LMFDB - Embedded newform 392.6.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,6,Mod(1,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("392.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	392.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$62.8704573667$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 328x^{4} - 1328x^{3} + 25933x^{2} + 205840x + 390334 $ x^6 - 328x^4 - 1328x^3 + 25933x^2 + 205840x + 390334
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$12.9780$ of defining polynomial
Character	$\chi$	$=$	392.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-17.5252 q^{3} +11.0064 q^{5} +64.1343 q^{9} +O(q^{10})$	$q-17.5252 q^{3} +11.0064 q^{5} +64.1343 q^{9} -95.0461 q^{11} +157.485 q^{13} -192.890 q^{15} -100.530 q^{17} +664.804 q^{19} +2141.54 q^{23} -3003.86 q^{25} +3134.67 q^{27} -3050.77 q^{29} +3084.25 q^{31} +1665.71 q^{33} +11872.7 q^{37} -2759.96 q^{39} +4013.50 q^{41} -10366.6 q^{43} +705.888 q^{45} +22647.4 q^{47} +1761.81 q^{51} -27712.6 q^{53} -1046.12 q^{55} -11650.9 q^{57} +23383.9 q^{59} -10242.6 q^{61} +1733.34 q^{65} +6555.18 q^{67} -37531.0 q^{69} -21269.5 q^{71} +313.509 q^{73} +52643.4 q^{75} -59231.2 q^{79} -70520.4 q^{81} -107561. q^{83} -1106.47 q^{85} +53465.4 q^{87} -88357.3 q^{89} -54052.3 q^{93} +7317.11 q^{95} -164187. q^{97} -6095.71 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 122 q^{9}+O(q^{10}) $ 6 * q - 122 * q^9	$ 6 q - 122 q^{9} + 248 q^{11} - 376 q^{15} - 2904 q^{23} + 1510 q^{25} - 408 q^{29} - 18648 q^{37} - 15240 q^{39} - 10584 q^{43} + 3016 q^{51} - 66772 q^{53} - 56264 q^{57} - 55316 q^{65} - 169632 q^{67} - 54880 q^{71} + 50576 q^{79} - 270194 q^{81} - 406076 q^{85} - 164944 q^{93} - 551912 q^{95} - 173512 q^{99}+O(q^{100}) $ 6 * q - 122 * q^9 + 248 * q^11 - 376 * q^15 - 2904 * q^23 + 1510 * q^25 - 408 * q^29 - 18648 * q^37 - 15240 * q^39 - 10584 * q^43 + 3016 * q^51 - 66772 * q^53 - 56264 * q^57 - 55316 * q^65 - 169632 * q^67 - 54880 * q^71 + 50576 * q^79 - 270194 * q^81 - 406076 * q^85 - 164944 * q^93 - 551912 * q^95 - 173512 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−17.5252	−1.12425	−0.562123	−	0.827054i	$-0.690015\pi$
$3$	−17.5252	−1.12425	−0.562123	+	0.827054i	$0.690015\pi$
$4$	0	0
$5$	11.0064	0.196889	0.0984443	−	0.995143i	$-0.468613\pi$
$5$	11.0064	0.196889	0.0984443	+	0.995143i	$0.468613\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	64.1343	0.263927
$10$	0	0
$11$	−95.0461	−0.236839	−0.118419	−	0.992964i	$-0.537783\pi$
$11$	−95.0461	−0.236839	−0.118419	+	0.992964i	$0.537783\pi$
$12$	0	0
$13$	157.485	0.258452	0.129226	−	0.991615i	$-0.458751\pi$
$13$	157.485	0.258452	0.129226	+	0.991615i	$0.458751\pi$
$14$	0	0
$15$	−192.890	−0.221351
$16$	0	0
$17$	−100.530	−0.0843669	−0.0421834	−	0.999110i	$-0.513431\pi$
$17$	−100.530	−0.0843669	−0.0421834	+	0.999110i	$0.513431\pi$
$18$	0	0
$19$	664.804	0.422483	0.211242	−	0.977434i	$-0.432249\pi$
$19$	664.804	0.422483	0.211242	+	0.977434i	$0.432249\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2141.54	0.844125	0.422062	−	0.906567i	$-0.361306\pi$
$23$	2141.54	0.844125	0.422062	+	0.906567i	$0.361306\pi$
$24$	0	0
$25$	−3003.86	−0.961235
$26$	0	0
$27$	3134.67	0.827526
$28$	0	0
$29$	−3050.77	−0.673619	−0.336809	−	0.941573i	$-0.609348\pi$
$29$	−3050.77	−0.673619	−0.336809	+	0.941573i	$0.609348\pi$
$30$	0	0
$31$	3084.25	0.576429	0.288215	−	0.957566i	$-0.406938\pi$
$31$	3084.25	0.576429	0.288215	+	0.957566i	$0.406938\pi$
$32$	0	0
$33$	1665.71	0.266265
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11872.7	1.42576	0.712879	−	0.701287i	$-0.247391\pi$
$37$	11872.7	1.42576	0.712879	+	0.701287i	$0.247391\pi$
$38$	0	0
$39$	−2759.96	−0.290564
$40$	0	0
$41$	4013.50	0.372875	0.186438	−	0.982467i	$-0.440306\pi$
$41$	4013.50	0.372875	0.186438	+	0.982467i	$0.440306\pi$
$42$	0	0
$43$	−10366.6	−0.854995	−0.427497	−	0.904017i	$-0.640605\pi$
$43$	−10366.6	−0.854995	−0.427497	+	0.904017i	$0.640605\pi$
$44$	0	0
$45$	705.888	0.0519642
$46$	0	0
$47$	22647.4	1.49546	0.747729	−	0.664004i	$-0.231144\pi$
$47$	22647.4	1.49546	0.747729	+	0.664004i	$0.231144\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1761.81	0.0948491
$52$	0	0
$53$	−27712.6	−1.35515	−0.677576	−	0.735453i	$-0.736969\pi$
$53$	−27712.6	−1.35515	−0.677576	+	0.735453i	$0.736969\pi$
$54$	0	0
$55$	−1046.12	−0.0466308
$56$	0	0
$57$	−11650.9	−0.474975
$58$	0	0
$59$	23383.9	0.874553	0.437276	−	0.899327i	$-0.355943\pi$
$59$	23383.9	0.874553	0.437276	+	0.899327i	$0.355943\pi$
$60$	0	0
$61$	−10242.6	−0.352440	−0.176220	−	0.984351i	$-0.556387\pi$
$61$	−10242.6	−0.352440	−0.176220	+	0.984351i	$0.556387\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1733.34	0.0508863
$66$	0	0
$67$	6555.18	0.178401	0.0892006	−	0.996014i	$-0.471569\pi$
$67$	6555.18	0.178401	0.0892006	+	0.996014i	$0.471569\pi$
$68$	0	0
$69$	−37531.0	−0.949003
$70$	0	0
$71$	−21269.5	−0.500739	−0.250370	−	0.968150i	$-0.580552\pi$
$71$	−21269.5	−0.500739	−0.250370	+	0.968150i	$0.580552\pi$
$72$	0	0
$73$	313.509	0.00688562	0.00344281	−	0.999994i	$-0.498904\pi$
$73$	313.509	0.00688562	0.00344281	+	0.999994i	$0.498904\pi$
$74$	0	0
$75$	52643.4	1.08066
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−59231.2	−1.06778	−0.533891	−	0.845553i	$-0.679271\pi$
$79$	−59231.2	−1.06778	−0.533891	+	0.845553i	$0.679271\pi$
$80$	0	0
$81$	−70520.4	−1.19427
$82$	0	0
$83$	−107561.	−1.71380	−0.856901	−	0.515481i	$-0.827613\pi$
$83$	−107561.	−1.71380	−0.856901	+	0.515481i	$0.827613\pi$
$84$	0	0
$85$	−1106.47	−0.0166109
$86$	0	0
$87$	53465.4	0.757312
$88$	0	0
$89$	−88357.3	−1.18241	−0.591204	−	0.806522i	$-0.701347\pi$
$89$	−88357.3	−1.18241	−0.591204	+	0.806522i	$0.701347\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−54052.3	−0.648048
$94$	0	0
$95$	7317.11	0.0831822
$96$	0	0
$97$	−164187.	−1.77178	−0.885891	−	0.463894i	$-0.846452\pi$
$97$	−164187.	−1.77178	−0.885891	+	0.463894i	$0.846452\pi$
$98$	0	0
$99$	−6095.71	−0.0625081
$100$	0	0
$101$	29974.5	0.292381	0.146190	−	0.989256i	$-0.453299\pi$
$101$	29974.5	0.292381	0.146190	+	0.989256i	$0.453299\pi$
$102$	0	0
$103$	81692.8	0.758736	0.379368	−	0.925246i	$-0.376141\pi$
$103$	81692.8	0.758736	0.379368	+	0.925246i	$0.376141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−56710.6	−0.478856	−0.239428	−	0.970914i	$-0.576960\pi$
$107$	−56710.6	−0.478856	−0.239428	+	0.970914i	$0.576960\pi$
$108$	0	0
$109$	−185773.	−1.49767	−0.748837	−	0.662754i	$-0.769387\pi$
$109$	−185773.	−1.49767	−0.748837	+	0.662754i	$0.769387\pi$
$110$	0	0
$111$	−208072.	−1.60290
$112$	0	0
$113$	−47906.7	−0.352940	−0.176470	−	0.984306i	$-0.556468\pi$
$113$	−47906.7	−0.352940	−0.176470	+	0.984306i	$0.556468\pi$
$114$	0	0
$115$	23570.7	0.166199
$116$	0	0
$117$	10100.2	0.0682125
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−152017.	−0.943907
$122$	0	0
$123$	−70337.5	−0.419203
$124$	0	0
$125$	−67456.7	−0.386145
$126$	0	0
$127$	−30596.7	−0.168331	−0.0841656	−	0.996452i	$-0.526822\pi$
$127$	−30596.7	−0.168331	−0.0841656	+	0.996452i	$0.526822\pi$
$128$	0	0
$129$	181676.	0.961224
$130$	0	0
$131$	138321.	0.704223	0.352111	−	0.935958i	$-0.385464\pi$
$131$	138321.	0.704223	0.352111	+	0.935958i	$0.385464\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	34501.4	0.162931
$136$	0	0
$137$	−147583.	−0.671793	−0.335897	−	0.941899i	$-0.609039\pi$
$137$	−147583.	−0.671793	−0.335897	+	0.941899i	$0.609039\pi$
$138$	0	0
$139$	−167389.	−0.734834	−0.367417	−	0.930056i	$-0.619758\pi$
$139$	−167389.	−0.734834	−0.367417	+	0.930056i	$0.619758\pi$
$140$	0	0
$141$	−396902.	−1.68126
$142$	0	0
$143$	−14968.3	−0.0612115
$144$	0	0
$145$	−33578.0	−0.132628
$146$	0	0
$147$	0	0
$148$	0	0
$149$	149761.	0.552627	0.276314	−	0.961068i	$-0.410887\pi$
$149$	149761.	0.552627	0.276314	+	0.961068i	$0.410887\pi$
$150$	0	0
$151$	−320225.	−1.14291	−0.571456	−	0.820633i	$-0.693621\pi$
$151$	−320225.	−1.14291	−0.571456	+	0.820633i	$0.693621\pi$
$152$	0	0
$153$	−6447.40	−0.0222667
$154$	0	0
$155$	33946.6	0.113492
$156$	0	0
$157$	335799.	1.08725	0.543626	−	0.839328i	$-0.317051\pi$
$157$	335799.	1.08725	0.543626	+	0.839328i	$0.317051\pi$
$158$	0	0
$159$	485670.	1.52352
$160$	0	0
$161$	0	0
$162$	0	0
$163$	494956.	1.45914	0.729571	−	0.683905i	$-0.239720\pi$
$163$	494956.	1.45914	0.729571	+	0.683905i	$0.239720\pi$
$164$	0	0
$165$	18333.4	0.0524245
$166$	0	0
$167$	−162081.	−0.449717	−0.224859	−	0.974391i	$-0.572192\pi$
$167$	−162081.	−0.449717	−0.224859	+	0.974391i	$0.572192\pi$
$168$	0	0
$169$	−346492.	−0.933202
$170$	0	0
$171$	42636.7	0.111505
$172$	0	0
$173$	162681.	0.413257	0.206629	−	0.978419i	$-0.433751\pi$
$173$	162681.	0.413257	0.206629	+	0.978419i	$0.433751\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−409808.	−0.983212
$178$	0	0
$179$	−461348.	−1.07621	−0.538104	−	0.842879i	$-0.680859\pi$
$179$	−461348.	−1.07621	−0.538104	+	0.842879i	$0.680859\pi$
$180$	0	0
$181$	603891.	1.37013	0.685065	−	0.728482i	$-0.259774\pi$
$181$	603891.	1.37013	0.685065	+	0.728482i	$0.259774\pi$
$182$	0	0
$183$	179504.	0.396229
$184$	0	0
$185$	130676.	0.280716
$186$	0	0
$187$	9554.95	0.0199813
$188$	0	0
$189$	0	0
$190$	0	0
$191$	560655.	1.11202	0.556010	−	0.831176i	$-0.312332\pi$
$191$	560655.	1.11202	0.556010	+	0.831176i	$0.312332\pi$
$192$	0	0
$193$	165122.	0.319089	0.159544	−	0.987191i	$-0.448997\pi$
$193$	165122.	0.319089	0.159544	+	0.987191i	$0.448997\pi$
$194$	0	0
$195$	−30377.2	−0.0572087
$196$	0	0
$197$	−678116.	−1.24491	−0.622456	−	0.782655i	$-0.713865\pi$
$197$	−678116.	−1.24491	−0.622456	+	0.782655i	$0.713865\pi$
$198$	0	0
$199$	−370360.	−0.662965	−0.331483	−	0.943461i	$-0.607549\pi$
$199$	−370360.	−0.662965	−0.331483	+	0.943461i	$0.607549\pi$
$200$	0	0
$201$	−114881.	−0.200567
$202$	0	0
$203$	0	0
$204$	0	0
$205$	44174.2	0.0734148
$206$	0	0
$207$	137346.	0.222787
$208$	0	0
$209$	−63187.0	−0.100060
$210$	0	0
$211$	884876.	1.36828	0.684142	−	0.729349i	$-0.260177\pi$
$211$	884876.	1.36828	0.684142	+	0.729349i	$0.260177\pi$
$212$	0	0
$213$	372754.	0.562954
$214$	0	0
$215$	−114099.	−0.168339
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5494.32	−0.00774112
$220$	0	0
$221$	−15831.9	−0.0218048
$222$	0	0
$223$	−307029.	−0.413444	−0.206722	−	0.978400i	$-0.566280\pi$
$223$	−307029.	−0.413444	−0.206722	+	0.978400i	$0.566280\pi$
$224$	0	0
$225$	−192650.	−0.253696
$226$	0	0
$227$	−1.05427e6	−1.35797	−0.678983	−	0.734154i	$-0.737579\pi$
$227$	−1.05427e6	−1.35797	−0.678983	+	0.734154i	$0.737579\pi$
$228$	0	0
$229$	−768119.	−0.967921	−0.483960	−	0.875090i	$-0.660802\pi$
$229$	−768119.	−0.967921	−0.483960	+	0.875090i	$0.660802\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.16012e6	−1.39995	−0.699976	−	0.714166i	$-0.746806\pi$
$233$	−1.16012e6	−1.39995	−0.699976	+	0.714166i	$0.746806\pi$
$234$	0	0
$235$	249267.	0.294439
$236$	0	0
$237$	1.03804e6	1.20045
$238$	0	0
$239$	1.14943e6	1.30163	0.650816	−	0.759235i	$-0.274427\pi$
$239$	1.14943e6	1.30163	0.650816	+	0.759235i	$0.274427\pi$
$240$	0	0
$241$	−112291.	−0.124538	−0.0622691	−	0.998059i	$-0.519834\pi$
$241$	−112291.	−0.124538	−0.0622691	+	0.998059i	$0.519834\pi$
$242$	0	0
$243$	474164.	0.515125
$244$	0	0
$245$	0	0
$246$	0	0
$247$	104697.	0.109192
$248$	0	0
$249$	1.88504e6	1.92673
$250$	0	0
$251$	−214195.	−0.214598	−0.107299	−	0.994227i	$-0.534220\pi$
$251$	−214195.	−0.214598	−0.107299	+	0.994227i	$0.534220\pi$
$252$	0	0
$253$	−203545.	−0.199921
$254$	0	0
$255$	19391.2	0.0186747
$256$	0	0
$257$	−2.03227e6	−1.91932	−0.959662	−	0.281156i	$-0.909282\pi$
$257$	−2.03227e6	−1.91932	−0.959662	+	0.281156i	$0.909282\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−195659.	−0.177786
$262$	0	0
$263$	1.09547e6	0.976584	0.488292	−	0.872680i	$-0.337620\pi$
$263$	1.09547e6	0.976584	0.488292	+	0.872680i	$0.337620\pi$
$264$	0	0
$265$	−305016.	−0.266814
$266$	0	0
$267$	1.54848e6	1.32932
$268$	0	0
$269$	−311944.	−0.262843	−0.131421	−	0.991327i	$-0.541954\pi$
$269$	−311944.	−0.262843	−0.131421	+	0.991327i	$0.541954\pi$
$270$	0	0
$271$	−67932.4	−0.0561893	−0.0280947	−	0.999605i	$-0.508944\pi$
$271$	−67932.4	−0.0561893	−0.0280947	+	0.999605i	$0.508944\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	285505.	0.227658
$276$	0	0
$277$	−498271.	−0.390181	−0.195090	−	0.980785i	$-0.562500\pi$
$277$	−498271.	−0.390181	−0.195090	+	0.980785i	$0.562500\pi$
$278$	0	0
$279$	197807.	0.152135
$280$	0	0
$281$	1.47068e6	1.11110	0.555549	−	0.831484i	$-0.312508\pi$
$281$	1.47068e6	1.11110	0.555549	+	0.831484i	$0.312508\pi$
$282$	0	0
$283$	−1.34284e6	−0.996682	−0.498341	−	0.866981i	$-0.666057\pi$
$283$	−1.34284e6	−0.996682	−0.498341	+	0.866981i	$0.666057\pi$
$284$	0	0
$285$	−128234.	−0.0935172
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.40975e6	−0.992882
$290$	0	0
$291$	2.87742e6	1.99192
$292$	0	0
$293$	2.20931e6	1.50345	0.751724	−	0.659478i	$-0.229223\pi$
$293$	2.20931e6	1.50345	0.751724	+	0.659478i	$0.229223\pi$
$294$	0	0
$295$	257372.	0.172189
$296$	0	0
$297$	−297938.	−0.195990
$298$	0	0
$299$	337260.	0.218166
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−525311.	−0.328708
$304$	0	0
$305$	−112734.	−0.0693914
$306$	0	0
$307$	1.47223e6	0.891516	0.445758	−	0.895153i	$-0.352934\pi$
$307$	1.47223e6	0.891516	0.445758	+	0.895153i	$0.352934\pi$
$308$	0	0
$309$	−1.43169e6	−0.853005
$310$	0	0
$311$	−687625.	−0.403135	−0.201568	−	0.979475i	$-0.564604\pi$
$311$	−687625.	−0.403135	−0.201568	+	0.979475i	$0.564604\pi$
$312$	0	0
$313$	−3.27656e6	−1.89041	−0.945206	−	0.326473i	$-0.894140\pi$
$313$	−3.27656e6	−1.89041	−0.945206	+	0.326473i	$0.894140\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.35932e6	1.31868	0.659338	−	0.751847i	$-0.270837\pi$
$317$	2.35932e6	1.31868	0.659338	+	0.751847i	$0.270837\pi$
$318$	0	0
$319$	289963.	0.159539
$320$	0	0
$321$	993867.	0.538351
$322$	0	0
$323$	−66832.6	−0.0356436
$324$	0	0
$325$	−473062.	−0.248433
$326$	0	0
$327$	3.25572e6	1.68375
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.14714e6	0.575502	0.287751	−	0.957705i	$-0.407092\pi$
$331$	1.14714e6	0.575502	0.287751	+	0.957705i	$0.407092\pi$
$332$	0	0
$333$	761449.	0.376296
$334$	0	0
$335$	72149.0	0.0351251
$336$	0	0
$337$	1.34477e6	0.645020	0.322510	−	0.946566i	$-0.395473\pi$
$337$	1.34477e6	0.645020	0.322510	+	0.946566i	$0.395473\pi$
$338$	0	0
$339$	839578.	0.396791
$340$	0	0
$341$	−293146.	−0.136521
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−413082.	−0.186848
$346$	0	0
$347$	769358.	0.343008	0.171504	−	0.985183i	$-0.445137\pi$
$347$	769358.	0.343008	0.171504	+	0.985183i	$0.445137\pi$
$348$	0	0
$349$	3.34745e6	1.47113	0.735564	−	0.677455i	$-0.236917\pi$
$349$	3.34745e6	1.47113	0.735564	+	0.677455i	$0.236917\pi$
$350$	0	0
$351$	493662.	0.213876
$352$	0	0
$353$	−1.72600e6	−0.737231	−0.368615	−	0.929582i	$-0.620168\pi$
$353$	−1.72600e6	−0.737231	−0.368615	+	0.929582i	$0.620168\pi$
$354$	0	0
$355$	−234101.	−0.0985899
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.25940e6	0.925245	0.462623	−	0.886555i	$-0.346909\pi$
$359$	2.25940e6	0.925245	0.462623	+	0.886555i	$0.346909\pi$
$360$	0	0
$361$	−2.03413e6	−0.821508
$362$	0	0
$363$	2.66414e6	1.06118
$364$	0	0
$365$	3450.61	0.00135570
$366$	0	0
$367$	837273.	0.324490	0.162245	−	0.986750i	$-0.448126\pi$
$367$	837273.	0.324490	0.162245	+	0.986750i	$0.448126\pi$
$368$	0	0
$369$	257403.	0.0984118
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3.69908e6	1.37665	0.688323	−	0.725405i	$-0.258347\pi$
$373$	3.69908e6	1.37665	0.688323	+	0.725405i	$0.258347\pi$
$374$	0	0
$375$	1.18220e6	0.434121
$376$	0	0
$377$	−480449.	−0.174098
$378$	0	0
$379$	−752171.	−0.268979	−0.134490	−	0.990915i	$-0.542939\pi$
$379$	−752171.	−0.268979	−0.134490	+	0.990915i	$0.542939\pi$
$380$	0	0
$381$	536214.	0.189246
$382$	0	0
$383$	396510.	0.138120	0.0690601	−	0.997613i	$-0.478000\pi$
$383$	396510.	0.138120	0.0690601	+	0.997613i	$0.478000\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−664852.	−0.225656
$388$	0	0
$389$	−3.46111e6	−1.15969	−0.579844	−	0.814727i	$-0.696887\pi$
$389$	−3.46111e6	−1.15969	−0.579844	+	0.814727i	$0.696887\pi$
$390$	0	0
$391$	−215288.	−0.0712162
$392$	0	0
$393$	−2.42411e6	−0.791719
$394$	0	0
$395$	−651923.	−0.210234
$396$	0	0
$397$	−424548.	−0.135192	−0.0675959	−	0.997713i	$-0.521533\pi$
$397$	−424548.	−0.135192	−0.0675959	+	0.997713i	$0.521533\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−520438.	−0.161625	−0.0808124	−	0.996729i	$-0.525751\pi$
$401$	−520438.	−0.161625	−0.0808124	+	0.996729i	$0.525751\pi$
$402$	0	0
$403$	485723.	0.148979
$404$	0	0
$405$	−776177.	−0.235138
$406$	0	0
$407$	−1.12846e6	−0.337675
$408$	0	0
$409$	964439.	0.285080	0.142540	−	0.989789i	$-0.454473\pi$
$409$	964439.	0.285080	0.142540	+	0.989789i	$0.454473\pi$
$410$	0	0
$411$	2.58643e6	0.755260
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.18386e6	−0.337428
$416$	0	0
$417$	2.93353e6	0.826134
$418$	0	0
$419$	−588781.	−0.163840	−0.0819198	−	0.996639i	$-0.526105\pi$
$419$	−588781.	−0.163840	−0.0819198	+	0.996639i	$0.526105\pi$
$420$	0	0
$421$	29013.6	0.00797804	0.00398902	−	0.999992i	$-0.498730\pi$
$421$	29013.6	0.00797804	0.00398902	+	0.999992i	$0.498730\pi$
$422$	0	0
$423$	1.45248e6	0.394692
$424$	0	0
$425$	301977.	0.0810964
$426$	0	0
$427$	0	0
$428$	0	0
$429$	262323.	0.0688167
$430$	0	0
$431$	−5.08256e6	−1.31792	−0.658960	−	0.752178i	$-0.729004\pi$
$431$	−5.08256e6	−1.31792	−0.658960	+	0.752178i	$0.729004\pi$
$432$	0	0
$433$	−4.61686e6	−1.18339	−0.591694	−	0.806163i	$-0.701540\pi$
$433$	−4.61686e6	−1.18339	−0.591694	+	0.806163i	$0.701540\pi$
$434$	0	0
$435$	588462.	0.149106
$436$	0	0
$437$	1.42370e6	0.356629
$438$	0	0
$439$	−3.42017e6	−0.847005	−0.423503	−	0.905895i	$-0.639200\pi$
$439$	−3.42017e6	−0.847005	−0.423503	+	0.905895i	$0.639200\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−549597.	−0.133056	−0.0665281	−	0.997785i	$-0.521192\pi$
$443$	−549597.	−0.133056	−0.0665281	+	0.997785i	$0.521192\pi$
$444$	0	0
$445$	−972496.	−0.232803
$446$	0	0
$447$	−2.62459e6	−0.621289
$448$	0	0
$449$	2.98225e6	0.698116	0.349058	−	0.937101i	$-0.386502\pi$
$449$	2.98225e6	0.698116	0.349058	+	0.937101i	$0.386502\pi$
$450$	0	0
$451$	−381467.	−0.0883112
$452$	0	0
$453$	5.61202e6	1.28491
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.65175e6	−1.26588	−0.632939	−	0.774201i	$-0.718152\pi$
$457$	−5.65175e6	−1.26588	−0.632939	+	0.774201i	$0.718152\pi$
$458$	0	0
$459$	−315127.	−0.0698158
$460$	0	0
$461$	−3.17668e6	−0.696179	−0.348089	−	0.937461i	$-0.613169\pi$
$461$	−3.17668e6	−0.696179	−0.348089	+	0.937461i	$0.613169\pi$
$462$	0	0
$463$	−5.69966e6	−1.23565	−0.617826	−	0.786315i	$-0.711987\pi$
$463$	−5.69966e6	−1.23565	−0.617826	+	0.786315i	$0.711987\pi$
$464$	0	0
$465$	−594922.	−0.127593
$466$	0	0
$467$	−4.63943e6	−0.984402	−0.492201	−	0.870481i	$-0.663808\pi$
$467$	−4.63943e6	−0.984402	−0.492201	+	0.870481i	$0.663808\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5.88496e6	−1.22234
$472$	0	0
$473$	985301.	0.202496
$474$	0	0
$475$	−1.99698e6	−0.406106
$476$	0	0
$477$	−1.77733e6	−0.357661
$478$	0	0
$479$	2.20916e6	0.439934	0.219967	−	0.975507i	$-0.429405\pi$
$479$	2.20916e6	0.439934	0.219967	+	0.975507i	$0.429405\pi$
$480$	0	0
$481$	1.86977e6	0.368490
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.80711e6	−0.348844
$486$	0	0
$487$	−7.82289e6	−1.49467	−0.747334	−	0.664449i	$-0.768666\pi$
$487$	−7.82289e6	−1.49467	−0.747334	+	0.664449i	$0.768666\pi$
$488$	0	0
$489$	−8.67422e6	−1.64043
$490$	0	0
$491$	4.08903e6	0.765449	0.382724	−	0.923863i	$-0.374986\pi$
$491$	4.08903e6	0.765449	0.382724	+	0.923863i	$0.374986\pi$
$492$	0	0
$493$	306693.	0.0568311
$494$	0	0
$495$	−67091.9	−0.0123071
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−911248.	−0.163827	−0.0819135	−	0.996639i	$-0.526103\pi$
$499$	−911248.	−0.163827	−0.0819135	+	0.996639i	$0.526103\pi$
$500$	0	0
$501$	2.84050e6	0.505593
$502$	0	0
$503$	−1.10725e7	−1.95130	−0.975649	−	0.219336i	$-0.929611\pi$
$503$	−1.10725e7	−1.95130	−0.975649	+	0.219336i	$0.929611\pi$
$504$	0	0
$505$	329912.	0.0575664
$506$	0	0
$507$	6.07235e6	1.04915
$508$	0	0
$509$	6.80487e6	1.16419	0.582097	−	0.813119i	$-0.302232\pi$
$509$	6.80487e6	1.16419	0.582097	+	0.813119i	$0.302232\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.08394e6	0.349616
$514$	0	0
$515$	899144.	0.149386
$516$	0	0
$517$	−2.15255e6	−0.354182
$518$	0	0
$519$	−2.85102e6	−0.464603
$520$	0	0
$521$	4.44538e6	0.717487	0.358744	−	0.933436i	$-0.383205\pi$
$521$	4.44538e6	0.717487	0.358744	+	0.933436i	$0.383205\pi$
$522$	0	0
$523$	−8.72959e6	−1.39553	−0.697766	−	0.716326i	$-0.745822\pi$
$523$	−8.72959e6	−1.39553	−0.697766	+	0.716326i	$0.745822\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−310059.	−0.0486315
$528$	0	0
$529$	−1.85015e6	−0.287453
$530$	0	0
$531$	1.49971e6	0.230818
$532$	0	0
$533$	632065.	0.0963703
$534$	0	0
$535$	−624180.	−0.0942813
$536$	0	0
$537$	8.08524e6	1.20992
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.73878e6	−0.549208	−0.274604	−	0.961557i	$-0.588547\pi$
$541$	−3.73878e6	−0.549208	−0.274604	+	0.961557i	$0.588547\pi$
$542$	0	0
$543$	−1.05833e7	−1.54036
$544$	0	0
$545$	−2.04470e6	−0.294875
$546$	0	0
$547$	−8.95708e6	−1.27996	−0.639982	−	0.768390i	$-0.721058\pi$
$547$	−8.95708e6	−1.27996	−0.639982	+	0.768390i	$0.721058\pi$
$548$	0	0
$549$	−656901.	−0.0930184
$550$	0	0
$551$	−2.02816e6	−0.284593
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.29013e6	−0.315593
$556$	0	0
$557$	−3.40749e6	−0.465368	−0.232684	−	0.972552i	$-0.574751\pi$
$557$	−3.40749e6	−0.465368	−0.232684	+	0.972552i	$0.574751\pi$
$558$	0	0
$559$	−1.63257e6	−0.220975
$560$	0	0
$561$	−167453.	−0.0224639
$562$	0	0
$563$	−7.39433e6	−0.983169	−0.491584	−	0.870830i	$-0.663582\pi$
$563$	−7.39433e6	−0.983169	−0.491584	+	0.870830i	$0.663582\pi$
$564$	0	0
$565$	−527281.	−0.0694898
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.39745e6	0.569404	0.284702	−	0.958616i	$-0.408105\pi$
$569$	4.39745e6	0.569404	0.284702	+	0.958616i	$0.408105\pi$
$570$	0	0
$571$	−4.11677e6	−0.528405	−0.264202	−	0.964467i	$-0.585109\pi$
$571$	−4.11677e6	−0.528405	−0.264202	+	0.964467i	$0.585109\pi$
$572$	0	0
$573$	−9.82562e6	−1.25018
$574$	0	0
$575$	−6.43289e6	−0.811402
$576$	0	0
$577$	5.12253e6	0.640538	0.320269	−	0.947327i	$-0.396227\pi$
$577$	5.12253e6	0.640538	0.320269	+	0.947327i	$0.396227\pi$
$578$	0	0
$579$	−2.89380e6	−0.358734
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.63398e6	0.320952
$584$	0	0
$585$	111167.	0.0134303
$586$	0	0
$587$	1.04610e7	1.25308	0.626538	−	0.779391i	$-0.284471\pi$
$587$	1.04610e7	1.25308	0.626538	+	0.779391i	$0.284471\pi$
$588$	0	0
$589$	2.05043e6	0.243532
$590$	0	0
$591$	1.18842e7	1.39959
$592$	0	0
$593$	1.38124e7	1.61300	0.806498	−	0.591237i	$-0.201360\pi$
$593$	1.38124e7	1.61300	0.806498	+	0.591237i	$0.201360\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.49064e6	0.745336
$598$	0	0
$599$	−3.74704e6	−0.426699	−0.213349	−	0.976976i	$-0.568437\pi$
$599$	−3.74704e6	−0.426699	−0.213349	+	0.976976i	$0.568437\pi$
$600$	0	0
$601$	9.77914e6	1.10437	0.552185	−	0.833722i	$-0.313794\pi$
$601$	9.77914e6	1.10437	0.552185	+	0.833722i	$0.313794\pi$
$602$	0	0
$603$	420412.	0.0470849
$604$	0	0
$605$	−1.67316e6	−0.185845
$606$	0	0
$607$	334361.	0.0368336	0.0184168	−	0.999830i	$-0.494137\pi$
$607$	334361.	0.0368336	0.0184168	+	0.999830i	$0.494137\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.56662e6	0.386504
$612$	0	0
$613$	1.08435e6	0.116551	0.0582756	−	0.998301i	$-0.481440\pi$
$613$	1.08435e6	0.116551	0.0582756	+	0.998301i	$0.481440\pi$
$614$	0	0
$615$	−774164.	−0.0825363
$616$	0	0
$617$	7.42021e6	0.784699	0.392350	−	0.919816i	$-0.371662\pi$
$617$	7.42021e6	0.784699	0.392350	+	0.919816i	$0.371662\pi$
$618$	0	0
$619$	2.44590e6	0.256574	0.128287	−	0.991737i	$-0.459052\pi$
$619$	2.44590e6	0.256574	0.128287	+	0.991737i	$0.459052\pi$
$620$	0	0
$621$	6.71301e6	0.698536
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8.64460e6	0.885207
$626$	0	0
$627$	1.10737e6	0.112492
$628$	0	0
$629$	−1.19356e6	−0.120287
$630$	0	0
$631$	−1.54761e7	−1.54735	−0.773673	−	0.633585i	$-0.781583\pi$
$631$	−1.54761e7	−1.54735	−0.773673	+	0.633585i	$0.781583\pi$
$632$	0	0
$633$	−1.55077e7	−1.53829
$634$	0	0
$635$	−336759.	−0.0331425
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.36411e6	−0.132159
$640$	0	0
$641$	−8.75649e6	−0.841753	−0.420877	−	0.907118i	$-0.638277\pi$
$641$	−8.75649e6	−0.841753	−0.420877	+	0.907118i	$0.638277\pi$
$642$	0	0
$643$	1.29737e7	1.23747	0.618736	−	0.785599i	$-0.287645\pi$
$643$	1.29737e7	1.23747	0.618736	+	0.785599i	$0.287645\pi$
$644$	0	0
$645$	1.99961e6	0.189254
$646$	0	0
$647$	4.82476e6	0.453121	0.226561	−	0.973997i	$-0.427252\pi$
$647$	4.82476e6	0.453121	0.226561	+	0.973997i	$0.427252\pi$
$648$	0	0
$649$	−2.22254e6	−0.207128
$650$	0	0
$651$	0	0
$652$	0	0
$653$	781133.	0.0716873	0.0358436	−	0.999357i	$-0.488588\pi$
$653$	781133.	0.0716873	0.0358436	+	0.999357i	$0.488588\pi$
$654$	0	0
$655$	1.52242e6	0.138653
$656$	0	0
$657$	20106.7	0.00181730
$658$	0	0
$659$	−3.55442e6	−0.318827	−0.159414	−	0.987212i	$-0.550960\pi$
$659$	−3.55442e6	−0.318827	−0.159414	+	0.987212i	$0.550960\pi$
$660$	0	0
$661$	9.88767e6	0.880219	0.440109	−	0.897944i	$-0.354940\pi$
$661$	9.88767e6	0.880219	0.440109	+	0.897944i	$0.354940\pi$
$662$	0	0
$663$	277458.	0.0245139
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.53334e6	−0.568618
$668$	0	0
$669$	5.38075e6	0.464813
$670$	0	0
$671$	973517.	0.0834713
$672$	0	0
$673$	−9.64825e6	−0.821128	−0.410564	−	0.911832i	$-0.634668\pi$
$673$	−9.64825e6	−0.821128	−0.410564	+	0.911832i	$0.634668\pi$
$674$	0	0
$675$	−9.41609e6	−0.795447
$676$	0	0
$677$	−434057.	−0.0363978	−0.0181989	−	0.999834i	$-0.505793\pi$
$677$	−434057.	−0.0363978	−0.0181989	+	0.999834i	$0.505793\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.84764e7	1.52669
$682$	0	0
$683$	2.20366e7	1.80756	0.903778	−	0.428001i	$-0.140782\pi$
$683$	2.20366e7	1.80756	0.903778	+	0.428001i	$0.140782\pi$
$684$	0	0
$685$	−1.62436e6	−0.132268
$686$	0	0
$687$	1.34615e7	1.08818
$688$	0	0
$689$	−4.36431e6	−0.350242
$690$	0	0
$691$	−1.00504e7	−0.800737	−0.400369	−	0.916354i	$-0.631118\pi$
$691$	−1.00504e7	−0.800737	−0.400369	+	0.916354i	$0.631118\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.84235e6	−0.144680
$696$	0	0
$697$	−403476.	−0.0314583
$698$	0	0
$699$	2.03314e7	1.57389
$700$	0	0
$701$	2.39421e7	1.84021	0.920106	−	0.391668i	$-0.128102\pi$
$701$	2.39421e7	1.84021	0.920106	+	0.391668i	$0.128102\pi$
$702$	0	0
$703$	7.89303e6	0.602359
$704$	0	0
$705$	−4.36846e6	−0.331021
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.05493e7	−0.788148	−0.394074	−	0.919079i	$-0.628935\pi$
$709$	−1.05493e7	−0.788148	−0.394074	+	0.919079i	$0.628935\pi$
$710$	0	0
$711$	−3.79875e6	−0.281817
$712$	0	0
$713$	6.60506e6	0.486578
$714$	0	0
$715$	−164747.	−0.0120518
$716$	0	0
$717$	−2.01441e7	−1.46335
$718$	0	0
$719$	9.30506e6	0.671270	0.335635	−	0.941992i	$-0.391049\pi$
$719$	9.30506e6	0.671270	0.335635	+	0.941992i	$0.391049\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.96793e6	0.140012
$724$	0	0
$725$	9.16407e6	0.647506
$726$	0	0
$727$	−3.88455e6	−0.272587	−0.136294	−	0.990668i	$-0.543519\pi$
$727$	−3.88455e6	−0.272587	−0.136294	+	0.990668i	$0.543519\pi$
$728$	0	0
$729$	8.82662e6	0.615142
$730$	0	0
$731$	1.04215e6	0.0721332
$732$	0	0
$733$	−2.15755e7	−1.48320	−0.741602	−	0.670840i	$-0.765934\pi$
$733$	−2.15755e7	−1.48320	−0.741602	+	0.670840i	$0.765934\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−623044.	−0.0422523
$738$	0	0
$739$	2.79027e7	1.87947	0.939733	−	0.341909i	$-0.111074\pi$
$739$	2.79027e7	1.87947	0.939733	+	0.341909i	$0.111074\pi$
$740$	0	0
$741$	−1.83483e6	−0.122758
$742$	0	0
$743$	−2.11208e7	−1.40358	−0.701790	−	0.712384i	$-0.747616\pi$
$743$	−2.11208e7	−1.40358	−0.701790	+	0.712384i	$0.747616\pi$
$744$	0	0
$745$	1.64833e6	0.108806
$746$	0	0
$747$	−6.89837e6	−0.452319
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.86917e7	−1.85634	−0.928168	−	0.372161i	$-0.878617\pi$
$751$	−2.86917e7	−1.85634	−0.928168	+	0.372161i	$0.878617\pi$
$752$	0	0
$753$	3.75382e6	0.241261
$754$	0	0
$755$	−3.52452e6	−0.225026
$756$	0	0
$757$	1.39601e7	0.885421	0.442710	−	0.896665i	$-0.354017\pi$
$757$	1.39601e7	0.885421	0.442710	+	0.896665i	$0.354017\pi$
$758$	0	0
$759$	3.56718e6	0.224761
$760$	0	0
$761$	2.41380e6	0.151091	0.0755456	−	0.997142i	$-0.475930\pi$
$761$	2.41380e6	0.151091	0.0755456	+	0.997142i	$0.475930\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−70962.7	−0.00438406
$766$	0	0
$767$	3.68260e6	0.226030
$768$	0	0
$769$	2.00276e7	1.22128	0.610638	−	0.791910i	$-0.290913\pi$
$769$	2.00276e7	1.22128	0.610638	+	0.791910i	$0.290913\pi$
$770$	0	0
$771$	3.56160e7	2.15779
$772$	0	0
$773$	−2.75771e7	−1.65997	−0.829983	−	0.557789i	$-0.811650\pi$
$773$	−2.75771e7	−1.65997	−0.829983	+	0.557789i	$0.811650\pi$
$774$	0	0
$775$	−9.26467e6	−0.554084
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.66819e6	0.157534
$780$	0	0
$781$	2.02158e6	0.118594
$782$	0	0
$783$	−9.56313e6	−0.557437
$784$	0	0
$785$	3.69594e6	0.214068
$786$	0	0
$787$	−1.74369e7	−1.00354	−0.501769	−	0.865002i	$-0.667317\pi$
$787$	−1.74369e7	−1.00354	−0.501769	+	0.865002i	$0.667317\pi$
$788$	0	0
$789$	−1.91983e7	−1.09792
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1.61305e6	−0.0910888
$794$	0	0
$795$	5.34549e6	0.299964
$796$	0	0
$797$	−6.96631e6	−0.388469	−0.194235	−	0.980955i	$-0.562222\pi$
$797$	−6.96631e6	−0.388469	−0.194235	+	0.980955i	$0.562222\pi$
$798$	0	0
$799$	−2.27674e6	−0.126167
$800$	0	0
$801$	−5.66673e6	−0.312069
$802$	0	0
$803$	−29797.8	−0.00163078
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.46690e6	0.295500
$808$	0	0
$809$	9.66666e6	0.519285	0.259642	−	0.965705i	$-0.416395\pi$
$809$	9.66666e6	0.519285	0.259642	+	0.965705i	$0.416395\pi$
$810$	0	0
$811$	3.47649e7	1.85604	0.928022	−	0.372525i	$-0.121508\pi$
$811$	3.47649e7	1.85604	0.928022	+	0.372525i	$0.121508\pi$
$812$	0	0
$813$	1.19053e6	0.0631706
$814$	0	0
$815$	5.44768e6	0.287288
$816$	0	0
$817$	−6.89173e6	−0.361221
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.42770e7	−0.739231	−0.369616	−	0.929185i	$-0.620511\pi$
$821$	−1.42770e7	−0.739231	−0.369616	+	0.929185i	$0.620511\pi$
$822$	0	0
$823$	9.54227e6	0.491080	0.245540	−	0.969386i	$-0.421035\pi$
$823$	9.54227e6	0.491080	0.245540	+	0.969386i	$0.421035\pi$
$824$	0	0
$825$	−5.00355e6	−0.255943
$826$	0	0
$827$	3.08484e7	1.56844	0.784222	−	0.620480i	$-0.213062\pi$
$827$	3.08484e7	1.56844	0.784222	+	0.620480i	$0.213062\pi$
$828$	0	0
$829$	1.18676e7	0.599757	0.299878	−	0.953977i	$-0.403054\pi$
$829$	1.18676e7	0.599757	0.299878	+	0.953977i	$0.403054\pi$
$830$	0	0
$831$	8.73232e6	0.438659
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.78392e6	−0.0885442
$836$	0	0
$837$	9.66811e6	0.477010
$838$	0	0
$839$	−2.05321e7	−1.00700	−0.503499	−	0.863996i	$-0.667954\pi$
$839$	−2.05321e7	−1.00700	−0.503499	+	0.863996i	$0.667954\pi$
$840$	0	0
$841$	−1.12040e7	−0.546238
$842$	0	0
$843$	−2.57740e7	−1.24915
$844$	0	0
$845$	−3.81363e6	−0.183737
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.35335e7	1.12052
$850$	0	0
$851$	2.54259e7	1.20352
$852$	0	0
$853$	−543335.	−0.0255679	−0.0127840	−	0.999918i	$-0.504069\pi$
$853$	−543335.	−0.0255679	−0.0127840	+	0.999918i	$0.504069\pi$
$854$	0	0
$855$	469277.	0.0219540
$856$	0	0
$857$	1.85917e7	0.864704	0.432352	−	0.901705i	$-0.357684\pi$
$857$	1.85917e7	0.864704	0.432352	+	0.901705i	$0.357684\pi$
$858$	0	0
$859$	−2.15483e7	−0.996394	−0.498197	−	0.867064i	$-0.666004\pi$
$859$	−2.15483e7	−0.996394	−0.498197	+	0.867064i	$0.666004\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.80274e7	0.823958	0.411979	−	0.911193i	$-0.364838\pi$
$863$	1.80274e7	0.823958	0.411979	+	0.911193i	$0.364838\pi$
$864$	0	0
$865$	1.79053e6	0.0813657
$866$	0	0
$867$	2.47062e7	1.11624
$868$	0	0
$869$	5.62969e6	0.252892
$870$	0	0
$871$	1.03234e6	0.0461082
$872$	0	0
$873$	−1.05300e7	−0.467621
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.98502e7	−1.31053	−0.655267	−	0.755397i	$-0.727444\pi$
$877$	−2.98502e7	−1.31053	−0.655267	+	0.755397i	$0.727444\pi$
$878$	0	0
$879$	−3.87188e7	−1.69024
$880$	0	0
$881$	3.90709e7	1.69595	0.847975	−	0.530036i	$-0.177822\pi$
$881$	3.90709e7	1.69595	0.847975	+	0.530036i	$0.177822\pi$
$882$	0	0
$883$	−3.74333e7	−1.61568	−0.807842	−	0.589399i	$-0.799365\pi$
$883$	−3.74333e7	−1.61568	−0.807842	+	0.589399i	$0.799365\pi$
$884$	0	0
$885$	−4.51051e6	−0.193583
$886$	0	0
$887$	3.25735e7	1.39013	0.695064	−	0.718948i	$-0.255376\pi$
$887$	3.25735e7	1.39013	0.695064	+	0.718948i	$0.255376\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.70269e6	0.282849
$892$	0	0
$893$	1.50561e7	0.631806
$894$	0	0
$895$	−5.07779e6	−0.211893
$896$	0	0
$897$	−5.91056e6	−0.245272
$898$	0	0
$899$	−9.40934e6	−0.388293
$900$	0	0
$901$	2.78594e6	0.114330
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.64667e6	0.269763
$906$	0	0
$907$	−3.29342e6	−0.132932	−0.0664659	−	0.997789i	$-0.521172\pi$
$907$	−3.29342e6	−0.132932	−0.0664659	+	0.997789i	$0.521172\pi$
$908$	0	0
$909$	1.92239e6	0.0771672
$910$	0	0
$911$	2.47725e7	0.988950	0.494475	−	0.869192i	$-0.335360\pi$
$911$	2.47725e7	0.988950	0.494475	+	0.869192i	$0.335360\pi$
$912$	0	0
$913$	1.02233e7	0.405895
$914$	0	0
$915$	1.97569e6	0.0780129
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.50473e7	0.587718	0.293859	−	0.955849i	$-0.405060\pi$
$919$	1.50473e7	0.587718	0.293859	+	0.955849i	$0.405060\pi$
$920$	0	0
$921$	−2.58012e7	−1.00228
$922$	0	0
$923$	−3.34962e6	−0.129417
$924$	0	0
$925$	−3.56640e7	−1.37049
$926$	0	0
$927$	5.23931e6	0.200251
$928$	0	0
$929$	3.20341e7	1.21779	0.608897	−	0.793250i	$-0.291612\pi$
$929$	3.20341e7	1.21779	0.608897	+	0.793250i	$0.291612\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.20508e7	0.453223
$934$	0	0
$935$	105166.	0.00393410
$936$	0	0
$937$	3.28148e7	1.22101	0.610507	−	0.792011i	$-0.290966\pi$
$937$	3.28148e7	1.22101	0.610507	+	0.792011i	$0.290966\pi$
$938$	0	0
$939$	5.74224e7	2.12529
$940$	0	0
$941$	2.23532e7	0.822935	0.411467	−	0.911424i	$-0.365016\pi$
$941$	2.23532e7	0.822935	0.411467	+	0.911424i	$0.365016\pi$
$942$	0	0
$943$	8.59507e6	0.314753
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.84264e7	1.75472	0.877360	−	0.479834i	$-0.159303\pi$
$947$	4.84264e7	1.75472	0.877360	+	0.479834i	$0.159303\pi$
$948$	0	0
$949$	49372.9	0.00177960
$950$	0	0
$951$	−4.13476e7	−1.48251
$952$	0	0
$953$	−3.85648e7	−1.37549	−0.687747	−	0.725951i	$-0.741400\pi$
$953$	−3.85648e7	−1.37549	−0.687747	+	0.725951i	$0.741400\pi$
$954$	0	0
$955$	6.17080e6	0.218944
$956$	0	0
$957$	−5.08168e6	−0.179361
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.91165e7	−0.667729
$962$	0	0
$963$	−3.63709e6	−0.126383
$964$	0	0
$965$	1.81740e6	0.0628249
$966$	0	0
$967$	−4.03667e7	−1.38821	−0.694107	−	0.719871i	$-0.744201\pi$
$967$	−4.03667e7	−1.38821	−0.694107	+	0.719871i	$0.744201\pi$
$968$	0	0
$969$	1.17126e6	0.0400722
$970$	0	0
$971$	7.51470e6	0.255778	0.127889	−	0.991788i	$-0.459180\pi$
$971$	7.51470e6	0.255778	0.127889	+	0.991788i	$0.459180\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	8.29053e6	0.279300
$976$	0	0
$977$	−4.57026e6	−0.153181	−0.0765904	−	0.997063i	$-0.524403\pi$
$977$	−4.57026e6	−0.153181	−0.0765904	+	0.997063i	$0.524403\pi$
$978$	0	0
$979$	8.39801e6	0.280040
$980$	0	0
$981$	−1.19144e7	−0.395277
$982$	0	0
$983$	−3.62539e7	−1.19666	−0.598330	−	0.801250i	$-0.704169\pi$
$983$	−3.62539e7	−1.19666	−0.598330	+	0.801250i	$0.704169\pi$
$984$	0	0
$985$	−7.46362e6	−0.245109
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.22004e7	−0.721722
$990$	0	0
$991$	−5.45804e7	−1.76544	−0.882719	−	0.469901i	$-0.844289\pi$
$991$	−5.45804e7	−1.76544	−0.882719	+	0.469901i	$0.844289\pi$
$992$	0	0
$993$	−2.01039e7	−0.647006
$994$	0	0
$995$	−4.07633e6	−0.130530
$996$	0	0
$997$	5.72786e7	1.82496	0.912482	−	0.409116i	$-0.134163\pi$
$997$	5.72786e7	1.82496	0.912482	+	0.409116i	$0.134163\pi$
$998$	0	0
$999$	3.72170e7	1.17985

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.6.a.m.1.1	✓	6
4.3	odd	2		784.6.a.bn.1.6		6
7.2	even	3		392.6.i.q.361.6		12
7.3	odd	6		392.6.i.q.177.1		12
7.4	even	3		392.6.i.q.177.6		12
7.5	odd	6		392.6.i.q.361.1		12
7.6	odd	2	inner	392.6.a.m.1.6	yes	6
28.27	even	2		784.6.a.bn.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.6.a.m.1.1	✓	6	1.1	even	1	trivial
392.6.a.m.1.6	yes	6	7.6	odd	2	inner
392.6.i.q.177.1		12	7.3	odd	6
392.6.i.q.177.6		12	7.4	even	3
392.6.i.q.361.1		12	7.5	odd	6
392.6.i.q.361.6		12	7.2	even	3
784.6.a.bn.1.1		6	28.27	even	2
784.6.a.bn.1.6		6	4.3	odd	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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

\(f(q)\)	\(=\)	\(q-17.5252 q^{3} +11.0064 q^{5} +64.1343 q^{9} +O(q^{10})\)	\(q-17.5252 q^{3} +11.0064 q^{5} +64.1343 q^{9} -95.0461 q^{11} +157.485 q^{13} -192.890 q^{15} -100.530 q^{17} +664.804 q^{19} +2141.54 q^{23} -3003.86 q^{25} +3134.67 q^{27} -3050.77 q^{29} +3084.25 q^{31} +1665.71 q^{33} +11872.7 q^{37} -2759.96 q^{39} +4013.50 q^{41} -10366.6 q^{43} +705.888 q^{45} +22647.4 q^{47} +1761.81 q^{51} -27712.6 q^{53} -1046.12 q^{55} -11650.9 q^{57} +23383.9 q^{59} -10242.6 q^{61} +1733.34 q^{65} +6555.18 q^{67} -37531.0 q^{69} -21269.5 q^{71} +313.509 q^{73} +52643.4 q^{75} -59231.2 q^{79} -70520.4 q^{81} -107561. q^{83} -1106.47 q^{85} +53465.4 q^{87} -88357.3 q^{89} -54052.3 q^{93} +7317.11 q^{95} -164187. q^{97} -6095.71 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 122 q^{9}+O(q^{10}) \) 6 * q - 122 * q^9	\( 6 q - 122 q^{9} + 248 q^{11} - 376 q^{15} - 2904 q^{23} + 1510 q^{25} - 408 q^{29} - 18648 q^{37} - 15240 q^{39} - 10584 q^{43} + 3016 q^{51} - 66772 q^{53} - 56264 q^{57} - 55316 q^{65} - 169632 q^{67} - 54880 q^{71} + 50576 q^{79} - 270194 q^{81} - 406076 q^{85} - 164944 q^{93} - 551912 q^{95} - 173512 q^{99}+O(q^{100}) \) 6 * q - 122 * q^9 + 248 * q^11 - 376 * q^15 - 2904 * q^23 + 1510 * q^25 - 408 * q^29 - 18648 * q^37 - 15240 * q^39 - 10584 * q^43 + 3016 * q^51 - 66772 * q^53 - 56264 * q^57 - 55316 * q^65 - 169632 * q^67 - 54880 * q^71 + 50576 * q^79 - 270194 * q^81 - 406076 * q^85 - 164944 * q^93 - 551912 * q^95 - 173512 * q^99

Introduction

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