LMFDB - Embedded newform 168.2.i.d.125.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,2,Mod(125,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("168.125");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	168.i (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:	$1.34148675396$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.10070523904.11
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{4} + 81 $ x^8 - 10*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			125.3
Root			$-0.420861 - 1.68014i$ of defining polynomial
Character	$\chi$	$=$	168.125
Dual form			168.2.i.d.125.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} +(0.420861 + 1.68014i) q^{3} -2.00000 q^{4} +2.16991i q^{5} +(2.37608 - 0.595188i) q^{6} +2.64575 q^{7} +2.82843i q^{8} +(-2.64575 + 1.41421i) q^{9} +O(q^{10})$	\(q-1.41421i q^{2} +(0.420861 + 1.68014i) q^{3} -2.00000 q^{4} +2.16991i q^{5} +(2.37608 - 0.595188i) q^{6} +2.64575 q^{7} +2.82843i q^{8} +(-2.64575 + 1.41421i) q^{9} +3.06871 q^{10} +(-0.841723 - 3.36028i) q^{12} +5.59388 q^{13} -3.74166i q^{14} +(-3.64575 + 0.913230i) q^{15} +4.00000 q^{16} +(2.00000 + 3.74166i) q^{18} -8.66259 q^{19} -4.33981i q^{20} +(1.11349 + 4.44524i) q^{21} -7.48331i q^{23} +(-4.75216 + 1.19038i) q^{24} +0.291503 q^{25} -7.91094i q^{26} +(-3.48957 - 3.85005i) q^{27} -5.29150 q^{28} +(1.29150 + 5.15587i) q^{30} -5.65685i q^{32} +5.74103i q^{35} +(5.29150 - 2.82843i) q^{36} +12.2508i q^{38} +(2.35425 + 9.39851i) q^{39} -6.13742 q^{40} +(6.28651 - 1.57472i) q^{42} +(-3.06871 - 5.74103i) q^{45} -10.5830 q^{46} +(1.68345 + 6.72057i) q^{48} +7.00000 q^{49} -0.412247i q^{50} -11.1878 q^{52} +(-5.44479 + 4.93500i) q^{54} +7.48331i q^{56} +(-3.64575 - 14.5544i) q^{57} -14.4207i q^{59} +(7.29150 - 1.82646i) q^{60} +0.543544 q^{61} +(-7.00000 + 3.74166i) q^{63} -8.00000 q^{64} +12.1382i q^{65} +(12.5730 - 3.14944i) q^{69} +8.11905 q^{70} +5.65685i q^{71} +(-4.00000 - 7.48331i) q^{72} +(0.122682 + 0.489766i) q^{75} +17.3252 q^{76} +(13.2915 - 3.32941i) q^{78} +5.29150 q^{79} +8.67963i q^{80} +(5.00000 - 7.48331i) q^{81} -1.40122i q^{83} +(-2.22699 - 8.89047i) q^{84} +(-8.11905 + 4.33981i) q^{90} +14.8000 q^{91} +14.9666i q^{92} -18.7970i q^{95} +(9.50432 - 2.38075i) q^{96} -9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4}+O(q^{10}) $ 8 * q - 16 * q^4	$ 8 q - 16 q^{4} - 8 q^{15} + 32 q^{16} + 16 q^{18} - 40 q^{25} - 32 q^{30} + 40 q^{39} + 56 q^{49} - 8 q^{57} + 16 q^{60} - 56 q^{63} - 64 q^{64} - 32 q^{72} + 64 q^{78} + 40 q^{81}+O(q^{100}) $ 8 * q - 16 * q^4 - 8 * q^15 + 32 * q^16 + 16 * q^18 - 40 * q^25 - 32 * q^30 + 40 * q^39 + 56 * q^49 - 8 * q^57 + 16 * q^60 - 56 * q^63 - 64 * q^64 - 32 * q^72 + 64 * q^78 + 40 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$		−	1.41421i		−	1.00000i
$2$		−	1.41421i		−	1.00000i
$3$	0.420861	+	1.68014i	0.242984	+	0.970030i
$3$	0.420861	+	1.68014i	0.242984	+	0.970030i
$4$	−2.00000			−1.00000
$5$			2.16991i			0.970412i	0.874400	+	0.485206i	$0.161255\pi$
$5$			2.16991i			0.970412i	−0.874400	+	0.485206i	$0.838745\pi$
$6$	2.37608	−	0.595188i	0.970030	−	0.242984i
$7$	2.64575			1.00000
$7$	2.64575			1.00000
$8$			2.82843i			1.00000i
$9$	−2.64575	+	1.41421i	−0.881917	+	0.471405i
$10$	3.06871			0.970412
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−0.841723	−	3.36028i	−0.242984	−	0.970030i
$13$	5.59388			1.55146			0.775732	−	0.631063i	$-0.217381\pi$
$13$	5.59388			1.55146			0.775732	+	0.631063i	$0.217381\pi$
$14$		−	3.74166i		−	1.00000i
$15$	−3.64575	+	0.913230i	−0.941329	+	0.235795i
$16$	4.00000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	2.00000	+	3.74166i	0.471405	+	0.881917i
$19$	−8.66259			−1.98734			−0.993668	−	0.112360i	$-0.964159\pi$
$19$	−8.66259			−1.98734			−0.993668	+	0.112360i	$0.964159\pi$
$20$		−	4.33981i		−	0.970412i
$21$	1.11349	+	4.44524i	0.242984	+	0.970030i
$22$		0			0
$23$		−	7.48331i		−	1.56038i	−0.625543	−	0.780189i	$-0.715123\pi$
$23$		−	7.48331i		−	1.56038i	0.625543	−	0.780189i	$-0.284877\pi$
$24$	−4.75216	+	1.19038i	−0.970030	+	0.242984i
$25$	0.291503			0.0583005
$26$		−	7.91094i		−	1.55146i
$27$	−3.48957	−	3.85005i	−0.671569	−	0.740942i
$28$	−5.29150			−1.00000
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	1.29150	+	5.15587i	0.235795	+	0.941329i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		−	5.65685i		−	1.00000i
$33$		0			0
$34$		0			0
$35$			5.74103i			0.970412i
$36$	5.29150	−	2.82843i	0.881917	−	0.471405i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$			12.2508i			1.98734i
$39$	2.35425	+	9.39851i	0.376981	+	1.50497i
$40$	−6.13742			−0.970412
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	6.28651	−	1.57472i	0.970030	−	0.242984i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−3.06871	−	5.74103i	−0.457457	−	0.855823i
$46$	−10.5830			−1.56038
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	1.68345	+	6.72057i	0.242984	+	0.970030i
$49$	7.00000			1.00000
$50$		−	0.412247i		−	0.0583005i
$51$		0			0
$52$	−11.1878			−1.55146
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−5.44479	+	4.93500i	−0.740942	+	0.671569i
$55$		0			0
$56$			7.48331i			1.00000i
$57$	−3.64575	−	14.5544i	−0.482892	−	1.92778i
$58$		0			0
$59$		−	14.4207i		−	1.87741i	−0.344721	−	0.938705i	$-0.612026\pi$
$59$		−	14.4207i		−	1.87741i	0.344721	−	0.938705i	$-0.387974\pi$
$60$	7.29150	−	1.82646i	0.941329	−	0.235795i
$61$	0.543544			0.0695936			0.0347968	−	0.999394i	$-0.488922\pi$
$61$	0.543544			0.0695936			0.0347968	+	0.999394i	$0.488922\pi$
$62$		0			0
$63$	−7.00000	+	3.74166i	−0.881917	+	0.471405i
$64$	−8.00000			−1.00000
$65$			12.1382i			1.50556i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	12.5730	−	3.14944i	1.51361	−	0.379148i
$70$	8.11905			0.970412
$71$			5.65685i			0.671345i	0.941979	+	0.335673i	$0.108964\pi$
$71$			5.65685i			0.671345i	−0.941979	+	0.335673i	$0.891036\pi$
$72$	−4.00000	−	7.48331i	−0.471405	−	0.881917i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	0.122682	+	0.489766i	0.0141661	+	0.0565533i
$76$	17.3252			1.98734
$77$		0			0
$78$	13.2915	−	3.32941i	1.50497	−	0.376981i
$79$	5.29150			0.595341			0.297670	−	0.954669i	$-0.403790\pi$
$79$	5.29150			0.595341			0.297670	+	0.954669i	$0.403790\pi$
$80$			8.67963i			0.970412i
$81$	5.00000	−	7.48331i	0.555556	−	0.831479i
$82$		0			0
$83$		−	1.40122i		−	0.153804i	−0.997039	−	0.0769020i	$-0.975497\pi$
$83$		−	1.40122i		−	0.153804i	0.997039	−	0.0769020i	$-0.0245028\pi$
$84$	−2.22699	−	8.89047i	−0.242984	−	0.970030i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−8.11905	+	4.33981i	−0.855823	+	0.457457i
$91$	14.8000			1.55146
$92$			14.9666i			1.56038i
$93$		0			0
$94$		0			0
$95$		−	18.7970i		−	1.92853i
$96$	9.50432	−	2.38075i	0.970030	−	0.242984i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		−	9.89949i		−	1.00000i
$99$		0			0
$100$	−0.583005			−0.0583005
$101$		−	17.9918i		−	1.79025i	−0.445815	−	0.895125i	$-0.647086\pi$
$101$		−	17.9918i		−	1.79025i	0.445815	−	0.895125i	$-0.352914\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$			15.8219i			1.55146i
$105$	−9.64575	+	2.41618i	−0.941329	+	0.235795i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	6.97915	+	7.70010i	0.671569	+	0.740942i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	10.5830			1.00000
$113$			14.1421i			1.33038i	0.746674	+	0.665190i	$0.231650\pi$
$113$			14.1421i			1.33038i	−0.746674	+	0.665190i	$0.768350\pi$
$114$	−20.5830	+	5.15587i	−1.92778	+	0.482892i
$115$	16.2381			1.51421
$116$		0			0
$117$	−14.8000	+	7.91094i	−1.36826	+	0.731367i
$118$	−20.3939			−1.87741
$119$		0			0
$120$	−2.58301	−	10.3117i	−0.235795	−	0.941329i
$121$	−11.0000			−1.00000
$122$		−	0.768687i		−	0.0695936i
$123$		0			0
$124$		0			0
$125$			11.4821i			1.02699i
$126$	5.29150	+	9.89949i	0.471405	+	0.881917i
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$			11.3137i			1.00000i
$129$		0			0
$130$	17.1660			1.50556
$131$			18.7605i			1.63911i	0.573000	+	0.819555i	$0.305779\pi$
$131$			18.7605i			1.63911i	−0.573000	+	0.819555i	$0.694221\pi$
$132$		0			0
$133$	−22.9191			−1.98734
$134$		0			0
$135$	8.35425	−	7.57205i	0.719019	−	0.651698i
$136$		0			0
$137$			14.9666i			1.27869i	0.768922	+	0.639343i	$0.220793\pi$
$137$			14.9666i			1.27869i	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−4.45398	−	17.7809i	−0.379148	−	1.51361i
$139$	−13.7129			−1.16312			−0.581558	−	0.813505i	$-0.697557\pi$
$139$	−13.7129			−1.16312			−0.581558	+	0.813505i	$0.697557\pi$
$140$		−	11.4821i		−	0.970412i
$141$		0			0
$142$	8.00000			0.671345
$143$		0			0
$144$	−10.5830	+	5.65685i	−0.881917	+	0.471405i
$145$		0			0
$146$		0			0
$147$	2.94603	+	11.7610i	0.242984	+	0.970030i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	0.692633	−	0.173499i	0.0565533	−	0.0141661i
$151$	−10.0000			−0.813788			−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000			−0.813788			−0.406894	+	0.913475i	$0.633388\pi$
$152$		−	24.5015i		−	1.98734i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−4.70850	−	18.7970i	−0.376981	−	1.50497i
$157$	−17.8687			−1.42608			−0.713040	−	0.701123i	$-0.752682\pi$
$157$	−17.8687			−1.42608			−0.713040	+	0.701123i	$0.752682\pi$
$158$		−	7.48331i		−	0.595341i
$159$		0			0
$160$	12.2748			0.970412
$161$		−	19.7990i		−	1.56038i
$162$	−10.5830	−	7.07107i	−0.831479	−	0.555556i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	−1.98162			−0.153804
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$	−12.5730	+	3.14944i	−0.970030	+	0.242984i
$169$	18.2915			1.40704
$170$		0			0
$171$	22.9191	−	12.2508i	1.75266	−	0.936839i
$172$		0			0
$173$			9.31216i			0.707991i	0.935247	+	0.353995i	$0.115177\pi$
$173$			9.31216i			0.707991i	−0.935247	+	0.353995i	$0.884823\pi$
$174$		0			0
$175$	0.771243			0.0583005
$176$		0			0
$177$	24.2288	−	6.06910i	1.82114	−	0.456182i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	6.13742	+	11.4821i	0.457457	+	0.855823i
$181$	10.6442			0.791178			0.395589	−	0.918428i	$-0.370540\pi$
$181$	10.6442			0.791178			0.395589	+	0.918428i	$0.370540\pi$
$182$		−	20.9304i		−	1.55146i
$183$	0.228757	+	0.913230i	0.0169102	+	0.0675079i
$184$	21.1660			1.56038
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−9.23254	−	10.1863i	−0.671569	−	0.740942i
$190$	−26.5830			−1.92853
$191$			22.6274i			1.63726i	0.574320	+	0.818631i	$0.305267\pi$
$191$			22.6274i			1.63726i	−0.574320	+	0.818631i	$0.694733\pi$
$192$	−3.36689	−	13.4411i	−0.242984	−	0.970030i
$193$	−26.4575			−1.90445			−0.952227	−	0.305392i	$-0.901213\pi$
$193$	−26.4575			−1.90445			−0.952227	+	0.305392i	$0.901213\pi$
$194$		0			0
$195$	−20.3939	+	5.10850i	−1.46044	+	0.365827i
$196$	−14.0000			−1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			0.824494i			0.0583005i
$201$		0			0
$202$	−25.4442			−1.79025
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	10.5830	+	19.7990i	0.735570	+	1.37612i
$208$	22.3755			1.55146
$209$		0			0
$210$	3.41699	+	13.6412i	0.235795	+	0.941329i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	−9.50432	+	2.38075i	−0.651225	+	0.163126i
$214$		0			0
$215$		0			0
$216$	10.8896	−	9.87000i	0.740942	−	0.671569i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		−	14.9666i		−	1.00000i
$225$	−0.771243	+	0.412247i	−0.0514162	+	0.0274831i
$226$	20.0000			1.33038
$227$			25.9027i			1.71922i	0.510947	+	0.859612i	$0.329295\pi$
$227$			25.9027i			1.71922i	−0.510947	+	0.859612i	$0.670705\pi$
$228$	7.29150	+	29.1088i	0.482892	+	1.92778i
$229$	29.0565			1.92011			0.960053	−	0.279817i	$-0.0902736\pi$
$229$	29.0565			1.92011			0.960053	+	0.279817i	$0.0902736\pi$
$230$		−	22.9641i		−	1.51421i
$231$		0			0
$232$		0			0
$233$		−	29.9333i		−	1.96099i	−0.196537	−	0.980497i	$-0.562969\pi$
$233$		−	29.9333i		−	1.96099i	0.196537	−	0.980497i	$-0.437031\pi$
$234$	11.1878	+	20.9304i	0.731367	+	1.36826i
$235$		0			0
$236$			28.8413i			1.87741i
$237$	2.22699	+	8.89047i	0.144659	+	0.577498i
$238$		0			0
$239$		−	7.48331i		−	0.484055i	−0.970269	−	0.242028i	$-0.922188\pi$
$239$		−	7.48331i		−	0.484055i	0.970269	−	0.242028i	$-0.0778125\pi$
$240$	−14.5830	+	3.65292i	−0.941329	+	0.235795i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$			15.5563i			1.00000i
$243$	14.6773	+	5.25127i	0.941551	+	0.336869i
$244$	−1.08709			−0.0695936
$245$			15.1894i			0.970412i
$246$		0			0
$247$	−48.4575			−3.08328
$248$		0			0
$249$	2.35425	−	0.589720i	0.149194	−	0.0373720i
$250$	16.2381			1.02699
$251$		−	21.5629i		−	1.36104i	−0.732730	−	0.680520i	$-0.761754\pi$
$251$		−	21.5629i		−	1.36104i	0.732730	−	0.680520i	$-0.238246\pi$
$252$	14.0000	−	7.48331i	0.881917	−	0.471405i
$253$		0			0
$254$		−	2.82843i		−	0.177471i
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		−	24.2764i		−	1.50556i
$261$		0			0
$262$	26.5313			1.63911
$263$		−	28.2843i		−	1.74408i	−0.489432	−	0.872041i	$-0.662796\pi$
$263$		−	28.2843i		−	1.74408i	0.489432	−	0.872041i	$-0.337204\pi$
$264$		0			0
$265$		0			0
$266$			32.4125i			1.98734i
$267$		0			0
$268$		0			0
$269$		−	4.97235i		−	0.303169i	−0.988444	−	0.151585i	$-0.951562\pi$
$269$		−	4.97235i		−	0.303169i	0.988444	−	0.151585i	$-0.0484376\pi$
$270$	−10.7085	−	11.8147i	−0.651698	−	0.719019i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	6.22876	+	24.8661i	0.376981	+	1.50497i
$274$	21.1660			1.27869
$275$		0			0
$276$	−25.1461	+	6.29888i	−1.51361	+	0.379148i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$			19.3930i			1.16312i
$279$		0			0
$280$	−16.2381			−0.970412
$281$			14.9666i			0.892834i	0.894825	+	0.446417i	$0.147300\pi$
$281$			14.9666i			0.892834i	−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$	9.74968			0.579558			0.289779	−	0.957094i	$-0.406418\pi$
$283$	9.74968			0.579558			0.289779	+	0.957094i	$0.406418\pi$
$284$		−	11.3137i		−	0.671345i
$285$	31.5817	−	7.91094i	1.87074	−	0.468604i
$286$		0			0
$287$		0			0
$288$	8.00000	+	14.9666i	0.471405	+	0.881917i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	31.0112i		−	1.81170i	−0.423603	−	0.905848i	$-0.639235\pi$
$293$		−	31.0112i		−	1.81170i	0.423603	−	0.905848i	$-0.360765\pi$
$294$	16.6326	−	4.16632i	0.970030	−	0.242984i
$295$	31.2915			1.82186
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	41.8608i		−	2.42087i
$300$	−0.245364	−	0.979531i	−0.0141661	−	0.0565533i
$301$		0			0
$302$			14.1421i			0.813788i
$303$	30.2288	−	7.57205i	1.73660	−	0.435003i
$304$	−34.6504			−1.98734
$305$			1.17944i			0.0675345i
$306$		0			0
$307$	−32.1252			−1.83348			−0.916741	−	0.399482i	$-0.869190\pi$
$307$	−32.1252			−1.83348			−0.916741	+	0.399482i	$0.869190\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$	−26.5830	+	6.65882i	−1.50497	+	0.376981i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$			25.2702i			1.42608i
$315$	−8.11905	−	15.1894i	−0.457457	−	0.855823i
$316$	−10.5830			−0.595341
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		−	17.3593i		−	0.970412i
$321$		0			0
$322$	−28.0000			−1.56038
$323$		0			0
$324$	−10.0000	+	14.9666i	−0.555556	+	0.831479i
$325$	1.63063			0.0904511
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			2.80244i			0.153804i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	4.45398	+	17.7809i	0.242984	+	0.970030i
$337$	−26.4575			−1.44123			−0.720616	−	0.693334i	$-0.756141\pi$
$337$	−26.4575			−1.44123			−0.720616	+	0.693334i	$0.756141\pi$
$338$		−	25.8681i		−	1.40704i
$339$	−23.7608	+	5.95188i	−1.29051	+	0.323262i
$340$		0			0
$341$		0			0
$342$	−17.3252	−	32.4125i	−0.936839	−	1.75266i
$343$	18.5203			1.00000
$344$		0			0
$345$	6.83399	+	27.2823i	0.367930	+	1.46883i
$346$	13.1694			0.707991
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	24.0062			1.28502			0.642510	−	0.766277i	$-0.277893\pi$
$349$	24.0062			1.28502			0.642510	+	0.766277i	$0.277893\pi$
$350$		−	1.09070i		−	0.0583005i
$351$	−19.5203	−	21.5367i	−1.04191	−	1.14954i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$	−8.58301	−	34.2646i	−0.456182	−	1.82114i
$355$	−12.2748			−0.651481
$356$		0			0
$357$		0			0
$358$		0			0
$359$			37.4166i			1.97477i	0.158334	+	0.987386i	$0.449388\pi$
$359$			37.4166i			1.97477i	−0.158334	+	0.987386i	$0.550612\pi$
$360$	16.2381	−	8.67963i	0.855823	−	0.457457i
$361$	56.0405			2.94950
$362$		−	15.0532i		−	0.791178i
$363$	−4.62948	−	18.4816i	−0.242984	−	0.970030i
$364$	−29.6000			−1.55146
$365$		0			0
$366$	1.29150	−	0.323511i	0.0675079	−	0.0169102i
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		−	29.9333i		−	1.56038i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−19.2915	+	4.83236i	−0.996209	+	0.249542i
$376$		0			0
$377$		0			0
$378$	−14.4056	+	13.0568i	−0.740942	+	0.671569i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$			37.5940i			1.92853i
$381$	0.841723	+	3.36028i	0.0431228	+	0.172153i
$382$	32.0000			1.63726
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−19.0086	+	4.76150i	−0.970030	+	0.242984i
$385$		0			0
$386$			37.4166i			1.90445i
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$	7.22451	+	28.8413i	0.365827	+	1.46044i
$391$		0			0
$392$			19.7990i			1.00000i
$393$	−31.5203	+	7.89556i	−1.58999	+	0.398278i
$394$		0			0
$395$			11.4821i			0.577726i
$396$		0			0
$397$	−12.8184			−0.643337			−0.321668	−	0.946852i	$-0.604244\pi$
$397$	−12.8184			−0.643337			−0.321668	+	0.946852i	$0.604244\pi$
$398$		0			0
$399$	−9.64575	−	38.5073i	−0.482892	−	1.92778i
$400$	1.16601			0.0583005
$401$		−	36.7696i		−	1.83618i	−0.396368	−	0.918092i	$-0.629729\pi$
$401$		−	36.7696i		−	1.83618i	0.396368	−	0.918092i	$-0.370271\pi$
$402$		0			0
$403$		0			0
$404$			35.9836i			1.79025i
$405$	16.2381	+	10.8495i	0.806878	+	0.539118i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−25.1461	+	6.29888i	−1.24036	+	0.310701i
$412$		0			0
$413$		−	38.1535i		−	1.87741i
$414$	28.0000	−	14.9666i	1.37612	−	0.735570i
$415$	3.04052			0.149253
$416$		−	31.6438i		−	1.55146i
$417$	−5.77124	−	23.0397i	−0.282619	−	1.12826i
$418$		0			0
$419$			12.8833i			0.629390i	0.949193	+	0.314695i	$0.101902\pi$
$419$			12.8833i			0.629390i	−0.949193	+	0.314695i	$0.898098\pi$
$420$	19.2915	−	4.83236i	0.941329	−	0.235795i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	3.36689	+	13.4411i	0.163126	+	0.651225i
$427$	1.43808			0.0695936
$428$		0			0
$429$		0			0
$430$		0			0
$431$			37.4166i			1.80229i	0.433515	+	0.901146i	$0.357273\pi$
$431$			37.4166i			1.80229i	−0.433515	+	0.901146i	$0.642727\pi$
$432$	−13.9583	−	15.4002i	−0.671569	−	0.740942i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			64.8249i			3.10100i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−18.5203	+	9.89949i	−0.881917	+	0.471405i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−21.1660			−1.00000
$449$		−	29.9333i		−	1.41264i	−0.707894	−	0.706319i	$-0.750354\pi$
$449$		−	29.9333i		−	1.41264i	0.707894	−	0.706319i	$-0.249646\pi$
$450$	0.583005	+	1.09070i	0.0274831	+	0.0514162i
$451$		0			0
$452$		−	28.2843i		−	1.33038i
$453$	−4.20861	−	16.8014i	−0.197738	−	0.789399i
$454$	36.6320			1.71922
$455$			32.1147i			1.50556i
$456$	41.1660	−	10.3117i	1.92778	−	0.482892i
$457$	5.29150			0.247526			0.123763	−	0.992312i	$-0.460504\pi$
$457$	5.29150			0.247526			0.123763	+	0.992312i	$0.460504\pi$
$458$		−	41.0921i		−	1.92011i
$459$		0			0
$460$	−32.4762			−1.51421
$461$			42.4933i			1.97911i	0.144154	+	0.989555i	$0.453954\pi$
$461$			42.4933i			1.97911i	−0.144154	+	0.989555i	$0.546046\pi$
$462$		0			0
$463$	−26.4575			−1.22958			−0.614792	−	0.788689i	$-0.710760\pi$
$463$	−26.4575			−1.22958			−0.614792	+	0.788689i	$0.710760\pi$
$464$		0			0
$465$		0			0
$466$	−42.3320			−1.96099
$467$			11.6182i			0.537627i	0.963192	+	0.268814i	$0.0866316\pi$
$467$			11.6182i			0.537627i	−0.963192	+	0.268814i	$0.913368\pi$
$468$	29.6000	−	15.8219i	1.36826	−	0.731367i
$469$		0			0
$470$		0			0
$471$	−7.52026	−	30.0220i	−0.346515	−	1.38334i
$472$	40.7878			1.87741
$473$		0			0
$474$	12.5730	−	3.14944i	0.577498	−	0.144659i
$475$	−2.52517			−0.115863
$476$		0			0
$477$		0			0
$478$	−10.5830			−0.484055
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	5.16601	+	20.6235i	0.235795	+	0.941329i
$481$		0			0
$482$		0			0
$483$	33.2651	−	8.33263i	1.51361	−	0.379148i
$484$	22.0000			1.00000
$485$		0			0
$486$	7.42642	−	20.7569i	0.336869	−	0.941551i
$487$	38.0000			1.72194			0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000			1.72194			0.860972	+	0.508652i	$0.169856\pi$
$488$			1.53737i			0.0695936i
$489$		0			0
$490$	21.4810			0.970412
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$			68.5293i			3.08328i
$495$		0			0
$496$		0			0
$497$			14.9666i			0.671345i
$498$	−0.833990	−	3.32941i	−0.0373720	−	0.149194i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		−	22.9641i		−	1.02699i
$501$		0			0
$502$	−30.4946			−1.36104
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−10.5830	−	19.7990i	−0.471405	−	0.881917i
$505$	39.0405			1.73728
$506$		0			0
$507$	7.69819	+	30.7323i	0.341889	+	1.36487i
$508$	−4.00000			−0.177471
$509$		−	25.1340i		−	1.11405i	−0.830497	−	0.557024i	$-0.811943\pi$
$509$		−	25.1340i		−	1.11405i	0.830497	−	0.557024i	$-0.188057\pi$
$510$		0			0
$511$		0			0
$512$		−	22.6274i		−	1.00000i
$513$	30.2288	+	33.3514i	1.33463	+	1.47250i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−15.6458	+	3.91913i	−0.686772	+	0.172031i
$520$	−34.3320			−1.50556
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	38.2626			1.67311			0.836554	−	0.547885i	$-0.184567\pi$
$523$	38.2626			1.67311			0.836554	+	0.547885i	$0.184567\pi$
$524$		−	37.5210i		−	1.63911i
$525$	0.324587	+	1.29580i	0.0141661	+	0.0565533i
$526$	−40.0000			−1.74408
$527$		0			0
$528$		0			0
$529$	−33.0000			−1.43478
$530$		0			0
$531$	20.3939	+	38.1535i	0.885020	+	1.65572i
$532$	45.8381			1.98734
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−7.03196			−0.303169
$539$		0			0
$540$	−16.7085	+	15.1441i	−0.719019	+	0.651698i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	4.47974	+	17.8838i	0.192244	+	0.767467i
$544$		0			0
$545$		0			0
$546$	35.1660	−	8.80879i	1.50497	−	0.376981i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		−	29.9333i		−	1.27869i
$549$	−1.43808	+	0.768687i	−0.0613758	+	0.0328068i
$550$		0			0
$551$		0			0
$552$	8.90796	+	35.5619i	0.379148	+	1.51361i
$553$	14.0000			0.595341
$554$		0			0
$555$		0			0
$556$	27.4259			1.16312
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$			22.9641i			0.970412i
$561$		0			0
$562$	21.1660			0.892834
$563$		−	8.54348i		−	0.360065i	−0.983661	−	0.180032i	$-0.942380\pi$
$563$		−	8.54348i		−	0.360065i	0.983661	−	0.180032i	$-0.0576202\pi$
$564$		0			0
$565$	−30.6871			−1.29102
$566$		−	13.7881i		−	0.579558i
$567$	13.2288	−	19.7990i	0.555556	−	0.831479i
$568$	−16.0000			−0.671345
$569$		−	2.82843i		−	0.118574i	−0.998241	−	0.0592869i	$-0.981117\pi$
$569$		−	2.82843i		−	0.118574i	0.998241	−	0.0592869i	$-0.0188827\pi$
$570$	−11.1878	−	44.6632i	−0.468604	−	1.87074i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−38.0173	+	9.52301i	−1.58819	+	0.397829i
$574$		0			0
$575$		−	2.18141i		−	0.0909709i
$576$	21.1660	−	11.3137i	0.881917	−	0.471405i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			24.0416i			1.00000i
$579$	−11.1349	−	44.4524i	−0.462753	−	1.84738i
$580$		0			0
$581$		−	3.70728i		−	0.153804i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−17.1660	−	32.1147i	−0.709727	−	1.32778i
$586$	−43.8565			−1.81170
$587$			31.7799i			1.31170i	0.754892	+	0.655849i	$0.227689\pi$
$587$			31.7799i			1.31170i	−0.754892	+	0.655849i	$0.772311\pi$
$588$	−5.89206	−	23.5220i	−0.242984	−	0.970030i
$589$		0			0
$590$		−	44.2529i		−	1.82186i
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−59.2001			−2.42087
$599$		−	11.3137i		−	0.462266i	−0.972922	−	0.231133i	$-0.925757\pi$
$599$		−	11.3137i		−	0.462266i	0.972922	−	0.231133i	$-0.0742432\pi$
$600$	−1.38527	+	0.346998i	−0.0565533	+	0.0141661i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	20.0000			0.813788
$605$		−	23.8690i		−	0.970412i
$606$	−10.7085	−	42.7499i	−0.435003	−	1.73660i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$			49.0030i			1.98734i
$609$		0			0
$610$	1.66798			0.0675345
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			45.4319i			1.83348i
$615$		0			0
$616$		0			0
$617$			14.1421i			0.569341i	0.958625	+	0.284670i	$0.0918842\pi$
$617$			14.1421i			0.569341i	−0.958625	+	0.284670i	$0.908116\pi$
$618$		0			0
$619$	−42.2259			−1.69720			−0.848601	−	0.529034i	$-0.822554\pi$
$619$	−42.2259			−1.69720			−0.848601	+	0.529034i	$0.822554\pi$
$620$		0			0
$621$	−28.8111	+	26.1136i	−1.15615	+	1.04790i
$622$		0			0
$623$		0			0
$624$	9.41699	+	37.5940i	0.376981	+	1.50497i
$625$	−23.4575			−0.938301
$626$		0			0
$627$		0			0
$628$	35.7375			1.42608
$629$		0			0
$630$	−21.4810	+	11.4821i	−0.855823	+	0.457457i
$631$	37.0405			1.47456			0.737280	−	0.675587i	$-0.236110\pi$
$631$	37.0405			1.47456			0.737280	+	0.675587i	$0.236110\pi$
$632$			14.9666i			0.595341i
$633$		0			0
$634$		0			0
$635$			4.33981i			0.172220i
$636$		0			0
$637$	39.1572			1.55146
$638$		0			0
$639$	−8.00000	−	14.9666i	−0.316475	−	0.592071i
$640$	−24.5497			−0.970412
$641$			48.0833i			1.89917i	0.313503	+	0.949587i	$0.398498\pi$
$641$			48.0833i			1.89917i	−0.313503	+	0.949587i	$0.601502\pi$
$642$		0			0
$643$	48.3633			1.90726			0.953631	−	0.300978i	$-0.0973129\pi$
$643$	48.3633			1.90726			0.953631	+	0.300978i	$0.0973129\pi$
$644$			39.5980i			1.56038i
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	21.1660	+	14.1421i	0.831479	+	0.555556i
$649$		0			0
$650$		−	2.30606i		−	0.0904511i
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$	−40.7085			−1.59061
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−41.3313			−1.60760			−0.803801	−	0.594898i	$-0.797193\pi$
$661$	−41.3313			−1.60760			−0.803801	+	0.594898i	$0.797193\pi$
$662$		0			0
$663$		0			0
$664$	3.96325			0.153804
$665$		−	49.7322i		−	1.92853i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	25.1461	−	6.29888i	0.970030	−	0.242984i
$673$	26.0000			1.00223			0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000			1.00223			0.501113	+	0.865382i	$0.332924\pi$
$674$			37.4166i			1.44123i
$675$	−1.01722	−	1.12230i	−0.0391528	−	0.0431973i
$676$	−36.5830			−1.40704
$677$		−	51.1729i		−	1.96674i	−0.181625	−	0.983368i	$-0.558136\pi$
$677$		−	51.1729i		−	1.96674i	0.181625	−	0.983368i	$-0.441864\pi$
$678$	8.41723	+	33.6028i	0.323262	+	1.29051i
$679$		0			0
$680$		0			0
$681$	−43.5203	+	10.9015i	−1.66770	+	0.417745i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−45.8381	+	24.5015i	−1.75266	+	0.936839i
$685$	−32.4762			−1.24085
$686$		−	26.1916i		−	1.00000i
$687$	12.2288	+	48.8190i	0.466556	+	1.86256i
$688$		0			0
$689$		0			0
$690$	38.5830	−	9.66472i	1.46883	−	0.367930i
$691$	4.69934			0.178771			0.0893857	−	0.995997i	$-0.471510\pi$
$691$	4.69934			0.178771			0.0893857	+	0.995997i	$0.471510\pi$
$692$		−	18.6243i		−	0.707991i
$693$		0			0
$694$		0			0
$695$		−	29.7558i		−	1.12870i
$696$		0			0
$697$		0			0
$698$		−	33.9498i		−	1.28502i
$699$	50.2921	−	12.5978i	1.90222	−	0.476491i
$700$	−1.54249			−0.0583005
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	−30.4575	+	27.6058i	−1.14954	+	1.04191i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	47.6018i		−	1.79025i
$708$	−48.4575	+	12.1382i	−1.82114	+	0.456182i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$			17.3593i			0.651481i
$711$	−14.0000	+	7.48331i	−0.525041	+	0.280646i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	12.5730	−	3.14944i	0.469548	−	0.117618i
$718$	52.9150			1.97477
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−12.2748	−	22.9641i	−0.457457	−	0.855823i
$721$		0			0
$722$		−	79.2533i		−	2.94950i
$723$		0			0
$724$	−21.2884			−0.791178
$725$		0			0
$726$	−26.1369	+	6.54707i	−0.970030	+	0.242984i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$			41.8608i			1.55146i
$729$	−2.64575	+	26.8701i	−0.0979908	+	0.995187i
$730$		0			0
$731$		0			0
$732$	−0.457513	−	1.82646i	−0.0169102	−	0.0675079i
$733$	−33.0197			−1.21961			−0.609806	−	0.792551i	$-0.708753\pi$
$733$	−33.0197			−1.21961			−0.609806	+	0.792551i	$0.708753\pi$
$734$		0			0
$735$	−25.5203	+	6.39261i	−0.941329	+	0.235795i
$736$	−42.3320			−1.56038
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$	−20.3939	−	81.4155i	−0.749189	−	2.99087i
$742$		0			0
$743$		−	7.48331i		−	0.274536i	−0.990534	−	0.137268i	$-0.956168\pi$
$743$		−	7.48331i		−	0.274536i	0.990534	−	0.137268i	$-0.0438322\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.98162	+	3.70728i	0.0725039	+	0.135642i
$748$		0			0
$749$		0			0
$750$	6.83399	+	27.2823i	0.249542	+	0.996209i
$751$	50.0000			1.82453			0.912263	−	0.409605i	$-0.134333\pi$
$751$	50.0000			1.82453			0.912263	+	0.409605i	$0.134333\pi$
$752$		0			0
$753$	36.2288	−	9.07500i	1.32025	−	0.330711i
$754$		0			0
$755$		−	21.6991i		−	0.789710i
$756$	18.4651	+	20.3725i	0.671569	+	0.740942i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$	53.1660			1.92853
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$	4.75216	−	1.19038i	0.172153	−	0.0431228i
$763$		0			0
$764$		−	45.2548i		−	1.63726i
$765$		0			0
$766$		0			0
$767$		−	80.6675i		−	2.91273i
$768$	6.73378	+	26.8823i	0.242984	+	0.970030i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	52.9150			1.90445
$773$			16.4544i			0.591824i	0.955215	+	0.295912i	$0.0956236\pi$
$773$			16.4544i			0.591824i	−0.955215	+	0.295912i	$0.904376\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	40.7878	−	10.2170i	1.46044	−	0.365827i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$		−	38.7735i		−	1.38389i
$786$	11.1660	+	44.5764i	0.398278	+	1.58999i
$787$	33.2123			1.18389			0.591945	−	0.805978i	$-0.298360\pi$
$787$	33.2123			1.18389			0.591945	+	0.805978i	$0.298360\pi$
$788$		0			0
$789$	47.5216	−	11.9038i	1.69181	−	0.423785i
$790$	16.2381			0.577726
$791$			37.4166i			1.33038i
$792$		0			0
$793$	3.04052			0.107972
$794$			18.1279i			0.643337i
$795$		0			0
$796$		0			0
$797$			8.04710i			0.285043i	0.989792	+	0.142521i	$0.0455210\pi$
$797$			8.04710i			0.285043i	−0.989792	+	0.142521i	$0.954479\pi$
$798$	−54.4575	+	13.6412i	−1.92778	+	0.482892i
$799$		0			0
$800$		−	1.64899i		−	0.0583005i
$801$		0			0
$802$	−52.0000			−1.83618
$803$		0			0
$804$		0			0
$805$	42.9620			1.51421
$806$		0			0
$807$	8.35425	−	2.09267i	0.294083	−	0.0736655i
$808$	50.8885			1.79025
$809$			31.1127i			1.09386i	0.837177	+	0.546932i	$0.184204\pi$
$809$			31.1127i			1.09386i	−0.837177	+	0.546932i	$0.815796\pi$
$810$	15.3436	−	22.9641i	0.539118	−	0.806878i
$811$	29.9510			1.05172			0.525861	−	0.850570i	$-0.323743\pi$
$811$	29.9510			1.05172			0.525861	+	0.850570i	$0.323743\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−39.1572	+	20.9304i	−1.36826	+	0.731367i
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	8.90796	+	35.5619i	0.310701	+	1.24036i
$823$	−26.4575			−0.922251			−0.461125	−	0.887335i	$-0.652554\pi$
$823$	−26.4575			−0.922251			−0.461125	+	0.887335i	$0.652554\pi$
$824$		0			0
$825$		0			0
$826$	−53.9572			−1.87741
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−21.1660	−	39.5980i	−0.735570	−	1.37612i
$829$	57.5694			1.99947			0.999735	−	0.0230361i	$-0.00733328\pi$
$829$	57.5694			1.99947			0.999735	+	0.0230361i	$0.00733328\pi$
$830$		−	4.29994i		−	0.149253i
$831$		0			0
$832$	−44.7510			−1.55146
$833$		0			0
$834$	−32.5830	+	8.16177i	−1.12826	+	0.282619i
$835$		0			0
$836$		0			0
$837$		0			0
$838$	18.2197			0.629390
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	−6.83399	−	27.2823i	−0.235795	−	0.941329i
$841$	−29.0000			−1.00000
$842$		0			0
$843$	−25.1461	+	6.29888i	−0.866076	+	0.216945i
$844$		0			0
$845$			39.6909i			1.36541i
$846$		0			0
$847$	−29.1033			−1.00000
$848$		0			0
$849$	4.10326	+	16.3808i	0.140824	+	0.562189i
$850$		0			0
$851$		0			0
$852$	19.0086	−	4.76150i	0.651225	−	0.163126i
$853$	−7.76806			−0.265973			−0.132987	−	0.991118i	$-0.542457\pi$
$853$	−7.76806			−0.265973			−0.132987	+	0.991118i	$0.542457\pi$
$854$		−	2.03375i		−	0.0695936i
$855$	26.5830	+	49.7322i	0.909120	+	1.70081i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−22.0245			−0.751467			−0.375734	−	0.926728i	$-0.622609\pi$
$859$	−22.0245			−0.751467			−0.375734	+	0.926728i	$0.622609\pi$
$860$		0			0
$861$		0			0
$862$	52.9150			1.80229
$863$			56.5685i			1.92562i	0.270187	+	0.962808i	$0.412914\pi$
$863$			56.5685i			1.92562i	−0.270187	+	0.962808i	$0.587086\pi$
$864$	−21.7792	+	19.7400i	−0.740942	+	0.671569i
$865$	−20.2065			−0.687043
$866$		0			0
$867$	−7.15464	−	28.5624i	−0.242984	−	0.970030i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	91.6763			3.10100
$875$			30.3787i			1.02699i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$	52.1033	−	13.0514i	1.75740	−	0.440214i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	14.0000	+	26.1916i	0.471405	+	0.881917i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	13.1694	+	52.5742i	0.442684	+	1.76726i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	5.29150			0.177471
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$			29.9333i			1.00000i
$897$	70.3320	−	17.6176i	2.34832	−	0.588234i
$898$	−42.3320			−1.41264
$899$		0			0
$900$	1.54249	−	0.824494i	0.0514162	−	0.0274831i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−40.0000			−1.33038
$905$			23.0970i			0.767769i
$906$	−23.7608	+	5.95188i	−0.789399	+	0.197738i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		−	51.8055i		−	1.71922i
$909$	25.4442	+	47.6018i	0.843932	+	1.57885i
$910$	45.4170			1.50556
$911$		−	52.3832i		−	1.73553i	−0.496972	−	0.867766i	$-0.665555\pi$
$911$		−	52.3832i		−	1.73553i	0.496972	−	0.867766i	$-0.334445\pi$
$912$	−14.5830	−	58.2175i	−0.482892	−	1.92778i
$913$		0			0
$914$		−	7.48331i		−	0.247526i
$915$	−1.98162	+	0.496381i	−0.0655105	+	0.0164098i
$916$	−58.1130			−1.92011
$917$			49.6356i			1.63911i
$918$		0			0
$919$	−58.2065			−1.92006			−0.960028	−	0.279904i	$-0.909697\pi$
$919$	−58.2065			−1.92006			−0.960028	+	0.279904i	$0.909697\pi$
$920$			45.9283i			1.51421i
$921$	−13.5203	−	53.9749i	−0.445508	−	1.77853i
$922$	60.0946			1.97911
$923$			31.6438i			1.04157i
$924$		0			0
$925$		0			0
$926$			37.4166i			1.22958i
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−60.6382			−1.98734
$932$			59.8665i			1.96099i
$933$		0			0
$934$	16.4306			0.537627
$935$		0			0
$936$	−22.3755	−	41.8608i	−0.731367	−	1.36826i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	58.3152i		−	1.90102i	−0.310693	−	0.950510i	$-0.600561\pi$
$941$		−	58.3152i		−	1.90102i	0.310693	−	0.950510i	$-0.399439\pi$
$942$	−42.4575	+	10.6353i	−1.38334	+	0.346515i
$943$		0			0
$944$		−	57.6827i		−	1.87741i
$945$	22.1033	−	20.0338i	0.719019	−	0.651698i
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$	−4.45398	−	17.7809i	−0.144659	−	0.577498i
$949$		0			0
$950$			3.57113i			0.115863i
$951$		0			0
$952$		0			0
$953$		−	29.9333i		−	0.969633i	−0.874616	−	0.484817i	$-0.838886\pi$
$953$		−	29.9333i		−	0.969633i	0.874616	−	0.484817i	$-0.161114\pi$
$954$		0			0
$955$	−49.0994			−1.58882
$956$			14.9666i			0.484055i
$957$		0			0
$958$		0			0
$959$			39.5980i			1.27869i
$960$	29.1660	−	7.30584i	0.941329	−	0.235795i
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	57.4103i		−	1.84810i
$966$	−11.7841	−	47.0440i	−0.379148	−	1.51361i
$967$	−58.0000			−1.86515			−0.932577	−	0.360971i	$-0.882445\pi$
$967$	−58.0000			−1.86515			−0.932577	+	0.360971i	$0.882445\pi$
$968$		−	31.1127i		−	1.00000i
$969$		0			0
$970$		0			0
$971$		−	0.136153i		−	0.00436936i	−0.999998	−	0.00218468i	$-0.999305\pi$
$971$		−	0.136153i		−	0.00436936i	0.999998	−	0.00218468i	$-0.000695406\pi$
$972$	−29.3547	−	10.5025i	−0.941551	−	0.336869i
$973$	−36.2810			−1.16312
$974$		−	53.7401i		−	1.72194i
$975$	0.686270	+	2.73969i	0.0219782	+	0.0877403i
$976$	2.17417			0.0695936
$977$			59.8665i			1.91530i	0.287936	+	0.957650i	$0.407031\pi$
$977$			59.8665i			1.91530i	−0.287936	+	0.957650i	$0.592969\pi$
$978$		0			0
$979$		0			0
$980$		−	30.3787i		−	0.970412i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	96.9150			3.08328
$989$		0			0
$990$		0			0
$991$	37.0405			1.17663			0.588315	−	0.808632i	$-0.299791\pi$
$991$	37.0405			1.17663			0.588315	+	0.808632i	$0.299791\pi$
$992$		0			0
$993$		0			0
$994$	21.1660			0.671345
$995$		0			0
$996$	−4.70850	+	1.17944i	−0.149194	+	0.0373720i
$997$	47.4688			1.50335			0.751675	−	0.659533i	$-0.229246\pi$
$997$	47.4688			1.50335			0.751675	+	0.659533i	$0.229246\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.i.d.125.3	yes	8
3.2	odd	2	inner	168.2.i.d.125.7	yes	8
4.3	odd	2		672.2.i.e.209.3		8
7.6	odd	2	inner	168.2.i.d.125.2	✓	8
8.3	odd	2		672.2.i.e.209.6		8
8.5	even	2	inner	168.2.i.d.125.2	✓	8
12.11	even	2		672.2.i.e.209.4		8
21.20	even	2	inner	168.2.i.d.125.6	yes	8
24.5	odd	2	inner	168.2.i.d.125.6	yes	8
24.11	even	2		672.2.i.e.209.5		8
28.27	even	2		672.2.i.e.209.6		8
56.13	odd	2	CM	168.2.i.d.125.3	yes	8
56.27	even	2		672.2.i.e.209.3		8
84.83	odd	2		672.2.i.e.209.5		8
168.83	odd	2		672.2.i.e.209.4		8
168.125	even	2	inner	168.2.i.d.125.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.i.d.125.2	✓	8	7.6	odd	2	inner
168.2.i.d.125.2	✓	8	8.5	even	2	inner
168.2.i.d.125.3	yes	8	1.1	even	1	trivial
168.2.i.d.125.3	yes	8	56.13	odd	2	CM
168.2.i.d.125.6	yes	8	21.20	even	2	inner
168.2.i.d.125.6	yes	8	24.5	odd	2	inner
168.2.i.d.125.7	yes	8	3.2	odd	2	inner
168.2.i.d.125.7	yes	8	168.125	even	2	inner
672.2.i.e.209.3		8	4.3	odd	2
672.2.i.e.209.3		8	56.27	even	2
672.2.i.e.209.4		8	12.11	even	2
672.2.i.e.209.4		8	168.83	odd	2
672.2.i.e.209.5		8	24.11	even	2
672.2.i.e.209.5		8	84.83	odd	2
672.2.i.e.209.6		8	8.3	odd	2
672.2.i.e.209.6		8	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	168.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(0.420861 + 1.68014i) q^{3} -2.00000 q^{4} +2.16991i q^{5} +(2.37608 - 0.595188i) q^{6} +2.64575 q^{7} +2.82843i q^{8} +(-2.64575 + 1.41421i) q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} +(0.420861 + 1.68014i) q^{3} -2.00000 q^{4} +2.16991i q^{5} +(2.37608 - 0.595188i) q^{6} +2.64575 q^{7} +2.82843i q^{8} +(-2.64575 + 1.41421i) q^{9} +3.06871 q^{10} +(-0.841723 - 3.36028i) q^{12} +5.59388 q^{13} -3.74166i q^{14} +(-3.64575 + 0.913230i) q^{15} +4.00000 q^{16} +(2.00000 + 3.74166i) q^{18} -8.66259 q^{19} -4.33981i q^{20} +(1.11349 + 4.44524i) q^{21} -7.48331i q^{23} +(-4.75216 + 1.19038i) q^{24} +0.291503 q^{25} -7.91094i q^{26} +(-3.48957 - 3.85005i) q^{27} -5.29150 q^{28} +(1.29150 + 5.15587i) q^{30} -5.65685i q^{32} +5.74103i q^{35} +(5.29150 - 2.82843i) q^{36} +12.2508i q^{38} +(2.35425 + 9.39851i) q^{39} -6.13742 q^{40} +(6.28651 - 1.57472i) q^{42} +(-3.06871 - 5.74103i) q^{45} -10.5830 q^{46} +(1.68345 + 6.72057i) q^{48} +7.00000 q^{49} -0.412247i q^{50} -11.1878 q^{52} +(-5.44479 + 4.93500i) q^{54} +7.48331i q^{56} +(-3.64575 - 14.5544i) q^{57} -14.4207i q^{59} +(7.29150 - 1.82646i) q^{60} +0.543544 q^{61} +(-7.00000 + 3.74166i) q^{63} -8.00000 q^{64} +12.1382i q^{65} +(12.5730 - 3.14944i) q^{69} +8.11905 q^{70} +5.65685i q^{71} +(-4.00000 - 7.48331i) q^{72} +(0.122682 + 0.489766i) q^{75} +17.3252 q^{76} +(13.2915 - 3.32941i) q^{78} +5.29150 q^{79} +8.67963i q^{80} +(5.00000 - 7.48331i) q^{81} -1.40122i q^{83} +(-2.22699 - 8.89047i) q^{84} +(-8.11905 + 4.33981i) q^{90} +14.8000 q^{91} +14.9666i q^{92} -18.7970i q^{95} +(9.50432 - 2.38075i) q^{96} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4}+O(q^{10}) \) 8 * q - 16 * q^4	\( 8 q - 16 q^{4} - 8 q^{15} + 32 q^{16} + 16 q^{18} - 40 q^{25} - 32 q^{30} + 40 q^{39} + 56 q^{49} - 8 q^{57} + 16 q^{60} - 56 q^{63} - 64 q^{64} - 32 q^{72} + 64 q^{78} + 40 q^{81}+O(q^{100}) \) 8 * q - 16 * q^4 - 8 * q^15 + 32 * q^16 + 16 * q^18 - 40 * q^25 - 32 * q^30 + 40 * q^39 + 56 * q^49 - 8 * q^57 + 16 * q^60 - 56 * q^63 - 64 * q^64 - 32 * q^72 + 64 * q^78 + 40 * q^81

Introduction

L-functions

Modular forms

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Database

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