	$0.772940\pi$
$182$	0	0
$183$	972.000	0.392636
$184$	0	0
$185$	−2315.47	−0.920197
$186$	0	0
$187$	2130.85	0.833277
$188$	0	0
$189$	−605.400	−0.232996
$190$	0	0
$191$	1279.56	0.484740	0.242370	−	0.970184i	$-0.422075\pi$
$191$	1279.56	0.484740	0.242370	+	0.970184i	$0.422075\pi$
$192$	0	0
$193$	−4836.84	−1.80396	−0.901978	−	0.431783i	$-0.857885\pi$
$193$	−4836.84	−1.80396	−0.901978	+	0.431783i	$0.857885\pi$
$194$	0	0
$195$	395.465	0.145230
$196$	0	0
$197$	2869.31	1.03771	0.518857	−	0.854861i	$-0.326358\pi$
$197$	2869.31	1.03771	0.518857	+	0.854861i	$0.326358\pi$
$198$	0	0
$199$	652.242	0.232343	0.116171	−	0.993229i	$-0.462938\pi$
$199$	652.242	0.232343	0.116171	+	0.993229i	$0.462938\pi$
$200$	0	0
$201$	1392.80	0.488758
$202$	0	0
$203$	−4614.49	−1.59544
$204$	0	0
$205$	−9329.82	−3.17865
$206$	0	0
$207$	892.400	0.299643
$208$	0	0
$209$	6748.08	2.23337
$210$	0	0
$211$	−537.511	−0.175373	−0.0876866	−	0.996148i	$-0.527947\pi$
$211$	−537.511	−0.175373	−0.0876866	+	0.996148i	$0.527947\pi$
$212$	0	0
$213$	−3158.53	−1.01605
$214$	0	0
$215$	−7621.06	−2.41745
$216$	0	0
$217$	3307.02	1.03454
$218$	0	0
$219$	−3066.80	−0.946278
$220$	0	0
$221$	−283.997	−0.0864421
$222$	0	0
$223$	−4041.98	−1.21377	−0.606885	−	0.794790i	$-0.707581\pi$
$223$	−4041.98	−1.21377	−0.606885	+	0.794790i	$0.707581\pi$
$224$	0	0
$225$	1929.40	0.571674
$226$	0	0
$227$	−3070.22	−0.897699	−0.448850	−	0.893607i	$-0.648166\pi$
$227$	−3070.22	−0.897699	−0.448850	+	0.893607i	$0.648166\pi$
$228$	0	0
$229$	205.110	0.0591881	0.0295940	−	0.999562i	$-0.490579\pi$
$229$	205.110	0.0591881	0.0295940	+	0.999562i	$0.490579\pi$
$230$	0	0
$231$	−3611.47	−1.02864
$232$	0	0
$233$	13.3776	0.00376137	0.00188068	−	0.999998i	$-0.499401\pi$
$233$	13.3776	0.00376137	0.00188068	+	0.999998i	$0.499401\pi$
$234$	0	0
$235$	−5782.93	−1.60526
$236$	0	0
$237$	1808.73	0.495737
$238$	0	0
$239$	4327.99	1.17136	0.585679	−	0.810543i	$-0.300828\pi$
$239$	4327.99	1.17136	0.585679	+	0.810543i	$0.300828\pi$
$240$	0	0
$241$	−1508.31	−0.403150	−0.201575	−	0.979473i	$-0.564606\pi$
$241$	−1508.31	−0.403150	−0.201575	+	0.979473i	$0.564606\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	2943.04	0.767446
$246$	0	0
$247$	−899.378	−0.231684
$248$	0	0
$249$	47.4652	0.0120803
$250$	0	0
$251$	4871.47	1.22504	0.612518	−	0.790456i	$-0.290157\pi$
$251$	4871.47	1.22504	0.612518	+	0.790456i	$0.290157\pi$
$252$	0	0
$253$	5323.55	1.32288
$254$	0	0
$255$	−2193.47	−0.538668
$256$	0	0
$257$	−1665.55	−0.404258	−0.202129	−	0.979359i	$-0.564786\pi$
$257$	−1665.55	−0.404258	−0.202129	+	0.979359i	$0.564786\pi$
$258$	0	0
$259$	−2818.22	−0.676122
$260$	0	0
$261$	−1852.20	−0.439265
$262$	0	0
$263$	7167.64	1.68052	0.840258	−	0.542188i	$-0.182404\pi$
$263$	7167.64	1.68052	0.840258	+	0.542188i	$0.182404\pi$
$264$	0	0
$265$	816.713	0.189322
$266$	0	0
$267$	−1144.13	−0.262246
$268$	0	0
$269$	5453.84	1.23616	0.618079	−	0.786116i	$-0.287911\pi$
$269$	5453.84	1.23616	0.618079	+	0.786116i	$0.287911\pi$
$270$	0	0
$271$	−5416.20	−1.21406	−0.607031	−	0.794678i	$-0.707640\pi$
$271$	−5416.20	−1.21406	−0.607031	+	0.794678i	$0.707640\pi$
$272$	0	0
$273$	481.332	0.106709
$274$	0	0
$275$	11509.7	2.52385
$276$	0	0
$277$	2648.75	0.574542	0.287271	−	0.957849i	$-0.407252\pi$
$277$	2648.75	0.574542	0.287271	+	0.957849i	$0.407252\pi$
$278$	0	0
$279$	1327.40	0.284836
$280$	0	0
$281$	6664.57	1.41486	0.707429	−	0.706784i	$-0.249855\pi$
$281$	6664.57	1.41486	0.707429	+	0.706784i	$0.249855\pi$
$282$	0	0
$283$	5630.84	1.18275	0.591376	−	0.806396i	$-0.298585\pi$
$283$	5630.84	1.18275	0.591376	+	0.806396i	$0.298585\pi$
$284$	0	0
$285$	−6946.40	−1.44375
$286$	0	0
$287$	−11355.6	−2.33554
$288$	0	0
$289$	−3337.80	−0.679381
$290$	0	0
$291$	−1977.46	−0.398354
$292$	0	0
$293$	−908.374	−0.181119	−0.0905593	−	0.995891i	$-0.528865\pi$
$293$	−908.374	−0.181119	−0.0905593	+	0.995891i	$0.528865\pi$
$294$	0	0
$295$	−5968.79	−1.17802
$296$	0	0
$297$	−1449.60	−0.283213
$298$	0	0
$299$	−709.517	−0.137232
$300$	0	0
$301$	−9275.82	−1.77624
$302$	0	0
$303$	1495.53	0.283551
$304$	0	0
$305$	−5968.79	−1.12056
$306$	0	0
$307$	414.671	0.0770896	0.0385448	−	0.999257i	$-0.487728\pi$
$307$	414.671	0.0770896	0.0385448	+	0.999257i	$0.487728\pi$
$308$	0	0
$309$	590.463	0.108706
$310$	0	0
$311$	1615.91	0.294629	0.147315	−	0.989090i	$-0.452937\pi$
$311$	1615.91	0.294629	0.147315	+	0.989090i	$0.452937\pi$
$312$	0	0
$313$	−8479.33	−1.53125	−0.765623	−	0.643289i	$-0.777569\pi$
$313$	−8479.33	−1.53125	−0.765623	+	0.643289i	$0.777569\pi$
$314$	0	0
$315$	3717.60	0.664962
$316$	0	0
$317$	6774.73	1.20034	0.600169	−	0.799873i	$-0.295100\pi$
$317$	6774.73	1.20034	0.600169	+	0.799873i	$0.295100\pi$
$318$	0	0
$319$	−11049.2	−1.93929
$320$	0	0
$321$	1078.13	0.187463
$322$	0	0
$323$	4988.44	0.859332
$324$	0	0
$325$	−1534.00	−0.261818
$326$	0	0
$327$	5909.20	0.999325
$328$	0	0
$329$	−7038.57	−1.17948
$330$	0	0
$331$	−9292.36	−1.54306	−0.771532	−	0.636191i	$-0.780509\pi$
$331$	−9292.36	−1.54306	−0.771532	+	0.636191i	$0.780509\pi$
$332$	0	0
$333$	−1131.20	−0.186154
$334$	0	0
$335$	−8552.80	−1.39489
$336$	0	0
$337$	6563.78	1.06098	0.530492	−	0.847690i	$-0.322007\pi$
$337$	6563.78	1.06098	0.530492	+	0.847690i	$0.322007\pi$
$338$	0	0
$339$	2080.93	0.333394
$340$	0	0
$341$	7918.49	1.25751
$342$	0	0
$343$	−4108.75	−0.646798
$344$	0	0
$345$	−5479.99	−0.855168
$346$	0	0
$347$	3870.93	0.598855	0.299427	−	0.954119i	$-0.403204\pi$
$347$	3870.93	0.598855	0.299427	+	0.954119i	$0.403204\pi$
$348$	0	0
$349$	−3474.57	−0.532922	−0.266461	−	0.963846i	$-0.585854\pi$
$349$	−3474.57	−0.532922	−0.266461	+	0.963846i	$0.585854\pi$
$350$	0	0
$351$	193.201	0.0293798
$352$	0	0
$353$	−4308.58	−0.649639	−0.324820	−	0.945776i	$-0.605304\pi$
$353$	−4308.58	−0.649639	−0.324820	+	0.945776i	$0.605304\pi$
$354$	0	0
$355$	19395.7	2.89977
$356$	0	0
$357$	−2669.73	−0.395791
$358$	0	0
$359$	−8161.19	−1.19981	−0.599904	−	0.800072i	$-0.704795\pi$
$359$	−8161.19	−1.19981	−0.599904	+	0.800072i	$0.704795\pi$
$360$	0	0
$361$	8938.68	1.30320
$362$	0	0
$363$	−4654.47	−0.672992
$364$	0	0
$365$	18832.4	2.70064
$366$	0	0
$367$	4427.66	0.629760	0.314880	−	0.949132i	$-0.398036\pi$
$367$	4427.66	0.629760	0.314880	+	0.949132i	$0.398036\pi$
$368$	0	0
$369$	−4558.00	−0.643035
$370$	0	0
$371$	994.045	0.139106
$372$	0	0
$373$	−11278.5	−1.56562	−0.782812	−	0.622258i	$-0.786215\pi$
$373$	−11278.5	−1.56562	−0.782812	+	0.622258i	$0.786215\pi$
$374$	0	0
$375$	−4939.60	−0.680213
$376$	0	0
$377$	1472.62	0.201177
$378$	0	0
$379$	−709.683	−0.0961846	−0.0480923	−	0.998843i	$-0.515314\pi$
$379$	−709.683	−0.0961846	−0.0480923	+	0.998843i	$0.515314\pi$
$380$	0	0
$381$	−7970.33	−1.07174
$382$	0	0
$383$	−1233.69	−0.164592	−0.0822962	−	0.996608i	$-0.526225\pi$
$383$	−1233.69	−0.164592	−0.0822962	+	0.996608i	$0.526225\pi$
$384$	0	0
$385$	22177.1	2.93571
$386$	0	0
$387$	−3723.20	−0.489046
$388$	0	0
$389$	6830.06	0.890226	0.445113	−	0.895474i	$-0.353164\pi$
$389$	6830.06	0.890226	0.445113	+	0.895474i	$0.353164\pi$
$390$	0	0
$391$	3935.37	0.509003
$392$	0	0
$393$	1847.20	0.237097
$394$	0	0
$395$	−11106.9	−1.41481
$396$	0	0
$397$	11289.7	1.42724	0.713618	−	0.700535i	$-0.247055\pi$
$397$	11289.7	1.42724	0.713618	+	0.700535i	$0.247055\pi$
$398$	0	0
$399$	−8454.66	−1.06081
$400$	0	0
$401$	−3055.59	−0.380521	−0.190261	−	0.981734i	$-0.560933\pi$
$401$	−3055.59	−0.380521	−0.190261	+	0.981734i	$0.560933\pi$
$402$	0	0
$403$	−1055.37	−0.130451
$404$	0	0
$405$	1492.20	0.183081
$406$	0	0
$407$	−6748.08	−0.821843
$408$	0	0
$409$	4089.01	0.494349	0.247175	−	0.968971i	$-0.420498\pi$
$409$	4089.01	0.494349	0.247175	+	0.968971i	$0.420498\pi$
$410$	0	0
$411$	1839.87	0.220813
$412$	0	0
$413$	−7264.79	−0.865562
$414$	0	0
$415$	−291.471	−0.0344765
$416$	0	0
$417$	−5697.86	−0.669126
$418$	0	0
$419$	−15397.0	−1.79520	−0.897602	−	0.440806i	$-0.854693\pi$
$419$	−15397.0	−1.79520	−0.897602	+	0.440806i	$0.854693\pi$
$420$	0	0
$421$	−1034.45	−0.119753	−0.0598766	−	0.998206i	$-0.519071\pi$
$421$	−1034.45	−0.119753	−0.0598766	+	0.998206i	$0.519071\pi$
$422$	0	0
$423$	−2825.20	−0.324742
$424$	0	0
$425$	8508.40	0.971101
$426$	0	0
$427$	−7264.79	−0.823344
$428$	0	0
$429$	1152.53	0.129707
$430$	0	0
$431$	−4943.86	−0.552523	−0.276261	−	0.961083i	$-0.589096\pi$
$431$	−4943.86	−0.552523	−0.276261	+	0.961083i	$0.589096\pi$
$432$	0	0
$433$	337.202	0.0374247	0.0187124	−	0.999825i	$-0.494043\pi$
$433$	337.202	0.0374247	0.0187124	+	0.999825i	$0.494043\pi$
$434$	0	0
$435$	11373.9	1.25364
$436$	0	0
$437$	12462.7	1.36424
$438$	0	0
$439$	4493.93	0.488573	0.244286	−	0.969703i	$-0.421446\pi$
$439$	4493.93	0.488573	0.244286	+	0.969703i	$0.421446\pi$
$440$	0	0
$441$	1437.80	0.155253
$442$	0	0
$443$	−4292.26	−0.460341	−0.230171	−	0.973150i	$-0.573928\pi$
$443$	−4292.26	−0.460341	−0.230171	+	0.973150i	$0.573928\pi$
$444$	0	0
$445$	7025.82	0.748440
$446$	0	0
$447$	−2929.13	−0.309940
$448$	0	0
$449$	−4167.96	−0.438081	−0.219040	−	0.975716i	$-0.570293\pi$
$449$	−4167.96	−0.438081	−0.219040	+	0.975716i	$0.570293\pi$
$450$	0	0
$451$	−27190.4	−2.83890
$452$	0	0
$453$	−2049.40	−0.212559
$454$	0	0
$455$	−2955.73	−0.304543
$456$	0	0
$457$	301.643	0.0308759	0.0154380	−	0.999881i	$-0.495086\pi$
$457$	301.643	0.0308759	0.0154380	+	0.999881i	$0.495086\pi$
$458$	0	0
$459$	−1071.60	−0.108971
$460$	0	0
$461$	−9611.88	−0.971085	−0.485542	−	0.874213i	$-0.661378\pi$
$461$	−9611.88	−0.971085	−0.485542	+	0.874213i	$0.661378\pi$
$462$	0	0
$463$	−13251.0	−1.33008	−0.665041	−	0.746807i	$-0.731586\pi$
$463$	−13251.0	−1.33008	−0.665041	+	0.746807i	$0.731586\pi$
$464$	0	0
$465$	−8151.20	−0.812909
$466$	0	0
$467$	−4432.00	−0.439161	−0.219581	−	0.975594i	$-0.570469\pi$
$467$	−4432.00	−0.439161	−0.219581	+	0.975594i	$0.570469\pi$
$468$	0	0
$469$	−10409.9	−1.02491
$470$	0	0
$471$	−1533.33	−0.150004
$472$	0	0
$473$	−22210.5	−2.15907
$474$	0	0
$475$	26944.9	2.60277
$476$	0	0
$477$	398.998	0.0382995
$478$	0	0
$479$	9076.49	0.865794	0.432897	−	0.901443i	$-0.357491\pi$
$479$	9076.49	0.865794	0.432897	+	0.901443i	$0.357491\pi$
$480$	0	0
$481$	899.378	0.0852559
$482$	0	0
$483$	−6669.86	−0.628342
$484$	0	0
$485$	12143.1	1.13688
$486$	0	0
$487$	−3343.89	−0.311142	−0.155571	−	0.987825i	$-0.549722\pi$
$487$	−3343.89	−0.311142	−0.155571	+	0.987825i	$0.549722\pi$
$488$	0	0
$489$	7277.86	0.673040
$490$	0	0
$491$	−2423.73	−0.222773	−0.111386	−	0.993777i	$-0.535529\pi$
$491$	−2423.73	−0.222773	−0.111386	+	0.993777i	$0.535529\pi$
$492$	0	0
$493$	−8167.95	−0.746179
$494$	0	0
$495$	8901.60	0.808277
$496$	0	0
$497$	23607.1	2.13063
$498$	0	0
$499$	811.819	0.0728296	0.0364148	−	0.999337i	$-0.488406\pi$
$499$	811.819	0.0728296	0.0364148	+	0.999337i	$0.488406\pi$
$500$	0	0
$501$	−1012.00	−0.0902449
$502$	0	0
$503$	18192.4	1.61264	0.806320	−	0.591479i	$-0.201456\pi$
$503$	18192.4	1.61264	0.806320	+	0.591479i	$0.201456\pi$
$504$	0	0
$505$	−9183.65	−0.809242
$506$	0	0
$507$	6437.39	0.563895
$508$	0	0
$509$	−5645.44	−0.491610	−0.245805	−	0.969319i	$-0.579052\pi$
$509$	−5645.44	−0.491610	−0.245805	+	0.969319i	$0.579052\pi$
$510$	0	0
$511$	22921.5	1.98432
$512$	0	0
$513$	−3393.60	−0.292068
$514$	0	0
$515$	−3625.88	−0.310243
$516$	0	0
$517$	−16853.5	−1.43369
$518$	0	0
$519$	−7944.87	−0.671948
$520$	0	0
$521$	12338.7	1.03756	0.518780	−	0.854908i	$-0.326386\pi$
$521$	12338.7	1.03756	0.518780	+	0.854908i	$0.326386\pi$
$522$	0	0
$523$	−10609.8	−0.887062	−0.443531	−	0.896259i	$-0.646274\pi$
$523$	−10609.8	−0.887062	−0.443531	+	0.896259i	$0.646274\pi$
$524$	0	0
$525$	−14420.5	−1.19878
$526$	0	0
$527$	5853.65	0.483850
$528$	0	0
$529$	−2335.17	−0.191926
$530$	0	0
$531$	−2916.00	−0.238312
$532$	0	0
$533$	3623.91	0.294501
$534$	0	0
$535$	−6620.53	−0.535010
$536$	0	0
$537$	8721.86	0.700887
$538$	0	0
$539$	8577.07	0.685419
$540$	0	0
$541$	4035.42	0.320696	0.160348	−	0.987061i	$-0.448738\pi$
$541$	4035.42	0.320696	0.160348	+	0.987061i	$0.448738\pi$
$542$	0	0
$543$	11048.4	0.873171
$544$	0	0
$545$	−36286.8	−2.85203
$546$	0	0
$547$	7407.45	0.579012	0.289506	−	0.957176i	$-0.406509\pi$
$547$	7407.45	0.579012	0.289506	+	0.957176i	$0.406509\pi$
$548$	0	0
$549$	−2916.00	−0.226688
$550$	0	0
$551$	−25866.7	−1.99993
$552$	0	0
$553$	−13518.6	−1.03954
$554$	0	0
$555$	6946.40	0.531276
$556$	0	0
$557$	−9500.77	−0.722730	−0.361365	−	0.932424i	$-0.617689\pi$
$557$	−9500.77	−0.722730	−0.361365	+	0.932424i	$0.617689\pi$
$558$	0	0
$559$	2960.19	0.223976
$560$	0	0
$561$	−6392.54	−0.481093
$562$	0	0
$563$	2700.26	0.202136	0.101068	−	0.994880i	$-0.467774\pi$
$563$	2700.26	0.202136	0.101068	+	0.994880i	$0.467774\pi$
$564$	0	0
$565$	−12778.4	−0.951492
$566$	0	0
$567$	1816.20	0.134521
$568$	0	0
$569$	15904.9	1.17183	0.585913	−	0.810374i	$-0.300736\pi$
$569$	15904.9	1.17183	0.585913	+	0.810374i	$0.300736\pi$
$570$	0	0
$571$	18234.0	1.33638	0.668188	−	0.743992i	$-0.267070\pi$
$571$	18234.0	1.33638	0.668188	+	0.743992i	$0.267070\pi$
$572$	0	0
$573$	−3838.67	−0.279865
$574$	0	0
$575$	21256.7	1.54168
$576$	0	0
$577$	4869.57	0.351339	0.175670	−	0.984449i	$-0.443791\pi$
$577$	4869.57	0.351339	0.175670	+	0.984449i	$0.443791\pi$
$578$	0	0
$579$	14510.5	1.04151
$580$	0	0
$581$	−354.758	−0.0253319
$582$	0	0
$583$	2380.19	0.169086
$584$	0	0
$585$	−1186.40	−0.0838486
$586$	0	0
$587$	−1616.99	−0.113697	−0.0568485	−	0.998383i	$-0.518105\pi$
$587$	−1616.99	−0.113697	−0.0568485	+	0.998383i	$0.518105\pi$
$588$	0	0
$589$	18537.7	1.29683
$590$	0	0
$591$	−8607.93	−0.599125
$592$	0	0
$593$	8117.01	0.562101	0.281050	−	0.959693i	$-0.409317\pi$
$593$	8117.01	0.562101	0.281050	+	0.959693i	$0.409317\pi$
$594$	0	0
$595$	16394.1	1.12957
$596$	0	0
$597$	−1956.73	−0.134143
$598$	0	0
$599$	9536.40	0.650495	0.325248	−	0.945629i	$-0.394552\pi$
$599$	9536.40	0.650495	0.325248	+	0.945629i	$0.394552\pi$
$600$	0	0
$601$	−16247.8	−1.10276	−0.551381	−	0.834253i	$-0.685899\pi$
$601$	−16247.8	−1.10276	−0.551381	+	0.834253i	$0.685899\pi$
$602$	0	0
$603$	−4178.39	−0.282185
$604$	0	0
$605$	28581.9	1.92069
$606$	0	0
$607$	27725.7	1.85396	0.926980	−	0.375111i	$-0.122395\pi$
$607$	27725.7	1.85396	0.926980	+	0.375111i	$0.122395\pi$
$608$	0	0
$609$	13843.5	0.921125
$610$	0	0
$611$	2246.22	0.148727
$612$	0	0
$613$	927.190	0.0610911	0.0305456	−	0.999533i	$-0.490276\pi$
$613$	927.190	0.0610911	0.0305456	+	0.999533i	$0.490276\pi$
$614$	0	0
$615$	27989.5	1.83519
$616$	0	0
$617$	−18727.8	−1.22196	−0.610982	−	0.791644i	$-0.709225\pi$
$617$	−18727.8	−1.22196	−0.610982	+	0.791644i	$0.709225\pi$
$618$	0	0
$619$	3210.22	0.208449	0.104224	−	0.994554i	$-0.466764\pi$
$619$	3210.22	0.208449	0.104224	+	0.994554i	$0.466764\pi$
$620$	0	0
$621$	−2677.20	−0.172999
$622$	0	0
$623$	8551.33	0.549922
$624$	0	0
$625$	3535.57	0.226276
$626$	0	0
$627$	−20244.3	−1.28944
$628$	0	0
$629$	−4988.44	−0.316220
$630$	0	0
$631$	−11911.2	−0.751468	−0.375734	−	0.926728i	$-0.622609\pi$
$631$	−11911.2	−0.751468	−0.375734	+	0.926728i	$0.622609\pi$
$632$	0	0
$633$	1612.53	0.101252
$634$	0	0
$635$	48943.7	3.05869
$636$	0	0
$637$	−1143.14	−0.0711036
$638$	0	0
$639$	9475.60	0.586618
$640$	0	0
$641$	17232.6	1.06185	0.530924	−	0.847419i	$-0.321845\pi$
$641$	17232.6	1.06185	0.530924	+	0.847419i	$0.321845\pi$
$642$	0	0
$643$	−12754.6	−0.782262	−0.391131	−	0.920335i	$-0.627916\pi$
$643$	−12754.6	−0.782262	−0.391131	+	0.920335i	$0.627916\pi$
$644$	0	0
$645$	22863.2	1.39572
$646$	0	0
$647$	9441.73	0.573714	0.286857	−	0.957973i	$-0.407390\pi$
$647$	9441.73	0.573714	0.286857	+	0.957973i	$0.407390\pi$
$648$	0	0
$649$	−17395.2	−1.05211
$650$	0	0
$651$	−9921.06	−0.597292
$652$	0	0
$653$	−5198.99	−0.311565	−0.155783	−	0.987791i	$-0.549790\pi$
$653$	−5198.99	−0.311565	−0.155783	+	0.987791i	$0.549790\pi$
$654$	0	0
$655$	−11343.2	−0.676664
$656$	0	0
$657$	9200.39	0.546334
$658$	0	0
$659$	6508.01	0.384698	0.192349	−	0.981327i	$-0.438389\pi$
$659$	6508.01	0.384698	0.192349	+	0.981327i	$0.438389\pi$
$660$	0	0
$661$	25280.2	1.48757	0.743785	−	0.668419i	$-0.233028\pi$
$661$	25280.2	1.48757	0.743785	+	0.668419i	$0.233028\pi$
$662$	0	0
$663$	851.991	0.0499074
$664$	0	0
$665$	51917.8	3.02750
$666$	0	0
$667$	−20406.2	−1.18460
$668$	0	0
$669$	12125.9	0.700770
$670$	0	0
$671$	−17395.2	−1.00079
$672$	0	0
$673$	−5525.89	−0.316504	−0.158252	−	0.987399i	$-0.550586\pi$
$673$	−5525.89	−0.316504	−0.158252	+	0.987399i	$0.550586\pi$
$674$	0	0
$675$	−5788.20	−0.330056
$676$	0	0
$677$	−6293.21	−0.357264	−0.178632	−	0.983916i	$-0.557167\pi$
$677$	−6293.21	−0.357264	−0.178632	+	0.983916i	$0.557167\pi$
$678$	0	0
$679$	14779.7	0.835335
$680$	0	0
$681$	9210.66	0.518287
$682$	0	0
$683$	−5675.91	−0.317984	−0.158992	−	0.987280i	$-0.550824\pi$
$683$	−5675.91	−0.317984	−0.158992	+	0.987280i	$0.550824\pi$
$684$	0	0
$685$	−11298.2	−0.630190
$686$	0	0
$687$	−615.331	−0.0341722
$688$	0	0
$689$	−317.229	−0.0175406
$690$	0	0
$691$	−3617.79	−0.199171	−0.0995854	−	0.995029i	$-0.531752\pi$
$691$	−3617.79	−0.199171	−0.0995854	+	0.995029i	$0.531752\pi$
$692$	0	0
$693$	10834.4	0.593888
$694$	0	0
$695$	34989.1	1.90966
$696$	0	0
$697$	−20100.2	−1.09232
$698$	0	0
$699$	−40.1329	−0.00217163
$700$	0	0
$701$	−7938.43	−0.427718	−0.213859	−	0.976865i	$-0.568603\pi$
$701$	−7938.43	−0.427718	−0.213859	+	0.976865i	$0.568603\pi$
$702$	0	0
$703$	−15797.7	−0.847540
$704$	0	0
$705$	17348.8	0.926799
$706$	0	0
$707$	−11177.7	−0.594597
$708$	0	0
$709$	−25691.6	−1.36088	−0.680442	−	0.732802i	$-0.738212\pi$
$709$	−25691.6	−1.36088	−0.680442	+	0.732802i	$0.738212\pi$
$710$	0	0
$711$	−5426.19	−0.286214
$712$	0	0
$713$	14624.3	0.768142
$714$	0	0
$715$	−7077.35	−0.370179
$716$	0	0
$717$	−12984.0	−0.676284
$718$	0	0
$719$	36803.3	1.90895	0.954473	−	0.298298i	$-0.0964190\pi$
$719$	36803.3	1.90895	0.954473	+	0.298298i	$0.0964190\pi$
$720$	0	0
$721$	−4413.16	−0.227954
$722$	0	0
$723$	4524.94	0.232759
$724$	0	0
$725$	−44118.9	−2.26005
$726$	0	0
$727$	28333.2	1.44542	0.722709	−	0.691152i	$-0.242897\pi$
$727$	28333.2	1.44542	0.722709	+	0.691152i	$0.242897\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−16418.8	−0.830742
$732$	0	0
$733$	−10767.9	−0.542592	−0.271296	−	0.962496i	$-0.587452\pi$
$733$	−10767.9	−0.542592	−0.271296	+	0.962496i	$0.587452\pi$
$734$	0	0
$735$	−8829.13	−0.443085
$736$	0	0
$737$	−24925.9	−1.24580
$738$	0	0
$739$	−17301.4	−0.861221	−0.430610	−	0.902538i	$-0.641702\pi$
$739$	−17301.4	−0.861221	−0.430610	+	0.902538i	$0.641702\pi$
$740$	0	0
$741$	2698.13	0.133763
$742$	0	0
$743$	24110.8	1.19050	0.595248	−	0.803542i	$-0.297054\pi$
$743$	24110.8	1.19050	0.595248	+	0.803542i	$0.297054\pi$
$744$	0	0
$745$	17987.0	0.884555
$746$	0	0
$747$	−142.396	−0.00697455
$748$	0	0
$749$	−8058.04	−0.393103
$750$	0	0
$751$	30052.8	1.46024	0.730121	−	0.683318i	$-0.239464\pi$
$751$	30052.8	1.46024	0.730121	+	0.683318i	$0.239464\pi$
$752$	0	0
$753$	−14614.4	−0.707275
$754$	0	0
$755$	12584.8	0.606633
$756$	0	0
$757$	25599.4	1.22910	0.614549	−	0.788879i	$-0.289338\pi$
$757$	25599.4	1.22910	0.614549	+	0.788879i	$0.289338\pi$
$758$	0	0
$759$	−15970.6	−0.763765
$760$	0	0
$761$	−28904.5	−1.37685	−0.688427	−	0.725306i	$-0.741698\pi$
$761$	−28904.5	−1.37685	−0.688427	+	0.725306i	$0.741698\pi$
$762$	0	0
$763$	−44165.7	−2.09555
$764$	0	0
$765$	6580.40	0.311000
$766$	0	0
$767$	2318.41	0.109143
$768$	0	0
$769$	−8756.13	−0.410603	−0.205302	−	0.978699i	$-0.565818\pi$
$769$	−8756.13	−0.410603	−0.205302	+	0.978699i	$0.565818\pi$
$770$	0	0
$771$	4996.66	0.233399
$772$	0	0
$773$	28289.8	1.31632	0.658159	−	0.752879i	$-0.271335\pi$
$773$	28289.8	1.31632	0.658159	+	0.752879i	$0.271335\pi$
$774$	0	0
$775$	31618.3	1.46550
$776$	0	0
$777$	8454.66	0.390359
$778$	0	0
$779$	−63654.4	−2.92767
$780$	0	0
$781$	56526.0	2.58983
$782$	0	0
$783$	5556.60	0.253610
$784$	0	0
$785$	9415.75	0.428105
$786$	0	0
$787$	4859.74	0.220116	0.110058	−	0.993925i	$-0.464896\pi$
$787$	4859.74	0.220116	0.110058	+	0.993925i	$0.464896\pi$
$788$	0	0
$789$	−21502.9	−0.970246
$790$	0	0
$791$	−15553.0	−0.699117
$792$	0	0
$793$	2318.41	0.103820
$794$	0	0
$795$	−2450.14	−0.109305
$796$	0	0
$797$	17361.7	0.771623	0.385811	−	0.922578i	$-0.373922\pi$
$797$	17361.7	0.771623	0.385811	+	0.922578i	$0.373922\pi$
$798$	0	0
$799$	−12458.8	−0.551638
$800$	0	0
$801$	3432.40	0.151408
$802$	0	0
$803$	54884.2	2.41198
$804$	0	0
$805$	40957.8	1.79326
$806$	0	0
$807$	−16361.5	−0.713696
$808$	0	0
$809$	−24475.3	−1.06367	−0.531833	−	0.846849i	$-0.678497\pi$
$809$	−24475.3	−1.06367	−0.531833	+	0.846849i	$0.678497\pi$
$810$	0	0
$811$	19875.4	0.860566	0.430283	−	0.902694i	$-0.358414\pi$
$811$	19875.4	0.860566	0.430283	+	0.902694i	$0.358414\pi$
$812$	0	0
$813$	16248.6	0.700939
$814$	0	0
$815$	−44691.4	−1.92083
$816$	0	0
$817$	−51996.1	−2.22658
$818$	0	0
$819$	−1444.00	−0.0616085
$820$	0	0
$821$	21682.1	0.921693	0.460846	−	0.887480i	$-0.347546\pi$
$821$	21682.1	0.921693	0.460846	+	0.887480i	$0.347546\pi$
$822$	0	0
$823$	6698.17	0.283698	0.141849	−	0.989888i	$-0.454695\pi$
$823$	6698.17	0.283698	0.141849	+	0.989888i	$0.454695\pi$
$824$	0	0
$825$	−34529.0	−1.45715
$826$	0	0
$827$	−3390.85	−0.142577	−0.0712886	−	0.997456i	$-0.522711\pi$
$827$	−3390.85	−0.142577	−0.0712886	+	0.997456i	$0.522711\pi$
$828$	0	0
$829$	−40093.4	−1.67974	−0.839869	−	0.542789i	$-0.817368\pi$
$829$	−40093.4	−1.67974	−0.839869	+	0.542789i	$0.817368\pi$
$830$	0	0
$831$	−7946.25	−0.331712
$832$	0	0
$833$	6340.50	0.263728
$834$	0	0
$835$	6214.40	0.257555
$836$	0	0
$837$	−3982.19	−0.164450
$838$	0	0
$839$	−6172.12	−0.253975	−0.126988	−	0.991904i	$-0.540531\pi$
$839$	−6172.12	−0.253975	−0.126988	+	0.991904i	$0.540531\pi$
$840$	0	0
$841$	17964.6	0.736585
$842$	0	0
$843$	−19993.7	−0.816869
$844$	0	0
$845$	−39530.3	−1.60933
$846$	0	0
$847$	34787.8	1.41124
$848$	0	0
$849$	−16892.5	−0.682862
$850$	0	0
$851$	−12462.7	−0.502018
$852$	0	0
$853$	−276.632	−0.0111040	−0.00555198	−	0.999985i	$-0.501767\pi$
$853$	−276.632	−0.0111040	−0.00555198	+	0.999985i	$0.501767\pi$
$854$	0	0
$855$	20839.2	0.833550
$856$	0	0
$857$	3704.41	0.147655	0.0738274	−	0.997271i	$-0.476479\pi$
$857$	3704.41	0.147655	0.0738274	+	0.997271i	$0.476479\pi$
$858$	0	0
$859$	26915.5	1.06909	0.534544	−	0.845141i	$-0.320484\pi$
$859$	26915.5	1.06909	0.534544	+	0.845141i	$0.320484\pi$
$860$	0	0
$861$	34066.8	1.34842
$862$	0	0
$863$	−23623.9	−0.931828	−0.465914	−	0.884830i	$-0.654274\pi$
$863$	−23623.9	−0.931828	−0.465914	+	0.884830i	$0.654274\pi$
$864$	0	0
$865$	48787.3	1.91771
$866$	0	0
$867$	10013.4	0.392241
$868$	0	0
$869$	−32369.5	−1.26359
$870$	0	0
$871$	3322.10	0.129236
$872$	0	0
$873$	5932.39	0.229990
$874$	0	0
$875$	36918.9	1.42638
$876$	0	0
$877$	14094.0	0.542667	0.271333	−	0.962485i	$-0.412535\pi$
$877$	14094.0	0.542667	0.271333	+	0.962485i	$0.412535\pi$
$878$	0	0
$879$	2725.12	0.104569
$880$	0	0
$881$	18967.3	0.725341	0.362671	−	0.931917i	$-0.381865\pi$
$881$	18967.3	0.725341	0.362671	+	0.931917i	$0.381865\pi$
$882$	0	0
$883$	32886.0	1.25334	0.626672	−	0.779283i	$-0.284417\pi$
$883$	32886.0	1.25334	0.626672	+	0.779283i	$0.284417\pi$
$884$	0	0
$885$	17906.4	0.680132
$886$	0	0
$887$	−27945.6	−1.05786	−0.528929	−	0.848666i	$-0.677406\pi$
$887$	−27945.6	−1.05786	−0.528929	+	0.848666i	$0.677406\pi$
$888$	0	0
$889$	59570.8	2.24740
$890$	0	0
$891$	4348.79	0.163513
$892$	0	0
$893$	−39455.1	−1.47852
$894$	0	0
$895$	−53558.6	−2.00030
$896$	0	0
$897$	2128.55	0.0792310
$898$	0	0
$899$	−30353.1	−1.12607
$900$	0	0
$901$	1759.53	0.0650592
$902$	0	0
$903$	27827.4	1.02551
$904$	0	0
$905$	−67845.2	−2.49199
$906$	0	0
$907$	−5544.89	−0.202993	−0.101497	−	0.994836i	$-0.532363\pi$
$907$	−5544.89	−0.202993	−0.101497	+	0.994836i	$0.532363\pi$
$908$	0	0
$909$	−4486.59	−0.163708
$910$	0	0
$911$	19638.9	0.714233	0.357116	−	0.934060i	$-0.383760\pi$
$911$	19638.9	0.714233	0.357116	+	0.934060i	$0.383760\pi$
$912$	0	0
$913$	−849.451	−0.0307916
$914$	0	0
$915$	17906.4	0.646958
$916$	0	0
$917$	−13806.1	−0.497184
$918$	0	0
$919$	22128.7	0.794297	0.397149	−	0.917754i	$-0.370000\pi$
$919$	22128.7	0.794297	0.397149	+	0.917754i	$0.370000\pi$
$920$	0	0
$921$	−1244.01	−0.0445077
$922$	0	0
$923$	−7533.72	−0.268663
$924$	0	0
$925$	−26944.9	−0.957775
$926$	0	0
$927$	−1771.39	−0.0627616
$928$	0	0
$929$	17599.3	0.621544	0.310772	−	0.950484i	$-0.399412\pi$
$929$	17599.3	0.621544	0.310772	+	0.950484i	$0.399412\pi$
$930$	0	0
$931$	20079.5	0.706850
$932$	0	0
$933$	−4847.72	−0.170104
$934$	0	0
$935$	39254.9	1.37302
$936$	0	0
$937$	441.299	0.0153859	0.00769297	−	0.999970i	$-0.497551\pi$
$937$	441.299	0.0153859	0.00769297	+	0.999970i	$0.497551\pi$
$938$	0	0
$939$	25438.0	0.884065
$940$	0	0
$941$	5408.01	0.187350	0.0936749	−	0.995603i	$-0.470139\pi$
$941$	5408.01	0.187350	0.0936749	+	0.995603i	$0.470139\pi$
$942$	0	0
$943$	−50216.8	−1.73413
$944$	0	0
$945$	−11152.8	−0.383916
$946$	0	0
$947$	−34237.1	−1.17482	−0.587410	−	0.809289i	$-0.699852\pi$
$947$	−34237.1	−1.17482	−0.587410	+	0.809289i	$0.699852\pi$
$948$	0	0
$949$	−7314.92	−0.250213
$950$	0	0
$951$	−20324.2	−0.693015
$952$	0	0
$953$	21410.7	0.727766	0.363883	−	0.931445i	$-0.381451\pi$
$953$	21410.7	0.727766	0.363883	+	0.931445i	$0.381451\pi$
$954$	0	0
$955$	23572.2	0.798722
$956$	0	0
$957$	33147.5	1.11965
$958$	0	0
$959$	−13751.3	−0.463038
$960$	0	0
$961$	−8038.09	−0.269816
$962$	0	0
$963$	−3234.40	−0.108232
$964$	0	0
$965$	−89105.3	−2.97243
$966$	0	0
$967$	−53874.6	−1.79161	−0.895806	−	0.444445i	$-0.853401\pi$
$967$	−53874.6	−1.79161	−0.895806	+	0.444445i	$0.853401\pi$
$968$	0	0
$969$	−14965.3	−0.496136
$970$	0	0
$971$	42901.5	1.41789	0.708947	−	0.705262i	$-0.249171\pi$
$971$	42901.5	1.41789	0.708947	+	0.705262i	$0.249171\pi$
$972$	0	0
$973$	42586.2	1.40314
$974$	0	0
$975$	4602.00	0.151161
$976$	0	0
$977$	58636.5	1.92011	0.960055	−	0.279812i	$-0.0902721\pi$
$977$	58636.5	1.92011	0.960055	+	0.279812i	$0.0902721\pi$
$978$	0	0
$979$	20475.7	0.668444
$980$	0	0
$981$	−17727.6	−0.576961
$982$	0	0
$983$	−25296.7	−0.820793	−0.410396	−	0.911907i	$-0.634610\pi$
$983$	−25296.7	−0.820793	−0.410396	+	0.911907i	$0.634610\pi$
$984$	0	0
$985$	52859.0	1.70988
$986$	0	0
$987$	21115.7	0.680973
$988$	0	0
$989$	−41019.6	−1.31885
$990$	0	0
$991$	10605.2	0.339944	0.169972	−	0.985449i	$-0.445632\pi$
$991$	10605.2	0.339944	0.169972	+	0.985449i	$0.445632\pi$
$992$	0	0
$993$	27877.1	0.890888
$994$	0	0
$995$	12015.7	0.382839
$996$	0	0
$997$	−5770.19	−0.183294	−0.0916469	−	0.995792i	$-0.529213\pi$
$997$	−5770.19	−0.183294	−0.0916469	+	0.995792i	$0.529213\pi$
$998$	0	0
$999$	3393.60	0.107476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.a.j.1.2		2
3.2	odd	2		2304.4.a.t.1.1		2
4.3	odd	2		768.4.a.p.1.2		2
8.3	odd	2		768.4.a.e.1.1		2
8.5	even	2		768.4.a.k.1.1		2
12.11	even	2		2304.4.a.s.1.1		2
16.3	odd	4		384.4.d.e.193.3	yes	4
16.5	even	4		384.4.d.c.193.4	yes	4
16.11	odd	4		384.4.d.e.193.2	yes	4
16.13	even	4		384.4.d.c.193.1	✓	4
24.5	odd	2		2304.4.a.bq.1.2		2
24.11	even	2		2304.4.a.bp.1.2		2
48.5	odd	4		1152.4.d.i.577.1		4
48.11	even	4		1152.4.d.o.577.1		4
48.29	odd	4		1152.4.d.i.577.4		4
48.35	even	4		1152.4.d.o.577.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.c.193.1	✓	4	16.13	even	4
384.4.d.c.193.4	yes	4	16.5	even	4
384.4.d.e.193.2	yes	4	16.11	odd	4
384.4.d.e.193.3	yes	4	16.3	odd	4
768.4.a.e.1.1		2	8.3	odd	2
768.4.a.j.1.2		2	1.1	even	1	trivial
768.4.a.k.1.1		2	8.5	even	2
768.4.a.p.1.2		2	4.3	odd	2
1152.4.d.i.577.1		4	48.5	odd	4
1152.4.d.i.577.4		4	48.29	odd	4
1152.4.d.o.577.1		4	48.11	even	4
1152.4.d.o.577.4		4	48.35	even	4
2304.4.a.s.1.1		2	12.11	even	2
2304.4.a.t.1.1		2	3.2	odd	2
2304.4.a.bp.1.2		2	24.11	even	2
2304.4.a.bq.1.2		2	24.5	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +18.4222 q^{5} +22.4222 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +18.4222 q^{5} +22.4222 q^{7} +9.00000 q^{9} +53.6888 q^{11} -7.15559 q^{13} -55.2666 q^{15} +39.6888 q^{17} +125.689 q^{19} -67.2666 q^{21} +99.1556 q^{23} +214.378 q^{25} -27.0000 q^{27} -205.800 q^{29} +147.489 q^{31} -161.066 q^{33} +413.066 q^{35} -125.689 q^{37} +21.4668 q^{39} -506.444 q^{41} -413.689 q^{43} +165.800 q^{45} -313.911 q^{47} +159.755 q^{49} -119.066 q^{51} +44.3331 q^{53} +989.066 q^{55} -377.066 q^{57} -324.000 q^{59} -324.000 q^{61} +201.800 q^{63} -131.822 q^{65} -464.266 q^{67} -297.467 q^{69} +1052.84 q^{71} +1022.27 q^{73} -643.133 q^{75} +1203.82 q^{77} -602.910 q^{79} +81.0000 q^{81} -15.8217 q^{83} +731.156 q^{85} +617.400 q^{87} +381.378 q^{89} -160.444 q^{91} -442.466 q^{93} +2315.47 q^{95} +659.154 q^{97} +483.199 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 8 q^{5} + 16 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 8 * q^5 + 16 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} + 8 q^{5} + 16 q^{7} + 18 q^{9} - 8 q^{11} - 72 q^{13} - 24 q^{15} - 36 q^{17} + 136 q^{19} - 48 q^{21} + 256 q^{23} + 198 q^{25} - 54 q^{27} - 152 q^{29} - 80 q^{31} + 24 q^{33} + 480 q^{35} - 136 q^{37} + 216 q^{39} - 436 q^{41} - 712 q^{43} + 72 q^{45} - 224 q^{47} - 142 q^{49} + 108 q^{51} - 344 q^{53} + 1632 q^{55} - 408 q^{57} - 648 q^{59} - 648 q^{61} + 144 q^{63} + 544 q^{65} + 456 q^{67} - 768 q^{69} + 2048 q^{71} + 660 q^{73} - 594 q^{75} + 1600 q^{77} + 496 q^{79} + 162 q^{81} + 776 q^{83} + 1520 q^{85} + 456 q^{87} + 532 q^{89} + 256 q^{91} + 240 q^{93} + 2208 q^{95} - 1220 q^{97} - 72 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 8 * q^5 + 16 * q^7 + 18 * q^9 - 8 * q^11 - 72 * q^13 - 24 * q^15 - 36 * q^17 + 136 * q^19 - 48 * q^21 + 256 * q^23 + 198 * q^25 - 54 * q^27 - 152 * q^29 - 80 * q^31 + 24 * q^33 + 480 * q^35 - 136 * q^37 + 216 * q^39 - 436 * q^41 - 712 * q^43 + 72 * q^45 - 224 * q^47 - 142 * q^49 + 108 * q^51 - 344 * q^53 + 1632 * q^55 - 408 * q^57 - 648 * q^59 - 648 * q^61 + 144 * q^63 + 544 * q^65 + 456 * q^67 - 768 * q^69 + 2048 * q^71 + 660 * q^73 - 594 * q^75 + 1600 * q^77 + 496 * q^79 + 162 * q^81 + 776 * q^83 + 1520 * q^85 + 456 * q^87 + 532 * q^89 + 256 * q^91 + 240 * q^93 + 2208 * q^95 - 1220 * q^97 - 72 * q^99

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