LMFDB - Embedded newform 900.6.j.c.557.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [900,6,Mod(557,900)] 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("900.557"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	900.j (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$144.345437832$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			557.1
Character	$\chi$	$=$	900.557
Dual form			900.6.j.c.593.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-123.937 - 123.937i) q^{7} +296.299i q^{11} +(16.2336 - 16.2336i) q^{13} +(-547.265 + 547.265i) q^{17} +262.818i q^{19} +(-2434.99 - 2434.99i) q^{23} -4327.66 q^{29} +8473.00 q^{31} +(-100.890 - 100.890i) q^{37} +14606.6i q^{41} +(-9268.83 + 9268.83i) q^{43} +(-1485.67 + 1485.67i) q^{47} +13913.6i q^{49} +(-9264.63 - 9264.63i) q^{53} -2053.40 q^{59} -10864.1 q^{61} +(9222.90 + 9222.90i) q^{67} +10577.6i q^{71} +(20495.6 - 20495.6i) q^{73} +(36722.4 - 36722.4i) q^{77} -58845.3i q^{79} +(61864.1 + 61864.1i) q^{83} +43592.7 q^{89} -4023.88 q^{91} +(-75406.9 - 75406.9i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 13848 q^{31} - 28200 q^{61} + 908328 q^{91}+O(q^{100}) $ 24 * q + 13848 * q^31 - 28200 * q^61 + 908328 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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
$7$	−123.937	−	123.937i	−0.955994	−	0.955994i	0.0430780	−	0.999072i	$-0.486284\pi$
$7$	−123.937	−	123.937i	−0.955994	−	0.955994i	−0.999072	+	0.0430780i	$0.986284\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			296.299i			0.738327i	0.929364	+	0.369164i	$0.120356\pi$
$11$			296.299i			0.738327i	−0.929364	+	0.369164i	$0.879644\pi$
$12$		0			0
$13$	16.2336	−	16.2336i	0.0266414	−	0.0266414i	−0.693661	−	0.720302i	$-0.744003\pi$
$13$	16.2336	−	16.2336i	0.0266414	−	0.0266414i	0.720302	+	0.693661i	$0.244003\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−547.265	+	547.265i	−0.459278	+	0.459278i	−0.898418	−	0.439140i	$-0.855283\pi$
$17$	−547.265	+	547.265i	−0.459278	+	0.459278i	0.439140	+	0.898418i	$0.355283\pi$
$18$		0			0
$19$			262.818i			0.167021i	0.996507	+	0.0835104i	$0.0266132\pi$
$19$			262.818i			0.167021i	−0.996507	+	0.0835104i	$0.973387\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2434.99	−	2434.99i	−0.959793	−	0.959793i	0.0394291	−	0.999222i	$-0.487446\pi$
$23$	−2434.99	−	2434.99i	−0.959793	−	0.959793i	−0.999222	+	0.0394291i	$0.987446\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4327.66			−0.955561			−0.477781	−	0.878479i	$-0.658559\pi$
$29$	−4327.66			−0.955561			−0.477781	+	0.878479i	$0.658559\pi$
$30$		0			0
$31$	8473.00			1.58355			0.791777	−	0.610810i	$-0.209156\pi$
$31$	8473.00			1.58355			0.791777	+	0.610810i	$0.209156\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−100.890	−	100.890i	−0.0121155	−	0.0121155i	0.701023	−	0.713139i	$-0.252727\pi$
$37$	−100.890	−	100.890i	−0.0121155	−	0.0121155i	−0.713139	+	0.701023i	$0.752727\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			14606.6i			1.35703i	0.734589	+	0.678513i	$0.237375\pi$
$41$			14606.6i			1.35703i	−0.734589	+	0.678513i	$0.762625\pi$
$42$		0			0
$43$	−9268.83	+	9268.83i	−0.764459	+	0.764459i	−0.977125	−	0.212666i	$-0.931785\pi$
$43$	−9268.83	+	9268.83i	−0.764459	+	0.764459i	0.212666	+	0.977125i	$0.431785\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1485.67	+	1485.67i	−0.0981017	+	0.0981017i	−0.754454	−	0.656353i	$-0.772098\pi$
$47$	−1485.67	+	1485.67i	−0.0981017	+	0.0981017i	0.656353	+	0.754454i	$0.272098\pi$
$48$		0			0
$49$			13913.6i			0.827848i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9264.63	−	9264.63i	−0.453042	−	0.453042i	0.443321	−	0.896363i	$-0.353800\pi$
$53$	−9264.63	−	9264.63i	−0.453042	−	0.453042i	−0.896363	+	0.443321i	$0.853800\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2053.40			−0.0767970			−0.0383985	−	0.999263i	$-0.512226\pi$
$59$	−2053.40			−0.0767970			−0.0383985	+	0.999263i	$0.512226\pi$
$60$		0			0
$61$	−10864.1			−0.373826			−0.186913	−	0.982376i	$-0.559848\pi$
$61$	−10864.1			−0.373826			−0.186913	+	0.982376i	$0.559848\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	9222.90	+	9222.90i	0.251004	+	0.251004i	0.821382	−	0.570378i	$-0.193203\pi$
$67$	9222.90	+	9222.90i	0.251004	+	0.251004i	−0.570378	+	0.821382i	$0.693203\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			10577.6i			0.249024i	0.992218	+	0.124512i	$0.0397365\pi$
$71$			10577.6i			0.249024i	−0.992218	+	0.124512i	$0.960263\pi$
$72$		0			0
$73$	20495.6	−	20495.6i	0.450146	−	0.450146i	−0.445257	−	0.895403i	$-0.646888\pi$
$73$	20495.6	−	20495.6i	0.450146	−	0.450146i	0.895403	+	0.445257i	$0.146888\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	36722.4	−	36722.4i	0.705836	−	0.705836i
$78$		0			0
$79$		−	58845.3i		−	1.06083i	−0.847740	−	0.530413i	$-0.822037\pi$
$79$		−	58845.3i		−	1.06083i	0.847740	−	0.530413i	$-0.177963\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	61864.1	+	61864.1i	0.985698	+	0.985698i	0.999899	−	0.0142015i	$-0.00452061\pi$
$83$	61864.1	+	61864.1i	0.985698	+	0.985698i	−0.0142015	+	0.999899i	$0.504521\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	43592.7			0.583363			0.291681	−	0.956516i	$-0.405785\pi$
$89$	43592.7			0.583363			0.291681	+	0.956516i	$0.405785\pi$
$90$		0			0
$91$	−4023.88			−0.0509380
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−75406.9	−	75406.9i	−0.813733	−	0.813733i	0.171459	−	0.985191i	$-0.445152\pi$
$97$	−75406.9	−	75406.9i	−0.813733	−	0.813733i	−0.985191	+	0.171459i	$0.945152\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	136828.i		−	1.33466i	−0.744763	−	0.667329i	$-0.767437\pi$
$101$		−	136828.i		−	1.33466i	0.744763	−	0.667329i	$-0.232563\pi$
$102$		0			0
$103$	68031.2	−	68031.2i	0.631851	−	0.631851i	−0.316681	−	0.948532i	$-0.602568\pi$
$103$	68031.2	−	68031.2i	0.631851	−	0.631851i	0.948532	+	0.316681i	$0.102568\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−21604.1	+	21604.1i	−0.182422	+	0.182422i	−0.792410	−	0.609988i	$-0.791174\pi$
$107$	−21604.1	+	21604.1i	−0.182422	+	0.182422i	0.609988	+	0.792410i	$0.291174\pi$
$108$		0			0
$109$			56892.1i			0.458655i	0.973349	+	0.229327i	$0.0736526\pi$
$109$			56892.1i			0.458655i	−0.973349	+	0.229327i	$0.926347\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	61980.0	+	61980.0i	0.456621	+	0.456621i	0.897544	−	0.440924i	$-0.145349\pi$
$113$	61980.0	+	61980.0i	0.456621	+	0.456621i	−0.440924	+	0.897544i	$0.645349\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	135653.			0.878134
$120$		0			0
$121$	73257.7			0.454873
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−59893.7	−	59893.7i	−0.329512	−	0.329512i	0.522889	−	0.852401i	$-0.324854\pi$
$127$	−59893.7	−	59893.7i	−0.329512	−	0.329512i	−0.852401	+	0.522889i	$0.824854\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			317743.i			1.61770i	0.588017	+	0.808849i	$0.299909\pi$
$131$			317743.i			1.61770i	−0.588017	+	0.808849i	$0.700091\pi$
$132$		0			0
$133$	32572.8	−	32572.8i	0.159671	−	0.159671i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	283098.	−	283098.i	1.28865	−	1.28865i	0.353045	−	0.935606i	$-0.385146\pi$
$137$	283098.	−	283098.i	1.28865	−	1.28865i	0.935606	−	0.353045i	$-0.114854\pi$
$138$		0			0
$139$		−	215426.i		−	0.945716i	−0.881139	−	0.472858i	$-0.843222\pi$
$139$		−	215426.i		−	0.945716i	0.881139	−	0.472858i	$-0.156778\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4810.01	+	4810.01i	0.0196701	+	0.0196701i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−369959.			−1.36517			−0.682587	−	0.730804i	$-0.739145\pi$
$149$	−369959.			−1.36517			−0.682587	+	0.730804i	$0.739145\pi$
$150$		0			0
$151$	280532.			1.00124			0.500621	−	0.865666i	$-0.333105\pi$
$151$	280532.			1.00124			0.500621	+	0.865666i	$0.333105\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−30811.3	−	30811.3i	−0.0997610	−	0.0997610i	0.655465	−	0.755226i	$-0.272473\pi$
$157$	−30811.3	−	30811.3i	−0.0997610	−	0.0997610i	−0.755226	+	0.655465i	$0.772473\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			603570.i			1.83511i
$162$		0			0
$163$	−96397.3	+	96397.3i	−0.284181	+	0.284181i	−0.834774	−	0.550593i	$-0.814402\pi$
$163$	−96397.3	+	96397.3i	−0.284181	+	0.284181i	0.550593	+	0.834774i	$0.314402\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	35684.1	−	35684.1i	0.0990109	−	0.0990109i	−0.655866	−	0.754877i	$-0.727696\pi$
$167$	35684.1	−	35684.1i	0.0990109	−	0.0990109i	0.754877	+	0.655866i	$0.227696\pi$
$168$		0			0
$169$			370766.i			0.998580i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−76138.1	−	76138.1i	−0.193414	−	0.193414i	0.603756	−	0.797169i	$-0.293670\pi$
$173$	−76138.1	−	76138.1i	−0.193414	−	0.193414i	−0.797169	+	0.603756i	$0.793670\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	411064.			0.958907			0.479454	−	0.877567i	$-0.340835\pi$
$179$	411064.			0.958907			0.479454	+	0.877567i	$0.340835\pi$
$180$		0			0
$181$	272889.			0.619141			0.309571	−	0.950876i	$-0.399815\pi$
$181$	272889.			0.619141			0.309571	+	0.950876i	$0.399815\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−162154.	−	162154.i	−0.339098	−	0.339098i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			868065.i			1.72175i	0.508820	+	0.860873i	$0.330082\pi$
$191$			868065.i			1.72175i	−0.508820	+	0.860873i	$0.669918\pi$
$192$		0			0
$193$	23668.8	−	23668.8i	0.0457387	−	0.0457387i	−0.683868	−	0.729606i	$-0.739703\pi$
$193$	23668.8	−	23668.8i	0.0457387	−	0.0457387i	0.729606	+	0.683868i	$0.239703\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	629819.	−	629819.i	1.15625	−	1.15625i	0.170969	−	0.985276i	$-0.445310\pi$
$197$	629819.	−	629819.i	1.15625	−	1.15625i	0.985276	−	0.170969i	$-0.0546898\pi$
$198$		0			0
$199$			120544.i			0.215780i	0.994163	+	0.107890i	$0.0344094\pi$
$199$			120544.i			0.215780i	−0.994163	+	0.107890i	$0.965591\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	536357.	+	536357.i	0.913511	+	0.913511i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−77872.7			−0.123316
$210$		0			0
$211$	1.16641e6			1.80362			0.901809	−	0.432135i	$-0.142240\pi$
$211$	1.16641e6			1.80362			0.901809	+	0.432135i	$0.142240\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.05012e6	−	1.05012e6i	−1.51387	−	1.51387i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			17768.2i			0.0244716i
$222$		0			0
$223$	794666.	−	794666.i	1.07010	−	1.07010i	0.0727445	−	0.997351i	$-0.476824\pi$
$223$	794666.	−	794666.i	1.07010	−	1.07010i	0.997351	−	0.0727445i	$-0.0231758\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	614503.	−	614503.i	0.791515	−	0.791515i	−0.190225	−	0.981740i	$-0.560922\pi$
$227$	614503.	−	614503.i	0.791515	−	0.791515i	0.981740	+	0.190225i	$0.0609218\pi$
$228$		0			0
$229$		−	1.41946e6i		−	1.78869i	−0.447377	−	0.894346i	$-0.647642\pi$
$229$		−	1.41946e6i		−	1.78869i	0.447377	−	0.894346i	$-0.352358\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	580216.	+	580216.i	0.700165	+	0.700165i	0.964446	−	0.264281i	$-0.0851346\pi$
$233$	580216.	+	580216.i	0.700165	+	0.700165i	−0.264281	+	0.964446i	$0.585135\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	438358.			0.496402			0.248201	−	0.968709i	$-0.420161\pi$
$239$	438358.			0.496402			0.248201	+	0.968709i	$0.420161\pi$
$240$		0			0
$241$	−61779.3			−0.0685173			−0.0342586	−	0.999413i	$-0.510907\pi$
$241$	−61779.3			−0.0685173			−0.0342586	+	0.999413i	$0.510907\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	4266.48	+	4266.48i	0.00444967	+	0.00444967i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			152110.i			0.152396i	0.997093	+	0.0761978i	$0.0242780\pi$
$251$			152110.i			0.152396i	−0.997093	+	0.0761978i	$0.975722\pi$
$252$		0			0
$253$	721486.	−	721486.i	0.708642	−	0.708642i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	794182.	−	794182.i	0.750045	−	0.750045i	−0.224442	−	0.974487i	$-0.572056\pi$
$257$	794182.	−	794182.i	0.750045	−	0.750045i	0.974487	+	0.224442i	$0.0720560\pi$
$258$		0			0
$259$			25007.9i			0.0231648i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	846425.	+	846425.i	0.754570	+	0.754570i	0.975328	−	0.220759i	$-0.0708534\pi$
$263$	846425.	+	846425.i	0.754570	+	0.754570i	−0.220759	+	0.975328i	$0.570853\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.38502e6			1.16701			0.583507	−	0.812108i	$-0.301680\pi$
$269$	1.38502e6			1.16701			0.583507	+	0.812108i	$0.301680\pi$
$270$		0			0
$271$	−450412.			−0.372552			−0.186276	−	0.982497i	$-0.559642\pi$
$271$	−450412.			−0.372552			−0.186276	+	0.982497i	$0.559642\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.25916e6	−	1.25916e6i	−0.986012	−	0.986012i	0.0138917	−	0.999904i	$-0.495578\pi$
$277$	−1.25916e6	−	1.25916e6i	−0.986012	−	0.986012i	−0.999904	+	0.0138917i	$0.995578\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1.96636e6i			1.48558i	0.669522	+	0.742792i	$0.266499\pi$
$281$			1.96636e6i			1.48558i	−0.669522	+	0.742792i	$0.733501\pi$
$282$		0			0
$283$	963548.	−	963548.i	0.715167	−	0.715167i	−0.252445	−	0.967611i	$-0.581235\pi$
$283$	963548.	−	963548.i	0.715167	−	0.715167i	0.967611	+	0.252445i	$0.0812346\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.81029e6	−	1.81029e6i	1.29731	−	1.29731i
$288$		0			0
$289$			820858.i			0.578127i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.78311e6	+	1.78311e6i	1.21342	+	1.21342i	0.969895	+	0.243522i	$0.0783026\pi$
$293$	1.78311e6	+	1.78311e6i	1.21342	+	1.21342i	0.243522	+	0.969895i	$0.421697\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−79057.4			−0.0511405
$300$		0			0
$301$	2.29750e6			1.46164
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.26197e6	−	1.26197e6i	−0.764193	−	0.764193i	0.212885	−	0.977077i	$-0.431714\pi$
$307$	−1.26197e6	−	1.26197e6i	−0.764193	−	0.764193i	−0.977077	+	0.212885i	$0.931714\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			2.26700e6i			1.32908i	0.747253	+	0.664540i	$0.231372\pi$
$311$			2.26700e6i			1.32908i	−0.747253	+	0.664540i	$0.768628\pi$
$312$		0			0
$313$	−1.03818e6	+	1.03818e6i	−0.598979	+	0.598979i	−0.940041	−	0.341062i	$-0.889213\pi$
$313$	−1.03818e6	+	1.03818e6i	−0.598979	+	0.598979i	0.341062	+	0.940041i	$0.389213\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	895082.	−	895082.i	0.500282	−	0.500282i	−0.411244	−	0.911526i	$-0.634905\pi$
$317$	895082.	−	895082.i	0.500282	−	0.500282i	0.911526	+	0.411244i	$0.134905\pi$
$318$		0			0
$319$		−	1.28228e6i		−	0.705517i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−143831.	−	143831.i	−0.0767090	−	0.0767090i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	368257.			0.187569
$330$		0			0
$331$	1.59270e6			0.799031			0.399515	−	0.916727i	$-0.369178\pi$
$331$	1.59270e6			0.799031			0.399515	+	0.916727i	$0.369178\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.14157e6	+	2.14157e6i	1.02721	+	1.02721i	0.999619	+	0.0275860i	$0.00878203\pi$
$337$	2.14157e6	+	2.14157e6i	1.02721	+	1.02721i	0.0275860	+	0.999619i	$0.491218\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			2.51054e6i			1.16918i
$342$		0			0
$343$	−358594.	+	358594.i	−0.164576	+	0.164576i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.35898e6	−	1.35898e6i	0.605883	−	0.605883i	−0.335984	−	0.941868i	$-0.609069\pi$
$347$	1.35898e6	−	1.35898e6i	0.605883	−	0.605883i	0.941868	+	0.335984i	$0.109069\pi$
$348$		0			0
$349$		−	331480.i		−	0.145678i	−0.997344	−	0.0728390i	$-0.976794\pi$
$349$		−	331480.i		−	0.145678i	0.997344	−	0.0728390i	$-0.0232059\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.53861e6	−	1.53861e6i	−0.657192	−	0.657192i	0.297523	−	0.954715i	$-0.403840\pi$
$353$	−1.53861e6	−	1.53861e6i	−0.657192	−	0.657192i	−0.954715	+	0.297523i	$0.903840\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.46509e6			−1.41899			−0.709493	−	0.704712i	$-0.751076\pi$
$359$	−3.46509e6			−1.41899			−0.709493	+	0.704712i	$0.751076\pi$
$360$		0			0
$361$	2.40703e6			0.972104
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−2.95418e6	−	2.95418e6i	−1.14491	−	1.14491i	−0.987539	−	0.157373i	$-0.949697\pi$
$367$	−2.95418e6	−	2.95418e6i	−1.14491	−	1.14491i	−0.157373	−	0.987539i	$-0.550303\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			2.29646e6i			0.866211i
$372$		0			0
$373$	1.99621e6	−	1.99621e6i	0.742906	−	0.742906i	−0.230231	−	0.973136i	$-0.573948\pi$
$373$	1.99621e6	−	1.99621e6i	0.742906	−	0.742906i	0.973136	+	0.230231i	$0.0739480\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−70253.6	+	70253.6i	−0.0254575	+	0.0254575i
$378$		0			0
$379$			3.85257e6i			1.37769i	0.724907	+	0.688847i	$0.241883\pi$
$379$			3.85257e6i			1.37769i	−0.724907	+	0.688847i	$0.758117\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	114173.	+	114173.i	0.0397709	+	0.0397709i	0.726713	−	0.686942i	$-0.241047\pi$
$383$	114173.	+	114173.i	0.0397709	+	0.0397709i	−0.686942	+	0.726713i	$0.741047\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.51434e6			1.17752			0.588762	−	0.808307i	$-0.299616\pi$
$389$	3.51434e6			1.17752			0.588762	+	0.808307i	$0.299616\pi$
$390$		0			0
$391$	2.66517e6			0.881624
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3.18815e6	−	3.18815e6i	−1.01523	−	1.01523i	−0.999882	−	0.0153441i	$-0.995116\pi$
$397$	−3.18815e6	−	3.18815e6i	−1.01523	−	1.01523i	−0.0153441	−	0.999882i	$-0.504884\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			3.76155e6i			1.16817i	0.811692	+	0.584086i	$0.198547\pi$
$401$			3.76155e6i			1.16817i	−0.811692	+	0.584086i	$0.801453\pi$
$402$		0			0
$403$	137547.	−	137547.i	0.0421881	−	0.0421881i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	29893.6	−	29893.6i	0.00894524	−	0.00894524i
$408$		0			0
$409$		−	5.01805e6i		−	1.48329i	−0.670792	−	0.741645i	$-0.734046\pi$
$409$		−	5.01805e6i		−	1.48329i	0.670792	−	0.741645i	$-0.265954\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	254492.	+	254492.i	0.0734174	+	0.0734174i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.71304e6			−0.476685			−0.238342	−	0.971181i	$-0.576604\pi$
$419$	−1.71304e6			−0.476685			−0.238342	+	0.971181i	$0.576604\pi$
$420$		0			0
$421$	−1.12412e6			−0.309105			−0.154552	−	0.987985i	$-0.549394\pi$
$421$	−1.12412e6			−0.309105			−0.154552	+	0.987985i	$0.549394\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.34646e6	+	1.34646e6i	0.357375	+	0.357375i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	3.16531e6i		−	0.820773i	−0.911912	−	0.410386i	$-0.865394\pi$
$431$		−	3.16531e6i		−	0.820773i	0.911912	−	0.410386i	$-0.134606\pi$
$432$		0			0
$433$	−520810.	+	520810.i	−0.133493	+	0.133493i	−0.770696	−	0.637203i	$-0.780091\pi$
$433$	−520810.	+	520810.i	−0.133493	+	0.133493i	0.637203	+	0.770696i	$0.280091\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	639959.	−	639959.i	0.160305	−	0.160305i
$438$		0			0
$439$			2.63513e6i			0.652589i	0.945268	+	0.326295i	$0.105800\pi$
$439$			2.63513e6i			0.652589i	−0.945268	+	0.326295i	$0.894200\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.89945e6	+	4.89945e6i	1.18615	+	1.18615i	0.978124	+	0.208023i	$0.0667028\pi$
$443$	4.89945e6	+	4.89945e6i	1.18615	+	1.18615i	0.208023	+	0.978124i	$0.433297\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.20484e6			0.516133			0.258066	−	0.966127i	$-0.416915\pi$
$449$	2.20484e6			0.516133			0.258066	+	0.966127i	$0.416915\pi$
$450$		0			0
$451$	−4.32791e6			−1.00193
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.27070e6	+	3.27070e6i	0.732571	+	0.732571i	0.971128	−	0.238558i	$-0.0766746\pi$
$457$	3.27070e6	+	3.27070e6i	0.732571	+	0.732571i	−0.238558	+	0.971128i	$0.576675\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	7.76391e6i		−	1.70148i	−0.525583	−	0.850742i	$-0.676153\pi$
$461$		−	7.76391e6i		−	1.70148i	0.525583	−	0.850742i	$-0.323847\pi$
$462$		0			0
$463$	171191.	−	171191.i	0.0371133	−	0.0371133i	−0.688307	−	0.725420i	$-0.741646\pi$
$463$	171191.	−	171191.i	0.0371133	−	0.0371133i	0.725420	+	0.688307i	$0.241646\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	921034.	−	921034.i	0.195426	−	0.195426i	−0.602610	−	0.798036i	$-0.705872\pi$
$467$	921034.	−	921034.i	0.195426	−	0.195426i	0.798036	+	0.602610i	$0.205872\pi$
$468$		0			0
$469$		−	2.28611e6i		−	0.479917i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.74635e6	−	2.74635e6i	−0.564421	−	0.564421i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	157207.			0.0313064			0.0156532	−	0.999877i	$-0.495017\pi$
$479$	157207.			0.0313064			0.0156532	+	0.999877i	$0.495017\pi$
$480$		0			0
$481$	−3275.61			−0.000645550
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	690885.	+	690885.i	0.132003	+	0.132003i	0.770021	−	0.638018i	$-0.220246\pi$
$487$	690885.	+	690885.i	0.132003	+	0.132003i	−0.638018	+	0.770021i	$0.720246\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	3.17012e6i		−	0.593434i	−0.954965	−	0.296717i	$-0.904108\pi$
$491$		−	3.17012e6i		−	0.593434i	0.954965	−	0.296717i	$-0.0958918\pi$
$492$		0			0
$493$	2.36838e6	−	2.36838e6i	0.438868	−	0.438868i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.31095e6	−	1.31095e6i	0.238065	−	0.238065i
$498$		0			0
$499$		−	3.10049e6i		−	0.557415i	−0.960376	−	0.278708i	$-0.910094\pi$
$499$		−	3.10049e6i		−	0.557415i	0.960376	−	0.278708i	$-0.0899061\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.20877e6	+	3.20877e6i	0.565481	+	0.565481i	0.930859	−	0.365378i	$-0.119060\pi$
$503$	3.20877e6	+	3.20877e6i	0.565481	+	0.565481i	−0.365378	+	0.930859i	$0.619060\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.43063e6			1.10017			0.550084	−	0.835109i	$-0.314596\pi$
$509$	6.43063e6			1.10017			0.550084	+	0.835109i	$0.314596\pi$
$510$		0			0
$511$	−5.08032e6			−0.860674
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−440202.	−	440202.i	−0.0724312	−	0.0724312i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			3.25599e6i			0.525519i	0.964861	+	0.262759i	$0.0846325\pi$
$521$			3.25599e6i			0.525519i	−0.964861	+	0.262759i	$0.915367\pi$
$522$		0			0
$523$	−259544.	+	259544.i	−0.0414912	+	0.0414912i	−0.727548	−	0.686057i	$-0.759340\pi$
$523$	−259544.	+	259544.i	−0.0414912	+	0.0414912i	0.686057	+	0.727548i	$0.259340\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.63698e6	+	4.63698e6i	−0.727292	+	0.727292i
$528$		0			0
$529$			5.42202e6i			0.842406i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	237117.	+	237117.i	0.0361531	+	0.0361531i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.12260e6			−0.611223
$540$		0			0
$541$	−8.89249e6			−1.30626			−0.653131	−	0.757245i	$-0.726545\pi$
$541$	−8.89249e6			−1.30626			−0.653131	+	0.757245i	$0.726545\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.90236e6	+	1.90236e6i	0.271846	+	0.271846i	0.829843	−	0.557997i	$-0.188430\pi$
$547$	1.90236e6	+	1.90236e6i	0.271846	+	0.271846i	−0.557997	+	0.829843i	$0.688430\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.13739e6i		−	0.159599i
$552$		0			0
$553$	−7.29309e6	+	7.29309e6i	−1.01414	+	1.01414i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.92347e6	+	6.92347e6i	−0.945553	+	0.945553i	−0.998592	−	0.0530399i	$-0.983109\pi$
$557$	−6.92347e6	+	6.92347e6i	−0.945553	+	0.945553i	0.0530399	+	0.998592i	$0.483109\pi$
$558$		0			0
$559$			300933.i			0.0407325i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9.52384e6	−	9.52384e6i	−1.26631	−	1.26631i	−0.947979	−	0.318334i	$-0.896877\pi$
$563$	−9.52384e6	−	9.52384e6i	−1.26631	−	1.26631i	−0.318334	−	0.947979i	$-0.603123\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.04760e7			−1.35648			−0.678241	−	0.734839i	$-0.737257\pi$
$569$	−1.04760e7			−1.35648			−0.678241	+	0.734839i	$0.737257\pi$
$570$		0			0
$571$	2.27504e6			0.292011			0.146006	−	0.989284i	$-0.453358\pi$
$571$	2.27504e6			0.292011			0.146006	+	0.989284i	$0.453358\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	541993.	+	541993.i	0.0677726	+	0.0677726i	0.740181	−	0.672408i	$-0.234740\pi$
$577$	541993.	+	541993.i	0.0677726	+	0.0677726i	−0.672408	+	0.740181i	$0.734740\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	1.53345e7i		−	1.88464i
$582$		0			0
$583$	2.74510e6	−	2.74510e6i	0.334493	−	0.334493i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.04755e7	+	1.04755e7i	−1.25482	+	1.25482i	−0.301286	+	0.953534i	$0.597416\pi$
$587$	−1.04755e7	+	1.04755e7i	−1.25482	+	1.25482i	−0.953534	+	0.301286i	$0.902584\pi$
$588$		0			0
$589$			2.22686e6i			0.264487i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9.39119e6	−	9.39119e6i	−1.09669	−	1.09669i	−0.994795	−	0.101894i	$-0.967510\pi$
$593$	−9.39119e6	−	9.39119e6i	−1.09669	−	1.09669i	−0.101894	−	0.994795i	$-0.532490\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.38639e6			−0.385629			−0.192814	−	0.981235i	$-0.561762\pi$
$599$	−3.38639e6			−0.385629			−0.192814	+	0.981235i	$0.561762\pi$
$600$		0			0
$601$	7.38299e6			0.833770			0.416885	−	0.908959i	$-0.363122\pi$
$601$	7.38299e6			0.833770			0.416885	+	0.908959i	$0.363122\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1.09377e7	+	1.09377e7i	1.20491	+	1.20491i	0.972654	+	0.232259i	$0.0746116\pi$
$607$	1.09377e7	+	1.09377e7i	1.20491	+	1.20491i	0.232259	+	0.972654i	$0.425388\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			48235.5i			0.00522713i
$612$		0			0
$613$	−9.57139e6	+	9.57139e6i	−1.02878	+	1.02878i	−0.0292102	+	0.999573i	$0.509299\pi$
$613$	−9.57139e6	+	9.57139e6i	−1.02878	+	1.02878i	−0.999573	+	0.0292102i	$0.990701\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.65393e6	−	2.65393e6i	0.280657	−	0.280657i	−0.552714	−	0.833371i	$-0.686408\pi$
$617$	2.65393e6	−	2.65393e6i	0.280657	−	0.280657i	0.833371	+	0.552714i	$0.186408\pi$
$618$		0			0
$619$			2.23480e6i			0.234429i	0.993107	+	0.117214i	$0.0373965\pi$
$619$			2.23480e6i			0.234429i	−0.993107	+	0.117214i	$0.962604\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.40274e6	−	5.40274e6i	−0.557691	−	0.557691i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	110427.			0.0111288
$630$		0			0
$631$	9.17502e6			0.917347			0.458674	−	0.888605i	$-0.348325\pi$
$631$	9.17502e6			0.917347			0.458674	+	0.888605i	$0.348325\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	225869.	+	225869.i	0.0220550	+	0.0220550i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			949377.i			0.0912628i	0.998958	+	0.0456314i	$0.0145300\pi$
$641$			949377.i			0.0912628i	−0.998958	+	0.0456314i	$0.985470\pi$
$642$		0			0
$643$	2.12833e6	−	2.12833e6i	0.203007	−	0.203007i	−0.598280	−	0.801287i	$-0.704149\pi$
$643$	2.12833e6	−	2.12833e6i	0.203007	−	0.203007i	0.801287	+	0.598280i	$0.204149\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.02885e7	+	1.02885e7i	−0.966253	+	0.966253i	−0.999449	−	0.0331962i	$-0.989431\pi$
$647$	−1.02885e7	+	1.02885e7i	−0.966253	+	0.966253i	0.0331962	+	0.999449i	$0.489431\pi$
$648$		0			0
$649$		−	608422.i		−	0.0567013i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4.59428e6	−	4.59428e6i	−0.421633	−	0.421633i	0.464133	−	0.885766i	$-0.346366\pi$
$653$	−4.59428e6	−	4.59428e6i	−0.421633	−	0.421633i	−0.885766	+	0.464133i	$0.846366\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5.22204e6			−0.468411			−0.234205	−	0.972187i	$-0.575249\pi$
$659$	−5.22204e6			−0.468411			−0.234205	+	0.972187i	$0.575249\pi$
$660$		0			0
$661$	−2.16231e7			−1.92493			−0.962465	−	0.271405i	$-0.912512\pi$
$661$	−2.16231e7			−1.92493			−0.962465	+	0.271405i	$0.912512\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.05378e7	+	1.05378e7i	0.917141	+	0.917141i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	3.21903e6i		−	0.276006i
$672$		0			0
$673$	−1.51265e7	+	1.51265e7i	−1.28736	+	1.28736i	−0.350974	+	0.936385i	$0.614150\pi$
$673$	−1.51265e7	+	1.51265e7i	−1.28736	+	1.28736i	−0.936385	+	0.350974i	$0.885850\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.02380e6	+	2.02380e6i	−0.169705	+	0.169705i	−0.786850	−	0.617144i	$-0.788289\pi$
$677$	−2.02380e6	+	2.02380e6i	−0.169705	+	0.169705i	0.617144	+	0.786850i	$0.288289\pi$
$678$		0			0
$679$			1.86914e7i			1.55585i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8.21866e6	+	8.21866e6i	0.674138	+	0.674138i	0.958667	−	0.284529i	$-0.0918372\pi$
$683$	8.21866e6	+	8.21866e6i	0.674138	+	0.674138i	−0.284529	+	0.958667i	$0.591837\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−300797.			−0.0241393
$690$		0			0
$691$	−1.62147e7			−1.29185			−0.645926	−	0.763400i	$-0.723529\pi$
$691$	−1.62147e7			−1.29185			−0.645926	+	0.763400i	$0.723529\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−7.99366e6	−	7.99366e6i	−0.623252	−	0.623252i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	2.49796e7i		−	1.91995i	−0.280081	−	0.959976i	$-0.590361\pi$
$701$		−	2.49796e7i		−	1.91995i	0.280081	−	0.959976i	$-0.409639\pi$
$702$		0			0
$703$	26515.6	−	26515.6i	0.00202355	−	0.00202355i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.69580e7	+	1.69580e7i	−1.27592	+	1.27592i
$708$		0			0
$709$		−	1.44518e7i		−	1.07971i	−0.841760	−	0.539853i	$-0.818480\pi$
$709$		−	1.44518e7i		−	1.07971i	0.841760	−	0.539853i	$-0.181520\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.06317e7	−	2.06317e7i	−1.51989	−	1.51989i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.32849e6			0.312258			0.156129	−	0.987737i	$-0.450098\pi$
$719$	4.32849e6			0.312258			0.156129	+	0.987737i	$0.450098\pi$
$720$		0			0
$721$	−1.68631e7			−1.20809
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.66934e7	+	1.66934e7i	1.17141	+	1.17141i	0.981873	+	0.189538i	$0.0606989\pi$
$727$	1.66934e7	+	1.66934e7i	1.17141	+	1.17141i	0.189538	+	0.981873i	$0.439301\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	1.01450e7i		−	0.702198i
$732$		0			0
$733$	7.29281e6	−	7.29281e6i	0.501343	−	0.501343i	−0.410512	−	0.911855i	$-0.634650\pi$
$733$	7.29281e6	−	7.29281e6i	0.501343	−	0.501343i	0.911855	+	0.410512i	$0.134650\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.73274e6	+	2.73274e6i	−0.185323	+	0.185323i
$738$		0			0
$739$		−	7.99531e6i		−	0.538548i	−0.963064	−	0.269274i	$-0.913216\pi$
$739$		−	7.99531e6i		−	0.538548i	0.963064	−	0.269274i	$-0.0867837\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.74977e7	+	1.74977e7i	1.16281	+	1.16281i	0.983858	+	0.178953i	$0.0572712\pi$
$743$	1.74977e7	+	1.74977e7i	1.16281	+	1.16281i	0.178953	+	0.983858i	$0.442729\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.35509e6			0.348788
$750$		0			0
$751$	−1.18625e7			−0.767494			−0.383747	−	0.923438i	$-0.625366\pi$
$751$	−1.18625e7			−0.767494			−0.383747	+	0.923438i	$0.625366\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.18681e7	+	1.18681e7i	0.752733	+	0.752733i	0.974989	−	0.222255i	$-0.0713418\pi$
$757$	1.18681e7	+	1.18681e7i	0.752733	+	0.752733i	−0.222255	+	0.974989i	$0.571342\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1.36406e7i		−	0.853832i	−0.904291	−	0.426916i	$-0.859600\pi$
$761$		−	1.36406e7i		−	0.853832i	0.904291	−	0.426916i	$-0.140400\pi$
$762$		0			0
$763$	7.05102e6	−	7.05102e6i	0.438471	−	0.438471i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−33334.2	+	33334.2i	−0.00204598	+	0.00204598i
$768$		0			0
$769$			8.00583e6i			0.488192i	0.969751	+	0.244096i	$0.0784912\pi$
$769$			8.00583e6i			0.488192i	−0.969751	+	0.244096i	$0.921509\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2.94684e6	−	2.94684e6i	−0.177381	−	0.177381i	0.612832	−	0.790213i	$-0.290030\pi$
$773$	−2.94684e6	−	2.94684e6i	−0.177381	−	0.177381i	−0.790213	+	0.612832i	$0.790030\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.83886e6			−0.226652
$780$		0			0
$781$	−3.13414e6			−0.183861
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−5.07512e6	−	5.07512e6i	−0.292085	−	0.292085i	0.545818	−	0.837904i	$-0.316219\pi$
$787$	−5.07512e6	−	5.07512e6i	−0.292085	−	0.292085i	−0.837904	+	0.545818i	$0.816219\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1.53632e7i		−	0.873053i
$792$		0			0
$793$	−176364.	+	176364.i	−0.00995924	+	0.00995924i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.32317e7	−	1.32317e7i	0.737855	−	0.737855i	−0.234308	−	0.972162i	$-0.575282\pi$
$797$	1.32317e7	−	1.32317e7i	0.737855	−	0.737855i	0.972162	+	0.234308i	$0.0752824\pi$
$798$		0			0
$799$		−	1.62611e6i		−	0.0901120i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6.07284e6	+	6.07284e6i	0.332355	+	0.332355i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.80318e6			0.204303			0.102152	−	0.994769i	$-0.467427\pi$
$809$	3.80318e6			0.204303			0.102152	+	0.994769i	$0.467427\pi$
$810$		0			0
$811$	1.12914e7			0.602829			0.301414	−	0.953493i	$-0.402541\pi$
$811$	1.12914e7			0.602829			0.301414	+	0.953493i	$0.402541\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.43601e6	−	2.43601e6i	−0.127681	−	0.127681i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	2.15427e7i		−	1.11543i	−0.830032	−	0.557716i	$-0.811678\pi$
$821$		−	2.15427e7i		−	1.11543i	0.830032	−	0.557716i	$-0.188322\pi$
$822$		0			0
$823$	−8.73412e6	+	8.73412e6i	−0.449489	+	0.449489i	−0.895185	−	0.445695i	$-0.852956\pi$
$823$	−8.73412e6	+	8.73412e6i	−0.449489	+	0.449489i	0.445695	+	0.895185i	$0.352956\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.59254e7	+	1.59254e7i	−0.809705	+	0.809705i	−0.984589	−	0.174884i	$-0.944045\pi$
$827$	−1.59254e7	+	1.59254e7i	−0.809705	+	0.809705i	0.174884	+	0.984589i	$0.444045\pi$
$828$		0			0
$829$		−	2.78306e7i		−	1.40649i	−0.710948	−	0.703244i	$-0.751734\pi$
$829$		−	2.78306e7i		−	1.40649i	0.710948	−	0.703244i	$-0.248266\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−7.61445e6	−	7.61445e6i	−0.380212	−	0.380212i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−8.50255e6			−0.417008			−0.208504	−	0.978022i	$-0.566859\pi$
$839$	−8.50255e6			−0.417008			−0.208504	+	0.978022i	$0.566859\pi$
$840$		0			0
$841$	−1.78247e6			−0.0869026
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−9.07933e6	−	9.07933e6i	−0.434856	−	0.434856i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			491332.i			0.0232568i
$852$		0			0
$853$	−4.51666e6	+	4.51666e6i	−0.212542	+	0.212542i	−0.805346	−	0.592805i	$-0.798021\pi$
$853$	−4.51666e6	+	4.51666e6i	−0.212542	+	0.212542i	0.592805	+	0.805346i	$0.298021\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.32536e7	+	2.32536e7i	−1.08153	+	1.08153i	−0.0851602	+	0.996367i	$0.527140\pi$
$857$	−2.32536e7	+	2.32536e7i	−1.08153	+	1.08153i	−0.996367	+	0.0851602i	$0.972860\pi$
$858$		0			0
$859$			3.86671e7i			1.78796i	0.448105	+	0.893981i	$0.352099\pi$
$859$			3.86671e7i			1.78796i	−0.448105	+	0.893981i	$0.647901\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.41644e7	−	1.41644e7i	−0.647400	−	0.647400i	0.304964	−	0.952364i	$-0.401356\pi$
$863$	−1.41644e7	−	1.41644e7i	−0.647400	−	0.647400i	−0.952364	+	0.304964i	$0.901356\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.74358e7			0.783236
$870$		0			0
$871$	299442.			0.0133742
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−6.20885e6	−	6.20885e6i	−0.272591	−	0.272591i	0.557551	−	0.830143i	$-0.311741\pi$
$877$	−6.20885e6	−	6.20885e6i	−0.272591	−	0.272591i	−0.830143	+	0.557551i	$0.811741\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1.69249e7i			0.734660i	0.930091	+	0.367330i	$0.119728\pi$
$881$			1.69249e7i			0.734660i	−0.930091	+	0.367330i	$0.880272\pi$
$882$		0			0
$883$	154530.	−	154530.i	0.00666976	−	0.00666976i	−0.703764	−	0.710434i	$-0.748499\pi$
$883$	154530.	−	154530.i	0.00666976	−	0.00666976i	0.710434	+	0.703764i	$0.248499\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.66600e7	+	1.66600e7i	−0.710992	+	0.710992i	−0.966743	−	0.255751i	$-0.917677\pi$
$887$	−1.66600e7	+	1.66600e7i	−0.710992	+	0.710992i	0.255751	+	0.966743i	$0.417677\pi$
$888$		0			0
$889$			1.48461e7i			0.630024i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−390460.	−	390460.i	−0.0163850	−	0.0163850i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.66683e7			−1.51318
$900$		0			0
$901$	1.01404e7			0.416145
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.88373e7	+	1.88373e7i	0.760328	+	0.760328i	0.976382	−	0.216053i	$-0.0693185\pi$
$907$	1.88373e7	+	1.88373e7i	0.760328	+	0.760328i	−0.216053	+	0.976382i	$0.569319\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			2.97112e7i			1.18611i	0.805163	+	0.593054i	$0.202078\pi$
$911$			2.97112e7i			1.18611i	−0.805163	+	0.593054i	$0.797922\pi$
$912$		0			0
$913$	−1.83303e7	+	1.83303e7i	−0.727767	+	0.727767i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.93800e7	−	3.93800e7i	1.54651	−	1.54651i
$918$		0			0
$919$		−	1.68807e7i		−	0.659329i	−0.944098	−	0.329665i	$-0.893064\pi$
$919$		−	1.68807e7i		−	0.659329i	0.944098	−	0.329665i	$-0.106936\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	171713.	+	171713.i	0.00663435	+	0.00663435i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.25540e7			1.23756			0.618779	−	0.785565i	$-0.287628\pi$
$929$	3.25540e7			1.23756			0.618779	+	0.785565i	$0.287628\pi$
$930$		0			0
$931$	−3.65675e6			−0.138268
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	6.56214e6	+	6.56214e6i	0.244172	+	0.244172i	0.818574	−	0.574401i	$-0.194765\pi$
$937$	6.56214e6	+	6.56214e6i	0.244172	+	0.244172i	−0.574401	+	0.818574i	$0.694765\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			4.46801e7i			1.64490i	0.568836	+	0.822451i	$0.307394\pi$
$941$			4.46801e7i			1.64490i	−0.568836	+	0.822451i	$0.692606\pi$
$942$		0			0
$943$	3.55668e7	−	3.55668e7i	1.30246	−	1.30246i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.74895e7	−	2.74895e7i	0.996075	−	0.996075i	−0.00391761	−	0.999992i	$-0.501247\pi$
$947$	2.74895e7	−	2.74895e7i	0.996075	−	0.996075i	0.999992	+	0.00391761i	$0.00124702\pi$
$948$		0			0
$949$		−	665436.i		−	0.0239851i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−3.00343e7	−	3.00343e7i	−1.07123	−	1.07123i	−0.997260	−	0.0739747i	$-0.976432\pi$
$953$	−3.00343e7	−	3.00343e7i	−1.07123	−	1.07123i	−0.0739747	−	0.997260i	$-0.523568\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7.01725e7			−2.46389
$960$		0			0
$961$	4.31626e7			1.50765
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.78773e7	−	1.78773e7i	−0.614803	−	0.614803i	0.329391	−	0.944194i	$-0.393157\pi$
$967$	−1.78773e7	−	1.78773e7i	−0.614803	−	0.614803i	−0.944194	+	0.329391i	$0.893157\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1.75062e7i			0.595861i	0.954588	+	0.297930i	$0.0962963\pi$
$971$			1.75062e7i			0.595861i	−0.954588	+	0.297930i	$0.903704\pi$
$972$		0			0
$973$	−2.66992e7	+	2.66992e7i	−0.904098	+	0.904098i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.05445e6	+	6.05445e6i	−0.202926	+	0.202926i	−0.801253	−	0.598326i	$-0.795833\pi$
$977$	−6.05445e6	+	6.05445e6i	−0.202926	+	0.202926i	0.598326	+	0.801253i	$0.295833\pi$
$978$		0			0
$979$			1.29165e7i			0.430713i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.92434e7	+	3.92434e7i	1.29534	+	1.29534i	0.931439	+	0.363898i	$0.118554\pi$
$983$	3.92434e7	+	3.92434e7i	1.29534	+	1.29534i	0.363898	+	0.931439i	$0.381446\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.51390e7			1.46744
$990$		0			0
$991$	9.21716e6			0.298135			0.149068	−	0.988827i	$-0.452373\pi$
$991$	9.21716e6			0.298135			0.149068	+	0.988827i	$0.452373\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.03016e6	+	3.03016e6i	0.0965446	+	0.0965446i	0.753729	−	0.657185i	$-0.228253\pi$
$997$	3.03016e6	+	3.03016e6i	0.0965446	+	0.0965446i	−0.657185	+	0.753729i	$0.728253\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.j.c.557.1	✓	24
3.2	odd	2	inner	900.6.j.c.557.2	yes	24
5.2	odd	4	inner	900.6.j.c.593.12	yes	24
5.3	odd	4	inner	900.6.j.c.593.2	yes	24
5.4	even	2	inner	900.6.j.c.557.11	yes	24
15.2	even	4	inner	900.6.j.c.593.11	yes	24
15.8	even	4	inner	900.6.j.c.593.1	yes	24
15.14	odd	2	inner	900.6.j.c.557.12	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.6.j.c.557.1	✓	24	1.1	even	1	trivial
900.6.j.c.557.2	yes	24	3.2	odd	2	inner
900.6.j.c.557.11	yes	24	5.4	even	2	inner
900.6.j.c.557.12	yes	24	15.14	odd	2	inner
900.6.j.c.593.1	yes	24	15.8	even	4	inner
900.6.j.c.593.2	yes	24	5.3	odd	4	inner
900.6.j.c.593.11	yes	24	15.2	even	4	inner
900.6.j.c.593.12	yes	24	5.2	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-123.937 - 123.937i) q^{7} +296.299i q^{11} +(16.2336 - 16.2336i) q^{13} +(-547.265 + 547.265i) q^{17} +262.818i q^{19} +(-2434.99 - 2434.99i) q^{23} -4327.66 q^{29} +8473.00 q^{31} +(-100.890 - 100.890i) q^{37} +14606.6i q^{41} +(-9268.83 + 9268.83i) q^{43} +(-1485.67 + 1485.67i) q^{47} +13913.6i q^{49} +(-9264.63 - 9264.63i) q^{53} -2053.40 q^{59} -10864.1 q^{61} +(9222.90 + 9222.90i) q^{67} +10577.6i q^{71} +(20495.6 - 20495.6i) q^{73} +(36722.4 - 36722.4i) q^{77} -58845.3i q^{79} +(61864.1 + 61864.1i) q^{83} +43592.7 q^{89} -4023.88 q^{91} +(-75406.9 - 75406.9i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 13848 q^{31} - 28200 q^{61} + 908328 q^{91}+O(q^{100}) \) 24 * q + 13848 * q^31 - 28200 * q^61 + 908328 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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