LMFDB - Embedded newform 76.6.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,6,Mod(1,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("76.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	76.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:	$12.1891703058$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.272193.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 74x + 168 $ x^3 - x^2 - 74*x + 168
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-9.12596$ of defining polynomial
Character	$\chi$	$=$	76.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.17335 q^{3} -47.6772 q^{5} +121.691 q^{7} -225.583 q^{9} +O(q^{10})$	$q+4.17335 q^{3} -47.6772 q^{5} +121.691 q^{7} -225.583 q^{9} -110.606 q^{11} -708.611 q^{13} -198.974 q^{15} -1010.36 q^{17} +361.000 q^{19} +507.859 q^{21} -1306.84 q^{23} -851.887 q^{25} -1955.56 q^{27} -7054.68 q^{29} -4106.31 q^{31} -461.600 q^{33} -5801.87 q^{35} +10147.1 q^{37} -2957.28 q^{39} +7240.23 q^{41} +17767.2 q^{43} +10755.2 q^{45} -6596.02 q^{47} -1998.35 q^{49} -4216.60 q^{51} -4584.15 q^{53} +5273.40 q^{55} +1506.58 q^{57} +28297.7 q^{59} +57651.8 q^{61} -27451.4 q^{63} +33784.6 q^{65} -29043.3 q^{67} -5453.88 q^{69} +40502.4 q^{71} -18993.7 q^{73} -3555.22 q^{75} -13459.8 q^{77} -50178.2 q^{79} +46655.4 q^{81} -55752.2 q^{83} +48171.3 q^{85} -29441.7 q^{87} -35164.6 q^{89} -86231.4 q^{91} -17137.1 q^{93} -17211.5 q^{95} -13078.6 q^{97} +24951.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{3} - 9 q^{5} - 13 q^{7} + 189 q^{9}+O(q^{10}) $ 3 * q - 8 * q^3 - 9 * q^5 - 13 * q^7 + 189 * q^9	$ 3 q - 8 q^{3} - 9 q^{5} - 13 q^{7} + 189 q^{9} - 1229 q^{11} - 56 q^{13} - 1698 q^{15} - 3149 q^{17} + 1083 q^{19} - 6186 q^{21} - 3212 q^{23} - 4422 q^{25} - 11582 q^{27} + 2514 q^{29} - 10784 q^{31} + 13538 q^{33} + 3081 q^{35} - 526 q^{37} - 9668 q^{39} - 14246 q^{41} - 77 q^{43} + 34155 q^{45} + 2893 q^{47} + 41766 q^{49} + 46628 q^{51} + 29600 q^{53} - 27351 q^{55} - 2888 q^{57} - 2612 q^{59} + 59895 q^{61} + 36095 q^{63} + 50592 q^{65} - 3050 q^{67} + 78908 q^{69} - 33562 q^{71} + 44027 q^{73} - 30698 q^{75} - 3527 q^{77} + 24944 q^{79} + 36231 q^{81} - 220156 q^{83} - 51021 q^{85} - 175128 q^{87} - 116120 q^{89} - 105286 q^{91} + 131212 q^{93} - 3249 q^{95} + 171204 q^{97} - 294437 q^{99}+O(q^{100}) $ 3 * q - 8 * q^3 - 9 * q^5 - 13 * q^7 + 189 * q^9 - 1229 * q^11 - 56 * q^13 - 1698 * q^15 - 3149 * q^17 + 1083 * q^19 - 6186 * q^21 - 3212 * q^23 - 4422 * q^25 - 11582 * q^27 + 2514 * q^29 - 10784 * q^31 + 13538 * q^33 + 3081 * q^35 - 526 * q^37 - 9668 * q^39 - 14246 * q^41 - 77 * q^43 + 34155 * q^45 + 2893 * q^47 + 41766 * q^49 + 46628 * q^51 + 29600 * q^53 - 27351 * q^55 - 2888 * q^57 - 2612 * q^59 + 59895 * q^61 + 36095 * q^63 + 50592 * q^65 - 3050 * q^67 + 78908 * q^69 - 33562 * q^71 + 44027 * q^73 - 30698 * q^75 - 3527 * q^77 + 24944 * q^79 + 36231 * q^81 - 220156 * q^83 - 51021 * q^85 - 175128 * q^87 - 116120 * q^89 - 105286 * q^91 + 131212 * q^93 - 3249 * q^95 + 171204 * q^97 - 294437 * 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
$3$	4.17335	0.267721	0.133860	−	0.991000i	$-0.457263\pi$
$3$	4.17335	0.267721	0.133860	+	0.991000i	$0.457263\pi$
$4$	0	0
$5$	−47.6772	−0.852875	−0.426438	−	0.904517i	$-0.640232\pi$
$5$	−47.6772	−0.852875	−0.426438	+	0.904517i	$0.640232\pi$
$6$	0	0
$7$	121.691	0.938669	0.469335	−	0.883020i	$-0.344494\pi$
$7$	121.691	0.938669	0.469335	+	0.883020i	$0.344494\pi$
$8$	0	0
$9$	−225.583	−0.928326
$10$	0	0
$11$	−110.606	−0.275612	−0.137806	−	0.990459i	$-0.544005\pi$
$11$	−110.606	−0.275612	−0.137806	+	0.990459i	$0.544005\pi$
$12$	0	0
$13$	−708.611	−1.16292	−0.581460	−	0.813575i	$-0.697518\pi$
$13$	−708.611	−1.16292	−0.581460	+	0.813575i	$0.697518\pi$
$14$	0	0
$15$	−198.974	−0.228332
$16$	0	0
$17$	−1010.36	−0.847920	−0.423960	−	0.905681i	$-0.639360\pi$
$17$	−1010.36	−0.847920	−0.423960	+	0.905681i	$0.639360\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	507.859	0.251301
$22$	0	0
$23$	−1306.84	−0.515111	−0.257556	−	0.966263i	$-0.582917\pi$
$23$	−1306.84	−0.515111	−0.257556	+	0.966263i	$0.582917\pi$
$24$	0	0
$25$	−851.887	−0.272604
$26$	0	0
$27$	−1955.56	−0.516253
$28$	0	0
$29$	−7054.68	−1.55770	−0.778848	−	0.627213i	$-0.784196\pi$
$29$	−7054.68	−1.55770	−0.778848	+	0.627213i	$0.784196\pi$
$30$	0	0
$31$	−4106.31	−0.767445	−0.383722	−	0.923448i	$-0.625358\pi$
$31$	−4106.31	−0.767445	−0.383722	+	0.923448i	$0.625358\pi$
$32$	0	0
$33$	−461.600	−0.0737872
$34$	0	0
$35$	−5801.87	−0.800568
$36$	0	0
$37$	10147.1	1.21854	0.609270	−	0.792963i	$-0.291463\pi$
$37$	10147.1	1.21854	0.609270	+	0.792963i	$0.291463\pi$
$38$	0	0
$39$	−2957.28	−0.311338
$40$	0	0
$41$	7240.23	0.672656	0.336328	−	0.941745i	$-0.390815\pi$
$41$	7240.23	0.672656	0.336328	+	0.941745i	$0.390815\pi$
$42$	0	0
$43$	17767.2	1.46537	0.732685	−	0.680568i	$-0.238267\pi$
$43$	17767.2	1.46537	0.732685	+	0.680568i	$0.238267\pi$
$44$	0	0
$45$	10755.2	0.791746
$46$	0	0
$47$	−6596.02	−0.435550	−0.217775	−	0.975999i	$-0.569880\pi$
$47$	−6596.02	−0.435550	−0.217775	+	0.975999i	$0.569880\pi$
$48$	0	0
$49$	−1998.35	−0.118900
$50$	0	0
$51$	−4216.60	−0.227006
$52$	0	0
$53$	−4584.15	−0.224166	−0.112083	−	0.993699i	$-0.535752\pi$
$53$	−4584.15	−0.224166	−0.112083	+	0.993699i	$0.535752\pi$
$54$	0	0
$55$	5273.40	0.235063
$56$	0	0
$57$	1506.58	0.0614193
$58$	0	0
$59$	28297.7	1.05833	0.529165	−	0.848519i	$-0.322505\pi$
$59$	28297.7	1.05833	0.529165	+	0.848519i	$0.322505\pi$
$60$	0	0
$61$	57651.8	1.98376	0.991879	−	0.127187i	$-0.0405948\pi$
$61$	57651.8	1.98376	0.991879	+	0.127187i	$0.0405948\pi$
$62$	0	0
$63$	−27451.4	−0.871391
$64$	0	0
$65$	33784.6	0.991825
$66$	0	0
$67$	−29043.3	−0.790422	−0.395211	−	0.918590i	$-0.629329\pi$
$67$	−29043.3	−0.790422	−0.395211	+	0.918590i	$0.629329\pi$
$68$	0	0
$69$	−5453.88	−0.137906
$70$	0	0
$71$	40502.4	0.953530	0.476765	−	0.879031i	$-0.341809\pi$
$71$	40502.4	0.953530	0.476765	+	0.879031i	$0.341809\pi$
$72$	0	0
$73$	−18993.7	−0.417159	−0.208579	−	0.978005i	$-0.566884\pi$
$73$	−18993.7	−0.417159	−0.208579	+	0.978005i	$0.566884\pi$
$74$	0	0
$75$	−3555.22	−0.0729817
$76$	0	0
$77$	−13459.8	−0.258709
$78$	0	0
$79$	−50178.2	−0.904581	−0.452290	−	0.891871i	$-0.649393\pi$
$79$	−50178.2	−0.904581	−0.452290	+	0.891871i	$0.649393\pi$
$80$	0	0
$81$	46655.4	0.790114
$82$	0	0
$83$	−55752.2	−0.888315	−0.444157	−	0.895949i	$-0.646497\pi$
$83$	−55752.2	−0.888315	−0.444157	+	0.895949i	$0.646497\pi$
$84$	0	0
$85$	48171.3	0.723170
$86$	0	0
$87$	−29441.7	−0.417027
$88$	0	0
$89$	−35164.6	−0.470577	−0.235288	−	0.971926i	$-0.575603\pi$
$89$	−35164.6	−0.470577	−0.235288	+	0.971926i	$0.575603\pi$
$90$	0	0
$91$	−86231.4	−1.09160
$92$	0	0
$93$	−17137.1	−0.205461
$94$	0	0
$95$	−17211.5	−0.195663
$96$	0	0
$97$	−13078.6	−0.141134	−0.0705669	−	0.997507i	$-0.522481\pi$
$97$	−13078.6	−0.141134	−0.0705669	+	0.997507i	$0.522481\pi$
$98$	0	0
$99$	24951.0	0.255858
$100$	0	0
$101$	19775.7	0.192898	0.0964492	−	0.995338i	$-0.469251\pi$
$101$	19775.7	0.192898	0.0964492	+	0.995338i	$0.469251\pi$
$102$	0	0
$103$	−44008.6	−0.408738	−0.204369	−	0.978894i	$-0.565514\pi$
$103$	−44008.6	−0.408738	−0.204369	+	0.978894i	$0.565514\pi$
$104$	0	0
$105$	−24213.3	−0.214329
$106$	0	0
$107$	−191516.	−1.61713	−0.808566	−	0.588405i	$-0.799756\pi$
$107$	−191516.	−1.61713	−0.808566	+	0.588405i	$0.799756\pi$
$108$	0	0
$109$	77143.3	0.621916	0.310958	−	0.950424i	$-0.399350\pi$
$109$	77143.3	0.621916	0.310958	+	0.950424i	$0.399350\pi$
$110$	0	0
$111$	42347.6	0.326228
$112$	0	0
$113$	−152243.	−1.12160	−0.560802	−	0.827950i	$-0.689507\pi$
$113$	−152243.	−1.12160	−0.560802	+	0.827950i	$0.689507\pi$
$114$	0	0
$115$	62306.2	0.439326
$116$	0	0
$117$	159851.	1.07957
$118$	0	0
$119$	−122952.	−0.795917
$120$	0	0
$121$	−148817.	−0.924038
$122$	0	0
$123$	30216.1	0.180084
$124$	0	0
$125$	189607.	1.08537
$126$	0	0
$127$	85457.3	0.470153	0.235077	−	0.971977i	$-0.424466\pi$
$127$	85457.3	0.470153	0.235077	+	0.971977i	$0.424466\pi$
$128$	0	0
$129$	74148.7	0.392310
$130$	0	0
$131$	−233440.	−1.18849	−0.594247	−	0.804282i	$-0.702550\pi$
$131$	−233440.	−1.18849	−0.594247	+	0.804282i	$0.702550\pi$
$132$	0	0
$133$	43930.4	0.215346
$134$	0	0
$135$	93235.7	0.440299
$136$	0	0
$137$	11157.7	0.0507894	0.0253947	−	0.999678i	$-0.491916\pi$
$137$	11157.7	0.0507894	0.0253947	+	0.999678i	$0.491916\pi$
$138$	0	0
$139$	−56726.0	−0.249026	−0.124513	−	0.992218i	$-0.539737\pi$
$139$	−56726.0	−0.249026	−0.124513	+	0.992218i	$0.539737\pi$
$140$	0	0
$141$	−27527.5	−0.116606
$142$	0	0
$143$	78377.0	0.320515
$144$	0	0
$145$	336347.	1.32852
$146$	0	0
$147$	−8339.82	−0.0318320
$148$	0	0
$149$	82403.9	0.304076	0.152038	−	0.988375i	$-0.451416\pi$
$149$	82403.9	0.304076	0.152038	+	0.988375i	$0.451416\pi$
$150$	0	0
$151$	271673.	0.969627	0.484814	−	0.874618i	$-0.338887\pi$
$151$	271673.	0.969627	0.484814	+	0.874618i	$0.338887\pi$
$152$	0	0
$153$	227921.	0.787146
$154$	0	0
$155$	195777.	0.654535
$156$	0	0
$157$	−77060.4	−0.249507	−0.124753	−	0.992188i	$-0.539814\pi$
$157$	−77060.4	−0.249507	−0.124753	+	0.992188i	$0.539814\pi$
$158$	0	0
$159$	−19131.3	−0.0600138
$160$	0	0
$161$	−159030.	−0.483519
$162$	0	0
$163$	480416.	1.41628	0.708139	−	0.706073i	$-0.249535\pi$
$163$	480416.	1.41628	0.708139	+	0.706073i	$0.249535\pi$
$164$	0	0
$165$	22007.8	0.0629312
$166$	0	0
$167$	153139.	0.424908	0.212454	−	0.977171i	$-0.431854\pi$
$167$	153139.	0.424908	0.212454	+	0.977171i	$0.431854\pi$
$168$	0	0
$169$	130837.	0.352381
$170$	0	0
$171$	−81435.5	−0.212973
$172$	0	0
$173$	−695523.	−1.76683	−0.883417	−	0.468587i	$-0.844763\pi$
$173$	−695523.	−1.76683	−0.883417	+	0.468587i	$0.844763\pi$
$174$	0	0
$175$	−103667.	−0.255885
$176$	0	0
$177$	118096.	0.283337
$178$	0	0
$179$	−68472.0	−0.159728	−0.0798638	−	0.996806i	$-0.525449\pi$
$179$	−68472.0	−0.159728	−0.0798638	+	0.996806i	$0.525449\pi$
$180$	0	0
$181$	−52950.0	−0.120135	−0.0600675	−	0.998194i	$-0.519132\pi$
$181$	−52950.0	−0.120135	−0.0600675	+	0.998194i	$0.519132\pi$
$182$	0	0
$183$	240601.	0.531093
$184$	0	0
$185$	−483787.	−1.03926
$186$	0	0
$187$	111753.	0.233697
$188$	0	0
$189$	−237974.	−0.484591
$190$	0	0
$191$	−704076.	−1.39648	−0.698242	−	0.715862i	$-0.746034\pi$
$191$	−704076.	−1.39648	−0.698242	+	0.715862i	$0.746034\pi$
$192$	0	0
$193$	153801.	0.297212	0.148606	−	0.988896i	$-0.452521\pi$
$193$	153801.	0.297212	0.148606	+	0.988896i	$0.452521\pi$
$194$	0	0
$195$	140995.	0.265532
$196$	0	0
$197$	29576.3	0.0542974	0.0271487	−	0.999631i	$-0.491357\pi$
$197$	29576.3	0.0542974	0.0271487	+	0.999631i	$0.491357\pi$
$198$	0	0
$199$	389532.	0.697285	0.348643	−	0.937256i	$-0.386643\pi$
$199$	389532.	0.697285	0.348643	+	0.937256i	$0.386643\pi$
$200$	0	0
$201$	−121208.	−0.211612
$202$	0	0
$203$	−858490.	−1.46216
$204$	0	0
$205$	−345194.	−0.573692
$206$	0	0
$207$	294800.	0.478191
$208$	0	0
$209$	−39928.9	−0.0632298
$210$	0	0
$211$	−174196.	−0.269359	−0.134680	−	0.990889i	$-0.543001\pi$
$211$	−174196.	−0.269359	−0.134680	+	0.990889i	$0.543001\pi$
$212$	0	0
$213$	169031.	0.255280
$214$	0	0
$215$	−847089.	−1.24978
$216$	0	0
$217$	−499700.	−0.720377
$218$	0	0
$219$	−79267.2	−0.111682
$220$	0	0
$221$	715954.	0.986063
$222$	0	0
$223$	752862.	1.01380	0.506901	−	0.862004i	$-0.330791\pi$
$223$	752862.	1.01380	0.506901	+	0.862004i	$0.330791\pi$
$224$	0	0
$225$	192171.	0.253065
$226$	0	0
$227$	1.34882e6	1.73736	0.868680	−	0.495373i	$-0.164969\pi$
$227$	1.34882e6	1.73736	0.868680	+	0.495373i	$0.164969\pi$
$228$	0	0
$229$	−824074.	−1.03843	−0.519215	−	0.854644i	$-0.673776\pi$
$229$	−824074.	−1.03843	−0.519215	+	0.854644i	$0.673776\pi$
$230$	0	0
$231$	−56172.5	−0.0692617
$232$	0	0
$233$	−1.52837e6	−1.84433	−0.922165	−	0.386797i	$-0.873581\pi$
$233$	−1.52837e6	−1.84433	−0.922165	+	0.386797i	$0.873581\pi$
$234$	0	0
$235$	314480.	0.371469
$236$	0	0
$237$	−209411.	−0.242175
$238$	0	0
$239$	−1.29445e6	−1.46586	−0.732929	−	0.680305i	$-0.761847\pi$
$239$	−1.29445e6	−1.46586	−0.732929	+	0.680305i	$0.761847\pi$
$240$	0	0
$241$	−895918.	−0.993632	−0.496816	−	0.867856i	$-0.665498\pi$
$241$	−895918.	−0.993632	−0.496816	+	0.867856i	$0.665498\pi$
$242$	0	0
$243$	669911.	0.727783
$244$	0	0
$245$	95275.7	0.101407
$246$	0	0
$247$	−255809.	−0.266792
$248$	0	0
$249$	−232674.	−0.237820
$250$	0	0
$251$	−274821.	−0.275337	−0.137669	−	0.990478i	$-0.543961\pi$
$251$	−274821.	−0.275337	−0.137669	+	0.990478i	$0.543961\pi$
$252$	0	0
$253$	144544.	0.141971
$254$	0	0
$255$	201036.	0.193608
$256$	0	0
$257$	−180556.	−0.170522	−0.0852608	−	0.996359i	$-0.527172\pi$
$257$	−180556.	−0.170522	−0.0852608	+	0.996359i	$0.527172\pi$
$258$	0	0
$259$	1.23481e6	1.14381
$260$	0	0
$261$	1.59142e6	1.44605
$262$	0	0
$263$	412116.	0.367392	0.183696	−	0.982983i	$-0.441194\pi$
$263$	412116.	0.367392	0.183696	+	0.982983i	$0.441194\pi$
$264$	0	0
$265$	218559.	0.191186
$266$	0	0
$267$	−146754.	−0.125983
$268$	0	0
$269$	1.68747e6	1.42185	0.710927	−	0.703266i	$-0.248276\pi$
$269$	1.68747e6	1.42185	0.710927	+	0.703266i	$0.248276\pi$
$270$	0	0
$271$	2.03675e6	1.68467	0.842334	−	0.538955i	$-0.181181\pi$
$271$	2.03675e6	1.68467	0.842334	+	0.538955i	$0.181181\pi$
$272$	0	0
$273$	−359874.	−0.292243
$274$	0	0
$275$	94224.2	0.0751330
$276$	0	0
$277$	1.39488e6	1.09229	0.546146	−	0.837690i	$-0.316094\pi$
$277$	1.39488e6	1.09229	0.546146	+	0.837690i	$0.316094\pi$
$278$	0	0
$279$	926314.	0.712439
$280$	0	0
$281$	−1.78789e6	−1.35075	−0.675374	−	0.737475i	$-0.736018\pi$
$281$	−1.78789e6	−1.35075	−0.675374	+	0.737475i	$0.736018\pi$
$282$	0	0
$283$	−1.44595e6	−1.07322	−0.536608	−	0.843832i	$-0.680295\pi$
$283$	−1.44595e6	−1.07322	−0.536608	+	0.843832i	$0.680295\pi$
$284$	0	0
$285$	−71829.5	−0.0523830
$286$	0	0
$287$	881070.	0.631401
$288$	0	0
$289$	−399024.	−0.281031
$290$	0	0
$291$	−54581.5	−0.0377844
$292$	0	0
$293$	342517.	0.233084	0.116542	−	0.993186i	$-0.462819\pi$
$293$	342517.	0.233084	0.116542	+	0.993186i	$0.462819\pi$
$294$	0	0
$295$	−1.34915e6	−0.902623
$296$	0	0
$297$	216298.	0.142286
$298$	0	0
$299$	926038.	0.599033
$300$	0	0
$301$	2.16210e6	1.37550
$302$	0	0
$303$	82531.0	0.0516429
$304$	0	0
$305$	−2.74868e6	−1.69190
$306$	0	0
$307$	1.99971e6	1.21093	0.605467	−	0.795870i	$-0.292986\pi$
$307$	1.99971e6	1.21093	0.605467	+	0.795870i	$0.292986\pi$
$308$	0	0
$309$	−183664.	−0.109428
$310$	0	0
$311$	−1.50842e6	−0.884344	−0.442172	−	0.896930i	$-0.645792\pi$
$311$	−1.50842e6	−0.884344	−0.442172	+	0.896930i	$0.645792\pi$
$312$	0	0
$313$	−1.07754e6	−0.621688	−0.310844	−	0.950461i	$-0.600612\pi$
$313$	−1.07754e6	−0.621688	−0.310844	+	0.950461i	$0.600612\pi$
$314$	0	0
$315$	1.30880e6	0.743188
$316$	0	0
$317$	1.95476e6	1.09256	0.546279	−	0.837603i	$-0.316044\pi$
$317$	1.95476e6	1.09256	0.546279	+	0.837603i	$0.316044\pi$
$318$	0	0
$319$	780294.	0.429320
$320$	0	0
$321$	−799264.	−0.432940
$322$	0	0
$323$	−364741.	−0.194526
$324$	0	0
$325$	603656.	0.317016
$326$	0	0
$327$	321946.	0.166500
$328$	0	0
$329$	−802675.	−0.408837
$330$	0	0
$331$	−325820.	−0.163459	−0.0817294	−	0.996655i	$-0.526044\pi$
$331$	−325820.	−0.163459	−0.0817294	+	0.996655i	$0.526044\pi$
$332$	0	0
$333$	−2.28903e6	−1.13120
$334$	0	0
$335$	1.38470e6	0.674131
$336$	0	0
$337$	1.94430e6	0.932586	0.466293	−	0.884630i	$-0.345589\pi$
$337$	1.94430e6	0.932586	0.466293	+	0.884630i	$0.345589\pi$
$338$	0	0
$339$	−635362.	−0.300277
$340$	0	0
$341$	454184.	0.211517
$342$	0	0
$343$	−2.28844e6	−1.05028
$344$	0	0
$345$	260026.	0.117617
$346$	0	0
$347$	−3.02066e6	−1.34672	−0.673361	−	0.739314i	$-0.735150\pi$
$347$	−3.02066e6	−1.34672	−0.673361	+	0.739314i	$0.735150\pi$
$348$	0	0
$349$	−2.07033e6	−0.909864	−0.454932	−	0.890526i	$-0.650336\pi$
$349$	−2.07033e6	−0.909864	−0.454932	+	0.890526i	$0.650336\pi$
$350$	0	0
$351$	1.38573e6	0.600360
$352$	0	0
$353$	1.49504e6	0.638583	0.319291	−	0.947657i	$-0.396555\pi$
$353$	1.49504e6	0.638583	0.319291	+	0.947657i	$0.396555\pi$
$354$	0	0
$355$	−1.93104e6	−0.813242
$356$	0	0
$357$	−513122.	−0.213083
$358$	0	0
$359$	−4.84808e6	−1.98533	−0.992667	−	0.120878i	$-0.961429\pi$
$359$	−4.84808e6	−1.98533	−0.992667	+	0.120878i	$0.961429\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	−621067.	−0.247384
$364$	0	0
$365$	905564.	0.355784
$366$	0	0
$367$	−4.97659e6	−1.92871	−0.964355	−	0.264611i	$-0.914756\pi$
$367$	−4.97659e6	−1.92871	−0.964355	+	0.264611i	$0.914756\pi$
$368$	0	0
$369$	−1.63327e6	−0.624444
$370$	0	0
$371$	−557849.	−0.210418
$372$	0	0
$373$	1.24121e6	0.461925	0.230963	−	0.972963i	$-0.425812\pi$
$373$	1.24121e6	0.461925	0.230963	+	0.972963i	$0.425812\pi$
$374$	0	0
$375$	791296.	0.290577
$376$	0	0
$377$	4.99903e6	1.81147
$378$	0	0
$379$	−5.14902e6	−1.84131	−0.920654	−	0.390379i	$-0.872344\pi$
$379$	−5.14902e6	−1.84131	−0.920654	+	0.390379i	$0.872344\pi$
$380$	0	0
$381$	356643.	0.125870
$382$	0	0
$383$	2.44998e6	0.853424	0.426712	−	0.904388i	$-0.359672\pi$
$383$	2.44998e6	0.853424	0.426712	+	0.904388i	$0.359672\pi$
$384$	0	0
$385$	641725.	0.220646
$386$	0	0
$387$	−4.00798e6	−1.36034
$388$	0	0
$389$	3.26816e6	1.09504	0.547520	−	0.836793i	$-0.315572\pi$
$389$	3.26816e6	1.09504	0.547520	+	0.836793i	$0.315572\pi$
$390$	0	0
$391$	1.32038e6	0.436773
$392$	0	0
$393$	−974228.	−0.318185
$394$	0	0
$395$	2.39236e6	0.771495
$396$	0	0
$397$	4.84703e6	1.54347	0.771737	−	0.635942i	$-0.219388\pi$
$397$	4.84703e6	1.54347	0.771737	+	0.635942i	$0.219388\pi$
$398$	0	0
$399$	183337.	0.0576525
$400$	0	0
$401$	−3.69052e6	−1.14611	−0.573055	−	0.819517i	$-0.694242\pi$
$401$	−3.69052e6	−1.14611	−0.573055	+	0.819517i	$0.694242\pi$
$402$	0	0
$403$	2.90977e6	0.892476
$404$	0	0
$405$	−2.22440e6	−0.673869
$406$	0	0
$407$	−1.12234e6	−0.335845
$408$	0	0
$409$	−6.39197e6	−1.88941	−0.944705	−	0.327921i	$-0.893652\pi$
$409$	−6.39197e6	−1.88941	−0.944705	+	0.327921i	$0.893652\pi$
$410$	0	0
$411$	46565.0	0.0135974
$412$	0	0
$413$	3.44357e6	0.993421
$414$	0	0
$415$	2.65811e6	0.757622
$416$	0	0
$417$	−236738.	−0.0666695
$418$	0	0
$419$	−6.07039e6	−1.68920	−0.844601	−	0.535397i	$-0.820162\pi$
$419$	−6.07039e6	−1.68920	−0.844601	+	0.535397i	$0.820162\pi$
$420$	0	0
$421$	6.00891e6	1.65231	0.826153	−	0.563446i	$-0.190525\pi$
$421$	6.00891e6	1.65231	0.826153	+	0.563446i	$0.190525\pi$
$422$	0	0
$423$	1.48795e6	0.404332
$424$	0	0
$425$	860715.	0.231146
$426$	0	0
$427$	7.01570e6	1.86209
$428$	0	0
$429$	327095.	0.0858085
$430$	0	0
$431$	−1.35952e6	−0.352527	−0.176264	−	0.984343i	$-0.556401\pi$
$431$	−1.35952e6	−0.352527	−0.176264	+	0.984343i	$0.556401\pi$
$432$	0	0
$433$	−1.79635e6	−0.460439	−0.230220	−	0.973139i	$-0.573944\pi$
$433$	−1.79635e6	−0.460439	−0.230220	+	0.973139i	$0.573944\pi$
$434$	0	0
$435$	1.40370e6	0.355672
$436$	0	0
$437$	−471767.	−0.118175
$438$	0	0
$439$	−1.91763e6	−0.474901	−0.237451	−	0.971400i	$-0.576312\pi$
$439$	−1.91763e6	−0.474901	−0.237451	+	0.971400i	$0.576312\pi$
$440$	0	0
$441$	450794.	0.110378
$442$	0	0
$443$	87948.5	0.0212921	0.0106461	−	0.999943i	$-0.496611\pi$
$443$	87948.5	0.0212921	0.0106461	+	0.999943i	$0.496611\pi$
$444$	0	0
$445$	1.67655e6	0.401343
$446$	0	0
$447$	343901.	0.0814075
$448$	0	0
$449$	7.99009e6	1.87040	0.935202	−	0.354114i	$-0.115217\pi$
$449$	7.99009e6	1.87040	0.935202	+	0.354114i	$0.115217\pi$
$450$	0	0
$451$	−800817.	−0.185392
$452$	0	0
$453$	1.13379e6	0.259589
$454$	0	0
$455$	4.11127e6	0.930996
$456$	0	0
$457$	−3.87005e6	−0.866815	−0.433408	−	0.901198i	$-0.642689\pi$
$457$	−3.87005e6	−0.866815	−0.433408	+	0.901198i	$0.642689\pi$
$458$	0	0
$459$	1.97583e6	0.437741
$460$	0	0
$461$	−1.78119e6	−0.390354	−0.195177	−	0.980768i	$-0.562528\pi$
$461$	−1.78119e6	−0.390354	−0.195177	+	0.980768i	$0.562528\pi$
$462$	0	0
$463$	−4.12821e6	−0.894971	−0.447486	−	0.894291i	$-0.647680\pi$
$463$	−4.12821e6	−0.894971	−0.447486	+	0.894291i	$0.647680\pi$
$464$	0	0
$465$	817047.	0.175233
$466$	0	0
$467$	8.45975e6	1.79500	0.897502	−	0.441011i	$-0.145380\pi$
$467$	8.45975e6	1.79500	0.897502	+	0.441011i	$0.145380\pi$
$468$	0	0
$469$	−3.53430e6	−0.741945
$470$	0	0
$471$	−321600.	−0.0667981
$472$	0	0
$473$	−1.96517e6	−0.403874
$474$	0	0
$475$	−307531.	−0.0625396
$476$	0	0
$477$	1.03411e6	0.208099
$478$	0	0
$479$	−2.19827e6	−0.437767	−0.218883	−	0.975751i	$-0.570241\pi$
$479$	−2.19827e6	−0.437767	−0.218883	+	0.975751i	$0.570241\pi$
$480$	0	0
$481$	−7.19038e6	−1.41706
$482$	0	0
$483$	−663687.	−0.129448
$484$	0	0
$485$	623549.	0.120370
$486$	0	0
$487$	5.99739e6	1.14588	0.572941	−	0.819597i	$-0.305802\pi$
$487$	5.99739e6	1.14588	0.572941	+	0.819597i	$0.305802\pi$
$488$	0	0
$489$	2.00495e6	0.379167
$490$	0	0
$491$	4.52742e6	0.847515	0.423758	−	0.905776i	$-0.360711\pi$
$491$	4.52742e6	0.847515	0.423758	+	0.905776i	$0.360711\pi$
$492$	0	0
$493$	7.12779e6	1.32080
$494$	0	0
$495$	−1.18959e6	−0.218215
$496$	0	0
$497$	4.92877e6	0.895050
$498$	0	0
$499$	−3.49223e6	−0.627843	−0.313922	−	0.949449i	$-0.601643\pi$
$499$	−3.49223e6	−0.627843	−0.313922	+	0.949449i	$0.601643\pi$
$500$	0	0
$501$	639104.	0.113757
$502$	0	0
$503$	−6.67074e6	−1.17558	−0.587792	−	0.809012i	$-0.700003\pi$
$503$	−6.67074e6	−1.17558	−0.587792	+	0.809012i	$0.700003\pi$
$504$	0	0
$505$	−942850.	−0.164518
$506$	0	0
$507$	546028.	0.0943398
$508$	0	0
$509$	2.78722e6	0.476845	0.238423	−	0.971161i	$-0.423370\pi$
$509$	2.78722e6	0.476845	0.238423	+	0.971161i	$0.423370\pi$
$510$	0	0
$511$	−2.31135e6	−0.391574
$512$	0	0
$513$	−705958.	−0.118437
$514$	0	0
$515$	2.09821e6	0.348602
$516$	0	0
$517$	729563.	0.120043
$518$	0	0
$519$	−2.90266e6	−0.473018
$520$	0	0
$521$	2.64026e6	0.426141	0.213070	−	0.977037i	$-0.431654\pi$
$521$	2.64026e6	0.426141	0.213070	+	0.977037i	$0.431654\pi$
$522$	0	0
$523$	−1.01209e7	−1.61795	−0.808977	−	0.587840i	$-0.799978\pi$
$523$	−1.01209e7	−1.61795	−0.808977	+	0.587840i	$0.799978\pi$
$524$	0	0
$525$	−432638.	−0.0685057
$526$	0	0
$527$	4.14886e6	0.650732
$528$	0	0
$529$	−4.72853e6	−0.734660
$530$	0	0
$531$	−6.38348e6	−0.982474
$532$	0	0
$533$	−5.13051e6	−0.782244
$534$	0	0
$535$	9.13094e6	1.37921
$536$	0	0
$537$	−285758.	−0.0427624
$538$	0	0
$539$	221030.	0.0327703
$540$	0	0
$541$	−9.02774e6	−1.32613	−0.663065	−	0.748562i	$-0.730745\pi$
$541$	−9.02774e6	−1.32613	−0.663065	+	0.748562i	$0.730745\pi$
$542$	0	0
$543$	−220979.	−0.0321626
$544$	0	0
$545$	−3.67797e6	−0.530417
$546$	0	0
$547$	−2.76232e6	−0.394735	−0.197368	−	0.980330i	$-0.563239\pi$
$547$	−2.76232e6	−0.394735	−0.197368	+	0.980330i	$0.563239\pi$
$548$	0	0
$549$	−1.30053e7	−1.84157
$550$	0	0
$551$	−2.54674e6	−0.357360
$552$	0	0
$553$	−6.10623e6	−0.849102
$554$	0	0
$555$	−2.01902e6	−0.278232
$556$	0	0
$557$	5.55335e6	0.758433	0.379217	−	0.925308i	$-0.376194\pi$
$557$	5.55335e6	0.758433	0.379217	+	0.925308i	$0.376194\pi$
$558$	0	0
$559$	−1.25900e7	−1.70411
$560$	0	0
$561$	466383.	0.0625656
$562$	0	0
$563$	9.82920e6	1.30691	0.653457	−	0.756964i	$-0.273318\pi$
$563$	9.82920e6	1.30691	0.653457	+	0.756964i	$0.273318\pi$
$564$	0	0
$565$	7.25849e6	0.956589
$566$	0	0
$567$	5.67754e6	0.741656
$568$	0	0
$569$	1.04098e7	1.34791	0.673956	−	0.738771i	$-0.264594\pi$
$569$	1.04098e7	1.34791	0.673956	+	0.738771i	$0.264594\pi$
$570$	0	0
$571$	−9.66015e6	−1.23992	−0.619960	−	0.784633i	$-0.712851\pi$
$571$	−9.66015e6	−1.23992	−0.619960	+	0.784633i	$0.712851\pi$
$572$	0	0
$573$	−2.93836e6	−0.373868
$574$	0	0
$575$	1.11328e6	0.140421
$576$	0	0
$577$	−3.43353e6	−0.429340	−0.214670	−	0.976687i	$-0.568868\pi$
$577$	−3.43353e6	−0.429340	−0.214670	+	0.976687i	$0.568868\pi$
$578$	0	0
$579$	641866.	0.0795698
$580$	0	0
$581$	−6.78453e6	−0.833834
$582$	0	0
$583$	507037.	0.0617829
$584$	0	0
$585$	−7.62123e6	−0.920737
$586$	0	0
$587$	5.30248e6	0.635161	0.317580	−	0.948231i	$-0.397130\pi$
$587$	5.30248e6	0.635161	0.317580	+	0.948231i	$0.397130\pi$
$588$	0	0
$589$	−1.48238e6	−0.176064
$590$	0	0
$591$	123433.	0.0145365
$592$	0	0
$593$	−1.27514e7	−1.48909	−0.744545	−	0.667572i	$-0.767333\pi$
$593$	−1.27514e7	−1.48909	−0.744545	+	0.667572i	$0.767333\pi$
$594$	0	0
$595$	5.86200e6	0.678818
$596$	0	0
$597$	1.62565e6	0.186678
$598$	0	0
$599$	9.36698e6	1.06668	0.533338	−	0.845902i	$-0.320937\pi$
$599$	9.36698e6	1.06668	0.533338	+	0.845902i	$0.320937\pi$
$600$	0	0
$601$	2.45920e6	0.277720	0.138860	−	0.990312i	$-0.455656\pi$
$601$	2.45920e6	0.277720	0.138860	+	0.990312i	$0.455656\pi$
$602$	0	0
$603$	6.55168e6	0.733769
$604$	0	0
$605$	7.09518e6	0.788089
$606$	0	0
$607$	1.35715e7	1.49505	0.747526	−	0.664233i	$-0.231242\pi$
$607$	1.35715e7	1.49505	0.747526	+	0.664233i	$0.231242\pi$
$608$	0	0
$609$	−3.58278e6	−0.391451
$610$	0	0
$611$	4.67402e6	0.506509
$612$	0	0
$613$	−7.18311e6	−0.772078	−0.386039	−	0.922482i	$-0.626157\pi$
$613$	−7.18311e6	−0.772078	−0.386039	+	0.922482i	$0.626157\pi$
$614$	0	0
$615$	−1.44062e6	−0.153589
$616$	0	0
$617$	3.99195e6	0.422156	0.211078	−	0.977469i	$-0.432303\pi$
$617$	3.99195e6	0.422156	0.211078	+	0.977469i	$0.432303\pi$
$618$	0	0
$619$	4.02966e6	0.422710	0.211355	−	0.977409i	$-0.432212\pi$
$619$	4.02966e6	0.422710	0.211355	+	0.977409i	$0.432212\pi$
$620$	0	0
$621$	2.55560e6	0.265928
$622$	0	0
$623$	−4.27921e6	−0.441716
$624$	0	0
$625$	−6.37777e6	−0.653083
$626$	0	0
$627$	−166638.	−0.0169279
$628$	0	0
$629$	−1.02523e7	−1.03322
$630$	0	0
$631$	1.66190e7	1.66162	0.830808	−	0.556559i	$-0.187879\pi$
$631$	1.66190e7	1.66162	0.830808	+	0.556559i	$0.187879\pi$
$632$	0	0
$633$	−726981.	−0.0721131
$634$	0	0
$635$	−4.07436e6	−0.400982
$636$	0	0
$637$	1.41605e6	0.138271
$638$	0	0
$639$	−9.13665e6	−0.885187
$640$	0	0
$641$	1.07181e7	1.03032	0.515160	−	0.857094i	$-0.327733\pi$
$641$	1.07181e7	1.03032	0.515160	+	0.857094i	$0.327733\pi$
$642$	0	0
$643$	2.56952e6	0.245089	0.122545	−	0.992463i	$-0.460895\pi$
$643$	2.56952e6	0.245089	0.122545	+	0.992463i	$0.460895\pi$
$644$	0	0
$645$	−3.53520e6	−0.334592
$646$	0	0
$647$	6.57455e6	0.617455	0.308728	−	0.951151i	$-0.400097\pi$
$647$	6.57455e6	0.617455	0.308728	+	0.951151i	$0.400097\pi$
$648$	0	0
$649$	−3.12991e6	−0.291689
$650$	0	0
$651$	−2.08542e6	−0.192860
$652$	0	0
$653$	266145.	0.0244251	0.0122125	−	0.999925i	$-0.496113\pi$
$653$	266145.	0.0244251	0.0122125	+	0.999925i	$0.496113\pi$
$654$	0	0
$655$	1.11298e7	1.01364
$656$	0	0
$657$	4.28465e6	0.387259
$658$	0	0
$659$	−8.14176e6	−0.730306	−0.365153	−	0.930948i	$-0.618983\pi$
$659$	−8.14176e6	−0.730306	−0.365153	+	0.930948i	$0.618983\pi$
$660$	0	0
$661$	1.18434e7	1.05433	0.527163	−	0.849764i	$-0.323256\pi$
$661$	1.18434e7	1.05433	0.527163	+	0.849764i	$0.323256\pi$
$662$	0	0
$663$	2.98793e6	0.263990
$664$	0	0
$665$	−2.09448e6	−0.183663
$666$	0	0
$667$	9.21931e6	0.802387
$668$	0	0
$669$	3.14196e6	0.271416
$670$	0	0
$671$	−6.37667e6	−0.546748
$672$	0	0
$673$	7.03337e6	0.598585	0.299293	−	0.954161i	$-0.403249\pi$
$673$	7.03337e6	0.598585	0.299293	+	0.954161i	$0.403249\pi$
$674$	0	0
$675$	1.66592e6	0.140732
$676$	0	0
$677$	1.72487e7	1.44638	0.723192	−	0.690647i	$-0.242674\pi$
$677$	1.72487e7	1.44638	0.723192	+	0.690647i	$0.242674\pi$
$678$	0	0
$679$	−1.59154e6	−0.132478
$680$	0	0
$681$	5.62911e6	0.465127
$682$	0	0
$683$	−1.20215e7	−0.986066	−0.493033	−	0.870010i	$-0.664112\pi$
$683$	−1.20215e7	−0.986066	−0.493033	+	0.870010i	$0.664112\pi$
$684$	0	0
$685$	−531968.	−0.0433170
$686$	0	0
$687$	−3.43915e6	−0.278009
$688$	0	0
$689$	3.24838e6	0.260687
$690$	0	0
$691$	8.81754e6	0.702509	0.351255	−	0.936280i	$-0.385755\pi$
$691$	8.81754e6	0.702509	0.351255	+	0.936280i	$0.385755\pi$
$692$	0	0
$693$	3.03630e6	0.240166
$694$	0	0
$695$	2.70453e6	0.212388
$696$	0	0
$697$	−7.31526e6	−0.570359
$698$	0	0
$699$	−6.37843e6	−0.493765
$700$	0	0
$701$	4.52181e6	0.347550	0.173775	−	0.984785i	$-0.444403\pi$
$701$	4.52181e6	0.347550	0.173775	+	0.984785i	$0.444403\pi$
$702$	0	0
$703$	3.66312e6	0.279552
$704$	0	0
$705$	1.31244e6	0.0994501
$706$	0	0
$707$	2.40652e6	0.181068
$708$	0	0
$709$	−1.80572e7	−1.34907	−0.674536	−	0.738242i	$-0.735656\pi$
$709$	−1.80572e7	−1.34907	−0.674536	+	0.738242i	$0.735656\pi$
$710$	0	0
$711$	1.13194e7	0.839746
$712$	0	0
$713$	5.36627e6	0.395320
$714$	0	0
$715$	−3.73679e6	−0.273359
$716$	0	0
$717$	−5.40221e6	−0.392440
$718$	0	0
$719$	−6.03823e6	−0.435599	−0.217800	−	0.975994i	$-0.569888\pi$
$719$	−6.03823e6	−0.435599	−0.217800	+	0.975994i	$0.569888\pi$
$720$	0	0
$721$	−5.35545e6	−0.383670
$722$	0	0
$723$	−3.73898e6	−0.266016
$724$	0	0
$725$	6.00979e6	0.424634
$726$	0	0
$727$	7.33157e6	0.514471	0.257235	−	0.966349i	$-0.417188\pi$
$727$	7.33157e6	0.514471	0.257235	+	0.966349i	$0.417188\pi$
$728$	0	0
$729$	−8.54150e6	−0.595272
$730$	0	0
$731$	−1.79513e7	−1.24252
$732$	0	0
$733$	8.65763e6	0.595167	0.297584	−	0.954696i	$-0.403819\pi$
$733$	8.65763e6	0.595167	0.297584	+	0.954696i	$0.403819\pi$
$734$	0	0
$735$	397619.	0.0271487
$736$	0	0
$737$	3.21238e6	0.217850
$738$	0	0
$739$	8.98966e6	0.605525	0.302762	−	0.953066i	$-0.402091\pi$
$739$	8.98966e6	0.605525	0.302762	+	0.953066i	$0.402091\pi$
$740$	0	0
$741$	−1.06758e6	−0.0714257
$742$	0	0
$743$	−1.73519e7	−1.15312	−0.576562	−	0.817054i	$-0.695606\pi$
$743$	−1.73519e7	−1.15312	−0.576562	+	0.817054i	$0.695606\pi$
$744$	0	0
$745$	−3.92879e6	−0.259339
$746$	0	0
$747$	1.25768e7	0.824645
$748$	0	0
$749$	−2.33057e7	−1.51795
$750$	0	0
$751$	−2.69336e7	−1.74258	−0.871292	−	0.490764i	$-0.836718\pi$
$751$	−2.69336e7	−1.74258	−0.871292	+	0.490764i	$0.836718\pi$
$752$	0	0
$753$	−1.14692e6	−0.0737135
$754$	0	0
$755$	−1.29526e7	−0.826971
$756$	0	0
$757$	−2.48338e7	−1.57509	−0.787543	−	0.616260i	$-0.788647\pi$
$757$	−2.48338e7	−1.57509	−0.787543	+	0.616260i	$0.788647\pi$
$758$	0	0
$759$	603235.	0.0380086
$760$	0	0
$761$	1.20747e7	0.755811	0.377905	−	0.925844i	$-0.376644\pi$
$761$	1.20747e7	0.755811	0.377905	+	0.925844i	$0.376644\pi$
$762$	0	0
$763$	9.38762e6	0.583774
$764$	0	0
$765$	−1.08666e7	−0.671338
$766$	0	0
$767$	−2.00520e7	−1.23075
$768$	0	0
$769$	1.66647e7	1.01621	0.508104	−	0.861296i	$-0.330347\pi$
$769$	1.66647e7	1.01621	0.508104	+	0.861296i	$0.330347\pi$
$770$	0	0
$771$	−753524.	−0.0456521
$772$	0	0
$773$	2.14767e7	1.29276	0.646380	−	0.763016i	$-0.276282\pi$
$773$	2.14767e7	1.29276	0.646380	+	0.763016i	$0.276282\pi$
$774$	0	0
$775$	3.49811e6	0.209208
$776$	0	0
$777$	5.15332e6	0.306221
$778$	0	0
$779$	2.61372e6	0.154318
$780$	0	0
$781$	−4.47982e6	−0.262805
$782$	0	0
$783$	1.37959e7	0.804165
$784$	0	0
$785$	3.67402e6	0.212798
$786$	0	0
$787$	1.88829e7	1.08676	0.543379	−	0.839487i	$-0.317145\pi$
$787$	1.88829e7	1.08676	0.543379	+	0.839487i	$0.317145\pi$
$788$	0	0
$789$	1.71990e6	0.0983585
$790$	0	0
$791$	−1.85265e7	−1.05282
$792$	0	0
$793$	−4.08527e7	−2.30695
$794$	0	0
$795$	912126.	0.0511843
$796$	0	0
$797$	−5.13580e6	−0.286393	−0.143197	−	0.989694i	$-0.545738\pi$
$797$	−5.13580e6	−0.286393	−0.143197	+	0.989694i	$0.545738\pi$
$798$	0	0
$799$	6.66438e6	0.369311
$800$	0	0
$801$	7.93254e6	0.436848
$802$	0	0
$803$	2.10082e6	0.114974
$804$	0	0
$805$	7.58209e6	0.412382
$806$	0	0
$807$	7.04240e6	0.380660
$808$	0	0
$809$	1.57396e7	0.845516	0.422758	−	0.906243i	$-0.361062\pi$
$809$	1.57396e7	0.845516	0.422758	+	0.906243i	$0.361062\pi$
$810$	0	0
$811$	−1.60257e7	−0.855589	−0.427794	−	0.903876i	$-0.640709\pi$
$811$	−1.60257e7	−0.855589	−0.427794	+	0.903876i	$0.640709\pi$
$812$	0	0
$813$	8.50008e6	0.451021
$814$	0	0
$815$	−2.29049e7	−1.20791
$816$	0	0
$817$	6.41395e6	0.336179
$818$	0	0
$819$	1.94524e7	1.01336
$820$	0	0
$821$	−1.87672e7	−0.971722	−0.485861	−	0.874036i	$-0.661494\pi$
$821$	−1.87672e7	−0.971722	−0.485861	+	0.874036i	$0.661494\pi$
$822$	0	0
$823$	−1.71146e7	−0.880780	−0.440390	−	0.897807i	$-0.645160\pi$
$823$	−1.71146e7	−0.880780	−0.440390	+	0.897807i	$0.645160\pi$
$824$	0	0
$825$	393231.	0.0201147
$826$	0	0
$827$	−2.42297e7	−1.23193	−0.615963	−	0.787775i	$-0.711233\pi$
$827$	−2.42297e7	−1.23193	−0.615963	+	0.787775i	$0.711233\pi$
$828$	0	0
$829$	−2.84347e7	−1.43702	−0.718508	−	0.695518i	$-0.755175\pi$
$829$	−2.84347e7	−1.43702	−0.718508	+	0.695518i	$0.755175\pi$
$830$	0	0
$831$	5.82134e6	0.292429
$832$	0	0
$833$	2.01906e6	0.100818
$834$	0	0
$835$	−7.30125e6	−0.362394
$836$	0	0
$837$	8.03014e6	0.396196
$838$	0	0
$839$	−2.96937e7	−1.45633	−0.728164	−	0.685403i	$-0.759626\pi$
$839$	−2.96937e7	−1.45633	−0.728164	+	0.685403i	$0.759626\pi$
$840$	0	0
$841$	2.92574e7	1.42642
$842$	0	0
$843$	−7.46149e6	−0.361623
$844$	0	0
$845$	−6.23792e6	−0.300537
$846$	0	0
$847$	−1.81097e7	−0.867366
$848$	0	0
$849$	−6.03446e6	−0.287322
$850$	0	0
$851$	−1.32606e7	−0.627684
$852$	0	0
$853$	1.75810e7	0.827316	0.413658	−	0.910432i	$-0.364251\pi$
$853$	1.75810e7	0.827316	0.413658	+	0.910432i	$0.364251\pi$
$854$	0	0
$855$	3.88262e6	0.181639
$856$	0	0
$857$	3.33397e7	1.55063	0.775317	−	0.631572i	$-0.217590\pi$
$857$	3.33397e7	1.55063	0.775317	+	0.631572i	$0.217590\pi$
$858$	0	0
$859$	8.42465e6	0.389555	0.194778	−	0.980847i	$-0.437602\pi$
$859$	8.42465e6	0.389555	0.194778	+	0.980847i	$0.437602\pi$
$860$	0	0
$861$	3.67702e6	0.169039
$862$	0	0
$863$	−2.90077e6	−0.132583	−0.0662913	−	0.997800i	$-0.521117\pi$
$863$	−2.90077e6	−0.132583	−0.0662913	+	0.997800i	$0.521117\pi$
$864$	0	0
$865$	3.31605e7	1.50689
$866$	0	0
$867$	−1.66527e6	−0.0752378
$868$	0	0
$869$	5.55003e6	0.249314
$870$	0	0
$871$	2.05804e7	0.919197
$872$	0	0
$873$	2.95031e6	0.131018
$874$	0	0
$875$	2.30734e7	1.01881
$876$	0	0
$877$	−684417.	−0.0300484	−0.0150242	−	0.999887i	$-0.504783\pi$
$877$	−684417.	−0.0300484	−0.0150242	+	0.999887i	$0.504783\pi$
$878$	0	0
$879$	1.42944e6	0.0624014
$880$	0	0
$881$	2.71140e7	1.17694	0.588469	−	0.808520i	$-0.299731\pi$
$881$	2.71140e7	1.17694	0.588469	+	0.808520i	$0.299731\pi$
$882$	0	0
$883$	−1.00088e7	−0.431995	−0.215997	−	0.976394i	$-0.569300\pi$
$883$	−1.00088e7	−0.431995	−0.215997	+	0.976394i	$0.569300\pi$
$884$	0	0
$885$	−5.63049e6	−0.241651
$886$	0	0
$887$	3.23264e7	1.37959	0.689793	−	0.724007i	$-0.257702\pi$
$887$	3.23264e7	1.37959	0.689793	+	0.724007i	$0.257702\pi$
$888$	0	0
$889$	1.03994e7	0.441319
$890$	0	0
$891$	−5.16039e6	−0.217765
$892$	0	0
$893$	−2.38116e6	−0.0999219
$894$	0	0
$895$	3.26455e6	0.136228
$896$	0	0
$897$	3.86468e6	0.160374
$898$	0	0
$899$	2.89687e7	1.19545
$900$	0	0
$901$	4.63166e6	0.190075
$902$	0	0
$903$	9.02322e6	0.368249
$904$	0	0
$905$	2.52451e6	0.102460
$906$	0	0
$907$	2.06863e7	0.834958	0.417479	−	0.908687i	$-0.362914\pi$
$907$	2.06863e7	0.834958	0.417479	+	0.908687i	$0.362914\pi$
$908$	0	0
$909$	−4.46107e6	−0.179073
$910$	0	0
$911$	−2.88279e7	−1.15085	−0.575423	−	0.817856i	$-0.695163\pi$
$911$	−2.88279e7	−1.15085	−0.575423	+	0.817856i	$0.695163\pi$
$912$	0	0
$913$	6.16656e6	0.244831
$914$	0	0
$915$	−1.14712e7	−0.452956
$916$	0	0
$917$	−2.84075e7	−1.11560
$918$	0	0
$919$	2.39153e7	0.934087	0.467043	−	0.884234i	$-0.345319\pi$
$919$	2.39153e7	0.934087	0.467043	+	0.884234i	$0.345319\pi$
$920$	0	0
$921$	8.34549e6	0.324192
$922$	0	0
$923$	−2.87004e7	−1.10888
$924$	0	0
$925$	−8.64422e6	−0.332178
$926$	0	0
$927$	9.92761e6	0.379442
$928$	0	0
$929$	1.34725e7	0.512162	0.256081	−	0.966655i	$-0.417569\pi$
$929$	1.34725e7	0.512162	0.256081	+	0.966655i	$0.417569\pi$
$930$	0	0
$931$	−721404.	−0.0272775
$932$	0	0
$933$	−6.29517e6	−0.236757
$934$	0	0
$935$	−5.32805e6	−0.199315
$936$	0	0
$937$	3.28999e7	1.22418	0.612090	−	0.790788i	$-0.290329\pi$
$937$	3.28999e7	1.22418	0.612090	+	0.790788i	$0.290329\pi$
$938$	0	0
$939$	−4.49696e6	−0.166439
$940$	0	0
$941$	3.02152e7	1.11238	0.556188	−	0.831057i	$-0.312264\pi$
$941$	3.02152e7	1.11238	0.556188	+	0.831057i	$0.312264\pi$
$942$	0	0
$943$	−9.46179e6	−0.346493
$944$	0	0
$945$	1.13459e7	0.413295
$946$	0	0
$947$	−5.40125e7	−1.95713	−0.978564	−	0.205941i	$-0.933974\pi$
$947$	−5.40125e7	−1.95713	−0.978564	+	0.205941i	$0.933974\pi$
$948$	0	0
$949$	1.34591e7	0.485122
$950$	0	0
$951$	8.15789e6	0.292500
$952$	0	0
$953$	−5.18417e7	−1.84904	−0.924522	−	0.381129i	$-0.875535\pi$
$953$	−5.18417e7	−1.84904	−0.924522	+	0.381129i	$0.875535\pi$
$954$	0	0
$955$	3.35684e7	1.19103
$956$	0	0
$957$	3.25644e6	0.114938
$958$	0	0
$959$	1.35779e6	0.0476745
$960$	0	0
$961$	−1.17674e7	−0.411028
$962$	0	0
$963$	4.32028e7	1.50123
$964$	0	0
$965$	−7.33280e6	−0.253485
$966$	0	0
$967$	5.05765e6	0.173933	0.0869666	−	0.996211i	$-0.472283\pi$
$967$	5.05765e6	0.173933	0.0869666	+	0.996211i	$0.472283\pi$
$968$	0	0
$969$	−1.52219e6	−0.0520787
$970$	0	0
$971$	5.45669e6	0.185730	0.0928648	−	0.995679i	$-0.470398\pi$
$971$	5.45669e6	0.185730	0.0928648	+	0.995679i	$0.470398\pi$
$972$	0	0
$973$	−6.90303e6	−0.233753
$974$	0	0
$975$	2.51927e6	0.0848718
$976$	0	0
$977$	1.78375e7	0.597857	0.298928	−	0.954276i	$-0.403371\pi$
$977$	1.78375e7	0.597857	0.298928	+	0.954276i	$0.403371\pi$
$978$	0	0
$979$	3.88943e6	0.129697
$980$	0	0
$981$	−1.74022e7	−0.577341
$982$	0	0
$983$	−4.86307e6	−0.160519	−0.0802596	−	0.996774i	$-0.525575\pi$
$983$	−4.86307e6	−0.160519	−0.0802596	+	0.996774i	$0.525575\pi$
$984$	0	0
$985$	−1.41012e6	−0.0463089
$986$	0	0
$987$	−3.34985e6	−0.109454
$988$	0	0
$989$	−2.32188e7	−0.754829
$990$	0	0
$991$	−5.03163e7	−1.62751	−0.813756	−	0.581206i	$-0.802581\pi$
$991$	−5.03163e7	−1.62751	−0.813756	+	0.581206i	$0.802581\pi$
$992$	0	0
$993$	−1.35976e6	−0.0437613
$994$	0	0
$995$	−1.85718e7	−0.594697
$996$	0	0
$997$	−2.17021e7	−0.691455	−0.345727	−	0.938335i	$-0.612368\pi$
$997$	−2.17021e7	−0.691455	−0.345727	+	0.938335i	$0.612368\pi$
$998$	0	0
$999$	−1.98434e7	−0.629074

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.a.a.1.2	✓	3
3.2	odd	2		684.6.a.b.1.3		3
4.3	odd	2		304.6.a.j.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.a.a.1.2	✓	3	1.1	even	1	trivial
304.6.a.j.1.2		3	4.3	odd	2
684.6.a.b.1.3		3	3.2	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:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	76.a (trivial)

\(f(q)\)	\(=\)	\(q+4.17335 q^{3} -47.6772 q^{5} +121.691 q^{7} -225.583 q^{9} +O(q^{10})\)	\(q+4.17335 q^{3} -47.6772 q^{5} +121.691 q^{7} -225.583 q^{9} -110.606 q^{11} -708.611 q^{13} -198.974 q^{15} -1010.36 q^{17} +361.000 q^{19} +507.859 q^{21} -1306.84 q^{23} -851.887 q^{25} -1955.56 q^{27} -7054.68 q^{29} -4106.31 q^{31} -461.600 q^{33} -5801.87 q^{35} +10147.1 q^{37} -2957.28 q^{39} +7240.23 q^{41} +17767.2 q^{43} +10755.2 q^{45} -6596.02 q^{47} -1998.35 q^{49} -4216.60 q^{51} -4584.15 q^{53} +5273.40 q^{55} +1506.58 q^{57} +28297.7 q^{59} +57651.8 q^{61} -27451.4 q^{63} +33784.6 q^{65} -29043.3 q^{67} -5453.88 q^{69} +40502.4 q^{71} -18993.7 q^{73} -3555.22 q^{75} -13459.8 q^{77} -50178.2 q^{79} +46655.4 q^{81} -55752.2 q^{83} +48171.3 q^{85} -29441.7 q^{87} -35164.6 q^{89} -86231.4 q^{91} -17137.1 q^{93} -17211.5 q^{95} -13078.6 q^{97} +24951.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{3} - 9 q^{5} - 13 q^{7} + 189 q^{9}+O(q^{10}) \) 3 * q - 8 * q^3 - 9 * q^5 - 13 * q^7 + 189 * q^9	\( 3 q - 8 q^{3} - 9 q^{5} - 13 q^{7} + 189 q^{9} - 1229 q^{11} - 56 q^{13} - 1698 q^{15} - 3149 q^{17} + 1083 q^{19} - 6186 q^{21} - 3212 q^{23} - 4422 q^{25} - 11582 q^{27} + 2514 q^{29} - 10784 q^{31} + 13538 q^{33} + 3081 q^{35} - 526 q^{37} - 9668 q^{39} - 14246 q^{41} - 77 q^{43} + 34155 q^{45} + 2893 q^{47} + 41766 q^{49} + 46628 q^{51} + 29600 q^{53} - 27351 q^{55} - 2888 q^{57} - 2612 q^{59} + 59895 q^{61} + 36095 q^{63} + 50592 q^{65} - 3050 q^{67} + 78908 q^{69} - 33562 q^{71} + 44027 q^{73} - 30698 q^{75} - 3527 q^{77} + 24944 q^{79} + 36231 q^{81} - 220156 q^{83} - 51021 q^{85} - 175128 q^{87} - 116120 q^{89} - 105286 q^{91} + 131212 q^{93} - 3249 q^{95} + 171204 q^{97} - 294437 q^{99}+O(q^{100}) \) 3 * q - 8 * q^3 - 9 * q^5 - 13 * q^7 + 189 * q^9 - 1229 * q^11 - 56 * q^13 - 1698 * q^15 - 3149 * q^17 + 1083 * q^19 - 6186 * q^21 - 3212 * q^23 - 4422 * q^25 - 11582 * q^27 + 2514 * q^29 - 10784 * q^31 + 13538 * q^33 + 3081 * q^35 - 526 * q^37 - 9668 * q^39 - 14246 * q^41 - 77 * q^43 + 34155 * q^45 + 2893 * q^47 + 41766 * q^49 + 46628 * q^51 + 29600 * q^53 - 27351 * q^55 - 2888 * q^57 - 2612 * q^59 + 59895 * q^61 + 36095 * q^63 + 50592 * q^65 - 3050 * q^67 + 78908 * q^69 - 33562 * q^71 + 44027 * q^73 - 30698 * q^75 - 3527 * q^77 + 24944 * q^79 + 36231 * q^81 - 220156 * q^83 - 51021 * q^85 - 175128 * q^87 - 116120 * q^89 - 105286 * q^91 + 131212 * q^93 - 3249 * q^95 + 171204 * q^97 - 294437 * q^99

Introduction

L-functions

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