LMFDB - Embedded newform 768.5.g.d.511.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,5,Mod(511,768)]

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([1, 0, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("768.511");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	768.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$79.3881316484$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 3 $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.1
Root			$-1.63746 + 1.52274i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.5.g.d.511.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615i q^{3} -30.1993 q^{5} -52.3068i q^{7} -27.0000 q^{9} +O(q^{10})$	$q-5.19615i q^{3} -30.1993 q^{5} -52.3068i q^{7} -27.0000 q^{9} -90.0666i q^{11} +60.3987 q^{13} +156.920i q^{15} -338.000 q^{17} +6.92820i q^{19} -271.794 q^{21} -732.295i q^{23} +287.000 q^{25} +140.296i q^{27} -1298.57 q^{29} -1307.67i q^{31} -468.000 q^{33} +1579.63i q^{35} -241.595 q^{37} -313.841i q^{39} -578.000 q^{41} -2029.96i q^{43} +815.382 q^{45} -2196.89i q^{47} -335.000 q^{49} +1756.30i q^{51} +2446.15 q^{53} +2719.95i q^{55} +36.0000 q^{57} -1198.58i q^{59} +6402.26 q^{61} +1412.28i q^{63} -1824.00 q^{65} +8265.35i q^{67} -3805.12 q^{69} -4289.16i q^{71} -8734.00 q^{73} -1491.30i q^{75} -4711.10 q^{77} +11246.0i q^{79} +729.000 q^{81} +13198.2i q^{83} +10207.4 q^{85} +6747.58i q^{87} -910.000 q^{89} -3159.26i q^{91} -6794.85 q^{93} -209.227i q^{95} +5422.00 q^{97} +2431.80i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 108 q^{9}+O(q^{10}) $ 4 * q - 108 * q^9	$ 4 q - 108 q^{9} - 1352 q^{17} + 1148 q^{25} - 1872 q^{33} - 2312 q^{41} - 1340 q^{49} + 144 q^{57} - 7296 q^{65} - 34936 q^{73} + 2916 q^{81} - 3640 q^{89} + 21688 q^{97}+O(q^{100}) $ 4 * q - 108 * q^9 - 1352 * q^17 + 1148 * q^25 - 1872 * q^33 - 2312 * q^41 - 1340 * q^49 + 144 * q^57 - 7296 * q^65 - 34936 * q^73 + 2916 * q^81 - 3640 * q^89 + 21688 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 5.19615i	− 0.577350i
$3$	− 5.19615i	− 0.577350i
$4$	0	0
$5$	−30.1993	−1.20797	−0.603987	−	0.796994i	$-0.706422\pi$
$5$	−30.1993	−1.20797	−0.603987	+	0.796994i	$0.706422\pi$
$6$	0	0
$7$	− 52.3068i	− 1.06749i	−0.845647	−	0.533743i	$-0.820785\pi$
$7$	− 52.3068i	− 1.06749i	0.845647	−	0.533743i	$-0.179215\pi$
$8$	0	0
$9$	−27.0000	−0.333333
$10$	0	0
$11$	− 90.0666i	− 0.744352i	−0.928162	−	0.372176i	$-0.878612\pi$
$11$	− 90.0666i	− 0.744352i	0.928162	−	0.372176i	$-0.121388\pi$
$12$	0	0
$13$	60.3987	0.357389	0.178694	−	0.983905i	$-0.442813\pi$
$13$	60.3987	0.357389	0.178694	+	0.983905i	$0.442813\pi$
$14$	0	0
$15$	156.920i	0.697424i
$16$	0	0
$17$	−338.000	−1.16955	−0.584775	−	0.811195i	$-0.698817\pi$
$17$	−338.000	−1.16955	−0.584775	+	0.811195i	$0.698817\pi$
$18$	0	0
$19$	6.92820i	0.0191917i	0.999954	+	0.00959585i	$0.00305450\pi$
$19$	6.92820i	0.0191917i	−0.999954	+	0.00959585i	$0.996945\pi$
$20$	0	0
$21$	−271.794	−0.616313
$22$	0	0
$23$	− 732.295i	− 1.38430i	−0.721753	−	0.692150i	$-0.756664\pi$
$23$	− 732.295i	− 1.38430i	0.721753	−	0.692150i	$-0.243336\pi$
$24$	0	0
$25$	287.000	0.459200
$26$	0	0
$27$	140.296i	0.192450i
$28$	0	0
$29$	−1298.57	−1.54408	−0.772040	−	0.635574i	$-0.780764\pi$
$29$	−1298.57	−1.54408	−0.772040	+	0.635574i	$0.780764\pi$
$30$	0	0
$31$	− 1307.67i	− 1.36074i	−0.732869	−	0.680369i	$-0.761819\pi$
$31$	− 1307.67i	− 1.36074i	0.732869	−	0.680369i	$-0.238181\pi$
$32$	0	0
$33$	−468.000	−0.429752
$34$	0	0
$35$	1579.63i	1.28949i
$36$	0	0
$37$	−241.595	−0.176475	−0.0882377	−	0.996099i	$-0.528123\pi$
$37$	−241.595	−0.176475	−0.0882377	+	0.996099i	$0.528123\pi$
$38$	0	0
$39$	− 313.841i	− 0.206338i
$40$	0	0
$41$	−578.000	−0.343843	−0.171921	−	0.985111i	$-0.554998\pi$
$41$	−578.000	−0.343843	−0.171921	+	0.985111i	$0.554998\pi$
$42$	0	0
$43$	− 2029.96i	− 1.09787i	−0.835865	−	0.548936i	$-0.815033\pi$
$43$	− 2029.96i	− 1.09787i	0.835865	−	0.548936i	$-0.184967\pi$
$44$	0	0
$45$	815.382	0.402658
$46$	0	0
$47$	− 2196.89i	− 0.994516i	−0.867603	−	0.497258i	$-0.834340\pi$
$47$	− 2196.89i	− 0.994516i	0.867603	−	0.497258i	$-0.165660\pi$
$48$	0	0
$49$	−335.000	−0.139525
$50$	0	0
$51$	1756.30i	0.675240i
$52$	0	0
$53$	2446.15	0.870825	0.435412	−	0.900231i	$-0.356603\pi$
$53$	2446.15	0.870825	0.435412	+	0.900231i	$0.356603\pi$
$54$	0	0
$55$	2719.95i	0.899158i
$56$	0	0
$57$	36.0000	0.0110803
$58$	0	0
$59$	− 1198.58i	− 0.344320i	−0.985069	−	0.172160i	$-0.944925\pi$
$59$	− 1198.58i	− 0.344320i	0.985069	−	0.172160i	$-0.0550747\pi$
$60$	0	0
$61$	6402.26	1.72058	0.860288	−	0.509809i	$-0.170284\pi$
$61$	6402.26	1.72058	0.860288	+	0.509809i	$0.170284\pi$
$62$	0	0
$63$	1412.28i	0.355828i
$64$	0	0
$65$	−1824.00	−0.431716
$66$	0	0
$67$	8265.35i	1.84124i	0.390454	+	0.920622i	$0.372318\pi$
$67$	8265.35i	1.84124i	−0.390454	+	0.920622i	$0.627682\pi$
$68$	0	0
$69$	−3805.12	−0.799226
$70$	0	0
$71$	− 4289.16i	− 0.850854i	−0.904993	−	0.425427i	$-0.860124\pi$
$71$	− 4289.16i	− 0.850854i	0.904993	−	0.425427i	$-0.139876\pi$
$72$	0	0
$73$	−8734.00	−1.63896	−0.819478	−	0.573110i	$-0.805737\pi$
$73$	−8734.00	−1.63896	−0.819478	+	0.573110i	$0.805737\pi$
$74$	0	0
$75$	− 1491.30i	− 0.265119i
$76$	0	0
$77$	−4711.10	−0.794585
$78$	0	0
$79$	11246.0i	1.80195i	0.433873	+	0.900974i	$0.357147\pi$
$79$	11246.0i	1.80195i	−0.433873	+	0.900974i	$0.642853\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	13198.2i	1.91584i	0.287034	+	0.957920i	$0.407331\pi$
$83$	13198.2i	1.91584i	−0.287034	+	0.957920i	$0.592669\pi$
$84$	0	0
$85$	10207.4	1.41279
$86$	0	0
$87$	6747.58i	0.891475i
$88$	0	0
$89$	−910.000	−0.114884	−0.0574422	−	0.998349i	$-0.518295\pi$
$89$	−910.000	−0.114884	−0.0574422	+	0.998349i	$0.518295\pi$
$90$	0	0
$91$	− 3159.26i	− 0.381507i
$92$	0	0
$93$	−6794.85	−0.785623
$94$	0	0
$95$	− 209.227i	− 0.0231831i
$96$	0	0
$97$	5422.00	0.576257	0.288128	−	0.957592i	$-0.406967\pi$
$97$	5422.00	0.576257	0.288128	+	0.957592i	$0.406967\pi$
$98$	0	0
$99$	2431.80i	0.248117i
$100$	0	0
$101$	−10962.4	−1.07464	−0.537318	−	0.843380i	$-0.680562\pi$
$101$	−10962.4	−1.07464	−0.537318	+	0.843380i	$0.680562\pi$
$102$	0	0
$103$	− 5387.60i	− 0.507833i	−0.967226	−	0.253916i	$-0.918281\pi$
$103$	− 5387.60i	− 0.507833i	0.967226	−	0.253916i	$-0.0817188\pi$
$104$	0	0
$105$	8208.00	0.744490
$106$	0	0
$107$	− 6436.30i	− 0.562171i	−0.959683	−	0.281086i	$-0.909305\pi$
$107$	− 6436.30i	− 0.562171i	0.959683	−	0.281086i	$-0.0906945\pi$
$108$	0	0
$109$	−60.3987	−0.00508364	−0.00254182	−	0.999997i	$-0.500809\pi$
$109$	−60.3987	−0.00508364	−0.00254182	+	0.999997i	$0.500809\pi$
$110$	0	0
$111$	1255.36i	0.101888i
$112$	0	0
$113$	−3166.00	−0.247944	−0.123972	−	0.992286i	$-0.539563\pi$
$113$	−3166.00	−0.247944	−0.123972	+	0.992286i	$0.539563\pi$
$114$	0	0
$115$	22114.8i	1.67220i
$116$	0	0
$117$	−1630.76	−0.119130
$118$	0	0
$119$	17679.7i	1.24848i
$120$	0	0
$121$	6529.00	0.445939
$122$	0	0
$123$	3003.38i	0.198518i
$124$	0	0
$125$	10207.4	0.653272
$126$	0	0
$127$	− 12919.8i	− 0.801028i	−0.916291	−	0.400514i	$-0.868832\pi$
$127$	− 12919.8i	− 0.801028i	0.916291	−	0.400514i	$-0.131168\pi$
$128$	0	0
$129$	−10548.0	−0.633856
$130$	0	0
$131$	22676.0i	1.32137i	0.750664	+	0.660684i	$0.229734\pi$
$131$	22676.0i	1.32137i	−0.750664	+	0.660684i	$0.770266\pi$
$132$	0	0
$133$	362.392	0.0204869
$134$	0	0
$135$	− 4236.85i	− 0.232475i
$136$	0	0
$137$	15550.0	0.828494	0.414247	−	0.910165i	$-0.364045\pi$
$137$	15550.0	0.828494	0.414247	+	0.910165i	$0.364045\pi$
$138$	0	0
$139$	31502.5i	1.63048i	0.579122	+	0.815241i	$0.303396\pi$
$139$	31502.5i	1.63048i	−0.579122	+	0.815241i	$0.696604\pi$
$140$	0	0
$141$	−11415.3	−0.574184
$142$	0	0
$143$	− 5439.91i	− 0.266023i
$144$	0	0
$145$	39216.0	1.86521
$146$	0	0
$147$	1740.71i	0.0805549i
$148$	0	0
$149$	−22619.3	−1.01884	−0.509421	−	0.860518i	$-0.670140\pi$
$149$	−22619.3	−1.01884	−0.509421	+	0.860518i	$0.670140\pi$
$150$	0	0
$151$	7689.10i	0.337226i	0.985682	+	0.168613i	$0.0539289\pi$
$151$	7689.10i	0.337226i	−0.985682	+	0.168613i	$0.946071\pi$
$152$	0	0
$153$	9126.00	0.389850
$154$	0	0
$155$	39490.8i	1.64374i
$156$	0	0
$157$	1207.97	0.0490070	0.0245035	−	0.999700i	$-0.492200\pi$
$157$	1207.97	0.0490070	0.0245035	+	0.999700i	$0.492200\pi$
$158$	0	0
$159$	− 12710.5i	− 0.502771i
$160$	0	0
$161$	−38304.0	−1.47772
$162$	0	0
$163$	26749.8i	1.00680i	0.864052	+	0.503402i	$0.167919\pi$
$163$	26749.8i	1.00680i	−0.864052	+	0.503402i	$0.832081\pi$
$164$	0	0
$165$	14133.3	0.519129
$166$	0	0
$167$	− 44983.8i	− 1.61296i	−0.591261	−	0.806480i	$-0.701370\pi$
$167$	− 44983.8i	− 1.61296i	0.591261	−	0.806480i	$-0.298630\pi$
$168$	0	0
$169$	−24913.0	−0.872273
$170$	0	0
$171$	− 187.061i	− 0.00639723i
$172$	0	0
$173$	4197.71	0.140256	0.0701278	−	0.997538i	$-0.477659\pi$
$173$	4197.71	0.140256	0.0701278	+	0.997538i	$0.477659\pi$
$174$	0	0
$175$	− 15012.0i	− 0.490189i
$176$	0	0
$177$	−6228.00	−0.198793
$178$	0	0
$179$	− 12491.6i	− 0.389861i	−0.980817	−	0.194931i	$-0.937552\pi$
$179$	− 12491.6i	− 0.389861i	0.980817	−	0.194931i	$-0.0624482\pi$
$180$	0	0
$181$	37870.0	1.15595	0.577973	−	0.816056i	$-0.303844\pi$
$181$	37870.0	1.15595	0.577973	+	0.816056i	$0.303844\pi$
$182$	0	0
$183$	− 33267.1i	− 0.993374i
$184$	0	0
$185$	7296.00	0.213178
$186$	0	0
$187$	30442.5i	0.870557i
$188$	0	0
$189$	7338.44	0.205438
$190$	0	0
$191$	32848.7i	0.900432i	0.892920	+	0.450216i	$0.148653\pi$
$191$	32848.7i	0.900432i	−0.892920	+	0.450216i	$0.851347\pi$
$192$	0	0
$193$	44830.0	1.20352	0.601761	−	0.798676i	$-0.294466\pi$
$193$	44830.0	1.20352	0.601761	+	0.798676i	$0.294466\pi$
$194$	0	0
$195$	9477.78i	0.249251i
$196$	0	0
$197$	42188.5	1.08708	0.543540	−	0.839383i	$-0.317084\pi$
$197$	42188.5	1.08708	0.543540	+	0.839383i	$0.317084\pi$
$198$	0	0
$199$	− 18778.1i	− 0.474183i	−0.971487	−	0.237092i	$-0.923806\pi$
$199$	− 18778.1i	− 0.474183i	0.971487	−	0.237092i	$-0.0761942\pi$
$200$	0	0
$201$	42948.0	1.06304
$202$	0	0
$203$	67924.1i	1.64828i
$204$	0	0
$205$	17455.2	0.415353
$206$	0	0
$207$	19772.0i	0.461434i
$208$	0	0
$209$	624.000	0.0142854
$210$	0	0
$211$	− 63704.8i	− 1.43089i	−0.698667	−	0.715447i	$-0.746223\pi$
$211$	− 63704.8i	− 1.43089i	0.698667	−	0.715447i	$-0.253777\pi$
$212$	0	0
$213$	−22287.1	−0.491241
$214$	0	0
$215$	61303.6i	1.32620i
$216$	0	0
$217$	−68400.0	−1.45257
$218$	0	0
$219$	45383.2i	0.946252i
$220$	0	0
$221$	−20414.8	−0.417984
$222$	0	0
$223$	− 19405.8i	− 0.390231i	−0.980780	−	0.195116i	$-0.937492\pi$
$223$	− 19405.8i	− 0.390231i	0.980780	−	0.195116i	$-0.0625082\pi$
$224$	0	0
$225$	−7749.00	−0.153067
$226$	0	0
$227$	12089.7i	0.234620i	0.993095	+	0.117310i	$0.0374271\pi$
$227$	12089.7i	0.234620i	−0.993095	+	0.117310i	$0.962573\pi$
$228$	0	0
$229$	−91020.8	−1.73568	−0.867840	−	0.496844i	$-0.834492\pi$
$229$	−91020.8	−1.73568	−0.867840	+	0.496844i	$0.834492\pi$
$230$	0	0
$231$	24479.6i	0.458754i
$232$	0	0
$233$	−45166.0	−0.831955	−0.415977	−	0.909375i	$-0.636560\pi$
$233$	−45166.0	−0.831955	−0.415977	+	0.909375i	$0.636560\pi$
$234$	0	0
$235$	66344.5i	1.20135i
$236$	0	0
$237$	58435.7	1.04036
$238$	0	0
$239$	12135.2i	0.212447i	0.994342	+	0.106223i	$0.0338759\pi$
$239$	12135.2i	0.212447i	−0.994342	+	0.106223i	$0.966124\pi$
$240$	0	0
$241$	85822.0	1.47763	0.738813	−	0.673910i	$-0.235387\pi$
$241$	85822.0	1.47763	0.738813	+	0.673910i	$0.235387\pi$
$242$	0	0
$243$	− 3788.00i	− 0.0641500i
$244$	0	0
$245$	10116.8	0.168543
$246$	0	0
$247$	418.454i	0.00685889i
$248$	0	0
$249$	68580.0	1.10611
$250$	0	0
$251$	58190.0i	0.923636i	0.886975	+	0.461818i	$0.152803\pi$
$251$	58190.0i	0.923636i	−0.886975	+	0.461818i	$0.847197\pi$
$252$	0	0
$253$	−65955.4	−1.03041
$254$	0	0
$255$	− 53039.1i	− 0.815672i
$256$	0	0
$257$	−121726.	−1.84297	−0.921483	−	0.388420i	$-0.873021\pi$
$257$	−121726.	−1.84297	−0.921483	+	0.388420i	$0.873021\pi$
$258$	0	0
$259$	12637.0i	0.188385i
$260$	0	0
$261$	35061.4	0.514693
$262$	0	0
$263$	97918.3i	1.41564i	0.706394	+	0.707819i	$0.250321\pi$
$263$	97918.3i	1.41564i	−0.706394	+	0.707819i	$0.749679\pi$
$264$	0	0
$265$	−73872.0	−1.05193
$266$	0	0
$267$	4728.50i	0.0663286i
$268$	0	0
$269$	−14465.5	−0.199907	−0.0999536	−	0.994992i	$-0.531869\pi$
$269$	−14465.5	−0.199907	−0.0999536	+	0.994992i	$0.531869\pi$
$270$	0	0
$271$	55497.5i	0.755675i	0.925872	+	0.377837i	$0.123332\pi$
$271$	55497.5i	0.755675i	−0.925872	+	0.377837i	$0.876668\pi$
$272$	0	0
$273$	−16416.0	−0.220263
$274$	0	0
$275$	− 25849.1i	− 0.341807i
$276$	0	0
$277$	−102013.	−1.32953	−0.664764	−	0.747053i	$-0.731468\pi$
$277$	−102013.	−1.32953	−0.664764	+	0.747053i	$0.731468\pi$
$278$	0	0
$279$	35307.1i	0.453579i
$280$	0	0
$281$	40082.0	0.507618	0.253809	−	0.967254i	$-0.418317\pi$
$281$	40082.0	0.507618	0.253809	+	0.967254i	$0.418317\pi$
$282$	0	0
$283$	48324.2i	0.603381i	0.953406	+	0.301691i	$0.0975510\pi$
$283$	48324.2i	0.603381i	−0.953406	+	0.301691i	$0.902449\pi$
$284$	0	0
$285$	−1087.18	−0.0133847
$286$	0	0
$287$	30233.3i	0.367047i
$288$	0	0
$289$	30723.0	0.367848
$290$	0	0
$291$	− 28173.5i	− 0.332702i
$292$	0	0
$293$	127592.	1.48624	0.743120	−	0.669158i	$-0.233345\pi$
$293$	127592.	1.48624	0.743120	+	0.669158i	$0.233345\pi$
$294$	0	0
$295$	36196.3i	0.415930i
$296$	0	0
$297$	12636.0	0.143251
$298$	0	0
$299$	− 44229.6i	− 0.494733i
$300$	0	0
$301$	−106181.	−1.17196
$302$	0	0
$303$	56962.1i	0.620441i
$304$	0	0
$305$	−193344.	−2.07841
$306$	0	0
$307$	− 39788.7i	− 0.422165i	−0.977468	−	0.211083i	$-0.932301\pi$
$307$	− 39788.7i	− 0.422165i	0.977468	−	0.211083i	$-0.0676989\pi$
$308$	0	0
$309$	−27994.8	−0.293197
$310$	0	0
$311$	− 174705.i	− 1.80627i	−0.429352	−	0.903137i	$-0.641258\pi$
$311$	− 174705.i	− 1.80627i	0.429352	−	0.903137i	$-0.358742\pi$
$312$	0	0
$313$	−26930.0	−0.274883	−0.137441	−	0.990510i	$-0.543888\pi$
$313$	−26930.0	−0.274883	−0.137441	+	0.990510i	$0.543888\pi$
$314$	0	0
$315$	− 42650.0i	− 0.429831i
$316$	0	0
$317$	−10841.6	−0.107888	−0.0539440	−	0.998544i	$-0.517179\pi$
$317$	−10841.6	−0.107888	−0.0539440	+	0.998544i	$0.517179\pi$
$318$	0	0
$319$	116958.i	1.14934i
$320$	0	0
$321$	−33444.0	−0.324570
$322$	0	0
$323$	− 2341.73i	− 0.0224457i
$324$	0	0
$325$	17334.4	0.164113
$326$	0	0
$327$	313.841i	0.00293504i
$328$	0	0
$329$	−114912.	−1.06163
$330$	0	0
$331$	− 127597.i	− 1.16462i	−0.812968	−	0.582309i	$-0.802149\pi$
$331$	− 127597.i	− 1.16462i	0.812968	−	0.582309i	$-0.197851\pi$
$332$	0	0
$333$	6523.06	0.0588251
$334$	0	0
$335$	− 249608.i	− 2.22417i
$336$	0	0
$337$	186482.	1.64201	0.821007	−	0.570917i	$-0.193412\pi$
$337$	186482.	1.64201	0.821007	+	0.570917i	$0.193412\pi$
$338$	0	0
$339$	16451.0i	0.143151i
$340$	0	0
$341$	−117777.	−1.01287
$342$	0	0
$343$	− 108066.i	− 0.918544i
$344$	0	0
$345$	114912.	0.965444
$346$	0	0
$347$	− 49224.9i	− 0.408814i	−0.978886	−	0.204407i	$-0.934473\pi$
$347$	− 49224.9i	− 0.408814i	0.978886	−	0.204407i	$-0.0655266\pi$
$348$	0	0
$349$	209825.	1.72269	0.861343	−	0.508023i	$-0.169624\pi$
$349$	209825.	1.72269	0.861343	+	0.508023i	$0.169624\pi$
$350$	0	0
$351$	8473.70i	0.0687795i
$352$	0	0
$353$	−67486.0	−0.541582	−0.270791	−	0.962638i	$-0.587285\pi$
$353$	−67486.0	−0.541582	−0.270791	+	0.962638i	$0.587285\pi$
$354$	0	0
$355$	129530.i	1.02781i
$356$	0	0
$357$	91866.4	0.720809
$358$	0	0
$359$	− 116016.i	− 0.900183i	−0.892983	−	0.450091i	$-0.851391\pi$
$359$	− 116016.i	− 0.900183i	0.892983	−	0.450091i	$-0.148609\pi$
$360$	0	0
$361$	130273.	0.999632
$362$	0	0
$363$	− 33925.7i	− 0.257463i
$364$	0	0
$365$	263761.	1.97982
$366$	0	0
$367$	− 77884.8i	− 0.578257i	−0.957290	−	0.289128i	$-0.906635\pi$
$367$	− 77884.8i	− 0.578257i	0.957290	−	0.289128i	$-0.0933654\pi$
$368$	0	0
$369$	15606.0	0.114614
$370$	0	0
$371$	− 127950.i	− 0.929593i
$372$	0	0
$373$	259835.	1.86758	0.933792	−	0.357816i	$-0.116479\pi$
$373$	259835.	1.86758	0.933792	+	0.357816i	$0.116479\pi$
$374$	0	0
$375$	− 53039.1i	− 0.377167i
$376$	0	0
$377$	−78432.0	−0.551837
$378$	0	0
$379$	− 116761.i	− 0.812867i	−0.913680	−	0.406433i	$-0.866772\pi$
$379$	− 116761.i	− 0.812867i	0.913680	−	0.406433i	$-0.133228\pi$
$380$	0	0
$381$	−67133.1	−0.462474
$382$	0	0
$383$	158803.i	1.08259i	0.840834	+	0.541293i	$0.182065\pi$
$383$	158803.i	1.08259i	−0.840834	+	0.541293i	$0.817935\pi$
$384$	0	0
$385$	142272.	0.959838
$386$	0	0
$387$	54809.0i	0.365957i
$388$	0	0
$389$	−121673.	−0.804073	−0.402037	−	0.915624i	$-0.631698\pi$
$389$	−121673.	−0.804073	−0.402037	+	0.915624i	$0.631698\pi$
$390$	0	0
$391$	247516.i	1.61901i
$392$	0	0
$393$	117828.	0.762893
$394$	0	0
$395$	− 339621.i	− 2.17671i
$396$	0	0
$397$	−237850.	−1.50911	−0.754557	−	0.656234i	$-0.772148\pi$
$397$	−237850.	−1.50911	−0.754557	+	0.656234i	$0.772148\pi$
$398$	0	0
$399$	− 1883.04i	− 0.0118281i
$400$	0	0
$401$	−58130.0	−0.361503	−0.180751	−	0.983529i	$-0.557853\pi$
$401$	−58130.0	−0.361503	−0.180751	+	0.983529i	$0.557853\pi$
$402$	0	0
$403$	− 78981.5i	− 0.486312i
$404$	0	0
$405$	−22015.3	−0.134219
$406$	0	0
$407$	21759.6i	0.131360i
$408$	0	0
$409$	−65186.0	−0.389680	−0.194840	−	0.980835i	$-0.562419\pi$
$409$	−65186.0	−0.389680	−0.194840	+	0.980835i	$0.562419\pi$
$410$	0	0
$411$	− 80800.2i	− 0.478331i
$412$	0	0
$413$	−62693.8	−0.367557
$414$	0	0
$415$	− 398578.i	− 2.31428i
$416$	0	0
$417$	163692.	0.941359
$418$	0	0
$419$	267519.i	1.52379i	0.647698	+	0.761897i	$0.275732\pi$
$419$	267519.i	1.52379i	−0.647698	+	0.761897i	$0.724268\pi$
$420$	0	0
$421$	−263399.	−1.48610	−0.743052	−	0.669233i	$-0.766623\pi$
$421$	−263399.	−1.48610	−0.743052	+	0.669233i	$0.766623\pi$
$422$	0	0
$423$	59315.9i	0.331505i
$424$	0	0
$425$	−97006.0	−0.537057
$426$	0	0
$427$	− 334882.i	− 1.83669i
$428$	0	0
$429$	−28266.6	−0.153588
$430$	0	0
$431$	− 105346.i	− 0.567104i	−0.958957	−	0.283552i	$-0.908487\pi$
$431$	− 105346.i	− 0.567104i	0.958957	−	0.283552i	$-0.0915129\pi$
$432$	0	0
$433$	83758.0	0.446736	0.223368	−	0.974734i	$-0.428295\pi$
$433$	83758.0	0.446736	0.223368	+	0.974734i	$0.428295\pi$
$434$	0	0
$435$	− 203772.i	− 1.07688i
$436$	0	0
$437$	5073.49	0.0265671
$438$	0	0
$439$	− 95983.0i	− 0.498041i	−0.968498	−	0.249020i	$-0.919891\pi$
$439$	− 95983.0i	− 0.498041i	0.968498	−	0.249020i	$-0.0801087\pi$
$440$	0	0
$441$	9045.00	0.0465084
$442$	0	0
$443$	− 222679.i	− 1.13468i	−0.823484	−	0.567339i	$-0.807973\pi$
$443$	− 222679.i	− 1.13468i	0.823484	−	0.567339i	$-0.192027\pi$
$444$	0	0
$445$	27481.4	0.138777
$446$	0	0
$447$	117533.i	0.588229i
$448$	0	0
$449$	−88658.0	−0.439770	−0.219885	−	0.975526i	$-0.570568\pi$
$449$	−88658.0	−0.439770	−0.219885	+	0.975526i	$0.570568\pi$
$450$	0	0
$451$	52058.5i	0.255940i
$452$	0	0
$453$	39953.7	0.194698
$454$	0	0
$455$	95407.6i	0.460851i
$456$	0	0
$457$	29086.0	0.139268	0.0696340	−	0.997573i	$-0.477817\pi$
$457$	29086.0	0.139268	0.0696340	+	0.997573i	$0.477817\pi$
$458$	0	0
$459$	− 47420.1i	− 0.225080i
$460$	0	0
$461$	−277139.	−1.30406	−0.652028	−	0.758195i	$-0.726082\pi$
$461$	−277139.	−1.30406	−0.652028	+	0.758195i	$0.726082\pi$
$462$	0	0
$463$	279371.i	1.30322i	0.758553	+	0.651611i	$0.225907\pi$
$463$	279371.i	1.30322i	−0.758553	+	0.651611i	$0.774093\pi$
$464$	0	0
$465$	205200.	0.949011
$466$	0	0
$467$	− 254702.i	− 1.16788i	−0.811797	−	0.583939i	$-0.801511\pi$
$467$	− 254702.i	− 1.16788i	0.811797	−	0.583939i	$-0.198489\pi$
$468$	0	0
$469$	432334.	1.96550
$470$	0	0
$471$	− 6276.81i	− 0.0282942i
$472$	0	0
$473$	−182832.	−0.817203
$474$	0	0
$475$	1988.39i	0.00881283i
$476$	0	0
$477$	−66046.0	−0.290275
$478$	0	0
$479$	52620.6i	0.229343i	0.993403	+	0.114671i	$0.0365815\pi$
$479$	52620.6i	0.229343i	−0.993403	+	0.114671i	$0.963418\pi$
$480$	0	0
$481$	−14592.0	−0.0630703
$482$	0	0
$483$	199033.i	0.853162i
$484$	0	0
$485$	−163741.	−0.696103
$486$	0	0
$487$	− 147557.i	− 0.622162i	−0.950383	−	0.311081i	$-0.899309\pi$
$487$	− 147557.i	− 0.622162i	0.950383	−	0.311081i	$-0.100691\pi$
$488$	0	0
$489$	138996.	0.581279
$490$	0	0
$491$	− 78240.2i	− 0.324539i	−0.986746	−	0.162270i	$-0.948119\pi$
$491$	− 78240.2i	− 0.324539i	0.986746	−	0.162270i	$-0.0518814\pi$
$492$	0	0
$493$	438917.	1.80588
$494$	0	0
$495$	− 73438.7i	− 0.299719i
$496$	0	0
$497$	−224352.	−0.908275
$498$	0	0
$499$	− 298044.i	− 1.19696i	−0.801138	−	0.598480i	$-0.795771\pi$
$499$	− 298044.i	− 1.19696i	0.801138	−	0.598480i	$-0.204229\pi$
$500$	0	0
$501$	−233743.	−0.931243
$502$	0	0
$503$	132964.i	0.525530i	0.964860	+	0.262765i	$0.0846344\pi$
$503$	132964.i	0.525530i	−0.964860	+	0.262765i	$0.915366\pi$
$504$	0	0
$505$	331056.	1.29813
$506$	0	0
$507$	129452.i	0.503607i
$508$	0	0
$509$	−12411.9	−0.0479075	−0.0239538	−	0.999713i	$-0.507625\pi$
$509$	−12411.9	−0.0479075	−0.0239538	+	0.999713i	$0.507625\pi$
$510$	0	0
$511$	456847.i	1.74956i
$512$	0	0
$513$	−972.000	−0.00369344
$514$	0	0
$515$	162702.i	0.613449i
$516$	0	0
$517$	−197866.	−0.740270
$518$	0	0
$519$	− 21811.9i	− 0.0809766i
$520$	0	0
$521$	−170018.	−0.626353	−0.313177	−	0.949695i	$-0.601393\pi$
$521$	−170018.	−0.626353	−0.313177	+	0.949695i	$0.601393\pi$
$522$	0	0
$523$	− 401621.i	− 1.46829i	−0.678991	−	0.734147i	$-0.737582\pi$
$523$	− 401621.i	− 1.46829i	0.678991	−	0.734147i	$-0.262418\pi$
$524$	0	0
$525$	−78004.9	−0.283011
$526$	0	0
$527$	441992.i	1.59145i
$528$	0	0
$529$	−256415.	−0.916288
$530$	0	0
$531$	32361.6i	0.114773i
$532$	0	0
$533$	−34910.4	−0.122886
$534$	0	0
$535$	194372.i	0.679088i
$536$	0	0
$537$	−64908.0	−0.225087
$538$	0	0
$539$	30172.3i	0.103856i
$540$	0	0
$541$	138857.	0.474430	0.237215	−	0.971457i	$-0.423765\pi$
$541$	138857.	0.474430	0.237215	+	0.971457i	$0.423765\pi$
$542$	0	0
$543$	− 196778.i	− 0.667386i
$544$	0	0
$545$	1824.00	0.00614090
$546$	0	0
$547$	− 99094.1i	− 0.331187i	−0.986194	−	0.165593i	$-0.947046\pi$
$547$	− 99094.1i	− 0.331187i	0.986194	−	0.165593i	$-0.0529539\pi$
$548$	0	0
$549$	−172861.	−0.573525
$550$	0	0
$551$	− 8996.77i	− 0.0296335i
$552$	0	0
$553$	588240.	1.92355
$554$	0	0
$555$	− 37911.1i	− 0.123078i
$556$	0	0
$557$	−199104.	−0.641756	−0.320878	−	0.947120i	$-0.603978\pi$
$557$	−199104.	−0.641756	−0.320878	+	0.947120i	$0.603978\pi$
$558$	0	0
$559$	− 122607.i	− 0.392367i
$560$	0	0
$561$	158184.	0.502617
$562$	0	0
$563$	− 572457.i	− 1.80603i	−0.429605	−	0.903017i	$-0.641347\pi$
$563$	− 572457.i	− 1.80603i	0.429605	−	0.903017i	$-0.358653\pi$
$564$	0	0
$565$	95611.1	0.299510
$566$	0	0
$567$	− 38131.6i	− 0.118609i
$568$	0	0
$569$	−563330.	−1.73996	−0.869978	−	0.493090i	$-0.835867\pi$
$569$	−563330.	−1.73996	−0.869978	+	0.493090i	$0.835867\pi$
$570$	0	0
$571$	113117.i	0.346940i	0.984839	+	0.173470i	$0.0554980\pi$
$571$	113117.i	0.346940i	−0.984839	+	0.173470i	$0.944502\pi$
$572$	0	0
$573$	170687.	0.519865
$574$	0	0
$575$	− 210169.i	− 0.635671i
$576$	0	0
$577$	155858.	0.468142	0.234071	−	0.972220i	$-0.424795\pi$
$577$	155858.	0.468142	0.234071	+	0.972220i	$0.424795\pi$
$578$	0	0
$579$	− 232944.i	− 0.694854i
$580$	0	0
$581$	690357.	2.04513
$582$	0	0
$583$	− 220316.i	− 0.648200i
$584$	0	0
$585$	49248.0	0.143905
$586$	0	0
$587$	270914.i	0.786239i	0.919487	+	0.393119i	$0.128604\pi$
$587$	270914.i	0.786239i	−0.919487	+	0.393119i	$0.871396\pi$
$588$	0	0
$589$	9059.80	0.0261149
$590$	0	0
$591$	− 219218.i	− 0.627626i
$592$	0	0
$593$	−418078.	−1.18891	−0.594454	−	0.804130i	$-0.702632\pi$
$593$	−418078.	−1.18891	−0.594454	+	0.804130i	$0.702632\pi$
$594$	0	0
$595$	− 533915.i	− 1.50813i
$596$	0	0
$597$	−97574.1	−0.273770
$598$	0	0
$599$	539701.i	1.50418i	0.659060	+	0.752090i	$0.270954\pi$
$599$	539701.i	1.50418i	−0.659060	+	0.752090i	$0.729046\pi$
$600$	0	0
$601$	439490.	1.21675	0.608373	−	0.793651i	$-0.291822\pi$
$601$	439490.	1.21675	0.608373	+	0.793651i	$0.291822\pi$
$602$	0	0
$603$	− 223164.i	− 0.613748i
$604$	0	0
$605$	−197171.	−0.538683
$606$	0	0
$607$	− 256879.i	− 0.697189i	−0.937274	−	0.348595i	$-0.886659\pi$
$607$	− 256879.i	− 0.697189i	0.937274	−	0.348595i	$-0.113341\pi$
$608$	0	0
$609$	352944.	0.951637
$610$	0	0
$611$	− 132689.i	− 0.355429i
$612$	0	0
$613$	222871.	0.593107	0.296553	−	0.955016i	$-0.404163\pi$
$613$	222871.	0.593107	0.296553	+	0.955016i	$0.404163\pi$
$614$	0	0
$615$	− 90700.0i	− 0.239804i
$616$	0	0
$617$	−588718.	−1.54645	−0.773227	−	0.634129i	$-0.781359\pi$
$617$	−588718.	−1.54645	−0.773227	+	0.634129i	$0.781359\pi$
$618$	0	0
$619$	417570.i	1.08980i	0.838500	+	0.544901i	$0.183433\pi$
$619$	417570.i	1.08980i	−0.838500	+	0.544901i	$0.816567\pi$
$620$	0	0
$621$	102738.	0.266409
$622$	0	0
$623$	47599.2i	0.122638i
$624$	0	0
$625$	−487631.	−1.24834
$626$	0	0
$627$	− 3242.40i	− 0.00824767i
$628$	0	0
$629$	81659.0	0.206397
$630$	0	0
$631$	− 204467.i	− 0.513529i	−0.966474	−	0.256765i	$-0.917344\pi$
$631$	− 204467.i	− 0.513529i	0.966474	−	0.256765i	$-0.0826565\pi$
$632$	0	0
$633$	−331020.	−0.826127
$634$	0	0
$635$	390169.i	0.967620i
$636$	0	0
$637$	−20233.6	−0.0498647
$638$	0	0
$639$	115807.i	0.283618i
$640$	0	0
$641$	−270578.	−0.658531	−0.329266	−	0.944237i	$-0.606801\pi$
$641$	−270578.	−0.658531	−0.329266	+	0.944237i	$0.606801\pi$
$642$	0	0
$643$	584969.i	1.41485i	0.706788	+	0.707426i	$0.250144\pi$
$643$	584969.i	1.41485i	−0.706788	+	0.707426i	$0.749856\pi$
$644$	0	0
$645$	318543.	0.765681
$646$	0	0
$647$	− 96976.8i	− 0.231664i	−0.993269	−	0.115832i	$-0.963047\pi$
$647$	− 96976.8i	− 0.231664i	0.993269	−	0.115832i	$-0.0369535\pi$
$648$	0	0
$649$	−107952.	−0.256296
$650$	0	0
$651$	355417.i	0.838641i
$652$	0	0
$653$	532082.	1.24782	0.623911	−	0.781496i	$-0.285543\pi$
$653$	532082.	1.24782	0.623911	+	0.781496i	$0.285543\pi$
$654$	0	0
$655$	− 684800.i	− 1.59618i
$656$	0	0
$657$	235818.	0.546319
$658$	0	0
$659$	− 253593.i	− 0.583938i	−0.956428	−	0.291969i	$-0.905690\pi$
$659$	− 253593.i	− 0.583938i	0.956428	−	0.291969i	$-0.0943104\pi$
$660$	0	0
$661$	−339441.	−0.776892	−0.388446	−	0.921471i	$-0.626988\pi$
$661$	−339441.	−0.776892	−0.388446	+	0.921471i	$0.626988\pi$
$662$	0	0
$663$	106078.i	0.241323i
$664$	0	0
$665$	−10944.0	−0.0247476
$666$	0	0
$667$	950937.i	2.13747i
$668$	0	0
$669$	−100836.	−0.225300
$670$	0	0
$671$	− 576630.i	− 1.28071i
$672$	0	0
$673$	−191570.	−0.422958	−0.211479	−	0.977383i	$-0.567828\pi$
$673$	−191570.	−0.422958	−0.211479	+	0.977383i	$0.567828\pi$
$674$	0	0
$675$	40265.0i	0.0883731i
$676$	0	0
$677$	−498923.	−1.08857	−0.544285	−	0.838900i	$-0.683199\pi$
$677$	−498923.	−1.08857	−0.544285	+	0.838900i	$0.683199\pi$
$678$	0	0
$679$	− 283607.i	− 0.615146i
$680$	0	0
$681$	62820.0	0.135458
$682$	0	0
$683$	− 96059.5i	− 0.205920i	−0.994685	−	0.102960i	$-0.967169\pi$
$683$	− 96059.5i	− 0.205920i	0.994685	−	0.102960i	$-0.0328314\pi$
$684$	0	0
$685$	−469600.	−1.00080
$686$	0	0
$687$	472958.i	1.00210i
$688$	0	0
$689$	147744.	0.311223
$690$	0	0
$691$	− 517474.i	− 1.08376i	−0.840456	−	0.541880i	$-0.817713\pi$
$691$	− 517474.i	− 1.08376i	0.840456	−	0.541880i	$-0.182287\pi$
$692$	0	0
$693$	127200.	0.264862
$694$	0	0
$695$	− 951356.i	− 1.96958i
$696$	0	0
$697$	195364.	0.402142
$698$	0	0
$699$	234689.i	0.480329i
$700$	0	0
$701$	214204.	0.435904	0.217952	−	0.975959i	$-0.430062\pi$
$701$	214204.	0.435904	0.217952	+	0.975959i	$0.430062\pi$
$702$	0	0
$703$	− 1673.82i	− 0.00338686i
$704$	0	0
$705$	344736.	0.693599
$706$	0	0
$707$	573406.i	1.14716i
$708$	0	0
$709$	−413550.	−0.822688	−0.411344	−	0.911480i	$-0.634941\pi$
$709$	−413550.	−0.822688	−0.411344	+	0.911480i	$0.634941\pi$
$710$	0	0
$711$	− 303641.i	− 0.600649i
$712$	0	0
$713$	−957600.	−1.88367
$714$	0	0
$715$	164282.i	0.321349i
$716$	0	0
$717$	63056.2	0.122656
$718$	0	0
$719$	734073.i	1.41998i	0.704213	+	0.709989i	$0.251300\pi$
$719$	734073.i	1.41998i	−0.704213	+	0.709989i	$0.748700\pi$
$720$	0	0
$721$	−281808.	−0.542104
$722$	0	0
$723$	− 445944.i	− 0.853108i
$724$	0	0
$725$	−372690.	−0.709042
$726$	0	0
$727$	1.03479e6i	1.95786i	0.204199	+	0.978929i	$0.434541\pi$
$727$	1.03479e6i	1.95786i	−0.204199	+	0.978929i	$0.565459\pi$
$728$	0	0
$729$	−19683.0	−0.0370370
$730$	0	0
$731$	686128.i	1.28402i
$732$	0	0
$733$	−200222.	−0.372652	−0.186326	−	0.982488i	$-0.559658\pi$
$733$	−200222.	−0.372652	−0.186326	+	0.982488i	$0.559658\pi$
$734$	0	0
$735$	− 52568.3i	− 0.0973082i
$736$	0	0
$737$	744432.	1.37053
$738$	0	0
$739$	526855.i	0.964722i	0.875972	+	0.482361i	$0.160221\pi$
$739$	526855.i	0.964722i	−0.875972	+	0.482361i	$0.839779\pi$
$740$	0	0
$741$	2174.35	0.00395998
$742$	0	0
$743$	− 380375.i	− 0.689024i	−0.938782	−	0.344512i	$-0.888044\pi$
$743$	− 380375.i	− 0.689024i	0.938782	−	0.344512i	$-0.111956\pi$
$744$	0	0
$745$	683088.	1.23073
$746$	0	0
$747$	− 356352.i	− 0.638614i
$748$	0	0
$749$	−336662.	−0.600110
$750$	0	0
$751$	591851.i	1.04938i	0.851293	+	0.524690i	$0.175819\pi$
$751$	591851.i	1.04938i	−0.851293	+	0.524690i	$0.824181\pi$
$752$	0	0
$753$	302364.	0.533261
$754$	0	0
$755$	− 232206.i	− 0.407360i
$756$	0	0
$757$	−373445.	−0.651681	−0.325840	−	0.945425i	$-0.605647\pi$
$757$	−373445.	−0.651681	−0.325840	+	0.945425i	$0.605647\pi$
$758$	0	0
$759$	342714.i	0.594906i
$760$	0	0
$761$	−37154.0	−0.0641558	−0.0320779	−	0.999485i	$-0.510212\pi$
$761$	−37154.0	−0.0641558	−0.0320779	+	0.999485i	$0.510212\pi$
$762$	0	0
$763$	3159.26i	0.00542671i
$764$	0	0
$765$	−275599.	−0.470929
$766$	0	0
$767$	− 72392.6i	− 0.123056i
$768$	0	0
$769$	−455330.	−0.769970	−0.384985	−	0.922923i	$-0.625793\pi$
$769$	−455330.	−0.769970	−0.384985	+	0.922923i	$0.625793\pi$
$770$	0	0
$771$	632507.i	1.06404i
$772$	0	0
$773$	57046.5	0.0954708	0.0477354	−	0.998860i	$-0.484800\pi$
$773$	57046.5	0.0954708	0.0477354	+	0.998860i	$0.484800\pi$
$774$	0	0
$775$	− 375301.i	− 0.624851i
$776$	0	0
$777$	65664.0	0.108764
$778$	0	0
$779$	− 4004.50i	− 0.00659893i
$780$	0	0
$781$	−386310.	−0.633335
$782$	0	0
$783$	− 182185.i	− 0.297158i
$784$	0	0
$785$	−36480.0	−0.0591992
$786$	0	0
$787$	− 99953.2i	− 0.161379i	−0.996739	−	0.0806895i	$-0.974288\pi$
$787$	− 99953.2i	− 0.161379i	0.996739	−	0.0806895i	$-0.0257122\pi$
$788$	0	0
$789$	508798.	0.817319
$790$	0	0
$791$	165603.i	0.264677i
$792$	0	0
$793$	386688.	0.614914
$794$	0	0
$795$	383850.i	0.607334i
$796$	0	0
$797$	517345.	0.814448	0.407224	−	0.913328i	$-0.366497\pi$
$797$	517345.	0.814448	0.407224	+	0.913328i	$0.366497\pi$
$798$	0	0
$799$	742547.i	1.16314i
$800$	0	0
$801$	24570.0	0.0382948
$802$	0	0
$803$	786642.i	1.21996i
$804$	0	0
$805$	1.15676e6	1.78505
$806$	0	0
$807$	75164.9i	0.115416i
$808$	0	0
$809$	570046.	0.870989	0.435495	−	0.900191i	$-0.356573\pi$
$809$	570046.	0.870989	0.435495	+	0.900191i	$0.356573\pi$
$810$	0	0
$811$	174418.i	0.265185i	0.991171	+	0.132592i	$0.0423301\pi$
$811$	174418.i	0.265185i	−0.991171	+	0.132592i	$0.957670\pi$
$812$	0	0
$813$	288373.	0.436289
$814$	0	0
$815$	− 807826.i	− 1.21619i
$816$	0	0
$817$	14064.0	0.0210700
$818$	0	0
$819$	85300.0i	0.127169i
$820$	0	0
$821$	−438827.	−0.651038	−0.325519	−	0.945535i	$-0.605539\pi$
$821$	−438827.	−0.651038	−0.325519	+	0.945535i	$0.605539\pi$
$822$	0	0
$823$	608380.i	0.898205i	0.893480	+	0.449102i	$0.148256\pi$
$823$	608380.i	0.898205i	−0.893480	+	0.449102i	$0.851744\pi$
$824$	0	0
$825$	−134316.	−0.197342
$826$	0	0
$827$	727898.i	1.06429i	0.846654	+	0.532144i	$0.178614\pi$
$827$	727898.i	1.06429i	−0.846654	+	0.532144i	$0.821386\pi$
$828$	0	0
$829$	−619509.	−0.901444	−0.450722	−	0.892664i	$-0.648833\pi$
$829$	−619509.	−0.901444	−0.450722	+	0.892664i	$0.648833\pi$
$830$	0	0
$831$	530077.i	0.767603i
$832$	0	0
$833$	113230.	0.163182
$834$	0	0
$835$	1.35848e6i	1.94841i
$836$	0	0
$837$	183461.	0.261874
$838$	0	0
$839$	1.10064e6i	1.56358i	0.623539	+	0.781792i	$0.285694\pi$
$839$	1.10064e6i	1.56358i	−0.623539	+	0.781792i	$0.714306\pi$
$840$	0	0
$841$	979007.	1.38418
$842$	0	0
$843$	− 208272.i	− 0.293073i
$844$	0	0
$845$	752356.	1.05368
$846$	0	0
$847$	− 341511.i	− 0.476034i
$848$	0	0
$849$	251100.	0.348362
$850$	0	0
$851$	176919.i	0.244295i
$852$	0	0
$853$	−634911.	−0.872599	−0.436299	−	0.899802i	$-0.643711\pi$
$853$	−634911.	−0.872599	−0.436299	+	0.899802i	$0.643711\pi$
$854$	0	0
$855$	5649.13i	0.00772769i
$856$	0	0
$857$	405310.	0.551856	0.275928	−	0.961178i	$-0.411015\pi$
$857$	405310.	0.551856	0.275928	+	0.961178i	$0.411015\pi$
$858$	0	0
$859$	100230.i	0.135835i	0.997691	+	0.0679177i	$0.0216355\pi$
$859$	100230.i	0.135835i	−0.997691	+	0.0679177i	$0.978364\pi$
$860$	0	0
$861$	157097.	0.211915
$862$	0	0
$863$	− 317816.i	− 0.426731i	−0.976972	−	0.213366i	$-0.931557\pi$
$863$	− 317816.i	− 0.426731i	0.976972	−	0.213366i	$-0.0684425\pi$
$864$	0	0
$865$	−126768.	−0.169425
$866$	0	0
$867$	− 159641.i	− 0.212377i
$868$	0	0
$869$	1.01289e6	1.34128
$870$	0	0
$871$	499216.i	0.658040i
$872$	0	0
$873$	−146394.	−0.192086
$874$	0	0
$875$	− 533915.i	− 0.697358i
$876$	0	0
$877$	323133.	0.420128	0.210064	−	0.977688i	$-0.432633\pi$
$877$	323133.	0.420128	0.210064	+	0.977688i	$0.432633\pi$
$878$	0	0
$879$	− 662989.i	− 0.858081i
$880$	0	0
$881$	−1.10419e6	−1.42263	−0.711315	−	0.702873i	$-0.751900\pi$
$881$	−1.10419e6	−1.42263	−0.711315	+	0.702873i	$0.751900\pi$
$882$	0	0
$883$	813877.i	1.04385i	0.852992	+	0.521924i	$0.174786\pi$
$883$	813877.i	1.04385i	−0.852992	+	0.521924i	$0.825214\pi$
$884$	0	0
$885$	188081.	0.240137
$886$	0	0
$887$	221781.i	0.281888i	0.990018	+	0.140944i	$0.0450138\pi$
$887$	221781.i	0.281888i	−0.990018	+	0.140944i	$0.954986\pi$
$888$	0	0
$889$	−675792.	−0.855085
$890$	0	0
$891$	− 65658.6i	− 0.0827058i
$892$	0	0
$893$	15220.5	0.0190864
$894$	0	0
$895$	377237.i	0.470942i
$896$	0	0
$897$	−229824.	−0.285634
$898$	0	0
$899$	1.69810e6i	2.10109i
$900$	0	0
$901$	−826797.	−1.01847
$902$	0	0
$903$	551732.i	0.676632i
$904$	0	0
$905$	−1.14365e6	−1.39635
$906$	0	0
$907$	− 577071.i	− 0.701479i	−0.936473	−	0.350739i	$-0.885930\pi$
$907$	− 577071.i	− 0.701479i	0.936473	−	0.350739i	$-0.114070\pi$
$908$	0	0
$909$	295984.	0.358212
$910$	0	0
$911$	922692.i	1.11178i	0.831255	+	0.555891i	$0.187623\pi$
$911$	922692.i	1.11178i	−0.831255	+	0.555891i	$0.812377\pi$
$912$	0	0
$913$	1.18872e6	1.42606
$914$	0	0
$915$	1.00464e6i	1.19997i
$916$	0	0
$917$	1.18611e6	1.41054
$918$	0	0
$919$	− 1.47845e6i	− 1.75056i	−0.483620	−	0.875278i	$-0.660678\pi$
$919$	− 1.47845e6i	− 1.75056i	0.483620	−	0.875278i	$-0.339322\pi$
$920$	0	0
$921$	−206748.	−0.243737
$922$	0	0
$923$	− 259059.i	− 0.304086i
$924$	0	0
$925$	−69337.7	−0.0810375
$926$	0	0
$927$	145465.i	0.169278i
$928$	0	0
$929$	241006.	0.279252	0.139626	−	0.990204i	$-0.455410\pi$
$929$	241006.	0.279252	0.139626	+	0.990204i	$0.455410\pi$
$930$	0	0
$931$	− 2320.95i	− 0.00267773i
$932$	0	0
$933$	−907792.	−1.04285
$934$	0	0
$935$	− 919344.i	− 1.05161i
$936$	0	0
$937$	931058.	1.06047	0.530234	−	0.847851i	$-0.322104\pi$
$937$	931058.	1.06047	0.530234	+	0.847851i	$0.322104\pi$
$938$	0	0
$939$	139932.i	0.158704i
$940$	0	0
$941$	−245490.	−0.277240	−0.138620	−	0.990346i	$-0.544267\pi$
$941$	−245490.	−0.277240	−0.138620	+	0.990346i	$0.544267\pi$
$942$	0	0
$943$	423267.i	0.475982i
$944$	0	0
$945$	−221616.	−0.248163
$946$	0	0
$947$	207853.i	0.231770i	0.993263	+	0.115885i	$0.0369703\pi$
$947$	207853.i	0.231770i	−0.993263	+	0.115885i	$0.963030\pi$
$948$	0	0
$949$	−527522.	−0.585744
$950$	0	0
$951$	56334.4i	0.0622892i
$952$	0	0
$953$	1.23446e6	1.35923	0.679613	−	0.733570i	$-0.262148\pi$
$953$	1.23446e6	1.35923	0.679613	+	0.733570i	$0.262148\pi$
$954$	0	0
$955$	− 992008.i	− 1.08770i
$956$	0	0
$957$	607731.	0.663572
$958$	0	0
$959$	− 813371.i	− 0.884405i
$960$	0	0
$961$	−786479.	−0.851609
$962$	0	0
$963$	173780.i	0.187390i
$964$	0	0
$965$	−1.35384e6	−1.45382
$966$	0	0
$967$	181034.i	0.193601i	0.995304	+	0.0968003i	$0.0308608\pi$
$967$	181034.i	0.193601i	−0.995304	+	0.0968003i	$0.969139\pi$
$968$	0	0
$969$	−12168.0	−0.0129590
$970$	0	0
$971$	97708.5i	0.103632i	0.998657	+	0.0518160i	$0.0165009\pi$
$971$	97708.5i	0.103632i	−0.998657	+	0.0518160i	$0.983499\pi$
$972$	0	0
$973$	1.64780e6	1.74052
$974$	0	0
$975$	− 90072.3i	− 0.0947506i
$976$	0	0
$977$	1.67313e6	1.75284	0.876419	−	0.481550i	$-0.159926\pi$
$977$	1.67313e6	1.75284	0.876419	+	0.481550i	$0.159926\pi$
$978$	0	0
$979$	81960.6i	0.0855145i
$980$	0	0
$981$	1630.76	0.00169455
$982$	0	0
$983$	− 1.03358e6i	− 1.06964i	−0.844966	−	0.534820i	$-0.820379\pi$
$983$	− 1.03358e6i	− 1.06964i	0.844966	−	0.534820i	$-0.179621\pi$
$984$	0	0
$985$	−1.27406e6	−1.31316
$986$	0	0
$987$	597100.i	0.612933i
$988$	0	0
$989$	−1.48653e6	−1.51978
$990$	0	0
$991$	− 907889.i	− 0.924454i	−0.886762	−	0.462227i	$-0.847050\pi$
$991$	− 907889.i	− 0.924454i	0.886762	−	0.462227i	$-0.152950\pi$
$992$	0	0
$993$	−663012.	−0.672393
$994$	0	0
$995$	567087.i	0.572801i
$996$	0	0
$997$	1.47844e6	1.48735	0.743675	−	0.668542i	$-0.233081\pi$
$997$	1.47844e6	1.48735	0.743675	+	0.668542i	$0.233081\pi$
$998$	0	0
$999$	− 33894.8i	− 0.0339627i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.5.g.d.511.1		4
4.3	odd	2	inner	768.5.g.d.511.3		4
8.3	odd	2	inner	768.5.g.d.511.2		4
8.5	even	2	inner	768.5.g.d.511.4		4
16.3	odd	4		384.5.b.a.319.2	yes	4
16.5	even	4		384.5.b.a.319.1	✓	4
16.11	odd	4		384.5.b.a.319.3	yes	4
16.13	even	4		384.5.b.a.319.4	yes	4
48.5	odd	4		1152.5.b.j.703.4		4
48.11	even	4		1152.5.b.j.703.3		4
48.29	odd	4		1152.5.b.j.703.2		4
48.35	even	4		1152.5.b.j.703.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.b.a.319.1	✓	4	16.5	even	4
384.5.b.a.319.2	yes	4	16.3	odd	4
384.5.b.a.319.3	yes	4	16.11	odd	4
384.5.b.a.319.4	yes	4	16.13	even	4
768.5.g.d.511.1		4	1.1	even	1	trivial
768.5.g.d.511.2		4	8.3	odd	2	inner
768.5.g.d.511.3		4	4.3	odd	2	inner
768.5.g.d.511.4		4	8.5	even	2	inner
1152.5.b.j.703.1		4	48.35	even	4
1152.5.b.j.703.2		4	48.29	odd	4
1152.5.b.j.703.3		4	48.11	even	4
1152.5.b.j.703.4		4	48.5	odd	4

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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	768.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.19615i q^{3} -30.1993 q^{5} -52.3068i q^{7} -27.0000 q^{9} +O(q^{10})\)	\(q-5.19615i q^{3} -30.1993 q^{5} -52.3068i q^{7} -27.0000 q^{9} -90.0666i q^{11} +60.3987 q^{13} +156.920i q^{15} -338.000 q^{17} +6.92820i q^{19} -271.794 q^{21} -732.295i q^{23} +287.000 q^{25} +140.296i q^{27} -1298.57 q^{29} -1307.67i q^{31} -468.000 q^{33} +1579.63i q^{35} -241.595 q^{37} -313.841i q^{39} -578.000 q^{41} -2029.96i q^{43} +815.382 q^{45} -2196.89i q^{47} -335.000 q^{49} +1756.30i q^{51} +2446.15 q^{53} +2719.95i q^{55} +36.0000 q^{57} -1198.58i q^{59} +6402.26 q^{61} +1412.28i q^{63} -1824.00 q^{65} +8265.35i q^{67} -3805.12 q^{69} -4289.16i q^{71} -8734.00 q^{73} -1491.30i q^{75} -4711.10 q^{77} +11246.0i q^{79} +729.000 q^{81} +13198.2i q^{83} +10207.4 q^{85} +6747.58i q^{87} -910.000 q^{89} -3159.26i q^{91} -6794.85 q^{93} -209.227i q^{95} +5422.00 q^{97} +2431.80i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 108 q^{9}+O(q^{10}) \) 4 * q - 108 * q^9	\( 4 q - 108 q^{9} - 1352 q^{17} + 1148 q^{25} - 1872 q^{33} - 2312 q^{41} - 1340 q^{49} + 144 q^{57} - 7296 q^{65} - 34936 q^{73} + 2916 q^{81} - 3640 q^{89} + 21688 q^{97}+O(q^{100}) \) 4 * q - 108 * q^9 - 1352 * q^17 + 1148 * q^25 - 1872 * q^33 - 2312 * q^41 - 1340 * q^49 + 144 * q^57 - 7296 * q^65 - 34936 * q^73 + 2916 * q^81 - 3640 * q^89 + 21688 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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