LMFDB - Embedded newform 2016.2.s.u.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(289,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("2016.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.s (of order $3$, degree $2$, 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:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.1156923.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 $ x^6 - 3x^5 + 12x^4 - 19x^3 + 27x^2 - 18*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.2
Root			$0.500000 + 1.51496i$ of defining polynomial
Character	$\chi$	$=$	2016.289
Dual form			2016.2.s.u.865.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.227452 - 0.393958i) q^{5} +(2.16908 + 1.51496i) q^{7} +O(q^{10})$	$q+(0.227452 - 0.393958i) q^{5} +(2.16908 + 1.51496i) q^{7} +(-2.89653 - 5.01694i) q^{11} -5.88325 q^{13} +(1.45490 + 2.51997i) q^{17} +(-2.94163 + 5.09505i) q^{19} +(-1.45490 + 2.51997i) q^{23} +(2.39653 + 4.15091i) q^{25} -3.54510 q^{29} +(2.16908 + 3.75696i) q^{31} +(1.09019 - 0.509947i) q^{35} +(-3.85144 + 6.67088i) q^{37} -9.58612 q^{41} +10.7931 q^{43} +(2.45490 - 4.25202i) q^{47} +(2.40981 + 6.57212i) q^{49} +(6.56561 + 11.3720i) q^{53} -2.63529 q^{55} +(-0.896531 - 1.55284i) q^{59} +(-2.33816 + 4.04981i) q^{61} +(-1.33816 + 2.31776i) q^{65} +(-3.94163 - 6.82710i) q^{67} +0.909808 q^{71} +(-2.60347 - 4.50934i) q^{73} +(1.31764 - 15.2703i) q^{77} +(1.37602 - 2.38333i) q^{79} -9.97345 q^{83} +1.32368 q^{85} +(-2.45490 + 4.25202i) q^{89} +(-12.7612 - 8.91288i) q^{91} +(1.33816 + 2.31776i) q^{95} -5.79306 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{7}+O(q^{10}) $ 6 * q - 3 * q^7	$ 6 q - 3 q^{7} - 6 q^{13} + 6 q^{17} - 3 q^{19} - 6 q^{23} - 3 q^{25} - 24 q^{29} - 3 q^{31} + 12 q^{35} - 3 q^{37} + 12 q^{41} + 30 q^{43} + 12 q^{47} + 9 q^{49} + 6 q^{53} - 24 q^{55} + 12 q^{59} + 18 q^{61} + 24 q^{65} - 9 q^{67} - 33 q^{73} + 12 q^{77} + 27 q^{79} - 36 q^{83} + 72 q^{85} - 12 q^{89} - 51 q^{91} - 24 q^{95}+O(q^{100}) $ 6 * q - 3 * q^7 - 6 * q^13 + 6 * q^17 - 3 * q^19 - 6 * q^23 - 3 * q^25 - 24 * q^29 - 3 * q^31 + 12 * q^35 - 3 * q^37 + 12 * q^41 + 30 * q^43 + 12 * q^47 + 9 * q^49 + 6 * q^53 - 24 * q^55 + 12 * q^59 + 18 * q^61 + 24 * q^65 - 9 * q^67 - 33 * q^73 + 12 * q^77 + 27 * q^79 - 36 * q^83 + 72 * q^85 - 12 * q^89 - 51 * q^91 - 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	0.227452	−	0.393958i	0.101720	−	0.176184i	−0.810674	−	0.585498i	$-0.800899\pi$
$5$	0.227452	−	0.393958i	0.101720	−	0.176184i	0.912393	+	0.409315i	$0.134232\pi$
$6$		0			0
$7$	2.16908	+	1.51496i	0.819835	+	0.572600i
$7$	2.16908	+	1.51496i	0.819835	+	0.572600i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.89653	−	5.01694i	−0.873337	−	1.51266i	−0.858524	−	0.512774i	$-0.828618\pi$
$11$	−2.89653	−	5.01694i	−0.873337	−	1.51266i	−0.0148132	−	0.999890i	$-0.504715\pi$
$12$		0			0
$13$	−5.88325			−1.63172			−0.815861	−	0.578249i	$-0.803736\pi$
$13$	−5.88325			−1.63172			−0.815861	+	0.578249i	$0.803736\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.45490	+	2.51997i	0.352866	+	0.611182i	0.986750	−	0.162246i	$-0.0518738\pi$
$17$	1.45490	+	2.51997i	0.352866	+	0.611182i	−0.633884	+	0.773428i	$0.718540\pi$
$18$		0			0
$19$	−2.94163	+	5.09505i	−0.674856	+	1.16888i	0.301656	+	0.953417i	$0.402461\pi$
$19$	−2.94163	+	5.09505i	−0.674856	+	1.16888i	−0.976511	+	0.215467i	$0.930873\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.45490	+	2.51997i	−0.303368	+	0.525450i	−0.976897	−	0.213712i	$-0.931445\pi$
$23$	−1.45490	+	2.51997i	−0.303368	+	0.525450i	0.673528	+	0.739161i	$0.264778\pi$
$24$		0			0
$25$	2.39653	+	4.15091i	0.479306	+	0.830183i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.54510			−0.658308			−0.329154	−	0.944276i	$-0.606763\pi$
$29$	−3.54510			−0.658308			−0.329154	+	0.944276i	$0.606763\pi$
$30$		0			0
$31$	2.16908	+	3.75696i	0.389578	+	0.674769i	0.992393	−	0.123112i	$-0.0392875\pi$
$31$	2.16908	+	3.75696i	0.389578	+	0.674769i	−0.602815	+	0.797881i	$0.705954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.09019	−	0.509947i	0.184276	−	0.0861968i
$36$		0			0
$37$	−3.85144	+	6.67088i	−0.633172	+	1.09669i	0.353727	+	0.935349i	$0.384914\pi$
$37$	−3.85144	+	6.67088i	−0.633172	+	1.09669i	−0.986899	+	0.161338i	$0.948419\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−9.58612			−1.49710			−0.748551	−	0.663078i	$-0.769250\pi$
$41$	−9.58612			−1.49710			−0.748551	+	0.663078i	$0.769250\pi$
$42$		0			0
$43$	10.7931			1.64593			0.822963	−	0.568095i	$-0.192319\pi$
$43$	10.7931			1.64593			0.822963	+	0.568095i	$0.192319\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.45490	−	4.25202i	0.358085	−	0.620221i	−0.629556	−	0.776955i	$-0.716763\pi$
$47$	2.45490	−	4.25202i	0.358085	−	0.620221i	0.987641	+	0.156734i	$0.0500966\pi$
$48$		0			0
$49$	2.40981	+	6.57212i	0.344258	+	0.938875i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.56561	+	11.3720i	0.901856	+	1.56206i	0.825083	+	0.565012i	$0.191128\pi$
$53$	6.56561	+	11.3720i	0.901856	+	1.56206i	0.0767730	+	0.997049i	$0.475538\pi$
$54$		0			0
$55$	−2.63529			−0.355342
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.896531	−	1.55284i	−0.116718	−	0.202162i	0.801747	−	0.597664i	$-0.203904\pi$
$59$	−0.896531	−	1.55284i	−0.116718	−	0.202162i	−0.918465	+	0.395501i	$0.870571\pi$
$60$		0			0
$61$	−2.33816	+	4.04981i	−0.299370	+	0.518525i	−0.975992	−	0.217806i	$-0.930110\pi$
$61$	−2.33816	+	4.04981i	−0.299370	+	0.518525i	0.676622	+	0.736331i	$0.263443\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.33816	+	2.31776i	−0.165978	+	0.287482i
$66$		0			0
$67$	−3.94163	−	6.82710i	−0.481546	−	0.834063i	0.518229	−	0.855242i	$-0.326591\pi$
$67$	−3.94163	−	6.82710i	−0.481546	−	0.834063i	−0.999776	+	0.0211789i	$0.993258\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.909808			0.107974			0.0539872	−	0.998542i	$-0.482807\pi$
$71$	0.909808			0.107974			0.0539872	+	0.998542i	$0.482807\pi$
$72$		0			0
$73$	−2.60347	−	4.50934i	−0.304713	−	0.527778i	0.672484	−	0.740111i	$-0.265227\pi$
$73$	−2.60347	−	4.50934i	−0.304713	−	0.527778i	−0.977197	+	0.212333i	$0.931894\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.31764	−	15.2703i	0.150159	−	1.74021i
$78$		0			0
$79$	1.37602	−	2.38333i	0.154814	−	0.268146i	−0.778177	−	0.628045i	$-0.783856\pi$
$79$	1.37602	−	2.38333i	0.154814	−	0.268146i	0.932991	+	0.359899i	$0.117189\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−9.97345			−1.09473			−0.547364	−	0.836895i	$-0.684369\pi$
$83$	−9.97345			−1.09473			−0.547364	+	0.836895i	$0.684369\pi$
$84$		0			0
$85$	1.32368			0.143574
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.45490	+	4.25202i	−0.260219	+	0.450713i	−0.966300	−	0.257418i	$-0.917128\pi$
$89$	−2.45490	+	4.25202i	−0.260219	+	0.450713i	0.706081	+	0.708131i	$0.250462\pi$
$90$		0			0
$91$	−12.7612	−	8.91288i	−1.33774	−	0.934324i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.33816	+	2.31776i	0.137292	+	0.237797i
$96$		0			0
$97$	−5.79306			−0.588196			−0.294098	−	0.955775i	$-0.595019\pi$
$97$	−5.79306			−0.588196			−0.294098	+	0.955775i	$0.595019\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.33816	+	14.4421i	0.829678	+	1.43704i	0.898291	+	0.439401i	$0.144809\pi$
$101$	8.33816	+	14.4421i	0.829678	+	1.43704i	−0.0686134	+	0.997643i	$0.521858\pi$
$102$		0			0
$103$	0.396531	−	0.686812i	0.0390714	−	0.0676736i	−0.845828	−	0.533455i	$-0.820893\pi$
$103$	0.396531	−	0.686812i	0.0390714	−	0.0676736i	0.884900	+	0.465781i	$0.154227\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.98672	−	10.3693i	0.578758	−	1.00244i	−0.416864	−	0.908969i	$-0.636871\pi$
$107$	5.98672	−	10.3693i	0.578758	−	1.00244i	0.995622	−	0.0934699i	$-0.0297959\pi$
$108$		0			0
$109$	2.30634	+	3.99470i	0.220907	+	0.382623i	0.955084	−	0.296336i	$-0.0957649\pi$
$109$	2.30634	+	3.99470i	0.220907	+	0.382623i	−0.734176	+	0.678959i	$0.762432\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.00000			−0.752577			−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000			−0.752577			−0.376288	+	0.926503i	$0.622800\pi$
$114$		0			0
$115$	0.661842	+	1.14634i	0.0617171	+	0.106897i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.661842	+	7.67013i	−0.0606709	+	0.703119i
$120$		0			0
$121$	−11.2798	+	19.5372i	−1.02544	+	1.77611i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.45490			0.398459
$126$		0			0
$127$	−1.24797			−0.110739			−0.0553696	−	0.998466i	$-0.517634\pi$
$127$	−1.24797			−0.110739			−0.0553696	+	0.998466i	$0.517634\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.89653	+	5.01694i	−0.253071	+	0.438332i	−0.964370	−	0.264558i	$-0.914774\pi$
$131$	−2.89653	+	5.01694i	−0.253071	+	0.438332i	0.711299	+	0.702890i	$0.248107\pi$
$132$		0			0
$133$	−14.0994	+	6.59512i	−1.22257	+	0.571870i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.54510	−	2.67618i	−0.132006	−	0.228642i	0.792443	−	0.609945i	$-0.208809\pi$
$137$	−1.54510	−	2.67618i	−0.132006	−	0.228642i	−0.924450	+	0.381303i	$0.875475\pi$
$138$		0			0
$139$	−5.70287			−0.483711			−0.241856	−	0.970312i	$-0.577756\pi$
$139$	−5.70287			−0.483711			−0.241856	+	0.970312i	$0.577756\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	17.0410	+	29.5159i	1.42504	+	2.46825i
$144$		0			0
$145$	−0.806339	+	1.39662i	−0.0669628	+	0.115983i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.54510	−	2.67618i	0.126579	−	0.219242i	−0.795770	−	0.605599i	$-0.792934\pi$
$149$	1.54510	−	2.67618i	0.126579	−	0.219242i	0.922349	+	0.386357i	$0.126267\pi$
$150$		0			0
$151$	0.862740	+	1.49431i	0.0702088	+	0.121605i	0.898993	−	0.437964i	$-0.144300\pi$
$151$	0.862740	+	1.49431i	0.0702088	+	0.121605i	−0.828784	+	0.559569i	$0.810967\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.97345			0.158511
$156$		0			0
$157$	2.79306	+	4.83773i	0.222911	+	0.386093i	0.955691	−	0.294373i	$-0.0951109\pi$
$157$	2.79306	+	4.83773i	0.222911	+	0.386093i	−0.732780	+	0.680466i	$0.761778\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.97345	+	3.26189i	−0.549585	+	0.257073i
$162$		0			0
$163$	4.42835	−	7.67013i	0.346855	−	0.600771i	−0.638834	−	0.769345i	$-0.720583\pi$
$163$	4.42835	−	7.67013i	0.346855	−	0.600771i	0.985689	+	0.168574i	$0.0539162\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	17.4057			1.34690			0.673448	−	0.739234i	$-0.264812\pi$
$167$	17.4057			1.34690			0.673448	+	0.739234i	$0.264812\pi$
$168$		0			0
$169$	21.6127			1.66251
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.33816	+	14.4421i	−0.633938	+	1.09801i	0.352801	+	0.935699i	$0.385229\pi$
$173$	−8.33816	+	14.4421i	−0.633938	+	1.09801i	−0.986739	+	0.162315i	$0.948104\pi$
$174$		0			0
$175$	−1.09019	+	12.6343i	−0.0824108	+	0.955064i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.09019	+	5.35237i	0.230972	+	0.400055i	0.958094	−	0.286453i	$-0.0924762\pi$
$179$	3.09019	+	5.35237i	0.230972	+	0.400055i	−0.727123	+	0.686508i	$0.759143\pi$
$180$		0			0
$181$	−3.20694			−0.238370			−0.119185	−	0.992872i	$-0.538028\pi$
$181$	−3.20694			−0.238370			−0.119185	+	0.992872i	$0.538028\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.75203	+	3.03461i	0.128812	+	0.223109i
$186$		0			0
$187$	8.42835	−	14.5983i	0.616342	−	1.06754i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.6763	+	18.4919i	−0.772511	+	1.33803i	0.163672	+	0.986515i	$0.447666\pi$
$191$	−10.6763	+	18.4919i	−0.772511	+	1.33803i	−0.936183	+	0.351514i	$0.885667\pi$
$192$		0			0
$193$	−8.20287	−	14.2078i	−0.590456	−	1.02270i	−0.994171	−	0.107814i	$-0.965615\pi$
$193$	−8.20287	−	14.2078i	−0.590456	−	1.02270i	0.403716	−	0.914885i	$-0.367719\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−24.6763			−1.75811			−0.879057	−	0.476716i	$-0.841827\pi$
$197$	−24.6763			−1.75811			−0.879057	+	0.476716i	$0.841827\pi$
$198$		0			0
$199$	6.90981	+	11.9681i	0.489823	+	0.848399i	0.999931	−	0.0117114i	$-0.00372795\pi$
$199$	6.90981	+	11.9681i	0.489823	+	0.848399i	−0.510108	+	0.860110i	$0.670395\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.68959	−	5.37067i	−0.539704	−	0.376947i
$204$		0			0
$205$	−2.18038	+	3.77654i	−0.152285	+	0.263765i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	34.0821			2.35751
$210$		0			0
$211$	−11.5861			−0.797622			−0.398811	−	0.917033i	$-0.630577\pi$
$211$	−11.5861			−0.797622			−0.398811	+	0.917033i	$0.630577\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.45490	−	4.25202i	0.167423	−	0.289985i
$216$		0			0
$217$	−0.986723	+	11.4352i	−0.0669831	+	0.776272i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.55957	−	14.8256i	−0.575779	−	0.997279i
$222$		0			0
$223$	−20.4549			−1.36976			−0.684881	−	0.728655i	$-0.740146\pi$
$223$	−20.4549			−1.36976			−0.684881	+	0.728655i	$0.740146\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.86998	−	15.3633i	−0.588721	−	1.01969i	−0.994400	−	0.105679i	$-0.966298\pi$
$227$	−8.86998	−	15.3633i	−0.588721	−	1.01969i	0.405679	−	0.914016i	$-0.367035\pi$
$228$		0			0
$229$	12.0994	−	20.9568i	0.799551	−	1.38486i	−0.120358	−	0.992731i	$-0.538404\pi$
$229$	12.0994	−	20.9568i	0.799551	−	1.38486i	0.919909	−	0.392132i	$-0.128262\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.3382	−	19.6383i	0.742787	−	1.28655i	−0.208434	−	0.978036i	$-0.566837\pi$
$233$	11.3382	−	19.6383i	0.742787	−	1.28655i	0.951221	−	0.308509i	$-0.0998300\pi$
$234$		0			0
$235$	−1.11675	−	1.93426i	−0.0728485	−	0.126177i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.90981			0.188220			0.0941099	−	0.995562i	$-0.469999\pi$
$239$	2.90981			0.188220			0.0941099	+	0.995562i	$0.469999\pi$
$240$		0			0
$241$	−10.8700	−	18.8274i	−0.700197	−	1.21278i	−0.968397	−	0.249413i	$-0.919762\pi$
$241$	−10.8700	−	18.8274i	−0.700197	−	1.21278i	0.268200	−	0.963363i	$-0.413571\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.13726	+	0.545479i	0.200432	+	0.0348494i
$246$		0			0
$247$	17.3063	−	29.9755i	1.10118	−	1.90729i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.02655			−0.506632			−0.253316	−	0.967384i	$-0.581521\pi$
$251$	−8.02655			−0.506632			−0.253316	+	0.967384i	$0.581521\pi$
$252$		0			0
$253$	16.8567			1.05977
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.45490	+	11.1802i	−0.402646	+	0.697403i	−0.994044	−	0.108976i	$-0.965243\pi$
$257$	−6.45490	+	11.1802i	−0.402646	+	0.697403i	0.591398	+	0.806379i	$0.298576\pi$
$258$		0			0
$259$	−18.4602	+	8.63491i	−1.14706	+	0.536547i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.45490	−	7.71612i	−0.274701	−	0.475796i	0.695359	−	0.718663i	$-0.255246\pi$
$263$	−4.45490	−	7.71612i	−0.274701	−	0.475796i	−0.970060	+	0.242867i	$0.921912\pi$
$264$		0			0
$265$	5.97345			0.366946
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.02051	−	5.23168i	−0.184164	−	0.318981i	0.759130	−	0.650938i	$-0.225624\pi$
$269$	−3.02051	−	5.23168i	−0.184164	−	0.318981i	−0.943295	+	0.331957i	$0.892291\pi$
$270$		0			0
$271$	6.68236	−	11.5742i	0.405924	−	0.703081i	−0.588504	−	0.808494i	$-0.700283\pi$
$271$	6.68236	−	11.5742i	0.405924	−	0.703081i	0.994429	+	0.105413i	$0.0336164\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	13.8833	−	24.0465i	0.837192	−	1.45006i
$276$		0			0
$277$	−2.39653	−	4.15091i	−0.143994	−	0.249404i	0.785003	−	0.619491i	$-0.212661\pi$
$277$	−2.39653	−	4.15091i	−0.143994	−	0.249404i	−0.928997	+	0.370087i	$0.879328\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.32368			−0.436894			−0.218447	−	0.975849i	$-0.570099\pi$
$281$	−7.32368			−0.436894			−0.218447	+	0.975849i	$0.570099\pi$
$282$		0			0
$283$	6.60347	+	11.4375i	0.392535	+	0.679891i	0.992783	−	0.119923i	$-0.0382647\pi$
$283$	6.60347	+	11.4375i	0.392535	+	0.679891i	−0.600248	+	0.799814i	$0.704931\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−20.7931	−	14.5226i	−1.22738	−	0.857240i
$288$		0			0
$289$	4.26651	−	7.38981i	0.250971	−	0.434695i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.54510			−0.440789			−0.220395	−	0.975411i	$-0.570735\pi$
$293$	−7.54510			−0.440789			−0.220395	+	0.975411i	$0.570735\pi$
$294$		0			0
$295$	−0.815671			−0.0474902
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.55957	−	14.8256i	0.495013	−	0.857387i
$300$		0			0
$301$	23.4110	+	16.3510i	1.34939	+	0.942458i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.06364	+	1.84227i	0.0609037	+	0.105488i
$306$		0			0
$307$	23.1086			1.31888			0.659439	−	0.751758i	$-0.270794\pi$
$307$	23.1086			1.31888			0.659439	+	0.751758i	$0.270794\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.70287	−	6.41356i	−0.209971	−	0.363680i	0.741734	−	0.670694i	$-0.234003\pi$
$311$	−3.70287	−	6.41356i	−0.209971	−	0.363680i	−0.951705	+	0.307014i	$0.900670\pi$
$312$		0			0
$313$	−7.29306	+	12.6320i	−0.412228	+	0.714000i	−0.995133	−	0.0985402i	$-0.968583\pi$
$313$	−7.29306	+	12.6320i	−0.412228	+	0.714000i	0.582905	+	0.812540i	$0.301916\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.682356	−	1.18188i	0.0383249	−	0.0663807i	−0.846227	−	0.532823i	$-0.821131\pi$
$317$	0.682356	−	1.18188i	0.0383249	−	0.0663807i	0.884552	+	0.466442i	$0.154464\pi$
$318$		0			0
$319$	10.2685	+	17.7855i	0.574925	+	0.995799i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−17.1191			−0.952534
$324$		0			0
$325$	−14.0994	−	24.4209i	−0.782094	−	1.35463i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	11.7665	−	5.50389i	0.648709	−	0.303439i
$330$		0			0
$331$	−5.48672	+	9.50328i	−0.301578	+	0.522348i	−0.976493	−	0.215547i	$-0.930847\pi$
$331$	−5.48672	+	9.50328i	−0.301578	+	0.522348i	0.674916	+	0.737895i	$0.264180\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.58612			−0.195931
$336$		0			0
$337$	−22.1722			−1.20780			−0.603900	−	0.797060i	$-0.706387\pi$
$337$	−22.1722			−1.20780			−0.603900	+	0.797060i	$0.706387\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.5656	−	21.7643i	0.680466	−	1.17860i
$342$		0			0
$343$	−4.72942	+	17.9062i	−0.255365	+	0.966845i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.116746	+	0.202210i	0.00626725	+	0.0108552i	0.869142	−	0.494563i	$-0.164672\pi$
$347$	0.116746	+	0.202210i	0.00626725	+	0.0108552i	−0.862875	+	0.505418i	$0.831338\pi$
$348$		0			0
$349$	7.35263			0.393577			0.196789	−	0.980446i	$-0.436949\pi$
$349$	7.35263			0.393577			0.196789	+	0.980446i	$0.436949\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.58612	+	16.6037i	0.510218	+	0.883723i	0.999930	+	0.0118391i	$0.00376859\pi$
$353$	9.58612	+	16.6037i	0.510218	+	0.883723i	−0.489712	+	0.871884i	$0.662898\pi$
$354$		0			0
$355$	0.206938	−	0.358427i	0.0109831	−	0.0190233i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.4694	−	18.1335i	0.552553	−	0.957049i	−0.445537	−	0.895264i	$-0.646987\pi$
$359$	10.4694	−	18.1335i	0.552553	−	0.957049i	0.998089	−	0.0617857i	$-0.0196795\pi$
$360$		0			0
$361$	−7.80634	−	13.5210i	−0.410860	−	0.711630i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.36866			−0.123981
$366$		0			0
$367$	12.5072	+	21.6632i	0.652872	+	1.13081i	0.982423	+	0.186670i	$0.0597696\pi$
$367$	12.5072	+	21.6632i	0.652872	+	1.13081i	−0.329550	+	0.944138i	$0.606897\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.98672	+	34.6133i	−0.155063	+	1.79703i
$372$		0			0
$373$	10.1630	−	17.6029i	0.526222	−	0.911444i	−0.473311	−	0.880895i	$-0.656941\pi$
$373$	10.1630	−	17.6029i	0.526222	−	0.911444i	0.999533	−	0.0305483i	$-0.00972533\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20.8567			1.07417
$378$		0			0
$379$	20.7931			1.06807			0.534034	−	0.845463i	$-0.320675\pi$
$379$	20.7931			1.06807			0.534034	+	0.845463i	$0.320675\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.5861	+	21.7998i	−0.643121	+	1.11392i	0.341611	+	0.939841i	$0.389027\pi$
$383$	−12.5861	+	21.7998i	−0.643121	+	1.11392i	−0.984732	+	0.174077i	$0.944306\pi$
$384$		0			0
$385$	−5.71615	−	3.99235i	−0.291322	−	0.203469i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	12.2214	+	21.1681i	0.619650	+	1.07327i	0.989549	+	0.144194i	$0.0460589\pi$
$389$	12.2214	+	21.1681i	0.619650	+	1.07327i	−0.369899	+	0.929072i	$0.620608\pi$
$390$		0			0
$391$	−8.46698			−0.428194
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.625956	−	1.08419i	−0.0314952	−	0.0545514i
$396$		0			0
$397$	4.39653	−	7.61502i	0.220656	−	0.382187i	−0.734352	−	0.678769i	$-0.762514\pi$
$397$	4.39653	−	7.61502i	0.220656	−	0.382187i	0.955007	+	0.296583i	$0.0958470\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.5596	−	28.6820i	0.826945	−	1.43231i	−0.0734778	−	0.997297i	$-0.523410\pi$
$401$	16.5596	−	28.6820i	0.826945	−	1.43231i	0.900423	−	0.435015i	$-0.143257\pi$
$402$		0			0
$403$	−12.7612	−	22.1031i	−0.635683	−	1.10103i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	44.6232			2.21189
$408$		0			0
$409$	3.47345	+	6.01618i	0.171751	+	0.297481i	0.939032	−	0.343830i	$-0.111724\pi$
$409$	3.47345	+	6.01618i	0.171751	+	0.297481i	−0.767281	+	0.641311i	$0.778391\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.407836	−	4.72643i	0.0200683	−	0.232573i
$414$		0			0
$415$	−2.26848	+	3.92912i	−0.111355	+	0.192873i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−29.2254			−1.42775			−0.713876	−	0.700272i	$-0.753062\pi$
$419$	−29.2254			−1.42775			−0.713876	+	0.700272i	$0.753062\pi$
$420$		0			0
$421$	33.2359			1.61982			0.809909	−	0.586556i	$-0.199516\pi$
$421$	33.2359			1.61982			0.809909	+	0.586556i	$0.199516\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.97345	+	12.0784i	−0.338262	+	0.585887i
$426$		0			0
$427$	−11.2069	+	5.24215i	−0.542342	+	0.253685i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.09019	−	5.35237i	−0.148849	−	0.257815i	0.781953	−	0.623337i	$-0.214224\pi$
$431$	−3.09019	−	5.35237i	−0.148849	−	0.257815i	−0.930802	+	0.365523i	$0.880890\pi$
$432$		0			0
$433$	16.6127			0.798354			0.399177	−	0.916874i	$-0.369296\pi$
$433$	16.6127			0.798354			0.399177	+	0.916874i	$0.369296\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.55957	−	14.8256i	−0.409460	−	0.709205i
$438$		0			0
$439$	−13.8136	+	23.9258i	−0.659286	+	1.14192i	0.321515	+	0.946905i	$0.395808\pi$
$439$	−13.8136	+	23.9258i	−0.659286	+	1.14192i	−0.980801	+	0.195012i	$0.937525\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.0133	−	17.3435i	0.475745	−	0.824015i	−0.523869	−	0.851799i	$-0.675512\pi$
$443$	10.0133	−	17.3435i	0.475745	−	0.824015i	0.999614	+	0.0277842i	$0.00884512\pi$
$444$		0			0
$445$	1.11675	+	1.93426i	0.0529388	+	0.0916927i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−15.0371			−0.709644			−0.354822	−	0.934934i	$-0.615459\pi$
$449$	−15.0371			−0.709644			−0.354822	+	0.934934i	$0.615459\pi$
$450$		0			0
$451$	27.7665	+	48.0930i	1.30747	+	2.26461i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.41388	+	3.00015i	−0.300687	+	0.140649i
$456$		0			0
$457$	21.0596	−	36.4762i	0.985125	−	1.70629i	0.343746	−	0.939063i	$-0.388304\pi$
$457$	21.0596	−	36.4762i	0.985125	−	1.70629i	0.641379	−	0.767224i	$-0.278363\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−20.9098			−0.973867			−0.486933	−	0.873439i	$-0.661885\pi$
$461$	−20.9098			−0.973867			−0.486933	+	0.873439i	$0.661885\pi$
$462$		0			0
$463$	38.1457			1.77278			0.886390	−	0.462939i	$-0.153205\pi$
$463$	38.1457			1.77278			0.886390	+	0.462939i	$0.153205\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.76651	+	9.98789i	−0.266842	+	0.462184i	−0.968045	−	0.250778i	$-0.919314\pi$
$467$	−5.76651	+	9.98789i	−0.266842	+	0.462184i	0.701202	+	0.712962i	$0.252647\pi$
$468$		0			0
$469$	1.79306	−	20.7799i	0.0827959	−	0.959527i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−31.2624	−	54.1481i	−1.43745	−	2.48973i
$474$		0			0
$475$	−28.1988			−1.29385
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.66184	+	6.34250i	0.167314	+	0.289796i	0.937475	−	0.348054i	$-0.113157\pi$
$479$	3.66184	+	6.34250i	0.167314	+	0.289796i	−0.770161	+	0.637850i	$0.779824\pi$
$480$		0			0
$481$	22.6590	−	39.2465i	1.03316	−	1.78949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.31764	+	2.28223i	−0.0598311	+	0.103631i
$486$		0			0
$487$	−13.0258	−	22.5613i	−0.590254	−	1.02235i	−0.994198	−	0.107567i	$-0.965694\pi$
$487$	−13.0258	−	22.5613i	−0.590254	−	1.02235i	0.403943	−	0.914784i	$-0.367639\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.37919			0.333018			0.166509	−	0.986040i	$-0.446751\pi$
$491$	7.37919			0.333018			0.166509	+	0.986040i	$0.446751\pi$
$492$		0			0
$493$	−5.15777	−	8.93353i	−0.232294	−	0.402346i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.97345	+	1.37832i	0.0885211	+	0.0618261i
$498$		0			0
$499$	0.175119	−	0.303315i	0.00783940	−	0.0135782i	−0.862079	−	0.506774i	$-0.830838\pi$
$499$	0.175119	−	0.303315i	0.00783940	−	0.0135782i	0.869918	+	0.493196i	$0.164171\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.9098			−0.664795			−0.332398	−	0.943139i	$-0.607858\pi$
$503$	−14.9098			−0.664795			−0.332398	+	0.943139i	$0.607858\pi$
$504$		0			0
$505$	7.58612			0.337578
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.44886	−	14.6339i	0.374489	−	0.648635i	−0.615761	−	0.787933i	$-0.711151\pi$
$509$	8.44886	−	14.6339i	0.374489	−	0.648635i	0.990250	+	0.139298i	$0.0444847\pi$
$510$		0			0
$511$	1.18433	−	13.7253i	0.0523916	−	0.607170i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.180384	−	0.312434i	−0.00794865	−	0.0137675i
$516$		0			0
$517$	−28.4428			−1.25091
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.79306	+	6.56978i	0.166177	+	0.287827i	0.937073	−	0.349134i	$-0.113524\pi$
$521$	3.79306	+	6.56978i	0.166177	+	0.287827i	−0.770896	+	0.636962i	$0.780191\pi$
$522$		0			0
$523$	21.2572	−	36.8185i	0.929511	−	1.60996i	0.145371	−	0.989377i	$-0.453562\pi$
$523$	21.2572	−	36.8185i	0.929511	−	1.60996i	0.784140	−	0.620584i	$-0.213104\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.31160	+	10.9320i	−0.274938	+	0.476206i
$528$		0			0
$529$	7.26651	+	12.5860i	0.315935	+	0.547216i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	56.3976			2.44285
$534$		0			0
$535$	−2.72338	−	4.71704i	−0.117742	−	0.203935i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	25.9919	−	31.1262i	1.11955	−	1.34070i
$540$		0			0
$541$	−7.03182	+	12.1795i	−0.302322	+	0.523636i	−0.976661	−	0.214785i	$-0.931095\pi$
$541$	−7.03182	+	12.1795i	−0.302322	+	0.523636i	0.674340	+	0.738421i	$0.264428\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.09833			0.0898824
$546$		0			0
$547$	−4.49593			−0.192232			−0.0961161	−	0.995370i	$-0.530642\pi$
$547$	−4.49593			−0.192232			−0.0961161	+	0.995370i	$0.530642\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10.4283	−	18.0624i	0.444263	−	0.769485i
$552$		0			0
$553$	6.59533	−	3.08503i	0.280462	−	0.131189i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.9303	+	27.5921i	0.674989	+	1.16912i	0.976472	+	0.215644i	$0.0691852\pi$
$557$	15.9303	+	27.5921i	0.674989	+	1.16912i	−0.301483	+	0.953472i	$0.597482\pi$
$558$		0			0
$559$	−63.4983			−2.68569
$560$		0			0
$561$		0			0
$562$		0			0
$563$	23.0688	+	39.9563i	0.972233	+	1.68396i	0.688779	+	0.724972i	$0.258147\pi$
$563$	23.0688	+	39.9563i	0.972233	+	1.68396i	0.283454	+	0.958986i	$0.408520\pi$
$564$		0			0
$565$	−1.81962	+	3.15167i	−0.0765518	+	0.132592i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.31160	+	14.3961i	−0.348441	+	0.603517i	−0.985973	−	0.166907i	$-0.946622\pi$
$569$	−8.31160	+	14.3961i	−0.348441	+	0.603517i	0.637532	+	0.770424i	$0.279955\pi$
$570$		0			0
$571$	−18.2532	−	31.6155i	−0.763874	−	1.32307i	−0.940840	−	0.338852i	$-0.889962\pi$
$571$	−18.2532	−	31.6155i	−0.763874	−	1.32307i	0.176966	−	0.984217i	$-0.443372\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−13.9469			−0.581626
$576$		0			0
$577$	7.59019	+	13.1466i	0.315984	+	0.547300i	0.979646	−	0.200732i	$-0.0643321\pi$
$577$	7.59019	+	13.1466i	0.315984	+	0.547300i	−0.663662	+	0.748032i	$0.730999\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−21.6332	−	15.1093i	−0.897496	−	0.626841i
$582$		0			0
$583$	38.0350	−	65.8785i	1.57525	−	2.72841i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.20694			−0.338737			−0.169368	−	0.985553i	$-0.554173\pi$
$587$	−8.20694			−0.338737			−0.169368	+	0.985553i	$0.554173\pi$
$588$		0			0
$589$	−25.5225			−1.05164
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.77859	+	4.81266i	−0.114103	+	0.197632i	−0.917421	−	0.397918i	$-0.869733\pi$
$593$	−2.77859	+	4.81266i	−0.114103	+	0.197632i	0.803318	+	0.595550i	$0.203066\pi$
$594$		0			0
$595$	2.87117	+	2.00532i	0.117707	+	0.0822103i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−17.7931	−	30.8185i	−0.727005	−	1.25921i	−0.958143	−	0.286288i	$-0.907578\pi$
$599$	−17.7931	−	30.8185i	−0.727005	−	1.25921i	0.231139	−	0.972921i	$-0.425755\pi$
$600$		0			0
$601$	1.18038			0.0481489			0.0240744	−	0.999710i	$-0.492336\pi$
$601$	1.18038			0.0481489			0.0240744	+	0.999710i	$0.492336\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.13122	+	8.88753i	0.208614	+	0.361330i
$606$		0			0
$607$	−6.16908	+	10.6852i	−0.250395	+	0.433697i	−0.963635	−	0.267223i	$-0.913894\pi$
$607$	−6.16908	+	10.6852i	−0.250395	+	0.433697i	0.713239	+	0.700920i	$0.247227\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−14.4428	+	25.0157i	−0.584294	+	1.01203i
$612$		0			0
$613$	10.5451	+	18.2646i	0.425912	+	0.737702i	0.996505	−	0.0835312i	$-0.0266198\pi$
$613$	10.5451	+	18.2646i	0.425912	+	0.737702i	−0.570593	+	0.821233i	$0.693287\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.3526			−0.618074			−0.309037	−	0.951050i	$-0.600007\pi$
$617$	−15.3526			−0.618074			−0.309037	+	0.951050i	$0.600007\pi$
$618$		0			0
$619$	9.21615	+	15.9628i	0.370428	+	0.641601i	0.989631	−	0.143630i	$-0.0458776\pi$
$619$	9.21615	+	15.9628i	0.370428	+	0.641601i	−0.619203	+	0.785231i	$0.712544\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.7665	+	5.50389i	−0.471415	+	0.220509i
$624$		0			0
$625$	−10.9694	+	18.9995i	−0.438775	+	0.759981i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−22.4139			−0.893700
$630$		0			0
$631$	−5.91375			−0.235423			−0.117711	−	0.993048i	$-0.537556\pi$
$631$	−5.91375			−0.235423			−0.117711	+	0.993048i	$0.537556\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.283853	+	0.491647i	−0.0112643	+	0.0195104i
$636$		0			0
$637$	−14.1775	−	38.6655i	−0.561734	−	1.53198i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	23.0821	+	39.9793i	0.911686	+	1.57909i	0.811681	+	0.584100i	$0.198553\pi$
$641$	23.0821	+	39.9793i	0.911686	+	1.57909i	0.100005	+	0.994987i	$0.468114\pi$
$642$		0			0
$643$	−31.0555			−1.22471			−0.612355	−	0.790583i	$-0.709778\pi$
$643$	−31.0555			−1.22471			−0.612355	+	0.790583i	$0.709778\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.51854	+	14.7545i	0.334898	+	0.580061i	0.983465	−	0.181097i	$-0.0579647\pi$
$647$	8.51854	+	14.7545i	0.334898	+	0.580061i	−0.648567	+	0.761158i	$0.724631\pi$
$648$		0			0
$649$	−5.19366	+	8.99568i	−0.203869	+	0.353111i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4.06968	+	7.04889i	−0.159259	+	0.275844i	−0.934602	−	0.355696i	$-0.884244\pi$
$653$	−4.06968	+	7.04889i	−0.159259	+	0.275844i	0.775343	+	0.631541i	$0.217577\pi$
$654$		0			0
$655$	1.31764	+	2.28223i	0.0514846	+	0.0891740i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	41.1191			1.60177			0.800887	−	0.598815i	$-0.204362\pi$
$659$	41.1191			1.60177			0.800887	+	0.598815i	$0.204362\pi$
$660$		0			0
$661$	6.67105	+	11.5546i	0.259474	+	0.449422i	0.966101	−	0.258164i	$-0.0831175\pi$
$661$	6.67105	+	11.5546i	0.259474	+	0.449422i	−0.706627	+	0.707586i	$0.749784\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.608734	+	7.05465i	−0.0236057	+	0.273568i
$666$		0			0
$667$	5.15777	−	8.93353i	0.199710	−	0.345908i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	27.0902			1.04581
$672$		0			0
$673$	9.75837			0.376158			0.188079	−	0.982154i	$-0.439774\pi$
$673$	9.75837			0.376158			0.188079	+	0.982154i	$0.439774\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	16.2275	−	28.1068i	0.623672	−	1.08023i	−0.365124	−	0.930959i	$-0.618974\pi$
$677$	16.2275	−	28.1068i	0.623672	−	1.08023i	0.988796	−	0.149272i	$-0.0476931\pi$
$678$		0			0
$679$	−12.5656	−	8.77624i	−0.482224	−	0.336801i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	18.4827	+	32.0129i	0.707219	+	1.22494i	0.965885	+	0.258973i	$0.0833839\pi$
$683$	18.4827	+	32.0129i	0.707219	+	1.22494i	−0.258666	+	0.965967i	$0.583283\pi$
$684$		0			0
$685$	−1.40574			−0.0537106
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−38.6272	−	66.9042i	−1.47158	−	2.54885i
$690$		0			0
$691$	−7.12596	+	12.3425i	−0.271084	+	0.469531i	−0.969140	−	0.246512i	$-0.920715\pi$
$691$	−7.12596	+	12.3425i	−0.271084	+	0.469531i	0.698056	+	0.716044i	$0.254049\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.29713	+	2.24669i	−0.0492029	+	0.0852220i
$696$		0			0
$697$	−13.9469	−	24.1567i	−0.528276	−	0.915001i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	51.3937			1.94111			0.970556	−	0.240876i	$-0.0774347\pi$
$701$	51.3937			1.94111			0.970556	+	0.240876i	$0.0774347\pi$
$702$		0			0
$703$	−22.6590	−	39.2465i	−0.854599	−	1.48021i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.79306	+	43.9580i	−0.142653	+	1.65321i
$708$		0			0
$709$	−12.5861	+	21.7998i	−0.472682	+	0.818709i	−0.999511	−	0.0312621i	$-0.990047\pi$
$709$	−12.5861	+	21.7998i	−0.472682	+	0.818709i	0.526829	+	0.849971i	$0.323381\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12.6232			−0.472743
$714$		0			0
$715$	15.5041			0.579819
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.09019	−	7.08442i	0.152538	−	0.264204i	−0.779622	−	0.626251i	$-0.784589\pi$
$719$	4.09019	−	7.08442i	0.152538	−	0.264204i	0.932160	+	0.362047i	$0.117922\pi$
$720$		0			0
$721$	1.90060	−	0.889022i	0.0707820	−	0.0331089i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.49593	−	14.7154i	−0.315531	−	0.546516i
$726$		0			0
$727$	26.9614			0.999942			0.499971	−	0.866042i	$-0.333344\pi$
$727$	26.9614			0.999942			0.499971	+	0.866042i	$0.333344\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	15.7029	+	27.1982i	0.580792	+	1.00596i
$732$		0			0
$733$	10.7347	−	18.5930i	0.396495	−	0.686749i	−0.596796	−	0.802393i	$-0.703560\pi$
$733$	10.7347	−	18.5930i	0.396495	−	0.686749i	0.993291	+	0.115644i	$0.0368931\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−22.8341	+	39.5498i	−0.841105	+	1.45684i
$738$		0			0
$739$	0.671052	+	1.16230i	0.0246850	+	0.0427557i	0.878104	−	0.478470i	$-0.158808\pi$
$739$	0.671052	+	1.16230i	0.0246850	+	0.0427557i	−0.853419	+	0.521226i	$0.825475\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−26.9919			−0.990236			−0.495118	−	0.868826i	$-0.664875\pi$
$743$	−26.9919			−0.990236			−0.495118	+	0.868826i	$0.664875\pi$
$744$		0			0
$745$	−0.702870	−	1.21741i	−0.0257512	−	0.0446024i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	28.6947	−	13.4222i	1.04848	−	0.490437i
$750$		0			0
$751$	6.44360	−	11.1606i	0.235130	−	0.407258i	−0.724180	−	0.689611i	$-0.757782\pi$
$751$	6.44360	−	11.1606i	0.235130	−	0.407258i	0.959311	+	0.282353i	$0.0911150\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.784928			0.0285664
$756$		0			0
$757$	−26.3897			−0.959151			−0.479575	−	0.877501i	$-0.659209\pi$
$757$	−26.3897			−0.959151			−0.479575	+	0.877501i	$0.659209\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15.2214	−	26.3643i	0.551776	−	0.955704i	−0.446371	−	0.894848i	$-0.647284\pi$
$761$	15.2214	−	26.3643i	0.551776	−	0.955704i	0.998147	−	0.0608556i	$-0.0193829\pi$
$762$		0			0
$763$	−1.04916	+	12.1588i	−0.0379823	+	0.440179i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.27452	+	9.13574i	0.190452	+	0.329872i
$768$		0			0
$769$	42.1191			1.51886			0.759428	−	0.650592i	$-0.225479\pi$
$769$	42.1191			1.51886			0.759428	+	0.650592i	$0.225479\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19.0145	+	32.9340i	0.683903	+	1.18455i	0.973780	+	0.227491i	$0.0730522\pi$
$773$	19.0145	+	32.9340i	0.683903	+	1.18455i	−0.289877	+	0.957064i	$0.593614\pi$
$774$		0			0
$775$	−10.3965	+	18.0073i	−0.373454	+	0.646842i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28.1988	−	48.8418i	1.01033	−	1.74994i
$780$		0			0
$781$	−2.63529	−	4.56445i	−0.0942980	−	0.163329i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.54115			0.0906976
$786$		0			0
$787$	15.7931	+	27.3544i	0.562962	+	0.975079i	0.997236	+	0.0742978i	$0.0236716\pi$
$787$	15.7931	+	27.3544i	0.562962	+	0.975079i	−0.434274	+	0.900781i	$0.642995\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−17.3526	−	12.1197i	−0.616989	−	0.430925i
$792$		0			0
$793$	13.7560	−	23.8261i	0.488489	−	0.846088i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−45.2133			−1.60154			−0.800768	−	0.598974i	$-0.795575\pi$
$797$	−45.2133			−1.60154			−0.800768	+	0.598974i	$0.795575\pi$
$798$		0			0
$799$	14.2866			0.505424
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−15.0821	+	26.1229i	−0.532234	+	0.921857i
$804$		0			0
$805$	−0.301075	+	3.48917i	−0.0106115	+	0.122977i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.495933	+	0.858981i	0.0174361	+	0.0302002i	0.874612	−	0.484824i	$-0.161116\pi$
$809$	0.495933	+	0.858981i	0.0174361	+	0.0302002i	−0.857176	+	0.515024i	$0.827783\pi$
$810$		0			0
$811$	23.6151			0.829237			0.414619	−	0.909995i	$-0.363915\pi$
$811$	23.6151			0.829237			0.414619	+	0.909995i	$0.363915\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.01447	−	3.48917i	−0.0705640	−	0.122220i
$816$		0			0
$817$	−31.7492	+	54.9912i	−1.11076	+	1.92390i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−17.6332	+	30.5416i	−0.615403	+	1.06591i	0.374911	+	0.927061i	$0.377673\pi$
$821$	−17.6332	+	30.5416i	−0.615403	+	1.06591i	−0.990314	+	0.138848i	$0.955660\pi$
$822$		0			0
$823$	5.27058	+	9.12890i	0.183721	+	0.318214i	0.943145	−	0.332382i	$-0.107852\pi$
$823$	5.27058	+	9.12890i	0.183721	+	0.318214i	−0.759424	+	0.650596i	$0.774519\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.4323			−1.09301			−0.546504	−	0.837456i	$-0.684042\pi$
$827$	−31.4323			−1.09301			−0.546504	+	0.837456i	$0.684042\pi$
$828$		0			0
$829$	12.7612	+	22.1031i	0.443216	+	0.767673i	0.997926	−	0.0643708i	$-0.0205040\pi$
$829$	12.7612	+	22.1031i	0.443216	+	0.767673i	−0.554710	+	0.832044i	$0.687171\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−13.0555	+	15.6345i	−0.452346	+	0.541702i
$834$		0			0
$835$	3.95897	−	6.85714i	0.137006	−	0.237301i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.78253			−0.0960637			−0.0480318	−	0.998846i	$-0.515295\pi$
$839$	−2.78253			−0.0960637			−0.0480318	+	0.998846i	$0.515295\pi$
$840$		0			0
$841$	−16.4323			−0.566631
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.91585	−	8.51450i	0.169110	−	0.292908i
$846$		0			0
$847$	−54.0647	+	25.2893i	−1.85769	+	0.868949i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−11.2069	−	19.4110i	−0.384169	−	0.665400i
$852$		0			0
$853$	18.7931			0.643462			0.321731	−	0.946831i	$-0.395735\pi$
$853$	18.7931			0.643462			0.321731	+	0.946831i	$0.395735\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10.4694	+	18.1335i	0.357627	+	0.619428i	0.987564	−	0.157218i	$-0.0502525\pi$
$857$	10.4694	+	18.1335i	0.357627	+	0.619428i	−0.629937	+	0.776646i	$0.716919\pi$
$858$		0			0
$859$	−5.79306	+	10.0339i	−0.197657	+	0.342352i	−0.947768	−	0.318960i	$-0.896666\pi$
$859$	−5.79306	+	10.0339i	−0.197657	+	0.342352i	0.750111	+	0.661311i	$0.230000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14.7665	−	25.5763i	0.502658	−	0.870629i	−0.497337	−	0.867557i	$-0.665689\pi$
$863$	14.7665	−	25.5763i	0.502658	−	0.870629i	0.999995	−	0.00307167i	$-0.000977744\pi$
$864$		0			0
$865$	3.79306	+	6.56978i	0.128968	+	0.223379i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−15.9427			−0.540819
$870$		0			0
$871$	23.1896	+	40.1656i	0.785749	+	1.36096i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.66304	+	6.74899i	0.326670	+	0.228158i
$876$		0			0
$877$	−2.61268	+	4.52529i	−0.0882239	+	0.152808i	−0.906760	−	0.421646i	$-0.861452\pi$
$877$	−2.61268	+	4.52529i	−0.0882239	+	0.152808i	0.818537	+	0.574454i	$0.194786\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	13.1433			0.442809			0.221405	−	0.975182i	$-0.428936\pi$
$881$	13.1433			0.442809			0.221405	+	0.975182i	$0.428936\pi$
$882$		0			0
$883$	7.96531			0.268054			0.134027	−	0.990978i	$-0.457209\pi$
$883$	7.96531			0.268054			0.134027	+	0.990978i	$0.457209\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	19.8977	−	34.4639i	0.668100	−	1.15718i	−0.310334	−	0.950627i	$-0.600441\pi$
$887$	19.8977	−	34.4639i	0.668100	−	1.15718i	0.978435	−	0.206556i	$-0.0662256\pi$
$888$		0			0
$889$	−2.70694	−	1.89062i	−0.0907878	−	0.0634092i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	14.4428	+	25.0157i	0.483311	+	0.837119i
$894$		0			0
$895$	2.81148			0.0939775
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−7.68959	−	13.3188i	−0.256462	−	0.444206i
$900$		0			0
$901$	−19.1047	+	33.0903i	−0.636469	+	1.10240i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.729425	+	1.26340i	−0.0242469	+	0.0419968i
$906$		0			0
$907$	5.79833	+	10.0430i	0.192530	+	0.333472i	0.946088	−	0.323909i	$-0.104997\pi$
$907$	5.79833	+	10.0430i	0.192530	+	0.333472i	−0.753558	+	0.657382i	$0.771664\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−34.4959			−1.14290			−0.571451	−	0.820636i	$-0.693619\pi$
$911$	−34.4959			−1.14290			−0.571451	+	0.820636i	$0.693619\pi$
$912$		0			0
$913$	28.8884	+	50.0362i	0.956066	+	1.65596i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−13.8833	+	6.49402i	−0.458465	+	0.214451i
$918$		0			0
$919$	0.927153	−	1.60588i	0.0305839	−	0.0529729i	−0.850328	−	0.526253i	$-0.823597\pi$
$919$	0.927153	−	1.60588i	0.0305839	−	0.0529729i	0.880912	+	0.473280i	$0.156930\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.35263			−0.176184
$924$		0			0
$925$	−36.9203			−1.21393
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−11.5596	+	20.0218i	−0.379257	+	0.656893i	−0.990954	−	0.134199i	$-0.957154\pi$
$929$	−11.5596	+	20.0218i	−0.379257	+	0.656893i	0.611697	+	0.791092i	$0.290487\pi$
$930$		0			0
$931$	−40.5740	−	7.05465i	−1.32976	−	0.231207i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.83409	−	6.64084i	−0.125388	−	0.217179i
$936$		0			0
$937$	−43.3445			−1.41600			−0.708002	−	0.706211i	$-0.750403\pi$
$937$	−43.3445			−1.41600			−0.708002	+	0.706211i	$0.750403\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.5656	−	25.2284i	−0.474825	−	0.822422i	0.524759	−	0.851251i	$-0.324155\pi$
$941$	−14.5656	−	25.2284i	−0.474825	−	0.822422i	−0.999584	+	0.0288292i	$0.990822\pi$
$942$		0			0
$943$	13.9469	−	24.1567i	0.454173	−	0.786651i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.97345	+	12.0784i	−0.226607	+	0.392494i	−0.956800	−	0.290746i	$-0.906096\pi$
$947$	−6.97345	+	12.0784i	−0.226607	+	0.392494i	0.730194	+	0.683240i	$0.239430\pi$
$948$		0			0
$949$	15.3169	+	26.5296i	0.497207	+	0.861187i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	39.7134			1.28644			0.643222	−	0.765680i	$-0.277597\pi$
$953$	39.7134			1.28644			0.643222	+	0.765680i	$0.277597\pi$
$954$		0			0
$955$	4.85670	+	8.41205i	0.157159	+	0.272208i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.702870	−	8.14561i	0.0226969	−	0.263035i
$960$		0			0
$961$	6.09019	−	10.5485i	0.196458	−	0.340275i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.46304			−0.240244
$966$		0			0
$967$	−12.5717			−0.404277			−0.202139	−	0.979357i	$-0.564789\pi$
$967$	−12.5717			−0.404277			−0.202139	+	0.979357i	$0.564789\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	7.07691	−	12.2576i	0.227109	−	0.393364i	−0.729841	−	0.683617i	$-0.760406\pi$
$971$	7.07691	−	12.2576i	0.227109	−	0.393364i	0.956950	+	0.290253i	$0.0937393\pi$
$972$		0			0
$973$	−12.3700	−	8.63961i	−0.396563	−	0.276973i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−23.2624	−	40.2917i	−0.744231	−	1.28905i	−0.950553	−	0.310562i	$-0.899483\pi$
$977$	−23.2624	−	40.2917i	−0.744231	−	1.28905i	0.206322	−	0.978484i	$-0.433851\pi$
$978$		0			0
$979$	28.4428			0.909037
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−25.2359	−	43.7098i	−0.804900	−	1.39413i	−0.916359	−	0.400358i	$-0.868886\pi$
$983$	−25.2359	−	43.7098i	−0.804900	−	1.39413i	0.111459	−	0.993769i	$-0.464448\pi$
$984$		0			0
$985$	−5.61268	+	9.72144i	−0.178835	+	0.309751i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−15.7029	+	27.1982i	−0.499322	+	0.864851i
$990$		0			0
$991$	1.98870	+	3.44452i	0.0631730	+	0.109419i	0.895882	−	0.444292i	$-0.146545\pi$
$991$	1.98870	+	3.44452i	0.0631730	+	0.109419i	−0.832709	+	0.553711i	$0.813211\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.28660			0.199299
$996$		0			0
$997$	22.0092	+	38.1211i	0.697039	+	1.20731i	0.969489	+	0.245136i	$0.0788328\pi$
$997$	22.0092	+	38.1211i	0.697039	+	1.20731i	−0.272450	+	0.962170i	$0.587834\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.s.u.289.2		6
3.2	odd	2		672.2.q.k.289.2	yes	6
4.3	odd	2		2016.2.s.v.289.2		6
7.4	even	3	inner	2016.2.s.u.865.2		6
12.11	even	2		672.2.q.l.289.2	yes	6
21.2	odd	6		4704.2.a.bu.1.2		3
21.5	even	6		4704.2.a.bt.1.2		3
21.11	odd	6		672.2.q.k.193.2	✓	6
24.5	odd	2		1344.2.q.z.961.2		6
24.11	even	2		1344.2.q.y.961.2		6
28.11	odd	6		2016.2.s.v.865.2		6
84.11	even	6		672.2.q.l.193.2	yes	6
84.23	even	6		4704.2.a.bs.1.2		3
84.47	odd	6		4704.2.a.bv.1.2		3
168.5	even	6		9408.2.a.ei.1.2		3
168.11	even	6		1344.2.q.y.193.2		6
168.53	odd	6		1344.2.q.z.193.2		6
168.107	even	6		9408.2.a.ej.1.2		3
168.131	odd	6		9408.2.a.eg.1.2		3
168.149	odd	6		9408.2.a.eh.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.q.k.193.2	✓	6	21.11	odd	6
672.2.q.k.289.2	yes	6	3.2	odd	2
672.2.q.l.193.2	yes	6	84.11	even	6
672.2.q.l.289.2	yes	6	12.11	even	2
1344.2.q.y.193.2		6	168.11	even	6
1344.2.q.y.961.2		6	24.11	even	2
1344.2.q.z.193.2		6	168.53	odd	6
1344.2.q.z.961.2		6	24.5	odd	2
2016.2.s.u.289.2		6	1.1	even	1	trivial
2016.2.s.u.865.2		6	7.4	even	3	inner
2016.2.s.v.289.2		6	4.3	odd	2
2016.2.s.v.865.2		6	28.11	odd	6
4704.2.a.bs.1.2		3	84.23	even	6
4704.2.a.bt.1.2		3	21.5	even	6
4704.2.a.bu.1.2		3	21.2	odd	6
4704.2.a.bv.1.2		3	84.47	odd	6
9408.2.a.eg.1.2		3	168.131	odd	6
9408.2.a.eh.1.2		3	168.149	odd	6
9408.2.a.ei.1.2		3	168.5	even	6
9408.2.a.ej.1.2		3	168.107	even	6

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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.227452 - 0.393958i) q^{5} +(2.16908 + 1.51496i) q^{7} +O(q^{10})\)	\(q+(0.227452 - 0.393958i) q^{5} +(2.16908 + 1.51496i) q^{7} +(-2.89653 - 5.01694i) q^{11} -5.88325 q^{13} +(1.45490 + 2.51997i) q^{17} +(-2.94163 + 5.09505i) q^{19} +(-1.45490 + 2.51997i) q^{23} +(2.39653 + 4.15091i) q^{25} -3.54510 q^{29} +(2.16908 + 3.75696i) q^{31} +(1.09019 - 0.509947i) q^{35} +(-3.85144 + 6.67088i) q^{37} -9.58612 q^{41} +10.7931 q^{43} +(2.45490 - 4.25202i) q^{47} +(2.40981 + 6.57212i) q^{49} +(6.56561 + 11.3720i) q^{53} -2.63529 q^{55} +(-0.896531 - 1.55284i) q^{59} +(-2.33816 + 4.04981i) q^{61} +(-1.33816 + 2.31776i) q^{65} +(-3.94163 - 6.82710i) q^{67} +0.909808 q^{71} +(-2.60347 - 4.50934i) q^{73} +(1.31764 - 15.2703i) q^{77} +(1.37602 - 2.38333i) q^{79} -9.97345 q^{83} +1.32368 q^{85} +(-2.45490 + 4.25202i) q^{89} +(-12.7612 - 8.91288i) q^{91} +(1.33816 + 2.31776i) q^{95} -5.79306 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{7}+O(q^{10}) \) 6 * q - 3 * q^7	\( 6 q - 3 q^{7} - 6 q^{13} + 6 q^{17} - 3 q^{19} - 6 q^{23} - 3 q^{25} - 24 q^{29} - 3 q^{31} + 12 q^{35} - 3 q^{37} + 12 q^{41} + 30 q^{43} + 12 q^{47} + 9 q^{49} + 6 q^{53} - 24 q^{55} + 12 q^{59} + 18 q^{61} + 24 q^{65} - 9 q^{67} - 33 q^{73} + 12 q^{77} + 27 q^{79} - 36 q^{83} + 72 q^{85} - 12 q^{89} - 51 q^{91} - 24 q^{95}+O(q^{100}) \) 6 * q - 3 * q^7 - 6 * q^13 + 6 * q^17 - 3 * q^19 - 6 * q^23 - 3 * q^25 - 24 * q^29 - 3 * q^31 + 12 * q^35 - 3 * q^37 + 12 * q^41 + 30 * q^43 + 12 * q^47 + 9 * q^49 + 6 * q^53 - 24 * q^55 + 12 * q^59 + 18 * q^61 + 24 * q^65 - 9 * q^67 - 33 * q^73 + 12 * q^77 + 27 * q^79 - 36 * q^83 + 72 * q^85 - 12 * q^89 - 51 * q^91 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more