LMFDB - Embedded newform 224.3.g.a.15.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,3,Mod(15,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("224.15");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.10355792167$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 16 $ x^4 + 6*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			15.4
Root			$-0.707107 - 1.87083i$ of defining polynomial
Character	$\chi$	$=$	224.15
Dual form			224.3.g.a.15.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.585786 q^{3} +9.03316i q^{5} -2.64575i q^{7} -8.65685 q^{9} +O(q^{10})$	$q-0.585786 q^{3} +9.03316i q^{5} -2.64575i q^{7} -8.65685 q^{9} -12.4853 q^{11} -9.03316i q^{13} -5.29150i q^{15} +12.3431 q^{17} -28.8701 q^{19} +1.54985i q^{21} +24.6418i q^{23} -56.5980 q^{25} +10.3431 q^{27} +22.4499i q^{29} +16.7824i q^{31} +7.31371 q^{33} +23.8995 q^{35} +16.2506i q^{37} +5.29150i q^{39} +6.97056 q^{41} +22.8284 q^{43} -78.1987i q^{45} +6.19938i q^{47} -7.00000 q^{49} -7.23045 q^{51} +8.01514i q^{53} -112.782i q^{55} +16.9117 q^{57} -30.4437 q^{59} -15.2325i q^{61} +22.9039i q^{63} +81.5980 q^{65} +78.6274 q^{67} -14.4348i q^{69} +17.5345i q^{71} +46.6863 q^{73} +33.1543 q^{75} +33.0329i q^{77} +81.0325i q^{79} +71.8528 q^{81} -40.3848 q^{83} +111.498i q^{85} -13.1509i q^{87} +111.941 q^{89} -23.8995 q^{91} -9.83089i q^{93} -260.788i q^{95} -164.108 q^{97} +108.083 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{3} - 12 q^{9}+O(q^{10}) $ 4 * q - 8 * q^3 - 12 * q^9	$ 4 q - 8 q^{3} - 12 q^{9} - 16 q^{11} + 72 q^{17} - 8 q^{19} - 68 q^{25} + 64 q^{27} - 16 q^{33} + 56 q^{35} - 40 q^{41} + 80 q^{43} - 28 q^{49} - 176 q^{51} - 136 q^{57} - 184 q^{59} + 168 q^{65} + 224 q^{67} + 232 q^{73} - 88 q^{75} - 52 q^{81} - 88 q^{83} + 312 q^{89} - 56 q^{91} - 136 q^{97} + 240 q^{99}+O(q^{100}) $ 4 * q - 8 * q^3 - 12 * q^9 - 16 * q^11 + 72 * q^17 - 8 * q^19 - 68 * q^25 + 64 * q^27 - 16 * q^33 + 56 * q^35 - 40 * q^41 + 80 * q^43 - 28 * q^49 - 176 * q^51 - 136 * q^57 - 184 * q^59 + 168 * q^65 + 224 * q^67 + 232 * q^73 - 88 * q^75 - 52 * q^81 - 88 * q^83 + 312 * q^89 - 56 * q^91 - 136 * q^97 + 240 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

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

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
$3$	−0.585786	−0.195262	−0.0976311	−	0.995223i	$-0.531127\pi$
$3$	−0.585786	−0.195262	−0.0976311	+	0.995223i	$0.531127\pi$
$4$	0	0
$5$	9.03316i	1.80663i	0.428976	+	0.903316i	$0.358875\pi$
$5$	9.03316i	1.80663i	−0.428976	+	0.903316i	$0.641125\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	−8.65685	−0.961873
$10$	0	0
$11$	−12.4853	−1.13503	−0.567513	−	0.823365i	$-0.692094\pi$
$11$	−12.4853	−1.13503	−0.567513	+	0.823365i	$0.692094\pi$
$12$	0	0
$13$	− 9.03316i	− 0.694858i	−0.937706	−	0.347429i	$-0.887055\pi$
$13$	− 9.03316i	− 0.694858i	0.937706	−	0.347429i	$-0.112945\pi$
$14$	0	0
$15$	− 5.29150i	− 0.352767i
$16$	0	0
$17$	12.3431	0.726067	0.363034	−	0.931776i	$-0.381741\pi$
$17$	12.3431	0.726067	0.363034	+	0.931776i	$0.381741\pi$
$18$	0	0
$19$	−28.8701	−1.51948	−0.759738	−	0.650229i	$-0.774673\pi$
$19$	−28.8701	−1.51948	−0.759738	+	0.650229i	$0.774673\pi$
$20$	0	0
$21$	1.54985i	0.0738022i
$22$	0	0
$23$	24.6418i	1.07138i	0.844414	+	0.535690i	$0.179949\pi$
$23$	24.6418i	1.07138i	−0.844414	+	0.535690i	$0.820051\pi$
$24$	0	0
$25$	−56.5980	−2.26392
$26$	0	0
$27$	10.3431	0.383079
$28$	0	0
$29$	22.4499i	0.774136i	0.922051	+	0.387068i	$0.126512\pi$
$29$	22.4499i	0.774136i	−0.922051	+	0.387068i	$0.873488\pi$
$30$	0	0
$31$	16.7824i	0.541367i	0.962668	+	0.270684i	$0.0872497\pi$
$31$	16.7824i	0.541367i	−0.962668	+	0.270684i	$0.912750\pi$
$32$	0	0
$33$	7.31371	0.221628
$34$	0	0
$35$	23.8995	0.682843
$36$	0	0
$37$	16.2506i	0.439204i	0.975589	+	0.219602i	$0.0704759\pi$
$37$	16.2506i	0.439204i	−0.975589	+	0.219602i	$0.929524\pi$
$38$	0	0
$39$	5.29150i	0.135680i
$40$	0	0
$41$	6.97056	0.170014	0.0850069	−	0.996380i	$-0.472909\pi$
$41$	6.97056	0.170014	0.0850069	+	0.996380i	$0.472909\pi$
$42$	0	0
$43$	22.8284	0.530894	0.265447	−	0.964126i	$-0.414481\pi$
$43$	22.8284	0.530894	0.265447	+	0.964126i	$0.414481\pi$
$44$	0	0
$45$	− 78.1987i	− 1.73775i
$46$	0	0
$47$	6.19938i	0.131902i	0.997823	+	0.0659509i	$0.0210081\pi$
$47$	6.19938i	0.131902i	−0.997823	+	0.0659509i	$0.978992\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	−7.23045	−0.141773
$52$	0	0
$53$	8.01514i	0.151229i	0.997137	+	0.0756145i	$0.0240918\pi$
$53$	8.01514i	0.151229i	−0.997137	+	0.0756145i	$0.975908\pi$
$54$	0	0
$55$	− 112.782i	− 2.05057i
$56$	0	0
$57$	16.9117	0.296696
$58$	0	0
$59$	−30.4437	−0.515994	−0.257997	−	0.966146i	$-0.583062\pi$
$59$	−30.4437	−0.515994	−0.257997	+	0.966146i	$0.583062\pi$
$60$	0	0
$61$	− 15.2325i	− 0.249714i	−0.992175	−	0.124857i	$-0.960153\pi$
$61$	− 15.2325i	− 0.249714i	0.992175	−	0.124857i	$-0.0398472\pi$
$62$	0	0
$63$	22.9039i	0.363554i
$64$	0	0
$65$	81.5980	1.25535
$66$	0	0
$67$	78.6274	1.17354	0.586772	−	0.809752i	$-0.300399\pi$
$67$	78.6274	1.17354	0.586772	+	0.809752i	$0.300399\pi$
$68$	0	0
$69$	− 14.4348i	− 0.209200i
$70$	0	0
$71$	17.5345i	0.246965i	0.992347	+	0.123482i	$0.0394062\pi$
$71$	17.5345i	0.246965i	−0.992347	+	0.123482i	$0.960594\pi$
$72$	0	0
$73$	46.6863	0.639538	0.319769	−	0.947495i	$-0.396395\pi$
$73$	46.6863	0.639538	0.319769	+	0.947495i	$0.396395\pi$
$74$	0	0
$75$	33.1543	0.442058
$76$	0	0
$77$	33.0329i	0.428999i
$78$	0	0
$79$	81.0325i	1.02573i	0.858470	+	0.512864i	$0.171416\pi$
$79$	81.0325i	1.02573i	−0.858470	+	0.512864i	$0.828584\pi$
$80$	0	0
$81$	71.8528	0.887072
$82$	0	0
$83$	−40.3848	−0.486564	−0.243282	−	0.969956i	$-0.578224\pi$
$83$	−40.3848	−0.486564	−0.243282	+	0.969956i	$0.578224\pi$
$84$	0	0
$85$	111.498i	1.31174i
$86$	0	0
$87$	− 13.1509i	− 0.151159i
$88$	0	0
$89$	111.941	1.25777	0.628883	−	0.777500i	$-0.283513\pi$
$89$	111.941	1.25777	0.628883	+	0.777500i	$0.283513\pi$
$90$	0	0
$91$	−23.8995	−0.262632
$92$	0	0
$93$	− 9.83089i	− 0.105709i
$94$	0	0
$95$	− 260.788i	− 2.74514i
$96$	0	0
$97$	−164.108	−1.69183	−0.845916	−	0.533317i	$-0.820945\pi$
$97$	−164.108	−1.69183	−0.845916	+	0.533317i	$0.820945\pi$
$98$	0	0
$99$	108.083	1.09175
$100$	0	0
$101$	12.1329i	0.120127i	0.998195	+	0.0600636i	$0.0191304\pi$
$101$	12.1329i	0.120127i	−0.998195	+	0.0600636i	$0.980870\pi$
$102$	0	0
$103$	106.582i	1.03478i	0.855750	+	0.517389i	$0.173096\pi$
$103$	106.582i	1.03478i	−0.855750	+	0.517389i	$0.826904\pi$
$104$	0	0
$105$	−14.0000	−0.133333
$106$	0	0
$107$	63.5980	0.594374	0.297187	−	0.954819i	$-0.403952\pi$
$107$	63.5980	0.594374	0.297187	+	0.954819i	$0.403952\pi$
$108$	0	0
$109$	− 130.848i	− 1.20044i	−0.799835	−	0.600220i	$-0.795080\pi$
$109$	− 130.848i	− 1.20044i	0.799835	−	0.600220i	$-0.204920\pi$
$110$	0	0
$111$	− 9.51936i	− 0.0857600i
$112$	0	0
$113$	−138.225	−1.22323	−0.611617	−	0.791154i	$-0.709481\pi$
$113$	−138.225	−1.22323	−0.611617	+	0.791154i	$0.709481\pi$
$114$	0	0
$115$	−222.593	−1.93559
$116$	0	0
$117$	78.1987i	0.668365i
$118$	0	0
$119$	− 32.6569i	− 0.274428i
$120$	0	0
$121$	34.8823	0.288283
$122$	0	0
$123$	−4.08326	−0.0331972
$124$	0	0
$125$	− 285.430i	− 2.28344i
$126$	0	0
$127$	114.442i	0.901114i	0.892748	+	0.450557i	$0.148775\pi$
$127$	114.442i	0.901114i	−0.892748	+	0.450557i	$0.851225\pi$
$128$	0	0
$129$	−13.3726	−0.103663
$130$	0	0
$131$	168.350	1.28512	0.642558	−	0.766237i	$-0.277873\pi$
$131$	168.350	1.28512	0.642558	+	0.766237i	$0.277873\pi$
$132$	0	0
$133$	76.3830i	0.574308i
$134$	0	0
$135$	93.4313i	0.692084i
$136$	0	0
$137$	34.6863	0.253185	0.126592	−	0.991955i	$-0.459596\pi$
$137$	34.6863	0.253185	0.126592	+	0.991955i	$0.459596\pi$
$138$	0	0
$139$	−107.664	−0.774561	−0.387281	−	0.921962i	$-0.626586\pi$
$139$	−107.664	−0.774561	−0.387281	+	0.921962i	$0.626586\pi$
$140$	0	0
$141$	− 3.63151i	− 0.0257554i
$142$	0	0
$143$	112.782i	0.788682i
$144$	0	0
$145$	−202.794	−1.39858
$146$	0	0
$147$	4.10051	0.0278946
$148$	0	0
$149$	252.176i	1.69246i	0.532819	+	0.846229i	$0.321133\pi$
$149$	252.176i	1.69246i	−0.532819	+	0.846229i	$0.678867\pi$
$150$	0	0
$151$	234.486i	1.55289i	0.630186	+	0.776444i	$0.282979\pi$
$151$	234.486i	1.55289i	−0.630186	+	0.776444i	$0.717021\pi$
$152$	0	0
$153$	−106.853	−0.698384
$154$	0	0
$155$	−151.598	−0.978051
$156$	0	0
$157$	− 10.0968i	− 0.0643109i	−0.999483	−	0.0321554i	$-0.989763\pi$
$157$	− 10.0968i	− 0.0643109i	0.999483	−	0.0321554i	$-0.0102372\pi$
$158$	0	0
$159$	− 4.69516i	− 0.0295293i
$160$	0	0
$161$	65.1960	0.404944
$162$	0	0
$163$	−104.534	−0.641313	−0.320657	−	0.947196i	$-0.603904\pi$
$163$	−104.534	−0.641313	−0.320657	+	0.947196i	$0.603904\pi$
$164$	0	0
$165$	66.0659i	0.400399i
$166$	0	0
$167$	− 296.765i	− 1.77703i	−0.458843	−	0.888517i	$-0.651736\pi$
$167$	− 296.765i	− 1.77703i	0.458843	−	0.888517i	$-0.348264\pi$
$168$	0	0
$169$	87.4020	0.517172
$170$	0	0
$171$	249.924	1.46154
$172$	0	0
$173$	− 40.0301i	− 0.231388i	−0.993285	−	0.115694i	$-0.963091\pi$
$173$	− 40.0301i	− 0.231388i	0.993285	−	0.115694i	$-0.0369091\pi$
$174$	0	0
$175$	149.744i	0.855681i
$176$	0	0
$177$	17.8335	0.100754
$178$	0	0
$179$	−294.794	−1.64689	−0.823447	−	0.567394i	$-0.807952\pi$
$179$	−294.794	−1.64689	−0.823447	+	0.567394i	$0.807952\pi$
$180$	0	0
$181$	− 40.4706i	− 0.223595i	−0.993731	−	0.111797i	$-0.964339\pi$
$181$	− 40.4706i	− 0.223595i	0.993731	−	0.111797i	$-0.0356608\pi$
$182$	0	0
$183$	8.92302i	0.0487596i
$184$	0	0
$185$	−146.794	−0.793481
$186$	0	0
$187$	−154.108	−0.824105
$188$	0	0
$189$	− 27.3654i	− 0.144790i
$190$	0	0
$191$	− 156.929i	− 0.821619i	−0.911721	−	0.410810i	$-0.865246\pi$
$191$	− 156.929i	− 0.821619i	0.911721	−	0.410810i	$-0.134754\pi$
$192$	0	0
$193$	−261.304	−1.35390	−0.676952	−	0.736027i	$-0.736700\pi$
$193$	−261.304	−1.35390	−0.676952	+	0.736027i	$0.736700\pi$
$194$	0	0
$195$	−47.7990	−0.245123
$196$	0	0
$197$	145.283i	0.737475i	0.929533	+	0.368738i	$0.120210\pi$
$197$	145.283i	0.737475i	−0.929533	+	0.368738i	$0.879790\pi$
$198$	0	0
$199$	− 390.508i	− 1.96235i	−0.193122	−	0.981175i	$-0.561861\pi$
$199$	− 390.508i	− 1.96235i	0.193122	−	0.981175i	$-0.438139\pi$
$200$	0	0
$201$	−46.0589	−0.229149
$202$	0	0
$203$	59.3970	0.292596
$204$	0	0
$205$	62.9662i	0.307152i
$206$	0	0
$207$	− 213.320i	− 1.03053i
$208$	0	0
$209$	360.451	1.72464
$210$	0	0
$211$	164.049	0.777482	0.388741	−	0.921347i	$-0.372910\pi$
$211$	164.049	0.777482	0.388741	+	0.921347i	$0.372910\pi$
$212$	0	0
$213$	− 10.2715i	− 0.0482229i
$214$	0	0
$215$	206.213i	0.959129i
$216$	0	0
$217$	44.4020	0.204618
$218$	0	0
$219$	−27.3482	−0.124878
$220$	0	0
$221$	− 111.498i	− 0.504514i
$222$	0	0
$223$	− 10.5830i	− 0.0474574i	−0.999718	−	0.0237287i	$-0.992446\pi$
$223$	− 10.5830i	− 0.0474574i	0.999718	−	0.0237287i	$-0.00755379\pi$
$224$	0	0
$225$	489.960	2.17760
$226$	0	0
$227$	213.806	0.941877	0.470939	−	0.882166i	$-0.343915\pi$
$227$	213.806	0.941877	0.470939	+	0.882166i	$0.343915\pi$
$228$	0	0
$229$	232.028i	1.01322i	0.862174	+	0.506612i	$0.169102\pi$
$229$	232.028i	1.01322i	−0.862174	+	0.506612i	$0.830898\pi$
$230$	0	0
$231$	− 19.3503i	− 0.0837673i
$232$	0	0
$233$	−192.863	−0.827738	−0.413869	−	0.910336i	$-0.635823\pi$
$233$	−192.863	−0.827738	−0.413869	+	0.910336i	$0.635823\pi$
$234$	0	0
$235$	−56.0000	−0.238298
$236$	0	0
$237$	− 47.4678i	− 0.200286i
$238$	0	0
$239$	327.917i	1.37204i	0.727583	+	0.686020i	$0.240644\pi$
$239$	327.917i	1.37204i	−0.727583	+	0.686020i	$0.759356\pi$
$240$	0	0
$241$	71.8721	0.298225	0.149112	−	0.988820i	$-0.452358\pi$
$241$	71.8721	0.298225	0.149112	+	0.988820i	$0.452358\pi$
$242$	0	0
$243$	−135.179	−0.556291
$244$	0	0
$245$	− 63.2321i	− 0.258090i
$246$	0	0
$247$	260.788i	1.05582i
$248$	0	0
$249$	23.6569	0.0950074
$250$	0	0
$251$	256.919	1.02358	0.511790	−	0.859110i	$-0.328982\pi$
$251$	256.919	1.02358	0.511790	+	0.859110i	$0.328982\pi$
$252$	0	0
$253$	− 307.659i	− 1.21604i
$254$	0	0
$255$	− 65.3138i	− 0.256133i
$256$	0	0
$257$	319.352	1.24262	0.621308	−	0.783566i	$-0.286602\pi$
$257$	319.352	1.24262	0.621308	+	0.783566i	$0.286602\pi$
$258$	0	0
$259$	42.9949	0.166004
$260$	0	0
$261$	− 194.346i	− 0.744620i
$262$	0	0
$263$	377.357i	1.43482i	0.696653	+	0.717408i	$0.254672\pi$
$263$	377.357i	1.43482i	−0.696653	+	0.717408i	$0.745328\pi$
$264$	0	0
$265$	−72.4020	−0.273215
$266$	0	0
$267$	−65.5736	−0.245594
$268$	0	0
$269$	28.1631i	0.104696i	0.998629	+	0.0523478i	$0.0166704\pi$
$269$	28.1631i	0.104696i	−0.998629	+	0.0523478i	$0.983330\pi$
$270$	0	0
$271$	− 399.715i	− 1.47496i	−0.675367	−	0.737482i	$-0.736015\pi$
$271$	− 399.715i	− 1.47496i	0.675367	−	0.737482i	$-0.263985\pi$
$272$	0	0
$273$	14.0000	0.0512821
$274$	0	0
$275$	706.642	2.56961
$276$	0	0
$277$	− 102.951i	− 0.371663i	−0.982582	−	0.185831i	$-0.940502\pi$
$277$	− 102.951i	− 0.371663i	0.982582	−	0.185831i	$-0.0594979\pi$
$278$	0	0
$279$	− 145.283i	− 0.520726i
$280$	0	0
$281$	−150.235	−0.534646	−0.267323	−	0.963607i	$-0.586139\pi$
$281$	−150.235	−0.534646	−0.267323	+	0.963607i	$0.586139\pi$
$282$	0	0
$283$	178.561	0.630959	0.315480	−	0.948932i	$-0.397835\pi$
$283$	178.561	0.630959	0.315480	+	0.948932i	$0.397835\pi$
$284$	0	0
$285$	152.766i	0.536021i
$286$	0	0
$287$	− 18.4424i	− 0.0642591i
$288$	0	0
$289$	−136.647	−0.472826
$290$	0	0
$291$	96.1320	0.330351
$292$	0	0
$293$	− 219.189i	− 0.748085i	−0.927411	−	0.374043i	$-0.877971\pi$
$293$	− 219.189i	− 0.748085i	0.927411	−	0.374043i	$-0.122029\pi$
$294$	0	0
$295$	− 275.002i	− 0.932211i
$296$	0	0
$297$	−129.137	−0.434805
$298$	0	0
$299$	222.593	0.744458
$300$	0	0
$301$	− 60.3983i	− 0.200659i
$302$	0	0
$303$	− 7.10726i	− 0.0234563i
$304$	0	0
$305$	137.598	0.451141
$306$	0	0
$307$	−316.669	−1.03150	−0.515748	−	0.856741i	$-0.672486\pi$
$307$	−316.669	−1.03150	−0.515748	+	0.856741i	$0.672486\pi$
$308$	0	0
$309$	− 62.4344i	− 0.202053i
$310$	0	0
$311$	− 72.2653i	− 0.232364i	−0.993228	−	0.116182i	$-0.962934\pi$
$311$	− 72.2653i	− 0.232364i	0.993228	−	0.116182i	$-0.0370656\pi$
$312$	0	0
$313$	81.9512	0.261825	0.130913	−	0.991394i	$-0.458209\pi$
$313$	81.9512	0.261825	0.130913	+	0.991394i	$0.458209\pi$
$314$	0	0
$315$	−206.894	−0.656808
$316$	0	0
$317$	− 109.150i	− 0.344322i	−0.985069	−	0.172161i	$-0.944925\pi$
$317$	− 109.150i	− 0.344322i	0.985069	−	0.172161i	$-0.0550749\pi$
$318$	0	0
$319$	− 280.294i	− 0.878664i
$320$	0	0
$321$	−37.2548	−0.116059
$322$	0	0
$323$	−356.347	−1.10324
$324$	0	0
$325$	511.259i	1.57310i
$326$	0	0
$327$	76.6489i	0.234400i
$328$	0	0
$329$	16.4020	0.0498542
$330$	0	0
$331$	−321.740	−0.972025	−0.486012	−	0.873952i	$-0.661549\pi$
$331$	−321.740	−0.972025	−0.486012	+	0.873952i	$0.661549\pi$
$332$	0	0
$333$	− 140.679i	− 0.422459i
$334$	0	0
$335$	710.254i	2.12016i
$336$	0	0
$337$	−164.049	−0.486792	−0.243396	−	0.969927i	$-0.578261\pi$
$337$	−164.049	−0.486792	−0.243396	+	0.969927i	$0.578261\pi$
$338$	0	0
$339$	80.9706	0.238851
$340$	0	0
$341$	− 209.533i	− 0.614466i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	130.392	0.377948
$346$	0	0
$347$	330.309	0.951898	0.475949	−	0.879473i	$-0.342105\pi$
$347$	330.309	0.951898	0.475949	+	0.879473i	$0.342105\pi$
$348$	0	0
$349$	262.402i	0.751869i	0.926646	+	0.375934i	$0.122678\pi$
$349$	262.402i	0.751869i	−0.926646	+	0.375934i	$0.877322\pi$
$350$	0	0
$351$	− 93.4313i	− 0.266186i
$352$	0	0
$353$	−578.098	−1.63767	−0.818835	−	0.574029i	$-0.805380\pi$
$353$	−578.098	−1.63767	−0.818835	+	0.574029i	$0.805380\pi$
$354$	0	0
$355$	−158.392	−0.446174
$356$	0	0
$357$	19.1300i	0.0535853i
$358$	0	0
$359$	− 365.114i	− 1.01703i	−0.861053	−	0.508515i	$-0.830195\pi$
$359$	− 365.114i	− 1.01703i	0.861053	−	0.508515i	$-0.169805\pi$
$360$	0	0
$361$	472.480	1.30881
$362$	0	0
$363$	−20.4335	−0.0562908
$364$	0	0
$365$	421.725i	1.15541i
$366$	0	0
$367$	520.071i	1.41709i	0.705666	+	0.708544i	$0.250648\pi$
$367$	520.071i	1.41709i	−0.705666	+	0.708544i	$0.749352\pi$
$368$	0	0
$369$	−60.3431	−0.163532
$370$	0	0
$371$	21.2061	0.0571592
$372$	0	0
$373$	526.711i	1.41210i	0.708164	+	0.706048i	$0.249524\pi$
$373$	526.711i	1.41210i	−0.708164	+	0.706048i	$0.750476\pi$
$374$	0	0
$375$	167.201i	0.445869i
$376$	0	0
$377$	202.794	0.537915
$378$	0	0
$379$	−121.976	−0.321835	−0.160918	−	0.986968i	$-0.551445\pi$
$379$	−121.976	−0.321835	−0.160918	+	0.986968i	$0.551445\pi$
$380$	0	0
$381$	− 67.0383i	− 0.175954i
$382$	0	0
$383$	− 316.427i	− 0.826179i	−0.910690	−	0.413089i	$-0.864450\pi$
$383$	− 316.427i	− 0.826179i	0.910690	−	0.413089i	$-0.135550\pi$
$384$	0	0
$385$	−298.392	−0.775044
$386$	0	0
$387$	−197.622	−0.510652
$388$	0	0
$389$	− 92.1474i	− 0.236883i	−0.992961	−	0.118441i	$-0.962210\pi$
$389$	− 92.1474i	− 0.236883i	0.992961	−	0.118441i	$-0.0377898\pi$
$390$	0	0
$391$	304.157i	0.777895i
$392$	0	0
$393$	−98.6173	−0.250935
$394$	0	0
$395$	−731.980	−1.85311
$396$	0	0
$397$	− 562.267i	− 1.41629i	−0.706068	−	0.708144i	$-0.749533\pi$
$397$	− 562.267i	− 1.41629i	0.706068	−	0.708144i	$-0.250467\pi$
$398$	0	0
$399$	− 44.7441i	− 0.112141i
$400$	0	0
$401$	81.2061	0.202509	0.101254	−	0.994861i	$-0.467714\pi$
$401$	81.2061	0.202509	0.101254	+	0.994861i	$0.467714\pi$
$402$	0	0
$403$	151.598	0.376174
$404$	0	0
$405$	649.058i	1.60261i
$406$	0	0
$407$	− 202.893i	− 0.498508i
$408$	0	0
$409$	450.735	1.10204	0.551021	−	0.834491i	$-0.314238\pi$
$409$	450.735	1.10204	0.551021	+	0.834491i	$0.314238\pi$
$410$	0	0
$411$	−20.3188	−0.0494374
$412$	0	0
$413$	80.5463i	0.195027i
$414$	0	0
$415$	− 364.802i	− 0.879041i
$416$	0	0
$417$	63.0681	0.151242
$418$	0	0
$419$	−624.988	−1.49162	−0.745809	−	0.666160i	$-0.767937\pi$
$419$	−624.988	−1.49162	−0.745809	+	0.666160i	$0.767937\pi$
$420$	0	0
$421$	− 566.476i	− 1.34555i	−0.739848	−	0.672774i	$-0.765103\pi$
$421$	− 566.476i	− 1.34555i	0.739848	−	0.672774i	$-0.234897\pi$
$422$	0	0
$423$	− 53.6671i	− 0.126873i
$424$	0	0
$425$	−698.597	−1.64376
$426$	0	0
$427$	−40.3015	−0.0943829
$428$	0	0
$429$	− 66.0659i	− 0.154000i
$430$	0	0
$431$	289.528i	0.671760i	0.941905	+	0.335880i	$0.109034\pi$
$431$	289.528i	0.671760i	−0.941905	+	0.335880i	$0.890966\pi$
$432$	0	0
$433$	−597.696	−1.38036	−0.690180	−	0.723638i	$-0.742468\pi$
$433$	−597.696	−1.38036	−0.690180	+	0.723638i	$0.742468\pi$
$434$	0	0
$435$	118.794	0.273090
$436$	0	0
$437$	− 711.409i	− 1.62794i
$438$	0	0
$439$	− 38.3890i	− 0.0874464i	−0.999044	−	0.0437232i	$-0.986078\pi$
$439$	− 38.3890i	− 0.0874464i	0.999044	−	0.0437232i	$-0.0139220\pi$
$440$	0	0
$441$	60.5980	0.137410
$442$	0	0
$443$	−599.058	−1.35228	−0.676138	−	0.736775i	$-0.736348\pi$
$443$	−599.058	−1.35228	−0.676138	+	0.736775i	$0.736348\pi$
$444$	0	0
$445$	1011.18i	2.27232i
$446$	0	0
$447$	− 147.721i	− 0.330473i
$448$	0	0
$449$	−460.039	−1.02459	−0.512293	−	0.858811i	$-0.671204\pi$
$449$	−460.039	−1.02459	−0.512293	+	0.858811i	$0.671204\pi$
$450$	0	0
$451$	−87.0294	−0.192970
$452$	0	0
$453$	− 137.359i	− 0.303220i
$454$	0	0
$455$	− 215.888i	− 0.474479i
$456$	0	0
$457$	266.323	0.582764	0.291382	−	0.956607i	$-0.405885\pi$
$457$	266.323	0.582764	0.291382	+	0.956607i	$0.405885\pi$
$458$	0	0
$459$	127.667	0.278142
$460$	0	0
$461$	763.123i	1.65537i	0.561196	+	0.827683i	$0.310341\pi$
$461$	763.123i	1.65537i	−0.561196	+	0.827683i	$0.689659\pi$
$462$	0	0
$463$	123.988i	0.267792i	0.990995	+	0.133896i	$0.0427488\pi$
$463$	123.988i	0.267792i	−0.990995	+	0.133896i	$0.957251\pi$
$464$	0	0
$465$	88.8040	0.190976
$466$	0	0
$467$	768.718	1.64608	0.823038	−	0.567986i	$-0.192277\pi$
$467$	768.718	1.64608	0.823038	+	0.567986i	$0.192277\pi$
$468$	0	0
$469$	− 208.029i	− 0.443558i
$470$	0	0
$471$	5.91457i	0.0125575i
$472$	0	0
$473$	−285.019	−0.602578
$474$	0	0
$475$	1633.99	3.43997
$476$	0	0
$477$	− 69.3859i	− 0.145463i
$478$	0	0
$479$	− 118.981i	− 0.248394i	−0.992258	−	0.124197i	$-0.960364\pi$
$479$	− 118.981i	− 0.248394i	0.992258	−	0.124197i	$-0.0396355\pi$
$480$	0	0
$481$	146.794	0.305185
$482$	0	0
$483$	−38.1909	−0.0790702
$484$	0	0
$485$	− 1482.41i	− 3.05652i
$486$	0	0
$487$	− 282.577i	− 0.580240i	−0.956990	−	0.290120i	$-0.906305\pi$
$487$	− 282.577i	− 0.580240i	0.956990	−	0.290120i	$-0.0936952\pi$
$488$	0	0
$489$	61.2346	0.125224
$490$	0	0
$491$	388.049	0.790323	0.395162	−	0.918612i	$-0.370689\pi$
$491$	388.049	0.790323	0.395162	+	0.918612i	$0.370689\pi$
$492$	0	0
$493$	277.103i	0.562075i
$494$	0	0
$495$	976.333i	1.97239i
$496$	0	0
$497$	46.3919	0.0933439
$498$	0	0
$499$	27.7157	0.0555425	0.0277713	−	0.999614i	$-0.491159\pi$
$499$	27.7157	0.0555425	0.0277713	+	0.999614i	$0.491159\pi$
$500$	0	0
$501$	173.841i	0.346988i
$502$	0	0
$503$	727.477i	1.44628i	0.690703	+	0.723138i	$0.257301\pi$
$503$	727.477i	1.44628i	−0.690703	+	0.723138i	$0.742699\pi$
$504$	0	0
$505$	−109.598	−0.217026
$506$	0	0
$507$	−51.1989	−0.100984
$508$	0	0
$509$	634.183i	1.24594i	0.782246	+	0.622969i	$0.214074\pi$
$509$	634.183i	1.24594i	−0.782246	+	0.622969i	$0.785926\pi$
$510$	0	0
$511$	− 123.520i	− 0.241723i
$512$	0	0
$513$	−298.607	−0.582080
$514$	0	0
$515$	−962.774	−1.86946
$516$	0	0
$517$	− 77.4010i	− 0.149712i
$518$	0	0
$519$	23.4491i	0.0451813i
$520$	0	0
$521$	833.127	1.59909	0.799546	−	0.600605i	$-0.205073\pi$
$521$	833.127	1.59909	0.799546	+	0.600605i	$0.205073\pi$
$522$	0	0
$523$	−876.434	−1.67578	−0.837891	−	0.545838i	$-0.816211\pi$
$523$	−876.434	−1.67578	−0.837891	+	0.545838i	$0.816211\pi$
$524$	0	0
$525$	− 87.7181i	− 0.167082i
$526$	0	0
$527$	207.147i	0.393069i
$528$	0	0
$529$	−78.2162	−0.147857
$530$	0	0
$531$	263.546	0.496321
$532$	0	0
$533$	− 62.9662i	− 0.118135i
$534$	0	0
$535$	574.491i	1.07381i
$536$	0	0
$537$	172.686	0.321576
$538$	0	0
$539$	87.3970	0.162147
$540$	0	0
$541$	− 405.915i	− 0.750305i	−0.926963	−	0.375152i	$-0.877590\pi$
$541$	− 405.915i	− 0.750305i	0.926963	−	0.375152i	$-0.122410\pi$
$542$	0	0
$543$	23.7072i	0.0436596i
$544$	0	0
$545$	1181.97	2.16875
$546$	0	0
$547$	606.024	1.10791	0.553953	−	0.832548i	$-0.313119\pi$
$547$	606.024	1.10791	0.553953	+	0.832548i	$0.313119\pi$
$548$	0	0
$549$	131.866i	0.240193i
$550$	0	0
$551$	− 648.131i	− 1.17628i
$552$	0	0
$553$	214.392	0.387689
$554$	0	0
$555$	85.9899	0.154937
$556$	0	0
$557$	36.4442i	0.0654294i	0.999465	+	0.0327147i	$0.0104153\pi$
$557$	36.4442i	0.0654294i	−0.999465	+	0.0327147i	$0.989585\pi$
$558$	0	0
$559$	− 206.213i	− 0.368896i
$560$	0	0
$561$	90.2742	0.160917
$562$	0	0
$563$	186.389	0.331064	0.165532	−	0.986204i	$-0.447066\pi$
$563$	186.389	0.331064	0.165532	+	0.986204i	$0.447066\pi$
$564$	0	0
$565$	− 1248.61i	− 2.20993i
$566$	0	0
$567$	− 190.105i	− 0.335282i
$568$	0	0
$569$	−670.891	−1.17907	−0.589536	−	0.807742i	$-0.700689\pi$
$569$	−670.891	−1.17907	−0.589536	+	0.807742i	$0.700689\pi$
$570$	0	0
$571$	677.082	1.18578	0.592892	−	0.805282i	$-0.297986\pi$
$571$	677.082	1.18578	0.592892	+	0.805282i	$0.297986\pi$
$572$	0	0
$573$	91.9271i	0.160431i
$574$	0	0
$575$	− 1394.67i	− 2.42552i
$576$	0	0
$577$	927.901	1.60815	0.804073	−	0.594530i	$-0.202662\pi$
$577$	927.901	1.60815	0.804073	+	0.594530i	$0.202662\pi$
$578$	0	0
$579$	153.068	0.264366
$580$	0	0
$581$	106.848i	0.183904i
$582$	0	0
$583$	− 100.071i	− 0.171649i
$584$	0	0
$585$	−706.382	−1.20749
$586$	0	0
$587$	321.120	0.547053	0.273526	−	0.961865i	$-0.411810\pi$
$587$	321.120	0.547053	0.273526	+	0.961865i	$0.411810\pi$
$588$	0	0
$589$	− 484.508i	− 0.822595i
$590$	0	0
$591$	− 85.1046i	− 0.144001i
$592$	0	0
$593$	−219.255	−0.369738	−0.184869	−	0.982763i	$-0.559186\pi$
$593$	−219.255	−0.369738	−0.184869	+	0.982763i	$0.559186\pi$
$594$	0	0
$595$	294.995	0.495790
$596$	0	0
$597$	228.754i	0.383173i
$598$	0	0
$599$	− 154.802i	− 0.258434i	−0.991616	−	0.129217i	$-0.958754\pi$
$599$	− 154.802i	− 0.258434i	0.991616	−	0.129217i	$-0.0412464\pi$
$600$	0	0
$601$	205.862	0.342533	0.171266	−	0.985225i	$-0.445214\pi$
$601$	205.862	0.342533	0.171266	+	0.985225i	$0.445214\pi$
$602$	0	0
$603$	−680.666	−1.12880
$604$	0	0
$605$	315.097i	0.520821i
$606$	0	0
$607$	− 790.663i	− 1.30258i	−0.758831	−	0.651288i	$-0.774229\pi$
$607$	− 790.663i	− 1.30258i	0.758831	−	0.651288i	$-0.225771\pi$
$608$	0	0
$609$	−34.7939	−0.0571329
$610$	0	0
$611$	56.0000	0.0916530
$612$	0	0
$613$	741.471i	1.20958i	0.796386	+	0.604789i	$0.206743\pi$
$613$	741.471i	1.20958i	−0.796386	+	0.604789i	$0.793257\pi$
$614$	0	0
$615$	− 36.8848i	− 0.0599752i
$616$	0	0
$617$	171.578	0.278084	0.139042	−	0.990286i	$-0.455598\pi$
$617$	171.578	0.278084	0.139042	+	0.990286i	$0.455598\pi$
$618$	0	0
$619$	540.198	0.872695	0.436347	−	0.899778i	$-0.356272\pi$
$619$	540.198	0.872695	0.436347	+	0.899778i	$0.356272\pi$
$620$	0	0
$621$	254.873i	0.410424i
$622$	0	0
$623$	− 296.168i	− 0.475391i
$624$	0	0
$625$	1163.38	1.86141
$626$	0	0
$627$	−211.147	−0.336758
$628$	0	0
$629$	200.583i	0.318892i
$630$	0	0
$631$	− 269.399i	− 0.426940i	−0.976950	−	0.213470i	$-0.931523\pi$
$631$	− 269.399i	− 0.426940i	0.976950	−	0.213470i	$-0.0684766\pi$
$632$	0	0
$633$	−96.0975	−0.151813
$634$	0	0
$635$	−1033.77	−1.62798
$636$	0	0
$637$	63.2321i	0.0992655i
$638$	0	0
$639$	− 151.794i	− 0.237549i
$640$	0	0
$641$	36.1867	0.0564535	0.0282268	−	0.999602i	$-0.491014\pi$
$641$	36.1867	0.0564535	0.0282268	+	0.999602i	$0.491014\pi$
$642$	0	0
$643$	−266.297	−0.414148	−0.207074	−	0.978325i	$-0.566394\pi$
$643$	−266.297	−0.414148	−0.207074	+	0.978325i	$0.566394\pi$
$644$	0	0
$645$	− 120.797i	− 0.187282i
$646$	0	0
$647$	1086.24i	1.67888i	0.543452	+	0.839440i	$0.317117\pi$
$647$	1086.24i	1.67888i	−0.543452	+	0.839440i	$0.682883\pi$
$648$	0	0
$649$	380.098	0.585666
$650$	0	0
$651$	−26.0101	−0.0399541
$652$	0	0
$653$	1195.35i	1.83055i	0.402832	+	0.915274i	$0.368026\pi$
$653$	1195.35i	1.83055i	−0.402832	+	0.915274i	$0.631974\pi$
$654$	0	0
$655$	1520.74i	2.32173i
$656$	0	0
$657$	−404.156	−0.615154
$658$	0	0
$659$	685.220	1.03979	0.519894	−	0.854231i	$-0.325971\pi$
$659$	685.220	1.03979	0.519894	+	0.854231i	$0.325971\pi$
$660$	0	0
$661$	− 993.382i	− 1.50285i	−0.659820	−	0.751423i	$-0.729368\pi$
$661$	− 993.382i	− 1.50285i	0.659820	−	0.751423i	$-0.270632\pi$
$662$	0	0
$663$	65.3138i	0.0985125i
$664$	0	0
$665$	−689.980	−1.03756
$666$	0	0
$667$	−553.206	−0.829394
$668$	0	0
$669$	6.19938i	0.00926664i
$670$	0	0
$671$	190.183i	0.283432i
$672$	0	0
$673$	106.569	0.158349	0.0791743	−	0.996861i	$-0.474772\pi$
$673$	106.569	0.158349	0.0791743	+	0.996861i	$0.474772\pi$
$674$	0	0
$675$	−585.401	−0.867261
$676$	0	0
$677$	1004.18i	1.48329i	0.670795	+	0.741643i	$0.265953\pi$
$677$	1004.18i	1.48329i	−0.670795	+	0.741643i	$0.734047\pi$
$678$	0	0
$679$	434.188i	0.639452i
$680$	0	0
$681$	−125.245	−0.183913
$682$	0	0
$683$	678.225	0.993009	0.496505	−	0.868034i	$-0.334617\pi$
$683$	678.225	0.993009	0.496505	+	0.868034i	$0.334617\pi$
$684$	0	0
$685$	313.327i	0.457411i
$686$	0	0
$687$	− 135.919i	− 0.197844i
$688$	0	0
$689$	72.4020	0.105083
$690$	0	0
$691$	365.175	0.528473	0.264236	−	0.964458i	$-0.414880\pi$
$691$	365.175	0.528473	0.264236	+	0.964458i	$0.414880\pi$
$692$	0	0
$693$	− 285.961i	− 0.412643i
$694$	0	0
$695$	− 972.546i	− 1.39935i
$696$	0	0
$697$	86.0387	0.123441
$698$	0	0
$699$	112.976	0.161626
$700$	0	0
$701$	940.292i	1.34136i	0.741748	+	0.670679i	$0.233997\pi$
$701$	940.292i	1.34136i	−0.741748	+	0.670679i	$0.766003\pi$
$702$	0	0
$703$	− 469.155i	− 0.667361i
$704$	0	0
$705$	32.8040	0.0465306
$706$	0	0
$707$	32.1005	0.0454038
$708$	0	0
$709$	− 1057.46i	− 1.49148i	−0.666239	−	0.745738i	$-0.732097\pi$
$709$	− 1057.46i	− 1.49148i	0.666239	−	0.745738i	$-0.267903\pi$
$710$	0	0
$711$	− 701.487i	− 0.986620i
$712$	0	0
$713$	−413.547	−0.580010
$714$	0	0
$715$	−1018.77	−1.42486
$716$	0	0
$717$	− 192.090i	− 0.267907i
$718$	0	0
$719$	1034.82i	1.43926i	0.694360	+	0.719628i	$0.255688\pi$
$719$	1034.82i	1.43926i	−0.694360	+	0.719628i	$0.744312\pi$
$720$	0	0
$721$	281.990	0.391109
$722$	0	0
$723$	−42.1017	−0.0582320
$724$	0	0
$725$	− 1270.62i	− 1.75258i
$726$	0	0
$727$	495.145i	0.681080i	0.940230	+	0.340540i	$0.110610\pi$
$727$	495.145i	0.681080i	−0.940230	+	0.340540i	$0.889390\pi$
$728$	0	0
$729$	−567.489	−0.778449
$730$	0	0
$731$	281.775	0.385465
$732$	0	0
$733$	− 567.494i	− 0.774207i	−0.922036	−	0.387103i	$-0.873476\pi$
$733$	− 567.494i	− 0.774207i	0.922036	−	0.387103i	$-0.126524\pi$
$734$	0	0
$735$	37.0405i	0.0503953i
$736$	0	0
$737$	−981.685	−1.33200
$738$	0	0
$739$	544.701	0.737078	0.368539	−	0.929612i	$-0.379858\pi$
$739$	544.701	0.737078	0.368539	+	0.929612i	$0.379858\pi$
$740$	0	0
$741$	− 152.766i	− 0.206162i
$742$	0	0
$743$	731.264i	0.984205i	0.870537	+	0.492102i	$0.163771\pi$
$743$	731.264i	0.984205i	−0.870537	+	0.492102i	$0.836229\pi$
$744$	0	0
$745$	−2277.95	−3.05765
$746$	0	0
$747$	349.605	0.468012
$748$	0	0
$749$	− 168.264i	− 0.224652i
$750$	0	0
$751$	666.262i	0.887166i	0.896233	+	0.443583i	$0.146293\pi$
$751$	666.262i	0.887166i	−0.896233	+	0.443583i	$0.853707\pi$
$752$	0	0
$753$	−150.500	−0.199867
$754$	0	0
$755$	−2118.15	−2.80550
$756$	0	0
$757$	238.623i	0.315222i	0.987501	+	0.157611i	$0.0503791\pi$
$757$	238.623i	0.315222i	−0.987501	+	0.157611i	$0.949621\pi$
$758$	0	0
$759$	180.223i	0.237447i
$760$	0	0
$761$	614.930	0.808055	0.404028	−	0.914747i	$-0.367610\pi$
$761$	614.930	0.808055	0.404028	+	0.914747i	$0.367610\pi$
$762$	0	0
$763$	−346.191	−0.453723
$764$	0	0
$765$	− 965.219i	− 1.26172i
$766$	0	0
$767$	275.002i	0.358543i
$768$	0	0
$769$	178.950	0.232705	0.116353	−	0.993208i	$-0.462880\pi$
$769$	178.950	0.232705	0.116353	+	0.993208i	$0.462880\pi$
$770$	0	0
$771$	−187.072	−0.242636
$772$	0	0
$773$	631.615i	0.817095i	0.912737	+	0.408548i	$0.133965\pi$
$773$	631.615i	0.817095i	−0.912737	+	0.408548i	$0.866035\pi$
$774$	0	0
$775$	− 949.849i	− 1.22561i
$776$	0	0
$777$	−25.1859	−0.0324142
$778$	0	0
$779$	−201.241	−0.258332
$780$	0	0
$781$	− 218.923i	− 0.280311i
$782$	0	0
$783$	232.203i	0.296556i
$784$	0	0
$785$	91.2061	0.116186
$786$	0	0
$787$	−456.655	−0.580247	−0.290124	−	0.956989i	$-0.593696\pi$
$787$	−456.655	−0.580247	−0.290124	+	0.956989i	$0.593696\pi$
$788$	0	0
$789$	− 221.050i	− 0.280165i
$790$	0	0
$791$	365.710i	0.462339i
$792$	0	0
$793$	−137.598	−0.173516
$794$	0	0
$795$	42.4121	0.0533486
$796$	0	0
$797$	− 218.566i	− 0.274236i	−0.990555	−	0.137118i	$-0.956216\pi$
$797$	− 218.566i	− 0.274236i	0.990555	−	0.137118i	$-0.0437839\pi$
$798$	0	0
$799$	76.5199i	0.0957695i
$800$	0	0
$801$	−969.058	−1.20981
$802$	0	0
$803$	−582.891	−0.725892
$804$	0	0
$805$	588.926i	0.731585i
$806$	0	0
$807$	− 16.4976i	− 0.0204431i
$808$	0	0
$809$	1347.46	1.66559	0.832794	−	0.553584i	$-0.186740\pi$
$809$	1347.46	1.66559	0.832794	+	0.553584i	$0.186740\pi$
$810$	0	0
$811$	672.620	0.829371	0.414686	−	0.909965i	$-0.363892\pi$
$811$	672.620	0.829371	0.414686	+	0.909965i	$0.363892\pi$
$812$	0	0
$813$	234.148i	0.288005i
$814$	0	0
$815$	− 944.273i	− 1.15862i
$816$	0	0
$817$	−659.058	−0.806681
$818$	0	0
$819$	206.894	0.252618
$820$	0	0
$821$	1162.57i	1.41604i	0.706190	+	0.708022i	$0.250412\pi$
$821$	1162.57i	1.41604i	−0.706190	+	0.708022i	$0.749588\pi$
$822$	0	0
$823$	− 1041.65i	− 1.26567i	−0.774286	−	0.632835i	$-0.781891\pi$
$823$	− 1041.65i	− 1.26567i	0.774286	−	0.632835i	$-0.218109\pi$
$824$	0	0
$825$	−413.941	−0.501747
$826$	0	0
$827$	−278.432	−0.336678	−0.168339	−	0.985729i	$-0.553840\pi$
$827$	−278.432	−0.336678	−0.168339	+	0.985729i	$0.553840\pi$
$828$	0	0
$829$	1065.74i	1.28557i	0.766046	+	0.642785i	$0.222221\pi$
$829$	1065.74i	1.28557i	−0.766046	+	0.642785i	$0.777779\pi$
$830$	0	0
$831$	60.3071i	0.0725717i
$832$	0	0
$833$	−86.4020	−0.103724
$834$	0	0
$835$	2680.72	3.21045
$836$	0	0
$837$	173.583i	0.207387i
$838$	0	0
$839$	305.844i	0.364533i	0.983249	+	0.182267i	$0.0583434\pi$
$839$	305.844i	0.364533i	−0.983249	+	0.182267i	$0.941657\pi$
$840$	0	0
$841$	337.000	0.400713
$842$	0	0
$843$	88.0059	0.104396
$844$	0	0
$845$	789.516i	0.934339i
$846$	0	0
$847$	− 92.2898i	− 0.108961i
$848$	0	0
$849$	−104.599	−0.123202
$850$	0	0
$851$	−400.442	−0.470555
$852$	0	0
$853$	164.018i	0.192283i	0.995368	+	0.0961417i	$0.0306502\pi$
$853$	164.018i	0.192283i	−0.995368	+	0.0961417i	$0.969350\pi$
$854$	0	0
$855$	2257.60i	2.64047i
$856$	0	0
$857$	851.068	0.993078	0.496539	−	0.868014i	$-0.334604\pi$
$857$	851.068	0.993078	0.496539	+	0.868014i	$0.334604\pi$
$858$	0	0
$859$	1179.69	1.37333	0.686666	−	0.726973i	$-0.259073\pi$
$859$	1179.69	1.37333	0.686666	+	0.726973i	$0.259073\pi$
$860$	0	0
$861$	10.8033i	0.0125474i
$862$	0	0
$863$	279.048i	0.323346i	0.986844	+	0.161673i	$0.0516890\pi$
$863$	279.048i	0.323346i	−0.986844	+	0.161673i	$0.948311\pi$
$864$	0	0
$865$	361.598	0.418032
$866$	0	0
$867$	80.0458	0.0923250
$868$	0	0
$869$	− 1011.71i	− 1.16423i
$870$	0	0
$871$	− 710.254i	− 0.815447i
$872$	0	0
$873$	1420.66	1.62733
$874$	0	0
$875$	−755.176	−0.863058
$876$	0	0
$877$	674.159i	0.768711i	0.923185	+	0.384355i	$0.125576\pi$
$877$	674.159i	0.768711i	−0.923185	+	0.384355i	$0.874424\pi$
$878$	0	0
$879$	128.398i	0.146073i
$880$	0	0
$881$	−1001.29	−1.13654	−0.568271	−	0.822841i	$-0.692387\pi$
$881$	−1001.29	−1.13654	−0.568271	+	0.822841i	$0.692387\pi$
$882$	0	0
$883$	−882.010	−0.998879	−0.499439	−	0.866349i	$-0.666461\pi$
$883$	−882.010	−0.998879	−0.499439	+	0.866349i	$0.666461\pi$
$884$	0	0
$885$	161.093i	0.182026i
$886$	0	0
$887$	7.08053i	0.00798256i	0.999992	+	0.00399128i	$0.00127047\pi$
$887$	7.08053i	0.00798256i	−0.999992	+	0.00399128i	$0.998730\pi$
$888$	0	0
$889$	302.784	0.340589
$890$	0	0
$891$	−897.103	−1.00685
$892$	0	0
$893$	− 178.976i	− 0.200422i
$894$	0	0
$895$	− 2662.92i	− 2.97533i
$896$	0	0
$897$	−130.392	−0.145364
$898$	0	0
$899$	−376.764	−0.419092
$900$	0	0
$901$	98.9320i	0.109802i
$902$	0	0
$903$	35.3805i	0.0391811i
$904$	0	0
$905$	365.578	0.403953
$906$	0	0
$907$	−450.372	−0.496551	−0.248275	−	0.968689i	$-0.579864\pi$
$907$	−450.372	−0.496551	−0.248275	+	0.968689i	$0.579864\pi$
$908$	0	0
$909$	− 105.032i	− 0.115547i
$910$	0	0
$911$	202.426i	0.222201i	0.993809	+	0.111101i	$0.0354376\pi$
$911$	202.426i	0.222201i	−0.993809	+	0.111101i	$0.964562\pi$
$912$	0	0
$913$	504.215	0.552262
$914$	0	0
$915$	−80.6030	−0.0880907
$916$	0	0
$917$	− 445.413i	− 0.485728i
$918$	0	0
$919$	1593.73i	1.73420i	0.498138	+	0.867098i	$0.334017\pi$
$919$	1593.73i	1.73420i	−0.498138	+	0.867098i	$0.665983\pi$
$920$	0	0
$921$	185.500	0.201412
$922$	0	0
$923$	158.392	0.171606
$924$	0	0
$925$	− 919.749i	− 0.994323i
$926$	0	0
$927$	− 922.666i	− 0.995325i
$928$	0	0
$929$	−1039.40	−1.11884	−0.559419	−	0.828885i	$-0.688976\pi$
$929$	−1039.40	−1.11884	−0.559419	+	0.828885i	$0.688976\pi$
$930$	0	0
$931$	202.090	0.217068
$932$	0	0
$933$	42.3320i	0.0453719i
$934$	0	0
$935$	− 1392.08i	− 1.48885i
$936$	0	0
$937$	881.765	0.941051	0.470525	−	0.882386i	$-0.344064\pi$
$937$	881.765	0.941051	0.470525	+	0.882386i	$0.344064\pi$
$938$	0	0
$939$	−48.0059	−0.0511245
$940$	0	0
$941$	− 953.344i	− 1.01312i	−0.862205	−	0.506559i	$-0.830917\pi$
$941$	− 953.344i	− 1.01312i	0.862205	−	0.506559i	$-0.169083\pi$
$942$	0	0
$943$	171.767i	0.182149i
$944$	0	0
$945$	247.196	0.261583
$946$	0	0
$947$	−16.8957	−0.0178413	−0.00892063	−	0.999960i	$-0.502840\pi$
$947$	−16.8957	−0.0178413	−0.00892063	+	0.999960i	$0.502840\pi$
$948$	0	0
$949$	− 421.725i	− 0.444389i
$950$	0	0
$951$	63.9386i	0.0672330i
$952$	0	0
$953$	1526.31	1.60159	0.800794	−	0.598940i	$-0.204411\pi$
$953$	1526.31	1.60159	0.800794	+	0.598940i	$0.204411\pi$
$954$	0	0
$955$	1417.57	1.48436
$956$	0	0
$957$	164.192i	0.171570i
$958$	0	0
$959$	− 91.7713i	− 0.0956948i
$960$	0	0
$961$	679.352	0.706921
$962$	0	0
$963$	−550.558	−0.571712
$964$	0	0
$965$	− 2360.40i	− 2.44601i
$966$	0	0
$967$	− 1410.39i	− 1.45852i	−0.684235	−	0.729262i	$-0.739864\pi$
$967$	− 1410.39i	− 1.45852i	0.684235	−	0.729262i	$-0.260136\pi$
$968$	0	0
$969$	208.743	0.215421
$970$	0	0
$971$	596.497	0.614312	0.307156	−	0.951659i	$-0.400623\pi$
$971$	596.497	0.614312	0.307156	+	0.951659i	$0.400623\pi$
$972$	0	0
$973$	284.852i	0.292757i
$974$	0	0
$975$	− 299.488i	− 0.307168i
$976$	0	0
$977$	146.686	0.150140	0.0750698	−	0.997178i	$-0.476082\pi$
$977$	146.686	0.150140	0.0750698	+	0.997178i	$0.476082\pi$
$978$	0	0
$979$	−1397.62	−1.42760
$980$	0	0
$981$	1132.73i	1.15467i
$982$	0	0
$983$	− 169.457i	− 0.172388i	−0.996278	−	0.0861939i	$-0.972530\pi$
$983$	− 169.457i	− 0.172388i	0.996278	−	0.0861939i	$-0.0274704\pi$
$984$	0	0
$985$	−1312.36	−1.33235
$986$	0	0
$987$	−9.60808	−0.00973463
$988$	0	0
$989$	562.533i	0.568789i
$990$	0	0
$991$	1686.90i	1.70222i	0.524988	+	0.851109i	$0.324070\pi$
$991$	1686.90i	1.70222i	−0.524988	+	0.851109i	$0.675930\pi$
$992$	0	0
$993$	188.471	0.189800
$994$	0	0
$995$	3527.52	3.54524
$996$	0	0
$997$	1736.14i	1.74136i	0.491849	+	0.870680i	$0.336321\pi$
$997$	1736.14i	1.74136i	−0.491849	+	0.870680i	$0.663679\pi$
$998$	0	0
$999$	168.082i	0.168250i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.g.a.15.4		4
3.2	odd	2		2016.3.g.a.1135.1		4
4.3	odd	2		56.3.g.a.43.2	yes	4
7.6	odd	2		1568.3.g.h.687.1		4
8.3	odd	2	inner	224.3.g.a.15.3		4
8.5	even	2		56.3.g.a.43.1	✓	4
12.11	even	2		504.3.g.a.379.3		4
16.3	odd	4		1792.3.d.g.1023.6		8
16.5	even	4		1792.3.d.g.1023.5		8
16.11	odd	4		1792.3.d.g.1023.3		8
16.13	even	4		1792.3.d.g.1023.4		8
24.5	odd	2		504.3.g.a.379.4		4
24.11	even	2		2016.3.g.a.1135.4		4
28.3	even	6		392.3.k.j.275.2		8
28.11	odd	6		392.3.k.i.275.2		8
28.19	even	6		392.3.k.j.67.4		8
28.23	odd	6		392.3.k.i.67.4		8
28.27	even	2		392.3.g.h.99.2		4
56.5	odd	6		392.3.k.j.67.2		8
56.13	odd	2		392.3.g.h.99.1		4
56.27	even	2		1568.3.g.h.687.2		4
56.37	even	6		392.3.k.i.67.2		8
56.45	odd	6		392.3.k.j.275.4		8
56.53	even	6		392.3.k.i.275.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.g.a.43.1	✓	4	8.5	even	2
56.3.g.a.43.2	yes	4	4.3	odd	2
224.3.g.a.15.3		4	8.3	odd	2	inner
224.3.g.a.15.4		4	1.1	even	1	trivial
392.3.g.h.99.1		4	56.13	odd	2
392.3.g.h.99.2		4	28.27	even	2
392.3.k.i.67.2		8	56.37	even	6
392.3.k.i.67.4		8	28.23	odd	6
392.3.k.i.275.2		8	28.11	odd	6
392.3.k.i.275.4		8	56.53	even	6
392.3.k.j.67.2		8	56.5	odd	6
392.3.k.j.67.4		8	28.19	even	6
392.3.k.j.275.2		8	28.3	even	6
392.3.k.j.275.4		8	56.45	odd	6
504.3.g.a.379.3		4	12.11	even	2
504.3.g.a.379.4		4	24.5	odd	2
1568.3.g.h.687.1		4	7.6	odd	2
1568.3.g.h.687.2		4	56.27	even	2
1792.3.d.g.1023.3		8	16.11	odd	4
1792.3.d.g.1023.4		8	16.13	even	4
1792.3.d.g.1023.5		8	16.5	even	4
1792.3.d.g.1023.6		8	16.3	odd	4
2016.3.g.a.1135.1		4	3.2	odd	2
2016.3.g.a.1135.4		4	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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.585786 q^{3} +9.03316i q^{5} -2.64575i q^{7} -8.65685 q^{9} +O(q^{10})\)	\(q-0.585786 q^{3} +9.03316i q^{5} -2.64575i q^{7} -8.65685 q^{9} -12.4853 q^{11} -9.03316i q^{13} -5.29150i q^{15} +12.3431 q^{17} -28.8701 q^{19} +1.54985i q^{21} +24.6418i q^{23} -56.5980 q^{25} +10.3431 q^{27} +22.4499i q^{29} +16.7824i q^{31} +7.31371 q^{33} +23.8995 q^{35} +16.2506i q^{37} +5.29150i q^{39} +6.97056 q^{41} +22.8284 q^{43} -78.1987i q^{45} +6.19938i q^{47} -7.00000 q^{49} -7.23045 q^{51} +8.01514i q^{53} -112.782i q^{55} +16.9117 q^{57} -30.4437 q^{59} -15.2325i q^{61} +22.9039i q^{63} +81.5980 q^{65} +78.6274 q^{67} -14.4348i q^{69} +17.5345i q^{71} +46.6863 q^{73} +33.1543 q^{75} +33.0329i q^{77} +81.0325i q^{79} +71.8528 q^{81} -40.3848 q^{83} +111.498i q^{85} -13.1509i q^{87} +111.941 q^{89} -23.8995 q^{91} -9.83089i q^{93} -260.788i q^{95} -164.108 q^{97} +108.083 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{3} - 12 q^{9}+O(q^{10}) \) 4 * q - 8 * q^3 - 12 * q^9	\( 4 q - 8 q^{3} - 12 q^{9} - 16 q^{11} + 72 q^{17} - 8 q^{19} - 68 q^{25} + 64 q^{27} - 16 q^{33} + 56 q^{35} - 40 q^{41} + 80 q^{43} - 28 q^{49} - 176 q^{51} - 136 q^{57} - 184 q^{59} + 168 q^{65} + 224 q^{67} + 232 q^{73} - 88 q^{75} - 52 q^{81} - 88 q^{83} + 312 q^{89} - 56 q^{91} - 136 q^{97} + 240 q^{99}+O(q^{100}) \) 4 * q - 8 * q^3 - 12 * q^9 - 16 * q^11 + 72 * q^17 - 8 * q^19 - 68 * q^25 + 64 * q^27 - 16 * q^33 + 56 * q^35 - 40 * q^41 + 80 * q^43 - 28 * q^49 - 176 * q^51 - 136 * q^57 - 184 * q^59 + 168 * q^65 + 224 * q^67 + 232 * q^73 - 88 * q^75 - 52 * q^81 - 88 * q^83 + 312 * q^89 - 56 * q^91 - 136 * q^97 + 240 * q^99

Introduction

L-functions

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