LMFDB - Embedded newform 400.6.n.g.143.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("400.143");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.n (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
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} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + \cdots + 57\!\cdots\!24 $ x^20 + 271x^18 + 109637x^16 + 25993614x^14 + 5522961902x^12 + 881545050522x^10 + 133816049059481x^8 + 14779507781220031x^6 + 824105698447750789x^4 + 12044868290803250652*x^2 + 579398322543528055824
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{67}\cdot 3^{4}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.1
Root			$-11.4741 - 7.80740i$ of defining polynomial
Character	$\chi$	$=$	400.143
Dual form			400.6.n.g.207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-20.3843 + 20.3843i) q^{3} +(-76.9082 - 76.9082i) q^{7} -588.037i q^{9} +O(q^{10})$	$q+(-20.3843 + 20.3843i) q^{3} +(-76.9082 - 76.9082i) q^{7} -588.037i q^{9} -556.846i q^{11} +(-141.317 - 141.317i) q^{13} +(477.013 - 477.013i) q^{17} -1608.20 q^{19} +3135.44 q^{21} +(-346.617 + 346.617i) q^{23} +(7033.34 + 7033.34i) q^{27} -7486.67i q^{29} +7927.33i q^{31} +(11350.9 + 11350.9i) q^{33} +(-3329.26 + 3329.26i) q^{37} +5761.27 q^{39} +18717.9 q^{41} +(-8253.12 + 8253.12i) q^{43} +(-5098.15 - 5098.15i) q^{47} -4977.25i q^{49} +19447.1i q^{51} +(-19488.7 - 19488.7i) q^{53} +(32782.1 - 32782.1i) q^{57} -108.893 q^{59} +14287.1 q^{61} +(-45224.9 + 45224.9i) q^{63} +(-28986.5 - 28986.5i) q^{67} -14131.1i q^{69} -982.591i q^{71} +(21571.6 + 21571.6i) q^{73} +(-42826.0 + 42826.0i) q^{77} +9383.55 q^{79} -143846. q^{81} +(9451.97 - 9451.97i) q^{83} +(152610. + 152610. i) q^{87} +8489.66i q^{89} +21736.8i q^{91} +(-161593. - 161593. i) q^{93} +(-122282. + 122282. i) q^{97} -327446. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q - 804 q^{13} + 2236 q^{17} - 4520 q^{21} + 11096 q^{33} - 44260 q^{37} - 6760 q^{41} - 182452 q^{53} + 34288 q^{57} - 41080 q^{61} - 264372 q^{73} - 399304 q^{77} - 520220 q^{81} - 713496 q^{93} - 374772 q^{97}+O(q^{100}) $ 20 * q - 804 * q^13 + 2236 * q^17 - 4520 * q^21 + 11096 * q^33 - 44260 * q^37 - 6760 * q^41 - 182452 * q^53 + 34288 * q^57 - 41080 * q^61 - 264372 * q^73 - 399304 * q^77 - 520220 * q^81 - 713496 * q^93 - 374772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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
$3$	−20.3843	+	20.3843i	−1.30765	+	1.30765i	−0.384546	+	0.923106i	$0.625642\pi$
$3$	−20.3843	+	20.3843i	−1.30765	+	1.30765i	−0.923106	+	0.384546i	$0.874358\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−76.9082	−	76.9082i	−0.593236	−	0.593236i	0.345268	−	0.938504i	$-0.387788\pi$
$7$	−76.9082	−	76.9082i	−0.593236	−	0.593236i	−0.938504	+	0.345268i	$0.887788\pi$
$8$		0			0
$9$		−	588.037i		−	2.41991i
$10$		0			0
$11$		−	556.846i		−	1.38756i	−0.720185	−	0.693782i	$-0.755943\pi$
$11$		−	556.846i		−	1.38756i	0.720185	−	0.693782i	$-0.244057\pi$
$12$		0			0
$13$	−141.317	−	141.317i	−0.231918	−	0.231918i	0.581575	−	0.813493i	$-0.302437\pi$
$13$	−141.317	−	141.317i	−0.231918	−	0.231918i	−0.813493	+	0.581575i	$0.802437\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	477.013	−	477.013i	0.400320	−	0.400320i	−0.478026	−	0.878346i	$-0.658647\pi$
$17$	477.013	−	477.013i	0.400320	−	0.400320i	0.878346	+	0.478026i	$0.158647\pi$
$18$		0			0
$19$	−1608.20			−1.02201			−0.511007	−	0.859577i	$-0.670727\pi$
$19$	−1608.20			−1.02201			−0.511007	+	0.859577i	$0.670727\pi$
$20$		0			0
$21$	3135.44			1.55149
$22$		0			0
$23$	−346.617	+	346.617i	−0.136625	+	0.136625i	−0.772112	−	0.635487i	$-0.780799\pi$
$23$	−346.617	+	346.617i	−0.136625	+	0.136625i	0.635487	+	0.772112i	$0.280799\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	7033.34	+	7033.34i	1.85674	+	1.85674i
$28$		0			0
$29$		−	7486.67i		−	1.65308i	−0.562879	−	0.826539i	$-0.690306\pi$
$29$		−	7486.67i		−	1.65308i	0.562879	−	0.826539i	$-0.309694\pi$
$30$		0			0
$31$			7927.33i			1.48157i	0.671742	+	0.740786i	$0.265547\pi$
$31$			7927.33i			1.48157i	−0.671742	+	0.740786i	$0.734453\pi$
$32$		0			0
$33$	11350.9	+	11350.9i	1.81445	+	1.81445i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3329.26	+	3329.26i	−0.399801	+	0.399801i	−0.878163	−	0.478362i	$-0.841231\pi$
$37$	−3329.26	+	3329.26i	−0.399801	+	0.399801i	0.478362	+	0.878163i	$0.341231\pi$
$38$		0			0
$39$	5761.27			0.606536
$40$		0			0
$41$	18717.9			1.73899			0.869497	−	0.493938i	$-0.164443\pi$
$41$	18717.9			1.73899			0.869497	+	0.493938i	$0.164443\pi$
$42$		0			0
$43$	−8253.12	+	8253.12i	−0.680686	+	0.680686i	−0.960155	−	0.279468i	$-0.909842\pi$
$43$	−8253.12	+	8253.12i	−0.680686	+	0.680686i	0.279468	+	0.960155i	$0.409842\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5098.15	−	5098.15i	−0.336642	−	0.336642i	0.518460	−	0.855102i	$-0.326506\pi$
$47$	−5098.15	−	5098.15i	−0.336642	−	0.336642i	−0.855102	+	0.518460i	$0.826506\pi$
$48$		0			0
$49$		−	4977.25i		−	0.296141i
$50$		0			0
$51$			19447.1i			1.04696i
$52$		0			0
$53$	−19488.7	−	19488.7i	−0.953001	−	0.953001i	0.0459428	−	0.998944i	$-0.485371\pi$
$53$	−19488.7	−	19488.7i	−0.953001	−	0.953001i	−0.998944	+	0.0459428i	$0.985371\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	32782.1	−	32782.1i	1.33644	−	1.33644i
$58$		0			0
$59$	−108.893			−0.00407257			−0.00203628	−	0.999998i	$-0.500648\pi$
$59$	−108.893			−0.00407257			−0.00203628	+	0.999998i	$0.500648\pi$
$60$		0			0
$61$	14287.1			0.491608			0.245804	−	0.969320i	$-0.420948\pi$
$61$	14287.1			0.491608			0.245804	+	0.969320i	$0.420948\pi$
$62$		0			0
$63$	−45224.9	+	45224.9i	−1.43558	+	1.43558i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−28986.5	−	28986.5i	−0.788875	−	0.788875i	0.192435	−	0.981310i	$-0.438362\pi$
$67$	−28986.5	−	28986.5i	−0.788875	−	0.788875i	−0.981310	+	0.192435i	$0.938362\pi$
$68$		0			0
$69$		−	14131.1i		−	0.357316i
$70$		0			0
$71$		−	982.591i		−	0.0231327i	−0.999933	−	0.0115664i	$-0.996318\pi$
$71$		−	982.591i		−	0.0231327i	0.999933	−	0.0115664i	$-0.00368177\pi$
$72$		0			0
$73$	21571.6	+	21571.6i	0.473778	+	0.473778i	0.903135	−	0.429357i	$-0.141260\pi$
$73$	21571.6	+	21571.6i	0.473778	+	0.473778i	−0.429357	+	0.903135i	$0.641260\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−42826.0	+	42826.0i	−0.823154	+	0.823154i
$78$		0			0
$79$	9383.55			0.169161			0.0845803	−	0.996417i	$-0.473045\pi$
$79$	9383.55			0.169161			0.0845803	+	0.996417i	$0.473045\pi$
$80$		0			0
$81$	−143846.			−2.43604
$82$		0			0
$83$	9451.97	−	9451.97i	0.150601	−	0.150601i	−0.627786	−	0.778386i	$-0.716039\pi$
$83$	9451.97	−	9451.97i	0.150601	−	0.150601i	0.778386	+	0.627786i	$0.216039\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	152610.	+	152610.i	2.16165	+	2.16165i
$88$		0			0
$89$			8489.66i			0.113610i	0.998385	+	0.0568049i	$0.0180913\pi$
$89$			8489.66i			0.113610i	−0.998385	+	0.0568049i	$0.981909\pi$
$90$		0			0
$91$			21736.8i			0.275165i
$92$		0			0
$93$	−161593.	−	161593.i	−1.93738	−	1.93738i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−122282.	+	122282.i	−1.31957	+	1.31957i	−0.405453	+	0.914116i	$0.632886\pi$
$97$	−122282.	+	122282.i	−1.31957	+	1.31957i	−0.914116	+	0.405453i	$0.867114\pi$
$98$		0			0
$99$	−327446.			−3.35778
$100$		0			0
$101$	−49322.0			−0.481102			−0.240551	−	0.970636i	$-0.577328\pi$
$101$	−49322.0			−0.481102			−0.240551	+	0.970636i	$0.577328\pi$
$102$		0			0
$103$	63162.5	−	63162.5i	0.586633	−	0.586633i	−0.350085	−	0.936718i	$-0.613847\pi$
$103$	63162.5	−	63162.5i	0.586633	−	0.586633i	0.936718	+	0.350085i	$0.113847\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−25175.2	−	25175.2i	−0.212575	−	0.212575i	0.592785	−	0.805361i	$-0.298028\pi$
$107$	−25175.2	−	25175.2i	−0.212575	−	0.212575i	−0.805361	+	0.592785i	$0.798028\pi$
$108$		0			0
$109$			79771.5i			0.643104i	0.946892	+	0.321552i	$0.104205\pi$
$109$			79771.5i			0.643104i	−0.946892	+	0.321552i	$0.895795\pi$
$110$		0			0
$111$		−	135729.i		−	1.04560i
$112$		0			0
$113$	−93876.0	−	93876.0i	−0.691606	−	0.691606i	0.270980	−	0.962585i	$-0.412652\pi$
$113$	−93876.0	−	93876.0i	−0.691606	−	0.691606i	−0.962585	+	0.270980i	$0.912652\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−83099.4	+	83099.4i	−0.561220	+	0.561220i
$118$		0			0
$119$	−73372.4			−0.474969
$120$		0			0
$121$	−149026.			−0.925336
$122$		0			0
$123$	−381551.	+	381551.i	−2.27400	+	2.27400i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−5616.82	−	5616.82i	−0.0309016	−	0.0309016i	0.691487	−	0.722389i	$-0.256956\pi$
$127$	−5616.82	−	5616.82i	−0.0309016	−	0.0309016i	−0.722389	+	0.691487i	$0.756956\pi$
$128$		0			0
$129$		−	336468.i		−	1.78020i
$130$		0			0
$131$			56715.0i			0.288749i	0.989523	+	0.144374i	$0.0461169\pi$
$131$			56715.0i			0.288749i	−0.989523	+	0.144374i	$0.953883\pi$
$132$		0			0
$133$	123684.	+	123684.i	0.606296	+	0.606296i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	95096.7	−	95096.7i	0.432877	−	0.432877i	−0.456729	−	0.889606i	$-0.650979\pi$
$137$	95096.7	−	95096.7i	0.432877	−	0.432877i	0.889606	+	0.456729i	$0.150979\pi$
$138$		0			0
$139$	−276110.			−1.21212			−0.606059	−	0.795420i	$-0.707251\pi$
$139$	−276110.			−1.21212			−0.606059	+	0.795420i	$0.707251\pi$
$140$		0			0
$141$	207844.			0.880421
$142$		0			0
$143$	−78691.5	+	78691.5i	−0.321801	+	0.321801i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	101458.	+	101458.i	0.387250	+	0.387250i
$148$		0			0
$149$			258844.i			0.955153i	0.878590	+	0.477577i	$0.158485\pi$
$149$			258844.i			0.955153i	−0.878590	+	0.477577i	$0.841515\pi$
$150$		0			0
$151$			399655.i			1.42640i	0.700958	+	0.713202i	$0.252756\pi$
$151$			399655.i			1.42640i	−0.700958	+	0.713202i	$0.747244\pi$
$152$		0			0
$153$	−280501.	−	280501.i	−0.968738	−	0.968738i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	174590.	−	174590.i	0.565289	−	0.565289i	−0.365516	−	0.930805i	$-0.619107\pi$
$157$	174590.	−	174590.i	0.565289	−	0.565289i	0.930805	+	0.365516i	$0.119107\pi$
$158$		0			0
$159$	794527.			2.49239
$160$		0			0
$161$	53315.4			0.162102
$162$		0			0
$163$	−1364.52	+	1364.52i	−0.00402264	+	0.00402264i	−0.709115	−	0.705093i	$-0.750905\pi$
$163$	−1364.52	+	1364.52i	−0.00402264	+	0.00402264i	0.705093	+	0.709115i	$0.250905\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	323133.	+	323133.i	0.896581	+	0.896581i	0.995132	−	0.0985507i	$-0.0314207\pi$
$167$	323133.	+	323133.i	0.896581	+	0.896581i	−0.0985507	+	0.995132i	$0.531421\pi$
$168$		0			0
$169$		−	331352.i		−	0.892428i
$170$		0			0
$171$			945684.i			2.47318i
$172$		0			0
$173$	173917.	+	173917.i	0.441800	+	0.441800i	0.892617	−	0.450816i	$-0.148867\pi$
$173$	173917.	+	173917.i	0.441800	+	0.441800i	−0.450816	+	0.892617i	$0.648867\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	2219.70	−	2219.70i	0.00532550	−	0.00532550i
$178$		0			0
$179$	60036.1			0.140049			0.0700245	−	0.997545i	$-0.477692\pi$
$179$	60036.1			0.140049			0.0700245	+	0.997545i	$0.477692\pi$
$180$		0			0
$181$	−71086.8			−0.161285			−0.0806423	−	0.996743i	$-0.525697\pi$
$181$	−71086.8			−0.161285			−0.0806423	+	0.996743i	$0.525697\pi$
$182$		0			0
$183$	−291232.	+	291232.i	−0.642852	+	0.642852i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−265622.	−	265622.i	−0.555470	−	0.555470i
$188$		0			0
$189$		−	1.08184e6i		−	2.20298i
$190$		0			0
$191$		−	898374.i		−	1.78186i	−0.454139	−	0.890931i	$-0.650053\pi$
$191$		−	898374.i		−	1.78186i	0.454139	−	0.890931i	$-0.349947\pi$
$192$		0			0
$193$	514879.	+	514879.i	0.994974	+	0.994974i	0.999987	−	0.00501311i	$-0.00159573\pi$
$193$	514879.	+	514879.i	0.994974	+	0.994974i	−0.00501311	+	0.999987i	$0.501596\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−320182.	+	320182.i	−0.587802	+	0.587802i	−0.937036	−	0.349234i	$-0.886442\pi$
$197$	−320182.	+	320182.i	−0.587802	+	0.587802i	0.349234	+	0.937036i	$0.386442\pi$
$198$		0			0
$199$	−458886.			−0.821433			−0.410717	−	0.911763i	$-0.634721\pi$
$199$	−458886.			−0.821433			−0.410717	+	0.911763i	$0.634721\pi$
$200$		0			0
$201$	1.18174e6			2.06315
$202$		0			0
$203$	−575786.	+	575786.i	−0.980666	+	0.980666i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	203824.	+	203824.i	0.330620	+	0.330620i
$208$		0			0
$209$			895521.i			1.41811i
$210$		0			0
$211$			561846.i			0.868783i	0.900724	+	0.434391i	$0.143037\pi$
$211$			561846.i			0.868783i	−0.900724	+	0.434391i	$0.856963\pi$
$212$		0			0
$213$	20029.4	+	20029.4i	0.0302495	+	0.0302495i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	609677.	−	609677.i	0.878922	−	0.878922i
$218$		0			0
$219$	−879443.			−1.23907
$220$		0			0
$221$	−134820.			−0.185683
$222$		0			0
$223$	821427.	−	821427.i	1.10613	−	1.10613i	0.112478	−	0.993654i	$-0.464121\pi$
$223$	821427.	−	821427.i	1.10613	−	1.10613i	0.993654	−	0.112478i	$-0.0358787\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−276085.	−	276085.i	−0.355613	−	0.355613i	0.506580	−	0.862193i	$-0.330910\pi$
$227$	−276085.	−	276085.i	−0.355613	−	0.355613i	−0.862193	+	0.506580i	$0.830910\pi$
$228$		0			0
$229$			503195.i			0.634085i	0.948411	+	0.317043i	$0.102690\pi$
$229$			503195.i			0.634085i	−0.948411	+	0.317043i	$0.897310\pi$
$230$		0			0
$231$		−	1.74596e6i		−	2.15280i
$232$		0			0
$233$	−594823.	−	594823.i	−0.717790	−	0.717790i	0.250362	−	0.968152i	$-0.419450\pi$
$233$	−594823.	−	594823.i	−0.717790	−	0.717790i	−0.968152	+	0.250362i	$0.919450\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−191277.	+	191277.i	−0.221203	+	0.221203i
$238$		0			0
$239$	−348307.			−0.394428			−0.197214	−	0.980360i	$-0.563189\pi$
$239$	−348307.			−0.394428			−0.197214	+	0.980360i	$0.563189\pi$
$240$		0			0
$241$	252895.			0.280477			0.140239	−	0.990118i	$-0.455213\pi$
$241$	252895.			0.280477			0.140239	+	0.990118i	$0.455213\pi$
$242$		0			0
$243$	1.22309e6	−	1.22309e6i	1.32875	−	1.32875i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	227266.	+	227266.i	0.237024	+	0.237024i
$248$		0			0
$249$			385343.i			0.393867i
$250$		0			0
$251$		−	493537.i		−	0.494465i	−0.968956	−	0.247232i	$-0.920479\pi$
$251$		−	493537.i		−	0.494465i	0.968956	−	0.247232i	$-0.0795211\pi$
$252$		0			0
$253$	193012.	+	193012.i	0.189576	+	0.189576i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.28934e6	+	1.28934e6i	−1.21768	+	1.21768i	−0.249242	+	0.968441i	$0.580181\pi$
$257$	−1.28934e6	+	1.28934e6i	−1.21768	+	1.21768i	−0.968441	+	0.249242i	$0.919819\pi$
$258$		0			0
$259$	512095.			0.474352
$260$		0			0
$261$	−4.40244e6			−4.00030
$262$		0			0
$263$	562452.	−	562452.i	0.501414	−	0.501414i	−0.410463	−	0.911877i	$-0.634633\pi$
$263$	562452.	−	562452.i	0.501414	−	0.501414i	0.911877	+	0.410463i	$0.134633\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−173056.	−	173056.i	−0.148562	−	0.148562i
$268$		0			0
$269$			305884.i			0.257737i	0.991662	+	0.128868i	$0.0411345\pi$
$269$			305884.i			0.257737i	−0.991662	+	0.128868i	$0.958866\pi$
$270$		0			0
$271$			1.98399e6i			1.64103i	0.571626	+	0.820514i	$0.306313\pi$
$271$			1.98399e6i			1.64103i	−0.571626	+	0.820514i	$0.693687\pi$
$272$		0			0
$273$	−443089.	−	443089.i	−0.359819	−	0.359819i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−669845.	+	669845.i	−0.524536	+	0.524536i	−0.918938	−	0.394402i	$-0.870952\pi$
$277$	−669845.	+	669845.i	−0.524536	+	0.524536i	0.394402	+	0.918938i	$0.370952\pi$
$278$		0			0
$279$	4.66157e6			3.58526
$280$		0			0
$281$	374638.			0.283039			0.141520	−	0.989935i	$-0.454801\pi$
$281$	374638.			0.283039			0.141520	+	0.989935i	$0.454801\pi$
$282$		0			0
$283$	894492.	−	894492.i	0.663912	−	0.663912i	−0.292388	−	0.956300i	$-0.594450\pi$
$283$	894492.	−	894492.i	0.663912	−	0.663912i	0.956300	+	0.292388i	$0.0944498\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.43956e6	−	1.43956e6i	−1.03163	−	1.03163i
$288$		0			0
$289$			964775.i			0.679488i
$290$		0			0
$291$		−	4.98525e6i		−	3.45107i
$292$		0			0
$293$	788207.	+	788207.i	0.536379	+	0.536379i	0.922463	−	0.386085i	$-0.126173\pi$
$293$	788207.	+	788207.i	0.536379	+	0.536379i	−0.386085	+	0.922463i	$0.626173\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	3.91649e6	−	3.91649e6i	2.57635	−	2.57635i
$298$		0			0
$299$	97965.5			0.0633717
$300$		0			0
$301$	1.26947e6			0.807616
$302$		0			0
$303$	1.00539e6	−	1.00539e6i	0.629114	−	0.629114i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−2.03742e6	−	2.03742e6i	−1.23377	−	1.23377i	−0.962505	−	0.271264i	$-0.912558\pi$
$307$	−2.03742e6	−	2.03742e6i	−1.23377	−	1.23377i	−0.271264	−	0.962505i	$-0.587442\pi$
$308$		0			0
$309$			2.57504e6i			1.53422i
$310$		0			0
$311$			2.55652e6i			1.49881i	0.662109	+	0.749407i	$0.269661\pi$
$311$			2.55652e6i			1.49881i	−0.662109	+	0.749407i	$0.730339\pi$
$312$		0			0
$313$	−985823.	−	985823.i	−0.568772	−	0.568772i	0.363012	−	0.931784i	$-0.381748\pi$
$313$	−985823.	−	985823.i	−0.568772	−	0.568772i	−0.931784	+	0.363012i	$0.881748\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	113582.	−	113582.i	0.0634836	−	0.0634836i	−0.674652	−	0.738136i	$-0.735706\pi$
$317$	113582.	−	113582.i	0.0634836	−	0.0634836i	0.738136	+	0.674652i	$0.235706\pi$
$318$		0			0
$319$	−4.16892e6			−2.29375
$320$		0			0
$321$	1.02636e6			0.555949
$322$		0			0
$323$	−767133.	+	767133.i	−0.409133	+	0.409133i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−1.62608e6	−	1.62608e6i	−0.840956	−	0.840956i
$328$		0			0
$329$			784180.i			0.399417i
$330$		0			0
$331$			327502.i			0.164303i	0.996620	+	0.0821513i	$0.0261791\pi$
$331$			327502.i			0.164303i	−0.996620	+	0.0821513i	$0.973821\pi$
$332$		0			0
$333$	1.95773e6	+	1.95773e6i	0.967480	+	0.967480i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.20574e6	+	1.20574e6i	−0.578335	+	0.578335i	−0.934444	−	0.356110i	$-0.884103\pi$
$337$	−1.20574e6	+	1.20574e6i	−0.578335	+	0.578335i	0.356110	+	0.934444i	$0.384103\pi$
$338$		0			0
$339$	3.82719e6			1.80876
$340$		0			0
$341$	4.41430e6			2.05578
$342$		0			0
$343$	−1.67539e6	+	1.67539e6i	−0.768918	+	0.768918i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.84834e6	+	1.84834e6i	0.824058	+	0.824058i	0.986687	−	0.162629i	$-0.0519975\pi$
$347$	1.84834e6	+	1.84834e6i	0.824058	+	0.824058i	−0.162629	+	0.986687i	$0.551998\pi$
$348$		0			0
$349$		−	28570.5i		−	0.0125561i	−0.999980	−	0.00627804i	$-0.998002\pi$
$349$		−	28570.5i		−	0.0125561i	0.999980	−	0.00627804i	$-0.00199838\pi$
$350$		0			0
$351$		−	1.98785e6i		−	0.861225i
$352$		0			0
$353$	1.77770e6	+	1.77770e6i	0.759316	+	0.759316i	0.976198	−	0.216882i	$-0.0695887\pi$
$353$	1.77770e6	+	1.77770e6i	0.759316	+	0.759316i	−0.216882	+	0.976198i	$0.569589\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	1.49564e6	−	1.49564e6i	0.621094	−	0.621094i
$358$		0			0
$359$	736136.			0.301454			0.150727	−	0.988575i	$-0.451838\pi$
$359$	736136.			0.301454			0.150727	+	0.988575i	$0.451838\pi$
$360$		0			0
$361$	110218.			0.0445130
$362$		0			0
$363$	3.03779e6	−	3.03779e6i	1.21002	−	1.21002i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	477935.	+	477935.i	0.185227	+	0.185227i	0.793629	−	0.608402i	$-0.208189\pi$
$367$	477935.	+	477935.i	0.185227	+	0.185227i	−0.608402	+	0.793629i	$0.708189\pi$
$368$		0			0
$369$		−	1.10068e7i		−	4.20820i
$370$		0			0
$371$			2.99769e6i			1.13071i
$372$		0			0
$373$	3.22422e6	+	3.22422e6i	1.19992	+	1.19992i	0.974189	+	0.225732i	$0.0724774\pi$
$373$	3.22422e6	+	3.22422e6i	1.19992	+	1.19992i	0.225732	+	0.974189i	$0.427523\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.05799e6	+	1.05799e6i	−0.383379	+	0.383379i
$378$		0			0
$379$	4.24297e6			1.51730			0.758652	−	0.651496i	$-0.225858\pi$
$379$	4.24297e6			1.51730			0.758652	+	0.651496i	$0.225858\pi$
$380$		0			0
$381$	228990.			0.0808171
$382$		0			0
$383$	654083.	−	654083.i	0.227843	−	0.227843i	−0.583948	−	0.811791i	$-0.698493\pi$
$383$	654083.	−	654083.i	0.227843	−	0.227843i	0.811791	+	0.583948i	$0.198493\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.85314e6	+	4.85314e6i	1.64720	+	1.64720i
$388$		0			0
$389$			2.46468e6i			0.825823i	0.910771	+	0.412911i	$0.135488\pi$
$389$			2.46468e6i			0.825823i	−0.910771	+	0.412911i	$0.864512\pi$
$390$		0			0
$391$			330681.i			0.109388i
$392$		0			0
$393$	−1.15609e6	−	1.15609e6i	−0.377583	−	0.377583i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−4.06922e6	+	4.06922e6i	−1.29579	+	1.29579i	−0.364645	+	0.931147i	$0.618810\pi$
$397$	−4.06922e6	+	4.06922e6i	−1.29579	+	1.29579i	−0.931147	+	0.364645i	$0.881190\pi$
$398$		0			0
$399$	−5.04242e6			−1.58565
$400$		0			0
$401$	3.14549e6			0.976849			0.488424	−	0.872606i	$-0.337572\pi$
$401$	3.14549e6			0.976849			0.488424	+	0.872606i	$0.337572\pi$
$402$		0			0
$403$	1.12026e6	−	1.12026e6i	0.343603	−	0.343603i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.85388e6	+	1.85388e6i	0.554749	+	0.554749i
$408$		0			0
$409$			4.42634e6i			1.30839i	0.756327	+	0.654194i	$0.226992\pi$
$409$			4.42634e6i			1.30839i	−0.756327	+	0.654194i	$0.773008\pi$
$410$		0			0
$411$			3.87696e6i			1.13210i
$412$		0			0
$413$	8374.74	+	8374.74i	0.00241600	+	0.00241600i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	5.62830e6	−	5.62830e6i	1.58503	−	1.58503i
$418$		0			0
$419$	−2.19963e6			−0.612090			−0.306045	−	0.952017i	$-0.599006\pi$
$419$	−2.19963e6			−0.612090			−0.306045	+	0.952017i	$0.599006\pi$
$420$		0			0
$421$	1240.34			0.000341063			0.000170532	−	1.00000i	$-0.499946\pi$
$421$	1240.34			0.000341063			0.000170532		1.00000i	$0.499946\pi$
$422$		0			0
$423$	−2.99791e6	+	2.99791e6i	−0.814642	+	0.814642i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.09879e6	−	1.09879e6i	−0.291640	−	0.291640i
$428$		0			0
$429$		−	3.20814e6i		−	0.841609i
$430$		0			0
$431$		−	1.29826e6i		−	0.336642i	−0.985732	−	0.168321i	$-0.946166\pi$
$431$		−	1.29826e6i		−	0.336642i	0.985732	−	0.168321i	$-0.0538344\pi$
$432$		0			0
$433$	−665935.	−	665935.i	−0.170691	−	0.170691i	0.616592	−	0.787283i	$-0.288513\pi$
$433$	−665935.	−	665935.i	−0.170691	−	0.170691i	−0.787283	+	0.616592i	$0.788513\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	557431.	−	557431.i	0.139633	−	0.139633i
$438$		0			0
$439$	−3.73469e6			−0.924898			−0.462449	−	0.886646i	$-0.653029\pi$
$439$	−3.73469e6			−0.924898			−0.462449	+	0.886646i	$0.653029\pi$
$440$		0			0
$441$	−2.92681e6			−0.716634
$442$		0			0
$443$	−2.91756e6	+	2.91756e6i	−0.706336	+	0.706336i	−0.965763	−	0.259427i	$-0.916466\pi$
$443$	−2.91756e6	+	2.91756e6i	−0.706336	+	0.706336i	0.259427	+	0.965763i	$0.416466\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−5.27635e6	−	5.27635e6i	−1.24901	−	1.24901i
$448$		0			0
$449$		−	992830.i		−	0.232412i	−0.993225	−	0.116206i	$-0.962927\pi$
$449$		−	992830.i		−	0.232412i	0.993225	−	0.116206i	$-0.0370733\pi$
$450$		0			0
$451$		−	1.04230e7i		−	2.41297i
$452$		0			0
$453$	−8.14667e6	−	8.14667e6i	−1.86524	−	1.86524i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.19086e6	−	4.19086e6i	0.938670	−	0.938670i	−0.0595554	−	0.998225i	$-0.518968\pi$
$457$	4.19086e6	−	4.19086e6i	0.938670	−	0.938670i	0.998225	+	0.0595554i	$0.0189683\pi$
$458$		0			0
$459$	6.70998e6			1.48658
$460$		0			0
$461$	−5.83166e6			−1.27803			−0.639013	−	0.769196i	$-0.720657\pi$
$461$	−5.83166e6			−1.27803			−0.639013	+	0.769196i	$0.720657\pi$
$462$		0			0
$463$	−4.17783e6	+	4.17783e6i	−0.905728	+	0.905728i	−0.995924	−	0.0901957i	$-0.971251\pi$
$463$	−4.17783e6	+	4.17783e6i	−0.905728	+	0.905728i	0.0901957	+	0.995924i	$0.471251\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.18280e6	−	2.18280e6i	−0.463150	−	0.463150i	0.436537	−	0.899686i	$-0.356205\pi$
$467$	−2.18280e6	−	2.18280e6i	−0.463150	−	0.463150i	−0.899686	+	0.436537i	$0.856205\pi$
$468$		0			0
$469$			4.45859e6i			0.935978i
$470$		0			0
$471$			7.11779e6i			1.47840i
$472$		0			0
$473$	4.59572e6	+	4.59572e6i	0.944497	+	0.944497i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.14601e7	+	1.14601e7i	−2.30617	+	2.30617i
$478$		0			0
$479$	−1.70781e6			−0.340095			−0.170047	−	0.985436i	$-0.554392\pi$
$479$	−1.70781e6			−0.340095			−0.170047	+	0.985436i	$0.554392\pi$
$480$		0			0
$481$	940959.			0.185442
$482$		0			0
$483$	−1.08680e6	+	1.08680e6i	−0.211973	+	0.211973i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−3.89197e6	−	3.89197e6i	−0.743613	−	0.743613i	0.229658	−	0.973271i	$-0.426239\pi$
$487$	−3.89197e6	−	3.89197e6i	−0.743613	−	0.743613i	−0.973271	+	0.229658i	$0.926239\pi$
$488$		0			0
$489$		−	55629.6i		−	0.0105204i
$490$		0			0
$491$			3.79408e6i			0.710236i	0.934821	+	0.355118i	$0.115559\pi$
$491$			3.79408e6i			0.710236i	−0.934821	+	0.355118i	$0.884441\pi$
$492$		0			0
$493$	−3.57123e6	−	3.57123e6i	−0.661761	−	0.661761i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−75569.3	+	75569.3i	−0.0137232	+	0.0137232i
$498$		0			0
$499$	−5.45318e6			−0.980388			−0.490194	−	0.871613i	$-0.663074\pi$
$499$	−5.45318e6			−0.980388			−0.490194	+	0.871613i	$0.663074\pi$
$500$		0			0
$501$	−1.31737e7			−2.34483
$502$		0			0
$503$	5.05453e6	−	5.05453e6i	0.890760	−	0.890760i	−0.103834	−	0.994595i	$-0.533111\pi$
$503$	5.05453e6	−	5.05453e6i	0.890760	−	0.890760i	0.994595	+	0.103834i	$0.0331112\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	6.75438e6	+	6.75438e6i	1.16699	+	1.16699i
$508$		0			0
$509$			5.72886e6i			0.980108i	0.871692	+	0.490054i	$0.163023\pi$
$509$			5.72886e6i			0.980108i	−0.871692	+	0.490054i	$0.836977\pi$
$510$		0			0
$511$		−	3.31807e6i		−	0.562125i
$512$		0			0
$513$	−1.13110e7	−	1.13110e7i	−1.89762	−	1.89762i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−2.83889e6	+	2.83889e6i	−0.467113	+	0.467113i
$518$		0			0
$519$	−7.09033e6			−1.15544
$520$		0			0
$521$	−6.55438e6			−1.05788			−0.528941	−	0.848659i	$-0.677411\pi$
$521$	−6.55438e6			−1.05788			−0.528941	+	0.848659i	$0.677411\pi$
$522$		0			0
$523$	−7.16908e6	+	7.16908e6i	−1.14606	+	1.14606i	−0.158745	+	0.987320i	$0.550745\pi$
$523$	−7.16908e6	+	7.16908e6i	−1.14606	+	1.14606i	−0.987320	+	0.158745i	$0.949255\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.78143e6	+	3.78143e6i	0.593103	+	0.593103i
$528$		0			0
$529$			6.19606e6i			0.962667i
$530$		0			0
$531$			64033.0i			0.00985524i
$532$		0			0
$533$	−2.64515e6	−	2.64515e6i	−0.403304	−	0.403304i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.22379e6	+	1.22379e6i	−0.183135	+	0.183135i
$538$		0			0
$539$	−2.77156e6			−0.410915
$540$		0			0
$541$	47552.7			0.00698526			0.00349263	−	0.999994i	$-0.498888\pi$
$541$	47552.7			0.00698526			0.00349263	+	0.999994i	$0.498888\pi$
$542$		0			0
$543$	1.44905e6	−	1.44905e6i	0.210904	−	0.210904i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	4.55854e6	+	4.55854e6i	0.651415	+	0.651415i	0.953334	−	0.301919i	$-0.0976272\pi$
$547$	4.55854e6	+	4.55854e6i	0.651415	+	0.651415i	−0.301919	+	0.953334i	$0.597627\pi$
$548$		0			0
$549$		−	8.40134e6i		−	1.18965i
$550$		0			0
$551$			1.20401e7i			1.68947i
$552$		0			0
$553$	−721672.	−	721672.i	−0.100352	−	0.100352i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.11785e6	+	3.11785e6i	−0.425811	+	0.425811i	−0.887199	−	0.461388i	$-0.847352\pi$
$557$	−3.11785e6	+	3.11785e6i	−0.425811	+	0.425811i	0.461388	+	0.887199i	$0.347352\pi$
$558$		0			0
$559$	2.33261e6			0.315727
$560$		0			0
$561$	1.08290e7			1.45272
$562$		0			0
$563$	8.10403e6	−	8.10403e6i	1.07753	−	1.07753i	0.0808018	−	0.996730i	$-0.474252\pi$
$563$	8.10403e6	−	8.10403e6i	1.07753	−	1.07753i	0.996730	−	0.0808018i	$-0.0257481\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.10629e7	+	1.10629e7i	1.44515	+	1.44515i
$568$		0			0
$569$		−	3.79477e6i		−	0.491366i	−0.969350	−	0.245683i	$-0.920988\pi$
$569$		−	3.79477e6i		−	0.491366i	0.969350	−	0.245683i	$-0.0790122\pi$
$570$		0			0
$571$		−	3.17124e6i		−	0.407042i	−0.979071	−	0.203521i	$-0.934761\pi$
$571$		−	3.17124e6i		−	0.407042i	0.979071	−	0.203521i	$-0.0652385\pi$
$572$		0			0
$573$	1.83127e7	+	1.83127e7i	2.33005	+	2.33005i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	407507.	−	407507.i	0.0509560	−	0.0509560i	−0.681170	−	0.732126i	$-0.738528\pi$
$577$	407507.	−	407507.i	0.0509560	−	0.0509560i	0.732126	+	0.681170i	$0.238528\pi$
$578$		0			0
$579$	−2.09909e7			−2.60216
$580$		0			0
$581$	−1.45387e6			−0.178684
$582$		0			0
$583$	−1.08522e7	+	1.08522e7i	−1.32235	+	1.32235i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3.38451e6	−	3.38451e6i	−0.405416	−	0.405416i	0.474721	−	0.880136i	$-0.342549\pi$
$587$	−3.38451e6	−	3.38451e6i	−0.405416	−	0.405416i	−0.880136	+	0.474721i	$0.842549\pi$
$588$		0			0
$589$		−	1.27488e7i		−	1.51419i
$590$		0			0
$591$		−	1.30533e7i		−	1.53728i
$592$		0			0
$593$	−1.57715e6	−	1.57715e6i	−0.184178	−	0.184178i	0.608996	−	0.793174i	$-0.291573\pi$
$593$	−1.57715e6	−	1.57715e6i	−0.184178	−	0.184178i	−0.793174	+	0.608996i	$0.791573\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	9.35406e6	−	9.35406e6i	1.07415	−	1.07415i
$598$		0			0
$599$	1.09992e7			1.25254			0.626272	−	0.779605i	$-0.284580\pi$
$599$	1.09992e7			1.25254			0.626272	+	0.779605i	$0.284580\pi$
$600$		0			0
$601$	1.44320e7			1.62982			0.814909	−	0.579589i	$-0.196787\pi$
$601$	1.44320e7			1.62982			0.814909	+	0.579589i	$0.196787\pi$
$602$		0			0
$603$	−1.70451e7	+	1.70451e7i	−1.90900	+	1.90900i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	4.21170e6	+	4.21170e6i	0.463966	+	0.463966i	0.899953	−	0.435987i	$-0.143601\pi$
$607$	4.21170e6	+	4.21170e6i	0.463966	+	0.463966i	−0.435987	+	0.899953i	$0.643601\pi$
$608$		0			0
$609$		−	2.34740e7i		−	2.56474i
$610$		0			0
$611$			1.44091e6i			0.156147i
$612$		0			0
$613$	−3.54916e6	−	3.54916e6i	−0.381482	−	0.381482i	0.490154	−	0.871636i	$-0.336941\pi$
$613$	−3.54916e6	−	3.54916e6i	−0.381482	−	0.381482i	−0.871636	+	0.490154i	$0.836941\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.56234e6	+	2.56234e6i	−0.270971	+	0.270971i	−0.829491	−	0.558520i	$-0.811369\pi$
$617$	−2.56234e6	+	2.56234e6i	−0.270971	+	0.270971i	0.558520	+	0.829491i	$0.311369\pi$
$618$		0			0
$619$	6.34177e6			0.665248			0.332624	−	0.943060i	$-0.392066\pi$
$619$	6.34177e6			0.665248			0.332624	+	0.943060i	$0.392066\pi$
$620$		0			0
$621$	−4.87575e6			−0.507356
$622$		0			0
$623$	652925.	−	652925.i	0.0673974	−	0.0673974i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−1.82546e7	−	1.82546e7i	−1.85440	−	1.85440i
$628$		0			0
$629$			3.17620e6i			0.320096i
$630$		0			0
$631$			4.54433e6i			0.454356i	0.973853	+	0.227178i	$0.0729500\pi$
$631$			4.54433e6i			0.454356i	−0.973853	+	0.227178i	$0.927050\pi$
$632$		0			0
$633$	−1.14528e7	−	1.14528e7i	−1.13607	−	1.13607i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−703367.	+	703367.i	−0.0686805	+	0.0686805i
$638$		0			0
$639$	−577800.			−0.0559790
$640$		0			0
$641$	6.59831e6			0.634290			0.317145	−	0.948377i	$-0.397276\pi$
$641$	6.59831e6			0.634290			0.317145	+	0.948377i	$0.397276\pi$
$642$		0			0
$643$	6.81404e6	−	6.81404e6i	0.649946	−	0.649946i	−0.303034	−	0.952980i	$-0.598000\pi$
$643$	6.81404e6	−	6.81404e6i	0.649946	−	0.649946i	0.952980	+	0.303034i	$0.0979996\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.11169e7	+	1.11169e7i	1.04405	+	1.04405i	0.998984	+	0.0450702i	$0.0143512\pi$
$647$	1.11169e7	+	1.11169e7i	1.04405	+	1.04405i	0.0450702	+	0.998984i	$0.485649\pi$
$648$		0			0
$649$			60636.4i			0.00565095i
$650$		0			0
$651$			2.48556e7i			2.29865i
$652$		0			0
$653$	1.27508e7	+	1.27508e7i	1.17019	+	1.17019i	0.982164	+	0.188025i	$0.0602087\pi$
$653$	1.27508e7	+	1.27508e7i	1.17019	+	1.17019i	0.188025	+	0.982164i	$0.439791\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.26849e7	−	1.26849e7i	1.14650	−	1.14650i
$658$		0			0
$659$	−9.10667e6			−0.816857			−0.408429	−	0.912790i	$-0.633923\pi$
$659$	−9.10667e6			−0.816857			−0.408429	+	0.912790i	$0.633923\pi$
$660$		0			0
$661$	1.89210e7			1.68438			0.842189	−	0.539182i	$-0.181266\pi$
$661$	1.89210e7			1.68438			0.842189	+	0.539182i	$0.181266\pi$
$662$		0			0
$663$	2.74820e6	−	2.74820e6i	0.242809	−	0.242809i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2.59501e6	+	2.59501e6i	0.225852	+	0.225852i
$668$		0			0
$669$			3.34884e7i			2.89287i
$670$		0			0
$671$		−	7.95570e6i		−	0.682138i
$672$		0			0
$673$	−3.24765e6	−	3.24765e6i	−0.276395	−	0.276395i	0.555273	−	0.831668i	$-0.312614\pi$
$673$	−3.24765e6	−	3.24765e6i	−0.276395	−	0.276395i	−0.831668	+	0.555273i	$0.812614\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.17838e7	+	1.17838e7i	−0.988128	+	0.988128i	−0.999930	−	0.0118027i	$-0.996243\pi$
$677$	−1.17838e7	+	1.17838e7i	−0.988128	+	0.988128i	0.0118027	+	0.999930i	$0.496243\pi$
$678$		0			0
$679$	1.88089e7			1.56563
$680$		0			0
$681$	1.12556e7			0.930036
$682$		0			0
$683$	8.18028e6	−	8.18028e6i	0.670991	−	0.670991i	−0.286954	−	0.957944i	$-0.592643\pi$
$683$	8.18028e6	−	8.18028e6i	0.670991	−	0.670991i	0.957944	+	0.286954i	$0.0926425\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−1.02573e7	−	1.02573e7i	−0.829163	−	0.829163i
$688$		0			0
$689$			5.50816e6i			0.442037i
$690$		0			0
$691$			1.23162e7i			0.981253i	0.871370	+	0.490626i	$0.163232\pi$
$691$			1.23162e7i			0.981253i	−0.871370	+	0.490626i	$0.836768\pi$
$692$		0			0
$693$	2.51833e7	+	2.51833e7i	1.99196	+	1.99196i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	8.92868e6	−	8.92868e6i	0.696154	−	0.696154i
$698$		0			0
$699$	2.42501e7			1.87724
$700$		0			0
$701$	−2.10018e7			−1.61421			−0.807107	−	0.590405i	$-0.798968\pi$
$701$	−2.10018e7			−1.61421			−0.807107	+	0.590405i	$0.798968\pi$
$702$		0			0
$703$	5.35413e6	−	5.35413e6i	0.408602	−	0.408602i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.79327e6	+	3.79327e6i	0.285407	+	0.285407i
$708$		0			0
$709$		−	1.87952e7i		−	1.40421i	−0.712074	−	0.702105i	$-0.752244\pi$
$709$		−	1.87952e7i		−	1.40421i	0.712074	−	0.702105i	$-0.247756\pi$
$710$		0			0
$711$		−	5.51788e6i		−	0.409353i
$712$		0			0
$713$	−2.74775e6	−	2.74775e6i	−0.202420	−	0.202420i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	7.09999e6	−	7.09999e6i	0.515774	−	0.515774i
$718$		0			0
$719$	2.33655e7			1.68559			0.842797	−	0.538232i	$-0.180908\pi$
$719$	2.33655e7			1.68559			0.842797	+	0.538232i	$0.180908\pi$
$720$		0			0
$721$	−9.71544e6			−0.696024
$722$		0			0
$723$	−5.15508e6	+	5.15508e6i	−0.366767	+	0.366767i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1.18759e7	−	1.18759e7i	−0.833355	−	0.833355i	0.154619	−	0.987974i	$-0.450585\pi$
$727$	−1.18759e7	−	1.18759e7i	−0.833355	−	0.833355i	−0.987974	+	0.154619i	$0.950585\pi$
$728$		0			0
$729$			1.49092e7i			1.03905i
$730$		0			0
$731$			7.87368e6i			0.544985i
$732$		0			0
$733$	−8.08731e6	−	8.08731e6i	−0.555961	−	0.555961i	0.372194	−	0.928155i	$-0.378606\pi$
$733$	−8.08731e6	−	8.08731e6i	−0.555961	−	0.555961i	−0.928155	+	0.372194i	$0.878606\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.61410e7	+	1.61410e7i	−1.09461	+	1.09461i
$738$		0			0
$739$	−1.24230e7			−0.836786			−0.418393	−	0.908266i	$-0.637407\pi$
$739$	−1.24230e7			−0.836786			−0.418393	+	0.908266i	$0.637407\pi$
$740$		0			0
$741$	−9.26530e6			−0.619889
$742$		0			0
$743$	−9.76492e6	+	9.76492e6i	−0.648928	+	0.648928i	−0.952734	−	0.303806i	$-0.901743\pi$
$743$	−9.76492e6	+	9.76492e6i	−0.648928	+	0.648928i	0.303806	+	0.952734i	$0.401743\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−5.55811e6	−	5.55811e6i	−0.364440	−	0.364440i
$748$		0			0
$749$			3.87236e6i			0.252215i
$750$		0			0
$751$		−	2.15742e7i		−	1.39584i	−0.716178	−	0.697918i	$-0.754110\pi$
$751$		−	2.15742e7i		−	1.39584i	0.716178	−	0.697918i	$-0.245890\pi$
$752$		0			0
$753$	1.00604e7	+	1.00604e7i	0.646588	+	0.646588i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.09605e7	−	1.09605e7i	0.695169	−	0.695169i	−0.268196	−	0.963364i	$-0.586427\pi$
$757$	1.09605e7	−	1.09605e7i	0.695169	−	0.695169i	0.963364	+	0.268196i	$0.0864274\pi$
$758$		0			0
$759$	−7.86883e6			−0.495799
$760$		0			0
$761$	−3.94007e6			−0.246628			−0.123314	−	0.992368i	$-0.539352\pi$
$761$	−3.94007e6			−0.246628			−0.123314	+	0.992368i	$0.539352\pi$
$762$		0			0
$763$	6.13508e6	−	6.13508e6i	0.381513	−	0.381513i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	15388.3	+	15388.3i	0.000944503	+	0.000944503i
$768$		0			0
$769$			1.05073e7i			0.640730i	0.947294	+	0.320365i	$0.103805\pi$
$769$			1.05073e7i			0.640730i	−0.947294	+	0.320365i	$0.896195\pi$
$770$		0			0
$771$		−	5.25645e7i		−	3.18461i
$772$		0			0
$773$	−1.29271e7	−	1.29271e7i	−0.778128	−	0.778128i	0.201384	−	0.979512i	$-0.435456\pi$
$773$	−1.29271e7	−	1.29271e7i	−0.778128	−	0.778128i	−0.979512	+	0.201384i	$0.935456\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1.04387e7	+	1.04387e7i	−0.620288	+	0.620288i
$778$		0			0
$779$	−3.01022e7			−1.77728
$780$		0			0
$781$	−547151.			−0.0320981
$782$		0			0
$783$	5.26563e7	−	5.26563e7i	3.06934	−	3.06934i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−2.08585e7	−	2.08585e7i	−1.20046	−	1.20046i	−0.974028	−	0.226427i	$-0.927295\pi$
$787$	−2.08585e7	−	2.08585e7i	−1.20046	−	1.20046i	−0.226427	−	0.974028i	$-0.572705\pi$
$788$		0			0
$789$			2.29304e7i			1.31135i
$790$		0			0
$791$			1.44397e7i			0.820571i
$792$		0			0
$793$	−2.01900e6	−	2.01900e6i	−0.114013	−	0.114013i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.06132e7	−	2.06132e7i	1.14947	−	1.14947i	0.162817	−	0.986656i	$-0.447942\pi$
$797$	2.06132e7	−	2.06132e7i	1.14947	−	1.14947i	0.986656	−	0.162817i	$-0.0520581\pi$
$798$		0			0
$799$	−4.86377e6			−0.269529
$800$		0			0
$801$	4.99224e6			0.274925
$802$		0			0
$803$	1.20121e7	−	1.20121e7i	0.657398	−	0.657398i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−6.23523e6	−	6.23523e6i	−0.337030	−	0.337030i
$808$		0			0
$809$		−	1.76429e7i		−	0.947761i	−0.880589	−	0.473881i	$-0.842853\pi$
$809$		−	1.76429e7i		−	0.947761i	0.880589	−	0.473881i	$-0.157147\pi$
$810$		0			0
$811$		−	2.55639e7i		−	1.36482i	−0.730970	−	0.682409i	$-0.760932\pi$
$811$		−	2.55639e7i		−	1.36482i	0.730970	−	0.682409i	$-0.239068\pi$
$812$		0			0
$813$	−4.04422e7	−	4.04422e7i	−2.14589	−	2.14589i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.32727e7	−	1.32727e7i	0.695671	−	0.695671i
$818$		0			0
$819$	1.27821e7			0.665873
$820$		0			0
$821$	−1.26661e7			−0.655818			−0.327909	−	0.944709i	$-0.606344\pi$
$821$	−1.26661e7			−0.655818			−0.327909	+	0.944709i	$0.606344\pi$
$822$		0			0
$823$	−4.72508e6	+	4.72508e6i	−0.243170	+	0.243170i	−0.818160	−	0.574990i	$-0.805006\pi$
$823$	−4.72508e6	+	4.72508e6i	−0.243170	+	0.243170i	0.574990	+	0.818160i	$0.305006\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.82843e7	+	1.82843e7i	0.929637	+	0.929637i	0.997682	−	0.0680450i	$-0.0216762\pi$
$827$	1.82843e7	+	1.82843e7i	0.929637	+	0.929637i	−0.0680450	+	0.997682i	$0.521676\pi$
$828$		0			0
$829$		−	2.47048e7i		−	1.24852i	−0.781216	−	0.624260i	$-0.785400\pi$
$829$		−	2.47048e7i		−	1.24852i	0.781216	−	0.624260i	$-0.214600\pi$
$830$		0			0
$831$		−	2.73086e7i		−	1.37182i
$832$		0			0
$833$	−2.37421e6	−	2.37421e6i	−0.118551	−	0.118551i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−5.57556e7	+	5.57556e7i	−2.75090	+	2.75090i
$838$		0			0
$839$	3.59284e7			1.76211			0.881054	−	0.473015i	$-0.156835\pi$
$839$	3.59284e7			1.76211			0.881054	+	0.473015i	$0.156835\pi$
$840$		0			0
$841$	−3.55390e7			−1.73267
$842$		0			0
$843$	−7.63673e6	+	7.63673e6i	−0.370117	+	0.370117i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1.14614e7	+	1.14614e7i	0.548943	+	0.548943i
$848$		0			0
$849$			3.64672e7i			1.73633i
$850$		0			0
$851$		−	2.30796e6i		−	0.109246i
$852$		0			0
$853$	2.48308e7	+	2.48308e7i	1.16847	+	1.16847i	0.982568	+	0.185905i	$0.0595217\pi$
$853$	2.48308e7	+	2.48308e7i	1.16847	+	1.16847i	0.185905	+	0.982568i	$0.440478\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.27271e7	−	2.27271e7i	1.05704	−	1.05704i	0.0587713	−	0.998271i	$-0.481282\pi$
$857$	2.27271e7	−	2.27271e7i	1.05704	−	1.05704i	0.998271	−	0.0587713i	$-0.0187183\pi$
$858$		0			0
$859$	−2.65352e7			−1.22698			−0.613492	−	0.789701i	$-0.710236\pi$
$859$	−2.65352e7			−1.22698			−0.613492	+	0.789701i	$0.710236\pi$
$860$		0			0
$861$	5.86889e7			2.69804
$862$		0			0
$863$	2.09059e7	−	2.09059e7i	0.955524	−	0.955524i	−0.0435285	−	0.999052i	$-0.513860\pi$
$863$	2.09059e7	−	2.09059e7i	0.955524	−	0.955524i	0.999052	+	0.0435285i	$0.0138599\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1.96662e7	−	1.96662e7i	−0.888533	−	0.888533i
$868$		0			0
$869$		−	5.22519e6i		−	0.234721i
$870$		0			0
$871$			8.19253e6i			0.365909i
$872$		0			0
$873$	7.19062e7	+	7.19062e7i	3.19323	+	3.19323i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.73327e7	+	1.73327e7i	−0.760968	+	0.760968i	−0.976497	−	0.215529i	$-0.930852\pi$
$877$	−1.73327e7	+	1.73327e7i	−0.760968	+	0.760968i	0.215529	+	0.976497i	$0.430852\pi$
$878$		0			0
$879$	−3.21341e7			−1.40279
$880$		0			0
$881$	−3.30949e7			−1.43655			−0.718276	−	0.695758i	$-0.755068\pi$
$881$	−3.30949e7			−1.43655			−0.718276	+	0.695758i	$0.755068\pi$
$882$		0			0
$883$	7.21332e6	−	7.21332e6i	0.311339	−	0.311339i	−0.534089	−	0.845428i	$-0.679345\pi$
$883$	7.21332e6	−	7.21332e6i	0.311339	−	0.311339i	0.845428	+	0.534089i	$0.179345\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2.63581e7	+	2.63581e7i	1.12488	+	1.12488i	0.990998	+	0.133879i	$0.0427433\pi$
$887$	2.63581e7	+	2.63581e7i	1.12488	+	1.12488i	0.133879	+	0.990998i	$0.457257\pi$
$888$		0			0
$889$			863960.i			0.0366639i
$890$		0			0
$891$			8.01000e7i			3.38017i
$892$		0			0
$893$	8.19887e6	+	8.19887e6i	0.344053	+	0.344053i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.99696e6	+	1.99696e6i	−0.0828681	+	0.0828681i
$898$		0			0
$899$	5.93493e7			2.44915
$900$		0			0
$901$	−1.85927e7			−0.763011
$902$		0			0
$903$	−2.58771e7	+	2.58771e7i	−1.05608	+	1.05608i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	5.10817e6	+	5.10817e6i	0.206180	+	0.206180i	0.802642	−	0.596462i	$-0.203427\pi$
$907$	5.10817e6	+	5.10817e6i	0.206180	+	0.206180i	−0.596462	+	0.802642i	$0.703427\pi$
$908$		0			0
$909$			2.90032e7i			1.16422i
$910$		0			0
$911$		−	1.09815e7i		−	0.438397i	−0.975680	−	0.219198i	$-0.929656\pi$
$911$		−	1.09815e7i		−	0.438397i	0.975680	−	0.219198i	$-0.0703442\pi$
$912$		0			0
$913$	−5.26329e6	−	5.26329e6i	−0.208968	−	0.208968i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.36185e6	−	4.36185e6i	0.171296	−	0.171296i
$918$		0			0
$919$	7.43231e6			0.290292			0.145146	−	0.989410i	$-0.453635\pi$
$919$	7.43231e6			0.290292			0.145146	+	0.989410i	$0.453635\pi$
$920$		0			0
$921$	8.30625e7			3.22668
$922$		0			0
$923$	−138856.	+	138856.i	−0.00536490	+	0.00536490i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−3.71419e7	−	3.71419e7i	−1.41960	−	1.41960i
$928$		0			0
$929$		−	4.06455e6i		−	0.154516i	−0.997011	−	0.0772580i	$-0.975383\pi$
$929$		−	4.06455e6i		−	0.154516i	0.997011	−	0.0772580i	$-0.0246165\pi$
$930$		0			0
$931$			8.00442e6i			0.302661i
$932$		0			0
$933$	−5.21128e7	−	5.21128e7i	−1.95993	−	1.95993i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	2.62414e7	−	2.62414e7i	0.976423	−	0.976423i	−0.0233056	−	0.999728i	$-0.507419\pi$
$937$	2.62414e7	−	2.62414e7i	0.976423	−	0.976423i	0.999728	+	0.0233056i	$0.00741907\pi$
$938$		0			0
$939$	4.01906e7			1.48751
$940$		0			0
$941$	2.31934e7			0.853867			0.426934	−	0.904283i	$-0.359594\pi$
$941$	2.31934e7			0.853867			0.426934	+	0.904283i	$0.359594\pi$
$942$		0			0
$943$	−6.48795e6	+	6.48795e6i	−0.237590	+	0.237590i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.30147e7	+	1.30147e7i	0.471585	+	0.471585i	0.902427	−	0.430842i	$-0.141783\pi$
$947$	1.30147e7	+	1.30147e7i	0.471585	+	0.471585i	−0.430842	+	0.902427i	$0.641783\pi$
$948$		0			0
$949$		−	6.09685e6i		−	0.219756i
$950$		0			0
$951$			4.63058e6i			0.166029i
$952$		0			0
$953$	−2.46875e6	−	2.46875e6i	−0.0880531	−	0.0880531i	0.661708	−	0.749761i	$-0.269832\pi$
$953$	−2.46875e6	−	2.46875e6i	−0.0880531	−	0.0880531i	−0.749761	+	0.661708i	$0.769832\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	8.49804e7	−	8.49804e7i	2.99943	−	2.99943i
$958$		0			0
$959$	−1.46274e7			−0.513596
$960$		0			0
$961$	−3.42134e7			−1.19505
$962$		0			0
$963$	−1.48039e7	+	1.48039e7i	−0.514413	+	0.514413i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−2.59267e7	−	2.59267e7i	−0.891624	−	0.891624i	0.103052	−	0.994676i	$-0.467139\pi$
$967$	−2.59267e7	−	2.59267e7i	−0.891624	−	0.891624i	−0.994676	+	0.103052i	$0.967139\pi$
$968$		0			0
$969$		−	3.12749e7i		−	1.07001i
$970$		0			0
$971$		−	5.51016e6i		−	0.187550i	−0.995593	−	0.0937748i	$-0.970107\pi$
$971$		−	5.51016e6i		−	0.187550i	0.995593	−	0.0937748i	$-0.0298934\pi$
$972$		0			0
$973$	2.12351e7	+	2.12351e7i	0.719073	+	0.719073i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.64191e7	+	3.64191e7i	−1.22065	+	1.22065i	−0.253256	+	0.967399i	$0.581501\pi$
$977$	−3.64191e7	+	3.64191e7i	−1.22065	+	1.22065i	−0.967399	+	0.253256i	$0.918499\pi$
$978$		0			0
$979$	4.72743e6			0.157641
$980$		0			0
$981$	4.69086e7			1.55625
$982$		0			0
$983$	−1.26298e7	+	1.26298e7i	−0.416881	+	0.416881i	−0.884127	−	0.467246i	$-0.845246\pi$
$983$	−1.26298e7	+	1.26298e7i	−0.416881	+	0.416881i	0.467246	+	0.884127i	$0.345246\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.59849e7	−	1.59849e7i	−0.522298	−	0.522298i
$988$		0			0
$989$		−	5.72135e6i		−	0.185998i
$990$		0			0
$991$			3.90456e7i			1.26295i	0.775395	+	0.631477i	$0.217551\pi$
$991$			3.90456e7i			1.26295i	−0.775395	+	0.631477i	$0.782449\pi$
$992$		0			0
$993$	−6.67590e6	−	6.67590e6i	−0.214851	−	0.214851i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	2.55989e6	−	2.55989e6i	0.0815612	−	0.0815612i	−0.665149	−	0.746710i	$-0.731632\pi$
$997$	2.55989e6	−	2.55989e6i	0.0815612	−	0.0815612i	0.746710	+	0.665149i	$0.231632\pi$
$998$		0			0
$999$	−4.68316e7			−1.48465

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.n.g.143.1		20
4.3	odd	2	inner	400.6.n.g.143.10		20
5.2	odd	4	inner	400.6.n.g.207.10		20
5.3	odd	4		80.6.n.d.47.1	✓	20
5.4	even	2		80.6.n.d.63.10	yes	20
20.3	even	4		80.6.n.d.47.10	yes	20
20.7	even	4	inner	400.6.n.g.207.1		20
20.19	odd	2		80.6.n.d.63.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.n.d.47.1	✓	20	5.3	odd	4
80.6.n.d.47.10	yes	20	20.3	even	4
80.6.n.d.63.1	yes	20	20.19	odd	2
80.6.n.d.63.10	yes	20	5.4	even	2
400.6.n.g.143.1		20	1.1	even	1	trivial
400.6.n.g.143.10		20	4.3	odd	2	inner
400.6.n.g.207.1		20	20.7	even	4	inner
400.6.n.g.207.10		20	5.2	odd	4	inner

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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-20.3843 + 20.3843i) q^{3} +(-76.9082 - 76.9082i) q^{7} -588.037i q^{9} +O(q^{10})\)	\(q+(-20.3843 + 20.3843i) q^{3} +(-76.9082 - 76.9082i) q^{7} -588.037i q^{9} -556.846i q^{11} +(-141.317 - 141.317i) q^{13} +(477.013 - 477.013i) q^{17} -1608.20 q^{19} +3135.44 q^{21} +(-346.617 + 346.617i) q^{23} +(7033.34 + 7033.34i) q^{27} -7486.67i q^{29} +7927.33i q^{31} +(11350.9 + 11350.9i) q^{33} +(-3329.26 + 3329.26i) q^{37} +5761.27 q^{39} +18717.9 q^{41} +(-8253.12 + 8253.12i) q^{43} +(-5098.15 - 5098.15i) q^{47} -4977.25i q^{49} +19447.1i q^{51} +(-19488.7 - 19488.7i) q^{53} +(32782.1 - 32782.1i) q^{57} -108.893 q^{59} +14287.1 q^{61} +(-45224.9 + 45224.9i) q^{63} +(-28986.5 - 28986.5i) q^{67} -14131.1i q^{69} -982.591i q^{71} +(21571.6 + 21571.6i) q^{73} +(-42826.0 + 42826.0i) q^{77} +9383.55 q^{79} -143846. q^{81} +(9451.97 - 9451.97i) q^{83} +(152610. + 152610. i) q^{87} +8489.66i q^{89} +21736.8i q^{91} +(-161593. - 161593. i) q^{93} +(-122282. + 122282. i) q^{97} -327446. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q - 804 q^{13} + 2236 q^{17} - 4520 q^{21} + 11096 q^{33} - 44260 q^{37} - 6760 q^{41} - 182452 q^{53} + 34288 q^{57} - 41080 q^{61} - 264372 q^{73} - 399304 q^{77} - 520220 q^{81} - 713496 q^{93} - 374772 q^{97}+O(q^{100}) \) 20 * q - 804 * q^13 + 2236 * q^17 - 4520 * q^21 + 11096 * q^33 - 44260 * q^37 - 6760 * q^41 - 182452 * q^53 + 34288 * q^57 - 41080 * q^61 - 264372 * q^73 - 399304 * q^77 - 520220 * q^81 - 713496 * q^93 - 374772 * q^97

Introduction

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