LMFDB - Embedded newform 448.6.f.b.447.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,6,Mod(447,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("448.447");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	448.f (of order $2$, degree $1$, not 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:	$71.8519512762$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{20}\cdot 7^{5} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			447.8
Root			$0.785404 + 3.50041i$ of defining polynomial
Character	$\chi$	$=$	448.447
Dual form			448.6.f.b.447.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+26.7378 q^{3} +100.497i q^{5} +(-89.7065 - 93.5935i) q^{7} +471.912 q^{9} +O(q^{10})$	$q+26.7378 q^{3} +100.497i q^{5} +(-89.7065 - 93.5935i) q^{7} +471.912 q^{9} +458.923i q^{11} +693.538i q^{13} +2687.08i q^{15} -813.920i q^{17} +373.543 q^{19} +(-2398.56 - 2502.49i) q^{21} +326.004i q^{23} -6974.68 q^{25} +6120.60 q^{27} +3155.47 q^{29} -4193.91 q^{31} +12270.6i q^{33} +(9405.88 - 9015.25i) q^{35} -8234.41 q^{37} +18543.7i q^{39} -3516.33i q^{41} +9770.29i q^{43} +47425.8i q^{45} +22547.1 q^{47} +(-712.471 + 16791.9i) q^{49} -21762.4i q^{51} -32148.7 q^{53} -46120.4 q^{55} +9987.73 q^{57} -24859.1 q^{59} -27316.4i q^{61} +(-42333.6 - 44167.9i) q^{63} -69698.6 q^{65} +67423.9i q^{67} +8716.65i q^{69} +48772.4i q^{71} +23068.9i q^{73} -186488. q^{75} +(42952.1 - 41168.4i) q^{77} -62780.8i q^{79} +48977.1 q^{81} +24828.4 q^{83} +81796.6 q^{85} +84370.4 q^{87} -27793.4i q^{89} +(64910.6 - 62214.9i) q^{91} -112136. q^{93} +37540.0i q^{95} -43926.1i q^{97} +216571. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 1192 q^{9}+O(q^{10}) $ 8 * q + 1192 * q^9	$ 8 q + 1192 q^{9} - 6272 q^{21} - 17048 q^{25} + 9744 q^{29} - 19376 q^{37} + 9800 q^{49} - 102192 q^{53} - 15680 q^{57} - 193344 q^{65} + 25872 q^{77} + 249736 q^{81} + 437376 q^{85} - 194432 q^{93}+O(q^{100}) $ 8 * q + 1192 * q^9 - 6272 * q^21 - 17048 * q^25 + 9744 * q^29 - 19376 * q^37 + 9800 * q^49 - 102192 * q^53 - 15680 * q^57 - 193344 * q^65 + 25872 * q^77 + 249736 * q^81 + 437376 * q^85 - 194432 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	26.7378			1.71523			0.857616	−	0.514290i	$-0.171944\pi$
$3$	26.7378			1.71523			0.857616	+	0.514290i	$0.171944\pi$
$4$		0			0
$5$			100.497i			1.79775i	0.438208	+	0.898874i	$0.355613\pi$
$5$			100.497i			1.79775i	−0.438208	+	0.898874i	$0.644387\pi$
$6$		0			0
$7$	−89.7065	−	93.5935i	−0.691957	−	0.721939i
$7$	−89.7065	−	93.5935i	−0.691957	−	0.721939i
$8$		0			0
$9$	471.912			1.94202
$10$		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
$15$			2687.08i			3.08356i
$16$		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
$21$	−2398.56	−	2502.49i	−1.18687	−	1.23829i
$22$		0			0
$23$			326.004i			0.128500i	0.997934	+	0.0642501i	$0.0204655\pi$
$23$			326.004i			0.128500i	−0.997934	+	0.0642501i	$0.979534\pi$
$24$		0			0
$25$	−6974.68			−2.23190
$26$		0			0
$27$	6120.60			1.61579
$28$		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	$-0.628185\pi$
$31$	−4193.91			−0.783817			−0.391908	+	0.920004i	$0.628185\pi$
$32$		0			0
$33$			12270.6i			1.96147i
$34$		0			0
$35$	9405.88	−	9015.25i	1.29786	−	1.24396i
$36$		0			0
$37$	−8234.41			−0.988845			−0.494423	−	0.869222i	$-0.664620\pi$
$37$	−8234.41			−0.988845			−0.494423	+	0.869222i	$0.664620\pi$
$38$		0			0
$39$			18543.7i			1.95225i
$40$		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	$0.132001\pi$
$43$			9770.29i			0.805817i	−0.915240	+	0.402908i	$0.867999\pi$
$44$		0			0
$45$			47425.8i			3.49127i
$46$		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
$51$		−	21762.4i		−	1.17161i
$52$		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
$57$	9987.73			0.407174
$58$		0			0
$59$	−24859.1			−0.929727			−0.464864	−	0.885382i	$-0.653897\pi$
$59$	−24859.1			−0.929727			−0.464864	+	0.885382i	$0.653897\pi$
$60$		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
$63$	−42333.6	−	44167.9i	−1.34380	−	1.40202i
$64$		0			0
$65$	−69698.6			−2.04616
$66$		0			0
$67$			67423.9i			1.83496i	0.397782	+	0.917480i	$0.369780\pi$
$67$			67423.9i			1.83496i	−0.397782	+	0.917480i	$0.630220\pi$
$68$		0			0
$69$			8716.65i			0.220408i
$70$		0			0
$71$			48772.4i			1.14823i	0.818775	+	0.574114i	$0.194654\pi$
$71$			48772.4i			1.14823i	−0.818775	+	0.574114i	$0.805346\pi$
$72$		0			0
$73$			23068.9i			0.506664i	0.967379	+	0.253332i	$0.0815265\pi$
$73$			23068.9i			0.506664i	−0.967379	+	0.253332i	$0.918473\pi$
$74$		0			0
$75$	−186488.			−3.82822
$76$		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
$81$	48977.1			0.829432
$82$		0			0
$83$	24828.4			0.395598			0.197799	−	0.980243i	$-0.436621\pi$
$83$	24828.4			0.395598			0.197799	+	0.980243i	$0.436621\pi$
$84$		0			0
$85$	81796.6			1.22797
$86$		0			0
$87$	84370.4			1.19507
$88$		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
$93$	−112136.			−1.34443
$94$		0			0
$95$			37540.0i			0.426762i
$96$		0			0
$97$		−	43926.1i		−	0.474017i	−0.971508	−	0.237008i	$-0.923833\pi$
$97$		−	43926.1i		−	0.474017i	0.971508	−	0.237008i	$-0.0761669\pi$
$98$		0			0
$99$			216571.i			2.22081i
$100$		0			0
$101$		−	122368.i		−	1.19362i	−0.802384	−	0.596809i	$-0.796435\pi$
$101$		−	122368.i		−	1.19362i	0.802384	−	0.596809i	$-0.203565\pi$
$102$		0			0
$103$	142684.			1.32520			0.662602	−	0.748972i	$-0.269452\pi$
$103$	142684.			1.32520			0.662602	+	0.748972i	$0.269452\pi$
$104$		0			0
$105$	251493.	−	241048.i	2.22614	−	2.13369i
$106$		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	$-0.372453\pi$
$109$	96768.1			0.780128			0.390064	+	0.920788i	$0.372453\pi$
$110$		0			0
$111$	−220170.			−1.69610
$112$		0			0
$113$	−48289.1			−0.355757			−0.177878	−	0.984052i	$-0.556923\pi$
$113$	−48289.1			−0.355757			−0.177878	+	0.984052i	$0.556923\pi$
$114$		0			0
$115$	−32762.5			−0.231011
$116$		0			0
$117$			327289.i			2.21038i
$118$		0			0
$119$	−76177.5	+	73013.9i	−0.493128	+	0.472648i
$120$		0			0
$121$	−49558.9			−0.307722
$122$		0			0
$123$		−	94018.9i		−	0.560341i
$124$		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
$129$			261236.i			1.38216i
$130$		0			0
$131$	65171.1			0.331800			0.165900	−	0.986143i	$-0.446947\pi$
$131$	65171.1			0.331800			0.165900	+	0.986143i	$0.446947\pi$
$132$		0			0
$133$	−33509.3	−	34961.2i	−0.164262	−	0.171379i
$134$		0			0
$135$			615103.i			2.90478i
$136$		0			0
$137$	31341.5			0.142665			0.0713327	−	0.997453i	$-0.477275\pi$
$137$	31341.5			0.142665			0.0713327	+	0.997453i	$0.477275\pi$
$138$		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
$141$	602860.			2.55369
$142$		0			0
$143$	−318280.			−1.30158
$144$		0			0
$145$			317116.i			1.25256i
$146$		0			0
$147$	−19049.9	+	448979.i	−0.0727109	+	1.71369i
$148$		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
$153$		−	384098.i		−	1.32652i
$154$		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
$159$	−859587.			−2.69648
$160$		0			0
$161$	30511.9	−	29244.7i	0.0927692	−	0.0889166i
$162$		0			0
$163$			152936.i			0.450859i	0.974259	+	0.225430i	$0.0723785\pi$
$163$			152936.i			0.450859i	−0.974259	+	0.225430i	$0.927621\pi$
$164$		0			0
$165$	−1.23316e6			−3.52622
$166$		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
$171$	176279.			0.461011
$172$		0			0
$173$			492322.i			1.25064i	0.780367	+	0.625322i	$0.215032\pi$
$173$			492322.i			1.25064i	−0.780367	+	0.625322i	$0.784968\pi$
$174$		0			0
$175$	625674.	+	652784.i	1.54438	+	1.61129i
$176$		0			0
$177$	−664679.			−1.59470
$178$		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
$183$		−	730382.i		−	1.61221i
$184$		0			0
$185$		−	827535.i		−	1.77769i
$186$		0			0
$187$	373526.			0.781118
$188$		0			0
$189$	−549058.	−	572849.i	−1.11806	−	1.16650i
$190$		0			0
$191$		−	558784.i		−	1.10831i	−0.832414	−	0.554154i	$-0.813042\pi$
$191$		−	558784.i		−	1.10831i	0.832414	−	0.554154i	$-0.186958\pi$
$192$		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
$195$	−1.86359e6			−3.50965
$196$		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	$-0.377176\pi$
$199$	420500.			0.752719			0.376360	+	0.926474i	$0.377176\pi$
$200$		0			0
$201$			1.80277e6i			3.14738i
$202$		0			0
$203$	−283066.	−	295331.i	−0.482112	−	0.503002i
$204$		0			0
$205$	353381.			0.587298
$206$		0			0
$207$			153845.i			0.249550i
$208$		0			0
$209$			171427.i			0.271465i
$210$		0			0
$211$			532870.i			0.823977i	0.911189	+	0.411988i	$0.135166\pi$
$211$			532870.i			0.823977i	−0.911189	+	0.411988i	$0.864834\pi$
$212$		0			0
$213$			1.30407e6i			1.96948i
$214$		0			0
$215$	−981886.			−1.44866
$216$		0			0
$217$	376221.	+	392522.i	0.542367	+	0.565868i
$218$		0			0
$219$			616813.i			0.869047i
$220$		0			0
$221$	564484.			0.777447
$222$		0			0
$223$	202475.			0.272652			0.136326	−	0.990664i	$-0.456471\pi$
$223$	202475.			0.272652			0.136326	+	0.990664i	$0.456471\pi$
$224$		0			0
$225$	−3.29143e6			−4.33440
$226$		0			0
$227$	990163.			1.27539			0.637693	−	0.770290i	$-0.279889\pi$
$227$	990163.			1.27539			0.637693	+	0.770290i	$0.279889\pi$
$228$		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
$231$	1.14845e6	−	1.10075e6i	1.41606	−	1.35725i
$232$		0			0
$233$	1.46530e6			1.76823			0.884113	−	0.467272i	$-0.154763\pi$
$233$	1.46530e6			1.76823			0.884113	+	0.467272i	$0.154763\pi$
$234$		0			0
$235$			2.26592e6i			2.67654i
$236$		0			0
$237$		−	1.67862e6i		−	1.94125i
$238$		0			0
$239$			989159.i			1.12014i	0.828446	+	0.560069i	$0.189225\pi$
$239$			989159.i			1.12014i	−0.828446	+	0.560069i	$0.810775\pi$
$240$		0			0
$241$			577035.i			0.639970i	0.947423	+	0.319985i	$0.103678\pi$
$241$			577035.i			0.639970i	−0.947423	+	0.319985i	$0.896322\pi$
$242$		0			0
$243$	−177764.			−0.193120
$244$		0			0
$245$	−1.68754e6	−	71601.3i	−1.79613	−	0.0762089i
$246$		0			0
$247$			259066.i			0.270189i
$248$		0			0
$249$	663859.			0.678543
$250$		0			0
$251$	−459977.			−0.460842			−0.230421	−	0.973091i	$-0.574010\pi$
$251$	−459977.			−0.460842			−0.230421	+	0.973091i	$0.574010\pi$
$252$		0			0
$253$	−149611.			−0.146947
$254$		0			0
$255$	2.18706e6			2.10626
$256$		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
$261$	1.48910e6			1.35308
$262$		0			0
$263$			1.27642e6i			1.13790i	0.822372	+	0.568951i	$0.192650\pi$
$263$			1.27642e6i			1.13790i	−0.822372	+	0.568951i	$0.807350\pi$
$264$		0			0
$265$		−	3.23085e6i		−	2.82620i
$266$		0			0
$267$		−	743134.i		−	0.637954i
$268$		0			0
$269$		−	161449.i		−	0.136036i	−0.997684	−	0.0680180i	$-0.978332\pi$
$269$		−	161449.i		−	0.136036i	0.997684	−	0.0680180i	$-0.0216675\pi$
$270$		0			0
$271$	939346.			0.776966			0.388483	−	0.921456i	$-0.372999\pi$
$271$	939346.			0.776966			0.388483	+	0.921456i	$0.372999\pi$
$272$		0			0
$273$	1.73557e6	−	1.66349e6i	1.40940	−	1.35087i
$274$		0			0
$275$		−	3.20084e6i		−	2.55230i
$276$		0			0
$277$	−1.06595e6			−0.834715			−0.417357	−	0.908742i	$-0.637044\pi$
$277$	−1.06595e6			−0.834715			−0.417357	+	0.908742i	$0.637044\pi$
$278$		0			0
$279$	−1.97915e6			−1.52219
$280$		0			0
$281$	−556041.			−0.420089			−0.210044	−	0.977692i	$-0.567361\pi$
$281$	−556041.			−0.420089			−0.210044	+	0.977692i	$0.567361\pi$
$282$		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
$285$			1.00374e6i			0.731996i
$286$		0			0
$287$	−329105.	+	315437.i	−0.235847	+	0.226052i
$288$		0			0
$289$	757392.			0.533428
$290$		0			0
$291$		−	1.17449e6i		−	0.813049i
$292$		0			0
$293$		−	634157.i		−	0.431546i	−0.976444	−	0.215773i	$-0.930773\pi$
$293$		−	634157.i		−	0.431546i	0.976444	−	0.215773i	$-0.0692272\pi$
$294$		0			0
$295$		−	2.49827e6i		−	1.67141i
$296$		0			0
$297$			2.80888e6i			1.84775i
$298$		0			0
$299$	−226096.			−0.146257
$300$		0			0
$301$	914435.	−	876459.i	0.581750	−	0.557591i
$302$		0			0
$303$		−	3.27186e6i		−	2.04733i
$304$		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
$309$	3.81506e6			2.27303
$310$		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	$-0.625996\pi$
$313$		−	3.19846e6i		−	1.84535i	0.385573	−	0.922677i	$-0.374004\pi$
$314$		0			0
$315$	4.43874e6	−	4.25440e6i	2.52048	−	2.41581i
$316$		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
$321$			4.27754e6i			2.31703i
$322$		0			0
$323$		−	304034.i		−	0.162150i
$324$		0			0
$325$		−	4.83720e6i		−	2.54030i
$326$		0			0
$327$	2.58737e6			1.33810
$328$		0			0
$329$	−2.02262e6	−	2.11026e6i	−1.03021	−	1.07485i
$330$		0			0
$331$		−	257593.i		−	0.129230i	−0.997910	−	0.0646152i	$-0.979418\pi$
$331$		−	257593.i		−	0.129230i	0.997910	−	0.0646152i	$-0.0205820\pi$
$332$		0			0
$333$	−3.88592e6			−1.92036
$334$		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
$339$	−1.29115e6			−0.610206
$340$		0			0
$341$		−	1.92468e6i		−	0.896339i
$342$		0			0
$343$	1.63552e6	−	1.43966e6i	0.750623	−	0.660731i
$344$		0			0
$345$	−875998.			−0.396237
$346$		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
$351$			4.24487e6i			1.83906i
$352$		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
$357$	−2.03682e6	+	1.95223e6i	−0.845829	+	0.810702i
$358$		0			0
$359$			2.62552e6i			1.07518i	0.843208	+	0.537588i	$0.180664\pi$
$359$			2.62552e6i			1.07518i	−0.843208	+	0.537588i	$0.819336\pi$
$360$		0			0
$361$	−2.33656e6			−0.943647
$362$		0			0
$363$	−1.32510e6			−0.527814
$364$		0			0
$365$	−2.31836e6			−0.910854
$366$		0			0
$367$	3.36188e6			1.30292			0.651459	−	0.758684i	$-0.274157\pi$
$367$	3.36188e6			1.30292			0.651459	+	0.758684i	$0.274157\pi$
$368$		0			0
$369$		−	1.65940e6i		−	0.634431i
$370$		0			0
$371$	2.88395e6	+	3.00891e6i	1.08781	+	1.13494i
$372$		0			0
$373$	−2.07818e6			−0.773412			−0.386706	−	0.922203i	$-0.626387\pi$
$373$	−2.07818e6			−0.773412			−0.386706	+	0.922203i	$0.626387\pi$
$374$		0			0
$375$		−	1.03444e7i		−	3.79862i
$376$		0			0
$377$			2.18844e6i			0.793014i
$378$		0			0
$379$		−	1.84637e6i		−	0.660267i	−0.943934	−	0.330134i	$-0.892906\pi$
$379$		−	1.84637e6i		−	0.660267i	0.943934	−	0.330134i	$-0.107094\pi$
$380$		0			0
$381$			6.70764e6i			2.36732i
$382$		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
$387$			4.61072e6i			1.56492i
$388$		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
$393$	1.74253e6			0.569115
$394$		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
$399$	−895965.	−	934787.i	−0.281747	−	0.293955i
$400$		0			0
$401$	739360.			0.229612			0.114806	−	0.993388i	$-0.463375\pi$
$401$	739360.			0.229612			0.114806	+	0.993388i	$0.463375\pi$
$402$		0			0
$403$		−	2.90863e6i		−	0.892126i
$404$		0			0
$405$			4.92206e6i			1.49111i
$406$		0			0
$407$		−	3.77896e6i		−	1.13080i
$408$		0			0
$409$			2.17984e6i			0.644340i	0.946682	+	0.322170i	$0.104412\pi$
$409$			2.17984e6i			0.644340i	−0.946682	+	0.322170i	$0.895588\pi$
$410$		0			0
$411$	838005.			0.244704
$412$		0			0
$413$	2.23002e6	+	2.32665e6i	0.643331	+	0.671206i
$414$		0			0
$415$			2.49519e6i			0.711186i
$416$		0			0
$417$	−6.67191e6			−1.87893
$418$		0			0
$419$	−3.07822e6			−0.856572			−0.428286	−	0.903643i	$-0.640883\pi$
$419$	−3.07822e6			−0.856572			−0.428286	+	0.903643i	$0.640883\pi$
$420$		0			0
$421$	4.99144e6			1.37253			0.686264	−	0.727353i	$-0.259250\pi$
$421$	4.99144e6			1.37253			0.686264	+	0.727353i	$0.259250\pi$
$422$		0			0
$423$	1.06402e7			2.89135
$424$		0			0
$425$			5.67683e6i			1.52452i
$426$		0			0
$427$	−2.55664e6	+	2.45046e6i	−0.678578	+	0.650396i
$428$		0			0
$429$	−8.51012e6			−2.23250
$430$		0			0
$431$			1.07037e6i			0.277551i	0.990324	+	0.138775i	$0.0443166\pi$
$431$			1.07037e6i			0.277551i	−0.990324	+	0.138775i	$0.955683\pi$
$432$		0			0
$433$		−	2.89323e6i		−	0.741590i	−0.928715	−	0.370795i	$-0.879085\pi$
$433$		−	2.89323e6i		−	0.741590i	0.928715	−	0.370795i	$-0.120915\pi$
$434$		0			0
$435$			8.47899e6i			2.14843i
$436$		0			0
$437$			121777.i			0.0305043i
$438$		0			0
$439$	1.15097e6			0.285038			0.142519	−	0.989792i	$-0.454480\pi$
$439$	1.15097e6			0.285038			0.142519	+	0.989792i	$0.454480\pi$
$440$		0			0
$441$	−336223.	+	7.92429e6i	−0.0823249	+	1.94028i
$442$		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
$447$	1.92578e6			0.455866
$448$		0			0
$449$	−1.77009e6			−0.414362			−0.207181	−	0.978303i	$-0.566429\pi$
$449$	−1.77009e6			−0.414362			−0.207181	+	0.978303i	$0.566429\pi$
$450$		0			0
$451$	1.61372e6			0.373583
$452$		0			0
$453$		−	4.46691e6i		−	1.02273i
$454$		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
$459$		−	4.98168e6i		−	1.10368i
$460$		0			0
$461$		−	2.88921e6i		−	0.633180i	−0.948563	−	0.316590i	$-0.897462\pi$
$461$		−	2.88921e6i		−	0.633180i	0.948563	−	0.316590i	$-0.102538\pi$
$462$		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
$465$		−	1.12693e7i		−	2.41694i
$466$		0			0
$467$	2.88406e6			0.611946			0.305973	−	0.952040i	$-0.401018\pi$
$467$	2.88406e6			0.611946			0.305973	+	0.952040i	$0.401018\pi$
$468$		0			0
$469$	6.31043e6	−	6.04836e6i	1.32473	−	1.26971i
$470$		0			0
$471$		−	1.61707e6i		−	0.335873i
$472$		0			0
$473$	−4.48381e6			−0.921497
$474$		0			0
$475$	−2.60534e6			−0.529823
$476$		0			0
$477$	−1.51714e7			−3.05301
$478$		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
$483$	815821.	−	781940.i	0.159121	−	0.152513i
$484$		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
$489$			4.08918e6i			0.773328i
$490$		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
$495$	−2.17648e7			−3.99246
$496$		0			0
$497$	4.56478e6	−	4.37520e6i	0.828951	−	0.794525i
$498$		0			0
$499$			6.47634e6i			1.16434i	0.813069	+	0.582168i	$0.197795\pi$
$499$			6.47634e6i			1.16434i	−0.813069	+	0.582168i	$0.802205\pi$
$500$		0			0
$501$	−7.64207e6			−1.36024
$502$		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
$507$	−2.93318e6			−0.506779
$508$		0			0
$509$		−	2.93740e6i		−	0.502538i	−0.967917	−	0.251269i	$-0.919152\pi$
$509$		−	2.93740e6i		−	0.502538i	0.967917	−	0.251269i	$-0.0808479\pi$
$510$		0			0
$511$	2.15910e6	−	2.06943e6i	0.365781	−	0.350590i
$512$		0			0
$513$	2.28631e6			0.383567
$514$		0			0
$515$			1.43393e7i			2.38238i
$516$		0			0
$517$			1.03474e7i			1.70256i
$518$		0			0
$519$			1.31636e7i			2.14515i
$520$		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
$525$	1.67292e7	+	1.74540e7i	2.64896	+	2.76374i
$526$		0			0
$527$			3.41350e6i			0.535394i
$528$		0			0
$529$	6.33006e6			0.983488
$530$		0			0
$531$	−1.17313e7			−1.80555
$532$		0			0
$533$	2.43870e6			0.371827
$534$		0			0
$535$	−1.60776e7			−2.42849
$536$		0			0
$537$		−	1.09859e7i		−	1.64400i
$538$		0			0
$539$	−7.70618e6	−	326969.i	−1.14253	−	0.0484768i
$540$		0			0
$541$	1.48725e6			0.218469			0.109235	−	0.994016i	$-0.465160\pi$
$541$	1.48725e6			0.218469			0.109235	+	0.994016i	$0.465160\pi$
$542$		0			0
$543$			1.06544e7i			1.55071i
$544$		0			0
$545$			9.72492e6i			1.40247i
$546$		0			0
$547$		−	1.35405e7i		−	1.93494i	−0.252986	−	0.967470i	$-0.581413\pi$
$547$		−	1.35405e7i		−	1.93494i	0.252986	−	0.967470i	$-0.418587\pi$
$548$		0			0
$549$		−	1.28909e7i		−	1.82538i
$550$		0			0
$551$	1.17870e6			0.165396
$552$		0			0
$553$	−5.87587e6	+	5.63185e6i	−0.817070	+	0.783137i
$554$		0			0
$555$		−	2.21265e7i		−	3.04916i
$556$		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
$561$	9.98728e6			1.33980
$562$		0			0
$563$	−7.35710e6			−0.978219			−0.489109	−	0.872223i	$-0.662678\pi$
$563$	−7.35710e6			−0.978219			−0.489109	+	0.872223i	$0.662678\pi$
$564$		0			0
$565$		−	4.85292e6i		−	0.639561i
$566$		0			0
$567$	−4.39357e6	−	4.58394e6i	−0.573931	−	0.598799i
$568$		0			0
$569$	1.25324e7			1.62275			0.811376	−	0.584525i	$-0.198719\pi$
$569$	1.25324e7			1.62275			0.811376	+	0.584525i	$0.198719\pi$
$570$		0			0
$571$			5.52022e6i			0.708543i	0.935143	+	0.354272i	$0.115271\pi$
$571$			5.52022e6i			0.708543i	−0.935143	+	0.354272i	$0.884729\pi$
$572$		0			0
$573$		−	1.49407e7i		−	1.90101i
$574$		0			0
$575$		−	2.27377e6i		−	0.286799i
$576$		0			0
$577$			1.31746e7i			1.64740i	0.567027	+	0.823699i	$0.308094\pi$
$577$			1.31746e7i			1.64740i	−0.567027	+	0.823699i	$0.691906\pi$
$578$		0			0
$579$	1.71341e7			2.12405
$580$		0			0
$581$	−2.22727e6	−	2.32378e6i	−0.273737	−	0.285598i
$582$		0			0
$583$		−	1.47538e7i		−	1.79776i
$584$		0			0
$585$	−3.28916e7			−3.97370
$586$		0			0
$587$	4.69162e6			0.561989			0.280994	−	0.959709i	$-0.409336\pi$
$587$	4.69162e6			0.561989			0.280994	+	0.959709i	$0.409336\pi$
$588$		0			0
$589$	−1.56661e6			−0.186068
$590$		0			0
$591$	1.01246e7			1.19237
$592$		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
$597$	1.12432e7			1.29109
$598$		0			0
$599$			1.37105e7i			1.56130i	0.624969	+	0.780650i	$0.285112\pi$
$599$			1.37105e7i			1.56130i	−0.624969	+	0.780650i	$0.714888\pi$
$600$		0			0
$601$		−	1.61956e7i		−	1.82899i	−0.404601	−	0.914493i	$-0.632590\pi$
$601$		−	1.61956e7i		−	1.82899i	0.404601	−	0.914493i	$-0.367410\pi$
$602$		0			0
$603$			3.18181e7i			3.56354i
$604$		0			0
$605$		−	4.98053e6i		−	0.553206i
$606$		0			0
$607$	925000.			0.101899			0.0509495	−	0.998701i	$-0.483775\pi$
$607$	925000.			0.101899			0.0509495	+	0.998701i	$0.483775\pi$
$608$		0			0
$609$	−7.56858e6	−	7.89652e6i	−0.826935	−	0.862765i
$610$		0			0
$611$			1.56372e7i			1.69456i
$612$		0			0
$613$	3.80091e6			0.408541			0.204271	−	0.978914i	$-0.434518\pi$
$613$	3.80091e6			0.408541			0.204271	+	0.978914i	$0.434518\pi$
$614$		0			0
$615$	9.44864e6			1.00735
$616$		0			0
$617$	1.13629e7			1.20164			0.600822	−	0.799383i	$-0.294840\pi$
$617$	1.13629e7			1.20164			0.600822	+	0.799383i	$0.294840\pi$
$618$		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
$621$			1.99534e6i			0.207629i
$622$		0			0
$623$	−2.60128e6	+	2.49325e6i	−0.268514	+	0.257362i
$624$		0			0
$625$	1.70846e7			1.74947
$626$		0			0
$627$			4.58360e6i			0.465626i
$628$		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
$633$			1.42478e7i			1.41331i
$634$		0			0
$635$	−2.52114e7			−2.48121
$636$		0			0
$637$	−1.16458e7	−	494125.i	−1.13716	−	0.0482490i
$638$		0			0
$639$			2.30163e7i			2.22989i
$640$		0			0
$641$	−1.00849e7			−0.969452			−0.484726	−	0.874666i	$-0.661081\pi$
$641$	−1.00849e7			−0.969452			−0.484726	+	0.874666i	$0.661081\pi$
$642$		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
$645$	−2.62535e7			−2.48478
$646$		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
$651$	1.00593e7	+	1.04952e7i	0.930287	+	0.970595i
$652$		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
$657$			1.08865e7i			0.983954i
$658$		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
$663$	1.50931e7			1.33350
$664$		0			0
$665$	3.51350e6	−	3.36759e6i	0.308096	−	0.295301i
$666$		0			0
$667$			1.02870e6i			0.0895309i
$668$		0			0
$669$	5.41373e6			0.467661
$670$		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
$675$	−4.26892e7			−3.60628
$676$		0			0
$677$			1.94176e7i			1.62826i	0.580683	+	0.814130i	$0.302785\pi$
$677$			1.94176e7i			1.62826i	−0.580683	+	0.814130i	$0.697215\pi$
$678$		0			0
$679$	−4.11120e6	+	3.94046e6i	−0.342211	+	0.327999i
$680$		0			0
$681$	2.64748e7			2.18758
$682$		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
$687$			1.41550e7i			1.14424i
$688$		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
$693$	2.02696e7	−	1.94278e7i	1.60329	−	1.53671i
$694$		0			0
$695$		−	2.50771e7i		−	1.96932i
$696$		0			0
$697$	−2.86201e6			−0.223146
$698$		0			0
$699$	3.91791e7			3.03292
$700$		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
$705$			6.05857e7i			4.59089i
$706$		0			0
$707$	−1.14529e7	+	1.09772e7i	−0.861719	+	0.825932i
$708$		0			0
$709$	4.21885e6			0.315195			0.157597	−	0.987503i	$-0.449625\pi$
$709$	4.21885e6			0.315195			0.157597	+	0.987503i	$0.449625\pi$
$710$		0			0
$711$		−	2.96270e7i		−	2.19793i
$712$		0			0
$713$		−	1.36723e6i		−	0.100721i
$714$		0			0
$715$		−	3.19862e7i		−	2.33990i
$716$		0			0
$717$			2.64480e7i			1.92130i
$718$		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
$723$			1.54287e7i			1.09770i
$724$		0			0
$725$	−2.20084e7			−1.55505
$726$		0			0
$727$	−9.39018e6			−0.658928			−0.329464	−	0.944168i	$-0.606868\pi$
$727$	−9.39018e6			−0.658928			−0.329464	+	0.944168i	$0.606868\pi$
$728$		0			0
$729$	−1.66545e7			−1.16068
$730$		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
$735$	−4.51211e7	−	1.91446e6i	−3.08078	−	0.130716i
$736$		0			0
$737$	−3.09423e7			−2.09838
$738$		0			0
$739$		−	8.27786e6i		−	0.557580i	−0.960352	−	0.278790i	$-0.910067\pi$
$739$		−	8.27786e6i		−	0.557580i	0.960352	−	0.278790i	$-0.0899333\pi$
$740$		0			0
$741$			6.92687e6i			0.463438i
$742$		0			0
$743$			861253.i			0.0572346i	0.999590	+	0.0286173i	$0.00911042\pi$
$743$			861253.i			0.0572346i	−0.999590	+	0.0286173i	$0.990890\pi$
$744$		0			0
$745$			7.23824e6i			0.477796i
$746$		0			0
$747$	1.17168e7			0.768261
$748$		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
$753$	−1.22988e7			−0.790451
$754$		0			0
$755$	1.67894e7			1.07193
$756$		0			0
$757$	−2.73727e7			−1.73611			−0.868056	−	0.496466i	$-0.834631\pi$
$757$	−2.73727e7			−1.73611			−0.868056	+	0.496466i	$0.834631\pi$
$758$		0			0
$759$	−4.00027e6			−0.252049
$760$		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
$765$	3.86008e7			2.38475
$766$		0			0
$767$		−	1.72407e7i		−	1.05820i
$768$		0			0
$769$			1.72452e7i			1.05160i	0.850607	+	0.525802i	$0.176235\pi$
$769$			1.72452e7i			1.05160i	−0.850607	+	0.525802i	$0.823765\pi$
$770$		0			0
$771$		−	1.05981e6i		−	0.0642085i
$772$		0			0
$773$			2.29735e6i			0.138286i	0.997607	+	0.0691430i	$0.0220265\pi$
$773$			2.29735e6i			0.138286i	−0.997607	+	0.0691430i	$0.977974\pi$
$774$		0			0
$775$	2.92511e7			1.74940
$776$		0			0
$777$	1.97507e7	+	2.06065e7i	1.17363	+	1.22448i
$778$		0			0
$779$		−	1.31350e6i		−	0.0775508i
$780$		0			0
$781$	−2.23828e7			−1.31306
$782$		0			0
$783$	1.93134e7			1.12578
$784$		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
$789$			3.41287e7i			1.95177i
$790$		0			0
$791$	4.33185e6	+	4.51955e6i	0.246169	+	0.256835i
$792$		0			0
$793$	1.89450e7			1.06982
$794$		0			0
$795$		−	8.63860e7i		−	4.84759i
$796$		0			0
$797$		−	1.53351e7i		−	0.855149i	−0.903980	−	0.427575i	$-0.859368\pi$
$797$		−	1.53351e7i		−	0.855149i	0.903980	−	0.427575i	$-0.140632\pi$
$798$		0			0
$799$		−	1.83515e7i		−	1.01696i
$800$		0			0
$801$		−	1.31160e7i		−	0.722305i
$802$		0			0
$803$	−1.05869e7			−0.579399
$804$		0			0
$805$	2.93901e6	+	3.06635e6i	0.159850	+	0.166776i
$806$		0			0
$807$		−	4.31679e6i		−	0.233333i
$808$		0			0
$809$	9.14411e6			0.491213			0.245607	−	0.969370i	$-0.421013\pi$
$809$	9.14411e6			0.491213			0.245607	+	0.969370i	$0.421013\pi$
$810$		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
$813$	2.51161e7			1.33268
$814$		0			0
$815$	−1.53696e7			−0.810531
$816$		0			0
$817$			3.64963e6i			0.191290i
$818$		0			0
$819$	3.06321e7	−	2.93599e7i	1.59576	−	1.52949i
$820$		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
$825$		−	8.55834e7i		−	4.37779i
$826$		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
$831$	−2.85012e7			−1.43173
$832$		0			0
$833$	1.36672e7	+	579894.i	0.682446	+	0.0289558i
$834$		0			0
$835$		−	2.87236e7i		−	1.42568i
$836$		0			0
$837$	−2.56692e7			−1.26648
$838$		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
$843$	−1.48673e7			−0.720550
$844$		0			0
$845$		−	1.10247e7i		−	0.531159i
$846$		0			0
$847$	4.44576e6	+	4.63839e6i	0.212930	+	0.222156i
$848$		0			0
$849$	6.27602e7			2.98824
$850$		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
$855$			1.77156e7i			0.828781i
$856$		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
$861$	−8.79956e6	+	8.43412e6i	−0.404532	+	0.387732i
$862$		0			0
$863$			4.00309e6i			0.182965i	0.995807	+	0.0914826i	$0.0291606\pi$
$863$			4.00309e6i			0.182965i	−0.995807	+	0.0914826i	$0.970839\pi$
$864$		0			0
$865$	−4.94769e7			−2.24834
$866$		0			0
$867$	2.02510e7			0.914954
$868$		0			0
$869$	2.88115e7			1.29425
$870$		0			0
$871$	−4.67610e7			−2.08852
$872$		0			0
$873$		−	2.07293e7i		−	0.920551i
$874$		0			0
$875$	−3.62096e7	+	3.47058e7i	−1.59883	+	1.53243i
$876$		0			0
$877$	1.53480e7			0.673835			0.336917	−	0.941534i	$-0.390616\pi$
$877$	1.53480e7			0.673835			0.336917	+	0.941534i	$0.390616\pi$
$878$		0			0
$879$		−	1.69560e7i		−	0.740203i
$880$		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	$0.0302367\pi$
$883$			4.39504e6i			0.189697i	−0.995492	+	0.0948487i	$0.969763\pi$
$884$		0			0
$885$		−	6.67983e7i		−	2.86687i
$886$		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
$891$			2.24767e7i			0.948503i
$892$		0			0
$893$	8.42230e6			0.353429
$894$		0			0
$895$	4.12918e7			1.72308
$896$		0			0
$897$	−6.04532e6			−0.250864
$898$		0			0
$899$	−1.32338e7			−0.546115
$900$		0			0
$901$			2.61665e7i			1.07382i
$902$		0			0
$903$	2.44500e7	−	2.34346e7i	0.997838	−	0.956398i
$904$		0			0
$905$	−4.00458e7			−1.62531
$906$		0			0
$907$			4.64251e7i			1.87385i	0.349529	+	0.936925i	$0.386341\pi$
$907$			4.64251e7i			1.87385i	−0.349529	+	0.936925i	$0.613659\pi$
$908$		0			0
$909$		−	5.77470e7i		−	2.31803i
$910$		0			0
$911$			1.27993e7i			0.510964i	0.966814	+	0.255482i	$0.0822342\pi$
$911$			1.27993e7i			0.510964i	−0.966814	+	0.255482i	$0.917766\pi$
$912$		0			0
$913$			1.13943e7i			0.452389i
$914$		0			0
$915$	7.34013e7			2.89835
$916$		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
$921$	9.31606e6			0.361895
$922$		0			0
$923$	−3.38255e7			−1.30689
$924$		0			0
$925$	5.74324e7			2.20700
$926$		0			0
$927$	6.73343e7			2.57358
$928$		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
$933$	2.69221e6			0.101252
$934$		0			0
$935$			3.75383e7i			1.40425i
$936$		0			0
$937$			4.02297e7i			1.49692i	0.663180	+	0.748460i	$0.269206\pi$
$937$			4.02297e7i			1.49692i	−0.663180	+	0.748460i	$0.730794\pi$
$938$		0			0
$939$		−	8.55198e7i		−	3.16521i
$940$		0			0
$941$		−	4.20545e7i		−	1.54824i	−0.633038	−	0.774121i	$-0.718192\pi$
$941$		−	4.20545e7i		−	1.54824i	0.633038	−	0.774121i	$-0.281808\pi$
$942$		0			0
$943$	1.14634e6			0.0419791
$944$		0			0
$945$	5.75696e7	−	5.51788e7i	2.09708	−	2.00998i
$946$		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
$951$	9.49144e7			3.40315
$952$		0			0
$953$	−54340.4			−0.00193816			−0.000969081	−	1.00000i	$-0.500308\pi$
$953$	−54340.4			−0.00193816			−0.000969081		1.00000i	$0.500308\pi$
$954$		0			0
$955$	5.61562e7			1.99246
$956$		0			0
$957$			3.87195e7i			1.36663i
$958$		0			0
$959$	−2.81154e6	−	2.93336e6i	−0.0987183	−	0.102996i
$960$		0			0
$961$	−1.10403e7			−0.385631
$962$		0			0
$963$			7.54968e7i			2.62339i
$964$		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
$969$		−	8.12921e6i		−	0.278124i
$970$		0			0
$971$	−2.84157e7			−0.967186			−0.483593	−	0.875293i	$-0.660668\pi$
$971$	−2.84157e7			−0.967186			−0.483593	+	0.875293i	$0.660668\pi$
$972$		0			0
$973$	2.23845e7	+	2.33545e7i	0.757995	+	0.790838i
$974$		0			0
$975$		−	1.29336e8i		−	4.35721i
$976$		0			0
$977$	−2.57873e7			−0.864308			−0.432154	−	0.901800i	$-0.642246\pi$
$977$	−2.57873e7			−0.864308			−0.432154	+	0.901800i	$0.642246\pi$
$978$		0			0
$979$	1.27550e7			0.425328
$980$		0			0
$981$	4.56660e7			1.51503
$982$		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
$987$	−5.40805e7	−	5.64237e7i	−1.76705	−	1.84361i
$988$		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
$993$		−	6.88748e6i		−	0.221660i
$994$		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
$999$	−5.03996e7			−1.59777

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.6.f.a.111.1	✓	8	8.5	even	2
112.6.f.a.111.2	yes	8	56.27	even	2
112.6.f.a.111.7	yes	8	8.3	odd	2
112.6.f.a.111.8	yes	8	56.13	odd	2
448.6.f.b.447.1		8	7.6	odd	2	inner
448.6.f.b.447.2		8	4.3	odd	2	inner
448.6.f.b.447.7		8	28.27	even	2	inner
448.6.f.b.447.8		8	1.1	even	1	trivial
1008.6.b.g.559.1		8	168.83	odd	2
1008.6.b.g.559.2		8	168.125	even	2
1008.6.b.g.559.7		8	24.5	odd	2
1008.6.b.g.559.8		8	24.11	even	2

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	448.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+26.7378 q^{3} +100.497i q^{5} +(-89.7065 - 93.5935i) q^{7} +471.912 q^{9} +O(q^{10})\)	\(q+26.7378 q^{3} +100.497i q^{5} +(-89.7065 - 93.5935i) q^{7} +471.912 q^{9} +458.923i q^{11} +693.538i q^{13} +2687.08i q^{15} -813.920i q^{17} +373.543 q^{19} +(-2398.56 - 2502.49i) q^{21} +326.004i q^{23} -6974.68 q^{25} +6120.60 q^{27} +3155.47 q^{29} -4193.91 q^{31} +12270.6i q^{33} +(9405.88 - 9015.25i) q^{35} -8234.41 q^{37} +18543.7i q^{39} -3516.33i q^{41} +9770.29i q^{43} +47425.8i q^{45} +22547.1 q^{47} +(-712.471 + 16791.9i) q^{49} -21762.4i q^{51} -32148.7 q^{53} -46120.4 q^{55} +9987.73 q^{57} -24859.1 q^{59} -27316.4i q^{61} +(-42333.6 - 44167.9i) q^{63} -69698.6 q^{65} +67423.9i q^{67} +8716.65i q^{69} +48772.4i q^{71} +23068.9i q^{73} -186488. q^{75} +(42952.1 - 41168.4i) q^{77} -62780.8i q^{79} +48977.1 q^{81} +24828.4 q^{83} +81796.6 q^{85} +84370.4 q^{87} -27793.4i q^{89} +(64910.6 - 62214.9i) q^{91} -112136. q^{93} +37540.0i q^{95} -43926.1i q^{97} +216571. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 1192 q^{9}+O(q^{10}) \) 8 * q + 1192 * q^9	\( 8 q + 1192 q^{9} - 6272 q^{21} - 17048 q^{25} + 9744 q^{29} - 19376 q^{37} + 9800 q^{49} - 102192 q^{53} - 15680 q^{57} - 193344 q^{65} + 25872 q^{77} + 249736 q^{81} + 437376 q^{85} - 194432 q^{93}+O(q^{100}) \) 8 * q + 1192 * q^9 - 6272 * q^21 - 17048 * q^25 + 9744 * q^29 - 19376 * q^37 + 9800 * q^49 - 102192 * q^53 - 15680 * q^57 - 193344 * q^65 + 25872 * q^77 + 249736 * q^81 + 437376 * q^85 - 194432 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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