LMFDB - Embedded newform 368.6.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [368,6,Mod(1,368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(368, 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("368.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 368 = 2^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	368.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:	$59.0212456912$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.7925.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 12 $ x^3 - x^2 - 13*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.917748$ of defining polynomial
Character	$\chi$	$=$	368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.43100 q^{3} -37.9865 q^{5} +43.8366 q^{7} -240.952 q^{9} +O(q^{10})$	$q+1.43100 q^{3} -37.9865 q^{5} +43.8366 q^{7} -240.952 q^{9} +163.213 q^{11} +430.356 q^{13} -54.3587 q^{15} +740.415 q^{17} +916.578 q^{19} +62.7302 q^{21} +529.000 q^{23} -1682.03 q^{25} -692.536 q^{27} -5112.96 q^{29} +5702.60 q^{31} +233.558 q^{33} -1665.20 q^{35} -10913.1 q^{37} +615.840 q^{39} +11092.6 q^{41} -5528.76 q^{43} +9152.92 q^{45} -14435.2 q^{47} -14885.4 q^{49} +1059.53 q^{51} +6852.79 q^{53} -6199.90 q^{55} +1311.62 q^{57} -4155.17 q^{59} -21911.4 q^{61} -10562.5 q^{63} -16347.7 q^{65} +15164.2 q^{67} +756.999 q^{69} -14031.4 q^{71} -23310.1 q^{73} -2406.98 q^{75} +7154.72 q^{77} -64276.8 q^{79} +57560.4 q^{81} -114349. q^{83} -28125.8 q^{85} -7316.65 q^{87} -63420.7 q^{89} +18865.4 q^{91} +8160.42 q^{93} -34817.6 q^{95} -91987.6 q^{97} -39326.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 20 q^{3} - 58 q^{5} + 282 q^{7} + 121 q^{9}+O(q^{10}) $ 3 * q + 20 * q^3 - 58 * q^5 + 282 * q^7 + 121 * q^9	$ 3 q + 20 q^{3} - 58 q^{5} + 282 q^{7} + 121 q^{9} - 136 q^{11} - 1116 q^{13} - 750 q^{15} - 896 q^{17} - 1654 q^{19} - 1670 q^{21} + 1587 q^{23} - 7347 q^{25} + 10700 q^{27} - 844 q^{29} + 3020 q^{31} - 7370 q^{33} - 1072 q^{35} + 8938 q^{37} - 16020 q^{39} - 12792 q^{41} + 16730 q^{43} - 3936 q^{45} - 22500 q^{47} + 2887 q^{49} - 50290 q^{51} + 17108 q^{53} + 436 q^{55} - 61960 q^{57} - 54176 q^{59} - 71324 q^{61} - 40696 q^{63} + 846 q^{65} + 62960 q^{67} + 10580 q^{69} - 98400 q^{71} - 81772 q^{73} - 44800 q^{75} - 304 q^{77} - 58224 q^{79} + 149947 q^{81} - 9892 q^{83} + 15536 q^{85} - 90500 q^{87} + 27542 q^{89} - 151974 q^{91} + 157330 q^{93} + 20644 q^{95} - 273672 q^{97} - 183082 q^{99}+O(q^{100}) $ 3 * q + 20 * q^3 - 58 * q^5 + 282 * q^7 + 121 * q^9 - 136 * q^11 - 1116 * q^13 - 750 * q^15 - 896 * q^17 - 1654 * q^19 - 1670 * q^21 + 1587 * q^23 - 7347 * q^25 + 10700 * q^27 - 844 * q^29 + 3020 * q^31 - 7370 * q^33 - 1072 * q^35 + 8938 * q^37 - 16020 * q^39 - 12792 * q^41 + 16730 * q^43 - 3936 * q^45 - 22500 * q^47 + 2887 * q^49 - 50290 * q^51 + 17108 * q^53 + 436 * q^55 - 61960 * q^57 - 54176 * q^59 - 71324 * q^61 - 40696 * q^63 + 846 * q^65 + 62960 * q^67 + 10580 * q^69 - 98400 * q^71 - 81772 * q^73 - 44800 * q^75 - 304 * q^77 - 58224 * q^79 + 149947 * q^81 - 9892 * q^83 + 15536 * q^85 - 90500 * q^87 + 27542 * q^89 - 151974 * q^91 + 157330 * q^93 + 20644 * q^95 - 273672 * q^97 - 183082 * 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
$3$	1.43100	0.0917987	0.0458994	−	0.998946i	$-0.485385\pi$
$3$	1.43100	0.0917987	0.0458994	+	0.998946i	$0.485385\pi$
$4$	0	0
$5$	−37.9865	−0.679523	−0.339761	−	0.940512i	$-0.610346\pi$
$5$	−37.9865	−0.679523	−0.339761	+	0.940512i	$0.610346\pi$
$6$	0	0
$7$	43.8366	0.338136	0.169068	−	0.985604i	$-0.445924\pi$
$7$	43.8366	0.338136	0.169068	+	0.985604i	$0.445924\pi$
$8$	0	0
$9$	−240.952	−0.991573
$10$	0	0
$11$	163.213	0.406700	0.203350	−	0.979106i	$-0.434817\pi$
$11$	163.213	0.406700	0.203350	+	0.979106i	$0.434817\pi$
$12$	0	0
$13$	430.356	0.706269	0.353134	−	0.935573i	$-0.385116\pi$
$13$	430.356	0.706269	0.353134	+	0.935573i	$0.385116\pi$
$14$	0	0
$15$	−54.3587	−0.0623793
$16$	0	0
$17$	740.415	0.621374	0.310687	−	0.950512i	$-0.399441\pi$
$17$	740.415	0.621374	0.310687	+	0.950512i	$0.399441\pi$
$18$	0	0
$19$	916.578	0.582486	0.291243	−	0.956649i	$-0.405931\pi$
$19$	916.578	0.582486	0.291243	+	0.956649i	$0.405931\pi$
$20$	0	0
$21$	62.7302	0.0310405
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−1682.03	−0.538249
$26$	0	0
$27$	−692.536	−0.182824
$28$	0	0
$29$	−5112.96	−1.12896	−0.564478	−	0.825448i	$-0.690923\pi$
$29$	−5112.96	−1.12896	−0.564478	+	0.825448i	$0.690923\pi$
$30$	0	0
$31$	5702.60	1.06578	0.532891	−	0.846184i	$-0.321105\pi$
$31$	5702.60	1.06578	0.532891	+	0.846184i	$0.321105\pi$
$32$	0	0
$33$	233.558	0.0373345
$34$	0	0
$35$	−1665.20	−0.229771
$36$	0	0
$37$	−10913.1	−1.31052	−0.655261	−	0.755402i	$-0.727441\pi$
$37$	−10913.1	−1.31052	−0.655261	+	0.755402i	$0.727441\pi$
$38$	0	0
$39$	615.840	0.0648346
$40$	0	0
$41$	11092.6	1.03056	0.515282	−	0.857021i	$-0.327687\pi$
$41$	11092.6	1.03056	0.515282	+	0.857021i	$0.327687\pi$
$42$	0	0
$43$	−5528.76	−0.455991	−0.227996	−	0.973662i	$-0.573217\pi$
$43$	−5528.76	−0.455991	−0.227996	+	0.973662i	$0.573217\pi$
$44$	0	0
$45$	9152.92	0.673796
$46$	0	0
$47$	−14435.2	−0.953189	−0.476594	−	0.879123i	$-0.658129\pi$
$47$	−14435.2	−0.953189	−0.476594	+	0.879123i	$0.658129\pi$
$48$	0	0
$49$	−14885.4	−0.885664
$50$	0	0
$51$	1059.53	0.0570413
$52$	0	0
$53$	6852.79	0.335102	0.167551	−	0.985863i	$-0.446414\pi$
$53$	6852.79	0.335102	0.167551	+	0.985863i	$0.446414\pi$
$54$	0	0
$55$	−6199.90	−0.276362
$56$	0	0
$57$	1311.62	0.0534715
$58$	0	0
$59$	−4155.17	−0.155403	−0.0777014	−	0.996977i	$-0.524758\pi$
$59$	−4155.17	−0.155403	−0.0777014	+	0.996977i	$0.524758\pi$
$60$	0	0
$61$	−21911.4	−0.753955	−0.376978	−	0.926222i	$-0.623037\pi$
$61$	−21911.4	−0.753955	−0.376978	+	0.926222i	$0.623037\pi$
$62$	0	0
$63$	−10562.5	−0.335287
$64$	0	0
$65$	−16347.7	−0.479925
$66$	0	0
$67$	15164.2	0.412699	0.206349	−	0.978478i	$-0.433842\pi$
$67$	15164.2	0.412699	0.206349	+	0.978478i	$0.433842\pi$
$68$	0	0
$69$	756.999	0.0191414
$70$	0	0
$71$	−14031.4	−0.330334	−0.165167	−	0.986266i	$-0.552816\pi$
$71$	−14031.4	−0.330334	−0.165167	+	0.986266i	$0.552816\pi$
$72$	0	0
$73$	−23310.1	−0.511962	−0.255981	−	0.966682i	$-0.582398\pi$
$73$	−23310.1	−0.511962	−0.255981	+	0.966682i	$0.582398\pi$
$74$	0	0
$75$	−2406.98	−0.0494106
$76$	0	0
$77$	7154.72	0.137520
$78$	0	0
$79$	−64276.8	−1.15874	−0.579371	−	0.815064i	$-0.696702\pi$
$79$	−64276.8	−1.15874	−0.579371	+	0.815064i	$0.696702\pi$
$80$	0	0
$81$	57560.4	0.974790
$82$	0	0
$83$	−114349.	−1.82195	−0.910974	−	0.412465i	$-0.864668\pi$
$83$	−114349.	−1.82195	−0.910974	+	0.412465i	$0.864668\pi$
$84$	0	0
$85$	−28125.8	−0.422238
$86$	0	0
$87$	−7316.65	−0.103637
$88$	0	0
$89$	−63420.7	−0.848703	−0.424351	−	0.905498i	$-0.639498\pi$
$89$	−63420.7	−0.848703	−0.424351	+	0.905498i	$0.639498\pi$
$90$	0	0
$91$	18865.4	0.238815
$92$	0	0
$93$	8160.42	0.0978375
$94$	0	0
$95$	−34817.6	−0.395813
$96$	0	0
$97$	−91987.6	−0.992658	−0.496329	−	0.868134i	$-0.665319\pi$
$97$	−91987.6	−0.992658	−0.496329	+	0.868134i	$0.665319\pi$
$98$	0	0
$99$	−39326.6	−0.403272
$100$	0	0
$101$	122133.	1.19132	0.595660	−	0.803236i	$-0.296890\pi$
$101$	122133.	1.19132	0.595660	+	0.803236i	$0.296890\pi$
$102$	0	0
$103$	145517.	1.35151	0.675756	−	0.737125i	$-0.263817\pi$
$103$	145517.	1.35151	0.675756	+	0.737125i	$0.263817\pi$
$104$	0	0
$105$	−2382.90	−0.0210927
$106$	0	0
$107$	129578.	1.09414	0.547068	−	0.837088i	$-0.315744\pi$
$107$	129578.	1.09414	0.547068	+	0.837088i	$0.315744\pi$
$108$	0	0
$109$	−10434.1	−0.0841178	−0.0420589	−	0.999115i	$-0.513392\pi$
$109$	−10434.1	−0.0841178	−0.0420589	+	0.999115i	$0.513392\pi$
$110$	0	0
$111$	−15616.7	−0.120304
$112$	0	0
$113$	−253268.	−1.86589	−0.932943	−	0.360025i	$-0.882768\pi$
$113$	−253268.	−1.86589	−0.932943	+	0.360025i	$0.882768\pi$
$114$	0	0
$115$	−20094.8	−0.141690
$116$	0	0
$117$	−103695.	−0.700317
$118$	0	0
$119$	32457.3	0.210109
$120$	0	0
$121$	−134412.	−0.834595
$122$	0	0
$123$	15873.6	0.0946044
$124$	0	0
$125$	182602.	1.04527
$126$	0	0
$127$	−235260.	−1.29431	−0.647156	−	0.762358i	$-0.724042\pi$
$127$	−235260.	−1.29431	−0.647156	+	0.762358i	$0.724042\pi$
$128$	0	0
$129$	−7911.65	−0.0418594
$130$	0	0
$131$	−62921.5	−0.320347	−0.160174	−	0.987089i	$-0.551205\pi$
$131$	−62921.5	−0.320347	−0.160174	+	0.987089i	$0.551205\pi$
$132$	0	0
$133$	40179.7	0.196960
$134$	0	0
$135$	26307.0	0.124233
$136$	0	0
$137$	−134562.	−0.612521	−0.306260	−	0.951948i	$-0.599078\pi$
$137$	−134562.	−0.612521	−0.306260	+	0.951948i	$0.599078\pi$
$138$	0	0
$139$	−287261.	−1.26107	−0.630535	−	0.776161i	$-0.717165\pi$
$139$	−287261.	−1.26107	−0.630535	+	0.776161i	$0.717165\pi$
$140$	0	0
$141$	−20656.8	−0.0875015
$142$	0	0
$143$	70239.9	0.287239
$144$	0	0
$145$	194223.	0.767152
$146$	0	0
$147$	−21300.9	−0.0813028
$148$	0	0
$149$	−418105.	−1.54284	−0.771418	−	0.636329i	$-0.780452\pi$
$149$	−418105.	−1.54284	−0.771418	+	0.636329i	$0.780452\pi$
$150$	0	0
$151$	−486983.	−1.73809	−0.869043	−	0.494736i	$-0.835265\pi$
$151$	−486983.	−1.73809	−0.869043	+	0.494736i	$0.835265\pi$
$152$	0	0
$153$	−178405.	−0.616138
$154$	0	0
$155$	−216622.	−0.724223
$156$	0	0
$157$	−439288.	−1.42233	−0.711165	−	0.703026i	$-0.751832\pi$
$157$	−439288.	−1.42233	−0.711165	+	0.703026i	$0.751832\pi$
$158$	0	0
$159$	9806.34	0.0307620
$160$	0	0
$161$	23189.6	0.0705063
$162$	0	0
$163$	125475.	0.369905	0.184952	−	0.982747i	$-0.440787\pi$
$163$	125475.	0.369905	0.184952	+	0.982747i	$0.440787\pi$
$164$	0	0
$165$	−8872.05	−0.0253696
$166$	0	0
$167$	626996.	1.73970	0.869848	−	0.493319i	$-0.164217\pi$
$167$	626996.	1.73970	0.869848	+	0.493319i	$0.164217\pi$
$168$	0	0
$169$	−186086.	−0.501185
$170$	0	0
$171$	−220852.	−0.577578
$172$	0	0
$173$	502364.	1.27615	0.638077	−	0.769972i	$-0.279730\pi$
$173$	502364.	1.27615	0.638077	+	0.769972i	$0.279730\pi$
$174$	0	0
$175$	−73734.4	−0.182002
$176$	0	0
$177$	−5946.05	−0.0142658
$178$	0	0
$179$	131681.	0.307178	0.153589	−	0.988135i	$-0.450917\pi$
$179$	131681.	0.307178	0.153589	+	0.988135i	$0.450917\pi$
$180$	0	0
$181$	343209.	0.778686	0.389343	−	0.921093i	$-0.372702\pi$
$181$	343209.	0.778686	0.389343	+	0.921093i	$0.372702\pi$
$182$	0	0
$183$	−31355.2	−0.0692121
$184$	0	0
$185$	414551.	0.890529
$186$	0	0
$187$	120846.	0.252713
$188$	0	0
$189$	−30358.4	−0.0618194
$190$	0	0
$191$	−831093.	−1.64841	−0.824207	−	0.566289i	$-0.808379\pi$
$191$	−831093.	−1.64841	−0.824207	+	0.566289i	$0.808379\pi$
$192$	0	0
$193$	498764.	0.963832	0.481916	−	0.876217i	$-0.339941\pi$
$193$	498764.	0.963832	0.481916	+	0.876217i	$0.339941\pi$
$194$	0	0
$195$	−23393.6	−0.0440565
$196$	0	0
$197$	415236.	0.762306	0.381153	−	0.924512i	$-0.375527\pi$
$197$	415236.	0.762306	0.381153	+	0.924512i	$0.375527\pi$
$198$	0	0
$199$	969194.	1.73491	0.867457	−	0.497512i	$-0.165753\pi$
$199$	969194.	1.73491	0.867457	+	0.497512i	$0.165753\pi$
$200$	0	0
$201$	21700.0	0.0378852
$202$	0	0
$203$	−224135.	−0.381741
$204$	0	0
$205$	−421370.	−0.700291
$206$	0	0
$207$	−127464.	−0.206757
$208$	0	0
$209$	149598.	0.236897
$210$	0	0
$211$	−372669.	−0.576259	−0.288129	−	0.957592i	$-0.593033\pi$
$211$	−372669.	−0.576259	−0.288129	+	0.957592i	$0.593033\pi$
$212$	0	0
$213$	−20078.9	−0.0303243
$214$	0	0
$215$	210018.	0.309856
$216$	0	0
$217$	249983.	0.360380
$218$	0	0
$219$	−33356.8	−0.0469974
$220$	0	0
$221$	318643.	0.438857
$222$	0	0
$223$	668196.	0.899791	0.449895	−	0.893081i	$-0.351461\pi$
$223$	668196.	0.899791	0.449895	+	0.893081i	$0.351461\pi$
$224$	0	0
$225$	405288.	0.533713
$226$	0	0
$227$	1.01295e6	1.30473	0.652367	−	0.757903i	$-0.273776\pi$
$227$	1.01295e6	1.30473	0.652367	+	0.757903i	$0.273776\pi$
$228$	0	0
$229$	−342003.	−0.430964	−0.215482	−	0.976508i	$-0.569132\pi$
$229$	−342003.	−0.430964	−0.215482	+	0.976508i	$0.569132\pi$
$230$	0	0
$231$	10238.4	0.0126242
$232$	0	0
$233$	−792950.	−0.956877	−0.478438	−	0.878121i	$-0.658797\pi$
$233$	−792950.	−0.956877	−0.478438	+	0.878121i	$0.658797\pi$
$234$	0	0
$235$	548343.	0.647713
$236$	0	0
$237$	−91980.2	−0.106371
$238$	0	0
$239$	−520763.	−0.589719	−0.294859	−	0.955541i	$-0.595273\pi$
$239$	−520763.	−0.589719	−0.294859	+	0.955541i	$0.595273\pi$
$240$	0	0
$241$	−867128.	−0.961702	−0.480851	−	0.876802i	$-0.659672\pi$
$241$	−867128.	−0.961702	−0.480851	+	0.876802i	$0.659672\pi$
$242$	0	0
$243$	250655.	0.272308
$244$	0	0
$245$	565442.	0.601829
$246$	0	0
$247$	394455.	0.411392
$248$	0	0
$249$	−163633.	−0.167252
$250$	0	0
$251$	590873.	0.591983	0.295992	−	0.955191i	$-0.404350\pi$
$251$	590873.	0.591983	0.295992	+	0.955191i	$0.404350\pi$
$252$	0	0
$253$	86339.8	0.0848027
$254$	0	0
$255$	−40248.0	−0.0387609
$256$	0	0
$257$	−448650.	−0.423716	−0.211858	−	0.977300i	$-0.567951\pi$
$257$	−448650.	−0.423716	−0.211858	+	0.977300i	$0.567951\pi$
$258$	0	0
$259$	−478394.	−0.443135
$260$	0	0
$261$	1.23198e6	1.11944
$262$	0	0
$263$	−434704.	−0.387529	−0.193765	−	0.981048i	$-0.562070\pi$
$263$	−434704.	−0.387529	−0.193765	+	0.981048i	$0.562070\pi$
$264$	0	0
$265$	−260313.	−0.227710
$266$	0	0
$267$	−90755.0	−0.0779098
$268$	0	0
$269$	873326.	0.735861	0.367930	−	0.929853i	$-0.380067\pi$
$269$	873326.	0.735861	0.367930	+	0.929853i	$0.380067\pi$
$270$	0	0
$271$	1.16572e6	0.964208	0.482104	−	0.876114i	$-0.339873\pi$
$271$	1.16572e6	0.964208	0.482104	+	0.876114i	$0.339873\pi$
$272$	0	0
$273$	26996.3	0.0219229
$274$	0	0
$275$	−274529.	−0.218906
$276$	0	0
$277$	−604025.	−0.472994	−0.236497	−	0.971632i	$-0.575999\pi$
$277$	−604025.	−0.472994	−0.236497	+	0.971632i	$0.575999\pi$
$278$	0	0
$279$	−1.37405e6	−1.05680
$280$	0	0
$281$	1.72154e6	1.30062	0.650310	−	0.759669i	$-0.274639\pi$
$281$	1.72154e6	1.30062	0.650310	+	0.759669i	$0.274639\pi$
$282$	0	0
$283$	2.01557e6	1.49600	0.748000	−	0.663699i	$-0.231014\pi$
$283$	2.01557e6	1.49600	0.748000	+	0.663699i	$0.231014\pi$
$284$	0	0
$285$	−49824.0	−0.0363351
$286$	0	0
$287$	486263.	0.348471
$288$	0	0
$289$	−871642.	−0.613894
$290$	0	0
$291$	−131634.	−0.0911248
$292$	0	0
$293$	1.45451e6	0.989798	0.494899	−	0.868951i	$-0.335205\pi$
$293$	1.45451e6	0.989798	0.494899	+	0.868951i	$0.335205\pi$
$294$	0	0
$295$	157840.	0.105600
$296$	0	0
$297$	−113031.	−0.0743544
$298$	0	0
$299$	227659.	0.147267
$300$	0	0
$301$	−242362.	−0.154187
$302$	0	0
$303$	174772.	0.109362
$304$	0	0
$305$	832337.	0.512329
$306$	0	0
$307$	−272463.	−0.164991	−0.0824957	−	0.996591i	$-0.526289\pi$
$307$	−272463.	−0.164991	−0.0824957	+	0.996591i	$0.526289\pi$
$308$	0	0
$309$	208235.	0.124067
$310$	0	0
$311$	2.33041e6	1.36625	0.683126	−	0.730301i	$-0.260620\pi$
$311$	2.33041e6	1.36625	0.683126	+	0.730301i	$0.260620\pi$
$312$	0	0
$313$	−1.67227e6	−0.964818	−0.482409	−	0.875946i	$-0.660238\pi$
$313$	−1.67227e6	−0.964818	−0.482409	+	0.875946i	$0.660238\pi$
$314$	0	0
$315$	401233.	0.227835
$316$	0	0
$317$	1.18782e6	0.663901	0.331950	−	0.943297i	$-0.392293\pi$
$317$	1.18782e6	0.663901	0.331950	+	0.943297i	$0.392293\pi$
$318$	0	0
$319$	−834503.	−0.459146
$320$	0	0
$321$	185426.	0.100440
$322$	0	0
$323$	678649.	0.361942
$324$	0	0
$325$	−723872.	−0.380148
$326$	0	0
$327$	−14931.2	−0.00772191
$328$	0	0
$329$	−632791.	−0.322308
$330$	0	0
$331$	2.09261e6	1.04983	0.524915	−	0.851155i	$-0.324097\pi$
$331$	2.09261e6	1.04983	0.524915	+	0.851155i	$0.324097\pi$
$332$	0	0
$333$	2.62954e6	1.29948
$334$	0	0
$335$	−576035.	−0.280438
$336$	0	0
$337$	2.13994e6	1.02642	0.513211	−	0.858262i	$-0.328456\pi$
$337$	2.13994e6	1.02642	0.513211	+	0.858262i	$0.328456\pi$
$338$	0	0
$339$	−362427.	−0.171286
$340$	0	0
$341$	930740.	0.433453
$342$	0	0
$343$	−1.38929e6	−0.637611
$344$	0	0
$345$	−28755.7	−0.0130070
$346$	0	0
$347$	−3.78048e6	−1.68548	−0.842739	−	0.538322i	$-0.819059\pi$
$347$	−3.78048e6	−1.68548	−0.842739	+	0.538322i	$0.819059\pi$
$348$	0	0
$349$	3.46473e6	1.52267	0.761334	−	0.648360i	$-0.224545\pi$
$349$	3.46473e6	1.52267	0.761334	+	0.648360i	$0.224545\pi$
$350$	0	0
$351$	−298037.	−0.129123
$352$	0	0
$353$	−2.39834e6	−1.02441	−0.512206	−	0.858863i	$-0.671171\pi$
$353$	−2.39834e6	−1.02441	−0.512206	+	0.858863i	$0.671171\pi$
$354$	0	0
$355$	533002.	0.224470
$356$	0	0
$357$	46446.4	0.0192878
$358$	0	0
$359$	2.04171e6	0.836098	0.418049	−	0.908424i	$-0.362714\pi$
$359$	2.04171e6	0.836098	0.418049	+	0.908424i	$0.362714\pi$
$360$	0	0
$361$	−1.63598e6	−0.660710
$362$	0	0
$363$	−192344.	−0.0766148
$364$	0	0
$365$	885469.	0.347889
$366$	0	0
$367$	−2.41363e6	−0.935417	−0.467709	−	0.883883i	$-0.654920\pi$
$367$	−2.41363e6	−0.935417	−0.467709	+	0.883883i	$0.654920\pi$
$368$	0	0
$369$	−2.67279e6	−1.02188
$370$	0	0
$371$	300403.	0.113310
$372$	0	0
$373$	1.83073e6	0.681321	0.340660	−	0.940186i	$-0.389349\pi$
$373$	1.83073e6	0.681321	0.340660	+	0.940186i	$0.389349\pi$
$374$	0	0
$375$	261304.	0.0959549
$376$	0	0
$377$	−2.20039e6	−0.797347
$378$	0	0
$379$	−5.24532e6	−1.87575	−0.937874	−	0.346977i	$-0.887208\pi$
$379$	−5.24532e6	−1.87575	−0.937874	+	0.346977i	$0.887208\pi$
$380$	0	0
$381$	−336657.	−0.118816
$382$	0	0
$383$	754501.	0.262823	0.131411	−	0.991328i	$-0.458049\pi$
$383$	754501.	0.262823	0.131411	+	0.991328i	$0.458049\pi$
$384$	0	0
$385$	−271782.	−0.0934479
$386$	0	0
$387$	1.33217e6	0.452148
$388$	0	0
$389$	2.05221e6	0.687618	0.343809	−	0.939040i	$-0.388283\pi$
$389$	2.05221e6	0.687618	0.343809	+	0.939040i	$0.388283\pi$
$390$	0	0
$391$	391680.	0.129565
$392$	0	0
$393$	−90040.7	−0.0294074
$394$	0	0
$395$	2.44165e6	0.787391
$396$	0	0
$397$	3.99917e6	1.27348	0.636742	−	0.771077i	$-0.280281\pi$
$397$	3.99917e6	1.27348	0.636742	+	0.771077i	$0.280281\pi$
$398$	0	0
$399$	57497.2	0.0180807
$400$	0	0
$401$	−5.10734e6	−1.58611	−0.793056	−	0.609149i	$-0.791511\pi$
$401$	−5.10734e6	−1.58611	−0.793056	+	0.609149i	$0.791511\pi$
$402$	0	0
$403$	2.45415e6	0.752729
$404$	0	0
$405$	−2.18652e6	−0.662392
$406$	0	0
$407$	−1.78116e6	−0.532989
$408$	0	0
$409$	−3.06823e6	−0.906942	−0.453471	−	0.891271i	$-0.649814\pi$
$409$	−3.06823e6	−0.906942	−0.453471	+	0.891271i	$0.649814\pi$
$410$	0	0
$411$	−192558.	−0.0562286
$412$	0	0
$413$	−182149.	−0.0525473
$414$	0	0
$415$	4.34370e6	1.23805
$416$	0	0
$417$	−411070.	−0.115765
$418$	0	0
$419$	−3.33601e6	−0.928308	−0.464154	−	0.885754i	$-0.653642\pi$
$419$	−3.33601e6	−0.928308	−0.464154	+	0.885754i	$0.653642\pi$
$420$	0	0
$421$	−2.75080e6	−0.756404	−0.378202	−	0.925723i	$-0.623458\pi$
$421$	−2.75080e6	−0.756404	−0.378202	+	0.925723i	$0.623458\pi$
$422$	0	0
$423$	3.47820e6	0.945156
$424$	0	0
$425$	−1.24540e6	−0.334454
$426$	0	0
$427$	−960521.	−0.254940
$428$	0	0
$429$	100513.	0.0263682
$430$	0	0
$431$	−926380.	−0.240213	−0.120106	−	0.992761i	$-0.538324\pi$
$431$	−926380.	−0.240213	−0.120106	+	0.992761i	$0.538324\pi$
$432$	0	0
$433$	680820.	0.174507	0.0872535	−	0.996186i	$-0.472191\pi$
$433$	680820.	0.174507	0.0872535	+	0.996186i	$0.472191\pi$
$434$	0	0
$435$	277934.	0.0704236
$436$	0	0
$437$	484870.	0.121457
$438$	0	0
$439$	−2.17406e6	−0.538406	−0.269203	−	0.963083i	$-0.586760\pi$
$439$	−2.17406e6	−0.538406	−0.269203	+	0.963083i	$0.586760\pi$
$440$	0	0
$441$	3.58666e6	0.878200
$442$	0	0
$443$	−4.17517e6	−1.01080	−0.505400	−	0.862885i	$-0.668655\pi$
$443$	−4.17517e6	−1.01080	−0.505400	+	0.862885i	$0.668655\pi$
$444$	0	0
$445$	2.40913e6	0.576713
$446$	0	0
$447$	−598308.	−0.141630
$448$	0	0
$449$	−4.68946e6	−1.09776	−0.548880	−	0.835901i	$-0.684945\pi$
$449$	−4.68946e6	−1.09776	−0.548880	+	0.835901i	$0.684945\pi$
$450$	0	0
$451$	1.81046e6	0.419130
$452$	0	0
$453$	−696873.	−0.159554
$454$	0	0
$455$	−716629.	−0.162280
$456$	0	0
$457$	−1.68983e6	−0.378487	−0.189244	−	0.981930i	$-0.560604\pi$
$457$	−1.68983e6	−0.378487	−0.189244	+	0.981930i	$0.560604\pi$
$458$	0	0
$459$	−512764.	−0.113602
$460$	0	0
$461$	−4.28765e6	−0.939652	−0.469826	−	0.882759i	$-0.655683\pi$
$461$	−4.28765e6	−0.939652	−0.469826	+	0.882759i	$0.655683\pi$
$462$	0	0
$463$	−5.26738e6	−1.14194	−0.570969	−	0.820972i	$-0.693432\pi$
$463$	−5.26738e6	−1.14194	−0.570969	+	0.820972i	$0.693432\pi$
$464$	0	0
$465$	−309986.	−0.0664828
$466$	0	0
$467$	3.66305e6	0.777231	0.388616	−	0.921400i	$-0.372953\pi$
$467$	3.66305e6	0.777231	0.388616	+	0.921400i	$0.372953\pi$
$468$	0	0
$469$	664748.	0.139548
$470$	0	0
$471$	−628621.	−0.130568
$472$	0	0
$473$	−902366.	−0.185451
$474$	0	0
$475$	−1.54171e6	−0.313523
$476$	0	0
$477$	−1.65119e6	−0.332279
$478$	0	0
$479$	−7.03353e6	−1.40067	−0.700333	−	0.713816i	$-0.746965\pi$
$479$	−7.03353e6	−1.40067	−0.700333	+	0.713816i	$0.746965\pi$
$480$	0	0
$481$	−4.69653e6	−0.925581
$482$	0	0
$483$	33184.3	0.00647239
$484$	0	0
$485$	3.49428e6	0.674534
$486$	0	0
$487$	−5.05170e6	−0.965195	−0.482598	−	0.875842i	$-0.660307\pi$
$487$	−5.05170e6	−0.965195	−0.482598	+	0.875842i	$0.660307\pi$
$488$	0	0
$489$	179555.	0.0339568
$490$	0	0
$491$	3.90292e6	0.730610	0.365305	−	0.930888i	$-0.380965\pi$
$491$	3.90292e6	0.730610	0.365305	+	0.930888i	$0.380965\pi$
$492$	0	0
$493$	−3.78571e6	−0.701505
$494$	0	0
$495$	1.49388e6	0.274033
$496$	0	0
$497$	−615087.	−0.111698
$498$	0	0
$499$	−1.13818e6	−0.204625	−0.102312	−	0.994752i	$-0.532624\pi$
$499$	−1.13818e6	−0.204625	−0.102312	+	0.994752i	$0.532624\pi$
$500$	0	0
$501$	897231.	0.159702
$502$	0	0
$503$	7.48030e6	1.31825	0.659127	−	0.752032i	$-0.270926\pi$
$503$	7.48030e6	1.31825	0.659127	+	0.752032i	$0.270926\pi$
$504$	0	0
$505$	−4.63939e6	−0.809529
$506$	0	0
$507$	−266290.	−0.0460081
$508$	0	0
$509$	3.03026e6	0.518424	0.259212	−	0.965821i	$-0.416537\pi$
$509$	3.03026e6	0.518424	0.259212	+	0.965821i	$0.416537\pi$
$510$	0	0
$511$	−1.02184e6	−0.173113
$512$	0	0
$513$	−634763.	−0.106492
$514$	0	0
$515$	−5.52767e6	−0.918383
$516$	0	0
$517$	−2.35602e6	−0.387661
$518$	0	0
$519$	718883.	0.117149
$520$	0	0
$521$	−1.90771e6	−0.307905	−0.153953	−	0.988078i	$-0.549200\pi$
$521$	−1.90771e6	−0.307905	−0.153953	+	0.988078i	$0.549200\pi$
$522$	0	0
$523$	−2.40656e6	−0.384718	−0.192359	−	0.981325i	$-0.561614\pi$
$523$	−2.40656e6	−0.384718	−0.192359	+	0.981325i	$0.561614\pi$
$524$	0	0
$525$	−105514.	−0.0167075
$526$	0	0
$527$	4.22229e6	0.662250
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	1.00120e6	0.154093
$532$	0	0
$533$	4.77378e6	0.727855
$534$	0	0
$535$	−4.92221e6	−0.743490
$536$	0	0
$537$	188435.	0.0281985
$538$	0	0
$539$	−2.42949e6	−0.360199
$540$	0	0
$541$	−711055.	−0.104450	−0.0522252	−	0.998635i	$-0.516631\pi$
$541$	−711055.	−0.104450	−0.0522252	+	0.998635i	$0.516631\pi$
$542$	0	0
$543$	491132.	0.0714823
$544$	0	0
$545$	396354.	0.0571600
$546$	0	0
$547$	−7.21528e6	−1.03106	−0.515531	−	0.856871i	$-0.672405\pi$
$547$	−7.21528e6	−1.03106	−0.515531	+	0.856871i	$0.672405\pi$
$548$	0	0
$549$	5.27960e6	0.747601
$550$	0	0
$551$	−4.68643e6	−0.657602
$552$	0	0
$553$	−2.81768e6	−0.391813
$554$	0	0
$555$	593222.	0.0817495
$556$	0	0
$557$	−920675.	−0.125739	−0.0628693	−	0.998022i	$-0.520025\pi$
$557$	−920675.	−0.125739	−0.0628693	+	0.998022i	$0.520025\pi$
$558$	0	0
$559$	−2.37934e6	−0.322052
$560$	0	0
$561$	172930.	0.0231987
$562$	0	0
$563$	7.13949e6	0.949284	0.474642	−	0.880179i	$-0.342578\pi$
$563$	7.13949e6	0.949284	0.474642	+	0.880179i	$0.342578\pi$
$564$	0	0
$565$	9.62077e6	1.26791
$566$	0	0
$567$	2.52325e6	0.329612
$568$	0	0
$569$	1.32562e7	1.71648	0.858239	−	0.513250i	$-0.171559\pi$
$569$	1.32562e7	1.71648	0.858239	+	0.513250i	$0.171559\pi$
$570$	0	0
$571$	8.74927e6	1.12300	0.561502	−	0.827475i	$-0.310224\pi$
$571$	8.74927e6	1.12300	0.561502	+	0.827475i	$0.310224\pi$
$572$	0	0
$573$	−1.18929e6	−0.151322
$574$	0	0
$575$	−889793.	−0.112233
$576$	0	0
$577$	2.28815e6	0.286118	0.143059	−	0.989714i	$-0.454306\pi$
$577$	2.28815e6	0.286118	0.143059	+	0.989714i	$0.454306\pi$
$578$	0	0
$579$	713731.	0.0884786
$580$	0	0
$581$	−5.01266e6	−0.616067
$582$	0	0
$583$	1.11847e6	0.136286
$584$	0	0
$585$	3.93902e6	0.475881
$586$	0	0
$587$	−1.48384e7	−1.77743	−0.888714	−	0.458463i	$-0.848400\pi$
$587$	−1.48384e7	−1.77743	−0.888714	+	0.458463i	$0.848400\pi$
$588$	0	0
$589$	5.22688e6	0.620804
$590$	0	0
$591$	594202.	0.0699787
$592$	0	0
$593$	−7.67836e6	−0.896668	−0.448334	−	0.893866i	$-0.647982\pi$
$593$	−7.67836e6	−0.896668	−0.448334	+	0.893866i	$0.647982\pi$
$594$	0	0
$595$	−1.23294e6	−0.142774
$596$	0	0
$597$	1.38692e6	0.159263
$598$	0	0
$599$	599539.	0.0682733	0.0341366	−	0.999417i	$-0.489132\pi$
$599$	599539.	0.0682733	0.0341366	+	0.999417i	$0.489132\pi$
$600$	0	0
$601$	−6.92017e6	−0.781503	−0.390751	−	0.920496i	$-0.627785\pi$
$601$	−6.92017e6	−0.781503	−0.390751	+	0.920496i	$0.627785\pi$
$602$	0	0
$603$	−3.65385e6	−0.409221
$604$	0	0
$605$	5.10585e6	0.567126
$606$	0	0
$607$	6.65094e6	0.732675	0.366337	−	0.930482i	$-0.380612\pi$
$607$	6.65094e6	0.732675	0.366337	+	0.930482i	$0.380612\pi$
$608$	0	0
$609$	−320737.	−0.0350434
$610$	0	0
$611$	−6.21229e6	−0.673207
$612$	0	0
$613$	−1.07456e7	−1.15499	−0.577497	−	0.816393i	$-0.695970\pi$
$613$	−1.07456e7	−1.15499	−0.577497	+	0.816393i	$0.695970\pi$
$614$	0	0
$615$	−602980.	−0.0642859
$616$	0	0
$617$	−1.66965e7	−1.76568	−0.882841	−	0.469673i	$-0.844372\pi$
$617$	−1.66965e7	−1.76568	−0.882841	+	0.469673i	$0.844372\pi$
$618$	0	0
$619$	1.06977e7	1.12218	0.561090	−	0.827755i	$-0.310382\pi$
$619$	1.06977e7	1.12218	0.561090	+	0.827755i	$0.310382\pi$
$620$	0	0
$621$	−366351.	−0.0381214
$622$	0	0
$623$	−2.78015e6	−0.286977
$624$	0	0
$625$	−1.68007e6	−0.172039
$626$	0	0
$627$	214074.	0.0217468
$628$	0	0
$629$	−8.08024e6	−0.814325
$630$	0	0
$631$	−1.31871e7	−1.31849	−0.659243	−	0.751930i	$-0.729123\pi$
$631$	−1.31871e7	−1.31849	−0.659243	+	0.751930i	$0.729123\pi$
$632$	0	0
$633$	−533290.	−0.0528998
$634$	0	0
$635$	8.93670e6	0.879514
$636$	0	0
$637$	−6.40601e6	−0.625517
$638$	0	0
$639$	3.38089e6	0.327551
$640$	0	0
$641$	−849150.	−0.0816280	−0.0408140	−	0.999167i	$-0.512995\pi$
$641$	−849150.	−0.0816280	−0.0408140	+	0.999167i	$0.512995\pi$
$642$	0	0
$643$	2.31164e6	0.220491	0.110246	−	0.993904i	$-0.464836\pi$
$643$	2.31164e6	0.220491	0.110246	+	0.993904i	$0.464836\pi$
$644$	0	0
$645$	300536.	0.0284444
$646$	0	0
$647$	1.51827e7	1.42590	0.712949	−	0.701216i	$-0.247359\pi$
$647$	1.51827e7	1.42590	0.712949	+	0.701216i	$0.247359\pi$
$648$	0	0
$649$	−678179.	−0.0632023
$650$	0	0
$651$	357725.	0.0330824
$652$	0	0
$653$	1.57201e7	1.44269	0.721343	−	0.692578i	$-0.243525\pi$
$653$	1.57201e7	1.44269	0.721343	+	0.692578i	$0.243525\pi$
$654$	0	0
$655$	2.39016e6	0.217683
$656$	0	0
$657$	5.61663e6	0.507647
$658$	0	0
$659$	1.71369e7	1.53716	0.768580	−	0.639754i	$-0.220964\pi$
$659$	1.71369e7	1.53716	0.768580	+	0.639754i	$0.220964\pi$
$660$	0	0
$661$	1.52598e7	1.35845	0.679227	−	0.733928i	$-0.262315\pi$
$661$	1.52598e7	1.35845	0.679227	+	0.733928i	$0.262315\pi$
$662$	0	0
$663$	455978.	0.0402865
$664$	0	0
$665$	−1.52628e6	−0.133839
$666$	0	0
$667$	−2.70476e6	−0.235404
$668$	0	0
$669$	956188.	0.0825996
$670$	0	0
$671$	−3.57623e6	−0.306633
$672$	0	0
$673$	−1.35466e7	−1.15290	−0.576451	−	0.817132i	$-0.695563\pi$
$673$	−1.35466e7	−1.15290	−0.576451	+	0.817132i	$0.695563\pi$
$674$	0	0
$675$	1.16486e6	0.0984048
$676$	0	0
$677$	1.13711e7	0.953520	0.476760	−	0.879034i	$-0.341811\pi$
$677$	1.13711e7	0.953520	0.476760	+	0.879034i	$0.341811\pi$
$678$	0	0
$679$	−4.03242e6	−0.335654
$680$	0	0
$681$	1.44953e6	0.119773
$682$	0	0
$683$	498823.	0.0409161	0.0204581	−	0.999791i	$-0.493488\pi$
$683$	498823.	0.0409161	0.0204581	+	0.999791i	$0.493488\pi$
$684$	0	0
$685$	5.11153e6	0.416222
$686$	0	0
$687$	−489406.	−0.0395619
$688$	0	0
$689$	2.94914e6	0.236672
$690$	0	0
$691$	−1.03020e7	−0.820783	−0.410392	−	0.911909i	$-0.634608\pi$
$691$	−1.03020e7	−0.820783	−0.410392	+	0.911909i	$0.634608\pi$
$692$	0	0
$693$	−1.72394e6	−0.136361
$694$	0	0
$695$	1.09120e7	0.856926
$696$	0	0
$697$	8.21315e6	0.640366
$698$	0	0
$699$	−1.13471e6	−0.0878401
$700$	0	0
$701$	−3.29623e6	−0.253351	−0.126675	−	0.991944i	$-0.540431\pi$
$701$	−3.29623e6	−0.253351	−0.126675	+	0.991944i	$0.540431\pi$
$702$	0	0
$703$	−1.00027e7	−0.763361
$704$	0	0
$705$	784679.	0.0594592
$706$	0	0
$707$	5.35388e6	0.402829
$708$	0	0
$709$	1.35819e7	1.01472	0.507359	−	0.861735i	$-0.330622\pi$
$709$	1.35819e7	1.01472	0.507359	+	0.861735i	$0.330622\pi$
$710$	0	0
$711$	1.54876e7	1.14898
$712$	0	0
$713$	3.01667e6	0.222231
$714$	0	0
$715$	−2.66816e6	−0.195185
$716$	0	0
$717$	−745212.	−0.0541354
$718$	0	0
$719$	8.30096e6	0.598834	0.299417	−	0.954122i	$-0.403208\pi$
$719$	8.30096e6	0.598834	0.299417	+	0.954122i	$0.403208\pi$
$720$	0	0
$721$	6.37896e6	0.456995
$722$	0	0
$723$	−1.24086e6	−0.0882831
$724$	0	0
$725$	8.60014e6	0.607660
$726$	0	0
$727$	5.01347e6	0.351806	0.175903	−	0.984408i	$-0.443716\pi$
$727$	5.01347e6	0.351806	0.175903	+	0.984408i	$0.443716\pi$
$728$	0	0
$729$	−1.36285e7	−0.949792
$730$	0	0
$731$	−4.09358e6	−0.283341
$732$	0	0
$733$	−1.48950e7	−1.02395	−0.511977	−	0.858999i	$-0.671087\pi$
$733$	−1.48950e7	−1.02395	−0.511977	+	0.858999i	$0.671087\pi$
$734$	0	0
$735$	809148.	0.0552471
$736$	0	0
$737$	2.47500e6	0.167844
$738$	0	0
$739$	−8.39705e6	−0.565608	−0.282804	−	0.959178i	$-0.591265\pi$
$739$	−8.39705e6	−0.565608	−0.282804	+	0.959178i	$0.591265\pi$
$740$	0	0
$741$	564466.	0.0377652
$742$	0	0
$743$	1.54362e7	1.02582	0.512908	−	0.858444i	$-0.328568\pi$
$743$	1.54362e7	1.02582	0.512908	+	0.858444i	$0.328568\pi$
$744$	0	0
$745$	1.58823e7	1.04839
$746$	0	0
$747$	2.75526e7	1.80659
$748$	0	0
$749$	5.68025e6	0.369967
$750$	0	0
$751$	5.26274e6	0.340496	0.170248	−	0.985401i	$-0.445543\pi$
$751$	5.26274e6	0.340496	0.170248	+	0.985401i	$0.445543\pi$
$752$	0	0
$753$	845539.	0.0543433
$754$	0	0
$755$	1.84988e7	1.18107
$756$	0	0
$757$	−2.58188e7	−1.63756	−0.818779	−	0.574109i	$-0.805348\pi$
$757$	−2.58188e7	−1.63756	−0.818779	+	0.574109i	$0.805348\pi$
$758$	0	0
$759$	123552.	0.00778478
$760$	0	0
$761$	4.74356e6	0.296922	0.148461	−	0.988918i	$-0.452568\pi$
$761$	4.74356e6	0.296922	0.148461	+	0.988918i	$0.452568\pi$
$762$	0	0
$763$	−457395.	−0.0284433
$764$	0	0
$765$	6.77697e6	0.418680
$766$	0	0
$767$	−1.78820e6	−0.109756
$768$	0	0
$769$	2.16055e7	1.31749	0.658746	−	0.752365i	$-0.271087\pi$
$769$	2.16055e7	1.31749	0.658746	+	0.752365i	$0.271087\pi$
$770$	0	0
$771$	−642018.	−0.0388966
$772$	0	0
$773$	4.60087e6	0.276944	0.138472	−	0.990366i	$-0.455781\pi$
$773$	4.60087e6	0.276944	0.138472	+	0.990366i	$0.455781\pi$
$774$	0	0
$775$	−9.59193e6	−0.573656
$776$	0	0
$777$	−684582.	−0.0406792
$778$	0	0
$779$	1.01673e7	0.600289
$780$	0	0
$781$	−2.29010e6	−0.134347
$782$	0	0
$783$	3.54091e6	0.206400
$784$	0	0
$785$	1.66870e7	0.966505
$786$	0	0
$787$	−3.20780e7	−1.84616	−0.923082	−	0.384603i	$-0.874338\pi$
$787$	−3.20780e7	−1.84616	−0.923082	+	0.384603i	$0.874338\pi$
$788$	0	0
$789$	−622062.	−0.0355747
$790$	0	0
$791$	−1.11024e7	−0.630924
$792$	0	0
$793$	−9.42971e6	−0.532495
$794$	0	0
$795$	−372508.	−0.0209035
$796$	0	0
$797$	950693.	0.0530145	0.0265072	−	0.999649i	$-0.491561\pi$
$797$	950693.	0.0530145	0.0265072	+	0.999649i	$0.491561\pi$
$798$	0	0
$799$	−1.06881e7	−0.592287
$800$	0	0
$801$	1.52813e7	0.841551
$802$	0	0
$803$	−3.80452e6	−0.208215
$804$	0	0
$805$	−880890.	−0.0479106
$806$	0	0
$807$	1.24973e6	0.0675511
$808$	0	0
$809$	1.89226e7	1.01650	0.508252	−	0.861208i	$-0.330292\pi$
$809$	1.89226e7	1.01650	0.508252	+	0.861208i	$0.330292\pi$
$810$	0	0
$811$	−4.78517e6	−0.255473	−0.127737	−	0.991808i	$-0.540771\pi$
$811$	−4.78517e6	−0.255473	−0.127737	+	0.991808i	$0.540771\pi$
$812$	0	0
$813$	1.66814e6	0.0885130
$814$	0	0
$815$	−4.76637e6	−0.251359
$816$	0	0
$817$	−5.06754e6	−0.265608
$818$	0	0
$819$	−4.54565e6	−0.236803
$820$	0	0
$821$	5.51231e6	0.285414	0.142707	−	0.989765i	$-0.454419\pi$
$821$	5.51231e6	0.285414	0.142707	+	0.989765i	$0.454419\pi$
$822$	0	0
$823$	1.44617e7	0.744249	0.372125	−	0.928183i	$-0.378629\pi$
$823$	1.44617e7	0.744249	0.372125	+	0.928183i	$0.378629\pi$
$824$	0	0
$825$	−392852.	−0.0200953
$826$	0	0
$827$	2.78469e7	1.41584	0.707919	−	0.706294i	$-0.249634\pi$
$827$	2.78469e7	1.41584	0.707919	+	0.706294i	$0.249634\pi$
$828$	0	0
$829$	1.89279e6	0.0956569	0.0478285	−	0.998856i	$-0.484770\pi$
$829$	1.89279e6	0.0956569	0.0478285	+	0.998856i	$0.484770\pi$
$830$	0	0
$831$	−864360.	−0.0434202
$832$	0	0
$833$	−1.10213e7	−0.550329
$834$	0	0
$835$	−2.38174e7	−1.18216
$836$	0	0
$837$	−3.94925e6	−0.194850
$838$	0	0
$839$	−8.81372e6	−0.432269	−0.216135	−	0.976364i	$-0.569345\pi$
$839$	−8.81372e6	−0.432269	−0.216135	+	0.976364i	$0.569345\pi$
$840$	0	0
$841$	5.63121e6	0.274544
$842$	0	0
$843$	2.46352e6	0.119395
$844$	0	0
$845$	7.06876e6	0.340566
$846$	0	0
$847$	−5.89219e6	−0.282207
$848$	0	0
$849$	2.88428e6	0.137331
$850$	0	0
$851$	−5.77304e6	−0.273263
$852$	0	0
$853$	4.21641e6	0.198413	0.0992064	−	0.995067i	$-0.468370\pi$
$853$	4.21641e6	0.198413	0.0992064	+	0.995067i	$0.468370\pi$
$854$	0	0
$855$	8.38937e6	0.392477
$856$	0	0
$857$	1.36923e7	0.636829	0.318415	−	0.947952i	$-0.396850\pi$
$857$	1.36923e7	0.636829	0.318415	+	0.947952i	$0.396850\pi$
$858$	0	0
$859$	−1.31118e7	−0.606291	−0.303145	−	0.952944i	$-0.598037\pi$
$859$	−1.31118e7	−0.606291	−0.303145	+	0.952944i	$0.598037\pi$
$860$	0	0
$861$	695843.	0.0319892
$862$	0	0
$863$	−5.67126e6	−0.259210	−0.129605	−	0.991566i	$-0.541371\pi$
$863$	−5.67126e6	−0.259210	−0.129605	+	0.991566i	$0.541371\pi$
$864$	0	0
$865$	−1.90830e7	−0.867176
$866$	0	0
$867$	−1.24732e6	−0.0563547
$868$	0	0
$869$	−1.04908e7	−0.471260
$870$	0	0
$871$	6.52602e6	0.291476
$872$	0	0
$873$	2.21646e7	0.984293
$874$	0	0
$875$	8.00465e6	0.353445
$876$	0	0
$877$	−2.51970e7	−1.10624	−0.553120	−	0.833101i	$-0.686563\pi$
$877$	−2.51970e7	−1.10624	−0.553120	+	0.833101i	$0.686563\pi$
$878$	0	0
$879$	2.08140e6	0.0908622
$880$	0	0
$881$	−2.99532e7	−1.30018	−0.650090	−	0.759857i	$-0.725269\pi$
$881$	−2.99532e7	−1.30018	−0.650090	+	0.759857i	$0.725269\pi$
$882$	0	0
$883$	2.98802e7	1.28968	0.644840	−	0.764318i	$-0.276924\pi$
$883$	2.98802e7	1.28968	0.644840	+	0.764318i	$0.276924\pi$
$884$	0	0
$885$	225869.	0.00969392
$886$	0	0
$887$	7.65547e6	0.326710	0.163355	−	0.986567i	$-0.447768\pi$
$887$	7.65547e6	0.326710	0.163355	+	0.986567i	$0.447768\pi$
$888$	0	0
$889$	−1.03130e7	−0.437654
$890$	0	0
$891$	9.39462e6	0.396447
$892$	0	0
$893$	−1.32310e7	−0.555219
$894$	0	0
$895$	−5.00208e6	−0.208734
$896$	0	0
$897$	325779.	0.0135189
$898$	0	0
$899$	−2.91572e7	−1.20322
$900$	0	0
$901$	5.07391e6	0.208224
$902$	0	0
$903$	−346820.	−0.0141542
$904$	0	0
$905$	−1.30373e7	−0.529134
$906$	0	0
$907$	3.50928e7	1.41644	0.708222	−	0.705990i	$-0.249498\pi$
$907$	3.50928e7	1.41644	0.708222	+	0.705990i	$0.249498\pi$
$908$	0	0
$909$	−2.94282e7	−1.18128
$910$	0	0
$911$	3.03942e7	1.21337	0.606687	−	0.794941i	$-0.292498\pi$
$911$	3.03942e7	1.21337	0.606687	+	0.794941i	$0.292498\pi$
$912$	0	0
$913$	−1.86632e7	−0.740985
$914$	0	0
$915$	1.19107e6	0.0470312
$916$	0	0
$917$	−2.75826e6	−0.108321
$918$	0	0
$919$	−3.59276e7	−1.40326	−0.701632	−	0.712540i	$-0.747545\pi$
$919$	−3.59276e7	−1.40326	−0.701632	+	0.712540i	$0.747545\pi$
$920$	0	0
$921$	−389894.	−0.0151460
$922$	0	0
$923$	−6.03848e6	−0.233305
$924$	0	0
$925$	1.83562e7	0.705387
$926$	0	0
$927$	−3.50626e7	−1.34012
$928$	0	0
$929$	−9.41890e6	−0.358064	−0.179032	−	0.983843i	$-0.557297\pi$
$929$	−9.41890e6	−0.358064	−0.179032	+	0.983843i	$0.557297\pi$
$930$	0	0
$931$	−1.36436e7	−0.515887
$932$	0	0
$933$	3.33481e6	0.125420
$934$	0	0
$935$	−4.59050e6	−0.171724
$936$	0	0
$937$	−2.13837e7	−0.795673	−0.397836	−	0.917456i	$-0.630239\pi$
$937$	−2.13837e7	−0.795673	−0.397836	+	0.917456i	$0.630239\pi$
$938$	0	0
$939$	−2.39302e6	−0.0885690
$940$	0	0
$941$	4.31706e7	1.58933	0.794666	−	0.607047i	$-0.207646\pi$
$941$	4.31706e7	1.58933	0.794666	+	0.607047i	$0.207646\pi$
$942$	0	0
$943$	5.86800e6	0.214887
$944$	0	0
$945$	1.15321e6	0.0420077
$946$	0	0
$947$	−1.75109e7	−0.634504	−0.317252	−	0.948341i	$-0.602760\pi$
$947$	−1.75109e7	−0.634504	−0.317252	+	0.948341i	$0.602760\pi$
$948$	0	0
$949$	−1.00317e7	−0.361582
$950$	0	0
$951$	1.69977e6	0.0609452
$952$	0	0
$953$	−4.05106e6	−0.144489	−0.0722447	−	0.997387i	$-0.523016\pi$
$953$	−4.05106e6	−0.144489	−0.0722447	+	0.997387i	$0.523016\pi$
$954$	0	0
$955$	3.15703e7	1.12013
$956$	0	0
$957$	−1.19417e6	−0.0421490
$958$	0	0
$959$	−5.89874e6	−0.207116
$960$	0	0
$961$	3.89048e6	0.135892
$962$	0	0
$963$	−3.12221e7	−1.08492
$964$	0	0
$965$	−1.89463e7	−0.654946
$966$	0	0
$967$	2.54319e7	0.874608	0.437304	−	0.899314i	$-0.355933\pi$
$967$	2.54319e7	0.874608	0.437304	+	0.899314i	$0.355933\pi$
$968$	0	0
$969$	971147.	0.0332258
$970$	0	0
$971$	4.42829e6	0.150726	0.0753630	−	0.997156i	$-0.475988\pi$
$971$	4.42829e6	0.150726	0.0753630	+	0.997156i	$0.475988\pi$
$972$	0	0
$973$	−1.25925e7	−0.426414
$974$	0	0
$975$	−1.03586e6	−0.0348971
$976$	0	0
$977$	5.60569e7	1.87885	0.939427	−	0.342750i	$-0.111358\pi$
$977$	5.60569e7	1.87885	0.939427	+	0.342750i	$0.111358\pi$
$978$	0	0
$979$	−1.03511e7	−0.345167
$980$	0	0
$981$	2.51412e6	0.0834090
$982$	0	0
$983$	−3.41997e7	−1.12886	−0.564428	−	0.825482i	$-0.690903\pi$
$983$	−3.41997e7	−1.12886	−0.564428	+	0.825482i	$0.690903\pi$
$984$	0	0
$985$	−1.57733e7	−0.518004
$986$	0	0
$987$	−905525.	−0.0295874
$988$	0	0
$989$	−2.92471e6	−0.0950807
$990$	0	0
$991$	904509.	0.0292569	0.0146285	−	0.999893i	$-0.495343\pi$
$991$	904509.	0.0292569	0.0146285	+	0.999893i	$0.495343\pi$
$992$	0	0
$993$	2.99453e6	0.0963730
$994$	0	0
$995$	−3.68163e7	−1.17891
$996$	0	0
$997$	−4.00447e7	−1.27587	−0.637936	−	0.770089i	$-0.720212\pi$
$997$	−4.00447e7	−1.27587	−0.637936	+	0.770089i	$0.720212\pi$
$998$	0	0
$999$	7.55772e6	0.239595

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.6.a.e.1.2		3
4.3	odd	2		23.6.a.a.1.2	✓	3
12.11	even	2		207.6.a.b.1.2		3
20.19	odd	2		575.6.a.b.1.2		3
92.91	even	2		529.6.a.a.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.6.a.a.1.2	✓	3	4.3	odd	2
207.6.a.b.1.2		3	12.11	even	2
368.6.a.e.1.2		3	1.1	even	1	trivial
529.6.a.a.1.2		3	92.91	even	2
575.6.a.b.1.2		3	20.19	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:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.43100 q^{3} -37.9865 q^{5} +43.8366 q^{7} -240.952 q^{9} +O(q^{10})\)	\(q+1.43100 q^{3} -37.9865 q^{5} +43.8366 q^{7} -240.952 q^{9} +163.213 q^{11} +430.356 q^{13} -54.3587 q^{15} +740.415 q^{17} +916.578 q^{19} +62.7302 q^{21} +529.000 q^{23} -1682.03 q^{25} -692.536 q^{27} -5112.96 q^{29} +5702.60 q^{31} +233.558 q^{33} -1665.20 q^{35} -10913.1 q^{37} +615.840 q^{39} +11092.6 q^{41} -5528.76 q^{43} +9152.92 q^{45} -14435.2 q^{47} -14885.4 q^{49} +1059.53 q^{51} +6852.79 q^{53} -6199.90 q^{55} +1311.62 q^{57} -4155.17 q^{59} -21911.4 q^{61} -10562.5 q^{63} -16347.7 q^{65} +15164.2 q^{67} +756.999 q^{69} -14031.4 q^{71} -23310.1 q^{73} -2406.98 q^{75} +7154.72 q^{77} -64276.8 q^{79} +57560.4 q^{81} -114349. q^{83} -28125.8 q^{85} -7316.65 q^{87} -63420.7 q^{89} +18865.4 q^{91} +8160.42 q^{93} -34817.6 q^{95} -91987.6 q^{97} -39326.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 20 q^{3} - 58 q^{5} + 282 q^{7} + 121 q^{9}+O(q^{10}) \) 3 * q + 20 * q^3 - 58 * q^5 + 282 * q^7 + 121 * q^9	\( 3 q + 20 q^{3} - 58 q^{5} + 282 q^{7} + 121 q^{9} - 136 q^{11} - 1116 q^{13} - 750 q^{15} - 896 q^{17} - 1654 q^{19} - 1670 q^{21} + 1587 q^{23} - 7347 q^{25} + 10700 q^{27} - 844 q^{29} + 3020 q^{31} - 7370 q^{33} - 1072 q^{35} + 8938 q^{37} - 16020 q^{39} - 12792 q^{41} + 16730 q^{43} - 3936 q^{45} - 22500 q^{47} + 2887 q^{49} - 50290 q^{51} + 17108 q^{53} + 436 q^{55} - 61960 q^{57} - 54176 q^{59} - 71324 q^{61} - 40696 q^{63} + 846 q^{65} + 62960 q^{67} + 10580 q^{69} - 98400 q^{71} - 81772 q^{73} - 44800 q^{75} - 304 q^{77} - 58224 q^{79} + 149947 q^{81} - 9892 q^{83} + 15536 q^{85} - 90500 q^{87} + 27542 q^{89} - 151974 q^{91} + 157330 q^{93} + 20644 q^{95} - 273672 q^{97} - 183082 q^{99}+O(q^{100}) \) 3 * q + 20 * q^3 - 58 * q^5 + 282 * q^7 + 121 * q^9 - 136 * q^11 - 1116 * q^13 - 750 * q^15 - 896 * q^17 - 1654 * q^19 - 1670 * q^21 + 1587 * q^23 - 7347 * q^25 + 10700 * q^27 - 844 * q^29 + 3020 * q^31 - 7370 * q^33 - 1072 * q^35 + 8938 * q^37 - 16020 * q^39 - 12792 * q^41 + 16730 * q^43 - 3936 * q^45 - 22500 * q^47 + 2887 * q^49 - 50290 * q^51 + 17108 * q^53 + 436 * q^55 - 61960 * q^57 - 54176 * q^59 - 71324 * q^61 - 40696 * q^63 + 846 * q^65 + 62960 * q^67 + 10580 * q^69 - 98400 * q^71 - 81772 * q^73 - 44800 * q^75 - 304 * q^77 - 58224 * q^79 + 149947 * q^81 - 9892 * q^83 + 15536 * q^85 - 90500 * q^87 + 27542 * q^89 - 151974 * q^91 + 157330 * q^93 + 20644 * q^95 - 273672 * q^97 - 183082 * q^99

Introduction

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