LMFDB - Embedded newform 448.3.d.d.127.6

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:	$6$
Coefficient field:	6.0.1539727.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{3} - 8x + 8 $ x^6 - 2x^5 + 3x^3 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.6
Root			$0.841985 - 1.13625i$ of defining polynomial
Character	$\chi$	$=$	448.127
Dual form			448.3.d.d.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.54500i q^{3} -1.36794 q^{5} -2.64575i q^{7} -11.6570 q^{9} +O(q^{10})$	$q+4.54500i q^{3} -1.36794 q^{5} -2.64575i q^{7} -11.6570 q^{9} -15.3073i q^{11} -19.9460 q^{13} -6.21727i q^{15} -4.73588 q^{17} -13.2765i q^{19} +12.0249 q^{21} -14.9487i q^{23} -23.1287 q^{25} -12.0760i q^{27} -6.20763 q^{29} +18.5385i q^{31} +69.5715 q^{33} +3.61922i q^{35} +27.5216 q^{37} -90.6547i q^{39} -11.1064 q^{41} -27.0248i q^{43} +15.9460 q^{45} -30.9731i q^{47} -7.00000 q^{49} -21.5245i q^{51} +12.7857 q^{53} +20.9394i q^{55} +60.3415 q^{57} +28.8289i q^{59} -6.52415 q^{61} +30.8415i q^{63} +27.2850 q^{65} -102.863i q^{67} +67.9420 q^{69} +45.8085i q^{71} -70.0997 q^{73} -105.120i q^{75} -40.4992 q^{77} -33.3739i q^{79} -50.0275 q^{81} +159.301i q^{83} +6.47839 q^{85} -28.2137i q^{87} -50.0417 q^{89} +52.7723i q^{91} -84.2575 q^{93} +18.1614i q^{95} +89.7343 q^{97} +178.437i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{5} - 10 q^{9}+O(q^{10}) $ 6 * q + 4 * q^5 - 10 * q^9	$ 6 q + 4 q^{5} - 10 q^{9} - 12 q^{13} - 4 q^{17} - 30 q^{25} + 36 q^{29} + 80 q^{33} - 28 q^{37} - 20 q^{41} - 12 q^{45} - 42 q^{49} - 92 q^{53} + 160 q^{57} + 164 q^{61} - 136 q^{65} + 48 q^{69} - 132 q^{73} - 112 q^{77} - 218 q^{81} + 232 q^{85} + 348 q^{89} - 288 q^{93} + 252 q^{97}+O(q^{100}) $ 6 * q + 4 * q^5 - 10 * q^9 - 12 * q^13 - 4 * q^17 - 30 * q^25 + 36 * q^29 + 80 * q^33 - 28 * q^37 - 20 * q^41 - 12 * q^45 - 42 * q^49 - 92 * q^53 + 160 * q^57 + 164 * q^61 - 136 * q^65 + 48 * q^69 - 132 * q^73 - 112 * q^77 - 218 * q^81 + 232 * q^85 + 348 * q^89 - 288 * q^93 + 252 * 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
$3$	4.54500i	1.51500i	0.652836	+	0.757499i	$0.273579\pi$
$3$	4.54500i	1.51500i	−0.652836	+	0.757499i	$0.726421\pi$
$4$	0	0
$5$	−1.36794	−0.273588	−0.136794	−	0.990600i	$-0.543680\pi$
$5$	−1.36794	−0.273588	−0.136794	+	0.990600i	$0.543680\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	−11.6570	−1.29522
$10$	0	0
$11$	− 15.3073i	− 1.39157i	−0.718250	−	0.695785i	$-0.755057\pi$
$11$	− 15.3073i	− 1.39157i	0.718250	−	0.695785i	$-0.244943\pi$
$12$	0	0
$13$	−19.9460	−1.53431	−0.767156	−	0.641461i	$-0.778329\pi$
$13$	−19.9460	−1.53431	−0.767156	+	0.641461i	$0.778329\pi$
$14$	0	0
$15$	− 6.21727i	− 0.414485i
$16$	0	0
$17$	−4.73588	−0.278581	−0.139290	−	0.990252i	$-0.544482\pi$
$17$	−4.73588	−0.278581	−0.139290	+	0.990252i	$0.544482\pi$
$18$	0	0
$19$	− 13.2765i	− 0.698761i	−0.936981	−	0.349380i	$-0.886392\pi$
$19$	− 13.2765i	− 0.698761i	0.936981	−	0.349380i	$-0.113608\pi$
$20$	0	0
$21$	12.0249	0.572616
$22$	0	0
$23$	− 14.9487i	− 0.649945i	−0.945723	−	0.324973i	$-0.894645\pi$
$23$	− 14.9487i	− 0.649945i	0.945723	−	0.324973i	$-0.105355\pi$
$24$	0	0
$25$	−23.1287	−0.925150
$26$	0	0
$27$	− 12.0760i	− 0.447260i
$28$	0	0
$29$	−6.20763	−0.214056	−0.107028	−	0.994256i	$-0.534133\pi$
$29$	−6.20763	−0.214056	−0.107028	+	0.994256i	$0.534133\pi$
$30$	0	0
$31$	18.5385i	0.598017i	0.954251	+	0.299008i	$0.0966558\pi$
$31$	18.5385i	0.598017i	−0.954251	+	0.299008i	$0.903344\pi$
$32$	0	0
$33$	69.5715	2.10823
$34$	0	0
$35$	3.61922i	0.103406i
$36$	0	0
$37$	27.5216	0.743827	0.371914	−	0.928267i	$-0.378702\pi$
$37$	27.5216	0.743827	0.371914	+	0.928267i	$0.378702\pi$
$38$	0	0
$39$	− 90.6547i	− 2.32448i
$40$	0	0
$41$	−11.1064	−0.270887	−0.135443	−	0.990785i	$-0.543246\pi$
$41$	−11.1064	−0.270887	−0.135443	+	0.990785i	$0.543246\pi$
$42$	0	0
$43$	− 27.0248i	− 0.628483i	−0.949343	−	0.314241i	$-0.898250\pi$
$43$	− 27.0248i	− 0.628483i	0.949343	−	0.314241i	$-0.101750\pi$
$44$	0	0
$45$	15.9460	0.354357
$46$	0	0
$47$	− 30.9731i	− 0.659001i	−0.944155	−	0.329501i	$-0.893120\pi$
$47$	− 30.9731i	− 0.659001i	0.944155	−	0.329501i	$-0.106880\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 21.5245i	− 0.422050i
$52$	0	0
$53$	12.7857	0.241240	0.120620	−	0.992699i	$-0.461512\pi$
$53$	12.7857	0.241240	0.120620	+	0.992699i	$0.461512\pi$
$54$	0	0
$55$	20.9394i	0.380716i
$56$	0	0
$57$	60.3415	1.05862
$58$	0	0
$59$	28.8289i	0.488626i	0.969696	+	0.244313i	$0.0785625\pi$
$59$	28.8289i	0.488626i	−0.969696	+	0.244313i	$0.921438\pi$
$60$	0	0
$61$	−6.52415	−0.106953	−0.0534767	−	0.998569i	$-0.517030\pi$
$61$	−6.52415	−0.106953	−0.0534767	+	0.998569i	$0.517030\pi$
$62$	0	0
$63$	30.8415i	0.489548i
$64$	0	0
$65$	27.2850	0.419769
$66$	0	0
$67$	− 102.863i	− 1.53526i	−0.640891	−	0.767632i	$-0.721435\pi$
$67$	− 102.863i	− 1.53526i	0.640891	−	0.767632i	$-0.278565\pi$
$68$	0	0
$69$	67.9420	0.984666
$70$	0	0
$71$	45.8085i	0.645190i	0.946537	+	0.322595i	$0.104555\pi$
$71$	45.8085i	0.645190i	−0.946537	+	0.322595i	$0.895445\pi$
$72$	0	0
$73$	−70.0997	−0.960270	−0.480135	−	0.877195i	$-0.659412\pi$
$73$	−70.0997	−0.960270	−0.480135	+	0.877195i	$0.659412\pi$
$74$	0	0
$75$	− 105.120i	− 1.40160i
$76$	0	0
$77$	−40.4992	−0.525964
$78$	0	0
$79$	− 33.3739i	− 0.422455i	−0.977437	−	0.211228i	$-0.932254\pi$
$79$	− 33.3739i	− 0.422455i	0.977437	−	0.211228i	$-0.0677461\pi$
$80$	0	0
$81$	−50.0275	−0.617623
$82$	0	0
$83$	159.301i	1.91930i	0.281206	+	0.959648i	$0.409266\pi$
$83$	159.301i	1.91930i	−0.281206	+	0.959648i	$0.590734\pi$
$84$	0	0
$85$	6.47839	0.0762163
$86$	0	0
$87$	− 28.2137i	− 0.324295i
$88$	0	0
$89$	−50.0417	−0.562266	−0.281133	−	0.959669i	$-0.590710\pi$
$89$	−50.0417	−0.562266	−0.281133	+	0.959669i	$0.590710\pi$
$90$	0	0
$91$	52.7723i	0.579915i
$92$	0	0
$93$	−84.2575	−0.905995
$94$	0	0
$95$	18.1614i	0.191172i
$96$	0	0
$97$	89.7343	0.925096	0.462548	−	0.886594i	$-0.346935\pi$
$97$	89.7343	0.925096	0.462548	+	0.886594i	$0.346935\pi$
$98$	0	0
$99$	178.437i	1.80239i
$100$	0	0
$101$	−162.881	−1.61269	−0.806343	−	0.591448i	$-0.798557\pi$
$101$	−162.881	−1.61269	−0.806343	+	0.591448i	$0.798557\pi$
$102$	0	0
$103$	− 86.1212i	− 0.836129i	−0.908417	−	0.418064i	$-0.862709\pi$
$103$	− 86.1212i	− 0.836129i	0.908417	−	0.418064i	$-0.137291\pi$
$104$	0	0
$105$	−16.4494	−0.156661
$106$	0	0
$107$	− 102.146i	− 0.954632i	−0.878732	−	0.477316i	$-0.841610\pi$
$107$	− 102.146i	− 0.954632i	0.878732	−	0.477316i	$-0.158390\pi$
$108$	0	0
$109$	−152.037	−1.39483	−0.697416	−	0.716667i	$-0.745667\pi$
$109$	−152.037	−1.39483	−0.697416	+	0.716667i	$0.745667\pi$
$110$	0	0
$111$	125.086i	1.12690i
$112$	0	0
$113$	−168.112	−1.48772	−0.743860	−	0.668335i	$-0.767007\pi$
$113$	−168.112	−1.48772	−0.743860	+	0.668335i	$0.767007\pi$
$114$	0	0
$115$	20.4489i	0.177817i
$116$	0	0
$117$	232.511	1.98727
$118$	0	0
$119$	12.5300i	0.105294i
$120$	0	0
$121$	−113.312	−0.936466
$122$	0	0
$123$	− 50.4783i	− 0.410393i
$124$	0	0
$125$	65.8372	0.526697
$126$	0	0
$127$	87.8953i	0.692089i	0.938218	+	0.346044i	$0.112475\pi$
$127$	87.8953i	0.692089i	−0.938218	+	0.346044i	$0.887525\pi$
$128$	0	0
$129$	122.827	0.952150
$130$	0	0
$131$	140.141i	1.06978i	0.844923	+	0.534888i	$0.179646\pi$
$131$	140.141i	1.06978i	−0.844923	+	0.534888i	$0.820354\pi$
$132$	0	0
$133$	−35.1262	−0.264107
$134$	0	0
$135$	16.5192i	0.122365i
$136$	0	0
$137$	118.725	0.866603	0.433301	−	0.901249i	$-0.357349\pi$
$137$	118.725	0.866603	0.433301	+	0.901249i	$0.357349\pi$
$138$	0	0
$139$	− 72.1046i	− 0.518738i	−0.965778	−	0.259369i	$-0.916485\pi$
$139$	− 72.1046i	− 0.518738i	0.965778	−	0.259369i	$-0.0835147\pi$
$140$	0	0
$141$	140.772	0.998386
$142$	0	0
$143$	305.319i	2.13510i
$144$	0	0
$145$	8.49165	0.0585631
$146$	0	0
$147$	− 31.8150i	− 0.216428i
$148$	0	0
$149$	−218.299	−1.46510	−0.732548	−	0.680716i	$-0.761669\pi$
$149$	−218.299	−1.46510	−0.732548	+	0.680716i	$0.761669\pi$
$150$	0	0
$151$	− 180.946i	− 1.19832i	−0.800629	−	0.599160i	$-0.795501\pi$
$151$	− 180.946i	− 1.19832i	0.800629	−	0.599160i	$-0.204499\pi$
$152$	0	0
$153$	55.2061	0.360824
$154$	0	0
$155$	− 25.3595i	− 0.163610i
$156$	0	0
$157$	191.081	1.21708	0.608538	−	0.793525i	$-0.291756\pi$
$157$	191.081	1.21708	0.608538	+	0.793525i	$0.291756\pi$
$158$	0	0
$159$	58.1111i	0.365479i
$160$	0	0
$161$	−39.5506	−0.245656
$162$	0	0
$163$	287.743i	1.76530i	0.470035	+	0.882648i	$0.344241\pi$
$163$	287.743i	1.76530i	−0.470035	+	0.882648i	$0.655759\pi$
$164$	0	0
$165$	−95.1695	−0.576785
$166$	0	0
$167$	124.859i	0.747659i	0.927497	+	0.373829i	$0.121955\pi$
$167$	124.859i	0.747659i	−0.927497	+	0.373829i	$0.878045\pi$
$168$	0	0
$169$	228.845	1.35411
$170$	0	0
$171$	154.764i	0.905050i
$172$	0	0
$173$	275.207	1.59079	0.795396	−	0.606090i	$-0.207263\pi$
$173$	275.207	1.59079	0.795396	+	0.606090i	$0.207263\pi$
$174$	0	0
$175$	61.1929i	0.349674i
$176$	0	0
$177$	−131.027	−0.740268
$178$	0	0
$179$	− 7.42479i	− 0.0414792i	−0.999785	−	0.0207396i	$-0.993398\pi$
$179$	− 7.42479i	− 0.0414792i	0.999785	−	0.0207396i	$-0.00660210\pi$
$180$	0	0
$181$	−99.5623	−0.550068	−0.275034	−	0.961435i	$-0.588689\pi$
$181$	−99.5623	−0.550068	−0.275034	+	0.961435i	$0.588689\pi$
$182$	0	0
$183$	− 29.6522i	− 0.162034i
$184$	0	0
$185$	−37.6479	−0.203502
$186$	0	0
$187$	72.4933i	0.387665i
$188$	0	0
$189$	−31.9501	−0.169048
$190$	0	0
$191$	58.2058i	0.304743i	0.988323	+	0.152371i	$0.0486909\pi$
$191$	58.2058i	0.304743i	−0.988323	+	0.152371i	$0.951309\pi$
$192$	0	0
$193$	−103.881	−0.538246	−0.269123	−	0.963106i	$-0.586734\pi$
$193$	−103.881	−0.538246	−0.269123	+	0.963106i	$0.586734\pi$
$194$	0	0
$195$	124.010i	0.635949i
$196$	0	0
$197$	181.201	0.919802	0.459901	−	0.887970i	$-0.347885\pi$
$197$	181.201	0.919802	0.459901	+	0.887970i	$0.347885\pi$
$198$	0	0
$199$	− 217.919i	− 1.09507i	−0.836783	−	0.547535i	$-0.815566\pi$
$199$	− 217.919i	− 1.09507i	0.836783	−	0.547535i	$-0.184434\pi$
$200$	0	0
$201$	467.510	2.32592
$202$	0	0
$203$	16.4238i	0.0809056i
$204$	0	0
$205$	15.1928	0.0741113
$206$	0	0
$207$	174.257i	0.841823i
$208$	0	0
$209$	−203.226	−0.972375
$210$	0	0
$211$	− 107.401i	− 0.509007i	−0.967072	−	0.254504i	$-0.918088\pi$
$211$	− 107.401i	− 0.509007i	0.967072	−	0.254504i	$-0.0819121\pi$
$212$	0	0
$213$	−208.199	−0.977462
$214$	0	0
$215$	36.9682i	0.171945i
$216$	0	0
$217$	49.0483	0.226029
$218$	0	0
$219$	− 318.603i	− 1.45481i
$220$	0	0
$221$	94.4620	0.427430
$222$	0	0
$223$	− 349.685i	− 1.56809i	−0.620702	−	0.784047i	$-0.713152\pi$
$223$	− 349.685i	− 1.56809i	0.620702	−	0.784047i	$-0.286848\pi$
$224$	0	0
$225$	269.612	1.19827
$226$	0	0
$227$	− 18.3421i	− 0.0808020i	−0.999184	−	0.0404010i	$-0.987136\pi$
$227$	− 18.3421i	− 0.0808020i	0.999184	−	0.0404010i	$-0.0128635\pi$
$228$	0	0
$229$	−68.5444	−0.299320	−0.149660	−	0.988737i	$-0.547818\pi$
$229$	−68.5444	−0.299320	−0.149660	+	0.988737i	$0.547818\pi$
$230$	0	0
$231$	− 184.069i	− 0.796835i
$232$	0	0
$233$	61.4788	0.263858	0.131929	−	0.991259i	$-0.457883\pi$
$233$	61.4788	0.263858	0.131929	+	0.991259i	$0.457883\pi$
$234$	0	0
$235$	42.3692i	0.180295i
$236$	0	0
$237$	151.684	0.640019
$238$	0	0
$239$	− 204.645i	− 0.856256i	−0.903718	−	0.428128i	$-0.859173\pi$
$239$	− 204.645i	− 0.856256i	0.903718	−	0.428128i	$-0.140827\pi$
$240$	0	0
$241$	−389.266	−1.61521	−0.807606	−	0.589723i	$-0.799237\pi$
$241$	−389.266	−1.61521	−0.807606	+	0.589723i	$0.799237\pi$
$242$	0	0
$243$	− 336.059i	− 1.38296i
$244$	0	0
$245$	9.57557	0.0390840
$246$	0	0
$247$	264.813i	1.07212i
$248$	0	0
$249$	−724.025	−2.90773
$250$	0	0
$251$	− 115.082i	− 0.458494i	−0.973368	−	0.229247i	$-0.926374\pi$
$251$	− 115.082i	− 0.458494i	0.973368	−	0.229247i	$-0.0736264\pi$
$252$	0	0
$253$	−228.824	−0.904444
$254$	0	0
$255$	29.4442i	0.115468i
$256$	0	0
$257$	337.233	1.31219	0.656094	−	0.754679i	$-0.272207\pi$
$257$	337.233	1.31219	0.656094	+	0.754679i	$0.272207\pi$
$258$	0	0
$259$	− 72.8153i	− 0.281140i
$260$	0	0
$261$	72.3623	0.277250
$262$	0	0
$263$	163.729i	0.622543i	0.950321	+	0.311271i	$0.100755\pi$
$263$	163.729i	0.622543i	−0.950321	+	0.311271i	$0.899245\pi$
$264$	0	0
$265$	−17.4901	−0.0660004
$266$	0	0
$267$	− 227.439i	− 0.851832i
$268$	0	0
$269$	−290.784	−1.08098	−0.540490	−	0.841350i	$-0.681761\pi$
$269$	−290.784	−1.08098	−0.540490	+	0.841350i	$0.681761\pi$
$270$	0	0
$271$	368.628i	1.36025i	0.733096	+	0.680126i	$0.238075\pi$
$271$	368.628i	1.36025i	−0.733096	+	0.680126i	$0.761925\pi$
$272$	0	0
$273$	−239.850	−0.878571
$274$	0	0
$275$	354.038i	1.28741i
$276$	0	0
$277$	−254.425	−0.918503	−0.459252	−	0.888306i	$-0.651882\pi$
$277$	−254.425	−0.918503	−0.459252	+	0.888306i	$0.651882\pi$
$278$	0	0
$279$	− 216.103i	− 0.774564i
$280$	0	0
$281$	146.631	0.521819	0.260909	−	0.965363i	$-0.415978\pi$
$281$	146.631	0.521819	0.260909	+	0.965363i	$0.415978\pi$
$282$	0	0
$283$	− 191.554i	− 0.676868i	−0.940990	−	0.338434i	$-0.890103\pi$
$283$	− 191.554i	− 0.676868i	0.940990	−	0.338434i	$-0.109897\pi$
$284$	0	0
$285$	−82.5434	−0.289626
$286$	0	0
$287$	29.3847i	0.102386i
$288$	0	0
$289$	−266.571	−0.922393
$290$	0	0
$291$	407.842i	1.40152i
$292$	0	0
$293$	−315.496	−1.07678	−0.538389	−	0.842696i	$-0.680967\pi$
$293$	−315.496	−1.07678	−0.538389	+	0.842696i	$0.680967\pi$
$294$	0	0
$295$	− 39.4362i	− 0.133682i
$296$	0	0
$297$	−184.851	−0.622393
$298$	0	0
$299$	298.168i	0.997218i
$300$	0	0
$301$	−71.5008	−0.237544
$302$	0	0
$303$	− 740.295i	− 2.44322i
$304$	0	0
$305$	8.92464	0.0292611
$306$	0	0
$307$	− 452.508i	− 1.47397i	−0.675911	−	0.736983i	$-0.736250\pi$
$307$	− 452.508i	− 1.47397i	0.675911	−	0.736983i	$-0.263750\pi$
$308$	0	0
$309$	391.421	1.26673
$310$	0	0
$311$	− 477.817i	− 1.53639i	−0.640217	−	0.768194i	$-0.721155\pi$
$311$	− 477.817i	− 1.53639i	0.640217	−	0.768194i	$-0.278845\pi$
$312$	0	0
$313$	209.435	0.669122	0.334561	−	0.942374i	$-0.391412\pi$
$313$	209.435	0.669122	0.334561	+	0.942374i	$0.391412\pi$
$314$	0	0
$315$	− 42.1893i	− 0.133934i
$316$	0	0
$317$	−1.74873	−0.00551651	−0.00275825	−	0.999996i	$-0.500878\pi$
$317$	−1.74873	−0.00551651	−0.00275825	+	0.999996i	$0.500878\pi$
$318$	0	0
$319$	95.0218i	0.297874i
$320$	0	0
$321$	464.251	1.44627
$322$	0	0
$323$	62.8757i	0.194662i
$324$	0	0
$325$	461.327	1.41947
$326$	0	0
$327$	− 691.006i	− 2.11317i
$328$	0	0
$329$	−81.9470	−0.249079
$330$	0	0
$331$	− 375.049i	− 1.13308i	−0.824035	−	0.566539i	$-0.808282\pi$
$331$	− 375.049i	− 1.13308i	0.824035	−	0.566539i	$-0.191718\pi$
$332$	0	0
$333$	−320.819	−0.963421
$334$	0	0
$335$	140.710i	0.420029i
$336$	0	0
$337$	254.740	0.755906	0.377953	−	0.925825i	$-0.376628\pi$
$337$	254.740	0.755906	0.377953	+	0.925825i	$0.376628\pi$
$338$	0	0
$339$	− 764.070i	− 2.25389i
$340$	0	0
$341$	283.774	0.832182
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	−92.9404	−0.269392
$346$	0	0
$347$	511.422i	1.47384i	0.675981	+	0.736919i	$0.263720\pi$
$347$	511.422i	1.47384i	−0.675981	+	0.736919i	$0.736280\pi$
$348$	0	0
$349$	383.129	1.09779	0.548895	−	0.835891i	$-0.315049\pi$
$349$	383.129	1.09779	0.548895	+	0.835891i	$0.315049\pi$
$350$	0	0
$351$	240.869i	0.686236i
$352$	0	0
$353$	217.384	0.615819	0.307909	−	0.951416i	$-0.400371\pi$
$353$	217.384	0.615819	0.307909	+	0.951416i	$0.400371\pi$
$354$	0	0
$355$	− 62.6632i	− 0.176516i
$356$	0	0
$357$	−56.9486	−0.159520
$358$	0	0
$359$	483.099i	1.34568i	0.739788	+	0.672840i	$0.234926\pi$
$359$	483.099i	1.34568i	−0.739788	+	0.672840i	$0.765074\pi$
$360$	0	0
$361$	184.736	0.511733
$362$	0	0
$363$	− 515.005i	− 1.41875i
$364$	0	0
$365$	95.8921	0.262718
$366$	0	0
$367$	− 470.944i	− 1.28323i	−0.767028	−	0.641614i	$-0.778265\pi$
$367$	− 470.944i	− 1.28323i	0.767028	−	0.641614i	$-0.221735\pi$
$368$	0	0
$369$	129.467	0.350858
$370$	0	0
$371$	− 33.8279i	− 0.0911803i
$372$	0	0
$373$	179.535	0.481326	0.240663	−	0.970609i	$-0.422635\pi$
$373$	179.535	0.481326	0.240663	+	0.970609i	$0.422635\pi$
$374$	0	0
$375$	299.230i	0.797946i
$376$	0	0
$377$	123.818	0.328429
$378$	0	0
$379$	− 87.8468i	− 0.231786i	−0.993262	−	0.115893i	$-0.963027\pi$
$379$	− 87.8468i	− 0.231786i	0.993262	−	0.115893i	$-0.0369729\pi$
$380$	0	0
$381$	−399.484	−1.04851
$382$	0	0
$383$	653.584i	1.70649i	0.521513	+	0.853243i	$0.325368\pi$
$383$	653.584i	1.70649i	−0.521513	+	0.853243i	$0.674632\pi$
$384$	0	0
$385$	55.4004	0.143897
$386$	0	0
$387$	315.027i	0.814024i
$388$	0	0
$389$	168.861	0.434090	0.217045	−	0.976162i	$-0.430358\pi$
$389$	168.861	0.434090	0.217045	+	0.976162i	$0.430358\pi$
$390$	0	0
$391$	70.7954i	0.181062i
$392$	0	0
$393$	−636.939	−1.62071
$394$	0	0
$395$	45.6535i	0.115578i
$396$	0	0
$397$	691.521	1.74187	0.870934	−	0.491401i	$-0.163515\pi$
$397$	691.521	1.74187	0.870934	+	0.491401i	$0.163515\pi$
$398$	0	0
$399$	− 159.648i	− 0.400121i
$400$	0	0
$401$	−399.834	−0.997091	−0.498546	−	0.866864i	$-0.666132\pi$
$401$	−399.834	−0.997091	−0.498546	+	0.866864i	$0.666132\pi$
$402$	0	0
$403$	− 369.770i	− 0.917544i
$404$	0	0
$405$	68.4345	0.168974
$406$	0	0
$407$	− 421.281i	− 1.03509i
$408$	0	0
$409$	111.079	0.271586	0.135793	−	0.990737i	$-0.456642\pi$
$409$	111.079	0.271586	0.135793	+	0.990737i	$0.456642\pi$
$410$	0	0
$411$	539.603i	1.31290i
$412$	0	0
$413$	76.2742	0.184683
$414$	0	0
$415$	− 217.915i	− 0.525095i
$416$	0	0
$417$	327.715	0.785888
$418$	0	0
$419$	641.986i	1.53219i	0.642729	+	0.766094i	$0.277802\pi$
$419$	641.986i	1.53219i	−0.642729	+	0.766094i	$0.722198\pi$
$420$	0	0
$421$	−151.970	−0.360975	−0.180487	−	0.983577i	$-0.557767\pi$
$421$	−151.970	−0.360975	−0.180487	+	0.983577i	$0.557767\pi$
$422$	0	0
$423$	361.053i	0.853553i
$424$	0	0
$425$	109.535	0.257729
$426$	0	0
$427$	17.2613i	0.0404246i
$428$	0	0
$429$	−1387.68	−3.23468
$430$	0	0
$431$	459.854i	1.06695i	0.845817	+	0.533473i	$0.179113\pi$
$431$	459.854i	1.06695i	−0.845817	+	0.533473i	$0.820887\pi$
$432$	0	0
$433$	534.370	1.23411	0.617056	−	0.786919i	$-0.288325\pi$
$433$	534.370	1.23411	0.617056	+	0.786919i	$0.288325\pi$
$434$	0	0
$435$	38.5945i	0.0887231i
$436$	0	0
$437$	−198.466	−0.454156
$438$	0	0
$439$	201.259i	0.458449i	0.973374	+	0.229224i	$0.0736189\pi$
$439$	201.259i	0.458449i	−0.973374	+	0.229224i	$0.926381\pi$
$440$	0	0
$441$	81.5989	0.185032
$442$	0	0
$443$	− 73.4928i	− 0.165898i	−0.996554	−	0.0829490i	$-0.973566\pi$
$443$	− 73.4928i	− 0.165898i	0.996554	−	0.0829490i	$-0.0264339\pi$
$444$	0	0
$445$	68.4539	0.153829
$446$	0	0
$447$	− 992.169i	− 2.21962i
$448$	0	0
$449$	−41.4382	−0.0922899	−0.0461450	−	0.998935i	$-0.514694\pi$
$449$	−41.4382	−0.0922899	−0.0461450	+	0.998935i	$0.514694\pi$
$450$	0	0
$451$	170.008i	0.376958i
$452$	0	0
$453$	822.401	1.81545
$454$	0	0
$455$	− 72.1892i	− 0.158658i
$456$	0	0
$457$	270.205	0.591258	0.295629	−	0.955303i	$-0.404471\pi$
$457$	270.205	0.591258	0.295629	+	0.955303i	$0.404471\pi$
$458$	0	0
$459$	57.1905i	0.124598i
$460$	0	0
$461$	735.133	1.59465	0.797324	−	0.603552i	$-0.206248\pi$
$461$	735.133	1.59465	0.797324	+	0.603552i	$0.206248\pi$
$462$	0	0
$463$	− 742.356i	− 1.60336i	−0.597753	−	0.801681i	$-0.703940\pi$
$463$	− 742.356i	− 1.60336i	0.597753	−	0.801681i	$-0.296060\pi$
$464$	0	0
$465$	115.259	0.247869
$466$	0	0
$467$	− 402.152i	− 0.861140i	−0.902557	−	0.430570i	$-0.858313\pi$
$467$	− 402.152i	− 0.861140i	0.902557	−	0.430570i	$-0.141687\pi$
$468$	0	0
$469$	−272.149	−0.580275
$470$	0	0
$471$	868.462i	1.84387i
$472$	0	0
$473$	−413.675	−0.874577
$474$	0	0
$475$	307.068i	0.646459i
$476$	0	0
$477$	−149.043	−0.312460
$478$	0	0
$479$	356.741i	0.744763i	0.928080	+	0.372381i	$0.121459\pi$
$479$	356.741i	0.744763i	−0.928080	+	0.372381i	$0.878541\pi$
$480$	0	0
$481$	−548.947	−1.14126
$482$	0	0
$483$	− 179.758i	− 0.372169i
$484$	0	0
$485$	−122.751	−0.253095
$486$	0	0
$487$	121.123i	0.248713i	0.992238	+	0.124356i	$0.0396866\pi$
$487$	121.123i	0.248713i	−0.992238	+	0.124356i	$0.960313\pi$
$488$	0	0
$489$	−1307.79	−2.67442
$490$	0	0
$491$	− 18.4535i	− 0.0375835i	−0.999823	−	0.0187917i	$-0.994018\pi$
$491$	− 18.4535i	− 0.0375835i	0.999823	−	0.0187917i	$-0.00598195\pi$
$492$	0	0
$493$	29.3986	0.0596320
$494$	0	0
$495$	− 244.090i	− 0.493112i
$496$	0	0
$497$	121.198	0.243859
$498$	0	0
$499$	149.571i	0.299742i	0.988706	+	0.149871i	$0.0478859\pi$
$499$	149.571i	0.299742i	−0.988706	+	0.149871i	$0.952114\pi$
$500$	0	0
$501$	−567.484	−1.13270
$502$	0	0
$503$	651.239i	1.29471i	0.762189	+	0.647354i	$0.224125\pi$
$503$	651.239i	1.29471i	−0.762189	+	0.647354i	$0.775875\pi$
$504$	0	0
$505$	222.812	0.441211
$506$	0	0
$507$	1040.10i	2.05148i
$508$	0	0
$509$	−817.972	−1.60702	−0.803509	−	0.595293i	$-0.797036\pi$
$509$	−817.972	−1.60702	−0.803509	+	0.595293i	$0.797036\pi$
$510$	0	0
$511$	185.466i	0.362948i
$512$	0	0
$513$	−160.327	−0.312528
$514$	0	0
$515$	117.809i	0.228754i
$516$	0	0
$517$	−474.113	−0.917046
$518$	0	0
$519$	1250.82i	2.41005i
$520$	0	0
$521$	−173.603	−0.333211	−0.166606	−	0.986024i	$-0.553281\pi$
$521$	−173.603	−0.333211	−0.166606	+	0.986024i	$0.553281\pi$
$522$	0	0
$523$	680.630i	1.30139i	0.759337	+	0.650697i	$0.225523\pi$
$523$	680.630i	1.30139i	−0.759337	+	0.650697i	$0.774477\pi$
$524$	0	0
$525$	−278.122	−0.529755
$526$	0	0
$527$	− 87.7961i	− 0.166596i
$528$	0	0
$529$	305.535	0.577571
$530$	0	0
$531$	− 336.059i	− 0.632879i
$532$	0	0
$533$	221.528	0.415624
$534$	0	0
$535$	139.729i	0.261175i
$536$	0	0
$537$	33.7456	0.0628410
$538$	0	0
$539$	107.151i	0.198796i
$540$	0	0
$541$	313.190	0.578909	0.289454	−	0.957192i	$-0.406526\pi$
$541$	313.190	0.578909	0.289454	+	0.957192i	$0.406526\pi$
$542$	0	0
$543$	− 452.510i	− 0.833352i
$544$	0	0
$545$	207.977	0.381609
$546$	0	0
$547$	− 352.784i	− 0.644944i	−0.946579	−	0.322472i	$-0.895486\pi$
$547$	− 352.784i	− 0.644944i	0.946579	−	0.322472i	$-0.104514\pi$
$548$	0	0
$549$	76.0520	0.138528
$550$	0	0
$551$	82.4153i	0.149574i
$552$	0	0
$553$	−88.2992	−0.159673
$554$	0	0
$555$	− 171.109i	− 0.308305i
$556$	0	0
$557$	135.925	0.244030	0.122015	−	0.992528i	$-0.461064\pi$
$557$	135.925	0.244030	0.122015	+	0.992528i	$0.461064\pi$
$558$	0	0
$559$	539.037i	0.964288i
$560$	0	0
$561$	−329.482	−0.587312
$562$	0	0
$563$	− 160.363i	− 0.284836i	−0.989807	−	0.142418i	$-0.954512\pi$
$563$	− 160.363i	− 0.284836i	0.989807	−	0.142418i	$-0.0454878\pi$
$564$	0	0
$565$	229.967	0.407022
$566$	0	0
$567$	132.360i	0.233440i
$568$	0	0
$569$	611.297	1.07434	0.537168	−	0.843475i	$-0.319494\pi$
$569$	611.297	1.07434	0.537168	+	0.843475i	$0.319494\pi$
$570$	0	0
$571$	286.952i	0.502542i	0.967917	+	0.251271i	$0.0808486\pi$
$571$	286.952i	0.502542i	−0.967917	+	0.251271i	$0.919151\pi$
$572$	0	0
$573$	−264.545	−0.461685
$574$	0	0
$575$	345.746i	0.601297i
$576$	0	0
$577$	−530.428	−0.919286	−0.459643	−	0.888104i	$-0.652023\pi$
$577$	−530.428	−0.919286	−0.459643	+	0.888104i	$0.652023\pi$
$578$	0	0
$579$	− 472.141i	− 0.815442i
$580$	0	0
$581$	421.472	0.725425
$582$	0	0
$583$	− 195.715i	− 0.335703i
$584$	0	0
$585$	−318.061	−0.543693
$586$	0	0
$587$	10.9359i	0.0186301i	0.999957	+	0.00931506i	$0.00296512\pi$
$587$	10.9359i	0.0186301i	−0.999957	+	0.00931506i	$0.997035\pi$
$588$	0	0
$589$	246.126	0.417871
$590$	0	0
$591$	823.558i	1.39350i
$592$	0	0
$593$	119.459	0.201449	0.100725	−	0.994914i	$-0.467884\pi$
$593$	119.459	0.201449	0.100725	+	0.994914i	$0.467884\pi$
$594$	0	0
$595$	− 17.1402i	− 0.0288071i
$596$	0	0
$597$	990.441	1.65903
$598$	0	0
$599$	− 1020.35i	− 1.70342i	−0.524014	−	0.851710i	$-0.675566\pi$
$599$	− 1020.35i	− 1.70342i	0.524014	−	0.851710i	$-0.324434\pi$
$600$	0	0
$601$	−109.785	−0.182671	−0.0913356	−	0.995820i	$-0.529114\pi$
$601$	−109.785	−0.182671	−0.0913356	+	0.995820i	$0.529114\pi$
$602$	0	0
$603$	1199.07i	1.98851i
$604$	0	0
$605$	155.004	0.256206
$606$	0	0
$607$	− 649.557i	− 1.07011i	−0.844817	−	0.535055i	$-0.820291\pi$
$607$	− 649.557i	− 1.07011i	0.844817	−	0.535055i	$-0.179709\pi$
$608$	0	0
$609$	−74.6463	−0.122572
$610$	0	0
$611$	617.790i	1.01111i
$612$	0	0
$613$	−802.148	−1.30856	−0.654281	−	0.756252i	$-0.727029\pi$
$613$	−802.148	−1.30856	−0.654281	+	0.756252i	$0.727029\pi$
$614$	0	0
$615$	69.0513i	0.112278i
$616$	0	0
$617$	1151.11	1.86565	0.932825	−	0.360329i	$-0.117336\pi$
$617$	1151.11	1.86565	0.932825	+	0.360329i	$0.117336\pi$
$618$	0	0
$619$	− 223.829i	− 0.361598i	−0.983520	−	0.180799i	$-0.942132\pi$
$619$	− 223.829i	− 0.361598i	0.983520	−	0.180799i	$-0.0578683\pi$
$620$	0	0
$621$	−180.521	−0.290694
$622$	0	0
$623$	132.398i	0.212517i
$624$	0	0
$625$	488.157	0.781052
$626$	0	0
$627$	− 923.663i	− 1.47315i
$628$	0	0
$629$	−130.339	−0.207216
$630$	0	0
$631$	− 385.557i	− 0.611026i	−0.952188	−	0.305513i	$-0.901172\pi$
$631$	− 385.557i	− 0.611026i	0.952188	−	0.305513i	$-0.0988279\pi$
$632$	0	0
$633$	488.135	0.771146
$634$	0	0
$635$	− 120.235i	− 0.189347i
$636$	0	0
$637$	139.622	0.219187
$638$	0	0
$639$	− 533.989i	− 0.835664i
$640$	0	0
$641$	−289.861	−0.452201	−0.226100	−	0.974104i	$-0.572598\pi$
$641$	−289.861	−0.452201	−0.226100	+	0.974104i	$0.572598\pi$
$642$	0	0
$643$	525.629i	0.817463i	0.912655	+	0.408731i	$0.134029\pi$
$643$	525.629i	0.817463i	−0.912655	+	0.408731i	$0.865971\pi$
$644$	0	0
$645$	−168.020	−0.260497
$646$	0	0
$647$	− 95.6042i	− 0.147765i	−0.997267	−	0.0738827i	$-0.976461\pi$
$647$	− 95.6042i	− 0.147765i	0.997267	−	0.0738827i	$-0.0235390\pi$
$648$	0	0
$649$	441.292	0.679958
$650$	0	0
$651$	222.924i	0.342434i
$652$	0	0
$653$	627.508	0.960962	0.480481	−	0.877005i	$-0.340462\pi$
$653$	627.508	0.960962	0.480481	+	0.877005i	$0.340462\pi$
$654$	0	0
$655$	− 191.704i	− 0.292677i
$656$	0	0
$657$	817.152	1.24376
$658$	0	0
$659$	− 220.070i	− 0.333945i	−0.985962	−	0.166972i	$-0.946601\pi$
$659$	− 220.070i	− 0.333945i	0.985962	−	0.166972i	$-0.0533991\pi$
$660$	0	0
$661$	−743.710	−1.12513	−0.562564	−	0.826754i	$-0.690185\pi$
$661$	−743.710	−1.12513	−0.562564	+	0.826754i	$0.690185\pi$
$662$	0	0
$663$	429.329i	0.647556i
$664$	0	0
$665$	48.0505	0.0722564
$666$	0	0
$667$	92.7962i	0.139125i
$668$	0	0
$669$	1589.32	2.37566
$670$	0	0
$671$	99.8669i	0.148833i
$672$	0	0
$673$	−1017.05	−1.51122	−0.755610	−	0.655022i	$-0.772659\pi$
$673$	−1017.05	−1.51122	−0.755610	+	0.655022i	$0.772659\pi$
$674$	0	0
$675$	279.303i	0.413782i
$676$	0	0
$677$	−861.905	−1.27312	−0.636562	−	0.771225i	$-0.719644\pi$
$677$	−861.905	−1.27312	−0.636562	+	0.771225i	$0.719644\pi$
$678$	0	0
$679$	− 237.415i	− 0.349653i
$680$	0	0
$681$	83.3646	0.122415
$682$	0	0
$683$	− 635.790i	− 0.930879i	−0.885080	−	0.465440i	$-0.845896\pi$
$683$	− 635.790i	− 0.930879i	0.885080	−	0.465440i	$-0.154104\pi$
$684$	0	0
$685$	−162.408	−0.237092
$686$	0	0
$687$	− 311.534i	− 0.453470i
$688$	0	0
$689$	−255.025	−0.370138
$690$	0	0
$691$	− 214.780i	− 0.310825i	−0.987850	−	0.155412i	$-0.950329\pi$
$691$	− 214.780i	− 0.310825i	0.987850	−	0.155412i	$-0.0496706\pi$
$692$	0	0
$693$	472.099	0.681240
$694$	0	0
$695$	98.6347i	0.141920i
$696$	0	0
$697$	52.5983	0.0754639
$698$	0	0
$699$	279.421i	0.399744i
$700$	0	0
$701$	271.746	0.387655	0.193828	−	0.981036i	$-0.437910\pi$
$701$	271.746	0.387655	0.193828	+	0.981036i	$0.437910\pi$
$702$	0	0
$703$	− 365.390i	− 0.519758i
$704$	0	0
$705$	−192.568	−0.273146
$706$	0	0
$707$	430.944i	0.609538i
$708$	0	0
$709$	616.894	0.870090	0.435045	−	0.900409i	$-0.356733\pi$
$709$	616.894	0.870090	0.435045	+	0.900409i	$0.356733\pi$
$710$	0	0
$711$	389.040i	0.547173i
$712$	0	0
$713$	277.127	0.388678
$714$	0	0
$715$	− 417.658i	− 0.584137i
$716$	0	0
$717$	930.112	1.29723
$718$	0	0
$719$	− 55.3428i	− 0.0769719i	−0.999259	−	0.0384859i	$-0.987747\pi$
$719$	− 55.3428i	− 0.0769719i	0.999259	−	0.0384859i	$-0.0122535\pi$
$720$	0	0
$721$	−227.855	−0.316027
$722$	0	0
$723$	− 1769.21i	− 2.44704i
$724$	0	0
$725$	143.575	0.198034
$726$	0	0
$727$	1046.07i	1.43888i	0.694554	+	0.719440i	$0.255602\pi$
$727$	1046.07i	1.43888i	−0.694554	+	0.719440i	$0.744398\pi$
$728$	0	0
$729$	1077.14	1.47756
$730$	0	0
$731$	127.986i	0.175083i
$732$	0	0
$733$	−851.328	−1.16143	−0.580715	−	0.814107i	$-0.697227\pi$
$733$	−851.328	−1.16143	−0.580715	+	0.814107i	$0.697227\pi$
$734$	0	0
$735$	43.5209i	0.0592121i
$736$	0	0
$737$	−1574.55	−2.13643
$738$	0	0
$739$	− 789.585i	− 1.06845i	−0.845342	−	0.534226i	$-0.820603\pi$
$739$	− 789.585i	− 1.06845i	0.845342	−	0.534226i	$-0.179397\pi$
$740$	0	0
$741$	−1203.57	−1.62426
$742$	0	0
$743$	892.788i	1.20160i	0.799400	+	0.600799i	$0.205151\pi$
$743$	892.788i	1.20160i	−0.799400	+	0.600799i	$0.794849\pi$
$744$	0	0
$745$	298.620	0.400832
$746$	0	0
$747$	− 1856.98i	− 2.48591i
$748$	0	0
$749$	−270.252	−0.360817
$750$	0	0
$751$	− 406.211i	− 0.540894i	−0.962735	−	0.270447i	$-0.912829\pi$
$751$	− 406.211i	− 0.540894i	0.962735	−	0.270447i	$-0.0871715\pi$
$752$	0	0
$753$	523.048	0.694619
$754$	0	0
$755$	247.524i	0.327846i
$756$	0	0
$757$	−89.9062	−0.118766	−0.0593832	−	0.998235i	$-0.518913\pi$
$757$	−89.9062	−0.118766	−0.0593832	+	0.998235i	$0.518913\pi$
$758$	0	0
$759$	− 1040.01i	− 1.37023i
$760$	0	0
$761$	−261.751	−0.343956	−0.171978	−	0.985101i	$-0.555016\pi$
$761$	−261.751	−0.343956	−0.171978	+	0.985101i	$0.555016\pi$
$762$	0	0
$763$	402.251i	0.527197i
$764$	0	0
$765$	−75.5185	−0.0987170
$766$	0	0
$767$	− 575.024i	− 0.749705i
$768$	0	0
$769$	−753.905	−0.980370	−0.490185	−	0.871618i	$-0.663071\pi$
$769$	−753.905	−0.980370	−0.490185	+	0.871618i	$0.663071\pi$
$770$	0	0
$771$	1532.72i	1.98796i
$772$	0	0
$773$	463.530	0.599650	0.299825	−	0.953994i	$-0.403072\pi$
$773$	463.530	0.599650	0.299825	+	0.953994i	$0.403072\pi$
$774$	0	0
$775$	− 428.773i	− 0.553255i
$776$	0	0
$777$	330.945	0.425927
$778$	0	0
$779$	147.453i	0.189285i
$780$	0	0
$781$	701.203	0.897827
$782$	0	0
$783$	74.9634i	0.0957388i
$784$	0	0
$785$	−261.387	−0.332977
$786$	0	0
$787$	− 618.821i	− 0.786304i	−0.919473	−	0.393152i	$-0.871385\pi$
$787$	− 618.821i	− 0.786304i	0.919473	−	0.393152i	$-0.128615\pi$
$788$	0	0
$789$	−744.146	−0.943151
$790$	0	0
$791$	444.784i	0.562305i
$792$	0	0
$793$	130.131	0.164100
$794$	0	0
$795$	− 79.4924i	− 0.0999905i
$796$	0	0
$797$	441.998	0.554577	0.277289	−	0.960787i	$-0.410564\pi$
$797$	441.998	0.554577	0.277289	+	0.960787i	$0.410564\pi$
$798$	0	0
$799$	146.685i	0.183585i
$800$	0	0
$801$	583.335	0.728259
$802$	0	0
$803$	1073.04i	1.33628i
$804$	0	0
$805$	54.1028	0.0672085
$806$	0	0
$807$	− 1321.61i	− 1.63768i
$808$	0	0
$809$	311.447	0.384978	0.192489	−	0.981299i	$-0.438344\pi$
$809$	311.447	0.384978	0.192489	+	0.981299i	$0.438344\pi$
$810$	0	0
$811$	965.110i	1.19002i	0.803717	+	0.595012i	$0.202853\pi$
$811$	965.110i	1.19002i	−0.803717	+	0.595012i	$0.797147\pi$
$812$	0	0
$813$	−1675.41	−2.06078
$814$	0	0
$815$	− 393.615i	− 0.482963i
$816$	0	0
$817$	−358.793	−0.439159
$818$	0	0
$819$	− 615.166i	− 0.751118i
$820$	0	0
$821$	−1464.81	−1.78417	−0.892087	−	0.451864i	$-0.850759\pi$
$821$	−1464.81	−1.78417	−0.892087	+	0.451864i	$0.850759\pi$
$822$	0	0
$823$	− 967.159i	− 1.17516i	−0.809165	−	0.587581i	$-0.800080\pi$
$823$	− 967.159i	− 1.17516i	0.809165	−	0.587581i	$-0.199920\pi$
$824$	0	0
$825$	−1609.10	−1.95043
$826$	0	0
$827$	− 1298.46i	− 1.57009i	−0.619441	−	0.785043i	$-0.712641\pi$
$827$	− 1298.46i	− 1.57009i	0.619441	−	0.785043i	$-0.287359\pi$
$828$	0	0
$829$	771.397	0.930515	0.465258	−	0.885175i	$-0.345962\pi$
$829$	771.397	0.930515	0.465258	+	0.885175i	$0.345962\pi$
$830$	0	0
$831$	− 1156.36i	− 1.39153i
$832$	0	0
$833$	33.1511	0.0397973
$834$	0	0
$835$	− 170.799i	− 0.204550i
$836$	0	0
$837$	223.871	0.267469
$838$	0	0
$839$	− 1088.04i	− 1.29683i	−0.761287	−	0.648415i	$-0.775432\pi$
$839$	− 1088.04i	− 1.29683i	0.761287	−	0.648415i	$-0.224568\pi$
$840$	0	0
$841$	−802.465	−0.954180
$842$	0	0
$843$	666.438i	0.790555i
$844$	0	0
$845$	−313.045	−0.370468
$846$	0	0
$847$	299.797i	0.353951i
$848$	0	0
$849$	870.611	1.02545
$850$	0	0
$851$	− 411.413i	− 0.483447i
$852$	0	0
$853$	−126.051	−0.147774	−0.0738870	−	0.997267i	$-0.523540\pi$
$853$	−126.051	−0.147774	−0.0738870	+	0.997267i	$0.523540\pi$
$854$	0	0
$855$	− 211.707i	− 0.247611i
$856$	0	0
$857$	−1135.93	−1.32547	−0.662736	−	0.748853i	$-0.730605\pi$
$857$	−1135.93	−1.32547	−0.662736	+	0.748853i	$0.730605\pi$
$858$	0	0
$859$	− 590.556i	− 0.687492i	−0.939063	−	0.343746i	$-0.888304\pi$
$859$	− 590.556i	− 0.687492i	0.939063	−	0.343746i	$-0.111696\pi$
$860$	0	0
$861$	−133.553	−0.155114
$862$	0	0
$863$	292.067i	0.338432i	0.985579	+	0.169216i	$0.0541235\pi$
$863$	292.067i	0.338432i	−0.985579	+	0.169216i	$0.945876\pi$
$864$	0	0
$865$	−376.466	−0.435221
$866$	0	0
$867$	− 1211.57i	− 1.39742i
$868$	0	0
$869$	−510.864	−0.587876
$870$	0	0
$871$	2051.70i	2.35557i
$872$	0	0
$873$	−1046.03	−1.19820
$874$	0	0
$875$	− 174.189i	− 0.199073i
$876$	0	0
$877$	−1087.23	−1.23971	−0.619856	−	0.784715i	$-0.712809\pi$
$877$	−1087.23	−1.23971	−0.619856	+	0.784715i	$0.712809\pi$
$878$	0	0
$879$	− 1433.93i	− 1.63132i
$880$	0	0
$881$	1217.53	1.38198	0.690991	−	0.722863i	$-0.257174\pi$
$881$	1217.53	1.38198	0.690991	+	0.722863i	$0.257174\pi$
$882$	0	0
$883$	− 1141.10i	− 1.29230i	−0.763212	−	0.646148i	$-0.776379\pi$
$883$	− 1141.10i	− 1.29230i	0.763212	−	0.646148i	$-0.223621\pi$
$884$	0	0
$885$	179.237	0.202528
$886$	0	0
$887$	368.518i	0.415466i	0.978186	+	0.207733i	$0.0666085\pi$
$887$	368.518i	0.415466i	−0.978186	+	0.207733i	$0.933391\pi$
$888$	0	0
$889$	232.549	0.261585
$890$	0	0
$891$	765.784i	0.859466i
$892$	0	0
$893$	−411.213	−0.460484
$894$	0	0
$895$	10.1566i	0.0113482i
$896$	0	0
$897$	−1355.17	−1.51078
$898$	0	0
$899$	− 115.080i	− 0.128009i
$900$	0	0
$901$	−60.5517	−0.0672050
$902$	0	0
$903$	− 324.971i	− 0.359879i
$904$	0	0
$905$	136.195	0.150492
$906$	0	0
$907$	341.525i	0.376544i	0.982117	+	0.188272i	$0.0602886\pi$
$907$	341.525i	0.376544i	−0.982117	+	0.188272i	$0.939711\pi$
$908$	0	0
$909$	1898.71	2.08879
$910$	0	0
$911$	− 536.979i	− 0.589439i	−0.955584	−	0.294719i	$-0.904774\pi$
$911$	− 536.979i	− 0.589439i	0.955584	−	0.294719i	$-0.0952262\pi$
$912$	0	0
$913$	2438.47	2.67083
$914$	0	0
$915$	40.5624i	0.0443305i
$916$	0	0
$917$	370.777	0.404337
$918$	0	0
$919$	987.383i	1.07441i	0.843452	+	0.537205i	$0.180520\pi$
$919$	987.383i	1.07441i	−0.843452	+	0.537205i	$0.819480\pi$
$920$	0	0
$921$	2056.65	2.23306
$922$	0	0
$923$	− 913.698i	− 0.989922i
$924$	0	0
$925$	−636.540	−0.688152
$926$	0	0
$927$	1003.91i	1.08297i
$928$	0	0
$929$	−1285.67	−1.38393	−0.691965	−	0.721932i	$-0.743255\pi$
$929$	−1285.67	−1.38393	−0.691965	+	0.721932i	$0.743255\pi$
$930$	0	0
$931$	92.9352i	0.0998230i
$932$	0	0
$933$	2171.68	2.32763
$934$	0	0
$935$	− 99.1664i	− 0.106060i
$936$	0	0
$937$	−887.657	−0.947339	−0.473670	−	0.880703i	$-0.657071\pi$
$937$	−887.657	−0.947339	−0.473670	+	0.880703i	$0.657071\pi$
$938$	0	0
$939$	951.882i	1.01372i
$940$	0	0
$941$	−380.526	−0.404385	−0.202192	−	0.979346i	$-0.564807\pi$
$941$	−380.526	−0.404385	−0.202192	+	0.979346i	$0.564807\pi$
$942$	0	0
$943$	166.026i	0.176061i
$944$	0	0
$945$	43.7058	0.0462495
$946$	0	0
$947$	− 363.169i	− 0.383494i	−0.981444	−	0.191747i	$-0.938585\pi$
$947$	− 363.169i	− 0.383494i	0.981444	−	0.191747i	$-0.0614153\pi$
$948$	0	0
$949$	1398.21	1.47335
$950$	0	0
$951$	− 7.94799i	− 0.00835750i
$952$	0	0
$953$	−679.479	−0.712990	−0.356495	−	0.934297i	$-0.616028\pi$
$953$	−679.479	−0.712990	−0.356495	+	0.934297i	$0.616028\pi$
$954$	0	0
$955$	− 79.6220i	− 0.0833738i
$956$	0	0
$957$	−431.874	−0.451279
$958$	0	0
$959$	− 314.116i	− 0.327545i
$960$	0	0
$961$	617.323	0.642376
$962$	0	0
$963$	1190.71i	1.23646i
$964$	0	0
$965$	142.103	0.147257
$966$	0	0
$967$	− 209.525i	− 0.216675i	−0.994114	−	0.108337i	$-0.965447\pi$
$967$	− 209.525i	− 0.216675i	0.994114	−	0.108337i	$-0.0345527\pi$
$968$	0	0
$969$	−285.770	−0.294912
$970$	0	0
$971$	538.867i	0.554961i	0.960731	+	0.277481i	$0.0894994\pi$
$971$	538.867i	0.554961i	−0.960731	+	0.277481i	$0.910501\pi$
$972$	0	0
$973$	−190.771	−0.196065
$974$	0	0
$975$	2096.73i	2.15049i
$976$	0	0
$977$	1028.30	1.05250	0.526252	−	0.850329i	$-0.323597\pi$
$977$	1028.30	1.05250	0.526252	+	0.850329i	$0.323597\pi$
$978$	0	0
$979$	766.001i	0.782432i
$980$	0	0
$981$	1772.29	1.80661
$982$	0	0
$983$	− 520.479i	− 0.529480i	−0.964320	−	0.264740i	$-0.914714\pi$
$983$	− 520.479i	− 0.529480i	0.964320	−	0.264740i	$-0.0852862\pi$
$984$	0	0
$985$	−247.872	−0.251646
$986$	0	0
$987$	− 372.449i	− 0.377355i
$988$	0	0
$989$	−403.986	−0.408479
$990$	0	0
$991$	1849.00i	1.86579i	0.360148	+	0.932895i	$0.382726\pi$
$991$	1849.00i	1.86579i	−0.360148	+	0.932895i	$0.617274\pi$
$992$	0	0
$993$	1704.60	1.71661
$994$	0	0
$995$	298.100i	0.299598i
$996$	0	0
$997$	−1010.23	−1.01327	−0.506635	−	0.862161i	$-0.669111\pi$
$997$	−1010.23	−1.01327	−0.506635	+	0.862161i	$0.669111\pi$
$998$	0	0
$999$	− 332.351i	− 0.332684i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.d.d.127.6		6
4.3	odd	2	inner	448.3.d.d.127.1		6
8.3	odd	2		28.3.c.a.15.1	✓	6
8.5	even	2		28.3.c.a.15.2	yes	6
16.3	odd	4		1792.3.g.g.127.12		12
16.5	even	4		1792.3.g.g.127.11		12
16.11	odd	4		1792.3.g.g.127.1		12
16.13	even	4		1792.3.g.g.127.2		12
24.5	odd	2		252.3.g.a.127.5		6
24.11	even	2		252.3.g.a.127.6		6
56.3	even	6		196.3.g.j.79.5		12
56.5	odd	6		196.3.g.j.67.5		12
56.11	odd	6		196.3.g.k.79.5		12
56.13	odd	2		196.3.c.g.99.2		6
56.19	even	6		196.3.g.j.67.4		12
56.27	even	2		196.3.c.g.99.1		6
56.37	even	6		196.3.g.k.67.5		12
56.45	odd	6		196.3.g.j.79.4		12
56.51	odd	6		196.3.g.k.67.4		12
56.53	even	6		196.3.g.k.79.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.3.c.a.15.1	✓	6	8.3	odd	2
28.3.c.a.15.2	yes	6	8.5	even	2
196.3.c.g.99.1		6	56.27	even	2
196.3.c.g.99.2		6	56.13	odd	2
196.3.g.j.67.4		12	56.19	even	6
196.3.g.j.67.5		12	56.5	odd	6
196.3.g.j.79.4		12	56.45	odd	6
196.3.g.j.79.5		12	56.3	even	6
196.3.g.k.67.4		12	56.51	odd	6
196.3.g.k.67.5		12	56.37	even	6
196.3.g.k.79.4		12	56.53	even	6
196.3.g.k.79.5		12	56.11	odd	6
252.3.g.a.127.5		6	24.5	odd	2
252.3.g.a.127.6		6	24.11	even	2
448.3.d.d.127.1		6	4.3	odd	2	inner
448.3.d.d.127.6		6	1.1	even	1	trivial
1792.3.g.g.127.1		12	16.11	odd	4
1792.3.g.g.127.2		12	16.13	even	4
1792.3.g.g.127.11		12	16.5	even	4
1792.3.g.g.127.12		12	16.3	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.54500i q^{3} -1.36794 q^{5} -2.64575i q^{7} -11.6570 q^{9} +O(q^{10})\)	\(q+4.54500i q^{3} -1.36794 q^{5} -2.64575i q^{7} -11.6570 q^{9} -15.3073i q^{11} -19.9460 q^{13} -6.21727i q^{15} -4.73588 q^{17} -13.2765i q^{19} +12.0249 q^{21} -14.9487i q^{23} -23.1287 q^{25} -12.0760i q^{27} -6.20763 q^{29} +18.5385i q^{31} +69.5715 q^{33} +3.61922i q^{35} +27.5216 q^{37} -90.6547i q^{39} -11.1064 q^{41} -27.0248i q^{43} +15.9460 q^{45} -30.9731i q^{47} -7.00000 q^{49} -21.5245i q^{51} +12.7857 q^{53} +20.9394i q^{55} +60.3415 q^{57} +28.8289i q^{59} -6.52415 q^{61} +30.8415i q^{63} +27.2850 q^{65} -102.863i q^{67} +67.9420 q^{69} +45.8085i q^{71} -70.0997 q^{73} -105.120i q^{75} -40.4992 q^{77} -33.3739i q^{79} -50.0275 q^{81} +159.301i q^{83} +6.47839 q^{85} -28.2137i q^{87} -50.0417 q^{89} +52.7723i q^{91} -84.2575 q^{93} +18.1614i q^{95} +89.7343 q^{97} +178.437i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{5} - 10 q^{9}+O(q^{10}) \) 6 * q + 4 * q^5 - 10 * q^9	\( 6 q + 4 q^{5} - 10 q^{9} - 12 q^{13} - 4 q^{17} - 30 q^{25} + 36 q^{29} + 80 q^{33} - 28 q^{37} - 20 q^{41} - 12 q^{45} - 42 q^{49} - 92 q^{53} + 160 q^{57} + 164 q^{61} - 136 q^{65} + 48 q^{69} - 132 q^{73} - 112 q^{77} - 218 q^{81} + 232 q^{85} + 348 q^{89} - 288 q^{93} + 252 q^{97}+O(q^{100}) \) 6 * q + 4 * q^5 - 10 * q^9 - 12 * q^13 - 4 * q^17 - 30 * q^25 + 36 * q^29 + 80 * q^33 - 28 * q^37 - 20 * q^41 - 12 * q^45 - 42 * q^49 - 92 * q^53 + 160 * q^57 + 164 * q^61 - 136 * q^65 + 48 * q^69 - 132 * q^73 - 112 * q^77 - 218 * q^81 + 232 * q^85 + 348 * q^89 - 288 * q^93 + 252 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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