LMFDB - Embedded newform 816.2.e.c.239.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [816,2,Mod(239,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("816.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	816.e (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:	$6.51579280494$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 2 x^{18} - 2 x^{17} + 11 x^{16} - 28 x^{15} + 36 x^{14} - 38 x^{13} + 61 x^{12} + \cdots + 1024 $ x^20 - 2x^19 + 2x^18 - 2x^17 + 11x^16 - 28x^15 + 36x^14 - 38x^13 + 61x^12 - 138x^11 + 226x^10 - 276x^9 + 244x^8 - 304x^7 + 576x^6 - 896x^5 + 704x^4 - 256x^3 + 512x^2 - 1024*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.12
Root			$0.0809031 + 1.41190i$ of defining polynomial
Character	$\chi$	$=$	816.239
Dual form			816.2.e.c.239.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.627306 + 1.61446i) q^{3} +4.11023i q^{5} +2.66199i q^{7} +(-2.21297 + 2.02552i) q^{9} +O(q^{10})$	\(q+(0.627306 + 1.61446i) q^{3} +4.11023i q^{5} +2.66199i q^{7} +(-2.21297 + 2.02552i) q^{9} +3.32327 q^{11} +4.92278 q^{13} +(-6.63581 + 2.57837i) q^{15} -1.00000i q^{17} -5.87898i q^{19} +(-4.29768 + 1.66988i) q^{21} +4.05744 q^{23} -11.8940 q^{25} +(-4.65834 - 2.30214i) q^{27} +6.15231i q^{29} -10.4424i q^{31} +(2.08471 + 5.36529i) q^{33} -10.9414 q^{35} -0.732705 q^{37} +(3.08809 + 7.94764i) q^{39} +6.92278i q^{41} -6.43775i q^{43} +(-8.32536 - 9.09583i) q^{45} -1.01943 q^{47} -0.0861847 q^{49} +(1.61446 - 0.627306i) q^{51} -14.0222i q^{53} +13.6594i q^{55} +(9.49139 - 3.68792i) q^{57} -1.85988 q^{59} +2.86360 q^{61} +(-5.39192 - 5.89091i) q^{63} +20.2337i q^{65} -2.34199i q^{67} +(2.54525 + 6.55058i) q^{69} +10.0509 q^{71} +4.25654 q^{73} +(-7.46117 - 19.2024i) q^{75} +8.84650i q^{77} -2.66199i q^{79} +(0.794509 - 8.96486i) q^{81} -8.03605 q^{83} +4.11023 q^{85} +(-9.93267 + 3.85938i) q^{87} +10.8008i q^{89} +13.1044i q^{91} +(16.8588 - 6.55058i) q^{93} +24.1640 q^{95} -9.10115 q^{97} +(-7.35430 + 6.73135i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{9}+O(q^{10}) $ 20 * q - 4 * q^9	$ 20 q - 4 q^{9} + 12 q^{13} - 16 q^{21} - 48 q^{25} + 12 q^{33} - 16 q^{45} - 20 q^{49} + 64 q^{57} - 48 q^{61} + 28 q^{69} + 64 q^{73} - 60 q^{81} + 4 q^{85} + 32 q^{93} + 32 q^{97}+O(q^{100}) $ 20 * q - 4 * q^9 + 12 * q^13 - 16 * q^21 - 48 * q^25 + 12 * q^33 - 16 * q^45 - 20 * q^49 + 64 * q^57 - 48 * q^61 + 28 * q^69 + 64 * q^73 - 60 * q^81 + 4 * q^85 + 32 * q^93 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/816\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$545$	$613$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	0.627306	+	1.61446i	0.362175	+	0.932110i
$3$	0.627306	+	1.61446i	0.362175	+	0.932110i
$4$		0			0
$5$			4.11023i			1.83815i	0.394082	+	0.919075i	$0.371062\pi$
$5$			4.11023i			1.83815i	−0.394082	+	0.919075i	$0.628938\pi$
$6$		0			0
$7$			2.66199i			1.00614i	0.864246	+	0.503069i	$0.167796\pi$
$7$			2.66199i			1.00614i	−0.864246	+	0.503069i	$0.832204\pi$
$8$		0			0
$9$	−2.21297	+	2.02552i	−0.737658	+	0.675174i
$10$		0			0
$11$	3.32327			1.00200			0.501001	−	0.865447i	$-0.332965\pi$
$11$	3.32327			1.00200			0.501001	+	0.865447i	$0.332965\pi$
$12$		0			0
$13$	4.92278			1.36533			0.682666	−	0.730730i	$-0.260820\pi$
$13$	4.92278			1.36533			0.682666	+	0.730730i	$0.260820\pi$
$14$		0			0
$15$	−6.63581	+	2.57837i	−1.71336	+	0.665733i
$16$		0			0
$17$		−	1.00000i		−	0.242536i
$17$		−	1.00000i		−	0.242536i
$18$		0			0
$19$		−	5.87898i		−	1.34873i	−0.738398	−	0.674366i	$-0.764417\pi$
$19$		−	5.87898i		−	1.34873i	0.738398	−	0.674366i	$-0.235583\pi$
$20$		0			0
$21$	−4.29768	+	1.66988i	−0.937831	+	0.364398i
$22$		0			0
$23$	4.05744			0.846034			0.423017	−	0.906122i	$-0.360971\pi$
$23$	4.05744			0.846034			0.423017	+	0.906122i	$0.360971\pi$
$24$		0			0
$25$	−11.8940			−2.37880
$26$		0			0
$27$	−4.65834	−	2.30214i	−0.896498	−	0.443047i
$28$		0			0
$29$			6.15231i			1.14246i	0.820791	+	0.571228i	$0.193533\pi$
$29$			6.15231i			1.14246i	−0.820791	+	0.571228i	$0.806467\pi$
$30$		0			0
$31$		−	10.4424i		−	1.87551i	−0.347299	−	0.937754i	$-0.612901\pi$
$31$		−	10.4424i		−	1.87551i	0.347299	−	0.937754i	$-0.387099\pi$
$32$		0			0
$33$	2.08471	+	5.36529i	0.362901	+	0.933977i
$34$		0			0
$35$	−10.9414			−1.84943
$36$		0			0
$37$	−0.732705			−0.120456			−0.0602280	−	0.998185i	$-0.519183\pi$
$37$	−0.732705			−0.120456			−0.0602280	+	0.998185i	$0.519183\pi$
$38$		0			0
$39$	3.08809	+	7.94764i	0.494490	+	1.27264i
$40$		0			0
$41$			6.92278i			1.08116i	0.841294	+	0.540578i	$0.181794\pi$
$41$			6.92278i			1.08116i	−0.841294	+	0.540578i	$0.818206\pi$
$42$		0			0
$43$		−	6.43775i		−	0.981748i	−0.871230	−	0.490874i	$-0.836677\pi$
$43$		−	6.43775i		−	0.981748i	0.871230	−	0.490874i	$-0.163323\pi$
$44$		0			0
$45$	−8.32536	−	9.09583i	−1.24107	−	1.35593i
$46$		0			0
$47$	−1.01943			−0.148700			−0.0743499	−	0.997232i	$-0.523688\pi$
$47$	−1.01943			−0.148700			−0.0743499	+	0.997232i	$0.523688\pi$
$48$		0			0
$49$	−0.0861847			−0.0123121
$50$		0			0
$51$	1.61446	−	0.627306i	0.226070	−	0.0878404i
$52$		0			0
$53$		−	14.0222i		−	1.92610i	−0.269315	−	0.963052i	$-0.586797\pi$
$53$		−	14.0222i		−	1.92610i	0.269315	−	0.963052i	$-0.413203\pi$
$54$		0			0
$55$			13.6594i			1.84183i
$56$		0			0
$57$	9.49139	−	3.68792i	1.25717	−	0.488477i
$58$		0			0
$59$	−1.85988			−0.242135			−0.121068	−	0.992644i	$-0.538632\pi$
$59$	−1.85988			−0.242135			−0.121068	+	0.992644i	$0.538632\pi$
$60$		0			0
$61$	2.86360			0.366646			0.183323	−	0.983053i	$-0.441315\pi$
$61$	2.86360			0.366646			0.183323	+	0.983053i	$0.441315\pi$
$62$		0			0
$63$	−5.39192	−	5.89091i	−0.679318	−	0.742185i
$64$		0			0
$65$			20.2337i			2.50969i
$66$		0			0
$67$		−	2.34199i		−	0.286120i	−0.989714	−	0.143060i	$-0.954306\pi$
$67$		−	2.34199i		−	0.286120i	0.989714	−	0.143060i	$-0.0456941\pi$
$68$		0			0
$69$	2.54525	+	6.55058i	0.306413	+	0.788597i
$70$		0			0
$71$	10.0509			1.19282			0.596408	−	0.802681i	$-0.296594\pi$
$71$	10.0509			1.19282			0.596408	+	0.802681i	$0.296594\pi$
$72$		0			0
$73$	4.25654			0.498190			0.249095	−	0.968479i	$-0.419867\pi$
$73$	4.25654			0.498190			0.249095	+	0.968479i	$0.419867\pi$
$74$		0			0
$75$	−7.46117	−	19.2024i	−0.861541	−	2.21730i
$76$		0			0
$77$			8.84650i			1.00815i
$78$		0			0
$79$		−	2.66199i		−	0.299497i	−0.988724	−	0.149749i	$-0.952154\pi$
$79$		−	2.66199i		−	0.299497i	0.988724	−	0.149749i	$-0.0478464\pi$
$80$		0			0
$81$	0.794509	−	8.96486i	0.0882788	−	0.996096i
$82$		0			0
$83$	−8.03605			−0.882071			−0.441036	−	0.897490i	$-0.645389\pi$
$83$	−8.03605			−0.882071			−0.441036	+	0.897490i	$0.645389\pi$
$84$		0			0
$85$	4.11023			0.445817
$86$		0			0
$87$	−9.93267	+	3.85938i	−1.06489	+	0.413769i
$88$		0			0
$89$			10.8008i			1.14489i	0.819944	+	0.572444i	$0.194005\pi$
$89$			10.8008i			1.14489i	−0.819944	+	0.572444i	$0.805995\pi$
$90$		0			0
$91$			13.1044i			1.37371i
$92$		0			0
$93$	16.8588	−	6.55058i	1.74818	−	0.679263i
$94$		0			0
$95$	24.1640			2.47917
$96$		0			0
$97$	−9.10115			−0.924082			−0.462041	−	0.886859i	$-0.652883\pi$
$97$	−9.10115			−0.924082			−0.462041	+	0.886859i	$0.652883\pi$
$98$		0			0
$99$	−7.35430	+	6.73135i	−0.739135	+	0.676527i
$100$		0			0
$101$			10.6825i			1.06295i	0.847075	+	0.531474i	$0.178362\pi$
$101$			10.6825i			1.06295i	−0.847075	+	0.531474i	$0.821638\pi$
$102$		0			0
$103$			2.60357i			0.256537i	0.991739	+	0.128269i	$0.0409419\pi$
$103$			2.60357i			0.256537i	−0.991739	+	0.128269i	$0.959058\pi$
$104$		0			0
$105$	−6.86360	−	17.6644i	−0.669818	−	1.72387i
$106$		0			0
$107$	0.866841			0.0838007			0.0419004	−	0.999122i	$-0.486659\pi$
$107$	0.866841			0.0838007			0.0419004	+	0.999122i	$0.486659\pi$
$108$		0			0
$109$	−2.23216			−0.213802			−0.106901	−	0.994270i	$-0.534093\pi$
$109$	−2.23216			−0.213802			−0.106901	+	0.994270i	$0.534093\pi$
$110$		0			0
$111$	−0.459630	−	1.18292i	−0.0436262	−	0.112278i
$112$		0			0
$113$		−	9.72795i		−	0.915129i	−0.889177	−	0.457564i	$-0.848722\pi$
$113$		−	9.72795i		−	0.915129i	0.889177	−	0.457564i	$-0.151278\pi$
$114$		0			0
$115$			16.6770i			1.55514i
$116$		0			0
$117$	−10.8940	+	9.97120i	−1.00715	+	0.921838i
$118$		0			0
$119$	2.66199			0.244024
$120$		0			0
$121$	0.0441006			0.00400914
$122$		0			0
$123$	−11.1766	+	4.34270i	−1.00776	+	0.391568i
$124$		0			0
$125$		−	28.3358i		−	2.53443i
$126$		0			0
$127$			7.40439i			0.657033i	0.944498	+	0.328517i	$0.106549\pi$
$127$			7.40439i			0.657033i	−0.944498	+	0.328517i	$0.893451\pi$
$128$		0			0
$129$	10.3935	−	4.03844i	0.915097	−	0.355565i
$130$		0			0
$131$	−4.53379			−0.396119			−0.198060	−	0.980190i	$-0.563464\pi$
$131$	−4.53379			−0.396119			−0.198060	+	0.980190i	$0.563464\pi$
$132$		0			0
$133$	15.6498			1.35701
$134$		0			0
$135$	9.46232	−	19.1469i	0.814387	−	1.64790i
$136$		0			0
$137$		−	6.64641i		−	0.567841i	−0.958848	−	0.283920i	$-0.908365\pi$
$137$		−	6.64641i		−	0.567841i	0.958848	−	0.283920i	$-0.0916351\pi$
$138$		0			0
$139$		−	4.64446i		−	0.393938i	−0.980410	−	0.196969i	$-0.936890\pi$
$139$		−	4.64446i		−	0.393938i	0.980410	−	0.196969i	$-0.0631099\pi$
$140$		0			0
$141$	−0.639497	−	1.64584i	−0.0538554	−	0.138605i
$142$		0			0
$143$	16.3597			1.36807
$144$		0			0
$145$	−25.2874			−2.10001
$146$		0			0
$147$	−0.0540642	−	0.139142i	−0.00445914	−	0.0114762i
$148$		0			0
$149$		−	13.5505i		−	1.11010i	−0.831817	−	0.555050i	$-0.812699\pi$
$149$		−	13.5505i		−	1.11010i	0.831817	−	0.555050i	$-0.187301\pi$
$150$		0			0
$151$			9.50198i			0.773261i	0.922235	+	0.386630i	$0.126361\pi$
$151$			9.50198i			0.773261i	−0.922235	+	0.386630i	$0.873639\pi$
$152$		0			0
$153$	2.02552	+	2.21297i	0.163754	+	0.178908i
$154$		0			0
$155$	42.9206			3.44747
$156$		0			0
$157$	1.38185			0.110283			0.0551417	−	0.998479i	$-0.482439\pi$
$157$	1.38185			0.110283			0.0551417	+	0.998479i	$0.482439\pi$
$158$		0			0
$159$	22.6384	−	8.79624i	1.79534	−	0.697587i
$160$		0			0
$161$			10.8008i			0.851226i
$162$		0			0
$163$			3.79226i			0.297033i	0.988910	+	0.148517i	$0.0474498\pi$
$163$			3.79226i			0.297033i	−0.988910	+	0.148517i	$0.952550\pi$
$164$		0			0
$165$	−22.0526	+	8.56862i	−1.71679	+	0.667066i
$166$		0			0
$167$	1.02506			0.0793213			0.0396606	−	0.999213i	$-0.487372\pi$
$167$	1.02506			0.0793213			0.0396606	+	0.999213i	$0.487372\pi$
$168$		0			0
$169$	11.2337			0.864134
$170$		0			0
$171$	11.9080	+	13.0100i	0.910629	+	0.994902i
$172$		0			0
$173$		−	10.6714i		−	0.811331i	−0.914022	−	0.405665i	$-0.867040\pi$
$173$		−	10.6714i		−	0.811331i	0.914022	−	0.405665i	$-0.132960\pi$
$174$		0			0
$175$		−	31.6616i		−	2.39340i
$176$		0			0
$177$	−1.16671	−	3.00270i	−0.0876954	−	0.225697i
$178$		0			0
$179$	3.74120			0.279631			0.139815	−	0.990178i	$-0.455349\pi$
$179$	3.74120			0.279631			0.139815	+	0.990178i	$0.455349\pi$
$180$		0			0
$181$	4.02438			0.299130			0.149565	−	0.988752i	$-0.452213\pi$
$181$	4.02438			0.299130			0.149565	+	0.988752i	$0.452213\pi$
$182$		0			0
$183$	1.79635	+	4.62317i	0.132790	+	0.341754i
$184$		0			0
$185$		−	3.01159i		−	0.221416i
$186$		0			0
$187$		−	3.32327i		−	0.243021i
$188$		0			0
$189$	6.12827	−	12.4005i	0.445766	−	0.902000i
$190$		0			0
$191$	−13.8949			−1.00540			−0.502701	−	0.864460i	$-0.667660\pi$
$191$	−13.8949			−1.00540			−0.502701	+	0.864460i	$0.667660\pi$
$192$		0			0
$193$	−7.70927			−0.554925			−0.277463	−	0.960736i	$-0.589493\pi$
$193$	−7.70927			−0.554925			−0.277463	+	0.960736i	$0.589493\pi$
$194$		0			0
$195$	−32.6666	+	12.6928i	−2.33930	+	0.908947i
$196$		0			0
$197$			19.1538i			1.36465i	0.731049	+	0.682325i	$0.239031\pi$
$197$			19.1538i			1.36465i	−0.731049	+	0.682325i	$0.760969\pi$
$198$		0			0
$199$			18.0467i			1.27930i	0.768667	+	0.639649i	$0.220920\pi$
$199$			18.0467i			1.27930i	−0.768667	+	0.639649i	$0.779080\pi$
$200$		0			0
$201$	3.78105	−	1.46914i	0.266695	−	0.103625i
$202$		0			0
$203$	−16.3774			−1.14947
$204$		0			0
$205$	−28.4542			−1.98733
$206$		0			0
$207$	−8.97900	+	8.21843i	−0.624084	+	0.571221i
$208$		0			0
$209$		−	19.5374i		−	1.35143i
$210$		0			0
$211$		−	16.0516i		−	1.10504i	−0.833500	−	0.552520i	$-0.813666\pi$
$211$		−	16.0516i		−	1.10504i	0.833500	−	0.552520i	$-0.186334\pi$
$212$		0			0
$213$	6.30496	+	16.2267i	0.432009	+	1.11184i
$214$		0			0
$215$	26.4606			1.80460
$216$		0			0
$217$	27.7975			1.88702
$218$		0			0
$219$	2.67015	+	6.87202i	0.180432	+	0.464368i
$220$		0			0
$221$		−	4.92278i		−	0.331142i
$222$		0			0
$223$			13.3421i			0.893453i	0.894671	+	0.446727i	$0.147410\pi$
$223$			13.3421i			0.893453i	−0.894671	+	0.446727i	$0.852590\pi$
$224$		0			0
$225$	26.3211	−	24.0915i	1.75474	−	1.60610i
$226$		0			0
$227$	−17.0912			−1.13438			−0.567190	−	0.823587i	$-0.691970\pi$
$227$	−17.0912			−1.13438			−0.567190	+	0.823587i	$0.691970\pi$
$228$		0			0
$229$	−1.50673			−0.0995678			−0.0497839	−	0.998760i	$-0.515853\pi$
$229$	−1.50673			−0.0995678			−0.0497839	+	0.998760i	$0.515853\pi$
$230$		0			0
$231$	−14.2823	+	5.54946i	−0.939709	+	0.365128i
$232$		0			0
$233$		−	13.7459i		−	0.900522i	−0.892897	−	0.450261i	$-0.851331\pi$
$233$		−	13.7459i		−	0.900522i	0.892897	−	0.450261i	$-0.148669\pi$
$234$		0			0
$235$		−	4.19011i		−	0.273333i
$236$		0			0
$237$	4.29768	−	1.66988i	0.279164	−	0.108470i
$238$		0			0
$239$	−2.59472			−0.167838			−0.0839192	−	0.996473i	$-0.526744\pi$
$239$	−2.59472			−0.167838			−0.0839192	+	0.996473i	$0.526744\pi$
$240$		0			0
$241$	17.1926			1.10747			0.553736	−	0.832692i	$-0.313202\pi$
$241$	17.1926			1.10747			0.553736	+	0.832692i	$0.313202\pi$
$242$		0			0
$243$	14.9718	−	4.34101i	0.960443	−	0.278476i
$244$		0			0
$245$		−	0.354239i		−	0.0226315i
$246$		0			0
$247$		−	28.9409i		−	1.84147i
$248$		0			0
$249$	−5.04107	−	12.9739i	−0.319465	−	0.822188i
$250$		0			0
$251$	1.60663			0.101410			0.0507049	−	0.998714i	$-0.483853\pi$
$251$	1.60663			0.101410			0.0507049	+	0.998714i	$0.483853\pi$
$252$		0			0
$253$	13.4839			0.847728
$254$		0			0
$255$	2.57837	+	6.63581i	0.161464	+	0.415550i
$256$		0			0
$257$			4.63373i			0.289044i	0.989502	+	0.144522i	$0.0461644\pi$
$257$			4.63373i			0.289044i	−0.989502	+	0.144522i	$0.953836\pi$
$258$		0			0
$259$		−	1.95045i		−	0.121195i
$260$		0			0
$261$	−12.4617	−	13.6149i	−0.771357	−	0.842742i
$262$		0			0
$263$	21.2594			1.31091			0.655457	−	0.755233i	$-0.272476\pi$
$263$	21.2594			1.31091			0.655457	+	0.755233i	$0.272476\pi$
$264$		0			0
$265$	57.6347			3.54047
$266$		0			0
$267$	−17.4376	+	6.77544i	−1.06716	+	0.414650i
$268$		0			0
$269$			4.25380i			0.259359i	0.991556	+	0.129679i	$0.0413948\pi$
$269$			4.25380i			0.259359i	−0.991556	+	0.129679i	$0.958605\pi$
$270$		0			0
$271$			11.0856i			0.673403i	0.941612	+	0.336701i	$0.109311\pi$
$271$			11.0856i			0.673403i	−0.941612	+	0.336701i	$0.890689\pi$
$272$		0			0
$273$	−21.1565	+	8.22046i	−1.28045	+	0.497525i
$274$		0			0
$275$	−39.5269			−2.38356
$276$		0			0
$277$	6.26635			0.376509			0.188254	−	0.982120i	$-0.439717\pi$
$277$	6.26635			0.376509			0.188254	+	0.982120i	$0.439717\pi$
$278$		0			0
$279$	21.1513	+	23.1087i	1.26630	+	1.38348i
$280$		0			0
$281$		−	1.94241i		−	0.115874i	−0.998320	−	0.0579371i	$-0.981548\pi$
$281$		−	1.94241i		−	0.115874i	0.998320	−	0.0579371i	$-0.0184523\pi$
$282$		0			0
$283$		−	21.4158i		−	1.27304i	−0.771261	−	0.636518i	$-0.780374\pi$
$283$		−	21.4158i		−	1.27304i	0.771261	−	0.636518i	$-0.219626\pi$
$284$		0			0
$285$	15.1582	+	39.0118i	0.897894	+	2.31086i
$286$		0			0
$287$	−18.4284			−1.08779
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	−5.70921	−	14.6935i	−0.334680	−	0.861346i
$292$		0			0
$293$		−	5.44841i		−	0.318299i	−0.987254	−	0.159150i	$-0.949125\pi$
$293$		−	5.44841i		−	0.318299i	0.987254	−	0.159150i	$-0.0508752\pi$
$294$		0			0
$295$		−	7.64452i		−	0.445081i
$296$		0			0
$297$	−15.4809	−	7.65062i	−0.898294	−	0.443934i
$298$		0			0
$299$	19.9739			1.15512
$300$		0			0
$301$	17.1372			0.987774
$302$		0			0
$303$	−17.2465	+	6.70119i	−0.990784	+	0.384973i
$304$		0			0
$305$			11.7700i			0.673950i
$306$		0			0
$307$		−	29.4440i		−	1.68046i	−0.542232	−	0.840229i	$-0.682421\pi$
$307$		−	29.4440i		−	1.68046i	0.542232	−	0.840229i	$-0.317579\pi$
$308$		0			0
$309$	−4.20336	+	1.63323i	−0.239121	+	0.0929114i
$310$		0			0
$311$	−11.7317			−0.665246			−0.332623	−	0.943060i	$-0.607934\pi$
$311$	−11.7317			−0.665246			−0.332623	+	0.943060i	$0.607934\pi$
$312$		0			0
$313$	−18.1556			−1.02621			−0.513107	−	0.858325i	$-0.671506\pi$
$313$	−18.1556			−1.02621			−0.513107	+	0.858325i	$0.671506\pi$
$314$		0			0
$315$	24.2130	−	22.1620i	1.36425	−	1.24869i
$316$		0			0
$317$			20.5805i			1.15592i	0.816066	+	0.577959i	$0.196151\pi$
$317$			20.5805i			1.15592i	−0.816066	+	0.577959i	$0.803849\pi$
$318$		0			0
$319$			20.4458i			1.14474i
$320$		0			0
$321$	0.543775	+	1.39948i	0.0303506	+	0.0781115i
$322$		0			0
$323$	−5.87898			−0.327115
$324$		0			0
$325$	−58.5514			−3.24785
$326$		0			0
$327$	−1.40025	−	3.60373i	−0.0774338	−	0.199287i
$328$		0			0
$329$		−	2.71372i		−	0.149612i
$330$		0			0
$331$		−	17.3146i		−	0.951695i	−0.879528	−	0.475848i	$-0.842141\pi$
$331$		−	17.3146i		−	0.951695i	0.879528	−	0.475848i	$-0.157859\pi$
$332$		0			0
$333$	1.62146	−	1.48411i	0.0888553	−	0.0813288i
$334$		0			0
$335$	9.62611			0.525931
$336$		0			0
$337$	−20.6634			−1.12561			−0.562804	−	0.826590i	$-0.690277\pi$
$337$	−20.6634			−1.12561			−0.562804	+	0.826590i	$0.690277\pi$
$338$		0			0
$339$	15.7054	−	6.10240i	0.853001	−	0.331437i
$340$		0			0
$341$		−	34.7029i		−	1.87926i
$342$		0			0
$343$			18.4045i			0.993750i
$344$		0			0
$345$	−26.9244	+	10.4616i	−1.44956	+	0.563232i
$346$		0			0
$347$	−10.4270			−0.559753			−0.279876	−	0.960036i	$-0.590293\pi$
$347$	−10.4270			−0.559753			−0.279876	+	0.960036i	$0.590293\pi$
$348$		0			0
$349$	−1.74407			−0.0933578			−0.0466789	−	0.998910i	$-0.514864\pi$
$349$	−1.74407			−0.0933578			−0.0466789	+	0.998910i	$0.514864\pi$
$350$		0			0
$351$	−22.9320	−	11.3329i	−1.22402	−	0.604907i
$352$		0			0
$353$			23.5441i			1.25313i	0.779371	+	0.626563i	$0.215539\pi$
$353$			23.5441i			1.25313i	−0.779371	+	0.626563i	$0.784461\pi$
$354$		0			0
$355$			41.3113i			2.19258i
$356$		0			0
$357$	1.66988	+	4.29768i	0.0883795	+	0.227457i
$358$		0			0
$359$	−33.6387			−1.77538			−0.887691	−	0.460441i	$-0.847691\pi$
$359$	−33.6387			−1.77538			−0.887691	+	0.460441i	$0.847691\pi$
$360$		0			0
$361$	−15.5624			−0.819076
$362$		0			0
$363$	0.0276646	+	0.0711987i	0.00145201	+	0.00373696i
$364$		0			0
$365$			17.4953i			0.915748i
$366$		0			0
$367$			4.40692i			0.230039i	0.993363	+	0.115020i	$0.0366931\pi$
$367$			4.40692i			0.230039i	−0.993363	+	0.115020i	$0.963307\pi$
$368$		0			0
$369$	−14.0222	−	15.3199i	−0.729969	−	0.797524i
$370$		0			0
$371$	37.3271			1.93793
$372$		0			0
$373$	−4.86817			−0.252064			−0.126032	−	0.992026i	$-0.540224\pi$
$373$	−4.86817			−0.252064			−0.126032	+	0.992026i	$0.540224\pi$
$374$		0			0
$375$	45.7471	−	17.7752i	2.36237	−	0.917910i
$376$		0			0
$377$			30.2865i			1.55983i
$378$		0			0
$379$			27.4159i			1.40826i	0.710071	+	0.704130i	$0.248663\pi$
$379$			27.4159i			1.40826i	−0.710071	+	0.704130i	$0.751337\pi$
$380$		0			0
$381$	−11.9541	+	4.64482i	−0.612427	+	0.237961i
$382$		0			0
$383$	−31.9905			−1.63464			−0.817319	−	0.576186i	$-0.804540\pi$
$383$	−31.9905			−1.63464			−0.817319	+	0.576186i	$0.804540\pi$
$384$		0			0
$385$	−36.3611			−1.85313
$386$		0			0
$387$	13.0398	+	14.2466i	0.662851	+	0.724195i
$388$		0			0
$389$			14.3323i			0.726675i	0.931658	+	0.363338i	$0.118363\pi$
$389$			14.3323i			0.726675i	−0.931658	+	0.363338i	$0.881637\pi$
$390$		0			0
$391$		−	4.05744i		−	0.205193i
$392$		0			0
$393$	−2.84408	−	7.31964i	−0.143465	−	0.369227i
$394$		0			0
$395$	10.9414			0.550521
$396$		0			0
$397$	−26.6570			−1.33787			−0.668937	−	0.743319i	$-0.733250\pi$
$397$	−26.6570			−1.33787			−0.668937	+	0.743319i	$0.733250\pi$
$398$		0			0
$399$	9.81721	+	25.2660i	0.491475	+	1.26488i
$400$		0			0
$401$			30.3966i			1.51793i	0.651129	+	0.758967i	$0.274296\pi$
$401$			30.3966i			1.51793i	−0.651129	+	0.758967i	$0.725704\pi$
$402$		0			0
$403$		−	51.4056i		−	2.56069i
$404$		0			0
$405$	36.8476	+	3.26561i	1.83097	+	0.162270i
$406$		0			0
$407$	−2.43497			−0.120697
$408$		0			0
$409$	23.2328			1.14879			0.574394	−	0.818579i	$-0.305238\pi$
$409$	23.2328			1.14879			0.574394	+	0.818579i	$0.305238\pi$
$410$		0			0
$411$	10.7304	−	4.16933i	0.529290	−	0.205658i
$412$		0			0
$413$		−	4.95097i		−	0.243621i
$414$		0			0
$415$		−	33.0300i		−	1.62138i
$416$		0			0
$417$	7.49831	−	2.91350i	0.367194	−	0.142675i
$418$		0			0
$419$	22.6030			1.10423			0.552114	−	0.833769i	$-0.313821\pi$
$419$	22.6030			1.10423			0.552114	+	0.833769i	$0.313821\pi$
$420$		0			0
$421$	37.1210			1.80917			0.904585	−	0.426294i	$-0.140181\pi$
$421$	37.1210			1.80917			0.904585	+	0.426294i	$0.140181\pi$
$422$		0			0
$423$	2.25598	−	2.06489i	0.109690	−	0.100398i
$424$		0			0
$425$			11.8940i			0.576943i
$426$		0			0
$427$			7.62286i			0.368896i
$428$		0			0
$429$	10.2625	+	26.4121i	0.495480	+	1.27519i
$430$		0			0
$431$	−35.1898			−1.69503			−0.847516	−	0.530769i	$-0.821903\pi$
$431$	−35.1898			−1.69503			−0.847516	+	0.530769i	$0.821903\pi$
$432$		0			0
$433$	37.8450			1.81872			0.909358	−	0.416015i	$-0.136574\pi$
$433$	37.8450			1.81872			0.909358	+	0.416015i	$0.136574\pi$
$434$		0			0
$435$	−15.8629	−	40.8256i	−0.760570	−	1.95744i
$436$		0			0
$437$		−	23.8536i		−	1.14107i
$438$		0			0
$439$			39.9169i			1.90513i	0.304338	+	0.952564i	$0.401565\pi$
$439$			39.9169i			1.90513i	−0.304338	+	0.952564i	$0.598435\pi$
$440$		0			0
$441$	0.190725	−	0.174569i	0.00908212	−	0.00831282i
$442$		0			0
$443$	−34.7269			−1.64993			−0.824963	−	0.565187i	$-0.808804\pi$
$443$	−34.7269			−1.64993			−0.824963	+	0.565187i	$0.808804\pi$
$444$		0			0
$445$	−44.3940			−2.10448
$446$		0			0
$447$	21.8768	−	8.50031i	1.03474	−	0.402051i
$448$		0			0
$449$		−	9.43661i		−	0.445341i	−0.974894	−	0.222671i	$-0.928523\pi$
$449$		−	9.43661i		−	0.445341i	0.974894	−	0.222671i	$-0.0714775\pi$
$450$		0			0
$451$			23.0062i			1.08332i
$452$		0			0
$453$	−15.3406	+	5.96065i	−0.720764	+	0.280056i
$454$		0			0
$455$	−53.8620			−2.52509
$456$		0			0
$457$	−0.719325			−0.0336486			−0.0168243	−	0.999858i	$-0.505356\pi$
$457$	−0.719325			−0.0336486			−0.0168243	+	0.999858i	$0.505356\pi$
$458$		0			0
$459$	−2.30214	+	4.65834i	−0.107455	+	0.217433i
$460$		0			0
$461$		−	5.36741i		−	0.249985i	−0.992158	−	0.124993i	$-0.960109\pi$
$461$		−	5.36741i		−	0.249985i	0.992158	−	0.124993i	$-0.0398907\pi$
$462$		0			0
$463$			14.3195i			0.665483i	0.943018	+	0.332741i	$0.107974\pi$
$463$			14.3195i			0.665483i	−0.943018	+	0.332741i	$0.892026\pi$
$464$		0			0
$465$	26.9244	+	69.2937i	1.24859	+	3.21342i
$466$		0			0
$467$	−18.3067			−0.847135			−0.423567	−	0.905865i	$-0.639222\pi$
$467$	−18.3067			−0.847135			−0.423567	+	0.905865i	$0.639222\pi$
$468$		0			0
$469$	6.23435			0.287876
$470$		0			0
$471$	0.866841	+	2.23094i	0.0399419	+	0.102796i
$472$		0			0
$473$		−	21.3944i		−	0.983714i
$474$		0			0
$475$			69.9245i			3.20836i
$476$		0			0
$477$	28.4024	+	31.0309i	1.30046	+	1.42081i
$478$		0			0
$479$	31.5888			1.44333			0.721663	−	0.692244i	$-0.243378\pi$
$479$	31.5888			1.44333			0.721663	+	0.692244i	$0.243378\pi$
$480$		0			0
$481$	−3.60694			−0.164462
$482$		0			0
$483$	−17.4376	+	6.77544i	−0.793436	+	0.308293i
$484$		0			0
$485$		−	37.4078i		−	1.69860i
$486$		0			0
$487$		−	32.7717i		−	1.48503i	−0.669832	−	0.742513i	$-0.733634\pi$
$487$		−	32.7717i		−	1.48503i	0.669832	−	0.742513i	$-0.266366\pi$
$488$		0			0
$489$	−6.12247	+	2.37891i	−0.276867	+	0.107578i
$490$		0			0
$491$	−14.5703			−0.657549			−0.328774	−	0.944408i	$-0.606636\pi$
$491$	−14.5703			−0.657549			−0.328774	+	0.944408i	$0.606636\pi$
$492$		0			0
$493$	6.15231			0.277086
$494$		0			0
$495$	−27.6674	−	30.2279i	−1.24356	−	1.35864i
$496$		0			0
$497$			26.7553i			1.20014i
$498$		0			0
$499$		−	3.87611i		−	0.173519i	−0.996229	−	0.0867593i	$-0.972349\pi$
$499$		−	3.87611i		−	0.173519i	0.996229	−	0.0867593i	$-0.0276511\pi$
$500$		0			0
$501$	0.643024	+	1.65491i	0.0287282	+	0.0739361i
$502$		0			0
$503$	1.14190			0.0509148			0.0254574	−	0.999676i	$-0.491896\pi$
$503$	1.14190			0.0509148			0.0254574	+	0.999676i	$0.491896\pi$
$504$		0			0
$505$	−43.9075			−1.95386
$506$		0			0
$507$	7.04700	+	18.1365i	0.312968	+	0.805468i
$508$		0			0
$509$		−	36.1545i		−	1.60252i	−0.598316	−	0.801260i	$-0.704163\pi$
$509$		−	36.1545i		−	1.60252i	0.598316	−	0.801260i	$-0.295837\pi$
$510$		0			0
$511$			11.3309i			0.501248i
$512$		0			0
$513$	−13.5342	+	27.3863i	−0.597551	+	1.20914i
$514$		0			0
$515$	−10.7013			−0.471554
$516$		0			0
$517$	−3.38785			−0.148998
$518$		0			0
$519$	17.2286	−	6.69423i	0.756250	−	0.293844i
$520$		0			0
$521$			21.9290i			0.960725i	0.877070	+	0.480363i	$0.159495\pi$
$521$			21.9290i			0.960725i	−0.877070	+	0.480363i	$0.840505\pi$
$522$		0			0
$523$		−	24.8075i		−	1.08476i	−0.840135	−	0.542378i	$-0.817524\pi$
$523$		−	24.8075i		−	1.08476i	0.840135	−	0.542378i	$-0.182476\pi$
$524$		0			0
$525$	51.1165	−	19.8615i	2.23091	−	0.866829i
$526$		0			0
$527$	−10.4424			−0.454878
$528$		0			0
$529$	−6.53721			−0.284227
$530$		0			0
$531$	4.11586	−	3.76722i	0.178613	−	0.163484i
$532$		0			0
$533$			34.0793i			1.47614i
$534$		0			0
$535$			3.56292i			0.154038i
$536$		0			0
$537$	2.34688	+	6.04003i	0.101275	+	0.260647i
$538$		0			0
$539$	−0.286415			−0.0123368
$540$		0			0
$541$	16.7271			0.719153			0.359577	−	0.933116i	$-0.382921\pi$
$541$	16.7271			0.719153			0.359577	+	0.933116i	$0.382921\pi$
$542$		0			0
$543$	2.52452	+	6.49721i	0.108337	+	0.278822i
$544$		0			0
$545$		−	9.17468i		−	0.393000i
$546$		0			0
$547$			21.0101i			0.898327i	0.893450	+	0.449163i	$0.148278\pi$
$547$			21.0101i			0.898327i	−0.893450	+	0.449163i	$0.851722\pi$
$548$		0			0
$549$	−6.33706	+	5.80028i	−0.270459	+	0.247550i
$550$		0			0
$551$	36.1693			1.54087
$552$		0			0
$553$	7.08618			0.301335
$554$		0			0
$555$	4.86209	−	1.88919i	0.206384	−	0.0801915i
$556$		0			0
$557$			18.8882i			0.800318i	0.916446	+	0.400159i	$0.131045\pi$
$557$			18.8882i			0.800318i	−0.916446	+	0.400159i	$0.868955\pi$
$558$		0			0
$559$		−	31.6916i		−	1.34041i
$560$		0			0
$561$	5.36529	−	2.08471i	0.226523	−	0.0880163i
$562$		0			0
$563$	11.3736			0.479341			0.239670	−	0.970854i	$-0.422961\pi$
$563$	11.3736			0.479341			0.239670	+	0.970854i	$0.422961\pi$
$564$		0			0
$565$	39.9841			1.68214
$566$		0			0
$567$	23.8644	+	2.11497i	1.00221	+	0.0888206i
$568$		0			0
$569$		−	24.5430i		−	1.02890i	−0.857521	−	0.514448i	$-0.827997\pi$
$569$		−	24.5430i		−	1.02890i	0.857521	−	0.514448i	$-0.172003\pi$
$570$		0			0
$571$			0.913466i			0.0382274i	0.999817	+	0.0191137i	$0.00608445\pi$
$571$			0.913466i			0.0382274i	−0.999817	+	0.0191137i	$0.993916\pi$
$572$		0			0
$573$	−8.71638	−	22.4329i	−0.364132	−	0.937146i
$574$		0			0
$575$	−48.2591			−2.01254
$576$		0			0
$577$	16.4084			0.683091			0.341545	−	0.939865i	$-0.389050\pi$
$577$	16.4084			0.683091			0.341545	+	0.939865i	$0.389050\pi$
$578$		0			0
$579$	−4.83607	−	12.4463i	−0.200980	−	0.517251i
$580$		0			0
$581$		−	21.3919i		−	0.887485i
$582$		0			0
$583$		−	46.5997i		−	1.92996i
$584$		0			0
$585$	−40.9839	−	44.7768i	−1.69448	−	1.85129i
$586$		0			0
$587$	47.6644			1.96732			0.983660	−	0.180038i	$-0.0576221\pi$
$587$	47.6644			1.96732			0.983660	+	0.180038i	$0.0576221\pi$
$588$		0			0
$589$	−61.3906			−2.52956
$590$		0			0
$591$	−30.9231	+	12.0153i	−1.27200	+	0.494243i
$592$		0			0
$593$		−	42.4727i		−	1.74415i	−0.489376	−	0.872073i	$-0.662775\pi$
$593$		−	42.4727i		−	1.74415i	0.489376	−	0.872073i	$-0.337225\pi$
$594$		0			0
$595$			10.9414i			0.448553i
$596$		0			0
$597$	−29.1357	+	11.3208i	−1.19245	+	0.463330i
$598$		0			0
$599$	25.0794			1.02472			0.512358	−	0.858772i	$-0.328772\pi$
$599$	25.0794			1.02472			0.512358	+	0.858772i	$0.328772\pi$
$600$		0			0
$601$	−17.6303			−0.719157			−0.359578	−	0.933115i	$-0.617079\pi$
$601$	−17.6303			−0.719157			−0.359578	+	0.933115i	$0.617079\pi$
$602$		0			0
$603$	4.74375	+	5.18276i	0.193181	+	0.211058i
$604$		0			0
$605$			0.181264i			0.00736941i
$606$		0			0
$607$			6.27614i			0.254741i	0.991855	+	0.127370i	$0.0406537\pi$
$607$			6.27614i			0.254741i	−0.991855	+	0.127370i	$0.959346\pi$
$608$		0			0
$609$	−10.2736	−	26.4407i	−0.416309	−	1.07143i
$610$		0			0
$611$	−5.01845			−0.203025
$612$		0			0
$613$	4.14433			0.167388			0.0836940	−	0.996492i	$-0.473328\pi$
$613$	4.14433			0.167388			0.0836940	+	0.996492i	$0.473328\pi$
$614$		0			0
$615$	−17.8495	−	45.9382i	−0.719761	−	1.85241i
$616$		0			0
$617$			31.9669i			1.28694i	0.765471	+	0.643470i	$0.222506\pi$
$617$			31.9669i			1.28694i	−0.765471	+	0.643470i	$0.777494\pi$
$618$		0			0
$619$			5.72690i			0.230184i	0.993355	+	0.115092i	$0.0367162\pi$
$619$			5.72690i			0.230184i	−0.993355	+	0.115092i	$0.963284\pi$
$620$		0			0
$621$	−18.9009	−	9.34078i	−0.758468	−	0.374833i
$622$		0			0
$623$	−28.7517			−1.15191
$624$		0			0
$625$	56.9969			2.27987
$626$		0			0
$627$	31.5424	−	12.2559i	1.25968	−	0.489455i
$628$		0			0
$629$			0.732705i			0.0292149i
$630$		0			0
$631$		−	47.8136i		−	1.90343i	−0.306983	−	0.951715i	$-0.599319\pi$
$631$		−	47.8136i		−	1.90343i	0.306983	−	0.951715i	$-0.400681\pi$
$632$		0			0
$633$	25.9148	−	10.0693i	1.03002	−	0.400218i
$634$		0			0
$635$	−30.4337			−1.20773
$636$		0			0
$637$	−0.424268			−0.0168101
$638$		0			0
$639$	−22.2423	+	20.3582i	−0.879891	+	0.805360i
$640$		0			0
$641$		−	9.60959i		−	0.379556i	−0.981827	−	0.189778i	$-0.939223\pi$
$641$		−	9.60959i		−	0.379556i	0.981827	−	0.189778i	$-0.0607768\pi$
$642$		0			0
$643$			1.19724i			0.0472147i	0.999721	+	0.0236074i	$0.00751515\pi$
$643$			1.19724i			0.0472147i	−0.999721	+	0.0236074i	$0.992485\pi$
$644$		0			0
$645$	16.5989	+	42.7197i	0.653582	+	1.68209i
$646$		0			0
$647$	−22.8133			−0.896882			−0.448441	−	0.893812i	$-0.648021\pi$
$647$	−22.8133			−0.896882			−0.448441	+	0.893812i	$0.648021\pi$
$648$		0			0
$649$	−6.18086			−0.242620
$650$		0			0
$651$	17.4376	+	44.8781i	0.683432	+	1.75891i
$652$		0			0
$653$		−	36.2910i		−	1.42018i	−0.704112	−	0.710089i	$-0.748655\pi$
$653$		−	36.2910i		−	1.42018i	0.704112	−	0.710089i	$-0.251345\pi$
$654$		0			0
$655$		−	18.6349i		−	0.728127i
$656$		0			0
$657$	−9.41961	+	8.62172i	−0.367494	+	0.336365i
$658$		0			0
$659$	−14.4060			−0.561178			−0.280589	−	0.959828i	$-0.590530\pi$
$659$	−14.4060			−0.561178			−0.280589	+	0.959828i	$0.590530\pi$
$660$		0			0
$661$	9.37637			0.364699			0.182349	−	0.983234i	$-0.441630\pi$
$661$	9.37637			0.364699			0.182349	+	0.983234i	$0.441630\pi$
$662$		0			0
$663$	7.94764	−	3.08809i	0.308661	−	0.119931i
$664$		0			0
$665$			64.3242i			2.49439i
$666$		0			0
$667$			24.9626i			0.966556i
$668$		0			0
$669$	−21.5403	+	8.36958i	−0.832797	+	0.323587i
$670$		0			0
$671$	9.51649			0.367380
$672$		0			0
$673$	−42.6687			−1.64476			−0.822378	−	0.568942i	$-0.807353\pi$
$673$	−42.6687			−1.64476			−0.822378	+	0.568942i	$0.807353\pi$
$674$		0			0
$675$	55.4062	+	27.3816i	2.13259	+	1.05392i
$676$		0			0
$677$		−	0.460441i		−	0.0176962i	−0.999961	−	0.00884810i	$-0.997184\pi$
$677$		−	0.460441i		−	0.0176962i	0.999961	−	0.00884810i	$-0.00281647\pi$
$678$		0			0
$679$		−	24.2272i		−	0.929753i
$680$		0			0
$681$	−10.7214	−	27.5930i	−0.410845	−	1.05737i
$682$		0			0
$683$	30.9119			1.18281			0.591405	−	0.806375i	$-0.298574\pi$
$683$	30.9119			1.18281			0.591405	+	0.806375i	$0.298574\pi$
$684$		0			0
$685$	27.3182			1.04378
$686$		0			0
$687$	−0.945184	−	2.43257i	−0.0360610	−	0.0928082i
$688$		0			0
$689$		−	69.0284i		−	2.62977i
$690$		0			0
$691$		−	42.0407i		−	1.59930i	−0.600465	−	0.799651i	$-0.705018\pi$
$691$		−	42.0407i		−	1.59930i	0.600465	−	0.799651i	$-0.294982\pi$
$692$		0			0
$693$	−17.9188	−	19.5771i	−0.680679	−	0.743671i
$694$		0			0
$695$	19.0898			0.724118
$696$		0			0
$697$	6.92278			0.262219
$698$		0			0
$699$	22.1922	−	8.62287i	0.839386	−	0.326147i
$700$		0			0
$701$		−	33.4495i		−	1.26337i	−0.775224	−	0.631686i	$-0.782363\pi$
$701$		−	33.4495i		−	1.26337i	0.775224	−	0.631686i	$-0.217637\pi$
$702$		0			0
$703$			4.30756i			0.162463i
$704$		0			0
$705$	6.76477	−	2.62848i	0.254776	−	0.0989943i
$706$		0			0
$707$	−28.4367			−1.06947
$708$		0			0
$709$	47.5089			1.78423			0.892116	−	0.451806i	$-0.149220\pi$
$709$	47.5089			1.78423			0.892116	+	0.451806i	$0.149220\pi$
$710$		0			0
$711$	5.39192	+	5.89091i	0.202213	+	0.220926i
$712$		0			0
$713$		−	42.3693i		−	1.58674i
$714$		0			0
$715$			67.2421i			2.51471i
$716$		0			0
$717$	−1.62768	−	4.18908i	−0.0607869	−	0.156444i
$718$		0			0
$719$	−36.7881			−1.37196			−0.685982	−	0.727618i	$-0.740627\pi$
$719$	−36.7881			−1.37196			−0.685982	+	0.727618i	$0.740627\pi$
$720$		0			0
$721$	−6.93067			−0.258112
$722$		0			0
$723$	10.7850	+	27.7568i	0.401099	+	1.03229i
$724$		0			0
$725$		−	73.1755i		−	2.71767i
$726$		0			0
$727$		−	16.9860i		−	0.629977i	−0.949095	−	0.314989i	$-0.897999\pi$
$727$		−	16.9860i		−	0.629977i	0.949095	−	0.314989i	$-0.102001\pi$
$728$		0			0
$729$	16.4003	+	21.4483i	0.607419	+	0.794382i
$730$		0			0
$731$	−6.43775			−0.238109
$732$		0			0
$733$	−9.89364			−0.365430			−0.182715	−	0.983166i	$-0.558489\pi$
$733$	−9.89364			−0.365430			−0.182715	+	0.983166i	$0.558489\pi$
$734$		0			0
$735$	0.571905	−	0.222216i	0.0210950	−	0.00819657i
$736$		0			0
$737$		−	7.78306i		−	0.286693i
$738$		0			0
$739$		−	51.8260i		−	1.90645i	−0.302259	−	0.953226i	$-0.597741\pi$
$739$		−	51.8260i		−	1.90645i	0.302259	−	0.953226i	$-0.402259\pi$
$740$		0			0
$741$	46.7240	−	18.1548i	1.71645	−	0.666934i
$742$		0			0
$743$	42.6654			1.56524			0.782621	−	0.622498i	$-0.213882\pi$
$743$	42.6654			1.56524			0.782621	+	0.622498i	$0.213882\pi$
$744$		0			0
$745$	55.6957			2.04053
$746$		0			0
$747$	17.7836	−	16.2772i	0.650667	−	0.595552i
$748$		0			0
$749$			2.30752i			0.0843150i
$750$		0			0
$751$			10.8120i			0.394534i	0.980350	+	0.197267i	$0.0632066\pi$
$751$			10.8120i			0.394534i	−0.980350	+	0.197267i	$0.936793\pi$
$752$		0			0
$753$	1.00785	+	2.59385i	0.0367281	+	0.0945251i
$754$		0			0
$755$	−39.0553			−1.42137
$756$		0			0
$757$	−29.1698			−1.06019			−0.530097	−	0.847937i	$-0.677845\pi$
$757$	−29.1698			−1.06019			−0.530097	+	0.847937i	$0.677845\pi$
$758$		0			0
$759$	8.45856	+	21.7693i	0.307026	+	0.790176i
$760$		0			0
$761$			48.8898i			1.77226i	0.463441	+	0.886128i	$0.346615\pi$
$761$			48.8898i			1.77226i	−0.463441	+	0.886128i	$0.653385\pi$
$762$		0			0
$763$		−	5.94198i		−	0.215114i
$764$		0			0
$765$	−9.09583	+	8.32536i	−0.328860	+	0.301004i
$766$		0			0
$767$	−9.15576			−0.330595
$768$		0			0
$769$	26.0333			0.938785			0.469393	−	0.882990i	$-0.344473\pi$
$769$	26.0333			0.938785			0.469393	+	0.882990i	$0.344473\pi$
$770$		0			0
$771$	−7.48097	+	2.90676i	−0.269421	+	0.104685i
$772$		0			0
$773$			19.7728i			0.711177i	0.934643	+	0.355588i	$0.115720\pi$
$773$			19.7728i			0.711177i	−0.934643	+	0.355588i	$0.884280\pi$
$774$		0			0
$775$			124.202i			4.46145i
$776$		0			0
$777$	3.14893	−	1.22353i	0.112967	−	0.0438939i
$778$		0			0
$779$	40.6989			1.45819
$780$		0			0
$781$	33.4017			1.19521
$782$		0			0
$783$	14.1635	−	28.6596i	0.506162	−	1.02421i
$784$		0			0
$785$			5.67971i			0.202718i
$786$		0			0
$787$		−	37.8699i		−	1.34992i	−0.737855	−	0.674959i	$-0.764161\pi$
$787$		−	37.8699i		−	1.34992i	0.737855	−	0.674959i	$-0.235839\pi$
$788$		0			0
$789$	13.3362	+	34.3226i	0.474780	+	1.22192i
$790$		0			0
$791$	25.8957			0.920745
$792$		0			0
$793$	14.0968			0.500594
$794$		0			0
$795$	36.1546	+	93.0490i	1.28227	+	3.30011i
$796$		0			0
$797$			5.61719i			0.198971i	0.995039	+	0.0994855i	$0.0317197\pi$
$797$			5.61719i			0.198971i	−0.995039	+	0.0994855i	$0.968280\pi$
$798$		0			0
$799$			1.01943i			0.0360650i
$800$		0			0
$801$	−21.8774	−	23.9020i	−0.772999	−	0.844536i
$802$		0			0
$803$	14.1456			0.499188
$804$		0			0
$805$	−44.3940			−1.56468
$806$		0			0
$807$	−6.86760	+	2.66844i	−0.241751	+	0.0939334i
$808$		0			0
$809$			24.0674i			0.846166i	0.906091	+	0.423083i	$0.139052\pi$
$809$			24.0674i			0.846166i	−0.906091	+	0.423083i	$0.860948\pi$
$810$		0			0
$811$		−	3.73438i		−	0.131132i	−0.997848	−	0.0655660i	$-0.979115\pi$
$811$		−	3.73438i		−	0.131132i	0.997848	−	0.0655660i	$-0.0208853\pi$
$812$		0			0
$813$	−17.8973	+	6.95407i	−0.627685	+	0.243890i
$814$		0			0
$815$	−15.5871			−0.545991
$816$		0			0
$817$	−37.8475			−1.32411
$818$		0			0
$819$	−26.5432	−	28.9997i	−0.927496	−	1.01333i
$820$		0			0
$821$		−	4.80957i		−	0.167855i	−0.996472	−	0.0839275i	$-0.973254\pi$
$821$		−	4.80957i		−	0.167855i	0.996472	−	0.0839275i	$-0.0267464\pi$
$822$		0			0
$823$			12.3481i			0.430427i	0.976567	+	0.215214i	$0.0690448\pi$
$823$			12.3481i			0.430427i	−0.976567	+	0.215214i	$0.930955\pi$
$824$		0			0
$825$	−24.7954	−	63.8146i	−0.863267	−	2.22174i
$826$		0			0
$827$	−19.8824			−0.691379			−0.345689	−	0.938349i	$-0.612355\pi$
$827$	−19.8824			−0.691379			−0.345689	+	0.938349i	$0.612355\pi$
$828$		0			0
$829$	41.4593			1.43994			0.719970	−	0.694005i	$-0.244156\pi$
$829$	41.4593			1.43994			0.719970	+	0.694005i	$0.244156\pi$
$830$		0			0
$831$	3.93092	+	10.1168i	0.136362	+	0.350947i
$832$		0			0
$833$			0.0861847i			0.00298612i
$834$		0			0
$835$			4.21322i			0.145804i
$836$		0			0
$837$	−24.0398	+	48.6442i	−0.830938	+	1.68139i
$838$		0			0
$839$	30.3881			1.04911			0.524557	−	0.851375i	$-0.324231\pi$
$839$	30.3881			1.04911			0.524557	+	0.851375i	$0.324231\pi$
$840$		0			0
$841$	−8.85095			−0.305205
$842$		0			0
$843$	3.13594	−	1.21848i	0.108007	−	0.0419668i
$844$		0			0
$845$			46.1733i			1.58841i
$846$		0			0
$847$			0.117395i			0.00403375i
$848$		0			0
$849$	34.5750	−	13.4343i	1.18661	−	0.461063i
$850$		0			0
$851$	−2.97290			−0.101910
$852$		0			0
$853$	19.0799			0.653285			0.326642	−	0.945148i	$-0.394083\pi$
$853$	19.0799			0.653285			0.326642	+	0.945148i	$0.394083\pi$
$854$		0			0
$855$	−53.4742	+	48.9447i	−1.82878	+	1.67387i
$856$		0			0
$857$			14.7352i			0.503346i	0.967812	+	0.251673i	$0.0809808\pi$
$857$			14.7352i			0.503346i	−0.967812	+	0.251673i	$0.919019\pi$
$858$		0			0
$859$			8.97642i			0.306272i	0.988205	+	0.153136i	$0.0489372\pi$
$859$			8.97642i			0.306272i	−0.988205	+	0.153136i	$0.951063\pi$
$860$		0			0
$861$	−11.5602	−	29.7519i	−0.393971	−	1.01394i
$862$		0			0
$863$	32.6138			1.11019			0.555093	−	0.831788i	$-0.312682\pi$
$863$	32.6138			1.11019			0.555093	+	0.831788i	$0.312682\pi$
$864$		0			0
$865$	43.8619			1.49135
$866$		0			0
$867$	−0.627306	−	1.61446i	−0.0213044	−	0.0548300i
$868$		0			0
$869$		−	8.84650i		−	0.300097i
$870$		0			0
$871$		−	11.5291i		−	0.390648i
$872$		0			0
$873$	20.1406	−	18.4346i	0.681656	−	0.623917i
$874$		0			0
$875$	75.4297			2.54999
$876$		0			0
$877$	1.39227			0.0470136			0.0235068	−	0.999724i	$-0.492517\pi$
$877$	1.39227			0.0470136			0.0235068	+	0.999724i	$0.492517\pi$
$878$		0			0
$879$	8.79624	−	3.41782i	0.296690	−	0.115280i
$880$		0			0
$881$		−	28.2287i		−	0.951050i	−0.879702	−	0.475525i	$-0.842258\pi$
$881$		−	28.2287i		−	0.951050i	0.879702	−	0.475525i	$-0.157742\pi$
$882$		0			0
$883$			16.9394i			0.570057i	0.958519	+	0.285028i	$0.0920030\pi$
$883$			16.9394i			0.570057i	−0.958519	+	0.285028i	$0.907997\pi$
$884$		0			0
$885$	12.3418	−	4.79545i	0.414864	−	0.161197i
$886$		0			0
$887$	−38.3446			−1.28749			−0.643743	−	0.765242i	$-0.722619\pi$
$887$	−38.3446			−1.28749			−0.643743	+	0.765242i	$0.722619\pi$
$888$		0			0
$889$	−19.7104			−0.661066
$890$		0			0
$891$	2.64037	−	29.7926i	0.0884556	−	0.998091i
$892$		0			0
$893$			5.99324i			0.200556i
$894$		0			0
$895$			15.3772i			0.514003i
$896$		0			0
$897$	12.5297	+	32.2470i	0.418355	+	1.07670i
$898$		0			0
$899$	64.2449			2.14269
$900$		0			0
$901$	−14.0222			−0.467149
$902$		0			0
$903$	10.7503	+	27.6674i	0.357747	+	0.920714i
$904$		0			0
$905$			16.5411i			0.549846i
$906$		0			0
$907$		−	22.9397i		−	0.761699i	−0.924637	−	0.380849i	$-0.875632\pi$
$907$		−	22.9397i		−	0.761699i	0.924637	−	0.380849i	$-0.124368\pi$
$908$		0			0
$909$	−21.6376	−	23.6401i	−0.717675	−	0.784091i
$910$		0			0
$911$	−18.1313			−0.600716			−0.300358	−	0.953827i	$-0.597106\pi$
$911$	−18.1313			−0.600716			−0.300358	+	0.953827i	$0.597106\pi$
$912$		0			0
$913$	−26.7059			−0.883838
$914$		0			0
$915$	−19.0023	+	7.38342i	−0.628196	+	0.244088i
$916$		0			0
$917$		−	12.0689i		−	0.398551i
$918$		0			0
$919$		−	51.8314i		−	1.70976i	−0.518826	−	0.854880i	$-0.673631\pi$
$919$		−	51.8314i		−	1.70976i	0.518826	−	0.854880i	$-0.326369\pi$
$920$		0			0
$921$	47.5362	−	18.4704i	1.56637	−	0.608620i
$922$		0			0
$923$	49.4781			1.62859
$924$		0			0
$925$	8.71478			0.286540
$926$		0			0
$927$	−5.27359	−	5.76163i	−0.173207	−	0.189237i
$928$		0			0
$929$			3.25080i			0.106655i	0.998577	+	0.0533277i	$0.0169828\pi$
$929$			3.25080i			0.106655i	−0.998577	+	0.0533277i	$0.983017\pi$
$930$		0			0
$931$			0.506678i			0.0166057i
$932$		0			0
$933$	−7.35939	−	18.9404i	−0.240936	−	0.620082i
$934$		0			0
$935$	13.6594			0.446710
$936$		0			0
$937$	−1.56569			−0.0511490			−0.0255745	−	0.999673i	$-0.508142\pi$
$937$	−1.56569			−0.0511490			−0.0255745	+	0.999673i	$0.508142\pi$
$938$		0			0
$939$	−11.3891	−	29.3115i	−0.371669	−	0.956544i
$940$		0			0
$941$			30.3242i			0.988539i	0.869309	+	0.494270i	$0.164564\pi$
$941$			30.3242i			0.988539i	−0.869309	+	0.494270i	$0.835436\pi$
$942$		0			0
$943$			28.0887i			0.914695i
$944$		0			0
$945$	50.9687	+	25.1886i	1.65801	+	0.819385i
$946$		0			0
$947$	−7.80544			−0.253643			−0.126821	−	0.991926i	$-0.540478\pi$
$947$	−7.80544			−0.253643			−0.126821	+	0.991926i	$0.540478\pi$
$948$		0			0
$949$	20.9540			0.680195
$950$		0			0
$951$	−33.2265	+	12.9103i	−1.07744	+	0.418645i
$952$		0			0
$953$			1.68838i			0.0546921i	0.999626	+	0.0273460i	$0.00870560\pi$
$953$			1.68838i			0.0546921i	−0.999626	+	0.0273460i	$0.991294\pi$
$954$		0			0
$955$		−	57.1114i		−	1.84808i
$956$		0			0
$957$	−33.0089	+	12.8258i	−1.06703	+	0.414598i
$958$		0			0
$959$	17.6927			0.571326
$960$		0			0
$961$	−78.0435			−2.51753
$962$		0			0
$963$	−1.91830	+	1.75581i	−0.0618163	+	0.0565801i
$964$		0			0
$965$		−	31.6869i		−	1.02004i
$966$		0			0
$967$			57.1018i			1.83627i	0.396268	+	0.918135i	$0.370305\pi$
$967$			57.1018i			1.83627i	−0.396268	+	0.918135i	$0.629695\pi$
$968$		0			0
$969$	−3.68792	−	9.49139i	−0.118473	−	0.304908i
$970$		0			0
$971$	25.6701			0.823792			0.411896	−	0.911231i	$-0.364867\pi$
$971$	25.6701			0.823792			0.411896	+	0.911231i	$0.364867\pi$
$972$		0			0
$973$	12.3635			0.396356
$974$		0			0
$975$	−36.7297	−	94.5290i	−1.17629	−	3.02735i
$976$		0			0
$977$		−	43.7861i		−	1.40084i	−0.713731	−	0.700420i	$-0.752996\pi$
$977$		−	43.7861i		−	1.40084i	0.713731	−	0.700420i	$-0.247004\pi$
$978$		0			0
$979$			35.8941i			1.14718i
$980$		0			0
$981$	4.93971	−	4.52129i	0.157713	−	0.144354i
$982$		0			0
$983$	−12.4684			−0.397679			−0.198839	−	0.980032i	$-0.563717\pi$
$983$	−12.4684			−0.397679			−0.198839	+	0.980032i	$0.563717\pi$
$984$		0			0
$985$	−78.7264			−2.50843
$986$		0			0
$987$	4.38120	−	1.70233i	0.139455	−	0.0541859i
$988$		0			0
$989$		−	26.1208i		−	0.830592i
$990$		0			0
$991$		−	5.67456i		−	0.180258i	−0.995930	−	0.0901292i	$-0.971272\pi$
$991$		−	5.67456i		−	0.180258i	0.995930	−	0.0901292i	$-0.0287280\pi$
$992$		0			0
$993$	27.9537	−	10.8615i	0.887085	−	0.344681i
$994$		0			0
$995$	−74.1762			−2.35154
$996$		0			0
$997$	−30.4395			−0.964029			−0.482015	−	0.876163i	$-0.660095\pi$
$997$	−30.4395			−0.964029			−0.482015	+	0.876163i	$0.660095\pi$
$998$		0			0
$999$	3.41319	+	1.68679i	0.107989	+	0.0533676i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.2.e.c.239.12	yes	20
3.2	odd	2	inner	816.2.e.c.239.10	yes	20
4.3	odd	2	inner	816.2.e.c.239.9	✓	20
12.11	even	2	inner	816.2.e.c.239.11	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
816.2.e.c.239.9	✓	20	4.3	odd	2	inner
816.2.e.c.239.10	yes	20	3.2	odd	2	inner
816.2.e.c.239.11	yes	20	12.11	even	2	inner
816.2.e.c.239.12	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.627306 + 1.61446i) q^{3} +4.11023i q^{5} +2.66199i q^{7} +(-2.21297 + 2.02552i) q^{9} +O(q^{10})\)	\(q+(0.627306 + 1.61446i) q^{3} +4.11023i q^{5} +2.66199i q^{7} +(-2.21297 + 2.02552i) q^{9} +3.32327 q^{11} +4.92278 q^{13} +(-6.63581 + 2.57837i) q^{15} -1.00000i q^{17} -5.87898i q^{19} +(-4.29768 + 1.66988i) q^{21} +4.05744 q^{23} -11.8940 q^{25} +(-4.65834 - 2.30214i) q^{27} +6.15231i q^{29} -10.4424i q^{31} +(2.08471 + 5.36529i) q^{33} -10.9414 q^{35} -0.732705 q^{37} +(3.08809 + 7.94764i) q^{39} +6.92278i q^{41} -6.43775i q^{43} +(-8.32536 - 9.09583i) q^{45} -1.01943 q^{47} -0.0861847 q^{49} +(1.61446 - 0.627306i) q^{51} -14.0222i q^{53} +13.6594i q^{55} +(9.49139 - 3.68792i) q^{57} -1.85988 q^{59} +2.86360 q^{61} +(-5.39192 - 5.89091i) q^{63} +20.2337i q^{65} -2.34199i q^{67} +(2.54525 + 6.55058i) q^{69} +10.0509 q^{71} +4.25654 q^{73} +(-7.46117 - 19.2024i) q^{75} +8.84650i q^{77} -2.66199i q^{79} +(0.794509 - 8.96486i) q^{81} -8.03605 q^{83} +4.11023 q^{85} +(-9.93267 + 3.85938i) q^{87} +10.8008i q^{89} +13.1044i q^{91} +(16.8588 - 6.55058i) q^{93} +24.1640 q^{95} -9.10115 q^{97} +(-7.35430 + 6.73135i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{9}+O(q^{10}) \) 20 * q - 4 * q^9	\( 20 q - 4 q^{9} + 12 q^{13} - 16 q^{21} - 48 q^{25} + 12 q^{33} - 16 q^{45} - 20 q^{49} + 64 q^{57} - 48 q^{61} + 28 q^{69} + 64 q^{73} - 60 q^{81} + 4 q^{85} + 32 q^{93} + 32 q^{97}+O(q^{100}) \) 20 * q - 4 * q^9 + 12 * q^13 - 16 * q^21 - 48 * q^25 + 12 * q^33 - 16 * q^45 - 20 * q^49 + 64 * q^57 - 48 * q^61 + 28 * q^69 + 64 * q^73 - 60 * q^81 + 4 * q^85 + 32 * q^93 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more