LMFDB - Embedded newform 672.2.b.b.223.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(223,672)]

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([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("672.223");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.b (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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.836829184.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 $ x^8 + 14x^6 + 61x^4 + 84*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.5
Root			$1.65222i$ of defining polynomial
Character	$\chi$	$=$	672.223
Dual form			672.2.b.b.223.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.922382i q^{5} +(-2.06644 + 1.65222i) q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.922382i q^{5} +(-2.06644 + 1.65222i) q^{7} +1.00000 q^{9} +1.61794i q^{11} +2.82843i q^{13} +0.922382i q^{15} +0.922382i q^{17} -0.540319 q^{19} +(-2.06644 + 1.65222i) q^{21} +7.05526i q^{23} +4.14921 q^{25} +1.00000 q^{27} +1.01634 q^{29} -8.67319 q^{31} +1.61794i q^{33} +(-1.52398 - 1.90604i) q^{35} +8.19717 q^{37} +2.82843i q^{39} +6.57924i q^{41} -4.96130i q^{43} +0.922382i q^{45} -2.60889 q^{47} +(1.54032 - 6.82843i) q^{49} +0.922382i q^{51} +5.59255 q^{53} -1.49236 q^{55} -0.540319 q^{57} -1.65685 q^{59} -7.01634i q^{61} +(-2.06644 + 1.65222i) q^{63} -2.60889 q^{65} -6.80606i q^{67} +7.05526i q^{69} +5.21049i q^{71} -7.50162i q^{73} +4.14921 q^{75} +(-2.67319 - 3.34336i) q^{77} +7.11654i q^{79} +1.00000 q^{81} +17.9226 q^{83} -0.850789 q^{85} +1.01634 q^{87} +17.9523i q^{89} +(-4.67319 - 5.84476i) q^{91} -8.67319 q^{93} -0.498381i q^{95} -4.18791i q^{97} +1.61794i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 4 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + 4 * q^7 + 8 * q^9	$ 8 q + 8 q^{3} + 4 q^{7} + 8 q^{9} + 8 q^{19} + 4 q^{21} - 16 q^{25} + 8 q^{27} - 16 q^{31} - 8 q^{35} + 8 q^{37} + 16 q^{47} + 16 q^{53} - 8 q^{55} + 8 q^{57} + 32 q^{59} + 4 q^{63} + 16 q^{65} - 16 q^{75} + 32 q^{77} + 8 q^{81} + 16 q^{83} - 56 q^{85} + 16 q^{91} - 16 q^{93}+O(q^{100}) $ 8 * q + 8 * q^3 + 4 * q^7 + 8 * q^9 + 8 * q^19 + 4 * q^21 - 16 * q^25 + 8 * q^27 - 16 * q^31 - 8 * q^35 + 8 * q^37 + 16 * q^47 + 16 * q^53 - 8 * q^55 + 8 * q^57 + 32 * q^59 + 4 * q^63 + 16 * q^65 - 16 * q^75 + 32 * q^77 + 8 * q^81 + 16 * q^83 - 56 * q^85 + 16 * q^91 - 16 * 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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$			0.922382i			0.412502i	0.978499	+	0.206251i	$0.0661263\pi$
$5$			0.922382i			0.412502i	−0.978499	+	0.206251i	$0.933874\pi$
$6$		0			0
$7$	−2.06644	+	1.65222i	−0.781040	+	0.624482i
$7$	−2.06644	+	1.65222i	−0.781040	+	0.624482i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			1.61794i			0.487826i	0.969797	+	0.243913i	$0.0784312\pi$
$11$			1.61794i			0.487826i	−0.969797	+	0.243913i	$0.921569\pi$
$12$		0			0
$13$			2.82843i			0.784465i	0.919866	+	0.392232i	$0.128297\pi$
$13$			2.82843i			0.784465i	−0.919866	+	0.392232i	$0.871703\pi$
$14$		0			0
$15$			0.922382i			0.238158i
$16$		0			0
$17$			0.922382i			0.223711i	0.993725	+	0.111855i	$0.0356793\pi$
$17$			0.922382i			0.223711i	−0.993725	+	0.111855i	$0.964321\pi$
$18$		0			0
$19$	−0.540319			−0.123958			−0.0619789	−	0.998077i	$-0.519741\pi$
$19$	−0.540319			−0.123958			−0.0619789	+	0.998077i	$0.519741\pi$
$20$		0			0
$21$	−2.06644	+	1.65222i	−0.450933	+	0.360545i
$22$		0			0
$23$			7.05526i			1.47112i	0.677458	+	0.735561i	$0.263081\pi$
$23$			7.05526i			1.47112i	−0.677458	+	0.735561i	$0.736919\pi$
$24$		0			0
$25$	4.14921			0.829842
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	1.01634			0.188729			0.0943646	−	0.995538i	$-0.469918\pi$
$29$	1.01634			0.188729			0.0943646	+	0.995538i	$0.469918\pi$
$30$		0			0
$31$	−8.67319			−1.55775			−0.778876	−	0.627178i	$-0.784210\pi$
$31$	−8.67319			−1.55775			−0.778876	+	0.627178i	$0.784210\pi$
$32$		0			0
$33$			1.61794i			0.281647i
$34$		0			0
$35$	−1.52398	−	1.90604i	−0.257600	−	0.322180i
$36$		0			0
$37$	8.19717			1.34761			0.673804	−	0.738911i	$-0.264659\pi$
$37$	8.19717			1.34761			0.673804	+	0.738911i	$0.264659\pi$
$38$		0			0
$39$			2.82843i			0.452911i
$40$		0			0
$41$			6.57924i			1.02750i	0.857939	+	0.513752i	$0.171745\pi$
$41$			6.57924i			1.02750i	−0.857939	+	0.513752i	$0.828255\pi$
$42$		0			0
$43$		−	4.96130i		−	0.756591i	−0.925685	−	0.378296i	$-0.876510\pi$
$43$		−	4.96130i		−	0.756591i	0.925685	−	0.378296i	$-0.123490\pi$
$44$		0			0
$45$			0.922382i			0.137501i
$46$		0			0
$47$	−2.60889			−0.380546			−0.190273	−	0.981731i	$-0.560937\pi$
$47$	−2.60889			−0.380546			−0.190273	+	0.981731i	$0.560937\pi$
$48$		0			0
$49$	1.54032	−	6.82843i	0.220046	−	0.975490i
$50$		0			0
$51$			0.922382i			0.129159i
$52$		0			0
$53$	5.59255			0.768196			0.384098	−	0.923292i	$-0.374512\pi$
$53$	5.59255			0.768196			0.384098	+	0.923292i	$0.374512\pi$
$54$		0			0
$55$	−1.49236			−0.201229
$56$		0			0
$57$	−0.540319			−0.0715670
$58$		0			0
$59$	−1.65685			−0.215704			−0.107852	−	0.994167i	$-0.534397\pi$
$59$	−1.65685			−0.215704			−0.107852	+	0.994167i	$0.534397\pi$
$60$		0			0
$61$		−	7.01634i		−	0.898350i	−0.893444	−	0.449175i	$-0.851718\pi$
$61$		−	7.01634i		−	0.898350i	0.893444	−	0.449175i	$-0.148282\pi$
$62$		0			0
$63$	−2.06644	+	1.65222i	−0.260347	+	0.208161i
$64$		0			0
$65$	−2.60889			−0.323593
$66$		0			0
$67$		−	6.80606i		−	0.831493i	−0.909480	−	0.415747i	$-0.863520\pi$
$67$		−	6.80606i		−	0.831493i	0.909480	−	0.415747i	$-0.136480\pi$
$68$		0			0
$69$			7.05526i			0.849353i
$70$		0			0
$71$			5.21049i			0.618371i	0.951002	+	0.309186i	$0.100056\pi$
$71$			5.21049i			0.618371i	−0.951002	+	0.309186i	$0.899944\pi$
$72$		0			0
$73$		−	7.50162i		−	0.877998i	−0.898488	−	0.438999i	$-0.855333\pi$
$73$		−	7.50162i		−	0.877998i	0.898488	−	0.438999i	$-0.144667\pi$
$74$		0			0
$75$	4.14921			0.479110
$76$		0			0
$77$	−2.67319	−	3.34336i	−0.304639	−	0.381012i
$78$		0			0
$79$			7.11654i			0.800673i	0.916368	+	0.400336i	$0.131107\pi$
$79$			7.11654i			0.800673i	−0.916368	+	0.400336i	$0.868893\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	17.9226			1.96726			0.983630	−	0.180197i	$-0.0576737\pi$
$83$	17.9226			1.96726			0.983630	+	0.180197i	$0.0576737\pi$
$84$		0			0
$85$	−0.850789			−0.0922811
$86$		0			0
$87$	1.01634			0.108963
$88$		0			0
$89$			17.9523i			1.90294i	0.307749	+	0.951468i	$0.400424\pi$
$89$			17.9523i			1.90294i	−0.307749	+	0.951468i	$0.599576\pi$
$90$		0			0
$91$	−4.67319	−	5.84476i	−0.489884	−	0.612698i
$92$		0			0
$93$	−8.67319			−0.899368
$94$		0			0
$95$		−	0.498381i		−	0.0511328i
$96$		0			0
$97$		−	4.18791i		−	0.425218i	−0.977137	−	0.212609i	$-0.931804\pi$
$97$		−	4.18791i		−	0.425218i	0.977137	−	0.212609i	$-0.0681960\pi$
$98$		0			0
$99$			1.61794i			0.162609i
$100$		0			0
$101$		−	12.2361i		−	1.21754i	−0.793348	−	0.608768i	$-0.791664\pi$
$101$		−	12.2361i		−	1.21754i	0.793348	−	0.608768i	$-0.208336\pi$
$102$		0			0
$103$	0.673192			0.0663316			0.0331658	−	0.999450i	$-0.489441\pi$
$103$	0.673192			0.0663316			0.0331658	+	0.999450i	$0.489441\pi$
$104$		0			0
$105$	−1.52398	−	1.90604i	−0.148725	−	0.186011i
$106$		0			0
$107$		−	8.22683i		−	0.795318i	−0.917533	−	0.397659i	$-0.869823\pi$
$107$		−	8.22683i		−	0.795318i	0.917533	−	0.397659i	$-0.130177\pi$
$108$		0			0
$109$	−10.2657			−0.983280			−0.491640	−	0.870799i	$-0.663602\pi$
$109$	−10.2657			−0.983280			−0.491640	+	0.870799i	$0.663602\pi$
$110$		0			0
$111$	8.19717			0.778041
$112$		0			0
$113$	−17.9553			−1.68909			−0.844545	−	0.535484i	$-0.820129\pi$
$113$	−17.9553			−1.68909			−0.844545	+	0.535484i	$0.820129\pi$
$114$		0			0
$115$	−6.50764			−0.606841
$116$		0			0
$117$			2.82843i			0.261488i
$118$		0			0
$119$	−1.52398	−	1.90604i	−0.139703	−	0.174727i
$120$		0			0
$121$	8.38228			0.762026
$122$		0			0
$123$			6.57924i			0.593230i
$124$		0			0
$125$			8.43907i			0.754813i
$126$		0			0
$127$		−	12.1972i		−	1.08232i	−0.840918	−	0.541162i	$-0.817985\pi$
$127$		−	12.1972i		−	1.08232i	0.840918	−	0.541162i	$-0.182015\pi$
$128$		0			0
$129$		−	4.96130i		−	0.436818i
$130$		0			0
$131$	10.7375			0.938139			0.469070	−	0.883161i	$-0.344589\pi$
$131$	10.7375			0.938139			0.469070	+	0.883161i	$0.344589\pi$
$132$		0			0
$133$	1.11654	−	0.892728i	0.0968159	−	0.0774093i
$134$		0			0
$135$			0.922382i			0.0793860i
$136$		0			0
$137$	−2.95204			−0.252210			−0.126105	−	0.992017i	$-0.540248\pi$
$137$	−2.95204			−0.252210			−0.126105	+	0.992017i	$0.540248\pi$
$138$		0			0
$139$	7.31371			0.620341			0.310170	−	0.950681i	$-0.399614\pi$
$139$	7.31371			0.620341			0.310170	+	0.950681i	$0.399614\pi$
$140$		0			0
$141$	−2.60889			−0.219708
$142$		0			0
$143$	−4.57622			−0.382682
$144$		0			0
$145$			0.937452i			0.0778512i
$146$		0			0
$147$	1.54032	−	6.82843i	0.127043	−	0.563199i
$148$		0			0
$149$	2.54459			0.208461			0.104231	−	0.994553i	$-0.466762\pi$
$149$	2.54459			0.208461			0.104231	+	0.994553i	$0.466762\pi$
$150$		0			0
$151$		−	19.6988i		−	1.60306i	−0.597951	−	0.801532i	$-0.704018\pi$
$151$		−	19.6988i		−	1.60306i	0.597951	−	0.801532i	$-0.295982\pi$
$152$		0			0
$153$			0.922382i			0.0745702i
$154$		0			0
$155$		−	8.00000i		−	0.642575i
$156$		0			0
$157$		−	18.3300i		−	1.46290i	−0.681897	−	0.731448i	$-0.738845\pi$
$157$		−	18.3300i		−	1.46290i	0.681897	−	0.731448i	$-0.261155\pi$
$158$		0			0
$159$	5.59255			0.443518
$160$		0			0
$161$	−11.6569	−	14.5792i	−0.918689	−	1.14900i
$162$		0			0
$163$		−	3.49236i		−	0.273542i	−0.990603	−	0.136771i	$-0.956327\pi$
$163$		−	3.49236i		−	0.273542i	0.990603	−	0.136771i	$-0.0436725\pi$
$164$		0			0
$165$	−1.49236			−0.116180
$166$		0			0
$167$	17.6121			1.36287			0.681434	−	0.731880i	$-0.261357\pi$
$167$	17.6121			1.36287			0.681434	+	0.731880i	$0.261357\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$	−0.540319			−0.0413192
$172$		0			0
$173$		−	22.5792i		−	1.71667i	−0.513092	−	0.858334i	$-0.671500\pi$
$173$		−	22.5792i		−	1.71667i	0.513092	−	0.858334i	$-0.328500\pi$
$174$		0			0
$175$	−8.57408	+	6.85542i	−0.648140	+	0.518221i
$176$		0			0
$177$	−1.65685			−0.124537
$178$		0			0
$179$			16.8030i			1.25592i	0.778246	+	0.627959i	$0.216110\pi$
$179$			16.8030i			1.25592i	−0.778246	+	0.627959i	$0.783890\pi$
$180$		0			0
$181$			23.6704i			1.75941i	0.475523	+	0.879703i	$0.342259\pi$
$181$			23.6704i			1.75941i	−0.475523	+	0.879703i	$0.657741\pi$
$182$		0			0
$183$		−	7.01634i		−	0.518663i
$184$		0			0
$185$			7.56093i			0.555891i
$186$		0			0
$187$	−1.49236			−0.109132
$188$		0			0
$189$	−2.06644	+	1.65222i	−0.150311	+	0.120182i
$190$		0			0
$191$		−	13.1658i		−	0.952641i	−0.879272	−	0.476321i	$-0.841970\pi$
$191$		−	13.1658i		−	0.952641i	0.879272	−	0.476321i	$-0.158030\pi$
$192$		0			0
$193$	−16.4629			−1.18503			−0.592513	−	0.805561i	$-0.701864\pi$
$193$	−16.4629			−1.18503			−0.592513	+	0.805561i	$0.701864\pi$
$194$		0			0
$195$	−2.60889			−0.186827
$196$		0			0
$197$	11.2494			0.801487			0.400744	−	0.916190i	$-0.368752\pi$
$197$	11.2494			0.801487			0.400744	+	0.916190i	$0.368752\pi$
$198$		0			0
$199$	−5.47926			−0.388414			−0.194207	−	0.980961i	$-0.562213\pi$
$199$	−5.47926			−0.388414			−0.194207	+	0.980961i	$0.562213\pi$
$200$		0			0
$201$		−	6.80606i		−	0.480063i
$202$		0			0
$203$	−2.10020	+	1.67922i	−0.147405	+	0.117858i
$204$		0			0
$205$	−6.06857			−0.423847
$206$		0			0
$207$			7.05526i			0.490374i
$208$		0			0
$209$		−	0.874202i		−	0.0604698i
$210$		0			0
$211$			7.88066i			0.542527i	0.962505	+	0.271264i	$0.0874415\pi$
$211$			7.88066i			0.542527i	−0.962505	+	0.271264i	$0.912558\pi$
$212$		0			0
$213$			5.21049i			0.357017i
$214$		0			0
$215$	4.57622			0.312095
$216$		0			0
$217$	17.9226	−	14.3300i	1.21667	−	0.972787i
$218$		0			0
$219$		−	7.50162i		−	0.506912i
$220$		0			0
$221$	−2.60889			−0.175493
$222$		0			0
$223$	23.4466			1.57010			0.785050	−	0.619433i	$-0.212637\pi$
$223$	23.4466			1.57010			0.785050	+	0.619433i	$0.212637\pi$
$224$		0			0
$225$	4.14921			0.276614
$226$		0			0
$227$	−21.3464			−1.41681			−0.708405	−	0.705807i	$-0.750585\pi$
$227$	−21.3464			−1.41681			−0.708405	+	0.705807i	$0.750585\pi$
$228$		0			0
$229$		−	1.48204i		−	0.0979362i	−0.998800	−	0.0489681i	$-0.984407\pi$
$229$		−	1.48204i		−	0.0979362i	0.998800	−	0.0489681i	$-0.0155933\pi$
$230$		0			0
$231$	−2.67319	−	3.34336i	−0.175883	−	0.219977i
$232$		0			0
$233$	−11.6569			−0.763666			−0.381833	−	0.924231i	$-0.624707\pi$
$233$	−11.6569			−0.763666			−0.381833	+	0.924231i	$0.624707\pi$
$234$		0			0
$235$		−	2.40640i		−	0.156976i
$236$		0			0
$237$			7.11654i			0.462269i
$238$		0			0
$239$			7.87015i			0.509078i	0.967063	+	0.254539i	$0.0819236\pi$
$239$			7.87015i			0.509078i	−0.967063	+	0.254539i	$0.918076\pi$
$240$		0			0
$241$		−	14.6867i		−	0.946055i	−0.881048	−	0.473028i	$-0.843161\pi$
$241$		−	14.6867i		−	0.946055i	0.881048	−	0.473028i	$-0.156839\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	6.29842	+	1.42076i	0.402391	+	0.0907692i
$246$		0			0
$247$		−	1.52825i		−	0.0972404i
$248$		0			0
$249$	17.9226			1.13580
$250$		0			0
$251$	−8.95204			−0.565048			−0.282524	−	0.959260i	$-0.591172\pi$
$251$	−8.95204			−0.565048			−0.282524	+	0.959260i	$0.591172\pi$
$252$		0			0
$253$	−11.4150			−0.717652
$254$		0			0
$255$	−0.850789			−0.0532785
$256$		0			0
$257$			26.2688i			1.63860i	0.573365	+	0.819300i	$0.305638\pi$
$257$			26.2688i			1.63860i	−0.573365	+	0.819300i	$0.694362\pi$
$258$		0			0
$259$	−16.9389	+	13.5436i	−1.05253	+	0.841556i
$260$		0			0
$261$	1.01634			0.0629097
$262$		0			0
$263$		−	6.47904i		−	0.399515i	−0.979845	−	0.199757i	$-0.935985\pi$
$263$		−	6.47904i		−	0.399515i	0.979845	−	0.199757i	$-0.0640154\pi$
$264$		0			0
$265$			5.15847i			0.316883i
$266$		0			0
$267$			17.9523i			1.09866i
$268$		0			0
$269$		−	16.1075i		−	0.982091i	−0.871134	−	0.491045i	$-0.836615\pi$
$269$		−	16.1075i		−	0.982091i	0.871134	−	0.491045i	$-0.163385\pi$
$270$		0			0
$271$	13.8910			0.843817			0.421908	−	0.906639i	$-0.361360\pi$
$271$	13.8910			0.843817			0.421908	+	0.906639i	$0.361360\pi$
$272$		0			0
$273$	−4.67319	−	5.84476i	−0.282834	−	0.353741i
$274$		0			0
$275$			6.71316i			0.404819i
$276$		0			0
$277$	−6.16450			−0.370389			−0.185194	−	0.982702i	$-0.559291\pi$
$277$	−6.16450			−0.370389			−0.185194	+	0.982702i	$0.559291\pi$
$278$		0			0
$279$	−8.67319			−0.519250
$280$		0			0
$281$	25.2690			1.50742			0.753711	−	0.657206i	$-0.228262\pi$
$281$	25.2690			1.50742			0.753711	+	0.657206i	$0.228262\pi$
$282$		0			0
$283$	6.57299			0.390724			0.195362	−	0.980731i	$-0.437412\pi$
$283$	6.57299			0.390724			0.195362	+	0.980731i	$0.437412\pi$
$284$		0			0
$285$		−	0.498381i		−	0.0295215i
$286$		0			0
$287$	−10.8704	−	13.5956i	−0.641657	−	0.802521i
$288$		0			0
$289$	16.1492			0.949954
$290$		0			0
$291$		−	4.18791i		−	0.245500i
$292$		0			0
$293$			24.4240i			1.42687i	0.700724	+	0.713433i	$0.252861\pi$
$293$			24.4240i			1.42687i	−0.700724	+	0.713433i	$0.747139\pi$
$294$		0			0
$295$		−	1.52825i		−	0.0889783i
$296$		0			0
$297$			1.61794i			0.0938822i
$298$		0			0
$299$	−19.9553			−1.15404
$300$		0			0
$301$	8.19717	+	10.2522i	0.472477	+	0.590928i
$302$		0			0
$303$		−	12.2361i		−	0.702945i
$304$		0			0
$305$	6.47175			0.370571
$306$		0			0
$307$	11.5883			0.661378			0.330689	−	0.943740i	$-0.392719\pi$
$307$	11.5883			0.661378			0.330689	+	0.943740i	$0.392719\pi$
$308$		0			0
$309$	0.673192			0.0382966
$310$		0			0
$311$	7.56093			0.428741			0.214370	−	0.976752i	$-0.431230\pi$
$311$	7.56093			0.428741			0.214370	+	0.976752i	$0.431230\pi$
$312$		0			0
$313$			32.9073i			1.86003i	0.367520	+	0.930016i	$0.380207\pi$
$313$			32.9073i			1.86003i	−0.367520	+	0.930016i	$0.619793\pi$
$314$		0			0
$315$	−1.52398	−	1.90604i	−0.0858666	−	0.107393i
$316$		0			0
$317$	−4.64052			−0.260637			−0.130319	−	0.991472i	$-0.541600\pi$
$317$	−4.64052			−0.260637			−0.130319	+	0.991472i	$0.541600\pi$
$318$		0			0
$319$			1.64437i			0.0920671i
$320$		0			0
$321$		−	8.22683i		−	0.459177i
$322$		0			0
$323$		−	0.498381i		−	0.0277307i
$324$		0			0
$325$			11.7357i			0.650982i
$326$		0			0
$327$	−10.2657			−0.567697
$328$		0			0
$329$	5.39111	−	4.31047i	0.297221	−	0.237644i
$330$		0			0
$331$			30.8839i			1.69753i	0.528768	+	0.848766i	$0.322654\pi$
$331$			30.8839i			1.69753i	−0.528768	+	0.848766i	$0.677346\pi$
$332$		0			0
$333$	8.19717			0.449202
$334$		0			0
$335$	6.27779			0.342993
$336$		0			0
$337$	−16.2331			−0.884272			−0.442136	−	0.896948i	$-0.645779\pi$
$337$	−16.2331			−0.884272			−0.442136	+	0.896948i	$0.645779\pi$
$338$		0			0
$339$	−17.9553			−0.975197
$340$		0			0
$341$		−	14.0327i		−	0.759912i
$342$		0			0
$343$	8.09911	+	16.6555i	0.437311	+	0.899310i
$344$		0			0
$345$	−6.50764			−0.350360
$346$		0			0
$347$			30.2780i			1.62541i	0.582676	+	0.812705i	$0.302006\pi$
$347$			30.2780i			1.62541i	−0.582676	+	0.812705i	$0.697994\pi$
$348$		0			0
$349$			28.4913i			1.52511i	0.646926	+	0.762553i	$0.276054\pi$
$349$			28.4913i			1.52511i	−0.646926	+	0.762553i	$0.723946\pi$
$350$		0			0
$351$			2.82843i			0.150970i
$352$		0			0
$353$			1.04494i			0.0556167i	0.999613	+	0.0278083i	$0.00885281\pi$
$353$			1.04494i			0.0556167i	−0.999613	+	0.0278083i	$0.991147\pi$
$354$		0			0
$355$	−4.80606			−0.255079
$356$		0			0
$357$	−1.52398	−	1.90604i	−0.0806576	−	0.100879i
$358$		0			0
$359$		−	0.0705425i		−	0.00372309i	−0.999998	−	0.00186155i	$-0.999407\pi$
$359$		−	0.0705425i		−	0.00372309i	0.999998	−	0.00186155i	$-0.000592549\pi$
$360$		0			0
$361$	−18.7081			−0.984634
$362$		0			0
$363$	8.38228			0.439956
$364$		0			0
$365$	6.91936			0.362176
$366$		0			0
$367$	0.00427190			0.000222991			0.000111496	−	1.00000i	$-0.499965\pi$
$367$	0.00427190			0.000222991			0.000111496		1.00000i	$0.499965\pi$
$368$		0			0
$369$			6.57924i			0.342501i
$370$		0			0
$371$	−11.5567	+	9.24015i	−0.599992	+	0.479724i
$372$		0			0
$373$	15.1524			0.784563			0.392282	−	0.919845i	$-0.371686\pi$
$373$	15.1524			0.784563			0.392282	+	0.919845i	$0.371686\pi$
$374$		0			0
$375$			8.43907i			0.435792i
$376$		0			0
$377$			2.87464i			0.148051i
$378$		0			0
$379$			29.7315i			1.52720i	0.645688	+	0.763601i	$0.276571\pi$
$379$			29.7315i			1.52720i	−0.645688	+	0.763601i	$0.723429\pi$
$380$		0			0
$381$		−	12.1972i		−	0.624880i
$382$		0			0
$383$	3.42378			0.174947			0.0874736	−	0.996167i	$-0.472121\pi$
$383$	3.42378			0.174947			0.0874736	+	0.996167i	$0.472121\pi$
$384$		0			0
$385$	3.08386	−	2.46571i	0.157168	−	0.125664i
$386$		0			0
$387$		−	4.96130i		−	0.252197i
$388$		0			0
$389$	35.9749			1.82400			0.911999	−	0.410193i	$-0.134539\pi$
$389$	35.9749			1.82400			0.911999	+	0.410193i	$0.134539\pi$
$390$		0			0
$391$	−6.50764			−0.329106
$392$		0			0
$393$	10.7375			0.541635
$394$		0			0
$395$	−6.56417			−0.330279
$396$		0			0
$397$		−	9.35948i		−	0.469739i	−0.972027	−	0.234870i	$-0.924534\pi$
$397$		−	9.35948i		−	0.469739i	0.972027	−	0.234870i	$-0.0754663\pi$
$398$		0			0
$399$	1.11654	−	0.892728i	0.0558967	−	0.0446923i
$400$		0			0
$401$	−17.5795			−0.877876			−0.438938	−	0.898517i	$-0.644645\pi$
$401$	−17.5795			−0.877876			−0.438938	+	0.898517i	$0.644645\pi$
$402$		0			0
$403$		−	24.5315i		−	1.22200i
$404$		0			0
$405$			0.922382i			0.0458336i
$406$		0			0
$407$			13.2625i			0.657398i
$408$		0			0
$409$			32.4089i			1.60252i	0.598317	+	0.801259i	$0.295836\pi$
$409$			32.4089i			1.60252i	−0.598317	+	0.801259i	$0.704164\pi$
$410$		0			0
$411$	−2.95204			−0.145613
$412$		0			0
$413$	3.42378	−	2.73749i	0.168473	−	0.134703i
$414$		0			0
$415$			16.5315i			0.811499i
$416$		0			0
$417$	7.31371			0.358154
$418$		0			0
$419$	8.31047			0.405993			0.202997	−	0.979179i	$-0.434932\pi$
$419$	8.31047			0.405993			0.202997	+	0.979179i	$0.434932\pi$
$420$		0			0
$421$	−2.22985			−0.108676			−0.0543381	−	0.998523i	$-0.517305\pi$
$421$	−2.22985			−0.108676			−0.0543381	+	0.998523i	$0.517305\pi$
$422$		0			0
$423$	−2.60889			−0.126849
$424$		0			0
$425$			3.82716i			0.185644i
$426$		0			0
$427$	11.5926	+	14.4988i	0.561003	+	0.701647i
$428$		0			0
$429$	−4.57622			−0.220942
$430$		0			0
$431$		−	19.8194i		−	0.954666i	−0.878722	−	0.477333i	$-0.841604\pi$
$431$		−	19.8194i		−	0.954666i	0.878722	−	0.477333i	$-0.158396\pi$
$432$		0			0
$433$		−	11.4956i		−	0.552442i	−0.961094	−	0.276221i	$-0.910918\pi$
$433$		−	11.4956i		−	0.552442i	0.961094	−	0.276221i	$-0.0890822\pi$
$434$		0			0
$435$			0.937452i			0.0449474i
$436$		0			0
$437$		−	3.81209i		−	0.182357i
$438$		0			0
$439$	−3.31798			−0.158359			−0.0791793	−	0.996860i	$-0.525230\pi$
$439$	−3.31798			−0.158359			−0.0791793	+	0.996860i	$0.525230\pi$
$440$		0			0
$441$	1.54032	−	6.82843i	0.0733485	−	0.325163i
$442$		0			0
$443$		−	32.5765i		−	1.54775i	−0.633336	−	0.773877i	$-0.718315\pi$
$443$		−	32.5765i		−	1.54775i	0.633336	−	0.773877i	$-0.281685\pi$
$444$		0			0
$445$	−16.5588			−0.784965
$446$		0			0
$447$	2.54459			0.120355
$448$		0			0
$449$	18.0839			0.853431			0.426715	−	0.904386i	$-0.359671\pi$
$449$	18.0839			0.853431			0.426715	+	0.904386i	$0.359671\pi$
$450$		0			0
$451$	−10.6448			−0.501244
$452$		0			0
$453$		−	19.6988i		−	0.925530i
$454$		0			0
$455$	5.39111	−	4.31047i	0.252739	−	0.202078i
$456$		0			0
$457$	−10.1371			−0.474196			−0.237098	−	0.971486i	$-0.576196\pi$
$457$	−10.1371			−0.474196			−0.237098	+	0.971486i	$0.576196\pi$
$458$		0			0
$459$			0.922382i			0.0430531i
$460$		0			0
$461$		−	9.23889i		−	0.430298i	−0.976581	−	0.215149i	$-0.930976\pi$
$461$		−	9.23889i		−	0.430298i	0.976581	−	0.215149i	$-0.0690237\pi$
$462$		0			0
$463$		−	3.66568i		−	0.170359i	−0.996366	−	0.0851793i	$-0.972854\pi$
$463$		−	3.66568i		−	0.170359i	0.996366	−	0.0851793i	$-0.0271463\pi$
$464$		0			0
$465$		−	8.00000i		−	0.370991i
$466$		0			0
$467$	9.48353			0.438845			0.219423	−	0.975630i	$-0.429583\pi$
$467$	9.48353			0.438845			0.219423	+	0.975630i	$0.429583\pi$
$468$		0			0
$469$	11.2451	+	14.0643i	0.519252	+	0.649429i
$470$		0			0
$471$		−	18.3300i		−	0.844604i
$472$		0			0
$473$	8.02707			0.369085
$474$		0			0
$475$	−2.24190			−0.102865
$476$		0			0
$477$	5.59255			0.256065
$478$		0			0
$479$	−39.0032			−1.78210			−0.891052	−	0.453902i	$-0.850032\pi$
$479$	−39.0032			−1.78210			−0.891052	+	0.453902i	$0.850032\pi$
$480$		0			0
$481$			23.1851i			1.05715i
$482$		0			0
$483$	−11.6569	−	14.5792i	−0.530405	−	0.663378i
$484$		0			0
$485$	3.86286			0.175403
$486$		0			0
$487$		−	8.91658i		−	0.404049i	−0.979381	−	0.202024i	$-0.935248\pi$
$487$		−	8.91658i		−	0.404049i	0.979381	−	0.202024i	$-0.0647520\pi$
$488$		0			0
$489$		−	3.49236i		−	0.157930i
$490$		0			0
$491$		−	26.5700i		−	1.19909i	−0.800342	−	0.599543i	$-0.795349\pi$
$491$		−	26.5700i		−	1.19909i	0.800342	−	0.599543i	$-0.204651\pi$
$492$		0			0
$493$			0.937452i			0.0422207i
$494$		0			0
$495$	−1.49236			−0.0670764
$496$		0			0
$497$	−8.60889	−	10.7671i	−0.386162	−	0.482973i
$498$		0			0
$499$			17.4210i			0.779871i	0.920842	+	0.389935i	$0.127503\pi$
$499$			17.4210i			0.779871i	−0.920842	+	0.389935i	$0.872497\pi$
$500$		0			0
$501$	17.6121			0.786852
$502$		0			0
$503$	−15.2952			−0.681979			−0.340989	−	0.940067i	$-0.610762\pi$
$503$	−15.2952			−0.681979			−0.340989	+	0.940067i	$0.610762\pi$
$504$		0			0
$505$	11.2864			0.502236
$506$		0			0
$507$	5.00000			0.222058
$508$		0			0
$509$		−	6.76111i		−	0.299681i	−0.988710	−	0.149840i	$-0.952124\pi$
$509$		−	6.76111i		−	0.299681i	0.988710	−	0.149840i	$-0.0478760\pi$
$510$		0			0
$511$	12.3943	+	15.5016i	0.548294	+	0.685751i
$512$		0			0
$513$	−0.540319			−0.0238557
$514$		0			0
$515$			0.620940i			0.0273619i
$516$		0			0
$517$		−	4.22102i		−	0.185640i
$518$		0			0
$519$		−	22.5792i		−	0.991118i
$520$		0			0
$521$		−	25.4539i		−	1.11515i	−0.830125	−	0.557577i	$-0.811731\pi$
$521$		−	25.4539i		−	1.11515i	0.830125	−	0.557577i	$-0.188269\pi$
$522$		0			0
$523$	18.5642			0.811754			0.405877	−	0.913928i	$-0.366966\pi$
$523$	18.5642			0.811754			0.405877	+	0.913928i	$0.366966\pi$
$524$		0			0
$525$	−8.57408	+	6.85542i	−0.374204	+	0.299195i
$526$		0			0
$527$		−	8.00000i		−	0.348485i
$528$		0			0
$529$	−26.7766			−1.16420
$530$		0			0
$531$	−1.65685			−0.0719014
$532$		0			0
$533$	−18.6089			−0.806041
$534$		0			0
$535$	7.58828			0.328070
$536$		0			0
$537$			16.8030i			0.725105i
$538$		0			0
$539$	11.0480	+	2.49214i	0.475869	+	0.107344i
$540$		0			0
$541$	−14.9161			−0.641295			−0.320647	−	0.947199i	$-0.603900\pi$
$541$	−14.9161			−0.641295			−0.320647	+	0.947199i	$0.603900\pi$
$542$		0			0
$543$			23.6704i			1.01579i
$544$		0			0
$545$		−	9.46894i		−	0.405605i
$546$		0			0
$547$		−	22.4302i		−	0.959048i	−0.877529	−	0.479524i	$-0.840809\pi$
$547$		−	22.4302i		−	0.959048i	0.877529	−	0.479524i	$-0.159191\pi$
$548$		0			0
$549$		−	7.01634i		−	0.299450i
$550$		0			0
$551$	−0.549147			−0.0233944
$552$		0			0
$553$	−11.7581	−	14.7059i	−0.500005	−	0.625357i
$554$		0			0
$555$			7.56093i			0.320944i
$556$		0			0
$557$	−10.1417			−0.429718			−0.214859	−	0.976645i	$-0.568929\pi$
$557$	−10.1417			−0.429718			−0.214859	+	0.976645i	$0.568929\pi$
$558$		0			0
$559$	14.0327			0.593519
$560$		0			0
$561$	−1.49236			−0.0630073
$562$		0			0
$563$	41.8332			1.76306			0.881529	−	0.472131i	$-0.156515\pi$
$563$	41.8332			1.76306			0.881529	+	0.472131i	$0.156515\pi$
$564$		0			0
$565$		−	16.5616i		−	0.696753i
$566$		0			0
$567$	−2.06644	+	1.65222i	−0.0867822	+	0.0693868i
$568$		0			0
$569$	22.2210			0.931554			0.465777	−	0.884902i	$-0.345775\pi$
$569$	22.2210			0.931554			0.465777	+	0.884902i	$0.345775\pi$
$570$		0			0
$571$		−	42.1198i		−	1.76266i	−0.472503	−	0.881329i	$-0.656650\pi$
$571$		−	42.1198i		−	1.76266i	0.472503	−	0.881329i	$-0.343350\pi$
$572$		0			0
$573$		−	13.1658i		−	0.550008i
$574$		0			0
$575$			29.2737i			1.22080i
$576$		0			0
$577$			3.87140i			0.161168i	0.996748	+	0.0805842i	$0.0256786\pi$
$577$			3.87140i			0.161168i	−0.996748	+	0.0805842i	$0.974321\pi$
$578$		0			0
$579$	−16.4629			−0.684176
$580$		0			0
$581$	−37.0359	+	29.6121i	−1.53651	+	1.22852i
$582$		0			0
$583$			9.04840i			0.374746i
$584$		0			0
$585$	−2.60889			−0.107864
$586$		0			0
$587$	−38.7646			−1.59998			−0.799992	−	0.600010i	$-0.795163\pi$
$587$	−38.7646			−1.59998			−0.799992	+	0.600010i	$0.795163\pi$
$588$		0			0
$589$	4.68629			0.193095
$590$		0			0
$591$	11.2494			0.462739
$592$		0			0
$593$		−	22.6123i		−	0.928578i	−0.885684	−	0.464289i	$-0.846310\pi$
$593$		−	22.6123i		−	0.928578i	0.885684	−	0.464289i	$-0.153690\pi$
$594$		0			0
$595$	1.75810	−	1.40569i	0.0720752	−	0.0576278i
$596$		0			0
$597$	−5.47926			−0.224251
$598$		0			0
$599$			31.5274i			1.28818i	0.764951	+	0.644088i	$0.222763\pi$
$599$			31.5274i			1.28818i	−0.764951	+	0.644088i	$0.777237\pi$
$600$		0			0
$601$		−	11.5077i		−	0.469407i	−0.972067	−	0.234704i	$-0.924588\pi$
$601$		−	11.5077i		−	0.469407i	0.972067	−	0.234704i	$-0.0754120\pi$
$602$		0			0
$603$		−	6.80606i		−	0.277164i
$604$		0			0
$605$			7.73167i			0.314337i
$606$		0			0
$607$	18.2288			0.739884			0.369942	−	0.929055i	$-0.379378\pi$
$607$	18.2288			0.739884			0.369942	+	0.929055i	$0.379378\pi$
$608$		0			0
$609$	−2.10020	+	1.67922i	−0.0851043	+	0.0680453i
$610$		0			0
$611$		−	7.37906i		−	0.298525i
$612$		0			0
$613$	−46.9258			−1.89532			−0.947658	−	0.319286i	$-0.896557\pi$
$613$	−46.9258			−1.89532			−0.947658	+	0.319286i	$0.896557\pi$
$614$		0			0
$615$	−6.06857			−0.244708
$616$		0			0
$617$	−38.2395			−1.53947			−0.769733	−	0.638366i	$-0.779611\pi$
$617$	−38.2395			−1.53947			−0.769733	+	0.638366i	$0.779611\pi$
$618$		0			0
$619$	33.7493			1.35650			0.678249	−	0.734832i	$-0.262739\pi$
$619$	33.7493			1.35650			0.678249	+	0.734832i	$0.262739\pi$
$620$		0			0
$621$			7.05526i			0.283118i
$622$		0			0
$623$	−29.6611	−	37.0972i	−1.18835	−	1.48627i
$624$		0			0
$625$	12.9620			0.518480
$626$		0			0
$627$		−	0.874202i		−	0.0349123i
$628$		0			0
$629$			7.56093i			0.301474i
$630$		0			0
$631$		−	21.7254i		−	0.864876i	−0.901664	−	0.432438i	$-0.857654\pi$
$631$		−	21.7254i		−	0.864876i	0.901664	−	0.432438i	$-0.142346\pi$
$632$		0			0
$633$			7.88066i			0.313228i
$634$		0			0
$635$	11.2505			0.446461
$636$		0			0
$637$	19.3137	+	4.35668i	0.765237	+	0.172618i
$638$		0			0
$639$			5.21049i			0.206124i
$640$		0			0
$641$	28.4988			1.12564			0.562818	−	0.826581i	$-0.309717\pi$
$641$	28.4988			1.12564			0.562818	+	0.826581i	$0.309717\pi$
$642$		0			0
$643$	29.7581			1.17354			0.586772	−	0.809752i	$-0.300398\pi$
$643$	29.7581			1.17354			0.586772	+	0.809752i	$0.300398\pi$
$644$		0			0
$645$	4.57622			0.180188
$646$		0			0
$647$	36.6601			1.44126			0.720628	−	0.693321i	$-0.243853\pi$
$647$	36.6601			1.44126			0.720628	+	0.693321i	$0.243853\pi$
$648$		0			0
$649$		−	2.68069i		−	0.105226i
$650$		0			0
$651$	17.9226	−	14.3300i	0.702442	−	0.561639i
$652$		0			0
$653$	−36.2853			−1.41995			−0.709977	−	0.704225i	$-0.751295\pi$
$653$	−36.2853			−1.41995			−0.709977	+	0.704225i	$0.751295\pi$
$654$		0			0
$655$			9.90407i			0.386984i
$656$		0			0
$657$		−	7.50162i		−	0.292666i
$658$		0			0
$659$		−	29.7018i		−	1.15702i	−0.815676	−	0.578509i	$-0.803635\pi$
$659$		−	29.7018i		−	1.15702i	0.815676	−	0.578509i	$-0.196365\pi$
$660$		0			0
$661$		−	11.8583i		−	0.461234i	−0.973045	−	0.230617i	$-0.925925\pi$
$661$		−	11.8583i		−	0.461234i	0.973045	−	0.230617i	$-0.0740745\pi$
$662$		0			0
$663$	−2.60889			−0.101321
$664$		0			0
$665$	0.823436	+	1.02987i	0.0319315	+	0.0399367i
$666$		0			0
$667$			7.17052i			0.277644i
$668$		0			0
$669$	23.4466			0.906497
$670$		0			0
$671$	11.3520			0.438239
$672$		0			0
$673$	−23.3105			−0.898553			−0.449277	−	0.893393i	$-0.648318\pi$
$673$	−23.3105			−0.898553			−0.449277	+	0.893393i	$0.648318\pi$
$674$		0			0
$675$	4.14921			0.159703
$676$		0			0
$677$		−	16.7998i		−	0.645670i	−0.946455	−	0.322835i	$-0.895364\pi$
$677$		−	16.7998i		−	0.645670i	0.946455	−	0.322835i	$-0.104636\pi$
$678$		0			0
$679$	6.91936	+	8.65405i	0.265541	+	0.332112i
$680$		0			0
$681$	−21.3464			−0.817995
$682$		0			0
$683$		−	31.9434i		−	1.22228i	−0.791522	−	0.611141i	$-0.790711\pi$
$683$		−	31.9434i		−	1.22228i	0.791522	−	0.611141i	$-0.209289\pi$
$684$		0			0
$685$		−	2.72291i		−	0.104037i
$686$		0			0
$687$		−	1.48204i		−	0.0565435i
$688$		0			0
$689$			15.8181i			0.602623i
$690$		0			0
$691$	10.9108			0.415067			0.207534	−	0.978228i	$-0.433456\pi$
$691$	10.9108			0.415067			0.207534	+	0.978228i	$0.433456\pi$
$692$		0			0
$693$	−2.67319	−	3.34336i	−0.101546	−	0.127004i
$694$		0			0
$695$			6.74604i			0.255892i
$696$		0			0
$697$	−6.06857			−0.229864
$698$		0			0
$699$	−11.6569			−0.440903
$700$		0			0
$701$	−3.23088			−0.122029			−0.0610144	−	0.998137i	$-0.519434\pi$
$701$	−3.23088			−0.122029			−0.0610144	+	0.998137i	$0.519434\pi$
$702$		0			0
$703$	−4.42909			−0.167046
$704$		0			0
$705$		−	2.40640i		−	0.0906301i
$706$		0			0
$707$	20.2167	+	25.2851i	0.760329	+	0.950944i
$708$		0			0
$709$	28.5642			1.07275			0.536375	−	0.843980i	$-0.319793\pi$
$709$	28.5642			1.07275			0.536375	+	0.843980i	$0.319793\pi$
$710$		0			0
$711$			7.11654i			0.266891i
$712$		0			0
$713$		−	61.1916i		−	2.29164i
$714$		0			0
$715$		−	4.22102i		−	0.157857i
$716$		0			0
$717$			7.87015i			0.293916i
$718$		0			0
$719$	−23.0032			−0.857876			−0.428938	−	0.903334i	$-0.641112\pi$
$719$	−23.0032			−0.857876			−0.428938	+	0.903334i	$0.641112\pi$
$720$		0			0
$721$	−1.39111	+	1.11226i	−0.0518076	+	0.0414228i
$722$		0			0
$723$		−	14.6867i		−	0.546205i
$724$		0			0
$725$	4.21700			0.156615
$726$		0			0
$727$	−43.9869			−1.63138			−0.815692	−	0.578487i	$-0.803644\pi$
$727$	−43.9869			−1.63138			−0.815692	+	0.578487i	$0.803644\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	4.57622			0.169257
$732$		0			0
$733$		−	32.3958i		−	1.19657i	−0.801284	−	0.598284i	$-0.795850\pi$
$733$		−	32.3958i		−	1.19657i	0.801284	−	0.598284i	$-0.204150\pi$
$734$		0			0
$735$	6.29842	+	1.42076i	0.232321	+	0.0524056i
$736$		0			0
$737$	11.0118			0.405624
$738$		0			0
$739$		−	30.4302i		−	1.11939i	−0.828697	−	0.559697i	$-0.810917\pi$
$739$		−	30.4302i		−	1.11939i	0.828697	−	0.559697i	$-0.189083\pi$
$740$		0			0
$741$		−	1.52825i		−	0.0561418i
$742$		0			0
$743$			41.4908i			1.52215i	0.648664	+	0.761075i	$0.275328\pi$
$743$			41.4908i			1.52215i	−0.648664	+	0.761075i	$0.724672\pi$
$744$		0			0
$745$			2.34709i			0.0859906i
$746$		0			0
$747$	17.9226			0.655754
$748$		0			0
$749$	13.5926	+	17.0002i	0.496661	+	0.621174i
$750$		0			0
$751$			1.40037i			0.0511003i	0.999674	+	0.0255501i	$0.00813375\pi$
$751$			1.40037i			0.0511003i	−0.999674	+	0.0255501i	$0.991866\pi$
$752$		0			0
$753$	−8.95204			−0.326230
$754$		0			0
$755$	18.1698			0.661267
$756$		0			0
$757$	−25.1851			−0.915368			−0.457684	−	0.889115i	$-0.651321\pi$
$757$	−25.1851			−0.915368			−0.457684	+	0.889115i	$0.651321\pi$
$758$		0			0
$759$	−11.4150			−0.414337
$760$		0			0
$761$			8.39089i			0.304170i	0.988367	+	0.152085i	$0.0485987\pi$
$761$			8.39089i			0.304170i	−0.988367	+	0.152085i	$0.951401\pi$
$762$		0			0
$763$	21.2135	−	16.9613i	0.767981	−	0.614040i
$764$		0			0
$765$	−0.850789			−0.0307604
$766$		0			0
$767$		−	4.68629i		−	0.169212i
$768$		0			0
$769$			8.90731i			0.321206i	0.987019	+	0.160603i	$0.0513439\pi$
$769$			8.90731i			0.321206i	−0.987019	+	0.160603i	$0.948656\pi$
$770$		0			0
$771$			26.2688i			0.946047i
$772$		0			0
$773$			13.3885i			0.481552i	0.970581	+	0.240776i	$0.0774019\pi$
$773$			13.3885i			0.481552i	−0.970581	+	0.240776i	$0.922598\pi$
$774$		0			0
$775$	−35.9869			−1.29269
$776$		0			0
$777$	−16.9389	+	13.5436i	−0.607681	+	0.485872i
$778$		0			0
$779$		−	3.55489i		−	0.127367i
$780$		0			0
$781$	−8.43024			−0.301658
$782$		0			0
$783$	1.01634			0.0363209
$784$		0			0
$785$	16.9073			0.603448
$786$		0			0
$787$	−25.4117			−0.905831			−0.452915	−	0.891554i	$-0.649616\pi$
$787$	−25.4117			−0.905831			−0.452915	+	0.891554i	$0.649616\pi$
$788$		0			0
$789$		−	6.47904i		−	0.230660i
$790$		0			0
$791$	37.1034	−	29.6661i	1.31925	−	1.05481i
$792$		0			0
$793$	19.8452			0.704724
$794$		0			0
$795$			5.15847i			0.182952i
$796$		0			0
$797$			28.1728i			0.997933i	0.866621	+	0.498967i	$0.166287\pi$
$797$			28.1728i			0.997933i	−0.866621	+	0.498967i	$0.833713\pi$
$798$		0			0
$799$		−	2.40640i		−	0.0851322i
$800$		0			0
$801$			17.9523i			0.634312i
$802$		0			0
$803$	12.1371			0.428311
$804$		0			0
$805$	13.4476	−	10.7521i	0.473967	−	0.378961i
$806$		0			0
$807$		−	16.1075i		−	0.567011i
$808$		0			0
$809$	−8.68069			−0.305197			−0.152598	−	0.988288i	$-0.548764\pi$
$809$	−8.68069			−0.305197			−0.152598	+	0.988288i	$0.548764\pi$
$810$		0			0
$811$	−43.4423			−1.52547			−0.762733	−	0.646714i	$-0.776143\pi$
$811$	−43.4423			−1.52547			−0.762733	+	0.646714i	$0.776143\pi$
$812$		0			0
$813$	13.8910			0.487178
$814$		0			0
$815$	3.22129			0.112837
$816$		0			0
$817$			2.68069i			0.0937853i
$818$		0			0
$819$	−4.67319	−	5.84476i	−0.163295	−	0.204233i
$820$		0			0
$821$	−8.24617			−0.287793			−0.143897	−	0.989593i	$-0.545963\pi$
$821$	−8.24617			−0.287793			−0.143897	+	0.989593i	$0.545963\pi$
$822$		0			0
$823$			2.54886i			0.0888478i	0.999013	+	0.0444239i	$0.0141452\pi$
$823$			2.54886i			0.0888478i	−0.999013	+	0.0444239i	$0.985855\pi$
$824$		0			0
$825$			6.71316i			0.233722i
$826$		0			0
$827$		−	24.7252i		−	0.859780i	−0.902881	−	0.429890i	$-0.858552\pi$
$827$		−	24.7252i		−	0.859780i	0.902881	−	0.429890i	$-0.141448\pi$
$828$		0			0
$829$			25.9241i			0.900381i	0.892933	+	0.450190i	$0.148644\pi$
$829$			25.9241i			0.900381i	−0.892933	+	0.450190i	$0.851356\pi$
$830$		0			0
$831$	−6.16450			−0.213844
$832$		0			0
$833$	6.29842	+	1.42076i	0.218227	+	0.0492265i
$834$		0			0
$835$			16.2451i			0.562186i
$836$		0			0
$837$	−8.67319			−0.299789
$838$		0			0
$839$	−53.1198			−1.83390			−0.916949	−	0.399003i	$-0.869356\pi$
$839$	−53.1198			−1.83390			−0.916949	+	0.399003i	$0.869356\pi$
$840$		0			0
$841$	−27.9671			−0.964381
$842$		0			0
$843$	25.2690			0.870310
$844$		0			0
$845$			4.61191i			0.158655i
$846$		0			0
$847$	−17.3215	+	13.8494i	−0.595172	+	0.475871i
$848$		0			0
$849$	6.57299			0.225585
$850$		0			0
$851$			57.8332i			1.98249i
$852$		0			0
$853$		−	45.7362i		−	1.56598i	−0.622036	−	0.782988i	$-0.713694\pi$
$853$		−	45.7362i		−	1.56598i	0.622036	−	0.782988i	$-0.286306\pi$
$854$		0			0
$855$		−	0.498381i		−	0.0170443i
$856$		0			0
$857$			3.70460i			0.126547i	0.997996	+	0.0632734i	$0.0201540\pi$
$857$			3.70460i			0.126547i	−0.997996	+	0.0632734i	$0.979846\pi$
$858$		0			0
$859$	−31.1677			−1.06343			−0.531715	−	0.846924i	$-0.678452\pi$
$859$	−31.1677			−1.06343			−0.531715	+	0.846924i	$0.678452\pi$
$860$		0			0
$861$	−10.8704	−	13.5956i	−0.370461	−	0.463336i
$862$		0			0
$863$			13.1512i			0.447671i	0.974627	+	0.223836i	$0.0718579\pi$
$863$			13.1512i			0.447671i	−0.974627	+	0.223836i	$0.928142\pi$
$864$		0			0
$865$	20.8267			0.708129
$866$		0			0
$867$	16.1492			0.548456
$868$		0			0
$869$	−11.5141			−0.390589
$870$		0			0
$871$	19.2505			0.652277
$872$		0			0
$873$		−	4.18791i		−	0.141739i
$874$		0			0
$875$	−13.9432	−	17.4388i	−0.471367	−	0.589539i
$876$		0			0
$877$	56.6751			1.91378			0.956891	−	0.290446i	$-0.0938038\pi$
$877$	56.6751			1.91378			0.956891	+	0.290446i	$0.0938038\pi$
$878$		0			0
$879$			24.4240i			0.823801i
$880$		0			0
$881$		−	20.7676i		−	0.699678i	−0.936810	−	0.349839i	$-0.886236\pi$
$881$		−	20.7676i		−	0.699678i	0.936810	−	0.349839i	$-0.113764\pi$
$882$		0			0
$883$		−	15.6541i		−	0.526801i	−0.964687	−	0.263401i	$-0.915156\pi$
$883$		−	15.6541i		−	0.526801i	0.964687	−	0.263401i	$-0.0848441\pi$
$884$		0			0
$885$		−	1.52825i		−	0.0513717i
$886$		0			0
$887$	29.6775			0.996472			0.498236	−	0.867041i	$-0.333981\pi$
$887$	29.6775			0.996472			0.498236	+	0.867041i	$0.333981\pi$
$888$		0			0
$889$	20.1524	+	25.2047i	0.675891	+	0.845338i
$890$		0			0
$891$			1.61794i			0.0542029i
$892$		0			0
$893$	1.40963			0.0471716
$894$		0			0
$895$	−15.4988			−0.518069
$896$		0			0
$897$	−19.9553			−0.666287
$898$		0			0
$899$	−8.81489			−0.293993
$900$		0			0
$901$			5.15847i			0.171854i
$902$		0			0
$903$	8.19717	+	10.2522i	0.272785	+	0.341172i
$904$		0			0
$905$	−21.8332			−0.725758
$906$		0			0
$907$		−	9.52897i		−	0.316404i	−0.987407	−	0.158202i	$-0.949430\pi$
$907$		−	9.52897i		−	0.316404i	0.987407	−	0.158202i	$-0.0505698\pi$
$908$		0			0
$909$		−	12.2361i		−	0.405846i
$910$		0			0
$911$			4.67900i			0.155022i	0.996991	+	0.0775111i	$0.0246973\pi$
$911$			4.67900i			0.155022i	−0.996991	+	0.0775111i	$0.975303\pi$
$912$		0			0
$913$			28.9976i			0.959682i
$914$		0			0
$915$	6.47175			0.213949
$916$		0			0
$917$	−22.1883	+	17.7407i	−0.732724	+	0.585851i
$918$		0			0
$919$			29.0899i			0.959587i	0.877381	+	0.479794i	$0.159288\pi$
$919$			29.0899i			0.959587i	−0.877381	+	0.479794i	$0.840712\pi$
$920$		0			0
$921$	11.5883			0.381847
$922$		0			0
$923$	−14.7375			−0.485090
$924$		0			0
$925$	34.0118			1.11830
$926$		0			0
$927$	0.673192			0.0221105
$928$		0			0
$929$			3.79702i			0.124576i	0.998058	+	0.0622881i	$0.0198398\pi$
$929$			3.79702i			0.124576i	−0.998058	+	0.0622881i	$0.980160\pi$
$930$		0			0
$931$	−0.832264	+	3.68953i	−0.0272763	+	0.120919i
$932$		0			0
$933$	7.56093			0.247534
$934$		0			0
$935$		−	1.37652i		−	0.0450171i
$936$		0			0
$937$		−	57.0630i		−	1.86417i	−0.362244	−	0.932083i	$-0.617989\pi$
$937$		−	57.0630i		−	1.86417i	0.362244	−	0.932083i	$-0.382011\pi$
$938$		0			0
$939$			32.9073i			1.07389i
$940$		0			0
$941$			54.4898i			1.77632i	0.459538	+	0.888158i	$0.348015\pi$
$941$			54.4898i			1.77632i	−0.459538	+	0.888158i	$0.651985\pi$
$942$		0			0
$943$	−46.4182			−1.51158
$944$		0			0
$945$	−1.52398	−	1.90604i	−0.0495751	−	0.0620036i
$946$		0			0
$947$			16.5579i			0.538060i	0.963132	+	0.269030i	$0.0867031\pi$
$947$			16.5579i			0.538060i	−0.963132	+	0.269030i	$0.913297\pi$
$948$		0			0
$949$	21.2178			0.688758
$950$		0			0
$951$	−4.64052			−0.150479
$952$		0			0
$953$	33.5141			1.08563			0.542814	−	0.839853i	$-0.317359\pi$
$953$	33.5141			1.08563			0.542814	+	0.839853i	$0.317359\pi$
$954$		0			0
$955$	12.1439			0.392966
$956$		0			0
$957$			1.64437i			0.0531549i
$958$		0			0
$959$	6.10020	−	4.87742i	0.196986	−	0.157500i
$960$		0			0
$961$	44.2243			1.42659
$962$		0			0
$963$		−	8.22683i		−	0.265106i
$964$		0			0
$965$		−	15.1851i		−	0.488826i
$966$		0			0
$967$		−	32.4182i		−	1.04250i	−0.853404	−	0.521249i	$-0.825466\pi$
$967$		−	32.4182i		−	1.04250i	0.853404	−	0.521249i	$-0.174534\pi$
$968$		0			0
$969$		−	0.498381i		−	0.0160103i
$970$		0			0
$971$	−32.9344			−1.05691			−0.528457	−	0.848960i	$-0.677229\pi$
$971$	−32.9344			−1.05691			−0.528457	+	0.848960i	$0.677229\pi$
$972$		0			0
$973$	−15.1133	+	12.0839i	−0.484511	+	0.387391i
$974$		0			0
$975$			11.7357i			0.375844i
$976$		0			0
$977$	29.7613			0.952149			0.476075	−	0.879405i	$-0.342059\pi$
$977$	29.7613			0.952149			0.476075	+	0.879405i	$0.342059\pi$
$978$		0			0
$979$	−29.0456			−0.928302
$980$		0			0
$981$	−10.2657			−0.327760
$982$		0			0
$983$	32.8605			1.04809			0.524043	−	0.851692i	$-0.324423\pi$
$983$	32.8605			1.04809			0.524043	+	0.851692i	$0.324423\pi$
$984$		0			0
$985$			10.3763i			0.330615i
$986$		0			0
$987$	5.39111	−	4.31047i	0.171601	−	0.137204i
$988$		0			0
$989$	35.0032			1.11304
$990$		0			0
$991$		−	12.2303i		−	0.388508i	−0.980951	−	0.194254i	$-0.937771\pi$
$991$		−	12.2303i		−	0.388508i	0.980951	−	0.194254i	$-0.0622286\pi$
$992$		0			0
$993$			30.8839i			0.980071i
$994$		0			0
$995$		−	5.05397i		−	0.160222i
$996$		0			0
$997$		−	31.4846i		−	0.997127i	−0.866853	−	0.498563i	$-0.833861\pi$
$997$		−	31.4846i		−	0.997127i	0.866853	−	0.498563i	$-0.166139\pi$
$998$		0			0
$999$	8.19717			0.259347

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.b.b.223.5	yes	8
3.2	odd	2		2016.2.b.c.1567.4		8
4.3	odd	2		672.2.b.a.223.5	yes	8
7.6	odd	2		672.2.b.a.223.4	✓	8
8.3	odd	2		1344.2.b.h.895.4		8
8.5	even	2		1344.2.b.g.895.4		8
12.11	even	2		2016.2.b.a.1567.4		8
21.20	even	2		2016.2.b.a.1567.5		8
24.5	odd	2		4032.2.b.q.3583.5		8
24.11	even	2		4032.2.b.o.3583.5		8
28.27	even	2	inner	672.2.b.b.223.4	yes	8
56.13	odd	2		1344.2.b.h.895.5		8
56.27	even	2		1344.2.b.g.895.5		8
84.83	odd	2		2016.2.b.c.1567.5		8
168.83	odd	2		4032.2.b.q.3583.4		8
168.125	even	2		4032.2.b.o.3583.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.b.a.223.4	✓	8	7.6	odd	2
672.2.b.a.223.5	yes	8	4.3	odd	2
672.2.b.b.223.4	yes	8	28.27	even	2	inner
672.2.b.b.223.5	yes	8	1.1	even	1	trivial
1344.2.b.g.895.4		8	8.5	even	2
1344.2.b.g.895.5		8	56.27	even	2
1344.2.b.h.895.4		8	8.3	odd	2
1344.2.b.h.895.5		8	56.13	odd	2
2016.2.b.a.1567.4		8	12.11	even	2
2016.2.b.a.1567.5		8	21.20	even	2
2016.2.b.c.1567.4		8	3.2	odd	2
2016.2.b.c.1567.5		8	84.83	odd	2
4032.2.b.o.3583.4		8	168.125	even	2
4032.2.b.o.3583.5		8	24.11	even	2
4032.2.b.q.3583.4		8	168.83	odd	2
4032.2.b.q.3583.5		8	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.922382i q^{5} +(-2.06644 + 1.65222i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.922382i q^{5} +(-2.06644 + 1.65222i) q^{7} +1.00000 q^{9} +1.61794i q^{11} +2.82843i q^{13} +0.922382i q^{15} +0.922382i q^{17} -0.540319 q^{19} +(-2.06644 + 1.65222i) q^{21} +7.05526i q^{23} +4.14921 q^{25} +1.00000 q^{27} +1.01634 q^{29} -8.67319 q^{31} +1.61794i q^{33} +(-1.52398 - 1.90604i) q^{35} +8.19717 q^{37} +2.82843i q^{39} +6.57924i q^{41} -4.96130i q^{43} +0.922382i q^{45} -2.60889 q^{47} +(1.54032 - 6.82843i) q^{49} +0.922382i q^{51} +5.59255 q^{53} -1.49236 q^{55} -0.540319 q^{57} -1.65685 q^{59} -7.01634i q^{61} +(-2.06644 + 1.65222i) q^{63} -2.60889 q^{65} -6.80606i q^{67} +7.05526i q^{69} +5.21049i q^{71} -7.50162i q^{73} +4.14921 q^{75} +(-2.67319 - 3.34336i) q^{77} +7.11654i q^{79} +1.00000 q^{81} +17.9226 q^{83} -0.850789 q^{85} +1.01634 q^{87} +17.9523i q^{89} +(-4.67319 - 5.84476i) q^{91} -8.67319 q^{93} -0.498381i q^{95} -4.18791i q^{97} +1.61794i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + 4 * q^7 + 8 * q^9	\( 8 q + 8 q^{3} + 4 q^{7} + 8 q^{9} + 8 q^{19} + 4 q^{21} - 16 q^{25} + 8 q^{27} - 16 q^{31} - 8 q^{35} + 8 q^{37} + 16 q^{47} + 16 q^{53} - 8 q^{55} + 8 q^{57} + 32 q^{59} + 4 q^{63} + 16 q^{65} - 16 q^{75} + 32 q^{77} + 8 q^{81} + 16 q^{83} - 56 q^{85} + 16 q^{91} - 16 q^{93}+O(q^{100}) \) 8 * q + 8 * q^3 + 4 * q^7 + 8 * q^9 + 8 * q^19 + 4 * q^21 - 16 * q^25 + 8 * q^27 - 16 * q^31 - 8 * q^35 + 8 * q^37 + 16 * q^47 + 16 * q^53 - 8 * q^55 + 8 * q^57 + 32 * q^59 + 4 * q^63 + 16 * q^65 - 16 * q^75 + 32 * q^77 + 8 * q^81 + 16 * q^83 - 56 * q^85 + 16 * q^91 - 16 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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