LMFDB - Embedded newform 1008.6.b.g.559.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,6,Mod(559,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1008.559");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1008.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$161.666890371$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 208x^{6} + 15517x^{4} + 287808x^{2} + 1830609 $ x^8 + 208x^6 + 15517x^4 + 287808*x^2 + 1830609
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{2}\cdot 7^{5} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.2
Root			$-0.785404 - 3.50041i$ of defining polynomial
Character	$\chi$	$=$	1008.559
Dual form			1008.6.b.g.559.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-100.497i q^{5} +(89.7065 - 93.5935i) q^{7} +O(q^{10})$	$q-100.497i q^{5} +(89.7065 - 93.5935i) q^{7} +458.923i q^{11} +693.538i q^{13} -813.920i q^{17} +373.543 q^{19} -326.004i q^{23} -6974.68 q^{25} +3155.47 q^{29} +4193.91 q^{31} +(-9405.88 - 9015.25i) q^{35} +8234.41 q^{37} -3516.33i q^{41} -9770.29i q^{43} +22547.1 q^{47} +(-712.471 - 16791.9i) q^{49} -32148.7 q^{53} +46120.4 q^{55} +24859.1 q^{59} -27316.4i q^{61} +69698.6 q^{65} -67423.9i q^{67} -48772.4i q^{71} -23068.9i q^{73} +(42952.1 + 41168.4i) q^{77} -62780.8i q^{79} -24828.4 q^{83} -81796.6 q^{85} -27793.4i q^{89} +(64910.6 + 62214.9i) q^{91} -37540.0i q^{95} +43926.1i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 17048 q^{25} + 9744 q^{29} + 19376 q^{37} + 9800 q^{49} - 102192 q^{53} + 193344 q^{65} + 25872 q^{77} - 437376 q^{85}+O(q^{100}) $ 8 * q - 17048 * q^25 + 9744 * q^29 + 19376 * q^37 + 9800 * q^49 - 102192 * q^53 + 193344 * q^65 + 25872 * q^77 - 437376 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	100.497i		−	1.79775i	−0.438208	−	0.898874i	$-0.644387\pi$
$5$		−	100.497i		−	1.79775i	0.438208	−	0.898874i	$-0.355613\pi$
$6$		0			0
$7$	89.7065	−	93.5935i	0.691957	−	0.721939i
$7$	89.7065	−	93.5935i	0.691957	−	0.721939i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			458.923i			1.14356i	0.820408	+	0.571778i	$0.193746\pi$
$11$			458.923i			1.14356i	−0.820408	+	0.571778i	$0.806254\pi$
$12$		0			0
$13$			693.538i			1.13818i	0.822275	+	0.569091i	$0.192705\pi$
$13$			693.538i			1.13818i	−0.822275	+	0.569091i	$0.807295\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	813.920i		−	0.683060i	−0.939871	−	0.341530i	$-0.889055\pi$
$17$		−	813.920i		−	0.683060i	0.939871	−	0.341530i	$-0.110945\pi$
$18$		0			0
$19$	373.543			0.237387			0.118693	−	0.992931i	$-0.462129\pi$
$19$	373.543			0.237387			0.118693	+	0.992931i	$0.462129\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	326.004i		−	0.128500i	−0.997934	−	0.0642501i	$-0.979534\pi$
$23$		−	326.004i		−	0.128500i	0.997934	−	0.0642501i	$-0.0204655\pi$
$24$		0			0
$25$	−6974.68			−2.23190
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3155.47			0.696737			0.348369	−	0.937358i	$-0.386736\pi$
$29$	3155.47			0.696737			0.348369	+	0.937358i	$0.386736\pi$
$30$		0			0
$31$	4193.91			0.783817			0.391908	−	0.920004i	$-0.371815\pi$
$31$	4193.91			0.783817			0.391908	+	0.920004i	$0.371815\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9405.88	−	9015.25i	−1.29786	−	1.24396i
$36$		0			0
$37$	8234.41			0.988845			0.494423	−	0.869222i	$-0.335380\pi$
$37$	8234.41			0.988845			0.494423	+	0.869222i	$0.335380\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	3516.33i		−	0.326685i	−0.986569	−	0.163343i	$-0.947772\pi$
$41$		−	3516.33i		−	0.326685i	0.986569	−	0.163343i	$-0.0522276\pi$
$42$		0			0
$43$		−	9770.29i		−	0.805817i	−0.915240	−	0.402908i	$-0.867999\pi$
$43$		−	9770.29i		−	0.805817i	0.915240	−	0.402908i	$-0.132001\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	22547.1			1.48883			0.744416	−	0.667716i	$-0.232728\pi$
$47$	22547.1			1.48883			0.744416	+	0.667716i	$0.232728\pi$
$48$		0			0
$49$	−712.471	−	16791.9i	−0.0423913	−	0.999101i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−32148.7			−1.57208			−0.786038	−	0.618178i	$-0.787871\pi$
$53$	−32148.7			−1.57208			−0.786038	+	0.618178i	$0.787871\pi$
$54$		0			0
$55$	46120.4			2.05583
$56$		0			0
$57$		0			0
$58$		0			0
$59$	24859.1			0.929727			0.464864	−	0.885382i	$-0.346103\pi$
$59$	24859.1			0.929727			0.464864	+	0.885382i	$0.346103\pi$
$60$		0			0
$61$		−	27316.4i		−	0.939938i	−0.882683	−	0.469969i	$-0.844265\pi$
$61$		−	27316.4i		−	0.939938i	0.882683	−	0.469969i	$-0.155735\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	69698.6			2.04616
$66$		0			0
$67$		−	67423.9i		−	1.83496i	−0.397782	−	0.917480i	$-0.630220\pi$
$67$		−	67423.9i		−	1.83496i	0.397782	−	0.917480i	$-0.369780\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	48772.4i		−	1.14823i	−0.818775	−	0.574114i	$-0.805346\pi$
$71$		−	48772.4i		−	1.14823i	0.818775	−	0.574114i	$-0.194654\pi$
$72$		0			0
$73$		−	23068.9i		−	0.506664i	−0.967379	−	0.253332i	$-0.918473\pi$
$73$		−	23068.9i		−	0.506664i	0.967379	−	0.253332i	$-0.0815265\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	42952.1	+	41168.4i	0.825578	+	0.791292i
$78$		0			0
$79$		−	62780.8i		−	1.13177i	−0.824484	−	0.565886i	$-0.808534\pi$
$79$		−	62780.8i		−	1.13177i	0.824484	−	0.565886i	$-0.191466\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−24828.4			−0.395598			−0.197799	−	0.980243i	$-0.563379\pi$
$83$	−24828.4			−0.395598			−0.197799	+	0.980243i	$0.563379\pi$
$84$		0			0
$85$	−81796.6			−1.22797
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	27793.4i		−	0.371934i	−0.982556	−	0.185967i	$-0.940458\pi$
$89$		−	27793.4i		−	0.371934i	0.982556	−	0.185967i	$-0.0595418\pi$
$90$		0			0
$91$	64910.6	+	62214.9i	0.821698	+	0.787573i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	37540.0i		−	0.426762i
$96$		0			0
$97$			43926.1i			0.474017i	0.971508	+	0.237008i	$0.0761669\pi$
$97$			43926.1i			0.474017i	−0.971508	+	0.237008i	$0.923833\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			122368.i			1.19362i	0.802384	+	0.596809i	$0.203565\pi$
$101$			122368.i			1.19362i	−0.802384	+	0.596809i	$0.796435\pi$
$102$		0			0
$103$	−142684.			−1.32520			−0.662602	−	0.748972i	$-0.730548\pi$
$103$	−142684.			−1.32520			−0.662602	+	0.748972i	$0.730548\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			159981.i			1.35085i	0.737427	+	0.675427i	$0.236040\pi$
$107$			159981.i			1.35085i	−0.737427	+	0.675427i	$0.763960\pi$
$108$		0			0
$109$	−96768.1			−0.780128			−0.390064	−	0.920788i	$-0.627547\pi$
$109$	−96768.1			−0.780128			−0.390064	+	0.920788i	$0.627547\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	48289.1			0.355757			0.177878	−	0.984052i	$-0.443077\pi$
$113$	48289.1			0.355757			0.177878	+	0.984052i	$0.443077\pi$
$114$		0			0
$115$	−32762.5			−0.231011
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−76177.5	−	73013.9i	−0.493128	−	0.472648i
$120$		0			0
$121$	−49558.9			−0.307722
$122$		0			0
$123$		0			0
$124$		0			0
$125$			386881.i			2.21464i
$126$		0			0
$127$			250867.i			1.38018i	0.723726	+	0.690088i	$0.242428\pi$
$127$			250867.i			1.38018i	−0.723726	+	0.690088i	$0.757572\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−65171.1			−0.331800			−0.165900	−	0.986143i	$-0.553053\pi$
$131$	−65171.1			−0.331800			−0.165900	+	0.986143i	$0.553053\pi$
$132$		0			0
$133$	33509.3	−	34961.2i	0.164262	−	0.171379i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−31341.5			−0.142665			−0.0713327	−	0.997453i	$-0.522725\pi$
$137$	−31341.5			−0.142665			−0.0713327	+	0.997453i	$0.522725\pi$
$138$		0			0
$139$	−249531.			−1.09544			−0.547718	−	0.836663i	$-0.684503\pi$
$139$	−249531.			−1.09544			−0.547718	+	0.836663i	$0.684503\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−318280.			−1.30158
$144$		0			0
$145$		−	317116.i		−	1.25256i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	72024.4			0.265775			0.132887	−	0.991131i	$-0.457575\pi$
$149$	72024.4			0.265775			0.132887	+	0.991131i	$0.457575\pi$
$150$		0			0
$151$		−	167063.i		−	0.596264i	−0.954525	−	0.298132i	$-0.903637\pi$
$151$		−	167063.i		−	0.596264i	0.954525	−	0.298132i	$-0.0963635\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	421476.i		−	1.40910i
$156$		0			0
$157$		−	60478.6i		−	0.195818i	−0.995195	−	0.0979089i	$-0.968785\pi$
$157$		−	60478.6i		−	0.195818i	0.995195	−	0.0979089i	$-0.0312154\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−30511.9	−	29244.7i	−0.0927692	−	0.0889166i
$162$		0			0
$163$		−	152936.i		−	0.450859i	−0.974259	−	0.225430i	$-0.927621\pi$
$163$		−	152936.i		−	0.450859i	0.974259	−	0.225430i	$-0.0723785\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−285815.			−0.793037			−0.396519	−	0.918027i	$-0.629782\pi$
$167$	−285815.			−0.793037			−0.396519	+	0.918027i	$0.629782\pi$
$168$		0			0
$169$	−109702.			−0.295458
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	492322.i		−	1.25064i	−0.780367	−	0.625322i	$-0.784968\pi$
$173$		−	492322.i		−	1.25064i	0.780367	−	0.625322i	$-0.215032\pi$
$174$		0			0
$175$	−625674.	+	652784.i	−1.54438	+	1.61129i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	410876.i		−	0.958468i	−0.877687	−	0.479234i	$-0.840914\pi$
$179$		−	410876.i		−	0.958468i	0.877687	−	0.479234i	$-0.159086\pi$
$180$		0			0
$181$			398477.i			0.904080i	0.891998	+	0.452040i	$0.149304\pi$
$181$			398477.i			0.904080i	−0.891998	+	0.452040i	$0.850696\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	827535.i		−	1.77769i
$186$		0			0
$187$	373526.			0.781118
$188$		0			0
$189$		0			0
$190$		0			0
$191$			558784.i			1.10831i	0.832414	+	0.554154i	$0.186958\pi$
$191$			558784.i			1.10831i	−0.832414	+	0.554154i	$0.813042\pi$
$192$		0			0
$193$	640819.			1.23835			0.619174	−	0.785254i	$-0.287468\pi$
$193$	640819.			1.23835			0.619174	+	0.785254i	$0.287468\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	378663.			0.695165			0.347582	−	0.937649i	$-0.387003\pi$
$197$	378663.			0.695165			0.347582	+	0.937649i	$0.387003\pi$
$198$		0			0
$199$	−420500.			−0.752719			−0.376360	−	0.926474i	$-0.622824\pi$
$199$	−420500.			−0.752719			−0.376360	+	0.926474i	$0.622824\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	283066.	−	295331.i	0.482112	−	0.503002i
$204$		0			0
$205$	−353381.			−0.587298
$206$		0			0
$207$		0			0
$208$		0			0
$209$			171427.i			0.271465i
$210$		0			0
$211$		−	532870.i		−	0.823977i	−0.911189	−	0.411988i	$-0.864834\pi$
$211$		−	532870.i		−	0.823977i	0.911189	−	0.411988i	$-0.135166\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−981886.			−1.44866
$216$		0			0
$217$	376221.	−	392522.i	0.542367	−	0.565868i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	564484.			0.777447
$222$		0			0
$223$	−202475.			−0.272652			−0.136326	−	0.990664i	$-0.543529\pi$
$223$	−202475.			−0.272652			−0.136326	+	0.990664i	$0.543529\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−990163.			−1.27539			−0.637693	−	0.770290i	$-0.720111\pi$
$227$	−990163.			−1.27539			−0.637693	+	0.770290i	$0.720111\pi$
$228$		0			0
$229$			529398.i			0.667104i	0.942732	+	0.333552i	$0.108247\pi$
$229$			529398.i			0.667104i	−0.942732	+	0.333552i	$0.891753\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.46530e6			−1.76823			−0.884113	−	0.467272i	$-0.845237\pi$
$233$	−1.46530e6			−1.76823			−0.884113	+	0.467272i	$0.845237\pi$
$234$		0			0
$235$		−	2.26592e6i		−	2.67654i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	989159.i		−	1.12014i	−0.828446	−	0.560069i	$-0.810775\pi$
$239$		−	989159.i		−	1.12014i	0.828446	−	0.560069i	$-0.189225\pi$
$240$		0			0
$241$		−	577035.i		−	0.639970i	−0.947423	−	0.319985i	$-0.896322\pi$
$241$		−	577035.i		−	0.639970i	0.947423	−	0.319985i	$-0.103678\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.68754e6	+	71601.3i	−1.79613	+	0.0762089i
$246$		0			0
$247$			259066.i			0.270189i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	459977.			0.460842			0.230421	−	0.973091i	$-0.425990\pi$
$251$	459977.			0.460842			0.230421	+	0.973091i	$0.425990\pi$
$252$		0			0
$253$	149611.			0.146947
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	39637.2i		−	0.0374343i	−0.999825	−	0.0187171i	$-0.994042\pi$
$257$		−	39637.2i		−	0.0374343i	0.999825	−	0.0187171i	$-0.00595820\pi$
$258$		0			0
$259$	738681.	−	770687.i	0.684238	−	0.713886i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1.27642e6i		−	1.13790i	−0.822372	−	0.568951i	$-0.807350\pi$
$263$		−	1.27642e6i		−	1.13790i	0.822372	−	0.568951i	$-0.192650\pi$
$264$		0			0
$265$			3.23085e6i			2.82620i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			161449.i			0.136036i	0.997684	+	0.0680180i	$0.0216675\pi$
$269$			161449.i			0.136036i	−0.997684	+	0.0680180i	$0.978332\pi$
$270$		0			0
$271$	−939346.			−0.776966			−0.388483	−	0.921456i	$-0.627001\pi$
$271$	−939346.			−0.776966			−0.388483	+	0.921456i	$0.627001\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	3.20084e6i		−	2.55230i
$276$		0			0
$277$	1.06595e6			0.834715			0.417357	−	0.908742i	$-0.362956\pi$
$277$	1.06595e6			0.834715			0.417357	+	0.908742i	$0.362956\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	556041.			0.420089			0.210044	−	0.977692i	$-0.432639\pi$
$281$	556041.			0.420089			0.210044	+	0.977692i	$0.432639\pi$
$282$		0			0
$283$	2.34724e6			1.74218			0.871088	−	0.491127i	$-0.163415\pi$
$283$	2.34724e6			1.74218			0.871088	+	0.491127i	$0.163415\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−329105.	−	315437.i	−0.235847	−	0.226052i
$288$		0			0
$289$	757392.			0.533428
$290$		0			0
$291$		0			0
$292$		0			0
$293$			634157.i			0.431546i	0.976444	+	0.215773i	$0.0692272\pi$
$293$			634157.i			0.431546i	−0.976444	+	0.215773i	$0.930773\pi$
$294$		0			0
$295$		−	2.49827e6i		−	1.67141i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	226096.			0.146257
$300$		0			0
$301$	−914435.	−	876459.i	−0.581750	−	0.557591i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.74522e6			−1.68977
$306$		0			0
$307$	348422.			0.210989			0.105495	−	0.994420i	$-0.466357\pi$
$307$	348422.			0.210989			0.105495	+	0.994420i	$0.466357\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	100689.			0.0590313			0.0295156	−	0.999564i	$-0.490604\pi$
$311$	100689.			0.0590313			0.0295156	+	0.999564i	$0.490604\pi$
$312$		0			0
$313$			3.19846e6i			1.84535i	0.385573	+	0.922677i	$0.374004\pi$
$313$			3.19846e6i			1.84535i	−0.385573	+	0.922677i	$0.625996\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.54982e6			1.98407			0.992037	−	0.125949i	$-0.0401976\pi$
$317$	3.54982e6			1.98407			0.992037	+	0.125949i	$0.0401976\pi$
$318$		0			0
$319$			1.44812e6i			0.796759i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	304034.i		−	0.162150i
$324$		0			0
$325$		−	4.83720e6i		−	2.54030i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.02262e6	−	2.11026e6i	1.03021	−	1.07485i
$330$		0			0
$331$			257593.i			0.129230i	0.997910	+	0.0646152i	$0.0205820\pi$
$331$			257593.i			0.129230i	−0.997910	+	0.0646152i	$0.979418\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.77591e6			−3.29880
$336$		0			0
$337$	−977553.			−0.468884			−0.234442	−	0.972130i	$-0.575326\pi$
$337$	−977553.			−0.468884			−0.234442	+	0.972130i	$0.575326\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			1.92468e6i			0.896339i
$342$		0			0
$343$	−1.63552e6	−	1.43966e6i	−0.750623	−	0.660731i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	2.46779e6i		−	1.10023i	−0.835088	−	0.550117i	$-0.814583\pi$
$347$		−	2.46779e6i		−	1.10023i	0.835088	−	0.550117i	$-0.185417\pi$
$348$		0			0
$349$			2.28274e6i			1.00321i	0.865096	+	0.501606i	$0.167257\pi$
$349$			2.28274e6i			1.00321i	−0.865096	+	0.501606i	$0.832743\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2.93371e6i		−	1.25309i	−0.779387	−	0.626543i	$-0.784469\pi$
$353$		−	2.93371e6i		−	1.25309i	0.779387	−	0.626543i	$-0.215531\pi$
$354$		0			0
$355$	−4.90149e6			−2.06423
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	2.62552e6i		−	1.07518i	−0.843208	−	0.537588i	$-0.819336\pi$
$359$		−	2.62552e6i		−	1.07518i	0.843208	−	0.537588i	$-0.180664\pi$
$360$		0			0
$361$	−2.33656e6			−0.943647
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.31836e6			−0.910854
$366$		0			0
$367$	−3.36188e6			−1.30292			−0.651459	−	0.758684i	$-0.725843\pi$
$367$	−3.36188e6			−1.30292			−0.651459	+	0.758684i	$0.725843\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.88395e6	+	3.00891e6i	−1.08781	+	1.13494i
$372$		0			0
$373$	2.07818e6			0.773412			0.386706	−	0.922203i	$-0.373613\pi$
$373$	2.07818e6			0.773412			0.386706	+	0.922203i	$0.373613\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			2.18844e6i			0.793014i
$378$		0			0
$379$			1.84637e6i			0.660267i	0.943934	+	0.330134i	$0.107094\pi$
$379$			1.84637e6i			0.660267i	−0.943934	+	0.330134i	$0.892906\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.17912e6			0.759076			0.379538	−	0.925176i	$-0.376083\pi$
$383$	2.17912e6			0.759076			0.379538	+	0.925176i	$0.376083\pi$
$384$		0			0
$385$	4.13730e6	−	4.31657e6i	1.42254	−	1.48418i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	32242.4			0.0108032			0.00540161	−	0.999985i	$-0.498281\pi$
$389$	32242.4			0.0108032			0.00540161	+	0.999985i	$0.498281\pi$
$390$		0			0
$391$	−265341.			−0.0877734
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.30929e6			−2.03464
$396$		0			0
$397$		−	2.21494e6i		−	0.705319i	−0.935752	−	0.352660i	$-0.885277\pi$
$397$		−	2.21494e6i		−	0.705319i	0.935752	−	0.352660i	$-0.114723\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−739360.			−0.229612			−0.114806	−	0.993388i	$-0.536625\pi$
$401$	−739360.			−0.229612			−0.114806	+	0.993388i	$0.536625\pi$
$402$		0			0
$403$			2.90863e6i			0.892126i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.77896e6i			1.13080i
$408$		0			0
$409$		−	2.17984e6i		−	0.644340i	−0.946682	−	0.322170i	$-0.895588\pi$
$409$		−	2.17984e6i		−	0.644340i	0.946682	−	0.322170i	$-0.104412\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.23002e6	−	2.32665e6i	0.643331	−	0.671206i
$414$		0			0
$415$			2.49519e6i			0.711186i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.07822e6			0.856572			0.428286	−	0.903643i	$-0.359117\pi$
$419$	3.07822e6			0.856572			0.428286	+	0.903643i	$0.359117\pi$
$420$		0			0
$421$	−4.99144e6			−1.37253			−0.686264	−	0.727353i	$-0.740750\pi$
$421$	−4.99144e6			−1.37253			−0.686264	+	0.727353i	$0.740750\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			5.67683e6i			1.52452i
$426$		0			0
$427$	−2.55664e6	−	2.45046e6i	−0.678578	−	0.650396i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	1.07037e6i		−	0.277551i	−0.990324	−	0.138775i	$-0.955683\pi$
$431$		−	1.07037e6i		−	0.277551i	0.990324	−	0.138775i	$-0.0443166\pi$
$432$		0			0
$433$			2.89323e6i			0.741590i	0.928715	+	0.370795i	$0.120915\pi$
$433$			2.89323e6i			0.741590i	−0.928715	+	0.370795i	$0.879085\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	121777.i		−	0.0305043i
$438$		0			0
$439$	−1.15097e6			−0.285038			−0.142519	−	0.989792i	$-0.545520\pi$
$439$	−1.15097e6			−0.285038			−0.142519	+	0.989792i	$0.545520\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	913636.i		−	0.221189i	−0.993866	−	0.110595i	$-0.964724\pi$
$443$		−	913636.i		−	0.221189i	0.993866	−	0.110595i	$-0.0352755\pi$
$444$		0			0
$445$	−2.79315e6			−0.668644
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.77009e6			0.414362			0.207181	−	0.978303i	$-0.433571\pi$
$449$	1.77009e6			0.414362			0.207181	+	0.978303i	$0.433571\pi$
$450$		0			0
$451$	1.61372e6			0.373583
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.25242e6	−	6.52333e6i	1.41586	−	1.47721i
$456$		0			0
$457$	−5.72173e6			−1.28155			−0.640777	−	0.767727i	$-0.721388\pi$
$457$	−5.72173e6			−1.28155			−0.640777	+	0.767727i	$0.721388\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			2.88921e6i			0.633180i	0.948563	+	0.316590i	$0.102538\pi$
$461$			2.88921e6i			0.633180i	−0.948563	+	0.316590i	$0.897462\pi$
$462$		0			0
$463$		−	5.03032e6i		−	1.09054i	−0.838260	−	0.545271i	$-0.816427\pi$
$463$		−	5.03032e6i		−	1.09054i	0.838260	−	0.545271i	$-0.183573\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.88406e6			−0.611946			−0.305973	−	0.952040i	$-0.598982\pi$
$467$	−2.88406e6			−0.611946			−0.305973	+	0.952040i	$0.598982\pi$
$468$		0			0
$469$	−6.31043e6	−	6.04836e6i	−1.32473	−	1.26971i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.48381e6			0.921497
$474$		0			0
$475$	−2.60534e6			−0.529823
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.78719e6			−1.94903			−0.974516	−	0.224317i	$-0.927985\pi$
$479$	−9.78719e6			−1.94903			−0.974516	+	0.224317i	$0.927985\pi$
$480$		0			0
$481$			5.71087e6i			1.12549i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.41445e6			0.852162
$486$		0			0
$487$			2.62742e6i			0.502003i	0.967987	+	0.251002i	$0.0807600\pi$
$487$			2.62742e6i			0.502003i	−0.967987	+	0.251002i	$0.919240\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	8.05557e6i		−	1.50797i	−0.656892	−	0.753985i	$-0.728129\pi$
$491$		−	8.05557e6i		−	1.50797i	0.656892	−	0.753985i	$-0.271871\pi$
$492$		0			0
$493$		−	2.56830e6i		−	0.475914i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.56478e6	−	4.37520e6i	−0.828951	−	0.794525i
$498$		0			0
$499$		−	6.47634e6i		−	1.16434i	−0.813069	−	0.582168i	$-0.802205\pi$
$499$		−	6.47634e6i		−	1.16434i	0.813069	−	0.582168i	$-0.197795\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2.53406e6			0.446578			0.223289	−	0.974752i	$-0.428321\pi$
$503$	2.53406e6			0.446578			0.223289	+	0.974752i	$0.428321\pi$
$504$		0			0
$505$	1.22977e7			2.14582
$506$		0			0
$507$		0			0
$508$		0			0
$509$			2.93740e6i			0.502538i	0.967917	+	0.251269i	$0.0808479\pi$
$509$			2.93740e6i			0.502538i	−0.967917	+	0.251269i	$0.919152\pi$
$510$		0			0
$511$	−2.15910e6	−	2.06943e6i	−0.365781	−	0.350590i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			1.43393e7i			2.38238i
$516$		0			0
$517$			1.03474e7i			1.70256i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	1.31766e6i		−	0.212672i	−0.994330	−	0.106336i	$-0.966088\pi$
$521$		−	1.31766e6i		−	0.212672i	0.994330	−	0.106336i	$-0.0339119\pi$
$522$		0			0
$523$	4.66634e6			0.745971			0.372986	−	0.927837i	$-0.378334\pi$
$523$	4.66634e6			0.745971			0.372986	+	0.927837i	$0.378334\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	3.41350e6i		−	0.535394i
$528$		0			0
$529$	6.33006e6			0.983488
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.43870e6			0.371827
$534$		0			0
$535$	1.60776e7			2.42849
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7.70618e6	−	326969.i	1.14253	−	0.0484768i
$540$		0			0
$541$	−1.48725e6			−0.218469			−0.109235	−	0.994016i	$-0.534840\pi$
$541$	−1.48725e6			−0.218469			−0.109235	+	0.994016i	$0.534840\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			9.72492e6i			1.40247i
$546$		0			0
$547$			1.35405e7i			1.93494i	0.252986	+	0.967470i	$0.418587\pi$
$547$			1.35405e7i			1.93494i	−0.252986	+	0.967470i	$0.581413\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.17870e6			0.165396
$552$		0			0
$553$	−5.87587e6	−	5.63185e6i	−0.817070	−	0.783137i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.99906e6			0.682732			0.341366	−	0.939930i	$-0.389110\pi$
$557$	4.99906e6			0.682732			0.341366	+	0.939930i	$0.389110\pi$
$558$		0			0
$559$	6.77607e6			0.917166
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.35710e6			0.978219			0.489109	−	0.872223i	$-0.337322\pi$
$563$	7.35710e6			0.978219			0.489109	+	0.872223i	$0.337322\pi$
$564$		0			0
$565$		−	4.85292e6i		−	0.639561i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.25324e7			−1.62275			−0.811376	−	0.584525i	$-0.801281\pi$
$569$	−1.25324e7			−1.62275			−0.811376	+	0.584525i	$0.801281\pi$
$570$		0			0
$571$		−	5.52022e6i		−	0.708543i	−0.935143	−	0.354272i	$-0.884729\pi$
$571$		−	5.52022e6i		−	0.708543i	0.935143	−	0.354272i	$-0.115271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			2.27377e6i			0.286799i
$576$		0			0
$577$		−	1.31746e7i		−	1.64740i	−0.567027	−	0.823699i	$-0.691906\pi$
$577$		−	1.31746e7i		−	1.64740i	0.567027	−	0.823699i	$-0.308094\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.22727e6	+	2.32378e6i	−0.273737	+	0.285598i
$582$		0			0
$583$		−	1.47538e7i		−	1.79776i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.69162e6			−0.561989			−0.280994	−	0.959709i	$-0.590664\pi$
$587$	−4.69162e6			−0.561989			−0.280994	+	0.959709i	$0.590664\pi$
$588$		0			0
$589$	1.56661e6			0.186068
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	8.57996e6i		−	1.00196i	−0.865460	−	0.500978i	$-0.832974\pi$
$593$		−	8.57996e6i		−	1.00196i	0.865460	−	0.500978i	$-0.167026\pi$
$594$		0			0
$595$	−7.33769e6	+	7.65563e6i	−0.849703	+	0.886519i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1.37105e7i		−	1.56130i	−0.624969	−	0.780650i	$-0.714888\pi$
$599$		−	1.37105e7i		−	1.56130i	0.624969	−	0.780650i	$-0.285112\pi$
$600$		0			0
$601$			1.61956e7i			1.82899i	0.404601	+	0.914493i	$0.367410\pi$
$601$			1.61956e7i			1.82899i	−0.404601	+	0.914493i	$0.632590\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			4.98053e6i			0.553206i
$606$		0			0
$607$	−925000.			−0.101899			−0.0509495	−	0.998701i	$-0.516225\pi$
$607$	−925000.			−0.101899			−0.0509495	+	0.998701i	$0.516225\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1.56372e7i			1.69456i
$612$		0			0
$613$	−3.80091e6			−0.408541			−0.204271	−	0.978914i	$-0.565482\pi$
$613$	−3.80091e6			−0.408541			−0.204271	+	0.978914i	$0.565482\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.13629e7			−1.20164			−0.600822	−	0.799383i	$-0.705160\pi$
$617$	−1.13629e7			−1.20164			−0.600822	+	0.799383i	$0.705160\pi$
$618$		0			0
$619$	1.27388e7			1.33630			0.668149	−	0.744028i	$-0.267087\pi$
$619$	1.27388e7			1.33630			0.668149	+	0.744028i	$0.267087\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.60128e6	−	2.49325e6i	−0.268514	−	0.257362i
$624$		0			0
$625$	1.70846e7			1.74947
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	6.70215e6i		−	0.675441i
$630$		0			0
$631$		−	1.04172e7i		−	1.04155i	−0.853695	−	0.520773i	$-0.825644\pi$
$631$		−	1.04172e7i		−	1.04155i	0.853695	−	0.520773i	$-0.174356\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.52114e7			2.48121
$636$		0			0
$637$	1.16458e7	−	494125.i	1.13716	−	0.0482490i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.00849e7			0.969452			0.484726	−	0.874666i	$-0.338919\pi$
$641$	1.00849e7			0.969452			0.484726	+	0.874666i	$0.338919\pi$
$642$		0			0
$643$	2.23262e6			0.212955			0.106477	−	0.994315i	$-0.466043\pi$
$643$	2.23262e6			0.212955			0.106477	+	0.994315i	$0.466043\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.25193e6			−0.587156			−0.293578	−	0.955935i	$-0.594846\pi$
$647$	−6.25193e6			−0.587156			−0.293578	+	0.955935i	$0.594846\pi$
$648$		0			0
$649$			1.14084e7i			1.06320i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.22646e6			0.112557			0.0562783	−	0.998415i	$-0.482077\pi$
$653$	1.22646e6			0.112557			0.0562783	+	0.998415i	$0.482077\pi$
$654$		0			0
$655$			6.54951e6i			0.596493i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			3.73955e6i			0.335433i	0.985835	+	0.167716i	$0.0536393\pi$
$659$			3.73955e6i			0.335433i	−0.985835	+	0.167716i	$0.946361\pi$
$660$		0			0
$661$			1.38831e7i			1.23590i	0.786217	+	0.617950i	$0.212037\pi$
$661$			1.38831e7i			1.23590i	−0.786217	+	0.617950i	$0.787963\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.51350e6	−	3.36759e6i	−0.308096	−	0.295301i
$666$		0			0
$667$		−	1.02870e6i		−	0.0895309i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.25361e7			1.07487
$672$		0			0
$673$	−1.81112e7			−1.54138			−0.770689	−	0.637211i	$-0.780088\pi$
$673$	−1.81112e7			−1.54138			−0.770689	+	0.637211i	$0.780088\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1.94176e7i		−	1.62826i	−0.580683	−	0.814130i	$-0.697215\pi$
$677$		−	1.94176e7i		−	1.62826i	0.580683	−	0.814130i	$-0.302785\pi$
$678$		0			0
$679$	4.11120e6	+	3.94046e6i	0.342211	+	0.327999i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.41477e7i			1.16047i	0.814450	+	0.580233i	$0.197039\pi$
$683$			1.41477e7i			1.16047i	−0.814450	+	0.580233i	$0.802961\pi$
$684$		0			0
$685$			3.14973e6i			0.256476i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	2.22963e7i		−	1.78931i
$690$		0			0
$691$	−2.20407e7			−1.75602			−0.878012	−	0.478639i	$-0.841130\pi$
$691$	−2.20407e7			−1.75602			−0.878012	+	0.478639i	$0.841130\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			2.50771e7i			1.96932i
$696$		0			0
$697$	−2.86201e6			−0.223146
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.92655e7			−1.48076			−0.740380	−	0.672189i	$-0.765354\pi$
$701$	−1.92655e7			−1.48076			−0.740380	+	0.672189i	$0.765354\pi$
$702$		0			0
$703$	3.07591e6			0.234739
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.14529e7	+	1.09772e7i	0.861719	+	0.825932i
$708$		0			0
$709$	−4.21885e6			−0.315195			−0.157597	−	0.987503i	$-0.550375\pi$
$709$	−4.21885e6			−0.315195			−0.157597	+	0.987503i	$0.550375\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1.36723e6i		−	0.100721i
$714$		0			0
$715$			3.19862e7i			2.33990i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.50522e7			1.08587			0.542935	−	0.839775i	$-0.317313\pi$
$719$	1.50522e7			1.08587			0.542935	+	0.839775i	$0.317313\pi$
$720$		0			0
$721$	−1.27997e7	+	1.33543e7i	−0.916984	+	0.956716i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.20084e7			−1.55505
$726$		0			0
$727$	9.39018e6			0.658928			0.329464	−	0.944168i	$-0.393132\pi$
$727$	9.39018e6			0.658928			0.329464	+	0.944168i	$0.393132\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.95223e6			−0.550422
$732$		0			0
$733$		−	1.81434e7i		−	1.24727i	−0.781717	−	0.623634i	$-0.785656\pi$
$733$		−	1.81434e7i		−	1.24727i	0.781717	−	0.623634i	$-0.214344\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.09423e7			2.09838
$738$		0			0
$739$			8.27786e6i			0.557580i	0.960352	+	0.278790i	$0.0899333\pi$
$739$			8.27786e6i			0.557580i	−0.960352	+	0.278790i	$0.910067\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	861253.i		−	0.0572346i	−0.999590	−	0.0286173i	$-0.990890\pi$
$743$		−	861253.i		−	0.0572346i	0.999590	−	0.0286173i	$-0.00911042\pi$
$744$		0			0
$745$		−	7.23824e6i		−	0.477796i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.49731e7	+	1.43513e7i	0.975233	+	0.934732i
$750$		0			0
$751$			2.96424e7i			1.91784i	0.283671	+	0.958922i	$0.408448\pi$
$751$			2.96424e7i			1.91784i	−0.283671	+	0.958922i	$0.591552\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.67894e7			−1.07193
$756$		0			0
$757$	2.73727e7			1.73611			0.868056	−	0.496466i	$-0.165369\pi$
$757$	2.73727e7			1.73611			0.868056	+	0.496466i	$0.165369\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			4.11268e6i			0.257432i	0.991681	+	0.128716i	$0.0410856\pi$
$761$			4.11268e6i			0.257432i	−0.991681	+	0.128716i	$0.958914\pi$
$762$		0			0
$763$	−8.68073e6	+	9.05686e6i	−0.539815	+	0.563205i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1.72407e7i			1.05820i
$768$		0			0
$769$		−	1.72452e7i		−	1.05160i	−0.850607	−	0.525802i	$-0.823765\pi$
$769$		−	1.72452e7i		−	1.05160i	0.850607	−	0.525802i	$-0.176235\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	2.29735e6i		−	0.138286i	−0.997607	−	0.0691430i	$-0.977974\pi$
$773$		−	2.29735e6i		−	0.138286i	0.997607	−	0.0691430i	$-0.0220265\pi$
$774$		0			0
$775$	−2.92511e7			−1.74940
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1.31350e6i		−	0.0775508i
$780$		0			0
$781$	2.23828e7			1.31306
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.07792e6			−0.352031
$786$		0			0
$787$	−4.09252e6			−0.235534			−0.117767	−	0.993041i	$-0.537574\pi$
$787$	−4.09252e6			−0.235534			−0.117767	+	0.993041i	$0.537574\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4.33185e6	−	4.51955e6i	0.246169	−	0.256835i
$792$		0			0
$793$	1.89450e7			1.06982
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1.53351e7i			0.855149i	0.903980	+	0.427575i	$0.140632\pi$
$797$			1.53351e7i			0.855149i	−0.903980	+	0.427575i	$0.859368\pi$
$798$		0			0
$799$		−	1.83515e7i		−	1.01696i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.05869e7			0.579399
$804$		0			0
$805$	−2.93901e6	+	3.06635e6i	−0.159850	+	0.166776i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−9.14411e6			−0.491213			−0.245607	−	0.969370i	$-0.578987\pi$
$809$	−9.14411e6			−0.491213			−0.245607	+	0.969370i	$0.578987\pi$
$810$		0			0
$811$	2.81005e7			1.50024			0.750121	−	0.661300i	$-0.229995\pi$
$811$	2.81005e7			1.50024			0.750121	+	0.661300i	$0.229995\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.53696e7			−0.810531
$816$		0			0
$817$		−	3.64963e6i		−	0.191290i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6.88425e6			−0.356450			−0.178225	−	0.983990i	$-0.557035\pi$
$821$	−6.88425e6			−0.356450			−0.178225	+	0.983990i	$0.557035\pi$
$822$		0			0
$823$		−	9.01181e6i		−	0.463780i	−0.972742	−	0.231890i	$-0.925509\pi$
$823$		−	9.01181e6i		−	0.463780i	0.972742	−	0.231890i	$-0.0744910\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			3.47775e7i			1.76821i	0.467288	+	0.884105i	$0.345231\pi$
$827$			3.47775e7i			1.76821i	−0.467288	+	0.884105i	$0.654769\pi$
$828$		0			0
$829$			1.87819e7i			0.949190i	0.880204	+	0.474595i	$0.157405\pi$
$829$			1.87819e7i			0.949190i	−0.880204	+	0.474595i	$0.842595\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.36672e7	+	579894.i	−0.682446	+	0.0289558i
$834$		0			0
$835$			2.87236e7i			1.42568i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−748198.			−0.0366954			−0.0183477	−	0.999832i	$-0.505841\pi$
$839$	−748198.			−0.0366954			−0.0183477	+	0.999832i	$0.505841\pi$
$840$		0			0
$841$	−1.05542e7			−0.514557
$842$		0			0
$843$		0			0
$844$		0			0
$845$			1.10247e7i			0.531159i
$846$		0			0
$847$	−4.44576e6	+	4.63839e6i	−0.212930	+	0.222156i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	2.68445e6i		−	0.127067i
$852$		0			0
$853$		−	2.85379e7i		−	1.34292i	−0.741042	−	0.671458i	$-0.765668\pi$
$853$		−	2.85379e7i		−	1.34292i	0.741042	−	0.671458i	$-0.234332\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.51196e7i			0.703217i	0.936147	+	0.351609i	$0.114365\pi$
$857$			1.51196e7i			0.703217i	−0.936147	+	0.351609i	$0.885635\pi$
$858$		0			0
$859$	3.29949e7			1.52568			0.762842	−	0.646585i	$-0.223804\pi$
$859$	3.29949e7			1.52568			0.762842	+	0.646585i	$0.223804\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	4.00309e6i		−	0.182965i	−0.995807	−	0.0914826i	$-0.970839\pi$
$863$		−	4.00309e6i		−	0.182965i	0.995807	−	0.0914826i	$-0.0291606\pi$
$864$		0			0
$865$	−4.94769e7			−2.24834
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.88115e7			1.29425
$870$		0			0
$871$	4.67610e7			2.08852
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.62096e7	+	3.47058e7i	1.59883	+	1.53243i
$876$		0			0
$877$	−1.53480e7			−0.673835			−0.336917	−	0.941534i	$-0.609384\pi$
$877$	−1.53480e7			−0.673835			−0.336917	+	0.941534i	$0.609384\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			5.67933e6i			0.246523i	0.992374	+	0.123261i	$0.0393354\pi$
$881$			5.67933e6i			0.246523i	−0.992374	+	0.123261i	$0.960665\pi$
$882$		0			0
$883$		−	4.39504e6i		−	0.189697i	−0.995492	−	0.0948487i	$-0.969763\pi$
$883$		−	4.39504e6i		−	0.189697i	0.995492	−	0.0948487i	$-0.0302367\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.93661e7			0.826483			0.413242	−	0.910621i	$-0.364396\pi$
$887$	1.93661e7			0.826483			0.413242	+	0.910621i	$0.364396\pi$
$888$		0			0
$889$	2.34795e7	+	2.25044e7i	0.996402	+	0.955022i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.42230e6			0.353429
$894$		0			0
$895$	−4.12918e7			−1.72308
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.32338e7			0.546115
$900$		0			0
$901$			2.61665e7i			1.07382i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.00458e7			1.62531
$906$		0			0
$907$		−	4.64251e7i		−	1.87385i	−0.349529	−	0.936925i	$-0.613659\pi$
$907$		−	4.64251e7i		−	1.87385i	0.349529	−	0.936925i	$-0.386341\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1.27993e7i		−	0.510964i	−0.966814	−	0.255482i	$-0.917766\pi$
$911$		−	1.27993e7i		−	0.510964i	0.966814	−	0.255482i	$-0.0822342\pi$
$912$		0			0
$913$		−	1.13943e7i		−	0.452389i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−5.84627e6	+	6.09959e6i	−0.229592	+	0.239540i
$918$		0			0
$919$			2.20868e7i			0.862669i	0.902192	+	0.431334i	$0.141957\pi$
$919$			2.20868e7i			0.862669i	−0.902192	+	0.431334i	$0.858043\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3.38255e7			1.30689
$924$		0			0
$925$	−5.74324e7			−2.20700
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1.83124e7i			0.696157i	0.937465	+	0.348078i	$0.113166\pi$
$929$			1.83124e7i			0.696157i	−0.937465	+	0.348078i	$0.886834\pi$
$930$		0			0
$931$	−266138.	−	6.27250e6i	−0.0100631	−	0.237174i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	3.75383e7i		−	1.40425i
$936$		0			0
$937$		−	4.02297e7i		−	1.49692i	−0.663180	−	0.748460i	$-0.730794\pi$
$937$		−	4.02297e7i		−	1.49692i	0.663180	−	0.748460i	$-0.269206\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			4.20545e7i			1.54824i	0.633038	+	0.774121i	$0.281808\pi$
$941$			4.20545e7i			1.54824i	−0.633038	+	0.774121i	$0.718192\pi$
$942$		0			0
$943$	−1.14634e6			−0.0419791
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	3.13108e7i		−	1.13454i	−0.823532	−	0.567270i	$-0.808000\pi$
$947$		−	3.13108e7i		−	1.13454i	0.823532	−	0.567270i	$-0.192000\pi$
$948$		0			0
$949$	1.59992e7			0.576676
$950$		0			0
$951$		0			0
$952$		0			0
$953$	54340.4			0.00193816			0.000969081	−	1.00000i	$-0.499692\pi$
$953$	54340.4			0.00193816			0.000969081		1.00000i	$0.499692\pi$
$954$		0			0
$955$	5.61562e7			1.99246
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.81154e6	+	2.93336e6i	−0.0987183	+	0.102996i
$960$		0			0
$961$	−1.10403e7			−0.385631
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	6.44005e7i		−	2.22624i
$966$		0			0
$967$		−	3.73789e7i		−	1.28546i	−0.766091	−	0.642732i	$-0.777801\pi$
$967$		−	3.73789e7i		−	1.28546i	0.766091	−	0.642732i	$-0.222199\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.84157e7			0.967186			0.483593	−	0.875293i	$-0.339332\pi$
$971$	2.84157e7			0.967186			0.483593	+	0.875293i	$0.339332\pi$
$972$		0			0
$973$	−2.23845e7	+	2.33545e7i	−0.757995	+	0.790838i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.57873e7			0.864308			0.432154	−	0.901800i	$-0.357754\pi$
$977$	2.57873e7			0.864308			0.432154	+	0.901800i	$0.357754\pi$
$978$		0			0
$979$	1.27550e7			0.425328
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.81345e7			−0.928656			−0.464328	−	0.885663i	$-0.653704\pi$
$983$	−2.81345e7			−0.928656			−0.464328	+	0.885663i	$0.653704\pi$
$984$		0			0
$985$		−	3.80546e7i		−	1.24973i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.18516e6			−0.103548
$990$		0			0
$991$			8.83390e6i			0.285738i	0.989742	+	0.142869i	$0.0456328\pi$
$991$			8.83390e6i			0.285738i	−0.989742	+	0.142869i	$0.954367\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			4.22590e7i			1.35320i
$996$		0			0
$997$			4.56085e7i			1.45314i	0.687092	+	0.726570i	$0.258887\pi$
$997$			4.56085e7i			1.45314i	−0.687092	+	0.726570i	$0.741113\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.6.b.g.559.2		8
3.2	odd	2		112.6.f.a.111.8	yes	8
4.3	odd	2	inner	1008.6.b.g.559.1		8
7.6	odd	2	inner	1008.6.b.g.559.7		8
12.11	even	2		112.6.f.a.111.2	yes	8
21.20	even	2		112.6.f.a.111.1	✓	8
24.5	odd	2		448.6.f.b.447.1		8
24.11	even	2		448.6.f.b.447.7		8
28.27	even	2	inner	1008.6.b.g.559.8		8
84.83	odd	2		112.6.f.a.111.7	yes	8
168.83	odd	2		448.6.f.b.447.2		8
168.125	even	2		448.6.f.b.447.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.6.f.a.111.1	✓	8	21.20	even	2
112.6.f.a.111.2	yes	8	12.11	even	2
112.6.f.a.111.7	yes	8	84.83	odd	2
112.6.f.a.111.8	yes	8	3.2	odd	2
448.6.f.b.447.1		8	24.5	odd	2
448.6.f.b.447.2		8	168.83	odd	2
448.6.f.b.447.7		8	24.11	even	2
448.6.f.b.447.8		8	168.125	even	2
1008.6.b.g.559.1		8	4.3	odd	2	inner
1008.6.b.g.559.2		8	1.1	even	1	trivial
1008.6.b.g.559.7		8	7.6	odd	2	inner
1008.6.b.g.559.8		8	28.27	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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1008.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-100.497i q^{5} +(89.7065 - 93.5935i) q^{7} +O(q^{10})\)	\(q-100.497i q^{5} +(89.7065 - 93.5935i) q^{7} +458.923i q^{11} +693.538i q^{13} -813.920i q^{17} +373.543 q^{19} -326.004i q^{23} -6974.68 q^{25} +3155.47 q^{29} +4193.91 q^{31} +(-9405.88 - 9015.25i) q^{35} +8234.41 q^{37} -3516.33i q^{41} -9770.29i q^{43} +22547.1 q^{47} +(-712.471 - 16791.9i) q^{49} -32148.7 q^{53} +46120.4 q^{55} +24859.1 q^{59} -27316.4i q^{61} +69698.6 q^{65} -67423.9i q^{67} -48772.4i q^{71} -23068.9i q^{73} +(42952.1 + 41168.4i) q^{77} -62780.8i q^{79} -24828.4 q^{83} -81796.6 q^{85} -27793.4i q^{89} +(64910.6 + 62214.9i) q^{91} -37540.0i q^{95} +43926.1i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 17048 q^{25} + 9744 q^{29} + 19376 q^{37} + 9800 q^{49} - 102192 q^{53} + 193344 q^{65} + 25872 q^{77} - 437376 q^{85}+O(q^{100}) \) 8 * q - 17048 * q^25 + 9744 * q^29 + 19376 * q^37 + 9800 * q^49 - 102192 * q^53 + 193344 * q^65 + 25872 * q^77 - 437376 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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