LMFDB - Embedded newform 208.12.f.b.129.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,12,Mod(129,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("208.129");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	208.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:	$159.815381556$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 18433 x^{10} + 121088056 x^{8} + 340607607312 x^{6} + 380893885719552 x^{4} + \cdots + 14\!\cdots\!56 $ x^12 + 18433x^10 + 121088056x^8 + 340607607312x^6 + 380893885719552x^4 + 134825856231997440*x^2 + 1497425476589715456
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{31}\cdot 3^{4}\cdot 13^{4} $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.6
Root			$65.0925i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.12.f.b.129.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.7424 q^{3} +6676.77i q^{5} -81596.2i q^{7} -176899. q^{9} +O(q^{10})$	$q-15.7424 q^{3} +6676.77i q^{5} -81596.2i q^{7} -176899. q^{9} -304131. i q^{11} +(1.14489e6 - 693820. i) q^{13} -105108. i q^{15} +640791. q^{17} -1.18006e7i q^{19} +1.28452e6i q^{21} +4.56803e7 q^{23} +4.24881e6 q^{25} +5.57352e6 q^{27} +4.74967e6 q^{29} -1.52916e8i q^{31} +4.78773e6i q^{33} +5.44799e8 q^{35} +2.70326e8i q^{37} +(-1.80233e7 + 1.09224e7i) q^{39} -9.31884e7i q^{41} -1.44702e9 q^{43} -1.18112e9i q^{45} +2.06068e9i q^{47} -4.68061e9 q^{49} -1.00876e7 q^{51} +2.76573e9 q^{53} +2.03061e9 q^{55} +1.85770e8i q^{57} +2.69401e9i q^{59} -6.51310e9 q^{61} +1.44343e10i q^{63} +(4.63248e9 + 7.64418e9i) q^{65} +9.67530e8i q^{67} -7.19116e8 q^{69} -1.34162e10i q^{71} -8.89945e8i q^{73} -6.68862e7 q^{75} -2.48159e10 q^{77} +3.87272e10 q^{79} +3.12494e10 q^{81} -4.24441e10i q^{83} +4.27842e9i q^{85} -7.47710e7 q^{87} -8.88675e10i q^{89} +(-5.66131e10 - 9.34187e10i) q^{91} +2.40726e9i q^{93} +7.87901e10 q^{95} -3.90545e10i q^{97} +5.38005e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 488 q^{3} + 654644 q^{9}+O(q^{10}) $ 12 * q + 488 * q^3 + 654644 * q^9	$ 12 q + 488 q^{3} + 654644 q^{9} + 3208868 q^{13} + 12198768 q^{17} - 5810592 q^{23} + 6102388 q^{25} + 52613336 q^{27} - 244463112 q^{29} + 562027560 q^{35} + 2199109744 q^{39} - 2294519976 q^{43} - 3573617796 q^{49} - 7713246552 q^{51} - 4602062760 q^{53} + 6178744976 q^{55} - 13775649944 q^{61} - 7598401512 q^{65} - 25419983328 q^{69} - 68016370832 q^{75} - 80478036048 q^{77} - 18046097296 q^{79} - 132677486692 q^{81} - 94507900752 q^{87} - 104793638664 q^{91} - 145093149648 q^{95}+O(q^{100}) $ 12 * q + 488 * q^3 + 654644 * q^9 + 3208868 * q^13 + 12198768 * q^17 - 5810592 * q^23 + 6102388 * q^25 + 52613336 * q^27 - 244463112 * q^29 + 562027560 * q^35 + 2199109744 * q^39 - 2294519976 * q^43 - 3573617796 * q^49 - 7713246552 * q^51 - 4602062760 * q^53 + 6178744976 * q^55 - 13775649944 * q^61 - 7598401512 * q^65 - 25419983328 * q^69 - 68016370832 * q^75 - 80478036048 * q^77 - 18046097296 * q^79 - 132677486692 * q^81 - 94507900752 * q^87 - 104793638664 * q^91 - 145093149648 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−15.7424			−0.0374027			−0.0187013	−	0.999825i	$-0.505953\pi$
$3$	−15.7424			−0.0374027			−0.0187013	+	0.999825i	$0.505953\pi$
$4$		0			0
$5$			6676.77i			0.955502i	0.878495	+	0.477751i	$0.158548\pi$
$5$			6676.77i			0.955502i	−0.878495	+	0.477751i	$0.841452\pi$
$6$		0			0
$7$		−	81596.2i		−	1.83498i	−0.397762	−	0.917489i	$-0.630213\pi$
$7$		−	81596.2i		−	1.83498i	0.397762	−	0.917489i	$-0.369787\pi$
$8$		0			0
$9$	−176899.			−0.998601
$10$		0			0
$11$		−	304131.i		−	0.569378i	−0.958620	−	0.284689i	$-0.908110\pi$
$11$		−	304131.i		−	0.569378i	0.958620	−	0.284689i	$-0.0918903\pi$
$12$		0			0
$13$	1.14489e6	−	693820.i	0.855215	−	0.518273i
$13$	1.14489e6	−	693820.i	0.855215	−	0.518273i
$14$		0			0
$15$		−	105108.i		−	0.0357384i
$16$		0			0
$17$	640791.			0.109458			0.0547290	−	0.998501i	$-0.482571\pi$
$17$	640791.			0.109458			0.0547290	+	0.998501i	$0.482571\pi$
$18$		0			0
$19$		−	1.18006e7i		−	1.09335i	−0.837344	−	0.546676i	$-0.815893\pi$
$19$		−	1.18006e7i		−	1.09335i	0.837344	−	0.546676i	$-0.184107\pi$
$20$		0			0
$21$			1.28452e6i			0.0686331i
$22$		0			0
$23$	4.56803e7			1.47988			0.739939	−	0.672673i	$-0.234854\pi$
$23$	4.56803e7			1.47988			0.739939	+	0.672673i	$0.234854\pi$
$24$		0			0
$25$	4.24881e6			0.0870155
$26$		0			0
$27$	5.57352e6			0.0747531
$28$		0			0
$29$	4.74967e6			0.0430006			0.0215003	−	0.999769i	$-0.493156\pi$
$29$	4.74967e6			0.0430006			0.0215003	+	0.999769i	$0.493156\pi$
$30$		0			0
$31$		−	1.52916e8i		−	0.959319i	−0.877455	−	0.479660i	$-0.840760\pi$
$31$		−	1.52916e8i		−	0.959319i	0.877455	−	0.479660i	$-0.159240\pi$
$32$		0			0
$33$			4.78773e6i			0.0212963i
$34$		0			0
$35$	5.44799e8			1.75332
$36$		0			0
$37$			2.70326e8i			0.640883i	0.947268	+	0.320441i	$0.103831\pi$
$37$			2.70326e8i			0.640883i	−0.947268	+	0.320441i	$0.896169\pi$
$38$		0			0
$39$	−1.80233e7	+	1.09224e7i	−0.0319874	+	0.0193848i
$40$		0			0
$41$		−	9.31884e7i		−	0.125618i	−0.998026	−	0.0628088i	$-0.979994\pi$
$41$		−	9.31884e7i		−	0.125618i	0.998026	−	0.0628088i	$-0.0200058\pi$
$42$		0			0
$43$	−1.44702e9			−1.50106			−0.750530	−	0.660836i	$-0.770202\pi$
$43$	−1.44702e9			−1.50106			−0.750530	+	0.660836i	$0.770202\pi$
$44$		0			0
$45$		−	1.18112e9i		−	0.954165i
$46$		0			0
$47$			2.06068e9i			1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$47$			2.06068e9i			1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$48$		0			0
$49$	−4.68061e9			−2.36714
$50$		0			0
$51$	−1.00876e7			−0.00409402
$52$		0			0
$53$	2.76573e9			0.908431			0.454215	−	0.890892i	$-0.349920\pi$
$53$	2.76573e9			0.908431			0.454215	+	0.890892i	$0.349920\pi$
$54$		0			0
$55$	2.03061e9			0.544042
$56$		0			0
$57$			1.85770e8i			0.0408943i
$58$		0			0
$59$			2.69401e9i			0.490584i	0.969449	+	0.245292i	$0.0788838\pi$
$59$			2.69401e9i			0.490584i	−0.969449	+	0.245292i	$0.921116\pi$
$60$		0			0
$61$	−6.51310e9			−0.987355			−0.493678	−	0.869645i	$-0.664348\pi$
$61$	−6.51310e9			−0.987355			−0.493678	+	0.869645i	$0.664348\pi$
$62$		0			0
$63$			1.44343e10i			1.83241i
$64$		0			0
$65$	4.63248e9	+	7.64418e9i	0.495211	+	0.817160i
$66$		0			0
$67$			9.67530e8i			0.0875494i	0.999041	+	0.0437747i	$0.0139384\pi$
$67$			9.67530e8i			0.0875494i	−0.999041	+	0.0437747i	$0.986062\pi$
$68$		0			0
$69$	−7.19116e8			−0.0553515
$70$		0			0
$71$		−	1.34162e10i		−	0.882485i	−0.897388	−	0.441242i	$-0.854538\pi$
$71$		−	1.34162e10i		−	0.882485i	0.897388	−	0.441242i	$-0.145462\pi$
$72$		0			0
$73$		−	8.89945e8i		−	0.0502443i	−0.999684	−	0.0251222i	$-0.992003\pi$
$73$		−	8.89945e8i		−	0.0502443i	0.999684	−	0.0251222i	$-0.00799748\pi$
$74$		0			0
$75$	−6.68862e7			−0.00325462
$76$		0			0
$77$	−2.48159e10			−1.04480
$78$		0			0
$79$	3.87272e10			1.41601			0.708007	−	0.706206i	$-0.249595\pi$
$79$	3.87272e10			1.41601			0.708007	+	0.706206i	$0.249595\pi$
$80$		0			0
$81$	3.12494e10			0.995805
$82$		0			0
$83$		−	4.24441e10i		−	1.18274i	−0.806402	−	0.591368i	$-0.798588\pi$
$83$		−	4.24441e10i		−	1.18274i	0.806402	−	0.591368i	$-0.201412\pi$
$84$		0			0
$85$			4.27842e9i			0.104587i
$86$		0			0
$87$	−7.47710e7			−0.00160834
$88$		0			0
$89$		−	8.88675e10i		−	1.68693i	−0.537182	−	0.843467i	$-0.680511\pi$
$89$		−	8.88675e10i		−	1.68693i	0.537182	−	0.843467i	$-0.319489\pi$
$90$		0			0
$91$	−5.66131e10	−	9.34187e10i	−0.951019	−	1.56930i
$92$		0			0
$93$			2.40726e9i			0.0358811i
$94$		0			0
$95$	7.87901e10			1.04470
$96$		0			0
$97$		−	3.90545e10i		−	0.461771i	−0.972981	−	0.230885i	$-0.925838\pi$
$97$		−	3.90545e10i		−	0.461771i	0.972981	−	0.230885i	$-0.0741623\pi$
$98$		0			0
$99$			5.38005e10i			0.568581i
$100$		0			0
$101$	1.40076e11			1.32616			0.663082	−	0.748547i	$-0.269248\pi$
$101$	1.40076e11			1.32616			0.663082	+	0.748547i	$0.269248\pi$
$102$		0			0
$103$	−1.65638e11			−1.40785			−0.703925	−	0.710275i	$-0.748571\pi$
$103$	−1.65638e11			−1.40785			−0.703925	+	0.710275i	$0.748571\pi$
$104$		0			0
$105$	−8.57643e9			−0.0655791
$106$		0			0
$107$	−1.17721e11			−0.811417			−0.405708	−	0.914003i	$-0.632975\pi$
$107$	−1.17721e11			−0.811417			−0.405708	+	0.914003i	$0.632975\pi$
$108$		0			0
$109$			1.41466e11i			0.880659i	0.897836	+	0.440329i	$0.145138\pi$
$109$			1.41466e11i			0.880659i	−0.897836	+	0.440329i	$0.854862\pi$
$110$		0			0
$111$		−	4.25557e9i		−	0.0239707i
$112$		0			0
$113$	−2.66858e10			−0.136254			−0.0681269	−	0.997677i	$-0.521702\pi$
$113$	−2.66858e10			−0.136254			−0.0681269	+	0.997677i	$0.521702\pi$
$114$		0			0
$115$			3.04997e11i			1.41403i
$116$		0			0
$117$	−2.02530e11	+	1.22736e11i	−0.854019	+	0.517548i
$118$		0			0
$119$		−	5.22862e10i		−	0.200853i
$120$		0			0
$121$	1.92816e11			0.675809
$122$		0			0
$123$			1.46700e9i			0.00469844i
$124$		0			0
$125$			3.54383e11i			1.03865i
$126$		0			0
$127$	2.53671e11			0.681317			0.340659	−	0.940187i	$-0.389350\pi$
$127$	2.53671e11			0.681317			0.340659	+	0.940187i	$0.389350\pi$
$128$		0			0
$129$	2.27795e10			0.0561437
$130$		0			0
$131$	−7.42174e11			−1.68079			−0.840395	−	0.541974i	$-0.817677\pi$
$131$	−7.42174e11			−1.68079			−0.840395	+	0.541974i	$0.817677\pi$
$132$		0			0
$133$	−9.62886e11			−2.00628
$134$		0			0
$135$			3.72131e10i			0.0714267i
$136$		0			0
$137$		−	3.81055e11i		−	0.674566i	−0.941403	−	0.337283i	$-0.890492\pi$
$137$		−	3.81055e11i		−	0.674566i	0.941403	−	0.337283i	$-0.109508\pi$
$138$		0			0
$139$	2.95068e11			0.482326			0.241163	−	0.970485i	$-0.422471\pi$
$139$	2.95068e11			0.482326			0.241163	+	0.970485i	$0.422471\pi$
$140$		0			0
$141$		−	3.24399e10i		−	0.0490201i
$142$		0			0
$143$	−2.11012e11	−	3.48196e11i	−0.295093	−	0.486941i
$144$		0			0
$145$			3.17125e10i			0.0410872i
$146$		0			0
$147$	7.36839e10			0.0885375
$148$		0			0
$149$			1.33510e11i			0.148933i	0.997224	+	0.0744663i	$0.0237253\pi$
$149$			1.33510e11i			0.148933i	−0.997224	+	0.0744663i	$0.976275\pi$
$150$		0			0
$151$		−	9.83412e11i		−	1.01944i	−0.860340	−	0.509721i	$-0.829749\pi$
$151$		−	9.83412e11i		−	1.01944i	0.860340	−	0.509721i	$-0.170251\pi$
$152$		0			0
$153$	−1.13355e11			−0.109305
$154$		0			0
$155$	1.02098e12			0.916632
$156$		0			0
$157$	−3.26970e11			−0.273565			−0.136782	−	0.990601i	$-0.543676\pi$
$157$	−3.26970e11			−0.273565			−0.136782	+	0.990601i	$0.543676\pi$
$158$		0			0
$159$	−4.35390e10			−0.0339778
$160$		0			0
$161$		−	3.72734e12i		−	2.71554i
$162$		0			0
$163$		−	3.86655e11i		−	0.263204i	−0.991303	−	0.131602i	$-0.957988\pi$
$163$		−	3.86655e11i		−	0.263204i	0.991303	−	0.131602i	$-0.0420120\pi$
$164$		0			0
$165$	−3.19666e10			−0.0203486
$166$		0			0
$167$		−	1.44061e12i		−	0.858234i	−0.903249	−	0.429117i	$-0.858825\pi$
$167$		−	1.44061e12i		−	0.858234i	0.903249	−	0.429117i	$-0.141175\pi$
$168$		0			0
$169$	8.29388e11	−	1.58870e12i	0.462787	−	0.886470i
$170$		0			0
$171$			2.08752e12i			1.09182i
$172$		0			0
$173$	−1.57606e12			−0.773248			−0.386624	−	0.922237i	$-0.626359\pi$
$173$	−1.57606e12			−0.773248			−0.386624	+	0.922237i	$0.626359\pi$
$174$		0			0
$175$		−	3.46686e11i		−	0.159672i
$176$		0			0
$177$		−	4.24101e10i		−	0.0183492i
$178$		0			0
$179$	−4.37692e12			−1.78023			−0.890117	−	0.455732i	$-0.849378\pi$
$179$	−4.37692e12			−1.78023			−0.890117	+	0.455732i	$0.849378\pi$
$180$		0			0
$181$	−8.19825e11			−0.313681			−0.156841	−	0.987624i	$-0.550131\pi$
$181$	−8.19825e11			−0.313681			−0.156841	+	0.987624i	$0.550131\pi$
$182$		0			0
$183$	1.02531e11			0.0369297
$184$		0			0
$185$	−1.80491e12			−0.612365
$186$		0			0
$187$		−	1.94884e11i		−	0.0623230i
$188$		0			0
$189$		−	4.54778e11i		−	0.137170i
$190$		0			0
$191$	4.42652e11			0.126003			0.0630013	−	0.998013i	$-0.479933\pi$
$191$	4.42652e11			0.126003			0.0630013	+	0.998013i	$0.479933\pi$
$192$		0			0
$193$			6.48567e12i			1.74337i	0.490066	+	0.871686i	$0.336973\pi$
$193$			6.48567e12i			1.74337i	−0.490066	+	0.871686i	$0.663027\pi$
$194$		0			0
$195$	−7.29261e10	−	1.20337e11i	−0.0185222	−	0.0305640i
$196$		0			0
$197$			3.34101e11i			0.0802257i	0.999195	+	0.0401128i	$0.0127717\pi$
$197$			3.34101e11i			0.0802257i	−0.999195	+	0.0401128i	$0.987228\pi$
$198$		0			0
$199$	−1.51855e12			−0.344935			−0.172467	−	0.985015i	$-0.555174\pi$
$199$	−1.51855e12			−0.344935			−0.172467	+	0.985015i	$0.555174\pi$
$200$		0			0
$201$		−	1.52312e10i		−	0.00327458i
$202$		0			0
$203$		−	3.87555e11i		−	0.0789051i
$204$		0			0
$205$	6.22198e11			0.120028
$206$		0			0
$207$	−8.08081e12			−1.47781
$208$		0			0
$209$	−3.58893e12			−0.622530
$210$		0			0
$211$	−4.76322e12			−0.784055			−0.392028	−	0.919953i	$-0.628226\pi$
$211$	−4.76322e12			−0.784055			−0.392028	+	0.919953i	$0.628226\pi$
$212$		0			0
$213$			2.11202e11i			0.0330073i
$214$		0			0
$215$		−	9.66143e12i		−	1.43427i
$216$		0			0
$217$	−1.24774e13			−1.76033
$218$		0			0
$219$			1.40098e10i			0.00187927i
$220$		0			0
$221$	7.33636e11	−	4.44594e11i	0.0936102	−	0.0567291i
$222$		0			0
$223$		−	3.11138e12i		−	0.377812i	−0.981995	−	0.188906i	$-0.939506\pi$
$223$		−	3.11138e12i		−	0.377812i	0.981995	−	0.188906i	$-0.0604941\pi$
$224$		0			0
$225$	−7.51610e11			−0.0868938
$226$		0			0
$227$		−	1.30368e12i		−	0.143558i	−0.997421	−	0.0717790i	$-0.977132\pi$
$227$		−	1.30368e12i		−	0.143558i	0.997421	−	0.0717790i	$-0.0228676\pi$
$228$		0			0
$229$		−	1.11715e13i		−	1.17224i	−0.810224	−	0.586120i	$-0.800655\pi$
$229$		−	1.11715e13i		−	1.17224i	0.810224	−	0.586120i	$-0.199345\pi$
$230$		0			0
$231$	3.90661e11			0.0390782
$232$		0			0
$233$	2.95745e12			0.282137			0.141068	−	0.990000i	$-0.454946\pi$
$233$	2.95745e12			0.282137			0.141068	+	0.990000i	$0.454946\pi$
$234$		0			0
$235$	−1.37587e13			−1.25229
$236$		0			0
$237$	−6.09658e11			−0.0529627
$238$		0			0
$239$			4.27758e12i			0.354821i	0.984137	+	0.177411i	$0.0567721\pi$
$239$			4.27758e12i			0.354821i	−0.984137	+	0.177411i	$0.943228\pi$
$240$		0			0
$241$			1.02218e13i			0.809903i	0.914338	+	0.404952i	$0.132712\pi$
$241$			1.02218e13i			0.809903i	−0.914338	+	0.404952i	$0.867288\pi$
$242$		0			0
$243$	−1.47927e12			−0.111999
$244$		0			0
$245$		−	3.12514e13i		−	2.26181i
$246$		0			0
$247$	−8.18751e12	−	1.35104e13i	−0.566655	−	0.935051i
$248$		0			0
$249$			6.68170e11i			0.0442375i
$250$		0			0
$251$	−1.19608e13			−0.757800			−0.378900	−	0.925438i	$-0.623698\pi$
$251$	−1.19608e13			−0.757800			−0.378900	+	0.925438i	$0.623698\pi$
$252$		0			0
$253$		−	1.38928e13i		−	0.842610i
$254$		0			0
$255$		−	6.73524e10i		−	0.00391185i
$256$		0			0
$257$	−2.11863e13			−1.17875			−0.589376	−	0.807859i	$-0.700626\pi$
$257$	−2.11863e13			−1.17875			−0.589376	+	0.807859i	$0.700626\pi$
$258$		0			0
$259$	2.20576e13			1.17601
$260$		0			0
$261$	−8.40213e11			−0.0429404
$262$		0			0
$263$	1.09311e13			0.535680			0.267840	−	0.963463i	$-0.413690\pi$
$263$	1.09311e13			0.535680			0.267840	+	0.963463i	$0.413690\pi$
$264$		0			0
$265$			1.84661e13i			0.868008i
$266$		0			0
$267$			1.39898e12i			0.0630958i
$268$		0			0
$269$	2.15359e13			0.932237			0.466119	−	0.884722i	$-0.345652\pi$
$269$	2.15359e13			0.932237			0.466119	+	0.884722i	$0.345652\pi$
$270$		0			0
$271$		−	1.02752e13i		−	0.427031i	−0.976940	−	0.213515i	$-0.931509\pi$
$271$		−	1.02752e13i		−	0.427031i	0.976940	−	0.213515i	$-0.0684913\pi$
$272$		0			0
$273$	8.91223e11	+	1.47063e12i	0.0355707	+	0.0586961i
$274$		0			0
$275$		−	1.29219e12i		−	0.0495447i
$276$		0			0
$277$	−5.29184e13			−1.94970			−0.974850	−	0.222863i	$-0.928460\pi$
$277$	−5.29184e13			−1.94970			−0.974850	+	0.222863i	$0.928460\pi$
$278$		0			0
$279$			2.70507e13i			0.957977i
$280$		0			0
$281$		−	4.70128e13i		−	1.60078i	−0.599481	−	0.800389i	$-0.704626\pi$
$281$		−	4.70128e13i		−	1.60078i	0.599481	−	0.800389i	$-0.295374\pi$
$282$		0			0
$283$	−3.09520e13			−1.01359			−0.506797	−	0.862066i	$-0.669170\pi$
$283$	−3.09520e13			−1.01359			−0.506797	+	0.862066i	$0.669170\pi$
$284$		0			0
$285$	−1.24034e12			−0.0390746
$286$		0			0
$287$	−7.60382e12			−0.230506
$288$		0			0
$289$	−3.38613e13			−0.988019
$290$		0			0
$291$			6.14810e11i			0.0172715i
$292$		0			0
$293$			4.56525e12i			0.123507i	0.998091	+	0.0617537i	$0.0196693\pi$
$293$			4.56525e12i			0.123507i	−0.998091	+	0.0617537i	$0.980331\pi$
$294$		0			0
$295$	−1.79873e13			−0.468754
$296$		0			0
$297$		−	1.69508e12i		−	0.0425627i
$298$		0			0
$299$	5.22990e13	−	3.16939e13i	1.26562	−	0.766981i
$300$		0			0
$301$			1.18071e14i			2.75441i
$302$		0			0
$303$	−2.20513e12			−0.0496021
$304$		0			0
$305$		−	4.34865e13i		−	0.943420i
$306$		0			0
$307$		−	5.10388e13i		−	1.06817i	−0.845432	−	0.534084i	$-0.820657\pi$
$307$		−	5.10388e13i		−	1.06817i	0.845432	−	0.534084i	$-0.179343\pi$
$308$		0			0
$309$	2.60754e12			0.0526573
$310$		0			0
$311$	−2.06535e12			−0.0402542			−0.0201271	−	0.999797i	$-0.506407\pi$
$311$	−2.06535e12			−0.0402542			−0.0201271	+	0.999797i	$0.506407\pi$
$312$		0			0
$313$	−7.61036e13			−1.43190			−0.715948	−	0.698153i	$-0.754005\pi$
$313$	−7.61036e13			−1.43190			−0.715948	+	0.698153i	$0.754005\pi$
$314$		0			0
$315$	−9.63746e13			−1.75087
$316$		0			0
$317$		−	7.00724e13i		−	1.22948i	−0.788730	−	0.614740i	$-0.789261\pi$
$317$		−	7.00724e13i		−	1.22948i	0.788730	−	0.614740i	$-0.210739\pi$
$318$		0			0
$319$		−	1.44452e12i		−	0.0244836i
$320$		0			0
$321$	1.85321e12			0.0303492
$322$		0			0
$323$		−	7.56174e12i		−	0.119676i
$324$		0			0
$325$	4.86442e12	−	2.94791e12i	0.0744170	−	0.0450978i
$326$		0			0
$327$		−	2.22702e12i		−	0.0329390i
$328$		0			0
$329$	1.68143e14			2.40493
$330$		0			0
$331$			2.10419e13i			0.291092i	0.989351	+	0.145546i	$0.0464939\pi$
$331$			2.10419e13i			0.291092i	−0.989351	+	0.145546i	$0.953506\pi$
$332$		0			0
$333$		−	4.78205e13i		−	0.639986i
$334$		0			0
$335$	−6.45998e12			−0.0836536
$336$		0			0
$337$	1.10208e14			1.38118			0.690590	−	0.723246i	$-0.257351\pi$
$337$	1.10208e14			1.38118			0.690590	+	0.723246i	$0.257351\pi$
$338$		0			0
$339$	4.20097e11			0.00509626
$340$		0			0
$341$	−4.65064e13			−0.546215
$342$		0			0
$343$			2.20578e14i			2.50868i
$344$		0			0
$345$		−	4.80138e12i		−	0.0528884i
$346$		0			0
$347$	9.07341e13			0.968185			0.484092	−	0.875017i	$-0.339150\pi$
$347$	9.07341e13			0.968185			0.484092	+	0.875017i	$0.339150\pi$
$348$		0			0
$349$			1.54141e14i			1.59359i	0.604248	+	0.796796i	$0.293474\pi$
$349$			1.54141e14i			1.59359i	−0.604248	+	0.796796i	$0.706526\pi$
$350$		0			0
$351$	6.38107e12	−	3.86702e12i	0.0639300	−	0.0387425i
$352$		0			0
$353$			4.87738e13i			0.473616i	0.971557	+	0.236808i	$0.0761012\pi$
$353$			4.87738e13i			0.473616i	−0.971557	+	0.236808i	$0.923899\pi$
$354$		0			0
$355$	8.95767e13			0.843216
$356$		0			0
$357$			8.23107e11i			0.00751244i
$358$		0			0
$359$		−	1.48417e14i		−	1.31361i	−0.754063	−	0.656803i	$-0.771908\pi$
$359$		−	1.48417e14i		−	1.31361i	0.754063	−	0.656803i	$-0.228092\pi$
$360$		0			0
$361$	−2.27644e13			−0.195419
$362$		0			0
$363$	−3.03538e12			−0.0252771
$364$		0			0
$365$	5.94196e12			0.0480086
$366$		0			0
$367$	−1.70331e14			−1.33546			−0.667730	−	0.744404i	$-0.732734\pi$
$367$	−1.70331e14			−1.33546			−0.667730	+	0.744404i	$0.732734\pi$
$368$		0			0
$369$			1.64850e13i			0.125442i
$370$		0			0
$371$		−	2.25673e14i		−	1.66695i
$372$		0			0
$373$	1.78164e14			1.27768			0.638840	−	0.769340i	$-0.279415\pi$
$373$	1.78164e14			1.27768			0.638840	+	0.769340i	$0.279415\pi$
$374$		0			0
$375$		−	5.57882e12i		−	0.0388481i
$376$		0			0
$377$	5.43785e12	−	3.29542e12i	0.0367748	−	0.0222860i
$378$		0			0
$379$			1.98465e14i			1.30367i	0.758359	+	0.651837i	$0.226002\pi$
$379$			1.98465e14i			1.30367i	−0.758359	+	0.651837i	$0.773998\pi$
$380$		0			0
$381$	−3.99337e12			−0.0254831
$382$		0			0
$383$			6.17365e13i			0.382779i	0.981514	+	0.191390i	$0.0612994\pi$
$383$			6.17365e13i			0.382779i	−0.981514	+	0.191390i	$0.938701\pi$
$384$		0			0
$385$		−	1.65690e14i		−	0.998304i
$386$		0			0
$387$	2.55977e14			1.49896
$388$		0			0
$389$	−2.12605e14			−1.21018			−0.605091	−	0.796156i	$-0.706863\pi$
$389$	−2.12605e14			−1.21018			−0.605091	+	0.796156i	$0.706863\pi$
$390$		0			0
$391$	2.92716e13			0.161985
$392$		0			0
$393$	1.16836e13			0.0628661
$394$		0			0
$395$			2.58573e14i			1.35300i
$396$		0			0
$397$			2.30303e14i			1.17207i	0.810287	+	0.586033i	$0.199311\pi$
$397$			2.30303e14i			1.17207i	−0.810287	+	0.586033i	$0.800689\pi$
$398$		0			0
$399$	1.51581e13			0.0750401
$400$		0			0
$401$			1.73401e14i			0.835137i	0.908645	+	0.417569i	$0.137118\pi$
$401$			1.73401e14i			0.835137i	−0.908645	+	0.417569i	$0.862882\pi$
$402$		0			0
$403$	−1.06096e14	−	1.75072e14i	−0.497189	−	0.820425i
$404$		0			0
$405$			2.08645e14i			0.951494i
$406$		0			0
$407$	8.22145e13			0.364904
$408$		0			0
$409$		−	1.79552e14i		−	0.775731i	−0.921716	−	0.387865i	$-0.873213\pi$
$409$		−	1.79552e14i		−	0.775731i	0.921716	−	0.387865i	$-0.126787\pi$
$410$		0			0
$411$			5.99870e12i			0.0252306i
$412$		0			0
$413$	2.19821e14			0.900210
$414$		0			0
$415$	2.83390e14			1.13011
$416$		0			0
$417$	−4.64507e12			−0.0180403
$418$		0			0
$419$	4.29126e14			1.62333			0.811667	−	0.584120i	$-0.198560\pi$
$419$	4.29126e14			1.62333			0.811667	+	0.584120i	$0.198560\pi$
$420$		0			0
$421$		−	6.60885e13i		−	0.243542i	−0.992558	−	0.121771i	$-0.961143\pi$
$421$		−	6.60885e13i		−	0.243542i	0.992558	−	0.121771i	$-0.0388574\pi$
$422$		0			0
$423$		−	3.64532e14i		−	1.30877i
$424$		0			0
$425$	2.72260e12			0.00952455
$426$		0			0
$427$			5.31444e14i			1.81177i
$428$		0			0
$429$	3.32183e12	+	5.48143e12i	0.0110373	+	0.0182129i
$430$		0			0
$431$			4.92076e14i			1.59370i	0.604176	+	0.796851i	$0.293503\pi$
$431$			4.92076e14i			1.59370i	−0.604176	+	0.796851i	$0.706497\pi$
$432$		0			0
$433$	−3.74971e14			−1.18390			−0.591949	−	0.805976i	$-0.701641\pi$
$433$	−3.74971e14			−1.18390			−0.591949	+	0.805976i	$0.701641\pi$
$434$		0			0
$435$		−	4.99229e11i		−	0.00153677i
$436$		0			0
$437$		−	5.39056e14i		−	1.61803i
$438$		0			0
$439$	−2.95835e14			−0.865955			−0.432977	−	0.901405i	$-0.642537\pi$
$439$	−2.95835e14			−0.865955			−0.432977	+	0.901405i	$0.642537\pi$
$440$		0			0
$441$	8.27997e14			2.36383
$442$		0			0
$443$	2.72243e14			0.758116			0.379058	−	0.925373i	$-0.376248\pi$
$443$	2.72243e14			0.758116			0.379058	+	0.925373i	$0.376248\pi$
$444$		0			0
$445$	5.93348e14			1.61187
$446$		0			0
$447$		−	2.10177e12i		−	0.00557048i
$448$		0			0
$449$		−	3.73650e13i		−	0.0966295i	−0.998832	−	0.0483147i	$-0.984615\pi$
$449$		−	3.73650e13i		−	0.0966295i	0.998832	−	0.0483147i	$-0.0153851\pi$
$450$		0			0
$451$	−2.83415e13			−0.0715239
$452$		0			0
$453$			1.54812e13i			0.0381299i
$454$		0			0
$455$	6.23736e14	−	3.77993e14i	1.49947	−	0.908701i
$456$		0			0
$457$			5.95864e14i			1.39832i	0.714963	+	0.699162i	$0.246443\pi$
$457$			5.95864e14i			1.39832i	−0.714963	+	0.699162i	$0.753557\pi$
$458$		0			0
$459$	3.57146e12			0.00818232
$460$		0			0
$461$		−	2.60395e14i		−	0.582474i	−0.956651	−	0.291237i	$-0.905933\pi$
$461$		−	2.60395e14i		−	0.582474i	0.956651	−	0.291237i	$-0.0940669\pi$
$462$		0			0
$463$		−	1.41646e14i		−	0.309391i	−0.987962	−	0.154696i	$-0.950560\pi$
$463$		−	1.41646e14i		−	0.309391i	0.987962	−	0.154696i	$-0.0494397\pi$
$464$		0			0
$465$	−1.60727e13			−0.0342845
$466$		0			0
$467$	−3.41616e14			−0.711697			−0.355848	−	0.934544i	$-0.615808\pi$
$467$	−3.41616e14			−0.711697			−0.355848	+	0.934544i	$0.615808\pi$
$468$		0			0
$469$	7.89468e13			0.160651
$470$		0			0
$471$	5.14728e12			0.0102321
$472$		0			0
$473$			4.40084e14i			0.854671i
$474$		0			0
$475$		−	5.01385e13i		−	0.0951386i
$476$		0			0
$477$	−4.89255e14			−0.907160
$478$		0			0
$479$			7.61265e14i			1.37940i	0.724095	+	0.689700i	$0.242258\pi$
$479$			7.61265e14i			1.37940i	−0.724095	+	0.689700i	$0.757742\pi$
$480$		0			0
$481$	1.87558e14	+	3.09494e14i	0.332152	+	0.548093i
$482$		0			0
$483$			5.86771e13i			0.101569i
$484$		0			0
$485$	2.60758e14			0.441223
$486$		0			0
$487$		−	4.24385e14i		−	0.702021i	−0.936371	−	0.351011i	$-0.885838\pi$
$487$		−	4.24385e14i		−	0.702021i	0.936371	−	0.351011i	$-0.114162\pi$
$488$		0			0
$489$			6.08686e12i			0.00984453i
$490$		0			0
$491$	−8.13160e14			−1.28596			−0.642981	−	0.765882i	$-0.722302\pi$
$491$	−8.13160e14			−1.28596			−0.642981	+	0.765882i	$0.722302\pi$
$492$		0			0
$493$	3.04355e12			0.00470676
$494$		0			0
$495$	−3.59214e14			−0.543281
$496$		0			0
$497$	−1.09471e15			−1.61934
$498$		0			0
$499$			8.35107e14i			1.20834i	0.796856	+	0.604170i	$0.206495\pi$
$499$			8.35107e14i			1.20834i	−0.796856	+	0.604170i	$0.793505\pi$
$500$		0			0
$501$			2.26786e13i			0.0321003i
$502$		0			0
$503$	5.51271e14			0.763381			0.381691	−	0.924290i	$-0.375342\pi$
$503$	5.51271e14			0.763381			0.381691	+	0.924290i	$0.375342\pi$
$504$		0			0
$505$			9.35258e14i			1.26715i
$506$		0			0
$507$	−1.30565e13	+	2.50098e13i	−0.0173095	+	0.0331563i
$508$		0			0
$509$			8.16981e14i			1.05990i	0.848030	+	0.529949i	$0.177789\pi$
$509$			8.16981e14i			1.05990i	−0.848030	+	0.529949i	$0.822211\pi$
$510$		0			0
$511$	−7.26161e13			−0.0921972
$512$		0			0
$513$		−	6.57710e13i		−	0.0817314i
$514$		0			0
$515$		−	1.10593e15i		−	1.34520i
$516$		0			0
$517$	6.26715e14			0.746229
$518$		0			0
$519$	2.48109e13			0.0289216
$520$		0			0
$521$	−3.30739e14			−0.377466			−0.188733	−	0.982028i	$-0.560438\pi$
$521$	−3.30739e14			−0.377466			−0.188733	+	0.982028i	$0.560438\pi$
$522$		0			0
$523$	−2.28272e14			−0.255090			−0.127545	−	0.991833i	$-0.540710\pi$
$523$	−2.28272e14			−0.255090			−0.127545	+	0.991833i	$0.540710\pi$
$524$		0			0
$525$			5.45766e12i			0.00597215i
$526$		0			0
$527$		−	9.79872e13i		−	0.105005i
$528$		0			0
$529$	1.13388e15			1.19004
$530$		0			0
$531$		−	4.76568e14i		−	0.489897i
$532$		0			0
$533$	−6.46560e13	−	1.06691e14i	−0.0651042	−	0.107430i
$534$		0			0
$535$		−	7.85998e14i		−	0.775310i
$536$		0			0
$537$	6.89031e13			0.0665856
$538$		0			0
$539$			1.42352e15i			1.34780i
$540$		0			0
$541$		−	1.68001e15i		−	1.55857i	−0.626671	−	0.779284i	$-0.715583\pi$
$541$		−	1.68001e15i		−	1.55857i	0.626671	−	0.779284i	$-0.284417\pi$
$542$		0			0
$543$	1.29060e13			0.0117325
$544$		0			0
$545$	−9.44540e14			−0.841471
$546$		0			0
$547$	1.06736e15			0.931925			0.465962	−	0.884805i	$-0.345708\pi$
$547$	1.06736e15			0.931925			0.465962	+	0.884805i	$0.345708\pi$
$548$		0			0
$549$	1.15216e15			0.985974
$550$		0			0
$551$		−	5.60491e13i		−	0.0470148i
$552$		0			0
$553$		−	3.15999e15i		−	2.59835i
$554$		0			0
$555$	2.84135e13			0.0229041
$556$		0			0
$557$			2.56120e14i			0.202413i	0.994865	+	0.101207i	$0.0322703\pi$
$557$			2.56120e14i			0.202413i	−0.994865	+	0.101207i	$0.967730\pi$
$558$		0			0
$559$	−1.65668e15	+	1.00397e15i	−1.28373	+	0.777959i
$560$		0			0
$561$			3.06794e12i			0.00233105i
$562$		0			0
$563$	7.05243e14			0.525463			0.262732	−	0.964869i	$-0.415377\pi$
$563$	7.05243e14			0.525463			0.262732	+	0.964869i	$0.415377\pi$
$564$		0			0
$565$		−	1.78175e14i		−	0.130191i
$566$		0			0
$567$		−	2.54983e15i		−	1.82728i
$568$		0			0
$569$	7.99734e14			0.562119			0.281059	−	0.959690i	$-0.409314\pi$
$569$	7.99734e14			0.562119			0.281059	+	0.959690i	$0.409314\pi$
$570$		0			0
$571$	−5.32271e13			−0.0366973			−0.0183487	−	0.999832i	$-0.505841\pi$
$571$	−5.32271e13			−0.0366973			−0.0183487	+	0.999832i	$0.505841\pi$
$572$		0			0
$573$	−6.96839e12			−0.00471283
$574$		0			0
$575$	1.94087e14			0.128772
$576$		0			0
$577$			7.32962e14i			0.477106i	0.971130	+	0.238553i	$0.0766730\pi$
$577$			7.32962e14i			0.477106i	−0.971130	+	0.238553i	$0.923327\pi$
$578$		0			0
$579$		−	1.02100e14i		−	0.0652068i
$580$		0			0
$581$	−3.46328e15			−2.17029
$582$		0			0
$583$		−	8.41142e14i		−	0.517240i
$584$		0			0
$585$	−8.19482e14	−	1.35225e15i	−0.494518	−	0.816017i
$586$		0			0
$587$			2.27831e14i			0.134928i	0.997722	+	0.0674641i	$0.0214908\pi$
$587$			2.27831e14i			0.134928i	−0.997722	+	0.0674641i	$0.978509\pi$
$588$		0			0
$589$	−1.80450e15			−1.04887
$590$		0			0
$591$		−	5.25953e12i		−	0.00300066i
$592$		0			0
$593$		−	1.68795e15i		−	0.945275i	−0.881257	−	0.472638i	$-0.843302\pi$
$593$		−	1.68795e15i		−	0.945275i	0.881257	−	0.472638i	$-0.156698\pi$
$594$		0			0
$595$	3.49103e14			0.191916
$596$		0			0
$597$	2.39055e13			0.0129015
$598$		0			0
$599$	1.22653e15			0.649874			0.324937	−	0.945736i	$-0.394657\pi$
$599$	1.22653e15			0.649874			0.324937	+	0.945736i	$0.394657\pi$
$600$		0			0
$601$	6.60303e14			0.343505			0.171753	−	0.985140i	$-0.445057\pi$
$601$	6.60303e14			0.343505			0.171753	+	0.985140i	$0.445057\pi$
$602$		0			0
$603$		−	1.71155e14i		−	0.0874269i
$604$		0			0
$605$			1.28739e15i			0.645737i
$606$		0			0
$607$	−2.60465e15			−1.28296			−0.641479	−	0.767141i	$-0.721679\pi$
$607$	−2.60465e15			−1.28296			−0.641479	+	0.767141i	$0.721679\pi$
$608$		0			0
$609$			6.10103e12i			0.00295126i
$610$		0			0
$611$	1.42974e15	+	2.35925e15i	0.679250	+	1.12085i
$612$		0			0
$613$		−	1.09532e15i		−	0.511104i	−0.966795	−	0.255552i	$-0.917743\pi$
$613$		−	1.09532e15i		−	0.511104i	0.966795	−	0.255552i	$-0.0822572\pi$
$614$		0			0
$615$	−9.79486e12			−0.00448937
$616$		0			0
$617$			3.38587e15i			1.52441i	0.647337	+	0.762204i	$0.275883\pi$
$617$			3.38587e15i			1.52441i	−0.647337	+	0.762204i	$0.724117\pi$
$618$		0			0
$619$		−	4.06337e14i		−	0.179717i	−0.995955	−	0.0898583i	$-0.971359\pi$
$619$		−	4.06337e14i		−	0.179717i	0.995955	−	0.0898583i	$-0.0286414\pi$
$620$		0			0
$621$	2.54600e14			0.110625
$622$		0			0
$623$	−7.25125e15			−3.09548
$624$		0			0
$625$	−2.15867e15			−0.905413
$626$		0			0
$627$	5.64982e13			0.0232843
$628$		0			0
$629$			1.73223e14i			0.0701498i
$630$		0			0
$631$		−	1.54091e15i		−	0.613219i	−0.951835	−	0.306610i	$-0.900805\pi$
$631$		−	1.54091e15i		−	0.613219i	0.951835	−	0.306610i	$-0.0991946\pi$
$632$		0			0
$633$	7.49842e13			0.0293258
$634$		0			0
$635$			1.69370e15i			0.651000i
$636$		0			0
$637$	−5.35879e15	+	3.24750e15i	−2.02442	+	1.22683i
$638$		0			0
$639$			2.37331e15i			0.881250i
$640$		0			0
$641$	−2.72635e15			−0.995089			−0.497545	−	0.867438i	$-0.665765\pi$
$641$	−2.72635e15			−0.995089			−0.497545	+	0.867438i	$0.665765\pi$
$642$		0			0
$643$		−	2.92050e15i		−	1.04784i	−0.851766	−	0.523922i	$-0.824468\pi$
$643$		−	2.92050e15i		−	1.04784i	0.851766	−	0.523922i	$-0.175532\pi$
$644$		0			0
$645$			1.52094e14i			0.0536454i
$646$		0			0
$647$	−2.63310e15			−0.913047			−0.456523	−	0.889711i	$-0.650906\pi$
$647$	−2.63310e15			−0.913047			−0.456523	+	0.889711i	$0.650906\pi$
$648$		0			0
$649$	8.19331e14			0.279328
$650$		0			0
$651$	1.96423e14			0.0658411
$652$		0			0
$653$	9.48678e14			0.312677			0.156339	−	0.987704i	$-0.450031\pi$
$653$	9.48678e14			0.312677			0.156339	+	0.987704i	$0.450031\pi$
$654$		0			0
$655$		−	4.95533e15i		−	1.60600i
$656$		0			0
$657$			1.57430e14i			0.0501741i
$658$		0			0
$659$	1.08598e15			0.340369			0.170185	−	0.985412i	$-0.445564\pi$
$659$	1.08598e15			0.340369			0.170185	+	0.985412i	$0.445564\pi$
$660$		0			0
$661$			5.81904e14i			0.179367i	0.995970	+	0.0896837i	$0.0285856\pi$
$661$			5.81904e14i			0.179367i	−0.995970	+	0.0896837i	$0.971414\pi$
$662$		0			0
$663$	−1.15492e13	+	6.99896e12i	−0.00350127	+	0.00212182i
$664$		0			0
$665$		−	6.42897e15i		−	1.91700i
$666$		0			0
$667$	2.16967e14			0.0636357
$668$		0			0
$669$			4.89804e13i			0.0141312i
$670$		0			0
$671$			1.98083e15i			0.562178i
$672$		0			0
$673$	2.47370e15			0.690659			0.345329	−	0.938482i	$-0.387767\pi$
$673$	2.47370e15			0.690659			0.345329	+	0.938482i	$0.387767\pi$
$674$		0			0
$675$	2.36808e13			0.00650468
$676$		0			0
$677$	8.27505e14			0.223632			0.111816	−	0.993729i	$-0.464333\pi$
$677$	8.27505e14			0.223632			0.111816	+	0.993729i	$0.464333\pi$
$678$		0			0
$679$	−3.18670e15			−0.847339
$680$		0			0
$681$			2.05229e13i			0.00536945i
$682$		0			0
$683$			5.10259e14i			0.131364i	0.997841	+	0.0656821i	$0.0209223\pi$
$683$			5.10259e14i			0.131364i	−0.997841	+	0.0656821i	$0.979078\pi$
$684$		0			0
$685$	2.54422e15			0.644549
$686$		0			0
$687$			1.75866e14i			0.0438450i
$688$		0			0
$689$	3.16645e15	−	1.91892e15i	0.776904	−	0.470815i
$690$		0			0
$691$			1.51204e15i			0.365118i	0.983195	+	0.182559i	$0.0584380\pi$
$691$			1.51204e15i			0.365118i	−0.983195	+	0.182559i	$0.941562\pi$
$692$		0			0
$693$	4.38991e15			1.04333
$694$		0			0
$695$			1.97010e15i			0.460864i
$696$		0			0
$697$		−	5.97143e13i		−	0.0137499i
$698$		0			0
$699$	−4.65572e13			−0.0105527
$700$		0			0
$701$	−2.10442e15			−0.469551			−0.234776	−	0.972050i	$-0.575436\pi$
$701$	−2.10442e15			−0.469551			−0.234776	+	0.972050i	$0.575436\pi$
$702$		0			0
$703$	3.19002e15			0.700710
$704$		0			0
$705$	2.16594e14			0.0468388
$706$		0			0
$707$		−	1.14297e16i		−	2.43348i
$708$		0			0
$709$		−	6.96665e15i		−	1.46039i	−0.683238	−	0.730196i	$-0.739429\pi$
$709$		−	6.96665e15i		−	1.46039i	0.683238	−	0.730196i	$-0.260571\pi$
$710$		0			0
$711$	−6.85081e15			−1.41403
$712$		0			0
$713$		−	6.98525e15i		−	1.41968i
$714$		0			0
$715$	2.32483e15	−	1.40888e15i	0.465273	−	0.281962i
$716$		0			0
$717$		−	6.73392e13i		−	0.0132713i
$718$		0			0
$719$	2.02513e15			0.393047			0.196523	−	0.980499i	$-0.437035\pi$
$719$	2.02513e15			0.393047			0.196523	+	0.980499i	$0.437035\pi$
$720$		0			0
$721$			1.35155e16i			2.58337i
$722$		0			0
$723$		−	1.60915e14i		−	0.0302926i
$724$		0			0
$725$	2.01804e13			0.00374172
$726$		0			0
$727$	3.03377e15			0.554044			0.277022	−	0.960864i	$-0.410653\pi$
$727$	3.03377e15			0.554044			0.277022	+	0.960864i	$0.410653\pi$
$728$		0			0
$729$	−5.51245e15			−0.991616
$730$		0			0
$731$	−9.27239e14			−0.164303
$732$		0			0
$733$		−	6.13119e15i		−	1.07022i	−0.844783	−	0.535110i	$-0.820270\pi$
$733$		−	6.13119e15i		−	1.07022i	0.844783	−	0.535110i	$-0.179730\pi$
$734$		0			0
$735$			4.91971e14i			0.0845978i
$736$		0			0
$737$	2.94256e14			0.0498487
$738$		0			0
$739$		−	1.73197e14i		−	0.0289065i	−0.999896	−	0.0144533i	$-0.995399\pi$
$739$		−	1.73197e14i		−	0.0289065i	0.999896	−	0.0144533i	$-0.00460078\pi$
$740$		0			0
$741$	1.28891e14	+	2.12686e14i	0.0211944	+	0.0349734i
$742$		0			0
$743$		−	5.86059e15i		−	0.949517i	−0.880116	−	0.474758i	$-0.842535\pi$
$743$		−	5.86059e15i		−	0.949517i	0.880116	−	0.474758i	$-0.157465\pi$
$744$		0			0
$745$	−8.91418e14			−0.142305
$746$		0			0
$747$			7.50832e15i			1.18108i
$748$		0			0
$749$			9.60560e15i			1.48893i
$750$		0			0
$751$	−6.38438e15			−0.975213			−0.487606	−	0.873064i	$-0.662130\pi$
$751$	−6.38438e15			−0.975213			−0.487606	+	0.873064i	$0.662130\pi$
$752$		0			0
$753$	1.88291e14			0.0283438
$754$		0			0
$755$	6.56602e15			0.974079
$756$		0			0
$757$	−1.18323e15			−0.172998			−0.0864991	−	0.996252i	$-0.527568\pi$
$757$	−1.18323e15			−0.172998			−0.0864991	+	0.996252i	$0.527568\pi$
$758$		0			0
$759$			2.18705e14i			0.0315159i
$760$		0			0
$761$		−	3.38183e15i		−	0.480327i	−0.970732	−	0.240163i	$-0.922799\pi$
$761$		−	3.38183e15i		−	0.480327i	0.970732	−	0.240163i	$-0.0772010\pi$
$762$		0			0
$763$	1.15431e16			1.61599
$764$		0			0
$765$		−	7.56849e14i		−	0.104441i
$766$		0			0
$767$	1.86916e15	+	3.08435e15i	0.254256	+	0.419555i
$768$		0			0
$769$			1.09082e15i			0.146271i	0.997322	+	0.0731353i	$0.0233005\pi$
$769$			1.09082e15i			0.146271i	−0.997322	+	0.0731353i	$0.976700\pi$
$770$		0			0
$771$	3.33522e14			0.0440885
$772$		0			0
$773$			1.19830e15i			0.156163i	0.996947	+	0.0780814i	$0.0248794\pi$
$773$			1.19830e15i			0.156163i	−0.996947	+	0.0780814i	$0.975121\pi$
$774$		0			0
$775$		−	6.49710e14i		−	0.0834757i
$776$		0			0
$777$	−3.47238e14			−0.0439858
$778$		0			0
$779$	−1.09968e15			−0.137344
$780$		0			0
$781$	−4.08027e15			−0.502467
$782$		0			0
$783$	2.64724e13			0.00321443
$784$		0			0
$785$		−	2.18311e15i		−	0.261392i
$786$		0			0
$787$			1.26330e16i			1.49158i	0.666183	+	0.745789i	$0.267927\pi$
$787$			1.26330e16i			1.49158i	−0.666183	+	0.745789i	$0.732073\pi$
$788$		0			0
$789$	−1.72081e14			−0.0200359
$790$		0			0
$791$			2.17746e15i			0.250023i
$792$		0			0
$793$	−7.45678e15	+	4.51892e15i	−0.844401	+	0.511719i
$794$		0			0
$795$		−	2.90700e14i		−	0.0324658i
$796$		0			0
$797$	−1.40771e16			−1.55058			−0.775288	−	0.631608i	$-0.782395\pi$
$797$	−1.40771e16			−1.55058			−0.775288	+	0.631608i	$0.782395\pi$
$798$		0			0
$799$			1.32046e15i			0.143456i
$800$		0			0
$801$			1.57206e16i			1.68457i
$802$		0			0
$803$	−2.70659e14			−0.0286080
$804$		0			0
$805$	2.48866e16			2.59471
$806$		0			0
$807$	−3.39027e14			−0.0348682
$808$		0			0
$809$	1.30932e16			1.32840			0.664198	−	0.747556i	$-0.268773\pi$
$809$	1.30932e16			1.32840			0.664198	+	0.747556i	$0.268773\pi$
$810$		0			0
$811$			1.00710e16i			1.00800i	0.863704	+	0.503999i	$0.168138\pi$
$811$			1.00710e16i			1.00800i	−0.863704	+	0.503999i	$0.831862\pi$
$812$		0			0
$813$			1.61756e14i			0.0159721i
$814$		0			0
$815$	2.58161e15			0.251492
$816$		0			0
$817$			1.70757e16i			1.64119i
$818$		0			0
$819$	1.00148e16	+	1.65257e16i	0.949688	+	1.56711i
$820$		0			0
$821$		−	1.15503e16i		−	1.08070i	−0.841440	−	0.540351i	$-0.818292\pi$
$821$		−	1.15503e16i		−	1.08070i	0.841440	−	0.540351i	$-0.181708\pi$
$822$		0			0
$823$	−8.26319e15			−0.762867			−0.381433	−	0.924396i	$-0.624569\pi$
$823$	−8.26319e15			−0.762867			−0.381433	+	0.924396i	$0.624569\pi$
$824$		0			0
$825$			2.03422e13i			0.00185311i
$826$		0			0
$827$			5.02924e15i			0.452087i	0.974117	+	0.226044i	$0.0725792\pi$
$827$			5.02924e15i			0.452087i	−0.974117	+	0.226044i	$0.927421\pi$
$828$		0			0
$829$	1.35377e16			1.20087			0.600434	−	0.799674i	$-0.294994\pi$
$829$	1.35377e16			1.20087			0.600434	+	0.799674i	$0.294994\pi$
$830$		0			0
$831$	8.33060e14			0.0729240
$832$		0			0
$833$	−2.99930e15			−0.259103
$834$		0			0
$835$	9.61862e15			0.820044
$836$		0			0
$837$		−	8.52280e14i		−	0.0717120i
$838$		0			0
$839$		−	3.06953e15i		−	0.254906i	−0.991845	−	0.127453i	$-0.959320\pi$
$839$		−	3.06953e15i		−	0.254906i	0.991845	−	0.127453i	$-0.0406803\pi$
$840$		0			0
$841$	−1.21780e16			−0.998151
$842$		0			0
$843$			7.40092e14i			0.0598734i
$844$		0			0
$845$	1.06074e16	+	5.53764e15i	0.847024	+	0.442194i
$846$		0			0
$847$		−	1.57331e16i		−	1.24009i
$848$		0			0
$849$	4.87258e14			0.0379111
$850$		0			0
$851$			1.23486e16i			0.948429i
$852$		0			0
$853$		−	9.87001e15i		−	0.748338i	−0.927360	−	0.374169i	$-0.877928\pi$
$853$		−	9.87001e15i		−	0.748338i	0.927360	−	0.374169i	$-0.122072\pi$
$854$		0			0
$855$	−1.39379e16			−1.04324
$856$		0			0
$857$	1.39695e16			1.03225			0.516126	−	0.856513i	$-0.327374\pi$
$857$	1.39695e16			1.03225			0.516126	+	0.856513i	$0.327374\pi$
$858$		0			0
$859$	1.26531e16			0.923068			0.461534	−	0.887122i	$-0.347299\pi$
$859$	1.26531e16			0.923068			0.461534	+	0.887122i	$0.347299\pi$
$860$		0			0
$861$	1.19702e14			0.00862153
$862$		0			0
$863$		−	2.40050e16i		−	1.70704i	−0.521064	−	0.853518i	$-0.674465\pi$
$863$		−	2.40050e16i		−	1.70704i	0.521064	−	0.853518i	$-0.325535\pi$
$864$		0			0
$865$		−	1.05230e16i		−	0.738840i
$866$		0			0
$867$	5.33056e14			0.0369546
$868$		0			0
$869$		−	1.17781e16i		−	0.806247i
$870$		0			0
$871$	6.71292e14	+	1.10772e15i	0.0453745	+	0.0748736i
$872$		0			0
$873$			6.90871e15i			0.461125i
$874$		0			0
$875$	2.89163e16			1.90589
$876$		0			0
$877$			6.86231e15i			0.446655i	0.974743	+	0.223328i	$0.0716920\pi$
$877$			6.86231e15i			0.446655i	−0.974743	+	0.223328i	$0.928308\pi$
$878$		0			0
$879$		−	7.18678e13i		−	0.00461951i
$880$		0			0
$881$	1.84971e16			1.17418			0.587092	−	0.809520i	$-0.300273\pi$
$881$	1.84971e16			1.17418			0.587092	+	0.809520i	$0.300273\pi$
$882$		0			0
$883$	−3.82588e15			−0.239855			−0.119927	−	0.992783i	$-0.538266\pi$
$883$	−3.82588e15			−0.239855			−0.119927	+	0.992783i	$0.538266\pi$
$884$		0			0
$885$	2.83162e14			0.0175327
$886$		0			0
$887$	−1.49960e16			−0.917053			−0.458527	−	0.888681i	$-0.651623\pi$
$887$	−1.49960e16			−0.917053			−0.458527	+	0.888681i	$0.651623\pi$
$888$		0			0
$889$		−	2.06986e16i		−	1.25020i
$890$		0			0
$891$		−	9.50391e15i		−	0.566989i
$892$		0			0
$893$	2.43173e16			1.43295
$894$		0			0
$895$		−	2.92237e16i		−	1.70102i
$896$		0			0
$897$	−8.23309e14	+	4.98937e14i	−0.0473374	+	0.0286871i
$898$		0			0
$899$		−	7.26300e14i		−	0.0412513i
$900$		0			0
$901$	1.77225e15			0.0994351
$902$		0			0
$903$		−	1.85872e15i		−	0.103022i
$904$		0			0
$905$		−	5.47378e15i		−	0.299723i
$906$		0			0
$907$	1.93699e16			1.04782			0.523910	−	0.851774i	$-0.324473\pi$
$907$	1.93699e16			1.04782			0.523910	+	0.851774i	$0.324473\pi$
$908$		0			0
$909$	−2.47794e16			−1.32431
$910$		0			0
$911$	−2.24608e16			−1.18597			−0.592987	−	0.805212i	$-0.702051\pi$
$911$	−2.24608e16			−1.18597			−0.592987	+	0.805212i	$0.702051\pi$
$912$		0			0
$913$	−1.29085e16			−0.673423
$914$		0			0
$915$			6.84580e14i			0.0352864i
$916$		0			0
$917$			6.05586e16i			3.08421i
$918$		0			0
$919$	1.07251e15			0.0539714			0.0269857	−	0.999636i	$-0.491409\pi$
$919$	1.07251e15			0.0539714			0.0269857	+	0.999636i	$0.491409\pi$
$920$		0			0
$921$			8.03471e14i			0.0399523i
$922$		0			0
$923$	−9.30840e15	−	1.53600e16i	−0.457368	−	0.754715i
$924$		0			0
$925$			1.14856e15i			0.0557668i
$926$		0			0
$927$	2.93013e16			1.40588
$928$		0			0
$929$		−	7.25989e15i		−	0.344226i	−0.985077	−	0.172113i	$-0.944941\pi$
$929$		−	7.25989e15i		−	0.344226i	0.985077	−	0.172113i	$-0.0550594\pi$
$930$		0			0
$931$			5.52341e16i			2.58812i
$932$		0			0
$933$	3.25134e13			0.00150562
$934$		0			0
$935$	1.30120e15			0.0595497
$936$		0			0
$937$	2.40207e16			1.08647			0.543235	−	0.839581i	$-0.317199\pi$
$937$	2.40207e16			1.08647			0.543235	+	0.839581i	$0.317199\pi$
$938$		0			0
$939$	1.19805e15			0.0535568
$940$		0			0
$941$			4.02038e16i			1.77633i	0.459525	+	0.888165i	$0.348020\pi$
$941$			4.02038e16i			1.77633i	−0.459525	+	0.888165i	$0.651980\pi$
$942$		0			0
$943$		−	4.25688e15i		−	0.185899i
$944$		0			0
$945$	3.03645e15			0.131066
$946$		0			0
$947$			2.20959e16i			0.942729i	0.881938	+	0.471365i	$0.156238\pi$
$947$			2.20959e16i			0.942729i	−0.881938	+	0.471365i	$0.843762\pi$
$948$		0			0
$949$	−6.17461e14	−	1.01889e15i	−0.0260403	−	0.0429697i
$950$		0			0
$951$			1.10310e15i			0.0459858i
$952$		0			0
$953$	2.83727e16			1.16920			0.584600	−	0.811321i	$-0.301251\pi$
$953$	2.83727e16			1.16920			0.584600	+	0.811321i	$0.301251\pi$
$954$		0			0
$955$			2.95549e15i			0.120396i
$956$		0			0
$957$			2.27402e13i			0.000915752i
$958$		0			0
$959$	−3.10926e16			−1.23781
$960$		0			0
$961$	2.02522e15			0.0797064
$962$		0			0
$963$	2.08248e16			0.810282
$964$		0			0
$965$	−4.33034e16			−1.66579
$966$		0			0
$967$			4.38431e16i			1.66746i	0.552171	+	0.833731i	$0.313800\pi$
$967$			4.38431e16i			1.66746i	−0.552171	+	0.833731i	$0.686200\pi$
$968$		0			0
$969$			1.19040e14i			0.00447621i
$970$		0			0
$971$	3.03701e16			1.12912			0.564561	−	0.825392i	$-0.309046\pi$
$971$	3.03701e16			1.12912			0.564561	+	0.825392i	$0.309046\pi$
$972$		0			0
$973$		−	2.40764e16i		−	0.885058i
$974$		0			0
$975$	−7.65774e13	+	4.64070e13i	−0.00278340	+	0.00168678i
$976$		0			0
$977$		−	2.09090e16i		−	0.751472i	−0.926727	−	0.375736i	$-0.877390\pi$
$977$		−	2.09090e16i		−	0.751472i	0.926727	−	0.375736i	$-0.122610\pi$
$978$		0			0
$979$	−2.70273e16			−0.960502
$980$		0			0
$981$		−	2.50253e16i		−	0.879427i
$982$		0			0
$983$			2.29870e15i			0.0798800i	0.999202	+	0.0399400i	$0.0127167\pi$
$983$			2.29870e15i			0.0798800i	−0.999202	+	0.0399400i	$0.987283\pi$
$984$		0			0
$985$	−2.23072e15			−0.0766558
$986$		0			0
$987$	−2.64697e15			−0.0899508
$988$		0			0
$989$	−6.61004e16			−2.22139
$990$		0			0
$991$	−4.44842e16			−1.47843			−0.739215	−	0.673470i	$-0.764803\pi$
$991$	−4.44842e16			−1.47843			−0.739215	+	0.673470i	$0.764803\pi$
$992$		0			0
$993$		−	3.31248e14i		−	0.0108876i
$994$		0			0
$995$		−	1.01390e16i		−	0.329586i
$996$		0			0
$997$	−8.07503e15			−0.259609			−0.129805	−	0.991540i	$-0.541435\pi$
$997$	−8.07503e15			−0.259609			−0.129805	+	0.991540i	$0.541435\pi$
$998$		0			0
$999$			1.50667e15i			0.0479079i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.12.f.b.129.6		12
4.3	odd	2		13.12.b.a.12.3	✓	12
12.11	even	2		117.12.b.b.64.10		12
13.12	even	2	inner	208.12.f.b.129.5		12
52.31	even	4		169.12.a.e.1.3		12
52.47	even	4		169.12.a.e.1.10		12
52.51	odd	2		13.12.b.a.12.10	yes	12
156.155	even	2		117.12.b.b.64.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.b.a.12.3	✓	12	4.3	odd	2
13.12.b.a.12.10	yes	12	52.51	odd	2
117.12.b.b.64.3		12	156.155	even	2
117.12.b.b.64.10		12	12.11	even	2
169.12.a.e.1.3		12	52.31	even	4
169.12.a.e.1.10		12	52.47	even	4
208.12.f.b.129.5		12	13.12	even	2	inner
208.12.f.b.129.6		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-15.7424 q^{3} +6676.77i q^{5} -81596.2i q^{7} -176899. q^{9} +O(q^{10})\)	\(q-15.7424 q^{3} +6676.77i q^{5} -81596.2i q^{7} -176899. q^{9} -304131. i q^{11} +(1.14489e6 - 693820. i) q^{13} -105108. i q^{15} +640791. q^{17} -1.18006e7i q^{19} +1.28452e6i q^{21} +4.56803e7 q^{23} +4.24881e6 q^{25} +5.57352e6 q^{27} +4.74967e6 q^{29} -1.52916e8i q^{31} +4.78773e6i q^{33} +5.44799e8 q^{35} +2.70326e8i q^{37} +(-1.80233e7 + 1.09224e7i) q^{39} -9.31884e7i q^{41} -1.44702e9 q^{43} -1.18112e9i q^{45} +2.06068e9i q^{47} -4.68061e9 q^{49} -1.00876e7 q^{51} +2.76573e9 q^{53} +2.03061e9 q^{55} +1.85770e8i q^{57} +2.69401e9i q^{59} -6.51310e9 q^{61} +1.44343e10i q^{63} +(4.63248e9 + 7.64418e9i) q^{65} +9.67530e8i q^{67} -7.19116e8 q^{69} -1.34162e10i q^{71} -8.89945e8i q^{73} -6.68862e7 q^{75} -2.48159e10 q^{77} +3.87272e10 q^{79} +3.12494e10 q^{81} -4.24441e10i q^{83} +4.27842e9i q^{85} -7.47710e7 q^{87} -8.88675e10i q^{89} +(-5.66131e10 - 9.34187e10i) q^{91} +2.40726e9i q^{93} +7.87901e10 q^{95} -3.90545e10i q^{97} +5.38005e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 488 q^{3} + 654644 q^{9}+O(q^{10}) \) 12 * q + 488 * q^3 + 654644 * q^9	\( 12 q + 488 q^{3} + 654644 q^{9} + 3208868 q^{13} + 12198768 q^{17} - 5810592 q^{23} + 6102388 q^{25} + 52613336 q^{27} - 244463112 q^{29} + 562027560 q^{35} + 2199109744 q^{39} - 2294519976 q^{43} - 3573617796 q^{49} - 7713246552 q^{51} - 4602062760 q^{53} + 6178744976 q^{55} - 13775649944 q^{61} - 7598401512 q^{65} - 25419983328 q^{69} - 68016370832 q^{75} - 80478036048 q^{77} - 18046097296 q^{79} - 132677486692 q^{81} - 94507900752 q^{87} - 104793638664 q^{91} - 145093149648 q^{95}+O(q^{100}) \) 12 * q + 488 * q^3 + 654644 * q^9 + 3208868 * q^13 + 12198768 * q^17 - 5810592 * q^23 + 6102388 * q^25 + 52613336 * q^27 - 244463112 * q^29 + 562027560 * q^35 + 2199109744 * q^39 - 2294519976 * q^43 - 3573617796 * q^49 - 7713246552 * q^51 - 4602062760 * q^53 + 6178744976 * q^55 - 13775649944 * q^61 - 7598401512 * q^65 - 25419983328 * q^69 - 68016370832 * q^75 - 80478036048 * q^77 - 18046097296 * q^79 - 132677486692 * q^81 - 94507900752 * q^87 - 104793638664 * q^91 - 145093149648 * q^95

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