LMFDB - Embedded newform 672.4.k.c.545.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,4,Mod(545,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("672.545");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.k (of order $2$, degree $1$, 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:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.36004060626969600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 28x^{14} + 308x^{12} - 1710x^{10} + 5156x^{8} - 7740x^{6} + 9473x^{4} + 368x^{2} + 256 $ x^16 - 28x^14 + 308x^12 - 1710x^10 + 5156x^8 - 7740x^6 + 9473x^4 + 368*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{60} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			545.8
Root			$-1.12308 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	672.545
Dual form			672.4.k.c.545.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.46410 + 3.87298i) q^{3} +14.3792 q^{5} +(-15.9613 - 9.39352i) q^{7} +(-3.00000 - 26.8328i) q^{9} +O(q^{10})$	\(q+(-3.46410 + 3.87298i) q^{3} +14.3792 q^{5} +(-15.9613 - 9.39352i) q^{7} +(-3.00000 - 26.8328i) q^{9} +7.69047i q^{11} -31.6168i q^{13} +(-49.8111 + 55.6905i) q^{15} -47.5885 q^{17} +40.8692i q^{19} +(91.6724 - 29.2776i) q^{21} +103.690i q^{23} +81.7619 q^{25} +(114.315 + 81.3327i) q^{27} +24.3625i q^{29} +160.354i q^{31} +(-29.7850 - 26.6406i) q^{33} +(-229.510 - 135.071i) q^{35} +376.762 q^{37} +(122.451 + 109.524i) q^{39} +67.4452 q^{41} +154.374 q^{43} +(-43.1376 - 385.835i) q^{45} -116.900 q^{47} +(166.524 + 299.865i) q^{49} +(164.851 - 184.310i) q^{51} +524.645i q^{53} +110.583i q^{55} +(-158.286 - 141.575i) q^{57} -849.639 q^{59} +492.782i q^{61} +(-204.171 + 456.466i) q^{63} -454.624i q^{65} +348.677 q^{67} +(-401.591 - 359.194i) q^{69} +128.262i q^{71} -289.225i q^{73} +(-283.231 + 316.662i) q^{75} +(72.2405 - 122.750i) q^{77} +953.821 q^{79} +(-711.000 + 160.997i) q^{81} -601.984 q^{83} -684.286 q^{85} +(-94.3557 - 84.3943i) q^{87} +1080.15 q^{89} +(-296.993 + 504.644i) q^{91} +(-621.047 - 555.482i) q^{93} +587.667i q^{95} +959.226i q^{97} +(206.357 - 23.0714i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9}+O(q^{10}) $ 16 * q - 48 * q^9	$ 16 q - 48 q^{9} - 240 q^{21} - 336 q^{25} + 4384 q^{37} - 624 q^{49} + 2400 q^{57} - 11376 q^{81} - 6016 q^{85} - 3360 q^{93}+O(q^{100}) $ 16 * q - 48 * q^9 - 240 * q^21 - 336 * q^25 + 4384 * q^37 - 624 * q^49 + 2400 * q^57 - 11376 * q^81 - 6016 * q^85 - 3360 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	−3.46410	+	3.87298i	−0.666667	+	0.745356i
$3$	−3.46410	+	3.87298i	−0.666667	+	0.745356i
$4$		0			0
$5$	14.3792			1.28612			0.643058	−	0.765817i	$-0.277665\pi$
$5$	14.3792			1.28612			0.643058	+	0.765817i	$0.277665\pi$
$6$		0			0
$7$	−15.9613	−	9.39352i	−0.861827	−	0.507202i
$7$	−15.9613	−	9.39352i	−0.861827	−	0.507202i
$8$		0			0
$9$	−3.00000	−	26.8328i	−0.111111	−	0.993808i
$10$		0			0
$11$			7.69047i			0.210797i	0.994430	+	0.105398i	$0.0336117\pi$
$11$			7.69047i			0.210797i	−0.994430	+	0.105398i	$0.966388\pi$
$12$		0			0
$13$		−	31.6168i		−	0.674532i	−0.941409	−	0.337266i	$-0.890498\pi$
$13$		−	31.6168i		−	0.674532i	0.941409	−	0.337266i	$-0.109502\pi$
$14$		0			0
$15$	−49.8111	+	55.6905i	−0.857411	+	0.958614i
$16$		0			0
$17$	−47.5885			−0.678936			−0.339468	−	0.940618i	$-0.610247\pi$
$17$	−47.5885			−0.678936			−0.339468	+	0.940618i	$0.610247\pi$
$18$		0			0
$19$			40.8692i			0.493475i	0.969082	+	0.246738i	$0.0793586\pi$
$19$			40.8692i			0.493475i	−0.969082	+	0.246738i	$0.920641\pi$
$20$		0			0
$21$	91.6724	−	29.2776i	0.952598	−	0.304233i
$22$		0			0
$23$			103.690i			0.940042i	0.882655	+	0.470021i	$0.155754\pi$
$23$			103.690i			0.940042i	−0.882655	+	0.470021i	$0.844246\pi$
$24$		0			0
$25$	81.7619			0.654095
$26$		0			0
$27$	114.315	+	81.3327i	0.814815	+	0.579721i
$28$		0			0
$29$			24.3625i			0.156000i	0.996953	+	0.0780002i	$0.0248535\pi$
$29$			24.3625i			0.156000i	−0.996953	+	0.0780002i	$0.975147\pi$
$30$		0			0
$31$			160.354i			0.929045i	0.885561	+	0.464522i	$0.153774\pi$
$31$			160.354i			0.929045i	−0.885561	+	0.464522i	$0.846226\pi$
$32$		0			0
$33$	−29.7850	−	26.6406i	−0.157119	−	0.140531i
$34$		0			0
$35$	−229.510	−	135.071i	−1.10841	−	0.652321i
$36$		0			0
$37$	376.762			1.67403			0.837017	−	0.547177i	$-0.184297\pi$
$37$	376.762			1.67403			0.837017	+	0.547177i	$0.184297\pi$
$38$		0			0
$39$	122.451	+	109.524i	0.502766	+	0.449688i
$40$		0			0
$41$	67.4452			0.256907			0.128453	−	0.991716i	$-0.458999\pi$
$41$	67.4452			0.256907			0.128453	+	0.991716i	$0.458999\pi$
$42$		0			0
$43$	154.374			0.547483			0.273742	−	0.961803i	$-0.411739\pi$
$43$	154.374			0.547483			0.273742	+	0.961803i	$0.411739\pi$
$44$		0			0
$45$	−43.1376	−	385.835i	−0.142902	−	1.27815i
$46$		0			0
$47$	−116.900			−0.362800			−0.181400	−	0.983409i	$-0.558063\pi$
$47$	−116.900			−0.362800			−0.181400	+	0.983409i	$0.558063\pi$
$48$		0			0
$49$	166.524	+	299.865i	0.485492	+	0.874241i
$50$		0			0
$51$	164.851	−	184.310i	0.452624	−	0.506049i
$52$		0			0
$53$			524.645i			1.35973i	0.733338	+	0.679864i	$0.237961\pi$
$53$			524.645i			1.35973i	−0.733338	+	0.679864i	$0.762039\pi$
$54$		0			0
$55$			110.583i			0.271109i
$56$		0			0
$57$	−158.286	−	141.575i	−0.367815	−	0.328983i
$58$		0			0
$59$	−849.639			−1.87481			−0.937404	−	0.348245i	$-0.886778\pi$
$59$	−849.639			−1.87481			−0.937404	+	0.348245i	$0.886778\pi$
$60$		0			0
$61$			492.782i			1.03433i	0.855885	+	0.517166i	$0.173013\pi$
$61$			492.782i			1.03433i	−0.855885	+	0.517166i	$0.826987\pi$
$62$		0			0
$63$	−204.171	+	456.466i	−0.408303	+	0.912846i
$64$		0			0
$65$		−	454.624i		−	0.867526i
$66$		0			0
$67$	348.677			0.635787			0.317894	−	0.948126i	$-0.397025\pi$
$67$	348.677			0.635787			0.317894	+	0.948126i	$0.397025\pi$
$68$		0			0
$69$	−401.591	−	359.194i	−0.700666	−	0.626694i
$70$		0			0
$71$			128.262i			0.214392i	0.994238	+	0.107196i	$0.0341873\pi$
$71$			128.262i			0.214392i	−0.994238	+	0.107196i	$0.965813\pi$
$72$		0			0
$73$		−	289.225i		−	0.463716i	−0.972750	−	0.231858i	$-0.925520\pi$
$73$		−	289.225i		−	0.463716i	0.972750	−	0.231858i	$-0.0744804\pi$
$74$		0			0
$75$	−283.231	+	316.662i	−0.436063	+	0.487534i
$76$		0			0
$77$	72.2405	−	122.750i	0.106916	−	0.181670i
$78$		0			0
$79$	953.821			1.35840			0.679198	−	0.733955i	$-0.262328\pi$
$79$	953.821			1.35840			0.679198	+	0.733955i	$0.262328\pi$
$80$		0			0
$81$	−711.000	+	160.997i	−0.975309	+	0.220846i
$82$		0			0
$83$	−601.984			−0.796100			−0.398050	−	0.917364i	$-0.630313\pi$
$83$	−601.984			−0.796100			−0.398050	+	0.917364i	$0.630313\pi$
$84$		0			0
$85$	−684.286			−0.873190
$86$		0			0
$87$	−94.3557	−	84.3943i	−0.116276	−	0.104000i
$88$		0			0
$89$	1080.15			1.28647			0.643234	−	0.765670i	$-0.277592\pi$
$89$	1080.15			1.28647			0.643234	+	0.765670i	$0.277592\pi$
$90$		0			0
$91$	−296.993	+	504.644i	−0.342124	+	0.581330i
$92$		0			0
$93$	−621.047	−	555.482i	−0.692469	−	0.619363i
$94$		0			0
$95$			587.667i			0.634666i
$96$		0			0
$97$			959.226i			1.00407i	0.864848	+	0.502034i	$0.167415\pi$
$97$			959.226i			1.00407i	−0.864848	+	0.502034i	$0.832585\pi$
$98$		0			0
$99$	206.357	−	23.0714i	0.209491	−	0.0234218i
$100$		0			0
$101$	−468.354			−0.461416			−0.230708	−	0.973023i	$-0.574104\pi$
$101$	−468.354			−0.461416			−0.230708	+	0.973023i	$0.574104\pi$
$102$		0			0
$103$			1606.66i			1.53698i	0.639862	+	0.768490i	$0.278992\pi$
$103$			1606.66i			1.53698i	−0.639862	+	0.768490i	$0.721008\pi$
$104$		0			0
$105$	1318.18	−	420.989i	1.22515	−	0.391279i
$106$		0			0
$107$			1333.12i			1.20446i	0.798322	+	0.602231i	$0.205721\pi$
$107$			1333.12i			1.20446i	−0.798322	+	0.602231i	$0.794279\pi$
$108$		0			0
$109$	−2174.19			−1.91055			−0.955273	−	0.295724i	$-0.904439\pi$
$109$	−2174.19			−1.91055			−0.955273	+	0.295724i	$0.904439\pi$
$110$		0			0
$111$	−1305.14	+	1459.19i	−1.11602	+	1.24775i
$112$		0			0
$113$			732.834i			0.610081i	0.952339	+	0.305041i	$0.0986701\pi$
$113$			732.834i			0.610081i	−0.952339	+	0.305041i	$0.901330\pi$
$114$		0			0
$115$			1490.99i			1.20900i
$116$		0			0
$117$	−848.367	+	94.8503i	−0.670355	+	0.0749480i
$118$		0			0
$119$	759.573	+	447.023i	0.585125	+	0.344358i
$120$		0			0
$121$	1271.86			0.955565
$122$		0			0
$123$	−233.637	+	261.214i	−0.171271	+	0.191487i
$124$		0			0
$125$	−621.731			−0.444874
$126$		0			0
$127$	−2032.67			−1.42024			−0.710120	−	0.704081i	$-0.751359\pi$
$127$	−2032.67			−1.42024			−0.710120	+	0.704081i	$0.751359\pi$
$128$		0			0
$129$	−534.766	+	597.887i	−0.364989	+	0.408070i
$130$		0			0
$131$	−721.413			−0.481146			−0.240573	−	0.970631i	$-0.577335\pi$
$131$	−721.413			−0.481146			−0.240573	+	0.970631i	$0.577335\pi$
$132$		0			0
$133$	383.905	−	652.323i	0.250292	−	0.425290i
$134$		0			0
$135$	1643.77	+	1169.50i	1.04795	+	0.745589i
$136$		0			0
$137$			2665.39i			1.66219i	0.556131	+	0.831095i	$0.312285\pi$
$137$			2665.39i			1.66219i	−0.556131	+	0.831095i	$0.687715\pi$
$138$		0			0
$139$		−	855.868i		−	0.522257i	−0.965304	−	0.261129i	$-0.915905\pi$
$139$		−	855.868i		−	0.522257i	0.965304	−	0.261129i	$-0.0840947\pi$
$140$		0			0
$141$	404.953	−	452.751i	0.241866	−	0.270415i
$142$		0			0
$143$	243.148			0.142189
$144$		0			0
$145$			350.314i			0.200635i
$146$		0			0
$147$	−1738.23	−	393.818i	−0.975282	−	0.220963i
$148$		0			0
$149$			1403.06i			0.771429i	0.922618	+	0.385715i	$0.126045\pi$
$149$			1403.06i			0.771429i	−0.922618	+	0.385715i	$0.873955\pi$
$150$		0			0
$151$	1531.20			0.825211			0.412606	−	0.910910i	$-0.364619\pi$
$151$	1531.20			0.825211			0.412606	+	0.910910i	$0.364619\pi$
$152$		0			0
$153$	142.766	+	1276.93i	0.0754373	+	0.674732i
$154$		0			0
$155$			2305.76i			1.19486i
$156$		0			0
$157$		−	1655.95i		−	0.841778i	−0.907112	−	0.420889i	$-0.861718\pi$
$157$		−	1655.95i		−	0.841778i	0.907112	−	0.420889i	$-0.138282\pi$
$158$		0			0
$159$	−2031.94	−	1817.42i	−1.01348	−	0.906485i
$160$		0			0
$161$	974.018	−	1655.03i	0.476791	−	0.810153i
$162$		0			0
$163$	3790.99			1.82167			0.910837	−	0.412765i	$-0.135437\pi$
$163$	3790.99			1.82167			0.910837	+	0.412765i	$0.135437\pi$
$164$		0			0
$165$	−428.286	−	383.070i	−0.202073	−	0.180739i
$166$		0			0
$167$	−2360.87			−1.09395			−0.546974	−	0.837149i	$-0.684220\pi$
$167$	−2360.87			−1.09395			−0.546974	+	0.837149i	$0.684220\pi$
$168$		0			0
$169$	1197.38			0.545007
$170$		0			0
$171$	1096.63	−	122.607i	0.490420	−	0.0548306i
$172$		0			0
$173$	−1110.63			−0.488091			−0.244045	−	0.969764i	$-0.578475\pi$
$173$	−1110.63			−0.488091			−0.244045	+	0.969764i	$0.578475\pi$
$174$		0			0
$175$	−1305.02	−	768.031i	−0.563717	−	0.331758i
$176$		0			0
$177$	2943.24	−	3290.64i	1.24987	−	1.39740i
$178$		0			0
$179$		−	1084.02i		−	0.452647i	−0.974052	−	0.226323i	$-0.927329\pi$
$179$		−	1084.02i		−	0.452647i	0.974052	−	0.226323i	$-0.0726706\pi$
$180$		0			0
$181$		−	2485.02i		−	1.02050i	−0.860026	−	0.510250i	$-0.829553\pi$
$181$		−	2485.02i		−	1.02050i	0.860026	−	0.510250i	$-0.170447\pi$
$182$		0			0
$183$	−1908.54	−	1707.05i	−0.770946	−	0.689555i
$184$		0			0
$185$	5417.54			2.15300
$186$		0			0
$187$		−	365.978i		−	0.143117i
$188$		0			0
$189$	−1060.62	−	2371.99i	−0.408194	−	0.912895i
$190$		0			0
$191$		−	3480.12i		−	1.31839i	−0.751972	−	0.659195i	$-0.770897\pi$
$191$		−	3480.12i		−	1.31839i	0.751972	−	0.659195i	$-0.229103\pi$
$192$		0			0
$193$	−2079.43			−0.775547			−0.387773	−	0.921755i	$-0.626756\pi$
$193$	−2079.43			−0.775547			−0.387773	+	0.921755i	$0.626756\pi$
$194$		0			0
$195$	1760.75	+	1574.87i	0.646616	+	0.578351i
$196$		0			0
$197$			4014.87i			1.45202i	0.687685	+	0.726009i	$0.258627\pi$
$197$			4014.87i			1.45202i	−0.687685	+	0.726009i	$0.741373\pi$
$198$		0			0
$199$			43.8696i			0.0156273i	0.999969	+	0.00781364i	$0.00248719\pi$
$199$			43.8696i			0.0156273i	−0.999969	+	0.00781364i	$0.997513\pi$
$200$		0			0
$201$	−1207.85	+	1350.42i	−0.423858	+	0.473888i
$202$		0			0
$203$	228.850	−	388.857i	0.0791237	−	0.134445i
$204$		0			0
$205$	969.809			0.330412
$206$		0			0
$207$	2782.31	−	311.071i	0.934221	−	0.104449i
$208$		0			0
$209$	−314.303			−0.104023
$210$		0			0
$211$	407.964			0.133106			0.0665531	−	0.997783i	$-0.478800\pi$
$211$	407.964			0.133106			0.0665531	+	0.997783i	$0.478800\pi$
$212$		0			0
$213$	−496.755	−	444.311i	−0.159799	−	0.142928i
$214$		0			0
$215$	2219.77			0.704127
$216$		0			0
$217$	1506.29	−	2559.45i	0.471213	−	0.800676i
$218$		0			0
$219$	1120.16	+	1001.91i	0.345633	+	0.309144i
$220$		0			0
$221$			1504.60i			0.457964i
$222$		0			0
$223$		−	2341.89i		−	0.703249i	−0.936141	−	0.351624i	$-0.885629\pi$
$223$		−	2341.89i		−	0.703249i	0.936141	−	0.351624i	$-0.114371\pi$
$224$		0			0
$225$	−245.286	−	2193.90i	−0.0726772	−	0.650045i
$226$		0			0
$227$	−1720.06			−0.502928			−0.251464	−	0.967867i	$-0.580912\pi$
$227$	−1720.06			−0.502928			−0.251464	+	0.967867i	$0.580912\pi$
$228$		0			0
$229$		−	6091.98i		−	1.75794i	−0.476873	−	0.878972i	$-0.658230\pi$
$229$		−	6091.98i		−	1.75794i	0.476873	−	0.878972i	$-0.341770\pi$
$230$		0			0
$231$	225.158	+	705.003i	0.0641313	+	0.200804i
$232$		0			0
$233$		−	1651.28i		−	0.464288i	−0.972681	−	0.232144i	$-0.925426\pi$
$233$		−	1651.28i		−	0.464288i	0.972681	−	0.232144i	$-0.0745741\pi$
$234$		0			0
$235$	−1680.93			−0.466602
$236$		0			0
$237$	−3304.13	+	3694.13i	−0.905597	+	1.01249i
$238$		0			0
$239$		−	6336.59i		−	1.71498i	−0.514501	−	0.857490i	$-0.672023\pi$
$239$		−	6336.59i		−	1.71498i	0.514501	−	0.857490i	$-0.327977\pi$
$240$		0			0
$241$			2862.23i			0.765030i	0.923949	+	0.382515i	$0.124942\pi$
$241$			2862.23i			0.765030i	−0.923949	+	0.382515i	$0.875058\pi$
$242$		0			0
$243$	1839.44	−	3311.40i	0.485597	−	0.874183i
$244$		0			0
$245$	2394.48	+	4311.82i	0.624399	+	1.12438i
$246$		0			0
$247$	1292.15			0.332865
$248$		0			0
$249$	2085.33	−	2331.47i	0.530733	−	0.593378i
$250$		0			0
$251$	6421.12			1.61473			0.807366	−	0.590051i	$-0.200892\pi$
$251$	6421.12			1.61473			0.807366	+	0.590051i	$0.200892\pi$
$252$		0			0
$253$	−797.428			−0.198158
$254$		0			0
$255$	2370.43	−	2650.23i	0.582127	−	0.650838i
$256$		0			0
$257$	1074.67			0.260840			0.130420	−	0.991459i	$-0.458367\pi$
$257$	1074.67			0.260840			0.130420	+	0.991459i	$0.458367\pi$
$258$		0			0
$259$	−6013.59	−	3539.12i	−1.44273	−	0.849074i
$260$		0			0
$261$	653.715	−	73.0876i	0.155034	−	0.0173334i
$262$		0			0
$263$			4840.93i			1.13500i	0.823374	+	0.567499i	$0.192089\pi$
$263$			4840.93i			1.13500i	−0.823374	+	0.567499i	$0.807911\pi$
$264$		0			0
$265$			7543.99i			1.74877i
$266$		0			0
$267$	−3741.75	+	4183.40i	−0.857645	+	0.958877i
$268$		0			0
$269$	3063.46			0.694359			0.347179	−	0.937799i	$-0.387139\pi$
$269$	3063.46			0.694359			0.347179	+	0.937799i	$0.387139\pi$
$270$		0			0
$271$			5062.98i			1.13489i	0.823412	+	0.567444i	$0.192067\pi$
$271$			5062.98i			1.13489i	−0.823412	+	0.567444i	$0.807933\pi$
$272$		0			0
$273$	−925.663	−	2898.38i	−0.205215	−	0.642558i
$274$		0			0
$275$			628.787i			0.137881i
$276$		0			0
$277$	1869.71			0.405560			0.202780	−	0.979224i	$-0.435002\pi$
$277$	1869.71			0.405560			0.202780	+	0.979224i	$0.435002\pi$
$278$		0			0
$279$	4302.74	−	481.061i	0.923292	−	0.103227i
$280$		0			0
$281$			1453.83i			0.308641i	0.988021	+	0.154321i	$0.0493188\pi$
$281$			1453.83i			0.308641i	−0.988021	+	0.154321i	$0.950681\pi$
$282$		0			0
$283$		−	3945.43i		−	0.828732i	−0.910110	−	0.414366i	$-0.864003\pi$
$283$		−	3945.43i		−	0.828732i	0.910110	−	0.414366i	$-0.135997\pi$
$284$		0			0
$285$	−2276.02	−	2035.74i	−0.473052	−	0.423111i
$286$		0			0
$287$	−1076.51	−	633.548i	−0.221409	−	0.130304i
$288$		0			0
$289$	−2648.33			−0.539046
$290$		0			0
$291$	−3715.07	−	3322.86i	−0.748388	−	0.669379i
$292$		0			0
$293$	−3625.64			−0.722908			−0.361454	−	0.932390i	$-0.617719\pi$
$293$	−3625.64			−0.722908			−0.361454	+	0.932390i	$0.617719\pi$
$294$		0			0
$295$	−12217.1			−2.41122
$296$		0			0
$297$	−625.486	+	879.138i	−0.122203	+	0.171760i
$298$		0			0
$299$	3278.36			0.634088
$300$		0			0
$301$	−2464.00	−	1450.11i	−0.471836	−	0.277685i
$302$		0			0
$303$	1622.43	−	1813.93i	0.307611	−	0.343919i
$304$		0			0
$305$			7085.82i			1.33027i
$306$		0			0
$307$		−	6166.82i		−	1.14645i	−0.819399	−	0.573223i	$-0.805693\pi$
$307$		−	6166.82i		−	1.14645i	0.819399	−	0.573223i	$-0.194307\pi$
$308$		0			0
$309$	−6222.57	−	5565.64i	−1.14560	−	1.02465i
$310$		0			0
$311$	75.7688			0.0138150			0.00690748	−	0.999976i	$-0.497801\pi$
$311$	75.7688			0.0138150			0.00690748	+	0.999976i	$0.497801\pi$
$312$		0			0
$313$			9077.21i			1.63921i	0.572926	+	0.819607i	$0.305808\pi$
$313$			9077.21i			1.63921i	−0.572926	+	0.819607i	$0.694192\pi$
$314$		0			0
$315$	−2935.81	+	6563.63i	−0.525125	+	1.17403i
$316$		0			0
$317$			9388.16i			1.66338i	0.555240	+	0.831690i	$0.312627\pi$
$317$			9388.16i			1.66338i	−0.555240	+	0.831690i	$0.687373\pi$
$318$		0			0
$319$	−187.359			−0.0328843
$320$		0			0
$321$	−5163.15	−	4618.06i	−0.897753	−	0.802975i
$322$		0			0
$323$		−	1944.90i		−	0.335038i
$324$		0			0
$325$		−	2585.05i		−	0.441208i
$326$		0			0
$327$	7531.61	−	8420.60i	1.27370	−	1.42404i
$328$		0			0
$329$	1865.87	+	1098.10i	0.312670	+	0.184013i
$330$		0			0
$331$	8714.17			1.44705			0.723526	−	0.690297i	$-0.242520\pi$
$331$	8714.17			1.44705			0.723526	+	0.690297i	$0.242520\pi$
$332$		0			0
$333$	−1130.29	−	10109.6i	−0.186004	−	1.66367i
$334$		0			0
$335$	5013.71			0.817696
$336$		0			0
$337$	452.668			0.0731703			0.0365852	−	0.999331i	$-0.488352\pi$
$337$	452.668			0.0731703			0.0365852	+	0.999331i	$0.488352\pi$
$338$		0			0
$339$	−2838.25	−	2538.61i	−0.454728	−	0.406721i
$340$		0			0
$341$	−1233.20			−0.195839
$342$		0			0
$343$	158.855	−	6350.46i	0.0250069	−	0.999687i
$344$		0			0
$345$	−5774.57	−	5164.93i	−0.901137	−	0.806002i
$346$		0			0
$347$		−	8509.93i		−	1.31653i	−0.752785	−	0.658266i	$-0.771290\pi$
$347$		−	8509.93i		−	1.31653i	0.752785	−	0.658266i	$-0.228710\pi$
$348$		0			0
$349$			11388.8i			1.74679i	0.487014	+	0.873394i	$0.338086\pi$
$349$			11388.8i			1.74679i	−0.487014	+	0.873394i	$0.661914\pi$
$350$		0			0
$351$	2571.48	−	3614.28i	0.391041	−	0.549619i
$352$		0			0
$353$	7242.67			1.09203			0.546017	−	0.837774i	$-0.316143\pi$
$353$	7242.67			1.09203			0.546017	+	0.837774i	$0.316143\pi$
$354$		0			0
$355$			1844.30i			0.275733i
$356$		0			0
$357$	−4362.55	+	1393.28i	−0.646753	+	0.206555i
$358$		0			0
$359$			5355.21i			0.787291i	0.919262	+	0.393645i	$0.128786\pi$
$359$			5355.21i			0.787291i	−0.919262	+	0.393645i	$0.871214\pi$
$360$		0			0
$361$	5188.71			0.756482
$362$		0			0
$363$	−4405.84	+	4925.88i	−0.637043	+	0.712236i
$364$		0			0
$365$		−	4158.83i		−	0.596392i
$366$		0			0
$367$			602.758i			0.0857322i	0.999081	+	0.0428661i	$0.0136489\pi$
$367$			602.758i			0.0857322i	−0.999081	+	0.0428661i	$0.986351\pi$
$368$		0			0
$369$	−202.336	−	1809.74i	−0.0285452	−	0.255316i
$370$		0			0
$371$	4928.26	−	8374.00i	0.689657	−	1.17185i
$372$		0			0
$373$	−11412.7			−1.58425			−0.792125	−	0.610359i	$-0.791025\pi$
$373$	−11412.7			−1.58425			−0.792125	+	0.610359i	$0.791025\pi$
$374$		0			0
$375$	2153.74	−	2407.95i	0.296583	−	0.331590i
$376$		0			0
$377$	770.265			0.105227
$378$		0			0
$379$	3872.04			0.524785			0.262392	−	0.964961i	$-0.415489\pi$
$379$	3872.04			0.524785			0.262392	+	0.964961i	$0.415489\pi$
$380$		0			0
$381$	7041.38	−	7872.50i	0.946826	−	1.05858i
$382$		0			0
$383$	735.160			0.0980807			0.0490403	−	0.998797i	$-0.484384\pi$
$383$	735.160			0.0980807			0.0490403	+	0.998797i	$0.484384\pi$
$384$		0			0
$385$	1038.76	−	1765.04i	0.137507	−	0.233649i
$386$		0			0
$387$	−463.121	−	4142.28i	−0.0608315	−	0.544093i
$388$		0			0
$389$		−	11411.4i		−	1.48735i	−0.668542	−	0.743675i	$-0.733081\pi$
$389$		−	11411.4i		−	1.48735i	0.668542	−	0.743675i	$-0.266919\pi$
$390$		0			0
$391$		−	4934.48i		−	0.638228i
$392$		0			0
$393$	2499.05	−	2794.02i	0.320764	−	0.358625i
$394$		0			0
$395$	13715.2			1.74705
$396$		0			0
$397$			10035.9i			1.26874i	0.773030	+	0.634370i	$0.218740\pi$
$397$			10035.9i			1.26874i	−0.773030	+	0.634370i	$0.781260\pi$
$398$		0			0
$399$	1196.55	+	3746.57i	0.150132	+	0.470083i
$400$		0			0
$401$			6414.15i			0.798771i	0.916783	+	0.399386i	$0.130777\pi$
$401$			6414.15i			0.798771i	−0.916783	+	0.399386i	$0.869223\pi$
$402$		0			0
$403$	5069.87			0.626670
$404$		0			0
$405$	−10223.6	+	2315.01i	−1.25436	+	0.284034i
$406$		0			0
$407$			2897.47i			0.352881i
$408$		0			0
$409$		−	14875.6i		−	1.79842i	−0.437522	−	0.899208i	$-0.644144\pi$
$409$		−	14875.6i		−	1.79842i	0.437522	−	0.899208i	$-0.355856\pi$
$410$		0			0
$411$	−10323.0	−	9233.19i	−1.23892	−	1.10813i
$412$		0			0
$413$	13561.3	+	7981.10i	1.61576	+	0.950906i
$414$		0			0
$415$	−8656.05			−1.02388
$416$		0			0
$417$	3314.76	+	2964.81i	0.389268	+	0.348172i
$418$		0			0
$419$	−8899.11			−1.03759			−0.518795	−	0.854899i	$-0.673619\pi$
$419$	−8899.11			−1.03759			−0.518795	+	0.854899i	$0.673619\pi$
$420$		0			0
$421$	9459.43			1.09507			0.547535	−	0.836783i	$-0.315566\pi$
$421$	9459.43			1.09507			0.547535	+	0.836783i	$0.315566\pi$
$422$		0			0
$423$	350.699	+	3136.75i	0.0403111	+	0.360553i
$424$		0			0
$425$	−3890.93			−0.444089
$426$		0			0
$427$	4628.96	−	7865.42i	0.524616	−	0.891416i
$428$		0			0
$429$	−842.288	+	941.707i	−0.0947927	+	0.105981i
$430$		0			0
$431$			5120.07i			0.572216i	0.958197	+	0.286108i	$0.0923616\pi$
$431$			5120.07i			0.572216i	−0.958197	+	0.286108i	$0.907638\pi$
$432$		0			0
$433$			854.204i			0.0948047i	0.998876	+	0.0474023i	$0.0150943\pi$
$433$			854.204i			0.0948047i	−0.998876	+	0.0474023i	$0.984906\pi$
$434$		0			0
$435$	−1356.76	−	1213.52i	−0.149544	−	0.133756i
$436$		0			0
$437$	−4237.74			−0.463887
$438$		0			0
$439$			11684.2i			1.27028i	0.772396	+	0.635142i	$0.219058\pi$
$439$			11684.2i			1.27028i	−0.772396	+	0.635142i	$0.780942\pi$
$440$		0			0
$441$	7546.64	−	5367.89i	0.814884	−	0.579624i
$442$		0			0
$443$		−	8303.83i		−	0.890580i	−0.895386	−	0.445290i	$-0.853101\pi$
$443$		−	8303.83i		−	0.890580i	0.895386	−	0.445290i	$-0.146899\pi$
$444$		0			0
$445$	15531.7			1.65455
$446$		0			0
$447$	−5434.02	−	4860.33i	−0.574989	−	0.514286i
$448$		0			0
$449$			10362.1i			1.08913i	0.838718	+	0.544566i	$0.183305\pi$
$449$			10362.1i			1.08913i	−0.838718	+	0.544566i	$0.816695\pi$
$450$		0			0
$451$			518.685i			0.0541550i
$452$		0			0
$453$	−5304.22	+	5930.29i	−0.550141	+	0.615076i
$454$		0			0
$455$	−4270.52	+	7256.38i	−0.440011	+	0.747658i
$456$		0			0
$457$	7079.33			0.724633			0.362316	−	0.932055i	$-0.381986\pi$
$457$	7079.33			0.724633			0.362316	+	0.932055i	$0.381986\pi$
$458$		0			0
$459$	−5440.10	−	3870.50i	−0.553207	−	0.393594i
$460$		0			0
$461$	−7297.84			−0.737298			−0.368649	−	0.929569i	$-0.620180\pi$
$461$	−7297.84			−0.737298			−0.368649	+	0.929569i	$0.620180\pi$
$462$		0			0
$463$	801.811			0.0804823			0.0402411	−	0.999190i	$-0.487187\pi$
$463$	801.811			0.0804823			0.0402411	+	0.999190i	$0.487187\pi$
$464$		0			0
$465$	−8930.18	−	7987.39i	−0.890596	−	0.796573i
$466$		0			0
$467$	−8342.32			−0.826631			−0.413316	−	0.910588i	$-0.635629\pi$
$467$	−8342.32			−0.826631			−0.413316	+	0.910588i	$0.635629\pi$
$468$		0			0
$469$	−5565.33	−	3275.31i	−0.547939	−	0.322473i
$470$		0			0
$471$	6413.47	+	5736.38i	0.627425	+	0.561186i
$472$		0			0
$473$			1187.21i			0.115408i
$474$		0			0
$475$			3341.54i			0.322780i
$476$		0			0
$477$	14077.7	−	1573.94i	1.35131	−	0.151081i
$478$		0			0
$479$	3929.61			0.374840			0.187420	−	0.982280i	$-0.439987\pi$
$479$	3929.61			0.374840			0.187420	+	0.982280i	$0.439987\pi$
$480$		0			0
$481$		−	11912.0i		−	1.12919i
$482$		0			0
$483$	3035.81	+	9505.55i	0.285992	+	0.895481i
$484$		0			0
$485$			13792.9i			1.29135i
$486$		0			0
$487$	−6907.85			−0.642761			−0.321381	−	0.946950i	$-0.604147\pi$
$487$	−6907.85			−0.642761			−0.321381	+	0.946950i	$0.604147\pi$
$488$		0			0
$489$	−13132.4	+	14682.4i	−1.21445	+	1.35780i
$490$		0			0
$491$			9858.16i			0.906095i	0.891486	+	0.453048i	$0.149663\pi$
$491$			9858.16i			0.906095i	−0.891486	+	0.453048i	$0.850337\pi$
$492$		0			0
$493$		−	1159.38i		−	0.105914i
$494$		0			0
$495$	2967.25	−	331.749i	0.269430	−	0.0301232i
$496$		0			0
$497$	1204.83	−	2047.22i	0.108740	−	0.184769i
$498$		0			0
$499$	−5143.62			−0.461443			−0.230721	−	0.973020i	$-0.574109\pi$
$499$	−5143.62			−0.461443			−0.230721	+	0.973020i	$0.574109\pi$
$500$		0			0
$501$	8178.28	−	9143.60i	0.729299	−	0.815381i
$502$		0			0
$503$	−7897.38			−0.700053			−0.350027	−	0.936740i	$-0.613827\pi$
$503$	−7897.38			−0.700053			−0.350027	+	0.936740i	$0.613827\pi$
$504$		0			0
$505$	−6734.57			−0.593435
$506$		0			0
$507$	−4147.84	+	4637.43i	−0.363338	+	0.406224i
$508$		0			0
$509$	19845.3			1.72815			0.864075	−	0.503363i	$-0.167904\pi$
$509$	19845.3			1.72815			0.864075	+	0.503363i	$0.167904\pi$
$510$		0			0
$511$	−2716.84	+	4616.40i	−0.235198	+	0.399643i
$512$		0			0
$513$	−3324.00	+	4671.97i	−0.286078	+	0.402091i
$514$		0			0
$515$			23102.5i			1.97674i
$516$		0			0
$517$		−	899.013i		−	0.0764769i
$518$		0			0
$519$	3847.34	−	4301.45i	0.325394	−	0.363801i
$520$		0			0
$521$	14257.7			1.19893			0.599463	−	0.800402i	$-0.295381\pi$
$521$	14257.7			1.19893			0.599463	+	0.800402i	$0.295381\pi$
$522$		0			0
$523$			8935.03i			0.747040i	0.927622	+	0.373520i	$0.121849\pi$
$523$			8935.03i			0.747040i	−0.927622	+	0.373520i	$0.878151\pi$
$524$		0			0
$525$	7495.30	−	2393.79i	0.623089	−	0.198997i
$526$		0			0
$527$		−	7631.00i		−	0.630762i
$528$		0			0
$529$	1415.29			0.116322
$530$		0			0
$531$	2548.92	+	22798.2i	0.208312	+	1.86320i
$532$		0			0
$533$		−	2132.40i		−	0.173292i
$534$		0			0
$535$			19169.2i			1.54908i
$536$		0			0
$537$	4198.41	+	3755.17i	0.337383	+	0.301764i
$538$		0			0
$539$	−2306.10	+	1280.64i	−0.184287	+	0.102340i
$540$		0			0
$541$	8966.47			0.712567			0.356284	−	0.934378i	$-0.384044\pi$
$541$	8966.47			0.712567			0.356284	+	0.934378i	$0.384044\pi$
$542$		0			0
$543$	9624.46	+	8608.38i	0.760635	+	0.680333i
$544$		0			0
$545$	−31263.1			−2.45719
$546$		0			0
$547$	18010.4			1.40781			0.703903	−	0.710296i	$-0.251439\pi$
$547$	18010.4			1.40781			0.703903	+	0.710296i	$0.251439\pi$
$548$		0			0
$549$	13222.7	−	1478.35i	1.02793	−	0.114926i
$550$		0			0
$551$	−995.676			−0.0769823
$552$		0			0
$553$	−15224.2	−	8959.73i	−1.17070	−	0.688981i
$554$		0			0
$555$	−18766.9	+	20982.0i	−1.43534	+	1.60475i
$556$		0			0
$557$		−	18001.6i		−	1.36939i	−0.728829	−	0.684696i	$-0.759935\pi$
$557$		−	18001.6i		−	1.36939i	0.728829	−	0.684696i	$-0.240065\pi$
$558$		0			0
$559$		−	4880.80i		−	0.369295i
$560$		0			0
$561$	1417.43	+	1267.78i	0.106673	+	0.0954116i
$562$		0			0
$563$	−17141.0			−1.28314			−0.641571	−	0.767064i	$-0.721717\pi$
$563$	−17141.0			−1.28314			−0.641571	+	0.767064i	$0.721717\pi$
$564$		0			0
$565$			10537.6i			0.784636i
$566$		0			0
$567$	12860.8	+	4109.08i	0.952561	+	0.304347i
$568$		0			0
$569$		−	17320.5i		−	1.27612i	−0.769987	−	0.638059i	$-0.779738\pi$
$569$		−	17320.5i		−	1.27612i	0.769987	−	0.638059i	$-0.220262\pi$
$570$		0			0
$571$	6668.31			0.488722			0.244361	−	0.969684i	$-0.421422\pi$
$571$	6668.31			0.488722			0.244361	+	0.969684i	$0.421422\pi$
$572$		0			0
$573$	13478.4	+	12055.5i	0.982670	+	0.878927i
$574$		0			0
$575$			8477.93i			0.614876i
$576$		0			0
$577$		−	25884.8i		−	1.86759i	−0.357810	−	0.933794i	$-0.616476\pi$
$577$		−	25884.8i		−	1.86759i	0.357810	−	0.933794i	$-0.383524\pi$
$578$		0			0
$579$	7203.35	−	8053.59i	0.517031	−	0.578058i
$580$		0			0
$581$	9608.42	+	5654.74i	0.686100	+	0.403784i
$582$		0			0
$583$	−4034.77			−0.286626
$584$		0			0
$585$	−12198.9	+	1363.87i	−0.862155	+	0.0963918i
$586$		0			0
$587$	3549.55			0.249583			0.124792	−	0.992183i	$-0.460174\pi$
$587$	3549.55			0.249583			0.124792	+	0.992183i	$0.460174\pi$
$588$		0			0
$589$	−6553.52			−0.458461
$590$		0			0
$591$	−15549.5	−	13907.9i	−1.08227	−	0.968012i
$592$		0			0
$593$	−26252.7			−1.81799			−0.908997	−	0.416802i	$-0.863151\pi$
$593$	−26252.7			−1.81799			−0.908997	+	0.416802i	$0.863151\pi$
$594$		0			0
$595$	10922.1	+	6427.85i	0.752539	+	0.442884i
$596$		0			0
$597$	−169.906	−	151.969i	−0.0116479	−	0.0104182i
$598$		0			0
$599$		−	18394.2i		−	1.25470i	−0.778736	−	0.627351i	$-0.784139\pi$
$599$		−	18394.2i		−	1.25470i	0.778736	−	0.627351i	$-0.215861\pi$
$600$		0			0
$601$			21397.8i			1.45230i	0.687534	+	0.726152i	$0.258693\pi$
$601$			21397.8i			1.45230i	−0.687534	+	0.726152i	$0.741307\pi$
$602$		0			0
$603$	−1046.03	−	9356.00i	−0.0706430	−	0.631850i
$604$		0			0
$605$	18288.3			1.22897
$606$		0			0
$607$			26974.0i			1.80369i	0.432060	+	0.901845i	$0.357787\pi$
$607$			26974.0i			1.80369i	−0.432060	+	0.901845i	$0.642213\pi$
$608$		0			0
$609$	713.277	+	2233.37i	0.0474605	+	0.148606i
$610$		0			0
$611$			3695.99i			0.244720i
$612$		0			0
$613$	−20634.8			−1.35959			−0.679796	−	0.733401i	$-0.737932\pi$
$613$	−20634.8			−1.35959			−0.679796	+	0.733401i	$0.737932\pi$
$614$		0			0
$615$	−3359.52	+	3756.06i	−0.220275	+	0.246274i
$616$		0			0
$617$			1152.71i			0.0752126i	0.999293	+	0.0376063i	$0.0119733\pi$
$617$			1152.71i			0.0752126i	−0.999293	+	0.0376063i	$0.988027\pi$
$618$		0			0
$619$		−	11769.7i		−	0.764237i	−0.924113	−	0.382118i	$-0.875195\pi$
$619$		−	11769.7i		−	0.764237i	0.924113	−	0.382118i	$-0.124805\pi$
$620$		0			0
$621$	−8433.42	+	11853.4i	−0.544962	+	0.765960i
$622$		0			0
$623$	−17240.6	−	10146.4i	−1.10871	−	0.652499i
$624$		0			0
$625$	−19160.2			−1.22625
$626$		0			0
$627$	1088.78	−	1217.29i	0.0693486	−	0.0775341i
$628$		0			0
$629$	−17929.5			−1.13656
$630$		0			0
$631$	14989.1			0.945654			0.472827	−	0.881155i	$-0.343233\pi$
$631$	14989.1			0.945654			0.472827	+	0.881155i	$0.343233\pi$
$632$		0			0
$633$	−1413.23	+	1580.04i	−0.0887375	+	0.0992115i
$634$		0			0
$635$	−29228.2			−1.82659
$636$		0			0
$637$	9480.76	−	5264.94i	0.589704	−	0.327480i
$638$		0			0
$639$	3441.62	−	384.785i	0.213065	−	0.0238214i
$640$		0			0
$641$			21138.6i			1.30254i	0.758848	+	0.651268i	$0.225763\pi$
$641$			21138.6i			1.30254i	−0.758848	+	0.651268i	$0.774237\pi$
$642$		0			0
$643$			2403.51i			0.147411i	0.997280	+	0.0737055i	$0.0234825\pi$
$643$			2403.51i			0.147411i	−0.997280	+	0.0737055i	$0.976518\pi$
$644$		0			0
$645$	−7689.52	+	8597.15i	−0.469418	+	0.524825i
$646$		0			0
$647$	−21287.0			−1.29347			−0.646736	−	0.762714i	$-0.723866\pi$
$647$	−21287.0			−1.29347			−0.646736	+	0.762714i	$0.723866\pi$
$648$		0			0
$649$		−	6534.12i		−	0.395203i
$650$		0			0
$651$	4694.77	+	14700.0i	0.282646	+	0.885006i
$652$		0			0
$653$		−	11934.5i		−	0.715209i	−0.933873	−	0.357605i	$-0.883594\pi$
$653$		−	11934.5i		−	0.715209i	0.933873	−	0.357605i	$-0.116406\pi$
$654$		0			0
$655$	−10373.4			−0.618810
$656$		0			0
$657$	−7760.72	+	867.675i	−0.460844	+	0.0515240i
$658$		0			0
$659$			4047.31i			0.239243i	0.992820	+	0.119621i	$0.0381680\pi$
$659$			4047.31i			0.239243i	−0.992820	+	0.119621i	$0.961832\pi$
$660$		0			0
$661$		−	8874.75i		−	0.522221i	−0.965309	−	0.261110i	$-0.915911\pi$
$661$		−	8874.75i		−	0.522221i	0.965309	−	0.261110i	$-0.0840886\pi$
$662$		0			0
$663$	−5827.27	−	5212.07i	−0.341346	−	0.305309i
$664$		0			0
$665$	5520.25	−	9379.90i	0.321904	−	0.546973i
$666$		0			0
$667$	−2526.16			−0.146647
$668$		0			0
$669$	9070.09	+	8112.54i	0.524171	+	0.468832i
$670$		0			0
$671$	−3789.72			−0.218034
$672$		0			0
$673$	−16833.7			−0.964178			−0.482089	−	0.876122i	$-0.660122\pi$
$673$	−16833.7			−0.964178			−0.482089	+	0.876122i	$0.660122\pi$
$674$		0			0
$675$	9346.64	+	6649.91i	0.532966	+	0.379193i
$676$		0			0
$677$	−9642.23			−0.547387			−0.273693	−	0.961817i	$-0.588245\pi$
$677$	−9642.23			−0.547387			−0.273693	+	0.961817i	$0.588245\pi$
$678$		0			0
$679$	9010.50	−	15310.5i	0.509266	−	0.865333i
$680$		0			0
$681$	5958.47	−	6661.78i	0.335285	−	0.374860i
$682$		0			0
$683$			9601.31i			0.537897i	0.963155	+	0.268949i	$0.0866762\pi$
$683$			9601.31i			0.537897i	−0.963155	+	0.268949i	$0.913324\pi$
$684$		0			0
$685$			38326.3i			2.13777i
$686$		0			0
$687$	23594.1	+	21103.2i	1.31029	+	1.17196i
$688$		0			0
$689$	16587.6			0.917180
$690$		0			0
$691$		−	22579.7i		−	1.24309i	−0.783380	−	0.621543i	$-0.786506\pi$
$691$		−	22579.7i		−	1.24309i	0.783380	−	0.621543i	$-0.213494\pi$
$692$		0			0
$693$	−3510.44	−	1570.17i	−0.192425	−	0.0860689i
$694$		0			0
$695$		−	12306.7i		−	0.671684i
$696$		0			0
$697$	−3209.62			−0.174423
$698$		0			0
$699$	6395.39	+	5720.21i	0.346060	+	0.309525i
$700$		0			0
$701$		−	13689.8i		−	0.737596i	−0.929510	−	0.368798i	$-0.879769\pi$
$701$		−	13689.8i		−	0.737596i	0.929510	−	0.368798i	$-0.120231\pi$
$702$		0			0
$703$			15397.9i			0.826094i
$704$		0			0
$705$	5822.90	−	6510.20i	0.311068	−	0.347785i
$706$		0			0
$707$	7475.53	+	4399.50i	0.397661	+	0.234031i
$708$		0			0
$709$	13558.0			0.718168			0.359084	−	0.933305i	$-0.383089\pi$
$709$	13558.0			0.718168			0.359084	+	0.933305i	$0.383089\pi$
$710$		0			0
$711$	−2861.46	−	25593.7i	−0.150933	−	1.34998i
$712$		0			0
$713$	−16627.2			−0.873341
$714$		0			0
$715$	3496.27			0.182872
$716$		0			0
$717$	24541.5	+	21950.6i	1.27827	+	1.14332i
$718$		0			0
$719$	19937.9			1.03416			0.517079	−	0.855938i	$-0.327019\pi$
$719$	19937.9			1.03416			0.517079	+	0.855938i	$0.327019\pi$
$720$		0			0
$721$	15092.2	−	25644.3i	0.779560	−	1.32461i
$722$		0			0
$723$	−11085.4	−	9915.05i	−0.570220	−	0.510020i
$724$		0			0
$725$			1991.93i			0.102039i
$726$		0			0
$727$		−	4991.04i		−	0.254618i	−0.991863	−	0.127309i	$-0.959366\pi$
$727$		−	4991.04i		−	0.254618i	0.991863	−	0.127309i	$-0.0406340\pi$
$728$		0			0
$729$	6453.00	+	18595.1i	0.327846	+	0.944731i
$730$		0			0
$731$	−7346.42			−0.371706
$732$		0			0
$733$			9750.45i			0.491325i	0.969355	+	0.245662i	$0.0790055\pi$
$733$			9750.45i			0.491325i	−0.969355	+	0.245662i	$0.920995\pi$
$734$		0			0
$735$	−24994.3	−	5662.80i	−1.25433	−	0.284184i
$736$		0			0
$737$			2681.49i			0.134022i
$738$		0			0
$739$	13449.7			0.669494			0.334747	−	0.942308i	$-0.391349\pi$
$739$	13449.7			0.669494			0.334747	+	0.942308i	$0.391349\pi$
$740$		0			0
$741$	−4476.14	+	5004.48i	−0.221910	+	0.248103i
$742$		0			0
$743$			5609.12i			0.276956i	0.990365	+	0.138478i	$0.0442211\pi$
$743$			5609.12i			0.276956i	−0.990365	+	0.138478i	$0.955779\pi$
$744$		0			0
$745$			20174.9i			0.992148i
$746$		0			0
$747$	1805.95	+	16152.9i	0.0884555	+	0.791170i
$748$		0			0
$749$	12522.7	−	21278.3i	0.610906	−	1.03804i
$750$		0			0
$751$	−32476.8			−1.57802			−0.789010	−	0.614380i	$-0.789406\pi$
$751$	−32476.8			−1.57802			−0.789010	+	0.614380i	$0.789406\pi$
$752$		0			0
$753$	−22243.4	+	24868.9i	−1.07649	+	1.20355i
$754$		0			0
$755$	22017.4			1.06132
$756$		0			0
$757$	322.197			0.0154696			0.00773478	−	0.999970i	$-0.497538\pi$
$757$	322.197			0.0154696			0.00773478	+	0.999970i	$0.497538\pi$
$758$		0			0
$759$	2762.37	−	3088.43i	0.132105	−	0.147698i
$760$		0			0
$761$	−12316.5			−0.586694			−0.293347	−	0.956006i	$-0.594769\pi$
$761$	−12316.5			−0.586694			−0.293347	+	0.956006i	$0.594769\pi$
$762$		0			0
$763$	34702.8	+	20423.3i	1.64656	+	0.969034i
$764$		0			0
$765$	2052.86	+	18361.3i	0.0970212	+	0.867784i
$766$		0			0
$767$			26862.9i			1.26462i
$768$		0			0
$769$			2805.05i			0.131538i	0.997835	+	0.0657690i	$0.0209500\pi$
$769$			2805.05i			0.131538i	−0.997835	+	0.0657690i	$0.979050\pi$
$770$		0			0
$771$	−3722.75	+	4162.16i	−0.173893	+	0.194418i
$772$		0			0
$773$	4.86859			0.000226534			0.000113267	−	1.00000i	$-0.499964\pi$
$773$	4.86859			0.000226534			0.000113267		1.00000i	$0.499964\pi$
$774$		0			0
$775$			13110.8i			0.607683i
$776$		0			0
$777$	34538.6	−	11030.7i	1.59468	−	0.509297i
$778$		0			0
$779$			2756.43i			0.126777i
$780$		0			0
$781$	−986.392			−0.0451932
$782$		0			0
$783$	−1981.47	+	2785.01i	−0.0904367	+	0.127111i
$784$		0			0
$785$		−	23811.3i		−	1.08262i
$786$		0			0
$787$			13302.1i			0.602501i	0.953545	+	0.301251i	$0.0974040\pi$
$787$			13302.1i			0.602501i	−0.953545	+	0.301251i	$0.902596\pi$
$788$		0			0
$789$	−18748.8	−	16769.5i	−0.845977	−	0.756665i
$790$		0			0
$791$	6883.89	−	11697.0i	0.309435	−	0.525785i
$792$		0			0
$793$	15580.2			0.697690
$794$		0			0
$795$	−29217.7	−	26133.1i	−1.30345	−	1.16585i
$796$		0			0
$797$	−39271.0			−1.74536			−0.872680	−	0.488293i	$-0.837620\pi$
$797$	−39271.0			−1.74536			−0.872680	+	0.488293i	$0.837620\pi$
$798$		0			0
$799$	5563.09			0.246318
$800$		0			0
$801$	−3240.45	−	28983.5i	−0.142941	−	1.27850i
$802$		0			0
$803$	2224.28			0.0977497
$804$		0			0
$805$	14005.6	−	23798.0i	0.613209	−	1.04195i
$806$		0			0
$807$	−10612.1	+	11864.7i	−0.462906	+	0.517545i
$808$		0			0
$809$			10604.2i			0.460847i	0.973090	+	0.230424i	$0.0740112\pi$
$809$			10604.2i			0.460847i	−0.973090	+	0.230424i	$0.925989\pi$
$810$		0			0
$811$		−	3095.55i		−	0.134032i	−0.997752	−	0.0670158i	$-0.978652\pi$
$811$		−	3095.55i		−	0.134032i	0.997752	−	0.0670158i	$-0.0213478\pi$
$812$		0			0
$813$	−19608.9	−	17538.7i	−0.845895	−	0.756591i
$814$		0			0
$815$	54511.4			2.34289
$816$		0			0
$817$			6309.13i			0.270169i
$818$		0			0
$819$	14432.0	+	6455.22i	0.615744	+	0.275413i
$820$		0			0
$821$		−	36565.8i		−	1.55439i	−0.629259	−	0.777195i	$-0.716642\pi$
$821$		−	36565.8i		−	1.55439i	0.629259	−	0.777195i	$-0.283358\pi$
$822$		0			0
$823$	−29612.4			−1.25422			−0.627110	−	0.778931i	$-0.715762\pi$
$823$	−29612.4			−1.25422			−0.627110	+	0.778931i	$0.715762\pi$
$824$		0			0
$825$	−2435.28	−	2178.18i	−0.102770	−	0.0919206i
$826$		0			0
$827$		−	43562.2i		−	1.83169i	−0.401534	−	0.915844i	$-0.631523\pi$
$827$		−	43562.2i		−	1.83169i	0.401534	−	0.915844i	$-0.368477\pi$
$828$		0			0
$829$		−	5497.69i		−	0.230329i	−0.993346	−	0.115164i	$-0.963261\pi$
$829$		−	5497.69i		−	0.230329i	0.993346	−	0.115164i	$-0.0367395\pi$
$830$		0			0
$831$	−6476.88	+	7241.37i	−0.270374	+	0.302287i
$832$		0			0
$833$	−7924.62	−	14270.1i	−0.329618	−	0.593554i
$834$		0			0
$835$	−33947.4			−1.40694
$836$		0			0
$837$	−13042.0	+	18330.9i	−0.538587	+	0.756999i
$838$		0			0
$839$	−17898.3			−0.736493			−0.368247	−	0.929728i	$-0.620042\pi$
$839$	−17898.3			−0.736493			−0.368247	+	0.929728i	$0.620042\pi$
$840$		0			0
$841$	23795.5			0.975664
$842$		0			0
$843$	−5630.65	−	5036.21i	−0.230047	−	0.205761i
$844$		0			0
$845$	17217.4			0.700942
$846$		0			0
$847$	−20300.4	−	11947.2i	−0.823532	−	0.484665i
$848$		0			0
$849$	15280.6	+	13667.4i	0.617701	+	0.552488i
$850$		0			0
$851$			39066.6i			1.57366i
$852$		0			0
$853$			29437.6i			1.18162i	0.806810	+	0.590811i	$0.201192\pi$
$853$			29437.6i			1.18162i	−0.806810	+	0.590811i	$0.798808\pi$
$854$		0			0
$855$	15768.7	−	1763.00i	0.630737	−	0.0705185i
$856$		0			0
$857$	27320.9			1.08899			0.544494	−	0.838765i	$-0.316722\pi$
$857$	27320.9			1.08899			0.544494	+	0.838765i	$0.316722\pi$
$858$		0			0
$859$		−	47069.4i		−	1.86960i	−0.355176	−	0.934799i	$-0.615579\pi$
$859$		−	47069.4i		−	1.86960i	0.355176	−	0.934799i	$-0.384421\pi$
$860$		0			0
$861$	6182.86	−	1974.63i	0.244729	−	0.0781595i
$862$		0			0
$863$			4821.40i			0.190176i	0.995469	+	0.0950882i	$0.0303133\pi$
$863$			4821.40i			0.190176i	−0.995469	+	0.0950882i	$0.969687\pi$
$864$		0			0
$865$	−15970.0			−0.627741
$866$		0			0
$867$	9174.09	−	10256.9i	0.359364	−	0.401781i
$868$		0			0
$869$			7335.33i			0.286345i
$870$		0			0
$871$		−	11024.1i		−	0.428859i
$872$		0			0
$873$	25738.7	−	2877.68i	0.997851	−	0.111563i
$874$		0			0
$875$	9923.60	+	5840.24i	0.383405	+	0.225641i
$876$		0			0
$877$	6612.09			0.254589			0.127294	−	0.991865i	$-0.459371\pi$
$877$	6612.09			0.254589			0.127294	+	0.991865i	$0.459371\pi$
$878$		0			0
$879$	12559.6	−	14042.0i	0.481938	−	0.538823i
$880$		0			0
$881$	−5909.57			−0.225991			−0.112996	−	0.993595i	$-0.536045\pi$
$881$	−5909.57			−0.225991			−0.112996	+	0.993595i	$0.536045\pi$
$882$		0			0
$883$	−8720.59			−0.332357			−0.166179	−	0.986096i	$-0.553143\pi$
$883$	−8720.59			−0.332357			−0.166179	+	0.986096i	$0.553143\pi$
$884$		0			0
$885$	42321.4	−	47316.8i	1.60748	−	1.79722i
$886$		0			0
$887$	43301.9			1.63916			0.819581	−	0.572964i	$-0.194206\pi$
$887$	43301.9			1.63916			0.819581	+	0.572964i	$0.194206\pi$
$888$		0			0
$889$	32444.0	+	19093.9i	1.22400	+	0.720348i
$890$		0			0
$891$	−1238.14	−	5467.92i	−0.0465536	−	0.205592i
$892$		0			0
$893$		−	4777.59i		−	0.179033i
$894$		0			0
$895$		−	15587.4i		−	0.582156i
$896$		0			0
$897$	−11356.6	+	12697.0i	−0.422725	+	0.472621i
$898$		0			0
$899$	−3906.62			−0.144931
$900$		0			0
$901$		−	24967.1i		−	0.923168i
$902$		0			0
$903$	14151.8	−	4519.69i	0.521531	−	0.166563i
$904$		0			0
$905$		−	35732.7i		−	1.31248i
$906$		0			0
$907$	−3305.86			−0.121025			−0.0605123	−	0.998167i	$-0.519273\pi$
$907$	−3305.86			−0.121025			−0.0605123	+	0.998167i	$0.519273\pi$
$908$		0			0
$909$	1405.06	+	12567.3i	0.0512684	+	0.458559i
$910$		0			0
$911$			19748.8i			0.718230i	0.933293	+	0.359115i	$0.116921\pi$
$911$			19748.8i			0.718230i	−0.933293	+	0.359115i	$0.883079\pi$
$912$		0			0
$913$		−	4629.53i		−	0.167815i
$914$		0			0
$915$	−27443.3	−	24546.0i	−0.991526	−	0.886848i
$916$		0			0
$917$	11514.7	+	6776.60i	0.414665	+	0.244038i
$918$		0			0
$919$	−1151.83			−0.0413441			−0.0206721	−	0.999786i	$-0.506581\pi$
$919$	−1151.83			−0.0413441			−0.0206721	+	0.999786i	$0.506581\pi$
$920$		0			0
$921$	23884.0	+	21362.5i	0.854510	+	0.764297i
$922$		0			0
$923$	4055.22			0.144614
$924$		0			0
$925$	30804.8			1.09498
$926$		0			0
$927$	43111.2	−	4819.98i	1.52746	−	0.170776i
$928$		0			0
$929$	−52478.2			−1.85334			−0.926671	−	0.375875i	$-0.877342\pi$
$929$	−52478.2			−1.85334			−0.926671	+	0.375875i	$0.877342\pi$
$930$		0			0
$931$	−12255.2	+	6805.68i	−0.431416	+	0.239578i
$932$		0			0
$933$	−262.471	+	293.451i	−0.00920997	+	0.0102971i
$934$		0			0
$935$		−	5262.47i		−	0.184066i
$936$		0			0
$937$			4715.86i			0.164419i	0.996615	+	0.0822094i	$0.0261976\pi$
$937$			4715.86i			0.164419i	−0.996615	+	0.0822094i	$0.973802\pi$
$938$		0			0
$939$	−35155.9	−	31444.4i	−1.22180	−	1.09281i
$940$		0			0
$941$	−21247.6			−0.736080			−0.368040	−	0.929810i	$-0.619971\pi$
$941$	−21247.6			−0.736080			−0.368040	+	0.929810i	$0.619971\pi$
$942$		0			0
$943$			6993.43i			0.241503i
$944$		0			0
$945$	−15250.9	−	34107.4i	−0.524984	−	1.17409i
$946$		0			0
$947$		−	54586.7i		−	1.87310i	−0.350530	−	0.936551i	$-0.613999\pi$
$947$		−	54586.7i		−	1.87310i	0.350530	−	0.936551i	$-0.386001\pi$
$948$		0			0
$949$	−9144.36			−0.312791
$950$		0			0
$951$	−36360.2	−	32521.5i	−1.23981	−	1.10892i
$952$		0			0
$953$			9928.31i			0.337471i	0.985661	+	0.168735i	$0.0539683\pi$
$953$			9928.31i			0.337471i	−0.985661	+	0.168735i	$0.946032\pi$
$954$		0			0
$955$		−	50041.4i		−	1.69560i
$956$		0			0
$957$	649.031	−	725.639i	0.0219229	−	0.0245105i
$958$		0			0
$959$	25037.4	−	42543.0i	0.843066	−	1.43252i
$960$		0			0
$961$	4077.67			0.136876
$962$		0			0
$963$	35771.3	−	3999.36i	1.19700	−	0.133829i
$964$		0			0
$965$	−29900.5			−0.997443
$966$		0			0
$967$	13701.3			0.455640			0.227820	−	0.973703i	$-0.426840\pi$
$967$	13701.3			0.455640			0.227820	+	0.973703i	$0.426840\pi$
$968$		0			0
$969$	7532.58	+	6737.34i	0.249723	+	0.223359i
$970$		0			0
$971$	4920.66			0.162628			0.0813139	−	0.996689i	$-0.474088\pi$
$971$	4920.66			0.162628			0.0813139	+	0.996689i	$0.474088\pi$
$972$		0			0
$973$	−8039.61	+	13660.7i	−0.264890	+	0.450096i
$974$		0			0
$975$	10011.8	+	8954.86i	0.328857	+	0.294139i
$976$		0			0
$977$		−	38183.9i		−	1.25037i	−0.780476	−	0.625185i	$-0.785023\pi$
$977$		−	38183.9i		−	1.25037i	0.780476	−	0.625185i	$-0.214977\pi$
$978$		0			0
$979$			8306.86i			0.271183i
$980$		0			0
$981$	6522.57	+	58339.6i	0.212283	+	1.89872i
$982$		0			0
$983$	539.085			0.0174915			0.00874575	−	0.999962i	$-0.497216\pi$
$983$	539.085			0.0174915			0.00874575	+	0.999962i	$0.497216\pi$
$984$		0			0
$985$			57730.7i			1.86746i
$986$		0			0
$987$	−10716.5	+	3422.54i	−0.345602	+	0.110376i
$988$		0			0
$989$			16007.1i			0.514657i
$990$		0			0
$991$	6668.49			0.213755			0.106878	−	0.994272i	$-0.465915\pi$
$991$	6668.49			0.213755			0.106878	+	0.994272i	$0.465915\pi$
$992$		0			0
$993$	−30186.8	+	33749.8i	−0.964701	+	1.07857i
$994$		0			0
$995$			630.810i			0.0200985i
$996$		0			0
$997$		−	46441.5i		−	1.47524i	−0.675214	−	0.737621i	$-0.735949\pi$
$997$		−	46441.5i		−	1.47524i	0.675214	−	0.737621i	$-0.264051\pi$
$998$		0			0
$999$	43069.7	+	30643.0i	1.36403	+	0.970473i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.k.c.545.8	yes	16
3.2	odd	2	inner	672.4.k.c.545.13	yes	16
4.3	odd	2	inner	672.4.k.c.545.12	yes	16
7.6	odd	2	inner	672.4.k.c.545.9	yes	16
12.11	even	2	inner	672.4.k.c.545.1	✓	16
21.20	even	2	inner	672.4.k.c.545.4	yes	16
28.27	even	2	inner	672.4.k.c.545.5	yes	16
84.83	odd	2	inner	672.4.k.c.545.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.k.c.545.1	✓	16	12.11	even	2	inner
672.4.k.c.545.4	yes	16	21.20	even	2	inner
672.4.k.c.545.5	yes	16	28.27	even	2	inner
672.4.k.c.545.8	yes	16	1.1	even	1	trivial
672.4.k.c.545.9	yes	16	7.6	odd	2	inner
672.4.k.c.545.12	yes	16	4.3	odd	2	inner
672.4.k.c.545.13	yes	16	3.2	odd	2	inner
672.4.k.c.545.16	yes	16	84.83	odd	2	inner

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.46410 + 3.87298i) q^{3} +14.3792 q^{5} +(-15.9613 - 9.39352i) q^{7} +(-3.00000 - 26.8328i) q^{9} +O(q^{10})\)	\(q+(-3.46410 + 3.87298i) q^{3} +14.3792 q^{5} +(-15.9613 - 9.39352i) q^{7} +(-3.00000 - 26.8328i) q^{9} +7.69047i q^{11} -31.6168i q^{13} +(-49.8111 + 55.6905i) q^{15} -47.5885 q^{17} +40.8692i q^{19} +(91.6724 - 29.2776i) q^{21} +103.690i q^{23} +81.7619 q^{25} +(114.315 + 81.3327i) q^{27} +24.3625i q^{29} +160.354i q^{31} +(-29.7850 - 26.6406i) q^{33} +(-229.510 - 135.071i) q^{35} +376.762 q^{37} +(122.451 + 109.524i) q^{39} +67.4452 q^{41} +154.374 q^{43} +(-43.1376 - 385.835i) q^{45} -116.900 q^{47} +(166.524 + 299.865i) q^{49} +(164.851 - 184.310i) q^{51} +524.645i q^{53} +110.583i q^{55} +(-158.286 - 141.575i) q^{57} -849.639 q^{59} +492.782i q^{61} +(-204.171 + 456.466i) q^{63} -454.624i q^{65} +348.677 q^{67} +(-401.591 - 359.194i) q^{69} +128.262i q^{71} -289.225i q^{73} +(-283.231 + 316.662i) q^{75} +(72.2405 - 122.750i) q^{77} +953.821 q^{79} +(-711.000 + 160.997i) q^{81} -601.984 q^{83} -684.286 q^{85} +(-94.3557 - 84.3943i) q^{87} +1080.15 q^{89} +(-296.993 + 504.644i) q^{91} +(-621.047 - 555.482i) q^{93} +587.667i q^{95} +959.226i q^{97} +(206.357 - 23.0714i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9}+O(q^{10}) \) 16 * q - 48 * q^9	\( 16 q - 48 q^{9} - 240 q^{21} - 336 q^{25} + 4384 q^{37} - 624 q^{49} + 2400 q^{57} - 11376 q^{81} - 6016 q^{85} - 3360 q^{93}+O(q^{100}) \) 16 * q - 48 * q^9 - 240 * q^21 - 336 * q^25 + 4384 * q^37 - 624 * q^49 + 2400 * q^57 - 11376 * q^81 - 6016 * q^85 - 3360 * q^93

Introduction

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