LMFDB - Embedded newform 384.6.c.d.383.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,6,Mod(383,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("384.383");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	384.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$61.5873868082$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 306 x^{18} + 37827 x^{16} + 2442168 x^{14} + 88368509 x^{12} + 1774000974 x^{10} + \cdots + 2870280625 $ x^20 + 306x^18 + 37827x^16 + 2442168x^14 + 88368509x^12 + 1774000974x^10 + 18093172325x^8 + 74958811500x^6 + 79355888475x^4 + 28497132750*x^2 + 2870280625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{88}\cdot 3^{14}\cdot 41^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.20
Root			$7.62048i$ of defining polynomial
Character	$\chi$	$=$	384.383
Dual form			384.6.c.d.383.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(15.4378 + 2.16198i) q^{3} +81.6092i q^{5} -91.7503i q^{7} +(233.652 + 66.7525i) q^{9} +O(q^{10})$	\(q+(15.4378 + 2.16198i) q^{3} +81.6092i q^{5} -91.7503i q^{7} +(233.652 + 66.7525i) q^{9} +710.534 q^{11} -666.057 q^{13} +(-176.438 + 1259.87i) q^{15} +2172.09i q^{17} -1126.10i q^{19} +(198.362 - 1416.42i) q^{21} +3120.07 q^{23} -3535.05 q^{25} +(3462.75 + 1535.66i) q^{27} -371.221i q^{29} -2506.88i q^{31} +(10969.1 + 1536.16i) q^{33} +7487.66 q^{35} +6270.56 q^{37} +(-10282.5 - 1440.00i) q^{39} +15413.6i q^{41} -11515.9i q^{43} +(-5447.62 + 19068.1i) q^{45} -21034.7 q^{47} +8388.89 q^{49} +(-4696.01 + 33532.2i) q^{51} +23609.2i q^{53} +57986.1i q^{55} +(2434.62 - 17384.6i) q^{57} +4375.95 q^{59} -53029.6 q^{61} +(6124.56 - 21437.6i) q^{63} -54356.3i q^{65} +24727.7i q^{67} +(48167.0 + 6745.53i) q^{69} +27045.3 q^{71} +15536.8 q^{73} +(-54573.5 - 7642.72i) q^{75} -65191.7i q^{77} +42768.6i q^{79} +(50137.2 + 31193.7i) q^{81} -29099.1 q^{83} -177262. q^{85} +(802.572 - 5730.83i) q^{87} +18718.0i q^{89} +61110.9i q^{91} +(5419.82 - 38700.7i) q^{93} +91900.4 q^{95} -14698.6 q^{97} +(166017. + 47429.9i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{3}+O(q^{10}) $ 20 * q + 2 * q^3	$ 20 q + 2 q^{3} + 948 q^{11} + 852 q^{15} - 1640 q^{21} + 328 q^{23} - 12500 q^{25} + 2030 q^{27} + 2836 q^{33} - 7184 q^{35} - 15056 q^{37} - 12980 q^{39} - 11800 q^{45} + 36640 q^{47} - 33388 q^{49} + 1936 q^{51} + 15404 q^{57} + 62908 q^{59} - 73264 q^{61} + 23608 q^{63} + 84024 q^{69} + 34888 q^{71} + 52568 q^{73} + 115698 q^{75} + 55444 q^{81} - 225172 q^{83} + 30112 q^{85} - 225700 q^{87} + 148016 q^{93} + 418616 q^{95} + 7600 q^{97} + 378260 q^{99}+O(q^{100}) $ 20 * q + 2 * q^3 + 948 * q^11 + 852 * q^15 - 1640 * q^21 + 328 * q^23 - 12500 * q^25 + 2030 * q^27 + 2836 * q^33 - 7184 * q^35 - 15056 * q^37 - 12980 * q^39 - 11800 * q^45 + 36640 * q^47 - 33388 * q^49 + 1936 * q^51 + 15404 * q^57 + 62908 * q^59 - 73264 * q^61 + 23608 * q^63 + 84024 * q^69 + 34888 * q^71 + 52568 * q^73 + 115698 * q^75 + 55444 * q^81 - 225172 * q^83 + 30112 * q^85 - 225700 * q^87 + 148016 * q^93 + 418616 * q^95 + 7600 * q^97 + 378260 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	15.4378	+	2.16198i	0.990336	+	0.138691i
$3$	15.4378	+	2.16198i	0.990336	+	0.138691i
$4$		0			0
$5$			81.6092i			1.45987i	0.683517	+	0.729934i	$0.260449\pi$
$5$			81.6092i			1.45987i	−0.683517	+	0.729934i	$0.739551\pi$
$6$		0			0
$7$		−	91.7503i		−	0.707721i	−0.935298	−	0.353861i	$-0.884869\pi$
$7$		−	91.7503i		−	0.707721i	0.935298	−	0.353861i	$-0.115131\pi$
$8$		0			0
$9$	233.652	+	66.7525i	0.961530	+	0.274702i
$10$		0			0
$11$	710.534			1.77053			0.885265	−	0.465088i	$-0.153977\pi$
$11$	710.534			1.77053			0.885265	+	0.465088i	$0.153977\pi$
$12$		0			0
$13$	−666.057			−1.09308			−0.546541	−	0.837432i	$-0.684056\pi$
$13$	−666.057			−1.09308			−0.546541	+	0.837432i	$0.684056\pi$
$14$		0			0
$15$	−176.438	+	1259.87i	−0.202471	+	1.44576i
$16$		0			0
$17$			2172.09i			1.82287i	0.411449	+	0.911433i	$0.365023\pi$
$17$			2172.09i			1.82287i	−0.411449	+	0.911433i	$0.634977\pi$
$18$		0			0
$19$		−	1126.10i		−	0.715640i	−0.933791	−	0.357820i	$-0.883520\pi$
$19$		−	1126.10i		−	0.715640i	0.933791	−	0.357820i	$-0.116480\pi$
$20$		0			0
$21$	198.362	−	1416.42i	0.0981547	−	0.700882i
$22$		0			0
$23$	3120.07			1.22983			0.614914	−	0.788594i	$-0.289191\pi$
$23$	3120.07			1.22983			0.614914	+	0.788594i	$0.289191\pi$
$24$		0			0
$25$	−3535.05			−1.13122
$26$		0			0
$27$	3462.75	+	1535.66i	0.914138	+	0.405403i
$28$		0			0
$29$		−	371.221i		−	0.0819666i	−0.999160	−	0.0409833i	$-0.986951\pi$
$29$		−	371.221i		−	0.0819666i	0.999160	−	0.0409833i	$-0.0130490\pi$
$30$		0			0
$31$		−	2506.88i		−	0.468520i	−0.972174	−	0.234260i	$-0.924733\pi$
$31$		−	2506.88i		−	0.468520i	0.972174	−	0.234260i	$-0.0752668\pi$
$32$		0			0
$33$	10969.1	+	1536.16i	1.75342	+	0.245557i
$34$		0			0
$35$	7487.66			1.03318
$36$		0			0
$37$	6270.56			0.753013			0.376506	−	0.926414i	$-0.377125\pi$
$37$	6270.56			0.753013			0.376506	+	0.926414i	$0.377125\pi$
$38$		0			0
$39$	−10282.5	−	1440.00i	−1.08252	−	0.151601i
$40$		0			0
$41$			15413.6i			1.43200i	0.698098	+	0.716002i	$0.254030\pi$
$41$			15413.6i			1.43200i	−0.698098	+	0.716002i	$0.745970\pi$
$42$		0			0
$43$		−	11515.9i		−	0.949786i	−0.880044	−	0.474893i	$-0.842487\pi$
$43$		−	11515.9i		−	0.949786i	0.880044	−	0.474893i	$-0.157513\pi$
$44$		0			0
$45$	−5447.62	+	19068.1i	−0.401028	+	1.40371i
$46$		0			0
$47$	−21034.7			−1.38896			−0.694482	−	0.719510i	$-0.744366\pi$
$47$	−21034.7			−1.38896			−0.694482	+	0.719510i	$0.744366\pi$
$48$		0			0
$49$	8388.89			0.499131
$50$		0			0
$51$	−4696.01	+	33532.2i	−0.252815	+	1.80525i
$52$		0			0
$53$			23609.2i			1.15450i	0.816569	+	0.577248i	$0.195873\pi$
$53$			23609.2i			1.15450i	−0.816569	+	0.577248i	$0.804127\pi$
$54$		0			0
$55$			57986.1i			2.58474i
$56$		0			0
$57$	2434.62	−	17384.6i	0.0992529	−	0.708724i
$58$		0			0
$59$	4375.95			0.163660			0.0818300	−	0.996646i	$-0.473924\pi$
$59$	4375.95			0.163660			0.0818300	+	0.996646i	$0.473924\pi$
$60$		0			0
$61$	−53029.6			−1.82471			−0.912355	−	0.409399i	$-0.865738\pi$
$61$	−53029.6			−1.82471			−0.912355	+	0.409399i	$0.865738\pi$
$62$		0			0
$63$	6124.56	−	21437.6i	0.194412	−	0.680495i
$64$		0			0
$65$		−	54356.3i		−	1.59576i
$66$		0			0
$67$			24727.7i			0.672971i	0.941689	+	0.336485i	$0.109238\pi$
$67$			24727.7i			0.672971i	−0.941689	+	0.336485i	$0.890762\pi$
$68$		0			0
$69$	48167.0	+	6745.53i	1.21794	+	0.170566i
$70$		0			0
$71$	27045.3			0.636717			0.318359	−	0.947970i	$-0.396868\pi$
$71$	27045.3			0.636717			0.318359	+	0.947970i	$0.396868\pi$
$72$		0			0
$73$	15536.8			0.341236			0.170618	−	0.985337i	$-0.445424\pi$
$73$	15536.8			0.341236			0.170618	+	0.985337i	$0.445424\pi$
$74$		0			0
$75$	−54573.5	−	7642.72i	−1.12028	−	0.156890i
$76$		0			0
$77$		−	65191.7i		−	1.25304i
$78$		0			0
$79$			42768.6i			0.771005i	0.922707	+	0.385503i	$0.125972\pi$
$79$			42768.6i			0.771005i	−0.922707	+	0.385503i	$0.874028\pi$
$80$		0			0
$81$	50137.2	+	31193.7i	0.849078	+	0.528268i
$82$		0			0
$83$	−29099.1			−0.463644			−0.231822	−	0.972758i	$-0.574469\pi$
$83$	−29099.1			−0.463644			−0.231822	+	0.972758i	$0.574469\pi$
$84$		0			0
$85$	−177262.			−2.66115
$86$		0			0
$87$	802.572	−	5730.83i	0.0113681	−	0.0811745i
$88$		0			0
$89$			18718.0i			0.250487i	0.992126	+	0.125243i	$0.0399711\pi$
$89$			18718.0i			0.250487i	−0.992126	+	0.125243i	$0.960029\pi$
$90$		0			0
$91$			61110.9i			0.773598i
$92$		0			0
$93$	5419.82	−	38700.7i	0.0649797	−	0.463993i
$94$		0			0
$95$	91900.4			1.04474
$96$		0			0
$97$	−14698.6			−0.158615			−0.0793077	−	0.996850i	$-0.525271\pi$
$97$	−14698.6			−0.158615			−0.0793077	+	0.996850i	$0.525271\pi$
$98$		0			0
$99$	166017.	+	47429.9i	1.70242	+	0.486367i
$100$		0			0
$101$		−	27203.5i		−	0.265352i	−0.991159	−	0.132676i	$-0.957643\pi$
$101$		−	27203.5i		−	0.265352i	0.991159	−	0.132676i	$-0.0423569\pi$
$102$		0			0
$103$			138397.i			1.28539i	0.766124	+	0.642693i	$0.222183\pi$
$103$			138397.i			1.28539i	−0.766124	+	0.642693i	$0.777817\pi$
$104$		0			0
$105$	115593.	+	16188.2i	1.02320	+	0.143293i
$106$		0			0
$107$	48368.3			0.408414			0.204207	−	0.978928i	$-0.434538\pi$
$107$	48368.3			0.408414			0.204207	+	0.978928i	$0.434538\pi$
$108$		0			0
$109$	3391.57			0.0273423			0.0136711	−	0.999907i	$-0.495648\pi$
$109$	3391.57			0.0273423			0.0136711	+	0.999907i	$0.495648\pi$
$110$		0			0
$111$	96803.7	+	13556.8i	0.745735	+	0.104436i
$112$		0			0
$113$		−	56024.3i		−	0.412744i	−0.978474	−	0.206372i	$-0.933834\pi$
$113$		−	56024.3i		−	0.412744i	0.978474	−	0.206372i	$-0.0661656\pi$
$114$		0			0
$115$			254626.i			1.79539i
$116$		0			0
$117$	−155625.	−	44461.0i	−1.05103	−	0.300272i
$118$		0			0
$119$	199289.			1.29008
$120$		0			0
$121$	343808.			2.13477
$122$		0			0
$123$	−33323.9	+	237952.i	−0.198606	+	1.41816i
$124$		0			0
$125$		−	33464.2i		−	0.191560i
$126$		0			0
$127$			58547.1i			0.322104i	0.986946	+	0.161052i	$0.0514887\pi$
$127$			58547.1i			0.322104i	−0.986946	+	0.161052i	$0.948511\pi$
$128$		0			0
$129$	24897.1	−	177780.i	0.131727	−	0.940607i
$130$		0			0
$131$	−343480.			−1.74873			−0.874367	−	0.485265i	$-0.838723\pi$
$131$	−343480.			−1.74873			−0.874367	+	0.485265i	$0.838723\pi$
$132$		0			0
$133$	−103320.			−0.506473
$134$		0			0
$135$	−125324.	+	282592.i	−0.591835	+	1.33452i
$136$		0			0
$137$		−	123940.i		−	0.564170i	−0.959389	−	0.282085i	$-0.908974\pi$
$137$		−	123940.i		−	0.564170i	0.959389	−	0.282085i	$-0.0910260\pi$
$138$		0			0
$139$		−	178910.i		−	0.785411i	−0.919664	−	0.392705i	$-0.871539\pi$
$139$		−	178910.i		−	0.785411i	0.919664	−	0.392705i	$-0.128461\pi$
$140$		0			0
$141$	−324729.	−	45476.5i	−1.37554	−	0.192637i
$142$		0			0
$143$	−473256.			−1.93533
$144$		0			0
$145$	30295.0			0.119661
$146$		0			0
$147$	129506.	+	18136.6i	0.494307	+	0.0692250i
$148$		0			0
$149$			465521.i			1.71781i	0.512139	+	0.858903i	$0.328853\pi$
$149$			465521.i			1.71781i	−0.512139	+	0.858903i	$0.671147\pi$
$150$		0			0
$151$		−	81121.6i		−	0.289531i	−0.989466	−	0.144765i	$-0.953757\pi$
$151$		−	81121.6i		−	0.289531i	0.989466	−	0.144765i	$-0.0462427\pi$
$152$		0			0
$153$	−144992.	+	507512.i	−0.500744	+	1.75274i
$154$		0			0
$155$	204584.			0.683978
$156$		0			0
$157$	106710.			0.345506			0.172753	−	0.984965i	$-0.444734\pi$
$157$	106710.			0.345506			0.172753	+	0.984965i	$0.444734\pi$
$158$		0			0
$159$	−51042.7	+	364475.i	−0.160118	+	1.14334i
$160$		0			0
$161$		−	286267.i		−	0.870376i
$162$		0			0
$163$		−	15721.2i		−	0.0463466i	−0.999731	−	0.0231733i	$-0.992623\pi$
$163$		−	15721.2i		−	0.0463466i	0.999731	−	0.0231733i	$-0.00737695\pi$
$164$		0			0
$165$	−125365.	+	895178.i	−0.358481	+	2.55976i
$166$		0			0
$167$	5039.69			0.0139834			0.00699170	−	0.999976i	$-0.497774\pi$
$167$	5039.69			0.0139834			0.00699170	+	0.999976i	$0.497774\pi$
$168$		0			0
$169$	72338.7			0.194829
$170$		0			0
$171$	75170.3	−	263116.i	0.196587	−	0.688109i
$172$		0			0
$173$		−	177743.i		−	0.451520i	−0.974183	−	0.225760i	$-0.927514\pi$
$173$		−	177743.i		−	0.451520i	0.974183	−	0.225760i	$-0.0724865\pi$
$174$		0			0
$175$			324342.i			0.800587i
$176$		0			0
$177$	67555.1	+	9460.73i	0.162078	+	0.0226982i
$178$		0			0
$179$	240296.			0.560550			0.280275	−	0.959920i	$-0.409574\pi$
$179$	240296.			0.560550			0.280275	+	0.959920i	$0.409574\pi$
$180$		0			0
$181$	572922.			1.29987			0.649933	−	0.759991i	$-0.274797\pi$
$181$	572922.			1.29987			0.649933	+	0.759991i	$0.274797\pi$
$182$		0			0
$183$	−818661.	−	114649.i	−1.80708	−	0.253071i
$184$		0			0
$185$			511735.i			1.09930i
$186$		0			0
$187$			1.54334e6i			3.22744i
$188$		0			0
$189$	140897.	−	317708.i	0.286912	−	0.646955i
$190$		0			0
$191$	342586.			0.679494			0.339747	−	0.940517i	$-0.389659\pi$
$191$	342586.			0.679494			0.339747	+	0.940517i	$0.389659\pi$
$192$		0			0
$193$	782820.			1.51275			0.756377	−	0.654136i	$-0.226967\pi$
$193$	782820.			1.51275			0.756377	+	0.654136i	$0.226967\pi$
$194$		0			0
$195$	117517.	−	839143.i	0.221317	−	1.58034i
$196$		0			0
$197$		−	548639.i		−	1.00721i	−0.863933	−	0.503606i	$-0.832006\pi$
$197$		−	548639.i		−	1.00721i	0.863933	−	0.503606i	$-0.167994\pi$
$198$		0			0
$199$		−	626664.i		−	1.12177i	−0.827895	−	0.560883i	$-0.810462\pi$
$199$		−	626664.i		−	1.12177i	0.827895	−	0.560883i	$-0.189538\pi$
$200$		0			0
$201$	−53460.8	+	381741.i	−0.0933351	+	0.666467i
$202$		0			0
$203$	−34059.6			−0.0580095
$204$		0			0
$205$	−1.25789e6			−2.09054
$206$		0			0
$207$	729009.	+	208272.i	1.18252	+	0.337836i
$208$		0			0
$209$		−	800135.i		−	1.26706i
$210$		0			0
$211$		−	291994.i		−	0.451511i	−0.974184	−	0.225755i	$-0.927515\pi$
$211$		−	291994.i		−	0.451511i	0.974184	−	0.225755i	$-0.0724850\pi$
$212$		0			0
$213$	417521.	+	58471.5i	0.630564	+	0.0883071i
$214$		0			0
$215$	939801.			1.38656
$216$		0			0
$217$	−230006.			−0.331582
$218$		0			0
$219$	239854.	+	33590.3i	0.337938	+	0.0473264i
$220$		0			0
$221$		−	1.44673e6i		−	1.99254i
$222$		0			0
$223$		−	314665.i		−	0.423727i	−0.977299	−	0.211864i	$-0.932047\pi$
$223$		−	314665.i		−	0.423727i	0.977299	−	0.211864i	$-0.0679533\pi$
$224$		0			0
$225$	−825971.	−	235974.i	−1.08770	−	0.310747i
$226$		0			0
$227$	−446955.			−0.575704			−0.287852	−	0.957675i	$-0.592941\pi$
$227$	−446955.			−0.575704			−0.287852	+	0.957675i	$0.592941\pi$
$228$		0			0
$229$	522511.			0.658426			0.329213	−	0.944256i	$-0.393217\pi$
$229$	522511.			0.658426			0.329213	+	0.944256i	$0.393217\pi$
$230$		0			0
$231$	140943.	−	1.00642e6i	0.173786	−	1.24093i
$232$		0			0
$233$			1.06954e6i			1.29064i	0.763911	+	0.645321i	$0.223276\pi$
$233$			1.06954e6i			1.29064i	−0.763911	+	0.645321i	$0.776724\pi$
$234$		0			0
$235$		−	1.71662e6i		−	2.02770i
$236$		0			0
$237$	−92465.0	+	660253.i	−0.106932	+	0.763554i
$238$		0			0
$239$	1.08235e6			1.22567			0.612837	−	0.790209i	$-0.290028\pi$
$239$	1.08235e6			1.22567			0.612837	+	0.790209i	$0.290028\pi$
$240$		0			0
$241$	−372598.			−0.413235			−0.206618	−	0.978422i	$-0.566246\pi$
$241$	−372598.			−0.413235			−0.206618	+	0.978422i	$0.566246\pi$
$242$		0			0
$243$	706568.	+	589958.i	0.767606	+	0.640922i
$244$		0			0
$245$			684610.i			0.728665i
$246$		0			0
$247$			750049.i			0.782253i
$248$		0			0
$249$	−449227.	−	62911.8i	−0.459163	−	0.0643034i
$250$		0			0
$251$	−888643.			−0.890314			−0.445157	−	0.895453i	$-0.646852\pi$
$251$	−888643.			−0.890314			−0.445157	+	0.895453i	$0.646852\pi$
$252$		0			0
$253$	2.21692e6			2.17745
$254$		0			0
$255$	−2.73654e6	−	383237.i	−2.63543	−	0.369077i
$256$		0			0
$257$		−	1.28879e6i		−	1.21716i	−0.793492	−	0.608581i	$-0.791739\pi$
$257$		−	1.28879e6i		−	1.21716i	0.793492	−	0.608581i	$-0.208261\pi$
$258$		0			0
$259$		−	575326.i		−	0.532923i
$260$		0			0
$261$	24779.9	−	86736.3i	0.0225164	−	0.0788133i
$262$		0			0
$263$	−572092.			−0.510007			−0.255004	−	0.966940i	$-0.582077\pi$
$263$	−572092.			−0.510007			−0.255004	+	0.966940i	$0.582077\pi$
$264$		0			0
$265$	−1.92673e6			−1.68541
$266$		0			0
$267$	−40468.0	+	288965.i	−0.0347403	+	0.248066i
$268$		0			0
$269$		−	2.32783e6i		−	1.96142i	−0.195467	−	0.980710i	$-0.562622\pi$
$269$		−	2.32783e6i		−	1.96142i	0.195467	−	0.980710i	$-0.437378\pi$
$270$		0			0
$271$		−	2.15421e6i		−	1.78182i	−0.454177	−	0.890912i	$-0.650066\pi$
$271$		−	2.15421e6i		−	1.78182i	0.454177	−	0.890912i	$-0.349934\pi$
$272$		0			0
$273$	−132121.	+	943418.i	−0.107291	+	0.766121i
$274$		0			0
$275$	−2.51178e6			−2.00285
$276$		0			0
$277$	−473172.			−0.370526			−0.185263	−	0.982689i	$-0.559314\pi$
$277$	−473172.			−0.370526			−0.185263	+	0.982689i	$0.559314\pi$
$278$		0			0
$279$	167340.	−	585736.i	0.128703	−	0.450496i
$280$		0			0
$281$		−	1.42241e6i		−	1.07463i	−0.843381	−	0.537315i	$-0.819439\pi$
$281$		−	1.42241e6i		−	1.07463i	0.843381	−	0.537315i	$-0.180561\pi$
$282$		0			0
$283$		−	2.03901e6i		−	1.51340i	−0.653764	−	0.756698i	$-0.726811\pi$
$283$		−	2.03901e6i		−	1.51340i	0.653764	−	0.756698i	$-0.273189\pi$
$284$		0			0
$285$	1.41874e6	+	198687.i	1.03464	+	0.144896i
$286$		0			0
$287$	1.41420e6			1.01346
$288$		0			0
$289$	−3.29810e6			−2.32284
$290$		0			0
$291$	−226914.	−	31778.0i	−0.157083	−	0.0219986i
$292$		0			0
$293$			2.17933e6i			1.48304i	0.670930	+	0.741521i	$0.265895\pi$
$293$			2.17933e6i			1.48304i	−0.670930	+	0.741521i	$0.734105\pi$
$294$		0			0
$295$			357118.i			0.238922i
$296$		0			0
$297$	2.46040e6	+	1.09114e6i	1.61851	+	0.717777i
$298$		0			0
$299$	−2.07814e6			−1.34430
$300$		0			0
$301$	−1.05658e6			−0.672184
$302$		0			0
$303$	58813.6	−	419963.i	0.0368020	−	0.262787i
$304$		0			0
$305$		−	4.32770e6i		−	2.66384i
$306$		0			0
$307$		−	695268.i		−	0.421024i	−0.977591	−	0.210512i	$-0.932487\pi$
$307$		−	695268.i		−	0.421024i	0.977591	−	0.210512i	$-0.0675131\pi$
$308$		0			0
$309$	−299212.	+	2.13654e6i	−0.178272	+	1.27296i
$310$		0			0
$311$	12139.6			0.00711713			0.00355857	−	0.999994i	$-0.498867\pi$
$311$	12139.6			0.00711713			0.00355857	+	0.999994i	$0.498867\pi$
$312$		0			0
$313$	509977.			0.294232			0.147116	−	0.989119i	$-0.453001\pi$
$313$	509977.			0.294232			0.147116	+	0.989119i	$0.453001\pi$
$314$		0			0
$315$	1.74950e6	+	499820.i	0.993433	+	0.283816i
$316$		0			0
$317$		−	1.95242e6i		−	1.09125i	−0.838028	−	0.545627i	$-0.816292\pi$
$317$		−	1.95242e6i		−	1.09125i	0.838028	−	0.545627i	$-0.183708\pi$
$318$		0			0
$319$		−	263765.i		−	0.145124i
$320$		0			0
$321$	746700.	+	104571.i	0.404467	+	0.0566435i
$322$		0			0
$323$	2.44599e6			1.30452
$324$		0			0
$325$	2.35455e6			1.23651
$326$		0			0
$327$	52358.4	+	7332.52i	0.0270780	+	0.00379214i
$328$		0			0
$329$			1.92994e6i			0.982999i
$330$		0			0
$331$		−	1.46773e6i		−	0.736337i	−0.929759	−	0.368168i	$-0.879985\pi$
$331$		−	1.46773e6i		−	0.736337i	0.929759	−	0.368168i	$-0.120015\pi$
$332$		0			0
$333$	1.46513e6	+	418576.i	0.724044	+	0.206854i
$334$		0			0
$335$	−2.01800e6			−0.982449
$336$		0			0
$337$	−2.60707e6			−1.25048			−0.625241	−	0.780432i	$-0.714999\pi$
$337$	−2.60707e6			−1.25048			−0.625241	+	0.780432i	$0.714999\pi$
$338$		0			0
$339$	121124.	−	864892.i	0.0572439	−	0.408755i
$340$		0			0
$341$		−	1.78122e6i		−	0.829529i
$342$		0			0
$343$		−	2.31173e6i		−	1.06097i
$344$		0			0
$345$	−550497.	+	3.93087e6i	−0.249005	+	1.77804i
$346$		0			0
$347$	−568898.			−0.253636			−0.126818	−	0.991926i	$-0.540476\pi$
$347$	−568898.			−0.253636			−0.126818	+	0.991926i	$0.540476\pi$
$348$		0			0
$349$	1.85024e6			0.813140			0.406570	−	0.913620i	$-0.366725\pi$
$349$	1.85024e6			0.813140			0.406570	+	0.913620i	$0.366725\pi$
$350$		0			0
$351$	−2.30639e6	−	1.02284e6i	−0.999228	−	0.443138i
$352$		0			0
$353$			3.50974e6i			1.49913i	0.661933	+	0.749563i	$0.269736\pi$
$353$			3.50974e6i			1.49913i	−0.661933	+	0.749563i	$0.730264\pi$
$354$		0			0
$355$			2.20715e6i			0.929524i
$356$		0			0
$357$	3.07659e6	+	430860.i	1.27761	+	0.178923i
$358$		0			0
$359$	−429267.			−0.175789			−0.0878944	−	0.996130i	$-0.528014\pi$
$359$	−429267.			−0.175789			−0.0878944	+	0.996130i	$0.528014\pi$
$360$		0			0
$361$	1.20799e6			0.487860
$362$		0			0
$363$	5.30763e6	+	743306.i	2.11414	+	0.296074i
$364$		0			0
$365$			1.26795e6i			0.498160i
$366$		0			0
$367$		−	127386.i		−	0.0493691i	−0.999695	−	0.0246846i	$-0.992142\pi$
$367$		−	127386.i		−	0.0493691i	0.999695	−	0.0246846i	$-0.00785814\pi$
$368$		0			0
$369$	−1.02890e6	+	3.60141e6i	−0.393374	+	1.37691i
$370$		0			0
$371$	2.16615e6			0.817061
$372$		0			0
$373$	−3.94120e6			−1.46675			−0.733375	−	0.679825i	$-0.762056\pi$
$373$	−3.94120e6			−1.46675			−0.733375	+	0.679825i	$0.762056\pi$
$374$		0			0
$375$	72349.0	−	516614.i	0.0265677	−	0.189709i
$376$		0			0
$377$			247254.i			0.0895963i
$378$		0			0
$379$		−	4.33847e6i		−	1.55145i	−0.631069	−	0.775726i	$-0.717384\pi$
$379$		−	4.33847e6i		−	1.55145i	0.631069	−	0.775726i	$-0.282616\pi$
$380$		0			0
$381$	−126578.	+	903838.i	−0.0446730	+	0.318991i
$382$		0			0
$383$	3.67132e6			1.27887			0.639433	−	0.768847i	$-0.279169\pi$
$383$	3.67132e6			1.27887			0.639433	+	0.768847i	$0.279169\pi$
$384$		0			0
$385$	5.32024e6			1.82928
$386$		0			0
$387$	768713.	−	2.69070e6i	0.260908	−	0.913247i
$388$		0			0
$389$		−	3.46691e6i		−	1.16163i	−0.814035	−	0.580816i	$-0.802734\pi$
$389$		−	3.46691e6i		−	1.16163i	0.814035	−	0.580816i	$-0.197266\pi$
$390$		0			0
$391$			6.77706e6i			2.24181i
$392$		0			0
$393$	−5.30258e6	−	742598.i	−1.73183	−	0.242534i
$394$		0			0
$395$	−3.49031e6			−1.12557
$396$		0			0
$397$	−1.70987e6			−0.544487			−0.272244	−	0.962228i	$-0.587766\pi$
$397$	−1.70987e6			−0.544487			−0.272244	+	0.962228i	$0.587766\pi$
$398$		0			0
$399$	−1.59504e6	−	223377.i	−0.501579	−	0.0702434i
$400$		0			0
$401$		−	2.72203e6i		−	0.845340i	−0.906284	−	0.422670i	$-0.861093\pi$
$401$		−	2.72203e6i		−	0.845340i	0.906284	−	0.422670i	$-0.138907\pi$
$402$		0			0
$403$			1.66972e6i			0.512131i
$404$		0			0
$405$	−2.54569e6	+	4.09165e6i	−0.771201	+	1.23954i
$406$		0			0
$407$	4.45545e6			1.33323
$408$		0			0
$409$	−1.55419e6			−0.459405			−0.229702	−	0.973261i	$-0.573775\pi$
$409$	−1.55419e6			−0.459405			−0.229702	+	0.973261i	$0.573775\pi$
$410$		0			0
$411$	267956.	−	1.91336e6i	0.0782454	−	0.558717i
$412$		0			0
$413$		−	401495.i		−	0.115826i
$414$		0			0
$415$		−	2.37476e6i		−	0.676860i
$416$		0			0
$417$	386800.	−	2.76197e6i	0.108930	−	0.777820i
$418$		0			0
$419$	−2.75545e6			−0.766756			−0.383378	−	0.923592i	$-0.625239\pi$
$419$	−2.75545e6			−0.766756			−0.383378	+	0.923592i	$0.625239\pi$
$420$		0			0
$421$	3.16875e6			0.871329			0.435665	−	0.900109i	$-0.356513\pi$
$421$	3.16875e6			0.871329			0.435665	+	0.900109i	$0.356513\pi$
$422$		0			0
$423$	−4.91478e6	−	1.40412e6i	−1.33553	−	0.381551i
$424$		0			0
$425$		−	7.67844e6i		−	2.06206i
$426$		0			0
$427$			4.86548e6i			1.29139i
$428$		0			0
$429$	−7.30603e6	−	1.02317e6i	−1.91663	−	0.268414i
$430$		0			0
$431$	5.51741e6			1.43068			0.715339	−	0.698778i	$-0.246272\pi$
$431$	5.51741e6			1.43068			0.715339	+	0.698778i	$0.246272\pi$
$432$		0			0
$433$	5.72171e6			1.46658			0.733290	−	0.679916i	$-0.237984\pi$
$433$	5.72171e6			1.46658			0.733290	+	0.679916i	$0.237984\pi$
$434$		0			0
$435$	467688.	+	65497.2i	0.118504	+	0.0165959i
$436$		0			0
$437$		−	3.51352e6i		−	0.880115i
$438$		0			0
$439$			3.22530e6i			0.798747i	0.916788	+	0.399373i	$0.130772\pi$
$439$			3.22530e6i			0.798747i	−0.916788	+	0.399373i	$0.869228\pi$
$440$		0			0
$441$	1.96008e6	+	559979.i	0.479929	+	0.137112i
$442$		0			0
$443$	−5.40747e6			−1.30913			−0.654567	−	0.756004i	$-0.727149\pi$
$443$	−5.40747e6			−1.30913			−0.654567	+	0.756004i	$0.727149\pi$
$444$		0			0
$445$	−1.52756e6			−0.365678
$446$		0			0
$447$	−1.00645e6	+	7.18663e6i	−0.238244	+	1.70120i
$448$		0			0
$449$			4.02319e6i			0.941792i	0.882189	+	0.470896i	$0.156069\pi$
$449$			4.02319e6i			0.941792i	−0.882189	+	0.470896i	$0.843931\pi$
$450$		0			0
$451$			1.09519e7i			2.53541i
$452$		0			0
$453$	175384.	−	1.25234e6i	0.0401553	−	0.286732i
$454$		0			0
$455$	−4.98721e6			−1.12935
$456$		0			0
$457$	4.00035e6			0.896000			0.448000	−	0.894034i	$-0.352137\pi$
$457$	4.00035e6			0.896000			0.448000	+	0.894034i	$0.352137\pi$
$458$		0			0
$459$	−3.33559e6	+	7.52139e6i	−0.738994	+	1.66635i
$460$		0			0
$461$			5.80933e6i			1.27313i	0.771222	+	0.636567i	$0.219646\pi$
$461$			5.80933e6i			1.27313i	−0.771222	+	0.636567i	$0.780354\pi$
$462$		0			0
$463$		−	1.00938e6i		−	0.218828i	−0.993996	−	0.109414i	$-0.965103\pi$
$463$		−	1.00938e6i		−	0.218828i	0.993996	−	0.109414i	$-0.0348974\pi$
$464$		0			0
$465$	3.15833e6	+	442307.i	0.677368	+	0.0948618i
$466$		0			0
$467$	−1.13001e6			−0.239766			−0.119883	−	0.992788i	$-0.538252\pi$
$467$	−1.13001e6			−0.239766			−0.119883	+	0.992788i	$0.538252\pi$
$468$		0			0
$469$	2.26877e6			0.476275
$470$		0			0
$471$	1.64737e6	+	230705.i	0.342167	+	0.0479186i
$472$		0			0
$473$		−	8.18242e6i		−	1.68162i
$474$		0			0
$475$			3.98084e6i			0.809544i
$476$		0			0
$477$	−1.57598e6	+	5.51634e6i	−0.317142	+	1.11008i
$478$		0			0
$479$	−2.14181e6			−0.426522			−0.213261	−	0.976995i	$-0.568409\pi$
$479$	−2.14181e6			−0.426522			−0.213261	+	0.976995i	$0.568409\pi$
$480$		0			0
$481$	−4.17655e6			−0.823105
$482$		0			0
$483$	618904.	−	4.41934e6i	0.120713	−	0.861964i
$484$		0			0
$485$		−	1.19954e6i		−	0.231558i
$486$		0			0
$487$		−	7.73961e6i		−	1.47876i	−0.673290	−	0.739378i	$-0.735120\pi$
$487$		−	7.73961e6i		−	1.47876i	0.673290	−	0.739378i	$-0.264880\pi$
$488$		0			0
$489$	33989.0	−	242701.i	0.00642786	−	0.0458987i
$490$		0			0
$491$	8.02474e6			1.50220			0.751099	−	0.660190i	$-0.229524\pi$
$491$	8.02474e6			1.50220			0.751099	+	0.660190i	$0.229524\pi$
$492$		0			0
$493$	806323.			0.149414
$494$		0			0
$495$	−3.87072e6	+	1.35485e7i	−0.710033	+	2.48530i
$496$		0			0
$497$		−	2.48142e6i		−	0.450618i
$498$		0			0
$499$		−	9.08779e6i		−	1.63383i	−0.576757	−	0.816916i	$-0.695682\pi$
$499$		−	9.08779e6i		−	1.63383i	0.576757	−	0.816916i	$-0.304318\pi$
$500$		0			0
$501$	77801.7	+	10895.7i	0.0138483	+	0.00193937i
$502$		0			0
$503$	9.15721e6			1.61378			0.806888	−	0.590705i	$-0.201150\pi$
$503$	9.15721e6			1.61378			0.806888	+	0.590705i	$0.201150\pi$
$504$		0			0
$505$	2.22006e6			0.387379
$506$		0			0
$507$	1.11675e6	+	156395.i	0.192946	+	0.0270211i
$508$		0			0
$509$		−	6.77792e6i		−	1.15958i	−0.814765	−	0.579792i	$-0.803134\pi$
$509$		−	6.77792e6i		−	1.15958i	0.814765	−	0.579792i	$-0.196866\pi$
$510$		0			0
$511$		−	1.42551e6i		−	0.241500i
$512$		0			0
$513$	1.72932e6	−	3.89942e6i	0.290122	−	0.654194i
$514$		0			0
$515$	−1.12945e7			−1.87649
$516$		0			0
$517$	−1.49458e7			−2.45920
$518$		0			0
$519$	384277.	−	2.74396e6i	0.0626218	−	0.447156i
$520$		0			0
$521$			4.57819e6i			0.738924i	0.929246	+	0.369462i	$0.120458\pi$
$521$			4.57819e6i			0.738924i	−0.929246	+	0.369462i	$0.879542\pi$
$522$		0			0
$523$			8.98245e6i			1.43595i	0.696067	+	0.717977i	$0.254932\pi$
$523$			8.98245e6i			1.43595i	−0.696067	+	0.717977i	$0.745068\pi$
$524$		0			0
$525$	−701222.	+	5.00713e6i	−0.111034	+	0.792849i
$526$		0			0
$527$	5.44515e6			0.854050
$528$		0			0
$529$	3.29849e6			0.512479
$530$		0			0
$531$	1.02245e6	+	292106.i	0.157364	+	0.0449577i
$532$		0			0
$533$		−	1.02663e7i		−	1.56530i
$534$		0			0
$535$			3.94729e6i			0.596232i
$536$		0			0
$537$	3.70965e6	+	519516.i	0.555133	+	0.0777434i
$538$		0			0
$539$	5.96059e6			0.883726
$540$		0			0
$541$	1.25763e6			0.184739			0.0923697	−	0.995725i	$-0.470556\pi$
$541$	1.25763e6			0.184739			0.0923697	+	0.995725i	$0.470556\pi$
$542$		0			0
$543$	8.84465e6	+	1.23865e6i	1.28730	+	0.180280i
$544$		0			0
$545$			276783.i			0.0399162i
$546$		0			0
$547$			2.63358e6i			0.376338i	0.982137	+	0.188169i	$0.0602553\pi$
$547$			2.63358e6i			0.376338i	−0.982137	+	0.188169i	$0.939745\pi$
$548$		0			0
$549$	−1.23905e7	−	3.53986e6i	−1.75451	−	0.501251i
$550$		0			0
$551$	−418033.			−0.0586586
$552$		0			0
$553$	3.92403e6			0.545657
$554$		0			0
$555$	−1.10636e6	+	7.90007e6i	−0.152463	+	1.08868i
$556$		0			0
$557$		−	3.67084e6i		−	0.501334i	−0.968073	−	0.250667i	$-0.919350\pi$
$557$		−	3.67084e6i		−	0.501334i	0.968073	−	0.250667i	$-0.0806500\pi$
$558$		0			0
$559$			7.67022e6i			1.03819i
$560$		0			0
$561$	−3.33668e6	+	2.38258e7i	−0.447617	+	3.19625i
$562$		0			0
$563$	337916.			0.0449302			0.0224651	−	0.999748i	$-0.492849\pi$
$563$	337916.			0.0449302			0.0224651	+	0.999748i	$0.492849\pi$
$564$		0			0
$565$	4.57210e6			0.602552
$566$		0			0
$567$	2.86203e6	−	4.60010e6i	0.373866	−	0.600910i
$568$		0			0
$569$		−	5.56446e6i		−	0.720514i	−0.932853	−	0.360257i	$-0.882689\pi$
$569$		−	5.56446e6i		−	0.720514i	0.932853	−	0.360257i	$-0.117311\pi$
$570$		0			0
$571$		−	1.29060e7i		−	1.65654i	−0.560331	−	0.828269i	$-0.689326\pi$
$571$		−	1.29060e7i		−	1.65654i	0.560331	−	0.828269i	$-0.310674\pi$
$572$		0			0
$573$	5.28877e6	+	740664.i	0.672927	+	0.0942398i
$574$		0			0
$575$	−1.10296e7			−1.39120
$576$		0			0
$577$	4.16281e6			0.520531			0.260266	−	0.965537i	$-0.416190\pi$
$577$	4.16281e6			0.520531			0.260266	+	0.965537i	$0.416190\pi$
$578$		0			0
$579$	1.20850e7	+	1.69244e6i	1.49813	+	0.209806i
$580$		0			0
$581$			2.66985e6i			0.328131i
$582$		0			0
$583$			1.67752e7i			2.04407i
$584$		0			0
$585$	3.62842e6	−	1.27004e7i	0.438357	−	1.53437i
$586$		0			0
$587$	6.04823e6			0.724492			0.362246	−	0.932083i	$-0.382010\pi$
$587$	6.04823e6			0.724492			0.362246	+	0.932083i	$0.382010\pi$
$588$		0			0
$589$	−2.82300e6			−0.335292
$590$		0			0
$591$	1.18615e6	−	8.46978e6i	0.139691	−	0.997478i
$592$		0			0
$593$			7.92156e6i			0.925069i	0.886601	+	0.462534i	$0.153060\pi$
$593$			7.92156e6i			0.925069i	−0.886601	+	0.462534i	$0.846940\pi$
$594$		0			0
$595$			1.62638e7i			1.88335i
$596$		0			0
$597$	1.35484e6	−	9.67432e6i	0.155579	−	1.11092i
$598$		0			0
$599$	3.14090e6			0.357673			0.178837	−	0.983879i	$-0.442767\pi$
$599$	3.14090e6			0.357673			0.178837	+	0.983879i	$0.442767\pi$
$600$		0			0
$601$	1.13195e6			0.127832			0.0639161	−	0.997955i	$-0.479641\pi$
$601$	1.13195e6			0.127832			0.0639161	+	0.997955i	$0.479641\pi$
$602$		0			0
$603$	−1.65063e6	+	5.77766e6i	−0.184866	+	0.647081i
$604$		0			0
$605$			2.80578e7i			3.11649i
$606$		0			0
$607$			4.91816e6i			0.541790i	0.962609	+	0.270895i	$0.0873195\pi$
$607$			4.91816e6i			0.541790i	−0.962609	+	0.270895i	$0.912680\pi$
$608$		0			0
$609$	−525805.	−	73636.2i	−0.0574489	−	0.00804541i
$610$		0			0
$611$	1.40103e7			1.51825
$612$		0			0
$613$	−7.34835e6			−0.789839			−0.394920	−	0.918716i	$-0.629228\pi$
$613$	−7.34835e6			−0.789839			−0.394920	+	0.918716i	$0.629228\pi$
$614$		0			0
$615$	−1.94191e7	−	2.71954e6i	−2.07033	−	0.289939i
$616$		0			0
$617$			2.88590e6i			0.305189i	0.988289	+	0.152594i	$0.0487628\pi$
$617$			2.88590e6i			0.305189i	−0.988289	+	0.152594i	$0.951237\pi$
$618$		0			0
$619$		−	1.44988e7i		−	1.52091i	−0.649389	−	0.760456i	$-0.724975\pi$
$619$		−	1.44988e7i		−	1.52091i	0.649389	−	0.760456i	$-0.275025\pi$
$620$		0			0
$621$	1.08040e7	+	4.79138e6i	1.12423	+	0.498576i
$622$		0			0
$623$	1.71738e6			0.177275
$624$		0			0
$625$	−8.31606e6			−0.851565
$626$		0			0
$627$	1.72988e6	−	1.23523e7i	0.175730	−	1.25482i
$628$		0			0
$629$			1.36202e7i			1.37264i
$630$		0			0
$631$			1.30715e7i			1.30693i	0.756956	+	0.653465i	$0.226685\pi$
$631$			1.30715e7i			1.30693i	−0.756956	+	0.653465i	$0.773315\pi$
$632$		0			0
$633$	631286.	−	4.50775e6i	0.0626206	−	0.447147i
$634$		0			0
$635$	−4.77798e6			−0.470229
$636$		0			0
$637$	−5.58748e6			−0.545591
$638$		0			0
$639$	6.31919e6	+	1.80534e6i	0.612222	+	0.174907i
$640$		0			0
$641$			1.76946e7i			1.70096i	0.526004	+	0.850482i	$0.323690\pi$
$641$			1.76946e7i			1.70096i	−0.526004	+	0.850482i	$0.676310\pi$
$642$		0			0
$643$		−	3.17000e6i		−	0.302365i	−0.988506	−	0.151182i	$-0.951692\pi$
$643$		−	3.17000e6i		−	0.302365i	0.988506	−	0.151182i	$-0.0483081\pi$
$644$		0			0
$645$	1.45085e7	+	2.03183e6i	1.37316	+	0.192304i
$646$		0			0
$647$	−1.04323e7			−0.979756			−0.489878	−	0.871791i	$-0.662959\pi$
$647$	−1.04323e7			−0.979756			−0.489878	+	0.871791i	$0.662959\pi$
$648$		0			0
$649$	3.10926e6			0.289765
$650$		0			0
$651$	−3.55080e6	−	497270.i	−0.328377	−	0.0459875i
$652$		0			0
$653$			2.86705e6i			0.263119i	0.991308	+	0.131559i	$0.0419984\pi$
$653$			2.86705e6i			0.263119i	−0.991308	+	0.131559i	$0.958002\pi$
$654$		0			0
$655$		−	2.80311e7i		−	2.55292i
$656$		0			0
$657$	3.63020e6	+	1.03712e6i	0.328108	+	0.0937381i
$658$		0			0
$659$	−8.99951e6			−0.807245			−0.403622	−	0.914926i	$-0.632249\pi$
$659$	−8.99951e6			−0.807245			−0.403622	+	0.914926i	$0.632249\pi$
$660$		0			0
$661$	−1.10916e6			−0.0987392			−0.0493696	−	0.998781i	$-0.515721\pi$
$661$	−1.10916e6			−0.0987392			−0.0493696	+	0.998781i	$0.515721\pi$
$662$		0			0
$663$	3.12781e6	−	2.23344e7i	0.276348	−	1.97329i
$664$		0			0
$665$		−	8.43189e6i		−	0.739385i
$666$		0			0
$667$		−	1.15823e6i		−	0.100805i
$668$		0			0
$669$	680300.	−	4.85774e6i	0.0587673	−	0.419632i
$670$		0			0
$671$	−3.76794e7			−3.23070
$672$		0			0
$673$	−1.74919e6			−0.148867			−0.0744335	−	0.997226i	$-0.523715\pi$
$673$	−1.74919e6			−0.148867			−0.0744335	+	0.997226i	$0.523715\pi$
$674$		0			0
$675$	−1.22410e7	−	5.42865e6i	−1.03409	−	0.458598i
$676$		0			0
$677$		−	1.67923e7i		−	1.40812i	−0.710143	−	0.704058i	$-0.751370\pi$
$677$		−	1.67923e7i		−	1.40812i	0.710143	−	0.704058i	$-0.248630\pi$
$678$		0			0
$679$			1.34860e6i			0.112256i
$680$		0			0
$681$	−6.90000e6	−	966309.i	−0.570140	−	0.0798451i
$682$		0			0
$683$	−1.94379e7			−1.59440			−0.797201	−	0.603714i	$-0.793687\pi$
$683$	−1.94379e7			−1.59440			−0.797201	+	0.603714i	$0.793687\pi$
$684$		0			0
$685$	1.01146e7			0.823614
$686$		0			0
$687$	8.06642e6	+	1.12966e6i	0.652062	+	0.0913178i
$688$		0			0
$689$		−	1.57251e7i		−	1.26196i
$690$		0			0
$691$			9.94513e6i			0.792347i	0.918176	+	0.396173i	$0.129662\pi$
$691$			9.94513e6i			0.792347i	−0.918176	+	0.396173i	$0.870338\pi$
$692$		0			0
$693$	4.35171e6	−	1.52321e7i	0.344213	−	1.20484i
$694$		0			0
$695$	1.46007e7			1.14660
$696$		0			0
$697$	−3.34797e7			−2.61035
$698$		0			0
$699$	−2.31232e6	+	1.65113e7i	−0.179001	+	1.27817i
$700$		0			0
$701$			1.96927e6i			0.151360i	0.997132	+	0.0756798i	$0.0241127\pi$
$701$			1.96927e6i			0.151360i	−0.997132	+	0.0756798i	$0.975887\pi$
$702$		0			0
$703$		−	7.06131e6i		−	0.538886i
$704$		0			0
$705$	3.71130e6	−	2.65009e7i	0.281225	−	2.00811i
$706$		0			0
$707$	−2.49593e6			−0.187795
$708$		0			0
$709$	1.75807e7			1.31347			0.656736	−	0.754121i	$-0.271937\pi$
$709$	1.75807e7			1.31347			0.656736	+	0.754121i	$0.271937\pi$
$710$		0			0
$711$	−2.85491e6	+	9.99296e6i	−0.211796	+	0.741344i
$712$		0			0
$713$		−	7.82163e6i		−	0.576200i
$714$		0			0
$715$		−	3.86220e7i		−	2.82533i
$716$		0			0
$717$	1.67092e7	+	2.34003e6i	1.21383	+	0.169990i
$718$		0			0
$719$	2.74937e7			1.98340			0.991700	−	0.128572i	$-0.0410393\pi$
$719$	2.74937e7			1.98340			0.991700	+	0.128572i	$0.0410393\pi$
$720$		0			0
$721$	1.26980e7			0.909695
$722$		0			0
$723$	−5.75209e6	−	805549.i	−0.409242	−	0.0573121i
$724$		0			0
$725$			1.31229e6i			0.0927221i
$726$		0			0
$727$		−	1.98468e7i		−	1.39269i	−0.717706	−	0.696346i	$-0.754808\pi$
$727$		−	1.98468e7i		−	1.39269i	0.717706	−	0.696346i	$-0.245192\pi$
$728$		0			0
$729$	9.63239e6	+	1.06352e7i	0.671298	+	0.741188i
$730$		0			0
$731$	2.50135e7			1.73133
$732$		0			0
$733$	−1.78657e7			−1.22818			−0.614088	−	0.789238i	$-0.710476\pi$
$733$	−1.78657e7			−1.22818			−0.614088	+	0.789238i	$0.710476\pi$
$734$		0			0
$735$	−1.48011e6	+	1.05689e7i	−0.101059	+	0.721623i
$736$		0			0
$737$			1.75698e7i			1.19151i
$738$		0			0
$739$			1.32601e6i			0.0893172i	0.999002	+	0.0446586i	$0.0142200\pi$
$739$			1.32601e6i			0.0893172i	−0.999002	+	0.0446586i	$0.985780\pi$
$740$		0			0
$741$	−1.62159e6	+	1.15791e7i	−0.108492	+	0.774693i
$742$		0			0
$743$	−6.81899e6			−0.453156			−0.226578	−	0.973993i	$-0.572754\pi$
$743$	−6.81899e6			−0.453156			−0.226578	+	0.973993i	$0.572754\pi$
$744$		0			0
$745$	−3.79908e7			−2.50777
$746$		0			0
$747$	−6.79906e6	−	1.94244e6i	−0.445808	−	0.127364i
$748$		0			0
$749$		−	4.43780e6i		−	0.289044i
$750$		0			0
$751$		−	1.64460e6i		−	0.106405i	−0.998584	−	0.0532024i	$-0.983057\pi$
$751$		−	1.64460e6i		−	0.106405i	0.998584	−	0.0532024i	$-0.0169428\pi$
$752$		0			0
$753$	−1.37187e7	−	1.92123e6i	−0.881710	−	0.123479i
$754$		0			0
$755$	6.62027e6			0.422677
$756$		0			0
$757$	1.33078e7			0.844050			0.422025	−	0.906584i	$-0.361319\pi$
$757$	1.33078e7			0.844050			0.422025	+	0.906584i	$0.361319\pi$
$758$		0			0
$759$	3.42243e7	+	4.79293e6i	2.15640	+	0.301993i
$760$		0			0
$761$			2.10386e7i			1.31690i	0.752623	+	0.658452i	$0.228789\pi$
$761$			2.10386e7i			1.31690i	−0.752623	+	0.658452i	$0.771211\pi$
$762$		0			0
$763$		−	311178.i		−	0.0193507i
$764$		0			0
$765$	−4.14176e7	−	1.18327e7i	−2.55877	−	0.731021i
$766$		0			0
$767$	−2.91463e6			−0.178894
$768$		0			0
$769$	−1.69563e7			−1.03399			−0.516993	−	0.855990i	$-0.672949\pi$
$769$	−1.69563e7			−1.03399			−0.516993	+	0.855990i	$0.672949\pi$
$770$		0			0
$771$	2.78633e6	−	1.98960e7i	0.168810	−	1.20540i
$772$		0			0
$773$		−	1.16231e7i		−	0.699635i	−0.936818	−	0.349817i	$-0.886244\pi$
$773$		−	1.16231e7i		−	0.699635i	0.936818	−	0.349817i	$-0.113756\pi$
$774$		0			0
$775$			8.86194e6i			0.529999i
$776$		0			0
$777$	1.24384e6	−	8.88177e6i	0.0739117	−	0.527773i
$778$		0			0
$779$	1.73573e7			1.02480
$780$		0			0
$781$	1.92166e7			1.12733
$782$		0			0
$783$	570070.	−	1.28544e6i	0.0332295	−	0.0749288i
$784$		0			0
$785$			8.70850e6i			0.504393i
$786$		0			0
$787$		−	7.57942e6i		−	0.436214i	−0.975925	−	0.218107i	$-0.930012\pi$
$787$		−	7.57942e6i		−	0.436214i	0.975925	−	0.218107i	$-0.0699881\pi$
$788$		0			0
$789$	−8.83184e6	−	1.23685e6i	−0.505078	−	0.0707335i
$790$		0			0
$791$	−5.14024e6			−0.292107
$792$		0			0
$793$	3.53207e7			1.99456
$794$		0			0
$795$	−2.97445e7	−	4.16555e6i	−1.66912	−	0.233752i
$796$		0			0
$797$		−	8.62153e6i		−	0.480771i	−0.970677	−	0.240386i	$-0.922726\pi$
$797$		−	8.62153e6i		−	0.480771i	0.970677	−	0.240386i	$-0.0772739\pi$
$798$		0			0
$799$		−	4.56891e7i		−	2.53189i
$800$		0			0
$801$	−1.24947e6	+	4.37349e6i	−0.0688091	+	0.240850i
$802$		0			0
$803$	1.10394e7			0.604168
$804$		0			0
$805$	2.33620e7			1.27063
$806$		0			0
$807$	5.03273e6	−	3.59366e7i	0.272032	−	1.94246i
$808$		0			0
$809$			6.09025e6i			0.327163i	0.986530	+	0.163581i	$0.0523046\pi$
$809$			6.09025e6i			0.327163i	−0.986530	+	0.163581i	$0.947695\pi$
$810$		0			0
$811$		−	1.47569e7i		−	0.787849i	−0.919143	−	0.393925i	$-0.871117\pi$
$811$		−	1.47569e7i		−	0.787849i	0.919143	−	0.393925i	$-0.128883\pi$
$812$		0			0
$813$	4.65736e6	−	3.32563e7i	0.247123	−	1.76460i
$814$		0			0
$815$	1.28300e6			0.0676599
$816$		0			0
$817$	−1.29681e7			−0.679705
$818$		0			0
$819$	−4.07931e6	+	1.42787e7i	−0.212509	+	0.743837i
$820$		0			0
$821$			1.26636e6i			0.0655689i	0.999462	+	0.0327845i	$0.0104375\pi$
$821$			1.26636e6i			0.0655689i	−0.999462	+	0.0327845i	$0.989563\pi$
$822$		0			0
$823$			2.36418e7i			1.21669i	0.793672	+	0.608346i	$0.208167\pi$
$823$			2.36418e7i			1.21669i	−0.793672	+	0.608346i	$0.791833\pi$
$824$		0			0
$825$	−3.87763e7	−	5.43042e6i	−1.98350	−	0.277778i
$826$		0			0
$827$	−4.79378e6			−0.243733			−0.121867	−	0.992546i	$-0.538888\pi$
$827$	−4.79378e6			−0.243733			−0.121867	+	0.992546i	$0.538888\pi$
$828$		0			0
$829$	1.79432e7			0.906802			0.453401	−	0.891307i	$-0.350211\pi$
$829$	1.79432e7			0.906802			0.453401	+	0.891307i	$0.350211\pi$
$830$		0			0
$831$	−7.30473e6	−	1.02299e6i	−0.366946	−	0.0513888i
$832$		0			0
$833$			1.82214e7i			0.909848i
$834$		0			0
$835$			411285.i			0.0204139i
$836$		0			0
$837$	3.84972e6	−	8.68069e6i	0.189939	−	0.428292i
$838$		0			0
$839$	−3.39791e7			−1.66651			−0.833253	−	0.552892i	$-0.813524\pi$
$839$	−3.39791e7			−1.66651			−0.833253	+	0.552892i	$0.813524\pi$
$840$		0			0
$841$	2.03733e7			0.993281
$842$		0			0
$843$	3.07523e6	−	2.19589e7i	0.149042	−	1.06424i
$844$		0			0
$845$			5.90350e6i			0.284425i
$846$		0			0
$847$		−	3.15444e7i		−	1.51083i
$848$		0			0
$849$	4.40830e6	−	3.14778e7i	0.209895	−	1.49877i
$850$		0			0
$851$	1.95646e7			0.926077
$852$		0			0
$853$	3.85782e6			0.181539			0.0907693	−	0.995872i	$-0.471067\pi$
$853$	3.85782e6			0.181539			0.0907693	+	0.995872i	$0.471067\pi$
$854$		0			0
$855$	2.14727e7	+	6.13458e6i	1.00455	+	0.286992i
$856$		0			0
$857$		−	8.53164e6i		−	0.396808i	−0.980120	−	0.198404i	$-0.936424\pi$
$857$		−	8.53164e6i		−	0.396808i	0.980120	−	0.198404i	$-0.0635758\pi$
$858$		0			0
$859$			7.87397e6i			0.364092i	0.983290	+	0.182046i	$0.0582719\pi$
$859$			7.87397e6i			0.364092i	−0.983290	+	0.182046i	$0.941728\pi$
$860$		0			0
$861$	2.18322e7	+	3.05748e6i	1.00367	+	0.140558i
$862$		0			0
$863$	1.31086e7			0.599141			0.299570	−	0.954074i	$-0.403157\pi$
$863$	1.31086e7			0.599141			0.299570	+	0.954074i	$0.403157\pi$
$864$		0			0
$865$	1.45054e7			0.659160
$866$		0			0
$867$	−5.09154e7	−	7.13043e6i	−2.30039	−	0.322157i
$868$		0			0
$869$			3.03885e7i			1.36509i
$870$		0			0
$871$		−	1.64700e7i		−	0.735612i
$872$		0			0
$873$	−3.43434e6	−	981166.i	−0.152513	−	0.0435719i
$874$		0			0
$875$	−3.07035e6			−0.135571
$876$		0			0
$877$	3.26413e7			1.43307			0.716536	−	0.697550i	$-0.245727\pi$
$877$	3.26413e7			1.43307			0.716536	+	0.697550i	$0.245727\pi$
$878$		0			0
$879$	−4.71167e6	+	3.36440e7i	−0.205685	+	1.46871i
$880$		0			0
$881$		−	1.98226e7i		−	0.860443i	−0.902723	−	0.430221i	$-0.858436\pi$
$881$		−	1.98226e7i		−	0.860443i	0.902723	−	0.430221i	$-0.141564\pi$
$882$		0			0
$883$			3.27609e7i			1.41401i	0.707207	+	0.707007i	$0.249955\pi$
$883$			3.27609e7i			1.41401i	−0.707207	+	0.707007i	$0.750045\pi$
$884$		0			0
$885$	−772082.	+	5.51312e6i	−0.0331364	+	0.236613i
$886$		0			0
$887$	3.48920e7			1.48907			0.744537	−	0.667581i	$-0.232670\pi$
$887$	3.48920e7			1.48907			0.744537	+	0.667581i	$0.232670\pi$
$888$		0			0
$889$	5.37171e6			0.227960
$890$		0			0
$891$	3.56242e7	+	2.21642e7i	1.50332	+	0.935313i
$892$		0			0
$893$			2.36872e7i			0.993998i
$894$		0			0
$895$			1.96104e7i			0.818330i
$896$		0			0
$897$	−3.20820e7	−	4.49291e6i	−1.33131	−	0.186443i
$898$		0			0
$899$	−930604.			−0.0384030
$900$		0			0
$901$	−5.12813e7			−2.10449
$902$		0			0
$903$	−1.63113e7	−	2.28432e6i	−0.665687	−	0.0932259i
$904$		0			0
$905$			4.67556e7i			1.89763i
$906$		0			0
$907$			1.81630e7i			0.733111i	0.930396	+	0.366555i	$0.119463\pi$
$907$			1.81630e7i			0.733111i	−0.930396	+	0.366555i	$0.880537\pi$
$908$		0			0
$909$	1.81590e6	−	6.35615e6i	0.0728926	−	0.255144i
$910$		0			0
$911$	−3.36338e7			−1.34270			−0.671352	−	0.741139i	$-0.734286\pi$
$911$	−3.36338e7			−1.34270			−0.671352	+	0.741139i	$0.734286\pi$
$912$		0			0
$913$	−2.06759e7			−0.820896
$914$		0			0
$915$	9.35642e6	−	6.68102e7i	0.369451	−	2.63809i
$916$		0			0
$917$			3.15144e7i			1.23762i
$918$		0			0
$919$		−	1.72283e7i		−	0.672906i	−0.941700	−	0.336453i	$-0.890773\pi$
$919$		−	1.72283e7i		−	0.672906i	0.941700	−	0.336453i	$-0.109227\pi$
$920$		0			0
$921$	1.50316e6	−	1.07334e7i	0.0583923	−	0.416955i
$922$		0			0
$923$	−1.80137e7			−0.695984
$924$		0			0
$925$	−2.21668e7			−0.851821
$926$		0			0
$927$	−9.23834e6	+	3.23367e7i	−0.353098	+	1.23594i
$928$		0			0
$929$		−	1.18690e7i		−	0.451207i	−0.974219	−	0.225603i	$-0.927565\pi$
$929$		−	1.18690e7i		−	0.451207i	0.974219	−	0.225603i	$-0.0724353\pi$
$930$		0			0
$931$		−	9.44676e6i		−	0.357198i
$932$		0			0
$933$	187409.	+	26245.7i	0.00704835	+	0.000987083i
$934$		0			0
$935$	−1.25951e8			−4.71164
$936$		0			0
$937$	1.80420e7			0.671329			0.335664	−	0.941982i	$-0.391039\pi$
$937$	1.80420e7			0.671329			0.335664	+	0.941982i	$0.391039\pi$
$938$		0			0
$939$	7.87292e6	+	1.10256e6i	0.291388	+	0.0408073i
$940$		0			0
$941$			2.10812e7i			0.776107i	0.921637	+	0.388053i	$0.126852\pi$
$941$			2.10812e7i			0.776107i	−0.921637	+	0.388053i	$0.873148\pi$
$942$		0			0
$943$			4.80915e7i			1.76112i
$944$		0			0
$945$	2.59279e7	+	1.14985e7i	0.944470	+	0.418854i
$946$		0			0
$947$	5.72764e6			0.207539			0.103770	−	0.994601i	$-0.466910\pi$
$947$	5.72764e6			0.207539			0.103770	+	0.994601i	$0.466910\pi$
$948$		0			0
$949$	−1.03484e7			−0.372999
$950$		0			0
$951$	4.22110e6	−	3.01411e7i	0.151347	−	1.08071i
$952$		0			0
$953$			3.09351e7i			1.10336i	0.834054	+	0.551682i	$0.186014\pi$
$953$			3.09351e7i			1.10336i	−0.834054	+	0.551682i	$0.813986\pi$
$954$		0			0
$955$			2.79581e7i			0.991972i
$956$		0			0
$957$	570255.	−	4.07195e6i	0.0201275	−	0.143722i
$958$		0			0
$959$	−1.13715e7			−0.399275
$960$		0			0
$961$	2.23447e7			0.780489
$962$		0			0
$963$	1.13013e7	+	3.22870e6i	0.392703	+	0.112192i
$964$		0			0
$965$			6.38853e7i			2.20842i
$966$		0			0
$967$			3.89231e7i			1.33857i	0.743006	+	0.669285i	$0.233400\pi$
$967$			3.89231e7i			1.33857i	−0.743006	+	0.669285i	$0.766600\pi$
$968$		0			0
$969$	3.77608e7	+	5.28820e6i	1.29191	+	0.180925i
$970$		0			0
$971$	7.31321e6			0.248920			0.124460	−	0.992225i	$-0.460280\pi$
$971$	7.31321e6			0.248920			0.124460	+	0.992225i	$0.460280\pi$
$972$		0			0
$973$	−1.64150e7			−0.555852
$974$		0			0
$975$	3.63490e7	+	5.09049e6i	1.22456	+	0.171494i
$976$		0			0
$977$		−	7.25205e6i		−	0.243066i	−0.992587	−	0.121533i	$-0.961219\pi$
$977$		−	7.25205e6i		−	0.243066i	0.992587	−	0.121533i	$-0.0387810\pi$
$978$		0			0
$979$			1.32998e7i			0.443494i
$980$		0			0
$981$	792447.	+	226396.i	0.0262904	+	0.00751097i
$982$		0			0
$983$	4.95103e7			1.63422			0.817112	−	0.576479i	$-0.195574\pi$
$983$	4.95103e7			1.63422			0.817112	+	0.576479i	$0.195574\pi$
$984$		0			0
$985$	4.47740e7			1.47040
$986$		0			0
$987$	−4.17249e6	+	2.97940e7i	−0.136333	+	0.973499i
$988$		0			0
$989$		−	3.59303e7i		−	1.16807i
$990$		0			0
$991$			1.25306e7i			0.405311i	0.979250	+	0.202655i	$0.0649571\pi$
$991$			1.25306e7i			0.405311i	−0.979250	+	0.202655i	$0.935043\pi$
$992$		0			0
$993$	3.17321e6	−	2.26585e7i	0.102123	−	0.729220i
$994$		0			0
$995$	5.11415e7			1.63763
$996$		0			0
$997$	−1.84608e7			−0.588183			−0.294092	−	0.955777i	$-0.595017\pi$
$997$	−1.84608e7			−0.588183			−0.294092	+	0.955777i	$0.595017\pi$
$998$		0			0
$999$	2.17134e7	+	9.62947e6i	0.688358	+	0.305273i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.6.c.d.383.20	yes	20
3.2	odd	2		384.6.c.a.383.2	yes	20
4.3	odd	2		384.6.c.a.383.1	✓	20
8.3	odd	2		384.6.c.c.383.20	yes	20
8.5	even	2		384.6.c.b.383.1	yes	20
12.11	even	2	inner	384.6.c.d.383.19	yes	20
24.5	odd	2		384.6.c.c.383.19	yes	20
24.11	even	2		384.6.c.b.383.2	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.c.a.383.1	✓	20	4.3	odd	2
384.6.c.a.383.2	yes	20	3.2	odd	2
384.6.c.b.383.1	yes	20	8.5	even	2
384.6.c.b.383.2	yes	20	24.11	even	2
384.6.c.c.383.19	yes	20	24.5	odd	2
384.6.c.c.383.20	yes	20	8.3	odd	2
384.6.c.d.383.19	yes	20	12.11	even	2	inner
384.6.c.d.383.20	yes	20	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	384.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(15.4378 + 2.16198i) q^{3} +81.6092i q^{5} -91.7503i q^{7} +(233.652 + 66.7525i) q^{9} +O(q^{10})\)	\(q+(15.4378 + 2.16198i) q^{3} +81.6092i q^{5} -91.7503i q^{7} +(233.652 + 66.7525i) q^{9} +710.534 q^{11} -666.057 q^{13} +(-176.438 + 1259.87i) q^{15} +2172.09i q^{17} -1126.10i q^{19} +(198.362 - 1416.42i) q^{21} +3120.07 q^{23} -3535.05 q^{25} +(3462.75 + 1535.66i) q^{27} -371.221i q^{29} -2506.88i q^{31} +(10969.1 + 1536.16i) q^{33} +7487.66 q^{35} +6270.56 q^{37} +(-10282.5 - 1440.00i) q^{39} +15413.6i q^{41} -11515.9i q^{43} +(-5447.62 + 19068.1i) q^{45} -21034.7 q^{47} +8388.89 q^{49} +(-4696.01 + 33532.2i) q^{51} +23609.2i q^{53} +57986.1i q^{55} +(2434.62 - 17384.6i) q^{57} +4375.95 q^{59} -53029.6 q^{61} +(6124.56 - 21437.6i) q^{63} -54356.3i q^{65} +24727.7i q^{67} +(48167.0 + 6745.53i) q^{69} +27045.3 q^{71} +15536.8 q^{73} +(-54573.5 - 7642.72i) q^{75} -65191.7i q^{77} +42768.6i q^{79} +(50137.2 + 31193.7i) q^{81} -29099.1 q^{83} -177262. q^{85} +(802.572 - 5730.83i) q^{87} +18718.0i q^{89} +61110.9i q^{91} +(5419.82 - 38700.7i) q^{93} +91900.4 q^{95} -14698.6 q^{97} +(166017. + 47429.9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3}+O(q^{10}) \) 20 * q + 2 * q^3	\( 20 q + 2 q^{3} + 948 q^{11} + 852 q^{15} - 1640 q^{21} + 328 q^{23} - 12500 q^{25} + 2030 q^{27} + 2836 q^{33} - 7184 q^{35} - 15056 q^{37} - 12980 q^{39} - 11800 q^{45} + 36640 q^{47} - 33388 q^{49} + 1936 q^{51} + 15404 q^{57} + 62908 q^{59} - 73264 q^{61} + 23608 q^{63} + 84024 q^{69} + 34888 q^{71} + 52568 q^{73} + 115698 q^{75} + 55444 q^{81} - 225172 q^{83} + 30112 q^{85} - 225700 q^{87} + 148016 q^{93} + 418616 q^{95} + 7600 q^{97} + 378260 q^{99}+O(q^{100}) \) 20 * q + 2 * q^3 + 948 * q^11 + 852 * q^15 - 1640 * q^21 + 328 * q^23 - 12500 * q^25 + 2030 * q^27 + 2836 * q^33 - 7184 * q^35 - 15056 * q^37 - 12980 * q^39 - 11800 * q^45 + 36640 * q^47 - 33388 * q^49 + 1936 * q^51 + 15404 * q^57 + 62908 * q^59 - 73264 * q^61 + 23608 * q^63 + 84024 * q^69 + 34888 * q^71 + 52568 * q^73 + 115698 * q^75 + 55444 * q^81 - 225172 * q^83 + 30112 * q^85 - 225700 * q^87 + 148016 * q^93 + 418616 * q^95 + 7600 * q^97 + 378260 * q^99

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