LMFDB - Embedded newform 1575.3.f.a.449.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,3,Mod(449,1575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1575.449"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1575.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)

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

Self dual:	no
Analytic conductor:	$42.9156416367$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.157351936.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.5
Root			$-1.28897 - 0.581861i$ of defining polynomial
Character	$\chi$	$=$	1575.449
Dual form			1575.3.f.a.449.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.16372 q^{2} -2.64575 q^{4} -2.64575i q^{7} -7.73381 q^{8} -0.412247i q^{11} -4.58301i q^{13} -3.07892i q^{14} +1.58301 q^{16} -32.4382 q^{17} +20.4575 q^{19} -0.479741i q^{22} +35.1779 q^{23} -5.33334i q^{26} +7.00000i q^{28} -25.0436i q^{29} -43.2915 q^{31} +32.7774 q^{32} -37.7490 q^{34} +21.4170i q^{37} +23.8069 q^{38} +59.5430i q^{41} +51.7490i q^{43} +1.09070i q^{44} +40.9373 q^{46} -5.47938 q^{47} -7.00000 q^{49} +12.1255i q^{52} -40.6572 q^{53} +20.4617i q^{56} -29.1438i q^{58} +73.3616i q^{59} +88.5385 q^{61} -50.3793 q^{62} +31.8118 q^{64} +80.0810i q^{67} +85.8233 q^{68} +24.2191i q^{71} -33.0405i q^{73} +24.9234i q^{74} -54.1255 q^{76} -1.09070 q^{77} +22.0000 q^{79} +69.2915i q^{82} +30.9352 q^{83} +60.2215i q^{86} +3.18824i q^{88} -73.2156i q^{89} -12.1255 q^{91} -93.0719 q^{92} -6.37648 q^{94} -34.7085i q^{97} -8.14605 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 72 q^{16} - 48 q^{19} - 304 q^{31} - 48 q^{34} + 264 q^{46} - 56 q^{49} - 96 q^{61} + 64 q^{64} - 560 q^{76} + 176 q^{79} - 224 q^{91} - 432 q^{94}+O(q^{100}) $ 8 * q - 72 * q^16 - 48 * q^19 - 304 * q^31 - 48 * q^34 + 264 * q^46 - 56 * q^49 - 96 * q^61 + 64 * q^64 - 560 * q^76 + 176 * q^79 - 224 * q^91 - 432 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	1.16372	0.581861	0.290930	−	0.956744i	$-0.406035\pi$
$2$	1.16372	0.581861	0.290930	+	0.956744i	$0.406035\pi$
$3$	0	0
$3$	0	0
$4$	−2.64575	−0.661438
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	−7.73381	−0.966726
$9$	0	0
$10$	0	0
$11$	− 0.412247i	− 0.0374770i	−0.999824	−	0.0187385i	$-0.994035\pi$
$11$	− 0.412247i	− 0.0374770i	0.999824	−	0.0187385i	$-0.00596500\pi$
$12$	0	0
$13$	− 4.58301i	− 0.352539i	−0.984342	−	0.176269i	$-0.943597\pi$
$13$	− 4.58301i	− 0.352539i	0.984342	−	0.176269i	$-0.0564030\pi$
$14$	− 3.07892i	− 0.219923i
$15$	0	0
$16$	1.58301	0.0989378
$17$	−32.4382	−1.90813	−0.954064	−	0.299603i	$-0.903146\pi$
$17$	−32.4382	−1.90813	−0.954064	+	0.299603i	$0.903146\pi$
$18$	0	0
$19$	20.4575	1.07671	0.538356	−	0.842718i	$-0.319046\pi$
$19$	20.4575	1.07671	0.538356	+	0.842718i	$0.319046\pi$
$20$	0	0
$21$	0	0
$22$	− 0.479741i	− 0.0218064i
$23$	35.1779	1.52947	0.764736	−	0.644343i	$-0.222869\pi$
$23$	35.1779	1.52947	0.764736	+	0.644343i	$0.222869\pi$
$24$	0	0
$25$	0	0
$26$	− 5.33334i	− 0.205129i
$27$	0	0
$28$	7.00000i	0.250000i
$29$	− 25.0436i	− 0.863572i	−0.901976	−	0.431786i	$-0.857884\pi$
$29$	− 25.0436i	− 0.863572i	0.901976	−	0.431786i	$-0.142116\pi$
$30$	0	0
$31$	−43.2915	−1.39650	−0.698250	−	0.715854i	$-0.746038\pi$
$31$	−43.2915	−1.39650	−0.698250	+	0.715854i	$0.746038\pi$
$32$	32.7774	1.02429
$33$	0	0
$34$	−37.7490	−1.11027
$35$	0	0
$36$	0	0
$37$	21.4170i	0.578838i	0.957203	+	0.289419i	$0.0934620\pi$
$37$	21.4170i	0.578838i	−0.957203	+	0.289419i	$0.906538\pi$
$38$	23.8069	0.626496
$39$	0	0
$40$	0	0
$41$	59.5430i	1.45227i	0.687553	+	0.726134i	$0.258685\pi$
$41$	59.5430i	1.45227i	−0.687553	+	0.726134i	$0.741315\pi$
$42$	0	0
$43$	51.7490i	1.20347i	0.798698	+	0.601733i	$0.205523\pi$
$43$	51.7490i	1.20347i	−0.798698	+	0.601733i	$0.794477\pi$
$44$	1.09070i	0.0247887i
$45$	0	0
$46$	40.9373	0.889940
$47$	−5.47938	−0.116583	−0.0582913	−	0.998300i	$-0.518565\pi$
$47$	−5.47938	−0.116583	−0.0582913	+	0.998300i	$0.518565\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	12.1255i	0.233183i
$53$	−40.6572	−0.767118	−0.383559	−	0.923516i	$-0.625302\pi$
$53$	−40.6572	−0.767118	−0.383559	+	0.923516i	$0.625302\pi$
$54$	0	0
$55$	0	0
$56$	20.4617i	0.365388i
$57$	0	0
$58$	− 29.1438i	− 0.502479i
$59$	73.3616i	1.24342i	0.783249	+	0.621709i	$0.213561\pi$
$59$	73.3616i	1.24342i	−0.783249	+	0.621709i	$0.786439\pi$
$60$	0	0
$61$	88.5385	1.45145	0.725726	−	0.687984i	$-0.241504\pi$
$61$	88.5385	1.45145	0.725726	+	0.687984i	$0.241504\pi$
$62$	−50.3793	−0.812569
$63$	0	0
$64$	31.8118	0.497059
$65$	0	0
$66$	0	0
$67$	80.0810i	1.19524i	0.801780	+	0.597620i	$0.203887\pi$
$67$	80.0810i	1.19524i	−0.801780	+	0.597620i	$0.796113\pi$
$68$	85.8233	1.26211
$69$	0	0
$70$	0	0
$71$	24.2191i	0.341114i	0.985348	+	0.170557i	$0.0545567\pi$
$71$	24.2191i	0.341114i	−0.985348	+	0.170557i	$0.945443\pi$
$72$	0	0
$73$	− 33.0405i	− 0.452610i	−0.974057	−	0.226305i	$-0.927335\pi$
$73$	− 33.0405i	− 0.452610i	0.974057	−	0.226305i	$-0.0726646\pi$
$74$	24.9234i	0.336803i
$75$	0	0
$76$	−54.1255	−0.712178
$77$	−1.09070	−0.0141650
$78$	0	0
$79$	22.0000	0.278481	0.139241	−	0.990259i	$-0.455534\pi$
$79$	22.0000	0.278481	0.139241	+	0.990259i	$0.455534\pi$
$80$	0	0
$81$	0	0
$82$	69.2915i	0.845018i
$83$	30.9352	0.372714	0.186357	−	0.982482i	$-0.440332\pi$
$83$	30.9352	0.372714	0.186357	+	0.982482i	$0.440332\pi$
$84$	0	0
$85$	0	0
$86$	60.2215i	0.700250i
$87$	0	0
$88$	3.18824i	0.0362300i
$89$	− 73.2156i	− 0.822647i	−0.911489	−	0.411324i	$-0.865067\pi$
$89$	− 73.2156i	− 0.822647i	0.911489	−	0.411324i	$-0.134933\pi$
$90$	0	0
$91$	−12.1255	−0.133247
$92$	−93.0719	−1.01165
$93$	0	0
$94$	−6.37648	−0.0678349
$95$	0	0
$96$	0	0
$97$	− 34.7085i	− 0.357820i	−0.983865	−	0.178910i	$-0.942743\pi$
$97$	− 34.7085i	− 0.357820i	0.983865	−	0.178910i	$-0.0572570\pi$
$98$	−8.14605	−0.0831230
$99$	0	0
$100$	0	0
$101$	185.757i	1.83918i	0.392877	+	0.919591i	$0.371480\pi$
$101$	185.757i	1.83918i	−0.392877	+	0.919591i	$0.628520\pi$
$102$	0	0
$103$	125.875i	1.22208i	0.791599	+	0.611041i	$0.209249\pi$
$103$	125.875i	1.22208i	−0.791599	+	0.611041i	$0.790751\pi$
$104$	35.4441i	0.340808i
$105$	0	0
$106$	−47.3137	−0.446356
$107$	−11.3710	−0.106271	−0.0531356	−	0.998587i	$-0.516922\pi$
$107$	−11.3710	−0.106271	−0.0531356	+	0.998587i	$0.516922\pi$
$108$	0	0
$109$	1.66798	0.0153026	0.00765128	−	0.999971i	$-0.497564\pi$
$109$	1.66798	0.0153026	0.00765128	+	0.999971i	$0.497564\pi$
$110$	0	0
$111$	0	0
$112$	− 4.18824i	− 0.0373950i
$113$	0.944665	0.00835987	0.00417993	−	0.999991i	$-0.498669\pi$
$113$	0.944665	0.00835987	0.00417993	+	0.999991i	$0.498669\pi$
$114$	0	0
$115$	0	0
$116$	66.2591i	0.571199i
$117$	0	0
$118$	85.3725i	0.723496i
$119$	85.8233i	0.721205i
$120$	0	0
$121$	120.830	0.998595
$122$	103.034	0.844543
$123$	0	0
$124$	114.539	0.923698
$125$	0	0
$126$	0	0
$127$	154.413i	1.21585i	0.793994	+	0.607925i	$0.207998\pi$
$127$	154.413i	1.21585i	−0.793994	+	0.607925i	$0.792002\pi$
$128$	−94.0896	−0.735075
$129$	0	0
$130$	0	0
$131$	162.045i	1.23698i	0.785791	+	0.618492i	$0.212256\pi$
$131$	162.045i	1.23698i	−0.785791	+	0.618492i	$0.787744\pi$
$132$	0	0
$133$	− 54.1255i	− 0.406959i
$134$	93.1921i	0.695463i
$135$	0	0
$136$	250.871	1.84464
$137$	−203.819	−1.48773	−0.743864	−	0.668331i	$-0.767009\pi$
$137$	−203.819	−1.48773	−0.743864	+	0.668331i	$0.767009\pi$
$138$	0	0
$139$	197.166	1.41846	0.709230	−	0.704977i	$-0.249043\pi$
$139$	197.166	1.41846	0.709230	+	0.704977i	$0.249043\pi$
$140$	0	0
$141$	0	0
$142$	28.1843i	0.198481i
$143$	−1.88933	−0.0132121
$144$	0	0
$145$	0	0
$146$	− 38.4500i	− 0.263356i
$147$	0	0
$148$	− 56.6640i	− 0.382865i
$149$	102.528i	0.688105i	0.938950	+	0.344053i	$0.111800\pi$
$149$	102.528i	0.688105i	−0.938950	+	0.344053i	$0.888200\pi$
$150$	0	0
$151$	−131.749	−0.872510	−0.436255	−	0.899823i	$-0.643695\pi$
$151$	−131.749	−0.872510	−0.436255	+	0.899823i	$0.643695\pi$
$152$	−158.214	−1.04088
$153$	0	0
$154$	−1.26927	−0.00824204
$155$	0	0
$156$	0	0
$157$	− 80.7895i	− 0.514583i	−0.966334	−	0.257292i	$-0.917170\pi$
$157$	− 80.7895i	− 0.514583i	0.966334	−	0.257292i	$-0.0828301\pi$
$158$	25.6019	0.162037
$159$	0	0
$160$	0	0
$161$	− 93.0719i	− 0.578086i
$162$	0	0
$163$	− 64.5830i	− 0.396215i	−0.980180	−	0.198107i	$-0.936521\pi$
$163$	− 64.5830i	− 0.396215i	0.980180	−	0.198107i	$-0.0634795\pi$
$164$	− 157.536i	− 0.960585i
$165$	0	0
$166$	36.0000	0.216867
$167$	−187.793	−1.12451	−0.562254	−	0.826965i	$-0.690066\pi$
$167$	−187.793	−1.12451	−0.562254	+	0.826965i	$0.690066\pi$
$168$	0	0
$169$	147.996	0.875716
$170$	0	0
$171$	0	0
$172$	− 136.915i	− 0.796018i
$173$	−166.846	−0.964426	−0.482213	−	0.876054i	$-0.660167\pi$
$173$	−166.846	−0.964426	−0.482213	+	0.876054i	$0.660167\pi$
$174$	0	0
$175$	0	0
$176$	− 0.652589i	− 0.00370789i
$177$	0	0
$178$	− 85.2026i	− 0.478666i
$179$	250.608i	1.40004i	0.714121	+	0.700022i	$0.246826\pi$
$179$	250.608i	1.40004i	−0.714121	+	0.700022i	$0.753174\pi$
$180$	0	0
$181$	128.170	0.708121	0.354061	−	0.935222i	$-0.384801\pi$
$181$	128.170	0.708121	0.354061	+	0.935222i	$0.384801\pi$
$182$	−14.1107	−0.0775313
$183$	0	0
$184$	−272.059	−1.47858
$185$	0	0
$186$	0	0
$187$	13.3725i	0.0715109i
$188$	14.4971	0.0771121
$189$	0	0
$190$	0	0
$191$	136.761i	0.716026i	0.933717	+	0.358013i	$0.116546\pi$
$191$	136.761i	0.716026i	−0.933717	+	0.358013i	$0.883454\pi$
$192$	0	0
$193$	− 96.5020i	− 0.500010i	−0.968245	−	0.250005i	$-0.919568\pi$
$193$	− 96.5020i	− 0.500010i	0.968245	−	0.250005i	$-0.0804323\pi$
$194$	− 40.3910i	− 0.208201i
$195$	0	0
$196$	18.5203	0.0944911
$197$	35.4699	0.180050	0.0900252	−	0.995939i	$-0.471305\pi$
$197$	35.4699	0.180050	0.0900252	+	0.995939i	$0.471305\pi$
$198$	0	0
$199$	21.4170	0.107623	0.0538115	−	0.998551i	$-0.482863\pi$
$199$	21.4170	0.107623	0.0538115	+	0.998551i	$0.482863\pi$
$200$	0	0
$201$	0	0
$202$	216.170i	1.07015i
$203$	−66.2591	−0.326400
$204$	0	0
$205$	0	0
$206$	146.483i	0.711082i
$207$	0	0
$208$	− 7.25492i	− 0.0348794i
$209$	− 8.43355i	− 0.0403519i
$210$	0	0
$211$	364.162	1.72589	0.862943	−	0.505301i	$-0.168618\pi$
$211$	364.162	1.72589	0.862943	+	0.505301i	$0.168618\pi$
$212$	107.569	0.507401
$213$	0	0
$214$	−13.2327	−0.0618350
$215$	0	0
$216$	0	0
$217$	114.539i	0.527827i
$218$	1.94106	0.00890396
$219$	0	0
$220$	0	0
$221$	148.664i	0.672689i
$222$	0	0
$223$	155.911i	0.699153i	0.936908	+	0.349576i	$0.113675\pi$
$223$	155.911i	0.699153i	−0.936908	+	0.349576i	$0.886325\pi$
$224$	− 86.7209i	− 0.387147i
$225$	0	0
$226$	1.09933	0.00486428
$227$	−18.5678	−0.0817965	−0.0408983	−	0.999163i	$-0.513022\pi$
$227$	−18.5678	−0.0817965	−0.0408983	+	0.999163i	$0.513022\pi$
$228$	0	0
$229$	−229.247	−1.00108	−0.500539	−	0.865714i	$-0.666865\pi$
$229$	−229.247	−1.00108	−0.500539	+	0.865714i	$0.666865\pi$
$230$	0	0
$231$	0	0
$232$	193.682i	0.834838i
$233$	230.339	0.988581	0.494290	−	0.869297i	$-0.335428\pi$
$233$	230.339	0.988581	0.494290	+	0.869297i	$0.335428\pi$
$234$	0	0
$235$	0	0
$236$	− 194.097i	− 0.822443i
$237$	0	0
$238$	99.8745i	0.419641i
$239$	− 272.766i	− 1.14128i	−0.821201	−	0.570640i	$-0.806695\pi$
$239$	− 272.766i	− 1.14128i	0.821201	−	0.570640i	$-0.193305\pi$
$240$	0	0
$241$	−51.6235	−0.214205	−0.107103	−	0.994248i	$-0.534157\pi$
$241$	−51.6235	−0.214205	−0.107103	+	0.994248i	$0.534157\pi$
$242$	140.613	0.581044
$243$	0	0
$244$	−234.251	−0.960045
$245$	0	0
$246$	0	0
$247$	− 93.7569i	− 0.379583i
$248$	334.808	1.35003
$249$	0	0
$250$	0	0
$251$	50.6713i	0.201878i	0.994893	+	0.100939i	$0.0321847\pi$
$251$	50.6713i	0.201878i	−0.994893	+	0.100939i	$0.967815\pi$
$252$	0	0
$253$	− 14.5020i	− 0.0573200i
$254$	179.694i	0.707456i
$255$	0	0
$256$	−236.741	−0.924770
$257$	−320.697	−1.24785	−0.623925	−	0.781484i	$-0.714463\pi$
$257$	−320.697	−1.24785	−0.623925	+	0.781484i	$0.714463\pi$
$258$	0	0
$259$	56.6640	0.218780
$260$	0	0
$261$	0	0
$262$	188.575i	0.719752i
$263$	−108.832	−0.413808	−0.206904	−	0.978361i	$-0.566339\pi$
$263$	−108.832	−0.413808	−0.206904	+	0.978361i	$0.566339\pi$
$264$	0	0
$265$	0	0
$266$	− 62.9870i	− 0.236793i
$267$	0	0
$268$	− 211.875i	− 0.790577i
$269$	331.364i	1.23184i	0.787810	+	0.615918i	$0.211215\pi$
$269$	331.364i	1.23184i	−0.787810	+	0.615918i	$0.788785\pi$
$270$	0	0
$271$	137.535	0.507508	0.253754	−	0.967269i	$-0.418335\pi$
$271$	137.535	0.507508	0.253754	+	0.967269i	$0.418335\pi$
$272$	−51.3498	−0.188786
$273$	0	0
$274$	−237.188	−0.865651
$275$	0	0
$276$	0	0
$277$	− 334.162i	− 1.20636i	−0.797605	−	0.603181i	$-0.793900\pi$
$277$	− 334.162i	− 1.20636i	0.797605	−	0.603181i	$-0.206100\pi$
$278$	229.446	0.825347
$279$	0	0
$280$	0	0
$281$	− 175.649i	− 0.625085i	−0.949904	−	0.312543i	$-0.898819\pi$
$281$	− 175.649i	− 0.625085i	0.949904	−	0.312543i	$-0.101181\pi$
$282$	0	0
$283$	− 428.458i	− 1.51398i	−0.653424	−	0.756992i	$-0.726668\pi$
$283$	− 428.458i	− 1.51398i	0.653424	−	0.756992i	$-0.273332\pi$
$284$	− 64.0777i	− 0.225626i
$285$	0	0
$286$	−2.19865	−0.00768760
$287$	157.536	0.548906
$288$	0	0
$289$	763.235	2.64095
$290$	0	0
$291$	0	0
$292$	87.4170i	0.299373i
$293$	−390.469	−1.33266	−0.666329	−	0.745658i	$-0.732135\pi$
$293$	−390.469	−1.33266	−0.666329	+	0.745658i	$0.732135\pi$
$294$	0	0
$295$	0	0
$296$	− 165.635i	− 0.559577i
$297$	0	0
$298$	119.314i	0.400382i
$299$	− 161.220i	− 0.539198i
$300$	0	0
$301$	136.915	0.454867
$302$	−153.319	−0.507680
$303$	0	0
$304$	32.3844	0.106527
$305$	0	0
$306$	0	0
$307$	− 436.029i	− 1.42029i	−0.704056	−	0.710144i	$-0.748630\pi$
$307$	− 436.029i	− 1.42029i	0.704056	−	0.710144i	$-0.251370\pi$
$308$	2.88573	0.00936925
$309$	0	0
$310$	0	0
$311$	− 170.238i	− 0.547389i	−0.961817	−	0.273695i	$-0.911754\pi$
$311$	− 170.238i	− 0.547389i	0.961817	−	0.273695i	$-0.0882458\pi$
$312$	0	0
$313$	− 228.745i	− 0.730815i	−0.930848	−	0.365407i	$-0.880930\pi$
$313$	− 228.745i	− 0.730815i	0.930848	−	0.365407i	$-0.119070\pi$
$314$	− 94.0166i	− 0.299416i
$315$	0	0
$316$	−58.2065	−0.184198
$317$	89.4393	0.282143	0.141071	−	0.989999i	$-0.454945\pi$
$317$	89.4393	0.282143	0.141071	+	0.989999i	$0.454945\pi$
$318$	0	0
$319$	−10.3241	−0.0323641
$320$	0	0
$321$	0	0
$322$	− 108.310i	− 0.336366i
$323$	−663.604	−2.05450
$324$	0	0
$325$	0	0
$326$	− 75.1567i	− 0.230542i
$327$	0	0
$328$	− 460.494i	− 1.40395i
$329$	14.4971i	0.0440641i
$330$	0	0
$331$	340.988	1.03018	0.515088	−	0.857137i	$-0.327759\pi$
$331$	340.988	1.03018	0.515088	+	0.857137i	$0.327759\pi$
$332$	−81.8469	−0.246527
$333$	0	0
$334$	−218.539	−0.654307
$335$	0	0
$336$	0	0
$337$	− 38.9229i	− 0.115498i	−0.998331	−	0.0577491i	$-0.981608\pi$
$337$	− 38.9229i	− 0.115498i	0.998331	−	0.0577491i	$-0.0183923\pi$
$338$	172.226	0.509545
$339$	0	0
$340$	0	0
$341$	17.8468i	0.0523366i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	− 400.217i	− 1.16342i
$345$	0	0
$346$	−194.162	−0.561162
$347$	257.444	0.741914	0.370957	−	0.928650i	$-0.379030\pi$
$347$	257.444	0.741914	0.370957	+	0.928650i	$0.379030\pi$
$348$	0	0
$349$	−491.535	−1.40841	−0.704204	−	0.709997i	$-0.748696\pi$
$349$	−491.535	−1.40841	−0.704204	+	0.709997i	$0.748696\pi$
$350$	0	0
$351$	0	0
$352$	− 13.5124i	− 0.0383875i
$353$	275.557	0.780615	0.390308	−	0.920685i	$-0.372369\pi$
$353$	275.557	0.780615	0.390308	+	0.920685i	$0.372369\pi$
$354$	0	0
$355$	0	0
$356$	193.710i	0.544130i
$357$	0	0
$358$	291.638i	0.814631i
$359$	− 215.018i	− 0.598935i	−0.954106	−	0.299468i	$-0.903191\pi$
$359$	− 215.018i	− 0.598935i	0.954106	−	0.299468i	$-0.0968091\pi$
$360$	0	0
$361$	57.5098	0.159307
$362$	149.154	0.412028
$363$	0	0
$364$	32.0810	0.0881347
$365$	0	0
$366$	0	0
$367$	− 436.575i	− 1.18958i	−0.803882	−	0.594789i	$-0.797236\pi$
$367$	− 436.575i	− 1.18958i	0.803882	−	0.594789i	$-0.202764\pi$
$368$	55.6867	0.151323
$369$	0	0
$370$	0	0
$371$	107.569i	0.289943i
$372$	0	0
$373$	370.988i	0.994606i	0.867577	+	0.497303i	$0.165676\pi$
$373$	370.988i	0.994606i	−0.867577	+	0.497303i	$0.834324\pi$
$374$	15.5619i	0.0416094i
$375$	0	0
$376$	42.3765	0.112703
$377$	−114.775	−0.304443
$378$	0	0
$379$	163.158	0.430496	0.215248	−	0.976559i	$-0.430944\pi$
$379$	163.158	0.430496	0.215248	+	0.976559i	$0.430944\pi$
$380$	0	0
$381$	0	0
$382$	159.152i	0.416627i
$383$	−230.511	−0.601857	−0.300929	−	0.953647i	$-0.597297\pi$
$383$	−230.511	−0.601857	−0.300929	+	0.953647i	$0.597297\pi$
$384$	0	0
$385$	0	0
$386$	− 112.301i	− 0.290936i
$387$	0	0
$388$	91.8301i	0.236675i
$389$	71.5925i	0.184042i	0.995757	+	0.0920212i	$0.0293327\pi$
$389$	71.5925i	0.184042i	−0.995757	+	0.0920212i	$0.970667\pi$
$390$	0	0
$391$	−1141.11	−2.91843
$392$	54.1366	0.138104
$393$	0	0
$394$	41.2771	0.104764
$395$	0	0
$396$	0	0
$397$	750.782i	1.89114i	0.325422	+	0.945569i	$0.394494\pi$
$397$	750.782i	1.89114i	−0.325422	+	0.945569i	$0.605506\pi$
$398$	24.9234	0.0626217
$399$	0	0
$400$	0	0
$401$	− 546.047i	− 1.36171i	−0.732417	−	0.680857i	$-0.761608\pi$
$401$	− 546.047i	− 1.36171i	0.732417	−	0.680857i	$-0.238392\pi$
$402$	0	0
$403$	198.405i	0.492321i
$404$	− 491.468i	− 1.21650i
$405$	0	0
$406$	−77.1072	−0.189919
$407$	8.82909	0.0216931
$408$	0	0
$409$	206.871	0.505796	0.252898	−	0.967493i	$-0.418616\pi$
$409$	206.871	0.505796	0.252898	+	0.967493i	$0.418616\pi$
$410$	0	0
$411$	0	0
$412$	− 333.033i	− 0.808332i
$413$	194.097	0.469968
$414$	0	0
$415$	0	0
$416$	− 150.219i	− 0.361103i
$417$	0	0
$418$	− 9.81430i	− 0.0234792i
$419$	504.514i	1.20409i	0.798462	+	0.602045i	$0.205647\pi$
$419$	504.514i	1.20409i	−0.798462	+	0.602045i	$0.794353\pi$
$420$	0	0
$421$	−568.996	−1.35153	−0.675767	−	0.737115i	$-0.736188\pi$
$421$	−568.996	−1.35153	−0.675767	+	0.737115i	$0.736188\pi$
$422$	423.783	1.00423
$423$	0	0
$424$	314.435	0.741593
$425$	0	0
$426$	0	0
$427$	− 234.251i	− 0.548597i
$428$	30.0849	0.0702917
$429$	0	0
$430$	0	0
$431$	− 627.018i	− 1.45480i	−0.686215	−	0.727399i	$-0.740729\pi$
$431$	− 627.018i	− 1.45480i	0.686215	−	0.727399i	$-0.259271\pi$
$432$	0	0
$433$	475.417i	1.09796i	0.835835	+	0.548980i	$0.184984\pi$
$433$	475.417i	1.09796i	−0.835835	+	0.548980i	$0.815016\pi$
$434$	133.291i	0.307122i
$435$	0	0
$436$	−4.41306	−0.0101217
$437$	719.652	1.64680
$438$	0	0
$439$	−186.834	−0.425590	−0.212795	−	0.977097i	$-0.568257\pi$
$439$	−186.834	−0.425590	−0.212795	+	0.977097i	$0.568257\pi$
$440$	0	0
$441$	0	0
$442$	173.004i	0.391412i
$443$	−168.125	−0.379515	−0.189757	−	0.981831i	$-0.560770\pi$
$443$	−168.125	−0.379515	−0.189757	+	0.981831i	$0.560770\pi$
$444$	0	0
$445$	0	0
$446$	181.437i	0.406810i
$447$	0	0
$448$	− 84.1660i	− 0.187871i
$449$	742.806i	1.65436i	0.561940	+	0.827178i	$0.310055\pi$
$449$	742.806i	1.65436i	−0.561940	+	0.827178i	$0.689945\pi$
$450$	0	0
$451$	24.5464	0.0544267
$452$	−2.49935	−0.00552953
$453$	0	0
$454$	−21.6078	−0.0475942
$455$	0	0
$456$	0	0
$457$	39.3202i	0.0860398i	0.999074	+	0.0430199i	$0.0136979\pi$
$457$	39.3202i	0.0860398i	−0.999074	+	0.0430199i	$0.986302\pi$
$458$	−266.780	−0.582489
$459$	0	0
$460$	0	0
$461$	464.174i	1.00689i	0.864028	+	0.503443i	$0.167934\pi$
$461$	464.174i	1.00689i	−0.864028	+	0.503443i	$0.832066\pi$
$462$	0	0
$463$	85.9268i	0.185587i	0.995685	+	0.0927936i	$0.0295797\pi$
$463$	85.9268i	0.185587i	−0.995685	+	0.0927936i	$0.970420\pi$
$464$	− 39.6441i	− 0.0854400i
$465$	0	0
$466$	268.051	0.575217
$467$	−344.014	−0.736648	−0.368324	−	0.929698i	$-0.620068\pi$
$467$	−344.014	−0.736648	−0.368324	+	0.929698i	$0.620068\pi$
$468$	0	0
$469$	211.875	0.451758
$470$	0	0
$471$	0	0
$472$	− 567.365i	− 1.20204i
$473$	21.3334	0.0451023
$474$	0	0
$475$	0	0
$476$	− 227.067i	− 0.477032i
$477$	0	0
$478$	− 317.423i	− 0.664066i
$479$	− 344.599i	− 0.719412i	−0.933066	−	0.359706i	$-0.882877\pi$
$479$	− 344.599i	− 0.719412i	0.933066	−	0.359706i	$-0.117123\pi$
$480$	0	0
$481$	98.1542	0.204063
$482$	−60.0754	−0.124638
$483$	0	0
$484$	−319.686	−0.660509
$485$	0	0
$486$	0	0
$487$	− 334.405i	− 0.686664i	−0.939214	−	0.343332i	$-0.888444\pi$
$487$	− 334.405i	− 0.686664i	0.939214	−	0.343332i	$-0.111556\pi$
$488$	−684.740	−1.40316
$489$	0	0
$490$	0	0
$491$	− 367.616i	− 0.748709i	−0.927286	−	0.374354i	$-0.877864\pi$
$491$	− 367.616i	− 0.748709i	0.927286	−	0.374354i	$-0.122136\pi$
$492$	0	0
$493$	812.369i	1.64781i
$494$	− 109.107i	− 0.220864i
$495$	0	0
$496$	−68.5307	−0.138167
$497$	64.0777	0.128929
$498$	0	0
$499$	−232.915	−0.466764	−0.233382	−	0.972385i	$-0.574979\pi$
$499$	−232.915	−0.466764	−0.233382	+	0.972385i	$0.574979\pi$
$500$	0	0
$501$	0	0
$502$	58.9674i	0.117465i
$503$	327.576	0.651245	0.325623	−	0.945500i	$-0.394426\pi$
$503$	327.576	0.651245	0.325623	+	0.945500i	$0.394426\pi$
$504$	0	0
$505$	0	0
$506$	− 16.8763i	− 0.0333523i
$507$	0	0
$508$	− 408.539i	− 0.804210i
$509$	1.26261i	0.00248057i	0.999999	+	0.00124028i	$0.000394794\pi$
$509$	1.26261i	0.00248057i	−0.999999	+	0.00124028i	$0.999605\pi$
$510$	0	0
$511$	−87.4170	−0.171070
$512$	100.857	0.196987
$513$	0	0
$514$	−373.203	−0.726075
$515$	0	0
$516$	0	0
$517$	2.25886i	0.00436917i
$518$	65.9412	0.127300
$519$	0	0
$520$	0	0
$521$	667.237i	1.28069i	0.768089	+	0.640343i	$0.221208\pi$
$521$	667.237i	1.28069i	−0.768089	+	0.640343i	$0.778792\pi$
$522$	0	0
$523$	282.154i	0.539492i	0.962932	+	0.269746i	$0.0869397\pi$
$523$	282.154i	0.539492i	−0.962932	+	0.269746i	$0.913060\pi$
$524$	− 428.730i	− 0.818188i
$525$	0	0
$526$	−126.650	−0.240779
$527$	1404.30	2.66470
$528$	0	0
$529$	708.482	1.33929
$530$	0	0
$531$	0	0
$532$	143.203i	0.269178i
$533$	272.886	0.511981
$534$	0	0
$535$	0	0
$536$	− 619.331i	− 1.15547i
$537$	0	0
$538$	385.616i	0.716758i
$539$	2.88573i	0.00535386i
$540$	0	0
$541$	238.988	0.441753	0.220876	−	0.975302i	$-0.429108\pi$
$541$	238.988	0.441753	0.220876	+	0.975302i	$0.429108\pi$
$542$	160.052	0.295299
$543$	0	0
$544$	−1063.24	−1.95448
$545$	0	0
$546$	0	0
$547$	378.842i	0.692581i	0.938127	+	0.346291i	$0.112559\pi$
$547$	378.842i	0.692581i	−0.938127	+	0.346291i	$0.887441\pi$
$548$	539.253	0.984039
$549$	0	0
$550$	0	0
$551$	− 512.330i	− 0.929818i
$552$	0	0
$553$	− 58.2065i	− 0.105256i
$554$	− 388.872i	− 0.701935i
$555$	0	0
$556$	−521.652	−0.938223
$557$	933.845	1.67656	0.838281	−	0.545239i	$-0.183561\pi$
$557$	933.845	1.67656	0.838281	+	0.545239i	$0.183561\pi$
$558$	0	0
$559$	237.166	0.424268
$560$	0	0
$561$	0	0
$562$	− 204.407i	− 0.363713i
$563$	−360.900	−0.641030	−0.320515	−	0.947243i	$-0.603856\pi$
$563$	−360.900	−0.641030	−0.320515	+	0.947243i	$0.603856\pi$
$564$	0	0
$565$	0	0
$566$	− 498.605i	− 0.880928i
$567$	0	0
$568$	− 187.306i	− 0.329764i
$569$	176.130i	0.309542i	0.987950	+	0.154771i	$0.0494641\pi$
$569$	176.130i	0.309542i	−0.987950	+	0.154771i	$0.950536\pi$
$570$	0	0
$571$	−732.478	−1.28280	−0.641400	−	0.767207i	$-0.721646\pi$
$571$	−732.478	−1.28280	−0.641400	+	0.767207i	$0.721646\pi$
$572$	4.99870	0.00873898
$573$	0	0
$574$	183.328	0.319387
$575$	0	0
$576$	0	0
$577$	− 449.911i	− 0.779742i	−0.920869	−	0.389871i	$-0.872520\pi$
$577$	− 449.911i	− 0.779742i	0.920869	−	0.389871i	$-0.127480\pi$
$578$	888.194	1.53667
$579$	0	0
$580$	0	0
$581$	− 81.8469i	− 0.140872i
$582$	0	0
$583$	16.7608i	0.0287493i
$584$	255.529i	0.437550i
$585$	0	0
$586$	−454.397	−0.775422
$587$	−832.674	−1.41852	−0.709262	−	0.704944i	$-0.750972\pi$
$587$	−832.674	−1.41852	−0.709262	+	0.704944i	$0.750972\pi$
$588$	0	0
$589$	−885.636	−1.50363
$590$	0	0
$591$	0	0
$592$	33.9032i	0.0572689i
$593$	−200.890	−0.338770	−0.169385	−	0.985550i	$-0.554178\pi$
$593$	−200.890	−0.338770	−0.169385	+	0.985550i	$0.554178\pi$
$594$	0	0
$595$	0	0
$596$	− 271.263i	− 0.455139i
$597$	0	0
$598$	− 187.616i	− 0.313739i
$599$	866.358i	1.44634i	0.690670	+	0.723170i	$0.257316\pi$
$599$	866.358i	1.44634i	−0.690670	+	0.723170i	$0.742684\pi$
$600$	0	0
$601$	−357.393	−0.594665	−0.297332	−	0.954774i	$-0.596097\pi$
$601$	−357.393	−0.594665	−0.297332	+	0.954774i	$0.596097\pi$
$602$	159.331	0.264669
$603$	0	0
$604$	348.575	0.577111
$605$	0	0
$606$	0	0
$607$	364.753i	0.600911i	0.953796	+	0.300455i	$0.0971387\pi$
$607$	364.753i	0.600911i	−0.953796	+	0.300455i	$0.902861\pi$
$608$	670.544	1.10287
$609$	0	0
$610$	0	0
$611$	25.1120i	0.0410999i
$612$	0	0
$613$	672.907i	1.09773i	0.835912	+	0.548864i	$0.184939\pi$
$613$	672.907i	1.09773i	−0.835912	+	0.548864i	$0.815061\pi$
$614$	− 507.416i	− 0.826411i
$615$	0	0
$616$	8.43529	0.0136936
$617$	−536.874	−0.870137	−0.435068	−	0.900397i	$-0.643276\pi$
$617$	−536.874	−0.870137	−0.435068	+	0.900397i	$0.643276\pi$
$618$	0	0
$619$	573.563	0.926597	0.463298	−	0.886202i	$-0.346666\pi$
$619$	573.563	0.926597	0.463298	+	0.886202i	$0.346666\pi$
$620$	0	0
$621$	0	0
$622$	− 198.110i	− 0.318504i
$623$	−193.710	−0.310931
$624$	0	0
$625$	0	0
$626$	− 266.196i	− 0.425233i
$627$	0	0
$628$	213.749i	0.340365i
$629$	− 694.728i	− 1.10450i
$630$	0	0
$631$	661.733	1.04871	0.524353	−	0.851501i	$-0.324307\pi$
$631$	661.733	1.04871	0.524353	+	0.851501i	$0.324307\pi$
$632$	−170.144	−0.269215
$633$	0	0
$634$	104.082	0.164168
$635$	0	0
$636$	0	0
$637$	32.0810i	0.0503627i
$638$	−12.0144	−0.0188314
$639$	0	0
$640$	0	0
$641$	1074.95i	1.67699i	0.544908	+	0.838496i	$0.316565\pi$
$641$	1074.95i	1.67699i	−0.544908	+	0.838496i	$0.683435\pi$
$642$	0	0
$643$	− 359.616i	− 0.559278i	−0.960105	−	0.279639i	$-0.909785\pi$
$643$	− 359.616i	− 0.559278i	0.960105	−	0.279639i	$-0.0902148\pi$
$644$	246.245i	0.382368i
$645$	0	0
$646$	−772.251	−1.19543
$647$	134.064	0.207208	0.103604	−	0.994619i	$-0.466962\pi$
$647$	134.064	0.207208	0.103604	+	0.994619i	$0.466962\pi$
$648$	0	0
$649$	30.2431	0.0465996
$650$	0	0
$651$	0	0
$652$	170.871i	0.262071i
$653$	−332.850	−0.509725	−0.254862	−	0.966977i	$-0.582030\pi$
$653$	−332.850	−0.509725	−0.254862	+	0.966977i	$0.582030\pi$
$654$	0	0
$655$	0	0
$656$	94.2569i	0.143684i
$657$	0	0
$658$	16.8706i	0.0256392i
$659$	− 1139.67i	− 1.72940i	−0.502291	−	0.864699i	$-0.667509\pi$
$659$	− 1139.67i	− 1.72940i	0.502291	−	0.864699i	$-0.332491\pi$
$660$	0	0
$661$	−45.7203	−0.0691684	−0.0345842	−	0.999402i	$-0.511011\pi$
$661$	−45.7203	−0.0691684	−0.0345842	+	0.999402i	$0.511011\pi$
$662$	396.815	0.599419
$663$	0	0
$664$	−239.247	−0.360312
$665$	0	0
$666$	0	0
$667$	− 880.980i	− 1.32081i
$668$	496.853	0.743792
$669$	0	0
$670$	0	0
$671$	− 36.4997i	− 0.0543960i
$672$	0	0
$673$	− 525.830i	− 0.781323i	−0.920535	−	0.390661i	$-0.872246\pi$
$673$	− 525.830i	− 0.781323i	0.920535	−	0.390661i	$-0.127754\pi$
$674$	− 45.2954i	− 0.0672039i
$675$	0	0
$676$	−391.561	−0.579232
$677$	−955.925	−1.41200	−0.706001	−	0.708211i	$-0.749503\pi$
$677$	−955.925	−1.41200	−0.706001	+	0.708211i	$0.749503\pi$
$678$	0	0
$679$	−91.8301	−0.135243
$680$	0	0
$681$	0	0
$682$	20.7687i	0.0304526i
$683$	219.724	0.321705	0.160852	−	0.986978i	$-0.448576\pi$
$683$	219.724	0.321705	0.160852	+	0.986978i	$0.448576\pi$
$684$	0	0
$685$	0	0
$686$	21.5524i	0.0314175i
$687$	0	0
$688$	81.9190i	0.119068i
$689$	186.332i	0.270439i
$690$	0	0
$691$	−142.243	−0.205851	−0.102926	−	0.994689i	$-0.532820\pi$
$691$	−142.243	−0.205851	−0.102926	+	0.994689i	$0.532820\pi$
$692$	441.432	0.637908
$693$	0	0
$694$	299.593	0.431691
$695$	0	0
$696$	0	0
$697$	− 1931.47i	− 2.77111i
$698$	−572.010	−0.819498
$699$	0	0
$700$	0	0
$701$	491.597i	0.701280i	0.936510	+	0.350640i	$0.114036\pi$
$701$	491.597i	0.701280i	−0.936510	+	0.350640i	$0.885964\pi$
$702$	0	0
$703$	438.138i	0.623241i
$704$	− 13.1143i	− 0.0186283i
$705$	0	0
$706$	320.672	0.454210
$707$	491.468	0.695145
$708$	0	0
$709$	590.154	0.832375	0.416188	−	0.909279i	$-0.363366\pi$
$709$	590.154	0.832375	0.416188	+	0.909279i	$0.363366\pi$
$710$	0	0
$711$	0	0
$712$	566.235i	0.795274i
$713$	−1522.90	−2.13591
$714$	0	0
$715$	0	0
$716$	− 663.046i	− 0.926042i
$717$	0	0
$718$	− 250.221i	− 0.348497i
$719$	− 1024.93i	− 1.42550i	−0.701419	−	0.712749i	$-0.747450\pi$
$719$	− 1024.93i	− 1.42550i	0.701419	−	0.712749i	$-0.252550\pi$
$720$	0	0
$721$	333.033	0.461904
$722$	66.9255	0.0926945
$723$	0	0
$724$	−339.106	−0.468378
$725$	0	0
$726$	0	0
$727$	570.125i	0.784217i	0.919919	+	0.392108i	$0.128254\pi$
$727$	570.125i	0.784217i	−0.919919	+	0.392108i	$0.871746\pi$
$728$	93.7762	0.128813
$729$	0	0
$730$	0	0
$731$	− 1678.64i	− 2.29637i
$732$	0	0
$733$	− 153.660i	− 0.209632i	−0.994492	−	0.104816i	$-0.966575\pi$
$733$	− 153.660i	− 0.209632i	0.994492	−	0.104816i	$-0.0334253\pi$
$734$	− 508.052i	− 0.692169i
$735$	0	0
$736$	1153.04	1.56663
$737$	33.0132	0.0447940
$738$	0	0
$739$	−453.069	−0.613084	−0.306542	−	0.951857i	$-0.599172\pi$
$739$	−453.069	−0.613084	−0.306542	+	0.951857i	$0.599172\pi$
$740$	0	0
$741$	0	0
$742$	125.180i	0.168707i
$743$	393.312	0.529357	0.264678	−	0.964337i	$-0.414734\pi$
$743$	393.312	0.529357	0.264678	+	0.964337i	$0.414734\pi$
$744$	0	0
$745$	0	0
$746$	431.727i	0.578723i
$747$	0	0
$748$	− 35.3804i	− 0.0473000i
$749$	30.0849i	0.0401667i
$750$	0	0
$751$	−174.818	−0.232781	−0.116390	−	0.993204i	$-0.537132\pi$
$751$	−174.818	−0.232781	−0.116390	+	0.993204i	$0.537132\pi$
$752$	−8.67389	−0.0115344
$753$	0	0
$754$	−133.566	−0.177143
$755$	0	0
$756$	0	0
$757$	277.474i	0.366545i	0.983062	+	0.183272i	$0.0586690\pi$
$757$	277.474i	0.366545i	−0.983062	+	0.183272i	$0.941331\pi$
$758$	189.871	0.250489
$759$	0	0
$760$	0	0
$761$	− 267.553i	− 0.351580i	−0.984428	−	0.175790i	$-0.943752\pi$
$761$	− 267.553i	− 0.351580i	0.984428	−	0.175790i	$-0.0562480\pi$
$762$	0	0
$763$	− 4.41306i	− 0.00578382i
$764$	− 361.835i	− 0.473606i
$765$	0	0
$766$	−268.251	−0.350197
$767$	336.217	0.438353
$768$	0	0
$769$	−1077.00	−1.40052	−0.700258	−	0.713890i	$-0.746931\pi$
$769$	−1077.00	−1.40052	−0.700258	+	0.713890i	$0.746931\pi$
$770$	0	0
$771$	0	0
$772$	255.320i	0.330726i
$773$	325.301	0.420829	0.210414	−	0.977612i	$-0.432519\pi$
$773$	325.301	0.420829	0.210414	+	0.977612i	$0.432519\pi$
$774$	0	0
$775$	0	0
$776$	268.429i	0.345913i
$777$	0	0
$778$	83.3137i	0.107087i
$779$	1218.10i	1.56367i
$780$	0	0
$781$	9.98425	0.0127839
$782$	−1327.93	−1.69812
$783$	0	0
$784$	−11.0810	−0.0141340
$785$	0	0
$786$	0	0
$787$	1290.82i	1.64018i	0.572238	+	0.820088i	$0.306075\pi$
$787$	1290.82i	1.64018i	−0.572238	+	0.820088i	$0.693925\pi$
$788$	−93.8446	−0.119092
$789$	0	0
$790$	0	0
$791$	− 2.49935i	− 0.00315973i
$792$	0	0
$793$	− 405.773i	− 0.511693i
$794$	873.701i	1.10038i
$795$	0	0
$796$	−56.6640	−0.0711860
$797$	53.9602	0.0677041	0.0338520	−	0.999427i	$-0.489223\pi$
$797$	53.9602	0.0677041	0.0338520	+	0.999427i	$0.489223\pi$
$798$	0	0
$799$	177.741	0.222454
$800$	0	0
$801$	0	0
$802$	− 635.447i	− 0.792328i
$803$	−13.6209	−0.0169625
$804$	0	0
$805$	0	0
$806$	230.888i	0.286462i
$807$	0	0
$808$	− 1436.61i	− 1.77798i
$809$	812.784i	1.00468i	0.864671	+	0.502339i	$0.167527\pi$
$809$	812.784i	1.00468i	−0.864671	+	0.502339i	$0.832473\pi$
$810$	0	0
$811$	−359.749	−0.443587	−0.221793	−	0.975094i	$-0.571191\pi$
$811$	−359.749	−0.443587	−0.221793	+	0.975094i	$0.571191\pi$
$812$	175.305	0.215893
$813$	0	0
$814$	10.2746	0.0126224
$815$	0	0
$816$	0	0
$817$	1058.66i	1.29578i
$818$	240.740	0.294303
$819$	0	0
$820$	0	0
$821$	− 555.889i	− 0.677088i	−0.940951	−	0.338544i	$-0.890066\pi$
$821$	− 555.889i	− 0.677088i	0.940951	−	0.338544i	$-0.109934\pi$
$822$	0	0
$823$	− 1020.14i	− 1.23954i	−0.784785	−	0.619768i	$-0.787227\pi$
$823$	− 1020.14i	− 1.23954i	0.784785	−	0.619768i	$-0.212773\pi$
$824$	− 973.489i	− 1.18142i
$825$	0	0
$826$	225.875	0.273456
$827$	497.986	0.602160	0.301080	−	0.953599i	$-0.402653\pi$
$827$	497.986	0.602160	0.301080	+	0.953599i	$0.402653\pi$
$828$	0	0
$829$	1191.98	1.43785	0.718924	−	0.695088i	$-0.244635\pi$
$829$	1191.98	1.43785	0.718924	+	0.695088i	$0.244635\pi$
$830$	0	0
$831$	0	0
$832$	− 145.793i	− 0.175233i
$833$	227.067	0.272590
$834$	0	0
$835$	0	0
$836$	22.3131i	0.0266903i
$837$	0	0
$838$	587.114i	0.700613i
$839$	1160.56i	1.38327i	0.722249	+	0.691633i	$0.243109\pi$
$839$	1160.56i	1.38327i	−0.722249	+	0.691633i	$0.756891\pi$
$840$	0	0
$841$	213.818	0.254243
$842$	−662.153	−0.786405
$843$	0	0
$844$	−963.482	−1.14157
$845$	0	0
$846$	0	0
$847$	− 319.686i	− 0.377434i
$848$	−64.3606	−0.0758970
$849$	0	0
$850$	0	0
$851$	753.404i	0.885316i
$852$	0	0
$853$	− 64.3765i	− 0.0754707i	−0.999288	−	0.0377353i	$-0.987986\pi$
$853$	− 64.3765i	− 0.0754707i	0.999288	−	0.0377353i	$-0.0120144\pi$
$854$	− 272.603i	− 0.319207i
$855$	0	0
$856$	87.9412	0.102735
$857$	−255.048	−0.297606	−0.148803	−	0.988867i	$-0.547542\pi$
$857$	−255.048	−0.297606	−0.148803	+	0.988867i	$0.547542\pi$
$858$	0	0
$859$	219.114	0.255080	0.127540	−	0.991833i	$-0.459292\pi$
$859$	219.114	0.255080	0.127540	+	0.991833i	$0.459292\pi$
$860$	0	0
$861$	0	0
$862$	− 729.674i	− 0.846490i
$863$	−16.7987	−0.0194654	−0.00973271	−	0.999953i	$-0.503098\pi$
$863$	−16.7987	−0.0194654	−0.00973271	+	0.999953i	$0.503098\pi$
$864$	0	0
$865$	0	0
$866$	553.253i	0.638860i
$867$	0	0
$868$	− 303.041i	− 0.349125i
$869$	− 9.06943i	− 0.0104366i
$870$	0	0
$871$	367.012	0.421368
$872$	−12.8998	−0.0147934
$873$	0	0
$874$	837.474	0.958209
$875$	0	0
$876$	0	0
$877$	1083.56i	1.23553i	0.786361	+	0.617767i	$0.211963\pi$
$877$	1083.56i	1.23553i	−0.786361	+	0.617767i	$0.788037\pi$
$878$	−217.423	−0.247634
$879$	0	0
$880$	0	0
$881$	422.573i	0.479651i	0.970816	+	0.239825i	$0.0770902\pi$
$881$	422.573i	0.479651i	−0.970816	+	0.239825i	$0.922910\pi$
$882$	0	0
$883$	1413.73i	1.60105i	0.599301	+	0.800524i	$0.295445\pi$
$883$	1413.73i	1.60105i	−0.599301	+	0.800524i	$0.704555\pi$
$884$	− 393.329i	− 0.444942i
$885$	0	0
$886$	−195.651	−0.220825
$887$	−1128.46	−1.27222	−0.636109	−	0.771599i	$-0.719457\pi$
$887$	−1128.46	−1.27222	−0.636109	+	0.771599i	$0.719457\pi$
$888$	0	0
$889$	408.539	0.459548
$890$	0	0
$891$	0	0
$892$	− 412.502i	− 0.462446i
$893$	−112.095	−0.125526
$894$	0	0
$895$	0	0
$896$	248.938i	0.277832i
$897$	0	0
$898$	864.420i	0.962605i
$899$	1084.17i	1.20598i
$900$	0	0
$901$	1318.85	1.46376
$902$	28.5652	0.0316687
$903$	0	0
$904$	−7.30586	−0.00808170
$905$	0	0
$906$	0	0
$907$	99.2470i	0.109423i	0.998502	+	0.0547117i	$0.0174240\pi$
$907$	99.2470i	0.109423i	−0.998502	+	0.0547117i	$0.982576\pi$
$908$	49.1258	0.0541033
$909$	0	0
$910$	0	0
$911$	598.711i	0.657202i	0.944469	+	0.328601i	$0.106577\pi$
$911$	598.711i	0.657202i	−0.944469	+	0.328601i	$0.893423\pi$
$912$	0	0
$913$	− 12.7530i	− 0.0139682i
$914$	45.7578i	0.0500632i
$915$	0	0
$916$	606.531	0.662151
$917$	428.730	0.467536
$918$	0	0
$919$	−73.2707	−0.0797287	−0.0398643	−	0.999205i	$-0.512693\pi$
$919$	−73.2707	−0.0797287	−0.0398643	+	0.999205i	$0.512693\pi$
$920$	0	0
$921$	0	0
$922$	540.170i	0.585868i
$923$	110.996	0.120256
$924$	0	0
$925$	0	0
$926$	99.9949i	0.107986i
$927$	0	0
$928$	− 820.864i	− 0.884552i
$929$	− 885.277i	− 0.952936i	−0.879192	−	0.476468i	$-0.841917\pi$
$929$	− 885.277i	− 0.952936i	0.879192	−	0.476468i	$-0.158083\pi$
$930$	0	0
$931$	−143.203	−0.153816
$932$	−609.421	−0.653885
$933$	0	0
$934$	−400.337	−0.428626
$935$	0	0
$936$	0	0
$937$	149.676i	0.159739i	0.996805	+	0.0798697i	$0.0254504\pi$
$937$	149.676i	0.159739i	−0.996805	+	0.0798697i	$0.974550\pi$
$938$	246.563	0.262860
$939$	0	0
$940$	0	0
$941$	68.5607i	0.0728594i	0.999336	+	0.0364297i	$0.0115985\pi$
$941$	68.5607i	0.0728594i	−0.999336	+	0.0364297i	$0.988401\pi$
$942$	0	0
$943$	2094.60i	2.22120i
$944$	116.132i	0.123021i
$945$	0	0
$946$	24.8261	0.0262433
$947$	1764.25	1.86299	0.931496	−	0.363753i	$-0.118505\pi$
$947$	1764.25	1.86299	0.931496	+	0.363753i	$0.118505\pi$
$948$	0	0
$949$	−151.425	−0.159563
$950$	0	0
$951$	0	0
$952$	− 663.741i	− 0.697207i
$953$	−686.467	−0.720322	−0.360161	−	0.932890i	$-0.617278\pi$
$953$	−686.467	−0.720322	−0.360161	+	0.932890i	$0.617278\pi$
$954$	0	0
$955$	0	0
$956$	721.670i	0.754885i
$957$	0	0
$958$	− 401.017i	− 0.418598i
$959$	539.253i	0.562308i
$960$	0	0
$961$	913.154	0.950212
$962$	114.224	0.118736
$963$	0	0
$964$	136.583	0.141684
$965$	0	0
$966$	0	0
$967$	1616.57i	1.67173i	0.548932	+	0.835867i	$0.315035\pi$
$967$	1616.57i	1.67173i	−0.548932	+	0.835867i	$0.684965\pi$
$968$	−934.476	−0.965368
$969$	0	0
$970$	0	0
$971$	634.799i	0.653758i	0.945066	+	0.326879i	$0.105997\pi$
$971$	634.799i	0.653758i	−0.945066	+	0.326879i	$0.894003\pi$
$972$	0	0
$973$	− 521.652i	− 0.536128i
$974$	− 389.155i	− 0.399543i
$975$	0	0
$976$	140.157	0.143603
$977$	−19.6159	−0.0200777	−0.0100389	−	0.999950i	$-0.503196\pi$
$977$	−19.6159	−0.0200777	−0.0100389	+	0.999950i	$0.503196\pi$
$978$	0	0
$979$	−30.1829	−0.0308303
$980$	0	0
$981$	0	0
$982$	− 427.803i	− 0.435644i
$983$	−1462.10	−1.48738	−0.743691	−	0.668523i	$-0.766927\pi$
$983$	−1462.10	−1.48738	−0.743691	+	0.668523i	$0.766927\pi$
$984$	0	0
$985$	0	0
$986$	945.371i	0.958794i
$987$	0	0
$988$	248.057i	0.251070i
$989$	1820.42i	1.84067i
$990$	0	0
$991$	280.235	0.282780	0.141390	−	0.989954i	$-0.454843\pi$
$991$	280.235	0.282780	0.141390	+	0.989954i	$0.454843\pi$
$992$	−1418.98	−1.43043
$993$	0	0
$994$	74.5687	0.0750188
$995$	0	0
$996$	0	0
$997$	− 810.797i	− 0.813237i	−0.913598	−	0.406619i	$-0.866708\pi$
$997$	− 810.797i	− 0.813237i	0.913598	−	0.406619i	$-0.133292\pi$
$998$	−271.048	−0.271592
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.3.f.a.449.5		8
3.2	odd	2	inner	1575.3.f.a.449.3		8
5.2	odd	4		63.3.b.a.8.3	yes	4
5.3	odd	4		1575.3.c.a.701.2		4
5.4	even	2	inner	1575.3.f.a.449.4		8
15.2	even	4		63.3.b.a.8.2	✓	4
15.8	even	4		1575.3.c.a.701.3		4
15.14	odd	2	inner	1575.3.f.a.449.6		8
20.7	even	4		1008.3.d.d.449.3		4
35.2	odd	12		441.3.q.b.116.3		8
35.12	even	12		441.3.q.a.116.3		8
35.17	even	12		441.3.q.a.422.2		8
35.27	even	4		441.3.b.b.197.3		4
35.32	odd	12		441.3.q.b.422.2		8
40.27	even	4		4032.3.d.c.449.2		4
40.37	odd	4		4032.3.d.b.449.2		4
45.2	even	12		567.3.r.a.134.2		8
45.7	odd	12		567.3.r.a.134.3		8
45.22	odd	12		567.3.r.a.512.2		8
45.32	even	12		567.3.r.a.512.3		8
60.47	odd	4		1008.3.d.d.449.2		4
105.2	even	12		441.3.q.b.116.2		8
105.17	odd	12		441.3.q.a.422.3		8
105.32	even	12		441.3.q.b.422.3		8
105.47	odd	12		441.3.q.a.116.2		8
105.62	odd	4		441.3.b.b.197.2		4
120.77	even	4		4032.3.d.b.449.3		4
120.107	odd	4		4032.3.d.c.449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.3.b.a.8.2	✓	4	15.2	even	4
63.3.b.a.8.3	yes	4	5.2	odd	4
441.3.b.b.197.2		4	105.62	odd	4
441.3.b.b.197.3		4	35.27	even	4
441.3.q.a.116.2		8	105.47	odd	12
441.3.q.a.116.3		8	35.12	even	12
441.3.q.a.422.2		8	35.17	even	12
441.3.q.a.422.3		8	105.17	odd	12
441.3.q.b.116.2		8	105.2	even	12
441.3.q.b.116.3		8	35.2	odd	12
441.3.q.b.422.2		8	35.32	odd	12
441.3.q.b.422.3		8	105.32	even	12
567.3.r.a.134.2		8	45.2	even	12
567.3.r.a.134.3		8	45.7	odd	12
567.3.r.a.512.2		8	45.22	odd	12
567.3.r.a.512.3		8	45.32	even	12
1008.3.d.d.449.2		4	60.47	odd	4
1008.3.d.d.449.3		4	20.7	even	4
1575.3.c.a.701.2		4	5.3	odd	4
1575.3.c.a.701.3		4	15.8	even	4
1575.3.f.a.449.3		8	3.2	odd	2	inner
1575.3.f.a.449.4		8	5.4	even	2	inner
1575.3.f.a.449.5		8	1.1	even	1	trivial
1575.3.f.a.449.6		8	15.14	odd	2	inner
4032.3.d.b.449.2		4	40.37	odd	4
4032.3.d.b.449.3		4	120.77	even	4
4032.3.d.c.449.2		4	40.27	even	4
4032.3.d.c.449.3		4	120.107	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1575.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.16372 q^{2} -2.64575 q^{4} -2.64575i q^{7} -7.73381 q^{8} -0.412247i q^{11} -4.58301i q^{13} -3.07892i q^{14} +1.58301 q^{16} -32.4382 q^{17} +20.4575 q^{19} -0.479741i q^{22} +35.1779 q^{23} -5.33334i q^{26} +7.00000i q^{28} -25.0436i q^{29} -43.2915 q^{31} +32.7774 q^{32} -37.7490 q^{34} +21.4170i q^{37} +23.8069 q^{38} +59.5430i q^{41} +51.7490i q^{43} +1.09070i q^{44} +40.9373 q^{46} -5.47938 q^{47} -7.00000 q^{49} +12.1255i q^{52} -40.6572 q^{53} +20.4617i q^{56} -29.1438i q^{58} +73.3616i q^{59} +88.5385 q^{61} -50.3793 q^{62} +31.8118 q^{64} +80.0810i q^{67} +85.8233 q^{68} +24.2191i q^{71} -33.0405i q^{73} +24.9234i q^{74} -54.1255 q^{76} -1.09070 q^{77} +22.0000 q^{79} +69.2915i q^{82} +30.9352 q^{83} +60.2215i q^{86} +3.18824i q^{88} -73.2156i q^{89} -12.1255 q^{91} -93.0719 q^{92} -6.37648 q^{94} -34.7085i q^{97} -8.14605 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{16} - 48 q^{19} - 304 q^{31} - 48 q^{34} + 264 q^{46} - 56 q^{49} - 96 q^{61} + 64 q^{64} - 560 q^{76} + 176 q^{79} - 224 q^{91} - 432 q^{94}+O(q^{100}) \) 8 * q - 72 * q^16 - 48 * q^19 - 304 * q^31 - 48 * q^34 + 264 * q^46 - 56 * q^49 - 96 * q^61 + 64 * q^64 - 560 * q^76 + 176 * q^79 - 224 * q^91 - 432 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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