LMFDB - Embedded newform 1575.4.a.bg.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,4,Mod(1,1575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-4,0,36,0,0,28,-27,0,0,-100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 32x^{2} - 35x + 120 $ x^4 - 32x^2 - 35x + 120
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.50478$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.504784 q^{2} -7.74519 q^{4} +7.00000 q^{7} -7.94792 q^{8} -54.8800 q^{11} +16.0073 q^{13} +3.53349 q^{14} +57.9496 q^{16} +0.422056 q^{17} +127.501 q^{19} -27.7025 q^{22} -51.1101 q^{23} +8.08024 q^{26} -54.2164 q^{28} -41.4750 q^{29} +192.354 q^{31} +92.8353 q^{32} +0.213047 q^{34} -189.232 q^{37} +64.3605 q^{38} +76.3187 q^{41} +294.499 q^{43} +425.056 q^{44} -25.7996 q^{46} -540.297 q^{47} +49.0000 q^{49} -123.980 q^{52} +661.316 q^{53} -55.6354 q^{56} -20.9359 q^{58} -410.312 q^{59} +46.0495 q^{61} +97.0974 q^{62} -416.735 q^{64} +10.4074 q^{67} -3.26890 q^{68} +491.117 q^{71} -814.540 q^{73} -95.5215 q^{74} -987.522 q^{76} -384.160 q^{77} -858.725 q^{79} +38.5244 q^{82} +1055.80 q^{83} +148.658 q^{86} +436.182 q^{88} -341.567 q^{89} +112.051 q^{91} +395.858 q^{92} -272.733 q^{94} -1417.21 q^{97} +24.7344 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 36 q^{4} + 28 q^{7} - 27 q^{8} - 100 q^{11} + 44 q^{13} - 28 q^{14} + 160 q^{16} + 53 q^{17} - 29 q^{19} - 152 q^{22} - 295 q^{23} - 700 q^{26} + 252 q^{28} - 129 q^{29} + 114 q^{31} + 310 q^{32}+ \cdots - 196 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 36 * q^4 + 28 * q^7 - 27 * q^8 - 100 * q^11 + 44 * q^13 - 28 * q^14 + 160 * q^16 + 53 * q^17 - 29 * q^19 - 152 * q^22 - 295 * q^23 - 700 * q^26 + 252 * q^28 - 129 * q^29 + 114 * q^31 + 310 * q^32 + 203 * q^34 + 403 * q^37 - 555 * q^38 - 671 * q^41 - 411 * q^43 - 438 * q^44 - 997 * q^46 + 8 * q^47 + 196 * q^49 + 74 * q^52 - 90 * q^53 - 189 * q^56 + 673 * q^58 - 1018 * q^59 + 50 * q^61 + 1626 * q^62 - 2421 * q^64 + 424 * q^67 + 617 * q^68 - 215 * q^71 - 1207 * q^73 - 623 * q^74 - 3257 * q^76 - 700 * q^77 - 951 * q^79 + 1695 * q^82 + 3035 * q^83 + 99 * q^86 + 163 * q^88 - 2819 * q^89 + 308 * q^91 - 3073 * q^92 - 3056 * q^94 - 1100 * q^97 - 196 * q^98

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 $

$2$	0.504784	0.178468	0.0892340	−	0.996011i	$-0.471558\pi$
$2$	0.504784	0.178468	0.0892340	+	0.996011i	$0.471558\pi$
$3$	0	0
$3$	0	0
$4$	−7.74519	−0.968149
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−7.94792	−0.351252
$9$	0	0
$10$	0	0
$11$	−54.8800	−1.50427	−0.752134	−	0.659010i	$-0.770975\pi$
$11$	−54.8800	−1.50427	−0.752134	+	0.659010i	$0.770975\pi$
$12$	0	0
$13$	16.0073	0.341510	0.170755	−	0.985313i	$-0.445379\pi$
$13$	16.0073	0.341510	0.170755	+	0.985313i	$0.445379\pi$
$14$	3.53349	0.0674546
$15$	0	0
$16$	57.9496	0.905462
$17$	0.422056	0.00602139	0.00301069	−	0.999995i	$-0.499042\pi$
$17$	0.422056	0.00602139	0.00301069	+	0.999995i	$0.499042\pi$
$18$	0	0
$19$	127.501	1.53952	0.769758	−	0.638336i	$-0.220377\pi$
$19$	127.501	1.53952	0.769758	+	0.638336i	$0.220377\pi$
$20$	0	0
$21$	0	0
$22$	−27.7025	−0.268464
$23$	−51.1101	−0.463357	−0.231678	−	0.972792i	$-0.574422\pi$
$23$	−51.1101	−0.463357	−0.231678	+	0.972792i	$0.574422\pi$
$24$	0	0
$25$	0	0
$26$	8.08024	0.0609487
$27$	0	0
$28$	−54.2164	−0.365926
$29$	−41.4750	−0.265576	−0.132788	−	0.991144i	$-0.542393\pi$
$29$	−41.4750	−0.265576	−0.132788	+	0.991144i	$0.542393\pi$
$30$	0	0
$31$	192.354	1.11445	0.557224	−	0.830362i	$-0.311866\pi$
$31$	192.354	1.11445	0.557224	+	0.830362i	$0.311866\pi$
$32$	92.8353	0.512848
$33$	0	0
$34$	0.213047	0.00107462
$35$	0	0
$36$	0	0
$37$	−189.232	−0.840801	−0.420400	−	0.907339i	$-0.638110\pi$
$37$	−189.232	−0.840801	−0.420400	+	0.907339i	$0.638110\pi$
$38$	64.3605	0.274754
$39$	0	0
$40$	0	0
$41$	76.3187	0.290707	0.145353	−	0.989380i	$-0.453568\pi$
$41$	76.3187	0.290707	0.145353	+	0.989380i	$0.453568\pi$
$42$	0	0
$43$	294.499	1.04443	0.522216	−	0.852813i	$-0.325105\pi$
$43$	294.499	1.04443	0.522216	+	0.852813i	$0.325105\pi$
$44$	425.056	1.45636
$45$	0	0
$46$	−25.7996	−0.0826943
$47$	−540.297	−1.67682	−0.838408	−	0.545043i	$-0.816513\pi$
$47$	−540.297	−1.67682	−0.838408	+	0.545043i	$0.816513\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−123.980	−0.330633
$53$	661.316	1.71394	0.856969	−	0.515368i	$-0.172345\pi$
$53$	661.316	1.71394	0.856969	+	0.515368i	$0.172345\pi$
$54$	0	0
$55$	0	0
$56$	−55.6354	−0.132761
$57$	0	0
$58$	−20.9359	−0.0473969
$59$	−410.312	−0.905390	−0.452695	−	0.891665i	$-0.649537\pi$
$59$	−410.312	−0.905390	−0.452695	+	0.891665i	$0.649537\pi$
$60$	0	0
$61$	46.0495	0.0966563	0.0483281	−	0.998832i	$-0.484611\pi$
$61$	46.0495	0.0966563	0.0483281	+	0.998832i	$0.484611\pi$
$62$	97.0974	0.198893
$63$	0	0
$64$	−416.735	−0.813935
$65$	0	0
$66$	0	0
$67$	10.4074	0.0189771	0.00948854	−	0.999955i	$-0.496980\pi$
$67$	10.4074	0.0189771	0.00948854	+	0.999955i	$0.496980\pi$
$68$	−3.26890	−0.00582960
$69$	0	0
$70$	0	0
$71$	491.117	0.820913	0.410456	−	0.911880i	$-0.365369\pi$
$71$	491.117	0.820913	0.410456	+	0.911880i	$0.365369\pi$
$72$	0	0
$73$	−814.540	−1.30595	−0.652977	−	0.757378i	$-0.726480\pi$
$73$	−814.540	−1.30595	−0.652977	+	0.757378i	$0.726480\pi$
$74$	−95.5215	−0.150056
$75$	0	0
$76$	−987.522	−1.49048
$77$	−384.160	−0.568560
$78$	0	0
$79$	−858.725	−1.22296	−0.611482	−	0.791258i	$-0.709426\pi$
$79$	−858.725	−1.22296	−0.611482	+	0.791258i	$0.709426\pi$
$80$	0	0
$81$	0	0
$82$	38.5244	0.0518818
$83$	1055.80	1.39626	0.698129	−	0.715972i	$-0.254016\pi$
$83$	1055.80	1.39626	0.698129	+	0.715972i	$0.254016\pi$
$84$	0	0
$85$	0	0
$86$	148.658	0.186398
$87$	0	0
$88$	436.182	0.528376
$89$	−341.567	−0.406809	−0.203405	−	0.979095i	$-0.565201\pi$
$89$	−341.567	−0.406809	−0.203405	+	0.979095i	$0.565201\pi$
$90$	0	0
$91$	112.051	0.129079
$92$	395.858	0.448598
$93$	0	0
$94$	−272.733	−0.299258
$95$	0	0
$96$	0	0
$97$	−1417.21	−1.48346	−0.741731	−	0.670697i	$-0.765995\pi$
$97$	−1417.21	−1.48346	−0.741731	+	0.670697i	$0.765995\pi$
$98$	24.7344	0.0254954
$99$	0	0
$100$	0	0
$101$	121.051	0.119258	0.0596289	−	0.998221i	$-0.481008\pi$
$101$	121.051	0.119258	0.0596289	+	0.998221i	$0.481008\pi$
$102$	0	0
$103$	−655.594	−0.627161	−0.313581	−	0.949562i	$-0.601529\pi$
$103$	−655.594	−0.627161	−0.313581	+	0.949562i	$0.601529\pi$
$104$	−127.225	−0.119956
$105$	0	0
$106$	333.821	0.305883
$107$	−2069.81	−1.87006	−0.935031	−	0.354567i	$-0.884628\pi$
$107$	−2069.81	−1.87006	−0.935031	+	0.354567i	$0.884628\pi$
$108$	0	0
$109$	904.218	0.794573	0.397286	−	0.917695i	$-0.369952\pi$
$109$	904.218	0.794573	0.397286	+	0.917695i	$0.369952\pi$
$110$	0	0
$111$	0	0
$112$	405.647	0.342232
$113$	843.350	0.702086	0.351043	−	0.936359i	$-0.385827\pi$
$113$	843.350	0.702086	0.351043	+	0.936359i	$0.385827\pi$
$114$	0	0
$115$	0	0
$116$	321.232	0.257117
$117$	0	0
$118$	−207.119	−0.161583
$119$	2.95439	0.00227587
$120$	0	0
$121$	1680.81	1.26282
$122$	23.2450	0.0172501
$123$	0	0
$124$	−1489.82	−1.07895
$125$	0	0
$126$	0	0
$127$	2030.50	1.41872	0.709362	−	0.704845i	$-0.248983\pi$
$127$	2030.50	1.41872	0.709362	+	0.704845i	$0.248983\pi$
$128$	−953.043	−0.658109
$129$	0	0
$130$	0	0
$131$	−2872.47	−1.91579	−0.957897	−	0.287113i	$-0.907305\pi$
$131$	−2872.47	−1.91579	−0.957897	+	0.287113i	$0.907305\pi$
$132$	0	0
$133$	892.509	0.581882
$134$	5.25348	0.00338680
$135$	0	0
$136$	−3.35446	−0.00211502
$137$	1621.62	1.01127	0.505636	−	0.862747i	$-0.331258\pi$
$137$	1621.62	1.01127	0.505636	+	0.862747i	$0.331258\pi$
$138$	0	0
$139$	2617.72	1.59735	0.798677	−	0.601760i	$-0.205534\pi$
$139$	2617.72	1.59735	0.798677	+	0.601760i	$0.205534\pi$
$140$	0	0
$141$	0	0
$142$	247.908	0.146507
$143$	−878.482	−0.513723
$144$	0	0
$145$	0	0
$146$	−411.166	−0.233071
$147$	0	0
$148$	1465.64	0.814021
$149$	−2252.63	−1.23854	−0.619270	−	0.785178i	$-0.712571\pi$
$149$	−2252.63	−1.23854	−0.619270	+	0.785178i	$0.712571\pi$
$150$	0	0
$151$	−1849.61	−0.996813	−0.498407	−	0.866943i	$-0.666081\pi$
$151$	−1849.61	−0.996813	−0.498407	+	0.866943i	$0.666081\pi$
$152$	−1013.37	−0.540757
$153$	0	0
$154$	−193.918	−0.101470
$155$	0	0
$156$	0	0
$157$	−3337.59	−1.69661	−0.848307	−	0.529505i	$-0.822378\pi$
$157$	−3337.59	−1.69661	−0.848307	+	0.529505i	$0.822378\pi$
$158$	−433.471	−0.218260
$159$	0	0
$160$	0	0
$161$	−357.771	−0.175132
$162$	0	0
$163$	−471.099	−0.226376	−0.113188	−	0.993574i	$-0.536106\pi$
$163$	−471.099	−0.226376	−0.113188	+	0.993574i	$0.536106\pi$
$164$	−591.103	−0.281447
$165$	0	0
$166$	532.952	0.249187
$167$	3676.72	1.70367	0.851836	−	0.523809i	$-0.175489\pi$
$167$	3676.72	1.70367	0.851836	+	0.523809i	$0.175489\pi$
$168$	0	0
$169$	−1940.77	−0.883371
$170$	0	0
$171$	0	0
$172$	−2280.95	−1.01117
$173$	−1009.11	−0.443475	−0.221737	−	0.975106i	$-0.571173\pi$
$173$	−1009.11	−0.443475	−0.221737	+	0.975106i	$0.571173\pi$
$174$	0	0
$175$	0	0
$176$	−3180.27	−1.36206
$177$	0	0
$178$	−172.417	−0.0726024
$179$	−3201.27	−1.33673	−0.668364	−	0.743835i	$-0.733005\pi$
$179$	−3201.27	−1.33673	−0.668364	+	0.743835i	$0.733005\pi$
$180$	0	0
$181$	−1970.14	−0.809056	−0.404528	−	0.914526i	$-0.632564\pi$
$181$	−1970.14	−0.809056	−0.404528	+	0.914526i	$0.632564\pi$
$182$	56.5617	0.0230364
$183$	0	0
$184$	406.219	0.162755
$185$	0	0
$186$	0	0
$187$	−23.1624	−0.00905778
$188$	4184.70	1.62341
$189$	0	0
$190$	0	0
$191$	−3953.39	−1.49768	−0.748842	−	0.662749i	$-0.769390\pi$
$191$	−3953.39	−1.49768	−0.748842	+	0.662749i	$0.769390\pi$
$192$	0	0
$193$	2425.45	0.904598	0.452299	−	0.891866i	$-0.350604\pi$
$193$	2425.45	0.904598	0.452299	+	0.891866i	$0.350604\pi$
$194$	−715.385	−0.264751
$195$	0	0
$196$	−379.514	−0.138307
$197$	−3126.50	−1.13073	−0.565366	−	0.824840i	$-0.691265\pi$
$197$	−3126.50	−1.13073	−0.565366	+	0.824840i	$0.691265\pi$
$198$	0	0
$199$	−3566.13	−1.27033	−0.635166	−	0.772376i	$-0.719068\pi$
$199$	−3566.13	−1.27033	−0.635166	+	0.772376i	$0.719068\pi$
$200$	0	0
$201$	0	0
$202$	61.1046	0.0212837
$203$	−290.325	−0.100378
$204$	0	0
$205$	0	0
$206$	−330.933	−0.111928
$207$	0	0
$208$	927.618	0.309225
$209$	−6997.27	−2.31584
$210$	0	0
$211$	−3699.18	−1.20693	−0.603465	−	0.797389i	$-0.706214\pi$
$211$	−3699.18	−1.20693	−0.603465	+	0.797389i	$0.706214\pi$
$212$	−5122.02	−1.65935
$213$	0	0
$214$	−1044.81	−0.333746
$215$	0	0
$216$	0	0
$217$	1346.48	0.421222
$218$	456.435	0.141806
$219$	0	0
$220$	0	0
$221$	6.75599	0.00205637
$222$	0	0
$223$	−102.041	−0.0306419	−0.0153210	−	0.999883i	$-0.504877\pi$
$223$	−102.041	−0.0306419	−0.0153210	+	0.999883i	$0.504877\pi$
$224$	649.847	0.193838
$225$	0	0
$226$	425.709	0.125300
$227$	−5739.91	−1.67829	−0.839144	−	0.543909i	$-0.816943\pi$
$227$	−5739.91	−1.67829	−0.839144	+	0.543909i	$0.816943\pi$
$228$	0	0
$229$	−843.125	−0.243298	−0.121649	−	0.992573i	$-0.538818\pi$
$229$	−843.125	−0.243298	−0.121649	+	0.992573i	$0.538818\pi$
$230$	0	0
$231$	0	0
$232$	329.640	0.0932841
$233$	2017.65	0.567300	0.283650	−	0.958928i	$-0.408455\pi$
$233$	2017.65	0.567300	0.283650	+	0.958928i	$0.408455\pi$
$234$	0	0
$235$	0	0
$236$	3177.94	0.876553
$237$	0	0
$238$	1.49133	0.000406170 0
$239$	−243.685	−0.0659526	−0.0329763	−	0.999456i	$-0.510499\pi$
$239$	−243.685	−0.0659526	−0.0329763	+	0.999456i	$0.510499\pi$
$240$	0	0
$241$	−3060.58	−0.818048	−0.409024	−	0.912524i	$-0.634131\pi$
$241$	−3060.58	−0.818048	−0.409024	+	0.912524i	$0.634131\pi$
$242$	848.448	0.225373
$243$	0	0
$244$	−356.662	−0.0935777
$245$	0	0
$246$	0	0
$247$	2040.95	0.525760
$248$	−1528.82	−0.391452
$249$	0	0
$250$	0	0
$251$	6594.14	1.65824	0.829120	−	0.559070i	$-0.188842\pi$
$251$	6594.14	1.65824	0.829120	+	0.559070i	$0.188842\pi$
$252$	0	0
$253$	2804.92	0.697012
$254$	1024.96	0.253197
$255$	0	0
$256$	2852.80	0.696484
$257$	−512.646	−0.124428	−0.0622139	−	0.998063i	$-0.519816\pi$
$257$	−512.646	−0.124428	−0.0622139	+	0.998063i	$0.519816\pi$
$258$	0	0
$259$	−1324.63	−0.317793
$260$	0	0
$261$	0	0
$262$	−1449.98	−0.341908
$263$	3560.04	0.834681	0.417341	−	0.908750i	$-0.362962\pi$
$263$	3560.04	0.834681	0.417341	+	0.908750i	$0.362962\pi$
$264$	0	0
$265$	0	0
$266$	450.524	0.103847
$267$	0	0
$268$	−80.6072	−0.0183726
$269$	549.744	0.124604	0.0623020	−	0.998057i	$-0.480156\pi$
$269$	549.744	0.124604	0.0623020	+	0.998057i	$0.480156\pi$
$270$	0	0
$271$	−3944.91	−0.884267	−0.442134	−	0.896949i	$-0.645778\pi$
$271$	−3944.91	−0.884267	−0.442134	+	0.896949i	$0.645778\pi$
$272$	24.4580	0.00545214
$273$	0	0
$274$	818.567	0.180480
$275$	0	0
$276$	0	0
$277$	−1254.67	−0.272150	−0.136075	−	0.990699i	$-0.543449\pi$
$277$	−1254.67	−0.272150	−0.136075	+	0.990699i	$0.543449\pi$
$278$	1321.38	0.285077
$279$	0	0
$280$	0	0
$281$	681.496	0.144679	0.0723393	−	0.997380i	$-0.476954\pi$
$281$	681.496	0.144679	0.0723393	+	0.997380i	$0.476954\pi$
$282$	0	0
$283$	−5946.88	−1.24914	−0.624568	−	0.780970i	$-0.714725\pi$
$283$	−5946.88	−1.24914	−0.624568	+	0.780970i	$0.714725\pi$
$284$	−3803.79	−0.794766
$285$	0	0
$286$	−443.444	−0.0916831
$287$	534.231	0.109877
$288$	0	0
$289$	−4912.82	−0.999964
$290$	0	0
$291$	0	0
$292$	6308.77	1.26436
$293$	−2171.03	−0.432877	−0.216438	−	0.976296i	$-0.569444\pi$
$293$	−2171.03	−0.432877	−0.216438	+	0.976296i	$0.569444\pi$
$294$	0	0
$295$	0	0
$296$	1504.00	0.295333
$297$	0	0
$298$	−1137.09	−0.221040
$299$	−818.137	−0.158241
$300$	0	0
$301$	2061.49	0.394759
$302$	−933.651	−0.177899
$303$	0	0
$304$	7388.64	1.39397
$305$	0	0
$306$	0	0
$307$	−3644.59	−0.677549	−0.338775	−	0.940868i	$-0.610012\pi$
$307$	−3644.59	−0.677549	−0.338775	+	0.940868i	$0.610012\pi$
$308$	2975.39	0.550451
$309$	0	0
$310$	0	0
$311$	−2584.19	−0.471177	−0.235589	−	0.971853i	$-0.575702\pi$
$311$	−2584.19	−0.471177	−0.235589	+	0.971853i	$0.575702\pi$
$312$	0	0
$313$	6693.63	1.20878	0.604388	−	0.796690i	$-0.293418\pi$
$313$	6693.63	1.20878	0.604388	+	0.796690i	$0.293418\pi$
$314$	−1684.76	−0.302791
$315$	0	0
$316$	6650.99	1.18401
$317$	3843.36	0.680962	0.340481	−	0.940251i	$-0.389410\pi$
$317$	3843.36	0.680962	0.340481	+	0.940251i	$0.389410\pi$
$318$	0	0
$319$	2276.15	0.399498
$320$	0	0
$321$	0	0
$322$	−180.597	−0.0312555
$323$	53.8126	0.00927002
$324$	0	0
$325$	0	0
$326$	−237.803	−0.0404009
$327$	0	0
$328$	−606.575	−0.102111
$329$	−3782.08	−0.633777
$330$	0	0
$331$	−821.922	−0.136486	−0.0682431	−	0.997669i	$-0.521739\pi$
$331$	−821.922	−0.136486	−0.0682431	+	0.997669i	$0.521739\pi$
$332$	−8177.39	−1.35179
$333$	0	0
$334$	1855.95	0.304051
$335$	0	0
$336$	0	0
$337$	1032.55	0.166904	0.0834520	−	0.996512i	$-0.473405\pi$
$337$	1032.55	0.166904	0.0834520	+	0.996512i	$0.473405\pi$
$338$	−979.667	−0.157653
$339$	0	0
$340$	0	0
$341$	−10556.4	−1.67643
$342$	0	0
$343$	343.000	0.0539949
$344$	−2340.65	−0.366859
$345$	0	0
$346$	−509.382	−0.0791461
$347$	2587.70	0.400332	0.200166	−	0.979762i	$-0.435852\pi$
$347$	2587.70	0.400332	0.200166	+	0.979762i	$0.435852\pi$
$348$	0	0
$349$	−8454.24	−1.29669	−0.648345	−	0.761346i	$-0.724539\pi$
$349$	−8454.24	−1.29669	−0.648345	+	0.761346i	$0.724539\pi$
$350$	0	0
$351$	0	0
$352$	−5094.80	−0.771460
$353$	4129.63	0.622658	0.311329	−	0.950302i	$-0.399226\pi$
$353$	4129.63	0.622658	0.311329	+	0.950302i	$0.399226\pi$
$354$	0	0
$355$	0	0
$356$	2645.50	0.393852
$357$	0	0
$358$	−1615.95	−0.238563
$359$	4536.69	0.666957	0.333478	−	0.942758i	$-0.391778\pi$
$359$	4536.69	0.666957	0.333478	+	0.942758i	$0.391778\pi$
$360$	0	0
$361$	9397.56	1.37011
$362$	−994.492	−0.144390
$363$	0	0
$364$	−867.859	−0.124968
$365$	0	0
$366$	0	0
$367$	1643.00	0.233689	0.116844	−	0.993150i	$-0.462722\pi$
$367$	1643.00	0.233689	0.116844	+	0.993150i	$0.462722\pi$
$368$	−2961.81	−0.419552
$369$	0	0
$370$	0	0
$371$	4629.21	0.647808
$372$	0	0
$373$	9019.53	1.25205	0.626024	−	0.779804i	$-0.284681\pi$
$373$	9019.53	1.25205	0.626024	+	0.779804i	$0.284681\pi$
$374$	−11.6920	−0.00161652
$375$	0	0
$376$	4294.23	0.588984
$377$	−663.904	−0.0906971
$378$	0	0
$379$	−2463.63	−0.333900	−0.166950	−	0.985965i	$-0.553392\pi$
$379$	−2463.63	−0.333900	−0.166950	+	0.985965i	$0.553392\pi$
$380$	0	0
$381$	0	0
$382$	−1995.61	−0.267289
$383$	11007.2	1.46852	0.734261	−	0.678867i	$-0.237529\pi$
$383$	11007.2	1.46852	0.734261	+	0.678867i	$0.237529\pi$
$384$	0	0
$385$	0	0
$386$	1224.33	0.161442
$387$	0	0
$388$	10976.6	1.43621
$389$	−6162.79	−0.803254	−0.401627	−	0.915803i	$-0.631555\pi$
$389$	−6162.79	−0.803254	−0.401627	+	0.915803i	$0.631555\pi$
$390$	0	0
$391$	−21.5713	−0.00279005
$392$	−389.448	−0.0501788
$393$	0	0
$394$	−1578.21	−0.201799
$395$	0	0
$396$	0	0
$397$	−15137.9	−1.91373	−0.956864	−	0.290537i	$-0.906166\pi$
$397$	−15137.9	−1.91373	−0.956864	+	0.290537i	$0.906166\pi$
$398$	−1800.12	−0.226714
$399$	0	0
$400$	0	0
$401$	−3302.29	−0.411243	−0.205622	−	0.978632i	$-0.565922\pi$
$401$	−3302.29	−0.411243	−0.205622	+	0.978632i	$0.565922\pi$
$402$	0	0
$403$	3079.08	0.380596
$404$	−937.564	−0.115459
$405$	0	0
$406$	−146.551	−0.0179143
$407$	10385.1	1.26479
$408$	0	0
$409$	4576.40	0.553273	0.276636	−	0.960975i	$-0.410780\pi$
$409$	4576.40	0.553273	0.276636	+	0.960975i	$0.410780\pi$
$410$	0	0
$411$	0	0
$412$	5077.70	0.607186
$413$	−2872.18	−0.342205
$414$	0	0
$415$	0	0
$416$	1486.05	0.175143
$417$	0	0
$418$	−3532.11	−0.413304
$419$	6778.05	0.790285	0.395142	−	0.918620i	$-0.370695\pi$
$419$	6778.05	0.790285	0.395142	+	0.918620i	$0.370695\pi$
$420$	0	0
$421$	−3952.68	−0.457582	−0.228791	−	0.973476i	$-0.573477\pi$
$421$	−3952.68	−0.457582	−0.228791	+	0.973476i	$0.573477\pi$
$422$	−1867.29	−0.215398
$423$	0	0
$424$	−5256.08	−0.602024
$425$	0	0
$426$	0	0
$427$	322.346	0.0365326
$428$	16031.1	1.81050
$429$	0	0
$430$	0	0
$431$	3435.10	0.383905	0.191952	−	0.981404i	$-0.438518\pi$
$431$	3435.10	0.383905	0.191952	+	0.981404i	$0.438518\pi$
$432$	0	0
$433$	−5457.82	−0.605742	−0.302871	−	0.953032i	$-0.597945\pi$
$433$	−5457.82	−0.605742	−0.302871	+	0.953032i	$0.597945\pi$
$434$	679.682	0.0751746
$435$	0	0
$436$	−7003.35	−0.769265
$437$	−6516.61	−0.713344
$438$	0	0
$439$	4188.10	0.455323	0.227662	−	0.973740i	$-0.426892\pi$
$439$	4188.10	0.455323	0.227662	+	0.973740i	$0.426892\pi$
$440$	0	0
$441$	0	0
$442$	3.41031	0.000366996 0
$443$	−2097.50	−0.224955	−0.112478	−	0.993654i	$-0.535879\pi$
$443$	−2097.50	−0.224955	−0.112478	+	0.993654i	$0.535879\pi$
$444$	0	0
$445$	0	0
$446$	−51.5085	−0.00546860
$447$	0	0
$448$	−2917.14	−0.307639
$449$	−11318.2	−1.18962	−0.594809	−	0.803867i	$-0.702772\pi$
$449$	−11318.2	−1.18962	−0.594809	+	0.803867i	$0.702772\pi$
$450$	0	0
$451$	−4188.37	−0.437301
$452$	−6531.91	−0.679724
$453$	0	0
$454$	−2897.41	−0.299521
$455$	0	0
$456$	0	0
$457$	−11696.4	−1.19723	−0.598617	−	0.801036i	$-0.704283\pi$
$457$	−11696.4	−1.19723	−0.598617	+	0.801036i	$0.704283\pi$
$458$	−425.596	−0.0434209
$459$	0	0
$460$	0	0
$461$	−10685.0	−1.07951	−0.539753	−	0.841823i	$-0.681482\pi$
$461$	−10685.0	−1.07951	−0.539753	+	0.841823i	$0.681482\pi$
$462$	0	0
$463$	−10407.2	−1.04463	−0.522315	−	0.852752i	$-0.674932\pi$
$463$	−10407.2	−1.04463	−0.522315	+	0.852752i	$0.674932\pi$
$464$	−2403.46	−0.240469
$465$	0	0
$466$	1018.48	0.101245
$467$	2978.24	0.295111	0.147555	−	0.989054i	$-0.452860\pi$
$467$	2978.24	0.295111	0.147555	+	0.989054i	$0.452860\pi$
$468$	0	0
$469$	72.8517	0.00717266
$470$	0	0
$471$	0	0
$472$	3261.12	0.318020
$473$	−16162.1	−1.57111
$474$	0	0
$475$	0	0
$476$	−22.8823	−0.00220338
$477$	0	0
$478$	−123.008	−0.0117704
$479$	14214.3	1.35588	0.677940	−	0.735117i	$-0.262873\pi$
$479$	14214.3	1.35588	0.677940	+	0.735117i	$0.262873\pi$
$480$	0	0
$481$	−3029.11	−0.287142
$482$	−1544.93	−0.145995
$483$	0	0
$484$	−13018.2	−1.22260
$485$	0	0
$486$	0	0
$487$	18097.0	1.68389	0.841946	−	0.539562i	$-0.181410\pi$
$487$	18097.0	1.68389	0.841946	+	0.539562i	$0.181410\pi$
$488$	−365.998	−0.0339507
$489$	0	0
$490$	0	0
$491$	13750.8	1.26388	0.631940	−	0.775017i	$-0.282259\pi$
$491$	13750.8	1.26388	0.631940	+	0.775017i	$0.282259\pi$
$492$	0	0
$493$	−17.5048	−0.00159914
$494$	1030.24	0.0938314
$495$	0	0
$496$	11146.9	1.00909
$497$	3437.82	0.310276
$498$	0	0
$499$	15684.2	1.40706	0.703528	−	0.710667i	$-0.251607\pi$
$499$	15684.2	1.40706	0.703528	+	0.710667i	$0.251607\pi$
$500$	0	0
$501$	0	0
$502$	3328.61	0.295943
$503$	−977.937	−0.0866880	−0.0433440	−	0.999060i	$-0.513801\pi$
$503$	−977.937	−0.0866880	−0.0433440	+	0.999060i	$0.513801\pi$
$504$	0	0
$505$	0	0
$506$	1415.88	0.124394
$507$	0	0
$508$	−15726.6	−1.37354
$509$	19664.6	1.71242	0.856208	−	0.516632i	$-0.172814\pi$
$509$	19664.6	1.71242	0.856208	+	0.516632i	$0.172814\pi$
$510$	0	0
$511$	−5701.78	−0.493604
$512$	9064.39	0.782409
$513$	0	0
$514$	−258.775	−0.0222064
$515$	0	0
$516$	0	0
$517$	29651.5	2.52238
$518$	−668.650	−0.0567158
$519$	0	0
$520$	0	0
$521$	−12522.8	−1.05304	−0.526519	−	0.850163i	$-0.676503\pi$
$521$	−12522.8	−1.05304	−0.526519	+	0.850163i	$0.676503\pi$
$522$	0	0
$523$	1104.42	0.0923382	0.0461691	−	0.998934i	$-0.485299\pi$
$523$	1104.42	0.0923382	0.0461691	+	0.998934i	$0.485299\pi$
$524$	22247.8	1.85477
$525$	0	0
$526$	1797.05	0.148964
$527$	81.1843	0.00671052
$528$	0	0
$529$	−9554.75	−0.785301
$530$	0	0
$531$	0	0
$532$	−6912.65	−0.563349
$533$	1221.66	0.0992794
$534$	0	0
$535$	0	0
$536$	−82.7170	−0.00666573
$537$	0	0
$538$	277.502	0.0222378
$539$	−2689.12	−0.214895
$540$	0	0
$541$	−2126.45	−0.168989	−0.0844945	−	0.996424i	$-0.526928\pi$
$541$	−2126.45	−0.168989	−0.0844945	+	0.996424i	$0.526928\pi$
$542$	−1991.33	−0.157813
$543$	0	0
$544$	39.1817	0.00308805
$545$	0	0
$546$	0	0
$547$	−2137.04	−0.167044	−0.0835220	−	0.996506i	$-0.526617\pi$
$547$	−2137.04	−0.167044	−0.0835220	+	0.996506i	$0.526617\pi$
$548$	−12559.8	−0.979063
$549$	0	0
$550$	0	0
$551$	−5288.11	−0.408859
$552$	0	0
$553$	−6011.08	−0.462237
$554$	−633.335	−0.0485701
$555$	0	0
$556$	−20274.8	−1.54648
$557$	19435.1	1.47844	0.739219	−	0.673465i	$-0.235195\pi$
$557$	19435.1	1.47844	0.739219	+	0.673465i	$0.235195\pi$
$558$	0	0
$559$	4714.14	0.356685
$560$	0	0
$561$	0	0
$562$	344.008	0.0258205
$563$	−1467.52	−0.109855	−0.0549276	−	0.998490i	$-0.517493\pi$
$563$	−1467.52	−0.109855	−0.0549276	+	0.998490i	$0.517493\pi$
$564$	0	0
$565$	0	0
$566$	−3001.89	−0.222931
$567$	0	0
$568$	−3903.35	−0.288347
$569$	−15080.3	−1.11107	−0.555536	−	0.831493i	$-0.687487\pi$
$569$	−15080.3	−1.11107	−0.555536	+	0.831493i	$0.687487\pi$
$570$	0	0
$571$	15800.8	1.15804	0.579022	−	0.815312i	$-0.303435\pi$
$571$	15800.8	1.15804	0.579022	+	0.815312i	$0.303435\pi$
$572$	6804.02	0.497361
$573$	0	0
$574$	269.671	0.0196095
$575$	0	0
$576$	0	0
$577$	6165.64	0.444851	0.222425	−	0.974950i	$-0.428603\pi$
$577$	6165.64	0.444851	0.222425	+	0.974950i	$0.428603\pi$
$578$	−2479.91	−0.178461
$579$	0	0
$580$	0	0
$581$	7390.62	0.527736
$582$	0	0
$583$	−36293.0	−2.57822
$584$	6473.89	0.458719
$585$	0	0
$586$	−1095.90	−0.0772547
$587$	−23104.0	−1.62454	−0.812268	−	0.583285i	$-0.801767\pi$
$587$	−23104.0	−1.62454	−0.812268	+	0.583285i	$0.801767\pi$
$588$	0	0
$589$	24525.4	1.71571
$590$	0	0
$591$	0	0
$592$	−10965.9	−0.761313
$593$	−6544.43	−0.453200	−0.226600	−	0.973988i	$-0.572761\pi$
$593$	−6544.43	−0.453200	−0.226600	+	0.973988i	$0.572761\pi$
$594$	0	0
$595$	0	0
$596$	17447.0	1.19909
$597$	0	0
$598$	−412.982	−0.0282410
$599$	−3215.71	−0.219349	−0.109675	−	0.993968i	$-0.534981\pi$
$599$	−3215.71	−0.219349	−0.109675	+	0.993968i	$0.534981\pi$
$600$	0	0
$601$	−234.653	−0.0159263	−0.00796313	−	0.999968i	$-0.502535\pi$
$601$	−234.653	−0.0159263	−0.00796313	+	0.999968i	$0.502535\pi$
$602$	1040.61	0.0704517
$603$	0	0
$604$	14325.6	0.965064
$605$	0	0
$606$	0	0
$607$	−14528.7	−0.971501	−0.485750	−	0.874098i	$-0.661454\pi$
$607$	−14528.7	−0.971501	−0.485750	+	0.874098i	$0.661454\pi$
$608$	11836.6	0.789537
$609$	0	0
$610$	0	0
$611$	−8648.71	−0.572650
$612$	0	0
$613$	11332.7	0.746696	0.373348	−	0.927691i	$-0.378210\pi$
$613$	11332.7	0.746696	0.373348	+	0.927691i	$0.378210\pi$
$614$	−1839.73	−0.120921
$615$	0	0
$616$	3053.27	0.199708
$617$	17400.2	1.13534	0.567672	−	0.823255i	$-0.307844\pi$
$617$	17400.2	1.13534	0.567672	+	0.823255i	$0.307844\pi$
$618$	0	0
$619$	−11987.8	−0.778402	−0.389201	−	0.921153i	$-0.627249\pi$
$619$	−11987.8	−0.778402	−0.389201	+	0.921153i	$0.627249\pi$
$620$	0	0
$621$	0	0
$622$	−1304.46	−0.0840900
$623$	−2390.97	−0.153759
$624$	0	0
$625$	0	0
$626$	3378.84	0.215728
$627$	0	0
$628$	25850.3	1.64258
$629$	−79.8667	−0.00506279
$630$	0	0
$631$	−3800.38	−0.239764	−0.119882	−	0.992788i	$-0.538252\pi$
$631$	−3800.38	−0.239764	−0.119882	+	0.992788i	$0.538252\pi$
$632$	6825.08	0.429568
$633$	0	0
$634$	1940.07	0.121530
$635$	0	0
$636$	0	0
$637$	784.359	0.0487872
$638$	1148.96	0.0712976
$639$	0	0
$640$	0	0
$641$	−12294.2	−0.757552	−0.378776	−	0.925488i	$-0.623655\pi$
$641$	−12294.2	−0.757552	−0.378776	+	0.925488i	$0.623655\pi$
$642$	0	0
$643$	14615.4	0.896382	0.448191	−	0.893938i	$-0.352068\pi$
$643$	14615.4	0.896382	0.448191	+	0.893938i	$0.352068\pi$
$644$	2771.01	0.169554
$645$	0	0
$646$	27.1637	0.00165440
$647$	16047.0	0.975071	0.487536	−	0.873103i	$-0.337896\pi$
$647$	16047.0	0.975071	0.487536	+	0.873103i	$0.337896\pi$
$648$	0	0
$649$	22517.9	1.36195
$650$	0	0
$651$	0	0
$652$	3648.75	0.219166
$653$	−623.425	−0.0373607	−0.0186803	−	0.999826i	$-0.505946\pi$
$653$	−623.425	−0.0373607	−0.0186803	+	0.999826i	$0.505946\pi$
$654$	0	0
$655$	0	0
$656$	4422.63	0.263224
$657$	0	0
$658$	−1909.13	−0.113109
$659$	−963.459	−0.0569515	−0.0284758	−	0.999594i	$-0.509065\pi$
$659$	−963.459	−0.0569515	−0.0284758	+	0.999594i	$0.509065\pi$
$660$	0	0
$661$	24183.6	1.42304	0.711522	−	0.702664i	$-0.248006\pi$
$661$	24183.6	1.42304	0.711522	+	0.702664i	$0.248006\pi$
$662$	−414.893	−0.0243584
$663$	0	0
$664$	−8391.43	−0.490438
$665$	0	0
$666$	0	0
$667$	2119.79	0.123057
$668$	−28476.9	−1.64941
$669$	0	0
$670$	0	0
$671$	−2527.20	−0.145397
$672$	0	0
$673$	14167.0	0.811437	0.405718	−	0.913998i	$-0.367021\pi$
$673$	14167.0	0.811437	0.405718	+	0.913998i	$0.367021\pi$
$674$	521.215	0.0297870
$675$	0	0
$676$	15031.6	0.855235
$677$	19397.1	1.10117	0.550584	−	0.834780i	$-0.314405\pi$
$677$	19397.1	1.10117	0.550584	+	0.834780i	$0.314405\pi$
$678$	0	0
$679$	−9920.47	−0.560696
$680$	0	0
$681$	0	0
$682$	−5328.71	−0.299189
$683$	−20984.8	−1.17564	−0.587820	−	0.808992i	$-0.700014\pi$
$683$	−20984.8	−1.17564	−0.587820	+	0.808992i	$0.700014\pi$
$684$	0	0
$685$	0	0
$686$	173.141	0.00963636
$687$	0	0
$688$	17066.1	0.945694
$689$	10585.9	0.585328
$690$	0	0
$691$	−18079.2	−0.995320	−0.497660	−	0.867372i	$-0.665807\pi$
$691$	−18079.2	−0.995320	−0.497660	+	0.867372i	$0.665807\pi$
$692$	7815.75	0.429350
$693$	0	0
$694$	1306.23	0.0714465
$695$	0	0
$696$	0	0
$697$	32.2107	0.00175046
$698$	−4267.56	−0.231418
$699$	0	0
$700$	0	0
$701$	−18660.6	−1.00542	−0.502711	−	0.864455i	$-0.667664\pi$
$701$	−18660.6	−1.00542	−0.502711	+	0.864455i	$0.667664\pi$
$702$	0	0
$703$	−24127.4	−1.29443
$704$	22870.4	1.22438
$705$	0	0
$706$	2084.57	0.111125
$707$	847.358	0.0450752
$708$	0	0
$709$	−24545.4	−1.30017	−0.650085	−	0.759862i	$-0.725267\pi$
$709$	−24545.4	−1.30017	−0.650085	+	0.759862i	$0.725267\pi$
$710$	0	0
$711$	0	0
$712$	2714.74	0.142892
$713$	−9831.26	−0.516387
$714$	0	0
$715$	0	0
$716$	24794.5	1.29415
$717$	0	0
$718$	2290.05	0.119030
$719$	7300.32	0.378659	0.189330	−	0.981914i	$-0.439369\pi$
$719$	7300.32	0.378659	0.189330	+	0.981914i	$0.439369\pi$
$720$	0	0
$721$	−4589.16	−0.237045
$722$	4743.74	0.244520
$723$	0	0
$724$	15259.1	0.783286
$725$	0	0
$726$	0	0
$727$	15466.0	0.788998	0.394499	−	0.918896i	$-0.370918\pi$
$727$	15466.0	0.788998	0.394499	+	0.918896i	$0.370918\pi$
$728$	−890.574	−0.0453391
$729$	0	0
$730$	0	0
$731$	124.295	0.00628893
$732$	0	0
$733$	−13830.5	−0.696917	−0.348459	−	0.937324i	$-0.613295\pi$
$733$	−13830.5	−0.696917	−0.348459	+	0.937324i	$0.613295\pi$
$734$	829.358	0.0417060
$735$	0	0
$736$	−4744.83	−0.237631
$737$	−571.157	−0.0285466
$738$	0	0
$739$	11714.5	0.583118	0.291559	−	0.956553i	$-0.405826\pi$
$739$	11714.5	0.583118	0.291559	+	0.956553i	$0.405826\pi$
$740$	0	0
$741$	0	0
$742$	2336.75	0.115613
$743$	−28963.4	−1.43010	−0.715050	−	0.699074i	$-0.753596\pi$
$743$	−28963.4	−1.43010	−0.715050	+	0.699074i	$0.753596\pi$
$744$	0	0
$745$	0	0
$746$	4552.91	0.223450
$747$	0	0
$748$	179.397	0.00876928
$749$	−14488.7	−0.706817
$750$	0	0
$751$	6331.99	0.307667	0.153833	−	0.988097i	$-0.450838\pi$
$751$	6331.99	0.307667	0.153833	+	0.988097i	$0.450838\pi$
$752$	−31310.0	−1.51829
$753$	0	0
$754$	−335.128	−0.0161865
$755$	0	0
$756$	0	0
$757$	−6256.80	−0.300406	−0.150203	−	0.988655i	$-0.547993\pi$
$757$	−6256.80	−0.300406	−0.150203	+	0.988655i	$0.547993\pi$
$758$	−1243.60	−0.0595905
$759$	0	0
$760$	0	0
$761$	−13953.7	−0.664679	−0.332340	−	0.943160i	$-0.607838\pi$
$761$	−13953.7	−0.664679	−0.332340	+	0.943160i	$0.607838\pi$
$762$	0	0
$763$	6329.53	0.300320
$764$	30619.8	1.44998
$765$	0	0
$766$	5556.28	0.262084
$767$	−6568.00	−0.309200
$768$	0	0
$769$	34278.1	1.60741	0.803707	−	0.595025i	$-0.202858\pi$
$769$	34278.1	1.60741	0.803707	+	0.595025i	$0.202858\pi$
$770$	0	0
$771$	0	0
$772$	−18785.5	−0.875786
$773$	−8769.31	−0.408034	−0.204017	−	0.978967i	$-0.565400\pi$
$773$	−8769.31	−0.408034	−0.204017	+	0.978967i	$0.565400\pi$
$774$	0	0
$775$	0	0
$776$	11263.9	0.521069
$777$	0	0
$778$	−3110.88	−0.143355
$779$	9730.73	0.447547
$780$	0	0
$781$	−26952.5	−1.23487
$782$	−10.8889	−0.000497934 0
$783$	0	0
$784$	2839.53	0.129352
$785$	0	0
$786$	0	0
$787$	−3546.15	−0.160618	−0.0803092	−	0.996770i	$-0.525591\pi$
$787$	−3546.15	−0.160618	−0.0803092	+	0.996770i	$0.525591\pi$
$788$	24215.4	1.09472
$789$	0	0
$790$	0	0
$791$	5903.45	0.265363
$792$	0	0
$793$	737.130	0.0330091
$794$	−7641.37	−0.341539
$795$	0	0
$796$	27620.3	1.22987
$797$	18169.4	0.807520	0.403760	−	0.914865i	$-0.367703\pi$
$797$	18169.4	0.807520	0.403760	+	0.914865i	$0.367703\pi$
$798$	0	0
$799$	−228.035	−0.0100968
$800$	0	0
$801$	0	0
$802$	−1666.94	−0.0733938
$803$	44702.0	1.96451
$804$	0	0
$805$	0	0
$806$	1554.27	0.0679241
$807$	0	0
$808$	−962.104	−0.0418895
$809$	−11853.0	−0.515115	−0.257557	−	0.966263i	$-0.582918\pi$
$809$	−11853.0	−0.515115	−0.257557	+	0.966263i	$0.582918\pi$
$810$	0	0
$811$	−11921.1	−0.516161	−0.258080	−	0.966123i	$-0.583090\pi$
$811$	−11921.1	−0.516161	−0.258080	+	0.966123i	$0.583090\pi$
$812$	2248.62	0.0971813
$813$	0	0
$814$	5242.22	0.225724
$815$	0	0
$816$	0	0
$817$	37548.9	1.60792
$818$	2310.09	0.0987415
$819$	0	0
$820$	0	0
$821$	42094.1	1.78940	0.894698	−	0.446671i	$-0.147390\pi$
$821$	42094.1	1.78940	0.894698	+	0.446671i	$0.147390\pi$
$822$	0	0
$823$	1622.17	0.0687064	0.0343532	−	0.999410i	$-0.489063\pi$
$823$	1622.17	0.0687064	0.0343532	+	0.999410i	$0.489063\pi$
$824$	5210.61	0.220291
$825$	0	0
$826$	−1449.83	−0.0610727
$827$	−28171.7	−1.18455	−0.592277	−	0.805735i	$-0.701771\pi$
$827$	−28171.7	−1.18455	−0.592277	+	0.805735i	$0.701771\pi$
$828$	0	0
$829$	−9244.23	−0.387293	−0.193646	−	0.981071i	$-0.562031\pi$
$829$	−9244.23	−0.387293	−0.193646	+	0.981071i	$0.562031\pi$
$830$	0	0
$831$	0	0
$832$	−6670.81	−0.277967
$833$	20.6807	0.000860198 0
$834$	0	0
$835$	0	0
$836$	54195.2	2.24208
$837$	0	0
$838$	3421.45	0.141041
$839$	43012.5	1.76991	0.884956	−	0.465674i	$-0.154188\pi$
$839$	43012.5	1.76991	0.884956	+	0.465674i	$0.154188\pi$
$840$	0	0
$841$	−22668.8	−0.929469
$842$	−1995.25	−0.0816637
$843$	0	0
$844$	28650.9	1.16849
$845$	0	0
$846$	0	0
$847$	11765.7	0.477302
$848$	38323.0	1.55191
$849$	0	0
$850$	0	0
$851$	9671.70	0.389591
$852$	0	0
$853$	−19756.9	−0.793041	−0.396521	−	0.918026i	$-0.629783\pi$
$853$	−19756.9	−0.793041	−0.396521	+	0.918026i	$0.629783\pi$
$854$	162.715	0.00651991
$855$	0	0
$856$	16450.7	0.656862
$857$	−325.658	−0.0129805	−0.00649024	−	0.999979i	$-0.502066\pi$
$857$	−325.658	−0.0129805	−0.00649024	+	0.999979i	$0.502066\pi$
$858$	0	0
$859$	3275.22	0.130092	0.0650461	−	0.997882i	$-0.479281\pi$
$859$	3275.22	0.130092	0.0650461	+	0.997882i	$0.479281\pi$
$860$	0	0
$861$	0	0
$862$	1733.98	0.0685147
$863$	−44585.0	−1.75862	−0.879311	−	0.476247i	$-0.841997\pi$
$863$	−44585.0	−1.75862	−0.879311	+	0.476247i	$0.841997\pi$
$864$	0	0
$865$	0	0
$866$	−2755.02	−0.108106
$867$	0	0
$868$	−10428.8	−0.407805
$869$	47126.9	1.83967
$870$	0	0
$871$	166.594	0.00648087
$872$	−7186.65	−0.279095
$873$	0	0
$874$	−3289.48	−0.127309
$875$	0	0
$876$	0	0
$877$	16433.8	0.632761	0.316380	−	0.948632i	$-0.397532\pi$
$877$	16433.8	0.632761	0.316380	+	0.948632i	$0.397532\pi$
$878$	2114.08	0.0812607
$879$	0	0
$880$	0	0
$881$	39080.2	1.49449	0.747245	−	0.664549i	$-0.231376\pi$
$881$	39080.2	1.49449	0.747245	+	0.664549i	$0.231376\pi$
$882$	0	0
$883$	48548.2	1.85026	0.925129	−	0.379652i	$-0.123956\pi$
$883$	48548.2	1.85026	0.925129	+	0.379652i	$0.123956\pi$
$884$	−52.3264	−0.00199087
$885$	0	0
$886$	−1058.78	−0.0401473
$887$	13447.1	0.509029	0.254514	−	0.967069i	$-0.418084\pi$
$887$	13447.1	0.509029	0.254514	+	0.967069i	$0.418084\pi$
$888$	0	0
$889$	14213.5	0.536227
$890$	0	0
$891$	0	0
$892$	790.325	0.0296660
$893$	−68888.5	−2.58148
$894$	0	0
$895$	0	0
$896$	−6671.30	−0.248742
$897$	0	0
$898$	−5713.24	−0.212309
$899$	−7977.90	−0.295971
$900$	0	0
$901$	279.112	0.0103203
$902$	−2114.22	−0.0780442
$903$	0	0
$904$	−6702.87	−0.246609
$905$	0	0
$906$	0	0
$907$	15750.3	0.576604	0.288302	−	0.957539i	$-0.406909\pi$
$907$	15750.3	0.576604	0.288302	+	0.957539i	$0.406909\pi$
$908$	44456.7	1.62483
$909$	0	0
$910$	0	0
$911$	−31449.7	−1.14377	−0.571886	−	0.820333i	$-0.693788\pi$
$911$	−31449.7	−1.14377	−0.571886	+	0.820333i	$0.693788\pi$
$912$	0	0
$913$	−57942.4	−2.10034
$914$	−5904.16	−0.213668
$915$	0	0
$916$	6530.17	0.235549
$917$	−20107.3	−0.724102
$918$	0	0
$919$	−30355.6	−1.08960	−0.544798	−	0.838567i	$-0.683394\pi$
$919$	−30355.6	−1.08960	−0.544798	+	0.838567i	$0.683394\pi$
$920$	0	0
$921$	0	0
$922$	−5393.64	−0.192657
$923$	7861.46	0.280350
$924$	0	0
$925$	0	0
$926$	−5253.39	−0.186433
$927$	0	0
$928$	−3850.34	−0.136200
$929$	28533.8	1.00771	0.503856	−	0.863788i	$-0.331914\pi$
$929$	28533.8	1.00771	0.503856	+	0.863788i	$0.331914\pi$
$930$	0	0
$931$	6247.56	0.219931
$932$	−15627.1	−0.549231
$933$	0	0
$934$	1503.37	0.0526678
$935$	0	0
$936$	0	0
$937$	−19269.1	−0.671817	−0.335908	−	0.941895i	$-0.609043\pi$
$937$	−19269.1	−0.671817	−0.335908	+	0.941895i	$0.609043\pi$
$938$	36.7743	0.00128009
$939$	0	0
$940$	0	0
$941$	2097.84	0.0726756	0.0363378	−	0.999340i	$-0.488431\pi$
$941$	2097.84	0.0726756	0.0363378	+	0.999340i	$0.488431\pi$
$942$	0	0
$943$	−3900.66	−0.134701
$944$	−23777.4	−0.819797
$945$	0	0
$946$	−8158.36	−0.280392
$947$	−5072.14	−0.174047	−0.0870235	−	0.996206i	$-0.527736\pi$
$947$	−5072.14	−0.174047	−0.0870235	+	0.996206i	$0.527736\pi$
$948$	0	0
$949$	−13038.6	−0.445997
$950$	0	0
$951$	0	0
$952$	−23.4813	−0.000799403 0
$953$	23464.1	0.797564	0.398782	−	0.917046i	$-0.369433\pi$
$953$	23464.1	0.797564	0.398782	+	0.917046i	$0.369433\pi$
$954$	0	0
$955$	0	0
$956$	1887.39	0.0638519
$957$	0	0
$958$	7175.13	0.241981
$959$	11351.3	0.382225
$960$	0	0
$961$	7209.24	0.241994
$962$	−1529.04	−0.0512457
$963$	0	0
$964$	23704.8	0.791992
$965$	0	0
$966$	0	0
$967$	−6994.27	−0.232596	−0.116298	−	0.993214i	$-0.537103\pi$
$967$	−6994.27	−0.232596	−0.116298	+	0.993214i	$0.537103\pi$
$968$	−13359.0	−0.443568
$969$	0	0
$970$	0	0
$971$	−37752.5	−1.24772	−0.623860	−	0.781536i	$-0.714436\pi$
$971$	−37752.5	−1.24772	−0.623860	+	0.781536i	$0.714436\pi$
$972$	0	0
$973$	18324.0	0.603743
$974$	9135.09	0.300521
$975$	0	0
$976$	2668.55	0.0875186
$977$	6380.21	0.208926	0.104463	−	0.994529i	$-0.466688\pi$
$977$	6380.21	0.208926	0.104463	+	0.994529i	$0.466688\pi$
$978$	0	0
$979$	18745.2	0.611950
$980$	0	0
$981$	0	0
$982$	6941.18	0.225562
$983$	−17330.0	−0.562299	−0.281150	−	0.959664i	$-0.590716\pi$
$983$	−17330.0	−0.562299	−0.281150	+	0.959664i	$0.590716\pi$
$984$	0	0
$985$	0	0
$986$	−8.83612	−0.000285395 0
$987$	0	0
$988$	−15807.6	−0.509014
$989$	−15051.9	−0.483945
$990$	0	0
$991$	32191.8	1.03189	0.515947	−	0.856621i	$-0.327440\pi$
$991$	32191.8	1.03189	0.515947	+	0.856621i	$0.327440\pi$
$992$	17857.3	0.571542
$993$	0	0
$994$	1735.35	0.0553743
$995$	0	0
$996$	0	0
$997$	−2678.20	−0.0850748	−0.0425374	−	0.999095i	$-0.513544\pi$
$997$	−2678.20	−0.0850748	−0.0425374	+	0.999095i	$0.513544\pi$
$998$	7917.13	0.251114
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bg.1.3		4
3.2	odd	2		175.4.a.h.1.2	yes	4
5.4	even	2		1575.4.a.bl.1.2		4
15.2	even	4		175.4.b.f.99.4		8
15.8	even	4		175.4.b.f.99.5		8
15.14	odd	2		175.4.a.g.1.3	✓	4
21.20	even	2		1225.4.a.bd.1.2		4
105.104	even	2		1225.4.a.z.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.g.1.3	✓	4	15.14	odd	2
175.4.a.h.1.2	yes	4	3.2	odd	2
175.4.b.f.99.4		8	15.2	even	4
175.4.b.f.99.5		8	15.8	even	4
1225.4.a.z.1.3		4	105.104	even	2
1225.4.a.bd.1.2		4	21.20	even	2
1575.4.a.bg.1.3		4	1.1	even	1	trivial
1575.4.a.bl.1.2		4	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+0.504784 q^{2} -7.74519 q^{4} +7.00000 q^{7} -7.94792 q^{8} -54.8800 q^{11} +16.0073 q^{13} +3.53349 q^{14} +57.9496 q^{16} +0.422056 q^{17} +127.501 q^{19} -27.7025 q^{22} -51.1101 q^{23} +8.08024 q^{26} -54.2164 q^{28} -41.4750 q^{29} +192.354 q^{31} +92.8353 q^{32} +0.213047 q^{34} -189.232 q^{37} +64.3605 q^{38} +76.3187 q^{41} +294.499 q^{43} +425.056 q^{44} -25.7996 q^{46} -540.297 q^{47} +49.0000 q^{49} -123.980 q^{52} +661.316 q^{53} -55.6354 q^{56} -20.9359 q^{58} -410.312 q^{59} +46.0495 q^{61} +97.0974 q^{62} -416.735 q^{64} +10.4074 q^{67} -3.26890 q^{68} +491.117 q^{71} -814.540 q^{73} -95.5215 q^{74} -987.522 q^{76} -384.160 q^{77} -858.725 q^{79} +38.5244 q^{82} +1055.80 q^{83} +148.658 q^{86} +436.182 q^{88} -341.567 q^{89} +112.051 q^{91} +395.858 q^{92} -272.733 q^{94} -1417.21 q^{97} +24.7344 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 36 q^{4} + 28 q^{7} - 27 q^{8} - 100 q^{11} + 44 q^{13} - 28 q^{14} + 160 q^{16} + 53 q^{17} - 29 q^{19} - 152 q^{22} - 295 q^{23} - 700 q^{26} + 252 q^{28} - 129 q^{29} + 114 q^{31} + 310 q^{32}+ \cdots - 196 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 36 * q^4 + 28 * q^7 - 27 * q^8 - 100 * q^11 + 44 * q^13 - 28 * q^14 + 160 * q^16 + 53 * q^17 - 29 * q^19 - 152 * q^22 - 295 * q^23 - 700 * q^26 + 252 * q^28 - 129 * q^29 + 114 * q^31 + 310 * q^32 + 203 * q^34 + 403 * q^37 - 555 * q^38 - 671 * q^41 - 411 * q^43 - 438 * q^44 - 997 * q^46 + 8 * q^47 + 196 * q^49 + 74 * q^52 - 90 * q^53 - 189 * q^56 + 673 * q^58 - 1018 * q^59 + 50 * q^61 + 1626 * q^62 - 2421 * q^64 + 424 * q^67 + 617 * q^68 - 215 * q^71 - 1207 * q^73 - 623 * q^74 - 3257 * q^76 - 700 * q^77 - 951 * q^79 + 1695 * q^82 + 3035 * q^83 + 99 * q^86 + 163 * q^88 - 2819 * q^89 + 308 * q^91 - 3073 * q^92 - 3056 * q^94 - 1100 * q^97 - 196 * q^98

Introduction

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