LMFDB - Embedded newform 448.3.d.e.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,3,Mod(127,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("448.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.d (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:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1997017344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 56*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$-1.92812i$ of defining polynomial
Character	$\chi$	$=$	448.127
Dual form			448.3.d.e.127.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.85623i q^{3} -0.490168 q^{5} +2.64575i q^{7} -5.87054 q^{9} +O(q^{10})$	$q-3.85623i q^{3} -0.490168 q^{5} +2.64575i q^{7} -5.87054 q^{9} -15.5633i q^{11} -3.50983 q^{13} +1.89020i q^{15} -24.1463 q^{17} -3.56870i q^{19} +10.2026 q^{21} -19.5741i q^{23} -24.7597 q^{25} -12.0679i q^{27} +10.9803 q^{29} +21.1767i q^{31} -60.0159 q^{33} -1.29686i q^{35} -58.4212 q^{37} +13.5347i q^{39} +54.1285 q^{41} +35.6420i q^{43} +2.87755 q^{45} +64.2248i q^{47} -7.00000 q^{49} +93.1140i q^{51} -87.4015 q^{53} +7.62865i q^{55} -13.7617 q^{57} -66.6954i q^{59} -16.8615 q^{61} -15.5320i q^{63} +1.72041 q^{65} -21.2420i q^{67} -75.4822 q^{69} -64.2140i q^{71} +99.4587 q^{73} +95.4793i q^{75} +41.1767 q^{77} -139.441i q^{79} -99.3716 q^{81} -6.03134i q^{83} +11.8358 q^{85} -42.3427i q^{87} -23.9821 q^{89} -9.28614i q^{91} +81.6624 q^{93} +1.74926i q^{95} +171.509 q^{97} +91.3652i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 40 q^{9}+O(q^{10}) $ 8 * q - 40 * q^9	$ 8 q - 40 q^{9} - 32 q^{13} - 16 q^{17} + 104 q^{25} + 80 q^{29} + 176 q^{37} + 144 q^{41} - 256 q^{45} - 56 q^{49} - 48 q^{53} - 400 q^{57} + 192 q^{61} - 304 q^{65} - 576 q^{69} + 272 q^{73} + 112 q^{77} + 504 q^{81} + 160 q^{85} - 80 q^{89} + 608 q^{93} + 528 q^{97}+O(q^{100}) $ 8 * q - 40 * q^9 - 32 * q^13 - 16 * q^17 + 104 * q^25 + 80 * q^29 + 176 * q^37 + 144 * q^41 - 256 * q^45 - 56 * q^49 - 48 * q^53 - 400 * q^57 + 192 * q^61 - 304 * q^65 - 576 * q^69 + 272 * q^73 + 112 * q^77 + 504 * q^81 + 160 * q^85 - 80 * q^89 + 608 * q^93 + 528 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 3.85623i	− 1.28541i	−0.766113	−	0.642706i	$-0.777812\pi$
$3$	− 3.85623i	− 1.28541i	0.766113	−	0.642706i	$-0.222188\pi$
$4$	0	0
$5$	−0.490168	−0.0980336	−0.0490168	−	0.998798i	$-0.515609\pi$
$5$	−0.490168	−0.0980336	−0.0490168	+	0.998798i	$0.515609\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−5.87054	−0.652282
$10$	0	0
$11$	− 15.5633i	− 1.41485i	−0.706789	−	0.707425i	$-0.749857\pi$
$11$	− 15.5633i	− 1.41485i	0.706789	−	0.707425i	$-0.250143\pi$
$12$	0	0
$13$	−3.50983	−0.269987	−0.134994	−	0.990846i	$-0.543101\pi$
$13$	−3.50983	−0.269987	−0.134994	+	0.990846i	$0.543101\pi$
$14$	0	0
$15$	1.89020i	0.126013i
$16$	0	0
$17$	−24.1463	−1.42037	−0.710187	−	0.704013i	$-0.751389\pi$
$17$	−24.1463	−1.42037	−0.710187	+	0.704013i	$0.751389\pi$
$18$	0	0
$19$	− 3.56870i	− 0.187826i	−0.995580	−	0.0939132i	$-0.970062\pi$
$19$	− 3.56870i	− 0.187826i	0.995580	−	0.0939132i	$-0.0299376\pi$
$20$	0	0
$21$	10.2026	0.485840
$22$	0	0
$23$	− 19.5741i	− 0.851046i	−0.904948	−	0.425523i	$-0.860090\pi$
$23$	− 19.5741i	− 0.851046i	0.904948	−	0.425523i	$-0.139910\pi$
$24$	0	0
$25$	−24.7597	−0.990389
$26$	0	0
$27$	− 12.0679i	− 0.446961i
$28$	0	0
$29$	10.9803	0.378632	0.189316	−	0.981916i	$-0.439373\pi$
$29$	10.9803	0.378632	0.189316	+	0.981916i	$0.439373\pi$
$30$	0	0
$31$	21.1767i	0.683120i	0.939860	+	0.341560i	$0.110955\pi$
$31$	21.1767i	0.683120i	−0.939860	+	0.341560i	$0.889045\pi$
$32$	0	0
$33$	−60.0159	−1.81866
$34$	0	0
$35$	− 1.29686i	− 0.0370532i
$36$	0	0
$37$	−58.4212	−1.57895	−0.789475	−	0.613783i	$-0.789647\pi$
$37$	−58.4212	−1.57895	−0.789475	+	0.613783i	$0.789647\pi$
$38$	0	0
$39$	13.5347i	0.347044i
$40$	0	0
$41$	54.1285	1.32021	0.660103	−	0.751175i	$-0.270513\pi$
$41$	54.1285	1.32021	0.660103	+	0.751175i	$0.270513\pi$
$42$	0	0
$43$	35.6420i	0.828884i	0.910076	+	0.414442i	$0.136023\pi$
$43$	35.6420i	0.828884i	−0.910076	+	0.414442i	$0.863977\pi$
$44$	0	0
$45$	2.87755	0.0639455
$46$	0	0
$47$	64.2248i	1.36648i	0.730192	+	0.683242i	$0.239431\pi$
$47$	64.2248i	1.36648i	−0.730192	+	0.683242i	$0.760569\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	93.1140i	1.82576i
$52$	0	0
$53$	−87.4015	−1.64908	−0.824542	−	0.565800i	$-0.808567\pi$
$53$	−87.4015	−1.64908	−0.824542	+	0.565800i	$0.808567\pi$
$54$	0	0
$55$	7.62865i	0.138703i
$56$	0	0
$57$	−13.7617	−0.241434
$58$	0	0
$59$	− 66.6954i	− 1.13043i	−0.824944	−	0.565215i	$-0.808793\pi$
$59$	− 66.6954i	− 1.13043i	0.824944	−	0.565215i	$-0.191207\pi$
$60$	0	0
$61$	−16.8615	−0.276418	−0.138209	−	0.990403i	$-0.544135\pi$
$61$	−16.8615	−0.276418	−0.138209	+	0.990403i	$0.544135\pi$
$62$	0	0
$63$	− 15.5320i	− 0.246539i
$64$	0	0
$65$	1.72041	0.0264678
$66$	0	0
$67$	− 21.2420i	− 0.317045i	−0.987355	−	0.158523i	$-0.949327\pi$
$67$	− 21.2420i	− 0.317045i	0.987355	−	0.158523i	$-0.0506731\pi$
$68$	0	0
$69$	−75.4822	−1.09394
$70$	0	0
$71$	− 64.2140i	− 0.904423i	−0.891911	−	0.452212i	$-0.850635\pi$
$71$	− 64.2140i	− 0.904423i	0.891911	−	0.452212i	$-0.149365\pi$
$72$	0	0
$73$	99.4587	1.36245	0.681224	−	0.732075i	$-0.261448\pi$
$73$	99.4587	1.36245	0.681224	+	0.732075i	$0.261448\pi$
$74$	0	0
$75$	95.4793i	1.27306i
$76$	0	0
$77$	41.1767	0.534763
$78$	0	0
$79$	− 139.441i	− 1.76507i	−0.470243	−	0.882537i	$-0.655834\pi$
$79$	− 139.441i	− 1.76507i	0.470243	−	0.882537i	$-0.344166\pi$
$80$	0	0
$81$	−99.3716	−1.22681
$82$	0	0
$83$	− 6.03134i	− 0.0726668i	−0.999340	−	0.0363334i	$-0.988432\pi$
$83$	− 6.03134i	− 0.0726668i	0.999340	−	0.0363334i	$-0.0115678\pi$
$84$	0	0
$85$	11.8358	0.139244
$86$	0	0
$87$	− 42.3427i	− 0.486698i
$88$	0	0
$89$	−23.9821	−0.269462	−0.134731	−	0.990882i	$-0.543017\pi$
$89$	−23.9821	−0.269462	−0.134731	+	0.990882i	$0.543017\pi$
$90$	0	0
$91$	− 9.28614i	− 0.102046i
$92$	0	0
$93$	81.6624	0.878091
$94$	0	0
$95$	1.74926i	0.0184133i
$96$	0	0
$97$	171.509	1.76813	0.884064	−	0.467365i	$-0.154797\pi$
$97$	171.509	1.76813	0.884064	+	0.467365i	$0.154797\pi$
$98$	0	0
$99$	91.3652i	0.922881i
$100$	0	0
$101$	146.427	1.44977	0.724887	−	0.688867i	$-0.241892\pi$
$101$	146.427	1.44977	0.724887	+	0.688867i	$0.241892\pi$
$102$	0	0
$103$	− 118.849i	− 1.15388i	−0.816787	−	0.576939i	$-0.804247\pi$
$103$	− 118.849i	− 1.15388i	0.816787	−	0.576939i	$-0.195753\pi$
$104$	0	0
$105$	−5.00100	−0.0476286
$106$	0	0
$107$	− 142.434i	− 1.33116i	−0.746326	−	0.665581i	$-0.768184\pi$
$107$	− 142.434i	− 1.33116i	0.746326	−	0.665581i	$-0.231816\pi$
$108$	0	0
$109$	−170.835	−1.56730	−0.783649	−	0.621204i	$-0.786644\pi$
$109$	−170.835	−1.56730	−0.783649	+	0.621204i	$0.786644\pi$
$110$	0	0
$111$	225.286i	2.02960i
$112$	0	0
$113$	24.9436	0.220740	0.110370	−	0.993891i	$-0.464796\pi$
$113$	24.9436	0.220740	0.110370	+	0.993891i	$0.464796\pi$
$114$	0	0
$115$	9.59458i	0.0834311i
$116$	0	0
$117$	20.6046	0.176108
$118$	0	0
$119$	− 63.8852i	− 0.536851i
$120$	0	0
$121$	−121.218	−1.00180
$122$	0	0
$123$	− 208.732i	− 1.69701i
$124$	0	0
$125$	24.3906	0.195125
$126$	0	0
$127$	68.8755i	0.542326i	0.962533	+	0.271163i	$0.0874083\pi$
$127$	68.8755i	0.542326i	−0.962533	+	0.271163i	$0.912592\pi$
$128$	0	0
$129$	137.444	1.06546
$130$	0	0
$131$	101.326i	0.773481i	0.922189	+	0.386741i	$0.126399\pi$
$131$	101.326i	0.773481i	−0.922189	+	0.386741i	$0.873601\pi$
$132$	0	0
$133$	9.44190	0.0709917
$134$	0	0
$135$	5.91532i	0.0438172i
$136$	0	0
$137$	63.8820	0.466292	0.233146	−	0.972442i	$-0.425098\pi$
$137$	63.8820	0.466292	0.233146	+	0.972442i	$0.425098\pi$
$138$	0	0
$139$	− 199.256i	− 1.43349i	−0.697333	−	0.716747i	$-0.745630\pi$
$139$	− 199.256i	− 1.43349i	0.697333	−	0.716747i	$-0.254370\pi$
$140$	0	0
$141$	247.666	1.75649
$142$	0	0
$143$	54.6247i	0.381991i
$144$	0	0
$145$	−5.38221	−0.0371187
$146$	0	0
$147$	26.9936i	0.183630i
$148$	0	0
$149$	271.585	1.82272	0.911359	−	0.411612i	$-0.135034\pi$
$149$	271.585	1.82272	0.911359	+	0.411612i	$0.135034\pi$
$150$	0	0
$151$	− 131.329i	− 0.869729i	−0.900496	−	0.434864i	$-0.856796\pi$
$151$	− 131.329i	− 0.869729i	0.900496	−	0.434864i	$-0.143204\pi$
$152$	0	0
$153$	141.752	0.926484
$154$	0	0
$155$	− 10.3802i	− 0.0669687i
$156$	0	0
$157$	−133.685	−0.851494	−0.425747	−	0.904842i	$-0.639989\pi$
$157$	−133.685	−0.851494	−0.425747	+	0.904842i	$0.639989\pi$
$158$	0	0
$159$	337.041i	2.11975i
$160$	0	0
$161$	51.7881	0.321665
$162$	0	0
$163$	29.8167i	0.182925i	0.995809	+	0.0914623i	$0.0291541\pi$
$163$	29.8167i	0.182925i	−0.995809	+	0.0914623i	$0.970846\pi$
$164$	0	0
$165$	29.4179	0.178290
$166$	0	0
$167$	259.708i	1.55514i	0.628796	+	0.777570i	$0.283548\pi$
$167$	259.708i	1.55514i	−0.628796	+	0.777570i	$0.716452\pi$
$168$	0	0
$169$	−156.681	−0.927107
$170$	0	0
$171$	20.9502i	0.122516i
$172$	0	0
$173$	84.9760	0.491190	0.245595	−	0.969372i	$-0.421017\pi$
$173$	84.9760	0.491190	0.245595	+	0.969372i	$0.421017\pi$
$174$	0	0
$175$	− 65.5081i	− 0.374332i
$176$	0	0
$177$	−257.193	−1.45307
$178$	0	0
$179$	19.3648i	0.108183i	0.998536	+	0.0540916i	$0.0172263\pi$
$179$	19.3648i	0.108183i	−0.998536	+	0.0540916i	$0.982774\pi$
$180$	0	0
$181$	183.350	1.01298	0.506491	−	0.862245i	$-0.330942\pi$
$181$	183.350	1.01298	0.506491	+	0.862245i	$0.330942\pi$
$182$	0	0
$183$	65.0219i	0.355311i
$184$	0	0
$185$	28.6362	0.154790
$186$	0	0
$187$	375.798i	2.00961i
$188$	0	0
$189$	31.9288	0.168935
$190$	0	0
$191$	− 93.1822i	− 0.487865i	−0.969792	−	0.243932i	$-0.921562\pi$
$191$	− 93.1822i	− 0.487865i	0.969792	−	0.243932i	$-0.0784375\pi$
$192$	0	0
$193$	276.855	1.43448	0.717242	−	0.696824i	$-0.245404\pi$
$193$	276.855	1.43448	0.717242	+	0.696824i	$0.245404\pi$
$194$	0	0
$195$	− 6.63429i	− 0.0340220i
$196$	0	0
$197$	177.712	0.902089	0.451045	−	0.892501i	$-0.351052\pi$
$197$	177.712	0.902089	0.451045	+	0.892501i	$0.351052\pi$
$198$	0	0
$199$	− 227.421i	− 1.14282i	−0.820666	−	0.571408i	$-0.806397\pi$
$199$	− 227.421i	− 1.14282i	0.820666	−	0.571408i	$-0.193603\pi$
$200$	0	0
$201$	−81.9143	−0.407534
$202$	0	0
$203$	29.0512i	0.143110i
$204$	0	0
$205$	−26.5320	−0.129425
$206$	0	0
$207$	114.910i	0.555122i
$208$	0	0
$209$	−55.5409	−0.265746
$210$	0	0
$211$	325.518i	1.54274i	0.636387	+	0.771370i	$0.280428\pi$
$211$	325.518i	1.54274i	−0.636387	+	0.771370i	$0.719572\pi$
$212$	0	0
$213$	−247.624	−1.16256
$214$	0	0
$215$	− 17.4706i	− 0.0812584i
$216$	0	0
$217$	−56.0284	−0.258195
$218$	0	0
$219$	− 383.536i	− 1.75131i
$220$	0	0
$221$	84.7496	0.383482
$222$	0	0
$223$	− 106.001i	− 0.475342i	−0.971346	−	0.237671i	$-0.923616\pi$
$223$	− 106.001i	− 0.475342i	0.971346	−	0.237671i	$-0.0763840\pi$
$224$	0	0
$225$	145.353	0.646013
$226$	0	0
$227$	− 282.765i	− 1.24566i	−0.782357	−	0.622830i	$-0.785983\pi$
$227$	− 282.765i	− 1.24566i	0.782357	−	0.622830i	$-0.214017\pi$
$228$	0	0
$229$	−144.716	−0.631949	−0.315975	−	0.948768i	$-0.602331\pi$
$229$	−144.716	−0.631949	−0.315975	+	0.948768i	$0.602331\pi$
$230$	0	0
$231$	− 158.787i	− 0.687390i
$232$	0	0
$233$	36.9174	0.158444	0.0792219	−	0.996857i	$-0.474756\pi$
$233$	36.9174	0.158444	0.0792219	+	0.996857i	$0.474756\pi$
$234$	0	0
$235$	− 31.4809i	− 0.133961i
$236$	0	0
$237$	−537.716	−2.26885
$238$	0	0
$239$	65.5138i	0.274116i	0.990563	+	0.137058i	$0.0437647\pi$
$239$	65.5138i	0.274116i	−0.990563	+	0.137058i	$0.956235\pi$
$240$	0	0
$241$	−155.844	−0.646656	−0.323328	−	0.946287i	$-0.604802\pi$
$241$	−155.844	−0.646656	−0.323328	+	0.946287i	$0.604802\pi$
$242$	0	0
$243$	274.589i	1.12999i
$244$	0	0
$245$	3.43118	0.0140048
$246$	0	0
$247$	12.5255i	0.0507107i
$248$	0	0
$249$	−23.2583	−0.0934067
$250$	0	0
$251$	− 230.946i	− 0.920105i	−0.887892	−	0.460052i	$-0.847831\pi$
$251$	− 230.946i	− 0.920105i	0.887892	−	0.460052i	$-0.152169\pi$
$252$	0	0
$253$	−304.638	−1.20410
$254$	0	0
$255$	− 45.6415i	− 0.178986i
$256$	0	0
$257$	337.195	1.31204	0.656022	−	0.754742i	$-0.272238\pi$
$257$	337.195	1.31204	0.656022	+	0.754742i	$0.272238\pi$
$258$	0	0
$259$	− 154.568i	− 0.596787i
$260$	0	0
$261$	−64.4605	−0.246975
$262$	0	0
$263$	− 395.803i	− 1.50495i	−0.658619	−	0.752476i	$-0.728859\pi$
$263$	− 395.803i	− 1.50495i	0.658619	−	0.752476i	$-0.271141\pi$
$264$	0	0
$265$	42.8414	0.161666
$266$	0	0
$267$	92.4806i	0.346369i
$268$	0	0
$269$	−31.6881	−0.117800	−0.0588998	−	0.998264i	$-0.518759\pi$
$269$	−31.6881	−0.117800	−0.0588998	+	0.998264i	$0.518759\pi$
$270$	0	0
$271$	69.0163i	0.254673i	0.991860	+	0.127336i	$0.0406428\pi$
$271$	69.0163i	0.254673i	−0.991860	+	0.127336i	$0.959357\pi$
$272$	0	0
$273$	−35.8095	−0.131170
$274$	0	0
$275$	385.344i	1.40125i
$276$	0	0
$277$	−54.5783	−0.197034	−0.0985168	−	0.995135i	$-0.531410\pi$
$277$	−54.5783	−0.197034	−0.0985168	+	0.995135i	$0.531410\pi$
$278$	0	0
$279$	− 124.319i	− 0.445587i
$280$	0	0
$281$	−140.453	−0.499831	−0.249916	−	0.968268i	$-0.580403\pi$
$281$	−140.453	−0.499831	−0.249916	+	0.968268i	$0.580403\pi$
$282$	0	0
$283$	35.7205i	0.126221i	0.998007	+	0.0631105i	$0.0201021\pi$
$283$	35.7205i	0.126221i	−0.998007	+	0.0631105i	$0.979898\pi$
$284$	0	0
$285$	6.74557	0.0236687
$286$	0	0
$287$	143.210i	0.498991i
$288$	0	0
$289$	294.046	1.01746
$290$	0	0
$291$	− 661.377i	− 2.27277i
$292$	0	0
$293$	−73.9179	−0.252279	−0.126140	−	0.992012i	$-0.540259\pi$
$293$	−73.9179	−0.252279	−0.126140	+	0.992012i	$0.540259\pi$
$294$	0	0
$295$	32.6919i	0.110820i
$296$	0	0
$297$	−187.817	−0.632382
$298$	0	0
$299$	68.7017i	0.229771i
$300$	0	0
$301$	−94.2999	−0.313289
$302$	0	0
$303$	− 564.658i	− 1.86356i
$304$	0	0
$305$	8.26498	0.0270983
$306$	0	0
$307$	507.360i	1.65264i	0.563202	+	0.826319i	$0.309569\pi$
$307$	507.360i	1.65264i	−0.563202	+	0.826319i	$0.690431\pi$
$308$	0	0
$309$	−458.311	−1.48321
$310$	0	0
$311$	− 242.948i	− 0.781185i	−0.920564	−	0.390592i	$-0.872270\pi$
$311$	− 242.948i	− 0.781185i	0.920564	−	0.390592i	$-0.127730\pi$
$312$	0	0
$313$	−208.239	−0.665301	−0.332651	−	0.943050i	$-0.607943\pi$
$313$	−208.239	−0.665301	−0.332651	+	0.943050i	$0.607943\pi$
$314$	0	0
$315$	7.61328i	0.0241691i
$316$	0	0
$317$	10.6307	0.0335355	0.0167677	−	0.999859i	$-0.494662\pi$
$317$	10.6307	0.0335355	0.0167677	+	0.999859i	$0.494662\pi$
$318$	0	0
$319$	− 170.891i	− 0.535708i
$320$	0	0
$321$	−549.260	−1.71109
$322$	0	0
$323$	86.1711i	0.266784i
$324$	0	0
$325$	86.9025	0.267392
$326$	0	0
$327$	658.781i	2.01462i
$328$	0	0
$329$	−169.923	−0.516482
$330$	0	0
$331$	− 122.813i	− 0.371035i	−0.982641	−	0.185517i	$-0.940604\pi$
$331$	− 122.813i	− 0.371035i	0.982641	−	0.185517i	$-0.0593961\pi$
$332$	0	0
$333$	342.964	1.02992
$334$	0	0
$335$	10.4122i	0.0310811i
$336$	0	0
$337$	−518.410	−1.53831	−0.769154	−	0.639063i	$-0.779322\pi$
$337$	−518.410	−1.53831	−0.769154	+	0.639063i	$0.779322\pi$
$338$	0	0
$339$	− 96.1885i	− 0.283742i
$340$	0	0
$341$	329.581	0.966512
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	36.9989	0.107243
$346$	0	0
$347$	231.720i	0.667780i	0.942612	+	0.333890i	$0.108361\pi$
$347$	231.720i	0.667780i	−0.942612	+	0.333890i	$0.891639\pi$
$348$	0	0
$349$	−170.720	−0.489169	−0.244584	−	0.969628i	$-0.578652\pi$
$349$	−170.720	−0.489169	−0.244584	+	0.969628i	$0.578652\pi$
$350$	0	0
$351$	42.3564i	0.120674i
$352$	0	0
$353$	−200.237	−0.567244	−0.283622	−	0.958936i	$-0.591536\pi$
$353$	−200.237	−0.567244	−0.283622	+	0.958936i	$0.591536\pi$
$354$	0	0
$355$	31.4757i	0.0886638i
$356$	0	0
$357$	−246.356	−0.690074
$358$	0	0
$359$	− 86.9601i	− 0.242229i	−0.992639	−	0.121114i	$-0.961353\pi$
$359$	− 86.9601i	− 0.242229i	0.992639	−	0.121114i	$-0.0386468\pi$
$360$	0	0
$361$	348.264	0.964721
$362$	0	0
$363$	467.443i	1.28772i
$364$	0	0
$365$	−48.7515	−0.133566
$366$	0	0
$367$	672.949i	1.83365i	0.399290	+	0.916825i	$0.369257\pi$
$367$	672.949i	1.83365i	−0.399290	+	0.916825i	$0.630743\pi$
$368$	0	0
$369$	−317.763	−0.861147
$370$	0	0
$371$	− 231.243i	− 0.623295i
$372$	0	0
$373$	−692.328	−1.85611	−0.928053	−	0.372448i	$-0.878519\pi$
$373$	−692.328	−1.85611	−0.928053	+	0.372448i	$0.878519\pi$
$374$	0	0
$375$	− 94.0560i	− 0.250816i
$376$	0	0
$377$	−38.5391	−0.102226
$378$	0	0
$379$	− 193.386i	− 0.510253i	−0.966908	−	0.255126i	$-0.917883\pi$
$379$	− 193.386i	− 0.510253i	0.966908	−	0.255126i	$-0.0821171\pi$
$380$	0	0
$381$	265.600	0.697112
$382$	0	0
$383$	518.606i	1.35406i	0.735954	+	0.677031i	$0.236734\pi$
$383$	518.606i	1.35406i	−0.735954	+	0.677031i	$0.763266\pi$
$384$	0	0
$385$	−20.1835	−0.0524247
$386$	0	0
$387$	− 209.238i	− 0.540666i
$388$	0	0
$389$	531.999	1.36761	0.683803	−	0.729666i	$-0.260325\pi$
$389$	531.999	1.36761	0.683803	+	0.729666i	$0.260325\pi$
$390$	0	0
$391$	472.642i	1.20880i
$392$	0	0
$393$	390.737	0.994242
$394$	0	0
$395$	68.3494i	0.173036i
$396$	0	0
$397$	490.068	1.23443	0.617214	−	0.786795i	$-0.288261\pi$
$397$	490.068	1.23443	0.617214	+	0.786795i	$0.288261\pi$
$398$	0	0
$399$	− 36.4102i	− 0.0912535i
$400$	0	0
$401$	712.938	1.77790	0.888951	−	0.458003i	$-0.151435\pi$
$401$	712.938	1.77790	0.888951	+	0.458003i	$0.151435\pi$
$402$	0	0
$403$	− 74.3268i	− 0.184434i
$404$	0	0
$405$	48.7088	0.120269
$406$	0	0
$407$	909.228i	2.23398i
$408$	0	0
$409$	−307.818	−0.752612	−0.376306	−	0.926495i	$-0.622806\pi$
$409$	−307.818	−0.752612	−0.376306	+	0.926495i	$0.622806\pi$
$410$	0	0
$411$	− 246.344i	− 0.599377i
$412$	0	0
$413$	176.459	0.427262
$414$	0	0
$415$	2.95637i	0.00712378i
$416$	0	0
$417$	−768.377	−1.84263
$418$	0	0
$419$	176.023i	0.420103i	0.977690	+	0.210051i	$0.0673631\pi$
$419$	176.023i	0.420103i	−0.977690	+	0.210051i	$0.932637\pi$
$420$	0	0
$421$	−263.167	−0.625101	−0.312550	−	0.949901i	$-0.601183\pi$
$421$	−263.167	−0.625101	−0.312550	+	0.949901i	$0.601183\pi$
$422$	0	0
$423$	− 377.034i	− 0.891333i
$424$	0	0
$425$	597.857	1.40672
$426$	0	0
$427$	− 44.6114i	− 0.104476i
$428$	0	0
$429$	210.646	0.491016
$430$	0	0
$431$	256.629i	0.595426i	0.954655	+	0.297713i	$0.0962238\pi$
$431$	256.629i	0.595426i	−0.954655	+	0.297713i	$0.903776\pi$
$432$	0	0
$433$	−636.797	−1.47066	−0.735331	−	0.677708i	$-0.762973\pi$
$433$	−636.797	−1.47066	−0.735331	+	0.677708i	$0.762973\pi$
$434$	0	0
$435$	20.7551i	0.0477128i
$436$	0	0
$437$	−69.8540	−0.159849
$438$	0	0
$439$	64.4215i	0.146746i	0.997305	+	0.0733730i	$0.0233763\pi$
$439$	64.4215i	0.146746i	−0.997305	+	0.0733730i	$0.976624\pi$
$440$	0	0
$441$	41.0938	0.0931831
$442$	0	0
$443$	− 449.411i	− 1.01447i	−0.861807	−	0.507236i	$-0.830667\pi$
$443$	− 449.411i	− 1.01447i	0.861807	−	0.507236i	$-0.169333\pi$
$444$	0	0
$445$	11.7553	0.0264163
$446$	0	0
$447$	− 1047.30i	− 2.34294i
$448$	0	0
$449$	362.900	0.808241	0.404120	−	0.914706i	$-0.367578\pi$
$449$	362.900	0.808241	0.404120	+	0.914706i	$0.367578\pi$
$450$	0	0
$451$	− 842.420i	− 1.86789i
$452$	0	0
$453$	−506.435	−1.11796
$454$	0	0
$455$	4.55177i	0.0100039i
$456$	0	0
$457$	−61.1286	−0.133761	−0.0668803	−	0.997761i	$-0.521305\pi$
$457$	−61.1286	−0.133761	−0.0668803	+	0.997761i	$0.521305\pi$
$458$	0	0
$459$	291.397i	0.634851i
$460$	0	0
$461$	564.752	1.22506	0.612530	−	0.790448i	$-0.290152\pi$
$461$	564.752	1.22506	0.612530	+	0.790448i	$0.290152\pi$
$462$	0	0
$463$	− 581.095i	− 1.25506i	−0.778590	−	0.627532i	$-0.784065\pi$
$463$	− 581.095i	− 1.25506i	0.778590	−	0.627532i	$-0.215935\pi$
$464$	0	0
$465$	−40.0283	−0.0860824
$466$	0	0
$467$	125.780i	0.269336i	0.990891	+	0.134668i	$0.0429968\pi$
$467$	125.780i	0.269336i	−0.990891	+	0.134668i	$0.957003\pi$
$468$	0	0
$469$	56.2012	0.119832
$470$	0	0
$471$	515.519i	1.09452i
$472$	0	0
$473$	554.709	1.17275
$474$	0	0
$475$	88.3601i	0.186021i
$476$	0	0
$477$	513.094	1.07567
$478$	0	0
$479$	− 393.505i	− 0.821513i	−0.911745	−	0.410756i	$-0.865265\pi$
$479$	− 393.505i	− 0.821513i	0.911745	−	0.410756i	$-0.134735\pi$
$480$	0	0
$481$	205.048	0.426296
$482$	0	0
$483$	− 199.707i	− 0.413472i
$484$	0	0
$485$	−84.0680	−0.173336
$486$	0	0
$487$	− 149.606i	− 0.307198i	−0.988133	−	0.153599i	$-0.950914\pi$
$487$	− 149.606i	− 0.307198i	0.988133	−	0.153599i	$-0.0490865\pi$
$488$	0	0
$489$	114.980	0.235133
$490$	0	0
$491$	− 79.1826i	− 0.161268i	−0.996744	−	0.0806340i	$-0.974305\pi$
$491$	− 79.1826i	− 0.161268i	0.996744	−	0.0806340i	$-0.0256945\pi$
$492$	0	0
$493$	−265.135	−0.537799
$494$	0	0
$495$	− 44.7843i	− 0.0904733i
$496$	0	0
$497$	169.894	0.341840
$498$	0	0
$499$	− 244.391i	− 0.489762i	−0.969553	−	0.244881i	$-0.921251\pi$
$499$	− 244.391i	− 0.489762i	0.969553	−	0.244881i	$-0.0787489\pi$
$500$	0	0
$501$	1001.50	1.99899
$502$	0	0
$503$	− 278.539i	− 0.553755i	−0.960905	−	0.276878i	$-0.910700\pi$
$503$	− 278.539i	− 0.553755i	0.960905	−	0.276878i	$-0.0892997\pi$
$504$	0	0
$505$	−71.7739	−0.142127
$506$	0	0
$507$	604.199i	1.19171i
$508$	0	0
$509$	−444.338	−0.872963	−0.436481	−	0.899713i	$-0.643775\pi$
$509$	−444.338	−0.872963	−0.436481	+	0.899713i	$0.643775\pi$
$510$	0	0
$511$	263.143i	0.514957i
$512$	0	0
$513$	−43.0669	−0.0839510
$514$	0	0
$515$	58.2562i	0.113119i
$516$	0	0
$517$	999.552	1.93337
$518$	0	0
$519$	− 327.687i	− 0.631382i
$520$	0	0
$521$	127.704	0.245113	0.122557	−	0.992462i	$-0.460891\pi$
$521$	127.704	0.245113	0.122557	+	0.992462i	$0.460891\pi$
$522$	0	0
$523$	937.135i	1.79184i	0.444211	+	0.895922i	$0.353484\pi$
$523$	937.135i	1.79184i	−0.444211	+	0.895922i	$0.646516\pi$
$524$	0	0
$525$	−252.615	−0.481171
$526$	0	0
$527$	− 511.341i	− 0.970286i
$528$	0	0
$529$	145.856	0.275721
$530$	0	0
$531$	391.538i	0.737359i
$532$	0	0
$533$	−189.982	−0.356439
$534$	0	0
$535$	69.8167i	0.130499i
$536$	0	0
$537$	74.6751	0.139060
$538$	0	0
$539$	108.943i	0.202121i
$540$	0	0
$541$	540.845	0.999713	0.499857	−	0.866108i	$-0.333386\pi$
$541$	540.845	0.999713	0.499857	+	0.866108i	$0.333386\pi$
$542$	0	0
$543$	− 707.040i	− 1.30210i
$544$	0	0
$545$	83.7380	0.153648
$546$	0	0
$547$	− 667.995i	− 1.22120i	−0.791940	−	0.610599i	$-0.790929\pi$
$547$	− 667.995i	− 1.22120i	0.791940	−	0.610599i	$-0.209071\pi$
$548$	0	0
$549$	98.9862	0.180303
$550$	0	0
$551$	− 39.1855i	− 0.0711171i
$552$	0	0
$553$	368.926	0.667135
$554$	0	0
$555$	− 110.428i	− 0.198969i
$556$	0	0
$557$	−746.470	−1.34016	−0.670081	−	0.742288i	$-0.733741\pi$
$557$	−746.470	−1.34016	−0.670081	+	0.742288i	$0.733741\pi$
$558$	0	0
$559$	− 125.097i	− 0.223788i
$560$	0	0
$561$	1449.16	2.58318
$562$	0	0
$563$	518.437i	0.920846i	0.887700	+	0.460423i	$0.152302\pi$
$563$	518.437i	0.920846i	−0.887700	+	0.460423i	$0.847698\pi$
$564$	0	0
$565$	−12.2266	−0.0216399
$566$	0	0
$567$	− 262.913i	− 0.463691i
$568$	0	0
$569$	−215.471	−0.378683	−0.189341	−	0.981911i	$-0.560635\pi$
$569$	−215.471	−0.378683	−0.189341	+	0.981911i	$0.560635\pi$
$570$	0	0
$571$	4.73397i	0.00829067i	0.999991	+	0.00414534i	$0.00131951\pi$
$571$	4.73397i	0.00829067i	−0.999991	+	0.00414534i	$0.998680\pi$
$572$	0	0
$573$	−359.332	−0.627107
$574$	0	0
$575$	484.649i	0.842867i
$576$	0	0
$577$	−72.0226	−0.124823	−0.0624113	−	0.998051i	$-0.519879\pi$
$577$	−72.0226	−0.124823	−0.0624113	+	0.998051i	$0.519879\pi$
$578$	0	0
$579$	− 1067.62i	− 1.84390i
$580$	0	0
$581$	15.9574	0.0274655
$582$	0	0
$583$	1360.26i	2.33321i
$584$	0	0
$585$	−10.0997	−0.0172645
$586$	0	0
$587$	272.118i	0.463575i	0.972766	+	0.231787i	$0.0744573\pi$
$587$	272.118i	0.463575i	−0.972766	+	0.231787i	$0.925543\pi$
$588$	0	0
$589$	75.5734	0.128308
$590$	0	0
$591$	− 685.297i	− 1.15956i
$592$	0	0
$593$	842.019	1.41993	0.709966	−	0.704236i	$-0.248710\pi$
$593$	842.019	1.41993	0.709966	+	0.704236i	$0.248710\pi$
$594$	0	0
$595$	31.3145i	0.0526294i
$596$	0	0
$597$	−876.987	−1.46899
$598$	0	0
$599$	− 478.633i	− 0.799054i	−0.916721	−	0.399527i	$-0.869174\pi$
$599$	− 478.633i	− 0.799054i	0.916721	−	0.399527i	$-0.130826\pi$
$600$	0	0
$601$	206.471	0.343546	0.171773	−	0.985137i	$-0.445050\pi$
$601$	206.471	0.343546	0.171773	+	0.985137i	$0.445050\pi$
$602$	0	0
$603$	124.702i	0.206803i
$604$	0	0
$605$	59.4170	0.0982099
$606$	0	0
$607$	− 292.279i	− 0.481514i	−0.970585	−	0.240757i	$-0.922604\pi$
$607$	− 292.279i	− 0.481514i	0.970585	−	0.240757i	$-0.0773956\pi$
$608$	0	0
$609$	112.028	0.183955
$610$	0	0
$611$	− 225.418i	− 0.368933i
$612$	0	0
$613$	−827.863	−1.35051	−0.675255	−	0.737584i	$-0.735967\pi$
$613$	−827.863	−1.35051	−0.675255	+	0.737584i	$0.735967\pi$
$614$	0	0
$615$	102.314i	0.166364i
$616$	0	0
$617$	−667.348	−1.08160	−0.540801	−	0.841151i	$-0.681879\pi$
$617$	−667.348	−1.08160	−0.540801	+	0.841151i	$0.681879\pi$
$618$	0	0
$619$	750.661i	1.21270i	0.795198	+	0.606350i	$0.207367\pi$
$619$	750.661i	1.21270i	−0.795198	+	0.606350i	$0.792633\pi$
$620$	0	0
$621$	−236.219	−0.380384
$622$	0	0
$623$	− 63.4507i	− 0.101847i
$624$	0	0
$625$	607.038	0.971261
$626$	0	0
$627$	214.179i	0.341593i
$628$	0	0
$629$	1410.66	2.24270
$630$	0	0
$631$	1076.70i	1.70633i	0.521637	+	0.853167i	$0.325321\pi$
$631$	1076.70i	1.70633i	−0.521637	+	0.853167i	$0.674679\pi$
$632$	0	0
$633$	1255.27	1.98305
$634$	0	0
$635$	− 33.7605i	− 0.0531662i
$636$	0	0
$637$	24.5688	0.0385696
$638$	0	0
$639$	376.971i	0.589939i
$640$	0	0
$641$	−298.632	−0.465885	−0.232942	−	0.972491i	$-0.574835\pi$
$641$	−298.632	−0.465885	−0.232942	+	0.972491i	$0.574835\pi$
$642$	0	0
$643$	− 504.242i	− 0.784202i	−0.919922	−	0.392101i	$-0.871748\pi$
$643$	− 504.242i	− 0.784202i	0.919922	−	0.392101i	$-0.128252\pi$
$644$	0	0
$645$	−67.3706	−0.104451
$646$	0	0
$647$	− 76.2691i	− 0.117881i	−0.998261	−	0.0589406i	$-0.981228\pi$
$647$	− 76.2691i	− 0.117881i	0.998261	−	0.0589406i	$-0.0187723\pi$
$648$	0	0
$649$	−1038.00	−1.59939
$650$	0	0
$651$	216.058i	0.331887i
$652$	0	0
$653$	199.148	0.304974	0.152487	−	0.988305i	$-0.451272\pi$
$653$	199.148	0.304974	0.152487	+	0.988305i	$0.451272\pi$
$654$	0	0
$655$	− 49.6668i	− 0.0758272i
$656$	0	0
$657$	−583.876	−0.888700
$658$	0	0
$659$	488.851i	0.741807i	0.928671	+	0.370904i	$0.120952\pi$
$659$	488.851i	0.741807i	−0.928671	+	0.370904i	$0.879048\pi$
$660$	0	0
$661$	477.111	0.721802	0.360901	−	0.932604i	$-0.382469\pi$
$661$	477.111	0.721802	0.360901	+	0.932604i	$0.382469\pi$
$662$	0	0
$663$	− 326.814i	− 0.492933i
$664$	0	0
$665$	−4.62811	−0.00695957
$666$	0	0
$667$	− 214.930i	− 0.322234i
$668$	0	0
$669$	−408.765	−0.611010
$670$	0	0
$671$	262.422i	0.391090i
$672$	0	0
$673$	−437.908	−0.650680	−0.325340	−	0.945597i	$-0.605479\pi$
$673$	−437.908	−0.650680	−0.325340	+	0.945597i	$0.605479\pi$
$674$	0	0
$675$	298.799i	0.442665i
$676$	0	0
$677$	−509.653	−0.752810	−0.376405	−	0.926455i	$-0.622840\pi$
$677$	−509.653	−0.752810	−0.376405	+	0.926455i	$0.622840\pi$
$678$	0	0
$679$	453.769i	0.668290i
$680$	0	0
$681$	−1090.41	−1.60119
$682$	0	0
$683$	794.509i	1.16326i	0.813452	+	0.581632i	$0.197586\pi$
$683$	794.509i	1.16326i	−0.813452	+	0.581632i	$0.802414\pi$
$684$	0	0
$685$	−31.3129	−0.0457123
$686$	0	0
$687$	558.060i	0.812315i
$688$	0	0
$689$	306.765	0.445232
$690$	0	0
$691$	19.0808i	0.0276133i	0.999905	+	0.0138066i	$0.00439493\pi$
$691$	19.0808i	0.0276133i	−0.999905	+	0.0138066i	$0.995605\pi$
$692$	0	0
$693$	−241.730	−0.348816
$694$	0	0
$695$	97.6688i	0.140531i
$696$	0	0
$697$	−1307.00	−1.87519
$698$	0	0
$699$	− 142.362i	− 0.203665i
$700$	0	0
$701$	−333.075	−0.475143	−0.237572	−	0.971370i	$-0.576351\pi$
$701$	−333.075	−0.475143	−0.237572	+	0.971370i	$0.576351\pi$
$702$	0	0
$703$	208.488i	0.296568i
$704$	0	0
$705$	−121.398	−0.172195
$706$	0	0
$707$	387.410i	0.547963i
$708$	0	0
$709$	692.721	0.977039	0.488519	−	0.872553i	$-0.337537\pi$
$709$	692.721	0.977039	0.488519	+	0.872553i	$0.337537\pi$
$710$	0	0
$711$	818.593i	1.15133i
$712$	0	0
$713$	414.515	0.581367
$714$	0	0
$715$	− 26.7753i	− 0.0374479i
$716$	0	0
$717$	252.636	0.352352
$718$	0	0
$719$	− 187.332i	− 0.260545i	−0.991478	−	0.130272i	$-0.958415\pi$
$719$	− 187.332i	− 0.260545i	0.991478	−	0.130272i	$-0.0415852\pi$
$720$	0	0
$721$	314.446	0.436125
$722$	0	0
$723$	600.971i	0.831219i
$724$	0	0
$725$	−271.870	−0.374993
$726$	0	0
$727$	− 669.583i	− 0.921022i	−0.887654	−	0.460511i	$-0.847666\pi$
$727$	− 669.583i	− 0.921022i	0.887654	−	0.460511i	$-0.152334\pi$
$728$	0	0
$729$	164.534	0.225698
$730$	0	0
$731$	− 860.624i	− 1.17732i
$732$	0	0
$733$	585.685	0.799024	0.399512	−	0.916728i	$-0.369180\pi$
$733$	585.685	0.799024	0.399512	+	0.916728i	$0.369180\pi$
$734$	0	0
$735$	− 13.2314i	− 0.0180019i
$736$	0	0
$737$	−330.597	−0.448571
$738$	0	0
$739$	− 318.797i	− 0.431389i	−0.976461	−	0.215695i	$-0.930798\pi$
$739$	− 318.797i	− 0.431389i	0.976461	−	0.215695i	$-0.0692016\pi$
$740$	0	0
$741$	48.3014	0.0651841
$742$	0	0
$743$	− 1440.64i	− 1.93895i	−0.245185	−	0.969476i	$-0.578849\pi$
$743$	− 1440.64i	− 1.93895i	0.245185	−	0.969476i	$-0.421151\pi$
$744$	0	0
$745$	−133.122	−0.178688
$746$	0	0
$747$	35.4072i	0.0473992i
$748$	0	0
$749$	376.846	0.503132
$750$	0	0
$751$	238.758i	0.317920i	0.987285	+	0.158960i	$0.0508141\pi$
$751$	238.758i	0.317920i	−0.987285	+	0.158960i	$0.949186\pi$
$752$	0	0
$753$	−890.583	−1.18271
$754$	0	0
$755$	64.3733i	0.0852626i
$756$	0	0
$757$	975.208	1.28825	0.644127	−	0.764919i	$-0.277221\pi$
$757$	975.208	1.28825	0.644127	+	0.764919i	$0.277221\pi$
$758$	0	0
$759$	1174.75i	1.54777i
$760$	0	0
$761$	−135.752	−0.178386	−0.0891929	−	0.996014i	$-0.528429\pi$
$761$	−135.752	−0.178386	−0.0891929	+	0.996014i	$0.528429\pi$
$762$	0	0
$763$	− 451.988i	− 0.592383i
$764$	0	0
$765$	−69.4823	−0.0908265
$766$	0	0
$767$	234.090i	0.305202i
$768$	0	0
$769$	−1375.28	−1.78841	−0.894203	−	0.447661i	$-0.852257\pi$
$769$	−1375.28	−1.78841	−0.894203	+	0.447661i	$0.852257\pi$
$770$	0	0
$771$	− 1300.30i	− 1.68652i
$772$	0	0
$773$	844.987	1.09313	0.546564	−	0.837418i	$-0.315936\pi$
$773$	844.987	1.09313	0.546564	+	0.837418i	$0.315936\pi$
$774$	0	0
$775$	− 524.330i	− 0.676555i
$776$	0	0
$777$	−596.050	−0.767117
$778$	0	0
$779$	− 193.168i	− 0.247970i
$780$	0	0
$781$	−999.385	−1.27962
$782$	0	0
$783$	− 132.510i	− 0.169234i
$784$	0	0
$785$	65.5279	0.0834750
$786$	0	0
$787$	− 706.124i	− 0.897235i	−0.893724	−	0.448617i	$-0.851917\pi$
$787$	− 706.124i	− 0.897235i	0.893724	−	0.448617i	$-0.148083\pi$
$788$	0	0
$789$	−1526.31	−1.93448
$790$	0	0
$791$	65.9947i	0.0834319i
$792$	0	0
$793$	59.1811	0.0746294
$794$	0	0
$795$	− 165.206i	− 0.207807i
$796$	0	0
$797$	−1103.57	−1.38465	−0.692325	−	0.721586i	$-0.743414\pi$
$797$	−1103.57	−1.38465	−0.692325	+	0.721586i	$0.743414\pi$
$798$	0	0
$799$	− 1550.79i	− 1.94092i
$800$	0	0
$801$	140.788	0.175765
$802$	0	0
$803$	− 1547.91i	− 1.92766i
$804$	0	0
$805$	−25.3849	−0.0315340
$806$	0	0
$807$	122.197i	0.151421i
$808$	0	0
$809$	−890.340	−1.10054	−0.550272	−	0.834986i	$-0.685476\pi$
$809$	−890.340	−1.10054	−0.550272	+	0.834986i	$0.685476\pi$
$810$	0	0
$811$	− 326.603i	− 0.402717i	−0.979518	−	0.201358i	$-0.935464\pi$
$811$	− 326.603i	− 0.402717i	0.979518	−	0.201358i	$-0.0645356\pi$
$812$	0	0
$813$	266.143	0.327359
$814$	0	0
$815$	− 14.6152i	− 0.0179328i
$816$	0	0
$817$	127.196	0.155686
$818$	0	0
$819$	54.5147i	0.0665625i
$820$	0	0
$821$	−177.210	−0.215847	−0.107924	−	0.994159i	$-0.534420\pi$
$821$	−177.210	−0.215847	−0.107924	+	0.994159i	$0.534420\pi$
$822$	0	0
$823$	1474.36i	1.79144i	0.444614	+	0.895722i	$0.353341\pi$
$823$	1474.36i	1.79144i	−0.444614	+	0.895722i	$0.646659\pi$
$824$	0	0
$825$	1485.98	1.80118
$826$	0	0
$827$	− 339.749i	− 0.410821i	−0.978676	−	0.205411i	$-0.934147\pi$
$827$	− 339.749i	− 0.410821i	0.978676	−	0.205411i	$-0.0658530\pi$
$828$	0	0
$829$	−1063.05	−1.28233	−0.641166	−	0.767402i	$-0.721549\pi$
$829$	−1063.05	−1.28233	−0.641166	+	0.767402i	$0.721549\pi$
$830$	0	0
$831$	210.467i	0.253269i
$832$	0	0
$833$	169.024	0.202910
$834$	0	0
$835$	− 127.301i	− 0.152456i
$836$	0	0
$837$	255.559	0.305328
$838$	0	0
$839$	670.497i	0.799162i	0.916698	+	0.399581i	$0.130844\pi$
$839$	670.497i	0.799162i	−0.916698	+	0.399581i	$0.869156\pi$
$840$	0	0
$841$	−720.432	−0.856638
$842$	0	0
$843$	541.618i	0.642489i
$844$	0	0
$845$	76.8000	0.0908876
$846$	0	0
$847$	− 320.712i	− 0.378644i
$848$	0	0
$849$	137.747	0.162246
$850$	0	0
$851$	1143.54i	1.34376i
$852$	0	0
$853$	165.395	0.193898	0.0969489	−	0.995289i	$-0.469092\pi$
$853$	165.395	0.193898	0.0969489	+	0.995289i	$0.469092\pi$
$854$	0	0
$855$	− 10.2691i	− 0.0120107i
$856$	0	0
$857$	−1088.22	−1.26981	−0.634904	−	0.772591i	$-0.718960\pi$
$857$	−1088.22	−1.26981	−0.634904	+	0.772591i	$0.718960\pi$
$858$	0	0
$859$	− 335.213i	− 0.390236i	−0.980780	−	0.195118i	$-0.937491\pi$
$859$	− 335.213i	− 0.390236i	0.980780	−	0.195118i	$-0.0625091\pi$
$860$	0	0
$861$	552.253	0.641409
$862$	0	0
$863$	− 527.699i	− 0.611471i	−0.952117	−	0.305735i	$-0.901098\pi$
$863$	− 527.699i	− 0.611471i	0.952117	−	0.305735i	$-0.0989023\pi$
$864$	0	0
$865$	−41.6525	−0.0481532
$866$	0	0
$867$	− 1133.91i	− 1.30785i
$868$	0	0
$869$	−2170.16	−2.49731
$870$	0	0
$871$	74.5560i	0.0855982i
$872$	0	0
$873$	−1006.85	−1.15332
$874$	0	0
$875$	64.5315i	0.0737503i
$876$	0	0
$877$	−1257.89	−1.43431	−0.717154	−	0.696915i	$-0.754556\pi$
$877$	−1257.89	−1.43431	−0.717154	+	0.696915i	$0.754556\pi$
$878$	0	0
$879$	285.045i	0.324283i
$880$	0	0
$881$	1111.12	1.26120	0.630599	−	0.776109i	$-0.282809\pi$
$881$	1111.12	1.26120	0.630599	+	0.776109i	$0.282809\pi$
$882$	0	0
$883$	358.772i	0.406310i	0.979147	+	0.203155i	$0.0651195\pi$
$883$	358.772i	0.406310i	−0.979147	+	0.203155i	$0.934880\pi$
$884$	0	0
$885$	126.068	0.142449
$886$	0	0
$887$	− 1570.56i	− 1.77064i	−0.464978	−	0.885322i	$-0.653938\pi$
$887$	− 1570.56i	− 1.77064i	0.464978	−	0.885322i	$-0.346062\pi$
$888$	0	0
$889$	−182.227	−0.204980
$890$	0	0
$891$	1546.55i	1.73575i
$892$	0	0
$893$	229.199	0.256662
$894$	0	0
$895$	− 9.49200i	− 0.0106056i
$896$	0	0
$897$	264.930	0.295351
$898$	0	0
$899$	232.528i	0.258651i
$900$	0	0
$901$	2110.43	2.34232
$902$	0	0
$903$	363.642i	0.402705i
$904$	0	0
$905$	−89.8722	−0.0993063
$906$	0	0
$907$	− 901.249i	− 0.993659i	−0.867848	−	0.496829i	$-0.834497\pi$
$907$	− 901.249i	− 0.993659i	0.867848	−	0.496829i	$-0.165503\pi$
$908$	0	0
$909$	−859.607	−0.945662
$910$	0	0
$911$	901.747i	0.989843i	0.868938	+	0.494922i	$0.164803\pi$
$911$	901.747i	0.989843i	−0.868938	+	0.494922i	$0.835197\pi$
$912$	0	0
$913$	−93.8678	−0.102813
$914$	0	0
$915$	− 31.8717i	− 0.0348324i
$916$	0	0
$917$	−268.084	−0.292348
$918$	0	0
$919$	− 1165.98i	− 1.26875i	−0.773025	−	0.634375i	$-0.781257\pi$
$919$	− 1165.98i	− 1.26875i	0.773025	−	0.634375i	$-0.218743\pi$
$920$	0	0
$921$	1956.50	2.12432
$922$	0	0
$923$	225.380i	0.244183i
$924$	0	0
$925$	1446.49	1.56378
$926$	0	0
$927$	697.710i	0.752654i
$928$	0	0
$929$	−1400.58	−1.50762	−0.753811	−	0.657091i	$-0.771787\pi$
$929$	−1400.58	−1.50762	−0.753811	+	0.657091i	$0.771787\pi$
$930$	0	0
$931$	24.9809i	0.0268323i
$932$	0	0
$933$	−936.866	−1.00414
$934$	0	0
$935$	− 184.204i	− 0.197010i
$936$	0	0
$937$	1252.37	1.33657	0.668285	−	0.743905i	$-0.267028\pi$
$937$	1252.37	1.33657	0.668285	+	0.743905i	$0.267028\pi$
$938$	0	0
$939$	803.020i	0.855186i
$940$	0	0
$941$	335.582	0.356622	0.178311	−	0.983974i	$-0.442937\pi$
$941$	335.582	0.356622	0.178311	+	0.983974i	$0.442937\pi$
$942$	0	0
$943$	− 1059.51i	− 1.12356i
$944$	0	0
$945$	−15.6505	−0.0165613
$946$	0	0
$947$	1588.48i	1.67738i	0.544610	+	0.838689i	$0.316678\pi$
$947$	1588.48i	1.67738i	−0.544610	+	0.838689i	$0.683322\pi$
$948$	0	0
$949$	−349.083	−0.367843
$950$	0	0
$951$	− 40.9946i	− 0.0431069i
$952$	0	0
$953$	−99.1693	−0.104060	−0.0520301	−	0.998646i	$-0.516569\pi$
$953$	−99.1693	−0.104060	−0.0520301	+	0.998646i	$0.516569\pi$
$954$	0	0
$955$	45.6749i	0.0478271i
$956$	0	0
$957$	−658.995	−0.688605
$958$	0	0
$959$	169.016i	0.176242i
$960$	0	0
$961$	512.546	0.533347
$962$	0	0
$963$	836.166i	0.868292i
$964$	0	0
$965$	−135.706	−0.140628
$966$	0	0
$967$	− 1322.78i	− 1.36792i	−0.729518	−	0.683962i	$-0.760256\pi$
$967$	− 1322.78i	− 1.36792i	0.729518	−	0.683962i	$-0.239744\pi$
$968$	0	0
$969$	332.296	0.342927
$970$	0	0
$971$	− 1571.76i	− 1.61870i	−0.587327	−	0.809350i	$-0.699820\pi$
$971$	− 1571.76i	− 1.61870i	0.587327	−	0.809350i	$-0.300180\pi$
$972$	0	0
$973$	527.181	0.541810
$974$	0	0
$975$	− 335.116i	− 0.343709i
$976$	0	0
$977$	−15.9043	−0.0162787	−0.00813936	−	0.999967i	$-0.502591\pi$
$977$	−15.9043	−0.0162787	−0.00813936	+	0.999967i	$0.502591\pi$
$978$	0	0
$979$	373.242i	0.381248i
$980$	0	0
$981$	1002.90	1.02232
$982$	0	0
$983$	202.919i	0.206428i	0.994659	+	0.103214i	$0.0329127\pi$
$983$	202.919i	0.206428i	−0.994659	+	0.103214i	$0.967087\pi$
$984$	0	0
$985$	−87.1085	−0.0884350
$986$	0	0
$987$	655.262i	0.663892i
$988$	0	0
$989$	697.659	0.705418
$990$	0	0
$991$	760.920i	0.767830i	0.923368	+	0.383915i	$0.125424\pi$
$991$	760.920i	0.767830i	−0.923368	+	0.383915i	$0.874576\pi$
$992$	0	0
$993$	−473.594	−0.476932
$994$	0	0
$995$	111.474i	0.112034i
$996$	0	0
$997$	−860.114	−0.862702	−0.431351	−	0.902184i	$-0.641963\pi$
$997$	−860.114	−0.862702	−0.431351	+	0.902184i	$0.641963\pi$
$998$	0	0
$999$	705.023i	0.705728i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.d.e.127.2		8
4.3	odd	2	inner	448.3.d.e.127.7		8
8.3	odd	2		224.3.d.b.127.2	✓	8
8.5	even	2		224.3.d.b.127.7	yes	8
16.3	odd	4		1792.3.g.f.127.2		8
16.5	even	4		1792.3.g.f.127.1		8
16.11	odd	4		1792.3.g.d.127.7		8
16.13	even	4		1792.3.g.d.127.8		8
24.5	odd	2		2016.3.m.c.127.4		8
24.11	even	2		2016.3.m.c.127.3		8
56.13	odd	2		1568.3.d.n.1471.2		8
56.27	even	2		1568.3.d.n.1471.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.d.b.127.2	✓	8	8.3	odd	2
224.3.d.b.127.7	yes	8	8.5	even	2
448.3.d.e.127.2		8	1.1	even	1	trivial
448.3.d.e.127.7		8	4.3	odd	2	inner
1568.3.d.n.1471.2		8	56.13	odd	2
1568.3.d.n.1471.7		8	56.27	even	2
1792.3.g.d.127.7		8	16.11	odd	4
1792.3.g.d.127.8		8	16.13	even	4
1792.3.g.f.127.1		8	16.5	even	4
1792.3.g.f.127.2		8	16.3	odd	4
2016.3.m.c.127.3		8	24.11	even	2
2016.3.m.c.127.4		8	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.85623i q^{3} -0.490168 q^{5} +2.64575i q^{7} -5.87054 q^{9} +O(q^{10})\)	\(q-3.85623i q^{3} -0.490168 q^{5} +2.64575i q^{7} -5.87054 q^{9} -15.5633i q^{11} -3.50983 q^{13} +1.89020i q^{15} -24.1463 q^{17} -3.56870i q^{19} +10.2026 q^{21} -19.5741i q^{23} -24.7597 q^{25} -12.0679i q^{27} +10.9803 q^{29} +21.1767i q^{31} -60.0159 q^{33} -1.29686i q^{35} -58.4212 q^{37} +13.5347i q^{39} +54.1285 q^{41} +35.6420i q^{43} +2.87755 q^{45} +64.2248i q^{47} -7.00000 q^{49} +93.1140i q^{51} -87.4015 q^{53} +7.62865i q^{55} -13.7617 q^{57} -66.6954i q^{59} -16.8615 q^{61} -15.5320i q^{63} +1.72041 q^{65} -21.2420i q^{67} -75.4822 q^{69} -64.2140i q^{71} +99.4587 q^{73} +95.4793i q^{75} +41.1767 q^{77} -139.441i q^{79} -99.3716 q^{81} -6.03134i q^{83} +11.8358 q^{85} -42.3427i q^{87} -23.9821 q^{89} -9.28614i q^{91} +81.6624 q^{93} +1.74926i q^{95} +171.509 q^{97} +91.3652i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{9}+O(q^{10}) \) 8 * q - 40 * q^9	\( 8 q - 40 q^{9} - 32 q^{13} - 16 q^{17} + 104 q^{25} + 80 q^{29} + 176 q^{37} + 144 q^{41} - 256 q^{45} - 56 q^{49} - 48 q^{53} - 400 q^{57} + 192 q^{61} - 304 q^{65} - 576 q^{69} + 272 q^{73} + 112 q^{77} + 504 q^{81} + 160 q^{85} - 80 q^{89} + 608 q^{93} + 528 q^{97}+O(q^{100}) \) 8 * q - 40 * q^9 - 32 * q^13 - 16 * q^17 + 104 * q^25 + 80 * q^29 + 176 * q^37 + 144 * q^41 - 256 * q^45 - 56 * q^49 - 48 * q^53 - 400 * q^57 + 192 * q^61 - 304 * q^65 - 576 * q^69 + 272 * q^73 + 112 * q^77 + 504 * q^81 + 160 * q^85 - 80 * q^89 + 608 * q^93 + 528 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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