LMFDB - Embedded newform 552.2.m.a.137.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [552,2,Mod(137,552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(552, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("552.137");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	552.m (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:	$4.40774219157$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			137.8
Root			$1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	552.137
Dual form			552.2.m.a.137.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.618034 + 1.61803i) q^{3} +2.28825 q^{5} -1.41421i q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})$	\(q+(0.618034 + 1.61803i) q^{3} +2.28825 q^{5} -1.41421i q^{7} +(-2.23607 + 2.00000i) q^{9} +1.95440 q^{11} +1.23607 q^{13} +(1.41421 + 3.70246i) q^{15} +7.73877 q^{17} +1.20788i q^{19} +(2.28825 - 0.874032i) q^{21} +(-4.24264 + 2.23607i) q^{23} +0.236068 q^{25} +(-4.61803 - 2.38197i) q^{27} +0.763932i q^{29} -6.00000 q^{31} +(1.20788 + 3.16228i) q^{33} -3.23607i q^{35} -7.61125i q^{37} +(0.763932 + 2.00000i) q^{39} +8.47214i q^{41} +5.78437i q^{43} +(-5.11667 + 4.57649i) q^{45} -1.70820i q^{47} +5.00000 q^{49} +(4.78282 + 12.5216i) q^{51} -2.28825 q^{53} +4.47214 q^{55} +(-1.95440 + 0.746512i) q^{57} -8.00000i q^{59} -8.27895i q^{61} +(2.82843 + 3.16228i) q^{63} +2.82843 q^{65} +3.36861i q^{67} +(-6.24013 - 5.48277i) q^{69} +10.1803i q^{71} -2.76393 q^{73} +(0.145898 + 0.381966i) q^{75} -2.76393i q^{77} -12.3153i q^{79} +(1.00000 - 8.94427i) q^{81} -4.78282 q^{83} +17.7082 q^{85} +(-1.23607 + 0.472136i) q^{87} -5.99070 q^{89} -1.74806i q^{91} +(-3.70820 - 9.70820i) q^{93} +2.76393i q^{95} -16.5579i q^{97} +(-4.37016 + 3.90879i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3}+O(q^{10}) $ 8 * q - 4 * q^3	$ 8 q - 4 q^{3} - 8 q^{13} - 16 q^{25} - 28 q^{27} - 48 q^{31} + 24 q^{39} + 40 q^{49} - 20 q^{69} - 40 q^{73} + 28 q^{75} + 8 q^{81} + 88 q^{85} + 8 q^{87} + 24 q^{93}+O(q^{100}) $ 8 * q - 4 * q^3 - 8 * q^13 - 16 * q^25 - 28 * q^27 - 48 * q^31 + 24 * q^39 + 40 * q^49 - 20 * q^69 - 40 * q^73 + 28 * q^75 + 8 * q^81 + 88 * q^85 + 8 * q^87 + 24 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$.

$n$	$97$	$185$	$277$	$415$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	0.618034	+	1.61803i	0.356822	+	0.934172i
$3$	0.618034	+	1.61803i	0.356822	+	0.934172i
$4$		0			0
$5$	2.28825			1.02333			0.511667	−	0.859184i	$-0.329028\pi$
$5$	2.28825			1.02333			0.511667	+	0.859184i	$0.329028\pi$
$6$		0			0
$7$		−	1.41421i		−	0.534522i	−0.963624	−	0.267261i	$-0.913881\pi$
$7$		−	1.41421i		−	0.534522i	0.963624	−	0.267261i	$-0.0861187\pi$
$8$		0			0
$9$	−2.23607	+	2.00000i	−0.745356	+	0.666667i
$10$		0			0
$11$	1.95440			0.589272			0.294636	−	0.955610i	$-0.404802\pi$
$11$	1.95440			0.589272			0.294636	+	0.955610i	$0.404802\pi$
$12$		0			0
$13$	1.23607			0.342824			0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607			0.342824			0.171412	+	0.985199i	$0.445167\pi$
$14$		0			0
$15$	1.41421	+	3.70246i	0.365148	+	0.955971i
$16$		0			0
$17$	7.73877			1.87693			0.938464	−	0.345378i	$-0.112249\pi$
$17$	7.73877			1.87693			0.938464	+	0.345378i	$0.112249\pi$
$18$		0			0
$19$			1.20788i			0.277107i	0.990355	+	0.138554i	$0.0442453\pi$
$19$			1.20788i			0.277107i	−0.990355	+	0.138554i	$0.955755\pi$
$20$		0			0
$21$	2.28825	−	0.874032i	0.499336	−	0.190729i
$22$		0			0
$23$	−4.24264	+	2.23607i	−0.884652	+	0.466252i
$23$	−4.24264	+	2.23607i	−0.884652	+	0.466252i
$24$		0			0
$25$	0.236068			0.0472136
$26$		0			0
$27$	−4.61803	−	2.38197i	−0.888741	−	0.458410i
$28$		0			0
$29$			0.763932i			0.141859i	0.997481	+	0.0709293i	$0.0225965\pi$
$29$			0.763932i			0.141859i	−0.997481	+	0.0709293i	$0.977404\pi$
$30$		0			0
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$		0			0
$33$	1.20788	+	3.16228i	0.210265	+	0.550482i
$34$		0			0
$35$		−	3.23607i		−	0.546995i
$36$		0			0
$37$		−	7.61125i		−	1.25128i	−0.780111	−	0.625641i	$-0.784838\pi$
$37$		−	7.61125i		−	1.25128i	0.780111	−	0.625641i	$-0.215162\pi$
$38$		0			0
$39$	0.763932	+	2.00000i	0.122327	+	0.320256i
$40$		0			0
$41$			8.47214i			1.32313i	0.749890	+	0.661563i	$0.230106\pi$
$41$			8.47214i			1.32313i	−0.749890	+	0.661563i	$0.769894\pi$
$42$		0			0
$43$			5.78437i			0.882109i	0.897480	+	0.441054i	$0.145395\pi$
$43$			5.78437i			0.882109i	−0.897480	+	0.441054i	$0.854605\pi$
$44$		0			0
$45$	−5.11667	+	4.57649i	−0.762749	+	0.682223i
$46$		0			0
$47$		−	1.70820i		−	0.249167i	−0.992209	−	0.124584i	$-0.960241\pi$
$47$		−	1.70820i		−	0.249167i	0.992209	−	0.124584i	$-0.0397595\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$	4.78282	+	12.5216i	0.669729	+	1.75337i
$52$		0			0
$53$	−2.28825			−0.314315			−0.157157	−	0.987574i	$-0.550233\pi$
$53$	−2.28825			−0.314315			−0.157157	+	0.987574i	$0.550233\pi$
$54$		0			0
$55$	4.47214			0.603023
$56$		0			0
$57$	−1.95440	+	0.746512i	−0.258866	+	0.0988780i
$58$		0			0
$59$		−	8.00000i		−	1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$		−	8.00000i		−	1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$		0			0
$61$		−	8.27895i		−	1.06001i	−0.847994	−	0.530005i	$-0.822190\pi$
$61$		−	8.27895i		−	1.06001i	0.847994	−	0.530005i	$-0.177810\pi$
$62$		0			0
$63$	2.82843	+	3.16228i	0.356348	+	0.398410i
$64$		0			0
$65$	2.82843			0.350823
$66$		0			0
$67$			3.36861i			0.411541i	0.978600	+	0.205771i	$0.0659700\pi$
$67$			3.36861i			0.411541i	−0.978600	+	0.205771i	$0.934030\pi$
$68$		0			0
$69$	−6.24013	−	5.48277i	−0.751223	−	0.660048i
$70$		0			0
$71$			10.1803i			1.20818i	0.796915	+	0.604092i	$0.206464\pi$
$71$			10.1803i			1.20818i	−0.796915	+	0.604092i	$0.793536\pi$
$72$		0			0
$73$	−2.76393			−0.323494			−0.161747	−	0.986832i	$-0.551713\pi$
$73$	−2.76393			−0.323494			−0.161747	+	0.986832i	$0.551713\pi$
$74$		0			0
$75$	0.145898	+	0.381966i	0.0168469	+	0.0441056i
$76$		0			0
$77$		−	2.76393i		−	0.314979i
$78$		0			0
$79$		−	12.3153i		−	1.38558i	−0.721142	−	0.692788i	$-0.756382\pi$
$79$		−	12.3153i		−	1.38558i	0.721142	−	0.692788i	$-0.243618\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$	−4.78282			−0.524983			−0.262491	−	0.964934i	$-0.584544\pi$
$83$	−4.78282			−0.524983			−0.262491	+	0.964934i	$0.584544\pi$
$84$		0			0
$85$	17.7082			1.92072
$86$		0			0
$87$	−1.23607	+	0.472136i	−0.132520	+	0.0506183i
$88$		0			0
$89$	−5.99070			−0.635013			−0.317507	−	0.948256i	$-0.602846\pi$
$89$	−5.99070			−0.635013			−0.317507	+	0.948256i	$0.602846\pi$
$90$		0			0
$91$		−	1.74806i		−	0.183247i
$92$		0			0
$93$	−3.70820	−	9.70820i	−0.384523	−	1.00669i
$94$		0			0
$95$			2.76393i			0.283573i
$96$		0			0
$97$		−	16.5579i		−	1.68120i	−0.541656	−	0.840600i	$-0.682203\pi$
$97$		−	16.5579i		−	1.68120i	0.541656	−	0.840600i	$-0.317797\pi$
$98$		0			0
$99$	−4.37016	+	3.90879i	−0.439218	+	0.392848i
$100$		0			0
$101$			0.472136i			0.0469793i	0.999724	+	0.0234896i	$0.00747767\pi$
$101$			0.472136i			0.0469793i	−0.999724	+	0.0234896i	$0.992522\pi$
$102$		0			0
$103$		−	7.07107i		−	0.696733i	−0.937358	−	0.348367i	$-0.886736\pi$
$103$		−	7.07107i		−	0.696733i	0.937358	−	0.348367i	$-0.113264\pi$
$104$		0			0
$105$	5.23607	−	2.00000i	0.510988	−	0.195180i
$106$		0			0
$107$	−19.1800			−1.85420			−0.927100	−	0.374814i	$-0.877707\pi$
$107$	−19.1800			−1.85420			−0.927100	+	0.374814i	$0.877707\pi$
$108$		0			0
$109$		−	5.45052i		−	0.522065i	−0.965330	−	0.261033i	$-0.915937\pi$
$109$		−	5.45052i		−	0.522065i	0.965330	−	0.261033i	$-0.0840631\pi$
$110$		0			0
$111$	12.3153	−	4.70401i	1.16891	−	0.446485i
$112$		0			0
$113$	9.89949			0.931266			0.465633	−	0.884978i	$-0.345827\pi$
$113$	9.89949			0.931266			0.465633	+	0.884978i	$0.345827\pi$
$114$		0			0
$115$	−9.70820	+	5.11667i	−0.905295	+	0.477132i
$116$		0			0
$117$	−2.76393	+	2.47214i	−0.255526	+	0.228549i
$118$		0			0
$119$		−	10.9443i		−	1.00326i
$120$		0			0
$121$	−7.18034			−0.652758
$122$		0			0
$123$	−13.7082	+	5.23607i	−1.23603	+	0.472120i
$124$		0			0
$125$	−10.9010			−0.975019
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$	−9.35931	+	3.57494i	−0.824042	+	0.314756i
$130$		0			0
$131$		−	7.23607i		−	0.632218i	−0.948723	−	0.316109i	$-0.897623\pi$
$131$		−	7.23607i		−	0.632218i	0.948723	−	0.316109i	$-0.102377\pi$
$132$		0			0
$133$	1.70820			0.148120
$134$		0			0
$135$	−10.5672	−	5.45052i	−0.909479	−	0.469106i
$136$		0			0
$137$	10.9799			0.938073			0.469036	−	0.883179i	$-0.344601\pi$
$137$	10.9799			0.938073			0.469036	+	0.883179i	$0.344601\pi$
$138$		0			0
$139$	−2.47214			−0.209684			−0.104842	−	0.994489i	$-0.533434\pi$
$139$	−2.47214			−0.209684			−0.104842	+	0.994489i	$0.533434\pi$
$140$		0			0
$141$	2.76393	−	1.05573i	0.232765	−	0.0889083i
$142$		0			0
$143$	2.41577			0.202016
$144$		0			0
$145$			1.74806i			0.145169i
$146$		0			0
$147$	3.09017	+	8.09017i	0.254873	+	0.667266i
$148$		0			0
$149$	−6.19704			−0.507681			−0.253840	−	0.967246i	$-0.581694\pi$
$149$	−6.19704			−0.507681			−0.253840	+	0.967246i	$0.581694\pi$
$150$		0			0
$151$	18.9443			1.54166			0.770831	−	0.637039i	$-0.219841\pi$
$151$	18.9443			1.54166			0.770831	+	0.637039i	$0.219841\pi$
$152$		0			0
$153$	−17.3044	+	15.4775i	−1.39898	+	1.25128i
$154$		0			0
$155$	−13.7295			−1.10278
$156$		0			0
$157$			23.0888i			1.84269i	0.388751	+	0.921343i	$0.372907\pi$
$157$			23.0888i			1.84269i	−0.388751	+	0.921343i	$0.627093\pi$
$158$		0			0
$159$	−1.41421	−	3.70246i	−0.112154	−	0.293624i
$160$		0			0
$161$	3.16228	+	6.00000i	0.249222	+	0.472866i
$162$		0			0
$163$	4.94427			0.387265			0.193633	−	0.981074i	$-0.437973\pi$
$163$	4.94427			0.387265			0.193633	+	0.981074i	$0.437973\pi$
$164$		0			0
$165$	2.76393	+	7.23607i	0.215172	+	0.563327i
$166$		0			0
$167$		−	23.1246i		−	1.78944i	−0.446631	−	0.894718i	$-0.647376\pi$
$167$		−	23.1246i		−	1.78944i	0.446631	−	0.894718i	$-0.352624\pi$
$168$		0			0
$169$	−11.4721			−0.882472
$170$		0			0
$171$	−2.41577	−	2.70091i	−0.184738	−	0.206544i
$172$		0			0
$173$		−	12.1803i		−	0.926054i	−0.886344	−	0.463027i	$-0.846763\pi$
$173$		−	12.1803i		−	0.926054i	0.886344	−	0.463027i	$-0.153237\pi$
$174$		0			0
$175$		−	0.333851i		−	0.0252367i
$176$		0			0
$177$	12.9443	−	4.94427i	0.972951	−	0.371634i
$178$		0			0
$179$			8.94427i			0.668526i	0.942480	+	0.334263i	$0.108487\pi$
$179$			8.94427i			0.668526i	−0.942480	+	0.334263i	$0.891513\pi$
$180$		0			0
$181$		−	10.0270i		−	0.745302i	−0.927972	−	0.372651i	$-0.878449\pi$
$181$		−	10.0270i		−	0.745302i	0.927972	−	0.372651i	$-0.121551\pi$
$182$		0			0
$183$	13.3956	−	5.11667i	0.990233	−	0.378235i
$184$		0			0
$185$		−	17.4164i		−	1.28048i
$186$		0			0
$187$	15.1246			1.10602
$188$		0			0
$189$	−3.36861	+	6.53089i	−0.245030	+	0.475052i
$190$		0			0
$191$	−6.06952			−0.439175			−0.219587	−	0.975593i	$-0.570471\pi$
$191$	−6.06952			−0.439175			−0.219587	+	0.975593i	$0.570471\pi$
$192$		0			0
$193$	−23.4164			−1.68555			−0.842775	−	0.538266i	$-0.819080\pi$
$193$	−23.4164			−1.68555			−0.842775	+	0.538266i	$0.819080\pi$
$194$		0			0
$195$	1.74806	+	4.57649i	0.125181	+	0.327729i
$196$		0			0
$197$			16.4721i			1.17359i	0.809735	+	0.586796i	$0.199611\pi$
$197$			16.4721i			1.17359i	−0.809735	+	0.586796i	$0.800389\pi$
$198$		0			0
$199$		−	17.3044i		−	1.22668i	−0.789820	−	0.613339i	$-0.789826\pi$
$199$		−	17.3044i		−	1.22668i	0.789820	−	0.613339i	$-0.210174\pi$
$200$		0			0
$201$	−5.45052	+	2.08191i	−0.384450	+	0.146847i
$202$		0			0
$203$	1.08036			0.0758266
$204$		0			0
$205$			19.3863i			1.35400i
$206$		0			0
$207$	5.01470	−	13.4853i	0.348546	−	0.937292i
$208$		0			0
$209$			2.36068i			0.163292i
$210$		0			0
$211$	−16.6525			−1.14640			−0.573202	−	0.819414i	$-0.694299\pi$
$211$	−16.6525			−1.14640			−0.573202	+	0.819414i	$0.694299\pi$
$212$		0			0
$213$	−16.4721	+	6.29180i	−1.12865	+	0.431107i
$214$		0			0
$215$			13.2361i			0.902692i
$216$		0			0
$217$			8.48528i			0.576018i
$218$		0			0
$219$	−1.70820	−	4.47214i	−0.115430	−	0.302199i
$220$		0			0
$221$	9.56564			0.643455
$222$		0			0
$223$	1.05573			0.0706968			0.0353484	−	0.999375i	$-0.488746\pi$
$223$	1.05573			0.0706968			0.0353484	+	0.999375i	$0.488746\pi$
$224$		0			0
$225$	−0.527864	+	0.472136i	−0.0351909	+	0.0314757i
$226$		0			0
$227$	8.27895			0.549493			0.274747	−	0.961517i	$-0.411406\pi$
$227$	8.27895			0.549493			0.274747	+	0.961517i	$0.411406\pi$
$228$		0			0
$229$			12.1877i			0.805389i	0.915335	+	0.402694i	$0.131926\pi$
$229$			12.1877i			0.805389i	−0.915335	+	0.402694i	$0.868074\pi$
$230$		0			0
$231$	4.47214	−	1.70820i	0.294245	−	0.112392i
$232$		0			0
$233$			13.8885i			0.909869i	0.890525	+	0.454934i	$0.150337\pi$
$233$			13.8885i			0.909869i	−0.890525	+	0.454934i	$0.849663\pi$
$234$		0			0
$235$		−	3.90879i		−	0.254981i
$236$		0			0
$237$	19.9265	−	7.61125i	1.29437	−	0.494404i
$238$		0			0
$239$		−	2.00000i		−	0.129369i	−0.997906	−	0.0646846i	$-0.979396\pi$
$239$		−	2.00000i		−	0.129369i	0.997906	−	0.0646846i	$-0.0206041\pi$
$240$		0			0
$241$		−	3.24109i		−	0.208777i	−0.994537	−	0.104388i	$-0.966711\pi$
$241$		−	3.24109i		−	0.208777i	0.994537	−	0.104388i	$-0.0332885\pi$
$242$		0			0
$243$	15.0902	−	3.90983i	0.968035	−	0.250816i
$244$		0			0
$245$	11.4412			0.730953
$246$		0			0
$247$			1.49302i			0.0949989i
$248$		0			0
$249$	−2.95595	−	7.73877i	−0.187326	−	0.490425i
$250$		0			0
$251$	18.5123			1.16849			0.584243	−	0.811579i	$-0.301392\pi$
$251$	18.5123			1.16849			0.584243	+	0.811579i	$0.301392\pi$
$252$		0			0
$253$	−8.29180	+	4.37016i	−0.521301	+	0.274750i
$254$		0			0
$255$	10.9443	+	28.6525i	0.685357	+	1.79429i
$256$		0			0
$257$			24.4721i			1.52653i	0.646086	+	0.763265i	$0.276405\pi$
$257$			24.4721i			1.52653i	−0.646086	+	0.763265i	$0.723595\pi$
$258$		0			0
$259$	−10.7639			−0.668838
$260$		0			0
$261$	−1.52786	−	1.70820i	−0.0945724	−	0.105735i
$262$		0			0
$263$	7.65996			0.472333			0.236167	−	0.971713i	$-0.424109\pi$
$263$	7.65996			0.472333			0.236167	+	0.971713i	$0.424109\pi$
$264$		0			0
$265$	−5.23607			−0.321649
$266$		0			0
$267$	−3.70246	−	9.69316i	−0.226587	−	0.593212i
$268$		0			0
$269$		−	16.4721i		−	1.00432i	−0.864774	−	0.502162i	$-0.832538\pi$
$269$		−	16.4721i		−	1.00432i	0.864774	−	0.502162i	$-0.167462\pi$
$270$		0			0
$271$	−4.00000			−0.242983			−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000			−0.242983			−0.121491	+	0.992592i	$0.538768\pi$
$272$		0			0
$273$	2.82843	−	1.08036i	0.171184	−	0.0653865i
$274$		0			0
$275$	0.461370			0.0278217
$276$		0			0
$277$	−22.3607			−1.34352			−0.671762	−	0.740767i	$-0.734462\pi$
$277$	−22.3607			−1.34352			−0.671762	+	0.740767i	$0.734462\pi$
$278$		0			0
$279$	13.4164	−	12.0000i	0.803219	−	0.718421i
$280$		0			0
$281$	14.8886			0.888182			0.444091	−	0.895982i	$-0.353527\pi$
$281$	14.8886			0.888182			0.444091	+	0.895982i	$0.353527\pi$
$282$		0			0
$283$			25.1707i			1.49624i	0.663562	+	0.748121i	$0.269044\pi$
$283$			25.1707i			1.49624i	−0.663562	+	0.748121i	$0.730956\pi$
$284$		0			0
$285$	−4.47214	+	1.70820i	−0.264906	+	0.101185i
$286$		0			0
$287$	11.9814			0.707240
$288$		0			0
$289$	42.8885			2.52286
$290$		0			0
$291$	26.7912	−	10.2333i	1.57053	−	0.599889i
$292$		0			0
$293$	7.94510			0.464158			0.232079	−	0.972697i	$-0.425447\pi$
$293$	7.94510			0.464158			0.232079	+	0.972697i	$0.425447\pi$
$294$		0			0
$295$		−	18.3060i		−	1.06581i
$296$		0			0
$297$	−9.02546	−	4.65530i	−0.523710	−	0.270128i
$298$		0			0
$299$	−5.24419	+	2.76393i	−0.303279	+	0.159842i
$300$		0			0
$301$	8.18034			0.471507
$302$		0			0
$303$	−0.763932	+	0.291796i	−0.0438867	+	0.0167632i
$304$		0			0
$305$		−	18.9443i		−	1.08475i
$306$		0			0
$307$	−24.3607			−1.39034			−0.695169	−	0.718847i	$-0.744670\pi$
$307$	−24.3607			−1.39034			−0.695169	+	0.718847i	$0.744670\pi$
$308$		0			0
$309$	11.4412	−	4.37016i	0.650869	−	0.248610i
$310$		0			0
$311$			20.0000i			1.13410i	0.823685	+	0.567048i	$0.191915\pi$
$311$			20.0000i			1.13410i	−0.823685	+	0.567048i	$0.808085\pi$
$312$		0			0
$313$			26.3786i			1.49101i	0.666502	+	0.745503i	$0.267791\pi$
$313$			26.3786i			1.49101i	−0.666502	+	0.745503i	$0.732209\pi$
$314$		0			0
$315$	6.47214	+	7.23607i	0.364664	+	0.407706i
$316$		0			0
$317$			9.70820i			0.545267i	0.962118	+	0.272634i	$0.0878947\pi$
$317$			9.70820i			0.545267i	−0.962118	+	0.272634i	$0.912105\pi$
$318$		0			0
$319$			1.49302i			0.0835934i
$320$		0			0
$321$	−11.8539	−	31.0339i	−0.661620	−	1.73214i
$322$		0			0
$323$			9.34752i			0.520110i
$324$		0			0
$325$	0.291796			0.0161859
$326$		0			0
$327$	8.81913	−	3.36861i	0.487699	−	0.186284i
$328$		0			0
$329$	−2.41577			−0.133185
$330$		0			0
$331$	16.6525			0.915303			0.457651	−	0.889132i	$-0.348691\pi$
$331$	16.6525			0.915303			0.457651	+	0.889132i	$0.348691\pi$
$332$		0			0
$333$	15.2225	+	17.0193i	0.834188	+	0.932650i
$334$		0			0
$335$			7.70820i			0.421144i
$336$		0			0
$337$			16.9706i			0.924445i	0.886764	+	0.462223i	$0.152948\pi$
$337$			16.9706i			0.924445i	−0.886764	+	0.462223i	$0.847052\pi$
$338$		0			0
$339$	6.11822	+	16.0177i	0.332296	+	0.869963i
$340$		0			0
$341$	−11.7264			−0.635019
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$	−14.2790	−	12.5459i	−0.768753	−	0.675450i
$346$		0			0
$347$			26.8328i			1.44046i	0.693735	+	0.720231i	$0.255964\pi$
$347$			26.8328i			1.44046i	−0.693735	+	0.720231i	$0.744036\pi$
$348$		0			0
$349$	0.291796			0.0156195			0.00780974	−	0.999970i	$-0.497514\pi$
$349$	0.291796			0.0156195			0.00780974	+	0.999970i	$0.497514\pi$
$350$		0			0
$351$	−5.70820	−	2.94427i	−0.304681	−	0.157154i
$352$		0			0
$353$			9.52786i			0.507117i	0.967320	+	0.253559i	$0.0816010\pi$
$353$			9.52786i			0.507117i	−0.967320	+	0.253559i	$0.918399\pi$
$354$		0			0
$355$			23.2951i			1.23638i
$356$		0			0
$357$	17.7082	−	6.76393i	0.937218	−	0.357985i
$358$		0			0
$359$	−8.74032			−0.461296			−0.230648	−	0.973037i	$-0.574085\pi$
$359$	−8.74032			−0.461296			−0.230648	+	0.973037i	$0.574085\pi$
$360$		0			0
$361$	17.5410			0.923212
$362$		0			0
$363$	−4.43769	−	11.6180i	−0.232919	−	0.609789i
$364$		0			0
$365$	−6.32456			−0.331042
$366$		0			0
$367$			30.2086i			1.57687i	0.615115	+	0.788437i	$0.289109\pi$
$367$			30.2086i			1.57687i	−0.615115	+	0.788437i	$0.710891\pi$
$368$		0			0
$369$	−16.9443	−	18.9443i	−0.882084	−	0.986199i
$370$		0			0
$371$			3.23607i			0.168008i
$372$		0			0
$373$			26.9976i			1.39788i	0.715180	+	0.698941i	$0.246345\pi$
$373$			26.9976i			1.39788i	−0.715180	+	0.698941i	$0.753655\pi$
$374$		0			0
$375$	−6.73722	−	17.6383i	−0.347908	−	0.910836i
$376$		0			0
$377$			0.944272i			0.0486325i
$378$		0			0
$379$		−	6.45207i		−	0.331421i	−0.986174	−	0.165710i	$-0.947008\pi$
$379$		−	6.45207i		−	0.331421i	0.986174	−	0.165710i	$-0.0529917\pi$
$380$		0			0
$381$	7.41641	+	19.4164i	0.379954	+	0.994733i
$382$		0			0
$383$	22.4698			1.14815			0.574076	−	0.818802i	$-0.305361\pi$
$383$	22.4698			1.14815			0.574076	+	0.818802i	$0.305361\pi$
$384$		0			0
$385$		−	6.32456i		−	0.322329i
$386$		0			0
$387$	−11.5687	−	12.9343i	−0.588072	−	0.657485i
$388$		0			0
$389$	−16.8430			−0.853976			−0.426988	−	0.904257i	$-0.640425\pi$
$389$	−16.8430			−0.853976			−0.426988	+	0.904257i	$0.640425\pi$
$390$		0			0
$391$	−32.8328	+	17.3044i	−1.66043	+	0.875122i
$392$		0			0
$393$	11.7082	−	4.47214i	0.590601	−	0.225589i
$394$		0			0
$395$		−	28.1803i		−	1.41791i
$396$		0			0
$397$	−31.3050			−1.57115			−0.785575	−	0.618766i	$-0.787633\pi$
$397$	−31.3050			−1.57115			−0.785575	+	0.618766i	$0.787633\pi$
$398$		0			0
$399$	1.05573	+	2.76393i	0.0528525	+	0.138370i
$400$		0			0
$401$	−9.48683			−0.473750			−0.236875	−	0.971540i	$-0.576123\pi$
$401$	−9.48683			−0.473750			−0.236875	+	0.971540i	$0.576123\pi$
$402$		0			0
$403$	−7.41641			−0.369438
$404$		0			0
$405$	2.28825	−	20.4667i	0.113704	−	1.01700i
$406$		0			0
$407$		−	14.8754i		−	0.737346i
$408$		0			0
$409$	−6.47214			−0.320027			−0.160013	−	0.987115i	$-0.551154\pi$
$409$	−6.47214			−0.320027			−0.160013	+	0.987115i	$0.551154\pi$
$410$		0			0
$411$	6.78593	+	17.7658i	0.334725	+	0.876321i
$412$		0			0
$413$	−11.3137			−0.556711
$414$		0			0
$415$	−10.9443			−0.537233
$416$		0			0
$417$	−1.52786	−	4.00000i	−0.0748198	−	0.195881i
$418$		0			0
$419$	−33.9898			−1.66051			−0.830256	−	0.557382i	$-0.811806\pi$
$419$	−33.9898			−1.66051			−0.830256	+	0.557382i	$0.811806\pi$
$420$		0			0
$421$		−	36.8183i		−	1.79441i	−0.441612	−	0.897206i	$-0.645593\pi$
$421$		−	36.8183i		−	1.79441i	0.441612	−	0.897206i	$-0.354407\pi$
$422$		0			0
$423$	3.41641	+	3.81966i	0.166111	+	0.185718i
$424$		0			0
$425$	1.82688			0.0886165
$426$		0			0
$427$	−11.7082			−0.566600
$428$		0			0
$429$	1.49302	+	3.90879i	0.0720839	+	0.188718i
$430$		0			0
$431$	16.5579			0.797566			0.398783	−	0.917045i	$-0.369433\pi$
$431$	16.5579			0.797566			0.398783	+	0.917045i	$0.369433\pi$
$432$		0			0
$433$		−	1.08036i		−	0.0519189i	−0.999663	−	0.0259595i	$-0.991736\pi$
$433$		−	1.08036i		−	0.0519189i	0.999663	−	0.0259595i	$-0.00826408\pi$
$434$		0			0
$435$	−2.82843	+	1.08036i	−0.135613	+	0.0517994i
$436$		0			0
$437$	−2.70091	−	5.12461i	−0.129202	−	0.245143i
$438$		0			0
$439$	23.4164			1.11760			0.558802	−	0.829301i	$-0.311261\pi$
$439$	23.4164			1.11760			0.558802	+	0.829301i	$0.311261\pi$
$440$		0			0
$441$	−11.1803	+	10.0000i	−0.532397	+	0.476190i
$442$		0			0
$443$		−	21.1246i		−	1.00366i	−0.864966	−	0.501830i	$-0.832660\pi$
$443$		−	21.1246i		−	1.00366i	0.864966	−	0.501830i	$-0.167340\pi$
$444$		0			0
$445$	−13.7082			−0.649831
$446$		0			0
$447$	−3.82998	−	10.0270i	−0.181152	−	0.474262i
$448$		0			0
$449$		−	24.8328i		−	1.17193i	−0.810335	−	0.585967i	$-0.800715\pi$
$449$		−	24.8328i		−	1.17193i	0.810335	−	0.585967i	$-0.199285\pi$
$450$		0			0
$451$			16.5579i			0.779681i
$452$		0			0
$453$	11.7082	+	30.6525i	0.550099	+	1.44018i
$454$		0			0
$455$		−	4.00000i		−	0.187523i
$456$		0			0
$457$		−	10.9010i		−	0.509929i	−0.966950	−	0.254965i	$-0.917936\pi$
$457$		−	10.9010i		−	0.509929i	0.966950	−	0.254965i	$-0.0820639\pi$
$458$		0			0
$459$	−35.7379	−	18.4335i	−1.66810	−	0.860401i
$460$		0			0
$461$		−	7.88854i		−	0.367406i	−0.982982	−	0.183703i	$-0.941192\pi$
$461$		−	7.88854i		−	0.367406i	0.982982	−	0.183703i	$-0.0588085\pi$
$462$		0			0
$463$	32.3607			1.50393			0.751964	−	0.659204i	$-0.229107\pi$
$463$	32.3607			1.50393			0.751964	+	0.659204i	$0.229107\pi$
$464$		0			0
$465$	−8.48528	−	22.2148i	−0.393496	−	1.03018i
$466$		0			0
$467$	14.3485			0.663968			0.331984	−	0.943285i	$-0.392282\pi$
$467$	14.3485			0.663968			0.331984	+	0.943285i	$0.392282\pi$
$468$		0			0
$469$	4.76393			0.219978
$470$		0			0
$471$	−37.3584	+	14.2697i	−1.72139	+	0.657511i
$472$		0			0
$473$			11.3050i			0.519802i
$474$		0			0
$475$			0.285142i			0.0130832i
$476$		0			0
$477$	5.11667	−	4.57649i	0.234276	−	0.209543i
$478$		0			0
$479$	−23.9628			−1.09489			−0.547445	−	0.836842i	$-0.684399\pi$
$479$	−23.9628			−1.09489			−0.547445	+	0.836842i	$0.684399\pi$
$480$		0			0
$481$		−	9.40802i		−	0.428969i
$482$		0			0
$483$	−7.75381	+	8.82488i	−0.352811	+	0.401546i
$484$		0			0
$485$		−	37.8885i		−	1.72043i
$486$		0			0
$487$	−3.05573			−0.138468			−0.0692341	−	0.997600i	$-0.522056\pi$
$487$	−3.05573			−0.138468			−0.0692341	+	0.997600i	$0.522056\pi$
$488$		0			0
$489$	3.05573	+	8.00000i	0.138185	+	0.361773i
$490$		0			0
$491$		−	30.4721i		−	1.37519i	−0.726095	−	0.687594i	$-0.758667\pi$
$491$		−	30.4721i		−	1.37519i	0.726095	−	0.687594i	$-0.241333\pi$
$492$		0			0
$493$			5.91189i			0.266258i
$494$		0			0
$495$	−10.0000	+	8.94427i	−0.449467	+	0.402015i
$496$		0			0
$497$	14.3972			0.645802
$498$		0			0
$499$	13.5279			0.605590			0.302795	−	0.953056i	$-0.402080\pi$
$499$	13.5279			0.605590			0.302795	+	0.953056i	$0.402080\pi$
$500$		0			0
$501$	37.4164	−	14.2918i	1.67164	−	0.638510i
$502$		0			0
$503$	15.2225			0.678738			0.339369	−	0.940653i	$-0.389787\pi$
$503$	15.2225			0.678738			0.339369	+	0.940653i	$0.389787\pi$
$504$		0			0
$505$			1.08036i			0.0480755i
$506$		0			0
$507$	−7.09017	−	18.5623i	−0.314886	−	0.824381i
$508$		0			0
$509$			39.5967i			1.75509i	0.479490	+	0.877547i	$0.340822\pi$
$509$			39.5967i			1.75509i	−0.479490	+	0.877547i	$0.659178\pi$
$510$		0			0
$511$			3.90879i			0.172915i
$512$		0			0
$513$	2.87714	−	5.57804i	0.127029	−	0.246277i
$514$		0			0
$515$		−	16.1803i		−	0.712991i
$516$		0			0
$517$		−	3.33851i		−	0.146827i
$518$		0			0
$519$	19.7082	−	7.52786i	0.865094	−	0.330437i
$520$		0			0
$521$	19.7202			0.863957			0.431978	−	0.901884i	$-0.357816\pi$
$521$	19.7202			0.863957			0.431978	+	0.901884i	$0.357816\pi$
$522$		0			0
$523$		−	23.1676i		−	1.01305i	−0.862226	−	0.506524i	$-0.830930\pi$
$523$		−	23.1676i		−	1.01305i	0.862226	−	0.506524i	$-0.169070\pi$
$524$		0			0
$525$	0.540182	−	0.206331i	0.0235755	−	0.00900502i
$526$		0			0
$527$	−46.4326			−2.02264
$528$		0			0
$529$	13.0000	−	18.9737i	0.565217	−	0.824942i
$530$		0			0
$531$	16.0000	+	17.8885i	0.694341	+	0.776297i
$532$		0			0
$533$			10.4721i			0.453599i
$534$		0			0
$535$	−43.8885			−1.89747
$536$		0			0
$537$	−14.4721	+	5.52786i	−0.624519	+	0.238545i
$538$		0			0
$539$	9.77198			0.420909
$540$		0			0
$541$	−2.76393			−0.118831			−0.0594154	−	0.998233i	$-0.518924\pi$
$541$	−2.76393			−0.118831			−0.0594154	+	0.998233i	$0.518924\pi$
$542$		0			0
$543$	16.2241	−	6.19704i	0.696241	−	0.265940i
$544$		0			0
$545$		−	12.4721i		−	0.534248i
$546$		0			0
$547$	26.1803			1.11939			0.559695	−	0.828699i	$-0.310918\pi$
$547$	26.1803			1.11939			0.559695	+	0.828699i	$0.310918\pi$
$548$		0			0
$549$	16.5579	+	18.5123i	0.706674	+	0.790086i
$550$		0			0
$551$	−0.922740			−0.0393101
$552$		0			0
$553$	−17.4164			−0.740621
$554$		0			0
$555$	28.1803	−	10.7639i	1.19619	−	0.456903i
$556$		0			0
$557$	−31.2402			−1.32369			−0.661845	−	0.749640i	$-0.730226\pi$
$557$	−31.2402			−1.32369			−0.661845	+	0.749640i	$0.730226\pi$
$558$		0			0
$559$			7.14988i			0.302408i
$560$		0			0
$561$	9.34752	+	24.4721i	0.394653	+	1.03321i
$562$		0			0
$563$	5.45052			0.229712			0.114856	−	0.993382i	$-0.463359\pi$
$563$	5.45052			0.229712			0.114856	+	0.993382i	$0.463359\pi$
$564$		0			0
$565$	22.6525			0.952997
$566$		0			0
$567$	−12.6491	−	1.41421i	−0.531213	−	0.0593914i
$568$		0			0
$569$	−35.1977			−1.47557			−0.737783	−	0.675038i	$-0.764127\pi$
$569$	−35.1977			−1.47557			−0.737783	+	0.675038i	$0.764127\pi$
$570$		0			0
$571$			27.9991i			1.17173i	0.810410	+	0.585864i	$0.199245\pi$
$571$			27.9991i			1.17173i	−0.810410	+	0.585864i	$0.800755\pi$
$572$		0			0
$573$	−3.75117	−	9.82068i	−0.156707	−	0.410265i
$574$		0			0
$575$	−1.00155	+	0.527864i	−0.0417676	+	0.0220135i
$576$		0			0
$577$	42.6525			1.77565			0.887823	−	0.460185i	$-0.152217\pi$
$577$	42.6525			1.77565			0.887823	+	0.460185i	$0.152217\pi$
$578$		0			0
$579$	−14.4721	−	37.8885i	−0.601441	−	1.57459i
$580$		0			0
$581$			6.76393i			0.280615i
$582$		0			0
$583$	−4.47214			−0.185217
$584$		0			0
$585$	−6.32456	+	5.65685i	−0.261488	+	0.233882i
$586$		0			0
$587$			33.1246i			1.36720i	0.729857	+	0.683600i	$0.239586\pi$
$587$			33.1246i			1.36720i	−0.729857	+	0.683600i	$0.760414\pi$
$588$		0			0
$589$		−	7.24730i		−	0.298620i
$590$		0			0
$591$	−26.6525	+	10.1803i	−1.09634	+	0.418763i
$592$		0			0
$593$			38.9443i			1.59925i	0.600500	+	0.799625i	$0.294968\pi$
$593$			38.9443i			1.59925i	−0.600500	+	0.799625i	$0.705032\pi$
$594$		0			0
$595$		−	25.0432i		−	1.02667i
$596$		0			0
$597$	27.9991	−	10.6947i	1.14593	−	0.437706i
$598$		0			0
$599$			24.4721i			0.999904i	0.866053	+	0.499952i	$0.166649\pi$
$599$			24.4721i			0.999904i	−0.866053	+	0.499952i	$0.833351\pi$
$600$		0			0
$601$	2.29180			0.0934843			0.0467422	−	0.998907i	$-0.485116\pi$
$601$	2.29180			0.0934843			0.0467422	+	0.998907i	$0.485116\pi$
$602$		0			0
$603$	−6.73722	−	7.53244i	−0.274361	−	0.306745i
$604$		0			0
$605$	−16.4304			−0.667990
$606$		0			0
$607$	10.0000			0.405887			0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000			0.405887			0.202944	+	0.979190i	$0.434949\pi$
$608$		0			0
$609$	0.667701	+	1.74806i	0.0270566	+	0.0708351i
$610$		0			0
$611$		−	2.11146i		−	0.0854204i
$612$		0			0
$613$		−	30.2387i		−	1.22133i	−0.791890	−	0.610664i	$-0.790903\pi$
$613$		−	30.2387i		−	1.22133i	0.791890	−	0.610664i	$-0.209097\pi$
$614$		0			0
$615$	−31.3677	+	11.9814i	−1.26487	+	0.483137i
$616$		0			0
$617$	3.41732			0.137576			0.0687880	−	0.997631i	$-0.478087\pi$
$617$	3.41732			0.137576			0.0687880	+	0.997631i	$0.478087\pi$
$618$		0			0
$619$		−	1.62054i		−	0.0651352i	−0.999470	−	0.0325676i	$-0.989632\pi$
$619$		−	1.62054i		−	0.0651352i	0.999470	−	0.0325676i	$-0.0103684\pi$
$620$		0			0
$621$	24.9189	−	0.220412i	0.999961	−	0.00884483i
$622$		0			0
$623$			8.47214i			0.339429i
$624$		0			0
$625$	−26.1246			−1.04498
$626$		0			0
$627$	−3.81966	+	1.45898i	−0.152543	+	0.0582661i
$628$		0			0
$629$		−	58.9017i		−	2.34856i
$630$		0			0
$631$			6.24574i			0.248639i	0.992242	+	0.124320i	$0.0396748\pi$
$631$			6.24574i			0.248639i	−0.992242	+	0.124320i	$0.960325\pi$
$632$		0			0
$633$	−10.2918	−	26.9443i	−0.409062	−	1.07094i
$634$		0			0
$635$	27.4589			1.08968
$636$		0			0
$637$	6.18034			0.244874
$638$		0			0
$639$	−20.3607	−	22.7639i	−0.805456	−	0.900527i
$640$		0			0
$641$	28.4605			1.12412			0.562061	−	0.827096i	$-0.310009\pi$
$641$	28.4605			1.12412			0.562061	+	0.827096i	$0.310009\pi$
$642$		0			0
$643$		−	10.3609i		−	0.408593i	−0.978909	−	0.204296i	$-0.934509\pi$
$643$		−	10.3609i		−	0.408593i	0.978909	−	0.204296i	$-0.0654906\pi$
$644$		0			0
$645$	−21.4164	+	8.18034i	−0.843270	+	0.322101i
$646$		0			0
$647$		−	3.05573i		−	0.120133i	−0.998194	−	0.0600665i	$-0.980869\pi$
$647$		−	3.05573i		−	0.120133i	0.998194	−	0.0600665i	$-0.0191313\pi$
$648$		0			0
$649$		−	15.6352i		−	0.613734i
$650$		0			0
$651$	−13.7295	+	5.24419i	−0.538100	+	0.205536i
$652$		0			0
$653$		−	34.0000i		−	1.33052i	−0.746611	−	0.665261i	$-0.768320\pi$
$653$		−	34.0000i		−	1.33052i	0.746611	−	0.665261i	$-0.231680\pi$
$654$		0			0
$655$		−	16.5579i		−	0.646971i
$656$		0			0
$657$	6.18034	−	5.52786i	0.241118	−	0.215663i
$658$		0			0
$659$	−28.3330			−1.10370			−0.551848	−	0.833945i	$-0.686077\pi$
$659$	−28.3330			−1.10370			−0.551848	+	0.833945i	$0.686077\pi$
$660$		0			0
$661$		−	15.4288i		−	0.600112i	−0.953922	−	0.300056i	$-0.902995\pi$
$661$		−	15.4288i		−	0.600112i	0.953922	−	0.300056i	$-0.0970054\pi$
$662$		0			0
$663$	5.91189	+	15.4775i	0.229599	+	0.601098i
$664$		0			0
$665$	3.90879			0.151576
$666$		0			0
$667$	−1.70820	−	3.24109i	−0.0661419	−	0.125495i
$668$		0			0
$669$	0.652476	+	1.70820i	0.0252262	+	0.0660430i
$670$		0			0
$671$		−	16.1803i		−	0.624635i
$672$		0			0
$673$	−6.65248			−0.256434			−0.128217	−	0.991746i	$-0.540925\pi$
$673$	−6.65248			−0.256434			−0.128217	+	0.991746i	$0.540925\pi$
$674$		0			0
$675$	−1.09017	−	0.562306i	−0.0419607	−	0.0216432i
$676$		0			0
$677$	15.6051			0.599751			0.299876	−	0.953978i	$-0.403055\pi$
$677$	15.6051			0.599751			0.299876	+	0.953978i	$0.403055\pi$
$678$		0			0
$679$	−23.4164			−0.898639
$680$		0			0
$681$	5.11667	+	13.3956i	0.196071	+	0.513321i
$682$		0			0
$683$			43.0132i			1.64585i	0.568148	+	0.822926i	$0.307660\pi$
$683$			43.0132i			1.64585i	−0.568148	+	0.822926i	$0.692340\pi$
$684$		0			0
$685$	25.1246			0.959962
$686$		0			0
$687$	−19.7202	+	7.53244i	−0.752372	+	0.287380i
$688$		0			0
$689$	−2.82843			−0.107754
$690$		0			0
$691$	−22.7639			−0.865981			−0.432990	−	0.901399i	$-0.642542\pi$
$691$	−22.7639			−0.865981			−0.432990	+	0.901399i	$0.642542\pi$
$692$		0			0
$693$	5.52786	+	6.18034i	0.209986	+	0.234772i
$694$		0			0
$695$	−5.65685			−0.214577
$696$		0			0
$697$			65.5639i			2.48341i
$698$		0			0
$699$	−22.4721	+	8.58359i	−0.849974	+	0.324661i
$700$		0			0
$701$	−48.2108			−1.82090			−0.910448	−	0.413623i	$-0.864263\pi$
$701$	−48.2108			−1.82090			−0.910448	+	0.413623i	$0.864263\pi$
$702$		0			0
$703$	9.19350			0.346739
$704$		0			0
$705$	6.32456	−	2.41577i	0.238197	−	0.0909830i
$706$		0			0
$707$	0.667701			0.0251115
$708$		0			0
$709$		−	36.4056i		−	1.36724i	−0.729838	−	0.683620i	$-0.760404\pi$
$709$		−	36.4056i		−	1.36724i	0.729838	−	0.683620i	$-0.239596\pi$
$710$		0			0
$711$	24.6305	+	27.5378i	0.923717	+	1.03275i
$712$		0			0
$713$	25.4558	−	13.4164i	0.953329	−	0.502448i
$714$		0			0
$715$	5.52786			0.206730
$716$		0			0
$717$	3.23607	−	1.23607i	0.120853	−	0.0461618i
$718$		0			0
$719$		−	28.0000i		−	1.04422i	−0.852877	−	0.522112i	$-0.825144\pi$
$719$		−	28.0000i		−	1.04422i	0.852877	−	0.522112i	$-0.174856\pi$
$720$		0			0
$721$	−10.0000			−0.372419
$722$		0			0
$723$	5.24419	−	2.00310i	0.195034	−	0.0744962i
$724$		0			0
$725$			0.180340i			0.00669766i
$726$		0			0
$727$			47.1791i			1.74978i	0.484324	+	0.874888i	$0.339065\pi$
$727$			47.1791i			1.74978i	−0.484324	+	0.874888i	$0.660935\pi$
$728$		0			0
$729$	15.6525	+	22.0000i	0.579721	+	0.814815i
$730$		0			0
$731$			44.7639i			1.65565i
$732$		0			0
$733$			22.2635i			0.822320i	0.911563	+	0.411160i	$0.134876\pi$
$733$			22.2635i			0.822320i	−0.911563	+	0.411160i	$0.865124\pi$
$734$		0			0
$735$	7.07107	+	18.5123i	0.260820	+	0.682836i
$736$		0			0
$737$			6.58359i			0.242510i
$738$		0			0
$739$	39.4853			1.45249			0.726245	−	0.687436i	$-0.241264\pi$
$739$	39.4853			1.45249			0.726245	+	0.687436i	$0.241264\pi$
$740$		0			0
$741$	−2.41577	+	0.922740i	−0.0887454	+	0.0338977i
$742$		0			0
$743$	28.2843			1.03765			0.518825	−	0.854881i	$-0.326370\pi$
$743$	28.2843			1.03765			0.518825	+	0.854881i	$0.326370\pi$
$744$		0			0
$745$	−14.1803			−0.519527
$746$		0			0
$747$	10.6947	−	9.56564i	0.391299	−	0.349989i
$748$		0			0
$749$			27.1246i			0.991112i
$750$		0			0
$751$			45.8437i			1.67286i	0.548073	+	0.836431i	$0.315362\pi$
$751$			45.8437i			1.67286i	−0.548073	+	0.836431i	$0.684638\pi$
$752$		0			0
$753$	11.4412	+	29.9535i	0.416942	+	1.09157i
$754$		0			0
$755$	43.3491			1.57764
$756$		0			0
$757$		−	39.8043i		−	1.44671i	−0.690475	−	0.723356i	$-0.742599\pi$
$757$		−	39.8043i		−	1.44671i	0.690475	−	0.723356i	$-0.257401\pi$
$758$		0			0
$759$	−12.1957	−	10.7155i	−0.442675	−	0.388948i
$760$		0			0
$761$		−	24.9443i		−	0.904229i	−0.891960	−	0.452115i	$-0.850670\pi$
$761$		−	24.9443i		−	0.904229i	0.891960	−	0.452115i	$-0.149330\pi$
$762$		0			0
$763$	−7.70820			−0.279056
$764$		0			0
$765$	−39.5967	+	35.4164i	−1.43162	+	1.28048i
$766$		0			0
$767$		−	9.88854i		−	0.357055i
$768$		0			0
$769$		−	30.7000i		−	1.10707i	−0.832825	−	0.553536i	$-0.813278\pi$
$769$		−	30.7000i		−	1.10707i	0.832825	−	0.553536i	$-0.186722\pi$
$770$		0			0
$771$	−39.5967	+	15.1246i	−1.42604	+	0.544700i
$772$		0			0
$773$	−2.70091			−0.0971449			−0.0485724	−	0.998820i	$-0.515467\pi$
$773$	−2.70091			−0.0971449			−0.0485724	+	0.998820i	$0.515467\pi$
$774$		0			0
$775$	−1.41641			−0.0508789
$776$		0			0
$777$	−6.65248	−	17.4164i	−0.238656	−	0.624810i
$778$		0			0
$779$	−10.2333			−0.366648
$780$		0			0
$781$			19.8964i			0.711949i
$782$		0			0
$783$	1.81966	−	3.52786i	0.0650293	−	0.126076i
$784$		0			0
$785$			52.8328i			1.88568i
$786$		0			0
$787$		−	12.1089i		−	0.431637i	−0.976434	−	0.215818i	$-0.930758\pi$
$787$		−	12.1089i		−	0.431637i	0.976434	−	0.215818i	$-0.0692419\pi$
$788$		0			0
$789$	4.73411	+	12.3941i	0.168539	+	0.441241i
$790$		0			0
$791$		−	14.0000i		−	0.497783i
$792$		0			0
$793$		−	10.2333i		−	0.363397i
$794$		0			0
$795$	−3.23607	−	8.47214i	−0.114772	−	0.300476i
$796$		0			0
$797$	−22.3423			−0.791404			−0.395702	−	0.918379i	$-0.629499\pi$
$797$	−22.3423			−0.791404			−0.395702	+	0.918379i	$0.629499\pi$
$798$		0			0
$799$		−	13.2194i		−	0.467669i
$800$		0			0
$801$	13.3956	−	11.9814i	0.473311	−	0.423342i
$802$		0			0
$803$	−5.40182			−0.190626
$804$		0			0
$805$	7.23607	+	13.7295i	0.255038	+	0.483900i
$806$		0			0
$807$	26.6525	−	10.1803i	0.938212	−	0.358365i
$808$		0			0
$809$			8.00000i			0.281265i	0.990062	+	0.140633i	$0.0449136\pi$
$809$			8.00000i			0.281265i	−0.990062	+	0.140633i	$0.955086\pi$
$810$		0			0
$811$	16.6525			0.584748			0.292374	−	0.956304i	$-0.405555\pi$
$811$	16.6525			0.584748			0.292374	+	0.956304i	$0.405555\pi$
$812$		0			0
$813$	−2.47214	−	6.47214i	−0.0867016	−	0.226988i
$814$		0			0
$815$	11.3137			0.396302
$816$		0			0
$817$	−6.98684			−0.244439
$818$		0			0
$819$	3.49613	+	3.90879i	0.122165	+	0.136584i
$820$		0			0
$821$			7.88854i			0.275312i	0.990480	+	0.137656i	$0.0439569\pi$
$821$			7.88854i			0.275312i	−0.990480	+	0.137656i	$0.956043\pi$
$822$		0			0
$823$	19.5279			0.680699			0.340349	−	0.940299i	$-0.389455\pi$
$823$	19.5279			0.680699			0.340349	+	0.940299i	$0.389455\pi$
$824$		0			0
$825$	0.285142	+	0.746512i	0.00992738	+	0.0259902i
$826$		0			0
$827$	45.5586			1.58423			0.792114	−	0.610374i	$-0.208981\pi$
$827$	45.5586			1.58423			0.792114	+	0.610374i	$0.208981\pi$
$828$		0			0
$829$	−38.1803			−1.32606			−0.663029	−	0.748594i	$-0.730729\pi$
$829$	−38.1803			−1.32606			−0.663029	+	0.748594i	$0.730729\pi$
$830$		0			0
$831$	−13.8197	−	36.1803i	−0.479399	−	1.25508i
$832$		0			0
$833$	38.6938			1.34066
$834$		0			0
$835$		−	52.9148i		−	1.83119i
$836$		0			0
$837$	27.7082	+	14.2918i	0.957736	+	0.493997i
$838$		0			0
$839$	20.4667			0.706589			0.353294	−	0.935512i	$-0.385061\pi$
$839$	20.4667			0.706589			0.353294	+	0.935512i	$0.385061\pi$
$840$		0			0
$841$	28.4164			0.979876
$842$		0			0
$843$	9.20169	+	24.0903i	0.316923	+	0.829715i
$844$		0			0
$845$	−26.2511			−0.903064
$846$		0			0
$847$			10.1545i			0.348914i
$848$		0			0
$849$	−40.7271	+	15.5563i	−1.39775	+	0.533893i
$850$		0			0
$851$	17.0193	+	32.2918i	0.583413	+	1.10695i
$852$		0			0
$853$	50.7214			1.73667			0.868333	−	0.495981i	$-0.165192\pi$
$853$	50.7214			1.73667			0.868333	+	0.495981i	$0.165192\pi$
$854$		0			0
$855$	−5.52786	−	6.18034i	−0.189049	−	0.211363i
$856$		0			0
$857$			11.4164i			0.389977i	0.980806	+	0.194989i	$0.0624670\pi$
$857$			11.4164i			0.389977i	−0.980806	+	0.194989i	$0.937533\pi$
$858$		0			0
$859$	−1.59675			−0.0544803			−0.0272402	−	0.999629i	$-0.508672\pi$
$859$	−1.59675			−0.0544803			−0.0272402	+	0.999629i	$0.508672\pi$
$860$		0			0
$861$	7.40492	+	19.3863i	0.252359	+	0.660684i
$862$		0			0
$863$			37.8885i			1.28974i	0.764292	+	0.644871i	$0.223089\pi$
$863$			37.8885i			1.28974i	−0.764292	+	0.644871i	$0.776911\pi$
$864$		0			0
$865$		−	27.8716i		−	0.947663i
$866$		0			0
$867$	26.5066	+	69.3951i	0.900211	+	2.35678i
$868$		0			0
$869$		−	24.0689i		−	0.816481i
$870$		0			0
$871$			4.16383i			0.141086i
$872$		0			0
$873$	33.1158	+	37.0246i	1.12080	+	1.25309i
$874$		0			0
$875$			15.4164i			0.521170i
$876$		0			0
$877$	−5.41641			−0.182899			−0.0914495	−	0.995810i	$-0.529150\pi$
$877$	−5.41641			−0.182899			−0.0914495	+	0.995810i	$0.529150\pi$
$878$		0			0
$879$	4.91034	+	12.8554i	0.165622	+	0.433603i
$880$		0			0
$881$	−21.2132			−0.714691			−0.357345	−	0.933972i	$-0.616318\pi$
$881$	−21.2132			−0.714691			−0.357345	+	0.933972i	$0.616318\pi$
$882$		0			0
$883$	33.5967			1.13062			0.565310	−	0.824878i	$-0.308756\pi$
$883$	33.5967			1.13062			0.565310	+	0.824878i	$0.308756\pi$
$884$		0			0
$885$	29.6197	−	11.3137i	0.995654	−	0.380306i
$886$		0			0
$887$			25.1246i			0.843602i	0.906688	+	0.421801i	$0.138602\pi$
$887$			25.1246i			0.843602i	−0.906688	+	0.421801i	$0.861398\pi$
$888$		0			0
$889$		−	16.9706i		−	0.569174i
$890$		0			0
$891$	1.95440	−	17.4806i	0.0654747	−	0.585624i
$892$		0			0
$893$	2.06331			0.0690460
$894$		0			0
$895$			20.4667i			0.684126i
$896$		0			0
$897$	−7.71323	−	6.77708i	−0.257537	−	0.226280i
$898$		0			0
$899$		−	4.58359i		−	0.152871i
$900$		0			0
$901$	−17.7082			−0.589946
$902$		0			0
$903$	5.05573	+	13.2361i	0.168244	+	0.440469i
$904$		0			0
$905$		−	22.9443i		−	0.762693i
$906$		0			0
$907$		−	46.7178i		−	1.55124i	−0.631202	−	0.775619i	$-0.717438\pi$
$907$		−	46.7178i		−	1.55124i	0.631202	−	0.775619i	$-0.282562\pi$
$908$		0			0
$909$	−0.944272	−	1.05573i	−0.0313195	−	0.0350163i
$910$		0			0
$911$	3.90879			0.129504			0.0647520	−	0.997901i	$-0.479374\pi$
$911$	3.90879			0.129504			0.0647520	+	0.997901i	$0.479374\pi$
$912$		0			0
$913$	−9.34752			−0.309358
$914$		0			0
$915$	30.6525	−	11.7082i	1.01334	−	0.387061i
$916$		0			0
$917$	−10.2333			−0.337935
$918$		0			0
$919$		−	45.0184i		−	1.48502i	−0.669835	−	0.742510i	$-0.733635\pi$
$919$		−	45.0184i		−	1.48502i	0.669835	−	0.742510i	$-0.266365\pi$
$920$		0			0
$921$	−15.0557	−	39.4164i	−0.496103	−	1.29881i
$922$		0			0
$923$			12.5836i			0.414194i
$924$		0			0
$925$		−	1.79677i		−	0.0590775i
$926$		0			0
$927$	14.1421	+	15.8114i	0.464489	+	0.519314i
$928$		0			0
$929$			10.3607i			0.339923i	0.985451	+	0.169961i	$0.0543643\pi$
$929$			10.3607i			0.339923i	−0.985451	+	0.169961i	$0.945636\pi$
$930$		0			0
$931$			6.03941i			0.197934i
$932$		0			0
$933$	−32.3607	+	12.3607i	−1.05944	+	0.404670i
$934$		0			0
$935$	34.6088			1.13183
$936$		0			0
$937$			36.1019i			1.17940i	0.807624	+	0.589698i	$0.200753\pi$
$937$			36.1019i			1.17940i	−0.807624	+	0.589698i	$0.799247\pi$
$938$		0			0
$939$	−42.6814	+	16.3029i	−1.39286	+	0.532024i
$940$		0			0
$941$	48.4658			1.57994			0.789970	−	0.613145i	$-0.210096\pi$
$941$	48.4658			1.57994			0.789970	+	0.613145i	$0.210096\pi$
$942$		0			0
$943$	−18.9443	−	35.9442i	−0.616910	−	1.17051i
$944$		0			0
$945$	−7.70820	+	14.9443i	−0.250748	+	0.486137i
$946$		0			0
$947$		−	58.2492i		−	1.89285i	−0.322930	−	0.946423i	$-0.604668\pi$
$947$		−	58.2492i		−	1.89285i	0.322930	−	0.946423i	$-0.395332\pi$
$948$		0			0
$949$	−3.41641			−0.110901
$950$		0			0
$951$	−15.7082	+	6.00000i	−0.509373	+	0.194563i
$952$		0			0
$953$	24.2967			0.787046			0.393523	−	0.919315i	$-0.371256\pi$
$953$	24.2967			0.787046			0.393523	+	0.919315i	$0.371256\pi$
$954$		0			0
$955$	−13.8885			−0.449423
$956$		0			0
$957$	−2.41577	+	0.922740i	−0.0780906	+	0.0298280i
$958$		0			0
$959$		−	15.5279i		−	0.501421i
$960$		0			0
$961$	5.00000			0.161290
$962$		0			0
$963$	42.8878	−	38.3600i	1.38204	−	1.23613i
$964$		0			0
$965$	−53.5825			−1.72488
$966$		0			0
$967$	−47.8885			−1.53999			−0.769996	−	0.638049i	$-0.779742\pi$
$967$	−47.8885			−1.53999			−0.769996	+	0.638049i	$0.779742\pi$
$968$		0			0
$969$	−15.1246	+	5.77709i	−0.485873	+	0.185587i
$970$		0			0
$971$	34.8152			1.11727			0.558636	−	0.829413i	$-0.311325\pi$
$971$	34.8152			1.11727			0.558636	+	0.829413i	$0.311325\pi$
$972$		0			0
$973$			3.49613i			0.112081i
$974$		0			0
$975$	0.180340	+	0.472136i	0.00577550	+	0.0151205i
$976$		0			0
$977$	20.5455			0.657309			0.328654	−	0.944450i	$-0.393405\pi$
$977$	20.5455			0.657309			0.328654	+	0.944450i	$0.393405\pi$
$978$		0			0
$979$	−11.7082			−0.374196
$980$		0			0
$981$	10.9010	+	12.1877i	0.348044	+	0.389125i
$982$		0			0
$983$	32.0354			1.02177			0.510886	−	0.859649i	$-0.329317\pi$
$983$	32.0354			1.02177			0.510886	+	0.859649i	$0.329317\pi$
$984$		0			0
$985$			37.6923i			1.20098i
$986$		0			0
$987$	−1.49302	−	3.90879i	−0.0475235	−	0.124418i
$988$		0			0
$989$	−12.9343	−	24.5410i	−0.411285	−	0.780359i
$990$		0			0
$991$	2.58359			0.0820705			0.0410353	−	0.999158i	$-0.486934\pi$
$991$	2.58359			0.0820705			0.0410353	+	0.999158i	$0.486934\pi$
$992$		0			0
$993$	10.2918	+	26.9443i	0.326600	+	0.855051i
$994$		0			0
$995$		−	39.5967i		−	1.25530i
$996$		0			0
$997$	29.5967			0.937338			0.468669	−	0.883374i	$-0.344734\pi$
$997$	29.5967			0.937338			0.468669	+	0.883374i	$0.344734\pi$
$998$		0			0
$999$	−18.1297	+	35.1490i	−0.573600	+	1.11207i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.m.a.137.8	yes	8
3.2	odd	2	inner	552.2.m.a.137.5	✓	8
4.3	odd	2		1104.2.m.c.689.2		8
12.11	even	2		1104.2.m.c.689.3		8
23.22	odd	2	inner	552.2.m.a.137.7	yes	8
69.68	even	2	inner	552.2.m.a.137.6	yes	8
92.91	even	2		1104.2.m.c.689.1		8
276.275	odd	2		1104.2.m.c.689.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.m.a.137.5	✓	8	3.2	odd	2	inner
552.2.m.a.137.6	yes	8	69.68	even	2	inner
552.2.m.a.137.7	yes	8	23.22	odd	2	inner
552.2.m.a.137.8	yes	8	1.1	even	1	trivial
1104.2.m.c.689.1		8	92.91	even	2
1104.2.m.c.689.2		8	4.3	odd	2
1104.2.m.c.689.3		8	12.11	even	2
1104.2.m.c.689.4		8	276.275	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:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	552.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.618034 + 1.61803i) q^{3} +2.28825 q^{5} -1.41421i q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})\)	\(q+(0.618034 + 1.61803i) q^{3} +2.28825 q^{5} -1.41421i q^{7} +(-2.23607 + 2.00000i) q^{9} +1.95440 q^{11} +1.23607 q^{13} +(1.41421 + 3.70246i) q^{15} +7.73877 q^{17} +1.20788i q^{19} +(2.28825 - 0.874032i) q^{21} +(-4.24264 + 2.23607i) q^{23} +0.236068 q^{25} +(-4.61803 - 2.38197i) q^{27} +0.763932i q^{29} -6.00000 q^{31} +(1.20788 + 3.16228i) q^{33} -3.23607i q^{35} -7.61125i q^{37} +(0.763932 + 2.00000i) q^{39} +8.47214i q^{41} +5.78437i q^{43} +(-5.11667 + 4.57649i) q^{45} -1.70820i q^{47} +5.00000 q^{49} +(4.78282 + 12.5216i) q^{51} -2.28825 q^{53} +4.47214 q^{55} +(-1.95440 + 0.746512i) q^{57} -8.00000i q^{59} -8.27895i q^{61} +(2.82843 + 3.16228i) q^{63} +2.82843 q^{65} +3.36861i q^{67} +(-6.24013 - 5.48277i) q^{69} +10.1803i q^{71} -2.76393 q^{73} +(0.145898 + 0.381966i) q^{75} -2.76393i q^{77} -12.3153i q^{79} +(1.00000 - 8.94427i) q^{81} -4.78282 q^{83} +17.7082 q^{85} +(-1.23607 + 0.472136i) q^{87} -5.99070 q^{89} -1.74806i q^{91} +(-3.70820 - 9.70820i) q^{93} +2.76393i q^{95} -16.5579i q^{97} +(-4.37016 + 3.90879i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3}+O(q^{10}) \) 8 * q - 4 * q^3	\( 8 q - 4 q^{3} - 8 q^{13} - 16 q^{25} - 28 q^{27} - 48 q^{31} + 24 q^{39} + 40 q^{49} - 20 q^{69} - 40 q^{73} + 28 q^{75} + 8 q^{81} + 88 q^{85} + 8 q^{87} + 24 q^{93}+O(q^{100}) \) 8 * q - 4 * q^3 - 8 * q^13 - 16 * q^25 - 28 * q^27 - 48 * q^31 + 24 * q^39 + 40 * q^49 - 20 * q^69 - 40 * q^73 + 28 * q^75 + 8 * q^81 + 88 * q^85 + 8 * q^87 + 24 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more