LMFDB - Embedded newform 1040.2.f.g.129.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(129,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1040.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.f (of order $2$, degree $1$, not 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:	$8.30444181021$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 11x^{8} + 36x^{6} + 42x^{4} + 13x^{2} + 1 $ x^10 + 11x^8 + 36x^6 + 42x^4 + 13x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.6
Root			$-0.341517i$ of defining polynomial
Character	$\chi$	$=$	1040.129
Dual form			1040.2.f.g.129.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.773218i q^{3} +(-1.98940 + 1.02093i) q^{5} +1.12600 q^{7} +2.40213 q^{9} +O(q^{10})$	$q+0.773218i q^{3} +(-1.98940 + 1.02093i) q^{5} +1.12600 q^{7} +2.40213 q^{9} +2.52383i q^{11} +(1.37666 - 3.33239i) q^{13} +(-0.789402 - 1.53824i) q^{15} -0.870643i q^{17} +6.27610i q^{19} +0.870643i q^{21} +2.81508i q^{23} +(2.91540 - 4.06207i) q^{25} +4.17703i q^{27} -0.176669 q^{29} -0.388669i q^{31} -1.95147 q^{33} +(-2.24006 + 1.14957i) q^{35} -3.12600 q^{37} +(2.57666 + 1.06446i) q^{39} +7.25350i q^{41} +4.52548i q^{43} +(-4.77880 + 2.45241i) q^{45} -1.12600 q^{47} -5.73212 q^{49} +0.673197 q^{51} +9.83455i q^{53} +(-2.57666 - 5.02091i) q^{55} -4.85279 q^{57} +12.1011i q^{59} +10.3863 q^{61} +2.70480 q^{63} +(0.663401 + 8.03492i) q^{65} -6.70480 q^{67} -2.17667 q^{69} -4.37084i q^{71} +4.70480 q^{73} +(3.14087 + 2.25424i) q^{75} +2.84184i q^{77} +8.31093 q^{79} +3.97665 q^{81} -9.15573 q^{83} +(0.888866 + 1.73206i) q^{85} -0.136604i q^{87} -6.15295i q^{89} +(1.55012 - 3.75227i) q^{91} +0.300526 q^{93} +(-6.40746 - 12.4857i) q^{95} -1.55760 q^{97} +6.06259i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9}+O(q^{10}) $ 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9	$ 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9} + 2 q^{13} + 13 q^{15} - q^{25} + 8 q^{29} + 8 q^{33} + 3 q^{35} - 22 q^{37} + 12 q^{39} - 4 q^{45} - 2 q^{47} + 12 q^{49} + 30 q^{51} - 12 q^{55} - 12 q^{57} - 16 q^{61} - 24 q^{63} - 5 q^{65} - 16 q^{67} - 12 q^{69} - 4 q^{73} - 21 q^{75} - 28 q^{79} + 22 q^{81} + 4 q^{83} - 25 q^{85} - 2 q^{91} + 12 q^{93} + 10 q^{95} + 72 q^{97}+O(q^{100}) $ 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9 + 2 * q^13 + 13 * q^15 - q^25 + 8 * q^29 + 8 * q^33 + 3 * q^35 - 22 * q^37 + 12 * q^39 - 4 * q^45 - 2 * q^47 + 12 * q^49 + 30 * q^51 - 12 * q^55 - 12 * q^57 - 16 * q^61 - 24 * q^63 - 5 * q^65 - 16 * q^67 - 12 * q^69 - 4 * q^73 - 21 * q^75 - 28 * q^79 + 22 * q^81 + 4 * q^83 - 25 * q^85 - 2 * q^91 + 12 * q^93 + 10 * q^95 + 72 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$.

$n$	$261$	$417$	$561$	$911$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			0.773218i			0.446418i	0.974771	+	0.223209i	$0.0716532\pi$
$3$			0.773218i			0.446418i	−0.974771	+	0.223209i	$0.928347\pi$
$4$		0			0
$5$	−1.98940	+	1.02093i	−0.889685	+	0.456574i
$5$	−1.98940	+	1.02093i	−0.889685	+	0.456574i
$6$		0			0
$7$	1.12600			0.425588			0.212794	−	0.977097i	$-0.431744\pi$
$7$	1.12600			0.425588			0.212794	+	0.977097i	$0.431744\pi$
$8$		0			0
$9$	2.40213			0.800711
$10$		0			0
$11$			2.52383i			0.760965i	0.924788	+	0.380482i	$0.124242\pi$
$11$			2.52383i			0.760965i	−0.924788	+	0.380482i	$0.875758\pi$
$12$		0			0
$13$	1.37666	−	3.33239i	0.381818	−	0.924238i
$13$	1.37666	−	3.33239i	0.381818	−	0.924238i
$14$		0			0
$15$	−0.789402	−	1.53824i	−0.203823	−	0.397171i
$16$		0			0
$17$		−	0.870643i		−	0.211162i	−0.994411	−	0.105581i	$-0.966330\pi$
$17$		−	0.870643i		−	0.211162i	0.994411	−	0.105581i	$-0.0336702\pi$
$18$		0			0
$19$			6.27610i			1.43984i	0.694059	+	0.719918i	$0.255821\pi$
$19$			6.27610i			1.43984i	−0.694059	+	0.719918i	$0.744179\pi$
$20$		0			0
$21$			0.870643i			0.189990i
$22$		0			0
$23$			2.81508i			0.586985i	0.955961	+	0.293492i	$0.0948175\pi$
$23$			2.81508i			0.586985i	−0.955961	+	0.293492i	$0.905183\pi$
$24$		0			0
$25$	2.91540	−	4.06207i	0.583080	−	0.812414i
$26$		0			0
$27$			4.17703i			0.803869i
$28$		0			0
$29$	−0.176669			−0.0328066			−0.0164033	−	0.999865i	$-0.505222\pi$
$29$	−0.176669			−0.0328066			−0.0164033	+	0.999865i	$0.505222\pi$
$30$		0			0
$31$		−	0.388669i		−	0.0698071i	−0.999391	−	0.0349035i	$-0.988888\pi$
$31$		−	0.388669i		−	0.0698071i	0.999391	−	0.0349035i	$-0.0111124\pi$
$32$		0			0
$33$	−1.95147			−0.339708
$34$		0			0
$35$	−2.24006	+	1.14957i	−0.378640	+	0.194312i
$36$		0			0
$37$	−3.12600			−0.513911			−0.256956	−	0.966423i	$-0.582719\pi$
$37$	−3.12600			−0.513911			−0.256956	+	0.966423i	$0.582719\pi$
$38$		0			0
$39$	2.57666	+	1.06446i	0.412596	+	0.170450i
$40$		0			0
$41$			7.25350i			1.13281i	0.824128	+	0.566403i	$0.191666\pi$
$41$			7.25350i			1.13281i	−0.824128	+	0.566403i	$0.808334\pi$
$42$		0			0
$43$			4.52548i			0.690130i	0.938579	+	0.345065i	$0.112143\pi$
$43$			4.52548i			0.690130i	−0.938579	+	0.345065i	$0.887857\pi$
$44$		0			0
$45$	−4.77880	+	2.45241i	−0.712381	+	0.365584i
$46$		0			0
$47$	−1.12600			−0.164244			−0.0821220	−	0.996622i	$-0.526170\pi$
$47$	−1.12600			−0.164244			−0.0821220	+	0.996622i	$0.526170\pi$
$48$		0			0
$49$	−5.73212			−0.818875
$50$		0			0
$51$	0.673197			0.0942664
$52$		0			0
$53$			9.83455i			1.35088i	0.737415	+	0.675440i	$0.236046\pi$
$53$			9.83455i			1.35088i	−0.737415	+	0.675440i	$0.763954\pi$
$54$		0			0
$55$	−2.57666	−	5.02091i	−0.347437	−	0.677019i
$56$		0			0
$57$	−4.85279			−0.642768
$58$		0			0
$59$			12.1011i			1.57543i	0.616040	+	0.787715i	$0.288736\pi$
$59$			12.1011i			1.57543i	−0.616040	+	0.787715i	$0.711264\pi$
$60$		0			0
$61$	10.3863			1.32982			0.664912	−	0.746922i	$-0.268469\pi$
$61$	10.3863			1.32982			0.664912	+	0.746922i	$0.268469\pi$
$62$		0			0
$63$	2.70480			0.340773
$64$		0			0
$65$	0.663401	+	8.03492i	0.0822848	+	0.996609i
$66$		0			0
$67$	−6.70480			−0.819122			−0.409561	−	0.912283i	$-0.634318\pi$
$67$	−6.70480			−0.819122			−0.409561	+	0.912283i	$0.634318\pi$
$68$		0			0
$69$	−2.17667			−0.262040
$70$		0			0
$71$		−	4.37084i		−	0.518724i	−0.965780	−	0.259362i	$-0.916488\pi$
$71$		−	4.37084i		−	0.518724i	0.965780	−	0.259362i	$-0.0835122\pi$
$72$		0			0
$73$	4.70480			0.550656			0.275328	−	0.961350i	$-0.411214\pi$
$73$	4.70480			0.550656			0.275328	+	0.961350i	$0.411214\pi$
$74$		0			0
$75$	3.14087	+	2.25424i	0.362676	+	0.260297i
$76$		0			0
$77$			2.84184i			0.323858i
$78$		0			0
$79$	8.31093			0.935052			0.467526	−	0.883979i	$-0.345145\pi$
$79$	8.31093			0.935052			0.467526	+	0.883979i	$0.345145\pi$
$80$		0			0
$81$	3.97665			0.441850
$82$		0			0
$83$	−9.15573			−1.00497			−0.502486	−	0.864585i	$-0.667581\pi$
$83$	−9.15573			−1.00497			−0.502486	+	0.864585i	$0.667581\pi$
$84$		0			0
$85$	0.888866	+	1.73206i	0.0964111	+	0.187868i
$86$		0			0
$87$		−	0.136604i		−	0.0146455i
$88$		0			0
$89$		−	6.15295i		−	0.652211i	−0.945333	−	0.326106i	$-0.894263\pi$
$89$		−	6.15295i		−	0.652211i	0.945333	−	0.326106i	$-0.105737\pi$
$90$		0			0
$91$	1.55012	−	3.75227i	0.162497	−	0.393344i
$92$		0			0
$93$	0.300526			0.0311631
$94$		0			0
$95$	−6.40746	−	12.4857i	−0.657392	−	1.28100i
$96$		0			0
$97$	−1.55760			−0.158150			−0.0790750	−	0.996869i	$-0.525197\pi$
$97$	−1.55760			−0.158150			−0.0790750	+	0.996869i	$0.525197\pi$
$98$		0			0
$99$			6.06259i			0.609313i
$100$		0			0
$101$	−10.5099			−1.04577			−0.522885	−	0.852403i	$-0.675144\pi$
$101$	−10.5099			−1.04577			−0.522885	+	0.852403i	$0.675144\pi$
$102$		0			0
$103$			5.21576i			0.513924i	0.966421	+	0.256962i	$0.0827216\pi$
$103$			5.21576i			0.513924i	−0.966421	+	0.256962i	$0.917278\pi$
$104$		0			0
$105$	−0.888866	−	1.73206i	−0.0867445	−	0.169031i
$106$		0			0
$107$		−	16.9282i		−	1.63651i	−0.574855	−	0.818256i	$-0.694941\pi$
$107$		−	16.9282i		−	1.63651i	0.574855	−	0.818256i	$-0.305059\pi$
$108$		0			0
$109$		−	12.2785i		−	1.17607i	−0.808835	−	0.588035i	$-0.799902\pi$
$109$		−	12.2785i		−	1.17607i	0.808835	−	0.588035i	$-0.200098\pi$
$110$		0			0
$111$		−	2.41708i		−	0.229419i
$112$		0			0
$113$			18.5138i			1.74163i	0.491608	+	0.870817i	$0.336409\pi$
$113$			18.5138i			1.74163i	−0.491608	+	0.870817i	$0.663591\pi$
$114$		0			0
$115$	−2.87400	−	5.60031i	−0.268002	−	0.522232i
$116$		0			0
$117$	3.30693	−	8.00484i	0.305726	−	0.740047i
$118$		0			0
$119$		−	0.980345i		−	0.0898681i
$120$		0			0
$121$	4.63026			0.420932
$122$		0			0
$123$	−5.60854			−0.505705
$124$		0			0
$125$	−1.65280	+	11.0575i	−0.147831	+	0.989013i
$126$		0			0
$127$		−	6.94881i		−	0.616607i	−0.951288	−	0.308303i	$-0.900239\pi$
$127$		−	6.94881i		−	0.616607i	0.951288	−	0.308303i	$-0.0997612\pi$
$128$		0			0
$129$	−3.49919			−0.308086
$130$		0			0
$131$	18.5927			1.62445			0.812224	−	0.583345i	$-0.198257\pi$
$131$	18.5927			1.62445			0.812224	+	0.583345i	$0.198257\pi$
$132$		0			0
$133$			7.06689i			0.612777i
$134$		0			0
$135$	−4.26445	−	8.30976i	−0.367026	−	0.715191i
$136$		0			0
$137$	11.4563			0.978774			0.489387	−	0.872067i	$-0.337221\pi$
$137$	11.4563			0.978774			0.489387	+	0.872067i	$0.337221\pi$
$138$		0			0
$139$	−2.93105			−0.248609			−0.124304	−	0.992244i	$-0.539670\pi$
$139$	−2.93105			−0.248609			−0.124304	+	0.992244i	$0.539670\pi$
$140$		0			0
$141$		−	0.870643i		−	0.0733214i
$142$		0			0
$143$	8.41039	+	3.47447i	0.703312	+	0.290550i
$144$		0			0
$145$	0.351465	−	0.180367i	0.0291876	−	0.0149787i
$146$		0			0
$147$		−	4.43218i		−	0.365560i
$148$		0			0
$149$		−	2.83560i		−	0.232301i	−0.993232	−	0.116151i	$-0.962944\pi$
$149$		−	2.83560i		−	0.232301i	0.993232	−	0.116151i	$-0.0370555\pi$
$150$		0			0
$151$		−	15.3580i		−	1.24981i	−0.780700	−	0.624906i	$-0.785137\pi$
$151$		−	15.3580i		−	1.24981i	0.780700	−	0.624906i	$-0.214863\pi$
$152$		0			0
$153$		−	2.09140i		−	0.169080i
$154$		0			0
$155$	0.396804	+	0.773218i	0.0318721	+	0.0621064i
$156$		0			0
$157$		−	6.33683i		−	0.505734i	−0.967501	−	0.252867i	$-0.918626\pi$
$157$		−	6.33683i		−	0.505734i	0.967501	−	0.252867i	$-0.0813735\pi$
$158$		0			0
$159$	−7.60425			−0.603056
$160$		0			0
$161$			3.16978i			0.249814i
$162$		0			0
$163$	20.3680			1.59534			0.797672	−	0.603092i	$-0.206065\pi$
$163$	20.3680			1.59534			0.797672	+	0.603092i	$0.206065\pi$
$164$		0			0
$165$	3.88226	−	1.99232i	0.302233	−	0.155102i
$166$		0			0
$167$	6.20080			0.479833			0.239916	−	0.970794i	$-0.422880\pi$
$167$	6.20080			0.479833			0.239916	+	0.970794i	$0.422880\pi$
$168$		0			0
$169$	−9.20959	−	9.17516i	−0.708430	−	0.705781i
$170$		0			0
$171$			15.0760i			1.15289i
$172$		0			0
$173$			6.58786i			0.500866i	0.968134	+	0.250433i	$0.0805730\pi$
$173$			6.58786i			0.500866i	−0.968134	+	0.250433i	$0.919427\pi$
$174$		0			0
$175$	3.28274	−	4.57389i	0.248152	−	0.345754i
$176$		0			0
$177$	−9.35679			−0.703300
$178$		0			0
$179$	23.7460			1.77486			0.887429	−	0.460944i	$-0.152489\pi$
$179$	23.7460			1.77486			0.887429	+	0.460944i	$0.152489\pi$
$180$		0			0
$181$	−7.13160			−0.530088			−0.265044	−	0.964236i	$-0.585386\pi$
$181$	−7.13160			−0.530088			−0.265044	+	0.964236i	$0.585386\pi$
$182$		0			0
$183$			8.03084i			0.593657i
$184$		0			0
$185$	6.21886	−	3.19143i	0.457219	−	0.234639i
$186$		0			0
$187$	2.19736			0.160687
$188$		0			0
$189$			4.70333i			0.342117i
$190$		0			0
$191$	−20.4192			−1.47748			−0.738740	−	0.673991i	$-0.764579\pi$
$191$	−20.4192			−1.47748			−0.738740	+	0.673991i	$0.764579\pi$
$192$		0			0
$193$	12.1629			0.875507			0.437754	−	0.899095i	$-0.355774\pi$
$193$	12.1629			0.875507			0.437754	+	0.899095i	$0.355774\pi$
$194$		0			0
$195$	−6.21274	+	0.512954i	−0.444904	+	0.0367334i
$196$		0			0
$197$	9.37346			0.667831			0.333916	−	0.942603i	$-0.391630\pi$
$197$	9.37346			0.667831			0.333916	+	0.942603i	$0.391630\pi$
$198$		0			0
$199$	−18.1125			−1.28396			−0.641982	−	0.766720i	$-0.721888\pi$
$199$	−18.1125			−1.28396			−0.641982	+	0.766720i	$0.721888\pi$
$200$		0			0
$201$		−	5.18427i		−	0.365670i
$202$		0			0
$203$	−0.198930			−0.0139621
$204$		0			0
$205$	−7.40532	−	14.4301i	−0.517210	−	1.00784i
$206$		0			0
$207$			6.76220i			0.470005i
$208$		0			0
$209$	−15.8398			−1.09567
$210$		0			0
$211$	−23.7460			−1.63474			−0.817370	−	0.576113i	$-0.804569\pi$
$211$	−23.7460			−1.63474			−0.817370	+	0.576113i	$0.804569\pi$
$212$		0			0
$213$	3.37962			0.231567
$214$		0			0
$215$	−4.62021	−	9.00299i	−0.315095	−	0.613999i
$216$		0			0
$217$		−	0.437642i		−	0.0297091i
$218$		0			0
$219$			3.63784i			0.245822i
$220$		0			0
$221$	−2.90132	−	1.19858i	−0.195164	−	0.0806255i
$222$		0			0
$223$	25.0412			1.67688			0.838441	−	0.544993i	$-0.183468\pi$
$223$	25.0412			1.67688			0.838441	+	0.544993i	$0.183468\pi$
$224$		0			0
$225$	7.00319	−	9.75764i	0.466879	−	0.650510i
$226$		0			0
$227$	−25.9171			−1.72018			−0.860088	−	0.510147i	$-0.829591\pi$
$227$	−25.9171			−1.72018			−0.860088	+	0.510147i	$0.829591\pi$
$228$		0			0
$229$		−	22.6577i		−	1.49726i	−0.662986	−	0.748631i	$-0.730711\pi$
$229$		−	22.6577i		−	1.49726i	0.662986	−	0.748631i	$-0.269289\pi$
$230$		0			0
$231$	−2.19736			−0.144576
$232$		0			0
$233$			2.11253i			0.138396i	0.997603	+	0.0691981i	$0.0220441\pi$
$233$			2.11253i			0.138396i	−0.997603	+	0.0691981i	$0.977956\pi$
$234$		0			0
$235$	2.24006	−	1.14957i	0.146126	−	0.0749895i
$236$		0			0
$237$			6.42616i			0.417424i
$238$		0			0
$239$		−	17.0410i		−	1.10229i	−0.834409	−	0.551145i	$-0.814191\pi$
$239$		−	17.0410i		−	1.10229i	0.834409	−	0.551145i	$-0.185809\pi$
$240$		0			0
$241$			7.06689i			0.455219i	0.973753	+	0.227609i	$0.0730909\pi$
$241$			7.06689i			0.455219i	−0.973753	+	0.227609i	$0.926909\pi$
$242$		0			0
$243$			15.6059i			1.00112i
$244$		0			0
$245$	11.4035	−	5.85210i	0.728541	−	0.373877i
$246$		0			0
$247$	20.9144	+	8.64009i	1.33075	+	0.549756i
$248$		0			0
$249$		−	7.07938i		−	0.448637i
$250$		0			0
$251$	3.35068			0.211493			0.105747	−	0.994393i	$-0.466277\pi$
$251$	3.35068			0.211493			0.105747	+	0.994393i	$0.466277\pi$
$252$		0			0
$253$	−7.10479			−0.446675
$254$		0			0
$255$	−1.33926	+	0.687287i	−0.0838675	+	0.0430396i
$256$		0			0
$257$		−	13.3728i		−	0.834175i	−0.908866	−	0.417087i	$-0.863051\pi$
$257$		−	13.3728i		−	0.834175i	0.908866	−	0.417087i	$-0.136949\pi$
$258$		0			0
$259$	−3.51988			−0.218715
$260$		0			0
$261$	−0.424383			−0.0262687
$262$		0			0
$263$		−	1.96083i		−	0.120910i	−0.998171	−	0.0604550i	$-0.980745\pi$
$263$		−	1.96083i		−	0.120910i	0.998171	−	0.0604550i	$-0.0192552\pi$
$264$		0			0
$265$	−10.0404	−	19.5648i	−0.616776	−	1.20186i
$266$		0			0
$267$	4.75757			0.291159
$268$		0			0
$269$	24.8165			1.51309			0.756544	−	0.653942i	$-0.226886\pi$
$269$	24.8165			1.51309			0.756544	+	0.653942i	$0.226886\pi$
$270$		0			0
$271$		−	19.2675i		−	1.17042i	−0.810882	−	0.585210i	$-0.801012\pi$
$271$		−	19.2675i		−	1.17042i	0.810882	−	0.585210i	$-0.198988\pi$
$272$		0			0
$273$	2.90132	+	1.19858i	0.175596	+	0.0725416i
$274$		0			0
$275$	10.2520	+	7.35799i	0.618219	+	0.443704i
$276$		0			0
$277$		−	11.1861i		−	0.672110i	−0.941842	−	0.336055i	$-0.890907\pi$
$277$		−	11.1861i		−	0.672110i	0.941842	−	0.336055i	$-0.109093\pi$
$278$		0			0
$279$		−	0.933636i		−	0.0558953i
$280$		0			0
$281$		−	19.2315i		−	1.14725i	−0.819117	−	0.573626i	$-0.805536\pi$
$281$		−	19.2315i		−	1.14725i	0.819117	−	0.573626i	$-0.194464\pi$
$282$		0			0
$283$			25.6512i			1.52481i	0.647102	+	0.762404i	$0.275981\pi$
$283$			25.6512i			1.52481i	−0.647102	+	0.762404i	$0.724019\pi$
$284$		0			0
$285$	9.65413	−	4.95437i	0.571862	−	0.293471i
$286$		0			0
$287$			8.16744i			0.482109i
$288$		0			0
$289$	16.2420			0.955411
$290$		0			0
$291$		−	1.20436i		−	0.0706010i
$292$		0			0
$293$	14.8422			0.867093			0.433547	−	0.901131i	$-0.357262\pi$
$293$	14.8422			0.867093			0.433547	+	0.901131i	$0.357262\pi$
$294$		0			0
$295$	−12.3544	−	24.0739i	−0.719301	−	1.40164i
$296$		0			0
$297$	−10.5421			−0.611716
$298$		0			0
$299$	9.38093	+	3.87542i	0.542513	+	0.224121i
$300$		0			0
$301$			5.09570i			0.293711i
$302$		0			0
$303$		−	8.12641i		−	0.466850i
$304$		0			0
$305$	−20.6624	+	10.6036i	−1.18313	+	0.607163i
$306$		0			0
$307$	−11.2634			−0.642839			−0.321419	−	0.946937i	$-0.604160\pi$
$307$	−11.2634			−0.642839			−0.321419	+	0.946937i	$0.604160\pi$
$308$		0			0
$309$	−4.03292			−0.229425
$310$		0			0
$311$	2.05307			0.116419			0.0582095	−	0.998304i	$-0.481461\pi$
$311$	2.05307			0.116419			0.0582095	+	0.998304i	$0.481461\pi$
$312$		0			0
$313$		−	14.9879i		−	0.847169i	−0.905857	−	0.423584i	$-0.860772\pi$
$313$		−	14.9879i		−	0.847169i	0.905857	−	0.423584i	$-0.139228\pi$
$314$		0			0
$315$	−5.38093	+	2.76142i	−0.303181	+	0.155588i
$316$		0			0
$317$	−18.7064			−1.05065			−0.525327	−	0.850900i	$-0.676057\pi$
$317$	−18.7064			−1.05065			−0.525327	+	0.850900i	$0.676057\pi$
$318$		0			0
$319$		−	0.445884i		−	0.0249647i
$320$		0			0
$321$	13.0892			0.730567
$322$		0			0
$323$	5.46425			0.304039
$324$		0			0
$325$	−9.52286	−	15.3074i	−0.528233	−	0.849099i
$326$		0			0
$327$	9.49398			0.525018
$328$		0			0
$329$	−1.26788			−0.0699003
$330$		0			0
$331$		−	8.13533i		−	0.447158i	−0.974686	−	0.223579i	$-0.928226\pi$
$331$		−	8.13533i		−	0.447158i	0.974686	−	0.223579i	$-0.0717741\pi$
$332$		0			0
$333$	−7.50907			−0.411495
$334$		0			0
$335$	13.3385	−	6.84514i	0.728761	−	0.373990i
$336$		0			0
$337$			1.19858i			0.0652910i	0.999467	+	0.0326455i	$0.0103932\pi$
$337$			1.19858i			0.0652910i	−0.999467	+	0.0326455i	$0.989607\pi$
$338$		0			0
$339$	−14.3152			−0.777496
$340$		0			0
$341$	0.980938			0.0531207
$342$		0			0
$343$	−14.3364			−0.774091
$344$		0			0
$345$	4.33026	−	2.22223i	0.233133	−	0.119641i
$346$		0			0
$347$		−	29.9696i		−	1.60885i	−0.594053	−	0.804426i	$-0.702473\pi$
$347$		−	29.9696i		−	1.60885i	0.594053	−	0.804426i	$-0.297527\pi$
$348$		0			0
$349$			24.5528i			1.31428i	0.753768	+	0.657141i	$0.228234\pi$
$349$			24.5528i			1.31428i	−0.753768	+	0.657141i	$0.771766\pi$
$350$		0			0
$351$	13.9195	+	5.75036i	0.742966	+	0.306932i
$352$		0			0
$353$	2.50744			0.133458			0.0667289	−	0.997771i	$-0.478744\pi$
$353$	2.50744			0.133458			0.0667289	+	0.997771i	$0.478744\pi$
$354$		0			0
$355$	4.46233	+	8.69535i	0.236836	+	0.461501i
$356$		0			0
$357$	0.758020			0.0401187
$358$		0			0
$359$			0.303519i			0.0160191i	0.999968	+	0.00800957i	$0.00254955\pi$
$359$			0.303519i			0.0160191i	−0.999968	+	0.00800957i	$0.997450\pi$
$360$		0			0
$361$	−20.3895			−1.07313
$362$		0			0
$363$			3.58020i			0.187912i
$364$		0			0
$365$	−9.35972	+	4.80328i	−0.489910	+	0.251415i
$366$		0			0
$367$			23.8940i			1.24726i	0.781721	+	0.623629i	$0.214342\pi$
$367$			23.8940i			1.24726i	−0.781721	+	0.623629i	$0.785658\pi$
$368$		0			0
$369$			17.4239i			0.907051i
$370$		0			0
$371$			11.0737i			0.574918i
$372$		0			0
$373$			2.39444i			0.123979i	0.998077	+	0.0619897i	$0.0197446\pi$
$373$			2.39444i			0.123979i	−0.998077	+	0.0619897i	$0.980255\pi$
$374$		0			0
$375$	−8.54985	−	1.27797i	−0.441513	−	0.0659943i
$376$		0			0
$377$	−0.243214	+	0.588730i	−0.0125262	+	0.0303211i
$378$		0			0
$379$		−	15.1088i		−	0.776089i	−0.921641	−	0.388044i	$-0.873151\pi$
$379$		−	15.1088i		−	0.776089i	0.921641	−	0.388044i	$-0.126849\pi$
$380$		0			0
$381$	5.37294			0.275264
$382$		0			0
$383$	−9.04118			−0.461983			−0.230991	−	0.972956i	$-0.574197\pi$
$383$	−9.04118			−0.461983			−0.230991	+	0.972956i	$0.574197\pi$
$384$		0			0
$385$	−2.90132	−	5.65355i	−0.147865	−	0.288131i
$386$		0			0
$387$			10.8708i			0.552595i
$388$		0			0
$389$	0.306640			0.0155473			0.00777364	−	0.999970i	$-0.497526\pi$
$389$	0.306640			0.0155473			0.00777364	+	0.999970i	$0.497526\pi$
$390$		0			0
$391$	2.45093			0.123949
$392$		0			0
$393$			14.3762i			0.725182i
$394$		0			0
$395$	−16.5337	+	8.48488i	−0.831902	+	0.426921i
$396$		0			0
$397$	24.6200			1.23564			0.617821	−	0.786319i	$-0.288016\pi$
$397$	24.6200			1.23564			0.617821	+	0.786319i	$0.288016\pi$
$398$		0			0
$399$	−5.46425			−0.273555
$400$		0			0
$401$		−	0.327941i		−	0.0163766i	−0.999966	−	0.00818829i	$-0.997394\pi$
$401$		−	0.327941i		−	0.0163766i	0.999966	−	0.00818829i	$-0.00260644\pi$
$402$		0			0
$403$	−1.29520	−	0.535067i	−0.0645183	−	0.0266536i
$404$		0			0
$405$	−7.91114	+	4.05988i	−0.393108	+	0.201737i
$406$		0			0
$407$		−	7.88951i		−	0.391068i
$408$		0			0
$409$		−	11.6980i		−	0.578427i	−0.957265	−	0.289213i	$-0.906606\pi$
$409$		−	11.6980i		−	0.578427i	0.957265	−	0.289213i	$-0.0933937\pi$
$410$		0			0
$411$			8.85818i			0.436942i
$412$		0			0
$413$			13.6259i			0.670484i
$414$		0			0
$415$	18.2144	−	9.34737i	0.894109	−	0.458844i
$416$		0			0
$417$		−	2.26634i		−	0.110983i
$418$		0			0
$419$	−5.69080			−0.278014			−0.139007	−	0.990291i	$-0.544391\pi$
$419$	−5.69080			−0.278014			−0.139007	+	0.990291i	$0.544391\pi$
$420$		0			0
$421$		−	22.3739i		−	1.09044i	−0.838294	−	0.545218i	$-0.816447\pi$
$421$		−	22.3739i		−	1.09044i	0.838294	−	0.545218i	$-0.183553\pi$
$422$		0			0
$423$	−2.70480			−0.131512
$424$		0			0
$425$	−3.53662	−	2.53828i	−0.171551	−	0.123124i
$426$		0			0
$427$	11.6949			0.565957
$428$		0			0
$429$	−2.68653	+	6.50306i	−0.129707	+	0.313971i
$430$		0			0
$431$			23.6433i			1.13886i	0.822040	+	0.569430i	$0.192836\pi$
$431$			23.6433i			1.13886i	−0.822040	+	0.569430i	$0.807164\pi$
$432$		0			0
$433$			30.1685i			1.44980i	0.688852	+	0.724902i	$0.258115\pi$
$433$			30.1685i			1.44980i	−0.688852	+	0.724902i	$0.741885\pi$
$434$		0			0
$435$	0.139463	+	0.271759i	0.00668674	+	0.0130299i
$436$		0			0
$437$	−17.6677			−0.845162
$438$		0			0
$439$	11.1002			0.529786			0.264893	−	0.964278i	$-0.414663\pi$
$439$	11.1002			0.529786			0.264893	+	0.964278i	$0.414663\pi$
$440$		0			0
$441$	−13.7693			−0.655682
$442$		0			0
$443$			20.5289i			0.975358i	0.873023	+	0.487679i	$0.162156\pi$
$443$			20.5289i			0.975358i	−0.873023	+	0.487679i	$0.837844\pi$
$444$		0			0
$445$	6.28173	+	12.2407i	0.297783	+	0.580263i
$446$		0			0
$447$	2.19253			0.103703
$448$		0			0
$449$			40.5254i			1.91251i	0.292535	+	0.956255i	$0.405501\pi$
$449$			40.5254i			1.91251i	−0.292535	+	0.956255i	$0.594499\pi$
$450$		0			0
$451$	−18.3066			−0.862026
$452$		0			0
$453$	11.8750			0.557938
$454$		0			0
$455$	0.746990	+	9.04732i	0.0350194	+	0.424145i
$456$		0			0
$457$	33.5152			1.56777			0.783887	−	0.620903i	$-0.213234\pi$
$457$	33.5152			1.56777			0.783887	+	0.620903i	$0.213234\pi$
$458$		0			0
$459$	3.63670			0.169747
$460$		0			0
$461$		−	0.190873i		−	0.00888983i	−0.999990	−	0.00444492i	$-0.998585\pi$
$461$		−	0.190873i		−	0.00888983i	0.999990	−	0.00444492i	$-0.00141487\pi$
$462$		0			0
$463$	−10.7048			−0.497494			−0.248747	−	0.968568i	$-0.580019\pi$
$463$	−10.7048			−0.497494			−0.248747	+	0.968568i	$0.580019\pi$
$464$		0			0
$465$	−0.597866	+	0.306816i	−0.0277254	+	0.0142283i
$466$		0			0
$467$		−	29.1759i		−	1.35010i	−0.737773	−	0.675049i	$-0.764123\pi$
$467$		−	29.1759i		−	1.35010i	0.737773	−	0.675049i	$-0.235877\pi$
$468$		0			0
$469$	−7.54961			−0.348609
$470$		0			0
$471$	4.89975			0.225769
$472$		0			0
$473$	−11.4216			−0.525165
$474$		0			0
$475$	25.4940	+	18.2974i	1.16974	+	0.839540i
$476$		0			0
$477$			23.6239i			1.08166i
$478$		0			0
$479$		−	1.45210i		−	0.0663481i	−0.999450	−	0.0331740i	$-0.989438\pi$
$479$		−	1.45210i		−	0.0663481i	0.999450	−	0.0331740i	$-0.0105616\pi$
$480$		0			0
$481$	−4.30345	+	10.4170i	−0.196221	+	0.474976i
$482$		0			0
$483$	−2.45093			−0.111521
$484$		0			0
$485$	3.09868	−	1.59020i	0.140704	−	0.0722072i
$486$		0			0
$487$	−31.6067			−1.43223			−0.716117	−	0.697980i	$-0.754082\pi$
$487$	−31.6067			−1.43223			−0.716117	+	0.697980i	$0.754082\pi$
$488$		0			0
$489$			15.7489i			0.712189i
$490$		0			0
$491$	18.1374			0.818531			0.409266	−	0.912415i	$-0.365785\pi$
$491$	18.1374			0.818531			0.409266	+	0.912415i	$0.365785\pi$
$492$		0			0
$493$			0.153816i			0.00692752i
$494$		0			0
$495$	−6.18948	−	12.0609i	−0.278197	−	0.542097i
$496$		0			0
$497$		−	4.92157i		−	0.220763i
$498$		0			0
$499$		−	15.2537i		−	0.682848i	−0.939909	−	0.341424i	$-0.889091\pi$
$499$		−	15.2537i		−	0.682848i	0.939909	−	0.341424i	$-0.110909\pi$
$500$		0			0
$501$			4.79457i			0.214206i
$502$		0			0
$503$		−	26.0741i		−	1.16259i	−0.813694	−	0.581293i	$-0.802547\pi$
$503$		−	26.0741i		−	1.16259i	0.813694	−	0.581293i	$-0.197453\pi$
$504$		0			0
$505$	20.9083	−	10.7298i	0.930406	−	0.477471i
$506$		0			0
$507$	7.09439	−	7.12102i	0.315073	−	0.316256i
$508$		0			0
$509$			35.9439i			1.59319i	0.604515	+	0.796594i	$0.293367\pi$
$509$			35.9439i			1.59319i	−0.604515	+	0.796594i	$0.706633\pi$
$510$		0			0
$511$	5.29761			0.234352
$512$		0			0
$513$	−26.2154			−1.15744
$514$		0			0
$515$	−5.32493	−	10.3762i	−0.234644	−	0.457231i
$516$		0			0
$517$		−	2.84184i		−	0.124984i
$518$		0			0
$519$	−5.09385			−0.223595
$520$		0			0
$521$	−10.9634			−0.480317			−0.240158	−	0.970734i	$-0.577199\pi$
$521$	−10.9634			−0.480317			−0.240158	+	0.970734i	$0.577199\pi$
$522$		0			0
$523$		−	18.6440i		−	0.815247i	−0.913150	−	0.407623i	$-0.866358\pi$
$523$		−	18.6440i		−	0.815247i	0.913150	−	0.407623i	$-0.133642\pi$
$524$		0			0
$525$	3.53662	+	2.53828i	0.154351	+	0.110779i
$526$		0			0
$527$	−0.338392			−0.0147406
$528$		0			0
$529$	15.0753			0.655449
$530$		0			0
$531$			29.0685i			1.26147i
$532$		0			0
$533$	24.1715	+	9.98564i	1.04698	+	0.432526i
$534$		0			0
$535$	17.2825	+	33.6769i	0.747188	+	1.45598i
$536$		0			0
$537$			18.3608i			0.792328i
$538$		0			0
$539$		−	14.4669i		−	0.623135i
$540$		0			0
$541$			39.2339i			1.68680i	0.537287	+	0.843399i	$0.319449\pi$
$541$			39.2339i			1.68680i	−0.537287	+	0.843399i	$0.680551\pi$
$542$		0			0
$543$		−	5.51428i		−	0.236640i
$544$		0			0
$545$	12.5355	+	24.4269i	0.536963	+	1.04633i
$546$		0			0
$547$		−	34.7580i		−	1.48614i	−0.669211	−	0.743072i	$-0.733368\pi$
$547$		−	34.7580i		−	1.48614i	0.669211	−	0.743072i	$-0.266632\pi$
$548$		0			0
$549$	24.9492			1.06481
$550$		0			0
$551$		−	1.10879i		−	0.0472362i
$552$		0			0
$553$	9.35811			0.397947
$554$		0			0
$555$	2.46767	+	4.80853i	0.104747	+	0.204111i
$556$		0			0
$557$	−37.9452			−1.60779			−0.803895	−	0.594771i	$-0.797243\pi$
$557$	−37.9452			−1.60779			−0.803895	+	0.594771i	$0.797243\pi$
$558$		0			0
$559$	15.0807	+	6.23007i	0.637844	+	0.263504i
$560$		0			0
$561$			1.69904i			0.0717335i
$562$		0			0
$563$			35.3136i			1.48829i	0.668018	+	0.744145i	$0.267143\pi$
$563$			35.3136i			1.48829i	−0.668018	+	0.744145i	$0.732857\pi$
$564$		0			0
$565$	−18.9013	−	36.8313i	−0.795185	−	1.54951i
$566$		0			0
$567$	4.47771			0.188046
$568$		0			0
$569$	14.1088			0.591472			0.295736	−	0.955270i	$-0.404435\pi$
$569$	14.1088			0.591472			0.295736	+	0.955270i	$0.404435\pi$
$570$		0			0
$571$	3.97612			0.166396			0.0831978	−	0.996533i	$-0.473487\pi$
$571$	3.97612			0.166396			0.0831978	+	0.996533i	$0.473487\pi$
$572$		0			0
$573$		−	15.7885i		−	0.659573i
$574$		0			0
$575$	11.4351	+	8.20709i	0.476875	+	0.342259i
$576$		0			0
$577$	−33.5256			−1.39569			−0.697844	−	0.716250i	$-0.745857\pi$
$577$	−33.5256			−1.39569			−0.697844	+	0.716250i	$0.745857\pi$
$578$		0			0
$579$			9.40460i			0.390842i
$580$		0			0
$581$	−10.3094			−0.427704
$582$		0			0
$583$	−24.8208			−1.02797
$584$		0			0
$585$	1.59358	+	19.3009i	0.0658864	+	0.797996i
$586$		0			0
$587$	−36.1691			−1.49286			−0.746428	−	0.665466i	$-0.768233\pi$
$587$	−36.1691			−1.49286			−0.746428	+	0.665466i	$0.768233\pi$
$588$		0			0
$589$	2.43933			0.100511
$590$		0			0
$591$			7.24772i			0.298131i
$592$		0			0
$593$	−4.34692			−0.178507			−0.0892533	−	0.996009i	$-0.528448\pi$
$593$	−4.34692			−0.178507			−0.0892533	+	0.996009i	$0.528448\pi$
$594$		0			0
$595$	1.00086	+	1.95029i	0.0410314	+	0.0799543i
$596$		0			0
$597$		−	14.0049i		−	0.573184i
$598$		0			0
$599$	8.05947			0.329301			0.164650	−	0.986352i	$-0.447350\pi$
$599$	8.05947			0.329301			0.164650	+	0.986352i	$0.447350\pi$
$600$		0			0
$601$	22.4527			0.915863			0.457932	−	0.888987i	$-0.348590\pi$
$601$	22.4527			0.915863			0.457932	+	0.888987i	$0.348590\pi$
$602$		0			0
$603$	−16.1058			−0.655880
$604$		0			0
$605$	−9.21142	+	4.72717i	−0.374497	+	0.192187i
$606$		0			0
$607$			21.5799i			0.875902i	0.898999	+	0.437951i	$0.144296\pi$
$607$			21.5799i			0.875902i	−0.898999	+	0.437951i	$0.855704\pi$
$608$		0			0
$609$		−	0.153816i		−	0.00623293i
$610$		0			0
$611$	−1.55012	+	3.75227i	−0.0627113	+	0.151800i
$612$		0			0
$613$	14.7221			0.594621			0.297310	−	0.954781i	$-0.403910\pi$
$613$	14.7221			0.594621			0.297310	+	0.954781i	$0.403910\pi$
$614$		0			0
$615$	11.1576	−	5.72593i	0.449918	−	0.230892i
$616$		0			0
$617$	18.1947			0.732491			0.366245	−	0.930518i	$-0.380643\pi$
$617$	18.1947			0.732491			0.366245	+	0.930518i	$0.380643\pi$
$618$		0			0
$619$		−	25.2834i		−	1.01623i	−0.861290	−	0.508113i	$-0.830343\pi$
$619$		−	25.2834i		−	1.01623i	0.861290	−	0.508113i	$-0.169657\pi$
$620$		0			0
$621$	−11.7587			−0.471859
$622$		0			0
$623$		−	6.92822i		−	0.277573i
$624$		0			0
$625$	−8.00086	−	23.6851i	−0.320035	−	0.947406i
$626$		0			0
$627$		−	12.2477i		−	0.489124i
$628$		0			0
$629$			2.72163i			0.108519i
$630$		0			0
$631$			11.2180i			0.446583i	0.974752	+	0.223292i	$0.0716803\pi$
$631$			11.2180i			0.446583i	−0.974752	+	0.223292i	$0.928320\pi$
$632$		0			0
$633$		−	18.3608i		−	0.729777i
$634$		0			0
$635$	7.09425	+	13.8239i	0.281527	+	0.548586i
$636$		0			0
$637$	−7.89121	+	19.1016i	−0.312661	+	0.756835i
$638$		0			0
$639$		−	10.4994i		−	0.415348i
$640$		0			0
$641$	20.1125			0.794397			0.397199	−	0.917733i	$-0.369982\pi$
$641$	20.1125			0.794397			0.397199	+	0.917733i	$0.369982\pi$
$642$		0			0
$643$	20.3133			0.801080			0.400540	−	0.916279i	$-0.368823\pi$
$643$	20.3133			0.801080			0.400540	+	0.916279i	$0.368823\pi$
$644$		0			0
$645$	6.96127	−	3.57243i	0.274100	−	0.140664i
$646$		0			0
$647$			32.9110i			1.29386i	0.762548	+	0.646932i	$0.223948\pi$
$647$			32.9110i			1.29386i	−0.762548	+	0.646932i	$0.776052\pi$
$648$		0			0
$649$	−30.5412			−1.19885
$650$		0			0
$651$	0.338392			0.0132626
$652$		0			0
$653$		−	35.5250i		−	1.39020i	−0.718913	−	0.695100i	$-0.755360\pi$
$653$		−	35.5250i		−	1.39020i	0.718913	−	0.695100i	$-0.244640\pi$
$654$		0			0
$655$	−36.9882	+	18.9818i	−1.44525	+	0.741681i
$656$		0			0
$657$	11.3016			0.440916
$658$		0			0
$659$	−3.26586			−0.127220			−0.0636099	−	0.997975i	$-0.520261\pi$
$659$	−3.26586			−0.127220			−0.0636099	+	0.997975i	$0.520261\pi$
$660$		0			0
$661$		−	9.30703i		−	0.362002i	−0.983483	−	0.181001i	$-0.942066\pi$
$661$		−	9.30703i		−	0.362002i	0.983483	−	0.181001i	$-0.0579337\pi$
$662$		0			0
$663$	0.926767	−	2.24335i	0.0359926	−	0.0871246i
$664$		0			0
$665$	−7.21481	−	14.0589i	−0.279778	−	0.545179i
$666$		0			0
$667$		−	0.497338i		−	0.0192570i
$668$		0			0
$669$			19.3623i			0.748589i
$670$		0			0
$671$			26.2132i			1.01195i
$672$		0			0
$673$		−	1.02446i		−	0.0394900i	−0.999805	−	0.0197450i	$-0.993715\pi$
$673$		−	1.02446i		−	0.0394900i	0.999805	−	0.0197450i	$-0.00628544\pi$
$674$		0			0
$675$	16.9674	+	12.1777i	0.653075	+	0.468720i
$676$		0			0
$677$		−	19.5277i		−	0.750511i	−0.926921	−	0.375255i	$-0.877555\pi$
$677$		−	19.5277i		−	0.750511i	0.926921	−	0.375255i	$-0.122445\pi$
$678$		0			0
$679$	−1.75386			−0.0673068
$680$		0			0
$681$		−	20.0395i		−	0.767916i
$682$		0			0
$683$	35.1786			1.34607			0.673036	−	0.739609i	$-0.264990\pi$
$683$	35.1786			1.34607			0.673036	+	0.739609i	$0.264990\pi$
$684$		0			0
$685$	−22.7911	+	11.6960i	−0.870801	+	0.446883i
$686$		0			0
$687$	17.5193			0.668404
$688$		0			0
$689$	32.7725	+	13.5389i	1.24853	+	0.515790i
$690$		0			0
$691$		−	43.3023i		−	1.64730i	−0.567101	−	0.823648i	$-0.691935\pi$
$691$		−	43.3023i		−	1.64730i	0.567101	−	0.823648i	$-0.308065\pi$
$692$		0			0
$693$			6.82648i			0.259316i
$694$		0			0
$695$	5.83103	−	2.99240i	0.221184	−	0.113508i
$696$		0			0
$697$	6.31521			0.239206
$698$		0			0
$699$	−1.63344			−0.0617825
$700$		0			0
$701$	20.3603			0.768996			0.384498	−	0.923126i	$-0.374375\pi$
$701$	20.3603			0.768996			0.384498	+	0.923126i	$0.374375\pi$
$702$		0			0
$703$		−	19.6191i		−	0.739948i
$704$		0			0
$705$	0.888866	+	1.73206i	0.0334766	+	0.0652330i
$706$		0			0
$707$	−11.8341			−0.445067
$708$		0			0
$709$		−	36.0978i		−	1.35568i	−0.735210	−	0.677840i	$-0.762916\pi$
$709$		−	36.0978i		−	1.35568i	0.735210	−	0.677840i	$-0.237084\pi$
$710$		0			0
$711$	19.9640			0.748707
$712$		0			0
$713$	1.09414			0.0409757
$714$		0			0
$715$	−20.2788	+	1.67432i	−0.758384	+	0.0626159i
$716$		0			0
$717$	13.1764			0.492082
$718$		0			0
$719$	−36.5731			−1.36395			−0.681973	−	0.731378i	$-0.738878\pi$
$719$	−36.5731			−1.36395			−0.681973	+	0.731378i	$0.738878\pi$
$720$		0			0
$721$			5.87295i			0.218720i
$722$		0			0
$723$	−5.46425			−0.203218
$724$		0			0
$725$	−0.515062	+	0.717643i	−0.0191289	+	0.0266526i
$726$		0			0
$727$			37.4385i			1.38852i	0.719725	+	0.694259i	$0.244268\pi$
$727$			37.4385i			1.38852i	−0.719725	+	0.694259i	$0.755732\pi$
$728$		0			0
$729$	−0.136805			−0.00506686
$730$		0			0
$731$	3.94008			0.145729
$732$		0			0
$733$	−53.0734			−1.96031			−0.980156	−	0.198227i	$-0.936482\pi$
$733$	−53.0734			−1.96031			−0.980156	+	0.198227i	$0.936482\pi$
$734$		0			0
$735$	4.52495	+	8.81737i	0.166905	+	0.325233i
$736$		0			0
$737$		−	16.9218i		−	0.623323i
$738$		0			0
$739$		−	13.7572i		−	0.506069i	−0.967457	−	0.253034i	$-0.918571\pi$
$739$		−	13.7572i		−	0.506069i	0.967457	−	0.253034i	$-0.0814285\pi$
$740$		0			0
$741$	−6.68067	+	16.1714i	−0.245421	+	0.594071i
$742$		0			0
$743$	11.6316			0.426721			0.213360	−	0.976974i	$-0.431559\pi$
$743$	11.6316			0.426721			0.213360	+	0.976974i	$0.431559\pi$
$744$		0			0
$745$	2.89495	+	5.64113i	0.106063	+	0.206675i
$746$		0			0
$747$	−21.9933			−0.804693
$748$		0			0
$749$		−	19.0612i		−	0.696480i
$750$		0			0
$751$	−33.2712			−1.21408			−0.607041	−	0.794670i	$-0.707644\pi$
$751$	−33.2712			−1.21408			−0.607041	+	0.794670i	$0.707644\pi$
$752$		0			0
$753$			2.59081i			0.0944142i
$754$		0			0
$755$	15.6794	+	30.5531i	0.570632	+	1.11194i
$756$		0			0
$757$			8.54473i			0.310563i	0.987870	+	0.155282i	$0.0496285\pi$
$757$			8.54473i			0.310563i	−0.987870	+	0.155282i	$0.950371\pi$
$758$		0			0
$759$		−	5.49355i		−	0.199403i
$760$		0			0
$761$		−	13.1264i		−	0.475833i	−0.971286	−	0.237917i	$-0.923535\pi$
$761$		−	13.1264i		−	0.475833i	0.971286	−	0.237917i	$-0.0764645\pi$
$762$		0			0
$763$		−	13.8256i		−	0.500521i
$764$		0			0
$765$	2.13518	+	4.16063i	0.0771975	+	0.150428i
$766$		0			0
$767$	40.3256	+	16.6592i	1.45607	+	0.601528i
$768$		0			0
$769$			26.7910i			0.966107i	0.875591	+	0.483053i	$0.160472\pi$
$769$			26.7910i			0.966107i	−0.875591	+	0.483053i	$0.839528\pi$
$770$		0			0
$771$	10.3401			0.372390
$772$		0			0
$773$	−46.5001			−1.67249			−0.836246	−	0.548354i	$-0.815255\pi$
$773$	−46.5001			−1.67249			−0.836246	+	0.548354i	$0.815255\pi$
$774$		0			0
$775$	−1.57880	−	1.13313i	−0.0567123	−	0.0407031i
$776$		0			0
$777$		−	2.72163i		−	0.0976380i
$778$		0			0
$779$	−45.5237			−1.63106
$780$		0			0
$781$	11.0313			0.394731
$782$		0			0
$783$		−	0.737952i		−	0.0263722i
$784$		0			0
$785$	6.46946	+	12.6065i	0.230905	+	0.449944i
$786$		0			0
$787$	40.6477			1.44893			0.724467	−	0.689310i	$-0.242086\pi$
$787$	40.6477			1.44893			0.724467	+	0.689310i	$0.242086\pi$
$788$		0			0
$789$	1.51615			0.0539764
$790$		0			0
$791$			20.8466i			0.741218i
$792$		0			0
$793$	14.2984	−	34.6110i	0.507751	−	1.22907i
$794$		0			0
$795$	15.1279	−	7.76341i	0.536530	−	0.275340i
$796$		0			0
$797$			6.10094i			0.216107i	0.994145	+	0.108053i	$0.0344617\pi$
$797$			6.10094i			0.216107i	−0.994145	+	0.108053i	$0.965538\pi$
$798$		0			0
$799$			0.980345i			0.0346821i
$800$		0			0
$801$		−	14.7802i		−	0.522233i
$802$		0			0
$803$			11.8741i			0.419030i
$804$		0			0
$805$	−3.23612	−	6.30595i	−0.114058	−	0.222256i
$806$		0			0
$807$			19.1886i			0.675469i
$808$		0			0
$809$	−19.0229			−0.668810			−0.334405	−	0.942430i	$-0.608535\pi$
$809$	−19.0229			−0.668810			−0.334405	+	0.942430i	$0.608535\pi$
$810$		0			0
$811$		−	35.0059i		−	1.22922i	−0.788830	−	0.614611i	$-0.789313\pi$
$811$		−	35.0059i		−	1.22922i	0.788830	−	0.614611i	$-0.210687\pi$
$812$		0			0
$813$	14.8980			0.522496
$814$		0			0
$815$	−40.5200	+	20.7943i	−1.41935	+	0.728392i
$816$		0			0
$817$	−28.4024			−0.993674
$818$		0			0
$819$	3.72361	−	9.01345i	0.130113	−	0.314955i
$820$		0			0
$821$		−	41.0849i		−	1.43387i	−0.697138	−	0.716937i	$-0.745544\pi$
$821$		−	41.0849i		−	1.43387i	0.697138	−	0.716937i	$-0.254456\pi$
$822$		0			0
$823$			4.02182i			0.140192i	0.997540	+	0.0700959i	$0.0223305\pi$
$823$			4.02182i			0.140192i	−0.997540	+	0.0700959i	$0.977669\pi$
$824$		0			0
$825$	−5.68933	+	7.92703i	−0.198077	+	0.275984i
$826$		0			0
$827$	−22.4757			−0.781556			−0.390778	−	0.920485i	$-0.627794\pi$
$827$	−22.4757			−0.781556			−0.390778	+	0.920485i	$0.627794\pi$
$828$		0			0
$829$	22.1836			0.770468			0.385234	−	0.922819i	$-0.374121\pi$
$829$	22.1836			0.770468			0.385234	+	0.922819i	$0.374121\pi$
$830$		0			0
$831$	8.64932			0.300042
$832$		0			0
$833$			4.99064i			0.172915i
$834$		0			0
$835$	−12.3359	+	6.33059i	−0.426900	+	0.219079i
$836$		0			0
$837$	1.62348			0.0561158
$838$		0			0
$839$			4.77343i			0.164797i	0.996599	+	0.0823985i	$0.0262580\pi$
$839$			4.77343i			0.164797i	−0.996599	+	0.0823985i	$0.973742\pi$
$840$		0			0
$841$	−28.9688			−0.998924
$842$		0			0
$843$	14.8701			0.512154
$844$		0			0
$845$	27.6887	+	8.85068i	0.952521	+	0.304473i
$846$		0			0
$847$	5.21367			0.179144
$848$		0			0
$849$	−19.8340			−0.680701
$850$		0			0
$851$		−	8.79994i		−	0.301658i
$852$		0			0
$853$	−0.590247			−0.0202097			−0.0101048	−	0.999949i	$-0.503217\pi$
$853$	−0.590247			−0.0202097			−0.0101048	+	0.999949i	$0.503217\pi$
$854$		0			0
$855$	−15.3916	−	29.9922i	−0.526381	−	1.02571i
$856$		0			0
$857$		−	25.8548i		−	0.883185i	−0.897216	−	0.441592i	$-0.854414\pi$
$857$		−	25.8548i		−	0.883185i	0.897216	−	0.441592i	$-0.145586\pi$
$858$		0			0
$859$	−35.0144			−1.19468			−0.597338	−	0.801990i	$-0.703775\pi$
$859$	−35.0144			−1.19468			−0.597338	+	0.801990i	$0.703775\pi$
$860$		0			0
$861$	−6.31521			−0.215222
$862$		0			0
$863$	−21.0449			−0.716378			−0.358189	−	0.933649i	$-0.616606\pi$
$863$	−21.0449			−0.716378			−0.358189	+	0.933649i	$0.616606\pi$
$864$		0			0
$865$	−6.72575	−	13.1059i	−0.228682	−	0.445613i
$866$		0			0
$867$			12.5586i			0.426512i
$868$		0			0
$869$			20.9754i			0.711542i
$870$		0			0
$871$	−9.23027	+	22.3430i	−0.312756	+	0.757063i
$872$		0			0
$873$	−3.74156			−0.126633
$874$		0			0
$875$	−1.86105	+	12.4507i	−0.0629150	+	0.420912i
$876$		0			0
$877$	50.2516			1.69688			0.848438	−	0.529295i	$-0.177543\pi$
$877$	50.2516			1.69688			0.848438	+	0.529295i	$0.177543\pi$
$878$		0			0
$879$			11.4763i			0.387086i
$880$		0			0
$881$	7.85306			0.264576			0.132288	−	0.991211i	$-0.457768\pi$
$881$	7.85306			0.264576			0.132288	+	0.991211i	$0.457768\pi$
$882$		0			0
$883$		−	39.9529i		−	1.34452i	−0.740315	−	0.672261i	$-0.765323\pi$
$883$		−	39.9529i		−	1.34452i	0.740315	−	0.672261i	$-0.234677\pi$
$884$		0			0
$885$	18.6144	−	9.55264i	0.625716	−	0.321108i
$886$		0			0
$887$			27.4085i			0.920286i	0.887845	+	0.460143i	$0.152202\pi$
$887$			27.4085i			0.920286i	−0.887845	+	0.460143i	$0.847798\pi$
$888$		0			0
$889$		−	7.82436i		−	0.262421i
$890$		0			0
$891$			10.0364i			0.336232i
$892$		0			0
$893$		−	7.06689i		−	0.236485i
$894$		0			0
$895$	−47.2402	+	24.2430i	−1.57907	+	0.810354i
$896$		0			0
$897$	−2.99654	+	7.25350i	−0.100052	+	0.242187i
$898$		0			0
$899$			0.0686659i			0.00229014i
$900$		0			0
$901$	8.56239			0.285254
$902$		0			0
$903$	−3.94008			−0.131118
$904$		0			0
$905$	14.1876	−	7.28087i	0.471611	−	0.242024i
$906$		0			0
$907$		−	42.5322i		−	1.41226i	−0.708083	−	0.706129i	$-0.750440\pi$
$907$		−	42.5322i		−	1.41226i	0.708083	−	0.706129i	$-0.249560\pi$
$908$		0			0
$909$	−25.2461			−0.837360
$910$		0			0
$911$	3.01332			0.0998357			0.0499178	−	0.998753i	$-0.484104\pi$
$911$	3.01332			0.0998357			0.0499178	+	0.998753i	$0.484104\pi$
$912$		0			0
$913$		−	23.1076i		−	0.764749i
$914$		0			0
$915$	−8.19893	−	15.9765i	−0.271048	−	0.528168i
$916$		0			0
$917$	20.9353			0.691346
$918$		0			0
$919$	46.6282			1.53812			0.769062	−	0.639174i	$-0.220724\pi$
$919$	46.6282			1.53812			0.769062	+	0.639174i	$0.220724\pi$
$920$		0			0
$921$		−	8.70910i		−	0.286974i
$922$		0			0
$923$	−14.5653	−	6.01719i	−0.479424	−	0.198058i
$924$		0			0
$925$	−9.11355	+	12.6980i	−0.299652	+	0.417509i
$926$		0			0
$927$			12.5290i			0.411505i
$928$		0			0
$929$		−	52.6774i		−	1.72829i	−0.503242	−	0.864145i	$-0.667860\pi$
$929$		−	52.6774i		−	1.72829i	0.503242	−	0.864145i	$-0.332140\pi$
$930$		0			0
$931$		−	35.9754i		−	1.17905i
$932$		0			0
$933$			1.58747i			0.0519715i
$934$		0			0
$935$	−4.37142	+	2.24335i	−0.142961	+	0.0733655i
$936$		0			0
$937$			44.2590i			1.44588i	0.690912	+	0.722939i	$0.257209\pi$
$937$			44.2590i			1.44588i	−0.690912	+	0.722939i	$0.742791\pi$
$938$		0			0
$939$	11.5889			0.378191
$940$		0			0
$941$		−	22.2970i		−	0.726860i	−0.931621	−	0.363430i	$-0.881606\pi$
$941$		−	22.2970i		−	0.726860i	0.931621	−	0.363430i	$-0.118394\pi$
$942$		0			0
$943$	−20.4192			−0.664940
$944$		0			0
$945$	−4.80178	−	9.35680i	−0.156202	−	0.304377i
$946$		0			0
$947$	58.6162			1.90477			0.952386	−	0.304895i	$-0.0986215\pi$
$947$	58.6162			1.90477			0.952386	+	0.304895i	$0.0986215\pi$
$948$		0			0
$949$	6.47694	−	15.6782i	0.210250	−	0.508937i
$950$		0			0
$951$		−	14.4641i		−	0.469031i
$952$		0			0
$953$		−	41.3632i		−	1.33989i	−0.742413	−	0.669943i	$-0.766319\pi$
$953$		−	41.3632i		−	1.33989i	0.742413	−	0.669943i	$-0.233681\pi$
$954$		0			0
$955$	40.6219	−	20.8466i	1.31449	−	0.674579i
$956$		0			0
$957$	0.344765			0.0111447
$958$		0			0
$959$	12.8998			0.416555
$960$		0			0
$961$	30.8489			0.995127
$962$		0			0
$963$		−	40.6638i		−	1.31037i
$964$		0			0
$965$	−24.1969	+	12.4175i	−0.778926	+	0.399734i
$966$		0			0
$967$	14.4761			0.465521			0.232761	−	0.972534i	$-0.425224\pi$
$967$	14.4761			0.465521			0.232761	+	0.972534i	$0.425224\pi$
$968$		0			0
$969$			4.22505i			0.135728i
$970$		0			0
$971$	−5.86256			−0.188138			−0.0940692	−	0.995566i	$-0.529987\pi$
$971$	−5.86256			−0.188138			−0.0940692	+	0.995566i	$0.529987\pi$
$972$		0			0
$973$	−3.30037			−0.105805
$974$		0			0
$975$	11.8359	−	7.36325i	0.379053	−	0.235813i
$976$		0			0
$977$	23.1557			0.740818			0.370409	−	0.928869i	$-0.379218\pi$
$977$	23.1557			0.740818			0.370409	+	0.928869i	$0.379218\pi$
$978$		0			0
$979$	15.5290			0.496310
$980$		0			0
$981$		−	29.4947i		−	0.941693i
$982$		0			0
$983$	24.8471			0.792499			0.396249	−	0.918143i	$-0.370312\pi$
$983$	24.8471			0.792499			0.396249	+	0.918143i	$0.370312\pi$
$984$		0			0
$985$	−18.6475	+	9.56965i	−0.594160	+	0.304914i
$986$		0			0
$987$		−	0.980345i		−	0.0312047i
$988$		0			0
$989$	−12.7396			−0.405096
$990$		0			0
$991$	18.9061			0.600574			0.300287	−	0.953849i	$-0.402918\pi$
$991$	18.9061			0.600574			0.300287	+	0.953849i	$0.402918\pi$
$992$		0			0
$993$	6.29038			0.199619
$994$		0			0
$995$	36.0330	−	18.4916i	1.14232	−	0.586224i
$996$		0			0
$997$		−	17.4382i		−	0.552272i	−0.961119	−	0.276136i	$-0.910946\pi$
$997$		−	17.4382i		−	0.552272i	0.961119	−	0.276136i	$-0.0890541\pi$
$998$		0			0
$999$		−	13.0574i		−	0.413117i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.f.g.129.6		10
4.3	odd	2		520.2.f.b.129.5	yes	10
5.4	even	2		1040.2.f.f.129.5		10
13.12	even	2		1040.2.f.f.129.6		10
20.3	even	4		2600.2.k.f.2001.12		20
20.7	even	4		2600.2.k.f.2001.9		20
20.19	odd	2		520.2.f.a.129.6	yes	10
52.51	odd	2		520.2.f.a.129.5	✓	10
65.64	even	2	inner	1040.2.f.g.129.5		10
260.103	even	4		2600.2.k.f.2001.11		20
260.207	even	4		2600.2.k.f.2001.10		20
260.259	odd	2		520.2.f.b.129.6	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.f.a.129.5	✓	10	52.51	odd	2
520.2.f.a.129.6	yes	10	20.19	odd	2
520.2.f.b.129.5	yes	10	4.3	odd	2
520.2.f.b.129.6	yes	10	260.259	odd	2
1040.2.f.f.129.5		10	5.4	even	2
1040.2.f.f.129.6		10	13.12	even	2
1040.2.f.g.129.5		10	65.64	even	2	inner
1040.2.f.g.129.6		10	1.1	even	1	trivial
2600.2.k.f.2001.9		20	20.7	even	4
2600.2.k.f.2001.10		20	260.207	even	4
2600.2.k.f.2001.11		20	260.103	even	4
2600.2.k.f.2001.12		20	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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.773218i q^{3} +(-1.98940 + 1.02093i) q^{5} +1.12600 q^{7} +2.40213 q^{9} +O(q^{10})\)	\(q+0.773218i q^{3} +(-1.98940 + 1.02093i) q^{5} +1.12600 q^{7} +2.40213 q^{9} +2.52383i q^{11} +(1.37666 - 3.33239i) q^{13} +(-0.789402 - 1.53824i) q^{15} -0.870643i q^{17} +6.27610i q^{19} +0.870643i q^{21} +2.81508i q^{23} +(2.91540 - 4.06207i) q^{25} +4.17703i q^{27} -0.176669 q^{29} -0.388669i q^{31} -1.95147 q^{33} +(-2.24006 + 1.14957i) q^{35} -3.12600 q^{37} +(2.57666 + 1.06446i) q^{39} +7.25350i q^{41} +4.52548i q^{43} +(-4.77880 + 2.45241i) q^{45} -1.12600 q^{47} -5.73212 q^{49} +0.673197 q^{51} +9.83455i q^{53} +(-2.57666 - 5.02091i) q^{55} -4.85279 q^{57} +12.1011i q^{59} +10.3863 q^{61} +2.70480 q^{63} +(0.663401 + 8.03492i) q^{65} -6.70480 q^{67} -2.17667 q^{69} -4.37084i q^{71} +4.70480 q^{73} +(3.14087 + 2.25424i) q^{75} +2.84184i q^{77} +8.31093 q^{79} +3.97665 q^{81} -9.15573 q^{83} +(0.888866 + 1.73206i) q^{85} -0.136604i q^{87} -6.15295i q^{89} +(1.55012 - 3.75227i) q^{91} +0.300526 q^{93} +(-6.40746 - 12.4857i) q^{95} -1.55760 q^{97} +6.06259i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9}+O(q^{10}) \) 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9	\( 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9} + 2 q^{13} + 13 q^{15} - q^{25} + 8 q^{29} + 8 q^{33} + 3 q^{35} - 22 q^{37} + 12 q^{39} - 4 q^{45} - 2 q^{47} + 12 q^{49} + 30 q^{51} - 12 q^{55} - 12 q^{57} - 16 q^{61} - 24 q^{63} - 5 q^{65} - 16 q^{67} - 12 q^{69} - 4 q^{73} - 21 q^{75} - 28 q^{79} + 22 q^{81} + 4 q^{83} - 25 q^{85} - 2 q^{91} + 12 q^{93} + 10 q^{95} + 72 q^{97}+O(q^{100}) \) 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9 + 2 * q^13 + 13 * q^15 - q^25 + 8 * q^29 + 8 * q^33 + 3 * q^35 - 22 * q^37 + 12 * q^39 - 4 * q^45 - 2 * q^47 + 12 * q^49 + 30 * q^51 - 12 * q^55 - 12 * q^57 - 16 * q^61 - 24 * q^63 - 5 * q^65 - 16 * q^67 - 12 * q^69 - 4 * q^73 - 21 * q^75 - 28 * q^79 + 22 * q^81 + 4 * q^83 - 25 * q^85 - 2 * q^91 + 12 * q^93 + 10 * q^95 + 72 * q^97

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