LMFDB - Embedded newform 1120.2.g.b.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(449,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1120.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.g (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:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.65174749855744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 $ x^10 + 13x^8 + 56x^6 + 97x^4 + 61x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$-0.271831i$ of defining polynomial
Character	$\chi$	$=$	1120.449
Dual form			1120.2.g.b.449.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.19794i q^{3} +(1.64514 - 1.51444i) q^{5} +1.00000i q^{7} -1.83094 q^{9} +O(q^{10})$	$q-2.19794i q^{3} +(1.64514 - 1.51444i) q^{5} +1.00000i q^{7} -1.83094 q^{9} -1.37460 q^{11} +2.74160i q^{13} +(-3.32864 - 3.61591i) q^{15} -6.94455i q^{17} -1.29028 q^{19} +2.19794 q^{21} -8.31915i q^{23} +(0.412961 - 4.98292i) q^{25} -2.56953i q^{27} +8.40089 q^{29} -9.49323 q^{31} +3.02128i q^{33} +(1.51444 + 1.64514i) q^{35} -1.73860i q^{37} +6.02587 q^{39} -5.30855 q^{41} +7.83136i q^{43} +(-3.01214 + 2.77284i) q^{45} -3.48822i q^{47} -1.00000 q^{49} -15.2637 q^{51} +6.13448i q^{53} +(-2.26140 + 2.08174i) q^{55} +2.83595i q^{57} -6.26070 q^{59} +6.59883 q^{61} -1.83094i q^{63} +(4.15198 + 4.51031i) q^{65} -1.66187i q^{67} -18.2850 q^{69} +16.0236 q^{71} +2.13448i q^{73} +(-10.9521 - 0.907662i) q^{75} -1.37460i q^{77} +5.45132 q^{79} -11.1405 q^{81} +2.54624i q^{83} +(-10.5171 - 11.4248i) q^{85} -18.4646i q^{87} -1.43548 q^{89} -2.74160 q^{91} +20.8655i q^{93} +(-2.12268 + 1.95404i) q^{95} +9.69375i q^{97} +2.51680 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{5} - 14 q^{9}+O(q^{10}) $ 10 * q - 2 * q^5 - 14 * q^9	$ 10 q - 2 q^{5} - 14 q^{9} - 8 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 6 q^{25} + 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} + 10 q^{45} - 10 q^{49} - 24 q^{51} - 16 q^{55} + 32 q^{59} - 20 q^{61} - 8 q^{65} - 8 q^{69} - 8 q^{71} - 64 q^{75} + 64 q^{79} + 2 q^{81} - 12 q^{85} - 4 q^{89} - 60 q^{95} + 80 q^{99}+O(q^{100}) $ 10 * q - 2 * q^5 - 14 * q^9 - 8 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^21 + 6 * q^25 + 24 * q^29 - 24 * q^31 + 64 * q^39 - 4 * q^41 + 10 * q^45 - 10 * q^49 - 24 * q^51 - 16 * q^55 + 32 * q^59 - 20 * q^61 - 8 * q^65 - 8 * q^69 - 8 * q^71 - 64 * q^75 + 64 * q^79 + 2 * q^81 - 12 * q^85 - 4 * q^89 - 60 * q^95 + 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	2.19794i		−	1.26898i	−0.772931	−	0.634490i	$-0.781210\pi$
$3$		−	2.19794i		−	1.26898i	0.772931	−	0.634490i	$-0.218790\pi$
$4$		0			0
$5$	1.64514	−	1.51444i	0.735728	−	0.677277i
$5$	1.64514	−	1.51444i	0.735728	−	0.677277i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.83094			−0.610312
$10$		0			0
$11$	−1.37460			−0.414457			−0.207228	−	0.978293i	$-0.566444\pi$
$11$	−1.37460			−0.414457			−0.207228	+	0.978293i	$0.566444\pi$
$12$		0			0
$13$			2.74160i			0.760383i	0.924908	+	0.380192i	$0.124142\pi$
$13$			2.74160i			0.760383i	−0.924908	+	0.380192i	$0.875858\pi$
$14$		0			0
$15$	−3.32864	−	3.61591i	−0.859451	−	0.933625i
$16$		0			0
$17$		−	6.94455i		−	1.68430i	−0.539242	−	0.842151i	$-0.681289\pi$
$17$		−	6.94455i		−	1.68430i	0.539242	−	0.842151i	$-0.318711\pi$
$18$		0			0
$19$	−1.29028			−0.296010			−0.148005	−	0.988987i	$-0.547285\pi$
$19$	−1.29028			−0.296010			−0.148005	+	0.988987i	$0.547285\pi$
$20$		0			0
$21$	2.19794			0.479630
$22$		0			0
$23$		−	8.31915i		−	1.73466i	−0.497731	−	0.867331i	$-0.665833\pi$
$23$		−	8.31915i		−	1.73466i	0.497731	−	0.867331i	$-0.334167\pi$
$24$		0			0
$25$	0.412961	−	4.98292i	0.0825921	−	0.996583i
$26$		0			0
$27$		−	2.56953i		−	0.494507i
$28$		0			0
$29$	8.40089			1.56001			0.780003	−	0.625775i	$-0.215217\pi$
$29$	8.40089			1.56001			0.780003	+	0.625775i	$0.215217\pi$
$30$		0			0
$31$	−9.49323			−1.70503			−0.852517	−	0.522699i	$-0.824925\pi$
$31$	−9.49323			−1.70503			−0.852517	+	0.522699i	$0.824925\pi$
$32$		0			0
$33$			3.02128i			0.525937i
$34$		0			0
$35$	1.51444	+	1.64514i	0.255987	+	0.278079i
$36$		0			0
$37$		−	1.73860i		−	0.285824i	−0.989735	−	0.142912i	$-0.954353\pi$
$37$		−	1.73860i		−	0.285824i	0.989735	−	0.142912i	$-0.0456465\pi$
$38$		0			0
$39$	6.02587			0.964912
$40$		0			0
$41$	−5.30855			−0.829057			−0.414528	−	0.910036i	$-0.636053\pi$
$41$	−5.30855			−0.829057			−0.414528	+	0.910036i	$0.636053\pi$
$42$		0			0
$43$			7.83136i			1.19427i	0.802140	+	0.597136i	$0.203695\pi$
$43$			7.83136i			1.19427i	−0.802140	+	0.597136i	$0.796305\pi$
$44$		0			0
$45$	−3.01214	+	2.77284i	−0.449024	+	0.413350i
$46$		0			0
$47$		−	3.48822i		−	0.508809i	−0.967098	−	0.254404i	$-0.918121\pi$
$47$		−	3.48822i		−	0.508809i	0.967098	−	0.254404i	$-0.0818794\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−15.2637			−2.13735
$52$		0			0
$53$			6.13448i			0.842635i	0.906913	+	0.421317i	$0.138432\pi$
$53$			6.13448i			0.842635i	−0.906913	+	0.421317i	$0.861568\pi$
$54$		0			0
$55$	−2.26140	+	2.08174i	−0.304927	+	0.280702i
$56$		0			0
$57$			2.83595i			0.375631i
$58$		0			0
$59$	−6.26070			−0.815074			−0.407537	−	0.913189i	$-0.633612\pi$
$59$	−6.26070			−0.815074			−0.407537	+	0.913189i	$0.633612\pi$
$60$		0			0
$61$	6.59883			0.844894			0.422447	−	0.906388i	$-0.361171\pi$
$61$	6.59883			0.844894			0.422447	+	0.906388i	$0.361171\pi$
$62$		0			0
$63$		−	1.83094i		−	0.230676i
$64$		0			0
$65$	4.15198	+	4.51031i	0.514990	+	0.559435i
$66$		0			0
$67$		−	1.66187i		−	0.203030i	−0.994834	−	0.101515i	$-0.967631\pi$
$67$		−	1.66187i		−	0.203030i	0.994834	−	0.101515i	$-0.0323690\pi$
$68$		0			0
$69$	−18.2850			−2.20125
$70$		0			0
$71$	16.0236			1.90165			0.950825	−	0.309730i	$-0.100239\pi$
$71$	16.0236			1.90165			0.950825	+	0.309730i	$0.100239\pi$
$72$		0			0
$73$			2.13448i			0.249821i	0.992168	+	0.124911i	$0.0398644\pi$
$73$			2.13448i			0.249821i	−0.992168	+	0.124911i	$0.960136\pi$
$74$		0			0
$75$	−10.9521	−	0.907662i	−1.26465	−	0.104808i
$76$		0			0
$77$		−	1.37460i		−	0.156650i
$78$		0			0
$79$	5.45132			0.613322			0.306661	−	0.951819i	$-0.400788\pi$
$79$	5.45132			0.613322			0.306661	+	0.951819i	$0.400788\pi$
$80$		0			0
$81$	−11.1405			−1.23783
$82$		0			0
$83$			2.54624i			0.279486i	0.990188	+	0.139743i	$0.0446277\pi$
$83$			2.54624i			0.279486i	−0.990188	+	0.139743i	$0.955372\pi$
$84$		0			0
$85$	−10.5171	−	11.4248i	−1.14074	−	1.23919i
$86$		0			0
$87$		−	18.4646i		−	1.97962i
$88$		0			0
$89$	−1.43548			−0.152161			−0.0760804	−	0.997102i	$-0.524241\pi$
$89$	−1.43548			−0.152161			−0.0760804	+	0.997102i	$0.524241\pi$
$90$		0			0
$91$	−2.74160			−0.287398
$92$		0			0
$93$			20.8655i			2.16366i
$94$		0			0
$95$	−2.12268	+	1.95404i	−0.217783	+	0.200481i
$96$		0			0
$97$			9.69375i			0.984251i	0.870524	+	0.492125i	$0.163780\pi$
$97$			9.69375i			0.984251i	−0.870524	+	0.492125i	$0.836220\pi$
$98$		0			0
$99$	2.51680			0.252948
$100$		0			0
$101$	8.25067			0.820973			0.410486	−	0.911867i	$-0.365359\pi$
$101$	8.25067			0.820973			0.410486	+	0.911867i	$0.365359\pi$
$102$		0			0
$103$		−	13.1501i		−	1.29572i	−0.761761	−	0.647858i	$-0.775665\pi$
$103$		−	13.1501i		−	1.29572i	0.761761	−	0.647858i	$-0.224335\pi$
$104$		0			0
$105$	3.61591	−	3.32864i	0.352877	−	0.324842i
$106$		0			0
$107$		−	0.153454i		−	0.0148349i	−0.999972	−	0.00741746i	$-0.997639\pi$
$107$		−	0.153454i		−	0.0148349i	0.999972	−	0.00741746i	$-0.00236107\pi$
$108$		0			0
$109$	8.23741			0.789001			0.394500	−	0.918896i	$-0.370918\pi$
$109$	8.23741			0.789001			0.394500	+	0.918896i	$0.370918\pi$
$110$		0			0
$111$	−3.82133			−0.362705
$112$		0			0
$113$			11.0532i			1.03979i	0.854229	+	0.519897i	$0.174030\pi$
$113$			11.0532i			1.03979i	−0.854229	+	0.519897i	$0.825970\pi$
$114$		0			0
$115$	−12.5988	−	13.6862i	−1.17485	−	1.27624i
$116$		0			0
$117$		−	5.01969i		−	0.464071i
$118$		0			0
$119$	6.94455			0.636606
$120$		0			0
$121$	−9.11048			−0.828226
$122$		0			0
$123$			11.6679i			1.05206i
$124$		0			0
$125$	−6.86694	−	8.82299i	−0.614198	−	0.789152i
$126$		0			0
$127$			7.74460i			0.687222i	0.939112	+	0.343611i	$0.111650\pi$
$127$			7.74460i			0.687222i	−0.939112	+	0.343611i	$0.888350\pi$
$128$		0			0
$129$	17.2128			1.51551
$130$		0			0
$131$	−4.75228			−0.415209			−0.207605	−	0.978213i	$-0.566567\pi$
$131$	−4.75228			−0.415209			−0.207605	+	0.978213i	$0.566567\pi$
$132$		0			0
$133$		−	1.29028i		−	0.111881i
$134$		0			0
$135$	−3.89140	−	4.22724i	−0.334918	−	0.363823i
$136$		0			0
$137$		−	18.3002i		−	1.56349i	−0.623599	−	0.781745i	$-0.714330\pi$
$137$		−	18.3002i		−	1.56349i	0.623599	−	0.781745i	$-0.285670\pi$
$138$		0			0
$139$	18.9370			1.60621			0.803106	−	0.595836i	$-0.203179\pi$
$139$	18.9370			1.60621			0.803106	+	0.595836i	$0.203179\pi$
$140$		0			0
$141$	−7.66688			−0.645668
$142$		0			0
$143$		−	3.76860i		−	0.315146i
$144$		0			0
$145$	13.8206	−	12.7226i	1.14774	−	1.05656i
$146$		0			0
$147$			2.19794i			0.181283i
$148$		0			0
$149$	−10.1155			−0.828694			−0.414347	−	0.910119i	$-0.635990\pi$
$149$	−10.1155			−0.828694			−0.414347	+	0.910119i	$0.635990\pi$
$150$		0			0
$151$	−0.567654			−0.0461951			−0.0230975	−	0.999733i	$-0.507353\pi$
$151$	−0.567654			−0.0461951			−0.0230975	+	0.999733i	$0.507353\pi$
$152$		0			0
$153$			12.7150i			1.02795i
$154$		0			0
$155$	−15.6177	+	14.3769i	−1.25444	+	1.15478i
$156$		0			0
$157$		−	2.87542i		−	0.229484i	−0.993395	−	0.114742i	$-0.963396\pi$
$157$		−	2.87542i		−	0.229484i	0.993395	−	0.114742i	$-0.0366041\pi$
$158$		0			0
$159$	13.4832			1.06929
$160$		0			0
$161$	8.31915			0.655641
$162$		0			0
$163$			10.2212i			0.800589i	0.916387	+	0.400294i	$0.131092\pi$
$163$			10.2212i			0.800589i	−0.916387	+	0.400294i	$0.868908\pi$
$164$		0			0
$165$	4.57554	+	4.97042i	0.356205	+	0.386947i
$166$		0			0
$167$			19.7672i			1.52963i	0.644249	+	0.764816i	$0.277170\pi$
$167$			19.7672i			1.52963i	−0.644249	+	0.764816i	$0.722830\pi$
$168$		0			0
$169$	5.48363			0.421817
$170$		0			0
$171$	2.36241			0.180658
$172$		0			0
$173$		−	21.7281i		−	1.65195i	−0.563704	−	0.825977i	$-0.690624\pi$
$173$		−	21.7281i		−	1.65195i	0.563704	−	0.825977i	$-0.309376\pi$
$174$		0			0
$175$	4.98292	+	0.412961i	0.376673	+	0.0312169i
$176$		0			0
$177$			13.7606i			1.03431i
$178$		0			0
$179$	21.1611			1.58166			0.790828	−	0.612039i	$-0.209650\pi$
$179$	21.1611			1.58166			0.790828	+	0.612039i	$0.209650\pi$
$180$		0			0
$181$	11.8396			0.880031			0.440016	−	0.897990i	$-0.354973\pi$
$181$	11.8396			0.880031			0.440016	+	0.897990i	$0.354973\pi$
$182$		0			0
$183$		−	14.5038i		−	1.07215i
$184$		0			0
$185$	−2.63300	−	2.86023i	−0.193582	−	0.210289i
$186$		0			0
$187$			9.54596i			0.698070i
$188$		0			0
$189$	2.56953			0.186906
$190$		0			0
$191$	1.82068			0.131739			0.0658697	−	0.997828i	$-0.479018\pi$
$191$	1.82068			0.131739			0.0658697	+	0.997828i	$0.479018\pi$
$192$		0			0
$193$		−	3.62770i		−	0.261128i	−0.991440	−	0.130564i	$-0.958321\pi$
$193$		−	3.62770i		−	0.261128i	0.991440	−	0.130564i	$-0.0416788\pi$
$194$		0			0
$195$	9.91339	−	9.12580i	0.709913	−	0.653512i
$196$		0			0
$197$			6.49380i			0.462664i	0.972875	+	0.231332i	$0.0743084\pi$
$197$			6.49380i			0.462664i	−0.972875	+	0.231332i	$0.925692\pi$
$198$		0			0
$199$	4.69145			0.332568			0.166284	−	0.986078i	$-0.446823\pi$
$199$	4.69145			0.332568			0.166284	+	0.986078i	$0.446823\pi$
$200$		0			0
$201$	−3.65269			−0.257641
$202$		0			0
$203$			8.40089i			0.589627i
$204$		0			0
$205$	−8.73331	+	8.03947i	−0.609961	+	0.561501i
$206$		0			0
$207$			15.2318i			1.05869i
$208$		0			0
$209$	1.77361			0.122683
$210$		0			0
$211$	17.0159			1.17142			0.585710	−	0.810521i	$-0.300816\pi$
$211$	17.0159			1.17142			0.585710	+	0.810521i	$0.300816\pi$
$212$		0			0
$213$		−	35.2189i		−	2.41316i
$214$		0			0
$215$	11.8601	+	12.8837i	0.808852	+	0.878659i
$216$		0			0
$217$		−	9.49323i		−	0.644442i
$218$		0			0
$219$	4.69145			0.317019
$220$		0			0
$221$	19.0392			1.28071
$222$		0			0
$223$			9.39251i			0.628969i	0.949263	+	0.314485i	$0.101832\pi$
$223$			9.39251i			0.628969i	−0.949263	+	0.314485i	$0.898168\pi$
$224$		0			0
$225$	−0.756104	+	9.12340i	−0.0504069	+	0.608227i
$226$		0			0
$227$			14.3134i			0.950016i	0.879981	+	0.475008i	$0.157555\pi$
$227$			14.3134i			0.950016i	−0.879981	+	0.475008i	$0.842445\pi$
$228$		0			0
$229$	8.78351			0.580430			0.290215	−	0.956961i	$-0.406273\pi$
$229$	8.78351			0.580430			0.290215	+	0.956961i	$0.406273\pi$
$230$		0			0
$231$	−3.02128			−0.198786
$232$		0			0
$233$		−	14.7706i		−	0.967652i	−0.875164	−	0.483826i	$-0.839247\pi$
$233$		−	14.7706i		−	0.967652i	0.875164	−	0.483826i	$-0.160753\pi$
$234$		0			0
$235$	−5.28268	−	5.73860i	−0.344604	−	0.374345i
$236$		0			0
$237$		−	11.9817i		−	0.778294i
$238$		0			0
$239$	25.0525			1.62051			0.810256	−	0.586076i	$-0.199328\pi$
$239$	25.0525			1.62051			0.810256	+	0.586076i	$0.199328\pi$
$240$		0			0
$241$	3.32975			0.214488			0.107244	−	0.994233i	$-0.465797\pi$
$241$	3.32975			0.214488			0.107244	+	0.994233i	$0.465797\pi$
$242$		0			0
$243$			16.7775i			1.07628i
$244$		0			0
$245$	−1.64514	+	1.51444i	−0.105104	+	0.0967538i
$246$		0			0
$247$		−	3.53742i		−	0.225081i
$248$		0			0
$249$	5.59648			0.354663
$250$		0			0
$251$	−9.89135			−0.624336			−0.312168	−	0.950027i	$-0.601055\pi$
$251$	−9.89135			−0.624336			−0.312168	+	0.950027i	$0.601055\pi$
$252$		0			0
$253$			11.4355i			0.718943i
$254$		0			0
$255$	−25.1109	+	23.1159i	−1.57251	+	1.44758i
$256$		0			0
$257$			13.6117i			0.849073i	0.905411	+	0.424536i	$0.139563\pi$
$257$			13.6117i			0.849073i	−0.905411	+	0.424536i	$0.860437\pi$
$258$		0			0
$259$	1.73860			0.108031
$260$		0			0
$261$	−15.3815			−0.952090
$262$		0			0
$263$			4.11550i			0.253772i	0.991917	+	0.126886i	$0.0404983\pi$
$263$			4.11550i			0.253772i	−0.991917	+	0.126886i	$0.959502\pi$
$264$		0			0
$265$	9.29028	+	10.0921i	0.570697	+	0.619950i
$266$		0			0
$267$			3.15510i			0.193089i
$268$		0			0
$269$	−10.7409			−0.654887			−0.327443	−	0.944871i	$-0.606187\pi$
$269$	−10.7409			−0.654887			−0.327443	+	0.944871i	$0.606187\pi$
$270$		0			0
$271$	−32.7438			−1.98904			−0.994521	−	0.104535i	$-0.966665\pi$
$271$	−32.7438			−1.98904			−0.994521	+	0.104535i	$0.966665\pi$
$272$		0			0
$273$			6.02587i			0.364702i
$274$		0			0
$275$	−0.567654	+	6.84950i	−0.0342309	+	0.413041i
$276$		0			0
$277$			6.92623i			0.416157i	0.978112	+	0.208078i	$0.0667209\pi$
$277$			6.92623i			0.416157i	−0.978112	+	0.208078i	$0.933279\pi$
$278$		0			0
$279$	17.3815			1.04060
$280$		0			0
$281$	−2.61209			−0.155824			−0.0779122	−	0.996960i	$-0.524825\pi$
$281$	−2.61209			−0.155824			−0.0779122	+	0.996960i	$0.524825\pi$
$282$		0			0
$283$		−	6.99972i		−	0.416090i	−0.978119	−	0.208045i	$-0.933290\pi$
$283$		−	6.99972i		−	0.416090i	0.978119	−	0.208045i	$-0.0667101\pi$
$284$		0			0
$285$	4.29487	+	4.66553i	0.254406	+	0.276362i
$286$		0			0
$287$		−	5.30855i		−	0.313354i
$288$		0			0
$289$	−31.2268			−1.83687
$290$		0			0
$291$	21.3063			1.24900
$292$		0			0
$293$			14.7099i			0.859362i	0.902981	+	0.429681i	$0.141374\pi$
$293$			14.7099i			0.859362i	−0.902981	+	0.429681i	$0.858626\pi$
$294$		0			0
$295$	−10.2997	+	9.48144i	−0.599673	+	0.552031i
$296$		0			0
$297$			3.53207i			0.204952i
$298$		0			0
$299$	22.8078			1.31901
$300$		0			0
$301$	−7.83136			−0.451392
$302$		0			0
$303$		−	18.1345i		−	1.04180i
$304$		0			0
$305$	10.8560	−	9.99351i	0.621612	−	0.572227i
$306$		0			0
$307$		−	32.5567i		−	1.85811i	−0.369942	−	0.929055i	$-0.620623\pi$
$307$		−	32.5567i		−	1.85811i	0.369942	−	0.929055i	$-0.379377\pi$
$308$		0			0
$309$	−28.9031			−1.64424
$310$		0			0
$311$	32.4065			1.83760			0.918802	−	0.394719i	$-0.129158\pi$
$311$	32.4065			1.83760			0.918802	+	0.394719i	$0.129158\pi$
$312$		0			0
$313$		−	21.2347i		−	1.20026i	−0.799904	−	0.600128i	$-0.795116\pi$
$313$		−	21.2347i		−	1.20026i	0.799904	−	0.600128i	$-0.204884\pi$
$314$		0			0
$315$	−2.77284	−	3.01214i	−0.156232	−	0.169715i
$316$		0			0
$317$			33.1847i			1.86384i	0.362667	+	0.931919i	$0.381866\pi$
$317$			33.1847i			1.86384i	−0.362667	+	0.931919i	$0.618134\pi$
$318$		0			0
$319$	−11.5478			−0.646555
$320$		0			0
$321$	−0.337282			−0.0188252
$322$		0			0
$323$			8.96040i			0.498570i
$324$		0			0
$325$	13.6612	+	1.13217i	0.757785	+	0.0628017i
$326$		0			0
$327$		−	18.1053i		−	1.00123i
$328$		0			0
$329$	3.48822			0.192312
$330$		0			0
$331$	−8.38585			−0.460928			−0.230464	−	0.973081i	$-0.574024\pi$
$331$	−8.38585			−0.460928			−0.230464	+	0.973081i	$0.574024\pi$
$332$		0			0
$333$			3.18326i			0.174442i
$334$		0			0
$335$	−2.51680	−	2.73401i	−0.137507	−	0.149375i
$336$		0			0
$337$		−	8.59953i		−	0.468446i	−0.972183	−	0.234223i	$-0.924745\pi$
$337$		−	8.59953i		−	0.468446i	0.972183	−	0.234223i	$-0.0752546\pi$
$338$		0			0
$339$	24.2942			1.31948
$340$		0			0
$341$	13.0494			0.706663
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−30.0813	+	27.6915i	−1.61952	+	1.49086i
$346$		0			0
$347$		−	5.37773i		−	0.288692i	−0.989527	−	0.144346i	$-0.953892\pi$
$347$		−	5.37773i		−	0.288692i	0.989527	−	0.144346i	$-0.0461078\pi$
$348$		0			0
$349$	11.1582			0.597284			0.298642	−	0.954365i	$-0.403466\pi$
$349$	11.1582			0.597284			0.298642	+	0.954365i	$0.403466\pi$
$350$		0			0
$351$	7.04463			0.376015
$352$		0			0
$353$		−	25.1040i		−	1.33615i	−0.744093	−	0.668076i	$-0.767118\pi$
$353$		−	25.1040i		−	1.33615i	0.744093	−	0.668076i	$-0.232882\pi$
$354$		0			0
$355$	26.3610	−	24.2667i	1.39910	−	1.28794i
$356$		0			0
$357$		−	15.2637i		−	0.807841i
$358$		0			0
$359$	24.4545			1.29066			0.645329	−	0.763905i	$-0.276720\pi$
$359$	24.4545			1.29066			0.645329	+	0.763905i	$0.276720\pi$
$360$		0			0
$361$	−17.3352			−0.912378
$362$		0			0
$363$			20.0243i			1.05100i
$364$		0			0
$365$	3.23253	+	3.51151i	0.169198	+	0.183801i
$366$		0			0
$367$			22.6016i			1.17979i	0.807478	+	0.589897i	$0.200832\pi$
$367$			22.6016i			1.17979i	−0.807478	+	0.589897i	$0.799168\pi$
$368$		0			0
$369$	9.71962			0.505983
$370$		0			0
$371$	−6.13448			−0.318486
$372$		0			0
$373$			11.4377i			0.592221i	0.955154	+	0.296111i	$0.0956897\pi$
$373$			11.4377i			0.592221i	−0.955154	+	0.296111i	$0.904310\pi$
$374$		0			0
$375$	−19.3924	+	15.0931i	−1.00142	+	0.779405i
$376$		0			0
$377$			23.0319i			1.18620i
$378$		0			0
$379$	−22.5122			−1.15638			−0.578188	−	0.815904i	$-0.696240\pi$
$379$	−22.5122			−1.15638			−0.578188	+	0.815904i	$0.696240\pi$
$380$		0			0
$381$	17.0222			0.872072
$382$		0			0
$383$			20.0106i			1.02249i	0.859434	+	0.511247i	$0.170816\pi$
$383$			20.0106i			1.02249i	−0.859434	+	0.511247i	$0.829184\pi$
$384$		0			0
$385$	−2.08174	−	2.26140i	−0.106095	−	0.115252i
$386$		0			0
$387$		−	14.3387i		−	0.728878i
$388$		0			0
$389$	22.5476			1.14321			0.571605	−	0.820529i	$-0.306321\pi$
$389$	22.5476			1.14321			0.571605	+	0.820529i	$0.306321\pi$
$390$		0			0
$391$	−57.7728			−2.92170
$392$		0			0
$393$			10.4452i			0.526892i
$394$		0			0
$395$	8.96818	−	8.25569i	0.451238	−	0.415389i
$396$		0			0
$397$			9.35871i			0.469700i	0.972032	+	0.234850i	$0.0754599\pi$
$397$			9.35871i			0.469700i	−0.972032	+	0.234850i	$0.924540\pi$
$398$		0			0
$399$	−2.83595			−0.141975
$400$		0			0
$401$	−31.2475			−1.56042			−0.780212	−	0.625516i	$-0.784889\pi$
$401$	−31.2475			−1.56042			−0.780212	+	0.625516i	$0.784889\pi$
$402$		0			0
$403$		−	26.0266i		−	1.29648i
$404$		0			0
$405$	−18.3276	+	16.8716i	−0.910707	+	0.838355i
$406$		0			0
$407$			2.38987i			0.118462i
$408$		0			0
$409$	28.0098			1.38499			0.692497	−	0.721421i	$-0.256511\pi$
$409$	28.0098			1.38499			0.692497	+	0.721421i	$0.256511\pi$
$410$		0			0
$411$	−40.2227			−1.98404
$412$		0			0
$413$		−	6.26070i		−	0.308069i
$414$		0			0
$415$	3.85612	+	4.18892i	0.189290	+	0.205626i
$416$		0			0
$417$		−	41.6223i		−	2.03825i
$418$		0			0
$419$	16.8212			0.821769			0.410885	−	0.911687i	$-0.365220\pi$
$419$	16.8212			0.821769			0.410885	+	0.911687i	$0.365220\pi$
$420$		0			0
$421$	−0.354309			−0.0172680			−0.00863398	−	0.999963i	$-0.502748\pi$
$421$	−0.354309			−0.0172680			−0.00863398	+	0.999963i	$0.502748\pi$
$422$		0			0
$423$			6.38670i			0.310532i
$424$		0			0
$425$	−34.6041	−	2.86783i	−1.67855	−	0.139110i
$426$		0			0
$427$			6.59883i			0.319340i
$428$		0			0
$429$	−8.28315			−0.399914
$430$		0			0
$431$	14.8578			0.715675			0.357837	−	0.933784i	$-0.383514\pi$
$431$	14.8578			0.715675			0.357837	+	0.933784i	$0.383514\pi$
$432$		0			0
$433$		−	13.6602i		−	0.656469i	−0.944596	−	0.328235i	$-0.893546\pi$
$433$		−	13.6602i		−	0.656469i	0.944596	−	0.328235i	$-0.106454\pi$
$434$		0			0
$435$	−27.9635	−	30.3769i	−1.34075	−	1.45646i
$436$		0			0
$437$			10.7340i			0.513477i
$438$		0			0
$439$	25.3663			1.21067			0.605334	−	0.795972i	$-0.293040\pi$
$439$	25.3663			1.21067			0.605334	+	0.795972i	$0.293040\pi$
$440$		0			0
$441$	1.83094			0.0871874
$442$		0			0
$443$			32.8555i			1.56101i	0.625148	+	0.780506i	$0.285039\pi$
$443$			32.8555i			1.56101i	−0.625148	+	0.780506i	$0.714961\pi$
$444$		0			0
$445$	−2.36157	+	2.17395i	−0.111949	+	0.103055i
$446$		0			0
$447$			22.2332i			1.05160i
$448$		0			0
$449$	−34.2268			−1.61526			−0.807632	−	0.589687i	$-0.799251\pi$
$449$	−34.2268			−1.61526			−0.807632	+	0.589687i	$0.799251\pi$
$450$		0			0
$451$	7.29712			0.343608
$452$		0			0
$453$			1.24767i			0.0586206i
$454$		0			0
$455$	−4.51031	+	4.15198i	−0.211447	+	0.194648i
$456$		0			0
$457$			21.8661i			1.02285i	0.859327	+	0.511427i	$0.170883\pi$
$457$			21.8661i			1.02285i	−0.859327	+	0.511427i	$0.829117\pi$
$458$		0			0
$459$	−17.8443			−0.832899
$460$		0			0
$461$	−32.8358			−1.52932			−0.764659	−	0.644436i	$-0.777092\pi$
$461$	−32.8358			−1.52932			−0.764659	+	0.644436i	$0.777092\pi$
$462$		0			0
$463$		−	20.9970i		−	0.975811i	−0.872896	−	0.487906i	$-0.837761\pi$
$463$		−	20.9970i		−	0.975811i	0.872896	−	0.487906i	$-0.162239\pi$
$464$		0			0
$465$	31.5995	+	34.3267i	1.46539	+	1.59186i
$466$		0			0
$467$			9.88718i			0.457524i	0.973482	+	0.228762i	$0.0734678\pi$
$467$			9.88718i			0.457524i	−0.973482	+	0.228762i	$0.926532\pi$
$468$		0			0
$469$	1.66187			0.0767381
$470$		0			0
$471$	−6.32000			−0.291210
$472$		0			0
$473$		−	10.7650i		−	0.494974i
$474$		0			0
$475$	−0.532833	+	6.42934i	−0.0244481	+	0.294998i
$476$		0			0
$477$		−	11.2318i		−	0.514270i
$478$		0			0
$479$	−18.1115			−0.827534			−0.413767	−	0.910383i	$-0.635787\pi$
$479$	−18.1115			−0.827534			−0.413767	+	0.910383i	$0.635787\pi$
$480$		0			0
$481$	4.76654			0.217336
$482$		0			0
$483$		−	18.2850i		−	0.831996i
$484$		0			0
$485$	14.6806	+	15.9476i	0.666610	+	0.724141i
$486$		0			0
$487$		−	23.4907i		−	1.06447i	−0.846597	−	0.532234i	$-0.821353\pi$
$487$		−	23.4907i		−	1.06447i	0.846597	−	0.532234i	$-0.178647\pi$
$488$		0			0
$489$	22.4656			1.01593
$490$		0			0
$491$	25.1693			1.13588			0.567938	−	0.823072i	$-0.307742\pi$
$491$	25.1693			1.13588			0.567938	+	0.823072i	$0.307742\pi$
$492$		0			0
$493$		−	58.3404i		−	2.62752i
$494$		0			0
$495$	4.14048	−	3.81153i	0.186101	−	0.171316i
$496$		0			0
$497$			16.0236i			0.718756i
$498$		0			0
$499$	17.5266			0.784601			0.392300	−	0.919837i	$-0.371679\pi$
$499$	17.5266			0.784601			0.392300	+	0.919837i	$0.371679\pi$
$500$		0			0
$501$	43.4471			1.94107
$502$		0			0
$503$		−	33.0352i		−	1.47297i	−0.676456	−	0.736483i	$-0.736485\pi$
$503$		−	33.0352i		−	1.47297i	0.676456	−	0.736483i	$-0.263515\pi$
$504$		0			0
$505$	13.5735	−	12.4951i	0.604013	−	0.556026i
$506$		0			0
$507$		−	12.0527i		−	0.535278i
$508$		0			0
$509$	21.4535			0.950909			0.475455	−	0.879740i	$-0.342284\pi$
$509$	21.4535			0.950909			0.475455	+	0.879740i	$0.342284\pi$
$510$		0			0
$511$	−2.13448			−0.0944236
$512$		0			0
$513$			3.31541i			0.146379i
$514$		0			0
$515$	−19.9150	−	21.6337i	−0.877559	−	0.953295i
$516$		0			0
$517$			4.79489i			0.210879i
$518$		0			0
$519$	−47.7569			−2.09630
$520$		0			0
$521$	23.4249			1.02626			0.513132	−	0.858310i	$-0.328485\pi$
$521$	23.4249			1.02626			0.513132	+	0.858310i	$0.328485\pi$
$522$		0			0
$523$		−	15.4736i		−	0.676611i	−0.941036	−	0.338306i	$-0.890146\pi$
$523$		−	15.4736i		−	0.676611i	0.941036	−	0.338306i	$-0.109854\pi$
$524$		0			0
$525$	0.907662	−	10.9521i	0.0396136	−	0.477991i
$526$		0			0
$527$			65.9262i			2.87179i
$528$		0			0
$529$	−46.2083			−2.00906
$530$		0			0
$531$	11.4629			0.497449
$532$		0			0
$533$		−	14.5539i		−	0.630401i
$534$		0			0
$535$	−0.232396	−	0.252452i	−0.0100473	−	0.0109145i
$536$		0			0
$537$		−	46.5108i		−	2.00709i
$538$		0			0
$539$	1.37460			0.0592081
$540$		0			0
$541$	−6.24274			−0.268396			−0.134198	−	0.990955i	$-0.542846\pi$
$541$	−6.24274			−0.268396			−0.134198	+	0.990955i	$0.542846\pi$
$542$		0			0
$543$		−	26.0227i		−	1.11674i
$544$		0			0
$545$	13.5517	−	12.4750i	0.580490	−	0.534372i
$546$		0			0
$547$		−	33.8997i		−	1.44945i	−0.689040	−	0.724724i	$-0.741967\pi$
$547$		−	33.8997i		−	1.44945i	0.689040	−	0.724724i	$-0.258033\pi$
$548$		0			0
$549$	−12.0820			−0.515649
$550$		0			0
$551$	−10.8395			−0.461777
$552$		0			0
$553$			5.45132i			0.231814i
$554$		0			0
$555$	−6.28662	+	5.78717i	−0.266852	+	0.245651i
$556$		0			0
$557$		−	0.161941i		−	0.00686165i	−0.999994	−	0.00343083i	$-0.998908\pi$
$557$		−	0.161941i		−	0.00686165i	0.999994	−	0.00343083i	$-0.00109207\pi$
$558$		0			0
$559$	−21.4705			−0.908104
$560$		0			0
$561$	20.9814			0.885837
$562$		0			0
$563$			38.6514i			1.62896i	0.580189	+	0.814482i	$0.302979\pi$
$563$			38.6514i			1.62896i	−0.580189	+	0.814482i	$0.697021\pi$
$564$		0			0
$565$	16.7393	+	18.1840i	0.704228	+	0.765006i
$566$		0			0
$567$		−	11.1405i		−	0.467856i
$568$		0			0
$569$	26.9523			1.12990			0.564949	−	0.825126i	$-0.308896\pi$
$569$	26.9523			1.12990			0.564949	+	0.825126i	$0.308896\pi$
$570$		0			0
$571$	−24.2525			−1.01493			−0.507467	−	0.861671i	$-0.669418\pi$
$571$	−24.2525			−1.01493			−0.507467	+	0.861671i	$0.669418\pi$
$572$		0			0
$573$		−	4.00173i		−	0.167175i
$574$		0			0
$575$	−41.4536	−	3.43548i	−1.72874	−	0.143269i
$576$		0			0
$577$			14.2848i			0.594684i	0.954771	+	0.297342i	$0.0961001\pi$
$577$			14.2848i			0.594684i	−0.954771	+	0.297342i	$0.903900\pi$
$578$		0			0
$579$	−7.97347			−0.331366
$580$		0			0
$581$	−2.54624			−0.105636
$582$		0			0
$583$		−	8.43243i		−	0.349236i
$584$		0			0
$585$	−7.60201	−	8.25809i	−0.314304	−	0.341430i
$586$		0			0
$587$		−	30.6314i		−	1.26429i	−0.774849	−	0.632146i	$-0.782174\pi$
$587$		−	30.6314i		−	1.26429i	0.774849	−	0.632146i	$-0.217826\pi$
$588$		0			0
$589$	12.2489			0.504707
$590$		0			0
$591$	14.2730			0.587112
$592$		0			0
$593$		−	37.4568i		−	1.53817i	−0.639149	−	0.769083i	$-0.720713\pi$
$593$		−	37.4568i		−	1.53817i	0.639149	−	0.769083i	$-0.279287\pi$
$594$		0			0
$595$	11.4248	−	10.5171i	0.468369	−	0.431159i
$596$		0			0
$597$		−	10.3115i		−	0.422022i
$598$		0			0
$599$	−18.9394			−0.773843			−0.386921	−	0.922113i	$-0.626462\pi$
$599$	−18.9394			−0.773843			−0.386921	+	0.922113i	$0.626462\pi$
$600$		0			0
$601$	−8.49158			−0.346379			−0.173189	−	0.984889i	$-0.555407\pi$
$601$	−8.49158			−0.346379			−0.173189	+	0.984889i	$0.555407\pi$
$602$		0			0
$603$			3.04278i			0.123911i
$604$		0			0
$605$	−14.9880	+	13.7973i	−0.609349	+	0.560938i
$606$		0			0
$607$			23.4717i			0.952688i	0.879259	+	0.476344i	$0.158038\pi$
$607$			23.4717i			0.952688i	−0.879259	+	0.476344i	$0.841962\pi$
$608$		0			0
$609$	18.4646			0.748225
$610$		0			0
$611$	9.56329			0.386889
$612$		0			0
$613$			33.9078i			1.36952i	0.728767	+	0.684762i	$0.240094\pi$
$613$			33.9078i			1.36952i	−0.728767	+	0.684762i	$0.759906\pi$
$614$		0			0
$615$	17.6703	+	19.1953i	0.712534	+	0.774028i
$616$		0			0
$617$			23.7015i			0.954186i	0.878853	+	0.477093i	$0.158310\pi$
$617$			23.7015i			0.954186i	−0.878853	+	0.477093i	$0.841690\pi$
$618$		0			0
$619$	17.5275			0.704491			0.352246	−	0.935908i	$-0.385418\pi$
$619$	17.5275			0.704491			0.352246	+	0.935908i	$0.385418\pi$
$620$		0			0
$621$	−21.3763			−0.857803
$622$		0			0
$623$		−	1.43548i		−	0.0575113i
$624$		0			0
$625$	−24.6589	−	4.11550i	−0.986357	−	0.164620i
$626$		0			0
$627$		−	3.89829i		−	0.155683i
$628$		0			0
$629$	−12.0738			−0.481413
$630$		0			0
$631$	13.9892			0.556902			0.278451	−	0.960450i	$-0.410179\pi$
$631$	13.9892			0.556902			0.278451	+	0.960450i	$0.410179\pi$
$632$		0			0
$633$		−	37.3998i		−	1.48651i
$634$		0			0
$635$	11.7287	+	12.7409i	0.465440	+	0.505609i
$636$		0			0
$637$		−	2.74160i		−	0.108626i
$638$		0			0
$639$	−29.3381			−1.16060
$640$		0			0
$641$	38.7939			1.53227			0.766133	−	0.642682i	$-0.222178\pi$
$641$	38.7939			1.53227			0.766133	+	0.642682i	$0.222178\pi$
$642$		0			0
$643$		−	40.9729i		−	1.61581i	−0.589310	−	0.807907i	$-0.700600\pi$
$643$		−	40.9729i		−	1.61581i	0.589310	−	0.807907i	$-0.299400\pi$
$644$		0			0
$645$	28.3175	−	26.0678i	1.11500	−	1.02642i
$646$		0			0
$647$		−	7.17647i		−	0.282136i	−0.990000	−	0.141068i	$-0.954946\pi$
$647$		−	7.17647i		−	0.282136i	0.990000	−	0.141068i	$-0.0450536\pi$
$648$		0			0
$649$	8.60594			0.337813
$650$		0			0
$651$	−20.8655			−0.817785
$652$		0			0
$653$		−	21.6869i		−	0.848672i	−0.905505	−	0.424336i	$-0.860508\pi$
$653$		−	21.6869i		−	0.848672i	0.905505	−	0.424336i	$-0.139492\pi$
$654$		0			0
$655$	−7.81817	+	7.19704i	−0.305481	+	0.281211i
$656$		0			0
$657$		−	3.90809i		−	0.152469i
$658$		0			0
$659$	5.56012			0.216592			0.108296	−	0.994119i	$-0.465461\pi$
$659$	5.56012			0.216592			0.108296	+	0.994119i	$0.465461\pi$
$660$		0			0
$661$	−35.4543			−1.37901			−0.689507	−	0.724279i	$-0.742173\pi$
$661$	−35.4543			−1.37901			−0.689507	+	0.724279i	$0.742173\pi$
$662$		0			0
$663$		−	41.8470i		−	1.62520i
$664$		0			0
$665$	−1.95404	−	2.12268i	−0.0757745	−	0.0823141i
$666$		0			0
$667$		−	69.8883i		−	2.70609i
$668$		0			0
$669$	20.6442			0.798149
$670$		0			0
$671$	−9.07073			−0.350172
$672$		0			0
$673$			28.8018i			1.11023i	0.831775	+	0.555114i	$0.187325\pi$
$673$			28.8018i			1.11023i	−0.831775	+	0.555114i	$0.812675\pi$
$674$		0			0
$675$	−12.8038	−	1.06112i	−0.492817	−	0.0408424i
$676$		0			0
$677$			4.41481i			0.169675i	0.996395	+	0.0848375i	$0.0270371\pi$
$677$			4.41481i			0.169675i	−0.996395	+	0.0848375i	$0.972963\pi$
$678$		0			0
$679$	−9.69375			−0.372012
$680$		0			0
$681$	31.4601			1.20555
$682$		0			0
$683$			29.9092i			1.14444i	0.820099	+	0.572221i	$0.193918\pi$
$683$			29.9092i			1.14444i	−0.820099	+	0.572221i	$0.806082\pi$
$684$		0			0
$685$	−27.7145	−	30.1063i	−1.05892	−	1.15030i
$686$		0			0
$687$		−	19.3056i		−	0.736555i
$688$		0			0
$689$	−16.8183			−0.640726
$690$		0			0
$691$	2.79869			0.106467			0.0532337	−	0.998582i	$-0.483047\pi$
$691$	2.79869			0.106467			0.0532337	+	0.998582i	$0.483047\pi$
$692$		0			0
$693$			2.51680i			0.0956053i
$694$		0			0
$695$	31.1539	−	28.6788i	1.18174	−	1.08785i
$696$		0			0
$697$			36.8655i			1.39638i
$698$		0			0
$699$	−32.4648			−1.22793
$700$		0			0
$701$	−45.3081			−1.71126			−0.855632	−	0.517584i	$-0.826832\pi$
$701$	−45.3081			−1.71126			−0.855632	+	0.517584i	$0.826832\pi$
$702$		0			0
$703$			2.24327i			0.0846066i
$704$		0			0
$705$	−12.6131	+	11.6110i	−0.475036	+	0.437296i
$706$		0			0
$707$			8.25067i			0.310298i
$708$		0			0
$709$	−11.2038			−0.420768			−0.210384	−	0.977619i	$-0.567471\pi$
$709$	−11.2038			−0.420768			−0.210384	+	0.977619i	$0.567471\pi$
$710$		0			0
$711$	−9.98102			−0.374318
$712$		0			0
$713$			78.9756i			2.95766i
$714$		0			0
$715$	−5.70730	−	6.19986i	−0.213441	−	0.231862i
$716$		0			0
$717$		−	55.0639i		−	2.05640i
$718$		0			0
$719$	−22.1695			−0.826782			−0.413391	−	0.910554i	$-0.635656\pi$
$719$	−22.1695			−0.826782			−0.413391	+	0.910554i	$0.635656\pi$
$720$		0			0
$721$	13.1501			0.489735
$722$		0			0
$723$		−	7.31858i		−	0.272181i
$724$		0			0
$725$	3.46924	−	41.8609i	0.128844	−	1.55468i
$726$		0			0
$727$		−	17.4515i		−	0.647241i	−0.946187	−	0.323620i	$-0.895100\pi$
$727$		−	17.4515i		−	0.647241i	0.946187	−	0.323620i	$-0.104900\pi$
$728$		0			0
$729$	3.45447			0.127943
$730$		0			0
$731$	54.3853			2.01151
$732$		0			0
$733$			26.0171i			0.960962i	0.877005	+	0.480481i	$0.159538\pi$
$733$			26.0171i			0.960962i	−0.877005	+	0.480481i	$0.840462\pi$
$734$		0			0
$735$	3.32864	+	3.61591i	0.122779	+	0.133375i
$736$		0			0
$737$			2.28440i			0.0841471i
$738$		0			0
$739$	−2.81093			−0.103402			−0.0517008	−	0.998663i	$-0.516464\pi$
$739$	−2.81093			−0.103402			−0.0517008	+	0.998663i	$0.516464\pi$
$740$		0			0
$741$	−7.77504			−0.285623
$742$		0			0
$743$			22.8466i			0.838159i	0.907950	+	0.419080i	$0.137647\pi$
$743$			22.8466i			0.838159i	−0.907950	+	0.419080i	$0.862353\pi$
$744$		0			0
$745$	−16.6414	+	15.3193i	−0.609693	+	0.561255i
$746$		0			0
$747$		−	4.66200i		−	0.170574i
$748$		0			0
$749$	0.153454			0.00560707
$750$		0			0
$751$	−51.7651			−1.88893			−0.944467	−	0.328607i	$-0.893421\pi$
$751$	−51.7651			−1.88893			−0.944467	+	0.328607i	$0.893421\pi$
$752$		0			0
$753$			21.7406i			0.792271i
$754$		0			0
$755$	−0.933870	+	0.859677i	−0.0339870	+	0.0312868i
$756$		0			0
$757$			8.13016i			0.295496i	0.989025	+	0.147748i	$0.0472024\pi$
$757$			8.13016i			0.295496i	−0.989025	+	0.147748i	$0.952798\pi$
$758$		0			0
$759$	25.1345			0.912324
$760$		0			0
$761$	31.2494			1.13279			0.566395	−	0.824134i	$-0.308338\pi$
$761$	31.2494			1.13279			0.566395	+	0.824134i	$0.308338\pi$
$762$		0			0
$763$			8.23741i			0.298214i
$764$		0			0
$765$	19.2561	+	20.9180i	0.696206	+	0.756291i
$766$		0			0
$767$		−	17.1643i		−	0.619769i
$768$		0			0
$769$	−28.7711			−1.03751			−0.518757	−	0.854922i	$-0.673605\pi$
$769$	−28.7711			−1.03751			−0.518757	+	0.854922i	$0.673605\pi$
$770$		0			0
$771$	29.9176			1.07746
$772$		0			0
$773$			15.7175i			0.565318i	0.959220	+	0.282659i	$0.0912165\pi$
$773$			15.7175i			0.565318i	−0.959220	+	0.282659i	$0.908783\pi$
$774$		0			0
$775$	−3.92033	+	47.3040i	−0.140822	+	1.69921i
$776$		0			0
$777$		−	3.82133i		−	0.137089i
$778$		0			0
$779$	6.84950			0.245409
$780$		0			0
$781$	−22.0260			−0.788151
$782$		0			0
$783$		−	21.5864i		−	0.771434i
$784$		0			0
$785$	−4.35464	−	4.73046i	−0.155424	−	0.168838i
$786$		0			0
$787$			28.8516i			1.02845i	0.857656	+	0.514224i	$0.171920\pi$
$787$			28.8516i			1.02845i	−0.857656	+	0.514224i	$0.828080\pi$
$788$		0			0
$789$	9.04561			0.322032
$790$		0			0
$791$	−11.0532			−0.393005
$792$		0			0
$793$			18.0914i			0.642443i
$794$		0			0
$795$	22.1817	−	20.4195i	0.786705	−	0.724204i
$796$		0			0
$797$		−	5.81776i		−	0.206076i	−0.994677	−	0.103038i	$-0.967144\pi$
$797$		−	5.81776i		−	0.206076i	0.994677	−	0.103038i	$-0.0328563\pi$
$798$		0			0
$799$	−24.2241			−0.856987
$800$		0			0
$801$	2.62827			0.0928655
$802$		0			0
$803$		−	2.93404i		−	0.103540i
$804$		0			0
$805$	13.6862	−	12.5988i	0.482374	−	0.444050i
$806$		0			0
$807$			23.6079i			0.831039i
$808$		0			0
$809$	−26.5157			−0.932242			−0.466121	−	0.884721i	$-0.654349\pi$
$809$	−26.5157			−0.932242			−0.466121	+	0.884721i	$0.654349\pi$
$810$		0			0
$811$	−55.5069			−1.94911			−0.974556	−	0.224144i	$-0.928041\pi$
$811$	−55.5069			−1.94911			−0.974556	+	0.224144i	$0.928041\pi$
$812$		0			0
$813$			71.9688i			2.52406i
$814$		0			0
$815$	15.4794	+	16.8153i	0.542220	+	0.589016i
$816$		0			0
$817$		−	10.1046i		−	0.353516i
$818$		0			0
$819$	5.01969			0.175402
$820$		0			0
$821$	−44.1325			−1.54024			−0.770118	−	0.637902i	$-0.779803\pi$
$821$	−44.1325			−1.54024			−0.770118	+	0.637902i	$0.779803\pi$
$822$		0			0
$823$			19.1816i			0.668629i	0.942461	+	0.334315i	$0.108505\pi$
$823$			19.1816i			0.668629i	−0.942461	+	0.334315i	$0.891495\pi$
$824$		0			0
$825$	15.0548	+	1.24767i	0.524141	+	0.0434383i
$826$		0			0
$827$		−	1.36085i		−	0.0473214i	−0.999720	−	0.0236607i	$-0.992468\pi$
$827$		−	1.36085i		−	0.0473214i	0.999720	−	0.0236607i	$-0.00753214\pi$
$828$		0			0
$829$	−7.04654			−0.244737			−0.122368	−	0.992485i	$-0.539049\pi$
$829$	−7.04654			−0.244737			−0.122368	+	0.992485i	$0.539049\pi$
$830$		0			0
$831$	15.2234			0.528095
$832$		0			0
$833$			6.94455i			0.240615i
$834$		0			0
$835$	29.9362	+	32.5198i	1.03598	+	1.12539i
$836$		0			0
$837$			24.3932i			0.843151i
$838$		0			0
$839$	14.4808			0.499934			0.249967	−	0.968254i	$-0.419580\pi$
$839$	14.4808			0.499934			0.249967	+	0.968254i	$0.419580\pi$
$840$		0			0
$841$	41.5750			1.43362
$842$		0			0
$843$			5.74122i			0.197738i
$844$		0			0
$845$	9.02132	−	8.30461i	0.310343	−	0.285687i
$846$		0			0
$847$		−	9.11048i		−	0.313040i
$848$		0			0
$849$	−15.3850			−0.528011
$850$		0			0
$851$	−14.4637			−0.495808
$852$		0			0
$853$		−	13.3025i		−	0.455470i	−0.973723	−	0.227735i	$-0.926868\pi$
$853$		−	13.3025i		−	0.455470i	0.973723	−	0.227735i	$-0.0731319\pi$
$854$		0			0
$855$	3.88650	−	3.57773i	0.132915	−	0.122356i
$856$		0			0
$857$			36.0508i			1.23147i	0.787953	+	0.615736i	$0.211141\pi$
$857$			36.0508i			1.23147i	−0.787953	+	0.615736i	$0.788859\pi$
$858$		0			0
$859$	6.42934			0.219366			0.109683	−	0.993967i	$-0.465016\pi$
$859$	6.42934			0.219366			0.109683	+	0.993967i	$0.465016\pi$
$860$		0			0
$861$	−11.6679			−0.397640
$862$		0			0
$863$		−	2.28639i		−	0.0778295i	−0.999243	−	0.0389148i	$-0.987610\pi$
$863$		−	2.28639i		−	0.0778295i	0.999243	−	0.0389148i	$-0.0123901\pi$
$864$		0			0
$865$	−32.9058	−	35.7457i	−1.11883	−	1.21539i
$866$		0			0
$867$			68.6347i			2.33096i
$868$		0			0
$869$	−7.49337			−0.254195
$870$		0			0
$871$	4.55619			0.154380
$872$		0			0
$873$		−	17.7486i		−	0.600700i
$874$		0			0
$875$	8.82299	−	6.86694i	0.298272	−	0.232145i
$876$		0			0
$877$			28.7188i			0.969764i	0.874579	+	0.484882i	$0.161138\pi$
$877$			28.7188i			0.969764i	−0.874579	+	0.484882i	$0.838862\pi$
$878$		0			0
$879$	32.3315			1.09051
$880$		0			0
$881$	−36.4536			−1.22815			−0.614077	−	0.789246i	$-0.710472\pi$
$881$	−36.4536			−1.22815			−0.614077	+	0.789246i	$0.710472\pi$
$882$		0			0
$883$			22.6608i			0.762597i	0.924452	+	0.381298i	$0.124523\pi$
$883$			22.6608i			0.762597i	−0.924452	+	0.381298i	$0.875477\pi$
$884$		0			0
$885$	20.8396	+	22.6382i	0.700516	+	0.760973i
$886$		0			0
$887$		−	13.7308i		−	0.461035i	−0.973068	−	0.230517i	$-0.925958\pi$
$887$		−	13.7308i		−	0.461035i	0.973068	−	0.230517i	$-0.0740418\pi$
$888$		0			0
$889$	−7.74460			−0.259746
$890$		0			0
$891$	15.3137			0.513027
$892$		0			0
$893$			4.50076i			0.150612i
$894$		0			0
$895$	34.8129	−	32.0472i	1.16367	−	1.07122i
$896$		0			0
$897$		−	50.1301i		−	1.67380i
$898$		0			0
$899$	−79.7516			−2.65986
$900$		0			0
$901$	42.6012			1.41925
$902$		0			0
$903$			17.2128i			0.572808i
$904$		0			0
$905$	19.4778	−	17.9303i	0.647464	−	0.596025i
$906$		0			0
$907$			48.7472i			1.61863i	0.587378	+	0.809313i	$0.300160\pi$
$907$			48.7472i			1.61863i	−0.587378	+	0.809313i	$0.699840\pi$
$908$		0			0
$909$	−15.1064			−0.501049
$910$		0			0
$911$	54.1839			1.79519			0.897596	−	0.440820i	$-0.145312\pi$
$911$	54.1839			1.79519			0.897596	+	0.440820i	$0.145312\pi$
$912$		0			0
$913$		−	3.50006i		−	0.115835i
$914$		0			0
$915$	−21.9651	−	23.8608i	−0.726145	−	0.788814i
$916$		0			0
$917$		−	4.75228i		−	0.156934i
$918$		0			0
$919$	29.3450			0.968003			0.484002	−	0.875067i	$-0.339183\pi$
$919$	29.3450			0.968003			0.484002	+	0.875067i	$0.339183\pi$
$920$		0			0
$921$	−71.5577			−2.35790
$922$		0			0
$923$			43.9303i			1.44598i
$924$		0			0
$925$	−8.66329	−	0.717972i	−0.284847	−	0.0236068i
$926$		0			0
$927$			24.0770i			0.790791i
$928$		0			0
$929$	−36.9346			−1.21179			−0.605893	−	0.795546i	$-0.707184\pi$
$929$	−36.9346			−1.21179			−0.605893	+	0.795546i	$0.707184\pi$
$930$		0			0
$931$	1.29028			0.0422871
$932$		0			0
$933$		−	71.2275i		−	2.33188i
$934$		0			0
$935$	14.4568	+	15.7044i	0.472787	+	0.513590i
$936$		0			0
$937$		−	19.3625i		−	0.632544i	−0.948668	−	0.316272i	$-0.897569\pi$
$937$		−	19.3625i		−	0.632544i	0.948668	−	0.316272i	$-0.102431\pi$
$938$		0			0
$939$	−46.6726			−1.52310
$940$		0			0
$941$	−13.6807			−0.445979			−0.222990	−	0.974821i	$-0.571582\pi$
$941$	−13.6807			−0.445979			−0.222990	+	0.974821i	$0.571582\pi$
$942$		0			0
$943$			44.1627i			1.43813i
$944$		0			0
$945$	4.22724	−	3.89140i	0.137512	−	0.126587i
$946$		0			0
$947$		−	12.0479i		−	0.391504i	−0.980653	−	0.195752i	$-0.937285\pi$
$947$		−	12.0479i		−	0.391504i	0.980653	−	0.195752i	$-0.0627147\pi$
$948$		0			0
$949$	−5.85188			−0.189960
$950$		0			0
$951$	72.9379			2.36517
$952$		0			0
$953$			34.6708i			1.12310i	0.827443	+	0.561549i	$0.189794\pi$
$953$			34.6708i			1.12310i	−0.827443	+	0.561549i	$0.810206\pi$
$954$		0			0
$955$	2.99526	−	2.75730i	0.0969244	−	0.0892241i
$956$		0			0
$957$			25.3815i			0.820466i
$958$		0			0
$959$	18.3002			0.590943
$960$		0			0
$961$	59.1214			1.90714
$962$		0			0
$963$			0.280964i			0.00905393i
$964$		0			0
$965$	−5.49393	−	5.96808i	−0.176856	−	0.192119i
$966$		0			0
$967$			27.6483i			0.889111i	0.895751	+	0.444555i	$0.146638\pi$
$967$			27.6483i			0.889111i	−0.895751	+	0.444555i	$0.853362\pi$
$968$		0			0
$969$	19.6944			0.632675
$970$		0			0
$971$	50.6217			1.62453			0.812264	−	0.583290i	$-0.198235\pi$
$971$	50.6217			1.62453			0.812264	+	0.583290i	$0.198235\pi$
$972$		0			0
$973$			18.9370i			0.607091i
$974$		0			0
$975$	2.48845	−	30.0264i	0.0796941	−	0.961615i
$976$		0			0
$977$		−	27.3458i		−	0.874869i	−0.899250	−	0.437435i	$-0.855887\pi$
$977$		−	27.3458i		−	0.874869i	0.899250	−	0.437435i	$-0.144113\pi$
$978$		0			0
$979$	1.97321			0.0630640
$980$		0			0
$981$	−15.0822			−0.481536
$982$		0			0
$983$		−	32.6694i		−	1.04199i	−0.853559	−	0.520996i	$-0.825561\pi$
$983$		−	32.6694i		−	1.04199i	0.853559	−	0.520996i	$-0.174439\pi$
$984$		0			0
$985$	9.83445	+	10.6832i	0.313352	+	0.340395i
$986$		0			0
$987$		−	7.66688i		−	0.244040i
$988$		0			0
$989$	65.1503			2.07166
$990$		0			0
$991$	4.85330			0.154170			0.0770850	−	0.997025i	$-0.475439\pi$
$991$	4.85330			0.154170			0.0770850	+	0.997025i	$0.475439\pi$
$992$		0			0
$993$			18.4316i			0.584909i
$994$		0			0
$995$	7.71808	−	7.10490i	0.244680	−	0.225240i
$996$		0			0
$997$			41.2053i			1.30498i	0.757796	+	0.652492i	$0.226276\pi$
$997$			41.2053i			1.30498i	−0.757796	+	0.652492i	$0.773724\pi$
$998$		0			0
$999$	−4.46738			−0.141342

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.g.b.449.2	✓	10
4.3	odd	2		1120.2.g.c.449.9	yes	10
5.2	odd	4		5600.2.a.bw.1.1		5
5.3	odd	4		5600.2.a.bu.1.5		5
5.4	even	2	inner	1120.2.g.b.449.9	yes	10
8.3	odd	2		2240.2.g.n.449.2		10
8.5	even	2		2240.2.g.o.449.9		10
20.3	even	4		5600.2.a.bx.1.1		5
20.7	even	4		5600.2.a.bv.1.5		5
20.19	odd	2		1120.2.g.c.449.2	yes	10
40.19	odd	2		2240.2.g.n.449.9		10
40.29	even	2		2240.2.g.o.449.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.g.b.449.2	✓	10	1.1	even	1	trivial
1120.2.g.b.449.9	yes	10	5.4	even	2	inner
1120.2.g.c.449.2	yes	10	20.19	odd	2
1120.2.g.c.449.9	yes	10	4.3	odd	2
2240.2.g.n.449.2		10	8.3	odd	2
2240.2.g.n.449.9		10	40.19	odd	2
2240.2.g.o.449.2		10	40.29	even	2
2240.2.g.o.449.9		10	8.5	even	2
5600.2.a.bu.1.5		5	5.3	odd	4
5600.2.a.bv.1.5		5	20.7	even	4
5600.2.a.bw.1.1		5	5.2	odd	4
5600.2.a.bx.1.1		5	20.3	even	4

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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.19794i q^{3} +(1.64514 - 1.51444i) q^{5} +1.00000i q^{7} -1.83094 q^{9} +O(q^{10})\)	\(q-2.19794i q^{3} +(1.64514 - 1.51444i) q^{5} +1.00000i q^{7} -1.83094 q^{9} -1.37460 q^{11} +2.74160i q^{13} +(-3.32864 - 3.61591i) q^{15} -6.94455i q^{17} -1.29028 q^{19} +2.19794 q^{21} -8.31915i q^{23} +(0.412961 - 4.98292i) q^{25} -2.56953i q^{27} +8.40089 q^{29} -9.49323 q^{31} +3.02128i q^{33} +(1.51444 + 1.64514i) q^{35} -1.73860i q^{37} +6.02587 q^{39} -5.30855 q^{41} +7.83136i q^{43} +(-3.01214 + 2.77284i) q^{45} -3.48822i q^{47} -1.00000 q^{49} -15.2637 q^{51} +6.13448i q^{53} +(-2.26140 + 2.08174i) q^{55} +2.83595i q^{57} -6.26070 q^{59} +6.59883 q^{61} -1.83094i q^{63} +(4.15198 + 4.51031i) q^{65} -1.66187i q^{67} -18.2850 q^{69} +16.0236 q^{71} +2.13448i q^{73} +(-10.9521 - 0.907662i) q^{75} -1.37460i q^{77} +5.45132 q^{79} -11.1405 q^{81} +2.54624i q^{83} +(-10.5171 - 11.4248i) q^{85} -18.4646i q^{87} -1.43548 q^{89} -2.74160 q^{91} +20.8655i q^{93} +(-2.12268 + 1.95404i) q^{95} +9.69375i q^{97} +2.51680 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{5} - 14 q^{9}+O(q^{10}) \) 10 * q - 2 * q^5 - 14 * q^9	\( 10 q - 2 q^{5} - 14 q^{9} - 8 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 6 q^{25} + 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} + 10 q^{45} - 10 q^{49} - 24 q^{51} - 16 q^{55} + 32 q^{59} - 20 q^{61} - 8 q^{65} - 8 q^{69} - 8 q^{71} - 64 q^{75} + 64 q^{79} + 2 q^{81} - 12 q^{85} - 4 q^{89} - 60 q^{95} + 80 q^{99}+O(q^{100}) \) 10 * q - 2 * q^5 - 14 * q^9 - 8 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^21 + 6 * q^25 + 24 * q^29 - 24 * q^31 + 64 * q^39 - 4 * q^41 + 10 * q^45 - 10 * q^49 - 24 * q^51 - 16 * q^55 + 32 * q^59 - 20 * q^61 - 8 * q^65 - 8 * q^69 - 8 * q^71 - 64 * q^75 + 64 * q^79 + 2 * q^81 - 12 * q^85 - 4 * q^89 - 60 * q^95 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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