LMFDB - Embedded newform 912.6.a.v.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [912,6,Mod(1,912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("912.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$N$	$=$	$912 = 2^{4} \cdot 3 \cdot 19$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	912.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-45,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$146.270043669$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - 2x^{4} - 7184x^{3} - 76134x^{2} + 12883743x + 275533272$ x^5 - 2x^4 - 7184x^3 - 76134x^2 + 12883743x + 275533272
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{4}\cdot 3$
Twist minimal:	no (minimal twist has level 228)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-57.2380$ of defining polynomial
Character	$\chi$	$=$	912.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -69.8269 q^{5} +147.923 q^{7} +81.0000 q^{9} -294.555 q^{11} -1111.41 q^{13} +628.442 q^{15} +1895.99 q^{17} -361.000 q^{19} -1331.31 q^{21} -828.618 q^{23} +1750.80 q^{25} -729.000 q^{27} +1436.16 q^{29} -10277.5 q^{31} +2650.99 q^{33} -10329.0 q^{35} +14291.3 q^{37} +10002.7 q^{39} -13724.9 q^{41} +701.259 q^{43} -5655.98 q^{45} +3566.06 q^{47} +5074.23 q^{49} -17063.9 q^{51} -29427.5 q^{53} +20567.9 q^{55} +3249.00 q^{57} -30626.3 q^{59} +40326.1 q^{61} +11981.8 q^{63} +77606.1 q^{65} -63762.8 q^{67} +7457.56 q^{69} +60885.7 q^{71} -25978.1 q^{73} -15757.2 q^{75} -43571.4 q^{77} -39210.9 q^{79} +6561.00 q^{81} -91156.3 q^{83} -132391. q^{85} -12925.5 q^{87} +17330.6 q^{89} -164403. q^{91} +92497.7 q^{93} +25207.5 q^{95} +125870. q^{97} -23858.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$5 q - 45 q^{3} - 6 q^{5} - 54 q^{7} + 405 q^{9} - 272 q^{11} + 440 q^{13} + 54 q^{15} + 1940 q^{17} - 1805 q^{19} + 486 q^{21} - 3224 q^{23} + 7313 q^{25} - 3645 q^{27} + 7524 q^{29} - 11774 q^{31} + 2448 q^{33}+ \cdots - 22032 q^{99}+O(q^{100})$ 5 * q - 45 * q^3 - 6 * q^5 - 54 * q^7 + 405 * q^9 - 272 * q^11 + 440 * q^13 + 54 * q^15 + 1940 * q^17 - 1805 * q^19 + 486 * q^21 - 3224 * q^23 + 7313 * q^25 - 3645 * q^27 + 7524 * q^29 - 11774 * q^31 + 2448 * q^33 - 12834 * q^35 + 17760 * q^37 - 3960 * q^39 + 10736 * q^41 - 21914 * q^43 - 486 * q^45 - 10190 * q^47 + 1503 * q^49 - 17460 * q^51 - 1676 * q^53 - 36474 * q^55 + 16245 * q^57 - 54920 * q^59 + 39268 * q^61 - 4374 * q^63 + 39684 * q^65 - 104168 * q^67 + 29016 * q^69 - 50620 * q^71 + 98616 * q^73 - 65817 * q^75 - 9418 * q^77 - 115782 * q^79 + 32805 * q^81 - 143266 * q^83 + 113334 * q^85 - 67716 * q^87 + 88112 * q^89 - 247372 * q^91 + 105966 * q^93 + 2166 * q^95 + 186490 * q^97 - 22032 * q^99

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−69.8269	−1.24910	−0.624551	−	0.780984i	$-0.714718\pi$
$5$	−69.8269	−1.24910	−0.624551	+	0.780984i	$0.714718\pi$
$6$	0	0
$7$	147.923	1.14101	0.570507	−	0.821293i	$-0.306747\pi$
$7$	147.923	1.14101	0.570507	+	0.821293i	$0.306747\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−294.555	−0.733980	−0.366990	−	0.930225i	$-0.619612\pi$
$11$	−294.555	−0.733980	−0.366990	+	0.930225i	$0.619612\pi$
$12$	0	0
$13$	−1111.41	−1.82396	−0.911978	−	0.410240i	$-0.865445\pi$
$13$	−1111.41	−1.82396	−0.911978	+	0.410240i	$0.865445\pi$
$14$	0	0
$15$	628.442	0.721170
$16$	0	0
$17$	1895.99	1.59116	0.795578	−	0.605851i	$-0.207167\pi$
$17$	1895.99	1.59116	0.795578	+	0.605851i	$0.207167\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	0	0
$21$	−1331.31	−0.658764
$22$	0	0
$23$	−828.618	−0.326614	−0.163307	−	0.986575i	$-0.552216\pi$
$23$	−828.618	−0.326614	−0.163307	+	0.986575i	$0.552216\pi$
$24$	0	0
$25$	1750.80	0.560257
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	1436.16	0.317109	0.158555	−	0.987350i	$-0.449317\pi$
$29$	1436.16	0.317109	0.158555	+	0.987350i	$0.449317\pi$
$30$	0	0
$31$	−10277.5	−1.92081	−0.960404	−	0.278610i	$-0.910126\pi$
$31$	−10277.5	−1.92081	−0.960404	+	0.278610i	$0.910126\pi$
$32$	0	0
$33$	2650.99	0.423764
$34$	0	0
$35$	−10329.0	−1.42524
$36$	0	0
$37$	14291.3	1.71620	0.858101	−	0.513480i	$-0.171644\pi$
$37$	14291.3	1.71620	0.858101	+	0.513480i	$0.171644\pi$
$38$	0	0
$39$	10002.7	1.05306
$40$	0	0
$41$	−13724.9	−1.27511	−0.637556	−	0.770404i	$-0.720055\pi$
$41$	−13724.9	−1.27511	−0.637556	+	0.770404i	$0.720055\pi$
$42$	0	0
$43$	701.259	0.0578372	0.0289186	−	0.999582i	$-0.490794\pi$
$43$	701.259	0.0578372	0.0289186	+	0.999582i	$0.490794\pi$
$44$	0	0
$45$	−5655.98	−0.416367
$46$	0	0
$47$	3566.06	0.235475	0.117737	−	0.993045i	$-0.462436\pi$
$47$	3566.06	0.235475	0.117737	+	0.993045i	$0.462436\pi$
$48$	0	0
$49$	5074.23	0.301912
$50$	0	0
$51$	−17063.9	−0.918655
$52$	0	0
$53$	−29427.5	−1.43901	−0.719505	−	0.694487i	$-0.755631\pi$
$53$	−29427.5	−1.43901	−0.719505	+	0.694487i	$0.755631\pi$
$54$	0	0
$55$	20567.9	0.916816
$56$	0	0
$57$	3249.00	0.132453
$58$	0	0
$59$	−30626.3	−1.14542	−0.572710	−	0.819758i	$-0.694108\pi$
$59$	−30626.3	−1.14542	−0.572710	+	0.819758i	$0.694108\pi$
$60$	0	0
$61$	40326.1	1.38759	0.693796	−	0.720171i	$-0.255937\pi$
$61$	40326.1	1.38759	0.693796	+	0.720171i	$0.255937\pi$
$62$	0	0
$63$	11981.8	0.380338
$64$	0	0
$65$	77606.1	2.27831
$66$	0	0
$67$	−63762.8	−1.73532	−0.867662	−	0.497155i	$-0.834378\pi$
$67$	−63762.8	−1.73532	−0.867662	+	0.497155i	$0.834378\pi$
$68$	0	0
$69$	7457.56	0.188571
$70$	0	0
$71$	60885.7	1.43341	0.716703	−	0.697379i	$-0.245650\pi$
$71$	60885.7	1.43341	0.716703	+	0.697379i	$0.245650\pi$
$72$	0	0
$73$	−25978.1	−0.570559	−0.285280	−	0.958444i	$-0.592086\pi$
$73$	−25978.1	−0.570559	−0.285280	+	0.958444i	$0.592086\pi$
$74$	0	0
$75$	−15757.2	−0.323464
$76$	0	0
$77$	−43571.4	−0.837481
$78$	0	0
$79$	−39210.9	−0.706870	−0.353435	−	0.935459i	$-0.614986\pi$
$79$	−39210.9	−0.706870	−0.353435	+	0.935459i	$0.614986\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−91156.3	−1.45242	−0.726209	−	0.687474i	$-0.758720\pi$
$83$	−91156.3	−1.45242	−0.726209	+	0.687474i	$0.758720\pi$
$84$	0	0
$85$	−132391.	−1.98752
$86$	0	0
$87$	−12925.5	−0.183083
$88$	0	0
$89$	17330.6	0.231920	0.115960	−	0.993254i	$-0.463006\pi$
$89$	17330.6	0.231920	0.115960	+	0.993254i	$0.463006\pi$
$90$	0	0
$91$	−164403.	−2.08116
$92$	0	0
$93$	92497.7	1.10898
$94$	0	0
$95$	25207.5	0.286564
$96$	0	0
$97$	125870.	1.35830	0.679148	−	0.734001i	$-0.262349\pi$
$97$	125870.	1.35830	0.679148	+	0.734001i	$0.262349\pi$
$98$	0	0
$99$	−23858.9	−0.244660
$100$	0	0
$101$	−19817.1	−0.193302	−0.0966509	−	0.995318i	$-0.530813\pi$
$101$	−19817.1	−0.193302	−0.0966509	+	0.995318i	$0.530813\pi$
$102$	0	0
$103$	−11835.7	−0.109926	−0.0549632	−	0.998488i	$-0.517504\pi$
$103$	−11835.7	−0.109926	−0.0549632	+	0.998488i	$0.517504\pi$
$104$	0	0
$105$	92961.1	0.822864
$106$	0	0
$107$	−124281.	−1.04941	−0.524706	−	0.851283i	$-0.675825\pi$
$107$	−124281.	−1.04941	−0.524706	+	0.851283i	$0.675825\pi$
$108$	0	0
$109$	−185517.	−1.49561	−0.747804	−	0.663920i	$-0.768892\pi$
$109$	−185517.	−1.49561	−0.747804	+	0.663920i	$0.768892\pi$
$110$	0	0
$111$	−128622.	−0.990850
$112$	0	0
$113$	4260.89	0.0313909	0.0156955	−	0.999877i	$-0.495004\pi$
$113$	4260.89	0.0313909	0.0156955	+	0.999877i	$0.495004\pi$
$114$	0	0
$115$	57859.8	0.407974
$116$	0	0
$117$	−90023.9	−0.607985
$118$	0	0
$119$	280460.	1.81553
$120$	0	0
$121$	−74288.5	−0.461273
$122$	0	0
$123$	123524.	0.736187
$124$	0	0
$125$	95956.0	0.549284
$126$	0	0
$127$	−213355.	−1.17380	−0.586900	−	0.809659i	$-0.699652\pi$
$127$	−213355.	−1.17380	−0.586900	+	0.809659i	$0.699652\pi$
$128$	0	0
$129$	−6311.33	−0.0333923
$130$	0	0
$131$	230700.	1.17454	0.587272	−	0.809389i	$-0.300202\pi$
$131$	230700.	1.17454	0.587272	+	0.809389i	$0.300202\pi$
$132$	0	0
$133$	−53400.2	−0.261766
$134$	0	0
$135$	50903.8	0.240390
$136$	0	0
$137$	274860.	1.25115	0.625575	−	0.780164i	$-0.284864\pi$
$137$	274860.	1.25115	0.625575	+	0.780164i	$0.284864\pi$
$138$	0	0
$139$	254088.	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$139$	254088.	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$140$	0	0
$141$	−32094.6	−0.135951
$142$	0	0
$143$	327370.	1.33875
$144$	0	0
$145$	−100283.	−0.396102
$146$	0	0
$147$	−45668.1	−0.174309
$148$	0	0
$149$	137476.	0.507295	0.253648	−	0.967297i	$-0.418370\pi$
$149$	137476.	0.507295	0.253648	+	0.967297i	$0.418370\pi$
$150$	0	0
$151$	131801.	0.470411	0.235206	−	0.971946i	$-0.424424\pi$
$151$	131801.	0.470411	0.235206	+	0.971946i	$0.424424\pi$
$152$	0	0
$153$	153575.	0.530386
$154$	0	0
$155$	717648.	2.39929
$156$	0	0
$157$	67744.1	0.219342	0.109671	−	0.993968i	$-0.465020\pi$
$157$	67744.1	0.219342	0.109671	+	0.993968i	$0.465020\pi$
$158$	0	0
$159$	264848.	0.830813
$160$	0	0
$161$	−122572.	−0.372671
$162$	0	0
$163$	−79847.8	−0.235393	−0.117697	−	0.993050i	$-0.537551\pi$
$163$	−79847.8	−0.235393	−0.117697	+	0.993050i	$0.537551\pi$
$164$	0	0
$165$	−185111.	−0.529324
$166$	0	0
$167$	576203.	1.59876	0.799382	−	0.600823i	$-0.205160\pi$
$167$	576203.	1.59876	0.799382	+	0.600823i	$0.205160\pi$
$168$	0	0
$169$	863930.	2.32681
$170$	0	0
$171$	−29241.0	−0.0764719
$172$	0	0
$173$	548390.	1.39308	0.696538	−	0.717520i	$-0.254723\pi$
$173$	548390.	1.39308	0.696538	+	0.717520i	$0.254723\pi$
$174$	0	0
$175$	258984.	0.639260
$176$	0	0
$177$	275637.	0.661309
$178$	0	0
$179$	655964.	1.53020	0.765099	−	0.643913i	$-0.222690\pi$
$179$	655964.	1.53020	0.765099	+	0.643913i	$0.222690\pi$
$180$	0	0
$181$	−289707.	−0.657298	−0.328649	−	0.944452i	$-0.606593\pi$
$181$	−289707.	−0.657298	−0.328649	+	0.944452i	$0.606593\pi$
$182$	0	0
$183$	−362935.	−0.801127
$184$	0	0
$185$	−997920.	−2.14371
$186$	0	0
$187$	−558472.	−1.16788
$188$	0	0
$189$	−107836.	−0.219588
$190$	0	0
$191$	370453.	0.734766	0.367383	−	0.930070i	$-0.380254\pi$
$191$	370453.	0.734766	0.367383	+	0.930070i	$0.380254\pi$
$192$	0	0
$193$	−179153.	−0.346203	−0.173102	−	0.984904i	$-0.555379\pi$
$193$	−179153.	−0.346203	−0.173102	+	0.984904i	$0.555379\pi$
$194$	0	0
$195$	−698455.	−1.31538
$196$	0	0
$197$	−228074.	−0.418707	−0.209353	−	0.977840i	$-0.567136\pi$
$197$	−228074.	−0.418707	−0.209353	+	0.977840i	$0.567136\pi$
$198$	0	0
$199$	54667.3	0.0978576	0.0489288	−	0.998802i	$-0.484419\pi$
$199$	54667.3	0.0978576	0.0489288	+	0.998802i	$0.484419\pi$
$200$	0	0
$201$	573865.	1.00189
$202$	0	0
$203$	212442.	0.361826
$204$	0	0
$205$	958366.	1.59275
$206$	0	0
$207$	−67118.0	−0.108871
$208$	0	0
$209$	106334.	0.168387
$210$	0	0
$211$	−467628.	−0.723094	−0.361547	−	0.932354i	$-0.617751\pi$
$211$	−467628.	−0.723094	−0.361547	+	0.932354i	$0.617751\pi$
$212$	0	0
$213$	−547971.	−0.827577
$214$	0	0
$215$	−48966.8	−0.0722446
$216$	0	0
$217$	−1.52028e6	−2.19167
$218$	0	0
$219$	233803.	0.329413
$220$	0	0
$221$	−2.10721e6	−2.90220
$222$	0	0
$223$	−296402.	−0.399134	−0.199567	−	0.979884i	$-0.563954\pi$
$223$	−296402.	−0.399134	−0.199567	+	0.979884i	$0.563954\pi$
$224$	0	0
$225$	141815.	0.186752
$226$	0	0
$227$	−41549.9	−0.0535186	−0.0267593	−	0.999642i	$-0.508519\pi$
$227$	−41549.9	−0.0535186	−0.0267593	+	0.999642i	$0.508519\pi$
$228$	0	0
$229$	1.42922e6	1.80098	0.900491	−	0.434874i	$-0.143207\pi$
$229$	1.42922e6	1.80098	0.900491	+	0.434874i	$0.143207\pi$
$230$	0	0
$231$	392143.	0.483520
$232$	0	0
$233$	657069.	0.792905	0.396453	−	0.918055i	$-0.370241\pi$
$233$	657069.	0.792905	0.396453	+	0.918055i	$0.370241\pi$
$234$	0	0
$235$	−249007.	−0.294132
$236$	0	0
$237$	352898.	0.408111
$238$	0	0
$239$	843896.	0.955640	0.477820	−	0.878458i	$-0.341427\pi$
$239$	843896.	0.955640	0.477820	+	0.878458i	$0.341427\pi$
$240$	0	0
$241$	−183013.	−0.202974	−0.101487	−	0.994837i	$-0.532360\pi$
$241$	−183013.	−0.202974	−0.101487	+	0.994837i	$0.532360\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−354318.	−0.377119
$246$	0	0
$247$	401217.	0.418444
$248$	0	0
$249$	820407.	0.838554
$250$	0	0
$251$	162400.	0.162706	0.0813528	−	0.996685i	$-0.474076\pi$
$251$	162400.	0.162706	0.0813528	+	0.996685i	$0.474076\pi$
$252$	0	0
$253$	244073.	0.239728
$254$	0	0
$255$	1.19152e6	1.14749
$256$	0	0
$257$	1.35301e6	1.27782	0.638908	−	0.769283i	$-0.279386\pi$
$257$	1.35301e6	1.27782	0.638908	+	0.769283i	$0.279386\pi$
$258$	0	0
$259$	2.11402e6	1.95821
$260$	0	0
$261$	116329.	0.105703
$262$	0	0
$263$	−2.03595e6	−1.81500	−0.907501	−	0.420051i	$-0.862012\pi$
$263$	−2.03595e6	−1.81500	−0.907501	+	0.420051i	$0.862012\pi$
$264$	0	0
$265$	2.05483e6	1.79747
$266$	0	0
$267$	−155975.	−0.133899
$268$	0	0
$269$	1.84402e6	1.55376	0.776882	−	0.629646i	$-0.216800\pi$
$269$	1.84402e6	1.55376	0.776882	+	0.629646i	$0.216800\pi$
$270$	0	0
$271$	1.20142e6	0.993738	0.496869	−	0.867826i	$-0.334483\pi$
$271$	1.20142e6	0.993738	0.496869	+	0.867826i	$0.334483\pi$
$272$	0	0
$273$	1.47962e6	1.20156
$274$	0	0
$275$	−515707.	−0.411217
$276$	0	0
$277$	−862594.	−0.675472	−0.337736	−	0.941241i	$-0.609661\pi$
$277$	−862594.	−0.675472	−0.337736	+	0.941241i	$0.609661\pi$
$278$	0	0
$279$	−832479.	−0.640270
$280$	0	0
$281$	−1.11324e6	−0.841051	−0.420525	−	0.907281i	$-0.638154\pi$
$281$	−1.11324e6	−0.841051	−0.420525	+	0.907281i	$0.638154\pi$
$282$	0	0
$283$	−2.38199e6	−1.76797	−0.883984	−	0.467518i	$-0.845148\pi$
$283$	−2.38199e6	−1.76797	−0.883984	+	0.467518i	$0.845148\pi$
$284$	0	0
$285$	−226868.	−0.165448
$286$	0	0
$287$	−2.03022e6	−1.45492
$288$	0	0
$289$	2.17491e6	1.53178
$290$	0	0
$291$	−1.13283e6	−0.784213
$292$	0	0
$293$	1.51039e6	1.02783	0.513915	−	0.857841i	$-0.328195\pi$
$293$	1.51039e6	1.02783	0.513915	+	0.857841i	$0.328195\pi$
$294$	0	0
$295$	2.13854e6	1.43075
$296$	0	0
$297$	214730.	0.141255
$298$	0	0
$299$	920930.	0.595729
$300$	0	0
$301$	103732.	0.0659930
$302$	0	0
$303$	178354.	0.111603
$304$	0	0
$305$	−2.81585e6	−1.73325
$306$	0	0
$307$	771102.	0.466945	0.233472	−	0.972363i	$-0.424991\pi$
$307$	771102.	0.466945	0.233472	+	0.972363i	$0.424991\pi$
$308$	0	0
$309$	106521.	0.0634660
$310$	0	0
$311$	1.81434e6	1.06370	0.531849	−	0.846840i	$-0.321498\pi$
$311$	1.81434e6	1.06370	0.531849	+	0.846840i	$0.321498\pi$
$312$	0	0
$313$	846275.	0.488259	0.244130	−	0.969743i	$-0.421498\pi$
$313$	846275.	0.488259	0.244130	+	0.969743i	$0.421498\pi$
$314$	0	0
$315$	−836650.	−0.475081
$316$	0	0
$317$	−773899.	−0.432550	−0.216275	−	0.976333i	$-0.569391\pi$
$317$	−773899.	−0.432550	−0.216275	+	0.976333i	$0.569391\pi$
$318$	0	0
$319$	−423029.	−0.232752
$320$	0	0
$321$	1.11853e6	0.605879
$322$	0	0
$323$	−684451.	−0.365036
$324$	0	0
$325$	−1.94585e6	−1.02188
$326$	0	0
$327$	1.66965e6	0.863490
$328$	0	0
$329$	527503.	0.268680
$330$	0	0
$331$	−1.38964e6	−0.697162	−0.348581	−	0.937279i	$-0.613336\pi$
$331$	−1.38964e6	−0.697162	−0.348581	+	0.937279i	$0.613336\pi$
$332$	0	0
$333$	1.15760e6	0.572068
$334$	0	0
$335$	4.45236e6	2.16760
$336$	0	0
$337$	−2.61928e6	−1.25634	−0.628169	−	0.778077i	$-0.716195\pi$
$337$	−2.61928e6	−1.25634	−0.628169	+	0.778077i	$0.716195\pi$
$338$	0	0
$339$	−38348.0	−0.0181236
$340$	0	0
$341$	3.02729e6	1.40984
$342$	0	0
$343$	−1.73555e6	−0.796528
$344$	0	0
$345$	−520739.	−0.235544
$346$	0	0
$347$	3.70097e6	1.65003	0.825016	−	0.565110i	$-0.191166\pi$
$347$	3.70097e6	1.65003	0.825016	+	0.565110i	$0.191166\pi$
$348$	0	0
$349$	3.79563e6	1.66809	0.834046	−	0.551695i	$-0.186019\pi$
$349$	3.79563e6	1.66809	0.834046	+	0.551695i	$0.186019\pi$
$350$	0	0
$351$	810215.	0.351020
$352$	0	0
$353$	4.64314e6	1.98324	0.991619	−	0.129196i	$-0.0412397\pi$
$353$	4.64314e6	1.98324	0.991619	+	0.129196i	$0.0412397\pi$
$354$	0	0
$355$	−4.25146e6	−1.79047
$356$	0	0
$357$	−2.52414e6	−1.04820
$358$	0	0
$359$	765033.	0.313288	0.156644	−	0.987655i	$-0.449932\pi$
$359$	765033.	0.313288	0.156644	+	0.987655i	$0.449932\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	668597.	0.266316
$364$	0	0
$365$	1.81397e6	0.712687
$366$	0	0
$367$	85881.0	0.0332837	0.0166419	−	0.999862i	$-0.494702\pi$
$367$	85881.0	0.0332837	0.0166419	+	0.999862i	$0.494702\pi$
$368$	0	0
$369$	−1.11171e6	−0.425038
$370$	0	0
$371$	−4.35301e6	−1.64193
$372$	0	0
$373$	2.23504e6	0.831789	0.415895	−	0.909413i	$-0.363469\pi$
$373$	2.23504e6	0.831789	0.415895	+	0.909413i	$0.363469\pi$
$374$	0	0
$375$	−863604.	−0.317130
$376$	0	0
$377$	−1.59616e6	−0.578393
$378$	0	0
$379$	−2.08723e6	−0.746400	−0.373200	−	0.927751i	$-0.621739\pi$
$379$	−2.08723e6	−0.746400	−0.373200	+	0.927751i	$0.621739\pi$
$380$	0	0
$381$	1.92020e6	0.677694
$382$	0	0
$383$	3.93781e6	1.37170	0.685848	−	0.727745i	$-0.259431\pi$
$383$	3.93781e6	1.37170	0.685848	+	0.727745i	$0.259431\pi$
$384$	0	0
$385$	3.04246e6	1.04610
$386$	0	0
$387$	56802.0	0.0192791
$388$	0	0
$389$	−160927.	−0.0539207	−0.0269604	−	0.999637i	$-0.508583\pi$
$389$	−160927.	−0.0539207	−0.0269604	+	0.999637i	$0.508583\pi$
$390$	0	0
$391$	−1.57105e6	−0.519694
$392$	0	0
$393$	−2.07630e6	−0.678124
$394$	0	0
$395$	2.73798e6	0.882953
$396$	0	0
$397$	−859353.	−0.273650	−0.136825	−	0.990595i	$-0.543690\pi$
$397$	−859353.	−0.273650	−0.136825	+	0.990595i	$0.543690\pi$
$398$	0	0
$399$	480602.	0.151131
$400$	0	0
$401$	1.29983e6	0.403668	0.201834	−	0.979420i	$-0.435310\pi$
$401$	1.29983e6	0.403668	0.201834	+	0.979420i	$0.435310\pi$
$402$	0	0
$403$	1.14225e7	3.50347
$404$	0	0
$405$	−458135.	−0.138789
$406$	0	0
$407$	−4.20958e6	−1.25966
$408$	0	0
$409$	3.21987e6	0.951765	0.475883	−	0.879509i	$-0.342129\pi$
$409$	3.21987e6	0.951765	0.475883	+	0.879509i	$0.342129\pi$
$410$	0	0
$411$	−2.47374e6	−0.722352
$412$	0	0
$413$	−4.53034e6	−1.30694
$414$	0	0
$415$	6.36517e6	1.81422
$416$	0	0
$417$	−2.28680e6	−0.644002
$418$	0	0
$419$	−2.22436e6	−0.618971	−0.309486	−	0.950904i	$-0.600157\pi$
$419$	−2.22436e6	−0.618971	−0.309486	+	0.950904i	$0.600157\pi$
$420$	0	0
$421$	648855.	0.178420	0.0892098	−	0.996013i	$-0.471566\pi$
$421$	648855.	0.178420	0.0892098	+	0.996013i	$0.471566\pi$
$422$	0	0
$423$	288851.	0.0784916
$424$	0	0
$425$	3.31950e6	0.891456
$426$	0	0
$427$	5.96516e6	1.58326
$428$	0	0
$429$	−2.94633e6	−0.772926
$430$	0	0
$431$	−5.39777e6	−1.39965	−0.699827	−	0.714312i	$-0.746740\pi$
$431$	−5.39777e6	−1.39965	−0.699827	+	0.714312i	$0.746740\pi$
$432$	0	0
$433$	−759488.	−0.194671	−0.0973355	−	0.995252i	$-0.531032\pi$
$433$	−759488.	−0.194671	−0.0973355	+	0.995252i	$0.531032\pi$
$434$	0	0
$435$	902546.	0.228690
$436$	0	0
$437$	299131.	0.0749304
$438$	0	0
$439$	−1.77828e6	−0.440393	−0.220196	−	0.975456i	$-0.570670\pi$
$439$	−1.77828e6	−0.440393	−0.220196	+	0.975456i	$0.570670\pi$
$440$	0	0
$441$	411013.	0.100637
$442$	0	0
$443$	−80380.9	−0.0194600	−0.00973002	−	0.999953i	$-0.503097\pi$
$443$	−80380.9	−0.0194600	−0.00973002	+	0.999953i	$0.503097\pi$
$444$	0	0
$445$	−1.21014e6	−0.289691
$446$	0	0
$447$	−1.23728e6	−0.292887
$448$	0	0
$449$	−3.09442e6	−0.724375	−0.362187	−	0.932105i	$-0.617970\pi$
$449$	−3.09442e6	−0.724375	−0.362187	+	0.932105i	$0.617970\pi$
$450$	0	0
$451$	4.04273e6	0.935907
$452$	0	0
$453$	−1.18621e6	−0.271592
$454$	0	0
$455$	1.14797e7	2.59958
$456$	0	0
$457$	−1.34871e6	−0.302084	−0.151042	−	0.988527i	$-0.548263\pi$
$457$	−1.34871e6	−0.302084	−0.151042	+	0.988527i	$0.548263\pi$
$458$	0	0
$459$	−1.38217e6	−0.306218
$460$	0	0
$461$	1.53370e6	0.336114	0.168057	−	0.985777i	$-0.446251\pi$
$461$	1.53370e6	0.336114	0.168057	+	0.985777i	$0.446251\pi$
$462$	0	0
$463$	−367343.	−0.0796379	−0.0398189	−	0.999207i	$-0.512678\pi$
$463$	−367343.	−0.0796379	−0.0398189	+	0.999207i	$0.512678\pi$
$464$	0	0
$465$	−6.45883e6	−1.38523
$466$	0	0
$467$	−3.59900e6	−0.763642	−0.381821	−	0.924236i	$-0.624703\pi$
$467$	−3.59900e6	−0.763642	−0.381821	+	0.924236i	$0.624703\pi$
$468$	0	0
$469$	−9.43199e6	−1.98003
$470$	0	0
$471$	−609697.	−0.126637
$472$	0	0
$473$	−206559.	−0.0424513
$474$	0	0
$475$	−632040.	−0.128532
$476$	0	0
$477$	−2.38363e6	−0.479670
$478$	0	0
$479$	−6.73909e6	−1.34203	−0.671015	−	0.741444i	$-0.734141\pi$
$479$	−6.73909e6	−1.34203	−0.671015	+	0.741444i	$0.734141\pi$
$480$	0	0
$481$	−1.58835e7	−3.13028
$482$	0	0
$483$	1.10314e6	0.215162
$484$	0	0
$485$	−8.78915e6	−1.69665
$486$	0	0
$487$	−1.37815e6	−0.263314	−0.131657	−	0.991295i	$-0.542030\pi$
$487$	−1.37815e6	−0.263314	−0.131657	+	0.991295i	$0.542030\pi$
$488$	0	0
$489$	718630.	0.135904
$490$	0	0
$491$	8.87480e6	1.66133	0.830663	−	0.556775i	$-0.187961\pi$
$491$	8.87480e6	1.66133	0.830663	+	0.556775i	$0.187961\pi$
$492$	0	0
$493$	2.72295e6	0.504571
$494$	0	0
$495$	1.66600e6	0.305605
$496$	0	0
$497$	9.00639e6	1.63554
$498$	0	0
$499$	−5.26279e6	−0.946161	−0.473080	−	0.881019i	$-0.656858\pi$
$499$	−5.26279e6	−0.946161	−0.473080	+	0.881019i	$0.656858\pi$
$500$	0	0
$501$	−5.18583e6	−0.923047
$502$	0	0
$503$	1.53134e6	0.269868	0.134934	−	0.990855i	$-0.456918\pi$
$503$	1.53134e6	0.269868	0.134934	+	0.990855i	$0.456918\pi$
$504$	0	0
$505$	1.38376e6	0.241454
$506$	0	0
$507$	−7.77537e6	−1.34339
$508$	0	0
$509$	3.44968e6	0.590180	0.295090	−	0.955469i	$-0.404650\pi$
$509$	3.44968e6	0.590180	0.295090	+	0.955469i	$0.404650\pi$
$510$	0	0
$511$	−3.84277e6	−0.651016
$512$	0	0
$513$	263169.	0.0441511
$514$	0	0
$515$	826452.	0.137309
$516$	0	0
$517$	−1.05040e6	−0.172834
$518$	0	0
$519$	−4.93551e6	−0.804293
$520$	0	0
$521$	7.01548e6	1.13230	0.566152	−	0.824301i	$-0.308432\pi$
$521$	7.01548e6	1.13230	0.566152	+	0.824301i	$0.308432\pi$
$522$	0	0
$523$	−87222.8	−0.0139436	−0.00697181	−	0.999976i	$-0.502219\pi$
$523$	−87222.8	−0.0139436	−0.00697181	+	0.999976i	$0.502219\pi$
$524$	0	0
$525$	−2.33086e6	−0.369077
$526$	0	0
$527$	−1.94860e7	−3.05631
$528$	0	0
$529$	−5.74974e6	−0.893323
$530$	0	0
$531$	−2.48073e6	−0.381807
$532$	0	0
$533$	1.52539e7	2.32575
$534$	0	0
$535$	8.67819e6	1.31082
$536$	0	0
$537$	−5.90368e6	−0.883460
$538$	0	0
$539$	−1.49464e6	−0.221597
$540$	0	0
$541$	−1.51464e6	−0.222493	−0.111246	−	0.993793i	$-0.535484\pi$
$541$	−1.51464e6	−0.222493	−0.111246	+	0.993793i	$0.535484\pi$
$542$	0	0
$543$	2.60736e6	0.379491
$544$	0	0
$545$	1.29541e7	1.86817
$546$	0	0
$547$	−3.81361e6	−0.544964	−0.272482	−	0.962161i	$-0.587845\pi$
$547$	−3.81361e6	−0.544964	−0.272482	+	0.962161i	$0.587845\pi$
$548$	0	0
$549$	3.26642e6	0.462531
$550$	0	0
$551$	−518455.	−0.0727499
$552$	0	0
$553$	−5.80020e6	−0.806548
$554$	0	0
$555$	8.98128e6	1.23767
$556$	0	0
$557$	1.21459e6	0.165879	0.0829397	−	0.996555i	$-0.473569\pi$
$557$	1.21459e6	0.165879	0.0829397	+	0.996555i	$0.473569\pi$
$558$	0	0
$559$	−779383.	−0.105492
$560$	0	0
$561$	5.02625e6	0.674274
$562$	0	0
$563$	−6.04498e6	−0.803756	−0.401878	−	0.915693i	$-0.631642\pi$
$563$	−6.04498e6	−0.803756	−0.401878	+	0.915693i	$0.631642\pi$
$564$	0	0
$565$	−297525.	−0.0392105
$566$	0	0
$567$	970523.	0.126779
$568$	0	0
$569$	−6.78575e6	−0.878652	−0.439326	−	0.898328i	$-0.644783\pi$
$569$	−6.78575e6	−0.878652	−0.439326	+	0.898328i	$0.644783\pi$
$570$	0	0
$571$	−7.53396e6	−0.967015	−0.483508	−	0.875340i	$-0.660637\pi$
$571$	−7.53396e6	−0.967015	−0.483508	+	0.875340i	$0.660637\pi$
$572$	0	0
$573$	−3.33407e6	−0.424217
$574$	0	0
$575$	−1.45075e6	−0.182988
$576$	0	0
$577$	3.02231e6	0.377919	0.188960	−	0.981985i	$-0.439488\pi$
$577$	3.02231e6	0.377919	0.188960	+	0.981985i	$0.439488\pi$
$578$	0	0
$579$	1.61238e6	0.199881
$580$	0	0
$581$	−1.34841e7	−1.65723
$582$	0	0
$583$	8.66801e6	1.05620
$584$	0	0
$585$	6.28609e6	0.759436
$586$	0	0
$587$	1.08580e7	1.30064	0.650319	−	0.759661i	$-0.274635\pi$
$587$	1.08580e7	1.30064	0.650319	+	0.759661i	$0.274635\pi$
$588$	0	0
$589$	3.71018e6	0.440664
$590$	0	0
$591$	2.05267e6	0.241741
$592$	0	0
$593$	−6.83093e6	−0.797706	−0.398853	−	0.917015i	$-0.630592\pi$
$593$	−6.83093e6	−0.797706	−0.398853	+	0.917015i	$0.630592\pi$
$594$	0	0
$595$	−1.95837e7	−2.26778
$596$	0	0
$597$	−492005.	−0.0564981
$598$	0	0
$599$	1.17238e7	1.33506	0.667531	−	0.744582i	$-0.267351\pi$
$599$	1.17238e7	1.33506	0.667531	+	0.744582i	$0.267351\pi$
$600$	0	0
$601$	3.62060e6	0.408878	0.204439	−	0.978879i	$-0.434463\pi$
$601$	3.62060e6	0.408878	0.204439	+	0.978879i	$0.434463\pi$
$602$	0	0
$603$	−5.16479e6	−0.578441
$604$	0	0
$605$	5.18734e6	0.576177
$606$	0	0
$607$	−9.15043e6	−1.00802	−0.504011	−	0.863697i	$-0.668143\pi$
$607$	−9.15043e6	−1.00802	−0.504011	+	0.863697i	$0.668143\pi$
$608$	0	0
$609$	−1.91198e6	−0.208900
$610$	0	0
$611$	−3.96334e6	−0.429496
$612$	0	0
$613$	8.84111e6	0.950289	0.475145	−	0.879908i	$-0.342396\pi$
$613$	8.84111e6	0.950289	0.475145	+	0.879908i	$0.342396\pi$
$614$	0	0
$615$	−8.62529e6	−0.919573
$616$	0	0
$617$	4.07070e6	0.430484	0.215242	−	0.976561i	$-0.430946\pi$
$617$	4.07070e6	0.430484	0.215242	+	0.976561i	$0.430946\pi$
$618$	0	0
$619$	4.77320e6	0.500706	0.250353	−	0.968155i	$-0.419453\pi$
$619$	4.77320e6	0.500706	0.250353	+	0.968155i	$0.419453\pi$
$620$	0	0
$621$	604062.	0.0628569
$622$	0	0
$623$	2.56359e6	0.264623
$624$	0	0
$625$	−1.21716e7	−1.24637
$626$	0	0
$627$	−957008.	−0.0972180
$628$	0	0
$629$	2.70962e7	2.73075
$630$	0	0
$631$	−4.06936e6	−0.406867	−0.203434	−	0.979089i	$-0.565210\pi$
$631$	−4.06936e6	−0.406867	−0.203434	+	0.979089i	$0.565210\pi$
$632$	0	0
$633$	4.20866e6	0.417479
$634$	0	0
$635$	1.48980e7	1.46620
$636$	0	0
$637$	−5.63953e6	−0.550674
$638$	0	0
$639$	4.93174e6	0.477802
$640$	0	0
$641$	−5.38965e6	−0.518103	−0.259051	−	0.965864i	$-0.583410\pi$
$641$	−5.38965e6	−0.518103	−0.259051	+	0.965864i	$0.583410\pi$
$642$	0	0
$643$	2.47088e6	0.235681	0.117841	−	0.993033i	$-0.462403\pi$
$643$	2.47088e6	0.235681	0.117841	+	0.993033i	$0.462403\pi$
$644$	0	0
$645$	440701.	0.0417104
$646$	0	0
$647$	−1.00297e7	−0.941950	−0.470975	−	0.882147i	$-0.656098\pi$
$647$	−1.00297e7	−0.941950	−0.470975	+	0.882147i	$0.656098\pi$
$648$	0	0
$649$	9.02113e6	0.840716
$650$	0	0
$651$	1.36825e7	1.26536
$652$	0	0
$653$	−7.76255e6	−0.712396	−0.356198	−	0.934411i	$-0.615927\pi$
$653$	−7.76255e6	−0.712396	−0.356198	+	0.934411i	$0.615927\pi$
$654$	0	0
$655$	−1.61091e7	−1.46713
$656$	0	0
$657$	−2.10423e6	−0.190186
$658$	0	0
$659$	4.07681e6	0.365685	0.182842	−	0.983142i	$-0.441470\pi$
$659$	4.07681e6	0.365685	0.182842	+	0.983142i	$0.441470\pi$
$660$	0	0
$661$	3.21715e6	0.286397	0.143198	−	0.989694i	$-0.454261\pi$
$661$	3.21715e6	0.286397	0.143198	+	0.989694i	$0.454261\pi$
$662$	0	0
$663$	1.89649e7	1.67559
$664$	0	0
$665$	3.72877e6	0.326973
$666$	0	0
$667$	−1.19003e6	−0.103572
$668$	0	0
$669$	2.66762e6	0.230440
$670$	0	0
$671$	−1.18783e7	−1.01847
$672$	0	0
$673$	1.93484e7	1.64667	0.823337	−	0.567552i	$-0.192110\pi$
$673$	1.93484e7	1.64667	0.823337	+	0.567552i	$0.192110\pi$
$674$	0	0
$675$	−1.27633e6	−0.107821
$676$	0	0
$677$	−1.51780e7	−1.27275	−0.636376	−	0.771379i	$-0.719567\pi$
$677$	−1.51780e7	−1.27275	−0.636376	+	0.771379i	$0.719567\pi$
$678$	0	0
$679$	1.86191e7	1.54983
$680$	0	0
$681$	373949.	0.0308990
$682$	0	0
$683$	−9.14664e6	−0.750257	−0.375128	−	0.926973i	$-0.622401\pi$
$683$	−9.14664e6	−0.750257	−0.375128	+	0.926973i	$0.622401\pi$
$684$	0	0
$685$	−1.91926e7	−1.56281
$686$	0	0
$687$	−1.28630e7	−1.03980
$688$	0	0
$689$	3.27059e7	2.62469
$690$	0	0
$691$	−1.52065e7	−1.21153	−0.605764	−	0.795644i	$-0.707133\pi$
$691$	−1.52065e7	−1.21153	−0.605764	+	0.795644i	$0.707133\pi$
$692$	0	0
$693$	−3.52929e6	−0.279160
$694$	0	0
$695$	−1.77422e7	−1.39330
$696$	0	0
$697$	−2.60222e7	−2.02890
$698$	0	0
$699$	−5.91362e6	−0.457784
$700$	0	0
$701$	−1.78791e7	−1.37420	−0.687100	−	0.726563i	$-0.741117\pi$
$701$	−1.78791e7	−1.37420	−0.687100	+	0.726563i	$0.741117\pi$
$702$	0	0
$703$	−5.15917e6	−0.393724
$704$	0	0
$705$	2.24107e6	0.169817
$706$	0	0
$707$	−2.93140e6	−0.220560
$708$	0	0
$709$	1.79619e7	1.34195	0.670976	−	0.741479i	$-0.265875\pi$
$709$	1.79619e7	1.34195	0.670976	+	0.741479i	$0.265875\pi$
$710$	0	0
$711$	−3.17608e6	−0.235623
$712$	0	0
$713$	8.51613e6	0.627363
$714$	0	0
$715$	−2.28592e7	−1.67223
$716$	0	0
$717$	−7.59507e6	−0.551739
$718$	0	0
$719$	1.49503e7	1.07852	0.539260	−	0.842139i	$-0.318704\pi$
$719$	1.49503e7	1.07852	0.539260	+	0.842139i	$0.318704\pi$
$720$	0	0
$721$	−1.75078e6	−0.125427
$722$	0	0
$723$	1.64712e6	0.117187
$724$	0	0
$725$	2.51444e6	0.177663
$726$	0	0
$727$	5.08924e6	0.357122	0.178561	−	0.983929i	$-0.442856\pi$
$727$	5.08924e6	0.357122	0.178561	+	0.983929i	$0.442856\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.32958e6	0.0920280
$732$	0	0
$733$	−2.25539e7	−1.55046	−0.775231	−	0.631678i	$-0.782367\pi$
$733$	−2.25539e7	−1.55046	−0.775231	+	0.631678i	$0.782367\pi$
$734$	0	0
$735$	3.18886e6	0.217730
$736$	0	0
$737$	1.87816e7	1.27369
$738$	0	0
$739$	−719515.	−0.0484650	−0.0242325	−	0.999706i	$-0.507714\pi$
$739$	−719515.	−0.0484650	−0.0242325	+	0.999706i	$0.507714\pi$
$740$	0	0
$741$	−3.61096e6	−0.241589
$742$	0	0
$743$	1.68089e7	1.11703	0.558517	−	0.829493i	$-0.311370\pi$
$743$	1.68089e7	1.11703	0.558517	+	0.829493i	$0.311370\pi$
$744$	0	0
$745$	−9.59952e6	−0.633664
$746$	0	0
$747$	−7.38366e6	−0.484139
$748$	0	0
$749$	−1.83841e7	−1.19739
$750$	0	0
$751$	1.82371e7	1.17993	0.589964	−	0.807430i	$-0.299142\pi$
$751$	1.82371e7	1.17993	0.589964	+	0.807430i	$0.299142\pi$
$752$	0	0
$753$	−1.46160e6	−0.0939381
$754$	0	0
$755$	−9.20329e6	−0.587592
$756$	0	0
$757$	2.73545e7	1.73496	0.867478	−	0.497476i	$-0.165740\pi$
$757$	2.73545e7	1.73496	0.867478	+	0.497476i	$0.165740\pi$
$758$	0	0
$759$	−2.19666e6	−0.138407
$760$	0	0
$761$	6.01078e6	0.376244	0.188122	−	0.982146i	$-0.439760\pi$
$761$	6.01078e6	0.376244	0.188122	+	0.982146i	$0.439760\pi$
$762$	0	0
$763$	−2.74423e7	−1.70651
$764$	0	0
$765$	−1.07237e7	−0.662506
$766$	0	0
$767$	3.40383e7	2.08920
$768$	0	0
$769$	−1.64306e7	−1.00193	−0.500966	−	0.865467i	$-0.667022\pi$
$769$	−1.64306e7	−1.00193	−0.500966	+	0.865467i	$0.667022\pi$
$770$	0	0
$771$	−1.21771e7	−0.737748
$772$	0	0
$773$	3.17331e7	1.91013	0.955066	−	0.296392i	$-0.0957836\pi$
$773$	3.17331e7	1.91013	0.955066	+	0.296392i	$0.0957836\pi$
$774$	0	0
$775$	−1.79939e7	−1.07615
$776$	0	0
$777$	−1.90262e7	−1.13057
$778$	0	0
$779$	4.95468e6	0.292531
$780$	0	0
$781$	−1.79342e7	−1.05209
$782$	0	0
$783$	−1.04696e6	−0.0610277
$784$	0	0
$785$	−4.73036e6	−0.273981
$786$	0	0
$787$	1.82166e6	0.104841	0.0524205	−	0.998625i	$-0.483306\pi$
$787$	1.82166e6	0.104841	0.0524205	+	0.998625i	$0.483306\pi$
$788$	0	0
$789$	1.83235e7	1.04789
$790$	0	0
$791$	630283.	0.0358175
$792$	0	0
$793$	−4.48187e7	−2.53091
$794$	0	0
$795$	−1.84935e7	−1.03777
$796$	0	0
$797$	−2.21473e7	−1.23502	−0.617510	−	0.786563i	$-0.711859\pi$
$797$	−2.21473e7	−1.23502	−0.617510	+	0.786563i	$0.711859\pi$
$798$	0	0
$799$	6.76121e6	0.374677
$800$	0	0
$801$	1.40378e6	0.0773066
$802$	0	0
$803$	7.65198e6	0.418779
$804$	0	0
$805$	8.55880e6	0.465504
$806$	0	0
$807$	−1.65962e7	−0.897066
$808$	0	0
$809$	−2.68213e7	−1.44082	−0.720409	−	0.693550i	$-0.756046\pi$
$809$	−2.68213e7	−1.44082	−0.720409	+	0.693550i	$0.756046\pi$
$810$	0	0
$811$	−7.22178e6	−0.385560	−0.192780	−	0.981242i	$-0.561750\pi$
$811$	−7.22178e6	−0.385560	−0.192780	+	0.981242i	$0.561750\pi$
$812$	0	0
$813$	−1.08128e7	−0.573735
$814$	0	0
$815$	5.57553e6	0.294030
$816$	0	0
$817$	−253154.	−0.0132688
$818$	0	0
$819$	−1.33166e7	−0.693719
$820$	0	0
$821$	−5.87682e6	−0.304288	−0.152144	−	0.988358i	$-0.548618\pi$
$821$	−5.87682e6	−0.304288	−0.152144	+	0.988358i	$0.548618\pi$
$822$	0	0
$823$	1.51965e6	0.0782068	0.0391034	−	0.999235i	$-0.487550\pi$
$823$	1.51965e6	0.0782068	0.0391034	+	0.999235i	$0.487550\pi$
$824$	0	0
$825$	4.64136e6	0.237416
$826$	0	0
$827$	3.19385e7	1.62387	0.811935	−	0.583748i	$-0.198414\pi$
$827$	3.19385e7	1.62387	0.811935	+	0.583748i	$0.198414\pi$
$828$	0	0
$829$	2.87925e6	0.145510	0.0727551	−	0.997350i	$-0.476821\pi$
$829$	2.87925e6	0.145510	0.0727551	+	0.997350i	$0.476821\pi$
$830$	0	0
$831$	7.76335e6	0.389984
$832$	0	0
$833$	9.62067e6	0.480389
$834$	0	0
$835$	−4.02345e7	−1.99702
$836$	0	0
$837$	7.49231e6	0.369660
$838$	0	0
$839$	1.08804e7	0.533627	0.266814	−	0.963748i	$-0.414029\pi$
$839$	1.08804e7	0.533627	0.266814	+	0.963748i	$0.414029\pi$
$840$	0	0
$841$	−1.84486e7	−0.899442
$842$	0	0
$843$	1.00191e7	0.485581
$844$	0	0
$845$	−6.03256e7	−2.90643
$846$	0	0
$847$	−1.09890e7	−0.526319
$848$	0	0
$849$	2.14379e7	1.02074
$850$	0	0
$851$	−1.18421e7	−0.560536
$852$	0	0
$853$	1.60726e7	0.756332	0.378166	−	0.925738i	$-0.376555\pi$
$853$	1.60726e7	0.756332	0.378166	+	0.925738i	$0.376555\pi$
$854$	0	0
$855$	2.04181e6	0.0955212
$856$	0	0
$857$	−2.42652e7	−1.12858	−0.564288	−	0.825578i	$-0.690849\pi$
$857$	−2.42652e7	−1.12858	−0.564288	+	0.825578i	$0.690849\pi$
$858$	0	0
$859$	−1.65351e7	−0.764583	−0.382291	−	0.924042i	$-0.624865\pi$
$859$	−1.65351e7	−0.764583	−0.382291	+	0.924042i	$0.624865\pi$
$860$	0	0
$861$	1.82720e7	0.839999
$862$	0	0
$863$	7795.72	0.000356311 0	0.000178156	−	1.00000i	$-0.499943\pi$
$863$	7795.72	0.000356311 0	0.000178156		1.00000i	$0.499943\pi$
$864$	0	0
$865$	−3.82924e7	−1.74009
$866$	0	0
$867$	−1.95742e7	−0.884373
$868$	0	0
$869$	1.15498e7	0.518828
$870$	0	0
$871$	7.08664e7	3.16515
$872$	0	0
$873$	1.01955e7	0.452765
$874$	0	0
$875$	1.41941e7	0.626741
$876$	0	0
$877$	3.96611e7	1.74127	0.870635	−	0.491929i	$-0.163708\pi$
$877$	3.96611e7	1.74127	0.870635	+	0.491929i	$0.163708\pi$
$878$	0	0
$879$	−1.35935e7	−0.593418
$880$	0	0
$881$	−3.11077e7	−1.35029	−0.675147	−	0.737684i	$-0.735920\pi$
$881$	−3.11077e7	−1.35029	−0.675147	+	0.737684i	$0.735920\pi$
$882$	0	0
$883$	3.85517e7	1.66396	0.831979	−	0.554808i	$-0.187208\pi$
$883$	3.85517e7	1.66396	0.831979	+	0.554808i	$0.187208\pi$
$884$	0	0
$885$	−1.92469e7	−0.826042
$886$	0	0
$887$	3.26547e7	1.39360	0.696798	−	0.717267i	$-0.254607\pi$
$887$	3.26547e7	1.39360	0.696798	+	0.717267i	$0.254607\pi$
$888$	0	0
$889$	−3.15602e7	−1.33932
$890$	0	0
$891$	−1.93257e6	−0.0815533
$892$	0	0
$893$	−1.28735e6	−0.0540216
$894$	0	0
$895$	−4.58040e7	−1.91137
$896$	0	0
$897$	−8.28837e6	−0.343944
$898$	0	0
$899$	−1.47602e7	−0.609106
$900$	0	0
$901$	−5.57942e7	−2.28969
$902$	0	0
$903$	−933591.	−0.0381011
$904$	0	0
$905$	2.02293e7	0.821032
$906$	0	0
$907$	1.60479e7	0.647739	0.323870	−	0.946102i	$-0.395016\pi$
$907$	1.60479e7	0.647739	0.323870	+	0.946102i	$0.395016\pi$
$908$	0	0
$909$	−1.60518e6	−0.0644339
$910$	0	0
$911$	−8.51674e6	−0.339999	−0.169999	−	0.985444i	$-0.554377\pi$
$911$	−8.51674e6	−0.339999	−0.169999	+	0.985444i	$0.554377\pi$
$912$	0	0
$913$	2.68505e7	1.06605
$914$	0	0
$915$	2.53427e7	1.00069
$916$	0	0
$917$	3.41259e7	1.34017
$918$	0	0
$919$	−1.63723e7	−0.639470	−0.319735	−	0.947507i	$-0.603594\pi$
$919$	−1.63723e7	−0.639470	−0.319735	+	0.947507i	$0.603594\pi$
$920$	0	0
$921$	−6.93992e6	−0.269591
$922$	0	0
$923$	−6.76687e7	−2.61447
$924$	0	0
$925$	2.50213e7	0.961514
$926$	0	0
$927$	−958693.	−0.0366421
$928$	0	0
$929$	−2.59014e7	−0.984655	−0.492327	−	0.870410i	$-0.663854\pi$
$929$	−2.59014e7	−0.984655	−0.492327	+	0.870410i	$0.663854\pi$
$930$	0	0
$931$	−1.83180e6	−0.0692633
$932$	0	0
$933$	−1.63291e7	−0.614126
$934$	0	0
$935$	3.89964e7	1.45880
$936$	0	0
$937$	−3.18342e7	−1.18453	−0.592263	−	0.805745i	$-0.701765\pi$
$937$	−3.18342e7	−1.18453	−0.592263	+	0.805745i	$0.701765\pi$
$938$	0	0
$939$	−7.61647e6	−0.281897
$940$	0	0
$941$	−2.05656e7	−0.757123	−0.378561	−	0.925576i	$-0.623581\pi$
$941$	−2.05656e7	−0.757123	−0.378561	+	0.925576i	$0.623581\pi$
$942$	0	0
$943$	1.13727e7	0.416469
$944$	0	0
$945$	7.52985e6	0.274288
$946$	0	0
$947$	8.81182e6	0.319294	0.159647	−	0.987174i	$-0.448964\pi$
$947$	8.81182e6	0.319294	0.159647	+	0.987174i	$0.448964\pi$
$948$	0	0
$949$	2.88722e7	1.04067
$950$	0	0
$951$	6.96509e6	0.249733
$952$	0	0
$953$	−3.31312e6	−0.118169	−0.0590846	−	0.998253i	$-0.518818\pi$
$953$	−3.31312e6	−0.118169	−0.0590846	+	0.998253i	$0.518818\pi$
$954$	0	0
$955$	−2.58676e7	−0.917798
$956$	0	0
$957$	3.80726e6	0.134379
$958$	0	0
$959$	4.06581e7	1.42758
$960$	0	0
$961$	7.69983e7	2.68951
$962$	0	0
$963$	−1.00668e7	−0.349804
$964$	0	0
$965$	1.25097e7	0.432443
$966$	0	0
$967$	−5.19189e6	−0.178550	−0.0892749	−	0.996007i	$-0.528455\pi$
$967$	−5.19189e6	−0.178550	−0.0892749	+	0.996007i	$0.528455\pi$
$968$	0	0
$969$	6.16006e6	0.210754
$970$	0	0
$971$	−2.47890e7	−0.843745	−0.421872	−	0.906655i	$-0.638627\pi$
$971$	−2.47890e7	−0.843745	−0.421872	+	0.906655i	$0.638627\pi$
$972$	0	0
$973$	3.75855e7	1.27274
$974$	0	0
$975$	1.75127e7	0.589985
$976$	0	0
$977$	1.03796e7	0.347893	0.173946	−	0.984755i	$-0.444348\pi$
$977$	1.03796e7	0.347893	0.173946	+	0.984755i	$0.444348\pi$
$978$	0	0
$979$	−5.10480e6	−0.170224
$980$	0	0
$981$	−1.50269e7	−0.498536
$982$	0	0
$983$	5.17548e7	1.70831	0.854155	−	0.520019i	$-0.174075\pi$
$983$	5.17548e7	1.70831	0.854155	+	0.520019i	$0.174075\pi$
$984$	0	0
$985$	1.59257e7	0.523008
$986$	0	0
$987$	−4.74753e6	−0.155122
$988$	0	0
$989$	−581075.	−0.0188904
$990$	0	0
$991$	575035.	0.0185999	0.00929994	−	0.999957i	$-0.497040\pi$
$991$	575035.	0.0185999	0.00929994	+	0.999957i	$0.497040\pi$
$992$	0	0
$993$	1.25068e7	0.402507
$994$	0	0
$995$	−3.81725e6	−0.122234
$996$	0	0
$997$	−2.29151e7	−0.730102	−0.365051	−	0.930988i	$-0.618948\pi$
$997$	−2.29151e7	−0.730102	−0.365051	+	0.930988i	$0.618948\pi$
$998$	0	0
$999$	−1.04184e7	−0.330283

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.6.a.v.1.2		5
4.3	odd	2		228.6.a.d.1.2	✓	5
12.11	even	2		684.6.a.f.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.6.a.d.1.2	✓	5	4.3	odd	2
684.6.a.f.1.4		5	12.11	even	2
912.6.a.v.1.2		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	912.6.a.v.1.2

Level	$912$
Weight	$6$
Character	912.1
Self dual	yes
Analytic conductor	$146.270$
Analytic rank	$0$
Dimension	$5$
CM	no
Inner twists	$1$