LMFDB - Embedded newform 180.6.d.c.109.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [180,6,Mod(109,180)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("180.109"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	180.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$28.8690875663$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-61}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 61 $ x^2 + 61
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.2
Root			$-7.81025i$ of defining polynomial
Character	$\chi$	$=$	180.109
Dual form			180.6.d.c.109.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(40.0000 + 39.0512i) q^{5} -124.964i q^{7} +80.0000 q^{11} -374.892i q^{13} -546.717i q^{17} +12.0000 q^{19} -2030.66i q^{23} +(75.0000 + 3124.10i) q^{25} +4560.00 q^{29} -344.000 q^{31} +(4880.00 - 4998.56i) q^{35} +4373.74i q^{37} +14240.0 q^{41} -18994.5i q^{43} -24524.2i q^{47} +1191.00 q^{49} -27414.0i q^{53} +(3200.00 + 3124.10i) q^{55} +38000.0 q^{59} -8206.00 q^{61} +(14640.0 - 14995.7i) q^{65} +13246.2i q^{67} +48480.0 q^{71} +42487.8i q^{73} -9997.12i q^{77} +9264.00 q^{79} -33427.9i q^{83} +(21350.0 - 21868.7i) q^{85} -24320.0 q^{89} -46848.0 q^{91} +(480.000 + 468.615i) q^{95} -136711. i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 80 q^{5} + 160 q^{11} + 24 q^{19} + 150 q^{25} + 9120 q^{29} - 688 q^{31} + 9760 q^{35} + 28480 q^{41} + 2382 q^{49} + 6400 q^{55} + 76000 q^{59} - 16412 q^{61} + 29280 q^{65} + 96960 q^{71} + 18528 q^{79}+ \cdots + 960 q^{95}+O(q^{100}) $ 2 * q + 80 * q^5 + 160 * q^11 + 24 * q^19 + 150 * q^25 + 9120 * q^29 - 688 * q^31 + 9760 * q^35 + 28480 * q^41 + 2382 * q^49 + 6400 * q^55 + 76000 * q^59 - 16412 * q^61 + 29280 * q^65 + 96960 * q^71 + 18528 * q^79 + 42700 * q^85 - 48640 * q^89 - 93696 * q^91 + 960 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$.

$n$	$37$	$91$	$101$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	40.0000	+	39.0512i	0.715542	+	0.698570i
$5$	40.0000	+	39.0512i	0.715542	+	0.698570i
$6$		0			0
$7$		−	124.964i		−	0.963917i	−0.876194	−	0.481959i	$-0.839926\pi$
$7$		−	124.964i		−	0.963917i	0.876194	−	0.481959i	$-0.160074\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	80.0000			0.199346			0.0996732	−	0.995020i	$-0.468220\pi$
$11$	80.0000			0.199346			0.0996732	+	0.995020i	$0.468220\pi$
$12$		0			0
$13$		−	374.892i		−	0.615245i	−0.951509	−	0.307622i	$-0.900467\pi$
$13$		−	374.892i		−	0.615245i	0.951509	−	0.307622i	$-0.0995333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	546.717i		−	0.458818i	−0.973330	−	0.229409i	$-0.926321\pi$
$17$		−	546.717i		−	0.458818i	0.973330	−	0.229409i	$-0.0736794\pi$
$18$		0			0
$19$	12.0000			0.00762601			0.00381300	−	0.999993i	$-0.498786\pi$
$19$	12.0000			0.00762601			0.00381300	+	0.999993i	$0.498786\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2030.66i		−	0.800421i	−0.916423	−	0.400211i	$-0.868937\pi$
$23$		−	2030.66i		−	0.800421i	0.916423	−	0.400211i	$-0.131063\pi$
$24$		0			0
$25$	75.0000	+	3124.10i	0.0240000	+	0.999712i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4560.00			1.00686			0.503431	−	0.864036i	$-0.332071\pi$
$29$	4560.00			1.00686			0.503431	+	0.864036i	$0.332071\pi$
$30$		0			0
$31$	−344.000			−0.0642916			−0.0321458	−	0.999483i	$-0.510234\pi$
$31$	−344.000			−0.0642916			−0.0321458	+	0.999483i	$0.510234\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4880.00	−	4998.56i	0.673364	−	0.689723i
$36$		0			0
$37$			4373.74i			0.525229i	0.964901	+	0.262614i	$0.0845847\pi$
$37$			4373.74i			0.525229i	−0.964901	+	0.262614i	$0.915415\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	14240.0			1.32297			0.661486	−	0.749958i	$-0.269926\pi$
$41$	14240.0			1.32297			0.661486	+	0.749958i	$0.269926\pi$
$42$		0			0
$43$		−	18994.5i		−	1.56660i	−0.621646	−	0.783299i	$-0.713536\pi$
$43$		−	18994.5i		−	1.56660i	0.621646	−	0.783299i	$-0.286464\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	24524.2i		−	1.61938i	−0.586855	−	0.809692i	$-0.699634\pi$
$47$		−	24524.2i		−	1.61938i	0.586855	−	0.809692i	$-0.300366\pi$
$48$		0			0
$49$	1191.00			0.0708633
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	27414.0i		−	1.34055i	−0.742114	−	0.670274i	$-0.766177\pi$
$53$		−	27414.0i		−	1.34055i	0.742114	−	0.670274i	$-0.233823\pi$
$54$		0			0
$55$	3200.00	+	3124.10i	0.142641	+	0.139257i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	38000.0			1.42119			0.710597	−	0.703599i	$-0.248425\pi$
$59$	38000.0			1.42119			0.710597	+	0.703599i	$0.248425\pi$
$60$		0			0
$61$	−8206.00			−0.282362			−0.141181	−	0.989984i	$-0.545090\pi$
$61$	−8206.00			−0.282362			−0.141181	+	0.989984i	$0.545090\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	14640.0	−	14995.7i	0.429791	−	0.440233i
$66$		0			0
$67$			13246.2i			0.360499i	0.983621	+	0.180249i	$0.0576905\pi$
$67$			13246.2i			0.360499i	−0.983621	+	0.180249i	$0.942310\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	48480.0			1.14134			0.570672	−	0.821178i	$-0.306683\pi$
$71$	48480.0			1.14134			0.570672	+	0.821178i	$0.306683\pi$
$72$		0			0
$73$			42487.8i			0.933161i	0.884479	+	0.466581i	$0.154514\pi$
$73$			42487.8i			0.933161i	−0.884479	+	0.466581i	$0.845486\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	9997.12i		−	0.192153i
$78$		0			0
$79$	9264.00			0.167006			0.0835028	−	0.996508i	$-0.473389\pi$
$79$	9264.00			0.167006			0.0835028	+	0.996508i	$0.473389\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	33427.9i		−	0.532615i	−0.963888	−	0.266308i	$-0.914196\pi$
$83$		−	33427.9i		−	0.532615i	0.963888	−	0.266308i	$-0.0858037\pi$
$84$		0			0
$85$	21350.0	−	21868.7i	0.320517	−	0.328304i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−24320.0			−0.325453			−0.162727	−	0.986671i	$-0.552029\pi$
$89$	−24320.0			−0.325453			−0.162727	+	0.986671i	$0.552029\pi$
$90$		0			0
$91$	−46848.0			−0.593045
$92$		0			0
$93$		0			0
$94$		0			0
$95$	480.000	+	468.615i	0.00545673	+	0.00532730i
$96$		0			0
$97$		−	136711.i		−	1.47527i	−0.675197	−	0.737637i	$-0.735941\pi$
$97$		−	136711.i		−	1.47527i	0.675197	−	0.737637i	$-0.264059\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−105200.			−1.02615			−0.513077	−	0.858343i	$-0.671494\pi$
$101$	−105200.			−1.02615			−0.513077	+	0.858343i	$0.671494\pi$
$102$		0			0
$103$			68105.4i			0.632541i	0.948669	+	0.316270i	$0.102431\pi$
$103$			68105.4i			0.632541i	−0.948669	+	0.316270i	$0.897569\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			21556.3i			0.182018i	0.995850	+	0.0910090i	$0.0290092\pi$
$107$			21556.3i			0.182018i	−0.995850	+	0.0910090i	$0.970991\pi$
$108$		0			0
$109$	41438.0			0.334066			0.167033	−	0.985951i	$-0.446581\pi$
$109$	41438.0			0.334066			0.167033	+	0.985951i	$0.446581\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			214391.i			1.57947i	0.613449	+	0.789735i	$0.289782\pi$
$113$			214391.i			1.57947i	−0.613449	+	0.789735i	$0.710218\pi$
$114$		0			0
$115$	79300.0	−	81226.6i	0.559150	−	0.572735i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−68320.0			−0.442263
$120$		0			0
$121$	−154651.			−0.960261
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−119000.	+	127893.i	−0.681196	+	0.732101i
$126$		0			0
$127$			244305.i			1.34407i	0.740519	+	0.672036i	$0.234580\pi$
$127$			244305.i			1.34407i	−0.740519	+	0.672036i	$0.765420\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−124560.			−0.634162			−0.317081	−	0.948398i	$-0.602703\pi$
$131$	−124560.			−0.634162			−0.317081	+	0.948398i	$0.602703\pi$
$132$		0			0
$133$		−	1499.57i		−	0.00735084i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			363880.i			1.65637i	0.560458	+	0.828183i	$0.310625\pi$
$137$			363880.i			1.65637i	−0.560458	+	0.828183i	$0.689375\pi$
$138$		0			0
$139$	89036.0			0.390867			0.195433	−	0.980717i	$-0.437389\pi$
$139$	89036.0			0.390867			0.195433	+	0.980717i	$0.437389\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	29991.4i		−	0.122647i
$144$		0			0
$145$	182400.	+	178074.i	0.720452	+	0.703363i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−438640.			−1.61861			−0.809306	−	0.587388i	$-0.800156\pi$
$149$	−438640.			−1.61861			−0.809306	+	0.587388i	$0.800156\pi$
$150$		0			0
$151$	−351704.			−1.25526			−0.627632	−	0.778510i	$-0.715976\pi$
$151$	−351704.			−1.25526			−0.627632	+	0.778510i	$0.715976\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−13760.0	−	13433.6i	−0.0460033	−	0.0449122i
$156$		0			0
$157$		−	82351.3i		−	0.266637i	−0.991073	−	0.133319i	$-0.957437\pi$
$157$		−	82351.3i		−	0.266637i	0.991073	−	0.133319i	$-0.0425634\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−253760.			−0.771540
$162$		0			0
$163$		−	520100.i		−	1.53327i	−0.642085	−	0.766634i	$-0.721930\pi$
$163$		−	520100.i		−	1.53327i	0.642085	−	0.766634i	$-0.278070\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	126995.i		−	0.352366i	−0.984357	−	0.176183i	$-0.943625\pi$
$167$		−	126995.i		−	0.352366i	0.984357	−	0.176183i	$-0.0563751\pi$
$168$		0			0
$169$	230749.			0.621474
$170$		0			0
$171$		0			0
$172$		0			0
$173$			323266.i			0.821193i	0.911817	+	0.410596i	$0.134679\pi$
$173$			323266.i			0.821193i	−0.911817	+	0.410596i	$0.865321\pi$
$174$		0			0
$175$	390400.	−	9372.30i	0.963640	−	0.0231340i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	495920.			1.15686			0.578428	−	0.815734i	$-0.303667\pi$
$179$	495920.			1.15686			0.578428	+	0.815734i	$0.303667\pi$
$180$		0			0
$181$	−683014.			−1.54965			−0.774824	−	0.632177i	$-0.782162\pi$
$181$	−683014.			−1.54965			−0.774824	+	0.632177i	$0.782162\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−170800.	+	174950.i	−0.366909	+	0.375823i
$186$		0			0
$187$		−	43737.4i		−	0.0914637i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	625440.			1.24052			0.620258	−	0.784398i	$-0.287028\pi$
$191$	625440.			1.24052			0.620258	+	0.784398i	$0.287028\pi$
$192$		0			0
$193$			70229.8i			0.135715i	0.997695	+	0.0678575i	$0.0216163\pi$
$193$			70229.8i			0.135715i	−0.997695	+	0.0678575i	$0.978384\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	179870.i		−	0.330212i	−0.986276	−	0.165106i	$-0.947203\pi$
$197$		−	179870.i		−	0.330212i	0.986276	−	0.165106i	$-0.0527967\pi$
$198$		0			0
$199$	295728.			0.529371			0.264685	−	0.964335i	$-0.414732\pi$
$199$	295728.			0.529371			0.264685	+	0.964335i	$0.414732\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	569836.i		−	0.970532i
$204$		0			0
$205$	569600.	+	556090.i	0.946641	+	0.924188i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	960.000			0.00152022
$210$		0			0
$211$	−1.04824e6			−1.62089			−0.810444	−	0.585816i	$-0.800774\pi$
$211$	−1.04824e6			−1.62089			−0.810444	+	0.585816i	$0.800774\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	741760.	−	759781.i	1.09438	−	1.12097i
$216$		0			0
$217$			42987.6i			0.0619718i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−204960.			−0.282285
$222$		0			0
$223$			799395.i			1.07646i	0.842797	+	0.538231i	$0.180907\pi$
$223$			799395.i			1.07646i	−0.842797	+	0.538231i	$0.819093\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	1.00315e6i		−	1.29211i	−0.763289	−	0.646057i	$-0.776417\pi$
$227$		−	1.00315e6i		−	1.29211i	0.763289	−	0.646057i	$-0.223583\pi$
$228$		0			0
$229$	593002.			0.747253			0.373626	−	0.927579i	$-0.378114\pi$
$229$	593002.			0.747253			0.373626	+	0.927579i	$0.378114\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	630834.i		−	0.761246i	−0.924730	−	0.380623i	$-0.875710\pi$
$233$		−	630834.i		−	0.761246i	0.924730	−	0.380623i	$-0.124290\pi$
$234$		0			0
$235$	957700.	−	980967.i	1.13125	−	1.15874i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.00240e6			1.13513			0.567566	−	0.823328i	$-0.307885\pi$
$239$	1.00240e6			1.13513			0.567566	+	0.823328i	$0.307885\pi$
$240$		0			0
$241$	−1.58637e6			−1.75938			−0.879692	−	0.475543i	$-0.842251\pi$
$241$	−1.58637e6			−1.75938			−0.879692	+	0.475543i	$0.842251\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	47640.0	+	46510.0i	0.0507057	+	0.0495030i
$246$		0			0
$247$		−	4498.70i		−	0.00469186i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.62552e6			1.62858			0.814288	−	0.580461i	$-0.197128\pi$
$251$	1.62552e6			1.62858			0.814288	+	0.580461i	$0.197128\pi$
$252$		0			0
$253$		−	162453.i		−	0.159561i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			668479.i			0.631328i	0.948871	+	0.315664i	$0.102227\pi$
$257$			668479.i			0.631328i	−0.948871	+	0.315664i	$0.897773\pi$
$258$		0			0
$259$	546560.			0.506277
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1.09672e6i		−	0.977698i	−0.872369	−	0.488849i	$-0.837417\pi$
$263$		−	1.09672e6i		−	0.977698i	0.872369	−	0.488849i	$-0.162583\pi$
$264$		0			0
$265$	1.07055e6	−	1.09656e6i	0.936467	−	0.959218i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	562800.			0.474213			0.237106	−	0.971484i	$-0.423801\pi$
$269$	562800.			0.474213			0.237106	+	0.971484i	$0.423801\pi$
$270$		0			0
$271$	−1.87645e6			−1.55208			−0.776039	−	0.630685i	$-0.782774\pi$
$271$	−1.87645e6			−1.55208			−0.776039	+	0.630685i	$0.782774\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6000.00	+	249928.i	0.00478431	+	0.199289i
$276$		0			0
$277$		−	768654.i		−	0.601910i	−0.953638	−	0.300955i	$-0.902695\pi$
$277$		−	768654.i		−	0.601910i	0.953638	−	0.300955i	$-0.0973053\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−687360.			−0.519300			−0.259650	−	0.965703i	$-0.583607\pi$
$281$	−687360.			−0.519300			−0.259650	+	0.965703i	$0.583607\pi$
$282$		0			0
$283$			607575.i			0.450956i	0.974248	+	0.225478i	$0.0723943\pi$
$283$			607575.i			0.450956i	−0.974248	+	0.225478i	$0.927606\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	1.77949e6i		−	1.27523i
$288$		0			0
$289$	1.12096e6			0.789486
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.87001e6i			1.27255i	0.771463	+	0.636274i	$0.219525\pi$
$293$			1.87001e6i			1.27255i	−0.771463	+	0.636274i	$0.780475\pi$
$294$		0			0
$295$	1.52000e6	+	1.48395e6i	1.01692	+	0.992804i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−761280.			−0.492455
$300$		0			0
$301$	−2.37363e6			−1.51007
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−328240.	−	320455.i	−0.202042	−	0.197250i
$306$		0			0
$307$			874248.i			0.529406i	0.964330	+	0.264703i	$0.0852739\pi$
$307$			874248.i			0.529406i	−0.964330	+	0.264703i	$0.914726\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.02096e6			−0.598560			−0.299280	−	0.954165i	$-0.596746\pi$
$311$	−1.02096e6			−0.598560			−0.299280	+	0.954165i	$0.596746\pi$
$312$		0			0
$313$			532097.i			0.306994i	0.988149	+	0.153497i	$0.0490535\pi$
$313$			532097.i			0.306994i	−0.988149	+	0.153497i	$0.950946\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			1.75535e6i			0.981107i	0.871411	+	0.490554i	$0.163205\pi$
$317$			1.75535e6i			0.981107i	−0.871411	+	0.490554i	$0.836795\pi$
$318$		0			0
$319$	364800.			0.200714
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	6560.61i		−	0.00349895i
$324$		0			0
$325$	1.17120e6	−	28116.9i	0.615067	−	0.0147659i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.06464e6			−1.56095
$330$		0			0
$331$	−2.24395e6			−1.12575			−0.562876	−	0.826541i	$-0.690305\pi$
$331$	−2.24395e6			−1.12575			−0.562876	+	0.826541i	$0.690305\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−517280.	+	529847.i	−0.251834	+	0.257952i
$336$		0			0
$337$			927733.i			0.444988i	0.974934	+	0.222494i	$0.0714197\pi$
$337$			927733.i			0.444988i	−0.974934	+	0.222494i	$0.928580\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−27520.0			−0.0128163
$342$		0			0
$343$		−	2.24910e6i		−	1.03222i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			4.12569e6i			1.83938i	0.392640	+	0.919692i	$0.371562\pi$
$347$			4.12569e6i			1.83938i	−0.392640	+	0.919692i	$0.628438\pi$
$348$		0			0
$349$	1.63749e6			0.719638			0.359819	−	0.933022i	$-0.382838\pi$
$349$	1.63749e6			0.719638			0.359819	+	0.933022i	$0.382838\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			1.98310e6i			0.847048i	0.905885	+	0.423524i	$0.139207\pi$
$353$			1.98310e6i			0.847048i	−0.905885	+	0.423524i	$0.860793\pi$
$354$		0			0
$355$	1.93920e6	+	1.89320e6i	0.816680	+	0.797309i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.14720e6			0.469789			0.234895	−	0.972021i	$-0.424526\pi$
$359$	1.14720e6			0.469789			0.234895	+	0.972021i	$0.424526\pi$
$360$		0			0
$361$	−2.47596e6			−0.999942
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.65920e6	+	1.69951e6i	−0.651878	+	0.667716i
$366$		0			0
$367$		−	4.09395e6i		−	1.58663i	−0.608808	−	0.793317i	$-0.708352\pi$
$367$		−	4.09395e6i		−	1.58663i	0.608808	−	0.793317i	$-0.291648\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.42576e6			−1.29218
$372$		0			0
$373$		−	2.99126e6i		−	1.11322i	−0.830773	−	0.556612i	$-0.812101\pi$
$373$		−	2.99126e6i		−	1.11322i	0.830773	−	0.556612i	$-0.187899\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.70951e6i		−	0.619466i
$378$		0			0
$379$	−98228.0			−0.0351267			−0.0175633	−	0.999846i	$-0.505591\pi$
$379$	−98228.0			−0.0351267			−0.0175633	+	0.999846i	$0.505591\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			2.70969e6i			0.943892i	0.881627	+	0.471946i	$0.156448\pi$
$383$			2.70969e6i			0.943892i	−0.881627	+	0.471946i	$0.843552\pi$
$384$		0			0
$385$	390400.	−	399885.i	0.134233	−	0.137494i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.03064e6			−0.345329			−0.172664	−	0.984981i	$-0.555238\pi$
$389$	−1.03064e6			−0.345329			−0.172664	+	0.984981i	$0.555238\pi$
$390$		0			0
$391$	−1.11020e6			−0.367248
$392$		0			0
$393$		0			0
$394$		0			0
$395$	370560.	+	361771.i	0.119499	+	0.116665i
$396$		0			0
$397$			6.10312e6i			1.94346i	0.236097	+	0.971730i	$0.424132\pi$
$397$			6.10312e6i			1.94346i	−0.236097	+	0.971730i	$0.575868\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.42192e6			−0.752140			−0.376070	−	0.926591i	$-0.622725\pi$
$401$	−2.42192e6			−0.752140			−0.376070	+	0.926591i	$0.622725\pi$
$402$		0			0
$403$			128963.i			0.0395551i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			349899.i			0.104702i
$408$		0			0
$409$	1.32812e6			0.392581			0.196291	−	0.980546i	$-0.437110\pi$
$409$	1.32812e6			0.392581			0.196291	+	0.980546i	$0.437110\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	4.74863e6i		−	1.36991i
$414$		0			0
$415$	1.30540e6	−	1.33711e6i	0.372069	−	0.381108i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.65944e6			−1.29658			−0.648289	−	0.761394i	$-0.724515\pi$
$419$	−4.65944e6			−1.29658			−0.648289	+	0.761394i	$0.724515\pi$
$420$		0			0
$421$	−1.39140e6			−0.382602			−0.191301	−	0.981531i	$-0.561271\pi$
$421$	−1.39140e6			−0.382602			−0.191301	+	0.981531i	$0.561271\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.70800e6	−	41003.8i	0.458686	−	0.0110116i
$426$		0			0
$427$			1.02545e6i			0.272174i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.76096e6			1.23453			0.617265	−	0.786756i	$-0.288241\pi$
$431$	4.76096e6			1.23453			0.617265	+	0.786756i	$0.288241\pi$
$432$		0			0
$433$			5.14627e6i			1.31908i	0.751668	+	0.659542i	$0.229250\pi$
$433$			5.14627e6i			1.31908i	−0.751668	+	0.659542i	$0.770750\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	24368.0i		−	0.00610402i
$438$		0			0
$439$	−1.30171e6			−0.322369			−0.161185	−	0.986924i	$-0.551531\pi$
$439$	−1.30171e6			−0.322369			−0.161185	+	0.986924i	$0.551531\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			3.02069e6i			0.731303i	0.930752	+	0.365651i	$0.119154\pi$
$443$			3.02069e6i			0.731303i	−0.930752	+	0.365651i	$0.880846\pi$
$444$		0			0
$445$	−972800.	−	949726.i	−0.232875	−	0.227352i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8.07504e6			1.89029			0.945146	−	0.326649i	$-0.105919\pi$
$449$	8.07504e6			1.89029			0.945146	+	0.326649i	$0.105919\pi$
$450$		0			0
$451$	1.13920e6			0.263729
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.87392e6	−	1.82947e6i	−0.424348	−	0.414283i
$456$		0			0
$457$		−	4.78487e6i		−	1.07172i	−0.844308	−	0.535858i	$-0.819988\pi$
$457$		−	4.78487e6i		−	1.07172i	0.844308	−	0.535858i	$-0.180012\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5.02104e6			1.10038			0.550188	−	0.835041i	$-0.314556\pi$
$461$	5.02104e6			1.10038			0.550188	+	0.835041i	$0.314556\pi$
$462$		0			0
$463$			2.46117e6i			0.533566i	0.963757	+	0.266783i	$0.0859608\pi$
$463$			2.46117e6i			0.533566i	−0.963757	+	0.266783i	$0.914039\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	2.38244e6i		−	0.505510i	−0.967530	−	0.252755i	$-0.918663\pi$
$467$		−	2.38244e6i		−	0.505510i	0.967530	−	0.252755i	$-0.0813366\pi$
$468$		0			0
$469$	1.65530e6			0.347491
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	1.51956e6i		−	0.312295i
$474$		0			0
$475$	900.000	+	37489.2i	0.000183024	+	0.00762381i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.79568e6			0.556735			0.278368	−	0.960475i	$-0.410207\pi$
$479$	2.79568e6			0.556735			0.278368	+	0.960475i	$0.410207\pi$
$480$		0			0
$481$	1.63968e6			0.323144
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.33872e6	−	5.46842e6i	1.03058	−	1.05562i
$486$		0			0
$487$		−	3.99372e6i		−	0.763055i	−0.924358	−	0.381527i	$-0.875398\pi$
$487$		−	3.99372e6i		−	0.763055i	0.924358	−	0.381527i	$-0.124602\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.04552e6			−0.944501			−0.472250	−	0.881465i	$-0.656558\pi$
$491$	−5.04552e6			−0.944501			−0.472250	+	0.881465i	$0.656558\pi$
$492$		0			0
$493$		−	2.49303e6i		−	0.461967i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	6.05825e6i		−	1.10016i
$498$		0			0
$499$	−4.80013e6			−0.862982			−0.431491	−	0.902117i	$-0.642012\pi$
$499$	−4.80013e6			−0.862982			−0.431491	+	0.902117i	$0.642012\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9.24718e6i		−	1.62963i	−0.579720	−	0.814816i	$-0.696838\pi$
$503$		−	9.24718e6i		−	1.62963i	0.579720	−	0.814816i	$-0.303162\pi$
$504$		0			0
$505$	−4.20800e6	−	4.10819e6i	−0.734256	−	0.716840i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.71816e6			0.293947			0.146974	−	0.989140i	$-0.453047\pi$
$509$	1.71816e6			0.293947			0.146974	+	0.989140i	$0.453047\pi$
$510$		0			0
$511$	5.30944e6			0.899490
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.65960e6	+	2.72422e6i	−0.441874	+	0.452609i
$516$		0			0
$517$		−	1.96193e6i		−	0.322818i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.20789e7			1.94954			0.974770	−	0.223210i	$-0.0716534\pi$
$521$	1.20789e7			1.94954			0.974770	+	0.223210i	$0.0716534\pi$
$522$		0			0
$523$			5.19325e6i			0.830205i	0.909775	+	0.415102i	$0.136254\pi$
$523$			5.19325e6i			0.830205i	−0.909775	+	0.415102i	$0.863746\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			188071.i			0.0294982i
$528$		0			0
$529$	2.31274e6			0.359326
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	5.33846e6i		−	0.813951i
$534$		0			0
$535$	−841800.	+	862252.i	−0.127152	+	0.130242i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	95280.0			0.0141263
$540$		0			0
$541$	4.68747e6			0.688566			0.344283	−	0.938866i	$-0.388122\pi$
$541$	4.68747e6			0.688566			0.344283	+	0.938866i	$0.388122\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.65752e6	+	1.61821e6i	0.239038	+	0.233369i
$546$		0			0
$547$			6.27819e6i			0.897152i	0.893745	+	0.448576i	$0.148069\pi$
$547$			6.27819e6i			0.897152i	−0.893745	+	0.448576i	$0.851931\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	54720.0			0.00767834
$552$		0			0
$553$		−	1.15767e6i		−	0.160980i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	8.89408e6i		−	1.21468i	−0.794441	−	0.607341i	$-0.792236\pi$
$557$		−	8.89408e6i		−	1.21468i	0.794441	−	0.607341i	$-0.207764\pi$
$558$		0			0
$559$	−7.12090e6			−0.963840
$560$		0			0
$561$		0			0
$562$		0			0
$563$			1.18135e7i			1.57075i	0.619021	+	0.785374i	$0.287529\pi$
$563$			1.18135e7i			1.57075i	−0.619021	+	0.785374i	$0.712471\pi$
$564$		0			0
$565$	−8.37225e6	+	8.57565e6i	−1.10337	+	1.13018i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.89744e6			0.375175			0.187587	−	0.982248i	$-0.439933\pi$
$569$	2.89744e6			0.375175			0.187587	+	0.982248i	$0.439933\pi$
$570$		0			0
$571$	1.01834e7			1.30709			0.653543	−	0.756889i	$-0.273282\pi$
$571$	1.01834e7			1.30709			0.653543	+	0.756889i	$0.273282\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.34400e6	−	152300.i	0.800191	−	0.0192101i
$576$		0			0
$577$			1.79898e6i			0.224951i	0.993655	+	0.112475i	$0.0358779\pi$
$577$			1.79898e6i			0.224951i	−0.993655	+	0.112475i	$0.964122\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.17728e6			−0.513397
$582$		0			0
$583$		−	2.19312e6i		−	0.267233i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1.29178e7i			1.54737i	0.633569	+	0.773686i	$0.281589\pi$
$587$			1.29178e7i			1.54737i	−0.633569	+	0.773686i	$0.718411\pi$
$588$		0			0
$589$	−4128.00			−0.000490288
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	5.13907e6i		−	0.600133i	−0.953918	−	0.300066i	$-0.902991\pi$
$593$		−	5.13907e6i		−	0.600133i	0.953918	−	0.300066i	$-0.0970089\pi$
$594$		0			0
$595$	−2.73280e6	−	2.66798e6i	−0.316458	−	0.308952i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−5.62272e6			−0.640294			−0.320147	−	0.947368i	$-0.603732\pi$
$599$	−5.62272e6			−0.640294			−0.320147	+	0.947368i	$0.603732\pi$
$600$		0			0
$601$	7.48833e6			0.845665			0.422833	−	0.906208i	$-0.361036\pi$
$601$	7.48833e6			0.845665			0.422833	+	0.906208i	$0.361036\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.18604e6	−	6.03931e6i	−0.687107	−	0.670810i
$606$		0			0
$607$			7.03610e6i			0.775104i	0.921848	+	0.387552i	$0.126679\pi$
$607$			7.03610e6i			0.775104i	−0.921848	+	0.387552i	$0.873321\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.19392e6			−0.996317
$612$		0			0
$613$		−	1.54499e7i		−	1.66064i	−0.557288	−	0.830319i	$-0.688158\pi$
$613$		−	1.54499e7i		−	1.66064i	0.557288	−	0.830319i	$-0.311842\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			7.86750e6i			0.832001i	0.909364	+	0.416000i	$0.136569\pi$
$617$			7.86750e6i			0.832001i	−0.909364	+	0.416000i	$0.863431\pi$
$618$		0			0
$619$	−8.69805e6			−0.912421			−0.456211	−	0.889872i	$-0.650794\pi$
$619$	−8.69805e6			−0.912421			−0.456211	+	0.889872i	$0.650794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			3.03912e6i			0.313710i
$624$		0			0
$625$	−9.75438e6	+	468615.i	−0.998848	+	0.0479862i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.39120e6			0.240985
$630$		0			0
$631$	1.50252e7			1.50226			0.751132	−	0.660152i	$-0.229508\pi$
$631$	1.50252e7			1.50226			0.751132	+	0.660152i	$0.229508\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.54040e6	+	9.77218e6i	−0.938928	+	0.961739i
$636$		0			0
$637$		−	446496.i		−	0.0435983i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.88048e6			−0.661414			−0.330707	−	0.943733i	$-0.607287\pi$
$641$	−6.88048e6			−0.661414			−0.330707	+	0.943733i	$0.607287\pi$
$642$		0			0
$643$		−	1.18953e7i		−	1.13462i	−0.823506	−	0.567308i	$-0.807985\pi$
$643$		−	1.18953e7i		−	1.13462i	0.823506	−	0.567308i	$-0.192015\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	5.03339e6i		−	0.472716i	−0.971666	−	0.236358i	$-0.924046\pi$
$647$		−	5.03339e6i		−	0.472716i	0.971666	−	0.236358i	$-0.0759538\pi$
$648$		0			0
$649$	3.04000e6			0.283310
$650$		0			0
$651$		0			0
$652$		0			0
$653$			1.12571e7i			1.03311i	0.856255	+	0.516554i	$0.172785\pi$
$653$			1.12571e7i			1.03311i	−0.856255	+	0.516554i	$0.827215\pi$
$654$		0			0
$655$	−4.98240e6	−	4.86422e6i	−0.453770	−	0.443007i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.84982e7			−1.65926			−0.829631	−	0.558312i	$-0.811449\pi$
$659$	−1.84982e7			−1.65926			−0.829631	+	0.558312i	$0.811449\pi$
$660$		0			0
$661$	1.13143e7			1.00722			0.503610	−	0.863931i	$-0.332005\pi$
$661$	1.13143e7			1.00722			0.503610	+	0.863931i	$0.332005\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	58560.0	−	59982.7i	0.00513508	−	0.00525983i
$666$		0			0
$667$		−	9.25983e6i		−	0.805914i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−656480.			−0.0562879
$672$		0			0
$673$			2.04878e7i			1.74365i	0.489820	+	0.871824i	$0.337063\pi$
$673$			2.04878e7i			1.74365i	−0.489820	+	0.871824i	$0.662937\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			6.31873e6i			0.529856i	0.964268	+	0.264928i	$0.0853482\pi$
$677$			6.31873e6i			0.529856i	−0.964268	+	0.264928i	$0.914652\pi$
$678$		0			0
$679$	−1.70839e7			−1.42204
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.41028e7i			1.15679i	0.815757	+	0.578394i	$0.196320\pi$
$683$			1.41028e7i			1.15679i	−0.815757	+	0.578394i	$0.803680\pi$
$684$		0			0
$685$	−1.42100e7	+	1.45552e7i	−1.15709	+	1.18520i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.02773e7			−0.824765
$690$		0			0
$691$	1.49779e7			1.19332			0.596660	−	0.802494i	$-0.296494\pi$
$691$	1.49779e7			1.19332			0.596660	+	0.802494i	$0.296494\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.56144e6	+	3.47697e6i	0.279681	+	0.273048i
$696$		0			0
$697$		−	7.78526e6i		−	0.607003i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.93112e6			0.148427			0.0742137	−	0.997242i	$-0.476355\pi$
$701$	1.93112e6			0.148427			0.0742137	+	0.997242i	$0.476355\pi$
$702$		0			0
$703$			52484.9i			0.00400540i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1.31462e7i			0.989127i
$708$		0			0
$709$	7.55428e6			0.564388			0.282194	−	0.959357i	$-0.408938\pi$
$709$	7.55428e6			0.564388			0.282194	+	0.959357i	$0.408938\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			698549.i			0.0514604i
$714$		0			0
$715$	1.17120e6	−	1.19965e6i	0.0856773	−	0.0877589i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.20608e6			0.664129			0.332065	−	0.943257i	$-0.392255\pi$
$719$	9.20608e6			0.664129			0.332065	+	0.943257i	$0.392255\pi$
$720$		0			0
$721$	8.51072e6			0.609717
$722$		0			0
$723$		0			0
$724$		0			0
$725$	342000.	+	1.42459e7i	0.0241647	+	1.00657i
$726$		0			0
$727$			2.02918e7i			1.42392i	0.702223	+	0.711958i	$0.252191\pi$
$727$			2.02918e7i			1.42392i	−0.702223	+	0.711958i	$0.747809\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.03846e7			−0.718783
$732$		0			0
$733$			1.93413e7i			1.32961i	0.747015	+	0.664807i	$0.231486\pi$
$733$			1.93413e7i			1.32961i	−0.747015	+	0.664807i	$0.768514\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.05969e6i			0.0718641i
$738$		0			0
$739$	−1.45942e7			−0.983033			−0.491517	−	0.870868i	$-0.663557\pi$
$739$	−1.45942e7			−0.983033			−0.491517	+	0.870868i	$0.663557\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1.82896e7i		−	1.21543i	−0.794153	−	0.607717i	$-0.792085\pi$
$743$		−	1.82896e7i		−	1.21543i	0.794153	−	0.607717i	$-0.207915\pi$
$744$		0			0
$745$	−1.75456e7	−	1.71294e7i	−1.15818	−	1.13071i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.69376e6			0.175450
$750$		0			0
$751$	1.98542e7			1.28456			0.642279	−	0.766471i	$-0.277989\pi$
$751$	1.98542e7			1.28456			0.642279	+	0.766471i	$0.277989\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.40682e7	−	1.37345e7i	−0.898194	−	0.876889i
$756$		0			0
$757$		−	1.30876e7i		−	0.830081i	−0.909803	−	0.415040i	$-0.863767\pi$
$757$		−	1.30876e7i		−	0.830081i	0.909803	−	0.415040i	$-0.136233\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5.17392e6			−0.323861			−0.161930	−	0.986802i	$-0.551772\pi$
$761$	−5.17392e6			−0.323861			−0.161930	+	0.986802i	$0.551772\pi$
$762$		0			0
$763$		−	5.17826e6i		−	0.322012i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	1.42459e7i		−	0.874382i
$768$		0			0
$769$	3.02498e6			0.184462			0.0922309	−	0.995738i	$-0.470600\pi$
$769$	3.02498e6			0.184462			0.0922309	+	0.995738i	$0.470600\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	3.16901e6i		−	0.190754i	−0.995441	−	0.0953772i	$-0.969594\pi$
$773$		−	3.16901e6i		−	0.190754i	0.995441	−	0.0953772i	$-0.0304057\pi$
$774$		0			0
$775$	−25800.0	−	1.07469e6i	−0.00154300	−	0.0642731i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	170880.			0.0100890
$780$		0			0
$781$	3.87840e6			0.227523
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3.21592e6	−	3.29405e6i	0.186265	−	0.190790i
$786$		0			0
$787$		−	1.71533e7i		−	0.987213i	−0.869685	−	0.493607i	$-0.835678\pi$
$787$		−	1.71533e7i		−	0.987213i	0.869685	−	0.493607i	$-0.164322\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.67912e7			1.52248
$792$		0			0
$793$			3.07636e6i			0.173722i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	4.40084e6i		−	0.245409i	−0.992443	−	0.122704i	$-0.960843\pi$
$797$		−	4.40084e6i		−	0.245409i	0.992443	−	0.122704i	$-0.0391567\pi$
$798$		0			0
$799$	−1.34078e7			−0.743003
$800$		0			0
$801$		0			0
$802$		0			0
$803$			3.39902e6i			0.186022i
$804$		0			0
$805$	−1.01504e7	−	9.90964e6i	−0.552069	−	0.538975i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.61328e6			0.0866639			0.0433320	−	0.999061i	$-0.486203\pi$
$809$	1.61328e6			0.0866639			0.0433320	+	0.999061i	$0.486203\pi$
$810$		0			0
$811$	1.18501e7			0.632658			0.316329	−	0.948650i	$-0.397550\pi$
$811$	1.18501e7			0.632658			0.316329	+	0.948650i	$0.397550\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.03106e7	−	2.08040e7i	1.07109	−	1.09712i
$816$		0			0
$817$		−	227934.i		−	0.0119469i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.22232e6			0.270399			0.135200	−	0.990818i	$-0.456832\pi$
$821$	5.22232e6			0.270399			0.135200	+	0.990818i	$0.456832\pi$
$822$		0			0
$823$			1.19744e7i			0.616247i	0.951346	+	0.308124i	$0.0997011\pi$
$823$			1.19744e7i			0.616247i	−0.951346	+	0.308124i	$0.900299\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	2.34245e6i		−	0.119099i	−0.998225	−	0.0595493i	$-0.981034\pi$
$827$		−	2.34245e6i		−	0.119099i	0.998225	−	0.0595493i	$-0.0189663\pi$
$828$		0			0
$829$	1.62647e7			0.821976			0.410988	−	0.911641i	$-0.365184\pi$
$829$	1.62647e7			0.821976			0.410988	+	0.911641i	$0.365184\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	651141.i		−	0.0325134i
$834$		0			0
$835$	4.95930e6	−	5.07979e6i	0.246152	−	0.252133i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3.72685e7			1.82783			0.913917	−	0.405901i	$-0.133042\pi$
$839$	3.72685e7			1.82783			0.913917	+	0.405901i	$0.133042\pi$
$840$		0			0
$841$	282451.			0.0137706
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.22996e6	+	9.01104e6i	0.444691	+	0.434143i
$846$		0			0
$847$			1.93258e7i			0.925612i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8.88160e6			0.420404
$852$		0			0
$853$		−	3.75696e7i		−	1.76792i	−0.467559	−	0.883962i	$-0.654867\pi$
$853$		−	3.75696e7i		−	1.76792i	0.467559	−	0.883962i	$-0.345133\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.34720e7i		−	0.626584i	−0.949657	−	0.313292i	$-0.898568\pi$
$857$		−	1.34720e7i		−	0.626584i	0.949657	−	0.313292i	$-0.101432\pi$
$858$		0			0
$859$	−2.75912e7			−1.27582			−0.637908	−	0.770112i	$-0.720200\pi$
$859$	−2.75912e7			−1.27582			−0.637908	+	0.770112i	$0.720200\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	2.91240e7i		−	1.33114i	−0.746336	−	0.665569i	$-0.768189\pi$
$863$		−	2.91240e7i		−	1.33114i	0.746336	−	0.665569i	$-0.231811\pi$
$864$		0			0
$865$	−1.26240e7	+	1.29306e7i	−0.573661	+	0.587598i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	741120.			0.0332919
$870$		0			0
$871$	4.96589e6			0.221795
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.59820e7	+	1.48707e7i	0.705685	+	0.656616i
$876$		0			0
$877$			2.59561e7i			1.13957i	0.821794	+	0.569785i	$0.192974\pi$
$877$			2.59561e7i			1.13957i	−0.821794	+	0.569785i	$0.807026\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.36704e6			−0.189560			−0.0947802	−	0.995498i	$-0.530215\pi$
$881$	−4.36704e6			−0.189560			−0.0947802	+	0.995498i	$0.530215\pi$
$882$		0			0
$883$			2.04414e7i			0.882283i	0.897437	+	0.441142i	$0.145426\pi$
$883$			2.04414e7i			0.882283i	−0.897437	+	0.441142i	$0.854574\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	4.38694e7i		−	1.87220i	−0.351732	−	0.936101i	$-0.614407\pi$
$887$		−	4.38694e7i		−	1.87220i	0.351732	−	0.936101i	$-0.385593\pi$
$888$		0			0
$889$	3.05293e7			1.29557
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	294290.i		−	0.0123494i
$894$		0			0
$895$	1.98368e7	+	1.93663e7i	0.827778	+	0.808144i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.56864e6			−0.0647327
$900$		0			0
$901$	−1.49877e7			−0.615068
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.73206e7	−	2.66725e7i	−1.10884	−	1.08254i
$906$		0			0
$907$		−	982217.i		−	0.0396451i	−0.999804	−	0.0198225i	$-0.993690\pi$
$907$		−	982217.i		−	0.0396451i	0.999804	−	0.0198225i	$-0.00631012\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.14842e7			1.25689			0.628443	−	0.777855i	$-0.283692\pi$
$911$	3.14842e7			1.25689			0.628443	+	0.777855i	$0.283692\pi$
$912$		0			0
$913$		−	2.67423e6i		−	0.106175i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1.55655e7i			0.611280i
$918$		0			0
$919$	−1.84427e7			−0.720338			−0.360169	−	0.932887i	$-0.617281\pi$
$919$	−1.84427e7			−0.720338			−0.360169	+	0.932887i	$0.617281\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	1.81748e7i		−	0.702206i
$924$		0			0
$925$	−1.36640e7	+	328030.i	−0.525078	+	0.0126055i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.20322e7			−0.457409			−0.228704	−	0.973496i	$-0.573449\pi$
$929$	−1.20322e7			−0.457409			−0.228704	+	0.973496i	$0.573449\pi$
$930$		0			0
$931$	14292.0			0.000540404
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.70800e6	−	1.74950e6i	0.0638938	−	0.0654461i
$936$		0			0
$937$		−	2.32728e7i		−	0.865963i	−0.901403	−	0.432982i	$-0.857461\pi$
$937$		−	2.32728e7i		−	0.865963i	0.901403	−	0.432982i	$-0.142539\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4.15912e6			−0.153118			−0.0765592	−	0.997065i	$-0.524393\pi$
$941$	−4.15912e6			−0.153118			−0.0765592	+	0.997065i	$0.524393\pi$
$942$		0			0
$943$		−	2.89167e7i		−	1.05893i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1.55721e7i			0.564250i	0.959378	+	0.282125i	$0.0910393\pi$
$947$			1.55721e7i			0.564250i	−0.959378	+	0.282125i	$0.908961\pi$
$948$		0			0
$949$	1.59283e7			0.574122
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1.97834e7i			0.705618i	0.935695	+	0.352809i	$0.114773\pi$
$953$			1.97834e7i			0.705618i	−0.935695	+	0.352809i	$0.885227\pi$
$954$		0			0
$955$	2.50176e7	+	2.44242e7i	0.887641	+	0.866587i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.54718e7			1.59660
$960$		0			0
$961$	−2.85108e7			−0.995867
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.74256e6	+	2.80919e6i	−0.0948064	+	0.0971098i
$966$		0			0
$967$		−	3.84196e7i		−	1.32125i	−0.750715	−	0.660627i	$-0.770291\pi$
$967$		−	3.84196e7i		−	1.32125i	0.750715	−	0.660627i	$-0.229709\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.37188e7			−1.14769			−0.573844	−	0.818964i	$-0.694549\pi$
$971$	−3.37188e7			−1.14769			−0.573844	+	0.818964i	$0.694549\pi$
$972$		0			0
$973$		−	1.11263e7i		−	0.376763i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			3.32199e7i			1.11343i	0.830705	+	0.556713i	$0.187938\pi$
$977$			3.32199e7i			1.11343i	−0.830705	+	0.556713i	$0.812062\pi$
$978$		0			0
$979$	−1.94560e6			−0.0648779
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1.71493e7i		−	0.566059i	−0.959111	−	0.283030i	$-0.908660\pi$
$983$		−	1.71493e7i		−	0.566059i	0.959111	−	0.283030i	$-0.0913395\pi$
$984$		0			0
$985$	7.02415e6	−	7.19480e6i	0.230676	−	0.236281i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.85715e7			−1.25394
$990$		0			0
$991$	−9.28074e6			−0.300191			−0.150096	−	0.988671i	$-0.547958\pi$
$991$	−9.28074e6			−0.300191			−0.150096	+	0.988671i	$0.547958\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.18291e7	+	1.15485e7i	0.378787	+	0.369802i
$996$		0			0
$997$			3.71124e7i			1.18245i	0.806508	+	0.591223i	$0.201355\pi$
$997$			3.71124e7i			1.18245i	−0.806508	+	0.591223i	$0.798645\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.6.d.c.109.2	yes	2
3.2	odd	2		180.6.d.a.109.1	✓	2
4.3	odd	2		720.6.f.e.289.2		2
5.2	odd	4		900.6.a.p.1.2		2
5.3	odd	4		900.6.a.p.1.1		2
5.4	even	2	inner	180.6.d.c.109.1	yes	2
12.11	even	2		720.6.f.b.289.1		2
15.2	even	4		900.6.a.o.1.2		2
15.8	even	4		900.6.a.o.1.1		2
15.14	odd	2		180.6.d.a.109.2	yes	2
20.19	odd	2		720.6.f.e.289.1		2
60.59	even	2		720.6.f.b.289.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.6.d.a.109.1	✓	2	3.2	odd	2
180.6.d.a.109.2	yes	2	15.14	odd	2
180.6.d.c.109.1	yes	2	5.4	even	2	inner
180.6.d.c.109.2	yes	2	1.1	even	1	trivial
720.6.f.b.289.1		2	12.11	even	2
720.6.f.b.289.2		2	60.59	even	2
720.6.f.e.289.1		2	20.19	odd	2
720.6.f.e.289.2		2	4.3	odd	2
900.6.a.o.1.1		2	15.8	even	4
900.6.a.o.1.2		2	15.2	even	4
900.6.a.p.1.1		2	5.3	odd	4
900.6.a.p.1.2		2	5.2	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	180.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(40.0000 + 39.0512i) q^{5} -124.964i q^{7} +80.0000 q^{11} -374.892i q^{13} -546.717i q^{17} +12.0000 q^{19} -2030.66i q^{23} +(75.0000 + 3124.10i) q^{25} +4560.00 q^{29} -344.000 q^{31} +(4880.00 - 4998.56i) q^{35} +4373.74i q^{37} +14240.0 q^{41} -18994.5i q^{43} -24524.2i q^{47} +1191.00 q^{49} -27414.0i q^{53} +(3200.00 + 3124.10i) q^{55} +38000.0 q^{59} -8206.00 q^{61} +(14640.0 - 14995.7i) q^{65} +13246.2i q^{67} +48480.0 q^{71} +42487.8i q^{73} -9997.12i q^{77} +9264.00 q^{79} -33427.9i q^{83} +(21350.0 - 21868.7i) q^{85} -24320.0 q^{89} -46848.0 q^{91} +(480.000 + 468.615i) q^{95} -136711. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 80 q^{5} + 160 q^{11} + 24 q^{19} + 150 q^{25} + 9120 q^{29} - 688 q^{31} + 9760 q^{35} + 28480 q^{41} + 2382 q^{49} + 6400 q^{55} + 76000 q^{59} - 16412 q^{61} + 29280 q^{65} + 96960 q^{71} + 18528 q^{79}+ \cdots + 960 q^{95}+O(q^{100}) \) 2 * q + 80 * q^5 + 160 * q^11 + 24 * q^19 + 150 * q^25 + 9120 * q^29 - 688 * q^31 + 9760 * q^35 + 28480 * q^41 + 2382 * q^49 + 6400 * q^55 + 76000 * q^59 - 16412 * q^61 + 29280 * q^65 + 96960 * q^71 + 18528 * q^79 + 42700 * q^85 - 48640 * q^89 - 93696 * q^91 + 960 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more