LMFDB - Embedded newform 900.6.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,6,Mod(1,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("900.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	900.a (trivial)

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:	yes
Analytic conductor:	$144.345437832$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{61}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 15 $ x^2 - x - 15
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.40512$ of defining polynomial
Character	$\chi$	$=$	900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-124.964 q^{7} +O(q^{10})$	$q-124.964 q^{7} +80.0000 q^{11} +374.892 q^{13} -546.717 q^{17} -12.0000 q^{19} +2030.66 q^{23} -4560.00 q^{29} -344.000 q^{31} +4373.74 q^{37} +14240.0 q^{41} +18994.5 q^{43} -24524.2 q^{47} -1191.00 q^{49} +27414.0 q^{53} -38000.0 q^{59} -8206.00 q^{61} +13246.2 q^{67} +48480.0 q^{71} -42487.8 q^{73} -9997.12 q^{77} -9264.00 q^{79} +33427.9 q^{83} +24320.0 q^{89} -46848.0 q^{91} -136711. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 160 q^{11} - 24 q^{19} - 9120 q^{29} - 688 q^{31} + 28480 q^{41} - 2382 q^{49} - 76000 q^{59} - 16412 q^{61} + 96960 q^{71} - 18528 q^{79} + 48640 q^{89} - 93696 q^{91}+O(q^{100}) $ 2 * q + 160 * q^11 - 24 * q^19 - 9120 * q^29 - 688 * q^31 + 28480 * q^41 - 2382 * q^49 - 76000 * q^59 - 16412 * q^61 + 96960 * q^71 - 18528 * q^79 + 48640 * q^89 - 93696 * q^91

Display 100 coefficients 10 coefficients

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−124.964	−0.963917	−0.481959	−	0.876194i	$-0.660074\pi$
$7$	−124.964	−0.963917	−0.481959	+	0.876194i	$0.660074\pi$
$8$	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
$13$	374.892	0.615245	0.307622	−	0.951509i	$-0.400467\pi$
$13$	374.892	0.615245	0.307622	+	0.951509i	$0.400467\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−546.717	−0.458818	−0.229409	−	0.973330i	$-0.573679\pi$
$17$	−546.717	−0.458818	−0.229409	+	0.973330i	$0.573679\pi$
$18$	0	0
$19$	−12.0000	−0.00762601	−0.00381300	−	0.999993i	$-0.501214\pi$
$19$	−12.0000	−0.00762601	−0.00381300	+	0.999993i	$0.501214\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2030.66	0.800421	0.400211	−	0.916423i	$-0.368937\pi$
$23$	2030.66	0.800421	0.400211	+	0.916423i	$0.368937\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4560.00	−1.00686	−0.503431	−	0.864036i	$-0.667929\pi$
$29$	−4560.00	−1.00686	−0.503431	+	0.864036i	$0.667929\pi$
$30$	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
$35$	0	0
$36$	0	0
$37$	4373.74	0.525229	0.262614	−	0.964901i	$-0.415415\pi$
$37$	4373.74	0.525229	0.262614	+	0.964901i	$0.415415\pi$
$38$	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
$43$	18994.5	1.56660	0.783299	−	0.621646i	$-0.213536\pi$
$43$	18994.5	1.56660	0.783299	+	0.621646i	$0.213536\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−24524.2	−1.61938	−0.809692	−	0.586855i	$-0.800366\pi$
$47$	−24524.2	−1.61938	−0.809692	+	0.586855i	$0.800366\pi$
$48$	0	0
$49$	−1191.00	−0.0708633
$50$	0	0
$51$	0	0
$52$	0	0
$53$	27414.0	1.34055	0.670274	−	0.742114i	$-0.266177\pi$
$53$	27414.0	1.34055	0.670274	+	0.742114i	$0.266177\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−38000.0	−1.42119	−0.710597	−	0.703599i	$-0.751575\pi$
$59$	−38000.0	−1.42119	−0.710597	+	0.703599i	$0.751575\pi$
$60$	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
$65$	0	0
$66$	0	0
$67$	13246.2	0.360499	0.180249	−	0.983621i	$-0.442310\pi$
$67$	13246.2	0.360499	0.180249	+	0.983621i	$0.442310\pi$
$68$	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
$73$	−42487.8	−0.933161	−0.466581	−	0.884479i	$-0.654514\pi$
$73$	−42487.8	−0.933161	−0.466581	+	0.884479i	$0.654514\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9997.12	−0.192153
$78$	0	0
$79$	−9264.00	−0.167006	−0.0835028	−	0.996508i	$-0.526611\pi$
$79$	−9264.00	−0.167006	−0.0835028	+	0.996508i	$0.526611\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	33427.9	0.532615	0.266308	−	0.963888i	$-0.414196\pi$
$83$	33427.9	0.532615	0.266308	+	0.963888i	$0.414196\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	24320.0	0.325453	0.162727	−	0.986671i	$-0.447971\pi$
$89$	24320.0	0.325453	0.162727	+	0.986671i	$0.447971\pi$
$90$	0	0
$91$	−46848.0	−0.593045
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−136711.	−1.47527	−0.737637	−	0.675197i	$-0.764059\pi$
$97$	−136711.	−1.47527	−0.737637	+	0.675197i	$0.764059\pi$
$98$	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
$103$	−68105.4	−0.632541	−0.316270	−	0.948669i	$-0.602431\pi$
$103$	−68105.4	−0.632541	−0.316270	+	0.948669i	$0.602431\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	21556.3	0.182018	0.0910090	−	0.995850i	$-0.470991\pi$
$107$	21556.3	0.182018	0.0910090	+	0.995850i	$0.470991\pi$
$108$	0	0
$109$	−41438.0	−0.334066	−0.167033	−	0.985951i	$-0.553419\pi$
$109$	−41438.0	−0.334066	−0.167033	+	0.985951i	$0.553419\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−214391.	−1.57947	−0.789735	−	0.613449i	$-0.789782\pi$
$113$	−214391.	−1.57947	−0.789735	+	0.613449i	$0.789782\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	0	0
$127$	244305.	1.34407	0.672036	−	0.740519i	$-0.265420\pi$
$127$	244305.	1.34407	0.672036	+	0.740519i	$0.265420\pi$
$128$	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
$133$	1499.57	0.00735084
$134$	0	0
$135$	0	0
$136$	0	0
$137$	363880.	1.65637	0.828183	−	0.560458i	$-0.189375\pi$
$137$	363880.	1.65637	0.828183	+	0.560458i	$0.189375\pi$
$138$	0	0
$139$	−89036.0	−0.390867	−0.195433	−	0.980717i	$-0.562611\pi$
$139$	−89036.0	−0.390867	−0.195433	+	0.980717i	$0.562611\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	29991.4	0.122647
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	438640.	1.61861	0.809306	−	0.587388i	$-0.199844\pi$
$149$	438640.	1.61861	0.809306	+	0.587388i	$0.199844\pi$
$150$	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
$155$	0	0
$156$	0	0
$157$	−82351.3	−0.266637	−0.133319	−	0.991073i	$-0.542563\pi$
$157$	−82351.3	−0.266637	−0.133319	+	0.991073i	$0.542563\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−253760.	−0.771540
$162$	0	0
$163$	520100.	1.53327	0.766634	−	0.642085i	$-0.221930\pi$
$163$	520100.	1.53327	0.766634	+	0.642085i	$0.221930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−126995.	−0.352366	−0.176183	−	0.984357i	$-0.556375\pi$
$167$	−126995.	−0.352366	−0.176183	+	0.984357i	$0.556375\pi$
$168$	0	0
$169$	−230749.	−0.621474
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−323266.	−0.821193	−0.410596	−	0.911817i	$-0.634679\pi$
$173$	−323266.	−0.821193	−0.410596	+	0.911817i	$0.634679\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−495920.	−1.15686	−0.578428	−	0.815734i	$-0.696333\pi$
$179$	−495920.	−1.15686	−0.578428	+	0.815734i	$0.696333\pi$
$180$	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
$185$	0	0
$186$	0	0
$187$	−43737.4	−0.0914637
$188$	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
$193$	−70229.8	−0.135715	−0.0678575	−	0.997695i	$-0.521616\pi$
$193$	−70229.8	−0.135715	−0.0678575	+	0.997695i	$0.521616\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−179870.	−0.330212	−0.165106	−	0.986276i	$-0.552797\pi$
$197$	−179870.	−0.330212	−0.165106	+	0.986276i	$0.552797\pi$
$198$	0	0
$199$	−295728.	−0.529371	−0.264685	−	0.964335i	$-0.585268\pi$
$199$	−295728.	−0.529371	−0.264685	+	0.964335i	$0.585268\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	569836.	0.970532
$204$	0	0
$205$	0	0
$206$	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
$215$	0	0
$216$	0	0
$217$	42987.6	0.0619718
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−204960.	−0.282285
$222$	0	0
$223$	−799395.	−1.07646	−0.538231	−	0.842797i	$-0.680907\pi$
$223$	−799395.	−1.07646	−0.538231	+	0.842797i	$0.680907\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.00315e6	−1.29211	−0.646057	−	0.763289i	$-0.723583\pi$
$227$	−1.00315e6	−1.29211	−0.646057	+	0.763289i	$0.723583\pi$
$228$	0	0
$229$	−593002.	−0.747253	−0.373626	−	0.927579i	$-0.621886\pi$
$229$	−593002.	−0.747253	−0.373626	+	0.927579i	$0.621886\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	630834.	0.761246	0.380623	−	0.924730i	$-0.375710\pi$
$233$	630834.	0.761246	0.380623	+	0.924730i	$0.375710\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.00240e6	−1.13513	−0.567566	−	0.823328i	$-0.692115\pi$
$239$	−1.00240e6	−1.13513	−0.567566	+	0.823328i	$0.692115\pi$
$240$	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
$245$	0	0
$246$	0	0
$247$	−4498.70	−0.00469186
$248$	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.	0.159561
$254$	0	0
$255$	0	0
$256$	0	0
$257$	668479.	0.631328	0.315664	−	0.948871i	$-0.397773\pi$
$257$	668479.	0.631328	0.315664	+	0.948871i	$0.397773\pi$
$258$	0	0
$259$	−546560.	−0.506277
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.09672e6	0.977698	0.488849	−	0.872369i	$-0.337417\pi$
$263$	1.09672e6	0.977698	0.488849	+	0.872369i	$0.337417\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−562800.	−0.474213	−0.237106	−	0.971484i	$-0.576199\pi$
$269$	−562800.	−0.474213	−0.237106	+	0.971484i	$0.576199\pi$
$270$	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
$275$	0	0
$276$	0	0
$277$	−768654.	−0.601910	−0.300955	−	0.953638i	$-0.597305\pi$
$277$	−768654.	−0.601910	−0.300955	+	0.953638i	$0.597305\pi$
$278$	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.	−0.450956	−0.225478	−	0.974248i	$-0.572394\pi$
$283$	−607575.	−0.450956	−0.225478	+	0.974248i	$0.572394\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.77949e6	−1.27523
$288$	0	0
$289$	−1.12096e6	−0.789486
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.87001e6	−1.27255	−0.636274	−	0.771463i	$-0.719525\pi$
$293$	−1.87001e6	−1.27255	−0.636274	+	0.771463i	$0.719525\pi$
$294$	0	0
$295$	0	0
$296$	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
$305$	0	0
$306$	0	0
$307$	874248.	0.529406	0.264703	−	0.964330i	$-0.414726\pi$
$307$	874248.	0.529406	0.264703	+	0.964330i	$0.414726\pi$
$308$	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.	−0.306994	−0.153497	−	0.988149i	$-0.549054\pi$
$313$	−532097.	−0.306994	−0.153497	+	0.988149i	$0.549054\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.75535e6	0.981107	0.490554	−	0.871411i	$-0.336795\pi$
$317$	1.75535e6	0.981107	0.490554	+	0.871411i	$0.336795\pi$
$318$	0	0
$319$	−364800.	−0.200714
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6560.61	0.00349895
$324$	0	0
$325$	0	0
$326$	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
$335$	0	0
$336$	0	0
$337$	927733.	0.444988	0.222494	−	0.974934i	$-0.428580\pi$
$337$	927733.	0.444988	0.222494	+	0.974934i	$0.428580\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−27520.0	−0.0128163
$342$	0	0
$343$	2.24910e6	1.03222
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.12569e6	1.83938	0.919692	−	0.392640i	$-0.128438\pi$
$347$	4.12569e6	1.83938	0.919692	+	0.392640i	$0.128438\pi$
$348$	0	0
$349$	−1.63749e6	−0.719638	−0.359819	−	0.933022i	$-0.617162\pi$
$349$	−1.63749e6	−0.719638	−0.359819	+	0.933022i	$0.617162\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.98310e6	−0.847048	−0.423524	−	0.905885i	$-0.639207\pi$
$353$	−1.98310e6	−0.847048	−0.423524	+	0.905885i	$0.639207\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.14720e6	−0.469789	−0.234895	−	0.972021i	$-0.575474\pi$
$359$	−1.14720e6	−0.469789	−0.234895	+	0.972021i	$0.575474\pi$
$360$	0	0
$361$	−2.47596e6	−0.999942
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.09395e6	−1.58663	−0.793317	−	0.608808i	$-0.791648\pi$
$367$	−4.09395e6	−1.58663	−0.793317	+	0.608808i	$0.791648\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.42576e6	−1.29218
$372$	0	0
$373$	2.99126e6	1.11322	0.556612	−	0.830773i	$-0.312101\pi$
$373$	2.99126e6	1.11322	0.556612	+	0.830773i	$0.312101\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.70951e6	−0.619466
$378$	0	0
$379$	98228.0	0.0351267	0.0175633	−	0.999846i	$-0.494409\pi$
$379$	98228.0	0.0351267	0.0175633	+	0.999846i	$0.494409\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.70969e6	−0.943892	−0.471946	−	0.881627i	$-0.656448\pi$
$383$	−2.70969e6	−0.943892	−0.471946	+	0.881627i	$0.656448\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.03064e6	0.345329	0.172664	−	0.984981i	$-0.444762\pi$
$389$	1.03064e6	0.345329	0.172664	+	0.984981i	$0.444762\pi$
$390$	0	0
$391$	−1.11020e6	−0.367248
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.10312e6	1.94346	0.971730	−	0.236097i	$-0.0758683\pi$
$397$	6.10312e6	1.94346	0.971730	+	0.236097i	$0.0758683\pi$
$398$	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.	−0.0395551
$404$	0	0
$405$	0	0
$406$	0	0
$407$	349899.	0.104702
$408$	0	0
$409$	−1.32812e6	−0.392581	−0.196291	−	0.980546i	$-0.562890\pi$
$409$	−1.32812e6	−0.392581	−0.196291	+	0.980546i	$0.562890\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.74863e6	1.36991
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.65944e6	1.29658	0.648289	−	0.761394i	$-0.275485\pi$
$419$	4.65944e6	1.29658	0.648289	+	0.761394i	$0.275485\pi$
$420$	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
$425$	0	0
$426$	0	0
$427$	1.02545e6	0.272174
$428$	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
$433$	−5.14627e6	−1.31908	−0.659542	−	0.751668i	$-0.729250\pi$
$433$	−5.14627e6	−1.31908	−0.659542	+	0.751668i	$0.729250\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−24368.0	−0.00610402
$438$	0	0
$439$	1.30171e6	0.322369	0.161185	−	0.986924i	$-0.448469\pi$
$439$	1.30171e6	0.322369	0.161185	+	0.986924i	$0.448469\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.02069e6	−0.731303	−0.365651	−	0.930752i	$-0.619154\pi$
$443$	−3.02069e6	−0.731303	−0.365651	+	0.930752i	$0.619154\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8.07504e6	−1.89029	−0.945146	−	0.326649i	$-0.894081\pi$
$449$	−8.07504e6	−1.89029	−0.945146	+	0.326649i	$0.894081\pi$
$450$	0	0
$451$	1.13920e6	0.263729
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.78487e6	−1.07172	−0.535858	−	0.844308i	$-0.680012\pi$
$457$	−4.78487e6	−1.07172	−0.535858	+	0.844308i	$0.680012\pi$
$458$	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
$463$	−2.46117e6	−0.533566	−0.266783	−	0.963757i	$-0.585961\pi$
$463$	−2.46117e6	−0.533566	−0.266783	+	0.963757i	$0.585961\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.38244e6	−0.505510	−0.252755	−	0.967530i	$-0.581337\pi$
$467$	−2.38244e6	−0.505510	−0.252755	+	0.967530i	$0.581337\pi$
$468$	0	0
$469$	−1.65530e6	−0.347491
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.51956e6	0.312295
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.79568e6	−0.556735	−0.278368	−	0.960475i	$-0.589793\pi$
$479$	−2.79568e6	−0.556735	−0.278368	+	0.960475i	$0.589793\pi$
$480$	0	0
$481$	1.63968e6	0.323144
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3.99372e6	−0.763055	−0.381527	−	0.924358i	$-0.624602\pi$
$487$	−3.99372e6	−0.763055	−0.381527	+	0.924358i	$0.624602\pi$
$488$	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
$493$	2.49303e6	0.461967
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.05825e6	−1.10016
$498$	0	0
$499$	4.80013e6	0.862982	0.431491	−	0.902117i	$-0.357988\pi$
$499$	4.80013e6	0.862982	0.431491	+	0.902117i	$0.357988\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.24718e6	1.62963	0.814816	−	0.579720i	$-0.196838\pi$
$503$	9.24718e6	1.62963	0.814816	+	0.579720i	$0.196838\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.71816e6	−0.293947	−0.146974	−	0.989140i	$-0.546953\pi$
$509$	−1.71816e6	−0.293947	−0.146974	+	0.989140i	$0.546953\pi$
$510$	0	0
$511$	5.30944e6	0.899490
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.96193e6	−0.322818
$518$	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
$523$	−5.19325e6	−0.830205	−0.415102	−	0.909775i	$-0.636254\pi$
$523$	−5.19325e6	−0.830205	−0.415102	+	0.909775i	$0.636254\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	188071.	0.0294982
$528$	0	0
$529$	−2.31274e6	−0.359326
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.33846e6	0.813951
$534$	0	0
$535$	0	0
$536$	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
$545$	0	0
$546$	0	0
$547$	6.27819e6	0.897152	0.448576	−	0.893745i	$-0.351931\pi$
$547$	6.27819e6	0.897152	0.448576	+	0.893745i	$0.351931\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	54720.0	0.00767834
$552$	0	0
$553$	1.15767e6	0.160980
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.89408e6	−1.21468	−0.607341	−	0.794441i	$-0.707764\pi$
$557$	−8.89408e6	−1.21468	−0.607341	+	0.794441i	$0.707764\pi$
$558$	0	0
$559$	7.12090e6	0.963840
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.18135e7	−1.57075	−0.785374	−	0.619021i	$-0.787529\pi$
$563$	−1.18135e7	−1.57075	−0.785374	+	0.619021i	$0.787529\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.89744e6	−0.375175	−0.187587	−	0.982248i	$-0.560067\pi$
$569$	−2.89744e6	−0.375175	−0.187587	+	0.982248i	$0.560067\pi$
$570$	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
$575$	0	0
$576$	0	0
$577$	1.79898e6	0.224951	0.112475	−	0.993655i	$-0.464122\pi$
$577$	1.79898e6	0.224951	0.112475	+	0.993655i	$0.464122\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.17728e6	−0.513397
$582$	0	0
$583$	2.19312e6	0.267233
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.29178e7	1.54737	0.773686	−	0.633569i	$-0.218411\pi$
$587$	1.29178e7	1.54737	0.773686	+	0.633569i	$0.218411\pi$
$588$	0	0
$589$	4128.00	0.000490288 0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.13907e6	0.600133	0.300066	−	0.953918i	$-0.402991\pi$
$593$	5.13907e6	0.600133	0.300066	+	0.953918i	$0.402991\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.62272e6	0.640294	0.320147	−	0.947368i	$-0.396268\pi$
$599$	5.62272e6	0.640294	0.320147	+	0.947368i	$0.396268\pi$
$600$	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
$605$	0	0
$606$	0	0
$607$	7.03610e6	0.775104	0.387552	−	0.921848i	$-0.373321\pi$
$607$	7.03610e6	0.775104	0.387552	+	0.921848i	$0.373321\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.19392e6	−0.996317
$612$	0	0
$613$	1.54499e7	1.66064	0.830319	−	0.557288i	$-0.188158\pi$
$613$	1.54499e7	1.66064	0.830319	+	0.557288i	$0.188158\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.86750e6	0.832001	0.416000	−	0.909364i	$-0.363431\pi$
$617$	7.86750e6	0.832001	0.416000	+	0.909364i	$0.363431\pi$
$618$	0	0
$619$	8.69805e6	0.912421	0.456211	−	0.889872i	$-0.349206\pi$
$619$	8.69805e6	0.912421	0.456211	+	0.889872i	$0.349206\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.03912e6	−0.313710
$624$	0	0
$625$	0	0
$626$	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
$635$	0	0
$636$	0	0
$637$	−446496.	−0.0435983
$638$	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
$643$	1.18953e7	1.13462	0.567308	−	0.823506i	$-0.307985\pi$
$643$	1.18953e7	1.13462	0.567308	+	0.823506i	$0.307985\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.03339e6	−0.472716	−0.236358	−	0.971666i	$-0.575954\pi$
$647$	−5.03339e6	−0.472716	−0.236358	+	0.971666i	$0.575954\pi$
$648$	0	0
$649$	−3.04000e6	−0.283310
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.12571e7	−1.03311	−0.516554	−	0.856255i	$-0.672785\pi$
$653$	−1.12571e7	−1.03311	−0.516554	+	0.856255i	$0.672785\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.84982e7	1.65926	0.829631	−	0.558312i	$-0.188551\pi$
$659$	1.84982e7	1.65926	0.829631	+	0.558312i	$0.188551\pi$
$660$	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
$665$	0	0
$666$	0	0
$667$	−9.25983e6	−0.805914
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−656480.	−0.0562879
$672$	0	0
$673$	−2.04878e7	−1.74365	−0.871824	−	0.489820i	$-0.837063\pi$
$673$	−2.04878e7	−1.74365	−0.871824	+	0.489820i	$0.837063\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.31873e6	0.529856	0.264928	−	0.964268i	$-0.414652\pi$
$677$	6.31873e6	0.529856	0.264928	+	0.964268i	$0.414652\pi$
$678$	0	0
$679$	1.70839e7	1.42204
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.41028e7	−1.15679	−0.578394	−	0.815757i	$-0.696320\pi$
$683$	−1.41028e7	−1.15679	−0.578394	+	0.815757i	$0.696320\pi$
$684$	0	0
$685$	0	0
$686$	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
$695$	0	0
$696$	0	0
$697$	−7.78526e6	−0.607003
$698$	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
$703$	−52484.9	−0.00400540
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.31462e7	0.989127
$708$	0	0
$709$	−7.55428e6	−0.564388	−0.282194	−	0.959357i	$-0.591062\pi$
$709$	−7.55428e6	−0.564388	−0.282194	+	0.959357i	$0.591062\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−698549.	−0.0514604
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9.20608e6	−0.664129	−0.332065	−	0.943257i	$-0.607745\pi$
$719$	−9.20608e6	−0.664129	−0.332065	+	0.943257i	$0.607745\pi$
$720$	0	0
$721$	8.51072e6	0.609717
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.02918e7	1.42392	0.711958	−	0.702223i	$-0.247809\pi$
$727$	2.02918e7	1.42392	0.711958	+	0.702223i	$0.247809\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.03846e7	−0.718783
$732$	0	0
$733$	−1.93413e7	−1.32961	−0.664807	−	0.747015i	$-0.731486\pi$
$733$	−1.93413e7	−1.32961	−0.664807	+	0.747015i	$0.731486\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.05969e6	0.0718641
$738$	0	0
$739$	1.45942e7	0.983033	0.491517	−	0.870868i	$-0.336443\pi$
$739$	1.45942e7	0.983033	0.491517	+	0.870868i	$0.336443\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.82896e7	1.21543	0.607717	−	0.794153i	$-0.292085\pi$
$743$	1.82896e7	1.21543	0.607717	+	0.794153i	$0.292085\pi$
$744$	0	0
$745$	0	0
$746$	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
$755$	0	0
$756$	0	0
$757$	−1.30876e7	−0.830081	−0.415040	−	0.909803i	$-0.636233\pi$
$757$	−1.30876e7	−0.830081	−0.415040	+	0.909803i	$0.636233\pi$
$758$	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
$763$	5.17826e6	0.322012
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.42459e7	−0.874382
$768$	0	0
$769$	−3.02498e6	−0.184462	−0.0922309	−	0.995738i	$-0.529400\pi$
$769$	−3.02498e6	−0.184462	−0.0922309	+	0.995738i	$0.529400\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.16901e6	0.190754	0.0953772	−	0.995441i	$-0.469594\pi$
$773$	3.16901e6	0.190754	0.0953772	+	0.995441i	$0.469594\pi$
$774$	0	0
$775$	0	0
$776$	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
$785$	0	0
$786$	0	0
$787$	−1.71533e7	−0.987213	−0.493607	−	0.869685i	$-0.664322\pi$
$787$	−1.71533e7	−0.987213	−0.493607	+	0.869685i	$0.664322\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.67912e7	1.52248
$792$	0	0
$793$	−3.07636e6	−0.173722
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.40084e6	−0.245409	−0.122704	−	0.992443i	$-0.539157\pi$
$797$	−4.40084e6	−0.245409	−0.122704	+	0.992443i	$0.539157\pi$
$798$	0	0
$799$	1.34078e7	0.743003
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.39902e6	−0.186022
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.61328e6	−0.0866639	−0.0433320	−	0.999061i	$-0.513797\pi$
$809$	−1.61328e6	−0.0866639	−0.0433320	+	0.999061i	$0.513797\pi$
$810$	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
$815$	0	0
$816$	0	0
$817$	−227934.	−0.0119469
$818$	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
$823$	−1.19744e7	−0.616247	−0.308124	−	0.951346i	$-0.599701\pi$
$823$	−1.19744e7	−0.616247	−0.308124	+	0.951346i	$0.599701\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.34245e6	−0.119099	−0.0595493	−	0.998225i	$-0.518966\pi$
$827$	−2.34245e6	−0.119099	−0.0595493	+	0.998225i	$0.518966\pi$
$828$	0	0
$829$	−1.62647e7	−0.821976	−0.410988	−	0.911641i	$-0.634816\pi$
$829$	−1.62647e7	−0.821976	−0.410988	+	0.911641i	$0.634816\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	651141.	0.0325134
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.72685e7	−1.82783	−0.913917	−	0.405901i	$-0.866958\pi$
$839$	−3.72685e7	−1.82783	−0.913917	+	0.405901i	$0.866958\pi$
$840$	0	0
$841$	282451.	0.0137706
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.93258e7	0.925612
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.88160e6	0.420404
$852$	0	0
$853$	3.75696e7	1.76792	0.883962	−	0.467559i	$-0.154867\pi$
$853$	3.75696e7	1.76792	0.883962	+	0.467559i	$0.154867\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.34720e7	−0.626584	−0.313292	−	0.949657i	$-0.601432\pi$
$857$	−1.34720e7	−0.626584	−0.313292	+	0.949657i	$0.601432\pi$
$858$	0	0
$859$	2.75912e7	1.27582	0.637908	−	0.770112i	$-0.279800\pi$
$859$	2.75912e7	1.27582	0.637908	+	0.770112i	$0.279800\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.91240e7	1.33114	0.665569	−	0.746336i	$-0.268189\pi$
$863$	2.91240e7	1.33114	0.665569	+	0.746336i	$0.268189\pi$
$864$	0	0
$865$	0	0
$866$	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
$875$	0	0
$876$	0	0
$877$	2.59561e7	1.13957	0.569785	−	0.821794i	$-0.307026\pi$
$877$	2.59561e7	1.13957	0.569785	+	0.821794i	$0.307026\pi$
$878$	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
$883$	−2.04414e7	−0.882283	−0.441142	−	0.897437i	$-0.645426\pi$
$883$	−2.04414e7	−0.882283	−0.441142	+	0.897437i	$0.645426\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.38694e7	−1.87220	−0.936101	−	0.351732i	$-0.885593\pi$
$887$	−4.38694e7	−1.87220	−0.936101	+	0.351732i	$0.885593\pi$
$888$	0	0
$889$	−3.05293e7	−1.29557
$890$	0	0
$891$	0	0
$892$	0	0
$893$	294290.	0.0123494
$894$	0	0
$895$	0	0
$896$	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
$905$	0	0
$906$	0	0
$907$	−982217.	−0.0396451	−0.0198225	−	0.999804i	$-0.506310\pi$
$907$	−982217.	−0.0396451	−0.0198225	+	0.999804i	$0.506310\pi$
$908$	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
$913$	2.67423e6	0.106175
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.55655e7	0.611280
$918$	0	0
$919$	1.84427e7	0.720338	0.360169	−	0.932887i	$-0.382719\pi$
$919$	1.84427e7	0.720338	0.360169	+	0.932887i	$0.382719\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.81748e7	0.702206
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.20322e7	0.457409	0.228704	−	0.973496i	$-0.426551\pi$
$929$	1.20322e7	0.457409	0.228704	+	0.973496i	$0.426551\pi$
$930$	0	0
$931$	14292.0	0.000540404 0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.32728e7	−0.865963	−0.432982	−	0.901403i	$-0.642539\pi$
$937$	−2.32728e7	−0.865963	−0.432982	+	0.901403i	$0.642539\pi$
$938$	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
$943$	2.89167e7	1.05893
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.55721e7	0.564250	0.282125	−	0.959378i	$-0.408961\pi$
$947$	1.55721e7	0.564250	0.282125	+	0.959378i	$0.408961\pi$
$948$	0	0
$949$	−1.59283e7	−0.574122
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.97834e7	−0.705618	−0.352809	−	0.935695i	$-0.614773\pi$
$953$	−1.97834e7	−0.705618	−0.352809	+	0.935695i	$0.614773\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	0	0
$967$	−3.84196e7	−1.32125	−0.660627	−	0.750715i	$-0.729709\pi$
$967$	−3.84196e7	−1.32125	−0.660627	+	0.750715i	$0.729709\pi$
$968$	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
$973$	1.11263e7	0.376763
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.32199e7	1.11343	0.556713	−	0.830705i	$-0.312062\pi$
$977$	3.32199e7	1.11343	0.556713	+	0.830705i	$0.312062\pi$
$978$	0	0
$979$	1.94560e6	0.0648779
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.71493e7	0.566059	0.283030	−	0.959111i	$-0.408660\pi$
$983$	1.71493e7	0.566059	0.283030	+	0.959111i	$0.408660\pi$
$984$	0	0
$985$	0	0
$986$	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
$995$	0	0
$996$	0	0
$997$	3.71124e7	1.18245	0.591223	−	0.806508i	$-0.298645\pi$
$997$	3.71124e7	1.18245	0.591223	+	0.806508i	$0.298645\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.6.d.a.109.1	✓	2	15.2	even	4
180.6.d.a.109.2	yes	2	15.8	even	4
180.6.d.c.109.1	yes	2	5.3	odd	4
180.6.d.c.109.2	yes	2	5.2	odd	4
720.6.f.b.289.1		2	60.47	odd	4
720.6.f.b.289.2		2	60.23	odd	4
720.6.f.e.289.1		2	20.3	even	4
720.6.f.e.289.2		2	20.7	even	4
900.6.a.o.1.1		2	3.2	odd	2
900.6.a.o.1.2		2	15.14	odd	2
900.6.a.p.1.1		2	1.1	even	1	trivial
900.6.a.p.1.2		2	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)

\(f(q)\)	\(=\)	\(q-124.964 q^{7} +O(q^{10})\)	\(q-124.964 q^{7} +80.0000 q^{11} +374.892 q^{13} -546.717 q^{17} -12.0000 q^{19} +2030.66 q^{23} -4560.00 q^{29} -344.000 q^{31} +4373.74 q^{37} +14240.0 q^{41} +18994.5 q^{43} -24524.2 q^{47} -1191.00 q^{49} +27414.0 q^{53} -38000.0 q^{59} -8206.00 q^{61} +13246.2 q^{67} +48480.0 q^{71} -42487.8 q^{73} -9997.12 q^{77} -9264.00 q^{79} +33427.9 q^{83} +24320.0 q^{89} -46848.0 q^{91} -136711. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 160 q^{11} - 24 q^{19} - 9120 q^{29} - 688 q^{31} + 28480 q^{41} - 2382 q^{49} - 76000 q^{59} - 16412 q^{61} + 96960 q^{71} - 18528 q^{79} + 48640 q^{89} - 93696 q^{91}+O(q^{100}) \) 2 * q + 160 * q^11 - 24 * q^19 - 9120 * q^29 - 688 * q^31 + 28480 * q^41 - 2382 * q^49 - 76000 * q^59 - 16412 * q^61 + 96960 * q^71 - 18528 * q^79 + 48640 * q^89 - 93696 * q^91

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