LMFDB - Embedded newform 560.4.g.f.449.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,4,Mod(449,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("560.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	560.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:	$33.0410696032$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 55x^{8} + 983x^{6} + 6409x^{4} + 13560x^{2} + 3600 $ x^10 + 55x^8 + 983x^6 + 6409x^4 + 13560x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.9
Root			$5.04851i$ of defining polynomial
Character	$\chi$	$=$	560.449
Dual form			560.4.g.f.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.52749i q^{3} +(6.15172 + 9.33576i) q^{5} -7.00000i q^{7} -15.6081 q^{9} -6.78210 q^{11} -48.9221i q^{13} +(-60.9390 + 40.1552i) q^{15} +92.4381i q^{17} -125.574 q^{19} +45.6924 q^{21} -32.2681i q^{23} +(-49.3128 + 114.862i) q^{25} +74.3607i q^{27} -282.778 q^{29} -205.434 q^{31} -44.2701i q^{33} +(65.3503 - 43.0620i) q^{35} +190.627i q^{37} +319.338 q^{39} +123.269 q^{41} +35.0202i q^{43} +(-96.0164 - 145.713i) q^{45} -419.030i q^{47} -49.0000 q^{49} -603.388 q^{51} +0.365379i q^{53} +(-41.7215 - 63.3160i) q^{55} -819.683i q^{57} +328.317 q^{59} -515.707 q^{61} +109.256i q^{63} +(456.725 - 300.955i) q^{65} +828.957i q^{67} +210.630 q^{69} +496.231 q^{71} -701.132i q^{73} +(-749.759 - 321.888i) q^{75} +47.4747i q^{77} +199.388 q^{79} -906.806 q^{81} -194.923i q^{83} +(-862.980 + 568.653i) q^{85} -1845.83i q^{87} -137.406 q^{89} -342.454 q^{91} -1340.97i q^{93} +(-772.496 - 1172.33i) q^{95} +220.440i q^{97} +105.855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6 q^{5} - 46 q^{9} - 84 q^{11} - 8 q^{15} - 72 q^{19} + 140 q^{21} - 362 q^{25} + 88 q^{29} - 120 q^{31} + 28 q^{35} - 212 q^{39} - 852 q^{41} - 510 q^{45} - 490 q^{49} - 1276 q^{51} + 1136 q^{55}+ \cdots + 5304 q^{99}+O(q^{100}) $ 10 * q + 6 * q^5 - 46 * q^9 - 84 * q^11 - 8 * q^15 - 72 * q^19 + 140 * q^21 - 362 * q^25 + 88 * q^29 - 120 * q^31 + 28 * q^35 - 212 * q^39 - 852 * q^41 - 510 * q^45 - 490 * q^49 - 1276 * q^51 + 1136 * q^55 - 864 * q^59 - 884 * q^61 + 520 * q^65 + 4808 * q^69 - 880 * q^71 - 720 * q^75 + 3580 * q^79 - 302 * q^81 + 1572 * q^85 - 1492 * q^89 - 476 * q^91 + 1628 * q^95 + 5304 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$			6.52749i			1.25622i	0.778127	+	0.628108i	$0.216170\pi$
$3$			6.52749i			1.25622i	−0.778127	+	0.628108i	$0.783830\pi$
$4$		0			0
$5$	6.15172	+	9.33576i	0.550226	+	0.835016i
$5$	6.15172	+	9.33576i	0.550226	+	0.835016i
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$		0			0
$9$	−15.6081			−0.578076
$10$		0			0
$11$	−6.78210			−0.185898			−0.0929491	−	0.995671i	$-0.529629\pi$
$11$	−6.78210			−0.185898			−0.0929491	+	0.995671i	$0.529629\pi$
$12$		0			0
$13$		−	48.9221i		−	1.04373i	−0.853027	−	0.521867i	$-0.825236\pi$
$13$		−	48.9221i		−	1.04373i	0.853027	−	0.521867i	$-0.174764\pi$
$14$		0			0
$15$	−60.9390	+	40.1552i	−1.04896	+	0.691202i
$16$		0			0
$17$			92.4381i			1.31880i	0.751794	+	0.659398i	$0.229189\pi$
$17$			92.4381i			1.31880i	−0.751794	+	0.659398i	$0.770811\pi$
$18$		0			0
$19$	−125.574			−1.51625			−0.758123	−	0.652112i	$-0.773883\pi$
$19$	−125.574			−1.51625			−0.758123	+	0.652112i	$0.773883\pi$
$20$		0			0
$21$	45.6924			0.474805
$22$		0			0
$23$		−	32.2681i		−	0.292538i	−0.989245	−	0.146269i	$-0.953274\pi$
$23$		−	32.2681i		−	0.292538i	0.989245	−	0.146269i	$-0.0467265\pi$
$24$		0			0
$25$	−49.3128	+	114.862i	−0.394502	+	0.918895i
$26$		0			0
$27$			74.3607i			0.530027i
$28$		0			0
$29$	−282.778			−1.81071			−0.905354	−	0.424659i	$-0.860394\pi$
$29$	−282.778			−1.81071			−0.905354	+	0.424659i	$0.860394\pi$
$30$		0			0
$31$	−205.434			−1.19023			−0.595115	−	0.803641i	$-0.702893\pi$
$31$	−205.434			−1.19023			−0.595115	+	0.803641i	$0.702893\pi$
$32$		0			0
$33$		−	44.2701i		−	0.233528i
$34$		0			0
$35$	65.3503	−	43.0620i	0.315606	−	0.207966i
$36$		0			0
$37$			190.627i			0.846998i	0.905897	+	0.423499i	$0.139198\pi$
$37$			190.627i			0.846998i	−0.905897	+	0.423499i	$0.860802\pi$
$38$		0			0
$39$	319.338			1.31115
$40$		0			0
$41$	123.269			0.469546			0.234773	−	0.972050i	$-0.424565\pi$
$41$	123.269			0.469546			0.234773	+	0.972050i	$0.424565\pi$
$42$		0			0
$43$			35.0202i			0.124198i	0.998070	+	0.0620991i	$0.0197795\pi$
$43$			35.0202i			0.124198i	−0.998070	+	0.0620991i	$0.980221\pi$
$44$		0			0
$45$	−96.0164	−	145.713i	−0.318073	−	0.482703i
$46$		0			0
$47$		−	419.030i		−	1.30046i	−0.759736	−	0.650231i	$-0.774672\pi$
$47$		−	419.030i		−	1.30046i	0.759736	−	0.650231i	$-0.225328\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	−603.388			−1.65669
$52$		0			0
$53$			0.365379i			0.000946956i	1.00000		0.000473478i	$0.000150713\pi$
$53$			0.365379i			0.000946956i	−1.00000		0.000473478i	$0.999849\pi$
$54$		0			0
$55$	−41.7215	−	63.3160i	−0.102286	−	0.155228i
$56$		0			0
$57$		−	819.683i		−	1.90473i
$58$		0			0
$59$	328.317			0.724461			0.362231	−	0.932088i	$-0.382015\pi$
$59$	328.317			0.724461			0.362231	+	0.932088i	$0.382015\pi$
$60$		0			0
$61$	−515.707			−1.08245			−0.541226	−	0.840877i	$-0.682040\pi$
$61$	−515.707			−1.08245			−0.541226	+	0.840877i	$0.682040\pi$
$62$		0			0
$63$			109.256i			0.218492i
$64$		0			0
$65$	456.725	−	300.955i	0.871534	−	0.574290i
$66$		0			0
$67$			828.957i			1.51154i	0.654837	+	0.755770i	$0.272737\pi$
$67$			828.957i			1.51154i	−0.654837	+	0.755770i	$0.727263\pi$
$68$		0			0
$69$	210.630			0.367490
$70$		0			0
$71$	496.231			0.829462			0.414731	−	0.909944i	$-0.363876\pi$
$71$	496.231			0.829462			0.414731	+	0.909944i	$0.363876\pi$
$72$		0			0
$73$		−	701.132i		−	1.12413i	−0.827094	−	0.562064i	$-0.810008\pi$
$73$		−	701.132i		−	1.12413i	0.827094	−	0.562064i	$-0.189992\pi$
$74$		0			0
$75$	−749.759	−	321.888i	−1.15433	−	0.495580i
$76$		0			0
$77$			47.4747i			0.0702629i
$78$		0			0
$79$	199.388			0.283961			0.141981	−	0.989869i	$-0.454653\pi$
$79$	199.388			0.283961			0.141981	+	0.989869i	$0.454653\pi$
$80$		0			0
$81$	−906.806			−1.24390
$82$		0			0
$83$		−	194.923i		−	0.257778i	−0.991659	−	0.128889i	$-0.958859\pi$
$83$		−	194.923i		−	0.257778i	0.991659	−	0.128889i	$-0.0411411\pi$
$84$		0			0
$85$	−862.980	+	568.653i	−1.10122	+	0.725636i
$86$		0			0
$87$		−	1845.83i		−	2.27464i
$88$		0			0
$89$	−137.406			−0.163651			−0.0818257	−	0.996647i	$-0.526075\pi$
$89$	−137.406			−0.163651			−0.0818257	+	0.996647i	$0.526075\pi$
$90$		0			0
$91$	−342.454			−0.394494
$92$		0			0
$93$		−	1340.97i		−	1.49518i
$94$		0			0
$95$	−772.496	−	1172.33i	−0.834278	−	1.26609i
$96$		0			0
$97$			220.440i			0.230745i	0.993322	+	0.115372i	$0.0368062\pi$
$97$			220.440i			0.230745i	−0.993322	+	0.115372i	$0.963194\pi$
$98$		0			0
$99$	105.855			0.107463
$100$		0			0
$101$	−591.358			−0.582597			−0.291298	−	0.956632i	$-0.594087\pi$
$101$	−591.358			−0.582597			−0.291298	+	0.956632i	$0.594087\pi$
$102$		0			0
$103$		−	476.494i		−	0.455829i	−0.973681	−	0.227914i	$-0.926809\pi$
$103$		−	476.494i		−	0.455829i	0.973681	−	0.227914i	$-0.0731906\pi$
$104$		0			0
$105$	281.087	+	426.573i	0.261250	+	0.396469i
$106$		0			0
$107$			225.584i			0.203813i	0.994794	+	0.101907i	$0.0324943\pi$
$107$			225.584i			0.203813i	−0.994794	+	0.101907i	$0.967506\pi$
$108$		0			0
$109$	1627.65			1.43028			0.715142	−	0.698979i	$-0.246362\pi$
$109$	1627.65			1.43028			0.715142	+	0.698979i	$0.246362\pi$
$110$		0			0
$111$	−1244.32			−1.06401
$112$		0			0
$113$		−	357.040i		−	0.297235i	−0.988895	−	0.148617i	$-0.952518\pi$
$113$		−	357.040i		−	0.297235i	0.988895	−	0.148617i	$-0.0474823\pi$
$114$		0			0
$115$	301.247	−	198.504i	0.244274	−	0.160962i
$116$		0			0
$117$			763.579i			0.603358i
$118$		0			0
$119$	647.067			0.498458
$120$		0			0
$121$	−1285.00			−0.965442
$122$		0			0
$123$			804.637i			0.589851i
$124$		0			0
$125$	−1375.68	+	246.225i	−0.984357	+	0.176185i
$126$		0			0
$127$			1728.25i			1.20754i	0.797158	+	0.603771i	$0.206336\pi$
$127$			1728.25i			1.20754i	−0.797158	+	0.603771i	$0.793664\pi$
$128$		0			0
$129$	−228.594			−0.156020
$130$		0			0
$131$	−1461.19			−0.974543			−0.487272	−	0.873250i	$-0.662008\pi$
$131$	−1461.19			−0.974543			−0.487272	+	0.873250i	$0.662008\pi$
$132$		0			0
$133$			879.018i			0.573087i
$134$		0			0
$135$	−694.214	+	457.446i	−0.442581	+	0.291635i
$136$		0			0
$137$			1892.96i			1.18049i	0.807226	+	0.590243i	$0.200968\pi$
$137$			1892.96i			1.18049i	−0.807226	+	0.590243i	$0.799032\pi$
$138$		0			0
$139$	−1715.03			−1.04652			−0.523262	−	0.852172i	$-0.675285\pi$
$139$	−1715.03			−1.04652			−0.523262	+	0.852172i	$0.675285\pi$
$140$		0			0
$141$	2735.21			1.63366
$142$		0			0
$143$			331.794i			0.194028i
$144$		0			0
$145$	−1739.57	−	2639.94i	−0.996299	−	1.51197i
$146$		0			0
$147$		−	319.847i		−	0.179459i
$148$		0			0
$149$	2798.32			1.53857			0.769286	−	0.638904i	$-0.220612\pi$
$149$	2798.32			1.53857			0.769286	+	0.638904i	$0.220612\pi$
$150$		0			0
$151$	2867.49			1.54538			0.772692	−	0.634781i	$-0.218910\pi$
$151$	2867.49			1.54538			0.772692	+	0.634781i	$0.218910\pi$
$152$		0			0
$153$		−	1442.78i		−	0.762365i
$154$		0			0
$155$	−1263.77	−	1917.89i	−0.654896	−	0.993860i
$156$		0			0
$157$		−	783.696i		−	0.398381i	−0.979961	−	0.199190i	$-0.936169\pi$
$157$		−	783.696i		−	0.398381i	0.979961	−	0.199190i	$-0.0638312\pi$
$158$		0			0
$159$	−2.38501			−0.00118958
$160$		0			0
$161$	−225.877			−0.110569
$162$		0			0
$163$			2416.93i			1.16140i	0.814116	+	0.580702i	$0.197222\pi$
$163$			2416.93i			1.16140i	−0.814116	+	0.580702i	$0.802778\pi$
$164$		0			0
$165$	413.295	−	272.337i	0.195000	−	0.128493i
$166$		0			0
$167$			704.424i			0.326407i	0.986592	+	0.163204i	$0.0521827\pi$
$167$			704.424i			0.326407i	−0.986592	+	0.163204i	$0.947817\pi$
$168$		0			0
$169$	−196.369			−0.0893803
$170$		0			0
$171$	1959.97			0.876506
$172$		0			0
$173$			1398.49i			0.614598i	0.951613	+	0.307299i	$0.0994252\pi$
$173$			1398.49i			0.614598i	−0.951613	+	0.307299i	$0.900575\pi$
$174$		0			0
$175$	804.033	+	345.189i	0.347310	+	0.149108i
$176$		0			0
$177$			2143.08i			0.910079i
$178$		0			0
$179$	368.688			0.153950			0.0769749	−	0.997033i	$-0.475474\pi$
$179$	368.688			0.153950			0.0769749	+	0.997033i	$0.475474\pi$
$180$		0			0
$181$	−315.621			−0.129613			−0.0648064	−	0.997898i	$-0.520643\pi$
$181$	−315.621			−0.129613			−0.0648064	+	0.997898i	$0.520643\pi$
$182$		0			0
$183$		−	3366.27i		−	1.35979i
$184$		0			0
$185$	−1779.65	+	1172.69i	−0.707257	+	0.466041i
$186$		0			0
$187$		−	626.924i		−	0.245162i
$188$		0			0
$189$	520.525			0.200331
$190$		0			0
$191$	151.629			0.0574424			0.0287212	−	0.999587i	$-0.490857\pi$
$191$	151.629			0.0574424			0.0287212	+	0.999587i	$0.490857\pi$
$192$		0			0
$193$			690.689i			0.257600i	0.991671	+	0.128800i	$0.0411126\pi$
$193$			690.689i			0.257600i	−0.991671	+	0.128800i	$0.958887\pi$
$194$		0			0
$195$	1964.48	+	2981.26i	0.721431	+	1.09483i
$196$		0			0
$197$		−	834.136i		−	0.301674i	−0.988559	−	0.150837i	$-0.951803\pi$
$197$		−	834.136i		−	0.301674i	0.988559	−	0.150837i	$-0.0481968\pi$
$198$		0			0
$199$	387.269			0.137954			0.0689769	−	0.997618i	$-0.478027\pi$
$199$	387.269			0.137954			0.0689769	+	0.997618i	$0.478027\pi$
$200$		0			0
$201$	−5411.00			−1.89882
$202$		0			0
$203$			1979.44i			0.684383i
$204$		0			0
$205$	758.316	+	1150.81i	0.258357	+	0.392078i
$206$		0			0
$207$			503.643i			0.169109i
$208$		0			0
$209$	851.656			0.281867
$210$		0			0
$211$	−3070.54			−1.00182			−0.500912	−	0.865498i	$-0.667002\pi$
$211$	−3070.54			−1.00182			−0.500912	+	0.865498i	$0.667002\pi$
$212$		0			0
$213$			3239.14i			1.04198i
$214$		0			0
$215$	−326.940	+	215.434i	−0.103707	+	0.0683371i
$216$		0			0
$217$			1438.04i			0.449865i
$218$		0			0
$219$	4576.63			1.41215
$220$		0			0
$221$	4522.26			1.37647
$222$		0			0
$223$			1004.46i			0.301631i	0.988562	+	0.150816i	$0.0481900\pi$
$223$			1004.46i			0.301631i	−0.988562	+	0.150816i	$0.951810\pi$
$224$		0			0
$225$	769.677	−	1792.77i	0.228052	−	0.531192i
$226$		0			0
$227$			5374.22i			1.57136i	0.618631	+	0.785681i	$0.287687\pi$
$227$			5374.22i			1.57136i	−0.618631	+	0.785681i	$0.712313\pi$
$228$		0			0
$229$	−3650.97			−1.05355			−0.526775	−	0.850005i	$-0.676599\pi$
$229$	−3650.97			−1.05355			−0.526775	+	0.850005i	$0.676599\pi$
$230$		0			0
$231$	−309.890			−0.0882653
$232$		0			0
$233$		−	4582.88i		−	1.28856i	−0.764789	−	0.644280i	$-0.777157\pi$
$233$		−	4582.88i		−	1.28856i	0.764789	−	0.644280i	$-0.222843\pi$
$234$		0			0
$235$	3911.96	−	2577.75i	1.08591	−	0.715549i
$236$		0			0
$237$			1301.50i			0.356716i
$238$		0			0
$239$	696.769			0.188578			0.0942892	−	0.995545i	$-0.469942\pi$
$239$	696.769			0.188578			0.0942892	+	0.995545i	$0.469942\pi$
$240$		0			0
$241$	5082.82			1.35856			0.679281	−	0.733878i	$-0.262292\pi$
$241$	5082.82			1.35856			0.679281	+	0.733878i	$0.262292\pi$
$242$		0			0
$243$		−	3911.42i		−	1.03258i
$244$		0			0
$245$	−301.434	−	457.452i	−0.0786037	−	0.119288i
$246$		0			0
$247$			6143.34i			1.58256i
$248$		0			0
$249$	1272.36			0.323825
$250$		0			0
$251$	1207.32			0.303606			0.151803	−	0.988411i	$-0.451492\pi$
$251$	1207.32			0.303606			0.151803	+	0.988411i	$0.451492\pi$
$252$		0			0
$253$			218.846i			0.0543822i
$254$		0			0
$255$	−3711.87	−	5633.09i	−0.911555	−	1.38336i
$256$		0			0
$257$			510.936i			0.124013i	0.998076	+	0.0620064i	$0.0197499\pi$
$257$			510.936i			0.124013i	−0.998076	+	0.0620064i	$0.980250\pi$
$258$		0			0
$259$	1334.39			0.320135
$260$		0			0
$261$	4413.61			1.04673
$262$		0			0
$263$			2269.04i			0.531997i	0.963974	+	0.265998i	$0.0857015\pi$
$263$			2269.04i			0.531997i	−0.963974	+	0.265998i	$0.914298\pi$
$264$		0			0
$265$	−3.41109	+	2.24771i	−0.000790723	+	0.000521040i
$266$		0			0
$267$		−	896.913i		−	0.205581i
$268$		0			0
$269$	7836.10			1.77612			0.888058	−	0.459731i	$-0.152054\pi$
$269$	7836.10			1.77612			0.888058	+	0.459731i	$0.152054\pi$
$270$		0			0
$271$	−1466.28			−0.328673			−0.164336	−	0.986404i	$-0.552548\pi$
$271$	−1466.28			−0.328673			−0.164336	+	0.986404i	$0.552548\pi$
$272$		0			0
$273$		−	2235.37i		−	0.495570i
$274$		0			0
$275$	334.444	−	779.005i	0.0733372	−	0.170821i
$276$		0			0
$277$			4154.17i			0.901083i	0.892755	+	0.450542i	$0.148769\pi$
$277$			4154.17i			0.901083i	−0.892755	+	0.450542i	$0.851231\pi$
$278$		0			0
$279$	3206.43			0.688044
$280$		0			0
$281$	−3490.40			−0.740996			−0.370498	−	0.928833i	$-0.620813\pi$
$281$	−3490.40			−0.740996			−0.370498	+	0.928833i	$0.620813\pi$
$282$		0			0
$283$		−	3125.29i		−	0.656464i	−0.944597	−	0.328232i	$-0.893547\pi$
$283$		−	3125.29i		−	0.656464i	0.944597	−	0.328232i	$-0.106453\pi$
$284$		0			0
$285$	7652.36	−	5042.46i	1.59048	−	1.04803i
$286$		0			0
$287$		−	862.883i		−	0.177472i
$288$		0			0
$289$	−3631.80			−0.739223
$290$		0			0
$291$	−1438.92			−0.289865
$292$		0			0
$293$			1447.69i			0.288652i	0.989530	+	0.144326i	$0.0461014\pi$
$293$			1447.69i			0.288652i	−0.989530	+	0.144326i	$0.953899\pi$
$294$		0			0
$295$	2019.71	+	3065.09i	0.398618	+	0.604936i
$296$		0			0
$297$		−	504.322i		−	0.0985310i
$298$		0			0
$299$	−1578.62			−0.305332
$300$		0			0
$301$	245.141			0.0469425
$302$		0			0
$303$		−	3860.08i		−	0.731867i
$304$		0			0
$305$	−3172.48	−	4814.52i	−0.595593	−	0.903864i
$306$		0			0
$307$		−	1591.43i		−	0.295856i	−0.988998	−	0.147928i	$-0.952740\pi$
$307$		−	1591.43i		−	0.295856i	0.988998	−	0.147928i	$-0.0472603\pi$
$308$		0			0
$309$	3110.31			0.572619
$310$		0			0
$311$	8584.92			1.56529			0.782647	−	0.622466i	$-0.213869\pi$
$311$	8584.92			1.56529			0.782647	+	0.622466i	$0.213869\pi$
$312$		0			0
$313$			7210.05i			1.30203i	0.759064	+	0.651016i	$0.225657\pi$
$313$			7210.05i			1.30203i	−0.759064	+	0.651016i	$0.774343\pi$
$314$		0			0
$315$	−1019.99	+	672.115i	−0.182445	+	0.120220i
$316$		0			0
$317$			7787.24i			1.37973i	0.723938	+	0.689865i	$0.242330\pi$
$317$			7787.24i			1.37973i	−0.723938	+	0.689865i	$0.757670\pi$
$318$		0			0
$319$	1917.83			0.336607
$320$		0			0
$321$	−1472.49			−0.256033
$322$		0			0
$323$		−	11607.8i		−	1.99962i
$324$		0			0
$325$	5619.28	+	2412.48i	0.959082	+	0.411755i
$326$		0			0
$327$			10624.5i			1.79674i
$328$		0			0
$329$	−2933.21			−0.491529
$330$		0			0
$331$	1729.78			0.287243			0.143621	−	0.989633i	$-0.454125\pi$
$331$	1729.78			0.287243			0.143621	+	0.989633i	$0.454125\pi$
$332$		0			0
$333$		−	2975.32i		−	0.489630i
$334$		0			0
$335$	−7738.94	+	5099.51i	−1.26216	+	0.831689i
$336$		0			0
$337$		−	7815.06i		−	1.26325i	−0.775276	−	0.631623i	$-0.782389\pi$
$337$		−	7815.06i		−	1.26325i	0.775276	−	0.631623i	$-0.217611\pi$
$338$		0			0
$339$	2330.57			0.373391
$340$		0			0
$341$	1393.28			0.221262
$342$		0			0
$343$			343.000i			0.0539949i
$344$		0			0
$345$	1295.73	+	1966.39i	0.202203	+	0.306860i
$346$		0			0
$347$			2359.77i			0.365070i	0.983199	+	0.182535i	$0.0584303\pi$
$347$			2359.77i			0.365070i	−0.983199	+	0.182535i	$0.941570\pi$
$348$		0			0
$349$	−3212.73			−0.492761			−0.246380	−	0.969173i	$-0.579241\pi$
$349$	−3212.73			−0.492761			−0.246380	+	0.969173i	$0.579241\pi$
$350$		0			0
$351$	3637.88			0.553207
$352$		0			0
$353$		−	5582.04i		−	0.841649i	−0.907142	−	0.420824i	$-0.861741\pi$
$353$		−	5582.04i		−	0.841649i	0.907142	−	0.420824i	$-0.138259\pi$
$354$		0			0
$355$	3052.67	+	4632.69i	0.456392	+	0.692614i
$356$		0			0
$357$			4223.72i			0.626171i
$358$		0			0
$359$	−10630.6			−1.56285			−0.781425	−	0.623999i	$-0.785507\pi$
$359$	−10630.6			−1.56285			−0.781425	+	0.623999i	$0.785507\pi$
$360$		0			0
$361$	8909.84			1.29900
$362$		0			0
$363$		−	8387.84i		−	1.21280i
$364$		0			0
$365$	6545.60	−	4313.17i	0.938664	−	0.618524i
$366$		0			0
$367$			4514.69i			0.642139i	0.947056	+	0.321070i	$0.104042\pi$
$367$			4514.69i			0.642139i	−0.947056	+	0.321070i	$0.895958\pi$
$368$		0			0
$369$	−1923.99			−0.271434
$370$		0			0
$371$	2.55765			0.000357916
$372$		0			0
$373$			11445.8i			1.58885i	0.607361	+	0.794426i	$0.292228\pi$
$373$			11445.8i			1.58885i	−0.607361	+	0.794426i	$0.707772\pi$
$374$		0			0
$375$	−1607.23	−	8979.74i	−0.221326	−	1.23656i
$376$		0			0
$377$			13834.1i			1.88990i
$378$		0			0
$379$	8146.48			1.10411			0.552054	−	0.833809i	$-0.313844\pi$
$379$	8146.48			1.10411			0.552054	+	0.833809i	$0.313844\pi$
$380$		0			0
$381$	−11281.2			−1.51693
$382$		0			0
$383$			5261.56i			0.701967i	0.936382	+	0.350983i	$0.114153\pi$
$383$			5261.56i			0.701967i	−0.936382	+	0.350983i	$0.885847\pi$
$384$		0			0
$385$	−443.212	+	292.051i	−0.0586706	+	0.0386605i
$386$		0			0
$387$		−	546.597i		−	0.0717961i
$388$		0			0
$389$	−13207.1			−1.72141			−0.860706	−	0.509103i	$-0.829977\pi$
$389$	−13207.1			−1.72141			−0.860706	+	0.509103i	$0.829977\pi$
$390$		0			0
$391$	2982.80			0.385798
$392$		0			0
$393$		−	9537.93i		−	1.22424i
$394$		0			0
$395$	1226.58	+	1861.44i	0.156243	+	0.237112i
$396$		0			0
$397$			5663.15i			0.715933i	0.933734	+	0.357967i	$0.116530\pi$
$397$			5663.15i			0.715933i	−0.933734	+	0.357967i	$0.883470\pi$
$398$		0			0
$399$	−5737.78			−0.719920
$400$		0			0
$401$	−11989.0			−1.49302			−0.746512	−	0.665372i	$-0.768273\pi$
$401$	−11989.0			−1.49302			−0.746512	+	0.665372i	$0.768273\pi$
$402$		0			0
$403$			10050.3i			1.24228i
$404$		0			0
$405$	−5578.41	−	8465.72i	−0.684429	−	1.03868i
$406$		0			0
$407$		−	1292.85i		−	0.157455i
$408$		0			0
$409$	−5249.67			−0.634669			−0.317334	−	0.948314i	$-0.602788\pi$
$409$	−5249.67			−0.634669			−0.317334	+	0.948314i	$0.602788\pi$
$410$		0			0
$411$	−12356.3			−1.48294
$412$		0			0
$413$		−	2298.22i		−	0.273821i
$414$		0			0
$415$	1819.76	−	1199.11i	0.215249	−	0.141836i
$416$		0			0
$417$		−	11194.8i		−	1.31466i
$418$		0			0
$419$	−14948.9			−1.74297			−0.871484	−	0.490424i	$-0.836842\pi$
$419$	−14948.9			−1.74297			−0.871484	+	0.490424i	$0.836842\pi$
$420$		0			0
$421$	5840.31			0.676103			0.338051	−	0.941128i	$-0.390232\pi$
$421$	5840.31			0.676103			0.338051	+	0.941128i	$0.390232\pi$
$422$		0			0
$423$			6540.24i			0.751767i
$424$		0			0
$425$	−10617.6	−	4558.38i	−1.21184	−	0.520268i
$426$		0			0
$427$			3609.95i			0.409128i
$428$		0			0
$429$	−2165.78			−0.243741
$430$		0			0
$431$	−7439.21			−0.831402			−0.415701	−	0.909501i	$-0.636464\pi$
$431$	−7439.21			−0.831402			−0.415701	+	0.909501i	$0.636464\pi$
$432$		0			0
$433$		−	877.657i		−	0.0974077i	−0.998813	−	0.0487038i	$-0.984491\pi$
$433$		−	877.657i		−	0.0974077i	0.998813	−	0.0487038i	$-0.0155090\pi$
$434$		0			0
$435$	17232.2	−	11355.0i	1.89936	−	1.25157i
$436$		0			0
$437$			4052.04i			0.443559i
$438$		0			0
$439$	−10855.8			−1.18023			−0.590114	−	0.807320i	$-0.700917\pi$
$439$	−10855.8			−1.18023			−0.590114	+	0.807320i	$0.700917\pi$
$440$		0			0
$441$	764.795			0.0825823
$442$		0			0
$443$			10797.0i			1.15798i	0.815336	+	0.578988i	$0.196552\pi$
$443$			10797.0i			1.15798i	−0.815336	+	0.578988i	$0.803448\pi$
$444$		0			0
$445$	−845.281	−	1282.79i	−0.0900453	−	0.136651i
$446$		0			0
$447$			18266.0i			1.93278i
$448$		0			0
$449$	−8621.70			−0.906198			−0.453099	−	0.891460i	$-0.649682\pi$
$449$	−8621.70			−0.906198			−0.453099	+	0.891460i	$0.649682\pi$
$450$		0			0
$451$	−836.023			−0.0872878
$452$		0			0
$453$			18717.5i			1.94134i
$454$		0			0
$455$	−2106.68	−	3197.07i	−0.217061	−	0.329409i
$456$		0			0
$457$			3785.90i			0.387520i	0.981049	+	0.193760i	$0.0620684\pi$
$457$			3785.90i			0.387520i	−0.981049	+	0.193760i	$0.937932\pi$
$458$		0			0
$459$	−6873.76			−0.698997
$460$		0			0
$461$	5760.18			0.581948			0.290974	−	0.956731i	$-0.406021\pi$
$461$	5760.18			0.581948			0.290974	+	0.956731i	$0.406021\pi$
$462$		0			0
$463$		−	10760.9i		−	1.08014i	−0.841621	−	0.540068i	$-0.818398\pi$
$463$		−	10760.9i		−	1.08014i	0.841621	−	0.540068i	$-0.181602\pi$
$464$		0			0
$465$	12519.0	−	8249.27i	1.24850	−	0.822690i
$466$		0			0
$467$			2153.58i			0.213395i	0.994292	+	0.106698i	$0.0340277\pi$
$467$			2153.58i			0.213395i	−0.994292	+	0.106698i	$0.965972\pi$
$468$		0			0
$469$	5802.70			0.571309
$470$		0			0
$471$	5115.57			0.500452
$472$		0			0
$473$		−	237.510i		−	0.0230882i
$474$		0			0
$475$	6192.40	−	14423.7i	0.598162	−	1.39327i
$476$		0			0
$477$		−	5.70286i		−	0.000547413i
$478$		0			0
$479$	−6890.26			−0.657253			−0.328626	−	0.944460i	$-0.606586\pi$
$479$	−6890.26			−0.657253			−0.328626	+	0.944460i	$0.606586\pi$
$480$		0			0
$481$	9325.88			0.884041
$482$		0			0
$483$		−	1474.41i		−	0.138898i
$484$		0			0
$485$	−2057.97	+	1356.08i	−0.192676	+	0.126962i
$486$		0			0
$487$		−	19006.4i		−	1.76850i	−0.467011	−	0.884251i	$-0.654669\pi$
$487$		−	19006.4i		−	1.76850i	0.467011	−	0.884251i	$-0.345331\pi$
$488$		0			0
$489$	−15776.5			−1.45897
$490$		0			0
$491$	−4530.05			−0.416371			−0.208186	−	0.978089i	$-0.566756\pi$
$491$	−4530.05			−0.416371			−0.208186	+	0.978089i	$0.566756\pi$
$492$		0			0
$493$		−	26139.4i		−	2.38795i
$494$		0			0
$495$	651.193	+	988.241i	0.0591292	+	0.0897336i
$496$		0			0
$497$		−	3473.62i		−	0.313507i
$498$		0			0
$499$	3620.18			0.324773			0.162386	−	0.986727i	$-0.448081\pi$
$499$	3620.18			0.324773			0.162386	+	0.986727i	$0.448081\pi$
$500$		0			0
$501$	−4598.12			−0.410038
$502$		0			0
$503$		−	11761.8i		−	1.04261i	−0.853372	−	0.521303i	$-0.825446\pi$
$503$		−	11761.8i		−	1.04261i	0.853372	−	0.521303i	$-0.174554\pi$
$504$		0			0
$505$	−3637.86	−	5520.77i	−0.320560	−	0.486478i
$506$		0			0
$507$		−	1281.79i		−	0.112281i
$508$		0			0
$509$	5254.47			0.457564			0.228782	−	0.973478i	$-0.426526\pi$
$509$	5254.47			0.457564			0.228782	+	0.973478i	$0.426526\pi$
$510$		0			0
$511$	−4907.92			−0.424880
$512$		0			0
$513$		−	9337.77i		−	0.803651i
$514$		0			0
$515$	4448.43	−	2931.26i	0.380624	−	0.250809i
$516$		0			0
$517$			2841.90i			0.241754i
$518$		0			0
$519$	−9128.64			−0.772067
$520$		0			0
$521$	−15511.8			−1.30439			−0.652193	−	0.758053i	$-0.726151\pi$
$521$	−15511.8			−1.30439			−0.652193	+	0.758053i	$0.726151\pi$
$522$		0			0
$523$		−	3814.73i		−	0.318942i	−0.987203	−	0.159471i	$-0.949021\pi$
$523$		−	3814.73i		−	0.318942i	0.987203	−	0.159471i	$-0.0509788\pi$
$524$		0			0
$525$	−2253.22	+	5248.31i	−0.187311	+	0.436296i
$526$		0			0
$527$		−	18990.0i		−	1.56967i
$528$		0			0
$529$	11125.8			0.914422
$530$		0			0
$531$	−5124.39			−0.418794
$532$		0			0
$533$		−	6030.58i		−	0.490081i
$534$		0			0
$535$	−2106.00	+	1387.73i	−0.170187	+	0.112143i
$536$		0			0
$537$			2406.60i			0.193394i
$538$		0			0
$539$	332.323			0.0265569
$540$		0			0
$541$	−4573.77			−0.363478			−0.181739	−	0.983347i	$-0.558173\pi$
$541$	−4573.77			−0.363478			−0.181739	+	0.983347i	$0.558173\pi$
$542$		0			0
$543$		−	2060.21i		−	0.162822i
$544$		0			0
$545$	10012.9	+	15195.4i	0.786980	+	1.19431i
$546$		0			0
$547$			13327.6i			1.04177i	0.853627	+	0.520885i	$0.174398\pi$
$547$			13327.6i			1.04177i	−0.853627	+	0.520885i	$0.825602\pi$
$548$		0			0
$549$	8049.19			0.625740
$550$		0			0
$551$	35509.5			2.74548
$552$		0			0
$553$		−	1395.72i		−	0.107327i
$554$		0			0
$555$	−7654.68	−	11616.6i	−0.585447	−	0.888467i
$556$		0			0
$557$		−	12096.1i		−	0.920155i	−0.887879	−	0.460077i	$-0.847822\pi$
$557$		−	12096.1i		−	0.920155i	0.887879	−	0.460077i	$-0.152178\pi$
$558$		0			0
$559$	1713.26			0.129630
$560$		0			0
$561$	4092.24			0.307976
$562$		0			0
$563$		−	22943.6i		−	1.71751i	−0.512387	−	0.858755i	$-0.671239\pi$
$563$		−	22943.6i		−	1.71751i	0.512387	−	0.858755i	$-0.328761\pi$
$564$		0			0
$565$	3333.24	−	2196.41i	0.248196	−	0.163546i
$566$		0			0
$567$			6347.64i			0.470152i
$568$		0			0
$569$	−485.307			−0.0357560			−0.0178780	−	0.999840i	$-0.505691\pi$
$569$	−485.307			−0.0357560			−0.0178780	+	0.999840i	$0.505691\pi$
$570$		0			0
$571$	−12271.8			−0.899401			−0.449701	−	0.893179i	$-0.648469\pi$
$571$	−12271.8			−0.899401			−0.449701	+	0.893179i	$0.648469\pi$
$572$		0			0
$573$			989.756i			0.0721600i
$574$		0			0
$575$	3706.38	+	1591.23i	0.268811	+	0.115407i
$576$		0			0
$577$			14122.8i			1.01896i	0.860484	+	0.509478i	$0.170162\pi$
$577$			14122.8i			1.01896i	−0.860484	+	0.509478i	$0.829838\pi$
$578$		0			0
$579$	−4508.46			−0.323601
$580$		0			0
$581$	−1364.46			−0.0974310
$582$		0			0
$583$		−	2.47804i		−	0.000176037i
$584$		0			0
$585$	−7128.59	+	4697.32i	−0.503813	+	0.331983i
$586$		0			0
$587$			11005.3i			0.773826i	0.922116	+	0.386913i	$0.126459\pi$
$587$			11005.3i			0.773826i	−0.922116	+	0.386913i	$0.873541\pi$
$588$		0			0
$589$	25797.2			1.80468
$590$		0			0
$591$	5444.81			0.378967
$592$		0			0
$593$		−	11233.1i		−	0.777889i	−0.921261	−	0.388944i	$-0.872840\pi$
$593$		−	11233.1i		−	0.777889i	0.921261	−	0.388944i	$-0.127160\pi$
$594$		0			0
$595$	3980.57	+	6040.86i	0.274265	+	0.416220i
$596$		0			0
$597$			2527.89i			0.173300i
$598$		0			0
$599$	14855.4			1.01331			0.506656	−	0.862149i	$-0.330882\pi$
$599$	14855.4			1.01331			0.506656	+	0.862149i	$0.330882\pi$
$600$		0			0
$601$	25358.8			1.72115			0.860573	−	0.509327i	$-0.170106\pi$
$601$	25358.8			1.72115			0.860573	+	0.509327i	$0.170106\pi$
$602$		0			0
$603$		−	12938.4i		−	0.873786i
$604$		0			0
$605$	−7904.97	−	11996.5i	−0.531211	−	0.806159i
$606$		0			0
$607$			14393.9i			0.962487i	0.876587	+	0.481243i	$0.159815\pi$
$607$			14393.9i			0.962487i	−0.876587	+	0.481243i	$0.840185\pi$
$608$		0			0
$609$	−12920.8			−0.859732
$610$		0			0
$611$	−20499.8			−1.35734
$612$		0			0
$613$			4769.94i			0.314284i	0.987576	+	0.157142i	$0.0502280\pi$
$613$			4769.94i			0.314284i	−0.987576	+	0.157142i	$0.949772\pi$
$614$		0			0
$615$	−7511.90	+	4949.90i	−0.492535	+	0.324551i
$616$		0			0
$617$			7769.86i			0.506974i	0.967339	+	0.253487i	$0.0815775\pi$
$617$			7769.86i			0.506974i	−0.967339	+	0.253487i	$0.918423\pi$
$618$		0			0
$619$	−5680.75			−0.368867			−0.184433	−	0.982845i	$-0.559045\pi$
$619$	−5680.75			−0.368867			−0.184433	+	0.982845i	$0.559045\pi$
$620$		0			0
$621$	2399.48			0.155053
$622$		0			0
$623$			961.840i			0.0618544i
$624$		0			0
$625$	−10761.5	−	11328.3i	−0.688736	−	0.725012i
$626$		0			0
$627$			5559.17i			0.354086i
$628$		0			0
$629$	−17621.2			−1.11702
$630$		0			0
$631$	10395.8			0.655864			0.327932	−	0.944701i	$-0.393648\pi$
$631$	10395.8			0.655864			0.327932	+	0.944701i	$0.393648\pi$
$632$		0			0
$633$		−	20042.9i		−	1.25851i
$634$		0			0
$635$	−16134.6	+	10631.7i	−1.00832	+	0.664421i
$636$		0			0
$637$			2397.18i			0.149105i
$638$		0			0
$639$	−7745.21			−0.479492
$640$		0			0
$641$	−8194.28			−0.504921			−0.252461	−	0.967607i	$-0.581240\pi$
$641$	−8194.28			−0.504921			−0.252461	+	0.967607i	$0.581240\pi$
$642$		0			0
$643$			32118.2i			1.96985i	0.172970	+	0.984927i	$0.444664\pi$
$643$			32118.2i			1.96985i	−0.172970	+	0.984927i	$0.555336\pi$
$644$		0			0
$645$	−1406.24	−	2134.09i	−0.0858461	−	0.130279i
$646$		0			0
$647$		−	22299.0i		−	1.35496i	−0.735539	−	0.677482i	$-0.763071\pi$
$647$		−	22299.0i		−	1.35496i	0.735539	−	0.677482i	$-0.236929\pi$
$648$		0			0
$649$	−2226.68			−0.134676
$650$		0			0
$651$	−9386.79			−0.565127
$652$		0			0
$653$			920.410i			0.0551584i	0.999620	+	0.0275792i	$0.00877984\pi$
$653$			920.410i			0.0551584i	−0.999620	+	0.0275792i	$0.991220\pi$
$654$		0			0
$655$	−8988.86	−	13641.4i	−0.536219	−	0.813759i
$656$		0			0
$657$			10943.3i			0.649832i
$658$		0			0
$659$	17824.3			1.05362			0.526812	−	0.849982i	$-0.323387\pi$
$659$	17824.3			1.05362			0.526812	+	0.849982i	$0.323387\pi$
$660$		0			0
$661$	11343.8			0.667510			0.333755	−	0.942660i	$-0.391684\pi$
$661$	11343.8			0.667510			0.333755	+	0.942660i	$0.391684\pi$
$662$		0			0
$663$			29519.0i			1.72915i
$664$		0			0
$665$	−8206.30	+	5407.47i	−0.478536	+	0.315327i
$666$		0			0
$667$			9124.71i			0.529700i
$668$		0			0
$669$	−6556.61			−0.378914
$670$		0			0
$671$	3497.58			0.201226
$672$		0			0
$673$		−	13422.9i		−	0.768816i	−0.923163	−	0.384408i	$-0.874406\pi$
$673$		−	13422.9i		−	0.768816i	0.923163	−	0.384408i	$-0.125594\pi$
$674$		0			0
$675$	−8541.21	−	3666.93i	−0.487039	−	0.209097i
$676$		0			0
$677$			1066.77i			0.0605602i	0.999541	+	0.0302801i	$0.00963993\pi$
$677$			1066.77i			0.0605602i	−0.999541	+	0.0302801i	$0.990360\pi$
$678$		0			0
$679$	1543.08			0.0872134
$680$		0			0
$681$	−35080.1			−1.97397
$682$		0			0
$683$			19090.6i			1.06952i	0.845005	+	0.534759i	$0.179598\pi$
$683$			19090.6i			1.06952i	−0.845005	+	0.534759i	$0.820402\pi$
$684$		0			0
$685$	−17672.2	+	11645.0i	−0.985724	+	0.649534i
$686$		0			0
$687$		−	23831.6i		−	1.32348i
$688$		0			0
$689$	17.8751			0.000988370
$690$		0			0
$691$	16878.4			0.929208			0.464604	−	0.885518i	$-0.346197\pi$
$691$	16878.4			0.929208			0.464604	+	0.885518i	$0.346197\pi$
$692$		0			0
$693$		−	740.988i		−	0.0406173i
$694$		0			0
$695$	−10550.4	−	16011.1i	−0.575825	−	0.873864i
$696$		0			0
$697$			11394.8i			0.619236i
$698$		0			0
$699$	29914.7			1.61871
$700$		0			0
$701$	30272.6			1.63107			0.815535	−	0.578707i	$-0.196443\pi$
$701$	30272.6			1.63107			0.815535	+	0.578707i	$0.196443\pi$
$702$		0			0
$703$		−	23937.8i		−	1.28426i
$704$		0			0
$705$	16826.2	+	25535.3i	0.898883	+	1.36413i
$706$		0			0
$707$			4139.50i			0.220201i
$708$		0			0
$709$	6593.32			0.349248			0.174624	−	0.984635i	$-0.444129\pi$
$709$	6593.32			0.349248			0.174624	+	0.984635i	$0.444129\pi$
$710$		0			0
$711$	−3112.06			−0.164151
$712$		0			0
$713$			6628.99i			0.348187i
$714$		0			0
$715$	−3097.55	+	2041.10i	−0.162017	+	0.106759i
$716$		0			0
$717$			4548.15i			0.236895i
$718$		0			0
$719$	3293.72			0.170842			0.0854208	−	0.996345i	$-0.472777\pi$
$719$	3293.72			0.170842			0.0854208	+	0.996345i	$0.472777\pi$
$720$		0			0
$721$	−3335.46			−0.172287
$722$		0			0
$723$			33178.1i			1.70665i
$724$		0			0
$725$	13944.6	−	32480.4i	0.714328	−	1.66385i
$726$		0			0
$727$			27757.8i			1.41606i	0.706181	+	0.708032i	$0.250417\pi$
$727$			27757.8i			1.41606i	−0.706181	+	0.708032i	$0.749583\pi$
$728$		0			0
$729$	1048.00			0.0532439
$730$		0			0
$731$	−3237.20			−0.163792
$732$		0			0
$733$			38324.6i			1.93117i	0.260080	+	0.965587i	$0.416251\pi$
$733$			38324.6i			1.93117i	−0.260080	+	0.965587i	$0.583749\pi$
$734$		0			0
$735$	2986.01	−	1967.61i	0.149851	−	0.0987432i
$736$		0			0
$737$		−	5622.07i		−	0.280993i
$738$		0			0
$739$	21957.3			1.09298			0.546490	−	0.837466i	$-0.315964\pi$
$739$	21957.3			1.09298			0.546490	+	0.837466i	$0.315964\pi$
$740$		0			0
$741$	−40100.6			−1.98803
$742$		0			0
$743$			14695.0i			0.725580i	0.931871	+	0.362790i	$0.118176\pi$
$743$			14695.0i			0.725580i	−0.931871	+	0.362790i	$0.881824\pi$
$744$		0			0
$745$	17214.5	+	26124.4i	0.846563	+	1.28473i
$746$		0			0
$747$			3042.37i			0.149016i
$748$		0			0
$749$	1579.09			0.0770341
$750$		0			0
$751$	21439.1			1.04171			0.520855	−	0.853645i	$-0.325613\pi$
$751$	21439.1			1.04171			0.520855	+	0.853645i	$0.325613\pi$
$752$		0			0
$753$			7880.74i			0.381395i
$754$		0			0
$755$	17640.0	+	26770.2i	0.850311	+	1.29042i
$756$		0			0
$757$			23896.8i			1.14735i	0.819083	+	0.573675i	$0.194483\pi$
$757$			23896.8i			1.14735i	−0.819083	+	0.573675i	$0.805517\pi$
$758$		0			0
$759$	−1428.51			−0.0683158
$760$		0			0
$761$	−24436.9			−1.16404			−0.582022	−	0.813173i	$-0.697738\pi$
$761$	−24436.9			−1.16404			−0.582022	+	0.813173i	$0.697738\pi$
$762$		0			0
$763$		−	11393.6i		−	0.540597i
$764$		0			0
$765$	13469.4	−	8875.57i	0.636587	−	0.419473i
$766$		0			0
$767$		−	16061.9i		−	0.756145i
$768$		0			0
$769$	30689.3			1.43912			0.719560	−	0.694430i	$-0.244343\pi$
$769$	30689.3			1.43912			0.719560	+	0.694430i	$0.244343\pi$
$770$		0			0
$771$	−3335.13			−0.155787
$772$		0			0
$773$		−	5110.74i		−	0.237801i	−0.992906	−	0.118901i	$-0.962063\pi$
$773$		−	5110.74i		−	0.237801i	0.992906	−	0.118901i	$-0.0379370\pi$
$774$		0			0
$775$	10130.5	−	23596.6i	0.469548	−	1.09370i
$776$		0			0
$777$			8710.22i			0.402159i
$778$		0			0
$779$	−15479.4			−0.711947
$780$		0			0
$781$	−3365.49			−0.154195
$782$		0			0
$783$		−	21027.6i		−	0.959723i
$784$		0			0
$785$	7316.40	−	4821.08i	0.332654	−	0.219199i
$786$		0			0
$787$			6991.30i			0.316662i	0.987386	+	0.158331i	$0.0506113\pi$
$787$			6991.30i			0.316662i	−0.987386	+	0.158331i	$0.949389\pi$
$788$		0			0
$789$	−14811.1			−0.668302
$790$		0			0
$791$	−2499.28			−0.112344
$792$		0			0
$793$			25229.5i			1.12979i
$794$		0			0
$795$	−14.6719	−	22.2658i	−0.000654539	−	0.000993319i
$796$		0			0
$797$		−	4020.31i		−	0.178678i	−0.996001	−	0.0893392i	$-0.971524\pi$
$797$		−	4020.31i		−	0.178678i	0.996001	−	0.0893392i	$-0.0284755\pi$
$798$		0			0
$799$	38734.3			1.71505
$800$		0			0
$801$	2144.64			0.0946030
$802$		0			0
$803$			4755.15i			0.208973i
$804$		0			0
$805$	−1389.53	−	2108.73i	−0.0608379	−	0.0923267i
$806$		0			0
$807$			51150.0i			2.23118i
$808$		0			0
$809$	41608.1			1.80824			0.904119	−	0.427281i	$-0.140528\pi$
$809$	41608.1			1.80824			0.904119	+	0.427281i	$0.140528\pi$
$810$		0			0
$811$	−42271.3			−1.83027			−0.915133	−	0.403152i	$-0.867914\pi$
$811$	−42271.3			−1.83027			−0.915133	+	0.403152i	$0.867914\pi$
$812$		0			0
$813$		−	9571.13i		−	0.412884i
$814$		0			0
$815$	−22563.9	+	14868.3i	−0.969790	+	0.639035i
$816$		0			0
$817$		−	4397.62i		−	0.188315i
$818$		0			0
$819$	5345.05			0.228048
$820$		0			0
$821$	27184.7			1.15561			0.577804	−	0.816176i	$-0.303910\pi$
$821$	27184.7			1.15561			0.577804	+	0.816176i	$0.303910\pi$
$822$		0			0
$823$			12967.9i			0.549250i	0.961551	+	0.274625i	$0.0885538\pi$
$823$			12967.9i			0.549250i	−0.961551	+	0.274625i	$0.911446\pi$
$824$		0			0
$825$	5084.94	+	2183.08i	0.214588	+	0.0921273i
$826$		0			0
$827$		−	33111.2i		−	1.39225i	−0.717921	−	0.696124i	$-0.754906\pi$
$827$		−	33111.2i		−	1.39225i	0.717921	−	0.696124i	$-0.245094\pi$
$828$		0			0
$829$	−3715.75			−0.155673			−0.0778367	−	0.996966i	$-0.524801\pi$
$829$	−3715.75			−0.155673			−0.0778367	+	0.996966i	$0.524801\pi$
$830$		0			0
$831$	−27116.3			−1.13195
$832$		0			0
$833$		−	4529.47i		−	0.188399i
$834$		0			0
$835$	−6576.33	+	4333.42i	−0.272555	+	0.179598i
$836$		0			0
$837$		−	15276.3i		−	0.630854i
$838$		0			0
$839$	−1460.56			−0.0601005			−0.0300502	−	0.999548i	$-0.509567\pi$
$839$	−1460.56			−0.0601005			−0.0300502	+	0.999548i	$0.509567\pi$
$840$		0			0
$841$	55574.2			2.27866
$842$		0			0
$843$		−	22783.6i		−	0.930851i
$844$		0			0
$845$	−1208.00	−	1833.25i	−0.0491794	−	0.0746339i
$846$		0			0
$847$			8995.02i			0.364903i
$848$		0			0
$849$	20400.3			0.824660
$850$		0			0
$851$	6151.19			0.247779
$852$		0			0
$853$		−	36027.4i		−	1.44613i	−0.690777	−	0.723067i	$-0.742732\pi$
$853$		−	36027.4i		−	1.44613i	0.690777	−	0.723067i	$-0.257268\pi$
$854$		0			0
$855$	12057.2	+	18297.8i	0.482276	+	0.731896i
$856$		0			0
$857$		−	23500.7i		−	0.936719i	−0.883538	−	0.468359i	$-0.844845\pi$
$857$		−	23500.7i		−	0.936719i	0.883538	−	0.468359i	$-0.155155\pi$
$858$		0			0
$859$	5551.11			0.220491			0.110245	−	0.993904i	$-0.464836\pi$
$859$	5551.11			0.220491			0.110245	+	0.993904i	$0.464836\pi$
$860$		0			0
$861$	5632.46			0.222943
$862$		0			0
$863$		−	27721.4i		−	1.09345i	−0.837312	−	0.546725i	$-0.815874\pi$
$863$		−	27721.4i		−	1.09345i	0.837312	−	0.546725i	$-0.184126\pi$
$864$		0			0
$865$	−13056.0	+	8603.13i	−0.513199	+	0.338168i
$866$		0			0
$867$		−	23706.6i		−	0.928624i
$868$		0			0
$869$	−1352.27			−0.0527878
$870$		0			0
$871$	40554.3			1.57765
$872$		0			0
$873$		−	3440.64i		−	0.133388i
$874$		0			0
$875$	1723.58	+	9629.77i	0.0665915	+	0.372052i
$876$		0			0
$877$		−	47255.6i		−	1.81951i	−0.415149	−	0.909754i	$-0.636270\pi$
$877$		−	47255.6i		−	1.81951i	0.415149	−	0.909754i	$-0.363730\pi$
$878$		0			0
$879$	−9449.79			−0.362609
$880$		0			0
$881$	32267.0			1.23394			0.616972	−	0.786985i	$-0.288359\pi$
$881$	32267.0			1.23394			0.616972	+	0.786985i	$0.288359\pi$
$882$		0			0
$883$		−	5062.08i		−	0.192925i	−0.995337	−	0.0964623i	$-0.969247\pi$
$883$		−	5062.08i		−	0.192925i	0.995337	−	0.0964623i	$-0.0307527\pi$
$884$		0			0
$885$	−20007.3	+	13183.6i	−0.759930	+	0.500749i
$886$		0			0
$887$			7626.08i			0.288679i	0.989528	+	0.144340i	$0.0461058\pi$
$887$			7626.08i			0.288679i	−0.989528	+	0.144340i	$0.953894\pi$
$888$		0			0
$889$	12097.8			0.456408
$890$		0			0
$891$	6150.05			0.231239
$892$		0			0
$893$			52619.2i			1.97182i
$894$		0			0
$895$	2268.06	+	3441.98i	0.0847072	+	0.128550i
$896$		0			0
$897$		−	10304.4i		−	0.383562i
$898$		0			0
$899$	58092.3			2.15516
$900$		0			0
$901$	−33.7750			−0.00124884
$902$		0			0
$903$			1600.16i			0.0589699i
$904$		0			0
$905$	−1941.61	−	2946.56i	−0.0713164	−	0.108229i
$906$		0			0
$907$		−	8546.31i		−	0.312873i	−0.987688	−	0.156436i	$-0.949999\pi$
$907$		−	8546.31i		−	0.312873i	0.987688	−	0.156436i	$-0.0500006\pi$
$908$		0			0
$909$	9229.95			0.336786
$910$		0			0
$911$	−8778.92			−0.319274			−0.159637	−	0.987176i	$-0.551032\pi$
$911$	−8778.92			−0.319274			−0.159637	+	0.987176i	$0.551032\pi$
$912$		0			0
$913$			1321.99i			0.0479205i
$914$		0			0
$915$	31426.7	−	20708.3i	1.13545	−	0.748193i
$916$		0			0
$917$			10228.4i			0.368343i
$918$		0			0
$919$	21626.0			0.776253			0.388126	−	0.921606i	$-0.373122\pi$
$919$	21626.0			0.776253			0.388126	+	0.921606i	$0.373122\pi$
$920$		0			0
$921$	10388.0			0.371658
$922$		0			0
$923$		−	24276.7i		−	0.865738i
$924$		0			0
$925$	−21895.8	−	9400.36i	−0.778302	−	0.334143i
$926$		0			0
$927$			7437.15i			0.263504i
$928$		0			0
$929$	−12262.6			−0.433073			−0.216536	−	0.976275i	$-0.569476\pi$
$929$	−12262.6			−0.433073			−0.216536	+	0.976275i	$0.569476\pi$
$930$		0			0
$931$	6153.13			0.216606
$932$		0			0
$933$			56037.9i			1.96634i
$934$		0			0
$935$	5852.82	−	3856.66i	0.204714	−	0.134894i
$936$		0			0
$937$			13362.6i			0.465889i	0.972490	+	0.232945i	$0.0748361\pi$
$937$			13362.6i			0.465889i	−0.972490	+	0.232945i	$0.925164\pi$
$938$		0			0
$939$	−47063.5			−1.63563
$940$		0			0
$941$	−46330.1			−1.60501			−0.802506	−	0.596643i	$-0.796501\pi$
$941$	−46330.1			−1.60501			−0.802506	+	0.596643i	$0.796501\pi$
$942$		0			0
$943$		−	3977.66i		−	0.137360i
$944$		0			0
$945$	3202.12	+	4859.49i	0.110228	+	0.167280i
$946$		0			0
$947$		−	17387.6i		−	0.596643i	−0.954465	−	0.298321i	$-0.903573\pi$
$947$		−	17387.6i		−	0.596643i	0.954465	−	0.298321i	$-0.0964267\pi$
$948$		0			0
$949$	−34300.8			−1.17329
$950$		0			0
$951$	−50831.1			−1.73324
$952$		0			0
$953$		−	42384.6i		−	1.44068i	−0.693619	−	0.720342i	$-0.743985\pi$
$953$		−	42384.6i		−	1.44068i	0.693619	−	0.720342i	$-0.256015\pi$
$954$		0			0
$955$	932.779	+	1415.57i	0.0316063	+	0.0479653i
$956$		0			0
$957$			12518.6i			0.422851i
$958$		0			0
$959$	13250.7			0.446182
$960$		0			0
$961$	12412.3			0.416647
$962$		0			0
$963$		−	3520.92i		−	0.117820i
$964$		0			0
$965$	−6448.10	+	4248.92i	−0.215100	+	0.141738i
$966$		0			0
$967$		−	15771.6i		−	0.524489i	−0.965001	−	0.262245i	$-0.915537\pi$
$967$		−	15771.6i		−	0.524489i	0.965001	−	0.262245i	$-0.0844627\pi$
$968$		0			0
$969$	75769.9			2.51195
$970$		0			0
$971$	−36370.5			−1.20204			−0.601022	−	0.799232i	$-0.705240\pi$
$971$	−36370.5			−1.20204			−0.601022	+	0.799232i	$0.705240\pi$
$972$		0			0
$973$			12005.2i			0.395549i
$974$		0			0
$975$	−15747.4	+	36679.8i	−0.517253	+	1.20481i
$976$		0			0
$977$			35122.8i			1.15013i	0.818108	+	0.575065i	$0.195023\pi$
$977$			35122.8i			1.15013i	−0.818108	+	0.575065i	$0.804977\pi$
$978$		0			0
$979$	931.899			0.0304225
$980$		0			0
$981$	−25404.5			−0.826814
$982$		0			0
$983$			12798.9i			0.415282i	0.978205	+	0.207641i	$0.0665785\pi$
$983$			12798.9i			0.415282i	−0.978205	+	0.207641i	$0.933421\pi$
$984$		0			0
$985$	7787.30	−	5131.37i	0.251902	−	0.165989i
$986$		0			0
$987$		−	19146.5i		−	0.617466i
$988$		0			0
$989$	1130.03			0.0363327
$990$		0			0
$991$	46594.9			1.49358			0.746789	−	0.665061i	$-0.231594\pi$
$991$	46594.9			1.49358			0.746789	+	0.665061i	$0.231594\pi$
$992$		0			0
$993$			11291.1i			0.360838i
$994$		0			0
$995$	2382.37	+	3615.45i	0.0759058	+	0.115194i
$996$		0			0
$997$		−	47534.1i		−	1.50995i	−0.655753	−	0.754975i	$-0.727649\pi$
$997$		−	47534.1i		−	1.50995i	0.655753	−	0.754975i	$-0.272351\pi$
$998$		0			0
$999$	−14175.2			−0.448932

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.4.g.f.449.9		10
4.3	odd	2		35.4.b.a.29.2	✓	10
5.4	even	2	inner	560.4.g.f.449.2		10
12.11	even	2		315.4.d.c.64.9		10
20.3	even	4		175.4.a.j.1.1		5
20.7	even	4		175.4.a.i.1.5		5
20.19	odd	2		35.4.b.a.29.9	yes	10
28.3	even	6		245.4.j.f.79.9		20
28.11	odd	6		245.4.j.e.79.9		20
28.19	even	6		245.4.j.f.214.2		20
28.23	odd	6		245.4.j.e.214.2		20
28.27	even	2		245.4.b.d.99.2		10
60.23	odd	4		1575.4.a.bn.1.5		5
60.47	odd	4		1575.4.a.bq.1.1		5
60.59	even	2		315.4.d.c.64.2		10
140.19	even	6		245.4.j.f.214.9		20
140.27	odd	4		1225.4.a.be.1.5		5
140.39	odd	6		245.4.j.e.79.2		20
140.59	even	6		245.4.j.f.79.2		20
140.79	odd	6		245.4.j.e.214.9		20
140.83	odd	4		1225.4.a.bh.1.1		5
140.139	even	2		245.4.b.d.99.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.2	✓	10	4.3	odd	2
35.4.b.a.29.9	yes	10	20.19	odd	2
175.4.a.i.1.5		5	20.7	even	4
175.4.a.j.1.1		5	20.3	even	4
245.4.b.d.99.2		10	28.27	even	2
245.4.b.d.99.9		10	140.139	even	2
245.4.j.e.79.2		20	140.39	odd	6
245.4.j.e.79.9		20	28.11	odd	6
245.4.j.e.214.2		20	28.23	odd	6
245.4.j.e.214.9		20	140.79	odd	6
245.4.j.f.79.2		20	140.59	even	6
245.4.j.f.79.9		20	28.3	even	6
245.4.j.f.214.2		20	28.19	even	6
245.4.j.f.214.9		20	140.19	even	6
315.4.d.c.64.2		10	60.59	even	2
315.4.d.c.64.9		10	12.11	even	2
560.4.g.f.449.2		10	5.4	even	2	inner
560.4.g.f.449.9		10	1.1	even	1	trivial
1225.4.a.be.1.5		5	140.27	odd	4
1225.4.a.bh.1.1		5	140.83	odd	4
1575.4.a.bn.1.5		5	60.23	odd	4
1575.4.a.bq.1.1		5	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	560.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.52749i q^{3} +(6.15172 + 9.33576i) q^{5} -7.00000i q^{7} -15.6081 q^{9} -6.78210 q^{11} -48.9221i q^{13} +(-60.9390 + 40.1552i) q^{15} +92.4381i q^{17} -125.574 q^{19} +45.6924 q^{21} -32.2681i q^{23} +(-49.3128 + 114.862i) q^{25} +74.3607i q^{27} -282.778 q^{29} -205.434 q^{31} -44.2701i q^{33} +(65.3503 - 43.0620i) q^{35} +190.627i q^{37} +319.338 q^{39} +123.269 q^{41} +35.0202i q^{43} +(-96.0164 - 145.713i) q^{45} -419.030i q^{47} -49.0000 q^{49} -603.388 q^{51} +0.365379i q^{53} +(-41.7215 - 63.3160i) q^{55} -819.683i q^{57} +328.317 q^{59} -515.707 q^{61} +109.256i q^{63} +(456.725 - 300.955i) q^{65} +828.957i q^{67} +210.630 q^{69} +496.231 q^{71} -701.132i q^{73} +(-749.759 - 321.888i) q^{75} +47.4747i q^{77} +199.388 q^{79} -906.806 q^{81} -194.923i q^{83} +(-862.980 + 568.653i) q^{85} -1845.83i q^{87} -137.406 q^{89} -342.454 q^{91} -1340.97i q^{93} +(-772.496 - 1172.33i) q^{95} +220.440i q^{97} +105.855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{5} - 46 q^{9} - 84 q^{11} - 8 q^{15} - 72 q^{19} + 140 q^{21} - 362 q^{25} + 88 q^{29} - 120 q^{31} + 28 q^{35} - 212 q^{39} - 852 q^{41} - 510 q^{45} - 490 q^{49} - 1276 q^{51} + 1136 q^{55}+ \cdots + 5304 q^{99}+O(q^{100}) \) 10 * q + 6 * q^5 - 46 * q^9 - 84 * q^11 - 8 * q^15 - 72 * q^19 + 140 * q^21 - 362 * q^25 + 88 * q^29 - 120 * q^31 + 28 * q^35 - 212 * q^39 - 852 * q^41 - 510 * q^45 - 490 * q^49 - 1276 * q^51 + 1136 * q^55 - 864 * q^59 - 884 * q^61 + 520 * q^65 + 4808 * q^69 - 880 * q^71 - 720 * q^75 + 3580 * q^79 - 302 * q^81 + 1572 * q^85 - 1492 * q^89 - 476 * q^91 + 1628 * q^95 + 5304 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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