LMFDB - Embedded newform 768.4.a.f.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:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 192)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} +3.46410 q^{5} +24.2487 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +3.46410 q^{5} +24.2487 q^{7} +9.00000 q^{9} -48.0000 q^{11} +41.5692 q^{13} -10.3923 q^{15} +54.0000 q^{17} +4.00000 q^{19} -72.7461 q^{21} +173.205 q^{23} -113.000 q^{25} -27.0000 q^{27} +162.813 q^{29} -58.8897 q^{31} +144.000 q^{33} +84.0000 q^{35} -325.626 q^{37} -124.708 q^{39} +294.000 q^{41} -188.000 q^{43} +31.1769 q^{45} -505.759 q^{47} +245.000 q^{49} -162.000 q^{51} +744.782 q^{53} -166.277 q^{55} -12.0000 q^{57} -252.000 q^{59} -90.0666 q^{61} +218.238 q^{63} +144.000 q^{65} -628.000 q^{67} -519.615 q^{69} -6.92820 q^{71} +1006.00 q^{73} +339.000 q^{75} -1163.94 q^{77} +1340.61 q^{79} +81.0000 q^{81} +720.000 q^{83} +187.061 q^{85} -488.438 q^{87} +1482.00 q^{89} +1008.00 q^{91} +176.669 q^{93} +13.8564 q^{95} +1822.00 q^{97} -432.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 18 * q^9	$ 2 q - 6 q^{3} + 18 q^{9} - 96 q^{11} + 108 q^{17} + 8 q^{19} - 226 q^{25} - 54 q^{27} + 288 q^{33} + 168 q^{35} + 588 q^{41} - 376 q^{43} + 490 q^{49} - 324 q^{51} - 24 q^{57} - 504 q^{59} + 288 q^{65} - 1256 q^{67} + 2012 q^{73} + 678 q^{75} + 162 q^{81} + 1440 q^{83} + 2964 q^{89} + 2016 q^{91} + 3644 q^{97} - 864 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 18 * q^9 - 96 * q^11 + 108 * q^17 + 8 * q^19 - 226 * q^25 - 54 * q^27 + 288 * q^33 + 168 * q^35 + 588 * q^41 - 376 * q^43 + 490 * q^49 - 324 * q^51 - 24 * q^57 - 504 * q^59 + 288 * q^65 - 1256 * q^67 + 2012 * q^73 + 678 * q^75 + 162 * q^81 + 1440 * q^83 + 2964 * q^89 + 2016 * q^91 + 3644 * q^97 - 864 * 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
$5$	3.46410	0.309839	0.154919	−	0.987927i	$-0.450488\pi$
$5$	3.46410	0.309839	0.154919	+	0.987927i	$0.450488\pi$
$6$	0	0
$7$	24.2487	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	24.2487	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−48.0000	−1.31569	−0.657843	−	0.753155i	$-0.728531\pi$
$11$	−48.0000	−1.31569	−0.657843	+	0.753155i	$0.728531\pi$
$12$	0	0
$13$	41.5692	0.886864	0.443432	−	0.896308i	$-0.353761\pi$
$13$	41.5692	0.886864	0.443432	+	0.896308i	$0.353761\pi$
$14$	0	0
$15$	−10.3923	−0.178885
$16$	0	0
$17$	54.0000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	0	0
$19$	4.00000	0.0482980	0.0241490	−	0.999708i	$-0.492312\pi$
$19$	4.00000	0.0482980	0.0241490	+	0.999708i	$0.492312\pi$
$20$	0	0
$21$	−72.7461	−0.755929
$22$	0	0
$23$	173.205	1.57025	0.785125	−	0.619337i	$-0.212599\pi$
$23$	173.205	1.57025	0.785125	+	0.619337i	$0.212599\pi$
$24$	0	0
$25$	−113.000	−0.904000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	162.813	1.04254	0.521269	−	0.853393i	$-0.325459\pi$
$29$	162.813	1.04254	0.521269	+	0.853393i	$0.325459\pi$
$30$	0	0
$31$	−58.8897	−0.341191	−0.170595	−	0.985341i	$-0.554569\pi$
$31$	−58.8897	−0.341191	−0.170595	+	0.985341i	$0.554569\pi$
$32$	0	0
$33$	144.000	0.759612
$34$	0	0
$35$	84.0000	0.405674
$36$	0	0
$37$	−325.626	−1.44682	−0.723412	−	0.690416i	$-0.757427\pi$
$37$	−325.626	−1.44682	−0.723412	+	0.690416i	$0.757427\pi$
$38$	0	0
$39$	−124.708	−0.512031
$40$	0	0
$41$	294.000	1.11988	0.559940	−	0.828533i	$-0.310824\pi$
$41$	294.000	1.11988	0.559940	+	0.828533i	$0.310824\pi$
$42$	0	0
$43$	−188.000	−0.666738	−0.333369	−	0.942796i	$-0.608185\pi$
$43$	−188.000	−0.666738	−0.333369	+	0.942796i	$0.608185\pi$
$44$	0	0
$45$	31.1769	0.103280
$46$	0	0
$47$	−505.759	−1.56963	−0.784814	−	0.619731i	$-0.787242\pi$
$47$	−505.759	−1.56963	−0.784814	+	0.619731i	$0.787242\pi$
$48$	0	0
$49$	245.000	0.714286
$50$	0	0
$51$	−162.000	−0.444795
$52$	0	0
$53$	744.782	1.93026	0.965129	−	0.261775i	$-0.0843080\pi$
$53$	744.782	1.93026	0.965129	+	0.261775i	$0.0843080\pi$
$54$	0	0
$55$	−166.277	−0.407650
$56$	0	0
$57$	−12.0000	−0.0278849
$58$	0	0
$59$	−252.000	−0.556061	−0.278031	−	0.960572i	$-0.589682\pi$
$59$	−252.000	−0.556061	−0.278031	+	0.960572i	$0.589682\pi$
$60$	0	0
$61$	−90.0666	−0.189047	−0.0945234	−	0.995523i	$-0.530133\pi$
$61$	−90.0666	−0.189047	−0.0945234	+	0.995523i	$0.530133\pi$
$62$	0	0
$63$	218.238	0.436436
$64$	0	0
$65$	144.000	0.274785
$66$	0	0
$67$	−628.000	−1.14511	−0.572555	−	0.819866i	$-0.694048\pi$
$67$	−628.000	−1.14511	−0.572555	+	0.819866i	$0.694048\pi$
$68$	0	0
$69$	−519.615	−0.906584
$70$	0	0
$71$	−6.92820	−0.0115807	−0.00579033	−	0.999983i	$-0.501843\pi$
$71$	−6.92820	−0.0115807	−0.00579033	+	0.999983i	$0.501843\pi$
$72$	0	0
$73$	1006.00	1.61292	0.806462	−	0.591286i	$-0.201380\pi$
$73$	1006.00	1.61292	0.806462	+	0.591286i	$0.201380\pi$
$74$	0	0
$75$	339.000	0.521925
$76$	0	0
$77$	−1163.94	−1.72264
$78$	0	0
$79$	1340.61	1.90924	0.954621	−	0.297824i	$-0.0962607\pi$
$79$	1340.61	1.90924	0.954621	+	0.297824i	$0.0962607\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	720.000	0.952172	0.476086	−	0.879399i	$-0.342055\pi$
$83$	720.000	0.952172	0.476086	+	0.879399i	$0.342055\pi$
$84$	0	0
$85$	187.061	0.238702
$86$	0	0
$87$	−488.438	−0.601909
$88$	0	0
$89$	1482.00	1.76508	0.882538	−	0.470242i	$-0.155833\pi$
$89$	1482.00	1.76508	0.882538	+	0.470242i	$0.155833\pi$
$90$	0	0
$91$	1008.00	1.16118
$92$	0	0
$93$	176.669	0.196986
$94$	0	0
$95$	13.8564	0.0149646
$96$	0	0
$97$	1822.00	1.90718	0.953588	−	0.301114i	$-0.0973586\pi$
$97$	1822.00	1.90718	0.953588	+	0.301114i	$0.0973586\pi$
$98$	0	0
$99$	−432.000	−0.438562
$100$	0	0
$101$	911.059	0.897562	0.448781	−	0.893642i	$-0.351858\pi$
$101$	911.059	0.897562	0.448781	+	0.893642i	$0.351858\pi$
$102$	0	0
$103$	−453.797	−0.434116	−0.217058	−	0.976159i	$-0.569646\pi$
$103$	−453.797	−0.434116	−0.217058	+	0.976159i	$0.569646\pi$
$104$	0	0
$105$	−252.000	−0.234216
$106$	0	0
$107$	1188.00	1.07335	0.536674	−	0.843790i	$-0.319680\pi$
$107$	1188.00	1.07335	0.536674	+	0.843790i	$0.319680\pi$
$108$	0	0
$109$	471.118	0.413990	0.206995	−	0.978342i	$-0.433632\pi$
$109$	471.118	0.413990	0.206995	+	0.978342i	$0.433632\pi$
$110$	0	0
$111$	976.877	0.835325
$112$	0	0
$113$	−390.000	−0.324674	−0.162337	−	0.986735i	$-0.551903\pi$
$113$	−390.000	−0.324674	−0.162337	+	0.986735i	$0.551903\pi$
$114$	0	0
$115$	600.000	0.486524
$116$	0	0
$117$	374.123	0.295621
$118$	0	0
$119$	1309.43	1.00870
$120$	0	0
$121$	973.000	0.731029
$122$	0	0
$123$	−882.000	−0.646563
$124$	0	0
$125$	−824.456	−0.589933
$126$	0	0
$127$	−606.218	−0.423568	−0.211784	−	0.977317i	$-0.567927\pi$
$127$	−606.218	−0.423568	−0.211784	+	0.977317i	$0.567927\pi$
$128$	0	0
$129$	564.000	0.384941
$130$	0	0
$131$	1380.00	0.920391	0.460195	−	0.887818i	$-0.347779\pi$
$131$	1380.00	0.920391	0.460195	+	0.887818i	$0.347779\pi$
$132$	0	0
$133$	96.9948	0.0632370
$134$	0	0
$135$	−93.5307	−0.0596285
$136$	0	0
$137$	1158.00	0.722150	0.361075	−	0.932537i	$-0.382410\pi$
$137$	1158.00	0.722150	0.361075	+	0.932537i	$0.382410\pi$
$138$	0	0
$139$	−1180.00	−0.720045	−0.360023	−	0.932944i	$-0.617231\pi$
$139$	−1180.00	−0.720045	−0.360023	+	0.932944i	$0.617231\pi$
$140$	0	0
$141$	1517.28	0.906225
$142$	0	0
$143$	−1995.32	−1.16683
$144$	0	0
$145$	564.000	0.323018
$146$	0	0
$147$	−735.000	−0.412393
$148$	0	0
$149$	−2171.99	−1.19420	−0.597102	−	0.802165i	$-0.703681\pi$
$149$	−2171.99	−1.19420	−0.597102	+	0.802165i	$0.703681\pi$
$150$	0	0
$151$	−142.028	−0.0765436	−0.0382718	−	0.999267i	$-0.512185\pi$
$151$	−142.028	−0.0765436	−0.0382718	+	0.999267i	$0.512185\pi$
$152$	0	0
$153$	486.000	0.256802
$154$	0	0
$155$	−204.000	−0.105714
$156$	0	0
$157$	1337.14	0.679717	0.339859	−	0.940476i	$-0.389621\pi$
$157$	1337.14	0.679717	0.339859	+	0.940476i	$0.389621\pi$
$158$	0	0
$159$	−2234.35	−1.11443
$160$	0	0
$161$	4200.00	2.05594
$162$	0	0
$163$	−1748.00	−0.839963	−0.419981	−	0.907533i	$-0.637963\pi$
$163$	−1748.00	−0.839963	−0.419981	+	0.907533i	$0.637963\pi$
$164$	0	0
$165$	498.831	0.235357
$166$	0	0
$167$	13.8564	0.00642060	0.00321030	−	0.999995i	$-0.498978\pi$
$167$	13.8564	0.00642060	0.00321030	+	0.999995i	$0.498978\pi$
$168$	0	0
$169$	−469.000	−0.213473
$170$	0	0
$171$	36.0000	0.0160993
$172$	0	0
$173$	599.290	0.263371	0.131685	−	0.991292i	$-0.457961\pi$
$173$	599.290	0.263371	0.131685	+	0.991292i	$0.457961\pi$
$174$	0	0
$175$	−2740.10	−1.18361
$176$	0	0
$177$	756.000	0.321042
$178$	0	0
$179$	3228.00	1.34789	0.673944	−	0.738782i	$-0.264599\pi$
$179$	3228.00	1.34789	0.673944	+	0.738782i	$0.264599\pi$
$180$	0	0
$181$	−2023.04	−0.830779	−0.415390	−	0.909644i	$-0.636355\pi$
$181$	−2023.04	−0.830779	−0.415390	+	0.909644i	$0.636355\pi$
$182$	0	0
$183$	270.200	0.109146
$184$	0	0
$185$	−1128.00	−0.448282
$186$	0	0
$187$	−2592.00	−1.01361
$188$	0	0
$189$	−654.715	−0.251976
$190$	0	0
$191$	3477.96	1.31757	0.658786	−	0.752330i	$-0.271070\pi$
$191$	3477.96	1.31757	0.658786	+	0.752330i	$0.271070\pi$
$192$	0	0
$193$	−766.000	−0.285689	−0.142844	−	0.989745i	$-0.545625\pi$
$193$	−766.000	−0.285689	−0.142844	+	0.989745i	$0.545625\pi$
$194$	0	0
$195$	−432.000	−0.158647
$196$	0	0
$197$	−2899.45	−1.04862	−0.524308	−	0.851529i	$-0.675676\pi$
$197$	−2899.45	−1.04862	−0.524308	+	0.851529i	$0.675676\pi$
$198$	0	0
$199$	1735.51	0.618228	0.309114	−	0.951025i	$-0.399968\pi$
$199$	1735.51	0.618228	0.309114	+	0.951025i	$0.399968\pi$
$200$	0	0
$201$	1884.00	0.661130
$202$	0	0
$203$	3948.00	1.36500
$204$	0	0
$205$	1018.45	0.346982
$206$	0	0
$207$	1558.85	0.523417
$208$	0	0
$209$	−192.000	−0.0635451
$210$	0	0
$211$	1100.00	0.358896	0.179448	−	0.983767i	$-0.442569\pi$
$211$	1100.00	0.358896	0.179448	+	0.983767i	$0.442569\pi$
$212$	0	0
$213$	20.7846	0.00668609
$214$	0	0
$215$	−651.251	−0.206581
$216$	0	0
$217$	−1428.00	−0.446723
$218$	0	0
$219$	−3018.00	−0.931222
$220$	0	0
$221$	2244.74	0.683246
$222$	0	0
$223$	391.443	0.117547	0.0587735	−	0.998271i	$-0.481281\pi$
$223$	391.443	0.117547	0.0587735	+	0.998271i	$0.481281\pi$
$224$	0	0
$225$	−1017.00	−0.301333
$226$	0	0
$227$	−3336.00	−0.975410	−0.487705	−	0.873008i	$-0.662166\pi$
$227$	−3336.00	−0.975410	−0.487705	+	0.873008i	$0.662166\pi$
$228$	0	0
$229$	5999.82	1.73135	0.865676	−	0.500605i	$-0.166889\pi$
$229$	5999.82	1.73135	0.865676	+	0.500605i	$0.166889\pi$
$230$	0	0
$231$	3491.81	0.994565
$232$	0	0
$233$	−318.000	−0.0894115	−0.0447057	−	0.999000i	$-0.514235\pi$
$233$	−318.000	−0.0894115	−0.0447057	+	0.999000i	$0.514235\pi$
$234$	0	0
$235$	−1752.00	−0.486331
$236$	0	0
$237$	−4021.82	−1.10230
$238$	0	0
$239$	859.097	0.232512	0.116256	−	0.993219i	$-0.462911\pi$
$239$	859.097	0.232512	0.116256	+	0.993219i	$0.462911\pi$
$240$	0	0
$241$	−2710.00	−0.724342	−0.362171	−	0.932112i	$-0.617964\pi$
$241$	−2710.00	−0.724342	−0.362171	+	0.932112i	$0.617964\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	848.705	0.221313
$246$	0	0
$247$	166.277	0.0428338
$248$	0	0
$249$	−2160.00	−0.549737
$250$	0	0
$251$	5136.00	1.29156	0.645780	−	0.763524i	$-0.276532\pi$
$251$	5136.00	1.29156	0.645780	+	0.763524i	$0.276532\pi$
$252$	0	0
$253$	−8313.84	−2.06596
$254$	0	0
$255$	−561.184	−0.137815
$256$	0	0
$257$	−4398.00	−1.06747	−0.533735	−	0.845652i	$-0.679212\pi$
$257$	−4398.00	−1.06747	−0.533735	+	0.845652i	$0.679212\pi$
$258$	0	0
$259$	−7896.00	−1.89434
$260$	0	0
$261$	1465.31	0.347512
$262$	0	0
$263$	−6817.35	−1.59839	−0.799194	−	0.601073i	$-0.794740\pi$
$263$	−6817.35	−1.59839	−0.799194	+	0.601073i	$0.794740\pi$
$264$	0	0
$265$	2580.00	0.598068
$266$	0	0
$267$	−4446.00	−1.01907
$268$	0	0
$269$	4624.58	1.04820	0.524099	−	0.851657i	$-0.324402\pi$
$269$	4624.58	1.04820	0.524099	+	0.851657i	$0.324402\pi$
$270$	0	0
$271$	3883.26	0.870447	0.435223	−	0.900322i	$-0.356669\pi$
$271$	3883.26	0.870447	0.435223	+	0.900322i	$0.356669\pi$
$272$	0	0
$273$	−3024.00	−0.670406
$274$	0	0
$275$	5424.00	1.18938
$276$	0	0
$277$	−1524.20	−0.330616	−0.165308	−	0.986242i	$-0.552862\pi$
$277$	−1524.20	−0.330616	−0.165308	+	0.986242i	$0.552862\pi$
$278$	0	0
$279$	−530.008	−0.113730
$280$	0	0
$281$	−4398.00	−0.933675	−0.466838	−	0.884343i	$-0.654607\pi$
$281$	−4398.00	−0.933675	−0.466838	+	0.884343i	$0.654607\pi$
$282$	0	0
$283$	−4372.00	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$283$	−4372.00	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$284$	0	0
$285$	−41.5692	−0.00863982
$286$	0	0
$287$	7129.12	1.46627
$288$	0	0
$289$	−1997.00	−0.406473
$290$	0	0
$291$	−5466.00	−1.10111
$292$	0	0
$293$	3571.49	0.712111	0.356056	−	0.934465i	$-0.384121\pi$
$293$	3571.49	0.712111	0.356056	+	0.934465i	$0.384121\pi$
$294$	0	0
$295$	−872.954	−0.172289
$296$	0	0
$297$	1296.00	0.253204
$298$	0	0
$299$	7200.00	1.39260
$300$	0	0
$301$	−4558.76	−0.872965
$302$	0	0
$303$	−2733.18	−0.518207
$304$	0	0
$305$	−312.000	−0.0585740
$306$	0	0
$307$	4172.00	0.775598	0.387799	−	0.921744i	$-0.373235\pi$
$307$	4172.00	0.775598	0.387799	+	0.921744i	$0.373235\pi$
$308$	0	0
$309$	1361.39	0.250637
$310$	0	0
$311$	−6470.94	−1.17985	−0.589925	−	0.807458i	$-0.700843\pi$
$311$	−6470.94	−1.17985	−0.589925	+	0.807458i	$0.700843\pi$
$312$	0	0
$313$	74.0000	0.0133633	0.00668167	−	0.999978i	$-0.497873\pi$
$313$	74.0000	0.0133633	0.00668167	+	0.999978i	$0.497873\pi$
$314$	0	0
$315$	756.000	0.135225
$316$	0	0
$317$	1964.15	0.348004	0.174002	−	0.984745i	$-0.444330\pi$
$317$	1964.15	0.348004	0.174002	+	0.984745i	$0.444330\pi$
$318$	0	0
$319$	−7815.01	−1.37165
$320$	0	0
$321$	−3564.00	−0.619698
$322$	0	0
$323$	216.000	0.0372092
$324$	0	0
$325$	−4697.32	−0.801725
$326$	0	0
$327$	−1413.35	−0.239017
$328$	0	0
$329$	−12264.0	−2.05513
$330$	0	0
$331$	7556.00	1.25473	0.627365	−	0.778726i	$-0.284134\pi$
$331$	7556.00	1.25473	0.627365	+	0.778726i	$0.284134\pi$
$332$	0	0
$333$	−2930.63	−0.482275
$334$	0	0
$335$	−2175.46	−0.354800
$336$	0	0
$337$	−4106.00	−0.663703	−0.331852	−	0.943332i	$-0.607673\pi$
$337$	−4106.00	−0.663703	−0.331852	+	0.943332i	$0.607673\pi$
$338$	0	0
$339$	1170.00	0.187450
$340$	0	0
$341$	2826.71	0.448900
$342$	0	0
$343$	−2376.37	−0.374088
$344$	0	0
$345$	−1800.00	−0.280895
$346$	0	0
$347$	5256.00	0.813132	0.406566	−	0.913621i	$-0.366726\pi$
$347$	5256.00	0.813132	0.406566	+	0.913621i	$0.366726\pi$
$348$	0	0
$349$	10385.4	1.59288	0.796442	−	0.604715i	$-0.206713\pi$
$349$	10385.4	1.59288	0.796442	+	0.604715i	$0.206713\pi$
$350$	0	0
$351$	−1122.37	−0.170677
$352$	0	0
$353$	−3942.00	−0.594367	−0.297183	−	0.954820i	$-0.596047\pi$
$353$	−3942.00	−0.594367	−0.297183	+	0.954820i	$0.596047\pi$
$354$	0	0
$355$	−24.0000	−0.00358813
$356$	0	0
$357$	−3928.29	−0.582373
$358$	0	0
$359$	6644.15	0.976782	0.488391	−	0.872625i	$-0.337584\pi$
$359$	6644.15	0.976782	0.488391	+	0.872625i	$0.337584\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	0	0
$363$	−2919.00	−0.422060
$364$	0	0
$365$	3484.89	0.499746
$366$	0	0
$367$	2906.38	0.413384	0.206692	−	0.978406i	$-0.433730\pi$
$367$	2906.38	0.413384	0.206692	+	0.978406i	$0.433730\pi$
$368$	0	0
$369$	2646.00	0.373293
$370$	0	0
$371$	18060.0	2.52730
$372$	0	0
$373$	10246.8	1.42241	0.711206	−	0.702983i	$-0.248149\pi$
$373$	10246.8	1.42241	0.711206	+	0.702983i	$0.248149\pi$
$374$	0	0
$375$	2473.37	0.340598
$376$	0	0
$377$	6768.00	0.924588
$378$	0	0
$379$	−13844.0	−1.87630	−0.938151	−	0.346226i	$-0.887463\pi$
$379$	−13844.0	−1.87630	−0.938151	+	0.346226i	$0.887463\pi$
$380$	0	0
$381$	1818.65	0.244547
$382$	0	0
$383$	−7163.76	−0.955747	−0.477874	−	0.878429i	$-0.658592\pi$
$383$	−7163.76	−0.955747	−0.477874	+	0.878429i	$0.658592\pi$
$384$	0	0
$385$	−4032.00	−0.533740
$386$	0	0
$387$	−1692.00	−0.222246
$388$	0	0
$389$	−12993.8	−1.69361	−0.846805	−	0.531904i	$-0.821477\pi$
$389$	−12993.8	−1.69361	−0.846805	+	0.531904i	$0.821477\pi$
$390$	0	0
$391$	9353.07	1.20973
$392$	0	0
$393$	−4140.00	−0.531388
$394$	0	0
$395$	4644.00	0.591557
$396$	0	0
$397$	−117.779	−0.0148896	−0.00744481	−	0.999972i	$-0.502370\pi$
$397$	−117.779	−0.0148896	−0.00744481	+	0.999972i	$0.502370\pi$
$398$	0	0
$399$	−290.985	−0.0365099
$400$	0	0
$401$	−5418.00	−0.674718	−0.337359	−	0.941376i	$-0.609534\pi$
$401$	−5418.00	−0.674718	−0.337359	+	0.941376i	$0.609534\pi$
$402$	0	0
$403$	−2448.00	−0.302589
$404$	0	0
$405$	280.592	0.0344265
$406$	0	0
$407$	15630.0	1.90357
$408$	0	0
$409$	−11450.0	−1.38427	−0.692135	−	0.721768i	$-0.743330\pi$
$409$	−11450.0	−1.38427	−0.692135	+	0.721768i	$0.743330\pi$
$410$	0	0
$411$	−3474.00	−0.416934
$412$	0	0
$413$	−6110.68	−0.728055
$414$	0	0
$415$	2494.15	0.295020
$416$	0	0
$417$	3540.00	0.415718
$418$	0	0
$419$	−1176.00	−0.137115	−0.0685577	−	0.997647i	$-0.521840\pi$
$419$	−1176.00	−0.137115	−0.0685577	+	0.997647i	$0.521840\pi$
$420$	0	0
$421$	−10032.0	−1.16136	−0.580679	−	0.814133i	$-0.697213\pi$
$421$	−10032.0	−1.16136	−0.580679	+	0.814133i	$0.697213\pi$
$422$	0	0
$423$	−4551.83	−0.523209
$424$	0	0
$425$	−6102.00	−0.696448
$426$	0	0
$427$	−2184.00	−0.247520
$428$	0	0
$429$	5985.97	0.673672
$430$	0	0
$431$	838.313	0.0936893	0.0468447	−	0.998902i	$-0.485083\pi$
$431$	838.313	0.0936893	0.0468447	+	0.998902i	$0.485083\pi$
$432$	0	0
$433$	4318.00	0.479237	0.239619	−	0.970867i	$-0.422978\pi$
$433$	4318.00	0.479237	0.239619	+	0.970867i	$0.422978\pi$
$434$	0	0
$435$	−1692.00	−0.186495
$436$	0	0
$437$	692.820	0.0758400
$438$	0	0
$439$	1610.81	0.175124	0.0875622	−	0.996159i	$-0.472092\pi$
$439$	1610.81	0.175124	0.0875622	+	0.996159i	$0.472092\pi$
$440$	0	0
$441$	2205.00	0.238095
$442$	0	0
$443$	1032.00	0.110681	0.0553406	−	0.998468i	$-0.482376\pi$
$443$	1032.00	0.110681	0.0553406	+	0.998468i	$0.482376\pi$
$444$	0	0
$445$	5133.80	0.546889
$446$	0	0
$447$	6515.98	0.689474
$448$	0	0
$449$	726.000	0.0763075	0.0381537	−	0.999272i	$-0.487852\pi$
$449$	726.000	0.0763075	0.0381537	+	0.999272i	$0.487852\pi$
$450$	0	0
$451$	−14112.0	−1.47341
$452$	0	0
$453$	426.084	0.0441925
$454$	0	0
$455$	3491.81	0.359778
$456$	0	0
$457$	−8666.00	−0.887042	−0.443521	−	0.896264i	$-0.646271\pi$
$457$	−8666.00	−0.887042	−0.443521	+	0.896264i	$0.646271\pi$
$458$	0	0
$459$	−1458.00	−0.148265
$460$	0	0
$461$	−14684.3	−1.48355	−0.741776	−	0.670648i	$-0.766016\pi$
$461$	−14684.3	−1.48355	−0.741776	+	0.670648i	$0.766016\pi$
$462$	0	0
$463$	−4998.70	−0.501748	−0.250874	−	0.968020i	$-0.580718\pi$
$463$	−4998.70	−0.501748	−0.250874	+	0.968020i	$0.580718\pi$
$464$	0	0
$465$	612.000	0.0610340
$466$	0	0
$467$	16824.0	1.66707	0.833535	−	0.552466i	$-0.186313\pi$
$467$	16824.0	1.66707	0.833535	+	0.552466i	$0.186313\pi$
$468$	0	0
$469$	−15228.2	−1.49930
$470$	0	0
$471$	−4011.43	−0.392435
$472$	0	0
$473$	9024.00	0.877218
$474$	0	0
$475$	−452.000	−0.0436614
$476$	0	0
$477$	6703.04	0.643419
$478$	0	0
$479$	−10953.5	−1.04484	−0.522419	−	0.852689i	$-0.674970\pi$
$479$	−10953.5	−1.04484	−0.522419	+	0.852689i	$0.674970\pi$
$480$	0	0
$481$	−13536.0	−1.28314
$482$	0	0
$483$	−12600.0	−1.18700
$484$	0	0
$485$	6311.59	0.590917
$486$	0	0
$487$	−10714.5	−0.996959	−0.498479	−	0.866902i	$-0.666108\pi$
$487$	−10714.5	−0.996959	−0.498479	+	0.866902i	$0.666108\pi$
$488$	0	0
$489$	5244.00	0.484953
$490$	0	0
$491$	−852.000	−0.0783100	−0.0391550	−	0.999233i	$-0.512467\pi$
$491$	−852.000	−0.0783100	−0.0391550	+	0.999233i	$0.512467\pi$
$492$	0	0
$493$	8791.89	0.803178
$494$	0	0
$495$	−1496.49	−0.135883
$496$	0	0
$497$	−168.000	−0.0151626
$498$	0	0
$499$	11156.0	1.00082	0.500412	−	0.865787i	$-0.333182\pi$
$499$	11156.0	1.00082	0.500412	+	0.865787i	$0.333182\pi$
$500$	0	0
$501$	−41.5692	−0.00370694
$502$	0	0
$503$	−14999.6	−1.32962	−0.664808	−	0.747014i	$-0.731487\pi$
$503$	−14999.6	−1.32962	−0.664808	+	0.747014i	$0.731487\pi$
$504$	0	0
$505$	3156.00	0.278099
$506$	0	0
$507$	1407.00	0.123249
$508$	0	0
$509$	−9287.26	−0.808743	−0.404372	−	0.914595i	$-0.632510\pi$
$509$	−9287.26	−0.808743	−0.404372	+	0.914595i	$0.632510\pi$
$510$	0	0
$511$	24394.2	2.11181
$512$	0	0
$513$	−108.000	−0.00929496
$514$	0	0
$515$	−1572.00	−0.134506
$516$	0	0
$517$	24276.4	2.06514
$518$	0	0
$519$	−1797.87	−0.152057
$520$	0	0
$521$	2766.00	0.232592	0.116296	−	0.993215i	$-0.462898\pi$
$521$	2766.00	0.232592	0.116296	+	0.993215i	$0.462898\pi$
$522$	0	0
$523$	18988.0	1.58755	0.793774	−	0.608213i	$-0.208113\pi$
$523$	18988.0	1.58755	0.793774	+	0.608213i	$0.208113\pi$
$524$	0	0
$525$	8220.31	0.683360
$526$	0	0
$527$	−3180.05	−0.262856
$528$	0	0
$529$	17833.0	1.46569
$530$	0	0
$531$	−2268.00	−0.185354
$532$	0	0
$533$	12221.4	0.993181
$534$	0	0
$535$	4115.35	0.332565
$536$	0	0
$537$	−9684.00	−0.778204
$538$	0	0
$539$	−11760.0	−0.939776
$540$	0	0
$541$	−12997.3	−1.03290	−0.516449	−	0.856318i	$-0.672746\pi$
$541$	−12997.3	−1.03290	−0.516449	+	0.856318i	$0.672746\pi$
$542$	0	0
$543$	6069.11	0.479651
$544$	0	0
$545$	1632.00	0.128270
$546$	0	0
$547$	−21188.0	−1.65619	−0.828093	−	0.560591i	$-0.810574\pi$
$547$	−21188.0	−1.65619	−0.828093	+	0.560591i	$0.810574\pi$
$548$	0	0
$549$	−810.600	−0.0630156
$550$	0	0
$551$	651.251	0.0503525
$552$	0	0
$553$	32508.0	2.49978
$554$	0	0
$555$	3384.00	0.258816
$556$	0	0
$557$	−12231.7	−0.930477	−0.465238	−	0.885185i	$-0.654031\pi$
$557$	−12231.7	−0.930477	−0.465238	+	0.885185i	$0.654031\pi$
$558$	0	0
$559$	−7815.01	−0.591306
$560$	0	0
$561$	7776.00	0.585210
$562$	0	0
$563$	−504.000	−0.0377284	−0.0188642	−	0.999822i	$-0.506005\pi$
$563$	−504.000	−0.0377284	−0.0188642	+	0.999822i	$0.506005\pi$
$564$	0	0
$565$	−1351.00	−0.100596
$566$	0	0
$567$	1964.15	0.145479
$568$	0	0
$569$	20358.0	1.49992	0.749958	−	0.661486i	$-0.230074\pi$
$569$	20358.0	1.49992	0.749958	+	0.661486i	$0.230074\pi$
$570$	0	0
$571$	−13300.0	−0.974760	−0.487380	−	0.873190i	$-0.662047\pi$
$571$	−13300.0	−0.974760	−0.487380	+	0.873190i	$0.662047\pi$
$572$	0	0
$573$	−10433.9	−0.760700
$574$	0	0
$575$	−19572.2	−1.41951
$576$	0	0
$577$	4606.00	0.332323	0.166161	−	0.986099i	$-0.446863\pi$
$577$	4606.00	0.332323	0.166161	+	0.986099i	$0.446863\pi$
$578$	0	0
$579$	2298.00	0.164942
$580$	0	0
$581$	17459.1	1.24669
$582$	0	0
$583$	−35749.5	−2.53961
$584$	0	0
$585$	1296.00	0.0915949
$586$	0	0
$587$	13980.0	0.982992	0.491496	−	0.870880i	$-0.336450\pi$
$587$	13980.0	0.982992	0.491496	+	0.870880i	$0.336450\pi$
$588$	0	0
$589$	−235.559	−0.0164788
$590$	0	0
$591$	8698.36	0.605419
$592$	0	0
$593$	−12486.0	−0.864652	−0.432326	−	0.901717i	$-0.642307\pi$
$593$	−12486.0	−0.864652	−0.432326	+	0.901717i	$0.642307\pi$
$594$	0	0
$595$	4536.00	0.312534
$596$	0	0
$597$	−5206.54	−0.356934
$598$	0	0
$599$	8778.03	0.598766	0.299383	−	0.954133i	$-0.403219\pi$
$599$	8778.03	0.598766	0.299383	+	0.954133i	$0.403219\pi$
$600$	0	0
$601$	−6986.00	−0.474151	−0.237076	−	0.971491i	$-0.576189\pi$
$601$	−6986.00	−0.474151	−0.237076	+	0.971491i	$0.576189\pi$
$602$	0	0
$603$	−5652.00	−0.381704
$604$	0	0
$605$	3370.57	0.226501
$606$	0	0
$607$	−4596.86	−0.307382	−0.153691	−	0.988119i	$-0.549116\pi$
$607$	−4596.86	−0.307382	−0.153691	+	0.988119i	$0.549116\pi$
$608$	0	0
$609$	−11844.0	−0.788084
$610$	0	0
$611$	−21024.0	−1.39205
$612$	0	0
$613$	−11092.1	−0.730838	−0.365419	−	0.930843i	$-0.619074\pi$
$613$	−11092.1	−0.730838	−0.365419	+	0.930843i	$0.619074\pi$
$614$	0	0
$615$	−3055.34	−0.200330
$616$	0	0
$617$	2850.00	0.185959	0.0929795	−	0.995668i	$-0.470361\pi$
$617$	2850.00	0.185959	0.0929795	+	0.995668i	$0.470361\pi$
$618$	0	0
$619$	−20116.0	−1.30619	−0.653094	−	0.757277i	$-0.726529\pi$
$619$	−20116.0	−1.30619	−0.653094	+	0.757277i	$0.726529\pi$
$620$	0	0
$621$	−4676.54	−0.302195
$622$	0	0
$623$	35936.6	2.31103
$624$	0	0
$625$	11269.0	0.721216
$626$	0	0
$627$	576.000	0.0366878
$628$	0	0
$629$	−17583.8	−1.11464
$630$	0	0
$631$	7271.15	0.458732	0.229366	−	0.973340i	$-0.426335\pi$
$631$	7271.15	0.458732	0.229366	+	0.973340i	$0.426335\pi$
$632$	0	0
$633$	−3300.00	−0.207209
$634$	0	0
$635$	−2100.00	−0.131238
$636$	0	0
$637$	10184.5	0.633474
$638$	0	0
$639$	−62.3538	−0.00386022
$640$	0	0
$641$	7230.00	0.445504	0.222752	−	0.974875i	$-0.428496\pi$
$641$	7230.00	0.445504	0.222752	+	0.974875i	$0.428496\pi$
$642$	0	0
$643$	−2948.00	−0.180805	−0.0904026	−	0.995905i	$-0.528815\pi$
$643$	−2948.00	−0.180805	−0.0904026	+	0.995905i	$0.528815\pi$
$644$	0	0
$645$	1953.75	0.119270
$646$	0	0
$647$	17161.2	1.04277	0.521387	−	0.853320i	$-0.325415\pi$
$647$	17161.2	1.04277	0.521387	+	0.853320i	$0.325415\pi$
$648$	0	0
$649$	12096.0	0.731602
$650$	0	0
$651$	4284.00	0.257916
$652$	0	0
$653$	10381.9	0.622168	0.311084	−	0.950382i	$-0.399308\pi$
$653$	10381.9	0.622168	0.311084	+	0.950382i	$0.399308\pi$
$654$	0	0
$655$	4780.46	0.285173
$656$	0	0
$657$	9054.00	0.537641
$658$	0	0
$659$	10308.0	0.609321	0.304661	−	0.952461i	$-0.401457\pi$
$659$	10308.0	0.609321	0.304661	+	0.952461i	$0.401457\pi$
$660$	0	0
$661$	15803.2	0.929916	0.464958	−	0.885333i	$-0.346069\pi$
$661$	15803.2	0.929916	0.464958	+	0.885333i	$0.346069\pi$
$662$	0	0
$663$	−6734.21	−0.394472
$664$	0	0
$665$	336.000	0.0195933
$666$	0	0
$667$	28200.0	1.63704
$668$	0	0
$669$	−1174.33	−0.0678658
$670$	0	0
$671$	4323.20	0.248726
$672$	0	0
$673$	30910.0	1.77042	0.885210	−	0.465191i	$-0.154014\pi$
$673$	30910.0	1.77042	0.885210	+	0.465191i	$0.154014\pi$
$674$	0	0
$675$	3051.00	0.173975
$676$	0	0
$677$	14802.1	0.840312	0.420156	−	0.907452i	$-0.361975\pi$
$677$	14802.1	0.840312	0.420156	+	0.907452i	$0.361975\pi$
$678$	0	0
$679$	44181.2	2.49708
$680$	0	0
$681$	10008.0	0.563153
$682$	0	0
$683$	−528.000	−0.0295803	−0.0147902	−	0.999891i	$-0.504708\pi$
$683$	−528.000	−0.0295803	−0.0147902	+	0.999891i	$0.504708\pi$
$684$	0	0
$685$	4011.43	0.223750
$686$	0	0
$687$	−17999.5	−0.999596
$688$	0	0
$689$	30960.0	1.71188
$690$	0	0
$691$	9052.00	0.498342	0.249171	−	0.968460i	$-0.419842\pi$
$691$	9052.00	0.498342	0.249171	+	0.968460i	$0.419842\pi$
$692$	0	0
$693$	−10475.4	−0.574212
$694$	0	0
$695$	−4087.64	−0.223098
$696$	0	0
$697$	15876.0	0.862764
$698$	0	0
$699$	954.000	0.0516217
$700$	0	0
$701$	32600.7	1.75650	0.878252	−	0.478197i	$-0.158710\pi$
$701$	32600.7	1.75650	0.878252	+	0.478197i	$0.158710\pi$
$702$	0	0
$703$	−1302.50	−0.0698788
$704$	0	0
$705$	5256.00	0.280784
$706$	0	0
$707$	22092.0	1.17518
$708$	0	0
$709$	−27227.8	−1.44226	−0.721130	−	0.692799i	$-0.756377\pi$
$709$	−27227.8	−1.44226	−0.721130	+	0.692799i	$0.756377\pi$
$710$	0	0
$711$	12065.5	0.636414
$712$	0	0
$713$	−10200.0	−0.535755
$714$	0	0
$715$	−6912.00	−0.361530
$716$	0	0
$717$	−2577.29	−0.134241
$718$	0	0
$719$	−685.892	−0.0355764	−0.0177882	−	0.999842i	$-0.505662\pi$
$719$	−685.892	−0.0355764	−0.0177882	+	0.999842i	$0.505662\pi$
$720$	0	0
$721$	−11004.0	−0.568392
$722$	0	0
$723$	8130.00	0.418199
$724$	0	0
$725$	−18397.8	−0.942453
$726$	0	0
$727$	−20192.2	−1.03011	−0.515054	−	0.857158i	$-0.672228\pi$
$727$	−20192.2	−1.03011	−0.515054	+	0.857158i	$0.672228\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−10152.0	−0.513660
$732$	0	0
$733$	−35236.8	−1.77558	−0.887792	−	0.460246i	$-0.847761\pi$
$733$	−35236.8	−1.77558	−0.887792	+	0.460246i	$0.847761\pi$
$734$	0	0
$735$	−2546.11	−0.127775
$736$	0	0
$737$	30144.0	1.50661
$738$	0	0
$739$	13940.0	0.693899	0.346949	−	0.937884i	$-0.387218\pi$
$739$	13940.0	0.693899	0.346949	+	0.937884i	$0.387218\pi$
$740$	0	0
$741$	−498.831	−0.0247301
$742$	0	0
$743$	−11002.0	−0.543235	−0.271618	−	0.962405i	$-0.587559\pi$
$743$	−11002.0	−0.543235	−0.271618	+	0.962405i	$0.587559\pi$
$744$	0	0
$745$	−7524.00	−0.370011
$746$	0	0
$747$	6480.00	0.317391
$748$	0	0
$749$	28807.5	1.40534
$750$	0	0
$751$	−33342.0	−1.62006	−0.810031	−	0.586388i	$-0.800550\pi$
$751$	−33342.0	−1.62006	−0.810031	+	0.586388i	$0.800550\pi$
$752$	0	0
$753$	−15408.0	−0.745682
$754$	0	0
$755$	−492.000	−0.0237162
$756$	0	0
$757$	−17445.2	−0.837592	−0.418796	−	0.908080i	$-0.637548\pi$
$757$	−17445.2	−0.837592	−0.418796	+	0.908080i	$0.637548\pi$
$758$	0	0
$759$	24941.5	1.19278
$760$	0	0
$761$	30222.0	1.43961	0.719807	−	0.694174i	$-0.244230\pi$
$761$	30222.0	1.43961	0.719807	+	0.694174i	$0.244230\pi$
$762$	0	0
$763$	11424.0	0.542040
$764$	0	0
$765$	1683.55	0.0795673
$766$	0	0
$767$	−10475.4	−0.493150
$768$	0	0
$769$	−11758.0	−0.551371	−0.275686	−	0.961248i	$-0.588905\pi$
$769$	−11758.0	−0.551371	−0.275686	+	0.961248i	$0.588905\pi$
$770$	0	0
$771$	13194.0	0.616304
$772$	0	0
$773$	−1874.08	−0.0872004	−0.0436002	−	0.999049i	$-0.513883\pi$
$773$	−1874.08	−0.0872004	−0.0436002	+	0.999049i	$0.513883\pi$
$774$	0	0
$775$	6654.54	0.308436
$776$	0	0
$777$	23688.0	1.09370
$778$	0	0
$779$	1176.00	0.0540880
$780$	0	0
$781$	332.554	0.0152365
$782$	0	0
$783$	−4395.94	−0.200636
$784$	0	0
$785$	4632.00	0.210603
$786$	0	0
$787$	31012.0	1.40465	0.702324	−	0.711857i	$-0.252146\pi$
$787$	31012.0	1.40465	0.702324	+	0.711857i	$0.252146\pi$
$788$	0	0
$789$	20452.1	0.922829
$790$	0	0
$791$	−9457.00	−0.425097
$792$	0	0
$793$	−3744.00	−0.167659
$794$	0	0
$795$	−7740.00	−0.345295
$796$	0	0
$797$	−7091.02	−0.315153	−0.157576	−	0.987507i	$-0.550368\pi$
$797$	−7091.02	−0.315153	−0.157576	+	0.987507i	$0.550368\pi$
$798$	0	0
$799$	−27311.0	−1.20925
$800$	0	0
$801$	13338.0	0.588358
$802$	0	0
$803$	−48288.0	−2.12210
$804$	0	0
$805$	14549.2	0.637010
$806$	0	0
$807$	−13873.7	−0.605178
$808$	0	0
$809$	−40650.0	−1.76660	−0.883299	−	0.468810i	$-0.844683\pi$
$809$	−40650.0	−1.76660	−0.883299	+	0.468810i	$0.844683\pi$
$810$	0	0
$811$	−8372.00	−0.362492	−0.181246	−	0.983438i	$-0.558013\pi$
$811$	−8372.00	−0.362492	−0.181246	+	0.983438i	$0.558013\pi$
$812$	0	0
$813$	−11649.8	−0.502553
$814$	0	0
$815$	−6055.25	−0.260253
$816$	0	0
$817$	−752.000	−0.0322021
$818$	0	0
$819$	9072.00	0.387059
$820$	0	0
$821$	−9370.39	−0.398330	−0.199165	−	0.979966i	$-0.563823\pi$
$821$	−9370.39	−0.398330	−0.199165	+	0.979966i	$0.563823\pi$
$822$	0	0
$823$	21668.0	0.917737	0.458868	−	0.888504i	$-0.348255\pi$
$823$	21668.0	0.917737	0.458868	+	0.888504i	$0.348255\pi$
$824$	0	0
$825$	−16272.0	−0.686689
$826$	0	0
$827$	−6684.00	−0.281046	−0.140523	−	0.990077i	$-0.544878\pi$
$827$	−6684.00	−0.281046	−0.140523	+	0.990077i	$0.544878\pi$
$828$	0	0
$829$	24359.6	1.02056	0.510279	−	0.860009i	$-0.329542\pi$
$829$	24359.6	1.02056	0.510279	+	0.860009i	$0.329542\pi$
$830$	0	0
$831$	4572.61	0.190881
$832$	0	0
$833$	13230.0	0.550291
$834$	0	0
$835$	48.0000	0.00198935
$836$	0	0
$837$	1590.02	0.0656622
$838$	0	0
$839$	−19613.7	−0.807082	−0.403541	−	0.914962i	$-0.632221\pi$
$839$	−19613.7	−0.807082	−0.403541	+	0.914962i	$0.632221\pi$
$840$	0	0
$841$	2119.00	0.0868834
$842$	0	0
$843$	13194.0	0.539058
$844$	0	0
$845$	−1624.66	−0.0661422
$846$	0	0
$847$	23594.0	0.957142
$848$	0	0
$849$	13116.0	0.530200
$850$	0	0
$851$	−56400.0	−2.27188
$852$	0	0
$853$	20569.8	0.825671	0.412836	−	0.910806i	$-0.364538\pi$
$853$	20569.8	0.825671	0.412836	+	0.910806i	$0.364538\pi$
$854$	0	0
$855$	124.708	0.00498820
$856$	0	0
$857$	27222.0	1.08505	0.542524	−	0.840040i	$-0.317469\pi$
$857$	27222.0	1.08505	0.542524	+	0.840040i	$0.317469\pi$
$858$	0	0
$859$	−3548.00	−0.140927	−0.0704634	−	0.997514i	$-0.522448\pi$
$859$	−3548.00	−0.140927	−0.0704634	+	0.997514i	$0.522448\pi$
$860$	0	0
$861$	−21387.4	−0.846550
$862$	0	0
$863$	45047.2	1.77685	0.888426	−	0.459019i	$-0.151799\pi$
$863$	45047.2	1.77685	0.888426	+	0.459019i	$0.151799\pi$
$864$	0	0
$865$	2076.00	0.0816024
$866$	0	0
$867$	5991.00	0.234677
$868$	0	0
$869$	−64349.2	−2.51196
$870$	0	0
$871$	−26105.5	−1.01556
$872$	0	0
$873$	16398.0	0.635725
$874$	0	0
$875$	−19992.0	−0.772403
$876$	0	0
$877$	−29022.2	−1.11746	−0.558729	−	0.829350i	$-0.688711\pi$
$877$	−29022.2	−1.11746	−0.558729	+	0.829350i	$0.688711\pi$
$878$	0	0
$879$	−10714.5	−0.411138
$880$	0	0
$881$	−48318.0	−1.84776	−0.923879	−	0.382685i	$-0.875000\pi$
$881$	−48318.0	−1.84776	−0.923879	+	0.382685i	$0.875000\pi$
$882$	0	0
$883$	14380.0	0.548047	0.274024	−	0.961723i	$-0.411645\pi$
$883$	14380.0	0.548047	0.274024	+	0.961723i	$0.411645\pi$
$884$	0	0
$885$	2618.86	0.0994712
$886$	0	0
$887$	−34086.8	−1.29033	−0.645164	−	0.764044i	$-0.723211\pi$
$887$	−34086.8	−1.29033	−0.645164	+	0.764044i	$0.723211\pi$
$888$	0	0
$889$	−14700.0	−0.554581
$890$	0	0
$891$	−3888.00	−0.146187
$892$	0	0
$893$	−2023.04	−0.0758100
$894$	0	0
$895$	11182.1	0.417628
$896$	0	0
$897$	−21600.0	−0.804017
$898$	0	0
$899$	−9588.00	−0.355704
$900$	0	0
$901$	40218.2	1.48708
$902$	0	0
$903$	13676.3	0.504007
$904$	0	0
$905$	−7008.00	−0.257408
$906$	0	0
$907$	31252.0	1.14411	0.572054	−	0.820216i	$-0.306147\pi$
$907$	31252.0	1.14411	0.572054	+	0.820216i	$0.306147\pi$
$908$	0	0
$909$	8199.53	0.299187
$910$	0	0
$911$	−13080.4	−0.475713	−0.237857	−	0.971300i	$-0.576445\pi$
$911$	−13080.4	−0.475713	−0.237857	+	0.971300i	$0.576445\pi$
$912$	0	0
$913$	−34560.0	−1.25276
$914$	0	0
$915$	936.000	0.0338177
$916$	0	0
$917$	33463.2	1.20507
$918$	0	0
$919$	−8843.85	−0.317445	−0.158722	−	0.987323i	$-0.550737\pi$
$919$	−8843.85	−0.317445	−0.158722	+	0.987323i	$0.550737\pi$
$920$	0	0
$921$	−12516.0	−0.447792
$922$	0	0
$923$	−288.000	−0.0102705
$924$	0	0
$925$	36795.7	1.30793
$926$	0	0
$927$	−4084.18	−0.144705
$928$	0	0
$929$	29622.0	1.04614	0.523071	−	0.852289i	$-0.324786\pi$
$929$	29622.0	1.04614	0.523071	+	0.852289i	$0.324786\pi$
$930$	0	0
$931$	980.000	0.0344986
$932$	0	0
$933$	19412.8	0.681187
$934$	0	0
$935$	−8978.95	−0.314057
$936$	0	0
$937$	23210.0	0.809218	0.404609	−	0.914490i	$-0.367408\pi$
$937$	23210.0	0.809218	0.404609	+	0.914490i	$0.367408\pi$
$938$	0	0
$939$	−222.000	−0.00771533
$940$	0	0
$941$	−19728.1	−0.683439	−0.341720	−	0.939802i	$-0.611009\pi$
$941$	−19728.1	−0.683439	−0.341720	+	0.939802i	$0.611009\pi$
$942$	0	0
$943$	50922.3	1.75849
$944$	0	0
$945$	−2268.00	−0.0780720
$946$	0	0
$947$	−1236.00	−0.0424125	−0.0212062	−	0.999775i	$-0.506751\pi$
$947$	−1236.00	−0.0424125	−0.0212062	+	0.999775i	$0.506751\pi$
$948$	0	0
$949$	41818.6	1.43044
$950$	0	0
$951$	−5892.44	−0.200920
$952$	0	0
$953$	−18402.0	−0.625498	−0.312749	−	0.949836i	$-0.601250\pi$
$953$	−18402.0	−0.625498	−0.312749	+	0.949836i	$0.601250\pi$
$954$	0	0
$955$	12048.0	0.408235
$956$	0	0
$957$	23445.0	0.791923
$958$	0	0
$959$	28080.0	0.945517
$960$	0	0
$961$	−26323.0	−0.883589
$962$	0	0
$963$	10692.0	0.357783
$964$	0	0
$965$	−2653.50	−0.0885174
$966$	0	0
$967$	41836.0	1.39127	0.695633	−	0.718398i	$-0.255124\pi$
$967$	41836.0	1.39127	0.695633	+	0.718398i	$0.255124\pi$
$968$	0	0
$969$	−648.000	−0.0214827
$970$	0	0
$971$	8832.00	0.291897	0.145949	−	0.989292i	$-0.453377\pi$
$971$	8832.00	0.291897	0.145949	+	0.989292i	$0.453377\pi$
$972$	0	0
$973$	−28613.5	−0.942761
$974$	0	0
$975$	14092.0	0.462876
$976$	0	0
$977$	−23034.0	−0.754271	−0.377136	−	0.926158i	$-0.623091\pi$
$977$	−23034.0	−0.754271	−0.377136	+	0.926158i	$0.623091\pi$
$978$	0	0
$979$	−71136.0	−2.32228
$980$	0	0
$981$	4240.06	0.137997
$982$	0	0
$983$	17791.6	0.577278	0.288639	−	0.957438i	$-0.406797\pi$
$983$	17791.6	0.577278	0.288639	+	0.957438i	$0.406797\pi$
$984$	0	0
$985$	−10044.0	−0.324902
$986$	0	0
$987$	36792.0	1.18653
$988$	0	0
$989$	−32562.6	−1.04695
$990$	0	0
$991$	3072.66	0.0984926	0.0492463	−	0.998787i	$-0.484318\pi$
$991$	3072.66	0.0984926	0.0492463	+	0.998787i	$0.484318\pi$
$992$	0	0
$993$	−22668.0	−0.724418
$994$	0	0
$995$	6012.00	0.191551
$996$	0	0
$997$	−52273.3	−1.66049	−0.830247	−	0.557396i	$-0.811800\pi$
$997$	−52273.3	−1.66049	−0.830247	+	0.557396i	$0.811800\pi$
$998$	0	0
$999$	8791.89	0.278442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.a.f.1.2		2
3.2	odd	2		2304.4.a.bk.1.1		2
4.3	odd	2		768.4.a.o.1.2		2
8.3	odd	2	inner	768.4.a.f.1.1		2
8.5	even	2		768.4.a.o.1.1		2
12.11	even	2		2304.4.a.x.1.1		2
16.3	odd	4		192.4.d.c.97.3	yes	4
16.5	even	4		192.4.d.c.97.4	yes	4
16.11	odd	4		192.4.d.c.97.2	yes	4
16.13	even	4		192.4.d.c.97.1	✓	4
24.5	odd	2		2304.4.a.x.1.2		2
24.11	even	2		2304.4.a.bk.1.2		2
48.5	odd	4		576.4.d.c.289.1		4
48.11	even	4		576.4.d.c.289.2		4
48.29	odd	4		576.4.d.c.289.3		4
48.35	even	4		576.4.d.c.289.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.4.d.c.97.1	✓	4	16.13	even	4
192.4.d.c.97.2	yes	4	16.11	odd	4
192.4.d.c.97.3	yes	4	16.3	odd	4
192.4.d.c.97.4	yes	4	16.5	even	4
576.4.d.c.289.1		4	48.5	odd	4
576.4.d.c.289.2		4	48.11	even	4
576.4.d.c.289.3		4	48.29	odd	4
576.4.d.c.289.4		4	48.35	even	4
768.4.a.f.1.1		2	8.3	odd	2	inner
768.4.a.f.1.2		2	1.1	even	1	trivial
768.4.a.o.1.1		2	8.5	even	2
768.4.a.o.1.2		2	4.3	odd	2
2304.4.a.x.1.1		2	12.11	even	2
2304.4.a.x.1.2		2	24.5	odd	2
2304.4.a.bk.1.1		2	3.2	odd	2
2304.4.a.bk.1.2		2	24.11	even	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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +3.46410 q^{5} +24.2487 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +3.46410 q^{5} +24.2487 q^{7} +9.00000 q^{9} -48.0000 q^{11} +41.5692 q^{13} -10.3923 q^{15} +54.0000 q^{17} +4.00000 q^{19} -72.7461 q^{21} +173.205 q^{23} -113.000 q^{25} -27.0000 q^{27} +162.813 q^{29} -58.8897 q^{31} +144.000 q^{33} +84.0000 q^{35} -325.626 q^{37} -124.708 q^{39} +294.000 q^{41} -188.000 q^{43} +31.1769 q^{45} -505.759 q^{47} +245.000 q^{49} -162.000 q^{51} +744.782 q^{53} -166.277 q^{55} -12.0000 q^{57} -252.000 q^{59} -90.0666 q^{61} +218.238 q^{63} +144.000 q^{65} -628.000 q^{67} -519.615 q^{69} -6.92820 q^{71} +1006.00 q^{73} +339.000 q^{75} -1163.94 q^{77} +1340.61 q^{79} +81.0000 q^{81} +720.000 q^{83} +187.061 q^{85} -488.438 q^{87} +1482.00 q^{89} +1008.00 q^{91} +176.669 q^{93} +13.8564 q^{95} +1822.00 q^{97} -432.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 18 * q^9	\( 2 q - 6 q^{3} + 18 q^{9} - 96 q^{11} + 108 q^{17} + 8 q^{19} - 226 q^{25} - 54 q^{27} + 288 q^{33} + 168 q^{35} + 588 q^{41} - 376 q^{43} + 490 q^{49} - 324 q^{51} - 24 q^{57} - 504 q^{59} + 288 q^{65} - 1256 q^{67} + 2012 q^{73} + 678 q^{75} + 162 q^{81} + 1440 q^{83} + 2964 q^{89} + 2016 q^{91} + 3644 q^{97} - 864 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 18 * q^9 - 96 * q^11 + 108 * q^17 + 8 * q^19 - 226 * q^25 - 54 * q^27 + 288 * q^33 + 168 * q^35 + 588 * q^41 - 376 * q^43 + 490 * q^49 - 324 * q^51 - 24 * q^57 - 504 * q^59 + 288 * q^65 - 1256 * q^67 + 2012 * q^73 + 678 * q^75 + 162 * q^81 + 1440 * q^83 + 2964 * q^89 + 2016 * q^91 + 3644 * q^97 - 864 * q^99

Introduction

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