LMFDB - Embedded newform 672.4.h.a.575.30

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,4,Mod(575,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("672.575"); S:= CuspForms(chi, 4); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [36,0,0,0,0,0,0,0,-132] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.30
Character	$\chi$	$=$	672.575
Dual form			672.4.h.a.575.29

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(3.68812 + 3.66030i) q^{3} -1.63405i q^{5} -7.00000i q^{7} +(0.204455 + 26.9992i) q^{9} +12.7716 q^{11} +4.43850 q^{13} +(5.98113 - 6.02659i) q^{15} -115.854i q^{17} -126.414i q^{19} +(25.6221 - 25.8168i) q^{21} +80.3969 q^{23} +122.330 q^{25} +(-98.0711 + 100.325i) q^{27} +249.311i q^{29} +119.670i q^{31} +(47.1033 + 46.7479i) q^{33} -11.4384 q^{35} +436.818 q^{37} +(16.3697 + 16.2462i) q^{39} -331.031i q^{41} -126.975i q^{43} +(44.1182 - 0.334090i) q^{45} -135.834 q^{47} -49.0000 q^{49} +(424.060 - 427.283i) q^{51} -395.400i q^{53} -20.8695i q^{55} +(462.714 - 466.231i) q^{57} +329.339 q^{59} -364.278 q^{61} +(188.995 - 1.43118i) q^{63} -7.25275i q^{65} +996.255i q^{67} +(296.513 + 294.276i) q^{69} +349.047 q^{71} +211.437 q^{73} +(451.167 + 447.764i) q^{75} -89.4014i q^{77} +240.806i q^{79} +(-728.916 + 11.0402i) q^{81} +580.873 q^{83} -189.312 q^{85} +(-912.553 + 919.490i) q^{87} -1291.93i q^{89} -31.0695i q^{91} +(-438.029 + 441.358i) q^{93} -206.568 q^{95} +1175.97 q^{97} +(2.61122 + 344.824i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 132 q^{9} - 120 q^{13} + 28 q^{21} - 756 q^{25} + 40 q^{33} + 672 q^{37} + 304 q^{45} - 1764 q^{49} + 1624 q^{57} + 2472 q^{61} - 1224 q^{69} - 2376 q^{73} + 468 q^{81} + 5160 q^{85} - 648 q^{93}+ \cdots - 4488 q^{97}+O(q^{100}) $ 36 * q - 132 * q^9 - 120 * q^13 + 28 * q^21 - 756 * q^25 + 40 * q^33 + 672 * q^37 + 304 * q^45 - 1764 * q^49 + 1624 * q^57 + 2472 * q^61 - 1224 * q^69 - 2376 * q^73 + 468 * q^81 + 5160 * q^85 - 648 * q^93 - 4488 * q^97

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
$3$	3.68812	+	3.66030i	0.709779	+	0.704424i
$3$	3.68812	+	3.66030i	0.709779	+	0.704424i
$4$		0			0
$5$		−	1.63405i		−	0.146154i	−0.997326	−	0.0730772i	$-0.976718\pi$
$5$		−	1.63405i		−	0.146154i	0.997326	−	0.0730772i	$-0.0232819\pi$
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$		0			0
$9$	0.204455	+	26.9992i	0.00757240	+	0.999971i
$10$		0			0
$11$	12.7716			0.350072			0.175036	−	0.984562i	$-0.443996\pi$
$11$	12.7716			0.350072			0.175036	+	0.984562i	$0.443996\pi$
$12$		0			0
$13$	4.43850			0.0946938			0.0473469	−	0.998879i	$-0.484923\pi$
$13$	4.43850			0.0946938			0.0473469	+	0.998879i	$0.484923\pi$
$14$		0			0
$15$	5.98113	−	6.02659i	0.102955	−	0.103737i
$16$		0			0
$17$		−	115.854i		−	1.65287i	−0.563035	−	0.826433i	$-0.690367\pi$
$17$		−	115.854i		−	1.65287i	0.563035	−	0.826433i	$-0.309633\pi$
$18$		0			0
$19$		−	126.414i		−	1.52639i	−0.646168	−	0.763195i	$-0.723630\pi$
$19$		−	126.414i		−	1.52639i	0.646168	−	0.763195i	$-0.276370\pi$
$20$		0			0
$21$	25.6221	−	25.8168i	0.266247	−	0.268271i
$22$		0			0
$23$	80.3969			0.728865			0.364433	−	0.931230i	$-0.381263\pi$
$23$	80.3969			0.728865			0.364433	+	0.931230i	$0.381263\pi$
$24$		0			0
$25$	122.330			0.978639
$26$		0			0
$27$	−98.0711	+	100.325i	−0.699030	+	0.715093i
$28$		0			0
$29$			249.311i			1.59641i	0.602385	+	0.798206i	$0.294217\pi$
$29$			249.311i			1.59641i	−0.602385	+	0.798206i	$0.705783\pi$
$30$		0			0
$31$			119.670i			0.693336i	0.937988	+	0.346668i	$0.112687\pi$
$31$			119.670i			0.693336i	−0.937988	+	0.346668i	$0.887313\pi$
$32$		0			0
$33$	47.1033	+	46.7479i	0.248474	+	0.246599i
$34$		0			0
$35$	−11.4384			−0.0552411
$36$		0			0
$37$	436.818			1.94088			0.970438	−	0.241352i	$-0.0775909\pi$
$37$	436.818			1.94088			0.970438	+	0.241352i	$0.0775909\pi$
$38$		0			0
$39$	16.3697	+	16.2462i	0.0672116	+	0.0667046i
$40$		0			0
$41$		−	331.031i		−	1.26094i	−0.776215	−	0.630469i	$-0.782863\pi$
$41$		−	331.031i		−	1.26094i	0.776215	−	0.630469i	$-0.217137\pi$
$42$		0			0
$43$		−	126.975i		−	0.450315i	−0.974322	−	0.225158i	$-0.927710\pi$
$43$		−	126.975i		−	0.450315i	0.974322	−	0.225158i	$-0.0722897\pi$
$44$		0			0
$45$	44.1182	−	0.334090i	0.146150	−	0.00110674i
$46$		0			0
$47$	−135.834			−0.421563			−0.210781	−	0.977533i	$-0.567601\pi$
$47$	−135.834			−0.421563			−0.210781	+	0.977533i	$0.567601\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	424.060	−	427.283i	1.16432	−	1.17317i
$52$		0			0
$53$		−	395.400i		−	1.02476i	−0.858759	−	0.512380i	$-0.828764\pi$
$53$		−	395.400i		−	1.02476i	0.858759	−	0.512380i	$-0.171236\pi$
$54$		0			0
$55$		−	20.8695i		−	0.0511645i
$56$		0			0
$57$	462.714	−	466.231i	1.07523	−	1.08340i
$58$		0			0
$59$	329.339			0.726717			0.363358	−	0.931649i	$-0.381630\pi$
$59$	329.339			0.726717			0.363358	+	0.931649i	$0.381630\pi$
$60$		0			0
$61$	−364.278			−0.764606			−0.382303	−	0.924037i	$-0.624869\pi$
$61$	−364.278			−0.764606			−0.382303	+	0.924037i	$0.624869\pi$
$62$		0			0
$63$	188.995	−	1.43118i	0.377954	−	0.00286210i
$64$		0			0
$65$		−	7.25275i		−	0.0138399i
$66$		0			0
$67$			996.255i			1.81660i	0.418324	+	0.908298i	$0.362618\pi$
$67$			996.255i			1.81660i	−0.418324	+	0.908298i	$0.637382\pi$
$68$		0			0
$69$	296.513	+	294.276i	0.517333	+	0.513431i
$70$		0			0
$71$	349.047			0.583440			0.291720	−	0.956504i	$-0.405772\pi$
$71$	349.047			0.583440			0.291720	+	0.956504i	$0.405772\pi$
$72$		0			0
$73$	211.437			0.338997			0.169498	−	0.985530i	$-0.445785\pi$
$73$	211.437			0.338997			0.169498	+	0.985530i	$0.445785\pi$
$74$		0			0
$75$	451.167	+	447.764i	0.694617	+	0.689377i
$76$		0			0
$77$		−	89.4014i		−	0.132315i
$78$		0			0
$79$			240.806i			0.342947i	0.985189	+	0.171474i	$0.0548528\pi$
$79$			240.806i			0.342947i	−0.985189	+	0.171474i	$0.945147\pi$
$80$		0			0
$81$	−728.916	+	11.0402i	−0.999885	+	0.0151444i
$82$		0			0
$83$	580.873			0.768182			0.384091	−	0.923295i	$-0.374515\pi$
$83$	580.873			0.768182			0.384091	+	0.923295i	$0.374515\pi$
$84$		0			0
$85$	−189.312			−0.241573
$86$		0			0
$87$	−912.553	+	919.490i	−1.12455	+	1.13310i
$88$		0			0
$89$		−	1291.93i		−	1.53871i	−0.638824	−	0.769353i	$-0.720579\pi$
$89$		−	1291.93i		−	1.53871i	0.638824	−	0.769353i	$-0.279421\pi$
$90$		0			0
$91$		−	31.0695i		−	0.0357909i
$92$		0			0
$93$	−438.029	+	441.358i	−0.488403	+	0.492115i
$94$		0			0
$95$	−206.568			−0.223088
$96$		0			0
$97$	1175.97			1.23095			0.615475	−	0.788157i	$-0.288964\pi$
$97$	1175.97			1.23095			0.615475	+	0.788157i	$0.288964\pi$
$98$		0			0
$99$	2.61122	+	344.824i	0.00265088	+	0.350062i
$100$		0			0
$101$		−	470.743i		−	0.463769i	−0.972743	−	0.231885i	$-0.925511\pi$
$101$		−	470.743i		−	0.463769i	0.972743	−	0.231885i	$-0.0744892\pi$
$102$		0			0
$103$			1443.29i			1.38069i	0.723479	+	0.690347i	$0.242542\pi$
$103$			1443.29i			1.38069i	−0.723479	+	0.690347i	$0.757458\pi$
$104$		0			0
$105$	−42.1861	−	41.8679i	−0.0392090	−	0.0389132i
$106$		0			0
$107$	564.535			0.510053			0.255027	−	0.966934i	$-0.417916\pi$
$107$	564.535			0.510053			0.255027	+	0.966934i	$0.417916\pi$
$108$		0			0
$109$	1889.97			1.66079			0.830396	−	0.557174i	$-0.188114\pi$
$109$	1889.97			1.66079			0.830396	+	0.557174i	$0.188114\pi$
$110$		0			0
$111$	1611.04	+	1598.88i	1.37759	+	1.36720i
$112$		0			0
$113$		−	579.748i		−	0.482638i	−0.970446	−	0.241319i	$-0.922420\pi$
$113$		−	579.748i		−	0.482638i	0.970446	−	0.241319i	$-0.0775800\pi$
$114$		0			0
$115$		−	131.373i		−	0.106527i
$116$		0			0
$117$	0.907473	+	119.836i	0.000717059	+	0.0946910i
$118$		0			0
$119$	−810.978			−0.624724
$120$		0			0
$121$	−1167.89			−0.877450
$122$		0			0
$123$	1211.67	−	1220.88i	0.888235	−	0.894987i
$124$		0			0
$125$		−	404.151i		−	0.289187i
$126$		0			0
$127$			526.748i			0.368042i	0.982922	+	0.184021i	$0.0589114\pi$
$127$			526.748i			0.368042i	−0.982922	+	0.184021i	$0.941089\pi$
$128$		0			0
$129$	464.767	−	468.300i	0.317213	−	0.319624i
$130$		0			0
$131$	−363.833			−0.242658			−0.121329	−	0.992612i	$-0.538716\pi$
$131$	−363.833			−0.242658			−0.121329	+	0.992612i	$0.538716\pi$
$132$		0			0
$133$	−884.900			−0.576921
$134$		0			0
$135$	163.936	+	160.254i	0.104514	+	0.102166i
$136$		0			0
$137$			1419.52i			0.885236i	0.896710	+	0.442618i	$0.145950\pi$
$137$			1419.52i			0.885236i	−0.896710	+	0.442618i	$0.854050\pi$
$138$		0			0
$139$			341.346i			0.208292i	0.994562	+	0.104146i	$0.0332109\pi$
$139$			341.346i			0.208292i	−0.994562	+	0.104146i	$0.966789\pi$
$140$		0			0
$141$	−500.973	−	497.193i	−0.299216	−	0.296959i
$142$		0			0
$143$	56.6869			0.0331496
$144$		0			0
$145$	407.388			0.233322
$146$		0			0
$147$	−180.718	−	179.355i	−0.101397	−	0.100632i
$148$		0			0
$149$		−	2006.58i		−	1.10326i	−0.834090	−	0.551628i	$-0.814007\pi$
$149$		−	2006.58i		−	1.10326i	0.834090	−	0.551628i	$-0.185993\pi$
$150$		0			0
$151$			2571.21i			1.38571i	0.721078	+	0.692854i	$0.243647\pi$
$151$			2571.21i			1.38571i	−0.721078	+	0.692854i	$0.756353\pi$
$152$		0			0
$153$	3127.97	−	23.6869i	1.65282	−	0.0125162i
$154$		0			0
$155$	195.548			0.101334
$156$		0			0
$157$	859.216			0.436770			0.218385	−	0.975863i	$-0.429921\pi$
$157$	859.216			0.436770			0.218385	+	0.975863i	$0.429921\pi$
$158$		0			0
$159$	1447.28	−	1458.28i	0.721866	−	0.727354i
$160$		0			0
$161$		−	562.778i		−	0.275485i
$162$		0			0
$163$		−	1936.14i		−	0.930371i	−0.885213	−	0.465186i	$-0.845988\pi$
$163$		−	1936.14i		−	0.930371i	0.885213	−	0.465186i	$-0.154012\pi$
$164$		0			0
$165$	76.3887	−	76.9693i	0.0360415	−	0.0363155i
$166$		0			0
$167$	−3869.27			−1.79289			−0.896447	−	0.443151i	$-0.853860\pi$
$167$	−3869.27			−1.79289			−0.896447	+	0.443151i	$0.853860\pi$
$168$		0			0
$169$	−2177.30			−0.991033
$170$		0			0
$171$	3413.09	−	25.8460i	1.52635	−	0.0115584i
$172$		0			0
$173$		−	1342.38i		−	0.589940i	−0.955506	−	0.294970i	$-0.904690\pi$
$173$		−	1342.38i		−	0.589940i	0.955506	−	0.294970i	$-0.0953097\pi$
$174$		0			0
$175$		−	856.309i		−	0.369891i
$176$		0			0
$177$	1214.64	+	1205.48i	0.515808	+	0.511917i
$178$		0			0
$179$	2598.39			1.08499			0.542493	−	0.840060i	$-0.317480\pi$
$179$	2598.39			1.08499			0.542493	+	0.840060i	$0.317480\pi$
$180$		0			0
$181$	2220.03			0.911677			0.455838	−	0.890063i	$-0.349339\pi$
$181$	2220.03			0.911677			0.455838	+	0.890063i	$0.349339\pi$
$182$		0			0
$183$	−1343.50	−	1333.36i	−0.542701	−	0.538607i
$184$		0			0
$185$		−	713.784i		−	0.283667i
$186$		0			0
$187$		−	1479.64i		−	0.578621i
$188$		0			0
$189$	702.273	+	686.498i	0.270280	+	0.264208i
$190$		0			0
$191$	−2615.71			−0.990921			−0.495460	−	0.868630i	$-0.665001\pi$
$191$	−2615.71			−0.990921			−0.495460	+	0.868630i	$0.665001\pi$
$192$		0			0
$193$	−236.696			−0.0882783			−0.0441392	−	0.999025i	$-0.514054\pi$
$193$	−236.696			−0.0882783			−0.0441392	+	0.999025i	$0.514054\pi$
$194$		0			0
$195$	26.5472	−	26.7490i	0.00974916	−	0.00982327i
$196$		0			0
$197$		−	728.396i		−	0.263432i	−0.991287	−	0.131716i	$-0.957951\pi$
$197$		−	728.396i		−	0.263432i	0.991287	−	0.131716i	$-0.0420487\pi$
$198$		0			0
$199$		−	1533.63i		−	0.546313i	−0.961970	−	0.273156i	$-0.911932\pi$
$199$		−	1533.63i		−	0.546313i	0.961970	−	0.273156i	$-0.0880677\pi$
$200$		0			0
$201$	−3646.59	+	3674.31i	−1.27965	+	1.28938i
$202$		0			0
$203$	1745.18			0.603387
$204$		0			0
$205$	−540.923			−0.184291
$206$		0			0
$207$	16.4375	+	2170.65i	0.00551926	+	0.728844i
$208$		0			0
$209$		−	1614.51i		−	0.534346i
$210$		0			0
$211$		−	4846.33i		−	1.58121i	−0.612327	−	0.790605i	$-0.709766\pi$
$211$		−	4846.33i		−	1.58121i	0.612327	−	0.790605i	$-0.290234\pi$
$212$		0			0
$213$	1287.33	+	1277.62i	0.414113	+	0.410989i
$214$		0			0
$215$	−207.485			−0.0658155
$216$		0			0
$217$	837.692			0.262056
$218$		0			0
$219$	779.803	+	773.921i	0.240613	+	0.238798i
$220$		0			0
$221$		−	514.218i		−	0.156516i
$222$		0			0
$223$			4607.48i			1.38359i	0.722096	+	0.691793i	$0.243179\pi$
$223$			4607.48i			1.38359i	−0.722096	+	0.691793i	$0.756821\pi$
$224$		0			0
$225$	25.0109	+	3302.81i	0.00741064	+	0.978611i
$226$		0			0
$227$	−4313.18			−1.26113			−0.630564	−	0.776137i	$-0.717176\pi$
$227$	−4313.18			−1.26113			−0.630564	+	0.776137i	$0.717176\pi$
$228$		0			0
$229$	343.924			0.0992451			0.0496225	−	0.998768i	$-0.484198\pi$
$229$	343.924			0.0992451			0.0496225	+	0.998768i	$0.484198\pi$
$230$		0			0
$231$	327.236	−	329.723i	0.0932057	−	0.0939142i
$232$		0			0
$233$			86.4325i			0.0243021i	0.999926	+	0.0121510i	$0.00386789\pi$
$233$			86.4325i			0.0243021i	−0.999926	+	0.0121510i	$0.996132\pi$
$234$		0			0
$235$			221.960i			0.0616132i
$236$		0			0
$237$	−881.423	+	888.123i	−0.241580	+	0.243417i
$238$		0			0
$239$	−1035.70			−0.280308			−0.140154	−	0.990130i	$-0.544760\pi$
$239$	−1035.70			−0.280308			−0.140154	+	0.990130i	$0.544760\pi$
$240$		0			0
$241$	3035.68			0.811392			0.405696	−	0.914008i	$-0.367029\pi$
$241$	3035.68			0.811392			0.405696	+	0.914008i	$0.367029\pi$
$242$		0			0
$243$	−2728.74	−	2627.33i	−0.720366	−	0.693595i
$244$		0			0
$245$			80.0687i			0.0208792i
$246$		0			0
$247$		−	561.090i		−	0.144540i
$248$		0			0
$249$	2142.33	+	2126.17i	0.545240	+	0.541126i
$250$		0			0
$251$	−2264.04			−0.569342			−0.284671	−	0.958625i	$-0.591884\pi$
$251$	−2264.04			−0.569342			−0.284671	+	0.958625i	$0.591884\pi$
$252$		0			0
$253$	1026.80			0.255155
$254$		0			0
$255$	−698.204	−	692.937i	−0.171464	−	0.170170i
$256$		0			0
$257$			6365.21i			1.54495i	0.635048	+	0.772473i	$0.280980\pi$
$257$			6365.21i			1.54495i	−0.635048	+	0.772473i	$0.719020\pi$
$258$		0			0
$259$		−	3057.72i		−	0.733582i
$260$		0			0
$261$	−6731.21	+	50.9729i	−1.59637	+	0.0120887i
$262$		0			0
$263$	−4360.24			−1.02230			−0.511148	−	0.859492i	$-0.670780\pi$
$263$	−4360.24			−1.02230			−0.511148	+	0.859492i	$0.670780\pi$
$264$		0			0
$265$	−646.105			−0.149773
$266$		0			0
$267$	4728.86	−	4764.81i	1.08390	−	1.09214i
$268$		0			0
$269$		−	15.6838i		−	0.00355487i	−0.999998	−	0.00177743i	$-0.999434\pi$
$269$		−	15.6838i		−	0.00355487i	0.999998	−	0.00177743i	$-0.000565775\pi$
$270$		0			0
$271$		−	2503.89i		−	0.561255i	−0.959817	−	0.280628i	$-0.909457\pi$
$271$		−	2503.89i		−	0.561255i	0.959817	−	0.280628i	$-0.0905426\pi$
$272$		0			0
$273$	113.724	−	114.588i	0.0252120	−	0.0254036i
$274$		0			0
$275$	1562.35			0.342594
$276$		0			0
$277$	−5570.51			−1.20830			−0.604151	−	0.796870i	$-0.706488\pi$
$277$	−5570.51			−1.20830			−0.604151	+	0.796870i	$0.706488\pi$
$278$		0			0
$279$	−3231.00	+	24.4672i	−0.693316	+	0.00525021i
$280$		0			0
$281$		−	6987.62i		−	1.48344i	−0.670710	−	0.741720i	$-0.734011\pi$
$281$		−	6987.62i		−	1.48344i	0.670710	−	0.741720i	$-0.265989\pi$
$282$		0			0
$283$		−	2067.00i		−	0.434172i	−0.976153	−	0.217086i	$-0.930345\pi$
$283$		−	2067.00i		−	0.434172i	0.976153	−	0.217086i	$-0.0696552\pi$
$284$		0			0
$285$	−761.847	−	756.099i	−0.158344	−	0.157149i
$286$		0			0
$287$	−2317.22			−0.476589
$288$		0			0
$289$	−8509.14			−1.73196
$290$		0			0
$291$	4337.13	+	4304.41i	0.873702	+	0.867111i
$292$		0			0
$293$		−	6164.53i		−	1.22913i	−0.788865	−	0.614566i	$-0.789331\pi$
$293$		−	6164.53i		−	1.22913i	0.788865	−	0.614566i	$-0.210669\pi$
$294$		0			0
$295$		−	538.158i		−	0.106213i
$296$		0			0
$297$	−1252.53	+	1281.31i	−0.244710	+	0.250334i
$298$		0			0
$299$	356.842			0.0690190
$300$		0			0
$301$	−888.827			−0.170203
$302$		0			0
$303$	1723.06	−	1736.16i	0.326690	−	0.329174i
$304$		0			0
$305$			595.250i			0.111751i
$306$		0			0
$307$			4343.59i			0.807497i	0.914870	+	0.403749i	$0.132293\pi$
$307$			4343.59i			0.807497i	−0.914870	+	0.403749i	$0.867707\pi$
$308$		0			0
$309$	−5282.86	+	5323.02i	−0.972594	+	0.979987i
$310$		0			0
$311$	−6682.90			−1.21850			−0.609249	−	0.792979i	$-0.708529\pi$
$311$	−6682.90			−1.21850			−0.609249	+	0.792979i	$0.708529\pi$
$312$		0			0
$313$	−7508.56			−1.35594			−0.677969	−	0.735090i	$-0.737140\pi$
$313$	−7508.56			−1.35594			−0.677969	+	0.735090i	$0.737140\pi$
$314$		0			0
$315$	−2.33863	−	308.827i	−0.000418308	−	0.0552396i
$316$		0			0
$317$			5241.90i			0.928753i	0.885638	+	0.464376i	$0.153722\pi$
$317$			5241.90i			0.928753i	−0.885638	+	0.464376i	$0.846278\pi$
$318$		0			0
$319$			3184.11i			0.558859i
$320$		0			0
$321$	2082.07	+	2066.37i	0.362025	+	0.359294i
$322$		0			0
$323$	−14645.6			−2.52292
$324$		0			0
$325$	542.961			0.0926710
$326$		0			0
$327$	6970.43	+	6917.85i	1.17879	+	1.16990i
$328$		0			0
$329$			950.839i			0.159336i
$330$		0			0
$331$		−	4927.68i		−	0.818278i	−0.912472	−	0.409139i	$-0.865829\pi$
$331$		−	4927.68i		−	0.818278i	0.912472	−	0.409139i	$-0.134171\pi$
$332$		0			0
$333$	89.3094	+	11793.7i	0.0146971	+	1.94082i
$334$		0			0
$335$	1627.93			0.265503
$336$		0			0
$337$	−2987.62			−0.482926			−0.241463	−	0.970410i	$-0.577627\pi$
$337$	−2987.62			−0.482926			−0.241463	+	0.970410i	$0.577627\pi$
$338$		0			0
$339$	2122.05	−	2138.18i	0.339982	−	0.342566i
$340$		0			0
$341$			1528.38i			0.242717i
$342$		0			0
$343$			343.000i			0.0539949i
$344$		0			0
$345$	480.864	−	484.519i	0.0750401	−	0.0756105i
$346$		0			0
$347$	3228.35			0.499443			0.249722	−	0.968318i	$-0.419661\pi$
$347$	3228.35			0.499443			0.249722	+	0.968318i	$0.419661\pi$
$348$		0			0
$349$	9278.67			1.42314			0.711570	−	0.702615i	$-0.247984\pi$
$349$	9278.67			1.42314			0.711570	+	0.702615i	$0.247984\pi$
$350$		0			0
$351$	−435.289	+	445.292i	−0.0661937	+	0.0677148i
$352$		0			0
$353$		−	3101.21i		−	0.467595i	−0.972285	−	0.233797i	$-0.924885\pi$
$353$		−	3101.21i		−	0.467595i	0.972285	−	0.233797i	$-0.0751152\pi$
$354$		0			0
$355$		−	570.362i		−	0.0852723i
$356$		0			0
$357$	−2990.98	−	2968.42i	−0.443416	−	0.440071i
$358$		0			0
$359$	−13340.9			−1.96129			−0.980646	−	0.195788i	$-0.937274\pi$
$359$	−13340.9			−1.96129			−0.980646	+	0.195788i	$0.937274\pi$
$360$		0			0
$361$	−9121.55			−1.32987
$362$		0			0
$363$	−4307.30	−	4274.81i	−0.622795	−	0.618097i
$364$		0			0
$365$		−	345.499i		−	0.0495459i
$366$		0			0
$367$			3668.79i			0.521824i	0.965363	+	0.260912i	$0.0840233\pi$
$367$			3668.79i			0.521824i	−0.965363	+	0.260912i	$0.915977\pi$
$368$		0			0
$369$	8937.59	−	67.6809i	1.26090	−	0.00954832i
$370$		0			0
$371$	−2767.80			−0.387323
$372$		0			0
$373$	−33.6447			−0.00467040			−0.00233520	−	0.999997i	$-0.500743\pi$
$373$	−33.6447			−0.00467040			−0.00233520	+	0.999997i	$0.500743\pi$
$374$		0			0
$375$	1479.31	−	1490.56i	0.203710	−	0.205259i
$376$		0			0
$377$			1106.57i			0.151170i
$378$		0			0
$379$			4344.58i			0.588828i	0.955678	+	0.294414i	$0.0951245\pi$
$379$			4344.58i			0.588828i	−0.955678	+	0.294414i	$0.904876\pi$
$380$		0			0
$381$	−1928.05	+	1942.71i	−0.259258	+	0.261228i
$382$		0			0
$383$	7664.00			1.02249			0.511243	−	0.859436i	$-0.329185\pi$
$383$	7664.00			1.02249			0.511243	+	0.859436i	$0.329185\pi$
$384$		0			0
$385$	−146.087			−0.0193384
$386$		0			0
$387$	3428.23	−	25.9607i	0.450302	−	0.00340996i
$388$		0			0
$389$			9812.91i			1.27901i	0.768788	+	0.639504i	$0.220860\pi$
$389$			9812.91i			1.27901i	−0.768788	+	0.639504i	$0.779140\pi$
$390$		0			0
$391$		−	9314.29i		−	1.20472i
$392$		0			0
$393$	−1341.86	−	1331.74i	−0.172234	−	0.170934i
$394$		0			0
$395$	393.491			0.0501232
$396$		0			0
$397$	−10638.6			−1.34493			−0.672466	−	0.740128i	$-0.734765\pi$
$397$	−10638.6			−1.34493			−0.672466	+	0.740128i	$0.734765\pi$
$398$		0			0
$399$	−3263.62	−	3238.99i	−0.409487	−	0.406397i
$400$		0			0
$401$		−	72.6321i		−	0.00904507i	−0.999990	−	0.00452254i	$-0.998560\pi$
$401$		−	72.6321i		−	0.00904507i	0.999990	−	0.00452254i	$-0.00143957\pi$
$402$		0			0
$403$			531.157i			0.0656546i
$404$		0			0
$405$	18.0404	+	1191.09i	0.00221341	+	0.146138i
$406$		0			0
$407$	5578.87			0.679446
$408$		0			0
$409$	−2811.49			−0.339900			−0.169950	−	0.985453i	$-0.554361\pi$
$409$	−2811.49			−0.339900			−0.169950	+	0.985453i	$0.554361\pi$
$410$		0			0
$411$	−5195.85	+	5235.34i	−0.623582	+	0.628322i
$412$		0			0
$413$		−	2305.37i		−	0.274673i
$414$		0			0
$415$		−	949.179i		−	0.112273i
$416$		0			0
$417$	−1249.43	+	1258.92i	−0.146726	+	0.147841i
$418$		0			0
$419$	−2892.92			−0.337299			−0.168650	−	0.985676i	$-0.553941\pi$
$419$	−2892.92			−0.337299			−0.168650	+	0.985676i	$0.553941\pi$
$420$		0			0
$421$	11794.5			1.36539			0.682696	−	0.730703i	$-0.260807\pi$
$421$	11794.5			1.36539			0.682696	+	0.730703i	$0.260807\pi$
$422$		0			0
$423$	−27.7719	−	3667.42i	−0.00319224	−	0.421551i
$424$		0			0
$425$		−	14172.4i		−	1.61756i
$426$		0			0
$427$			2549.94i			0.288994i
$428$		0			0
$429$	209.068	+	207.491i	0.0235289	+	0.0233514i
$430$		0			0
$431$	1662.13			0.185759			0.0928795	−	0.995677i	$-0.470393\pi$
$431$	1662.13			0.185759			0.0928795	+	0.995677i	$0.470393\pi$
$432$		0			0
$433$	11414.8			1.26688			0.633440	−	0.773792i	$-0.281642\pi$
$433$	11414.8			1.26688			0.633440	+	0.773792i	$0.281642\pi$
$434$		0			0
$435$	1502.50	+	1491.16i	0.165607	+	0.164358i
$436$		0			0
$437$		−	10163.3i		−	1.11253i
$438$		0			0
$439$		−	4118.63i		−	0.447771i	−0.974615	−	0.223885i	$-0.928126\pi$
$439$		−	4118.63i		−	0.447771i	0.974615	−	0.223885i	$-0.0718741\pi$
$440$		0			0
$441$	−10.0183	−	1322.96i	−0.00108177	−	0.142853i
$442$		0			0
$443$	12153.2			1.30342			0.651711	−	0.758467i	$-0.274052\pi$
$443$	12153.2			1.30342			0.651711	+	0.758467i	$0.274052\pi$
$444$		0			0
$445$	−2111.09			−0.224888
$446$		0			0
$447$	7344.67	−	7400.49i	0.777161	−	0.783068i
$448$		0			0
$449$		−	1682.19i		−	0.176810i	−0.996085	−	0.0884049i	$-0.971823\pi$
$449$		−	1682.19i		−	0.176810i	0.996085	−	0.0884049i	$-0.0281769\pi$
$450$		0			0
$451$		−	4227.81i		−	0.441418i
$452$		0			0
$453$	−9411.39	+	9482.93i	−0.976127	+	0.983547i
$454$		0			0
$455$	−50.7693			−0.00523099
$456$		0			0
$457$	2659.67			0.272241			0.136121	−	0.990692i	$-0.456537\pi$
$457$	2659.67			0.272241			0.136121	+	0.990692i	$0.456537\pi$
$458$		0			0
$459$	11623.0	+	11361.9i	1.18195	+	1.15540i
$460$		0			0
$461$			3117.33i			0.314942i	0.987524	+	0.157471i	$0.0503341\pi$
$461$			3117.33i			0.314942i	−0.987524	+	0.157471i	$0.949666\pi$
$462$		0			0
$463$		−	13481.9i		−	1.35325i	−0.736326	−	0.676627i	$-0.763441\pi$
$463$		−	13481.9i		−	1.35325i	0.736326	−	0.676627i	$-0.236559\pi$
$464$		0			0
$465$	721.204	+	715.763i	0.0719248	+	0.0713822i
$466$		0			0
$467$	−12649.6			−1.25343			−0.626717	−	0.779247i	$-0.715602\pi$
$467$	−12649.6			−1.25343			−0.626717	+	0.779247i	$0.715602\pi$
$468$		0			0
$469$	6973.78			0.686609
$470$		0			0
$471$	3168.89	+	3144.99i	0.310010	+	0.307672i
$472$		0			0
$473$		−	1621.68i		−	0.157643i
$474$		0			0
$475$		−	15464.2i		−	1.49378i
$476$		0			0
$477$	10675.5	−	80.8413i	1.02473	−	0.00775989i
$478$		0			0
$479$	−15116.4			−1.44194			−0.720968	−	0.692969i	$-0.756302\pi$
$479$	−15116.4			−1.44194			−0.720968	+	0.692969i	$0.756302\pi$
$480$		0			0
$481$	1938.82			0.183789
$482$		0			0
$483$	2059.93	−	2075.59i	0.194059	−	0.195534i
$484$		0			0
$485$		−	1921.61i		−	0.179909i
$486$		0			0
$487$			8537.59i			0.794405i	0.917731	+	0.397202i	$0.130019\pi$
$487$			8537.59i			0.794405i	−0.917731	+	0.397202i	$0.869981\pi$
$488$		0			0
$489$	7086.86	−	7140.73i	0.655376	−	0.660358i
$490$		0			0
$491$	−7424.84			−0.682441			−0.341220	−	0.939983i	$-0.610840\pi$
$491$	−7424.84			−0.682441			−0.341220	+	0.939983i	$0.610840\pi$
$492$		0			0
$493$	28883.7			2.63865
$494$		0			0
$495$	563.461	−	4.26687i	0.0511630	−	0.000387438i
$496$		0			0
$497$		−	2443.33i		−	0.220520i
$498$		0			0
$499$			16808.7i			1.50794i	0.656908	+	0.753970i	$0.271864\pi$
$499$			16808.7i			1.50794i	−0.656908	+	0.753970i	$0.728136\pi$
$500$		0			0
$501$	−14270.3	−	14162.7i	−1.27256	−	1.26296i
$502$		0			0
$503$	−2365.94			−0.209725			−0.104863	−	0.994487i	$-0.533440\pi$
$503$	−2365.94			−0.209725			−0.104863	+	0.994487i	$0.533440\pi$
$504$		0			0
$505$	−769.220			−0.0677819
$506$		0			0
$507$	−8030.14	−	7969.56i	−0.703414	−	0.698108i
$508$		0			0
$509$			2691.97i			0.234419i	0.993107	+	0.117210i	$0.0373949\pi$
$509$			2691.97i			0.234419i	−0.993107	+	0.117210i	$0.962605\pi$
$510$		0			0
$511$		−	1480.06i		−	0.128129i
$512$		0			0
$513$	12682.5	+	12397.6i	1.09151	+	1.06699i
$514$		0			0
$515$	2358.41			0.201794
$516$		0			0
$517$	−1734.82			−0.147577
$518$		0			0
$519$	4913.53	−	4950.88i	0.415568	−	0.418727i
$520$		0			0
$521$		−	10208.0i		−	0.858390i	−0.903212	−	0.429195i	$-0.858797\pi$
$521$		−	10208.0i		−	0.858390i	0.903212	−	0.429195i	$-0.141203\pi$
$522$		0			0
$523$			12291.9i			1.02770i	0.857880	+	0.513850i	$0.171781\pi$
$523$			12291.9i			1.02770i	−0.857880	+	0.513850i	$0.828219\pi$
$524$		0			0
$525$	3134.35	−	3158.17i	0.260560	−	0.262541i
$526$		0			0
$527$	13864.3			1.14599
$528$		0			0
$529$	−5703.35			−0.468755
$530$		0			0
$531$	67.3349	+	8891.90i	0.00550299	+	0.726696i
$532$		0			0
$533$		−	1469.28i		−	0.119403i
$534$		0			0
$535$		−	922.482i		−	0.0745465i
$536$		0			0
$537$	9583.16	+	9510.86i	0.770100	+	0.764291i
$538$		0			0
$539$	−625.810			−0.0500102
$540$		0			0
$541$	2973.08			0.236271			0.118135	−	0.992997i	$-0.462308\pi$
$541$	2973.08			0.236271			0.118135	+	0.992997i	$0.462308\pi$
$542$		0			0
$543$	8187.74	+	8125.97i	0.647089	+	0.642207i
$544$		0			0
$545$		−	3088.31i		−	0.242732i
$546$		0			0
$547$			3140.53i			0.245484i	0.992439	+	0.122742i	$0.0391687\pi$
$547$			3140.53i			0.245484i	−0.992439	+	0.122742i	$0.960831\pi$
$548$		0			0
$549$	−74.4783	−	9835.22i	−0.00578990	−	0.764584i
$550$		0			0
$551$	31516.5			2.43675
$552$		0			0
$553$	1685.64			0.129622
$554$		0			0
$555$	2612.66	−	2632.52i	0.199822	−	0.201341i
$556$		0			0
$557$			9414.66i			0.716180i	0.933687	+	0.358090i	$0.116572\pi$
$557$			9414.66i			0.716180i	−0.933687	+	0.358090i	$0.883428\pi$
$558$		0			0
$559$		−	563.580i		−	0.0426420i
$560$		0			0
$561$	5415.93	−	5457.10i	0.407595	−	0.410693i
$562$		0			0
$563$	7650.56			0.572705			0.286352	−	0.958124i	$-0.407557\pi$
$563$	7650.56			0.572705			0.286352	+	0.958124i	$0.407557\pi$
$564$		0			0
$565$	−947.339			−0.0705396
$566$		0			0
$567$	77.2817	+	5102.41i	0.00572403	+	0.377921i
$568$		0			0
$569$			7371.22i			0.543089i	0.962426	+	0.271545i	$0.0875344\pi$
$569$			7371.22i			0.543089i	−0.962426	+	0.271545i	$0.912466\pi$
$570$		0			0
$571$			21587.2i			1.58213i	0.611730	+	0.791066i	$0.290474\pi$
$571$			21587.2i			1.58213i	−0.611730	+	0.791066i	$0.709526\pi$
$572$		0			0
$573$	−9647.04	−	9574.26i	−0.703335	−	0.698029i
$574$		0			0
$575$	9834.94			0.713296
$576$		0			0
$577$	−16742.1			−1.20794			−0.603970	−	0.797007i	$-0.706415\pi$
$577$	−16742.1			−1.20794			−0.603970	+	0.797007i	$0.706415\pi$
$578$		0			0
$579$	−872.961	−	866.376i	−0.0626581	−	0.0621854i
$580$		0			0
$581$		−	4066.11i		−	0.290346i
$582$		0			0
$583$		−	5049.89i		−	0.358740i
$584$		0			0
$585$	195.819	−	1.48286i	0.0138395	−	0.000104801i
$586$		0			0
$587$	−22992.2			−1.61668			−0.808339	−	0.588718i	$-0.799633\pi$
$587$	−22992.2			−1.61668			−0.808339	+	0.588718i	$0.799633\pi$
$588$		0			0
$589$	15128.0			1.05830
$590$		0			0
$591$	2666.15	−	2686.41i	0.185568	−	0.186978i
$592$		0			0
$593$			18637.7i			1.29066i	0.763906	+	0.645328i	$0.223279\pi$
$593$			18637.7i			1.29066i	−0.763906	+	0.645328i	$0.776721\pi$
$594$		0			0
$595$			1325.18i			0.0913062i
$596$		0			0
$597$	5613.55	−	5656.22i	0.384836	−	0.387761i
$598$		0			0
$599$	18988.8			1.29526			0.647630	−	0.761955i	$-0.275760\pi$
$599$	18988.8			1.29526			0.647630	+	0.761955i	$0.275760\pi$
$600$		0			0
$601$	−17998.7			−1.22160			−0.610801	−	0.791784i	$-0.709152\pi$
$601$	−17998.7			−1.22160			−0.610801	+	0.791784i	$0.709152\pi$
$602$		0			0
$603$	−26898.1	+	203.689i	−1.81654	+	0.0137560i
$604$		0			0
$605$			1908.39i			0.128243i
$606$		0			0
$607$			21235.7i			1.41999i	0.704208	+	0.709994i	$0.251302\pi$
$607$			21235.7i			1.41999i	−0.704208	+	0.709994i	$0.748698\pi$
$608$		0			0
$609$	6436.43	+	6387.87i	0.428271	+	0.425040i
$610$		0			0
$611$	−602.900			−0.0399194
$612$		0			0
$613$	−2865.49			−0.188803			−0.0944013	−	0.995534i	$-0.530094\pi$
$613$	−2865.49			−0.188803			−0.0944013	+	0.995534i	$0.530094\pi$
$614$		0			0
$615$	−1994.99	−	1979.94i	−0.130806	−	0.129819i
$616$		0			0
$617$		−	20566.3i		−	1.34192i	−0.741492	−	0.670962i	$-0.765881\pi$
$617$		−	20566.3i		−	1.34192i	0.741492	−	0.670962i	$-0.234119\pi$
$618$		0			0
$619$			20485.5i			1.33018i	0.746764	+	0.665090i	$0.231607\pi$
$619$			20485.5i			1.33018i	−0.746764	+	0.665090i	$0.768393\pi$
$620$		0			0
$621$	−7884.61	+	8065.79i	−0.509498	+	0.521206i
$622$		0			0
$623$	−9043.54			−0.581576
$624$		0			0
$625$	14630.8			0.936373
$626$		0			0
$627$	5909.60	−	5954.52i	0.376406	−	0.379268i
$628$		0			0
$629$		−	50607.1i		−	3.20801i
$630$		0			0
$631$			14167.4i			0.893810i	0.894581	+	0.446905i	$0.147474\pi$
$631$			14167.4i			0.893810i	−0.894581	+	0.446905i	$0.852526\pi$
$632$		0			0
$633$	17739.0	−	17873.9i	1.11384	−	1.12231i
$634$		0			0
$635$	860.734			0.0537909
$636$		0			0
$637$	−217.487			−0.0135277
$638$		0			0
$639$	71.3643	+	9423.99i	0.00441804	+	0.583423i
$640$		0			0
$641$			25353.5i			1.56225i	0.624372	+	0.781127i	$0.285355\pi$
$641$			25353.5i			1.56225i	−0.624372	+	0.781127i	$0.714645\pi$
$642$		0			0
$643$		−	9662.86i		−	0.592638i	−0.955089	−	0.296319i	$-0.904241\pi$
$643$		−	9662.86i		−	0.592638i	0.955089	−	0.296319i	$-0.0957591\pi$
$644$		0			0
$645$	−765.228	−	759.455i	−0.0467144	−	0.0463620i
$646$		0			0
$647$	11168.8			0.678656			0.339328	−	0.940668i	$-0.389800\pi$
$647$	11168.8			0.678656			0.339328	+	0.940668i	$0.389800\pi$
$648$		0			0
$649$	4206.19			0.254403
$650$		0			0
$651$	3089.51	+	3066.20i	0.186002	+	0.184599i
$652$		0			0
$653$		−	11125.4i		−	0.666725i	−0.942799	−	0.333363i	$-0.891817\pi$
$653$		−	11125.4i		−	0.666725i	0.942799	−	0.333363i	$-0.108183\pi$
$654$		0			0
$655$			594.523i			0.0354656i
$656$		0			0
$657$	43.2292	+	5708.62i	0.00256702	+	0.338987i
$658$		0			0
$659$	30142.5			1.78177			0.890883	−	0.454234i	$-0.150087\pi$
$659$	30142.5			1.78177			0.890883	+	0.454234i	$0.150087\pi$
$660$		0			0
$661$	−4282.37			−0.251989			−0.125995	−	0.992031i	$-0.540212\pi$
$661$	−4282.37			−0.251989			−0.125995	+	0.992031i	$0.540212\pi$
$662$		0			0
$663$	1882.19	−	1896.50i	0.110254	−	0.111092i
$664$		0			0
$665$			1445.97i			0.0843195i
$666$		0			0
$667$			20043.8i			1.16357i
$668$		0			0
$669$	−16864.7	+	16992.9i	−0.974631	+	0.982040i
$670$		0			0
$671$	−4652.42			−0.267667
$672$		0			0
$673$	4880.18			0.279520			0.139760	−	0.990185i	$-0.455367\pi$
$673$	4880.18			0.279520			0.139760	+	0.990185i	$0.455367\pi$
$674$		0			0
$675$	−11997.0	+	12272.7i	−0.684097	+	0.699818i
$676$		0			0
$677$		−	32138.5i		−	1.82449i	−0.409641	−	0.912247i	$-0.634346\pi$
$677$		−	32138.5i		−	1.82449i	0.409641	−	0.912247i	$-0.365654\pi$
$678$		0			0
$679$		−	8231.82i		−	0.465255i
$680$		0			0
$681$	−15907.5	−	15787.5i	−0.895122	−	0.888369i
$682$		0			0
$683$	23535.1			1.31852			0.659258	−	0.751917i	$-0.270870\pi$
$683$	23535.1			1.31852			0.659258	+	0.751917i	$0.270870\pi$
$684$		0			0
$685$	2319.57			0.129381
$686$		0			0
$687$	1268.43	+	1258.86i	0.0704421	+	0.0699107i
$688$		0			0
$689$		−	1754.98i		−	0.0970384i
$690$		0			0
$691$			20492.5i			1.12818i	0.825714	+	0.564089i	$0.190773\pi$
$691$			20492.5i			1.12818i	−0.825714	+	0.564089i	$0.809227\pi$
$692$		0			0
$693$	2413.77	−	18.2785i	0.132311	−	0.00100194i
$694$		0			0
$695$	557.777			0.0304427
$696$		0			0
$697$	−38351.3			−2.08416
$698$		0			0
$699$	−316.368	+	318.773i	−0.0171190	+	0.0172491i
$700$		0			0
$701$			14101.2i			0.759763i	0.925035	+	0.379881i	$0.124035\pi$
$701$			14101.2i			0.759763i	−0.925035	+	0.379881i	$0.875965\pi$
$702$		0			0
$703$		−	55220.0i		−	2.96253i
$704$		0			0
$705$	−812.441	+	818.617i	−0.0434018	+	0.0437318i
$706$		0			0
$707$	−3295.20			−0.175288
$708$		0			0
$709$	−17337.2			−0.918352			−0.459176	−	0.888345i	$-0.651855\pi$
$709$	−17337.2			−0.918352			−0.459176	+	0.888345i	$0.651855\pi$
$710$		0			0
$711$	−6501.59	+	49.2340i	−0.342937	+	0.00259693i
$712$		0			0
$713$			9621.11i			0.505349i
$714$		0			0
$715$		−	92.6294i		−	0.00484496i
$716$		0			0
$717$	−3819.78	−	3790.96i	−0.198957	−	0.197456i
$718$		0			0
$719$	25469.6			1.32108			0.660539	−	0.750791i	$-0.270328\pi$
$719$	25469.6			1.32108			0.660539	+	0.750791i	$0.270328\pi$
$720$		0			0
$721$	10103.0			0.521853
$722$		0			0
$723$	11196.0	+	11111.5i	0.575909	+	0.571564i
$724$		0			0
$725$			30498.2i			1.56231i
$726$		0			0
$727$		−	34982.4i		−	1.78463i	−0.451412	−	0.892316i	$-0.649079\pi$
$727$		−	34982.4i		−	1.78463i	0.451412	−	0.892316i	$-0.350921\pi$
$728$		0			0
$729$	−447.108	−	19677.9i	−0.0227155	−	0.999742i
$730$		0			0
$731$	−14710.6			−0.744310
$732$		0			0
$733$	37078.1			1.86836			0.934182	−	0.356797i	$-0.116131\pi$
$733$	37078.1			1.86836			0.934182	+	0.356797i	$0.116131\pi$
$734$		0			0
$735$	−293.075	+	295.303i	−0.0147078	+	0.0148196i
$736$		0			0
$737$			12723.8i			0.635939i
$738$		0			0
$739$			4104.04i			0.204289i	0.994770	+	0.102144i	$0.0325704\pi$
$739$			4104.04i			0.204289i	−0.994770	+	0.102144i	$0.967430\pi$
$740$		0			0
$741$	2053.75	−	2069.37i	0.101817	−	0.102591i
$742$		0			0
$743$	−14476.2			−0.714779			−0.357389	−	0.933955i	$-0.616333\pi$
$743$	−14476.2			−0.714779			−0.357389	+	0.933955i	$0.616333\pi$
$744$		0			0
$745$	−3278.86			−0.161246
$746$		0			0
$747$	118.762	+	15683.1i	0.00581698	+	0.768160i
$748$		0			0
$749$		−	3951.75i		−	0.192782i
$750$		0			0
$751$			34702.8i			1.68618i	0.537771	+	0.843091i	$0.319267\pi$
$751$			34702.8i			1.68618i	−0.537771	+	0.843091i	$0.680733\pi$
$752$		0			0
$753$	−8350.05	−	8287.06i	−0.404107	−	0.401059i
$754$		0			0
$755$	4201.50			0.202527
$756$		0			0
$757$	29592.9			1.42083			0.710417	−	0.703781i	$-0.248506\pi$
$757$	29592.9			1.42083			0.710417	+	0.703781i	$0.248506\pi$
$758$		0			0
$759$	3786.96	+	3758.39i	0.181104	+	0.179738i
$760$		0			0
$761$			15694.7i			0.747613i	0.927507	+	0.373807i	$0.121948\pi$
$761$			15694.7i			0.747613i	−0.927507	+	0.373807i	$0.878052\pi$
$762$		0			0
$763$		−	13229.8i		−	0.627720i
$764$		0			0
$765$	−38.7057	−	5111.27i	−0.00182929	−	0.241566i
$766$		0			0
$767$	1461.77			0.0688155
$768$		0			0
$769$	22310.0			1.04619			0.523094	−	0.852275i	$-0.324778\pi$
$769$	22310.0			1.04619			0.523094	+	0.852275i	$0.324778\pi$
$770$		0			0
$771$	−23298.6	+	23475.7i	−1.08830	+	1.09657i
$772$		0			0
$773$			18819.8i			0.875682i	0.899052	+	0.437841i	$0.144257\pi$
$773$			18819.8i			0.875682i	−0.899052	+	0.437841i	$0.855743\pi$
$774$		0			0
$775$			14639.2i			0.678526i
$776$		0			0
$777$	11192.2	−	11277.3i	0.516753	−	0.520681i
$778$		0			0
$779$	−41847.1			−1.92468
$780$		0			0
$781$	4457.89			0.204246
$782$		0			0
$783$	−25012.1	−	24450.2i	−1.14158	−	1.11594i
$784$		0			0
$785$		−	1404.01i		−	0.0638359i
$786$		0			0
$787$		−	34626.0i		−	1.56834i	−0.620545	−	0.784171i	$-0.713089\pi$
$787$		−	34626.0i		−	1.56834i	0.620545	−	0.784171i	$-0.286911\pi$
$788$		0			0
$789$	−16081.1	−	15959.8i	−0.725605	−	0.720131i
$790$		0			0
$791$	−4058.23			−0.182420
$792$		0			0
$793$	−1616.85			−0.0724034
$794$		0			0
$795$	−2382.91	−	2364.93i	−0.106306	−	0.105504i
$796$		0			0
$797$			21365.3i			0.949558i	0.880105	+	0.474779i	$0.157472\pi$
$797$			21365.3i			0.949558i	−0.880105	+	0.474779i	$0.842528\pi$
$798$		0			0
$799$			15736.9i			0.696786i
$800$		0			0
$801$	34881.2	−	264.142i	1.53866	−	0.0116517i
$802$		0			0
$803$	2700.39			0.118673
$804$		0			0
$805$	−919.610			−0.0402633
$806$		0			0
$807$	57.4075	−	57.8438i	0.00250414	−	0.00252317i
$808$		0			0
$809$			7749.36i			0.336777i	0.985721	+	0.168389i	$0.0538564\pi$
$809$			7749.36i			0.336777i	−0.985721	+	0.168389i	$0.946144\pi$
$810$		0			0
$811$		−	22749.9i		−	0.985025i	−0.870305	−	0.492513i	$-0.836079\pi$
$811$		−	22749.9i		−	0.985025i	0.870305	−	0.492513i	$-0.163921\pi$
$812$		0			0
$813$	9164.96	−	9234.63i	0.395362	−	0.398367i
$814$		0			0
$815$	−3163.77			−0.135978
$816$		0			0
$817$	−16051.5			−0.687356
$818$		0			0
$819$	838.853	−	6.35231i	0.0357898	−	0.000271023i
$820$		0			0
$821$			41536.1i			1.76568i	0.469677	+	0.882838i	$0.344370\pi$
$821$			41536.1i			1.76568i	−0.469677	+	0.882838i	$0.655630\pi$
$822$		0			0
$823$			8365.48i			0.354316i	0.984182	+	0.177158i	$0.0566904\pi$
$823$			8365.48i			0.354316i	−0.984182	+	0.177158i	$0.943310\pi$
$824$		0			0
$825$	5762.14	+	5718.67i	0.243166	+	0.241331i
$826$		0			0
$827$	−3152.02			−0.132535			−0.0662675	−	0.997802i	$-0.521109\pi$
$827$	−3152.02			−0.132535			−0.0662675	+	0.997802i	$0.521109\pi$
$828$		0			0
$829$	10968.4			0.459529			0.229765	−	0.973246i	$-0.426204\pi$
$829$	10968.4			0.459529			0.229765	+	0.973246i	$0.426204\pi$
$830$		0			0
$831$	−20544.7	−	20389.7i	−0.857627	−	0.851157i
$832$		0			0
$833$			5676.84i			0.236124i
$834$		0			0
$835$			6322.60i			0.262039i
$836$		0			0
$837$	−12005.9	−	11736.2i	−0.495800	−	0.484662i
$838$		0			0
$839$	40012.2			1.64645			0.823226	−	0.567714i	$-0.192172\pi$
$839$	40012.2			1.64645			0.823226	+	0.567714i	$0.192172\pi$
$840$		0			0
$841$	−37767.1			−1.54853
$842$		0			0
$843$	25576.8	−	25771.2i	1.04497	−	1.05291i
$844$		0			0
$845$			3557.83i			0.144844i
$846$		0			0
$847$			8175.20i			0.331645i
$848$		0			0
$849$	7565.85	−	7623.36i	0.305841	−	0.308166i
$850$		0			0
$851$	35118.8			1.41464
$852$		0			0
$853$	−43396.7			−1.74194			−0.870970	−	0.491336i	$-0.836509\pi$
$853$	−43396.7			−1.74194			−0.870970	+	0.491336i	$0.836509\pi$
$854$		0			0
$855$	−42.2338	−	5577.17i	−0.00168931	−	0.223082i
$856$		0			0
$857$			1435.44i			0.0572153i	0.999591	+	0.0286077i	$0.00910734\pi$
$857$			1435.44i			0.0572153i	−0.999591	+	0.0286077i	$0.990893\pi$
$858$		0			0
$859$		−	9540.54i		−	0.378951i	−0.981885	−	0.189476i	$-0.939321\pi$
$859$		−	9540.54i		−	0.378951i	0.981885	−	0.189476i	$-0.0606788\pi$
$860$		0			0
$861$	−8546.18	−	8481.71i	−0.338273	−	0.335721i
$862$		0			0
$863$	−9834.33			−0.387908			−0.193954	−	0.981011i	$-0.562131\pi$
$863$	−9834.33			−0.387908			−0.193954	+	0.981011i	$0.562131\pi$
$864$		0			0
$865$	−2193.53			−0.0862223
$866$		0			0
$867$	−31382.7	−	31146.0i	−1.22931	−	1.22004i
$868$		0			0
$869$			3075.49i			0.120056i
$870$		0			0
$871$			4421.88i			0.172020i
$872$		0			0
$873$	240.433	+	31750.4i	0.00932124	+	1.23091i
$874$		0			0
$875$	−2829.05			−0.109302
$876$		0			0
$877$	4647.38			0.178941			0.0894703	−	0.995989i	$-0.471483\pi$
$877$	4647.38			0.178941			0.0894703	+	0.995989i	$0.471483\pi$
$878$		0			0
$879$	22564.0	−	22735.5i	0.865831	−	0.872412i
$880$		0			0
$881$		−	34914.2i		−	1.33518i	−0.744531	−	0.667588i	$-0.767327\pi$
$881$		−	34914.2i		−	1.33518i	0.744531	−	0.667588i	$-0.232673\pi$
$882$		0			0
$883$		−	32912.2i		−	1.25434i	−0.778881	−	0.627171i	$-0.784213\pi$
$883$		−	32912.2i		−	1.25434i	0.778881	−	0.627171i	$-0.215787\pi$
$884$		0			0
$885$	1969.82	−	1984.79i	0.0748189	−	0.0753876i
$886$		0			0
$887$	13103.7			0.496029			0.248015	−	0.968756i	$-0.420222\pi$
$887$	13103.7			0.496029			0.248015	+	0.968756i	$0.420222\pi$
$888$		0			0
$889$	3687.23			0.139107
$890$		0			0
$891$	−9309.45	+	141.002i	−0.350032	+	0.00530161i
$892$		0			0
$893$			17171.4i			0.643469i
$894$		0			0
$895$		−	4245.91i		−	0.158575i
$896$		0			0
$897$	1316.07	+	1306.15i	0.0489882	+	0.0486187i
$898$		0			0
$899$	−29835.1			−1.10685
$900$		0			0
$901$	−45808.6			−1.69379
$902$		0			0
$903$	−3278.10	−	3253.37i	−0.120807	−	0.119895i
$904$		0			0
$905$		−	3627.65i		−	0.133245i
$906$		0			0
$907$			12655.0i			0.463288i	0.972801	+	0.231644i	$0.0744104\pi$
$907$			12655.0i			0.463288i	−0.972801	+	0.231644i	$0.925590\pi$
$908$		0			0
$909$	12709.7	−	96.2457i	0.463756	−	0.00351185i
$910$		0			0
$911$	−21875.4			−0.795569			−0.397784	−	0.917479i	$-0.630221\pi$
$911$	−21875.4			−0.795569			−0.397784	+	0.917479i	$0.630221\pi$
$912$		0			0
$913$	7418.70			0.268919
$914$		0			0
$915$	−2178.79	+	2195.35i	−0.0787198	+	0.0793182i
$916$		0			0
$917$			2546.83i			0.0917162i
$918$		0			0
$919$		−	28815.5i		−	1.03432i	−0.855890	−	0.517158i	$-0.826990\pi$
$919$		−	28815.5i		−	1.03432i	0.855890	−	0.517158i	$-0.173010\pi$
$920$		0			0
$921$	−15898.8	+	16019.7i	−0.568821	+	0.573144i
$922$		0			0
$923$	1549.24			0.0552481
$924$		0			0
$925$	53435.8			1.89942
$926$		0			0
$927$	−38967.7	+	295.087i	−1.38065	+	0.0104552i
$928$		0			0
$929$			29988.8i			1.05909i	0.848280	+	0.529547i	$0.177638\pi$
$929$			29988.8i			1.05909i	−0.848280	+	0.529547i	$0.822362\pi$
$930$		0			0
$931$			6194.30i			0.218056i
$932$		0			0
$933$	−24647.4	−	24461.4i	−0.864864	−	0.858339i
$934$		0			0
$935$	−2417.82			−0.0845680
$936$		0			0
$937$	3294.71			0.114870			0.0574352	−	0.998349i	$-0.481708\pi$
$937$	3294.71			0.114870			0.0574352	+	0.998349i	$0.481708\pi$
$938$		0			0
$939$	−27692.5	−	27483.6i	−0.962417	−	0.955156i
$940$		0			0
$941$			48458.9i			1.67876i	0.543543	+	0.839381i	$0.317082\pi$
$941$			48458.9i			1.67876i	−0.543543	+	0.839381i	$0.682918\pi$
$942$		0			0
$943$		−	26613.9i		−	0.919053i
$944$		0			0
$945$	1121.78	−	1147.55i	0.0386152	−	0.0395025i
$946$		0			0
$947$	56011.9			1.92201			0.961004	−	0.276536i	$-0.0891866\pi$
$947$	56011.9			1.92201			0.961004	+	0.276536i	$0.0891866\pi$
$948$		0			0
$949$	938.462			0.0321009
$950$		0			0
$951$	−19186.9	+	19332.8i	−0.654236	+	0.659209i
$952$		0			0
$953$		−	36959.0i		−	1.25626i	−0.778107	−	0.628132i	$-0.783820\pi$
$953$		−	36959.0i		−	1.25626i	0.778107	−	0.628132i	$-0.216180\pi$
$954$		0			0
$955$			4274.21i			0.144827i
$956$		0			0
$957$	−11654.8	+	11743.4i	−0.393674	+	0.396666i
$958$		0			0
$959$	9936.61			0.334588
$960$		0			0
$961$	15470.0			0.519285
$962$		0			0
$963$	115.422	+	15242.0i	0.00386233	+	0.510039i
$964$		0			0
$965$			386.773i			0.0129023i
$966$		0			0
$967$		−	39513.6i		−	1.31404i	−0.753875	−	0.657018i	$-0.771818\pi$
$967$		−	39513.6i		−	1.31404i	0.753875	−	0.657018i	$-0.228182\pi$
$968$		0			0
$969$	−54014.7	−	53607.2i	−1.79071	−	1.77720i
$970$		0			0
$971$	19526.8			0.645361			0.322681	−	0.946508i	$-0.395416\pi$
$971$	19526.8			0.645361			0.322681	+	0.946508i	$0.395416\pi$
$972$		0			0
$973$	2389.42			0.0787269
$974$		0			0
$975$	2002.51	+	1987.40i	0.0657759	+	0.0652797i
$976$		0			0
$977$			46524.0i			1.52348i	0.647885	+	0.761738i	$0.275654\pi$
$977$			46524.0i			1.52348i	−0.647885	+	0.761738i	$0.724346\pi$
$978$		0			0
$979$		−	16500.1i		−	0.538657i
$980$		0			0
$981$	386.413	+	51027.7i	0.0125762	+	1.66074i
$982$		0			0
$983$	−15597.6			−0.506089			−0.253044	−	0.967455i	$-0.581432\pi$
$983$	−15597.6			−0.506089			−0.253044	+	0.967455i	$0.581432\pi$
$984$		0			0
$985$	−1190.24			−0.0385017
$986$		0			0
$987$	−3480.35	+	3506.81i	−0.112240	+	0.113093i
$988$		0			0
$989$		−	10208.4i		−	0.328219i
$990$		0			0
$991$			18646.1i			0.597692i	0.954301	+	0.298846i	$0.0966017\pi$
$991$			18646.1i			0.597692i	−0.954301	+	0.298846i	$0.903398\pi$
$992$		0			0
$993$	18036.8	−	18173.9i	0.576415	−	0.580797i
$994$		0			0
$995$	−2506.04			−0.0798460
$996$		0			0
$997$	25040.3			0.795422			0.397711	−	0.917511i	$-0.369805\pi$
$997$	25040.3			0.795422			0.397711	+	0.917511i	$0.369805\pi$
$998$		0			0
$999$	−42839.2	+	43823.6i	−1.35673	+	1.38791i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.h.a.575.30	yes	36
3.2	odd	2	inner	672.4.h.a.575.8	yes	36
4.3	odd	2	inner	672.4.h.a.575.7	✓	36
12.11	even	2	inner	672.4.h.a.575.29	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.h.a.575.7	✓	36	4.3	odd	2	inner
672.4.h.a.575.8	yes	36	3.2	odd	2	inner
672.4.h.a.575.29	yes	36	12.11	even	2	inner
672.4.h.a.575.30	yes	36	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(3.68812 + 3.66030i) q^{3} -1.63405i q^{5} -7.00000i q^{7} +(0.204455 + 26.9992i) q^{9} +12.7716 q^{11} +4.43850 q^{13} +(5.98113 - 6.02659i) q^{15} -115.854i q^{17} -126.414i q^{19} +(25.6221 - 25.8168i) q^{21} +80.3969 q^{23} +122.330 q^{25} +(-98.0711 + 100.325i) q^{27} +249.311i q^{29} +119.670i q^{31} +(47.1033 + 46.7479i) q^{33} -11.4384 q^{35} +436.818 q^{37} +(16.3697 + 16.2462i) q^{39} -331.031i q^{41} -126.975i q^{43} +(44.1182 - 0.334090i) q^{45} -135.834 q^{47} -49.0000 q^{49} +(424.060 - 427.283i) q^{51} -395.400i q^{53} -20.8695i q^{55} +(462.714 - 466.231i) q^{57} +329.339 q^{59} -364.278 q^{61} +(188.995 - 1.43118i) q^{63} -7.25275i q^{65} +996.255i q^{67} +(296.513 + 294.276i) q^{69} +349.047 q^{71} +211.437 q^{73} +(451.167 + 447.764i) q^{75} -89.4014i q^{77} +240.806i q^{79} +(-728.916 + 11.0402i) q^{81} +580.873 q^{83} -189.312 q^{85} +(-912.553 + 919.490i) q^{87} -1291.93i q^{89} -31.0695i q^{91} +(-438.029 + 441.358i) q^{93} -206.568 q^{95} +1175.97 q^{97} +(2.61122 + 344.824i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 132 q^{9} - 120 q^{13} + 28 q^{21} - 756 q^{25} + 40 q^{33} + 672 q^{37} + 304 q^{45} - 1764 q^{49} + 1624 q^{57} + 2472 q^{61} - 1224 q^{69} - 2376 q^{73} + 468 q^{81} + 5160 q^{85} - 648 q^{93}+ \cdots - 4488 q^{97}+O(q^{100}) \) 36 * q - 132 * q^9 - 120 * q^13 + 28 * q^21 - 756 * q^25 + 40 * q^33 + 672 * q^37 + 304 * q^45 - 1764 * q^49 + 1624 * q^57 + 2472 * q^61 - 1224 * q^69 - 2376 * q^73 + 468 * q^81 + 5160 * q^85 - 648 * q^93 - 4488 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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