LMFDB - Embedded newform 208.12.f.b.129.8

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			129.8
Root			$-37.0312i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.12.f.b.129.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+143.287 q^{3} +5266.14i q^{5} -16188.3i q^{7} -156616. q^{9} +O(q^{10})$	$q+143.287 q^{3} +5266.14i q^{5} -16188.3i q^{7} -156616. q^{9} +447110. i q^{11} +(-1.33862e6 - 16101.7i) q^{13} +754569. i q^{15} -6.84716e6 q^{17} -1.68794e7i q^{19} -2.31957e6i q^{21} -1.49304e7 q^{23} +2.10958e7 q^{25} -4.78238e7 q^{27} +3.19957e7 q^{29} -2.68660e7i q^{31} +6.40650e7i q^{33} +8.52500e7 q^{35} +5.43137e8i q^{37} +(-1.91806e8 - 2.30716e6i) q^{39} -9.19462e8i q^{41} +2.47741e8 q^{43} -8.24762e8i q^{45} -1.05508e9i q^{47} +1.71526e9 q^{49} -9.81107e8 q^{51} +3.13264e9 q^{53} -2.35455e9 q^{55} -2.41860e9i q^{57} +2.01659e9i q^{59} +1.47077e9 q^{61} +2.53535e9i q^{63} +(8.47940e7 - 7.04936e9i) q^{65} -1.34287e10i q^{67} -2.13932e9 q^{69} +2.43380e9i q^{71} +2.78909e10i q^{73} +3.02276e9 q^{75} +7.23797e9 q^{77} -1.03269e10 q^{79} +2.08915e10 q^{81} +3.76857e10i q^{83} -3.60581e10i q^{85} +4.58456e9 q^{87} +9.91225e10i q^{89} +(-2.60660e8 + 2.16700e10i) q^{91} -3.84954e9i q^{93} +8.88896e10 q^{95} -4.38508e10i q^{97} -7.00246e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	143.287			0.340439			0.170219	−	0.985406i	$-0.445552\pi$
$3$	143.287			0.340439			0.170219	+	0.985406i	$0.445552\pi$
$4$		0			0
$5$			5266.14i			0.753629i	0.926289	+	0.376815i	$0.122981\pi$
$5$			5266.14i			0.753629i	−0.926289	+	0.376815i	$0.877019\pi$
$6$		0			0
$7$		−	16188.3i		−	0.364051i	−0.983294	−	0.182026i	$-0.941735\pi$
$7$		−	16188.3i		−	0.364051i	0.983294	−	0.182026i	$-0.0582654\pi$
$8$		0			0
$9$	−156616.			−0.884101
$10$		0			0
$11$			447110.i			0.837057i	0.908204	+	0.418528i	$0.137454\pi$
$11$			447110.i			0.837057i	−0.908204	+	0.418528i	$0.862546\pi$
$12$		0			0
$13$	−1.33862e6	−	16101.7i	−0.999928	−	0.0120277i
$13$	−1.33862e6	−	16101.7i	−0.999928	−	0.0120277i
$14$		0			0
$15$			754569.i			0.256565i
$16$		0			0
$17$	−6.84716e6			−1.16961			−0.584805	−	0.811174i	$-0.698829\pi$
$17$	−6.84716e6			−1.16961			−0.584805	+	0.811174i	$0.698829\pi$
$18$		0			0
$19$		−	1.68794e7i		−	1.56392i	−0.623331	−	0.781958i	$-0.714221\pi$
$19$		−	1.68794e7i		−	1.56392i	0.623331	−	0.781958i	$-0.285779\pi$
$20$		0			0
$21$		−	2.31957e6i		−	0.123937i
$22$		0			0
$23$	−1.49304e7			−0.483690			−0.241845	−	0.970315i	$-0.577753\pi$
$23$	−1.49304e7			−0.483690			−0.241845	+	0.970315i	$0.577753\pi$
$24$		0			0
$25$	2.10958e7			0.432043
$26$		0			0
$27$	−4.78238e7			−0.641421
$28$		0			0
$29$	3.19957e7			0.289670			0.144835	−	0.989456i	$-0.453735\pi$
$29$	3.19957e7			0.289670			0.144835	+	0.989456i	$0.453735\pi$
$30$		0			0
$31$		−	2.68660e7i		−	0.168544i	−0.996443	−	0.0842721i	$-0.973144\pi$
$31$		−	2.68660e7i		−	0.168544i	0.996443	−	0.0842721i	$-0.0268565\pi$
$32$		0			0
$33$			6.40650e7i			0.284967i
$34$		0			0
$35$	8.52500e7			0.274360
$36$		0			0
$37$			5.43137e8i			1.28766i	0.765170	+	0.643828i	$0.222655\pi$
$37$			5.43137e8i			1.28766i	−0.765170	+	0.643828i	$0.777345\pi$
$38$		0			0
$39$	−1.91806e8	−	2.30716e6i	−0.340414	−	0.00409471i
$40$		0			0
$41$		−	9.19462e8i		−	1.23943i	−0.784826	−	0.619716i	$-0.787248\pi$
$41$		−	9.19462e8i		−	1.23943i	0.784826	−	0.619716i	$-0.212752\pi$
$42$		0			0
$43$	2.47741e8			0.256993			0.128496	−	0.991710i	$-0.458985\pi$
$43$	2.47741e8			0.256993			0.128496	+	0.991710i	$0.458985\pi$
$44$		0			0
$45$		−	8.24762e8i		−	0.666285i
$46$		0			0
$47$		−	1.05508e9i		−	0.671036i	−0.942034	−	0.335518i	$-0.891089\pi$
$47$		−	1.05508e9i		−	0.671036i	0.942034	−	0.335518i	$-0.108911\pi$
$48$		0			0
$49$	1.71526e9			0.867467
$50$		0			0
$51$	−9.81107e8			−0.398181
$52$		0			0
$53$	3.13264e9			1.02895			0.514474	−	0.857506i	$-0.327987\pi$
$53$	3.13264e9			1.02895			0.514474	+	0.857506i	$0.327987\pi$
$54$		0			0
$55$	−2.35455e9			−0.630831
$56$		0			0
$57$		−	2.41860e9i		−	0.532418i
$58$		0			0
$59$			2.01659e9i			0.367224i	0.982999	+	0.183612i	$0.0587790\pi$
$59$			2.01659e9i			0.367224i	−0.982999	+	0.183612i	$0.941221\pi$
$60$		0			0
$61$	1.47077e9			0.222961			0.111481	−	0.993767i	$-0.464441\pi$
$61$	1.47077e9			0.222961			0.111481	+	0.993767i	$0.464441\pi$
$62$		0			0
$63$			2.53535e9i			0.321858i
$64$		0			0
$65$	8.47940e7	−	7.04936e9i	0.00906445	−	0.753575i
$66$		0			0
$67$		−	1.34287e10i		−	1.21513i	−0.794269	−	0.607566i	$-0.792146\pi$
$67$		−	1.34287e10i		−	1.21513i	0.794269	−	0.607566i	$-0.207854\pi$
$68$		0			0
$69$	−2.13932e9			−0.164667
$70$		0			0
$71$			2.43380e9i			0.160090i	0.996791	+	0.0800449i	$0.0255064\pi$
$71$			2.43380e9i			0.160090i	−0.996791	+	0.0800449i	$0.974494\pi$
$72$		0			0
$73$			2.78909e10i			1.57466i	0.616531	+	0.787331i	$0.288538\pi$
$73$			2.78909e10i			1.57466i	−0.616531	+	0.787331i	$0.711462\pi$
$74$		0			0
$75$	3.02276e9			0.147084
$76$		0			0
$77$	7.23797e9			0.304732
$78$		0			0
$79$	−1.03269e10			−0.377591			−0.188795	−	0.982016i	$-0.560458\pi$
$79$	−1.03269e10			−0.377591			−0.188795	+	0.982016i	$0.560458\pi$
$80$		0			0
$81$	2.08915e10			0.665736
$82$		0			0
$83$			3.76857e10i			1.05014i	0.851059	+	0.525070i	$0.175961\pi$
$83$			3.76857e10i			1.05014i	−0.851059	+	0.525070i	$0.824039\pi$
$84$		0			0
$85$		−	3.60581e10i		−	0.881453i
$86$		0			0
$87$	4.58456e9			0.0986148
$88$		0			0
$89$			9.91225e10i			1.88160i	0.338963	+	0.940800i	$0.389924\pi$
$89$			9.91225e10i			1.88160i	−0.338963	+	0.940800i	$0.610076\pi$
$90$		0			0
$91$	−2.60660e8	+	2.16700e10i	−0.00437871	+	0.364025i
$92$		0			0
$93$		−	3.84954e9i		−	0.0573790i
$94$		0			0
$95$	8.88896e10			1.17861
$96$		0			0
$97$		−	4.38508e10i		−	0.518481i	−0.965813	−	0.259240i	$-0.916528\pi$
$97$		−	4.38508e10i		−	0.518481i	0.965813	−	0.259240i	$-0.0834722\pi$
$98$		0			0
$99$		−	7.00246e10i		−	0.740043i
$100$		0			0
$101$	−6.46974e10			−0.612519			−0.306259	−	0.951948i	$-0.599077\pi$
$101$	−6.46974e10			−0.612519			−0.306259	+	0.951948i	$0.599077\pi$
$102$		0			0
$103$	6.05203e9			0.0514395			0.0257197	−	0.999669i	$-0.491812\pi$
$103$	6.05203e9			0.0514395			0.0257197	+	0.999669i	$0.491812\pi$
$104$		0			0
$105$	1.22152e10			0.0934027
$106$		0			0
$107$	1.83061e11			1.26179			0.630893	−	0.775869i	$-0.282688\pi$
$107$	1.83061e11			1.26179			0.630893	+	0.775869i	$0.282688\pi$
$108$		0			0
$109$		−	2.18877e11i		−	1.36255i	−0.732026	−	0.681277i	$-0.761425\pi$
$109$		−	2.18877e11i		−	1.36255i	0.732026	−	0.681277i	$-0.238575\pi$
$110$		0			0
$111$			7.78243e10i			0.438368i
$112$		0			0
$113$	−2.05501e11			−1.04926			−0.524630	−	0.851330i	$-0.675796\pi$
$113$	−2.05501e11			−1.04926			−0.524630	+	0.851330i	$0.675796\pi$
$114$		0			0
$115$		−	7.86255e10i		−	0.364523i
$116$		0			0
$117$	2.09649e11	+	2.52179e9i	0.884037	+	0.0106337i
$118$		0			0
$119$			1.10844e11i			0.425798i
$120$		0			0
$121$	8.54040e10			0.299336
$122$		0			0
$123$		−	1.31747e11i		−	0.421951i
$124$		0			0
$125$			3.68230e11i			1.07923i
$126$		0			0
$127$	−5.86461e11			−1.57514			−0.787569	−	0.616226i	$-0.788661\pi$
$127$	−5.86461e11			−1.57514			−0.787569	+	0.616226i	$0.788661\pi$
$128$		0			0
$129$	3.54980e10			0.0874903
$130$		0			0
$131$	1.08205e11			0.245051			0.122525	−	0.992465i	$-0.460901\pi$
$131$	1.08205e11			0.245051			0.122525	+	0.992465i	$0.460901\pi$
$132$		0			0
$133$	−2.73250e11			−0.569346
$134$		0			0
$135$		−	2.51847e11i		−	0.483394i
$136$		0			0
$137$		−	4.94496e11i		−	0.875386i	−0.899125	−	0.437693i	$-0.855796\pi$
$137$		−	4.94496e11i		−	0.875386i	0.899125	−	0.437693i	$-0.144204\pi$
$138$		0			0
$139$	4.85658e11			0.793869			0.396934	−	0.917847i	$-0.370074\pi$
$139$	4.85658e11			0.793869			0.396934	+	0.917847i	$0.370074\pi$
$140$		0			0
$141$		−	1.51178e11i		−	0.228447i
$142$		0			0
$143$	7.19925e9	−	5.98510e11i	0.0100679	−	0.836996i
$144$		0			0
$145$			1.68494e11i			0.218304i
$146$		0			0
$147$	2.45775e11			0.295319
$148$		0			0
$149$			3.26280e11i			0.363971i	0.983301	+	0.181985i	$0.0582524\pi$
$149$			3.26280e11i			0.363971i	−0.983301	+	0.181985i	$0.941748\pi$
$150$		0			0
$151$			9.77744e11i			1.01357i	0.862074	+	0.506783i	$0.169165\pi$
$151$			9.77744e11i			1.01357i	−0.862074	+	0.506783i	$0.830835\pi$
$152$		0			0
$153$	1.07237e12			1.03405
$154$		0			0
$155$	1.41480e11			0.127020
$156$		0			0
$157$	1.82267e12			1.52497			0.762483	−	0.647008i	$-0.223980\pi$
$157$	1.82267e12			1.52497			0.762483	+	0.647008i	$0.223980\pi$
$158$		0			0
$159$	4.48866e11			0.350294
$160$		0			0
$161$			2.41698e11i			0.176088i
$162$		0			0
$163$		−	2.23968e12i		−	1.52459i	−0.647228	−	0.762296i	$-0.724072\pi$
$163$		−	2.23968e12i		−	1.52459i	0.647228	−	0.762296i	$-0.275928\pi$
$164$		0			0
$165$	−3.37376e11			−0.214759
$166$		0			0
$167$			2.94541e12i			1.75471i	0.479843	+	0.877355i	$0.340694\pi$
$167$			2.94541e12i			1.75471i	−0.479843	+	0.877355i	$0.659306\pi$
$168$		0			0
$169$	1.79164e12	+	4.31081e10i	0.999711	+	0.0240537i
$170$		0			0
$171$			2.64359e12i			1.38266i
$172$		0			0
$173$	6.82552e11			0.334875			0.167437	−	0.985883i	$-0.446451\pi$
$173$	6.82552e11			0.334875			0.167437	+	0.985883i	$0.446451\pi$
$174$		0			0
$175$		−	3.41506e11i		−	0.157286i
$176$		0			0
$177$			2.88951e11i			0.125017i
$178$		0			0
$179$	−3.04733e12			−1.23945			−0.619723	−	0.784820i	$-0.712755\pi$
$179$	−3.04733e12			−1.23945			−0.619723	+	0.784820i	$0.712755\pi$
$180$		0			0
$181$	2.76647e11			0.105851			0.0529254	−	0.998598i	$-0.483145\pi$
$181$	2.76647e11			0.105851			0.0529254	+	0.998598i	$0.483145\pi$
$182$		0			0
$183$	2.10741e11			0.0759047
$184$		0			0
$185$	−2.86024e12			−0.970415
$186$		0			0
$187$		−	3.06143e12i		−	0.979030i
$188$		0			0
$189$			7.74187e11i			0.233510i
$190$		0			0
$191$	6.98096e12			1.98716			0.993578	−	0.113153i	$-0.0360949\pi$
$191$	6.98096e12			1.98716			0.993578	+	0.113153i	$0.0360949\pi$
$192$		0			0
$193$			2.26651e12i			0.609245i	0.952473	+	0.304622i	$0.0985302\pi$
$193$			2.26651e12i			0.609245i	−0.952473	+	0.304622i	$0.901470\pi$
$194$		0			0
$195$	1.21499e10	−	1.01008e12i	0.00308589	−	0.256546i
$196$		0			0
$197$		−	3.61395e12i		−	0.867798i	−0.900962	−	0.433899i	$-0.857138\pi$
$197$		−	3.61395e12i		−	0.867798i	0.900962	−	0.433899i	$-0.142862\pi$
$198$		0			0
$199$	7.55025e12			1.71502			0.857511	−	0.514466i	$-0.172010\pi$
$199$	7.55025e12			1.71502			0.857511	+	0.514466i	$0.172010\pi$
$200$		0			0
$201$		−	1.92416e12i		−	0.413678i
$202$		0			0
$203$		−	5.17957e11i		−	0.105455i
$204$		0			0
$205$	4.84202e12			0.934072
$206$		0			0
$207$	2.33833e12			0.427631
$208$		0			0
$209$	7.54697e12			1.30909
$210$		0			0
$211$	7.88114e12			1.29729			0.648643	−	0.761093i	$-0.275337\pi$
$211$	7.88114e12			1.29729			0.648643	+	0.761093i	$0.275337\pi$
$212$		0			0
$213$			3.48731e11i			0.0545008i
$214$		0			0
$215$			1.30464e12i			0.193677i
$216$		0			0
$217$	−4.34916e11			−0.0613587
$218$		0			0
$219$			3.99640e12i			0.536076i
$220$		0			0
$221$	9.16573e12	+	1.10251e11i	1.16953	+	0.0140678i
$222$		0			0
$223$		−	1.04892e13i		−	1.27370i	−0.770988	−	0.636849i	$-0.780237\pi$
$223$		−	1.04892e13i		−	1.27370i	0.770988	−	0.636849i	$-0.219763\pi$
$224$		0			0
$225$	−3.30394e12			−0.381970
$226$		0			0
$227$		−	1.15289e13i		−	1.26954i	−0.772702	−	0.634769i	$-0.781095\pi$
$227$		−	1.15289e13i		−	1.26954i	0.772702	−	0.634769i	$-0.218905\pi$
$228$		0			0
$229$			1.27585e12i			0.133876i	0.997757	+	0.0669381i	$0.0213230\pi$
$229$			1.27585e12i			0.133876i	−0.997757	+	0.0669381i	$0.978677\pi$
$230$		0			0
$231$	1.03710e12			0.103743
$232$		0			0
$233$	7.27631e12			0.694151			0.347075	−	0.937837i	$-0.387175\pi$
$233$	7.27631e12			0.694151			0.347075	+	0.937837i	$0.387175\pi$
$234$		0			0
$235$	5.55618e12			0.505712
$236$		0			0
$237$	−1.47971e12			−0.128547
$238$		0			0
$239$			1.50484e13i			1.24825i	0.781325	+	0.624125i	$0.214544\pi$
$239$			1.50484e13i			1.24825i	−0.781325	+	0.624125i	$0.785456\pi$
$240$		0			0
$241$		−	1.40977e13i		−	1.11700i	−0.829504	−	0.558501i	$-0.811377\pi$
$241$		−	1.40977e13i		−	1.11700i	0.829504	−	0.558501i	$-0.188623\pi$
$242$		0			0
$243$	1.14653e13			0.868064
$244$		0			0
$245$			9.03283e12i			0.653748i
$246$		0			0
$247$	−2.71788e11	+	2.25951e13i	−0.0188104	+	1.56380i
$248$		0			0
$249$			5.39986e12i			0.357508i
$250$		0			0
$251$	7.86696e12			0.498427			0.249213	−	0.968449i	$-0.419828\pi$
$251$	7.86696e12			0.498427			0.249213	+	0.968449i	$0.419828\pi$
$252$		0			0
$253$		−	6.67552e12i		−	0.404876i
$254$		0			0
$255$		−	5.16665e12i		−	0.300081i
$256$		0			0
$257$	1.15670e13			0.643560			0.321780	−	0.946814i	$-0.395719\pi$
$257$	1.15670e13			0.643560			0.321780	+	0.946814i	$0.395719\pi$
$258$		0			0
$259$	8.79247e12			0.468773
$260$		0			0
$261$	−5.01104e12			−0.256097
$262$		0			0
$263$	−1.58415e13			−0.776319			−0.388160	−	0.921592i	$-0.626889\pi$
$263$	−1.58415e13			−0.776319			−0.388160	+	0.921592i	$0.626889\pi$
$264$		0			0
$265$			1.64969e13i			0.775445i
$266$		0			0
$267$			1.42029e13i			0.640570i
$268$		0			0
$269$	2.84481e13			1.23145			0.615723	−	0.787963i	$-0.288864\pi$
$269$	2.84481e13			1.23145			0.615723	+	0.787963i	$0.288864\pi$
$270$		0			0
$271$		−	4.04902e13i		−	1.68275i	−0.540453	−	0.841374i	$-0.681747\pi$
$271$		−	4.04902e13i		−	1.68275i	0.540453	−	0.841374i	$-0.318253\pi$
$272$		0			0
$273$	−3.73491e10	+	3.10502e12i	−0.00149068	+	0.123928i
$274$		0			0
$275$			9.43217e12i			0.361644i
$276$		0			0
$277$	2.83125e13			1.04313			0.521565	−	0.853211i	$-0.325348\pi$
$277$	2.83125e13			1.04313			0.521565	+	0.853211i	$0.325348\pi$
$278$		0			0
$279$			4.20764e12i			0.149010i
$280$		0			0
$281$		−	3.39651e12i		−	0.115651i	−0.998327	−	0.0578254i	$-0.981583\pi$
$281$		−	3.39651e12i		−	0.115651i	0.998327	−	0.0578254i	$-0.0184167\pi$
$282$		0			0
$283$	4.66025e12			0.152610			0.0763052	−	0.997085i	$-0.475688\pi$
$283$	4.66025e12			0.152610			0.0763052	+	0.997085i	$0.475688\pi$
$284$		0			0
$285$	1.27367e13			0.401246
$286$		0			0
$287$	−1.48845e13			−0.451217
$288$		0			0
$289$	1.26116e13			0.367988
$290$		0			0
$291$		−	6.28324e12i		−	0.176511i
$292$		0			0
$293$		−	2.66585e12i		−	0.0721214i	−0.999350	−	0.0360607i	$-0.988519\pi$
$293$		−	2.66585e12i		−	0.0721214i	0.999350	−	0.0360607i	$-0.0114810\pi$
$294$		0			0
$295$	−1.06197e13			−0.276751
$296$		0			0
$297$		−	2.13825e13i		−	0.536906i
$298$		0			0
$299$	1.99861e13	+	2.40405e11i	0.483655	+	0.00581770i
$300$		0			0
$301$		−	4.01050e12i		−	0.0935585i
$302$		0			0
$303$	−9.27028e12			−0.208525
$304$		0			0
$305$			7.74527e12i			0.168030i
$306$		0			0
$307$		−	3.69671e13i		−	0.773667i	−0.922150	−	0.386833i	$-0.873569\pi$
$307$		−	3.69671e13i		−	0.773667i	0.922150	−	0.386833i	$-0.126431\pi$
$308$		0			0
$309$	8.67176e11			0.0175120
$310$		0			0
$311$	7.88387e13			1.53659			0.768294	−	0.640097i	$-0.221106\pi$
$311$	7.88387e13			1.53659			0.768294	+	0.640097i	$0.221106\pi$
$312$		0			0
$313$	−7.62878e13			−1.43536			−0.717681	−	0.696372i	$-0.754796\pi$
$313$	−7.62878e13			−1.43536			−0.717681	+	0.696372i	$0.754796\pi$
$314$		0			0
$315$	−1.33515e13			−0.242562
$316$		0			0
$317$			6.93621e13i			1.21702i	0.793548	+	0.608508i	$0.208232\pi$
$317$			6.93621e13i			1.21702i	−0.793548	+	0.608508i	$0.791768\pi$
$318$		0			0
$319$			1.43056e13i			0.242470i
$320$		0			0
$321$	2.62303e13			0.429561
$322$		0			0
$323$			1.15576e14i			1.82917i
$324$		0			0
$325$	−2.82393e13	−	3.39679e11i	−0.432012	−	0.00519650i
$326$		0			0
$327$		−	3.13621e13i		−	0.463866i
$328$		0			0
$329$	−1.70799e13			−0.244291
$330$		0			0
$331$		−	2.88136e13i		−	0.398606i	−0.979938	−	0.199303i	$-0.936132\pi$
$331$		−	2.88136e13i		−	0.398606i	0.979938	−	0.199303i	$-0.0638677\pi$
$332$		0			0
$333$		−	8.50639e13i		−	1.13842i
$334$		0			0
$335$	7.07176e13			0.915759
$336$		0			0
$337$	5.86242e13			0.734704			0.367352	−	0.930082i	$-0.380264\pi$
$337$	5.86242e13			0.734704			0.367352	+	0.930082i	$0.380264\pi$
$338$		0			0
$339$	−2.94456e13			−0.357209
$340$		0			0
$341$	1.20121e13			0.141081
$342$		0			0
$343$		−	5.97769e13i		−	0.679854i
$344$		0			0
$345$		−	1.12660e13i		−	0.124098i
$346$		0			0
$347$	−1.02722e14			−1.09610			−0.548052	−	0.836444i	$-0.684631\pi$
$347$	−1.02722e14			−1.09610			−0.548052	+	0.836444i	$0.684631\pi$
$348$		0			0
$349$			1.14493e13i			0.118369i	0.998247	+	0.0591845i	$0.0188500\pi$
$349$			1.14493e13i			0.118369i	−0.998247	+	0.0591845i	$0.981150\pi$
$350$		0			0
$351$	6.40179e13	+	7.70046e11i	0.641375	+	0.00771485i
$352$		0			0
$353$		−	1.33798e14i		−	1.29924i	−0.760258	−	0.649621i	$-0.774928\pi$
$353$		−	1.33798e14i		−	1.29924i	0.760258	−	0.649621i	$-0.225072\pi$
$354$		0			0
$355$	−1.28167e13			−0.120648
$356$		0			0
$357$			1.58825e13i			0.144958i
$358$		0			0
$359$		−	1.72020e13i		−	0.152251i	−0.997098	−	0.0761253i	$-0.975745\pi$
$359$		−	1.72020e13i		−	0.152251i	0.997098	−	0.0761253i	$-0.0242549\pi$
$360$		0			0
$361$	−1.68425e14			−1.44583
$362$		0			0
$363$	1.22373e13			0.101906
$364$		0			0
$365$	−1.46878e14			−1.18671
$366$		0			0
$367$	−1.09729e14			−0.860319			−0.430159	−	0.902753i	$-0.641543\pi$
$367$	−1.09729e14			−0.860319			−0.430159	+	0.902753i	$0.641543\pi$
$368$		0			0
$369$			1.44002e14i			1.09578i
$370$		0			0
$371$		−	5.07122e13i		−	0.374590i
$372$		0			0
$373$	1.05991e14			0.760100			0.380050	−	0.924966i	$-0.375907\pi$
$373$	1.05991e14			0.760100			0.380050	+	0.924966i	$0.375907\pi$
$374$		0			0
$375$			5.27625e13i			0.367412i
$376$		0			0
$377$	−4.28301e13	−	5.15186e11i	−0.289649	−	0.00348407i
$378$		0			0
$379$			1.76509e14i			1.15944i	0.814814	+	0.579722i	$0.196839\pi$
$379$			1.76509e14i			1.15944i	−0.814814	+	0.579722i	$0.803161\pi$
$380$		0			0
$381$	−8.40321e13			−0.536238
$382$		0			0
$383$		−	1.13316e14i		−	0.702583i	−0.936266	−	0.351292i	$-0.885743\pi$
$383$		−	1.13316e14i		−	0.702583i	0.936266	−	0.351292i	$-0.114257\pi$
$384$		0			0
$385$			3.81162e13i			0.229655i
$386$		0			0
$387$	−3.88001e13			−0.227208
$388$		0			0
$389$	9.60873e13			0.546944			0.273472	−	0.961880i	$-0.411828\pi$
$389$	9.60873e13			0.546944			0.273472	+	0.961880i	$0.411828\pi$
$390$		0			0
$391$	1.02231e14			0.565729
$392$		0			0
$393$	1.55044e13			0.0834248
$394$		0			0
$395$		−	5.43830e13i		−	0.284564i
$396$		0			0
$397$		−	1.05375e14i		−	0.536276i	−0.963381	−	0.268138i	$-0.913592\pi$
$397$		−	1.05375e14i		−	0.536276i	0.963381	−	0.268138i	$-0.0864083\pi$
$398$		0			0
$399$	−3.91531e13			−0.193827
$400$		0			0
$401$			1.95797e14i			0.942999i	0.881866	+	0.471499i	$0.156287\pi$
$401$			1.95797e14i			0.942999i	−0.881866	+	0.471499i	$0.843713\pi$
$402$		0			0
$403$	−4.32589e11	+	3.59634e13i	−0.00202721	+	0.168532i
$404$		0			0
$405$			1.10018e14i			0.501719i
$406$		0			0
$407$	−2.42842e14			−1.07784
$408$		0			0
$409$			1.68257e14i			0.726936i	0.931607	+	0.363468i	$0.118407\pi$
$409$			1.68257e14i			0.726936i	−0.931607	+	0.363468i	$0.881593\pi$
$410$		0			0
$411$		−	7.08547e13i		−	0.298015i
$412$		0			0
$413$	3.26452e13			0.133688
$414$		0			0
$415$	−1.98458e14			−0.791416
$416$		0			0
$417$	6.95883e13			0.270264
$418$		0			0
$419$	−3.97425e14			−1.50341			−0.751706	−	0.659498i	$-0.770769\pi$
$419$	−3.97425e14			−1.50341			−0.751706	+	0.659498i	$0.770769\pi$
$420$		0			0
$421$		−	4.38503e14i		−	1.61593i	−0.589233	−	0.807963i	$-0.700570\pi$
$421$		−	4.38503e14i		−	1.61593i	0.589233	−	0.807963i	$-0.299430\pi$
$422$		0			0
$423$			1.65242e14i			0.593264i
$424$		0			0
$425$	−1.44447e14			−0.505322
$426$		0			0
$427$		−	2.38092e13i		−	0.0811694i
$428$		0			0
$429$	1.03156e12	−	8.57586e13i	0.00342750	−	0.284946i
$430$		0			0
$431$			1.83108e14i			0.593037i	0.955027	+	0.296519i	$0.0958257\pi$
$431$			1.83108e14i			0.593037i	−0.955027	+	0.296519i	$0.904174\pi$
$432$		0			0
$433$	−1.30715e14			−0.412706			−0.206353	−	0.978478i	$-0.566159\pi$
$433$	−1.30715e14			−0.412706			−0.206353	+	0.978478i	$0.566159\pi$
$434$		0			0
$435$			2.41430e13i			0.0743190i
$436$		0			0
$437$			2.52016e14i			0.756451i
$438$		0			0
$439$	−3.54362e14			−1.03727			−0.518636	−	0.854995i	$-0.673560\pi$
$439$	−3.54362e14			−1.03727			−0.518636	+	0.854995i	$0.673560\pi$
$440$		0			0
$441$	−2.68638e14			−0.766928
$442$		0			0
$443$	−3.54802e14			−0.988021			−0.494010	−	0.869456i	$-0.664470\pi$
$443$	−3.54802e14			−0.988021			−0.494010	+	0.869456i	$0.664470\pi$
$444$		0			0
$445$	−5.21993e14			−1.41803
$446$		0			0
$447$			4.67517e13i			0.123910i
$448$		0			0
$449$		−	6.30849e14i		−	1.63144i	−0.578449	−	0.815718i	$-0.696342\pi$
$449$		−	6.30849e14i		−	1.63144i	0.578449	−	0.815718i	$-0.303658\pi$
$450$		0			0
$451$	4.11101e14			1.03747
$452$		0			0
$453$			1.40098e14i			0.345057i
$454$		0			0
$455$	−1.14117e14	−	1.37267e12i	−0.274340	−	0.00329993i
$456$		0			0
$457$			2.37509e14i			0.557366i	0.960383	+	0.278683i	$0.0898979\pi$
$457$			2.37509e14i			0.557366i	−0.960383	+	0.278683i	$0.910102\pi$
$458$		0			0
$459$	3.27457e14			0.750213
$460$		0			0
$461$			2.34445e14i			0.524428i	0.965010	+	0.262214i	$0.0844527\pi$
$461$			2.34445e14i			0.524428i	−0.965010	+	0.262214i	$0.915547\pi$
$462$		0			0
$463$		−	5.27378e14i		−	1.15193i	−0.817474	−	0.575966i	$-0.804626\pi$
$463$		−	5.27378e14i		−	1.15193i	0.817474	−	0.575966i	$-0.195374\pi$
$464$		0			0
$465$	2.02723e13			0.0432425
$466$		0			0
$467$	−5.77169e14			−1.20243			−0.601215	−	0.799087i	$-0.705316\pi$
$467$	−5.77169e14			−1.20243			−0.601215	+	0.799087i	$0.705316\pi$
$468$		0			0
$469$	−2.17388e14			−0.442370
$470$		0			0
$471$	2.61165e14			0.519158
$472$		0			0
$473$			1.10767e14i			0.215117i
$474$		0			0
$475$		−	3.56086e14i		−	0.675679i
$476$		0			0
$477$	−4.90621e14			−0.909694
$478$		0			0
$479$			9.38721e13i			0.170095i	0.996377	+	0.0850474i	$0.0271042\pi$
$479$			9.38721e13i			0.170095i	−0.996377	+	0.0850474i	$0.972896\pi$
$480$		0			0
$481$	8.74544e12	−	7.27053e14i	0.0154876	−	1.28756i
$482$		0			0
$483$			3.46321e13i			0.0599472i
$484$		0			0
$485$	2.30925e14			0.390742
$486$		0			0
$487$			5.12071e14i			0.847073i	0.905879	+	0.423536i	$0.139211\pi$
$487$			5.12071e14i			0.847073i	−0.905879	+	0.423536i	$0.860789\pi$
$488$		0			0
$489$		−	3.20916e14i		−	0.519031i
$490$		0			0
$491$	−1.08104e15			−1.70959			−0.854795	−	0.518967i	$-0.826317\pi$
$491$	−1.08104e15			−1.70959			−0.854795	+	0.518967i	$0.826317\pi$
$492$		0			0
$493$	−2.19080e14			−0.338801
$494$		0			0
$495$	3.68760e14			0.557718
$496$		0			0
$497$	3.93991e13			0.0582809
$498$		0			0
$499$		−	6.34438e14i		−	0.917986i	−0.888440	−	0.458993i	$-0.848210\pi$
$499$		−	6.34438e14i		−	0.917986i	0.888440	−	0.458993i	$-0.151790\pi$
$500$		0			0
$501$			4.22038e14i			0.597371i
$502$		0			0
$503$	3.95519e14			0.547701			0.273850	−	0.961772i	$-0.411703\pi$
$503$	3.95519e14			0.547701			0.273850	+	0.961772i	$0.411703\pi$
$504$		0			0
$505$		−	3.40706e14i		−	0.461612i
$506$		0			0
$507$	2.56719e14	+	6.17683e12i	0.340340	+	0.00818883i
$508$		0			0
$509$			5.60027e14i			0.726542i	0.931683	+	0.363271i	$0.118340\pi$
$509$			5.60027e14i			0.726542i	−0.931683	+	0.363271i	$0.881660\pi$
$510$		0			0
$511$	4.51507e14			0.573258
$512$		0			0
$513$			8.07239e14i			1.00313i
$514$		0			0
$515$			3.18709e13i			0.0387663i
$516$		0			0
$517$	4.71735e14			0.561695
$518$		0			0
$519$	9.78007e13			0.114004
$520$		0			0
$521$	1.68889e14			0.192750			0.0963750	−	0.995345i	$-0.469275\pi$
$521$	1.68889e14			0.192750			0.0963750	+	0.995345i	$0.469275\pi$
$522$		0			0
$523$	1.18986e15			1.32965			0.664825	−	0.746999i	$-0.268506\pi$
$523$	1.18986e15			1.32965			0.664825	+	0.746999i	$0.268506\pi$
$524$		0			0
$525$		−	4.89333e13i		−	0.0535462i
$526$		0			0
$527$			1.83956e14i			0.197131i
$528$		0			0
$529$	−7.29894e14			−0.766044
$530$		0			0
$531$		−	3.15830e14i		−	0.324663i
$532$		0			0
$533$	−1.48049e13	+	1.23081e15i	−0.0149076	+	1.23934i
$534$		0			0
$535$			9.64028e14i			0.950920i
$536$		0			0
$537$	−4.36642e14			−0.421956
$538$		0			0
$539$			7.66913e14i			0.726119i
$540$		0			0
$541$		−	1.55433e15i		−	1.44197i	−0.692948	−	0.720987i	$-0.743689\pi$
$541$		−	1.55433e15i		−	1.44197i	0.692948	−	0.720987i	$-0.256311\pi$
$542$		0			0
$543$	3.96399e13			0.0360358
$544$		0			0
$545$	1.15264e15			1.02686
$546$		0			0
$547$	−1.40215e14			−0.122424			−0.0612118	−	0.998125i	$-0.519496\pi$
$547$	−1.40215e14			−0.122424			−0.0612118	+	0.998125i	$0.519496\pi$
$548$		0			0
$549$	−2.30345e14			−0.197120
$550$		0			0
$551$		−	5.40070e14i		−	0.453019i
$552$		0			0
$553$			1.67175e14i			0.137462i
$554$		0			0
$555$	−4.09834e14			−0.330367
$556$		0			0
$557$		−	6.89987e14i		−	0.545303i	−0.962113	−	0.272651i	$-0.912099\pi$
$557$		−	6.89987e14i		−	0.545303i	0.962113	−	0.272651i	$-0.0879005\pi$
$558$		0			0
$559$	−3.31630e14	−	3.98905e12i	−0.256974	−	0.00309104i
$560$		0			0
$561$		−	4.38663e14i		−	0.333300i
$562$		0			0
$563$	7.54180e14			0.561926			0.280963	−	0.959719i	$-0.409346\pi$
$563$	7.54180e14			0.561926			0.280963	+	0.959719i	$0.409346\pi$
$564$		0			0
$565$		−	1.08220e15i		−	0.790753i
$566$		0			0
$567$		−	3.38199e14i		−	0.242362i
$568$		0			0
$569$	2.05848e15			1.44687			0.723434	−	0.690393i	$-0.242563\pi$
$569$	2.05848e15			1.44687			0.723434	+	0.690393i	$0.242563\pi$
$570$		0			0
$571$	−1.50848e15			−1.04002			−0.520009	−	0.854161i	$-0.674072\pi$
$571$	−1.50848e15			−1.04002			−0.520009	+	0.854161i	$0.674072\pi$
$572$		0			0
$573$	1.00028e15			0.676505
$574$		0			0
$575$	−3.14969e14			−0.208975
$576$		0			0
$577$			2.36925e15i			1.54221i	0.636706	+	0.771107i	$0.280297\pi$
$577$			2.36925e15i			1.54221i	−0.636706	+	0.771107i	$0.719703\pi$
$578$		0			0
$579$			3.24760e14i			0.207411i
$580$		0			0
$581$	6.10068e14			0.382305
$582$		0			0
$583$			1.40064e15i			0.861288i
$584$		0			0
$585$	−1.32801e13	+	1.10404e15i	−0.00801390	+	0.666236i
$586$		0			0
$587$		−	1.64925e15i		−	0.976738i	−0.872637	−	0.488369i	$-0.837592\pi$
$587$		−	1.64925e15i		−	0.976738i	0.872637	−	0.488369i	$-0.162408\pi$
$588$		0			0
$589$	−4.53483e14			−0.263589
$590$		0			0
$591$		−	5.17832e14i		−	0.295432i
$592$		0			0
$593$		−	2.18612e15i		−	1.22426i	−0.790759	−	0.612128i	$-0.790314\pi$
$593$		−	2.18612e15i		−	1.22426i	0.790759	−	0.612128i	$-0.209686\pi$
$594$		0			0
$595$	−5.83720e14			−0.320894
$596$		0			0
$597$	1.08185e15			0.583860
$598$		0			0
$599$	3.27562e15			1.73559			0.867794	−	0.496924i	$-0.165537\pi$
$599$	3.27562e15			1.73559			0.867794	+	0.496924i	$0.165537\pi$
$600$		0			0
$601$	3.53256e15			1.83772			0.918861	−	0.394583i	$-0.129111\pi$
$601$	3.53256e15			1.83772			0.918861	+	0.394583i	$0.129111\pi$
$602$		0			0
$603$			2.10315e15i			1.07430i
$604$		0			0
$605$			4.49750e14i			0.225588i
$606$		0			0
$607$	2.05879e15			1.01408			0.507041	−	0.861922i	$-0.330739\pi$
$607$	2.05879e15			1.01408			0.507041	+	0.861922i	$0.330739\pi$
$608$		0			0
$609$		−	7.42164e13i		−	0.0359009i
$610$		0			0
$611$	−1.69885e13	+	1.41235e15i	−0.00807104	+	0.670987i
$612$		0			0
$613$			8.47376e14i			0.395406i	0.980262	+	0.197703i	$0.0633482\pi$
$613$			8.47376e14i			0.395406i	−0.980262	+	0.197703i	$0.936652\pi$
$614$		0			0
$615$	6.93797e14			0.317994
$616$		0			0
$617$			2.17604e15i			0.979714i	0.871803	+	0.489857i	$0.162951\pi$
$617$			2.17604e15i			0.979714i	−0.871803	+	0.489857i	$0.837049\pi$
$618$		0			0
$619$		−	1.76976e15i		−	0.782739i	−0.920234	−	0.391369i	$-0.872002\pi$
$619$		−	1.76976e15i		−	0.782739i	0.920234	−	0.391369i	$-0.127998\pi$
$620$		0			0
$621$	7.14027e14			0.310249
$622$		0			0
$623$	1.60463e15			0.684999
$624$		0			0
$625$	−9.09081e14			−0.381296
$626$		0			0
$627$	1.08138e15			0.445664
$628$		0			0
$629$		−	3.71894e15i		−	1.50606i
$630$		0			0
$631$		−	4.92537e15i		−	1.96010i	−0.198759	−	0.980048i	$-0.563691\pi$
$631$		−	4.92537e15i		−	1.96010i	0.198759	−	0.980048i	$-0.436309\pi$
$632$		0			0
$633$	1.12926e15			0.441647
$634$		0			0
$635$		−	3.08839e15i		−	1.18707i
$636$		0			0
$637$	−2.29609e15	−	2.76187e13i	−0.867404	−	0.0104337i
$638$		0			0
$639$		−	3.81171e14i		−	0.141536i
$640$		0			0
$641$	−1.22830e14			−0.0448318			−0.0224159	−	0.999749i	$-0.507136\pi$
$641$	−1.22830e14			−0.0448318			−0.0224159	+	0.999749i	$0.507136\pi$
$642$		0			0
$643$			3.16693e15i			1.13626i	0.822939	+	0.568130i	$0.192333\pi$
$643$			3.16693e15i			1.13626i	−0.822939	+	0.568130i	$0.807667\pi$
$644$		0			0
$645$			1.86937e14i			0.0659353i
$646$		0			0
$647$	2.65721e15			0.921408			0.460704	−	0.887554i	$-0.347597\pi$
$647$	2.65721e15			0.921408			0.460704	+	0.887554i	$0.347597\pi$
$648$		0			0
$649$	−9.01638e14			−0.307388
$650$		0			0
$651$	−6.23177e13			−0.0208889
$652$		0			0
$653$	4.35077e15			1.43398			0.716992	−	0.697082i	$-0.245519\pi$
$653$	4.35077e15			1.43398			0.716992	+	0.697082i	$0.245519\pi$
$654$		0			0
$655$			5.69824e14i			0.184677i
$656$		0			0
$657$		−	4.36816e15i		−	1.39216i
$658$		0			0
$659$	−4.68870e15			−1.46954			−0.734771	−	0.678315i	$-0.762711\pi$
$659$	−4.68870e15			−1.46954			−0.734771	+	0.678315i	$0.762711\pi$
$660$		0			0
$661$			4.44733e15i			1.37085i	0.728142	+	0.685427i	$0.240384\pi$
$661$			4.44733e15i			1.37085i	−0.728142	+	0.685427i	$0.759616\pi$
$662$		0			0
$663$	1.31333e15	+	1.57975e13i	0.398152	+	0.00478921i
$664$		0			0
$665$		−	1.43897e15i		−	0.429075i
$666$		0			0
$667$	−4.77708e14			−0.140110
$668$		0			0
$669$		−	1.50297e15i		−	0.433616i
$670$		0			0
$671$			6.57595e14i			0.186631i
$672$		0			0
$673$	1.30699e15			0.364913			0.182457	−	0.983214i	$-0.441595\pi$
$673$	1.30699e15			0.364913			0.182457	+	0.983214i	$0.441595\pi$
$674$		0			0
$675$	−1.00888e15			−0.277122
$676$		0			0
$677$	−6.24197e15			−1.68688			−0.843439	−	0.537224i	$-0.819473\pi$
$677$	−6.24197e15			−1.68688			−0.843439	+	0.537224i	$0.819473\pi$
$678$		0			0
$679$	−7.09870e14			−0.188754
$680$		0			0
$681$		−	1.65194e15i		−	0.432200i
$682$		0			0
$683$			1.60803e15i			0.413980i	0.978343	+	0.206990i	$0.0663668\pi$
$683$			1.60803e15i			0.413980i	−0.978343	+	0.206990i	$0.933633\pi$
$684$		0			0
$685$	2.60409e15			0.659716
$686$		0			0
$687$			1.82812e14i			0.0455767i
$688$		0			0
$689$	−4.19341e15	−	5.04409e13i	−1.02887	−	0.0123759i
$690$		0			0
$691$			1.96543e15i			0.474601i	0.971436	+	0.237301i	$0.0762627\pi$
$691$			1.96543e15i			0.474601i	−0.971436	+	0.237301i	$0.923737\pi$
$692$		0			0
$693$	−1.13358e15			−0.269414
$694$		0			0
$695$			2.55754e15i			0.598283i
$696$		0			0
$697$			6.29570e15i			1.44965i
$698$		0			0
$699$	1.04260e15			0.236316
$700$		0			0
$701$	3.16026e15			0.705138			0.352569	−	0.935786i	$-0.385308\pi$
$701$	3.16026e15			0.705138			0.352569	+	0.935786i	$0.385308\pi$
$702$		0			0
$703$	9.16785e15			2.01378
$704$		0			0
$705$	7.96128e14			0.172164
$706$		0			0
$707$			1.04734e15i			0.222988i
$708$		0			0
$709$			5.15618e15i			1.08087i	0.841385	+	0.540436i	$0.181741\pi$
$709$			5.15618e15i			1.08087i	−0.841385	+	0.540436i	$0.818259\pi$
$710$		0			0
$711$	1.61736e15			0.333829
$712$		0			0
$713$			4.01119e14i			0.0815232i
$714$		0			0
$715$	3.15184e15	+	3.79123e13i	0.630785	+	0.00758746i
$716$		0			0
$717$			2.15623e15i			0.424953i
$718$		0			0
$719$	6.74859e14			0.130980			0.0654899	−	0.997853i	$-0.479139\pi$
$719$	6.74859e14			0.130980			0.0654899	+	0.997853i	$0.479139\pi$
$720$		0			0
$721$		−	9.79722e13i		−	0.0187266i
$722$		0			0
$723$		−	2.02001e15i		−	0.380271i
$724$		0			0
$725$	6.74977e14			0.125150
$726$		0			0
$727$	8.65251e14			0.158017			0.0790083	−	0.996874i	$-0.474825\pi$
$727$	8.65251e14			0.158017			0.0790083	+	0.996874i	$0.474825\pi$
$728$		0			0
$729$	−2.05804e15			−0.370214
$730$		0			0
$731$	−1.69632e15			−0.300581
$732$		0			0
$733$		−	1.01376e16i		−	1.76956i	−0.466009	−	0.884780i	$-0.654309\pi$
$733$		−	1.01376e16i		−	1.76956i	0.466009	−	0.884780i	$-0.345691\pi$
$734$		0			0
$735$			1.29429e15i			0.222561i
$736$		0			0
$737$	6.00412e15			1.01713
$738$		0			0
$739$			1.18425e16i			1.97652i	0.152793	+	0.988258i	$0.451173\pi$
$739$			1.18425e16i			1.97652i	−0.152793	+	0.988258i	$0.548827\pi$
$740$		0			0
$741$	−3.89436e13	+	3.23759e15i	−0.00640378	+	0.532379i
$742$		0			0
$743$		−	8.20805e15i		−	1.32985i	−0.746911	−	0.664923i	$-0.768464\pi$
$743$		−	8.20805e15i		−	1.32985i	0.746911	−	0.664923i	$-0.231536\pi$
$744$		0			0
$745$	−1.71824e15			−0.274299
$746$		0			0
$747$		−	5.90218e15i		−	0.928430i
$748$		0			0
$749$		−	2.96346e15i		−	0.459355i
$750$		0			0
$751$	−8.13107e15			−1.24202			−0.621009	−	0.783803i	$-0.713277\pi$
$751$	−8.13107e15			−1.24202			−0.621009	+	0.783803i	$0.713277\pi$
$752$		0			0
$753$	1.12723e15			0.169684
$754$		0			0
$755$	−5.14894e15			−0.763853
$756$		0			0
$757$	2.28113e15			0.333521			0.166760	−	0.985997i	$-0.446669\pi$
$757$	2.28113e15			0.333521			0.166760	+	0.985997i	$0.446669\pi$
$758$		0			0
$759$		−	9.56514e14i		−	0.137836i
$760$		0			0
$761$		−	6.48483e14i		−	0.0921050i	−0.998939	−	0.0460525i	$-0.985336\pi$
$761$		−	6.48483e14i		−	0.0921050i	0.998939	−	0.0460525i	$-0.0146641\pi$
$762$		0			0
$763$	−3.54324e15			−0.496039
$764$		0			0
$765$			5.64727e15i			0.779293i
$766$		0			0
$767$	3.24706e13	−	2.69944e15i	0.00441688	−	0.367198i
$768$		0			0
$769$		−	3.11073e15i		−	0.417126i	−0.978009	−	0.208563i	$-0.933121\pi$
$769$		−	3.11073e15i		−	0.417126i	0.978009	−	0.208563i	$-0.0668786\pi$
$770$		0			0
$771$	1.65740e15			0.219093
$772$		0			0
$773$			7.17757e14i			0.0935385i	0.998906	+	0.0467693i	$0.0148926\pi$
$773$			7.17757e14i			0.0935385i	−0.998906	+	0.0467693i	$0.985107\pi$
$774$		0			0
$775$		−	5.66761e14i		−	0.0728183i
$776$		0			0
$777$	1.25985e15			0.159588
$778$		0			0
$779$	−1.55200e16			−1.93837
$780$		0			0
$781$	−1.08818e15			−0.134004
$782$		0			0
$783$	−1.53016e15			−0.185800
$784$		0			0
$785$			9.59845e15i			1.14926i
$786$		0			0
$787$		−	1.47799e16i		−	1.74506i	−0.488557	−	0.872532i	$-0.662476\pi$
$787$		−	1.47799e16i		−	1.74506i	0.488557	−	0.872532i	$-0.337524\pi$
$788$		0			0
$789$	−2.26988e15			−0.264289
$790$		0			0
$791$			3.32672e15i			0.381984i
$792$		0			0
$793$	−1.96880e15	−	2.36819e13i	−0.222945	−	0.00268172i
$794$		0			0
$795$			2.36379e15i			0.263992i
$796$		0			0
$797$	−1.57649e14			−0.0173648			−0.00868240	−	0.999962i	$-0.502764\pi$
$797$	−1.57649e14			−0.0173648			−0.00868240	+	0.999962i	$0.502764\pi$
$798$		0			0
$799$			7.22427e15i			0.784850i
$800$		0			0
$801$		−	1.55242e16i		−	1.66352i
$802$		0			0
$803$	−1.24703e16			−1.31808
$804$		0			0
$805$	−1.27281e15			−0.132705
$806$		0			0
$807$	4.07623e15			0.419232
$808$		0			0
$809$	−7.98389e15			−0.810023			−0.405012	−	0.914312i	$-0.632733\pi$
$809$	−7.98389e15			−0.810023			−0.405012	+	0.914312i	$0.632733\pi$
$810$		0			0
$811$		−	6.94088e15i		−	0.694704i	−0.937735	−	0.347352i	$-0.887081\pi$
$811$		−	6.94088e15i		−	0.694704i	0.937735	−	0.347352i	$-0.112919\pi$
$812$		0			0
$813$		−	5.80171e15i		−	0.572873i
$814$		0			0
$815$	1.17945e16			1.14898
$816$		0			0
$817$		−	4.18172e15i		−	0.401915i
$818$		0			0
$819$	4.08235e13	−	3.39387e15i	0.00387123	−	0.321835i
$820$		0			0
$821$		−	7.52996e15i		−	0.704539i	−0.935899	−	0.352270i	$-0.885410\pi$
$821$		−	7.52996e15i		−	0.704539i	0.935899	−	0.352270i	$-0.114590\pi$
$822$		0			0
$823$	1.12488e16			1.03850			0.519251	−	0.854622i	$-0.326211\pi$
$823$	1.12488e16			1.03850			0.519251	+	0.854622i	$0.326211\pi$
$824$		0			0
$825$			1.35151e15i			0.123118i
$826$		0			0
$827$			1.66909e16i			1.50038i	0.661224	+	0.750188i	$0.270037\pi$
$827$			1.66909e16i			1.50038i	−0.661224	+	0.750188i	$0.729963\pi$
$828$		0			0
$829$	6.61191e14			0.0586512			0.0293256	−	0.999570i	$-0.490664\pi$
$829$	6.61191e14			0.0586512			0.0293256	+	0.999570i	$0.490664\pi$
$830$		0			0
$831$	4.05680e15			0.355122
$832$		0			0
$833$	−1.17447e16			−1.01460
$834$		0			0
$835$	−1.55110e16			−1.32240
$836$		0			0
$837$			1.28483e15i			0.108108i
$838$		0			0
$839$			4.53131e15i			0.376299i	0.982140	+	0.188149i	$0.0602489\pi$
$839$			4.53131e15i			0.376299i	−0.982140	+	0.188149i	$0.939751\pi$
$840$		0			0
$841$	−1.11768e16			−0.916092
$842$		0			0
$843$		−	4.86675e14i		−	0.0393720i
$844$		0			0
$845$	−2.27014e14	+	9.43505e15i	−0.0181276	+	0.753411i
$846$		0			0
$847$		−	1.38255e15i		−	0.108974i
$848$		0			0
$849$	6.67753e14			0.0519545
$850$		0			0
$851$		−	8.10923e15i		−	0.622826i
$852$		0			0
$853$			1.83259e16i			1.38946i	0.719270	+	0.694731i	$0.244477\pi$
$853$			1.83259e16i			1.38946i	−0.719270	+	0.694731i	$0.755523\pi$
$854$		0			0
$855$	−1.39215e16			−1.04201
$856$		0			0
$857$	1.43741e15			0.106215			0.0531076	−	0.998589i	$-0.483087\pi$
$857$	1.43741e15			0.106215			0.0531076	+	0.998589i	$0.483087\pi$
$858$		0			0
$859$	−1.66971e16			−1.21809			−0.609043	−	0.793137i	$-0.708446\pi$
$859$	−1.66971e16			−1.21809			−0.609043	+	0.793137i	$0.708446\pi$
$860$		0			0
$861$	−2.13276e15			−0.153612
$862$		0			0
$863$			6.56479e14i			0.0466833i	0.999728	+	0.0233416i	$0.00743055\pi$
$863$			6.56479e14i			0.0466833i	−0.999728	+	0.0233416i	$0.992569\pi$
$864$		0			0
$865$			3.59442e15i			0.252371i
$866$		0			0
$867$	1.80708e15			0.125277
$868$		0			0
$869$		−	4.61727e15i		−	0.316065i
$870$		0			0
$871$	−2.16226e14	+	1.79759e16i	−0.0146153	+	1.21504i
$872$		0			0
$873$			6.86773e15i			0.458390i
$874$		0			0
$875$	5.96102e15			0.392895
$876$		0			0
$877$		−	1.38289e16i		−	0.900097i	−0.893004	−	0.450049i	$-0.851407\pi$
$877$		−	1.38289e16i		−	0.900097i	0.893004	−	0.450049i	$-0.148593\pi$
$878$		0			0
$879$		−	3.81981e14i		−	0.0245529i
$880$		0			0
$881$	−8.15095e15			−0.517417			−0.258709	−	0.965955i	$-0.583297\pi$
$881$	−8.15095e15			−0.517417			−0.258709	+	0.965955i	$0.583297\pi$
$882$		0			0
$883$	−2.00627e16			−1.25778			−0.628891	−	0.777494i	$-0.716491\pi$
$883$	−2.00627e16			−1.25778			−0.628891	+	0.777494i	$0.716491\pi$
$884$		0			0
$885$	−1.52166e15			−0.0942168
$886$		0			0
$887$	1.58308e16			0.968104			0.484052	−	0.875039i	$-0.339165\pi$
$887$	1.58308e16			0.968104			0.484052	+	0.875039i	$0.339165\pi$
$888$		0			0
$889$			9.49382e15i			0.573431i
$890$		0			0
$891$			9.34081e15i			0.557259i
$892$		0			0
$893$	−1.78091e16			−1.04944
$894$		0			0
$895$		−	1.60477e16i		−	0.934083i
$896$		0			0
$897$	2.86374e15	+	3.44468e13i	0.164655	+	0.00198057i
$898$		0			0
$899$		−	8.59597e14i		−	0.0488221i
$900$		0			0
$901$	−2.14497e16			−1.20347
$902$		0			0
$903$		−	5.74652e14i		−	0.0318510i
$904$		0			0
$905$			1.45687e15i			0.0797723i
$906$		0			0
$907$	−3.23895e16			−1.75212			−0.876059	−	0.482203i	$-0.839837\pi$
$907$	−3.23895e16			−1.75212			−0.876059	+	0.482203i	$0.839837\pi$
$908$		0			0
$909$	1.01326e16			0.541528
$910$		0			0
$911$	2.16663e16			1.14402			0.572009	−	0.820247i	$-0.306164\pi$
$911$	2.16663e16			1.14402			0.572009	+	0.820247i	$0.306164\pi$
$912$		0			0
$913$	−1.68497e16			−0.879026
$914$		0			0
$915$			1.10979e15i			0.0572040i
$916$		0			0
$917$		−	1.75166e15i		−	0.0892110i
$918$		0			0
$919$	2.33762e16			1.17635			0.588177	−	0.808732i	$-0.299846\pi$
$919$	2.33762e16			1.17635			0.588177	+	0.808732i	$0.299846\pi$
$920$		0			0
$921$		−	5.29689e15i		−	0.263386i
$922$		0			0
$923$	3.91883e13	−	3.25793e15i	0.00192552	−	0.160078i
$924$		0			0
$925$			1.14579e16i			0.556322i
$926$		0			0
$927$	−9.47844e14			−0.0454777
$928$		0			0
$929$			1.37774e16i			0.653254i	0.945153	+	0.326627i	$0.105912\pi$
$929$			1.37774e16i			0.653254i	−0.945153	+	0.326627i	$0.894088\pi$
$930$		0			0
$931$		−	2.89527e16i		−	1.35664i
$932$		0			0
$933$	1.12965e16			0.523114
$934$		0			0
$935$	1.61220e16			0.737826
$936$		0			0
$937$	−5.30380e15			−0.239894			−0.119947	−	0.992780i	$-0.538272\pi$
$937$	−5.30380e15			−0.239894			−0.119947	+	0.992780i	$0.538272\pi$
$938$		0			0
$939$	−1.09310e16			−0.488653
$940$		0			0
$941$		−	8.71178e15i		−	0.384914i	−0.981305	−	0.192457i	$-0.938354\pi$
$941$		−	8.71178e15i		−	0.384914i	0.981305	−	0.192457i	$-0.0616457\pi$
$942$		0			0
$943$			1.37279e16i			0.599501i
$944$		0			0
$945$	−4.07698e15			−0.175980
$946$		0			0
$947$		−	3.10725e16i		−	1.32572i	−0.748744	−	0.662860i	$-0.769343\pi$
$947$		−	3.10725e16i		−	1.32572i	0.748744	−	0.662860i	$-0.230657\pi$
$948$		0			0
$949$	4.49092e14	−	3.73353e16i	0.0189396	−	1.57455i
$950$		0			0
$951$			9.93867e15i			0.414319i
$952$		0			0
$953$	−4.00695e15			−0.165121			−0.0825605	−	0.996586i	$-0.526310\pi$
$953$	−4.00695e15			−0.165121			−0.0825605	+	0.996586i	$0.526310\pi$
$954$		0			0
$955$			3.67628e16i			1.49758i
$956$		0			0
$957$			2.04981e15i			0.0825462i
$958$		0			0
$959$	−8.00506e15			−0.318685
$960$		0			0
$961$	2.46867e16			0.971593
$962$		0			0
$963$	−2.86703e16			−1.11555
$964$		0			0
$965$	−1.19357e16			−0.459145
$966$		0			0
$967$			1.13777e16i			0.432724i	0.976313	+	0.216362i	$0.0694191\pi$
$967$			1.13777e16i			0.432724i	−0.976313	+	0.216362i	$0.930581\pi$
$968$		0			0
$969$			1.65605e16i			0.622721i
$970$		0			0
$971$	8.13823e15			0.302569			0.151284	−	0.988490i	$-0.451659\pi$
$971$	8.13823e15			0.302569			0.151284	+	0.988490i	$0.451659\pi$
$972$		0			0
$973$		−	7.86198e15i		−	0.289009i
$974$		0			0
$975$	−4.04632e15	−	4.86716e13i	−0.147074	−	0.00176909i
$976$		0			0
$977$		−	1.57114e16i		−	0.564671i	−0.959316	−	0.282335i	$-0.908891\pi$
$977$		−	1.57114e16i		−	0.564671i	0.959316	−	0.282335i	$-0.0911091\pi$
$978$		0			0
$979$	−4.43187e16			−1.57501
$980$		0			0
$981$			3.42795e16i			1.20464i
$982$		0			0
$983$			7.00897e15i			0.243562i	0.992557	+	0.121781i	$0.0388606\pi$
$983$			7.00897e15i			0.243562i	−0.992557	+	0.121781i	$0.961139\pi$
$984$		0			0
$985$	1.90316e16			0.653998
$986$		0			0
$987$	−2.44733e15			−0.0831663
$988$		0			0
$989$	−3.69886e15			−0.124305
$990$		0			0
$991$	1.80634e16			0.600335			0.300167	−	0.953887i	$-0.402957\pi$
$991$	1.80634e16			0.600335			0.300167	+	0.953887i	$0.402957\pi$
$992$		0			0
$993$		−	4.12861e15i		−	0.135701i
$994$		0			0
$995$			3.97607e16i			1.29249i
$996$		0			0
$997$	4.85471e16			1.56077			0.780387	−	0.625296i	$-0.215022\pi$
$997$	4.85471e16			1.56077			0.780387	+	0.625296i	$0.215022\pi$
$998$		0			0
$999$		−	2.59749e16i		−	0.825930i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.12.f.b.129.8		12
4.3	odd	2		13.12.b.a.12.9	yes	12
12.11	even	2		117.12.b.b.64.4		12
13.12	even	2	inner	208.12.f.b.129.7		12
52.31	even	4		169.12.a.e.1.9		12
52.47	even	4		169.12.a.e.1.4		12
52.51	odd	2		13.12.b.a.12.4	✓	12
156.155	even	2		117.12.b.b.64.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.b.a.12.4	✓	12	52.51	odd	2
13.12.b.a.12.9	yes	12	4.3	odd	2
117.12.b.b.64.4		12	12.11	even	2
117.12.b.b.64.9		12	156.155	even	2
169.12.a.e.1.4		12	52.47	even	4
169.12.a.e.1.9		12	52.31	even	4
208.12.f.b.129.7		12	13.12	even	2	inner
208.12.f.b.129.8		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)

\(f(q)\)	\(=\)	\(q+143.287 q^{3} +5266.14i q^{5} -16188.3i q^{7} -156616. q^{9} +O(q^{10})\)	\(q+143.287 q^{3} +5266.14i q^{5} -16188.3i q^{7} -156616. q^{9} +447110. i q^{11} +(-1.33862e6 - 16101.7i) q^{13} +754569. i q^{15} -6.84716e6 q^{17} -1.68794e7i q^{19} -2.31957e6i q^{21} -1.49304e7 q^{23} +2.10958e7 q^{25} -4.78238e7 q^{27} +3.19957e7 q^{29} -2.68660e7i q^{31} +6.40650e7i q^{33} +8.52500e7 q^{35} +5.43137e8i q^{37} +(-1.91806e8 - 2.30716e6i) q^{39} -9.19462e8i q^{41} +2.47741e8 q^{43} -8.24762e8i q^{45} -1.05508e9i q^{47} +1.71526e9 q^{49} -9.81107e8 q^{51} +3.13264e9 q^{53} -2.35455e9 q^{55} -2.41860e9i q^{57} +2.01659e9i q^{59} +1.47077e9 q^{61} +2.53535e9i q^{63} +(8.47940e7 - 7.04936e9i) q^{65} -1.34287e10i q^{67} -2.13932e9 q^{69} +2.43380e9i q^{71} +2.78909e10i q^{73} +3.02276e9 q^{75} +7.23797e9 q^{77} -1.03269e10 q^{79} +2.08915e10 q^{81} +3.76857e10i q^{83} -3.60581e10i q^{85} +4.58456e9 q^{87} +9.91225e10i q^{89} +(-2.60660e8 + 2.16700e10i) q^{91} -3.84954e9i q^{93} +8.88896e10 q^{95} -4.38508e10i q^{97} -7.00246e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 488 q^{3} + 654644 q^{9}+O(q^{10}) \) 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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