LMFDB - Embedded newform 1400.3.p.a.1049.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,3,Mod(1049,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1400.1049");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1400.p (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$38.1472370104$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1049.1
Root			$-0.382683 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	1400.1049
Dual form			1400.3.p.a.1049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.69552 q^{3} +(6.75699 - 1.82843i) q^{7} +4.65685 q^{9} +O(q^{10})$	$q-3.69552 q^{3} +(6.75699 - 1.82843i) q^{7} +4.65685 q^{9} +13.3137 q^{11} +21.5391 q^{13} +6.12293 q^{17} -19.7457i q^{19} +(-24.9706 + 6.75699i) q^{21} +28.6274i q^{23} +16.0502 q^{27} -37.3137 q^{29} +20.9050i q^{31} -49.2011 q^{33} -23.9411i q^{37} -79.5980 q^{39} -70.1061i q^{41} +46.0000i q^{43} -1.05053 q^{47} +(42.3137 - 24.7093i) q^{49} -22.6274 q^{51} +15.3726i q^{53} +72.9706i q^{57} -14.8909i q^{59} +86.7903i q^{61} +(31.4663 - 8.51472i) q^{63} +24.6274i q^{67} -105.793i q^{69} -45.0294 q^{71} +51.7373 q^{73} +(89.9605 - 24.3431i) q^{77} +33.7157 q^{79} -101.225 q^{81} +77.3883 q^{83} +137.893 q^{87} -78.7652i q^{89} +(145.539 - 39.3826i) q^{91} -77.2548i q^{93} +90.1781 q^{97} +62.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} + 16 q^{11} - 64 q^{21} - 208 q^{29} - 320 q^{39} + 248 q^{49} - 496 q^{71} + 496 q^{79} - 312 q^{81} + 576 q^{91} + 496 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 + 16 * q^11 - 64 * q^21 - 208 * q^29 - 320 * q^39 + 248 * q^49 - 496 * q^71 + 496 * q^79 - 312 * q^81 + 576 * q^91 + 496 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	−3.69552			−1.23184			−0.615920	−	0.787809i	$-0.711215\pi$
$3$	−3.69552			−1.23184			−0.615920	+	0.787809i	$0.711215\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	6.75699	−	1.82843i	0.965284	−	0.261204i
$7$	6.75699	−	1.82843i	0.965284	−	0.261204i
$8$		0			0
$9$	4.65685			0.517428
$10$		0			0
$11$	13.3137			1.21034			0.605169	−	0.796097i	$-0.293106\pi$
$11$	13.3137			1.21034			0.605169	+	0.796097i	$0.293106\pi$
$12$		0			0
$13$	21.5391			1.65685			0.828425	−	0.560100i	$-0.189237\pi$
$13$	21.5391			1.65685			0.828425	+	0.560100i	$0.189237\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.12293			0.360173			0.180086	−	0.983651i	$-0.442362\pi$
$17$	6.12293			0.360173			0.180086	+	0.983651i	$0.442362\pi$
$18$		0			0
$19$		−	19.7457i		−	1.03925i	−0.854395	−	0.519623i	$-0.826072\pi$
$19$		−	19.7457i		−	1.03925i	0.854395	−	0.519623i	$-0.173928\pi$
$20$		0			0
$21$	−24.9706	+	6.75699i	−1.18907	+	0.321761i
$22$		0			0
$23$			28.6274i			1.24467i	0.782751	+	0.622335i	$0.213816\pi$
$23$			28.6274i			1.24467i	−0.782751	+	0.622335i	$0.786184\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	16.0502			0.594451
$28$		0			0
$29$	−37.3137			−1.28668			−0.643340	−	0.765581i	$-0.722452\pi$
$29$	−37.3137			−1.28668			−0.643340	+	0.765581i	$0.722452\pi$
$30$		0			0
$31$			20.9050i			0.674355i	0.941441	+	0.337178i	$0.109472\pi$
$31$			20.9050i			0.674355i	−0.941441	+	0.337178i	$0.890528\pi$
$32$		0			0
$33$	−49.2011			−1.49094
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	23.9411i		−	0.647057i	−0.946218	−	0.323529i	$-0.895131\pi$
$37$		−	23.9411i		−	0.647057i	0.946218	−	0.323529i	$-0.104869\pi$
$38$		0			0
$39$	−79.5980			−2.04097
$40$		0			0
$41$		−	70.1061i		−	1.70990i	−0.518707	−	0.854952i	$-0.673587\pi$
$41$		−	70.1061i		−	1.70990i	0.518707	−	0.854952i	$-0.326413\pi$
$42$		0			0
$43$			46.0000i			1.06977i	0.844926	+	0.534884i	$0.179645\pi$
$43$			46.0000i			1.06977i	−0.844926	+	0.534884i	$0.820355\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.05053			−0.0223517			−0.0111758	−	0.999938i	$-0.503557\pi$
$47$	−1.05053			−0.0223517			−0.0111758	+	0.999938i	$0.503557\pi$
$48$		0			0
$49$	42.3137	−	24.7093i	0.863545	−	0.504272i
$50$		0			0
$51$	−22.6274			−0.443675
$52$		0			0
$53$			15.3726i			0.290049i	0.989428	+	0.145024i	$0.0463261\pi$
$53$			15.3726i			0.290049i	−0.989428	+	0.145024i	$0.953674\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			72.9706i			1.28019i
$58$		0			0
$59$		−	14.8909i		−	0.252387i	−0.992006	−	0.126194i	$-0.959724\pi$
$59$		−	14.8909i		−	0.252387i	0.992006	−	0.126194i	$-0.0402761\pi$
$60$		0			0
$61$			86.7903i			1.42279i	0.702792	+	0.711396i	$0.251937\pi$
$61$			86.7903i			1.42279i	−0.702792	+	0.711396i	$0.748063\pi$
$62$		0			0
$63$	31.4663	−	8.51472i	0.499465	−	0.135154i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			24.6274i			0.367573i	0.982966	+	0.183787i	$0.0588356\pi$
$67$			24.6274i			0.367573i	−0.982966	+	0.183787i	$0.941164\pi$
$68$		0			0
$69$		−	105.793i		−	1.53323i
$70$		0			0
$71$	−45.0294			−0.634217			−0.317109	−	0.948389i	$-0.602712\pi$
$71$	−45.0294			−0.634217			−0.317109	+	0.948389i	$0.602712\pi$
$72$		0			0
$73$	51.7373			0.708730			0.354365	−	0.935107i	$-0.384697\pi$
$73$	51.7373			0.708730			0.354365	+	0.935107i	$0.384697\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	89.9605	−	24.3431i	1.16832	−	0.316145i
$78$		0			0
$79$	33.7157			0.426781			0.213391	−	0.976967i	$-0.431549\pi$
$79$	33.7157			0.426781			0.213391	+	0.976967i	$0.431549\pi$
$80$		0			0
$81$	−101.225			−1.24970
$82$		0			0
$83$	77.3883			0.932389			0.466195	−	0.884682i	$-0.345625\pi$
$83$	77.3883			0.932389			0.466195	+	0.884682i	$0.345625\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	137.893			1.58498
$88$		0			0
$89$		−	78.7652i		−	0.885002i	−0.896768	−	0.442501i	$-0.854091\pi$
$89$		−	78.7652i		−	0.885002i	0.896768	−	0.442501i	$-0.145909\pi$
$90$		0			0
$91$	145.539	−	39.3826i	1.59933	−	0.432776i
$92$		0			0
$93$		−	77.2548i		−	0.830697i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	90.1781			0.929671			0.464836	−	0.885397i	$-0.346113\pi$
$97$	90.1781			0.929671			0.464836	+	0.885397i	$0.346113\pi$
$98$		0			0
$99$	62.0000			0.626263
$100$		0			0
$101$		−	129.651i		−	1.28367i	−0.766842	−	0.641836i	$-0.778173\pi$
$101$		−	129.651i		−	1.28367i	0.766842	−	0.641836i	$-0.221827\pi$
$102$		0			0
$103$	−34.4190			−0.334165			−0.167082	−	0.985943i	$-0.553435\pi$
$103$	−34.4190			−0.334165			−0.167082	+	0.985943i	$0.553435\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			149.882i			1.40077i	0.713766	+	0.700384i	$0.246988\pi$
$107$			149.882i			1.40077i	−0.713766	+	0.700384i	$0.753012\pi$
$108$		0			0
$109$	67.2548			0.617017			0.308508	−	0.951222i	$-0.400170\pi$
$109$	67.2548			0.617017			0.308508	+	0.951222i	$0.400170\pi$
$110$		0			0
$111$			88.4749i			0.797071i
$112$		0			0
$113$		−	70.2843i		−	0.621985i	−0.950412	−	0.310992i	$-0.899339\pi$
$113$		−	70.2843i		−	0.621985i	0.950412	−	0.310992i	$-0.100661\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	100.304			0.857301
$118$		0			0
$119$	41.3726	−	11.1953i	0.347669	−	0.0940785i
$120$		0			0
$121$	56.2548			0.464916
$122$		0			0
$123$			259.078i			2.10633i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	93.7645i		−	0.738303i	−0.929369	−	0.369152i	$-0.879648\pi$
$127$		−	93.7645i		−	0.738303i	0.929369	−	0.369152i	$-0.120352\pi$
$128$		0			0
$129$		−	169.994i		−	1.31778i
$130$		0			0
$131$			146.226i			1.11623i	0.829763	+	0.558116i	$0.188475\pi$
$131$			146.226i			1.11623i	−0.829763	+	0.558116i	$0.811525\pi$
$132$		0			0
$133$	−36.1036	−	133.421i	−0.271455	−	1.00317i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	149.765i		−	1.09317i	−0.837403	−	0.546586i	$-0.815927\pi$
$137$		−	149.765i		−	1.09317i	0.837403	−	0.546586i	$-0.184073\pi$
$138$		0			0
$139$			120.249i			0.865100i	0.901610	+	0.432550i	$0.142386\pi$
$139$			120.249i			0.865100i	−0.901610	+	0.432550i	$0.857614\pi$
$140$		0			0
$141$	3.88225			0.0275337
$142$		0			0
$143$	286.765			2.00535
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−156.371	+	91.3137i	−1.06375	+	0.621182i
$148$		0			0
$149$	−33.7645			−0.226607			−0.113304	−	0.993560i	$-0.536143\pi$
$149$	−33.7645			−0.226607			−0.113304	+	0.993560i	$0.536143\pi$
$150$		0			0
$151$	144.627			0.957797			0.478899	−	0.877870i	$-0.341036\pi$
$151$	144.627			0.957797			0.478899	+	0.877870i	$0.341036\pi$
$152$		0			0
$153$	28.5136			0.186363
$154$		0			0
$155$		0			0
$156$		0			0
$157$	128.818			0.820496			0.410248	−	0.911974i	$-0.365442\pi$
$157$	128.818			0.820496			0.410248	+	0.911974i	$0.365442\pi$
$158$		0			0
$159$		−	56.8097i		−	0.357293i
$160$		0			0
$161$	52.3431	+	193.435i	0.325113	+	1.20146i
$162$		0			0
$163$		−	57.4315i		−	0.352340i	−0.984360	−	0.176170i	$-0.943629\pi$
$163$		−	57.4315i		−	0.352340i	0.984360	−	0.176170i	$-0.0563709\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	229.122			1.37199			0.685994	−	0.727607i	$-0.259367\pi$
$167$	229.122			1.37199			0.685994	+	0.727607i	$0.259367\pi$
$168$		0			0
$169$	294.931			1.74515
$170$		0			0
$171$		−	91.9528i		−	0.537736i
$172$		0			0
$173$	−123.310			−0.712776			−0.356388	−	0.934338i	$-0.615992\pi$
$173$	−123.310			−0.712776			−0.356388	+	0.934338i	$0.615992\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			55.0294i			0.310901i
$178$		0			0
$179$	271.941			1.51922			0.759612	−	0.650376i	$-0.225389\pi$
$179$	271.941			1.51922			0.759612	+	0.650376i	$0.225389\pi$
$180$		0			0
$181$		−	30.6333i		−	0.169245i	−0.996413	−	0.0846225i	$-0.973032\pi$
$181$		−	30.6333i		−	0.169245i	0.996413	−	0.0846225i	$-0.0269684\pi$
$182$		0			0
$183$		−	320.735i		−	1.75265i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	81.5190			0.435930
$188$		0			0
$189$	108.451	−	29.3466i	0.573814	−	0.155273i
$190$		0			0
$191$	−69.5980			−0.364387			−0.182194	−	0.983263i	$-0.558320\pi$
$191$	−69.5980			−0.364387			−0.182194	+	0.983263i	$0.558320\pi$
$192$		0			0
$193$		−	279.421i		−	1.44778i	−0.689916	−	0.723890i	$-0.742353\pi$
$193$		−	279.421i		−	1.44778i	0.689916	−	0.723890i	$-0.257647\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			102.451i			0.520055i	0.965601	+	0.260027i	$0.0837316\pi$
$197$			102.451i			0.520055i	−0.965601	+	0.260027i	$0.916268\pi$
$198$		0			0
$199$			161.950i			0.813820i	0.913468	+	0.406910i	$0.133394\pi$
$199$			161.950i			0.813820i	−0.913468	+	0.406910i	$0.866606\pi$
$200$		0			0
$201$		−	91.0111i		−	0.452791i
$202$		0			0
$203$	−252.128	+	68.2254i	−1.24201	+	0.336086i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			133.314i			0.644028i
$208$		0			0
$209$		−	262.888i		−	1.25784i
$210$		0			0
$211$	−51.0193			−0.241798			−0.120899	−	0.992665i	$-0.538578\pi$
$211$	−51.0193			−0.241798			−0.120899	+	0.992665i	$0.538578\pi$
$212$		0			0
$213$	166.407			0.781254
$214$		0			0
$215$		0			0
$216$		0			0
$217$	38.2233	+	141.255i	0.176144	+	0.650944i
$218$		0			0
$219$	−191.196			−0.873041
$220$		0			0
$221$	131.882			0.596752
$222$		0			0
$223$	221.296			0.992358			0.496179	−	0.868220i	$-0.334736\pi$
$223$	221.296			0.992358			0.496179	+	0.868220i	$0.334736\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	101.047			0.445141			0.222571	−	0.974917i	$-0.428555\pi$
$227$	101.047			0.445141			0.222571	+	0.974917i	$0.428555\pi$
$228$		0			0
$229$		−	170.193i		−	0.743200i	−0.928393	−	0.371600i	$-0.878809\pi$
$229$		−	170.193i		−	0.743200i	0.928393	−	0.371600i	$-0.121191\pi$
$230$		0			0
$231$	−332.451	+	89.9605i	−1.43918	+	0.389440i
$232$		0			0
$233$			451.019i			1.93571i	0.251518	+	0.967853i	$0.419070\pi$
$233$			451.019i			1.93571i	−0.251518	+	0.967853i	$0.580930\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−124.597			−0.525726
$238$		0			0
$239$	−219.137			−0.916892			−0.458446	−	0.888722i	$-0.651594\pi$
$239$	−219.137			−0.916892			−0.458446	+	0.888722i	$0.651594\pi$
$240$		0			0
$241$		−	325.386i		−	1.35015i	−0.737750	−	0.675074i	$-0.764112\pi$
$241$		−	325.386i		−	1.35015i	0.737750	−	0.675074i	$-0.235888\pi$
$242$		0			0
$243$	229.629			0.944974
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	425.304i		−	1.72188i
$248$		0			0
$249$	−285.990			−1.14855
$250$		0			0
$251$			260.063i			1.03611i	0.855348	+	0.518054i	$0.173343\pi$
$251$			260.063i			1.03611i	−0.855348	+	0.518054i	$0.826657\pi$
$252$		0			0
$253$			381.137i			1.50647i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−184.558			−0.718126			−0.359063	−	0.933313i	$-0.616904\pi$
$257$	−184.558			−0.718126			−0.359063	+	0.933313i	$0.616904\pi$
$258$		0			0
$259$	−43.7746	−	161.770i	−0.169014	−	0.624594i
$260$		0			0
$261$	−173.765			−0.665764
$262$		0			0
$263$		−	246.853i		−	0.938604i	−0.883038	−	0.469302i	$-0.844506\pi$
$263$		−	246.853i		−	0.938604i	0.883038	−	0.469302i	$-0.155494\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			291.078i			1.09018i
$268$		0			0
$269$		−	105.159i		−	0.390926i	−0.980711	−	0.195463i	$-0.937379\pi$
$269$		−	105.159i		−	0.390926i	0.980711	−	0.195463i	$-0.0626210\pi$
$270$		0			0
$271$		−	71.8093i		−	0.264979i	−0.991184	−	0.132489i	$-0.957703\pi$
$271$		−	71.8093i		−	0.264979i	0.991184	−	0.132489i	$-0.0422971\pi$
$272$		0			0
$273$	−537.842	+	145.539i	−1.97012	+	0.533110i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			402.098i			1.45162i	0.687898	+	0.725808i	$0.258534\pi$
$277$			402.098i			1.45162i	−0.687898	+	0.725808i	$0.741466\pi$
$278$		0			0
$279$			97.3516i			0.348930i
$280$		0			0
$281$	321.529			1.14423			0.572116	−	0.820173i	$-0.306123\pi$
$281$	321.529			1.14423			0.572116	+	0.820173i	$0.306123\pi$
$282$		0			0
$283$	485.561			1.71576			0.857882	−	0.513847i	$-0.171780\pi$
$283$	485.561			1.71576			0.857882	+	0.513847i	$0.171780\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−128.184	−	473.706i	−0.446634	−	1.65054i
$288$		0			0
$289$	−251.510			−0.870276
$290$		0			0
$291$	−333.255			−1.14521
$292$		0			0
$293$	−480.218			−1.63897			−0.819485	−	0.573100i	$-0.805741\pi$
$293$	−480.218			−1.63897			−0.819485	+	0.573100i	$0.805741\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	213.687			0.719486
$298$		0			0
$299$			616.608i			2.06223i
$300$		0			0
$301$	84.1076	+	310.821i	0.279427	+	1.03263i
$302$		0			0
$303$			479.127i			1.58128i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−13.1876			−0.0429564			−0.0214782	−	0.999769i	$-0.506837\pi$
$307$	−13.1876			−0.0429564			−0.0214782	+	0.999769i	$0.506837\pi$
$308$		0			0
$309$	127.196			0.411637
$310$		0			0
$311$		−	393.211i		−	1.26434i	−0.774829	−	0.632171i	$-0.782164\pi$
$311$		−	393.211i		−	1.26434i	0.774829	−	0.632171i	$-0.217836\pi$
$312$		0			0
$313$	−221.694			−0.708287			−0.354143	−	0.935191i	$-0.615228\pi$
$313$	−221.694			−0.708287			−0.354143	+	0.935191i	$0.615228\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	94.5685i		−	0.298323i	−0.988813	−	0.149162i	$-0.952342\pi$
$317$		−	94.5685i		−	0.298323i	0.988813	−	0.149162i	$-0.0476575\pi$
$318$		0			0
$319$	−496.784			−1.55732
$320$		0			0
$321$		−	553.893i		−	1.72552i
$322$		0			0
$323$		−	120.902i		−	0.374308i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−248.541			−0.760066
$328$		0			0
$329$	−7.09841	+	1.92082i	−0.0215757	+	0.00583835i
$330$		0			0
$331$	−190.333			−0.575024			−0.287512	−	0.957777i	$-0.592828\pi$
$331$	−190.333			−0.575024			−0.287512	+	0.957777i	$0.592828\pi$
$332$		0			0
$333$		−	111.490i		−	0.334806i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	430.735i		−	1.27815i	−0.769146	−	0.639073i	$-0.779318\pi$
$337$		−	430.735i		−	1.27815i	0.769146	−	0.639073i	$-0.220682\pi$
$338$		0			0
$339$			259.737i			0.766185i
$340$		0			0
$341$			278.323i			0.816197i
$342$		0			0
$343$	240.734	−	244.328i	0.701848	−	0.712327i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	251.255i		−	0.724077i	−0.932163	−	0.362039i	$-0.882081\pi$
$347$		−	251.255i		−	0.724077i	0.932163	−	0.362039i	$-0.117919\pi$
$348$		0			0
$349$			78.3487i			0.224495i	0.993680	+	0.112247i	$0.0358049\pi$
$349$			78.3487i			0.224495i	−0.993680	+	0.112247i	$0.964195\pi$
$350$		0			0
$351$	345.706			0.984916
$352$		0			0
$353$	43.9111			0.124394			0.0621970	−	0.998064i	$-0.480189\pi$
$353$	43.9111			0.124394			0.0621970	+	0.998064i	$0.480189\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−152.893	+	41.3726i	−0.428272	+	0.115890i
$358$		0			0
$359$	−546.431			−1.52209			−0.761045	−	0.648699i	$-0.775314\pi$
$359$	−546.431			−1.52209			−0.761045	+	0.648699i	$0.775314\pi$
$360$		0			0
$361$	−28.8924			−0.0800342
$362$		0			0
$363$	−207.891			−0.572702
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−481.505			−1.31200			−0.656002	−	0.754759i	$-0.727754\pi$
$367$	−481.505			−1.31200			−0.656002	+	0.754759i	$0.727754\pi$
$368$		0			0
$369$		−	326.474i		−	0.884753i
$370$		0			0
$371$	28.1076	+	103.872i	0.0757619	+	0.279979i
$372$		0			0
$373$			432.843i			1.16044i	0.814461	+	0.580218i	$0.197033\pi$
$373$			432.843i			1.16044i	−0.814461	+	0.580218i	$0.802967\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−803.702			−2.13184
$378$		0			0
$379$	512.607			1.35253			0.676263	−	0.736660i	$-0.263598\pi$
$379$	512.607			1.35253			0.676263	+	0.736660i	$0.263598\pi$
$380$		0			0
$381$			346.508i			0.909471i
$382$		0			0
$383$	−651.897			−1.70208			−0.851040	−	0.525100i	$-0.824028\pi$
$383$	−651.897			−1.70208			−0.851040	+	0.525100i	$0.824028\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			214.215i			0.553528i
$388$		0			0
$389$	523.352			1.34538			0.672689	−	0.739925i	$-0.265139\pi$
$389$	523.352			1.34538			0.672689	+	0.739925i	$0.265139\pi$
$390$		0			0
$391$			175.284i			0.448296i
$392$		0			0
$393$		−	540.382i		−	1.37502i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−15.8513			−0.0399276			−0.0199638	−	0.999801i	$-0.506355\pi$
$397$	−15.8513			−0.0399276			−0.0199638	+	0.999801i	$0.506355\pi$
$398$		0			0
$399$	133.421	+	493.061i	0.334389	+	1.23574i
$400$		0			0
$401$	11.7746			0.0293631			0.0146816	−	0.999892i	$-0.495327\pi$
$401$	11.7746			0.0293631			0.0146816	+	0.999892i	$0.495327\pi$
$402$		0			0
$403$			450.274i			1.11731i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	318.745i		−	0.783158i
$408$		0			0
$409$			492.483i			1.20411i	0.798453	+	0.602057i	$0.205652\pi$
$409$			492.483i			1.20411i	−0.798453	+	0.602057i	$0.794348\pi$
$410$		0			0
$411$			553.457i			1.34661i
$412$		0			0
$413$	−27.2268	−	100.617i	−0.0659246	−	0.243625i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	444.382i		−	1.06566i
$418$		0			0
$419$			276.946i			0.660970i	0.943811	+	0.330485i	$0.107212\pi$
$419$			276.946i			0.660970i	−0.943811	+	0.330485i	$0.892788\pi$
$420$		0			0
$421$	14.3532			0.0340932			0.0170466	−	0.999855i	$-0.494574\pi$
$421$	14.3532			0.0340932			0.0170466	+	0.999855i	$0.494574\pi$
$422$		0			0
$423$	−4.89216			−0.0115654
$424$		0			0
$425$		0			0
$426$		0			0
$427$	158.690	+	586.441i	0.371639	+	1.37340i
$428$		0			0
$429$	−1059.74			−2.47027
$430$		0			0
$431$	250.471			0.581139			0.290570	−	0.956854i	$-0.406155\pi$
$431$	250.471			0.581139			0.290570	+	0.956854i	$0.406155\pi$
$432$		0			0
$433$	656.534			1.51625			0.758123	−	0.652112i	$-0.226117\pi$
$433$	656.534			1.51625			0.758123	+	0.652112i	$0.226117\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	565.268			1.29352
$438$		0			0
$439$		−	356.398i		−	0.811841i	−0.913908	−	0.405921i	$-0.866951\pi$
$439$		−	356.398i		−	0.811841i	0.913908	−	0.405921i	$-0.133049\pi$
$440$		0			0
$441$	197.049	−	115.068i	0.446823	−	0.260924i
$442$		0			0
$443$		−	253.785i		−	0.572877i	−0.958099	−	0.286439i	$-0.907529\pi$
$443$		−	253.785i		−	0.572877i	0.958099	−	0.286439i	$-0.0924715\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	124.777			0.279144
$448$		0			0
$449$	107.921			0.240358			0.120179	−	0.992752i	$-0.461653\pi$
$449$	107.921			0.240358			0.120179	+	0.992752i	$0.461653\pi$
$450$		0			0
$451$		−	933.372i		−	2.06956i
$452$		0			0
$453$	−534.473			−1.17985
$454$		0			0
$455$		0			0
$456$		0			0
$457$			143.990i			0.315076i	0.987513	+	0.157538i	$0.0503557\pi$
$457$			143.990i			0.315076i	−0.987513	+	0.157538i	$0.949644\pi$
$458$		0			0
$459$	98.2742			0.214105
$460$		0			0
$461$		−	380.728i		−	0.825875i	−0.910759	−	0.412938i	$-0.864503\pi$
$461$		−	380.728i		−	0.825875i	0.910759	−	0.412938i	$-0.135497\pi$
$462$		0			0
$463$			227.990i			0.492419i	0.969217	+	0.246209i	$0.0791851\pi$
$463$			227.990i			0.492419i	−0.969217	+	0.246209i	$0.920815\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−502.917			−1.07691			−0.538455	−	0.842654i	$-0.680992\pi$
$467$	−502.917			−1.07691			−0.538455	+	0.842654i	$0.680992\pi$
$468$		0			0
$469$	45.0294	+	166.407i	0.0960116	+	0.354813i
$470$		0			0
$471$	−476.049			−1.01072
$472$		0			0
$473$			612.431i			1.29478i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			71.5879i			0.150079i
$478$		0			0
$479$		−	212.637i		−	0.443918i	−0.975056	−	0.221959i	$-0.928755\pi$
$479$		−	212.637i		−	0.443918i	0.975056	−	0.221959i	$-0.0712451\pi$
$480$		0			0
$481$		−	515.669i		−	1.07208i
$482$		0			0
$483$	−193.435	−	714.843i	−0.400487	−	1.48001i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	121.411i		−	0.249304i	−0.992200	−	0.124652i	$-0.960218\pi$
$487$		−	121.411i		−	0.249304i	0.992200	−	0.124652i	$-0.0397815\pi$
$488$		0			0
$489$			212.239i			0.434027i
$490$		0			0
$491$	−130.353			−0.265485			−0.132743	−	0.991151i	$-0.542378\pi$
$491$	−130.353			−0.265485			−0.132743	+	0.991151i	$0.542378\pi$
$492$		0			0
$493$	−228.469			−0.463427
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−304.263	+	82.3330i	−0.612200	+	0.165660i
$498$		0			0
$499$	−157.980			−0.316593			−0.158296	−	0.987392i	$-0.550600\pi$
$499$	−157.980			−0.316593			−0.158296	+	0.987392i	$0.550600\pi$
$500$		0			0
$501$	−846.725			−1.69007
$502$		0			0
$503$	323.502			0.643146			0.321573	−	0.946885i	$-0.395788\pi$
$503$	323.502			0.643146			0.321573	+	0.946885i	$0.395788\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1089.92			−2.14975
$508$		0			0
$509$			505.723i			0.993563i	0.867876	+	0.496781i	$0.165485\pi$
$509$			505.723i			0.993563i	−0.867876	+	0.496781i	$0.834515\pi$
$510$		0			0
$511$	349.588	−	94.5978i	0.684125	−	0.185123i
$512$		0			0
$513$		−	316.922i		−	0.617781i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−13.9864			−0.0270531
$518$		0			0
$519$	455.696			0.878026
$520$		0			0
$521$			827.833i			1.58893i	0.607309	+	0.794466i	$0.292249\pi$
$521$			827.833i			1.58893i	−0.607309	+	0.794466i	$0.707751\pi$
$522$		0			0
$523$	−114.306			−0.218559			−0.109279	−	0.994011i	$-0.534854\pi$
$523$	−114.306			−0.218559			−0.109279	+	0.994011i	$0.534854\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			128.000i			0.242884i
$528$		0			0
$529$	−290.529			−0.549204
$530$		0			0
$531$		−	69.3446i		−	0.130592i
$532$		0			0
$533$		−	1510.02i		−	2.83306i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1004.96			−1.87144
$538$		0			0
$539$	563.352	−	328.973i	1.04518	−	0.610339i
$540$		0			0
$541$	529.098			0.978001			0.489000	−	0.872284i	$-0.337362\pi$
$541$	529.098			0.978001			0.489000	+	0.872284i	$0.337362\pi$
$542$		0			0
$543$			113.206i			0.208483i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	281.314i		−	0.514285i	−0.966374	−	0.257142i	$-0.917219\pi$
$547$		−	281.314i		−	0.514285i	0.966374	−	0.257142i	$-0.0827809\pi$
$548$		0			0
$549$			404.170i			0.736193i
$550$		0			0
$551$			736.785i			1.33718i
$552$		0			0
$553$	227.817	−	61.6468i	0.411965	−	0.111477i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	198.471i		−	0.356321i	−0.984001	−	0.178161i	$-0.942985\pi$
$557$		−	198.471i		−	0.356321i	0.984001	−	0.178161i	$-0.0570147\pi$
$558$		0			0
$559$			990.797i			1.77244i
$560$		0			0
$561$	−301.255			−0.536996
$562$		0			0
$563$	98.2933			0.174588			0.0872942	−	0.996183i	$-0.472178\pi$
$563$	98.2933			0.174588			0.0872942	+	0.996183i	$0.472178\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−683.979	+	185.083i	−1.20631	+	0.326426i
$568$		0			0
$569$	709.029			1.24610			0.623049	−	0.782183i	$-0.285894\pi$
$569$	709.029			1.24610			0.623049	+	0.782183i	$0.285894\pi$
$570$		0			0
$571$	462.118			0.809313			0.404657	−	0.914469i	$-0.367391\pi$
$571$	462.118			0.809313			0.404657	+	0.914469i	$0.367391\pi$
$572$		0			0
$573$	257.201			0.448867
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−586.571			−1.01659			−0.508294	−	0.861184i	$-0.669724\pi$
$577$	−586.571			−1.01659			−0.508294	+	0.861184i	$0.669724\pi$
$578$		0			0
$579$			1032.61i			1.78343i
$580$		0			0
$581$	522.912	−	141.499i	0.900020	−	0.243544i
$582$		0			0
$583$			204.666i			0.351057i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	441.613			0.752322			0.376161	−	0.926554i	$-0.377244\pi$
$587$	441.613			0.752322			0.376161	+	0.926554i	$0.377244\pi$
$588$		0			0
$589$	412.784			0.700821
$590$		0			0
$591$		−	378.609i		−	0.640624i
$592$		0			0
$593$	−87.0265			−0.146756			−0.0733782	−	0.997304i	$-0.523378\pi$
$593$	−87.0265			−0.146756			−0.0733782	+	0.997304i	$0.523378\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	598.489i		−	1.00249i
$598$		0			0
$599$	−484.108			−0.808193			−0.404097	−	0.914716i	$-0.632414\pi$
$599$	−484.108			−0.808193			−0.404097	+	0.914716i	$0.632414\pi$
$600$		0			0
$601$		−	904.895i		−	1.50565i	−0.658221	−	0.752825i	$-0.728691\pi$
$601$		−	904.895i		−	1.50565i	0.658221	−	0.752825i	$-0.271309\pi$
$602$		0			0
$603$			114.686i			0.190193i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	940.980			1.55021			0.775107	−	0.631830i	$-0.217696\pi$
$607$	940.980			1.55021			0.775107	+	0.631830i	$0.217696\pi$
$608$		0			0
$609$	931.744	−	252.128i	1.52996	−	0.414004i
$610$		0			0
$611$	−22.6274			−0.0370334
$612$		0			0
$613$		−	240.627i		−	0.392541i	−0.980550	−	0.196270i	$-0.937117\pi$
$613$		−	240.627i		−	0.392541i	0.980550	−	0.196270i	$-0.0628830\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			781.716i			1.26696i	0.773758	+	0.633481i	$0.218375\pi$
$617$			781.716i			1.26696i	−0.773758	+	0.633481i	$0.781625\pi$
$618$		0			0
$619$		−	9.99870i		−	0.0161530i	−0.999967	−	0.00807649i	$-0.997429\pi$
$619$		−	9.99870i		−	0.0161530i	0.999967	−	0.00807649i	$-0.00257085\pi$
$620$		0			0
$621$			459.475i			0.739895i
$622$		0			0
$623$	−144.016	−	532.215i	−0.231166	−	0.854278i
$624$		0			0
$625$		0			0
$626$		0			0
$627$			971.509i			1.54946i
$628$		0			0
$629$		−	146.590i		−	0.233052i
$630$		0			0
$631$	908.538			1.43984			0.719919	−	0.694058i	$-0.244179\pi$
$631$	908.538			1.43984			0.719919	+	0.694058i	$0.244179\pi$
$632$		0			0
$633$	188.543			0.297856
$634$		0			0
$635$		0			0
$636$		0			0
$637$	911.397	−	532.215i	1.43077	−	0.835503i
$638$		0			0
$639$	−209.696			−0.328162
$640$		0			0
$641$	193.460			0.301810			0.150905	−	0.988548i	$-0.451781\pi$
$641$	193.460			0.301810			0.150905	+	0.988548i	$0.451781\pi$
$642$		0			0
$643$	−362.412			−0.563627			−0.281814	−	0.959469i	$-0.590936\pi$
$643$	−362.412			−0.563627			−0.281814	+	0.959469i	$0.590936\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	359.625			0.555834			0.277917	−	0.960605i	$-0.410356\pi$
$647$	359.625			0.555834			0.277917	+	0.960605i	$0.410356\pi$
$648$		0			0
$649$		−	198.253i		−	0.305474i
$650$		0			0
$651$	−141.255	−	522.010i	−0.216981	−	0.801858i
$652$		0			0
$653$			715.685i			1.09600i	0.836480	+	0.547998i	$0.184610\pi$
$653$			715.685i			1.09600i	−0.836480	+	0.547998i	$0.815390\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	240.933			0.366717
$658$		0			0
$659$	390.431			0.592459			0.296230	−	0.955117i	$-0.404271\pi$
$659$	390.431			0.592459			0.296230	+	0.955117i	$0.404271\pi$
$660$		0			0
$661$			350.474i			0.530218i	0.964218	+	0.265109i	$0.0854080\pi$
$661$			350.474i			0.530218i	−0.964218	+	0.265109i	$0.914592\pi$
$662$		0			0
$663$	−487.373			−0.735103
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	1068.20i		−	1.60149i
$668$		0			0
$669$	−817.803			−1.22243
$670$		0			0
$671$			1155.50i			1.72206i
$672$		0			0
$673$		−	487.214i		−	0.723944i	−0.932189	−	0.361972i	$-0.882104\pi$
$673$		−	487.214i		−	0.723944i	0.932189	−	0.361972i	$-0.117896\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−451.742			−0.667271			−0.333635	−	0.942702i	$-0.608275\pi$
$677$	−451.742			−0.667271			−0.333635	+	0.942702i	$0.608275\pi$
$678$		0			0
$679$	609.332	−	164.884i	0.897396	−	0.242834i
$680$		0			0
$681$	−373.421			−0.548343
$682$		0			0
$683$			224.745i			0.329056i	0.986372	+	0.164528i	$0.0526101\pi$
$683$			224.745i			0.329056i	−0.986372	+	0.164528i	$0.947390\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			628.950i			0.915503i
$688$		0			0
$689$			331.111i			0.480567i
$690$		0			0
$691$		−	56.0855i		−	0.0811657i	−0.999176	−	0.0405828i	$-0.987079\pi$
$691$		−	56.0855i		−	0.0811657i	0.999176	−	0.0405828i	$-0.0129215\pi$
$692$		0			0
$693$	418.933	−	113.362i	0.604521	−	0.163582i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	429.255i		−	0.615861i
$698$		0			0
$699$		−	1666.75i		−	2.38448i
$700$		0			0
$701$	−769.647			−1.09793			−0.548963	−	0.835846i	$-0.684977\pi$
$701$	−769.647			−1.09793			−0.548963	+	0.835846i	$0.684977\pi$
$702$		0			0
$703$	−472.734			−0.672452
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−237.057	−	876.049i	−0.335300	−	1.23911i
$708$		0			0
$709$	−465.960			−0.657207			−0.328603	−	0.944468i	$-0.606578\pi$
$709$	−465.960			−0.657207			−0.328603	+	0.944468i	$0.606578\pi$
$710$		0			0
$711$	157.009			0.220829
$712$		0			0
$713$	−598.456			−0.839350
$714$		0			0
$715$		0			0
$716$		0			0
$717$	809.825			1.12946
$718$		0			0
$719$		−	1051.88i		−	1.46298i	−0.681852	−	0.731490i	$-0.738825\pi$
$719$		−	1051.88i		−	1.46298i	0.681852	−	0.731490i	$-0.261175\pi$
$720$		0			0
$721$	−232.569	+	62.9326i	−0.322564	+	0.0872852i
$722$		0			0
$723$			1202.47i			1.66317i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−161.080			−0.221568			−0.110784	−	0.993845i	$-0.535336\pi$
$727$	−161.080			−0.221568			−0.110784	+	0.993845i	$0.535336\pi$
$728$		0			0
$729$	62.4315			0.0856399
$730$		0			0
$731$			281.655i			0.385301i
$732$		0			0
$733$	1438.70			1.96275			0.981376	−	0.192097i	$-0.0615287\pi$
$733$	1438.70			1.96275			0.981376	+	0.192097i	$0.0615287\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			327.882i			0.444888i
$738$		0			0
$739$	−1137.39			−1.53909			−0.769547	−	0.638590i	$-0.779518\pi$
$739$	−1137.39			−1.53909			−0.769547	+	0.638590i	$0.779518\pi$
$740$		0			0
$741$			1571.72i			2.12108i
$742$		0			0
$743$			745.882i			1.00388i	0.864903	+	0.501940i	$0.167380\pi$
$743$			745.882i			1.00388i	−0.864903	+	0.501940i	$0.832620\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	360.386			0.482445
$748$		0			0
$749$	274.049	+	1012.75i	0.365886	+	1.35214i
$750$		0			0
$751$	−153.882			−0.204903			−0.102452	−	0.994738i	$-0.532669\pi$
$751$	−153.882			−0.204903			−0.102452	+	0.994738i	$0.532669\pi$
$752$		0			0
$753$		−	961.068i		−	1.27632i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	119.137i		−	0.157381i	−0.996899	−	0.0786903i	$-0.974926\pi$
$757$		−	119.137i		−	0.157381i	0.996899	−	0.0786903i	$-0.0250738\pi$
$758$		0			0
$759$		−	1408.50i		−	1.85573i
$760$		0			0
$761$		−	1385.20i		−	1.82024i	−0.414348	−	0.910119i	$-0.635990\pi$
$761$		−	1385.20i		−	1.82024i	0.414348	−	0.910119i	$-0.364010\pi$
$762$		0			0
$763$	454.440	−	122.971i	0.595596	−	0.161167i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	320.735i		−	0.418168i
$768$		0			0
$769$			55.9020i			0.0726945i	0.999339	+	0.0363472i	$0.0115722\pi$
$769$			55.9020i			0.0726945i	−0.999339	+	0.0363472i	$0.988428\pi$
$770$		0			0
$771$	682.039			0.884616
$772$		0			0
$773$	−179.759			−0.232548			−0.116274	−	0.993217i	$-0.537095\pi$
$773$	−179.759			−0.232548			−0.116274	+	0.993217i	$0.537095\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	161.770	+	597.823i	0.208198	+	0.769399i
$778$		0			0
$779$	−1384.29			−1.77701
$780$		0			0
$781$	−599.509			−0.767617
$782$		0			0
$783$	−598.892			−0.764868
$784$		0			0
$785$		0			0
$786$		0			0
$787$	208.363			0.264756			0.132378	−	0.991199i	$-0.457739\pi$
$787$	208.363			0.264756			0.132378	+	0.991199i	$0.457739\pi$
$788$		0			0
$789$			912.249i			1.15621i
$790$		0			0
$791$	−128.510	−	474.910i	−0.162465	−	0.600392i
$792$		0			0
$793$			1869.38i			2.35735i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−237.327			−0.297776			−0.148888	−	0.988854i	$-0.547569\pi$
$797$	−237.327			−0.297776			−0.148888	+	0.988854i	$0.547569\pi$
$798$		0			0
$799$	−6.43232			−0.00805047
$800$		0			0
$801$		−	366.798i		−	0.457925i
$802$		0			0
$803$	688.815			0.857802
$804$		0			0
$805$		0			0
$806$		0			0
$807$			388.617i			0.481558i
$808$		0			0
$809$	−513.793			−0.635097			−0.317548	−	0.948242i	$-0.602860\pi$
$809$	−513.793			−0.635097			−0.317548	+	0.948242i	$0.602860\pi$
$810$		0			0
$811$		−	452.373i		−	0.557797i	−0.960321	−	0.278898i	$-0.910031\pi$
$811$		−	452.373i		−	0.557797i	0.960321	−	0.278898i	$-0.0899692\pi$
$812$		0			0
$813$			265.373i			0.326412i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	908.302			1.11175
$818$		0			0
$819$	677.754	−	183.399i	0.827539	−	0.223930i
$820$		0			0
$821$	−1441.04			−1.75522			−0.877611	−	0.479373i	$-0.840864\pi$
$821$	−1441.04			−1.75522			−0.877611	+	0.479373i	$0.840864\pi$
$822$		0			0
$823$			1603.66i			1.94855i	0.225362	+	0.974275i	$0.427644\pi$
$823$			1603.66i			1.94855i	−0.225362	+	0.974275i	$0.572356\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			933.549i			1.12884i	0.825488	+	0.564419i	$0.190900\pi$
$827$			933.549i			1.12884i	−0.825488	+	0.564419i	$0.809100\pi$
$828$		0			0
$829$		−	418.809i		−	0.505198i	−0.967571	−	0.252599i	$-0.918715\pi$
$829$		−	418.809i		−	0.505198i	0.967571	−	0.252599i	$-0.0812853\pi$
$830$		0			0
$831$		−	1485.96i		−	1.78816i
$832$		0			0
$833$	259.084	−	151.294i	0.311025	−	0.181625i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			335.529i			0.400871i
$838$		0			0
$839$			995.689i			1.18676i	0.804924	+	0.593378i	$0.202206\pi$
$839$			995.689i			1.18676i	−0.804924	+	0.593378i	$0.797794\pi$
$840$		0			0
$841$	551.313			0.655544
$842$		0			0
$843$	−1188.22			−1.40951
$844$		0			0
$845$		0			0
$846$		0			0
$847$	380.113	−	102.858i	0.448776	−	0.121438i
$848$		0			0
$849$	−1794.40			−2.11355
$850$		0			0
$851$	685.373			0.805373
$852$		0			0
$853$	315.117			0.369422			0.184711	−	0.982793i	$-0.440865\pi$
$853$	315.117			0.369422			0.184711	+	0.982793i	$0.440865\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−218.287			−0.254711			−0.127355	−	0.991857i	$-0.540649\pi$
$857$	−218.287			−0.254711			−0.127355	+	0.991857i	$0.540649\pi$
$858$		0			0
$859$		−	1441.76i		−	1.67842i	−0.543811	−	0.839208i	$-0.683019\pi$
$859$		−	1441.76i		−	1.67842i	0.543811	−	0.839208i	$-0.316981\pi$
$860$		0			0
$861$	473.706	+	1750.59i	0.550181	+	2.03320i
$862$		0			0
$863$		−	874.873i		−	1.01376i	−0.862017	−	0.506879i	$-0.830799\pi$
$863$		−	874.873i		−	1.01376i	0.862017	−	0.506879i	$-0.169201\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	929.459			1.07204
$868$		0			0
$869$	448.881			0.516549
$870$		0			0
$871$			530.451i			0.609014i
$872$		0			0
$873$	419.946			0.481038
$874$		0			0
$875$		0			0
$876$		0			0
$877$			1702.65i			1.94144i	0.240207	+	0.970722i	$0.422785\pi$
$877$			1702.65i			1.94144i	−0.240207	+	0.970722i	$0.577215\pi$
$878$		0			0
$879$	1774.66			2.01895
$880$		0			0
$881$		−	1018.58i		−	1.15617i	−0.815978	−	0.578083i	$-0.803801\pi$
$881$		−	1018.58i		−	1.15617i	0.815978	−	0.578083i	$-0.196199\pi$
$882$		0			0
$883$		−	1646.04i		−	1.86414i	−0.362276	−	0.932071i	$-0.618000\pi$
$883$		−	1646.04i		−	1.86414i	0.362276	−	0.932071i	$-0.382000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	688.597			0.776321			0.388161	−	0.921592i	$-0.373111\pi$
$887$	688.597			0.776321			0.388161	+	0.921592i	$0.373111\pi$
$888$		0			0
$889$	−171.442	−	633.565i	−0.192848	−	0.712672i
$890$		0			0
$891$	−1347.69			−1.51255
$892$		0			0
$893$			20.7434i			0.0232289i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	2278.68i		−	2.54034i
$898$		0			0
$899$		−	780.043i		−	0.867679i
$900$		0			0
$901$			94.1253i			0.104468i
$902$		0			0
$903$	−310.821	−	1148.65i	−0.344210	−	1.27203i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	212.018i		−	0.233758i	−0.993146	−	0.116879i	$-0.962711\pi$
$907$		−	212.018i		−	0.233758i	0.993146	−	0.116879i	$-0.0372890\pi$
$908$		0			0
$909$		−	603.765i		−	0.664208i
$910$		0			0
$911$	−601.882			−0.660683			−0.330342	−	0.943861i	$-0.607164\pi$
$911$	−601.882			−0.660683			−0.330342	+	0.943861i	$0.607164\pi$
$912$		0			0
$913$	1030.33			1.12851
$914$		0			0
$915$		0			0
$916$		0			0
$917$	267.364	+	988.049i	0.291564	+	1.07748i
$918$		0			0
$919$	−1118.34			−1.21691			−0.608456	−	0.793588i	$-0.708211\pi$
$919$	−1118.34			−1.21691			−0.608456	+	0.793588i	$0.708211\pi$
$920$		0			0
$921$	48.7351			0.0529154
$922$		0			0
$923$	−969.892			−1.05080
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−160.284			−0.172906
$928$		0			0
$929$		−	401.907i		−	0.432623i	−0.976324	−	0.216312i	$-0.930597\pi$
$929$		−	401.907i		−	0.432623i	0.976324	−	0.216312i	$-0.0694027\pi$
$930$		0			0
$931$	−487.902	−	835.513i	−0.524063	−	0.897437i
$932$		0			0
$933$			1453.12i			1.55747i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1597.84			−1.70527			−0.852635	−	0.522507i	$-0.824997\pi$
$937$	−1597.84			−1.70527			−0.852635	+	0.522507i	$0.824997\pi$
$938$		0			0
$939$	819.273			0.872496
$940$		0			0
$941$			1537.71i			1.63413i	0.576547	+	0.817064i	$0.304400\pi$
$941$			1537.71i			1.63413i	−0.576547	+	0.817064i	$0.695600\pi$
$942$		0			0
$943$	2006.96			2.12827
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	320.274i		−	0.338199i	−0.985599	−	0.169099i	$-0.945914\pi$
$947$		−	320.274i		−	0.338199i	0.985599	−	0.169099i	$-0.0540859\pi$
$948$		0			0
$949$	1114.37			1.17426
$950$		0			0
$951$			349.480i			0.367487i
$952$		0			0
$953$			734.861i			0.771103i	0.922686	+	0.385552i	$0.125989\pi$
$953$			734.861i			0.771103i	−0.922686	+	0.385552i	$0.874011\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	1835.87			1.91836
$958$		0			0
$959$	−273.833	−	1011.96i	−0.285541	−	1.05522i
$960$		0			0
$961$	523.981			0.545245
$962$		0			0
$963$			697.980i			0.724797i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1446.98i			1.49636i	0.663497	+	0.748179i	$0.269072\pi$
$967$			1446.98i			1.49636i	−0.663497	+	0.748179i	$0.730928\pi$
$968$		0			0
$969$			446.794i			0.461088i
$970$		0			0
$971$		−	966.084i		−	0.994937i	−0.867482	−	0.497469i	$-0.834263\pi$
$971$		−	966.084i		−	0.994937i	0.867482	−	0.497469i	$-0.165737\pi$
$972$		0			0
$973$	219.866	+	812.520i	0.225967	+	0.835067i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	609.606i		−	0.623957i	−0.950089	−	0.311979i	$-0.899008\pi$
$977$		−	609.606i		−	0.623957i	0.950089	−	0.311979i	$-0.100992\pi$
$978$		0			0
$979$		−	1048.66i		−	1.07115i
$980$		0			0
$981$	313.196			0.319262
$982$		0			0
$983$	−1277.27			−1.29936			−0.649679	−	0.760209i	$-0.725097\pi$
$983$	−1277.27			−1.29936			−0.649679	+	0.760209i	$0.725097\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	26.2323	−	7.09841i	0.0265778	−	0.00719191i
$988$		0			0
$989$	−1316.86			−1.33151
$990$		0			0
$991$	−1669.99			−1.68515			−0.842577	−	0.538575i	$-0.818963\pi$
$991$	−1669.99			−1.68515			−0.842577	+	0.538575i	$0.818963\pi$
$992$		0			0
$993$	703.379			0.708338
$994$		0			0
$995$		0			0
$996$		0			0
$997$	789.952			0.792329			0.396164	−	0.918180i	$-0.370341\pi$
$997$	789.952			0.792329			0.396164	+	0.918180i	$0.370341\pi$
$998$		0			0
$999$		−	384.259i		−	0.384644i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.3.p.a.1049.1		8
5.2	odd	4		1400.3.f.a.601.4		4
5.3	odd	4		56.3.c.a.41.1	✓	4
5.4	even	2	inner	1400.3.p.a.1049.8		8
7.6	odd	2	inner	1400.3.p.a.1049.7		8
15.8	even	4		504.3.f.a.433.3		4
20.3	even	4		112.3.c.c.97.4		4
35.3	even	12		392.3.o.b.313.4		8
35.13	even	4		56.3.c.a.41.4	yes	4
35.18	odd	12		392.3.o.b.313.1		8
35.23	odd	12		392.3.o.b.129.4		8
35.27	even	4		1400.3.f.a.601.1		4
35.33	even	12		392.3.o.b.129.1		8
35.34	odd	2	inner	1400.3.p.a.1049.2		8
40.3	even	4		448.3.c.e.321.1		4
40.13	odd	4		448.3.c.f.321.4		4
60.23	odd	4		1008.3.f.h.433.3		4
105.83	odd	4		504.3.f.a.433.2		4
140.3	odd	12		784.3.s.f.705.1		8
140.23	even	12		784.3.s.f.129.1		8
140.83	odd	4		112.3.c.c.97.1		4
140.103	odd	12		784.3.s.f.129.4		8
140.123	even	12		784.3.s.f.705.4		8
280.13	even	4		448.3.c.f.321.1		4
280.83	odd	4		448.3.c.e.321.4		4
420.83	even	4		1008.3.f.h.433.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.c.a.41.1	✓	4	5.3	odd	4
56.3.c.a.41.4	yes	4	35.13	even	4
112.3.c.c.97.1		4	140.83	odd	4
112.3.c.c.97.4		4	20.3	even	4
392.3.o.b.129.1		8	35.33	even	12
392.3.o.b.129.4		8	35.23	odd	12
392.3.o.b.313.1		8	35.18	odd	12
392.3.o.b.313.4		8	35.3	even	12
448.3.c.e.321.1		4	40.3	even	4
448.3.c.e.321.4		4	280.83	odd	4
448.3.c.f.321.1		4	280.13	even	4
448.3.c.f.321.4		4	40.13	odd	4
504.3.f.a.433.2		4	105.83	odd	4
504.3.f.a.433.3		4	15.8	even	4
784.3.s.f.129.1		8	140.23	even	12
784.3.s.f.129.4		8	140.103	odd	12
784.3.s.f.705.1		8	140.3	odd	12
784.3.s.f.705.4		8	140.123	even	12
1008.3.f.h.433.2		4	420.83	even	4
1008.3.f.h.433.3		4	60.23	odd	4
1400.3.f.a.601.1		4	35.27	even	4
1400.3.f.a.601.4		4	5.2	odd	4
1400.3.p.a.1049.1		8	1.1	even	1	trivial
1400.3.p.a.1049.2		8	35.34	odd	2	inner
1400.3.p.a.1049.7		8	7.6	odd	2	inner
1400.3.p.a.1049.8		8	5.4	even	2	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:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1400.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.69552 q^{3} +(6.75699 - 1.82843i) q^{7} +4.65685 q^{9} +O(q^{10})\)	\(q-3.69552 q^{3} +(6.75699 - 1.82843i) q^{7} +4.65685 q^{9} +13.3137 q^{11} +21.5391 q^{13} +6.12293 q^{17} -19.7457i q^{19} +(-24.9706 + 6.75699i) q^{21} +28.6274i q^{23} +16.0502 q^{27} -37.3137 q^{29} +20.9050i q^{31} -49.2011 q^{33} -23.9411i q^{37} -79.5980 q^{39} -70.1061i q^{41} +46.0000i q^{43} -1.05053 q^{47} +(42.3137 - 24.7093i) q^{49} -22.6274 q^{51} +15.3726i q^{53} +72.9706i q^{57} -14.8909i q^{59} +86.7903i q^{61} +(31.4663 - 8.51472i) q^{63} +24.6274i q^{67} -105.793i q^{69} -45.0294 q^{71} +51.7373 q^{73} +(89.9605 - 24.3431i) q^{77} +33.7157 q^{79} -101.225 q^{81} +77.3883 q^{83} +137.893 q^{87} -78.7652i q^{89} +(145.539 - 39.3826i) q^{91} -77.2548i q^{93} +90.1781 q^{97} +62.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} + 16 q^{11} - 64 q^{21} - 208 q^{29} - 320 q^{39} + 248 q^{49} - 496 q^{71} + 496 q^{79} - 312 q^{81} + 576 q^{91} + 496 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 + 16 * q^11 - 64 * q^21 - 208 * q^29 - 320 * q^39 + 248 * q^49 - 496 * q^71 + 496 * q^79 - 312 * q^81 + 576 * q^91 + 496 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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