LMFDB - Embedded newform 2736.3.b.e.2735.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(2735,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2736.2735");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2735.9
Character	$\chi$	$=$	2736.2735
Dual form			2736.3.b.e.2735.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.26257i q^{5} +8.06963i q^{7} +O(q^{10})$	$q-1.26257i q^{5} +8.06963i q^{7} -8.70004 q^{11} -16.8236i q^{13} -4.74680i q^{17} +(-5.61192 + 18.1523i) q^{19} -16.7960 q^{23} +23.4059 q^{25} +51.4472 q^{29} +27.6566 q^{31} +10.1885 q^{35} +24.8546i q^{37} -13.7463 q^{41} -5.53897i q^{43} -47.0118 q^{47} -16.1189 q^{49} -105.283 q^{53} +10.9844i q^{55} -2.94791i q^{59} -1.19469 q^{61} -21.2411 q^{65} -123.333 q^{67} +99.1425i q^{71} -79.3625 q^{73} -70.2061i q^{77} +122.293 q^{79} -90.1340 q^{83} -5.99318 q^{85} +10.0459 q^{89} +135.760 q^{91} +(22.9186 + 7.08546i) q^{95} +84.1666i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 96 q^{25} + 336 q^{49} - 256 q^{61} - 560 q^{73} - 752 q^{85}+O(q^{100}) $ 32 * q + 96 * q^25 + 336 * q^49 - 256 * q^61 - 560 * q^73 - 752 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	1.26257i		−	0.252515i	−0.991998	−	0.126257i	$-0.959703\pi$
$5$		−	1.26257i		−	0.252515i	0.991998	−	0.126257i	$-0.0402965\pi$
$6$		0			0
$7$			8.06963i			1.15280i	0.817166	+	0.576402i	$0.195544\pi$
$7$			8.06963i			1.15280i	−0.817166	+	0.576402i	$0.804456\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−8.70004			−0.790913			−0.395456	−	0.918485i	$-0.629414\pi$
$11$	−8.70004			−0.790913			−0.395456	+	0.918485i	$0.629414\pi$
$12$		0			0
$13$		−	16.8236i		−	1.29413i	−0.762437	−	0.647063i	$-0.775997\pi$
$13$		−	16.8236i		−	1.29413i	0.762437	−	0.647063i	$-0.224003\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.74680i		−	0.279223i	−0.990206	−	0.139612i	$-0.955415\pi$
$17$		−	4.74680i		−	0.279223i	0.990206	−	0.139612i	$-0.0445854\pi$
$18$		0			0
$19$	−5.61192	+	18.1523i	−0.295364	+	0.955385i
$19$	−5.61192	+	18.1523i	−0.295364	+	0.955385i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−16.7960			−0.730260			−0.365130	−	0.930957i	$-0.618975\pi$
$23$	−16.7960			−0.730260			−0.365130	+	0.930957i	$0.618975\pi$
$24$		0			0
$25$	23.4059			0.936236
$26$		0			0
$27$		0			0
$28$		0			0
$29$	51.4472			1.77404			0.887021	−	0.461730i	$-0.152771\pi$
$29$	51.4472			1.77404			0.887021	+	0.461730i	$0.152771\pi$
$30$		0			0
$31$	27.6566			0.892150			0.446075	−	0.894996i	$-0.352822\pi$
$31$	27.6566			0.892150			0.446075	+	0.894996i	$0.352822\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	10.1885			0.291100
$36$		0			0
$37$			24.8546i			0.671746i	0.941907	+	0.335873i	$0.109031\pi$
$37$			24.8546i			0.671746i	−0.941907	+	0.335873i	$0.890969\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−13.7463			−0.335276			−0.167638	−	0.985849i	$-0.553614\pi$
$41$	−13.7463			−0.335276			−0.167638	+	0.985849i	$0.553614\pi$
$42$		0			0
$43$		−	5.53897i		−	0.128813i	−0.997924	−	0.0644067i	$-0.979485\pi$
$43$		−	5.53897i		−	0.128813i	0.997924	−	0.0644067i	$-0.0205155\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−47.0118			−1.00025			−0.500125	−	0.865953i	$-0.666713\pi$
$47$	−47.0118			−1.00025			−0.500125	+	0.865953i	$0.666713\pi$
$48$		0			0
$49$	−16.1189			−0.328957
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−105.283			−1.98648			−0.993238	−	0.116098i	$-0.962961\pi$
$53$	−105.283			−1.98648			−0.993238	+	0.116098i	$0.962961\pi$
$54$		0			0
$55$			10.9844i			0.199717i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	2.94791i		−	0.0499646i	−0.999688	−	0.0249823i	$-0.992047\pi$
$59$		−	2.94791i		−	0.0499646i	0.999688	−	0.0249823i	$-0.00795294\pi$
$60$		0			0
$61$	−1.19469			−0.0195851			−0.00979257	−	0.999952i	$-0.503117\pi$
$61$	−1.19469			−0.0195851			−0.00979257	+	0.999952i	$0.503117\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−21.2411			−0.326786
$66$		0			0
$67$	−123.333			−1.84079			−0.920397	−	0.390986i	$-0.872134\pi$
$67$	−123.333			−1.84079			−0.920397	+	0.390986i	$0.872134\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			99.1425i			1.39637i	0.715916	+	0.698187i	$0.246010\pi$
$71$			99.1425i			1.39637i	−0.715916	+	0.698187i	$0.753990\pi$
$72$		0			0
$73$	−79.3625			−1.08716			−0.543579	−	0.839358i	$-0.682931\pi$
$73$	−79.3625			−1.08716			−0.543579	+	0.839358i	$0.682931\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	70.2061i		−	0.911767i
$78$		0			0
$79$	122.293			1.54801			0.774007	−	0.633177i	$-0.218249\pi$
$79$	122.293			1.54801			0.774007	+	0.633177i	$0.218249\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−90.1340			−1.08595			−0.542976	−	0.839748i	$-0.682703\pi$
$83$	−90.1340			−1.08595			−0.542976	+	0.839748i	$0.682703\pi$
$84$		0			0
$85$	−5.99318			−0.0705080
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.0459			0.112875			0.0564375	−	0.998406i	$-0.482026\pi$
$89$	10.0459			0.112875			0.0564375	+	0.998406i	$0.482026\pi$
$90$		0			0
$91$	135.760			1.49187
$92$		0			0
$93$		0			0
$94$		0			0
$95$	22.9186	+	7.08546i	0.241249	+	0.0745838i
$96$		0			0
$97$			84.1666i			0.867697i	0.900986	+	0.433848i	$0.142845\pi$
$97$			84.1666i			0.867697i	−0.900986	+	0.433848i	$0.857155\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	158.381i		−	1.56813i	−0.620681	−	0.784063i	$-0.713144\pi$
$101$		−	158.381i		−	1.56813i	0.620681	−	0.784063i	$-0.286856\pi$
$102$		0			0
$103$	−134.194			−1.30285			−0.651426	−	0.758712i	$-0.725829\pi$
$103$	−134.194			−1.30285			−0.651426	+	0.758712i	$0.725829\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	103.561i		−	0.967862i	−0.875106	−	0.483931i	$-0.839209\pi$
$107$		−	103.561i		−	0.967862i	0.875106	−	0.483931i	$-0.160791\pi$
$108$		0			0
$109$			25.6647i			0.235456i	0.993046	+	0.117728i	$0.0375611\pi$
$109$			25.6647i			0.235456i	−0.993046	+	0.117728i	$0.962439\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	47.7467			0.422537			0.211269	−	0.977428i	$-0.432240\pi$
$113$	47.7467			0.422537			0.211269	+	0.977428i	$0.432240\pi$
$114$		0			0
$115$			21.2062i			0.184401i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	38.3049			0.321890
$120$		0			0
$121$	−45.3093			−0.374457
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	61.1160i		−	0.488928i
$126$		0			0
$127$	27.9359			0.219968			0.109984	−	0.993933i	$-0.464920\pi$
$127$	27.9359			0.219968			0.109984	+	0.993933i	$0.464920\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.98464			−0.0380507			−0.0190253	−	0.999819i	$-0.506056\pi$
$131$	−4.98464			−0.0380507			−0.0190253	+	0.999819i	$0.506056\pi$
$132$		0			0
$133$	−146.482	−	45.2861i	−1.10137	−	0.340497i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			28.2287i			0.206049i	0.994679	+	0.103024i	$0.0328519\pi$
$137$			28.2287i			0.206049i	−0.994679	+	0.103024i	$0.967148\pi$
$138$		0			0
$139$		−	155.316i		−	1.11738i	−0.829375	−	0.558692i	$-0.811303\pi$
$139$		−	155.316i		−	1.11738i	0.829375	−	0.558692i	$-0.188697\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			146.366i			1.02354i
$144$		0			0
$145$		−	64.9559i		−	0.447972i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	247.641i		−	1.66202i	−0.556256	−	0.831011i	$-0.687762\pi$
$149$		−	247.641i		−	1.66202i	0.556256	−	0.831011i	$-0.312238\pi$
$150$		0			0
$151$	−133.915			−0.886851			−0.443426	−	0.896311i	$-0.646237\pi$
$151$	−133.915			−0.886851			−0.443426	+	0.896311i	$0.646237\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	34.9186i		−	0.225281i
$156$		0			0
$157$	−233.833			−1.48938			−0.744692	−	0.667409i	$-0.767404\pi$
$157$	−233.833			−1.48938			−0.744692	+	0.667409i	$0.767404\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	135.537i		−	0.841847i
$162$		0			0
$163$			159.126i			0.976233i	0.872778	+	0.488117i	$0.162316\pi$
$163$			159.126i			0.976233i	−0.872778	+	0.488117i	$0.837684\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	230.137i		−	1.37807i	−0.724730	−	0.689033i	$-0.758035\pi$
$167$		−	230.137i		−	1.37807i	0.724730	−	0.689033i	$-0.241965\pi$
$168$		0			0
$169$	−114.035			−0.674761
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−247.028			−1.42791			−0.713953	−	0.700193i	$-0.753097\pi$
$173$	−247.028			−1.42791			−0.713953	+	0.700193i	$0.753097\pi$
$174$		0			0
$175$			188.877i			1.07930i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	208.995i		−	1.16757i	−0.811909	−	0.583784i	$-0.801572\pi$
$179$		−	208.995i		−	1.16757i	0.811909	−	0.583784i	$-0.198428\pi$
$180$		0			0
$181$			246.275i			1.36064i	0.732917	+	0.680319i	$0.238159\pi$
$181$			246.275i			1.36064i	−0.732917	+	0.680319i	$0.761841\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	31.3808			0.169626
$186$		0			0
$187$			41.2973i			0.220841i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−56.5720			−0.296188			−0.148094	−	0.988973i	$-0.547314\pi$
$191$	−56.5720			−0.296188			−0.148094	+	0.988973i	$0.547314\pi$
$192$		0			0
$193$		−	297.205i		−	1.53992i	−0.638090	−	0.769962i	$-0.720275\pi$
$193$		−	297.205i		−	1.53992i	0.638090	−	0.769962i	$-0.279725\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	324.486i		−	1.64714i	−0.567216	−	0.823569i	$-0.691980\pi$
$197$		−	324.486i		−	1.64714i	0.567216	−	0.823569i	$-0.308020\pi$
$198$		0			0
$199$			240.153i			1.20680i	0.797440	+	0.603399i	$0.206187\pi$
$199$			240.153i			1.20680i	−0.797440	+	0.603399i	$0.793813\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			415.160i			2.04512i
$204$		0			0
$205$			17.3558i			0.0846623i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	48.8239	−	157.926i	0.233607	−	0.755626i
$210$		0			0
$211$	120.368			0.570463			0.285231	−	0.958459i	$-0.407930\pi$
$211$	120.368			0.570463			0.285231	+	0.958459i	$0.407930\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.99336			−0.0325273
$216$		0			0
$217$			223.179i			1.02847i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−79.8584			−0.361350
$222$		0			0
$223$	262.781			1.17839			0.589195	−	0.807991i	$-0.299445\pi$
$223$	262.781			1.17839			0.589195	+	0.807991i	$0.299445\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			312.674i			1.37742i	0.725037	+	0.688710i	$0.241823\pi$
$227$			312.674i			1.37742i	−0.725037	+	0.688710i	$0.758177\pi$
$228$		0			0
$229$	−252.921			−1.10446			−0.552230	−	0.833692i	$-0.686223\pi$
$229$	−252.921			−1.10446			−0.552230	+	0.833692i	$0.686223\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			112.804i			0.484137i	0.970259	+	0.242068i	$0.0778258\pi$
$233$			112.804i			0.484137i	−0.970259	+	0.242068i	$0.922174\pi$
$234$		0			0
$235$			59.3559i			0.252578i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−432.191			−1.80833			−0.904165	−	0.427183i	$-0.859506\pi$
$239$	−432.191			−1.80833			−0.904165	+	0.427183i	$0.859506\pi$
$240$		0			0
$241$			262.386i			1.08874i	0.838846	+	0.544369i	$0.183231\pi$
$241$			262.386i			1.08874i	−0.838846	+	0.544369i	$0.816769\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			20.3513i			0.0830665i
$246$		0			0
$247$	305.388	+	94.4129i	1.23639	+	0.382238i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−93.0469			−0.370705			−0.185352	−	0.982672i	$-0.559343\pi$
$251$	−93.0469			−0.370705			−0.185352	+	0.982672i	$0.559343\pi$
$252$		0			0
$253$	146.126			0.577572
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−354.307			−1.37863			−0.689313	−	0.724464i	$-0.742087\pi$
$257$	−354.307			−1.37863			−0.689313	+	0.724464i	$0.742087\pi$
$258$		0			0
$259$	−200.567			−0.774392
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−425.725			−1.61873			−0.809363	−	0.587309i	$-0.800188\pi$
$263$	−425.725			−1.61873			−0.809363	+	0.587309i	$0.800188\pi$
$264$		0			0
$265$			132.928i			0.501614i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	284.588			1.05795			0.528974	−	0.848638i	$-0.322577\pi$
$269$	284.588			1.05795			0.528974	+	0.848638i	$0.322577\pi$
$270$		0			0
$271$			167.744i			0.618983i	0.950902	+	0.309491i	$0.100159\pi$
$271$			167.744i			0.618983i	−0.950902	+	0.309491i	$0.899841\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−203.632			−0.740481
$276$		0			0
$277$	−283.432			−1.02322			−0.511610	−	0.859218i	$-0.670951\pi$
$277$	−283.432			−1.02322			−0.511610	+	0.859218i	$0.670951\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−151.397			−0.538781			−0.269390	−	0.963031i	$-0.586822\pi$
$281$	−151.397			−0.538781			−0.269390	+	0.963031i	$0.586822\pi$
$282$		0			0
$283$		−	208.913i		−	0.738208i	−0.929388	−	0.369104i	$-0.879665\pi$
$283$		−	208.913i		−	0.738208i	0.929388	−	0.369104i	$-0.120335\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	110.928i		−	0.386508i
$288$		0			0
$289$	266.468			0.922034
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−177.412			−0.605503			−0.302752	−	0.953070i	$-0.597905\pi$
$293$	−177.412			−0.605503			−0.302752	+	0.953070i	$0.597905\pi$
$294$		0			0
$295$	−3.72195			−0.0126168
$296$		0			0
$297$		0			0
$298$		0			0
$299$			282.569i			0.945048i
$300$		0			0
$301$	44.6975			0.148497
$302$		0			0
$303$		0			0
$304$		0			0
$305$			1.50839i			0.00494554i
$306$		0			0
$307$	66.0105			0.215018			0.107509	−	0.994204i	$-0.465713\pi$
$307$	66.0105			0.215018			0.107509	+	0.994204i	$0.465713\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−398.103			−1.28007			−0.640036	−	0.768345i	$-0.721081\pi$
$311$	−398.103			−1.28007			−0.640036	+	0.768345i	$0.721081\pi$
$312$		0			0
$313$	−270.829			−0.865270			−0.432635	−	0.901569i	$-0.642416\pi$
$313$	−270.829			−0.865270			−0.432635	+	0.901569i	$0.642416\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	245.673			0.774994			0.387497	−	0.921871i	$-0.373340\pi$
$317$	245.673			0.774994			0.387497	+	0.921871i	$0.373340\pi$
$318$		0			0
$319$	−447.593			−1.40311
$320$		0			0
$321$		0			0
$322$		0			0
$323$	86.1654	+	26.6386i	0.266766	+	0.0824726i
$324$		0			0
$325$		−	393.772i		−	1.21161i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	379.368i		−	1.15309i
$330$		0			0
$331$	−115.633			−0.349345			−0.174673	−	0.984627i	$-0.555887\pi$
$331$	−115.633			−0.349345			−0.174673	+	0.984627i	$0.555887\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			155.717i			0.464828i
$336$		0			0
$337$		−	190.240i		−	0.564509i	−0.959340	−	0.282255i	$-0.908918\pi$
$337$		−	190.240i		−	0.564509i	0.959340	−	0.282255i	$-0.0910823\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−240.614			−0.705613
$342$		0			0
$343$			265.338i			0.773581i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	203.327			0.585958			0.292979	−	0.956119i	$-0.405353\pi$
$347$	203.327			0.585958			0.292979	+	0.956119i	$0.405353\pi$
$348$		0			0
$349$	−566.664			−1.62368			−0.811839	−	0.583882i	$-0.801533\pi$
$349$	−566.664			−1.62368			−0.811839	+	0.583882i	$0.801533\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	433.568i		−	1.22824i	−0.789213	−	0.614119i	$-0.789511\pi$
$353$		−	433.568i		−	1.22824i	0.789213	−	0.614119i	$-0.210489\pi$
$354$		0			0
$355$	125.175			0.352605
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−87.5008			−0.243735			−0.121867	−	0.992546i	$-0.538888\pi$
$359$	−87.5008			−0.243735			−0.121867	+	0.992546i	$0.538888\pi$
$360$		0			0
$361$	−298.013	−	203.739i	−0.825520	−	0.564373i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			100.201i			0.274523i
$366$		0			0
$367$			57.5734i			0.156876i	0.996919	+	0.0784379i	$0.0249932\pi$
$367$			57.5734i			0.156876i	−0.996919	+	0.0784379i	$0.975007\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	849.596i		−	2.29002i
$372$		0			0
$373$			473.751i			1.27011i	0.772467	+	0.635054i	$0.219022\pi$
$373$			473.751i			1.27011i	−0.772467	+	0.635054i	$0.780978\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	865.529i		−	2.29583i
$378$		0			0
$379$	214.278			0.565377			0.282689	−	0.959212i	$-0.408774\pi$
$379$	214.278			0.565377			0.282689	+	0.959212i	$0.408774\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			391.583i			1.02241i	0.859459	+	0.511205i	$0.170801\pi$
$383$			391.583i			1.02241i	−0.859459	+	0.511205i	$0.829199\pi$
$384$		0			0
$385$	−88.6404			−0.230235
$386$		0			0
$387$		0			0
$388$		0			0
$389$			8.21192i			0.0211103i	0.999944	+	0.0105552i	$0.00335988\pi$
$389$			8.21192i			0.0211103i	−0.999944	+	0.0105552i	$0.996640\pi$
$390$		0			0
$391$			79.7271i			0.203906i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	154.404i		−	0.390897i
$396$		0			0
$397$	583.868			1.47070			0.735350	−	0.677687i	$-0.237018\pi$
$397$	583.868			1.47070			0.735350	+	0.677687i	$0.237018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	418.320			1.04319			0.521596	−	0.853193i	$-0.325337\pi$
$401$	418.320			1.04319			0.521596	+	0.853193i	$0.325337\pi$
$402$		0			0
$403$		−	465.285i		−	1.15455i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	216.236i		−	0.531293i
$408$		0			0
$409$		−	367.032i		−	0.897389i	−0.893685	−	0.448695i	$-0.851889\pi$
$409$		−	367.032i		−	0.897389i	0.893685	−	0.448695i	$-0.148111\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	23.7885			0.0575994
$414$		0			0
$415$			113.801i			0.274219i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	428.636			1.02300			0.511499	−	0.859284i	$-0.329090\pi$
$419$	428.636			1.02300			0.511499	+	0.859284i	$0.329090\pi$
$420$		0			0
$421$			665.331i			1.58036i	0.612876	+	0.790179i	$0.290013\pi$
$421$			665.331i			1.58036i	−0.612876	+	0.790179i	$0.709987\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	111.103i		−	0.261419i
$426$		0			0
$427$		−	9.64073i		−	0.0225778i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			305.584i			0.709012i	0.935054	+	0.354506i	$0.115351\pi$
$431$			305.584i			0.709012i	−0.935054	+	0.354506i	$0.884649\pi$
$432$		0			0
$433$			561.349i			1.29642i	0.761462	+	0.648209i	$0.224482\pi$
$433$			561.349i			1.29642i	−0.761462	+	0.648209i	$0.775518\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	94.2577	−	304.886i	0.215693	−	0.697679i
$438$		0			0
$439$	27.8653			0.0634745			0.0317372	−	0.999496i	$-0.489896\pi$
$439$	27.8653			0.0634745			0.0317372	+	0.999496i	$0.489896\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−29.5128			−0.0666203			−0.0333102	−	0.999445i	$-0.510605\pi$
$443$	−29.5128			−0.0666203			−0.0333102	+	0.999445i	$0.510605\pi$
$444$		0			0
$445$		−	12.6837i		−	0.0285026i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−115.556			−0.257363			−0.128682	−	0.991686i	$-0.541075\pi$
$449$	−115.556			−0.257363			−0.128682	+	0.991686i	$0.541075\pi$
$450$		0			0
$451$	119.594			0.265174
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	171.408i		−	0.376720i
$456$		0			0
$457$	−337.852			−0.739283			−0.369641	−	0.929175i	$-0.620519\pi$
$457$	−337.852			−0.739283			−0.369641	+	0.929175i	$0.620519\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			435.457i			0.944593i	0.881440	+	0.472296i	$0.156575\pi$
$461$			435.457i			0.944593i	−0.881440	+	0.472296i	$0.843425\pi$
$462$		0			0
$463$			206.395i			0.445779i	0.974844	+	0.222889i	$0.0715488\pi$
$463$			206.395i			0.445779i	−0.974844	+	0.222889i	$0.928451\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	727.030			1.55681			0.778405	−	0.627763i	$-0.216029\pi$
$467$	727.030			1.55681			0.778405	+	0.627763i	$0.216029\pi$
$468$		0			0
$469$		−	995.253i		−	2.12207i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			48.1893i			0.101880i
$474$		0			0
$475$	−131.352	+	424.871i	−0.276531	+	0.894466i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	485.062			1.01266			0.506328	−	0.862341i	$-0.331002\pi$
$479$	485.062			1.01266			0.506328	+	0.862341i	$0.331002\pi$
$480$		0			0
$481$	418.145			0.869324
$482$		0			0
$483$		0			0
$484$		0			0
$485$	106.267			0.219106
$486$		0			0
$487$	23.9347			0.0491472			0.0245736	−	0.999698i	$-0.492177\pi$
$487$	23.9347			0.0491472			0.0245736	+	0.999698i	$0.492177\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−370.960			−0.755519			−0.377760	−	0.925904i	$-0.623305\pi$
$491$	−370.960			−0.755519			−0.377760	+	0.925904i	$0.623305\pi$
$492$		0			0
$493$		−	244.209i		−	0.495354i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−800.043			−1.60975
$498$		0			0
$499$		−	227.390i		−	0.455691i	−0.973697	−	0.227846i	$-0.926832\pi$
$499$		−	227.390i		−	0.455691i	0.973697	−	0.227846i	$-0.0731681\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−262.806			−0.522476			−0.261238	−	0.965274i	$-0.584131\pi$
$503$	−262.806			−0.522476			−0.261238	+	0.965274i	$0.584131\pi$
$504$		0			0
$505$	−199.967			−0.395975
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−170.015			−0.334018			−0.167009	−	0.985955i	$-0.553411\pi$
$509$	−170.015			−0.334018			−0.167009	+	0.985955i	$0.553411\pi$
$510$		0			0
$511$		−	640.426i		−	1.25328i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			169.430i			0.328990i
$516$		0			0
$517$	409.004			0.791111
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−624.724			−1.19909			−0.599543	−	0.800342i	$-0.704651\pi$
$521$	−624.724			−1.19909			−0.599543	+	0.800342i	$0.704651\pi$
$522$		0			0
$523$	−7.61339			−0.0145572			−0.00727858	−	0.999974i	$-0.502317\pi$
$523$	−7.61339			−0.0145572			−0.00727858	+	0.999974i	$0.502317\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	131.281i		−	0.249109i
$528$		0			0
$529$	−246.895			−0.466720
$530$		0			0
$531$		0			0
$532$		0			0
$533$			231.263i			0.433890i
$534$		0			0
$535$	−130.754			−0.244399
$536$		0			0
$537$		0			0
$538$		0			0
$539$	140.235			0.260176
$540$		0			0
$541$	262.599			0.485395			0.242697	−	0.970102i	$-0.421968\pi$
$541$	262.599			0.485395			0.242697	+	0.970102i	$0.421968\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	32.4036			0.0594561
$546$		0			0
$547$	−699.002			−1.27788			−0.638942	−	0.769255i	$-0.720628\pi$
$547$	−699.002			−1.27788			−0.638942	+	0.769255i	$0.720628\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−288.717	+	933.885i	−0.523988	+	1.69489i
$552$		0			0
$553$			986.860i			1.78456i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			845.806i			1.51850i	0.650798	+	0.759251i	$0.274434\pi$
$557$			845.806i			1.51850i	−0.650798	+	0.759251i	$0.725566\pi$
$558$		0			0
$559$	−93.1857			−0.166701
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	529.105i		−	0.939796i	−0.882721	−	0.469898i	$-0.844291\pi$
$563$		−	529.105i		−	0.939796i	0.882721	−	0.469898i	$-0.155709\pi$
$564$		0			0
$565$		−	60.2838i		−	0.106697i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	833.171			1.46427			0.732136	−	0.681158i	$-0.238523\pi$
$569$	833.171			1.46427			0.732136	+	0.681158i	$0.238523\pi$
$570$		0			0
$571$		−	558.316i		−	0.977786i	−0.872344	−	0.488893i	$-0.837401\pi$
$571$		−	558.316i		−	0.977786i	0.872344	−	0.488893i	$-0.162599\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−393.125			−0.683696
$576$		0			0
$577$	−619.903			−1.07435			−0.537177	−	0.843469i	$-0.680509\pi$
$577$	−619.903			−1.07435			−0.537177	+	0.843469i	$0.680509\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	727.348i		−	1.25189i
$582$		0			0
$583$	915.968			1.57113
$584$		0			0
$585$		0			0
$586$		0			0
$587$	871.024			1.48386			0.741928	−	0.670479i	$-0.233911\pi$
$587$	871.024			1.48386			0.741928	+	0.670479i	$0.233911\pi$
$588$		0			0
$589$	−155.207	+	502.032i	−0.263509	+	0.852346i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			1027.34i			1.73244i	0.499661	+	0.866221i	$0.333458\pi$
$593$			1027.34i			1.73244i	−0.499661	+	0.866221i	$0.666542\pi$
$594$		0			0
$595$		−	48.3628i		−	0.0812820i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	218.920i		−	0.365477i	−0.983162	−	0.182738i	$-0.941504\pi$
$599$		−	218.920i		−	0.365477i	0.983162	−	0.182738i	$-0.0584961\pi$
$600$		0			0
$601$		−	457.162i		−	0.760668i	−0.924849	−	0.380334i	$-0.875809\pi$
$601$		−	457.162i		−	0.760668i	0.924849	−	0.380334i	$-0.124191\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			57.2063i			0.0945559i
$606$		0			0
$607$	811.872			1.33752			0.668758	−	0.743480i	$-0.266826\pi$
$607$	811.872			1.33752			0.668758	+	0.743480i	$0.266826\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			790.909i			1.29445i
$612$		0			0
$613$	559.871			0.913329			0.456665	−	0.889639i	$-0.349044\pi$
$613$	559.871			0.913329			0.456665	+	0.889639i	$0.349044\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	417.042i		−	0.675919i	−0.941161	−	0.337959i	$-0.890263\pi$
$617$		−	417.042i		−	0.675919i	0.941161	−	0.337959i	$-0.109737\pi$
$618$		0			0
$619$		−	220.784i		−	0.356679i	−0.983969	−	0.178340i	$-0.942927\pi$
$619$		−	220.784i		−	0.356679i	0.983969	−	0.178340i	$-0.0570726\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			81.0665i			0.130123i
$624$		0			0
$625$	507.984			0.812775
$626$		0			0
$627$		0			0
$628$		0			0
$629$	117.980			0.187567
$630$		0			0
$631$			603.444i			0.956329i	0.878270	+	0.478165i	$0.158698\pi$
$631$			603.444i			0.956329i	−0.878270	+	0.478165i	$0.841302\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	35.2712i		−	0.0555451i
$636$		0			0
$637$			271.178i			0.425712i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	826.584			1.28952			0.644761	−	0.764384i	$-0.276957\pi$
$641$	826.584			1.28952			0.644761	+	0.764384i	$0.276957\pi$
$642$		0			0
$643$		−	293.326i		−	0.456183i	−0.973640	−	0.228091i	$-0.926751\pi$
$643$		−	293.326i		−	0.456183i	0.973640	−	0.228091i	$-0.0732485\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−388.399			−0.600308			−0.300154	−	0.953891i	$-0.597038\pi$
$647$	−388.399			−0.600308			−0.300154	+	0.953891i	$0.597038\pi$
$648$		0			0
$649$			25.6469i			0.0395176i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	23.6198i		−	0.0361712i	−0.999836	−	0.0180856i	$-0.994243\pi$
$653$		−	23.6198i		−	0.0361712i	0.999836	−	0.0180856i	$-0.00575714\pi$
$654$		0			0
$655$			6.29348i			0.00960836i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1180.14i		−	1.79080i	−0.445258	−	0.895402i	$-0.646888\pi$
$659$		−	1180.14i		−	1.79080i	0.445258	−	0.895402i	$-0.353112\pi$
$660$		0			0
$661$		−	633.074i		−	0.957752i	−0.877883	−	0.478876i	$-0.841044\pi$
$661$		−	633.074i		−	0.957752i	0.877883	−	0.478876i	$-0.158956\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−57.1770	+	184.945i	−0.0859805	+	0.278113i
$666$		0			0
$667$	−864.106			−1.29551
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.3939			0.0154901
$672$		0			0
$673$		−	235.149i		−	0.349404i	−0.984621	−	0.174702i	$-0.944104\pi$
$673$		−	235.149i		−	0.349404i	0.984621	−	0.174702i	$-0.0558963\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−245.673			−0.362885			−0.181442	−	0.983402i	$-0.558077\pi$
$677$	−245.673			−0.362885			−0.181442	+	0.983402i	$0.558077\pi$
$678$		0			0
$679$	−679.193			−1.00028
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	918.996i		−	1.34553i	−0.739857	−	0.672764i	$-0.765107\pi$
$683$		−	918.996i		−	1.34553i	0.739857	−	0.672764i	$-0.234893\pi$
$684$		0			0
$685$	35.6408			0.0520303
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1771.25i			2.57075i
$690$		0			0
$691$			523.353i			0.757386i	0.925522	+	0.378693i	$0.123626\pi$
$691$			523.353i			0.757386i	−0.925522	+	0.378693i	$0.876374\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−196.099			−0.282156
$696$		0			0
$697$			65.2511i			0.0936170i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1167.81i			1.66592i	0.553333	+	0.832960i	$0.313356\pi$
$701$			1167.81i			1.66592i	−0.553333	+	0.832960i	$0.686644\pi$
$702$		0			0
$703$	−451.169	−	139.482i	−0.641776	−	0.198410i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1278.07			1.80774
$708$		0			0
$709$	597.363			0.842544			0.421272	−	0.906934i	$-0.361584\pi$
$709$	597.363			0.842544			0.421272	+	0.906934i	$0.361584\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−464.520			−0.651501
$714$		0			0
$715$	184.798			0.258459
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1145.94			−1.59379			−0.796897	−	0.604115i	$-0.793527\pi$
$719$	−1145.94			−1.59379			−0.796897	+	0.604115i	$0.793527\pi$
$720$		0			0
$721$		−	1082.89i		−	1.50193i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1204.17			1.66092
$726$		0			0
$727$			798.756i			1.09870i	0.835592	+	0.549351i	$0.185125\pi$
$727$			798.756i			1.09870i	−0.835592	+	0.549351i	$0.814875\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−26.2924			−0.0359677
$732$		0			0
$733$	434.360			0.592578			0.296289	−	0.955098i	$-0.404251\pi$
$733$	434.360			0.592578			0.296289	+	0.955098i	$0.404251\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1073.00			1.45591
$738$		0			0
$739$		−	827.915i		−	1.12032i	−0.828385	−	0.560159i	$-0.810740\pi$
$739$		−	827.915i		−	1.12032i	0.828385	−	0.560159i	$-0.189260\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			699.042i			0.940837i	0.882443	+	0.470419i	$0.155897\pi$
$743$			699.042i			0.940837i	−0.882443	+	0.470419i	$0.844103\pi$
$744$		0			0
$745$	−312.665			−0.419685
$746$		0			0
$747$		0			0
$748$		0			0
$749$	835.700			1.11575
$750$		0			0
$751$	198.322			0.264078			0.132039	−	0.991245i	$-0.457848\pi$
$751$	198.322			0.264078			0.132039	+	0.991245i	$0.457848\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			169.077i			0.223943i
$756$		0			0
$757$	206.137			0.272308			0.136154	−	0.990688i	$-0.456526\pi$
$757$	206.137			0.272308			0.136154	+	0.990688i	$0.456526\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			768.017i			1.00922i	0.863347	+	0.504610i	$0.168364\pi$
$761$			768.017i			1.00922i	−0.863347	+	0.504610i	$0.831636\pi$
$762$		0			0
$763$	−207.105			−0.271435
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−49.5946			−0.0646605
$768$		0			0
$769$	1123.13			1.46050			0.730252	−	0.683178i	$-0.239403\pi$
$769$	1123.13			1.46050			0.730252	+	0.683178i	$0.239403\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	119.490			0.154580			0.0772899	−	0.997009i	$-0.475373\pi$
$773$	119.490			0.154580			0.0772899	+	0.997009i	$0.475373\pi$
$774$		0			0
$775$	647.329			0.835263
$776$		0			0
$777$		0			0
$778$		0			0
$779$	77.1433	−	249.528i	0.0990286	−	0.320318i
$780$		0			0
$781$		−	862.544i		−	1.10441i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			295.232i			0.376091i
$786$		0			0
$787$	245.366			0.311773			0.155887	−	0.987775i	$-0.450177\pi$
$787$	245.366			0.311773			0.155887	+	0.987775i	$0.450177\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			385.298i			0.487103i
$792$		0			0
$793$			20.0991i			0.0253456i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1175.97			−1.47550			−0.737748	−	0.675076i	$-0.764111\pi$
$797$	−1175.97			−1.47550			−0.737748	+	0.675076i	$0.764111\pi$
$798$		0			0
$799$			223.155i			0.279293i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	690.457			0.859847
$804$		0			0
$805$	−171.126			−0.212579
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	52.0914i		−	0.0643899i	−0.999482	−	0.0321950i	$-0.989750\pi$
$809$		−	52.0914i		−	0.0643899i	0.999482	−	0.0321950i	$-0.0102497\pi$
$810$		0			0
$811$	−67.1541			−0.0828040			−0.0414020	−	0.999143i	$-0.513182\pi$
$811$	−67.1541			−0.0828040			−0.0414020	+	0.999143i	$0.513182\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	200.908			0.246513
$816$		0			0
$817$	100.545	+	31.0843i	0.123066	+	0.0380468i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			1465.76i			1.78533i	0.450718	+	0.892667i	$0.351168\pi$
$821$			1465.76i			1.78533i	−0.450718	+	0.892667i	$0.648832\pi$
$822$		0			0
$823$		−	187.326i		−	0.227614i	−0.993503	−	0.113807i	$-0.963695\pi$
$823$		−	187.326i		−	0.227614i	0.993503	−	0.113807i	$-0.0363045\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1356.19i			1.63989i	0.572440	+	0.819947i	$0.305997\pi$
$827$			1356.19i			1.63989i	−0.572440	+	0.819947i	$0.694003\pi$
$828$		0			0
$829$			358.872i			0.432898i	0.976294	+	0.216449i	$0.0694475\pi$
$829$			358.872i			0.432898i	−0.976294	+	0.216449i	$0.930553\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			76.5131i			0.0918525i
$834$		0			0
$835$	−290.565			−0.347982
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1317.98i		−	1.57089i	−0.618933	−	0.785444i	$-0.712435\pi$
$839$		−	1317.98i		−	1.57089i	0.618933	−	0.785444i	$-0.287565\pi$
$840$		0			0
$841$	1805.81			2.14722
$842$		0			0
$843$		0			0
$844$		0			0
$845$			143.977i			0.170387i
$846$		0			0
$847$		−	365.629i		−	0.431676i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	417.458i		−	0.490549i
$852$		0			0
$853$	−71.8677			−0.0842529			−0.0421264	−	0.999112i	$-0.513413\pi$
$853$	−71.8677			−0.0842529			−0.0421264	+	0.999112i	$0.513413\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−37.1925			−0.0433984			−0.0216992	−	0.999765i	$-0.506908\pi$
$857$	−37.1925			−0.0433984			−0.0216992	+	0.999765i	$0.506908\pi$
$858$		0			0
$859$		−	1233.82i		−	1.43634i	−0.695868	−	0.718170i	$-0.744980\pi$
$859$		−	1233.82i		−	1.43634i	0.695868	−	0.718170i	$-0.255020\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	640.499i		−	0.742177i	−0.928597	−	0.371089i	$-0.878985\pi$
$863$		−	640.499i		−	0.742177i	0.928597	−	0.371089i	$-0.121015\pi$
$864$		0			0
$865$			311.891i			0.360567i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1063.96			−1.22434
$870$		0			0
$871$			2074.91i			2.38222i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	493.184			0.563638
$876$		0			0
$877$		−	1021.29i		−	1.16453i	−0.812999	−	0.582265i	$-0.802167\pi$
$877$		−	1021.29i		−	1.16453i	0.812999	−	0.582265i	$-0.197833\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			62.6481i			0.0711102i	0.999368	+	0.0355551i	$0.0113199\pi$
$881$			62.6481i			0.0711102i	−0.999368	+	0.0355551i	$0.988680\pi$
$882$		0			0
$883$		−	1051.94i		−	1.19133i	−0.803234	−	0.595664i	$-0.796889\pi$
$883$		−	1051.94i		−	1.19133i	0.803234	−	0.595664i	$-0.203111\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			654.171i			0.737510i	0.929527	+	0.368755i	$0.120216\pi$
$887$			654.171i			0.737510i	−0.929527	+	0.368755i	$0.879784\pi$
$888$		0			0
$889$			225.432i			0.253580i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	263.826	−	853.373i	0.295438	−	0.955624i
$894$		0			0
$895$	−263.871			−0.294828
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1422.86			1.58271
$900$		0			0
$901$			499.758i			0.554671i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	310.941			0.343581
$906$		0			0
$907$	610.951			0.673595			0.336798	−	0.941577i	$-0.390656\pi$
$907$	610.951			0.673595			0.336798	+	0.941577i	$0.390656\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			696.657i			0.764716i	0.924014	+	0.382358i	$0.124888\pi$
$911$			696.657i			0.764716i	−0.924014	+	0.382358i	$0.875112\pi$
$912$		0			0
$913$	784.170			0.858893
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	40.2242i		−	0.0438650i
$918$		0			0
$919$		−	1036.71i		−	1.12809i	−0.825745	−	0.564043i	$-0.809245\pi$
$919$		−	1036.71i		−	1.12809i	0.825745	−	0.564043i	$-0.190755\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1667.94			1.80708
$924$		0			0
$925$			581.745i			0.628913i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			264.280i			0.284477i	0.989832	+	0.142239i	$0.0454301\pi$
$929$			264.280i			0.284477i	−0.989832	+	0.142239i	$0.954570\pi$
$930$		0			0
$931$	90.4579	−	292.595i	0.0971621	−	0.314281i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	52.1409			0.0557657
$936$		0			0
$937$	240.698			0.256882			0.128441	−	0.991717i	$-0.459003\pi$
$937$	240.698			0.256882			0.128441	+	0.991717i	$0.459003\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	626.633			0.665923			0.332961	−	0.942940i	$-0.391952\pi$
$941$	626.633			0.665923			0.332961	+	0.942940i	$0.391952\pi$
$942$		0			0
$943$	230.883			0.244839
$944$		0			0
$945$		0			0
$946$		0			0
$947$	628.381			0.663549			0.331775	−	0.943359i	$-0.392353\pi$
$947$	628.381			0.663549			0.331775	+	0.943359i	$0.392353\pi$
$948$		0			0
$949$			1335.17i			1.40692i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	95.0580			0.0997461			0.0498730	−	0.998756i	$-0.484118\pi$
$953$	95.0580			0.0997461			0.0498730	+	0.998756i	$0.484118\pi$
$954$		0			0
$955$			71.4263i			0.0747919i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−227.795			−0.237534
$960$		0			0
$961$	−196.110			−0.204069
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−375.244			−0.388854
$966$		0			0
$967$		−	1402.53i		−	1.45039i	−0.688543	−	0.725195i	$-0.741750\pi$
$967$		−	1402.53i		−	1.45039i	0.688543	−	0.725195i	$-0.258250\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1559.37i		−	1.60594i	−0.596018	−	0.802971i	$-0.703251\pi$
$971$		−	1559.37i		−	1.60594i	0.596018	−	0.802971i	$-0.296749\pi$
$972$		0			0
$973$	1253.35			1.28813
$974$		0			0
$975$		0			0
$976$		0			0
$977$	313.089			0.320460			0.160230	−	0.987080i	$-0.448776\pi$
$977$	313.089			0.320460			0.160230	+	0.987080i	$0.448776\pi$
$978$		0			0
$979$	−87.3995			−0.0892743
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	178.376i		−	0.181461i	−0.995875	−	0.0907304i	$-0.971080\pi$
$983$		−	178.376i		−	0.181461i	0.995875	−	0.0907304i	$-0.0289202\pi$
$984$		0			0
$985$	−409.688			−0.415927
$986$		0			0
$987$		0			0
$988$		0			0
$989$			93.0325i			0.0940672i
$990$		0			0
$991$	−622.297			−0.627949			−0.313974	−	0.949431i	$-0.601661\pi$
$991$	−622.297			−0.627949			−0.313974	+	0.949431i	$0.601661\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	303.211			0.304734
$996$		0			0
$997$	−92.7916			−0.0930708			−0.0465354	−	0.998917i	$-0.514818\pi$
$997$	−92.7916			−0.0930708			−0.0465354	+	0.998917i	$0.514818\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.b.e.2735.9	yes	32
3.2	odd	2	inner	2736.3.b.e.2735.22	yes	32
4.3	odd	2	inner	2736.3.b.e.2735.3	✓	32
12.11	even	2	inner	2736.3.b.e.2735.8	yes	32
19.18	odd	2	inner	2736.3.b.e.2735.7	yes	32
57.56	even	2	inner	2736.3.b.e.2735.4	yes	32
76.75	even	2	inner	2736.3.b.e.2735.21	yes	32
228.227	odd	2	inner	2736.3.b.e.2735.10	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.3.b.e.2735.3	✓	32	4.3	odd	2	inner
2736.3.b.e.2735.4	yes	32	57.56	even	2	inner
2736.3.b.e.2735.7	yes	32	19.18	odd	2	inner
2736.3.b.e.2735.8	yes	32	12.11	even	2	inner
2736.3.b.e.2735.9	yes	32	1.1	even	1	trivial
2736.3.b.e.2735.10	yes	32	228.227	odd	2	inner
2736.3.b.e.2735.21	yes	32	76.75	even	2	inner
2736.3.b.e.2735.22	yes	32	3.2	odd	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.26257i q^{5} +8.06963i q^{7} +O(q^{10})\)	\(q-1.26257i q^{5} +8.06963i q^{7} -8.70004 q^{11} -16.8236i q^{13} -4.74680i q^{17} +(-5.61192 + 18.1523i) q^{19} -16.7960 q^{23} +23.4059 q^{25} +51.4472 q^{29} +27.6566 q^{31} +10.1885 q^{35} +24.8546i q^{37} -13.7463 q^{41} -5.53897i q^{43} -47.0118 q^{47} -16.1189 q^{49} -105.283 q^{53} +10.9844i q^{55} -2.94791i q^{59} -1.19469 q^{61} -21.2411 q^{65} -123.333 q^{67} +99.1425i q^{71} -79.3625 q^{73} -70.2061i q^{77} +122.293 q^{79} -90.1340 q^{83} -5.99318 q^{85} +10.0459 q^{89} +135.760 q^{91} +(22.9186 + 7.08546i) q^{95} +84.1666i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 96 q^{25} + 336 q^{49} - 256 q^{61} - 560 q^{73} - 752 q^{85}+O(q^{100}) \) 32 * q + 96 * q^25 + 336 * q^49 - 256 * q^61 - 560 * q^73 - 752 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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