LMFDB - Embedded newform 765.2.g.b.271.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [765,2,Mod(271,765)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(765, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("765.271");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 765 = 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	765.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:	$6.10855575463$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.5
Root			$-0.854638 + 0.854638i$ of defining polynomial
Character	$\chi$	$=$	765.271
Dual form			765.2.g.b.271.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.17009 q^{2} +2.70928 q^{4} -1.00000i q^{5} -4.87936i q^{7} +1.53919 q^{8} -2.17009i q^{10} +3.17009i q^{11} +2.63090 q^{13} -10.5886i q^{14} -2.07838 q^{16} +(3.24846 - 2.53919i) q^{17} +1.07838 q^{19} -2.70928i q^{20} +6.87936i q^{22} -5.21953i q^{23} -1.00000 q^{25} +5.70928 q^{26} -13.2195i q^{28} +2.92162i q^{29} +4.09171i q^{31} -7.58864 q^{32} +(7.04945 - 5.51026i) q^{34} -4.87936 q^{35} +5.26180i q^{37} +2.34017 q^{38} -1.53919i q^{40} +5.60197i q^{41} +3.36910 q^{43} +8.58864i q^{44} -11.3268i q^{46} +6.78765 q^{47} -16.8082 q^{49} -2.17009 q^{50} +7.12783 q^{52} -3.75872 q^{53} +3.17009 q^{55} -7.51026i q^{56} +6.34017i q^{58} +2.34017 q^{59} +12.2557i q^{61} +8.87936i q^{62} -12.3112 q^{64} -2.63090i q^{65} +10.2062 q^{67} +(8.80098 - 6.87936i) q^{68} -10.5886 q^{70} -4.06505i q^{71} +11.0784i q^{73} +11.4186i q^{74} +2.92162 q^{76} +15.4680 q^{77} +6.92881i q^{79} +2.07838i q^{80} +12.1568i q^{82} -8.23287 q^{83} +(-2.53919 - 3.24846i) q^{85} +7.31124 q^{86} +4.87936i q^{88} -7.15449 q^{89} -12.8371i q^{91} -14.1412i q^{92} +14.7298 q^{94} -1.07838i q^{95} -8.18342i q^{97} -36.4752 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 8 q^{13} - 6 q^{16} + 2 q^{17} - 6 q^{25} + 20 q^{26} - 6 q^{32} + 6 q^{34} - 4 q^{35} - 8 q^{38} + 28 q^{43} + 20 q^{47} - 14 q^{49} - 2 q^{50} + 28 q^{53} + 8 q^{55}+ \cdots - 86 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 + 8 * q^13 - 6 * q^16 + 2 * q^17 - 6 * q^25 + 20 * q^26 - 6 * q^32 + 6 * q^34 - 4 * q^35 - 8 * q^38 + 28 * q^43 + 20 * q^47 - 14 * q^49 - 2 * q^50 + 28 * q^53 + 8 * q^55 - 8 * q^59 - 22 * q^64 + 12 * q^67 + 34 * q^68 - 24 * q^70 + 24 * q^76 + 28 * q^77 - 4 * q^83 - 12 * q^85 - 8 * q^86 - 4 * q^89 + 8 * q^94 - 86 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/765\mathbb{Z}\right)^\times$.

$n$	$307$	$496$	$596$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	2.17009			1.53448			0.767241	−	0.641358i	$-0.221629\pi$
$2$	2.17009			1.53448			0.767241	+	0.641358i	$0.221629\pi$
$3$		0			0
$3$		0			0
$4$	2.70928			1.35464
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	4.87936i		−	1.84423i	−0.386921	−	0.922113i	$-0.626462\pi$
$7$		−	4.87936i		−	1.84423i	0.386921	−	0.922113i	$-0.373538\pi$
$8$	1.53919			0.544185
$9$		0			0
$10$		−	2.17009i		−	0.686242i
$11$			3.17009i			0.955817i	0.878410	+	0.477909i	$0.158605\pi$
$11$			3.17009i			0.955817i	−0.878410	+	0.477909i	$0.841395\pi$
$12$		0			0
$13$	2.63090			0.729680			0.364840	−	0.931070i	$-0.381124\pi$
$13$	2.63090			0.729680			0.364840	+	0.931070i	$0.381124\pi$
$14$		−	10.5886i		−	2.82993i
$15$		0			0
$16$	−2.07838			−0.519594
$17$	3.24846	−	2.53919i	0.787868	−	0.615844i
$17$	3.24846	−	2.53919i	0.787868	−	0.615844i
$18$		0			0
$19$	1.07838			0.247397			0.123698	−	0.992320i	$-0.460524\pi$
$19$	1.07838			0.247397			0.123698	+	0.992320i	$0.460524\pi$
$20$		−	2.70928i		−	0.605812i
$21$		0			0
$22$			6.87936i			1.46668i
$23$		−	5.21953i		−	1.08835i	−0.838972	−	0.544174i	$-0.816843\pi$
$23$		−	5.21953i		−	1.08835i	0.838972	−	0.544174i	$-0.183157\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	5.70928			1.11968
$27$		0			0
$28$		−	13.2195i		−	2.49826i
$29$			2.92162i			0.542532i	0.962504	+	0.271266i	$0.0874422\pi$
$29$			2.92162i			0.542532i	−0.962504	+	0.271266i	$0.912558\pi$
$30$		0			0
$31$			4.09171i			0.734893i	0.930045	+	0.367446i	$0.119768\pi$
$31$			4.09171i			0.734893i	−0.930045	+	0.367446i	$0.880232\pi$
$32$	−7.58864			−1.34149
$33$		0			0
$34$	7.04945	−	5.51026i	1.20897	−	0.945002i
$35$	−4.87936			−0.824763
$36$		0			0
$37$			5.26180i			0.865034i	0.901626	+	0.432517i	$0.142374\pi$
$37$			5.26180i			0.865034i	−0.901626	+	0.432517i	$0.857626\pi$
$38$	2.34017			0.379626
$39$		0			0
$40$		−	1.53919i		−	0.243367i
$41$			5.60197i			0.874880i	0.899247	+	0.437440i	$0.144115\pi$
$41$			5.60197i			0.874880i	−0.899247	+	0.437440i	$0.855885\pi$
$42$		0			0
$43$	3.36910			0.513783			0.256892	−	0.966440i	$-0.417302\pi$
$43$	3.36910			0.513783			0.256892	+	0.966440i	$0.417302\pi$
$44$			8.58864i			1.29479i
$45$		0			0
$46$		−	11.3268i		−	1.67005i
$47$	6.78765			0.990081			0.495040	−	0.868870i	$-0.335153\pi$
$47$	6.78765			0.990081			0.495040	+	0.868870i	$0.335153\pi$
$48$		0			0
$49$	−16.8082			−2.40117
$50$	−2.17009			−0.306897
$51$		0			0
$52$	7.12783			0.988452
$53$	−3.75872			−0.516300			−0.258150	−	0.966105i	$-0.583113\pi$
$53$	−3.75872			−0.516300			−0.258150	+	0.966105i	$0.583113\pi$
$54$		0			0
$55$	3.17009			0.427454
$56$		−	7.51026i		−	1.00360i
$57$		0			0
$58$			6.34017i			0.832505i
$59$	2.34017			0.304665			0.152332	−	0.988329i	$-0.451322\pi$
$59$	2.34017			0.304665			0.152332	+	0.988329i	$0.451322\pi$
$60$		0			0
$61$			12.2557i			1.56918i	0.620018	+	0.784588i	$0.287125\pi$
$61$			12.2557i			1.56918i	−0.620018	+	0.784588i	$0.712875\pi$
$62$			8.87936i			1.12768i
$63$		0			0
$64$	−12.3112			−1.53891
$65$		−	2.63090i		−	0.326323i
$66$		0			0
$67$	10.2062			1.24689			0.623443	−	0.781869i	$-0.285733\pi$
$67$	10.2062			1.24689			0.623443	+	0.781869i	$0.285733\pi$
$68$	8.80098	−	6.87936i	1.06728	−	0.834245i
$69$		0			0
$70$	−10.5886			−1.26558
$71$		−	4.06505i		−	0.482432i	−0.970471	−	0.241216i	$-0.922454\pi$
$71$		−	4.06505i		−	0.482432i	0.970471	−	0.241216i	$-0.0775463\pi$
$72$		0			0
$73$			11.0784i			1.29663i	0.761374	+	0.648313i	$0.224525\pi$
$73$			11.0784i			1.29663i	−0.761374	+	0.648313i	$0.775475\pi$
$74$			11.4186i			1.32738i
$75$		0			0
$76$	2.92162			0.335133
$77$	15.4680			1.76274
$78$		0			0
$79$			6.92881i			0.779552i	0.920910	+	0.389776i	$0.127448\pi$
$79$			6.92881i			0.779552i	−0.920910	+	0.389776i	$0.872552\pi$
$80$			2.07838i			0.232370i
$81$		0			0
$82$			12.1568i			1.34249i
$83$	−8.23287			−0.903674			−0.451837	−	0.892100i	$-0.649231\pi$
$83$	−8.23287			−0.903674			−0.451837	+	0.892100i	$0.649231\pi$
$84$		0			0
$85$	−2.53919	−	3.24846i	−0.275414	−	0.352345i
$86$	7.31124			0.788392
$87$		0			0
$88$			4.87936i			0.520142i
$89$	−7.15449			−0.758374			−0.379187	−	0.925320i	$-0.623796\pi$
$89$	−7.15449			−0.758374			−0.379187	+	0.925320i	$0.623796\pi$
$90$		0			0
$91$		−	12.8371i		−	1.34569i
$92$		−	14.1412i		−	1.47432i
$93$		0			0
$94$	14.7298			1.51926
$95$		−	1.07838i		−	0.110639i
$96$		0			0
$97$		−	8.18342i		−	0.830900i	−0.909616	−	0.415450i	$-0.863624\pi$
$97$		−	8.18342i		−	0.830900i	0.909616	−	0.415450i	$-0.136376\pi$
$98$	−36.4752			−3.68455
$99$		0			0
$100$	−2.70928			−0.270928
$101$	2.47414			0.246186			0.123093	−	0.992395i	$-0.460719\pi$
$101$	2.47414			0.246186			0.123093	+	0.992395i	$0.460719\pi$
$102$		0			0
$103$	−19.6514			−1.93631			−0.968156	−	0.250348i	$-0.919455\pi$
$103$	−19.6514			−1.93631			−0.968156	+	0.250348i	$0.919455\pi$
$104$	4.04945			0.397081
$105$		0			0
$106$	−8.15676			−0.792254
$107$			12.6381i			1.22177i	0.791719	+	0.610885i	$0.209186\pi$
$107$			12.6381i			1.22177i	−0.791719	+	0.610885i	$0.790814\pi$
$108$		0			0
$109$		−	8.15676i		−	0.781275i	−0.920544	−	0.390638i	$-0.872255\pi$
$109$		−	8.15676i		−	0.781275i	0.920544	−	0.390638i	$-0.127745\pi$
$110$	6.87936			0.655921
$111$		0			0
$112$			10.1412i			0.958249i
$113$		−	17.0205i		−	1.60116i	−0.599229	−	0.800578i	$-0.704526\pi$
$113$		−	17.0205i		−	1.60116i	0.599229	−	0.800578i	$-0.295474\pi$
$114$		0			0
$115$	−5.21953			−0.486724
$116$			7.91548i			0.734934i
$117$		0			0
$118$	5.07838			0.467503
$119$	−12.3896	−	15.8504i	−1.13575	−	1.45301i
$120$		0			0
$121$	0.950552			0.0864138
$122$			26.5958i			2.40787i
$123$		0			0
$124$			11.0856i			0.995513i
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	−8.04945			−0.714273			−0.357137	−	0.934052i	$-0.616247\pi$
$127$	−8.04945			−0.714273			−0.357137	+	0.934052i	$0.616247\pi$
$128$	−11.5392			−1.01993
$129$		0			0
$130$		−	5.70928i		−	0.500737i
$131$		−	11.6937i		−	1.02168i	−0.859675	−	0.510841i	$-0.829334\pi$
$131$		−	11.6937i		−	1.02168i	0.859675	−	0.510841i	$-0.170666\pi$
$132$		0			0
$133$		−	5.26180i		−	0.456256i
$134$	22.1483			1.91333
$135$		0			0
$136$	5.00000	−	3.90829i	0.428746	−	0.335133i
$137$	−1.95055			−0.166647			−0.0833234	−	0.996523i	$-0.526553\pi$
$137$	−1.95055			−0.166647			−0.0833234	+	0.996523i	$0.526553\pi$
$138$		0			0
$139$		−	2.00719i		−	0.170247i	−0.996370	−	0.0851237i	$-0.972871\pi$
$139$		−	2.00719i		−	0.170247i	0.996370	−	0.0851237i	$-0.0271286\pi$
$140$	−13.2195			−1.11725
$141$		0			0
$142$		−	8.82150i		−	0.740284i
$143$			8.34017i			0.697440i
$144$		0			0
$145$	2.92162			0.242628
$146$			24.0410i			1.98965i
$147$		0			0
$148$			14.2557i			1.17181i
$149$	14.2823			1.17005			0.585026	−	0.811014i	$-0.301084\pi$
$149$	14.2823			1.17005			0.585026	+	0.811014i	$0.301084\pi$
$150$		0			0
$151$	12.8638			1.04684			0.523419	−	0.852075i	$-0.324656\pi$
$151$	12.8638			1.04684			0.523419	+	0.852075i	$0.324656\pi$
$152$	1.65983			0.134630
$153$		0			0
$154$	33.5669			2.70490
$155$	4.09171			0.328654
$156$		0			0
$157$	3.75872			0.299979			0.149989	−	0.988688i	$-0.452076\pi$
$157$	3.75872			0.299979			0.149989	+	0.988688i	$0.452076\pi$
$158$			15.0361i			1.19621i
$159$		0			0
$160$			7.58864i			0.599934i
$161$	−25.4680			−2.00716
$162$		0			0
$163$		−	8.69594i		−	0.681119i	−0.940223	−	0.340559i	$-0.889384\pi$
$163$		−	8.69594i		−	0.681119i	0.940223	−	0.340559i	$-0.110616\pi$
$164$			15.1773i			1.18515i
$165$		0			0
$166$	−17.8660			−1.38667
$167$		−	1.37629i		−	0.106501i	−0.998581	−	0.0532503i	$-0.983042\pi$
$167$		−	1.37629i		−	0.106501i	0.998581	−	0.0532503i	$-0.0169581\pi$
$168$		0			0
$169$	−6.07838			−0.467568
$170$	−5.51026	−	7.04945i	−0.422618	−	0.540668i
$171$		0			0
$172$	9.12783			0.695990
$173$			17.3607i			1.31991i	0.751306	+	0.659954i	$0.229424\pi$
$173$			17.3607i			1.31991i	−0.751306	+	0.659954i	$0.770576\pi$
$174$		0			0
$175$			4.87936i			0.368845i
$176$		−	6.58864i		−	0.496637i
$177$		0			0
$178$	−15.5259			−1.16371
$179$	−6.83710			−0.511029			−0.255514	−	0.966805i	$-0.582245\pi$
$179$	−6.83710			−0.511029			−0.255514	+	0.966805i	$0.582245\pi$
$180$		0			0
$181$			15.0205i			1.11647i	0.829684	+	0.558233i	$0.188521\pi$
$181$			15.0205i			1.11647i	−0.829684	+	0.558233i	$0.811479\pi$
$182$		−	27.8576i		−	2.06494i
$183$		0			0
$184$		−	8.03385i		−	0.592263i
$185$	5.26180			0.386855
$186$		0			0
$187$	8.04945	+	10.2979i	0.588634	+	0.753058i
$188$	18.3896			1.34120
$189$		0			0
$190$		−	2.34017i		−	0.169774i
$191$	24.2823			1.75701			0.878503	−	0.477736i	$-0.158543\pi$
$191$	24.2823			1.75701			0.878503	+	0.477736i	$0.158543\pi$
$192$		0			0
$193$		−	11.8576i		−	0.853530i	−0.904363	−	0.426765i	$-0.859653\pi$
$193$		−	11.8576i		−	0.853530i	0.904363	−	0.426765i	$-0.140347\pi$
$194$		−	17.7587i		−	1.27500i
$195$		0			0
$196$	−45.5380			−3.25271
$197$			18.2557i			1.30066i	0.759651	+	0.650331i	$0.225370\pi$
$197$			18.2557i			1.30066i	−0.759651	+	0.650331i	$0.774630\pi$
$198$		0			0
$199$			3.72487i			0.264049i	0.991246	+	0.132025i	$0.0421478\pi$
$199$			3.72487i			0.264049i	−0.991246	+	0.132025i	$0.957852\pi$
$200$	−1.53919			−0.108837
$201$		0			0
$202$	5.36910			0.377769
$203$	14.2557			1.00055
$204$		0			0
$205$	5.60197			0.391258
$206$	−42.6453			−2.97124
$207$		0			0
$208$	−5.46800			−0.379138
$209$			3.41855i			0.236466i
$210$		0			0
$211$		−	22.2485i		−	1.53165i	−0.643051	−	0.765824i	$-0.722332\pi$
$211$		−	22.2485i		−	1.53165i	0.643051	−	0.765824i	$-0.277668\pi$
$212$	−10.1834			−0.699400
$213$		0			0
$214$			27.4257i			1.87478i
$215$		−	3.36910i		−	0.229771i
$216$		0			0
$217$	19.9649			1.35531
$218$		−	17.7009i		−	1.19885i
$219$		0			0
$220$	8.58864			0.579046
$221$	8.54638	−	6.68035i	0.574892	−	0.449369i
$222$		0			0
$223$	−2.19183			−0.146776			−0.0733878	−	0.997303i	$-0.523381\pi$
$223$	−2.19183			−0.146776			−0.0733878	+	0.997303i	$0.523381\pi$
$224$			37.0277i			2.47402i
$225$		0			0
$226$		−	36.9360i		−	2.45695i
$227$		−	9.55971i		−	0.634500i	−0.948342	−	0.317250i	$-0.897241\pi$
$227$		−	9.55971i		−	0.634500i	0.948342	−	0.317250i	$-0.102759\pi$
$228$		0			0
$229$	7.36910			0.486964			0.243482	−	0.969905i	$-0.421710\pi$
$229$	7.36910			0.486964			0.243482	+	0.969905i	$0.421710\pi$
$230$	−11.3268			−0.746870
$231$		0			0
$232$			4.49693i			0.295238i
$233$		−	9.44521i		−	0.618776i	−0.950936	−	0.309388i	$-0.899876\pi$
$233$		−	9.44521i		−	0.618776i	0.950936	−	0.309388i	$-0.100124\pi$
$234$		0			0
$235$		−	6.78765i		−	0.442778i
$236$	6.34017			0.412710
$237$		0			0
$238$	−26.8865	−	34.3968i	−1.74280	−	2.22961i
$239$	−6.25565			−0.404644			−0.202322	−	0.979319i	$-0.564849\pi$
$239$	−6.25565			−0.404644			−0.202322	+	0.979319i	$0.564849\pi$
$240$		0			0
$241$		−	2.49693i		−	0.160841i	−0.996761	−	0.0804207i	$-0.974374\pi$
$241$		−	2.49693i		−	0.160841i	0.996761	−	0.0804207i	$-0.0256264\pi$
$242$	2.06278			0.132600
$243$		0			0
$244$			33.2039i			2.12566i
$245$			16.8082i			1.07383i
$246$		0			0
$247$	2.83710			0.180520
$248$			6.29791i			0.399918i
$249$		0			0
$250$			2.17009i			0.137248i
$251$	11.8166			0.745856			0.372928	−	0.927860i	$-0.378354\pi$
$251$	11.8166			0.745856			0.372928	+	0.927860i	$0.378354\pi$
$252$		0			0
$253$	16.5464			1.04026
$254$	−17.4680			−1.09604
$255$		0			0
$256$	−0.418551			−0.0261594
$257$	−14.9444			−0.932207			−0.466103	−	0.884730i	$-0.654342\pi$
$257$	−14.9444			−0.932207			−0.466103	+	0.884730i	$0.654342\pi$
$258$		0			0
$259$	25.6742			1.59532
$260$		−	7.12783i		−	0.442049i
$261$		0			0
$262$		−	25.3763i		−	1.56775i
$263$	12.9444			0.798186			0.399093	−	0.916910i	$-0.369325\pi$
$263$	12.9444			0.798186			0.399093	+	0.916910i	$0.369325\pi$
$264$		0			0
$265$			3.75872i			0.230897i
$266$		−	11.4186i		−	0.700116i
$267$		0			0
$268$	27.6514			1.68908
$269$			7.47641i			0.455845i	0.973679	+	0.227922i	$0.0731932\pi$
$269$			7.47641i			0.455845i	−0.973679	+	0.227922i	$0.926807\pi$
$270$		0			0
$271$	−2.15676			−0.131014			−0.0655068	−	0.997852i	$-0.520866\pi$
$271$	−2.15676			−0.131014			−0.0655068	+	0.997852i	$0.520866\pi$
$272$	−6.75154	+	5.27739i	−0.409372	+	0.319989i
$273$		0			0
$274$	−4.23287			−0.255717
$275$		−	3.17009i		−	0.191163i
$276$		0			0
$277$			12.1568i			0.730429i	0.930923	+	0.365214i	$0.119004\pi$
$277$			12.1568i			0.730429i	−0.930923	+	0.365214i	$0.880996\pi$
$278$		−	4.35577i		−	0.261242i
$279$		0			0
$280$	−7.51026			−0.448824
$281$	−13.1194			−0.782639			−0.391319	−	0.920255i	$-0.627981\pi$
$281$	−13.1194			−0.782639			−0.391319	+	0.920255i	$0.627981\pi$
$282$		0			0
$283$		−	13.9577i		−	0.829701i	−0.909889	−	0.414851i	$-0.863834\pi$
$283$		−	13.9577i		−	0.829701i	0.909889	−	0.414851i	$-0.136166\pi$
$284$		−	11.0133i		−	0.653521i
$285$		0			0
$286$			18.0989i			1.07021i
$287$	27.3340			1.61348
$288$		0			0
$289$	4.10504	−	16.4969i	0.241473	−	0.970408i
$290$	6.34017			0.372308
$291$		0			0
$292$			30.0144i			1.75646i
$293$	−4.73820			−0.276809			−0.138404	−	0.990376i	$-0.544197\pi$
$293$	−4.73820			−0.276809			−0.138404	+	0.990376i	$0.544197\pi$
$294$		0			0
$295$		−	2.34017i		−	0.136250i
$296$			8.09890i			0.470739i
$297$		0			0
$298$	30.9939			1.79543
$299$		−	13.7321i		−	0.794146i
$300$		0			0
$301$		−	16.4391i		−	0.947532i
$302$	27.9155			1.60636
$303$		0			0
$304$	−2.24128			−0.128546
$305$	12.2557			0.701757
$306$		0			0
$307$	−21.5936			−1.23241			−0.616205	−	0.787586i	$-0.711331\pi$
$307$	−21.5936			−1.23241			−0.616205	+	0.787586i	$0.711331\pi$
$308$	41.9071			2.38788
$309$		0			0
$310$	8.87936			0.504314
$311$		−	24.2628i		−	1.37582i	−0.725796	−	0.687910i	$-0.758528\pi$
$311$		−	24.2628i		−	1.37582i	0.725796	−	0.687910i	$-0.241472\pi$
$312$		0			0
$313$			4.07223i			0.230176i	0.993355	+	0.115088i	$0.0367151\pi$
$313$			4.07223i			0.230176i	−0.993355	+	0.115088i	$0.963285\pi$
$314$	8.15676			0.460312
$315$		0			0
$316$			18.7721i			1.05601i
$317$		−	8.05786i		−	0.452574i	−0.974061	−	0.226287i	$-0.927341\pi$
$317$		−	8.05786i		−	0.452574i	0.974061	−	0.226287i	$-0.0726588\pi$
$318$		0			0
$319$	−9.26180			−0.518561
$320$			12.3112i			0.688219i
$321$		0			0
$322$	−55.2678			−3.07995
$323$	3.50307	−	2.73820i	0.194916	−	0.152358i
$324$		0			0
$325$	−2.63090			−0.145936
$326$		−	18.8710i		−	1.04517i
$327$		0			0
$328$			8.62249i			0.476097i
$329$		−	33.1194i		−	1.82593i
$330$		0			0
$331$	−10.0722			−0.553620			−0.276810	−	0.960925i	$-0.589277\pi$
$331$	−10.0722			−0.553620			−0.276810	+	0.960925i	$0.589277\pi$
$332$	−22.3051			−1.22415
$333$		0			0
$334$		−	2.98667i		−	0.163423i
$335$		−	10.2062i		−	0.557624i
$336$		0			0
$337$		−	11.2351i		−	0.612017i	−0.952029	−	0.306008i	$-0.901006\pi$
$337$		−	11.2351i		−	0.612017i	0.952029	−	0.306008i	$-0.0989936\pi$
$338$	−13.1906			−0.717474
$339$		0			0
$340$	−6.87936	−	8.80098i	−0.373086	−	0.477300i
$341$	−12.9711			−0.702423
$342$		0			0
$343$			47.8576i			2.58407i
$344$	5.18568			0.279593
$345$		0			0
$346$			37.6742i			2.02538i
$347$			8.74927i			0.469685i	0.972033	+	0.234843i	$0.0754575\pi$
$347$			8.74927i			0.469685i	−0.972033	+	0.234843i	$0.924543\pi$
$348$		0			0
$349$	−26.9093			−1.44042			−0.720212	−	0.693754i	$-0.755955\pi$
$349$	−26.9093			−1.44042			−0.720212	+	0.693754i	$0.755955\pi$
$350$			10.5886i			0.565986i
$351$		0			0
$352$		−	24.0566i		−	1.28222i
$353$	−18.3135			−0.974730			−0.487365	−	0.873198i	$-0.662042\pi$
$353$	−18.3135			−0.974730			−0.487365	+	0.873198i	$0.662042\pi$
$354$		0			0
$355$	−4.06505			−0.215750
$356$	−19.3835			−1.02732
$357$		0			0
$358$	−14.8371			−0.784165
$359$	9.57531			0.505365			0.252683	−	0.967549i	$-0.418687\pi$
$359$	9.57531			0.505365			0.252683	+	0.967549i	$0.418687\pi$
$360$		0			0
$361$	−17.8371			−0.938795
$362$			32.5958i			1.71320i
$363$		0			0
$364$		−	34.7792i		−	1.82293i
$365$	11.0784			0.579869
$366$		0			0
$367$			12.1145i			0.632371i	0.948697	+	0.316186i	$0.102402\pi$
$367$			12.1145i			0.632371i	−0.948697	+	0.316186i	$0.897598\pi$
$368$			10.8482i			0.565500i
$369$		0			0
$370$	11.4186			0.593622
$371$			18.3402i			0.952174i
$372$		0			0
$373$	30.4619			1.57726			0.788628	−	0.614871i	$-0.210792\pi$
$373$	30.4619			1.57726			0.788628	+	0.614871i	$0.210792\pi$
$374$	17.4680	+	22.3474i	0.903249	+	1.15555i
$375$		0			0
$376$	10.4475			0.538788
$377$			7.68649i			0.395874i
$378$		0			0
$379$		−	0.986669i		−	0.0506818i	−0.999679	−	0.0253409i	$-0.991933\pi$
$379$		−	0.986669i		−	0.0506818i	0.999679	−	0.0253409i	$-0.00806712\pi$
$380$		−	2.92162i		−	0.149876i
$381$		0			0
$382$	52.6947			2.69610
$383$	24.9588			1.27533			0.637667	−	0.770312i	$-0.279900\pi$
$383$	24.9588			1.27533			0.637667	+	0.770312i	$0.279900\pi$
$384$		0			0
$385$		−	15.4680i		−	0.788322i
$386$		−	25.7321i		−	1.30973i
$387$		0			0
$388$		−	22.1711i		−	1.12557i
$389$	−33.8082			−1.71414			−0.857071	−	0.515198i	$-0.827718\pi$
$389$	−33.8082			−1.71414			−0.857071	+	0.515198i	$0.827718\pi$
$390$		0			0
$391$	−13.2534	−	16.9555i	−0.670252	−	0.857475i
$392$	−25.8710			−1.30668
$393$		0			0
$394$			39.6163i			1.99584i
$395$	6.92881			0.348626
$396$		0			0
$397$			28.5236i			1.43156i	0.698327	+	0.715779i	$0.253928\pi$
$397$			28.5236i			1.43156i	−0.698327	+	0.715779i	$0.746072\pi$
$398$			8.08330i			0.405179i
$399$		0			0
$400$	2.07838			0.103919
$401$			33.0928i			1.65257i	0.563250	+	0.826287i	$0.309551\pi$
$401$			33.0928i			1.65257i	−0.563250	+	0.826287i	$0.690449\pi$
$402$		0			0
$403$			10.7649i			0.536236i
$404$	6.70313			0.333493
$405$		0			0
$406$	30.9360			1.53533
$407$	−16.6803			−0.826814
$408$		0			0
$409$	−30.1978			−1.49318			−0.746592	−	0.665282i	$-0.768311\pi$
$409$	−30.1978			−1.49318			−0.746592	+	0.665282i	$0.768311\pi$
$410$	12.1568			0.600379
$411$		0			0
$412$	−53.2411			−2.62300
$413$		−	11.4186i		−	0.561870i
$414$		0			0
$415$			8.23287i			0.404135i
$416$	−19.9649			−0.978861
$417$		0			0
$418$			7.41855i			0.362853i
$419$		−	18.1639i		−	0.887367i	−0.896184	−	0.443683i	$-0.853671\pi$
$419$		−	18.1639i		−	0.887367i	0.896184	−	0.443683i	$-0.146329\pi$
$420$		0			0
$421$	0.760991			0.0370884			0.0185442	−	0.999828i	$-0.494097\pi$
$421$	0.760991			0.0370884			0.0185442	+	0.999828i	$0.494097\pi$
$422$		−	48.2811i		−	2.35029i
$423$		0			0
$424$	−5.78539			−0.280963
$425$	−3.24846	+	2.53919i	−0.157574	+	0.123169i
$426$		0			0
$427$	59.7998			2.89391
$428$			34.2401i			1.65506i
$429$		0			0
$430$		−	7.31124i		−	0.352579i
$431$		−	6.34736i		−	0.305742i	−0.988246	−	0.152871i	$-0.951148\pi$
$431$		−	6.34736i		−	0.305742i	0.988246	−	0.152871i	$-0.0488518\pi$
$432$		0			0
$433$	3.62475			0.174195			0.0870973	−	0.996200i	$-0.472241\pi$
$433$	3.62475			0.174195			0.0870973	+	0.996200i	$0.472241\pi$
$434$	43.3256			2.07970
$435$		0			0
$436$		−	22.0989i		−	1.05835i
$437$		−	5.62863i		−	0.269254i
$438$		0			0
$439$		−	6.40522i		−	0.305704i	−0.988249	−	0.152852i	$-0.951154\pi$
$439$		−	6.40522i		−	0.305704i	0.988249	−	0.152852i	$-0.0488459\pi$
$440$	4.87936			0.232614
$441$		0			0
$442$	18.5464	−	14.4969i	0.882161	−	0.689549i
$443$	−27.4824			−1.30573			−0.652864	−	0.757476i	$-0.726432\pi$
$443$	−27.4824			−1.30573			−0.652864	+	0.757476i	$0.726432\pi$
$444$		0			0
$445$			7.15449i			0.339155i
$446$	−4.75646			−0.225225
$447$		0			0
$448$			60.0710i			2.83809i
$449$			28.7526i			1.35692i	0.734638	+	0.678459i	$0.237352\pi$
$449$			28.7526i			1.35692i	−0.734638	+	0.678459i	$0.762648\pi$
$450$		0			0
$451$	−17.7587			−0.836226
$452$		−	46.1133i		−	2.16899i
$453$		0			0
$454$		−	20.7454i		−	0.973630i
$455$	−12.8371			−0.601813
$456$		0			0
$457$	−31.4101			−1.46930			−0.734652	−	0.678444i	$-0.762655\pi$
$457$	−31.4101			−1.46930			−0.734652	+	0.678444i	$0.762655\pi$
$458$	15.9916			0.747238
$459$		0			0
$460$	−14.1412			−0.659335
$461$	7.75872			0.361360			0.180680	−	0.983542i	$-0.442170\pi$
$461$	7.75872			0.361360			0.180680	+	0.983542i	$0.442170\pi$
$462$		0			0
$463$	24.2329			1.12620			0.563098	−	0.826390i	$-0.309609\pi$
$463$	24.2329			1.12620			0.563098	+	0.826390i	$0.309609\pi$
$464$		−	6.07223i		−	0.281896i
$465$		0			0
$466$		−	20.4969i		−	0.949502i
$467$	12.3174			0.569981			0.284990	−	0.958530i	$-0.408010\pi$
$467$	12.3174			0.569981			0.284990	+	0.958530i	$0.408010\pi$
$468$		0			0
$469$		−	49.7998i		−	2.29954i
$470$		−	14.7298i		−	0.679435i
$471$		0			0
$472$	3.60197			0.165794
$473$			10.6803i			0.491083i
$474$		0			0
$475$	−1.07838			−0.0494794
$476$	−33.5669	−	42.9432i	−1.53854	−	1.96830i
$477$		0			0
$478$	−13.5753			−0.620920
$479$		−	2.14957i		−	0.0982162i	−0.998793	−	0.0491081i	$-0.984362\pi$
$479$		−	2.14957i		−	0.0982162i	0.998793	−	0.0491081i	$-0.0156379\pi$
$480$		0			0
$481$			13.8432i			0.631198i
$482$		−	5.41855i		−	0.246808i
$483$		0			0
$484$	2.57531			0.117059
$485$	−8.18342			−0.371590
$486$		0			0
$487$		−	40.0833i		−	1.81635i	−0.418593	−	0.908174i	$-0.637477\pi$
$487$		−	40.0833i		−	1.81635i	0.418593	−	0.908174i	$-0.362523\pi$
$488$			18.8638i			0.853922i
$489$		0			0
$490$			36.4752i			1.64778i
$491$	2.25565			0.101796			0.0508981	−	0.998704i	$-0.483792\pi$
$491$	2.25565			0.101796			0.0508981	+	0.998704i	$0.483792\pi$
$492$		0			0
$493$	7.41855	+	9.49079i	0.334115	+	0.427443i
$494$	6.15676			0.277006
$495$		0			0
$496$		−	8.50412i		−	0.381846i
$497$	−19.8348			−0.889714
$498$		0			0
$499$		−	42.5452i		−	1.90458i	−0.305190	−	0.952291i	$-0.598720\pi$
$499$		−	42.5452i		−	1.90458i	0.305190	−	0.952291i	$-0.401280\pi$
$500$			2.70928i			0.121162i
$501$		0			0
$502$	25.6430			1.14450
$503$			9.55971i			0.426246i	0.977025	+	0.213123i	$0.0683636\pi$
$503$			9.55971i			0.426246i	−0.977025	+	0.213123i	$0.931636\pi$
$504$		0			0
$505$		−	2.47414i		−	0.110098i
$506$	35.9071			1.59626
$507$		0			0
$508$	−21.8082			−0.967581
$509$	−28.3545			−1.25679			−0.628397	−	0.777893i	$-0.716288\pi$
$509$	−28.3545			−1.25679			−0.628397	+	0.777893i	$0.716288\pi$
$510$		0			0
$511$	54.0554			2.39127
$512$	22.1701			0.979789
$513$		0			0
$514$	−32.4307			−1.43046
$515$			19.6514i			0.865945i
$516$		0			0
$517$			21.5174i			0.946336i
$518$	55.7152			2.44799
$519$		0			0
$520$		−	4.04945i		−	0.177580i
$521$		−	15.1050i		−	0.661764i	−0.943672	−	0.330882i	$-0.892654\pi$
$521$		−	15.1050i		−	0.661764i	0.943672	−	0.330882i	$-0.107346\pi$
$522$		0			0
$523$	−10.8865			−0.476036			−0.238018	−	0.971261i	$-0.576498\pi$
$523$	−10.8865			−0.476036			−0.238018	+	0.971261i	$0.576498\pi$
$524$		−	31.6814i		−	1.38401i
$525$		0			0
$526$	28.0905			1.22480
$527$	10.3896	+	13.2918i	0.452579	+	0.578999i
$528$		0			0
$529$	−4.24354			−0.184502
$530$			8.15676i			0.354307i
$531$		0			0
$532$		−	14.2557i		−	0.618061i
$533$			14.7382i			0.638383i
$534$		0			0
$535$	12.6381			0.546392
$536$	15.7093			0.678537
$537$		0			0
$538$			16.2245i			0.699486i
$539$		−	53.2834i		−	2.29508i
$540$		0			0
$541$			6.86830i			0.295291i	0.989040	+	0.147646i	$0.0471695\pi$
$541$			6.86830i			0.295291i	−0.989040	+	0.147646i	$0.952830\pi$
$542$	−4.68035			−0.201038
$543$		0			0
$544$	−24.6514	+	19.2690i	−1.05692	+	0.826151i
$545$	−8.15676			−0.349397
$546$		0			0
$547$		−	5.89988i		−	0.252261i	−0.992014	−	0.126130i	$-0.959744\pi$
$547$		−	5.89988i		−	0.252261i	0.992014	−	0.126130i	$-0.0402558\pi$
$548$	−5.28458			−0.225746
$549$		0			0
$550$		−	6.87936i		−	0.293337i
$551$			3.15061i			0.134221i
$552$		0			0
$553$	33.8082			1.43767
$554$			26.3812i			1.12083i
$555$		0			0
$556$		−	5.43802i		−	0.230624i
$557$	17.7359			0.751496			0.375748	−	0.926722i	$-0.377386\pi$
$557$	17.7359			0.751496			0.375748	+	0.926722i	$0.377386\pi$
$558$		0			0
$559$	8.86376			0.374897
$560$	10.1412			0.428542
$561$		0			0
$562$	−28.4703			−1.20095
$563$	38.8020			1.63531			0.817655	−	0.575708i	$-0.195274\pi$
$563$	38.8020			1.63531			0.817655	+	0.575708i	$0.195274\pi$
$564$		0			0
$565$	−17.0205			−0.716059
$566$		−	30.2895i		−	1.27316i
$567$		0			0
$568$		−	6.25687i		−	0.262533i
$569$	−12.1568			−0.509638			−0.254819	−	0.966989i	$-0.582016\pi$
$569$	−12.1568			−0.509638			−0.254819	+	0.966989i	$0.582016\pi$
$570$		0			0
$571$		−	15.4569i		−	0.646853i	−0.946253	−	0.323426i	$-0.895165\pi$
$571$		−	15.4569i		−	0.646853i	0.946253	−	0.323426i	$-0.104835\pi$
$572$			22.5958i			0.944779i
$573$		0			0
$574$	59.3172			2.47585
$575$			5.21953i			0.217670i
$576$		0			0
$577$	−2.36296			−0.0983713			−0.0491856	−	0.998790i	$-0.515663\pi$
$577$	−2.36296			−0.0983713			−0.0491856	+	0.998790i	$0.515663\pi$
$578$	8.90829	−	35.7998i	0.370536	−	1.48907i
$579$		0			0
$580$	7.91548			0.328672
$581$			40.1711i			1.66658i
$582$		0			0
$583$		−	11.9155i		−	0.493489i
$584$			17.0517i			0.705605i
$585$		0			0
$586$	−10.2823			−0.424758
$587$	3.65142			0.150710			0.0753550	−	0.997157i	$-0.475991\pi$
$587$	3.65142			0.150710			0.0753550	+	0.997157i	$0.475991\pi$
$588$		0			0
$589$			4.41241i			0.181810i
$590$		−	5.07838i		−	0.209074i
$591$		0			0
$592$		−	10.9360i		−	0.449467i
$593$	1.38735			0.0569718			0.0284859	−	0.999594i	$-0.490931\pi$
$593$	1.38735			0.0569718			0.0284859	+	0.999594i	$0.490931\pi$
$594$		0			0
$595$	−15.8504	+	12.3896i	−0.649804	+	0.507925i
$596$	38.6947			1.58500
$597$		0			0
$598$		−	29.7998i		−	1.21860i
$599$	0.451356			0.0184419			0.00922095	−	0.999957i	$-0.497065\pi$
$599$	0.451356			0.0184419			0.00922095	+	0.999957i	$0.497065\pi$
$600$		0			0
$601$			22.1301i			0.902705i	0.892346	+	0.451353i	$0.149058\pi$
$601$			22.1301i			0.902705i	−0.892346	+	0.451353i	$0.850942\pi$
$602$		−	35.6742i		−	1.45397i
$603$		0			0
$604$	34.8515			1.41809
$605$		−	0.950552i		−	0.0386454i
$606$		0			0
$607$			10.2667i			0.416713i	0.978053	+	0.208357i	$0.0668114\pi$
$607$			10.2667i			0.416713i	−0.978053	+	0.208357i	$0.933189\pi$
$608$	−8.18342			−0.331881
$609$		0			0
$610$	26.5958			1.07683
$611$	17.8576			0.722442
$612$		0			0
$613$	−9.05172			−0.365595			−0.182798	−	0.983151i	$-0.558515\pi$
$613$	−9.05172			−0.365595			−0.182798	+	0.983151i	$0.558515\pi$
$614$	−46.8599			−1.89111
$615$		0			0
$616$	23.8082			0.959259
$617$		−	17.8166i		−	0.717269i	−0.933478	−	0.358634i	$-0.883243\pi$
$617$		−	17.8166i		−	0.717269i	0.933478	−	0.358634i	$-0.116757\pi$
$618$		0			0
$619$			38.3884i			1.54296i	0.636254	+	0.771480i	$0.280483\pi$
$619$			38.3884i			1.54296i	−0.636254	+	0.771480i	$0.719517\pi$
$620$	11.0856			0.445207
$621$		0			0
$622$		−	52.6525i		−	2.11117i
$623$			34.9093i			1.39861i
$624$		0			0
$625$	1.00000			0.0400000
$626$			8.83710i			0.353202i
$627$		0			0
$628$	10.1834			0.406363
$629$	13.3607	+	17.0928i	0.532726	+	0.681533i
$630$		0			0
$631$	−27.4863			−1.09421			−0.547105	−	0.837064i	$-0.684270\pi$
$631$	−27.4863			−1.09421			−0.547105	+	0.837064i	$0.684270\pi$
$632$			10.6647i			0.424221i
$633$		0			0
$634$		−	17.4863i		−	0.694468i
$635$			8.04945i			0.319433i
$636$		0			0
$637$	−44.2206			−1.75208
$638$	−20.0989			−0.795723
$639$		0			0
$640$			11.5392i			0.456126i
$641$		−	9.79976i		−	0.387067i	−0.981094	−	0.193534i	$-0.938005\pi$
$641$		−	9.79976i		−	0.387067i	0.981094	−	0.193534i	$-0.0619949\pi$
$642$		0			0
$643$			16.9372i			0.667939i	0.942584	+	0.333969i	$0.108388\pi$
$643$			16.9372i			0.667939i	−0.942584	+	0.333969i	$0.891612\pi$
$644$	−68.9998			−2.71897
$645$		0			0
$646$	7.60197	−	5.94214i	0.299095	−	0.233790i
$647$	2.98545			0.117370			0.0586850	−	0.998277i	$-0.481309\pi$
$647$	2.98545			0.117370			0.0586850	+	0.998277i	$0.481309\pi$
$648$		0			0
$649$			7.41855i			0.291204i
$650$	−5.70928			−0.223936
$651$		0			0
$652$		−	23.5597i		−	0.922669i
$653$		−	40.1978i		−	1.57306i	−0.617551	−	0.786531i	$-0.711875\pi$
$653$		−	40.1978i		−	1.57306i	0.617551	−	0.786531i	$-0.288125\pi$
$654$		0			0
$655$	−11.6937			−0.456910
$656$		−	11.6430i		−	0.454583i
$657$		0			0
$658$		−	71.8720i		−	2.80186i
$659$	−43.9832			−1.71334			−0.856671	−	0.515864i	$-0.827471\pi$
$659$	−43.9832			−1.71334			−0.856671	+	0.515864i	$0.827471\pi$
$660$		0			0
$661$	26.1133			1.01569			0.507844	−	0.861449i	$-0.330443\pi$
$661$	26.1133			1.01569			0.507844	+	0.861449i	$0.330443\pi$
$662$	−21.8576			−0.849521
$663$		0			0
$664$	−12.6719			−0.491766
$665$	−5.26180			−0.204044
$666$		0			0
$667$	15.2495			0.590463
$668$		−	3.72875i		−	0.144270i
$669$		0			0
$670$		−	22.1483i		−	0.855665i
$671$	−38.8515			−1.49984
$672$		0			0
$673$		−	13.3340i		−	0.513989i	−0.966413	−	0.256995i	$-0.917268\pi$
$673$		−	13.3340i		−	0.513989i	0.966413	−	0.256995i	$-0.0827322\pi$
$674$		−	24.3812i		−	0.939129i
$675$		0			0
$676$	−16.4680			−0.633385
$677$			26.5113i			1.01891i	0.860497	+	0.509456i	$0.170153\pi$
$677$			26.5113i			1.01891i	−0.860497	+	0.509456i	$0.829847\pi$
$678$		0			0
$679$	−39.9299			−1.53237
$680$	−3.90829	−	5.00000i	−0.149876	−	0.191741i
$681$		0			0
$682$	−28.1483			−1.07786
$683$		−	5.71646i		−	0.218734i	−0.994001	−	0.109367i	$-0.965118\pi$
$683$		−	5.71646i		−	0.218734i	0.994001	−	0.109367i	$-0.0348824\pi$
$684$		0			0
$685$			1.95055i			0.0745267i
$686$			103.855i			3.96521i
$687$		0			0
$688$	−7.00227			−0.266959
$689$	−9.88882			−0.376734
$690$		0			0
$691$		−	43.8504i		−	1.66815i	−0.551652	−	0.834075i	$-0.686002\pi$
$691$		−	43.8504i		−	1.66815i	0.551652	−	0.834075i	$-0.313998\pi$
$692$			47.0349i			1.78800i
$693$		0			0
$694$			18.9867i			0.720724i
$695$	−2.00719			−0.0761370
$696$		0			0
$697$	14.2245	+	18.1978i	0.538790	+	0.689291i
$698$	−58.3956			−2.21031
$699$		0			0
$700$			13.2195i			0.499651i
$701$	−0.0806452			−0.00304593			−0.00152296	−	0.999999i	$-0.500485\pi$
$701$	−0.0806452			−0.00304593			−0.00152296	+	0.999999i	$0.500485\pi$
$702$		0			0
$703$			5.67420i			0.214007i
$704$		−	39.0277i		−	1.47091i
$705$		0			0
$706$	−39.7419			−1.49571
$707$		−	12.0722i		−	0.454023i
$708$		0			0
$709$		−	10.8227i		−	0.406456i	−0.979131	−	0.203228i	$-0.934857\pi$
$709$		−	10.8227i		−	0.406456i	0.979131	−	0.203228i	$-0.0651433\pi$
$710$	−8.82150			−0.331065
$711$		0			0
$712$	−11.0121			−0.412696
$713$	21.3568			0.799819
$714$		0			0
$715$	8.34017			0.311905
$716$	−18.5236			−0.692259
$717$		0			0
$718$	20.7792			0.775474
$719$			43.7659i			1.63219i	0.577916	+	0.816097i	$0.303866\pi$
$719$			43.7659i			1.63219i	−0.577916	+	0.816097i	$0.696134\pi$
$720$		0			0
$721$			95.8864i			3.57100i
$722$	−38.7081			−1.44056
$723$		0			0
$724$			40.6947i			1.51241i
$725$		−	2.92162i		−	0.108506i
$726$		0			0
$727$	−3.59809			−0.133446			−0.0667229	−	0.997772i	$-0.521254\pi$
$727$	−3.59809			−0.133446			−0.0667229	+	0.997772i	$0.521254\pi$
$728$		−	19.7587i		−	0.732307i
$729$		0			0
$730$	24.0410			0.889799
$731$	10.9444	−	8.55479i	0.404794	−	0.316410i
$732$		0			0
$733$	−39.8264			−1.47102			−0.735511	−	0.677513i	$-0.763058\pi$
$733$	−39.8264			−1.47102			−0.735511	+	0.677513i	$0.763058\pi$
$734$			26.2895i			0.970363i
$735$		0			0
$736$			39.6092i			1.46001i
$737$			32.3545i			1.19180i
$738$		0			0
$739$	13.7587			0.506123			0.253061	−	0.967450i	$-0.418563\pi$
$739$	13.7587			0.506123			0.253061	+	0.967450i	$0.418563\pi$
$740$	14.2557			0.524048
$741$		0			0
$742$			39.7998i			1.46110i
$743$			9.34963i			0.343005i	0.985184	+	0.171502i	$0.0548621\pi$
$743$			9.34963i			0.343005i	−0.985184	+	0.171502i	$0.945138\pi$
$744$		0			0
$745$		−	14.2823i		−	0.523264i
$746$	66.1049			2.42027
$747$		0			0
$748$	21.8082	+	27.8999i	0.797386	+	1.02012i
$749$	61.6658			2.25322
$750$		0			0
$751$			53.7392i			1.96097i	0.196586	+	0.980487i	$0.437014\pi$
$751$			53.7392i			1.96097i	−0.196586	+	0.980487i	$0.562986\pi$
$752$	−14.1073			−0.514441
$753$		0			0
$754$			16.6803i			0.607462i
$755$		−	12.8638i		−	0.468160i
$756$		0			0
$757$	−41.5136			−1.50884			−0.754418	−	0.656394i	$-0.772081\pi$
$757$	−41.5136			−1.50884			−0.754418	+	0.656394i	$0.772081\pi$
$758$		−	2.14116i		−	0.0777703i
$759$		0			0
$760$		−	1.65983i		−	0.0602083i
$761$	18.0372			0.653847			0.326923	−	0.945051i	$-0.393988\pi$
$761$	18.0372			0.653847			0.326923	+	0.945051i	$0.393988\pi$
$762$		0			0
$763$	−39.7998			−1.44085
$764$	65.7875			2.38011
$765$		0			0
$766$	54.1627			1.95698
$767$	6.15676			0.222308
$768$		0			0
$769$	−23.0843			−0.832443			−0.416221	−	0.909263i	$-0.636646\pi$
$769$	−23.0843			−0.832443			−0.416221	+	0.909263i	$0.636646\pi$
$770$		−	33.5669i		−	1.20967i
$771$		0			0
$772$		−	32.1256i		−	1.15622i
$773$	27.4101			0.985874			0.492937	−	0.870065i	$-0.335923\pi$
$773$	27.4101			0.985874			0.492937	+	0.870065i	$0.335923\pi$
$774$		0			0
$775$		−	4.09171i		−	0.146979i
$776$		−	12.5958i		−	0.452164i
$777$		0			0
$778$	−73.3667			−2.63032
$779$			6.04104i			0.216443i
$780$		0			0
$781$	12.8865			0.461117
$782$	−28.7610	−	36.7948i	−1.02849	−	1.31578i
$783$		0			0
$784$	34.9337			1.24763
$785$		−	3.75872i		−	0.134155i
$786$		0			0
$787$			30.6069i			1.09102i	0.838105	+	0.545509i	$0.183664\pi$
$787$			30.6069i			1.09102i	−0.838105	+	0.545509i	$0.816336\pi$
$788$			49.4596i			1.76192i
$789$		0			0
$790$	15.0361			0.534961
$791$	−83.0493			−2.95289
$792$		0			0
$793$			32.2434i			1.14500i
$794$			61.8987i			2.19670i
$795$		0			0
$796$			10.0917i			0.357691i
$797$	−22.9770			−0.813888			−0.406944	−	0.913453i	$-0.633406\pi$
$797$	−22.9770			−0.813888			−0.406944	+	0.913453i	$0.633406\pi$
$798$		0			0
$799$	22.0494	−	17.2351i	0.780053	−	0.609735i
$800$	7.58864			0.268299
$801$		0			0
$802$			71.8141i			2.53585i
$803$	−35.1194			−1.23934
$804$		0			0
$805$			25.4680i			0.897629i
$806$			23.3607i			0.822845i
$807$		0			0
$808$	3.80817			0.133971
$809$		−	42.7670i		−	1.50361i	−0.659388	−	0.751803i	$-0.729184\pi$
$809$		−	42.7670i		−	1.50361i	0.659388	−	0.751803i	$-0.270816\pi$
$810$		0			0
$811$			9.41136i			0.330478i	0.986254	+	0.165239i	$0.0528395\pi$
$811$			9.41136i			0.330478i	−0.986254	+	0.165239i	$0.947161\pi$
$812$	38.6225			1.35538
$813$		0			0
$814$	−36.1978			−1.26873
$815$	−8.69594			−0.304606
$816$		0			0
$817$	3.63317			0.127108
$818$	−65.5318			−2.29127
$819$		0			0
$820$	15.1773			0.530013
$821$			32.6681i			1.14012i	0.821602	+	0.570062i	$0.193081\pi$
$821$			32.6681i			1.14012i	−0.821602	+	0.570062i	$0.806919\pi$
$822$		0			0
$823$		−	15.9265i		−	0.555164i	−0.960702	−	0.277582i	$-0.910467\pi$
$823$		−	15.9265i		−	0.555164i	0.960702	−	0.277582i	$-0.0895331\pi$
$824$	−30.2472			−1.05371
$825$		0			0
$826$		−	24.7792i		−	0.862180i
$827$			6.01560i			0.209183i	0.994515	+	0.104591i	$0.0333535\pi$
$827$			6.01560i			0.209183i	−0.994515	+	0.104591i	$0.966647\pi$
$828$		0			0
$829$	11.0472			0.383684			0.191842	−	0.981426i	$-0.438554\pi$
$829$	11.0472			0.383684			0.191842	+	0.981426i	$0.438554\pi$
$830$			17.8660i			0.620139i
$831$		0			0
$832$	−32.3896			−1.12291
$833$	−54.6007	+	42.6791i	−1.89180	+	1.47874i
$834$		0			0
$835$	−1.37629			−0.0476285
$836$			9.26180i			0.320326i
$837$		0			0
$838$		−	39.4173i		−	1.36165i
$839$			35.9805i			1.24219i	0.783737	+	0.621093i	$0.213311\pi$
$839$			35.9805i			1.24219i	−0.783737	+	0.621093i	$0.786689\pi$
$840$		0			0
$841$	20.4641			0.705659
$842$	1.65142			0.0569116
$843$		0			0
$844$		−	60.2772i		−	2.07483i
$845$			6.07838i			0.209103i
$846$		0			0
$847$		−	4.63809i		−	0.159367i
$848$	7.81205			0.268267
$849$		0			0
$850$	−7.04945	+	5.51026i	−0.241794	+	0.189000i
$851$	27.4641			0.941458
$852$		0			0
$853$		−	28.7792i		−	0.985382i	−0.870204	−	0.492691i	$-0.836013\pi$
$853$		−	28.7792i		−	0.985382i	0.870204	−	0.492691i	$-0.163987\pi$
$854$	129.771			4.44066
$855$		0			0
$856$			19.4524i			0.664869i
$857$			17.0661i			0.582967i	0.956576	+	0.291483i	$0.0941488\pi$
$857$			17.0661i			0.582967i	−0.956576	+	0.291483i	$0.905851\pi$
$858$		0			0
$859$	−18.9360			−0.646088			−0.323044	−	0.946384i	$-0.604706\pi$
$859$	−18.9360			−0.646088			−0.323044	+	0.946384i	$0.604706\pi$
$860$		−	9.12783i		−	0.311256i
$861$		0			0
$862$		−	13.7743i		−	0.469155i
$863$	−30.8332			−1.04958			−0.524788	−	0.851233i	$-0.675855\pi$
$863$	−30.8332			−1.04958			−0.524788	+	0.851233i	$0.675855\pi$
$864$		0			0
$865$	17.3607			0.590281
$866$	7.86603			0.267299
$867$		0			0
$868$	54.0905			1.83595
$869$	−21.9649			−0.745109
$870$		0			0
$871$	26.8515			0.909828
$872$		−	12.5548i		−	0.425159i
$873$		0			0
$874$		−	12.2146i		−	0.413165i
$875$	4.87936			0.164953
$876$		0			0
$877$			4.30898i			0.145504i	0.997350	+	0.0727519i	$0.0231781\pi$
$877$			4.30898i			0.145504i	−0.997350	+	0.0727519i	$0.976822\pi$
$878$		−	13.8999i		−	0.469098i
$879$		0			0
$880$	−6.58864			−0.222103
$881$			12.2245i			0.411852i	0.978568	+	0.205926i	$0.0660207\pi$
$881$			12.2245i			0.411852i	−0.978568	+	0.205926i	$0.933979\pi$
$882$		0			0
$883$	−28.2329			−0.950112			−0.475056	−	0.879956i	$-0.657572\pi$
$883$	−28.2329			−0.950112			−0.475056	+	0.879956i	$0.657572\pi$
$884$	23.1545	−	18.0989i	0.778770	−	0.608732i
$885$		0			0
$886$	−59.6391			−2.00362
$887$		−	5.17396i		−	0.173725i	−0.996220	−	0.0868623i	$-0.972316\pi$
$887$		−	5.17396i		−	0.173725i	0.996220	−	0.0868623i	$-0.0276840\pi$
$888$		0			0
$889$			39.2762i			1.31728i
$890$			15.5259i			0.520428i
$891$		0			0
$892$	−5.93827			−0.198828
$893$	7.31965			0.244943
$894$		0			0
$895$			6.83710i			0.228539i
$896$			56.3039i			1.88098i
$897$		0			0
$898$			62.3956i			2.08217i
$899$	−11.9544			−0.398702
$900$		0			0
$901$	−12.2101	+	9.54411i	−0.406777	+	0.317960i
$902$	−38.5380			−1.28317
$903$		0			0
$904$		−	26.1978i		−	0.871326i
$905$	15.0205			0.499299
$906$		0			0
$907$		−	6.32457i		−	0.210004i	−0.994472	−	0.105002i	$-0.966515\pi$
$907$		−	6.32457i		−	0.210004i	0.994472	−	0.105002i	$-0.0334849\pi$
$908$		−	25.8999i		−	0.859518i
$909$		0			0
$910$	−27.8576			−0.923471
$911$		−	27.6526i		−	0.916173i	−0.888908	−	0.458086i	$-0.848535\pi$
$911$		−	27.6526i		−	0.916173i	0.888908	−	0.458086i	$-0.151465\pi$
$912$		0			0
$913$		−	26.0989i		−	0.863747i
$914$	−68.1627			−2.25462
$915$		0			0
$916$	19.9649			0.659660
$917$	−57.0577			−1.88421
$918$		0			0
$919$	−47.5318			−1.56793			−0.783965	−	0.620805i	$-0.786806\pi$
$919$	−47.5318			−1.56793			−0.783965	+	0.620805i	$0.786806\pi$
$920$	−8.03385			−0.264868
$921$		0			0
$922$	16.8371			0.554500
$923$		−	10.6947i		−	0.352021i
$924$		0			0
$925$		−	5.26180i		−	0.173007i
$926$	52.5874			1.72813
$927$		0			0
$928$		−	22.1711i		−	0.727803i
$929$			43.7875i			1.43662i	0.695723	+	0.718310i	$0.255084\pi$
$929$			43.7875i			1.43662i	−0.695723	+	0.718310i	$0.744916\pi$
$930$		0			0
$931$	−18.1256			−0.594041
$932$		−	25.5897i		−	0.838218i
$933$		0			0
$934$	26.7298			0.874626
$935$	10.2979	−	8.04945i	0.336778	−	0.263245i
$936$		0			0
$937$	14.2367			0.465094			0.232547	−	0.972585i	$-0.425294\pi$
$937$	14.2367			0.465094			0.232547	+	0.972585i	$0.425294\pi$
$938$		−	108.070i		−	3.52860i
$939$		0			0
$940$		−	18.3896i		−	0.599803i
$941$		−	35.2183i		−	1.14808i	−0.818826	−	0.574042i	$-0.805375\pi$
$941$		−	35.2183i		−	1.14808i	0.818826	−	0.574042i	$-0.194625\pi$
$942$		0			0
$943$	29.2397			0.952175
$944$	−4.86376			−0.158302
$945$		0			0
$946$			23.1773i			0.753558i
$947$		−	49.1227i		−	1.59627i	−0.602476	−	0.798137i	$-0.705819\pi$
$947$		−	49.1227i		−	1.59627i	0.602476	−	0.798137i	$-0.294181\pi$
$948$		0			0
$949$			29.1461i			0.946122i
$950$	−2.34017			−0.0759252
$951$		0			0
$952$	−19.0700	−	24.3968i	−0.618061	−	0.790705i
$953$	17.0556			0.552485			0.276242	−	0.961088i	$-0.410911\pi$
$953$	17.0556			0.552485			0.276242	+	0.961088i	$0.410911\pi$
$954$		0			0
$955$		−	24.2823i		−	0.785757i
$956$	−16.9483			−0.548147
$957$		0			0
$958$		−	4.66475i		−	0.150711i
$959$			9.51745i			0.307334i
$960$		0			0
$961$	14.2579			0.459933
$962$			30.0410i			0.968562i
$963$		0			0
$964$		−	6.76487i		−	0.217882i
$965$	−11.8576			−0.381710
$966$		0			0
$967$	−25.1955			−0.810233			−0.405117	−	0.914265i	$-0.632769\pi$
$967$	−25.1955			−0.810233			−0.405117	+	0.914265i	$0.632769\pi$
$968$	1.46308			0.0470251
$969$		0			0
$970$	−17.7587			−0.570198
$971$	−1.67420			−0.0537277			−0.0268639	−	0.999639i	$-0.508552\pi$
$971$	−1.67420			−0.0537277			−0.0268639	+	0.999639i	$0.508552\pi$
$972$		0			0
$973$	−9.79380			−0.313975
$974$		−	86.9842i		−	2.78715i
$975$		0			0
$976$		−	25.4719i		−	0.815335i
$977$	39.9109			1.27686			0.638432	−	0.769678i	$-0.279583\pi$
$977$	39.9109			1.27686			0.638432	+	0.769678i	$0.279583\pi$
$978$		0			0
$979$		−	22.6803i		−	0.724867i
$980$			45.5380i			1.45466i
$981$		0			0
$982$	4.89496			0.156204
$983$			5.46081i			0.174173i	0.996201	+	0.0870864i	$0.0277556\pi$
$983$			5.46081i			0.174173i	−0.996201	+	0.0870864i	$0.972244\pi$
$984$		0			0
$985$	18.2557			0.581673
$986$	16.0989	+	20.5958i	0.512693	+	0.655905i
$987$		0			0
$988$	7.68649			0.244540
$989$		−	17.5851i		−	0.559175i
$990$		0			0
$991$		−	42.0749i		−	1.33655i	−0.743913	−	0.668276i	$-0.767032\pi$
$991$		−	42.0749i		−	1.33655i	0.743913	−	0.668276i	$-0.232968\pi$
$992$		−	31.0505i		−	0.985854i
$993$		0			0
$994$	−43.0433			−1.36525
$995$	3.72487			0.118086
$996$		0			0
$997$		−	54.6681i		−	1.73135i	−0.500602	−	0.865677i	$-0.666888\pi$
$997$		−	54.6681i		−	1.73135i	0.500602	−	0.865677i	$-0.333112\pi$
$998$		−	92.3267i		−	2.92255i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	765.2.g.b.271.5		6
3.2	odd	2		85.2.d.a.16.1	✓	6
12.11	even	2		1360.2.c.f.1121.4		6
15.2	even	4		425.2.c.b.424.1		6
15.8	even	4		425.2.c.a.424.6		6
15.14	odd	2		425.2.d.c.101.6		6
17.16	even	2	inner	765.2.g.b.271.6		6
51.38	odd	4		1445.2.a.k.1.3		3
51.47	odd	4		1445.2.a.j.1.3		3
51.50	odd	2		85.2.d.a.16.2	yes	6
204.203	even	2		1360.2.c.f.1121.3		6
255.89	odd	4		7225.2.a.r.1.1		3
255.149	odd	4		7225.2.a.q.1.1		3
255.152	even	4		425.2.c.a.424.1		6
255.203	even	4		425.2.c.b.424.6		6
255.254	odd	2		425.2.d.c.101.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.d.a.16.1	✓	6	3.2	odd	2
85.2.d.a.16.2	yes	6	51.50	odd	2
425.2.c.a.424.1		6	255.152	even	4
425.2.c.a.424.6		6	15.8	even	4
425.2.c.b.424.1		6	15.2	even	4
425.2.c.b.424.6		6	255.203	even	4
425.2.d.c.101.5		6	255.254	odd	2
425.2.d.c.101.6		6	15.14	odd	2
765.2.g.b.271.5		6	1.1	even	1	trivial
765.2.g.b.271.6		6	17.16	even	2	inner
1360.2.c.f.1121.3		6	204.203	even	2
1360.2.c.f.1121.4		6	12.11	even	2
1445.2.a.j.1.3		3	51.47	odd	4
1445.2.a.k.1.3		3	51.38	odd	4
7225.2.a.q.1.1		3	255.149	odd	4
7225.2.a.r.1.1		3	255.89	odd	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:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	765.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.17009 q^{2} +2.70928 q^{4} -1.00000i q^{5} -4.87936i q^{7} +1.53919 q^{8} -2.17009i q^{10} +3.17009i q^{11} +2.63090 q^{13} -10.5886i q^{14} -2.07838 q^{16} +(3.24846 - 2.53919i) q^{17} +1.07838 q^{19} -2.70928i q^{20} +6.87936i q^{22} -5.21953i q^{23} -1.00000 q^{25} +5.70928 q^{26} -13.2195i q^{28} +2.92162i q^{29} +4.09171i q^{31} -7.58864 q^{32} +(7.04945 - 5.51026i) q^{34} -4.87936 q^{35} +5.26180i q^{37} +2.34017 q^{38} -1.53919i q^{40} +5.60197i q^{41} +3.36910 q^{43} +8.58864i q^{44} -11.3268i q^{46} +6.78765 q^{47} -16.8082 q^{49} -2.17009 q^{50} +7.12783 q^{52} -3.75872 q^{53} +3.17009 q^{55} -7.51026i q^{56} +6.34017i q^{58} +2.34017 q^{59} +12.2557i q^{61} +8.87936i q^{62} -12.3112 q^{64} -2.63090i q^{65} +10.2062 q^{67} +(8.80098 - 6.87936i) q^{68} -10.5886 q^{70} -4.06505i q^{71} +11.0784i q^{73} +11.4186i q^{74} +2.92162 q^{76} +15.4680 q^{77} +6.92881i q^{79} +2.07838i q^{80} +12.1568i q^{82} -8.23287 q^{83} +(-2.53919 - 3.24846i) q^{85} +7.31124 q^{86} +4.87936i q^{88} -7.15449 q^{89} -12.8371i q^{91} -14.1412i q^{92} +14.7298 q^{94} -1.07838i q^{95} -8.18342i q^{97} -36.4752 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 8 q^{13} - 6 q^{16} + 2 q^{17} - 6 q^{25} + 20 q^{26} - 6 q^{32} + 6 q^{34} - 4 q^{35} - 8 q^{38} + 28 q^{43} + 20 q^{47} - 14 q^{49} - 2 q^{50} + 28 q^{53} + 8 q^{55}+ \cdots - 86 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 + 8 * q^13 - 6 * q^16 + 2 * q^17 - 6 * q^25 + 20 * q^26 - 6 * q^32 + 6 * q^34 - 4 * q^35 - 8 * q^38 + 28 * q^43 + 20 * q^47 - 14 * q^49 - 2 * q^50 + 28 * q^53 + 8 * q^55 - 8 * q^59 - 22 * q^64 + 12 * q^67 + 34 * q^68 - 24 * q^70 + 24 * q^76 + 28 * q^77 - 4 * q^83 - 12 * q^85 - 8 * q^86 - 4 * q^89 + 8 * q^94 - 86 * q^98

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